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+{"text":"\\section*{Introduction}\n\nSet $[n]:=\\{x_1,\\ldots,x_n\\}$. Let $K$ be a field and\n$S=K[x_1,\\ldots,x_n]$, a polynomial ring over $K$. Let $\\Delta$ be a\nsimplicial complex over $[n]$. For an integer $t\\geq 0$, Haghighi,\nYassemi and Zaare-Nahandi introduced the concept of ${\\mathrm{CM}}_t$-ness\nwhich is the pure version of simplicial complexes\n\\emph{Cohen-Macaulay in codimension $t$} studied in \\cite{MiNoSw}. A\nreason for the importance of ${\\mathrm{CM}}_t$ simplicial complexes is that\nthey generalizes two notions for simplicial complexes: being\nCohen-Macaulay and Buchsbaum. In particular, by the results from\n\\cite{Re,Sh}, ${\\mathrm{CM}}_0$ is the same as Cohen-Macaulayness and ${\\mathrm{CM}}_1$\nis identical with Buchsbaum property.\n\nIn \\cite{HaYaZa1}, the authors described some combinatorial properties\nof ${\\mathrm{CM}}_t$ simplicial complexes and gave some characterizations of them\nand generalized some results of \\cite{Hi,Mi}. Then, in \\cite{HaYaZa2},\nthey generalized a characterization of Cohen-Macaulay bipartite graphs\nfrom \\cite{HeHi} and \\cite{CoNa} on unmixed Buchsbaum graphs.\n\nBayati and Herzog defined the expansion functor in the category of\nfinitely generated multigraded $S$-modules and studied some homological\nbehaviors of this functor (see \\cite{BaHe}). The expansion functor helps\nus to present other multigraded $S$-modules from a given finitely generated\nmultigraded $S$-module which may have some of algebraic properties of the\nprimary module. This allows to introduce new structures of a given multigraded\n$S$-module with the same properties and especially to extend some homological\nor algebraic results for larger classes (see for example \\cite[Theorem 4.2]{BaHe}.\nThere are some combinatorial versions of expansion functor which we will recall in this paper.\n\nThe purpose of this paper is the study of behaviors of expansion\nfunctor on ${\\mathrm{CM}}_t$ complexes. We first recall some notations and\ndefinitions of ${\\mathrm{CM}}_t$ simplicial complexes in Section 1. In the\nnext section we describe the expansion functor in three contexts,\nthe expansion of a simplicial complex, the expansion of a simple\ngraph and the expansion of a monomial ideal. We show that there is a\nclose relationship between these three contexts. In Section 3 we\nprove that the expansion of a ${\\mathrm{CM}}_t$ complex $\\Delta$ with respect to\n$\\alpha$ is ${\\mathrm{CM}}_{t+e-k+1}$ but it is not ${\\mathrm{CM}}_{t+e-k}$ where\n$e=\\dim(\\Delta^\\alpha)+1$ and $k$ is the minimum of the components of\n$\\alpha$ (see Theorem \\ref{main}). In Section 4, we introduce a new\nfunctor, called contraction, which acts in contrast to expansion\nfunctor. As a main result of this section we show that if the\ncontraction of a ${\\mathrm{CM}}_t$ complex is pure and all components of the\nvector obtained from contraction are greater than or equal to $t$\nthen it is Buchsbaum (see Theorem \\ref{contract,CM-t}). The section\nis finished with a view towards the contraction of simple graphs.\n\n\\section{Preliminaries}\n\nLet $t$ be a non-negative integer. We recall from \\cite{HaYaZa1}\nthat a simplicial complex $\\Delta$ is called ${\\mathrm{CM}}_t$ or\n\\emph{Cohen-Macaulay in codimension $t$} if it is pure and for every\nface $F\\in\\Delta$ with $\\#(F)\\geq t$, $\\mathrm{link}_\\Delta(F)$ is Cohen-Macaulay.\nEvery ${\\mathrm{CM}}_t$ complex is also ${\\mathrm{CM}}_r$ for all $r\\geq t$. For $t<0$,\n${\\mathrm{CM}}_t$ means ${\\mathrm{CM}}_0$. The properties ${\\mathrm{CM}}_0$ and ${\\mathrm{CM}}_1$ are the\nsame as Cohen-Macaulay-ness and Buchsbaum-ness, respectively.\n\nThe link of a face $F$ in a simplicial complex $\\Delta$ is denoted by\n$\\mathrm{link}_\\Delta(F)$ and is $$\\mathrm{link}_\\Delta(F)=\\{G\\in\\Delta: G\\cap F=\\emptyset,\nG\\cup F\\in\\Delta\\}.$$ The following lemma is useful for checking the\n${\\mathrm{CM}}_t$ property of simplicial complexes:\n\n\\begin{lem}\\label{CM-t eq}\n(\\cite[Lemma 2.3]{HaYaZa1}) Let $t\\geq 1$ and let $\\Delta$ be a nonempty complex. Then $\\Delta$ is ${\\mathrm{CM}}_t$ if and only if $\\Delta$ is pure and $\\mathrm{link}_\\Delta(v)$ is ${\\mathrm{CM}}_{t-1}$ for every vertex $v\\in\\Delta$.\n\\end{lem}\n\nLet $\\mathcal G=(V(\\mathcal G),E(\\mathcal G))$ be a simple graph with vertex set $V$ and edge set $E$. The \\emph{independence complex} of $\\mathcal G$ is the complex $\\Delta_\\mathcal G$ with vertex set $V$ and with faces consisting of independent sets of vertices of $\\mathcal G$. Thus $F$ is a face of $\\Delta_\\mathcal G$ if and only if there is no edge of $\\mathcal G$ joining any two\nvertices of $F$.\n\nThe \\emph{edge ideal} of a simple graph $\\mathcal G$, denoted by $I(\\mathcal G)$, is an ideal of $S$ generated by all squarefree monomials $x_ix_j$ with $x_ix_j\\in E(\\mathcal G)$.\n\nA simple graph $\\mathcal G$ is called ${\\mathrm{CM}}_t$ if $\\Delta_\\mathcal G$ is ${\\mathrm{CM}}_t$ and it is called \\emph{unmixed} if $\\Delta_\\mathcal G$ is pure.\n\nFor a monomial ideal $I\\subset S$, We denote by $G(I)$ the unique minimal set of monomial generators of $I$.\n\n\\section{The expansion functor in combinatorial and algebraic concepts}\n\nIn this section we define the expansion of a simplicial complex and recall the expansion of a simple graph from \\cite{Sc} and the expansion of a monomial ideal from \\cite{BaHe}. We show that these concepts are intimately related to each other.\n\n(1) Let $\\alpha=(k_1,\\ldots,k_n)\\in\\mathbb N^n$. For $F=\\{x_{i_1},\\ldots,x_{i_r}\\}\\subseteq \\{x_1,\\ldots,x_n\\}$ define\n$$F^\\alpha=\\{x_{i_11},\\ldots,x_{i_1k_{i_1}},\\ldots,x_{i_r1},\\ldots,x_{i_rk_{i_r}}\\}$$\nas a subset of $[n]^\\alpha:=\\{x_{11},\\ldots,x_{1k_1},\\ldots,x_{n1},\\ldots,x_{nk_n}\\}$. $F^\\alpha$ is called \\emph{the expansion of $F$ with respect to $\\alpha$.}\n\nFor a simplicial complex $\\Delta=\\langle F_1,\\ldots,F_r\\rangle$ on $[n]$, we define \\emph{the expansion of $\\Delta$ with respect to $\\alpha$} as the simplicial complex\n$$\\Delta^\\alpha=\\langle F^\\alpha_1,\\ldots,F^\\alpha_r\\rangle.$$\n\n(2) The \\emph{duplication} of a vertex $x_i$ of a simple graph $\\mathcal G$ was first introduced by Schrijver \\cite{Sc} and it means extending its vertex set $V(\\mathcal G)$ by a new vertex $x'_i$ and replacing $E(\\mathcal G)$ by\n$$E(\\mathcal G)\\cup\\{(e\\backslash\\{x_i\\})\\cup\\{x'_i\\}:x_i\\in e\\in E(\\mathcal G)\\}.$$\nFor the $n$-tuple $\\alpha=(k_1,\\ldots,k_n)\\in\\mathbb N^n$, with positive integer entries, the \\emph{expansion} of the simple graph $\\mathcal G$ is denoted by $\\mathcal G^\\alpha$ and it is obtained from $\\mathcal G$ by successively duplicating $k_i-1$ times every vertex $x_i$.\n\n(3) In \\cite{BaHe} Bayati and Herzog defined the expansion functor in the category of finitely generated multigraded $S$-modules and studied some homological behaviors of this functor. We recall the expansion functor defined by them only in the category of monomial ideals and refer the reader to \\cite{BaHe} for more general case in the category of finitely generated multigraded $S$-modules.\n\nLet $S^\\alpha$ be a polynomial ring over $K$ in the variables\n$$x_{11},\\ldots,x_{1k_1},\\ldots,x_{n1},\\ldots,x_{nk_n}.$$\nWhenever $I\\subset S$ is a monomial ideal minimally generated by\n$u_1,\\ldots,u_r$, the expansion of $I$ with respect to $\\alpha$ is\ndefined by\n$$I^\\alpha=\\sum^r_{i=1}P^{\\nu_1(u_i)}_1\\ldots P^{\\nu_n(u_i)}_n\\subset S^\\alpha$$\nwhere $P_j=(x_{j1},\\ldots,x_{jk_j})$ is a prime ideal of $S^\\alpha$ and $\\nu_j(u_i)$ is the exponent of $x_j$ in $u_i$.\n\nIt was shown in \\cite{BaHe} that the expansion functor is exact and\nso $(S\/I)^\\alpha=S^\\alpha\/I^\\alpha$. In the following lemmas we\ndescribe the relations between the above three concepts of expansion\nfunctor.\n\n\\begin{lem}\\label{epansion s-R}\nFor a simplicial complex $\\Delta$ we have $I^\\alpha_\\Delta=I_{\\Delta^\\alpha}$. In particular, $K[\\Delta]^\\alpha=K[\\Delta^\\alpha]$.\n\\end{lem}\n\\begin{proof}\nLet $\\Delta=\\langle F_1,\\ldots,F_r\\rangle$. Since $I_\\Delta=\\bigcap^r_{i=1}P_{F^c_i}$, it follows from Lemma 1.1 in \\cite{BaHe} that $I^\\alpha_\\Delta=\\bigcap^r_{i=1}P^\\alpha_{F^c_i}$. The result is obtained by the fact that $P^\\alpha_{F^c_i}=P_{(F^\\alpha_i)^c}$.\n\\end{proof}\n\nLet $u=x_{i_1}\\ldots x_{i_t}\\in S$ be a monomial and $\\alpha=(k_1,\\ldots,k_n)\\in\\mathbb N^n$. We set $u^\\alpha=G((u)^\\alpha)$\nand for a set $A$ of monomials in $S$, $A^\\alpha$ is defined $$A^\\alpha=\\bigcup_{u\\in A} u^\\alpha.$$\nOne can easily obtain the following lemma.\n\n\\begin{lem}\nLet $I\\subset S$ be a monomial ideal and $\\alpha\\in\\mathbb N^n$. Then $G(I^\\alpha)=G(I)^\\alpha$.\n\\end{lem}\n\n\\begin{lem}\nFor a simple graph $\\mathcal G$ on the vertex set $[n]$ and $\\alpha\\in\\mathbb N^n$ we have $I(\\mathcal G^\\alpha)=I(\\mathcal G)^\\alpha$.\n\\end{lem}\n\\begin{proof}\nLet $\\alpha=(k_1,\\ldots,k_n)$ and $P_j=(x_{j1},\\ldots,x_{jk_j})$. Then it follows from Lemma 11(ii,iii) of \\cite{BaHe} that\n$$I(\\mathcal G^\\alpha)=(x_{ir}x_{js}:x_ix_j\\in E(\\mathcal G), 1\\leq r\\leq k_i,1\\leq s\\leq k_j)=\\sum_{x_ix_j\\in E(\\mathcal G)}P_iP_j$$\n$$=\\sum_{x_ix_j\\in E(\\mathcal G)}(x_i)^\\alpha (x_j)^\\alpha=(\\sum_{x_ix_j\\in E(\\mathcal G)}(x_i)(x_j))^\\alpha=I(\\mathcal G)^\\alpha.$$\n\\end{proof}\n\n\\section{The expansion of a ${\\mathrm{CM}}_t$ complex}\n\nThe following proposition gives us some information about the\nexpansion of a simplicial complex which are useful in the proof of\nthe next results.\n\n\\begin{prop}\\label{ex indepen}\nLet $\\Delta$ be a simplicial complex and let $\\alpha\\in\\mathbb N^n$.\n\\begin{enumerate}[\\upshape (i)]\n  \\item For all $i\\leq\\dim(\\Delta)$, there exists an epimorphism $\\theta:\\tilde{H}_{i}(\\Delta^\\alpha;K)\\rightarrow\\tilde{H}_{i}(\\Delta;K)$.\n\nIn particular in this case\n$$\\tilde{H}_{i}(\\Delta^\\alpha;K)\/\\ker(\\theta)\\cong\\tilde{H}_{i}(\\Delta;K);$$\n  \\item For $F\\in\\Delta^\\alpha$ such that $F=G^\\alpha$ for some $G\\in\\Delta$, we have\n$$\\mathrm{link}_{\\Delta^\\alpha}(F)=(\\mathrm{link}_\\Delta (G))^\\alpha;$$\n  \\item For $F\\in\\Delta^\\alpha$ such that $F\\neq G^\\alpha$ for every $G\\in\\Delta$, we have\n$$\\mathrm{link}_{\\Delta^\\alpha}F=\\langle U^\\alpha\\backslash F\\rangle\\ast \\mathrm{link}_{\\Delta^\\alpha}U^\\alpha$$\nfor some $U\\in\\Delta$ with $F\\subseteq U^\\alpha$. Here $\\ast$ means the join of two simplicial complexes.\n\nIn the third case, $\\mathrm{link}_{\\Delta^\\alpha}F$ is a cone and so acyclic,\ni.e., $\\tilde{H}_i(\\mathrm{link}_{\\Delta^\\alpha}F;K)=0$ for all $i>0$.\n\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(i) Consider the map $\\pi:[n]^\\alpha\\rightarrow [n]$ by $\\pi(x_{ij})=x_i$ for all $i,j$. Let the simplicial map $\\varphi:\\Delta^\\alpha\\rightarrow\\Delta$ be defined by $\\varphi(\\{x_{i_1j_1},\\ldots,x_{i_qj_q}\\})=\\{\\pi(x_{i_1j_1}),\\ldots,\\pi(x_{i_qj_q})\\}=\\{x_{i_1},\\ldots,x_{i_q}\\}$. Actually, $\\varphi$ is an extension of $\\pi$ to $\\Delta^\\alpha$ by linearity. Define $\\varphi_\\#:\\tilde{\\mathcal C}_q(\\Delta^\\alpha;K)\\rightarrow\\tilde{\\mathcal C}_q(\\Delta;K)$, for each $q$, by\n$$\\varphi_\\#([x_{i_0j_0},\\ldots,x_{i_qj_q}])=\\left\\{\n  \\begin{array}{ll}\n    0 & \\mbox{if for some indices}\\ i_r=i_t \\\\\n    \\left[\\varphi(\\{x_{i_0j_0}\\}),\\ldots,\\varphi(\\{x_{i_qj_q}\\})\\right] & \\mbox{otherwise}.\n  \\end{array}\n\\right.\n$$\nIt is clear from the definitions of $\\tilde{\\mathcal C}_q(\\Delta^\\alpha;K)$ and $\\tilde{\\mathcal C}_q(\\Delta;K)$ that $\\varphi_\\#$ is well-defined. Also, define $\\varphi_\\alpha:\\tilde{H}_{i}(\\Delta^\\alpha;K)\\rightarrow\\tilde{H}_{i}(\\Delta;K)$ by\n$$\\varphi_\\alpha:z+B_i(\\Delta^\\alpha)\\rightarrow \\varphi_\\#(z)+B_i(\\Delta).$$\nIt is trivial that $\\varphi_\\alpha$ is onto.\n\n(ii) The inclusion $\\mathrm{link}_{\\Delta^\\alpha}(F)\\supseteq(\\mathrm{link}_\\Delta (G))^\\alpha$ is trivial. So we show the reverse inclusion. Let $\\sigma\\in\\mathrm{link}_{\\Delta^\\alpha}(G^\\alpha)$. Then $\\sigma\\cap G^\\alpha=\\emptyset$ and $\\sigma\\cup G^\\alpha\\in\\Delta^\\alpha$. We want to show $\\pi(\\sigma)\\in\\mathrm{link}_\\Delta (G)$. Because in this case, $\\pi(\\sigma)^\\alpha\\in(\\mathrm{link}_\\Delta (G))^\\alpha$ and since that $\\sigma\\subseteq \\pi(\\sigma)^\\alpha$, we can conclude that $\\sigma\\in (\\mathrm{link}_\\Delta (G))^\\alpha$.\n\nClearly, $\\pi(\\sigma)\\cup G\\in\\Delta$. To show that $\\pi(\\sigma)\\cap G=\\emptyset$, suppose, on the contrary, that $x_i\\in \\pi(\\sigma)\\cap G$. Then $x_{ij}\\in \\sigma$ for some $j$. Especially, $x_{ij}\\in G^\\alpha$. Therefore $\\sigma\\cap G^\\alpha\\neq\\emptyset$, a contradiction.\n\n(iii) Let $\\tau\\in\\mathrm{link}_{\\Delta^\\alpha}F$. Let $\\tau\\cap \\pi(F)^\\alpha=\\emptyset$. It follows from $\\tau\\cup F\\in \\Delta^\\alpha$ that $\\pi(\\tau)^\\alpha\\cup\\pi(F)^\\alpha\\in \\Delta^\\alpha$. Now by $\\tau\\subset\\pi(\\tau)^\\alpha$ it follows that $\\tau\\cup\\pi(F)^\\alpha\\in \\Delta^\\alpha$. Hence $\\tau\\in\\mathrm{link}_{\\Delta^\\alpha}(\\pi(F)^\\alpha)$. So we suppose that $\\tau\\cap \\pi(F)^\\alpha\\neq\\emptyset$. We write\n$\\tau=(\\tau\\cap \\pi(F)^\\alpha)\\cup (\\tau\\backslash \\pi(F)^\\alpha)$. It is clear that $\\tau\\cap \\pi(F)^\\alpha\\subset \\pi(F)^\\alpha\\backslash F$ and $\\tau\\backslash \\pi(F)^\\alpha\\in\\mathrm{link}_{\\Delta^\\alpha}\\pi(F)^\\alpha$. The reverse inclusion is trivial.\n\\end{proof}\n\n\\begin{rem}\\label{pure expan}\nLet $\\Delta=\\langle x_1x_2,x_2x_3\\rangle$ be a complex on $[3]$ and\n$\\alpha=(2,1,1)\\in\\mathbb N^3$. Then $\\Delta^\\alpha=\\langle\nx_{11}x_{12}x_{21},x_{21}x_{31}\\rangle$ is a complex on\n$\\{x_{11},x_{12},x_{21},x_{31}\\}$. Notice that $\\Delta$ is pure but\n$\\Delta^\\alpha$ is not. Therefore, the expansion of a pure simplicial\ncomplex is not necessarily pure.\n\\end{rem}\n\n\\begin{thm}\\label{main}\nLet $\\Delta$ be a simplicial complex on $[n]$ of dimension $d-1$ and\nlet $t\\geq 0$ be the least integer that $\\Delta$ is ${\\mathrm{CM}}_t$. Suppose\nthat $\\alpha=(k_1,\\ldots,k_n)\\in\\mathbb N^n$ such that $k_i>1$ for some\n$i$ and $\\Delta^\\alpha$ is pure. Then $\\Delta^\\alpha$ is ${\\mathrm{CM}}_{t+e-k+1}$\nbut it is not ${\\mathrm{CM}}_{t+e-k}$, where $e=\\dim(\\Delta^\\alpha)+1$ and\n$k=\\min\\{k_i:k_i>1\\}$ .\n\\end{thm}\n\\begin{proof}\nWe use induction on $e\\geq 2$. If $e=2$, then $\\dim(\\Delta^\\alpha)=1$ and $\\Delta$ should be only in form $\\Delta=\\langle x_1,\\ldots,x_n\\rangle$. In particular, $\\Delta^\\alpha$ is of the form\n$$\\Delta^\\alpha=\\langle \\{x_{i_11},x_{i_12}\\},\\{x_{i_21},x_{i_22}\\},\\ldots,\\{x_{i_r1},x_{i_r2}\\}\\rangle.$$\nIt is clear that $\\Delta^\\alpha$ is ${\\mathrm{CM}}_1$ but it is not Cohen-Macaulay.\n\nAssume that $e>2$. Let $\\{x_{ij}\\}\\in\\Delta^\\alpha$. We want to show that $\\mathrm{link}_{\\Delta^\\alpha}(x_{ij})$ is ${\\mathrm{CM}}_{e-k}$. Consider the following cases:\n\nCase 1: $k_i>1$. Then\n$$\\mathrm{link}_{\\Delta^\\alpha}(x_{ij})=\\langle\\{x_i\\}^\\alpha\\backslash x_{ij}\\rangle\\ast(\\mathrm{link}_{\\Delta}(x_i))^\\alpha.$$\n$(\\mathrm{link}_{\\Delta}(x_i))^\\alpha$ is of dimension $e-k_i-1$ and, by induction hypothesis, it is ${\\mathrm{CM}}_{t+e-k_i-k+1}$. On the other hand, $\\langle\\{x_i\\}^\\alpha\\backslash x_{ij}\\rangle$ is Cohen-Macaulay of dimension $k_i-2$. Therefore, it follows from Theorem 1.1(i) of \\cite{HaYaZa2} that $\\mathrm{link}_{\\Delta^\\alpha}(x_{ij})$ is ${\\mathrm{CM}}_{t+e-k}$.\n\nCase 2: $k_i=1$. Then\n$$\\mathrm{link}_{\\Delta^\\alpha}(x_{ij})=(\\mathrm{link}_{\\Delta}(x_i))^\\alpha$$\nwhich is of dimension $e-2$ and, by induction, it is ${\\mathrm{CM}}_{t+e-k}$.\n\nNow suppose that $e>2$ and $k_s=k$ for some $s\\in [n]$. Let $F$ be a facet of $\\Delta$ such that $x_s$ belongs to $F$.\n\nIf $\\dim(\\Delta)=0$, then $k_l=k$ for all $l\\in [n]$. In particular, $e=k$. It is clear that $\\Delta^\\alpha$ is not ${\\mathrm{CM}}_{t+e-k}$ (or Cohen-Macaulay). So suppose that $\\dim(\\Delta)>0$. Choose $x_i\\in F\\backslash x_s$. Then\n$$\\mathrm{link}_{\\Delta^\\alpha}(x_{ij})=\\langle \\{x_i\\}^\\alpha\\backslash x_{ij}\\rangle\\ast (\\mathrm{link}_\\Delta(x_i))^\\alpha.$$\nBy induction hypothesis, $(\\mathrm{link}_\\Delta(x_i))^\\alpha$ is not ${\\mathrm{CM}}_{t+e-k_i-k}$. It follows from Theorem 3.1(ii) of \\cite{HaYaZa2} that $\\mathrm{link}_{\\Delta^\\alpha}(x_{ij})$ is not ${\\mathrm{CM}}_{t+e-k-1}$. Therefore $\\Delta^\\alpha$ is not ${\\mathrm{CM}}_{t+e-k}$.\n\\end{proof}\n\n\\begin{cor}\nLet $\\Delta$ be a non-empty Cohen-Macaulay simplicial complex on\n$[n]$. Then for any $\\alpha\\in\\mathbb N^n$, with $\\alpha\\neq\\mathbf 1$,\n$\\Delta^\\alpha$ can never be Cohen-Macaulay.\n\\end{cor}\n\n\\section{The contraction functor}\n\nLet $\\Delta=\\langle F_1,\\ldots,F_r\\rangle$ be a simplicial complex on $[n]$. Consider the equivalence relation `$\\sim$' on the vertices of $\\Delta$ given by\n$$x_i\\sim x_j\\Leftrightarrow\\langle x_i\\rangle\\ast\\mathrm{link}_\\Delta(x_i)=\\langle x_j\\rangle\\ast\\mathrm{link}_\\Delta(x_j).$$\nIn fact $\\langle x_i\\rangle\\ast\\mathrm{link}_\\Delta(x_i)$ is the cone over\n$\\mathrm{link}_\\Delta(x_i)$, and the elements of\n$\\langle x_i\\rangle\\ast\\mathrm{link}_\\Delta(x_i)$ are those faces of $\\Delta$,\nwhich contain $x_i$. Hence  $\\langle x_i\\rangle\\ast\\mathrm{link}_\\Delta(x_i)\n=\\langle x_j\\rangle\\ast\\mathrm{link}_\\Delta(x_j)$, means the cone with vertex $x_i$ is equal to the cone with vertex $x_j$. In other words,\n$x_i\\sim x_j$ is equivalent to saying that for a facet $F\\in\\Delta$, $F$ contains $x_i$ if and only if it contains $x_j$.\n\nLet $[\\bar{m}]=\\{\\bar{y}_1,\\ldots,\\bar{y}_m\\}$ be the set of\nequivalence classes under $\\sim$. Let\n$\\bar{y}_i=\\{x_{i1},\\ldots,x_{ia_i}\\}$. Set\n$\\alpha=(a_1,\\ldots,a_m)$. For $F_t\\in\\Delta$, define\n$G_t=\\{\\bar{y}_i:\\bar{y}_i\\subset F_t\\}$ and let $\\Gamma$ be a\nsimplicial complex on the vertex set $[m]$ with facets\n$G_1,\\ldots,G_r$. We call $\\Gamma$  the \\emph{contraction of $\\Delta$\nby $\\alpha$} and $\\alpha$ is called \\emph{the vector obtained from\ncontraction}.\n\nFor example, consider the simplicial complex $\\Delta=\\langle x_1x_2x_3,x_2x_3x_4,x_1x_4x_5,x_2x_3x_5\\rangle$\non the vertex set $[5]=\\{x_1,\\ldots,x_5\\}$. Then $\\bar{y}_1=\\{x_1\\}$, $\\bar{y}_2=\\{x_2,x_3\\}$, $\\bar{y}_3=\\{x_4\\}$,\n$\\bar{y}_4=\\{x_5\\}$ and $\\alpha=(1,2,1,1)$. Therefore, the contraction of $\\Delta$ by $\\alpha$ is\n$\\Gamma=\\langle \\bar{y}_1\\bar{y}_2,\\bar{y}_2\\bar{y}_3,\\bar{y}_1\\bar{y}_3\\bar{y}_4,\\bar{y}_2\\bar{y}_4\\rangle$ a\ncomplex on the vertex set $[\\bar{4}]=\\{\\bar{y}_1,\\ldots,\\bar{y}_4\\}$.\n\n\\begin{rem}\nNote that if $\\Delta$ is a pure simplicial complex then the contraction of $\\Delta$ is not necessarily pure (see the above example). In the special case where the vector $\\alpha=(k_1,\\dots,k_n)\\in\\mathbb N^n$ and $k_i=k_j$ for all $i,j$, it is easy to check that in this case $\\Delta$ is pure if and only if $\\Delta^\\alpha$ is pure. Another case is introduced in the following proposition.\n\\end{rem}\n\n\\begin{prop}\nLet $\\Delta$ be a simplicial complex on $[n]$ and assume that\n$\\alpha=(k_1,\\dots,k_n)\\in\\mathbb N^n$ satisfies the following condition:\n\n$(\\dag)$ for all facets $F,G\\in\\Delta$, if $x_i\\in F\\backslash G$ and $x_j\\in G\\backslash F$ then $k_i=k_j$.\n\nThen $\\Delta$ is pure if and only if $\\Delta^\\alpha$ is pure.\n\\end{prop}\n\\begin{proof}\nLet $\\Delta$ be a pure simplicial complex and let $F,G\\in\\Delta$ be two facets of $\\Delta$. Then $$|F^\\alpha|-|G^\\alpha|=\\sum_{x_i\\in F}k_i-\\sum_{x_i\\in G}k_i=\\sum_{x_i\\in F\\backslash G}k_i-\\sum_{x_i\\in G\\backslash F}k_i.$$\nNow the condition $(\\dag)$ implies that $|F^\\alpha|=|G^\\alpha|$. This means that all facets of $\\Delta^\\alpha$ have the same cardinality.\n\nLet $\\Delta^\\alpha$ be pure. Suppose that $F,G$ are two facets in\n$\\Delta$. If $|F|>|G|$ then $|F\\backslash G|>|G\\backslash F|$.\nTherefore $\\sum_{x_i\\in F\\backslash G}k_i>\\sum_{x_i\\in G\\backslash\nF}k_i$. This concludes that $|F^\\alpha|=\\sum_{x_i\\in\nF}k_i>\\sum_{x_i\\in G}k_i=|G^\\alpha|$, a contradiction.\n\\end{proof}\n\nThere is a close relationship between a simplicial complex and its contraction. In fact, the expansion of the contraction of a simplicial complex is the same complex. The precise statement is the following.\n\n\\begin{lem}\nLet $\\Gamma$ be the contraction of $\\Delta$ by $\\alpha$. Then $\\Gamma^\\alpha\\cong \\Delta$.\n\\end{lem}\n\\begin{proof}\nSuppose that $\\Delta$ and $\\Gamma$ are on the vertex sets $[n]=\\{x_1,\\ldots,x_n\\}$ and $[\\bar{m}]=\\{\\bar{y}_1,\\ldots,\\bar{y}_m\\}$, respectively. Let $\\alpha=(a_1,\\ldots,a_m)$. For $\\bar{y}_i\\in\\Gamma$, suppose that $\\{\\bar{y}_i\\}^\\alpha=\\{\\bar{y}_{i1},\\ldots,\\bar{y}_{ia_i}\\}$. So $\\Gamma^\\alpha$ is a simplicial complex on the vertex set $[\\bar{m}]^\\alpha=\\{\\bar{y}_{ij}:i=1,\\ldots,m,\\ j=1,\\ldots,a_i\\}$. Now define $\\varphi:[\\bar{m}]^\\alpha\\rightarrow [n]$ by $\\varphi(\\bar{y}_{ij})=x_{ij}$. Extending $\\varphi$, we obtain the isomorphism $\\varphi:\\Gamma^\\alpha\\rightarrow \\Delta$.\n\\end{proof}\n\n\\begin{prop}\\label{CM indepen}\nLet $\\Delta$ be a simplicial complex and assume that $\\Delta^\\alpha$ is\nCohen-Macaulay for some $\\alpha\\in\\mathbb N^n$. Then $\\Delta$ is\nCohen-Macaulay.\n\\end{prop}\n\\begin{proof}\nBy Lemma \\ref{ex indepen}(i), for all $i\\leq \\dim(\\mathrm{link}_\\Delta F)$ and all $F\\in\\Delta$ there exists an epimorphism $\\theta:\\mathrm{link}_{\\Delta^\\alpha} F^\\alpha\\rightarrow \\mathrm{link}_\\Delta F$ such that\n$$\\tilde{H}_{i}(\\mathrm{link}_{\\Delta^\\alpha} F^\\alpha;K)\/\\ker(\\theta)\\cong\\tilde{H}_{i}(\\mathrm{link}_\\Delta F;K).$$\nNow suppose that $i<\\dim(\\mathrm{link}_\\Delta F)$. Then $i<\\dim(\\mathrm{link}_{\\Delta^\\alpha} G^\\alpha)$ and by Cohen-Macaulayness of $\\Delta^\\alpha$, $\\tilde{H}_{i}(\\mathrm{link}_{\\Delta^\\alpha} F^\\alpha;K)=0$. Therefor $\\tilde{H}_{i}(\\mathrm{link}_\\Delta F;K)=0$. This means that $\\Delta$ is Cohen-Macaulay.\n\\end{proof}\n\nIt follows from Proposition \\ref{CM indepen} that:\n\n\\begin{cor}\\label{CM-ness}\nThe contraction of a Cohen-Macaulay simplicial complex $\\Delta$ is Cohen-Macaulay.\n\\end{cor}\n\nThis can be generalized in the following theorem.\n\n\\begin{thm}\\label{contract,CM-t}\nLet $\\Gamma$ be the contraction of a ${\\mathrm{CM}}_t$ simplicial complex\n$\\Delta$, for some $t\\geq 0$, by $\\alpha=(k_1,\\ldots,k_n)$. If\n$k_i\\geq t$ for all $i$ and $\\Gamma$ is pure, then $\\Gamma$ is\nBuchsbaum.\n\\end{thm}\n\\begin{proof}\nIf $t=0$, then we saw in Corollary \\ref{CM-ness} that $\\Gamma$ is Cohen-Macaulay and so it is ${\\mathrm{CM}}_t$. Hence assume that $t>0$. Let $\\Delta=\\langle F_1,\\ldots,F_r\\rangle$. We have to show that $\\tilde{H}_i(\\mathrm{link}_\\Gamma G;K)=0$, for all faces $G\\in\\Gamma$ with $|G|\\geq 1$ and all $i<\\dim(\\mathrm{link}_\\Gamma G)$.\n\nLet $G\\in\\Gamma$ with $|G|\\geq 1$. Then $|G^\\alpha|\\geq t$. It follows from Lemma \\ref{CM-t eq} and ${\\mathrm{CM}}_t$-ness of $\\Delta$ that\n$$\\tilde{H}_{i}(\\mathrm{link}_\\Gamma G;K)\\cong\\tilde{H}_{i}(\\mathrm{link}_\\Delta G^\\alpha;K)=0$$\nfor $i<\\dim(\\mathrm{link}_\\Delta G^\\alpha)$ and, particularly, for $i<\\dim(\\mathrm{link}_\\Gamma G)$. Therefore $\\Gamma$ is Buchsbaum.\n\\end{proof}\n\n\\begin{cor}\nLet $\\Gamma$ be the contraction of a Buchsbaum simplicial complex\n$\\Delta$. If $\\Gamma$ is pure, then $\\Gamma$ is also Buchsbaum.\n\\end{cor}\n\nLet $\\mathcal G$ be a simple graph on the vertex set $[n]$ and let $\\Delta_\\mathcal G$ be its independence complex on $[n]$, i.e., a simplicial complex whose faces are the independent vertex sets of $G$. Let $\\Gamma$ be the contraction of $\\Delta_\\mathcal G$. In the following we show that $\\Gamma$ is the independence complex of a simple graph $\\mathcal H$. We call $\\mathcal H$ the \\emph{contraction} of $\\mathcal G$.\n\n\\begin{lem}\nLet $\\mathcal G$ be a simple graph. The contraction of $\\Delta_\\mathcal G$ is the independence complex of a simple graph $\\mathcal H$.\n\\end{lem}\n\\begin{proof}\nIt suffices to show that $I_\\Gamma$ is a squarefree monomial ideal generated in degree 2. Let $\\Gamma$ be the contraction of $\\Delta_\\mathcal G$ and let $\\alpha=(k_1,\\ldots,k_n)$ be the vector obtained from the contraction. Let $[n]=\\{x_1,\\ldots,x_n\\}$ be the vertex set of $\\Gamma$. Suppose that $u=x_{i_1}\\ldots x_{i_t}\\in G(I_\\Gamma)$. Then $u^\\alpha\\subset G(I_\\Gamma)^\\alpha=G(I_{\\Delta_\\mathcal G})=G(I(\\mathcal G)$. Since $u^\\alpha=\\{x_{i_1j_1}\\ldots x_{i_tj_t}:1\\leq j_l\\leq k_{i_l},1\\leq l\\leq t\\}$ we have $t=2$ and the proof is completed.\n\\end{proof}\n\n\\begin{exam}\nLet $\\mathcal G_1$ and $\\mathcal G_2$ be, respectively, from left to right the following graphs:\n\n$$\\begin{array}{cccc}\n\\begin{tikzpicture}\n\\coordinate (a) at (0,0);\\fill (0,0) circle (1pt);\n\\coordinate (b) at (0,1);\\fill (0,1) circle (1pt);\n\\coordinate (c) at (1,0);\\fill (1,0) circle (1pt);\n\\coordinate (d) at (0,-1);\\fill (0,-1) circle (1pt);\n\\coordinate (e) at (-1,0);\\fill (-1,0) circle (1pt);\n\\draw[black] (a) -- (c) -- (d) -- (e) -- (b);\n\\end{tikzpicture}\n&&&\n\\begin{tikzpicture}\n\\coordinate (a) at (0,0);\\fill (0,0) circle (1pt);\n\\coordinate (b) at (0,1);\\fill (0,1) circle (1pt);\n\\coordinate (c) at (1,0);\\fill (1,0) circle (1pt);\n\\coordinate (d) at (0,-1);\\fill (0,-1) circle (1pt);\n\\coordinate (e) at (-1,0);\\fill (-1,0) circle (1pt);\n\\draw[black] (e) -- (a) -- (c) -- (d) -- (e) -- (b) -- (c);\n\\end{tikzpicture}\n\\end{array}$$\n\nThe contraction of $\\mathcal G_1$ and $\\mathcal G_2$ are\n\n$$\\begin{array}{cccc}\n\\begin{tikzpicture}\n\\coordinate (a) at (0,0);\\fill (0,0) circle (1pt);\n\\coordinate (b) at (0,1);\\fill (0,1) circle (1pt);\n\\coordinate (c) at (1,0);\\fill (1,0) circle (1pt);\n\\coordinate (d) at (0,-1);\\fill (0,-1) circle (1pt);\n\\coordinate (e) at (-1,0);\\fill (-1,0) circle (1pt);\n\\draw[black] (a) -- (c) -- (d) -- (e) -- (b);\n\\end{tikzpicture}\n&&&\n\\begin{tikzpicture}\n\\coordinate (a) at (0,0);\\fill (0,0) circle (1pt);\n\\coordinate (b) at (1,0);\\fill (1,0) circle (1pt);\n\\draw[black] (a) -- (b);\n\\end{tikzpicture}\n\\end{array}$$\nThe contraction of $\\mathcal G_1$ is equal to itself but $\\mathcal G_2$ is contracted to an edge and the vector obtained from contraction is $\\alpha=(2,3)$.\n\\end{exam}\n\nWe recall that a simple graph is ${\\mathrm{CM}}_t$ for some $t\\geq 0$, if the associated independence complex is ${\\mathrm{CM}}_t$.\n\n\\begin{rem}\nThe simple graph $\\mathcal G'$ obtained from $\\mathcal G$ in Lemma 4.3 and Theorem\n4.4 of \\cite{HaYaZa2} is the expansion of $\\mathcal G$. Actually, suppose\nthat $\\mathcal G$ is a bipartite graph on the vertex set $V(\\mathcal G)=V\\cup W$\nwhere $V=\\{x_1,\\ldots,x_d\\}$ and $W=\\{x_{d+1},\\ldots,x_{2d}\\}$. Then\nfor $\\alpha=(n_1,\\ldots,n_d,n_1,\\ldots,n_d)$ we have\n$\\mathcal G'=\\mathcal G^\\alpha$. It follows from Theorem \\ref{main} that if $\\mathcal G$ is\n${\\mathrm{CM}}_t$ for some $t\\geq 0$ then $\\mathcal G'$ is ${\\mathrm{CM}}_{t+n-n_{i_0}+1}$ where\n$n=\\sum^d_{i=1}n_i$ and $n_{i_0}=\\min\\{n_i>1:i=1,\\ldots,d\\}$. This\nimplies that the first part of Theorem 4.4 of \\cite{HaYaZa2} is an\nobvious consequence of Theorem \\ref{main} for $t=0$.\n\\end{rem}\n\n\\subsection*{Acknowledgment}\n\nThe author would like to thank Hassan Haghighi from K. N. Toosi University of Technology and Rahim Zaare-Nahandi from University of Tehran for careful\nreading an earlier version of this article and for their helpful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nRandom unitary operators have often been used to approximate chaotic dynamics. Notably, in the context of black holes Hayden and Preskill used a model of random dynamics to show such systems can be efficient information scramblers; initially localized information will quickly thermalize and become thoroughly mixed across the entire system \\cite{Hayden07}.  It was later conjectured \\cite{Sekino08} and then proven \\cite{Shenker:2013pqa,Maldacena:2015waa} that black holes are the fastest scramblers in nature, suggesting their dynamics must have much in common with random unitary evolution. Such ``scrambling'' is a byproduct of strongly-coupled chaotic dynamics \\cite{Lashkari13,Almheiri13b,Hosur:2015ylk}, and so the work of \\cite{Hayden07} suggests there should be strong quantitative connection between such chaos and pseudorandomness.\\footnote{\n    Scrambling has a close connection to the decoupling theorem, see e.g.~\\cite{Berta, Brown15}. Also, see \\cite{Chamon14,Hayden:2016cfa,Nahum:2016muy} for studies connecting randomness to entanglement.\n}\n\n\n\nThe connection between pseudorandomness and chaotic dynamics can also be understood at the level of operators. For example, consider $W$ a local operator of low weight (e.g. a Pauli operator acting on a single spin). With a chaotic Hamiltonian $H$, the operator $W(t)=e^{iHt} W e^{-iHt}$ will be a complicated nonlocal operator that has an expansion as a sum of products of many local operators with an exponential number of terms each with a pseudorandom coefficient \\cite{Roberts:2014isa}. We can gain intuition for this by considering the Baker-Campbell-Hausdorff expansion of $W(t)$\n\\begin{equation}\n   W(t) = \\sum_{j=0}^\\infty \\frac{(it)^j}{j!}\\underbrace{[H, \\dots[H}_j,W\\underbrace{] \\dots]}_j. \\label{eq:BCH}\n\\end{equation}\nIf $H$ is $q$-local and sufficiently ``generic'' to contain all possible $q$-local interactions, then the $j$th term will consist of a sum of roughly $\\sim (n\/q)^{qj}$ terms each of weight ranging from $1$ to $\\sim j(q-1)$, where we assume the system consists of $n$ spins so that the Hilbert space is of dimension $d=2^n$: \n\\begin{itemize}\n    \\item At roughly $j \\sim n \/ (q-1)$,  there will be many terms in the sum of weight $n$. These terms are delocalized over the entire system. For a system without spatial locality, the relationship between time $t$ and when the $j$th term becomes $O(1)$ is roughly $t \\sim \\log j$. The timescale $t \\sim O(\\log n)$ for the operator to cover the entire system is indicative of fast-scrambling behavior.\n    \\item At around $j \\sim 2n \/ q \\log(n\/q)$, the total number of terms will reach $2^{2n}$, equal to the total number of orthogonal linear operators acting on the Hilbert space.\\footnote{The number of terms is actually not a well defined quantity, since it can change under local rotations of spin, and should possibly instead consider the minimum number of terms under all local changes of basis. For chaotic systems, the distinction should not be important, since there will not exist a local change of basis where the number of terms drastically simplifies. We thank Douglas Stanford who thanks Juan Maldacena for raising this point.} Even after the operator covers the entire system, it continues to spread over the unitary group (though possibly only until a time roughly $O(\\log n) +$ a constant). \n\\end{itemize}\nFurthermore, the coefficient of any given term will be incredibly complicated, depending on the details of the path through the interaction graph and the time. \nOver time, $W(t)$ should cover the entire unitary group (possibly quotiented by a group set by the symmetries of the Hamiltonian). At sufficiently large $t$, one might even suspect that for many purposes $W(t)$ can be approximated by a random operator $\\tilde{W}\\equiv U^\\dagger W U$, with $U$ sampled randomly from the unitary group.\\footnote{Of course, for a less generic and sparser Hamiltonian e.g. a free system, the expansion of $W(t)$ will organize itself in a way such that the commutator in Eq.~\\eqref{eq:BCH} does not produce an exponential number of terms. Said another way, the terms in the expansion of $W(t)$ will only cover a small subset of the unitary group, and the assumption of uniform randomness will not hold.} If this is true, then we would say that $W(t)$ behaves pseudorandomly.\n\n\n\n\n\n\\subsubsection*{Chaos}\n\nThis pattern of growth of $W(t)$ can be measured by a second local operator $V$. For example, the group commutator of $W(t)$ with $V$, given by $W(t)^\\dagger\\, V^\\dagger \\, W(t)\\, V$, measures the effect of the small perturbation $V$ on a later measurement of $W$. In other words, it is a measure of the butterfly effect and the strength of chaos:\n\\begin{itemize}\n    \\item If $W(t)$ is of low weight and few terms, then $W(t)$ and $V$ approximately commute $[W(t),V]\\approx 0$, and the operator $W(t)^\\dagger\\, V^\\dagger \\, W(t)\\, V$ is close to the identity. \n    \\item If instead the dynamics are strongly chaotic, $W(t)$ will grow to eventually have a large commutator with all other local operators in the system (in fact, just about all other operators), and so $W(t)^\\dagger\\, V^\\dagger \\, W(t)\\, V$ will be nearly random and have a small expectation in most states.\n\\end{itemize}\nThus, the decay of out-of-time-order (OTO) four-point functions of the form\n\\begin{equation}\n    \\langle W(t)^\\dagger\\, V^\\dagger \\, W(t)\\, V \\rangle = \\langle U(t)^\\dagger W^\\dagger U(t) ~ V^\\dagger ~ U(t)^\\dagger W U(t) ~ V \\rangle, \\label{oto-4pt-def}\n\\end{equation} \ncan act as a simple diagnostic of quantum chaos, where \n$U(t)=e^{-iHt}$ is the unitary time evolution operator, and the correlator is usually evaluated on the thermal state $\\langle \\cdot \\rangle \\equiv \\text{tr} \\, \\{ e^{-\\beta H} \\, \\cdot \\, \\} \/ \\text{tr} \\, e^{-\\beta H}$ \\cite{Larkin:1969abc,Almheiri13b,Shenker:2013pqa,Kitaev:2014t1,Maldacena:2015waa}.\\footnote{Usually $W, V$ are taken to be Hermitian, but here we will more generally allow them to be unitary.} For further discussion, please see a selection (but by all means not a complete set) of recent work on out-of-time-order four-point functions and chaos \\cite{Shenker:2013pqa,Shenker:2013yza,Roberts:2014isa,Roberts:2014ifa,Kitaev:2014t1,Shenker:2014cwa,Maldacena:2015waa,Hosur:2015ylk,Stanford:2015owe,Fitzpatrick:2016thx,Gu:2016hoy,Caputa:2016tgt,Swingle:2016var,Perlmutter:2016pkf,Blake:2016wvh,Roberts:2016wdl,Blake:2016sud,Swingle:2016jdj,Huang:2016knw,fan2016out,Halpern:2016zcm}.\n\n\n\nFor sufficiently chaotic systems and sufficiently large times, the correlators Eq.~\\eqref{oto-4pt-def} will reach a floor value equivalent to the substitution, $W(t) \\to U^\\dagger W U$ with $U$ chosen randomly. \nFurthermore, it can be shown \\cite{Hosur:2015ylk} that the decay of correlators Eq.~\\eqref{oto-4pt-def} implies the sort of information-theoretic scrambling studied by Hayden and Preskill in \\cite{Hayden07}. This explains why the random dynamics model of \\cite{Hayden07} was such a good approximation for strongly-chaotic systems, such as black holes. However, are out-of-time-order four-point functions Eq.~\\eqref{oto-4pt-def} actually a sufficient diagnostic of chaos?\n\n\n\nIn \\cite{Hayden07} the authors did not actually require the dynamics to be a uniformly random unitary operator sampled from the Haar measure on the unitary group. Instead, it would have been sufficient to sample from a simpler ensemble of operators that could reproduce only a few moments of the larger Haar ensemble.\\footnote{Inspection of Eq.~\\eqref{oto-4pt-def} suggests that only two moments are required since there are only two copies of $U$ and two copies of $U^\\dagger$ in the correlator. We will explain this more carefully in the rest of the paper.} Of course, there may be other finer-grained information theoretic properties of a system that are dependent on higher moments and would require a larger ensemble to replicate the statistics of the Haar random dynamics. If random dynamics is a valid approximation for computing some, but not all, of these finer-grained quantities then they can represent a measure of the degree of pseudorandomness of the underlying dynamics. In this paper, we will make some progress in developing some of these finer-grained quantities and therefore connect measures of chaos to measures of randomness.\n\n\\subsubsection*{Unitary design}\n\nThe extent to which an ensemble of operators behaves like the uniform distribution can be quantified by the notion of unitary $k$-designs \\cite{DiVincenzo02, Emerson05, Ambainis07, Gross07, Dankert09}.\\footnote{N.B. in the literature, these are often referred to as unitary $t$-designs. Here, we will reserve $t$ for time. For recent work on unitary designs, see~\\cite{Emerson03, Emerson05, Brown10, Brown15, Hayden07, Harrow09, Knill08, Brandao12, Kueng15, Webb15, Low_thesis, Scott08, Zhu15, Collins16,Nakata:2016blv}.} A unitary $k$-design is a subset of the unitary group that replicates the statistics of at least $k$ moments of the distribution. Consider a finite-dimensional Hilbert space $\\mathcal{H}^{\\otimes k}=(\\mathbb{C}^{d})^{\\otimes k}$ consisting of $k$ copies of $\\mathcal{H}=\\mathbb{C}^d$. Given an ensemble of unitary operators $\\mathcal{E}=\\{p_{j},U_{j}\\}$ acting on $\\mathcal{H}$ with probability distribution $p_{j}$, the ensemble $\\mathcal{E}$ is a unitary $k$-design if and only if\n    \\begin{align}\n        \\sum_{j}p_{j} (\\underbrace{U_{j}\\otimes \\cdots \\otimes U_{j}}_{k}) \\rho (\\underbrace{U_{j}^{\\dagger}\\otimes \\cdots \\otimes U_{j}^{\\dagger}}_{k}) = \\int_{\\text{Haar}}dU(\\underbrace{U\\otimes \\cdots \\otimes U}_{k}) \\rho (\\underbrace{U^{\\dagger}\\otimes \\cdots \\otimes U^{\\dagger}}_{k}), \n    \\end{align}\nfor all quantum states $\\rho$ in $\\mathcal{H}^{\\otimes k}$. Therefore, whether or not a given ensemble $\\mathcal{E}$ forms at most a $k$-design is a fine-grained notion of the randomness of the ensemble.\\footnote{With this definition, a $k+1$-design is automatically a $k$-design.}\n\n\n\nInspection of Eq.~\\eqref{oto-4pt-def} suggests that a $2$-design is sufficient for OTO four-point functions to decay. This was also the requirement for a system to appear scrambled \\cite{Page93,Hayden07,Sekino08}. However, the four-point functions Eq.~\\eqref{oto-4pt-def} are not the only observables related to chaos and the butterfly effect. In an attempt to understand the geometry and interior of a typical microstate of a holographic black hole, Shenker and Stanford constructed a series of black hole geometries by making multiple perturbations of an essentially maximally entangled state \\cite{Shenker:2013yza}. While they were unfortunately unsuccessful at constructing a generic microstate, studying correlation functions in these geometries leads one to a family of $2k$-point out-of-time-order correlation functions of the form\n    \\begin{equation}\n        \\T{\\mc{W}^\\dagger \\, V(0)^\\dagger \\, \\mc{W} \\, V(0) } = \\T{W_1(t_1)^\\dagger \\cdots W_{k-1}(t_{k-1})^\\dagger ~ V(0)^\\dagger ~ W_{k-1}(t_{k-1}) \\cdots W_1(t_1) ~ V(0) }, \\label{multiple-shocks-correlator}\n    \\end{equation}\nwhere $V(0)$ and the $W_j(0)$ are local operators, and $\\mc{W} \\equiv W_{k-1}(t_{k-1}) \\cdots W_1(t_1)$ is a composite operator that lets us understand how the correlator is organized. The existence of these correlators begs the question: do these contain any additional information about the system or are they redundant with the four-point functions? The fact that the correlators Eq.~\\eqref{multiple-shocks-correlator} involve many more copies of unitary time evolution, suggests a relationship between higher-point functions and unitary design. In fact, we will show that a generalization of the OTO higher-point correlators Eq.~\\eqref{multiple-shocks-correlator} are probes of unitary design.  \n\n\n\\subsubsection*{Complexity}\n\n\n\nProbes of chaos and unitary design can only indirectly address a sharp question of recent interest in holography: the existence and growth of the black hole interior via the time evolution of the state. To be precise, one cannot make use of unitary designs since evolution by a time-independent Hamiltonian does not define any ensemble. However, even in this setting we still expect that higher-point OTO should be able to see fine-grained properties of time evolved states that are not captured by four-point OTO correlators.\n\nOne fine-grained quantity is the quantum circuit complexity of a quantum state $|\\psi(t)\\rangle = U(t)|\\psi(0)\\rangle$. Consider a simple initial state, such as the product state $|\\psi(0)\\rangle= |0\\rangle^{\\otimes n}$, undergoing chaotic time evolution. After a short time thermalization time of $O(1)$, the system will evolve to an equilibrium in which local quantities will reach their thermodynamic values. Next, after the scrambling time of $O(\\log n)$, the initial state $|\\psi(0)\\rangle$ will be forgotten: the information will be distributed in such a way that measurements of even a large number of collections of local quantities of $|\\psi(t)\\rangle$ will not reveal the initial state. However, even after the scrambling time, the quantum circuit complexity of the time-evolved state $|\\psi(t)\\rangle$, as quantified by the number of elementary quantum gates necessary to reach it from the initial product state, will continue to evolve. In fact, it is expected to keep growing linearly in time until it saturates at a time exponential in the system size $e^{O(n)}$ \\cite{Knill95,Susskind:2015toa}. \n\n\n\n\n\n\\sbreak\n\nWe hope this presents an intuitive picture that chaos, pseudorandomness, and quantum circuit complexity should be related.  \nTo that end, having first established a connection between higher-point out-of-time-order correlators and pseudorandomness, we will next connect the randomness of an ensemble to computational complexity. Finally, we will use correlation functions to probe pseudorandomness by comparing different random averages to expectations from time evolution.\n\n\n\nBelow, we will summarize our main results, deferring the technical statements to the body and appendices.\n\n\n \n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\\subsection*{Main results}\n\nWe will focus on a particular form of $2k$-point correlation functions evaluated on the maximally mixed state $\\rho=\\frac{1}{d}I$\n    \\begin{equation}\n        \\langle A_{1} \\, U^{\\dagger}B_1U \\cdots A_{k} \\, U^{\\dagger}B_kU  \\rangle := \\frac{1}{2^n}\\text{tr} \\, \\{ A_{1}\\, U^{\\dagger}B_1U \\cdots A_{k}\\, U^{\\dagger}B_kU \\},  \\label{OCO-correlator}\n    \\end{equation}\nwhere any of the $A_{j}, B_{j}$ may be a product Pauli operators that act on a single spin.\\footnote{To be clear, this means that the operators we are correlating are not necessarily simple or local.} Note that each of the $B_j$ is conjugated by the same unitary $U$ (which is similar to picking all the time arguments in Eq.~\\eqref{multiple-shocks-correlator} to be either $0$ or $t$). Furthermore, $U$ will not necessarily represent Hamiltonian time evolution, and instead we will let $U$ be sampled from some ensemble.\\footnote{As a result, these correlators are not really out-of-\\emph{time}-order, since there may not be a notion of time. Instead, they are probably more accurately called \\emph{out-of-complexity-order} (OCO) since we might generalize the notion of time ordering to complexity ordering, where we put unitaries of smaller complexity to the right of unitaries of larger complexity. In order to (hopefully) avoid confusion, we will continue to call the $2k$-point functions in Eq.~\\eqref{OCO-correlator} out-of-time-order, despite there not necessarily being a notion of time.} From this point forward, we will use the notation\n\\begin{equation}\n\\tilde{B} \\equiv U^{\\dagger}BU,\n\\end{equation}\nto simplify expressions involving unitary conjugation. Therefore, we can represent the ensemble average of OTO $2k$-point correlation functions as\n\\begin{align}\n\\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\rangle_{\\mathcal{E}} := \\int_{\\mathcal{E}} dU \\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\rangle,\n\\end{align}\nwhere the integral is with respect to the probability distribution in an ensemble of unitary operators $\\mathcal{E}=\\{p_{j},U_{j}\\}$. Finally, the $k$-fold channel over the ensemble $\\mathcal{E}$ is \n\\begin{align}\n\\Phi_{\\mathcal{E}}^{(k)}(\\cdot) = \\int_{\\mathcal{E}} dU (U_{j}\\otimes \\cdots \\otimes U_{j}) (\\cdot )(U_{j}^{\\dagger}\\otimes \\cdots \\otimes U_{j}^{\\dagger}),\n\\end{align}\nwhich is a superoperator. \n\n\\subsubsection*{Chaos and $k$-designs}\n\nFirst,  we will prove a theorem stating that a particular set of $2k$-point OTO correlators, averaged over an ensemble $\\mathcal{E}$, is in a one-to-one correspondence with the $k$-fold channel~$\\Phi_{\\mathcal{E}}^{(k)}$ \n\\begin{align}\n\\text{$2k$-point OTO correlators}\\ \\leftrightarrow \\ \\text{$k$-fold channel $\\Phi_{\\mathcal{E}}^{(k)}$},\n\\end{align}\nand we provide a simple formula to convert from one to the other. Such an explicit relation between OTO correlators and the $k$-fold channel may have practical and experimental applications such as statistical testing (e.g. a quantum analog of the $\\chi^2$-test) and randomized bench marking~\\cite{Knill08}. \n\n\n\nNext, we prove that generic ``smallness'' of $2k$-point OTO correlators implies that the ensemble $\\mathcal{E}$ is close to $k$-design. We will make this statement precise by relating OTO correlators to a useful quantity known as the frame potential \n\\begin{align}\nF_{\\mathcal{E}}^{(k)}=\\frac{1}{|\\mathcal{E}|^{2}}\\sum_{U,V\\in\\mathcal{E}}\\big|\\text{tr}\\, \\{ U^{\\dagger}V \\} \\big|^{2k}.\n\\end{align}\nThis quantity, first introduced in \\cite{Scott08}, measures the ($2$-norm) distance between $\\Phi_{\\mathcal{E}}^{(k)}(\\cdot)$ and $\\Phi_{\\text{Haar}}^{(k)}(\\cdot)$, and has been shown to be minimized if and only if the ensemble $\\mathcal{E}$ is $k$-design. We will derive the following formula:\n\\begin{align}\n\\text{Average of $|\\text{$2k$-point OTO correlator}|^{2}$}\\ \\propto \\ \\text{$k$th frame potential $F_{\\mathcal{E}}^{(k)}$},\n\\end{align}\nwhich shows that $2k$-point OTO correlators $\\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\rangle_{\\mathcal{E}}$ are measures of whether an ensemble $\\mathcal{E}$ is a unitary $k$-design.\nThus, the decay of OTO correlators can be used to quantify an increase in pseudorandomness. \n\n\\subsubsection*{Chaos, randomness, and complexity}\n\n\n\nWe prove a lower bound on quantum circuit complexity needed to generate an ensemble of unitary operators $\\mathcal{E}$\n\\begin{align}\n\\text{Complexity of $\\mathcal{E}$}\\ \\geq \\ \\frac{2kn \\log(2) - \\log F_{\\mathcal{E}}^{(k)} }{\\log( \\text{choices})}.\\label{intro-complexity-lower-bound}\n\\end{align}\nThis bound is actually given by a rather simple counting argument. The denominator should be thought of as (the log of) the number of choices made at each step in generating the circuit.  For instance, if we have a set $\\mc{G}$ of cardinality $g$ of $q$-qubit quantum gates available and at each step randomly select $q$ qubits out of $n$ total and select one of the gates in $\\mc{G}$ to apply, then we would make $g\\binom{n}{q}$ choices at each step. Recalling our result relating OTO correlators and the frame potential, this result implies that generic smallness of OTO correlators leads to higher quantum circuit complexity. This is a direct and quantitative link between chaos and complexity. \n\nHowever, we caution the reader that in many cases Eq.~\\eqref{intro-complexity-lower-bound} may not be a very tight lower bound. We will provide some discussion of this point as well as a few examples, however further work is most likely required to better understand the utility of this bound.\n\n\\subsubsection*{Haar vs. simpler ensemble averages}\nFinally, we present calculations of the Haar average of some higher-point OTO correlators and compare them to averages in simpler ensembles. These results suggest that the floor value of OTO correlators of local operators might be good diagnostics of pseudorandomness.\n\nFor $4$-point OTO correlators, we find\n\\begin{equation}\n\\begin{split}\n&\\langle A\\tilde{B}C\\tilde{D} \\rangle_{\\text{Haar}} = \\langle AC \\rangle \\langle B \\rangle\\langle D \\rangle + \\langle A \\rangle\\langle C \\rangle \\langle BD \\rangle -  \\langle A\\rangle \\langle C \\rangle \\langle B \\rangle\\langle D \\rangle - \\frac{1}{d^2-1}\\langle\\!\\langle AC\\rangle\\!\\rangle \\langle\\!\\langle BD\\rangle\\!\\rangle,\n\\end{split} \\label{intro-4p-haar-average}\n\\end{equation}\nwhere $\\langle\\!\\langle AC\\rangle\\!\\rangle$ represents a connected correlator and $d=2^{n}$ is the total number of states. In contrast, if the $U$ are averaged over the Pauli operators (which form a $1$-design but not a $2$-design), we find\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D} \\rangle_{\\text{Pauli}}&= \n\\langle AC \\rangle \\langle B D \\rangle. \\label{intro-4p-pauli-average}\n\\end{align}\nWe will present intuitive explanations on this difference from the viewpoint of local thermal dissipations vs. global thermalization or scrambling. \n\nFor $8$-point OTO correlators with Pauli operators $A,B,C, D$, we compute averages over the unitary group and the Clifford group (which form a $3$-design on qubits but not a $4$-design)\n\\begin{equation}\n\\begin{split}\n&\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle_{\\text{Haar}} \\sim \\frac{1}{d^4},\\\\\n&\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle_{\\text{Clifford}} \\sim \\frac{1}{d^2}.\n\\end{split} \\label{intro-8p-averages}\n\\end{equation}\nThis suggests that forming a higher $k$-design leads to a lower value of the correlator.\n\n\nThe results Eq.~\\eqref{intro-4p-haar-average}-\\eqref{intro-8p-averages} are exact for any choice of operators. (The extended results for the correlators in Eq.~\\eqref{intro-8p-averages} is presented in Appendix~\\ref{sec:8-pt-functions}.) However, for a particular ordering of OTO $4m$-point functions where we average over choices of operators, we will also show that a Haar averaging over the unitary group scales as\n\\begin{equation}\n\\textrm{OTO}^{(4m)} \\sim \\frac{1}{d^{2m}}.\\label{intro-4m-averages}\n\\end{equation}\nThis result hints that these correlators continue to be probes of increasing pseudorandomnes. \n\n\n\n\\subsection*{Organization of the paper}\nFor convenience of the reader, we included a(n almost) self-contained introduction to Haar random unitary operators and unitary design in \\S\\ref{sec:review}. In \\S\\ref{sec:OTO_channel}, we establish a formal connection between chaos and unitary design by proving the theorems mentioned above. In \\S\\ref{sec:complexity-bound}, we connect complexity to unitary design by proving the complexity lower bound \\eqref{intro-complexity-lower-bound}.\nIn \\S\\ref{sec:haar-averages}, we include the explicit calculations of $2$-point and $4$-point functions averaged over different ensembles and discuss how these averages relate to expectations from time evolution with chaotic Hamiltonians. We also discuss results and expectations for higher-point functions.\nWe conclude in \\S\\ref{sec:discussion} with an extended discussion of these results, their relevance for physical situations, and outline some future research directions.\n\nDespite page counting to the contrary, this is actually a short paper. A knowledgeable reader may learn our results by simple reading \\S\\ref{sec:OTO_channel}, \\S\\ref{sec:complexity-bound} and \\S\\ref{sec:haar-averages}. On the other hand, a large number of extended calculations and digressions are relegated to the Appendices: \n\\begin{itemize}\n\\item In \\S\\ref{app:proof}, we collect some proofs that we felt interrupted the flow of the main discussion. \n\\item In \\S\\ref{sec:appendix:orthogonal}, we discuss the number of nearly orthogonal states in large-dimensional Hilbert spaces.\n\\item In \\S\\ref{sec:complexity-appendix}, we extend our complexity lower bound to minimum circuit depth by considering gates that may be applied in parallel. We also derive a bound on early-time complexity for evolution with an ensemble of Hamiltonians.\n\\item In \\S\\ref{sec:appendix-averages}, we hide the details of our $8$-point functions Haar averages and also derive the ${}\\sim d^{-2m}$ scaling of Haar averages of certain $4m$-point functions.\n\\item In \\S\\ref{sec:sub-space-randomization}, we provide a generalization of the frame potential that equals an average of the square of OTO correlators for arbitrary states rather than just the maximally mixed state.\n\\item Finally, in \\S\\ref{sec:more-chaos} we prove some extended results relating to our earlier work \\cite{Hosur:2015ylk} that are somewhat outside the main focus of the current paper.\n\\end{itemize}\n\n\\section{Measures of Haar}\\label{sec:review}\n\n\nThe goal of this section is to provide a review of the theory of Haar random unitary operators and unitary design in a self-contained manner. The presentation of this section owes a lot to a recent paper by Webb~\\cite{Webb15} as well as a course note by Kitaev \\cite{Kitaev-Haar}.\\footnote{We also would like to highlight a Master's thesis by Yinzheng Gu, which considers a certain matrix multiplication problem with conjugations by Haar random unitary operators. Although not directly relevant to the work, these are actually out-of-time-order correlation functions in a disguised form~\\cite{gu2013moments}.}\n\n\\subsection*{Haar random unitaries}\n\n\n\\subsubsection*{Schur-Weyl duality}\nConsider a finite-dimensional Hilbert space $\\mathcal{H}^{\\otimes k}=(\\mathbb{C}^{d})^{\\otimes k}$ consisting of $k$ copies of $\\mathcal{H}=\\mathbb{C}^d$. A permutation operator $W_{\\pi}$ with a permutation $\\pi=\\pi(1)\\ldots \\pi (k)$ is defined as follows\n\\begin{align}\nW_{\\pi} |a_{1},\\ldots,a_{k}  \\rangle = |a_{\\pi(1)},\\ldots,a_{\\pi(k)}  \\rangle,\n\\end{align}\nand $W_{\\pi}(A_{1}\\otimes \\cdots \\otimes A_{k})W_{\\pi}^{-1} = A_{\\pi(1)}\\otimes \\cdots \\otimes A_{\\pi(k)}$. If $\\pi$ is a cyclic permutation ($\\pi(j)=j+1$), then $W_{\\text{cyc}}$ acts as follows\n\\begin{align}\n\\includegraphics[width=0.35\\linewidth]{fig_permutation}.\n\\end{align}\n\n\n\\begin{theorem}\\emph{\\tb{[Schur-Weyl duality]}}\nLet $L(\\mathcal{H}^{\\otimes k})$ be the algebra of all the operators acting on $\\mathcal{H}^{\\otimes k}$. Let $U(\\mathcal{H})$ be the unitary group on $\\mathcal{H}$. An operator $A \\in L(\\mathcal{H}^{\\otimes k})$ commutes with all operators $V^{\\otimes k}$ with $V\\in U(\\mathcal{H})$ if and only if $A$ is a linear combination of permutation operators $W_{\\pi}$\n\\begin{align}\n[A,V^{\\otimes k}]=0, ~ \\forall V \\ \\Leftrightarrow \\ A = \\sum_{\\pi\\in S_k} c_{\\pi}\\cdot W_{\\pi}.\\label{Schur-Weyl-duality} \n\\end{align}\n\\end{theorem}\n\nWhen an operator $A$ is a linear combination of permutation operators, it is clear that $A$ commutes with $V^{\\otimes k}$ ($\\Leftarrow$). A difficult part is to prove the converse ($\\Rightarrow$), which relies on Von Neumann's double commutant theorem. \n\n\\subsubsection*{Pauli operators}\nPauli operators for $\\mathbb{C}^d$ (i.e. $d$-state spins or qudits) are defined by \n\\begin{align}\nX|j\\rangle = |j+1\\rangle, \\qquad Z|j\\rangle = \\omega^j|j\\rangle, \\label{eq:pauli-def}\n\\end{align}\nwhere $\\omega \\equiv e^{2\\pi i\/d}$. We note that Eq.~\\eqref{eq:pauli-def} implies $ZX = \\omega XZ$ and $X^d = Z^d = I$, and that for $d>2$, the Pauli operators are unitary and traceless, but not Hermitian.\n\nThe Pauli group is $\\tilde{\\mathcal{P}}=\\langle \\tilde{\\omega}I,X,Z \\rangle$, where $\\tilde{\\omega}=\\omega$ for odd $d$, and $\\tilde{\\omega}=e^{\\pi\/d}$ for even $d$. Since we are usually uninterested in global phases, we will consider the quotient of the group\n\\begin{align}\n\\mathcal{P} = \\tilde{\\mathcal{P}} \\backslash \\langle \\tilde{\\omega} I \\rangle.\n\\end{align}\nThere are $d^2$ (representative) Pauli operators in $\\mathcal{P}$. When the Hilbert space is built up from the space of $n$ qubits, we will denote the Pauli group by $\\mathcal{P}_{n}$. For such systems, (the representatives of) $\\mathcal{P}_{n}$ consist of tensor products of qubit Pauli operators, such as $X \\otimes Y \\otimes I \\otimes Z \\otimes \\cdots $ without any global phases.\n\nThe Pauli operators provide a basis for the space of linear operators acting on the Hilbert space. They are orthogonal, $\\text{tr} \\, \\{P_{i}^{\\dagger}P_{j} \\}=d\\delta_{ij}$ for $P_{i},P_{j}\\in \\mathcal{P}$, and therefore we can expand any operator $A$ acting on $\\mathcal{H}$ as\n\\begin{align}\nA = \\sum_{j} a_{j} P_{j},\\qquad a_{j}=\\frac{1}{d}\\text{tr} \\, \\{P_{j}^{\\dagger}A\\}.\n\\end{align}\nWith this property, the cyclic permutation operator $W_{\\text{cyc}}$ on $\\mathcal{H}^{\\otimes k}$ can be decomposed as\n\\begin{align}\nW_{\\text{cyc}} = \\frac{1}{d^{k-1}}\\sum_{P_{1},\\ldots,P_{k-1}\\in \\mathcal{P}} P_{1}\\otimes P_{2} \\otimes \\cdots P_{k-1}\\otimes Q^{\\dagger}, \n\\qquad Q = P_{1}P_{2}\\cdots P_{k-1}, \\label{eq:pauli-decomposition-of-cyc}\n\\end{align}\nwhere the sum is over $k-1$ copies of $\\mathcal{P}$.\n\nThe case of $k=2$ is particularly important, giving an operator that swaps two subsystems. Explicitly, we have $\\text{SWAP}=\\frac{1}{d}\\sum_{P} P\\otimes P^{\\dagger}$, or graphically (up to a multiplicative factor)\n\\begin{align}\n\\includegraphics[width=0.35\\linewidth]{fig_SWAP}.\\label{eq:fig_SWAP}\n\\end{align}\nHere and in what follows, a dotted line represents an average over all Pauli operators. For example, let us consider the Pauli channel\n\\begin{align}\n\\frac{1}{d^2}\\sum_{P \\in \\mathcal{P}}P^{\\dagger} A P = \\frac{1}{d}\\, \\text{tr}  \\{A \\},\\label{eq:Pauli_twirl}\n\\end{align}\nwhere $A$ is any operator on the system.\nThis equation can be derived graphically by applying Eq.~(\\ref{eq:fig_SWAP})\n\\begin{align}\n\\includegraphics[width=0.50\\linewidth]{fig_Pauli_twirl}.\n\\end{align}\n\n\n\\subsubsection*{$k$-fold channel}\nLet $A$ be an operator acting on $\\mathcal{H}^{\\otimes k}$. The $k$-fold channel of $A$ with respect to the unitary group is defined as\n\\begin{align}\n\\Phi_{\\text{Haar}}^{(k)}(A) := \\int_{\\text{Haar}} (U^{\\otimes k})^{\\dagger} A \\, U^{\\otimes k}  dU,\n\\end{align}\nwhere the integral is taken over the Haar measure. Note, this is sometimes referred to as the $k$-fold twirl of $A$. The Haar measure is the unique probability measure on the unitary group that is both left-invariant and right-invariant~\\cite{Watrous}\n\\begin{align}\n\\int_{\\text{Haar}} dU = 1,\\qquad \\int_{\\text{Haar}} f(VU)dU=\\int_{\\text{Haar}} f(UV)dU = \\int_{\\text{Haar}} f(U)dU, \\label{eq:def_Haar}\n\\end{align} \nfor all $V\\in U(\\mathcal{H})$, where $f$ is an arbitrary function. If we take $f(U)= (U^{\\otimes k})^{\\dagger}A\\,U^{\\otimes k}$, then we can show that the twirl of $A$ is invariant under $k$-fold unitary conjugation, \n\\begin{align}\n(V^{\\otimes k})^{\\dagger}\\big(\\Phi^{(k)}_{\\text{Haar}}(A)\\big) V^{\\otimes k}  &= \\int_{\\text{Haar}} f(UV)dU = \\Phi^{(k)}_{\\text{Haar}}(A),\\label{eq:duality1}\n\\end{align}\nand that twirl of the $k$-fold unitary conjugation of $A$ equals the twirl of $A$\n\\begin{align}\n\\Phi^{(k)}_{\\text{Haar}}\\big((V^{\\otimes k})^{\\dagger}A V^{\\otimes k}\\big)&=\\int_{\\text{Haar}} f(VU)dU = \\Phi^{(k)}_{\\text{Haar}}(A),\\label{eq:duality2}\n\\end{align}\nwhere for Eq.~\\eqref{eq:duality1} we used the right-invariance property, and for Eq.~\\eqref{eq:duality2} we used left-invariance.\n\n\\subsubsection*{Weingarten function}\nThe content of Eq.~(\\ref{eq:duality1}) is that $\\Phi^{(k)}_{\\text{Haar}}(A)$ commutes with all operators $V^{\\otimes k}$. Thus, we may use the Schur-Weyl duality Eq.~\\eqref{Schur-Weyl-duality} to rewrite it as\n\\begin{align}\n\\Phi^{(k)}_{\\text{Haar}}(A) = \\sum_{\\pi \\in S_k} W_{\\pi} \\cdot u_{\\pi}(A).\n\\end{align}\nHere, $S_k$ is the permutation group, and $u_{\\pi}(A)$ is some linear function of $A$. Since $u_{\\pi}(A)$ is a linear function, it can be written as\n\\begin{align}\nu_{\\pi}(A) = \\text{tr} \\{ C_{\\pi} A\\},\n\\end{align}\nfor some operators $C_{\\pi}$. From Eq.~(\\ref{eq:duality2}), we find that $C_{\\pi}$ commutes with all operators $V^{\\otimes k}$. Then again, by the Schur-Weyl duality, we have\n\\begin{align}\n\\Phi^{(k)}_{\\text{Haar}}(A) =\\sum_{\\pi,\\sigma\\in S_k} c_{\\pi,\\sigma} W_{\\pi}\\, \\text{tr} \\{ W_{\\sigma}A \\}.\n\\end{align}\nThe coefficients $c_{\\pi,\\sigma}$ are called the Weingarten matrix~\\cite{Collins03}. Since $\\Phi^{(k)}_{\\text{Haar}}(W_{\\lambda})=W_{\\lambda}$, we have $W_{\\lambda} = \\sum_{\\pi,\\sigma} c_{\\pi,\\sigma}W_{\\pi} \\, \\text{tr} \\{W_{\\sigma}W_{\\lambda} \\}$. Recalling that $\\text{tr} \\{ W_{\\sigma}W_{\\lambda}\\}=d^{\\# \\text{cycles}(\\sigma\\lambda)}$, we have\n\\begin{align}\n\\delta_{\\pi,\\lambda} = \\sum_{\\sigma\\in S_k}c_{\\pi,\\sigma} Q_{\\sigma,\\lambda},\\qquad Q_{\\sigma,\\lambda}:=d^{\\# \\text{cycles}(\\sigma\\lambda)}.\n\\end{align}\nSo finally, we find\n\\begin{align}\n\\Phi^{(k)}_{\\text{Haar}}(A) = \\sum_{\\pi,\\sigma\\in S_k} (Q^{-1})_{\\pi,\\sigma}  W_{\\pi} \\cdot \\text{tr} \\{W_{\\sigma}A\\}. \\label{eq:Weingarten_decomposition}\n\\end{align}\nHere, we assumed the presence of the inverse $Q^{-1}$ which is guaranteed for $k \\leq d$.\n\n\\subsection*{Examples}\nFor $k=1$, $Q_{I,I}=d$, so one has\n\\begin{align}\n\\Phi^{(1)}_{\\text{Haar}}(A) = \\frac{1}{d}  \\text{tr}\\{A\\}.\n\\end{align}\nFor $k=2$, so one has\n\\begin{align}\nQ = \\left(\n\\begin{array}{cc}\nQ_{I,I}, Q_{I,S} \\\\\nQ_{S,I} , Q_{S,S}\n\\end{array}\n\\right) =\n\\left(\n\\begin{array}{cc}\nd^2,&d \\\\\nd , &d^2\n\\end{array}\n\\right) \\quad \nC = \\left(\n\\begin{array}{cc}\nC_{I,I}, C_{I,S} \\\\\nC_{S,I} , C_{S,S}\n\\end{array}\n\\right) =\n\\left(\n\\begin{array}{cc}\n\\frac{1}{d^2-1} ,&  \\frac{-1}{d(d^2-1)} \\\\\n\\frac{-1}{d(d^2-1)} , &\\frac{1}{d^2-1}\n\\end{array}\n\\right).\n\\end{align}\nExplicitly, we have\n\\begin{align}\n\\Phi^{(2)}_{\\text{Haar}}(A) = \\frac{1}{d^2-1} \\left( I \\ \\text{tr} \\{A\\} + S\\ \\text{tr} \\{ SA\\} - \\frac{1}{d} S\\ \\text{tr} \\{A\\} - \\frac{1}{d} I\\ \\text{tr} \\{SA\\} \\right),\n\\end{align}\nwhere $S$ is the SWAP operator. \n\n\n\\subsubsection*{Haar random states}\nWe can also consider the $k$-fold average of a Haar random state. Define a random state by $|\\psi\\rangle = U|0\\rangle$, where $U$ is sampled uniformly from the unitary group. Then, we have\n\\begin{align}\n\\int_{\\text{Haar}} (|\\psi\\rangle \\langle \\psi| )^{\\otimes k} d\\psi := \\Phi^{(k)}_{\\text{Haar}}\\big((|0\\rangle\\langle 0 |)^{\\otimes k}\\big) = \\sum_{\\pi}c_{\\pi}W_{\\pi},\\label{eq:Haar_state}\n\\end{align}\nfor some coefficients $c_{\\pi}$.\nSince $\\int_{\\text{Haar}} (|\\psi\\rangle \\langle \\psi| )^{\\otimes k}$ commutes with $V^{\\otimes k}$, we again decomposed it with permutation operators. Furthermore, one has\n\\begin{align}\nW_{\\pi}\\int_{\\text{Haar}} (|\\psi\\rangle \\langle \\psi| )^{\\otimes k}=\\int_{\\text{Haar}} (|\\psi\\rangle \\langle \\psi| )^{\\otimes k}, \\qquad \\forall \\pi \\in S_{k}\n\\end{align}\nwhich implies $c_{\\pi}=c$ for all $\\pi$. By taking the trace in Eq.~(\\ref{eq:Haar_state}), we have\n\\begin{align}\nc_{\\pi}= \\frac{1}{\\sum_{\\pi\\in S_k} d^{\\text{\\#cycles($\\pi$)}}} = \\frac{k!}{\n\\Big(\\begin{array}{c}\nk+d-1 \\\\\nk\n\\end{array}\\Big)\n}.\n\\end{align}\nDefining the projector onto the symmetric subspace as $\\Pi_{\\text{sym}} = \\frac{1}{k!}\\sum_{\\pi\\in S_k}W_{\\pi}$, one has\n\\begin{align}\n\\int_{\\text{Haar}} (|\\psi\\rangle \\langle \\psi| )^{\\otimes k} d\\psi = \\frac{\\Pi_{\\text{sym}}}{\n\\Big(\\begin{array}{c}\nk+d-1 \\\\\nk\n\\end{array}\\Big) \\label{haar-random-states-equation}\n}.\n\\end{align}\n\n\n\n\\subsubsection*{Frame potential} In this paper, we will be particularly interested in the following quantity\n\\begin{align}\nF_{\\text{Haar}}^{(k)}=\\iint dU dV\\big|\\text{tr}\\, \\{ U^{\\dagger}V \\} \\big|^{2k}.\n\\end{align}\nUsing Eq.~\\eqref{eq:Weingarten_decomposition}, we can rewrite this\n\\begin{equation}\n\\begin{split}\nF_{\\text{Haar}}^{(k)} &= \\sum_{\\pi_{1},\\pi_{2},\\pi_{3},\\pi_{4}\\in S_{k}} \\text{tr} \\{ W_{\\pi_{1}}W_{\\pi_{2}} \\} (Q^{-1})_{\\pi_{2},\\pi_{3}} \\text{tr} \\{ W_{\\pi_{3}}W_{\\pi_{4}} \\} (Q^{-1})_{\\pi_{4},\\pi_{1}},  \\\\\n&= \\sum_{\\pi_{1},\\pi_{2},\\pi_{3},\\pi_{4}\\in S_{k}}Q_{\\pi_{1},\\pi_{2}} (Q^{-1})_{\\pi_{2},\\pi_{3}}Q_{\\pi_{3},\\pi_{4}}(Q^{-1})_{\\pi_{4},\\pi_{1}} = k!,\n\\end{split}\n\\end{equation}\nfor $k\\leq d$, where we used $C=Q^{-1}$ and $|S_{k}|=k!$.\n\n\n\\subsection*{Unitary design}\n\n\nConsider an ensemble of unitary operators $\\mathcal{E}=\\{p_{j},U_{j}\\}$ where $p_{j}$ are probability distributions such that $\\sum_{j}p_{j}=1$, and $U_{j}$ are unitary operators. The action of the $k$-fold channel with respect to the ensemble $\\mathcal{E}$ is given by \n\\begin{align}\n\\Phi^{(k)}_{\\mathcal{E}}(A) := \\sum_{j} p_{j}   (U_{j}^{\\otimes k})^{\\dagger}A (U_{j})^{\\otimes k },\n\\end{align}\nor for continuous distributions\n\\begin{align}\n\\Phi^{(k)}_{\\mathcal{E}}(A) := \\int_{\\mathcal{E}} dU   (U_{j}^{\\otimes k})^{\\dagger}A (U_{j})^{\\otimes k }.\n\\end{align}\nThe ensemble $\\mathcal{E}$ is a unitary $k$-design if and only if $ \\Phi^{(k)}_{\\mathcal{E}}(A)=\\Phi^{(k)}_{\\text{Haar}}(A)$ for all $A$. Intuitively, a unitary $k$-design is as random as the Haar ensemble up to the $k$th moment. That is, a unitary $k$-design is an ensemble which satisfies the definition of the Haar measure in Eq.~(\\ref{eq:def_Haar}) when $f(U)$ contains up to $k$th powers of $U$ and $U^{\\dagger}$ (i.e. balanced monomials of degree at most $k$). By this definition, if an ensemble is $k$-design, then it is also $k-1$-design. However, the converse is not true in general. \n\n\nIt is convenient to write the above definition of $k$-design in terms of Pauli operators. An ensemble $\\mathcal{E}$ is $k$-design if and only if $ \\Phi^{(k)}_{\\mathcal{E}}(P)=\\Phi^{(k)}_{\\text{Haar}}(P)$ for all Pauli operators $P\\in (\\mathcal{P})^{\\otimes k}$, since the Pauli operators are the basis of the operator space. Furthermore, for an arbitrary ensemble $\\mathcal{E}$\n\\begin{align}\n\\Phi_{\\text{Haar}}\\left( \\Phi_{\\mathcal{E}}\\big( A \\big) \\right) =\\Phi_{\\text{Haar}}\\left( A\\right), \\qquad \\forall \\mathcal{E},\\label{eq:ensemble-followed-by-Haar}\n\\end{align}\ndue to the left\/right-invariance of the Haar measure. By using Eq.~\\eqref{eq:ensemble-followed-by-Haar}, we can derive the following useful criteria for $k$-designs~\\cite{Webb15}\n\\begin{align}\n\\text{$\\mathcal{E}$ is $k$-design} \\quad \\Leftrightarrow \\quad \\text{$\\Phi_{\\mathcal{E}}(P)$ is a linear combination of $W_{\\pi}$ for all $P\\in (\\mathcal{P})^{\\otimes k}$.} \\label{eq:criteria}\n\\end{align}\nTo make use of this, we will look at some illustrative examples. \n\n\\subsubsection*{Pauli is a $1$-design}\nThe Pauli operators form a unitary $1$-design. In fact, we have already shown this in with Eq.~\\eqref{eq:Pauli_twirl}. We have shown that an average over Pauli operators gives $\\frac{1}{d^2}\\sum_{P\\in \\mathcal{P}}P^{\\dagger} A P = \\frac{1}{d}\\, \\text{tr}\\{A\\}$, and the Haar random channel gives $\\Phi^{(1)}_{\\text{Haar}}(A)=\\frac{1}{d}\\, \\text{tr}\\{A\\}$, so therefore\n\\begin{align}\n\\frac{1}{d^2}\\sum_{P\\in \\mathcal{P}}P^{\\dagger} \\rho P=\\Phi^{(1)}_{\\text{Haar}}(\\rho),\n\\end{align}\nfor all $\\rho$. For example, if $d=2$ and $A=X$ (Pauli $X$ operator), then we have\n\\begin{align}\n\\frac{1}{4}(IXI + XXX + YXY + ZXZ) = X + X - X - X =0,\n\\end{align}\nwhich is consistent with $\\text{tr} \\{X \\}=0$. \n\nThus, an average over Pauli operators is equivalent to talking a trace. Since Pauli operators can be written in a tensor product form: $P_{1}\\otimes P_{2}\\otimes \\ldots \\otimes P_{n}$ for a system of $n$ qubits, they do not create entanglement between different qubits and do not scramble. Instead, they can only mix quantum information locally. This implies some kind of relationship between $1$-designs and local thermalization that we will revisit in the discussion \\S\\ref{sec:discussion}.\n\n\\subsubsection*{Clifford is a $2$-design}\nThe Clifford operators form a unitary $2$-design. The Clifford group $\\mathcal{C}_{n}$ is a group of unitary operators acting on a system of $n$ qubits that transform a Pauli operator into another Pauli operator\n\\begin{align}\nC^{\\dagger} P C = Q, \\qquad P,Q\\in \\tilde{\\mathcal{P}}, \\quad C\\in \\mathcal{C}_{n}.\n\\end{align}\nClearly, Pauli operators are Clifford operators, since $PQP = e^{i\\theta} Q$ for any pairs of Pauli operators $P,Q$: Pauli operators transform a Pauli operator to itself up to a global phase. However, non-trivial Clifford operators are those which transform a Pauli operator into a different Pauli operator. An example of such an operator is the Control-$Z$ gate\n\\begin{align}\n\\text{C$Z$}|a,b\\rangle = (-1)^{ab}|a,b\\rangle, \\qquad a,b =0,1,\n\\end{align}\nwhere summation is modulo $2$. The conjugation of a Pauli operator with a Control-$Z$ gate is as follows\n\\begin{align}\nX\\otimes I \\rightarrow X \\otimes Z, \\qquad Z\\otimes I \\rightarrow Z \\otimes I, \n\\qquad\nI\\otimes X \\rightarrow Z \\otimes X, \\qquad I\\otimes Z \\rightarrow I \\otimes Z. \\label{eq:control-z}\n\\end{align}\n\nLet us prove that the Clifford group is $2$-design by using Eq.~(\\ref{eq:criteria})~\\cite{Webb15}. For qubit Pauli operators of the form $P\\otimes P$ ($P\\not=I$), the action of the Clifford $2$-fold channel is\n\\begin{align}\n\\sum_{C\\in \\mathcal{C}_{n} } (C^{\\dagger}\\otimes C^{\\dagger}) P\\otimes P (C\\otimes C) \\propto \\sum_{Q\\in \\mathcal{P}_{n}, Q\\not=I }Q\\otimes Q,\n\\end{align}\nbecause a random Clifford operator will transform $P$ into some other non-identity Pauli operator. Recalling the definition of the swap operator $\\text{SWAP}=\\frac{1}{d}\\sum_{P\\in \\mathcal{P}_{n}} P\\otimes P$, the RHS is a linear combination of $I\\otimes I$ and $\\text{SWAP}$. On the other hand, for other Pauli operators $P\\otimes Q$ with $P\\not=Q$, the action of the channel is\n\\begin{align}\n\\sum_{C\\in \\mathcal{C}_{n} } (C^{\\dagger}\\otimes C^{\\dagger}) P\\otimes Q (C\\otimes C) =0.\n\\end{align}\nThis can be seen by rewriting the sum as \n\\begin{equation}\n\\frac{1}{2}\\sum_{C\\in \\mathcal{C}_{n} } (C^{\\dagger}\\otimes C^{\\dagger}) P\\otimes Q (C\\otimes C) + \\frac{1}{2}\\sum_{RC\\in \\mathcal{C}_{n} } ( C^\\dagger R^{\\dagger}\\otimes C^\\dagger R^{\\dagger}) P\\otimes Q (RC\\otimes RC), \n\\end{equation}\nsince the Clifford group is invariant under element-wise multiplication by a Pauli $R$. Since we assumed the Pauli operators are different we have $PQ \\neq I$, and thus we can pick $R$ such that it anti-commutes with $PQ$. This implies $R^\\dagger P R \\otimes R^\\dagger Q R = - P \\otimes Q$, and therefore the two terms cancel.\nFinally, $I\\otimes I$ remains invariant. Thus, since the action of the channel on Pauli operators gives a linear combination of permutation operators, by Eq.~\\eqref{eq:criteria} the Clifford group is a unitary $2$-design. \n\nNote that Clifford operators do not have a tensor product form, in general. This means that unlike evolution restricted to Pauli operators, they can change the sizes of an operator as seen in the Control-$Z$ gate example Eq.~\\eqref{eq:control-z}. In other words they can grow local operators into a global operators, indicative of the butterfly effect. \n\nIn fact, Clifford operators can prepare a large class of interesting quantum states called the stabilizer states. Let $|\\psi_{0}\\rangle=|0\\rangle^{\\otimes n}$ be an initial product state. This state satisfies $Z_{j}|\\psi_{0}\\rangle = |\\psi_{0}\\rangle$. Let $U$ be an arbitrary Clifford operator and consider $|\\psi\\rangle = U |\\psi_{0}\\rangle$. This state $|\\psi\\rangle$ satisfies the following\n\\begin{align}\nS_{j}|\\psi\\rangle = |\\psi\\rangle, \\qquad S_{j}=UZ_{j}U^{\\dagger}.\n\\end{align}\nBy definition, $S_{j}$ are Pauli operators and will commute with each other. A quantum state that can be represented by a set of commuting Pauli operators $S_{j}$ is called a stabilizer state. Examples of stabilizer states include ground states of the toric code  and the perfect tensors used in construction of holographic quantum error-correcting codes \\cite{Pastawski15b}. The upshot is that Clifford operators can create a global entanglement and can scramble quantum information. We will return to this point again in the discussion \\S\\ref{sec:discussion}.\n\n\\subsubsection*{? is a higher-design}\nCurrently there is no known method of constructing an ensemble which forms an exact $k$-design for $k\\geq 4$ in a way which generalizes to large $d$. Instead, there are several constructions for preparing approximate $k$-design in an efficient manner~\\cite{Brandao12,Nakata:2016blv}.\n\n\\section{Measures of chaos and design}\\label{sec:OTO_channel}\n\nIn this section, we show that $2k$-point OTO correlators are probes of $k$-unitary designs. We will focus on a Hilbert space $\\mathcal{H}=\\mathbb{C}^d$ with $2k$-point correlators of the following form\n\\begin{align}\n\\big\\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\big\\rangle := \\frac{1}{d}\\text{tr}\\, \\{A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}}\\},\\label{eq-form-of-correlator}\n\\end{align}\nwhere $\\tilde{B_{j}}=U^{\\dagger}B_{j}U$. We can think of this as a correlator evaluated in a maximally mixed or  infinite temperature state $\\rho = \\frac{1}{d}I$. The trace can be rewritten as\n\\begin{align}\n\\text{tr} \\,\\big\\{ (A_{1} \\otimes \\cdots \\otimes A_{k}) (U^{\\dagger}\\otimes \\cdots\\otimes U^{\\dagger})\n(B_{1} \\otimes \\cdots \\otimes B_{k})(U\\otimes \\cdots\\otimes U)\n W_{\\pi_{\\text{cyc}}} \\big\\},\\label{eq:cyclic}\n\\end{align}\nby considering an enlarged Hilbert space $\\mathcal{H}^{\\otimes k}$ that consists of $k$ copies of the original Hilbert space $\\mathcal{H}$, where $W_{\\pi_{\\text{cyc}}}$ represents a cyclic permutation operator on $\\mathcal{H}^{\\otimes k}$. The action of $W_{\\text{cyc}}$ is to send the $j$th Hilbert space to the $(j+1)$th Hilbert space (modulo $k$), see Fig.~\\ref{fig_k-fold_twirl} for a graphical representation.\\footnote{N.B. this trick of using cyclic permutation operators is similar to the method used for computing R\\'enyi-$k$ entanglement entropies in quantum field theories via the insertion of twist operators.} \n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=0.40\\linewidth]{fig_k-fold_twirl}\n\\caption{Schematic form of the $2k$-point OTO correlation functions Eq.~\\eqref{eq-form-of-correlator}, interpreted as a correlation function on the enlarged $k$-copied system. The dotted line diagram surrounds the $k$-fold channel $\\Phi_{\\mathcal{E}}(B_{1} \\otimes \\cdots \\otimes B_{k})$, which is probed by $A_{1}\\otimes \\ldots \\otimes A_{k}$. (Periodic boundary conditions are implied to take the trace.)\n} \n\\label{fig_k-fold_twirl}\n\\end{figure}\n\nObserve that Eq.~(\\ref{eq:cyclic}) contains a $k$-fold unitary action \n\\begin{equation}\n(U\\otimes \\cdots\\otimes U){}\n(B_{1} \\otimes \\cdots \\otimes B_{k})\n(U^{\\dagger}\\otimes \\cdots\\otimes U^{\\dagger}), \\notag\n\\end{equation} \nsuggesting correlators of the form Eq.~\\eqref{eq-form-of-correlator} have the potential to be sensitive to whether an ensemble is or is not a $k$-design.\nUnitary $k$-designs concern the \\emph{$k$-fold channel} of an ensemble of unitary operators\n\\begin{align}\n\\Phi_{\\mathcal{E}}(B_{1} \\otimes \\cdots \\otimes B_{k}) = \\int_{\\mathcal{E}} dU (U\\otimes \\cdots\\otimes U)\n(B_{1} \\otimes \\cdots \\otimes B_{k})\n(U^{\\dagger}\\otimes \\cdots\\otimes U^{\\dagger}),\n\\end{align} \nwhere $\\mathcal{E}$ is an ensemble of unitary operators.  To further this point, let us consider an average of these correlators over an ensemble of unitary operators $\\mathcal{E}$\n\\begin{align}\n\\left|\\big\\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\big\\rangle\\right|_{\\mathcal{E}} := \\frac{1}{d}\\int_{\\mathcal{E}} dU\\, \\text{tr}\\, \\{ A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}}\\}. \\label{eq-form-of-average-oto}\n\\end{align}\nLooking back at Eq.~\\eqref{eq:cyclic}, the idea is that $A_{1},\\ldots,A_{k}$ operators probe the outcome of $k$-fold channel $\\Phi_{\\mathcal{E}}(B_{1} \\otimes \\cdots \\otimes B_{k})$. Indeed, the part of Fig.~\\ref{fig_k-fold_twirl} surrounded by a dotted line, is $\\Phi_{\\mathcal{E}}(B_{1} \\otimes \\cdots \\otimes B_{k})$. Below, we will make this intuition precise by proving that this set of OTO correlators Eq.~\\eqref{eq-form-of-correlator} completely determine the $k$-fold channel $\\Phi^{(k)}_{\\mathcal{E}}$. \n\n\n\n\\subsection{Chaos and $k$-designs}\n\nIn this subsection, we prove that $2k$-point OTO correlators completely determines the $k$-fold channel of an ensemble $\\mathcal{E}$, denoted by $\\Phi_{\\mathcal{E}}^{(k)}$. \nTo recap, we write the $k$-fold channel of an ensemble $\\mathcal{E}=\\{p_{j},U_{j}\\}$ by \n\\begin{align}\n\\Phi^{(k)}_{\\mathcal{E}}(\\rho) = \\sum_{j} p_{j}  (U_{j}^{\\dagger}\\otimes \\cdots \\otimes U_{j}^{\\dagger})\\rho (U_{j}\\otimes \\cdots \\otimes U_{j}),\n\\end{align}\nwhere $\\rho$ is defined over $k$ copies of the system, $\\mathcal{H}^{\\otimes k}$. The map is linear, and completely-positive and trace-preserving (CPTP), i.e. a quantum channel. For simplicity of discussion, we assume that the system $\\mathcal{H}$ is made of $n$ qubits so that $\\mathcal{H}=\\mathbb{C}^{d}$ with $d=2^n$. The input density matrix $\\rho$ can be expanded by Pauli operators \n\\begin{align}\n\\rho_{\\text{in}} = \\sum_{B_{1},\\ldots,B_{k}}\\beta_{B_{1},\\ldots,B_{k}}(B_{1}\\otimes \\cdots \\otimes B_{k}),\n\\end{align}\nwhere $\\beta_{B_{1},\\ldots,B_{k}}= d^{-k} \\, \\text{tr} \\, \\big\\{(B_{1}^{\\dagger}\\otimes \\cdots \\otimes B_{k}^{\\dagger}) \\rho\\big\\}$. The output density matrix is given by\n\\begin{align}\n\\rho_{\\text{out}} = \\sum_{B_{1},\\ldots,B_{k}}\\beta_{B_{1},\\ldots,B_{k}}\\Phi_{\\mathcal{E}}(B_{1}\\otimes \\cdots \\otimes B_{k}).\n\\end{align}\nFor a given Pauli operator $B_{1},\\ldots,B_{k}$, we would like to examine $\\Phi_{\\mathcal{E}}(B_{1}\\otimes \\cdots \\otimes B_{k})$. \n\nLet us fix Pauli operator $B_{1},\\ldots,B_{k}$ for the rest of the argument. Note that the output $\\Phi_{\\mathcal{E}}(B_{1}\\otimes \\cdots \\otimes B_{n})$ can be also expanded by Pauli operators\n\\begin{align}\n\\Phi_{\\mathcal{E}}(B_{1}\\otimes \\cdots \\otimes B_{k}) = \\sum_{C_{1},\\ldots,C_{k}}\\gamma_{C_{1},\\ldots,C_{k}}(C_{1}\\otimes \\cdots \\otimes C_{k}).\\label{eq:expansion}\n\\end{align}\nSince we have fixed $B_{1}\\otimes \\cdots \\otimes B_{k}$, for notational simplicity we have not included $B_{1}, \\cdots, B_{k}$ indices from the tensor $\\gamma$. In order to characterize the $k$-fold channel, we need to know values of $\\gamma_{C_{1},\\ldots,C_{n}}$ for a given $B_{1}, \\cdots, B_{k}$. We would like to show that we can determine the values of $\\gamma_{C_{1},\\ldots,C_{n}}$ by knowing a certain collection of OTO correlators. Consider a $2k$-point OTO correlator labeled by the set of $A$ operators, averaged over an ensemble $\\mathcal{E}$\n\\begin{align}\n\\alpha_{A_{1},\\ldots,A_{k}} = \\left|\\big\\langle A_{1} \\tilde{B_{1}} \\cdots A_{k}\\tilde{B_{k}} \\big\\rangle \\right|_{\\mathcal{E}}\\label{eq:definition},\n\\end{align}\nwhere as always $\\tilde{B_{j}}=U^{\\dagger}B_{j}U$ and $A_{1},\\ldots,A_{k}$ are Pauli operators. As before, for simplicity of notation we have not included $B_{1},\\ldots,B_{k}$ indices on $\\alpha$. Now that the notation is setup, the main question is whether one can determine the coefficients $\\gamma_{C_{1},\\ldots,C_{k}}$ from the numbers $\\alpha_{A_{1},\\ldots,A_{k}}$.\n\n Substituting Eq.~\\eqref{eq:expansion} into Eq.~\\eqref{eq:definition}, we see\n\\begin{align}\n\\alpha_{A_{1},\\ldots,A_{k}} = \\frac{1}{d}\\cdot M^{C_{1},\\ldots,C_{k}}_{A_{1},\\ldots,A_{k}} \\gamma_{C_{1},\\ldots,C_{k}},\n\\qquad\nM^{C_{1},\\ldots,C_{k}}_{A_{1},\\ldots,A_{k}} =\n \\text{tr} \\, \\{ A_{1}C_{1}\\cdots A_{k}C_{k} \\}, \\label{eq:normal}\n\\end{align}\nwhere tensor contractions are implicit following the Einstein summation convention.\nThis shows we can compute OTO correlators $\\alpha_{A_{1},\\ldots,A_{k}}$ the coefficients defining the $k$-fold channel $\\gamma_{C_{1},\\ldots,C_{k}}$. To establish the converse, we must prove that the tensor $M$ is invertible. \n\n\n\\begin{theorem}\\label{theorem:OTO_chaos}\nConsider the tensor $M^{C_{1},\\ldots,C_{k}}_{A_{1},\\ldots,A_{k}}$ \nand its conjugate transpose ${M^{\\dagger}}_{C_{1},\\ldots,C_{k}}^{A_{1},\\ldots,A_{k}}$. Then\n\\begin{align}\n\\sum_{C_{1},\\ldots,C_{k}}{M^{\\dagger}}_{C_{1},\\ldots,C_{k}}^{A_{1}',\\ldots,A_{k}'}M^{C_{1},\\ldots,C_{k}}_{A_{1},\\ldots,A_{k}} = d^{2k}\\cdot  \n\\delta^{A_{1}'}_{A_{1}}\\cdots \\delta^{A_{k}'}_{A_{k}},\\label{eq:tensor_M}\n\\end{align}\nwhere $\\delta^{P}_{Q}$ is the delta function for Pauli operators $P,Q$\n\\begin{align}\n\\delta^{P}_{Q} = 1, \\quad (P=Q), \\qquad \\delta^{P}_{Q} = 0, \\quad (P\\not=Q).\n\\end{align}\n\\end{theorem}\n\nThe proof of this theorem is sort of technical and has been relegated to Appendix~\\ref{sec:proof:oto-channnel}. Thus, from the OTO correlators $\\alpha_{C_{1},\\ldots,C_{k}}$, we can completely determine the $k$-fold channel $\\gamma_{A_{1},\\ldots,A_{k}}$\n\\begin{align}\n\\gamma_{C_{1},\\ldots,C_{k}}= \\frac{1}{d^{2k-1}} {M^{\\dagger}}_{C_{1},\\ldots,C_{k}}^{A_{1},\\ldots,A_{k}}\\alpha_{A_{1},\\ldots,A_{k}}.\n\\end{align}\nAs an obvious corollary, this means that $2k$-point OTO correlators can measure whether or not an ensemble forms $k$-design.\n\n\n\n\\subsection{Frame potentials}\n\n\nIn this section we introduce the \\emph{frame potential}, a single quantity that can measure whether an ensemble is a $k$-design. Furthermore, we show how the frame potential may be computed from OTO correlators.\n\nGiven an ensemble of unitary operators $\\mathcal{E}$, the $k$th frame potential is defined by the following double sum \\cite{Scott08}\n\\begin{align}\nF_{\\mathcal{E}}^{(k)} := \\frac{1}{|\\mathcal{E}|^{2}}\\sum_{U,V\\in \\mathcal{E}} \\left| \\text{tr} \\{U^{\\dagger}V\\} \\right|^{2k},\n\\end{align}\nwhere $|\\mathcal{E}|$ denotes the cardinality of $\\mathcal{E}$.Denote the frame potential for the Haar ensemble as $F^{(k)}_{\\text{Haar}}$. Then, the following theorem holds.\n\n\\begin{theorem}\nFor any ensemble $\\mathcal{E}$ of unitary operators, \n\\begin{align}\nF^{(k)}_{\\mathcal{E}} \\geq F^{(k)}_{\\mathrm{Haar}},\n\\end{align}\nwith equality if and only if $\\mathcal{E}$ is $k$-design.\n\\end{theorem}\n\nThe proof of this theorem is very insightful and beautiful, which we reprint from~\\cite{Scott08}. \n\n\\begin{proof}\nLetting $S=\\int_{\\mathcal{E}}(U^{\\dagger})^{\\otimes k}\\otimes U^{\\otimes k}  - \\int_{\\text{Haar}}(U^{\\dagger})^{\\otimes k}\\otimes U^{\\otimes k}$, we have\n\\begin{equation}\n\\begin{split}\n0 \\leq \\text{tr} \\{ S^{\\dagger}S\\} =  &\\int_{U\\in\\mathcal{E}}\\int_{V\\in\\mathcal{E}}dUdV\\, |\\text{tr} \\{U^{\\dagger}V \\} |^{2k} - 2 \\int_{U\\in\\mathcal{E}}\\int_{V\\in\\text{Haar}}dUdV \\, |\\text{tr} \\{U^{\\dagger}V \\} |^{2k} \\\\\n&+ \\iint_{U,V\\in\\text{Haar}}dUdV \\, |\\text{tr} \\{ U^{\\dagger}V \\} |^{2k}.\n\\end{split}\n\\end{equation}\nThe first term is $F^{(k)}_{\\mathcal{E}}$, and the third term is $F^{(k)}_{\\text{Haar}}$ by definition. The second term is equal to  $- 2F^{(k)}_{\\text{Haar}}$ by the left\/right-invariance of the Haar measure.  Thus, we see that\n\\begin{align}\n0 \\leq F^{(k)}_{\\mathcal{E}} - 2F^{(k)}_{\\text{Haar}} + F^{(k)}_{\\text{Haar}}\n=F^{(k)}_{\\mathcal{E}} - F^{(k)}_{\\text{Haar}},\n\\end{align}\nwith equality if and only if $\\mathcal{E}$ is $k$-design. \n\\end{proof}\n\nNote that we derived the minimal value of the frame potential in \\S\\ref{sec:review},\n\\begin{equation}\nF_{\\text{Haar}}^{(k)}=k!,\n\\end{equation}\n which holds for $k \\le d$. The frame potential quantifies the $2$-norm distance between the Haar ensemble and the $k$-fold $\\mathcal{E}$-channel.\\footnote{The distance between two quantum channels is typically measured by the diamond norm, while the frame potential measures the $2$-norm (or more precisely the Frobenius norm) distance between a given ensemble and the Haar ensemble. Importantly, the $2$-norm is weaker than the diamond norm.  For precise statements regarding the different norms and bounds relating them, see~\\cite{Low_thesis}.}  Here we show that the frame potential can be expressed as a certain average of OTO correlation functions. \n\n\\begin{theorem}\nFor any ensemble $\\mathcal{E}$ of unitary operators,\n\\begin{align}\n\\frac{1}{d^{4k}}\\sum_{A_{1},\\cdots,B_{1},\\cdots}\\left|\\big\\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\big\\rangle_{\\mathcal{E}}\\right|^{2} = \\frac{1}{d^{2(k+1)}}\\cdot F^{(k)}_{\\mathcal{E}}\\label{eq:OTO_frame},\n\\end{align}\nwhere summations are over all possible Pauli operators.\n\\end{theorem} \n\nThe LHS of the equation is the operator average of the $2$-norm of OTO correlators, and the RHS is the $k$th frame potential up to a constant factor. There are $d^{4k}$ Pauli operators $A_{1},\\ldots, A_k, B_{1},\\ldots, B_k$, which leads to $1\/d^{4k}$. The theorem implies that the quantitative effect of random unitary evolution is to decrease the frame potential, which is equivalent to the decay of  OTO correlators. \n\n\n\\begin{proof}\nWe take the averages over $A_{1},\\cdots,A_k, B_{1},\\cdots, B_k$ first. Expanding the LHS gives \n\\begin{align}\n\\frac{1}{d^{4k}}\\frac{1}{|\\mathcal{E}|^{2}d^2}\\sum_{U,V\\in \\mathcal{E}}\\sum_{A_{1},\\cdots,B_{1},\\cdots}\\text{tr} \\{A_{1}U^{\\dagger}B_{1}U\\cdots A_{k}U^{\\dagger}B_{k}U \\} \\cdot\\text{tr} \\{ V^{\\dagger}B_{k}^{\\dagger}VA_{k}^{\\dagger}\\cdots V^{\\dagger}B_{1}^{\\dagger}VA_{1}^{\\dagger}\\}.\n\\end{align}\nFor $k=2$, this can be depicted graphically as\n\\begin{align}\n\\includegraphics[width=0.40\\linewidth]{fig_frame}.\n\\end{align}\nRecall that a SWAP operator is given by $\\text{SWAP} = \\frac{1}{d}\\sum_{P} P \\otimes P^{\\dagger}$\n\\begin{align}\n\\includegraphics[width=0.35\\linewidth]{fig_SWAP}.\n\\end{align}\nThus, we replace each average of Pauli operators $A_{1},\\cdots,A_k, B_{1},\\cdots, B_k$ by SWAP operators. There are $2k$ loops, where $k$ of them contribute to $\\text{tr} \\{UV^{\\dagger}\\}$ and the remaining $k$ loops contribute to  $\\text{tr} \\{ VU^{\\dagger} \\}$. Keeping track of the number of factors of $d$, we find \n\\begin{align}\n\\frac{1}{d^{2(k+1)}|\\mathcal{E}|^2}\\sum_{U,V\\in\\mathcal{E}}\\text{tr} \\{ UV^{\\dagger} \\} ^k\\cdot \\text{tr} \\{ VU^{\\dagger}\\}^k = \\frac{1}{d^{2(k+1)}} \\cdot F^{(k)}_{\\mathcal{E}},\n\\end{align}\nwhich is the desired result.\n\\end{proof}\n\n\nLastly, we note that we can use the frame potential to lower bound the size of the ensemble. Since all the terms in the sum are positive, by taking the diagonal part of the double sum, we find\n\\begin{align}\nF_{\\mathcal{E}}^{(k)} \\geq \\frac{1}{|\\mathcal{E}|^2}\\sum_{U,V\\in \\mathcal{E},\\, U=V} \\big|\\text{tr} \\{I \\}\\big|^{2k}=\n \\frac{1}{|\\mathcal{E}|}d^{2k}, \\label{frame-potential-vs-size}\n\\end{align}\nmeaning the cardinality is lower bounded as $|\\mathcal{E}| \\ge d^{2k} \/ F_{\\mathcal{E}}^{(k)}$. Using the fact that $F^{(k)}_\\mathrm{Haar} = k!$, we can simply bound the size of a $k$-design\\footnote{Actually, one can slightly improve this lower bound for a $k$-design, see~\\cite{Roy:2009aa, Brandao12}.}\n\\begin{equation}\n\\mathcal{|E|} \\ge \\frac{d^{2k}}{k!}, \\qquad \\text{($k$-design)}. \\label{eq-cardinality-of-k-design}\n\\end{equation}\n\n\n\n\\subsubsection*{Large $k$}\n\n\nIn Appendix~\\ref{sec:appendix:orthogonal}, we provided an intuitive way of counting the number of nearly orthogonal states in a high-dimensional vector space $\\mathbb{C}^d$ by introducing a precision tolerance $\\epsilon$. A similar argument holds for the number of operators; if we can only distinguish operators up to some precision $\\epsilon$, then the total number of unitary operators is given by the volume of the unitary group measured in balls of size $\\epsilon$, which roughly goes like $\\sim \\epsilon^{-2^{2n}}$.\n\nThe key point is that if we have a tolerance $\\epsilon$, then the number of operators is finite even for a continuous ensemble. This means that there is a maximum size to any ensemble that is a subset of the unitary group. Looking at our bound Eq.~\\eqref{eq-cardinality-of-k-design}, we see that for $k\\sim d$ we begin to reach the maximum size for some $\\epsilon$. \n\nAs a corollary, this means that for large $k \\gtrsim d$, being a $k$-design implies that the ensemble is an approximate $k+1$-design. In this sense, there are really only $O(d)$ nontrivial moments of the Haar ensemble.\n\nAs an explicit example, we will consider the case of $d=2$. Here, for Haar we can compute the frame potential exactly for all $k$ \\cite{Scott08}\n\\begin{align}\nF^{(k)}_{\\text{Haar}} = \\frac{(2k)!}{k ! (k+1)!},\n\\end{align}\nand thus we have the following relationship\n\\begin{align}\n\\frac{F_{\\text{Haar}}^{(k+1)}}{F_{\\text{Haar}}^{(k)}} = 4 \\frac{k+1\/2}{k+2}.\n\\end{align}\nOn the other hand, for any ensemble the frame potential satisfies\n\\begin{align}\nF^{(k+1)} \\leq d^2 F^{(k)}=4F^{(k)}.\n\\end{align}\nTherefore, for $d=2$ we see that  if an ensemble is a $k$-design, for $k \\gtrsim 2$ it will automatically be close to being a  $k+1$-design.\n\n\n\n\n\\subsection{Ensemble vs. time averages}\\label{sec:F-time-average}\n\nIn this subsection, we consider the time average of the frame potential\nto ask whether the ensemble specified by sampling $U(t)=e^{-iHt}$ at different times $t$ will ever form a $k$-design.  In the classical statistical mechanics and chaos literature, this is the question of whether a system is ergodic---whether the average over phase space equals an average over the time evolution of some initial state.\n\nConsider the one-parameter ensemble of unitary matrices defined by evolving with a fixed time-independent Hamiltonian $H$, \n\\begin{equation}\n\\mathcal{E} = \\big\\{ e^{-iHt} \\big\\}_{t=0}^{\\infty}.\n\\end{equation} \nWe can compute the frame potential as\n\\begin{align}\nF_{\\mathcal{E}}^{(k)} &= \\lim_{T\\to\\infty} \\frac{1}{T^2}\\int_0^T dt_1 dt_2 \\, \\Big|\\trr{e^{-iH(t_1 - t_2)}}\\Big|^{2k}, \\\\\n               &= \\lim_{T\\to\\infty} \\frac{1}{T^2}\\int_0^T dt_1 dt_2 \\, \\sum_{\\ell_{1}\\dots \\ell_{k}} \\sum_{m_{1}\\dots m_{k}} \\exp \\Big\\{ -i (t_2 - t_1) \\sum_{i=1}^k E_{\\ell_{i}} + i(t_2-t_1)\\sum_{j=1}^k E_{m_{j}}  \\Big\\}. \\notag\n\\end{align}\nIf the spectrum is chaotic\/generic, then the energy levels are all incommensurate. The terms in the sum evaluate to zero unless we have $E_{\\ell_{i}} = E_{m_{j}}$ for all possible pairings $(i,j)$\n\\begin{align}\nF_{\\mathcal{E}}^{(k)} &= \\sum_{\\ell_{1}\\dots \\ell_{k}} \\sum_{m_{1}\\dots m_{k}} \\sum_{ij} \\delta_{\\ell_{i}m_{j}}, \\\\\n               &= k! \\Big(\\sum_{\\ell} I \\Big)^k = k!\\,d^k. \\notag\n\\end{align}\nThis is larger than $F_{\\text{Haar}}^{(k)}$ by a factor of $d^k$; the ensemble $\\mathcal{E}=\\{e^{-iHt}\\}$ does not form a $k$-design!\nRecall that $F_{\\mathcal{E}}=d^{2k}$ for a trivial ensemble $\\mathcal{E}=\\{I\\}$ while $F_{\\mathcal{E}}=k!$ for a Haar ensemble. The time average ensemble sits in the middle of these ensembles.\n\nThis ``half-randomness''can be understood in the following way. Let us write the Hamiltonian as\n\\begin{align}\nH = \\sum_{j} E_{j} |\\psi_{j}\\rangle \\langle \\psi_{j}|.\n\\end{align}\nRotating $H$ by a unitary operator does not affect the frame potential, so we can consider a classical Hamiltonian \n\\begin{align}\nH' = \\sum_{j} E_{j} |j\\rangle \\langle j|,\n\\end{align}\nwith the same frame potential. Even though $H'$ is classical, it has an ability to generate entanglement. Namely, if an initial state is $|+\\rangle=\\sum_{j}|j\\rangle$, then time-evolution will create entanglement. In terms of the original Hamiltonian $H$, we see that the system keeps evolving in a non-trivial manner as long as the initial state is ``generic'' and is different from eigenstates of the Hamiltonian $H$. This arguments suggests that the frame potential, in a time-average sense, can only see the spectrum distribution. \n\n\nThis fact, that Hamiltonian time evolution can never lead to random unitaries, can also be understood in terms of the distribution of the spacing of the eigenvalues.\\footnote{This argument has been made by Michael Berry, who related it to Steve Shenker, who related it to Douglas Stanford, who related it to us. As far as we know, it does not otherwise appear in the literature.} The phases of the Haar random unitary operators have Wigner-Dyson statistics; they are repulsed from being degenerate. The eigenvalues of a typical Hamiltonian $H$ also have this property. However, these eigenvalues live on the real line, while the phases of $e^{-iHt}$ live on the circle. In mapping the line to the circle, the eigenvalues of $H$ wrap many times. This means that the difference of neighboring phases of $e^{-iHt}$ will not be repulsed; $e^{-iHt}$ will have Poisson statistics! In this sense, the ensemble formed by sampling $e^{-iHt}$ over time may never become Haar random.\\footnote{Relatedly, the space of unitaries with a fixed Hamiltonian is $d$-dimensional, while the space of unitaries is $d^2$-dimensional.} The calculation in the beginning of this subsection provides an explicit calculation of the $2$-norm distance between such an ensemble and $k$-designs and quantifies the degree to which the ensemble average is ``too random'' as compared to the time average.\n\n\n\n\\section{Measures of complexity}\\label{sec:complexity-bound}\nFor an operator or state, the computational complexity is a measure of the minimum number of basic elementary gates necessary to generate the operator or the state from a simple reference operator (e.g. the identity operator) or reference state (e.g. the product state). With the assumption that the elementary gates are easy to apply, the complexity is a measure of how difficult it is to create the state or operator.\n\n\nOn the other hand, an ensemble contains many different operators with different weights or probabilities. In that case, the computational complexity of the ensemble should be understood as the number of steps it takes to generate the ensemble by probabilistic applications of elementary gates. For instance, to generate the ensemble of Pauli operators, we randomly choose with probability $1\/4$ a Pauli operator $I,X,Y,Z$ to apply to a qubit, and then repeat this procedure for all the qubits.\n\nThe complexity of an ensemble is related to the complexity of an operator in the following way. If an ensemble can be prepared in $\\mathcal{C}$ steps, then all the operators in the ensemble can be generated by applications of at most $\\mathcal{C}$ elementary gates. On the other hand, if an ensemble cannot be prepared (or approximated) in $\\mathcal{C}$ steps, then---for the sorts of ensembles we are interested in---most of the operators  cannot be generated by applications of $\\mathcal{C}$ elementary gates. \nFor example, generating the Haar ensemble will take exponential complexity since, on average, individual elements have exponential complexity.\n\n\n\nThe complexity of the ensemble can be lower bounded in terms of the number of elements or cardinality of the ensemble $|\\mathcal{E}|$. If all the elements are represented equally (with uniform probabilities), then clearly at least $\\mathcal{E}$ circuits need to be generated from probabilistic applications of the elementary gates. Making use of a fact introduced in the previous section that $F_{\\mathcal{E}}^{(k)}$ provides a lower bound on $|\\mathcal{E}|$, here we show that the frame potential provides a lower bound on a circuit complexity of generating $\\mathcal{E}$. We will also explain how this bound applies to ensembles that depend continuously on some parameters and thus have a divergent number of elements. \n\nTwo additional bounds that are somewhat outside the scope of the main presentation, one on circuit depth and one on the early-time complexity growth with a disordered ensemble of Hamiltonians, are relegated to Appendix~\\ref{sec:complexity-appendix}.\n\n\\subsection{Discrete ensembles}\\label{sec:complexity-discrete}\n\nConsider a system of $n$ qubits. Let $\\mathcal{G}$ denote an elementary gate set that consists of a finite number of two-qubit quantum gates. We denote the number of elementary two-qubit gates by $g:=|\\mathcal{G}|$. At each time step we assume that we can implement any of the gates from $\\mathcal{G}$. One typically chooses $\\mathcal{G}$ so that gates in $\\mathcal{G}$ enables a universal quantum computation. A well-known example is\n\\begin{align}\n\\mathcal{G} = \\{\\text{$2$-qubit Clifford}, T\\},  \n\\end{align}\nwhere $T$ is the $\\pi\/4$ phase shift operator; $T=\\text{diag}(1,e^{i\\pi\/4})$. (Of course, this is not the only choice of elementary gate sets.) \n\nOur goal is to generate an ensemble of unitary operators $\\mathcal{E}$ by sequentially implementing quantum gates from $\\mathcal{G}$. Let us denote the necessary number of steps (i.e. the circuit complexity) to generate $\\mathcal{E}$ by $\\mathcal{C}(\\mathcal{E})$. Then one has the following complexity lower bound. \n\\begin{theorem}\nLet $g$ be the number of distinct two-qubit gates from the elementary gate set. Then the circuit complexity $\\mathcal{C}(\\mathcal{E})$ to generate an ensemble $\\mathcal{E}$ is lower bounded by \n\\begin{align}\n\\mathcal{C}(\\mathcal{E}) \\geq \\frac{\\log |\\mathcal{E}| }{\\log(g n^2)}.\\label{eq-for-theorem-complexity-ensemble-size-bound}\n\\end{align}\n\\label{theorem-complexity-ensemble-size-bound}\n\\end{theorem}\n\nThe proof relies on an elementary counting argument. Arguments along this line of thought have been used commonly in the literature. \n\n\n\\begin{proof}\n At each step, we randomly pick a pair of qubits. Since there are $g$ implementable quantum gates and $\\binom{n}{2}$ qubit pairs, there are in total $\\simeq g n^2$ choices at each step. \nIf this procedure runs for $\\mathcal{C}$ steps, the number of unique circuits this procedure can implement is upper bounded by\n\\begin{align}\n\\text{$\\#$ of circuits}\\leq (g n^2)^{\\mathcal{C}}.\n\\end{align}\nSince there are $|\\mathcal{E}|$ unitary operators in an ensemble $\\mathcal{E}$, we must have\n\\begin{align}\n(g n^2)^{\\mathcal{C}} \\geq |\\mathcal{E}|, \n\\end{align}\nwhich implies\n\\begin{align}\n\\mathcal{C}(\\mathcal{E}) \\geq \\frac{\\log |\\mathcal{E}| }{\\log(g n^2)}. \n\\end{align} \n\\end{proof}\n\n\nIn Appendix~\\ref{sec:appendix:orthogonal}, we provide an intuitive way of counting the number of states in a $d=2^n$-dimensional Hilbert space (which was well known for a long time from \\cite{Knill95}). \nAs a sanity check on Theorem~\\ref{theorem-complexity-ensemble-size-bound}, if we substitute $|\\mathcal{E}| \\gtrsim 2^{2^n}$ into Eq.~\\eqref{eq-for-theorem-complexity-ensemble-size-bound} we see that the complexity \nof most states is exponential in the number of qubits $\\mathcal{C} > 2^n \\log 2$.\n\n\nFinally, let us examine the relation between the frame potential and the circuit complexity. Using Eq.~\\eqref{frame-potential-vs-size} to trade $|\\mathcal{E}|$ for $F_{\\mathcal{E}}^{(k)}$, we find\n\\begin{align}\n\\mathcal{C}(\\mathcal{E}) \\geq \\frac{2kn \\log(2) - \\log F_{\\mathcal{E}}^{(k)} }{\\log(g n^2)}.\n\\end{align}\nOf course, this bound obviously depends on the choice of basic elements $g$ and the fact that we are using two-qubit gates. If we had considered $q$-body gates, we would have found a denominator $\\log(g\\binom{n}{q}) \\approx \\log(gn^q)$ for $n\\gg q$.\nThis is no more than a choice of ``units'' with which to measure complexity. Thus, we state our key result as:\n\\begin{theorem}\\label{theorem-bound}\nFor an ensemble $\\mathcal{E}$ with the $k$th frame potential $F_{\\mathcal{E}}^{(k)}$, the circuit complexity is lower bounded by\n\\begin{align}\n\\mathcal{C}(\\mathcal{E}) \\geq \\frac{2kn \\log(2) - \\log F_{\\mathcal{E}}^{(k)} }{\\log( \\mathrm{choices})},\\label{eq:lower-bound}\n\\end{align}\n\\end{theorem}\nIn this context, $\\log( \\mathrm{choices})$ simply indicates the logarithm of the number of decisions that are made at each step. If we imagine we have some kind of decision tree for determining which gate to apply where we make a binary decision at each step (and use $\\log_2$), then we may set the denominator to unity and measure complexity in bits rather than gates, i.e. $\\mathcal{C}(\\mathcal{E}) \\geq 2kn  - \\log_2 F_{\\mathcal{E}}^{(k)}$.\n\n\nIn the above discussion, we glossed over a subtlety. Here, we considered the quantum circuit complexity to prepare or approximate an entire ensemble $\\mathcal{E}$. A closely related but different question concerns the quantum circuit complexity required to implement a typical unitary operator from the ensemble $\\mathcal{E}$. Nevertheless, in ordinary settings the typical operator complexity and the ensemble complexity are roughly of the same order. While establishing a rigorous result in this direction is beyond the scope of this paper, see~\\cite{Knill95} for some basic proof techniques that are useful in establishing this connection. \n\nAn important consequence of Theorem~\\ref{theorem-bound} is that the smallness of the frame potential (i.e. generic smallness of OTO correlators) implies increases in quantum circuit complexity of generating the ensemble $\\mathcal{E}$. As a corollary, we can simply rewrite Eq.~\\eqref{eq:lower-bound} as\n\\begin{align}\n\\mathcal{C}(\\mathcal{E}) \\geq(2k-1)2n   - \\log_2  \\sum_{A_{1},\\cdots,B_{1},\\cdots}\\left|\\big\\langle A_{1}\\tilde{B_{1}}\\cdots A_{k}\\tilde{B_{k}} \\big\\rangle_{\\mathcal{E}}\\right|^{2}.\\label{eq:lower-bound-oto}   \n\\end{align}\nIn this sense, we see how the decay of OTO correlators is directly related to an increase in the (lower-bound) of the complexity.\n\n\nNext, recall that the frame potential for a $k$-design is given by $F^{(k)}_{\\text{Haar}}=k!$, which does not grow with $n$. The complexity of a $k$-design is thus lower bounded as\n\\begin{equation}\n\\mathcal{C}(k\\text{-design}) \\ge 2kn  - k\\log_2 (k) + \\log_2 k + \\dots, \\label{eq:complexity-k-design}\n\\end{equation}\nwhich for large $n$ grows roughly linearly in $k$ and $n$ (at least). In \\S\\ref{sec:complexity:depth}, we show that the minimum depth circuit to make a $k$-design also growths linearly in $k$ (at least).\n\n\n\n\n\n\n\n\nFinally, we offer an additional information-theoretic interpretation for our lower bound and generalize it for ensembles with non-uniform probability distributions. Consider an ensemble $\\mathcal{E}=\\{ p_{j}, U_{j} \\}$ with probability distribution $\\{p_{j} \\}$ such that $\\sum_{j}p_{j}=1$. The second R\\`{e}nyi entropy of the distribution $\\{p_{j} \\}$ is defined as $S^{(2)}= - \\log \\big( \\sum_{j} p_{j}^2 \\big)$. In this more general situation, we can still bound the frame potential by considering the diagonal part of the sum\n\\begin{align}\nF^{(k)}_{\\mathcal{E}} = \\sum_{i,j} p_{i}p_{j} \\, \\big|\\text{tr} \\{ U_{i}U_{j}^{\\dagger} \\}\\big|^{2k} \\geq \\sum_{i} p_{i}^2 \\, \\big|\\text{tr} \\{ I \\} \\big|^{2k} = e^{-S^{(2)}}d^{2k}. \n\\end{align}\nSince the von Neumann entropy $S^{(1)} = - \\sum_{j} p_j \\log(p_j)$ is always greater than the second R\\`enyi entropy, $S^{(1)} \\geq S^{(2)}$, we can bound the von Neumann entropy as\n\\begin{align}\nS^{(1)} \\geq 2kn - \\log_2 F^{(k)}_{\\mathcal{E}}.\n\\end{align}\nThe entropy of the ensemble is a notion of complexity measured in bits.\\footnote{However, this is not to be confused with the entanglement entropy. The entropy of the ensemble is essentially the logarithm of the number of different operators, and therefore can be exponential in the size of the system. Instead, the entanglement entropy (as a measure of entanglement) can only be as large as (half) the size of the system.} \n\n\n\n\n\n\n\n\n\n\\subsection{Continuous ensembles}\\label{sec:continuous}\n\nMany interesting ensembles of unitary operators are defined by continuous parameters, e.g. a disordered system has a time evolution that may be modeled by an ensemble of Hamiltonians.\\footnote{A notable example of recent interest to the holography and condensed matter community is the ensemble implied by time evolving with the Sachdev-Ye-Kitaev Hamiltonian \\cite{Sachdev:1992fk,Kitaev:2014t2,Maldacena:2016hyu}.} While the counting argument in \\S\\ref{sec:complexity-discrete} is not directly applicable to these systems with continuous parameters, the complexity lower bound generalizes to such systems by allowing approximations of unitary operators. To be concrete, imagine that we wish to create some unitary operator $U_{0}$ by combining gates from the elementary gate set. In practice, we do not need to create an exact unitary operator $U_{0}$. Instead, we may be fine with preparing some $U$ that faithfully approximates $U_{0}$ to within a trace distance\n\\begin{align}\n||U_{0}-U||_{1}\\leq \\epsilon,\n\\end{align}\nwhere the notation $||\\cdot||_p \\equiv (\\text{tr} \\, \\{ |\\cdot|^p \\} )^{1\/p}$ specifies the $p$-norm of an operator.\n\nNow, let us derive a complexity lower bound up to an $\\epsilon$-tolerance. We begin by taking $N_s$ samples from the ensemble and use them to estimate the frame potential of the continuous distribution\n\\begin{equation}\nF_{\\mathcal{E}}^{(k)} \\approx \\frac{1}{N_s^2}\\sum_{i,j} \\big|\\text{tr}\\, \\{ U_i^{\\dagger}V_j \\} \\big|^{2k},\\label{sampled-frame-potential}\n\\end{equation}\nwhere the each of the two sums runs over all $N_s$ samples. We can lower bound Eq.~\\eqref{sampled-frame-potential} as follows\n\\begin{equation}\nF_{\\mathcal{E}}^{(k)} \\ge \\frac{1}{N_s^2}\\sum_{i} \\sum_{\\substack{j, \\\\ \\mathllap{||}U_i\\mathrlap{ - V_j||_1 < \\epsilon} } } \\big|\\text{tr}\\, \\{ U_i^{\\dagger}V_j \\} \\big|^{2k},\\label{epsilon-frame-potential}\n\\end{equation}\nwhere the sum over $i$ runs from $1$ to $N_s$. The sum over $j$ includes only a smaller subset $N_\\epsilon(U_i)$, which is the number of operators within a trace distance $\\epsilon$ of a particular $U_i$.\n\n\nTo continue, let's bound the summand. First, note that\n\\begin{equation}\n\\big| \\text{tr} \\{ U^\\dagger V \\} \\big| > \\text{Re} \\big\\{ \\, \\text{tr} \\, \\{  U^\\dagger V \\} \\big\\} = d - \\frac{1}{2}||U-V||_2^2.\n\\end{equation}\nThe $2$-norm is upper bounded by the $1$-norm as $|| \\mathcal{O} ||_2 \\le \\sqrt{d}  || \\mathcal{O}||_1$, which let's us rewrite this as\n\\begin{equation}\n\\big| \\text{tr}\\, \\{ U^{\\dagger}V \\} \\big|^{2k} > d^{2k}\\bigg(1 - \\frac{1}{2}||U-V||_1^2\\bigg)^{2k}.\\label{summand-bound}\n\\end{equation}\nFor this formula to be sensible, this approximation requires that $\\epsilon < \\sqrt{2}$. Substituting Eq.~\\eqref{summand-bound} into Eq.~\\eqref{epsilon-frame-potential}, we can bound the frame potential\n\\begin{equation}\nF_{\\mathcal{E}}^{(k)} \\ge  \\frac{1}{N_s} d^{2k}\\bigg(1 - \\frac{\\epsilon^2}{2}\\bigg)^{2k} \\bigg[ \\frac{1}{N_s}  \\sum_{i=1}^{N_s} N{_\\epsilon}(U_i) \\bigg] = d^{2k}\\bigg(\\frac{ \\overline{N_\\epsilon} }{N_s} \\bigg) \\bigg(1 - \\frac{\\epsilon^2}{2}\\bigg)^{2k}. \n\\end{equation}\nThe term in brackets in the middle expression is the average number of operators within a trace distance $\\epsilon$ of an operator in our sample set. In the final expression this is represented by the symbol $\\overline{N_\\epsilon}$.\n\n\nNow, let's run the counting argument again. If we want to make $N_s$ circuits exactly, then in $\\mathcal{C}$ steps we must have\n\\begin{equation}\n(\\text{choices})^\\mathcal{C} > N_s,\n\\end{equation}\nwhere as before $(\\text{choices})$ summarizes the information about our choice of gate set, etc. Instead, if you only care about making circuits to within an $\\epsilon$-accuracy, then in $\\mathcal{C}_\\epsilon$ steps $N_s$ instead satisfies\n\\begin{equation}\n(\\text{choices})^{\\mathcal{C}_\\epsilon} ~  \\overline{N_\\epsilon}  > N_s.\n\\end{equation}\nThis lets us lower bound the complexity of the ensemble at precision $\\epsilon$ as\n\\begin{equation}\n\\mathcal{C}_\\epsilon(\\mathcal{E}) > \\frac{2k \\log(d) - k \\epsilon^2 - \\log F_{\\mathcal{E}}^{(k)}}{\\log (\\text{choices}) }.  \\label{complexity-boudnd-eps}\n\\end{equation}\nWe then take the continuum limit by taking $N_s \\to \\infty$. The number of operators within an $\\epsilon$-ball of a given sample will also diverge, but the ratio $ N_s\/ \\overline{N_\\epsilon}$ should remain finite and converge to some value, roughly the volume of the ensemble as measured in balls of $\\epsilon$-radius.\\footnote{Note that $\\log(\\text{choices})$ diverges if the elementary gate set is continuous. This problem may be also fixed by employing the $\\epsilon$-tolerance.}\n\nFinally, in $\\S\\ref{sec:complexity:early}$, we further extend this notion of bounding complexity for continuous ensembles to show that the initial early-time growth of complexity for evolution with an ensemble of Hamiltonians grows initially as $t^2$ for a time $t < 1\/\\sqrt{\\log(d)}$.\n\n\n\n\n\n\n\n\\section{Measures of correlators}\\label{sec:haar-averages}\n\nWhile much of the focus of this paper has been on the behavior of ensembles, we were originally motivated by the following question: When is a random unitary operator an appropriate approximation to the underlying dynamics? In this section, we will attempt to return the focus to this question by computing random averages over correlation functions and comparing them to expectations for chaotic time evolution in physical systems.\n\n\\subsection{Haar random averages}\n\nIn this subsection, we will explicitly compute some ensemble averages of OTO correlators for different choices of ensembles. A particular goal will be to understand the asymptotic behavior of these averages in the limit of a large number of degrees of freedom $d\\to \\infty$. We present explicit calculations for $2$-point and $4$-point functions here and provide results for  $6$-point and $8$-point functions.  Additional calculations may be found in Appendix~\\ref{sec:appendix-averages}.\n\n\n\\subsubsection*{$2$-point functions}\n\nConsider a $2$-point correlator, averaged over Haar random unitary operators\n\\begin{align}\n\\langle A  \\tilde{B} \\rangle_{\\text{Haar}}=\\frac{1}{d}\\int_{\\text{Haar}} dU \\, \\text{tr} \\, \\{ A \\, U^{\\dagger}BU\\} .\n\\end{align}\nSince $U$ and $U^\\dagger$ each only appear in the expression once, we will obtain the same answer if the average is instead performed over a 1-design: $\\langle A  \\tilde{B} \\rangle_{\\text{Haar}} = \\langle A \\tilde{B} \\rangle_{\\text{1-design}}$. By using a formula from \\S\\ref{sec:review}, we can derive the following expression\n\\begin{align}\n\\langle A  \\tilde{B} \\rangle_{\\text{Haar}} = \\langle A\\rangle\\langle B \\rangle. \\label{2-pt-haar}\n\\end{align}\nGraphically, the calculation goes as follows\n\\begin{align}\n\\includegraphics[width=0.65\\linewidth]{fig_2point.pdf}.\n\\end{align}\nIt is often convenient to consider physical observables with zero mean by shifting $A \\rightarrow A -\\langle A\\rangle$. Then, we see that these $2$-point correlation function vanish. Of course, this always holds for Pauli operators\n\\begin{align}\n\\langle A  \\tilde{B} \\rangle_{\\text{Haar}} = 0, \\qquad A,B\\in \\mathcal{P},\\quad A,B\\not=I.\n\\end{align}\n\nNext, let us consider the norm squared of a $2$-point correlator averaged over Haar random unitary operators $|\\langle A  \\tilde{B} \\rangle|^2_{\\text{Haar}}=\\frac{1}{d^2}\\int_{\\text{Haar}} dU\\, \\text{tr} \\{ A\\,U^{\\dagger}BU \\} \\, \\text{tr} \\{ U^{\\dagger}B^\\dagger U \\, A^{\\dagger}\\}$. Note that we take the Haar average after squaring the correlator. Since there are two pairs of $U$ and $U^{\\dagger}$ appearing, we can perform the average over a $2$-design: $| \\langle A  \\tilde{B} \\rangle|^{2}_{\\text{Haar}} = |\\langle A  \\tilde{B} \\rangle|^{2}_{\\text{2-design}}$. Let us assume that $A,B$ are Pauli operators so that we can neglect contributions from $\\langle A\\rangle$ and $\\langle B\\rangle$. There are four terms, but only one term survives because the trace of non-identity Pauli operators is zero. We depict the calculation graphically as\n\\begin{align}\n\\includegraphics[width=0.6\\linewidth]{fig_2point_absolute.pdf}.\n\\end{align}\nwhere $C_{I}=1\/(d^2-1)$ comes from the Weingarten function as shown in \\S\\ref{sec:review}.  The final result is\n\\begin{align}\n|\\langle A  \\tilde{B}\\rangle|^2_{\\text{Haar}}=\\frac{1}{d^2-1},\\qquad A,B\\in \\mathcal{P},\\quad A,B\\not=I.\\label{eq:2-pt-variance}\n\\end{align}\nThus, the variance of the Haar averaged $2$-point function is exponentially small in the number of qubits. \n\n\\subsubsection*{4-point functions}\n\nNext, consider a $4$-point OTO correlator averaged over Haar random unitary operators: \n\\begin{align}\n\\langle A \\tilde{B} C \\tilde{D} \\rangle_{\\text{Haar}} = \\frac{1}{d}\\int_{\\text{Haar}} dU \\text{tr} \\{ A\\, U^\\dagger B U \\,C \\, U^\\dagger D U \\}. \n\\end{align}\nAs has already been explained, we will obtain the same answer if the average is performed with a 2-design: $\\langle A \\tilde{B} C \\tilde{D} \\rangle_{\\text{Haar}} = \\langle A \\tilde{B} C \\tilde{D} \\rangle_{\\text{2-design}}$. By using formulas from \\S\\ref{sec:review} we can derive the following expression\\footnote{Note: this formula was independently obtained by Kitaev.}\n\\begin{equation}\n\\langle A \\tilde{B} C \\tilde{D}  \\rangle_{\\text{Haar}} = \\langle AC \\rangle \\langle B \\rangle\\langle D \\rangle + \\langle A \\rangle\\langle C \\rangle \\langle BD \\rangle -  \\langle A\\rangle \\langle C \\rangle \\langle B \\rangle\\langle D \\rangle - \\frac{1}{d^2-1}\\langle\\!\\langle AC\\rangle\\!\\rangle \\langle\\!\\langle BD\\rangle\\!\\rangle, \\label{eq:OTO_formula}\n\\end{equation}\nwhere $d=2^n$ and $\\langle \\! \\langle AC\\rangle \\! \\rangle\\equiv\\langle AC\\rangle-\\langle A\\rangle\\langle C\\rangle$. In particular, for Pauli operators $A,B,C,D\\not=I$, one has\n\\begin{equation}\n\\begin{split}\n\\langle A \\tilde{B} C \\tilde{D} \\rangle_{\\text{Haar}} &= - \\frac{1}{d^2-1} \\qquad (A=C^{\\dagger},\\ B=D^{\\dagger} ), \\\\\n&= 0 \\qquad \\quad \\qquad (\\text{otherwise}).\n\\end{split}\n\\end{equation}\nWhen nonzero, the result is exponential small in the number of qubits $n$. The derivation of the aforementioned formula can be understood graphically as follows\n\\begin{equation}\n\\includegraphics[width=0.88\\linewidth]{fig_4pt_OTO}.\n\\end{equation}\nBy rewriting the expression in terms of connected correlators, we obtain the formula Eq.~\\eqref{eq:OTO_formula}.\n\nOf course, we can obtain the same result by instead averaging over the Clifford group. When $A\\not=C^{\\dagger}$ or $B\\not=D^{\\dagger}$, it is easy to show $\\langle A\\tilde{B}C\\tilde{D} \\rangle_{\\text{Clifford}}=0$.  When $A=C^{\\dagger}$ and $B=D^{\\dagger}$, the value of the correlator depends on the commutation relations between $A$ and~$B$\n\\begin{equation}\n\\begin{split}\n\\langle A\\tilde{B}A^{\\dagger}\\tilde{B}^{\\dagger} \\rangle&=1, \\quad \\qquad [A,B]=0,\\\\\n&=-1, \\qquad \\{A,B\\}=0.\n\\end{split}\n\\end{equation}\nSince by definition Clifford operators transform a Pauli operators to another Pauli operator, $\\tilde{B}$ is a random Pauli operator with $\\tilde{B}\\not=I$. There are $d^2-1$ non-identity Pauli operators. Among them, $d^2\/2-1$ Pauli operators commute with $A$, and $d^2\/2$ anti-commute with $A$. Therefore, we find\n\\begin{align}\n\\langle A\\tilde{B}A^{\\dagger}\\tilde{B}^{\\dagger} \\rangle_{\\text{Clifford}}\n= \\frac{d^2\/2-1}{d^2-1} - \\frac{d^2\/2}{d^2-1} = \\frac{-1}{d^2-1}.\n\\end{align}\nAs expected, the $4$-point OTO correlator ensemble average over the Clifford group equals the Haar average. Recalling our result from \\S\\ref{sec:OTO_channel} that $4$-point OTO values completely determine the $2$-fold channel, this explicit calculation gives an alternative proof that the Clifford group forms a unitary $2$-design.\n\nFinally, we will present one additional way of computing the Haar average of $4$-point OTO correlators. \nFor convenience, we introduce the following notation\n\\begin{align}\n\\text{OTO}^{(4)}(A,B) :=\\langle A \\tilde{B} A^{\\dagger}\\tilde{B}^{\\dagger} \\rangle_{\\text{Haar}}.\n\\end{align}\nSince $U$ is sampled uniformly over the unitary group, $\\text{OTO}^{(4)}(A,B)$ does not depend on $A,B$ as long as $A,B\\not=I$. Consider the average of OTO correlation functions over all Pauli operators $A$, including identity operators: $\\sum_{A} \\langle A \\tilde{B} A^{\\dagger}\\tilde{B}^{\\dagger} \\rangle_{\\text{Haar}}$. Since $\\frac{1}{d}\\sum_{A}A\\otimes A^{\\dagger}$ is the swap operator, we have\n\\begin{align}\n\\includegraphics[width=0.65\\linewidth]{fig_short_cut},\n\\end{align}\nwhere a dotted line represents an average over all the Pauli operators. This expression must be zero for $B\\not=I$. Thus, we have\n\\begin{align}\n\\sum_{A}\\text{OTO}^{(4)}(A,B) = 0, \\qquad B\\not=I. \n\\end{align}\nIf $A=I$, we have $\\text{OTO}(A,B)=1$. Since there are $d^2$ Pauli operators, we have \n\\begin{align}\n\\text{OTO}^{(4)}(A,B) = -\\frac{1}{d^2-1}, \\qquad A,B\\not=I. \n\\end{align}\nIn Appendix~\\ref{sec:k-point-averages}, we use this method to estimate the scaling of higher-point OTO correlators with $d$, finding $4m$-point functions of a related ordering scale as $\\sim 1\/d^{2m}$.\n\n\n\\subsubsection*{$6$-point functions}\n\nNext, consider the Haar average of $6$-point OTO correlators, $\\langle A \\tilde{B} C \\tilde{D} E \\tilde{F} \\rangle_{\\text{Haar}}$. We will assume that $A,\\ldots,F\\not=I$ are Pauli operators. In order to have non-zero contributions, we must have $ACE \\propto I$ and $BDF \\propto I$. Thus, we will only consider cases with $ACE \\propto I$ and $BDF \\propto I$.\n\nThe results depend on commutation relations between $A,C$ and $B,D$, but always have the following scaling\n\\begin{align}\n\\langle \\text{6-point} \\rangle_{\\text{Haar}} \\sim \\frac{1}{d^2}, \\qquad (A C E=I, \\quad B  D  F=I).\n\\end{align}\nExplicitly, when $[A,C]=0$ and $[B,D]=0$, we find\n\\begin{align}\n\\langle \\text{6-point} \\rangle_{\\text{Haar}}  = \\frac{2d^2}{(d^2-1)(d^2-4)}.\n\\end{align}\nThe more general expression is slightly complicated though has the same scaling.\nThus, the Haar average of $6$-point OTO correlators does not reach any lower a floor value than the Haar average of $4$-point OTO correlators. \n\n\\subsubsection*{$8$-point functions}\nFinally, we will study Haar averages of $8$-point OTO correlators. In this case, there are two different types of nontrivial out-of-time ordering, which behave differently at large $d$. These computations are annoyingly technical, and so the details are hidden in Appendix~\\ref{sec:8-pt-functions}.\n\n\nThe $8$-point OTO correlators of the first type can be written in the following manner\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{B}^{\\dagger} C^{\\dagger}\\tilde{D}^{\\dagger}\\rangle, \\qquad \\text{(non-commutator type)}.\\label{eq:non-commutator-type}\n\\end{align}\nFor Hermitian operators, this  essentially repeats $A\\tilde{B}C\\tilde{D}$ twice. For reasons that will subsequently become clear, we will call such OTO correlators ``non-commutator types.'' (However, the result does depends on the commutation relations between $A,C$ and $B,D$.) For these correlators, the scaling of the Haar average with respect to $d$ is\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{B}^{\\dagger} C^{\\dagger}\\tilde{D}^{\\dagger}\\rangle_{\\text{Haar}} \\sim \\frac{1}{d^2}, \\qquad \\text{(non-commutator type)},\n\\end{align}\nand this scaling does not depend on any commutation relations. Similar to what we found for the Haar average of the $6$-point functions, these non-commutator type $8$-point OTO correlators have the same scaling with $d$ as the Haar average of $4$-point OTO correlators. \n\nThe $8$-point OTO correlators of the second type take the form\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle, \\qquad \\text{(commutator type)},\n\\end{align}\nand are denoted ``commutator-type'' correlators.\nThese correlators have the property that they can be written in the form\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle = \\langle AKA^{\\dagger} K^{\\dagger}\\rangle,\n\\qquad K = \\tilde{B}C\\tilde{D},\n\\end{align}\ni.e. they are the expectation value of the group commutator of the operator $AKA^{\\dagger} K^{\\dagger}$.\nThe OTO correlators Eq.~\\eqref{eq:non-commutator-type} cannot be written in this way. As with the non-commutator types, the exact Haar average depends on commutation relations between $A,C$ and $B,D$. However, the scaling with respect to $d$ does not\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle_{\\text{Haar}} \\sim \\frac{1}{d^4}, \\qquad \\text{(commutator type)}.\n\\end{align}\nThe Haar average of these correlators is much smaller than the Haar average of the non-commutator types and the $4$- and $8$-point Haar averages! This suggests they might be a useful statistic for distinguishing ensembles that form a $4$-design and ensembles that form a $2$-design but do not form a higher design.\n\nTo test this idea, we can take an average of the commutator type $8$-point OTO correlators averaged over the Clifford group.\nSince we have assumed the operators $A, \\dots, D$ are Pauli operators, we find\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle_{\\text{Clifford}} =  \\langle A\\tilde{B}\\tilde{D}CA^{\\dagger} C^{\\dagger} \\tilde{D}^{\\dagger} \\tilde{B}^{\\dagger} \\rangle_{\\text{Clifford}}  = \\frac{K(A,C^{\\dagger})}{d}  \\, \\text{tr} \\{ A\\tilde{B}\\tilde{D}A^{\\dagger} \\tilde{D}^{\\dagger} \\tilde{B}^{\\dagger}\\},\n\\end{align}\nwhere in the first equality we commuted $C,\\tilde{D}$ and $C^{\\dagger},\\tilde{D}^{\\dagger}$, which holds because $C, D$ are Pauli operators, and in the second equality we\nhave defined $K(P,Q)$ by\n\\begin{equation}\nQ^{\\dagger}PQ = K(P,Q)P,\n\\end{equation} \nfor Pauli operators $P,Q$.  The final answer is\n\\begin{align}\n\\langle A\\tilde{B}C\\tilde{D}A^{\\dagger}  \\tilde{D}^{\\dagger} C^{\\dagger}\\tilde{B}^{\\dagger}\\rangle_{\\text{Clifford}} = \\frac{-K(A,C^{\\dagger})}{d^2-1}\\sim \\frac{1}{d^2}.\n\\end{align}\nRecall that the Clifford group is a unitary $2$-design, is not a $3$-design in general (except for a system of qubits) \\cite{Webb15}, and is never a $4$-design. Therefore, we see that commutator-type correlators may provide a statistical test of whether an ensemble forms a $k$-design but not a $k+1$-design. We explore this idea further in \\S\\ref{sec:k-point-averages}.\n\n\n\n\\subsection{Dissipation vs. scrambling}\nIn this subsection, we will return to the $2$- and $4$-point averages and compare them against expectations from time evolution. Furthermore, we will attempt to provide some physical intuition for the behavior of these averages over different ensembles. This will support our picture of chaotic time evolution leading to increased pseudorandomness.\n\nFor strongly coupled thermal systems, it is expected that the connected part of the $2$-point correlation functions decays exponentially within a time scale $t_{d}$ of order the inverse temperature $\\beta$\n\\begin{align}\n\\langle A(0)B(t) \\rangle \\rightarrow \\langle A\\rangle \\langle B \\rangle + O(e^{-t\/t_d}). \\label{2-pt-decay}\n\\end{align}\nThis time scale is often referred to as a ``dissipation'' or ``thermalization'' time and is related to the time it takes local thermodynamic quantities reaching equilibrium.\\footnote{For weakly coupled systems where the quasiparticle picture is valid (e.g. Fermi-liquids) the time scale is instead be given by $t_{d}\\sim \\beta^2$.} \nIt is suggestive that the results Eq.~\\eqref{2-pt-haar} and Eq.~\\eqref{2-pt-decay} are so similar. After a short time $t_d$, for these $2$-point functions the chaotic dynamics give the same results as the Haar random dynamics.\n\nNext, we turn to the variance of the $2$-point correlator $\\langle A(0)B(t) \\rangle$. For a closed system of finite number of degrees of freedom, the $2$-point function will be quasi-periodic with recurrences after a timescale $t_r \\sim e^{d}$ that is exponential in the dimension and doubly exponential in the number of degrees of freedom $d=2^n$.  As such, the long-time average of $|\\langle A(0)B(t) \\rangle|^2$ must be nonzero. This can be estimated by performing a time average and gives a well known result \\cite{Dyson:2002pf,Barbon:2003aq,Barbon:2014rma}\\footnote{N.B. there is an error in this calculation as presented in \\cite{Dyson:2002pf,Barbon:2003aq} and so interested readers should consult \\cite{Barbon:2014rma} for the actual details.}\n\\begin{align}\n\\lim_{T\\rightarrow \\infty}\\frac{1}{T}\\int_{0}^{T} |\\langle A(0)B(t)|^2 dt \\sim \\frac{1}{d^2}.\n\\end{align}\nComparing against our result for the Haar-averaged dynamics Eq.~\\eqref{eq:2-pt-variance}, we see that they coincide.\n\n\nNext, let's consider  $4$-point correlators in strongly-coupled theories with a large number of degrees of freedom $N$.\\footnote{We thank Douglas Stanford for conversations relating to dissipative behavior of $4$-point functions.} (For example, this can be thought of a system of $N$ qubits where all the qubits interact but the interactions are at most $q$-local, with $q\\ll N$ and $N\\to \\infty$.) First, let's consider the case of a \\emph{time-ordered} correlator. Similar to the case of the $2$-point functions, two of the three Wick contractions are expected to decay exponentially within a dissipation time $t_{d}$\n\\begin{equation}\n\\begin{split}\n\\langle A(0)C(0)B(t)D(t) \\rangle \\rightarrow \n\\langle AC \\rangle \\langle B D \\rangle + O(e^{-t\/t_d}),\n\\end{split}\n\\end{equation}\nwhich for qubits is analogous to considering a $2$-point function between the composite operators $AC$ and $B D$.\\footnote{Note that the exact timescale $t_{d}$ might depend on the operators being correlated and the particular contraction.} Thus, this correlator will equilibrate after a time $t_{d}$ with a late-time value that depends on the expectations $\\langle AC \\rangle$ and $\\langle B D \\rangle$.\n\nNow, let's consider the out-of-time-order $4$-point correlator in a large $N$ strongly interacting theory. For $t \\sim t_d$, this will behave similarly to the time-ordered correlator with two of the three Wick contractions decaying exponentially\n\\begin{equation}\n\\begin{split}\n\\langle A(0)B(t)C(0)D(t) \\rangle \\rightarrow \n\\langle AC \\rangle \\langle B D \\rangle + O(e^{-t\/t_d}), \\qquad t< t_d.\n\\end{split}\n\\end{equation}\nHowever, for $t > t_d$ the correlator obtains a exponentially growing connected component\n\\begin{equation}\n\\langle A(0)B(t)C(0)D(t) \\rangle \\rightarrow \n\\langle AC \\rangle \\langle B D \\rangle - O(e^{\\lambda (t-t_*)}), \\qquad t_d < t < t_*.\n\\end{equation}\nThis growth occurs in the regime $t_d < t < t_*$. The time scale $t_* = \\lambda^{-1} \\log N$, known as the fast scrambling time, is the time at which the exponentially growing piece of the correlator compensates its $1\/N$ suppression and  becomes $O(1)$. The coefficient $\\lambda$ has the interpretation of a new kind of Lyapunov exponent \\cite{Kitaev:2014t1} and is bounded from above by $2\\pi \/ \\beta$ \\cite{Maldacena:2015waa}. Finally, for $t > t_*$, these OTO $4$-point correlators are expected to decay to a small floor value that is  exponentially small in $N$. A natural guess for this floor is\n\\begin{equation}\n\\langle A(0)B(t)C(0)D(t) \\rangle \\rightarrow  e^{-O(N)}\\langle AC\\rangle \\langle BD\\rangle, \\qquad t > t_*,\n\\end{equation}\nwhich is reproduced from Eq.~\\eqref{eq:OTO_formula} with $1$-point functions assumed to be subtracted off.\n\n\nAs we mentioned, the $2$-point function Eq.~\\eqref{2-pt-decay} reached its Haar random value after a short dissipation time $t_d$. This is very suggestive of a picture where chaotic dynamics behave as a pseudo-$1$-design after a time $t_d$. Taking this point further, let's consider the 4-point OTO correlator averaged over Pauli operators, an ensemble that forms a $1$-design, but not a $2$-design. Furthermore, we will assume that the operators $A$ and $B$ have zero overlap. Under this assumption, we can show that \n\\begin{equation}\n\\begin{split}\n\\langle  A \\tilde{B} C \\tilde{D} \\rangle_{\\text{Pauli}} = \n\\langle AC \\rangle \\langle B D \\rangle. \\label{eq:some_result}\n\\end{split}\n\\end{equation}\nThe proof of this is relegated to \\S\\ref{sec:proof:pauli}.\nApparently, Pauli operators capture the behavior of the dynamics around $t\\sim t_d$, i.e. from after the dissipative regime until the scrambling regime, but then a $2$-design is required to capture the behavior after $t\\sim t_*$, i.e. the post-scrambling regime.\\footnote{Note that these observations depend on the ensemble we average over actually being the Pauli operators, and not just any ensemble that forms a $1$-design without forming a $2$-design. Furthermore, we assume that $A,B,C,D$ are simply few-body operators so that the correlator is of local operators. These choices are determined for us by the basis in which the Hamiltonian is $q$-local.}\n\nThus, we might say that after a time $\\sim t_d$, the system becomes a pseudo-$1$-design, and then after $\\sim t_*$ the system becomes a pseudo-$2$-design. (See Fig.~\\ref{fig_scrambling} for a cartoon of this behavior.) However, it remains an open question whether there are any additional meaningful timescales that can be probed with correlators after $t_*$, though we are hopeful that such timescales might be hiding in higher-point OTO correlators.\n\n\n \n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=0.70\\linewidth]{fig_plot2}\n\\caption{Thermalization and scrambling in the decay of a four-point OTO correlator. These correlators typically decay to $\\langle AC \\rangle \\langle B D \\rangle$ in a thermal dissipation time $t_d$, and then they decay to a floor value $\\sim d^{-2}$ at around the scrambling time $t_*$. These regimes are very well captured by replacing the dynamics with $1$-designs or $2$-designs, respectively. \n} \n\\label{fig_scrambling}\n\\end{figure}\n\n\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\nIn this paper, we have connected the related ideas of chaos and complexity to pseudorandomness and unitary design. A cartoon of these ideas is expressed nicely by Fig.~\\ref{fig-universe}. Operators can be thought of as being organized by increasing complexity. Regions defined by circles of larger and larger radius can be thought of as defining designs with increasing $k$.\\footnote{\n    While this picture is a cartoon, the manifold for the unitary group $U(n)$ has a dimension exponentially large in $n$. However, following \\cite{Dowling2006:gt}, one can find a metric in which the length of a minimal geodesic between points computes operator complexity and in which most sections are expected to be hyperbolic. Taking this further, \\cite{Brown:2016wib} considered an interesting analog system on a hyperbolic plane that captures many of the expected properties of complexity.\n} In the rest of the discussion, we will make some related points, tie up loose ends, and mention future work.\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[scale=.45]{fig_universe_rev4}\n\\caption{A cartoon of the unitary group, with operators arranged by design. We pick the identity operator to be the reference operator of zero complexity and place it at the center. Typical operators have exponential complexity and live near the edge. Operators closer to the center have lower complexity, which makes them both atypical and more physically realizable in a particular computational model.\n}\n\n \n\\label{fig-universe}\n\\end{figure}\n\n\n\n\n\\subsection*{Generalized frame potentials and designs}\n\nIn realistic physical systems, one usually does not have access to the full Hilbert space. For example, there may be some conserved quantities, such as energy or particle numbers, or the system may be at some finite temperature $\\beta$. In that case, one would be interested in understanding pseudorandomness inside a subspace of the Hilbert space, i.e restricted to some state $\\rho$. In Appendix~\\ref{sec:sub-space-randomization}, we generalize the frame potential for an arbitrary state $\\rho$ finding that the quantity\n\\begin{align}\n\\mathcal{F}^{(k)}_{\\mathcal{E}}(\\rho)&= \\iint dU dV \\ [\\text{tr} \\{ \\rho^{1\/k} UV^{\\dagger}\\} \\, \\text{tr} \\{ \\rho^{1\/k} VU^{\\dagger} \\}]^{k}, \\label{eq:discussion:generalized}\n\\end{align}\nhas all the useful properties desired of a frame potential. In particular, it is minimized by the Haar  ensemble, and it provides a lower bound on ensemble size and complexity.\n\nHowever, if the state $\\rho$ is the thermal density matrix, $\\propto e^{-\\beta H}$ and the ensemble is given by time evolution with an ensemble of Hamiltonians $\\mathcal{E}= \\{ e^{-iHt}\\}$, then we need to take into account the fact that the state itself depends on the ensemble. Instead, we can define a \\emph{thermal} frame potential\n\\begin{align}\n\\mathcal{W}^{(k)}_{\\beta}(t)=  \\iint dG dH \\ \\frac{ \n\\big|\\text{tr} \\, \\{e^{-(\\beta\/2k-it )G } e^{-(\\beta\/2k+it )H }\\}\\big|^{2k}\n}{\\text{tr} \\, \\{e^{-\\beta G} \\}\\text{tr} \\, \\{e^{-\\beta H} \\}}.\n\\end{align}\n In this case, even at $t=0$ one can derive an interesting bound on the complexity of the ensemble. We hope to return to this in the future to analyze the \\emph{complexity of formation}: the computational complexity of forming the thermal state $\\rho_{\\beta}$ from a suitable reference state.\\footnote{This is also a question of interest in holography, see e.g. \\cite{CofFormation}.}\n\nFinally, it would be similarly interesting to consider a different generalization of unitary designs where, under some physical constraints, we can only access some limited degrees of freedom in the system. In this sense, one could think of the unitary ensemble (as opposed the state) as being generated by tracing over these additional degrees of freedom. These ``subsystem designs'' would then be ``purified'' by integrating back in the original degrees of freedom.\\footnote{We thank Patrick Hayden and Michael Walter for initial discussions about this idea.}\n This interesting direction is a potential subject of future work.\n \n\n\\subsection*{More chaos in quantum channels}\n\nIn Appendix~\\ref{sec:more-chaos}, we revisit some ideas from our previous work \\cite{Hosur:2015ylk}, though these ideas are also relevant to the current work. In particular, in \\S\\ref{sec:more:HP} we reconsider the Hayden-Preskill notion of black hole scrambling \\cite{Hayden07}. In this thought experiment, Alice throws a secret quantum state into a black hole. Assuming Bob knows the initial state and dynamics of the black hole,  we show how the question of whether Bob can reconstruct Alice's secret is related to the decay of an average of a certain set of OTO four-point correlators.\n\n\n\n In \\S\\ref{sec:more:op}, we provide an operational interpretation to taking averages over OTO correlators. We show that this is related to a quantum game of ``catch'' where Alice may ``spit-on'' or otherwise perturb the ball before throwing it to Bob. The average over four-point OTO correlators gives the probability that Alice did not modify the ball. We also show that an average over higher-point OTO correlators can be interpreted as an ``iterated'' game of catch (i.e. what normally people just call ``catch'') where both Alice and Bob have the opportunity to modify the ball each turn. In this case, the OTO correlator average is related to the joint probability that neither Alice nor Bob perturb the ball.\n\n\n\n Finally, in \\S\\ref{sec:more:renyi} we show that an average over a particular ordering of $2k$-point OTO correlators can be related to the $k$th R\\'enyi entropy of the operator $U$ interpreted as a state. We find that\n\\begin{equation}\n-\\log( \\text{a certain average of $2k$-point OTO correlators}) \\propto S_{ \\text{subsystem}}^{(k)}(U),\\label{eq:summary-of-renyi-formula}\n\\end{equation}\nwhere $S_{\\text{subsystem} }^{(k)}$ is the R\\'enyi $k$-entropy of a particular subsystem of the density matrix $\\rho = \\ket{U}\\bra{U}$.\n\n\n\n\\subsection*{Volume of unitary operators}\n\nThe argument in \\S\\ref{sec:continuous} led to a bound on the ratio $ N_s \/ \\overline{N_\\epsilon}$, which can be interpreted as the volume of $\\mathcal{E}$ in terms of $\\epsilon$-balls. An interesting application of this bound might be to think about the volume of unitary operators in $U(d)$ that can be probed in a finite time scale $T$, i.e. the volume of operators with depth $\\mathcal{D} \\sim T$. (See \\S\\ref{sec:complexity:depth} for further discussion of a lower bound on circuit depth in terms of the frame potential.)\n\n\n\nIn fact, in certain situations (such as the Brownian circuit introduced in \\cite{Lashkari13} or the random circuit model), it is not difficult to show that the volume of unitary operators with depth $T$ grows at least $\\sim \\exp(\\text{const} \\cdot n \\, T)$ for some small $T$ and some constant independent of $n$ by computing the $k=1$ frame potential. This implies that  the space of unitary operators, with the metric being quantum gate complexity, has hyperbolic structure with constant curvature, as discussed in e.g. \\cite{Dowling2006:gt,Susskind:2014jwa,Brown:2016wib} (see also \\cite{Chemissany:2016qqq}). On the other hand, one can upper bound the volume of unitary operators with circuit depth $T$ by thinking about how the depth can grow $V(T) \\sim \\binom{n}{2}\\binom{n-2}{2}\\binom{n-4}{2}  \\cdots \\binom{2}{2} \\approx  \\exp(n \\log n \\cdot T)$. Thus, for small $T$ and large $n$, the lower bound seems to be reasonably tight.\n\nOnce a lower bound on the volume of unitary operators in an ensemble is obtained in the unit of $\\epsilon$-balls, we can also obtain a lower bound (of the same order) on the complexity of a typical operator in the ensemble. This seems possible by using the formal arguments given in \\cite{Knill95} even when the elementary gate set is not discrete, e.g. all the two-qubit gates.\n\nFinally, it's a curious fact that for systems with time-dependent Hamiltonian ensembles (such as the random circuit models or the Brownian circuit of \\cite{Lashkari13}) that we get an initial linear growth of the volume with $T$. As argued in \\S\\ref{sec:complexity:early} (and confirmed numerically), for time independent Hamiltonian evolution---e.g. in SYK or in the Gaussian unitary ensemble (GUE)---we get a lower bound  $V(T) \\sim \\exp(\\text{const} \\cdot n\\, T^2 )$, which persists for a short time $T \\sim 1\/\\sqrt{n}$. It would be very interesting to understand the difference in this scaling.\\footnote{\n    One might worry that this bound saturates at a value smaller than unity. However, we expect that there may be a continuous definition of complexity sensible for small complexities, see e.g. \\cite{Brown:2017jil}. \n}\n\n\n\n\n\n\\subsection*{Tightness of the complexity bound}\n\nWhile the frame potential provides a rigorous lower bound on the complexity of generating an ensemble of unitary operators, there may be a cost: the  bound may not be very tight when applied to time evolution by an ensemble of Hamiltonians.\\footnote{We have learned this by some numerical investigations of the frame potential.} \n\nLet us try to understand this better. \nTo be concrete, consider the $k=2$ frame potential for a strongly coupled spin systems that scrambles in $t_* \\sim \\log n$ time. In such a system, for local operators $W, V$ of unit weight, OTO four-point correlators $\\langle W(t)^{\\dagger}V^{\\dagger}W(t)V\\rangle$ will begin to decay after $t\\sim O(\\log n)$. Since the $k=2$ frame potential is the average of four-point OTO correlators, one might expect that the frame potential will also start to decay at $t\\sim O(\\log n)$. \n\nHowever, this is not quite right. We expect the decay time for more general correlators of larger operators to be reduced to $\\bar{t}_* \\sim t_*  -  O\\big(\\log(\\text{size}~W)\\big) - O\\big(\\log(\\text{size}~V)\\big)$, where $t_*\\sim O(\\log n)$ is the scrambling when $V$ and $W$ are low-weight operators. If we randomly select Pauli operators $W$ and $V$, they will typically be nonlocal operators with $O(n)$ weights, and therefore the OTO decay time $\\bar{t}_*$ will be reduced to $O(1)$ for $V$ and $W$ of typical sizes.\n\nIn fact, the above estimate suggests most of the correlators determining the complexity bound should begin to decay immediately. As we can see in Eq.~\\eqref{eq:lower-bound-oto}, each correlator itself only makes a logarithmic contribution to the complexity and so we shouldn't expect the remaining slow decaying local correlators to be dominant. (To be sure, a further investigation of this point is required.)\n\nOne possible way to fix this problem would be to generalize the frame potential by using $p$-norm with $p \\neq 2$ so that it is more sensitive to the slower decaying local correlators. We leave the study of such a generalization to the future.\n\n\n\n\n\n\\subsection*{Complexity and holography}\nFinally, we will return to the question of complexity and holography discussed in the introduction.\nIn the context of holography, computational complexity was ``introduced'' \\cite{Harlow:2013tf} as a possible resolution to the firewall paradox of \\cite{Almheiri13,Almheiri13b}. A direct connection between complexity and black hole geometry was first proposed by Susskind \\cite{Susskind:2014rva,Susskind:2014ira}, which culminated in proposals that the interior of the black hole geometry is holographically dual to the spatial volume \\cite{Stanford:2014jda} or the spacetime action \\cite{Brown:2015bva,Brown:2015lvg}. These proposals are motivated by the fact that the black hole interior continues to grow as the state evolves long past the time entropic quantities equilibrate \\cite{Hartman13}. While there is nice qualitative evidence for both of these proposals \\cite{Susskind:2014jwa,Roberts:2014isa,Brown:2016wib}, missing is a direct understanding of computational complexity in systems that evolve continuously in time with a time-independent Hamiltonian.\n\n\n\n    A hint can be obtained by considering some of the motivations for these holographic complexity proposals. In particular, building on the work of \\cite{Hartman13} and previous work of Swingle \\cite{Swingle12,Swingle12b}, Maldacena suggested that the black hole interior could be found in the boundary  theory by a tensor network construction of the state \\cite{Maldacena:2013t1}. A tensor network toy model of the evolution of the black hole interior was investigated in \\cite{Hosur:2015ylk}. In this toy model, the interior of the black hole was modeled as a flat tiling of perfect tensors, see Fig.~\\ref{tn-erb}. These tensors were elements of the Clifford group and acted as two-qubit unitary operators that highly entangled neighboring qubits at each time step. From the perspective of the boundary theory, this is a model for Hamiltonian time evolution.\\footnote{In addition to providing a model for the black hole interior, Swingle's identification of the ground state of AdS with a tensor network \\cite{Swingle12,Swingle12b} has led to numerous other quantum information insights in holography (e.g. quantum error correction and its relationship to bulk operator reconstruction \\cite{Almheiri14}) as well as additional toy models that demonstrate those features (e.g. \\cite{Pastawski15b,Hayden:2016cfa}).}\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\linewidth]{fig_circuit}\n\\caption{A $6$-qubit tensor network model for the geometry of the interior of a black hole. Via holography, the growth of the interior is expected to correspond to chaotic time evolution of a strongly coupled quantum theory. Here, each node corresponds to a perfect tensor and the numbers label the qubit. \n} \n\\label{tn-erb}\n\\end{figure}\n\n\n    This toy model captures some important features of the complexity growth of the black hole state. The number of tensors in the network grows linearly in time, by construction. Operators will grow ballistically, exhibiting the butterfly effect, and the network scrambles in linear time. Thus, this network captures the aspects black hole chaos related to local scrambling and ballistic operator growth discussed in \\cite{Roberts:2014isa} as well as aspects of complexity growth discussed in \\cite{Hartman13,Susskind:2014ira,Stanford:2014jda,Brown:2015bva}. \n\n\n\n    However, since in this model the perfect tensor is a repeated element of the Clifford group, the complexity can never actually grow to be very big. In fact, the quantum recurrence time of the model was investigated in \\cite{Hosur:2015ylk} and was found to be exponential in the entropy $\\sim e^{n}$ rather than doubly exponential $\\sim e^{e^n}$ as expected in a fully chaotic model. This is related to our oft stated fact that the Clifford group generally does not form a higher-than-2-design. In fact, this model can actually be mapped to a classical problem, and \n    by the Gottesman-Knill theorem its complexity can be no greater than $O(n^2)$ gates \\cite{Nielsen_Chuang}.\\footnote{Since the Clifford group is a group, the complexity of any particular circuit in our model is an element of the group. Thus, at most we should be able to reach it by a polynomial number of applications of $2$-qubit gates.} \n\n\n\n   These observations were the inspiration for this current work, since in this  toy model $4$-point OTO correlators behave chaotically, but higher-point OTO correlators do not. Nevertheless, this model can be ``improved'' by using random $2$-qubit tensors rather than a repeated perfect tensor.\\footnote{Note: the case of a averaging over a single random tensor is like time evolution with a time independent Hamiltonian. On the other hand, the case of averaging separately over all tensors has a continuum limit of time evolution with a time-dependent Hamiltonian with couplings that evolve at each time step. This model is known as the Brownian circuit \\cite{Lashkari13}.} In \\cite{Brandao12}, it was shown that this local random quantum circuit approaches a unitary $k$-design in a circuit depth that scales at most as $O(k^{10})$. Our complexity lower bound for a $k$-design Eq.~\\eqref{eq:complexity-k-design}  suggests that the time to become a $k$-design is lower bounded by $k$, and we suspect that this can be saturated.\\footnote{It is believed that this is actually saturated by the local random circuit of \\cite{Brandao12}. For example, there is some numerical evidence for this claim in \\cite{mozrzymas2013local}.} \n\nIt is in this sense that we speculate that the complexity growth of the chaotic black hole is pseudorandom. That is, we suspect that as the complexity of the black hole state increases linearly with time evolution $t$, the dynamics evolve to become pseudo-$k$-designs, with the value $k$ roughly scaling with $t$\n   \\begin{equation}\n   \\mathcal{C}(e^{-iHt}) \\sim t \\sim k,\n   \\end{equation}\n and that this may be quantified by either representative $2k$-point OTO correlators or by an appropriate generalization of unitary design. With this in mind, it would be interesting to see whether one could use the tools of unitary design to prove a version of the conjectures of \\cite{Brown:2015bva,Brown:2015lvg} suggesting that complexity (is greater than or) equal to action.\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nWe are grateful to Fernando Brandao, Adam Brown, Jordan Cotler, Guy Gur-Ari, Patrick Hayden, Alexei Kitaev, M\\'ark Mezei,  Xiao-Liang Qi, Steve Shenker, Lenny Susskind, Douglas Stanford, and Michael Walter for discussions. \n\nDR and BY both acknowledge support from the Simons Foundation through the ``It from Qubit'' collaboration.\nDR is supported by the Fannie and John Hertz Foundation, the National Science Foundation grant\nnumber PHY-1606531 and the Paul Dirac Fund, and is also very thankful for the hospitality of the Stanford Institute for Theoretical Physics and the Perimeter Institute of Theoretical Physics during the completion of this work. \nResearch at Center for Theoretical Physics at MIT is supported by the U.S. Department of Energy under cooperative research agreement Contract Number DESC0012567.\nResearch at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This paper was brought to you unitarily by the Haar measure.\\footnote{\n    Finally, we would like to thank one of our anonymous JHEP referees for pointing out the prior work \\cite{cbd:cd} and suggesting an acknowledgment. While we were unaware of \\cite{cbd:cd} at the time of submission, we are nevertheless quite happy to include a citation in our revision.\n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCreating realistic music pieces automatically has always been regarded as one of frontier subjects in the field of computational creativity. With recent advances in deep learning, deep generative model and its variants have been widely used in automatic music generation \\cite{HerremansCC17}\\cite{deepgenerationsurvey}. However, most of deep composition methods focus on Western music rather than Chinese music. How to employ deep learning to model the structure and style of Chinese music is a challenging but novel problem.\n\nChinese folk songs, an important part of traditional Chinese music, are improvised by local people and passed on from one generation to the next orally. Folk tunes from the same region exhibit similar style while tunes from different areas present different regional styles \\cite{Miao1985}\\cite{LiLDZY19}. For example, the songs named \\emph{Mo Li Hua} have different versions in many areas of China and show various music styles, though they share the same name and similar lyrics \\footnote{The Chorus of \\emph{Mo Li Hua Diao} from various regions in China by Central National Orchestra: \\url{http:\/\/ncpa-classic.cntv.cn\/2017\/05\/11\/VIDEEMEg82W5MuXUMM1jpEuL170511.shtml}.} . The regional characteristics of Chinese folk songs are not well explored and should be utilized to guide automatic composition for Chinese folk tunes. Furthermore, folk song composition based on regional style provides abundant potential materials for Chinese national music creation, and promotes the spread and development of Chinese national music and even Chinese culture in the world.\n\nThere are lots of studies on music style composition of Western Music\\cite{Dai2018}.  However, few studies employ deep generative model for Chinese music composition. There is a clear difference between Chinese and Western music. Unlike Western Music, which focuses on the vertical structure of music, Chinese music focuses on the horizontal structure, i.e., the development of melody, and the regional style of Chinese folk songs is mainly reflected in its rhythm and pitch interval patterns \\cite{Guan2014}. \n\nIn this paper, we propose a deep music generation model named MG-VAE to capture regional style of Chinese folk songs (\\emph{Min Ge}) and create novel tunes with controlled regional style. Firstly, a MIDI dataset with more than 2000 Chinese folk songs covering six regions is collected. After that, we encode the input music representations to the latent space and decode the latent space to reconstruct music notes. In detail, the latent space is divided into two parts to present the pitch features and rhythm features, namely, \\emph{pitch variable} and \\emph{rhythm variable}. Then we further divide the pitch latent space into \\emph{style variable} part and \\emph{content variable} part to present style feature and style-less feature in pitch variable, the same operation is launched in rhythm variable.  In order to capture the regional style of Chinese folk songs precisely and generate regional style songs in controllable way, we propose a method based on adversarial training for disentanglement of the four latent variables, where temporal supervision is employed in the separation of pitch and rhythm variable, and label supervision is used for the disentanglement the style and content variable. The experimental results and visualization of latent spaces show that our model is effective to disentangle latent variables and is able to generate folk songs with specific regional style. \n\nThe rest of the paper is structured as follows: after introducing related work on deep music generation in Section 2, we present our music representations and model in Section 3. Section 4 describes the experimental results and analysis of our methods. Conclusions and future work are presented in Section 5.\n\n\\section{Related Work}\nRNN (Recurrent Neural Network) is one of the most earliest models introduced into the domain of deep music generation. Researchers employ RNNs to model the music structure and generate different formats of music, including monophonic folk melodies \\cite{SturmSBK16}, rhythm composition \\cite{MakrisKKK19}, expressive music performance \\cite{Sageev19}, multi-part music harmonization \\cite{YanLVD18}. Other recent studies have started to combine convolutional structure and explore using VAE, GAN (Generative Adversarial Network) and Transformer for music generation. MidiNet \\cite{YangCY17} and MuseGAN \\cite{DongHYY18} combine CNN (Convolutional Neural Network) and GAN architecture to generate music with multiple MIDI tracks. MusicVAE \\cite{RobertsERHE18} introduces a hierarchical decoder into general VAE model to generate music note sequences with long-term structure. Due to the impressive results of Transformer in neural translation, Huang et al. modify this sequence model's relative attention mechanism and generate minutes of music clips with high long-range structural coherence \\cite{MusicTransformer}. In addition to the study of music structure, researchers also employ deep generative models to model music styles, such as producing jazz melodies through two LSTM networks \\cite{JohnsonKW17}, harmonizing a user-made melody in Bach's style \\cite{Coconet2019}. Most of them are trained on the specific style dataset. The music generated from these models can only mimic the single style embodied in the training data. \n\nMoreover, little attention has been paid to Chinese music generation with deep learning techniques, especially for modeling the music style of Chinese music, though some researchers utilize Seq2Seq model to create multi-track Chinese popular songs from scratch \\cite{ZhuLYQLZZWXC18} or generate melody of Chinese popular songs with given lyrics \\cite{Bao2018}. The existing generation algorithms for Chinese traditional songs are mostly based on non-deep models such as Markov models \\cite{HuangLNC16}, genetic algorithms \\cite{ZhengWLSGGW17}. These studies cannot break up the bottleneck in melody creation and style imitation.  \n\nSome latest work in the domain of music style transfer begins to generate music with mixed style or recombine music content and style.  For example, Mao et al. propose an end to end generative modal to produce music with mixture of different classical composer styles \\cite{MaoSC18}. Lu et al. study the deep style transfer between Bach chorales and Jazz \\cite{LuS18a}. Nakamura et al. complete melody style conversion among different music genres \\cite{NakamuraSNY19}. The above studies are based on the music data from different genres or composing periods. However, the regional style generation of Chinese folk songs studied here is modeling style within the same genre, which is more challenging.\n\n\\section{Approach}\n\n\\subsection{Music Representation}\nThe monophonic folk songs $M$ can be represented as a sequence of note tokens, which is a combination of its pitch, interval and rhythm. Pitch and rhythm are essential information for music. The interval is an important indicator to distinguish the regional music feature, especially for Han Chinese folk songs\\cite{Han1989}. The detail processing is described as below and shown in Fig.~\\ref{fig:1}.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=3.6in]{data_representation.png}\n\t\\caption{Chinese folk songs representation including pitch sequence, interval sequence, rhythm sequence.}\n\t\\label{fig:1}\n\\end{figure}\n\n\\begin{itemize}\n\\item \\textbf{Pitch Sequence} $P$: Sequence of pitch tokens which consists of the pitch type presented in melody sequence. Rest note is assigned a special token.\n\t\n\\item \\textbf{Interval Sequence} $I$: Sequence of interval tokens derived from $P$. Each interval token is represented as a deviation between the next pitch and current pitch in step of semitone.\n\t\n\\item \\textbf{Rhythm Sequence} $R$: Sequence of duration tokens comprised of the duration type presented in melody sequence.\n\\end{itemize}\n\n\n\\subsection{Model}\n\nAs mentioned in Section 1, the regional characteristics of Chinese folk songs are mainly reflected in their pitch patterns and rhythm patterns. In some areas, the regional characteristics of folk songs are more dependent on pitch feature, while the rhythm patterns in some areas are more distinctive. For example, in terms of pitch, folk songs in northern Shaanxi tend to use perfect forth, the Hunan folk songs often use the combination of major third and minor third \\cite{Miao1985}, while Uighur folk songs employ the non-pentatonic scale. In terms of rhythm, Korean folk songs have their special rhythm system named \\emph{Jangdan}, while Mongolian \\emph{Long Song}s generally prefer long duration notes \\cite{Du2014}. \n\nInspired by the above observations, it is necessary to further refine the style of folk songs both in pitch and rhythm. Therefore, we propose a VAE-based model to separate pitch space and rhythm space, and further disentangle the music style and content space from pitch and rhythm space, respectively. \n\n\\subsubsection{VAE and its Latent Space Division}\n\nThe VAE introduces a continuous latent variable $z$ from a Gaussian prior $p_{\\theta}(z)$, and then generates sequence $x$ from the distribution $p_{\\theta}(x|z)$ \\cite{KingmaW13}. Concisely, a VAE includes an encoder $q_{\\phi}(z|x)$, a decoder $p_{\\theta}(x|z)$  and latent variable $z$. The loss function of VAE is \n\n\\begin{equation}\nJ(\\phi, \\theta) = -\\mathbb{E}_{q_{\\phi}(z|x)}[{\\rm log}p_{\\theta}(x|z)] + \\beta KL(q_{\\phi}(z|x)\\|p_{\\theta}(z))\n\\label{eq:1}\n\\end{equation}\n\nwhere the first term denotes reconstruction loss, and the second term refers to the Kullback-Leibler (KL) divergence, which is added to regularize the latent space. Weight $\\beta$  is a hyperparameter to balance the two loss terms. By setting $\\beta<1$, we can improve the generation quality of the model \\cite{Higgins2017}. $p_{\\theta}(z)$  is the prior and generally obeys the standard normal distribution, i.e., $p_{\\theta}(z) = \\mathcal{N}(0,I)$. The posterior approximation $q_{\\phi}(z|x)$ is parameterized by encoder which is also assumed to be Gaussian and reparameterization trick is used to acquire its mean and variance.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=4.4in]{model1.png}\n\t\\caption{Architecture of our model, it consists of melody encoder $E$, pitch decoder $D_P$, rhythm decoder $D_R$ and melody decoder $D_M$.}\n\t\\label{fig:2}\n\\end{figure}\n\nWith the labeled data, we can disentangle the latent space of VAE in a way that different parts of the latent space correspond to different external attributes, which can enable the generation process in a more controllable way. In our case, we assume that the latent space can be firstly divided into two independent parts, i.e., \\emph{pitch variable} and \\emph{rhythm variable}. The pitch variable learns the pitch features of Chinese folk songs, while rhythm variable captures the rhythm patterns. Further, we assume both the pitch variable and rhythm variable consist of two independent parts, which refer to music \\emph{style variable} and music \\emph{content variable}, respectively. \n\nSpecifically, given a melody sequence $M=\\{m_1,m_2,\\cdots,m_n\\}$ as the input sequence with $n$ tokens (notes), where $m_k$ denotes the feature combination of the corresponding pitch token $p_k$, interval sequence $i_k$ and rhythm sequence $r_k$, we firstly encode $M$ and obtain four latent variables from the linear transformation of the encoder's output. The four latent variables are pitch style variable $Z_{P_s}$, pitch content variable $Z_{P_c}$, rhythm style variable $Z_{R_s}$  and rhythm content variable $Z_{R_c}$, respectively. Then, we concatenate $Z_{P_s}$ and $Z_{P_c}$ into the total pitch variable $Z_{P}$, which is used to predict the pitch sequence $\\hat{P}$. The same operation is launched in rhythm variable to predict $\\hat{R}$. Finally, all latent variables are concatenated to predict the total melody sequence  $\\hat{M}$. The architecture of our model is shown in Fig.~\\ref{fig:2}. \n\n\n\nBased the above assumption and operation, it is easy to extend the basic loss function:\n\\begin{equation}\nJ_{vae} = H(\\hat{P},P) + H(\\hat{R},R) + BCE(\\hat{M},M) + \\beta KL_{total}\n\\label{eq:2}\n\\end{equation}\n\nwhere $H(\\cdot,\\cdot)$ and $BCE(\\cdot,\\cdot)$ denote the cross entropy and binary cross entropy between prediction values and target values, respectively, and $KL_{total}$  denotes the sum KL loss of the four latent variables.\n\n\n\\subsubsection{Adversarial Training for Latent Spaces Disentanglement}\n\nHere, we propose an adversarial training based method to conduct the disentanglement of pitch and rhythm, music style and content. The detail processing is shown in Fig.~\\ref{fig:3}.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=4.5in]{model2.png}\n\t\\caption{Detail processing of latent spaces disentanglement. The\n\t\tdashed lines indicate the adversarial training parts.}\n\t\\label{fig:3}\n\\end{figure}\n\nAs shown in Fig.~\\ref{fig:2}, we use two parallel decoders to reconstruct pitch sequence and rhythm sequence, respectively. Ideally, we expect the pitch variable $Z_P$ and rhythm variable $Z_R$ should be independent of each other. However, the pitch feature may be implicit in rhythm variables actually, vice versa, since the two variables are sampled from the same encoder output. \n\nIn order to separate the pitch and rhythm variable explicitly, the temporal supervision is employed in the separation of pitch and rhythm, which is similar to the work of disentangled representation for pitch and timbre \\cite{IJCAI2019}. Specifically, we feed the latent variable to the wrong decoder deliberately and force the decoder to predict nothing, i.e., all zero sequence, resulting in the following two loss terms based on cross entropy:\n\n\\begin{equation}\nJ_{adv,P} = -\\Sigma[\\mathbf{0}\\cdot {\\rm log}\\hat{P}_{adv} + (1-\\mathbf{0})\\cdot {\\rm log}(1-\\hat{P}_{adv})]\n\\label{eq:3}\n\\end{equation}\n\n\\begin{equation}\nJ_{adv,R} = -\\Sigma[\\mathbf{0}\\cdot {\\rm log}\\hat{R}_{adv} + (1-\\mathbf{0})\\cdot {\\rm log}(1-\\hat{R}_{adv})]\n\\label{eq:4}\n\\end{equation}\n\nwhere $\\mathbf{0}$ denotes all zero sequence, $`\\cdot$' denotes the element-wise product.\n\nFor the disentanglement of music style and content, we firstly obtain the total music style variable $Z_s$ and content variable $Z_c$:\n\n\\begin{equation}\nZ_s = Z_{P_s} \\oplus Z_{R_s}, Z_c = Z_{P_c} \\oplus Z_{R_c}\n\\label{eq:5}\n\\end{equation}\n\nwhere $\\oplus$ means the concatenate operation. \n\nThen two classifiers are defined to force the separation of style and content in the latent space using the regional information. The style classifier ensures the style variable is discriminative for regional label, while the adversary classifier force the content variable is not distinctive for regional label. For style classifier is trained with the cross entropy defined by\n\\begin{equation}\nJ_{dis, Z_s} = -\\Sigma y {\\rm log}p(y|Z_s)\n\\label{eq:6}\n\\end{equation}\n\nwhere $y$ denotes the ground truth, $p(y|Z_s)$ is the predicted probability distributions from style classifier.\n\nFor adversary classifier, we train it by maximizing the empirical entropy of the adversary classifier's prediction \\cite{FuTPZY18}\\cite{Vineet2018}. The training processing is divided into two steps. Firstly, the parameters of the adversary classifier are trained independently, i.e., the gradients of the classifier don't propagate back to VAE. Secondly, we compute the empirical entropy based on the output from adversary classifier as defined by\n\n\\begin{equation}\nJ_{adv, Z_c} = -\\Sigma p(y|Z_c) {\\rm log}p(y|Z_c)\n\\label{eq:7}\n\\end{equation}\nwhere $p(y|Z_c)$ is the predicted probability distributions from adversary classifier. \n\nIn summary, the overall training objective of our model is the minimization the loss function defined by\n\n\\begin{equation}\nJ_{total} = J_{vae} + J_{adv,P} + J_{adv, R} + J_{dis, Z_s} - J_{adv, Z_c}\n\\label{eq:8}\n\\end{equation}\n\n\n\\section{Experimental Results and Analysis}\n\\subsection{Datasets and Preprocessing}\n\nThe lack of large-scale Chinese folk song datasets makes it impossible to apply deep learning methods for automatic generation and analysis of Chinese music. Therefore, we digitize more than 2000 Chinese folk songs in MIDI format from the record of \\emph{Chinese Folk Music Integration}\\footnote{\\emph{Chinese Folk Music Integration} is one of the major national cultural project leaded by the former Ministry of Culture, National Ethnic Affairs Commission and Chinese Musicians Association from 1984 to 2001. This set of book contains more than 40000 selected folk songs of different nationalities. The project website is \\url{http:\/\/www.cefla.org\/project\/book }.}. These songs contain Han folk songs from Wu dialect district, Xiang dialect district\\footnote{According to the analysis of Han Chinese folk songs \\cite{Han1989}\\cite{Du1993}, the folk song style of each region is closely related to the local dialects. Therefore, we classify Han folk songs based on dialect divisions. Wu dialect district here mainly includes Southern Jiangsu, Northern Zhejiang and Shanghai. Xiang dialect district here mainly includes Yiyang, Changsha, Hengyang, Loudi and Shaoyang in Hunan province.}  and northern Shaanxi, as well as three ethnic minority folk songs of Uygur in Xinjiang, Mongolian in Inner Mongolia and Korean in northeast China.\n\nAll melodies in datasets are transposed to C key. We use the Pretty-midi python toolkit \\cite{raffel2014intuitive} to process each MIDI file, and count the numbers of pitch token, interval token and rhythm token as the feature dimension of the corresponding sequence, which are 40, 46 and 58, respectively. Then pitch sequence, interval sequence and rhythm sequence are extracted from raw notes sequence with the overlapping window of length 32 tokens and a hop-size of 1. Finally, we get 65508 ternary sequences in total. The regional labels of the token sequences drawn from the same song are consistent.\n\n\\subsection{Experimental Setup}\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=2.8in]{encoder.png}\n\t\\caption{Encoder with residual connections.}\n\t\\label{fig:4}\n\\end{figure}\n\nIn order to extract melody feature into latent space effectively, we employ a bidirectional GRU model with the residual connection \\cite{HeZRS16} as encoder, which is illustrated in Fig.~\\ref{fig:4}. The decoder is a normal two-layers GRU. All recurrent hidden size in this paper is 128. Both style classifier and adversary classifier are one-layer linear layer with Softmax function. The size of pitch style variable and rhythm style variable is set to 32, while the size of pitch content variable and rhythm content variable is 96. During training period, the KL term coefficient $\\beta$ increases from 0.0 to 0.15 linearly to alleviate the impact of posterior collapse.\n\nAdam optimizer is employed with the initial learning rate of 0.01 for VAE training, and vanilla SGD optimizer with the initial learning rate of 0.005 for classifiers. All test models are trained for 30 epochs and the size of mini-batch is set to 50.\n\n\n\n\\subsection{Evaluation and Results Analysis}\n\nTo evaluate the generated music, we employ the following metrics from objective and subjective perspectives.\n\n\\begin{itemize}\n\t\\item \\textbf{Reconstruction Accuracy}: We calculate the accuracy between the target notes sequence and reconstructed notes sequence on our test set to evaluate the music generation quality.\n\t\\item \\textbf{Style Recognition Accuracy}: We train a separate style evaluation classifier using the architecture in Fig.~\\ref{fig:4} to predict the regional style of the tunes that are generated using different latent variables. The classifier achieves a reasonable regional accuracy on the independent test set, which is up to 82.71\\%. \n\t\\item \\textbf{Human Evaluation}:  As human should be the ultimate judge of creations, human evaluations are conducted to overcome the incoordinations between objective metrics and user studies. We invite three experts who are well educated and expertise in Chinese music. Each expert is asked to listen to the random selected five folk songs of each region on-site, and rate each song on a 5-point scale from 1 (very low) to 5 (very high) according to the following two criteria: a) \\emph{Musicality}: Does the song have a clear music pattern or structure? b) \\emph{Style Significance}: Does the songs' style match the given regional label?\n\\end{itemize}\n\n\\begin{table}[htb]\n\t\\centering\n\t\\caption{Results of automatic evaluations.}\n\t\\begin{tabular}{ccc}\n\t\t\\hline\n\t\tObjectives & Reconstruction Accuracy  & Style Recognition Accuracy  \\\\\n\t\t\\hline\n\t\t$J_{vae}$ &  0.7684 & 0.1726\/0.1772\/0.1814 \\\\\n\t\t\\hline\n\t\t$J_{vae}, J_{adv,P,R}$ & 0.7926  & 0.1835\/0.1867\/0.1901 \\\\\n\t\t\\hline\n\t\t$J_{vae}, J_{adv, P,R}, J_{adv, Z_c}$ &  0.7746  & 0.4797\/0.4315\/0.5107 \\\\\n\t\t\\hline\n\t\t$J_{vae}, J_{adv, P,R}, J_{dis, Z_s}$ & \\textbf{0.8079}   & 0.5774\/0.5483\/0.6025 \\\\\n\t\t\\hline\n\t\t$J_{total}$ &  0.7937   &\\textbf{0.6271}\/\\textbf{0.5648}\/\\textbf{0.6410} \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{tab:1}\n\\end{table}\t\t\n\n\n\nTab~\\ref{tab:1} shows all evaluation results of our models. The three values in the third column denote the accuracies derived from the following three kinds of latent variables: a) the concatenation of pitch style variable $Z_{P_s}$ and a random variable sampled from standard normal distribution; b)  the concatenation of rhythm style variable $Z_{R_s}$ and the random variable; c) the concatenation of total style variable $Z_s$ and the random variable. $J_{adv,P,R}$ denotes the sum of $J_{adv,P}$ and $J_{adv, R}$. \n\n\n\nThe model with $J_{total}$ achieves the best results in style recognition accuracy and a sub-optimal result in reconstruction accuracy. The model without any constraints performs poorly on the two objective metrics. The addition of $J_{adv,P,R}$ improves the the reconstruction accuracy but fails to bring meaningful improvement to style classification. With the addition of either $J_{adv, Z_c}$ or $J_{dis, Z_s}$, all the three recognition accuracies improve a lot, which indicates that the latent spaces are disentangled into style and content subspaces as expected. Moreover, only employing pitch style or rhythm style for style recognition can also obtain fair results, demonstrating the disentanglement of pitch and rhythm is effective.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=4.6in]{human_new.png}\n\t\\caption{Results of human evaluations including musicality and style significance. The heights of bars represent means of the ratings and the error bars represent the standard deviation.}\n\t\\label{fig:5_add}\n\\end{figure}\n\nThe result of human evaluations is shown in Fig.~\\ref{fig:5_add}. In terms of musicality, all test models have similar performance, which demonstrates the addition of extra loss function has no negative impact on the generation quality of original VAE. Moreover, the model with total objectives $J_{total}$ performs significantly better than other models in terms of style significance (two-tailed $t$-test, $p<0.05$), which is consistent with the results in Tab~\\ref{tab:1}.\n\n\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\subfigure[Pitch style latent space ]{\\includegraphics[width=2.2in]{ps_space.png} \\label{fig:5a}}~~\n\t\\subfigure[Rhythm style latent space ]{\\includegraphics[width=2.2in]{ds_space.png} \\label{fig:5b}}\n\t\\subfigure[Total style latent space]{\\includegraphics[width=2.2in]{style_space.png} \\label{fig:5c}}~~\n\t\\subfigure[Total content latent space ]{\\includegraphics[width=2.2in]{content_space.png} \\label{fig:5d}}\n\t\\caption{t-SNE visualization of model with $J_{total}$.}\n\t\\label{fig:5}      \n\\end{figure*}\n\nFig.~\\ref{fig:5} shows the t-SNE visualization\\cite{maaten2008visualizing} of our model with $J_{total}$. We can observe that music with different regional labels is noticeably separated in the pitch style space, rhythm style space and total style space, but looks chaos in content space.This further demonstrates the validity of our proposed methods to disentangle the pitch, rhythm, style and content.\n\nFinally, we present several examples \\footnote{Online Supplementary Material: \\url{https:\/\/csmt201986.github.io\/mgvaeresults\/}.} of generating folk song with given regional labels with our methods in Fig.~\\ref{fig:6}. As seen, we can create novel folk songs with dominated regional features such as long duration notes and large interval in Mongolian songs, the combination of major third and minor third in Hunan folk songs, and so on. However, there are still several failed examples. For instance, few generated songs repeat same melody pattern. More commonly, some songs don't show the correct regional feature, especially when the given regions belong to Han nationality areas. This may due to the fact that folk tunes in those regions share the same tonal system.\n\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=4.8in]{music_example.png}\n\t\\caption{Examples of folk songs generation given regional labels. In order to align each row, the scores of several regions are not completely displayed.}\n\t\\label{fig:6}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn this paper, we focus on how to capture the regional style of Chinese folk songs and generate novel folk songs with specific regional labels. We firstly collect a database including more than 2000 Chinese folk songs for analysis and generation. Then, inspired by the observation of the regional characteristics in Chinese folk songs, a model named MG-VAE based on adversarial learning is proposed to disentangle the pitch variable, rhythm variable, style variable and content variable in the latent space of VAE. Three metrics containing automatic and subjective evaluation in our experiments are used to evaluate the proposed model. Finally, the experimental results and t-SNE visualization show that the disentanglement of the four variables is successful and our model is able to generate folk songs with controllable regional style. In the future, we plan to expand the proposed model to generate longer melody sequence using more powerful model like Transformers, and explore the evolution of tune families like \\emph{Mo Li Hua Diao}, \\emph{Chun Diao} among different regions.\n\n\n\\bibliographystyle{spmpsci}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Analysis and Results}\n\\label{sec:results}\n\\subsection{RQ1- Influence of grammatical,  temporal  and  sentimental   sentence  characteristics on  FR\/NFR classification}\nFor the classification of functional and non-functional requirements, we used the approach of Hussain et al. \\cite{hussain2008using}. We applied this approach to the unprocessed data set of requirements as well as on the processed one resulting from our preprocessing.\n\n{\\it Classification Process:}\nFirstly, we clean up the respective data set by iteratively removing encoding and formatting errors to ensure the further processing. Subsequently, we apply the part-of-speech tagger of the Stanford Parser \\cite{Klein:2003:AUP:1075096.1075150} to assign parts of speech to each word in each requirement.\n\nBased on the tagging of all requirements, we extract the five syntactic features \\textit{number of adjectives}, \\textit{number of adverbs}, \\textit{number of adverbs that modify verbs}, \\textit{number of cardinals}, and \\textit{number of degree adjective\/adverbs}. For each feature, we determine its rank based on the feature's probability of occurrence in the requirements of the data set. According \nto Hussain et al.~\\cite{hussain2008using}, we selected a cutoff threshold of $>0.8$. Therefore, we determined \\textit{number of cardinals} and \\textit{number of degree of adjectives\/adverbs} as valid features among all five ones for the unprocessed data set. For the processed data set, we identified \\textit{number of cardinals} and \\textit{number of adverbs} as valid features.\n\nAfterwards, we extract the required keyword features for the nine defined part-of-speech keyword groups \\textit{adjective}, \\textit{adverb}, \\textit{modal}, \\textit{determiner}, \\textit{verb}, \\textit{preposition}, \\textit{singular noun}, and \\textit{plural noun}. For each keyword group, we calculate the smoothed probability measure and selected the respective cutoff threshold manually to determine the most discriminating keywords for each data set, corresponding to Hussain et al. \\cite{hussain2008using}.\n\nOur final feature list for the unprocessed data set consisted of the ten features \\textit{number of cardinals}, \\textit{number of degree of adjectives\/adverbs}, \\textit{adjective}, \\textit{adverb}, \\textit{modal}, \\textit{determiner}, \\textit{verb}, \\textit{preposition}, \\textit{singular noun}, and \\textit{plural noun}.\n\nOur final feature list for the processed data set consisted of the ten features \\textit{number of cardinals}, \\textit{number of adverbs}, \\textit{adjective}, \\textit{adverb}, \\textit{modal}, \\textit{determiner}, \\textit{verb}, \\textit{preposition}, \\textit{singular noun}, and \\textit{plural noun}.\n\nTo classify each requirement of the respective data set, we implemented a Java-based feature extraction prototype that parses all requirements from the data set and extracts the values for all ten features mentioned above. Subsequently, we used Weka \\cite{witten2016data} to train a C4.5 decision tree algorithm \\cite{quinlan2014c4} which comes with Weka as J48 implementation. According to Hussain et al. \\cite{hussain2008using}, we set the parameters for the minimum number of instances in a leaf to 6 to counter possible chances of over-fitting.\n\n\n\n\n\nSince the data set was not very large with 625 requirements, we performed a 10-fold-cross validation. In the following, we report our classification results for each data set.\\\\\n{\\bf Results:} The classification of the \\emph{unprocessed data set} results in $89.92\\%$ correctly classified requirements with a weighted average precision and recall of $0.90$. The classification of the \\emph{processed data set} results in $94.40\\%$ correctly classified requirements with a weighted average precision of $0.95$ and recall of $0.94$. \\tablename{ \\ref{tb:classification_unprocessed}} and  \\tablename{ \\ref{tb:classification_processed}} show the details. By applying our approach, we could achieve an improvement of $4.48\\%$ correctly classified requirements. In total, we could correctly classify $28$ additional requirements, which consist of $9$ functional and $19$ non-functional ones.\nWhen classifying NFRs into sub-categories, the influence of our preprocessing is much stronger. The last two columns of \\tablename{ \\ref{tab:compare}} show the overall precision and recall of six different machine learning algorithms for sub-classifying NFRs into the categories listed in columns 1--10 of the table. For all algorithms, results are dramatically better when using the preprocessed data (column Total P) compared to using the raw data (column Total UP).\n\n\n\\begin{table}[htbp]\n\t\\centering\n\t\\caption{Classification results of the unprocessed data set}\n\n\t\\label{tb:classification_unprocessed}\n\t\\resizebox{\\linewidth}{!}{\\begin{tabular}{c|c|c|c|c|c|c|}\n\t\t\\cline{2-7}\n\t\t& \\begin{tabular}[c]{@{}c@{}}Correctly \\\\ Classified\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Incorrectly \\\\ Classified\\end{tabular} & Precision & Recall & F-Measure & Kappa \\\\ \\hline\n\t\t\\multicolumn{1}{|c|}{NFR} & 325 (87.84\\%) & 45 (12.16\\%) & 0.95 & 0.88 & 0.91 & \\multirow{3}{*}{0.79} \\\\ \\cline{1-6}\n\t\t\\multicolumn{1}{|c|}{FR} & 237 (92.94\\%) & 18 (7.06\\%)& 0.84 & 0.93 & 0.88 &  \\\\ \\cline{1-6}\n\t\t\\multicolumn{1}{|c|}{Total} & 562 (89.92\\%) & 63 (10.08\\%) & 0.90 & 0.90 & 0.90 &  \\\\ \\hline\n\t\\end{tabular}}\n\\end{table}\n\n\n\n\n\n\n\\begin{table}[htbp]\n\t\\centering\n\t\\caption{Classification results of the processed data set}\n\n\t\\label{tb:classification_processed}\n\t\\resizebox{\\linewidth}{!}{\\begin{tabular}{c|c|c|c|c|c|c|}\n\t\t\t\\cline{2-7}\n\t\t\t& \\begin{tabular}[c]{@{}c@{}}Correctly \\\\ Classified\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Incorrectly \\\\ Classified\\end{tabular} & Precision & Recall & F-Measure & Kappa \\\\ \\hline\n\t\t\t\\multicolumn{1}{|c|}{NFR} & 344 (92.97\\%) & 26 (7.03\\%) & 0.98 & 0.93 & 0.95 & \\multirow{3}{*}{0.89} \\\\ \\cline{1-6}\n\t\t\t\\multicolumn{1}{|c|}{FR} & 246 (96.47\\%) & 9 (3.53\\%)& 0.90 & 0.97 & 0.93 &  \\\\ \\cline{1-6}\n\t\t\t\\multicolumn{1}{|c|}{Total} & 590 (94.40\\%) & 35 (5.60\\%) & 0.95 & 0.94 & 0.94 &  \\\\ \\hline\n\t\\end{tabular}}\n\n\\end{table}\n\n\\vspace{-3mm}\n\n\n\n\n\n\n\\subsection{RQ2- Classifying Non-functional Requirements}\nIn this section, we describe the machine learning algorithms we used to classify NFRs. The performance of each method is assessed in terms of its recall and precision.\n\n\n\\begin{figure*}[!ht]\n\\centering\n\n\\subfloat[{\\scriptsize Hopkins statistic to assess the clusterability of the data set (hopkins-stat = 0.1)}]{\\includegraphics[scale=0.26]{Figures\/Tendency}}\n\\subfloat[Hierarchical= 0.13]{\\includegraphics[scale=0.28]{Figures\/HCL}}\n\\subfloat[K-means= 0.1]{\\includegraphics[scale=0.28]{Figures\/Kmeans}}\n\\subfloat[Hybrid= 0.13]{\\includegraphics[scale=0.28]{Figures\/HB}}\n\\subfloat[A visual representation of the confusion matrix (BNB algorithm)]{\\includegraphics[scale=0.2]{Figures\/ConfMatrix}}\\\\\n\\caption{Detailed visual representation of classifying NFRs }\n\\label{fig:cluster}\n\\end{figure*}\n\n\n\n\n\\subsubsection{Topic Modeling}\nTopic modeling is an unsupervised text analysis technique that groups a small number of highly correlated words, over a large volume of unlabelled text \\cite{TM1}, into {\\it topics}.  \n\n\n{\\bf Algorithms:} The {\\it Latent Dirichlet Allocation (LDA)} algorithm classify documents based on the frequency of word co-occurrences. Unlike the LDA approach, the {\\it Biterm Topic Model} (BTM) method models topics based on the word co-occurrence patterns and learns topics by exploring word-word (i.e., biterm) patterns. Some recent studies on the application of topic modeling in classifying short text documents stated that the BTM approach has a better ability in modeling short and sparse text, as the ones typical for requirements specifications.\n\n\n\n\n\n\n\n\\begin{table*}\n\\centering\n\\footnotesize\n\\caption{Comparison between classification algorithms for classifying non-functional requirements [(U)P= (Un)Processed]}\n\\label{tab:compare}\n\n\\begin{tabular}{ |c|cc|cc|cc|cc|cc|cc|cc|cc|cc|cc|cc|cc| cc| }\n  \\hline\n{\\bf Algorithm}& \\multicolumn{2}{c|}{\\bf A}& \\multicolumn{2}{c|}{\\bf US}& \\multicolumn{2}{c|}{\\bf SE}& \\multicolumn{2}{c|}{\\bf SC}& \\multicolumn{2}{c|}{\\bf LF}& \\multicolumn{2}{c|}{\\bf L}& \\multicolumn{2}{c|}{\\bf MN}& \\multicolumn{2}{c|}{\\bf FT}& \\multicolumn{2}{c|}{\\bf O}& \\multicolumn{2}{c|}{\\bf PE} &  \\multicolumn{2}{c|}{\\bf PO}& \\multicolumn{2}{c|}{\\bf Total [P]}& \\multicolumn{2}{c|}{\\bf Total [UP]}\\\\ \\cline{2-27}\n&{\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P}   & {\\bf R}&{\\bf P}&   {\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P}   &{\\bf R}&{\\bf P} & {\\bf R}&{\\bf P} & {\\bf R}&{\\bf P}&{\\bf R}&{\\bf P}\\\\\\hline\n{\\bf LDA}&95&60   &61&76   & 87&87&   81&57   &60&85   &47&20   &70&52   &10&2   &35&70   &70&95 & -&-   &\\cellcolor{Gray}62&\\cellcolor{Gray}62&31&31\\\\\\cline{2-27}\n{\\bf BTM}&0&0   &6&12   & 13&18&   9&8   &5&7   &0&0   &0&0   &40&17   &0&0   &18&43 & -&-   &\\cellcolor{Gray}8&\\cellcolor{Gray}8&3&3\\\\\\cline{2-27}\n{\\bf Hierarchical }&13&14   &25&20   &24&17   &16&29   &5&3   &6&15   &19&35   &18&40   &32&29   &26&22   &-&-  & \\cellcolor{Gray}21&\\cellcolor{Gray}21&16&16\\\\ \\cline{2-27}\n{\\bf K-means}&10&23   &19&14   &29&18   &14&14   &21&21   &8&15   &22&47   &18&40   &26&30   &31&11&   -&-   &\\cellcolor{Gray}20&\\cellcolor{Gray}20&15&15\\\\ \\cline{2-27}\n{\\bf Hybrid}&15&14   &27&22&   29&18&   20&4&   26&24   &6&15   &17&35   &18&40   &22&27   &26&22&   -&-   &\\cellcolor{Gray}22&\\cellcolor{Gray}22&19&17\\\\\\cline{2-27}\n{\\bf Na\\\"{i}ve Bayes} &\\cellcolor{Gray}90&\\cellcolor{Gray}90&\\cellcolor{Gray}   97&\\cellcolor{Gray}77   &\\cellcolor{Gray}97&\\cellcolor{Gray}100   &\\cellcolor{Gray}83&\\cellcolor{Gray}83   &\\cellcolor{Gray}94&\\cellcolor{Gray}94   &\\cellcolor{Gray}75&\\cellcolor{Gray}100   &\\cellcolor{Gray}90&\\cellcolor{Gray}82   &\\cellcolor{Gray}97&\\cellcolor{Gray}90   &\\cellcolor{Gray}78&\\cellcolor{Gray}91   &\\cellcolor{Gray}90&\\cellcolor{Gray}100   &-&-   &\\cellcolor{Gray}91&\\cellcolor{Gray}90&45&45\\\\\\hline\n\\end{tabular}\n\n\\end{table*}\n{\\bf Results and Evaluation:} The modeled topics for both LDA and BTM, including the top frequent words and the NFR assigned to each topic are provided in our source code package\\footnote{http:\/\/wcm.ucalgary.ca\/zshakeri\/projects}. We determined each topic by the most probable words that are assigned to it. For instance, LDA yields the word set \\{user, access, allow, prior, and detail\\} for the topic describing the Fault Tolerance sub-category, while BTM yields the set \\{failure, tolerance, case, use and data\\}. \n\n\nGenerally, the word lists generated by BTM for each topic are more intuitive than those produced by LDA. This confirms previous research that BTM performs better than LDA in terms of modeling and generating the topics\/themes of a corpus consisting of short texts. However, surprisingly, BTM performed much worse than LDA for sub-classifying NFRs as shown in Table~\\ref{tab:compare}. This might be because BTM performs its modeling directly at the corpus level and biterms are generated independently from topics. \n\n\n \\noindent\\makebox[\\linewidth]{\\resizebox{0.3333\\linewidth}{1pt}{$\\bullet$}}\\bigskip \n  \\vspace{-3mm}\n\\subsubsection{Clustering}\nClustering is an unsupervised classification technique which categorizes documents into groups based on likeness \\cite{Cluster}. This likeness can be defined as the numerical distance between two documents \\(D_i\\) and \\(D_j\\) which is measured as:\n\\vspace{-3mm}\n\n\\begingroup\n\\everymath{\\scriptstyle}\n\\scriptsize\n\\[\nd(D_i, D_j)= \\sqrt{({d_i}_1-{d_j}_1)^2+ ({d_i}_2-{d_j}_2)^2 + ...+ ({d_i}_n-{d_j}_n)^2}\n\\] \\endgroup\n\nWhere \\(({d_i}_1, {d_i}_2, ..., {d_i}_n)\\) and \\(({d_j}_1, {d_j}_2, ..., {d_j}_n)\\) represent the coordinates (i.e., word frequencies) of the two documents. \n\n\n{\\bf Algorithms: }The {\\it Hierarchical} (i.e., Agglomerative) algorithm, first, assigns each document to its own cluster and iteratively merges clusters that are closest to each other until the entire corpus forms a single cluster. Despite the hierarchical approach in which we do not need to specify the number of clusters upfront, the {\\it K-means} algorithm assigns documents randomly to {\\it k} bins. This approach computes the location of the centroid of each bin and computes the distance between each document and each centroid. We defined the {\\it k=10} to run this algorithm. However, the k-means approach is highly sensitive to the initial random selection of cluster centroid (i.e., mean), which might lead to different results each time we run this algorithm. Thus, we used a {\\it Hybrid} algorithm, which combined hierarchical and K-means algorithms. This algorithm, first, computes the center (i.e., mean) of each cluster by applying the hierarchical approach. Then computes the K-means approach by using the set of defined clusters' centers. \n\n\n{\\bf Results and Evaluation: }Before applying clustering algorithms we used Hopkins (H) statistic to test the spatial randomness and assess the {\\it clustering tendency} (i.e., clusterability) of our data set. To this end, we raised the following null hypothesis: \\(\\big(\\){\\it \\(H_0\\): the NFR data set is uniformly distributed and has no meaningful clusters}\\(\\big)\\). As presented in Figure \\ref{fig:cluster} (a), the {\\it H-value} of this test is 0.1 (close to zero), which rejects this hypothesis and concludes that our data set is significantly clusterable. However, as presented in Table \\ref{tab:compare}, the clustering algorithms had poor performance at classifying NFRs. This may imply that the data set under study is quite unstructured and sub-categories of NFRs are not well separated. Thus, an unsupervised algorithm (e.g. Hierarchical or K-means) cannot accurately achieve segmentation.\n\nMoreover, we used Silhouette (s) analysis to assess the cohesion of resulted clusters. We used the function {\\it silhouette()} of {\\it cluster package} to compute the silhouette coefficient. Small {\\it s-value} (i.e., around 0) means that the observation lies between two clusters and has a low cohesion. The results of this test and the details of each cluster, including a number of requirements assigned to it, and its {\\it s-value} are illustrated in Figure \\ref{fig:cluster}(b-d). \n\n \\noindent\\makebox[\\linewidth]{\\resizebox{0.3333\\linewidth}{1pt}{$\\bullet$}}\\bigskip \n  \\vspace{-3mm}\n  \n\\subsubsection{Na\\\"{i}ve Bayes Classification}\n\nThis approach is a supervised learning method which predicts unseen data based on the {\\it bayes' theorem} \\cite{Naive} used to calculate conditional probability:\n\n\\begingroup\n\\everymath{\\scriptstyle}\n\\scriptsize\n\\[\nP(C=c_k\\mid F=f)= \\dfrac{P(F=f\\mid C=c_k) P(C=c_k)}{P(f)}\n\\] \\endgroup\n\nWhere \\(C=(c_1, c_2, ..., c_k)\\) represents classes and \\(F= (f_1, f_2, ..., f_d)\\) is a vector random variable, which includes one vector for each document. \n\n\n{\\bf Algorithm:} We use a variation of the multinomial Na\\\"{i}ve Bayes (BNB) algorithm known as {\\it Binarized Na\\\"{i}ve Bayes}. In this method, the term frequencies are replaced by Boolean presence\/absence features. The logic behind this is the higher importance of word occurrence than word frequency to sentiment classification.\n\n\n{\\bf Results and Evaluation:} To apply this algorithm we employed a 5-fold-cross validation. To reduce the data splitting bias, we run five runs of the 5-fold-cross validation. Overall accuracy is calculated at just over 90\\% with a {\\it p-value} of 2.2e-16. As illustrated in Table \\ref{tab:compare}, results obtained using the BNB algorithm were generally more accurate. All of the NFRs (except for PO) were recalled at relatively high values ranging from 75 (i.e., Legal requirements) to 97\\% (i.e., security and performance requirements). To represent more details about the performance of our classifier for each NFR, we visualized the confusion matrix resulted from applying the BNB algorithm (Figure \\ref{fig:cluster} (e)). Each column and row of this matrix represent the actual (i.e., reference) and the prediction data, respectively. The blocks are colored based on the frequency of the intersection between actual and predicted classes (e.g., the diagonal represents the correct predictions for the actual class). Since some of the NFRs in our data set occur more frequently, we normalized our data set before visualizing the confusion matrix. As illustrated in Figure \\ref{fig:cluster} (e), requirements in classes FT, L, MN, O, and SC were often assigned to class US. We can imply that {\\it the terminology we use for representing usability requirements is very general, which covers other NFRs that are indirectly related to usability. } This shows a clear need for additional (or better) sentimental patterns, which differentiate this category of NFR from other similar categories. \nto differentiate usability requirements from other types of NFRs.\n\n\n\\begin{tcolorbox}[colback=white, title= Findings]\n\\footnotesize\n  \\textcolor{white}{..................  }  \\\\\n\\vspace{-5mm}\n    \n{\\bf Finding 1:} Our preprocessing approach positively impacted the performance of the applied classification of functional and non-functional requirements. We could improve the accuracy from 89.92\\% to 95.04\\%.\\\\\n\\vspace{-2mm}\n\n{\\bf Finding 2:} Our preprocessing approach strongly impacted the performance of all applied sub-classification methods. For LDA and BNB, both precision and recall doubled. \\\\\n\\vspace{-2mm}\n\n{\\bf Finding 3:} Among the  machine learning algorithms LDA, BTM, Hierarchical, K-means, Hybrid and Binarized Na\\\"{i}ve Bayes (BNB), {\\it BNB} had the highest performance for sub-classifying NFRs.\\\\\n\\vspace{-2mm}\n\n\n{\\bf Finding 4:} While BTM generally works better than LDA for exploring the general themes and topics of a short-texts corpus, it did not perform well for sub-classifying NFRs. \\\\\n\\vspace{-2mm}\n\n{\\bf Finding 5:} There is a clear need for additional sentimental patterns\/sentence structures to differentiate usability requirements from other types of NFRs.\n\\end{tcolorbox}\n\\vspace{-2mm}\n\n\n\n\n\n\n\\section{Conclusion and Implications}\n\\label{sec:agenda}\n\n\n\n\nOur findings are summarized in the box at the end of Section~\\ref{sec:results}. \nIn particular, we conclude that using our preprocessing approach improves the performance of both classifying FR\/NFR and sub-classifying NFR into sub-categories. Further, we found that, among popular machine learning algorithms, Binarized Na\\\"{i}ve Bayes (BNB) performed best for the task of classifying NFR into sub-categories. Our results further show that, although BTM generally works better than LDA for extracting the topics of short-texts, BTM does not perform well for classifying NFRs into sub-categories. Finally, additional (or better) sentimental patterns and sentence structures are needed for differentiating usability requirements from other types of NFRs.\n\\vspace{-2mm}\n\\section{Introduction}\n \n In requirements engineering, classifying the requirements of a system by their kind into \\emph{functional requirements}, \\emph{quality requirements} and \\emph{constraints} (the latter two usually called \\emph{non-functional requirements}) \\cite{re_glossary} is a widely accepted standard practice today.\n \nWhile the different kinds of requirements are known and well-described today~\\cite{Martin}, automated classification of requirements written in natural language into functional requirements (FRs) and the various sub-categories of non-functional requirements (NFRs) is still a challenge \\cite{Ernst2010}. This is particularly due to the fact that stakeholders, as well as requirements engineers, use different terminologies and sentence structures to describe the same kind of requirements \\cite{RO, IWSPM}. The high level of inconsistency in documenting requirements makes automated classification more complicated and therefore error-prone.\n\nIn this paper, we investigate how automated classification algorithms for requirements can be improved and how well some of the frequently used machine learning approaches work in this context. We make two contributions. (1)~We investigate if and to which extent an existing decision tree learning algorithm~\\cite{hussain2008using} for classifying requirements into FRs and NFRs can be improved by preprocessing the requirements with a set of rules for (automated) standardizing and normalizing the requirements found in a requirements specification. (2)~We study how well several existing machine learning methods perform for automated classification of NFRs into sub-categories such as availability, security, or usability.\n\nWith this work, we address the \\emph{RE Data Challenge} posed by the 25th IEEE International Requirements Engineering Conference (RE'17). \n\n \n\n \n\n\n\n\n\n\n\\section{Limitations and Threats to Validity}\n\\label{sec:limits}\nIn this section, we discuss the potential threats to the validity\nof our findings in two main threads: \n\n{\\bf (1) Data Analysis Limitations:} The biggest threat to the validity of this work is the fact that our preprocessing model was developed on the basis of the data set given for the RE Data Challenge and that we had to use the same data set for evaluating our approach.\nWe mitigate this threat by using sentence structure features such as temporal, entity, and functional features which are applicable to sentences with different structures from different contexts. We also created a set of regular expressions which are less context-dependent and have been formed mainly based on the semantics of NFRs.\n\nAnother limiting factor is that our work depends on\nthe number and choice of the NFR sub-categories used by the creators of our data set.\nHowever, our preprocessing can be adapted to a different set of NFR sub-categories. In terms of the co-occurrence rules and regular expressions presented in Table \\ref{tab:reg}, we aim to\nexpand these rules by adding more NFR sub-categories in future work, as we gain additional insights from processing real world requirements specifications.\n\n{\\bf (2) Dataset Limitations:} Due to the nature of the RE'17 Data Challenge, we used the data set as is, although it has major data quality issues: (1)~Some requirements are incorrectly labeled. For example, R2.18 ``The product shall allow the user to view previously downloaded search results, CMA reports and appointments'' is labeled as NFR. Obviously, however, this is a functional requirement.\n(2) The important distinction between quality requirements and constraints is not properly reflected in the labeling. (3)~The selection of the requirements has some bias. For example, the data set does not contain any compatibility, compliance or safety requirements. Neither does it contain any cultural, environmental or physical constraints. (4)~Only one single requirement is classified as PO which makes this sub-category useless for our study. The repetition of our study on a data set of higher data quality is subject to future work.\n\nFurthermore, the {\\it unbalanced data set} we used for classifying the NFRs may affect the findings of this study. However, a study by Xue and Titterington \\cite{unbalanced} revealed that there is no reliable empirical\nevidence to support the claim that an unbalanced data\nset negatively impacts the performance of the LDA\/BTM approaches. Further, a recent study by L\\'{o}pez et al. \\cite{unbalanced2} shows that the unbalanced ratio by itself does not have the most significant effect on the classifiers'\nperformance, but there are other issues such as (a) the presence of small disjuncts, (b) the lack of density, (c) the class overlapping,\n(d) the noisy data, (e) the management of borderline examples, and (f) the dataset shift that must be taken into account. The pre-processing step we conducted before applying the classification algorithms helps discriminate the NFR sub-classes more precisely and mitigates the negative impact of the noisy data and borderline problems. Moreover, we employed the n-fold cross validation technique which helps generate enough positive class instances in different folds and reduces additional problems in the data distribution especially for highly unbalanced datasets. This technique, to a great extent, mitigates the negative impact of class overlapping, the dataset shift, and the presence of small disjuncts issues on the performance of the classification algorithms we applied in this study. \n \n\\section{Preprocessing of Requirements Specifications}\n\\label{sec:PP}\nIn this section, we describe the preprocessing\nwe applied to reduce the inconsistency of requirements specifications by leveraging rich sentence features and latent co-occurrence relations.\n\\subsection{Part Of Speech (POS) Tagging} We used the part-of-speech tagger of the Stanford Parser \\cite{Klein:2003:AUP:1075096.1075150} to assign parts of speech, such as noun, verb, adjective, etc. to each word in each requirement. \nThe POS tags\\footnote{Check  https:\/\/gist.github.com\/nlothian\/9240750 for a complete list of tags} are necessary to perform the FR\/NFR classification based on the approach of Hussain et al. \\cite{hussain2008using}.\n\\subsection{Entity Tagging}\nTo improve the generalization of input requirements, we used a ``supervised training data'' method in which all context-based products and users are blinded by assigning names as {\\it PRODUCT} and {\\it USER}, respectively. To this end, we used the LingPipe NLP toolkit\\footnote{http:\/\/alias-i.com\/lingpipe\/} and created the \\(SRS\\_dictionary\\) by defining project specific users\/customers and products  (e.g., program administrators, nursing staff members, realtor, or card member marked as USER), such as below:\n\\vspace{-2mm}\n\\begin{figure}[H]\n\\centering\n{\\includegraphics[scale=0.85]{Figures\/User}}\n\n\\caption*{}\n\n\\end{figure}\n\n\nThen, each sentence is tokenized and POS tagged with the developed SRS dictionary. All of the tokens associated with {\\it user} and {\\it product} were discarded and we only kept these two keywords to represent these two entities. Finally, we used the POS tagger of the Stanford Parser \\cite{Klein:2003:AUP:1075096.1075150} and replaced all Noun Phrases (NPs) including ``USER'' and ``PRODUCT'' with \\(USER\\) and \\(PRODUCT\\), respectively. For instance, \\mybox[fill=gray!30]{registered USER} in \\big<Only registered {\\bf USER} shall be able to access the {\\bf PRODUCT}\\big>, is replaced with \\(USER\\). \n\\subsection{Temporal Tagging} {\\it Time} is a key factor to characterize non-functional requirements, such as availability, fault tolerance, and performance. \n\nFor this purpose, we used  SUTime, a rule-based temporal tagger for recognizing and normalizing temporal expressions by TIMEX3 standard\\footnote{See http:\/\/www.timeml.org for details on the TIMEX3 tag}. SUTIME detects the following basic types of temporal objects \\cite{SUTIME}:\n\\begin{enumerate}\n\\item {\\it Time:}  A particular instance on a time scale. SUTIME also handles absolute times, such as {\\it Date}. As in: \n\n\\begin{figure}[H]\n\\centering\n{\\includegraphics[scale=0.67]{Figures\/TIME}}\n\n\\caption*{}\n\n\\end{figure}\n\n\\item {\\it Duration and Intervals:} The amount of intervening time in a time interval. As in:\n\n\\begin{figure}[H]\n\\centering\n{\\includegraphics[scale=0.67]{Figures\/Duration}}\n\n\\caption*{}\n\n\\end{figure}\n\nIntervals can be described as a range of time defined by a start and end time points. SUTime represents this type in the form of other types. \n\\item{\\it Set:} A set of temporals, representing times that occur with some frequency. As in:\n\n\\begin{figure}[H]\n\\centering\n{\\includegraphics[scale=0.67]{Figures\/SET1}}\n\n\\caption*{}\n\\end{figure}\n\\end{enumerate}\n\n\n\\begin{table*}[!htb]\n\\centering\n\\caption{\\small Proposed co-occurrence and regular expressions for preprocessing SRSs [(\\(CO(w)\\)): the set of terms co-occur with word \\(w\\)] }\n\n\\label{tab:reg}\n\\begin{tabular}{ |c|p{4.1cm}|p{5.1cm}p{5.2cm}| }\n  \\hline\n {\\bf NFR}&{\\bf Keywords} & {\\bf Part of Speech (POS) and Regular Expressions} & {\\bf Replacements}\\\\\\hline \n\n\n\n\n \n    \n  \n  \n \n  \n \n  \n \n  \n \n \n\n\n\n{\\bf Security [SE]} &protect, encrypt, policy, authenticate, prevent, malicious, login, logon, password, authorize, secure, ensure, access&\n\\( (\\mybox[fill=gray!30]{only}\/ ...\/\\mybox[fill=gray!30]{ nsubj}\/...\/\\mybox[fill=gray!30]{root}\/... )\\)\n\\(  \\mid\\mid (\\mybox[fill=gray!30]{only}\/ ...\/\\mybox[fill=gray!30]{ root}\/...\/\\mybox[fill=gray!30]{nmod: agent}\/... )\\)\n\\(\\wedge \\big(\\mybox[fill=gray!30]{root}\\text{ is a } VBP\\big)\\) \n\n\n& \\( \\Rightarrow \\begin{cases}\\mybox[fill=gray!30]{nsubj}\\mybox[fill=gray!30]{agent} \\gets\\text{\\it authorized user}&\\\\ \\mybox[fill=gray!30]{root} \\gets access& \\end{cases}\\)\n\n{\\raisebox{-\\totalheight}{\\includegraphics[scale=.43]{Figures\/SE2}}}\\\\\\cline{3-4}\n\n&&\\( \\forall \\omega \\in \\{\\text{\\it username \\& password, login, logon}\\}\\)\n\n \\(\\text{\\it security, privacy, right, integrity, polict}\\}\\)\n  \n  \\(\\wedge CO(\\omega)\\cap CO(NFR_{se}) \\neq \\emptyset\\)\n  &\\(\\Rightarrow \\omega \\gets authorization\\)\\\\\\cline{3-4}\n  \n  &&\\( \\forall \\omega \\in \\{\\text{\\it reach, enter, protect, input, interface}\\)\n  \n  \\(\\wedge \\text{product is obj }\\wedge CO(\\omega)\\cap CO(NFR_{se}) \\neq \\emptyset\\)\n  &\\(\\Rightarrow \\omega \\gets access\\)\\\\\\hline\n\n   \n   \\hline\n\\end{tabular}\n\\vspace{-6mm}\n\\end{table*}\n\\vspace{-6mm}\n\nAfter tagging a sample set of all the classified NFRs, we identified the following patterns and used them to normalize the entire data set.  To do this, first, we replaced all expressions in this format ``24[-\/\/*]7'' and ``\\(24\\times7\\times365\\)'' with ``24 hours per day 365 days per year'', and ``everyday'' with ``every day''. Likewise, we replaced all ``sec(s)'' and ``min(s)'' with  ``seconds'' and  ``minutes'', respectively. In the rest of this section, we present the rules we defined and applied to normalize the temporal expressions embedded in requirements' descriptions. \\\\\n\\vspace{-3mm}\n\\begin{tcolorbox}[colback=white, title= Temporal Rules]\n\\scriptsize\n\\begin{enumerate}\n\\item \\(\\forall \\text{   } [\\backslash exp]\\big<DURATION\\big>, exp \\gets within\\) \\\\\nwhere \\(exp \\in\\)\\{\\it no longer than, under, no more than, not be more than, no later, in, for less than, at a maximum\\} \\\\\n\\vspace{-3mm}\n\n\n \\vspace{2mm}\n\\item \\(\\forall \\text{   } [\\backslash DURATION \\backslash TIME \\backslash DATE]^+  \\gets alltimes\\) \\\\\n\n\\begin{minipage}[t]{1\\linewidth}\n     \\includegraphics[scale=.55]{Figures\/Temporal1}\\hfill\n        \\vspace{-7mm}\n        \\captionsetup{labelformat=empty}\n     { \\captionof{figure}{ }\n \\label{fig:Terminology}}\n    \\end{minipage}\\hfill\\\\\n    \\vspace{2mm}\n\n\n\\item  \\(within \\text{ } \\big<DURATION\\big> \\gets fast\\) \\\\\n\\(if \\big<DURATION\\big> == [\\backslash seconds \\backslash minutes]\\)\\\\\n \\vspace{-2mm}\n \n\\begin{minipage}[t]{1\\linewidth}\n\\vspace{-2mm}\n     \\includegraphics[scale=.6]{Figures\/Temporal2}\n        \\vspace{-3mm}\n        \\captionsetup{labelformat=empty}\n     { \\captionof{figure}{ }\n \\label{fig:Terminology}}\n    \\end{minipage}\\hfill\\\\\n\\vspace{-5mm}\n\n \\vspace{2mm}\n\\item \\(\\{timely, quick\\} \\mid\\mid [\\backslash \\text{\\it positive adj } \\backslash time] \\gets fast\\)\\\\\n\\item \\({[8\\text{-}9][0\\text{-}9][{\\bf \\backslash.}?[0\\text{-}9]?{\\bf \\%}?]}[IN\\mid DET]^* time \\gets alltimes\\)\n\n\\vspace{-1mm}\n \\noindent   \n\n\\end{enumerate}\n\\end{tcolorbox}\n\\vspace{-5mm}\n\n\n\n\n\n\n\n\n\\subsection{Co-occurrence and Regular Expressions}\nOnce the sentence features are utilized to reduce the complexity of the text, we used the co-occurrence and regular expressions to increase the weight of the influential words for each type of NFRs. To explore these rules we manually analyzed 6 to 10 requirements of each NFR and deployed different components of Stanford Parser such as part-of-speech, named entities, sentiment, and relations. Moreover, in this step, we recorded co-occurrence counts of each term within the provided NFR data set as the co-occurrence vector. We used this parameter as a supplement for exploring the SRS regular expressions. For instance, Table \\ref{tab:reg} represents the details of the rules we proposed for Security (SE) NFR. Please refer to the footnote\\footnote{ http:\/\/wcm.ucalgary.ca\/zshakeri\/projects} for the complete list of these rules containing regular expressions for all of the provided NFRs. \n\n\n\n\n\\section{Related Work}\n \\label{sec:RW}\n Software Requirements Specifications (SRSs) are written in natural language, with mixed statements of functional and non-functional requirements. There is a growing body of research studies that compare the effect of using manual and automatic approaches for classification of requirements~\\cite{nanniu,Ernst2010}. An efficient classification enables focused communication and prioritization of requirements~\\cite{janerej2}. Categorization of requirements allows filtering relevant requirements for a given important aspect. Our work is also closely related to the research on automatic classification of textual requirements.\n\nKnauss and Ott \\cite{knaussREFSQ} introduced a model of a socio-technical system for requirements classification. They evaluated their model in an industrial setting with a team of ten practitioners by comparing a manual, a semi-automatic, and a fully-automatic approach of requirements classification. \nThey reported that a semi-automatic approach offers the best ratio of quality and effort as well as the best learning performance. Therefore, it is the most promising approach of the three.\n\nCleland-Huang et al. \\cite{janere} investigated mining large requirements documents for non-functional requirements. Their results indicate that NFR-Classifier adequately distinguishes several types of NFRs. However, further work is needed to improve the results for some other NFR types such as 'look-and-feel'. Although their study is similar to ours as they trained a classifier to recognize a set of weighted indicator terms, indicative of each type of requirement, we used different classification algorithms and additionally assessed their precisions and recalls to compare their performance with each other.\n\nRahimi et al. \\cite{monaRE} present a set of machine learning and data mining methods for automatically extracting quality concerns from requirements, feature requests, and online forums. Then, they generate a basic goal model from the requirements specification. Each concern is modeled as a softgoal. For attaching topics to softgoals they used an LDA approach to estimate the similarity between each requirement and the discovered topics. In addition, they used LDA to identify the best sub-goal placement for each of the unattached requirements. However, in this research, we used LDA as one of our approaches for classifying the non-functional requirements. \n\n\nNa\\\"{i}ve  Bayes  classifier is used in several studies\\cite{KnaussRE, koj} for  automatic classification of requirements. Therefore, we included Na\\\"{i}ve Bayes in our study to be comparable with other classifiers.\n\\section{The Challenge and Research Questions}\n\\label{sec:RQ}\n\n\n\\subsection{Context and Data Set}\nThe challenge put forward by the Data Track of RE'17 consists of taking a given data set and performing an automated RE task on the data such as tracing, identifying\/classifying requirements or extracting knowledge. For this paper, we chose the task of automated classification of requirements.\n\nThe data set given for this task comes from the OpenScience tera-PROMISE repository\\footnote{https:\/\/terapromise.csc.ncsu.edu\/!\/\\#repo\/view\/head\/requirements\/nfr}. It consists of 625 labeled natural language requirements (255 FRs and 370 NFRs). The labels classify the requirements first into FR and NFR. Within the latter category, eleven sub-categories are defined: (a)~ten \\emph{quality requirement categories}: Availability (A), Look \\& Feel (LF), Maintainability (MN), Operability (O), Performance (PE), Scalability (SC), Security (SE), Usability (US), Fault Tolerance (FT), and Portability (PO); (b)~one \\emph{constraint category}: Legal \\& Licensing (L). These labels constitute the ground truth for our investigations.\n\n\n\\subsection{Research Questions}\nWe frame the goal of our study in two research questions:\n {\\it RQ1. How do grammatical, temporal and sentimental characteristics of a sentence affect the accuracy of classifying requirements into functional and non-functional ones?}\n \nWith this research question, we investigate whether our preprocessing approach, which addresses the aforementioned characteristics, has a positive impact on the classification into FRs and NFRs in terms of precision and recall.\n\n {\\it RQ2. To what extent is the performance of classifying NFRs into sub-categories influenced by the chosen machine learning classification method?}\n\n\nWith this research question, we study the effects of the chosen machine learning method on the precision and recall achieved when classifying the NFRs in the given data set into the sub-categories defined in the data set.\n\n\n\n\n \n\n \n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\nDeep neural networks have achieved state-of-the-art performances in various natural and medical-imaging problems \\cite{litjens2017survey}. However, they tend to under-perform when the test-image distribution is different from those seen during training. In medical imaging, this is due to, for instance, variations in imaging modalities and protocols, vendors, machines, clinical sites and subject populations. For semantic segmentation problems, labelling a large number of images for each different target distribution is impractical, time-consuming, and often impossible. To circumvent those impediments, methods learning robust networks with less supervision have triggered interest in medical imaging \\cite{Cheplygina2018Notsosupervised}.\n\nThis motivates {\\em Domain Adaptation} (DA) methods: DA amounts to adapting a model trained on an annotated source domain to another target domain, with no or minimal new annotations for the latter. Popular strategies involve minimizing the discrepancy between source and target distributions in the feature or output spaces \\cite{ADDA,tsai2018learning}; integrating a domain-specific module in the network \\cite{dou2018pnp}; translating images from one domain to the other \\cite{cyclegan}; or integrating a domain-discriminator module and penalizing\nits success in the loss function \\cite{ADDA}.\n\nIn medical applications, separating the source training and adaptation is critical for privacy and regulatory reasons, as the source and target data may come from different clinical sites. Therefore, it is crucial to develop adaptation methods, which neither assume access to the source data nor modify the pre-training stage. Standard DA methods, such as \\cite{dou2018pnp,tsai2018learning,ADDA,cyclegan}, do not comply with these restrictions. This has recently motivated \\emph{Source-Free Domain Adaptation} (SFDA) \\cite{Bateson2020,KARANI2021101907}, a setting where the source data (neither the images nor the ground-truth masks) is unavailable during the training of the adaptation phase. \n\nEvaluating SFDA methods consists in: (i) adapting on a dedicated training set \\textit{Tr} from the target domain; and (ii) measuring the generalization performance on an unseen test set \\textit{Te} in the target domain. However, emerging and very recent \\emph{Test-Time Adaptation} (TTA) works in machine learning \\cite{wang2021tent,Sun2020} and medical imaging \\cite{KARANI2021101907,Varsavsky} argue that this is not as useful as adapting directly to the test set \\textit{Te}. In various interesting applications, access to the target distribution might not be possible. This is particularly common in medical image segmentation when only a single target-domain subject is available for test-time inference. In the context of image classification, the authors of \\cite{wang2021tent} showed recently that simple adaptation of batch normalization's scale and bias parameters on a set of test-time samples can deal competitively with domain shifts.\n\nWith this context in mind, we propose a simple formulation for source-free and single-subject test-time adaptation of segmentation networks. During inference for a single testing subject, we optimize a loss integrating shape priors and the entropy of predictions with respect to the batch normalization's scale and bias parameters. Unlike the standard SFDA setting, we perform test-time adaptation on each subject separately, and forgo the use of target training set \\textit{Tr} during adaptation. Our setting is most similar to the image classification work in \\cite{wang2021tent}, which minimized a label-free entropy loss defined over test-time samples. Building on this entropy loss, we further guide segmentation adaptation with domain-invariant shape priors on the target regions, and show the substantial effect of such shape priors on TTA performances.  \nWe report comprehensive experiments and comparisons with state-of-the-art TTA, SFDA and DA methods, which show the effectiveness of our shape-guided entropy minimization in two different adaptation scenarios: cross-modality cardiac segmentation (from MRI to CT) and prostate segmentation in MRI\nacross different sites. Our method exhibits substantially better performances than the existing TTA methods. Surprisingly, it also fares better than various state-of-the-art SFDA and DA methods, although it does not train on source and additional target data during adaptation, but just performs joint inference and adaptation on a single 3D data point in the target domain.\nOur results and ablation studies question the usefulness of training on target set \\textit{Tr} during adaptation and points to the surprising and substantial effect of embedding shape priors during inference on domain-shifted testing data. \nOur framework can be readily used for integrating various priors and adapting any\nsegmentation network at test times. \n\n\n\n\\begin{comment}\n\\begin{figure}[t]\n    \\includegraphics[width=1\\linewidth]{figures\/no_ad3.png}\n      \\caption[]{Visualization of 2 aligned slice pairs in source (Water) and target modality (In-Phase): the domain shift in the target produces a drop in confidence and accuracy.}\n         \\label{fig:s_t_im}\n\n\\end{figure}    \n\\end{comment}\n\n\n\n\\begin{figure}[t]\n    \\includegraphics[width=1\\linewidth]{figures\/over2.png}\n      \\caption[]{Overview of our framework for Test-Time Adaptation with Shape Moments: we leverage entropy minimization and shape priors to adapt a segmentation network on a single subject at test-time.}\n      \\label{fig:overview}\n\n\\end{figure}\n\n\\section{Method} \n\nWe consider a set of $M$ source images ${I}_{m}: \\Omega_s\\subset \\mathbb R^{2} \\rightarrow {\\mathbb R}$, $m=1, \\dots, M$, and denote their ground-truth K-class segmentation for each pixel $i \\in \\Omega_s$ as a $K$-simplex vector ${\\mathbf y}_m (i) = \\left(y^{(1)}_{m} (i), \\dots, y^{(K)}_{m} (i)\\right) \\in \\{0,1\\}^K$. For each pixel $i$, its coordinates in the 2D space are represented by the tuple $\\left(u_{(i)}, v_{(i)}\\right) \\in \\mathbb{R}^{2}$.\n\n\\paragraph{Pre-training Phase} The network is first trained on the source domain only, by minimizing the cross-entropy loss with respect to network parameters $\\theta$:  \n\\begin{equation}\\label{eq:crossent}\n\\begin{aligned}\n\\min_{\\theta} \\frac{1}{\\left|\\Omega_{s}\\right|} \\sum_{{m}=1}^{M} \\ell\\left({\\mathbf y}_{m} (i), {\\mathbf s}_{m} (i, \\theta)\\right)\n  \\end{aligned}\n\\end{equation}\nwhere ${\\mathbf s}_{m} (i, \\theta) = (s^{(1)}_{m} (i,\\theta), \\dots, s^{(K)}_{m} (i, \\theta)) \\in [0,1]^K$ denotes the predicted softmax probability for class $k \\in\\{1, \\ldots, K\\}$.\n\\paragraph{Shape moments and descriptors}\nShape moments are well-known in classical computer vision \\cite{nosrati2016incorporating}, and were recently shown useful in the different context of supervised training \\cite{KervadecMIDL2021}. Each moment is parametrized by its orders $p, q \\in \\mathbb{N}$, and each order represents a different characteristic of the shape. For a given $p, q \\in \\mathbb{N}$ and class $k$, the shape moments of the segmentation prediction of an image $I_n$ can be computed as follows from the softmax matrix $\\mathrm{S}_{n}(\\theta)=\\left(\\mathrm{s}_{n}^{(k)}(\\theta)\\right)_{k=1\\ldots K}$ :\n$$\n\\mu_{p, q}\\left(\\mathrm{s}_{n}^{(k)}(\\theta)\\right)=\\sum_{i \\in \\Omega} s_{n}^{(k)}(i, \\theta) u_{(i)}^{p} v_{(i)}^{q}\n$$\n\\begin{comment}\n$$\n\\mu_{p, q}^{(k)}\\left({\\mathbf s}_{n}(\\theta)\\right)=\\sum_{i \\in \\Omega} s_{n}^{(k)}(i,\\theta) u_{(i)}^{p} v_{(i)}^{q}.\n$$    \n\\end{comment}\nCentral moments are derived from shape moments to guarantee translation invariance. They are computed as follows:\n\\begin{comment}\n    $$\n\\bar{\\mu}_{p, q}^{(k)}\\left({\\mathbf s}_{n,\\theta}\\right)=\\sum_{i \\in \\Omega} s_{n}^{k}(i,\\theta)\\left(u_{(i)}-\\frac{\\mu_{1,0}^{(k)}}{\\mu_{0,0}^{(k)}}\\right)^{p}\\left(v_{(i)}-\\frac{\\mu_{0,1}^{(k)}}{\\mu_{0,0}^{(k)}}\\right)^{q}.\n$$\n$$\n \\bar{\\mu}_{p, q}^{(k)}\\left({\\mathbf s}_{n,\\theta}\\right)=\\sum_{i \\in \\Omega} s_{n}^{k}(i,\\theta)\\left(u_{(i)}-\\bar{u}^{(k)}\\right)^{p}\\left(v_{(i)}-\\bar{v}^{(k)}\\right)^{q}.\n$$\n\\end{comment}\n$$\n \\bar{\\mu}_{p, q}\\left({\\mathbf s}_{n}^{(k)}(\\theta)\\right)=\\sum_{i \\in \\Omega} s_{n}^{k}(i,\\theta)\\left(u_{(i)}-\\bar{u}^{(k)}\\right)^{p}\\left(v_{(i)}-\\bar{v}^{(k)}\\right)^{q}.\n$$\nwhere \n\\begin{comment}\n    $\\left( \\bar{u}^{(k)}, \\bar{v}^{(k)}\\right)=\\left(\\frac{\\mu_{1,0}^{(k)}}{\\mu_{0,0}^{(k)}},\\frac{\\mu_{0,1}^{(k)}}{\\mu_{0,0}^{(k)}}\\right)$ \n\\end{comment}\n$\\left(\\frac{\\mu_{1,0}(s_{n}^{(k)}(\\theta))}{\\mu_{0,0}(s_{n}^{(k)}(\\theta))},\\frac{\\mu_{0,1}(s_{n}^{(k)}(\\theta))}{\\mu_{0,0}(s_{n}^{(k)}(\\theta))}\\right)$ are the components of the centroid. \nWe use the vectorized form onwards, e.g. $\\mu_{p, q}\\left(s_{n}(\\theta)\\right) =\\left ( \\mu_{p, q}(s_{n}^{(1)}(\\theta)), \\dots, \\mu_{p, q}(s_{n}^{(K)}(\\theta)) \\right )^\\top$.\n\\begin{comment}\n    $\\mu_{p, q}\\left(s_{n}\\right) = (\\mu_{p, q}^{(1)}(s_{n}), \\dots, \\mu_{p, q}^{(K)}(s_{n}))^\\top$.\n\\end{comment}\nBuilding from these definitions, we obtain 2D shape moments from the network predictions. We then derive the shape descriptors $\\mathcal{R} ,\\mathcal{C},\\mathcal{D}$ defined in Table \\ref{table:shapes}, which respectively inform on the size, position, and compactness of a shape.  \n\\begin{comment}\n  \\begin{table}[t]\n\\begin{tabular}{lll}\nShape Descriptor & Computation from the predicted segmentation $s_\\theta$ \\\\\n\\midrule\nClass-Ratio & $\\mathcal{R}^{(k)}(s_\\theta)=\\frac{1}{\\left| \\Omega_T  \\right|}\\mu_{0, 0}^{(k)}\\left(s_{\\theta}\\right) $ \\\\\nDistance to Centroid & $\\mathcal{D}^{(k)}\\left(s_{\\theta}\\right)=\\left(\\sqrt[2]{\\frac{\\overline{\\mu_{2,0}^{(k)}\\left(s_{\\theta}\\right)}}{\\mu_{0,0}^{(k)}\\left(s_{\\theta}\\right)}}, \\sqrt[2]{\\frac{\\bar{\\mu}_{0,2}^{(k)}\\left(s_{\\theta}\\right)}{\\mu_{0,0}^{(k)}\\left(s_{\\theta}\\right)}}\\right) $\\\\ \nEccentricity & $\\mathcal{E}^{(k)}\\left(s_{\\theta}\\right)={\\sqrt  {1-{\\frac  {\\lambda _{2}}{\\lambda _{1}}}}} \\text{ with } \\lambda _{i}={\\frac  {\\bar{\\mu}_{{20}}+\\bar{\\mu}_{{02}}}{2}}\\pm {\\frac  {{\\sqrt  {4{\\bar{\\mu}}_{{11}}^{2}+({\\bar{\\mu}}_{{20}}-{\\bar{\\mu}}_{{02}})^{2}}}}{2}}$ \\\\\nCompactness & $\\mathcal{C}^{(k)}\\left(s_{\\theta}\\right)=\\frac{\\sum_{i \\in \\Omega_T} \\sqrt{\\left(\\nabla s_{u_{i}}\\right)^{2}+\\left(\\nabla s_{v_{i}}\\right)^{2}}}{4 \\pi\\mu_{0, 0}^{(k)}\\left(s_{\\theta}\\right)}$ \\\\\n\\midrule\n\\label{table:shapes}\n\\end{tabular}\n\\end{table}  \n\\end{comment}\n\n\\vspace{-0.3em}\n\\begin{table}[h!!!]\n\\centering\n  \\caption{Examples of shape descriptors based on softmax predictions.}\n\\begin{tabular}{ll}\n\\toprule\nShape Descriptor & \\multicolumn{1}{c}{Definition} \\\\\n\\midrule\nClass-Ratio & $\\mathcal{R}(s):=\\frac{1}{\\left| \\Omega_T  \\right|}\\mu_{0, 0}\\left(s\\right) $ \\\\\nCentroid & $\\mathcal{C}\\left(s\\right):=\\left(\\frac{\\mu_{1,0}\\left(s\\right)}{\\mu_{0,0}\\left(s\\right)}, \\frac{\\mu_{0,1}\\left(s\\right)}{\\mu_{0,0}\\left(s\\right)}\\right)$\\\\ \nDistance to Centroid & $\\mathcal{D}\\left(s\\right):=\\left(\\sqrt[2]{\\frac{\\bar{\\mu}_{2,0}\\left(s\\right)}{\\mu_{0,0}\\left(s\\right)}}, \\sqrt[2]{\\frac{\\bar{\\mu}_{0,2}\\left(s\\right)}{\\mu_{0,0}\\left(s\\right)}}\\right) $\\\\ \n\\bottomrule\n\\label{table:shapes}\n\\end{tabular}\n\\end{table}\n\n\\vspace{-1.5em}\n\\paragraph{Test-time adaptation and inference with shape-prior constraints}\n\nGiven a single new subject in the target domain composed of $N$ 2D slices, ${I}_n: \\Omega_t\\subset \\mathbb R^{2} \\rightarrow {\\mathbb R}$, $n=1, \\ldots, N$, the first loss term in our adaptation phase is derived from \\cite{wang2021tent}, to encourage high confidence in the softmax predictions, by minimizing their weighted Shannon entropy: $\\ell_{ent}({\\mathbf s}_n (i,\\theta)) = - \\sum_k \\nu_k s^k_n (i,\\theta) \\log s^k_n (i, \\theta)$, where $\\nu_k, k=1 \\ldots K$, are class weights added to mitigate imbalanced class-ratios.\n\\begin{comment}\n\\begin{equation}\n\\label{eq:ent-target}\n\\ell_{ent}({\\mathbf s}_n (i,\\theta)) = - \\nu_k\\sum_k s^k_n (i,\\theta) \\log s^k_n (i, \\theta)\n\\end{equation}    \n\\end{comment}\n\n\nIdeally, to guide adaptation, for each slice $I_n$, we would penalize the deviations between the shape descriptors of the softmax predictions ${S}_{n}(\\theta)$ and those corresponding to the ground truth $\\mathbf{y_n}$. As the ground-truth labels are unavailable, instead, we estimate the shape descriptors using the predictions from the whole subject $  \\left\\{  S_{n}(\\theta), n=1,\\dots,N\\right\\}$, which we denote respectively $\\mathcal{\\bar{C}},\\mathcal{\\bar{D}}$.\n\n\nThe first shape moment we leverage is the simplest: a zero-order class-ratio $\\mathcal{R}$. Seeing these class ratios as distributions, we integrate a KL divergence with the Shannon entropy: \n\\begin{equation}\\label{eq:AdaMI}\n\\begin{aligned}\n \\mathcal{L}_{TTAS}(\\theta) = \\sum_n\\frac{1}{\\left|\\Omega_{n}\\right|} \\sum_{i \\in \\Omega_t} \\ell_{ent}({s}_n (i, \\theta))+ \\mbox{KL}(\\mathcal{R}(S_n (\\theta)),\\mathcal{\\bar{R}}).\n  \\end{aligned}\n\\end{equation}\nIt is worth noting that, unlike \\cite{Bateson2022}, which used a loss of the form in Eq~\\eqref{eq:AdaMI} for training on target data, here we use this term for inference on a test subject, as a part of our overall shape-based objective.\nAdditionally, we integrate the centroid ($\\mathcal{M}=\\mathcal{C}$) and the distance to centroid ($\\mathcal{M}=\\mathcal{D}$) to further guide adaptation to plausible solutions:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:ineq}\n   \\min_{\\theta} &\\quad\\mathcal{L}_{TTAS}(\\theta) \n   \\\\\n  &\\text{s.t. } \\left|\\mathcal{M}^{(k)}(S_n (\\theta))-\\mathcal{\\bar{M}}^{(k)} \\right|\\leq0.1, & k=\\{2,\\dots, K\\}, n = \\{1, \\dots, N\\}.\n  \\end{aligned}\n\\end{equation}\n\n\nImposing such hard constraints is typically handled through the minimization of the Lagrangian dual in standard convex-optimization. As this is computationally intractable in deep networks, inequality constraints such as Eq~\\eqref{eq:ineq} are typically relaxed to soft penalties \\cite{He2017,kervadec2019constrained,Jia2017}.\nTherefore, we experiment with the integration of $\\mathcal{C}$ and $\\mathcal{D}$ through a quadratic penalty, leading to the following unconstrained objectives for joint test-time adaptation and inference:\n\\begin{comment}\n  \\begin{equation}\\label{eq:TTAdS}\n\\begin{aligned}\n \\sum_n\\frac{1}{\\left|\\Omega_{n}\\right|} \\sum_{i \\in \\Omega_n} \\ell_{ent}({s}_n (i, \\theta))+ \\mbox{KL}(\\mathcal{R}(\\mathbf{s}_n (\\theta)),\\mathcal{\\bar{R}}) + \\lambda (\\mathcal{M}(\\mathbf{s}_n (\\theta))-0.9\\mathcal{\\bar{M}})]_+^2 +\\lambda (1.1\\mathcal{\\bar{M}}-\\mathcal{M}(\\mathbf{s}_n (\\theta)))]_+^2\n  \\end{aligned}\n\\end{equation}  \n\\end{comment}\n\\begin{equation}\\label{eq:TTAS}\n \\sum_n\\frac{1}{\\left|\\Omega_{t}\\right|} \\sum_{i \\in \\Omega_n} \\ell_{ent}({\\mathbf s}_n (i, \\theta))+ \\mbox{KL}(\\mathcal{R}(S_n (\\theta)),\\mathcal{\\bar{R}}) + \\lambda \\mathcal{F}(\\mathcal{M}(S_n (\\theta)),\\mathcal{\\bar{M}}),\n\\end{equation}\nwhere $\\mathcal{F}$ is a quadratic penalty function corresponding to the relaxation of Eq~\\eqref{eq:ineq}: $\\mathcal{F}(m_1,m_2)= [m_1-0.9m_2]_+^2 + [1.1m_2-m_1]_+^2$ and $[m]_+ = \\max (0,m)$, with $\\lambda$ denoting a weighting hyper-parameter.\nFollowing recent TTA methods \\cite{wang2021tent,KARANI2021101907}, we only optimize for the scale and bias parameters of batch normalization layers while the rest of the network is frozen. Figure \\ref{fig:overview} shows the overview of the proposed framework.\n\n\\section{Experiments}\n    \n\\subsection{Test-time Adaptation with shape descriptors} \n\n\\paragraph{\\textbf{Heart Application}} We employ the 2017 Multi-Modality Whole Heart Segmentation (MMWHS) Challenge dataset for cardiac segmentation \\cite{Zhuang2019}. The dataset consists of 20 MRI (source domain) and 20 CT volumes (target domain) of non-overlapping subjects, with their manual annotations of four cardiac structures: the Ascending Aorta (AA), the Left Atrium (LA), the Left Ventricle (LV) and the Myocardium (MYO). We employ the pre-processed data provided by \\cite{dou2018pnp}.\nThe scans were normalized as zero mean and unit variance, and data augmentation based on affine transformations was performed. For the domain adaptation benchmark methods (DA and SFDA), we use the data split in \\cite{dou2018pnp}: 14 subjects for training, 2 for validation, and 4 for testing. Each subject has $N=256$ slices.\n\n\\paragraph{\\textbf{Prostate Application}} We employ the dataset from the publicly available NCI-ISBI 2013 Challenge\\footnote{https:\/\/wiki.cancerimagingarchive.net}. It is composed of manually annotated T2-weighted MRI from two different sites: 30 samples from Boston Medical Center (source domain), and 30 samples from Radboud University Medical Center (target domain). For the DA and SFDA benchmark methods, 19 scans were used for training, one for validation, and 10 scans for testing. We used the pre-processed dataset from \\cite{SAML}, who resized each sample to $384\\times384$ in axial plane, and normalized it to zero mean and unit variance. We employed data augmentation based on affine transformations on the source domain. Each subject has $N \\in \\left [15,24 \\right ] $ slices.\n\n\n\\paragraph{\\textbf{Benchmark Methods}} Our first model denoted $TTAS_{\\mathcal{R}\\mathcal{C}}$ constrains the class-ratio $\\mathcal{R}$ and the centroid $\\mathcal{C}$ using Eq~\\eqref{eq:TTAS}; similarly, $TTAS_{\\mathcal{R}\\mathcal{D}}$ constrains $\\mathcal{R}$ and the distance-to-centroid $\\mathcal{D}$. We compare to two \\emph{TTA} methods: the method in \\cite{KARANI2021101907}, denoted $TTDAE$, where an auxiliary branch is used to denoise segmentation, and $Tent$ \\cite{wang2021tent}, which is based on the following loss: $\\min_{\\theta}\\sum_{n} \\sum_{i \\in \\Omega_n} \\ell_{ent}({\\mathbf s}_n (i, \\theta))$. Note that $Tent$ corresponds to performing an ablation of both shape moments terms in our loss. As an additional ablation study, $TTAS_{\\mathcal{R}}$ is trained with the class-ratio matching loss in Eq~\\eqref{eq:AdaMI} only.\nWe also compared to two \\emph{DA} methods based on class-ratio matching, \\textit{CDA} \\cite{Bateson2021}, and $CurDA$ \\cite{zhang2019curriculum}, and to the recent source-free domain adaptation (\\emph{SFDA}) method $AdaMI$ in \\cite{Bateson2022}. \nA model trained on the source only, \\textit{NoAdap}, was used as a lower bound. A model trained on the target domain with the cross-entropy loss, $Oracle$, served as an upper bound.\n\n\\paragraph{\\textbf{Estimating the shape descriptors}}\\label{sec:sizeprior} For the estimation of the class-ratio $\\mathcal{\\bar{R}}$, we employed the coarse estimation in \\cite{Bateson2021}, which is derived from anatomical knowledge available in the clinical literature.\nFor $\\mathcal{M} \\in \\left\\{ \\mathcal{C},\\mathcal{D}\\right\\}$, we estimate the target shape descriptor from the network prediction masks $\\mathbf{\\hat{y}_n}$ after each epoch: $\\mathcal{\\bar{M}}^{(k)} = \\frac{1}{\\left|V^k \\right|}\\sum_{v \\in V^{k}}v$, with  $V^{k} = \\left\\{ {\\mathcal{M}^{(k)}(\\mathbf{\\hat{y}_n}) \\text{ if }{\\mathcal{R}^k(\\mathbf{\\hat{y}_n})>\\epsilon^k}},n=1 \\cdots N \\right\\}$.\n\nNote that, for a fair comparison, we used exactly the same class-ratio priors and weak supervision employed in the benchmarks methods in \\cite{Bateson2021,Bateson2022,zhang2019curriculum}.\nWeak supervision takes the form of simple image-level tags by setting $\\mathcal{\\bar{R}}^{(k)}=\\textbf{0}$ and $\\lambda=0$ for the target images that do not contain structure $k$.   \n\n\\paragraph{\\textbf{Training and implementation details}} For all methods, the segmentation network employed was UNet \\cite{UNet}. A model trained on the source data with Eq~\\eqref{eq:crossent} for 150 epochs was used as initialization. Then, for TTA models, adaptation is performed on each test subject independently, without target training. Our model was initialized with Eq~\\eqref{eq:AdaMI} for 150 epochs, after which the additional shape constraint was added using Eq~\\eqref{eq:TTAS} for 200 epochs.  As there is no learning and validation set in the target domain, the hyper-parameters are set following those in the source training, and are fixed across experiments: we trained with the Adam optimizer \\cite{Adam}, a batch size of $min(N,22)$, an initial learning rate of $5\\times10^{-4}$, a learning rate decay of 0.9 every 20 epochs, and a weight decay of $10^{-4}$. The weights $\\nu_k$ are calculated as: $\\nu_k=\\frac{\\bar{\\mathcal{R}}_k^{-1}}{\\sum_k \\bar{\\mathcal{R}}_k^{-1}}$. We set $\\lambda=1\\times10^{-4}$.\n \n\n\\paragraph{\\textbf{Evaluation}} The 3D Dice similarity coefficient (DSC) and the 3D Average Surface distance (ASD) were used as evaluation metrics in our experiments. \n\n\n\\subsection{Results \\& discussion} \n\n\n\nTable~\\ref{table:resultswhs} and Table \\ref{table:resultspro} report quantitative metrics for the heart and prostate respectively. Among DA methods, the source-free $AdaMI$ achieves the best DSC improvement over the lower baseline \\textit{NoAdap}, with a mean DSC of 75.7\\% (cardiac) and 79.5\\% (prostate). Surprisingly though, in both applications, our method $TTAS_{\\mathcal{R}\\mathcal{D}}$ yields better scores: 76.5\\% DSC, 5.4 vox. ASD (cardiac) and 79.5\\% DSC, 3.9 vox. ASD (prostate); while $TTAS_{\\mathcal{R}\\mathcal{C}}$ achieves the best DSC across methods: 80.0\\% DSC and 5.3 vox. ASD (cardiac), 80.2\\% DSC and 3.79 ASD vox. (prostate).\nFinally, comparing to the TTA methods, both $TTAS_{\\mathcal{R}\\mathcal{C}}$ and $TTAS_{\\mathcal{R}\\mathcal{D}}$ widely outperform $TTADAE$, which yields 40.7\\% DSC, 12.9 vox. ASD (cardiac) and 73.2\\% DSC, 5.80 vox. ASD (prostate), and $Tent$, which reaches 48.2\\% DSC, 11.2 vox. ASD (cardiac) and 68.7\\% DSC, 5.87 vox. ASD (prostate). \n\n\n\n\n\n\\begin{table}[t]\n\\footnotesize\n    \\caption{Test-time metrics on the cardiac dataset, for our method and various \\textit{Domain Adaptation} (DA), \\textit{Source Free Domain Adaptation} (SFDA) and \\textit{Test Time Adaptation} (TTA) methods.}\n   \\centering\n    \\resizebox{\\textwidth}{!}{\n      \\begin{tabular}{lccccccc|cp{5pt}cccc|c}\n\n\\toprule\n\n\\multirow{2}{3em}{Methods} & \\multirow{2}{3em}{DA}& \\multirow{2}{3em}{SFDA} & \\multirow{2}{3em}{TTA} &  \n \\multicolumn{5}{c}{ DSC (\\%)} && \\multicolumn{5}{c}{ASD (vox)} \\\\\n \\cmidrule(lr){5-9}\\cmidrule(lr){10-15}\n& &  & & AA & LA & LV & Myo & Mean && AA & LA & LV & Myo & Mean\n\\\\\n\\hline\nNoAdap (lower b.)&&&& 49.8&62.0&21.1&22.1&38.8 && 19.8&13.0&13.3&12.4&14.6  \\\\\nOracle \\, \\,(upper b.)&&&&91.9  & 88.3  & 91.0  & 85.8  & 89.2 && 3.1 & 3.4 & 3.6 & 2.2 & 3.0 \\\\\n\\midrule\nCurDA \\cite{zhang2019curriculum}&$\\checkmark$&$\\times$& $\\times$&79.0  & 77.9   & 64.4 & 61.3 & 70.7&& 6.5 & 7.6 & 7.2 & 9.1 & 7.6 \\\\\nCDA \\cite{Bateson2021} & $\\checkmark$& $\\times$&$\\times$&77.3 & 72.8 & 73.7 & 61.9 &71.4 && \\textbf{4.1} &6.3 &  6.6 &6.6 & 5.9 \\\\\nAdaMI \\cite{Bateson2022} & $\\times$&$\\checkmark$&$\\times$ &83.1&78.2&74.5&66.8& 75.7&&5.6&\\textbf{4.2}&\\textbf{5.7}&6.9&5.6\\\\\nTTDAE \\cite{KARANI2021101907} &$\\times$& $\\times$& $\\checkmark$& 59.8 & 26.4 & 32.3& 44.4 & 40.7 && 15.1 & 11.7 & 13.6 &11.3 & 12.9 \\\\\nTent \\cite{wang2021tent} & $\\times$& $\\times$& $\\checkmark$& 55.4 & 33.4 &63.0 &41.1 & 48.2 && 18.0 & 8.7 & 8.1 & 10.1 & 11.2 \\\\\n\\rowcolor{Gray}\\begin{tabular}[c]{@{}l@{}}Proposed Method \\end{tabular}  &&  &  & & & & &&  & & & & &\\\\%u\n\\textbf{TTAS$_{\\mathcal{RC}}$} (Ours)  &$\\times$&$\\times$&$\\checkmark$&\\textbf{85.1}&\\textbf{82.6}&\\textbf{79.3}&\\textbf{73.2}& \\textbf{80.0}&&5.6&4.3&6.1&\\textbf{5.3}&\\textbf{5.3}\\\\\n\\textbf{TTAS$_{\\mathcal{RD}}$} (Ours) &$\\times$&$\\times$&$\\checkmark$&82.3&78.9&76.1&68.4& 76.5&&4.0&5.8&6.1&5.7&5.4\\\\\n\\rowcolor{Gray}\\begin{tabular}[c]{@{}l@{}}Ablation study \\end{tabular}&  &   &  & & & & &&  & & & & &\\\\%u\n\\textbf{TTAS$_{\\mathcal{R}}$}  &$\\times$&$\\times$&$\\checkmark$ &78.9&77.7&74.8&65.3& 74.2&& 5.2&4.9&7.0&7.6&6.2\\\\\n\\bottomrule\n  \\end{tabular}\n  }\n  \\label{table:resultswhs}\n\\end{table}\n\n\\vspace{-0.5em}\n\\begin{table}[h!]\n\\centering\n\\footnotesize\n\\caption{Test-time metrics on the prostate dataset.\n\n\\begin{tabular}{lccccc}\n\\toprule\nMethods & DA & SFDA &TTA & DSC (\\%) & ASD (vox) \\\\\n\n\\midrule\nNoAdap (lower bound) &&&    & 67.2 & 10.60\\\\%u\nOracle \\, \\,(upper bound)  &&&   & 88.9 & 1.88\\\\%u\n\\midrule\n\\begin{tabular}[c]{@{}l@{}}CurDA \\cite{zhang2019curriculum}\\end{tabular} &  $\\checkmark$ &$\\times$& $\\times$  & 76.3 & 3.93\\\\%u \/data\/users\/mathilde\/ccnn\/CDA\/results\/ivdsag\/CKLSizewTag \n\\begin{tabular}[c]{@{}l@{}}CDA \\cite{Bateson2021}\\end{tabular} &  $\\checkmark$&$\\times$  &$\\times$  & 77.9 & \\textbf{3.28}\\\\%u\n\\begin{tabular}[c]{@{}l@{}}AdaMI\\cite{Bateson2022}\\end{tabular} & $\\times$&$\\checkmark$& $\\times$& 79.5 & 3.92\\\\%u results\/ivdsag\/sfda2 results\/sal\/invsekllit9 \n\\begin{tabular}[c]{@{}l@{}}TTDAE \\cite{KARANI2021101907}\\end{tabular} &$\\times$ & $\\times$ & $\\checkmark$ & 73.2 & 5.80\\\\%u\n\\begin{tabular}[c]{@{}l@{}}Tent \\cite{wang2021tent}\\end{tabular} &$\\times$  &$\\times$ & $\\checkmark$ & 68.7 & 5.87\\\\%u \n\\rowcolor{Gray}\\begin{tabular}[c]{@{}l@{}}Proposed Method \\end{tabular}  & & & & & \\\\%u\n\\begin{tabular}[c]{@{}l@{}}TTAS$_{\\mathcal{RC}}$ (Ours)\\end{tabular} &  $\\times$& $\\times$ & $\\checkmark$& \\textbf{80.2} & 3.79\\\\\n\\begin{tabular}[c]{@{}l@{}}TTAS$_{\\mathcal{RD}}$ (Ours)\\end{tabular} &  $\\times$& $\\times$& $\\checkmark$ & 79.5 & 3.90\\\\\n\\rowcolor{Gray}\\begin{tabular}[c]{@{}l@{}}Ablation study \\end{tabular} & &  & & & \\\\%u\n\\begin{tabular}[c]{@{}l@{}}TTAS$_{\\mathcal{R}}$ (Ours)\\end{tabular} &  $\\times$& $\\times$& $\\checkmark$ & 75.3 & 5.06\\\\\n\\bottomrule\n\\end{tabular}\n\\label{table:resultspro}\n\\end{table}\n\n\nQualitative segmentations are depicted in Figure~\\ref{fig:seg}. These visuals results confirm that without adaptation, a model trained only on source data cannot properly segment the structures on the target images. The segmentation masks obtained using the TTA formulations $Tent$ \\cite{wang2021tent}, $TTADAE$ \\cite{KARANI2021101907} only show little improvement. Both methods are unable to recover existing structures when the initialization $NoAdap$ fails to detect them (see fourth and fifth row, Figure~\\ref{fig:seg}). On the contrary, those produced from our degraded model $TTAS_\\mathcal{R}$ show more regular edges and is closer to the ground truth. However, the improvement over $TTAS_\\mathcal{R}$ obtained by our two models $TTAS_{\\mathcal{R}\\mathcal{C}}$, $TTAS_{\\mathcal{R}\\mathcal{D}}$ is remarkable regarding the shape and position of each structures: the prediction masks show better centroid position (first row, Figure~\\ref{fig:seg}, see LA and LV) and better \ncompactness (third, fourth, fifth row, Figure~\\ref{fig:seg}).\n\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=\\textwidth]{figures\/segall8.png}\n      \\caption[]{Qualitative performance on cardiac images (top) and prostate images (bottom): examples of the segmentations achieved by our formulation ($TTAS_{\\mathcal{R}\\mathcal{C}},TTAS_{\\mathcal{R}\\mathcal{D}}$), and benchmark TTA models. The cardiac structures of MYO, LA, LV and AA are depicted in blue, red, green and yellow respectively. }\n         \\label{fig:seg}\n\\end{figure}    \n\n\n\n\\begin{comment}\n    \\begin{table}[]\n\\scriptsize\n\\begin{tabular}{lcccccccc}\n\\toprule\n           &  Adversarial & Constraint& Adv+Cons \\\\\n           \\midrule\nTraining Time (ms\/batch)      & 168 & 153  & 0  \\\\\n  \n\\bottomrule\n\\end{tabular}\n    \\caption[]{Training times for the diverse methods with a batch size of 1}\n    \\label{tab:res_source}\n\\end{table}\n\\end{comment}\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we proposed a simple formulation for \\emph{single-subject} test-time adaptation (TTA), which does not need access to the source data, nor the availability of a target training data. \nOur approach performs inference on a test subject by minimizing the entropy of predictions and a class-ratio prior over batchnorm parameters. To further guide adaptation, we integrate shape priors through penalty constraints.\nWe validate our method on two challenging tasks, the MRI-to-CT adaptation of cardiac segmentation and the cross-site adaptation of prostate segmentation. Our formulation achieved better performances than state-of-the-art TTA methods, with a 31.8\\% (resp. 7.0\\%) DSC improvement on cardiac and prostate images respectively. Surprisingly, it also fares better than various state-of-the-art DA and SFDA methods. These results highlight the effectiveness of shape priors on test-time inference, and question the usefulness of training on target data in segmentation adaptation. Future work will involve the introduction of higher-order shape moments, as well as the integration of multiple shapes moments in the adaptation loss. Our test-time adaptation framework is straightforward to use with any segmentation network architecture.\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{splncs04}\n\n\\section{Introduction}\n\nDeep convolutional neural networks, have demonstrated state-of-the-art performance in many natural and medical imaging problems \\cite{litjens2017survey}. However, deep-learning methods tend to under-perform when trained on a dataset with an underlying distribution different from the target images. In medical imaging, this is due for instance to variations in imaging modalities and protocols, vendors, machines and clinical sites (see Fig~\\ref{fig:s_t_im}). For semantic segmentation problems, labelling for each different target distribution is impractical,  time-consuming, and often impossible.\nTo circumvent those impediments, methods learning robust networks with less supervision have been popularized in computer vision.\n\n\\paragraph{Domain Adaptation} This motivates Domain Adaptation (DA) methods, to adapt a model trained on an annotated source domain to another target domain with no or minimal annotations. While some approaches to the problem require labelled data from the target domain, others adopt an unsupervised approach to domain adaptation (UDA). \n\n\\paragraph{Source Free Domain Adaptation} Additionally, a more recent line of works tackles source-free domain adaptation \\cite{Bateson2020} (SFDA), a setting where the source data is unavailable (neither images, nor labeled masks) during the training of the adaptation phase. In medical imaging, this may be the case when the source and target data come from different clinical sites, due to, for instance, privacy concerns or the loss or corruption of source data.\n\n\\paragraph{Test-Time Domain Adaptation} In the standard DA setting, the first step consists in fine-tuning or retraining the model with some (unlabeled) samples the target domain. Then, the evaluation consists of measuring the model's ability to generalise to unseen data\nin the target domain. However, the emerging field of Test-Time Adaptation (TTA) \\cite{wang2021tent} argues that this is not as useful as adapting to the test set directly. In some applications, it might not even be possible, such as when only a single target-domain subject is available. \n\n\\paragraph{}Our framework will therefore be Source-Free Test-Time Adaptation for Segmentation. We propose an evaluation framework where we perform test-time adaptation on each subject from the target data separately. This can be seen as an extreme case of the active field of \"learning with less data\", as our method should work with only a single sample from a new data distribution.\n\n\\begin{figure}[t]\n    \\includegraphics[width=1\\linewidth]{figures\/s_t_im_crop.png}\n      \\caption[]{Visualization of 2 aligned slice pairs in two different MRI modalities: Water and In-Phase, showing the different aspects of the structures to be segmented.}\n         \\label{fig:s_t_im}\n\\end{figure}\n\n\n\\section{Project} \n\nIn this project, we will propose a simple formulation for test-time adaptation (DA), which removes the need for a concurrent access to the source and target data, as well as the need for multiple target subjects, in the context of semantic segmentation. Our approach will substitute the standard supervised loss in the source domain by a direct minimization of the entropy of predictions in the target domain. To prevent trivial solutions, we will integrate the entropy loss with different shape moments, such as were introduced in the less challenging context of semi-supervised learning in  \\cite{KervadecMIDL2021}. Our formulation is usable with any segmentation network architecture, we will use \\cite{UNet}.\n\nTTA is a nascent field, with only a few works in segmentation \\cite{KARANI2021101907}. The question we wish to answer is the following : Can we leverage the inherent structure of anatomical shapes, to further guide test-time adaptation ?\n\n\\section{Datasets}\n    \n\\subsection{Test-time Adaption with shape moments} \n\n\\subsubsection{\\textbf{Prostate Application}} We will first evaluate the proposed method on the publicly available NCI-ISBI 2013 Challenge\\footnote{https:\/\/wiki.cancerimagingarchive.net\/display\/Public\/NCI-ISBI+2013+Challenge+-+Automated+Segmentation+of+Prostate+Structures} dataset. It is composed of manually annotated 3D T2-weighted magnetic resonance from two different sites, including 30 samples from Radboud University Nijmegen Medical Centre (Site A, target domain $S$), 30 samples from Boston Medical Center (Site B, source domain $T$).\n\n\\subsubsection{\\textbf{Heart Application}} We will employ the 2017 Multi-Modality Whole Heart Segmentation (MMWHS) Challenge dataset for cardiac segmentation \\cite{Zhuang2019}. The dataset consists of 20 MRI (source domain $S$) and 20 CT volumes (target domain $T$) of non-overlapping subjects, with their manual annotations. We will adapt the segmentation network for parsing four cardiac structures: the Ascending Aorta (AA), the Left Atrium blood cavity (LA), the Left Ventricle blood cavity (LV) and the Myocardium of the left ventricle (MYO). We will employ the pre-processed data provided by \\cite{dou2018pnp}.   \n\n\\subsubsection{\\textbf{Benchmark Methods}}\n\nA model trained on the source only, \\textit{NoAdaptation}, will be used as a lower bound. A model trained with the cross-entropy loss on the target domain, referred to as $Oracle$, will be the upper bound. Regarding Test Time adaptation methods, we will compare our method to \\cite{wang2021tent}, the seminal test-time adaptation method, as well as \\cite{KARANI2021101907}. Then, we will also compare the TTA setting to the DA setting, where a set of subjects from the target distribution is available for adapting the network. We will compare to both SFDA methods \\cite{Bateson2020,zhang2019curriculum} and standard DA methods, such as adversarial methods \\cite{tsai2018learning}.\n\n \n\\subsubsection{\\textbf{Evaluation.}}\nThe Dice similarity coefficient (DSC) and the Hausdorff distance (HD) will be used as evaluation metrics in our experiments. \n\n\n\n\n\\section{Outline} \n\nTable \\ref{tab:our} shows the outline of the project.\n\n    \\begin{table}[]\n\\begin{tabular}{ll}\n\\toprule\n1 week &   Feasibility study   \\\\\n2 weeks &  Literature study   \\\\\n1 week  &  Project proposal  \\\\\n1 week &   Literature Report   \\\\\n5 weeks & Implementation, evaluation, and report writing \\\\\n\\bottomrule\n\\end{tabular}\n    \\label{tab:our}\n\\end{table}\n\n\n\n\n\n\n\\section{Critical Points}\n\n\\paragraph{Experiments} We will introduce an adaptation loss in the form of entropy loss with a various shapes moments to constrain the structures to be segmented. We will investigate simple shape priors, suche as the size, the centroid, the distance to centroid, the eccentricity ... \n\n\\paragraph{Questions to be answered} How to derive good priors for the shape moments ? Should the whole network be adapted, or just the batch normalization parameters, as is advocated in recent works \\cite{KARANI2021101907,wang2021tent} ?\n\n\\paragraph{Code}\nThe code will build on previous work \\cite{Bateson2020}, using PyTorch.\n\n\\bibliographystyle{splncs04}\n\n\\section{Introduction}\n\nDeep convolutional neural networks, have demonstrated state-of-the-art performance in many natural and medical imaging problems \\cite{litjens2017survey}. However, deep-learning methods tend to under-perform when trained on a dataset with an underlying distribution different from the target images. In medical imaging, this is due for instance to variations in imaging modalities and protocols, vendors, machines and clinical sites. For semantic segmentation problems, labelling for each different target distribution is impractical,  time-consuming, and often impossible. To circumvent those impediments, methods learning robust networks with less supervision have been popularized in computer vision.\n\nThis motivates Domain Adaptation (DA) methods, to adapt a model trained on an annotated source domain to another target domain with no or minimal annotations. While some approaches to the problem require labelled data from the target domain, others adopt an unsupervised approach to domain adaptation (UDA). Evaluating UDA methods consists of measuring the model's ability to generalise to unseen data\nin the target domain. However, the emerging field of Test-Time Adaptation (TTA) \\citep{wang2021tent} argues that this is not as useful as adapting to the test set directly. In some applications, it might not even be possible, such as when only a single target-domain subject is available. \n\nMoreover, a more recent line of works tackles source-free domain adaptation \\citep{Bateson2020}, a setting where the source data is unavailable (neither images, nor labeled masks) during the training of the adaptation phase. In medical imaging, this may the case when the source and target data come from different clinical sites, due to, for instance, privacy concerns or the loss or corruption of source data.\n\nOur framework will therefore be source-free Test-Time Adaptation for Segmentation. We therefore propose an evaluation framework where we perform test-time adaptation on each subject separately.\n\n\n\n\\section{Project} \n\nIn this project, we will propose a simple formulation for test-time adaptation (DA), which removes the need for a concurrent access to the source and target data, as well as the need for multiple target subjects, in the context of semantic segmentation. Our approach will substitute the standard supervised loss in the source domain by a direct minimization of the entropy of predictions in the target domain. To prevent trivial solutions, we will integrate the entropy loss with different shape moments, such as in \\citep{KervadecMIDL2021,Bateson2020}. Our formulation is usable with any segmentation network architecture, we will use \\citep{UNet}.\n\n\\section{Datasets}\n    \n\\subsection{Test-time Adaption with shape moments} \n\n\\subsubsection{\\textbf{Prostate Application}} We will first evaluate the proposed method on the publicly available NCI-ISBI 2013 Challenge\\footnote{https:\/\/wiki.cancerimagingarchive.net\/display\/Public\/NCI-ISBI+2013+Challenge+-+Automated+Segmentation+of+Prostate+Structures} dataset. It is composed of manually annotated 3D T2-weighted magnetic resonance from two different sites, including 30 samples from Radboud University Nijmegen Medical Centre (Site A, target domain $S$), 30 samples from Boston Medical Center (Site B, source domain $T$).\n\n\\subsubsection{\\textbf{Heart Application}} We employ the 2017 Multi-Modality Whole Heart Segmentation (MMWHS) Challenge dataset for cardiac segmentation \\citep{Zhuang2019}. The dataset consists of 20 MRI (source domain $S$) and 20 CT volumes (target domain $T$) of non-overlapping subjects, with their manual annotations. We will adapt the segmentation network for parsing four cardiac structures: the Ascending Aorta (AA), the Left Atrium blood cavity (LA), the Left Ventricle blood cavity (LV) and the Myocardium of the left ventricle (MYO). We will employ the pre-processed data provided by \\citep{dou2018pnp}.   \n\n\\subsubsection{\\textbf{Benchmark Methods}}\n\nA model trained on the source only, \\textit{NoAdaptation}, will be used as a lower bound. A model trained with the cross-entropy loss on the target domain, referred to as $Oracle$, will be the upper bound. Regarding Test Time adaptation methods, we will compare our method to \\citep{wang2021tent}, the the seminal test-time adaptation method, as well as \\citep{KARANI2021101907}. Then, we will also compare the TTA setting to the DA setting, where a set of subjects from the target distribution is available for adaptation. We will compare to both SFDA methods \\citep{Bateson2020,zhang2019curriculum} and standard DA methods, such as adversarial methods \\citep{tsai2018learning}.\n\n \n\\subsubsection{\\textbf{Evaluation.}}\nThe Dice similarity coefficient (DSC) and the Hausdorff distance (HD) were used as evaluation metrics in our experiments. \n\n\n\n\n\\section{Outline} \n\nTable \\ref{tab:our} shows the outline of the project.\n\n    \\begin{table}[]\n\\begin{tabular}{ll}\n\\toprule\n1 week &   Feasibility study   \\\\\n2 weeks &  Literature study   \\\\\n1 week  &  Project proposal  \\\\\n1 week &   Literature Report   \\\\\n5 weeks & Implementation, evaluation, and report writing \\\\\n\\bottomrule\n\\end{tabular}\n    \\label{tab:our}\n\\end{table}\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we proposed a simple formulation for domain adaptation (DA), which removes the need for a concurrent access to the source and target data, in the context of semantic segmentation for multi-modal magnetic resonance images. Our approach substitutes the standard supervised loss in the source domain by a direct minimization of the entropy of predictions in the target domain. To prevent trivial solutions, we integrate the entropy loss with a class-ratio prior, which is built from an auxiliary network. Unlike the recent domain-adaptation techniques, our method tackles DA without resorting to source data during the adaptation phase. Interestingly, our formulation achieved better performances than related state-of-the-art methods with access to both source and target data. \nThis shows the effectiveness of our prior-aware entropy minimization and that, in several cases of interest where the domain shift is not too large, adaptation might not need access to the source data. \nOur proposed adaptation framework is usable with any segmentation network architecture.\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{splncs04}\n\n\\section{Introduction}\n\\lipsum[2]\n\\lipsum[3]\n\n\n\\section{Headings: first level}\n\\label{sec:headings}\n\n\\lipsum[4] See Section \\ref{sec:headings}.\n\n\\subsection{Headings: second level}\n\\lipsum[5]\n\\begin{equation}\n\\xi _{ij}(t)=P(x_{t}=i,x_{t+1}=j|y,v,w;\\theta)= {\\frac {\\alpha _{i}(t)a^{w_t}_{ij}\\beta _{j}(t+1)b^{v_{t+1}}_{j}(y_{t+1})}{\\sum _{i=1}^{N} \\sum _{j=1}^{N} \\alpha _{i}(t)a^{w_t}_{ij}\\beta _{j}(t+1)b^{v_{t+1}}_{j}(y_{t+1})}}\n\\end{equation}\n\n\\subsubsection{Headings: third level}\n\\lipsum[6]\n\n\\paragraph{Paragraph}\n\\lipsum[7]\n\n\\section{Examples of citations, figures, tables, references}\n\\label{sec:others}\n\\lipsum[8] \\cite{kour2014real,kour2014fast} and see \\cite{hadash2018estimate}.\n\nThe documentation for \\verb+natbib+ may be found at\n\\begin{center}\n  \\url{http:\/\/mirrors.ctan.org\/macros\/latex\/contrib\/natbib\/natnotes.pdf}\n\\end{center}\nOf note is the command \\verb+\\citet+, which produces citations\nappropriate for use in inline text.  For example,\n\\begin{verbatim}\n   \\citet{hasselmo} investigated\\dots\n\\end{verbatim}\nproduces\n\\begin{quote}\n  Hasselmo, et al.\\ (1995) investigated\\dots\n\\end{quote}\n\n\\begin{center}\n  \\url{https:\/\/www.ctan.org\/pkg\/booktabs}\n\\end{center}\n\n\n\\subsection{Figures}\n\\lipsum[10] \nSee Figure \\ref{fig:fig1}. Here is how you add footnotes. \\footnote{Sample of the first footnote.}\n\\lipsum[11] \n\n\\begin{figure}\n  \\centering\n  \\fbox{\\rule[-.5cm]{4cm}{4cm} \\rule[-.5cm]{4cm}{0cm}}\n  \\caption{Sample figure caption.}\n  \\label{fig:fig1}\n\\end{figure}\n\n\\subsection{Tables}\n\\lipsum[12]\nSee awesome Table~\\ref{tab:table}.\n\n\\begin{table}\n \\caption{Sample table title}\n  \\centering\n  \\begin{tabular}{lll}\n    \\toprule\n    \\multicolumn{2}{c}{Part}                   \\\\\n    \\cmidrule(r){1-2}\n    Name     & Description     & Size ($\\mu$m) \\\\\n    \\midrule\n    Dendrite & Input terminal  & $\\sim$100     \\\\\n    Axon     & Output terminal & $\\sim$10      \\\\\n    Soma     & Cell body       & up to $10^6$  \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\label{tab:table}\n\\end{table}\n\n\\subsection{Lists}\n\\begin{itemize}\n\\item Lorem ipsum dolor sit amet\n\\item consectetur adipiscing elit. \n\\item Aliquam dignissim blandit est, in dictum tortor gravida eget. In ac rutrum magna.\n\\end{itemize}\n\n\n\\section{Conclusion}\nYour conclusion here\n\n\\section*{Acknowledgments}\nThis was was supported in part by......\n\n\\bibliographystyle{unsrt}  \n\n\\section{Introduction}\n\\lipsum[2]\n\\lipsum[3]\n\n\n\\section{Headings: first level}\n\\label{sec:headings}\n\n\\lipsum[4] See Section \\ref{sec:headings}.\n\n\\subsection{Headings: second level}\n\\lipsum[5]\n\\begin{equation}\n\\xi _{ij}(t)=P(x_{t}=i,x_{t+1}=j|y,v,w;\\theta)= {\\frac {\\alpha _{i}(t)a^{w_t}_{ij}\\beta _{j}(t+1)b^{v_{t+1}}_{j}(y_{t+1})}{\\sum _{i=1}^{N} \\sum _{j=1}^{N} \\alpha _{i}(t)a^{w_t}_{ij}\\beta _{j}(t+1)b^{v_{t+1}}_{j}(y_{t+1})}}\n\\end{equation}\n\n\\subsubsection{Headings: third level}\n\\lipsum[6]\n\n\\paragraph{Paragraph}\n\\lipsum[7]\n\n\\section{Examples of citations, figures, tables, references}\n\\label{sec:others}\n\\lipsum[8] \\cite{kour2014real,kour2014fast} and see \\cite{hadash2018estimate}.\n\nThe documentation for \\verb+natbib+ may be found at\n\\begin{center}\n  \\url{http:\/\/mirrors.ctan.org\/macros\/latex\/contrib\/natbib\/natnotes.pdf}\n\\end{center}\nOf note is the command \\verb+\\citet+, which produces citations\nappropriate for use in inline text.  For example,\n\\begin{verbatim}\n   \\citet{hasselmo} investigated\\dots\n\\end{verbatim}\nproduces\n\\begin{quote}\n  Hasselmo, et al.\\ (1995) investigated\\dots\n\\end{quote}\n\n\\begin{center}\n  \\url{https:\/\/www.ctan.org\/pkg\/booktabs}\n\\end{center}\n\n\n\\subsection{Figures}\n\\lipsum[10] \nSee Figure \\ref{fig:fig1}. Here is how you add footnotes. \\footnote{Sample of the first footnote.}\n\\lipsum[11] \n\n\\begin{figure}\n  \\centering\n  \\fbox{\\rule[-.5cm]{4cm}{4cm} \\rule[-.5cm]{4cm}{0cm}}\n  \\caption{Sample figure caption.}\n  \\label{fig:fig1}\n\\end{figure}\n\n\\subsection{Tables}\n\\lipsum[12]\nSee awesome Table~\\ref{tab:table}.\n\n\\begin{table}\n \\caption{Sample table title}\n  \\centering\n  \\begin{tabular}{lll}\n    \\toprule\n    \\multicolumn{2}{c}{Part}                   \\\\\n    \\cmidrule(r){1-2}\n    Name     & Description     & Size ($\\mu$m) \\\\\n    \\midrule\n    Dendrite & Input terminal  & $\\sim$100     \\\\\n    Axon     & Output terminal & $\\sim$10      \\\\\n    Soma     & Cell body       & up to $10^6$  \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\label{tab:table}\n\\end{table}\n\n\\subsection{Lists}\n\\begin{itemize}\n\\item Lorem ipsum dolor sit amet\n\\item consectetur adipiscing elit. \n\\item Aliquam dignissim blandit est, in dictum tortor gravida eget. In ac rutrum magna.\n\\end{itemize}\n\n\n\\section{Conclusion}\nYour conclusion here\n\n\\section*{Acknowledgments}\nThis was was supported in part by......\n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nLet $\\A$ be a Banach algebra (over the complex field $\\mathbb{C}$), and $\\U$ be a Banach $\\A$-bimodule. A linear map $\\delta:\\A\\rightarrow\\U$ is called a \\textit{derivation} if $\\delta (ab)=a\\delta(b)+\\delta (a)b$ holds for all $a,b\\in\\A$. For any $x\\in\\U$, the map $id_x:\\A\\rightarrow\\U$ given by $id_x(a)=ax-xa$ is a continuous derivation called \\textit{inner derivation}. The set of all continuous derivations from $\\A$ into $\\U$ is denoted by $Z^{1}(\\A,\\U)$ while $N^{1}(\\A,\\U)$ denotes the set of all inner derivations from $\\A$ into $\\U$. $Z^{1}(\\A,\\U)$ is a linear subspace $\\mathbb{B}(\\A,\\U)$ (the space of all continuous linear maps from $\\A$ into $\\U$) and $N^{1}(\\A,\\U)$ is a linear subspace of $Z^{1}(\\A,\\U)$. We denote by $H^{1}(\\A,\\U)$, the quotient space $\\frac{Z^{1}(\\A,\\U)}{N^{1}(\\A,\\U)}$ which is called \\textit{the first cohomology group of $\\A$ with coefficients in $\\U$}. We call the the first cohomology group of $\\A$ with coefficients in $\\A$ briefly by first cohomology group of $\\A$. Derivations and cohomology group are important subjects in the study of Banach algebras. Among the most important problems related to them are these questions; Under what conditions a derivation $\\delta:\\A\\rightarrow\\U$ is continuous? Under what conditions one has $H^{1}(\\A,\\U)=(0)$? (i.e., every continuous derivation from $\\A$ into $\\U$ is inner). Also the calculation of $H^{1}(\\A,\\U)$, up to isomorphism of linear spaces, is a considerable problem.\n\\par \nThe problem of continuity of derivations is related to the subject of automatic continuity which is an important subject in mathematical analysis. Many studies have been performed in this regard and it has a long history. We may refer to \\cite{da} for more information which is a detailed source in this subject. Here we only review the most important obtained results concerning the automatic continuity of derivations. Johnson and Sinclair in \\cite{john1} have shown that every derivation on a semisimple Banach algebra is continuous. Ringrose in \\cite{rin} showed that every derivation from a $C^{*}$-algebra $\\A$ into Banach $\\A$-bimodule $\\U$ is continuous. In \\cite{chr}, Christian proved that every derivation from a nest algebra on Hilbert space $\\mathcal{H}$ into $\\mathbb{B}(\\mathcal{H})$ is continuous. Additionally, some results on automatic continuity of derivations on prime Banach algebras have been established by Villena in \\cite{vi1} and \\cite{vi2} .\n\\par \nThe study of the first cohomology group of Banach algebras is also a considerable topic which may be used to study the structure of Banach algebras. Johnson in \\cite{john2}, using the first cohomology group, has defined amenable Banach Algebras and then various types of amenability defined by using the first cohomology group. We may refer the reader to \\cite{run} for more information. Also among the interesting problems in the theory of derivations is either characterizing algebras on which every continuous derivation is inner, that is, the first  cohomology group is trivial or characterizing the first cohomology group of Banach algebras up to vector spaces isomorphism. Sakai in \\cite{sa} showed that every continuous derivation on a $W^{*}$-algebra is inner. Kadison \\cite{ka} proved that every derivation of a $C^{*}$-algebra on a Hilbert space $\\mathcal{H}$ is spatial (i.e., it is of the form $a\\mapsto ta-at$ for $t\\in \\mathbb{B}(\\mathcal{H})$) and in particular, every derivation on a von Neumann algebra is inner. Some results have also been obtained in the case of  non-self-adjoint operator algebras. Christian \\cite{chr} showed that every continuous derivation on a nest algebra on $\\mathcal{H}$ to itself and to $\\mathbb{B}(\\mathcal{H})$ is inner and then this result generalized to some other forms among which we may refer to \\cite{li} and the references therein. Gilfeather and Smith have calculated the first cohomology group of some operator algebras called joins(\\cite{gil1}, \\cite{gil2}). In \\cite{do}, the cohomology group of operator algebras called seminest algebras has been calculated. All of these operator algebras and nest algebras have a structure like triangluar Banach algebras, so motivated by these studies, Forrest and Marcoux in \\cite{for} verified the first cohomology group of triangular Banach algebras. The first cohomology group of triangular Banach algebras is not necessarily zero and in \\cite{for}, in fact it has been calculated under some special conditions and using the results, various examples of Banach algebras with non-trivial cohomology have been given and then the first cohomology group of those examples has been computed. Also in \\cite{for}, some results concerning the automatic continuity of derivations on triangluar Banach algebras are presented. A generalization of triangluar Banach algebras is module extensions of Banach algebras which Zhang \\cite{zh} has studied the weak amenability of them and then Medghalchi and Pourmahmood in \\cite{med} computed the first cohomology group of module extensions of Banach algebras and using those results, they gave various examples of Banach algebras  with non-trivial cohomology group and in fact they calculated the first cohomology group of the given examples.\n\\par \nAnother class of Banach algebras which considered during the last thirty years, is the class of algebras obtained by a special product called $\\theta$-Lau product. This product is firstly introduced by Lau \\cite{lau} for a special class of Banach algebras which are pre-dual of von Neumann algebras where the dual unit element (the unit element of the dual) is a multiplicative linear functional. Afterwards, various studies have been performed to it. For instance, Monfared in \\cite{mn} has verified the structure of this special product and in \\cite{gha} the amenability of Banach algebras equipped with the same product has been studied. For more information about this product the reader may refer to \\cite{gha}, \\cite{mn} and references therein.  \n\\par \nLet $\\mathcal{B}$ be a Banach algebra such that $\\mathcal{B} =\\A\\oplus\\U$ (as direct sum of Banach spaces) where $\\A$ is a closed subalgebra of $\\mathcal{B}$ and $\\U$ is a closed ideal in $\\mathcal{B}$. In this case, we say that $\\mathcal{B}$ is \\textit{semidirect product} of $\\A$ and $\\U$ and write $\\mathcal{B}=\\A\\ltimes \\U$. Semidirect product Banach algebras appear in the study of many classes of Banach algebras. For instance, in the strong Wedderburn decomposition of a Banach algebra $\\mathcal{B}$, it is assumed that $\\mathcal{B}=\\A\\ltimes Rad\\mathcal{B}$ where $Rad\\mathcal{B}$ is the Jacobson radical of $\\mathcal{B}$ and $\\A $ is a closed subalgebra of $\\mathcal{B}$ with $\\A\\cong \\frac{\\mathcal{B}}{Rad\\mathcal{B}}$ or for example, in \\cite{da2} using the structure of semidirect product, the authors have studied the amenability of measure algebras since every measure algebra has a decomposition to semidirect product of Banach algebras. In \\cite{tho2}, Thomas has verified the necessary conditions of decomposition of a commutative Banach algebra to semidirect product of a closed subalgebra and a principal ideal. We also may refer to \\cite{ba}, \\cite{ber} and \\cite{wh} where semidirect product Banach algebras are studied from different points of view. Equivalently, one may discuss semidirect product Banach algebras as follows; Let $\\A$ and $\\U$ be Banach algebras such that $\\U$ is a Banach $\\A$-bimodule with compatible actions and appropriate norm. Consider the multiplication on $\\A\\times\\U$ given by \n\\[(a,x)(b,y)=(ab,ay+xb+xy)\\quad\\quad  ((a,b)\\in \\A\\times\\U).\\]\nIt can be shown that with this multiplication and $\\l^{1}$-norm, $\\A\\times\\U$ is a Banach algebra where $\\A$ is a closed subalgebra of this Banach algebra and $\\U$ is a closed ideal of it. So in fact this Banach algebra is equivalent to $\\A\\ltimes\\U$. By considering different module actions or algebra multiplications, it can be seen that $\\A\\ltimes\\U$ is a generalization of direct products of Banach algebras, tivial extension Banach algebras, triangular Banach algebras or $\\theta$-Lau product Banach algebras. In this paper we consider semidirect product Banach algebras as mentioned above and study the derivations on this special product of Banach algebras. In fact we establish various results concerning the automatic continuity of derivations on $\\A\\ltimes\\U$ and the first cohomology group of it and we present these results in special cases of $\\A\\ltimes\\U$ and obtain various examples of Banach algebras with automatically continuous derivations and trivial first cohomology group or compute their first cohomology group.\n\\par\nThis paper is organized as follows. In section 2, we investigate the definition of semidirect product Banach algebras and we show that this product is a generalization of various types of Banach algebras. In section 3, the structure of derivations on semidirect product Banach algebras will be discussed and using that we obtain several results about the automatic continuity of derivations on these Banach algebras and also verify the decomposition of derivations into the sum of a continuous derivation and another derivation. In section 4, we consider the fist cohomology group of semidirect product Banach algebras and compute it under some different conditions and establish various results in this context. In section 5, we apply the obtained results in sections 3,4 to some special cases of semidirect products of Banach algebras. In fact we investigate the automatic continuity of the derivations and the first cohomology group of direct products of Banach algebras, module extension Banach algebras and $\\theta$-Lau products of Banach algebras and  establish some various results about the derivations on these Banach algebras. \n\\par \nAt the end of this section we introduce some used notations and expressions in the paper.\n\\par \nIf $\\mathcal{X}$ and $\\mathcal{Y}$ are Banach spaces, for a linear transform $T:\\mathcal{X}\\rightarrow \\mathcal{Y}$, define the separating space $\\mathfrak{S}(T)$ as \n\\[\\mathfrak{S}(T):=\\{y\\in \\mathcal{Y}\\, \\mid \\, \\text{there is}\\, \\, \\{x_n\\}\\subseteq \\mathcal{X}\\,\\, \\text{with}\\, \\, x_n\\rightarrow 0 , \\, T(x_n)\\rightarrow y\\} .\\]\nBy closed graph theorem, $T$ is continuous if and only if $\\mathfrak{S}(T)=(0)$.\n\\par \nLet $\\A$ be a Banach algebra and $\\U$ be a Banach $\\A$-bimodule. By $Z(\\A)$, we mean the center of $\\A$. Consider the set $ann_{\\A}\\U$ as\n\\[ann_{\\A}\\U:=\\{a\\in\\A\\, \\mid \\, a\\U=\\U a=(0)\\}.\\]\nIf $\\mathcal{N}$ is an $\\A$-submodule of $\\U$, we put \n\\[(\\mathcal{N}:\\U)_{\\A}:=\\{a\\in\\A \\, \\mid \\, a\\U\\subseteq \\mathcal{N}, \\, \\U a\\subseteq \\mathcal{N}\\}. \\] \nIt is clear that if $\\mathcal{N}=(0)$, then $((0):\\U)_{\\A}=ann_{\\A}\\U$. Let $\\U$ and $\\mathcal{V}$ be Banach $\\A$-bimodules. A linear map $\\phi :\\U\\rightarrow \\mathcal{V}$ is said to be a \\textit{left $\\A$-module homomorphism} if $\\phi (ax)=a\\phi (x)$ whenever $a\\in \\A$ and $x\\in\\U$ and it is a \\textit{right $\\A$-module homomorphism} if $\\phi (xa)=\\phi (x) a\\quad (a\\in \\A,x\\in\\U)$. The linear map $\\phi$ is called \\textit{$\\A$-module homomorphism}, if $ \\phi$ is both of left and right $\\A$-module homomorphism. The set of all continuous $\\A$-module homomorphisms from $\\U$ into $\\mathcal{V}$ is denoted by $Hom_{\\A}(\\U,\\mathcal{V})$. Note that if the spaces are the same, we just write $Z^{1}(\\A), N^{1}(\\A), H^{1}(\\A), Hom_{\\A}(\\U)$\n\\section{Semidirect products of Banach algebras}\nIn this section we introduce the notion of semidirect produts of Banach algebras and give some properties of this concept.\\\\\nLet $\\A$ and $\\U$ be Banach algebras such that $\\U$ is a Banach $\\A$-bimodule with\n\\[\\parallel ax \\parallel \\leq \\parallel a \\parallel \\parallel x \\parallel, \\quad \\parallel xa \\parallel \\leq \\parallel x \\parallel \\parallel a \\parallel \\quad\\quad (a\\in \\A, x\\in \\U), \\] \nand compatible actions, that is \n\\[ (a.x)y=a.(xy),\\,\\,\\, (xy).a=x(y.a), \\,\\,\\, (x.a)y=x(a.y)  \\quad\\quad (a \\in \\A, \\, x,y\\in \\U). \\]\nIf we equip the set $\\A \\times \\U$ with the usual $\\mathbb{C}$-module structure, then the multiplication \n\\[ (a,x)(b,y)=(ab, a.y+x.b+xy) \\]\nturns $\\A \\times \\U$ into an associative algebra. \n\\par \nIn continue we also denote the module actions by $ax$ and $xa$.\n\\par \nThe \\textit{semidirect product} of Banach algebras $\\A$ and $\\U$, denoted by $\\A \\ltimes \\U$, is defined as the space $\\A \\times \\U$ with the above algebra multiplication and with the norm \n\\[ \\parallel (a,x) \\parallel = \\parallel a \\parallel + \\parallel x \\parallel . \\]\nThe semidirect product $\\A \\ltimes \\U$ is a Banach algebra.\n\\begin{rem}\\label {1}\nIn $\\A \\ltimes \\U$ we identify $\\A \\times \\lbrace 0 \\rbrace$ with $\\A$, and $\\lbrace 0 \\rbrace \\times \\U $ with $\\U$. Then $\\A$ is a closed subalgebra while $\\U$ is a closed ideal of $\\A \\ltimes \\U$, and\n\\[ \\A \\ltimes \\U \/ \\U \\cong \\A \\quad (isometric \\, \\, isomorphism). \\]\nIndeed $\\A \\ltimes \\U$ is equal to direct sum of $\\A$ and $\\U$ as Banach spaces.\n\\par\nConversely, let $\\mathcal{B}$ be a Banach algebra which has the form $\\mathcal{B}=\\A \\oplus \\U$ as Banach spaces direct sum, where $\\A$ is a closed subalgebra of $\\mathcal{B}$ and $\\U$ is a closed ideal in $\\mathcal{B}$. In this case, the product of two elements $a+x$ and $b+y$ of  $\\mathcal{B}=\\A \\oplus \\U$ is given by \n\\[(a+x)(b+y)=ab+(ay+xb+xy), \\] where $ab\\in \\A$ and $ay+xb+xy\\in \\U$. Also the norm on $\\mathcal{B}$ is equivalent to the one given by \n\\[ \\parallel a+x \\parallel =\\parallel a \\parallel + \\parallel  x \\parallel \\quad\\quad (a\\in \\A, x \\in \\U)).\\]\nOn the other hand $\\A$ and $\\U$ are Banach algebras such that  $\\U$ is a Banach $\\A$-bimodule with compatible actions and\n\\[\\parallel ax \\parallel \\leq \\parallel a \\parallel \\parallel x \\parallel, \\quad \\parallel xa \\parallel \\leq \\parallel x \\parallel \\parallel a \\parallel \\quad\\quad (a\\in \\A, x\\in \\U). \\] \nBy above arguments we have \n\\[ \\mathcal{B}\\cong \\A \\ltimes \\U , \\]\nas isomorphism of Banach algebras.\n\\end{rem}\nThe following examples provides various\ntypes of semidirect product af Banach algebras.\n \\begin{exm}\nLet $\\A$ be a Banach algebra. Then the unitization of $\\A$ denoted by $\\A^{\\#}$ is in fact the semidirect product  $\\mathbb{C}\\ltimes \\A$.\n\\end{exm}\n\\begin{exm}\\label{dp}\nSuppose that the action  $\\A$ on  $\\U$ is trivial, that is, $ \\A\\U=\\U\\A=(0)$, then we obtain the usual $l^{1}$-direct product of Banach algebras $\\A$ and $\\U$. In this case, $\\A \\ltimes \\U= \\A \\times \\U$.\n\\end{exm}\n\\begin{exm}\\label{me}\nLet the algebra multiplication on $\\U$ be the trivial action, that is, $\\U^2=(0)$, then $\\A\\ltimes \\U$ is same as the module extension (or trivial extension) of $\\A$ by $\\U$ which we denote by $T(\\A,\\U)$.\n\\end{exm}\n\\begin{exm}\\label{tri}\nLet $\\A$ and $\\mathcal{B}$ be Banach algebras and $\\mathcal{M}$ be a Banach $(\\A,\\mathcal{B})$-bimodule. The triangular Banach algebra introduced in \\cite{for} is \n$Tri(\\A, \\mathcal{M}, \\mathcal{B}):=\\begin{pmatrix}\n \\A & \\mathcal{M}  \\\\\n   0 &\\mathcal{B} \n\\end{pmatrix}$ with the usual matrix operations and $l^1$-norm. If we trun $\\mathcal{M}$ into a  Banach $\\A \\times\\mathcal{B}$-bimodule with the actions \n$$(a,b)m=am\\quad ,\\quad m(a,b)=mb\\quad\\quad ((a,b)\\in \\A\\times\\mathcal{B}, m\\in \\mathcal{M} )$$\n( $\\A \\times \\mathcal{B}$  is considered with $l^1$-norm), then $Tri(\\A, \\mathcal{M}, \\mathcal{B})\\cong T(\\A\\times\\mathcal{B},\\mathcal{M})$ as isomorphism of Banach algebras. Therefore triangular Banach algebra is an example of semidirect products of Banach algebras.\n\\end{exm}\n\\begin{rem}\nLet $\\A$ and $\\U$ be Banach algebras such that $\\U$ is a Banach $\\A$-bimodule with the compatible actions and norm, and  let $\\overline{\\A\\U}=\\U$ or $\\overline{\\U\\A}=\\U$. Let the Banach algebra $\\A$ be the direct sum of its closed ideals $I_1$ and $I_2$, that is, $\\A=I_1 \\oplus I_2$. If $I_2\\U=\\U I_1=(0)$, then $\\U^2=(0)$. Because if $a\\in I_1$, $b\\in I_2$ and $x,y\\in \\U$ are arbitrary, then by the associativity we have $$x[(a+b)y]=[x(a+b)]y.$$\nSo $$x(ay)=(xb)y.$$\nIf $\\overline{\\A\\U}=\\U$ and we put $b=0$ in the above equation, then $xy=0$ and analogously, if $\\overline{\\U\\A}=\\U$, letting $a=0$ gives  $xy=0$.\n\\par \nBy the above arguments, hypothesis and according to Example \\ref{tri}, in this case we have \n\\begin{equation*}\n\\A\\ltimes \\U=T(\\A,\\U)\\cong\\begin{pmatrix}\n I_1 & \\U \\\\\n   0 &I_2\n\\end{pmatrix}\n\\end{equation*}\n as isomorphism of Banach algebras.\n\\end{rem}\n\\begin{exm}\\label{la}\nLet $\\A$ and $\\U$ be Banach algebras, $\\theta\\in \\Delta(\\A)$ where $\\Delta(\\A)$ is the set of all non-zero characters of $\\A$. With the following module actions, $\\U$ becomes a Banach  $\\A$-bimodule:\n$$ax=xa=\\theta(a)x\\quad\\quad (a\\in \\A ,x\\in \\U).$$\nThe norm and the actions on $\\U$ are compatible and one can consider $\\A\\ltimes \\U$ with the multiplication\n$$(a,x)(b,y)=(ab,\\theta(a)y+\\theta(b)x+xy).$$\nIn this case $\\A\\ltimes \\U$ is the $\\theta$-Lau product which is introduced in \\cite{lau}. \n\\end{exm}\n\\begin{exm}\nLet $G$ be a locally compact group. Then $M(G)=l^{1}(G)\\ltimes M_{c}(G)$ where $M_{c}(G)$ is a subspace of $M(G)$ containing all continuous measures such that for $\\mu\\in M(G)$, $\\mu\\in M_c(G)$ if and only if $\\mu(\\{s\\})=0$ $(s\\in G)$. We denote the subspace of discrete measures by $M_d({G})$ which is isomorphic to $l^{1}(G)$ and \n\\begin{equation*}M_{d}(G)=\\{\\mu =\\sum{\\alpha_{s}\\delta_{s}}:\\Vert\\mu\\Vert =\\sum_{s\\in G}{\\vert \\alpha_{s}\\vert }<\\infty \\}.\n\\end{equation*}\nIndeed $l^{1}(G)$ is a closed subspace of $M(G)$ and $M_c(G)$ is a closed ideal of $M(G)$. If $G$ is discrete, then $M(G)=l^{1}(G)$ and $M_{c}(G)=\\{0\\}$. But if $G$ is not discrete, then $M_{c}(G)\\neq\\{0\\}$ \n\\end{exm}\nIt is possible for an algebra $\\mathcal{B}$ with a closed ideal $\\mathcal{I}$, not to exist some closed subalgebra $\\A$ of $\\mathcal{B}$ such that $\\mathcal{B}=\\A\\ltimes \\mathcal{I}$. In the following we give an example of such a Banach algebra.\n\\begin{exm}\nLet $\\mathcal{B}:=C([0,1])$ be the Banach algebra of continuous complex-valued functions on $[0,1]$ and let $\\mathcal{I}:=\\{f\\in \\mathcal{B} :f(0)=f(1)=0\\}$. $\\mathcal{I}$ is a closed ideal of $\\mathcal{B}$. If $\\A $ is a closed subalgebra of $\\mathcal{B}$ satisfying  $\\mathcal{B}=\\A \\oplus \\mathcal{I}$ as Banach spaces direct sum, then for $f\\in \\mathcal{I}$ and $g\\in \\A$ with $f(x)+g(x)=x$ for $x\\in [0,1]$, we have $g(0)=0$, $g(1)=1$ and $g-g^2\\in \\A\\cap \\mathcal{I}$. But yet $g-g^2\\neq 0$.\n\\end{exm}\nNote that $\\A\\ltimes\\U$ is commutative if and only if both $\\A$ and $\\U$ are commutative Banach algebras and $\\U$ is a commutative $\\A$-bimodule.\n\\par \nIn the rest of this section we introduce some special maps which are used in next sections.\n\\par \nFor $a\\in\\A$ define the map $r_{a}:\\U\\rightarrow\\U$ by $r_{a}(x)=xa-ax$. Some properties of this map are given in the following remark.\n\\begin{rem}\\label{inn1}\nFor $a\\in\\A$, consider the map $r_{a}:\\U\\rightarrow\\U$.\n\\begin{enumerate}\n\\item[(i)]\n$r_a$ is a derivation on $\\U$.\n\\item[(ii)]\nFor every $b\\in\\A$ and $x\\in\\U$,\n\\[r_{a}(bx)=br_{a}(x)+id_{a}(b)x \\quad \\text{and} \\quad r_{a}(xb)=r_{a}(x)b+x id_{a}(b).\\]\n\\item[(iii)]\nFor $a\\in \\A$, if $id_{a}=0$, then $r_a$ is an $\\A$-bimodule homomorphism. Also if $ann_{\\A}\\U =(0)$ and $r_a$ is an $\\A$-bimodule homomorphism, then $id_{a}=0$.\n\\end{enumerate}\n\\end{rem}\nAlso inner derivations on $\\U$ have significant properties considered in determining the first cohomology group of $\\A\\ltimes\\U$ which are given in the next remark.\n\\par \nNote that both of inner derivations from $\\A$ to $\\U$ and inner derivations from $\\U$ to $\\U$  are denoted by $id_{x}$. So in order to avoid confusion, we denote by $id_{\\A , x}$, the inner derivations  from $\\A$ to $\\U$ while $id_{\\U , x}$ denotes the inner derivations from $\\U$ to $\\U$.\n\\begin{rem}\\label{inn2}\nFor $x_0\\in\\U$, consider the inner derivation $id_{\\U , x_0}:\\U\\rightarrow\\U$.\n\\begin{enumerate}\n\\item[(i)]\nFor every $a\\in\\A$ and $x\\in\\U$, \n\\[id_{\\U ,x_0}(ax)=a \\, id _{\\U ,x_0}(x)+id _{\\A , x_0}(a)x \\quad \\text{and} \\quad id_{\\U ,x_0}(xa)=id _{\\U ,x_0}(x)a+x \\, id _{\\A, x_0}(a).\\]\n\\item[(ii)]\nFor $x_{0}\\in \\U$, if $id_{\\A, x_0}=0$, then $id_{\\U , x_0}$ is an $\\A$-bimodule homomorphism. If $ann_{\\U}\\U =(0)$ and $id_{\\U , x_0}$ is an $\\A$-bimodule homomorphism, then $id_{\\A, x_0}=0$.\n\\end{enumerate}\n\\end{rem}\nThis following sets play an important role in determining the first cohomology group of $\\A\\ltimes\\U$.\n\\[R_{\\A}(\\U):=\\{r_{a}: \\U\\rightarrow\\U \\,  \\mid  \\, a\\in \\A\\};\\]\n\\[C_{\\A}(\\U):=\\{r_{a} :\\U\\rightarrow\\U \\,  \\mid  \\,  id_{a}=0 \\, \\, (a\\in \\A)\\};\\]\n\\[I(\\U):=\\{id_{\\U, x}:\\U\\rightarrow\\U \\,  \\mid  \\,  id_{\\A, x}=0 \\, \\, (x\\in \\U)\\};\\]\n\\indent In view of the above remarks, the set $R_{\\A}(\\U)$ is a linear subspace of $Z^{1}(\\U)$. Also $C_{\\A}(\\U)$ is a linear subspace of $Hom_{\\A}(\\U) \\cap R_{\\A}(\\U)$ and $I(\\U)$ is a linear subspace of $Hom_{\\A}(\\U) \\cap N^{1}(\\U)$. Indeed, we have the following inclusion linear subspaces;\n\\[ C_{\\A}(\\U)+I(\\U)\\subseteq Hom_{\\A}(\\U) \\cap (R_{\\A}(\\U)+N^{1}(\\U))\\subseteq Hom_{\\A}(\\U) \\cap Z^{1}(\\U).\\]\nIf $\\A$ is commutative, then $R_{\\A}(\\U)= C_{\\A}(\\U)$ and if $\\U$ is a commutative $\\A$-bimodule, then $N^{1}(\\U)=I(\\U)$. If $\\U ^{2}=(0)$, then $Z^{1} (\\U)=\\mathbb{B}(\\U)$ and $N^{1}(\\U)=I(\\U )=(0)$.\n\\section{Derivations on $\\A \\ltimes \\U$}\nIn this section we determine the structure of derivations on $\\A\\ltimes \\U$. According to which, we get some results concerning the automatic continuity of derivations on $\\A\\ltimes \\U$. Also we use the results of this section to determine the first cohomology group of $\\A\\ltimes \\U$ in the next sections.\n\\par \nThroughout this section we always assume that $\\A$ and $\\U$ are Banach algebras where $\\U$ is a Banach $\\A$-bimodule with the compatible actions and norm (As in the section 2). In other cases the conditions will be specified.\n\\par \nIn the following theorem the structure of derivations on $\\A\\ltimes \\U$ are determined.\n\\begin{thm}\\label{asll}\nLet $D:\\A\\ltimes \\U\\rightarrow \\A\\ltimes \\U$ be a map. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item[(i)] $D$ is a derivation.\n\\item[(ii)] \n\\[D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)\\]\nsuch that \n\\begin{enumerate}\n\\item[(a)] \n$\\delta_1:\\A\\rightarrow\\A$ is a derivation.\n\\item[(b)]\n$\\delta_2:\\A\\rightarrow \\U$ is a derivation.\n\\item[(c)]\n$\\tau_1 :\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism such that \n$$\\tau_1(xy)=0,$$\nfor all $x,y\\in \\U$.\n\\item[(d)]\n$\\tau_2 :\\U\\rightarrow \\U$ is a linear map such that for every $a\\in \\A$ and $x,y\\in \\U$ satisfies the following conditions\n\\begin{eqnarray*}\n\\tau_2 (ax)&=&a\\tau_2 (x)+\\delta_1 (a)x+\\delta_2 (a)x; \\\\\n\\tau_2 (xa)&=&\\tau_2 (x)a+x\\delta_1 (a)+x\\delta_2 (a); \\\\\n\\tau_2(xy)&=&x\\tau_1(y)+\\tau_1(x)y+x\\tau_2(y)+\\tau_2(x)y.\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{enumerate}\nMoreover, $D$ is an inner derivation if and only if $\\delta_1 ,\\delta_2$ are inner derivations, $\\tau_1 =0$ and if $\\delta_1 =ad_{a_0}$ and $ \\delta_2 =ad_{x_0}$, then $\\tau_2 =ad_{x_0}+r_{a_0}$.\n\\end{thm}\n\\begin{proof}\n$(i)\\implies (ii)$. Since $D$ is a linear map and $\\A\\ltimes \\U$ is the direct sum of the linear spaces $\\A$ and $\\U$, there are some linear maps $\\delta_1:\\A\\rightarrow \\A$, $\\delta_2:\\A\\rightarrow \\U$, $\\tau_1 :\\U\\rightarrow \\A$ and  $\\tau_2 :\\U\\rightarrow \\U$ such that for any $(a,x)\\in \\A\\ltimes \\U$ we have \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x)).$$\nBy applying $D$ on the equality $(a,0)(b,0)=(ab,0)$ we conclude that $\\delta_1$ and $\\delta_2$ are derivations. Analogously, applying $D$ on the equalities \n\\[ (a,0)(0,x)=(0,ax),\\,\\,\\, (0,x)(a,0)=(0,xa) \\,\\,\\, and \\,\\,\\, (0,x)(0,y)=(0,xy),\\]\n establishes the desired properties for the maps $\\tau_1$ and $\\tau_2$ given in parts $(c)$ and $(d)$ respectively.\n$(ii)\\implies (i)$ Clear.\n\\par \nThe equivalent conditions of inner-ness of $D$ can be obtained by a straightforward calculation.\n\\end{proof}\nIn the sequel for a derivation $D$ on $\\A\\ltimes \\U$, we always assume that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U)$$ in which the mentioned maps satisfy the conditions of the preceding theorem.\n\\par \nBy Theorem \\ref{asll}, if $D$ is an inner derivation on $\\A\\ltimes \\U$, then $\\tau_1 =0$. So this question is of interest for a given derivation $D$, under what conditions one has $\\tau_1 =0$?\nBy part $(ii)-(c)$ of Theorem \\ref{asll} if $\\U^2=\\U$ ($\\overline{\\U^2}=\\U$, if $D$ is continuous), then $\\tau_1 =0$. If $\\U$ has a bounded approximate identity, then by Cohen's factorization theorem we have $\\U^2=\\U$ and therefore in this case $\\tau_1 =0$. All unital Bnach algebras, $C^{*}$-algebras and group algebras have bounded approximate identity. Also if $\\U$ is a simple Banach algebra, then $\\U^{2}=\\U$.\n\\par \n The following corollary follows from Theorem \\ref{asll}.\n \\begin{cor}\\label{tak}\n Suppose that $\\delta_1:\\A\\rightarrow \\A$,  $\\delta_2:\\A\\rightarrow \\U$, $\\tau_1 :\\U\\rightarrow \\A$ and $\\tau_2 :\\U\\rightarrow \\U$ are linear  maps.\n \\begin{enumerate}\n \\item[(i)]\n $D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ defined by $D((a,x))=(\\delta_1(a),0)$ is a derivation if and only if $\\delta_1$ is a derivation and $\\delta_1 (\\A)\\subseteq ann_{\\A}\\U$. In this case if $\\delta_1 =id_{a_0}$ where $aa_0-a_0 a\\in ann_{\\A}\\U \\, \\, (a\\in \\A) $ and for any $x\\in \\U$, $a_0x =xa_0$, then $D$ is inner.\n \\item[(ii)]\n  $D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ with $D((a,x))=(0,\\delta_2(a))$ is a derivation if and only if $\\delta_2$ is a derivation and $\\delta_2 (A)\\subseteq ann_{\\U}\\U$. Moreover, if $\\delta_2 =id_{\\A, x_0}$ is inner, $ax_0 - x_0a\\in ann_{\\U}\\U$ (for all $a\\in \\A$) and $x_0\\in Z(\\U)$, then $D$ is inner.\n \\item[(iii)]\n$D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ with $D((a,x))=(\\tau_1(x),0)$ is a derivation if and only if $\\tau_1(xy)=0, x\\tau_1(y)+\\tau_1(x)y=0 (x,y\\in\\U)$. In this case $D$ is inner if and only if $\\tau_1 =0$.\n\\item[(iv)] $D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ with $D((a,x))=(0,\\tau_2(x))$ is a derivation if and only if $\\tau_2$ is a derivation and also an $\\A$-bimodule homomorphism. In this case $D$ is inner if and only if $\\tau_2(x)=r_{a_0}(x)+id_{\\U, x_0}(x)$ where $a_0\\in Z(\\A)$ and $ax_0 =x_0 a$ for all $a\\in \\A$.\n\\end{enumerate}\n \\end{cor}\nIn the following we give an example showing that the condition $\\tau_1 =0$ does not hold in general.\n\\begin{exm}\nLet $\\mathcal{B}$ be a Banach algebra and $\\A:=T(\\mathcal{B},\\mathcal{B})$ and let $\\U :=\\mathcal{B}$. The Banach algebra $\\U$ becomes a Banach $\\A$-bimodule by the following compatible module actions:\n\\[(a,b)x=ax,\\quad  x(a,b)= xa\\quad\\quad (a,b\\in \\A , x\\in \\U).\\]\nNow consider the Banach algebra $T(\\A,\\U)$ and define the map $\\tau_1:\\U\\rightarrow \\A$ by \n$$\\tau_1(x)=(0,x)\\quad\\quad (x\\in \\U)$$\nThen $\\tau_1\\neq 0$ is an $\\A$-bimodule homomorphism with $\\tau_1(\\U ^2)=(0)$ and for any $x,y\\in \\U$ we have \n$$x\\tau_1(y)+\\tau_1(x)y=x(0,y)+(0,x)y=0$$\nNow by Corollary \\ref{tak} it can be seen that the map $D$ on $T(\\A,\\U)$ defined by $D((a,b),x)=(\\tau_1(x),0)$ is a derivation in which $\\tau_1\\neq 0$.\n\\end{exm}\nIn this example $\\tau_1\\neq 0$ but $x\\tau_1(y)+\\tau_1(x)y=0$ $ (x,y\\in\\U)$. The next example shows that $\\tau_1$ does not satisfy the condition $x\\tau_1(y)+\\tau_1(x)y=0 \\,\\, (x,y\\in\\U)$ in general.\n\\begin{exm}\nLet $\\A$ be a Banach algebra, $\\mathcal{C}$ be a Banach $\\A$-bimodule and $\\gamma :\\mathcal{C}\\rightarrow \\A$ be a nonzero $\\A$-bimodule homomorphism such that \n\\[ c\\gamma (c')+\\gamma (c)c' =0\\]\nfor all $c,c'\\in\\mathcal{C}$.\n\\\\\nLet $\\U :=\\A\\times \\mathcal{C}$. With the usual actions, $\\U$ is a Banach $\\A$-bimodule. Consider the multiplication on $\\U$ given by\n\\[(x,y)(x',y')=(xx',0)\\quad\\quad ((x,y),(x',y')\\in\\U). \\]\nWith this multiplication $\\U$ is a Banach algebra such that the multiplication on $\\U$ is compatible with its module actions. Now we may consider the Banach algebra $\\A\\ltimes\\U$. Define the maps $\\tau_1:\\U\\rightarrow\\A$ and $\\tau_2:\\U\\rightarrow\\U$ by \n\\[\\tau_1 ((x,y))=\\gamma(y)\\quad , \\quad \\tau_2((x,y))=(-\\gamma (y),0).\\]\n$\\tau_1$ and $\\tau_2$ are $\\A$-bimodule homomorphisms such that $\\tau_1(\\U ^2)=0$, $\\tau_1\\,,\\tau_2\\neq 0$ and \n\\[0=\\tau_2((x,y)(x',y'))=(x,y)\\tau_1 ((x',y'))+(x,y)\\tau_2((x',y'))+\\tau_1 ((x,y)) (x',y')+\\tau_2((x,y))(x',y').\\]\nBy Theorem \\ref{asll} it can be seen that $D=(\\tau_1,\\tau_2)$ is a derivation on $\\A\\ltimes\\U$. If we assume further that $\\A$ is a unital algebra, then for any $y,y'\\in\\mathcal{C}$ with $\\gamma (y)\\neq 0$,\n\\begin{eqnarray*}\n(0,y)\\tau_1((1,y'))+\\tau_1((0,y))(1,y')&=&(0,y)\\gamma (y')+\\gamma (y)(1,y')\\\\&=&(\\gamma(y),y\\gamma (y')+\\gamma(y)y')\\\\&=&(\\gamma (y),0)\\neq 0.\n\\end{eqnarray*}\n \\end{exm}\nIn the following we investigate the automatic continuity of derivations on $\\A\\ltimes\\U$. It is clear that a derivation $D$ on $\\A\\ltimes \\U$ is continuous if and only if the maps $\\delta_1 , \\delta_2 , \\tau_1$ and $\\tau_2$ are continuous. \n\\par \n In the next theorem we state the relation between separating spaces of $\\delta_i$ and $\\tau_i$ ($i=1,2)$ and then by using this theorem, we obtain some results concerning the automatic continuity of the derivations on $\\A\\ltimes\\U$.\n\\begin{thm}\\label{joda}\nLet $D$ be a derivation on $\\A\\ltimes\\U$ such that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U),$$\nthen\n\\begin{enumerate}\n\\item[(i)]\n$\\mathfrak{S} (\\tau_1)$ is an ideal in $\\A$. If $\\tau_2$ is continuous, then $\\mathfrak{S} (\\tau_1)\\subseteq ann_{\\A}\\U$ and if $\\tau_2(\\U)\\subseteq ann_{\\U}\\U$, then $\\mathfrak{S} (\\tau_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\item[(ii)]\n$\\mathfrak{S} (\\tau_2)$ is an $\\A$-sub-bimodule of $\\U$ and if $\\tau_1$ is continuous or $\\tau_1 (\\U)\\subseteq ann_{\\A}\\U$, then $\\mathfrak{S} (\\tau_2)$ is an ideal in $\\U$.\n\\item[(iii)]\n$\\mathfrak{S} (\\delta_1)$ is an ideal in $\\A$ and if $\\delta_2$ is continuous or $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$, then $\\mathfrak{S} (\\delta_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\item[(iv)]\n$\\mathfrak{S} (\\delta_2)$ is an $\\A$-subbimodule of $\\U$ and if $\\delta_1$ is continuous or $\\delta_1 (\\A)\\subseteq ann_{\\A}\\U$, then $\\mathfrak{S} (\\delta_2)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\U$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n$(i)$ Let $a\\in \\mathfrak{S} (\\tau_1)$. Thus there is a sequence $\\{x_n\\}$ in $\\U$ such that \n$x_n\\rightarrow 0$ and $ \\tau_1(x_n)\\rightarrow a$.\nFor any $b\\in \\A$, $bx_n\\rightarrow 0$, so $\\tau_1(bx_n)=b\\tau_1 (x_n)\\rightarrow ba $ and hence $ba\\in \\mathfrak{S} (\\tau_1)$. Similarly, $ab\\in \\mathfrak{S} (\\tau_1)$. Therefore $\\mathfrak{S} (\\tau_1)$ is an ideal in $\\A$.\\\\\n Now for every $x\\in \\U$ we have  \n$$\\tau_2(xx_n)=x\\tau_1(x_n)+x\\tau_2(x_n)+\\tau_1(x)x_n+\\tau_2(x)x_n$$\nand\n$$\\tau_2(x_nx)=x_n\\tau_1(x)+x_n\\tau_2(x)+\\tau_1(x_n)x+\\tau_2(x_n)x.$$\nIf $\\tau_2$ is continuous, taking limit of the above equations gives $ax=xa=0$ ($x\\in \\U$). So $a\\in ann_{\\A}\\U$. Hence $\\mathfrak{S} (\\tau_1)\\subseteq ann_{\\A}\\U$. \n\\par \nIf $\\tau_2 (\\U)\\subseteq ann_{\\U}\\U$, by taking limit of the above equations we get \n$$\\tau_2(xx_n)\\rightarrow xa\\quad  \\text{and} \\quad \\tau_2(x_n x)\\rightarrow ax.$$ So $ax,xa\\in \\mathfrak{S} (\\tau_2)$ and hence $\\mathfrak{S} (\\tau_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\\\\n$(ii)$ If $x_n\\rightarrow 0$ and $\\tau_2(x_n)\\rightarrow x$, then for any $a\\in \\A$ we have \n$$\\tau_2(ax_n)=a\\tau_2(x_n)+\\delta_1(a)x_n+\\delta_2(a)x_n$$\nand\n$$\\tau_2(x_{n}a)=\\tau_2(x_n)a+x_n\\delta_1(a)+x_n\\delta_2(a).$$\nBy taking limits of these equations we get \n$$\\tau_2(ax_n)\\rightarrow ax\\quad\\text{and}\\quad \\tau_2(x_{n}a)\\rightarrow xa.$$\nSo $ax,xa\\in \\mathfrak{S} (\\tau_2)$ and hence $\\mathfrak{S} (\\tau_2)$ is an $\\A$-subbimodule of $\\U$.\n\\par \nFor any $y\\in \\U$ we have\n$$\\tau_2(yx_n)=\\tau_1(y)x_n+\\tau_2(y)x_n+y\\tau_1(x_n)+y\\tau_2(x_n)$$\n$$\\tau_2(x_n y)=\\tau_1(x_n)y+\\tau_2(x_n)y+x_n\\tau_1(y)+x_n\\tau_2(y).$$\nIf $\\tau_1$ is continuous or $\\tau_1(\\U)\\subseteq ann_{\\A} \\U$, by taking limit we obtain \n$$\\tau_2(yx_n)\\rightarrow yx\\quad \\text{and}\\quad \\tau_2(x_{n}y)\\rightarrow xy.$$\nTherefore $xy,yx\\in \\mathfrak{S} (\\tau_2)$ and $\\mathfrak{S} (\\tau_2)$ is an ideal.\n\\\\\n$(iii)$ Let $a_n\\rightarrow 0 $ and $\\delta_1 (a_n)\\rightarrow a$. Since $\\delta_1$ is a derivation, it follows that $\\mathfrak{S} (\\delta_1)$ is an ideal in $\\A$. For every $x\\in \\U$ we have \n$$\\tau_2(a_nx)=a_n\\tau_2(x)+\\delta_1(a_n)x+\\delta_2(a_n)x$$\nand\n$$\\tau_2(ax_n)=\\tau_2(x)a_n+x\\delta_1(a_n)+x\\delta_2(a_n).$$\nIf $\\delta_2$ is continuous or $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$, taking limit of the above equations yields \n$$\\tau_2(a_n x)\\rightarrow ax\\quad \\text{and}\\quad \\tau_2(x a_n)\\rightarrow xa.$$\nSince $xa_n, a_nx\\rightarrow 0$, it follows that $ax,xa\\in \\mathfrak{S} (\\tau_2)$. So $\\mathfrak{S} (\\delta_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\\\\n$(iv)$ The proof is similar to part $(iii)$.\n\\end{proof}\nNow we give some results of the preceding theorem.\n\\begin{prop}\\label{au1}\nLet $D$ be a derivation on $\\A\\ltimes\\U$ such that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U),$$\nthen \n\\begin{enumerate}\n\\item[(i)]\nif $ann_{\\A}\\U =(0)$, $\\tau_2$ and $\\delta_2$ are continuous, then $D$ is continuous.\n\\item[(ii)]\nif $ann_{\\U}\\U =(0)$, $\\tau_2$ and $\\delta_1$ are continuous, then $\\delta_2$ is continuous. If we also add the assumption of continuity of $\\tau_1$, then $D$ is continuous.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n$(i)$ Since $\\tau_2$ is continuous so $\\mathfrak{S} (\\tau_2)=0$. By Theorem \\ref{joda}-$(i)$ from the continuity of $\\tau_2$ we conclude that $\\mathfrak{S} (\\tau_1)\\subseteq ann_{\\A}\\U=(0)$, thus $\\tau_1$ is continuous. Moreover, since the conditions of Theorem \\ref{joda}-$(iii)$ hold, we have\n$$\\mathfrak{S} (\\delta_1)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_\\A=((0):\\U)_\\A =ann_{\\A}\\U=(0).$$ Therefore $\\delta_1$ is continuous. So $D$ is continuous.\n\\\\\n$(ii)$ Since the conditions of Theorem \\ref{joda}-$(iv)$ hold and $\\tau_2$ is continuous, it follows that\n $$\\mathfrak{S} (\\delta_2)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_\\U=((0):\\U)_\\U=ann_{\\U}\\U =(0).$$ So $\\delta_2$ is continuous . If $\\tau_1$ is continuous, then we have the continuity of all the maps and thus $D$ is continuous.\n\\end{proof}\nIf we add the condition $ann_{\\A}\\U=(0)$ to part $(ii)$ of the previous proposition, then as in part $(i)$, this implies that $\\tau_1$ is continuous and in this case $D$ is continuous as well.\n\\par \nNow by the preceding proposition, we obtain some results concerning the automatic continuity of the derivations on $\\A\\ltimes\\U$.\n\\begin{cor}\\label{n1}\nSuppose that for every derivation $D$ on $\\A\\ltimes \\U$ we have \n$x\\tau_1 (y)+\\tau_1(x)y=0$ $ (x,y\\in \\U)$. If $ann_{\\A}\\U=(0)$ and each derivation from $\\U$ to $\\U$ and each derivation from $\\A$ to $\\U$ is continuous, then every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{cor}\n\\begin{proof}\nBy Theorem \\ref{asll}, for any derivation $D=(\\delta_1 +\\tau_1,\\delta_2+\\tau_2)$ on $\\A\\ltimes \\U$, the maps $\\delta_2:\\A \\rightarrow\\U$ and $\\tau_2:\\U\\rightarrow \\U$ are derivations, since we have \n$x\\tau_1 (y)+\\tau_1(x)y=0$ $ (x,y\\in \\U)$.  So $\\delta_2$ and $\\tau_2$ are continuous by the hypothesis. Now the continuity of $D$ follows from Proposition \\ref{au1}-$(i)$.\n\\end{proof}\n\\begin{cor}\\label{n2}\nSuppose that for any derivation $D$ on $\\A\\ltimes \\U$ we have  $x\\tau_1 (y)+\\tau_1(x)y=0 $ $(x,y\\in \\U)$. If $ann_{\\U}\\U=(0)$, every derivation from $\\U$ to $\\U$ and every derivation from $\\A$ to $\\A$ is continuous and every $\\A$-bimodule homomorphism from $\\U$ to $\\A$ is continuous, then every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{cor}\n\\begin{proof}\nConsider the derivation $D=(\\delta_1 +\\tau_1,\\delta_2+\\tau_2)$. By Theorem \\ref{asll}, the maps \n $\\delta_1:\\A\\rightarrow \\A$ and  $\\tau_2:\\U\\rightarrow \\U$ are derivation and  $\\tau_1:\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism. By the hypothesis, $\\delta_1, \\tau_1$ and $\\tau_2$ are continuous. Therefore  $D$ is continuous by Proposition \\ref{au1}-$(ii)$.\n\\end{proof}\nBy Johnson's and Sinclair's theorem \\cite{john1}, every derivation on a semisimple Banach algebra is continuous, so we have the following corollary.\n\\begin{cor}\\label{semi}\nSuppose that $\\A$ and $\\U$ are semisimple Banach algebras such that $\\U$ has a bounded approximate identity. Then every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{cor}\n\\begin{proof}\nLet $D=(\\delta_1 +\\tau_1,\\delta_2+\\tau_2)$ be a derivation on $\\A\\ltimes \\U$. By the Cohen's factorization theorem, $\\U ^2 =\\U$ and so $\\tau_1=0$. Also from the hypothesis we conclude that $ann_{\\U}\\U =(0)$. By Johnson's and Sinclair's theorem \\cite{john1}, every derivation on $\\A$ and $\\U$ is continuous. Now by Proposition \\ref{au1}-$(ii)$, every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{proof}\nAll $C^{*}$-algebras, semigroup algebras, measure algebras and unital simple algebras are semisimple Banach algebras with bounded approximate identity. \n\\par \nIn next results we investigate some conditions under which we can express the derivation $D$ as the sum of two derivations one of which being continuous.\n\\begin{prop}\\label{taj1}\nLet $D$ be a derivation on $\\A\\ltimes\\U$ such that \n\\[D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U),\\]\nthen\n\\begin{enumerate}\n\\item[(i)]\nif $ann_{\\U}\\U =(0)$, $x\\tau_1 (y)+\\tau_1(x)y=0 $ $(x,y\\in \\U)$, $\\delta_1$ and $\\tau_2$ are continuous, then $D=D_1 +D_2$ in which \n$$D_1((a,x))=(\\delta_1(a),\\delta_2(a)+\\tau_2(x))\\quad\\text{and}\\quad D_2((a,x))=(\\tau_1(x),0),$$\nsuch that $D_1$ and $D_2$ are derivations on $\\A\\ltimes\\U$ and $D_1$ is continuous.\n\\item[(ii)]\nif $ann_{\\U}\\U =(0)$, $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$ and $\\tau_1, \\tau_2$ are continuous, then \\[D_1((a,x))=(\\tau_1(x),\\delta_2 (a)+\\tau_2 (x))\\] is a continuous derivation on $\\A\\ltimes \\U$ and $D=D_1 +D_2$ where $D_2((a,x))=(\\delta_1(a),0)$ is a derivation on $\\A\\ltimes \\U$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n$(i)$ By Proposition \\ref{au1}-$(ii)$, $\\delta_2$ is continuous. By Corollary \\ref{tak}, $D=D_1 +D_2$ where \n$$D_1((a,x))=(\\delta_1(a),\\delta_2(a)+\\tau_2(x)) \\quad \\text{and} \\quad D_2((a,x))=(\\tau_1(x),0)\\quad\\quad ((a,x)\\in \\A\\ltimes\\U)$$\nare derivations on $\\A\\ltimes \\U$. By the assumptions and that $\\delta_2$ is continuous, it follows that $D_1$ is a continuous derivation.\n\\\\\n$(ii)$ By Corollary \\ref{tak} \n$$D_1((a,x))=(\\tau_1(x),\\delta_2 (a)+\\tau_2 (x))\\quad\\text{and}\\quad D_2((a,x))=(\\delta_1(a),0)$$\nare derivations on $\\A\\ltimes \\U$. Now by the continuity of $\\tau_2$, $ann_{\\U}\\U =(0)$ and that $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$,  from Theorem \\ref{joda}-$(iv)$, it follows that \n$$\\mathfrak{S} (\\delta_2)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_\\U =((0):\\U)_\\U =ann_{\\U}\\U=(0).$$ \nTherefore $\\delta_2$ is continuous and hence so is $D_1$.\n\\end{proof}\nTo prove the next proposition we need the following lemma.\n\\begin{lem}(\\cite [Proposition 5.2.2]{da}).\\label{da}\nLet $\\mathcal{X}$, $\\mathcal{Y}$ and $\\mathcal{Z}$ be Banach spaces, and let $T:\\mathcal{X}\\rightarrow \\mathcal{Y}$ be linear.\n\\begin{enumerate}\n\\item[(i)] \nSuppose that $R:\\mathcal{Z}\\rightarrow \\mathcal{X}$ is a continuous surjective linear map. Then $ \\mathfrak{S} (TR)= \\mathfrak{S} (T). $\n\\item[(ii)]\n Suppose that $S:\\mathcal{Y}\\rightarrow \\mathcal{Z}$ is a continuous linear map. Then $ST$ is continuous if and only if $S(\\mathfrak{S} (T))=(0)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{prop}\\label{taj2}\nLet $D$ be a derivation on $\\A\\ltimes \\U$ such that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)$$\nand $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$. Then under any of the following conditions, \\[D_1((a,x))=(\\delta_1(a)+\\tau_1(x),\\tau_2(x))\\]\nis a continuous derivation on  $\\A\\ltimes \\U$ and $D=D_1+D_2$ where $D_2((a,x))=(0,\\delta_2(a))$ is a derivation on  $\\A\\ltimes \\U$.\n\\begin{enumerate}\n\\item[(i)]\n$ann_{\\A}\\U =(0)$ and $\\tau_2$ is continuous.\n\\item[(ii)]\n$\\A$ possesses a bounded right approximate identity, there is a surjective left $\\A$-module homomorphism  \n$\\phi :\\A\\rightarrow \\U$ and both $\\delta_1$ and $\\tau_1$ are continuous.\n\\item[(iii)]\n $\\A$ possesses a bounded right approximate identity, there is an injective left $\\A$-module homomorphism  \n$\\phi :\\A\\rightarrow \\U$ and both $\\tau_1$ and $\\tau_2$ are continuous.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n From Corollary \\ref{tak}-$(ii)$,\n\\[D_1((a,x))=(\\delta_1(a)+\\tau_1(x),\\tau_2(x)) \\quad \\text{and} \\quad\nD_2((a,x))=(0,\\delta_2(a))\\quad\\quad ((a,x))\\in \\A\\ltimes \\U) \\]\nare derivations on $\\A\\ltimes \\U$.\\\\\n$(i)$ Since $ann_{\\A}\\U =(0)$, as in Proposition \\ref{au1}-$(i)$, it can be proved that $\\tau_1$ is continuous. By continuity of $\\tau_2$, $ann_{\\A}\\U =(0)$, $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$ and Theorem \\ref{joda}-$(iii)$, we have \n$$\\mathfrak{S} (\\delta_1)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_{\\A}=((0),\\U)_{\\A}=ann_{\\A}\\U =(0).$$\nTherefore $\\delta_1$ is continuous and so is $D_1$.\\par \nBefore proving parts $(ii)$ and $(iii)$ we show that if $\\psi :\\A\\rightarrow \\U$ is any left $\\A$-module homomorphism, then it is continuous. Let $\\{a_k\\}$ be a sequence in $\\A$ such that $a_k\\rightarrow 0$. By the Cohen's factorization theorem there is some $c\\in \\A$ and some sequence $b_k$ in $\\A$ for which $b_k\\rightarrow 0$ and $a_k=b_k c \\, \\, (k\\in \\mathbb{N})$. Thus $\\psi (a_k)=b_k\\psi (c)\\rightarrow 0$ and hence $\\psi$ is continuous.\n\\\\\n$(ii)$ Let $\\psi :\\A\\rightarrow \\U$ be the map $\\psi = \\tau_2 o\\phi - \\phi o\\delta_1$. By the condition $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$ and the properties of $\\delta_1$ and $\\tau_2$ it can be shown that $\\psi$ is a left $\\A$-module homomorphism. Therefore $\\psi$ and $\\phi$ are continuous and hence $\\tau_2 o\\phi$ is continuous. So $\\mathfrak{S} (\\tau_2 o\\phi)=(0)$. On the other hand by Lemma \\ref{da}-$(i)$, since $\\phi$ is surjective, $\\mathfrak{S} (\\tau_2 o\\phi)=\\mathfrak{S} (\\tau_2)$. Thus $\\mathfrak{S} (\\tau_2)=0$. Therefore $\\tau_2$ is continuous. So by the hypothesis $D_1$ is continuous.\n\\\\\n$(iii)$\nAs in the previous part we put $\\psi = \\tau_2 o\\phi - \\phi o\\delta_1$ which is a left $\\A$-module homomorphism. Since $\\tau_2$ and $\\phi$ are continuous, it follows that $\\phi o\\delta_1 $ is continuous as well. By Lemma \\ref{da}-$(ii)$ we have $\\phi(\\mathfrak{S} (\\delta_1))=(0)$. Since $\\phi$ is injective, it follows that $\\mathfrak{S} (\\delta_1)=(0)$. So $\\delta_1$ is continuous and by the hypothesis $D_1$ is continuous as well.\n\\end{proof}\nNote that Propositions \\ref{taj2} also holds if \"bounded right approximate identity\" and \"left $\\A$-module homomorphism\" are replaced respectively by \"bounded left approximate identity\" and \"right $\\A$-module homomorphism\".\n\\begin{rem}\\label{taj11}\nIn Propositions \\ref{taj1} and \\ref{taj2} if we assume that $ann_{\\A}\\U =(0)$, then as in Proposition \\ref{au1}-$(i)$, the continuity of $\\tau_2$ implies the continuity of $\\tau_1$ or if we assume that $\\U$ has a bounded approximate identity, we conclude that $\\tau_1 =0$. Therefore any of the conditions $ann_{\\A}\\U =(0)$ or $\\U$ to have a bounded approximate identity implies that $\\tau_1$ is continuous and therefore so $D_1$ is continuous.\n\\end{rem}\n\\par \nIn the following example we show that the derivation $D_2$ in Propositions \\ref{taj1} and \\ref{taj2}, can be discontinuous.\n\\begin{exm}\nLet $\\mathcal{B}$ be a Banach algebra and $d:\\mathcal{B}\\rightarrow \\mathcal{B}$ be a discontinuous derivation.\n\\begin{enumerate}\n\\item[(i)]\nLet $\\A:=\\mathcal{B}\\times \\mathcal{B}$ be the direct product of Banach algebras and let $\\U :=\\mathcal{B}$ becomes a Banach $\\A$-bimodule with compatible actions, by the following module actions;\n\\[ \n(a,b)x=ax \\quad  \\text{and} \\quad \nx(a,b)= xa\\quad\\quad (a,b\\in \\A , x\\in \\U).\n\\]\nTherefore $(0)\\times \\mathcal{B}\\subseteq ann_{\\A}\\U$.\nDefine the map $\\delta_1: \\A\\rightarrow \\A$ by $\\delta_1((a,b))=(0,d(b))$. Then $\\delta_1$ is a discontinuous derivation on $\\A$ such that \n$$\\delta_1 (\\A)\\subseteq (0)\\times \\mathcal{B}\\subseteq ann_{\\A}\\U. $$\nSo the map $D_2: T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D_2((a,b),x)=(\\delta_1((a,b)),0)$ is discontinuous.\n\\item[(ii)]\nAssume that $\\U :=\\mathcal{B}\\times \\mathcal{B}$ as the direct product of Banach spaces which becomes a Banach algebra with thr product \n$$(x,y)(x',y')=(xx',0)\\quad \\quad ((x,y),(x',y')\\in \\U).$$\nLet $\\A:=\\mathcal{B}$ and $\\U$ is became a Banach $\\A$-bimodule with the pointwise module actions which are compatible  with its algebraic operations.\\par \nNow consider the map $\\delta_2: \\A\\rightarrow \\U$ defined by $\\delta_2(a)=(0,d(a))$. $\\delta_2$ is a discontinuous derivation such that \n$$\\delta_2(\\A)\\subseteq (0)\\times \\mathcal{B}\\subseteq ann_{\\U}\\U.$$\nHence the map $D_2: \\A \\ltimes \\U \\rightarrow \\A \\ltimes \\U$ defined by \n$$D_2((a,(x,y)))=(0,\\delta_2(a))$$\nis a discontinuous derivation on $\\A\\ltimes \\U$.\n\\item[(iii)]\nSuppose that $\\U$ is a Banach space and $T:\\U\\rightarrow \\mathbb{C}$ is a discontinuous linear functional. \nSet $\\A:=T(\\mathbb{C},\\mathbb{C})$ and we turn $\\U$ into a Banach $\\U$-bimodule by the actions below;\n\\[(a,b)x=ax \\quad \\text{and} \\quad x(a,b)=xb  \\quad\\quad ((a,b)\\in A, x\\in \\U).\\]\nConsider the Banach algebra $T(\\A,\\U)$ and the map $\\tau_1:\\U\\rightarrow \\A$ defined by \n$$\\tau_1(x)=(0,T(x)).$$\nSo $\\tau_1$ is an $\\A$-bimodule homomorphism such that \n$$x\\tau_1(x')+\\tau_1 (x)x'=0\\quad\\quad (x,x'\\in \\U).$$\nThus the map $D_2((a,b),x)=(\\tau_1(x),0)$ is a derivation on $T(\\A,\\U)$ which is discontinuous.\n\\end{enumerate}\n\\end{exm}\nNote that in Proposition \\ref{taj2} if we assume that $\\delta_2$ is continuous, then derivation $D$ is continuous as well.\n\n\\section{The first cohomology group of $\\A\\ltimes\\U$}\nIn this section we determine the first cohomology group of $\\A\\ltimes\\U$ in some special cases. Throughout this section  for two linear spaces $\\mathcal{X}$ and $\\mathcal{Y}$, we shall write $ \\mathcal{X} \\cong \\mathcal{Y}$ to indicate that the spaces are linearly isomorphic. \n\\par \nFrom this point up to the last section we assume that every derivation $D$ on $\\A\\ltimes\\U$ is of the form \n\\[D(a,x)=(\\delta_1 (a),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)\\]\nin which $\\delta_1:\\A\\rightarrow\\A$, $\\delta_2:\\A\\rightarrow\\U$ and $\\tau_2:\\U\\rightarrow\\U$\nare derivations such that \n\\[\n\\tau_2 (ax)=a\\tau_2 (x)+\\delta_1 (a)x+\\delta_2 (a)x \\,\\, \\text{and} \\,\\, \n\\tau_2 (xa)=\\tau_2 (x)a+x\\delta_1 (a)+x\\delta_2 (a) \\quad\\quad (a\\in \\A,x\\in \\U).\n\\]\nIndeed, we consider that for every derivation $D$ on $\\A\\ltimes\\U$ we have $\\tau_1=0$, where $\\tau_1$ is as in Theorem \\ref{asll}. \n\\par \nIf for all $\\delta\\in Z^{1}(\\A)$, $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$, then by Corollary \\ref{tak}, the map $D$ given by $D((a,x))=(\\delta(a),0)$ is a continuous derivation on $\\A\\ltimes\\U$ and for every continuous derivation $D$ on $\\A\\ltimes\\U$ the maps $D_1((a,x))=(0,\\delta_2 (a)+\\tau_2 (x))$ and $D_2((a,x))=(\\delta_1(a),0)$ are derivations in $Z^{1}(\\A\\ltimes\\U)$ and $D=D_1+D_2$. By these arguments, in this case,\n\\[Z^{1}(\\A\\ltimes\\U)\\cong \\mathcal{W} \\times Z^{1}(\\A),\\]\nwhere $\\mathcal{W}$ is a linear subspace consisting of all continuous derivations on $\\A\\ltimes\\U$ of the form \n$D(a,x)=(0,\\delta_2 (a)+\\tau_2 (x))$, such that $\\delta_2\\in Z^{1}(\\A,\\U)$, $\\tau_2\\in Z^{1}(\\U)$ and \n\\[\\tau_2 (ax)=a\\tau_2 (x)+\\delta_2 (a)x \\,\\, \\text{and} \\,\\,\\tau_2 (xa)=\\tau_2 (x)a+x\\delta_2 (a)\\quad\\quad (a\\in\\A,x\\in\\U).\\]\n\\begin{thm}\\label{11}\nLet for every derivation $\\delta\\in Z^{1}(\\A)$ we have $\\delta(\\A)\\subseteq ann_{\\A}\\U$. If $H^{1}(\\A,\\U)=(0)$, then \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]}{\\mathcal{E}},\\]\nwhere $\\mathcal{E}$ is the following linear subspace of $Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]$;\n\n\\[\\mathcal{E}=\\{(id_{a},r_{a}+id_{\\U,x})\\, \\mid \\, a\\in\\A, \\, id_{\\A , x}=0 \\, \\, (x\\in\\U)\\}.\\]\n\\end{thm}\n\\begin{proof}\nDefine the map \\[\\Phi:Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\nby $\\Phi ((\\delta,\\tau))=[D_{\\delta,\\tau}]$\nwhere $D_{\\delta,\\tau} ((a,x))=(\\delta (a),\\tau (x))$ and $[D_{\\delta,\\tau}]$ represents the equivalence class of $D_{\\delta,\\tau}$ in $H^{1}(\\A\\ltimes\\U).$ By Corollary \\ref{tak} the maps $D_1:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U$ and $D_2:\\A\\ltimes\\U\\rightarrow\\A\\ltimes\\U$ given respectively by $D_1((a,x))=(0,\\tau (x))$ and $D_2((a,x))=(\\delta(a),0)$, are continuous derivations on $\\A\\ltimes\\U$ and so $\\Phi$ is well-defined. Clearly $\\Phi$ is linear. Let $D\\in Z^{1}(\\A\\ltimes\\U)$. By the hypothesis and the discussion before the theorem, $D=D_1+D_2$ where  $D_1((a,x))=(\\delta_1(a),0)$ and $D_2((a,x))=(\\delta_1(a),0)$ are continuous derivations on $\\A\\ltimes\\U$ and $\\delta_1\\in Z^{1}(\\A)$, $\\delta_2\\in Z^{1}(\\A,\\U)$ and $\\tau_2\\in Z^{1}(\\U)$. Since $H^{1}(\\A,\\U)=(0)$, there is some $x_0\\in \\U$ such that $\\delta_2 =id_{\\A,x_0}$. Define the map $\\tau:\\U\\rightarrow \\U $ by $\\tau=\\tau_2 - id_{\\U,x_0}$. By the properties of $\\tau$ and Remark \\ref{inn2}, it follows that $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. Now we have\n\\[\nD((a,x))-D_{\\delta_1 ,\\tau} ((a,x))=(0,id_{\\A,x_0}(a)+id_{\\U,x_0}(x))= id_{(0,x_0)}(a,x) \\quad \\quad ((a,x)\\in\\A\\ltimes\\U).\\]\nSo $[D]=[D_{\\delta,\\tau}]$ and hence $\\Phi ((\\delta_1,\\tau))=[D_{\\delta_1 ,\\tau}]=[D]$. Thus $\\Phi$ is surjective.\n\\par\nIt can be easily checked that $\\mathcal{E}$ is a linear subspace and if $(\\delta,\\tau)\\in ker \\Phi$, then $D_{\\delta,\\tau}$ is an inner derivation on $\\A\\ltimes\\U$. So for some $(a_0,x_0)\\in \\A\\ltimes\\U$, \n\\[D_{\\delta,\\tau}((a,x))=id_{a_0,x_0}(a,x)=(id_{a_0}(a),id_{\\A,x_0}(a)+r_{a_0}(x)+id_{\\U,x_0}(x)),\\]\nwhich implies that \\[\\delta =id_{a_0}, \\quad \\tau (x)=r_{a_0}(x)+id_{\\U,x_0}(x)\\quad \\text {and} \\quad id_{\\A,x_0}=0 \\quad \\quad (a\\in\\A ,x\\in\\U).\\] Hence $(\\delta,\\tau)\\in \\mathcal{E}$. Conversely, Suppose that $(\\delta,\\tau)\\in \\mathcal{E}$. Then for $a_0\\in\\A$ and $x_0\\in\\U$ we have $\\delta =id_{a_0}$ and $ \\tau =r_{a_0}+id_{\\U,x_0}$ where $id_{\\A,x_0}=0$ for all $a\\in\\A$. So \n\\[\nD_{\\delta,\\tau}((a,x))=(id_{a_0}(a),r_{a_0}(x)+id_{\\U ,x_0}(x))\n= id_{(a_0,x_0)}((a,x)).\\]\nThus $D_{\\delta,\\tau}\\in N^{1}(\\A\\ltimes\\U)$ and hence $(\\delta,\\tau)\\in ker\\Phi$. So $\\mathcal{E}=ker\\Phi$. Therefore by the above arguments we have \n\\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]}{\\mathcal{E}}.\\]\n\\end{proof}\n\\begin{cor}\nSuppose that for every $\\delta\\in Z^{1}(\\A)$, $\\delta (\\A)\\subseteq ann_{\\A}\\U$. If $H^{1}(\\A,\\U)=(0)$ and  $H^{1}(\\A\\ltimes\\U)=(0)$, then $H^{1}(\\A)=(0)$ and $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)+I(\\U)}=(0).$\n\\end{cor}\n\\begin{proof}\nLet $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$ and $\\delta\\in Z^{1}(\\A)$. Since $\\delta (\\A)\\subseteq ann_{\\A}\\U$, it follows that \n\\[ (0,\\tau)\\,,\\, (\\delta,0)\\in Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)].\\]\nBy the hypothesis and according to the preceding theorem, there exist some $a_0\\in\\A$ and $x_0\\in\\U$ such that $id_{\\A,x_0}=0$ and \\[(\\delta,0)=(id_{a_0},r_{a_0}+id_{\\U,x_0})\\quad ,\\quad (0,\\tau)=(id_{a_0},r_{a_0}+id_{\\U,x_0}).\\]\nHence $\\delta=id_{a_0}$ and so $H^{1}(\\A)=(0)$. Moreover, $\\tau=r_{a_0}+id_{\\U,x_0}$ where $id_{a_0}=0$. Thus $\\tau\\in C_{\\A}(\\U)+I(\\U)$.\n \\end{proof}\n If we assume that $\\A$ is a Banach algebra with $ann_{\\A}\\A=(0)$ and $\\delta\\in Z^{1}(\\A)$ where $\\delta\\neq 0$, then $\\delta(\\A)\\not\\subseteq ann_{\\A}\\A =(0)$. So in this case, the condition  $\\delta (\\A)\\subseteq ann_{\\A}\\A$ is not stisfied on $\\A\\ltimes\\A$ for every derivation $\\delta\\in Z^{1}(\\A)$ and this shows that this condition does not hold in general. In the following we give an example of Banach algebras which are satisfying in conditions of Theorem \\ref{11}.\n \\begin{exm}\nLet $\\A$ be a semisimple commutative Banach algebra. By Thomas' theorem \\cite{tho}, we have $Z^{1}(\\A)=(0)$. So in this case, for every $\\delta\\in Z^{1}(\\A)$ and every Banach $\\A$-bimodule $\\U$, $\\delta (\\A)\\subseteq ann_{\\A}\\U$ (in fact $\\delta=0$). In particular for a semisimple commutative Banach algebra $\\A$, if $\\U$ is a Banach $\\A$-bimodule such that $H^{1}(\\A,\\U)=(0)$ and $\\A\\ltimes\\U$ is satisfying in conditions of Theorem \\ref{11}, then\n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{R_{\\A}(\\U)+ I(\\U)}.\\]\n \\end{exm}\n Note that if $\\A$ is a commutative Banach algebra with $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$, then above example holds again.\n \\par \nWe continue by characterizing the first cohomology group of $\\A\\ltimes\\U$ in another case.\n\\par \nIf for each $\\delta_2\\in Z^{1}(\\A,\\U)$, $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$, then by Corollary \\ref{tak}, the map $D ((a,x))=(0,\\delta_2(a))$ is a continuous derivation on $\\A\\ltimes\\U$ and for each derivation $D\\in Z^{1}(\\A\\ltimes\\U)$ we conclude that the maps $D_1 ((a,x))=(\\delta_1 (a),\\tau_2(x))$ and $D_2((a,x))=(0,\\delta_2(a))$ are derivations in $Z^{1}(\\A\\ltimes\\U)$ and $D=D_1+D_2$. So by this argument in this case,\n \\[Z^{1}(\\A\\ltimes\\U)\\cong Z^{1}(\\A ,\\U)\\times \\mathcal{W}\\]\n where $\\mathcal{W}$ is the linear subspace of all continuous derivations on $\\A\\ltimes\\U$ which are of the  form $D_1((a,x))=(\\delta_1 (a),\\tau_2(x))$ with $\\delta_1\\in Z^{1}(\\A)\\,,\\tau_2\\in Z^{1}(\\U)$ and \n \\[\\tau_2 (ax)=a\\tau_2 (x)+\\delta_1 (a)x\\quad \\text{and} \\quad \\tau_2 (xa)=\\tau_2 (x)a+x\\delta_1 (a)\\quad\\quad ((a,x)\\in \\A\\ltimes\\U).\\]\n  \\begin{thm}\\label{22}\n Suppose that for each derivation $\\delta\\in Z^{1}(\\A,\\U)$ we have $\\delta(\\A)\\subseteq ann_{\\U}\\U$. If $H^{1}(\\A)=(0)$, then \n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A,\\U)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]}{\\mathcal{F}},\\]\n where $\\mathcal{F}$ is the following linear subspace of $Z^{1}(\\A,\\U)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]$;\n \\[\\mathcal{F}=\\{ (id_{\\A,x},r_{a}+id_{\\U, x})\\, \\mid \\, x\\in\\U, \\, id_{a}=0 \\, \\, (a\\in\\A) \\}.\\]\n \\end{thm}\n \\begin{proof}\n Define the map \n \\[\\Phi: Z^{1}(\\A,\\U)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\n by $\\Phi ((\\delta,\\tau))=[D_{\\delta,\\tau}]$\n where $D_{\\delta,\\tau}((a,x))=(0,\\delta(a)+\\tau(x))$\n is a continuous derivation on $\\A\\ltimes\\U$. Clearly $\\Phi$ is a well-defined linear map. If \n $D\\in Z^{1}(\\A\\ltimes\\U)$, then $D=D_1+D_2$ where $D_1((a,x))=(\\delta_1 (a),\\tau_2(x))$ and $D_2((a,x))=(0,\\delta_2(a))$ are continuous derivations on $\\A\\ltimes\\U$ with $\\delta_1\\in Z^{1}(\\A), \\delta_2\\in Z^{1}(\\A,\\U)$ and $\\tau\\in Z^{1}(\\U)$. Since $H^{1}(\\A)=(0)$, there is some $a_0\\in \\A$ such that $\\delta_1=id_{a_0}$. Consider the map $\\tau=\\tau_2-r_{a_0}$ on $\\U$. We have  \n $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. Also\n \\[D((a,x))-D_{\\delta_2 ,\\tau}((a,x))=id_{(a_0,0)}((a,x)).\\]\n So $[D]=[D_{\\delta_2 ,\\tau}]$ and hence $\\Phi ((\\delta_2,\\tau))=[D_{\\delta_2,\\tau}]=[D]$.\n Therefore $\\Phi$ is surjective. A straightforward proceeding shows that $\\mathcal{F}$ is a vector subspace and $ker\\Phi =\\mathcal{F}$. This establishes the desired vector space isomorphism.  \n \\end{proof}\n  \\begin{cor}\\label{222}\n  Suppose that for any derivation $\\delta\\in Z^{1}(\\A,\\U)$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$. If \n  $H^{1}(\\A)=(0)$ and $H^{1}(\\A\\ltimes\\U)=(0)$, then $H^{1}(\\A,\\U)=(0)$ and \n  $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)+I(\\U)}=(0).$\n  \\end{cor}\n  \\begin{proof}\n  Let $\\delta\\in Z^{1}(\\A,\\U)$ and $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. By the hypothesis, \n$(\\delta,0),(0,\\tau)\\in  Z^{1}(\\A,\\U)\\times Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. Again by the hypothesis and the preceding theorem, there are $x_0\\in\\U$ and $a_0\\in \\A$ such that $id_{a_0}=0$ and \n\\[(\\delta,0)=(id_{\\A,x_0},r_{a_0}+id_{\\U,x_0})\\quad , \\quad (0,\\tau)=(id_{\\A,x_0},r_{a_0}+id_{\\U,x_0}).\\]\nNow the result follows from these equalities.\n  \\end{proof}\nIf we assume that $\\A$ is a Banach algebra with $ann_{\\A}\\A =(0)$ and $\\delta\\in Z^{1}(\\A)$ where $\\delta\\neq 0$, then by putting $\\U =\\A$ we have $\\delta(\\A)\\not\\subseteq ann_{\\A}\\A =ann_{\\A}\\U =(0)$. So in this case, the condition $\\delta(\\A)\\subseteq ann_{\\A}\\U$ is not satisfied on $\\A\\ltimes\\U$ for every derivation $\\delta\\in Z^{1}(\\A,\\U)$. This shows that this condition does not hold in general. In the following we give an example of Banach algebras which are satisfying in conditions of Theorem \\ref{22}.\n\\begin{exm}\nIf $\\A$ is a Banach algebra and $\\U$ is a commutative Banach $\\A$-bimodule such that $H^{1}(\\A,\\U)=(0)$, then $Z^{1}(\\A,\\U)=(0)$. So in this case, for every $\\delta\\in Z^{1}(\\A,\\U)$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$ (in fact $\\delta =0$). In this case if $H^{1}(\\A)=(0)$ and $\\A\\ltimes\\U$ satisfies in Theorem \\ref{22}, then\n  \\[ H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)\\cap N^{1}(\\U)}.\\]\n\\end{exm}\n Note that if $\\A$ is a super amenable Banach algebra and $\\U$ is a commutative Banach $\\A$-bimodule, then above example holds.\n \\par \n In the continuation we consider a case on $\\A\\ltimes\\U$ in which for any derivation $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$ we have $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$ and $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$.  In fact by these conditions on $\\A\\ltimes\\U$, for every $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U);$\n  \\[\\delta_1(a)x+\\delta_2(a)x=0\\quad \\text {and} \\quad x\\delta_1 (a)+x\\delta_2 (a)=0 \\quad\\quad (a\\in \\A,x\\in \\U).\\]\nIn this case, every derivation $D\\in Z^{1}(A\\ltimes\\U)$ can be written as $D=D_1+D_2$ where \n  \\[D_{1}((a,x))=(\\delta_1 (a),\\delta_2(a))\\quad \\text{and} \\quad D_{2}((a,x))=(0,\\tau_2(x))\\]\nare continuous derivations on $\\A\\ltimes\\U$ and $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. In this case, we conclude that \n$$R_{\\A}(\\U)+N^{1}(\\U)\\subseteq Hom_{\\A}(\\U)\\cap Z^{1}(\\U).$$  \n\\begin{thm}\\label{33}\n Suppose that for every $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$ we have \\[\\delta_1(\\A)\\subseteq ann_{\\A}\\U \\quad \\text{and} \\quad \\delta_{2}(\\A)\\subseteq ann _{\\U}\\U.\\]\nSuppose further that $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{R_{\\A}(\\U)+ N^{1}(\\U)}=(0)$. Then \n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A)\\times Z^{1}(\\A,\\U)}{\\mathcal{K}},\\]\n where $\\mathcal{K}$ is the following linear subspace of $Z^{1}(\\A)\\times Z^{1}(\\A,\\U)$;\n \\[\\mathcal{K}=\\{(id_{a},id_{\\A,x})\\, \\mid \\,  r_{a}+id_{\\U,x}=0\\,\\,(a\\in \\A,x\\in \\U)\\}.\\]\n \\end{thm}\n  \\begin{proof}\n Define the map \n \\[\\Phi: Z^{1}(\\A)\\times Z^{1}(\\A,\\U)\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\n by $\\Phi ((\\delta_1,\\delta_2))=[D_{\\delta_1 , \\delta_2}]$\n where $D_{\\delta_1 , \\delta_2} ((a,x))=(\\delta_1 (a),\\delta_2(a))$ is a continuous derivation on $\\A\\ltimes\\U$. The map $\\Phi$ is well-defined and linear. If $D\\in Z^{1}(\\A\\ltimes\\U)$, then $D((a,x))=(\\delta_1 (a),\\delta_2(a)+\\tau_2(x))$. By the hypothesis, $\\tau_2=r_{a}+id_{\\U,x}$ where $a\\in\\A$ and $x\\in\\U$. Define the derivations $d_1\\in Z^{1}(\\A)$ and $d_2\\in Z^{1}(\\A,\\U)$ by $d_1=\\delta_1 -id_{a}$ and $d_2=\\delta_2 -id_{\\A,x}$ respectively. Then $D-D_{d_1 ,d_2}=id_{(a ,x )}$. So $\\Phi ((\\delta_1,\\delta_2))=[D_{d_1 , d_2}]=[D]$.\nThus $\\Phi$ is surjective. It can be easily seen that $ker \\Phi =\\mathcal{K}$. So we have the desired vector spaces isomorphism.\n \\end{proof}\n \\begin{cor}\\label{333}\n Suppose that for every $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$ we have \\[\\delta_1(\\A)\\subseteq ann_{\\A}\\U \\quad \\text{and} \\quad \\delta_2(\\A)\\subseteq ann _{\\U}\\U.\\] Suppose further that \n  $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{{R_{\\A}(\\U)+ N^{1}(\\U)}}=(0)$. If $H^{1}(\\A\\ltimes\\U)=(0)$, then $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$.\n  \\end{cor}\n  \\begin{proof}\n  Let $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$. By the hypothesis \n  \\[(\\delta_1 , 0)\\,,\\, (0,\\delta_2)\\in Z^{1}(\\A)\\times Z^{1}(\\A,\\U).\\]\n Again by the hypothesis and the preceding theorem, there exists some $(id_{a},id_{\\A,x})\\in \\mathcal{K}$ such that \n  $(\\delta_1 , 0)=(id_{a},id_{\\A,x})$ and $(0,\\delta_2)=(id_{a},id_{\\A,x})$ and so $\\delta_1$ and $\\delta_2$ are inner.\n  \\end{proof} \nIf $H^{1}(\\U)=(0)$, since $R_{\\A}(\\U)+ N^{1}(\\U)\\subseteq Z^{1}(\\U)= N^{1}(\\U)$, It follows that \n $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{{R_{\\A}(\\U)+ N^{1}(\\U)}}=(0)$. So if $H^{1}(\\U)=(0)$, Theorem \\ref{33} and Corollary \\ref{333} hold again.  \n \\par \n Let $\\A$ be a Banach algebra with $ann_{\\A}\\A =(0)$ and $\\delta_1 ,\\delta_2\\in Z^{1}(\\A)$ such that $\\delta_1 +\\delta_2 \\neq 0$. If we put $\\tau =\\delta_1+\\delta_2$ and define linear map $D$ on $\\A\\ltimes\\A$ by \n \\[D((a,x))=(\\delta_1 (a),\\delta_2(a)+\\tau(x))\\quad\\quad ((a,x)\\in \\A\\ltimes\\A),\\]\n then $D\\in Z^{1}(\\A\\ltimes\\A )$. But for $a, x\\in\\A$, the equation $\\delta_1 (a)x+\\delta_2 (a)x=0$\n is not necessarily true. This example does not satisfy the conditions of Theorem \\ref{33}. \n \\par \nThe last case we consider is as follows.\n \\begin{thm}\\label{44}\n Let $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$. Then \n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)+I(\\U)}.\\]\n\\end{thm} \n\\begin{proof}\nDefine the map \n\\[\\Phi : Hom_{\\A}(\\U)\\cap Z^{1}(\\U)\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\nby $\\Phi(\\tau)=[D_{\\tau}]$ where $D_{\\tau}((a,x))=(0,\\tau (x))$ is a continuous derivation on $\\A\\ltimes\\U$. The map $\\Phi$ is well-defined linear. If $D\\in Z^{1}(\\A\\ltimes\\U )$, then $D=(\\delta_1 , \\delta_2 +\\tau_2)$. By hypotheses  $\\delta_1 =id _{a}$ and $\\delta_2 =id_{\\A, x}$ for some  $a\\in \\A$ and $x\\in \\U$. Define the map $\\tau:\\U\\rightarrow \\U$ by $\\tau=\\tau_2-r_{a}-id_{\\U, x}$. Then $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$ and so $D-D_{\\tau}=id_{(a , x)}$. Hence $\\Phi (\\tau)=[D_\\tau]=[D]$. Thus $\\Phi$ is surjective. By Corollary \\ref{tak}-$(iv)$, $ker \\Phi =C_{\\A}(\\U)+I(\\U)$. So the desired vector space isomorphism is established.\n\\end{proof}\nThe following corollary follows immediately from the preceding theorem.\n\\begin{cor}\nIf $H^{1}(\\A)=(0),H^{1}(\\A,\\U)=(0)$ and $H^{1}(\\A\\ltimes\\U)=(0)$, then for every $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$ there are some $r_{a}\\in C_{\\A}(\\U)$ and $id_{\\U , x}\\in I(\\U)$ such that $\\tau =r_{a}+id_{x}$.\n\\end{cor}\nIn the following an example of Banach algebras satisfying the conditions of Theorem \\ref{44}, is given.\n\\begin{exm}\nIf $\\A$ is a weakly amenable commutative Banach algebra, then for any commutative Banach $\\A$-bimodule $\\mathcal{X}$ we have $H^{1}(\\A,\\mathcal{X})=(0)$. So in this case, if $\\U$ is a commutative Banach $\\A$-bimodule satisfying the conditions of Theorem \\ref{44}, then $Z^{1}(\\A)=H^{1}(\\A)=(0)$ and $Z^{1}(\\A,\\U)=H^{1}(\\A,\\U)=(0)$ and hence \n\\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{R_{\\A}(\\U)+ N^{1}(\\U)}.\\]\n\\end{exm}\nThis example could be also derived from Theorem \\ref{22}.\n\\section{Applications}\nIn this section we investigate applications of the previous sections and give some examples.\n\\subsection*{Direct product of Banach algebras}\nLet $\\A$ and $\\U$ be Banach algebras. With trivial module actions $\\A\\U=\\U\\A =(0)$, as we saw in Example \\ref{dp}, $\\A\\ltimes\\U=\\A\\times\\U$ where $\\A\\times\\U$ is $l^{1}$-direct product of Banach algebras. In this case, $ann_{\\A}\\U =\\A$ and so for every derivation $\\delta:\\A\\rightarrow \\A$ we have $\\delta(\\A)\\subseteq ann_{\\A}\\U$. Also in this case, $R_{\\A}(\\U)=(0)$ and $Hom_{\\A}(\\U)=\\mathbb{B}(\\U)$. The following proposition follows from Theorem \\ref{asll}.\n\\begin{prop}\\label{dpd}\nLet $\\A$ and $\\U$ be Banach algebras and $D:\\A\\times\\U\\rightarrow\\A\\times\\U$ be a map. The following are equivalent.\n\\begin{enumerate}\n\\item[(i)]\n$D$ is a derivation.\n\\item[(ii)]\n\\[D((a,x))=(\\delta_1(a)+\\tau_1(x),\\delta_2(a)+\\tau_2(x))\\quad \\quad ((a,x)\\in\\A\\times\\U)\\]\nsuch that $\\delta_1:\\A\\rightarrow\\A,\\tau_2:\\U\\rightarrow\\U$ are derivations and $\\tau_1:\\U\\rightarrow\\A$ and $\\delta_2:\\A\\rightarrow\\U$ are linear maps satisfying the following conditions;\n\\[\\tau_1(\\U)\\subseteq ann_{\\A}\\A , \\,\\, \\delta_2(\\A)\\subseteq ann_{\\U}\\U, \\,\\,  \\tau_1(xy)=0, \\, \\text{and} \\,\\, \\delta_2(ab)=0 \\quad \\quad(a,b\\in\\A , x,y\\in \\U).\\]\n\\end{enumerate}\nMoreover, if $ann_{\\U}\\U=(0)$ or $\\A^{2}=\\A$ and $ann_{\\A}\\A=(0)$ or $\\U ^{2}=\\U$, then $\\delta_2=0$ and $\\tau_1=0$, respectively.\n\\end{prop}\nBy this proposition it is clear if $\\A$ or $\\U$ has a bounded approximate identity, then $\\delta_2=0$ and $\\tau_1=0$. Hence in this case every derivation $D$ on $\\A\\times\\U$ is of the form $D((a,x))=(\\delta(a),\\tau(x))$ where $\\delta$ and $\\tau$ are derivations on $\\A$ and $\\U$, respectively.\n\\begin{rem}\\label{d1}\nIf $\\A$ and $\\U$ are Banach algebras, then for any derivation $\\delta:\\A\\rightarrow\\A$ and $\\tau:\\U\\rightarrow\\U$ the maps $D_1$ and $D_2$ on $\\A\\times\\U$ given by \n\\[D_{1}((a,x))=(\\delta (a),0)\\quad \\text{and} \\quad D_2((a,x))=(0,\\tau (x)),\\]\nare derivations. According to this point, if every derivation on $\\A\\times\\U$ is continuous, then every derivation on $\\A$ and every derivation on $\\U$ is continuous.\n\\par \nAlso let  $\\A$ or $\\U$ have a bounded approximate identity. So by above arguments if every derivation on $\\A$ and every derivation on $\\U$ is continuous, then every derivation on $\\A\\times\\U$ is continuous.\n\\end{rem}\n\\begin{rem}\\label{d2}\nLet $\\A$ and $\\U$ be Banach algebras and $D\\in Z^{1}(\\A\\ltimes\\U)$. If $ann_{\\U}\\U =(0)$ or $\\overline{\\A^{2}}=\\A$ and $ann_{\\A}\\A =(0)$ or $\\overline{\\U ^{2}}=\\U$, then in the representation of  $D$ as in the preceding proposition, $\\delta_2=0$ and $\\tau_1=0$. So $D((a,x))=(\\delta(a),\\tau(x))$ where $\\delta \\in Z^{1}(\\A)$ and $\\tau\\in Z^{1}(\\U)$. In this case we can conclude\n\\[ Z^{1}(\\A\\times\\U)\\cong Z^{1}(\\A)\\times Z^{1}(\\U) \\quad \\text{and} \\quad N^{1}(\\A\\times\\U)\\cong N^{1}(\\A)\\times N^{1}(\\U).\\]\nMoreover\n\\[ H^{1}(\\A\\times\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U).\\]\nIn particular if $\\A$ or $\\U$ have a bounded approximate identity, then above observation holds.\n\\par \nIn the case of isomorphism of the first cohomology group by Theorem \\ref{11}, one can weaken the conditions. Indeed $\\A\\times\\U$ with $\\overline{\\U ^{2}}=\\U$ or $ann_{\\A}\\A =(0)$ satisfies the conditions of Theorem \\ref{11}. Since $Hom_{\\A}(\\U)=\\mathbb{B}(\\U)$ and $R_{\\A}(\\U)=(0)$, in this case  \n\\[H^{1}(\\A\\times\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U)\\]\n(In fact in this case, $\\mathcal{E}=N^{1}(\\A)\\times N^{1}(\\U)$).\n\\par \nIf $\\overline{\\A ^{2}}=\\A$ or $ann_{\\U}\\U =(0)$, by symmetry as above we obtain the same result again.\n\\end{rem}\nAs the next example confirms, it is not necessarily true that for any derivation on $\\A\\times\\U$, $\\tau_1 =0$ or $\\delta_1 =0$ in the decomposition of it. The example is given from \\cite{ess}.\n \\begin{exm}\n Let $\\A$ be a Banach algebra with $ann_{\\A}\\A\\neq 0$. Put $\\U:=ann_{\\A}\\A$. Then $\\U$ is a closed subalgebra of $\\A$. Define the map $D$ on $\\A\\times\\U$ by $D((a,x))=(x,0)$. Then $D$ is a derivation on $\\A\\times\\U$ such that in its representation the map $\\tau_1:\\U\\rightarrow\\A$ is given by $\\tau_1(x)=x$ with $\\tau_1\\neq 0$. \n \\end{exm}\nLet $\\A$ and $\\U$ be Banach algebras and $\\alpha:\\A\\rightarrow \\U$ be a continuous algebra homomorphism with $\\Vert \\alpha\\Vert \\leq 1$. Then the following module actions turn $\\U$ into a Banach $\\A$-bimodule with the compatible actions and norm;\n\\[\nax=\\alpha(a)x \\quad \\text{and} \\quad\nxa=x\\alpha(a)\\quad\\quad (a\\in \\A,x\\in \\U).\\]\nIn this case we can consider $\\A\\ltimes \\U$ with the multiplication given by \n$$(a,x)(b,y)=(ab,\\alpha(a) y+x\\alpha(b)+xy).$$\nWe denote this Banach algebra by $\\A\\ltimes_{\\alpha} \\U$ which is introduced in \\cite{bh}. In the next proposition we see that $\\A\\ltimes_{\\alpha} \\U$ is isomorphic as a Banach algebra to the direct product $\\A\\times \\U$.\n\\begin{prop}\\label{dal}\nLet $\\A$, $\\U$ be Banach algebras and $\\alpha:\\A\\rightarrow \\U$ be a continuous algebra homomorphism with $\\Vert \\alpha\\Vert \\leq 1$. Consider the Banach algebra $\\A\\ltimes_{\\alpha} \\U$ as above. Then $\\A\\ltimes_{\\alpha} \\U$ is isomorphic as a Banach algebra to  $\\A\\times \\U$.\n\\end{prop}\n\\begin{proof}\nDefine $\\theta:\\A\\times \\U\\rightarrow\\A\\ltimes_{\\alpha} \\U$ by $\\theta((a,x))=(a, x-\\alpha(a))$  $(a,x)\\in \\A\\times \\U$. The map $\\theta$ is linear and continuous. Also\n\\[\\theta((a,x)(b,y))=(a,x-\\alpha(a))(b,y-\\alpha(b))=(ab,xy-\\alpha(ab))=\\theta((a,x))\\theta((b,y)),\\]\nfor any $(a,x),(b,y)\\in \\A\\times \\U$. It is obvious that $\\theta$ is bijective. Hence $\\theta$ is a continuous algebra isomorphism.\n\\end{proof}\n\\begin{rem}\nLet $\\A$, $\\U$ and $\\alpha$ are as Proposition \\ref{dal}. By Remarks \\ref{d1}, \\ref{d2} and Proposition \\ref{dal} we have the following.\n\\par \nIf every derivation on $\\A\\ltimes_{\\alpha}\\U$ is continuous, then every derivation on $\\A$ and every derivation on $\\U$ is continuous. If  $\\A$ or $\\U$ have a bounded approximate identity and every derivation on $\\A$ and every derivation on $\\U$ is continuous, then every derivation on $\\A\\ltimes_{\\alpha}\\U$ is continuous. Also if $\\overline{\\U ^{2}}=\\U$ or $ann_{\\A}\\A =(0)$ ($\\overline{\\A ^{2}}=\\A$ or $ann_{\\U}\\U =(0)$), then $H^{1}(\\A\\ltimes_{\\alpha}\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U)$.\n\\end{rem}\nIf $\\U$ is a Banach algebra and $\\A$ is a closed subalgebra of it, then it is clear that the embedding map $i:\\A\\rightarrow\\U$ is continuous algebra homomorphism and hence we can conmsider $\\A\\ltimes_{i} \\U$ which is isomorphic as a Banach algebra to  $\\A\\times \\U$ by Proposition \\ref{dal}. So by above remark we have the following examples.\n\\begin{exm}\n\\begin{enumerate}\n\\item[(i)] If $\\A$ is a semisimle Banach algebra with a bounded approximate identity, then every derivation on $\\A\\ltimes_{i}\\A$ is continuous. \n\\item[(ii)] Consider the group algebra $L^{1}(G)$ as a closed ideal in the measure algebra $M(G)$. Every derivation on $M(G)$ and every derivation on $L^{1}(G)$ is continuous and $M(G)$ is unital. So every derivation on $L^{1}(G)\\ltimes_{i} M(G)$ is continuous.\n\\item[(iii)] Let $\\mathcal{H}$ be a Hilbert space and $\\mathcal{N}$ be a complete nest in $\\mathcal{H}$. The associated nest algebra $Alg\\mathcal{N}$ is a closed subalgebra of $\\mathbb{B}(\\mathcal{H})$ which is unital. By \\cite{chr} every derivation on $Alg\\mathcal{N}$ is  continuous. Also $\\mathbb{B}(\\mathcal{H})$ is a unital $C^{*}$-algebra. Hence every derivation on $Alg\\mathcal{N}\\ltimes_{i} Alg\\mathcal{N}$ and $Alg\\mathcal{N}\\ltimes_{i} \\mathbb{B}(\\mathcal{H})$ is continuous.\n\\end{enumerate}\n\\end{exm}\n\\begin{exm}\n\\begin{enumerate}\n\\item[(i)] Let $\\A$ be a weakly amenable commutative Banach algebra. Since $H^{1}(\\A)=(0)$ and $\\overline{\\A ^{2}}=\\A$, it follows that $H^{1}(\\A\\ltimes_{i}\\A)=(0)$.\n\\item[(ii)]\nSakai showed in \\cite{sa} that every continuous derivation on a $W^{*}$-algebra is inner. Every von Neumann algebra is a $W^{*}$-algebra which is unital. Let $\\A$ be a von Neumann algebra on a Hilbert space $\\mathcal{H}$. Hence $H^{1}(\\A\\ltimes_{i}\\A)=(0)$ and $H^{1}(\\A\\ltimes_{i}\\mathbb{B} (\\mathcal{H}))=(0)$.\n\\item[(iii)]\nLet $\\mathcal{H}$ be a Hilbert space and $\\mathcal{N}$ be a complete nest in $\\mathcal{H}$. In \\cite{chr}, Christensen proved that $H^{1}(Alg\\mathcal{N})=(0)$. Also $\\mathbb{B}(\\mathcal{H})$ is a von Neumann algebra. Hence  \\[ H^{1}(Alg\\mathcal{N}\\ltimes_{i} Alg\\mathcal{N})=(0) \\quad \\text{ and } \\quad H^{1}(Alg\\mathcal{N}\\ltimes_{i} \\mathbb{B}(\\mathcal{H}))=(0).\\]\n\\end{enumerate}\n\\end{exm}\n\\subsection*{Module extension Banach algebras}\nLet $\\A$ be a Banach algebra and $\\U$ be a Banach $\\A$-bimodule. With trivial product $\\U ^{2}=(0)$, as we saw in Example \\ref{me}, $\\A\\ltimes\\U$ is the same as module extension of $\\A$ by $\\U$, namely $T(\\A,\\U)$. In this case, $ann_{\\U}\\U =\\U$ and so for every derivation $\\delta:\\A\\rightarrow\\U$ we have $\\delta(\\A)\\subseteq ann_{\\U}\\U$. By these notes, Theorem \\ref{asll} and Corollary \\ref{tak}, we have the following proposition on derivations on $T(\\A,\\U)$.\n\\begin{prop}\\label{ttd}\n Let $D:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ be a linear map such that \n \\[D((a,x))=(\\delta_1(a)+\\tau_1(x),\\delta_2(a)+\\tau_2(x))\\quad\\quad ((a,x)\\in \\A\\ltimes\\U).\\]\n The following are equivalent.\n \\begin{enumerate}\n \\item[(i)]\n $D$ is a derivation. \n \\item[(ii)]\n $D=D_1+D_2$ where $D_1((a,x))=(\\delta_1(a)+\\tau_1(x),\\tau_2(x))$ and $D_{2}((a,x))=(0,\\delta_2(a))$ are derivations on $T(\\A,\\U)$.\n \\item[(iii)]\n $\\delta_1:\\A\\rightarrow \\A$, $\\delta_2:\\A\\rightarrow \\U$ are derivations, $\\tau_{2}:\\U\\rightarrow \\U$ is a linear map such that \n \\[\\tau_2 (ax)=a\\tau_2 (x)+\\delta_2 (a)x\\quad \\text{and} \\quad \\tau_2 (xa)=\\tau_2 (x)a+x\\delta_2(a)\\quad \\quad (a\\in\\A,x\\in\\U).\\] \n and $\\tau_1:\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism such that $x\\tau_1(y)+\\tau_1(x)y=0$ $ (x,y\\in\\U)$. \\\\\n Moreover, $D$ is an inner derivation if and only if $\\delta_1 ,\\delta_2$ are inner derivations, $\\tau_1 =0$ and if $\\delta_1 =ad_{a}$ and $ \\delta_2 =ad_{x}$, then $\\tau_2 =r_{a}$.\n \\end{enumerate}\n \\end{prop}\n  Proposition 2.2 of \\cite{med} is a consequence of this proposition. \n\\begin{rem} \\label{r0} \n By Proposition \\ref{ttd}, the linear map $\\delta:\\A\\rightarrow\\U$ is a derivation if and only if the linear map $D((a,x))=(0,\\delta (a))$ on $T(\\A,\\U)$ is a derivation. So any derivation $\\delta:\\A\\rightarrow\\U$ is continuous, if every derivation on $T(\\A,\\U)$ is continuous.\n \\end{rem} \n \\begin{prop}\\label{tau}\nSuppose that there are $\\A$-bimodule homomorphisms $\\phi:\\A\\rightarrow\\U$ and $\\psi:\\U\\rightarrow\\A$ such that $\\phi o\\psi = I_{\\U}$ ($I_\\U$ is the identity map on $\\U$). If every derivation on $T(\\A,\\U)$ is continuous, then every derivation on $\\A$ is continuous.\n\\end{prop}\n\\begin{proof}\nLet $\\delta$ be a derivation on $\\A$. Define the map $\\tau:\\U\\rightarrow\\U$ by $\\tau =\\phi o\\delta o\\psi$.\nThen for every $a\\in\\A,x\\in\\U$,\n\\[\\tau(ax)=a\\tau(x)+\\delta(a)x\\quad \\text{and} \\quad \\tau(xa)=\\tau(x)a+x\\delta(a).\\]\nSo the map $D:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D((a,x))=(\\delta(a),\\tau(x))$ is a derivation which is continuous by the hypothesis. Thus $\\delta$ is continuous.\n\\end{proof}\nIn the previous proposition the assumption of existence of $\\A$-bimodule homomorphisms $\\phi:\\A\\rightarrow\\U$ and $\\psi:\\U\\rightarrow\\A$ with $\\phi o \\psi =I_{\\U}$ is equivalent to that there exists a subbimodule $\\mathcal{V}$ of $\\A$ such that $\\A=\\U\\oplus \\mathcal{V}$ as $\\A$-bimodules direct sum. (In fact in this case, $\\U$ and $\\mathcal{V}$ are ideals of $\\A$). \n \\par \nSince for every derivation $\\delta:\\A\\rightarrow\\U$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$, in the stated cases in Proposition \\ref{taj2}, one can express any derivation $D$ on $T(\\A,\\U)$ as the sum of two derivations one of which being continuous. Also the Proposition \\ref{au1}-$(i)$ holds in the case of module extension Banach algebras. \n\\begin{rem}\\label{r1}\nIf $\\A$ is a Banach algebra and $\\mathcal{I}$ is a closed ideal on it, then $\\frac{\\A}{\\mathcal{I}}$ is a Banach $\\A$-bimodule and so we can consider $T(\\A, \\frac{\\A}{\\mathcal{I}})$. Suppose that $\\A$ possesses a bounded right (or left) approximate identity and every derivation on $\\A$ and every derivation from $\\A$ to $\\frac{\\A}{\\mathcal{I}}$ is continuous. Let $D$ be a derivation on $T(\\A, \\frac{\\A}{\\mathcal{I}})$ which has a structure as in Proposition \\ref{ttd}. Then $\\tau_1:\\frac{\\A}{\\mathcal{I}}\\rightarrow \\A$ is an $\\A$-bimodule homomorphism. Since $\\A$ has a bounded right(left) approximate identity, then so does $\\frac{\\A}{\\mathcal{I}}$. Hence $\\tau_1$ is continuous. Now from Proposition \\ref{taj2}-$(ii)$ it follows that $D$ is continuous. Hence in this case any derivation on $T(\\A, \\frac{\\A}{\\mathcal{I}})$ is continuous.\n\\end{rem}\n\\begin{rem}\\label{r2}\nIf $\\mathcal{I}$ is a closed ideal in a Banach algebra $\\A$ and $\\delta:\\A\\rightarrow\\A$ is a derivation such that $\\delta(\\mathcal{I})\\subseteq \\mathcal{I}$, then the map $\\tau: \\frac{\\A}{\\mathcal{I}}\\rightarrow \\frac{\\A}{\\mathcal{I}}$ defined by $\\tau(a+\\mathcal{I})=\\delta(a)+\\mathcal{I}$ is well-defined and linear and \n\\[\\tau (a(x+\\mathcal{I}))=a\\tau (x+\\mathcal{I})+\\delta (a)(x+\\mathcal{I})\\quad,\\quad \\tau((x+\\mathcal{I})a)=\\tau(x+\\mathcal{I})a+(x+\\mathcal{I})\\delta(a).\\]\n Therefore the map $D$ on  $T(\\A, \\frac{\\A}{\\mathcal{I}})$ defined by $D((a,x))=(\\delta (a),\\tau(x+\\mathcal{I}))$ is a derivation. So if every derivation on  $T(\\A, \\frac{\\A}{\\mathcal{I}})$ is continuous, then every derivation $\\delta:\\A\\rightarrow \\A $ with $\\delta(\\mathcal{I})\\subseteq \\mathcal{I}$ is continuous.\n\\end{rem}\n In Remarks \\ref{r1} and \\ref{r2}, if we let $\\mathcal{I}=(0)$, then we have the following corollary.\n \\begin{cor}\n  Let $\\A$ be a Banach algebra.\n\\begin{enumerate}\n\\item[(i)] If $\\A$ has a right (left) approximate identity and every derivation on $\\A$ is continuous, then any derivation on $T(\\A, \\A)$ is continuous. \n\\item[(ii)] If every derivation on $T(\\A, \\A)$ is continuous, then any derivation on $\\A$ is continuous.\n\\end{enumerate} \n \\end{cor}\nThe part (ii) of this corollary could also be derived from Remark \\ref{r0} or Proposition \\ref{tau}.\n\\par \nIn continue we give some results of Proposition \\ref{taj2} in the case of module extension Banach algebras.\n\\begin{cor}\\label{trs}\nLet $\\A$ be a semisimple Banach algebra which has a bounded approximate identity and $\\U$ be a Banach $\\A$-bimodule with $ann_{\\A}\\U=(0)$. Suppose that there exists a surjective left $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$ and every derivation from $\\A$ into $\\U$ is continuous, then every derivation on $T(\\A,\\U)$ is continuous.\n\\end{cor}\n\\begin{proof}\nLet $D$ be a derivation on $T(\\A,\\U)$ which has a structure as in Proposition \\ref{ttd}. Since $\\A$ is semisimple, every derivation from $\\A$ into $\\A$ is continuous. Now by Proposition \\ref{taj2}-$(ii)$ and Remark \\ref{taj11}, it follows that every derivation on $T(\\A,\\U)$ is continuous.\n\\end{proof}\nRingrose in \\cite{rin} proved that every derivation from a $C^{*}$-algebra into a Banach bimodule is continuous. So we have the following example which satisfies the conditions of the above corrolary.\n\\begin{exm}\nLet $\\A$ be a $C^{*}$-algebra and $\\U$ be a Banach $\\A$-bimodule with $ann_{\\A}\\U=(0)$. Suppose that there exists a surjective left $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$. Hence $\\A$ is a semisimple Banach algebra with a bounded approximate identity. So by \\cite{rin} and Corrolary \\ref{trs}, any derivation on $T(\\A,\\U)$ is continuous. \n\\end{exm}\nAn element $p$ in an algebra $\\A$ is called an \\textit{idempotent} if $p^{2}=p$.\n\\begin{cor}\nLet $\\A$ be a prime Banach algebra with a non-trivial idempotent\n$p$ (i.e. $p \\neq 0$) such that $\\A p$ is finite dimensional. Then every derivation\non $\\A$ is continuous.\n\\end{cor}\n\\begin{proof}\nLet $\\U:=\\A p$. Then $\\U$ is a closed left ideal in $\\A$. By the following make $\\U$ into a Banach\n$\\A$-bimodule:\n\\[ xa=0 \\quad \\quad (x\\in \\A p, a\\in \\A), \\]\nand the left multiplication is the usual multiplication of $\\A$. So we can consider $T(\\A,\\U)$ in this case. Since $\\A$ is prime, it follows that $ann_{\\A}\\U=(0)$. Let $\\delta:\\A \\rightarrow \\A$ be a derivation. Define the map $\\tau:\\U \\rightarrow \\U$ by $\\tau(ap)=\\delta(ap)p$ $(a\\in \\A)$. The map $\\tau$ is well-defined and linear. Also\n\\[\\tau (ax)=a\\tau (x)+\\delta (a)x\\quad \\text{and} \\quad \\tau (xa)=\\tau (x)a+x\\delta(a)\\quad \\quad (a\\in\\A,x\\in\\U).\\] \nSince $\\U$ is finite dimensional, it follows that $\\tau$ is continuous. By Proposition \\ref{ttd}, the mapping $D:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D((a,x))=(\\delta(a),\\tau(x))$ is a derivation. Now for $D$ the conditions of Proposition \\ref{taj2}-$(i)$ hold and hence $D$ is continuous. Therefore $\\delta$ is continuous.\n\\end{proof}\n Now we investigate the first cohomology group of $T(\\A ,\\U)$. \n \\par \n In module extension $T(\\A ,\\U)$ since $\\U ^{2}=(0)$, for every derivation $\\delta:\\A\\rightarrow\\U$ on $T(\\A ,\\U)$ we have $\\delta (\\A)\\subseteq ann_{\\U}\\U$ and also $Z^{1}(\\U)=\\mathbb{B}(\\U)$ and $N^{1}(\\U)=(0)$, we may conclude the following proposition from Theorem \\ref{22}. \n \\begin{prop}\\label{cte}\nConsider the module extension $T(\\A,\\U)$ of a Banach algebra $\\A$ and a Banach $\\A$-bimodule $\\U$. Suppose that $H^{1}(\\A)=(0)$ and the only continuous $\\A$-bimodule homomorphism $T:\\U\\rightarrow\\A$ satisfying $T(x)y+xT(y)=0$ is $T=0$. Then \n\\[H^{1}(T(\\A ,\\U))\\cong H^{1}(\\A ,\\U)\\times \\frac{Hom_{\\A}(\\U)}{C_{\\A}(\\U)}.\\]\n \\end{prop} \nIn fact with the assumptions of the previous proposition, in Theorem \\ref{22} we have $\\mathcal{F}=N^{1}(\\A)\\times C_{\\A}(\\U)$. Proposition \\ref{cte} is the same as Theorem 2.5 of \\cite{med}. So it can be said that Theorem \\ref{22} is a generalization of Theorem 2.5 in \\cite{med}. \n\\begin{rem}\n \\begin{enumerate}\n \\item[(i)]\n If $\\U$ is a closed ideal of a Banach algebra $\\A$ such that $\\overline{\\U^{2}}=\\U$, then for any continuous $\\A$-bimodule homomorphism $T:\\U\\rightarrow\\A$ with  $T(x)y+xT(y)=0\\, (x,y\\in\\U)$ we have $T(xy)=0$. Since $\\overline{\\U^{2}}=\\U$ and $T$ is continuous, it follows that $T=0$. Hence in this case, for any $\\delta\\in Z^{1}(T(\\A ,\\U))$ in its representation, $\\tau_1=0$. Also in this case, for any $r_a \\in C_{\\A}(\\U)$, since $a\\in Z(\\A)$, then $r_{a}=0$. Therefore if $H^{1}(\\A)=(0)$, in this case by Proposition \\ref{22} we have \n \\[H^{1}(T(\\A ,\\U))\\cong H^{1}(\\A,\\U)\\times Hom_{\\A}(\\U).\\]\n \\item[(ii)]\n If Banach algebra $\\A$ has no nonzero nilpotent elements, $\\U$ is a Banach $\\A$-bimodule and $T:\\U\\rightarrow\\A$ is an $\\A$-bimodule homomorphism such that  $T(x)y+xT(y)=0\\, (x,y\\in\\U)$, then $2T(x)T(y)=0\\, (x,y\\in \\U)$. Thus for any $x\\in \\U$, $T(x)^{2}=0$ and by the hypothesis $T(x)=0$. So in this case for any derivation on $T(\\A ,\\U)$, in its structure by Proposition \\ref{ttd} we have $\\tau_1 =0$.\n\\end{enumerate}\n \\end{rem} \n \\begin{exm}\n If $\\A$ is a weakly amenable commutative Banach algebra and $\\U$ is a commutative Banach $\\A$-bimodule such that for any continuous $\\A$-bimodule homomorphism $T:\\U\\rightarrow\\A$ with  $T(x)y+xT(y)=0\\, (x,y\\in\\U)$ implies that $T=0$, then $H^{1}(\\A)=(0)$, $H^{1}(\\A,\\U)=(0)$ and $C_{\\A}(\\U)=(0)$ and thus \n \\[H^{1}(T(\\A ,\\U))\\cong  Hom_{\\A}(\\U).\\]\n \\end{exm}\nVarious examples of the trivial extension of Banach algebras and computing their first cohomology group are given in \\cite{med}. \n\\par \nIf $\\delta\\in Z^{1}(\\A,\\U)$, then the map $D_{\\delta}:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D_{\\delta}((a,x))=(0,\\delta(a))$ is a continuous derivation. Now we may define the linear map $\\Phi :Z^{1}(\\A,\\U)\\rightarrow H^{1}(T(\\A ,\\U))$ by $\\Phi (\\delta)=[D_{\\delta}]$. By noting that $\\delta$ is inner if and only if $D_{\\delta}$ is inner (by Corollary \\ref{tak}), it can be seen that $ker\\Phi= N^{1}(T(\\A ,\\U))$. Thus $H^{1}(\\A ,\\U)$ is isomorphic to some subspace of  $H^{1}(T(\\A ,\\U))$. So we have the following corollary. \n \\begin{cor}\n Let $\\A$ be a Banach algebra and $\\U$ be a Banach $\\A$-bimodule. Then there is a linear isomorphism from  \n $H^{1}(\\A ,\\U)$ onto a subspace of $H^{1}(T(\\A ,\\U))$.\n \\end{cor}\n By the above corollary from $H^{1}(T(\\A ,\\U))=(0)$ we conclude that $H^{1}(\\A ,\\U)=(0)$. In particular, if $H^{1}(T(\\A ,\\A))=(0)$, then $H^{1}(\\A)=(0)$. We can also obtain this result from the fact that any derivation $\\delta:\\A\\rightarrow\\A$ gives rise to a derivation $D:T(\\A,\\A)\\rightarrow T(\\A,\\A)$ given by $D((a,x))=(\\delta(a),\\delta(x))$ (by Remark \\ref{r2}). So if $D$ is inner, then $\\delta$ is inner.\n\n \\subsection*{$\\theta$-Lau products of Banach algebras}  \nIn this subsection we assume that $0\\neq \\theta\\in \\Delta (\\A)$ and $\\U$ is a Banach algebra. By the module action given in Example \\ref{la} we turn $\\U$ into a Banach $\\A$-bimodule with comaptible actions and norm and if it is necessary we show this module by $ \\U_{\\theta}$. Note that $ann_{\\A}\\U_{\\theta} =ker \\theta$. Consider $\\A\\ltimes\\U$ and denote it by $\\A\\ltimes_{\\theta}\\U$ which is called $\\theta$-Lau product. In the continuation of this section we always consider $\\A\\ltimes_{\\theta}\\U$ as just mentioned. \n\\par \nThe following proposition characterizes the structure of derivations on $\\A\\ltimes_{\\theta}\\U$, which is obtained from Theorem \\ref{asll}.\n \\begin{prop}\\label{lau-der}\nLet $D:\\A\\ltimes_{\\theta} \\U\\rightarrow \\A \\ltimes_{\\theta}\\U$ be a map. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item[(i)] $D$ is a derivation.\n\\item[(ii)] \n\\[D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)\\]\nsuch that \n\\begin{enumerate}\n\\item[(a)]\n$\\delta_1 :\\A\\rightarrow\\A, \\delta_2 :\\A\\rightarrow\\U$ are derivations such that \n\\[\\theta (\\delta_1 (a))x+\\delta_2(a)x=0\\quad \\text{and}\\quad \\theta (\\delta_1 (a))x+x\\delta_2(a)=0\\quad (a\\in\\A,x\\in\\U).\\]\n\\item[(b)]\n$\\tau_1:\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism such that $\\tau_1(xy)=0\\quad (x,y\\in\\U)$.\n\\item[(c)]\n$\\tau_2:\\U\\rightarrow \\U$ is a linear map such that \n\\[\\tau_2(xy)=\\theta (\\tau_1(y))x+\\theta (\\tau_1(x))y+x\\tau_2(y)+\\tau_2(x)y\\quad \\quad (x,y\\in\\U).\\]\nAlso $D$ is inner if and only if $\\tau_1 =0, \\delta_2 =0, \\delta_1 =id_{a}$ and $\\tau_2 =id_{\\U,x}$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{prop}\nBy the above proposition, for a derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ we have \n\\[\\delta_2 (\\A)\\subseteq Z(\\U), \\quad \\theta (a)\\tau_1 (x) =a\\tau_1(x)=\\tau_1(x)a\\]\nand so $\\tau_1(\\U)\\subseteq Z(\\A)$. Also $x\\tau_1 (y)+\\tau_1 (x)y =0$ for all $x,y\\in \\U$ if and only if $\\tau_1 (\\U)\\subseteq ker \\theta$. Additionally, $\\delta_1 (\\A)\\subseteq ann_{\\A}\\U =ker\\theta$ if and only if $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$.\n\\begin{rem} \\label{cla}\nIf $\\A$ is a commutative Banach algebra, then by Thomas' theorem \\cite{tho}, for any derivation $\\delta:\\A\\rightarrow\\A$, $\\delta(\\A)\\subseteq rad (\\A)$. So in this case for any derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ we always have $\\delta_1 (\\A)\\subseteq rad (\\A) \\subseteq ker\\theta=ann_{\\A}\\U $ and hence $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$. Also $D=D_1+D_2+D_{3}$ where $D_1((a,x))=(\\delta_1(a),0)$, $D_{2}((a,x))=(0,\\delta_2(a))$ and $D_3((a,x))=(\\tau_1(x),\\tau_2(x))$ are derivations on $\\A\\ltimes_{\\theta}\\U$. In this case by Corollary \\ref{tak}-$(i)$, for every derivation $\\delta:\\A\\rightarrow\\A$ the map $D$ on $\\A\\ltimes_{\\theta}\\U$ defined by $D((a,x))=(\\delta(a),0)$ is a derivation.\n\\end{rem}\nIn the following we always assume that for any derivation $D\\in \\A\\ltimes_{\\theta}\\U$ we have $\\tau_1 =0$ and study the derivations under this condition. In this case, $\\tau_2$ is then always a derivation on $\\U$. \n\\par \nThe established results about the automatic continuity of derivations in section 3 obviously hold as well as in this special case $\\theta$-Lau product. Now we are ready to state some results concerning the automatic continuity of derivations on $\\A\\ltimes_{\\theta}\\U$.\n\\par \nBy the definition of $\\theta$-Lau product and Corollary \\ref{tak}-$(iv)$, any linear map $\\tau:\\U\\rightarrow\\U$ is a derivation if and only if the linear map $D((a,x))=(0,\\tau(x))$ on $\\A\\ltimes_{\\theta}\\U$ is a derivation. According to this point, the following corollary is clear.\n\\begin{cor}\nIf every derivation on $\\A\\ltimes_{\\theta}\\U$ is continuous, then every derivation on $\\U$ is continuous. In particular, if every derivation on $\\A\\ltimes_{\\theta}\\A$ is continuous, then every derivation on $\\A$ is continuous.\n\\end{cor}\nIf $\\A$ and $\\U$ are semisimple where $\\U$ has a bounded approximate identity, then by Proposition \\ref{semi} every derivatin on $\\A\\ltimes_{\\theta}\\U$ is continuous. In particular if $\\A$ is a semisimple Banach algebra with a bounded approximate identity, then all derivations on $\\A\\ltimes_{\\theta}\\A$ are continuous. In the case of $C^{*}$-algebras we can drop the semisimplicity of $\\U$. In fact by Ringrose's result \\cite{rin}, Proposition \\ref{lau-der} and preceding corollary we have the next corollary.\n\\begin{cor}\nLet $\\A$ be a $C^{*}$-algebra. Then every derivation on $\\A\\ltimes_{\\theta}\\U$ is continuous if and only if every derivation on $\\U$ is continuous.\n\\end{cor}\nFrom Proposition \\ref{taj1}-$(ii)$ and Remark \\ref{cla} we have the following corollary.\n\\begin{cor}\nLet $\\A$ be a commutative Banach algebra and $\\U$ a semisimple Banach algebra. Then every derivation $D$ on $ \\A\\ltimes_{\\theta}\\U $ is of the form $D=D_1+D_2$ where $D_1 ((a,x))=(0,\\delta_2 (a)+\\tau_2(x))$ is a continuous derivation on $\\A\\ltimes_{\\theta}\\U$ and $D_1 ((a,x))=(\\delta_1 (a),0)$ is a derivation on $\\A\\ltimes_{\\theta}\\U$. In particular, in this case, if every derivation on $\\A$ is continuous, then every derivation on $\\A\\ltimes_{\\theta}\\U$ is continuous.\n\\end{cor}\nNote that if $\\A$ is commutative, then $\\U$ is a commutative $\\A$-bimodule and hence the set of all left $\\A$-module homomorphisms, all right $\\A$-module homomorphisms and all $\\A$-module homomorphisms are the same. Now by Proposition \\ref{taj2} and Remark \\ref{cla} we obtain the next corollary.\n\\begin{cor}\nLet $\\A$ be a commutative Baanch algebra which has a bounded approximate identity. Then any derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ is of the form $D=D_1+D_2$ where $D_1 ((a,x))=(\\delta_1 (a),\\tau_2(x))$ and $D_2 ((a,x))=(0,\\delta_2(a))$ are derivations on $\\A\\ltimes_{\\theta}\\U$ and under any of the following conditions, $D_1$ is continuous.\n\\begin{enumerate}\n\\item[(i)]\nThere is a surjective $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$ and $\\delta_1$ is continuous.\n\\item[(ii)]\nThere is an injective $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$ and $\\tau_2$ is continuous.\n\\end{enumerate}\n\\end{cor}\nIn the continuation we assume that for any continuous derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ we have $\\tau_1 =0$. By the definition of $\\A\\ltimes_{\\theta}\\U$, in this case, $Hom_{\\A}(\\U)=\\mathbb{B}(\\U), N^{1}(\\A,\\U)=(0), Z^{1}(\\A,\\U)=H^{1}(\\A,\\U), R_{\\A}(\\U)=C_{\\A}(\\U)=(0)$ and $N^{1}(\\U)=I(\\U)$. \n\\begin{rem}\\label{pri}\nFor any derivation $\\delta\\in Z^{1}(\\A)$, we have $\\delta(\\A)\\subseteq ker \\theta=ann_{\\A}\\U $, since Sinclair's theorem \\cite{sinc} implies that $\\delta(P)\\subseteq P$ for any primitive ideal $P$ of $\\A$. So in this case for any derivation $D\\in Z^{1}(\\A\\ltimes_{\\theta}\\U)$ we always have $\\delta_1 (\\A)\\subseteq ker \\theta =ann_{\\A}\\U $ and hence $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$. Also $D=D_1+D_2+D_{3}$ where $D_1((a,x))=(\\delta_1(a),0)$, $D_{2}((a,x))=(0,\\delta_2(a))$ and $D_3((a,x))=(\\tau_1(x),\\tau_2(x))$ are in $Z^{1}(\\A\\ltimes_{\\theta}\\U)$. \n\\end{rem}\nNote that it is not necessarily true that $\\delta\\in Z^{1}(\\A,\\U)$ implies $\\delta(\\A)\\subseteq ann_{\\U}\\U$ as the next example shows this.\n\\begin{exm}\nAssume that $G$ is a non-discrete abelian group. In \\cite{br}, it has been shown that there is a nonzero continuous point derivation $d$ at a nonzero character $\\theta$ on $M(G)$. Now consider $M(G)\\ltimes_{\\theta} \\mathbb{C}$. Every derivation from $M(G)$ into $\\mathbb{C}_\\theta$ is a point derivation at $\\theta$. It is clear that $ann_{\\mathbb{C}}\\mathbb{C} =(0)$. But $d\\in Z^{1}(M(G),\\mathbb{C}_\\theta)$ is a nonzero derivation such that $d(M(G))\\not\\subseteq ann_{\\mathbb{C}}\\mathbb{C} =(0)$.\n\\end{exm}\n\\par \nNow we determine the first cohomology group of $\\A\\ltimes_{\\theta}\\U$ in some other cases.\n\\par \nLinear space $\\mathcal{E}$ in Theorem \\ref{11} in the case of $\\A\\ltimes_{\\theta}\\U$ is $\\mathcal{E}=N^{1}(\\A)\\times N^{1}(\\U)$. Therefore by theorem \\ref{11} and Remark \\ref{pri} we have the next proposition.\n\\begin{prop}\\label{a1}\nIf $H^{1}(\\A,\\U)=(0)$, then \n\\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U).\\]\n\\end{prop}\n\\begin{cor}\nLet $\\A$ be a Banach algebra with $H^{1}(\\A,\\A_{\\theta})=(0)$ and $\\overline{\\A^{2}}=\\A$. Then $H^{1}(\\A \\ltimes_{\\theta}\\A)=(0)$ if and only if $H^{1}(\\A)=(0)$.\n\\end{cor}\n\\par \nLinear space $\\mathcal{F}$ in Theorem \\ref{22} in the case of $\\A\\ltimes_{\\theta}\\U$ is $\\mathcal{F}=(0)\\times N^{1}(\\U)$. So we have the following proposition.\n\\begin{prop}\nLet for any $\\delta\\in Z^{1}(\\A,\\U)$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$. If $H^{1}(\\A)=(0)$, then \n\\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\A,\\U)\\times H^{1}(\\U).\\]\n\\end{prop}\n The linear space $\\mathcal{K}$ in Theorem \\ref{33} in the case of $\\A\\ltimes_{\\theta}\\U$ is $\\mathcal{K}=N^{1}(\\A)\\times (0)$. Therefore by Theorem \\ref{33} we have the next proposition.\n \\begin{prop}\\label{prop8}\nLet for every $\\delta\\in Z^{1}(\\A,\\U)$, $ \\delta(\\A)\\subseteq ann _{\\U}\\U$.\n If $H^{1}(\\U)=(0)$, then \n \\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\A)\\times H^{1}(\\A,\\U).\\]\n\\end{prop}\nThe following proposition follows directly from Theorem \\ref{44} and the properties of $\\A\\ltimes_{\\theta}\\U$.\n\\begin{prop}\\label{prop10}\nSuppose that $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$. Then\n\\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\U).\\] \n\\end{prop}\n\\begin{rem}\\label{akh}\nFrom Proposition \\ref{prop10} it follows that if $H^{1}(\\A)=(0), H^{1}(\\A,\\U)=(0)$ and $H^{1}(\\U)=(0)$, then $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$. Under some conditions the converse of is also true. That is, if for any $\\delta\\in Z^{1}(\\A,\\U)$ we have $\\delta (\\A)\\subseteq ann_{\\U}\\U$ and $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$, then $H^{1}(\\A)=(0), H^{1}(\\A,\\U)=(0)$ and $H^{1}(\\U)=(0)$. It does not yield from above propositions. We prove it directly as follows.  By the hypotheses, for any $\\delta_1\\in Z^{1}(\\A)$, $\\delta_2\\in Z^{1}(\\A,\\U)$ and $\\tau\\in Z^{1}(\\U)$ the map $D:\\A\\ltimes_{\\theta}\\U\\rightarrow \\A\\ltimes_{\\theta}\\U$ given by $D((a,x))=(\\delta_1(a),0)$, $D((a,x))=(0,\\delta_2(a))$ or $D((a,x))=(0,\\tau_2(x))$ is a continuous derivation. If $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$, then $\\delta_2=0$ and $\\delta_1, \\tau_2$ are inner.\n\\end{rem}\n \\begin{exm}\nLet $\\A$ be a weakly amenable commutative Banach algebra and $\\overline{\\U^{2}}=\\U$. So $Z^{1}(\\A)=(0)$, $Z^{1}(\\A,\\U)=(0)$ and from Remark \\ref{akh} we have $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$ if and only if $H^{1}(\\U)=(0)$.\n \\end{exm}\n In particular if $\\A$ is a weakly amenable commutative Banach algebra, above example implies that $H^{1}(\\A\\ltimes_{\\theta}\\A)=(0)$. Note that for a weakly amenable Banach algebra $\\A$ the equality $\\overline{\\A^{2}}=\\A$ always holds.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \\label{Sec_intro}\n\n  In this paper we present results on the longitudinal double spin\nasymmetry $A_1^{\\rho }$ for exclusive incoherent  $\\rho ^0$ production in the\nscattering of high energy muons on nucleons.\nThe experiment was carried out at CERN by the COMPASS collaboration\nusing the 160~GeV muon beam and the large $^{6}$LiD polarised target.\n\nThe studied reaction is \n\\begin{equation} \n \\mu  +  N  \\rightarrow  \\mu^{\\prime}  +  \\rho ^0  +  N^{\\prime},       \\label{murho}\n\\end{equation}\nwhere $N$ is a quasi-free nucleon from the polarised deuterons.\nThe reaction (\\ref{murho}) can be described in terms of the virtual photoproduction\nprocess\n\\begin{equation}\n \\gamma^{\\ast}  +  N  \\rightarrow  \\rho^0  +  N^{\\prime}.   \\label{phorho} \n\\end{equation}\nThe reaction (\\ref{phorho}) can be regarded as a fluctuation of the\nvirtual photon into a quark-antiquark pair (in partonic language),\nor an off-shell vector meson (in Vector Meson Dominance model), which then scatters\noff the target nucleon resulting in the production of an on-shell vector meson.\nAt high energies this is predominantly a diffractive process and plays\nan important role in the investigation of Pomeron exchange and its\ninterpretation in terms of multiple gluon exchange.\n\nMost of the presently available information on the spin structure \nof reaction (\\ref{phorho})\nstems from the $\\rho ^0$ spin density matrix elements,\nwhich are obtained from the analysis of angular distributions\nof  $\\rho ^0$ production and decay \\cite{Schilling73}. Experimental results\non $\\rho ^0$ spin density matrix elements \ncome from various experiments \\cite{NMC,E665-1,ZEUS,H1,HERMES00} including\nthe preliminary results from COMPASS \\cite{sandacz}.\n \nThe emerging picture of the spin structure of the considered process \nis the following. At low photon virtuality $Q^2$ the cross section by transverse virtual photons\n$\\sigma _T$\ndominates, while the relative contribution of the cross section by longitudinal\nphotons $\\sigma _L$ \nrapidly increases with $Q^2$. At $Q^2$ of about 2~(GeV\/{\\it c})$^2$ both\ncomponents become comparable and at a larger $Q^2$ the contribution of\n$\\sigma _L$ becomes dominant and continues to grow, although\nat lower rate than at low $Q^2$. \nApproximately, the so called $s$-channel helicity\nconservation (SCHC) is valid, i.e.\\ the helicity of the vector meson is the same\nas the helicity of the parent virtual photon. The data indicate that \nthe process can be described approximately by the \nexchange in the $t$-channel of an object with natural \nparity $P$.\nSmall deviations from SCHC are observed, also at the highest energies, whose\norigin is still to be understood. \nAn interesting suggestion was made in Ref.~\\cite{igivanov} that at high energies\nthe magnitudes of various helicity amplitudes for the reaction (\\ref{phorho})\nmay shed a light on the spin-orbital momentum structure of the vector meson. \n\nA complementary information can be obtained from measurements of the double\nspin cross section asymmetry, when the information on both the beam\nand target polarisation is used.\nThe asymmetry is defined as\n\\begin{equation}\n A_{1}^{\\rho}  = \n   \\frac{\n    \\sigma_{1\/2}  -  \\sigma_{3\/2}\n   }{\n    \\sigma_{1\/2}  +  \\sigma_{3\/2}}\n   ,\n   \\label{A1def}\n\\end{equation}\nwhere $\\sigma_{1\/2  (3\/2)}$ stands for the cross sections of the reaction (\\ref{phorho})\nand the subscripts denote the total virtual photon--nucleon angular momentum\ncomponent along the virtual photon\ndirection.\nIn the following we will also use the asymmetry\n\\ALL\\\nwhich is defined for reaction (\\ref{murho}) as the asymmetry of\nmuon--nucleon cross sections for antiparallel and parallel beam and\ntarget longitudinal spin orientations.\n\n\nIn the Regge approach \\cite{manaenkov} the longitudinal double spin asymmetry\n$A_1^{\\rho }$ can arise due \nto the interference of amplitudes for exchange in the $t$-channel of Reggeons with natural parity\n(Pomeron, $\\rho $, $\\omega $, $f$, $A_2$ ) with amplitudes for Reggeons with \nunnatural parity \n($\\pi , A_1$).\nNo significant asymmetry is expected \nwhen only a non-perturbative \nPomeron is exchanged because it has small spin-dependent couplings as found from \nhadron-nucleon data for cross sections and polarisations.\n\nSimilarly, in the approach of Fraas \\cite{fraas76}, \nassuming approximate validity of \nSCHC, the spin asymmetry $A_1^{\\rho }$ arises from the interference between\nparts of the helicity amplitudes for transverse photons corresponding\nto the natural and unnatural parity exchanges\nin the $t$ channel. While a measurable asymmetry can arise even from a small\ncontribution of the unnatural parity exchange, the latter may remain\nunmeasurable in the cross sections.\nA significant unnatural-parity contribution may indicate an exchange \nof certain Reggeons like $\\pi$, $A_{1}$ or in\npartonic terms an exchange of $q\\bar{q}$ pairs.\n\nIn the same reference a theoretical prediction for $A_1^{\\rho}$ was\npresented, which is based on the description of forward exclusive $\\rho^{0}$\nleptoproduction and inclusive inelastic lepton-nucleon\nscattering by the off-diagonal Generalised Vector Meson Dominance (GVMD) model,\napplied to the case of polarised lepton--nucleon scattering.\nAt the values of Bjorken variable $x < 0.2$, with additional assumptions\n\\cite{HERMES01}, \\Aor\\ can be related \nto the $A_{1}$ asymmetry for inclusive inelastic lepton scattering at the same\n\\xBj\\ as\n\\begin{equation}\n A_1^{\\rho }  =  \\frac{2 A_1}{1  +  (A_1)^2} .    \\label{A1-Fraas}\n\\end{equation} \nThis prediction is consistent with the HERMES results for both the proton\nand deuteron targets, although with rather large errors.\n\nIn perturbative QCD, there exists a general proof of factorisation \\cite{fact}\nfor exclusive vector meson production by longitudinal photons. \nIt allows a decomposition of the full amplitude for reaction (\\ref{phorho})\ninto three components:\na hard scattering amplitude for the exchange of quarks or gluons,\na distribution amplitude for the meson and \nthe non-perturbative description of the target nucleon in terms of\nthe generalised parton distributions (GPDs), which are related to the internal\nstructure\nof the nucleon.\nNo similar proof of factorisation exists for transverse virtual photons, and as a consequence\nthe interpretation of $A_{1}^{\\rho}$ in perturbative QCD is not possible\nat leading twist. However, a model including higher twist effects proposed\nby Martin et al. \\cite{mrt} describes the behaviour of both $\\sigma_{L}$\nas well as of $\\sigma _T$ reasonably well. An extension of this model by Ryskin\n\\cite{Ryskin} for the\nspin dependent cross sections allows to relate\n\\Aor\\ to the spin dependent GPDs of gluons and quarks in the nucleon.\nThe applicability of this model is limited to the range $Q^2 \\geq 4~(\\mathrm{GeV}\/c)^2$.\nMore recently another pQCD-inspired model involving GPDs has been proposed by Goloskokov and Kroll \n\\cite{krgo,gokr}. The non-leading twist asymmetry $A_{LL}$ results from the \ninterference between the dominant GPD $H_g$ and the helicity-dependent GPD\n$\\tilde{H}_g$. The asymmetry is estimated to be of the order \n$ k_T^2  \\tilde{H}_g \/ (Q^2  H_g  )$,\nwhere $k_T$ is the transverse momentum of the quark and the antiquark.\n\nUp to now little experimental information has been available on the\ndouble spin asymmetries for exclusive leptoproduction of vector mesons. \nThe first observation of a non-zero asymmetry $A_1^{\\rho }$ in polarised\nelectron--proton deep-inelastic scattering was reported by the HERMES experiment \n\\cite{HERMES01}.\nIn the deep inelastic region $(0.8 < Q^2 < 3~(\\mathrm{GeV}\/c)^2)$\nthe measured asymmetry is equal to 0.23 $\\pm$ 0.14 (stat) $\\pm$ 0.02 (syst)\n\\cite{HERMES03},\nwith little dependence on the kinematical variables. In contrast, for the `quasi-real photoproduction' data, with \n$\\langle Q^2\\rangle = 0.13~(\\mathrm{GeV}\/c)^2$, the asymmetry for the proton target is consistent with zero.\nOn the other hand the measured asymmetry $A_1^{\\rho }$ for the polarised deuteron target and the asymmetry $A_1^{\\phi }$ for exclusive production of $\\phi $\nmeson \neither on polarised protons or deuterons\n are consistent with zero both in the deep inelastic and in the \nquasi-real photoproduction regions\n \\cite{HERMES03}.\n\nThe HERMES result indicating a non-zero $A_1^{\\rho }$ for the proton\ntarget differs from the unpublished result of similar measurements by \nthe SMC experiment \\cite{ATrip-1} at comparable values of $Q^2$\\ but at\nabout three times higher values of the photon-nucleon centre of mass energy $W$, i.e.\\ at smaller \\xBj. The SMC measurements of \n\\ALL\\ \nin several bins of $Q^2$ are consistent with zero\nfor both proton and deuteron targets. \n\n\\section{The experimental set-up}\n \\label{Sec_exper}\n\nThe experiment \\cite{setup} was performed with\nthe high intensity positive muon beam from the CERN M2 beam line.\nThe $\\mu ^{+}$ beam intensity is $2 \\cdot 10^8$ per spill of 4.8 s with a cycle time of\n16.8~s. The average beam energy is 160~GeV and the momentum spread is $\\sigma _{p}\/p = 0.05$.\nThe momentum of each beam muon is measured upstream of the experimental area in a beam\nmomentum station consisting of several planes of scintillator strips or scintillating fibres\nwith a dipole magnet in between. The precision of the momentum determination is typically\n$\\Delta p\/p \\leq 0.003$. The $\\mu ^{+}$ beam is naturally polarised by the weak decays \nof the parent\nhadrons. The polarisation of the muon varies with its energy and the \naverage polarisation is $-0.76$.\n\nThe beam traverses the two cells of the polarised target, \neach 60~cm long, 3~cm in diameter and separated by 10~cm, which are placed one after the other. \nThe target cells are filled with $^{6}$LiD which is used as polarised deuteron target material\nand is longitudinally polarised by dynamic nuclear polarisation (DNP).\nThe two cells are polarised in opposite directions so that data from both spin directions\nare recorded at the same time. The typical values of polarisation are about 0.50.\nA mixture of liquid $^3$He\nand $^4$He, used to refrigerate the target, and a small amount of heavier nuclei are also\npresent in the target.  \nThe spin directions in the two target cells are reversed every 8 hours by rotating\nthe direction of the magnetic field in the target. \nIn this way fluxes and acceptances cancel in the calculation of spin asymmetries, provided that\nthe ratio of acceptances of the two cells remains unchanged after the reversal. \n\nThe COMPASS spectrometer is designed to reconstruct the scattered muons\nand the produced hadrons in wide momentum and angular ranges.\nIt is divided in two stages with two dipole magnets, SM1 and SM2. The first magnet, \nSM1, accepts charged particles of momenta larger than 0.4~GeV\/{\\it c}, and the second one, SM2, those\nlarger than 4~GeV\/{\\it c}. The angular acceptance of the spectrometer is limited by\nthe aperture of the polarised target magnet. For the upstream\nend of the target it is $\\pm 70$~mrad.\n\nTo match the expected particle flux at various locations in the spectrometer, COMPASS\nuses various tracking detectors. Small-angle tracking is provided by stations of\nscintillating fibres, silicon detectors, micromesh gaseous chambers and gas electron\nmultiplier chambers. Large-angle tracking devices are multiwire proportional chambers,\ndrift chambers and straw detectors. Muons are identified in large-area mini drift \ntubes and drift tubes placed downstream of hadron absorbers. Hadrons are detected by\ntwo large iron-scintillator sampling calorimeters installed in front of the absorbers\nand shielded to avoid electromagnetic contamination.\nThe identification of charged particles is possible with a RICH detector, although\nin this paper we have not utilised the information from the RICH.\n\nThe data recording system is activated by various triggers indicating the presence\nof a scattered muon and\/or an energy deposited by hadrons\nin the calorimeters. In addition to the inclusive trigger, in which the scattered muon\nis identified by coincidence signals in the trigger hodoscopes, several semi-inclusive\ntriggers were used. They select events fulfilling the requirement to detect\nthe scattered muon together with the energy deposited in the hadron calorimeters exceeding\na given threshold. In 2003 the acceptance was further extended towards high $Q^2$\nvalues by the addition of a standalone calorimetric trigger in which no condition \nis set for the scattered muon. \nThe COMPASS trigger system allows us to cover a wide range of  $Q^2$, from quasi-real\nphotoproduction to deep inelastic interactions.\n \nA more detailed description of the COMPASS apparatus can be found in Ref.~\\cite{setup}\n\n\\section{Event sample}\n \\label{Sec_sample}\n\nFor the present analysis the whole data sample taken in 2002 and 2003 with the \nlongitudinally polarised target is used. For an event to be accepted for further analysis it is required to originate in the target, have a reconstructed \nbeam track, a\nscattered  muon track, and only two additional tracks \nof oppositely charged hadrons associated to the primary vertex. \nThe fluxes of beam muons passing through each target cell are equalised using\nappropriate cuts on the position and angle of the beam tracks.\n\nThe charged\npion mass hypothesis is assigned to each hadron track and the invariant mass of two\npions, $m_{\\pi \\pi}$, calculated. A cut on the invariant mass of two pions, $0.5 < m_{\\pi \\pi} < 1~\\mathrm{GeV}\/c^2$, is applied to select\nthe $\\rho ^0$.\nAs slow recoil target particles are not detected, in order to select\nexclusive events we use the cut on the missing energy, $-2.5 < E_{miss} < 2.5~\\mathrm{GeV}$, and on the transverse momentum of $\\rho^0$ with respect to the direction of\nvirtual photon, $p_t^2 < 0.5~(\\mathrm{GeV}\/c)^2$. Here $E_{miss} = (M^{2}_{X} - M^{2}_{p})\/2M_{p}$, where $M_X$ is the missing mass of the unobserved recoiling\nsystem and $M_p$ is the proton mass.\nCoherent interactions on the\ntarget nuclei are removed by a cut $p_t^2 > 0.15~(\\rm{GeV}\/c)^2$. \nTo avoid large corrections for acceptance and misidentification of\nevents, additional cuts \n$\\nu > 30~\\mathrm{GeV}$ and $E_{\\mu'} > 20~\\mathrm{GeV}$ are applied.\n\n\nThe distributions of $m_{\\pi \\pi}$, $E_{miss}$ and $p_t^2$ are shown in \nFig.~\\ref{hadplots}. Each plot is obtained applying all cuts except\nthose corresponding to the displayed variable. \nOn the left top panel of Fig.~\\ref{hadplots} a clear peak of the \\rn\\ resonance, centred at 770~MeV\/$c^2$, is visible on the top of the small contribution of\nbackground of the non-resonant \\pip\\pin\\ pairs. Also the skewing of the resonance peak towards smaller values\nof \\mpipi, due to an interference with the non-resonant background, is noticeable. A small bump below 0.4~GeV\/$c^2$\nis due to assignment of the charged pion mass to the kaons from decays of $\\phi$ mesons.\nThe mass cuts eliminate the non-resonant background outside of the \\rn\\ peak, as well as the contribution of $\\phi$\nmesons.\n\n\\begin{figure}[t]\n \\begin{center}\n \\epsfig{figure=mpipi_0203.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=Emiss_0203.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=ptsq_0203.eps,height=5cm,width=7.5cm}\n \\caption{Distributions of \\mpipi\\ (top left), \\Emi\\ (top right) \nand \\pts\\ (bottom) for the exclusive sample. \nThe arrows show cuts imposed on each variable to define the final sample.}\n \\label{hadplots}\n \\end{center}\n\\end{figure}\n \nOn the right top panel of the figure the peak at $E_{miss} \\approx 0$ is the signal of \nexclusive \\rn\\ production. The width of the peak, $\\sigma \\approx 1.1~\\mathrm{GeV}$,\nis due to the spectrometer resolution. Non-exclusive events, where in addition to the recoil nucleon other undetected hadrons are produced, appear at $E_{miss} > 0$.\nDue to the finite resolution, however, they are not resolved from the exclusive peak.\nThis background\nconsists of two components: the double-diffractive events where additionally to \\rn\\ an excited nucleon state is produced in the\nnucleon vertex of reaction (\\ref{phorho}), and events with semi-inclusive \\rn\\ production, in which other\nhadrons are produced but escape detection. \n\nThe $p_t^2$ distribution shown on the bottom panel of the figure\nindicates a contribution from coherent production on target \nnuclei at small $p_t^2$ values. A three-exponential fit to this distribution \nwas performed, which indicates also \na contribution of non-exclusive background increasing\nwith $p_t^2$.\nTherefore to select the sample of exclusive incoherent \\rn\\ production, \nthe aforementioned $p_t^2$ cuts, indicated by arrows, were applied. \n\nAfter all selections the final sample consists of about 2.44 million events.\nThe distributions of $Q^2$, \\xBj\\ and $W$ are shown in Fig.~\\ref{kinplots}.\nThe data cover a wide range in $Q^2$\\ and \\xBj\\ which extends towards the small values \nby almost two orders of magnitude compared to the similar studies reported in \nRef.~\\cite{HERMES03}. The sharp edge of the $W$ distribution at the low $W$ values\nis a consequence of the cut applied on $\\nu $. For this sample \n$\\langle W \\rangle$ is equal to 10.2~GeV and $\\langle p_{t}^{2}\n\\rangle = 0.27(\\mathrm{GeV}\/c)^2$.\n\n\\begin{figure}[t]\n \\begin{center}\n \\epsfig{figure=Qsq_0203_incoh_liny.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=Qsq_0203_incoh_logy.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=xBj_0203_incoh.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=Wene_0203_incoh.eps,height=5cm,width=7.5cm}\n \n \\caption{Distributions of the kinematical variables for the final sample: $Q^2$\\ with linear and\n           logarithmic vertical axis scale (top left and right panels respectively), \\xBj\\ (bottom left), and the energy\n           $W$ (bottom right).}\n \\label{kinplots}\n \\end{center}\n\\end{figure}\n\n\\section{Extraction of asymmetry \\Aor}\n \\label{Sec_extract} \n\nThe cross section asymmetry $A_{LL}  =  \n(\\sigma_{\\uparrow \\downarrow}  -  \\sigma_{\\uparrow \\uparrow})\/\n(\\sigma_{\\uparrow \\downarrow}  +  \\sigma_{\\uparrow \\uparrow})$ for reaction\n(\\ref{murho}) , for antiparallel ($\\uparrow \\downarrow $) and parallel \n($\\uparrow \\uparrow $) spins of\nthe incoming muon and the target nucleon, is related to the virtual-photon\nnucleon asymmetry $A_{1}^{\\rho}$ by\n\\begin{equation}\n A_{LL}  =  D  \\left( A_{1}^{\\rho}  +  \\eta  A_{2}^{\\rho}  \\right) ,      \\label{A1rALL}\n\\end{equation}\nwhere the factors $D$ and $\\eta $ depend on the event kinematics and \n$A_{2}^{\\rho}$ is related to\nthe interference cross section for exclusive production by \nlongitudinal and transverse virtual photons. \nAs the presented results extend into the range of very small $Q^2$, \nthe exact formulae for the depolarisation factor $D$ and kinematical \nfactor $\\eta$ \\cite{JKir}\nare used without neglecting terms proportional to the lepton mass squared $m^2$.\nThe depolarisation factor is given by\n\\begin{equation}\n D(y, Q^{2}) = \n   \\frac{\n    y \\left[ (1 + \\gamma^{2} y\/2) (2  -  y) - \n    2 y^{2} m^{2} \/ Q^{2}  \\right]\n   }{\n    y^{2}  (1  -  2 m^{2} \/ Q^{2})  (1  +  \\gamma^{2})  + \n    2  (1  +  R)  (1  -  y  -  \\gamma^{2} y^{2}\/4)\n   },    \\label{depf}\n\\end{equation}\nwhere $R  =  \\sigma_{L} \/ \\sigma_{T}$, $\\sigma_{L(T)}$ is the cross section for reaction (\\ref{phorho})\ninitiated by longitudinally (transversely) polarised virtual photons, the fraction of the muon energy lost $y = \\nu \/E_{\\mu }$ and $\\gamma^{2}  =  Q^{2} \/ \\nu^{2}$.\nThe kinematical factor $\\eta (y, Q^{2})$  is the same as for the inclusive \nasymmetry.\n\nThe asymmetry $A_{2}^{\\rho}$  obeys the positivity limit $A_{2}^{\\rho} < \\sqrt{R}$, analogous to the one for the inclusive case.\nFor $Q^2 \\leq  0.1~(\\mathrm{GeV}\/c)^2$ the ratio $R$ for the reaction (\\ref{phorho}) is\nsmall, cf.\\ Fig.~\\ref{R-pict}, and the positivity limit constrains $A_{2}^{\\rho}$ to small values. Although for larger $Q^2$\\ the ratio $R$ for the process (\\ref{phorho}) increases\nwith $Q^2$, because of small values of $\\eta $ the product $\\eta \\sqrt{R}$ is small in the  whole $Q^2$\\ range of our sample.\nTherefore the second term in Eq.~\\ref{A1rALL} can be neglected, so that\n\\begin{equation}\n A_{1}^{\\rho} \\simeq \\frac{1}{D}  A_{LL},\n\\end{equation}\nand the effect of this approximation is included in the systematic uncertainty of \\Aor.\n\\begin{figure}[thb]\n\\begin{center}\n  \\epsfig{file=RLT.eps,height=5cm,width=7.4cm}\n  \\caption{The ratio $R= \\sigma_L\/\\sigma_T$ as a function of $Q^2$\\ measured in the E665 experiment.\n           The curve is a fit to the data described in the text.}\n  \\label{R-pict}\n \\end{center}\n\\end{figure}\n\nThe number of events $N_i$ collected from a given target cell in a given time interval\nis related to the spin-independent cross section $\\bar{\\sigma}$ for reaction (\\ref{phorho})\nand to the asymmetry $A_1^{\\rho}$ by\n\\begin{equation}\nN_i =  a_i \\phi _i n_i \\bar{\\sigma } (1+P_B P_T f D A_1^{\\rho }  ),\n\\label{nevents}\n\\end{equation}\nwhere $P_B$ and $P_T$ are the beam and target polarisations, $\\phi _i$ is the incoming\nmuon flux, $a_i$ the acceptance for the target cell, $n_i$ corresponding number of target nucleons, and $f$ the target dilution factor.\nThe asymmetry is extracted from the data sets taken before and after a reversal of the\ntarget spin directions. The four relations of Eq.~\\ref{nevents}, corresponding to the two \ncells ($u$ and $d$) and the two spin orientations (1 and 2) lead to a second-order\nequation in \\Aor\\ for the ratio $(N_{u,1}N_{d,2}\/N_{d,1}N_{u,2})$. Here fluxes cancel out\nas well as acceptances, if the ratio of acceptances for the two cells is the same before\nand after the reversal \\cite{SMClong}. In order to minimise the statistical error\nall quantities used in the asymmetry calculation are evaluated event by event with the\nweight factor $w=P_BfD$. The polarisation of the beam muon, $P_B$, is obtained from\na simulation of the beam line and parameterised as a function of the beam momentum. The target polarisation is not \nincluded in the event weight $w$ because it may vary in time and generate false\nasymmetries. An average $P_T$ is used for each target cell and each spin orientation.\n\n The ratio $R$, which enters the formula for $D$ and strongly depends on $Q^2$\\ for reaction (\\ref{phorho}), was calculated on an event-by-event basis\nusing the parameterisation \n\\begin{equation}\n R(Q^{2})  = \n   a_{0}  (Q^{2})^{a_{1}} ,\n\\end{equation}\nwith $a_{0} = 0.66 \\pm 0.05$, and $a_{1} = 0.61 \\pm 0.09$.\nThe parameterisation was obtained by the Fermilab E665 experiment from a fit\nto their $R$ measurements for exclusive $\\rho ^0$ muoproduction on protons \\cite{E665-1}.\nThese are shown in Fig.~\\ref{R-pict} together with the fitted $Q^2$-dependence.\nThe preliminary COMPASS results on $R$ for the incoherent exclusive $\\rho ^0$\nproduction on the nucleon \\cite{sandacz}, which cover a broader kinematic region in $Q^2$\n, agree reasonably well with this parameterisation.\nThe uncertainty of $a_{0}$ and $a_{1}$ is included in the \nsystematic error of $A_{1}^{\\rho}$.\n\nThe dilution factor $f$ gives the fraction of events of reaction (\\ref{phorho}) \noriginating from nucleons in polarised deuterons inside the target material.\nIt is calculated event-by-event using the formula\n\\begin{equation}\n f  =  C_1  \\cdot f_{0}  =  C_1  \\cdot \n   \\frac{\n    n_{\\rm D}\n   }{\n    n_{\\rm D} +  \\Sigma_{A}  n_{\\rm A} \n    (\\tilde{\\sigma}_{\\rm A} \/ \\tilde{\\sigma}_{\\rm D})\n   }.   \n   \\vspace*{3mm}\n\\end{equation}\nHere $n_{\\rm D}$ and $n_{\\rm A}$ denote numbers of nucleons in deuteron and nucleus of atomic mass $A$\nin the target, and \n$\\tilde{\\sigma}_{\\rm D}$ and $\\tilde{\\sigma}_{\\rm A}$ are the cross sections\n per nucleon for reaction (\\ref{phorho}) occurring on the deuteron and on the nucleus of atomic mass\n$A$, respectively. The sum runs over all nuclei present in the COMPASS target.\nThe factor $C_1$ takes into account that there are two polarised deuterons in\nthe $^6$LiD molecule,\nas the $^6$Li nucleus is in a first approximation composed of a deuteron and\nan $\\alpha $ particle.\n\nThe measurements of the $\\tilde{\\sigma}_{\\rm A} \/ \\tilde{\\sigma}_{\\rm D}$ \nfor incoherent exclusive $\\rho ^0$ production come from the NMC \\cite{NMC}, E665 \\cite{E665-2} and early experiments on \\rn\\ photoproduction \\cite{BPSY}. They were fitted in Ref.~\\cite{ATrip-2}\nwith the formula:\n\\begin{equation}\n \\tilde{\\sigma}_{\\rm A}  = \n   \\sigma_{\\rm p} \\cdot A^{\\alpha(Q^{2})  -  1} , \\hspace*{1cm}\n    {\\rm with}\\ \\ \\alpha(Q^{2})  -  1  =  -\\frac{1}{3}  \\exp\\{-Q^{2}\/Q^{2}_{0}\\},\n    \\label{alpha}\n\\end{equation}\nwhere $\\sigma_{\\rm p}$ is the cross section for reaction (\\ref{phorho})\non the free proton.\nThe value of the fitted parameter $Q^{2}_{0}$ is  equal to $9 \\pm 3~(\\mathrm{GeV}\/c)^2$.\nThe measured values of the parameter $\\alpha $ and the fitted curve $\\alpha (Q^2)$\nare shown on the left panel of \nFig.~\\ref{alpha-fig} taken from Ref.~\\cite{ATrip-2}.\nOn the right panel of the figure the average value of $f$ is plotted for\nthe various $Q^2$ bins used in the present analysis. The values of $f$\nare equal to about 0.36 in most of the $Q^2$ range, rising to about 0.38 at the\nhighest $Q^2$.\n\n\\begin{figure}[t]\n\\begin{center}\n  \\epsfig{file=alpha.eps,height=5cm,width=7.5cm}\n  \\epsfig{file=f_Qsq_0203.eps,height=5cm,width=7.5cm}\n  \\caption{(Left) Parameter $\\alpha$ of Eq.~\\ref{alpha} as a function of $Q^2$\\ (from Ref.~\\cite{ATrip-2}). The experimental points and the fitted curve are shown. See text for details. (Right) The dilution factor $f$ as a function of $Q^2$.}\n  \\label{alpha-fig}\n \\end{center}\n\\end{figure}\n\n  The radiative corrections (RC) have been neglected in the present analysis, in particular in the calculation of $f$, because\nthey are expected to be small for reaction (\\ref{murho}). They were evaluated \\cite{KKurek} to be\nof the order of 6\\% for the NMC exclusive \\rn\\ production analysis.\nThe small values of RC are mainly due to the requirement of event exclusivity via cuts on \\Emi\\ and $p_t^2$,\nwhich largely suppress the dominant external photon radiation.\nThe internal (infrared and virtual) RC were\nestimated in Ref.~\\cite{KKurek} to be of the order of 2\\%.  \n\n\\section{Systematic errors}\n \\label{Sec_syst}\n\nThe main systematic uncertainty of \\Aor\\ comes from an estimate of possible false asymmetries. \nIn order to improve the accuracy of this estimate,\nin addition to the standard sample of incoherent events, a second sample\nwas selected by changing the $p_t^2$ cuts to\n\\begin{equation}\n 0 < p_{t}^{2} < 0.5~(\\mathrm{GeV}\/c)^{2},    \\label{ptsext}\n\\end{equation}  and keeping all the remaining selections and cuts the same\nas for the `incoherent sample'.\nIn the following it will be\n referred to as the `extended $p_t^2$ sample'.\nSuch an extension of the \\pts\\ range allows one to obtain a sample which is\nabout\nfive times larger than the incoherent sample. \nHowever, in addition to incoherent events such a sample contains a large fraction of events originating from coherent \\rn\\ production.\nTherefore, for the estimate of the dilution factor $f$ a different nuclear dependence of the\nexclusive cross section was used, applicable for the sum of coherent and incoherent\ncross sections \\cite{NMC}.\nThe physics asymmetries \\Aor\\ for both samples are consistent within statistical errors.\n\nPossible, false experimental asymmetries were searched for by modifying the selection\nof data sets used for the asymmetry calculation. The grouping of the data into\nconfigurations with opposite target-polarisation was varied from large samples,\ncovering at most two weeks of data taking, into about 100 small samples, taken in time\nintervals of the order of 16 hours.\nA statistical test was performed on the distributions of asymmetries obtained\nfrom these small samples. In each of the $Q^2$\\ and \\xBj\\ bins the dispersion of the values of\n$A_{1}^{\\rho}$ around their mean agrees with the statistical error.\nTime-dependent effects which would lead to a broadening of these distributions\nwere thus not observed. Allowing the dispersion of $A_{1}^{\\rho}$ to vary within its\ntwo standard deviations we \nobtain for each bin an upper bound for the systematic error arising from\ntime-dependent effects\n\\begin{equation}\n \\sigma_{\\rm falseA,tdep}  < \n   0.56~\\sigma_{\\rm stat}.\n\\end{equation}\nHere $\\sigma _{\\rm stat}$ is the statistical error on $A_{1}^{\\rho}$ for the extended\n$p_t^2$ sample.\nThe uncertainty on the estimates of possible false asymmetries due to the time-dependent \neffects is the dominant contribution to the total systematic error in most of the kinematical\nregion.\n\nAsymmetries for configurations where spin effects cancel out were calculated to\ncheck the cancellation of effects due to fluxes and acceptances. They were found compatible\nwith zero within the statistical errors. Asymmetries obtained with different\nsettings of the microwave (MW) frequency, used for DNP, were compared in order to test possible\neffects related to the orientation of the target magnetic field. \nThe results for the extended $p_t^2$ sample tend to show that there is a small\ndifference between asymmetries for the two MW configurations.\nHowever, because the numbers of events of the data samples taken with each MW setting \nare approximately balanced, the effect of this difference on \\Aor\\ is\nnegligible for the total sample.\n \nThe systematic error on $A_{1}^{\\rho}$ also contains an overall scale uncertainty of\n6.5\\% due to uncertainties on $P_B$ and $P_T$. The uncertainty of the \nparameterisation of $R(Q^2)$ \naffects the depolarisation factor $D$. The uncertainty of the dilution factor\n$f$ is mostly due to uncertainty of the parameter $\\alpha (Q^2)$ which takes into account\nnuclear effects in the incoherent $\\rho ^0$ production. \nThe neglect of the $A_2^{\\rho }$ term mainly affects the highest bins of $Q^2$\\ and \\xBj.\n\nAnother source of systematic errors is due to the contribution of\nthe non-exclusive background to our sample.\nThis background originates from two sources.\nFirst one is due to the  production of\n\\rn\\ accompanied by the dissociation of the target nucleon,\nthe second one is the production of \\rn\\ in inclusive scattering.\nIn order to evaluate the amount of background in the sample of exclusive\nevents it is necessary to\ndetermine the $E_{miss}$ dependence for the non-exclusive background\nin the region under the exclusive peak (cf.\\ Fig.~\\ref{hadplots} ).\nFor this purpose complete Monte Carlo simulations of the experiment were \nused, with events \ngenerated by either the PYTHIA 6.2 or LEPTO 6.5.1 generators.\nEvents generated with LEPTO come only from deep inelastic scattering and cover the range of $Q^2 > 0.5~(\\mathrm{GeV}\/c)^2$. \nThose generated with PYTHIA cover the whole kinematical range of the experiment and include exclusive production of vector mesons and processes with diffractive excitation of the target nucleon or the vector meson, in addition to inelastic\nproduction.\n\nThe generated MC events were reconstructed\nand selected for the analysis using the same procedure as for the data.\nIn each bin of $Q^2$\\ the $E_{miss}$ distribution for the MC was \nnormalised to the corresponding one for the data\nin the range of large $E_{miss} > 7.5~\\mathrm{GeV}$. Then the normalised MC\ndistribution was used to estimate the number of background events under the exclusive peak in the data.  \nThe fraction of background events in the sample of\nincoherent exclusive $\\rho ^0$ production was estimated to be about $0.12 \\pm 0.06$\nin most of the kinematical range, except in the largest $Q^2$\\ region, where it is about\n$0.24 \\pm 0.12$. \nThe large uncertainties of these fractions reflect the differences between \nestimates from LEPTO and PYTHIA in the region where they overlap. In \nthe case of PYTHIA the\nuncertainties on the cross sections for diffractive photo- and\nelectroproduction of vector mesons also contribute.\nFor events generated with PYTHIA the $E_{miss}$  \ndistributions for various physics processes could be studied separately. \nIt was found\nthat events of \\rn\\ production with an\nexcitation of the target nucleon into $N^*$ resonances\nof small mass, $M < 2~\\mathrm{GeV}\/c^2$, cannot be resolved from the exclusive\npeak and therefore were not included                 \nin the estimates of number of background events.\n\nAn estimate of the asymmetry \\Aor\\ for the background was obtained using a non-exclusive sample,\nwhich was selected with the standard cuts used in this analysis,\nexcept the cut on $E_{miss}$ which was modified to $E_{miss} > 2.5$~GeV. In different high-$E_{miss}$ bins \\Aor\\ for this sample was\nfound compatible with zero.\n\nBecause no indication of a non-zero \\Aor\\ for the background was found,\nand also due to a large uncertainty of the estimated amount of background \nin the exclusive sample, no background corrections were made.\nInstead, the effect of background was treated as a source\nof systematic error. Its contribution \nto the total systematic error was not significant in most of the kinematical\nrange, except for the highest $Q^2$ and $x$.\n\nThe total systematic error on \\Aor\\ was obtained as a quadratic sum of the errors from\nall discussed sources. \nIts values for each $Q^2$\\ and \\xBj\\ bin are given\nin Tables~\\ref{binsq2-1}  and \\ref{binsx-1}. The total systematic error \namounts to about 40\\% of the \nstatistical error for most of the kinematical range. Both errors become\ncomparable in the highest bin of $Q^2$. \n\n\\section{Results}\n \\label{Sec_resu}\n\nThe COMPASS results on $A^{\\rho}_1$ are shown as a function of $Q^2$\\  and \\xBj\\ \nin Fig.~\\ref{A1-Com} and listed in \nTables~\\ref{binsq2-1} and \\ref{binsx-1}.\nThe statistical errors are represented by vertical bars and the total systematic\nerrors by shaded bands.\n\n\\begin{figure}[ht]\n\\begin{center}\n  \\epsfig{file=A1_Qsq_incoh_cons_0203.eps,height=5cm,width=7.5cm}\n  \\epsfig{file=A1_xBj_incoh_cons_0203.eps,height=5cm,width=7.5cm}\n  \\caption{\\Aor\\ as a function of $Q^2$\\ (left) and \\xBj\\ (right) from the present analysis.\nError bars correspond to statistical errors, while bands at the bottom represent the systematical errors.}\n  \\label{A1-Com}\n \\end{center}\n\\end{figure}\n\\begin{table}[ht]\n \\caption{Asymmetry $A_1^{\\rho }$ as a function of $Q^2$. Both the statistical errors (first) and the total systematic errors (second) are listed.}\n \\label{binsq2-1}\n \\small\n \\begin{center}\n  \\begin{tabular}{||c|c|c|c|c||}\n   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm]{$Q^2$\\ range}\\ \\  &\n                            $\\langle  Q^{2}  \\rangle$ [$(\\mathrm{GeV}\/c)^2$]  &  $\\langle  x  \\rangle$  &\n                            $\\langle  \\nu  \\rangle$ [GeV] & $A_1^{\\rho }$ \\\\   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.0004 - 0.005~$  & \\ \\ \\ 0.0031\\ \\ \\  & $4.0 \\cdot 10^{-5}$ & 42.8  & $-0.030 \\pm 0.045 \\pm 0.014$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.005 - 0.010$  &  ~0.0074 & $8.4 \\cdot 10^{-5}$ & 49.9             & $~~0.048 \\pm 0.038 \\pm 0.013$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.010  - 0.025$ &  0.017  & $1.8 \\cdot 10^{-4}$ & 55.6             & $~~0.063 \\pm 0.026 \\pm 0.014$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.025 - 0.050$  &  0.036  & $3.7 \\cdot 10^{-4}$ & 59.9             & $-0.035 \\pm 0.027 \\pm 0.009$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.05  - 0.10$   &  0.072  & $7.1 \\cdot 10^{-4}$ & 62.0             & $-0.010 \\pm 0.028 \\pm 0.008$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.10   - 0.25$  &  0.16~  & 0.0016 & 62.3                           & $-0.019 \\pm 0.029 \\pm 0.009$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.25  - 0.50$   &  0.35~  & 0.0036 & 60.3                           & $~~0.016 \\pm 0.045 \\pm 0.014$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.5   - 1~~$     &  0.69~  & 0.0074 & 58.6                           & $~~0.141 \\pm 0.069 \\pm 0.030$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $1     - 4$     &  1.7~~  & 0.018~~  & 59.7                           & $~~0.000 \\pm 0.098 \\pm 0.035$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $~4    - 50$  & 6.8~~  & 0.075~~  & 55.9                               & $-0.85  \\pm 0.50  \\pm 0.39 $\\\\\n   \\hline\n   \\hline\n  \\end{tabular}\n \\end{center}\n  \\normalsize\n\\end{table}\n\\begin{table}[ht]\n \\caption{Asymmetry $A_1^{\\rho }$ as a function of \\xBj. Both the statistical errors (first) and the total systematic errors (second) are listed.}\n \\label{binsx-1}\n \\small\n \\begin{center}\n  \\begin{tabular}{||c|c|c|c|c||}\n   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm]{\\xBj\\ range}\\ \\  &   $\\langle  x  \\rangle$  &  $\\langle  Q^{2}  \\rangle$ [$(\\mathrm{GeV}\/c)^2$]  &\n                           $\\langle  \\nu  \\rangle$ [GeV] & $A_1^{\\rho }$  \\\\   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $8 \\cdot 10^{-6} - 1 \\cdot 10^{-4}$   & $5.8 \\cdot 10^{-5}$ & ~0.0058 & 51.7 & $~~0.035 \\pm 0.026 \\pm 0.011$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $1 \\cdot 10^{-4} - 2.5 \\cdot 10^{-4}$  & $1.7 \\cdot 10^{-4}$ & 0.019 &  59.7  & $~~0.036 \\pm 0.024 \\pm 0.010$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $2.5 \\cdot 10^{-4} - 5 \\cdot 10^{-4}$   & $3.6 \\cdot 10^{-4}$ & 0.041 &  61.3  & $-0.039 \\pm 0.027 \\pm 0.012$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $5 \\cdot 10^{-4} - 0.001$ &  $7.1 \\cdot 10^{-4}$ & 0.082 & 60.8  & $-0.010 \\pm 0.030 \\pm 0.010$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.001 - 0.002$ &  0.0014 & 0.16~~ & 58.6 & $-0.005 \\pm 0.036 \\pm 0.013$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.002 - 0.004$ &  0.0028 & 0.29~~ & 54.8 & $ ~~0.032 \\pm 0.050 \\pm 0.019$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.004 - 0.01~~$  &  0.0062 & 0.59~~ & 50.7 & $ ~~0.019 \\pm 0.069 \\pm 0.026$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.01  - 0.025$ &  0.015~~  & 1.3~~~~  & 47.5 & $-0.03 \\pm 0.14 \\pm 0.06$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.025 - 0.8~~~~$  &   0.049~~  & 3.9~~~~  & 43.8 & $-0.27 \\pm 0.38 \\pm 0.19$  \\\\\n   \\hline\n   \\hline\n  \\end{tabular}\n \\end{center}\n \\normalsize\n\\end{table}\n\nThe wide range in $Q^2$\ncovers four orders of magnitude from $3 \\cdot 10^{-3}$ to 7~$(\\mathrm{GeV}\/c)^2$. The domain in \\xBj\\ which is strongly correlated\nwith $Q^2$, varies from $5 \\cdot 10^{-5}$ to about 0.05 (see Tables for more details).\nFor the whole kinematical range the \\Aor\\\nasymmetry measured by COMPASS is consistent with zero. As discussed in the introduction,\nthis indicates that the role of unnatural parity exchanges,\nlike $\\pi$- or $A_{1}$-Reggeon exchange, \nis small in that kinematical domain, which is to be expected if diffraction\nis the dominant process for reaction (\\ref{phorho}). \n\nIn Fig.~\\ref{A1-Com-Her} the COMPASS results are compared to the HERMES results on $A^{\\rho}_1$ obtained\non a deuteron target \\cite{HERMES03}.\nNote that the lowest $Q^2$\\ and \\xBj\\ HERMES points, referred to as `quasi-photoproduction', come from measurements where the kinematics of the small-angle scattered \nelectron was not measured but estimated from a MC simulation. This is in contrast to COMPASS,\nwhere scattered muon kinematics is measured even at the smallest $Q^2$.\n\\begin{figure}[ht]\n\\begin{center}\n  \\epsfig{file=A1_Qsq_incoh_cons_0203_wHermes.eps,height=5cm,width=7.5cm}\n  \\epsfig{file=A1_xBj_incoh_cons_0203_wHermes.eps,height=5cm,width=7.5cm}\n  \\caption{\\Aor\\ as a function of $Q^2$\\ (left) and \\xBj\\ (right) from the present analysis (circles)\ncompared to HERMES results on the deuteron target (triangles). For the COMPASS results inner bars represent statistical errors, while the outer bars correspond to the total error.\n For the HERMES results vertical bars represent the quadratic sum of statistical and systematic errors. The curve represents the prediction explained in the text.}\n   \\label{A1-Com-Her}\n\\end{center}\n\\end{figure}\n\nThe results from both experiments are consistent within errors. \nThe kinematical range covered by the present analysis extends further towards small\nvalues of \\xBj\\ and $Q^2$\\ by almost two orders of \nmagnitude. In each of the two experiments $A^{\\rho}_1$ is measured\nat different average  $W$, which is equal to about 10~GeV for\nCOMPASS and 5~GeV for HERMES. Thus, no significant $W$ dependence\nis observed for $A^{\\rho}_1$ on an isoscalar nucleon target.\n\nThe \\xBj\\ dependence of the measured \\Aor\\ is compared in Fig.~\\ref{A1-Com-Her} to the prediction given by Eq.~\\ref{A1-Fraas}, which\nrelates $A_1^{\\rho}$ to the asymmetry $A_{1}$ for the inclusive inelastic\nlepton-nucleon scattering. To produce the curve\nthe inclusive asymmetry $A_{1}$ was parameterised as\n$A_1(x) = (x^{\\alpha } - \\gamma^{\\alpha }) \\cdot (1- e^{-\\beta x})$ , where\n$\\alpha = 1.158 \\pm 0.024$, $\\beta = 125.1 \\pm 115.7$ and $\\gamma = 0.0180 \\pm 0.0038$.\nThe values of the parameters have been obtained\nfrom a fit of $A_{1}(x)$ to the world data\nfrom polarised deuteron targets \\cite{smc,e143,e155_d,smc_lowx,hermes_new,compass_a1_recent} including COMPASS measurements at very \nlow $Q^2$ and \\xBj\\ \\cite{compass_a1_lowq2}. Within the present accuracy the results on \\Aor\\ are consistent with this prediction.  \n\n\nIn the highest $Q^2$\\ bin,  $\\langle  Q^{2} \\rangle = 6.8~(\\mathrm{GeV}\/c)^2$, in the kinematical domain of applicability of pQCD-inspired models which relate\nthe asymmetry to the spin-dependent\nGPDs for gluons and quarks (cf.\\ Introduction), \none can observe a hint of a possible nonzero asymmetry, although with a\nlarge error.\nIt should be noted that in Ref.~\\cite{ATrip-1} a negative value of $A_{LL}$ different from zero by about 2 standard deviations\nwas reported at $\\langle  Q^{2}  \\rangle  = 7.7~(\\mathrm{GeV}\/c)^2$.\nAt COMPASS, including the data \ntaken with the longitudinally polarised deuteron target in 2004 and 2006 will result in an\nincrease of\nstatistics by a factor of about three compared to the present paper,\nand thus may help to clarify the issue.\n \nFor the whole $Q^2$ range future COMPASS data, to be taken with the polarised proton target, would be very valuable for checking if the role of the flavour-blind\nexchanges is indeed dominant, as expected for the Pomeron-mediated process.\n\n \n\\section{Summary}\n \\label{Sec_summ}\n\nThe longitudinal double spin asymmetry \\Aor\\ for the diffractive muoproduction of \\rn\\ meson, $\\mu + N \\rightarrow \n\\mu + N + \\rho$, has been measured by scattering longitudinally polarised muons off longitudinally polarised deuterons from the\n$^6$LiD target and selecting incoherent exclusive $\\rho^0$ production.\nThe presented results for the COMPASS 2002 and 2003 data cover a range of energy $W$ from about 7 to 15~GeV.\n\nThe $Q^2$\\ and \\xBj\\ dependence of \\Aor\\ is presented in a wide kinematical range\n$3 \\cdot 10^{-3} \\leq Q^{2} \\leq 7~(\\mathrm{GeV}\/c)^2$ and $5 \\cdot 10^{-5} \\leq x \\leq 0.05$.\nThese results extend the range in $Q^2$\\ and \\xBj\\ by two orders of magnitude down\nwith respect to the existing data from HERMES.\n\nThe asymmetry $A^{\\rho}_1$ is compatible with zero in the whole \\xBj\\ and\n$Q^2$\\ range.\nThis may indicate that the role of unnatural parity exchanges like $\\pi$- or $A_{1}$-Reggeon exchange is\nsmall in that kinematical domain.\n\nThe \\xBj\\ dependence of measured \\Aor\\ is consistent with the prediction of Ref.~\\cite{HERMES01} which relates $A^{\\rho}_1$\nto the asymmetry $A_{1}$ for the inclusive inelastic lepton--nucleon scattering.\n\n\\section{Acknowledgements}\n \\label{Sec_ack}\n\nWe gratefully acknowledge the support of the CERN management and staff and\nthe skill and effort of the technicians of our collaborating institutes.\nSpecial thanks are due to V. Anosov and V. Pesaro for their support during the\ninstallation and the running of the experiment. This work was made possible by \nthe financial support of our funding agencies.\n\n\n\\noindent\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nMany tasks in natural language processing, computational biology, reinforcement learning, and time series analysis rely on learning\nwith sequential data, i.e.\\  estimating functions defined over sequences of observations from training data.  \nWeighted  finite automata~(WFAs) and recurrent neural networks~(RNNs) are two powerful and flexible classes of models which can efficiently represent such functions.\nOn the one hand, WFAs are tractable, they encompass a wide range of machine learning models~(they can for example compute any probability distribution defined by a hidden Markov \nmodel~(HMM)~\\cite{denis2008rational} and can model the transition and observation behavior of\npartially observable Markov decision processes~\\cite{thon2015links}) and they offer appealing theoretical guarantees. In particular, \nthe so-called \n\\emph{spectral\nmethods} for learning HMMs~\\cite{hsu2009spectral}, WFAs~\\cite{bailly2009grammatical,balle2014spectral} and related models~\\cite{glaude2016pac,boots2011closing}, \nprovide an alternative to Expectation-Maximization based algorithms that is both computationally efficient and \nconsistent. \nOn the other hand, RNNs are \nremarkably expressive models --- they can represent any computable function~\\cite{siegelmann1992computational} --- and they have successfully\ntackled many practical problems in speech and audio recognition~\\cite{graves2013speech,mikolov2011extensions,gers2000learning}, but\ntheir theoretical  analysis is difficult. Even though recent work provides interesting results on their\nexpressive power~\\cite{khrulkov2018expressive,yu2017long} as well as alternative training algorithms coming with learning guarantees~\\cite{sedghi2016training},\nthe theoretical understanding of RNNs is still limited.\n\n\\footnotetext{\\footnotemark[1] Mila \\footnotemark[2] Universit\u00e9 de Montr\u00e9al \\footnotemark[3] McGill University}\n\n\\renewcommand*{\\thefootnote}{\\arabic{footnote}}\n\nIn this work, we bridge a gap between these two classes of models by unraveling a fundamental connection between WFAs and second-order RNNs~(2-RNNs): \n\\textit{when considering input sequences of discrete symbols, 2-RNNs with linear activation functions and WFAs are one and the same}, i.e.\\  they are expressively\nequivalent and there exists a one-to-one mapping between the two classes~(moreover, this mapping conserves model sizes).  While connections between\nfinite state machines~(e.g.\\  deterministic finite automata)  and recurrent neural networks have been noticed and investigated in the past~(see e.g.\\  \\cite{giles1992learning,omlin1996constructing}), to the\nbest of our knowledge this is the first time that such a \\rev{rigorous} equivalence between linear 2-RNNs and \\emph{weighted} automata is explicitly formalized. \n\\rev{More precisely, we pinpoint exactly the class of recurrent neural architectures to which weighted automata are equivalent, namely second-order RNNs with\nlinear activation functions.}\nThis result naturally leads to the observation that linear 2-RNNs are a natural generalization of WFAs~(which take sequences of \\emph{discrete} observations as\ninputs) to sequences of \\emph{continuous vectors}, and raises the question of whether the spectral learning algorithm for WFAs can be extended to linear 2-RNNs. \nThe second contribution of this paper is to show that the answer is in the positive: building upon the spectral learning algorithm for vector-valued WFAs introduced\nrecently in~\\cite{rabusseau2017multitask}, \\emph{we propose the first provable learning algorithm for second-order RNNs with linear activation functions}.\nOur learning algorithm relies on estimating  sub-blocks of the so-called Hankel tensor, from which the parameters of a 2-linear RNN can be recovered\nusing basic linear algebra operations. One of the key technical difficulties in designing this algorithm resides in estimating\nthese sub-blocks from training data where the inputs are sequences of \\emph{continuous} vectors. \nWe leverage multilinear properties of linear 2-RNNs and the fact that the Hankel sub-blocks can be reshaped into higher-order tensors of low tensor train rank~(a\nresult we believe is of independent interest) to perform this estimation efficiently using matrix sensing and tensor recovery techniques.\nAs a proof of concept, we validate our theoretical findings in a simulation study on toy examples where we experimentally compare  \ndifferent recovery methods and investigate the robustness of our algorithm to noise and rank mis-specification. \\rev{We also show that refining the estimator returned\nby our algorithm using stochastic gradient descent can lead to significant improvements.}\n\n\\rev{\n\\paragraph{Summary of contributions.} We formalize a \\emph{strict equivalence between weighted automata and second-order RNNs with linear activation\nfunctions}~(Section~\\ref{sec:WFAs.and.2RNNs}), showing that linear 2-RNNs can be seen as a natural extension of (vector-valued) weighted automata for input sequences of \\emph{continuous} vectors. We then\npropose a \\emph{consistent learning algorithm for linear 2-RNNs}~(Section~\\ref{sec:Spectral.learning.of.2RNNs}).\nThe relevance of our contributions can be seen from two perspectives.\nFirst, while learning feed-forward neural networks with linear activation functions is a trivial task (it reduces to linear or reduced-rank regression), this\nis not at all the case for recurrent architectures with linear activation functions; to the best of our knowledge, our algorithm is the \\emph{first consistent learning algorithm\nfor the class of functions computed by linear second-order recurrent networks}. Second, from the perspective of learning weighted automata, we propose a  natural extension of WFAs to continuous inputs and \\emph{our learning algorithm addresses the long-standing limitation of the spectral learning method to discrete inputs}.\n}\n\n\\paragraph{Related work.}\nCombining the spectral learning algorithm for WFAs with matrix completion techniques~(a problem which is closely related to matrix sensing) has\nbeen theoretically investigated in~\\cite{balle2012spectral}. An extension of probabilistic transducers to continuous inputs~(along with a spectral learning algorithm) has been proposed in~\\cite{recasens2013spectral}.\nThe connections between tensors  and RNNs have been previously leveraged to study the expressive power of RNNs in~\\cite{khrulkov2018expressive}\nand to achieve model compression in~\\cite{yu2017long,yang2017tensor,tjandra2017compressing}. \nExploring relationships between RNNs and automata has recently received a renewed interest~\\cite{peng2018rational,chen2018recurrent,li2018nonlinear}. In particular, such connections have been explored for interpretability purposes~\\cite{weiss2018extracting,ayache2018explaining} and the ability of RNNs to learn classes of formal languages\nhas been investigated in~\\cite{avcu2017subregular}.  \\rev{Connections between the tensor train decomposition and WFAs have been previously noticed in~\\cite{critch2013algebraic,critch2014algebraic,rabusseau2016thesis}.} \nThe predictive state RNN model introduced in~\\cite{downey2017predictive} is closely related to 2-RNNs and the authors propose\nto use the spectral learning algorithm for predictive state representations to initialize a gradient based algorithm; their approach however comes without\ntheoretical guarantees. Lastly, a provable algorithm for RNNs relying on the tensor method of moments has been proposed in~\\cite{sedghi2016training} but\nit is limited to first-order RNNs with quadratic activation functions~(which do not encompass linear 2-RNNs).\n\n\\emph{The proofs of the results given in the paper can be found in the supplementary material.}\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\nIn this section, we first present basic notions of tensor algebra before introducing  second-order recurrent neural \nnetwork, \nweighted finite automata and the spectral learning algorithm.\nWe start by introducing some notation.\nFor any integer $k$ we use $[k]$ to denote the set of integers from $1$ to $k$. We use\n$\\lceil l \\rceil$ to denote the smallest integer greater or equal to $l$.\nFor any set $\\Scal$, we denote by $\\Scal^*=\\bigcup_{k\\in\\Nbb}\\Scal^k$ the set of all\nfinite-length sequences of elements of $\\Scal$~(in particular, \n$\\Sigma^*$ will denote the set of strings on a finite alphabet $\\Sigma$). \nWe use lower case bold letters  for vectors (e.g.\\  $\\vec{v} \\in \\Rbb^{d_1}$),\nupper case bold letters for matrices (e.g.\\  $\\mat{M} \\in \\Rbb^{d_1 \\times d_2}$) and\nbold calligraphic letters for higher order tensors (e.g.\\  $\\ten{T} \\in \\Rbb^{d_1\n\\times d_2 \\times d_3}$). We use $\\ten_i$ to denote the $i$th canonical basis \nvector of $\\Rbb^d$~(where the dimension $d$ will always appear clearly from context).\nThe $d\\times d$ identity matrix will be written as $\\mat{I}_d$.\nThe $i$th row (resp. column) of a matrix $\\mat{M}$ will be denoted by\n$\\mat{M}_{i,:}$ (resp. $\\mat{M}_{:,i}$). This notation is extended to\nslices of a tensor in the straightforward way.\nIf $\\vec{v} \\in \\Rbb^{d_1}$ and $\\vec{v}' \\in \\Rbb^{d_2}$, we use $\\vec{v} \\otimes \\vec{v}' \\in \\Rbb^{d_1\n\\cdot d_2}$ to denote the Kronecker product between vectors, and its\nstraightforward extension to matrices and tensors.\nGiven a matrix $\\mat{M} \\in \\Rbb^{d_1 \\times d_2}$, we use $\\vectorize{\\mat{M}} \\in \\Rbb^{d_1\n\\cdot d_2}$ to denote the column vector obtained by concatenating the columns of\n$\\mat{M}$. The inverse of $\\mat{M}$ is denoted by $\\mat{M}^{-1}$, its Moore-Penrose pseudo-inverse\nby $\\mat{M}^\\dagger$, and the transpose of its inverse by $\\mat{M}^{-\\top}$; the Frobenius norm\nis denoted by $\\norm{\\mat{M}}_F$ and the nuclear norm by $\\norm{\\mat{M}}_*$.\n\n\\paragraph{Tensors.}\nWe first recall basic definitions of tensor algebra; more details can be found\nin~\\cite{Kolda09}. \nA \\emph{tensor} $\\ten{T}\\in \\Rbb^{d_1\\times\\cdots \\times d_p}$ can simply be seen\nas a multidimensional array $(\\ten{T}_{i_1,\\cdots,i_p}\\ : \\ i_n\\in [d_n], n\\in [p])$. The\n\\emph{mode-$n$} fibers of $\\ten{T}$ are the vectors obtained by fixing all\nindices except  the $n$th one, e.g.\\  $\\ten{T}_{:,i_2,\\cdots,i_p}\\in\\Rbb^{d_1}$.\nThe \\emph{$n$th mode matricization} of $\\ten{T}$ is the matrix having the\nmode-$n$ fibers of $\\ten{T}$ for columns and is denoted by\n$\\tenmat{T}{n}\\in \\Rbb^{d_n\\times d_1\\cdots d_{n-1}d_{n+1}\\cdots d_p}$.\nThe vectorization of a tensor is defined by $\\vectorize{\\ten{T}}=\\vectorize{\\tenmat{T}{1}}$.\nIn the following $\\ten{T}$ always denotes a tensor of size $d_1\\times\\cdots \\times d_p$.\n\nThe \\emph{mode-$n$ matrix product} of the tensor $\\ten{T}$ and a matrix\n$\\mat{X}\\in\\Rbb^{m\\times d_n}$ is a tensor  denoted by $\\ten{T}\\ttm{n}\\mat{X}$. It is \nof size $d_1\\times\\cdots \\times d_{n-1}\\times m \\times d_{n+1}\\times\n\\cdots \\times d_p$ and is defined by the relation \n$\\ten{Y} = \\ten{T}\\ttm{n}\\mat{X} \\Leftrightarrow \\tenmat{Y}{n} = \\mat{X}\\tenmat{T}{n}$.\nThe \\emph{mode-$n$ vector product} of the tensor $\\ten{T}$ and a vector\n$\\vec{v}\\in\\Rbb^{d_n}$ is a tensor defined by $\\ten{T}\\ttv{n}\\vec{v} = \\ten{T}\\ttm{n}\\vec{v}^\\top\n\\in \\Rbb^{d_1\\times\\cdots \\times d_{n-1}\\times d_{n+1}\\times\n\\cdots \\times d_p}$.\nIt is easy to check that the $n$-mode product satisfies $(\\ten{T}\\ttm{n}\\mat{A})\\ttm{n}\\mat{B} = \\ten{T}\\ttm{n}\\mat{BA}$\nwhere we assume compatible dimensions of the tensor $\\ten{T}$ and\nthe matrices $\\mat{A}$ and $\\mat{B}$.\n\n\nGiven strictly positive integers $n_1,\\cdots, n_k$ satisfying\n$\\sum_i n_i = p$, we use the notation $\\tenmatgen{\\ten{T}}{n_1,n_2,\\cdots,n_k}$ to denote the $k$th order tensor \nobtained by reshaping $\\ten{T}$ into a tensor\\footnote{Note that the specific ordering used to perform matricization, vectorization\nand such a reshaping is not relevant as long as it is consistent across all operations.} of size \n$(\\prod_{i_1=1}^{n_1} d_{i_1}) \\times (\\prod_{i_2=1}^{n_2} d_{n_1 + i_2}) \\times \\cdots \\times (\\prod_{i_k=1}^{n_k} d_{n_1+\\cdots+n_{k-1} + i_k})$.\nIn particular we have $\\tenmatgen{\\ten{T}}{p} = \\vectorize{\\ten{T}}$ and $\\tenmatgen{\\ten{T}}{1,p-1} = \\tenmat{\\ten{T}}{1}$.\n\n\nA rank $R$ \\emph{tensor train (TT) decomposition}~\\cite{oseledets2011tensor} of a tensor \n$\\ten{T}\\in\\Rbb^{d_1\\times \\cdots\\times d_p}$ consists in factorizing $\\ten{T}$ into the product of $p$ core tensors\n$\\ten{G}_1\\in\\Rbb^{d_1\\times R},\\ten{G}_2\\in\\Rbb^{R\\times d_2\\times R},\n\\cdots, \\ten{G}_{p-1}\\in\\Rbb^{R\\times d_{p-1} \\times R},\n\\ten{G}_p \\in \\Rbb^{R\\times d_p}$, and is defined\\footnote{The classical definition of the TT-decomposition allows the rank $R$ to be different\nfor each mode, but this definition is sufficient for the purpose of this paper.} by\n\\begin{align*}\n\\MoveEqLeft\\ten{T}_{i_1,\\cdots,i_p} =  \n&(\\ten{G}_1)_{i_1,:}(\\ten{G}_2)_{:,i_2,:}\\cdots \n (\\ten{G}_{p-1})_{:,i_{p-1},:}(\\ten{G}_p)_{:,i_p}\n\\end{align*}\n %\nfor all indices $i_1\\in[d_1],\\cdots,i_p\\in[d_p]$; we will use the notation $\\ten{T} = \\TT{\\ten{G}_1,\\cdots,\\ten{G}_p}$\nto denote such a decomposition. A tensor network representation of this decomposition is shown in Figure~\\ref{fig:tn.TT}.\n While the problem of finding the best approximation of  TT-rank $R$\nof a given tensor is NP-hard~\\cite{hillar2013most}, \na quasi-optimal SVD based compression algorithm~(TT-SVD) has been proposed \nin~\\cite{oseledets2011tensor}.\nIt is worth mentioning that the TT decomposition is invariant under change of basis: \nfor any invertible matrix $\\mat{M}$ and any core tensors $\\ten{G}_1,\\ten{G}_2,\\cdots,\\ten{G}_p$, we have\n$\\TT{\\ten{G}_1,\\cdots,\\ten{G}_p} = \\TT{\\ten{G}_1\\ttm{2}\\mat{M}^{-\\top},\\ten{G}_2\\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top},\\cdots,\n\\ten{G}_{p-1}\\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top},\\ten{G}_p\\ttm{1}\\mat{M}}$.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{0.08\\textwidth}{%\n\\begin{tikzpicture}\n\t\\input{tikz_tensor_networks}\n\t\\node[tensor](G1){$\\ten{G}_1$};\n\t\\node[draw = none,below=0.8cm of G1](G11){};\n\t\n\t\\node[tensor,right = 1cm of G1](G2){$\\ten{G}_2$};\n\t\\node[draw=none,below=0.8cm of G2](G22){};\n\t\n\t\\node[tensor,right = 1cm of G2](G3){$\\ten{G}_3$};\n\t\\node[draw=none,below=0.8cm of G3](G32){};\n\t\n\t\\node[tensor,right = 1cm of G3](G4){$\\ten{G}_4$};\n\t\\node[draw=none,below=0.8cm of G4](G41){};\n\t\n\t\\node[tensor,left=3cm of G1](T){$\\ten{T}$};\n\t\\node[draw=none,below left = 0.2cm and 1cm of T](T1){};\n\t\\node[draw=none,below left = 0.8cm and 0.2cm of T](T2){};\n\t\\node[draw=none,below right = 0.8cm and 0.2cm of T](T3){};\n\t\\node[draw=none,below right = 0.2cm and 1cm of T](T4){};\n\t\\node[draw=none,right=1.5cm of T](eq){$=$};\t\n\t\t\n\t\\edgeports{T}{1}{above left}{T1}{}{}{$d_1$};\n\t\\edgeports{T}{2}{below right}{T2}{}{}{$d_2$};\n\t\\edgeports{T}{3}{below right=-0.1cm and 0.1cm}{T3}{}{}{$d_3$};\n\t\\edgeports{T}{4}{above right}{T4}{}{}{$d_4$};\n\t\n\t\\edgeports{G1}{1}{below left = -0.1cm and 0.01cm }{G11}{}{}{$d_1$};\n\t\n\t\\edgeports{G2}{1}{below left}{G1}{2}{below right}{$R$};\t\n\t\\edgeports{G2}{2}{below left = -0.1cm and 0.01cm }{G22}{}{}{$d_2$};\n\t\n\t\\edgeports{G3}{1}{below left}{G2}{3}{below right}{$R$};\t\n\t\\edgeports{G3}{2}{below left = -0.1cm and 0.01cm }{G32}{}{}{$d_3$};\n\t\n\t\\edgeports{G4}{1}{below left}{G3}{3}{below right}{$R$};\t\n\t\\edgeports{G4}{2}{below left = -0.1cm and 0.01cm }{G41}{}{}{$d_4$};\n\n\t\n\\end{tikzpicture}\n}%\n\\end{center}\n\\caption{Tensor network representation of a rank $R$ tensor train decomposition~(nodes represent tensors and an edge between two nodes\nrepresents a contraction between the corresponding modes of the two tensors).}\n\\label{fig:tn.TT}\n\\end{figure}\n\n\\paragraph{Second-order RNNs.}\nA \\emph{second-order recurrent neural network} (2-RNN)~\\cite{giles1990higher,pollack1991induction,lee1986machine}\\footnote{Second-order reccurrent architectures have also been successfully used more recently, see e.g.\\  \\cite{sutskever2011generating} and \\cite{wu2016multiplicative}.} with $n$ hidden units can be defined as a tuple \n$M=(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ where $\\vec{h}_0\\in\\Rbb^n$ is the initial state, $\\Aten\\in\\Rbb^{ n\\times d \\times n }$ is the transition tensor, and\n$\\vecs{\\vvsinfsymbol}\\in \\Rbb^{p\\times n}$ is the output matrix,\nwith $d$ and $p$ being the input and output dimensions respectively. \nA 2-RNN maps any sequence of inputs $\\vec{x}_1,\\cdots,\\vec{x}_k\\in\\Rbb^d$ to\na sequence of outputs $\\vec{y}_1,\\cdots,\\vec{y}_k\\in\\Rbb^p$ defined for any $ t=1,\\cdots,k$ by\n\\begin{equation}\\label{eq:2RNN.definition}\n\\vec{y}_t = z_2(\\vecs{\\vvsinfsymbol}\\vec{h}_t) \\text{ with }\\vec{h}_t = z_1(\\Aten\\ttv{1}\\vec{x}_t\\ttv{2}\\vec{h}_{t-1})\n\\end{equation}\nwhere $z_1:\\Rbb^n\\to\\Rbb^n$ and $z_2:\\Rbb^p\\to\\Rbb^p$ are activation functions.\nAlternatively, one can think of a 2-RNN as computing\na function $f_M:(\\Rbb^d)^*\\to\\Rbb^p$ mapping each input sequence $\\vec{x}_1,\\cdots,\\vec{x}_k$ to the corresponding final output $\\vec{y}_k$.\nWhile $z_1$ and $z_2$ are usually non-linear\ncomponent-wise functions, we consider in this paper the case where both $z_1$ and $z_2$ are the identity, and we refer to\nthe resulting model as a \\emph{linear 2-RNN}.\nFor a linear 2-RNN $M$, the function $f_M$ is multilinear in the sense that, for any integer $l$, its restriction to the domain $(\\Rbb^d)^l$ is\nmultilinear. Another useful observation is that linear 2-RNNs are invariant under change of basis: for any invertible matrix\n$\\P$, the linear 2-RNN $\\tilde{M}=(\\P^{-\\top}\\vec{h}_0,\\Aten\\ttm{1}\\P\\ttm{3}\\P^{-\\top},\\P\\vecs{\\vvsinfsymbol})$ is such that $f_{\\tilde{M}}=f_M$. A linear 2-RNN $M$ with $n$ states is called \\emph{minimal} if its number of hidden units is minimal~(i.e.\\  any linear 2-RNN computing $f_M$ has\nat least $n$ hidden units).\n\n\n\\paragraph{Weighted automata and spectral learning.} \\emph{Vector-valued weighted finite automaton}~(vv-WFA) have\nbeen introduced in~\\cite{rabusseau2017multitask} as a natural generalization of weighted automata from scalar-valued functions\nto vector-valued ones. A $p$-dimensional vv-WFA with $n$ states is a tuple $A=\\vvwa$\nwhere $\\vecs{\\szerosymbol}\\in\\Rbb^n$ is the initial weights vector, $\\vecs{\\vvsinfsymbol}\\in\\Rbb^{p\\times n}$ is the matrix of final weights, \nand $\\mat{A}^\\sigma\\in\\Rbb^{n\\times n}$ is the transition matrix for each symbol $\\sigma$ in a finite alphabet $\\Sigma$.\nA vv-WFA $A$ computes a function $f_A:\\Sigma^*\\to\\Rbb^p$ defined by \n$$f_A(x) =\\vecs{\\vvsinfsymbol} (\\mat{A}^{x_1}\\mat{A}^{x_2}\\cdots\\mat{A}^{x_k})^\\top\\vecs{\\szerosymbol} $$\nfor each word $x=x_1x_2\\cdots x_k\\in\\Sigma^*$. We call a vv-WFA \\emph{minimal} if its number of states\nis minimal. Given a function $f:\\Sigma^*\\to \\Rbb^p$ we denote by $\\rank(f)$ the number of states of a minimal vv-WFA computing $f$~(which is\nset to $\\infty$ if $f$ cannot be computed by a vv-WFA).\n\nThe  spectral learning algorithm for  vv-WFAs relies on the following fundamental theorem relating\nthe rank of a function $f:\\Sigma^*\\to\\Rbb^d$ to its Hankel tensor $\\ten{H} \\in \\Rbb^{\\Sigma^*\\times\\Sigma^*\\times p}$, which is defined\nby $\\ten{H}_{u,v,:} = f(uv)$ for all $u,v\\in\\Sigma^*$.\n\\begin{theorem}[\\cite{rabusseau2017multitask}]\n\\label{thm:fliess-vvWFA}\nLet $f:\\Sigma^*\\to\\Rbb^d$ and let $\\ten{H}$ be its Hankel tensor. Then $\\rank(f) = \\rank(\\tenmat{H}{1})$.\n\\end{theorem}\nThe vv-WFA learning algorithm leverages the fact that the proof of this theorem is constructive: one can recover a vv-WFA computing $f$\nfrom any low rank factorization of $\\tenmat{H}{1}$. In practice, a finite sub-block $\\ten{H}_{\\Pcal,\\Scal} \\in \\Rbb^{\\Pcal\\times \\Scal\\times p}$ of the Hankel tensor  is used\nto recover the vv-WFA, where $\\Pcal,\\Scal\\subset\\Sigma^*$ are finite sets of prefixes and suffixes forming a \\emph{complete basis} for $f$, i.e.\\  such that\n$\\rank(\\tenmatpar{\\ten{H}_{\\Pcal,\\Scal}}{1}) = \\rank(\\tenmat{H}{1})$. More details can be found \nin~\\cite{rabusseau2017multitask}.\n\n\n\\section{A Fundamental Relation between WFAs and Linear 2-RNNs}\\label{sec:WFAs.and.2RNNs}\n\nWe start by unraveling a fundamental connection between vv-WFAs and linear 2-RNNs: vv-WFAs and\nlinear 2-RNNs are expressively equivalent for representing functions defined over sequences of\ndiscrete symbols. \\rev{Moreover, both models have the same capacity in the sense that there is a direct\ncorrespondence between the hidden units of a linear 2-RNN and the states of a vv-WFA computing the same function}. More formally, we have the following theorem. \n\\begin{theorem}\\label{thm:2RNN-vvWFA}\nAny function that can be computed by a vv-WFA with $n$ states can be computed by a linear 2-RNN with $n$ hidden units.\nConversely, any function that can be computed by a linear 2-RNN with $n$ hidden units on sequences of one-hot vectors~(i.e.\\  canonical basis \nvectors) can be computed by a WFA with $n$ states.\n\n\nMore precisely, the WFA $A=\\vvwa$ with $n$ states and the linear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ with\n$n$ hidden units, where $\\Aten\\in\\Rbb^{n\\times \\Sigma \\times n}$ is defined by $\\Aten_{:,\\sigma,:}=\\mat{A}^\\sigma$ for all $\\sigma\\in\\Sigma$, are\nsuch that\n$f_A(\\sigma_1\\sigma_2\\cdots\\sigma_k) = f_M(\\vec{x}_1,\\vec{x}_2,\\cdots,\\vec{x}_k)$ for all sequences of input symbols $\\sigma_1,\\cdots,\\sigma_k\\in\\Sigma$,\nwhere for each $i\\in[k]$ the input vector $\\vec{x}_i\\in\\Rbb^\\Sigma$ is\nthe one-hot encoding of the symbol $\\sigma_i$.\n\n\\end{theorem}\n\nThis result first implies that linear 2-RNNs defined over sequence of discrete symbols~(using one-hot encoding) \\emph{can be provably learned using the spectral\nlearning algorithm for WFAs\/vv-WFAs}; indeed, these  algorithms have been proved to return consistent estimators.\n\\rev{Let us stress again that, contrary to the case of feed-forward architectures, learning recurrent networks with linear activation functions is not a trivial task.}\nFurthermore, Theorem~\\ref{thm:2RNN-vvWFA} reveals that linear 2-RNNs are a natural generalization of classical weighted automata  to functions\ndefined over sequences of continuous vectors~(instead of discrete symbols). This spontaneously raises the question of whether the spectral learning algorithms\nfor WFAs and vv-WFAs can be extended to the general setting of linear 2-RNNs; we show that the answer is in the positive in the next section.\n\n\n\n \n\n\n\n\\section{Spectral Learning of Linear 2-RNNs}\\label{sec:Spectral.learning.of.2RNNs}\nIn this section, we extend the learning algorithm for vv-WFAs to linear 2-RNNs, \\rev{thus at the same time addressing the limitation of the spectral learning algorithm to discrete inputs and  providing the first consistent learning algorithm for linear second-order RNNs.}\n\n\n\n\\subsection{Recovering 2-RNNs from Hankel Tensors}\\label{subsec:SL-2RNN}\nWe first present an identifiability result showing how one can recover a linear 2-RNN computing a function $f:(\\Rbb^d)^*\\to \\Rbb^p$ from observable tensors extracted from some Hankel tensor\nassociated with $f$. Intuitively, we obtain this result by reducing the problem to the one of learning a vv-WFA. This is done by considering the restriction of\n$f$ to canonical basis vectors; loosely speaking, since the domain of this restricted function is isomorphic to $[d]^*$, this allows us to fall back onto the \nsetting of sequences of discrete symbols.\n\nGiven a function $f:(\\Rbb^d)^*\\to\\Rbb^p$, we define its Hankel tensor $\\ten{H}_f\\in \\Rbb^{[d]^* \\times [d]^* \\times p}$ by\n$$(\\ten{H}_f)_{i_1\\cdots i_s,  j_1\\cdots j_t,:} = f(\\ten_{i_1},\\cdots,\\ten_{i_s},\\ten_{j_1},\\cdots,\\ten_{j_t}),$$ \nfor all $i_1,\\cdots,i_s,j_1,\\cdots,j_t\\in [d]$, which is infinite in two\nof its modes. It is easy to see that $\\ten{H}_f$ is also the Hankel tensor associated with the function $\\tilde{f}:[d]^* \\to \\Rbb^p$ mapping any\nsequence $i_1i_2\\cdots i_k\\in[d]^*$ to $f(\\ten_{i_1},\\cdots,\\ten_{i_k})$. Moreover, in the special case where $f$ can be computed by a linear 2-RNN, one\ncan use the multilinearity of $f$ to show that $f(\\vec{x}_1,\\cdots,\\vec{x}_k) = \\sum_{i_1,\\cdots,i_k = 1}^d (\\vec{x}_1)_{i_1}\\cdots(\\vec{x}_l)_{i_k} \\tilde{f}(i_1\\cdots i_k)$,\n\\rev{giving us some intuition on how one could} learn $f$ by learning a vv-WFA computing $\\tilde{f}$ using the spectral learning algorithm. \nThat is, given a large enough sub-block $\\ten{H}_{\\Pcal,\\Scal}\\in \\Rbb^{\\Pcal\\times \\Scal\\times p}$ of $\\ten{H}_f$ for some prefix and suffix sets $\\Pcal,\\Scal\\subseteq [d]^*$, \none should be able to recover a vv-WFA computing $\\tilde{f}$ and consequently a linear 2-RNN computing $f$ using Theorem~\\ref{thm:2RNN-vvWFA}. \n\\rev{Before devoting the remaining of this section to formalize this intuition~(leading to Theorem~\\ref{thm:2RNN-SL}), it is worth \nobserving that while this approach is sound, it is not realistic since it requires observing entries of the Hankel tensor $\\ten{H}_f$, which implies having access to input\/output examples where the inputs are  \\emph{sequences of canonical basis vectors}; This issue will be discussed in more details and addressed in the \nnext section.} \n\n\n\n\\emph{For the sake of clarity, we present the learning algorithm for the particular case where there exists an $L$ such that \nthe prefix and suffix sets consisting of all sequences of length $L$, that is $\\Pcal = \\Scal = \n[d]^L$, forms a complete basis for $\\tilde{f}$}~\\rev{(i.e.\\  the sub-block $\\ten{H}_{\\Pcal,\\Scal}\\in\\Rbb^{[d]^L\\times [d]^L\\times p}$ of the Hankel tensor $\\ten{H}_f$ is\nsuch that $\\rank(\\tenmatpar{\\ten{H}_{\\Pcal,\\Scal}}{1}) = \\rank(\\tenmatpar{\\ten{H}_f}{1})$)}. This assumption allows us to present all the key elements of the algorithm  in a simpler way, the technical details\nneeded to lift this assumption are given in the supplementary material.\n\nFor any integer $l$, we define the finite tensor \n$\\ten{H}^{(l)}_f\\in \\Rbb^{ d\\times \\cdots \\times d\\times p}$ of order $l+1$ by\n$$  (\\ten{H}^{(l)}_f)_{i_1,\\cdots,i_l,:} = f(\\ten_{i_1},\\cdots,\\ten_{i_l}) \\ \\ \\ \\text{for all } i_1,\\cdots,i_l\\in [d].$$ \nObserve that for any integer $l$, the tensor $\\ten{H}^{(l)}_f$ can be obtained by reshaping a finite sub-block of the Hankel tensor $\\ten{H}_f$. \nWhen $f$ is computed by a linear $2$-RNN, we have the useful property that, for any integer $l$,\n\\begin{equation}\\label{eq:}\n f(\\vec{x}_1,\\cdots,\\vec{x}_l)  = \\ten{H}^{(l)}_f \\ttv{1} \\vec{x}_1 \\ttv{2} \\cdots \\ttv{l} \\vec{x}_l\n\\end{equation}\nfor any sequence of inputs $\\vec{x}_1,\\cdots,\\vec{x}_l\\in\\Rbb^d$~(which can be shown using the\nmultilinearity of $f$). Another fundamental property of the tensors $\\ten{H}^{(l)}_f$ is that they are of low tensor train rank. Indeed, for any $l$, one can check that\n$\\ten{H}^{(l)}_f = \\TT{\\Aten\\ttv{1}\\vecs{\\szerosymbol}, \\underbrace{\\Aten, \\cdots, \\Aten}_{l-1\\text{ times}}, \\vecs{\\vvsinfsymbol}^\\top}$~(the tensor network representation of this\ndecomposition is shown in Figure~\\ref{fig:Hl.TT.rank}).\nThis property will be particularly relevant to the learning algorithm we design in the following section, but it is also a fundamental relation\nthat deserves some attention on its own: it implies in particular that, beyond the classical relation between the rank of the Hankel matrix $\\H_f$ and the number states of\na minimal WFA computing $f$, the Hankel matrix possesses a deeper structure intrinsically connecting weighted automata to the tensor\ntrain decomposition. \n\\rev{We now state the main result of this section, showing that a (minimal) linear 2-RNN computing a function $f$ can be exactly recovered from sub-blocks of the Hankel tensor $\\ten{H}_f$.}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{0.08\\textwidth}{%\n\\begin{tikzpicture}\n\t\\input{tikz_tensor_networks}\n\t\\node[tensor](G1){$\\vecs{\\szerosymbol}$};\n\t\\node[draw=none,left of = G1](H){$\\ten{H}^{(4)}_f=\\ \\ \\ $};\n\t\\node[tensor,right = 1cm of G1](G2){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G2](G22){};\n\t\n\t\\node[tensor,right = 1cm of G2](G3){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G3](G32){};\n\t\n\n\t\\node[tensor,right = 1cm of G3](G4){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G4](G42){};\t\n\t\n\t\\node[tensor,right = 1cm of G4](G4-){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G4-](G42-){};\t\n\t\n\t\\node[tensor,right = 1cm of G4-](G5){$\\vecs{\\vvsinfsymbol}$};\n\t\\node[draw=none,below=0.8cm of G5](G51){};\n\t\n\n\t\n\t\\edgeports{G2}{1}{below left = 0.01cm }{G1}{1}{below right}{$n$};\t\n\t\\edgeports{G2}{2}{below left = -0.1cm and 0.01cm }{G22}{}{}{$d$};\n\t\n\t\\edgeports{G3}{1}{below left}{G2}{3}{below right}{$n$};\t\n\t\\edgeports{G3}{2}{below left = -0.1cm and 0.01cm }{G32}{}{}{$d$};\n\t\n\t\\edgeports{G4}{1}{below left}{G3}{3}{below right}{$n$};\t\n\t\\edgeports{G4}{2}{below left = -0.1cm and 0.01cm }{G42}{}{}{$d$};\n\t\n\t\\edgeports{G4-}{1}{below left}{G4}{3}{below right}{$n$};\t\n\t\\edgeports{G4-}{2}{below left = -0.1cm and 0.01cm }{G42-}{}{}{$d$};\n\t\n\t\\edgeports{G5}{1}{below left}{G4-}{3}{below right}{$n$};\t\n\t\\edgeports{G5}{2}{below left = -0.1cm and 0.01cm }{G51}{}{}{$p$};\n\n\t\n\\end{tikzpicture}\n}%\n\\end{center}\n\\caption{Tensor network representation of the TT decomposition of the Hankel tensor $\\ten{H}^{(4)}_f$ induced\nby a linear $2$-RNN $(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$.}\n\\label{fig:Hl.TT.rank}\n\\end{figure}\n\n\n\n\\begin{theorem}\\label{thm:2RNN-SL}\nLet $f:(\\Rbb^d)^*\\to \\Rbb^p$ be a function computed by a minimal linear $2$-RNN  with $n$ hidden units and let\n$L$ be an integer such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\n\nThen, for any $\\P\\in\\Rbb^{d^L\\times n}$ and $\\S\\in\\Rbb^{n\\times d^Lp}$ such that $\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P\\S$, the\nlinear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ defined by\n\n$$\\vecs{\\szerosymbol} = (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1}, \\ \\ \\ \\ \\vecs{\\vvsinfsymbol}^\\top = \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\n$$\n$$\\Aten = (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top$$\nis a minimal linear $2$-RNN computing $f$.\n\\end{theorem}\nFirst observe that such an integer $L$ exists under the assumption that $\\Pcal = \\Scal = \n[d]^L$ forms a complete basis for $\\tilde{f}$.\nIt is also worth mentioning that a necessary condition for $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$ is that\n$d^L\\geq n$, i.e.\\  $L$ must be of the order $\\log_d(n)$.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Hankel Tensors Recovery from Linear Measurements}\\label{subsec:Hankel.tensor.recovery}\n\n\n\nWe showed in the previous section that, given the Hankel tensors $\\ten{H}^{(L)}_f$, $\\ten{H}^{(2L)}_f$ and $\\ten{H}^{(2L+1)}_f$, one can recover \na linear 2-RNN computing $f$ if it exists. This first implies that the class of functions that can be computed by linear 2-RNNs is learnable  in Angluin's\nexact learning model~\\cite{angluin1988queries} where one has access to an oracle that can answer membership queries~(e.g.\\  \\textit{what is the value computed by the target $f$ \non~$(\\vec{x}_1,\\cdots,\\vec{x}_k)$?}) and  equivalence queries~(e.g.\\  \\textit{is my current hypothesis $h$ equal to the target $f$?}). While this fundamental result is \nof significant theoretical interest, assuming access to such an oracle is unrealistic. In this section, we show that a stronger learnability result can be obtained in a more realistic setting,\nwhere we  only\nassume access to randomly generated input\/output examples $((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)})\\in(\\Rbb^d)^*\\times\\Rbb^p$ where $\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)})$.\n\n\nThe key observation is that such an input\/output example $((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)})$ can be seen as a \\emph{linear measurement} of the \nHankel tensor $\\ten{H}^{(l)}$. Indeed, we have\n\\begin{align*}\n\\vec{y}^{(i)} &= f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}) = \\ten{H}^{(l)}_f \\ttv{1} \\vec{x}_1 \\ttv{2} \\cdots \\ttv{l} \\vec{x}_l \\\\\n&= \n\\tenmatgen{\\ten{H}^{(l)}}{l,1}^\\top \\vec{x}^{(i)}\n\\end{align*}\nwhere $\\vec{x}^{(i)} = \\vec{x}^{(i)}_1\\otimes\\cdots \\otimes \\vec{x}_{l}^{(i)}\\in\\Rbb^{d^l}$. Hence,  by regrouping $N$ output examples $\\vec{y}^{(i)}$ into\nthe matrix $\\Ymat\\in\\Rbb^{N\\times p}$ and the corresponding input vectors $\\vec{x}^{(i)}$ into the matrix $\\mat{X}\\in\\Rbb^{N\\times d^l}$,\none can recover $\\ten{H}^{(l)}$ by solving the linear system $\\Ymat = \\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1}$, which has a unique\nsolution whenever $\\mat{X}$ is of full column rank. \nThis  naturally leads to the following theorem, whose proof relies on the fact that\n$\\mat{X}$ will be of full column rank whenever $N\\geq d^l$ and  the components of each $\\vec{x}^{(i)}_j$ for $j\\in[l],i\\in[N]$\nare drawn independently from a continuous distribution over $\\Rbb^{d}$~(w.r.t. the Lebesgue measure).\n\n\n\n\n\n\n\n\\begin{theorem}\\label{thm:learning-2RNN}\nLet $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ be a minimal linear 2-RNN with $n$ hidden units computing a function $f:(\\Rbb^d)^*\\to \\Rbb^p$, and let $L$ be an integer\\footnote{Note that the theorem can be adapted if such an integer $L$ does not exists~(see supplementary material).}\nsuch that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\nSuppose we have access to $3$ datasets\n$D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ for $l\\in\\{L,2L,2L+1\\}$ \nwhere the entries of each $\\vec{x}^{(i)}_j$ are drawn independently from the standard normal distribution and where each\n$\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)})$.\n\nThen, if $N_l \\geq d^l$ for $l =L,\\ 2L,\\ 2L+1$,\nthe linear 2-RNN $M$ returned by Algorithm~\\ref{alg:2RNN-SL} with the least-squares method satisfies $f_M = f$ with probability one.\n\\end{theorem}\n\n\n\\begin{algorithm}[tb]\n   \\caption{\\texttt{2RNN-SL}: Spectral Learning of linear 2-RNNs }\n   \\label{alg:2RNN-SL}\n\\begin{algorithmic}[1]\n   \\REQUIRE Three training datasets $D_L,D_{2L},D_{2L+1}$ with input sequences of length $L$, $2L$ and $2L+1$ respectively, a \\texttt{recovery\\_method}, rank $R$ and learning rate $\\gamma$~(for IHT\/TIHT).\n  \n   \\FOR{$l\\in\\{L,2L,2L+1\\}$}\n   \\STATE\\label{alg.firstline.forloop} Use $D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ to build $\\mat{X}\\in\\Rbb^{N_l\\times d^l}$ with rows $\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}$ for $i\\in[N_l]$ and  $\\Ymat\\in\\Rbb^{N_l\\times p}$ with rows $\\vec{y}^{(i)}$ for $i\\in[N_l]$.\n   \\IF{\\texttt{recovery\\_method} = \"Least-Squares\"}\n   \\STATE\\label{alg.line.lst-sq} $\\ten{H}^{(l)} = \\displaystyle\\argmin_{\\ten{T}\\in \\Rbb^{d\\times\\cdots\\times d\\times p}} \\norm{\\mat{X}\\tenmatgen{\\ten{T}}{l,1} - \\Ymat}_F^2$.\n   \\ELSIF{\\texttt{recovery\\_method} = \"Nuclear Norm\"}\n   \\STATE $\\ten{H}^{(l)} = \\displaystyle\\argmin_{\\ten{T}\\in \\Rbb^{d\\times\\cdots\\times d\\times p}} \\norm{\\tenmatgen{\\ten{T}}{\\ceil{l\/2},l-\\ceil{l\/2} + 1}}_*$ subject to $\\mat{X} \\tenmatgen{\\ten{T}}{l,1} = \\Ymat$.\n   \\label{alg.line.nucnorm}\n   \\ELSIF{\\texttt{recovery\\_method} = \"(T)IHT\"}\n   \\STATE Initialize $\\ten{H}^{(l)} \\in \\Rbb^{d\\times\\cdots\\times d\\times p}$ to $\\mat{0}$.\n   \\REPEAT\\label{alg.line.iht.start}\n   \\STATE\\label{alg.line.iht.gradient} $\\tenmatgen{\\ten{H}^{(l)}}{l,1} = \\tenmatgen{\\ten{H}^{(l)} }{l,1} + \\gamma\\mat{X}^\\top(\\Ymat - \\mat{X}\\tenmatgen{\\ten{H}^{(l)} }{l,1})$\n  \n   \\STATE $\\ten{H}^{(l)} = \\texttt{project}(\\ten{H}^{(l)},R)$~(using either SVD for IHT or TT-SVD for TIHT)\n  \n   \\UNTIL{convergence}\\label{alg.line.iht.end}\n   \\ENDIF\\label{alg.lastline.forloop} \n  \n   \\ENDFOR\n   \\STATE\\label{alg.line.svd} Let $\\tenmatgen{\\ten{H}^{(2L)}}{L,L+1} = \\P\\S$ be a rank $R$ factorization.\n   \\STATE Return the linear 2-RNN $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ where \n   \\begin{align*}\n    \\vecs{\\szerosymbol}\\ &= (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1},\\ \\ \\ \\ \\vecs{\\vvsinfsymbol}^\\top = \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\\\\\n    \\Aten\\ &= (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\n\\end{align*}  \n\\end{algorithmic}\n\\end{algorithm}\n\nA few remarks on this theorem are in order.   The first observation is that the $3$\ndatasets $D_L$, $D_{2L}$ and $D_{2L+1}$ can either be drawn independently or not~(e.g.\\  the sequences in $D_{L}$ can\nbe prefixes of the sequences in $D_{2L}$ but it is not necessary). In particular, the result still holds when  the datasets $D_l$ are constructed from a unique dataset \n$S =\\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_T^{(i)}),(\\vec{y}^{(i)}_1,\\vec{y}^{(i)}_2,\\cdots,\\vec{y}^{(i)}_T)) \\}_{i=1}^{N}$\nof input\/output sequences with $T\\geq 2L+1$, where $\\vec{y}^{(i)}_t = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_t^{(i)})$ for any $t\\in[T]$.\nObserve that having access to such input\/output training sequences is not an unrealistic assumption: for example when training RNNs for\nlanguage modeling the output $\\vec{y}_t$ is the  conditional probability vector of the next symbol, and  for classification tasks the output is\nthe one-hot encoded label for all time steps. Lastly, when the outputs $\\vec{y}^{(i)}$ are noisy, one can solve the least-squares problem\n$\\norm{\\Ymat - \\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1}}^2_F$ to approximate the Hankel tensors; we will empirically evaluate this approach \nin Section~\\ref{sec:xp} and we defer its theoretical analysis  in the noisy setting to future work.\n\n\n\n\\subsection{\\rev{Leveraging the low rank structure of the Hankel tensors}}\nWhile the least-squares method is sufficient to obtain the theoretical guarantees of Theorem~\\ref{thm:learning-2RNN}, it does not leverage\nthe low rank structure of the Hankel tensors $\\ten{H}^{(L)}$, $\\ten{H}^{(2L)}$ and $\\ten{H}^{(2L+1)}$. We now  propose three alternative recovery\nmethods to leverage this structure, whose  sample efficiency will be assessed in a simulation study in Section~\\ref{sec:xp}~(deriving improved sample\ncomplexity guarantees using these methods is left for future work). In the noiseless setting, we first propose to replace solving the linear \nsystem $\\Ymat = \\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1}$ with a nuclear norm minimization problem~(see line~\\ref{alg.line.nucnorm} of Algorithm~\\ref{alg:2RNN-SL}), thus leveraging the\nfact that $\\tenmatgen{\\ten{H}^{(l)}}{\\ceil{l\/2},l-\\ceil{l\/2} + 1}$ is potentially of low matrix rank. We also propose to use iterative hard thresholding~(IHT)~\\cite{jain2010guaranteed}\nand its tensor counterpart TIHT~\\cite{rauhut2017low}, which are based on the classical projected gradient descent algorithm and have shown to be\nrobust to noise in practice. These two methods are implemented in lines~\\ref{alg.line.iht.start}-\\ref{alg.line.iht.end} of Algorithm~\\ref{alg:2RNN-SL}. There,\nthe \\texttt{project} method either projects $\\tenmatgen{\\ten{H}^{(l)}}{\\ceil{l\/2},l-\\ceil{l\/2} + 1}$ onto the manifold of low rank matrices \nusing SVD~(IHT) or projects  $\\ten{H}^{(l)}$ onto the manifold of tensors with TT-rank $R$~(TIHT). \n\n\\input{xp_figures.tex}\n\n\n\\rev{\nThe low rank structure of the Hankel tensors can also be leveraged to improve the scalability of the learning algorithm.\nOne can check that the computational complexity of Algorithm~\\ref{alg:2RNN-SL} is exponential in the maximum sequence length: indeed,\nbuilding the matrix $\\mat{X}$ in line~2 is already in $\\bigo{N_ld^l}$, where $l$ is in turn equal to $L,\\ 2L$ and $2L+1$.\nFocusing on the TIHT recovery method, a careful analysis shows that the computational complexity of the algorithm\nis in \n$$\\bigo{d^{2L+1}\\left(p(TN+R) +R^2\\right) + TL\\max(p,d)^{2L+3}}, $$\nwhere $N=\\max(N_L,N_{2L},N_{2L+1})$ and $T$ is the number of iterations of the loop on line~\\ref{alg.line.iht.start}.\nThus, in its present form, our approach cannot scale to high dimensional inputs and long sequences. However, one can  leverage\nthe low tensor train rank structure of the Hankel tensors to circumvent this issue:\nby storing \nboth the estimates of the Hankel tensors $\\ten{H}^{(l)}$ and the matrices $\\mat{X}$ in TT format~(with decompositions of ranks $R$ and $N$ respectively),\nall the operations needed to implement Algorithm~\\ref{alg:2RNN-SL} with the TIHT recovery method can be performed in time $\\bigo{T(N+R)^3(Ld + p)}$~(more details can be found in the supplementary\nmaterial). By leveraging the tensor train structure, one can thus lift the dependency on $d^{2L+1}$ by paying the price of an increased cubic complexity \nin the number of examples $N$ and the number of states $R$. While the\ndependency on the number of states is not a major issue~($R$ should be negligible w.r.t. $N$), the dependency on $N^3$ can quickly become prohibitive for realistic application scenario. \nFortunately, this issue can  be\ndealt with by using mini-batches of training data for the gradient updates on line~\\ref{alg.line.iht.gradient} instead of the whole dataset $D_l$, in which case the overall complexity\nof Algorithm~\\ref{alg:2RNN-SL} becomes $\\bigo{T(M+R)^3(Ld + p)}$ where $M$ is the mini-batch size~(the overall algorithm in TT format is summarized in Algorithm~\\ref{alg:2RNN-SL-TT} in the supplementary material).\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\input{xp}\n\n\n\n\n\n\\section{Conclusion and Future Directions}\n\nWe proposed the first provable learning algorithm for second-order RNNs with linear activation functions:\nwe showed that linear 2-RNNs are a natural extension of vv-WFAs to the setting of input sequences of \\emph{continuous vectors}~(rather than\ndiscrete symbol) and we extended the vv-WFA spectral learning\nalgorithm to this setting. We believe that the results presented in this paper open a number of exciting and promising research directions on both the\ntheoretical and practical perspectives. We first plan to use the spectral learning estimate as a starting point for\ngradient based methods to train non-linear 2-RNNs. More precisely, linear 2-RNNs can be thought of as 2-RNNs using LeakyRelu activation functions with negative slope $1$, therefore one could use \na linear 2-RNN as initialization before gradually reducing the negative slope parameter during training. The extension of the spectral method to linear 2-RNNs also\nopens the door to scaling up the classical spectral algorithm to problems with large discrete alphabets~(which is a known caveat of the spectral algorithm for WFAs) since\nit allows one to use low dimensional embeddings of large vocabularies~(using e.g.\\  word2vec or latent semantic analysis). From the theoretical perspective, we plan on \nderiving learning guarantees for  linear 2-RNNs in the noisy setting~(e.g.\\  using the PAC learnability framework). Even though it is intuitive that such guarantees should hold~(given\nthe continuity of all operations used in our algorithm), we believe that such an analysis may entail results of independent interest. In particular, analogously to the\nmatrix case  studied in~\\cite{cai2015rop}, obtaining rate optimal convergence rates for the recovery of the low TT-rank Hankel tensors from rank one measurements is an interesting\ndirection; such a result could for example allow one to improve the generalization bounds provided in~\\cite{balle2012spectral} for spectral learning of \ngeneral WFAs. \n\n\n\n\\newpage\n\\subsubsection*{Acknowledgements} This work was done while G. Rabusseau was an IVADO postdoctoral scholar at McGill University. \n{\n\\bibliographystyle{plain}\n\n\n\\section*{\\centering \\LARGE Connecting Weighted Automata and Recurrent Neural Networks through Spectral Learning \\\\ \\vspace*{0.3cm}(Supplementary Material)}\n\n\n\\section{Proofs}\n\\subsection{Proof of Theorem~\\ref{thm:2RNN-vvWFA}}\n\\begin{theorem*}\nAny function that can be computed by a vv-WFA with $n$ states can be computed by a linear 2-RNN with $n$ hidden units.\nConversely, any function that can be computed by a linear 2-RNN with $n$ hidden units on sequences of one-hot vectors~(i.e.\\  canonical basis \nvectors) can be computed by a WFA with $n$ states.\n\n\nMore precisely, the WFA $A=\\vvwa$ with $n$ states and the linear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ with\n$n$ hidden units, where $\\Aten\\in\\Rbb^{n\\times \\Sigma \\times n}$ is defined by $\\Aten_{:,\\sigma,:}=\\mat{A}^\\sigma$ for all $\\sigma\\in\\Sigma$, are\nsuch that\n$f_A(\\sigma_1\\sigma_2\\cdots\\sigma_k) = f_M(\\vec{x}_1,\\vec{x}_2,\\cdots,\\vec{x}_k)$ for all sequences of input symbols $\\sigma_1,\\cdots,\\sigma_k\\in\\Sigma$,\nwhere for each $i\\in[k]$ the input vector $\\vec{x}_i\\in\\Rbb^\\Sigma$ is\nthe one-hot encoding of the symbol $\\sigma_i$.\n\\end{theorem*}\n\\begin{proof}\nWe first show by induction on $k$ that, for any sequence $\\sigma_1\\cdots\\sigma_k\\in\\Sigma^*$, the hidden state $\\vec{h}_k$ computed by $M$~(see\nEq.~\\eqref{eq:2RNN.definition})\non the corresponding one-hot encoded sequence $\\vec{x}_1,\\cdots,\\vec{x}_k\\in\\Rbb^d$\nsatisfies $\\vec{h}_k = (\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_k})^\\top\\vecs{\\szerosymbol}$. The case $k=0$\nis immediate. Suppose the result true for sequences of length up to $k$. One can check easily check that $\\Aten\\ttv{2}\\vec{x}_i = \\mat{A}^{\\sigma_i}$\nfor any index $i$. Using the induction hypothesis it then follows that\n\\begin{align*}\n\\vec{h}_{k+1} &= \\Aten \\ttv{1}\\vec{h}_k \\ttv{2} \\vec{x}_{k+1} = \\mat{A}^{\\sigma_{k+1}}\\ttv{1} \\vec{h}_k = (\\mat{A}^{\\sigma_{k+1}})^\\top \\vec{h}_k\\\\\n&= (\\mat{A}^{\\sigma_{k+1}})^\\top (\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_k})^\\top\\vecs{\\szerosymbol} = (\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_{k+1}})^\\top\\vecs{\\szerosymbol} .\n\\end{align*} \nTo conclude, we thus have\n\\begin{equation*}\nf_M(\\vec{x}_1,\\vec{x}_2,\\cdots,\\vec{x}_k) = \\vecs{\\vvsinfsymbol}\\vec{h}_{k} = \\vecs{\\vvsinfsymbol}(\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_{k}})^\\top\\vecs{\\szerosymbol} = f_A(\\sigma_1\\sigma_2\\cdots\\sigma_k).\\qedhere\n\\end{equation*}\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:2RNN-SL}}\n\n\n\\begin{theorem*}\nLet $f:(\\Rbb^d)^*\\to \\Rbb^p$ be a function computed by a minimal linear $2$-RNN  with $n$ hidden units and let\n$L$ be an integer such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\n\nThen, for any $\\P\\in\\Rbb^{d^L\\times n}$ and $\\S\\in\\Rbb^{n\\times d^Lp}$ such that $\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P\\S$, the\nlinear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ defined by\n$$\\vecs{\\szerosymbol} = (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1},\\ \\ \\ \\ \\Aten = (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top,\\ \\ \\ \\ \n\\vecs{\\vvsinfsymbol}^\\top = \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}$$\nis a minimal linear $2$-RNN computing $f$.\n\\end{theorem*}\n\\begin{proof}\nLet $\\P\\in\\Rbb^{d^L\\times n}$ and $\\S\\in\\Rbb^{n\\times d^Lp}$ be such that $\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P\\S$\nDefine the tensors \n$$\\Pten^* = \\TT{\\Aten^\\star\\ttv{1}\\vecs{\\szerosymbol}^\\star, \\underbrace{\\Aten^\\star, \\cdots, \\Aten^\\star}_{L-1\\text{ times}}, \\mat{I}_n}\\in\\Rbb^{d\\times\\cdots\\times d\\times n}\\ \\ \\ \\  \n\\text{ and }\\ \\ \\ \\ \n\\Sten^* = \\TT{\\mat{I}_n,\\underbrace{\\Aten^\\star, \\cdots, \\Aten^\\star}_{L\\text{ times}}, \\vecs{\\vvsinfsymbol}^\\star}\\in\\Rbb^{n\\times d\\times\\cdots\\times d\\times p}$$ \nof order $L+1$ and $L+2$ respectively, and let $\\P^\\star = \\tenmatgen{\\Pten^*}{l,1} \\in\\Rbb^{d^l\\times n}$ \nand $\\S = \\tenmatgen{\\Sten^*}{1,L+1}  \\in\\Rbb^{n\\times d^lp}$. Using the identity $\\ten{H}^{(j)}_f = \\TT{\\Aten\\ttv{1}\\vecs{\\szerosymbol}, \\underbrace{\\Aten, \\cdots, \\Aten}_{j-1\\text{ times}}, \\vecs{\\vvsinfsymbol}^\\top}$ for\nany $j$, one can easily check the following identities:\n\\begin{gather*}\n\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P^\\star\\S^\\star,\\ \\  \\ \\ \\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1}= \\Aten^\\star \\ttm{1} \\P^\\star \\ttm{3} (\\S^\\star)^\\top,\\\\\n\\tenmatgen{\\ten{H}^{(L)}_f}{L,1} = \\P^\\star(\\vecs{\\vvsinfsymbol}^\\star)^\\top, \\ \\ \\ \\ \\ \\ \\\n\\tenmatgen{\\ten{H}^{(L)}_f}{L+1} = (\\S^\\star)^\\top\\vecs{\\szerosymbol}.\n\\end{gather*}\n\nLet $\\mat{M} = \\P^\\dagger\\P^\\star$. We will show that $\\vecs{\\szerosymbol} = \\mat{M}^{-\\top}\\vecs{\\szerosymbol}^\\star$, $\\Aten = \\Aten^\\star \\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top}$ and\n$\\vecs{\\vvsinfsymbol} = \\mat{M}\\vecs{\\vvsinfsymbol}^\\star$, which will entail the results since linear 2-RNN are invariant under change of basis~(see Section~\\ref{sec:prelim}). First observe that $\\mat{M}^{-1} = \\S^\\star\\S^\\dagger$. Indeed,\nwe have\n$\\P^\\dagger\\P^\\star\\S^\\star\\S^\\dagger = \\P^\\dagger\\tenmatgen{\\ten{H}^{(2l)}_f}{l,l+1}\\S^\\dagger = \\P^\\dagger\\P\\S\\S^\\dagger = \\mat{I}$\nwhere we used the fact that $\\P$~(resp. $\\S$) is of full column rank~(resp. row rank) for the last equality. \n\nThe following derivations then follow from basic tensor algebra:\n\\begin{align*}\n\\vecs{\\szerosymbol} \n&= \n(\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1} \n=\n(\\S^\\dagger)^\\top (\\S^\\star)^\\top\\vecs{\\szerosymbol}\n=\n(\\S^\\star\\S^\\dagger)^\\top\n=\n\\mat{M}^{-\\top}\\vecs{\\szerosymbol}^\\star,\\\\\n\\ \\\\\n\\Aten \n&= \n(\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\\\\\n&=\n(\\Aten^\\star \\ttm{1} \\P^\\star \\ttm{3} (\\S^\\star)^\\top)\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\\\\\n&=\n\\Aten^\\star \\ttm{1} \\P^\\dagger\\P^\\star \\ttm{3} (\\S^\\star\\S^\\dagger)^\\top =  \\Aten^\\star \\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top},\\\\\n\\ \\\\\n\\vecs{\\vvsinfsymbol}^\\top \n&= \n\\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\n=\n\\P^\\dagger\\P^\\star(\\vecs{\\vvsinfsymbol}^\\star)^\\top\n=\n\\mat{M}\\vecs{\\vvsinfsymbol}^\\star,\n\\end{align*}\nwhich concludes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:learning-2RNN}}\n\n\n\\begin{theorem*}\nLet $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ be a minimal linear 2-RNN with $n$ hidden units computing a function $f:(\\Rbb^d)^*\\to \\Rbb^p$, and let $L$ be an integer\\footnote{Note that the theorem can be adapted if such an integer $L$ does not exists~(see supplementary material).}\nsuch that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\n\nSuppose we have access to $3$ datasets\n$D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ for $l\\in\\{L,2L,2L+1\\}$ \nwhere the entries of each $\\vec{x}^{(i)}_j$ are drawn independently from the standard normal distribution and where each\n$\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)})$.\n\nThen, whenever $N_l \\geq d^l$ for each $l\\in\\{L,2L,2L+1\\}$,\nthe linear 2-RNN $M$ returned by Algorithm~\\ref{alg:2RNN-SL} with the least-squares method satisfies $f_M = f$ with probability one.\n\\end{theorem*}\n\\begin{proof}\nWe just need to show for each $l\\in \\{L,2L,2L+1\\}$ that, under the hypothesis of the Theorem, the Hankel tensors $\\hat{\\ten{H}}^{(l)}$ computed in line~\\ref{alg.line.lst-sq} of\nAlgorithm~\\ref{alg:2RNN-SL} are equal to the true Hankel tensors $\\ten{H}^{(l)}$ with probability one. Recall that these tensors are computed by solving the least-squares\nproblem\n$$\\hat{\\ten{H}}^{(l)} = \\argmin_{T\\in \\Rbb^{d\\times\\cdots\\times d\\times p}} \\norm{\\mat{X}\\tenmatgen{\\ten{T}}{l,1} - \\Ymat}_F^2$$\nwhere $\\mat{X}\\in\\Rbb^{N_l\\times d_l}$ is the matrix  with rows $\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}$ for each $i\\in[N_l]$. Since  $\\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1} = \\Ymat$ and since the solution\nof the least-squares problem is unique as soon as $\\mat{X}$ is of full column rank, we just need to show that this is the case with probability one\nwhen the entries of the vectors $\\vec{x}^{(i)}_j$ are drawn at random from a standard normal distribution. The result will  then directly follow\nby applying Theorem~\\ref{thm:2RNN-SL}.\n\nWe will show that the set \n$$\\Scal = \\{ (\\vec{x}_1^{(i)},\\cdots, \\vec{x}_l^{(i)}) \\mid \\ i\\in[N_l],\\ dim(span(\\{ \\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)} \\})) < d^l\\} $$\nhas Lebesgue measure $0$ in $((\\Rbb^d)^{l})^{N_l}\\simeq \\Rbb^{dlN_l}$ as soon as $N_l \\geq d^l$, which will imply that it has probability $0$ under any continuous probability, hence\nthe result. For any  $S=\\{(\\vec{x}_1^{(i)},\\cdots, \\vec{x}_l^{(i)})\\}_{i=1}^{N_l}$, we denote  by $\\mat{X}_S\\in\\Rbb^{N_l\\times d^l}$  the matrix  with rows $\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}$.\nOne can easily check that $S\\in\\Scal$ if and only if $\\mat{X}_S$ is of rank strictly less than $d^l$, which is equivalent to the determinant of \n$\\mat{X}_S^\\top\\mat{X}_S$ being equal to $0$. Since this determinant is a polynomial in the entries of the vectors $\\vec{x}_j^{(i)}$, $\\Scal$ is an algebraic\nsubvariety of $\\Rbb^{dlN_l}$.\nIt is then easy to check that the polynomial $det(\\mat{X}_S^\\top\\mat{X}_S)$ is not uniformly 0 when $N_l \\geq d^l$. Indeed, \nit suffices to choose the vectors $\\vec{x}_j^{(i)}$ such that the family  $(\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)})_{n=1}^{N_l}$ spans the whole space \n$\\Rbb^{d^l}$~(which is possible since we can choose arbitrarily any of the $N_l\\geq d^l$ elements of this family),  hence the result. \nIn conclusion, $\\Scal$ is a proper algebraic subvariety of $\\Rbb^{dlN_l}$ and hence has Lebesgue\nmeasure zero~\\cite[Section 2.6.5]{federer2014geometric}.\n\n\n\n\\end{proof}\n\n\n\\section{Lifting the simplifying assumption}\\label{app:lift}\nWe now show how all our results still hold when there does not exist an $L$ such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\nRecall that this simplifying assumption followed from assuming that the sets $\\Pcal=\\Scal=[d]^L$ form a complete basis for the function\n$\\tilde{f}:[d]^*\\to \\Rbb^p$ defined by $\\tilde{f}(i_1i_2\\cdots i_k) = f(\\ten_{i_1},\\ten_{i_2},\\cdots,\\ten_{i_k})$. \nWe first show that there always exists an integer $L$ such that $\\Pcal=\\Scal=\\cup_{i\\leq L} [d]^i$ forms a complete basis for $\\tilde{f}$.\n Let $M = (\\vecs{\\szerosymbol}^\\star,\\Aten^\\star,\\vecs{\\vvsinfsymbol}^\\star)$  be a linear 2-RNN with $n$ hidden units computing\n$f$~(i.e.\\  such that $f_M=f$). It follows from Theorem~\\ref{thm:2RNN-vvWFA} and from the discussion  at the beginning of Section~\\ref{subsec:SL-2RNN}\nthat there exists a vv-WFA computing $\\tilde{f}$ and it is easy to check that $\\rank(\\tilde{f}) = n$.  This implies $\\rank(\\tenmatpar{\\ten{H}_f}{1}) = n$ by\nTheorem~\\ref{thm:fliess-vvWFA}. Since  $\\Pcal=\\Scal=\\cup_{i\\leq l} [d]^i$ converges to $[d]^*$ as $l$ grows to infinity, there exists an $L$ such that \nthe finite sub-block $\\tilde{\\ten{H}}_f \\in \\Rbb^{\\Pcal\\times\\Scal\\times p}$ of $\\ten{H}_f\\in \\Rbb^{[d]^*\\times[d]^*\\times p}$\nsatisfies $\\rank(\\tenmatpar{\\tilde{\\ten{H}}_f}{1}) = n$, i.e.\\  such that\n$\\Pcal=\\Scal=\\cup_{i\\leq L} [d]^i$ forms a complete basis for $\\tilde{f}$. \n\nNow consider the finite sub-blocks $\\tilde{\\ten{H}}^{+}_f\\in \\Rbb^{\\Pcal \\times [d] \\times \\Scal \\times p}$ and  $\\tilde{\\H}^{-}_f\\in \\Rbb^{\\Pcal  \\times p}$ of $\\ten{H}_f$  defined by\n$$(\\tilde{\\ten{H}}^{+}_f)_{u,i,v,:}=\\tilde{f}(uiv),\\ \\ \\ \\text{and} (\\tilde{\\H}^{-}_f)_{u,:}=f(u)$$\nfor any $u\\in \\Pcal=\\Scal$ and any $i\\in [d]$. One can check that Theorem~\\ref{thm:2RNN-SL} holds by replacing \\emph{mutatis mutandi}\n$\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}$ by $\\tenmatpar{\\tilde{\\ten{H}}_f}{1}$, $\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1}$ by  $\\tilde{\\ten{H}}^{+}_f$, $\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}$ by $\\tilde{\\H}^{-}_f$\nand $\\tenmatgen{\\ten{H}^{(L)}_f}{L+1}$ by $\\vectorize{\\tilde{\\H}^{-}_f}$.\n\nTo conclude, it suffices to observe that both $\\tilde{\\ten{H}}^{+}_f$ and  $\\tilde{\\H}^{-}_f$ can be constructed\nfrom the entries for the tensors $\\ten{H}^{(l)}$ for $1\\leq l \\leq 2L+1$, which can be recovered~(or estimated in the noisy setting) using\nthe techniques described in Section~\\ref{subsec:Hankel.tensor.recovery}~(corresponding to lines \\ref{alg.firstline.forloop}-\\ref{alg.lastline.forloop} of\nAlgorithm~\\ref{alg:2RNN-SL}).\n\nWe thus showed that linear 2-RNNs can be provably learned even when  there does not exist an $L$ such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$. In this\nsetting, one needs to estimate enough of the tensors $\\ten{H}^{(l)}$ to reconstruct a complete sub-block $\\tilde{\\ten{H}}_f$ of the Hankel tensor \n$\\ten{H}$~(along with the corresponding tensor $\\tilde{\\ten{H}}^{+}_f$ and matrix $\\tilde{\\H}^{-}_f$)\nand recover the linear 2-RNN by applying Theorem~\\ref{thm:2RNN-SL}. In addition, one needs to have access to sufficiently large datasets \n$D_l$ for each $l\\in [2L+1]$ rather than only the three datasets mentioned in Theorem~\\ref{thm:learning-2RNN}.  However the data requirement remains the\nsame in the case where we assume that each of the datasets $D_l$ is constructed from a unique training  dataset \n$S =\\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_T^{(i)}),(\\vec{y}^{(i)}_1,\\vec{y}^{(i)}_2,\\cdots,\\vec{y}^{(i)}_T)) \\}_{i=1}^{N}$\nof input\/output sequences.\n\n\\section{Leveraging the tensor train structure for computational efficiency}\nThe overall learning algorithm using the TIHT recovery method in TT format is summarized in Algorithm~\\ref{alg:2RNN-SL-TT}. The key ingredients to improve the complexity of Algorithm~\\ref{alg:2RNN-SL} are (i) to estimate the gradient using mini-batches of data and  (ii) to directly use the TT format to represent and perform operations on the tensors $\\ten{H}^{(l)}$ and the tensors $\\Xten^{(l)}\\in\\Rbb^{M\\times d\\times \\cdots \\times d}$ defined by\n\\begin{equation}\\label{eq:Xten}\n \\Xten_{i,:,\\cdots,:}= \\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}\\ \\  \\ \\text{for }i\\in[M]\n\\end{equation}\nwhere $M$ is the size of a mini-batch of training data~($\\ten{H}^{(l)}$ is of TT-rank $R$ by design and it can easily be shown that $\\Xten^{(l)}$ is of TT-rank at most $M$, cf. Eq.~\\eqref{eq:XtenTT}). \nThen, all the operations of the algorithm can be expressed in terms of these tensors and performed efficiently in TT format.\nMore precisely, the products and sums needed to compute the gradient update on line~\\ref{algtt.line.iht.gradient} can be performed in~$\\bigo{(R+M)^2(ld+p)+(R+M)^3d}$. After the gradient update, the tensor $\\ten{H}^{(l)}$ has TT-rank at most $(M+R)$ but can be efficiently projected back to a tensor of TT-rank $R$ using the tensor train rounding operation~\\cite{oseledets2011tensor} in $\\bigo{(R+M)^3(ld+p)}$~(which is the operation dominating the complexity of the whole algorithm).\nThe subsequent operations on line~\\ref{line.algtt.return} can be performed efficiently in the TT format in~$\\bigo{R^3d+R^2p}$~(using the method described in~\\cite{klus2018tensor} to compute the pseudo-inverses of the matrices $\\P$ and $\\S$). \nThe overall complexity of Algorithm~2 is thus in~$\\bigo{T(R+M)^3(Ld+p)}$ where $T$ is the number of iterations of the inner loop.\n\n\n\\begin{algorithm}\n   \\caption{\\texttt{2RNN-SL-TT}: Spectral Learning of linear 2-RNNs  \\textbf{in tensor train format}}\n   \\label{alg:2RNN-SL-TT}\n\\begin{algorithmic}[1]\n   \\REQUIRE Three training datasets $D_L,D_{2L},D_{2L+1}$ with input sequences of length $L$, $2L$ and $2L+1$ respectively, rank $R$, learning rate $\\gamma$ and\n   mini-batch size $M$.\n  \n   \\FOR{$l\\in\\{L,2L,2L+1\\}$}\n   \\STATE Initialize all cores of the rank $R$ TT-decomposition $\\ten{H}^{(l)} = \\TT{\\Gten^{(l)}_1,\\cdots,\\Gten^{(l)}_{l+1}} \\in \\Rbb^{d\\times\\cdots\\times d\\times p}$ to $\\mat{0}$.\\\\\n   \/\/ \\emph{Note that all the updates of  $\\ten{H}^{(l)}$ stated below are in effect applied directly to the core tensors $\\Gten^{(l)}_k$, i.e. the tensor $\\ten{H}^{(l)}$ is never \n   explicitely constructed.}\n   \n   \\REPEAT\\label{algtt.line.iht.start}\n   \\STATE Subsample a minibatch $$\\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{M}\\subset(\\Rbb^d)^l\\times \\Rbb^p$$ of size $M$ from $D_l$.\n   \\STATE Compute the rank $M$ TT-decomposition of the tensor $\\Xten=\\Xten^{(l)}$~(defined in Eq.~\\eqref{eq:Xten}), which is given by\n   \\begin{equation}\\label{eq:XtenTT}\n       \\Xten = \\TT{\\mat{I}_M, \\Aten_1,\\cdots,\\Aten_l} \\text{ where the cores are defined by } (\\Aten_k)_{i,:,j}=\\delta_{ij}\\vec{x}^{(i)}_k \\ \\ \\text{ and } \\ \\  (\\Aten_l)_{i,:} = \\vec{x}^{(i)}_k\n   \\end{equation}\n   for all $1\\leq k <l$, $i,j\\in[M]$, where $\\delta$ is the Kroencker symbol.\n   \\STATE\\label{algtt.line.iht.gradient} Perform the gradient update using efficient addition and product operations in TT format~(see~\\cite{oseledets2011tensor}):\n   $$\\tenmatgen{\\ten{H}^{(l)}}{l,1} = \\tenmatgen{\\ten{H}^{(l)} }{l,1} + \\gamma\\tenmatgen{\\Xten}{1,l}^\\top(\\Ymat - \\tenmatgen{\\Xten}{1,l}\\tenmatgen{\\ten{H}^{(l)} }{l,1})$$\n   \\STATE Project the Hankel tensor $\\ten{H}^{(l)}$~(which is now of rank at most $R+M$) back onto the manifold of tensor of TT-rank $R$ using the TT rounding\n   operation~(see again~\\cite{oseledets2011tensor}):\n   $$\\ten{H}^{(l)} = \\texttt{TT-rounding}(\\ten{H}^{(l)},R)$$\n  \n   \\UNTIL{convergence}\\label{algtt.line.iht.end}\n  \n   \\ENDFOR\n   \\STATE\\label{algtt.line.svd} Let $\\P = \\tenmatgen{\\TT{\\Gten^{(2L)}_1,\\cdots,\\Gten^{(2L)}_L,\\mat{I}_R}}{L,1}$ and\n   $\\S = \\tenmatgen{\\TT{\\mat{I}_R,\\Gten^{(2L)}_{L+1},\\cdots,\\Gten^{(2L)}_{2L+1}}}{1,L+1}$~(observe that\n   $\\tenmatgen{\\ten{H}^{(2L)}}{L,L+1} = \\P\\S$ is a rank $R$ factorization).\n   \\STATE\\label{line.algtt.return} Return the linear 2-RNN $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ where \n   \\begin{align*}\n    \\vecs{\\szerosymbol}\\ &= (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1}\\\\\n    \\Aten\\ &= (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\\\\\n\\vecs{\\vvsinfsymbol}^\\top &= \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\n\\end{align*}  \nby performing efficient computations in TT format for the products~\\cite{oseledets2011tensor} and pseudo-inverses~(see e.g. \\cite{klus2018tensor}).\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Real Data Experiment on Wind Speed Prediction}\\label{app:real}\nBesides the synthetic data experiments we showed in the paper, we have also conducted experiments on real data. The data that we use for the these experiments is from TUDelft\\footnote{http:\/\/weather.tudelft.nl\/csv\/}. Specifically, we use the data from Rijnhaven station as described in~\\cite{lin2016short}, which proposed a regression automata model and performed various experiments on the dataset we mentioned above. The data contains wind speed and related information at the Rijnhaven station from  2013-04-22 at 14:55:00 to 2018-10-20 at 11:40:00 and was collected every five minutes. To compare to the results in~\\cite{lin2016short}, we strictly followed the data preprocessing procedure described in the paper. We use the data from 2013-04-23 to 2015-10-12 as training data and the rest as our testing data.  The paper uses SAX as a preprocessing method to discretize the data. However, as there is no need to discretize data for our algorithm, we did not perform this procedure. For our method, we set the length $L = 3$ and we use the general algorithm described in Appendix~\\ref{app:lift}. We calculate hourly averages of the wind speed, and predict one\/three\/six hour(s) ahead, as in~\\cite{lin2016short}.  For our methods we use a linear 2-RNN with 10 states. Averages over 5 runs of this experiment for one-hour-ahead, three-hour-ahead, six-hour-ahead prediction error can be found in Table~\\ref{one_hour}, \\ref{three_hours} and \\ref{six_hours}. The results for RA, RNN and persistence are taken directly from~\\cite{lin2016short}. We see that while TIHT+SGD performs slightly worse than ARIMA and RA for one-hour-ahead prediction, it outperforms all other methods for three-hours and six-hours ahead predictions~(and the superiority w.r.t. other methods increases as the prediction horizon gets longer). \n\n\n\n\\begin{table}[h]\n\\caption{One-hour-ahead Speed Prediction Performance Comparisons}\n\\centering\n\\begin{tabular}{c|cccccc}\nMethod & TIHT     & TIHT+SGD & \\begin{tabular}[c]{@{}c@{}}Regression \\\\ Automata\\end{tabular} & ARIMA & RNN & Persistence\\\\ \\hline\nRMSE   & 0.573  & 0.519   & 0.500    & \\textbf{0.496 }& 0.606 & 0.508                                                    \\\\\nMAPE   & 21.35  & 18.79    & \\textbf{18.58}   & 18.74 & 24.48 & 18.61                                                    \\\\\nMAE    & 0.412  & 0.376    & 0.363    & \\textbf{0.361} & 0.471 & 0.367                                                    \n\\end{tabular}%\n\\label{one_hour}\n\\end{table}\n\\begin{table}[h]\n\\caption{Three-hour-ahead Speed Prediction Performance Comparisons}\n\\centering\n\\begin{tabular}{c|cccccc}\nMethod & TIHT     & TIHT+SGD & \\begin{tabular}[c]{@{}c@{}}Regression \\\\ Automata\\end{tabular} & ARIMA & RNN & Persistence\\\\ \\hline\nRMSE   & 0.868  & \\textbf{0.854}    & 0.872  & 0.882 & 1.002 & 0.893                                                       \\\\\nMAPE   & 33.98  & \\textbf{31.70}    & 32.52 & 33.165 & 37.24 & 33.29                                                      \\\\\nMAE    & 0.632  & \\textbf{0.624}    & 0.632 &0.642 & 0.764 & 0.649                                                       \n\\end{tabular}\n\\label{three_hours}\n\\end{table}\n\\begin{table}[h]\n\\caption{Six-hour-ahead Speed Prediction Performance Comparisons}\n\\centering\n\\begin{tabular}{c|cccccc}\nMethod & TIHT   & TIHT+SGD & \\begin{tabular}[c]{@{}c@{}}Regression \\\\ Automata\\end{tabular} & ARIMA & RNN & Persistence\\\\ \\hline\nRMSE   & 1.234  & \\textbf{1.145}    & 1.205  & 1.227 & 1.261 & 1.234                                                      \\\\\nMAPE   & 49.08  & \\textbf{44.88}    & 46.809  &48.02 & 47.03 & 48.11                                                      \\\\\nMAE    & 0.940  & \\textbf{0.865}    & 0.898  & 0.919 & 0.944 & 0.923            \n\\end{tabular}\n\\label{six_hours}\n\\end{table}\n\n\n\n\n\n\n\\section{Experiments}\\label{sec:xp}\n\n\n\n\n\n\n\\rev{\nIn this section, we perform experiments\\footnote{\\url{https:\/\/github.com\/litianyu1993\/learning_2RNN}} on two toy examples to compare how the choice of the recovery method~(\\texttt{LeastSquares}, \\texttt{NuclearNorm}, \\texttt{IHT} and \\texttt{TIHT}) affects the sample efficiency of Algorithm~\\ref{alg:2RNN-SL}. Additional experiments on real data can be found in Appendix~\\ref{app:real}.\nWe also include comparisons with RNNs with long short term memory~(LSTM) units~\\cite{hochreiter1997long} and report the performances\nobtained by  refining  the solution returned by our algorithm~(with the TIHT recovery method) using stochastic gradient descent~(\\texttt{TIHT+SGD}). \n}\n\n\n\n\n\n\n\nWe perform two experiments. In the first one, we randomly generate a linear 2-RNN with $5$ units computing a function $f:\\Rbb^3\\to\\Rbb^2$ by \ndrawing the entries of all parameters  $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$  independently  from the normal distribution\n$\\Ncal(0,0.2)$.\nThe training data consists of $3$ independently drawn sets \n$D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ for $l\\in\\{L,2L,2L+1\\}$ with $L=2$,\nwhere  each $\\vec{x}^{(i)}_j\\sim\\Ncal(\\vec{0},\\mat{I})$ and where the outputs can be noisy, i.e.\\  \n$\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}) + \\vecs{\\xi}^{(i)}$ where $\\vecs{\\xi}^{(i)}\\sim\\Ncal(0,\\sigma^2)$ for some noise variance $\\sigma^2$. \nIn the second experiment, the goal is to learn a simple arithmetic function computing the sum of the running differences between the two components of a sequence\nof $2$-dimensional vectors, i.e.\\  $f(\\vec{x}_1,\\cdots,\\vec{x}_k) = \\sum_{i=1}^k \\v^\\top\\vec{x}_i$ where $\\v^\\top = (-1\\ \\ 1)$. The $3$ training datasets are generated \nusing the same process as above and a constant entry equal to one is added to all the input vectors to encode a bias \nterm~(one can check that the resulting function can be computed by a linear 2-RNN with $2$ hidden units). \n\nWe run the experiments for different sizes of training data ranging from $N=20$ to $N=20,000$~(we set $N_L=N_{2L}=N_{2L+1}=N$) \nand we compare the different methods in terms of mean squared error~(MSE)\n on a test set of $1,000$ sequences of length $6$ generated in the same\nway as the training data~(note that the training data only contains sequences of length up to $5$). \n\n\\rev{We report the performances of (non-linear) RNNs with a single layer of LSTM with $20$  hidden units~(with $\\mathrm{tanh}$ activation \nfunctions\\footnote{We also tried training LSTMs with linear recurrent activation functions on the two tasks but they always performed worse than non-linear ones.}) and one fully-connected output layer, trained using\nthe Adam optimizer~\\cite{kingma2014adam} with learning rate 0.001. We also use Adam with learning rate 0.001 to refine the models returned by TIHT with stochastic gradient\ndescent SGD (we tried directly training a linear 2-RNN from random initializations using SGD as well but this approach always failed to return a good model). \nThe IHT\/TIHT methods sometimes returned aberrant models~(due to numerical instabilities), we used the following scheme to circumvent this issue: \nwhen the training MSE of the hypothesis was greater than the one of the zero function,\nthe zero function was returned instead (we applied this scheme to all other methods in the experiments).}\n\n\nThe results are reported in Figure~\\ref{fig:xp.random} and~\\ref{fig:xp.addition} where we see that all recovery methods of Algorithm~\\ref{alg:2RNN-SL} lead to consistent estimates of the\ntarget function given enough training data. This is the case even in the presence of noise~(in which case more samples are needed to achieve the same accuracy, as expected).\nWe can also see that \\texttt{IHT} and \\texttt{TIHT} are overall more sample efficient than the other methods~(especially with noisy data), showing\nthat taking the low rank structure of the Hankel tensors into account is profitable. Moreover, \\texttt{TIHT} tends to perform better than its matrix counterpart, \nconfirming our intuition that leveraging the tensor train \nstructure is beneficial. \n\\rev{While LSTMs obtain good performances on the addition task, they struggle to recover the random linear 2-RNN in the first task~(despite our efforts at hyper-parameter tuning and architecture search). \nIn the meantime, refining the TIHT models using SGD almost always leads to significant improvements~(especially under the noisy setting), matching or outperforming the\nperformances of RNNs on the two tasks.\nLastly, we show the effect of rank mis-specification in Figure~\\ref{fig:xp.misrank}: as one can expect, when the rank parameter $R$ \nis over-estimated Algorithm~\\ref{alg:2RNN-SL} still converges to the target function but it requires  more samples~(when the rank parameter was underestimated all  algorithms did not learn at all). \n}\n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace*{0cm}\\includegraphics[width=0.5\\textwidth]{Mis_rank.jpeg}\n\\end{center}\n\\vspace*{-0.5cm}\n\\caption{ Comparison between different rank settings in terms of average MSE for the second experiment~(learning a simple arithmetic function) in the noisy\nsetting~($\\sigma^2=1$).}\n\\label{fig:xp.misrank}\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","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect_intro}\n\n \nCoronal mass ejections (CMEs) are associated with magnetic instabilities occurring in the solar corona, and they are\nexpelled during solar eruptive flare events.\nAs a consequence of these instabilities, large quantities of plasma and magnetic fields are expelled into the interplanetary space (IP), which can be observed by coronagraphs as CMEs. \nWhen their interplanetary manifestations (interplanetary CMEs or ICMEs) arrive at Earth, the observed intensity of energetic particles (\\textit{{\\it e.g.},}\\ Galactic Cosmic Rays, GCR) is modified.\nWhen they arrive at other planets (\\textit{{\\it e.g.},}\\ Mars) energetic particle fluxes can be also modified.\nICMEs\ncan also produce perturbations in the magnetosphere, triggering geomagnetic storms.\n \nA subset of ICMEs include a Magnetic Clouds (MC) which is characterised by a low proton temperature and an enhanced magnetic field intensity $|\\vec{B}|$ with a smooth rotation of its vector components, resembling that of a flux rope \\citep{burlaga81}. \nThe solar wind following an MC is expected to be perturbed, with characteristics similar to a wake.\n\n \nTypically, MCs travel faster than the local Alfv\\'en waves in the solar wind reference frame, producing a fast MHD shock ahead of them.\nThis shock produces an intermediate region of compressed plasma between the shock interface and the MC leading edge. \nThis region is characterised by high temperatures produced by the conversion of macroscopic to thermal energy at the shock, and therefore it generally presents high plasma $\\beta$ values.\nTypically, sheath regions also present large magnetic intensity and a high level of magnetic fluctuations.\n\n \nFluctuations around the shocks in the solar wind are\nobserved after the shock (downstream) as typically observed in fluids, but also upstream (before the arrival of the shock).  \nThis last case is mainly due to beam instabilities that are induced by shock-accelerated particles. \nThese instabilities can generate waves in the upstream region\n\\citep[\\textit{{\\it e.g.},}\\ ][]{blancocano_etal11,wang15,strumik15}, so that an increased level of fluctuations is expected before and after the shocks associated with ICMEs.\n\n  \nDuring their propagation away from the Sun, MCs interact with the plasma and magnetic field encountered in the interplanetary medium.\nThis interaction can induce reconnection between the sheath and the MC magnetic fields.  \nThis implies changes of the magnetic connectivity of the flux rope \\citep{mccomas88}, and a consequent peel off (erosion) of magnetic flux from the leading edge of the MC.\nThis also implies the formation of a 'back region' which involves field lines that were part of the MC before the erosion \\citep{dasso06,Ruffenach15}. \nThe back region is located in the MC wake and it typically has mixed plasma and magnetic field properties of ambient solar wind and MC. \n\n  \nSheaths in front of MCs significantly differ from planetary magneto-sheaths mainly because ICMEs not only propagate but also expand into the IP medium \\citep[\\textit{{\\it e.g.},}\\ ][]{Demoulin09,Gulisano10}.\nIn particular, the lateral deflection of the solar wind away from the nose of an MC is reduced due to the expansion, and the solar wind tends to pile up in front of an MC instead of flowing around it \\citep{Siscoe08}.\nThis effect is more important near the corona, where the expansion is stronger, than in the interplanetary medium \\citep{Das11}.\nMoreover, the drainage toward the sides can be enhanced due to magnetic reconnection between the magnetic field of the sheath and the flux rope.\n\n \nSummarising, a typical ICME is expected to be formed by the following sub-structures: \nslightly enhanced level of fluctuations (upstream waves turbulence), shock (driven by the MC), sheath (shocked, compressed, heated and turbulent material),  \nMC (flux rope), back (mixed flux rope and solar wind plasma when erosion has occurred), and wake (perturbed solar wind after the back region of the flux rope).\n\n \nTransient structures in the heliosphere affect the transport of energetic particles in the solar wind \\citep[\\textit{{\\it e.g.},}\\ ][and references therein]{masson12}. \nOn the one hand, acceleration at shocks driven by ICMEs is the main mechanism involved in the production of gradual energetic particle events in the inner heliosphere \\citep[\\textit{{\\it e.g.},}\\ ][]{vainio09}. \nOn the other hand, the decrease of the flux of energetic particles over a huge range of energies is associated with shocks and ICMEs. \nAt lower energy, this is observed in situ.  \nFor higher energies (\\textit{{\\it e.g.},}\\ galactic cosmic rays, GCRs) this is observed at ground level by neutron monitors \\citep[\\textit{{\\it e.g.},}\\ ][]{Simpson54}, by muon telescopes \\citep[\\textit{{\\it e.g.},}\\ ][]{arunbabu15}, or by water-Cherenkov detectors \\citep[]{auger11_scalers,Dasso12,Asorey16}.\n\n \nIn particular, the passage of ICMEs and their associated shocks have important effects on GCRs. \nIndeed, a Forbush decrease (FD) of galactic particles is typically observed during several days and in association with the passage of an ICME \\citep[\\textit{{\\it e.g.},}\\ see the review,][]{cane00}. \nAdditionally, a smaller amplitude and a higher-frequency variability in the GCR intensity compared with classical FDs can also be observed in situ\\ near Earth \\citep[\\textit{{\\it e.g.},}\\ ][]{mulligan09}. \nFurthermore, a decrease in the abundance of energetic particles associated with FD has also been in situ\\ observed by the Mars Science Laboratory's rover Curiosity, during its cruise phase from Earth to Mars \\citep{guo_etal15}.  \n\nThe full structure of ICMEs can perturb the transport of GCRs and has important effects both locally and globally on the density of GCRs.\nThese effects are associated respectively with \n  (1) strong changes of the local properties of the solar wind turbulence (mainly in the sheath) \n  which consequently affect the diffusion coefficients of these energetic particles and \n  (2) the presence of structures with smooth closed \\vec{B} field lines with typically high $|\\vec{B}|$ values inside MCs, which hardly allow diffusion transport across \\vec{B} \\citep[\\textit{{\\it e.g.},}\\ ][]{krittinatham09}.\n\n \nFrom a statistical study of interplanetary properties of ICMEs and muon data for cutoff rigidities \nbetween 14 and 24 GV (using the GRAPES-3 muon telescope), \\cite{arunbabu15} found that the enhancement of the interplanetary magnetic field inside the sheath regions is strongly associated with the observed FD profile.\nThey concluded that FDs are mainly caused by the cumulative diffusion of protons across $\\vec{B}$ in the sheath. \nHowever, it remains quantitatively unclear how FDs can be modelled, in particular how one can quantify the importance of $|\\vec{B}|$ and the level of turbulent fluctuations. \n\n \nIn this paper we characterise the mean properties of MCs at 1 AU and their relation to GCR transport.\nWe apply the superposed epoch analysis method on in situ\\ solar wind data and ground-based neutron monitor observations.\nThis method is powerful because it emphasises the common properties and removes peculiarities of some events. \nIt is similar to the superposed epoch developed by \\cite{lepping03b}, where they studied 19 MCs from 1995 to 1998 and created a profile of  magnetic field and plasma parameters of MCs, and even more to the method developed by \\cite{Yermolaev15} where they studied and compared the profiles of ejecta, MCs, their sheaths and corotating interaction regions.\nOur method is also similar to the one recently developed in \\cite{rodriguez16}, to analyse plasma, magnetic and composition properties, focussing on the physical mechanisms for the formation of structures inside and at the rear of MCs.\nIt is also close to the method used by \\cite{badruddin_etal16}, where the response of GCRs to corotating interaction regions and ICMEs was studied with a focus on their interfaces with the ambient solar wind.\n\nFinally, our study is related to the one of \\cite{belov_etal15} since they model the response of GCRs inside MCs as a function of the interplanetary conditions (plasma speed and magnetic field), and geomagnetic properties (Dst index).\nIn particular, they report the presence of\na local minimum of GCR inside MCs for strong-field events.\n\n\n\n \nIn \\sect{data_and_events} we present the ICMEs studied and the data used to analyse them.\nIn \\sect{superposed_epoch} the superposed epoch method, applied to our sample, provides the typical profiles of magnetic field and plasma parameters in ICMEs containing an MC.\nIn \\sect{splits_with_V} we further investigate how these profiles depend on the strength of the MC, in particular its mean velocity. \nFor that we split our sample in slow, mid and fast events.\nThe association of the event properties with the observed flux of GCRs is presented in \\sect{forbush}. \nThese results allow us to propose a novel quantitative model to describe the temporal evolution of GCR flux using parameters observed in the solar wind.\nFinally, in \\sect{conclusions}, we present our conclusions.\n\n\n\n\\section{Data and events selection}\n\\label{sect_data_and_events}\n\n\n\n\\begin{figure*}[ht]\n    \\sidecaption    \n    \\img{0.25}{example_01}  \n    \\img{0.25}{example_03}  \n    \\img{0.25}{example_02}  \n    \\caption{\n    Time profiles for different observables, from top to bottom: magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$, normalised magnetic-fluctuation density $rmsBoB$ (\\eq{rmsBoB}), and absolute magnetic fluctuation $rmsB$ (\\eq{rmsB}).\n    We show three different events; one per column.\nEach of them belongs to each of the groups analysed in \\sect{splits_with_V}; from left to right, the events belong to the slow, intermediate, and fast groups.  \n    The date of the shock arrival is shown at the top of each column in UT.\n    The passage of the sheath is marked in orange, and the passage of the MC in blue.\n    The fastest event shows stronger compression at the MC front, with respect to the others.\n    The slowest event presents higher density values within the sheath. See \\sect{quantities_examples} for a description of the main important features shown in each panel.\n    }\n\\label{fig_examples}\n\\end{figure*}\n\nIn this section we present the data used for the analysis, the sample of studied events, and the observed physical quantities we explore.\n\n\\subsection{Events selection}\n\\label{sect_event_selection}\n\n \nWe use data of interplanetary magnetic field and plasma from the MAG and SWEPAM experiments onboard the ACE spacecraft \\citep{ACE98, mccomas98_ace}, using a 64 seconds time cadence, which are available at \\url{http:\/\/www.srl.caltech.edu\/ACE\/ASC\/level2\/}.\nThe intensity of GCRs is analysed using data from McMurdo neutron monitors\nwith one hour time cadence; these data are available at \\url{http:\/\/neutronm.bartol.udel.edu\/~pyle\/bri_table.html}.\n\n \nWe studied events taken from the list of \\cite{Richardson10}, including only ICMEs flagged as MCs from 1998 to 2006, \\url{http:\/\/www.srl.caltech.edu\/ACE\/ASC\/DATA\/level3\/icmetable2.htm}.\nSpecifically, we use only those events satisfying the following conditions: \na) ICMEs with flag \\texttt{2} in the list, which means that they are included in the WIND MC list compiled by R.P. Lepping, \\url{http:\/\/lepmfi.gsfc.nasa.gov\/mfi\/mag_cloud_S1.html}.\nb) no multiple MCs, to avoid complex structures involving interactions of different events.\nc) MCs with a sheath and with an associated shock.\n\n \nWe selected the shock events from the shock list of \\cite{wang10}.\nWe intersect both ICME and shock catalogs, only selecting ICMEs for which associated shocks are less than 3 hours before the ICME starting time.\nThe above procedure defines 44 events, and we summarise their main characteristics in \\tab{selec_events} (on line version).\n\n\n \nBecause the times listed in the table of \\cite{Richardson10} are referenced at Earth and our data are referenced at L1, we shift these listed times at L1, where the ACE data are observed.  \nWe use a global time shift as defined by the time difference of the shock observed at L1 and at Earth.\nWe made this shift for each ICME by identifying the shock discontinuity in the time series of the magnetic field and plasma parameters.\nThe time shifts are written in the column $5$, shown in \\tab{selec_events}. \n\n\n\\subsection{Analysed physical quantities}\n\\label{sect_quantities_examples}\n\nIn this section we describe some characteristics of three individual events as a context for further discussions of our results on average profiles.\nIn \\fig{examples} we show three MCs: one with a slow bulk velocity, one with a mid velocity value, and a fast one (shown on the left, central, and right column, respectively).\n\n \nThe upper panels show that the magnetic field strength has a comparable magnitude for the three selected MCs ($B_{\\rm max} \\approx 20$ nT). \nThe compression of the magnetic field (enhanced intensity) is present in the MC sheath, especially for the mid and fast MCs. \nThe faster MC presents also the most asymmetric B profile while the slow MC present a very symmetric B profile. \n\n \nThe second row of panels shows a gradient in the velocity profile.  \nThis typical velocity profile is associated with the MC expansion as it moves away from the Sun.\nThe slow and fast events present a very different change on the velocity during the MC passage (so that $\\Delta V =V_{end}-V_{start}$ is larger for the fast event compared to the slow one, a result consistent with the results of \\cite{Gulisano10} for MCs observed in the inner heliosphere).  \nFinally, we note that the event chosen with intermediate speed looks perturbed with a break in the velocity profile around the half time of the MC.\n\n \nIn the third and fourth rows, we see enhanced proton density \\np\\ and temperature \\Tp\\ inside the sheaths, as expected because these regions are formed of compressed and heated plasma behind the shock.\n\n \nIn the fifth row, we show the plasma $\\beta$. \nIts value is estimated with the same method as in the OMNI database, \\textit{i.e.}\\ by assuming that the temperature of alpha particles, $T_\\alpha$, is proportional to the proton temperature with: $T_\\alpha = 4 T_p$, and that the electron temperature is constant $T_e = 1.4 \\times 10^5 K$ (see \\url{http:\/\/omniweb.gsfc.nasa.gov\/ftpbrowser\/magnetopause\/Reference.html}).\nThese examples show that $\\beta$ can be greater than 10 inside the sheath, while inside the MC, $\\beta \\ll 1$ as expected \\citep[\\textit{{\\it e.g.},}\\ ][]{Dasso05}.\n\n \nWe also present the fluctuations of the unit vector of the magnetic field $\\vec{B}$, which we call the normalised magnetic fluctuations \n(i.e., rms of B over its magnitude B) and write it as $rmsBoB$ hereon.\nWe define it with:\n   \\begin{align}\n    \\centering\n    rmsBoB(t) &= rmsB(t) \/ B(t) \\label{eq_rmsBoB} \\\\\n    rmsB(t)   &= \\sqrt{\\sum_{i=1}^{3} <(B_i- \\langle B_i \\rangle)^2>} \n    \\label{eq_rmsB}\n   \\end{align}\nwhere $rmsB$ is computed in time windows of 16 seconds, using a high time cadence of 3 vectors per second, and the mean value of each component $\\langle B_i \\rangle$ is computed inside this time window (as provided by the ACE webpage).\n$rmsBoB$ is much lower inside MCs, than outside (sixth row of \\fig{examples}).\nIn the next section, we will see that this magnitude is a very robust quantity within MCs.\n\n\n \nThe magnetic fluctuations $rmsB$ are shown in the seventh (bottom) row. \nThey have maximum values within the sheath, while inside MCs it is roughly similar to the values observed before the shock arrival.\n\n \nBesides these expected features in these three cases, we obtain below typical profiles of ICME events, and quantify them.\nThis is the motivation to implement a superposed epoch analysis in the next section.\n\n\\begin{figure*}\n\\centering\n    \\img{0.9}{figs_all.global}\n    \\caption{\n    Superposed epoch for different quantities derived with the method described in \\sect{superposed_method}.   \n    The sheath is shown with an orange background colour (time range $0<t<1$) and the MC with a blue background colour ($1<t<4$).\n    The time $t$ is normalised with the sheath duration for $t<1$ and with the MC duration for $t>1$ (the MC duration is three times longer than the sheath duration as observed in average).\n    The black dots, connected with a line, are averaged values for each bin on the time axis, the grey band represents the error of the mean and the red line represents the median values in each time bin.\n    The average profiles were computed filtering those events that had less than 20\\% of data-gaps in the transient structure (MC or sheath).\n    The number of events after filtering is shown in each panel.\nThe panels represent: magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$, normalised magnetic-fluctuation density $rmsBoB$ (\\eq{rmsBoB}), GCR intensity normalised to the pre-ICME level \\nGCR\\ and absolute magnetic fluctuation $rmsB$, \\eq{rmsB}.\n    A clear discontinuity of the magnetic fluctuation intensity, $rmsB$, is present at the shock position (front of the sheath).\n    The normalised magnetic fluctuations, $rmsBoB$, inside the MC are significantly lower than in the ambient solar wind (by a factor of $\\simeq$ 3).\n}\n    \\label{fig_avrg_profiles}\n\\end{figure*}\n\n\\section{Superposed epoch for the sheath and MC structures}\n\\label{sect_superposed_epoch}\n\n\\subsection{Method}\n\\label{sect_superposed_method}\n\nThe main aim of the superposed epoch is to obtain an average profile by taking a sample of individual profiles.\nIn order to obtain this average, each individual profile must have the same number of data points or bins in the time dimension.\nHowever, the primitive data have different durations for different events so we implement a re-binning such that within each MC we end up with 50 time bins. \nThen, the data within each bin are averaged to a single value per bin. \nIn the case when a primitive time series has a data gap in more than 20$\\%$ of the MC structure, we discard it from the average profile.\nWe also re-bin the data obtained after the MC, over the same time interval and with the same number of bins.\nNext, we average the data associated to the different events, by averaging bin by bin each quantity ($B$, $V$, $\\beta$, etc), which finally builds the average profiles. \n   \nWe repeat this procedure to obtain average profiles of the sheath structure.\nAgain, we resample all the time series to 50 bins inside each sheath.  \nSince the number of events having more than 20$\\%$ of gaps are not necessarily the same for the sheath analysis as for the MC analysis, we have indicated in each graph the total number of cases taken for both the sheath and the MC.\nThe same time normalisation as for the sheath is used to resample the data before the shock, over twice the time interval of each sheath. \n\nWe next build a combined profile which shows an average profile for: the background solar wind, the sheath, the MC and the MC wake. \nWe select the time origin at the shock.  \nFinally, we set the temporal lengths of the MC to sheath to a ratio 3:1 to better represent the typical relative durations of each structure. \n\nIn \\fig{avrg_profiles}, we show the average profiles (black lines) with the associated errors of the means (grey bands), and the median values (red lines).\nThe difference between the median and the mean profiles is a proxy of how non-symmetric the distributions of the quantities are.\nIndeed, in the central part of MCs, the observables are distributed in a log-normal manner \\citep{rodriguez16}.\nSimilar log-normal distributions also have been found for ICME observables in \\cite{guo_etal10} and \\cite{Mitsakou14}.\n\n\n\n\\begin{table*}\n\\centering\n\\caption{\n    Mean values of all quantities (column (1)) for all events (columns (2) and (3)), \nand for each group of events associated with MCs with low, mid and high velocities (columns (4) to (9)).\n    These mean values are calculated within sheaths and MCs.\n}\n\\begin{tabular}{c|cccccccccccccccrr}\n\\hline\n& & \\multicolumn{2}{c}{All} & &\\multicolumn{2}{c}{$V_{mc}^{low}$} & & \\multicolumn{2}{c}{$V_{mc}^{mid}$} & & \\multicolumn{2}{c}{$V_{mc}^{high}$} \\\\\n\\cline{3-4}\\cline{6-7}\\cline{9-10}\\cline{12-13}\nquantity                        &&sheath & MC    &&sheath & MC    &&sheath & MC    && sheath & MC    & \\\\\n (1) && (2) & (3) && (4) & (5) && (6) & (7) && (8) & (9)  \\\\\n\n\\hline\n$V$ [km\/s]                      && 561   & 525   && 441   & 402   && 520   & 493   &&  723   & 681 & \\\\\n$B$  [nT]                       && 15.4  & 14.3  && 11.4  & 14.0  && 13.4  & 11.9  &&  21.5  & 16.9  & \\\\\n$n_p$ [1$\/cm^3$]                && 14.3  & 6.5   && 18.5  & 9.5   && 14.1  & 5.3   &&  10.1  & 4.7   & \\\\\n\\Tp\\ [K] ($\\times 10^4$)        && 22.8  & 6.5   && 8.3   & 2.6   && 14    & 4.3   &&  46    & 12.4  & \\\\\n$\\beta$                         && 1.6   & 0.4   && 2.8   & 0.42  && 1.4   & 0.37  &&  0.7   & 0.25  & \\\\\n$rmsBoB$     ($\\times 10^{-2}$) && 6.8   & 2.7   && 6.3   & 2.3   && 6.3   & 3.0   &&  7.9   & 2.8   & \\\\\n$rmsB$ [nT]                     && 0.97  & 0.37  && 0.62  & 0.30  && 0.78  & 0.34  &&  1.53  & 0.47  & \\\\\n\\hline\n\\end{tabular}\n\\label{tab_avr_values}\n\\end{table*}\n\n\n\n\n\\subsection{Results for the average profiles}\n\\label{sect_profiles}\n\nFrom the mean profiles in \\fig{avrg_profiles}, we can clearly see sharp jumps at the arrival of the fast shock. \n$B$, $V$, $N_p$, and $T$ jump by factors of \n$\\sim$100$\\%$, $\\sim$30 $\\%$, $\\sim$150$\\%$, and $\\sim$300$\\%$, respectively.\nThen, all quantities behave differently whether in sheath or in the MC.\nWe describe shortly below the main results for each of the panels of \\fig{avrg_profiles} (except the bottom left one, which is described in \\sect{model_nGCR}).\nSome results in this subsection closely agree with those found in the sample of MCs studied in \\cite{Yermolaev15,rodriguez16}.\n\n \nThe upper left panel of \\fig{avrg_profiles} shows the piled-up magnetic field $B$ in the sheath. $B$ is enhanced by more than a factor 2 with respect to the pre-shock value.\nNext, the $B$ profile inside the MC is strongly asymmetric, with its maximum strongly shifted to the front of the MC.\nIndeed, due to the flux rope expansion a stronger $B$ is expected at the front compared to the rear since the spacecraft crosses the front earlier during the propagation of the MC than the rear \\citep[\\textit{{\\it e.g.},}\\ see Figures 3 and 4, and associated explanation in the main text of][]{Demoulin08}.  \nThis is the so-called aging effect.\nHowever, it was previously shown that expansion is typically not sufficient to explain such a $B$ asymmetry \\citep[see the upper panel of Figure 6 and associated main text of][]{Demoulin08}.\n\n \nThe speed profile (upper right panel of \\fig{avrg_profiles}) shows that across the shock there is a jump from 420-440 km\/s to 520-540 km\/s.\nThe first part of the sheath shows a fast compressing profile, while it later stabilises to be a roughly constant speed up to the boundary with the flux rope.\nInside the MC, the speed is a decreasing time function, in agreement with the expected typical expansion. \nThe linear decrease ends before the MC rear boundary. \nThis characteristic is present both in individual profiles and in other superposed epoch studies \\citep[\\textit{{\\it e.g.},}\\ ][]{lepping03b,rodriguez16}.\nA peak of $V$ is also present after the MC.\n\n\\begin{figure} \n    \\centering\n    \\img{0.6}{__paper__deviations_mix_Vmc.group}\n    \\caption{\n    Normalised values of average parameters corresponding to each subgroup (split according to the ordering parameter \\Vmc): \n    low \\Vmc\\ (blue triangle down), mid \\Vmc\\ (green circle), and high \\Vmc\\ (red triangle up).\n    Empty symbols are for sheaths and filled symbols for MCs.\n    For each quantity, we use the mean values of all events for the normalisation of each parameter (grey horizontal line).\n    From left to right the parameters are the means of: magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$, normalised magnetic-fluctuation density $rmsBoB$ (\\eq{rmsBoB}), absolute magnetic fluctuation $rmsB$ (\\eq{rmsB}) and GCR intensity normalised to the pre-ICME level \\nGCR.\n    }\n    \\label{fig_Vmc_ordering}\n\\end{figure}\n\nIn the second left panel of \\fig{avrg_profiles} the high proton density, \\np, in the sheath contrasts with its lower value in the sheath surroundings.\nHowever, \\np\\ has no discontinuity at the interface between the sheath and the MC.\nOn the contrary, \\np\\ starts to decrease from the sheath-MC interface to inside the MC, up to around 15-20\\% of the MC size where its value becomes constant.  \nThis is not an effect of smearing the MC boundary in the averaging since this effect is also present in some individual MCs (\\textit{{\\it e.g.},}\\ \\fig{examples}).\nFinally, a clear density peak is present just after the rear boundary. \nIts physical origin was investigated in \\cite{rodriguez16}, where it was shown that several possible mechanisms, such as compression from a fast overtaking stream, or intrinsic mechanisms, such as the eruption conditions at the Sun, could account for this density peak.\n\n \nWhile in the sheath, the magnetic field, velocity and density are roughly constant (besides fluctuations), the proton temperature \\Tp\\ (right second panel of \\fig{avrg_profiles}) has a steep maximum near the shock.  \nThis is a consequence of the macroscopic to thermal energy conversion produced at the shock. \nLater on \\Tp\\ progressively decreases towards the MC.\nAs observed for \\np\\, the temperature profile is continuous around the sheath-MC interface without a clear break, up to a $\\sim$ 15-20 \\% of the mean MC size.\nLater on, \\Tp\\ stabilises to a roughly constant value up to almost the end of the MC.\n\n \nIn the density and temperature profiles (\\fig{avrg_profiles}), the transition between the sheath region and the \nMC is not as abrupt as expected, considering that the MC environment is much different from the sheath region. \nThis transition can be explained as follows.\n\\cite{gosling_etal95} discuss a mechanism where part of the field lines at the MC periphery are magnetically connected to the outer heliosphere after reconnection occurred.\nCase studies \\citep{dasso06,dasso07,Feng13} and a statistical analysis of a large sample of events \\citep{Ruffenach15} indicate that an MC can suffer magnetic erosion at its front during its travel in the heliosphere, forming a back region with mixed properties of both MC and ambient solar wind \\citep{dasso06,Ruffenach12}.\nThus, because MCs are 3D structures, it is possible that this erosion only happens at various specific locations along the flux rope.\nThen, the transition region between the sheath and the MC, explored by the spacecraft, can be magnetically connected to another region where erosion has already occurred. \nThis gives a possibility for solar wind material and heat to diffuse along the reconnected field lines, which leads to a smoother profile at the interface between those sub-structures.  \nIn this paper we refer to this mechanism as '3D reconnection'.\n\n\n\\def 0.20 {0.22}\n\\begin{figure*}[ht]\n    \\sidecaption\n    \\img{0.8}{figs_splitted_1}\n    \\caption{\n    Average profiles (black lines) distributed in three columns for the three subsets of events. Left: slow ($V<450$ km\/s), middle: medium ($450$ km\/s$<V<550$ km\/s), and right: fast MCs ($V>550$ km\/s).\n    The black line represents the average value, the grey band represents the error of the mean and the red line represents the median values in each time bin.\n    As in \\fig{avrg_profiles}, for $t<1$, time is normalised with the sheath average radial duration\nand for $t>1$, time is normalised with the average MC radial duration.\n    Orange delimits the sheath region ($0<t<1$), and blue the MC region ($1<t<4$).   \n    See the caption of \\fig{avrg_profiles} for more information.\n    From top to bottom the parameters are the means of: \n    magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$ and normalised magnetic-fluctuation density $rmsBoB$, \\eq{rmsBoB}.\n    }\n    \\label{fig_split_i}\n\\end{figure*}\n\n\n \nThe plasma $\\beta$ and the normalised magnetic fluctuations $rmsBoB$ (left\/right third panels of \\fig{avrg_profiles}) are the quantities that most clearly mark the MC boundaries.\nBoth are roughly symmetric around the MC center with a flat minimum, with $rmsBoB$ being even more symmetric than $\\beta$.\n$rmsBoB$ shows a maximum at the shock, and then a smooth decrease\ntoward almost one-third of the MC size.\n\n \nThe level of absolute fluctuations $rmsB$ (right bottom panel of \\fig{avrg_profiles}) shows a sharp jump with a maximum at shock arrival, while it later continuously decreases to almost one-third of the MC.\nAfter the MC, it recovers towards typical solar wind values, but with slightly larger values. \n\n \nTo summarise this section, we describe the differences between mean values inside sheaths and MCs (columns 2 and 3 of \\tab{avr_values}). \nAll the quantities are higher in sheaths than in MCs, by: \n$7\\%$ for $V$, 8$\\%$ for $B$, $\\sim$ twice for $n_p$, $\\sim$ 3.5 times for $T_p$, \n$\\sim$ 4 times for $\\beta$, $\\sim$ 2.5 times for $rmsBoB$, and $\\sim$ 2.6 times for $rmsB$.\n\n\n\n\\section{Sheath and MC properties for slow and fast events}\n\\label{sect_splits_with_V}\n\n\\subsection{Criterion to define an ordering parameter}\n\\label{sect_criterium_split}\n\n \nThe mean velocity inside the ICME at 1 AU, its mean magnetic field intensity, and its time duration could be considered as different proxies for the strength of the event.\nMoreover, these values can be computed with a time window inside an ICME sub-structure such as the sheath or the MC; \nnamely \\Vsh, \\Bsh, \\dtsh, \\Vmc, \\Bmc, \\dtmc, where the suffixes ``sh'' and ``mc'' are associated to the sheath and the MC respectively.\n\n\n \nWhich is the most relevant parameter to order the events?\nTo answer this question, we first preliminarily consider one of the possible strength ordering parameter, \\Vmc . \nWe then split our sample of 44 events into 3 subgroups with roughly the same number of events: weak (slow), mid, and strong (fast).\nWe produce separate superposed epoch profiles for every physical quantity analysed (similar to those shown in \\fig{avrg_profiles}), \nand for each subgroup. \nFinally, we repeat this procedure for all the possible ordering parameters: \n\\Bmc, \\dtmc, \\Vsh, \\Bsh, and \\dtsh.\n\n \nThe criterion to select the best ordering parameter is as follows.\nFor every ordering parameter we construct a plot as the one shown in \n\\fig{Vmc_ordering} (where \\Vmc\\ was used in this case).\nThis figure shows normalised average values of all observables for each subgroup.\nThese values are derived from Table~\\ref{tab_avr_values}.\nThe average value of each subgroup is shown with: red triangle up \n(events with large velocities: \\Vmc $> 550$ km\/s), \ngreen circle (mid velocities events: $450$ km\/s < \\Vmc $<550$ km\/s), \nand blue triangle down (events with weak velocities: \\Vmc $<450$ km\/s).\nThese values are normalised with the global average using all events. \nThe criterion to select which strength parameter is the best is to observe: \ni) the largest separation between mean values of the 3 subgroups, and \nii) a coherent ordering between these values.\n\n \nAmong all the plots (not shown here) that we looked at, we realised that the best parameter to order the MCs is the mean MC speed \\Vmc\\ , which happened to be better for splitting our sample into three groups.\nAs is seen in \\fig{Vmc_ordering}, most of the parameters are well ordered (red, green and blue in succession for each parameter).  \n The same quality of ordering and spacing as \\fig{Vmc_ordering} was not found for any of the other explored ordering parameters, \\textit{{\\it e.g.},} \\Bmc.  \n\n \nThe parameters with largest spread are the proton density \\np\\ , the proton temperature \\Tp, the plasma $\\beta$, and the GCR percentage of depression \\nGCR\\  (see \\sect{forbush}).  \nOn the other hand, the parameter showing less split and no coherent order is $rmsBoB$.  \nThis implies that the normalised magnetic fluctuations are very similar in all groups within a $\\sim 20 \\%$ factor from the global mean.\nNext, we observe that when the splitting is large between the groups, the relative order and spread for the sheaths are typically the same as for the MCs (\\textit{{\\it e.g.},}\\ see $V$, \\np, and \\Tp\\ in \\fig{Vmc_ordering}).\nFinally, the fastest MCs (red triangle up) have: the largest $B$ intensity, the lowest proton density, the hottest temperature, the lowest plasma beta, the strongest magnetic fluctuations $rmsB$, and the deepest associated GCR depression.\n\n\n\\subsection{Superposed epoch for each MC subset}\n\\label{sect_results_split}\n\nWe analyze below the averaged profiles of the three subsets.\nIn the upper row of \\fig{split_i} we find large differences in the $B$ profiles: \nfor faster\/stronger events, $B$ is more peaked toward the start of the MC, while for slower\/weaker events, it is more peaked toward the MC center.\nAlso, $B$ jumps at the shock with $\\Delta B=7$, $8$, and $13$nT, for slow, mid, and fast events, respectively.\n\n \nIn the second row of \\fig{split_i}, the ambient solar wind before the front of the sheaths is slower for weaker\/slower events compared with stronger\/faster events, then they are not propagating in the same mean solar wind. \nNext, the speed profiles in the three groups have all a linear profile within the MCs, which indicates a common expansion process. \nFor sheaths, we also see expansion in slow events, but for fast events the sheaths have profiles showing compression.\nThis bimodal feature of the sheath, with an expansion and a compression, has not been reported before as far as we know, and it is shown pretty clearly in these superposed profiles with both processes having a comparable effect for MCs with mid velocities.\nThe fact that fast events are related to compressing sheaths is the result of the fast overtaking MC behind.\nFor the slow events, the magnetic field and the plasma had the time to reach a quasi-pressure balance.  \nThis implies an expansion similar to that of the flux rope \\citep[as modeled by ][]{Demoulin09}.\n\n \nIn the third row of \\fig{split_i}, slower events have higher proton density in all the sub-structures: sheath, MC, and its wake. \nBy contrast, the ambient solar wind at the front does not show systematic changes along the three groups.  \nWe interpret the higher density in the sheath of slow events as a selection effect due to the drag force, as follows.\nCMEs that are fast near the Sun and that later on encounter slow solar wind ahead, \nwhich is typically denser in the inner heliosphere \\citep[\\textit{{\\it e.g.},}\\ see][]{wolfe72}, accumulate more mass with a low velocity, resulting in a denser sheath and a slower MC by conservation of mechanic momentum.\nStudies in this direction have been made in \\cite{feng_etal15}, where they find that the SW pile-up of mass during the CME expulsion can be an important contribution to the mass increase determined by coronagraph observations.\nWe explore this observed feature in more details in \\sect{sigma_p_sheath}.\nNext, the low $n_p$ values observed in the wake of fast cases (right column of \\fig{split_i}), can be due to the sweeping of ambient solar wind plasma made by faster MCs \\citep[which are also typically larger, \\textit{{\\it e.g.},} , ][]{Janvier14c} combined with the difficulties to re-fill this space with new solar wind material just after the passage of the flux rope.\n\n \nIn the fourth row of \\fig{split_i} the proton temperature in the ambient solar wind is lower for slow events, which are also traveling in a slower solar wind. \nThese results are consistent because of the known direct correlation of speed and proton temperature in the quiet solar wind \\citep[\\textit{{\\it e.g.},}\\ ][]{Lopez86, Demoulin09d}.\nNext, the faster MCs present much hotter sheaths than slower ones; this is a consequence of the local heating near the shock.\nThe temperature in slow MCs is lower. \nThis is consistent with a relaxed flux rope which had time to adapt to the ambient solar wind pressure, to fully expand, and accordingly to cool.\n\n\\begin{figure}[ht!]\n    \\centering\n   \n    \\img{0.73}{Vmc__vs__sigma}\n    \\caption{\n    Left panel: Average proton surface densities of sheaths $\\sigma_{sh}$, \\eq{def_sigma}, as a function of the average MC speeds ($V_{mc}$) of all events. \n    The vertical dashed lines mark the thresholds between the three groups shown in \\figs{split_i}{split_ii}.\n    The quantity \\sigsh\\ is a proxy of the accumulated material ahead of MCs, within the sheaths.  \n    On average, slower MCs drag more material.\n    Right panel: histogram of \\sigsh\\ for the 44 events.\n    }\n\\label{fig_Vmc_vs_sigmas}\n\\end{figure}\n\n \nIn the fifth row of \\fig{split_i} the average plasma $\\beta$ values are higher for slower events, for all ICME sub-structures.\nEven in the ambient solar wind we note a tendency for slow events to travel in a less magnetised plasma (higher $\\beta$). \nWithin the sheaths, $\\beta$ is more constant and inside the MC the $\\beta$ profile is more symmetric for slow events.\nFor faster events $\\beta$ has a more asymmetric profile, according to the asymmetry of $B$ (first row of \\fig{split_i}).\n\n \nIn the last row of \\fig{split_i} we observe big differences between the $rmsBoB$ fluctuation profiles of slow and fast events.\nFor instance, the fluctuation intensity is much higher near the shock for fast events.\nAlso, these fluctuations start even before the shock arrival and with a clearer effect for faster events.\nThis can be related to foreshock waves excited by energetic particles near interplanetary shocks \\citep[\\textit{{\\it e.g.},}\\ see,][]{blancocano_etal11}.\nNext, the magnetic fluctuations have a steeper drop in the sheath-MC interface for the slow events. \nIn contrast, the fluctuations are roughly the same inside the MC for the three groups.\nFinally, the extension of low $rmsBoB$ values in the back suggests the presence of remnant MC-like structure due to magnetic reconnection at the front, as summarised in \\sect{profiles}.\n\n\n\\subsection{Surface density of protons in the sheath}\n\\label{sect_sigma_p_sheath}\n\nTo further explore the drag as the physical reason for finding some slow events, we estimate the accumulated material inside the sheath for every event by defining the ``surface density'' \\sigsh :\n \\begin{equation}\n \\label{eq_def_sigma}\n  \\sigma_{sh} = \\sum_{r~\\in\\ {\\rm sheath}} n_p \\Delta r  \n              = \\sum_{t~\\in\\ {\\rm sheath}} n_p V \\Delta t\n \\end{equation}\nwhere $\\Delta t$ is the time resolution of the data, and the sum approximates an integral along the radial direction in the ecliptic plane within the sheath boundaries.\n\\sigsh\\ characterises the total amount of material per unit surface (perpendicular to the radial from the Sun).\n\\sigsh\\ is associated with the material that has not escaped perpendicularly to the spacecraft crossing.\nThe right panel of \\fig{Vmc_vs_sigmas} shows an asymmetric distribution of \\sigsh, having more cases with \\sigsh\\ $\\lesssim 3 \\times 10^{14}$ cm$^{-2}$.\n\nIn the left side of \\fig{Vmc_vs_sigmas} we show a scatter plot of \\sigsh\\ versus the average MC speeds for all the studied events.\nFaster MCs tend to have less material ahead of them.\nThis result may seem surprising at first, since faster MCs are expected to overtake more solar wind and moreover the plasma has more difficulty to escape from the sides (faster and larger MCs).  \nHowever, this is consistent again with the drag scenario where the MCs are slowed down when they are ejected with a high speed in the corona and encounter a slow and dense solar wind.\nIn contrast, high speed-MCs encountering a fast solar wind ahead are less slowed down (lower mass pile-up with faster velocity). \nThese cases correspond to the lower right region of the scatter plot.\nFinally, the lower left part of the plot would correspond to MCs with a slow speed encountering a low solar wind density along their travel to 1 AU.\n\n\n\n\\section{Analysis of associated Forbush events}\n\\label{sect_forbush}\n\n\\subsection{Superposed epoch analysis}\n\\label{sect_Superposed_GCR}\n\n \nWe analyse below the cosmic rays observed with ground-based detectors.\nWe use data from McMurdo neutron monitors because they \nare close enough to one geomagnetic pole, and then they can measure particles with low energies. In particular, they can observe primary protons with energies as low as $\\sim$500 MeV \\cite[\\textit{{\\it e.g.},}\\ ][]{jordan11}.\nThe Larmor radius for these particles is $\\sim$5x10$^{-3}$ AU, when they are embedded in an interplanetary magnetic field of $\\sim$5 nT. \nFor normal conditions, particles observed by these detectors are mainly of galactic origin.\n\n \nAn additional treatment with respect to the previous quantities \n(\\sects{superposed_epoch}{splits_with_V}), is that we use the average GCR intensity before the MC sheath arrival as a reference level.\nIn essence, we take the data from 2 days prior the sheath arrival.\nIf a ground level enhancement (GLE) is present, we only take data from 6 hours prior to the sheath arrival (we verify for each event that during this time interval there is no GLE or intense noises).\nThen, we normalise the GCR flux of each event with this level taken before the MC sheath defining the percentage of variation \\nGCR.\nNext, we continue with the same treatment as in \\sect{superposed_method}.\n\n \nThe superposed epoch of all events is shown in the bottom left panel of \\fig{avrg_profiles}.  \nThere is a strong decrease of \\nGCR\\ in the sheath followed by a progressive recovery phase during and after the MC. \nThe importance of these effects depends on the MC speed as shown in the second row of \\fig{split_ii}. \nThe recovery time ($\\tau_{FD}$) and the amplitude ($A_{FD}$) of the GCR depressions are larger for the faster events; this result is respectively consistent with the following previous studies: \\cite{penna05} and \\cite{richardson11}.\nA minimum of \\nGCR\\ is present near the sheath rear for the three subsets. \nFor fast events, another minimum can be observed within the fluctuations, inside the MC window and near the MC center.\nThis second minimum is associated to the shielding of a strong magnetic field within the flux rope configuration (with closed field lines still connected to the Sun).\nThis result\nis consistent with \\cite{belov_etal15}, where they report that for strong-field ($>18$nT), there is a local minimum of \\nGCR\\ inside MCs.\n\n\n\\subsection{Model for the \\nGCR\\ typical profiles}\n\\label{sect_model_nGCR}\n\n\n\\def 0.20 {0.20}\n\\begin{figure*}[ht!]\n    \\centering\n    \\img{0.8}{figs_splitted_2}\n    \\caption{\n    Superposed epoch for the three subsets of events. Left: slow ($V<450$ km\/s), middle: medium ($450$ km\/s$<V<550$ km\/s), and right: fast MCs ($V>550$ km\/s).\n    {\\bf First row:} mean profiles of magnetic fluctuations $rmsB$, \\eq{rmsB}.\n    Typically, inside the MCs, the fluctuations are the same as in the ambient solar wind.\n    For slow and mid groups, $rmsB$ has higher values in the rear with respect to the ambient solar wind.\n    {\\bf Second row:} mean profiles of GCR intensity \\nGCR .\n    Before averaging events, the intensity was normalised by the pre-shock level.  \n    In the first and second rows the black line represents the average value, the grey band represents the error of the mean and the red line represents the median values in each time bin.\n    {\\bf Third row:} Fitted model (red lines) to the observed average profiles (see \\sect{model_nGCR}).\n     See the caption of \\fig{avrg_profiles} for more information.\n    }\n    \\label{fig_split_ii}\n\\end{figure*}\n\n\n\nA theoretical model to describe the profile of FD was developed by \\cite{wibberenz98}.\nHowever, they did not explicitly include small-scale interplanetary magnetic fluctuations, while it was shown that these can contribute to the variety of FD profiles \\citep[\\textit{{\\it e.g.},}\\ ][]{jordan11}.  \n\nIn order to quantitatively analyze the processes involved in the GCR shielding produced by ICMEs, we construct a mathematical model to reproduce the GCR profiles. \nIn particular, we use $B$ to account for the strong closed field lines of the MC, and $rmsB$ to account for the scattering turbulence.\n\n  \nFrom the superposed epoch profiles, we learned the following contributions (second row of \\fig{split_ii}):\\\\\ni) Inside the sheath region, \\nGCR\\ can be thought to be determined by an accumulation of interaction between the high energy particles and scattering centres. \nThe efficiency of the scattering is related to the level of turbulent fluctuations $rmsB$.\\\\\nii) The decrease of \\nGCR\\ is larger for a stronger magnetic field inside the MC.\\\\\niii) The recovery mechanism is effective after the passage of the scattering turbulence region.\\\\\niv) There is a jump in \\nGCR\\ at the interface between the sheath and the MC; this can also be seen in the individual profiles. \nThis feature can be related to the change of the cross diffusion coefficients associated to the plasma properties in these two different structures.\n\n  \nBy representing each of these mechanisms with a contributing term for the temporal evolution of the GCRs flux at Earth position, we propose the following model:\n\\begin{align}\n    \\centering\n    &\\frac{d n_{GCR}(t)}{d t} \n     &= \\text{ }& q_{\\xi} \\, \\xi(t)  & (i)  && \\nonumber \\\\\n    &&+ \\text{ }& q_{b} \\, b(t)      & (ii) && \\nonumber \\\\\n    &&+ \\text{ }& \\frac{-1}{\\tau_{FD}} n_{GCR}(t) \\, \\Theta(t-1) & (iii) &&\\nonumber\\\\\n    &&+ \\text{ }& \\Delta_o \\, \\delta (t-1)   & (iv) && \\forall t>0\n    \\label{eq_FD_model}\n\\end{align}\nwith,\n\\begin{align}\n    \\xi(t) &= \\textrm{max}(0, rmsB(t)-rmsB_o) \\nonumber \\\\\n    b(t)   &= \\textrm{max}(0, B(t)-B_o) \\,.\n\\end{align}\n$\\xi(t)$ is the $rmsB(t)$ value above its value in the quiet solar wind, $rmsB_o$, when $rmsB(t)$ is larger than $rmsB_o$. \n$b$ is the magnetic field above a reference value $B_o$ when $B>B_o$. \n$\\Theta$ is the Heaviside function indicating that the recovery mechanism starts at the beginning of the MC. \nThis simplification facilitates the mathematical fit and we checked that the recovery is negligible during the passage of the sheath.\n$\\delta$ is the Dirac delta function (more precisely distribution) which account for the observed jump of \\nGCR\\ at the sheath-MC interface. \nThere are 5 free parameters to fit: \n$q_{\\xi}$, $q_{b}$, $B_o$, $\\tau_{FD}$ and $\\Delta_o$.  \nThis model does not include explicitly the larger expansion of faster events nor a non local effect due to the sheath and MC spatial extension.  \nThe expansion rate is correlated to the field strength, so it is implicitly included, while we cannot include the effect of the global extension from local measurements.\n\n  \nIn order to perform the non-linear fit of the time integral of \\eq{FD_model} to the data, we employ the L-BFGS-B algorithm \\citep{zhu_etal97, byrd_etal95}.\nThis algorithm is suitable when working with a large number of fitting variables, and it uses the gradient information of a multiple variable-function to be minimised within given bounds.\nMainly, we use this method instead of other alternatives because we observe a better convergence for our problem.\n\n  \nThe third row of \\fig{split_ii} shows the fitted models to each of the average \\nGCR\\ profiles, where the detailed reproduction of the observations is notable.\nThe inclusion of the magnetic fluctuations allows the reproduction of the observed mean decrease of \\nGCR\\ within the sheath with only one adjusted parameter $q_{\\xi}$. \nMoreover, for the slow $V_{mc}$ group, we see a small decrease of \\nGCR\\ right after the MC passage, which according to our model, is due to the enhancement of $rmsB$ in the wake of the flux rope (see the left column of \\fig{split_ii}). \nThis behaviour is also present, but weaker, for the mid velocity group.\nThis result at the MC rear provides a further confirmation that magnetic fluctuations are locally affecting GCRs.\nIt is also in agreement with \\citet[][see their Fig. 7]{badruddin_etal16}, where they find a GCR decrease around the ICME trailing edge, and simultaneously find enhanced magnetic fluctuations.\nFinally, $|q_{\\xi}|$ is decreasing from the low to the fast velocity group (Table~\\ref{tab_fitted_vals}), which may be due to non-linearities in the diffusion process that are not taken into account in our model.\n\n  \nThe contribution of the magnetic field $B$ is important only for the group of fast \\Vmc , where \\nGCR\\ has a local minimum inside the MC. \nAs mentioned before, this can also be related to the larger expansion of the flux rope compared to the surrounding solar wind. \nThis leads to a decrease of the local density of the high energetic particles \\citep[\\textit{{\\it e.g.},}\\ ][]{Munakata06}.\n\nNext, the normalised time $\\tau_{FD}$, as well as the time $\\tau_{FD}$ in hours (taking into account the mean MC duration of each group) are both increasing by more than a factor 2 from the slow to the fast velocity group (Table~\\ref{tab_fitted_vals}).\nFinally, it is worth mentioning that without the inclusion of the jump parameter $\\Delta_o$, it is not possible to reproduce the recovery profile for the slow and fast $V_{mc}$ groups that clearly show this discontinuity.\n\n\n\n\\begin{table}\n\\centering\n\\caption{\n    Fitted values of the Forbush decrease model (see \\eq{FD_model}), for the three subset of events.\n    These are all free parameters in the model defined by \\eq{FD_model}.\n    \\tauFD\\ is normalised to the mean duration of the MCs for each group; see \\sect{model_nGCR} for informations.\n    The $B_o$ value in the first column is superfluous since $q_{b}=0$, in that case. \n}\n\\begin{tabular}{c|ccccccccccrr}\n\\hline\nfit parameter & \\multicolumn{1}{c}{$V_{mc}^{low}$} & \\multicolumn{1}{c}{$V_{mc}^{mid}$} & \\multicolumn{1}{c}{$V_{mc}^{high}$} \\\\\n\\hline\n$q_{\\xi}$  [nT$^{-1}$]    & -9.4   & -6.0   &  -5.5    & \\\\\n$q_{b}$  [nT$^{-1}$]      &~~0.0   & -0.9   &  -0.2    & \\\\\n$B_o$  [nT]               &~~--    & 11.9   &  14.5    & \\\\\n\\tauFD [1]                &~~2.4   &~~4.2   &~~5.8     & \\\\\n$\\Delta_o$  [\\%]          &~~0.9   &~~0.0   &~~1.0    & \\\\\n\\hline\n\\end{tabular}\n\\label{tab_fitted_vals}\n\\end{table}\n\n\n\n\\section{Summary and conclusions}\n\\label{sect_conclusions}\n\n  \nWe obtained superposed epoch profiles during the passage of MCs and their sheaths at 1~AU, for different physical quantities observed in situ. \nThis technique allows us to identify several phenomena that are common to most of the events.\nFirst, the magnetic field strength and the plasma velocity in the MC sheath are typically close to the values found at the beginning of the MC. \nSecond, the MC has a similar density than the background solar wind, while the sheath is a factor $\\approx 2.5$ denser. \n\n  \nWe next explored how the sheath and MC properties could be best differentiated according to the importance of the event.\nWe found that the mean MC velocity is the best parameter to order the sheath and MC properties. \nThis provides a measure of the strength of the events.\nThen, we separated our sample in three groups: slow\/intermediate\/fast events and compared their superposed epoch profiles.\n\n \nWe found that the slowest MCs have a larger proton density in their sheaths than faster events.\nWe next computed the density of protons per unit of surface along the sunward direction to better characterise the accumulated material that does not escape from the MC front to the sides. \nWe found that on average, slow events have more massive sheaths than fast events. \nWe attribute this result to a selection effect: a part of the slow MCs at 1 AU were initially fast events close to the Sun but they were slowed down as they encountered slow and dense plasma on their path to 1 AU.\nThen, such decelerated MCs have more massive sheaths at 1 AU.\n\nThe comparison of the three groups show that the slow MCs have properties compatible with a more relaxed configuration.\nFirst, they have an expanding sheath, with a very similar expansion rate as in the following MC. \nThen, the accumulated magnetic field in the sheath has the time to reach a quasi-equilibrium with the surrounding solar wind, which implies that the sheath expands with a similar rate as the driving flux rope.\nSecond, the slow MCs have a nearly symmetric magnetic profile around their center.\nFinally, they have a lower proton temperature (as they have the time to expand enough to be in quasi-equilibrium with the surrounding solar wind).\nAll these characteristics together indicate that the slower MCs are expected to be in a more relaxed force-free state than faster MCs. \n\n   \nThe proton density and temperature are significantly larger in sheaths than in MCs. \nStill, there is no sharp transition but a progressive sheath-MC transition which extends up to 20\\% of the MC duration. \nA very similar transition is also present for the amplitude of the magnetic fluctuations.  \nThese results are present for all three groups of events as well as in individual events, so they are not due to a smearing of sheath properties inside the MC due to inappropriate boundary definition. \nWe interpret these results as the consequence of magnetic reconnection at various locations along the flux rope between sheath and flux-rope magnetic fields.\nThen, while along the spacecraft trajectory the crossed magnetic field appears as belonging to the flux rope, some parts of the crossing can belong to a magnetic field region that has reconnected with the sheath magnetic field further away from the local spacecraft trajectory.\nThis allows plasma, heat and magnetic fluctuations to enter in this reconnected field implying, in average, sheath properties ``diffusing'' within the front of the MC.\n\n  \nWe next studied the typical effects that these sheaths and MCs have on the cosmic ray transport by studying the associated Forbush decrease profiles.\nFor all MCs a local minimum of the GCR flux is present inside the sheaths, while for fast MCs another minimum is also present within the MCs. \nIn the superposed epoch profiles of Forbush events, we found a discontinuity of the GCR flux at the sheath-MC interface, owing probably to the change of properties in the cross magnetic-field diffusion coefficients in these two structures.\nIndeed, GCRs encounter a structure with closed and intense magnetic field lines, where the high energy particles have difficulties to penetrate. \n\n\nWe finally derive a new semi-empirical model for the Earth observations of the Forbush profile that takes into account the former process \nas a simple superposition of \n(1) the enhancement of magnetic fluctuations over a threshold, \n(2) the intensity of the magnetic field over a threshold, \n(3) a jump at the time of the sheath-MC interface to consider the change of the magnetic connectivity, and \n(4) a recovery phase for times after the sheath-MC interface.\nThis model has five free parameters.  \nWe fit the model to each of the groups of events. \nBesides reproducing the first steep decrease of GCRs after the shock time and the recovery phase, we can reproduce a second decrease of GCRs in the wake of the slow MCs.\nThis second fall is due to the presence of scattering centres which we associate with the amplitude of the magnetic fluctuations. \nThis last result is in agreement with the recent study of \\cite{badruddin_etal16}.\n\nFurthermore, our model also reproduces a local \\nGCR\\ minima inside the MC for the sub-group of fast events.\nThis feature is consistent with \\cite{belov_etal15}, where they find that strong-field ($>18$ nT) are associated to local minima of \\nGCR\\ within MCs.\n\nFinally, we find that the recovery time \\tauFD\\ is increasing by a factor a bit larger than two from the slow to the fast events.\n\n\nThe results presented here improve the knowledge of MCs and their sheaths, their evolution in the solar wind, and also the relationship between MCs and their GCR shielding.\nIn particular, the semi-empirical model for the FD profile we presented will help to improve the understanding of energetic particle transport in the heliosphere, and can be used to put constrains on theoretical models that consider \n(1) global macro-scale magnetic configurations for MCs and (2) turbulent properties of the solar wind.\n\n\\onltab{\n\\begin{table*}\n\\centering\n\\caption{\n    List of the 44 events studied. Column: \n(1): event number,\n(2): time of shock arrival \\citep[as in][]{Richardson10},\n(3): time difference between MC leading edge and shock time (\\textit{i.e.}, the sheath duration),\n(4): MC duration, \n(5): difference between \\citeauthor{Richardson10} and the shifted shock time at L1 point, where the IP observations are compared,\n(6): amplitude of the associated GCR perturbation.\n}\n\\begin{tabular}{cccccc}\n\\hline\ncase & shock date        & sheath duration & MC duration & time shift & $A_{\\rm FD}$ \\\\\n     & [yyyy-mm-dd HH:MM]& [hours]         & [hours]     & [min]      & [\\% of the background] \\\\\n (1) &      (2)          & (3)    & (4) & (5)  &  (6)   \\\\\n\\hline\n   1 & 1998-03-04 11:56  &  ~~1.1 &  41 &  -60 & ~~-0.6 \\\\\n   2 & 1998-05-01 21:56  &   14.1 &  29 &  -30 & ~~-6.9 \\\\\n   3 & 1998-09-24 23:45  &   10.2 &  27 &  -30 &  -13.3 \\\\\n   4 & 1998-10-18 19:52  &  ~~8.1 &  10 &  -50 & ~~-2.1 \\\\\n   5 & 1998-11-08 04:51  &   20.1 &  24 &  -30 & ~~-6.5 \\\\\n   6 & 1999-02-18 02:46  &   11.2 &  22 &  -40 & ~~-5.6 \\\\\n   7 & 1999-04-16 11:25  &  ~~8.6 &  25 &  -45 & ~~-0.6 \\\\\n   8 & 1999-08-08 18:41  &   26.3 &  20 &  -60 & ~~-0.1 \\\\\n   9 & 2000-02-11 23:52  &   17.1 &   7 &  -40 & ~~-4.4 \\\\\n  10 & 2000-02-20 21:39  &   12.3 &  26 &  -60 & ~~-1.6 \\\\\n  11 & 2000-06-23 13:03  &   13.9 &  41 &  -40 & ~~-2.3 \\\\\n  12 & 2000-07-28 06:34  &   14.4 &  13 &  -50 & ~~-1.6 \\\\\n  13 & 2000-08-11 18:45  &   10.2 &  24 &  -40 & ~~-2.9 \\\\\n  14 & 2000-09-17 17:57  &  ~~8.1 &  16 &  -60 & ~~-5.2 \\\\\n  15 & 2000-10-03 00:54  &   16.1 &  21 &  -45 & ~~-1.9 \\\\\n  16 & 2000-10-28 09:54  &   11.1 &  25 &  -50 & ~~-6.5 \\\\\n  17 & 2000-11-06 09:48  &   12.2 &  20 &  -35 & ~~-6.4 \\\\\n  18 & 2001-04-04 14:55  &  ~~3.1 &  14 &  -40 & ~~-4.6 \\\\\n  19 & 2001-04-11 13:43  &   18.3 &  10 &  -30 & ~~-9.9 \\\\\n  20 & 2001-04-21 16:01  &  ~~7.0 &  26 &  -30 & ~~-1.3 \\\\\n  21 & 2001-04-28 05:01  &   21.0 &  11 &  -30 & ~~-7.7 \\\\\n  22 & 2001-05-27 14:59  &   21.0 &  22 &  -45 & ~~-3.5 \\\\\n  23 & 2001-10-31 13:48  &  ~~6.2 &  38 &  -60 &~~~~0.4 \\\\\n  24 & 2002-03-18 13:22  &   33.6 &  17 &  -60 & ~~-4.8 \\\\\n  25 & 2002-03-23 11:37  &   24.4 &  34 &  -45 & ~~-2.9 \\\\\n  26 & 2002-04-17 11:07  &   15.9 &  23 &  -45 & ~~-4.9 \\\\\n  27 & 2002-04-19 08:35  &   27.4 &  30 &  -30 & ~~-3.3 \\\\\n  28 & 2002-05-23 10:50  &   12.2 &  18 &  -40 & ~~-5.8 \\\\\n  29 & 2002-08-01 05:10  &  ~~6.8 &  11 &  -50 & ~~-1.3 \\\\\n  30 & 2002-08-01 23:09  &  ~~9.8 &  12 &  -50 & ~~-3.4 \\\\\n  31 & 2003-03-20 04:40  &  ~~7.3 &  10 &  -25 & ~~-2.5 \\\\\n  32 & 2003-08-17 14:21  &   20.6 &  18 &  -40 & ~~-1.9 \\\\\n  33 & 2003-11-20 08:03  &  ~~1.9 &  16 &  -35 & ~~-5.0 \\\\\n  34 & 2004-04-03 10:00  &   16.0 &  37 &  -60 & ~~-2.0 \\\\\n  35 & 2004-07-22 10:36  &  ~~4.4 &   6 &  -45 & ~~-4.6 \\\\\n  36 & 2004-07-26 22:49  &  ~~3.2 &  10 &  -25 &  -13.3 \\\\\n  37 & 2004-08-29 09:09  &  ~~9.8 &  27 &  -50 & ~~-0.6 \\\\\n  38 & 2004-11-07 18:27  &  ~~7.5 &  15 &  -30 & ~~-7.9 \\\\\n  39 & 2005-05-20 03:00  &  ~~4.0 &  22 &  -60 &~~~~0.3 \\\\\n  40 & 2005-06-12 07:45  &  ~~7.2 &  16 &  -60 & ~~-3.0 \\\\\n  41 & 2005-06-14 18:35  &   10.4 &  28 &  -45 & ~~-0.8 \\\\\n  42 & 2005-07-17 01:34  &   12.4 &  14 &  -60 & ~~-4.5 \\\\\n  43 & 2005-12-31 00:00  &   13.0 &  22 &  -40 & ~~-2.1 \\\\\n  44 & 2006-12-14 14:14  &  ~~7.8 &  22 &  -25 & ~~-8.0 \\\\\n\\hline\n\\end{tabular}\n\\label{tab_selec_events}\n\\end{table*}\n}\n\n\\begin{acknowledgements}\nThanks to the public service of NASA Space Physics Data Facility (SPDF) for the observations.\nWe thank Roger Clay and the anonymous referee for reading carefully the manuscript, and their valuable comments. \nThis work was partially supported by the Argentine grants UBACyT 20020120100220 (UBA), PICT-2013-1462 (FONCyT-ANPCyT), \nPIP-11220130100439CO (CONICET) and PIDDEF 2014-2017 nro. 8 (Ministerio de Defensa, Argentina), \nand by a one month invitation of SD by Paris Observatory.\nJJMM, SD, and LR acknowledge the cooperation project BE\/13\/07 (F.R.S.-FNRS\/MinCyT) between Belgium and Argentina.\nS.D. is member of the Carrera del Investigador Cien\\-t\\'\\i fi\\-co, CONICET. J.J. Mas\\'\\i as-Meza is a fellow of CONICET.\nL. R. acknowledges support from the Belgian Federal Science Policy Office through the ESA-PRODEX program. \nThis research has been partially funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7\/08 CHARM). \n\\end{acknowledgements}\n\n   \\bibliographystyle{aa}\n   ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Empirical accuracy of influence functions on constructed groups}\\label{sec:accuracy}\n\nHow well do influence functions estimate the effect of (removing) a group of training points?\nIf $n$ is large and we remove a subset $w$ uniformly at random, the new parameters $\\hat\\theta(\\bone - w)$ should remain close to $\\hat\\theta(\\bone)$ even when if fraction of removed points $\\alpha$ is non-negligible, so the influence error $|\\sI^{*}_f(w) - \\sI_f(w)|$ should be small.\nHowever, we are usually interested in removing coherent, non-random groups, e.g., all points from a data source or share some feature. In such settings, the parameters $\\hat\\theta(\\bone - w)$ and $\\hat\\theta(\\bone)$ might differ substantially, and the error $|\\sI^{*}_f(w) - \\sI_f(w)|$ could be large.\nPut another way, there could be a cluster of points such that removing one of those points would not change the model by much---so influence could be low---but removing all of them would.\n\nSurprisingly (to us), we found that even when removing large and coherent groups of points, the influence $\\sI_f(w)$ behaved consistently relative to the actual effect $\\sI^{*}_f(w)$ on test predictions, test losses, and self-loss, with two broad phenomena emerging:\n\\begin{enumerate}\n  \\item {\\bf Correlation}: $\\sI_f(w)$ and $\\sI^{*}_f(w)$ rank subsets of points $w$ similarly (e.g., high Spearman $\\rho$).\n  \\item {\\bf Underestimation}: $\\sI_f(w)$ and $\\sI^{*}_f(w)$ tend to have the same sign with $|\\sI_f(w)| < |\\sI^{*}_f(w)|$.\\footnote{\n  This holds with one exception: when measuring the change in test loss,\n  $f(\\theta) = \\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$,\n  underestimation only holds when actual effect $\\sI^{*}_f(w)$ is positive (\\reffig{acc}-Mid).\n  }\n\\end{enumerate}\nHere, we report results on 5 datasets chosen to span a range of applications, training set size $n$, and number of features $d$ (\\reftab{data-characteristics}).\\footnotemark{}\nIn an attempt to make the influence approximation as inaccurate as possible,\nwe constructed a variety of subsets, from small ($\\alpha = 0.25\\%$) to large ($\\alpha = 25\\%$), to be coherent and have considerable influence on the model.\nOn each dataset, we trained an $L_2$-regularized logistic regression model (or softmax for the multiclass tasks)\nand compared the influences and actual effects of these subsets.\n\n\\footnotetext{\nThe first 4 datasets involve hospital readmission prediction, spam classification, and object recognition, and were used in \\citet{koh2017understanding} to study the influence of individual points. The fifth dataset is a chemical-disease relationship (CDR) dataset \\citet{hancock2018babble}.\nIn \\refsec{applications}, we will also study the MultiNLI language inference dataset \\citep{williams2018broad}, which was omitted from the experiments here because its large size makes repeated retraining to compute the actual effect too expensive.\nSee \\refapp{ds-details} for dataset details.}\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{lrrrrrl}\nDataset          & Classes & $n$       & $d$     & $\\lambda\/n$ & Test acc. & Source \\\\\n\\hline\nDiabetes         &  $2$    & $20,000$  & $127$   & $2.2\\times 10^{-4}$    & $68.2\\%$  & \\citet{strack2014impact} \\\\\nEnron            &  $2$    & $4,137$   & $3,289$ & $1.0\\times 10^{-3}$    & $96.1\\%$  & \\citet{metsis2006spam} \\\\\nDogfish          &  $2$    & $1,800$   & $2,048$ & $2.2\\times 10^{-2}$    & $98.5\\%$  & \\citet{koh2017understanding} \\\\\nMNIST            &  $10$   & $55,000$  & $784$   & $1.0\\times 10^{-3}$    & $92.1\\%$  & \\citet{lecun1998gradient} \\\\\nCDR              &  $2$    & $24,177$  & $328$   & $1.0 \\times 10^{-4}$    & $67.4\\%$  & \\citet{hancock2018babble} \\\\\nMultiNLI         &  $3$    & $392,702$ & $600$     & $1.0 \\times 10^{-4}$    & $50.4\\%$  & \\citet{williams2018broad}\n\\end{tabular}\n\\caption{Dataset characteristics and the test accuracies that logistic regression achieves (with regularization $\\lambda$ selected by cross-validation). $n$ is the training set size and $d$ is the number of features.}\n\\label{tab:data-characteristics}\n\\end{table*}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig1.pdf}\n  \\vspace{3mm}\n  \\caption{\n  Influences vs. actual effects of coherent groups of points ranging from $0.25\\%$ to $25\\%$ in size. Each point corresponds to a group, and its color reflects how that group was constructed.\n  In Top and Mid, we show results for the test point with highest loss; other test points are similar (\\refapp{median}), though with more curvature for test loss (\\refapp{supp-curvature}).\n  The grey reference line has slope 1, and the red borders represent points that are not plotted because they are outside the x- or y-axis range.\n  We omit the top row for MNIST, as $\\theta^\\top {x_{\\mathrm{test}}}$ is not meaningful in the multi-class setting.\n  }\n  \\vspace{-12mm}\n  \\label{fig:acc}\n\\end{figure}\n\\paragraph{Group construction.}\nOur aim is to construct coherent groups that when removed will substantially change the model. To do so, we need to choose points that are similar in some way.\nSpecifically, for each dataset, we grouped points in 7 ways: 1) points that share feature values;\n2) points that cluster on their features or 3) on their gradients $\\nabla_\\theta\\ell(x, y, \\hat\\theta(\\bone))$);\n4) random points within the same class; 5) random points from any class.\nWe also grouped\n6) points with large positive and 7) negative influence on the test loss $\\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}, \\hat\\theta(\\bone))$,\nsince intuitively, training points that all have high influence on a test point should act together to change the model substantially.\nOverall, for each dataset, we constructed\n1,700 subsets ranging in size from $0.25\\%$ to $25\\%$ of the training points. See \\refapp{expt-details} for more details.\n\n\n\\vspace{-1mm}\n\\paragraph{Results.} \\reffig{acc} shows that the influences and actual effects of all of these subsets on test prediction (Top), test loss (Mid), and self-loss (Bot) are highly correlated (Spearman $\\rho$ of $0.89$ to $0.99$ across all plots),\neven though the absolute and relative errors of the influence approximation can be quite large.\nMoreover, the influence of a group tends to underestimate its actual effect in all settings except for groups with negative influence on test loss (the left side of each plot in \\reffig{acc}-Mid).\nThese trends held across a wide range of regularizations $\\lambda$, though correlation increased with $\\lambda$ (\\refapp{reg}).\n\nIn \\refsec{applications}, we will use the CDR dataset \\citep{hancock2018babble} and the MultiNLI \\citep{williams2018broad} dataset to show that correlation and underestimation also apply to groups of data that arise naturally, and that influence functions can therefore be used to derive insights about real datasets and applications.\nBefore that, we first attempt to develop some theoretical insight into the results above.\n\n\\subsubsection*{Acknowledgments}\nWe are grateful to\nZhenghao Chen,\nBrad Efron,\nJean Feng,\nTatsunori Hashimoto,\nRobin Jia,\nStephen Mussmann,\nAditi Raghunathan,\nMarco T\\'ulio Ribeiro,\nNoah Simon,\nJacob Steinhardt,\nand Jian Zhang\nfor helpful discussions and comments.\nWe are further indebted to\nRyan Giordano,\nRuoxi Jia,\nand Will Stephenson for discussion about prior work,\nand\nSamuel Bowman,\nBraden Hancock,\nEmma Pierson,\nand Pranav Rajpurkar\nfor their assistance with applications and datasets.\nThis work was funded by an Open Philanthropy Project Award.\nPWK was supported by the Facebook Fellowship Program.\n\n\\section{Applications of influence functions on natural groups of data}\\label{sec:applications}\n\nThe analysis in \\refsec{theory} shows that\nthe cone constraint between predicted and actual group effects need not always hold.\nNonetheless, our experiments in \\refsec{accuracy} demonstrate that on real datasets, the correlation is much stronger than the theory predicts.\nWe now turn to using influence functions to predict group effects in two case studies where groups arise naturally.\n\n\n\n\\paragraph{Chemical-disease relation (CDR).}\nThe CDR dataset tackles the following task: given text about the relationship between a chemical and a disease, predict if the chemical causes the disease.\nIt was collected via data programming, where users provide labeling functions (LFs)---instead of labels---that take in an unlabeled point and either abstain or output a heuristic label \\citep{ratner2016data}. Specifically, \\citet{hancock2018babble} collected natural language explanations of provided classifications; parsed those explanations into LFs; and used those LFs to label a large pool of data (\\refapp{bl-details}).\n\nWe used influence functions to study two important properties of LFs: \\emph{coverage}, the fraction of unlabeled points for which an LF outputs a non-abstaining label; and \\emph{precision}, the proportion of correct labels output.\nWe associated each LF with the group of points that it labeled, and computed its influence; as expected, these correlated with actual effects on overall test loss (Spearman $\\rho=1$; \\reffig{babble-acc}).\nLFs with higher coverage had more influence (\\reffig{combined-apps}-Left; see also \\reffig{babble-size-corr}), but surprisingly, LFs with higher precision did not (\\reffig{combined-apps}-Mid).\nThe association with coverage stems at least partially from class balance:\neach LF outputs either all positive or all negative labels,\nso removing an LF with high coverage changes the class balance and consequently improves test performance on one class at the expense of the other (\\reffig{combined-apps}-Left).\nWhile these findings are not causal claims, they suggest that the coverage of an LF, rather than its precision, might have a stronger effect on its overall contribution to test performance.\n\n\n\n\\paragraph{MultiNLI.}\nThe MultiNLI dataset deals with natural language inference: determining if a pair of sentences agree, contradict, or are neutral. \\citet{williams2018broad} presented\ncrowdworkers with initial sentences from five genres and asked them to generate follow-on sentences that were neutral or in agreement\/contradiction (\\refapp{multinli-details}).\nWe studied the effect that each crowdworker had on the model's test set performance by computing the influence of the examples they created on overall test loss (Spearman $\\rho$ of $0.77$ to $0.86$ with actual effects across different genres; see \\reffig{mnli-acc}).\n\n\nStudying the influence of each crowdworker reveals that the number of examples a crowdworker created was not predictive of influence on test performance: e.g., the most prolific crowdworker contributed 35,000 examples but had negative influence, and we verified that removing all of those examples and retraining the model indeed made overall test performance worse (\\reffig{combined-apps}-Right).\nCuriously, this effect was genre-specific: crowdworkers who improved performance on some genres would lower performance on others (\\reffig{mnli-genre-categories}), even though the number of examples they contributed to a genre did not correlate with their influence on it (\\reffig{mnli-genre-contrib}).\nWe note that these results are obtained on a baseline logistic regression model built on top of a continuous bag-of-words representation.\nIdentifying precisely what makes a crowdworker's contributions useful, especially on higher-performing models, could help us improve dataset collection and credit attribution as well as better understand the biases due to annotator effects \\citep{geva2019annotator}.\n\n\n\\begin{figure}[t]\n  \\centering\n  \\vspace{-4mm}\n  \\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={right,top},capbesidewidth=58mm}}]{figure}[\\FBwidth]\n  {\\caption{\n    In CDR, the influence of a labeling function (LF) on test performance is predicted by its coverage (Left) but not its precision (Mid).\n    However, in MultiNLI, the number of examples contributed by a crowdworkers is not predictive of its influence (Right).\n    For CDR, LFs output either all $+$ or all $-$ labels; we plot the influence of each LF on the test points of the same class.\n  }\\label{fig:combined-apps}}\n  {\\includegraphics[width=0.55\\textwidth]{figs\/plots\/paper-combined-apps.pdf}}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discussion}\n\nIn this paper, we showed empirically that the influences of groups of points are highly correlated with, and consistently underestimate, their actual effects across a range of datasets, types of groups, and sizes.\nThese phenomena allows us to use influence functions to better understand the ``different stories that different parts of the data tell,'' in the words of \\citet{hampel1986robust}.\nWe showed that we can gain insight into the effects of a labeling function in data programming, or a crowdworker in a crowdsourced dataset, by computing the influence of their corresponding group effects.\n\nWhile these applications involved predefined groups, influence functions could potentially also discover coherent, semantically-relevant groups in the data.\nThey can also be used to approximate Shapley values,\nwhich are a different but related way of measuring the effect of data points;\nsee, e.g., \\citet{jia2019towards} and \\citet{ghorbani2019data}.\nSeparately, influence functions can also estimate the effects of \\emph{adding} training points. In this context, underestimation turns into overestimation, i.e., the influence of adding a group of training points tends to overestimate the actual effect of adding that group. This raises the possibility of using influence functions to evaluate the vulnerability of a given dataset and model to data poisoning attacks \\citep{steinhardt2017certified}.\n\nOur theoretical analysis showed that while correlation and underestimation hold in some restricted settings, they need not hold in general, realistic settings. This gap between theory and experiments opens up important directions for future work:\nWhy do we observe such striking correlation between predicted and actual effects on real data?\nTo what extent is this due to the specific model, datasets, or subsets used?\nDo these trends hold for non-convex models like neural networks?\nOur work suggests that there could be distributional assumptions that hold for real data and give rise to the broad phenomena of correlation and underestimation.\nOne promising lead is the surprising observation that the Newton approximation is much more accurate than influence at predicting group effects,\nwhich holds out the hope that we can understand group effects using just low-order terms (since the Newton approximation only uses the first and second derivatives of the loss) without needing to account for the whole loss function through higher order terms (as in \\citet{giordano2019higher}).\n\n\\section{Dataset details}\\label{sec:ds-details}\nWe used the same versions of the Diabetes, Enron, Dogfish, and MNIST datasets as \\citet{koh2017understanding}, since the examination of the accuracy of influence functions for large perturbations is a natural extension of their studies of small perturbations. Additionally, we applied influence to more natural settings in CDR and MultiNLI; here, we discuss their preprocessing pipelines.\n\n\\subsection{CDR}\\label{sec:bl-details}\n\n\\citet{hancock2018babble} established the BabbleLabble framework for data programming, following the following pipeline: They took labeled examples with natural language explanations, parsed the explanations into programmatic labeling functions (LFs) via a semantic parser, and filtered out obviously incorrect LFs. Then, they applied the remaining LFs to unlabeled data to create a sparse label matrix, from which they learned a label aggregator that outputs a noisily labeled training set. Finally, they ran $L_2$-regularized logistic regression on a set of basic linguistic features with the noisy labels.\n\nThey demonstrated their method on three datasets: Spouse, CDR, and Protein. The Protein dataset was not publicly available, and the vast majority of Spouse was labeled by a single LF, hence we chose to use CDR. This dataset's associated task involved identifying whether, according to a given sentence, a given chemical causes a given disease. For instance, the sentence\n``Young women on replacement \\textit{estrogens} for ovarian failure after cancer therapy may also have increased risk of \\textit{endometrial carcinoma} and should be examined periodically.''\nwould be labeled True, since it indicates that estrogens may cause endometrial carcinoma \\citep{hancock2018babble}.\nThe sentences and ground truth labels were sourced from the 2015 BioCreative chemical-disease relation dataset \\citep{wei2015overview}.\n\nIn our application, we began with their 28 LFs and the corresponding label matrix. For simplicity, we did not learn a label aggregator; instead, if an example $x$ was given labels $y_{i_1},y_{i_2},\\ldots,y_{i_k}$ by $k$ LFs $i_1,\\ldots,i_k$, then we created $k$ copies of $x$, each with weight $1\/k$. The subset of points corresponding to LF $i_1$ then included one instance of $x$ with weight $1\/k$. This weighting was taken into account in model training as well as in calculations of influence and actual effect. In addition, we used $L_1$-regularization for feature selection, reducing the number of features to 328 while still achieving similar F1 score to \\citep{hancock2018babble}; they reported an F1 of 42.3, while we achieved 42.0. After feature selection, we remove the $L_1$-regularization and train a $L_2$-regularized logistic regression model. We assume that the feature selection step is static and not affected by removing groups of data (though in general this assumption is not true); we therefore do not include feature selection in our influence calculations.\n\nWe note that in BabbleLabble, a given LF can never output positive on one example but negative on another. Hence, some LFs are positive (unable to output negative and only able to abstain or output positive), while the others are negative (unable to output positive and only able to abstain or output negative).\n\n\\subsection{MultiNLI}\\label{sec:multinli-details}\n\n\\citet{williams2018broad} created the MultiNLI dataset for the task of natural language inference: determining if a pair of sentences agree, contradict, or are neutral. To do so, they presented crowdworkers with initial sentences and asked them to generate follow-on sentences that were neutral or in agreement\/contradiction. For example, a crowdworker may be presented with ``Met my first girlfriend that way.'' and write the contradicting sentence ``I didn't meet my first girlfriend until later.'' \\citep{williams2018broad}. Thus, each of the 380 crowdworkers generated a subset of the dataset. We used these subsets in our application of influence.\n\nThe training set consisted of 392,702 examples from five genres. The development set consisted of 10,000 ``matched'' examples from the same five genres as the training set, as well as 10,000 ``mismatched'' examples from five new genres. The test set was put on Kaggle as an open competition, hence we do not have its labels and could not use it; therefore, we use the development set as the test set.\n\nThe continuous bag-of-words baseline in \\citet{williams2018broad} first converted the raw text of each sentence in the pair into a vector by treating the sentence as a continuous bag of words and simply averaging the 300-dimensional GloVe vector embeddings. This converted a pair of sentences into vectors $a,b$. They then concatenated $[a,b,a-b,a \\odot b]$ into a 1200D vector, where $a \\odot b$ denotes the element-wise product. Finally, they treated this as input to a neural network with three hidden layers and fine-tuned the entire model, including word embeddings (more details in \\citep{williams2018broad}).\n\nFor our application, we truncated their baseline and just used the concatenation of $a$ and $b$ as the representation for every example. By running logistic regression on this, we achieved test accuracy of $50.4\\%$ (vs. their baseline's $64.7\\%$; the performance difference comes from the additional dimensions in their vector embeddings and the finetuning through the neural network). Future work could explore influence in the setting of more complex and higher-performing models.\n\n\\section{Experimental details for comparing influence vs. actual effects on constructed groups}\\label{sec:expt-details}\n\\subsection{Model training}\nFor all experiments in \\refsec{accuracy}, we trained a logistic regression model (or softmax for multiclass) using\n\\texttt{sklearn.linear\\_model.LogisticRegression.fit},\n fitting the intercept but only applying $L_2$-regularization to the weights.\nTo choose the regularization strength $\\lambda$, we conducted 5-fold\ncross-validation across 10 possible values of $\\lambda\/n$ logarithmically\nspaced between $1.0\\times 10^{-4}$ and $1.0\\times 10^{-1}$, inclusive,\nselecting the regularization that yielded the highest cross-validation accuracy\n(except on the CDR dataset, where we selected regularization based on\ncross-validation F1 score to account for class imbalance as per\n\\citet{hancock2018babble}'s procedure).\n\n\\subsection{Group construction}\\label{sec:subsetgen}\nFor each dataset, we constructed groups of various sizes relative to the entire dataset by considering 100 sizes linearly spaced between $0.25\\%$ and $25\\%$ of the dataset. For each of these 100 sizes, we constructed one group with each of the following methods:\n\\begin{enumerate}\n  \\item Shared features: We selected a single feature uniformly at random and sorted the dataset along this selected feature. Next, we selected an training point uniformly at random. We then constructed a group of size $s$ that consisted of the $s$ unique training points that were closest to the chosen point, as measured by their values in the selected feature. We randomly sampled a feature and initial training point for each different group constructed in this way.\n\n  \\item Feature clustering: We clustered the dataset with respect to raw features via \\texttt{scipy.cluster.hierarchy.fclusterdata} with \\texttt{t} set to $1$, as well as with \\texttt{sklearn.cluster.KMeans.fit} with \\texttt{n\\_clusters} taking on values $4,8,16,32,64,128$. Since hierarchical clustering determines cluster sizes automatically with a principled heuristic and we try a range of values for \\texttt{n\\_clusters} in $k$-means, this recovers clusters with a large range of sizes. The clustering with $\\texttt{n\\_clusters}=4$ also guarantees (via the pigeonhole principle) that there is at least one cluster which contains at least $25\\%$ of the dataset. From all the clusters that are at least the size of the desired group, we chose one uniformly at random and chose the group uniformly at random and without replacement from the training points in this cluster.\n\n  \\item Gradient clustering: We followed the same procedure as ``Feature clustering,'' except that we clustered the dataset with respect to $\\nabla_\\theta \\ell(x,y;\\hat\\theta(\\bone))$, i.e. each training point was represented by the gradient of the loss on that point.\n\n  \\item Random within class: We considered all classes with at least as many training points as the size of the desired group. From these classes, we chose one uniformly at random. Then, we chose the group uniformly at random and without replacement from all training points in this class.\n\n\t\\item Random: We picked a group uniformly at random and without replacement from the entire dataset.\n\\end{enumerate}\n\nThe above methods gave us a total of 500 groups (100 groups per method) for each dataset, with the exception of the ``random within class'' method for MNIST. Since MNIST has 10 classes, each with only $10\\%$ of the data, we skipped over groups of size $>10\\%$ just for the ``random within class'' groups.\n\nIn addition, we selected 3 random test points and the 3 test points with highest loss; we intend these to represent the average case and the more extreme case that may be relevant to model developers who want to debug errors that their model outputs. For each of these 6 test points, we selected groups that had large positive influence on its test loss. More specifically, we proceeded in 3 stages:\n\\begin{enumerate}\n\t\\item We considered 33 group sizes linearly spaced between $0.25\\%$ and $2.5\\%$ of the dataset, and for any size $s$ out of these 33, we selected a group uniformly at random and without replacement from training points in the top $1.5\\times 2.5\\%$ of the dataset, ordered according to their influence on the test point of interest.\n\t\\item This was similar to the first stage, but with 33 sizes spaced between $0.25\\%$ and $10\\%$ and groups chosen from the top $1.5\\times 10\\%$ of the dataset.\n\t\\item Finally, we considered 34 sizes spaced between $0.25\\%$ and $25\\%$, with groups chosen from the top $1.5\\times 25\\%$ of the dataset.\n\\end{enumerate}\nLarger groups tend to have lower average influence than smaller groups, since by necessity, the group must contain points farther from the top.\nThis multi-stage approach ensured that we would select small groups with both a high average influence and also with a low average influence, so that we could compare them to larger groups and mitigate confounding the group size with its average influence.\n\nFinally, we repeated this last method of group construction for groups with large negative influence on test point loss.\n\nUsing these 6 test points, we generated 1,200 groups (100 subsets per group, with 6 test points, and drawing from the positive and negative tails). In total, we therefore generated 1,700 groups per dataset (except MNIST).\n\n\n\\subsection{Comparison of influence and actual effect}\\label{sec:compare}\n\nTo produce \\reffig{acc}, we selected groups as described in \\refapp{subsetgen}. We retrained the model once for each group, excluding the group in order to calculate its actual effect. To compute all groups' influences, we first calculated the influence of every individual training point using the procedure of \\citet{koh2017understanding}. Then, to compute the influence on test prediction or loss of some group, we simply added the relevant individual influences (in CDR, we weighted these individual influences according to that point's weight; see \\refapp{bl-details}). To compute the influence on self-loss of some group, we summed up the gradients of the loss of each training point to compute $g_\\bone(w)$, we calculated the inverse Hessian vector product $H_{\\lambda, \\bone}^{-1} g_\\bone(w)$ and took its dot product with $g_\\bone(w)$ (again, we modified this with appropriate weighting for individual points in CDR).\n\n\\section{Introduction}\\label{sec:intro}\n\nInfluence functions \\citep{jaeckel1972infinitesimal, hampel1974influence, cook1977detection} estimate the effect of removing an\nindividual training point on a model's predictions without the computationally-prohibitive cost of retraining the model.\nTracing a model's output back to its training data can be useful:\ninfluence functions have been recently applied to explain predictions \\citep{koh2017understanding},\nproduce confidence intervals \\citep{schulam2019can},\ninvestigate model bias \\citep{brunet2018understanding, wang2019repairing},\nimprove human trust \\citep{zhou2019effects},\nand even craft data poisoning attacks \\citep{koh2019stronger}.\n\nInfluence functions are based on first-order Taylor approximations that are accurate for estimating small perturbations to the model,\nwhich makes them suitable for predicting the effects of removing individual training points on the model.\nHowever, we often want to study the effects of removing \\emph{groups} of points, which represent large perturbations to the data.\nFor example, we might wish to analyze the effect of data collected from different experimental batches \\citep{leek2010tackling} or demographic groups \\citep{chen2018my};\napportion credit between crowdworkers, each of whom generated part of the data \\citep{arrieta2018should};\nor, in a multi-party learning setting, ensure that no individual user has too much influence on the joint model \\citep{hayes2018contamination}.\nAre influence functions still accurate when predicting the effects of (removing) these larger groups?\n\nIn this paper, we first show empirically that on real datasets and across a broad variety of groups of data, the predicted and actual effects are strikingly \\emph{correlated} (Spearman $\\rho$ of $0.8$ to $1.0$),\nsuch that the groups with the largest actual effect also tend to have the largest predicted effect. Moreover, the predicted effect tends to \\emph{underestimate} the actual effect, suggesting that it could be an approximate lower bound in practice.\nUsing influence functions to predict the actual effect of removing large, coherent groups of data can therefore still be useful,\neven though the violation of the small-perturbation assumption can result in high absolute and relative errors between the predicted and actual effects.\n\nWhat explains these phenomena of correlation and underestimation?\nPrior theoretical work focused on establishing the conditions under which this influence approximation is accurate, i.e., the error between the actual and predicted effects is small \\citep{giordano2019swiss, rad2018scalable}.\nHowever, in our setting of removing large, coherent groups of data,\nthis error can be quite large.\nAs a first step towards understanding the behavior of the influence approximation in this regime,\nwe characterize the relationship between the predicted and actual effects of a group via the one-step Newton approximation \\citep{pregibon1981logistic},\nwhich we find is a surprisingly accurate approximation in practice.\nWe show that correlation and underestimation arise under certain settings\n(e.g., removing multiple copies of a single training point),\nbut need not hold in general,\nwhich opens up the intriguing question of why we observe those phenomena across a wide range of empirical settings.\n\nFinally, we exploit the correlation of predicted and actual group effects in two example case studies: a chemical-disease relationship (CDR) task, where the groups correspond to different labeling functions \\citep{hancock2018babble}, and a natural language inference (NLI) task \\citep{williams2018broad}, where the groups come from different crowdworkers.\nOn the CDR task, we find that the influence of each labeling function correlates with its size (the number of examples it labels) but not its average accuracy, which suggests that practitioners should focus on the coverage of the labeling functions they construct. In contrast, on the NLI task, we find that the influence of each crowdworker is uncorrelated with the number of examples they contibute, which suggests that practitioners should focus on how to elicit high-quality examples from crowdworkers over increasing quantity.\n\n\n\n\\section{Background and problem setup}\\label{sec:setting}\nConsider learning a predictive model with parameters $\\theta \\in \\Theta$ that\nmaps from an input space $\\sX$ to an output space $\\sY$.\nWe are given $n$ training points $\\{(x_1, y_1), \\ldots, (x_n, y_n)\\}$\nand a loss function $\\ell(x, y, \\theta)$ that is twice-differentiable and convex in $\\theta$.\nTo train the model, we select the model parameters\n\\begin{align}\n  \\hat\\theta(\\bone) = {\\arg\\min}_{\\theta\\in\\Theta}  \\left[ \\sum_{i=1}^n \\ell(x_i, y_i; \\theta) \\right] + \\frac{\\lambda}{2} \\|\\theta\\|_2^2\n\\end{align}\nthat minimize the $L_2$-regularized empirical risk,\nwhere $\\lambda > 0$ controls regularization strength.\nThe all-ones vector $\\bone$ in $\\hat\\theta(\\bone)$ denotes that the initial training points all have uniform sample weights.\n\nOur goal is to measure the effects of different groups of training data\non the model: if we removed a subset of training points $W$,\nhow much would the model $\\hat\\theta$ change?\nConcretely, we define a vector $w \\in \\{0,1\\}^n$ of sample weights with $w_i = \\1((x_i, y_i) \\in W)$\nand consider the modified parameters\n\\begin{align}\n\\hat\\theta(\\bone - w) = {\\arg\\min}_{\\theta\\in\\Theta}  \\left[\\sum_{i=1}^n (1 - w_i) \\ell(x_i, y_i; \\theta) \\right] + \\frac{\\lambda}{2} \\|\\theta\\|_2^2\n\\end{align}\ncorresponding to retraining the model after excluding $W$.\nWe refer to $w$ as the subset (corresponding to $W$);\nthe number of removed points as $\\|w\\|_1$;\nand the fraction of removed points as $\\alpha = \\|w\\|_1 \/ n$.\n\nThe \\emph{actual effect} $\\sI^{*}_f:[0,1]^n \\to \\R$ of the subset $w$ is\n\\begin{align}\n  \\sI^{*}_f(w) = f(\\hat\\theta(\\bone - w)) - f(\\hat\\theta(\\bone)),\n\\end{align}\nwhere the evaluation function $f:\\Theta\\to\\R$ measures a quantity of interest. Specifically, we study:\n\\begin{itemize}\n  \\item The \\emph{change in test prediction}, with $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$. Linear models (for regression or binary classification) make predictions that are functions of $\\theta^\\top {x_{\\mathrm{test}}}$, so this measures the effect that removing a subset will have on the model's prediction for some test point ${x_{\\mathrm{test}}}$.\n\n  \\item The \\emph{change in test loss}, with $f(\\theta) = \\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$, which is similar to the test prediction.\n\n  \\item The \\emph{change in self-loss}, with $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$, measures the increase in loss on the removed points $w$. Its average over all subsets of size $\\|w\\|_1$ is the estimated extra loss that leave-$\\|w\\|_1$-out cross-validation (CV) measures over the training loss.\n\n\\end{itemize}\n\n\n\\subsection{Influence functions}\nThe issue with computing the actual effect $\\sI^{*}_f(w)$ is that retraining the model to compute $\\hat\\theta(\\bone - w)$ for each subset $w$ can be prohibitively expensive.\nInfluence functions provide a relatively efficient first-order approximation to $\\sI^{*}_f(w)$ that avoids retraining.\n\nConsider the function $q_w: [0, 1] \\to \\R$ with\n$q_w(t) = f\\bigl(\\hat\\theta(\\bone - tw)\\bigr)$,\nsuch that the actual effect $\\sI^{*}_f(w)$ can be written as $q_w(1) - q_w(0)$.\nWe define the \\emph{predicted effect} of the subset $w$ to be\nits \\emph{influence} $\\sI_f(w) = q_w'(0) \\approx q_w(1) - q_w(0)$;\nin this paper, we use the term predicted effect interchangeably with influence.\nIntuitively, influence measures the effect of removing an infinitesimal weight from each point in $w$ and then linearly extrapolates to removing all of $w$.\\footnote{In the statistics literature, influence typically refers to the effect of \\emph{adding} weight, so the sign is flipped.} By taking a Taylor approximation (see, e.g., \\citet{hampel1986robust} for details), the influence can be computed as\n\\begin{align}\n\\sI_f(w) \\eqdef q_w'(0)\\nonumber\n&= \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top \\left[ \\frac{d}{dt}\\hat\\theta(\\bone - tw)\\Bigr|_{\\substack{t=0}} \\right]\\nonumber\\\\\n&= \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w),\n\\end{align}\nwhere\n$g_\\bone(w) = \\sum_{i=1}^n w_i\\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))$,\n$H_\\bone = \\sum_{i=1}^n \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))$,\nand $H_{\\lambda, \\bone} = H_\\bone + \\lambda I$.\nWhen measuring the change in test prediction or test loss, influence is additive: if $w = w_1 + w_2$, then $\\sI_f(w) = \\sI_f(w_1) + \\sI_f(w_2)$, i.e., the influence of a subset is the sum of influences of its constituent points,\nand we can efficiently compute the influence of any subset by pre-computing the influence of each individual point (e.g., by taking a single inverse Hessian-vector product, as in\n\\citet{koh2017understanding}).\nHowever, when measuring the change in self-loss, influence is not additive and requires a separate calculation for each subset removed.\n\n\\subsection{Relation to prior work}\\label{sec:related}\nInfluence functions---introduced in the seminal work of \\citet{hampel1974influence} and in \\citet{jaeckel1972infinitesimal}, where it was called the infinitesimal jackknife---have a rich history in robust statistics.\nThe use of influence functions in the ML community is more recent, though growing; in \\refsec{intro}, we provide references for several recent applications of influence functions in ML.\n\nRemoving a single training point, especially when the total number of points $n$ is large, represents a small perturbation to the training distribution,\nso we expect the first-order influence approximation to be accurate.\nIndeed, prior work on the accuracy of influence has focused on this regime: e.g.,\n\\citet{debruyne2008model, liu2014efficient, rad2018scalable, giordano2019swiss} give\nevidence that the influence on self-loss can approximate LOOCV, and \\citet{koh2017understanding} similarly examined the accuracy of estimating the change in test loss after removing single training points.\n\nHowever, removing a constant fraction $\\alpha$ of the training data represents a large perturbation to the training distribution.\nTo the best of our knowledge,\nthis setting has\nnot been empirically studied;\nperhaps the closest work is \\citet{khanna2019interpreting}'s use of Bayesian quadrature to estimate a maximally influential subset.\nInstead, older references have alluded to the phenomena of correlation and underestimation we observe:\n\\citet{pregibon1981logistic} note that influence tends to be conservative,\nwhile \\citet{hampel1986robust} say that ``bold extrapolations'' (i.e., large perturbations) are often still useful.\nOn the theoretical front, \\citet{giordano2019swiss} established\nfinite-sample error bounds that apply to groups, e.g., showing that the leave-$k$-out approximation is consistent as the fraction of removed points $\\alpha \\to 0$.\nOur focus is instead on the relationship of the actual effect $\\sI^{*}_f(w)$ and predicted effect (influence) $\\sI_f(w)$ in the regime where $\\alpha$ is constant and\nthe error $|\\sI^{*}_f(w) - \\sI_f(w)|$ is large.\n\n\\section{Proofs}\\label{sec:proofs}\n\\subsection{Notation}\\label{sec:notation}\nWe first review the notation given in \\refsec{setting} and introduce new definitions that will be useful in the sequel. We define the empirical risk as\n\\begin{align*}\n  L_s(\\theta) \\eqdef \\left[ \\sum_{i=1}^n s_i \\ell(x_i, y_i; \\theta)\\right] + \\frac{\\lambda}{2} \\|\\theta\\|_2^2,\n\\end{align*}\nsuch that the optimal parameters are $\\hat\\theta(s) \\eqdef {\\arg\\min}_{\\theta\\in\\Theta} L_s(\\theta)$.\n\nGiven sample weight vectors $r, s \\in \\R^n$, we define the derivatives\n\\begin{align*}\n  g_r(s) &\\eqdef \\sum_{i=1}^n s_i\\nabla_\\theta \\ell(x_i, y_i; \\hat\\theta(r))\\\\\n  H_r(s) &\\eqdef \\sum_{i=1}^n s_i \\nabla^2_\\theta \\ell(x_i, y_i; \\hat\\theta(r)).\n\\end{align*}\nIf the argument $s$ is omitted, it is assumed to be equal to $r$. For example,\n\\begin{align*}\n  H_\\bone \\eqdef \\sum_{i=1}^n \\nabla^2_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone)).\n\\end{align*}\nIf $H$ has a $\\lambda$ subscript, then we add $\\lambda I$. For example,\n\\begin{align*}\n  H_{\\lambda, \\bone} \\eqdef H_\\bone + \\lambda I.\n\\end{align*}\n\nFor a given dataset, we define the following constants:\n\\begin{align*}\n  C_\\ell &= \\max_{1 \\leq i \\leq n} \\twonorm{\\nabla_\\theta \\ell(x_i,y_i,\\hat\\theta(\\bone))},\\\\\n  \\sigma_{\\mathrm{min}} &= \\text{smallest singular value of}\\ H_\\bone,\\\\\n  \\sigma_{\\mathrm{max}} &= \\text{largest singular value of}\\ H_\\bone.\n\\end{align*}\n\nTo avoid confusion with the vector 2-norm, we will use the operator norm $\\opnorm{\\cdot}$ to denote the matrix 2-norm.\n\nIn the sequel, we study the order-3 tensor $\\nabla_\\theta^3 f(\\hat\\theta(\\bone))$. We define its product with a vector (which returns a matrix) as a contraction along the last dimension:\n\\begin{equation*}\n  \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone)), v \\right\\rangle_{ij} \\eqdef \\sum_k \\frac{\\partial^3 f(\\hat\\theta(\\bone))}{\\partial \\theta_i \\partial \\theta_j \\partial \\theta_k} v_k.\n\\end{equation*}\n\n\\subsection{Assumptions}\\label{sec:assumptions}\nWe make the following assumptions on the derivatives of the loss $\\ell(x, y, \\theta)$\nand the evaluation function $f(\\theta)$.\n\\begin{assumption}[Positive-definiteness and Lipschitz continuity of $H$]\\label{assump:l-const}\n  The loss $\\ell(x, y, \\theta)$ is convex and twice-differentiable in $\\theta$, with positive regularization $\\lambda > 0$.\n  Further, there exists $C_H\\in\\R$ such that\n  \\[\n    \\opnorm{\\nabla_\\theta^2 \\ell(x, y, \\theta_1) - \\nabla_\\theta^2\\ell(x, y, \\theta_2)} \\leq C_H \\twonorm{\\theta_1 - \\theta_2}\n  \\]\n  for all $(x, y) \\in \\sX \\times \\sY$ and $\\theta_1, \\theta_2 \\in \\Theta$. This is a bound on the third derivative of $\\ell$, if it exists.\n\\end{assumption}\n\\begin{assumption}[Bounded derivatives of $f$]\\label{assump:q-const}\n  $f(\\theta)$ is thrice-differentiable, with $C_f, C_{f, 3} \\in \\R$ such that\n  \\begin{align*}\n    C_{f} &= \\sup_{\\theta \\in \\Theta} \\twonorm{\\nabla_\\theta f(\\theta)},\n    &C_{f,3} &= \\sup_{v \\in \\Theta, \\twonorm{v} = 1} \\opnorm{\n    \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone)), v \\right\\rangle}.\n  \\end{align*}\n\\end{assumption}\nThese assumptions apply to all the results that follow below.\n\n\\subsection{Bounding the error of the one-step Newton approximation}\\label{sec:proof-newton}\n\n\n\\begin{repproposition}{prop:newton-actual}[Restated]\n  Let the Newton error be $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) \\eqdef \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$.\n  Then under \\refassumps{l-const}{q-const},\n  \\begin{align*}\n      \\abs{\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)} \\leq \\frac{n \\|w\\|_1^2 C_f C_H C_\\ell^2}{(\\sigma_{\\mathrm{min}} + \\lambda)^3}.\n  \\end{align*}\n  $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ only involves third-order or higher derivatives of the loss, so it is 0 for quadratic losses.\n\\end{repproposition}\n\\begin{proof}\n  This proof is adapted to our setting from the standard analysis of the Newton method in convex optimization \\citep{boyd2004convex}.\n\n  First, note that $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) = \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w) = f(\\hat\\theta(\\bone - w)) - f(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w}))$.\n  We will bound the norm of the difference of the parameters $\\twonorm{\\hat\\theta(\\bone - w) - \\hat\\theta_{\\mathrm{Nt}}({\\bone - w})}$; the desired bound on $f$ then follows from the assumption that $f$ has gradients bounded by $C_f$ and is therefore Lipschitz.\n\n  Since $L_{\\bone - w}(\\theta)$ is strongly convex (with parameter $\\sigma_{\\mathrm{min}} + \\lambda$) and minimized by $\\hat\\theta(\\bone - w)$, we can bound the distance $\\twonorm{\\hat\\theta(\\bone - w) - \\hat\\theta_{\\mathrm{Nt}}({\\bone - w})}$ in terms of the norm of the gradient at $\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})$:\n  \\begin{align*}\n    \\twonorm{\\hat\\theta(\\bone - w) - \\hat\\theta_{\\mathrm{Nt}}({\\bone - w})} &\\leq \\frac{2}{\\sigma_{\\mathrm{min}} + \\lambda}\\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})\\right)}.\n  \\end{align*}\n\n  Therefore, the problem reduces to bounding $\\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})\\right)}$.\n\n  We start by expressing the Newton step $\\Delta\\theta_{\\mathrm{Nt}}(w)$ in terms of the first and second derivatives of the empirical risk $L_{\\bone - w}(\\theta)$:\n  \\begin{align*}\n    g_\\bone(w) &= \\sum_{i=1}^n w_i \\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= -\\sum_{i=1}^n (1 - w_i) \\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= -\\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone)),\\\\\n    H_{\\lambda, \\bone}({\\bone - w}) &= \\sum_{i=1}^n (1 - w_i) \\nabla_\\theta^2 \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= \\nabla_\\theta^2 L_{\\bone - w}(\\hat\\theta(\\bone)),\\\\\n    \\Delta\\theta_{\\mathrm{Nt}}(w) &= H_{\\lambda, \\bone}({\\bone - w})^{-1} g_\\bone(w)\\\\\n    &= -\\left[ \\nabla_\\theta^2 L_{\\bone - w}(\\hat\\theta(\\bone)) \\right]^{-1} \\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone)),\n  \\end{align*}\n  where the second equality for $g_\\bone(w)$ comes from the fact that at the optimum $\\hat\\theta(\\bone)$, the sum of the gradients $\\sum_{i=1}^n \\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))$ is 0.\n\n  With these expressions, we bound the norm of the gradient $\\nabla_\\theta L_{\\bone - w}(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w}))$:\n  \\begin{align*}\n    &\\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})\\right)}\\\\\n    &= \\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)\\right)}\\\\\n    &= \\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)\\right)\n       - \\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta(\\bone)\\right)\n       - \\nabla_\\theta^2 L_{\\bone - w}\\left(\\hat\\theta(\\bone)\\right) \\Delta\\theta_{\\mathrm{Nt}}(w)}\\\\\n    &= \\twonorm{\\int_0^1 \\left(\\nabla_\\theta^2 L_{\\bone - w}\\left(\\hat\\theta(\\bone) + t\\Delta\\theta_{\\mathrm{Nt}}(w) \\right) - \\nabla_\\theta^2 L_{\\bone - w}\\left(\\hat\\theta(\\bone)\\right)\\right) \\Delta\\theta_{\\mathrm{Nt}}(w) \\ dt}\\\\\n    &\\leq \\frac{n C_H}{2} \\twonorm{\\Delta\\theta_{\\mathrm{Nt}}(w)}^2\\\\\n    &= \\frac{n C_H}{2} \\twonorm{\\left[ \\nabla_\\theta^2 L_{\\bone - w}(\\hat\\theta(\\bone)) \\right]^{-1} \\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone))}^2\\\\\n    &\\leq \\frac{n C_H}{2(\\sigma_{\\mathrm{min}} + \\lambda)^2} \\twonorm{\\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone))}^2\\\\\n    &\\leq \\frac{n \\onenorm{w}^2 C_H C_\\ell^2}{2(\\sigma_{\\mathrm{min}} + \\lambda)^2}.\n  \\end{align*}\n  Putting together the successive bounds gives the result.\n\\end{proof}\n\n\\subsection{Characterizing the difference between the Newton approximation and influence}\\label{sec:proof-decomp}\nBefore proving \\refprop{newton-inf}, we first prove a lemma about the spectrum of the error matrix $D(w)$.\n\\begin{lemma}\\label{lem:sv-bound-d}\nThe matrix $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ has singular values bounded between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$.\n\\end{lemma}\n\\begin{proof}\nWe first show that $H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ has singular values bounded between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}} + \\lambda}$.\nThe lower bound of 0 comes from the fact that $H_{\\lambda, \\bone}^{-\\inv{2}} H_\\bone(w) H_{\\lambda, \\bone}^{-\\inv{2}}$ is symmetric and $H_\\bone(w) \\succeq 0$.\n\nTo show the upper bound, first note that $H_\\bone(w) \\preceq H_\\bone(w) + H_\\bone(\\bone - w) = H_\\bone$ (recalling that $w\\in\\{0,1\\}^n$),\nand let $U \\Sigma U^\\top$ be the singular value decomposition of $H_\\bone$. Since $H_{\\lambda, \\bone} = H_\\bone + \\lambda I$, we have\n\\begin{align*}\n  H_{\\lambda, \\bone}^{-\\inv{2}} H_\\bone(w) H_{\\lambda, \\bone}^{-\\inv{2}} &=\n  \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}} H_\\bone(w) \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}}\\\\\n  &\\preceq \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}} H_\\bone \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}}\\\\\n  &= \\bigl(U (\\Sigma + \\lambda I) U^\\top\\bigr)^{-\\inv{2}} U \\Sigma U^\\top \\bigl(U (\\Sigma + \\lambda I) U^\\top\\bigr)^{-\\inv{2}}\\\\\n  &= U (\\Sigma + \\lambda I)^{-\\inv{2}} \\Sigma (\\Sigma + \\lambda I)^{-\\inv{2}} U^\\top,\n\\end{align*}\nso its maximum singular value is upper bounded by $\\frac{\\sigma_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}} + \\lambda}$.\n\nThe bound on the singular values of $H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ implies that the singular values of $I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ lie in $\\left[\\frac{\\lambda}{\\sigma_{\\mathrm{max}} + \\lambda}, 1\\right]$.\nIn turn, this implies that the singular values of $\\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1}$ lie in $\\left[1, \\frac{\\sigma_{\\mathrm{max}} + \\lambda}{\\lambda}\\right]$. Subtracting 1 from each end (for the identity matrix) gives the desired result.\n\\end{proof}\n\n\\begin{repproposition}{prop:newton-inf}[Restated]\n  Under \\refassumps{l-const}{q-const}, the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$ is\n  \\begin{align*}\n    \\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w) &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n    \\ + \\ \\underbrace{\\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) + \\mathrm{Err}_{f,3}(w),}_\\text{\n      Error from curvature of $f(\\cdot)$\n    }\n  \\end{align*}\n  with $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$\n  and $H_\\bone(w) \\eqdef \\sum_{i=1}^n w_i \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))$.\n  The \\emph{error matrix} $D(w)$ has eigenvalues between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$,\n  where $\\sigma_{\\mathrm{max}}$ is the largest eigenvalue of $H_\\bone$.\n  The residual term $\\mathrm{Err}_{f,3}(w)$ captures the error due to third-order derivatives of $f(\\cdot)$ and is bounded by $\\abs{\\mathrm{Err}_{f,3}(w)} \\leq {\\|w\\|_1^3 C_{f, 3}C_\\ell^3}\/{6(\\sigma_{\\mathrm{min}} + \\lambda)^3}$.\n\\end{repproposition}\n\\begin{proof}\n  From the second-order Taylor expansion of $f$ about $\\hat\\theta(\\bone)$, there exists $0 \\leq \\xi \\leq 1$ such that\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) &= f(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})) - f(\\hat\\theta(\\bone))\\\\\n    &= f(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)) - f(\\hat\\theta(\\bone))\\\\\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) +\\\\\n    &\\quad \\inv{6} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone) + \\xi\\Delta\\theta_{\\mathrm{Nt}}(w)), \\Delta\\theta_{\\mathrm{Nt}}(w) \\right\\rangle \\Delta\\theta_{\\mathrm{Nt}}(w)\\\\\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) + \\mathrm{Err}_{f,3}(w),\n    \\numberthis \\label{eqn:newton-inf-proof-1}\n  \\end{align*}\n  where we define $\\mathrm{Err}_{f,3}(w) \\eqdef \\inv{6} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone) + \\xi\\Delta\\theta_{\\mathrm{Nt}}(w)), \\Delta\\theta_{\\mathrm{Nt}}(w) \\right\\rangle \\Delta\\theta_{\\mathrm{Nt}}(w)$ to be the error due to third-order and higher derivatives of $f$.\n\n  We can express the difference between the first-order Taylor term $\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w)$ above and the first-order influence approximation $\\sI_f(w) = q'_w(0) = \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)$ as\n  \\begin{align*}\n    &\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) - \\sI_f(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) - \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\left(H_{\\lambda, \\bone}({\\bone - w})^{-1} - H_{\\lambda, \\bone}^{-1}\\right) g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\left( \\big(H_{\\lambda, \\bone} - H_\\bone(w)\\big)^{-1} - H_{\\lambda, \\bone}^{-1}\\right) g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left(H_{\\lambda, \\bone}^{\\inv{2}}\\big(H_{\\lambda, \\bone} - H_\\bone(w)\\big)^{-1}H_{\\lambda, \\bone}^{\\inv{2}} - I\\right) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left(\\big(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\big)^{-1} - I\\right) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n    \\numberthis \\label{eqn:newton-inf-proof-2}.\n  \\end{align*}\n  Substituting \\refeqn{newton-inf-proof-2} into \\refeqn{newton-inf-proof-1},\n  we have that\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) - \\sI_f(w)\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &\\quad + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w)\n    + \\mathrm{Err}_{f,3}(w),\n  \\end{align*}\n  as desired.\n\n  We can bound $\\mathrm{Err}_{f,3}(w)$ as follows:\n  \\begin{align*}\n    \\abs{\\mathrm{Err}_{f,3}(w)} &\\leq\n    \\frac{C_{f, 3}}{6}\\twonorm{\\Delta\\theta_{\\mathrm{Nt}}(w)}^3\\\\\n    &\\leq \\frac{\\onenorm{w}^3 C_{f, 3}C_\\ell^3}{6(\\sigma_{\\mathrm{min}} + \\lambda)^3}.\n  \\end{align*}\n  Applying \\reflem{sv-bound-d} to bound the spectrum of $D(w)$ completes the proof.\n\\end{proof}\n\n\n\\subsection{The influence on self-loss}\\label{sec:proof-selfloss}\nWe first state two linear algebra facts that will be useful in the sequel.\n\\begin{lemma}\\label{lem:sv-bound}\n  Let $A \\succ 0, B \\succeq 0 \\in \\R^{d \\times d}$ be a pair of symmetric positive-definite and positive-semidefinite matrices, respectively. Let $\\sigma_{A, 1}$ be the largest eigenvalue of $A$, $\\sigma_{A, d}$ the smallest eigenvalue of $A$, and similarly let $\\sigma_{B, 1}$ and $\\sigma_{B, d}$ be the largest and smallest eigenvalues of $B$, respectively. Then\n  \\begin{align*}\n    \\frac{\\sigma_{B, d}}{\\sigma_{A, 1}} I\n    \\preceq A^{-\\inv{2}} B A^{-\\inv{2}}\n    \\preceq \\frac{\\sigma_{B, 1}}{\\sigma_{A, d}} I.\n  \\end{align*}\n\\end{lemma}\n\\begin{proof}\n  Note that $\\inv{\\sigma_{A, 1}}$ is the smallest eigenvalue of $A^{-1}$, while $\\inv{\\sigma_{A, d}}$ is its largest. The lemma follows from the fact that the smallest singular value of the product of two matrices is lower bounded by the product of the smallest singular values of each matrix, and similarly the largest singular value of the product is upper bounded by the product of the largest singular values of each matrix.\n\\end{proof}\n\nThe next fact is a consequence of the variational definition of eigenvalues.\n\\begin{lemma}\\label{lem:quadform-bound}\n  Given a symmetric matrix $A \\in \\R^{d \\times d}$ and a vector $v \\in \\R^d$, we have the following bounds on the quadratic form $v^\\top A v$:\n  \\[\n  \\sigma_d \\twonorm{v}^2 \\leq v^\\top A v \\leq \\sigma_1 \\twonorm{v}^2,\n  \\]\n  where $\\sigma_d$ is the smallest eigenvalue of $A$, and $\\sigma_1$ is the largest.\n\\end{lemma}\n\nWe are now ready to analyze the effect of removing a subset $w$ of $k$ training points on the total loss on those $k$ points.\n\\begin{repproposition}{prop:selfloss}[Restated]\n  Under \\refassumps{l-const}{q-const}, the influence on the self-loss $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$ obeys\n  \\begin{align*}\n    \\sI_f(w) + \\mathrm{Err}_{f,3}(w) &\\leq \\sI^\\mathrm{Nt}_f(w)\n    \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w) + \\mathrm{Err}_{f,3}(w).\n  \\end{align*}\n\\end{repproposition}\n\\begin{proof}\n  Since $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$, we have that\n  \\begin{align*}\n    \\nabla_\\theta f(\\hat\\theta(\\bone)) &= \\sum_{i=1}^n w_i \\nabla_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= g_\\bone(w),\\\\\n    \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) &= \\sum_{i=1}^n w_i \\nabla_\\theta^2 \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= H_\\bone(w).\n  \\end{align*}\n  Substituting these and $\\Delta\\theta_{\\mathrm{Nt}}(w) = H_{\\lambda, \\bone}({\\bone - w})^{-1} g_\\bone(w)$ into \\refprop{newton-inf}, we obtain\n  \\begin{align*}\n    &\\sI^\\mathrm{Nt}_f(w) - \\sI_f(w) - \\mathrm{Err}_{f,3}(w)\\\\\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n      + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w)\\\\\n    &= g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n      + \\inv{2} g_\\bone(w)^\\top H_{\\lambda, \\bone}({\\bone - w})^{-1} H_\\bone(w) H_{\\lambda, \\bone}({\\bone - w})^{-1} g_\\bone(w)\\\\\n      &= g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left[ D(w) + \\underbrace{\\inv{2}H_{\\lambda, \\bone}^{\\inv{2}}H_{\\lambda, \\bone}({\\bone - w})^{-1} H_\\bone(w) H_{\\lambda, \\bone}({\\bone - w})^{-1} H_{\\lambda, \\bone}^{\\inv{2}}}_{\\eqdef \\Lambda(w)}\\right] H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w),\n  \\end{align*}\n  where $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ has singular values bounded between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$.\n  From \\reflem{sv-bound}, $\\Lambda(w)$ has singular values bounded between $0$ and\n  $\\frac{\\sigma_{\\mathrm{max}}(\\sigma_{\\mathrm{max}} + \\lambda)}{2\\lambda^2}$.\n\n  Applying \\reflem{quadform-bound} and using $\\sI_f(w) = g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)$, we obtain\n  \\begin{align*}\n    0 &\\leq\n    g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left[ D(w) + \\Lambda(w) \\right] H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &\\leq \\left(\\frac{\\sigma_{\\mathrm{max}}}{\\lambda} + \\frac{\\sigma_{\\mathrm{max}}(\\sigma_{\\mathrm{max}} + \\lambda)}{2\\lambda^2}\\right) g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)\\\\\n    &= \\left(\\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w),\n  \\end{align*}\n  which gives us\n  \\begin{align*}\n      \\sI_f(w) + \\mathrm{Err}_{f,3}(w) \\leq \\sI^\\mathrm{Nt}_f(w) \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w) + \\mathrm{Err}_{f,3}(w).\n  \\end{align*}\n  Note that $\\mathrm{Err}_{f,2}(w) \\eqdef \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\sI_f(w)$ can be bounded as\n  \\begin{align*}\n    \\abs{\\mathrm{Err}_{f,2}(w)} &= \\abs{\\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\sI_f(w)}\\\\\n    &\\leq \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\cdot \\abs{\\sI_f(w)}\\\\\n    &\\leq \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\cdot \\abs{g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)}\\\\\n    &\\leq \\frac{\\onenorm{w}^2 C_\\ell^2 \\sigma_{\\mathrm{max}}^2}{2 (\\sigma_{\\mathrm{min}} + \\lambda) \\lambda^2},\n  \\end{align*}\n  and $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) = \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$ grows as $O(1\/\\lambda^3)$ from \\refprop{newton-actual},\n  so we can also write\n  \\begin{align*}\n      \\sI_f(w) + O\\Bigl(\\inv{\\lambda^3}\\Bigr) \\leq \\sI^{*}_f(w) \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda}\\right) \\sI_f(w) + O\\Bigl(\\inv{\\lambda^3}\\Bigr).\n  \\end{align*}\n\n\\end{proof}\n\n\n\\subsection{The influence on a test point}\n\\begin{repcorollary}{cor:xtest}[Restated]\n  Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$,\n  and define ${v_{\\mathrm{test}}} \\eqdef H_{\\lambda, \\bone}^{-\\inv{2}} {x_{\\mathrm{test}}}$ and $v_w \\eqdef H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)$.\n  Then under \\refassumps{l-const}{q-const}, $\\sI^\\mathrm{Nt}_f(w) = \\sI_f(w) + {v_{\\mathrm{test}}}^\\top D(w) v_w$,\n  where $D(w) = \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ is the \\emph{error matrix} from \\refprop{newton-inf}.\n\\end{repcorollary}\n\\begin{proof}\n  Since $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$ is linear, we have that for any $\\theta \\in \\Theta$,\n  \\begin{align*}\n    \\nabla_\\theta f(\\theta) &= {x_{\\mathrm{test}}},\\\\\n    \\nabla_\\theta^2 f(\\theta) &= 0,\\\\\n    C_{f, 3} &= 0.\n  \\end{align*}\n  This in turn implies that $\\mathrm{Err}_{f,3}(w) = 0$. Substituting these expressions into  \\refprop{newton-inf} gives us the desired result.\n\\end{proof}\n\n\n\\begin{repproposition}{prop:single}[Restated]\n  Consider a binary classification setting with $y \\in \\{-1, +1\\}$ and a margin-based model with loss\n  $\\ell(x, y; \\theta) = \\phi(y\\theta^\\top x)$ for some $\\phi:\\R\\to\\R_+$.\n  Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$\n  and that the subset $w$ comprises $\\|w\\|_1$ identical copies of the training point $(x_w, y_w)$.\n  Then under \\refassumps{l-const}{q-const},\n  the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is related to the influence $\\sI_f(w)$ according to\n  \\begin{equation*}\n    \\sI^\\mathrm{Nt}_f(w) = \\frac{\\sI_f(w)}\n        {1 - \\onenorm{w} \\cdot \\phi''(y_w \\hat\\theta(\\bone)^\\top x_w) \\cdot x_w^\\top H_{\\lambda, \\bone}^{-1} x_w}.\n  \\end{equation*}\n  This implies the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is bounded between $\\sI_f(w)$ and\n  $\\bigl(1 + \\frac{\\sigma_{\\mathrm{max}}}{\\lambda}\\bigr) \\sI_f(w)$.\n\\end{repproposition}\n\\begin{proof}\n  From \\refcor{xtest},\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) &= \\sI_f(w) + {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w).\n  \\end{align*}\n  With the additional assumptions on $w$ and $\\ell(x, y; \\theta)$, we have that\n  \\begin{align*}\n    \\nabla_\\theta \\ell(x, y; \\theta) &= y \\phi'(y\\theta^\\top x) x,\\\\\n    g_\\bone(w) &= \\sum_{i=1}^n w_i \\nabla_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= \\sum_{i=1}^n w_i y_i \\phi'(y_i\\hat\\theta(\\bone)^\\top x_i) x_i\\\\\n    &= \\sum_{i=1}^n w_i y_i \\phi_i' x_i\\\\\n    &= \\onenorm{w} y_w \\phi_k' x_w,\n  \\end{align*}\n  where in the last equality we use the assumption that we are removing $\\onenorm{w}$ copies of the point $(x_w, y_w)$. Similarly,\n  \\begin{align*}\n    \\nabla_\\theta^2 \\ell(x, y; \\theta) &= \\phi''(y\\theta^\\top x) xx^\\top,\\\\\n    H_\\bone(w) &= \\sum_{i=1}^n w_i \\nabla^2_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= \\sum_{i=1}^n w_i \\phi''(y_i\\hat\\theta(\\bone)^\\top x_i) x_i x_i^\\top\\\\\n    &= \\onenorm{w} \\phi''_k x_w x_w^T.\n  \\end{align*}\n  We thus have\n  \\begin{align*}\n    D(w) &= \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I\\\\\n    &= \\Bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}\n    \\onenorm{w} \\phi''_k x_w x_w^T\n    H_{\\lambda, \\bone}^{-\\inv{2}}\\Bigr)^{-1} - I\\\\\n    &= \\frac{\n      H_{\\lambda, \\bone}^{-\\inv{2}}\n      \\onenorm{w} \\phi''_k x_w x_w^T\n      H_{\\lambda, \\bone}^{-\\inv{2}}}\n    {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n    &= \\frac{\n    \\onenorm{w} \\phi''_k H_{\\lambda, \\bone}^{-\\inv{2}}\n       x_w x_w^T\n      H_{\\lambda, \\bone}^{-\\inv{2}}}\n    {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w},\n  \\end{align*}\n  where the third equality comes from the Sherman-Morrison formula.\n  Substituting $D(w)$ into \\refcor{xtest}, we obtain\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) &= \\sI_f(w) + {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &= \\sI_f(w) +\n      \\frac{\n        \\onenorm{w} \\phi''_k {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-1}\n        x_w x_w^T\n        H_{\\lambda, \\bone}^{-1} g_\\bone(w)\n        }\n        {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n    &= \\sI_f(w) +\n      \\frac{\n        {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-1} \\onenorm{w} y_w \\phi_k' x_w\n        \\cdot \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w\n        }\n        {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n      &= \\sI_f(w) +\n        \\frac{\n          {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)\n          \\cdot \\onenorm{w} z_k^T H_{\\lambda, \\bone}^{-1} z_k\n          }\n          {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n        &= \\sI_f(w) +\n          \\frac{\n            \\sI_f(w)\n            \\cdot \\onenorm{w} z_k^T H_{\\lambda, \\bone}^{-1} z_k\n            }\n            {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n          &= \\frac{\\sI_f(w)}\n              {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}.\n  \\end{align*}\n  To bound the denominator, we first use the trace trick to rearrange terms\n  \\begin{align*}\n    \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w &=\n    \\tr\\left(\\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w\\right)\\\\\n    &= \\tr\\left( H_{\\lambda, \\bone}^{-\\inv{2}} \\onenorm{w} \\phi''_k x_w x_w^T H_{\\lambda, \\bone}^{-\\inv{2}}\\right)\\\\\n    &= \\tr\\left(H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}} \\right).\n  \\end{align*}\n  Since $H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ has rank one under our assumptions, it only has at most one non-zero eigenvalue. We can therefore apply \\reflem{sv-bound-d} to conclude that\n  \\begin{align*}\n    \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w &= \\tr\\left(H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}} \\right)\\\\\n    &\\leq \\frac{\\sigma_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}} + \\lambda},\n  \\end{align*}\n  which in turn implies that\n    $1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w\n    \\geq \\frac{\\lambda}{\\sigma_{\\mathrm{max}} + \\lambda}$,\n  so\n  \\begin{align*}\n    \\inv{1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\n    \\leq \\frac{\\sigma_{\\mathrm{max}} + \\lambda}{\\lambda}=1 + \\frac{\\sigma_{\\mathrm{max}}}{\\lambda}.\n  \\end{align*}\n\n\\end{proof}\n\n\\subsubsection*{Reproducibility}\nThe code for replicating our experiments is available in the GitHub repository \\url{https:\/\/github.com\/kohpangwei\/group-influence-release}.\nAn executable version of this paper is also available on CodaLab at \\url{https:\/\/worksheets.codalab.org\/worksheets\/0xfed2ae0b9e5b44b7a1af8096365592a5}.\n\n\\chapter{#2}\\label{chp:#1}}\n\\newcommand\\Section[2]{\\section{#2}\\label{sec:#1}}\n\\newcommand\\Subsection[2]{\\subsection{#2}\\label{sec:#1}}\n\\newcommand\\Subsubsection[2]{\\subsubsection{#2}\\label{sec:#1}}\n\\ifthenelse{\\isundefined{\\definition}}{\\newtheorem{definition}{Definition}}{}\n\\ifthenelse{\\isundefined{\\assumption}}{\\newtheorem{assumption}{Assumption}}{}\n\\ifthenelse{\\isundefined{\\hypothesis}}{\\newtheorem{hypothesis}{Hypothesis}}{}\n\\ifthenelse{\\isundefined{\\proposition}}{\\newtheorem{proposition}{Proposition}}{}\n\\ifthenelse{\\isundefined{\\theorem}}{\\newtheorem{theorem}{Theorem}}{}\n\\ifthenelse{\\isundefined{\\lemma}}{\\newtheorem{lemma}{Lemma}}{}\n\\ifthenelse{\\isundefined{\\corollary}}{\\newtheorem{corollary}{Corollary}}{}\n\\ifthenelse{\\isundefined{\\alg}}{\\newtheorem{alg}{Algorithm}}{}\n\\ifthenelse{\\isundefined{\\example}}{\\newtheorem{example}{Example}}{}\n\\newcommand\\cv{\\ensuremath{\\to}}\n\\newcommand\\cvL{\\ensuremath{\\xrightarrow{\\mathcal{L}}}}\n\\newcommand\\cvd{\\ensuremath{\\xrightarrow{d}}}\n\\newcommand\\cvP{\\ensuremath{\\xrightarrow{P}}}\n\\newcommand\\cvas{\\ensuremath{\\xrightarrow{a.s.}}}\n\\newcommand\\eqdistrib{\\ensuremath{\\stackrel{d}{=}}}\n\\newcommand{\\E}{\\ensuremath{\\mathbb{E}}}\n\\newcommand\\KL[2]{\\ensuremath{\\text{KL}\\left( #1 \\| #2 \\right)}}\n\n\\section{Additional analysis on influence vs. actual effect on a test point}\\label{sec:supp-theory}\n\n\\subsection{Counterexamples}\\label{sec:theory-cex}\n\nFor \\reffig{cex}, we constructed two binary datasets in which the influence of a\ncertain class of subsets on the test prediction of a single test point exhibits\npathological behavior.\n\n\\paragraph{Rotation effect.}\nIn \\reffig{cex}-Left, our aim was to show that there can be a dataset with subsets such that the cone constraint discussed in \\refsec{theory-test} does not hold.\n\nThe rotation effect described in \\refcor{xtest} is due to the\nangular difference between the change in parameters predicted by the influence approximation,\n$\\Delta\\theta_{\\mathrm{inf}}(w) \\eqdef H_{\\lambda, \\bone}^{-1} g_\\bone(w) = H_{\\lambda, \\bone}^{-\\frac 1 2} v_w$,\nand the change in parameters predicted by the Newton approximation,\n$\\Delta\\theta_{\\mathrm{Nt}}(w) = H_{\\lambda, \\bone}^{-\\frac 1 2} \\bigl(D(w) + I\\bigr) v_w$.\nIf $\\Delta\\theta_{\\mathrm{inf}}(w)$ and $\\Delta\\theta_{\\mathrm{Nt}}(w)$ are linearly independent,\nthen for any pair of target values $a, b \\in \\R$,\nwe can find some ${x_{\\mathrm{test}}}$ such that $\\sI_f(w) = {x_{\\mathrm{test}}}^\\top \\Delta\\theta_{\\mathrm{inf}}(w) = a$ and $\\sI^\\mathrm{Nt}_f(w  = {x_{\\mathrm{test}}}^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) = b$.\n\nTo exploit this, we constructed the \\emph{MoG} dataset as an equal mixture of two standard (identity covariance) Gaussian distributions in $\\R^{60}$, one for each class, and with means $(-1\/2, 0, \\ldots, 0)$ and $(1\/2, 0, \\ldots, 0)$, respectively. In particular:\n\\begin{enumerate}\n  \\item We sampled $60$ examples from each class for a total of $n = 120$ training points, and set the regularization strength $\\lambda = 0.001$.\n  \\item We then computed $\\Delta\\theta_{\\mathrm{inf}}(w)$ and $\\Delta\\theta_{\\mathrm{Nt}}(w)$ for each pair of training points and chose the $120$ pairs of training points with the largest angles between $\\Delta\\theta_{\\mathrm{inf}}(w)$ and $\\Delta\\theta_{\\mathrm{Nt}}(w)$.\n  \\item Finally, we solved a\n  least-squares optimization problem to find ${x_{\\mathrm{test}}}$ for which $\\sI_f(w)$ and\n  $\\sI^\\mathrm{Nt}_f(w)$ are approximately decorrelated.\n\\end{enumerate}\n\nNote that we adversarially chose which subsets to study in this counterexample, since our main goal was to show that there existed subsets for which the cone constraint did not hold.\nFor the next counterexample, we instead study all possible subsets in the restricted setting of removing copies of single points.\n\n\\paragraph{Scaling effect.} In \\reffig{cex}-Right, our aim was to construct a dataset such that even if we only removed subsets comprising copies of single distinct points, a low influence need not translate into a low actual effect.\n\nTo do so, we constructed the \\emph{Ortho} dataset that contains 2 repeated points of opposite classes on each of the 2 canonical axes of $\\R^2$ (for a total of 4 distinct points). By varying their\nrelative distances from the origin, we can control the influence of removing one of these points as well as the rate that the scaling factor $d(w)$ from \\refprop{single} grows as we remove more copies of the same point.\nFurthermore, because the axes are orthogonal, we can control $d(w)$ independently for each repeated\npoint. We fix the test point ${x_{\\mathrm{test}}} = (1, 1)$.\nMaximizing $d(w)$ for one axis\nand minimizing it for the other produces the two distinct lines in\n\\reffig{cex}-Right.\n\n\n\\subsection{Scaling effects when removing multiple points}\\label{sec:theory-D}\nIn the general setting of removing subsets of different points, the analogous\nfailure case to a varying scaling factor $d(w)$ (\\reffig{cex}-Right) is the\nvarying scaling effect that the error matrix $D(w)$ in \\refprop{newton-inf} can have on\ndifferent subsets $w$. The range of this effect is bounded by the spectral norm\nof $D(w)$. This norm is precisely equal to $d(w)$ in the single-point setting,\nand it is large when we remove a subset $w$ whose Hessian\n$H_\\bone(w)$ is almost as large as the full Hessian $H_{\\lambda, \\bone}$ in some direction.\nAs with $d(w)$, the spectral norm of $D(w)$ decreases with $\\lambda$ (\\refprop{newton-inf}), so as regularization increases, we expect that the influence of a group will track its actual effect more accurately.\n\n\\subsection{The relationship between influence and actual effect on the loss of a test point}\\label{sec:theory-loss}\nIn the margin-based setting, the loss $\\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$ is a monotone function of the linear prediction $\\theta^\\top {x_{\\mathrm{test}}}$. Thus, measuring $f(\\theta) = \\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$ will display the same rank correlation as measuring $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$ above,\nso the same results about correlation in the test prediction setting carry over.\n\nHowever, the second-order $f$-curvature term $\\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w)$ from \\refprop{newton-inf} is always non-negative, even if the influence is negative. Under the assumption that $\\mathrm{Err}_{f,3}(w)$ and $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ are both small\nbecause they decay as $O(1\/\\lambda^3)$,\nthis implies that underestimation is only preserved when the influence is positive, as we observed empirically in \\reffig{acc}-Mid.\n\n\\section{Additional experiments}\\label{sec:supp-experiments}\n\nAs in \\reffig{acc}, in each of the plots below, the grey reference line has slope 1, and the red borders represent points that are not plotted because they are outside the x- or y-axis range.\n\n\\subsection{Representative test points}\\label{sec:median}\n\n\\reffig{median} is similar to \\reffig{acc} in the main text:\nit shows the influences vs. actual effects of groups on test points,\nbut with test points that are closer to the median (within the 40th to 60th percentile) of the test loss distribution.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig5.pdf}\n  \\caption{\n    Influences vs. actual effects of the same coherent groups in \\reffig{acc},\n    but on test points closer to the median (within the 40th to 60th\n    percentile) of the test loss distribution. We consider these to represent\n    average test points. On these, influence on the test prediction remains\n    well-correlated with the actual effect.\n  }\n  \\label{fig:median}\n\\end{figure}\n\n\n\\subsection{Regularization}\\label{sec:reg}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{figs\/plots\/paper-fig3.pdf}\n  \\caption{\n    The effect of regularization for a representative test point.\n    Red frame lines indicate the existence of points exceeding those bounds. We did not include the test prediction for MNIST (small) because the margin is not well-defined for a multiclass model.\n  }\n  \\label{fig:regularization}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.95\\textwidth]{figs\/plots\/paper-regs-spearman.pdf}\n  \\caption{\n    As regularization increases, correlation increases between the influences and actual effects on test prediction (Left), test loss (Middle), and self-loss (Right).\n  }\n  \\label{fig:regs-spearman}\n\\end{figure}\n\nIn \\refsec{theory}, our bounds show that influence ought to be closer to actual effect as regularization increases. Here, we support this claim empirically on Diabetes, Enron, Dogfish, and MNIST (small).\\footnotemark{} To do so, for each dataset, we selected a range of values for $\\lambda\/n$, and we selected subsets as described in \\refapp{subsetgen}. We then computed the influence and actual effect of each of these subsets on a representative test point's prediction, that point's loss, and on self-loss (\\reffig{regularization}).\n\n\\footnotetext{This experiment required us to retrain the model for every value of $\\lambda$ and for every subset. Thus, for computational purposes, we omitted CDR and MultiNLI, and we selected a random $10\\%$ subset of MNIST's training set to use in place of all of MNIST.}\n\nIn \\reffig{regs-spearman}, we observe the trend that correlation generally increases as $\\lambda$ does. Specifically, we computed the Spearman $\\rho$ between the influence and actual effect for each dataset, each value of $\\lambda$, and each evaluation function $f(\\cdot)$ of interest (i.e., test prediction, test loss, or self-loss).\n\n\n\\subsection{The effect of loss curvature on the accuracy of influence}\\label{sec:supp-curvature}\n\nOne takeaway from the results on test loss in \\reffig{acc}-Mid is that the curvature of $f(\\theta)$ can significantly increase\napproximation error; this is expected since the influence $\\sI_f(w)$\nlinearizes $f(\\cdot)$ around $\\hat\\theta(\\bone)$.\nWhen possible, choosing a $f(\\cdot)$ that has low curvature (e.g., the linear prediction) will result in higher accuracy.\nWe can mitigate this by using influence to approximate the parameters $\\hat\\theta(\\bone - w)$\nand then plug that estimate into $f(\\cdot)$ (\\reffig{pparam}), though this can be more\ncomputationally expensive.\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig6.pdf}\n  \\vspace{-4mm}\n  \\caption{\n    The top 3 rows show influence on test loss (with the same test points as in \\reffig{acc}),\n    while the bottom 3 show self-loss.\n    Within each set, the first row shows the influence vs. actual effect (as in \\reffig{acc});\n    the second shows the predicted effect obtained by estimating the change in parameters via influence and then\n    evaluating $f(\\cdot)$ directly on those parameters;\n    and the third shows the Newton approximation.\n    }\n    \\vspace{-4mm}\n  \\label{fig:pparam}\n\\end{figure}\n\n\nNote that \\reffig{pparam} shows that this technique does not help much for measuring self-loss.\nHowever, in the context of LOOCV,\nthe computational complexity of the Newton approximation for self-loss (described in \\refsec{theory})\nis similar to that of the influence approximation,\nso we encourage the use of the Newton approximation for LOOCV (as in \\citet{rad2018scalable});\n\\reffig{pparam} shows that this leads to more accurate approximations for self-loss.\n\n\\subsection{Additional analysis of influence functions applied to natural groups of data}\\label{sec:supp-applications}\nIn \\refsec{applications}, we considered the CDR and MultiNLI datasets, which contain the natural subsets of LFs and crowdworkers, respectively.\nTo draw inferences about these subsets, we took the $L_2$-regularized logistic regression model described in \\refapp{expt-details}, calculated the influence of the LF\/crowdworker subsets, and retrained the model once for each LF\/crowdworker.\n\n\\paragraph{CDR.}\nAs discussed in \\refapp{bl-details}, an LF is either positive or negative, where a positive LF can only give positive labels or abstain, and similarly for negative LFs. Because of this stark class separation, we indicate whether an LF is positive or negative, and we consider LF influence on the positive test examples separately from their influence on the negative test examples. To measure an LF's influence and actual effect on a set of test points, we simply add up its influence and actual effect on the set's individual test points.\n\nIn \\reffig{babble-acc}, we note that influence is a good approximation of an LF's actual effect, just as with other kinds of subsets as well as other datasets (\\reffig{acc}). Furthermore, we observe that positive LFs improve the overall performance of the positively labeled portion of the test set while hurting the negatively labeled portion of the test set, and vice versa for negative LFs. This dichotomous effect further motivates the analysis of influence on the positive test set separately from the negative test set, since the process of adding these two influences to study the influence on the entire test set would obscure the full story.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.65\\textwidth]{figs\/plots\/paper-suppfig-babble-acc-test.pdf}\n  \\caption{We observe both correlation and underestimation for LFs on the positive and negative test sets. We also see that positive LFs help the positive test set and hurt the negative test set; vice versa for negative LFs.\n  }\n  \\label{fig:babble-acc}\n\\end{figure}\n\nNext, we define an LF's \\textit{coverage} to be the proportion of the examples that it does not abstain on, which can be measured through the number of examples in its corresponding subset. In \\reffig{babble-size-corr}, we observe that the magnitude of influence correlates strongly with coverage.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.65\\textwidth]{figs\/plots\/paper-suppfig-babble-sizes.pdf}\n  \\caption{The magnitude of LF influence correlates with coverage. This figure is an extension of \\reffig{combined-apps}-Left: there, we showed the influence of positive LFs on the positive test set and the influence of negative LFs on the negative test set. Here, we additionally show the influence of positive LFs on the negative test set and vice versa.\n  }\n  \\label{fig:babble-size-corr}\n\\end{figure}\n\nFinally, we define an LF's \\textit{precision} to be the number of examples it labels correctly divided by the number of examples it does not abstain on. Because the dataset had many more negative than positive examples, positive LFs had lower precision than negative LFs. Surprisingly, even when this effect was taken into account and we considered positive LFs separately from negative ones, precision did not correlate with influence (\\reffig{babble-acc-no-corr}).\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.65\\textwidth]{figs\/plots\/paper-suppfig-babble-accs.pdf}\n  \\caption{LF influence does not correlate with precision. Similar to \\reffig{babble-size-corr}, this figure is an extension of \\reffig{combined-apps}-Mid: there, we showed the influence of positive LFs on the positive test set and the influence of negative LFs on the negative test set. Here, we additionally show the influence of positive LFs on the negative test set and vice versa.\n  }\n  \\label{fig:babble-acc-no-corr}\n\\end{figure}\n\n\n\\paragraph{MultiNLI.}\nAs discussed in \\refapp{multinli-details}, the training set consisted of five genres, and the test set consisted of a matched portion with the same five genres, as well as a mismatched portion with five new genres. For succinctness, we refer to the influence\/actual effect of the set of examples generated by a single crowdworker as that crowdworker's influence\/actual effect.\n\nFirst, we note in \\reffig{mnli-acc} that influence is a good approximation of a crowdworker's actual effect for both matched and mismatched test sets, consistent with our findings in \\reffig{acc} for other subset types and datasets.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-suppfig-mnli-acc-test.pdf}\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-suppfig-mnli-acc-nonfires.pdf}\n  \\caption{We observe strong correlation between crowdworkers' influence and actual effects.\n  }\n  \\label{fig:mnli-acc}\n\\end{figure}\n\nUnlike in CDR (\\reffig{babble-size-corr}), we do not find strong correlation between a crowdworker's influence and the number of examples they contributed; it is possible to contribute many examples but have relatively little influence (\\reffig{mnli-size-no-corr}).\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.42\\textwidth]{figs\/plots\/paper-suppfig-mnli-size.pdf}\n  \\caption{Size does not correlate strongly with influence. The hidden point is the crowdworker that contributed 35,000 examples. This is the figure presented in \\reffig{combined-apps}-Right.\n  }\n  \\label{fig:mnli-size-no-corr}\n\\end{figure}\n\nThe most prolific crowdworker contributed 35,000 examples and had large negative influence on the test set. A closer analysis revealed that they had positive influence on the fiction genre but lowered performance on many other genres, despite contributing roughly equally to each genre. This genre-specific trend tended to hold more broadly among the workers: there appear to be two categories of genres (fiction, facetoface, nineeleven vs. travel, government, verbatim, letters, oup) such that each worker tended to have positive influence on all genres in one category and negative influence on all genres in the other (\\reffig{mnli-genre-categories}). Moreover, the number of examples a worker contributed to a given genre was not a good indicator for their influence on that genre (\\reffig{mnli-genre-contrib}).\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.55\\textwidth]{figs\/plots\/paper-suppfig-mnli-genre-effect.pdf}\n  \\caption{Workers tended to have positive influence on fiction, facetoface, and nineeleven and negative influence on travel, government, verbatim, letters, and oup (or vice versa). In this plot, we allowed for full color saturation when the magnitude of the total influence on the test set (matched) exceeded 0.8.\n  }\n  \\label{fig:mnli-genre-categories}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-suppfig-mnli-genre-contrib.pdf}\n  \\caption{Influence on a genre does not correlate with number of contributions to that genre.\n  }\n  \\label{fig:mnli-genre-contrib}\n\\end{figure}\n\n\\section{Theoretical analysis}\\label{sec:theory}\n\nThe experimental results above show that there is consistent underestimation and high correlation between the predicted effects, based on influence functions, and the actual effects of groups across a variety of datasets, despite the influence approximation incurring large absolute and relative error.\nAs we discussed in \\refsec{related}, this is outside the regime of existing theory.\n\nAs an initial step towards understanding the high-error regime,\nwe establish conditions under which the actual effect $\\sI^{*}_f(w)$ lies approximately between $\\sI_f(w)$ and $C_\\mathrm{max} \\sI_f(w)$ for some $C_\\mathrm{max} > 0$.\nThis cone constraint---so called because it implies that all points on the graph of influence vs. actual effect lie within a cone---implies underestimation and,\nif $C_\\mathrm{max}$ is small, some degree of correlation.\nWe first show that this constraint holds in restricted settings---when measuring self-loss, or when removing multiple copies of the same point---and that $C_\\mathrm{max}$ varies inversely with the regularization term $\\lambda$, which is expected since stronger regularization reduces the change in the model.\nHowever, the cone constraint is stronger than necessary because it bounds the degree of underestimation,\nand we construct counterexamples to show that it need not hold in more general settings.\n\nOur analysis centers on the \\emph{one-step Newton approximation}, which estimates the change in parameters\n\\begin{align}\n  \\hat\\theta(\\bone - w) - \\hat\\theta(\\bone) &\\approx \\Delta\\theta_{\\mathrm{Nt}}(w) \\nonumber\n  \\eqdef \\bigl(H_{\\lambda, \\bone}({\\bone - w})\\bigr)^{-1} g_\\bone(w),\n\\end{align}\nwhere $H_{\\lambda, \\bone}({\\bone - w}) = (\\sum_{i=1}^n (1 - w_i) \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))) + \\lambda I$ is the\nregularized empirical Hessian at $\\hat\\theta(\\bone)$ but reweighted after removing the subset $w$.\nThis change in parameters gives the Newton approximation of the effect\n$\\sI^\\mathrm{Nt}_f(w) = f\\bigl(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)\\bigr) - f(\\hat\\theta(\\bone)))$ and the corresponding Newton error $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) = \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$, which measures its gap from the actual effect. Specifically, we decompose the error between the actual effect $\\sI^{*}_f(w)$ and influence $\\sI_f(w)$ as\n\\begin{align}\n  \\sI^{*}_f(w) - \\sI_f(w) \\ =\\  \\underbrace{\\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)}_{\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)} \\ +\\\n  \\underbrace{\\sI^\\mathrm{Nt}_f(w) - \\sI_f(w).}_{\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)}\\label{eqn:decomp}\n\\end{align}\nIn \\refsec{theory-newton}, we first show that the Newton-actual error $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$\ndecays at a rate of $O\\bigl(1 \/ (\\sigma_{\\mathrm{min}} + \\lambda)^3\\bigr)$, where $\\lambda$ is regularization strength and $\\sigma_{\\mathrm{min}}$ is the smallest eigenvalue of the empirical Hessian $H_\\bone$. Empirically, this error is small on our datasets, so we focus on characterizing the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$ in \\refsec{theory-decompose}.\nWe use this characterization to study the behavior of influence relative to the actual effect on self-loss (\\refsec{theory-selfloss}) and test prediction (\\refsec{theory-test}). For margin-based models, the test loss is a monotone function of the test prediction, so the analysis is similar (\\refapp{theory-loss}).\n\n\n\\subsection{Bounding the error of the one-step Newton approximation}\\label{sec:theory-newton}\n\nThe Newton approximation is computationally expensive\nbecause it computes $(H_{\\lambda, \\bone}({\\bone - w}))^{-1}$ for each $w$\n(instead of the fixed $H_{\\lambda, \\bone}^{-1}$ in the influence calculation).\nHowever, it provides more accurate estimates (e.g., \\citet{pregibon1981logistic}, \\citet{rad2018scalable}),\nand we show that its error can be bounded as follows (all proofs in \\refapp{proofs}):\n\\begin{proposition}\\label{prop:newton-actual}\n  Let the Newton error be $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) \\eqdef \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$.\n  Assume that the evaluation function $f(\\theta)$ is $C_f$-Lipschitz\n  and that the Hessian $\\nabla_\\theta^2 \\ell(x, y, \\theta)$ is $C_H$-Lipschitz.\n  Then\n  \\begin{align*}\n      \\abs{\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)} \\leq \\frac{n \\|w\\|_1^2 C_f C_H C_\\ell^2}{(\\sigma_{\\mathrm{min}} + \\lambda)^3},\n  \\end{align*}\n  where we define\n  $C_\\ell \\eqdef \\max_{1 \\leq i \\leq n} \\|\\nabla_\\theta \\ell(x_i,y_i,\\hat\\theta(\\bone))\\|_2$ to be the largest norm of a training point's gradient at $\\hat\\theta(\\bone)$,\n  and $\\sigma_{\\mathrm{min}}$ to be the smallest eigenvalue of $H_\\bone$.\n  $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ only involves third-order or higher derivatives of the loss, so it is 0 for quadratic losses.\n\\end{proposition}\n\\refprop{newton-actual} tells us that the Newton approximation is accurate\nwhen $\\lambda$ is large or the third derivative of $\\ell(x, y; \\cdot)$ (controlled by $C_H$) is small. Empirically,\nthe Newton error $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ is strikingly small in most of our settings (\\reffig{newton}), even though the overall error of the influence approximation $\\sI^{*}_f(w) - \\sI_f(w)$ is still large.\nIn the remainder of this section, we therefore focus on characterizing the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$, under the assumption that the Newton approximation is similar to the actual effect (within a factor of $O(1\/\\lambda^3)$).\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig2.pdf}\n  \\caption{\n    The Newton approximation accurately captures the actual effect for our datasets (though there is more error on the Diabetes dataset),\n    with the same test point as in \\reffig{acc}-Top.\n    We omit MNIST and MultiNLI for computational reasons. See \\reffig{pparam} for plots of test loss and self-loss.\n  }\n  \\label{fig:newton}\n\\end{figure}\n\n\\subsection{Characterizing the difference between the Newton approximation and influence}\\label{sec:theory-decompose}\nWe next characterize the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w) = \\sI^\\mathrm{Nt}_f(w) - \\sI_f(w)$:\n\\begin{proposition}\\label{prop:newton-inf}\n  Under the assumptions of \\refprop{newton-actual} and the additional assumption that the third derivative of $f(\\theta)$ exists and is bounded in norm by $C_{f, 3}$, the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$ is\n  \\begin{align*}\n    \\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w) &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n    \\ + \\ \\underbrace{\\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) + \\mathrm{Err}_{f,3}(w),}_\\text{\n      Error from curvature of $f(\\cdot)$\n    }\n  \\end{align*}\n  with $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$\n  and $H_\\bone(w) \\eqdef \\sum_{i=1}^n w_i \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))$.\n  The \\emph{error matrix} $D(w)$ has eigenvalues between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$,\n  where $\\sigma_{\\mathrm{max}}$ is the largest eigenvalue of $H_\\bone$.\n  The residual term $\\mathrm{Err}_{f,3}(w)$ captures the error due to third-order derivatives of $f(\\cdot)$ and is bounded by $\\abs{\\mathrm{Err}_{f,3}(w)} \\leq {\\|w\\|_1^3 C_{f, 3}C_\\ell^3}\/{6(\\sigma_{\\mathrm{min}} + \\lambda)^3}$.\n\\end{proposition}\nWe can interpret \\refprop{newton-inf} as a formalization of \\citet{hampel1986robust}'s observation that influence approximations are accurate when the model is robust and the curvature of the loss is low.\nIn general, the error decreases as $\\lambda$ increases and $f(\\cdot)$ becomes less curved;\nin \\reffig{regularization}, we show that increasing $\\lambda$ reduces error and increases correlation in our experiments.\n\n\\subsection{The relationship between influence and actual effect on self-loss}\\label{sec:theory-selfloss}\nLet us now apply \\refprop{newton-inf} to analyze the behavior of influence under different choices of evaluation function $f(\\cdot)$. We start with the self-loss $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$, as its influences and actual effects are always non-negative, and it is the cleanest to characterize:\n\\begin{proposition}\\label{prop:selfloss}\n  Under the assumptions of \\refprop{newton-inf}, the influence on the self-loss obeys\n  \\begin{align*}\n    \\sI_f(w) + \\mathrm{Err}_{f,3}(w) &\\leq \\sI^\\mathrm{Nt}_f(w)\n    \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w) + \\mathrm{Err}_{f,3}(w).\n  \\end{align*}\n\\end{proposition}\nThe constraint in \\refprop{selfloss} implies that up to $O(1\/\\lambda^3)$ terms, influence underestimates the Newton approximation and therefore the actual effect.\nThis explains the previously-unexplained downward bias observed when using influence to approximate LOOCV \\citep{debruyne2008model, giordano2019swiss}.\nEquivalently, all points on the graph of influences vs. actual effects\nlie within the cone bounded by the lines with slope $1$ and slope $\\frac{\\lambda}{\\lambda + 3\\sigma_{\\mathrm{max}}\/2}$ lines, up to $O(1\/\\lambda^3)$ terms.\nAs $\\lambda$ grows, these lines will converge, and the error terms $\\mathrm{Err}_{f,3}(w)$ and $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ will decay at a rate of $O(1\/\\lambda^3)$,\nforcing the influences and actual effects to be equal.\n\nHowever, $\\lambda \/ \\sigma_{\\mathrm{max}}$ is quite small in our experiments in \\refsec{accuracy},\nso the actual correlation of influence is better than predicted by this theory:\nin \\reffig{acc}-Bot, the sizes of the theoretically-permissible cones can be quite large, but the points in the graphs nevertheless trace a tight curve through the cone.\n\n\n\\subsection{The relationship between influence and actual effect on a test point}\\label{sec:theory-test}\n\nWe now turn to measuring the test prediction $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$. Here, we show that correlation and underestimation need not hold, and that we cannot obtain a cone constraint similar to \\refprop{selfloss} except in a restricted setting.\nDefine ${v_{\\mathrm{test}}} = H_{\\lambda, \\bone}^{-\\inv{2}} {x_{\\mathrm{test}}}$ and $v_w = H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)$.\n\\refprop{newton-inf} gives:\n\\begin{corollary}\\label{cor:xtest}\n  Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$.\n  Then $\\sI^\\mathrm{Nt}_f(w) = \\sI_f(w) + {v_{\\mathrm{test}}}^\\top D(w) v_w$,\n  where $D(w) = \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ is the \\emph{error matrix} from \\refprop{newton-inf}.\n\\end{corollary}\nUnfortunately, \\refcor{xtest} implies that no cone constraint applies: in general, we can find ${x_{\\mathrm{test}}}$ such that the influence $\\sI_f(w) = {v_{\\mathrm{test}}}^\\top v_w = 0$ but the Newton approximation $\\sI^\\mathrm{Nt}_f(w) = {v_{\\mathrm{test}}}^\\top D(w) v_w$ is large.\nAs a counterexample, \\reffig{cex}-Left shows that on synthetic data, $\\sI_f(w)$ and $\\sI^\\mathrm{Nt}_f(w)$ can even have opposite signs on some subsets $w$.\n\n\\begin{figure}[t]\n  \\centering\n  \\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={right,top},capbesidewidth=6.5cm}}]{figure}[\\FBwidth]\n  {\\caption{\n    Influence $\\sI_f(w)$ vs. Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ on the test prediction on two  counterexamples detailed in \\refapp{theory-cex}.\n    Left: We adversarially choose a set of $w$'s such that $\\sI_f(w)$ and $\\sI^\\mathrm{Nt}_f(w)$ can have different signs and need not correlate.\n    Right: When we only remove copies of single points, underestimation holds. However, we can control the scaling factor $d(w)$ between $\\sI_f(w)$ and $\\sI^\\mathrm{Nt}_f(w)$ on different groups, so correlation need not hold.\n  }\\label{fig:cex}}\n  {\\includegraphics[width=0.5\\textwidth]{figs\/plots\/paper-fig4.pdf}}\n\\end{figure}\n\n\nWe can recover a cone constraint similar to \\refprop{selfloss} if we restrict our attention to\nthe special case where we use a margin-based model and remove (possibly multiple copies) of a single point:\n\\begin{proposition}\\label{prop:single}\n  Consider a binary classification setting with $y \\in \\{-1, +1\\}$ and a margin-based model with loss\n    $\\ell(x, y; \\theta) = \\phi(y\\theta^\\top x)$ for some $\\phi:\\R\\to\\R_+$.\n    Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$\n    and that the subset $w$ comprises $\\|w\\|_1$ identical copies of the training point $(x_w, y_w)$.\n    Then under the assumptions of \\refprop{newton-actual},\n    the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is related to the influence $\\sI_f(w)$ according to\n  \\begin{equation*}\n    \\sI^\\mathrm{Nt}_f(w) = \\frac{\\sI_f(w)}\n        {1 - \\onenorm{w} \\cdot \\phi''(y_w \\hat\\theta(\\bone)^\\top x_w) \\cdot x_w^\\top H_{\\lambda, \\bone}^{-1} x_w}.\n  \\end{equation*}\n  This implies the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is bounded between $\\sI_f(w)$ and\n  $\\bigl(1 + \\frac{\\sigma_{\\mathrm{max}}}{\\lambda}\\bigr) \\sI_f(w)$.\n\\end{proposition}\nSimilar to \\refprop{selfloss}, \\refprop{single} shows that up to $O(1\/\\lambda^3)$ terms, the influence underestimates the actual effect when removing copies of a single point.\nMoreover, all points on the graph of influences vs. actual effects lie within the cone bounded by the lines with slope $1$ and slope $\\lambda \/ (\\lambda + \\sigma_{\\mathrm{max}})$,\nup to $O(1\/\\lambda^3)$ terms.\nAs $\\lambda \/ \\sigma_{\\mathrm{max}}$ grows, the cone shrinks, and correlation increases.\n\nHowever, if $\\lambda \/ \\sigma_{\\mathrm{max}}$ is small (as in our experiments in \\refsec{accuracy}),\nthe cone is wide,\nand the scaling factor $d(w) = 1 \/ (1 - \\onenorm{w} \\cdot \\phi_k'' x_k^\\top H_{\\lambda, \\bone}^{-1} x_k)$ in \\refprop{single} can be quite large for some subsets $w$ but not for others.\nIn particular, $d(w)$ is large when there are few remaining points in the direction of the removed points.\nIn \\reffig{cex}-Right, we exploit this fact to show that the influence $\\sI_f(w)$ and Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ can exhibit low correlation (e.g., low $\\sI_f(w)$ need not mean low $\\sI^\\mathrm{Nt}_f(w)$),\neven in the simplified setting of removing copies of single points.\nWe comment on the analogue of $d(w)$ in the general multiple-point setting in \\refapp{theory-D}, and on the influence on test loss (instead of test prediction) in \\refapp{theory-loss}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nIn genome-wide association (GWA) studies the aim is to test for association between genetic markers and a phenotype. A large number of markers are tested, and it is important to control the overall Type I error rate. Our focus is on controlling the familywise error rate (FWER). Multiple testing correction methods may achieve this goal by estimating a local significance level for the individual tests. In this work we present a new method, {\\em the order $k$ FWER-approximation method}, for finding a local significance level in multiple hypothesis testing for correlated common variants, as is often observed in GWA studies.\n\nAssume that we have collected independent individual observations in a case--control, cohort or cross-sectional study. The phenotype of interest can be continuous or discrete. We consider biallelic genetic markers, giving three possible genotypes. For each genetic marker we specify a hypothesis situation, where the null hypothesis is of the type ``no association between the phenotype and genetic marker'' and we have a two sided alternative. We will model the data using a generalized linear regression model (GLM) with phenotype as response (outcome), genotype as the independent variable of interest (exposure), and possibly non-genetical, referred to as environmental, independent covariates (not of interest) in the model. In epidemiological studies, a confounder is a common factor which is associated with both the exposure and outcome. In GWA studies, population substructure may be associated with both the exposure (genotype) and outcome (phenotype) and therefore may be a confounding factor and need to be adjusted for in the analysis. Population stratification can be adjusted for by including principal components of the genotype covariance matrix of the individuals as covariates in the model \\citep{Price2006}. As test statistics for the multiple hypothesis problem we use the score test statistics to evaluate the genotype contribution to the model for each genetic marker separately. It is known that the vector of separate score test statistics asymptotically follows a multivariate normal distribution with a covariance matrix that can be estimated using key features of the fitted GLM model and the genetic markers \\citep{Schaid2002, SeamanMyhsok2005}. This has also been a key ingredient in the work of \\cite{ConneelyBoehnke2007}.\n\nFurther, we show that for the special case when no environmental covariates are present or when environmental and genetic covariates are observed to be independent, the estimated correlation matrix between score test statistics can be approximated by the estimated correlation matrix between the genetic markers.\n\nIn a multiple testing situation with $m$ tests the familywise error rate can be controlled at level $\\alpha$ by specifying a local $p$-value cut-off, $\\alpha_{\\text{loc}}$, to be used for all the $m$ hypothesis tests. Inspired from the work of \\cite{MoskvinaSchmidt2008} and \\cite{DickhausStange2013} we will use an approximation to the $m$-dimensional asymptotic simultaneous multivariate normal distribution of the score test statistics vector to estimate $\\alpha_{\\text{loc}}$. The $\\alpha_{\\text{loc}}$ estimate can be used to define an effective number of independent tests, and our FWER-approximation can be used to compute FWER-adjusted $p$-values.\n\nThe order $k$ FWER-approximation method is more powerful than the \\v Sid\\'ak  method (which assumes that the score test statistics are independent across markers) and the Bonferroni method (which is valid for all dependence structures between the score test statistics). Further, it is more efficient and more widely applicable than the method of \\cite{ConneelyBoehnke2007}. In Section \\ref{sec:discuss} we will see that the method of \\cite{ConneelyBoehnke2007}  is built on numerical integration in $m$ dimensions and is computationally intensive. \n\nThe Westfall--Young permutation procedure is known to have asymptotically optimal power for a broad class of problems, including block-dependent and sparse dependence structure \\citep{MeinshausenMaathuisBuhlman2011}. However, this method is computer intensive and to have a valid permutation test, the assumption of exchangeability needs to be satisfied \\citep{Commenges2003}. This assumption is in general not satisfied when environmental covariates are present in the model.\n\nWe will use two genetic data sets presented by \\cite{TOP1}, \\cite{TOP2}, \\cite{VO2max1} and \\cite{VO2max2} to illustrate our method applied to real data. \n\nThe paper is organized as follows. In Section \\ref{sec:background} we present statistical background on the score test, and derive expressions for the score test covariance matrix, which is of importance for the subsequent work. Our proposed method is outlined and presented in detail in Section \\ref{sec:multtest}, together with characteristics of our method. In Section \\ref{sec:invest} real data and an artificial correlation structure are used to evaluate our proposed model and compare to other methods. Finally, we discuss and conclude in Sections~\\ref{sec:discuss} and~6.\n\n\\section{Statistical background}\\label{sec:background}\n\nIn this section, we present notation and details on the score test in generalized linear models. \n\n\\subsection{Notation and data} \\label{sec:notation}\n\nWe assume that data -- phenotype, $m$ genetic covariates and $d$ environmental covariates -- from $n$ independent individuals are available in a case--control, cohort or cross-sectional study. Let $\\bm Y$ be an $n$-dimensional vector having the phenotype $Y_i$ of individual $i$ as its $i$th entry, $i=1$, \\ldots, $n$. Let $\\xe$ be an $n\\times d$ matrix having environmental covariates (the first one being 1 to allow for an intercept in the model presented below) for individual $i$ as its $i$th row, and let $\\xg$ be an $n\\times m$ matrix having genetic covariates, or genotypes, for individual~$i$ as its $i$th row, each column corresponding to a genetic marker.\n\nWe assume that the genetic data are from common variant biallelic genetic markers with alleles $a$ and $A$, where $A$ is the minor allele. We will use the additive coding 0, 1, 2 for the genotypes $aa$, $aA$, and $AA$, respectively, in the genetic covariate matrix $\\xg$, but other coding schemes are also possible. We denote the total design matrix $\\x=(\\xe\\ \\xg)$, which has the total covariate vector for individual $i$ as its $i$th row.\n\n\\subsection{Testing statistical hypotheses with the score test}\\label{sec:score}\n\nWe assume that the relationship between the phenotype $\\y$ and covariates $\\x$ can be modelled by a generalized linear model (GLM) \\citep{glm} with an $n$-dimensional vector $\\bm\\eta = \\xe\\be + \\xg\\bg = \\x\\bm\\beta$ of linear predictors, where $\\bm\\beta=(\\be^T\\ \\bg^T)^T$ is a $d+m$-dimensional parameter vector. Let $\\eta_i$ be the $i$th entry of $\\bm\\eta$, and let $\\bm\\mu$ be the $n$-dimensional vector having $\\mu_i=EY_i$ as its $i$th entry. We assume that the link function $g$ defined by $\\eta_i=g(\\mu_i)$ of the GLM is canonical, which implies that the log likelihood for individual~$i$ is $l_i=(Y_i\\eta_i-b(\\eta_i))\/\\phi_i+c(Y_i,\\phi_i)$, where $b$ and $c$ are functions defining the exponential family of the phenotypes and $\\phi_i$ the dispersion parameter. In our context $\\phi_i=\\phi$ will be equal for all observations. In general, $\\mu_i=b'(\\eta_i)$ and $\\var\\y[i]=\\sigma^2_i=\\phi b''(\\eta_i)$. For $Y_i$ normally distributed, this reduces to $\\sigma_i^2=\\sigma^2=\\phi$, and for $Y_i$ Bernoulli distributed,  $\\sigma_i^2=\\mu_i(1-\\mu_i)$ with $\\phi=1$.\n\nThe full $d+m$-dimensional score vector $\\sum_{i=1}^n\\nabla_{\\!\\bm\\beta}l_i$ can then be calculated to be\n\\[\n \\bm U=\\frac1\\phi\\x^T(\\y-\\bm\\mu),\n\\]\nwhich is asymptotically normal with mean $\\bm0$ and covariance matrix \n\\[\n V=\\frac1{\\phi^2}\\x^T\\Lambda\\x,\n\\]\nwhere $\\Lambda$ is the diagonal matrix having $\\sigma_i^2$ as its $i$th entry.\n\nPartition $\\bm U$ into its environmental and genetic components, $\\bm U^T=(\\bm U_\\text e^T\\ \\bm U_\\text g^T)$. Since $\\be$ are nuisance parameters and unknown, they are estimated by their maximum likelihood estimates under the null hypothesis of $\\bg=\\bm0$. In effect, $\\bm\\mu$ is to be replaced by $\\hat{\\bm\\mu}_\\text e$, the fitted values in a model with only environmental covariates $\\xe$ present, giving the statistic\n\\begin{equation}\n \\bm U_{\\text g\\mid\\text e}=\\frac1\\phi\\xg^T(\\y-\\hat{\\bm\\mu}_\\text e).\\label{u}\n\\end{equation}\nThen $\\bm U_{\\text g\\mid\\text e}$ has the conditional distribution of $\\bm U_\\text g$ given $\\bm U_\\text e=\\bm 0$, which is asymptotically normal with mean $\\bm 0$ and covariance matrix\n\\begin{equation}\n V_{\\text g\\mid\\text e}^{\\vphantom1}=V_\\text{gg}^{\\vphantom1}-V_\\text{ge}^{\\vphantom1}V_\\text{ee}^{-1}V_\\text{eg}^{\\vphantom1}\n =\\frac{1}{\\phi^2}\\xg^T\\big(\\Lambda-\\Lambda    \\xe(\\xe^T\\Lambda\\xe)^{-1}\\xe^T \\Lambda\\big)\\xg,\\label{var}\n\\end{equation}\nwhere $V_\\text{ee}$, $V_\\text{eg}$, $V_\\text{ge}$ and $V_\\text{gg}$ are the upper left $d\\times d$, upper right $d\\times m$, lower left $m\\times d$ and lower right $m\\times m$ submatrices of $V$, respectively \\citep[see][]{Smyth2003}.\n\nThe score test statistic $\\bm U_{\\text g\\mid\\text e}^TV_{\\text g\\mid\\text e}^{-1}\\bm U_{\\text g\\mid\\text e}$ with $\\bg=\\bm 0$ is asymptotically $\\chi^2$ distributed with $m$ degrees of freedom when the complete null hypothesis $\\bg=\\bm0$ is true \\citep[see][]{Smyth2003}. However, our interest lies not in the complete null hypothesis, but in the $m$ individual hypotheses $\\bg[j]=0$ for each component $\\bg[j]$ of $\\bg$, \\ $1\\leq j\\leq m$, against two-sided alternatives. We consider the standardized components of $\\bm U_{\\text g\\mid\\text e}$,\n\\begin{equation}\n T_j=\\frac{\\bm U_{\\text g\\mid\\text e\\,j}}{\\sqrt{V_{\\text g\\mid\\text e\\,jj}}}, \\label{eq:Tk}\n\\end{equation}\nwhere $\\bm U_{\\text g\\mid\\text e\\,j}$ denotes the $j$th entry of  $\\bm U_{\\text g\\mid\\text e}$ and $V_{\\text g\\mid\\text e\\,jk}$ the $jk$ entry of $V_{\\text g\\mid\\text e}$. Under the null hypothesis $H_j\\colon \\bg[j]=0$, \\ $T_j$ is asymptotically standard normally distributed, and $H_j$ will be rejected for large values of $\\lvert T_j\\rvert$. Under the complete null hypothesis, $\\bg=\\bm0$, the vector $\\bm T=(T_1,T_2,\\ldots,T_m)$ is asymptotically multivariate standard normally distributed with covariance matrix $R$, having\n\\begin{equation}\n \\cov(T_j,T_k)=\\frac{V_{\\text g\\mid\\text e\\,jk}}{\\sqrt{V_{\\text g\\mid\\text e\\,jj}V_{\\text g\\mid\\text e\\,kk}}}, \\label{eq:R}\n\\end{equation}\nas its $jk$ entry, all evaluated at $\\bg=\\bm0$. Note that the dispersion parameter $\\phi$ is cancelled from $\\bm T$ and the covariances. However, the $\\sigma_i^2$ of $\\Lambda$ will have to be estimated.\n\n\\subsection{Special cases} \\label{sec:corr}\n\nWe will now look at $\\bm U_{\\text g\\mid\\text e}$ and  $V_{\\text g\\mid\\text e}$ for some special cases.\n\n\\subsubsection{No environmental covariates}\\label{noenvcov}\n\nIf no evironmental covariates except the intercept are present in the GLM, then $\\xe=\\bm1$, the $n$-dimensional vector having all entries equal to 1, and $\\Lambda=\\sigma^2I$ under the null hypothesis, where $I$ is the $n\\times n$ identity matrix. Then\n\\[\n U_{\\text g\\mid\\text e}=\\frac1\\phi\\xg^T\\Big(I-\\frac{1}{n}\\bm1\\bm1^T\\Big)\\bm Y\\qquad\\text{and}\\qquad V_{\\text g\\mid\\text e}=\\frac{\\sigma^2}{\\phi^2}\\xg^T\\Big(I-\\frac{1}{n}\\bm1\\bm1^T\\Big)\\xg,\n\\]\nso that\n\\begin{equation}\nT_j=\\frac{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm Y}{\\sigma\\sqrt{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_j}},\\quad\\cov(T_j,T_k)=\\frac{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_k}{\\sqrt{\\bm x_j^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_j}\\sqrt{\\bm x_k^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm x_k}},\\label{Tk-noenv}\n\\end{equation}\nwhere $\\bm x_j$ is the $j$th column of $\\xg$, \\ $1\\leq j\\leq m$, \\ $1\\leq k\\leq m$. So $T_j$, the score test statistic for testing $\\bg[j]=0$, is $\\sqrt n$ times the Pearson correlation between $\\bm x_j$ and $\\bm Y$ when $\\sigma^2=\\var Y_i$ is replaced by the estimate $\\bm Y^T(I-\\frac{1}{n}\\bm1\\bm1^T)\\bm Y\/n$, and $\\cov(T_j,T_k)$ is the sample correlation between $\\bm x_j$ and $\\bm x_k$. Thus, for a GLM without adjustment for environmental covariates, the correlation between the score test statistics can be estimated by estimating the genotype correlation. The genotype correlation estimates twice the composite linkage disequilibrium if the genotypes are coded 0, 1,~2 \\citep{Weir2008}.\n\n\\subsubsection{Uncorrelated environmental and genetic covariates}\\label{uncorrenvgen}\n\nTwo $n$-dimensional vectors $\\bm X_1$ and $\\bm X_2$ of observations have zero Pearson correlation if their centered observations are orthogonal,\n\\[\n 0=(\\bm X_1-\\bar X_1\\bm1)^T(\\bm X_2-\\bar X_2\\bm1)=\\bm X_1^T\\Big(I-\\frac1n\\bm1\\bm1^T\\Big)\\bm X_2.\n\\]\nIf $X_1$ and $X_2$ are two matrices, then near zero Pearson correlation of each combination of a column of $X_1$ and a column of $X_2$ can be written compactly as\n\\begin{equation}\n X_1^T\\Big(I-\\frac1n\\bm1\\bm1^T\\Big)X_2\\approx\\bm0,\\qquad\\text{or}\\qquad\n   X_1^TX_2\\approx\\frac1nX_1^T\\bm1\\bm1^TX_2.\\label{centeredorth}\n\\end{equation}\n\nIf we consider genetic and environmental covariates to be random variables, and all pairs of an environmental and a genetic covariate to be independent, we would expect~\\eqref{centeredorth} to  hold for all $X_1$ having columns that are functions of genetic covariates and $X_2$ having columns that are functions of environmental covariates. In particular, we consider $X_1=\\xg$ and $X_2=\\Lambda\\xe$. Since $\\Lambda$ is a function of environmental covariates only under the null hypothesis, so is $X_2$. By~\\eqref{centeredorth}, $\\xg^T\\Lambda\\xe\\approx\\frac1n\\xg^T\\bm1\\bm1^T\\Lambda\\xe$, Then, from~\\eqref{var},\n\\begin{align*}\n \\phi^2V_{\\text g\\mid\\text e}\n  &\\approx\\xg^T\\Lambda\\xg\n   -\\frac1{n^2}\\xg^T\\bm1\\bm1^T\\Lambda\\xe(\\xe^T\\Lambda\\xe)^{-1}\\xe^T \\Lambda\\bm1\\bm1^T\\xg\\\\\n  &=\\xg^T\\Lambda\\xg\n   -\\frac1{n^2}\\xg^T\\bm1\\bm1^T\\Lambda^{1\/2}H\\Lambda^{1\/2}\\bm1\\bm1^T\\xg,\n\\end{align*}\nwhere $H=\\Lambda^{1\/2}\\xe(\\xe^T\\Lambda\\xe)^{-1}\\xe^T\\Lambda^{1\/2}$ will project onto the column space of $\\Lambda^{1\/2}\\xe$. Since $\\bm1$ is a column (the intercept) of $\\xe$, \\ $\\Lambda^{1\/2}\\bm1$ is in the column space of $\\Lambda^{1\/2}\\xe$, so that $H\\Lambda^{1\/2}\\bm1=\\Lambda^{1\/2}\\bm1$, and\n\\[\n \\phi^2V_{\\text g\\mid\\text e}\n  \\approx\\xg^T\\Lambda\\xg\n   -\\frac1{n^2}(\\tr\\Lambda)\\xg^T\\bm1\\bm1^T\\xg.\n\\]\n\nWe now turn to the term $\\xg^T\\Lambda\\xg$. Its $(j,k)$ entry is $\\bm X_1^T\\Lambda\\bm1$, where $\\bm X_1$ is the vector consisting of the entry-wise products of the $j$th and the $k$th column of $\\xg$. Letting $\\bm X_2=\\Lambda\\bm1$, by~\\eqref{centeredorth}, independence of environmental and genetic covariates yields $\\bm X_1^T\\Lambda\\bm1\\approx\\frac1n\\bm X_1^T\\bm1\\bm1^T\\Lambda\\bm1=\\frac1n(\\tr\\Lambda)\\bm X_1^T\\bm1$, which is the $(j,k)$ entry of $\\frac1n(\\tr\\Lambda)\\xg^T\\xg$. Thus $\\xg^T\\Lambda\\xg\\approx\\frac1n(\\tr\\Lambda)\\xg^T\\xg$, and we have\n\\[\n V_{\\text g\\mid\\text e} \\approx\\frac{\\tr\\Lambda}{n\\phi^2}\\xg^T\\Big(I-\\frac1n\\bm1\\bm1^T\\Big)\\xg,\n\\]\nwhich is the same expression as in the case of no environmental covariates with the exception that the common variance $\\sigma^2$ of the responses is replaced by their average variance $\\tr\\Lambda\/n=\\frac1n\\sum_{i=1}^n\\sigma_i^2$, where the $\\sigma_i^2$ are defined by the environmental covariates. The conclusion is that, if environmental and genetic covariates are uncorrelated, correlations of the score vector under the null hypothesis can be estimated more easily by estimating only correlations between genetic covariates instead \n\n\\subsubsection{The normal model}\n\nFor $Y_i$ normally distributed, $\\Lambda=\\sigma^2I$, where $I$ is the $n\\times n$ identity matrix. The score vector can then be written\n\\[\n  \\bm U_{\\text g \\mid\\text e}=\\frac1{\\sigma^2} \\xg^T (I-H)\\y,\n\\]\nand \\eqref{var} reduces to\n\\[\n  V_{\\text g\\mid\\text e}=\\frac{1}{\\sigma^2}\\xg^T(I-H)\\xg,\n\\]\nwhere $H=\\xe(\\xe^T\\xe)^{-1}\\xe^T$ is the idempotent matrix projecting onto the column space of $\\xe$. Then $I-H$ is the idempotent matrix projecting onto the orthogonal complement of the column space of $\\xe$, and $(I-H)\\bm Y$ are the residuals when fitting the multiple linear model with only the environmental covariates present. Note that $\\sigma^2$ enters into the test statistics $T_j$~\\eqref{eq:Tk}, and needs to be replaced by an estimate; we have used the residual sum of squares of a fitted model with only environmental covariates present (the null hypothesis), divided by $n-d$.\n\n\\subsubsection{The logistic model}\n\nFor $Y_i$ Bernoulli distributed, $\\phi=1$ and the $\\sigma_i^2$ of $\\Lambda$ are estimated by $\\hat \\mu_{\\text ei}(1-\\hat \\mu_{\\text ei})$, where $\\hat \\mu_{\\text ei}$ are the fitted values under the null hypothesis with only environmental covariates. Inference about $\\bg$ is valid also if data are collected in a case--control study since the canonical (logit) link is used \\citep[pp. 170--171]{agresti2002categorical}.\n\nIn the special case of no environmental covariates, that is, $\\xe=\\bm1$, each score test statistic, $T_j$ \\eqref{Tk-noenv}, is equal to the Cochran--Armitage trend test \\citep{Armitage1955,Cochran1954} statistic,\n\\[\n \\frac{\\sum_{i=0}^2s_i(n_2x_i-n_1y_i)}\n  {\\sqrt{n_1n_2\\big(\\sum_{i=0}^2s_i^2m_i-\\frac1n(\\sum_{i=0}^2s_im_i)^2\\big)}},\n\\]\nwhere $s_i$ are the possible values of the genetic covariates, $n_1$ and $n_2$ the number of 0 and 1 phenotypes $Y_i$, respectively, $x_i$ the number of observations having phenotype 1 and genotype $s_i$ at marker $k$, \\ $y_i$ the number of observations having phenotype 0 and genotype $s_i$, and $m_i=x_i+y_i$. The Cochran--Armitage test is used in disease--genotype association testing with scores $(s_0,s_1,s_2)=(0,s,1)$ \\citep{sasieni1997genotypes,slager2001case}, for example with $s=\\frac12$ for an additive genetic model.\n\n\\section{Familywise error rate control and approximations}\\label{sec:multtest}\n\nWe now turn to the topic of how to control the familywise error rate (FWER) by intersection approximations, and then apply this to our situation.\n\n\\subsection{Multiple hypothesis familywise error rate control}\\label{sec:fwer}\n\nWe have a collection of $m$ null hypotheses, $H_k\\colon \\bg[k]=0$ (no association between phenotype and genotype at marker $k$), $1\\leq k\\leq m$, against two-sided alternatives. We will present a method for multiple testing correction that controls the FWER -- the probability of making at least one type~I error. We adopt the notation of \\cite{MoskvinaSchmidt2008}, and denote by $O_k$ the event that the null hypothesis $H_k$ is not rejected, and by $\\bar O_k$ its complement, $1\\leq k\\leq m$. Then, if all $m$ null hypotheses are true,\n\\begin{equation}\n \\text{FWER} = P(\\bar O_1 \\cup \\cdots \\cup \\bar O_m) = 1-P(O_1 \\cap \\cdots \\cap O_m). \\label{eq:FWER}\n\\end{equation}\nIn our case, $O_k$ is an event of the form $|T_k| < c$, where $T_k$ is the test statistic of \\eqref{eq:Tk}. We will consider single-step multiple testing methods, and choose the same cut-off $c$ for each $k$. We denote by $\\alpha_{\\text{loc}}=2\\Phi(-c)=P(\\bar O_k)$, the asymptotic probability of false rejection of $H_k$, where $\\Phi$ is the univariate standard normal cumulative distribution function. When the joint distribution of the test statistics is known under the complete null hypothesis, or can be estimated, FWER control at the $\\alpha$ significance level can be achieved by solving the inequality $\\text{FWER} \\leq \\alpha$ for $\\alpha_{\\text{loc}}$, based on either the union or intersection formulation of~\\eqref{eq:FWER}. When $m$ is large, this involves evaluating high dimensional integrals over the acceptance or rejection regions, which is suggested by \\cite{ConneelyBoehnke2007}.\n\nTo avoid evalulating these costly integrals, we may instead control FWER by considering bounds based on~\\eqref{eq:FWER}. For example, the Bonferroni method is based on the Boole inequality applied to the union formulation of~\\eqref{eq:FWER},\n\\[\n \\text{FWER} = P(\\bar O_1 \\cup \\cdots \\cup \\bar O_m) \\leq \\sum_{k=1}^m P(\\bar O_k)=\\sum_{k=1}^m \\alpha_{\\text{loc}}=m\\alpha_{\\text{loc}},\n\\]\nfrom which it is seen that a local significance level of $\\alpha_{\\text{loc}}=\\alpha\/m$ guarantees $\\text{FWER}\\leq\\alpha$.\n\nWhen the FWER is calculated under the complete null hypothesis, so-called weak FWER control is achieved. However, in our situation, subset pivotality is satisfied, meaning that the distribution of any subvector $(T_k)_{k\\in K}$ is identical under $\\bigcap_{k\\in K}H_k$ and under the complete null hypothesis $\\bigcap_{k=1}^mH_k$, for all subsets $K\\subseteq\\{1,2,\\ldots, m\\}$. In particular, a subvector of $\\bm U_{\\text g\\mid\\text{e}}$~\\eqref{u} and a submatrix of $V_{\\text g\\mid\\text{e}}$~\\eqref{var} corresponding to $K$ only involves genetic covariates corresponding to~$K$. Then strong FWER control is achieved, meaning that $\\text{FWER}\\leq\\alpha$ regardless of which null hypotheses are true \\citep{WestfallYoung1993,westfall2008multiple}. \n\nThe focus in this work will be on the intersection formulation of~\\eqref{eq:FWER}. Background theory will be given next and new application in \\ref{sec:FWERkapprox}.\n\n\\subsection{Intersection approximations}\\label{sec:fwerk}\n\nFollowing \\cite{glazjohnson}, we define $k$th order product-type approximations to $P(O_1\\cap\\nobreak\\cdots\\cap O_m)$ by\n\\begin{equation}\n \\gamma_k=P(O_1\\cap\\cdots\\cap O_k)\\prod_{j=k+1}^mP(O_j\\mid O_{j-k+1}\\cap\\cdots\\cap O_{j-1})\n  =\\frac{\\prod_{j=k}^{m}P(O_{j-k+1}\\cap\\cdots\\cap O_j)}{\\prod_{j=k+1}^{m}P(O_{j-k+1}\\cap\\cdots\\cap O_{j-1})},\n\\label{gammak}\n\\end{equation}\n$1\\leq k\\leq m$, where probabilities are evaluated under the complete null hypothesis. This is similar to the usual multiplicative rule for the probability of intersection of events applied to $\\gamma_m=P(O_1\\cap\\nobreak\\cdots\\cap O_m)$, but with dimension of distributions limited to $k$. The idea is that the $\\gamma_k$ should constitute increasingly better approximations of $\\gamma_m$ as $k$ increases, and that calculation of $\\gamma_k$ is less costly than calculation of $\\gamma_m$ when $k<m$, since only $k$-variate distributions are involved in $\\gamma_k$.\n\n\\sloppy\nNote that the approximations depend on the order of the components of $\\bm T=(T_1,\\ldots T_m)$. We have used the order in which the $m$ markers are positioned along the genome, assuming that the largest correlations occur between close markers.\n\n\\fussy\nIn our case, $\\gamma_1=\\prod_{j=1}^m P(\\lvert T_j\\rvert<c)=(1-\\alpha_{\\text{loc}})^m$ and $\\gamma_m=P(\\lvert T_1\\rvert<c,\\ldots,\\lvert T_m\\rvert<\\nobreak c)=1-\\text{FWER}$. Since $\\bm T$ is asymptotically multivariate normally distributed with mean~$\\bm0$ under the complete null hypothesis, $\\gamma_1\\leq\\gamma_m$ asymptotically \\citep{Sidak1967}. Choosing $\\alpha_{\\text{loc}}$ such that $\\text{FWER}=1-\\gamma_m\\leq1-\\gamma_1=1-(1-\\alpha_{\\text{loc}})^m=\\alpha$ keeps FWER at the $\\alpha$ level. It is well known that the $\\alpha_{\\text{loc}}$ found by this method, the \\v Sid\\'ak  method, is slightly larger than the $\\alpha_{\\text{loc}}$ found by the Bonferroni method, thus the \\v Sid\\'ak  method will give slightly higher power.\n\nWe have seen that in our case, $\\gamma_1\\leq\\gamma_m$, meaning that the \\v Sid\\'ak  method can safely be used. If $\\gamma_k\\leq\\gamma_m$, then $\\text{FWER}=1-\\gamma_m\\leq1-\\gamma_k=\\alpha$ can be used to control FWER by solving the last equation for $\\alpha_{\\text{loc}}$ (choosing the greatest solution if not unique -- we have, however, never observed a $\\gamma_k$ that is not monotonically decreasing in $\\alpha_{\\text{loc}}$). If $\\gamma_k\\leq\\gamma_l$, then continuity of $\\gamma_k$ and of $\\gamma_l$ as functions of $\\alpha_{\\text{loc}}$ implies that the $\\alpha_{\\text{loc}}$ making $1-\\gamma_l=\\alpha$ is no less than the $\\alpha_{\\text{loc}}$ making $1-\\gamma_k=\\alpha$, so that the power obtained by the $l$th approximation is no less than the power obtained by the $k$th approximation.\n\n\\sloppy\nThe ideal property $\\gamma_1\\leq\\gamma_2\\leq\\cdots\\leq\\gamma_k\\leq\\gamma_m$ for all $\\alpha_{\\text{loc}}$ is ensured if $\\lvert\\bm T\\rvert=(\\lvert T_1\\rvert,\\ldots,\\lvert T_1\\rvert)$ is monotonically sub-Markovian of order $k$ ($\\text{MSM}_k$) with respect to $(-\\infty,c)^k$ for all $c$, \\ $2 \\leq k \\leq m-1$, as defined by \\cite{Block1992}. Unfortunately, our $\\lvert\\bm T\\rvert$ is not $\\text{MSM}_{m-1}$. It is possible to construct a trivariate normal distribution with mean $\\bm0$ such that $\\gamma_1<\\gamma_3<\\gamma_2$ for some~$\\alpha_{\\text{loc}}$. However, the violations of $\\text{MSM}_{m-1}$ we have observed have been very small and only for restricted ranges of $\\alpha_{\\text{loc}}$, and only for carefully constructed covariance matrices. We have not observed violations for covariance matrices estimated from real data, and will therefore proceed to apply $\\gamma_2$ and $\\gamma_3$ as better approximations to $\\gamma_m$ than $\\gamma_1$ (the latter giving \\v Sid\\'ak  cutoffs). A summary of concepts of positive dependence, like $\\text{MSM}$, was given by \\citet[pp.~58--61]{Dickhaus2014}.\n\n\\fussy\n\\subsection{Controlling FWER using $k$th order approximation for score tests}\\label{sec:FWERkapprox}\n\nAs we have seen, the vector $\\bm T$ of score test statistics is under the complete null hypothesis asymptotically standard multivariate normal with covariance matrix $R$ \\eqref{eq:R}. We denote by $O_j$ the event $\\lvert T_j\\rvert<c$ of non-rejection of $H_j$, which has probability $P(O_j)=1-\\alpha_{\\text{loc}}$ under the null hypothesis, with $\\alpha_{\\text{loc}}=2\\Phi(-c)$. We will detail how to find $\\alpha_{\\text{loc}}$ given by the second order approximation, $\\gamma_2$: Denote by $r_j$ the $(j-1,j)$ entry of $R$. Then\n\\[\n P(O_{j-1}\\cap O_j)= 1-\\alpha_{\\text{loc}}-\\sqrt{\\frac{2}{\\pi}} \\int_{-c}^{c}\\!\\!e^{-x^2\/2}\\,\\Phi\\Biggl(\\frac{r_j x-c}{\\sqrt{1-r_{\\vphantom i\\smash j}^2}}\\Biggr)dx,\n\\]\ngiving\n\\begin{align}\n \\gamma_2&=P(O_1\\cap O_2)\\prod_{j=3}^mP(O_j\\mid O_{j-1})\n  =\\frac{\\prod_{j=2}^m P(O_{j-1} \\cap O_j)}{\\prod_{j=3}^{m} P(O_{j-1})}\\nonumber\\\\\n &= \\frac{\\prod_{j=2}^{m}\\Bigl(1-\\alpha_{\\text{loc}}-\\sqrt{\\frac{2}{\\pi}}\\int_{-c}^{c}e^{-x^2\/2}\\,\\Phi\\Bigl(\\frac{r_{j}x-c}{\\sqrt{1-r_{\\vphantom i\\smash j}^2}}\\Bigr)dx\\Bigr)}{(1-\\alpha_{\\text{loc}})^{m-2}}\\nonumber\\\\\n &= (1-\\alpha_{\\text{loc}})\\prod_{j=2}^{m}\\Biggl(1-\\sqrt{\\frac{2}{\\pi}}\\frac{1}{1-\\alpha_{\\text{loc}}}\\int_{-c}^{c}\\!\\!e^{-x^2\/2}\\,\\Phi\\Biggr(\\frac{r_{j}x-c}{\\sqrt{1-r_{\\vphantom i\\smash j}^2}}\\Biggr)dx\\Biggr).\n\\label{eq:gamma2}\n\\end{align}\nFor a desired upper bound $\\alpha$ on FWER, the equation $1-\\gamma_2=\\alpha$ is solved with respect to $\\alpha_{\\text{loc}}$, which can be done numerically using for example a bisection algorithm. Note that $\\alpha_{\\text{loc}}$ enters into $c=-\\Phi^{-1}(\\alpha_{\\text{loc}}\/2)$.\n\nWe can control FWER by higher-order approximations by solving the equation $1-\\gamma_k=\\alpha$ for $\\alpha_{\\text{loc}}$ in a similar way, which we will henceforth refer to as order $k$ FWER approximation. By~\\eqref{gammak}, $\\gamma_k$ can be written as a ratio of products of $k$-dimensional and products of $k-1$-dimensional multivariate normal integrals. Good numerical methods for calculating multivariate normal integrals exist for small dimensions \\citep{genz2009computation}. We will illustrate using $k=2$ and $k=3$ for real data in Section \\ref{sec:invest}.\n\nThe procedure to find $\\alpha_{\\text{loc}}$ does not depend on the exact form of the test statistic, only that the vector $(T_1,\\ldots,T_m)$ of test statistics is asymptotically standard multivariate normal under the complete null hypothesis and $\\lvert T_j\\rvert\\geq c$ leads to rejection. In particular, \\eqref{eq:gamma2} is identical to what was found by \\cite{MoskvinaSchmidt2008} for an allelic test and correlations given by linkage disequilibria.\n\nIn practice, instead of calculating $\\alpha_{\\text{loc}}$, it may be preferable to calculate FWER-adjusted $p$-values: Replace $\\alpha_{\\text{loc}}$ with $p$, the unadjusted $p$-value for an individual test, in the calculation of $\\gamma_k$. Then $1-\\gamma_k$ is an FWER-adjusted $p$-value for the test, in the sense that if $1-\\gamma_k\\leq\\alpha$ (rejection based on adjusted $p$-value), then $p\\leq\\alpha_{\\text{loc}}$ (rejection based on local significance level).\n\n\\subsection{FWER control with independent blocks}\\label{sec:blocks}\n\nGenetic markers are distributed along the chromosomes and a common assumption is independence of genetic markers from different chromosomes.\n\nAs we have seen in Section \\ref{sec:corr}, if the genetic markers are independent and no environmental covariates that are correlated with the genetic markers are included, the score test statistics for these markers would also be independent. Within a chromosome, genetic markers can belong to different haplotype blocks, being highly correlated within a block and independent or nearly independent between the blocks \\citep{biologi}.\n\nAssume that the $m$ markers to be tested, and thus $\\{O_1,\\ldots,O_m\\}$, can be partitioned into $b$ independent blocks, $\\{O_1,\\ldots,O_{m_1}\\}$, \\ $\\{O_{m_1+1},\\ldots,O_{m_2}\\}$, \\ldots, $\\{O_{m_{b-1}+1},\\ldots,O_m\\}$,  so that $O_{j_1}$ and $O_{j_2}$ are independent if they belong to different blocks. Let $\\gamma_k^{(l)}$ be the $k$th order approximation given by~\\eqref{gammak} for the intersection of the events belonging to the $l$th block, \\ $1\\leq l\\leq b$, and let $\\gamma_k$ be the overall $k$th order approximation. Then it is easy to verify that $\\gamma_k=\\prod_{l=1}^b\\gamma_k^{(l)}$.\n\n\\subsection{The effective number of independent tests}\\label{sec:Meff}\n\nThe concept of an effective number of independent tests, $M_{\\text{eff}}$, in multiple\ntesting problems has been described and discussed by many authors, including \\cite{nyholt}, \\cite{gao1}, \\cite{MoskvinaSchmidt2008}, \\cite{LiJi2005}, \\cite{Galwey2009} and \\cite{ChenLiu2011}. All except \\cite{MoskvinaSchmidt2008} first estimate $M_{\\text{eff}}$, and then use $M_{\\text{eff}}$ in place of $m$ in the \\v Sid\\'ak  formula to calculate $\\alpha_{\\text{loc}}=1-(1-\\alpha)^{1\/M_{\\text{eff}}}$. An alternative formulation using the Bonferroni formula also exists.\n\nNone of these methods use the concept of FWER in the derivation of $M_{\\text{eff}}$, and there is no mathematical justification that FWER is controlled. All methods start with the linkage disequilibrium or composite linkage disequilibrium matrix, and there is no mention of the dependence of the $M_{\\text{eff}}$ estimate on the test statistics used for the hypothesis tests.\n\nThe method of \\cite{MoskvinaSchmidt2008} is based on an allelic test and controls the FWER using second order intersection approximations. As for our method, the main output of their method is an estimate of $\\alpha_{\\text{loc}}$. The above \\v Sid\\'ak  formula can then be used to define $M_{\\text{eff}}=\\ln(1-\\alpha)\/\\ln(1-\\alpha_{\\text{loc}})$. Note that $M_{\\text{eff}}$ depends on both $\\alpha_{\\text{loc}}$ and the FWER threshold~$\\alpha$. We will not consider $M_{\\text{eff}}$ further in this article. \n\n\\subsection{The maxT permutation method}\\label{maxT}\n\nWe will compare the local significance level $\\alpha_{\\text{loc}}$, as calculated by the FWER approximation method presented in section~\\ref{sec:FWERkapprox}, with the \\cite{WestfallYoung1993} maxT permutation method, and give a brief review of the latter.\n\nThe FWER is the probability that at least one of the $m$ null hypotheses is falsely rejected, which can be formulated as $P(\\max_j\\lvert T_j\\rvert\\geq c)=1-\\gamma_m$ under the complete null hypothesis. In the maxT method, the critical value $c$ is found empirically by permutation of the response variable in order to generate a sample from the distribution of the $\\max_j\\lvert T_j\\rvert$ statistic. If the FWER is to be controlled at the $\\alpha$ level and $b$ permutations are made, $c$ is estimated by the $(1-\\alpha)b$th order statistic of the $\\max_j\\lvert T_j\\rvert$ (the $(1-\\alpha)b$th smallest value), which is an estimate of the $1-\\alpha$ quantile of $\\max_j\\lvert T_j\\rvert$. The probability that the $k$th order statistic of a random sample of size $b$ is greater than the $1-\\alpha$ quantile is equal to the binomial cumulative distribution function with parameters $b$ and $1-\\alpha$ evaluated at $k-1$, which can be used to construct a confidence interval for $c$ (\\citeauthor{thompson1936confidence}, \\citeyear{thompson1936confidence}, see e.g. \\citeauthor{conover1980practical}, \\citeyear{conover1980practical}, p. 114). A confidence interval for $\\alpha_{\\text{loc}}$ is obtained by transforming the bounds via $\\alpha_{\\text{loc}}=2\\Phi(-c)$.\n\nThe success of the permutation method relies on the exchangeablity of the data, which in general does not hold for regression problems \\citep{Commenges2003}.  In our GLM the responses $\\bm Y$ are in general not exchangeable since their expected values are not equal when environmental covariates (which may not be independent of the genetic covariates) are present. Without environmental covariates (only intercept) the responses are exchangeable and permutation of $\\bm{Y}$ gives FWER control. With discrete environmental covariates permutation can be done in a stratified manner \\citep{Solari2014}. When the exchangeability assumption is not satisfied, there is no standard solution to how permutation testing can be performed. Asymptotic or second moment exchangeability may be obtained by different transformations of the data, but comparison with these methods is beyond the scope of this paper. \n\n\\section{Power and efficiency of the FWER approximation method}\\label{sec:invest}\n\nWe will compare the local significance level $\\alpha_{\\text{loc}}$, as calculated by the FWER approximation method presented in the previous section, with $\\alpha_{\\text{loc}}$ of the Bonferroni method and of the \\cite{WestfallYoung1993} maxT permutation method. We proceed to compare the $\\alpha_{\\text{loc}}$ calculated by FWER approximation in two cases were the ``true'' $\\alpha_{\\text{loc}}$ based on the entire joint distribution can be calculated; one artificial and one based on data.\n\n\\subsection{Illustration of methods: TOP and $\\text{VO}_2$-max data}\\label{sec:topvo2}\n\nOur two data sets (referred to as TOP and $\\text{VO}_2$-max) are of limited sample size, and our aim is to use the data to investigate the correlation structure of GWA-data and the effect this has on the estimation of the local significance level. We assume that our findings will hold in data sets with larger sample sizes. An increase in sample size will give more precise estimates of the score test statistics correlations, but the estimation of the local significance level is mainly dependent on score test statistics correlations under study (not the sample size).\n\nThe TOP data set is a case--control GWA data set, in which case is schizophrenia or bipolar disorder. The data set was collected with the aim to detect single-nucleotide polymorphisms (SNPs) associated with the schizophrenia or bipolar disorder \\citep{TOP1,TOP2}. The preprocessed TOP GWA data contain genetic information on 672972 SNPs (Affymetrix Genome-Wide Human SNP Array 6.0) for 1148 cases and 420 controls. Our dataset included individuals sampled until March 2013, and therefore the sample size is larger than in the cited papers. Preprosessing of the data was done as described in \\cite{TOP1} and \\cite{TOP2}.\n\nGenotype--phenotype association was assessed by fitting a logistic regression without any environmental covariates, so that score test correlations equal genotype correlations (Section~\\ref{noenvcov}).\n \nThe $\\text{VO}_2$-max data set comes from a cross-sectional GWA study \\citep{VO2max1, VO2max2}, in which the aim was to find SNPs associated with maximum oxygen uptake. The preprocessed $\\text{VO}_2$-max GWA data consist of 123497 SNPs \\citep[Illumina Cardio-MetaboChip,][]{VO2chip} for 2802 individuals. The $\\text{VO}_2$-max data were analysed using a normal linear regression model, including age, sex and physical activity score as covariates. For both datasets, some genotype data were missing. In the TOP data, mean imputation was done for $0.04\\%$ of the genotypes, and in the $\\text{VO}_2$-max data for $0.7\\%$ of the genotypes. \n\nFor the $\\text{VO}_2$ data, the local significance level controlling the FWER at level 0.05 was lowest for the Bonferroni method, slightly higher for the order 1 approximation (the \\v Sid\\'ak method), and further increasing through the order 2 and~3 FWER approximations (Table~\\ref{tab:resultater}).\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccccc}\\hline\n & \\multicolumn2c{TOP} && \\multicolumn2c{$\\text{VO}_2$-max}\\\\ \\cline{2-3}\\cline{5-6}\nMethod & $10^8\\alpha_{\\text{loc}}\\vphantom{\\big)}$ & Ratio & & $10^7\\alpha_{\\text{loc}}$ & Ratio \\\\\n\\hline\nBonferroni  & 7.43 & 1.00 && 4.05 & 1.00\\\\\nOrder 1 (\\v Sid\\'ak)   & 7.62 & 1.03 && 4.15 & 1.02\\\\\nOrder 2 & 8.62 & 1.16 && 4.70 & 1.16 \\\\\nOrder 3 & 9.07 & 1.22 && 5.02 & 1.24 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local significance level $\\alpha_{\\text{loc}}$ calculated by the Bonferroni method and by order 1--3 FWER approximations for the TOP and $\\text{VO}_2$-max data, controlling the FWER at level $0.05$, \nand ratio of $\\alpha_{\\text{loc}}$ to Bonferroni $\\alpha_{\\text{loc}}$.}\n\\label{tab:resultater}\n\\end{center}\n\\end{table}\n\nFor the TOP data, since no environmental covariates are included, permutation of the binary response vector is feasible\n(the exchangeability assumption is satisfied), and the maxT method can be used to estimate the local significance level controlling FWER at level $0.05$. Permutation of the responses, followed by calculation of the maximal score test statistics over the whole genome, is a time consuming task, and we will only present results on two of the smallest chromosomes (chromosome 21 and 22):\n\nThe local significance level controlling the FWER at level 0.05 was, as for the $\\text{VO}_2$ data, lowest for the Bonferroni method and increasing through order 1--3 FWER approximations (Table \\ref{tab:TOPres}). The highest level was obtained for the maxT method, and also the lower bound of the 95\\% confidence interval for $\\alpha_{\\text{loc}}$ of maxT was greater than the order 3 FWER approximation.\n\nOn a $4\\times6$-core Xeon 2.67 GHz computer (Intel CPU) running Linux (Ubuntu 14.0) using one thread, the analyses on chromosome 22 took 85 hours for maxT, 20 minutes for order 3 FWER and 10 seconds for order 2 FWER approximation.\n\nSmoothed frequency distributions of the estimated correlations between neighbouring SNPs along chromosomes are very similar across chromosomes (Figure \\ref{fig:densTOPandVO2}), and therefore, we would expect that the trends for chromosome 21 and 22 can be extended to the other chromosomes and to the whole genome.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{.\/neighbourcorr-top-4}\\hfill\n\\includegraphics[width=0.49\\textwidth]{.\/neighbourcorr-vo2-4}\n\\caption{Smoothed frequency distributions of absolute value of the estimated genotype correlations between neighbouring SNPs on each chromosome, one line per chromosome. TOP data (left) and $\\text{VO}_2$-max data (right). The plots are logspline density estimates \\citep{stone1997polynomial}, implemented by the R logspline package \\citep{logsplineR}.}\n\\label{fig:densTOPandVO2}\n\\end{figure}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccccc}\\hline\n & \\multicolumn5c{Chromosome}\\\\ \\cline{2-6}\n & \\multicolumn2c{21} && \\multicolumn2c{22}\\\\ \\cline{2-3}\\cline{5-6}\nMethod & $10^6\\alpha_{\\text{loc}}\\vphantom{\\big)}$ & Ratio && $10^6\\alpha_{\\text{loc}}$ & Ratio\\\\\n\\hline\nBonferroni & 5.10 & 1.00 && 5.57 & 1.00\\\\\nOrder 1 (\\v Sid\\'ak) & 5.23 & 1.03 && 5.72 & 1.03\\\\\nOrder 2 & 6.09 &1.19 && 6.46 & 1.16\\\\\nOrder 3 & 6.57 & 1.29 && 6.93 &1.24\\\\\nmaxT, lower & 7.34 & 1.44 && 7.68 & 1.38\\\\\nmaxT & 7.44 &1.46 && 7.79 &1.40\\\\\nmaxT, upper & 7.55 & 1.48 && 7.91 & 1.42\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Local significance level $\\alpha_{\\text{loc}}$ for the TOP data calculated by the Bonferroni method and by order 1--3 FWER approximations, and estimated by the maxT method, controlling FWER at level $0.05$, and ratio of $\\alpha_{\\text{loc}}$ to Bonferroni $\\alpha_{\\text{loc}}$. Chromosome 21 contained $9802$ SNPs and chromosome~22 contained $8970$ SNPs. The number of permutations for the maxT method was 500000. The lower and upper values for maxT are bounds of a 95\\% confidence interval for $\\alpha_{\\text{loc}}$ (see Section~\\ref{maxT}).}\n\\label{tab:TOPres}\n\\end{table}\n\n\\subsection{Correlation structure and local significance level}\\label{sec:alphaksim}\n\nConsider 100 markers and a multivariate normal test statistic $\\bm T$ having an AR1 correlation structure, that is, all entries on the main diagonal of the $100\\times100$ correlation matrix are equal to~1, on the sub- and superdiagonal $\\rho$, on the next diagonals $\\rho^2$, and so on. We investigated the effect of positive $\\rho$ on the local significance level $\\alpha_{\\text{loc}}$ found by order 1--4 FWER approximations to control FWER at the 0.05 level. Also, the ``true'' $\\alpha_{\\text{loc}}$ was calculated without approximation (that is, based on $\\gamma_{100}$; see Section~\\ref{sec:fwerk}), using the \\texttt{pmvnorm} function of the R \\citep{R} package mvtnorm \\citep{mvtnormR} using the Genz--Bretz algorithm \\citep{Genz1992,Genz1993,GenzBretz2002}. The \\texttt{pmvnorm} function can calculate multivariate normal probabilities with some accuracy for dimensions up to 1000.\n\nThe inverse of an AR1 correlation matrix contains only negative off-diagonal entries, which ensures a property called  $\\text{MTP}_2$ \\citep{karlin1981total} for the density of $\\lvert\\bm T\\rvert$, which implies that the product-type approximations $\\gamma_k$ of Section~\\ref{sec:fwerk} are non-decreasing in $k$ \\citep{glazjohnson}, making the $\\alpha_{\\text{loc}}$ of the order $k$ FWER approximations non-decreasing in $k$.\n\nThe effect of $\\rho$ on $\\alpha_{\\text{loc}}$ was small for $\\rho<0.4$ (Figure \\ref{fig:ar1}), so for the 100 markers considered, there would be no gain in using FWER approximation or even the true joint distribution of $\\bm T$ instead of \\v Sid\\'ak this case. For larger $\\rho$, order 2 FWER approximation provides an improvement compared to \\v Sid\\'ak. The increase in $\\alpha_{\\text{loc}}$ from \\v Sid\\'ak to order 2 FWER approximation was greater than the difference between higher orders.\n\nTo assess the order $k$ FWER approximation method for a more realistic correlation structure, we considered the empirical correlation matrix for the first 1000 markers on chromosome 22 of the TOP data. The order 1--4 approximations to control FWER at the 0.05 level gave an $\\alpha_{\\text{loc}}$ of 5.1 (\\v Sid\\'ak), 5.8, 6.2 and $6.4\\cdot10^{-5}$, respectively, whereas the $\\alpha_{\\text{loc}}$ calculated without approximation using the Genz--Bretz algorithm was $7.3\\cdot10^{-5}$.\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.49\\textwidth]{.\/ar1.pdf}\n\\caption{Local significance level $\\alpha_{\\text{loc}}$ for order 1--4 FWER approximations and $\\alpha_{\\text{loc}}$ based on true joint distribution of test statistic as a function of the parameter $\\rho$ of an AR1 correlation matrix for 100 markers. The horizontal line corresponds to \\v Sid\\'ak correction (order 1 FWER approximation), then $\\alpha_{\\text{loc}}$ is increasing with the order of the approximation (order 2--4; the three curves in the middle). The uppermost curve shows $\\alpha_{\\text{loc}}$ based on the true joint distribution.}\n\\label{fig:ar1}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discuss}\n\nWe have presented the order $k$ FWER approximation method for estimating the local significance level $\\alpha_{\\text{loc}}$ used to control FWER in a GWA study.  Our method takes the estimated correlation structure between the test statistics into account, and is applicable when environmental covariates are present. The relation between the phenotype response and the genetic and environmental covariates can be modelled by any generalized linear model (using the canonical link); in particular, both models with discrete and models with continuous phenotypes are allowed. We have applied the method to common genetic variants, but it can also be used for rare variants. However, since rare variants are less correlated than common variants, we expect the increase in $\\alpha_{\\text{loc}}$ from the \\v Sid\\'ak  method to be less than when analyzing common variants. \n\nThe order $k$ FWER approximation is based on conditioning on the previous $k-1$ neighbouring markers along the chromosome. A sufficient condition to have non-decreasing local significance levels when the order of the FWER approximation increases from 1 to $k$, and that the order $k$ order approximation gives valid FWER control, is that the test statistic has the $\\text{MSM}_k$ property. Even $\\text{MSM}_2$ and $\\text{MSM}_3$ are difficult to verify for our test statistic with GWA data, but it is reasonable to assume that they are satisfied (Section \\ref{sec:fwerk}).\n\nPopulation substructure can be associated with both the genotype and phenotype and is therefore a possible confounding factor in GWA studies. Population substructure can be adjusted for in the analysis using principal components of the covariance matrix of the individuals \\citep{Price2006} as covariates. In both the TOP data and the VO$_2$-max data, related individuals were removed in the preprocessing of the data, and no adjustment for population structure was done in our analysis. \n\nThe AR1 correlation structure (Section~\\ref{sec:alphaksim}) might not be a realistic model for genotype correlations, but the calculations nevertheless show potential for a significant improvement over the \\v Sid\\'ak method by applying the fast order 2 FWER approximation. Also, there is potential to get quite close to the local significance level given by the full joint distribution of the test statistic vector by using order 3 or 4 approximation. Calculations using the more realistic empirical correlation matrix of part of chromosome 22 of the TOP study confirm this impression.\n\nThe maxT method (Section~\\ref{maxT}) of \\cite{WestfallYoung1993} may give higher power than FWER approximation (Table~\\ref{tab:TOPres}). However, there is no general way of including environmental covariates using that method (Section~\\ref{maxT}). Also, computing time is much larger than for lower-order FWER approximation (Section~\\ref{sec:topvo2}), and the $\\alpha_{\\text{loc}}$ estimate would likely differ if a new set of permutations were made (see confidence limits of Table~\\ref{tab:TOPres}).\n\nAnother alternative is parametric bootstrap methods \\citep{SeamanMyhsok2005}, which  could be used to estimate the local significance level when the exchangeability assumption is not satisfied. It would be an efficient method, but to our knowledge it has not been proven that parametric bootstrap will control the overall error rate, since nuisance parameters need to be estimated.\n\n\\cite{ConneelyBoehnke2007} introduced a method for multiple testing correction for GLMs for multiple responses (traits) based on the estimated correlation matrix of the score vector. The focus of the method is to calculate FWER-adjusted $p$-values based on the multivariate integral arising from \\eqref{eq:FWER}. Currently, this integral can be computed numerically with some accuracy for dimensions smaller than or equal to 1000 using the \\texttt{pmvnorm} function of the R package mvtnorm (see Section~\\ref{sec:alphaksim} for details and references). Thus, the method of \\citeauthor{ConneelyBoehnke2007} is not applicable for larger problems, e.g. more than $m=1000$ hypothesis tests. \n\nFor our order $k$ FWER approximation method we have used standard R functions to compute the second order approximation given by~\\eqref{eq:gamma2}. For orders 3, 4 and 5 we have used the above-mentioned function \\texttt{pmvnorm} specifying the Miwa algorithm \\citep{Miwa2003} instead of the default Genz--Bretz algorithm. The Miwa algorithm can be used for small dimensions, and is deterministic, whereas the Genz--Bretz algorithm includes simulations that lead to inaccuracies, which accumulate to an intolerable level when used for the large number of factors in~\\eqref{gammak}. The research into better and faster integration of multivariate normal densities is ongoing, and \\cite{Botev2016} provides an interesting new approach, applicable for dimensions smaller than or equal to 100. This will enable our order $k$ FWER approximation method to be applied with larger values of~$k$ than what has been presented here.\n\n\\section{Conclusions}\\label{sec:conclude}\nWe have presented a new method for controlling the FWER for GWA data. The order~$k$ FWER approximation method can be used for generalized linear models and include adjustment for environmental covariates, possibly confounding, like population substructure or sex. We have applied the FWER approximation method to GWA data, and shown that our method is a powerful alternative to the Bonferroni and \\v Sid\\'ak  methods, especially in situations were permutation methods cannot be used (exchangeability assumption not satisfied).\n\nThe method provides a local significance level, $\\alpha_{\\text{loc}}$, for the individual tests, meaning that the null hypothesis of no association between phenotype and genetic marker should be rejected if the (unadjusted) $p$-value of a test is less than $\\alpha_{\\text{loc}}$. We found a substantial increase in $\\alpha_{\\text{loc}}$ already at the order~2 approximation, compared to the $\\alpha_{\\text{loc}}$ produced by the well-known Bonferroni and the \\v Sid\\'ak methods -- methods that does not take correlation structure between the test statistics of the markers into account (\\v Sid\\'ak assumes independence, but that could be considered worst-case for GWA data).\n\n\\section*{Software}\nThe statistical analysis were performed using R \\citep{R}, and the preprocessing of the genetic data were done using the software PLINK \\citep{plink}.  \n\n\\section*{Acknowledgements}\nThe authors would like to thank Dr. Jelle J. Goeman (Leiden University Medical Centre, Leiden, The Netherlands) for valuable comments. Part of the work was done while the last author was on sabbatical at Centre for the Genetic Origins of Health and Disease, University of Western Australia, Australia. The PhD position of the first author is founded by the Liaison Committee between the Central Norway Regional Health Authority (RHA) and the Norwegian University of Science and Technology (NTNU).\n\n\\emph{Conflict of Interest:} None declared.\n\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUltrarelativistic heavy-ion experiments have shown very rich physics \nwhich cannot be interpreted by mere extrapolation from elementary \nnucleon-nucleon collisions to nucleus-nucleus collisions. This is evidenced from\nthe suppression of high transverse momentum region of hadron spectra up to \na factor of 5 relative to nucleon-nucleon collisions,\nwhich is an indication for strong absorption of high-energy partons traversing \nthe medium~\\cite{Wang}. Also the inclusive production of charm quark bound \nstates are suppressed by a factor of 3-5 at both Super \nProton Synchrotron (SPS) and Relativistic Heavy Ion Collider (RHIC) experiments \nwhich hints their dissolution in the medium~\\cite{Rapp,Kluberg}.\nThese accumulated evidences indicate that a new form of matter\nhas been produced in the heavy-ion collision experiments, called quark-gluon \nplasma (QGP). Among different experimental \nobservations which may be served as the signals for the QGP formation, \nquarkonium suppression has been proposed long time ago as a clear probe of \nthe QGP formation in the collider experiments~\\cite{McLer,adam05}. \nFirst, in a pioneering work by Matsui and Satz~\\cite{matsui} and \nlater in a follow up quantitative calculation~\\cite{Karsch}, it\nwas shown that the suppression of $J\/\\psi$ yields could be explained\nby the (color) screening of the potential between a heavy quark and anti-quark \nby the surrounding deconfined light quarks and gluons. \nTheoretically, one can study the quarkonium states\nby using the effective field theories. Since the mass of heavy quark, $m_Q$\nis much larger than the intrinsic scale in the theory of \nQuantum Chromodynamics (QCD), $\\Lambda_{\\rm{QCD}}$, the \nheavy quark and anti-quark are expected to move slowly with \na relative velocity $v \\ll 1$ and results in the non-relativistic version of  \nQCD (NRQCD) \\cite{nrqcd1,nrqcd2}. However, NRQCD\ndoes not fully exploit the smallness of $v$ which further gives rise \nanother effective theory, known as potential non-relativistic QCD (pNRQCD)~\\cite{pNRQCD1,pNRQCD2}\nby integrating out the momentum scale.\n\nThe heavy quark pairs formed\nin relativistic nuclear collisions develop into the physical resonances \nand traverse the plasma and then hot hadronic matter\nbefore decaying into dilepton pairs. Even before the resonance is formed\nit may be absorbed by the nucleons streaming past it~\\cite{GH} and by the\ntime the resonance is formed, the screening of the color force\nin the plasma may inhibit the formation of bound states.\nThe resonance(s) could also be dissociated either\nby hard gluons~\\cite{xu,gdiss1,gdiss2,gdiss3,gdiss4} or by comoving\nhot hadrons~\\cite{vogt}. In order to disentangle these sequential\neffects~\\cite{dks}, we must\nknow how the properties of quarkonium states change in medium.\nThe basic tools of phenomenological approach to study the properties of \nquarkonium states are potential models where the possible relativistic effects \nfor excited states of charmonium may also be incorporated there.\nAt zero temperature, the potential model\nhas made great success. At finite temperature, \nthe essence of the potential model in the context of deconfinement is to use a\nfinite temperature extension of the potential. Quantitative understanding\nof the bound state properties needs the exact potential at finite temperature \nwhich, in principle, should be derived directly from QCD, like the Cornell\npotential at zero temperature has been derived from pNRQCD from the \nzeroth-order matching coefficient. Such derivations\nat finite temperature for weakly-coupled plasma have been recently come up in the \nliterature~\\cite{Brambilla05,Brambilla08} but they\nare, however, plagued by the existence of temperature-driven \nhard as well as soft scales, $T$, $gT$, $g^2 T$, respectively.\nDue to these difficulties in finite temperature extension in effective\nfield theories, the lattice-based potentials become popular. \nHowever, neither the free energy nor the internal energy \ncan be directly used as the potential. \nIn fact, what kind of screened potentials should be used \nin the Schr\\\"odinger equation which describes well the bound states at finite \ntemperature are still an open question. \nHowever, recently more involved calculations of quarkonium spectral \nfunctions and meson current correlators obtained from potential \nmodels have been performed and compared to \nfirst-principle QCD calculations performed numerically on \nlattices~\\cite{Asak04,Dat04,Hatsuda,Jakovac,Aarts11}.\nIn addition to the uncertainties of the correct form of the\nfinite-temperature potential, there are also arbitrariness of \nthe criteria of dissociation. Besides the binding energy \nfor a particular potential as an criterion of dissociation, the decay width \nis another important quantity \nto determine the dissociation of the bound states. \nThe calculation based on a real-valued potential model does not include the \ntrue width of a state. By performing \nan analytical continuation of the Euclidean Wilson loop to Minkowski space, \nthe potential has an imaginary part due to Landau damping which clearly \nbroadens the peak~\\cite{Brambilla08,Laine1} and facilitates the early\ndissociation of the bound states.\n\nIn the RHIC or LHC era (small $\\mu_{B}$), recent lattice studies have \nconfirmed that the transition from\nnuclear matter to QGP is not a phase transition, rather a\ncrossover\\cite{phaseT}.\nThe large distance property of the heavy quark\ninteraction is important for our understanding of the bulk properties of\nthe QCD plasma phase, {\\it e.g.} the screening property of the quark gluon\nplasma \\cite{Kar62+70}, the equation of state \\cite{Dum05,Dum2} etc. \nIn these studies, deviations from perturbative \ncalculations and the ideal gas behavior are found\nbeyond the deconfinement temperature. \nIt is then reasonable to assume that the string-tension\ndoes not vanish abruptly at the deconfinement point~\\cite{string1,string2,string3}, so\none should study its effects on heavy quark potential even above $T_c$.\nThis issue, usually overlooked in the\nliterature where only a screened Coulomb potential was\nassumed above $T_c$ and the linear\/string term was assumed zero,\nwas certainly worth investigation.\nRecently a heavy quark potential at finite\ntemperature was derived by correcting the full Cornell potential,\nnot its Coulomb part alone,\nwith a dielectric function encoding the effects of the deconfined \nmedium~\\cite{prc_vineet}. This was found to have an additional\nlong range Coulomb term, in addition to the conventional Yukawa term. \nIn the short distance limit, the potential is reduced to vacuum potential, \n{\\it i.e.}, $Q \\bar Q$ pair does not see the medium, giving rise the duality\nbetween $V(r,T=0)$ and $V(0,T)$. On the\nother hand, in the large distance limit (where the screening\noccurs), potential is reduced to a long-range Coulomb potential with\na dynamically screened-color charge. Thereafter the binding energies and \ndissociation temperatures\nof the ground and the lowest-lying states of charmonium and bottomonium\nspectra have been determined~\\cite{prc_vineet,chic_vineet} which \nmatches with the finding of recent works based on\npotential models~\\cite{cabrera,mocsyprl,digal} with the Debye mass\nextracted from the lattice free energy. However, when the Debye mass \nin leading-order was used, the results~\\cite{prc_vineet}  \nmatch with the lattice correlator studies or \nwith the stronger (lattice) binding potential, i.e. internal energy~\\cite{wong1,wong2}.\nIn a way, their findings address the reason of arbitrariness of the results \non dissociation temperatures~\\cite{Satz} and ensues a basic question about the nature of dissociation\nof quarkonium in a hot QCD medium.\n\n\nHowever, all the works described above was limited to an isotropic \nmedium but the partonic system generated in an ultra-relativistic \nheavy-ion collisions cannot be homogeneous and isotropic\nbecause at the very early time of collision, asymptotic weak-coupling enhances\nthe longitudinal expansion substantially than the radial expansion so\nthe system becomes colder in the longitudinal direction than in the \ntransverse direction. As a result, an anisotropy in the momentum space sets in and \ncauses the parton system produced\nunstable with respect to the chromomagnetic plasma modes~\\cite{Romatschke} \nwhich facilitate to isotropize the system~\\cite{Arnold05,Mrowczy93}.\nIn our work, we restricted ourselves to a {\\em weakly anisotropic medium}\nbecause by the time ($t_F$ =$\\gamma \\tau_F$, $\\tau_F$ is the quarkonium \nformation time in its rest frame) quarkonia are formed in the plasma, the \nplasma becomes almost equilibrated. Motivated with this preamble on\nthe anisotropy generated in the very stage of collision, we wish to\ninvestigate the effect of weak anisotropy on the heavy-quark potential\nand subsequently on the dissociation of quarkonia states in an anisotropic\nmedium.\n\n\nOur work is organized as follows. In Section~\\ref{poten}, we will discuss the \nmedium modifications to a heavy quark potential both in isotropic and\nanisotropic medium. In Sec.\\ref{iso}, we start with the in-medium modification\nto the heavy quark potential in an isotropic medium and then \nextend it to a medium which exhibits a local anisotropy in momentum \nspace in Sec.\\ref{aniso}. \nTo do that, we first obtain the self-energy tensor for an \nanisotropic medium to obtain the hard-loop resumed \ngluon propagator. Thereafter the dielectric permittivity be \nobtained\nin terms of retarded gluon propagator and its Fourier transform at vanishing frequency \ngives the desired non-relativistic potential at finite temperature. The\npotential thus obtained depends not only on the relative separation\nof $Q \\bar Q$ pair but also on their relative orientation\nwith respect to the direction of anisotropy and \nis found always deeper than in an isotropic medium.\nIt is found that in the weak-anisotropy limit, the correction\narising due to anisotropy to the isotropic part of the potential is small\nand thus has been treated as a perturbation.\nSo, using the first-order perturbation theory, \nwe estimate the shift in energy eigen values due to the small \nanisotropic correction \nto the energy eigen values from the spherically-symmetric\npart in isotropic medium and determine their dissociation \ntemperatures in Sec.3. Finally, we conclude \nin Section~\\ref{conclu}.\n\n\\section{Heavy-quark effective potential}\\label{poten}\n\\subsection{For isotropic medium {\\texorpdfstring{$(\\xi = 0)$}{xi=0}}}\\label{iso}\n\nPotential models are based on the assumption that the\ninteraction between a heavy quark and its anti-quark\ncan be described by a potential.\nAt $T=0$, the hierarchy of well separated energy-scales: \n$m_Q\\gg m_Qv\\gg m_Qv^2$,\nallows one to systematically integrate out the different\nscales and obtain the non-relativistic potential\nQCD (pNRQCD) where the Cornell potential indeed shows up as the\nzeroth-order matching coefficient~\\cite{pNRQCD1,pNRQCD2}. Inspired \nby its success at\nzero temperature, the potential model has been applied at finite\ntemperature, with the main assumption that medium effects can be accounted \nby a temperature-dependent potential.\n\n\nRecent lattice results~\\cite{phaseT} indicate the phase transition in full QCD\nappears to be a crossover rather than a `true' phase transition with the \nrelated singularities in thermodynamic observables.\nIn light of the above findings, one cannot simply\nignore the effects of string tension between the quark-anti-quark pairs\nbeyond $T_c$. This is indeed a very important effect which needs\nto be incorporated while setting up the criterion for the dissociation.\nRecently this issue\nhas successfully been addressed for the dissociation of quarkonium in\nQGP by one of us~\\cite{prc_vineet,chic_vineet}\nand we closely follow their work in brief.\nLet us now start with a heavy quark potential (Cornell potential) at T=0 \n\\begin{equation}\n\\label{eqc}\n V(r)=-\\frac{\\alpha}{r}+\\sigma r \\quad ,\n\\end{equation}\nwhere $\\alpha$ and $\\sigma$ are the phenomenological parameters. The former\naccounts for the effective coupling between a heavy quark and its anti-quark\nand the latter gives the string coupling.\nThe medium modification enters through the Fourier transform of\nheavy quark potential as \n\\begin{equation}\n\\label{eq1}\n\\tilde{V}(k)=\\frac{V(k)}{\\epsilon(k)} \\quad ,\n\\end{equation}\nwhere $V(k)$ is the Fourier transform (FT) of the Cornell potential\nwhich requires a regularization. We regulate both terms in the\npotential by multiplying with an exponential damping factor and\nis switched off after the FT is evaluated. \nThis can be done by assuming $r$- as distribution\n($r \\rightarrow$ $r \\exp(-\\gamma r))$. The FT of\nthe linear part $\\sigma r\\exp{(-\\gamma r)}$ is\n\\begin{eqnarray}\n\\label{eq-6-3}\n=-\\frac{i}{k\\sqrt{2\\pi}}\\left( \\frac{2}{(\\gamma-i k)^3}-\\frac{2}{(\\gamma+ik)^3}\n\\right).\n\\end{eqnarray}\nAfter putting $\\gamma=0$,  we obtain the FT of the linear term $\\sigma r$ \nas,\n\\begin{equation}\n\\label{eq-6-4}\n\\tilde{(\\sigma r)}=-\\frac{4\\sigma}{k^4\\sqrt{2\\pi}}\n\\end{equation}\nand for the full Cornell potential, the FT is \n\\begin{equation}\n\\label{vk}\n{V}(k)=-\\sqrt{(2\/\\pi)} \\frac{\\alpha}{k^2}-\\frac{4\\sigma}{\\sqrt{2 \\pi} k^4}.\n\\end{equation}\nThe dielectric permittivity, $\\epsilon(k)$ is given in terms\nof the static limit of the longitudinal part of the gluon\nself-energy~\\cite{schneider}. In isotropic case, it can be \ndecomposed into longitudinal ($\\Pi_L$) and transverse ($\\Pi_T$) \ncomponents which, in the static limit, are associated with \nthe screening of electric and magnetic fields, respectively. However, in the\nstatic limit, the transverse part\n$\\Pi_T (0,k\\rightarrow 0,T)$ vanishes, {\\em i.e.}, static magnetic \nfields are not screened.\nIn perturbation theory, the quantity that enters in the Fourier transform \nof the potential at finite temperature is the static limit of\nthe longitudinal gauge boson self-energy which\nwas calculated long ago~\\cite{Shur78} at one loop level \n\\begin{eqnarray}\n\\lim_{k\\rightarrow 0} \\Pi_L (0,k,T)=g^{2} T^{2}\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)\n\\equiv m_{{}_D}^2 \\quad ,\n\\label{debye}\n\\end{eqnarray}\nwhere $ m_{{}_D} $ is defined as the screening mass. Since the static \nlimit of the \nself-energy is momentum independent, the pole of the inverse of dielectric\npermittivity is simply the gauge invariant Debye \nmass $m_{{}_D} $, so it leads to an exponential damping of the\npotential $V(r)\\sim \\exp(-m_{{}_D}r)\/r$. In particular, this form of $ \\Pi_L $\nhas the consequence that gluons screen the strong interaction, in contrast \nto the zero temperature case, over long-distance scale. If one\nassumes non-perturbative effects such as the string\ntension which survives even above the deconfinement point~\\cite{bkp}\nthen the dependence of the dielectric function on\nthe Debye mass may get modified. \nHowever, we assume the same screening mass scale $m_{{}_D}$ which\nemerges in the Debye screened Coulomb potential also appears in the\nnon-perturbative long-distance contribution due to string.\nIn the following section, we take over this assumption to\nanisotropic medium too. However, different scales for the\nCoulomb and linear pieces of the T=0 potential, rather than a single \none, was already employed in Ref.~\\cite{megiasind,megiasprd}.\nMoreover, they developed a theoretical model to include non-perturbative \neffects beyond the deconfinement temperature through dimension-two gluon \ncondensates to calculate the heavy quark free energy.\nInterestingly, their model predicts a duality between the zero temperature \n$Q \\bar Q$ potential and the quark self energy and explains the lattice \ndata well.\n\n\\noindent Finally, one can define a dielectric permittivity\nin one-loop by\n\\begin{eqnarray}\n\\label{eqn2}\n\\epsilon(k)=\\left(1+\\frac{\\Pi_L (0,k,T)}{k^2}\\right)\n\\equiv \\left( 1+ \\frac{m_{{}_D}^2}{k^2} \\right)~.\n\\end{eqnarray}\nAfter substituting the dielectric permittivity in the Fourier \ntransform of Cornell potential (\\ref{eq1}) and then evaluating its \ninverse FT, one obtains the medium modified\npotential in the co-ordinate space~\\cite{prc_vineet,chic_vineet}\n\\begin{eqnarray}\n\\label{defn}\nV(r,T)&=&\\int \\frac{d^3\\mathbf k}{{(2\\pi)}^{3\/2}}\n e^{i\\mathbf{k} \\cdot \\mathbf{r}}~\n\\tilde{V}(k) \\nonumber\\\\\n&=&\\left(\\frac{2\\sigma}{m_D}-\\alpha m_D\\right)\\frac{\\exp{(-\\hat r)}}{\\hat r}\n-\\frac{2\\sigma}{m_D \\hat r}+\\frac{2\\sigma}{m_D}-\\alpha m_D\n\\end{eqnarray}\nwith the dimensionless variable $\\hat{r}=m_{{}_D} r$. The constant terms are introduced  to yield \nthe correct limit of $V(r,T)$ as $T\\rightarrow 0$. Such terms could arise\nnaturally from the basic computations of real time static potential\nin hot QCD~\\cite{Laine2} and also from the real and imaginary time\ncorrelators in a thermal QCD medium~\\cite{beraudo}.\nThe medium modified potential thus obtained has an additional \nlong range Coulomb term with an (reduced) effective charge, in addition\nto the conventional Yukawa term. \n\nIn the small distance limit, $r\\ll 1\/m_{{}_D}$, the above potential reduces to\nthe Cornell potential, {\\it i.e.} $Q \\bar Q$ does not see the medium.\nOn the other hand, in the screening region \n$r\\gg1\/m_{{}_D}$, the potential (\\ref{defn}) reduces to \n\\begin{eqnarray}\n\\label{lrp}\n{V(r,T)}\\sim -\\frac{2\\sigma}{m^2_{{}_D}r}-\\alpha m_{{}_D},\n\\end{eqnarray}\nwhich, apart from a constant term, looks like a Coulomb potential encountered\nin hydrogen-atom problem after identifying the fine structure constant\n$e^2$ with the effective charge $2 \\sigma\/m_{{}_D}^2$. \nThe binding energies and the dissociation temperatures for \nquarkonium states can thus be determined by solving the Schr\\\"odinger\nequation numerically either with the full potential (\\ref{defn}) or \nanalytically with the approximated form (\\ref{lrp}). \nIn addition, one can also exploit the advantage to demonstrate the flavor \ndependence of the dissociation process where the dissociation temperatures \nfor 2-flavor are found to be higher than \nthe 3-flavor case~\\cite{prc_vineet,chic_vineet}. \n\n\n\\subsection{For anisotropic medium {\\texorpdfstring{$(\\xi \\ne 0)$}{xi=0}}}\\label{aniso}\n\\subsubsection{Dielectric permittivity tensor}\nTo study the perturbative potential with an anisotropic parton \ndistribution, consider a hot QCD plasma which, due to expansion \nand finite (momentum) relaxation time, manifests a local anisotropy in\nmomentum space through the distribution function \n\\begin{equation}                                        \nf_{\\rm{aniso}} (\\mathbf{k}) = f_{\\rm{iso}}\\left( \\sqrt{\\mathbf{k}^{2} + \\xi(\\mathbf{k}.\\mathbf{n})^{2}}  \\right) \\quad,\n\\end{equation}                                            \n{\\it i.e.}, $f_{\\rm{aniso}} (\\mathbf{k})$ is obtained from an isotropic distribution $f_{iso} (|\\mathbf{k}|)$ \nby removing particles with a large momentum component along\nthe direction of anisotropy, $\\mathbf{n}$~\\cite{Romatschke}. We shall restrict \nourselves to a plasma close to equilibrium and so that $f_{\\rm{aniso}}(\\mathbf{k})$ \nis either a Bose-Einstein $n_{B} (k)$ or a Fermi-Dirac\n$n_{F} (k)$ distribution function. This may be true because by the \ntime quarkonia have been formed in the plasma medium from the\n$Q \\bar Q$ pairs produced at very early stages of the collision, the system \nmay not be then highly anisotropic \nrather closer to isotropic distribution. In the limit of small anisotropy,\nanisotropy parameter $ \\xi $ is related to the shear \nviscosity-to-entropy density ($\\eta\/s$) through the one-dimensional Navier \nStokes formula by\n\\begin{equation}\n\\xi= \\frac{10}{T \\tau}~\\frac{\\eta}{s},~                 \n\\end{equation}                                   \nwhere $1\/\\tau$ denotes the expansion rate of the fluid element.\nHowever, the degree of momentum-space anisotropy is generically \ndefined by the parameter,\n\\begin{equation}\n\\xi = \\frac{\\langle \\mathbf{k}_{T}^{2}\\rangle}{2\\langle k_{L}^{2}\\rangle}-1~,~\n\\label{anparameter}\n\\end{equation}\nwhere $ {k}_{L}= \\mathbf{k}.\\mathbf{n} $ and ${\\bf k}_T = \n\\mathbf{k}-\\mathbf{n}(\\mathbf{k}.\\mathbf{n}) $ are the components of momentum\nparallel and perpendicular to the direction of \nanisotropy, $\\mathbf{n}$, respectively.\nThe positive and negative values of $ \\xi$ corresponds to the \nsqueezing and the stretching of the distribution function in the direction of \nanisotropy, respectively. In the relativistic nucleus-nucleus \ncollisions, $\\xi$ is however, found to be positive.\nThe calculation of the real part of the potential at finite  anisotropy was \nfirst obtained in Ref. \\cite{Laine2,Romatschke,Dumitru08} and  was later \nextended to calculate the imaginary \npart~\\cite{Brambilla08,beraudo,Laine09,Dumitru09} which is seen as a generic\nfeature of the medium. \nTo study the effect of anisotropy on the in-medium potential, one need \nto calculate first the self-energy in an anisotropic medium. With the specified anisotropic \ndistribution function, we can compute the gluon self-energy \nanalytically~\\cite{Thoma62}. \nWe will restrict our consideration to the spatial part of the self-energy, $\\Pi^{\\mu\\nu} $ \nfor  simplicity and the time-like components can be easily obtained \nby using the symmetry and transversality of the gluon self-energy tensor.\nThe spatial components of the retarded self-energy tensor reads~ \\cite{Dumitru08}\n\\begin{equation}\n\\Pi^{ij}(P)= -g^{2}\\int d^{3}\\mathbf{k}~v^{i}\\frac{\\partial f(\\mathbf{k})}{\\partial \nk^{l}}\\bigg(\\delta^{jl}+\\frac{v^{j}p^{l}}{ P\\cdot V+i\\epsilon}\n\\bigg) \\quad,\n\\label{spatialselfeenrgy}\n\\end{equation}\nwhere $P^{\\mu}\\equiv(p_0,p)$ is the four momentum of the external gluon, \n$p=|{\\bf p}|=$ is the amplitude of the spatial momentum. The four-velocity,\n$V^\\mu$ $(1,{\\bf v}={\\bf k}\/|\\bf k|)$ is a light-like four vector\nwith $v=|{\\bf v}|$ and the partial-derivative, \n$\\partial f \/\\partial k^l$, in terms of new variable $\\tilde{k}$, is given by \n\\begin{equation}\n\\frac{\\partial f(\\mathbf{k})}{\\partial k^{l}}= \\frac{v^{l}+\\xi(\\mathbf{v} .\\mathbf{n})n^{l}}\n{\\sqrt{1+\\xi(\\mathbf{v} .\\mathbf{n})^{2}}}\\frac{\\partial f(\\tilde {k}^{2})}{\\partial\\tilde {k}},\n\\end{equation}\nwhere $\\tilde{k}^{2}= k^{2}(1+\\xi(\\mathbf{v} .\\mathbf{n})^{2})$.  After \nintegrating out over the modified momentum $\\tilde {k}$, the \nself-energy, $\\Pi_{ij}$ is simplified into~\\cite{Dumitru08}\n\\begin{equation}\n\\Pi^{ij}(P)= m_{{}_D}^{2}\\int \\frac{d\\Omega}{4\\pi}v^{i}\\frac{v^{l}+\\xi(\\mathbf{v} .\n\\mathbf{n})n^{l}}{(1+\\xi(\\mathbf{v} .\\mathbf{n})^{2})^{2}}\\bigg(\\delta^{jl}+\\frac{v^{j}p^{l}}{P\\cdot\n V+ i \\epsilon}\\bigg) \\quad,\n\\end{equation}\nwhere the square of the Debye mass is defined by\n\\begin{equation}\nm_{{}_D}^{2}= -\\frac{g^{2}}{2\\pi^{2}}\\int\\limits_{0}^{\\infty} dkk^{2}\\frac{d f_{iso}\n(k^{2})}{dk}.\n\\label{deb}\n\\end{equation}\nUnlike in isotropic medium, the self-energy $\\Pi^{\\rm{ij}}$ shows an \nextra dependence on the preferred anisotropic \ndirection ($\\mathbf{n}$), therefore, it can no longer be decomposed \ninto transverse and longitudinal parts rather it becomes a tensor \n\\cite{Romatschke,Strickland,Strick04} with more basis vectors. \nSo the self-energy tensor \nbe decomposed into four structure functions as~\\cite{Romatschke}\n\\begin{equation}\n\\Pi^{ij}= \\alpha A^{ij}+ \\beta B^{ij} + \\gamma C^{ij} + \\delta D^{ij}~,~\n\\end{equation}\nwhere the coefficients $\\alpha$, $\\beta$, $\\gamma$ and $\\delta$ can be \ndetermined for any value of $\\xi $~\\cite{Romatschke}. \nUsing these structure functions, one can find the gluon propagator in temporal axial gauge as~\\cite{Romatschke}\n\\begin{eqnarray}\n\\Delta_{ij}(\\omega,k)&=&\\frac{(A_{ij}-C_{ij})}{k^{2}-\\omega^{2}+\\alpha}\n+\\frac{(k^{2}- \\omega^{2}+\\alpha +\\gamma)B_{ij}}{(k^{2}-\\omega^{2}+\\alpha +\\gamma)\n(\\beta-\\omega^{2})-k^{2}{\\tilde n}^{2}\\delta^{2}}\\nonumber\\\\\n&&+\\frac{(\\beta-\\omega^{2})C_{ij}-\\delta D_{ij}}{(k^{2}-\\omega^{2}+\\alpha\n +\\gamma)(\\beta-\\omega^{2})-k^{2}{\\tilde n}^{2}\\delta^{2}}\n\\end{eqnarray}\nIn order to see how the anisotropy affects the\nresponse to static electric field, we examine the\npropagator in the  static limit ($\\omega\\rightarrow 0$). Defining the masses\nin the static limit~\\cite{Romatschke}\n\\begin{eqnarray}\nm_{\\alpha}^{2}&=&\\lim_{\\omega \\to 0}\\alpha,~~\nm_{\\beta}^{2}=\\lim_{\\omega \\to 0}-\\frac{k^{2}}{\\omega^{2}}\\beta \\nonumber\\\\\nm_{\\gamma}^{2}&=&\\lim_{\\omega \\to 0}\\gamma,~~\nm_{\\delta}^{2}=\\lim_{\\omega \\to 0}\\frac{\\tilde n k^{2}}{\\omega}{\\rm Im} \\ \\delta \\quad,\n\\label{mminus}\n\\end{eqnarray}\nthe gluon propagator becomes \n\\begin{eqnarray}\n\\label{epsaniso}\n\\lim_{\\omega \\rightarrow 0} \\Delta_{ij}(\\omega,k)=-\\frac{(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})\\ k_ik_j}\n{\\omega^2\\left[(k^{2}+ m_{\\alpha}^{2}+m_{\\gamma}^{2})(k^{2}+m_{\\beta}^{2})-\nm_{\\delta}^{4}\\right]}~.\n\\end{eqnarray}\nWe can now factorize the denominator of the gluon propagator as \n\\begin{eqnarray}\n(k^{2}+ m_{\\alpha}^2+m_{\\gamma}^2)(k^2+m_\\beta^2)- \nm_{\\delta}^4=(k^2+m_+^2) (k^2+m_-^2)\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n2 m_\\pm^2=M^2 \\pm \\sqrt{M^4-4(m_\\beta^2(m_\\alpha^2+m_\\gamma^2)-m_\\delta^4)}\\ \n,\\ \\ M^2=m_\\alpha^2+m_\\beta^2+m_\\gamma^2~.\n\\end{eqnarray}\nThus the dielectric permittivity for an anisotropic medium in the temporal \naxial gauge can be obtained from the definition~\\cite{kapusta}\n\\begin{eqnarray}\n\\epsilon^{-1}(k)=-\\lim_{\\omega \\to 0} {\\omega^2\\frac{k_ik_j}{k^2}\\Delta_{ij}(\\omega,k)}\n=\\frac{k^{2}(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})}\n{(k^2+ m_+^2)(k^{2}+ m_-^2)}~,\n\\label{permittivity}\n\\end{eqnarray}\nIf the anisotropy is small, the pole masses (\\ref{mminus}) can be simplified, \nby retaining only the linear term in $\\xi$, into~\\cite{Romatschke} \n\\begin{eqnarray}\nm_\\alpha^2&=&-\\frac{\\xi}{6}(1+\\cos 2 \\beta_n)~m_{{}_D}^2\\nonumber\\\\\nm_\\beta^2&=&\\left( 1+\\frac{\\xi}{6}(3\\cos 2 \\beta_n-1)\\right)~m_{{}_D}^2\\nonumber\\\\\nm_\\gamma^2&=&\\frac{\\xi}{3} \\sin^2\\beta_n~m_{{}_D}^2\\nonumber\\\\\nm_\\delta^2&=&-\\xi\\frac{\\pi}{4}\\sin \\beta_n \\cos\\beta_n~m_{{}_D}^2\n\\end{eqnarray}\nwhere $\\beta_n$ is the angle between ${\\bf k }$ and ${\\bf n}$ .\nSo the explicit dependencies of $m_\\pm$ on the anisotropy, in the small\n$\\xi$ limit, are given by\n\\begin{eqnarray}\nm_+^{2}&=&\\left(1+\\frac{\\xi}{6}(3\\ \\cos2\\beta_n-1)\\right)m_{{}_D}^{2},\\nonumber\\\\\nm_-^{2}&=&-\\frac{\\xi}{3}\\ \\cos2\\beta_n m_{{}_{D}}^{2}.\n\\label{mplus}\n\\end{eqnarray}\nIn the isotropic limit, all masses become zero except\none ($m_\\alpha^2 = m_\\gamma^2= m_\\delta^2= m_-^2=0, m_+^2=m_{{}_D}^2$),\nwhich is the only pole in the isotropic medium (\\ref{eqn2}).\n\\subsubsection{Medium-modification to heavy quark potential}\nOnce we have obtained the dielectric permittivity in anisotropic \nmedium (\\ref{permittivity}), we substitute it in the Fourier\ntransform (\\ref{eq1}) and then evaluate its inverse Fourier transform\nto obtain the medium modified\npotential in an anisotropic medium:\n\\begin{eqnarray}\nV(\\mathbf{r},\\xi,T)&=&\\frac{1}{{(2\\pi)}^{3\/2}}\n\\int d^{3}\\mathbf{k}~\\tilde{V}(k)~e^{i\\mathbf{k} \\cdot \\mathbf{r}}\\nonumber\\\\\n&=& -\\frac{\\alpha}{2\\pi^{2}}\\int d^{3}\\mathbf{k} \n~\\frac{(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})}{(k^{2}+m_+^2)\n(k^{2}+m_-^{2})}~e^{i\\mathbf{k} \\cdot \\mathbf{r}}\n\\nonumber\\\\\n&& -\\frac{4\\sigma}{(2\\pi)^{2}}\\int d^{3}\\mathbf{k}~ \n \\frac{(k^{2}+m_{\\alpha}^{2}+m_{\\gamma}^{2})}{k^{2}(k^{2}+m_+^{2})\n(k^{2}+m_-^{2})}~e^{i\\mathbf{k} \\cdot \\mathbf{r}}\n\\label{modipot}\n\\end{eqnarray}\nAfter substituting the pole masses, $m_+$ and $m_-$ (in the small $\\xi$ limit) from (\\ref{mplus}), the potential becomes\n\\begin{eqnarray}\nV({\\bf r},\\xi,T)&=&\\!\\!\\!\\!-\\frac{\\alpha}{2\\pi^{2}}\\int \\frac{d^{3}\\bf{k}  \n~e^{i\\bf{k} \\cdot\\bf{r}}}{\nk^2+m_{{}_D}^2\\left(1+\\frac{\\xi}{6}(3\\cos2\\beta_n-1)\\right)\n} \\nonumber\\\\\n&&\\!\\!\\!\\!-\\frac{4\\sigma}{(2\\pi)^{2}}\n\\int \\frac{d^{3}\\mathbf{k}~e^{i\\mathbf{k} \\cdot \\mathbf{r}}}\n{k^{2}\\left[k^{2}+m_{{}_D}^{2}\\left(1+\\frac{\\xi}{6}(3\\cos2\\beta_n-1)\\right)\\right]}\\nonumber\\\\\n&\\equiv &\\!\\!\\!\\!V_1({\\bf r},\\xi,T)+V_2({\\bf r},\\xi,T)\n\\label{v1}\\ ,\n\\end{eqnarray}\nwhere $V_1 ({\\bf r},\\xi,T)$ and $V_2(\\mathbf{r},\\xi,T)$ are the \nmedium-modified potential corresponding to short-distance Coulombic and \nlong-distance string term, respectively, can be rewritten as \n\\begin{eqnarray}\nV_1({\\bf r},\\xi,T) =-\\frac{\\alpha}{2\\pi^{2}}\\int d^{3}\\mathbf{k} \ne^{i\\mathbf{k} \\cdot \\mathbf{r}}\\frac{1}{\\left( k^{2}+{m_{D}}^{2}\\right)}\\left(1+\\frac{\\xi}{6}~\\frac{{m_{D}}^{2}}{\\left( k^{2}+{m_{D}}^{2}\\right)}~(3\\cos2\\beta_n-1)\\right)^{-1}\n\\end{eqnarray}\nExpanding the integrand in terms of the anisotropy parameter ($\\xi$) and \nretaining the term linear in $\\xi$ (weak-anisotropy limit, $\\xi<1$),\n$V_1({\\bf r},\\xi,T)$ can be written as\n\\begin{eqnarray}\nV_1({\\bf r},\\xi,T) &=&-\\frac{\\alpha}{2\\pi^{2}}\\int d^{3}\\mathbf{k} \ne^{i\\mathbf{k} \\cdot \\mathbf{r}}\\left[ \\frac{1}{\\left( k^{2}+{m_{D}}^{2}\\right)}-\\frac{\\xi}{6}~\\frac{{m_{D}}^{2}}{\\left( k^{2}+{m_{D}}^{2}\\right)^{2}}~(3\\cos 2\\beta_n-1)\\right]\\nonumber\\\\\n&\\equiv & V_1^{(1)}({ r},\\xi =0,T)+ V_1^{(2)}({\\bf r},\\xi,T)\n\\end{eqnarray}\nwhere $V_1^{(1)}({ r},\\xi=0,T)$ and $V_1^{(2)}({\\bf r},\\xi,T)$ are \nthe isotropic and anisotropic contributions due\nto the medium modification of the Coulomb term, respectively.\nSimilarly $V_2({\\bf r},\\xi,T)$ can be decomposed into isotropic and anisotropic\nparts :\n\\begin{eqnarray}\nV_2({\\bf r},\\xi,T)=&V_2^{(1)}({r},\\xi =0,T)+ V_2^{(2)}({\\bf r},\\xi,T)\n\\end{eqnarray}\nwhere $V_2^{(1)}({r},\\xi=0,T)$ and $V_2^{(2)}({\\bf{r}},\\xi,T)$ are the\nisotropic and anisotropic contributions due\nto the medium modification of the linear term, respectively.\nLet us now calculate them one-by-one. \n\nThe isotropic part, $V_1^{(1)}({ r},\\xi=0,T)$ of\nthe Coulomb term (already calculated in (\\ref{defn}))\nis given by ($\\hat{r}=r~m_D$)\n\\begin{eqnarray}\nV_1^{(1)}({r}, \\xi =0,T)\n=-\\frac{\\alpha m_{{}_D}}{\\hat{r}}e^{-\\hat{r}} -\\alpha m_{{}_D}\\ ,\n\\label{col1}\n\\end{eqnarray}\nand the anisotropic part, $V_1^{(2)}({\\bf r},\\xi,T)$ is given by \n\\begin{eqnarray}\nV_1^{(2)}({\\bf r},\\xi,T) = -\\xi~\\frac{\\alpha~ m_{{}_D}^2}{2 \\pi^2}\\int \n\\frac{d^3\\mathbf{k} e^{i \\mathbf{k} \\cdot \\mathbf{r}}}{(k^2+m_{{}_D}^2)^2}\n~\\left(\\frac{2}{3}-\\cos^2\\beta_n\\right). \n\\end{eqnarray}\nOne immediately observes that unlike the isotropic part $V_1^{(1)}(r, \\xi \n=0,T)$, momentum anisotropy ($\\xi \\ne 0$) causes the potential to depend\non angle, in addition to inter-particle distance (r).\nBefore deriving a general angular dependence we first illustrate the two \ncases which are especially interesting to grasp the effect of \nanisotropy on the heavy-quark potential.\nHowever they will be used further to derive the general angular dependence.\nFirst, we consider that $ \\mathbf{r} $ is parallel to the \ndirection of anisotropy, $\\mathbf{n}$, where we have taken the \ndirection of anisotropy $\\mathbf{n}$ along the $z$-axis and \nthe angle between the $r$ and $\\bf k$ is $\\theta$. \nSo the anisotropic part, $ V_1^{(2)}({\\bf r},\\xi,T)$ for the medium\nmodification to the Coulomb term becomes\n\\begin{eqnarray}\n V_1^{(2)}({\\bf r}||{\\bf n},\\xi,T)&=&-\\xi~\\frac{\\alpha m_{{}_D}^2}{\\pi}~\n~\\int \\frac{d^3 \\mathbf{k}~e^{i{\\bf k} \\cdot {\\bf r}}}{(k^2+m_{{}_D}^2)^2}~ \n\\left(\\frac{2}{3}-\\cos^2\\theta\\right).\n\\end{eqnarray}\nAfter the angular integration, it becomes simplified as\n\\begin{eqnarray}\n V_1^{(2)}({\\bf r}||{\\bf n},\\xi,T)\n&=& \\xi~V_1^{(1)}({ r},\\xi=0,T)\\left(2~\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\n\\frac{2}{\\hat{r}}-\\frac{\\hat{r}}{6}-1\\right).\n\\label{col2}\n\\end{eqnarray}\nThus, the complete in-medium modification \nto the Coulomb term (Eqs. \\ref{col1} and \\ref{col2}), for the parallel alignment \nbecomes\n\\begin{eqnarray}\n V_1({\\bf r}||{\\bf n}, \\xi,T) &=&V_1^{(1)}({ r},\\xi=0,T)+ \nV_1^{(2)} ({\\bf r}||{\\bf n},\\xi,T) \\nonumber\\\\\n&=&-\\frac{\\alpha m_{{}_D}}{\\hat{r}}e^{-\\hat{r}}-\\alpha m_{{}_D}-\\xi~\n\\frac{\\alpha m_{{}_D}}{\\hat{r}}~e^{-\\hat{r}}\\left(2\n\\frac{e^{\\hat r}-1}{{\\hat r}^2}-\\frac{2}{\\hat r}-\\frac{\\hat r}{6}-1\\right).\n\\label{col}\n\\end{eqnarray}\nNext we consider the other scenario, i.e.  when $ \\mathbf{r} $ is \ntransverse to the direction of anisotropy, ${\\bf n}$ \nwhere we take {\\bf r} along the $z$-axis and {\\bf n} lying in the $x$-$y$ plane, so\n$\\phi$ is the azimuthal angle and $\\phi_n$ is the angle between \n$\\mathbf{n}$ with $x$-axis.  Then $V_1^{(2)}({\\bf r} \\perp {\\bf n},\\xi,T)$ \nbecomes\n\\begin{eqnarray}\nV_1^{(2)}({\\bf r \\perp n},\\xi,T)\\!\\!\\!\\!\\!&=&\\!\\!\\!\\!\\!-\\xi\n~\\frac{\\alpha m_{{}_D}^2}{2\\pi^2}\\int \n\\frac{d^3\\mathbf{k}~e^{i {\\bf k} \\cdot {\\bf r}} }{(k^2+m_{{}_D}^2)^2}~\n\\left(\\frac{2}{3}-\\cos^2\n\\left(\\phi-\\phi_n\\right)\\sin^2\\theta\\right).\n\\end{eqnarray}\nAfter the angular integration (exploiting the cylindrical symmetry), it \nis simplified into\n\\begin{eqnarray}\nV_1^{(2)}({\\bf r \\perp n},\\xi,T)=\\xi V_1^{(1)}({ r},\\xi=0,T)\\left(\\frac{1-e^{\\hat{r}}}{{\\hat{r}}^2}\n+\\frac{1}{\\hat{r}} +\\frac{\\hat{r}}{3}+\\frac{1}{2}\\right)\n\\label{colperp}\n\\end{eqnarray}\nThus, the complete in-medium modification \nto the Coulomb term (Eqs. \\ref{col1} and \\ref{colperp}), for the transverse \nalignment becomes\n\\begin{eqnarray}\nV_1({\\bf r\\perp n},\\xi, T) &=& V_1^{(1)}({ r},\\xi=0,T)\n        + V_1^{(2)}({\\bf r\\perp n},\\xi,T) \\nonumber\\\\\n      &=&-\\frac{\\alpha m_{{}_D}}{\\hat{r}}e^{-\\hat{r}}-\\alpha m_{{}_D}\n       +\\xi~\\frac{\\alpha m_{{}_D}}{\\hat{r}} e^{-\\hat{r}}\n       \\left(\\frac{e^{\\hat{r}}-1}{{\\hat{r}}^2}-\\frac{1}{\\hat{r}}\n      -\\frac{\\hat{r}}{3} -\\frac{1}{2}\\right).\n\\label{couperp}\n\\end{eqnarray}\nNext we calculate the in-medium-modification to the linear term, \n$V_2({\\bf r},\\xi,T)$ where the isotropic part (from (\\ref{defn})) is given by \n\\begin{equation}\nV_2^{(1)}({r},\\xi =0,T)=\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\ne^{-\\hat{r}}-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}+\\frac{2\\sigma}{m_{{}_D}}\n\\label{lin1}\n\\end{equation}\nand the anisotropic contribution, when $ \\mathbf{r} $ is parallel to the \ndirection of anisotropy $\\bf n$, is given by\n\\begin{eqnarray}\nV_2^{(2)}({\\bf r || n},\\xi,T) &=&-\\xi~\\frac{4\\sigma}{(2\\pi)^2}\\int \n\\frac{d^3\\mathbf{k}~e^{i \\mathbf{k} \\cdot \\mathbf{r}}}{k^2~(k^2+m_{{}_D}^2)^2}\n~\\left(\\frac{2}{3}-\\cos^2\\beta_n\\right) .\n\\end{eqnarray}\nAfter the angular integration, it becomes\n\\begin{eqnarray}\nV_2^{(2)}({\\bf r || n},\\xi,T)\n&=& -\\xi \\frac{4\\sigma}{m_{{}_D} \\hat{r}}{e^{-\\hat{r}}}\\left(2\\frac{(1-e^{\\hat r})}{\\hat r^2}+\n\\frac{e^{\\hat r}+2}{3} +\\frac{2}{\\hat r}+\\frac{\\hat {r}}{12}\\right)\n\\label{lin2}\n\\end{eqnarray}\nThus, the complete in-medium modification to the linear term (Eqs. \\ref{lin1} \nand \\ref{lin2}), for $ \\mathbf{r} \\ || \\mathbf{n} $ becomes\n\\begin{eqnarray}\nV_2({\\bf r|| n}, \\xi, T) &=&\nV_2^{(1)}({r},\\xi=0,T)+ V_2^{(2)}({\\bf r|| n},\\xi,T)\n\\nonumber\\\\\n      &=&-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\n   + \\frac{2\\sigma}{m_{{}_D} \\hat r} e^{-\\hat{r}}+\\frac{2\\sigma}{m_{{}_D}}\\nonumber\\\\\n     &+&\\xi~\\frac{4\\sigma}{m_{{}_D} \\hat{r}}{e^{-\\hat{r}}}\\left(2~\n\\frac{e^{\\hat{r}}-1}{{\\hat{r}}^2}-\\frac{e^{\\hat{r}}+2}{3} -\\frac{2}{\\hat{r}}\n-\\frac{\\hat{r}}{12}\\right)\n\\label{lin}\n\\end{eqnarray}\nOn the other hand, when $ \\mathbf{r} $ is transverse to the direction of anisotropy,\n$\\mathbf{n}$, the anisotropic contribution to the linear term becomes\n\\begin{eqnarray}\nV_2^{(2)}({\\bf{r} \\perp\\bf{n}},\\xi,T) &=&-\\xi~\\frac{4\\sigma}{(2\\pi)^2}\\int \n\\frac{d^3\\mathbf{k}~e^{i \\mathbf{k} \\cdot \\mathbf{r}}}{k^2~(k^2+m_{{}_D}^2)^2}\n~ \\left(\\frac{2}{3}-\\cos^2 \\left(\\phi-\\phi_n\\right)\\sin^2\\theta\\right), \n\\end{eqnarray}\nwhich is simplified into\n\\begin{eqnarray}\nV_2^{(2)}({\\bf{r} \\perp\\bf{n}},\\xi,T) &=&\n-\\xi ~\\frac{4\\sigma}{m_{{}_D} \\hat{r}}{e^{-\\hat{r}}}\\left(\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n+\\frac{(e^{\\hat r}-7)}{12}-\\frac{1}{\\hat r}-\\frac{\\hat r}{6}\\right) ,\n\\label{linp}\n\\end{eqnarray}\nafter the angular integration. Thus, the complete in-medium modification to the linear term \n(Eqs. \\ref{lin1} and \\ref{linp})\nfor $ \\mathbf{r} \\perp \\mathbf{n} $ becomes\n\\begin{eqnarray}\nV_2({\\bf{r}\\perp\\bf{n}}, \\xi, T) &=&\nV_2^{(1)}({r},\\xi=0,T)+ V_2^{(2)}({\\bf r\\perp\\bf{n}},\\xi,T)\n\\nonumber\\\\\n&=&-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\n+ \\frac{2\\sigma}{m_{{}_D} \\hat{r}} e^{-\\hat{r}}+\\frac{2\\sigma}{m_{{}_D}}\\nonumber\\\\\n&-&\\xi ~\\frac{4\\sigma}{m_{{}_D}^2 r}{e^{-m_{{}_D} r}}\\left(\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n+\\frac{(e^{\\hat r}-7)}{12}-\\frac{1}{\\hat r}-\\frac{\\hat r}{6}\\right).\n\\label{linperp}\n\\end{eqnarray}\n\\par Therefore, the full medium-modified potential, consisting of Coulomb\nand linear term (Eqs. \\ref{col} and \\ref{lin}, respectively), for the\nquark pairs aligned to the direction of anisotropy, yields as \n\\begin{eqnarray}\nV({\\bf{r} \\parallel \\bf{n}},\\xi,T)\n&=& \\left(\\frac{2\\sigma}{m_{{}_D}}-\\alpha m_{{}_D} \\right)\\frac{e^{-\\hat r}}{\\hat{r}}\n-\\frac{2\\sigma}{m_{{}_D} \\hat{r}}\n+\\frac{2\\sigma}{m_{{}_D}}- \\alpha m_{{}_D} \\nonumber\\\\\n&+& \\xi~\\left[ \\frac{4\\sigma}{m_{{}_D} \\hat{r}}~e^{-\\hat r} \\left( {} 2~\\frac{e^{\\hat r}-1}\n{\\hat r^2}-\\frac{e^{\\hat r}+2}{3} -\\frac{2}{\\hat r}-\\frac{\\hat r}{12} \\right) \\right. \\nonumber\\\\\n&-&\\left. \\frac{\\alpha m_{{}_D} }{\\hat{r}} e^{-\\hat r}\\left( {} 2~\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n-\\frac{2}{\\hat r}-\\frac{\\hat r}{6} -1\\right) \\right] \\nonumber\\\\\n&\\equiv & V_{\\rm{iso}} (r,T)+ V_{\\rm{aniso}}^\\parallel (r,\\xi,T).\n\\label{fullpotpara}\n\\end{eqnarray}\nIn the short distance limit ($r \\ll 1\/m_{{}_D}$), the potential \nreduces to the vacuum potential (Cornell) for $\\xi=0$, {\\it i.e.} \n$Q\\bar Q$ pairs are not affected by \nthe medium. On the other hand, in the long-distance limit\n($r\\gg 1\/m_{{}_D}$) (where the screening occurs), we can \nneglect the Yukawa term and for large values of temperatures, the product\n$\\alpha m_{{}_D}$ will be much greater than $2\\sigma\/m_{{}_D}$. Thus\nthe potential is simplified into the following form:\n\\begin{eqnarray}\nV({\\mathbf r} \\parallel {\\mathbf n},\\xi,T) &\\stackrel{\\hat{r}\\gg 1}{\\simeq} & -\\frac{2\\sigma}{m_{{}_D}^{2}r} - \n\\alpha m_{{}_D} -\\frac{4\\xi}{6}~\\left(\\frac{2\\sigma}{m_{{}_D}^{2}r} \\right) \\nonumber\\\\\n&\\stackrel{\\hat{r}\\gg 1}{\\equiv} & V_{\\rm{iso}} (\\hat{r} \\gg 1,T)+ V_{\\rm{aniso}}^\\parallel\n(\\hat{r} \\gg 1,\\xi,T),\n\\label{paral}\n\\end{eqnarray}\nwhich clearly shows that the potential for $Q \\bar Q$ pairs aligned in the \ndirection of anisotropy gets screened less, {\\em i.e.} becomes stronger \ncompared to isotropic medium (\\ref{lrp}).\n\n\\noindent On the other hand, when the quark pairs are aligned transverse to the \ndirection of anisotropy ($ \\mathbf{r}  \\perp \\mathbf{n} $), \nthe medium modification to the Coulombic and linear terms together \n(Eqs. \\ref{couperp} and \\ref{linperp}, respectively) gives rise to \nthe following form:\n\\begin{eqnarray}\nV({\\bf r\\perp {\\bf n}},\\xi,T)&=& \n\\left(\\frac{2\\sigma}{m_{{}_D} }-\\alpha m_{{}_D} \\right) \n\\frac{e^{-\\hat r}}{\\hat{r}}-\\frac{2\\sigma}{m_{{}_D}\\hat{r}}  \n+\\frac{2\\sigma}{m_{{}_D}}- \\alpha m_{{}_D} \\nonumber\\\\\n&-&\\xi~\\left[ \\frac{4\\sigma}{m_{{}_D} \\hat{r}}~{e^{-\\hat{r}}} \n\\left(\\frac{(e^{\\hat r}-1)}{\\hat r^2}\n+\\frac{(e^{\\hat r}-7)}{12}-\\frac{1}{\\hat r}-\\frac{\\hat r}{6}\\right) \\right.\n\\nonumber\\\\\n&-& \\left. \\frac{\\alpha m_{{}_D}}{\\hat{r}} e^{-\\hat{r}}\n\\left(\\frac{e^{\\hat r}-1}{\\hat r^2}-\\frac{1}{\\hat r}-\\frac{1}{2}\n-\\frac{\\hat r}{3}\\right) \\right] \\nonumber\\\\\n&\\equiv & V_{\\rm{iso}} (r,T)+ V_{\\rm{aniso}}^\\perp (r,\\xi,T),\n\\label{fullpotperp}\n\\end{eqnarray}\nSimilarly, $Q \\bar Q$ pair does not see the medium, in the short-distance \nlimit whereas in the long-distance limit, the \npotential is simplified into a Coulombic form with\na dynamically screened color charge ($2\\sigma\/m_D^2$):\n\\begin{eqnarray}\nV({\\bf r} \\perp {\\bf n},\\xi,T)&\\stackrel{\\hat{r}\\gg1}{\\simeq}&-\\frac{2\\sigma}{m_{D}^{2}r}\n-\\alpha m_{D}-\\frac{\\xi}{6} \\left(\\frac{2\\sigma}{m_{D}^{2}r}\\right)\\nonumber\\\\\n&\\stackrel{\\hat{r}\\gg 1}{\\equiv} & V_{\\rm{iso}} (\\hat{r}\\gg 1,T)+ V_{\\rm{aniso}}^\\perp \n(\\hat{r}\\gg 1,\\xi,T),\n\\label{perp} \n\\end{eqnarray}\nwhich again shows that the potential for the transverse alignment is still\nstronger than in isotropic medium but less stronger than the former.\n\n\\par Until now we have demonstrated the effects of momentum anisotropy on\nthe heavy-quark interaction for the special cases {\\em viz} the inter-particle \nseparation (r) may be parallel or perpendicular to the direction of \nanisotropy (n). We would now like now to derive a potential which depends \non both the inter-particle separation (r) and the\nangle ($\\theta_n$) between ${\\bf r}$ and $n$, explicitly.\n\nLet us assume that ${\\bf r}$  is parallel to the z component of ${\\bf k}$\nand the direction of anisotropy $ {\\bf n} $ lies in the $ y$-$z$\nplane (cylindrical symmetry) in the momentum space.\nWe may further assume that given the weak anisotropy, the potential in \nthe anisotropic medium represents a perturbation to the central potential \nin the isotropic medium as : \n\\begin{eqnarray}\nV(r,\\theta_n, T)&=&V(r,T)+ V_{\\rm{tensor}}(r,\\theta_n,T)\\\\\n&=& V_{\\rm{iso}}(r,T)+\\xi F(r,\\theta_n,T).\n\\label{potgen}\n\\end{eqnarray}\nwhere the tensorial part $V_{\\rm{tensor}}(r,\\theta_n,T)$ represents a small\nperturbation to the central one $V(r;T)$ and $\\xi$ is the strength of \nnon-central component of the potential. The function $F(r,\\theta_n,T)$ can \nbe expanded as\n\\begin{equation}\nF(r,\\theta_n, T)=f_0(r,T)+f_1(r,T)\\cos 2\\theta_n~.\n\\end{equation}\nThe potentials for the angles $\\theta_n=0$ and $\\theta_n=\\pi\/2$ help us to \ndetermine the functions $ f_0(r,T) $ \nand $f_1(r,T)$ in terms\\footnote{The variable\n$\\hat{r}$ should not be confused with the usual notation of unit vector in\nthe coordinate system.} of $\\hat{r}$ (=$rm_D$) as\n\\begin{eqnarray}\nf_0(\\hat{r},T)&=&\\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}} \n\\left(\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{5 e^{\\hat{r}}}{12}-\n\\frac{1}{\\hat{r}}+\\frac{\\hat{r}}{12} -\\frac{1}{12}\\right)\\nonumber\\\\\n&-&\\frac{\\alpha m_{D}} {2}\\frac{e^{-\\hat{r}}}{\\hat{r}}\n\\left(\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{1} {\\hat{r}}+\n\\frac{\\hat{r}}{6}-\\frac{1}{2}\\right)\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nf_1(\\hat{r},T)&=&\\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}}\\left(\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{e^{\\hat{r}}}{4}-\n\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{4}-\\frac{5}{4}\\right)\\nonumber\\\\\n&-&\\frac{\\alpha m_{D}}{2} \\frac{e^{-\\hat{r}}}{\\hat{r}}\\left(\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{2}-\n\\frac{3}{2}\\right)\n\\end{eqnarray}\nSo after substituting $f_0$ and $f_1$ into the Eq.(\\ref{potgen}), we obtain\nthe complete angular dependence of the potential in \nthe limit of weak anisotropy ($\\xi \\ll 1$) :\n\\begin{eqnarray}\nV(r,\\theta_n,T) &=& \\left(\\frac{2\\sigma}{m_{{}_D} }-\\alpha m_{{}_D} \\right) \n\\frac{e^{-\\hat r}}{\\hat{r}}-\\frac{2\\sigma}{m_{{}_D}\\hat{r}}  \n+\\frac{2\\sigma}{m_{{}_D}}- \\alpha m_{{}_D} \\nonumber\\\\\n&+&  \\xi \\left( \\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}} \n\\left[\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{5 e^{\\hat{r}}}{12}-\n\\frac{1}{\\hat{r}}+\\frac{\\hat{r}}{12} -\\frac{1}{12}\\right] \\right. \\nonumber\\\\\n&-& \\left. \\frac{\\alpha m_{D}} {2}\\frac{e^{-\\hat{r}}}{\\hat{r}}\n\\left[\\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{1} {\\hat{r}}+\n\\frac{\\hat{r}}{6}-\\frac{1}{2}\\right] \\right. \\nonumber\\\\\n&+ & \\left( \\left. \\frac{2 \\sigma} {m_{D}}\\frac{e^{-\\hat{r}}}{\\hat{r}}\\left[\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{e^{\\hat{r}}}{4}-\n\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{4}-\\frac{5}{4}\\right] \\right.\\right.\\nonumber\\\\\n&-&\\frac{\\alpha m_{D}}{2} \\frac{e^{-\\hat{r}}}{\\hat{r}}\\left[ \\left.\\left.\n3 \\frac{e^{\\hat{r}}-1}{\\hat{r}^2}-\\frac{3}{\\hat{r}}-\\frac{\\hat{r}}{2}-\n\\frac{3}{2}\\right] \\right) \\cos~2 \\theta_n \\right) \\nonumber\\\\\n&=& V(r,T) + V_{\\rm{tensor}}(r,\\theta_n,T) \n\\label{fullp}\n\\end{eqnarray}\nThus the anisotropy in the momentum space introduces an angular \n($\\theta_n$) dependence, in addition to the inter-particle separation ($r$),\nto the potential in the coordinate space which was earlier only $r$-dependent \nin the isotropic medium. We can now identify the $\\xi$-independent \nterm with $V(r,T)$ in (50) which depends only on the separation ($r$) distance \nand the $\\xi$-dependent term with the tensorial component \n$V_{\\rm{tensor}}(r,\\theta_n,T)$ in (50) which depends on both \n$r$ and $\\theta_n$. Thus the full potential in an anisotropic \nmedium, $V(r, \\theta_n,T)$ needs to be solved by the three-dimensional \nSchr\\\"odinger equation. Since we are restricted in the small $\\xi$ limit, so \nthe tensorial component $V_{\\rm{tensor}}(r,\\theta_n,T)$ is much smaller than\nthe (isotropic) central component $V(r,T)$ and hence may be treated as the \nperturbation by a first-order perturbation theory in quantum mechanics.\nHowever, the isotropic component may be solved numerically by the \none-dimensional (radial) Schr\\\"odinger equation.\n\nThe heavy quark interaction at short and intermediate distances\n($rm_D \\le1$) are important for the understanding of in-medium\nmodification of heavy quark bound states and the large distance\nproperty ($rm_D>1$) helps to understand the bulk properties of\nQGP phase which also affect the in-medium properties\nof the quarkonium states. So we wish to \nto see how the potential in anisotropic medium behaves in these\n(short, intermediate and long) limiting cases.\nIn the short-distance limit, the vacuum contribution \ndominates over the medium contribution and this is exactly happens \nhere \n\\begin{eqnarray}\nV(r,\\theta_n,T)\\stackrel{\\hat{r}\\ll 1}{\\simeq} \\sigma r - \\frac{\\alpha}{r}\n\\label{vprime}\n\\end{eqnarray}                                        \nfor $\\xi=0$. On the other hand, in the long-distance limit ($\\hat{r}\\gg 1 $), the \npotential is reduced to a long-range Coulombic interaction after\nidentifying the factor $2\\sigma\/m_D^2$ with the coupling ($g_s^2$)\nof the interaction\n\\begin{eqnarray}\n\\label{largp}\nV(r,\\theta_n,T) &\\stackrel{\\hat{r}\\gg1}{\\simeq}& -\\frac{2\\sigma}{m^2_{{}_D}r}-\\alpha m_{{}_D}\n-\\frac{5\\xi}{12}~\\frac{2\\sigma}{m^2_{{}_D}r} \\left(1+\\frac{3}{5}\\cos 2\\theta_n \\right)\\nonumber\\\\\n&\\equiv & V_{\\rm{iso}} (\\hat{r} \\gg 1,T)+ V_{\\rm{tensor}} (\\hat{r} \\gg 1~.\n\\theta_n,T)~.\n\\end{eqnarray}\nSince the resulting potential is Coulombic plus a\nsubleading anisotropic contribution, it then has to satisfy\nthe condition: $a_0 m_D\\gg 1$, where $a_0$ is the Bohr radius and $m_D$ is\nthe Debye mass. Since the Bohr radius $a_0$ is proportional to \n$m_D^2\/(m_Q \\sigma)$,\nthe above condition for the long-distance limit implies that \n$m_D^3\/(m_Q \\sigma)$ \nshould be greater than 1. Thus this inequality results in a condition on the\nDebye mass and hence on the temperature. It is seen that the above condition \nis satisfied for the temperatures above the critical temperature ($>T_c$)\nfor the charmonium states and above 1.6$T_c$ \nfor the bottomonium states. The temperature ranges ($T_c$ and \n1.6$T_c$) for $c \\bar c$ and $b\\bar b$ states above which \nthe effective potential looks Coulombic are\nsmaller than their respective dissociation temperatures and thus seems \njustified to approximate the potential in the long-distance limit. In the intermediate distance ($rm_D \\simeq 1$) scale, the \ninteraction becomes complicated and thus the potential\ndoes not look simpler in contrast to the asymptotic limits, so\nthis limit needs to be dealt numerically with the full potential in\na Schr\\\"odinger equation.\n\n\\par We have thus noticed overall that in the short \ndistance limit, the potential have not been affected in the isotropic limit.\nOn the contrary, in the long-distance limit, the momentum anisotropy \ntranspires an angular dependence in the potential \nand gives rise a characteristic angular ($\\theta_n$) dependence \nbetween the relative \nseparation ($\\mathbf{r}$) and the direction of anisotropy ($\\mathbf{n}$). \nAs a corollary, the quark pairs aligned \nalong the direction of anisotropy feel more attraction than the transverse \nalignment because the inter-quark potential along\nthe direction of anisotropy is screened less than the transverse alignment.\nHowever, the potential in the anisotropic medium is always stronger than in \nisotropic medium. \n\n\nTo see the effects of anisotropy, we have shown the potentials \nfor $Q \\bar Q$ pairs in an anisotropic medium in Figures 1 and 2,\nfor $\\theta_n=0$ (parallel) and $\\theta_n=\\pi\/2$\n(perpendicular), respectively. The immediate observation\ncommon to all figures is that the inter-quark potential in anisotropic\nmedium is always more attractive than in isotropic medium. \nThis can be understood physically: In the small \nanisotropic limit, the anisotropic distribution function may be \nobtained from an isotropic distribution $f_{iso} (|\\mathbf{k}|)$\nby removing particles with a large momentum component along\n$\\mathbf{n}$ {\\it i.e.}\n$f_{iso}(\\sqrt{\\mathbf{k}^{2} + \\xi(\\mathbf{k}.\\mathbf{n})^{2}}$.\nThis transpires in the reduction of the number of partons (around\na static test heavy quark) than in isotropic medium {\\it i.e.} $n_{\\rm{aniso}} (\\xi)= \nn_{\\rm{iso}}\/\\sqrt{1+\\xi}$. Therefore, the (effective) Debye mass \nalways becomes smaller and results in less screening of the potential than \nin isotropic medium.\n\\begin{figure}\n\\begin{subfigure}{\n\\includegraphics[width=7.9cm,height=8.5cm]{para.eps}\n\\label{para_plot_a}}\n\\end{subfigure}\n \\hspace{-5mm}\n\\begin{subfigure}{\n \\includegraphics[width=7.9cm,height=8.5cm]{paraz1.eps}\n\\label{para_plot_b}}\n\\end{subfigure}\n\\caption{\\footnotesize The left panel represents the potential \ndivided by $(g^2 C_F m_{{}_D})$ and the right panel represents the \ncontribution of Coulomb, string  and both together as a function of \n$\\hat{r}$ (= $rm_{{}_D} $) for quark pairs parallel to the \ndirection of anisotropy, $\\mathbf{n}$.} \n\\end{figure}\n\n\n\\begin{figure}\n\\vspace{0mm}\n\\begin{subfigure}{\n\\includegraphics[width=7.9cm,height=8.5cm]{perp.eps}\n\\label{perp_plot_a}}\n\\end{subfigure}\n\\hspace{-5mm}\n\\begin{subfigure}{\n\\includegraphics[width=7.9cm,height=8.5cm]{perpz1.eps}\n\\label{perp_plot_b}}\n\\end{subfigure}\n\\caption{\\footnotesize The notations are same as in Figure 1\nbut for quark pairs perpendicular to the \ndirection of anisotropy, $\\mathbf{n}$.} \n\\vspace{0mm}\n\\end{figure}\nThe second observation is that the quark pairs aligned along ($\\theta_n=0$) the\ndirection of anisotropy are stronger than aligned perpendicular \n($\\theta_n=\\pi\/2$) to the direction of anisotropy because for the parallel alignment, the component of momentum to be removed is higher\nthan the transverse alignment so the distribution function \nfor the parallel alignment case contributing to the Debye mass is\nsmaller than the transverse alignment.\nHence the potential for parallel case will be screened less compared to \nthe transverse case. However the difference between the two scenario \nwill be not much different\nbecause the contributions to the Debye mass from the partons having higher \nmomenta are very small.\n \nTo understand the effect of linear term on the medium modified \npotential quantitatively, in addition to the Coulomb term, we have \nplotted separately the medium modifications to the linear term, \nthe Coulomb term and their sum in the right panels of \nFigure 1 and 2, for parallel and transverse case, respectively.\nMedium modification to the Cornell potential contains two parts: one is due to\nthe medium modifications of linear term ($\\sigma r$) and the other one is due \nto the medium modifications of\nCoulomb term. As usually done in the literature, medium \nmodification to the linear term does not arise because\nthe string tension was assumed to be \nzero~\\cite{mocsyprd,Satz,shro,Alb05} \nat or beyond deconfinement temperature~\\cite{Lusch}.\nSince string tension is found to be nonzero at $T_c$ rather it \napproaches zero much beyond $T_c$~\\cite{string1,string2,string3} and hence\nthe medium modification to the linear term may be\nnon-zero contribution to the potential even at temperatures beyond $T_c$, although it is very small. \nIn isotropic medium, medium modification to the linear term\nremains  positive up to 2-3 $T_c$, making the potential less attractive \ncompared to $T=0$. On contrast, in anisotropic case medium \nmodification to the linear term becomes negative\nand the overall full potential becomes more attractive.\n\nAs mentioned earlier we used the same screening scale for both the linear\nand Coulombic terms in our calculation which does not look plausible.\nIt would thus be interesting to see the effects of different scales for the \nCoulomb and linear pieces of the T=0 potential~\\cite{megiasind,megiasprd}.\nTo illustrate it graphically, we have compared our results with their results\n(in Figure 3) for the isotropic case. The difference in the large\ndistance limit arises due to the difference in the potential\nat infinity (-$\\sigma\/m_D$) so the \npotential in Ref.\\cite{megiasprd} is more attractive than our potential.\n\\begin{figure}\n\\vspace{0mm}\n\\includegraphics[width=7.9cm,height=8.5cm]{pot_emegias_our.eps} \n\\caption{\\footnotesize The dotted-circle represents the results from \nMegias et al. \\cite{megiasprd} where\ndifferent scales was used for linear and Coulomb terms separately whereas\nthe solid line represents our work.}\n\\vspace{0mm}\n\\end{figure}\n\n\\section{Properties of Quarkonium in an Anisotropic Medium}\\label{prop_q}\n\\subsection{Binding energy}\nTo understand the in-medium properties of the quarkonium states,\nwe need to model the heavy quark potential as a function of\ntemperature and solve the resulting Schr\\\"{o}dinger equation.\nThe potential thus obtained in anisotropic medium (\\ref{fullp}), \nin contrast to the (spherically symmetric) potential in isotropic medium,\nis non-spherical and so one cannot simply obtain the energy eigen values \nby solving the radial part of the Schr\\\"{o}dinger equation only because \nthe radial part is no longer sufficient due to \nthe angular dependence in the potential. Other way to understand\nis that because of the anisotropic\nscreening scale, the wave functions are no longer radially symmetric for\n$\\xi \\ne 0$.  So one has to solve the potential in anisotropic medium\nthrough the Schr\\\"odinger equation in three dimension. \nHowever, we have seen in the potential (55) that in the small $\\xi$-limit, \nthe spherically non-symmetric component\n$V_{\\rm{tensor}}(r,\\theta_n,T)$ is much smaller in comparison to spherically \nsymmetric (isotropic) component $V(r,T)$ and thus\ncan be treated as perturbation. This can be understood physically:\nThe tensorial (non-sphericity) nature\nof the potential in the co-ordinate space is arisen due to anisotropy \nin the momentum space. However, we are restricted to a plasma which\nis very much close to equilibrium because\nby the time quarkonium states are formed in the plasma\naround (1-2)$T_c$, the plasma becomes almost \nisotropized. Thus  this weak (momentum) anisotropy\n($\\xi \\ll 1$) transpires feeble angular dependence in the potential\nso the potential will be spherically abundant\nwith a tiny non-spherical component. So we could \ntreat the anisotropic component through the perturbation theory in quantum \nmechanics and the isotropic part should be\nhandled numerically by the one-dimensional radial\nSchr\\\"{o}dinger equation.\n\nThere are some numerical methods to solve the Schr\\\"odinger equation\neither in partial differential form (time-dependent) or eigen value form\n(time-independent\/stationary) by the\nfinite difference time domain method (FDTD) or matrix method, respectively.\nHowever, we choose the matrix method to solve the\nstationary Schr\\\"odinger equation with the isotropic part of the potential\n(55) in anisotropic medium. In this method, the Schr\\\"odinger equation\ncan be cast in a matrix form through a discrete basis, instead\nof the continuous real-space position basis spanned by the states\n$|\\overrightarrow{x}\\rangle$. Here the confining potential V is subdivided\ninto N discrete wells with potentials $V_{1},V_{2},...,V_{N+2}$ such that\nfor $i^{\\rm{th}}$ boundary potential, $V=V_{i}$ for $x_{i-1} < x < x_{i};~i=2, 3,..\n.,(N+1)$. Therefore for the existence of a bound state, there\nmust be exponentially decaying wave function\nin the region $x > x_{N+1}$ as $x \\rightarrow  \\infty $ and\nhas the form:\n\\begin{equation}\n\\Psi_{N+2}(x)=P_{{}_E} \\exp[-\\gamma_{{}_{N+2}}(x-x_{N+1})]+ \nQ_{{}_E} \\exp [\\gamma_{{}_{N+2}}(x-x_{N+1})] , \n\\end{equation}\nwhere, $P_{{}_E}= \\frac{1}{2}(A_{N+2}- B_{N+2})$,\n$Q_{{}_E}= \\frac{1}{2}(A_{N+2}+ B_{N+2}) $ and,\n$ \\gamma_{{}_{N+2}} = \\sqrt{2 \\mu(V_{N+2}-E)}$. The eigenvalues\ncan be obtained by identifying the zeros of $Q_{E} $.\n\n\nTherefore, the corrected energy eigen value comes\nfrom the solution of Schr\\\"odinger equation\nof the isotropic component $V_{\\rm{iso}}(r,T)$, using the \nabovementioned matrix method\nplus the first-order perturbation due to the anisotropic \ncomponent $V_{\\rm{aniso}}(r,\\theta;\\xi,T)$ (55) through\nthe quantum mechanical perturbation theory.\nThe variations of the binding energies with the \ntemperature are shown in figure~(\\ref{bind_jsi})\nfor $J\/\\psi$ and $\\Upsilon$ \nfor different values of anisotropy parameter $\\xi$,\nto see the effect of anisotropy on the binding energies compared to\nthe isotropic case.  \n\n\\begin{figure*}\n\\begin{subfigure}{\n\\includegraphics[trim = 1mm 1mm 1mm 1mm, clip,height=7cm,width=7cm]{charm.eps}\n}\n\\end{subfigure}\n\\hspace{5mm}\n\\begin{subfigure}{\n\\includegraphics[trim = 1mm 1mm 1mm 1mm, clip,height=7cm,width=7cm]{bot.eps}\n}\n\\end{subfigure}\n\\vspace{0mm}\n\\caption{\\footnotesize Variation of $J\/\\psi$ and $\\Upsilon$ binding \nenergy (in $GeV$) with the temperature in an anisotropic hot QCD medium.}\n\\label{bind_jsi}\n\\end{figure*}\n\nThere are mainly two observations: First, as the anisotropy increases, the binding of \n$Q \\bar Q$ pairs get\nstronger with respect to their isotropic counterpart because \nthe potential becomes deeper with the increase of anisotropy due to\nweaker screening. It seems \nthat the (effective) Debye mass $m_D(\\xi,T)$ in an anisotropic\nmedium is always smaller than in an isotropic medium. As a result\nthe screening of the Coulomb and string contribution are \nless accentuated and hence the quarkonium states \nbecome more stronger than in an isotropic medium.\nHowever, the effects of anisotropy on the \nexcited states are not \nso pronounced compared to the ground states\nbecause they are generically weakly bound.\nSecondly, there is\na strong decreasing trend with the temperature.\nThis is due to the fact that the screening becomes always stronger with the increase of\ntemperature, so the potential becomes weaker compared to $T=0$ and\nresults in early dissolution of quarkonia in the medium.\nOur results on the temperature dependence of the binding energies show\nan agreement with the similar variations in other\ncalculations~\\cite{Dumitru09}.\n\nIn our calculation, we use the Debye mass \n($m_{{}_D}^{\\rm L}=1.4 m^{\\rm LO}_{{}_D}$) obtained by fitting\nthe (color-singlet) free energy in lattice QCD~\\cite{mocsyprl} \nwhere both one and two-loop expression \n~\\cite{shro,shaung,ijmp} for coupling have been used to explore the\neffects of running coupling on the dissociation process.\n\nThus the study of the temperature dependence of the binding \nenergies are poised\nto provide a wealth of information about the dissociation\npattern of quarkonium states in an anisotropic thermal medium that can be\nused to determine the dissociation temperatures of different states\nin the next Section.\n\n\\subsection{Dissociation temperatures for heavy quarkonia}\\label{diss}\nDissociation of a two-body bound state in an thermal medium can be \nunderstood qualitatively: When the binding energy of a resonance state\ndrops below the mean thermal energy of a parton, \nthe state becomes feebly bound. The thermal\nfluctuation then can easily dissociate by exciting\nthem into the continuum.\nThe spectral function technique in potential models defines the dissociation\ntemperature as the temperature above which the quarkonium spectral\nfunction shows no resonance-like structures but the widths shown \nin spectral functions from current potential model calculations\nare not physical. The broadening of states with the increase in \ntemperature is not included in any of these models. \nIn Ref.\\cite{mocsyprl}, the authors\nargued that one need not to reach the binding energy ($E_{\\rm{bin}}$) to\nbe zero for the dissociation rather a weaker condition $E_{\\rm{bin}}<T$\ncauses a state weakly bound. In fact, when $E_{\\rm{bin}} \\simeq T$, the\nresonances have been broadened due to direct thermal activation, \nso the dissociation of the bound states may be expected to occur\nroughly around $E_{\\rm{bin}} \\simeq T$.\n\nUsing the binding energies calculated earlier in Sec.3.1, \nthe dissociation temperatures ($T_D$) (shown in Table 1)\nare found minimum for the isotropic case and increase\nwith the increase of anisotropy ($\\xi>0$) {\\em viz.} $J\/\\psi$ \nis dissociated at $1.38~T_c$ in an isotropic medium while in an \nanisotropic medium with the anisotropies $\\xi$=0.3 and 0.6, they will \nsurvive higher \ntemperatures, $1.41~T_c$ and $1.43~T_c$, respectively. Similarly \nthe dissociation temperatures of \n$\\Upsilon$ for $\\xi$=0.3 and 0.6 are $1.71~T_c$\nand $1.72~T_c$, respectively, corresponding to the value \n($1.70~T_c$) in an isotropic medium. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nState &$\\xi=0.0$ & $\\xi=0.3$ & $\\xi=0.6$\\\\\n\\hline\\hline\n$J \/ \\psi$ & 1.38& 1.41& 1.43  \\\\\n\\hline\n$\\Upsilon$ & 1.70 & 1.71 & 1.72 \\\\\n\\hline\n\\end{tabular} \n\\caption{\\footnotesize Dissociation temperatures ($T_D$)\nfor the quarkonium states with one-loop QCD coupling}\n\\label{tdpara1l}\n\\end{table}\n\\par Finally we wish to explore the effects of perturbative as well as\nnon-perturbative contributions on the dissociation of quarkonia states\nqualitatively in terms of the debye mass.\nInstead of lattice Debye mass ($m_D^L$), if we use the leading-order\nDebye mass ($m_D^{\\rm{LO}}<m_D^L$), the screening of the potential \nwill be much smaller and hence the binding energies ($1\/m_{{}_D}^4$) \nwill be enhanced substantially\nand results in the increase of the dissociation temperatures.\nOn the other hand, if we include the non-perturbative corrections \nof order $O(g^2T)$ and $O(g^3T)$ to the leading-order Debye mass~\\cite{kajantie},\nthe dissociation temperatures become unrealistically small\nwhich looks unfeasible. Thus, this study\nprovides us a handle to decipher the extent up to which and how much\nnon-perturbative effects should be incorporated into the Debye mass.\n\n\\section{Conclusions and Outlook}\\label{conclu}\nIn conclusion, we have studied the dissociation of quarkonia \nby correcting the full Cornell potential with a dielectric function \nembodying the effects of an weakly anisotropic medium \nwhere the in-medium modification causes less screening of the interaction\nand hence the potential gets stronger than in an isotropic medium. \nAnisotropy further introduces a characteristic\nangular ($\\theta$) dependence to the potential in the coordinate space, in addition to the \ninter-particle separation $r$, making it spherically non-symmetric \nand needs to be solved numerically by the three-dimensional\nSchr\\\"odinger equation. However, in the small $\\xi$-limit, the spherically \nnon-symmetric component is much smaller in comparison to spherically \nsymmetric component and can be treated in a perturbation theory and\nthe symmetric (isotropic) component is solved numerically by\none-dimensional radial Schr\\\"{o}dinger equation.\nSo the corrected binding energy is obtained by the direction-independent shift \ndue to the spherically non-symmetric component to the eigen values of \nspherically-symmetric potential.\n\nWe have observed that the quarkonia states are always more bound \nand as a consequence, they survive\nhigher temperature compared to the isotropic medium. \nOur results are found relatively higher compared to similar \ncalculation~\\cite{Dumitru08}, which\nmay be due to the absence of three-dimensional medium modification \nof the linear term in their calculation. In fact, one-dimensional Fourier\ntransform of the Cornell potential yields the similar form\nused in the lattice QCD in which one-dimensional color flux tube structure\nwas assumed~\\cite{dixit}.\nHowever, at finite temperature that may not be\nthe case since the flux tube structure may expand in more\ndimensions~\\cite{Satz}.\nTherefore, it would be better to consider the three-dimensional form of the medium\nmodified Cornell potential which has been done exactly in the present work.\n\n\n\nIn brief, $J \/ \\psi$ is found to be dissociated at 1.38 $T_c$ and $1.43~T_c $ \nfor $\\xi$=0 and 0.6, respectively\nwhereas the corresponding temperatures for the $\\Upsilon$ state are\n$1.70~T_c$ and $1.72~T_c$.\nMoreover we explore the effects of perturbative as well as\nnon-perturbative effects on the dissociation process qualitatively.\n{\\em For example}, the perturbative result of Debye mass \ngives much higher values of dissociation temperatures \nwhereas the inclusion of non-perturbative corrections to it\ngives unrealistically smaller values. It may be important to note that in \nthe weakly-coupled regime, the effects of (non-perturbative) terms {\\em viz.}\n$g^2T$, \n$g^3T$ etc. may be checked separately but in the strong-coupling regime, this \nmay not be possible because they are no longer uncoupled.\nThese findings envisage a basic question about the nature of dissociation \nof quarkonium in an anisotropic hot QCD medium.\n\nApart from the uncertainty of the correct form of the potential,\nthere is an arbitrariness in the definition of dissociation temperature. So,\nin future, we wish to investigate the dissociation \nthrough the decay width of quarkonium bound states calculated from the\nimaginary part of the potential because it is now well understood that potential\nin thermal medium has always an imaginary component~\\cite{Laine2,nora11}. \n\n\\noindent {\\bf Acknowledgments:}\nWe are thankful for some financial assistance from CSIR project (CSR-656-PHY),\nGovernment of India. We also thank M G Mustafa for some suggestion.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\renewcommand{\\theequation}{1.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\nString and M-theory all suggest that we may live in a world that \nhas more than three spatial dimensions \\cite{string1,string2,string3}. Because only three of these are \npresently observable, one has to explain why the others are hidden from \ndetection.  One such explanation is the so-called Kaluza-Klein (KK) \ncompactification, according to which the\nsize of the extra dimensions is very small (often taken to be on the order\nof the Planck length) \\cite{zcw1,zcw2}.  As a consequence, modes that have momentum in the\ndirections of the extra dimensions are excited at currently inaccessible\nenergies.\n \nIn such a frame of compactifications, one often finds that the low-dimensional \neffective theories are usually singular, and such singularities might disappear\nwhen we consider the problem in the original high dimensional spacetimes.\nIn particular, it was shown that the 4-dimensional extreme black hole with a\nspecial dilaton coupling constant can be interpreted as a completely non-singular,\nnon-dilatonic, black p-brane in (4+p)-dimensions, provided that $p$ is\nodd  \\cite{GHT95}.\n\nRecently, three of the current authors \\cite{TWW08} studied the problem in the frame \nof colliding two timelike branes in string theory. After developing the general \nformulas to describe such events, we studied a particular class of exact solutions \nfirst in the 5-dimensional effective theory, and then lifted it to the 10-dimensional \nspacetime. In general, the 5-dimensional spacetime is singular, due to the mutual \nfocus of the two \ncolliding 3-branes. Non-singular cases also exist, but with the price that both of the\ncolliding branes violate all the three energy conditions, weak, dominant, and strong\n\\cite{HE73}.\nAfter lifted to 10 dimensions,  we found that the spacetime remains singular,\nwhenever it is singular in the 5-dimensional effective theory. In the cases where\nno singularities are formed after the collision, we found that the two 8-branes necessarily\nviolate all the energy conditions. \n\nIn this paper we shall address the same problem, but for the sake of simplicity,\nwe consider only the case where no branes are present.  Specifically, the\npaper is organized as follows. In the next section, we will set up the model to be\nstudied in the framework of both $D+d$-dimensional  string theory and its $D$-dimensional \neffective theory. In Section III, we  study two classes of exact solutions, by paying\nparticular attention on their local and global singular behavior. In Section IV,\nwe first lift these solutions to the $D+d$-dimensional spacetimes of string theory, and \nthen study their singular behavior. Section V contains our conclusions and\ndiscussing remarks. \n\n \n\n\\section{Toroidal Compactification of the Effective Action}\n\n\\renewcommand{\\theequation}{2.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\nLet us consider the toroidal compactification of the NS-NS sector of the action \n in (D+d) dimensions, $\\hat{M}_{D+d} = M_{D}\\times M_{d}$, where for the string \n theory we have\n$D+d = 10$. Then, the action takes the form\n\\cite{LWC001,LWC002,LWC003},\n\\begin{eqnarray}\n\\label{2.1}\nS_{D+d} &=& - \\frac{1}{2\\kappa^{2}_{D+d}}\n\\int{d^{D+d}x\\sqrt{\\left|\\hat{g}_{D+d}\\right|}  e^{-\\hat{\\Gamma}} \\left\\{\n{\\hat{R}}_{D+d}[\\hat{g}]\\right.}\\nonumber\\\\\n& & \\left. + \\hat{g}^{AB}\\left(\\hat{\\nabla}_{A}\\hat{\\Gamma}\\right)\n\\left(\\hat{\\nabla}_{B}\\hat{\\Gamma}\\right) - \\frac{1}{12}{\\hat{H}}^{2}\\right\\},\n\\end{eqnarray}\nwhere in this paper we consider the $(D+d)$-dimensional spacetimes described by the\nmetric,\n\\begin{eqnarray}\n\\label{2.2}\nd{\\hat{s}}^{2}_{D+d} &=& \\hat{g}_{AB} dx^{A}dx^{B} = \n   \\gamma_{\\mu\\nu}\\left(x^{\\lambda}\\right) dx^{\\mu}\ndx^{\\nu} \\nonumber\\\\& & + \\hat{\\Phi}^{2}\\left(x^{\\lambda}\\right) \n\\hat{\\gamma}_{ab}\\left(z^{c}\\right)dz^{a} dz^{b},\n\\end{eqnarray}\nwith  $\\gamma_{\\mu\\nu}\\left(x^{\\lambda}\\right)$ and $\\hat{\\Phi}^{2}\n\\left(x^{\\lambda}\\right)$ depending only on the coordinates  $x^{\\lambda}$ of \nthe spacetime $M_{D}$, and $\\hat{\\gamma}_{ab}\\left(z^{c}\\right)$ only on the internal \ncoordinates $x^{c}$, where $\\mu, \\nu, \\lambda = 0, 1, 2, ...,\nD-1$; $a, b, c = D, D+1, ..., D+d - 1$; and $A, B, C = 0, 1, 2, ...,\nD+ d -1$. Assuming that matter fields are all independent of $z^{a}$, \none finds that the internal space $M_{d}$ must be Ricci flat $R[\\hat{\\gamma}] = 0$ \n\\cite{LWC001,LWC002,LWC003}. For the purpose of the current work, it is sufficient to assume \nthat $M_{d}$ is a $d-$dimensional torus, $T^{d} = S^{1} \\times\nS^{1} \\times ... \\times S^{1}$.  \nThen, we find that \n\\begin{eqnarray}\n\\label{2.3}\n\\hat{R}_{D+d}\\left[\\hat{g}\\right] &=& R_{D}\\left[{\\gamma}\\right] \n+ \\frac{d(d-1)}{\\hat{\\Phi}^{2}} \n\\gamma^{\\mu\\nu}\\left(\\nabla_{\\mu}\\hat{\\Phi}\\right)\\left(\\nabla_{\\nu}\\hat{\\Phi}\\right)\\nonumber\\\\\n& & - \\frac{2}{\\hat{\\Phi}^{d}}\\gamma^{\\mu\\nu} \\left(\\nabla_{\\mu}\\nabla_{\\nu}\\hat{\\Phi}^{d}\\right).\n\\end{eqnarray}\nIgnoring the dilaton  $\\hat{\\Gamma}$ and the form field  $\\hat{H}$, \nthe integration of the action (\\ref{2.1})\nover the internal space yields,\n\\begin{eqnarray}\n\\label{2.4}\nS_{D}^{eff.} &=& - \\frac{1}{2\\kappa^{2}_{D}} \\int{d^{D}x\\sqrt{\\left|\\gamma\\right|}\n \\hat{\\Phi}^{d}\\left\\{R_{D}\\left[\\gamma\\right] \\right.}\\nonumber\\\\\n& & \\left. \n+ \\frac{d(d-1)}{\\hat{\\Phi}^{2}} \n\\gamma^{\\mu\\nu}\\left(\\nabla_{\\mu}\\hat{\\Phi}\\right)\\left(\\nabla_{\\nu}\\hat{\\Phi}\\right)\n\\right\\},\n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{2.5}\n\\kappa_{D}^{2} \\equiv \\frac{\\kappa_{D+d}^{2}}{V_{s}},\n\\end{equation}\nand $V_{s}$ is defined as\n\\begin{equation}\n\\label{2.6}\nV_{s} \\equiv \\int{\\sqrt{\\hat{\\gamma}} d^{d}z}.\n\\end{equation}\nFor a string scale compactification, we have $V_{s} = \n\\left(2\\pi \\sqrt{\\alpha'}\\right)^{d}$, where $\\left(2\\pi  \\alpha'\\right)$\nis the inverse string tension.\n\nAfter the conformal transformation,\n\\begin{equation}\n\\label{2.7}\ng_{\\mu\\nu} = \\hat{\\Phi}^{\\frac{2d}{D-2}}\\gamma_{\\mu\\nu},\n\\end{equation}\nthe D-dimensional effective action of Eq.(\\ref{2.4}) can be cast in the\nminmally coupled form,\n\\begin{eqnarray}\n\\label{2.8}\nS_{D}^{eff.} &=&  - \\frac{1}{2\\kappa^{2}_{D}}\\int{d^{D}x\\sqrt{\\left|g_{D}\\right|}\n  \\left\\{ R_{D}\\left[g\\right] \n-  \\kappa^{2}_{D} \\left(\\nabla\\phi\\right)^{2}\\right\\}},\\nonumber\\\\ \n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{2.9}\n\\phi \\equiv \\pm \n  \\left(\\frac{(D+d-2)d}{\\kappa^{2}_{D}\\left(D-2\\right)}\\right)^{\\frac{1}{2}} \n  \\ln\\left(\\hat{\\Phi}\\right).\n\\end{equation}\n\nIn this paper, we refer the actions of Eqs. (\\ref{2.1}) and (\\ref{2.4})  to as  {\\em the string frames},\nand the one of Eq.(\\ref{2.8})   {\\em the Einstein frames}. It should be noted that\nsolutions related by this conformal transformation can have completely different \nphysical and goemetrical properties in the two frames. In particular, in one \nframe a solution can be singular, while in the other it can be totally free from \nany kind of singularities. A simple example is the conformally-flat spacetimes\n$g_{AB} = \\Omega^{2}(x) \\eta_{AB}$, where the spacetime described by $g_{AB}$\ncan have a completely different spacetime structure from that of the Minkowski,\n$\\eta_{AB}$.\n\nAlthough we are mainly interested in the string theory with the split\n$D = 5$ and $d = 5$, in most cases considered in this paper \nwe shall not impose these restrictions here,\nso that our results obtained in this paper can be applied to other situations.\n\n\n\\section{Solutions in D-dimensional Spacetimes in   the   Einstein Frame}\n\n\\renewcommand{\\theequation}{3.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\n \nThe variation of the action (\\ref{2.8}) with respect to $g_{\\mu\\nu}$ and\n$\\phi$ yields the D-dimensional Einstein-scalar field equations,\n\\begin{eqnarray}\n\\label{3.1a}\n& & R_{\\mu\\nu} = \\kappa^{2}_{D} \\phi_{,\\mu} \\phi_{,\\nu},\\\\\n\\label{3.1b}\n& & \\nabla_{\\lambda} \\nabla^{\\lambda} \\phi = 0,\n\\end{eqnarray}\nwhere   $(\\;)_{,\\mu} \\equiv\n\\partial (\\;)\/\\partial x^{\\mu}$.\n\nIn this paper, we consider the D-dimensional spacetimes described by the metric\n\\begin{equation}\n\\label{3.2}\nds^{2}_{D,E} = 2 e^{2\\sigma(u, v)}du dv - e^{2h(u, v)} d\\Sigma^{2}_{D-2},\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{3.3}\nd\\Sigma^{2}_{D-2} \\equiv \\sum_{i = 2}^{D-1}{\\left(dx^{i}\\right)^{2}}.\n\\end{equation}\nClearly, the $(D-2)$-dimensional space ${\\cal{S}}$, spaned by  $d\\Sigma^{2}_{D-2}$, \nis Ricci flat, and its topology remains unspecified. In this paper, we assume that \nthis space is compact. One example is that it is a $(D-2)$-dimensional torus.\n\nIt should be noted that metric (\\ref{3.2}) is invariant under the coordinate\ntransformation,\n\\begin{equation}\n\\label{3.8}\nu = f(\\bar{u}), \\;\\;\\;\\; v = g(\\bar{v}),\n\\end{equation}\nwhere $f(\\bar{u})$ and $g(\\bar{v})$ are arbitrary functions of their indicated\narguments. \n\nIntroducing the following two null vectors \\cite{Wang1,Wang2,Wang3,Wang4}, \n\\begin{eqnarray}\n\\label{3.8a}\nl_{\\mu} &\\equiv& \\frac{\\partial u}{\\partial x^{\\mu}} = \\delta^{u}_{\\mu},\\nonumber\\\\\nn_{\\mu} &\\equiv& \\frac{\\partial v}{\\partial x^{\\mu}} = \\delta^{v}_{\\mu}, \n\\end{eqnarray}\nwe can see that both of them are future-directed and orthogonal to ${\\cal{S}}$.\nIn addition, each of these two null vectors defines an affinely parameterized\nnull geodesic congruence,\n\\begin{equation}\n\\label{3.8b}\nl^{\\lambda}\\nabla_{\\lambda}l_{\\mu} = 0 = n^{\\lambda}\\nabla_{\\lambda}n_{\\mu}.\n\\end{equation}\nThen, the expansions of the null ray $u = Const.$ and the one $v = Const.$\nare defined, respectively, by\n\\begin{eqnarray}\n\\label{3.8c}\n\\theta_{l}   &\\equiv& \\nabla^{\\lambda}l_{\\lambda} \n                = \\left(D-2\\right)e^{-2\\sigma}h_{,v},\\nonumber\\\\\n\\theta_{n}   &\\equiv& \\nabla^{\\lambda}n_{\\lambda} \n                = \\left(D-2\\right)e^{-2\\sigma}h_{,u}. \n\\end{eqnarray}\nIt should be noted that the two null vectors are uniquely defined only upto a\nfactor \\cite{HE73,Wang1,Wang2,Wang3,Wang4}.  In fact,  \n\\begin{equation}\n\\label{3.8ca}\n\\bar{l}_{\\mu} = f(u)\\delta^{u}_{\\mu},\\;\\;\\;\n\\bar{n}_{\\mu} = g(v)\\delta^{v}_{\\mu},\n\\end{equation}\nrepresent another set of null vectors that also define affinely parameterized null \ngeodesics, and  the corresponding expansions\nare given by \n\\begin{equation}\n\\label{3.8cb}\n\\bar{\\theta}_{+} = f(u)\\theta_{+},\\;\\;\\;\n\\bar{\\theta}_{-} = g(v)\\theta_{-}. \n\\end{equation}\nHowever, since along each curve $u = Const.\\; \\left(v = Const.\\right)$, the function \n$f(u) \\;\\left(g(v)\\right)$ is constant, this does not affect the definition of \ntrapped surfaces in terms of the expansions (See \\cite{HE73,Wang1,Wang2,Wang3,Wang4} in details). Thus, \nwithout loss of generality, in the following definitions of trapped surfaces and \nblack holes we consider only the expressions given by Eq.(\\ref{3.8c}).\n \n\n{\\em Definitions} \\cite{Pen68,Hay941,Hay942,Wang1,Wang2,Wang3,Wang4}:  The spatial $(D-2)$-dimensional \nsurface ${\\cal{S}}$ of constant $u$ and $v$\nis said {\\em trapped, marginally trapped, or untrapped}, according to whether\n$\\left. \\theta_{l}\\theta_{n}\\right|_{{\\cal{S}}} > 0$,\n$\\; \\left. \\theta_{l}\\theta_{n}\\right|_{{\\cal{S}}} = 0$,\nor $\\left. \\theta_{l}\\theta_{n}\\right|_{{\\cal{S}}} < 0$.\nAssuming that on the marginally trapped surfaces ${\\cal{S}}$ we have\n$\\left.\\theta_{l}\\right|_{{\\cal{S}}} = 0$, then an {\\em apparent horizon} is\nthe closure $\\tilde{\\Sigma}$ of a three-surface $\\Sigma$ foliated by the trapped\nsurfaces $\\cal{S}$ on which $\\left.\\theta_{n}\\right|_{\\Sigma} \\not= 0$.\nIt is said {\\em outer, degenerate, or inner}, according to\nwhether $\\left.{\\cal{L}}_{n}\\theta_{l}\\right|_{\\Sigma} < 0$,\n$\\left.{\\cal{L}}_{n}\\theta_{l}\\right|_{\\Sigma} = 0$, or\n$\\left.{\\cal{L}}_{n}\\theta_{l}\\right|_{\\Sigma} > 0$, where ${\\cal{L}}_{n}$\ndenotes the Lie derivative along the normal direction\n${n}_{\\mu}$, given by,\n\\begin{eqnarray}\n\\label{3.8d}\n{\\cal{L}}_{n}\\theta_{l}   &=& \\left(D-2\\right)e^{-4\\sigma}\n\\left(h_{,uv} - 2\\sigma_{,u}h_{,v}\\right),\\nonumber\\\\\n{\\cal{L}}_{l}\\theta_{n}   &=& \\left(D-2\\right)e^{-4\\sigma}\n\\left(h_{,uv} - 2\\sigma_{,v}h_{,u}\\right). \n\\end{eqnarray}\nIn addition, if $\\left. \\theta_{n}\\right|_{\\Sigma} < 0$\nthen the apparent horizon is said {\\em future}, and if\n$\\left. \\theta_{n}\\right|_{\\Sigma} > 0$ it is said {\\em past}.\n\n{\\em Black holes} are usually defined by the existence of {\\em future outer\napparent horizons} \\cite{HE73,Hay941,Hay942,Ida00,Wang1,Wang2,Wang3,Wang4}. However, in a definition given by\nTipler \\cite{Tip77} the degenerate case was also included.\n\nFor the metric (\\ref{3.2}), we find that Eqs.(\\ref{3.1a}) and (\\ref{3.1b})  yield\n\\begin{eqnarray}\n\\label{3.4a}\n& &  h_{,uu} + {h_{,u}}^{2} - 2 h_{,u}\\sigma_{,u} = \n            - \\frac{\\kappa^{2}_{D}}{D-2}{\\phi_{,u}}^{2},\\\\\n\\label{3.4b}\n& & h_{,vv} + {h_{,v}}^{2} - 2 h_{,v}\\sigma_{,v} = \n           - \\frac{\\kappa^{2}_{D}}{D-2}{\\phi_{,v}}^{2},\\\\\n\\label{3.4c}\n& &  2\\sigma_{,uv} + (D-2)\\left(h_{,uv} + {h_{,u}}h_{,v}\\right) = \n  -  \\kappa^{2}_{D} \\phi_{,u}\\phi_{,v},\\nonumber\\\\\n\\\\\n\\label{3.4d}\n& &   h_{,uv} + (D-2)h_{,u}h_{,v} = 0,\\\\\n\\label{3.4e}\n& &  2\\phi_{,uv} + (D-2)\\left(h_{,u}\\phi_{,v} + h_{,v}\\phi_{,u}\\right) = 0.\n\\end{eqnarray}\nIt should be noted that Eqs.(\\ref{3.4a})-(\\ref{3.4e}) are not all  \nindependent. In fact,\nEq.(\\ref{3.4c}) is the integrability condition of Eqs.(\\ref{3.4a}) and (\\ref{3.4b}), and\ncan be obtained from Eqs.(\\ref{3.4a}), (\\ref{3.4b}) and (\\ref{3.4e}). Therefore,  \nthe field equations reduce to Eqs. (\\ref{3.4a}), (\\ref{3.4b}), (\\ref{3.4d}) and \n(\\ref{3.4e}). To find the solution, one may first integrate Eq.(\\ref{3.4d}) to \nfind $h$, which gives,\n\\begin{equation}\n\\label{3.5}\nh(u, v)  =  \\frac{1}{D-2}\\ln\\left(F(u) + G(v)\\right),\n\\end{equation}\nwhere $F(u)$ and $G(v)$ are arbitrary functions. Then, one can integrate \nEq.(\\ref{3.4e}) to find $\\phi$. Once $h$ and $\\phi$ are found, $\\sigma$ can be \nobtained by integrating  Eqs.(\\ref{3.4a}) and (\\ref{3.4b}). The general solutions\nfor $\\sigma$ and $\\phi$ are unknown. In the following, we shall consider some\nspecific solutions. In particular, we shall consider the three cases separately: \n(a) $F'(u) \\not= 0,\\; G'(v)  = 0$; \n(b) $F'(u) = 0,\\; G'(v)  \\not= 0$; and  (c) $F'(u) G'(v)  \\not= 0$, where a prime\ndenotes the ordinary differentiation.  The second case can be obtained from the first\none by exchanging the $u$ and $v$ coordinates. Thus, without loss of\ngenerality, we need consider only Cases (a) and (c). \n\nBefore proceesing further, we note that for the solution of Eq.(\\ref{3.5}), \n Eqs.(\\ref{3.8c}) and (\\ref{3.8d}) reduce to\n\\begin{eqnarray}\n\\label{3.5a}\n\\theta_{l}   &=&  e^{-2\\sigma}\\frac{G'(v)}{F(u) + G(v)},\\nonumber\\\\\n\\theta_{n}   &=& e^{-2\\sigma}\\frac{F'(u)}{F(u) + G(v)},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\label{3.5b}\n{\\cal{L}}_{n}\\theta_{l}   &=& - \\theta_{l}\\left(\\theta_{n} \n+ 2e^{-2\\sigma}\\sigma_{,u}\\right),\\nonumber\\\\ \n{\\cal{L}}_{l}\\theta_{n}   &=& - \\theta_{n}\\left(\\theta_{l} \n+ 2e^{-2\\sigma}\\sigma_{,v}\\right). \n\\end{eqnarray}\n\n\\subsection{ $F'(u) \\not= 0,\\; G'(v)  = 0$}\n\n\n\nIn this case, from Eq.(\\ref{3.4b}) we find that $\\phi = \\phi(u)$. Hence, Eq.(\\ref{3.4c}) \nyields\n\\begin{equation}\n\\label{3.6}\n\\sigma(u,v) = a(u) + b(u),\n\\end{equation}\nwhere $a(u)$ and $b(u)$ are other arbitrary functions. Using the gauge\nfreedom of Eq.(\\ref{3.8}), without loss of generality we can always set \n$a(u) = 0 = b(u)$, so that this class of solutions are given by\n\\begin{eqnarray}\n\\label{3.7}\n\\sigma(u,v) &=& 0,\\nonumber\\\\\nh(u,v) &=& \\ln\\alpha(u),\\nonumber\\\\\n\\phi(u,v) &=& \\pm \\sqrt{\\frac{D-2}{\\kappa^{2}_{D}}}\n\\int^{u}{\\left(- \\frac{\\alpha''(u')}{\\alpha(u')}\\right)^{1\/2}\ndu'} \\nonumber\\\\\n& & + \\phi_{0},\n\\end{eqnarray}\nwhere $\\alpha(u) \\equiv F(u)^{1\/(D-2)}$, and $\\phi_{0}$ is an integration constant.\nInserting Eq.(\\ref{3.7}) into Eq.(\\ref{3.5a}), we find that\n\\begin{equation}\n\\label{3.7a}\n\\theta_{l}   = 0,\\;\\;\\;  \n\\theta_{n}   = \\left(D-2\\right)\\frac{\\alpha'(u)}{\\alpha(u)},\n\\end{equation}\nfor which we have $\\theta_{l} \\theta_{n} = 0$ identically. Then, according the above\ndefinition, the $(D-2)$-surfaxe ${\\cal{S}}$ is   always marginally trapped. Since\n\\begin{eqnarray}\n\\label{3.7b}\n{\\cal{L}}_{n}\\theta_{l}   =  0 =  {\\cal{L}}_{l}\\theta_{n}, \n\\end{eqnarray}\n${\\cal{S}}$ is also degenerate.  \n\nTo study these solutions further, we notice that \nfor these solutions  all the scalars built from the Riemann \ncurvature tensor are zero, therefore, in the present case scalar curvature singularities\nare always absent \\cite{ES77}.  However, non-scalar curvature singularities might also exist. \nIn particular, tidal forces experienced by an observer may become infinitely large under\ncertain conditions \\cite{HWW02}. To see how this can happen, let us consider the timelike \ngeodesics in the $(u, v)$-plane, which in the present case are simply given by\n\\begin{equation}\n\\label{3.9}\n\\dot{u} = \\gamma_{0},\\;\\;\\;\\;\n\\dot{v} = \\frac{1}{2\\gamma_{0}},\\;\\;\\;\\;\n\\dot{x}^{i} = 0,\n\\end{equation}\nwhere $i = 2, ..., D-1$, $\\; \\gamma_{0}$ is an integration  constant, \nand an overdot denotes the\nordinary derivative with respect to the proper time, $\\lambda$, of the timelike\ngeodesics. Defining $e^{\\mu}_{(0)} = d{x}^{\\mu}\/d\\lambda$, we find that the  \nunit vectors, given by\n\\begin{eqnarray}\n\\label{3.10}\ne^{\\mu}_{(0)} &=& \\gamma_{0}\\delta^{\\mu}_{u} +\n\\frac{1}{2\\gamma_{0}}\\delta^{\\mu}_{v},\\nonumber\\\\\ne^{\\mu}_{(1)} &=& \\gamma_{0}\\delta^{\\mu}_{u} -\n\\frac{1}{2\\gamma_{0}} \\delta^{\\mu}_{v},\\nonumber\\\\\ne^{\\mu}_{(i)} &=&   \\frac{1}{\\alpha(u)} \\delta^{\\mu}_{i}, \n\\end{eqnarray}\nform a freely falling frame, \n\\begin{equation}\n\\label{3.11}\ne^{\\mu}_{(\\alpha)}e^{\\nu}_{(\\beta)} g_{\\mu\\nu} = \\eta_{\\alpha\\beta},\\;\\;\\;\ne^{\\mu}_{(\\alpha); \\nu}e^{\\nu}_{(0)}  = 0,\n\\end{equation}\nwhere $\\eta_{\\alpha\\beta} = {\\mbox{diag.}}\\;\\{- 1, \\;  1, ..., \\;  1\\}$.\nProjecting the Ricci tensor onto the above frame, we find that\n\\begin{eqnarray}\n\\label{3.12}\nR_{(\\alpha)(\\beta)} &\\equiv& R_{\\mu\\nu}e^{\\mu}_{(\\alpha)}e^{\\nu}_{(\\beta)} \\nonumber\\\\\n&=& - \\gamma^{2}_{0}(D-2)  \\left(\\frac{\\alpha''(u)}{\\alpha(u)}\\right) \n\\left(\\delta^{u}_{\\alpha}\\delta^{u}_{\\beta} \\right.\\nonumber\\\\\n& & \\left. -\n\\left(\\delta^{u}_{\\alpha}\\delta^{v}_{\\beta} \n+ \\delta^{v}_{\\alpha}\\delta^{u}_{\\beta}\\right)\n+ \\delta^{v}_{\\alpha}\\delta^{v}_{\\beta}\\right).\n\\end{eqnarray}\nClearly,  the tidal forces remain finite over the whole spacetime, as long as \n$\\alpha''\/\\alpha$ is finite. To see this clearly, let us consider the following\nsolutions,\n\\begin{equation}\n\\label{3.13} \n \\frac{\\alpha''(u)}{\\alpha(u)}  = \n- \\frac{\\omega^{2}}{(u - u_{0})^{\\gamma}},\n\\end{equation}\nfor which we have\n\\begin{eqnarray}\n\\label{3.15}\n\\phi(u,v) &=& \\phi_{0}\n   + \\left(\\frac{\\omega^{2}(D-2)}{\\kappa^{2}_{D}}\\right)^{1\/2}\n\\nonumber\\\\\n& & \\times \\cases{\\frac{2}{2-\\gamma}(u - u_{0})^{1 - \\gamma\/2}, & $ \\gamma \\not= 2$,\\cr\n\\ln{(u - u_{0})}, &  $ \\gamma = 2$,\\cr}\n\\end{eqnarray}\n where $u_{0}$ is an arbitrary constant, and without\nloss of generality, we can always set $u_{0} = 0$, an assumption we shall adopt in the \nfollowing discussions. The constants $\\omega$ and $\\gamma$ have to satisfy the  conditions\n$\\alpha(u) > 0$ and $\\alpha''(u)\/\\alpha(u) < 0$, so that  the metric has the correct \nsigns and the scalar field is real. \n\nFrom Eq.(\\ref{3.9}) we find that $ u   \\sim \n\\gamma_{0}\\lambda$, where the proper time $\\lambda$ was chosen such\nthat $u = 0$ corresponds to $\\lambda = 0$. Then, the distortion,\nwhich is proportional to the twice integrals of $R_{(\\alpha)(\\beta)}$ \nwith respect to the proper time $\\lambda$, is given by\n\\begin{eqnarray}\n\\label{3.14}\nD_{(\\alpha)(\\beta)} &\\equiv& \\int{d\\lambda\\int{R_{(\\alpha)(\\beta)} d\\lambda}} \\nonumber\\\\\n &\\sim& \\cases{ \\lambda\\left(\\ln\\lambda -1\\right), &\n$\\gamma = 1$,\\cr\n\\ln\\lambda, & $\\gamma = 2$,\\cr\n\\lambda^{2-\\gamma}, & $\\gamma \\not= 1, 2$,\\cr}\n\\end{eqnarray}\nas $\\lambda \\rightarrow 0$. To study these solutions further, let us consider\nthe following cases separately.\n\n\\subsubsection{$\\gamma < 0$}\n\nIn this case,  Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) show that both the tidal forces\nand distortions are finite at $u = 0$. Therfore, to have a geodesically maximal spacetime,\nwe need to extend the solutions across this surface to the region $u < 0$. When $\\gamma \n= - n$, where $n (= 1, 2, ...)$ is an integer, the extension is simple, and can be obtained\nby simply taking $u$ to be $u \\in (-\\infty, \\infty)$. However, When $- \\gamma$ is not an\ninteger, the solution is not anyalytical at $u = 0$, and the extension is not unique.\nOne possible extension is to replace $u$ by $|u|$.  Once such an extension is done, it can be\nseen that  both the tidal forces and distortions diverge at $|u| \\rightarrow \\infty$. Therefore,\nthe spacetimes are singular at the null infinities, and the nature of the singularities is\n{\\em strong}, as both of them diverge. The corresponding Penrose diagram is given by Fig. \\ref{fig1a}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1a}\n\\caption{The Penrose diagram for the case $\\gamma < 0$ of the solution (\\ref{3.13})\nin the Einstein frame. The double solid lines $AB$ and $CD$ represent null infinities\n$u = \\pm \\infty$ and denote strong spacetime singularities, where both the tidal \nforces and distortions exerting on a freely falling observer become unbound. The spacetime\nis regular at $u = 0$, and its extension to Region $II$ where $u < 0$ is unique only \nwhen $-\\gamma$ is an integer. \n}\n\\label{fig1a}\n\\end{figure} \n\n\n\n\\subsubsection{$\\gamma = 0$}\n\nIn this case,   $\\alpha(u)$ and $\\phi(u)$ can be given explicitly,  \n\\begin{eqnarray}\n\\label{3.12b}\n \\alpha(u) &=& \\alpha_{0} \\sin\\left(\\omega u + \\Delta\\right),  \\nonumber\\\\\n \\phi(u) &=& \\pm \\sqrt{\\frac{D-2}{\\kappa^{2}_{D}}} \\; \\omega u + \\phi_{0},\n\\end{eqnarray}\nwhere $\\Delta$ and $\\phi_{0}$ are the integration constants. From the\nabove we can see that the $(D-2)$-dimensional space ${\\cal{S}}$ collapses to\na point at $\\omega u + \\Delta = n \\pi$. This can also be seen from the \nexpansion given by  Eq.(\\ref{3.7a}), which  now reads,    \n\\begin{equation}\n\\label{3.12c}\n \\theta_{n} = \\omega \\left(D-2\\right) \n \\frac{\\cos\\left(\\omega u + \\Delta\\right)}{\\sin\\left(\\omega u + \\Delta\\right)}.\n\\end{equation}\nClearly, $\\theta_{n}$ diverges at $\\omega u + \\Delta = n \\pi$. However, a closer\ninvestigation of this solution shows that the spacetime is not singular\nat these points. For example, one may consider the tidal forces measured \nby observers that move along timelike geodesics not perpendicular to ${\\cal{S}}$.\nWithout loss of generality, let us consider the timelike geodesics, described\nby $u = u(\\lambda), \\; v = v(\\lambda), \\; x^{2} = x^{2}(\\lambda)$ and\n$  x^{i} = x^{i}_{0} = Const.$ Then, it can be shown that the timelike geodesical\nequation allows the first integration, and yields,\n\\begin{eqnarray}\n\\label{3.12d}\n \\dot{u} &=& \\gamma_{0}, \\;\\;\\;\n \\dot{v} = \\frac{1}{2\\gamma_{0}}\n           \\left(\\frac{\\beta^{2}_{0}}{\\alpha^{2}(u)} + 1\\right), \\nonumber\\\\\n\\dot{x}^{2} &=& \\frac{\\beta_{0}}{\\alpha^{2}(u)},\\;\\;\\;\n\\dot{x}^{i} = 0, \\; (i = 3, 4, ..., D-1),  \n\\end{eqnarray}\nwhere $\\gamma_{0}$ and $\\beta_{0}$ are two integration constants.\nFrom the above we find the following unit vectors,\n\\begin{eqnarray}\n\\label{3.10a}\ne^{\\mu}_{(0)} &\\equiv& \\frac{dx^{\\mu}}{d\\lambda\n=\\gamma_{0}\\delta^{\\mu}_{u} + \\frac{1}{2\\gamma_{0}}\n\\left(\\frac{\\beta^{2}_{0}}{\\alpha^{2}(u)} + 1\\right)\\delta^{\\mu}_{v}\n+ \\frac{\\beta_{0}}{\\alpha^{2}(u)}\\delta^{\\mu}_{2},\\nonumber\\\\\ne^{\\mu}_{(1)} &=& \\gamma_{0}\\delta^{\\mu}_{u} + \\frac{1}{2\\gamma_{0}}\n\\left(\\frac{\\beta^{2}_{0}}{\\alpha^{2}(u)} - 1\\right)\\delta^{\\mu}_{v}\n+ \\frac{\\beta_{0}}{\\alpha^{2}(u)}\\delta^{\\mu}_{2},\\nonumber\\\\\ne^{\\mu}_{(2)} &=& \\frac{\\beta_{0}}{\\gamma_{0}\\alpha(u)} \\delta^{\\mu}_{v} \n     + \\frac{1}{\\alpha(u)} \\delta^{\\mu}_{2},\\nonumber\\\\\ne^{\\mu}_{(i)} &=&   \\frac{1}{\\alpha(u)} \\delta^{\\mu}_{i}, \n\\; (i = 3, 4, ..., D-1).\n\\end{eqnarray}\nIt can be shown that they satisfy Eq.(\\ref{3.11}), that is, they  also\nform a freely-falling frame. Then, we find that\n\\begin{equation}\n\\label{3.10b}\nR_{(\\alpha)(\\beta)} \\equiv R_{\\mu\\nu}e^{\\mu}_{(\\alpha)}e^{\\nu}_{(\\beta)}  \n= \\omega^{2} \\left(D-2\\right)e^{u}_{(\\alpha)} e^{u}_{(\\beta)},\n\\end{equation}\nwhich is always finite, as can be seen from Eq.(\\ref{3.10a}). Therefore, although\nthe $(D-2)$-dimensional space ${\\cal{S}}$ focuses at $\\alpha(u) = 0$,  \nno spacetime singularities appear at these points, because no trapped compact\nsurface exists in the present case. The Hawking-Penrose singularity theorems\nrequire the existence of both a compact trapped surface and the focusing of\ngeodesics \\cite{HE73}. The extension across these focusing points can be done\nby simply taking $u$ as any real value, i.e.,  $u \\in (-\\infty, \\infty)$.\nOnce such an extension is made, Eq.(\\ref{3.12b}) shows that we may have\nspacetime singularities at $u = \\pm \\infty$.  As Eq.(\\ref{3.14}) shows that\nindeed the distortation at these null infinities diverge, although the \ntidal forces still remain finite. Then, the corresponding Penrsoe diagram\nis given by Fig. \\ref{fig1b}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1b}\n\\caption{The Penrose diagram for the case $\\gamma = 0$ of the solution (\\ref{3.12b})\nin the Einstein frame. The lines $AB$ and $CD$ represent null infinities\n$u = \\pm \\infty$ and the spacetime is singular there, in the sense that\nthe  distortation becomes unbounded along these null infinities, although \nthe tidal forces remain finite.  \n}\n\\label{fig1b}\n\\end{figure} \n\n \n\n\n\n\\subsubsection{$0 < \\gamma < 2$}\n\nIn this case, Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) show that  at\n$u = 0$ the tidal forces become unbounded, while the distortions remain finite.\nThis type of   singularities  is usually said   {\\em weak}, and the spacetime \nbeyond this surface may be extendible \\cite{Ori921,Ori922}, although it is still unclear\nhow to carry out specifically such extensions. As $u \\rightarrow \\infty$,\nfrom Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) we find that the tidal \nforces are bounded, but now the distortions become unbounded. The corresponding\nPenrose diagram is givne by Fig. \\ref{fig1c}.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1c}\n\\caption{The Penrose diagram for the case $0 < \\gamma < 2$ of the solution (\\ref{3.13}).\nThe spacetime is singular at $u = 0$ in the sense that the tidal forces become\nunbounded, while the distorations remain bounded. \nIt is also singular at the null infinity $u = \\infty$,  denoted by the line\n$CD$, where the tidal forces remain finite, but the distorations become\nunbounded.  \n}\n\\label{fig1c}\n\\end{figure} \n\n\n\\subsubsection{$ \\gamma = 2$}\n \nWhen $\\gamma = 2$,   Eq.(\\ref{3.13})   has the solution\n\\begin{equation}\n\\label{3.16}\n\\alpha(u) =  \\alpha_{0} u^{\\delta},  \n\\end{equation}\nwhere  $\\omega^{2} = \\delta(1-\\delta)$ with $0 < \\delta < 1$. For such a\nsolution, Eq.(\\ref{3.7a}) reads,    \n\\begin{equation}\n\\label{3.16a}\n \\theta_{n} = \\frac{\\delta \\left(D-2\\right)}{u}.\n\\end{equation}\nThen, from Eqs.(\\ref{3.12}), (\\ref{3.13}) and (\\ref{3.14}) we find that in \nthe present case both  the tidal forces and distortions   become unbound\nat $u = 0$, so a strong spacetime singularity appears at $u = 0$. In this\ncase, we also have $\\theta_{n}(u = 0) = \\infty$. However, at $u = \\infty$,\nthe tidal forces remain finite, while the distortions   become unbound.\nThe corresponding Penrose diagram in this case is given by Fig. \\ref{fig1d}.  \n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1d}\n\\caption{The Penrose diagram for the case $  \\gamma = 2$ of the solution (\\ref{3.13}).\nThe spacetime is singular at $u = 0$ in the sense that the tidal forces become\nunbounded, while the distorations remain bounded. \nIt is also singular at the null infinity $u = \\infty$,  denoted by the line\n$CD$, where the tidal forces remain finite, but the distorations become\nunbounded.  \n}\n\\label{fig1d}\n\\end{figure} \n\n\\subsubsection{$\\gamma > 2$}\n \n\nWhen $\\gamma >  2$, both the  tidal forces and the distortion\nbecome unbound at $u = 0$, so the spacetime has a strong singularity along \nthis surface. However, at $u = \\infty$ all of them remain finite. Therefore,\nin the present case the spacetime is free of spacetime singularity at the \nnull infinity $ u = \\infty$. The corresponding Penrose diagram in this case \nis given by  Fig. \\ref{fig1e}.\n\n \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1e}\n\\caption{The Penrose diagram for the case $  \\gamma > 2$ of the solution (\\ref{3.13}).\nThe spacetime is singular at $u = 0$ in the sense that both the tidal forces\nand distorations become unbounded. At the null infinity $u = \\infty$,  denoted \nby the line $CD$, it is non-singular, because now  both the tidal forces and\ndistorations  remain finite.  \n}\n\\label{fig1e}\n\\end{figure} \n\n\n\n\n\\subsection{ $F'(u) G'(v)  \\not= 0$}\n\nIn this case to solve Eqs.(\\ref{3.4a})-(\\ref{3.4e}), it is found convenient first to\nintruduce two new coordinates $\\bar{u}$ and $\\bar{v}$ via the relations \n$\\bar{u} \\equiv F(u)$ and $\\bar{v} \\equiv G(v)$,  using the gauge freedom (\\ref{3.8}).\nIn terms of these new coordinates, the metric (\\ref{3.2}) takes the form,\n\\begin{eqnarray}\n\\label{3.2a}\nds^{2}_{D,E} &=& \\bar{g}_{\\mu\\nu}d\\bar{x}^{\\mu}d\\bar{x}^{\\nu} \\nonumber\\\\\n&=& 2 e^{2\\Sigma(\\bar{u}, \\bar{v})}d\\bar{u} d\\bar{v} \n- e^{2H(\\bar{u}, \\bar{v})} d\\Sigma^{2}_{D-2},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{3.3a} \nH(\\bar{u}, \\bar{v}) &\\equiv&  h(u,v)  \n=  \\frac{1}{D-2}\\ln\\left(\\bar{u} + \\bar{v}\\right),\\nonumber\\\\\n\\Sigma(\\bar{u}, \\bar{v}) &\\equiv&     \\sigma({u}, {v}) - \\frac{1}{2}\\ln\\left[F'(u)G'(v)\\right]. \n\\end{eqnarray}\nThen, it can be shown that Eqs.(\\ref{3.4a})-(\\ref{3.4e}) reduce to\n\\begin{eqnarray}\n\\label{3.16aa}\nM_{,t} &=& \\frac{1}{2} t\\left({\\phi_{,t}}^{2} + {\\phi_{,y}}^{2}\\right),\\\\\n\\label{3.16ab}\nM_{,y} &=&   t{\\phi_{,t}}{\\phi_{,y}},\\\\\n\\label{3.16ac}\nM_{,tt} &-& M_{,yy}  = - \\frac{1}{2} \\left({\\phi_{,t}}^{2} - {\\phi_{,y}}^{2}\\right),\\\\\n\\label{3.16ad}\n\\phi_{,tt} &+& \\frac{1}{t} \\phi_{,t} - \\phi_{,yy} = 0,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.16ae}\n\\Sigma  &\\equiv& - \\frac{D-3}{2(D-2)} \\ln(t) + \\kappa^{2}_{D} M,\\nonumber\\\\\nt &\\equiv& \\bar{u} + \\bar{v},\\;\\;\\; y \\equiv \\bar{u} - \\bar{v}.\n\\end{eqnarray}\nEq.(\\ref{3.16ac}) is the integrability condition of Eqs.(\\ref{3.16aa}) and (\\ref{3.16ab}).\nThus, once a solution for $\\phi$ is found from Eq.(\\ref{3.16ad}), the remaining is to find $M$ \nfrom Eqs.(\\ref{3.16aa}) and (\\ref{3.16ab}) by quadratures. \n\nUsing the freedom on the choice of the two null vectors defined by Eq.(\\ref{3.8c}), for the\nmetric (\\ref{3.2a}) we define $\\bar{l}_{\\mu} = \\delta^{\\bar{u}}_{\\mu}$ and\n$ \\bar{n}_{\\mu} = \\delta^{\\bar{v}}_{\\mu}$. Then, we find that,\n\\begin{eqnarray}\n\\label{3.16af}\n\\bar{\\theta}_{l} &\\equiv& \\bar{g}^{\\alpha\\beta}\\bar{l}_{\\alpha;\\beta}\n= \\frac{e^{-2\\Sigma}}{t},\\nonumber\\\\\n\\bar{\\theta}_{n} &\\equiv& \\bar{g}^{\\alpha\\beta}\\bar{n}_{\\alpha;\\beta}\n= \\frac{e^{-2\\Sigma}}{t},\n\\end{eqnarray}\nfrom which we find that $\\bar{\\theta}_{l}\\bar{\\theta}_{n} \\ge 0$, where equality holds\nonly when $t = \\infty$ or\/and $\\Sigma = \\infty$. Therefore, in the present case, the\n$(D-2)$-dimensional surface ${\\cal{S}}$ is always trapped or marginally trapped. However,\nit must be noted that the coordinates $(t, y)$ or $(\\bar{u}, \\bar{v})$ do not always\ncover the whole spacetime. There exist cases where the hypersurface $\\Sigma = \\infty$\ndoes not represents the boundary of the spacetime. To have a geodesically maximal \nspacetime, extension beyond this surface is needed. In the extended region(s), one\nmay have $\\bar{\\theta}_{l}\\bar{\\theta}_{n} < 0$, that is, the spacetime is not trapped.\nTypical examples of this kind can be found in \\cite{Wang1,Wang2,Wang3,Wang4}. In the following, we consider \nthree classes of solutions, to be referred, respectively, to as, Class IIa, IIb, \nand IIc, and show explicitly that such cases happen here, too.  \n\n\\subsubsection{Class IIa Solutions}\n\n\nThis class of solutions is given by\n\\begin{eqnarray}\n\\label{3.16a1}\nM    &=& \\frac{1}{2} c^{2} \\ln(t) + M_{0},\\nonumber\\\\\n\\phi &=& c \\ln(t) + \\phi_{0},\n\\end{eqnarray}\nwhere $c,\\; \\phi_{0}$ and $M_{0}$ are integration constants. Then, from Eq.(\\ref{3.16af})\nwe find that\n\\begin{equation}\n\\label{3.16a2}\n\\bar{\\theta}_{l} = \\bar{\\theta}_{n} = \\frac{e^{-\\kappa^{2}_{D}M_{0}}}{t^{\\chi^{2} +\n\\frac{1}{D-2}}},\n\\end{equation}\nwhich means that $t = 0$ represents a focusing point, where \n$\\chi \\equiv c^{2}\\kappa_{D}^{2}$. Since now all the surfaces\n${\\cal{S}}$ of constant $t$ and $y$ are trapped, then, according to the singularity\ntheorems \\cite{HE73}, the spacetime must be singular at $t = 0$, as can be seen clearly\nfrom the expression,\n\\begin{equation}\n\\label{3.16a3}\nR_{D}[g] \\equiv   \\kappa_{D}^{2}g^{\\alpha\\beta}\\phi_{,\\alpha}\n\\phi_{,\\beta}= \\frac{A_{0}}{t^{\\chi^{2} + \\frac{D-1}{D-2}}},\n\\end{equation}\nwhere $A_{0} \\equiv 2\\chi^{2} e^{-2\\kappa^{2}_{D}M_{0}}$.\n The corresponding Penrose diagram is given by \nFig. \\ref{fig2a}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2a}\n\\caption{The Penrose diagram for the solutions given by Eq.(\\ref{3.16a1}) in D-dimensional\nspacetime. The horizontal line $t = 0$ represents a big bang  singularity.\nThe $(D-2)$-dimensional surfaces of constant $\\bar{u}$\nand $\\bar{v}$ are always trapped.}\n\\label{fig2a}\n\\end{figure} \n\n\n\\subsubsection{Class IIb Solutions}\n\nThis class of solutions is given by\n\\begin{eqnarray}\n\\label{3.16a4}\nM &=& \\frac{1}{2} c^{2} \\ln\\left(\\frac{t^{4}}{\\left(y^{2} - t^{2}\\right)\n\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2}}\\right) + M_{0},\\nonumber\\\\\n\\phi &=& c \\ln\\left(\\frac{t^{2}}{y + \\sqrt{y^{2} - t^{2}}}\\right) + \\phi_{0},\n\\end{eqnarray}\nfor which from Eq.(\\ref{3.16a2}) we find that\n\\begin{equation}\n\\label{3.16ah}\n\\bar{\\theta}_{l} = \\bar{\\theta}_{n} = \n\\frac{\\left\\{\\left(y^{2} - t^{2}\\right)\\left[y\n+ \\sqrt{y^{2} - t^{2}}\\right]^{2}\\right\\}^{\\chi^{2}}}\n{e^{2\\kappa^{2}_{D}M_{0}}t^{4\\chi^{2} +\n\\frac{1}{D-2}}}.\n\\end{equation}\nClearly, in the present case the hypersurface $t = 0$ still represents a focusing\npoint. In addition, on the hypersurfaces $y^{2} = t^{2}$, we have $\\Sigma = \\infty$\nand $\\bar{\\theta}_{l} \\bar{\\theta}_{n} = 0$. Thus, the surfaces ${\\cal{S}}$ now become\nmarginally trapped along $y^{2} = t^{2}$. If the spacetime is not singular on \nthese null surfaces, extension beyond them is needed. To study the singular behavior \nalong these surfaces, let us consider the quantity,\n\\begin{eqnarray}\n\\label{3.16ai}\nR_{D}[g] &\\equiv&   \\kappa_{D}^{2}g^{\\alpha\\beta}\\phi_{,\\alpha}\n\\phi_{,\\beta}\\nonumber\\\\\n&=&  \\frac{2A_{0}\\left(4y^{2} - 3 t^{2} + 2\\right)}\n{\\left[\\sqrt{y^{2} - t^{2}}\\left(y + \\sqrt{y^{2} - t^{2}}\\right)\\right]^{1-2\\chi^{2}}}\\nonumber\\\\\n& & \\times t^{- \\left(4\\chi^{2} + \\frac{D-1}{D-2}\\right)}.\n\\end{eqnarray}\nClearly, the spacetime is singular at $t = 0$. The singular behavior along the hypersurfaces\n$y^{2} = t^{2}$ depend on the values of $\\chi$.\n\n\n\\noindent{\\bf Case B.2.1: $\\chi^{2} < \\frac{1}{2}$}. In this case, \n it is also singular on the null hypersurfaces $y^{2} =  t^{2}$, and the \ncorresponding Penrose diagram is given by Fig. \\ref{fig2b}. \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2b}\n\\caption{The Penrose diagram in the D-dimensional spacetime \nfor the solutions given by Eq.(\\ref{3.16a4}),\nas well as for the ones given by Eq.(\\ref{3.16am}),\nfor $\\chi^{2} < 1\/2$. The horizontal line $t = 0$ \nrepresents a big bang  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\bar{u}$ and $\\bar{v}$ are always trapped in Regions $I$ and $I'$.\nThe spacetime along the line $0D$ and $0E$ are singular.   The two regions\n$I$ and $I'$ are physically disconnected in both cases. \n}\n\\label{fig2b}\n\\end{figure} \n\n\\noindent{\\bf Case B.2.2: $\\frac{1}{2} \\le \\chi^{2} < 1$.} In this case,  \nthe spacetime is not singular at  $y = \\pm t$, but the metric\ncoefficient $\\Sigma$ is. To extend the metric beyond these surfaces, \nbecause of the symmetry, it is suffcient to consider   the extension \nonly across the hypersurface $y = t$ from the region $\\bar{u} > 0,\\; \n\\bar{v} < 0$. The extension across the hypersurface $y = -t$ from the region \n$\\bar{u} < 0,\\; \\bar{v} > 0$ is similar, and can be obtained by exchanging the \nroles of $\\bar{u}$ and $\\bar{v}$. Then, using the gauge freedom \n(\\ref{3.8}), we  introduce two new coordinates \n$\\tilde{u}$ and $\\tilde{v}$  via the relations,\n\\begin{equation}\n\\label{3.16aj}\n \\bar{u} =  \\tilde{u}^{2n},  \\;\\;\\;\n \\bar{v} =  - \\left(-\\tilde{v}\\right)^{2n}, \n \\end{equation}\nwhere \n\\begin{equation}\n\\label{3.16ak}\nn \\equiv \\frac{1}{2(1- \\chi^{2})}.\n\\end{equation}\nIt can be shown that the solutions are analytical across the hypersurfaces $y = \\pm t$ \nonly when $n$ is an integer. Otherwise, the solutions are not analytical, and the extension\nacross $y = \\pm t$ in not unique. Therefore, in the following we shall consider only\nthe case where $n$ is an integer. Then, in terms of $\\tilde{u}$ and $\\tilde{v}$, the\nsolutions read\n\\begin{eqnarray}\n\\label{3.2aa}\nds^{2}_{D,E} &=& \\tilde{g}_{\\mu\\nu}d\\tilde{x}^{\\mu}d\\tilde{x}^{\\nu} \\nonumber\\\\\n&=& 2 e^{2\\tilde{\\Sigma}} d\\tilde{u} d\\tilde{v} \n- e^{2\\tilde{H}} d\\Sigma^{2}_{D-2},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{3.3a1} \n\\tilde{H} &=&    \\frac{1}{D-2}\\ln\\left(\\tilde{u}^{2n} - \\left(-\\tilde{v}\\right)^{2n}\\right),\\nonumber\\\\\n\\tilde{\\Sigma} &=&    \\left(2\\chi^{2} - \\frac{D-3}{2(D-2)}\\right)\n                     \\ln\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)\\nonumber\\\\\n\t& & - \\frac{D-3}{2(D-2)} \\ln\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)\n\t    + \\tilde{\\Sigma}_{0},\\nonumber\\\\\n\\phi &=&  \\frac{2\\chi}{\\kappa_{D}} \n                     \\ln\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)\n\t\t     + \\phi_{0}, \n\\end{eqnarray}\nwhere $\\tilde{\\Sigma}_{0} \\equiv \\kappa^{2}_{D}M_{0} + \\ln\\left(2^{3\/2 - \\chi^{2}} n\\right)$.\nClearly, the coordinate singularity at $y = t$ or $\\tilde{v} = 0$ disappears, and the solutions \ncan be considered as   valid for $\\tilde{v} > 0$.  To study the properties of the spacetime\nin the extended region, let us consider the quantities, \n\\begin{eqnarray}\n\\label{3.3aa} \n\\tilde{\\theta}_{l} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{v}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\left(-\\tilde{v}\\right)^{2n-1}}{\\tilde{u}^{2n}\n             - \\left(-\\tilde{v}\\right)^{2n}},\\nonumber\\\\\n\\tilde{\\theta}_{n} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{u}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\tilde{u}^{2n-1}}{\\tilde{u}^{2n}\n             - \\left(-\\tilde{v}\\right)^{2n}},\\nonumber\\\\\n{\\cal{L}}_{n}\\tilde{\\theta}_{l}\t &=& - 4n^{2}e^{-4\\tilde{\\Sigma}}\n\\frac{\\tilde{u}^{n-1}\\left(-\\tilde{v}\\right)^{2n-1}}{\\left(\\tilde{u}^{2n}\n             - \\left(-\\tilde{v}\\right)^{2n}\\right)^{2}}\\nonumber\\\\\n& & \\times\\left\\{\\left(2\\chi^{2} + \\frac{1}{D-2}\\right)\\tilde{u}^{n} \n    + 2\\chi^{2}\\left(-\\tilde{v}\\right)^{n}\\right\\},\\nonumber\\\\    \nR_{D}\\left[\\tilde{g}\\right] &=& 8 n^{2}\\chi^{2}e^{-2\\tilde{\\Sigma}_{0}}\\nonumber\\\\\n& & \\times \\frac{\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)^{\\frac{D-3}{D-2}}\n\\left(-\\tilde{u}\\tilde{v}\\right)^{n-1}}\n{\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)^{4\\chi^{2} + \\frac{D-1}{D-2}}}.\n\\end{eqnarray}\nThen, we find that\n\\begin{eqnarray}\n\\label{3.3ab}\n\\tilde{\\theta}_{l} \\tilde{\\theta}_{n} &=& 4n^{2}e^{-4\\tilde{\\Sigma}}\n\\frac{\\left(-\\tilde{u} \\tilde{v}\\right)^{2n-1}}\n{\\left(\\tilde{u}^{2n} - \\left(-\\tilde{v}\\right)^{2n}\\right)^{2}}\\nonumber\\\\\n&=&\\cases{< 0, \\tilde{u} \\tilde{v} >0,\\cr\n= 0, \\tilde{u} \\tilde{v} = 0,\\cr\n> 0, \\tilde{u} \\tilde{v} < 0.\\cr}\n\\end{eqnarray}\nThat is, now in the extended region where $\\tilde{u} \\tilde{v} >0$ the  spacetime\nbecomes untrapped. Across the hypersurface $\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$,\nwe have\n\\begin{eqnarray}\n\\label{3.3ac}\n& & \\tilde{\\theta}_{l}\\left(\\tilde{u}>0, \\tilde{v} = 0\\right) = 0,\\;\\;\\;\n\\tilde{\\theta}_{n}\\left(\\tilde{u}>0, \\tilde{v} =0 \\right)   >0,\\nonumber\\\\\n& & {\\cal{L}}_{n}\\tilde{\\theta}_{l} \\left(\\tilde{u}> 0, \\tilde{v} = 0\\right) = 0.\n\\end{eqnarray}\nTherefore, the half infinite line $\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$ represents\na past degenerate apparent horizon. \n\nTo study the singular behavior of the spacetime in the extended region,\nwe need to distinguish the case where $n$ is an even interger from the one where $n$\nis an odd integer.  When $n$ is an even interger, Eq.(\\ref{3.3aa}) shows that the spacetime\nis singular along the line $\\tilde{u} = \\tilde{v}$, denoted by the vertical line\n$0C$ in Fig. \\ref{fig2c}. \n\nBecause of the symmetry of the spacetime, one can make\na similar extension across the half line $\\bar{u} = 0$ and $\\bar{v} \\ge 0$, but\nnow with\n\\begin{equation}\n\\label{3.16ak2}\n \\bar{u} =  - \\left(-\\tilde{u}\\right)^{2n},  \\;\\;\\;\n \\bar{v} =   \\tilde{v}^{2n}.\n \\end{equation}\nThen, one finds that  this half line also  represents a past degenerate apparent\nhorizon, and the extended region $II'$ is not trapped, although it is disconnected\nwith Region $II$, because of the timelike singularity along the vertical line\n$0C$.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2c}\n\\caption{The Penrose diagram in the D-dimensional spacetime for $\\frac{1}{2} \\le \n\\chi^{2} < 1$ for the solutions given by Eq.(\\ref{3.2aa}) with\n$n$ being an even integer, as well as for the ones  given by Eq.(\\ref{3.16am}) with\n $n$ being an odd integer. The horizontal line $t = 0$ \nrepresents a big bang  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\bar{u}$ and $\\bar{v}$ are always trapped in Regions $I$ and $I'$,\nbut not in Regions $II$ and $II'$. The spacetime along the vertical\nline $0C$ is singular.   The two regions\n$II$ and $II'$ are physically disconnected. The lines $0E$ and $0D$ represent\n past degenerate apparent horizons. \n}\n\\label{fig2c}\n\\end{figure} \n\nWhen $n$ is an odd interger, Eq.(\\ref{3.3aa}) shows that the spacetime\nis not singular along the line $\\tilde{u} = \\tilde{v}$, as shown in Fig. \\ref{fig2d},\nwhere the lines $0E$ and $0D$ represent past degenerate apparent horizons.\nThen, Regions $I$ and $I'$ act as white holes. Since the solutions are symmetric\nwith respect to $t$, one may consider the case where $t \\le 0$. Then, the spacetime\nwill be given by the lower half part of Fig. \\ref{fig2d}, in which the $(D-2)$-dimensional\nsurfaces ${\\cal{S}}$ of constant $\\tilde{u}$ and $\\tilde{v}$ are trapped in\nRegions $III$ and $III'$, and is not trapped in   Region $IV$. The lines $0D'$\nand $0E'$ act as future degenerate apparent horizons, so  Regions $III$ and $III'$ \nnow represent black holes. the spacetimes of the lower half part is disconnected with \nthat of the upper half by the spacelike singularity located along $t = 0$.  \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2d}\n\\caption{The Penrose diagram in the D-dimensional spacetime for $\\frac{1}{2} \\le \n\\chi^{2} < 1$ for the solutions given by Eq.(\\ref{3.2aa}) with\n$n$ being an odd integer, as well as for the ones  given by Eq.(\\ref{3.16am}) with\n $n$ being an even integer. The horizontal line $t = 0$ \nrepresents a spacetime  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\tilde{u}$ and $\\tilde{v}$ are always trapped in Regions $I$,\n$I'$, $III$ and $III'$, but not in Regions $II$, and $IV$.   The upper\nhalf part is disconnected with the lower half part by the spacetime\nsingularity along the line $AB$ where $t=0$. The lines $0E$ and $0D$ represent\n past degenerate apparent horizons, while the ones   $0E'$ and $0D'$ represent\n future degenerate apparent horizons. Regions $I$ and $I'$ represent white holes,\n while Regions $III$ and $III'$ represent black holes.\n}\n\\label{fig2d}\n\\end{figure} \n\n\n\n\n\\noindent{\\bf Case B.2.3: $\\chi^{2} \\ge 1$.}\nWhen $\\chi^{2} \\ge 1$, the hypersurfaces $y = \\pm t$  represent spacetime null infinities, \nand the solutions are already geodesically maximal. Indeed, it is found that the null goedesics \n$\\bar{u} = $ Constant have the integral,\n\\begin{equation}\n\\label{3.16al}\n\\eta = \\cases{\\eta_{0} \\left(-\\bar{v}\\right)^{1- \\chi^{2}}, & $\\chi^{2} > 1$,\\cr\n\\eta_{0} \\ln(- \\bar{v}), & $\\chi^{2} = 1$,\\cr}\n\\end{equation}\nnear the hypersurface $ y = t\\; (\\bar{v} = 0)$, where $\\eta_{0}$ is an integration constant, and\n$\\eta$ denotes the affine parameter along the null geodesics.\nThus, as $\\bar{v} \\rightarrow 0^{-}$, we always have $|\\eta| \\rightarrow \\infty$. \n\nSimilar, it can be shown that the hypersurface $y = -t$ also represents a null infinity of the \nspacetime. Therefore, the \ncorresponding Penrose diagram  is given by  Fig. \\ref{fig2e}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2e}\n\\caption{The Penrose diagram for the solutions given by Eq.(\\ref{3.16a4}) and\nfor the ones  given by Eq.(\\ref{3.16am})\nin the D-dimensional spacetime for $\\chi^{2} \\ge 1$. The horizontal line $t = 0$ \nrepresents a big bang  singularity. The $(D-2)$-dimensional surfaces of \nconstant $\\bar{u}$ and $\\bar{v}$ are always trapped in Regions $I$ and $I'$,\nand are physically disconnected in both cases. The  line $0D$ and $0E$ represent \nspacetime null infinities, and are not singular.   \n}\n\\label{fig2e}\n\\end{figure} \n\n\\subsubsection{Class IIc Solutions}\n\nThis class of solutions is given by\n\\begin{eqnarray}\n\\label{3.16am}\nM &=& \\frac{1}{2} c^{2} \\ln\\left(\\frac{\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2}}\n{y^{2} - t^{2}}\\right) + M_{0},\\nonumber\\\\\n\\phi &=& c \\ln\\left(y + \\sqrt{y^{2} - t^{2}}\\right) + \\phi_{0},\n\\end{eqnarray}\nfor which we find\n\\begin{eqnarray}\n\\label{3.16ana}\n\\bar{\\theta}_{l} &=& \\bar{\\theta}_{n} = \\frac{e^{-2\\kappa^{2}_{D}M_{0}}\n         \\left(y^{2} - t^{2}\\right)^{\\chi^{2}}}\n\t {t^{\\frac{1}{D-2}} \\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2\\chi^{2}}}\\nonumber\\\\\n&\\propto& \\cases{ t^{-\\frac{1}{D-2}}, & $y > 0$,\\cr\nt^{-\\left(4\\chi^{2} + \\frac{1}{D-2}\\right)}, & $ y < 0$,\\cr}\n\\end{eqnarray}\nas $ t \\rightarrow 0$. Therefore, in this case the hypersurface $t =0$ represents a \nfocusing point, while the ones $y = \\pm t$ may represent some kind of horizons. \nTo see their exact natures, let us consider the quantity,\n\\begin{eqnarray}\n\\label{3.16an}\nR_{D}[g] &=&   \\kappa_{D}^{2}g^{\\alpha\\beta}\\phi_{,\\alpha}\n\\phi_{,\\beta}   \\nonumber\\\\\n&=& - \\frac{2A_{0}\\left(y^{2} -  t^{2}\\right)^{\\chi^{2} -1\/2}}\n{\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{2\\chi^{2}+1}\n\\; t^{\\frac{D-3}{D-2}}.\n\\end{eqnarray}\nThus, the nature of the spacetime singularity at $t = 0$ is different on the half line\n$y > 0$  from that on the other half line  $y < 0$. In particular, we find\n\\begin{eqnarray}\n\\label{3.16anb}\nR_{D}[g]  &\\propto& \\cases{ - t^{\\frac{D-3}{D-2}} \\rightarrow 0, & $y > 0$,\\cr\nt^{-\\left(4\\chi^{2} + \\frac{1}{D-2}\\right)} \\rightarrow - \\infty, & $ y < 0$,\\cr} \n\\end{eqnarray}\nas  $ t \\rightarrow 0$. Therefore, $R_{D}[g]$ is singular only on the half\nline $t = 0, \\; y \\le 0$. However, the studies of other scalars show that the\nother half line, $t = 0$ and $y > 0$, is also singular. For example, the \ncorresponding Kretschmann scalar is given by\n\\begin{eqnarray}\n\\label{Kretsch}\nI_{D} &\\equiv& R^{\\mu\\nu\\lambda\\sigma}R_{\\mu\\nu\\lambda\\sigma}\\nonumber\\\\\n&=&\\left. \\frac{\\left(y^{2} - t^{2}\\right)^{2\\chi^{2} -5}}{8 t^{3}\n\\left(y + \\sqrt{y^{2} - t^{2}}\\right)^{4\\chi^{2}}}I^{(0)}_{D}(t,y)\\right|_{D=4},\n\\end{eqnarray}\nwhere, as $t \\rightarrow 0$, we have\n\\begin{equation}\n\\label{Kretsch2}\nI^{(0)}_{4}  \\simeq \\cases{- 9y^{10}, & $y > 0$,\\cr\n- \\left(1024\\chi^{6} -320\\chi^{4} + 96 \\chi^{2} + 9\\right)y^{10}, & $y < 0$.\\cr}\n\\end{equation}\n \n \nOn the other hand, depending on the value of $\\chi$, the spacetime may or may not\nbe singular on the  two null surfaces $y = \\pm t$, similar to the last case.\nIn particular,  when $\\chi^{2} < 1\/2$, the spacetime is singular there,   \nand the the corresponding Penrose diagram is also given  by Fig. \\ref{fig2b}.\n\n \n \nWhen $ \\frac{1}{2} \\le \\chi^{2} < 1$,  the spacetime is not singular at  \n$y = \\pm t$, although the metric coefficients are. Then, extending the solutions\nto the region $\\tilde{v} > 0$ is needed. As in the last case, the extension\nacross this surface is uniqe only when $n$ defined by Eq.(\\ref{3.16ak}) is an\ninteger. Thus, in the following we shall consider only this case. Then, the \nextension can  be done by  introducing the new  coordinates  $\\tilde{u}$ and\n$\\tilde{v}$ defined by Eq.(\\ref{3.16aj}), for which we find the extended metric \ntakes the same form as that given by Eq.(\\ref{3.2aa}), but now with \n\\begin{eqnarray}\n\\label{3.3a1a} \n\\tilde{H} &=&    \\frac{1}{D-2}\\ln\\left(\\tilde{u}^{2n} - \\tilde{v}^{2n}\\right),\\nonumber\\\\\n\\tilde{\\Sigma} &=&    \\left(2\\chi^{2} - \\frac{D-3}{2(D-2)}\\right)\n                     \\ln\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)\\nonumber\\\\\n\t& & - \\frac{D-3}{2(D-2)} \\ln\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)\n\t    + \\tilde{\\Sigma}_{0},\\nonumber\\\\\n\\tilde{\\Sigma}_{0} &\\equiv& \\kappa^{2}_{D}M_{0} \n  + \\frac{1}{2}\\ln\\left(4^{1 - \\chi^{2}} n^{2}\\right),\\nonumber\\\\\n  \\phi &=&  2c    \\ln\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)\n\t\t     + \\phi_{0}.\t\t     \n\\end{eqnarray} \nThus, the coordinate singularity at  $\\tilde{v} = 0$ disappears, and the solutions \ncan be considered as   valid for $\\tilde{v} > 0$, too.  In terms of $\\tilde{u}$ and \n$\\tilde{v}$, we find   \n\\begin{eqnarray}\n\\label{3.3aa2} \n\\tilde{\\theta}_{l} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{v}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\left(-\\tilde{v}\\right)^{2n-1}}{\\tilde{u}^{2n}\n             - \\tilde{v}^{2n}},\\nonumber\\\\\n\\tilde{\\theta}_{n} &=& \\left(D-2\\right)e^{-2\\tilde{\\Sigma}}\\tilde{H}_{,\\tilde{u}}\n= 2n e^{-2\\tilde{\\Sigma}}\\frac{\\tilde{u}^{2n-1}}{\\tilde{u}^{2n}\n             - \\tilde{v}^{2n}},\\nonumber\\\\\n{\\cal{L}}_{n}\\tilde{\\theta}_{l}\t &=& - 4n^{2}e^{-4\\tilde{\\Sigma}}\n\\frac{\\tilde{u}^{n-1}\\left(-\\tilde{v}\\right)^{2n-1}}{\\left(\\tilde{u}^{2n}\n             - \\tilde{v}^{2n}\\right)^{2}}\\nonumber\\\\\n& & \\times\\left\\{\\left(2\\chi^{2} + \\frac{1}{D-2}\\right)\\tilde{u}^{n} \n    - 2\\chi^{2}\\left(-\\tilde{v}\\right)^{n}\\right\\},\\nonumber\\\\    \nR_{D}\\left[\\tilde{g}\\right] &=& - 4^{\\chi^{2} + \\frac{1}{2}}\n\\chi^{2}  e^{-2\\kappa^{2}_{D}M_{0}} \\nonumber\\\\\n& & \\times \\frac{\\left(\\tilde{u}^{n} - \\left(-\\tilde{v}\\right)^{n}\\right)^{\\frac{D-3}{D-2}}\n\\left(-\\tilde{u}\\tilde{v}\\right)^{n-1}}\n{\\left(\\tilde{u}^{n} + \\left(-\\tilde{v}\\right)^{n}\\right)^{4\\chi^{2} + \\frac{D-1}{D-2}}}.\n\\end{eqnarray}\nFrom the above expressions, we can see that  in the extended region where \n$\\tilde{u} \\tilde{v} >0$ the  spacetime\nbecomes untrapped. Across the hypersurface $\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$,\nwe have $\\tilde{\\theta}_{l} = 0,\\; \\tilde{\\theta}_{n}   >0$ and \n$ {\\cal{L}}_{n}\\tilde{\\theta}_{l}   = 0$. That is, the half infinite line \n$\\tilde{v} = 0,\\; \\tilde{u} \\ge 0$ acts as a past degenerate apparent horizon. \n\nIn addition, in contrast to the last case, the expression of $R_{D}\\left[\\tilde{g}\\right]$\ngiven above tells us that the spacetime is singular along the vertical line\n$\\tilde{u} = \\tilde{v}$ for $n$ being an odd integer, and not singular for \n$n$ being an even integer. The corresponding Penrose diagram is given, respectively, by\nFigs. \\ref{fig2b} and \\ref{fig2c}. \n\n \n\n\nWhen $\\chi^{2} \\ge 1$, the hypersurfaces $y = \\pm t$ already represent the null \ninfinities  of the spacetime, \nand the corresponding Penrose diagram  is that \nof Fig. \\ref{fig2e}, where the hypersurrfaces $\\bar{u} = 0$ and  \n$\\bar{v} = 0$  are not singular, and\nthe two regions, $II$ and $II'$, are still not connected.\n\n \n\n\n\\section{Solutions  in ($D+d$)-dimensional Spacetimes in the string Frame} \n\n\\renewcommand{\\theequation}{4.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n \nIn ($D+d$)-dimensions, the metric in the string frame takes the form of Eq. (\\ref{2.2}). Considering\nEq. (\\ref{3.2}), we find that it can be written in the form,\n\\begin{eqnarray}\n\\label{8.1}\nd\\hat{s}^{2}_{D+d} &=& \\hat{g}_{AB}dx^{A} dx^{B} = \\gamma_{\\mu\\nu}dx^{\\mu}dx^{\\nu} + \\hat{\\Phi}\\hat{\\gamma}_{ab}dz^{a}dz^{b}\\nonumber\\\\\n&=& 2e^{2\\hat{\\sigma}(\\hat{u}, \\hat{v})}d\\hat{u}d\\hat{v} - e^{2\\hat{h}(\\hat{u}, \\hat{v})}d\\Sigma_{D-2}^{2} \\nonumber\\\\\n& & \n- e^{2\\hat{g}(\\hat{u}, \\hat{v})}\\hat{\\gamma}_{ab}dz^{a}dz^{b},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.17a}\n\\gamma_{\\mu\\nu} &=& \\exp\\left\\{\\epsilon_{a}\n\\left(\\frac{4\\kappa^{2}_{D}d}{(D-2)(D+d-2)}\\right)^{1\/2}\\phi\\right\\}\n g_{\\mu\\nu},\\nonumber\\\\\n\\hat{\\Phi}  &=& e^{\\hat{g}} =  \\exp\\left\\{- \\epsilon\n\\left(\\frac{(D-2)\\kappa^{2}_{D}}{(D+d-2)d}\\right)^{1\/2}\\phi\\right\\},\n\\end{eqnarray}\n as can be seen from Eqs.(\\ref{2.7}) and (\\ref{2.9}), where $\\epsilon_{a} = \\pm 1$.\n \n Introducing the two null vectors, $\\hat{l}_{A}$ and $\\hat{n}_{A}$ via\n \\begin{equation}\n \\label{8.2}\n \\hat{l}_{A} = \\delta^{\\hat{u}}_{A}, \\;\\;\\;  \\hat{n}_{A} = \\delta^{\\hat{v}}_{A},\n \\end{equation}\n once can show that they also define two null affinely defined geodesic congruences,\n $ \\hat{l}_{A; B} \\hat{l}^{B} =  0 =  \\hat{n}_{A; B} \\hat{n}^{B}$. The corresponding expansions\n are given by\n \\begin{eqnarray}\n \\label{8.3}\n \\hat{\\theta}_{l} &\\equiv&  \\hat{l}_{A; B}\\hat{g}^{AB}\n    =e^{-2\\hat{\\sigma}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{v}} + d \\hat{g}_{,\\hat{v}}\\Big],\\nonumber\\\\\n  \\hat{\\theta}_{n} &\\equiv&  \\hat{n}_{A; B}\\hat{g}^{AB}\n   = e^{-2\\hat{\\sigma}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{u}} + d \\hat{g}_{,\\hat{u}}\\Big],\n \\end{eqnarray}\n from which we find\n  \\begin{eqnarray}\n \\label{8.4}\n{\\cal{L}}_{\\hat{n}}  \\hat{\\theta}_{l} &=&   e^{-4\\hat{\\sigma}}\\Bigg\\{\\big(D-2\\big)\\hat{h}_{,\\hat{u}\\hat{v}} + d \\hat{g}_{,\\hat{u}\\hat{v}}\\nonumber\\\\\n& & ~~~~~~~~\n - 2\\hat{\\sigma}_{,\\hat{u}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{v}} + d \\hat{g}_{,\\hat{v}} \\Big]\\Bigg\\},\\nonumber\\\\\n{\\cal{L}}_{\\hat{l}}  \\hat{\\theta}_{n} &=&   e^{-4\\hat{\\sigma}}\\Bigg\\{\\big(D-2\\big)\\hat{h}_{,\\hat{u}\\hat{v}} + d \\hat{g}_{,\\hat{u}\\hat{v}}\\nonumber\\\\\n& & ~~~~~~~~\n - 2\\hat{\\sigma}_{,\\hat{v}}\\Big[\\big(D-2\\big)\\hat{h}_{,\\hat{u}} + d \\hat{g}_{,\\hat{u}} \\Big]\\Bigg\\}.\n  \\end{eqnarray}\n\n \n\n\n\\subsection{$F'(u) \\not= 0,\\; G'(v)  = 0$}\n\nIn this case, the modulus  $\\phi(u,v)$ is given by Eq.(\\ref{3.15}). \nIn the $(D + d)$-dimensional \nspacetime, the metric can be written in the form\n\\begin{eqnarray}\n\\label{3.30}\nd{\\hat{s}}^{2}_{D+d} &=&  2d\\hat{u}d\\hat{v} \n- e^{2\\hat{h}(\\hat{u})}d^{2}\\Sigma_{D-2}\\nonumber\\\\\n& & \\;\n  - \\hat{\\Phi}^{2}(\\hat{u}) \\hat{\\gamma}_{ab}(z) dz^{a} dz^{b},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{3.30a}\nd\\hat{u}  &\\equiv& e^{2\\hat{\\sigma}}du,\\nonumber\\\\\n\\hat{\\sigma} &\\equiv& \\sigma - \\frac{d}{D-2}\\ln\\hat{\\Phi},\\nonumber\\\\\n\\hat{h} &\\equiv& h - \\frac{d}{D-2}\\ln\\hat{\\Phi}.\n\\end{eqnarray}\nFollowing what we did in the Einstein frame, we can construct\na free-falling frame in the $(D+d)$-dimensions, given by \n\\begin{eqnarray}\n\\label{3.34}\ne^{A}_{(0)} &=& \\hat{\\gamma}_{0}\\delta^{A}_{\\hat{u}} \n+ \\frac{1}{2\\hat{\\gamma}_{0}}\\delta^{A}_{v},\\nonumber\\\\\ne^{A}_{(1)} &=& \\hat{\\gamma}_{0}\\delta^{A}_{\\hat{u}} \n- \\frac{1}{2\\hat{\\gamma}_{0}}\\delta^{A}_{v},\\nonumber\\\\\ne^{A}_{(i)} &=& e^{-\\hat{h}}  \\delta^{A}_{i},\\nonumber\\\\\ne^{A}_{(b)} &=& \\hat{\\Phi}^{-1}  \\delta^{A}_{b}, \n\\end{eqnarray}\n which satisfies the relations,\n\\begin{equation}\n\\label{3.35}\ne^{A}_{(C)}e^{B}_{(D)} \\hat{g}_{AB} = \\eta_{CD},\\;\\;\\;\ne^{A}_{(C); B} e^{B}_{(0)} = 0,\n\\end{equation}\nwhere $\\hat{\\gamma}_{0}$ is another integration constant.\nThen, it can be shown that the Riemann tensor in this case has only two \nindependent components, given by\n\\begin{eqnarray}\n\\label{3.33}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=& \n- \\hat{\\gamma}_{0}^{2}\\left(\\hat{h}_{,\\hat{u}\\hat{u}}\n+  \\hat{h}_{,\\hat{u}}^{2}   \\right)\\delta^{i}_{j},\\nonumber\\\\\n\\hat{R}^{(a)}_{ \\;\\;\\;\\; (0)(b)(0)} &=& \n- \\hat{\\gamma}_{0}^{2}\\left(\\frac{\\hat{\\Phi}_{,\\hat{u}\\hat{u}}}\n{\\hat{\\Phi}}\\right)\\delta^{a}_{b}. \n\\end{eqnarray}\n\nIn addition, Eqs.(\\ref{8.3}) and (\\ref{8.4}) now read,\n\\begin{eqnarray}\n\\label{3.33aa}\n\\hat{\\theta}_{n} &=& \\big(D-2\\big) \\alpha'(u)\\exp\\Big(-a u^{(2-\\gamma)\/2}\\Big), \\nonumber\\\\\n\\hat{\\theta}_{l} &=& 0, \\;\\;\\; {\\cal{L}}_{\\hat{n}}  \\hat{\\theta}_{l} = 0 = {\\cal{L}}_{\\hat{n}}  \\hat{\\theta}_{l}.\n\\end{eqnarray}\nThen, according the previous\ndefinition, the $(D+d-2)$-surfaxe ${\\cal{S}}$ is   always marginally trapped and degenerate.  \n\n\n\nTo study the solutions further, it is found convenient to\nconsider the two cases $\\gamma \\not= 2$ and $\\gamma =2$ separately.\n\n\n\n\n\n\n\n\\subsubsection{$\\gamma \\not= 2$}\n\nIn this case, we have  \n\\begin{eqnarray}\n\\label{3.31}\nd{\\hat{u}}  &=& e^{au^{1-\\gamma\/2}}du,\\nonumber\\\\\n\\hat{h}   &=& \\frac{1}{2}a u^{1-\\gamma\/2} + \\alpha(u),\\nonumber\\\\\n\\hat{\\Phi}   &=& e^{b u^{1-\\gamma\/2}}, \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.31a}\na &\\equiv& \\frac{2\\epsilon_{a}}{2-\\gamma} \n\\left(\\frac{4\\omega^{2}d}{D+d-2}\\right)^{1\/2},\\nonumber\\\\\nb &\\equiv& - \\frac{2\\epsilon_{a}}{2-\\gamma} \n\\left(\\frac{\\omega^{2}(D-2)^{2}}{(D+d-2)d}\\right)^{1\/2}.\n\\end{eqnarray}\nThen, the non-vanishing frame components of the Riemann tensor are given by\n\\begin{eqnarray}\n\\label{3.33a}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=&  \n\\hat{\\gamma}_{0}^{2} e^{-2au^{1-\\gamma\/2}}\\nonumber\\\\\n& & \\times \\left\\{\n\\frac{a\\gamma(2-\\gamma)}{8 u^{\\gamma\/2+1}}\n+ \\frac{a^{2} (2-\\gamma)^{2} + 16\\omega^{2}}{16 u^{\\gamma}}\\right\\}\n\\delta^{i}_{j},\\nonumber\\\\\n\\hat{R}^{(a)}_{\\;\\;\\;\\; (0)(b)(0)} &=& \n\\hat{\\gamma}_{0}^{2} \\frac{b(2-\\gamma)}{4}e^{-2au^{1-\\gamma\/2}}\\nonumber\\\\\n& & \\times \\left\\{\n\\frac{ \\gamma }{u^{\\gamma\/2+1}}\n- \\frac{(b-a) (2-\\gamma)}{u^{\\gamma}}\\right\\}\n\\delta^{a}_{b}.\n\\end{eqnarray}\nTherefore, for the choice $\\epsilon_{a} = +1$ we have\n\\begin{eqnarray}\n\\label{3.36}\n\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)} &=& \\cases\n\\infty, & $\\gamma> 0$,\\cr\n{\\mbox{constant}}, & $\\gamma = 0$,\\cr\n\\infty, & $- 2 < \\gamma < 0$,\\cr\n{\\mbox{finite}}, & $\\gamma \\le -2$,\\cr}\\\\\n\\label{3.36a}\n\\hat{\\theta}_{n} &=& \\cases\n\\infty, & $\\gamma> 2$,\\cr\n{(D-2)\\alpha'(0)}, & $\\gamma \\le 2$,\\cr}\n\\end{eqnarray}\nas $u \\rightarrow 0$, but now with $A, B = i, a$. Thus, the\ntidal forces  experiencing by a free-falling observer remain finite\nin the string frame at $u = 0$ for all the cases, except for the ones \nwhere $0 < \\gamma$ or $ -2 < \\gamma < 0$. As a result,\nthe spacetime is singular at $u = 0$ for these latter solutions.\nHowever, the singularity is weak for $|\\gamma| < 2$, because the distortion exerting on the \nobserver is still finite,\n\\begin{eqnarray}\n\\label{3.37}\n \\int{d\\lambda \\int{\\hat{R}^{(A)}_{\\;\\;\\;\\; (0)(B)(0)} d\\lambda}} \n &\\sim & A_{1} \\lambda^{2-\\gamma} + A_{2} \\lambda^{1-\\gamma\/2} \\nonumber\\\\\n&\\sim &  {\\mbox{finite}},\n\\end{eqnarray}  \nas $ \\lambda \\rightarrow 0 \\; (\\mbox{or} \\; u \\rightarrow 0)$ when\n$|\\gamma| < 2$, where $A_{1}$ and $A_{2}$ \nare finite constants. \n\nTherefore, for the choice $\\epsilon_{a} = +1$ the strong singularities of \nthe solutions with $\\gamma > 2$ at $u = 0$ in the Einstein frame now \nremain in the string  frame. The singularities of the solutions with $0 < \\gamma < 2$ \nare weak in both of the two frames. The solutions with $ -2 < \\gamma < 0$ \nis free from singularity in the Einstein frame, while they become singular \nin the string frame, although the nature of the singularities is still weak. \nThe solutions with $ \\gamma = 0$ and $\\gamma \\le -2$ are free from singularity \nat $u = 0$ in both of the two frames.  \n\nNote that for $\\gamma = 0$ Eq.(\\ref{3.33a}) shows that \n\\begin{equation}\n\\label{3.37a}\n\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)} \\rightarrow 0,\n\\end{equation}\nas   $u \\rightarrow \\infty$. Thus, in this case the spacetime singularity \nat $u = \\infty$ appearing in the Einstein frame now disappears in the \n$(D+d)$-dimensional string frame, although the null infinity $u = - \\infty$ \nstill remains singular [cf. Fig. \\ref{fig1a}].\n \n\nWhen $\\epsilon_{a} = - 1$, from Eq.(\\ref{3.33}) we find that\n\\begin{equation}\n\\label{3.38}\n\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)} = \\cases{0, & $\\gamma > 2$,\\cr\n\\infty, & $ 2 > \\gamma > 0$,\\cr\n{\\mbox{constant}}, & $\\gamma = 0$,\\cr\n\\infty, & $- 2 < \\gamma < 0$,\\cr\n{\\mbox{finite}}, & $\\gamma \\le -2$,\\cr}\n\\end{equation}\nas $u \\rightarrow 0$. It can be shown that in this case the nature of the \nsingularities of the solutions remains the same in both of the two frames\nfor $2 > \\gamma \\ge  0$ and $\\gamma \\le -2$, that is, in both frames\nit is  \nweak for $ 0 < \\gamma < 2$, and free of singularities for  $\\gamma = 0$\nand $\\gamma \\le -2$.\nFor $-2 < \\gamma < 0$, the solutions are free of singularities  \nin the Einstein frame, but singular in the string frame with the nature\nof the singularities being still weak.     \nFor $ 2 < \\gamma$, on the other hand, the solutions are free of singularities  \nin the string frame, but singular in the Einstein frame with the nature\nof the singularities being   strong.   \n\nSimilarly, one can show that for $\\gamma = 0$ the spacetime singularity \nat $u = -\\infty$ appearing in the Einstein frame now disappears in the \n$(D+d)$-dimensional string frame, although the spacetime is still\nsingular at the null infinity $u = + \\infty$ [cf. Fig. \\ref{fig1a}].\n \n\n\\subsubsection{$\\gamma = 2$}\n\nWhen $\\gamma = 2$, the corresponding solutions in the Einstein frame\nare given by Eqs.(\\ref{3.15}) and (\\ref{3.16}). The solutions have\na strong singularity at $u = 0$. In the string frame, the corresponding\nsolutions are given by Eq.(\\ref{3.30}) but with\n\\begin{eqnarray}\n\\label{3.39}\n\\hat{h}(\\hat{u}) & =& \\frac{a+2\\delta}{2(1+a)}\\ln\\left|\\hat{u}\\right|,\\nonumber\\\\\n\\hat{\\Phi}(\\hat{u}) & =& \\left((1+a)\\hat{u}\\right)^{\\frac{b}{1+a}}, \n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{3.39a}\n\\hat{u} \\equiv \\frac{1}{1+a} u^{1+a} = \\cases{0, & $ a > -1$,\\cr  \n- \\infty, & $ a < -1$,\\cr}\n\\end{equation}\nas $u \\rightarrow 0$. Thus, when $a > -1$ the half plane $u \\ge 0$ is mapped\nto the half plane $\\hat{u} \\ge 0$, and the hypersurface $u = 0 \\; (u = \n\\infty)$ is mapped to the one $\\hat{u} = 0 \\; (\\hat{u} = \n\\infty)$. when $a < -1$ the half plane $u \\ge 0$ is mapped\nto the one $\\hat{u} \\le 0$, and the hypersurface $u = 0 \\; (u = \n\\infty)$ corresponds to the one $\\hat{u} = -\\infty \\; (\\hat{u} = \n0)$. \n\nIt can be shown that now we have  \n\\begin{eqnarray}\n\\label{3.40}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=&  \n- \\hat{\\gamma}_{0}^{2} \\frac{(a+2\\delta)(2\\delta - a -2)}\n{4(1+a)^{2}\\hat{u}^{2}}\\delta^{i}_{j},\\nonumber\\\\\n\\hat{R}^{(a)}_{\\;\\;\\;\\; (0)(b)(0)} &=& \n- \\hat{\\gamma}_{0}^{2} \\frac{b(b - a -1)}\n{(1+a)^{2}\\hat{u}^{2}} \\delta^{a}_{b}.\n\\end{eqnarray}\nClearly, the spacetime is singular at $\\hat{u} = 0$, and the nature\nof the singularity is strong, because\n\\begin{equation}\n\\label{3.41}\n\\int{d\\lambda\\int{\\hat{R}^{(A)}_{\\;\\;\\;\\;(0)(B)(0)}d\\lambda}} \\sim \\ln{\\lambda}\n\\rightarrow - \\infty,\n\\end{equation}\nas $ \\lambda \\rightarrow 0 \\; ({\\mbox{or}}\\; \\hat{u} \\rightarrow 0)$. \nNote that the distortion also becomes unbound as $|\\hat{u}| \\rightarrow    \\infty\n\\; (|\\lambda| \\rightarrow \\infty)$, although the tidal forces vanish there.\n \nIt should be noted that the above analysis is valid only for $a \\not= -1$.\nWhen $a = -1$, we find that\n\\begin{equation}\n\\label{3.42}\n\\omega^{2} = \\frac{D+d-2}{4d}, \\;\\; \\; \nb = \\frac{D-2}{2d}. \n\\end{equation}\nThe corresponding solutions are given by\n\\begin{eqnarray}\n\\label{3.43}\n\\hat{h}(\\hat{u}) &=& \\left(\\delta - \\frac{1}{2}\\right)\\hat{u},\\nonumber\\\\\n\\hat{\\Phi}(\\hat{u}) &=& e^{\\frac{D-2}{2d} \\hat{u}},\n\\end{eqnarray}\nwhere $u \\equiv  e^{\\hat{u}}$.\nThen, we find that\n\\begin{eqnarray}\n\\label{3.44}\n\\hat{R}^{(i)}_{\\;\\;\\;\\; (0)(j)(0)} &=&  \n- \\hat{\\gamma}_{0}^{2} \\left(\\delta - \\frac{1}{2}\\right)^{2}\\delta^{i}_{j}, \\nonumber\\\\\n\\hat{R}^{(a)}_{\\;\\;\\;\\; (0)(b)(0)} &=& \n- \\hat{\\gamma}_{0}^{2} \\left(\\frac{D-2}{2d}\\right)^{2}  \n\\delta^{a}_{b},\n\\end{eqnarray}\nwhich are finite (constants). However, at the null infinities \n$\\hat{u} = \\pm \\infty$, which correspond, respectively,  to $u = 0$ and $u = \\infty$,\nthe distortions are still unbound. As a result, the ($D+d$)-dimensional spacetimes  remain\nsingular on these surfaces. \n\nTherefore, when $\\gamma = 2$   the corresponding Penrose daigram of the ($D+d$)-dimensional \nspacetimes is that of Fig. \\ref{fig1a}, where the two hypersurfaces $u = 0$ and $u = \\infty$ \nremain singular.\n\n\n\n\n\n\n\\subsection{ $F'(u) G'(v)  \\not= 0$}\n\nIn this case, three classes of solutions were studied in the last section.  To have them \nmanagible, in this subsection we shall restrict ourselves only to $D = d = 5$. We generalize \nthe metric (\\ref{3.30}) and using $t=u+v$, $y=u-v$ we arrive at the form:\n\\begin{eqnarray}\n\\label{3.45}\nd{\\hat{s}}^{2}_{10} &=& \\frac{1}{2} e^{2A(t,y)} \\left(dt^2-dy^2\\right) - e^{2B(t,y)}d\\Sigma^{2}_{3} \\nonumber\\\\\n& &  - e^{2C(t,y)}d\\Sigma^{2}_{5,z},\n\\end{eqnarray}\nwhere $d\\Sigma^{2}_{5,z} \\equiv {\\hat{\\gamma}_{ab}\\left(z^{c}\\right)dz^{a}dz^{b}}\\;\n(a, b = 1, 2, ..., 5)$, and\n\\begin{eqnarray}\n\\label{3.46}\nA &=& \\sigma - \\frac{5}{3}\\beta \\phi = - \\frac{1}{3} \\ln(t) + \\kappa^{2}_{5}M -  \\frac{5}{3}\\beta \\phi,\\nonumber\\\\\nB &=& h - \\frac{5}{3}\\beta \\phi = \\frac{1}{3} \\ln(t) - \\frac{5}{3}\\beta \\phi,\\nonumber\\\\\nC &=&  \\beta \\phi, \\hspace{14pt}\n\\beta =  \\epsilon \\sqrt{\\frac{3\\kappa_{5}^{2}}{40}}, \\;\\;\\;\n\\epsilon_{a} = \\pm 1. \n\\end{eqnarray}\n\n\n\\subsubsection{Class IIa Solutions}\n\n\nIn this case, substituting the solution (\\ref{3.16af}) into Eq.(\\ref{3.46}) and setting \n$M_0=\\phi_0=0$ without loss of generality, we obtain that\n\\begin{eqnarray}\n\\label{3.47a}\nA  &=& \\frac{1}{2} \\left[\\left( c\\kappa_5 - \\epsilon \\sqrt{\\frac{5}{24}} \\right)^2  \n- \\frac{7}{8}\\right] \\ln(t), \\nonumber\\\\\nB  &=& \\left(\\frac{1}{3} - \\epsilon \\sqrt{\\frac{5}{24}} c\\kappa_5\\right) \\ln(t), \\nonumber\\\\\nC  &=& \\epsilon_{a} \\sqrt{\\frac{3}{40}} c\\kappa_5 \\ln(t).\n\\end{eqnarray}\nThe corresponding Kretschmann scalar is given by,\n\\begin{equation}\n\\label{3.48a}\nI_{10} \\equiv  R_{ABCD}R^{ABCD} = \\frac{\\tilde{I}_{10}}{t^{\\alpha_0}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\label{alpha0}\n\\alpha_0    &=& 2\\left(c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{24}} \\right)^2 + \\frac{9}{4} > 0, \\nonumber\\\\\n\\tilde{I}_{10} &=& \\frac{1}{45} \\left[9 \\chi^{2} \\left(40\\chi + 143\\right)  \n\t- \\epsilon_{a} 52 \\sqrt{30} \\chi^{3\/2}  \\left(3 \\chi + 4\\right)\\right.\\nonumber\\\\\n         & & \\left. + 80\\left(5 \\chi +2\\right)\\right],\n\\end{eqnarray}\nnow with $\\chi \\equiv c^{2}\\kappa^{2}_{5}$.\nClearly, the spacetime is always singular at $t = 0$ for any given $c$, similar to that in \nthe 5-dimensional case. Therefore, in the present case, the spacetime singularity remains even\nafter lifted  from the effective 5-dimensional spacetime to the 10 dimensional bulk. \n\n\n\\subsubsection{Class IIb Solutions}\n\n\nIn this case, the combination of Eqs.(\\ref{3.16ah}) and (\\ref{3.46}) yields\n\\begin{eqnarray}\n\\label{3.47b}\nA &=& \\left[2\\left(c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{96}}\\right)^2 \n- \\frac{7}{16}\\right] \\ln(t) \\nonumber\\\\\n  & & - \\left[\\left(c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{96}}\\right)^2 \n  - \\frac{5}{96}\\right] \\ln \\left (y+\\sqrt{y^2-t^2} \\right) \\nonumber\\\\\n  & & - \\frac{1}{2} \\chi^2 \\ln \\left( y^2 - t^2 \\right), \\nonumber\\\\\nB &=&  \\left( \\frac{1}{3} - \\epsilon_{a} \\sqrt{\\frac{5}{6}} c\\kappa_5 \\right) \\ln(t) \\nonumber\\\\\n  & & - \\epsilon_{a} \\sqrt{ \\frac{5}{24} } c \\kappa_5  \\ln \\left( y+\\sqrt{y^2-t^2} \\right),  \\nonumber\\\\\nC &=& \\epsilon_{a} \\sqrt{\\frac{3}{10}} c\\kappa_5 \\ln(t) \\nonumber\\\\\n  & & - \\epsilon_{a} \\sqrt{\\frac{3}{40}} c\\kappa_5 \\ln\\left( y+\\sqrt{y^2-t^2}\\right),\n\\end{eqnarray}\nfor which we find that\n\\begin{equation}\n\\label{3.48b}\nI_{10} = \\frac{\\tilde{I}_{10}}{t^{\\alpha_{0}}(y^2-t^2)^{\\alpha_{1}}(y+\\sqrt{y^2-t^2})^{\\alpha_{2}}}, \n\\end{equation}\nwhere $\\alpha_{0}$ is given by Eq.(\\ref{alpha0}),\n\\begin{eqnarray}\n\\label{alpha1}\n\\alpha_{1} &\\equiv& 2 \\left(\\frac{3}{4} - \\chi\\right),\\nonumber\\\\\n\\alpha_{2} &\\equiv& \\frac{101}{24} - \\left(2 c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{24}}\\right)^2,\n\\end{eqnarray} \nand $\\tilde{I}_{10} = \\tilde{I}_{10}(t,y)$, which is non-zero for $t = 0$ and\n$y^{2} = t^{2}$, but its  expression is too complicated to give it here explicitly. \n\nFrom the above expression, it can be seen that the spacetime is always singular at $t = 0$, but the\nstrenght of the singularity for $y > 0$ and $y < 0$ is different, because when $t = 0$ we have\n$y+\\sqrt{y^2-t^2} = 0$ for $y \\le 0$ and $y+\\sqrt{y^2-t^2} \\not= 0$ for $y > 0$.\nIn particular, when  $t = 0$ and $y > 0$, we find that\n\\begin{equation}\nI_{10} \\Big|_{t=0,\\; y>0} \\simeq \\frac{\\tilde{I}_{10}}{t^{\\alpha_{0}}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\label{aa}\n\\tilde{I}_{10} \\Big|_{t=0} & = & \\frac{256}{45} y^8 \\bigg[9\\chi^2 \\left(160 \\chi + 143\\right) \\nonumber\\\\\n    & & \n    - \\epsilon_{a} 104 \\sqrt{30} \\chi^{3\/2} \\left(3 \\chi + 1\\right) \\nonumber\\\\\n    & & \n    + 10 \\left(10 \\chi + 1\\right)\\bigg].\n\\end{eqnarray}\nOn the other hand, when $t = 0$ and $y < 0$, we find that  \n\\begin{equation}\nI_{10} \\Big|_{t=0, \\; y<0} \\simeq \\frac{y^{\\alpha_{2} - 2\\alpha_{1}} \\tilde{I}_{10}}\n{t^{{32}\/{3}}}, \n\\end{equation}\nwhere $\\tilde{I}_{10}$ is still given by Eq.(\\ref{aa}).  \n\nEqs.(\\ref{3.48b}) and (\\ref{alpha1}) also show that  the spacetime is singular when \n$y^{2} - t^{2} = 0$ for $\\chi < 3\/4$,   \n\\begin{equation}\nI_{10(b)} \\Big|_{t^2=y^2} \\propto \\frac{\\tilde{I}_{10}}{(y^2-t^2)^{\\alpha_{1}}},\n\\end{equation}\nbut now with\n\\begin{eqnarray}\n\\label{4.33}\n\\tilde{I}_{10} \\Big|_{t^2=y^2} &=& \\pm\\frac{8}{15}t^7\\chi \\bigg[20(6\\chi^{2}-\\chi-1) \\nonumber\\\\\n                    & & - 13\\epsilon_{a} \\sqrt{30} c\\kappa_5(2\\chi^2-1)\\bigg].\n\\end{eqnarray}\nThe corresponding Penrose diagram for $\\chi < 3\/4$ is given by Fig. \\ref{fig2b}.\nIt is remarkable to note that the solutions with $ 1\/2 \\le \\chi < 3\/4$ is not singular in the \n5-dimensional effective theory, as shown explicitly in Sec. III.B.2. \n\nSince the solution in the 10-dimensional bulk is not singular across the hypersurfaces $y^{2}\n= t^{2}$ for $\\chi \\ge 3\/4$, one must extend the solutions beyond these surfaces. The\nextension is quite similar to the 5-dimensional case for the ones with $\\chi \\ge 1\/2$.\nIn particular, for  $3\/4 \\le \\chi < 1$, setting \n\\begin{equation}\n\\label{4.33a}\n\\bar{u} = (y + t)^{2n},\\;\\;\\;\n\\bar{v} = (y - t)^{2n},\n\\end{equation}\nwhere $n$ is given by Eq.(\\ref{3.16ak}), one can show that the coordinate singularity at $y^{2} = t^{2}$\ndisappears in terms of   $\\bar{u}$ and $ \\bar{v}$. Then, the Penrose diagram for the extended\nsolutions is given exactly by Fig. \\ref{fig2a}. \n\nWhen $\\chi \\ge 1$, the hypersurfaces $y^{2} = t^{2}$ already  represent the null infinities, and\nthe corresponding Penrose diagram is given by Fig. \\ref{fig2b}, but  now the hypersurfaces \n$y^{2} = t^{2}$, represent,\nrespectively, by the lines $0D$ and $0E$, are non-singular. \nThe two regions $I$ and $I'$ are physically disconnected.\n\n\n\n\n\\subsubsection{Class IIc Solutions}\n\n\n\nIn this case, from Eq.(\\ref{3.16am}) we find that\n\\begin{eqnarray}\n\\label{3.47c}\nA &=& -\\frac{1}{3} \\ln(t) - \\frac{1}{2} \\chi^2 \\ln(y^2-t^2)  \\nonumber\\\\\n  & & + \\left[\\left( c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{96}} \\right)^2  \n  - \\frac{5}{96}\\right] \\ln \\left( y + \\sqrt{y^2-t^2}\\right),\\nonumber\\\\\nB &=& \\frac{1}{3} \\ln(t) - \\epsilon_{a} \\sqrt{\\frac{5}{24}} c \\kappa_5 \\ln\\left(y+\\sqrt{y^2-t^2}\\right), \\nonumber\\\\\nC &=& \\epsilon_{a} \\sqrt{\\frac{3}{40}} c\\kappa_5 \\ln \\left( y+\\sqrt{y^2-t^2} \\right),\n\\end{eqnarray}\nand that \n\\begin{equation}\n\\label{3.48c}\nI_{10} = \\frac{\\tilde{I}_{10}}{t^{{8}\/{3}}(y^2-t^2)^{\\alpha_{1}}(y+\\sqrt{y^2-t^2})^{\\alpha_{3}}}, \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{alpha3}\n\\alpha_{3} \\equiv \\frac{91}{24} + \\left( 2 c\\kappa_5 - \\epsilon_{a} \\sqrt{\\frac{5}{24}} \\right)^2 > 0,\n\\end{equation}\nand $\\tilde{I}_{10} = \\tilde{I}_{10}(t,y)$  is non-zero and finite at $t = 0$, but too complicated\nto be written out here. \nFrom the above expressions we can see that the spacetime is   singular at $t = 0$.\nThe strength of the singularities once again depends on $y < 0$\nand $y > 0$. In partucular, when  $t = 0$ and $y >0$, we find that\n\\begin{equation}\nI_{10} \\Big|_{t = 0, y > 0} \\propto \\frac{\\tilde{I}_{10}}{t^{8\/3}},\n\\end{equation}\nwith $\\tilde{I}_{10} \\simeq   {512}y^7\/9$. But for   $t = 0$ and $y < 0$, we find that\n\\begin{equation}\nI_{10} \\Big|_{t=0, y<0} = \\frac{ \\tilde{I}_{10}(y-\\sqrt{y^2-t^2})^{\\alpha_{3}}}\n{t^{{8}\/{3}+2\\alpha_{3}}(y^2-t^2)^{\\alpha_{1}}}, \n\\end{equation}\nwhere  $\\alpha_{3} >0$, as shown by Eq.(\\ref{alpha3}). Therefore, we again have a spacetime \nsingularity at $t=0$ for $y < 0$, but with more singular strength.\n\nThe singular behavior of the spacetime along the hypersurfaces $y = \\pm t$ are similar to the last\ncase. In particular, it is singular  for $\\chi < 3\/4$,  and corresponding Penrose diagram   is given \nby Fig. \\ref{fig2b}. It is interesting to note again that the solutions with $ 1\/2 \\le \\chi < 3\/4$ \nis not singular in the 5-dimensional effective theory. \n\nWhen $1> \\chi \\ge 3\/4$   the corresponding solutions in the 10-dimensional bulk is not singular \nacross the hypersurfaces $y^{2} = t^{2}$, one must extend the solutions beyond these surfaces. The\nextension is quite similar to the last case by simply introducing two null coordinates, defined by\nEq.(\\ref{4.33a}).  Then, the Penrose diagram for the extended\nsolutions is given exactly by Fig. \\ref{fig2a}. \n\nWhen $\\chi \\ge 1$, the hypersurfaces $y^{2} = t^{2}$ already  represent the null infinities, and\nthe corresponding Penrose diagram is given by Fig. \\ref{fig2b}.\n\n \n \n\\section{Conclusions and Discussing Remarks}\n\nAccording to the singularity theorems \\cite{HE73}, the 4-dimensional spacetimes are \ngenerically singular, if  matter filled the spacetimes satisfies certain energy condition(s).\nIt is a common belief that new physics involved with quantum mechanics will play an important\nrole    when the spacetime curvature is very high. The new physics gives us hope that\nthe classical singularities might be finally removed. \n\nHowever, it was shown that certain \nsingularities can be also removed by simply passing to a higher-dimensional theory of gravity, \nfor which spacetime is only effectively four-dimensional below some compactification scale \n \\cite{GHT95,RMS07}. This is because when lifted to higher-dimensions, the conditions required\n by the singularity theorems of Penrose and Hawking do not hold any more. \n\nIn this paper, we have investigated this  problem, by first studying the local and global properties\nof the spacetimes in the low dimensional effective theory, and then lifting them to the corresponding\n high dimensional spacetimes. We have shown explicitly that spacetime\nsingularities may or may not remain after lifted to higher dimensions, depending on the particular\nsolutions considered. We have also found that there exist cases in which the spacetime singularities of low \ndimensional effective theory occur in different surfaces from those of high dimensional spacetimes. \n\n\n \n \n\\begin{acknowledgements}\n\nOne of the authors (AW) would like to express his gratitude to Jianxin Lu for organizing  \nthe advanced workshop, ``Dark Energy and Fundamental Theory,\"  Xidi, Anhui, China,\nMay 27 - June 5, 2010, supported by the Special Fund for Theoretical Physics from the \nNational Natural Science Foundation of China with grant No. 10947203. Part of this work \nwas done during the  Workshop. He would also like to thank the other ten participants for \nvaluable discussions. The work of AW was supported in part by DOE Grant, DE-FG02-10ER41692.\n\n\\end{acknowledgements}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the 50's, Goeppert-Mayer~\\cite{mayer} and Haxel, Jensen, \nand Suess~\\cite{suess} were able to obtain a description of the nuclear magic\nnumbers by adding a strong spin-orbit interaction to a central mean field potential. This leads to the\ncreation of the so-called spin-orbit gaps with magic numbers 2, 8, 20, 28, 50, 82 and 126. Stable nuclei with magic number of protons and\/or neutrons were found to \nbe spherical, having a closed shell configuration. This simple view of indestructible magic numbers has been modified over the last 30 years due to new spectroscopic information on exotic nuclei such as $^{32}_{12}$Mg$_{20}$~\\cite{32Mg,32MgB}. A reduction of the spherical $N=20$ gap was invoked to explain the deformation of $^{32}_{12}$Mg where a gain in correlation energy, due to quadrupole excitations obtained when promoting nucleons across the shell gap, is larger than the loss in energy required to cross that shell gap. Therefore, if the spherical shell gap is reduced, and\/or the two orbits forming the gap are separated by two units of angular momentum, as for all spin-orbit magic numbers, the nuclear deformation is favored. These two conditions are fulfilled also in $^{42}$Si, which was found to be deformed~\\cite{beyhan, bazin}. One of the key nuclei to understand how deformation sets in for the N~=~28 isotones between the spherical $^{48}$Ca and $^{42}$Si is $^{44}$S. \nThis nucleus has been investigated using different experimental\n approaches, such as $\\beta$-decay~\\cite{sorlin,grevy1}, Coulomb excitation~\\cite{glas}, in-beam $\\gamma$-ray spectroscopy~\\cite{dora}, isomer studies~\\cite{grevy,force}, electron spectroscopy~\\cite{force} and the two proton knock-out reaction~\\cite{santiago}. The works of Ref.~\\cite{glas, dora} showed that $^{44}$S has a deformed ground state. \nThe study of the decay of the 0$^+_2$ level~\\cite{force} and the B(E2:~2$_1^+\\rightarrow$0$^+_1$) reduced transition probability~\\cite{glas} led to the conclusion, by means of shell model calculations, that $^{44}$S exhibits a prolate ground state and a 0$^+_2$ spherical isomeric level.\n In addition, shell-model calculations predicted the existence of a deformed band built on top of the ground state\n and a 2$^+_2$ level of spherical character above the 0$^+_2$ state. A candidate for the 2$^+_2$ level has been proposed in Ref.~\\cite{santiago}. The aim of the present work is to confirm the existence of this state, among others found in Ref.~\\cite{santiago}, and to discover new-high energy levels connecting the deformed and spherical bands in $^{44}$S. All this information is relevant to further understand shape~-~coexistence in this nucleus. The authors of Ref.~\\cite{santiago} used a two proton knock-out reaction from a secondary beam  of $^{46}$Ar to populate excited states in $^{44}$S while, in the present work, a one proton knock-out reaction $^{45}$Cl(-1p)$^{44}$S was employed. This different reaction mechanism is expected to populate additional states in $^{44}$S leading to a complementary understanding of the structure of this nucleus.\n\n\\section{Experimental method}\n\n\\begin{figure}\n\\includegraphics[width=0.50\\textwidth]{caceres_ID_rev.eps}\n\\caption{\\label{fig:z_aoq}(Color online) (a): Energy loss ($\\Delta$E) versus time of flight (TOF) particle identification plot of the ions produced in the fragmentation of $^{48}$Ca beam on C target and separated in the ALPHA spectrometer. (b): Energy loss ($\\Delta$E) versus A\/Q plot of the ions produced in the fragmentation of the $^{45}$Cl secondary beam in the Be secondary target and separated in the SPEG spectrometer.}\n\\end{figure}\n\nThe experiment was performed at the Grand Acc\\'el\\'erateur National d'Ions Lourds (GANIL, France) facility.\nA $^{48}$Ca$^{19+}$ primary beam at 60~MeV\/u and 3.8~$\\mu$A average intensity impinged on a 200~mg\/cm$^2$ C target located between the two superconductor solenoids of the SISSI~\\cite{sissi} device.\nThe reaction products were separated by means of the B$\\rho$-$\\Delta$E-B$\\rho$ method~\\cite{dufour, Anne} in the ALPHA\n spectrometer. The ion identification was performed on an event-by-event basis by measuring their time-of-flight (TOF) over an 80~m flight path using two micro-channel plates and energy loss ($\\Delta$E) in a 50~$\\mu$m Si detector located at the entrance of the SPEG spectrometer. An example of the particle identification plot for the reaction residues is shown in Fig.~\\ref{fig:z_aoq}~(a). \nAfterwards, the selected fragments underwent secondary reactions in a 195~mg\/cm$^2$ $^9$Be target placed at the Si detector position. The $\\Delta$E and positions of the secondary ions were measured at the final focal plane of SPEG in ionization and drift chambers, respectively. A plastic scintillator was used to determine the TOF and residual energy. All of these measurements permitted a full reconstruction of the mass-to-charge ratio of the produced secondary ions (Fig.~\\ref{fig:z_aoq}~(b)). \nAlthough, the magnetic rigidity of the spectrometer was optimized \nfor $^{42}$Si transmission, the large acceptance of SPEG ($\\Delta$p\/p$\\sim$7\\%) spectrometer permitted 50\\% of the $^{44}$S momentum distribution from the one proton knock-out reaction from $^{45}$Cl to be transmitted. This reaction is of particular interest as it provides spin and parity selectivity of the final states populated.\nThe inclusive $\\sigma_{1p}$($^{45}$Cl$\\rightarrow^{44}$S) cross section was measured to\n be 13(3)~mb. The prompt-$\\gamma$ rays detected in coincidence with the nuclei identified at the final focal plane of SPEG\n were measured in 70~BaF$_2$ detectors of the Ch\\^{a}teau de Cristal array positioned above and below \nthe secondary target at an average distance of 25~cm. The efficiency of the\n array was determined using standard sources of $^{60}$Co and $^{152}$Eu to be 24\\% at 1.33~MeV with an energy resolution of 15\\% at 800~keV. A time resolution of 800~ps was obtained. \n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{caceres_SPEC.eps\n\\caption{\\label{fig:energy} (a): Singles $\\gamma$-ray spectrum in coincidence with $^{44}$S ions.\n Coincidence spectra gated by the (b) 1321, (c) summed 977 and 1006, (d) 1979 and (e) 1198~keV transitions, respectively.}\n\\end{figure}\n\n\n\\section{results}\n\n\n\nThe $\\gamma$-ray singles spectrum in coincidence with $^{44}$S reaction products is shown in Fig.~\\ref{fig:energy}(a). Transitions previously reported in Ref.~\\cite{glas,dora} are confirmed with the exception of the $\\gamma$-ray at 2632~keV that is not visible in the present work. However, a broad $\\gamma$-ray distribution at high excitation energy is observed.\nThe construction of the level scheme shown in Fig.~\\ref{fig:ls} is based on a exhaustive $\\gamma\\gamma$-coincidence analysis described below. The fitting function employed to obtain the $\\gamma$-ray intensities and their energies has, as a fixed parameter, the energy-dependent peak-width calculated from the systematics of known transitions of other reaction channels measured in the same experiment.\n\n\\begin{figure}\n\\includegraphics[width=.50\\textwidth]{caceres_HE_rev.eps}\n\\caption{\\label{fig:fit} (Color online) High energy part of the spectra obtained (a) from the $\\gamma\\gamma$ matrix gated by the 1321~keV transition and (b) singles $\\gamma$-ray. The lines shows the result of the fit with two (a) and three (b) Gaussian functions, respectively. The p-value of the $\\chi^2$ goodness-of-fit test is represented by the letter P.}\n\\end{figure}\n\nThe spectrum obtained from the $\\gamma\\gamma$-matrix gated on the 2$^+_1\\rightarrow$0$^+_1$ 1321~keV $\\gamma$ transition is shown in Fig~\\ref{fig:energy}(b). Two well separated $\\gamma$ rays at high excitation energy are observed. A fit to the high~-~energy part of this spectrum with a two Gaussian functions yields 1979 and 2262~keV for the two $\\gamma$-rays, respectively (Fig.~\\ref{fig:fit} (a)). The p-value from the $\\chi^2$ goodness-of-fit test is 0.84.\nIt was not possible to fit the broad high energy distribution visible in the singles spectrum with two Gaussian functions if the 1979 and 2262~keV centroids are considered as fix parameters (p-value~$<$~0.07). Similarly, if the high energy part of the singles spectrum is fitted first with two Gaussian functions and then the energies obtained are used as a fixed parameter in the fitting procedure of the spectrum gated on the 1321~keV, the p-value of the latter fit yields 0.3. The energies of the two $\\gamma$ rays obtained by the two methods are thus not consistent.\nTherefore, this analysis suggests that the high energy and broad $\\gamma$-ray distribution visible in the single spectrum is actually composed of three different $\\gamma$-rays.  \nThe fit of this distribution using three Gaussian functions, in which the centroids and widths of the 1979 and 2262 keV transitions observed in Fig.~\\ref{fig:energy}(b) are fixed, yields 2156~keV as the energy of the third $\\gamma$-ray (Fig.~\\ref{fig:fit}(b)). These three transitions are visible with a significance level of more than 4~$\\sigma$. The 2156~keV $\\gamma$-ray is not in coincidence with the 1321~keV transition. Therefore, it is proposed that it decays directly from a level at 2156~keV to the ground state of $^{44}$S. The observation of a 1979~keV $\\gamma$ ray connected to the 1321~keV 2$^+_1$ state establishes a level at about 3300~keV excitation energy (see below).\nThe structure in the $\\gamma$-ray spectrum between 950 and 1050~keV is a doublet composed of two different transitions at 977 and 1006~keV [Fig~\\ref{fig:energy}(c)]. The sum of the 977 and 1006~keV $\\gamma$ energies leads to 1983(34)~keV which, within the experimental uncertainties, agrees with the energy of the 1979(19)~keV $\\gamma$ ray. Therefore, these three transitions most likely belong to two parallel branches that de-excites from a new state. This state decays by a direct 1979 keV $\\gamma$-ray and a cascade of 977 and 1006~keV transitions. The energy spacing between the proposed level and the 2$^+_1$ state at 1321~keV can be estimated using the weighted mean of the energies of the two decay branches and yields 3301(18)~keV. \n\nThe 977~keV $\\gamma$-ray is placed immediately above the 2$^+_1$ state based on the fact that its energy matches the one of 949(5)~keV reported in Ref.~\\cite{santiago}. The 1006~keV transition is placed above the 977~keV and this $\\gamma $-ray was not observed in Ref.~\\cite{santiago}. This establishes the level at 2298(25)~keV corresponding that of 2268(5)~keV from Ref.~\\cite{santiago}. \nThe spectra shown in Fig~\\ref{fig:energy}(b) and (c) reveal the possible existence of a new $\\gamma$ ray at 1198~keV. This transition is likely in coincidence with all other $\\gamma$ rays with the exception of the 2262 and 2156~keV peaks. It is placed above the 1979~keV $\\gamma$ line depopulating a state at 4499~keV excitation energy with spin and parity Y$^+$ (see below).\n\n\\begin{table*}\n\\caption{\\label{tab:inten} Experimental energies, tentative spin and parity assignments, relative intensities and transition energies for the excited states in $^{44}$S. The direct feeding to the measured levels have been normalized to the\n number of detected $^{44}$S in SPEG and was obtained considering the 2262~keV transition feeding the 1321~keV (D$_{A}$) or the (2$^+_3$) (D$_{B}$) state (see text). The width of the $\\gamma$-ray ($\\sigma$) used in the fitting function is shown in the last column.}\n\\begin{minipage}{10.3cm}\n \\begin{tabular}{cccccccccccccc}\n\\hline\n\\hline\nE$_i$ [keV] &I${_i^\\pi}$ $\\rightarrow$ I${_f^\\pi}$&E$_\\gamma$ [keV]& I$_\\gamma[\\%]$&D$_A$[\\%]&D$_B$[\\%]&$\\sigma$[keV]&&\\\\\n\\hline\n1321(10)&2$^+_1\\rightarrow$0$^+_1$&1321(10)&100(8)&14(6)&20(7)&69(1)&\\\\\n2156(49)&(2$^+_2$)$\\rightarrow$0$^+_1$&2156(49)&17(6)&12(4)&12(4)&99(1)&\\\\\n2298(25)&(2$^+_3$)$\\rightarrow$2$^+_1$&977(23)\\footnote[1]{Extracted from the $\\gamma\\gamma$ matrix gating on the 950~-~1050~keV distribution [Fig~\\ref{fig:energy}(c)].}&48(6)&25(4)&11(4)&57(1)&\\\\\n3301(18)&(2$^+_6$,1$^+_2$)$\\rightarrow$(2$^+_3$)&1006(25)$^b$&12(3)&12(2)&13(2)&58(1)&\\\\\n&(2$^+_6$,1$^+_2$)$\\rightarrow$(2$^+_1$)&1979(19)\\footnote[2]{Extracted from the $\\gamma\\gamma$ matrix gating on the 1321~keV transition [Fig~\\ref{fig:energy}(b)].}&24(5)&&&93(1)&\\\\\n3583(39)\\footnote[3]{State depopulated by the 2262~keV transition feeding the 2$^+_1$ level.};4560(45)\\footnote[4]{State depopulated by the 2262~keV transition feeding the (2$^+_3$) level. }&(X$^+$)$\\rightarrow$2$^+_1$;(2$^+_3$)&2262(38)$^c$&21(5)&15(3)&15(3)&103(1)&\\\\\n4499(37)&(Y$^+$)$\\rightarrow$(2$^+_6$,1$^+_2$)&1198(25)$^b$&18(3)&13(2)&13(2)&65(1)&\\\\\n\\hline\n\\hline\n\\end{tabular}\n \\end{minipage}\n\\end{table*}\n\n\nFrom the present data, it is not possible to place the 2262~keV $\\gamma$ ray in the level scheme. The coincidence relation with the 1321~keV line indicates that it must decay from a state above the 2$^+_1$ level, labeled X$^+$ in Table~\\ref{tab:inten}.\nThe broad structure observed around 200-300~keV in the spectrum of singles (Fig.\\ref{fig:energy}(a)) is not associated with $^{44}$S. This structure corresponds to a peak at constant energy in the non Doppler-corrected spectra and, consequently, must originate from a stopped source. As shown in  Fig.~\\ref{fig:ls}, the present level scheme agrees with the one of Ref.~\\cite{santiago} for the first three excited states. In addition, two new levels are proposed at higher energy in the present work that were not observed in Ref.~\\cite{santiago} and highlights the different selectivity of the complementary reactions used in the two experiments.\n\nThe direct feeding to the levels normalized to the total number of $^{44}$S detected in SPEG have been\n extracted (Tab.~\\ref{tab:inten}). They are based on the measured $\\gamma$-ray intensities and calculated branching ratios. As the precise location of the 2262~keV $\\gamma$-ray in the level scheme could not be determined unambiguously, the direct population to states in $^{44}$S were deduced considering two assumptions: (D$_A$) in which the 2262~keV transition feeds the 2$^+_1$ level or (D$_B$) in which the 2262~keV transition feeds the 2298~keV state. It is interesting to note that the direct feeding of all the states populated in $^{44}$S is nearly the same (between 10 and 20\\%). This feature is a direct consequence of the large configuration mixing between levels in $^{44}$S. In contrast, the fact that none of the newly observed transitions is in coincidence with the 2156~keV $\\gamma$ ray indicates a different configuration for the 2156~keV state.\n\n\\section{Discussion}\n\nThe tentative spin and parity assignment to the observed levels in $^{44}$S is based on the comparison between the calculated and experimental (i) excitation energies, (ii) branching ratios and (iii) direct population of the states following the one proton knock-out reaction. \nShell model calculations (Fig.~\\ref{fig:ls}) using the SDPF-U interaction~\\cite{SDPF},\n known to reproduce the spectroscopic information in the region, were performed. The full model space was considered,\n comprising the proton $\\pi$(d$_{5\/2}$, d$_{3\/2}$, s$_{1\/2}$) and neutron $\\nu$(f$_{7\/2}$, p$_{3\/2}$, p$_{1\/2}$,\n f$_{5\/2}$) orbitals. Calculations were performed with the ANTOINE code~\\cite{antoine1,antoine2}. $E2$ reduced transition probabilities were calculated using polarization charges of 0.35~$e$ for both protons\n and neutrons. Effective magnetic moments for the $M1$ transition rate calculations are taken from\n Ref.~\\cite{SM} and were derived from systematic studies in the $sd$ shell. They correspond to the bare values quenched by 0.75 for the spin operator while the orbital operator is set 1.1 and -0.1 for proton ($g_p$) and neutron ($g_n$) gyromagnetic moments, respectively. These values ensure a very good agreement with the measured g-factors in $^{44}$Cl~\\cite{rydt} \nand $^{43}$S~\\cite{43S}. \n\n\\begin{figure*}\n\\includegraphics[width=1.\\textwidth,height=100mm]{caceres_LS_rev.eps}\n\\caption{\\label{fig:ls}Experimental level scheme of $^{44}$S obtained from (a) Force:~Ref.~\\cite{force}, (b) Santiago-Gonzalez:~Ref.~\\cite{santiago} and (c) in this work. The width of the arrows in (c) indicates the relative $\\gamma$-ray intensities normalized to the 1321~keV transition. The energy uncertainties on the $\\gamma$-rays and levels are quoted in Table~\\ref{tab:inten}. (d) Shell model calculations performed using the SDPF-U interaction. The arrow length in the right hand side of the figure shows the error on the neutron separation energy (S$_n$~=~5220(440)~keV~\\cite{Sn}). The levels observed in this data are represented with thicker widths line.}\n\\end{figure*}\n\n\nTheoretical direct population of the levels in $^{44}$S has been obtained combining the eikonal-model calculations~\\cite{tostevin} for the single particle cross sections with the shell model spectroscopic factors (SF) following the prescription of Ref.~\\cite{gade08}. The ground state spin and parity of $^{45}$Cl are not firmly known experimentally~\\cite{sorlin2,gade2}. Shell model calculations predict an I$^\\pi$=1\/2$^+$ assignment although the 3\/2$^{+}$ level is calculated to be at only 132~keV excitation energy. Therefore, both spins (1\/2 or 3\/2) were considered in the SF calculations (Fig.~\\ref{fig:SF}). The range of possible states populated in $^{44}$S from the proton knock-out reaction of $^{45}$Cl are from 0$^{+}$~to~3$^{+}$ (0$^{+}$~to~4$^{+}$) assuming 1\/2$^+$ (3\/2$^+$) for the ground state of $^{45}$Cl. Therefore, all calculations were restricted to these spin states~(Fig.~\\ref{fig:ls}). \n\n\nThe calculated 2$^+_1$ level at 1172~keV excitation energy corresponds to the 1321~keV experimental state previously reported in Ref.~\\cite{glas}. \nBetween 2000 and 2800~keV five levels were calculated. Among them, only the 2$^+_2$ state at 2140~keV excitation energy has a spherical configuration with intrinsic quadrupole moment -0.3~efm$^2$~\\cite{force}. It is predicted to decay by 83\\% and 16\\% to the 0$^+_1$ and 2$^+_1$ levels, respectively. Its decay to the 0$^+_2$ level, although favored in terms of reduced transition probability (B(E2:~2$_2^+\\rightarrow$0$^+_2$)$\\simeq$~3.2~B(E2:~2$_2^+\\rightarrow$0$^+_1$)), amounts to only 1\\%. This is due to the energy dependence of the transition probabilities. Although, having comparable configurations, the 2$^+_2$ and 0$^+_2$ are thus only weakly connected to each other. \nThe experimental analogue of the 2140~keV state is the 2156~keV level and the non observation of the 835~keV $\\gamma$ ray associated with the 2$^+_2\\rightarrow$2$^+_1$ decay is in agreement with experimental results considering the intensity of the 2156~keV $\\gamma$-ray. The 2156~keV state is tentatively assigned to be the (2$^+_2$) level of spherical character based on the fact that if this state had had a more mixed configuration, higher lying states would have decayed through it and been detected in this experiment. \nAs shown in Fig.~\\ref{fig:SF}, the calculated direct population to the 2$^+_2$ state is 5\\% and 1\\% assuming 1\/2$^+$ and 3\/2$^+$ spin and parity assignment for the ground state of $^{45}$Cl, respectively. The theoretical direct feeding to the 4$^+_1$ state (calculated to be 2569~keV and observed at 2447(5)~keV in Ref.~\\cite{santiago}) is 0\\% in the case of a 1\/2$^+$ and 2\\% in the case of a 3\/2$^+$ spin and parity of the ground state of $^{45}$Cl, respectively. Both, the experimental feeding of 12~\\% to the 2$^+_2$ and the non observed feeding to the 4$^+_1$ state rather favor a I$^\\pi$~=~1\/2$^+$ ground state for $^{45}$Cl.\nThe other four calculated states between 2000 and 2800~keV (i.e. 2$^+_3$, 3$^+_1$, 4$^+_1$, 4$^+_2$) are predicted to decay mostly to the 2$^+_1$ level ($>$90\\%); only the 2$^+_3$ state is expected to be strongly populated in the reaction (Fig.~\\ref{fig:SF}), independent of the I$^\\pi$ value of the ground state of $^{45}$Cl. For this reason, an I$^\\pi$~=~(2$^+_3$) spin and parity is assigned to the 2298~keV level. Above 3000~keV there is a large level density is predicted. A study of the decay modes of each state was performed considering the theoretical reduced transition probabilities and the experimental excitation energies (when available), otherwise they were taken from theory. In the latter case, the branching ratio of each level was investigated considering a variation of 500~keV of the theoretical excitation energies. Only the 1$^+_2$ and 2$^+_6$ states decay by competing and parallel branches to the 2$^+_3$ and 2$^+_1$ levels in agreement with the experimental data.!\n  Therefore, both states are possible candidates to spin and parity assignment for the 3301~keV level. In addition, the theoretical cross section calculations further support the I$^\\pi$~=~(1$^+_2$, 2$^+_6$) assignment (Fig.~\\ref{fig:SF}). Due to the large number of calculated levels around 4000~keV, it was not possible to find a theoretical counterpart to the experimental 4499~keV level, therefore no spin assignment can be given.\n\n\\begin{figure}\n\\includegraphics[width=.550\\textwidth,height=60mm]{caceres_SF_rev.eps}\n\\caption{\\label{fig:SF} (Color online) Theoretical partial cross sections to the levels in $^{44}$S normalized to the strength below the neutron separation energy S$_n$~=~5220(440)~keV assuming 3\/2$^+$ (a) and  1\/2$^+$ (b) spin and parity assignment for the ground state of $^{45}$Cl.}\n\\end{figure}\n\n\n\\section{Summary}\nThe $^{44}$S nucleus has been investigated using in-beam $\\gamma$-ray spectroscopy following the one proton knock-out reaction from $^{45}$Cl. The large efficiency of the BaF$_2$ detectors of the Ch\\^{a}teau de Cristal array, together with the particle selection provided by the SPEG spectrometer, allowed the construction of the level scheme up to high excitation energy. Analysis of $\\gamma\\gamma$-coincidences relations have been realized in this nucleus and the experimental direct feedings of the states were extracted. State-of-the-art nuclear shell model calculations have been performed for comparison to experimental energies, deduced spin assignments and decay branches of the states, and estimate their direct feeding from the one proton knock-out reaction. The energies and tentative spin assignments of the three excited states (2$^+_1$, 2$^+_2$ and 2$^+_3$) determined in Refs.~\\cite{force,santiago} are confirmed. The 2$^+_2$ level at 2156(49)~keV is found to have a specifi!\n c configuration, and is likely spherical. This is based on the weak feeding observed from other excited states, its feeding intensity from the (-1p) knock-out reaction and its decay pattern.\nTwo new excited states have been proposed at higher excitation energy with the 3301~keV level tentatively assigned to have I$^\\pi$~=~(1$^+_2$, 2$^+_6$), while no spin and parity could be determined for the level at 4499~keV excitation energy. \nThe information from  the present data, and the one reported in Ref.~\\cite{force,santiago}, highlights the large density of states at low excitation energy in the semi-magic $^{44}$S nucleus. This confirms the complexity of its nuclear structure that combines both shape and configuration coexistence.\n\n\n\\begin{acknowledgments}\nThis work benefits from discussions with G.~F.~Grinyer. \nThe authors are thankful to the\nGANIL and LPC staffs. We thank support by BMBF\nNo. 06BN109, GA of Czech Republic No. 202\/040791,\nMEC-DGI-(Spain) No. BFM2003-1153, and by the EC\nthrough the Eurons Contract No. RII3-CT-3\/2004-506065\nand OTKA No.~K68801 and Bolyai J\\'anos Foundation. R. B., A. B.,\nM. L., S. I., F. N. and R. C., X. L. acknowledge the\nIN2P3\/CNRS and EPSRC support.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Introduction}\n\\setcounter{page}{1}\n\\section{Motivation}\nRecent Artificial Intelligence (AI) research has shown considerable interest in developing systems that can mimic human learning and thinking  \\cite{motivationintro}. Similar to human learning, there has been significant success in transferring knowledge between AI systems, like a ``student\" - ``teacher\" relationship \\cite{generalkt}. To simulate human learning, a ``student\" system should also be able to accumulate knowledge and combine the obtained information. For example, it is not expected of a Computer Science pupil to learn how to implement quicksort without first understanding the notions of median and partition. We can extend this expectation to an AI system. Instead of solving a complex task without prior knowledge, firstly, learn how to resolve more straightforward problems from the same area. This form of Machine Learning is referred to as multi-task learning \\cite{zhang2018overview}.\n\nInductive Logic Programming is an ideal candidate for the above use case \\cite{ILPat30}.\nInductive Logic Programming (ILP) is a form of machine learning that learns logic programs (also referred to as symbolic programs) based on given examples and background knowledge (BK). The BK set consists of a set of logic programs given by the user. Therefore, transferring knowledge becomes trivial \\cite{johnthesis}: if we want to reuse a solution in solving another task, it is sufficient to append it to the given BK. Returning to the ``student\" - ``teacher\" example, the previously mentioned approach would be equivalent to having a teacher provide his student with a new definition.\n\nAside from providing support for multi-task learning, ILP has been shown to provide extensive benefits compared to most ML models, such as Data Efficiency (being able to generalise from a small number of examples) and Explainability (providing outputs that are easily understandable by a human) \\cite{ILPat30}. \n\nWe will proceed by providing the reader with an example logic program (see \\autoref{fig:ILPexample}) written in Prolog in order to accommodate them with the syntax used throughout the project. Afterwards, we will describe four problems that can be solved by an ILP system and which would benefit from multi-task learning.\n\\begin{figure}[hbt!]\n    \\begin{minted}{prolog}\n   \n        isFather(\"John\", \"Dan\")\n        isFather(\"Paul\", \"John\")\n        isWife(\"Alice\", \"Paul\")\n        \n        isGrandmother(X,Y):-isWife(X,Z),\n                            isGrandfather(Z,Y)\n                                              \n                                              \n                                                \n        isGrandfather(X,Y):-isFather(X,Z),\n                            isFather(Z,Y) \n                                          \n                                          \n                                            \n   \n        ?-isGrandmother(\"Alice\",\"Dan\")\n    \\end{minted}\n    \\caption{Logic program example}\n    \\label{fig:ILPexample}\n\\end{figure}\n\n\\paragraph{Kinship relationships.} Kinship relationships allow an illustration of the problem solved by introducing multi-task learning in ILP systems. \nMoreover, the trivial nature of the problem allows for synthetic data generation to benchmark. In \\autoref{fig:kinship_example} we show a situation where a shorter program (\\texttt{isGrandmother(A,B) :- isWife(A,C), isGrandfather(C,B)} instead of \\texttt{isGrandmother(A,B) :- isWife(A,C), isFather(C, D), isFather(D,B)}) is learnt to explain the grandmother relationship. \n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{minted}{prolog}\n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Alice\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isGrandfather(C,B)\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isFather(C, D), isFather(D,B)\n    \\end{minted}\n    \\caption{Example for the kinship relationship problem}\n    \\label{fig:kinship_example}\n\\end{figure} \n\n\\paragraph{String transformation.} One of the most widely used examples to benchmark ILP systems are EXCEL type string transformation operations \\cite{ILPat30}. \nThrough multi-task learning, an ILP system can better capture situations where a complex transformation consists of sequential applications of more simple operations. In \\autoref{fig:string_example}, we present our initial set of example and their equivalent programs. The given functions perform the following operations:\n\\begin{itemize}\n    \\item \\texttt{f0} - capitalize each word of a string\n    \\item \\texttt{f1} - capitalize a single word\n    \\item \\texttt{f2} - drop the first word of a string and capitalize the second\n    \\item \\texttt{f3} - copy a string until the first non-letter character is encountered\n\\end{itemize}\nThe predicates \\texttt{inv0} and \\texttt{inv2} from the example are auxiliary functions invented during the learning process.\n\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{minted}{prolog}\n   \n        f0('gerson zaverucha', 'Gerson Zaverucha')\n        f0('jesse davis', 'Jesse Davis')\n        f0('paolo frasconi', 'Paolo Frasconi')\n        \n        f1('james', 'James')\n        f1('jozie', 'Jozie')\n        f1('paul', 'Paul')\n        \n        f2('dr wilson', 'Wilson')\n        f2('dr brown', 'Brown')\n        f2('dr wright', 'Wright')\n        \n        f3('artificial.', 'artificial')\n        f3('intelligence ', 'intelligence')\n        f3('systems1a', 'systems')\n\n   \n        f0(A,B):-f1(A,C),inv0(C,B).\n        inv0(A,B):-copyskip1(A,C),f1(C,B).\n        \n        f1(A,B):-mk_uppercase(A,C),f3(C,B).\n        \n        f2(A,B):-skip1(A,C),inv2(C,B).\n        inv2(A,B):-skip1(A,C),f1(C,B).\n        inv2(A,B):-skip1(A,C),inv2(C,B).\n        \n        f3(A,B):-copyskip1(A,B),not_letter(B).\n        f3(A,B):-copyskip1(A,C),f3(C,B).\n    \\end{minted}\n    \\caption{Example for the string transformation problem}\n    \\label{fig:string_example}\n\\end{figure} \n\\paragraph{Robot movement.} Robot movement can be represented as basic movements such as going up\/down\/left\/right on a grid. For traversing large distances or completing complex paths it would be more efficient to have common patterns such as corner turns, as part of the background knowledge. \n\nIn \\autoref{fig:robot_example}, we show a solution where a 5 step translation is reduced to only two literals.\n\\begin{figure}[hbt!]\n    \\centering\n    \\begin{minted}{prolog}\n   \n        f(State(X, Y, Xnew, Ynew, Width, Height))\n        \n   \n        turn(State(0, 1, 1, 2, 5, 2))\n        turn(State(0, 0, 1, 1, 5, 2))\n        \n        jump(State(0, 1, 1, 4, 5, 2))\n        jump(State(0, 0, 0, 3, 5, 2))\n        \n        turnAndJump(State(0, 0, 1, 4, 5, 2))\n\n   \n        turn(State(X, Y, Xnew, Ynew, W, H)) :- right(X, Y, X2, Y2),\n                                               down(X2, Y2, Xnew, Ynew)\n                                                    \n        jump(State(X, Y, Xnew, Ynew, W, H)) :- right(X, Y, X2, Y2),\n                                               right(X2, Y2, X3, Y3)\n                                               right(X3, Y3, Xnew, Ynew)\n                                                        \n        turnAndJump(State(X, Y, Xnew, Ynew, W, H)) :- \n                                                turn(X, Y, X2, Y2),\n                                                jump(X2, Y2, Xnew, Ynew)\n    \\end{minted}\n    \\caption{Example for the robot movement problem}\n    \\label{fig:robot_example}\n\\end{figure} \n\n\\paragraph{Drawing images.} Most graphic rendering techniques are performed by decomposing scenes into primitives such as lines, triangles or quads. Specific algorithms (such as Constructive Solid Geometry) also make use of more complex shapes, for which the primitive decomposition is computed in advance \\cite{Graphics21}.\n\nBased on the above, we formulate the following problem: given a set of functions that control a printer head (moving over the four axes, returning to (1,1) coordinates or drawing a pixel) the system has to learn to draw complex images. In \\autoref{fig:drawing-images} we show how a cube can be drawn by connecting four lines.\n\\begin{figure}[hbt!]\n    \\centering\n\n    \\begin{minted}{prolog}\n   \n        f(State(X, Y, Width, Height, Grid))\n        \n   \n        topLine(State(_, _, 3, 3, [1,1,1,0,0,0,0,0,0]))\n        bottomLine(State(_, _, 3, 3, [0,0,0,0,0,0,1,1,1]))\n        leftLine(State(_, _, 3, 3, [1,0,0,1,0,0,1,0,0]))\n        rightLine(State(_, _, 3, 3, [0,0,1,0,0,1,0,0,1]))\n        isCube(State(_, _, 3, 3, [1,1,1,1,0,1,1,1,1]))\n        \n   \n        topLine(S, S5) :- at_start(S), move_up(S, S2), move_up(S, S3), \n                             draw1(S3), move_right(S3, S4), draw1(S4), \n                             move_right(S4, S5), draw1(S5)\n        bottomLine(S, S3) :- at_start(S), draw1(S), move_right(S, S2), \n                                draw1(S2), move_right(S2, S3) draw1(S3)\n        leftLine(S, S3) :- at_start(S), draw1(S), move_up(S, S2), \n                              draw1(S2), move_up(S2, S3) draw1(S3)\n        rightLine(S, S5) :- at_start(S), move_right(S, S2), \n                               move_right(S, S3), draw1(S3), \n                               move_up(S3, S4), draw1(S4), \n                               move_up(S4, S5), draw1(S5)\n        isCube(S, S4) :- topLine(S, S2), returnStart(S2, S3), \n                         bottomLine(S3, S4), returnStart(S4, S5),\n                         leftLine(S5, S6), returnStart(S6, S7),\n                         rightLine(S7, S8), returnStart(S8, S9)\n    \\end{minted}\n    \\caption{Example for the image drawing problem}\n    \\label{fig:drawing-images}\n\\end{figure}\n\nAll the above problems are solvable by an ILP system, but certain limitations exist. ILP systems learn by searching through a hypothesis space of possible solutions, but even the most advanced search techniques face difficulties when the size of the hypothesis space increases significantly. The size of the hypothesis space is directly related to the maximum size a program can have \\cite{popper}. In order to better understand this statement, imagine the task of writing down all possible programs of size $n$ that contain a given list of predicates (allowing duplicates). One can easily observe that the size of the result grows exponentially with $n$. Therefore, the maximum size of possible solutions must be restricted such that searching through it is feasible. Consequently, if the solution for a task is too large, the task will become unsolvable. \n\nMulti-task learning mitigates the above issue by reusing learnt solutions when learning tasks from related domains. In the four previous examples, we can observe that several tasks can be represented as more compact solutions by reusing the solutions from some other tasks.\n\nThe state-of-the-art approach to ILP multi-task learning is based on the Metagol \\cite{metagol} system and was described by \\citet{lin2014bias} in 2014. It has been shown to approach the performance of commercial systems of the time and outperform an independent learning strategy (running Metagol on each task separately). In \\autoref{implementation} we will identify significant drawbacks of the current strategy that negatively impact its learning abilities. \n\nFinally, we introduce Popper \\cite{popper}, an ILP system which has superseded Metagol \\cite{metagol}. Popper uses a new approach to ILP, referred to as ``Learn from failures\": while searching through the hypothesis space, it generates constraints based on incorrect programs found. Those constraints are then used to prune the search space and improve learning time. Furthermore, in \\autoref{constraints} we detail how constraints can be further used to improve the learning rate for multi-task systems.\n\n\\section{Contribution} \n\nOur contribution to the field represents an enhanced version of Popper that supports multi-task learning and uses knowledge transfer. \n\nWe have decided to implement our approach and re-implement the method described by \\citet{lin2014bias} as extensions to the Popper ILP system in order to be able to evaluate the benefit that multi-task learning can obtain by preserving constraints.\n\nSuch, we summarise our contribution as follows:\n\\begin{itemize}\n    \\item Metagol$_{DF}$ performs iterative deepening search based on the maximum size of programs in the hypothesis space. We identify situations where increasing search size is detrimental to the learning process, virtually blocking the system from progressing further. In consequence, we provide an alternative search strategy that circumvents this obstacle and has a greatly improved learning ability.\n    \\item Between each stage of iterative deepening search, the Metagol$_{DF}$ does not maintain any information. Our implementation gives Popper the ability to preserve constraints when the maximum search size is modified. Such, we improve the learning rate and allow for a breadth first search approach to replace the iterative deepening approach. For example, let $C$ be the set of constraints obtained by searching through a hypothesis space of size $n$. This constraints will help prune a search space of size $n+1$. Without constraint preservation, the system would be unable to transfer information between runs. Therefore, it must search through a space size that includes both hypotheses of size $n$ and size $n+1$.\n    \\item We show that the performance of the system is strongly correlated to the order in which the user inputs the list of tasks to be solved \\cite{ordering}. We design two methods that order tasks based on how close the system is to solving them, aiming to achieve more consistent and closer to optimal performance.\n    \\item Metagol$_{DF}$ has been tested against a limited data set of 17 string transformation operations. We refine one external data set and synthetically generate two additional ones. We then proceed to empirically evaluate the performance of all proposed techniques and indicate how each performs in comparison to the original approach.\n\\end{itemize}\n\n\\chapter{Related work}\n\n\\section{Ability to learn}\nPrograms that expand their learning abilities (``have the ability to learn\") have been an area of interest in Machine Learning for multiple decades. In \\cite{Solomonoff1989ASF}, Solomonoff describes a system that, given a small set of concepts, is able to learn problems of increasing difficulty by extracting new, more complex concepts during each learning stage. A concept was defined as a sequence of instructions. The approach reduces most problems from science to two main categories: \n\n\\textbf{Inversion problems.} Given a machine $M$, that performs string transformation operations, and a string $s$, the goal of an inversion problem is to determine a string $x$, such as $M(x) = s$. \n\n\\textbf{Time limited optimisation problems.} Given a machine $M$, that takes a string and returns a real number, the goal of a time limited optimisation problem is to find a string $x$, within a fixed time limit $T$, such as $M(x)$ is as large as possible. \n\nFurthermore, the paper concludes with an overview of areas where such a system would benefit from further research, most notably: improvements in the method used to update the set of knowledge once a new task is learnt and designing a system that is able to work on an unordered set of programs.\n\n\\section{Multi-task learning}\nIn this thesis, we refer to multiple Machine Learning paradigms related to multi-task learning. \n\n\\textbf{Meta-learning.} Meta-learning (learning to learn) focuses on how machine learning approaches perform on certain tasks and then uses information obtained from previous runs in order to speed up learning during future tasks \\cite{Joaquin2018}. \n\n\\textbf{Curriculum learning.} Curriculum learning means that tasks will be learnt in a specific order, based on their perceived difficulty (similar to a human curriculum) \\cite{XinWang2020}. \n\n\\textbf{Transfer learning.} Transfer learning is a technique through which multiple learners (that focus on related tasks) transfer knowledge to each other.  \\cite{weiss2016survey}. \n\nThe above approaches have been empirically and theoretically shown to improve performance over traditional independent learning methods \\cite{zhang2018overview}. \n\nAlthough the fact that tasks from a similar domain may be related appears intuitive, it is necessary to outline both the benefits and downsides of multi-task learning. Therefore, depending on the similarity between the given set of tasks, it is crucial to evaluate if multi-task learning is expected to provide a guaranteed advantage over independent learning \\cite{ben2003exploiting}.\nThe expectation is that multi-task learning approaches perform significantly worse than conventional learning methods on unrelated tasks \\cite{pan2010}. Moreover, transfer learning systems that retain information from all learnt tasks as part of their knowledge set face difficulty when used to solve large data sets \\cite{ferri2001incremental}. This is a consequence of the fact that search speed is usually correlated to the size of the background knowledge.\nSeveral methods that attempt to circumvent the above issues have been developed. \n\nFor example, neural networks have been used to recognise which programs are expected to perform better on specific input\/output pairs \\cite{ellis2018learning}. \n\nIt has also been experimented with the idea of ``forgetting\", which refers to removing data from the background knowledge (or ``unlearning\") with the purpose of improving search time \\cite{cropper2020forgetting}.\n\\section{Multi-task learning in ILP}\nOne of the main advantages of ILP, in the context of multi-task learning, is the fact that ILP systems can perform knowledge transfer without significant overhead through the fact that outputs are reusable as inputs. Despite such considerations, very few studies explore multi-task learning approaches for ILP. \n\nMetagol is an ILP system based on meta-interpretive learning \\cite{metagol}. Lin et al. adapt the Metagol system to make use of multi-task learning (Metagol$_{DF}$) and benchmark it against multiple agents: a human agent, a commercial system (Microsoft FlashFill, introduced with Excel 2013), and a version of itself, which does not make use of multi-task learning. \n\nThe system has shown promising results and consistently achieved better or equal performance when compared with the standard approach. \n\nThe author concludes by highlighting a series of limitations of multi-task learning in the context of ILP.\nMost importantly, it observes that the system will become overwhelmed when run on multiple data sets. This claim is empirically confirmed by Cropper, in \\cite{cropper2020forgetting}, who also provides a solution for such use cases: introducing Forgetgol. This system can forget programs and reduce the background knowledge size. \n\nIn this project, we identify significant issues with the previous approach and propose an alternative solution. We build on the Popper ILP system \\cite{Cropper2019} and introduce the following elements of novelty: we modify how the hypothesis space is searched, we retain information between failed search attempts, and we introduce new search strategies that focus on analysing the difficulty of the tasks at run-time (similar to curriculum learning).\n\n\\chapter{Problem setting}\nIn this chapter, we will give an overview of ILP in order to familiarise the reader with specific terminology used throughout the project. \n\\section{Logic programming syntax}\nWe will define the syntax of Logic Programs by following definitions from \\cite{ILPat30}, and \\cite{kowalski1974}: \n\n\\textbf{Variable.} A variable is a string of characters that starts with an uppercase letter. \n\n\\textbf{Function.} A function symbol is a string of characters that starts with a lowercase letter. Examples: \\texttt{f1}, \\texttt{move\\_up}, \\texttt{draw1}. \n\n\\textbf{Predicate symbol.} A predicate symbol is a string of characters that start with a lowercase letter, similarly to a function. \n\n\\textbf{Arity.} The arity of a function\/predicate symbol \\texttt{f} is the number of arguments it takes and is denoted as \\texttt{f\/n}. \n\n\\textbf{Constant symbol} A constant symbol is a predicate symbol of arity zero. Examples: \\texttt{alice}, \\texttt{bob}. \n\n\\textbf{Term.} A term refers to either a variable, a constant symbol or a predicate symbol of arity $n$ followed by a tuple of size $n$. \n\n\\textbf{Ground term.} We say that a term is ground if it contains no variables. \n\n\\textbf{Atom.} An atom is a formula $f(x_1,..,x_n)$, such as $p$ is a predicate symbol of arity $n$ and $x_i$ is a term $\\forall i \\in \\{1,..,n\\}$. \n\n\\textbf{Ground atom.} We say that an atom is ground if all its terms are ground. \n\n\\textbf{Negation.} We say that an atom is false if it cannot be proven true. Throughout this paper, we will represent negation through the symbol $\\neg$. \n\n\\textbf{Literal.} A literal is either an atom or its negation. \n\n\\textbf{Clause.} A clause is a disjunction of atoms $h_1,h_2,..,h_n,b_1,b_2,..,b_m$, represented as:\n\\begin{align*}\n    h_1,h_2,..,h_n \\textrm{ :- } b_1,b_2,..,b_m\n\\end{align*}\nThe above definition is equivalent to:\n\\begin{align*}\n    (h_1\\land h_2\\land..\\land h_n) \\lor \\neg b_1 \\lor \\neg b_2 .. \\lor \\neg b_m\n\\end{align*}\n\nWe refer to $h_1,h_2,..,h_n$ as the \\textit{head literals} and  $b_1,b_2,..,b_m$ as the \\textit{body literals} of the clause. \n\nThe above representation is shorthand for ``$h$ holds true if all of $b_1,..,b_m$\" hold true. \n\n\\textbf{Horn clause} A Horn clause is a clause for which $n$ (number of head\/positive literals) is at most $1$. \n\nWe will only be concerned with Horn clauses since other types of clauses are not supported by basic Prolog \\cite{kowalski1988early}. \n\n\\textbf{Ground clause.} We say that a clause is ground if it contains no variables. \n\n\\textbf{Definite clause.} We say that a Horn clause is definite if the number of positive literals is exactly one. \n\n\\textbf{Definite logic program.} A definite logic program is a set of definite clauses. We will further refer to definite logic programs as programs.\n\\section{Inductive logic programming}\nInductive logic programming problems are given as triplets $(B,E^+,E^-)$, where $B$ represents the background knowledge (BK), $E^+$ represents the set of positive examples, and $E^-$ represents the set of negative examples. \n\n\\textbf{Background knowledge.} The background knowledge is a set of programs for which the definition is given completely. By the close world assumption, anything that is not explicitly stated true in the BK is considered false \\cite{REITER1981119}. We show a specimen from the BK of the string transformation problem: \n\n\\[\n\\texttt{B} = \\left\\{\n\\begin{array}{l}\n    \\texttt{is\\char`_empty([]).}\\\\\n    \\texttt{is\\char`_space([' '|\\char`_]).}\\\\\n    \\texttt{is\\char`_number(['0'|\\char`_]).}\\\\\n    \\texttt{...} \\\\\n    \\texttt{is\\char`_number(['9'|\\char`_]).}\\\\\n\\end{array}\n\\right\\}\n\\]\n\n\\textbf{Examples.} Examples are given as two sets (positive and negative examples) and are represented similarly to the BK, but all clauses must be ground. Moreover, there is no assumption on the function behaviour outside of the given use cases. Below we show examples of a function that turns the first letter of a string to uppercase. \n\n\\[\n\\texttt{E$^+$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).}\\\\\n\\texttt{f(['j', 'o', 'z', 'i', 'e'],['J', 'o', 'z', 'i', 'e']).}\\\\\n\\texttt{f(['b', 'e', 'n'],['B', 'e', 'n']).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['j', 'a', 'm', 'e', 's']).}\\\\\n\\texttt{f(['j', 'o', 'z', 'i', 'e'],['o', 'z', 'i', 'e']).}\\\\\n\\texttt{f(['b', 'e', 'n'],['B']).}\\\\\n\\end{array}\n\\right\\}\n\\]\n\n\\textbf{Hypothesis space.} The hypothesis space contains all the possible programs that can be built from the predicates given in the BK. Since the hypothesis space is infinite, it is important to enforce certain restrictions to make searching through it practical. Such, many machine learning makes use of inductive bias and language bias \\cite[p,64,106]{Mitchell97} to restrict the hypothesis space. \n\\section{Popper ILP System}\nIn \\autoref{implementation} we introduce our multi-task learning strategies, which build on the Popper ILP system. The Popper ILP system uses an approach known as ``Learning from failures\" \\cite{popper}. This technique divides the learning process into three stages: \\textbf{generate}, \\textbf{test}, and \\textbf{constrain}. \n\nIn the generate phase, a logic program (referred to as hypothesis) is selected, such that it respects the currently learnt \\textbf{hypothesis constraints}. Afterwards, the program is tested against the given examples. If all examples are covered, Popper will return the program as a valid answer. Otherwise, we proceed to the constrain stage, where information from the testing stage is used in order to augment the set of \\textbf{hypothesis constraints}. Those three stages are repeated until either a solution is found or there are no more programs in the hypothesis space. \n\nBelow we will describe the four types of constraints. \n\n\\textbf{Generalisation constraints.} A generalisation constraint is a constraint that eliminates generalisations of the current hypothesis. Example: \n\nGiven the following negative examples:\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).}\\\\\n\\texttt{f(['b', 'e', 'n'],['B', 'e', 'n']).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nAnd the hypothesis:\n\\[\n\\texttt{h} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nWe observe that a negative example holds for $h$. Such, we can prune generalisations, such as:\n\\[\n\\texttt{h'} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B).}\\\\\n\\texttt{f(A, B):-head(A,B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\textbf{Specialisation constraints.} A specialisation constraint is a constraint that prunes out specialisations of the current hypothesis. Example: \n\nGiven the following positive examples:\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).} \\\\\n\\texttt{f(['b', 'e', 'n',' ', 'g'],['B', 'e', 'n',' ', 'G']).} \\\\\n\\end{array}\n\\right\\} \n\\]\n\nAnd the hypothesis:\n\\[\n\\texttt{h} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nWe observe that the first positive example holds for $h$, but not the second. Such, we conclude that $h$ is too specific and prune out specialisations of $h$, such as:\n\\[\n\\texttt{h'} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B),empty(B).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\textbf{Elimination constraints.} An elimination constraint is a constraint that eliminates a hypothesis $h$ from any separable hypothesis $h_s$ that contains $h$. We refer to a hypothesis $h_s$ as separable if no predicate symbol from the head of a clause in $h_s$ appears in the body of a clause in $h_s$. Example: \n\nGiven the following positive examples:\n\\[\n\\texttt{E$^-$} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(['j', 'a', 'm', 'e', 's'],['J', 'a', 'm', 'e', 's']).} \\\\\n\\texttt{f(['b', 'e', 'n',' ', 'g'],['B', 'e', 'n',' ', 'G']).} \\\\\n\\end{array}\n\\right\\} \n\\]\n\nAnd the hypothesis:\n\\[\n\\texttt{h} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-empty(A).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\nWe observe that the no positive example holds for $h$. Such, we conclude that $h$ can be eliminated from any separable hypothesis, such as:\n\\[\n\\texttt{h'} = \\left\\{\n\\begin{array}{l}\n\\texttt{f(A, B):-mk\\_uppercase(A,B)}\\\\\n\\texttt{f(A, B):-empty(A).}\\\\\n\\end{array}\n\\right\\} \n\\]\n\n\\textbf{Banish constraints.} A banish constraint is a constraint that rules out one specific hypothesis. \n\nTo handle searching through a vast hypothesis space size, the Popper ILP system looks for solutions in increasing order of the number of literals in a hypothesis. The formula used to compute an upper bound on the size of the hypothesis space is given in \\cite{popper}: \n\\begin{itemize}\n    \\item Maximum arity $\\boldsymbol{a}$.\n    \\item Maximum number of unique variables in a clause $\\boldsymbol{v}$.\n    \\item Maximum number of body literals allowed in a clause $\\boldsymbol{m}$.\n    \\item Maximum number of clauses in a hypothesis $\\boldsymbol{n}$.\n    \\item Declaration bias $\\boldsymbol{D}=(D_h,D_b)$.\n\\end{itemize}\n\n\\begin{align*}\n    \\sum_{j=1}^{n}\\binom{|D_h| v^a \\sum_{i=1}^{m}\\binom{|D_b| v^a}{i}}{j}\n\\end{align*}\n\\label{ILPintro}\n\\chapter{Implementation}\n\\label{implementation}\nIn this section, we will describe several multi-task learning approaches. For each one, we will detail the advantages and disadvantages. We begin with the naive approach, followed by the current state-of-the-art strategy. Afterwards, we will provide a new search algorithm that focuses on resetting the size of the search space in order to overcome the limitations of the state-of-the-art approach. Finally, we will describe two refinements that can be applied to any of the described approaches: constraint preservation and automatic task ordering.\n\n\\section{Naive approach}\n\\noindent This naive approach to multi-task learning iterates through all the tasks in order and attempts to learn them by calling an ILP system (in our case Popper) \\footnote{We note that this search strategy can get blocked when working on unsolvable tasks. This behaviour does not impact our results as the naive approach is only used for reference and we will benchmark against the state-of-the-art strategy (which is discussed in \\autoref{sota-sub}). Other approaches that do not make use of multi-task learning can be designed to overcome this difficulty.}. Therefore, it does not make use of the fact that tasks learned together are expected to be correlated. \n\n\\paragraph{Advantages.} The main advantage of this approach is that it does not suffer from the additional overhead introduced by increasing the number of tasks in the BK \\cite{lin2014bias}. In \\autoref{fig:naive-example} the system is given two independent tasks: learning the \\texttt{isGrandfather} and \\texttt{isGrandmother} functions. In such a situation, transferring knowledge between the two will be redundant and only cause unnecessary overhead. Therefore, such use-cases benefit the naive approach.\n\n\\begin{figure}[hbt!]\n    \\centering\n\n    \\begin{minted}{prolog}\n    \n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isMother(\"Alice\",\"Dan\")\n        isMother(\"Ana\", \"Alice\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Ana\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isMother(A,C), isMother(C, B)\n        \n    \\end{minted}\n    \\caption{Advantage example for naive approach}\n    \\label{fig:naive-example}\n\\end{figure}\n\n\\paragraph{Disadvantages.} Although it performs well on uncorrelated data, multi-task learning systems have been shown repeatedly to out-perform traditional systems on tasks from the same domain \\cite{ILPat30}. In \\autoref{fig:bad-naive-example} the system is given the same two tasks, but the BK is modified such that the tasks are no longer independent: \n\n\\texttt{isGrandmother} can be learnt as a function of \\texttt{isGrandfather}. The naive approach has no provision to benefit from such situations.\n\n\\begin{figure}[hbt!]\n    \\centering\n\n    \\begin{minted}{prolog}\n    \n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Alice\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isGrandfather(C,B)\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isFather(C, D), isFather(D,B)\n        \n    \\end{minted}\n    \\caption{Disadvantage example for naive approach}\n    \\label{fig:bad-naive-example}\n\\end{figure}\n\nFinally, in \\autoref{alg:multi_popper_naive} we show an implementation of the naive multi-task learning approach.\n\n\\begin{algorithm}[hbt!]\n\\caption{Naive approach to multi-task learning through Popper}\\label{alg:multi_popper_naive}\n\\begin{algorithmic}\n\\Function{multipoppernaive}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\ForAll{$t$ in $T$}\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{size} \\gets 1$\n\\While{TRUE}\n    \\State $\\mathit{sol} \\gets$ \\Call{popper}{$t$, $BK$, $\\mathit{size}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\State \\textbf{break}\n    \\EndIf\n\\State $\\mathit{size} \\gets \\mathit{size}+1$ \n\\EndWhile\n\\EndFor\n\\State \\Return $S$\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Knowledge transfer: iterative deepening approach}\n\\label{sota-sub}\nAs previously discussed, the naive approach does not make use of knowledge transfer. To address this issue, in \\cite{lin2014bias}, Lin et al. propose a new multi-task learning approach based on the Metagol ILP system. This method is currently state-of-the-art in multi-task ILP. The described solution uses iterative deepening search, proceeding as follows:\n\\begin{enumerate}\n    \\item Set $\\mathit{maxSize} = 1$.\n    \\item For all tasks, try to find a solution with size at most $\\mathit{maxSize}$.\n    \\item If all tasks are solved, return.\n    \\item Otherwise, update the background knowledge with newly learnt tasks, increment $\\mathit{maxSize}$ by 1, and return at step 2.\n\\end{enumerate}\n\nPseudo-code for integrating this search strategy into the Popper ILP system is present in \\autoref{alg:multi_popper_soda}. \n\n\\begin{algorithm}[hbt!]\n\\caption{Iterative deepening search}\\label{alg:multi_popper_soda}\n\\begin{algorithmic}\n\\Function{multipopper}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{maxSize} \\gets 1$\n\\While{TRUE}\n\\For{$t$ in $T$}\n    \\State $\\mathit{sol} \\gets$ \\Call{popper}{$t$, $\\mathit{BK}$, $\\mathit{maxSize}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\EndIf\n\\EndFor\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\State $\\mathit{maxSize} \\gets \\mathit{maxSize} + 1$\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\paragraph{Advantages.} Through the introduction of knowledge transfer as part of the multi-task learning strategy, this solution improves performance over the naive approach when used on correlated data sets. For example, assume our background knowledge contains two primitive predicates \\texttt{bk0} and \\texttt{bk1} and we are given two tasks, \\texttt{f0} and \\texttt{f1}, such that \\linebreak \\texttt{f1(A,B,C):-f0(A),f0(B),f0(C)}. Learning \\texttt{f0} and re-using it's definition as part of an enhanced background knowledge allows us to learn \\texttt{f1} more quickly by reducing the size of its solution.\n\\paragraph{Disadvantages.} We observe that even when the BK is augmented, the search size continues to increase. On one hand, this particularity provides an advantage when solution sizes remain large even with the new BK, such as in \\autoref{ref:soda-example}. On the other hand, it gives rise to the following disadvantages: \n\n\\begin{figure}[hbt!]\n\\begin{minted}{prolog}\n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        isMother(\"Mary\",\"Alice\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandgrandmother(\"Mary\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandgrandmother(A,B) :- isMother(A,C), isWife(C,D),\n                                   isGrandfather(D, B)\n\\end{minted}\n\\caption{Positive example for iterative deepening search}\n\\label{ref:soda-example}\n\\end{figure}\n\nFirstly, after each increase of the $\\mathit{maxSize}$ variable, we will reevaluate all programs of size between $1$ and  $\\mathit{maxSize} - 1$ from the previous iteration. \n\nSecondly, we might search through an unnecessarily large hypothesis space when a BK augmentation might have allowed us to find a solution in a smaller space. \n\nMoreover, as a consequence of the second disadvantage, this approach does not learn the most compact solution possible. We will further describe this case below:\n\\begin{enumerate}\n    \\item We will use the BK from the robot movement problem. For simplicity, we will only be concerned with \\texttt{move\\_up(A,B)}.\n    \\item Assume we have two tasks, \\texttt{f0(A,B)} and \\texttt{f1(A,B)}, such that their solutions are:\n    \\begin{itemize}\n        \\item \\texttt{f0(A,B):-move\\_up(A,C),move\\_up(C,D),move\\_up(D,B)}\n        \\item \\texttt{f1(A,B):-move\\_up(A,C),move\\_up(C,D),move\\_up(D,E),move\\_up(E,B)}\n    \\end{itemize}\n    \\item In this approach, we will find the solution for \\texttt{f0} after having $\\mathit{maxSize}=4$, and then increment $\\mathit{maxSize}$ to 5, before augmenting the BK.\n    \\item When searching for programs up to size 5, we can identify two valid solutions for \\texttt{f1}:\n    \\begin{itemize}\n        \\item  \\texttt{f1(A,B):-move\\_up(A,C),move\\_up(C,D),move\\_up(D,E),move\\_up(E,B)}\n        \\item  \\texttt{f1(A,B):-f0(A,C),move\\_up(C,B)}\n    \\end{itemize}\n    \\item Such, we observe that the algorithm could return either solution, having no preference for the more compact program.\n\\end{enumerate}\n\nThirdly, this approach may lead to situations where learning does not conclude in a reasonable time frame because of the increase in the hypothesis space size. This can be encountered in a situation where functions a learnt in a chain, for example:\n\\begin{enumerate}\n    \\item Define $\\texttt{p(A,B)}^n \\texttt{:-} \\texttt{p(A,C),p(C,B)}^{n-1}$ for any predicate \\texttt{p} and natural number $n > 1$. Base case $\\texttt{p(A,B)}^1 \\texttt{:-} \\texttt{p(A,B)}$. Then, we have tasks \\texttt{f0,f1,f2,f3,f4}:\n    \\begin{itemize}\n        \\item $\\texttt{f0(A,B):-move\\_up(A,C)}^6$\n        \\item $\\texttt{f1(A,B):-move\\_up(A,C)}^{36}$\n        \\item $\\texttt{f2(A,B):-move\\_up(A,C)}^{216}$\n        \\item $\\texttt{f3(A,B):-move\\_up(A,C)}^{1296}$\n        \\item $\\texttt{f4(A,B):-move\\_up(A,C)}^{7776}$\n    \\end{itemize}\n    \\item By using this approach, \\texttt{f0} has to be learnt and added to the BK before \\texttt{f1}, \\texttt{f1} before \\texttt{f2}, \\texttt{f2} before \\texttt{f3}, \\texttt{f3} before \\texttt{f4}.\n    \\item Such, each task will be learnt by searching through hypothesis spaces of sizes as follows:\n    \\begin{itemize}\n        \\item \\texttt{f0} at size 7\n        \\item \\texttt{f1} at size 8\n        \\item \\texttt{f2} at size 9\n        \\item \\texttt{f3} at size 10\n        \\item \\texttt{f4} at size 11\n    \\end{itemize}\n    \\item But, all given tasks will have solutions of 6 body literals:\n    \\begin{itemize}\n        \\item $\\texttt{f0(A,B):-move\\_up(A,C)}^6$\n        \\item $\\texttt{f1(A,B):-f0(A,C)}^{6}$\n        \\item $\\texttt{f2(A,B):-f1(A,C)}^{6}$\n        \\item $\\texttt{f3(A,B):-f2(A,C)}^{6}$\n        \\item $\\texttt{f4(A,B):-f3(A,C)}^{6}$\n    \\end{itemize}\n    \\item Given the formula from \\autoref{ILPintro} and previous experiments we know that hypothesis spaces containing programs of more than 10 literals can be generally considered unsolvable within a reasonable time limit. Such, we observe the benefit of the new approach.\n\\end{enumerate}\n\nIn the rest of this chapter, we will propose and analyze multiple search strategies and improvements to overcome the aforementioned disadvantages.\n\n\\subsection{Knowledge transfer: resetting search size}\nWe proceed by bringing forward a new search strategy and providing an argument for its theoretical advantages over the current state-of-the-art approach.\n\nWhen concerned with programs that achieve a significant reduction in size through knowledge transfer, the previous approach performs redundant work by continuing to search for solutions of constantly increasing maximum size. We argue for restarting the search from maximum size to one after a solution is found to one of the tasks.\n\nBy doing this we reduce the amount of work required when programs become ``easy\" after knowledge transfer is performed. Below we will compute the difference in the number of candidate solutions between the two approaches:\n\\begin{enumerate}\n    \\item Let $\\mathit{BK}_0$ be our initial background knowledge.\n    \\item Let $\\mathit{BK}_1 = \\mathit{BK}_0 \\cup \\{\\texttt{f0}\\}$.\n    \\item Let \\texttt{f0}, \\texttt{f1} be two predicates such that:\n    \\begin{itemize}\n        \\item \\texttt{f0} has a solution of size $n$ in $\\mathit{BK}_0$.\n        \\item \\texttt{f1} has a solution of size $m > n$ in $\\mathit{BK}_0$.\n        \\item \\texttt{f1} has a solution of size $p < m$ in $\\mathit{BK}_1$.\n    \\end{itemize}\n    \\item Let $\\mathit{HS}(\\texttt{f}, \\mathit{size}, \\mathit{BK})$ be the size of the hypothesis space for task \\texttt{f}, given $\\mathit{size}$ and the background knowledge $\\mathit{BK}$.\n    \\item Let $\\mathit{HS}^*(\\texttt{f}, \\mathit{maxSize}, \\mathit{BK}) = \\sum\\limits_{\\mathit{size}=1}^{\\mathit{maxSize}}\\mathit{HS}(\\texttt{f}, \\mathit{size}, \\mathit{BK})$\n    \\item Then, the total number of programs generated can be computed as follows:\n    \\begin{itemize}\n        \\item Without reset we have: \\begin{dmath}\\sum\\limits_{i=1}^{n}(\\mathit{HS}^*(\\texttt{f0}, i, \\mathit{BK}_0) + \\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_0)) +\\\\ \\mathit{max}(\\sum\\limits_{i=n + 1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1),\\mathit{HS}^*(\\texttt{f1}, n + 1, \\mathit{BK}_1))\\end{dmath}\n        \\item With reset we have: \\begin{dmath}\\sum\\limits_{i=1}^{n}(\\mathit{HS}^*(\\texttt{f0}, i, \\mathit{BK}_0) + \\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_0)) + \\sum\\limits_{i=1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1)\n        \\end{dmath}\n    \\end{itemize}\n    \\item Therefore, we identify two cases:\n    \\begin{itemize}\n        \\item If $p \\leq n$ (4.1) and (4.2) can be written and compared as follows:\n                \\begin{dmath}\n                    \\mathit{HS}^*(\\texttt{f1}, n + 1, \\mathit{BK}_1) > \\sum\\limits_{i=1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1)\n                \\end{dmath}\n        \\item If $p > n$ (4.1) and (4.2) can be written and compared as follows:\n            \\begin{dmath}\n                \\sum\\limits_{i=n + 1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1) < \\sum\\limits_{i=1}^{p}\\mathit{HS}^*(\\texttt{f1}, i, \\mathit{BK}_1)\n            \\end{dmath}\n    \\end{itemize}\n    \\item By using the formula for $\\mathit{HS}$, given in \\autoref{ILPintro}, we argue that the performance increase in the case where $p \\leq n$ overcomes the performance decrease from the case $p > n$. Moreover, we can extend this argument to multiple tasks (e.g. introduce \\texttt{f2} such that \\texttt{f2} is very difficult, but has a much simpler solution that uses \\texttt{f1}) in order to strengthen the argument.\n\\end{enumerate}\n\nAs a result, we can observe one use case improvement of the reset strategy. \n\nA more concrete example can be seen in \\autoref{fig:reset-example}, where the original approach would have performed redundant work on learning the isGrandgrandmother task by evaluating solutions of size four when a solution can be found at size three. \n\n\\begin{figure}[hbt!]\n\\begin{minted}{prolog}\n   \n        isFather(\"John\",\"Dan\")\n        isFather(\"Paul\",\"John\")\n\n        isWife(\"Alice\",\"Paul\")\n        isMother(\"Olga\",\"Alice\")\n        \n   \n        isGrandfather(\"Paul\",\"Dan\")\n        \n        isGrandmother(\"Alice\",\"Dan\")\n        \n        isGrandgrandmother(\"Olga\",\"Dan\")\n        \n   \n        isGrandfather(A,B) :- isFather(A,C), isFather(C,B)\n        \n        isGrandmother(A,B) :- isWife(A,C), isGrandfather(C, B)\n        \n        isGrandgrandmother(A,B) :- isMother(A,C), isGrandmother(C,B)\n\\end{minted}\n\\caption{Example for iterative deepening search with search size reset}\n\\label{fig:reset-example}\n\\end{figure}\n\nMoreover, we will formalise this claim:\n\\begin{enumerate}\n    \\item Assume there exists some task $f$, such that a solution of size $n$ has been found, but there exists a simpler solution, of size $m$, $m < n$. Both solutions generated from the same BK.\n    \\item By our approach, if a solution of size $n$ has been found then Popper tried all solutions of size $s$, $s\\in \\{1,2,..,n-1\\}$ with the current BK.\n    \\item Then it means that no solution exists of size $s$, $s \\in \\{1,2,..,n-1\\}$.\n    \\item Contradiction with $m < n$.\n    \\item Therefore, $n$ is the smallest solution size that can be found with the current BK.\n\\end{enumerate}\n\nAn \\autoref{alg:multi_popper_reset} we showcase pseudo-code for the reset approach.\n\n\\begin{algorithm}[hbt!]\n\\caption{Iterative deepening search with search size reset}\\label{alg:multi_popper_reset}\n\\begin{algorithmic}\n\\Function{multipopperreset}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{maxSize} \\gets 1$\n\\While{TRUE}\n\\State $\\mathit{change} \\gets \\mathit{false}$\n\\For{$t$ in $T$}\n    \\State $\\mathit{sol} \\gets$ \\Call{popper}{$t$, $\\mathit{BK}$, $\\mathit{maxSize}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $\\mathit{change}\\gets \\mathit{true}$\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\EndIf\n\\EndFor\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\If{$\\mathit{change}==\\mathit{true}$}\n\\State $\\mathit{maxSize} \\gets 1$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\Else\n\\State $\\mathit{maxSize} \\gets \\mathit{maxSize} + 1$\n\\EndIf\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Knowledge transfer: removing redundant testing}\nRecall \\autoref{alg:multi_popper_reset}. Any call to Popper would search all solutions up to $\\mathit{maxSize}$, each time it is incremented. We can modify the Popper algorithm to only search for programs of a fixed size. Therefore, we will no longer evaluate programs twice unless a solution has been found to one of the tasks. An overview of this change is present in \\autoref{alg:multi_popper_reset+}. \n\\paragraph{Advantages.} The main advantage of this approach is that it does not reiterate through smaller search space sizes unless necessary (when the BK is augmented).\n\\paragraph{Disadvantages.} Because we do not re-search the smaller hypothesis space sizes, we do not keep constraint information between searches. This may result in a significant increase in the size of the hypothesis space evaluated. More specifically, if we are searching through a hypothesis space that contains only programs of size $n$, we will not make use of the constraint generated in the previous steps from programs of sizes between $1$ and $n - 1$.\n\\begin{algorithm}[hbt!]\n\\caption{Breadth fist search with size reset}\\label{alg:multi_popper_reset+}\n\\begin{algorithmic}\n\\Function{multipopperreset+}{}\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S$ $\\gets$ \\{\\}\n\\State $\\mathit{size} \\gets 1$\n\\While{TRUE}\n\\State $\\mathit{change} \\gets \\mathit{false}$\n\\For{$t$ in $T$}\n    \\State $\\mathit{sol} \\gets$ \\Call{fixedsizedpopper}{$t$, $\\mathit{BK}$, $\\mathit{size}$}\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $\\mathit{change}\\gets \\mathit{true}$\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\State \\textbf{break}\n    \\EndIf\n\\EndFor\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\If{$\\mathit{change}==\\mathit{true}$}\n\\State $\\mathit{size} \\gets 1$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\Else\n\\State $\\mathit{size} \\gets \\mathit{size} + 1$\n\\EndIf\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\\section{Preserving constraints}\n\\label{constraints}\nTransferring knowledge between tasks requires Popper to retry hypothesis spaces that contain programs with a smaller number of literals than the maximum tried until now (because the introduction of more programs in the background knowledge creates new possible solutions at the respective sizes). Therefore, all presented approaches will retry the same set of solutions multiple times. \n\nWhen modifying search size, the Metagol did not have the ability to transfer knowledge through a standard form, even when working with a single task, e.g. if you have not found a solution of maximum size up to $n$, you would need to restart the search from scratch when looking for a solution of maximum size up to $n + 1$. In this project, we circumvent this limitation by extending the Popper ILP system, which has the ability to retain information from failed attempts in the form of hypothesis constraints.\n\nMore specifically, Popper allows us to retain constraints between failed attempts and prune the hypothesis space as soon as we begin searching with different parameters (in our case, an increased maximum number of literals).\n\nTo reduce the amount of repeated work caused by search size resets, we adapt all previous methods to preserve constraints learnt by Popper for each task. \n\nWe showcase that necessary changes to implement constraint preservation in the improved reset approach (recall \\autoref{alg:multi_popper_reset+}) in \\autoref{alg:multi_popper_cons}. \n\\begin{algorithm}[hbt!]\n\\caption{Constraint preserving search}\\label{alg:multi_popper_cons}\n\\begin{algorithmic}\n\\Function{multipoppercons}{}\n\\For{$i$ in \\Call{range}{n}}\n    \\State $\\mathit{rules}[i] \\gets \\emptyset$\n\\EndFor\n\\State $T \\gets \\{t_1, t_2,..,t_n\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S \\gets \\emptyset$\n\\State $\\mathit{size} \\gets 1$\n\\While{TRUE}\n\\State $\\mathit{change} \\gets \\mathit{false}$\n\\For{$t$ in $T$}\n    \\State $(\\mathit{sol},\\mathit{cons}) \\gets$ \\Call{constraintpopper}{$t$, $\\mathit{BK}$, $\\mathit{size}$, $\\mathit{rules}[i]$}\n    \\State $\\mathit{rules}[i] \\gets \\mathit{rules}[i] \\cup \\mathit{cons}$\n    \\If{$\\mathit{sol} \\neq \\mathit{null}$}\n    \\State $\\mathit{change}\\gets \\mathit{true}$\n    \\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n    \\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n    \\State \\textbf{break}\n    \\EndIf\n\\EndFor\n\\If{$T =$ \\{\\}}\n\\State \\Return $S$\n\\EndIf\n\\If{$\\mathit{change}==\\mathit{true}$}\n\\State $\\mathit{size} \\gets 1$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\Else\n\\State $\\mathit{size} \\gets \\mathit{size} + 1$\n\\EndIf\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\\paragraph{Advantages.} By making use of constraint preservation, we remove the bottleneck created by resetting the search on a program and then retesting all candidate hypotheses. Moreover, this technique resolves the significant disadvantage of the breadth first search approach. Now that all constraints are preserved, there is no information lost by not reiterating through the smaller hypothesis spaces.\n\\paragraph{Disadvantages.} We argue that there do not exist significant disadvantages of preserving constraints. One could point out the fact that they may increase memory requirements and the overhead of the reset phase. However, the expectation is that unless a solution is found early, most of the constraints from previous stages would have also been learnt during the current iteration. \n\nFinally, we provide an argument for the fact that it is always correct to preserve constraints. \n\nWe are given the fact that any constraint generated by the system is sound \\cite{popper}. \n\nGiven a constraint $C$ learnt during an iteration of the learning algorithm with background knowledge $\\mathit{BK}$, we show that constraint $C$ would also hold for background knowledge $\\mathit{BK}' = \\mathit{BK} \\cup N$, where $N$ is some set of valid background knowledge. \n\nLet $h$ be the hypothesis that determined $C$. $h$ is generated from $\\mathit{BK}$, therefore  $h$ also generated from $\\mathit{BK'}$. Therefore, during a learning process based on $\\mathit{BK'}$, if $h$ will also be tested, it will generate the same sound constraint. \n\\section{Automatic computation of task learning order}\nWhen considering large data sets where some tasks may be unsolvable, the ordering of the tasks becomes important because the algorithm may not get to consider certain tasks that have a solution of the current search size. Consequently, we propose the introduction of task ordering methods, similar to curriculum learning. We focus on run-time evaluation of tasks, aiming to order tasks by how close our system is to finding a solution. For example, take the following scenario\\footnote{For this example we assume that the solution program is always found last in the hypothesis space}:\n\\begin{enumerate}\n    \\item We are given 100 tasks, \\texttt{fi} with \\texttt{i} $\\in \\{0,..,99\\}$.\n    \\item Task \\texttt{f99} has a solution of size seven.\n    \\item All other tasks have solutions of nine using the initial BK and solutions of size three if \\texttt{f99} is part of their BK.\n    \\item Recall the formula in \\autoref{ILPintro}. Based on it, we will approximate the number of programs in hypothesis space of size $x$ to $2^{2^x}$ \\footnote{The purpose of this approximation is to provide the reader with an overview of the rate at which the size of the hypothesis space increases with the number of literals. Providing a more accurate result would require a discussion around the values of other parameters from the formula in \\autoref{ILPintro}.}. \n    \\item Now we will compare the solving time for the reset approach of the best case and worst case orderings.\n    \\begin{itemize}\n        \\item Order \\texttt{f0}, \\texttt{f1},..,\\texttt{f99}\n        \\begin{dmath}\n            \\sum\\limits_{i=1}^{7}(100*2^{2^i})+\\sum\\limits_{i=1}^{3}(99*2^{2^i})\n        \\end{dmath}\n        \\item Order \\texttt{f99}, \\texttt{f0},\\texttt{f1},\\texttt{f2},..,\\texttt{f98} (\\texttt{f99} is computed first)\n        \\begin{dmath}\n            \\sum\\limits_{i=1}^{6}(100*2^{2^i})+2^{2^7}+\\sum\\limits_{i=1}^{3}(99*2^{2^i})\n        \\end{dmath}\n    \\end{itemize}\n    \\item We obtain that when the system is given the order \\texttt{f0}, \\texttt{f1},..,\\texttt{f99} it tests approximately 100 times more programs then when it is given the order \\texttt{f99}, \\texttt{f0},\\texttt{f1},\\texttt{f2},..,\\texttt{f98}.\n\\end{enumerate}\n\\paragraph{Advantages.} Given the above example, we know that task ordering is important for the performance of the system. Approaches that perform automatic task ordering can overcome this limitation, by identifying such scenarios and selecting and appropriate task sequence.\n\n\\paragraph{Possible disadvantages.} We outline a list of possible disadvantages in order to highlight the points we need to take into account when designing our heuristics. Firstly, the evaluation function is not guaranteed to provide a significant benefit. For example, a heuristic that orders by the number of examples entailed by the best program yet would have no effect on a data set where all tasks are given a single example. Secondly, wrongly estimating the importance of size in the evaluation function may significantly deter search time by testing more extensive programs when not necessary. To mitigate this issue, we experiment with a conservative evaluation function: firstly, ordering by size and then tie-breaking through our heuristic. \n\nGiven the above considerations, we propose two solutions for this problem: one that orders tasks by the number of examples currently solved and the other by the number of constraints generated by the system.\n\\subsection{Ordering by number of examples solved}\nThe first approach we experiment with is ordering tasks decreasingly by the number of examples currently solved. We motivate this finding on the fact that very often, entailing an example is equivalent to finding a program that has at least one clause which is part of a solution.\n\\subsection{Ordering by number of constraints}\nThe second approach we experiment with is ordering tasks by the number of constraints the system has generated for them. If the system has generated more constraints for a program it means that its hypothesis space can be pruned the most. Therefore, we expect this approach to identify tasks that are closed to being solved.\n\nIn \\autoref{alg:multi_popper_prio} we provide a generic implementation of the task order approach. It makes use of a priority queue data structure \\textit{TPQ}, which is initialised with the desired heuristic.\n\n\\begin{algorithm}[hbt!]\n\\caption{Priority searching of solutions}\n\\label{alg:multi_popper_prio}\n\\begin{algorithmic}\n\\Function{multipopperprio}{}\n\\For{$i$ in \\Call{range}{n}}\n    \\State $\\mathit{rules}[i] \\gets \\emptyset$\n\\EndFor\n\\State $TPQ \\gets \\{(t_1, 1, 1.0), (t_2, 1, 1.0),..,(t_n, 1, 1.0)\\}$\n\\State $\\mathit{BK} \\gets \\{$BACKGROUND KNOWLEDGE$\\}$\n\\State $S \\gets \\{\\}$\n\\While{\\textbf{not} TPQ \\textbf{is empty}}\n\\State $(t, size, \\_)$ $\\gets$ \\Call{pop}{$TPQ$}\n\\State $(\\mathit{partialsol},\\mathit{cons}) \\gets$ \\Call{popperpartial}{$t$, $BK$, $\\mathit{size}$, $\\mathit{rules}[i]$}\n\\State $\\mathit{rules}[i] \\gets \\mathit{rules}[i] \\cup \\mathit{cons}$\n\\If{\\Call{isvalidsol}{$\\mathit{partialsol}$,$t$}}\n\\State $S$ $\\gets$ $S$ $\\cup \\{\\mathit{sol}\\}$\n\\State $\\mathit{BK} \\gets \\mathit{BK} \\cup S$\n\\State \\Call{ResetPopper}{T}\n\\State $T$ $\\gets$ $T$ $\\setminus$ \\{$t$\\} \n\\State $TPQ \\gets \\{(t_i, 1, 1.0) \\vert t_i \\in T \\}$\n\\Else\n\\If{$partialsol \\neq null$}\n\\State $priority \\gets$ \\Call{computeprio}{$sol$}\n\\State \\Call{PUSH}{$TPQ$, $(t, size + 1, priority)$}\n\\EndIf\n\\EndIf\n\\EndWhile\n\\State \\Return $S$\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\\chapter{Experimental Results}\nThe goal of this chapter is to determine the performance of the approaches proposed in the previous chapter. We will perform experiments in order to obtain answers to the following questions: \n\n\\textbf{Q1.} Does constraint preservation offer an improvement in learning speed for approaches that repeat searching over the same hypothesis space? \n\nFor each strategy mentioned previously, we will evaluate the running time change resulting from enabling constraint preservation on multiple data sets that present a wide array of specific features. \n\n\\textbf{Q2.} Do the advantages of resetting search size overcome its disadvantages when compared to the state-of-the-art approach? \n\nWe will compare the two reset strategies with the current state-of-the-art iterative deepening search approach on data sets that are expected to benefit each method and compare the best case and worst case results for each algorithm. \n\n\\textbf{Q3.} Is there any benefit in heuristically ordering programs on the same size level during a search? \n\nWe will compare the two ordering strategies against a system that selects a random ordering at the beginning of the learning process and respects it throughout each stage. We have chosen this benchmark in order to remove any bias from a system that takes the given original order, which may be beneficial\/not beneficial. \n\nThe scope of this experiment will be to determine if heuristic program ordering is either beneficial, neutral, or provides a drawback through the added overhead.\n\\section{Data sets}\n\\textbf{Strings transformation.} We test our system using an existing data set of string transformation operations, out of which we select the ones that have been found solvable by Popper during previous experiments \\cite{string-data}.\n\n\\textbf{Robot movement.} We generate a set of tasks that focus on robot movement. The goal of each task is to learn to traverse a large area using only unit length movement instructions. This task will be used to compare the performance of the reset strategy and state-of-the-art strategy on a large set of tasks that can all be reduced to a size of three literals in a multi-task learning setting. \n\n\\textbf{Printer head.} We generate a set of tasks that simulate the movement of a printer head over a piece of paper. The printer head is able to perform one pixel size movements, fill a pixel with colour or return to an initial position. \n\nThis data set is an extension of the robot movement set, which also allows recursion. Instead of learning to traverse fixed distances, the printer head learns to work on sheets of arbitrary size. \n\nIn \\autoref{fig:zebra} and \\autoref{fig:cross} we show two examples of pixel images used in training the system.\n\\begin{figure}[!htb]\n    \\centering\n    \\begin{minipage}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.4\\linewidth]{images\/zebra.png}\n        \\caption{Zebra pattern image}\n        \\label{fig:zebra}\n    \\end{minipage}%\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.4\\linewidth]{images\/cross.png}\n    \\caption{Cross pattern image}\n    \\label{fig:cross}\n    \\end{minipage}\n\\end{figure}\n\\section{Testing environment}\n\\subsection{Specifications}\nAll experiments have been run on Amazon EC2 machines (c6i.large), which have the following specifications: \n\n\\textbf{Operating system:} Ubuntu 20.04.3 LTS\n\n\\textbf{Software versions:}\n\\begin{itemize}\n    \\item \\textbf{Python} 3.8.10\n    \\item \\textbf{PySwip} 0.2.11\n    \\item \\textbf{Clingo} 5.5.0\n    \\item \\textbf{SWI-Prolog} 8.4.2\n\\end{itemize}\n\\textbf{Processor:} Intel(R) Xeon(R) Platinum 8375C CPU @ 2.90GHz \n\n\\textbf{Number of cores:} 2 \n\n\\textbf{Threads per cores:} 2 \n\n\\textbf{RAM memory:} 4 GB\n\\subsection{Testing protocol}\nFor all data sets, each strategy was run ten times with a timeout limit of 6000 seconds. \n\nBefore each run, the ordering of the input tasks is randomly determined. This approach allows our results to ensure the reliability of our data and provide greater confidence that the observed performance of the system is consistent. \n\nThere are significant sources of noise that require experiments to be run multiple times. For example, some input orderings benefit some systems over others. Furthermore, the Popper system searches through a hypothesis space in a non-deterministic manner. Therefore, during different runs, specific tasks can be given different definitions that are not necessarily reusable in the same contexts.\n\\section{Results}\n\\subsection{Algorithms}\nIn this section, we will define a list of acronyms for each of our search strategies in order to make it easier for the reader to interpret the results. \n\n\\textbf{NAIVE} is the naive algorithm. \n\n\\textbf{ID} is the state-of-the-art, iterative deepening approach. \n\n\\textbf{RESET-ID} is the iterative deepening reset approach. \n\n\\textbf{RESET-BFS} is the breadth first search reset approach. \n\n\\textbf{PRIO-EX} is a version of RESET-BFS that sorts tasks on the same level decreasingly by the number of examples solved. \n\n\\textbf{PRIO-CONS} is a version of RESET-BFS that sorts tasks on the same level decreasingly by the number of constraints discovered. \n\nIf any algorithm name is followed by the PRESERVE keyword (e.g. ID PRESERVE), it means that constraint preservation option has been set true. Otherwise, we assume that constraint preservation has not been used. \n\nBecause the breadth first search based algorithms perform poorly without constraint preservation, we will choose always to enable constraint preservation when comparing the performance of multiple strategies.\n\\subsection{Printer head}\n\\subsubsection{Overview}\nIn \\autoref{fig:printer-results} we showcase the experimental results for all our algorithms as a percentage of tasks solved after several minutes. This data set contains many tasks that are solved relatively early and a few complicated tasks, such we attach an additional plot (see \\autoref{fig:printer-sec-results}) that highlights the behaviour of the strategies in the first 100 seconds. We consider that this metric better describes the performance of the systems because it more clearly showcases their differences and removes the redundant noise generated by task ordering in the higher hypothesis spaces.\n\\pgfplotsset{\n    myplotstyle\/.style={\n    legend style={draw=none, font=\\small},\n    legend cell align=left,\n    legend pos=outer north east,\n    ylabel style={align=center, font=\\bfseries\\boldmath},\n    xlabel style={align=center, font=\\bfseries\\boldmath},\n    x tick label style={font=\\bfseries\\boldmath},\n    y tick label style={font=\\bfseries\\boldmath},\n    ymax = 100,\n    scaled ticks=false,\n    every axis plot\/.append style={thick},\n    },\n}\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={NAIVE, ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE,\n    PRIO-EX, PRIO-EX PRESERVE, PRIO-CONS, PRIO-CONS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 80,\n    width=10cm\n]\n\\addplot+[mark = none, color = black, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=NAIVE, col sep=comma, y error = NAIVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO NO-PRESERVE, col sep=comma, y error = PRIO NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = diamond*, color = teal, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS NO-PRESERVE, col sep=comma, y error = PRIO CONS NO-PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Images_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Overview of printer head results (showing standard error)}\n    \\label{fig:printer-results}\n\\end{figure}\n\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={NAIVE, ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE,\n    PRIO-EX, PRIO-EX PRESERVE, PRIO-CONS, PRIO-CONS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = none, color = black, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=NAIVE, col sep=comma, y error = NAIVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = triangle*, color = magenta, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO NO-PRESERVE, col sep=comma, y error = PRIO NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = diamond*, color = teal, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS NO-PRESERVE, col sep=comma, y error = PRIO CONS NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Zoom in of printer head results for the first 100 seconds (showing standard error)}\n    \\label{fig:printer-sec-results}\n\\end{figure}\n\n \\begin{table}[hbt!]\n \\centering\n\\begin{tabular}{c|c|c|c}\n        Algorithm & \\multicolumn{1}{l|}{Min} & \\multicolumn{1}{l|}{Max} & \\multicolumn{1}{l}{Standard deviation} \\\\ \\hline\nNAIVE              & 1                        & 9                       & 1.97                                   \\\\\nPRIO-CONS          & 7                       & 10                       & 1.1                                   \\\\\nID               & 11                       & 13                       & 0.7                                   \\\\\nID PRESERVE      & 11                       & 13                       & 0.65                                    \\\\\nRESET-ID           & 11                       & 13                       & 0.6                                   \\\\\nRESET-ID PRESERVE  & 11                        & 13                       & 0.62                                  \\\\\nRESET-BFS PRESERVE & 11                       & 13                       & 0.64                                  \\\\\nPRIO-EX            & 11                       & 13                       & 0.66                                   \\\\\nPRIO-EX PRESERVE   & 11                       & 13                       & 0.63                                    \\\\\nPRIO-CONS PRESERVE & 12                       & 13                       & 0.46                                  \\\\\nRESET-BFS          & 11                       & 14                       & 0.81                                  \n\\end{tabular}\n    \\caption{Printer head algorithms results variance}\n    \\label{tab:printer-table}\n\\end{table}\n\n\\subsubsection{Question 1}\nIn \\autoref{fig:printer-results} and \\autoref{fig:q1-printer-results} we observe that by enabling the preserve option, all algorithms consistently outperform their default variants. Therefore, we conclude that the answer for \\textbf{Q1} is \\textbf{yes}. This result confirms our expectation that constraint preservation will result in at least a minor improvement in all use cases.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Preserve comparison of printer head results for the first 100 seconds (showing standard error)}\n    \\label{fig:q1-printer-results}\n\\end{figure}\n\\subsubsection{Question 2}\nBy analysing \\autoref{fig:q2-printer-results} and \\autoref{fig:printer-results} we observe that the reset approaches perform slightly worse when they have to perform reset operations quite often (solving simple tasks repeatedly), but end up performing similarly to the state-of-the-art approach in the long run. Therefore, we determine that the answer for \\textbf{Q2} is \\textbf{no}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-ID PRESERVE, RESET-BFS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph comparing state-of-the-art and reset strategies for printer data during the first 100 seconds (showing standard error)}\n    \\label{fig:q2-printer-results}\n\\end{figure}\n\\subsubsection{Question 3}\nWe do not observe any significant benefit of the priority approaches that can not be attributed to noise. Therefore, we conclude that the answer for \\textbf{Q3} is \\textbf{no}. It is worth mentioning that both priority algorithms compute the most difficult task faster on average than the other approaches. Their average times to learn 13 tasks are 661 and 671 seconds, whereas the state-of-the-art approach achieves a mean of 1951 seconds and the best reset approach 1096. Is it highly likely that these results are a consequence of variation in the initial ordering and do not reflect the actual performance of the search strategies.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-BFS PRESERVE,\n    PRIO-EX PRESERVE, PRIO-CONS PRESERVE},\n    xlabel={Seconds},\n    ylabel={Percentage of tasks solved},\n    ymax = 60,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Images_Sec.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Zoom in of printer head results for the first 100 seconds (showing standard error)}\n    \\label{fig:q3-printer-results}\n\\end{figure}\n\\subsection{String transformation}\n\\subsubsection{Overview}\nIn \\autoref{fig:string-results} we present an overview of the performance of each algorithm, expressed as the percentage of tasks solved after several minutes. We also outline the variance in \\autoref{tab:string-table}, by listing their worst and best performances, along with the standard deviation of the total number of tasks solved.\n\nWe observe the fact that no algorithm was able to solve all the tasks. We attribute this to the fact that we have used functional testing in order to validate if a generated program is correct, whereas the original experiment did not enable this option. Functional testing refers to evaluating predicates by disregarding their truth value and only considering their output \\cite{functional-definition}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={NAIVE, ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE,\n    PRIO-EX, PRIO-EX PRESERVE, PRIO-CONS, PRIO-CONS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = none, color = black, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=NAIVE, col sep=comma, y error = NAIVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO NO-PRESERVE, col sep=comma, y error = PRIO NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = diamond*, color = teal, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS NO-PRESERVE, col sep=comma, y error = PRIO CONS NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Overview of string transformation results (showing standard error)}\n    \\label{fig:string-results}\n\\end{figure}\n\n \\begin{table}[hbt!]\n \\centering\n\\begin{tabular}{c|c|c|c}\n        Algorithm & \\multicolumn{1}{l|}{Min} & \\multicolumn{1}{l|}{Max} & \\multicolumn{1}{l}{Standard deviation} \\\\ \\hline\nPRIO-CONS          & 13                       & 15                       & 0.54                                   \\\\\nPRIO-CONS PRESERVE & 14                       & 16                       & 0.83                                    \\\\\nID PRESERVE      & 12                       & 19                       & 2.5                                    \\\\\nNAIVE              & 0                        & 21                       & 6.28                                   \\\\\nID               & 12                       & 21                       & 2.45                                   \\\\\nRESET-BFS          & 11                       & 38                       & 11.41                                  \\\\\nRESET-ID           & 13                       & 39                       & 9.78                                   \\\\\nPRIO-EX            & 20                       & 39                       & 7.87                                   \\\\\nRESET-BFS PRESERVE & 15                       & 40                       & 10.39                                  \\\\\nPRIO-EX PRESERVE   & 35                       & 40                       & 1.5                                    \n\\end{tabular}\n    \\caption{String transformation algorithms results variance\n    }\n    \\label{tab:string-table}\n\\end{table}\n\\subsubsection{Question 1}\nFor the first question, we compare the results of the three main search strategies with the constraint preservation option both enabled and disabled. The result of the comparison is present in \\autoref{fig:q1-string-results}. We conclude that the answer to \\textbf{Q1} is \\textbf{yes}, as all strategies perform at least as well as their default version when constraint preservation is used.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma, y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph showing the effects of the preserve option for string data (showing standard error)}\n    \\label{fig:q1-string-results}\n\\end{figure}\n\\subsubsection{Question 2}\nIn \\autoref{fig:q2-string-results} we compare the results of the state-of-the-art strategy, versus the two reset strategies we suggested. Since both reset strategies perform better than the state-of-the-art approach we conclude that the answer for \\textbf{Q2} is \\textbf{yes}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-ID PRESERVE, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph comparing state-of-the-art and reset strategies for string data (showing standard error)}\n    \\label{fig:q2-string-results}\n\\end{figure}\n\\subsubsection{Question 3}\nIn \\autoref{fig:q3-string-results} we observe the effect of using a priority ordering for the breadth first search approach. We have also plotted the performance of the state-of-the-art system to reference as a benchmark. The PRIO-EX algorithm, which sorts by the precision of the best program found until now, outperforms the original reset approach. On the other hand, the PRIO-CONS algorithm, which sorts by the number of constraints, exhibits below par performance. We associate this with the fact that the PRIO-CONS algorithm uses a poor heuristic. Tasks for which we have accumulated more constraints have a smaller hypothesis space size, but it is not a good indicator of being closer to a solution. \nFurthermore, by analysing \\autoref{tab:string-table} we observe that priority ordering results in a lower variance in the performance of the system. We consider this a possible upside, as it improves the system's reliability. \nWe conclude that the answer for \\textbf{Q3} is \\textbf{yes}, but we outline the need for further research in this area, with the purpose of developing more accurate heuristics.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-BFS PRESERVE, PRIO-EX PRESERVE, PRIO-CONS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 90,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma,y error = SOTA PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = triangle*, color = magenta, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO PRESERVE, col sep=comma, y error = PRIO PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\addplot+[mark = diamond*, color = teal, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=PRIO CONS PRESERVE, col sep=comma, y error = PRIO CONS PRESERVE ERROR]{anc\/Results-Strings_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph showing the effects of priority ordering for string data (showing standard error)}\n    \\label{fig:q3-string-results}\n\\end{figure}\n\\subsection{Robot movement}\nThis data set contains single examples for each task because any positive examples generated would be part of the same equivalence class (e.g. moving two cells to the right is independent of starting position). Consequently, there would be no benefit in testing the PRIO-EX algorithm against this data set. Moreover, the previous two data sets have shown that the PRIO-CONS approach performs poorly in practice. Therefore, we decide to conclude our answer for \\textbf{Q3} based on the previous data and only test our claims for \\textbf{Q1} and \\textbf{Q2} against this data set.\n\\subsubsection{Overview}\nIn \\autoref{fig:robot-results} we present an overview of the performance of each algorithm, expressed as the percentage of tasks solved after several minutes.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID, ID PRESERVE, RESET-ID, RESET-ID PRESERVE, RESET-BFS, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 75,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA NO-PRESERVE, col sep=comma,y error = SOTA NO-PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma, y error = SOTA PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = star, color = blue, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET NO-PRESERVE, col sep=comma, y error = RESET NO-PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = *, color = green, solid,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP NO-PRESERVE, col sep=comma, y error = RESET-IMP NO-PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Overview of robot results (showing standard error)}\n    \\label{fig:robot-results}\n\\end{figure}\n\\subsubsection{Question 1}\nBy reviewing \\autoref{fig:robot-results}, we observe that using constraint preservation has improved the performance of two out of three of the tested strategies. As expected, RESET-BFS benefits the most from constraint preservation. On the other hand, the RESET-ID approach performs slightly worse when constraint preservation is enabled. By manually analysing the results of the data, we associate this finding with noise generated by order randomisation. Therefore, the answer for \\textbf{Q1} is \\textbf{yes}.\n\\subsubsection{Question 2}\n\\autoref{fig:q2-robot-results} showcases a comparison between the state-of-the-art approach and the two reset strategies. The current state-of-the-art approach solves the first few tasks much quicker because it does not reset the hypothesis space size on each found solution. After the 20 minutes mark, we observe that the state-of-the-art approach stops making progress because it has reached a large enough search size to be able to find any more solutions in a reasonable time frame. Whereas both reset approaches show that they are making constant progress, with their learning rate remaining relatively constant\\footnote{As mentioned in the introduction, we expect to observe a slight decay in learning rate as more tasks are added to the BK.}. Although the learning time of the first tasks is lower for the reset strategies, we evaluate that the ability to learn a higher percentage of tasks is more important when evaluating the performance of a system. Therefore, we conclude that the answer for \\textbf{Q2} is \\textbf{yes}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\n    myplotstyle,\n    legend pos=outer north east,\n    legend entries={ID PRESERVE, RESET-ID PRESERVE, RESET-BFS PRESERVE},\n    xlabel={Minutes},\n    ylabel={Percentage of tasks solved},\n    ymax = 75,\n    width=10cm\n]\n\\addplot+[mark = square*, color = red, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=SOTA PRESERVE, col sep=comma, y error = SOTA PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = star, color = blue, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET PRESERVE, col sep=comma, y error = RESET PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\addplot+[mark = *, color = green, dashed,error bars\/.cd, y dir=both,y explicit] table[x=Minutes,y=RESET-IMP PRESERVE, col sep=comma, y error = RESET-IMP PRESERVE ERROR]{anc\/Results-Robots_Min.csv};\n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Graph comparing state-of-the-art and reset strategies for robot data (showing standard error)}\n    \\label{fig:q2-robot-results}\n\\end{figure}\n\\chapter{Conclusions}\n\\section{Summary}\nAlthough an area that has shown high potential, there has been little development in terms of multi-task learning approaches in ILP. We have experimented with refining the current state-of-the-art approach with the expectation of obtaining substantially improved performance. We presented our assumptions' theoretical basis and then empirically confirmed them against three data sets. We have implemented both our new approach and the state-of-the-art technique based on the Popper ILP system. Our experimental results show that (1) constraint preservation improves learning speed in multi-task settings (\\textbf{yes} to \\textbf{Q1}), (2) the benefits of search size reset overcome it's disadvantages (\\textbf{yes} to \\textbf{Q2}), and (3) heuristic ordering of programs can improve learning rate (\\textbf{yes} to \\textbf{Q3}). \n\nIn the following sections, we present the limitations of our work and suggest future work that could improve the performance of Popper in multi-task environments. \n\\section{Limitations}\nWe have identified multiple factors that restricted the scope of our work.\n\\paragraph{Limited data.} The use of multi-task learning is relatively new to the field of ILP. Therefore, we highlight the lack of publicly available data sets used to benchmark such systems. We believe that our findings would have been better supported by testing in a wider variety of data, which have been previously found to uncover more use cases we might not have considered in our project.\n\\paragraph{Implementation improvement.} Our implementation of multi-task Popper has one specific drawback we have encountered during testing. On each reset operation, the underlying ILP environment is re-initialised, which often results in long reset times. There exists a simple solution to this issue, which would enforce the system always to preserve constraints between runs or to iterate through the constraint list and ``forget\" each of them one by one. Such an approach would have introduced bias in our experimental results by favouring constraint preservation. We argue that a more specialised implementation of Popper would circumvent this problem and further improve learning time (especially for the reset approach, where this part of the code is called more frequently).\n\\paragraph{Bias settings.} One significant challenge in ILP is selecting bias parameters \\cite{ade1995declarative}. Examples of bias parameters are the maximum number of variables allowed in a program or the maximum number of clauses allowed in a program. Appropriate selection of bias parameters has a significant impact on the learning rate by modifying the size of the possible hypothesis space. We argue that in the case of multi-task learning, the initial bias, which was set by the user with the initial BK in mind, may no longer be appropriate. For example, some tasks may have three clause solutions when solved by themselves, but only two clause solutions in the multi-task learning environment. Therefore, if our system could automatically compute bias parameters, it would improve the learning rate by constraining the hypothesis spaces size when the BK is augmented.\n\\section{Future work}\n\\paragraph{Explainable results.} Without multi-task learning, each program learnt is built from primitives, which are given in the BK and known to a human operator. When multi-task learning is introduced, program definitions may also contain other programs. This affects the ability of a human to understand the programs in two ways. Firstly, we note that, in a multi-task learning setting, knowledge of all the new predicates that appear in a solution is required before reading it. Secondly, we outline the fact that it is a common occurrence for more complicated tasks to be selected in favour of simpler tasks when they cannot be differentiated. For example see \\autoref{fig:explainable-anc\/Results-example}.\n\\begin{figure}[hbt!]\n\\centering\n\\begin{minted}{prolog}\n    f0(A,B) :- draw1(A,B)\n    f0(A,B) :- move_right(A,B),f0(A,B)\n   \n   \n    f1(A,B) :- f0(A,B) \n   \n    f1(A,B) :- draw1(A,B)\n\\end{minted}\n\\caption{Example of two different ways in which a program can be learnt}\n\\label{fig:explainable-anc\/Results-example}\n\\end{figure}\n\nIn the above case, both \\texttt{draw1} and \\texttt{f0} appear to behave identically for the system, in the absence of negative examples. In consequence, we propose the development of a predicate translator. A predicate translator would take the output of a multi-task ILP system and transform it such that all predicates will only make use of definitions from the BK, and any clause that does not affect the correctness of the task will be pruned out. \n\\paragraph{Task ordering.} In this project, we have shown the potential of task ordering in improving the learning capacity of multi-task ILP systems. We suggest further research in developing more accurate heuristics to predict which tasks are closer to being solved. One of the solutions we propose is the use of other Machine Learning techniques in order to identify more manageable tasks and give them priority. We also concede the fact that there are certain limitations to the applicability of such an approach, especially when there are very few examples given for each task. \n\\paragraph{Parallel computation.} When searching for solutions of a specific size, one could modify the current approach that attempts to solve tasks sequentially to try and solve multiple tasks concurrently, in a bag of tasks manner. This method would allow one to vastly increase the learning rate of multi-task Popper on multi-core systems. Moreover, a significant bottleneck observed during all experiments is that the system got stuck searching for solutions to unsolvable tasks. This issue will be mitigated, as data sets would require as many high difficulty tasks as threads to congest the system. \n\\newpage\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:sec1}\n\nThe first theoretical prediction of the famous phenomenon known as\nBose-Einstein condensation (BEC) was made in 1924 and 1925 by Bose\n\\cite{Bose24} and Einstein \\cite{Einstein 24-25}, respectively. In\na system of particles obeying Bose-Einstein statistics where the\ntotal number of particles is conserved, there should be a\ntemperature below which a finite fraction of all the particles\ncondense into the same one-particle state\n\\cite{Griffin95,Dalfavo99,Leggett01,Parkins98,Fetter98,Courteille01}.\nSeventy years later, in a remarkable experiment, Anderson \\emph{et\nal.} \\cite{Anderson95} have cooled magnetically trapped $^{87}$Rb\ngas to nanokelvin temperatures, and observed the BEC. This\ndiscovery has generated a huge amount of theoretical\ninvestigations\n\\cite{Baym96,Dalfavo96,Esry97,Fabrocini99,Fabrocini01,DuBois01,Minguzzi00,Naraschewski99,Pitaevski99}.\n\nThe main feature of the trapped alkali-metal and atomic hydrogen\nsystems (which obey the Bose-Einstein statistics) is that they are\ndilute. The crucial parameter defining the condition of diluteness\nis the gas parameter $\\chi=na^3$, where $n$ is the density of the\nsystem and $a$ is the s-wave scattering length \\cite{Fabrocini01}.\nThere are two ways to bring $\\chi$ outside the regime of validity\nof the mean field description. The first one is to increase the\ndensity, while the second one to change the effective size of the\natoms.\n\nThe diluteness of the gas is ensured when the effective atomic\nsize is small compared both to the trap size and to the\ninteratomic distance.  However, the effects of inter-particle\ninteractions are of fundamental importance in the study of the BEC\ndilute-gas where the physics should be dominated by two-body\ncollisions described in terms of the s-wave scattering length $a$.\nIn the case of positive $a$, it is equivalent to consider a very\ndilute  (atomic) system of hard spheres, whose diameter coincides\nwith the scattering length itself \\cite{Fabrocini99}. The natural\nstarting point for studying the behavior of those systems is the\ntheory of weakly interacting bosons which, for inhomogeneous\nsystems, takes the form of the Gross-Pitaevskii equation\n\\cite{Pitaevski61,Gross61}. This is a mean-field approach for the\norder parameter associated with the condensate \\cite{Dalfavo99}.\n\nIn the present work we study BEC in a phenomenological way where\nthe Bose gas is considered as a many-body system\n\\cite{Ketterle99}. In particular, we study the ground state of a\nsystem of correlated bosonic atoms at zero temperature, trapped by\na harmonic oscillator potential (HO). The key quantities for this\neffort is the one- and two-body density matrices \\cite{Lowdin}. As\nthe mean-field approach (non-interacting atoms) fails to\nincorporate the interparticle interactions which are necessary for\nthe description of the correlated Bose system, we introduce the\nrepulsive interactions among the atoms, through the Jastrow\ncorrelation functions $f\\left(|\\textbf{r}_1-\\textbf{r}_2| \\right)$\n\\cite{Moustakidis02}.\n\nWe focus our efforts to the calculation of the one- and two-body\ndensity and momentum distributions and the calculation of the\nstatic structure factor\n\\cite{Moustakidis02,MoustaChatz05,Moustakidis04,Massen05}.\nOne-body density and momentum distributions are complementary\ndescriptions of the Bose gas and related directly with the\nmean-square radius and mean kinetic energy of the trapped Bose gas\nrespectively. In addition the two-body density distribution is\nrelated with the calculation of the static structure factor, a\nquantity which gives information for the ground and excited states\nof the gas. Special effort has been devoted to the derivation of\nthe natural orbital and natural occupation numbers through the\ndiagonalization of the one-body density matrix.\n\n In recent years information-theoretic methods play an increasing\nrole for the study of quantum mechanical systems. An example is\nthe application of the Maximum Entropy Principle \\cite{Kapur89}\n(MEP) to the calculation of the wave function in a potential\n\\cite{Canosa92} using as constraints expectation values of simple\nobservables and reconstructing a quantum wave function from a\nlimited set of expectation values. The idea behind MEP is to\nchoose the least biased result, compatible with the constraints of\nthe problem. Thus the MEP provides the least biased description\nconsistent with the available relevant information. This is done\nby employing a suitably defined quantum entropy that measures the\nlack of information associated with the distribution of a quantum\nstate over a given known basis.\n\nInformation entropy is important for the study of quantum\nmechanical systems in two cases: first in the clarification of\nfundamental concepts of quantum mechanics and second in the\nsynthesis of probability densities in position and momentum space\n\\cite{Garbaczewski05}.\n\nIn the present work special effort is devoted for the calculation\nof various quantum information properties including Shannon\nentropy, Onicescu energy, Kullback-Leibler relative entropy and\nalso Jensen-Shannon divergence. The information properties are\ncalculated for the interatomic correlations. In addition the\nGross-Piatevskii equation is solved giving the density and\nmomentum distribution which are employed to calculate the above\nquantum information properties of the Bose gas\n\\cite{Massen05,Massen02}. The results are compared with those\ntaken in a phenomenological way in the framework of the Jastrow\ncorrelations.\n\nThe plan of the paper is the following: In Sec. \\ref{sec:sec2} the\ngeneral definitions related to the density matrices of a Bose\nsystem are considered. Details of the lowest-order cluster\nexpansion, analytical expressions and numerical results are\nreported in Sec. \\ref{sec:sec3}. In Sec. \\ref{sec:sec4} formulas\nfor the quantum information properties (both for the one-and\ntwo-body density matrices) are reviewed and analytical results are\npresented. Quantum information properties based on\nGross-Piatevskii equation are presented in Sec. \\ref{sec:sec5}\nwhile the summary of the work is given in Sec. \\ref{sec:sec6}.\n\n\\section{Definition of Density Matrices}\\label{sec:sec2}\n\nLet $\\Psi({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$ be the wave\nfunction describing the trapped Bose gases. In the case where this\nsystem is composed of non-interacting bosonic atoms at zero\ntemperature, all atoms occupy the same single-particle ground\nstate. The many body ground state wave function $\\Psi_0({\\bf\nr}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$ is then a product of $A$\nidentical single particle ground state wave functions. This ground\nstate wave function is therefore called the condensate wave\nfunction or macroscopic wave function and has the form\n\\cite{Ketterle99}\n\\begin{equation}\n\\Psi_0({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)= \\psi_0({\\bf\nr}_1)\\psi_0({\\bf r}_2) \\cdots \\psi_0({\\bf r}_A), \\label{WF-1}\n\\end{equation}\n\n\\noindent where $\\psi_0({\\bf r})$ is the normalized to one\nground-state single-particle wave function describing a bosonic\natom. It is worth to indicate that Eq. (\\ref{WF-1}) is valid even\nwhen\n weak interactions are included. In this case the wave function\n$\\Psi_0({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$ is still, to a very\ngood approximation, a product of $A$ single particle wave\nfunctions obtained now from the solution of a non-linear\nSchr\\\"odinger equation, the well known Gross-Pitaevskii equation.\nHowever, in the general case where interactions between atoms are\nincluded, the ground state wave function $\\Psi({\\bf r}_1,{\\bf\nr}_2,\\ldots,{\\bf r}_A)$ is modified from the simple form of Eq.\n(\\ref{WF-1}). In that case a percentage of atoms is moving from\nthe condensate orbit $\\psi_0$ to higher orbits.\n\n\nIn the present work we adopt the following normalization of the\nwave function $\\Psi({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$,\n\n\\begin{equation}\n \\int \\Psi^{*}({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)\n \\Psi({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)\n d\\textbf{r}_1 d\\textbf{r}_2 \\cdots d\\textbf{r}_A=1,\n\\end{equation}\n\n\\noindent where the integration is carried out over the radius\nvectors $\\textbf{r}_1$, $\\textbf{r}_2$, $\\ldots$, $\\textbf{r}_A$.\n\nA quantity characterizing very important aspects of a Bose gas (as\nwell a variety of quantum many-body systems) is the one-body\ndensity matrix defined as in \\cite{Lowdin}\n\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')=\\int \\Psi^{*}({\\bf r}_1,{\\bf\nr}_2,\\ldots,{\\bf r}_A) \\Psi({\\bf r}_1',{\\bf r}_2,\\ldots,{\\bf r}_A)\nd{\\bf r}_2 \\cdots d{\\bf r}_A. \\label{OBDM-1}\n\\end{equation}\n\nThe one-body density matrix is connected to the position- and\nmomentum-space properties of the Bose gas and in addition it is\nthe quantity which gives the percentage of the condensate of the\nsystem.\n\nThe two-body density matrix is a generalization of the one-body\ndensity matrix and is defined as\n\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf r}_2')= \\int\n\\Psi^{*}({\\bf r}_1,{\\bf r}_2,{\\bf r}_3,\\cdots,{\\bf r}_A) \\Psi({\\bf\nr}_1',{\\bf r}_2',{\\bf r}_3,\\cdots,{\\bf r}_A) d{\\bf r}_3 \\cdots\nd{\\bf r}_A. \\label{TBDM-1}\n\\end{equation}\nThe above density matrices are related by the following equation\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')=\\int \\rho({\\bf r}_1,{\\bf r}_2;{\\bf\nr}_1',{\\bf r}_2)  d{\\bf r}_2. \\label{O-T-1}\n\\end{equation}\n\nThe two-body density matrix is related directly to the interatomic\ninteraction and its diagonal part provides the two-body density\ndistribution $\\rho(\\textbf{r}_1,\\textbf{r}_2)$ (expresses the\njoint probability of finding two atoms at the positions ${\\bf\nr}_1$ and ${\\bf r}_2$, respectively), a key quantity of the\npresent work\n\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2)= \\rho({\\bf r}_1,{\\bf r}_2;{\\bf\nr}_1',{\\bf r}_2')\\mid_{{\\bf r}_1'={\\bf r}_1, {\\bf r}_2'={\\bf\nr}_2}. \\label{TBDD-1}\n\\end{equation}\n\nOn the other hand the diagonal part of the one-body density matrix\nis just the density distribution of the Bose gas and expresses the\nprobability of finding an atom at position $\\textbf{r}_1$\n\n\\begin{equation}\n\\rho({\\bf r}_1)=\\rho({\\bf r}_1,{\\bf r}_1')|_{{\\bf r}_1={\\bf\nr}_1'}. \\label{DD-1}\n\\end{equation}\n\nThe quantities $\\rho(\\textbf{r}_1)$ and\n$\\rho(\\textbf{r}_1,\\textbf{r}_2)$ are also related by the\nfollowing integral\n\n\\begin{equation}\n\\rho({\\bf r}_1)=\\int \\rho({\\bf r}_1,{\\bf r}_2)\n d{\\bf r}_2. \\label{DD-2}\n\\end{equation}\n\nVery interesting is also the description of the Bose gas in\nmomentum-space via the quantities of the one- and two-body\nmomentum distributions. The two-body momentum distribution\n$n(\\textbf{k}_1,\\textbf{k}_2)$ expresses the joint probability of\nfinding two atoms with momentum $\\textbf{k}_1$ and $\\textbf{k}_2$\nrespectively and is given by a particular Fourier transform of the\ncorresponding two-body density matrix $\\rho({\\bf r}_1,{\\bf\nr}_2;{\\bf r}_1',{\\bf r}_2')$\n\n\\begin{equation}\nn({\\bf k}_1,{\\bf k}_2)=\\frac{1}{(2\\pi)^6} \\int \\rho({\\bf r}_1,{\\bf\nr}_2;{\\bf r}_1',{\\bf r}_2') \\exp[i{\\bf k}_1({\\bf r}_1-{\\bf r}_1')]\n\\exp[i{\\bf k}_2({\\bf r}_2-{\\bf r}_2')] d {\\bf r}_1 d {\\bf r}_1' d\n{\\bf r}_2 d {\\bf r}_2'. \\label{TBMD-1}\n\\end{equation}\n\nThe one-body momentum distribution (or simply momentum\ndistribution) $n(\\textbf{k})$, expresses the probability of\nfinding an atom with momentum $\\textbf{k}$, and it is given by a\nparticular Fourier transform of the one-body density matrix\n$\\rho({\\bf r}_1,{\\bf r}_1')$\n\n\\begin{equation}\nn({\\bf k})=\\frac{1}{(2\\pi)^3} \\int \\rho({\\bf r}_1,{\\bf r}_1') \\exp\n\\left[i{\\bf k}({\\bf r}_1-{\\bf r}_1')\\right]\n d  {\\bf r}_1   d  {\\bf r}_1'. \\label{mom}\n\\end{equation}\n\nIt can be shown easily that in the case where the Bose gas is\ndescribed by the wave function of Eq. (\\ref{WF-1}) the two-body\ndensity matrix is given by\n\\begin{equation}\n\\rho_{0}({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf r}_2')= \\rho_{0}({\\bf\nr}_1,{\\bf r}_1')\\rho_{0}({\\bf r}_2,{\\bf r}_2'), \\label{TBDM-BG}\n\\end{equation}\nwhere\n\\begin{equation}\n\\rho_{0}({\\bf r}_1,{\\bf r}_1')=\\psi_{0}^{*}({\\bf r}_1)\n\\psi_{0}({\\bf r}_1'). \\label{eq:eq12new}\n\\end{equation}\n\nFrom Eqs. (\\ref{TBMD-1}), (\\ref{mom}) and (\\ref{TBDM-BG}) we get\n\\begin{eqnarray}\nn_0(\\textbf{k}_1,\\textbf{k}_2) &=& \\displaystyle{\n\\frac{1}{(2\\pi)^3}} \\int \\rho_0(\\textbf{r}_1,\\textbf{r}_1')\\,\n\\textrm{exp}\\left[ i \\textbf{k}_1 (\\textbf{r}_1-\\textbf{r}_1')\n\\right]\\,d\\textbf{r}_1\nd\\textbf{r}_1' \\nonumber \\\\\n& & \\times \\displaystyle{ \\frac{1}{(2\\pi)^3}} \\int\n\\rho_0(\\textbf{r}_2,\\textbf{r}_2')\\, \\textrm{exp}\\left[ i\n\\textbf{k}_2 (\\textbf{r}_2-\\textbf{r}_2') \\right]\\,d\\textbf{r}_2\nd\\textbf{r}_2' \\\\\n&=& n_0(\\textbf{k}_1)n_0(\\textbf{k}_2). \\nonumber\n\\end{eqnarray}\n\nFinally form Eqs. (\\ref{mom}) and (\\ref{eq:eq12new}) we get\n\n\\begin{eqnarray}\nn_0(\\textbf{k})&=& \\displaystyle{ \\frac{1}{(2\\pi)^{3\/2}}} \\int\n\\psi_{0}^{*}({\\bf r}_1) \\textrm{exp}\\left[ i \\textbf{k}\n\\textbf{r}_1 \\right] d\\textbf{r}_1 \\nonumber \\\\\n& & \\times \\displaystyle{ \\frac{1}{(2\\pi)^{3\/2}}} \\int\n\\psi_{0}^{*}({\\bf r}_1') \\textrm{exp}\\left[ i \\textbf{k}\n\\textbf{r}_1' \\right] d\\textbf{r}_1' \\label{eq:eq14new} \\\\\n&=& \\tilde{\\psi}_{0}^{*}({\\bf k})\\tilde{\\psi}_{0}({\\bf k}).\n\\nonumber\n\\end{eqnarray}\n\nFrom Eq. (\\ref{eq:eq14new}) it is obvious that\n$\\tilde{\\psi}_{0}^{*}({\\bf k})$ is the particular Fourier\ntransform of the single particle wave function $\\psi_{0}({\\bf\nr})$.\n\n\\subsection{Static Structure Factor}\n\nSpectroscopic studies have been used to assemble a complete\nunderstanding of the structure of atoms and simple molecules\n\\cite{Stamper-Kurn}. The static structure factor $S(k)$ is a\nfundamental quantity, connected with the atomic structure, and is\nthe Fourier transform of the radial distribution function $g(r)$.\n$S(k)$ gives the magnitude of the density fluctuation in the\nsystem (atomic, molecular, electronic or nuclear) at wavelength\n$2\\pi\/k$, where $k$ is the momentum transfer. In recent papers,\nthe Bragg spectroscopic method was used  to measure $S(k)$ either\nin the phonon regime \\cite{Stamper-Kurn} or\/and in the\nsingle-particle regime \\cite{Steinhauer}.\n\nThe static structure factor in a finite system is defined as\n\\cite{Zambelli}\n\n\\begin{equation}\nS({\\bf k})=1+ \\frac{1}{N} \\int e^{i{\\bf k}({\\bf r}_1-{\\bf r}_2)}\n\\left[ \\rho({\\bf r}_1,{\\bf r}_2) - \\rho({\\bf r}_1) \\rho({\\bf r}_2\n) \\right] d  {\\bf r}_1   d  {\\bf r}_2. \\label{eq:eq18}\n\\end{equation}\n\nIn the most general case the two-body density distribution\n$\\rho(\\textbf{r}_1,\\textbf{r}_2)$ and the one-body density\ndistribution $\\rho(\\textbf{r})$ are connected via the following\nrelation\n\n\\begin{equation}\n \\rho(\\textbf{r}_1,\\textbf{r}_2)=C\\rho(\\textbf{r}_1)\\rho(\\textbf{r}_2)f^2(r_{12})\n =C\\rho(\\textbf{r}_1)\\rho(\\textbf{r}_2)g(r_{12}), \\label{eq:eq19}\n\\end{equation}\nwhere $g(r_{12})$ is the radial distribution function and $C$ is\nthe normalization factor which ensures that\n\n\\begin{equation}\n  \\int \\rho(\\textbf{r}_1,\\textbf{r}_2) d\\textbf{r}_1\n  d\\textbf{r}_2=N(N-1). \\label{eq:eq20}\n\\end{equation}\n\nWe also consider that\n\\begin{equation}\n  \\int \\rho(\\textbf{r}_1) d\\textbf{r}_1=N, \\label{eq:eq21}\n\\end{equation}\nwhere $N$ is the number of the atoms of the Bose condensate.\n\nIn the uncorrelated case (non-interacting gas) the radial\ndistribution function is $g(r_{12})=1$ (absence of correlations),\nand the two-body density distribution becomes\n\n\\begin{equation}\n \\rho(\\textbf{r}_1,\\textbf{r}_2)=\n \\frac{N-1}{N}\\rho(\\textbf{r}_1)\\rho(\\textbf{r}_2).\n\\end{equation}\n\nUsing Eq. (\\ref{eq:eq19}), Eq. (\\ref{eq:eq18}) is written as\n\\begin{equation}\nS({\\bf k})=1+ \\frac{1}{N} \\int \\textrm{exp} \\left[i{\\bf k}({\\bf\nr}_1-{\\bf r}_2)\\right] \\rho({\\bf r}_1) \\rho({\\bf r}_2 ) [C\ng(r_{12})-1] d {\\bf r}_1   d {\\bf r}_2. \\label{str-fin2}\n\\end{equation}\n\nConditions (\\ref{eq:eq20}) and (\\ref{eq:eq21}) ensure that\n$S(0)=0$.\n\nThe integration in Eq. (\\ref{str-fin2}) can be performed if the\nfunction $g(r)$ is known. $g(r)$ must obey  the rules $g(r=0)=0$\nand $\\displaystyle{\\lim_{r \\rightarrow \\infty} g(r) \\rightarrow\n1}$. The first rule introduces the repulsive correlations between\nthe atoms and the second the absence of such correlations in long\ndistances. In general the form of $S(k)$ is affected appreciably\nfrom the form of $g(r)$. More specifically, the long range\nbehavior of $g(r)$ affects $S(k)$ for small values of $k$ while\nits short range behavior affects $S(k)$ for large values of $k$ as\na direct consequence of the Fourier transform theory.\n\n\n\\subsection{Natural Orbitals and Natural Occupation Numbers}\n\nIn the case of the inclusion of the inter-particle interactions\nbetween the atoms, which give rise to the depletion of the\ncondensate, the one-body density matrix is written\n\\cite{Stringari01}\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')= n_0 \\psi_{0}^{*}({\\bf r}_1)\n\\psi_{0}({\\bf r}_1')+ \\sum_{i\\neq 0}n_i \\psi_{i}^{*}({\\bf r}_1)\n\\psi_{i}({\\bf r}_1'),\n\\end{equation}\nwhere $\\displaystyle{\\sum_{i}n_i=1}$. The sum\n$\\displaystyle{\\sum_{i\\neq 0}n_i \\psi_{i}^{*}({\\bf r}_1)\n\\psi_{i}({\\bf r}_1')}$ is the contribution arising from the atoms\nout of the condensate. The eigenfunctions $\\psi_i({\\bf r}),$ which\nare called natural orbitals (NO's), and the eigenvalues $n_i$,\ncalled natural occupation numbers (NON's), are obtained by\ndiagonalizing the one-body density matrix through the eigenvalue\nequation\n\\begin{equation}\n\\int \\rho({\\bf r}_1,{\\bf r}_1') \\psi_i({\\bf r}_1')  d {\\bf r}_1'\n=n_i\\psi_i({\\bf r}_1). \\label{diag-rho}\n\\end{equation}\nThe condition, generally adopted, for the existence of\ncondensation is that there should be one eigenvalue $n_i$ which is\nof the order of the number of the particles in the trap.\n\nThe NO's $\\psi_{i}({\\bf r}_1)$ and the NON's $n_i$ are obtained by\ndiagonalizing the one-body density matrix through the eigenvalue\nequation (\\ref{diag-rho}) by expanding first the one-body density\nmatrix in a series of Legendre polynomials $P_l(x)$\n\\begin{equation}\n\\rho({\\bf r},{\\bf r}')=\\rho(r, r',\\cos\\omega_{rr'})=\n\\sum_{l=0}^{\\infty}\\rho_l(r,r') P_l(\\cos\\omega_{rr'}),\n\\label{ch4-NO4}\n\\end{equation}\nwhere $\\rho_l(r,r')$ are the  coefficients of the expansion\n\\begin{equation}\n\\rho_l(r,r')=\\frac{2l+1}{2}\\int_{-1}^{1} \\rho(r,\nr',\\cos\\omega_{rr'}) \\ P_l(\\cos\\omega_{rr'}) \\ {\\rm d}\n(\\cos\\omega_{rr'}). \\label{ch4-NO5}\n\\end{equation}\nFrom the Eqs. (\\ref{diag-rho}), (\\ref{ch4-NO4}) and\n(\\ref{ch4-NO5}) the eigenvalue equation is written\n\\begin{equation}\n4\\pi \\int_{0}^{\\infty}\\rho_{l}(r,r')\\varphi_{nl}^{NO}(r')\n{r'}^2{\\rm d} r'=n_{nl}^{NO}\\varphi_{nl}^{NO}(r), \\label{NO-3}\n\\end{equation}\nwhere $\\varphi_{nl}^{NO}(r)$ is the radial part of\n $\\psi_{i}({\\bf r})$\n($\\psi_{i}({\\bf r})=\\varphi_{nl}^{NO}(r)Y_{lm}(\\Omega_r) $).\n\n\n\n\\section{Jastrow type Correlated Properties of a Trapped Bose\nGas}\\label{sec:sec3}\n\n\n\\subsection{Correlated Density Matrices}\\label{sub:sub3-1}\n\nA dilute trapped Bose gas can be studied using the lowest-order\napproximation \\cite{Fabrocini99}. In this approximation the\ntwo-body density matrix has the form\n\\cite{Moustakidis02,MoustaChatz05}\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf r}_2')= N_0 \\rho_0({\\bf\nr}_1,{\\bf r}_1' ) \\rho_0({\\bf r}_2,{\\bf r}_2') f(|{\\bf r}_1' -\n{\\bf r}_2'|) f(|{\\bf r}_1-{\\bf r}_2|), \\label{TBDM-1}\n\\end{equation}\nwhere $f(|{\\bf r}_1-{\\bf r}_2|)$ is the Jastrow correlation\nfunction, which depends on the inter-particle distance and $N_0$\nis the normalization factor which ensures that\n\\[\\int \\rho({\\bf\nr}_1,{\\bf r}_2;{\\bf r}_1',{\\bf\nr}_2')|_{(\\textbf{r}_1=\\textbf{r}_1', \\textbf{r}_2=\\textbf{r}_2')}\n\\, d\\textbf{r}_1 d\\textbf{r}_2=1.\n\\]\n\nThe diagonal part of $\\rho({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf\nr}_2')$ that is the two body density distribution takes the form\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2)= N_0 \\rho_0({\\bf r}_1)\\rho_0({\\bf r}_2)\nf^2(r_{12}), \\label{TBDD-12}\n\\end{equation}\nwhile the one-body density matrix is given by the integral\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')= N_0 \\rho_0({\\bf r}_1,{\\bf r}_1') \\int\n\\rho_0({\\bf r}_2,{\\bf r}_2) f(|{\\bf r}_1' - {\\bf r}_2|)f(|{\\bf\nr}_1 - {\\bf r}_2|) d\\textbf{r}_2. \\label{OBDM-1}\n\\end{equation}\n\nThe density distribution, which is the diagonal part of $\\rho({\\bf\nr}_1,{\\bf r}_1')$, can also be obtained from the integral\n\\begin{eqnarray}\n\\rho({\\bf r})= \\int \\rho({\\bf r},{\\bf r}_2) d{\\bf r}_2 =N_0\n\\rho_0({\\bf r}) \\int \\rho_0({\\bf r}_2) f^2(|{\\bf r}-{\\bf r}_2|)  d\n{\\bf r}_2. \\label{cor-dd1}\n\\end{eqnarray}\n\nIn the present work we consider that the atoms are confined in an\nisotropic HO well where the normalized to $1$ ground state single\nparticle wave function $\\psi_0(r)$ has the form of a Gaussian\ngiven by the formula\n\\[\n  \\psi_0(r)=\\left( \\frac{1}{\\pi b^2} \\right)^{3\/4}\n  \\textrm{exp}\\left[-\\frac{r^2}{2 b^2}\\right], \\quad\n  \\mbox{\\textrm{where the width}}\\,\\, b=\\left( \\frac{\\hbar}{m\\omega}\n  \\right)^{1\/2},\n\\]\nwhile the density distribution has the form  $\\rho_0({\\bf r})=|\n\\psi_0({\\bf r}) |^2$. The correlation function $f(r_{12})$ is\ntaken to be of the form\n\\begin{equation}\nf(r)=1-\\exp\\left[-\\frac{y r^{2}}{b^2}\\right], \\label{case-1}\n\\end{equation}\nwhere $r=|{\\bf r}_1-{\\bf r}_2|$. The correlation function  $f(r)$\ngoes to $1$ for large values of $r$ and goes to $0$ for $r\n\\rightarrow 0$. It is obvious that the effect of the correlations\nintroduced by the function $f(r)$, becomes large when the\ncorrelation parameter $y$ becomes small and vice versa.\n\nThe above defined correlation function was used in\n\\cite{Moustakidis02,MoustaChatz05} to find analytical expressions\nof the one-body density matrices in position and momentum spaces\nand static structure factor, while the NO's and NON's are\ncalculated numerically employing Eq. (\\ref{NO-3}).\n\n\nThe analytical expression of the two-body density matrix obtained\nfrom Eq. (\\ref{TBDM-1}) has the form\n\n\\begin{eqnarray}\n \\rho(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)= &\n \\displaystyle{\\frac{N_0}{\\pi^3 b^6}} \\, \\Big(\n O_1(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)-\n O_{21}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)- \\nonumber \\\\\n &  -O_{22}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)+\n O_{23}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)\n \\Big),\n\\end{eqnarray}\nwhere $N_0$ is the normalization factor of the form\n\\cite{MoustaChatz05}\n\\begin{equation}\nN_0=\\left[1-\\frac{2}{(1+2y)^{3\/2}}+\\frac{1}{(1+4y)^{3\/2}}\\right]^{-1}\n\\label{norm-1b}\n\\end{equation}\nand\n\\[\n \\begin{array}{lcl}\n O_1(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{r_{1b}^2+r_{1b}^{'2}+r_{2b}^2+r_{2b}^{'2}}{2}}\\right], \\\\\n O_{21}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{r_{1b}^2+r_{2b}^2}{2}}\\right]\\,\n \\textrm{exp}\\left[\\displaystyle{-\\frac{(1+2y)(r^{'2}_{1b}+r^{'2}_{2b})}{2}}\\right]\\,\n \\textrm{exp}\\left[2y \\textbf{r}'_{1b} \\textbf{r}'_{2b}\\right], \\\\\n O_{22}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{r^{'2}_{1b}+r^{'2}_{2b}}{2}}\\right]\\,\n \\textrm{exp}\\left[\\displaystyle{-\\frac{(1+2y)(r_{1b}^2+r_{2b}^2)}{2}}\\right]\\,\n \\textrm{exp}\\left[2y \\textbf{r}_{1b} \\textbf{r}_{2b}\\right], \\\\\n O_{23}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{(1+2y)(r_{1b}^2+r_{1b}^{'2}+r_{2b}^2+r_{2b}^{'2})}{2}}\\right]\\,\n \\textrm{exp}\\left[2y(\\textbf{r}_{1b}\\textbf{r}_{2b}+\\textbf{r}'_{1b}\\textbf{r}'_{2b})\\right].\n\\end{array}\n\\]\nwhere $r_b=r\/b$.\n\nThe two-body density distribution  in accordance with Eq.\n(\\ref{TBDD-1}) is given by\n\n\\begin{equation}\n \\rho(\\textbf{r}_1,\\textbf{r}_2)=\\frac{N_0}{\\pi^3 b^6}\n \\,\\textrm{exp}[-r_{1b}^2]\\,\\textrm{exp}[-r_{2b}^2]\\,\n \\Big(1-\\textrm{exp}[-y({\\bf r}_{1b}-{\\bf r}_{2b})^2]\\Big)^2.\n\\end{equation}\n\nThe analytical expressions of the one-body density matrix obtained\nfrom Eq. (\\ref{OBDM-1}) has the form \\cite{Moustakidis02}\n\\begin{equation}\n\\rho({\\bf r},{\\bf r}')= \\frac{N_0}{\\pi^{3\/2}b^{3}}\\,\\left[O_1({\\bf\nr},{\\bf r}')-O_{21}({\\bf r},{\\bf r}')- O_{22}({\\bf r},{\\bf\nr}')+O_{23}({\\bf r},{\\bf r}')\\right], \\label{cluster-11}\n\\end{equation}\n\n\\noindent where the one- and the two-body terms of the expansion\nin the low order approximation have the forms\n\\begin{eqnarray}\n O_1({\\bf r},{\\bf r}')&=&\n\\exp\\left[-\\frac{r_b^2+{r_b'}^2}{2}\\right], \\\\\n& & \\nonumber\\\\\nO_{21}({\\bf r},{\\bf r}')&=& \\frac{1}{(1+y)^{3\/2}}\n\\exp\\left[-\\frac{1+3y}{1+y}\\frac{r_b^2}{2}-\\frac{{r_b'}^2}{2}\\right],  \\\\\n& & \\nonumber\\\\\nO_{22}({\\bf r},{\\bf r}')&=&O_{21}({\\bf r}',{\\bf r}), \\\\\n& & \\nonumber\\\\\nO_{23}({\\bf r},{\\bf r}')&=& \\frac{1}{(1+2y)^{3\/2}}\n\\exp\\left[-(1+2y)\\frac{r_b^2+{r_b'}^2}{2}\\right]  \\nonumber\\\\\n&& \\times \\exp\\left[\\frac{y^2}{1+2y}({\\bf r}_b+{\\bf\nr}_b')^2\\right], \\label{case1-o1r}\n\\end{eqnarray}\n\n\nThe analytical expression of the density distribution can be found\nfrom Eq. (\\ref{cluster-11}), putting ${\\bf r'}={\\bf r}$\n\\begin{eqnarray}\n\\rho(r)&=& \\frac{N_0}{\\pi^{3\/2} b^3} \\left(\n\\exp\\left[-r_b^2\\right] - \\frac{2}{(1+y)^{3\/2}}\n\\exp\\left[-\\frac{1+2y}{1+y}r_b^2\\right]  \\right.\n\\nonumber\\\\\n&& \\left. + \\frac{1}{(1+2y)^{3\/2}}\n\\exp\\left[-\\frac{1+4y}{1+2y}r_b^2\\right]    \\right).\n\\label{cluster-nr}\n\\end{eqnarray}\n\nThe two-body momentum distribution is calculated form the integral\nof Eq. (\\ref{TBMD-1}) and has the form\n\n\\begin{eqnarray}\n n(\\textbf{k}_1,\\textbf{k}_2)&=&\\frac{b^6}{\\pi^3} N_0 \\,\n \\textrm{exp}[-k_{1b}^2]\\,\\textrm{exp}[-k_{2b}^2]  \\nonumber \\\\\n & & \\times \\left( 1-\\frac{1}{(1+4y)^{3\/2}}\\,\n \\textrm{exp}\\left[-\\frac{y}{1+4y}(\\textbf{k}_{1b}-\\textbf{k}_{2b})^2\\right]\n \\right)^2,\n\\end{eqnarray}\n\n\\noindent while the momentum distribution can be found\nanalytically using Eq. (\\ref{mom}) and has the form\n\\begin{eqnarray}\nn(k)&=& \\frac{N_0 b^3}{\\pi^{3\/2}} \\, \\left(\n\\exp\\left[-k_b^2\\right] - \\frac{2}{(1+3y)^{3\/2}}\n\\exp\\left[-\\frac{1+2y}{1+3y}k_b^2\\right]\n\\nonumber \\right.\\\\\n&& \\left. + \\frac{1}{(1+2y)^{3\/2}(1+4y)^{3\/2}}\n\\exp\\left[-\\frac{1}{1+2y}k_b^2\\right]    \\right),\n\\label{cluster-nk}\n\\end{eqnarray}\nwhere $k_b =kb$.\n\nThe above analytical expressions of $\\rho(r)$ and $n(k)$ have been\nused to find the analytical expressions of the mean square radius\nand kinetic energy of the trapped gas. The expressions we found,\nfor $\\langle r^2 \\rangle$ and $\\langle T \\rangle$, are\n\\begin{equation}\n\\langle r^2 \\rangle=N_0 b^2 \\left[ \\frac{3}{2} -\n3\\frac{1+y}{(1+2y)^{5\/2}} +\\frac{3}{2}\\frac{1+2y}{(1+4y)^{5\/2}}\n\\right] \\label{eq:rad-1}\n\\end{equation}\nand\n\\begin{equation}\n\\langle T \\rangle= N_0 \\hbar \\omega  \\left[ \\frac{3}{4} -\n\\frac{3}{2}\\frac{1+3y}{(1+2y)^{5\/2}} +\n\\frac{3}{4}\\frac{1+2y}{(1+4y)^{3\/2}}\\right]. \\label{eq:kinetic-1}\n\\end{equation}\n\n\nThese expressions, which for a given HO trap are functions of the\ncorrelation parameter $y$, could be used to find the value of $y$\nfrom Eq. (\\ref{eq:rad-1}), if the rms radius of the trapped atoms\nis known and then to define  $\\langle T \\rangle$ from Eq.\n(\\ref{eq:kinetic-1}) and vice versa. For very large values of $y$\nEqs. (\\ref{eq:rad-1}) and (\\ref{eq:kinetic-1}) give the HO\nexpressions of $\\langle r^2 \\rangle$ and $\\langle T \\rangle$, i.e.\n$\\langle r^2 \\rangle = \\frac{3}{2} b^2$ and $\\langle T \\rangle =\n\\frac{3}{4}\\hbar\\omega$, respectively.\n\n\n\nThe calculation of the density distribution of a trapped Bose gas,\nconfined in an isotropic HO potential with length $b=10^4$ \\AA,\nhas been carried out on the basis of Eq. (\\ref{cluster-nr})\n\\cite{Moustakidis02}. The dependence of the density distribution\non the parameter $y$, including also the uncorrelated case\n($y=\\infty$), has been plotted in Fig. \\ref{fig:fig1}(a). It is\nseen that, the large values of $y$ ($y>10$) correspond to the\nGaussian distribution (HO case), while when $y$ becomes small\nenough ($y<1$) the density distribution spreads out as in\nGross-Pitaevskii's theory \\cite{Moustakidis02}. For $y>10$ the\neffect of correlations is small, while for very large correlations\n($y \\lesssim 0.1$) the density distribution is modified entirely\ncompared to the Gaussian form originating from the HO trap.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=4.0cm]{fig1a.eps}\n\\hspace{0.5cm}\n\\includegraphics[height=5.0cm,width=4.cm]{fig1b.eps} \\caption{(a) The\ndensity distribution $\\rho(r)$ versus $r_b=r\/b$ ($b=10^4$  \\AA)\n for various\nvalues of the parameter $y$. The normalization is $\\int \\rho({\\bf\nr})  d {\\bf r}=1$. (b) The rms radius, $\\sqrt{\\langle r_b ^{2}\n\\rangle}$, versus the parameter $y$.}\\label{fig:fig1}\n\\end{figure}\n\n\n\nThe rms radius of the Bose gas has also been calculated\nanalytically from Eq. (\\ref{eq:rad-1}) for various values of the\nparameter $y$. From Fig. \\ref{fig:fig1}(b), it is seen that\n$\\sqrt{\\langle r_{b}^2\\rangle}$ ($r_b=r\/b$) is a decreasing\nfunction of the parameter $y$ and for $y > 10$ approaches the rms\nradius of the uncorrelated case.\n\n\\subsection{Natural Orbitals and Natural Occupations\nNumbers}\\label{sub:3-1}\n\nThe NO's and the NON's were calculated \\cite{Moustakidis02}, by\ndiagonalizing the one-body density matrix through Eq.\n(\\ref{NO-3}). The NON $n_{1s}$, gives directly the condensation\nfraction $n_0$ as a result of the repulsive interaction between\nthe atoms of the Bose gas at zero temperature. The NON's for the\n$1s$, $1p$, $1d$ and $1f$ states are given in Table\n\\ref{tbl:table1}. It seems that, for strong correlations, a\nfraction of atoms spread out into many states. The condensation\nfraction $n_{1s}$, versus the parameter $\\frac{1}{y}$ is plotted\nin Fig. \\ref{fig:fig2}. From that figure and from Table\n\\ref{tbl:table1} it is seen that the effect of the correlations on\n$n_{1s}$ is small and all the atoms occupy the 1s ground state,\nwhen $y>10$. The effect of the correlations is prominent when\n$y<10$, while the decrease of the parameter $y$ (strong\ncorrelations) induces a significant depletion of the condensated\natoms spreading them into many states.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=5.0cm,width=4.0cm]{fig2.eps}\n\\caption{The condensate fraction $n_{1s}$, at zero temperature,\nfor interacting atoms versus $1\/y$.}\\label{fig:fig2}\n\\end{figure}\n\n\n\\begin{table}[pt]\n\\centering\n\\begin{tabular}{@{}c c c c c c@{}}\n\\hline $y$ & $n_{1s}$ & $n_{1p}$ & $n_{1d}$  & $n_{1f}$ & Sum \\\\\n\\hline\n100.00  & 0.99988  &   -       &    -     &  -         & 0.99988 \\\\\n\\hphantom{0}10.00   & 0.99634  & 0.00063   & 0.00042  &0.00042     & 0.99781  \\\\\n\\hphantom{00}5.00    & 0.99055  & 0.00273   & 0.00108  &0.00108     & 0.99544   \\\\\n\\hphantom{00}2.50  & 0.97771  & 0.00960   & 0.00186  &0.00186     & 0.99103   \\\\\n\\hphantom{00}1.00    & 0.94422  & 0.03462   & 0.00172  &0.00172     & 0.98228   \\\\\n\\hphantom{00}0.50  & 0.90815  & 0.06830   & 0.00082  &0.00082     & 0.97809   \\\\\n\\hphantom{00}0.10  & 0.83097  & 0.15185   & 0.00001  & 0.00001    & 0.98284  \\\\\n\\hphantom{00}0.01 & 0.79273  & 0.19414   &   -      & -          & 0.98687\\\\\n\\hline\n\\end{tabular}\n\\caption{The natural occupation numbers for various values of the\ncorrelation parameter $y$ \\cite{Moustakidis02}.}\\label{tbl:table1}\n\\end{table}\n\n\n\nThe NO's of the states $1s$, $1p$ and $1d$ for $y=1$ are shown in\nFig. \\ref{fig:fig3}. It is seen that the interatomic correlations\nin the $1s$-state spread out the ground state wave function and\nconsequently the condensation appears in the outer region of the\ntrap. From the same figure it is obvious that the NO's of the $1p$\nand $1d$ states are much more localized in coordinate space than\nthe equivalent HO orbitals.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig3a.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig3b.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig3c.eps}\n \\caption{The NO's (dot lines) versus $r_b$ ($r_b=r\/b$) of the\nstates (a) 1s, (b) 1p,  and  (c) 1d obtained by diagonalization of\nthe one-body density matrix (for $y$=1). The solid lines\ncorrespond to the HO wave-function with the trap length $b=10^4$\n\\AA.}\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig4a.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig4b.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig4c.eps}\n \\caption{(a) The normalized to $1$ momentum distribution versus\n$k_b$ ($k_b=kb$) for various values of the correlation parameter\n$y$. (b) The momentum distribution and the contribution to it from\nthe NO's of the $1s$ NO state and of the rest of the NO states\n(for $y=0.1$). (c) The mean kinetic energy per atom, $\\langle T_b\n\\rangle$ ($T_b=T\/(\\hbar\\omega)$), versus $y$. The solid curve\ncorresponds to the total values of $\\langle T_b \\rangle$, while\ndashed and dotted lines are the contributions to the total\n$\\langle T_b \\rangle$ of the NO's of the $1s$ NO state and of the\nrest of the NO states, respectively.}\\label{fig:fig4}\n\\end{figure}\n\nThe momentum distribution  can be calculated analytically from Eq.\n(\\ref{cluster-nk}) or by Fourier transform of the NO's. The\nmomentum distribution calculated analytically for various values\nof the parameter $y$ has been plotted in Fig. \\ref{fig:fig4}(a).\nIt is seen that the large values of $y$ ($y>10$) correspond to the\nGaussian distribution, while when $y$ becomes small enough ($y<1$)\nthe momentum distribution has a sharp maximum for $k=0$. The\nmomentum distribution of the $1s$ NO state as well as of the rest\nof the NO states for $y=0.1$ are shown and compared with the total\nmomentum distribution in Fig. \\ref{fig:fig4}(b). It is obvious\nthat although the $1s$ NO state gives the main contribution to the\nmomentum distribution, the additional NO states contribute to the\nmomentum distribution mainly in the large values of the momentum\n$k$.\n\nThe dependence of the mean kinetic energy $\\langle T \\rangle$ on\nthe parameter $y$  calculated analytically, using Eq.\n(\\ref{eq:kinetic-1}), is presented in Fig. \\ref{fig:fig4}(c). It\nis seen that $\\langle T \\rangle$ has a maximum for $y\\simeq2.5$.\nIt is interesting to note that for the same value of the parameter\n$y$ the NON's of the states $1d$  and $1f$ have the same value as\ncan be seen from Table \\ref{tbl:table1}. The contribution of the\n$1s$ NO state and of the rest of the NO states to $\\langle T\n\\rangle$ are shown in the same figure. It is seen that, for large\nvalues of the parameter $y$ (weak correlations) the main\ncontribution to $\\langle T \\rangle$ comes from the $1s$ NO state,\nwhile for strong correlations there is a significant contribution\ncoming from the NO's of the additional states.\n\n\n\nA few comments are appropriate. In this section we study the\nbehavior of various condensate quantities treated in the Jastrow\nmanner, which introduces one parameter. The determination of that\nparameter could be made by fit of the theoretically calculated\nquantities (density distribution, momentum distribution,\n$\\sqrt{\\langle r^2\\rangle}$, and $\\langle T \\rangle$) to the\nexperimental ones as we mentioned in the end of subsection\n\\ref{sub:sub3-1}, provided that there are experimental data for\nthe corresponding quantities. It could be determined also by using\nthe density distribution or the two-body density matrix as a trial\none and applying the variation principle to the ground state\nenergy of the system. The present approach is quite frequent in\nthe study of the quantum many body problem when the solution of\nthe Schr\\\"odinger equation is very difficult. It should be noted\nalso that in the present work there is not a direct dependence\nbetween the condensation and the number of the atoms. The\ninter-particle correlations are incorporated in the mean field\nonly by the correlation function which, in some way, depends on\nthe effective size of the atoms. That dependence can be found from\nthe information entropy $S$ using the linear dependence of $S$ on\n$\\ln (Na_b)$ and the linear dependence of $S$ on\n$\\ln(\\frac{1}{y})$ (see Sec. \\ref{sub:sub5-1}).\n\n\n\\subsection{Static Structure Factor}\n\nIn order to calculate the static structure factor in the framework\nof the atomic calculations we choose two trial forms for $g(r)$\n\\cite{Moustakidis04}. The first one is a gaussian type which has\nbeen extensively and successfully used for the study of similar\nproblems in atomic physics (Bose gas, liquid helium) as well in\nnuclear physics. The relevant $g(r)$ and the entailed $S(k)$ (Case\n1) are\n\\begin{eqnarray}\ng(r)&=& 1- \\exp[-\\beta r^2], \\nonumber \\\\\nS(k)&=&1+N(C_1-1)\\exp\\left[-\\frac{k_b^2}{2}\\right] \\nonumber \\\\\n& &-\\frac{N\nC_1}{(1+2y^2)^{3\/2}}\\exp\\left[-\\frac{k_b^2}{2}(1+2y)\\right],\n\\label{fin-cs-1}\n\\end{eqnarray}\nwhere $k_b=kb$, $y=\\beta b^2 $, $\\beta$ is the correlation\nparameter and $C_1$ is the normalization factor.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig5a.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig5b.eps}\n \\caption{The static structure factor $S(k)$ of the trapped Bose gas\n in various cases versus the momentum $k$,\n(a) in Case 1 for various values of the correlation parameter\n$\\beta$ as well as for the uncorrelated case (harmonic\noscillator), (b) in Case 2 for the least squares best fit value of\nthe parameter $a$. The experimental points are from reference\n\\cite{Steinhauer}. For the various cases see\ntext.}\\label{fig:fig5}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig6.eps}\n \\caption{The radial distribution function $g(r)$ for Case 1 and\n2 (corresponding to inhomogeneous Bose gas) with the best fit\nvalues of the correlation parameters.}\\label{fig:fig6}\n\\end{figure}\n\nThe second trial function $g(r)$ and the relevant $S(k)$ (Case 2)\nare of the form\n\\begin{eqnarray}\ng(r)&=& 1- \\frac{\\sin^4ar}{(ar)^4}, \\nonumber \\\\\nS(k)&=&1+N(C_2-1)\\exp\\left[-\\frac{k_b^2}{2}\\right] \\nonumber\\\\\n& &- \\frac{N C_2}{2^{15\/2} ab \\pi k_b} \\sum_{i=1}^{5}\n\\alpha_i\\left[\\beta_i\\exp\\left[-\\frac{\\beta_i^2}{4}\\right]+\n\\sqrt{\\pi}\\left(1+\\frac{\\beta_i^2}{2}\\right) {\\rm\nerf}\\left(\\frac{\\beta_i}{2}\\right) \\right], \\label{fin-cs-2}\n\\end{eqnarray}\nwhere $a$ is the correlation parameter, $\\alpha_i$ are known\ncoefficients, $\\beta_i=\\beta_i(a,b,k_b)$, ${\\rm\nerf}(z)=\\displaystyle{\\frac{2}{\\sqrt{\\pi}}} \\int_{0}^{z} e^{-t^2}\ndt$ and $C_2$ is the normalization factor \\cite{Moustakidis04}.\n\n\nThe behavior of S(k), in Case 1, for various values of the\ncorrelation parameter $\\beta$ is shown in Fig. \\ref{fig:fig5}(a).\nIt is obvious that the effect of correlations, induced by the\nfunction $g(r)$, becomes large when the parameter $\\beta$ becomes\nsmall and vice versa. The case where $\\beta\\rightarrow \\infty$,\ncorresponds to the uncorrelated case (HO). For the values of $k$\nemployed in the experiment of Ref. \\cite{Steinhauer} (hereafter\nEXP) the prediction of the HO model is always close to 1 for\n$S(k)$. When the correlation parameter $\\beta$  decreases\nconsiderably (strong correlations) the theoretical prediction of\n$S(k)$ is in good agreement with the experimental data. The value\n$\\beta$=5.3 ${\\rm \\mu m}^{-2}$ gives the best least squares fit in\nthat case. In general the gaussian form of $g(r)$, in spite of its\nsimplicity, reproduces fairly well the experimental data of EXP,\nboth in low and high values of the momentum $k$.  Within our\ntheoretical model, the gaussian type of $g(r)$ is flexible enough\nto obtain values for $S(k)$ in agreement with the experimental\ndata.\n\nFig. \\ref{fig:fig5}(b) displays the results in Case 2, which are\ncompared with those of the data of EXP. The model reproduces well\nthe experimental data in the range 1.5-3 ${\\rm \\mu m}^{-1}$ (with\nbest least squares fit value $a=1.34 \\  \\mu m^{-1}$), but fails in\nthe range $k>3$ ${\\rm \\mu m}^{-1}$. The main drawback of this\nmodel is the predicted negative values of  $S(k)$ in the range\nclose to $k=0$ when the correlation parameter $a$ decreases\nconsiderably (strong correlation case).\n\nThe correlation function $g(r)$ corresponding to Cases 1 and 2 for\nthe correlation parameters $\\beta$=5.3 ${\\rm \\mu m}^{-2}$ and\n$a=1.34 \\ \\mu m^{-1} $ respectively is sketched in Fig.\n\\ref{fig:fig6}. Those values of the parameters $\\beta$ and $a$\ngive the best $x^2$ value in the fit of the theoretical\nexpressions of $S(k)$ to the data of EXP. The most striking\nfeature in Case 2 is the existence of strong correlations,\nintroduced by  $g(r)$, in order to reproduce the experimental data\nof $S(k)$. It is worthwhile to point out that $g(r)$, in Case 2,\nexhibits fluctuations in the range $r>2$ ${\\rm \\mu m}$ but this is\nnot visible in Fig. \\ref{fig:fig6}.\n\nThe possibility of a linear dependence of $S(k)$ on $k$ for small\nvalues of $k$, as predicted from other works \\cite{Zambelli}, is\nprohibitive, on the basis of Eq. (\\ref{eq:eq18}) at least in the\ncase where the trap is an harmonic oscillator one. That can be\nseen considering  the ground state wave function to be the\nharmonic oscillator one and transforming ${\\bf r}_1$ and ${\\bf\nr}_2$ in Eq. (\\ref{eq:eq18}) into the coordinates of the relative\nmotion (${\\bf r}={\\bf r}_1-{\\bf r}_2$) and the center of mass\nmotion (${\\bf R}=({\\bf r}_1+{\\bf r}_2)\/2 $). After some algebra\n$S({\\bf k})$ takes the form\n\\begin{equation}\nS({\\bf k}) \\sim \\int \\textrm{exp} \\left[ i{\\bf k}{\\bf r} \\right]\n \\textrm{exp}\\left[ -r_b^2\\right] [Cg(r)-1] d {\\bf r}. \\label{sk-rel}\n\\end{equation}\n\nFor finite systems, as a trapped Bose gas, we can expand the\nexponential $\\textrm{exp} \\left[ i{\\bf k}{\\bf r} \\right]$, since\n${\\bf r}$ is bounded. Thus:\n\\begin{equation}\n\\textrm{exp} \\left[ i{\\bf k}{\\bf r} \\right]=1+i{\\bf k}{\\bf r}\n+\\frac{(i{\\bf k}{\\bf r})^2}{2 !}+ \\frac{(i{\\bf k}{\\bf r})^3}{3\n!}+\\cdots .\\label{expand}\n\\end{equation}\n\nSubstituting  Eq. (\\ref{expand}) into Eq. (\\ref{sk-rel}) and\ntaking into account that the terms with odd powers of $k$ do not\ncontribute to the integral, $S(k)$ takes the form\n\\begin{equation}\nS(k) \\sim  a_1k^2+a_2k^4+\\cdots .\\label{sk-rel-2}\n\\end{equation}\n\nHence, for small values of $k$, $S(k)$ depends linearly on $k^2$.\nThe gaussian factor $\\textrm{exp}\\left[ -r_b^2\\right]$,\noriginating from the harmonic oscillator wave function of the\ntrapped Bose gas, ensures the convergence of the integrals $a_i$\ncorresponding to the even powers of the expansion\n\\cite{Moustakidis04}.\n\n\n\n\\section{Quantum-Information properties of trapped\nBose gas}\\label{sec:sec4}\n\n\\subsection{Shannon Information Entropy}\n\nThe Boltzmann-Gibbs-Shannon information entropy\n\\cite{Shannon48,Halliwell93} of a finite probability distribution\n($p_1$,$p_2$,$\\cdot$,$p_k$) is defined as the quantity\n\\begin{equation}\nS=-\\sum_{i=1}^{k}p_i \\ln p_i, \\label{BGS}\n\\end{equation}\nwith the constraint: $\\displaystyle{\\sum_{i=1}^{k}p_i=1}$. $S$ is\nmeasured in bits if the base of the logarithm is 2 and nats\n(natural units of information) if the logarithm is natural.\n\n$S$ appears in different areas: information theory, ergodic theory\nand statistical mechanics. It is closely related to the entropy\nand disorder in thermodynamics. The maximum value of $S$ is\nobtained if $p_1=p_2=\\cdots=p_k=\\frac{1}{k}$ i.e. $S_{max}=\\ln k$.\nThe minimum value of $S$ is found when one of the $p_i$'s is equal\n$1$ and all the others are equal to $0$. Then $S_{min}=0$. The\nabove definition holds for discrete probability distributions\n\\cite{Chatzisavvas05}. In quantum mechanics we are often\ninterested in a continuous probability distribution $p(x)$. In\nthis case the obvious generalization of Eq. (\\ref{BGS}) is the\ninformation entropy\n\\begin{equation}\nS=-\\int p(x) \\ln p(x) dx, \\label{S-con}\n\\end{equation}\nwhere $\\int p(x) dx=1$. Now $p(x)$ is a quantum mechanical\nprobability distribution and $S$ may be called the quantum entropy\n\\cite{Ohya93}. $S$ indicates the amount of disorder or randomness\n(uncertainty) in a physical system. Shannon considered this\nuncertainty attached to the system as the amount of information\ncarried by the system. If a physical system has  a large\nuncertainty and one obtains information on the system by some\nprocedure, as a measurement, then the information is more valuable\nthan that received from a system having less uncertainty. Thus,\nbefore a measurement, the uncertainty of the position of a\nparticle is small for a localized probability distribution, while\nfor a diffuse distribution is large. The same holds for the\nmissing information due to a limited knowledge of the system via a\nprobability distribution. After the measurement the gain in\ninformation for a localized distribution is smaller than the\ncorresponding gain for a diffuse distribution.\n\nAn important step is the discovery of an entropic uncertainty\nrelation (EUR) \\cite{Bialynicki75}, which for a three-dimensional\nsystem has the form\n\\begin{equation}\\label{eq:equ2}\n    S=S_r+S_k\\geq 3\\,(1+\\ln{\\pi})\\simeq 6.434,\n\\end{equation}\nwhere $S_r$ is the information entropy in position-space of the\ndensity distribution $\\rho(\\textbf{r})$ of a quantum system\n\\begin{equation}\\label{eq:equ3}\n    S_r=-\\int\n    \\rho(\\textbf{r})\\,\\ln{\\rho(\\textbf{r})}\\,d\\textbf{r},\n\\end{equation}\nand $S_k$ is the information entropy in momentum-space of the\ncorresponding momentum distribution $n(\\textbf{k})$\n\\begin{equation}\\label{eq:equ4}\n    S_k=-\\int n(\\textbf{k})\\,\\ln{n(\\textbf{k})}\\,d\\textbf{k}.\n\\end{equation}\n\nThe total information entropy is given by\n\\begin{equation}\\label{eq:stot}\n    S=S_r+S_k.\n\\end{equation}\n\nThe density distributions $\\rho(\\textbf{r})$ and $n(\\textbf{k})$\nare normalized to one. Inequality (\\ref{eq:equ2}), for the\ninformation entropy sum in conjugate spaces, is a joint measure of\nuncertainty of a quantum mechanical distribution, since a highly\nlocalized $\\rho(\\textbf{r})$ is associated with a diffuse\n$n(\\textbf{k})$, leading to low $S_r$ and high $S_k$ and\nvice-versa. Expression (\\ref{eq:equ2}) is an\ninformation-theoretical relation stronger than Heisenberg's.\n\nIn previous work we proposed a universal property of $S$ for the\ndensity distributions of nuclei, electrons in atoms and valence\nelectrons in atomic clusters \\cite{Massen98}. This property has\nthe form\n\\begin{equation}\\label{eq:equ5}\n    S=a+b \\ln{N},\n\\end{equation}\nwhere $N$ is the number of particles of the system and the\nparameters $a, b$ depend on the system under consideration. It is\nnoted that recently we have obtained the same form for systems of\ncorrelated bosons in a trap \\cite{Massen02}. This concept was also\nfound to be useful in a different context. Using the formalism in\nphase-space of Ghosh, Berkowitz and Parr \\cite{Ghosh84}, we found\nthat the larger the information entropy, the better the quality of\nthe nuclear density distribution \\cite{Lalazissis98}. Recently the\nShannon information entropy has been applied successfully to the\nstudy of the free expansion of impenetrable bosons on the\none-dimensional optical lattices \\cite{Rigol05}.\n\n\\subsection{Onicescu's Information Entropy}\n\nOnicescu tried to define a finer measure of dispersion\ndistributions than that of Shannon's information entropy\n\\cite{Onicescu96}. Thus, he introduced the concept of information\nenergy $E$. For a discrete probability distribution\n$(p_1,p_2,\\ldots,p_k)$ the information energy $E$ is defined by\n\\begin{equation}\\label{eq:equ9}\n    E=\\sum_i^k p_i^2,\n\\end{equation}\nwhich is extended for a continuous density distribution $\\rho(x)$\nas\n\\begin{equation}\\label{eq:equ10}\n    E=\\int \\rho^2(x)\\,dx.\n\\end{equation}\nThe meaning of (\\ref{eq:equ10}) can be seen by the following\nsimple argument: For a Gaussian distribution of mean value $\\mu$,\nstandard deviation $\\sigma$ and normalized density\n\\begin{equation}\\label{eq:equ11}\n  \\rho(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\, \\textrm{exp}\n  \\left[-\\frac{(x-\\mu)^2}{2\\sigma^2} \\right],\n\\end{equation}\nrelation (\\ref{eq:equ10}) gives\n\\begin{equation}\\label{eq:equ12}\n  E=\\frac{1}{2\\pi\\sigma^2} \\int_{-\\infty}^{\\infty}\n   \\textrm{exp}\n  \\left[-\\frac{(x-\\mu)^2}{\\sigma^2}\n  \\right]\\,dx=\\frac{1}{2\\sigma\\sqrt{\\pi}}.\n\\end{equation}\n$E$ is maximum if one of the $p_i$'s equals 1 and all the others\nare equal to zero i.e. $E_{max}=1$, while $E$ is minimum when\n$p_1=p_2=\\ldots=p_k=\\frac{1}{k}$, hence $E_{min}=\\frac{1}{k}$\n(total disorder). $E$ has been called information energy, although\nit does not have the dimension of energy \\cite{Lepadatu03}. This\nis due to the fact that $E$ becomes minimum for equal\nprobabilities (total disorder), by analogy with thermodynamics.\n\nIt is seen from (\\ref{eq:equ12}) that the greater the information\nenergy, the more concentrated is the probability distribution,\nwhile the information content decreases. $E$ and information\ncontent are reciprocal, hence one can define the quantity\n\\cite{MoustaChatz05}\n\\begin{equation}\\label{eq:equ13}\n  O=\\frac{1}{E},\n\\end{equation}\nas a measure of the information content of a quantum system\ncorresponding to Onicescu's information energy.\n\nRelation (\\ref{eq:equ10}) is extended for a 3-dimensional\nspherically symmetric density  and momentum distribution as follow\n$\\rho(\\textbf{r})$\n\\begin{eqnarray}\\label{eq:equ14}\n  E_r=\\int \\rho^2(\\textbf{r})\\,d\\textbf{r} \\nonumber \\\\\n  E_k=\\int n^2(\\textbf{k})\\,d\\textbf{k}.\n\\end{eqnarray}\n\n\n$E_r$ has dimension of inverse volume, while $E_k$ of volume. Thus\nthe product $E_r E_k$ is dimensionless and can serve as a measure\nof concentration (or information content) of a quantum system. It\nis also seen from (\\ref{eq:equ12}),(\\ref{eq:equ13}) that $E$\nincreases as $\\sigma$ decreases (or concentration increases) and\nthe information (or uncertainty) decreases. Thus $O$ and $E$ are\nreciprocal. In order to be able to compare $O$ with Shannon's\nentropy $S$, we redifine $O$ as\n\\begin{equation}\\label{eq:equ15}\n  O=\\frac{1}{E_r E_k},\n\\end{equation}\nas a measure of the information content of a quantum system in\nboth position and momentum spaces, inspired by Onicescu's\ndefinition.\n\n\\subsection{Landsberg's Order Parameter}\n\nLandsberg \\cite{Landsberg84} defined the order parameter $\\Omega$\n(or disorder $\\Delta$) as\n\\begin{equation}\n \\Omega = 1-\\Delta = 1- \\frac{S}{S({\\rm max})},\n \\label{omega}\n \\end{equation}\n %\nwhere $S$ is the information entropy (actual) of the system and\n$S({\\rm max})$ the maximum entropy accessible to the system. Thus\nthe concepts of entropy and disorder are decoupled and it is\npossible for the entropy and order to increase simultaneously. It\nis noted that $\\Omega =1$ corresponds to perfect order and\npredictability, while $\\Omega =0$ means complete disorder and\nrandomness.\n\n\\subsection{Two-body information entropies}\n\nThe two-body Shannon information entropy both in position- and\nmomentum-space and in total are defined respectively\n\\cite{MoustaChatz05,Amovilli04,Cover91}\n\\begin{equation}\nS_{2r}=-\\int \\rho({\\bf r}_1,{\\bf r}_2) \\ln \\rho({\\bf r}_1,{\\bf\nr}_2) d {\\bf r}_1 d{\\bf r}_2 \\label{S2r-1}\n\\end{equation}\n\\begin{equation}\nS_{2k}=-\\int n({\\bf k}_1,{\\bf k}_2) \\ln n({\\bf k}_1,{\\bf k}_2)\nd{\\bf k}_1 d{\\bf k}_2, \\label{S2k-1}\n\\end{equation}\n\\begin{equation}\nS_2=S_{2r}+S_{2k}. \\label{S2-1}\n\\end{equation}\n\nThe one-body Onicescu's information entropy is already defined in\n(\\ref{eq:equ14}) and (\\ref{eq:equ15}), where the generalization to\nthe two-body information entropy is straightforward and is given\nby\n\\begin{equation}\nO_2= \\frac{1} {E_{2r} E_{2k}}, \\label{O1-2}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nE_{2r}&=&\\int \\rho^2({\\bf r}_1,{\\bf r}_2) d{\\bf r}_1 d{\\bf r}_2 \\nonumber \\\\\nE_{2k}&=&\\int n^2({\\bf k}_1,{\\bf k}_2) d{\\bf k}_1 d {\\bf k}_2.\n\\label{E2r-2k}\n\\end{eqnarray}\n\n\\subsection{Kullback-Leibler relative entropy and Jensen-Shannon\ndivergence}\n\nA well known measure of distance of two discrete probability\ndistributions $p_i^{(1)}, p_i^{(2)}$ is the Kullback-Leibler\nrelative entropy \\cite{Kullback59}\n\\begin{equation}\n    K(p_i^{(1)},p_i^{(2)})=\\sum_i\n    p_i^{(1)}\\,\\ln{\\frac{p_i^{(1)}}{p_i^{(2)}}}, \\label{eq:equ6}\n\\end{equation}\nwhich for continuous probability distributions $\\rho^{(1)},\n\\rho^{(2)}$ is defined as\n\\begin{equation}\n    K=\\int\n    \\rho^{(1)}(x)\\,\\ln{\\frac{\\rho^{(1)}(x)}{\\rho^{(2)}(x)}}\\,dx, \\label{eq:equ7}\n\\end{equation}\nwhich can be easily extended for 3-dimensional systems.\n\nOur aim is to calculate the relative entropy (distance) between\n$p^{(1)}$ (correlated) and $p^{(2)}$ (uncorrelated) densities both\nat the one- and the two-body levels in order to assess the\ninfluence of short range correlations (SRC) through the\ncorrelation parameter $y$, on the distance $K$\n\\cite{MoustaChatz05}. It is noted that this is done for both\nsystems under consideration: nuclei and trapped Bose gases. An\nalternative definition of distance of two probability\ndistributions was introduced by Rao and Lin \\cite{Rao87,Lin91},\ni.e. a symmetrized version of $K$, the Jensen-Shannon divergence\n$J$ \\cite{Majtey05}\n\\begin{equation}\n    J(p^{(1)},p^{(2)})=H\\left(\\frac{p^{(1)}+p^{(2)}}{2}\\right)-\\frac{1}{2}H\\left(p^{(1)}\\right)\n    -\\frac{1}{2}H\\left(p^{(2)}\\right), \\label{eq:equ8}\n\\end{equation}\nwhere $H(p)=\\displaystyle{-\\sum_i p_i \\ln{p_i}}$ stands for\nShannon's entropy. We expect for strong SRC the amount of\ndistinguishability of the correlated from the uncorrelated\ndistributions is larger than the corresponding one with small SRC.\nWe may also see the effect of SRC on the number of trials $L$\nneeded to distinguish $p^{(1)}$ and $p^{(2)}$ (in the sense\ndescribed in \\cite{Majtey05}).\n\nIn addition to the above considerations, we connect $S_r$ and\n$S_k$ with fundamental quantities i.e. the root mean square radius\nand kinetic energy respectively. We also argue on the effect of\nSRC on EUR and we propose a universal relation for $S$, by\nextending our formalism from the one- and two-body level to the\n$N$-body level, which holds exactly for uncorrelated densities in\ntrapped Bose gas and it is conjectured to hold approximately for\ncorrelated densities in  Bose gases (see Sec. \\ref{sub:sec4-7}).\n\nThe Kullback-Leibler relative information entropy $K$ for\ncontinuous distributions $\\rho_i^{(1)}$ and $\\rho_i^{(2)}$ is\ndefined by relation (\\ref{eq:equ7}). It measures the difference of\n$\\rho_i^{(1)}$ form the reference (or apriori) distribution\n$\\rho_i^{(2)}$. It satisfies: $K\\geq 0$ for any distributions\n$\\rho_i^{(1)}$ and $\\rho_i^{(2)}$. It is a measure which\nquantifies the distinguishability (or distance) of $\\rho_i^{(1)}$\nfrom $\\rho_i^{(2)}$, employing a well-known concept in standard\ninformation theory. In other words it describes how close\n$\\rho_i^{(1)}$ is to $\\rho_i^{(2)}$ by carrying out observations\nor coin tossing, namely $L$ trials (in the sense described in\n\\cite{Majtey05}). We expect for strong SRC the amount of\ndistinguishability of the correlated $\\rho_i^{(1)}$ and the\nuncorrelated distributions $\\rho_i^{(2)}$ is larger than the\ncorresponding one with small SRC.\n\nHowever, the distance $K$ does not satisfy the triangle inequality\nand in addition is i) not symmetric ii) unbounded and iii) not\nalways well defined \\cite{Majtey05}. To avoid these difficulties\nRao and Lin \\cite{Rao87,Lin91} introduced a symmetrized version of\n$K$ (recently discussed in \\cite{Majtey05}), the Jensen-Shannon\ndivergence $J$ defined by relation (\\ref{eq:equ8}). $J$ is minimum\nfor $\\rho^{(1)}=\\rho^{(2)}$ and maximum when $\\rho^{(1)}$ and\n$\\rho^{(2)}$ are two distinct distributions, when $J=\\ln{2}$. In\nour case  $J$ can be easily generalized for continuous density\ndistributions. For $J$ minimum the two states represented by\n$\\rho^{(1)}$ and $\\rho^{(2)}$ are completely indistinguishable,\nwhile for $J$ maximum they are completely distinguishable. It is\nexpected that for strong SRC the amount of distinguishability can\nbe further examined by using Wooter's criterion \\cite{Majtey05}.\nTwo probability distributions $\\rho^{(1)}$ and $\\rho^{(2)}$ are\ndistinguishable after $L$ trials $(L\\rightarrow \\infty)$ if and\nonly if $\\left( J(\\rho^{(1)},\\rho^{(2)})\n\\right)^{\\frac{1}{2}}>\\frac{1}{\\sqrt{2L}}$.\n\n\n\nThe relative entropy is a measure of distinguishability or\ndistance of two states. It is defined \\cite{MoustaChatz05},\ngeneralizing (\\ref{eq:equ7}), by\n\\begin{equation}\n    K=\\int \\psi^2(\\textbf{r})\n    \\ln{\\frac{\\psi^2(\\textbf{r})}{\\phi^2(\\textbf{r})}}\\,d\\textbf{r}. \\label{eq:equ16}\n\\end{equation}\nIn our case $\\psi(\\textbf{r})$ is the correlated case and\n$\\phi(\\textbf{r})$ the uncorrelated one. Thus\n\\begin{equation}\n    K_{1r}=\\int\n    \\rho(\\textbf{r})\\,\\ln{\\frac{\\rho(\\textbf{r})}{\\rho'(\\textbf{r})}} \\label{eq:equ17}\n    \\,d\\textbf{r},\n\\end{equation}\nwhere $\\rho(\\textbf{r})$ is the correlated one-body density and\n$\\rho'(\\textbf{r})$ is the uncorrelated one-body density.\n\nA corresponding formula holds in momentum-space\n\\begin{equation}\n     K_{1k}=\\int\n     n(\\textbf{k})\\,\\ln{\\frac{n(\\textbf{k})}{n'(\\textbf{k})}}\n     \\,d\\textbf{k},\n     \\label{eq:equ18}\n\\end{equation}\nwhere $n(\\textbf{k})$ is the correlated one-body density and\n$n'(\\textbf{k})$ is the uncorrelated one.\n\nFor the two-body case we have\n\\begin{equation}\n    K_{2r}=\\int \\rho(\\textbf{r}_1,\\textbf{r}_2)\\,\n    \\ln{\\frac{\\rho(\\textbf{r}_1,\\textbf{r}_2)}{\\rho'(\\textbf{r}_1,\\textbf{r}_2)}}\\,\n    d\\textbf{r}_1 d\\textbf{r}_2, \\label{eq:equ19}\n\\end{equation}\nwhere $\\rho(\\textbf{r}_1,\\textbf{r}_2)$ is the correlated two-body\ndensity in position-space and $\\rho'(\\textbf{r}_1,\\textbf{r}_2)$\nis the uncorrelated one.\n\nThe generalization to momentum-space is straightforward\n\\begin{equation}\n    K_{2k}=\\int n(\\textbf{k}_1,\\textbf{k}_2)\n    \\ln{\\frac{n(\\textbf{k}_1,\\textbf{k}_2)}{n'(\\textbf{k}_1,\\textbf{k}_2)}}\\,\n    d\\textbf{k}_1 d\\textbf{k}_2, \\label{eq:equ20}\n\\end{equation}\nwhere $n(\\textbf{k}_1,\\textbf{k}_2)$ is the correlated two-body\ndensity in momentum-space and $n'(\\textbf{k}_1,\\textbf{k}_2)$ is\nthe uncorrelated one.\n\nFor the Jensen-Shannon divergence $J$ we may write formulas for\n$J_1$ (one-body) and $J_2$ (two-body), employing definition\n(\\ref{eq:equ8}) and putting the corresponding correlated\n$\\rho^{(1)}$ and uncorrelated $\\rho^{(2)}$ distributions in\nposition- and momentum-spaces. We calculate $K$ and $J$ in\nposition- and momentum-spaces, for nuclei and bosons.\n\n\\subsection{Numerical Results and Discussion}\n\nFor the sake of symmetry and simplicity we put the width of the HO\npotential $b=1$. Actually for $b=1$ in the case of uncorrelated\ncase it is easy to see that $S_{1r}=S_{1k}$ and also\n$S_{2r}=S_{2k}$ (the same holds for Onicescu entropy), while when\n$b\\neq 1$ there is a shift of the values of $S_{1r}$ and $S_{1k}$\nby an additive factor $\\ln{b^3}$. However, the value of $b$ does\nnot affect directly the total information entropy $S$ (and also\n$O$). $S$ and $O$ are just functions of the correlation parameter\n$y$ \\cite{MoustaChatz05}.\n\n\nIn Fig. \\ref{fig:fig7} we present the Shannon information entropy\n$S_1$ using relation (\\ref{eq:stot}) and $S_2$ using relation\n(\\ref{S2-1}) in  trapped Bose gas as functions of the correlation\nparameter $\\ln{(\\frac{1}{y})}$. It is seen that $S_1$ and $S_2$\nincrease almost linearly with the strength of SRC i.e.\n$\\ln{(\\frac{1}{y})}$ in both systems. The relations $S_2=2 S_1$\nand $O_2=O_1^2$ hold exactly for the uncorrelated densities while\nthe above relations are almost exact for the uncorrelated\ndensities. For the sake of comparison we also present the\ndecomposition of $S$ in coordinate and momentum spaces i.e.\n$S_{1r}$, $S_{1k}$, $S_{2r}$, $S_{2k}$ employing (\\ref{eq:equ3}),\n(\\ref{eq:equ4}), (\\ref{S2r-1}), (\\ref{S2k-1}). The most striking\nfeature concluded from the above Figures is the similar behavior\nbetween $S_{1r}$ and $S_{2r}$ and also $S_{1k}$ and $S_{2k}$\nrespectively.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig7a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig7b.eps}\n \\caption{ (a) Shannon information entropy (one- and\n two-body) (b) Shannon information entropy (one- and two-body) both in\ncoordinate- and momentum-space, in a trapped Bose\ngas.}\\label{fig:fig7}\n\\end{figure}\n\n\nIn Fig. \\ref{fig:fig8} we plot the Onicescu information entropy\nboth one-body $(O_1)$ and two-body $(O_2)$ (relations\n(\\ref{eq:equ15}), (\\ref{O1-2})). We conclude by noting once again\nthe strong similarities of the behavior between one- and two-body\nOnicescu entropy.\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig8.eps}\n \\caption{The Onicescu information entropy (both one- and two-body) in a trapped Bose gas.}\\label{fig:fig8}\n\\end{figure}\n\nIt is interesting to observe the relation of the rms radii\n$\\sqrt{\\langle r^2 \\rangle}$ with $S_r$ as well as the\ncorresponding relation of the mean kinetic energy $\\langle T\n\\rangle$ with $S_k$, as functions of the strength of SRC,\n$\\ln{(\\frac{1}{y})}$. This is done in Fig. \\ref{fig:fig9} for\n$\\sqrt{\\langle r^2 \\rangle}$ and $\\langle T \\rangle$ after\napplying the suitable rescaling. The corresponding curves are\nsimilar for nuclei and trapped Bose gas.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig9a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig9b.eps}\n \\caption{ (a) Mean-square radius and the Shannon information entropy\n$S_{1r}$ (b) Mean kinetic energy $\\langle T \\rangle$ (in $\\hbar\n\\omega$ units) and the Shannon information entropy $S_{1k}$, as\nfunctions of the correlation parameter $\\ln{(\\frac{1}{y})}$ in a\ntrapped Bose gas.}\\label{fig:fig9}\n\\end{figure}\n\n\n\nA well-known concept in information theory is the distance between\nthe probability distributions $\\rho_i^{(1)}$ and $\\rho^{(2)}$, in\nour case the correlated and the uncorrelated distributions\nrespectively. A measure of distance is the Kullback-Leibler\nrelative entropy $K$ defined previously. The correlated and\nuncorrelated cases are compared for the one-body case $(K_1)$  and\nfor the two-body case $(K_2)$ in Fig. \\ref{fig:fig10}, decomposing\nin position- and momentum-spaces according to\n(\\ref{eq:equ17})-(\\ref{eq:equ20}). It is seen that $K_{1r}$,\n$K_{2r}$ increase as the strength of SRC increases, while\n$K_{1k}$, $K_{2k}$ have a maximum at a certain value of\n$\\ln{(\\frac{1}{y})}$ depending on the system under consideration.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig10a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig10b.eps}\n \\caption{(a) One- body Kullback-Leibler relative entropy both in\ncoordinate- and momentum-space (b) Two-body Kullback-Leibler\nrelative entropy both in coordinate- and momentum-space, in a\ntrapped Bose gas.}\\label{fig:fig10}\n\\end{figure}\n\nCalculations are also carried out for the Jensen-Shannon\ndivergence for one-body density distribution ($J_1$ entropy) as\nfunction of $\\ln{(\\frac{1}{y})}$, decomposed in position- and\nmomentum- spaces (Fig. \\ref{fig:fig11}). We observe again that\n$J_1$ increases with the strength of SRC in position-space, while\nin momentum-space there is a maximum for a certain value of\n$\\ln{(\\frac{1}{y})}$. It is verified that $0<J<\\ln{2}$ as expected\ntheoretically \\cite{Majtey05}.\n\nIt is noted that the dependence of the various kinds of\ninformation entropy on the correlation parameter\n$\\ln{(\\frac{1}{y})}$ is studied up to the value\n$\\ln{(\\frac{1}{y})}=0$ $(y=1)$, which is already unrealistic\ncorresponding to strong SRC. In addition, lowest order\napproximation does not work well beyond that value. In this case\nthree-body terms should be included but this prospect is out of\nthe scope of the present work.\n\nFor very strong SRC the momentum distribution $n(k)$ exhibits a\nsimilar behavior with the mean field $(y\\rightarrow \\infty)$. This\nis illustrated in Fig. \\ref{fig:fig5}, where we present $n(k)$ for\nvarious values of $\\ln{(\\frac{1}{y})}$. It is seen that for small\nand large SRC the tail of $n(k)$ disappears. That is why for small\nand large SRC the relative entropy ($K_{1k}$ and $J_{1k}$) is\nsmall, while in between shows a maximum (Fig. \\ref{fig:fig10}). A\nsimilar trend of $n(\\textbf{k}_1,\\textbf{k}_2)$ for large SRC\nexplains also the maximum of the relative entropy $K_{2k}$.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig11a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig11b.eps}\n \\caption{ (a) One-body Jensen-Shannon divergence entropy both in coordinate-\nand momentum-space, in a trapped Bose gas (b) Momentum\ndistribution $n(k)$ of $^4$He for various values\n of the correlation parameter $\\ln{(\\frac{1}{y})}$. The case MF (mean field)\n corresponds to the uncorrelated case $(y\\rightarrow \\infty).$ }\\label{fig:fig11}\n\\end{figure}\n\n\\subsection{Conclusions and comments}\\label{sub:sec4-7}\n\nOur main conclusions are the following \\cite{MoustaChatz05}\n\\begin{itemize}\n\\item[(i)] Increasing the SRC (i.e. the parameter\n$\\ln{(\\frac{1}{y})}$) the information entropies $S$, $O$, $K$ and\n$J$ increase. A comparison leads to the conclusion that the\ncorrelated Bose gas has larger values of entropies than the\nuncorrelated one.\n\n\\item[(ii)] There is a relation of $S_r$ with $\\sqrt{\\langle r^2\n\\rangle}$ and $S_k$ with $\\langle T \\rangle$  in the sense that\nthey have the same behavior as a function of the correlation\nparameter $\\ln{(\\frac{1}{y})}$. These results can lead us to\nrelate the theoretical quantities $S_r$ and $S_k$ with\nexperimental ones like charge density distribution, and momentum\ndistribution, radii, etc. A recent paper addresses this problem\n\\cite{Psonis05}.\n\n\\item[(iii)] The relations $S_2=2 S_1$ and $O_2=O_1^2$ hold\nexactly for the uncorrelated densities in trapped Bose gas while\nthe above relations are almost exact in the case of correlated\ndensities. In previous work we proposed the universal relation\n$S_1=S_r+S_k=a+b\\,\\ln{N}$ where $N$ is the number of particles of\nthe system either fermionic (nucleus, atom, atomic cluster) or\nbosonic (correlated atoms in a trap). Thereby, in the general case\n(including correlations among the atoms)\n\\[\n  S_2\\simeq 2 S_1=2(a+b\\,\\ln{N}).\n\\]\n\nFor the generalized $N$-body uncorrelated distributions\n$\\rho(\\textbf{r}_1,\\textbf{r}_2,\\ldots,\\textbf{r}_N)$ and\n$n(\\textbf{k}_1,\\textbf{k}_2,\\ldots,\\textbf{k}_N)$ the relation\n\\[\n  S_N=N S_1=N\\,(a+b\\,\\ln{N}).\n\\]\nholds exactly.\n\nIt is conjectured that it holds approximately for correlated\nsystems (which has still to be proved for $N\\geq 3$).\n\n\n\\item[(iv)] The entropic uncertainty relation (EUR) is\n\\[\n  S=S_r+S_k\\geq 6.434.\n\\]\nIt is well-known that the lower bound is attained for a Gaussian\ndistribution (i.e. the case of uncorrelated Bose gas). In all\ncases studied in the present work EUR is verified.\n\nA final comment seems appropriate. In general, the calculation of\n$\\rho({\\bf r}_1,{\\bf r}_2)$ and $n({\\bf k}_1,{\\bf k}_2)$ is a\nproblem very hard to be solved in the framework of short range\ncorrelations. In the present work we tried to treat the problem in\nan approximate but self-consistent way in the sense that the\ncalculations of $\\rho({\\bf r}_1,{\\bf r}_2)$ and $n({\\bf k}_1,{\\bf\nk}_2)$ are based in the same $\\rho({\\bf r}_1,{\\bf r}_2;{\\bf\nr}_1',{\\bf r}_2')$, which is the generating function of the above\nquantities. As a consequence the information entropy\n$S_2=S_{2r}+S_{2k}$ is derived also in a self-consistent way and\nthere is a direct link between $S_{2r}$ and $S_{2k}$, as well as\nthe other kinds of information entropies which are studied in the\npresent work.\n\\end{itemize}\n\n\n\n\n\n\\section{Quantum-Information properties based on Gross-Pitaevskii\nequation}\\label{sec:sec5}\n\nThe ground-state properties of the condensate, for weakly\ninteracting atoms, are explained quite successfully by the\nnon-linear equation, known as Gross-Pitaevskii  (GP) equation, of\nthe form\n\\begin{equation}\n\\left[-\\frac{\\hbar^2}{2m} \\nabla^2 + \\frac{1}{2}m\\omega^2 r^2+\nN\\frac{4\\pi \\hbar^2 a_0 }{m} | \\psi({\\bf r}) | ^2\\right] \\psi({\\bf\nr})=\\mu \\,\\psi({\\bf r}), \\label{gros-pit}\n\\end{equation}\nwhere $N$ is the number of the atoms, $m$ is the atomic mass,\n$a_0$ is the scattering length of the interaction and $\\mu$ is the\nchemical potential \\cite{Dalfavo99}. This equation has the form of\na non-linear stationary Schr\\\"odinger equation, and it has been\nsolved for several types of traps using various numerical methods\n\\cite{Dalfavo96,Edwards95,Cerboneschi}. The presence of the third\nterm, which is linear in $N$ is responsible for the dependence of\nthe gas parameter $\\chi$  on the density of the system.\n\n Eq. (\\ref{gros-pit}) was solved numerically\nin Ref. \\cite{Massen02} for trapped boson-alkali atoms in two\ncases. For a system of ${}^{87}$Rb atoms with parameter\n$b_0=12180$ \\AA $\\,$ (angular frequency $\\omega_0\/2\\pi=77.78$ Hz)\nand scattering length $a_0=52.9$ \\AA$\\,$ \\cite{Fabrocini99}, and\nfor a system of $^{133}$Cs atoms with parameters $b_0=27560$ \\AA\n$\\,$ ($\\omega_0\/2\\pi=10$ Hz) and $a_0=32$ \\AA$\\,$ \\cite{Schuck00}.\n In these cases the effective\natomic size is small compared both to the trap size and to the\ninteratomic distance ensuring the diluteness of the gas.\n\nThere is in atomic physics a connection of $S_{r}$ and $S_{k}$\nwith the total kinetic energy $T$ and mean square radius of the\nsystem through rigorous inequalities derived using the EUR\n\\cite{Gadre87,Gadre85}\n\\begin{eqnarray}\n \\label{Sr-ineq}\n &&S_r({\\rm min}) \\le S_r \\le S_r({\\rm max})\\\\\n \\label{Sk-ineq}\n &&S_k({\\rm min}) \\le S_k \\le S_k({\\rm max})\\\\\n \\label{S-ineq}\n &&S({\\rm min}) \\le S \\le S({\\rm max}).\n\\end{eqnarray}\n\nThe lower and upper limits are written here more conveniently in\nthe following form, for density distributions normalized to one:\n\\begin{eqnarray}\nS_r({\\rm min})&=& \\frac{3}{2}(1+\\ln \\pi)\n           -\\frac{3}{2} \\ln \\left( \\frac{4}{3} T \\right) \\nonumber\\\\\nS_r({\\rm max})&=& \\frac{3}{2}(1+\\ln \\pi)\n           +\\frac{3}{2}\\ln \\left(\\frac{2}{3}\\langle r^2 \\rangle\n           \\right),\n \\label{Sr-min}\n\\end{eqnarray}\n\\begin{eqnarray}\nS_k({\\rm min})&=& \\frac{3}{2}(1+\\ln \\pi)\n           -\\frac{3}{2}\\ln \\left( \\frac{2}{3}\\langle r ^2\\rangle \\right) \\nonumber\\\\\nS_k({\\rm max})&=& \\frac{3}{2}(1+\\ln \\pi)\n           +\\frac{3}{2}\\ln \\left(\\frac{4}{3} T \\right),\n            \\label{Sk-min}\n\\end{eqnarray}\n\\begin{eqnarray}\nS({\\rm min})&=& 3(1+\\ln \\pi)  \\nonumber\\\\\nS({\\rm max})&=& 3(1+\\ln \\pi)\n        + \\frac{3}{2}\\ln \\left(\\frac{8}{9}\\langle r^2\\rangle T\n        \\right).\n \\label{S-min}\n\\end{eqnarray}\n\nIn Ref. \\cite{Massen01} it was  verified numerically that the\nabove inequalities hold for nuclear density distributions and\nvalence electron distributions in atomic clusters. We also found a\nlink of $S$ with the total kinetic energy of the system $T$, and a\nrelationship of Shannon's information entropy in position-space\n$S_{r}$ with an experimental quantity i.e. the rms radius of\nnuclei and clusters.\n\n\nIt has been verified numerically \\cite{Massen02} that inequalities\n(\\ref{Sr-ineq}), (\\ref{Sk-ineq}) and (\\ref{S-ineq}) hold for\ncorrelated bosonic systems as well, i.e. trapped boson-alkali\natoms $^{87}$Rb and $^{133}$Cs. That is shown in Table\n\\ref{tbl:table2} for $^{87}$Rb bosonic system. An analogous table\ncan be displayed for $^{133}$Cs. It is noted that for large $N$,\n$S_k$ may become negative, but the important quantity is the net\ninformation content $S=S_r+S_k$ of the system which is positive.\nWe employed density distributions $\\rho({\\bf r})$ and $n({\\bf k})$\nfor bosons derived by solving numerically the GP equation\n(\\ref{gros-pit}). It is also seen that the right-hand-side of\ninequality (\\ref{Sr-ineq}) is nearly an equality. Thus there\nexists a relation between $S_r$ and $ T $ for bosons as well as\nfor nuclei \\cite{Massen01}.\n\n\n\\begin{table}[pt]\n\\centering\n\\begin{tabular}{@{}r c c c c c c c c c@{}}\n\\hline    $N$ & $S_r$& $S_r\\,\\,\\,$& $S_r$& $S_k$& $S_k\\,\\,\\,$\n& $S_k$& $S$ &$S\\,\\,\\,\\,$& $S$ \\\\\n\\hline\n5$\\times 10^2$ & 3.797 &3.834 &3.845 &2.590 &2.630 &2.637 &6.434 &6.465 &  6.482\\\\\n$10^3$ &4.027 &4.100 &4.120 & 2.314 & 2.394 & 2.408 &6.434 &6.494 &6.528\\\\\n3$\\times 10^3$ &4.437 &4.599 &4.640 & 1.794 & 1.963 & 1.997 &6.434 &6.562 &6.637 \\\\\n5$\\times 10^3$ &4.641 &4.855 &4.907 & 1.527 & 1.746 & 1.794 &6.434 &6.601 &6.701 \\\\\n7$\\times 10^3$ &4.778 &5.029 &5.090 & 1.345 & 1.598 & 1.657 &6.434 &6.627 &6.746 \\\\\n$10^4$ &4.925 &5.219 &5.287 & 1.148 & 1.437 & 1.509 &6.434 &6.655 &6.796 \\\\\n5$\\times 10^4$ &5.615 &6.113 &6.211 & 0.223 & 0.667 & 0.819 &6.434 &6.780 &7.030 \\\\\n$10^5$ &5.922 &6.511 &6.619 &-0.185 & 0.317 & 0.512 &6.434 &6.828 &7.132\\\\\n5$\\times 10^5$ & 6.654 &7.452 &7.577 &-1.142 &-0.533 &-0.220 &6.434 &6.919 &7.357 \\\\\n$10^6$ &6.993 &7.864 &7.992 &-1.557 &-0.920 &-0.560 &6.434 &6.943 &7.432 \\\\\n\\hline\n\\end{tabular}\n\\caption{Values of $S_r$, $S_k$ and $S$ versus the number of\nparticles $N$ for ${}^{87}$Rb bosonic system (see inequalities\n(\\ref{Sr-ineq}),\n (\\ref{Sk-ineq}), and (\\ref{S-ineq})) \\cite{Massen02}.}\\label{tbl:table2}\n\\end{table}\n\n\n\nIn Ref. \\cite{Massen02} we addressed the problem of finding\n$S_{r}$ and $S_{k}$ (i.e. the extent of $\\rho({\\bf r})$ and\n$n({\\bf k})$) for bosonic many-body systems in order to compare\nwith corresponding results for fermionic systems. First we review\nthe results of Ref. \\cite{Massen98} for systems of fermions where\nwe proposed a universal property for $S$ for the density\ndistributions of nucleons in nuclei, electrons in atoms and\nvalence electrons in atomic clusters. This property has the form\n\\begin{equation}\nS=a+b \\ln{N},\n \\label{S-ln1}\n\\end{equation}\nwhere the parameters $a$ and $b$ depend on the system under\nconsideration. The values of the parameters are the following\n\\begin{eqnarray}\na=5.325,& \\quad b=0.858 & \\quad ({\\rm nuclei}) \\nonumber\\\\\na=5.891,& \\quad b=0.849 & \\quad ({\\rm atomic \\ clusters})\\\\\na=6.257,& \\quad b=1.007 & \\quad ({\\rm atoms}).  \\nonumber\n\\label{eq-8}\n\\end{eqnarray}\nNext \\cite{Massen02}, we verified (\\ref{S-ln1}) employing\ndensities $\\rho (r)$ and $n(k)$ for trapped bosons solving the GP\nequation (\\ref{gros-pit}). $\\rho (r)$, derived in this way, and\n$n(k)$ derived by Fourier transform of $\\psi(r)$, were inserted\ninto equations (\\ref{eq:equ3}) and (\\ref{eq:equ4}) to find the\nvalues of $S_r$, $S_{k}$ and $S=S_{r}+S_{k}$ as functions of the\nnumber of bosons $N$. The results are shown in Fig.\n\\ref{fig:fig12}(a). The circles and the triangles correspond to\nthe calculated values for the bosonic systems ${}^{87}$Rb and\n$^{133}$Cs, respectively, while the lines to the fitted form of\nEq. (\\ref{S-ln1}) where\n\\begin{eqnarray}\na=6.0291, & \\quad b=0.0678, & \\quad 5\\times 10^2 < N < 10^6 \\,\\,\\, (^{87}{\\rm Rb})\\nonumber\\\\\na=5.9614,  & \\quad b=0.0661,  & \\quad 10^3 < N <5\\times 10^6\n\\,\\,\\,   (^{133}{\\rm Cs}). \\label{ab-SrSk-2}\n\\end{eqnarray}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=5.0cm,width=4.cm]{fig12a.eps}\n\\hspace{0.5cm}\n\\includegraphics[height=5.0cm,width=4.cm]{fig12b.eps}\n\\caption{The information entropy $S$ (a) versus $\\ln N$ and (b)\nversus $\\ln (Na_b)$ for the bosonic systems according to the\nfitted expressions (\\ref{S-ln1}) and (\\ref{SrSk-3}) respectively.\nThe open circles ($^{87}$Rb) and triangles ($^{133}$Cs) come from\nthe numerical solution of the GP equation.} \\label{fig:fig12}\n\\end{figure}\n\nFrom the values of the parameters $a$ and $b$, for the two systems\nwe examined, and from the strength of the interatomic interaction,\nwhich is proportional to $N a_0\/m$ we can conclude that for the\nsame trap (same value of $b_0$) and for different values of the\nscattering length or\/and  of the atomic mass, the linear\ndependence of $S$ on $\\ln N$ does not change. The only change will\nbe in the values of the parameter $a$ of Eq. (\\ref{S-ln1}). Thus\nthe change of the scattering length or\/and of the atomic mass will\nproduce a parallel displacement of the lines of Fig.\n\\ref{fig:fig12}(a). From the values of the parameter $b$ of Eq.\n(\\ref{S-ln1}) which give the slope of the lines of Fig.\n\\ref{fig:fig12}(a) corresponding to different sizes of the trap\n$b_0$ and because the GP equation can be written in the form\n\\begin{equation}\n\\left[-\\frac{1}{2} \\nabla_{r_b}^2 + \\frac{1}{2} r_b^2+ 4\\pi N a_b\n| \\psi_b({\\bf r}_b) | ^2\\right] \\psi_b({\\bf r}_b)=\\mu_{b}\n\\psi_b({\\bf r}_b), \\label{gros-pit2}\n\\end{equation}\nwhere $r_b=r\/{b_0}$, $a_b=a_0\/{b_0}$,  $\\mu_{b}=\\mu \/\n{\\hbar\\omega_0}$ and $\\psi_b({\\bf r}_b)=b_0^{3\/2} \\psi({\\bf r})$,\nwe should conclude that there is a parallel displacement of the\nlines for the various values of $b_0$. That leads us to fit the\nnumerical values of $S$, for the two systems we examined, using\nthe formula\n\\begin{equation}\nS= a+b \\ln\\left( N a_b \\right). \\label{SrSk-3}\n\\end{equation}\nThe new values of $a$ and $b$ are now $a=6.3976$ and  $b=0.0678$\nfor ${}^{87}$Rb and $a=6.4081$ and $b=0.0661$ for ${}^{133}$Cs.\nAs the two lines are almost the same we use the same parameters\n$a$ and $b$ for the two bosonic systems\n\\begin{equation}\na=6.4028,\\quad b=0.0669 \\label{ab-SrSk-3b}\n\\end{equation}\nwhich are the mean values of the corresponding parameters of the\ntwo systems. The results are shown in Fig. \\ref{fig:fig12}(b). It\nis seen that the numerical values of $S$ for the two bosonic\nsystems are very close to those calculated from Eq. (\\ref{SrSk-3})\nwith the parameters $a$ and $b$ given by Eq. (\\ref{ab-SrSk-3b}).\n\nThe results of $S$ and $S({\\rm max})$, displayed in Table\n\\ref{tbl:table2}, allowed us to calculate the order parameter\n$\\Omega$ (relation (\\ref{omega})) as function of the number of\nparticles $N$ in a system of trapped correlated boson-alkali\natoms. The dependence of $\\Omega$ on $N$ is shown  in Fig.\n\\ref{fig:fig13} for ${}^{87}$Rb (an analogous figure can be\ndisplayed for ${}^{133}$Cs). It is seen that $\\Omega$ is an\nincreasing function of $N$. A similar trend has been observed in\nFig. 1 of Ref. \\cite{Panos01b}, where $\\Omega(N)$ was calculated\nfor nucleons in nuclei and valence electrons in atomic clusters.\nAs stated in \\cite{Panos01b}, our result is in a way\ncounter-intuitive and indicates that as particles are added in a\ncorrelated quantum-mechanical system, the system becomes more\nordered. The authors in \\cite{Landsberg98} studied disorder and\ncomplexity in an ideal Fermi gas of electrons. They observed that\nfor a small number of electrons the order parameter $\\Omega$ is\nsmall, while $\\Omega$ increases as one pumps electrons into the\nsystem and the energy levels fill up.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=4.cm]{fig13.eps}\n \\caption{The order parameter $\\Omega$ as a function\nof the number of particles $N$ for ${}^{87}$Rb bosonic\nsystem.}\\label{fig:fig13}\n\\end{figure}\n\n\\subsection{Shannon's Information Entropy Using Density Distribution\nof Correlated Bosons, Results and Discussion}\\label{sub:sub5-1}\n\nFrom the analytical expressions of $\\rho(r)$ and $n(k)$  (Eqs.\n(\\ref{cluster-nr}) and (\\ref{cluster-nk}) respectively) for a\ncorrelated bosonic system, the information entropy $S$ can be\nfound using Eqs. (\\ref{eq:equ3}) and (\\ref{eq:equ4}). The values\nof $S$ as function of the correlation parameter $\\frac{1}{y}$ are\nshown in Fig. \\ref{fig:fig14}(a). The open squares correspond to\nthe calculated values of $S$ while the line to the fitting\nexpression\n\\begin{equation}\n S=a+b\\ln(1\/y),\n \\label{SrSk-4}\n \\end{equation}\n %\n where\n \\begin{eqnarray}\n a=6.6687,& \\quad b=0.0913,&\\,\\,\\, 0.05 \\le y \\le 10.\n  \\label{ab-srsk-4}\n \\end{eqnarray}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=4.0cm]{fig14a.eps}\n\\hspace{0.5cm}\n\\includegraphics[height=5.0cm,width=4.cm]{fig14b.eps}\n\\caption{(a) The information entropy $S$ versus the correlation\nparameter $y$ for a bosonic system. The line corresponds to the\nfitted expression (\\ref{SrSk-4}), while the open squares to values\nfound using the JCM. (b) The information entropy $S$ versus $\\ln\n(Na_b)$ for the bosonic systems according to the fitted expression\n(\\ref{SrSk-3}). The open circles (${}^{87}$Rb) and triangles\n(${}^{133}$Cs) come from the numerical solution of the GP\nequation, while the open squares correspond to the\nJCM.}\\label{fig:fig14}\n\\end{figure}\n\nWe have fitted our numerical results for $0.05 \\leq y \\leq 10$. As\nmentioned before large values of $y$ ($y > 10$) correspond to a\nGaussian distribution, while for $y\\lesssim 0.05$ (very strong\ncorrelations) higher order terms must be included in the expansion\nof the density.\n\nIt should be noted that in the present approach there is not a\ndirect dependence between the condensation and the number of the\natoms. The inter-particle correlations are incorporated in the\nmean field only by the correlation function which, in some way,\ndepends on the effective size of the atoms. We could find that\ndependence making the assumption that the correlated parameter $y$\ndepends on $N$ and $a_b$ through the relation\n\\begin{equation}\n\\frac{1}{y}= \\left(\\lambda_1 Na_b \\right)^{\\lambda_2}\n\\label{inv-y}\n\\end{equation}\n and try to find $\\lambda_1$ and $\\lambda_2$ equating the rhs of\n Eqs. (\\ref{SrSk-3}) and (\\ref{SrSk-4}). In this way $\\lambda_1$\n and  $\\lambda_2$ can be found as functions of the parameters $a$\n and $b$ of Eqs. (\\ref{SrSk-3}) and (\\ref{SrSk-4}) having the\n forms\n \\begin{equation}\n \\lambda_1 ={\\rm e}^{(a_1-a_2)\/b_1},\\quad\n \\lambda_2 =b_1\/b_2\n \\label{a12-b12}\n \\end{equation}\nwhere $a_1$ and $b_1$ are the values of the parameters $a$ and $b$\nof Eq. (\\ref{ab-SrSk-3b})\n and $a_2$ and $b_2$ the parameters of Eq. (\\ref{ab-srsk-4}).\n The numerical values of\n$\\lambda_1$ and $\\lambda_2$ are: $\\lambda_1=0.0188$ and\n$\\lambda_2=0.7330$.\n\nThe values of $S$ for the bosonic systems ${}^{87}$Rb and\n$^{133}$Cs found in Jastrow correlation method (JCM), versus $\\ln\n(Na_b)$ (calculated from Eqs. (\\ref{inv-y}) and (\\ref{a12-b12})),\nare shown in Fig. \\ref{fig:fig14}(b) with open squares. It is seen\nthat the two bosonic systems studied with the GP theory and the\nbosonic system studied with the JCM give very similar results for\n$S$. It seems that the information entropy $S$ for the bosonic\nsystems depends only on $\\ln(Na_b)$.\n\n\\section{Summary}\\label{sec:sec6}\nThe effect of the interparticle correlations between Bose atoms at\nzero temperature is examined using a phenomenological way to\nincorporate the atomic correlations. This is made by introducing\nthe Jastrow correlation function in the two-body density matrix.\nAnalytical expressions are found for the one- and the two-body\ndensity and momentum distribution, mean-square radius, kinetic\nenergy and static structure factor. The introduction of\ncorrelations changes the shape of the density and momentum\ndistributions compared with the Gaussian one, corresponding to the\nharmonic oscillator model. There is a decrease of the density\ndistribution in the central region of the atomic system while the\nmomentum distribution increases in the region of small $k$ and\nthus there is a decrease of the mean kinetic energy of the system.\nIn addition the natural orbitals and the natural occupation\nnumbers have been calculated and consequently the condensate\nfraction has been obtained for different values of the parameter\n$y$. A theoretical calculation of the static structure factor is\nreported also by applying two trial forms for the radial\ndistribution function. Our results are compared with recent\nexperimental data concerning trapped Bose gas. By applying\nsuitable parametrization the experimental data are reproduced\nquite well.\n\nVarious kinds of quantum information properties of the trapped\nBose gas are calculated i.e. the Shannon and Onicescu information\nmeasures for the correlated and uncorrelated cases which are\ncompared as functions of the strength of the short range\ncorrelations. It can be seen that increasing the short range\ncorrelations the information entropies $S$ and $O$ increase. There\nis a relation between $\\sqrt{\\langle r^2 \\rangle}$ and $S_r$ and\nbetween $\\langle T \\rangle$ and $S_k$. It is  also conjectured\nthat the relation $S_N=N(a+b\\,\\ln{N})$ holds approximately for the\ncorrelated system. The Gross-Pitavskii equation is solved in order\nto calculate the information properties of the Bose gas from\nanother point of view. It is concluded that the Shannon\ninformation entropy obeys the functional form $S=a+b\\,\\ln{N}$.\nFinally it is shown that Landsberg's order parameter $\\Omega$ is\nan increasing function of the number of Bose atoms $N$.\n\n\n\n\n\n\\section*{Acknowledgments}\n\nThe work of K.~Ch.~Chatzisavvas and C.~P.~Panos was supported by\nHerakleitos Research Scholarships (21866) of\n$\\textrm{E}\\Pi\\textrm{EAEK}$ and the European Union while the work\nof S.~E.~Massen, Ch.~C.~Moustakidis and C.~P.~Panos  was supported\nby the Pythagoras II Research project (80861) of\n$\\textrm{E}\\Pi\\textrm{EAEK}$ and the European Union.\n\n\\clearpage\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe idealized tasks on which machine learning models are benchmarked commonly involve a single data source and readily available labels. Many datasets, however,\nare composed of multiple sources, e.g., different MRI modalities~\\cite{calhoun2016multimodal, fedorov2020self} in medical imaging, LiDAR, and video for self-driving cars~\\cite{xiao2020multimodal}, and confounder-influenced data~\\cite{scholkopf2016modeling}. Additionally, labels can be scarce, incorrect, or too definitive, which leads to the need for self-supervised or at least semi-supervised learning. In this work, we will be addressing both these constraints simultaneously by proposing an analysis of self-supervised learning approaches to multimodal data and comparing them to more classical methods. The self-supervised learning methods we consider are based on contrastive Deep InfoMax (DIM)~\\cite{dim}.\n\nClassical approaches to multimodal data include \\emph{canonical correlation analysis} (CCA)~\\cite{hotelling1992relations}, which finds maximally correlated linear projections of two data sources. More recently, CCA has been extended to allow for representations obtained using neural networks in works such as \\emph{deep canonical correlation analysis} (DCCA)~\\cite{andrew2013deep} and \\emph{deep\ncanonically correlated autoencoders} (DCCAE)~\\cite{wang2015deep}. Another family of methods is based on variational autoencoders, such as the \\emph{multimodal mixture of experts VAE} (MMVAE)~\\cite{shi2019variational}, which has resulted in considerable performance improvements.\n\n\\begin{figure}[t]\n  \\center\n  \\includegraphics[width=\\linewidth]{dataset_scheme.png}\n  \\caption{Sample images from Two-View MNIST, MNIST-SVHN, OASIS-3, and a general scheme of all the possible ways to maximize the mutual information between two modalities. The arrows represent all possible combinations as coordinated connections between vectors of locations (along convolutional channels) or whole latent representations. The red arrow is Convolution-to-Representation (CR), the pink --- Convolution-Convolution (CC), the burgundy --- Cross-Convolution-to-Representation (XX), and the purple --- Representation-to-Representation (RR).}\n  \\label{fig:dataset_scheme}\n\\end{figure}\n\n\nContrastive objectives have in recent years become essential components for a large number of self-supervised learning methods. Mutual information estimation~\\cite{belghazi2018mine} has inspired a number of successful uses for single-view data, such as \\emph{deep infomax} (DIM)~\\cite{hjelm2018learning}, \\emph{contrastive predictive coding} (CPC)~\\cite{oord2018representation}) and for multi-view data, such as \\emph{augmented multiscale DIM} (AMDIM)~\\cite{bachman2019amdim}, \\emph{contrastive multiview coding} (CMC)~\\cite{tian2019contrastive}, SimCLR~\\cite{chen2020simple}) image classification, reinforcement learning (\\emph{spatio-temporal DIM} (ST-DIM)~\\cite{anand2019unsupervised}) and zero-shot learning (\\emph{class matching DIM} (CM-DIM)~\\cite{sylvain2019locality,sylvain2020zeroshot}). Such methods have resulted in large representation learning improvements by considering different views of the same instance. These and other related methods mostly operate in a self-supervised fashion, where the goal is to encourage similarity between transformed representations of a single instance. These objectives can also be readily applied to a multimodal context, where different sources can be understood as different views of the same instance.\nIn addition to the relative scarcity of the literature on self-supervised multimodal representation learning, we also note that there are no studies that consider explicit combinations of unsupervised and multimodal objectives. This work aims to contribute to both issues. Further, we propose a taxonomy for self-supervised learning that is readily applicable to newer models: contrasting with momentum encoders MoCo~\\cite{he2020momentum}, prototypical\/clustering approaches SwAV~\\cite{caron2020unsupervised}, PCL~\\cite{li2020prototypical}, and non-contrastive approaches, such as BYOL~\\cite{grill2020bootstrap} and SimSiam~\\cite{chen2020exploring}.\n\nOur contributions are as follows:\n\\begin{itemize}\n    \\item We unify contrastive learning methods under a systematic framework and perform a comparison of their merits. By analyzing the effect of adding different contrastive objectives, we show that correctly combining such objectives significantly impacts performance.\n    \\item We demonstrate empirically that multimodal contrastive learning has significant benefits over its unimodal counterpart.\n    \\item We propose novel models that emerge naturally from our systematic framework and taxonomy.\n    \\item We show that simultaneously maximizing the inter- and intra-modality mutual information is essential because intra-modality similarity maximization alone may lead to a collapse of the representations, which could, in turn, be caused by underspecification~\\cite{d2020underspecification}.\n    \\item We propose a new analysis with \\emph{centered kernel alignment} (CKA)~\\cite{kornblith2019similarity} to test the alignment of representations between modalities as a measure of their joint shared subspace.\n    \\item We propose solutions to the possible detrimental effects of a similarity metric on representations.\n    \\item We applied modern contrastive techniques to a complex multimodal medical setting and found a promising solution to uncover connections between modalities through a jointly shared subspace.\n\\end{itemize}\n\n\n\n\n\n\n\n\\begin{figure}[t]\n  \\center\n  \\includegraphics[width=1\\linewidth]{variants.pdf}\n  \\caption{Possible variants of connected pairs to maximize mutual information. The 4 main objectives are CR, XX, CC, RR. CR (Convolution-to-Representation) is a unimodal objective capturing the relation between the location in the Convolutional feature map and the Representation. XX corresponds to a relation between a location in the convolutional feature map of one modality and the representation of the other. CC (Convolution-to-Convolution) relates locations in the Convolutional feature map across modalities. RR captures a multimodal Representation-to-Representation relation. The other objectives are combinations of the first four (CR, XX, RR, and CC).}\n  \\label{fig:variants}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\center\n  \\includegraphics[width=\\linewidth]{variants2.pdf}\n  \\caption{Scheme for Supervised, AE, DCCAE, L-CCA, RR-AE and MMVAE.}\n  \\label{fig:others}\n\\end{figure}\n\n\\section{Methods}\n\n\\subsection{Problem setting}\n\nLet $\\{D_i = \\{x_1,...,x_N\\}\\}_{i=1,...,n}$ be a set of datasets $D_i$ with $N$ samples and $n$ modalities. For each $i$th modality $D_i$, we define a sampled image $x_i$, a CNN encoder $E_i$, a convolutional feature $c_i$ from a fixed layer $l$ in the encoder as $c_i = E^l_i(x_i)$, which is needed to define DIM~\\cite{dim}-based objective, and a latent representation $z_i$ defined as $z_i = E_i(x_i)$. The AE-based approaches also produce a reconstruction $x'_i$ of the original sample $x_i$.\n\nTo learn the set of encoders $\\{E_i\\}_{i=1,...,n}$ we want to maximize the following objective $L$:\n\\begin{equation*}\n  \\mathcal{L} = \\sum_{i=1}^n \\sum_{j=1}^n \\ell(D_i,D_j),\n\\end{equation*}\nwhere $\\ell(D_i,D_j)$ is a loss function between datasets $D_i$ and $D_j$.\nIn this study, we are specifically exploring the self-supervised contrastive objectives based on the maximization of mutual information, $\\ell(D_i, D_j)$ is therefore a loss function that maximizes the mutual information between the datasets of the two modalities.\n\n\\subsection{Contrastive mutual information maximization}\n\nTo maximize the lower bound of the mutual information, we utilize the InfoNCE~\\cite{oord2018representation} estimator $I$. We chose InfoNCE, because it is the common choice in the literature for self-supervised learning in natural images. There are, however, multiple estimators available (e.g., JSD and NWJ)~\\cite{dim, tschannen2019mutual, poole2019variational}. We define $I$ by adopting implementation from AMDIM~\\cite{bachman2019amdim}:\n\\begin{align*}\n  &\\ell(D_i,D_j) = I(D_i; D_j) \\\\\n  &\\ge \\frac{1}{N} \\sum_{l=1}^N \\log \\frac{e^{f(u^l_i,v^l_j)}}{\\sum_{k=1, k\\ne l}^N  e^{f(u^l_i,v^k_j)}  + e^{f^{c}(u^l_i,v^l_j)}},\n\\end{align*}\nwhere $f$ is a critic function, $u_i, v_j$ are two embeddings. The embeddings $u_i,v_j$ are obtained by projecting the location of a convolutional feature $c$ or a latent $z$ (e.g. $u_i = \\phi(c_i)$ and $v_j = \\psi(z_j)$). The neural networks that parametrize these projections $\\phi$ and $\\psi$ are also known as projection heads~\\cite{chen2020simple}.\n\nTo describe a critic function, we define positive and negative pairs. A pair $(u_i, v_j)$ is positive if it is sampled from a joint distribution $(u^l_i, v^l_j) \\sim p(D_i, D_j)$ and negative if it is sampled from a product of marginals $(u^l_i, v^k_j) \\sim p(D_i)p(D_j)$.\nA single entity could be represented differently in dataset $D_i$ than in dataset $D_j$. More specifically, the digit \"1\" can be represented by an image in multiple domains, for example, as a handwritten digit in MNIST and a house number in the SVHN dataset. The same number $(1_{\\text{MNIST}}, 1_{\\text{SVHN})}$ chosen from MNIST and SVHN will be a positive pair, whereas a digit in MNIST paired with a different digit in SVHN is a negative pair (such as $(1_{\\text{MNIST}}, 2_{\\text{SVHN}})$).\n\nThe idea behind a critic function $f(u,v)$ is to assign higher values to positive pairs and lower values to negative pairs. The critic used in this study is a separable critic $f(u,v) = \\frac{u^{\\intercal} v}{\\sqrt{d}}$, which is also used in the AMDIM~\\cite{bachman2019amdim} implementation (e.g. there are other possible choices such as bilinear, concatenated critics~\\cite{tschannen2019mutual}). Such a critic is equivalent to the scaled-dot product used in transformers~\\cite{vaswani2017attention}.\n\nAdditionally, we clip scores from critic function $f(x,y)$ to interval $[-c, c]$ by $c\\tanh(\\frac{f(x,y)}{c})$ with $c=20$, like in AMDIM~\\cite{bachman2019amdim}. Thus we need an additional term for a positive pair $e^{f^{c}(u^l_i,v^l_j)} = e^{-c}$. Lastly, we penalize $I(D_i; D_j)$ with the squared matching scores as $\\lambda f(x,y)^2$ with $\\lambda = 4\\mathrm{e}{-2}$.\n\n\n\\subsection{Objectives}\n\nMost contrastive self-supervised methods incorporate an estimator of mutual information. There are multiple ways of doing this, which are schematically shown in Figure~\\ref{fig:dataset_scheme}. The combinations of objectives shown in Figure~\\ref{fig:dataset_scheme} can be formulated as objectives based on Local DIM. In our taxonomy, we refer to the original Local DIM as CR since it captures a unimodal \\emph{Convolution-to-Representation} relation. This method maximizes the mutual information between the location in the convolutional feature map $c_i$ (where the representation is considered to be along the channels) and the latent representation $z_i$. Further, AMDIM~\\cite{bachman2019amdim}, ST-DIM~\\cite{anand2019unsupervised}, and CM-DIM~\\cite{sylvain2019locality} are referred to as XX, which captures a  \\emph{Cross-Convolution-to-Representation} relation and as CC, which captures a \\emph{Convolution-to-Convolution} relation. The XX objective pairs the latent $z_i$ with $c_j$, where $i \\ne j$. The CC objective pairs $c_i$ with $c_j$, where $i \\ne j$. The last type of pairing, RR, represents a \\emph{Representation-to-Representation} relation and was introduced with CMC~\\cite{tian2019contrastive} and SimCLR~\\cite{chen2020simple}. These two methods pair $z_i$ and $z_j$, where $i \\ne j$. The CR and XX objectives are more closely related to the Deep InfoMax principle, while the pairing of RR and CC is more closely related to CCA.\n\nResearchers can combine these four basic contrastive objectives as shown in Figure~\\ref{fig:variants} to create new objectives. Each combination of edges is a type of objective. The objective is equal to the total sum of all the edges. To create a full picture, we extend these objectives to other approaches such as autoencoders and DCCA~\\cite{andrew2013deep} by adding the contrastive terms to their respective objective functions. We call these models the \\emph{RR autoencoder} (RR-AE) and the \\emph{CR canonical correlation analysis} (CR-CCA). We show them schematically in Figure~\\ref{fig:others} alongside other baselines: a Supervised uni-modal model, a normal AE, a multimodal DCCAE~\\cite{wang2015deep}, and a MMVAE~\\cite{shi2019variational} with a loose \\emph{ importance weighted autoencoder} (IWAE) estimator. We introduced the supervised model baseline specifically to serve as a discriminative bound on the multimodal dataset.\n\n\n\n\n\\section{Experiments}\n\n\\subsection{Datasets}\nFor our experiments, we incorporate diverse datasets with two input sources (shown in Figure~\\ref{fig:dataset_scheme}). The first experiments are the most straightforward multi-view dataset, Two-View MNIST, and the more challenging multi-domain MNIST-SVHN dataset. These datasets allow us to validate our approach because the datasets gradually increase in multimodal dataset complexity. These datasets have the additional advantage of being tasks DCCAE, and MMVAE were directly evaluated in the original articles, allowing for a direct comparison to the author's reported results. The last dataset we evaluate is an Alzheimer's disease dataset, where we use a functional and structural view of the brain to learn representations that allow us to differentiate between healthy controls and patients with Alzheimer's disease.\n\n\\subsubsection{Multi-view dataset}\nTwo-View MNIST is inspired by~\\cite{wang2015deep}, where each view represents a corrupted version of an original MNIST digit. First, the intensities of the original images are rescaled to the unit interval. The images are then resized to $32 \\times 32$ to fit the DCGAN architecture. Lastly, to generate the first view, we rotate the image by a random angle within the $[-\\pi\/4, \\pi\/4]$ interval. For the second view, we add unit uniform noise and rescale the intensity to a unit interval again.\n\n\\subsubsection{Multi-domain dataset}\nThe multi-domain dataset MNIST-SVHN was used by the authors of the MMVAE~\\cite{shi2019variational}, where the first view is a grayscale MNIST digit and the second view is an RGB street view house number sampled from the SVHN dataset. The MNIST digits are modified by resizing the images to $32 \\times 32$, which is also the default SVHN image size, to use them with the DCGAN encoder. All intensities are scaled to a unit interval. This dataset is more complicated than two-View MNIST because the digit is represented in different underlying domains. It is also more similar to the neuroimaging dataset because the views occur more naturally than the Two-View MNIST, where the views are augmentations of the original MNIST dataset.\n\n\\subsubsection{Multi-modal dataset}\n\nThe multimodal MRI dataset that we use is OASIS3~\\cite{LaMontagne2019.12.13.19014902}. We use it to evaluate different representations for Alzheimer's disease (AD) classification. We use T1-weighted images to account for the anatomy of the brain. The T1-weighted image is brain masked using FSL~\\cite{fsl} (v 6.0.2). T1 is the first modality and captures the structural aspects of the brain. The second modality captures the functional aspects of the brain. The second modality we use is resting-state fMRI (rs-fMRI), which captures the brain's metabolic function. We preprocess rs-fMRI into fALFF (relative low-frequency power in the 0.01 to 0.1 Hz power band) using REST~\\cite{rest}.\nAll images are linearly converted to MNI space and resampled to 3mm isotropic voxel resolution. The final input volume is  $64\\times64\\times64$.\nCareful selection (removing poor images and trying to limit race as a confounder) resulted in a final subset of the OASIS-3 dataset with 826 non-Hispanic Caucasian subjects. For each subject, we combined their sMRI and fALFF scans to create 4021 multimodal pairs. We left 100 ($66\/22\/12$) subjects for hold-out and used others in a stratified (about $70\/15\/15\\%$) 5-folds for training and validation. We defined three groups: healthy cohort (HC), AD, and others (subjects with other brain problems). During pretraining, we employ all groups and pairs, whereas, during the linear evaluation, we only take one pair for each subject only use the HC or AD subjects. An additional preprocessing we applied is histogram standardization and z-normalization. During pretraining, we also use simple data augmentations, such as random crops and flips. These data augmentations were done using the TorchIO library~\\cite{perez_garcia_torchio_2020}. The classes in the dataset are highly unbalanced, so we utilize a class balanced data sampler~\\cite{balanced_data_sampler}.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{two_view_mnist.pdf}\n  \\caption{Test downstream perfomance with linear evaluation on Two-View MNIST. Cross-modal losses have a strong positive impact on performance.}\n  \\label{fig:two_view_mnist}\n\\end{figure}\n\n\\subsection{Evaluation}\n\\subsubsection{Linear evaluation on downstream task}\nTo evaluate representations on natural images, we employ the linear evaluation protocol, which is common for self-supervised approaches~\\cite{bachman2019amdim,chen2020simple}. It trains a linear mapping from the latent representations to the number of classes. The weights in the encoder that produces these representations are kept frozen. In this study, we evaluate the encoder for each modality separately.\n\n\\subsubsection{Measuring similarity between representations}\nTo better understand the underlying inductive bias of the specific objectives, we measure the similarity between the representations of the different modalities it produces. The measure of similarity we use is \\emph{canonical correlation analysis} CCA. CCA measures the average correlation of the aligned directions between the representations.  Additionally, we propose to use linear \\emph{centered kernel alignment} (CKA)~\\cite{kornblith2019similarity}, which has been shown to identify the relationship between representations of networks reliably.\n\nSpecifically, in our case, the CCA measure for a pair modalities $i$ and $j$ can be written as :\n\\begin{equation}\n    CCA(Z^i, Z^j) = \\frac{1}{d} ||Q^{\\mathrm{T}}_{Z^j}Q_{Z^i}||_*,\n\\end{equation}\nwhere $d$ is a dimension of latent representation, $Z$ is a $n \\times d$ matrix of $d$-dimensional representation for $n$ samples, $||\\cdot||_*$ is the nuclear norm, and $Q_{Z^i}$ is an orthonormal basis for $Z^i$.\n\nThe linear CKA measure is defined as:\n\\begin{equation}\nCKA(Z^i, Z^j) = \\left\\|Z^{j\\mathrm{T}} Z^i\\right\\|_{\\mathrm{F}}^{2} \/\\left(\\left\\|Z^{i\\mathrm{T}} Z^i\\right\\|_{\\mathrm{F}}\\left\\|Z^{j\\mathrm{T}} Z^j\\right\\|_{\\mathrm{F}}\\right),\n\\end{equation}\nwhere $||\\cdot||_F$ is the Frobenius norm.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{mnist_svhn.pdf}\n  \\caption{Downstream test performance with linear evaluation on MNIST-SVHN. We can see that overall, multimodal contrastive losses fare better than unimodal contrastive losses. }\n  \\label{fig:mnist_svhn}\n\\end{figure}\n\nAdditionally, we considered \\emph{singular value CCA} (SVCCA)~\\cite{NIPS2017_7188} and \\emph{projected weighted CCA} (PWCCA)~\\cite{morcos2018insights}. SVCCA is equivalent to CCA but performs an additional SVD-based dimensionality reduction. PWCCA is a weighted sum of the CCA vectors, where the weights are found through projection weighting. However, their results are similar to CCA.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{oasis_le.pdf}\n  \\caption{Holdout test downstream performance with linear evaluation on OASIS-3. The error lines show  $\\pm$ standard deviation over models trained on 5 different folds.}\n  \\label{fig:oasis}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n  \\centering\n  \\includegraphics[width=\\linewidth]{march14_dslw_oasis_results.pdf}\n  \\caption{CCA, SVCCA, PWCCA and CKA measures on OASIS-3 for the training and holdout test sets.}\n  \\label{fig:oasis_svcca_cka}\n\\end{figure*}\n\n\n\\subsection{Implementation details}\n\nThe architecture and the hyper-parameters for each encoder and decoder are entirely based on DCGAN~\\cite{radford2015unsupervised}. However, we removed one layer for experiments with natural images to be able to use an input size of $32x32$. The encoder produces a latent vector $z$, which in our case is $64$-dimensional. All the layers in the models are initialized with a uniform Xavier.\n\nThe projection head for the latent representation $z$ is identity. The convolutional features are taken from the layer with a feature side size of $8$ and passed through the convolutional projection head. The convolutional projection head is a ResNet like a block where one direction consists of $2$ convolutional layers (kernel size $1$, number of output and hidden features $64$, Xavier uniform initialization), and the second direction consists of one convolutional layer which (kernel size $1$, number of output features $64$, with initialization as identity). After summation of features from two directions, we apply a batch normalization layer to get the convolutional embeddings. The weights of the convolutional projection heads are shared across all contrastive objectives but separate for each modality.\n\nThe optimizer we use is RAdam~\\cite{liu2019variance}($ \\text{lr=}4\\mathrm{e}{-4}$) with a OneCycleLR scheduler~\\cite{smith2019super}($ \\text{max\\_lr=}0.01$). We pretrain on the natural images for $50$ epochs and on the volumes in OASIS-3 for $200$ epochs. The linear classification projection for natural images and OASIS-3 is trained for $50$ and $500$ epochs, respectively. All experiments were performed with a batch size of $64$. In some runs, we noticed that the CCA objective is unstable. The MMVAE did not converge for three folds out of 5 or completely collapsed due to the model's greater capacity. The greater capacity leads to GPU memory issues. Thus we had to decrease the batch size to 4. Thus we currently removed MMVAE from the OASIS-3 result.\n\nThe code was developed using PyTorch~\\cite{NEURIPS2019_9015} and the Catalyst framework~\\cite{catalyst}.\nFor data transforms of the brain images we utilized TorchIO~\\cite{perez_garcia_torchio_2020}, for CKA analysis of the representations we used code by anatome~\\cite{hataya2020anatome}, SVCCA~\\cite{NIPS2017_7188}, for AMDIM~\\cite{bachman2019amdim}, for DCCAE~\\cite{perry2020mvlearn}, for MMVAE and MNIST-SVHN~\\cite{shi2019variational}. The experiments were performed on an NVIDIA DGX-1 V100.\n\n\\section{Results}\n\n\\subsection{Two-View MNIST and MNIST-SVHN}\nWe can see in Figures~\\ref{fig:two_view_mnist} and~\\ref{fig:mnist_svhn} that cross-modal contrastive losses have a strong positive impact on downstream test performance across different architectural and model choices. We also note that the formulations have different performances across settings and datasets, leading to the conclusion that applying them in practice requires careful adaptation to a given problem.\n\nAlthough contrastive methods will result in the best performance for a simple multi-view case, reconstruction-based models, such as MMVAE and RR-AA, stand out in multi-domain experiments. The performance for most of the contrastive multimodal models is within $3\\%$ of RR-AE's performance. Thus one can choose decoder-free self-supervised approaches to reduce computational cost. It should also be noted that Uni-source AE, CR, multimodal DCCAE, and CR-CCA are not able to perform well on SVHN.\n\n\n\n\\subsection{OASIS-3 }\n\nFigure~\\ref{fig:oasis} shows the results on OASIS-3. Multimodal approaches exhibit strong performance, but the performance gain is less noticeable than with the previous tasks on natural images. Overall, accounting for robust performance on two modalities, the absolute leader in terms of the self-supervised methods is CR-XX-CC. The supervised baseline trained only on T1 modalities does still outperform this self-supervised method, however. In the case of the fALFF modality, the model that performs the best is XX-CC; notably, it is better than the Supervised model. This suggests that multimodal objectives can improve training for certain modalities that may be hard to extract discriminative representations from. Utilizing such an objective as some form of regularization may be beneficial.\n\nWe also want to note that the maximization of similarity between modalities by itself is not enough. The performance of the RR method shows that it might lead to a collapse of the representations. It might indicate that the T1 representation dominated over the fALFF representation during training because the RR objective model can learn meaningful T1 representations. By adding the reconstruction loss to the contrastive objective, we get RR-AE. The reconstruction loss improves the model. Analogous to the multi-domain experiment, the autoencoder is vital to learning the modality. Adding a reconstruction loss is not only the choice. The objective CR-XX-CC can, for example, be used, and it comes with reduced computational requirements because it does not require a decoder. The RR-AE model is also highly related to the idea and structure of the DCCAE. However, the typical implementation of CCA objectives in DCCAE is less numerically stable and requires the computation of eigenvalues and eigenvectors. Additionally, RR-AE is more robust and has a higher downstream performance.\n\nThe downstream performance might not be the main criteria to select a model, however. The multimodal models can be used to analyze the connection between modalities in neuroimaging~\\cite{calhoun2016multimodal}. Thus one will want to have a model with a representation that has a shared subspace. Based on our experiments on downstream performance in Figure~\\ref{fig:oasis} and similarity measures Figure~\\ref{fig:oasis_svcca_cka} we would advise CR-XX-CC and RR-AE as models that perform well and should be investigated further.\n\n\n\\subsection{Understanding similarity measures}\nMost self-supervised methods and the supervised model in Figure~\\ref{fig:oasis_svcca_cka} are noticeably worse than DCCAE, RR, RR-AE, and AE in terms of the CCA and SVCCA metrics. The lower performance is, however, not found to be significant for the PWCCA metric.\n\nVisually CCA, SVCCA, PWCCA measures of similarity behave comparably with a noticeable difference between training and testing sets. Interestingly, the CKA measure has very close values between training and testing sets. We hypothesize that the CKA measure shows the inductive bias hidden in representations through optimization, architecture, and the learned weights. The authors of the original CKA manuscript~\\cite{kornblith2019similarity} argue that methods with higher CKA have a higher similarity between representations and that more of the subspace is shared. The reason why CCA, SVCCA, PWCCA behave poorly might be related to their sensitivity to noise.\n\n\\section{Conclusion}\nIn this work, we proposed a unifying view on contrastive methods and conducted a detailed study of their performance on multimodal datasets. We\nbelieve that this unifying view will contribute to understanding how to learn powerful representations from multiple\nmodalities. Hopefully, instead of combining the similarities in various\nways and publishing the winning combinations as individual methods, the\nthe field will consider a broader perspective on the problem.\n\nWe empirically demonstrate that multimodal contrastive approaches result in performance improvements over methods that rely on a single modality for contrastive learning. We also show that downstream performance is highly dependent on the composition of such objectives. We argue that maximizing information similarity might not guarantee higher downstream performance. In some cases, it may weaken the representation or have a regularizing effect on the objective. However, high similarities between representations can be significant for other applications, i.e., multimodal analysis~\\cite{calhoun2016multimodal}.\n\nDIM-based methods have a smaller computational cost than autoencoder-based methods because they do not require a decoder to be trained. The smaller computational cost lowers the hardware requirements. While DIM-based methods do have comparable downstream performance, the lower hardware requirements can help democratize medical imaging. DIM-based methods can also be helpful in cases where labels do not exist or are inaccurate, a scenario that is quite common in neurological and mental disorder nosology.\n\nFor future work, we are interested in considering how the conclusions we draw here hold in different learning settings with scarcer data or annotations such as few-shot or zero-shot learning cases. Another goal is to study the joint shared subspace projects in brain space for visualization. The proposed interpretable joint learning approach can help advance work in our search for neuroimaging biomarkers.\n\n\\section{Acknowledgments}\n\\label{sec:acknowledgments}\nThis work is supported by NIH R01 EB006841.\n\nData were provided in part by OASIS-3: Principal Investigators: T. Benzinger, D. Marcus, J. Morris; NIH P50 AG00561, P30 NS09857781, P01 AG026276, P01 AG003991, R01 AG043434, UL1 TR000448, R01 EB009352. AV-45 doses were provided by Avid Radiopharmaceuticals, a wholly-owned subsidiary of Eli Lilly.\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMirror symmetry is a fascinating relation between a pair of Calabi-Yau manifolds originating from physics. Even though two mirror manifolds may look very different geometrically, they are predicted to give rise to equivalent quantum field theories and thus equivalent notions of physics.\\\\\n\n\nIt took mathematicians decades to understand this relation. For a pair of mirror Calabi-Yau manifolds $X$ and $X^\\vee$, the first thing to notice is that their Hodge diamonds are reflections of each other, i.e. $h^{p,q}(X)=h^{q,\\,p}(X^\\vee)$. In 1990, physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that the enumerative invariants and period integrals of a mirror pair were related to each other, which led to the calculations of Gromov-Witten invariants on quintic 3-folds. \\\\\n\nIn 1994, Maxim Kontsevich proposed the celebrated Homological Mirror Symmetry conjecture which provides a mathematically rigorous explanation for the mysterious mirror symmetry phenomenon. It asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves of its mirror dual (usually a complex manifold). \\\\\n\nOn the other hand, Strominger-Yau-Zaslow (SYZ)'s conjecture\\cite{SYZ} gives a geometric intuition for mirror symmetry. It conjectures that mirror symmetry is a torus duality between symplectic geometry (A-side) and complex geometry (B-side), which can be\ninterchanged by implementing a Fourier-transform on torus fibers. Within this geometric framework (following Fukaya\\cite{F} and Abouzaid\\cite{AbF}), one could construct a mirror functor from the Fukaya category of the A-side to the category of coherent sheaves of the B-side which induces the derived equivalence.\n\\subsection{Homological mirror symmetry and coisotropic branes}\nThe Fukaya category consists of Lagrangian branes (Lagrangian submanifolds equipped with flat connections) as objects and Lagrangian intersections as morphisms, and the Floer products (compositions of morphisms in the Fukaya category) are determined by counting holomorphic triangles with vertices at Lagrangian intersections and with edges in Lagrangian submanifolds.\nIn general, Lagrangian submanifolds do not generate the version of Fukaya category for which homological mirror symmetry holds as stated, certain objects are missing. For example, Let $E_\\tau$ be a elliptic curve with a Teichmuller parameter $\\tau$. Suppose $E_\\tau$ admits a complex multiplication, for example, $\\tau =i$, then multiplication by $i$ is an automorphism of $E_\\tau$ which is called a complex multiplication. Let $E=E_\\tau^2$ be the product abelian surface, then the mirror of $E$ is a 4 dimensional symplectic torus $(T,\\,\\omega)$. Considering the Grothendieck groups of the categories involved in homological mirror symmetry and their images under the Chern character map, we expect a commutative diagram.\n$$\n\\begin{tikzcd}\nK_0(D^bCoh(E)) \\arrow[rr] \\arrow[d, \"Ch\"] &  & K_0(Fuk(T,\\omega)) \\arrow[d, \"Ch\"] \\\\\nH^*(E) \\arrow[rr]                         &  & H^*(T).\n\\end{tikzcd}\n$$\nBy comparing the images of the Chern character maps, one finds that $Im(Ch)\\otimes \\mathbb{Q}$ for $K_0(D^bCoh(E))$ is 6 dimensional, yet $Im(Ch)\\otimes \\mathbb{Q}$ for $K_0(Fuk(T))$ is 5 dimensional, taking values in the kernel of \n$$  \\omega\\wedge:H^2(T;\\mathbb{Q})\\rightarrow H^4(T;\\mathbb{Q}),$$\nthus the two categories cannot be equivalent. This indicates that Lagrangians do not generate a sufficiently large category. Kapustin and Orlov \\cite{KO} suggest that the missing objects are coisotropic submanifolds equiped with a $U(1)$ connection satisfying certain conditions.\nThe difficulty of incorporating coisotropic submanifolds is to define the morphisms involving coisotropic objects and the product operations involving these morphisms. It is a long standing problem to define the appropriate Fukaya category including coisotropic branes proposed by Kapustin-Orlov \\cite{KO} and further studied by Aldi-Zaslow \\cite{AZ}, Chan-Leung-Zhang \\cite{CLZ}, Herbst \\cite{H}.\n\n\\begin{remark}\n\tAn alternative way to enlarge the Fukaya category is to take the split closure of the derived Fukaya category, i.e. enlarging the Fukaya category by adding formal direct summands of objects\\cite{S} \\cite{AS}.\n\\end{remark}\n\n\n\n\n\\subsection{Main result}\n\n\n\n\nOn a linear symplectic torus $(T,\\,\\omega)$, a new method is proposed to extend the Fukaya category of the torus to include coisotropic submanifolds alongside linear Lagrangian submanifolds as objects of the category. The approach is by considering a twisted doubling torus $T\\times T^\\vee$ (see definition \\ref{Double}) of $T$ and lifting (possibly coisotropic) objects into Lagrangians of $T\\times T^\\vee$.\n\\begin{theorem}\\label{LiftisLag}\\quad\n\t\\begin{enumerate}\n\t\t\\item The lift of a (possibly coisotropic) linear object on $T$ is a Lagrangian submanifold of $T\\times T^\\vee$.\n\t\t\\item The lift is a complex submanifold with respect to a canonical complex structure on $T\\times T^\\vee$.\n\t\\end{enumerate}\n\n\\end{theorem}\nThe Lagrangian Floer theory of $T$ is naturally related to the Lagrangian Floer theory of the twisted doubling torus $T\\times T^\\vee$ of $T$. However, the morphism spaces that we want to consider are only a certain subspace of the Floer cohomology in doubling torus, which we call the ``u-part\".\nThe main result, informally, is that the Floer cohomology of two objects in $T$ is isomorphic to the ``u-part\" Floer cohomology of the lifts in $T\\times T^\\vee$ and respects the Floer products.\n\\begin{theorem}\n\tFor a pair of Lagrangian branes $L,\\,L'$ which are mirror to a pair of line bundles $\\mathcal{L},\\,\\mathcal{L'}$, let $\\boldsymbol{L},\\,\\boldsymbol{L}'$ be the lifts in double torus. Suppose $\\mathcal{L'}\\otimes \\mathcal{L}^{-1}$ is ample. Then the ``u-part\" Floer cohomology $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')$ is isomorphic to $HF^*(L,\\,L')$. And for two such pair $L,L'$ and $L',L''$, the following diagram commutes\n\t$$\\begin{tikzcd}\n\t{HF^*(L',\\,L'')\\otimes HF^*(L,\\,L')} \\arrow[rr] \\arrow[d, \"\\cong\"] &  & {HF^*(L,\\,L'')} \\arrow[d, \"\\cong\"] \\\\\n\tHF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}'')\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\arrow[rr]                                 &  & HF_u(\\boldsymbol{L},\\boldsymbol{L}'').\n\t\\end{tikzcd}$$\n\\end{theorem}\n By Theorem \\ref{LiftisLag}, a coisotropic submanifold of $T$ lifts to a Lagrangian submanifold of the doubling torus which could be studied using Lagrangian Floer theory. Thus, the enlarged Fukaya category including coisotropic branes is realized as a non-full subcategory of the Fukaya category of the doubling torus. As a corollary of this doubling construction, the Fukaya category of $T$ is equivalent to the Fukaya category of the dual torus $T^\\vee$.\n\n\n\n\\section{HMS for tori}\n\nHomological mirror symmetry for tori was extensively studied by Polishchuk-Zaslow\\cite{PZ}, Fukaya\\cite{F}, Kontsevich and Soibelman\\cite{KS}. Here we will review the construction of the mirror manifold of a symplectic torus, and how Lagrangian branes correspond to coherent sheaves on the mirror.\n\n\n\\subsection{SYZ fibrations and construction of mirrors}\n\n\n\n$T$ be a torus equipped with a complex valued closed 2-form $B+i\\omega$, where real part is called B-field, and where imaginary part is a (non-degenerate) symplectic form.\n\\begin{definition}\n\tA SYZ fibration for $(T,\\,B+i\\omega)$ is a torus fibration such that each fiber is a Lagrangian submanifold.\n\\end{definition}\n\nGiven a SYZ fibration of $(T,\\,B+i\\omega)$ with base $Q$ and fibers $F_q$, $T_qQ$ is naturally identified with $H^1(F_q;\\, \\mathbb{R})$ by $v\\mapsto [\\iota_v\\omega]$, where we lift $v\\in T_qQ$ to a normal vector field along $F_q$, also denoted $v$. Let $T^{\\mathbb{Z}}Q\\subset TQ$ be the lattice corresponding to $H^1(F_q,\\, \\mathbb{Z})$, and $T^*_\\mathbb{Z}Q\\subset T^*Q$ the dual lattice, which is natrually isomorphic to $H_1(F;\\,\\mathbb{Z})$. The key property of this lattice $T^*_\\mathbb{Z}Q$ is that its sections are locally exact 1-forms on $Q$. Indeed, for a class $\\beta\\in H_1(F,\\,\\mathbb{Z})$, near a point $q_0\\in Q$, let $x_\\beta(q)=\\int_{(q-q_0)\\times \\beta}\\omega$, then $dx_\\beta(v)=\\int_{\\beta}\\iota_v\\omega$ is the corresponding section of $T_\\mathbb{Z}^*Q$. If $\\beta_1,...,\\,\\beta_n$ form a basis of $H_1(F;\\,\\mathbb{Z})$, then the local coordinates $x_{\\beta_1},...,x_{\\beta_n}:\\,Q\\rightarrow \\mathbb{R}^n$ induce an integral affine structure on $Q$, and this structure doesn't depend on the choice of basis of $H_1(F;\\,\\mathbb{Z})$.\n\\begin{assumption}\n\tThe restriction of the B-field $B$ to the SYZ fibers vanishes.\n\\end{assumption}\nThis assumption eliminates the case with noncommutative mirrors, and allows one to construct the mirror manifold $Y$ of $(T,\\,B+i\\omega)$ as the moduli space of SYZ fibers equipped with flat $U(1)$ connections. $Y$ naturally admits a complex structure, specified by local coordinates\n\\begin{equation}\\label{coord}\nz_\\beta(F_q,\\,\\nabla)=e^{2\\pi i \\int_{(q-q_0)\\times \\beta}(B+i\\omega)}hol_{\\nabla}(\\beta)\n\\end{equation}\nwhere $F_{q_0}$ is a fixed fiber, $\\nabla$ is a flat $U(1)$ connection on $F_q$, and $(q-q_0)\\times \\beta $ is a relative homology class in $H_2(T,\\,F_q\\cup F_{q_0};\\,\\mathbb{Z})$ with boundary $\\beta$ in $H_1(F_q;\\,\\mathbb{Z})$ and $-\\beta$ in $H_1(F_{q_0};\\,\\mathbb{Z})$.\nThe tangent space of $Y$ at $(F,\\,\\nabla)$ is the quotient of the set of all pairs $(v,\\,\\alpha)\\in C^{\\infty}(NF)\\oplus \\Omega^1(F,\\,\\mathbb{R})$ such that $v$ is an infinitesimal Lagrangian deformation, and $\\alpha$ is a closed $1$-form, viewed as an infinitesimal deformation of the flat connection, by the subspace consisting of Hamiltonian vector fields and  exact 1-forms (which correspond to trivial deformations).\n\n\\begin{lemma}\n\t$T_{(F_q,\\,\\nabla)}Y$ is identified with $H^1(F_q,\\,\\mathbb{C})$ via the map\n\t\\begin{equation}\n\t\\phi :(v,\\,\\alpha)\\mapsto \\iota_v (\\omega-iB)+i\\alpha.\n\t\\end{equation}\n\tAnd the map is complex linear.\n\\end{lemma} \n\n\\begin{proof}\n\tThe Arnold-Liouville theorem implies that there are canonical identifications \n\t\\begin{equation}\n\tT_qQ \\cong H^1(F_q;\\,\\mathbb{R}),\\, v\\mapsto [\\iota_v \\omega]\n\t\\end{equation}\n\twhere $v$ is lifted from $T_qQ$ to $C^{\\infty}(NF)$, so $v\\mapsto [\\iota_v \\omega]$ maps bijectively to $H^1(F_q;\\,\\mathbb{R})$. And an exact $1$-form $\\alpha$ is simply a gauge transformation which do not contribute to the deformation of connections, so the deformations of connections are classified by cohomology class of the 1-form $\\alpha$. So $\\phi$ is a bijection.\n\tTo verify $\\phi$ is complex linear, we see that\n\t\\begin{equation}\n\tdlog(z_\\beta)(v,\\,\\alpha)=-2\\pi  \\int_{\\beta}(\\iota_v (\\omega-iB)+i\\alpha)\n\t\\end{equation}\n\tis complex linear for every holomorphic coordinate function $z_\\beta$. So $\\phi$ is complex linear. \n\\end{proof}\n\nIn fact, using the coordinates from \\eqref{coord} and observing that interior product with $B+i\\omega$ defines an (injective) linear map $H_1(Q) \\rightarrow H^1(F;\\,\\mathbb{C})$, we can identify the mirror manifold $Y$ with\n\\begin{equation}\nH^1(F;\\,\\mathbb{C})\/H^1(F;\\,\\mathbb{Z})+(B+i\\omega) H_1(Q;\\,\\mathbb{Z}).\n\\end{equation} \n\\begin{example}\\label{stdex}\n\tFor $T=(\\mathbb{R}\/\\mathbb{Z})^{2n}$, $B+i\\omega=\\tau dr\\wedge d\\theta=\\sum \\tau_{jk}dr_j\\wedge d\\theta_k (\\tau \\in M_{n\\times n}(\\mathbb{C}))$, with SYZ fibers $F=\\{r\\}\\times T_\\theta ^n$, the mirror complex torus is $E=\\mathbb{C}^n\/(\\mathbb{Z}^n+\\tau^{T} (\\mathbb{Z}^n))$.\n\\end{example}\n\n\n\\begin{comment\n\\begin{example}\nThe dual torus $T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}, -\\omega^{-1}=-a^{-1}d\\hat{\\theta}\\wedge d\\hat{r} $, with Lagrangian fibers $F^\\vee =T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$, the mirror abelian variety is $E=\\mathbb{C}^n\/(\\mathbb{Z}^n-\\tau^{-1} (\\mathbb{Z}^n))$. \n\\end{example}\n\n\n\\begin{example}\nThe double torus $\\mathbb{T}=(\\mathbb{R}\/\\mathbb{Z})^{4n},\\mathbf{B}+i\\mathbb{\\omega}=\\frac{1}{2}(\\sigma_0+iadr\\wedge d\\theta+ia^{-1}d\\hat{r}\\wedge d\\hat{\\theta}),$\nwith Lagrangian torus fibers \n$\\mathbf{F}=\\{r\\}\\times T_\\theta^n\\times T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$. \nThe mirror is \n\n\\begin{equation*}\n\\mathbb{E}=\\mathbb{C}^{4n}\/(\\mathbb{Z}^{2n}+\\boldsymbol{\\tau}(\\mathbb{Z}^{2n})), \\boldsymbol{\\tau} =\\frac{1}{2}\n\\left(\\begin{array}{cc}\n&\\tau \\ \\ \\ \\mathbbm{1} \\\\\n&-\\mathbbm{1} \\ -\\tau^{-1}\n\\end{array}\\right)\n\\end{equation*}\nUsing coordinates $u=z+\\tau \\hat{z}, v=z-\\tau \\hat{z}$, we have: $\\mathbf{E}\\simeq E\\times E.$\n\\end{example}\n\\end{comment}\n\n\n\\subsection{Floer cohomology and sheaf cohomology}\n\n\\begin{figure}\n\t\\centering \n\t\\includegraphics[scale=0.8]{fig0506.png}\n\t\\caption{The trapezoid bound by $L_0$, $L_z$, $L$ and $L_{z'}$. }\\label{holo}\n\\end{figure}\n\n\\begin{definition}\n\tA linear Lagrangian brane in $(T,\\, B+i\\omega)$ is a linear Lagrangian submanifold $L\\subset (T,\\, \\omega)$ (with a choice of grading and spin structure) together with a complex line bundle over $L$ with a unitary connection $\\nabla$ whose curvature satisfies $F=-B|_L$.\n\\end{definition}\n\\begin{assumption}\n\tAssume there exists a linear Lagrangian brane $L_0$ with trivial local system such that $L_0\\cap F_q$ at only 1 point for each $q\\in Q$.\n\\end{assumption}\nThis $L_0$ will serve as the mirror of structure sheaf on $Y$. In the case of Example \\ref{stdex}, we simply choose $L_0=\\{\\theta=0\\}$. The points of the mirror torus parametrize Lagrangian branes $L_z=(F_q,\\, \\nabla)$ supported on SYZ fibers, and the generators $e_0(z)\\in CF(L_0,\\, L_z)$ are chosen to serve as evaluation map of the structure sheaf at $z$.\nGiven a linear Lagrangian brane $(L,\\, \\nabla)$ in $(T,\\, B+i\\omega)$, assume $L$ is transversal to $F_q$ for each $q\\in Q$. Now we construct its mirror sheaf to be a vector bundle\n\\begin{equation*}\n\\mathcal{L}=\\cup_{z\\in Y}CF^*(L_z,\\, L)\\rightarrow Y.\n\\end{equation*}\nWhen we move $L_z$, let $x(z)$ be a local continuous section of the intersection $L_z\\cap L$, and let $b(z)\\in CF(L_z,\\, L)$ be a rescaling of $x(z)\\in L_z\\cap L$. Then $b(z)$ is a local section of $\\mathcal{L}$.\n\\begin{definition}\n\t$\\mathcal{L}$ admits a natural holomorphic structure such that, if locally for any $z,z'\\in Y$, and N is a trapezoid bound by $L_0$, $L_z$, $L$, $L_{z'}$, see figure \\ref{holo},\n\t\\begin{equation*}\n\te^{2\\pi i\\int_{N}B+i\\omega}hol(\\partial N)\\equiv 1,\n\t\\end{equation*}\n\tthen $b(z)$ is a holomorphic section.\n\tHere $hol(\\partial N)$ is the composition of $e_0$, parallel transport along $L_z$, $b(z)$,parallel transport along $L$, $b(z')^{-1}$, parallel transport along $L_{z'}$, $e_0(z')^{-1}$ and parallel transport along $L_0$. \n\\end{definition}\n\n\\begin{remark}\n\tThese holomorphic sections uniquely determine the holomorphic structure of $\\mathcal{L}$. Here we only consider linear Lagrangians, in which case the Floer differentials are automatically zero, so the corresponding mirror sheaves are holomorphic vector bundles. In general, with Floer differential, one obtains chain complexes of locally free sheaves \\cite{F} \\cite{AbF}. \n\\end{remark}\n\nUnder homological mirror symmetry, we have \n\n\\begin{theorem}\\cite{F} \\cite{KS}\n\t\\begin{equation*}\n\tHF^*(L_1,L_2)\\cong Ext^*(\\mathcal{L}_1,\\mathcal{L}_2)\n\t\\end{equation*}\n\tAnd the following diagram commutes:  \\\\\n\t\n\t\\begin{equation*}\n\t\\begin{tikzcd}\n\t{HF^*(L_2,\\,L_3)\\otimes HF^*(L_1,\\,L_2)} \\arrow[rr, \"\\mu_2\"] \\arrow[d, \"\\cong\"]                 &  & {HF^*(L_1,L_3)} \\arrow[d, \"\\cong\"]    \\\\\n\t{Ext^*(\\mathcal{L}_2,\\,\\mathcal{L}_3)\\otimes Ext^*(\\mathcal{L}_1,\\,\\mathcal{L}_2)} \\arrow[rr] &  & {Ext^*(\\mathcal{L}_1,\\,\\mathcal{L}_3)}\n\t\\end{tikzcd}\n\t\\end{equation*}\n\n\\end{theorem}\n\nThe above theorem is best illustrated by the example of theta functions on the elliptic curve.\n\\begin{figure}\t\\label{line1}\n\t\\centering \n\t\\includegraphics[scale=1]{fig3.pdf}\n\t\t\\caption{Holomorphic triangles computing the Floer product $\\mu^2(e_1,s)\\in HF^*(L_0,L_x)$. }\\label{holo1}\n\\end{figure}\n\n\\begin{example}\n\tLet $T=\\mathbb{R}^{2}\/\\mathbb{Z}^{2}$, $B+i\\omega=\\tau dr\\wedge d\\theta$ be a symplectic two torus with coordinates $r,\\,\\theta$. Let $L_0$ be a horizontal Lagrangian $\\{\\theta=0\\}$ (mirror to the structure sheaf), $L_d$ be a slope $-d$ Lagrangian $\\{\\theta=-dr\\}$ (mirror to a degree d line bundle $\\mathcal{L}^d$), $L_z$ be a vertical Lagrangian with position $x$ and a connection with holonomy $e^{-2\\pi iy}$, $(\\{r=x\\},\\,\\nabla=d+2\\pi iyd\\theta)$, where $z=x+iy$. See figure \\ref{holo1} for the case $d=1$.\n\tThe generators $s_k=(\\frac{k}{d},\\,0)\\in L_0\\cap L_d$ of \n\t$HF(L_0,\\,L_d)$ correspond to the $\\vartheta$-basis of $H^0(E,\\,\\mathcal{L}^d)$\n\t\n\t$$\\vartheta_{d,\\,k} =\\sum_{n\\in \\mathbb{Z}} e^{\\pi\\tau id (n-\\frac{k}{d})^2}e^{2\\pi id(n-\\frac{k}{d}) z}.$$\n\\end{example}\n\n\n\n\\subsection{Coisotropic branes}\n\nKapustin and Orlov introduce the following notion of coisotropic brane, motivated by a string theoretical calculation.\n\n\\begin{definition}\n\tGiven a symplectic manifold $(X^{2n},\\,B+i\\omega)$, a coisotropic brane is a coisotropic submanifold $C^{n+k}$ equiped with a complex line bundle $(\\mathcal{L},\\,\\nabla)$ such that\n\t\\begin{enumerate}\n\t\t\\item   Let $-2\\pi iF$ be the curvature of $(\\mathcal{L},\\,\\nabla)$, then $F+B|_C$ vanishes on $TC_{iso}= ker\\ \\omega|_C$, where $F+B|_C$ is viewed as a bundle morphism $TC\\rightarrow TC^*$. In particular, $F=-B|_C$ along the isotropic leaves (foliated by $ker\\, \\omega|_C=TC_{iso}$).\n\t\t\\item  $\\omega^{-1}(F+B|_C)$ defines a transverse almost complex structure on $C$, i.e. an almost complex structure on $TC_{red}=TC\/TC_{iso}$. Equivalently, $\\omega+(F+B)\\omega^{-1}(F+B)=0$ on $TC$\n\t\\end{enumerate}\n\t\n\\end{definition}\n\\begin{remark}\n\tThe second condition implies that $F+B+i\\omega$ is a holomorphic symplectic form on the space of isotropic leaves, hence forces $k$ to be even. In the case $n=2,\\,k=2$ and $B=0$, we have a space filling coisotropic brane, and this condition is equivalent to $$\\omega\\wedge F=0,\\, \\omega\\wedge\\omega=F\\wedge F.$$\n\tSince $F\\wedge F$ represents an integral cohomology class, coisotropic branes can only arise for some special $\\omega$.\n\\end{remark}\n\\begin{remark}\n\tThe transverse almost complex structure arising from the geometry of the coisotropic brane is always integrable \\cite{KO}.\n\\end{remark}\n\n\\begin{example}\n\tLet $(T=\\mathbf{R}^4\/\\mathbf{Z}^4,\\, \\omega=dr_1\\wedge d\\theta_1 + dr_2\\wedge d\\theta_2)$ be the standard symplectic four torus. \n\tThen $(C=T,\\,\\nabla=d+2\\pi ir_1d\\theta_2-2\\pi ir_2d\\theta_1)$ is a coisotropic brane. And the induced complex stucture has complex coordinates $r_1-ir_2$, $\\theta_1+i\\theta_2$.\n\\end{example}\n\n\\noindent\nKapustin and Orlov have made a proposal for the endomorphisms of a coisotropic brane, namely $End(C)\\simeq H^{0,\\,*}(C)$, where Dolbeault cohomology is considered with respect to the transverse complex structure, but until now it was not understood how to define morphisms between different branes. In the next sections we propose a definition based on a ``doubling\" construction.\n\n\n\n\n\\section{Doubling and lifting}\n\n\\subsection{Construction of the twisted double torus and lift of coisotropic branes   }\n\\subsubsection{Construction for symplectic torus without B-field}\n\nLet $(T=V\/\\Lambda, \\,\\omega)$ be a linear symplectic torus. Let $(T^\\vee=V^\\vee \/\\Lambda^\\vee, \\,-\\omega^{-1})$ be the dual torus with the inverse symplectic form $-\\omega^{-1}(\\alpha, \\,\\beta):=\\alpha(\\omega^{-1}\\beta)$.\n\nWe introduce the following doubling procedure:\n\n\\begin{definition}\\label{Double}\n\tThe twisted doubling torus of $(T, \\,\\omega)$ is a symplectic torus with a B-field\n$$(T\\times T^\\vee, \\  \\frac{1}{2}\\omega\\oplus-\\frac{1}{2}\\omega^{-1}, \\ \\sigma_0=\\sum \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)$$\n\twhere $x_j$ are coordinates on $T$ and $\\hat{x_j}$ are the dual coordinates on $T^\\vee$. $\\sigma_0=\\sum \\frac{1}{2} dx_j\\wedge d\\hat{x}_j$ is the B-field which does not depend on the choice of coordinates.\n\\end{definition}\n\n\\begin{remark}\\label{AS}\n\tThe twisted doubling torus naturally comes with a complex structure $J$ which sends a tangent vector ${v}$ in $T$ to its symplectic dual $v \\lrcorner \\omega$ as a tangent vector of $T^\\vee$. In matrix notation,\n\t\n\t$$ J=\\left( \\begin{array}{ccc}\n\t0 & \\omega^{-1} \\\\\n\t-\\omega & 0\n\t\\end{array}\\right).$$\n\t\n\\end{remark}\n\n\\begin{example}\n\tLet $T=\\mathbb{R}^4\/\\mathbb{Z}^4$, $\\omega=dr_1\\wedge d\\theta_1+dr_2\\wedge d\\theta_2$. Its dual torus is $T^\\vee=\\mathbb{R}^4\/\\mathbb{Z}^4$, $-\\omega^{-1}=d\\hat{r}_1\\wedge d\\hat{\\theta}_1+d\\hat{r}_2\\wedge d\\hat{\\theta}_2$. The twisted double torus is $T\\times T^\\vee=\\mathbb{R}^4\/\\mathbb{Z}^4\\times \\mathbb{R}^4\/\\mathbb{Z}^4$, $ \\frac{1}{2}\\omega\\oplus-\\frac{1}{2}\\omega^{-1}=\\frac{1}{2}(dr_1\\wedge d\\theta_1+dr_2\\wedge d\\theta_2+d\\hat{r}_1\\wedge d\\hat{\\theta}_1+d\\hat{r}_2\\wedge d\\hat{\\theta}_2)$ and $B$ as above.\n\t\n\\end{example}\n\nOne can lift a linear Lagrangian brane or a coisotropic brane of $(T, \\,\\omega)$ to a Lagrangian brane in the doubling torus.\\\\\n\nFor a Lagrangian $(L, \\,\\nabla)$, its lift is $\\boldsymbol{L}=(L\\times L^\\perp, \\,\\nabla\\otimes \\mathds{1})$, where $L^\\perp$ is the conormal of $L$ translated by the holonomy of $\\nabla$.\\\\\n\nFor a coisotropic brane $(C, \\,\\nabla)$, its lift $\\boldsymbol{C}$ is a graph over $T$ in the doubling torus determined by the holonomy of $\\nabla$.\n\n\\begin{definition}\n\tThe lift of a coisotropic (posibly Lagrangian) brane $(C, \\,\\nabla)$ is defined to be\n\t$$\\{(x, \\,\\hat{x})\\in T\\times T^{\\vee}|x\\in C\\  \\text{and} \\\n\t\\langle\\hat{x}, \\,\\gamma_x\\rangle =(-1)^{\\xi(\\gamma_x)}hol_\\nabla(\\gamma_x),\\ \\forall\\ \\gamma_x \\in \\pi_1(C,x)\\},\\quad \\pi_T^*\\nabla$$\n\twhere $\\gamma_x$ is any linear circle passing through $x$ within $C$. And $\\xi:\\,H_1(C)\\rightarrow \\mathbb{Z}\/2$ is such that\n\t\\begin{equation*}\n\t\\xi(\\gamma+\\gamma')-\\xi(\\gamma)-\\xi(\\gamma')=c_1(\\nabla)(\\gamma\\wedge \\gamma') \\ mod \\ 2\n\t\\end{equation*}\n\\end{definition}\n\n\\begin{remark}\n\t$\\xi$ is introduced to make sure $(-1)^{\\xi(\\cdot)}hol_\\nabla(\\cdot)$ is a homomorphism: $\\pi_1(C, \\,x)\\rightarrow U(1)$.\n\t$\\xi$ has a similar role to the spin structure.\n\tThe space of different choices of $\\xi$ is an affine space over $H^1(C; \\,\\mathbb{Z}\/2)$.\n\\end{remark}\n\n\\begin{proposition}\\label{LiftisLag1}\n\tThe lift of a coisotropic (possibly Lagrangian) brane is a Lagrangian brane in the doubling torus. And it is also a $J$-complex (see remark \\ref{AS}) submanifold of $T\\times T^\\vee$.\n\\end{proposition}\n\\begin{proof}\n\tLet $(C, \\,\\nabla)$ be a coisotropic brane, $\\boldsymbol{C}$ be its lift. By linearizing the definition of the lift, we have\n\t$$T\\boldsymbol{C}=\\{u+Fu+\\omega v| u\\in TC, v \\in TC_{iso}\\}\n\t=\\left(\\begin{array}{ll}\n\t1\\ &0 \\\\\n\tF\\ &1\n\t\\end{array}\\right)\n\t\\left(\\begin{array}{ll}\n\t& TC \\\\\n\t& \\omega\\ TC_{iso}\n\t\\end{array}\\right), $$\n\twhere $F$ denote the curvature of $\\nabla$.\n\tGiven two tangent vectors $u_1+Fu_1+\\omega v_1$, $u_2+Fu_2+\\omega v_2$ of $T\\boldsymbol{C}$,\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t&\\omega\\oplus-\\omega^{-1}(u_1+Fu_1+\\omega v_1,\\ u_2+Fu_2+\\omega v_2) \\\\\n\t=&\\omega(u_1,\\,u_2)-\\omega^{-1}(Fu_1+\\omega v_1,\\,Fu_2+\\omega v_2) \\\\\t=&\\omega(u_1,\\,u_2)-\\omega^{-1}(Fu_1,\\,Fu_2)-\\omega^{-1}(\\omega v_1,\\,Fu_2)-\\omega^{-1}(Fu_1,\\,\\omega v_2)-\\omega^{-1}(\\omega v_1,\\,\\omega v_2)\\\\\n\t=&(\\omega+F\\omega^{-1}F)(u_1,\\,u_2)+F(v_1,\\,u_2)+F(u_1,\\,v_2)+\\omega(v_1,\\,v_2)\\\\\n\t=&0\n\t\\end{aligned}\n\t\\end{equation*}\n\tThe last line equals $0$ because $\\omega+F\\omega^{-1}F=0$ and $F$ is zero on $TC_{iso}$. So the lift $\\boldsymbol{C}$ is Lagrangian. \\\\\n\t\t\n\tTo prove that $\\boldsymbol{C}$ is a $J$-complex submanifold of $T\\times T^\\vee$,\n\twe observe that, for $u \\in TC$,\n\t$\\omega u +F\\omega^{-1}Fu $ vanishes on $TC$,\n\thence equals $\\omega v'$ for some $v'\\in TC_{iso}$. Therefore,\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t&J(u+Fu+\\omega v)=\\omega^{-1}Fu+v-\\omega u \\\\\n\t=&\\omega^{-1}Fu+v+F\\omega^{-1}F u -\\omega v'\\\\\n\t=&(\\omega^{-1}Fu+v)+F(\\omega^{-1}Fu+v) -\\omega v' \\in T\\boldsymbol{C}.\n\t\\end{aligned}\n\t\\end{equation*}\n\t\n\\end{proof}\n\n\\begin{remark}\n\tThere is an ambiguity when writing $Fu$ as a vector of $TT^\\vee$, in fact $Fu\\in T^*C$ whose lift to $T^*T$ is only unique up to a vector in $\\omega TC_{iso}$. In the above calculation, a lift of $Fu$ to $T^*T$ is chosen. Another way is to write $T\\boldsymbol{C}=\\{u+f|<f,\\,v>=F(u,\\,v)\\ \\forall v\\in TC\\}$.\n\\end{remark}\n\n\\subsubsection{Construction for symplectic torus with B-field}\nNow we deal with the more general situation when the symplectic torus starts with a B-field. Let $(T=V\/\\Lambda,\\, \\omega,\\, B)$ be a linear symplectic torus equipped with a B-field $B\\in H^2(T;\\,\\mathbb{R})$.\n\\begin{assumption}\\label{assump}\n\t$Id+(\\omega^{-1}B)^2$ is invertible.\n\\end{assumption}\nThe dual torus is  $(T^\\vee=V^\\vee \/\\Lambda^\\vee,\\ -(\\omega+B\\omega^{-1}B)^{-1},\\  (\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1})$.\n\n\\begin{remark}\n\tThese formular for the symplectic form and B-field on the dual torus are rather confusing at first sight. In fact, they are the imaginary part and real part of $(B+iw)^{-1}$. Indeed,\n\\begin{equation*}\\begin{aligned}\n(B+i\\omega)^{-1}&=-i(Id-i\\omega^{-1}B)^{-1}\\omega^{-1}=-i\\sum_{k\\geq 0} (i\\omega^{-1} B)^k \\omega^{-1}\\\\\n&=-i(\\omega+B\\omega^{-1}B)^{-1}+(\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1}. \\end{aligned}\\end{equation*}\n\\end{remark}\n\n\\begin{definition}\n\tThe twisted doubling torus of $(T,\\,\\omega,\\, B)$ is a symplectic torus with a B-field\n\t\\begin{equation*}\n\t(T\\times T^\\vee,\\quad  \\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t \\sigma_0=\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)\t\n\t\\end{equation*}\n\twhere $x_j$ are coordinates on $T$ and $\\hat{x}_j$ are the dual coordinates on $T^\\vee$. $\\sigma_0=\\sum \\frac{1}{2} dx_j\\wedge d\\hat{x}_j$ is the B-field which does not depend on the choice of coordinates.\n\\end{definition}\n\n\\begin{remark}\n\tIf we start with the dual torus  $(T^\\vee=V^\\vee \/\\Lambda^\\vee,\\,-(\\omega+B\\omega^{-1}B)^{-1},\\, (\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1})$, then the twisted double torus of the dual torus is\n\t\t\\begin{equation*}\n\t\t(T^\\vee\\times T,\\quad \\frac{1}{2}\n\t\t\\left(\\begin{array}{cc}\n\t\t-\\omega^{-1} & -\\omega^{-1}B\\\\\n\t\t B\\omega^{-1} & \\omega+B\\omega^{-1}B\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t-\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j).\t\n\t\t\\end{equation*}\n\tThis is different from the twisted double torus of $(T,\\omega, B)$ by a B-field twist. See Chapter 6 for more discussion.\n\\end{remark}\n\n\\begin{remark}\\label{AS2}\n\tThe twisted doubling torus still comes with a complex structure $J$ which is twisted by the B-field. In matrix notation,\n\t\n\t\\begin{equation*}\t\n\t J=\\left( \\begin{array}{cc}\n\t \\omega^{-1}B & \\omega^{-1} \\\\\n\t -\\omega-B\\omega^{-1}B & -B\\omega^{-1}\n\t\\end{array}\\right)\n\t=\\left( \\begin{array}{cc}\n\t 1 & 0 \\\\\n\t -B & 1\n\t\\end{array}\\right)\n\t\\left( \\begin{array}{cc}\n\t 0 & \\omega^{-1} \\\\\n\t -\\omega & 0\n\t\\end{array}\\right)\n\t\\left( \\begin{array}{cc}\n\t 1 & 0 \\\\\n\t B & 1\n\t\\end{array}\\right).\n\t\\end{equation*}\n\t\n\\end{remark}\n\nThe lift procedures are the same as in the case without B-field, the definition is copied:\n\\begin{definition}\n\tThe lift of a coisotropic (possibly Lagrangian) brane $(C,\\,\\nabla)$ is defined to be\n\t\\begin{equation*}\n\\{(x,\\,\\hat{x})\\in T\\times T^{\\vee}|x\\in C \\text{ and }\\ \\langle\\hat{x},\\,\\gamma_x\\rangle =(-1)^{\\xi(\\gamma_x)}hol_\\nabla(\\gamma_x),\\   \\forall\\  \\gamma_x \\in \\pi_1(C,\\,x) \\},\\quad \\pi_T^*\\nabla\n    \\end{equation*}\n\twhere $\\gamma_x$ is any linear circle passing through $x$ within $C$. And $\\xi:\\,H_1(C)\\rightarrow \\mathbb{Z}\/2$ such that\n\t\\begin{equation*} \\xi(\\gamma+\\gamma')-\\xi(\\gamma)-\\xi(\\gamma')=c_1(\\nabla)(\\gamma\\wedge \\gamma') \\ mod \\ 2\n\t\\end{equation*}\n\\end{definition}\nSimilarly to the case without B-field, the lifts are Lagrangian and complex submanifolds of $T\\times T^\\vee $:\n\n\\begin{proposition}\\label{LiftisLag2}\n\tThe lift of a coisotropic (possibly Lagrangian) brane is a Lagrangian brane in the doubling torus. And it is also a $J$-complex (see remark \\ref{AS2}) submanifold of $T\\times T^\\vee$.\n\\end{proposition}\n\\begin{proof}\nNote that\n$$\n\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right)\n=\\left(\\begin{array}{cc}\n\t1 & -B \\\\\n\t0 & 1\n\t\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\t\\omega & 0 \\\\\n\t0 & -\\omega^{-1}\n\t\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\t1 & 0 \\\\\n\tB & 1\n\t\\end{array}\\right).\n$$\nThe proof goes over the same as for Proposition \\ref{LiftisLag1} if we replace $F$ by $B+F$.\n\\end{proof}\n\n\n\\begin{remark}\nRecall that a space filling coisotropic brane $(C,\\,\\nabla)$ comes with a complex structure $\\omega^{-1}(F+B|_C)$. This complex structure coincides the complex structure \\ref{AS2} on the lift $\\boldsymbol{C}$ under the isomorphism $\\Pi_{T}:\\,\\boldsymbol{C} \\rightarrow C$, where $\\Pi_{T}:\\,T\\times T^\\vee \\rightarrow T$ is the projection to the $T$ factor.\n\\end{remark}\n\n\\subsection{Mirror of the twisted double torus}\n\nWe recall that the mirror of a symplectic torus with B-field $(T,\\,\\omega,\\,B)$ equipped with a SYZ fibration $F\\rightarrow T \\rightarrow Q$ is as follows:\n\\begin{equation}\nH^1(F;\\,\\mathbb{C})\/H^1(F;\\,\\mathbb{Z})+(B+i\\omega) H_1(Q;\\,\\mathbb{Z}).\n\\end{equation}\n\n\\begin{example}\\label{Tndouble}\n\tLet $(T,\\,B+i\\omega)=((\\mathbb{R}\/\\mathbb{Z})^{n}_r\\times (\\mathbb{R}\/\\mathbb{Z})^{n}_\\theta ,\\,\\tau dr \\wedge d\\theta)$ with SYZ fibers $F=T_\\theta \\times \\{r\\}$. The dual torus is $(T^\\vee,\\,(B+i\\omega)^{-1})=((\\mathbb{R}\/\\mathbb{Z})^{n}_{\\hat{\\theta}} \\times (\\mathbb{R}\/\\mathbb{Z})_{\\hat{r}}^{n} ,\\,\\tau^{-1} d\\hat{\\theta}\\wedge d\\hat{r})$ with dual SYZ fibers $F^\\vee =T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$.\n\tThe mirror complex torus for the dual torus is\n\t$$E_{-\\tau^{-1}}=\\mathbb{C}^n\/(\\mathbb{Z}^n-(\\tau^{T})^{-1} (\\mathbb{Z}^n)).$$\n\tThe twisted double torus is\n\t\\begin{equation*}\\begin{aligned}\n\t\\Big(T\\times T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}\\times (\\mathbb{R}\/\\mathbb{Z})^{2n},\n\\quad  \\boldsymbol{B}+i\\boldsymbol{\\omega}&=\n\t\\frac{i}{2}(Im\\tau +Re\\tau (Im\\tau)^{-1}Re\\tau)dr\\wedge d\\theta \\\\\n\t&-\\frac{i}{2}\\tau(Im\\tau)^{-1}dr\\wedge d\\hat{r}\\\\\n\t&+\\frac{i}{2}(Im\\tau)^{-1}\\tau d\\hat{\\theta}\\wedge d\\theta \\\\\n\t&-\\frac{i}{2} (Im\\tau)^{-1} d\\hat{\\theta}\\wedge d\\hat{r}\\Big).\n\t\\end{aligned}\\end{equation*}\n\twith SYZ fibers\n\t$\\boldsymbol{F}=\\{r\\}\\times T_\\theta^n\\times\\{\\hat{\\theta}\\}\\times T_{\\hat{r}}^n$.\n\tThe mirror is\n\t\n\t\\begin{equation*} \\mathbb{E}=\\mathbb{C}^{4n}\/(\\mathbb{Z}^{2n}+\\boldsymbol{\\tau}^{T}(\\mathbb{Z}^{2n})), \\quad\\boldsymbol{\\tau}^{T} =\\frac{i}{2}\n\t\\left(\\begin{array}{cc}\n\tIm\\tau^T+Re\\tau^T (Im\\tau^T)^{-1}Re\\tau^T & \\tau^{T}(Im\\tau^T)^{-1} \\\\\n\t-(Im\\tau^{T})^{-1}\\tau^T & -(Im\\tau^T)^{-1}\n\t\\end{array}\\right).\n\t\\end{equation*}\n\tEquipping SYZ fibers\n\t$\\boldsymbol{F}=\\{r\\}\\times T_\\theta^n\\times\\{\\hat{\\theta}\\}\\times T_{\\hat{r}}^n$\n\twith connection $\\nabla=d+2\\pi i (\\phi d\\theta +\\kappa d\\hat{r})$, local coordinates on the mirror are\n\t\\begin{equation*}\\begin{aligned}\n\t&\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\theta})\n\t=\\frac{i}{2}(Im\\tau^T+Re\\tau^T (Im\\tau^T)^{-1}Re\\tau^T)r\n\t+\\frac{i}{2}\\tau^{T}(Im\\tau^T)^{-1}\\hat{\\theta}-\\phi \\\\\n\t&\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\hat{r}})\n\t=-\\frac{i}{2}(Im\\tau^{T})^{-1}\\tau^T r\n\t-\\frac{i}{2}(Im\\tau^T)^{-1}\\hat{\\theta} -\\kappa.\n\t\\end{aligned}\\end{equation*}\n\tUsing alternative coordinates\n\t\\begin{equation*}\\begin{aligned}\n\t\t&u=\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\theta})\n\t\t+\\tau^T \\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\hat{r}})\n\t\t=\\tau^T r-\\tau^T \\kappa -\\phi \\\\\n\t\t&v=\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\theta})\n\t\t+\\bar{\\tau}^T \\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\hat{r}})\n\t\t=-\\hat{\\theta}-\\phi-\\bar{\\tau}^T\\kappa,\n\t\\end{aligned}\\end{equation*}\n\twe have: $\\boldsymbol{E}\\simeq E_\\tau \\times E_{-\\bar{\\tau}}.$\n\t\n\\end{example}\n\\begin{comment}\n\\begin{example}\n\tThe dual torus $T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}, -\\omega^{-1}=-a^{-1}d\\hat{\\theta}\\wedge d\\hat{r} $, with Lagrangian fibers $F^\\vee =T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$, the mirror abelian variety is $E=\\mathbb{C}^n\/(\\mathbb{Z}^n-\\tau^{-1} (\\mathbb{Z}^n))$.\n\\end{example}\n\\begin{example}\n\tThe twisted double torus $\\mathbb{T}=(\\mathbb{R}\/\\mathbb{Z})^{4n},\\mathbf{B}+i\\mathbb{\\omega}=\\frac{1}{2}(\\sigma_0+iadr\\wedge d\\theta+ia^{-1}d\\hat{r}\\wedge d\\hat{\\theta}),$\n\twith Lagrangian torus fibers\n\t$\\mathbf{F}=\\{r\\}\\times T_\\theta^n\\times T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$.\n\tThe mirror is\n\t\n\t\\begin{equation*}\n\t\\mathbb{E}=\\mathbb{C}^{4n}\/(\\mathbb{Z}^{2n}+\\boldsymbol{\\tau}(\\mathbb{Z}^{2n})), \\boldsymbol{\\tau} =\\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t&\\tau \\ \\ \\ \\mathbbm{1} \\\\\n\t&-\\mathbbm{1} \\ -\\tau^{-1}\n\t\\end{array}\\right)\n\t\\end{equation*}\n\tUsing coordinates $u=z+\\tau \\hat{z}, v=z-\\tau \\hat{z}$, we have: $\\mathbf{E}\\simeq E\\times E.$\n\\end{example}\n\\end{comment}\n\n\\begin{remark}\n\tThe mirror of the twisted double torus constructed above turns out to be isomorphic to the product of the original mirror with its complex conjugate $E\\times \\bar{E}$. This twisted double torus has the property that, even if a sheaf $\\mathcal{E}\\in Coh(E)$ corresponds to a coisotropic brane in $T$, a closely related sheaf on $E\\times \\bar{E}$ corresponds to a Lagrangian (which is the lift of the coisotropic brane) in $\\boldsymbol{T}$.\n\\end{remark}\nSYZ fibers $F\\subset T$ lift to fibers $\\boldsymbol{F}\\subset \\boldsymbol{T}$ which correspond to points in $E\\times 0$, i.e. $v=0$.\nIn example \\ref{Tndouble}, $(F=\\{r\\}\\times T^n_{\\theta},\\ \\nabla =d+2\\pi i\\phi d\\theta) $ is lifted to\n$(\\boldsymbol{F}=\\{r\\}\\times T^n_{\\theta}\\times\\{\\hat{\\theta}=-\\phi\\}\\times T^n_{\\hat{r}},\\ \\nabla=d+2\\pi i\\phi d\\theta)$ which corresponds to the point $u=\\tau^Tr-\\phi$, $v=0$.\\\\\nSimilarly, if a Lagrangian brane $L$ is mirror to a coherent sheaf $\\mathcal{E}$, then the lift $\\boldsymbol{L}$ is mirror to $\\mathcal{E}\\boxtimes \\mathcal{E}_0$ on $E\\times \\bar{E}$, where $\\mathcal{E}_0$ is a particular sheaf on $\\bar{E}$.\n\n\n\\section{Floer Theory}\n\n\nThe Floer theory of $(T,B+i\\omega)$ and its twisted double torus\n$$(T\\times T^{\\vee},\\quad \\frac{1}{2}\\sigma_0+\\frac{i}{2}\n\\left(\\begin{array}{cc}\n\\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n-\\omega^{-1}B & -\\omega^{-1}\n\\end{array}\\right))$$\nare deeply related to each other. In fact, the Floer cohomology between two Lagrangian brane $HF^*(L,\\; L')$ in $(T,\\; B+i\\omega)$ can be identified with a subspace of the Floer cohomology of the lifts $HF^*(\\boldsymbol{L},\\; \\boldsymbol{L}')$.\n\n\n\n\\subsection{Floer theory in 2-tori and their twisted doubles}\nLet $(T=\\mathbb{R}^2\/\\mathbb{Z}^2,\\  B+i\\omega=(b+ia)dr\\wedge d\\theta)$ equipped with SYZ fibration projecting to $r$ coordinate. \\\\\nThe coordinate on the mirror manifold is\n$$z_{\\theta}=e^{2\\pi i\\int{B+i\\omega} }hol_\\nabla(S^1_{\\theta})\n=e^{2\\pi i\\tau r}e^{-2\\pi i\\phi}=e^{2\\pi i(\\tau r-\\phi)}.$$\nThus the mirror is the elliptic curve $E=\\mathbb{C}\/\\mathbb{Z}+\\tau \\mathbb{Z}$, where $\\tau=b+ia$. \\\\\n\n\\subsubsection{Theta functions on Elliptic curves}\nTheta functions are holomorphic sections of holomorphic line bundles on elliptic curves. They can be constructed with the help of a holomorphic connection by periodizing a holomorphic section on the universal cover of the elliptic curve. If we begin with two gauge equivalent holomorphic connections on a degree $1$ line bundle, we get a priori different holomorhpic sections of the line bundle. The dimension of the space of holomorphic sections, which equals $1$, forces the two sections to be the same up to a constant. We can establish some magic formula for theta functions using this approach.\n\\begin{example}\t\n\tConsider the degree $1$ holomorphic line bundle on the elliptic curve $E_\\tau=\\mathbb{C}\/(\\mathbb{Z}+\\tau\\mathbb{Z})$ where $\\tau =b+ia$ with holomorphic connection $d+\\frac{2\\pi i}{a}ydx$. The transition function of the line bundle is given by\n\t             $$ s(z+1)=s(z),\\quad s(z+n\\tau)=e^{-\\pi in^2b}e^{-2\\pi inx}s(z).$$\n\tBy starting with the holomorphic section $e^{-\\frac{\\pi}{a}y^2}$ on the universal cover of the torus and by periodizing it, we get a section of the line bundle given by\n \t        $$s=\\sum_n e^{-\\frac{\\pi}{a}(y+na)^2}e^{2\\pi inx}e^{\\pi in^2b}=\\sum_n e^{\\pi in^2\\tau}e^{2\\pi in(x+iy)}e^{-\\frac{\\pi}{a}y^2}.$$\n\\end{example}\n\n\\begin{example}\n\tConsider a gauge equivalent connection $d+\\frac{\\pi i}{a}(ydx-xdy)$. The transition function of the line bundle is given by\n\t$$ s(z+m+n\\tau)=(-1)^{mn}e^{\\frac{\\pi i}{a}my}e^{\\frac{\\pi ib}{a}ny}e^{-\\pi inx}s(z).$$\n\tBy periodizing the holomorphic section $e^{-\\frac{\\pi}{2a}(x^2+y^2)}$ we get a holomorphic section\n\t\\[\n\t\\begin{aligned}\n\t&\\sum_{m,n} (-1)^{mn}e^{\\frac{\\pi i}{a}my}e^{\\frac{\\pi ib}{a}ny}e^{-\\pi inx}e^{-\\frac{\\pi}{2a}((x-m-nb)^2+(y-na)^2)}\\\\\n\t=&\\sum_{m,n}(-1)^{mn}e^{-\\frac{\\pi}{2a}(m+n\\tau)(m+n\\bar{\\tau})}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})(x+iy)}e^{-\\frac{\\pi}{2a}(x^2+y^2)}.\n\t\\end{aligned}\n\t\\]\n\\end{example}\n\n\\begin{proposition}\n\tThe holomorphic sections from the above examples are equal to each other up to a factor.\n\t      \\begin{equation}\\label{thetaformula1}\n\t\\begin{aligned}\n\t&\\sum_{m,n}(-1)^{mn}e^{-\\frac{\\pi}{2a}(m+n\\tau)(m+n\\bar{\\tau})}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})z}e^{-\\frac{\\pi}{2a}z^2}\\\\\n\t=&\\sum_{m,n}e^{-\\frac{\\pi}{2a}(z+m)^2}e^{-\\frac{\\pi}{a}n\\bar{\\tau}(z+m)}e^{-\\frac{\\pi}{2a}n^2\\tau\\bar{\\tau}} \\\\\n\t=&\\sqrt{2a}\\sum_{l}e^{-\\pi i\\bar{\\tau}l^2}\\sum_{k}e^{\\pi i\\tau k^2}e^{2\\pi ikz}.\n\t\\end{aligned}\n\t       \\end{equation}\n\t\n\\end{proposition}\n\\begin{remark}\n\tThe right hand side of equation \\eqref{thetaformula1} is (up to a constant factor) the standard formula for the theta function for a degree $1$ line bundle, while the left hand side natrually arises in the Floer products on the twisted double torus. And this formula is the key to relate Floer theory of $(T,\\; B+i\\omega)$ and Floer theory of its twisted doubling torus. We will provide an aternative proof of a generalization of this formula \\eqref{thetaformula2} using Fourier series.\n\\end{remark}\n\n\\subsubsection{Floer products on the 2-torus and on its twisted doubling torus}\n\\begin{example}\n\n\tLet $T=\\mathbb{R}^{2}\/\\mathbb{Z}^{2},\\; B+i\\omega=\\tau dr\\wedge d\\theta$ be a symplectic two torus with coordinates $r,\\theta$. Let $L_0$ be a horizontal Lagrangian $\\{\\theta=0\\}$ (mirror to the structure sheaf), $L_1$ be a slope $-1$ Lagrangian $\\{\\theta=-r\\}$ (mirror to a degree 1 line bundle $\\mathcal{L}^1$), $L_z$ be a vertical Lagrangian with position $r$ and a connection with holonomy $e^{-2\\pi i\\phi}$, $(L_z=\\{r\\}\\times S^1_\\theta,\\;  \\nabla=d+2\\pi i\\phi d\\theta)$, where $z=\\tau r-\\phi$.\n\tThe generator $s=(0,\\;0)\\in L_0\\cap L_1$ of\n\t$HF(L_0,\\; L_1)$ correspond to the $\\vartheta$-function in $H^0(E,\\; \\mathcal{L}^1)$\n\t\t\n\t\t$$\\vartheta =\\sum_{n\\in \\mathbb{Z}} e^{\\pi\\tau i n^2}e^{2\\pi in z}.$$\n\n\tThe doubling torus is given by\n\t    \\begin{equation*}\n\t     (T\\times T^{\\vee},\\; \\frac{1}{2}(\\sigma_0+i\\Omega)=\\frac{i}{2a}(\\tau\\bar{\\tau}dr\\wedge d\\theta+\\tau (d\\hat{r}\\wedge dr+d\\hat{\\theta}\\wedge d\\theta)+d\\hat{r}\\wedge d\\hat{\\theta}))\n\t     \\end{equation*}\n\t     Complex coordinates on the mirror are given by\n\t     \\begin{equation*}\\begin{aligned}\n\t    z_{\\frac{\\partial}{\\partial\\theta}}&=e^{2\\pi i \\frac{i}{2a}(\\tau\\bar{\\tau}r+\\tau \\hat{\\theta})}e^{-2\\pi i \\phi};\\\\\n\t    z_{\\frac{\\partial}{\\partial\\hat{r}}}&=e^{2\\pi i \\frac{-i}{2a}(\\tau r+\\hat{\\theta})}e^{-2\\pi i \\kappa};\n\t    \\end{aligned}\\end{equation*}\n\t    Note that\n\t    \\begin{equation*}\\begin{aligned}\n\t    z_{\\frac{\\partial}{\\partial\\theta}}(z_{\\frac{\\partial}{\\partial\\hat{r}}})^{\\tau}&=e^{2\\pi i(\\tau r-\\tau \\kappa -\\phi)};\\\\\n\t    z_{\\frac{\\partial}{\\partial\\theta}}(z_{\\frac{\\partial}{\\partial\\hat{r}}})^{\\bar{\\tau}}&=e^{2\\pi i(-\\hat{\\theta}-\\phi-\\bar{\\tau}\\kappa)}.\n\t    \\end{aligned}\\end{equation*}\n\t\n\tLet\n\t$$u=\\tau r-\\tau \\kappa -\\phi,\\quad v=-\\hat{\\theta}-\\phi -\\bar{\\tau}\\kappa$$\n\t be the new coordinates of the mirror manifold. We can see that the mirror manifold is isomorphic to $E_{\\tau}\\times E_{-\\bar{\\tau}}$.\n\t\n\tThe lifts $\\boldsymbol{L}_0=\\{\\theta=0,\\; \\hat{r}=0\\}$ and $\\boldsymbol{L}_1=\\{\\theta=-r,\\; \\hat{r}=\\hat{\\theta}\\}$ of $L_0$ and $L_1$ to the twisted doubling torus intersect in one point, which corresponds to a section of a line bundle on the mirror manifold. Considering the intersection with SYZ fibers $(\\boldsymbol{F}=\\{r\\}\\times S_\\theta^1\\times S_{\\hat{r}}^1\\times\\{\\hat{\\theta}\\},\\; \\nabla=d+2\\pi i \\phi d\\theta+2\\pi i \\kappa d\\hat{r})$, the Flor product $CF(\\boldsymbol{L}_0,\\;\\boldsymbol{L}_1)\\otimes CF(\\boldsymbol{L}_1,\\;\\boldsymbol{F})\\rightarrow CF(\\boldsymbol{L}_0,\\;\\boldsymbol{F})$ is given by the following expression, summing the contribution of holomorphic triangles of edge length $(n+r)$ in $T$ and $(m+\\hat{\\theta})$ in $T^\\vee$:\n\t\\begin{equation*}\\begin{aligned}\ns&=\\sum_{m,n}e^{2\\pi i (\\frac{i}{4a}\\tau\\bar{\\tau}(n+r)^2+\\frac{i}{4a}(m+\\hat{\\theta})^2+\\frac{i}{2a}\\tau(n+r)(m+\\hat{\\theta}))}e^{-2\\pi i(n+r)\\phi}e^{2\\pi i(m+\\hat{\\theta})\\kappa}\\\\\n\t&=\\sum_{m,n}e^{-\\frac{\\pi}{2a}(n^2\\tau\\bar{\\tau}+m^2+2\\tau mn)}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})u}e^{\\frac{\\pi}{a}(m+n\\tau)v}e^{-\\frac{\\pi}{2a}(\\tau\\bar{\\tau}r^2+\\hat{\\theta}^2+2\\tau r\\hat{\\theta})}e^{-2\\pi ir\\phi}e^{2\\pi i\\hat{\\theta}\\kappa}\n\\end{aligned}\\end{equation*}\n\tClaim:\n\t\\begin{equation}\\label{thetaformula2}\n\t\\begin{aligned}\n\t\t&\\sum_{m,n}e^{-\\frac{\\pi}{2a}(n^2\\tau\\bar{\\tau}+m^2+2\\tau mn)}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})u}e^{\\frac{\\pi}{a}(m+n\\tau)v}e^{-\\frac{\\pi}{2a}u^2}e^{\\frac{\\pi}{a}uv} e^{-\\frac{\\pi}{2a}v^2} \\\\\n\t\t=&\\sqrt{2a}\\sum_k e^{\\pi i\\tau k^2}e^{2\\pi iku}\\sum_l e^{-\\pi i\\bar{\\tau}l^2}e^{2\\pi ilv}\n\t\\end{aligned}\n\t\\end{equation}\n\n\t\\begin{proof}\n\t\t\n\t\tLet $\\displaystyle f(u)=\\sum_m e^{-\\frac{\\pi}{2a}(u+m)^2}e^{-\\frac{\\pi}{a}(n\\bar{\\tau}-v)(u+m)}$. \\\\\n\t\tThen $f(u+1)=f(u)$, hence $f$ is equal to its Fourier series\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\tf(u)&=\\sum_k\\int_{0}^{1}f(z)e^{-2\\pi ikz}dz\\ e^{2\\pi iku} \\\\\n\t\t&=\\sum_k \\sum_m\\int_{0}^{1}e^{-\\frac{\\pi}{2a}((z+m)^2+(2n\\bar{\\tau}-2v)(z+m))}e^{-2\\pi ikz}dz\\ e^{2\\pi iku}  \\\\\n\t\t&=\\sum_k e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2}\\sum_m\\int_{0}^{1}e^{-\\frac{\\pi}{2a}((z+m)^2+(2n\\bar{\\tau}-2v+4ika)(z+m)+(n\\bar{\\tau}-v+2ika)^2)}dz\\ e^{2\\pi ik u} \\\\\n\t\t&= \\sum_k e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2}\\int_{-\\infty}^{\\infty}e^{-\\frac{\\pi}{2a}((z+n\\bar{\\tau}-v+2ika)^2)}dz\\ e^{2\\pi ik u} \\\\\n\t\t&=\\sum_k \\sqrt{2a}\\ e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2} e^{2\\pi ik u}. \\\\\n\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tThen\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\tLHS &=\\sum_{m,n}e^{-\\frac{\\pi}{2a}(n^2\\tau\\bar{\\tau}+m^2+2\\tau mn)}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})u}e^{\\frac{\\pi}{a}(m+n\\tau)v}e^{-\\frac{\\pi}{2a}u^2}e^{\\frac{\\pi}{a}uv} \\\\\n\t\t&=\\sum_n f(u)e^{-\\frac{\\pi}{2a}\\tau\\bar{\\tau}n^2}e^{\\frac{\\pi}{a}n\\tau v} \\\\\n\t\t&=\\sum_n \\sum_k \\sqrt{2a}\\ e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2} e^{2\\pi ik u} e^{-\\frac{\\pi}{2a}\\tau\\bar{\\tau}n^2}e^{\\frac{\\pi}{a}n\\tau v} \\\\\n\t\t&=\\sum_k \\sum_n \\sqrt{2a}\\ e^{-\\pi i\\bar{\\tau}(n-k)^2}e^{\\pi i\\tau k^2}e^{2\\pi i(n-k)v}e^{2\\pi iku} \\\\\n\t\t&=\\sqrt{2a} \\sum_k e^{\\pi i\\tau k^2}e^{2\\pi iku}\\sum_l e^{-\\pi i\\bar{\\tau}l^2}e^{2\\pi ilv}\n\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\t\n\t\\end{proof}\n\t\n\\end{example}\n\n  This calculation shows that, up to a suitable rescaling (due to the discrepancy between the trivialization given by Floer complex and the usual holomorphic trivialization), the generator of $CF(\\boldsymbol{L}_0,\\,\\boldsymbol{L}_1)$ corresponds to the section $\\theta_\\tau(u)\\theta_{-\\bar{\\tau}}(v)$ of $\\mathcal{L}\\boxtimes\\mathcal{L}$ on $E_\\tau\\times E_{-\\bar{\\tau}}$.\n  \n  In general, let $L_0$ be a horizontal Lagrangian $\\{\\theta=0\\}$ (mirror to the structure sheaf), $L_d$ be a slope $-d$ Lagrangian $\\{\\theta=-dr\\}$ (mirror to the degree d line bundle $\\mathcal{L}^d$), $L_z$ be a vertical Lagrangian with position $r$ and a connection $\\nabla=d+2\\pi i\\phi d\\theta$, where $z=\\tau r-\\phi$.\nThe generators $s_k=(\\frac{k}{d},\\; 0)\\in L_0\\cap L_d$ of\n$HF(L_0,\\; L_d)$ correspond to the $\\vartheta$-basis of $H^0(E,\\; \\mathcal{L}^d)$\n\n$$\\vartheta_{k\/d} =\\sum_{n\\in \\mathbb{Z}} e^{\\pi\\tau id (n+\\frac{k}{d})^2}e^{2\\pi id(n+\\frac{k}{d}) z}.$$\n\nOn the doubling torus\n$(T\\times T^\\vee,\\;  \\frac{i}{2a}(\\tau\\bar{\\tau}dr\\wedge d\\theta+\\tau (d\\hat{r}\\wedge dr+d\\hat{\\theta}\\wedge d\\theta)+d\\hat{r}\\wedge d\\hat{\\theta}))$, we get lifts of the above Lagrangian branes:\n\\begin{equation*}\\begin{aligned}\n\\boldsymbol{L}_0&=\\{\\theta=0,\\  \\hat{r}=0\\}\\\\\n\\boldsymbol{L}_d&=\\{\\theta=-dr,\\  \\hat{r}=d\\hat{\\theta}\\}\\\\\n\\boldsymbol{L}_z&=(\\{r\\}\\times S^1_\\theta\\times S^1_{\\hat{r}}\\times\\{ \\hat{\\theta}=-\\phi\\},\\\n\\nabla=d+2\\pi i\\phi d\\theta).\n\\end{aligned}\\end{equation*}\n\nAn argument similar to that given above for the case $d=1$ shows that, in $T\\times T^\\vee$, the generator $s_j\\otimes \\hat{s}_k \\in CF(\\boldsymbol{L}_0,\\; \\boldsymbol{L}_d)$ given by the point of $\\boldsymbol{L}_0\\cap\\boldsymbol{L}_d$ with coordinate $r=j\/d$ and $\\hat{\\theta}=k\/d$ corresponds to\n\\begin{equation*}\n\\sum_{l\\in \\mathbb{Z}\/d}e^{2\\pi ikl\/d}\\vartheta_{(j-l)\/d}(u)\\vartheta_{l\/d}(v)\\in H^0(E_\\tau\\times E_{-\\bar{\\tau}},\\ \\mathcal{L}^{d}\\boxtimes \\mathcal{L}^{d}).\n\\end{equation*}\n\n\\subsection{The general case for $T^{2n}$}\n\nLet $T=(\\mathbb{R}\/\\mathbb{Z})^{2n}$, $ B+i\\omega=\\tau dr\\wedge d\\theta=\\sum \\tau_{jk}dr_j\\wedge d\\theta_k \\ (\\tau \\in M_{n\\times n}(\\mathbb{C}))$, with Lagrangian fibers $F=\\{r\\}\\times T_\\theta ^n$ and base $Q$, the mirror complex torus is\n$$E\\cong H^1(F;\\,\\mathbb{C})\/(H^1(F;\\,\\mathbb{Z})+(B+i\\omega)H_1(Q;\\,\\mathbb{Z}))\\cong \\mathbb{C}^n\/(\\mathbb{Z}^n+\\tau^{T} (\\mathbb{Z}^n)).$$\nWe consider the following three Lagrangian branes in $(T,\\,B+i\\omega)$:\n\\begin{itemize}\n\t\\item $L_0=\\{\\theta=0\\}$ with trivial connection. $L_0$ is mirror to the structure sheaf on $E$.\n\t\n\t\\item $L_z=(\\{r\\}\\times T_\\theta ^n$, $\\nabla=d+2\\pi i(\\phi d\\theta)$. $L_z$ is mirror to the skyscraper sheaf at $z=\\tau^{T}r-\\phi$ on $E$.\n\t\n\t\\item $L_D=(\\{\\theta=-Dr\\}$, $\\nabla=d-\\pi i(r^{T}(Re\\tau D-D^{T}Re\\tau^{T} )dr)$\n where $D\\in GL(n,\\;\\mathbb{Z})$ such that $Im\\tau D=D^{T}Im\\tau^{T}>0$ and $Re\\tau D-D^{T}Re\\tau^{T}\\in M_{n\\times n}(\\mathbb{Z})$. The transition of the line bundle is \n \\begin{equation} s(r+m)=(-1)^{\\xi(m)}e^{\\pi i m^{T}(Re\\tau D-D^{T}Re\\tau^{T})r}s(r), \\end{equation}\n  where $\\xi:H_1(T;\\mathbb{Z})\\rightarrow \\mathbb{Z}\/2$ such that \n \\begin{equation}\n \\xi(m_1+m_2)=\\xi(m_1)+\\xi(m_2)+m_1^{T}(Re\\tau D-D^{T}Re\\tau^{T})m_2 \\in \\mathbb{Z}\/2.\n \\end{equation}\n Note that $F=Re\\tau D dr\\wedge dr=-B|_{L_D}$ satisfying the B-field condition. $L_D$ is mirror to a line bundle $\\mathcal{L}_D$ on $E$ with first Chern class \n $ D^{T}dr \\wedge (d\\phi-Re\\tau^{T}dr)$, or equivalently $\\frac{i}{2}(Im\\tau)^{-1}D^{T}dz\\wedge d\\bar{z}$, where $z=\\tau^{T}r-\\phi$.\n\t\n\t\n\\end{itemize}\n\nThe intersections of $L_0$ and $L_D$ have coordinates $(D^{-1}k,\\;0)$, where $k\\in \\mathbb{Z}^n$. Hence\n$HF(L_0,\\;L_D)=CF(L_0,\\; L_D)=span_{\\mathbb{C}}\\{s_{(D,\\;k)}\\}_{k\\in \\mathbb{Z}^n}$. If $k\\equiv k'\\; mod\\, D\\mathbb{Z}^n$, then $s_{(D,k)}$ and $s_{(D,k')}$ correspond to the same intersection point, and the corresponding generators of $CF(L_0,L_D)$ coincide up to a multiplicative factor (which arise from the holonomy of the connection on $L_D$, see \\eqref{eq4.8}).\n\nDenote $e_D=(r,\\;-Dr)$ the generator of $HF(L_D,\\;L_z)$ and $e_0=(r,\\;0)$ the generator of $HF(L_0,\\;L_z)$. We will calculate the Floer product $\\mu_2:HF(L_D,\\;L_z)\\otimes HF(L_0,\\;L_D)\\rightarrow HF(L_0,\\;L_D)$.\\\\\nNote that \n$$hol(L_D)=e^{-\\pi i \\langle m-D^{-1}k,\\;(Re\\tau D-D^{T}Re\\tau^{T})r\\rangle }e^{\\pi i\\langle (D^{-1}k),\\;(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}(-1)^{\\xi(m)}$$ on the line segment from $s_{(D,\\;k)}$ to $e_D$ along the vector $(m+r-D^{-1}k,\\, -D(m+r-D^{-1}k))$, $m\\in\\mathbb{Z}^n$. (The first term is obtained by integrating the connection form, the rest come from the transition functions of the line bundle over $L_D$.)\n\\begin{equation*}\\begin{aligned}\ne_D\\circ s_{(D,\\,k)}\n&=\\sum_{ \\triangle\\in M(e_D,\\, S_{(D,k)},\\, e_0)} e^{2\\pi i\\int_{\\triangle}B+i\\omega}hol(\\partial \\triangle) e_0 \\\\\n&=\\quad\\sum_{m\\in \\mathbb{Z}^n}\\quad  e^{\\pi i(B+i\\omega)(m+r-D^{-1}k,\\,D(m+r-D^{-1}k))}e^{-2\\pi i\\langle D(m+r-D^{-1}k),\\,\\phi \\rangle}hol(L_D)e_0 \\\\\n&=\\quad\\sum_{m\\in \\mathbb{Z}^n}\\quad (-1)^{\\xi(m)}e^{\\pi i\\langle D^{-1}k,\\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}e^{\\pi i\\langle \\tau D(m-D^{-1}k),\\,m-D^{-1}k\\rangle} e^{2\\pi i \\langle Dm-k,\\,\\tau^{T}r-\\phi\\rangle}\\\\\n&\\qquad\\qquad\\qquad e^{\\pi i\\langle \\tau Dr,\\,r\\rangle}e^{-2\\pi i\\langle Dr,\\, \\phi \\rangle}e_0\n\\end{aligned}\\end{equation*}\n\nUp to a rescaling, whole expression coincide with the theta functions\n\\begin{equation*}\n\\vartheta_{D, \\,k}(z)=\\sum_{m\\in \\mathbb{Z}^n} (-1)^{\\xi(m)}e^{\\pi i\\langle D^{-1}k, \\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}e^{\\pi i\\langle \\tau D(m-D^{-1}k), \\,m-D^{-1}k\\rangle} e^{2\\pi i \\langle Dm-k, \\,z\\rangle}\n\\end{equation*}\ncorresponding to the sections of the line bundle $\\mathcal{L}_D$, which satisfy\n\\begin{equation*}\n\\vartheta_{D,\\,k}(z+\\tau^{T}h)=(-1)^{\\xi(h)}e^{-\\pi i\\langle \\tau Dh,\\,h\\rangle}e^{-2\\pi i \\langle Dh,\\,z\\rangle} \\vartheta_{D,\\,k}(z),\\; \\vartheta_{D,\\,k}(z+h)=\\vartheta_{D,\\,k}(z).\n\\end{equation*}\n\\begin{equation}\\label{eq4.8}\n\\vartheta_{D,\\,k+Ds}(z)=(-1)^{\\xi(s)}e^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})s,\\,D^{-1}k\\rangle} \\vartheta_{D,\\,k}(z)\n\\end{equation}\n\nThe twisted double torus is given by\n\\begin{equation*}\\begin{aligned}\n\\Big(T\\times T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}\\times (\\mathbb{R}\/\\mathbb{Z})^{2n},\\quad  \\frac{1}{2}\\sigma_0+\\frac{i}{2}\\Omega=&\n\\frac{i}{2}(Im\\tau +Re\\tau (Im\\tau)^{-1}Re\\tau)dr\\wedge d\\theta \\\\\n&-\\frac{i}{2}\\tau(Im\\tau)^{-1}dr\\wedge d\\hat{r}\\\\\n&+\\frac{i}{2}(Im\\tau)^{-1}\\tau d\\hat{\\theta}\\wedge d\\theta \\\\\n&-\\frac{i}{2} (Im\\tau)^{-1} d\\hat{\\theta}\\wedge d\\hat{r}\\Big)\n\\end{aligned}\\end{equation*}\n\nRecall that the mirror of the twisted double torus is\n\\begin{equation*}\n\\boldsymbol{E}\\cong \\mathbb{C}^n\/(\\mathbb{Z}^n+\\tau^{T}\\mathbb{Z}^n)\n\\times \\mathbb{C}^n\/(\\mathbb{Z}^n-\\bar{\\tau}^{T}\\mathbb{Z}^n)\n\\end{equation*}\nwith holomorphic coordinates\n\\begin{equation*}\nu=\\tau^{T}(r-\\kappa)-\\phi, \\qquad\nv=-\\bar{\\tau}^T\\kappa-\\hat{\\theta}-\\phi.\n\\end{equation*}\n\nThe lifts of the three Lagrangians above are\n\\begin{enumerate}\n\t\\item $\\boldsymbol{L}_0=\\{\\theta=0,\\;\\hat{r}=0\\}$ with trivial connection. $\\boldsymbol{L}_0$ is mirror to the structure sheaf.\n\t\n\t\\item $\\boldsymbol{L}_z=\\{\\{r\\}\\times T_{\\theta}^n\\times T_{\\hat{r}}\\times \\{\\hat{\\theta}\\} \\}$.\n\t\n\t\\item $\\boldsymbol{L}_D=\\Big(\\{\\theta=-Dr,\\;  \\hat{r}-D^{T}\\hat{\\theta}=-(Re\\tau D-D^{T}Re\\tau^{T})r \\},\\quad \\nabla=d-\\pi ir^{T}(Re\\tau D-D^{T}Re\\tau^{T} )dr\\Big)$ .\n\t\n\\end{enumerate}\n\n\n\\begin{equation*}\n\\boldsymbol{L}_0\\cap \\boldsymbol{L}_D\n=\\{(r=D^{-1}k,\\,\\theta=0,\\,\\hat{r}=0,\\ \\hat{\\theta}\n=(D^{T})^{-1}(Re\\tau D-D^{T} Re\\tau^{T})D^{-1}k+(D^{T})^{-1}l)\\}\n\\end{equation*}\n\nDenote $p=D^{-1}k,\\; q=(D^{T})^{-1}(Re\\tau D-D^{T} Re\\tau^{T})D^{-1}k+(D^{T})^{-1}l.$\\\\\nLet $s_{k,\\,l}=(p,\\,0,\\,0,\\,q) \\in CF(\\boldsymbol{L}_0,\\,\\boldsymbol{L}_D),\\;\ne_0=(r,\\,0,\\,0,\\,\\hat{\\theta}) \\in CF(\\boldsymbol{L}_0,\\;\\boldsymbol{L}_z)$ and $$e_{D}=(r,\\;-D(r-p),\\;D^{T}(\\hat{\\theta}-q)-(Re\\tau D-D^{T} Re\\tau^{T})(r-p),\\;\\hat{\\theta}) \\in CF(\\boldsymbol{L}_D,\\;\\boldsymbol{L}_z). $$\nThe coefficient of $e_0$ in $\\mu^2(e_D,\\, s_{k,l})$ is then\n\\begin{equation*}\\begin{aligned}\ns_{k, \\,l}=\n\\sum_{m, \\,n\\in \\mathbb{Z}^n}\n&e^{-\\frac{\\pi }{2}\n\\langle Im\\tau^{-1} D^{T}(\\hat{\\theta}+n-q),\\,\\hat{\\theta}+n-q\\rangle\n-\\frac{\\pi}{2}\n\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}(r+m-p),\\,r+m-p\\rangle\n-\\pi\\langle Im\\tau^{-1}D^{T}\\tau^{T}(r+m-p),\\,\\hat{\\theta}+n-q\\rangle} \\\\\n&e^{-2\\pi i\\langle D(r+m-p),\\,\\phi\\rangle +2\\pi i\\langle D^{T}(\\hat{\\theta}+n-q)-(Re\\tau D-D^{T}Re\\tau^{T})(r+m-p),\\,\\kappa\\rangle}\\\\\n& (-1)^{\\xi(m)}e^{-\\pi i\\langle m-p,\\,(Re\\tau D-D^{T}Re\\tau^{T})r\\rangle}e^{\\pi i\\langle p,\\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle} \\\\\n=\\sum_{m,\\,n\\in \\mathbb{Z}^n}\n& (-1)^{\\xi(m)}e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}(n-q),\\,n-q\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,m-p\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,n-q\\rangle} \\\\\n&e^{\\pi i \\langle p,\\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle} e^{-\\pi\\langle Im\\tau^{-1}D^{T}(n-q+\\bar{\\tau}^{T}(m-p)),\\,u\\rangle}e^{\\pi\\langle Im\\tau^{-1}D^{T}(n-q+\\tau^{T}(m-p)),\\,v\\rangle} \\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n=\\sum_{m,\\,n\\in \\mathbb{Z}^n}\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}(u+n-q),\\,u+n-q\\rangle}\ne^{-\\pi \\langle Im\\tau ^{-1}D^{T}(\\bar{\\tau}^{T}(m-p)-v),\\,u+n-q\\rangle}e^{-2\\pi i\\langle D(m-p),\\,n-q\\rangle } \\\\\n& (-1)^{\\xi(m)}e^{\\pi i\\langle p,\\, (Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}\ne^{-\\frac{\\pi}{2}\\langle \\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^T(m-p),\\,m-p\\rangle}\ne^{\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,v\\rangle} \\\\\n&e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}u,\\,u\\rangle}\ne^{-\\pi \\langle Im\\tau^{-1}D^{T}u,\\,v\\rangle }\ne^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}\\\\\n&e^{-2\\pi i\\langle Dr,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle}\n\\end{aligned}\\end{equation*}\n\nUsing the same Fourier series manipulation as in the proof of \\eqref{thetaformula2}, this can be rewritten as\n\n\\begin{equation*}\\begin{aligned}\ns_{k, \\,l}=\\sum_{m,\\,s\\in \\mathbb{Z}^n}\n&\\sqrt{\\frac{2}{det(Im\\tau^{-1}D^{T})}}\ne^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}(\\bar{\\tau}^{T}(m-p)-v)+2is,\\,\\bar{\\tau}^{T}(m-p)-v+2iIm\\tau^{T}D^{-1}s \\rangle}e^{2\\pi i \\langle s,u-q\\rangle}\\\\\n&(-1)^{\\xi(m)}e^{\\pi i\\langle p,\\, (Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}\ne^{-\\frac{\\pi}{2}\\langle \\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^T(m-p),\\,m-p\\rangle}\ne^{\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,v\\rangle} \\\\\n&e^{2\\pi i\\langle D(m-p),\\,q\\rangle}e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}u,\\,u\\rangle}\ne^{-\\pi \\langle Im\\tau^{-1}D^{T}u,\\,v\\rangle }\\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n\\end{aligned}\\end{equation*}\nAfter a change of variables $s\\mapsto Ds-t$ and rearranging, this becomes\n\\begin{equation*}\\begin{aligned}\ns_{k, \\,l}=\\sum_{t\\in \\mathbb{Z}^n\/D\\mathbb{Z}^n}\n&\\sqrt{\\frac{2}{det(Im\\ \\tau^{-1}D^{T})}}\ne^{-2\\pi i\\langle l,\\,D^{-1}(k-t)\\rangle}\ne^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})D^{-1}k,\\,D^{-1}t\\rangle}\n\\\\\n&\\sum_{s\\in \\mathbb{Z}^n}\n(-1)^{\\xi(s)}e^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})s,\\,D^{-1}t\\rangle}\ne^{\\pi i \\langle \\tau D(s-D^{-1}t),\\,s-D^{-1}t\\rangle}e^{2\\pi i \\langle Ds-t,u\\rangle} \\\\\n&\\sum_{m\\in \\mathbb{Z}^n}\n(-1)^{\\xi(m)}e^{-\\pi i \\langle (Re\\tau D-D^{T}Re\\tau^{T})m,\\,p-D^{-1}t\\rangle}\ne^{-\\pi i \\langle \\bar{\\tau}D(m-p+D^{-1}t),m-p+D^{-1}t\\rangle}\ne^{2\\pi i\\langle D(m-p+D^{-1}t),\\,v\\rangle} \\\\\n&e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}(u-v),\\,(u-v)\\rangle}\\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n=\\sum_{t\\in \\mathbb{Z}^n\/D\\mathbb{Z}^n}\n&\\sqrt{\\frac{2}{det(Im\\ \\tau^{-1}D^{T})}}\ne^{-2\\pi i\\langle l,\\,D^{-1}(k-t)\\rangle}\ne^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})D^{-1}k,\\,D^{-1}t\\rangle}\n\\vartheta_{D,t}(u)\\bar{\\vartheta}_{D,\\,k-t}(v)\\\\\n&e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}(u-v),\\,(u-v)\\rangle}\\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n\\end{aligned}\\end{equation*}\nThe factors on the last two lines correspond to the difference between the Floer basis and the usual holomorphic trivialization of $\\mathcal{L}^D\\boxtimes\\mathcal{L}_0^D$ on the mirror, and can be dropped.\nThis leads to the formula:\n\n\\begin{equation}\\label{usub}\n\\sum_{l\\in \\mathbb{Z}^n\/D^{T}\\mathbb{Z}^n}s_{k,l}\n=\\sqrt{2det(Im\\ \\tau D)}\\vartheta_{D,k}(u)\\bar{\\vartheta}_{D,0}(v) \\in H^0(E_\\tau\\times E_{-\\bar{\\tau}}; \\mathcal{L}^{D}\\boxtimes \\mathcal{L}^{D}_0)\n\\end{equation}\nBy evaluating  $v$ at $0$, we recover (up to a scaling factor) $\\vartheta_{D,k}(u)$ up to a constant factor which corresponds to the Floer product on the original torus $T$.\n\n\n\n\\section{The ``u-part\" Floer cohomology $HF_u(\\boldsymbol{L},\\,\\boldsymbol{L}')$}\n\nGiven two Lagrangian branes $L$ and $L'$ in $(T,\\,\\omega)$ with their lifts $\\boldsymbol{L}$ and $\\boldsymbol{L}'$, and assuming that $\\mathcal{L},\\,\\mathcal{L'}$ are the mirror sheaves of $L$ and $L'$,  we expect \n$$HF^*(\\boldsymbol{L},\\,\\boldsymbol{L}')\\cong Ext_{E}^*(\\mathcal{L},\\,\\mathcal{L}')\\boxtimes Ext_{\\bar{E}}^*(\\mathcal{L}_0,\\,\\mathcal{L}'_0).$$\nThe goal of this chapter is to define a subspace $HF_u(\\boldsymbol{L},\\,\\boldsymbol{L}')$ of $HF^*(\\boldsymbol{L},\\,\\boldsymbol{L}')$ which is isomorphic to $Ext_{E}^*(\\mathcal{L},\\,\\mathcal{L}')$, and thus to $HF(L,L')$.\n\\subsection{The ``u-part\" of $HF^*(\\boldsymbol{L},\\,\\boldsymbol{L})$}\nRecall that the twisted doubling torus $T\\times T^\\vee$ is equipped with a natural complex structure from Remark \\ref{AS2},\n\\begin{equation}\nJ=\\left( \\begin{array}{cc}\n\\omega^{-1}B & \\omega^{-1} \\\\\n-\\omega-B\\omega^{-1}B & -B\\omega^{-1}\n\\end{array}\\right)\n\\end{equation}\nwhich induces a complex structure on the cotangent bundle $T^*(T\\times T^\\vee)$, still denoted by $J$,\n\\begin{equation}\nJ=\\left( \\begin{array}{cc}\nB\\omega^{-1} & \\omega+B\\omega^{-1}B \\\\\n-\\omega^{-1}  & -\\omega^{-1}B\n\\end{array}\\right).\n\\end{equation}\n\nGiven a Lagrangian brane $(L,\\,\\nabla)$ in $(T,\\,\\omega)$ with its lift $(\\boldsymbol{L},\\,\\pi_T^*\\nabla)$, we can compare the first order deformation of the objects.\nA first order deformation of $(L,\\,\\nabla)$ is described by $(v;\\,\\alpha)$, where $v$ is a normal vector of $L$, and $\\alpha$ is a real $1$-form. $(v;\\,\\alpha)$ maps to \n$[\\iota_v (\\omega-iB)+i\\alpha]\\in H^1(L;\\,\\mathbb{C})\\cong HF(L,L)$. \nThe corresponding first order deformation of $(\\boldsymbol{L},\\,\\pi_T^*\\nabla)$ is given by \n$(v,\\,-\\tilde{\\alpha};\\,\\alpha,\\,0)$,where $\\tilde{\\alpha}$ is the image of $\\alpha$ under the identification $T^*T\\cong TT^\\vee$. \nThis maps to \n\\begin{equation}\n\\iota_{v-\\tilde{\\alpha}}(\\frac{1}{2}\n\\left(\\begin{array}{cc}\n\\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n-\\omega^{-1}B & -\\omega^{-1}\n\\end{array}\\right)\n-i\\sigma_0)+i\\alpha=\\frac{1}{2}(1+iJ)(\\iota_v (\\omega-iB)+i\\alpha)\n\\in H^1(\\boldsymbol{L};\\,\\mathbb{C}).\n\\end{equation}\n\nNote that the full first order deformation space of $(\\boldsymbol{L},\\,\\pi_T^*\\nabla)$ coincides with $H^1(\\boldsymbol{L};\\, \\mathbb{C})\\cong HF(\\boldsymbol{L},\\boldsymbol{L})$, and the first order deformations coming from lifts are exactly the $(0,1)$ part of $H^1(\\boldsymbol{L},\\,\\mathbb{C})\\cong HF(\\boldsymbol{L},\\,\\boldsymbol{L})$ with respect to $J$ mentioned above. Then \n\\begin{equation}\nHF^*(L,\\,L)\\cong H^*(L;\\,\\mathbb{C})=\\bigwedge H^1(L;\\,\\mathbb{C})\\cong \\bigwedge H^{0,\\,1}_{J}(\\boldsymbol{L})=H^{0,\\,*}_{J}(\\boldsymbol{L})\n\\end{equation}.\n\\begin{definition}\n\t$HF(\\boldsymbol{L},\\,\\boldsymbol{L})_u:=\\, H^{0,\\,*}_{J}(\\boldsymbol{L})\\subset H^*(\\boldsymbol{L};\\,\\mathbb{C})=HF(\\boldsymbol{L},\\,\\boldsymbol{L})$.\n\\end{definition}\n\n\\subsection{The case of transversal intersection $L\\cap L'$}\nAs in Chapter 4, consider a pair of Lagrangian branes $L,\\,L'$, which are mirror to a pair of line bundles $\\mathcal{L},\\,\\mathcal{L'}$. Let $\\boldsymbol{L},\\,\\boldsymbol{L}'$ be the lifts in the double torus. Suppose $\\mathcal{L'}\\otimes \\mathcal{L}^{-1}$ is ample, then the subspace of $HF^*(\\boldsymbol{L},\\,\\boldsymbol{L'})$ spanned by \n$$\\displaystyle \\sum_{l\\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n}s_{k,\\,l}=\\sqrt{2det(Im\\ \\tau (D'-D))}\\vartheta_{D'-D,k}(u)\\bar{\\vartheta}_{D'-D,0}(v)$$\nis isomorphic to $HF^*(L,L')$ by evaluation at $v=0$. \nWe define the ``u-part\" cohomology for transversal intersections as follows.\n\n\\begin{definition}\n    Suppose $L=\\{\\theta=-Dr+c\\} $ and $ L'=\\{\\theta=-D'r+c'\\}$ intersect transversally, assume $\\boldsymbol{L}\\cap \\boldsymbol{L}'=\\{s_{k,l}\\}$, \n    where $k\\in \\mathbb{Z}^n\/(D'-D)\\mathbb{Z}^n,\\,\n    l \\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n$. Then we define\n\n\\begin{equation}\n    HF_u(\\boldsymbol{L},\\boldsymbol{L}'):=\n    span\\Big\\{\\sum_{l\\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n}s_{k,l}\\Big\\}_{k\\in \\mathbb{Z}^n\/(D'-D)\\mathbb{Z}^n}\n    \\subset HF^*(\\boldsymbol{L},\\boldsymbol{L}').\n    \\end{equation}\n\\end{definition}\n\\begin{definition}\n    Let \n    \\begin{equation}\n    \\begin{aligned}\n    \\Pi_T:  HF^*(\\boldsymbol{L},\\boldsymbol{L}') &\\rightarrow \n    HF_u^*(\\boldsymbol{L},\\boldsymbol{L}') \\\\\n     s_j\\otimes s_h' &\\mapsto s_j\\otimes (\\frac{1}{det(D'-D)}\\sum_l s_l').\n    \\end{aligned}\n    \\end{equation}\n    Define the product structure to be\n    \\begin{equation}\n    \\begin{aligned}\n    \\mu^2_u: HF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}\")&\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\rightarrow HF_u(\\boldsymbol{L},\\boldsymbol{L}\") \\\\\n    x&\\otimes y \\longmapsto \\Pi_T(\\mu^2(x,y)),\n    \\end{aligned}\n    \\end{equation}\n    where $\\mu^2$ is the usual Floer product in $HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}')$.\n\\end{definition}\n\nWith these definitions, we state the main theorem:\n\\begin{theorem}\n\tFor a pair of Lagrangian branes $L,\\,L'$ which are mirror to a pair of line bundles $\\mathcal{L},\\,\\mathcal{L'}$, let $\\boldsymbol{L},\\,\\boldsymbol{L}'$ be the lifts in the twisted doubling torus. Suppose $\\mathcal{L'}\\otimes \\mathcal{L}^{-1}$ is ample. Then the ``u-part\" Floer cohomology $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')$ is isomorphic to $HF^*(L,\\,L')$. And for two such pairs $L,L'$ and $L',L''$, the following diagram commutes\n\t\\begin{equation}\\label{uspscomm}\n\t\\begin{tikzcd}\n\t{HF^*(L',\\,L'')\\otimes HF^*(L,\\,L')} \\arrow[rr] \\arrow[d, \"\\cong\"] &  & {HF^*(L,\\,L\")} \\arrow[d, \"\\cong\"] \\\\\n\tHF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}'')\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\arrow[rr]                                 &  & HF_u(\\boldsymbol{L},\\boldsymbol{L}'').\n\t\\end{tikzcd}\n\t\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe isomorphism $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')\\cong HF^*(L,\\,L')$ is given by\n\\begin{equation}\ns_k \\mapsto \n\\sum_{l\\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n}s_{k,l}\/(\\sqrt{2det(Im\\ \\tau (D'-D))}\\bar{\\vartheta}_{D'-D,0}(0))\n\\end{equation}\nThen the commutative diagram follows from \\eqref{usub} and formulas for $\\vartheta$-functions.\n\\end{proof}\n\n\\subsection{Proposal for general case}\n\nIn the general case when $L$ and $L'$ intersect non transversally, we expect there is still a ``u-part\" subspace $HF_u(\\boldsymbol{L},\\boldsymbol{L}')$ isomorphic to $HF(L,L)$, and a similar commutative diagram holds.\nIn a simple case, assume $T,\\,L,\\,L'$ admit a simultaneous  decomposition $T=T_1\\times T_2,\\, L=L_1\\times L_2,\\, L'=L_1'\\times L_2'$, such that $L_1,\\,L_2$ (resp. $L_1',\\,L_2'$) are Lagrangian submanifolds of $T_1$ (resp. $T_2$), and assume $L_1=L_1'$, while $L_2$ intersects $L_2'$ transversally.\nThen the twisted doubling torus and lifts of $L,\\,L'$ also admit product structures, and \n\\begin{equation}\n\\begin{aligned}\nHF^*(L,L')& \\cong  HF^*(L_1,L_1')\\otimes HF(L_2,L_2') \\\\\n&\\cong HF_u^*(\\boldsymbol{L}_1,\\boldsymbol{L}_1)\\otimes HF_u^*(\\boldsymbol{L}_2,\\boldsymbol{L}_2') \\\\\n&\\cong H^{0,*}(\\boldsymbol{L}_1)\\otimes HF_u(\\boldsymbol{L}_2,\\boldsymbol{L}_2')\n\\end{aligned}\n\\end{equation}\nInspired by this, and combining the definitions of ``u-part\" Floer cohomology in the previous sections, we make the following tentative definition.\n\n\\begin{definition}\n    For a pair of Lagrangian branes $L$ and $L'$, and their lift $\\boldsymbol{L}$ and $\\boldsymbol{L}'$, let\n\\begin{equation*} \nHF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') :=\nG- \\textbf{invariant part of } H^{0,*}(\\boldsymbol{L}\\cap \\boldsymbol{L}')\n\\end{equation*} \nWhere $G$ is the discrete group of translations in the direction of $T^\\vee$ acting on $\\boldsymbol{L}\\cap \\boldsymbol{L}'$.\n\\end{definition}\n\n\\begin{conjecture}\nFor a pair of Lagrangian branes $L,\\,L'$  with their lifts $\\boldsymbol{L},\\,\\boldsymbol{L}'$ in the twisted doubling torus, the \"u-part\" Floer cohomology $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')$ is isomorphic to $HF^*(L,\\,L')$. And for two such pair $L,L'$ and $L',L\"$, the following diagram commutes:\n\t\\begin{equation}\\label{uspscomm}\n\t\\begin{tikzcd}\n\t{HF^*(L',\\,L\")\\otimes HF^*(L,\\,L')} \\arrow[rr] \\arrow[d, \"\\cong\"] &  & {HF^*(L,\\,L\")} \\arrow[d, \"\\cong\"] \\\\\n\tHF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}\")\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\arrow[rr]                                 &  & HF_u(\\boldsymbol{L},\\boldsymbol{L}\").\n\t\\end{tikzcd}\n\t\\end{equation}\n\\end{conjecture}\n\n\n\n\\section{Equivalence of $T$ and $T^\\vee$ with B-field Twist}\n\nRecall that the dual torus of $(T=V\/\\Lambda,B+i\\omega)$ is $(T^\\vee=V^\\vee \/\\Lambda^\\vee,\\ (B+i\\omega)^{-1} )$ under assumption \\ref{assump}. Explicitly, $$(B+i\\omega)^{-1}=(\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1}-i(\\omega+B\\omega^{-1}B)^{-1}.$$\n\n\\begin{example}\nLet $T=\\mathbb{R}^2\/\\mathbb{Z}^2$ with $B+i\\omega=\\tau dr\\wedge d\\theta$, $\\tau=b+ia,\\ a>0$, then its dual torus is $T^\\vee=\\mathbb{R}^2\/\\mathbb{Z}^2$ with $(B+i\\omega)^{-1}=-\\tau^{-1}d\\hat{r}\\wedge d\\hat{\\theta}$. They are non symplectomorphic tori for generic $\\tau$. However, their mirror manifolds are isomorphic as complex manifold.\n\\begin{equation*} \n\\begin{tikzcd}\n\\mathbb{C}\/\\mathbb{Z}+\\tau \\mathbb{Z} &  & \\mathbb{C}\/\\mathbb{Z}-\\tau^{-1}\\mathbb{Z}. \\arrow[ll, \"\\times \\tau\"']\n\\end{tikzcd}\n\\end{equation*}\nThis implies that their Fukaya categories are equivalent.\n\\end{example}\nThe phenomenon that dual tori have equivalent Fukaya categories is not obvious without referring to Homological Mirror Symmetry. \nHowever, the twisted doubling tori of $(T,B+i\\omega)$ and $(T^\\vee,\\ (B+i\\omega)^{-1} )$ are the same up to a B-field twist.\nExplicitly, their twisted doubling tori are\n\\begin{equation*}\n\t(T\\times T^\\vee,\\quad  \\Omega=\\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t \\sigma_0=\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)\t\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t(T\\times T^\\vee,\\quad  \\Omega=\\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t -\\sigma_0=-\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)\t\n\t\\end{equation*}\n\tThe difference of B-field is an integral class in $H^2(T\\times T^\\vee; \\mathbb{R})$. Let\n\t\\begin{equation}\n\t\\nabla_0=d-2\\pi i (rd\\hat{r}+\\theta d\\hat{\\theta})\n\t\\end{equation}\n\tbe a $U(1)$ connection on $T\\times T^\\vee$ with curvature $2\\sigma_0$. Then we have an equivalence of Fukaya categories of the two doubling tori by a B-twist:\n\t\\begin{equation}\n\t\\begin{aligned} \n\tFuk(T\\times T^\\vee,\\Omega,\\sigma_0) &\\rightarrow Fuk(T\\times T^\\vee,\\Omega,-\\sigma_0) \\\\\n\t(L,\\nabla) &\\mapsto (L,\\nabla\\otimes \\nabla_0|_L)\n\t\\end{aligned} \n\t\\end{equation}\n\\begin{conjecture}\n\tThe Fukaya category of a torus $(T,B+i\\omega)$ is equivalent to the Fukaya category of its dual $(T^\\vee,(B+i\\omega)^{-1})$.\n\\end{conjecture}\n\n\n\n\\begin{remark}\n\t The isomorphism between morphism spaces in $(T,B+i\\omega)$ and $(T^\\vee,(B+i\\omega)^{-1})$ is not directly given by the above equivalence of doubling tori. It also involves the projection map associated with $HF(\\boldsymbol{C},\\boldsymbol{C}')_u$. \\\\\n\\end{remark}\n\n\n\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\nThis paper will address the question of existence of \\textit{conserved charges  on null hypersurfaces} (and their associated \\textit{conservation laws}) for the wave equation\n\\begin{equation}\n\\Box_{g}\\psi=0\n\\label{wave}\n\\end{equation}\non a general four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. \n\nThe simplest example of such a conserved charge arises in Minkowski space and is given by \\[char\\left[\\psi;S_{v}\\right]=\\int_{S_{v}}\\frac{1}{r^{2}}\\partial_{u}(r\\psi) \\]\n which for all solutions of \\eqref{wave} satisfies the conservation law \n\\[\\partial_{v}\\Big(char\\left[\\psi;S_{v}\\right]\\Big)=0.\\]\n Here $u,v$ are standard null coordinates and $S_{v}$ are the spherical sections of the standard null cones $\\left\\{u=c\\right\\}$.   Another example is the recently discovered conservation laws on the degenerate event horizons of extremal black hole spacetimes (see \\cite{aretakis4, hj2012, murata2012}). A third (limiting) example are the celebrated \\textit{Newman--Penrose constants} which are conserved along  null infinity in any asymptotically flat spacetime (see \\cite{np1,np2}).\n\nIn the present paper, we define a general notion of conserved charges (see Section \\ref{sec:ConservationLawsForTheWaveEquations}) encompassing all the above examples and give a characterization of  null hypersurfaces admitting such charges in terms of the kernel of an elliptic operator (defined for the first time here and in our companion paper \\cite{aretakiselliptic}).\\footnote{It will follow in particular from this characterization that  generic Lorentzian manifolds do not admit such charges.}    In fact,\nwe show that the only information that can be propagated by all solutions to the wave equation along null hypersurfaces is given precisely by these conserved charges.  For this,  we initiate the general study of \\textit{gluing constructions for the characteristic initial value problem} (see Section \\ref{sec:TheCharacteristicInitialValueProblem}) and we show that the only obstruction to gluing along a null hypersurface $\\mathcal{H}$ is the existence of conserved charges (in our sense) on $\\mathcal{H}$. \n\nPart of the importance of the conservation laws on degenerate horizons referred to  above lies in their role in the instability properties of the wave equation on extremal black holes  (see  Section \\ref{sec:Remarks}). This result led to the so-called ``horizon instability of extremal black holes''. The present general study may thus shed light on new aspects of the global evolution of the wave equation on more general backgrounds. \n\n\n\n\nThe statement of the main result can be found in Section \\ref{sec:TheMainResultxx}. Our proof introduces a new method which we hope will be relevant for applications to other linear and non-linear equations such as the Einstein equations. \n\n\n\\subsection{Conservation laws for the wave equation}\n\\label{sec:ConservationLawsForTheWaveEquations}\n\nWe first present some basic geometric definitions that will be useful for defining the notion of conservation laws on null hypersurfaces. For more details about the geometric setting see Section \\ref{sec:TheDoubleNullFoliation}; our notation follows  \\cite{DC09,christab}. \n\n\\paragraph{Null foliations\\medskip \\\\}\n\\label{sec:nullfoliations}\n\n\nLet $\\mathcal{H}$ be a regular null hypersurface of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. A foliation  $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ of $\\mathcal{H}$, that is a collection of sections $S_{v}$ which vary smoothly in $v$ such that $\\cup_{v}S_{v}=\\mathcal{H}$, can be uniquely determined by the choice of one section $S_{0}$, the choice of a smooth function $\\Omega$ on $\\mathcal{H}$ and the choice of a null geodesic vector field $L_{geod}$ tangential to the null generators of $\\mathcal{H}$ and such that \n\\[\\nabla_{L_{geod}}L_{geod}=0.\\]\nIndeed, if we define the vector field \n\\[L=\\Omega^{2}\\cdot L_{geod}\\]\non $\\mathcal{H}$ and consider the affine parameter $v$ of $L$ such that \n\\[Lv=1, \\text{ with }v=0 \\text{ on } S_{0},\\]\nthen the level sets $S_{v}$ of $v$ on $\\mathcal{H}$ are precisely the leaves of the foliation $\\mathcal{S}$. We use the notation\n\\begin{equation}\n\\mathcal{S}=\\Big\\langle S_{0},L_{geod}, \\Omega\\Big\\rangle.\n\\label{foliation}\n\\end{equation}\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.06]{glue1.png}\n\t\\label{fig:glue1}\n\\end{figure}\nWe will henceforth restrict to the case that all sections $S_{v}$ above are diffeomorphic (via a diffeomorphism $\\Phi$) to the 2-sphere.\\footnote{ Topologies with higher genus can be treated analogously. Our argument heavily relies  on the compactness of the sections and hence the non-compact case remains an open problem.} The flow of $L$ on $\\mathcal{H}$ provides a diffeomorphism $\\Phi_{v}$ between the sections $S_{v}$ and $S_{0}$. In addition to the induced metric on $S_{v}$, which we will denote by $\\mbox{$g \\mkern-8.8mu \/$\\,}$, we can also equip all sections with the standard metric on the unit sphere $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ (via $\\Phi$) such that it is invariant under the flow of $L$. The volume form on $S_{v}$ with respect to $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ will be denoted by $d\\mu_{_{\\mathbb{S}^{2}}}$.\n\nGiven any section $S_{v}$, there is a unique metric $\\hat{g}$ which is conformal to the induced metric $\\mbox{$g \\mkern-8.8mu \/$\\,}$ such that the volume form $d\\mu_{_{\\hat{g}}}$ with respect to $\\hat{g}$ and the volume form $d\\mu_{_{\\mathbb{S}^{2}}}$ with respect to $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ are equal:\n\\[d\\mu_{_{\\hat{g}}}=d\\mu_{_{\\mathbb{S}^{2}}}. \\]We denote by $\\phi$ the conformal factor:\n\\begin{equation}\n\\mbox{$g \\mkern-8.8mu \/$\\,}=\\phi^{2}\\cdot \\hat{g}. \n\\label{eq:theconformalfactorintroduction}\n\\end{equation}\n\nFurthermore, given a foliation $\\mathcal{S}$ we denote by $Y^{\\mathcal{S}}$ the unique null vector field   which is normal to the sections $S_{v}$, conjugate to $\\mathcal{H}$ and normalized such that \n\\begin{equation}\ng\\big(L_{geod},Y^{\\mathcal{S}}\\big)=-1.\n\\label{Y}\n\\end{equation}\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.06]{glue2.png}\n\t\\label{fig:glue2}\n\\end{figure}\nThe vector field $Y^{\\mathcal{S}}$ is transversal to $\\mathcal{H}$ and can be seen as the generator of an appropriately normalized ``retarded time'' $u$ such that $u=0$ on $\\mathcal{H}$. Specifically, one can construct an optical function $u$ such that $Y^{\\mathcal{S}}u=1$ on $\\mathcal{H}$ and the level sets of $u$ are ``outgoing'' null hypersurfaces $\\mathcal{H}_{u}$ (hence we assume here that $\\mathcal{H}$ is an ``outgoing'' null hypersurface). Note that in a similar  fashion as above we can define the conformal factor $\\phi$ of the section $S_{u,v}$ which are the intersections of $\\mathcal{H}_{u}$ and the ``incoming'' null hypersurfaces $\\underline{\\mathcal{H}}_{v}$ generated by the null geodesics normal to $S_{v}$ and conjugate to $\\mathcal{H}$.\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.07]{glue3.png}\n\t\\label{fig:glue3}\n\\end{figure}\n\n\\newpage \n\n\\paragraph{Conservation laws on $\\mathcal{H}$\\medskip \\\\}\n\\label{sec:conservation laws}\n\n\n\n\nConsider the linear space $\\mathcal{V}_{\\mathcal{H}}$ consisting of all smooth functions on $\\mathcal{H}$ which are constant along the null generators of $\\mathcal{H}$, i.e.\n\\begin{equation}\n\\mathcal{V}_{\\mathcal{H}}=\\Big\\{f\\in C^{\\infty}(\\mathcal{H})\\, :\\, Lf=0\\Big\\}.\n\\label{linearspace}\n\\end{equation}\nLet $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ be a foliation of $\\mathcal{H}$ and let $Y^{\\mathcal{S}}$ be the vector field and $\\phi$ the conformal factor defined above.  \nWe define the linear space $\\mathcal{W}^{\\mathcal{S}}$ to be the subspace of $\\mathcal{V}_{\\mathcal{H}}$ such that for all $\\Theta^{\\mathcal{S}}\\in \\mathcal{W}^{\\mathcal{S}}$ and for all solutions $\\psi$ to the wave equation \\eqref{wave} the integrals\n\\begin{equation}\n\\int_{S_{v}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta^{\\mathcal{S}} \\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:integrals}\n\\end{equation}\nare conserved, i.e.~independent of $v$. That is,\n\\begin{equation}\n\\mathcal{W}^{\\mathcal{S}}=\\left\\{\\Theta^{\\mathcal{S}}\\in C^{\\infty}(\\mathcal{H})\\, :\\, L\\Theta^{\\mathcal{S}}=0, \\ \\partial_{v}\\left(\\int_{S_{v}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta^{\\mathcal{S}} \\, d\\mu_{_{\\mathbb{S}^{2}}}\\right)=0\\right\\}\\subset \\mathcal{V}_{\\mathcal{H}}. \n\\label{eq:}\n\\end{equation}\n\n\n We make the following definition:\n\\begin{definition} \\textbf{(Conservation laws on $\\mathcal{H}$)}:\nWe say that a null hypersurface $\\mathcal{H}$ admits (first order) conservation laws  with respect to a foliation $\\mathcal{S}$ of $\\mathcal{H}$ if\n\\begin{equation}\n\\dim\\mathcal{W}^{\\mathcal{S}}\\geq 1.\n\\label{dimensionofw}\n\\end{equation} \nIf \\eqref{dimensionofw} holds then we will refer to the space $\\mathcal{W}^{\\mathcal{S}}$ and the number $\\dim \\mathcal{W}^{\\mathcal{S}}$ as the kernel and the dimension of the conservation laws, respectively. The integrals of the form \\eqref{eq:integrals} will be called conserved charges and will be denoted by $char\\big(S_{v}\\big)[\\psi; \\Theta^{\\mathcal{S}}]$.\n\\label{definitionconservationlaw}\n\\end{definition}\nA priori Definition \\ref{definitionconservationlaw} appears to be a very restrictive notion of conservation laws. However, as we shall show (see Theorem \\ref{theoremmainintro}), the conservation laws in the sense of Definition \\ref{definitionconservationlaw} are in fact the only type of ``first order'' conservation laws that a null hypersurface $\\mathcal{H}$ might admit.\n\n One could also define higher order conservation laws by considering higher derivatives of $\\phi\\cdot\\psi$ in \\eqref{eq:integrals}. In order to make our method clear, in the bulk of this paper we only consider first order conservation laws and for this reason we will simply refer to them as conservation laws (see however Section \\ref{sec:Genericity} where we consider higher order conservation laws for spherically symmetric geometries). \n\n \n\n\n\\paragraph{Examples \\\\}\n\\label{sec:examples}\n\nWe next consider three main examples of spacetimes admitting conserved charges for the wave equation.\n\n\\paragraph{\\small 1. Minkowski spacetime\\medskip \\\\}\n\\label{sec:MinkowskiSpacetimea}\n\n\\normalsize\n\n\nThe wave equation in double null coordinates $(u,v)$ on the Minkowski spacetime reads\n\\[\\partial_{v}\\partial_{u}(r\\psi)=\\frac{1}{2r^{2}}\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}(r\\psi),   \\]\nwhere $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ is the standard Laplacian on the unit sphere. Hence, \n\\[\\int_{S_{v}}\\partial_{v}\\partial_{u}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\frac{1}{2r^{2}}\\int_{S_{v}}\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0\\]\nand thus the integral \n\\[\\int_{S_{v}}\\partial_{u}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}\\, \\]\nis conserved along the null hypersurfaces $\\left\\{u=c\\right\\}$. This conserved charge can be written in  terms of Definition \\ref{definitionconservationlaw}; indeed, if we consider the foliation $\\mathcal{S}=\\left\\langle S_{0}, L_{geod}=\\partial_{v},\\Omega=1\\right\\rangle$ of $\\mathcal{H}$ then $\\phi=r$, $Y^{\\mathcal{S}}=\\partial_{u}$ and  $\\left\\langle 1\\right\\rangle\\subset \\mathcal{W}^{\\mathcal{S}}$ and hence $\\dim \\mathcal{W}^{\\mathcal{S}}\\geq 1$. \n\n\n\\paragraph{\\small 2. Extremal black holes\\medskip \\\\}\n\\label{sec:MinkowskiSpacetime4}\n\n\\normalsize\n\nConsider the coordinate vector fields\n\\[T=\\partial_{v}, \\ \\ \\ \\ \\ R=\\partial_{r}, \\ \\ \\ \\ \\ \\Phi=\\partial_{\\phi^{*}} \\]\nwith respect to the ingoing Eddington--Finkelstein coordinates  $(v,r,\\theta,\\phi^{*})$ on an extremal Kerr black hole with mass parameter equal to $M$. Then, the quantity\n\\begin{equation}\n\\int_{S_{v}} \\left[R\\psi+\\frac{\\sin^{2}\\theta}{4}\\cdot T\\psi+ \\frac{1}{2M}\\cdot\\psi\\right]\\,d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eextremalcharge}\n\\end{equation}\nis conserved along the horizon $\\mathcal{H}=\\left\\{r=M\\right\\}$, i.e.~it is independent of $v$. This conservation law was first found in \\cite{aretakis4} and was then generalized to all extremal black holes by Lucietti and Reall \\cite{hj2012} and Murata \\cite{murata2012}. \n\nThe above conserved charge can be written in  terms of  Definition \\ref{definitionconservationlaw} as follows: Consider the foliation $\\mathcal{S}=\\left\\langle S_{0}, L_{geod}=\\partial_{v},\\Omega=1\\right\\rangle$ of $\\mathcal{H}$. Then  \\[Y^{\\mathcal{S}}=\\frac{1}{\\frac{\\sin^{2}\\theta}{2}-1}\\cdot\\left[R+\\frac{\\sin^{2}\\theta}{4}\\cdot T+\\frac{3+\\cos^{2}\\theta}{8M}\\cdot\\Phi\\right]\\] and \n\\[\\left.\\phi\\right|_{\\mathcal{H}}=\\sqrt{2}\\cdot M,\\ \\ \\ \\ \\ \\left.Y^{\\mathcal{S}}\\phi\\right|_{\\mathcal{H}}=\\frac{\\sqrt{2}}{2}\\cdot\\frac{1}{\\frac{\\sin^{2}\\theta}{2}-1}, \\ \\ \\ \\ \\  \\left\\langle \n\\frac{\\sin^{2}\\theta}{2}-1\\right\\rangle\\subset \\mathcal{W}^{\\mathcal{S}} \\]\nand hence $\\dim \\mathcal{W}^{\\mathcal{S}}\\geq 1$. Note that since the integral curves of $\\Phi$ are closed, the $\\Phi$-derivative drops out from \\eqref{eextremalcharge}. \n\nFor more results based on this conservation law see \\ref{sec:Remarks}. \n\n\n\\paragraph{\\small 3. Null infinity of asymptotically flat spacetimes \\medskip \\\\}\n\\label{sec:MinkowskiSpacetime}\n\n\\normalsize\n\n Given sufficient smoothness for $\\psi$ at the null infinity $\\mathcal{I}^{+}$ of an asymptotically flat spacetime we can write\n\\[\\psi\\big(u,r,\\theta^{1},\\theta^{2}\\big)=\\frac{\\alpha_{1}\\big(u,\\theta^{1},\\theta^{2}\\big)}{r}+\\frac{\\alpha_{2}\\big(u,\\theta^{1},\\theta^{2}\\big)}{r^{2}}+O\\left(\\frac{1}{r^{3}}\\right)\\]\nwith respect to outgoing Eddington--Finkelstein coordinates $\\big(u,r,\\theta^{1},\\theta^{2}\\big)$. Here we identity $\\mathcal{I}^{+}=\\left\\{r=\\infty\\right\\}$. If $\\psi$ is a solution to the wave equation, then the quantity\n\\begin{equation}\n\\lim_{r\\rightarrow+\\infty}\\int_{S_{u}}r^{2}\\cdot\\partial_{r}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:npc}\n\\end{equation}\ndoes not depend on $u$. This charge as well as other charges involving higher order derivatives were found by Newman and Penrose \\cite{np1,np2} (see also \\cite{npexton}) and are known as Newman--Penrose constants.  \n\n\n\nThe origin of these peculiar constants has been the object of intense study. In particular, we mention the work of Goldberg \\cite{goldberg1,goldberg2} who showed that these constants do not arise from non-trivial transformation laws. The same author was able to rederive these constants in the flat case by using Green's theorem in appropriate regions in conjunction with the fundamental solution to the wave equation. See also the related work by Robinson \\cite{robinson}. Further work on the Newman--Penrose constants can be found in \\cite{chrugrav,goldberg3,pressnp,valientenp1, valientenp2,valientenp3} and references there-in. A nice geometric relation of the the Newman--Penrose constants and the  charges at the event horizon of extremal Kerr  was given by Bizon and Friedrich \\cite{bizon2012}  and independently by Lucietti et al \\cite{hm2012}.\n\n\nThe Newman--Penrose constants can be seen as a limiting example of the conserved charges given by Definition \\ref{definitionconservationlaw} as follows: Let $\\mathcal{I}_{S_{0}}$ be an incoming null hypersurface  and  $\\mathcal{S}=\\left\\langle S_{0}, L_{geod}, \\Omega=1\\right\\rangle$ be a foliation  of it such that $\\frac{1}{2}\\big(L_{geod}+Y^{\\mathcal{S}}\\big)\\left.\\!\\right|_{S_{0}}$ is the (unit timelike) binormal of $S_{0}$ (see Section \\ref{sec:TheNullInfinityMathcalI}). Let also $r=\\sqrt{A\/4\\pi}$  be the area-radius function of the sections of $\\mathcal{S}$ on $\\mathcal{I}_{S_{0}}$, where $A$ is the area of the sections. Then $\\mathcal{I}_{S_{0}}\\rightarrow \\mathcal{I}$ as $r\\big(S_{0}\\big)\\rightarrow+\\infty$ and the Newman--Penrose constants can be retrieved in the limit as $r\\rightarrow +\\infty$ by the conservation law of Definition \\ref{definitionconservationlaw} if we take $Y^{\\mathcal{S}}$ to be the limit of $Y^{\\mathcal{S}}\\left.\\right|_{\\mathcal{I}_{S_{0}}}$ and $\\Theta^{S}$ to be the limit of $r^{2}\\left.\\right|_{\\mathcal{I}_{S_{0}}}$ as $\\mathcal{I}_{S_{0}}\\rightarrow\\mathcal{I}$ .\n\nOur general theory (see Theorem \\ref{theo3}) will in particular show that  the conserved charge \\eqref{eq:npc} is the \\textbf{only} non-trivial conserved  charge along the null infinity $\\mathcal{I}$ which involves the 1-jet of $\\psi$.\n\n\n\n\n\n\n\n\n\n\\subsection{The characteristic gluing problem}\n\\label{sec:TheCharacteristicInitialValueProblem}\n\n\nThe characteristic gluing problem for the wave equation  provides a means to formally show that Definition \\ref{definitionconservationlaw} is the right notion of conservation laws on null hypersurfaces. We introduce this problem below. \n\n \nLet $(\\mathcal{M},g)$ be a four-dimensional Lorentzian manifold and $\\mathcal{H}, \\underline{\\mathcal{H}}$  be two regular null hypersurfaces intersecting at a two dimensional sphere $S_{0}$. Characteristic initial data for the wave equation \\eqref{wave} correspond to prescribing the restriction of $\\psi$ on  the union $\\mathcal{H}\\cup \\underline{\\mathcal{H}}$. \nIn fact, given smooth data at $\\mathcal{A},\\underline{\\mathcal{A}}$, as depicted below, there is a unique smooth solution to the wave equation in the domain of dependence $\\mathcal{R}$, depicted schematically below:\n\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.09]{picture01.png}\n\t\\label{fig:pfsdkfjapwoeijwe45}\n\\end{figure}\n\n\nThe problem of gluing constructions which we wish to formulate is the following: Consider a null hypersurface $\\mathcal{H}$ and two conjugate null hypersurfaces $\\underline{\\mathcal{H}}_{0}$ and $\\underline{\\mathcal{H}}_{1}$ intersecting $\\mathcal{H}$ at the two-dimensional spheres $S_{0}$ and $S_{1}$. We  prescribe initial data for the wave equation \\eqref{wave} on the hypersurfaces $\\mathcal{A}_{0},\\underline{\\mathcal{A}}_{0}$ and $\\mathcal{A}_{1},\\underline{\\mathcal{A}}_{1}$ depicted in the figure below\n\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.129]{picture02.png}\n\t\\label{figfdsp45picutre02}\n\\end{figure}\n\n\\noindent and we want to extend the data on the  truncated hypersurface $\\mathcal{G}$ such that there is a smooth solution $\\psi$ to the wave equation in the region $\\mathcal{R}_{0}\\cup\\mathcal{R}_{\\mathcal{G}}\\cup\\mathcal{R}_{1}$ such that $\\left.\\psi\\right|_{\\underline{A}_{0}\\cup\\underline{A}_{1}}$ and $\\left.\\psi\\right|_{\\mathcal{A}_{0}\\cup\\mathcal{A}_{1}}$ coincide with the prescribed data. \n\nIn this paper we address in fact a weaker version of the above gluing problem, which is however sufficient for the complete classification of all null hypersurfaces admitting conservation laws. We make the following definition\n\\begin{definition} We shall say that ``we can perform first order gluing along $\\mathcal{H}$'' of the characteristic data $(\\underline{\\mathcal{A}}_{0},\\mathcal{A}_{0})$, $(\\underline{\\mathcal{A}}_{1},\\mathcal{A}_{1})$ as defined above if we can  smoothly extend the data in $\\mathcal{G}$ such that the arising solutions\n\n\\begin{itemize}\n\t\\item $\\psi_{0}$ with data  given on $\\underline{\\mathcal{A}}_{0},\\mathcal{A}_{0}\\cup\\mathcal{G}$, and \n\t\\item $\\psi_{1}$ with data given on $\\underline{\\mathcal{A}}_{1},\\mathcal{A}_{1}$\n\\end{itemize} agree at $S_{1}$ to all orders tangential to  $\\mathcal{H}$ and up to first order in directions transversal to $\\mathcal{H}$; that is, $\\psi_{0}=\\psi_{1}$ at $S_{1}$ to all orders tangential to  $\\mathcal{H}$ and $Y\\psi_{0}=Y\\psi_{1}$ at $S_{1}$, where $Y$ is a smooth vector field transversal to $\\mathcal{H}$. \n\\label{firstordergluingdefinition}\n\\end{definition}\n\n\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.1]{picture03.png}\n\t\\label{figfdsp45picutre03}\n\\end{figure}\n\nIf $\\psi$ solves the wave equation then the transversal derivative $Y\\psi$ on $\\mathcal{H}$ is completely determined by the data $\\left.\\psi\\right|_{\\mathcal{H}}$ on  $\\mathcal{H}$ and the transversal derivative $Y\\psi$ at a section $S$ of $\\mathcal{H}$. For this reason, it is convenient to  ``forget'' about the incoming null hypersurfaces $\\underline{\\mathcal{H}}_{1},\\underline{\\mathcal{H}}_{2}$ and hence just ``keep'' the following data\n\\[\\psi\\left.\\right|_{\\mathcal{A}_{0}}, \\ \\ \\ Y\\psi\\left.\\right|_{S_{0}}\\]\nand \n\\[\\psi\\left.\\right|_{\\mathcal{A}_{1}}, \\ \\ \\ Y\\psi\\left.\\right|_{S_{1}}.\\]\nIn fact,\\textit{ we can simply think of the data as given at the two spheres $S_{0},S_{1}$ as follows\n\\[ \\text{Data}(S_{0})=\\left\\{  Y\\psi\\left.\\right|_{S_{0}}, \\ L^{n}\\psi\\left.\\right|_{S_{0}},\\, n\\geq 0\\right\\}\\]\nand \n\\[ \\text{Data}(S_{1})=\\left\\{  Y\\psi\\left.\\right|_{S_{1}}, \\ L^{n}\\psi\\left.\\right|_{S_{1}},\\, n\\geq 0\\right\\},\\]\nwhere $L$ is tangential to the null generator of $\\mathcal{H}$. \nThen, our problem is to smoothly extend $\\psi$ on $\\mathcal{H}$ between $S_{0}$ and $S_{1}$ such that the transversal derivative $Y\\psi$ is continuous on $\\mathcal{H}\\cap\\left\\{0\\leq v\\leq 1\\right\\}$ (and hence such that $\\psi$ is $C^{1}$ on $\\mathcal{H}\\cap\\left\\{0\\leq v\\leq 1\\right\\}$). \n}\n\n\n\\medskip\n\n\nNote that the $C^{k}$ case with $k>1$ (where one needs to ``glue'' transversal derivatives up to the $k$'th order) is addressed here, for simplicity, only in the spherically symmetric case; however the general case can be addressed by an amalgamation of the methods presented in Sections \\ref{sec:TheGeneralCase} and \\ref{sec:Genericity}. \n\n\n\nLet  $S_{0}$ and $S_{1}$ be two leaves of a foliation $\\mathcal{S}$ of $\\mathcal{H}$. Clearly, if $\\mathcal{H}$ admits non-trivial conservation laws with respect to the foliation $\\mathcal{S}$ in the sense of  Definition \\ref{definitionconservationlaw}, then gluing constructions of data at $S_{0}$ and $S_{1}$ are not always possible since the prescribed charges at $S_{0}$ and $S_{1}$ may not coincide. However, it is not a priori obvious if such charges are the only obstruction to gluing of characteristic data. On the other hand, if we can show that we can always glue to first order  characteristic data on $\\mathcal{H}$ then it immediately follows that $\\mathcal{H}$ does not admit any non-trivial charges. \n\n\n\\subsection{The main theorems}\n\\label{sec:TheMainResultxx}\n\nThe next theorem 1) characterizes the conservation laws on null hypersurfaces,  2) derives their role in the characteristic gluing problem, 3) provides necessary and sufficient conditions for their existence in terms of the kernel of the elliptic operator $\\mathcal{O}^{\\mathcal{S}}$ defined by \\eqref{adjoint}, 4) uncovers their behavior under change of foliations and 5) establishes their non-genericity. \n\n\\begin{mytheo}\nLet $\\mathcal{H}$ be a regular, free from conjugate or focal points null hypersurface of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. Let also  $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ be a foliation of $\\mathcal{H}$, such that $S_{v}$ are diffeomorphic to $\\mathbb{S}^{2}$, and $\\mathcal{O}^{\\mathcal{S}}$ be the associated elliptic operator given by \\eqref{adjoint}. Then we have the following\n\\begin{itemize}\n\n\t\\item \\textbf{Characteristic gluing constructions and conservations laws:}\t\n\tOne can perform first order gluing constructions on $\\mathcal{H}$ in the sense of  Definition \\ref{firstordergluingdefinition} for general characteristic data if and only if there are no first order conservation laws on $\\mathcal{H}$ in the sense of  Definition \\ref{definitionconservationlaw}, i.e. $\\mathcal{W}^{\\mathcal{S}}=\\left\\{0\\right\\}$. If $\\mathcal{H}$ admits conservation laws, then we can glue characteristic data if and only if their associated charges are equal. \n\t\t\n\t\\item  \\textbf{Classification of null hypersurfaces admitting conservation laws:} Consider the following linear space\n\t\\begin{equation}\n\t\\mathcal{U}^{\\mathcal{S}}=\\left\\{\\Theta^{\\mathcal{S}}\\in C^{\\infty}(\\mathcal{H})\\, :\\ L\\Theta^{\\mathcal{S}}=0, \\ \\mathcal{O}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot\\Theta^{\\mathcal{S}}\\right)=0 \\text{ on }\\mathcal{H}  \\right\\}\\subset \\mathcal{V}_{\\mathcal{H}},\n\t\\label{newdef}\n\t\\end{equation}\n\twhere $\\phi$ denotes the conformal factor of the sections of $\\mathcal{S}$. Then, the null hypersurface $\\mathcal{H}$ admits first order conservation laws with respect to $\\mathcal{S}$ for the wave equation \\eqref{wave} in the sense of Definition \\ref{definitionconservationlaw} if and only if   $\\mathcal{U}^{\\mathcal{S}}\\neq \\left\\{ 0\\right\\}$.  In fact, the kernel of the conservation laws satisfies $\\mathcal{W}^{\\mathcal{S}}=\\mathcal{U}^{\\mathcal{S}}$.\n\n\t\n\t\n\t\\item \\textbf{Classification of conservation laws on null hypersurfaces:} No conserved (linear or non-linear) quantities (or, more generally, monotonic in $v$ quantities) involving the 1-jet of solutions to the wave equation exist on $\\mathcal{H}$ apart from the conservation laws given precisely by  Definition \\ref{definitionconservationlaw}. Moreover, $\\mathcal{H}$ can only admit finitely many linearly independent conservation laws, i.e.~$\\dim \\mathcal{W}^{\\mathcal{S}}<\\infty$. \n\t\n\t\t\\item \\textbf{Conservation laws and change of foliation:} If $\\mathcal{H}$ admits a conservation law with respect to the foliation $\\mathcal{S}=\\left\\langle S_{0},L_{geod}, \\Omega\\right\\rangle$, then it also admits a conservation law with respect to any other foliation $\\mathcal{S}'=\\big\\langle S'_{0},L_{geod}',\\Omega'\\big\\rangle$.  Specifically, the kernels $\\mathcal{W}^{\\mathcal{S}},\\mathcal{W}^{\\mathcal{S}'}$ satisfy $\\mathcal{W}^{\\mathcal{S}'}=f^{2}\\cdot\\mathcal{W}^{\\mathcal{S}}=\\left\\{f^{2}\\cdot \\Theta^{\\mathcal{S}},\\,\\Theta^{\\mathcal{S}}\\in \\mathcal{W}^{\\mathcal{S}}\\right\\}$, where $f\\in\\mathcal{V}_{\\hh}$ such that $L_{geod}'=f^{2}\\cdot L_{geod}$, and so $\\dim \\mathcal{W}^{\\mathcal{S}'}=\\dim \\mathcal{W}^{\\mathcal{S}}$. Moreover, the value of the conserved charges is independent of the choice of foliation. \n\n\n\\item \\textbf{Non-genericity of conservation laws:} A null hypersurface does not admit conservation laws for generic ambient metrics. The same result holds even if we restrict to spacetimes $(\\mathcal{M},g)$ satisfying the Einstein-vacuum equations. \n\\end{itemize}\n\\label{theoremmainintro}\n\\end{mytheo}\n\n\nTheorem \\ref{theoremmainintro} can be used to show that the event horizon of a subextremal black hole does not admit conservation laws and that the event horizon of an extemal black hole admits a unique conservation law: \n\n\n\\begin{mytheo}\n \\textbf{Conservation laws on extremal black holes:}\tThe event horizon ${\\mathcal{H}}^{+}$ of \nan extremal black hole satisfies $\\dim \\mathcal{U}^{\\mathcal{S}}_{{\\mathcal{H}}^{+}} =1$ and the unique corresponding conservation law coincides with the conservation law on  extremal black holes found in \\cite{aretakis4, hj2012, murata2012}. On the other hand, any Killing horizon with positive surface gravity and negative transversal null mean curvature (such as the event horizon of any subextremal Kerr black hole) does not admit conservation laws.\n\n\\label{the2}\n\\end{mytheo}\n\t\n\tTheorem \\ref{theoremmainintro} applied in a limiting sense on null infinity of asymptotically flat spacetime recovers the Newman--Penrose constant and in fact shows that it is the \\textbf{only} conserved charge along null infinity:\n\t\n\t\\begin{mytheo}\n\t\\textbf{Conservation laws and the Newman--Penrose constant:}\n\tThe null infinity $\\mathcal{I}$ of an asympotically flat spacetime satisfies $\\dim\\mathcal{U}^{\\mathcal{S}}_{\\mathcal{I}}=1$ with respect to an appropriately rescaled operator $\\mathcal{O}_{\\mathcal{I}}^{\\ \\mathcal{S}}$, and the \\textbf{unique} corresponding conservation law  coincides with the  (first-order) Newman--Penrose constant on $\\mathcal{I}$.\n\t\\label{theo3}\n\t\\end{mytheo}\n\t\n\tFinally, the following theorem derives necessary and sufficient conditions for the existence of higher order conservation laws in the context of spherical symmetry.\n\n\t \\begin{mytheo}\n\t\\textbf{Higher order gluing constructions and conservation laws in spherical symmetry:} Let $(\\mathcal{M},g)$ be a spherically symmetric spacetime and $\\mathcal{H}$ be a spherically symmetric null hypersurface. Then, for all $k,l\\in\\mathbb{N}$ there is a unique expression $R_{k,l}$ which depends only on the geometry of $\\mathcal{H}$ such that if  $R_{i,l}\\neq 0$ for  $i=1,...,k-1$ and $R_{k,l}=0$ on $\\mathcal{H}$ then there is a conservation law involving the $k$-jet of solutions to the wave equation. The kernel of this conservation law consists of all the eigenfunctions of the standard spherical Laplacian $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ which correspond to the eigenvalue $-l(l+1)$. If, on the other hand, we have that for all $l\\in \\mathbb{N}$  $R_{i,l}\\neq 0,\\,  i=1,...,k$ almost everywhere on $\\mathcal{H}$ then we can glue characteristic data up to the $k$'th order. Moreover, the higher-order Newman--Penrose constants are limiting examples of the above charges. \n\n\\label{theo4}\n\\end{mytheo}\n\n\n\\subsection{PDE aspects of the conservation laws}\n\\label{sec:Remarks}\n\n\nThe existence of the charge \\eqref{eextremalcharge} implies that the solutions to the wave equation on extremal black holes do not disperse along the event horizon. This is in stark contrast to the result of Dafermos and Rodnianski \\cite{tria} who derived decay results for $\\psi$ and all its derivatives up to and including the event horizon for the general subextremal Kerr family $|a|<M$ (for decay results on slowly rotating Kerr black holes see \\cite{blukerr, lecturesMD, tataru2}). In fact, a closer analysis \\cite{aretakis1,aretakis2,aretakis3} showed that for generic initial data we have\n\\[\\left|\\psi\\right|\\rightarrow 0, \\ \\ \\left|Y\\psi\\right|\\nrightarrow 0, \\ \\ \\left|Y^{k}\\psi\\right|\\rightarrow+\\infty,\\text{ for }k\\geq 2, \\]\nalong the (degenerate) event horizon. Similar blow-up results were obtained for the higher order energies.  The asymptotic behavior of tails along the event horizon was studied by Lucietti et al \\cite{hm2012} and Ori \\cite{ori2013}.  Further applications were given by  Dain and Dotti  \\cite{dd2012}. A very recent numerical analysis of this instability in the context of the Einstein-Maxwell-scalar field system under spherical symmetry can be found in \\cite{harvey2013}.  A non-linear instability arrising from the conserved charges was given in \\cite{aretakis2013}.\n\n\nIn a broader content and at a slightly philosophical level, transversal derivatives to globally distinguished null hypersurfaces usually pose significant difficulties in proving dispersive estimates for wave type equations in view of the fact that one does not have an a priori equation for these derivatives along their direction (i.e.~we do not have an equation of the form $Y^{\\mathcal{S}}Y^{\\mathcal{S}}\\psi=...$).  For this reason, understanding the structure of the transversal derivatives in PDE is of fundamental importance; see for example the null condition introduced by Klainerman \\cite{SK86} which allowed him to obtain global existence of quasilinear equations with small initial data. It would be interesting to see further global repercussions of the conservation laws defined here to other linear and non-linear settings. \n\n\n\n\n\n\n\n\n\\subsection{Outline of the paper}\n\\label{sec:OutlineOfThePaper}\n\n\nAn outline of the paper is as follows: In Section \\ref{sec:TheGeometricSetting} we introduce the basic geometric concepts relevant to a double null foliation and in Section \\ref{sec:TheGeneralCase} we derive the relation between the  conservation laws and  characteristic gluing constructions on a general null hypersurface. In Section \\ref{sec:TheNullInfinityMathcalI} we restrict to the null infinity of asympotically flat spacetimes and recover the Newman--Penrose constants. In Section \\ref{sec:KillingHorizons} we apply the main theorem to Killing horizons with particular emphasis on the event horizon of black holes. Finally, in the Section \\ref{sec:Genericity} we derive necessary and sufficient conditions for higher order conservation laws and gluing constructions in spherical symmetry. \n\n\n\\section{The geometric setting}\n\\label{sec:TheGeometricSetting}\n\n\nLet $\\mathcal{H}$ be a regular null hypersurface of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. We will express the wave equation in an appropriate frame which will allow us to explicitly make use of the elliptic structure on $\\mathcal{H}$. For we work with canonical coordinates suitably adapted to a double null foliation of $\\mathcal{M}$ which we introduce in the next subsection. \n\n\n\\subsection{The double null foliation}\n\\label{sec:TheDoubleNullFoliation}\n\n\n\\paragraph{Null Foliations and Optical Functions\\medskip \\\\}\n\\label{sec:NullFoliationsandOpticalFunctions}\n\n\n\nA \\textit{foliation} $\\mathcal{S}$ of a null hypersurface $\\mathcal{H}$ in a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$ is a collection of sections $S_{v}$, smoothly varying in $v$, such that $\\cup_{v}S_{v}=\\mathcal{H}$. We assume that $S_{v}$ are diffeomorphic to the 2-sphere. We will show that any foliation is uniquely determined by the choice of one section, say $S_{0}$, the choice of a null tangential to $\\mathcal{H}$ vector field $\\left.L_{geod}\\right|_{S_{0}}$ restricted on $S_{0}$ and a function $\\Omega$ on $\\mathcal{H}$. We extend $\\left.L_{geod}\\right|_{S_{0}}$  to a  null vector field tangential to the null generators of $\\mathcal{H}$ such that\n\\[\\nabla_{{L}_{geod}}{{L}_{geod}}=0.\\]We then define the vector field\n\\begin{equation}\nL=\\Omega^{2}\\cdot L_{geod}\n\\label{eq:l}\n\\end{equation}\non $\\mathcal{H}$ and consider the affine parameter $v$ of $L$ such that \n\\[Lv=1, \\text{ with } v=0, \\text{ on } S_{0}.\\]\nLet $S_{v}$ denote the level sets of $v$ on $\\mathcal{H}$ which are the leaves of the foliaton $\\mathcal{S}$. We use the notation\n\\begin{equation}\n\\mathcal{S}=\\left\\langle S_{0}, \\left.L_{geod}\\right|_{S_{0}}, \\Omega \\right\\rangle.\n\\label{foliationdef}\n\\end{equation}\nWe can also extend the above construction to obtain a foliation of (an appropriate region of) $\\mathcal{M}$ by null hypersurfaces $C_{u},\\underline{C}_{v}$ intersecting at embedded 2-spheres $S_{u,v}$.  We denote $C_{0}=\\mathcal{H}$ and define $\\underline{C}_{0}$ to be the null hypersurface generated by null hypersurfaces normal to $S_{0}$ and conjugate to $C_{0}$. \n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.1]{p43.png}\n\t\\label{fig:p43}\n\\end{figure}\nWe then consider the vector field $\\underline{L}_{geod}\\!\\left.\\right|_{S_{0}}$ at $S_{0}$ tangential to the null generators of $\\underline{C}_{0}$ determined by the relation\n\\[g({L}_{geod},\\underline{L}_{geod})=-\\Omega^{-2}\\]\nand extend it on $\\underline{C}_{0}$ via the geodesic equation\n\\[ \\nabla_{\\underline{L}_{geod}}\\underline{L}_{geod}=0. \\]\nWe now extend $\\Omega$ to be a function on $\\underline{C}_{0}$ and, as before, consider the vector field\n\\begin{equation}\n\\begin{split}\n \\underline{L}=\\Omega^{2}\\cdot \\underline{L}_{geod}\n\\end{split}\n\\label{normalizedvf}\n\\end{equation}\nand similarly define the function $u$ on $\\underline{C}_{0}$ by \n\\begin{equation*}\n\\underline{L}u=1, \\ \\ \\text{with } u=0 \\text{ on }S_{0}.\n\\end{equation*}\nLet $\\underline{S}_{\\tau}$ be the embedded 2-surface on $\\underline{C}_{0}$ such that $u=\\tau$.\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.101]{p44.png}\n\t\\label{fig:p44}\n\\end{figure}\n\\noindent We also define $\\underline{L}_{geod}$ on $C_{0}$ such that $\\underline{L}_{geod}$ is null and normal to $S_{\\tau}$ and $g(L_{geod},\\underline{L}_{geod})=-\\Omega^{-2}$. We similarly extend $L_{geod}$ on $\\underline{C}_{0}$. \n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.11]{p445.png}\n\t\\label{fig:p44}\n\\end{figure}\nConsider the affinely parametrized null geodesics which emanate from the points on $S_{\\tau}$ with initial tangent vector $\\underline{L}_{geod}$. These geodesics, whose tangent we denote by $\\underline{L}_{geod}$, span  null hypersurfaces  which we will denote by $\\underline{C}_{\\tau}$. Hence, $C_{0}\\cap\\underline{C}_{\\tau}=S_{\\tau}$. Similarly, we define $L_{geod}$ globally (and the hypersurfaces $C_{\\tau}$ such that their normal is $L_{geod}$). Extend the vector field $L,\\underline{L}$ to global vector fields such that \n \\begin{equation*}\n\\begin{split}\nL=\\Omega^{2}\\cdot L_{geod}, \\ \\ \\ \\underline{L}=\\Omega^{2}\\cdot\\underline{L}_{geod}.\n\\end{split}\n\\end{equation*}\nWe will refer to $\\Omega$ as the null lapse function. We also extend the functions $u,v$ to global functions such that \n \\begin{equation*}\n\\begin{split}\nLu=0, \\ \\ \\ \\ \\ \\underline{L}\\,v=0.\n\\end{split}\n\\end{equation*}\nTherefore, \n \\begin{equation*}\n\\begin{split}\nC_{\\tau}=\\left\\{u=\\tau\\right\\}, \\ \\ \\ \\underline{C}_{\\tau}=\\left\\{v=\\tau\\right\\},\n\\end{split}\n\\end{equation*}\nand hence $u,v$ are optical functions.  \n\\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.087]{p42.png}\n\t\\label{fig:p42}\n\\end{figure}\n\n The importance of the renormalized vector fields $L,\\underline{L}$ is manifest from the following \n\\begin{proposition}\nThe optical functions $u,v$ satisfy the following relations\n\\[\\nabla v=-\\underline{L}_{geod}, \\ \\ \\ \\ \\nabla u=-L_{geod}\\]\nand \n\\[Lv=1, \\ \\ \\ \\ \\underline{L} u=1. \\]\n\\label{impo}\n\\end{proposition}\n\\begin{proof}\n\nSince $u,v$ are optical functions we have that $\\nabla u,\\nabla v$ satisfy the geodesic equation. Since $L_{geod},\\underline{L}_{geod}$ satisfy the geodesic equation as well, it suffices to show, for instance, that $\\nabla v=-\\underline{L}_{geod}$ on $C_{0}$. Expressing $\\nabla v$ in terms of the null frame $(L_{geod},\\underline{L}_{geod}, e_{1}, e_{2})$, where $e_{1},e_{2}$ is a local frame on the spheres $S_{\\tau}$, we obtain\n \\begin{equation*}\n(\\nabla v)^{\\underline{L}_{geod}}=g^{\\underline{L}_{geod}L_{geod}}\\cdot(L_{geod}v)+g^{\\underline{L}_{geod}\\underline{L}_{geod}}\\cdot(\\underline{L}_{geod}v)=-\\Omega^{2}\\cdot\\Omega^{-2}=-1,\n\\end{equation*}\non $C_{0}$, and similarly, we obtain that the remaining components with respect to the above frame are zero. This proves the first equation. For the second, it suffices to notice that \n\\[Lv=g(L,\\nabla v)=g(L,-\\underline{L}_{geod})=-\\Omega^{2}\\cdot g(L_{geod},\\underline{L}_{geod})=1.\\]\n\n\\end{proof}\nNote  that\n\\begin{equation}\ng\\big(\\underline{L},L_{geod}\\big)=-1,\n\\label{eq:unormalizationgeod}\n\\end{equation}\nwhich shows that $L_{geod}$ determines to first order the optical function $u$ on $\\mathcal{H}=C_{0}$. \n\n\n\n\\paragraph{Gauge Freedom\\medskip \\\\}\n\\label{sec:FixingAGauge}\n\n\n\n\nThe above analysis shows that a double null foliation $\\mathcal{D}$ can be completely determined by the following \n\\[\\mathcal{D}=\\left\\langle S_{0}, \\left. L_{geod}\\right|_{S_{0}}, \\left.\\Omega\\right|_{C_{0}},\\left.\\Omega\\right|_{\\underline{C}_{0}}\\right\\rangle.\\]\nNote that the freedom for the vector field $\\left. L_{geod}\\right|_{S_{0}}$ and the functions $\\left.\\Omega\\right|_{C_{0}},\\left.\\Omega\\right|_{\\underline{C}_{0}}$ reflects the freedom for the functions $u'=u'(u), v'=v'(v)$. Indeed, fixing $\\Omega$ and $L_{geod}$ determines up to additive constants the (optical) functions $u,v$. \n\n\n\\paragraph{The Diffeomorphisms $\\Phi_{u,v}$\\medskip \\\\}\n\\label{sec:coordinatescdiffeo}\n\n\n\n\n\n\nWe can construct a  diffeomorphism $\\Phi_{u,v}$ from any sphere $S_{u,v}$ to  $S_{0}$ as follows: If $p\\in S_{u,v}$ then we can consider the point $q\\in S_{0,v}$ which is the intersection of $C_{0}$ and the null generator of $\\underline{C}_{u}$ passing through $p$. We then let $\\Phi_{u,v}(p)$ to be the unique point of intersection of $S_{0}$ and the null generator of $C_{0}$ passing through $q$. We can also consider a diffeomorphism \n\\begin{equation}\n\\Phi: S_{0}\\rightarrow\\mathbb{S}^{2}\n\\label{phidiffeo}\n\\end{equation}\nand compose $\\Phi_{u,v}$ with $\\Phi$ to obtain a diffeomorphism from $S_{u,v}$ to $\\mathbb{S}^{2}$. This constructions allows us, for example, to equip all surfaces $S_{u,v}$ with the standard metric in a very precise way. See also the figure below.\n\n\n\n\\paragraph{The Canonical Coordinate System\\medskip \\\\}\n\\label{sec:coordinatesc}\nOne can consider at each point a \\textit{null frame} $(L,\\underline{L},e_{1},e_{2})$ adapted to the double null foliation, where $e_{1},e_{2}$ is a local frame for the spheres $S_{u,v}$. However, these vector fields do not correspond to a coordinate system. \n\nUsing the optical functions $u,v$ we will introduce a coordinate system suitably adapted to the corresponding double null foliation of the spacetime.\n\nIf $p\\in\\mathcal{M}$ then $p\\in C_{u_{0}}\\cap\\underline{C}_{v_{0}}$ and hence $u(p)=u_{0}, v(p)=v_{0}$. We next prescribe angular coordinates for the point $p$ on the 2-surface $C_{u_{0}}\\cap\\underline{C}_{v_{0}}$.\n\nLet $(\\theta^{1},\\theta^{2})$ denote a coordinate system on a domain of $S_{0}$.  Then  we assign to $p$ the angular coordinates of the point $\\Phi_{u_{0},v_{0}}(p)$ as depicted below\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.07]{p45paper.png}\n\t\\label{fig:p45}\n\\end{figure}\nBy construction we have everywhere: \\[\\frac{\\partial}{\\partial{u}}=\\underline{L}\\] and \n\\[\\frac{\\partial}{\\partial{\\theta^{1}}},\\frac{\\partial}{\\partial{\\theta^{2}}}\\in TS_{u,v},\\]\nwhereas\n\\[\\frac{\\partial}{\\partial{v}}=L \\text{ : on }C_{0}.\\]\nNote that the latter equation will not in general hold everywhere, as it is easily seen from the picture below:\n \\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.08]{p46paper.png}\n\t\\label{fig:p46}\n\\end{figure}\nFrom now on, for simplicity we denote $\\partial_{v}=\\frac{\\partial}{\\partial{v}}$, and so on.\n In general we have:\n\\[\\partial_{v}=L+b^{i}\\partial_{\\theta^{i}}.\\]\nBy virtue of the equations $\\underline{L}=\\partial_{u}$ and $[\\partial_{u},\\partial_{v}]=[\\partial_{u},\\partial_{\\theta^{i}}]=0$ we obtain:\n\\begin{equation*}\n[L,\\underline{L}]=-\\frac{\\partial{b^{i}}}{\\partial{u}}\\partial_{\\theta^{i}}\\in TS_{u,v},\n\\end{equation*}\nand therefore,\n\\begin{equation}\n\\frac{\\partial{b^{i}}}{\\partial{u}}=-d\\theta^{i}([L,\\underline{L}]), \\text{ and }b^{i}=0 \\text{ on }C_{0}=\\left\\{u=0\\right\\}. \n\\label{cootor}\n\\end{equation}\nHence, the S-tangent vector field $b=b^{i}\\partial_{\\theta^{i}}$ is the obstruction to the integrability of  $\\left\\langle L,\\underline{L}\\right\\rangle=\\big(TS_{u,v}\\big)^{\\perp}$. In order to compute  $b$ it suffices to compute $[L,\\underline{L}]$. Since $[L,\\underline{L}]=\\nabla_{L}\\underline{L}-\\nabla_{\\underline{L}}L$, it suffices to compute the connection coefficients (see below).\n\nThe metric $g$ with respect to the canonical coordinates is given by\n\\begin{equation}\ng=-2\\Omega^{2}dudv+(b^{i}\\, b^{j}\\,\\mbox{$g \\mkern-8.8mu \/$\\,}_{ij})dvdv-2(b^{i}\\, \\mbox{$g \\mkern-8.8mu \/$\\,}_{ij})d\\theta^{j}dv+\\mbox{$g \\mkern-8.8mu \/$\\,}_{ij}\\,d\\theta^{i}d\\theta^{j},\n\\label{metriccan}\n\\end{equation}\nwhere $\\mbox{$g \\mkern-8.8mu \/$\\,}$ denotes the induced metric on the 2-surfaces $S_{u,v}=C_{u}\\cap \\underline{C}_{v}$. We immediately obtain\n\\begin{equation}\n\\text{det}(g)=-\\Omega^{4}\\cdot \\text{det}(\\mbox{$g \\mkern-8.8mu \/$\\,}).\n\\label{detg}\n\\end{equation}\n\\bigskip\n\n\\noindent\\textbf{Null Frames}\n\\bigskip\n \n\\noindent From now on we denote $S_{u,v}=C_{u}\\cap \\underline{C}_{v}$. If $\\left\\{e_{1},e_{2}\\right\\}=\\big(e_{A}\\big)_{A=1,2}$ is an arbitrary frame on the spheres $S_{(u,v)}$ then we, in fact, have the following null frames:\n\\begin{itemize}\n\t\\item \n\\textbf{Geodesic frame:} $(e_{1},e_{2},L_{geod}, \\underline{L}_{geod})$,\n\\item\n\\textbf{Equivariant frame:} $(e_{1},e_{2},L,\\underline{L})$,\n\\item\n\\textbf{Normalized frame:} $(e_{1},e_{2}, e_{3},e_{4}),$\n\\item\n\\textbf{Coordinate frame:} $(\\partial_{\\theta^{1}},\\partial_{\\theta^{2}},\\partial_{v},\\partial_{u})$.\n\n\\end{itemize}\nwhere \\[e_{3}=\\Omega \\underline{L}_{geod}=\\frac{1}{\\Omega}\\underline{L}, \\ \\ \\ \\ e_{4}=\\Omega L_{geod}=\\frac{1}{\\Omega}{L}.\\]\n\\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.135]{3344.png}\n\t\\label{fig:p3344}\n\\end{figure} \nNote that $e_{3},e_{4}$ satisfy the normalization property\n\\begin{equation}\ng(e_{3},e_{4})=-1.\n\\label{eq:}\n\\end{equation}\n\n\n\\smallskip\n\n\\noindent\\textbf{The conformal geometry and conformal factor}\n\\bigskip\n \n\nThe conformal class of $\\mbox{$g \\mkern-8.8mu \/$\\,}$ contains a unique representative (metric) $\\hat{g}$ such that $\\sqrt{\\hat{g}}=\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}$. Here we consider the diffeomorphism $\\Phi\\circ\\Phi_{v}$ to identify the section $S_{v}$ with $\\mathbb{S}^{2}$. Equivalently, $\\mbox{$g \\mkern-8.8mu \/$\\,}$ and $h$ are such that the induced volume forms on $S_{v}$ are equal. Since $\\mbox{$g \\mkern-8.8mu \/$\\,}$ and $\\hat{g}$ are conformal there is a conformal factor such that $\\mbox{$g \\mkern-8.8mu \/$\\,}=\\phi^{2}\\cdot \\hat{g}$. Then, $\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}=\\phi^{2}\\sqrt{\\hat{g}}=\\phi^{2}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}$ and hence \n\\[\\phi^{2}= \\frac{\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}}{\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}}. \\]\n Clearly the above imply that \n\\begin{equation}\n\\phi=\\frac{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}}.\n\\label{phi}\n\\end{equation}\nNote that $\\phi$ is a smooth function on the sphere $S_{v}$ and does not depend on the choice of the coordinate system. Note also that for a spherically symmetric metric we have $\\phi=r$, where $r$ is the radius. \n\n\n\\bigskip\n\n\\noindent\\textbf{Connection Coefficients}\n\n\\bigskip\n \n\nWe consider the \\textbf{normalized frame} $(e_{1},e_{2},e_{3},e_{4})$ defined above.  We define the connection coefficients with respect to this frame to be the smooth functions $\\Gamma^{\\lambda}_{\\mu\\nu}$\nsuch that \n\\[\\nabla_{e_{\\mu}}e_{\\nu}=\\Gamma^{\\lambda}_{\\mu\\nu}e_{\\lambda}, \\ \\ \\lambda, \\mu, \\nu\\in\\left\\{1,2,3,4\\right\\}\\]\nHere $\\nabla$ denotes the connection of the spacetime metric $g$. We are mainly interested in the case where at least one of the indices $\\lambda,\\mu,\\nu$ is either 3 or 4 (otherwise, we obtain the Christoffel symbols with respect to the induced metric $\\mbox{$g \\mkern-8.8mu \/$\\,}$).  Following \\cite{DC09,christab}, these coefficients  are completely determined by the following components:\n\\medskip\n\n\\noindent\\textbf{The components} $\\chi,\\underline{\\chi},\\eta,\\underline{\\eta}, \\omega,\\underline{\\omega},\\zeta$:\n\\smallskip\n\\begin{equation}\n\\begin{split}\n&\\chi_{AB}=g(\\nabla_{A}e_{4},e_{B}), \\ \\ \\ \\ \\ \\underline{\\chi}_{AB}=g(\\nabla_{A}e_{3},e_{B}),\\\\\n&\\ \\eta_{A}=g(\\nabla_{3}e_{4},e_{A}),\\ \\ \\ \\ \\  \\ \\,  \\underline{\\eta}_{A}=g(\\nabla_{4}e_{3},e_{A}),\\\\\n&\\ \\ \\ \\omega=-g(\\nabla_{4}e_{4},e_{3}), \\ \\ \\ \\  \\underline{\\omega}=-g(\\nabla_{3}e_{3},e_{4}),\\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\  \\zeta_{A}=g(\\nabla_{A}e_{4},e_{3})\n\\end{split}\n\\label{concoef}\n\\end{equation}\nwhere $\\big(e_{A}\\big)_{A=1,2}$ is an arbitrary frame on the spheres $S_{v}$ and $\\nabla_{\\mu}=\\nabla_{e_{\\mu}}$. Note that $\\underline{\\zeta}=-\\zeta$. The covariant tensor fields $\\chi,\\underline{\\chi},\\eta,\\underline{\\eta},\\zeta$ are only defined on $T_{x}S_{v}$. \\textbf{We can naturally extend these to tensor fields to be defined on $T_{x}\\mathcal{M}$ by simply letting their value to be zero if they act on $e_{3}$ or $e_{4}$. Such tensor fields will in general be called \\underline{$S$-tensor fields}. Note that a vector field is an $S$-vector field if it is tangent to the spheres $S_{v}$.} The importance the $S$-tensors originates from the fact that one is interested in understanding the embedding of $S_{v}$ in $\\mathcal{H}$.  \n\n\n\n\nThe connection coefficients $\\Gamma$ can be recovered by the following relations:\n\\begin{equation}\n\\begin{split}\n\\nabla_{A}e_{B}=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{A}e_{B}+&\\chi_{AB}e_{3}+\\underline{\\chi}_{AB}e_{4},\\\\\n\\nabla_{3}e_{A}=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{3}e_{A}+\\eta_{A}e_{3},& \\ \\ \\ \\nabla_{4}e_{A}=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{4}e_{A}+\\underline{\\eta}_{A}e_{4},\\\\\n\\nabla_{A}e_{3}=\\underline{\\chi}_{A}^{\\ \\ \\sharp B}e_{B}+\\zeta_{A}e_{3},& \\ \\ \\ \\nabla_{A}e_{4}={\\chi}_{A}^{\\ \\ \\sharp B}e_{B}-\\zeta_{A}e_{4},\\\\\n\\nabla_{3}e_{4}=\\eta^{\\sharp A}e_{A}-\\underline{\\omega}e_{4},& \\ \\ \\ \\nabla_{4}e_{3}=\\underline{\\eta}^{\\sharp A}e_{A}-\\omega e_{3},\\\\\n\\nabla_{3}e_{3}=\\underline{\\omega}e_{3},& \\ \\ \\ \\nabla_{4}e_{4}=\\omega e_{4},\n\\end{split}\n\\label{connectioncoef}\n\\end{equation} \n\n\n\\bigskip\n\n\\noindent\\textbf{Curvature Components}\n\n\\bigskip\n \n\n\n\n \nWe next decompose the Riemann curvature $R$ in terms of the normalized null frame. First, we define the following components, which contain at most two S-tangential components (and hence at least 2 null components):\n\\begin{equation}\n\\begin{split}\n\\alpha_{AB}=R_{A4B4},&\\ \\ \\ \\ \\underline{\\alpha}_{AB}=R_{A3B3},\\\\\n\\beta_{A}=R_{A434},&\\ \\ \\ \\ \\underline{\\beta}_{A}=R_{A334},\\\\\n\\rho= R_{3434}, & \\ \\ \\ \\sigma=\\frac{1}{2}\\mbox{$\\epsilon \\mkern-7.4mu \/$\\,}^{AB}R_{AB34}.\n\\end{split}\n\\label{curvcompdeflist}\n\\end{equation}\nNote that $R(\\cdot,\\cdot, e_{3},e_{4})$, when restricted on $T_{x}S_{v}$, is an antisymmetric form and hence collinear to the volume form $\\mbox{$\\epsilon \\mkern-7.4mu \/$\\,}$ on $S_{v}$. Furthermore, if \n\\[(*R)_{3434}=*\\rho,\\]\nthen $*\\rho= 2\\sigma$. Here,  the dual $*R$ of the Riemann curvature is defined to be the (0,4) tensor:\n\\[(*R)_{\\alpha\\beta\\gamma\\delta}=\\epsilon_{\\mu\\nu\\alpha\\beta}\\,R^{\\mu\\nu}_{\\ \\ \\  \\gamma\\delta}.\\]\nClearly, the (0,2) $S$-tensor fields $\\alpha,\\underline{\\alpha}$ are symmetric. Note that if the Einstein equations $Ric(g)=0$ are satisfied, then all the remaining curvature components can be expressed in terms of the above components. \n\n\\bigskip\n\n\\noindent\\textbf{Remarks:}\n\\medskip\n\n\\textbf{1.} Recall, that the second fundamental form of a manifold $S$ embedded in a manifold $\\mathcal{M}$ is defined to be the symmetric $(0,2)$ tensor field $II$ such that for each $x\\in S$ we have\n\\[II_{x}:T_{x}S\\times T_{x}S\\rightarrow (T_{x}S)^{\\perp}, \\]\nwhere the $\\perp$ is defined via the decomposition $T_{x}\\mathcal{M}=T_{x}S\\oplus(T_{x}S)^{\\perp}$. Specifically, if $X,Y\\in T_{x}S$ then \\[II_{x}(X,Y)=\\big(\\nabla_{X}Y\\big)^{\\perp}\\] and hence\\[\\nabla_{X}Y=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{X}Y+II(X,Y).\\]\nHere $\\mbox{$\\nabla \\mkern-13mu \/$\\,}$ denotes the induced connection on $S$ (which is taken by projecting the spacetime connection $\\nabla$ on $T_{x}S$).\n\n The $S$-tensor fields $\\chi,\\underline{\\chi}$ give us the projections of $II_{AB}$ on $e_{3}$ and $e_{4}$, respectively. Indeed\n \\[II(X,Y)=\\chi(X,Y)e_{3}+\\underline{\\chi}(X,Y)e_{4}.\\]\n \n For this reason we will refer to \n $\\chi,\\underline{\\chi}$ as the \\textit{null second fundamental forms} of $S_{v}$.   One can easily verify that $\\chi$ and $\\underline{\\chi}$ are symmetric (0,2) $S$-tensor fields. Indeed, a simple calculation shows that if $X,Y$ are $S$-tangent vector fields then\n \\begin{equation*}\n \\begin{split}\n  \\chi(X,Y)-\\chi(Y,X)=g(e_{4},[X,Y]) ,\\\\ \\underline{\\chi}(X,Y)-\\underline{\\chi}(Y,X)=g(e_{3},[X,Y]).\n \\end{split}\n \\end{equation*}\n Hence, $\\chi,\\underline{\\chi}$ are symmetric if and only if $[X,Y]\\perp e_{3}$ and $[X,Y]\\perp e_{4}$ and thus if and only if $\\left\\langle e_{3},e_{4}\\right\\rangle^{\\perp}\\ni [X,Y]\\in TS_{v}$. The symmetry of $\\chi,\\underline{\\chi}$ is thus equivalent to the integrability of the orthogonal complement $\\left\\langle e_{3},e_{4}\\right\\rangle^{\\perp}$.\n \n  Furthermore, we can decompose $\\chi$ and $\\underline{\\chi}$ into their trace and traceless parts by\n \\begin{equation}\n\\chi=\\hat{\\chi}+\\frac{1}{2}(tr\\chi)\\mbox{$g \\mkern-8.8mu \/$\\,}, \\ \\ \\ \\\n\\underline{\\chi}=\\hat{\\underline{\\chi}}+\\frac{1}{2}(tr\\underline{\\chi})\\mbox{$g \\mkern-8.8mu \/$\\,}.\n\\label{tracechi}\n\\end{equation}\nThe trace of the $S$-tensor fields $\\chi,\\underline{\\chi}$ (and more general $S$-tensor fields) is taken with respect to the induced metric $\\mbox{$g \\mkern-8.8mu \/$\\,}$. The trace $tr\\chi$ is known as the  \\textit{expansion} and the component $\\hat{\\chi}$ is called the \\textit{shear} of $S_{v}$ with respect to $\\mathcal{H}$.\n\n\n\n\n\n\\medskip\n\n\\textbf{2.} The $S$ 1-form $\\zeta$ is known as the \\textit{torsion}. \nIf $\\mbox{$d \\mkern-9.2mu \/$\\,}$ denotes the exterior derivative on $S_{v}$ then the $S$ 1-forms $\\eta,\\underline{\\eta}$ are related to $\\zeta$ via\n\\begin{equation*}\n\\begin{split}\\eta=\\zeta+\\mbox{$d \\mkern-9.2mu \/$\\,}(\\log\\Omega),\\ \\ \\ \\ \\underline{\\eta}=-\\zeta+\\mbox{$d \\mkern-9.2mu \/$\\,}{\\log\\Omega},\n\\end{split}\n\\end{equation*}\nand hence\n\\[\\zeta= \\frac{1}{2}(\\eta-\\underline{\\eta}), \\ \\ \\ \\ \\mbox{$d \\mkern-9.2mu \/$\\,}\\log\\Omega=\\frac{1}{2}(\\eta+\\underline{\\eta}).\\]\nThe 1-forms $\\eta,\\underline{\\eta}$ can be thought of as the torsion of the null hypersurfaces with respect to the geodesic vector fields. Indeed, the previous relations imply\n\\[\\eta_{A}=\\Omega^{2}g(\\nabla_{A}L_{geod}, \\underline{L}_{geod}).\\]\nFurthermore, we have\n\\begin{equation}\n\\begin{split}\n[L,\\underline{L}]=-2\\Omega^{2}\\zeta^{\\sharp}.\n\\end{split}\n\\label{torsionll}\n\\end{equation}\nHence the torsion $\\zeta$ is the obstruction to the integrability of the timelike planes $\\left\\langle e_{3},e_{4}\\right\\rangle$ orthogonal to the spheres $S_{v}$.\n\n\n\\medskip\n\n\\textbf{3. } Let $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}$ denote the projection of the Lie derivative ${\\cal L}_{L}$ onto the spheres $S_{v}$. The \\textbf{first variation formula} then reads\n\\begin{equation}\n\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\mbox{$g \\mkern-8.8mu \/$\\,}=2\\Omega\\chi,\\ \\ \\ \\ \\ \\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}(\\mbox{$g \\mkern-8.8mu \/$\\,}^{-1})=-2\\Omega\\chi^{\\sharp\\sharp}.\n\\label{1stvarform}\n\\end{equation}Note that we use the induced metric $\\mbox{$g \\mkern-8.8mu \/$\\,}$ to raise and lower indices.\n Hence, since $[L,\\partial_{\\theta^{i}}]=0$ on $\\mathcal{H}$,\n\\[ L\\big(\\mbox{$g \\mkern-8.8mu \/$\\,}_{ij}\\big)=2\\Omega\\chi_{ij} \\]\nand hence\n\\begin{equation}\nL\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}=\\Omega tr\\chi\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\n\\label{derdet}\n\\end{equation}\non $\\mathcal{H}$. Similarly, since $\\partial_{u}=\\underline{L}$ we obtain\n\\[\\partial_{u}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}=\\Omega tr\\underline{\\chi}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\]\neverywhere. Therefore,\n\\begin{equation}\n\\partial_{u}\\partial_{v}\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}=\\left[\\frac{1}{2}\\partial_{v}(\\Omega tr\\underline{\\chi})+\\frac{1}{4}(\\Omega tr\\chi)(\\Omega tr\\underline{\\chi})\\right]\\cdot \\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\n\\label{needforw}\n\\end{equation}\non $\\mathcal{H}$. \n\n\n\n\\medskip\n\n\\textbf{4. } We denote by $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L},\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{L}$ the projection of ${\\cal L}_{L},\\nabla_{L}$ on the sections $S_{v}$ and by $\\mbox{$\\triangle \\mkern-13mu \/$\\,},\\mbox{$\\nabla \\mkern-13mu \/$\\,}$ the induced Laplacian and gradient of $(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,})$. \n\n\n\\subsection{The wave equation}\n\\label{sec:TheWaveEquation}\nLet $\\mathcal{D}$ be a double null foliation of $\\mathcal{M}$ such that $\\mathcal{H}=\\left\\{u=0\\right\\}$ as defined in Section \\ref{sec:TheDoubleNullFoliation}. Let $\\mathcal{S}$ be the associated foliation on $\\mathcal{H}$. Using the canonical coordinates $(u,v,\\theta^{1},\\theta^{2})$ and recalling that $b^{A}=0$ on $\\mathcal{H}$ we obtain\n\\begin{equation*}\n\\begin{split}\n\\Box_{g}\\psi&=\\frac{1}{\\sqrt{g}}\\partial_{a}\\Big(\\sqrt{g}\\cdot g^{ab}\\cdot \\partial_{b}\\psi\\Big)\\\\\n&=\\frac{1}{\\sqrt{g}}\\partial_{\\theta^{i}}\\Big(\\sqrt{g}\\cdot g^{ib}\\cdot \\partial_{b}\\psi\\Big)+\\frac{1}{\\sqrt{g}}\\partial_{v}\\Big(\\sqrt{g}\\cdot g^{vb}\\cdot \\partial_{b}\\psi\\Big)+\\frac{1}{\\sqrt{g}}\\partial_{u}\\Big(\\sqrt{g}\\cdot g^{ub}\\cdot \\partial_{b}\\psi\\Big),\n\\end{split}\n\\end{equation*} \nwhere for simplicity we denote $\\sqrt{g}=\\sqrt{|\\text{det}(g)|}$. Since the vector fields $\\partial_{\\theta^{i}},\\partial_{v}$ are tangential to $\\mathcal{H}$, it suffices to compute $g^{ib},g^{vb}$ only for the case where $b^{i}=0$. In this case we have\n\\begin{equation*}\ng^{vv}=g^{iv}=g^{iu}=0, \\ \\ \\ \\ \\ g^{ij}=\\mbox{$g \\mkern-8.8mu \/$\\,}^{ij}, \\ \\ \\ \\  \\ g^{vu}=-\\Omega^{-2}\\ : \\text{ on }\\mathcal{H}.\n\\end{equation*} \nWe also need to compute $g^{ub}$ everywhere (not just on $\\mathcal{H}$). We obtain\n\\begin{equation*}\ng^{uu}=O(b^{2}), \\ \\ \\ \\ \\ g^{uv}=-\\Omega^{-2},\\ \\ \\ \\ \\ g^{ui}=-\\frac{b^{i}}{\\Omega^{2}}.\n\\end{equation*}\nTherefore, using that $b^{i}=0$, we obtain on $\\mathcal{H}$: \n\\begin{equation}\n\\begin{split}\n\\Omega^{2}\\cdot &(\\Box_{g}\\psi)= \\\\ &=-\\frac{1}{\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}}\\partial_{v}\\Big(\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\cdot \\partial_{u}\\psi\\Big)-\\frac{1}{\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}}\\partial_{u}\\Big(\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\cdot \\partial_{v}\\psi\\Big)+\\frac{1}{\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}}\\partial_{\\theta^{i}}\\Big(\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\cdot\\big(\\Omega^{2}\\cdot\\mbox{$g \\mkern-8.8mu \/$\\,}^{ij}\\cdot\\partial_{j}\\psi\\big)\\Big)-(\\partial_{u}b^{i})\\cdot (\\partial_{\\theta^{i}}\\psi)\\\\\n&=-\\frac{1}{\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}}\\bigg(2\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\partial_{v}\\partial_{u}\\left(\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\psi\\right)-2\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\Big(\\partial_{v}\\partial_{u}\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\Big)\\cdot\\psi\\bigg)+\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\Big(\\Omega^{2}\\,\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi\\Big)-(\\partial_{u}b^{i})\\cdot (\\partial_{\\theta^{i}}\\psi),\\\\\n\\end{split}\n\\label{important}\n\\end{equation}\nwhere $\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,} $ denotes the divergence on the 2-surfaces with respect to the metric $\\mbox{$g \\mkern-8.8mu \/$\\,}$.\n\n\n\nRecalling \\eqref{torsionll} we can rewrite the restriction of the wave equation of $\\mathcal{H}$ with respect to a double null foliation $\\mathcal{D}$ as follows\n\\begin{equation}\n-2\\partial_{v}\\partial_{u}(\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\psi) +\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\cdot\\mathcal{Q}^{\\mathcal{S}}\\psi=0,\n\\label{wenull}\n\\end{equation}\non $\\mathcal{H}$, where the operator $\\mathcal{Q}^{\\mathcal{S}}:C^{\\infty}(\\mathcal{H})\\rightarrow\\mathbb{R}$ is defined by\n\\begin{equation}\n\\mathcal{Q}^{\\mathcal{S}}\\psi=\\Omega^{2}\\cdot\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+(\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\Omega^{2}-2\\Omega^{2}\\cdot\\zeta^{\\sharp})\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+w\\cdot\\psi,\n\\label{ovoperator}\n\\end{equation}\nwhere \n\\begin{equation}\nw=2\\frac{\\partial_{v}\\partial_{u}\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=\\bigg[\\partial_{v}(\\Omega tr\\underline{\\chi})+\\frac{1}{2}(\\Omega tr\\underline{\\chi})(\\Omega tr\\chi)\\bigg] \\text{ on }\\mathcal{H}.\n\\label{w}\n\\end{equation}\nClearly, the operator $\\mathcal{Q}^{\\mathcal{S}}$ depends on the foliation $\\mathcal{S}$ of $\\mathcal{H}$. \nIf we introduce the conformal factor $\\phi$ then, using that $\\partial_{u}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{_{\\mathbb{S}^{2}}}}=\\partial_{v}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{_{\\mathbb{S}^{2}}}}=0$, the wave equation can be rewritten as\n\\begin{equation}\n-2\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi) +\\phi\\cdot\\mathcal{Q}^{\\mathcal{S}}\\psi=0, \n\\label{wenull}\n\\end{equation}\non $\\mathcal{H}$. If we define the operator $\\mathcal{F}^{\\mathcal{D}}:C^{\\infty}(\\mathcal{M})\\rightarrow\\mathbb{R}$ such that\n\\begin{equation}\n\\mathcal{F}^{\\mathcal{D}}\\psi=\\frac{2}{\\phi}\\cdot \\partial_{v}\\partial_{u}(\\phi\\cdot\\psi),\n\\label{defF}\n\\end{equation} \nthen the wave equation on $\\mathcal{H}$ reads\n\\begin{equation}\n\\mathcal{F}^{\\mathcal{D}}\\psi=\\mathcal{Q}^{\\mathcal{S}}\\psi.\n\\label{wenull1}\n\\end{equation}\n\n\n\\section{Gluing constructions and conservation laws}\n\\label{sec:TheGeneralCase}\n\n\n\n\\subsection{Elliptic theory on $\\mathcal{H}$ and the operator $\\mathcal{O}^{\\mathcal{S}}$}\n\\label{sec:EllipticTheoryOnHh}\n\n In this subsection, we will introduce some basic facts about elliptic operators on the sections $S_{v}$ of a foliation $\\mathcal{S}$ of $\\mathcal{H}$ that will be needed for the proof of the main theorem. \n\n\nThe restriction of the operator $\\mathcal{Q}^{\\mathcal{S}}$ on a section $S_{v}$ gives rise to an elliptic operator\\footnote{For simplicity, we will often use the same notation for the operator $\\mathcal{Q}^{\\mathcal{S}}$ and its restriction  $\\mathcal{Q}^{\\mathcal{S}}_{v}$ on the section $S_{v}$ of $\\mathcal{S}$ on $\\mathcal{H}$. }\n\\begin{equation}\n\\mathcal{Q}^{\\mathcal{S}}_{v}:=\\mathcal{Q}^{\\mathcal{S}}\\left.\\right|_{S_{v}}:C^{\\infty}\\big(S_{v}\\big)\\rightarrow \\mathbb{R}\n\\label{restofq}\n\\end{equation} whose spectrum consists only of discrete eigenvalues with the property that the only limit (accumulation) point is infinity. Indeed, if we consider the operator\n\\[\\mathcal{Q}^{\\mathcal{S}}_{temp}\\psi=\n\\mathcal{Q}^{\\mathcal{S}}\\psi-\\frac{1}{\\epsilon}\\psi= \\Omega^{2}\\cdot\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+(\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\Omega^{2}+2\\Omega^{2}\\zeta^{\\sharp})\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+w\\cdot\\psi-\\frac{1}{\\epsilon}\\psi   \\]\nthen\n\\begin{equation*}\n\\begin{split}\n\\int_{S_{v}}\\mathcal{Q}^{\\mathcal{S}}_{temp}\\psi\\cdot\\psi=&-\\int_{S_{v}}\\psi\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\Omega^{2}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+\\Omega^{2}\\cdot|\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi|^{2}+\\Big(\\frac{1}{2}\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\Omega^{2}+Div(\\Omega^{2}\\zeta^{\\sharp})-w+\\frac{1}{\\epsilon}\\Big)\\psi^{2}\\\\\n\\leq &\n\\!-\\!\\!\\!\\int_{S_{v}}\\!(\\Omega^{2}-\\epsilon_{1})\\cdot|\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi|^{2}+\\Big(\\frac{1}{2}\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\Omega^{2}+Div(\\Omega^{2}\\zeta^{\\sharp})-w-\\frac{1}{\\epsilon_{1}}|\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\Omega^{2}|^{2}+\\frac{1}{\\epsilon} \\Big)\\psi^{2}\n \\end{split}\n\\end{equation*}\n\nWe first take $\\epsilon_{1}$ sufficiently small such that  $\\Omega^{2}>\\epsilon_{1}$ on $S_{v}$ and then take $\\epsilon$ sufficiently small so the coefficient of $\\psi^{2}$ is strictly positive. Hence, for these choices, the operator $\\mathcal{Q}^{\\mathcal{S}}_{temp}$ is negative definite and hence it has trivial kernel. By the Atiyah--Singer theorem  we have  $ind\\big(\\mathcal{Q}^{\\mathcal{S}}_{temp}\\big)=0$ and hence by the Fredholm alternative the operator $\\mathcal{Q}^{\\mathcal{S}}_{temp}$ is invertible. By the Poincar\\'{e} inequality and Rellich's theorem the inverse is a compact operator $\\big(\\mathcal{Q}^{\\mathcal{S}}_{temp}\\big)^{-1}:L^{2}(S_{v})\\rightarrow L^{2}(S_{v})$. The spectrum of this operator contains zero and only discrete eigenvalues whose limit point is zero. Denote this spectrum by $\\sigma_{temp}$. It follows that the spectrum of $\\mathcal{Q}^{\\mathcal{S}}_{temp}$ is the set $\\frac{1}{\\sigma_{temp}}$ which consists of discrete eigenvalues whose only limit point is infinity. Then, the spectrum of $\\mathcal{Q}^{\\mathcal{S}}$, restricted at the section $S_{v}$, is precisely the set $\\sigma_{v}=\\frac{1}{\\sigma_{temp}}+\\frac{1}{\\epsilon}$.\n\n\nNote also that if $f\\in C^{\\infty}(S_{v})$  then the equation \n\\begin{equation}\n\\mathcal{Q}^{\\mathcal{S}}\\psi=f_{v},\n\\label{tosolveelliptic}\n\\end{equation} \nhas a solution $\\psi$ on $S_{v}$ if and only if $f_{v}$ lies in the orthogonal complement of the kernel of the adjoint of $\\mathcal{Q}^{\\mathcal{S}}$ with respect to the space $\\big(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,}\\big)$. We have the following definition\n\\begin{definition}\nLet $\\mathcal{S}$ be a foliation of regular null hypersurface $\\mathcal{H}$ of a Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:TheDoubleNullFoliation}. We define the operator $\\mathcal{O}^{\\mathcal{S}}:C^{\\infty}(\\mathcal{H})\\rightarrow\\mathbb{R}$ given by \n\\begin{equation}\n\\begin{split}\n\\mathcal{O}^{\\mathcal{S}}\\psi=& \\Omega^{2}\\cdot\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+\\left[\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\Omega^{2}+2\\Omega^{2}\\cdot\\zeta^{\\sharp}\\right]\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi\\\\&+\\left[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\Big(\\Omega^{2}\\cdot\\zeta^{\\sharp}\\Big)+\\partial_{v}(\\Omega tr\\underline{\\chi})+\\frac{1}{2}(\\Omega tr\\underline{\\chi})(\\Omega tr\\chi)\\right]\\cdot\\psi.\n\\label{adjoint}\n\\end{split}\n\\end{equation}\nWe also denote by \n\\begin{equation}\n\\mathcal{O}^{\\mathcal{S}}_{v}:=\\mathcal{O}^{\\mathcal{S}}\\left.\\right|_{S_{v}}:C^{\\infty}(S_{v})\\rightarrow \\mathbb{R}\n\\label{restofo}\n\\end{equation}\nthe restriction of $\\mathcal{O}^{\\mathcal{S}}$ on a section $S_{v}$. \n\\label{definitiono}\n\\end{definition}\nThe operator $\\mathcal{O}^{\\mathcal{S}}_{v}$ is the adjoint of $\\mathcal{Q}^{\\mathcal{S}}_{v}$ with respect to the space $\\big(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,}\\big)$. Hence we have \n\\[ Im\\big(\\mathcal{Q}^{\\mathcal{S}}_{v}\\big)=\\left(Ker(\\mathcal{O}^{\\mathcal{S}}_{v})\\right)^{\\perp}. \\]\nTherefore, the equation \\eqref{tosolveelliptic} as a solution if and only if \n\\begin{equation}\nf_{v}\\in \\big(Ker(\\mathcal{O}^{\\mathcal{S}}_{v}) \\big)^{\\perp}.\n\\label{eq:integrability}\n\\end{equation}\nwhere the operator $\\mathcal{O}^{\\mathcal{S}}$ is the adjoint of $\\mathcal{Q}^{\\mathcal{S}}$ with respect to the space $\\big(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,}\\big)$ and is given by \n\nIf $f_{v}$ depend smoothly on $v$ for all $v\\in I=[v_{1},v_{2}]$  and \\eqref{eq:integrability} is satisfied in $I$, then $\\psi$ can be chosen to depend smoothly on $v$ too. \n\n\n\nFinally, if $S_{v}$ are endowed with the standard unit metric $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ (see Section \\ref{sec:TheDoubleNullFoliation}) and $f(v,\\theta^{1},\\theta^{2}):\\mathcal{H}\\rightarrow \\mathbb{R}$ is a smooth function then \n\\[f(v,\\theta^{1},\\theta^{2})=\\sum_{ml}f_{ml}(v)\\cdot Y_{ml}(\\theta^{1},\\theta^{2}),\\]\nwhere $Y_{ml}$ are the standard spherical harmonics. \n\n\n\n\\subsection{The gluing construction}\n\\label{sec:GeneralizedConservationLaws}\n\nLet $\\mathcal{D}$ be a regular double null foliation such that $\\mathcal{H}=\\left\\{u=0\\right\\}$. Clearly $\\mathcal{D}$ defines a foliation $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ of $\\mathcal{H}$. Let $S_{0}$ and $S_{1}$ be two sections of $\\mathcal{S}$. We will first show that we can always glue data on $S_{0}$ to data on $S_{1}$ in the sense of Definition \\ref{firstordergluingdefinition} if the operator $\\mathcal{Q}^{\\mathcal{S}}_{v}$ is surjective for some $v\\in[0,1]$. \n\n\n\\begin{proposition}\nLet $\\mathcal{S}$ be a foliation of a regular null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Let also $\\mathcal{O}_{v}^{\\mathcal{S}}$ be the elliptic operator given by \\eqref{adjoint}. If there is $v_{0}\\in[0,1]$ such that \n\\[Ker(\\mathcal{O}_{v_{0}}^{\\mathcal{S}})= \\left\\{0\\right\\},\\]\ni.e.~if $0$ is \\textbf{not} an eigenvalue of $\\mathcal{O}^{\\mathcal{S}}_{v_{0}}$, then we can glue arbitrary data at $S_{0}$ to arbitrary data at $S_{1}$ in the sense of Definition \\ref{firstordergluingdefinition}. \n\\label{pert1prop}\n\\end{proposition}\n\\begin{proof}\n\nSuppose that the spectrum $\\sigma(\\mathcal{O}_{v_{0}}^{\\mathcal{S}})$ does not contain zero for some $v_{0}\\in[0,1]$. Then zero is in the resolvent set $\\rho(\\mathcal{O}_{v_{0}}^{\\mathcal{S}})$. By Kato's upper semicontinuity of spectrum (see \\cite{kato}), the spectrum $\\sigma(\\mathcal{O}_{v}^{\\mathcal{S}})$ of the operator $\\sigma(\\mathcal{O}_{v}^{\\mathcal{S}})$ also does not contain zero for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$ with $\\epsilon>0$ sufficiently small. In other words, since the operators $\\mathcal{O}^{\\mathcal{S}}_{v}$ depend smoothly on $v$, there is a sufficiently small $\\epsilon$ such that the resolvent set of the operators $\\mathcal{O}_{v}^{\\mathcal{S}}$ contains zero for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$. In view of previous comments, these operators are also surjective.\n\nIf we integrate the wave equation \\eqref{wenull} along the null generators of $\\mathcal{H}$ we obtain\n\\begin{equation}\n2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{1}}}-2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{0}}} =\\int_{0}^{1}\\phi\\cdot\\mathcal{Q}_{v}^{\\mathcal{S}}\\psi \\ dv.\n\\label{malista}\n\\end{equation}\nIn the context of our gluing problem the first two terms are given. We smoothly extend $\\psi$  in the cylinders\n\\[(v,\\theta^{1},\\theta^{2})\\in[0,v_{0}-\\epsilon]\\times\\mathbb{S}^{2}, \\ \\ \\ (v,\\theta^{1},\\theta^{2})\\in[v_{0}+\\epsilon,1]\\times\\mathbb{S}^{2}.\\]\nsuch that $\\psi$ vanishes at all orders at the spheres $S_{v_{0}-\\epsilon}$ and $S_{v_{0}+\\epsilon}$. \nThen equation \\eqref{malista} is satisfied if \n\\[\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}\\phi(v,\\theta^{1},\\theta^{2})\\cdot\\mathcal{Q}_{v}^{\\mathcal{S}}\\psi(v,\\theta^{1},\\theta^{2}) \\ dv =\\rho(\\theta^{1},\\theta^{2}),     \\]\nwhere $\\rho$ is a given (prescribed) function of the sphere (which depends only on the initial data at $S_{0}$ and $S_{1}$ and  the extension of $\\psi$ in the complement of the cylinder for which $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$. )\n\n\nWe consider a smooth function $f(v,\\theta^{1},\\theta^{2})$,  such that  \n\\[\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}\\phi\\big(v,\\theta^{1},\\theta^{2}\\big)\\cdot f\\big(v,\\theta^{1},\\theta^{2}\\big)\\ dv=\\rho\\big(\\theta^{1},\\theta^{2}\\big) \\]\nand $f$  vanishes to all orders at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ (such a function clearly exists). Then, we simply have to solve the equations\n\\[\\mathcal{Q}_{v}^{\\mathcal{S}}\\psi=f\\big(v,\\cdot,\\cdot\\big)\\]\non $S_{v}$ for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$. This is clearly possible in view of the fact that the operators $\\mathcal{O}_{v}^{\\mathcal{S}}$ (and hence $\\mathcal{Q}_{v}^{\\mathcal{S}}$) are all invertible and the comments in the Section \\ref{sec:EllipticTheoryOnHh}. Moreover, $\\psi$ must necessarily vanish to all orders at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ and hence extends to a smooth function in the cylinder where $\\big(v,\\theta^{1},\\theta^{2}\\big)\\in [0,1]\\times\\mathbb{S}^{2}$. \n\n\\end{proof}\n\nHence, in order to have a conservation law along $\\mathcal{H}$ we must have that $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq \\left\\{0\\right\\}$ for all $v$  (otherwise we can perform gluing). The above result, however, does not exclude the possibility of gluing \\textit{general} characteristic data even if $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq \\left\\{0\\right\\}$ for all $v$. For we have the following general result\n\n\\begin{theorem}\nLet $\\mathcal{S}$ be a foliation of a regular null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:TheDoubleNullFoliation}. Let also $S_{0}$ and $S_{1}$ be two sections of $\\mathcal{S}$. Then\n\n\\begin{enumerate}\n\t\\item We can glue (to first order) general data on $S_{0}$ to general data on $S_{1}$ in the sense of Definition \\ref{firstordergluingdefinition} if and only if $\\mathcal{H}$ does not admit conservation laws with respect to $\\mathcal{S}$ in the sense of Definition \\ref{definitionconservationlaw}. If $\\mathcal{H}$ admits conservation laws with respect to $\\mathcal{S}$, then we can glue characteristic data if and only if their associated charges are equal, i.e. if and only if the data at $S_{0}$ and $S_{1}$ are such that\n\t\\[\\int_{S_{0}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot \\Theta^{\\mathcal{S}}\\,d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{1}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot \\Theta^{\\mathcal{S}}\\,d\\mu_{_{\\mathbb{S}^{2}}}, \\]\n\tfor all $\\Theta^{\\mathcal{S}}\\in  \\mathcal{W}^{\\mathcal{S}}$, where $\\mathcal{W}^{\\mathcal{S}}$ is the kernel of the conservation laws as defined in Section \\ref{sec:ConservationLawsForTheWaveEquations}.  Here $\\phi$ denotes the conformal factor of the sections of an associated double null foliation $\\mathcal{D}$.\n\t\n\t\\item The null hypersurface $\\mathcal{H}$ admits (first-order) conservation laws with respect to $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$  in the sense of Definition \\ref{definitionconservationlaw} if and only if there is a non-trivial linear space  $\\mathcal{U}^{\\mathcal{S}}\\subset  \\mathcal{V}_{\\mathcal{H}}$, where $\\mathcal{V}_{\\mathcal{H}}$ is the linear space defined in \\eqref{linearspace}, such that \n\t\t\\[\\mathcal{O}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot\\Theta^{\\mathcal{S}}\\right)=0 \\text{ on }\\mathcal{H},\\text{ for all }\\Theta^{\\mathcal{S}}\\in \\mathcal{U}^{\\mathcal{S}}.\\] Furthermore, the kernel of the conservation laws satisfies $\\mathcal{W}^{\\mathcal{S}}=\\mathcal{U}^{\\mathcal{S}}$, and moreover, $dim\\, \\mathcal{W}^{\\mathcal{S}}=dim\\, \\mathcal{U}^{\\mathcal{S}}<\\infty$.\n\n\t\\end{enumerate}\n\\label{theorem}\n\\end{theorem}\n\n\\begin{remark}\nThe conserved charges are independent of the choice of the diffeomorphism $\\Phi$ defined by \\eqref{phidiffeo}.  \n\\label{rem1giatheorema}\n\\end{remark}\n\n\\begin{remark}\nGluing is \\textbf{not} always possible even if we allow to freely choose the initial data in an angular neighborhood $\\mathcal{X}$ on  $S_{0}$ or $S_{1}$. Indeed, if the function  $\\Theta^{\\mathcal{S}}$ vanishes in $\\mathcal{X}$, then the charges cannot change value even if we change the data in that region $\\mathcal{X}$. Hence if the charges at $S_{0}$ and $S_{1}$ do not initially coincide then they will not coincide even after changing the data at $\\mathcal{X}$. \n\\label{re2}\n\\end{remark}\n\nBefore we give the proof of the Theorem \\ref{theorem} we present some lemmata. The first one concerns the kernels of the elliptic operators $\\mathcal{O}_{v}^{\\mathcal{S}}$.\n\\begin{lemma}\n\\textbf{(Variation analysis of the kernel of elliptic operators)} There is an upper bound for the dimension of the kernel $K(v)\\subset L^{2}\\big(\\mathbb{S}^{2}\\big)$ of the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$, given by \\eqref{adjoint}, for $v\\in[0,1]$. Moreover, there is a dense set $\\mathcal{Z}\\subseteq [0,1]$ of point $x$ for which there is an open neighborhood $V_{x}$ containing $x$ such that $K(v)$ varies smoothly for $v\\in V_{x}$ (and hence in particular the dimension $\\text{dim}\\big(K(v)\\big)$ is constant for all $v\\in V_{x}$). \n\\label{lemmagiaelliptickernel}\n\\end{lemma}\n\\begin{proof}\nFollowing the idea of Section \\ref{sec:EllipticTheoryOnHh} we have that for sufficiently large  $\\lambda >0$ the operator $\\mathcal{O}_{v}^{\\mathcal{S}}-\\lambda\\cdot I$ has a compact inverse $\\mathcal{C}_{v}:L^{2}(\\mathbb{S}^{2})\\rightarrow H^{1}(\\mathbb{S}^{2})\\subset L^{2}(\\mathbb{S}^{2})$. By the continuity of the resolvent theorem (see \\cite{extremumproblemsbook}, Chapter 2) we obtain that the operators $\\mathcal{C}_{v}$ vary continuously in $v$ with respect to the topology of the space $\\mathcal{L}\\big(L^{2},L^{2} \\big)$. \nThe kernel $K(v)$ of $\\mathcal{O}_{v}^{\\mathcal{S}}$ coincides with the kernel of the operator \n\\begin{equation}\n\\mathcal{P}_{v}=\\mathcal{C}_{v}-\\frac{1}{\\lambda}\\cdot I:L^{2}(\\mathbb{S}^{2})\\rightarrow  L^{2}(\\mathbb{S}^{2}),\n\\label{operatorp}\n\\end{equation}\nwhich also varies continuously in $v$. \nSince $\\mathcal{C}_{v}$ is compact it is easily seen that the operator \\[\\Big. \\mathcal{P}_{v}\\Big|_{\\big(K(v)\\big)^{\\perp}}   :\\big(K(v)\\big)^{\\perp}\\rightarrow  L^{2}(\\mathbb{S}^{2})\\]\nis bounded from below. This implies that the mapping\n\\[K:[0,1]\\rightarrow K(v):=ker\\big(\\mathcal{O}_{v}\\big)\\subset L^{2}\\big(\\mathbb{S}^{2}\\big)  \\]\nis upper semicontinuous, i.e.~if $B(1)$ is the unit ball in $L^{2}\\big(\\mathbb{S}^{2}\\big)$ then for all $\\epsilon>0$ there is $\\delta>0$ such that if $|v-v_{0}|<\\delta$ then \n\\[ K(v)\\cap B(1)\\subset B_{\\epsilon}\\Big( K(v_{0})\\cap B(1)  \\Big), \\] \nwhere $B_{\\epsilon}(S)$ denotes the set of points who distance from $S$ is at most  $\\epsilon$. It thus follows that \n\\begin{equation}\n\\limsup_{n}\\text{dim}\\big(K(v_{n})\\big)\\leq \\text{dim}\\big(K(v_{0})\\big).\n\\label{simantikodiastasi}\n\\end{equation}\nDefine now the sets\n\\begin{equation}\nA_{n}=\\Big\\{v\\in[0,1]\\, :\\, \\text{dim}\\big(K(v)\\big)\\geq n\\Big\\}.\n\\label{tasinolaa}\n\\end{equation}\nIn view of \\eqref{simantikodiastasi} the sets $A_{v}$ are closed in [0,1]. Moreover, $A_{n+1}\\subset A_{n}$. By the compactness of $[0,1]$ it follows that there is a $n_{0}\\in \\mathbb{N}$ such that $A_{n}=\\emptyset$ for all $n\\geq n_{0}$. Hence, there is an upper bound on the dimension of the kernel $K(v)$ for all $v\\in [0,1]$. We consider next the following sets\n\\begin{equation}\nB_{n}=A_{n}\/A_{n-1}=\\Big\\{v\\in[0,1]\\, :\\, \\text{dim}\\big(K(v)\\big)=n\\Big\\}.\n\\label{tasinolab}\n\\end{equation}\nClearly, \n\\begin{equation}\n\\bigcup_{n=0}^{n_{0}}B_{n}=[0,1].\n\\label{tasinolab1}\n\\end{equation}\nThe set $B_{0}=[0,1]\/A_{1}$ is open in $[0,1]$ and hence $B_{0}=\\text{int}\\big(B_{0}\\big)$. We will show that for all $1\\leq n\\leq n_{0}$ we have \n\\begin{equation}\n\\text{int}\\big(B_{n}\\big)=B_{n}\/\\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big).\n\\label{interiorofb}\n\\end{equation}\nThe inclusion $\\text{int}\\big(B_{n}\\big)\\subseteq B_{n}\/\\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big)$ is trivial. If now there is $x\\in B_{n}\/\\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big)$ such that $x\\notin \\text{int}\\big(B_{n}\\big)$ then there is a sequence $y_{k}\\rightarrow x$ with $y_{k}\\notin B_{n}$. Clearly, $y_{k}$ cannot have an infinite subsequence in either $B_{0}\\cup ...\\cup B_{n-1}$ since otherwise $x\\in \\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big)$.   Hence, $y_{k}\\in A_{n+1}$ and since $A_{n+1}$ is closed we have $x\\in A_{n+1}$ and hence $x\\notin B_{n}$, contradiction. \n\nDefine the set\n\\begin{equation}\n\\mathcal{Z}=\\bigcup_{n=1}^{n_{0}}\\text{int}\\big(B_{n}\\big).\n\\label{eq:thesetz}\n\\end{equation}\nThe set $\\mathcal{Z}$ is open in $[0,1]$ and dense. Indeed, in view of \\eqref{tasinolab1} and \\eqref{interiorofb} we have\n\\[\\text{clos}(\\mathcal{Z})= \\text{clos}\\big(B_{0}\\cup...\\cup B_{n_{0}}\\big)=[0,1] .  \\]\nIn other words, there is a dense set of points $x\\in [0,1]$ for which there is an open neighborhood $V_{x}$ such that $\\text{dim}\\big(K(v)\\big)$ is constant for all $v\\in V_{x}$. It remains to show that $K(v)$ varies smoothly in $v$ for $v\\in V_{x}$, i.e.~the curve $V_{x}\\ni x\\mapsto K(v)$ is smooth in the Grassmannian $Gr(L^{2},n)$ where $n=\\text{dim}\\big(K(v)\\big)$. Using an adaptation of the aforementioned result of \\cite{extremumproblemsbook} and the fact that the coefficients of $\\mathcal{O}_{v}^{\\mathcal{S}}$ depend smoothly in $v$ one can show that $\\mathcal{C}_{v}$, and hence $\\mathcal{P}_{v}$, varies smoothly in $v$ in the space $\\mathcal{L}\\big(L^{2},L^{2}\\big)$. In view of the fact that $K(v)$ is upper semicontinuous and has constant dimension we obtain that if $v_{0}\\in V_{x}$ then  \n\\[\\Big. \\text{proj}\\Big|_{K(v_{0})\\cap B(1)}\\Big(K(v)\\cap B(1)\\Big)= K(v_{0})\\cap B(1)  \\]\nfor all $v$ sufficiently close to $v_{0}$. Given $x_{v_{0}}\\in K(v_{0})\\cap B(1)$ there is a unique $x_{v}\\in B_{\\epsilon}\\Big(K(v)\\cap B(1) \\Big)$ such that $x_{v}=x_{v_{0}}+a_{v}^{\\perp}$. Since $\\mathcal{P}_{v}=\\mathcal{P}_{v_{0}}+(v-v_{0})\\cdot Z+O\\big((v-v_{0})^{2}\\big)$ and  $\\mathcal{P}_{v}(x_{v})=0$ we obtain $\\mathcal{P}_{v_{0}}(a_{v}^{\\perp})=(v_{0}-v)\\cdot Zx_{v}+O(v^{2})  $ and since $\\Big. \\mathcal{P}_{v_{0}}\\Big|_{\\big(K(v)\\big)^{\\perp}}:\\big(K(v)\\big)^{\\perp}\\rightarrow Im(\\mathcal{P}_{0})$ has a bounded inverse we obtain that $a_{v}^{\\perp}\\rightarrow 0$ in $L^{2}$ as $v\\rightarrow v_{0}$ in a differentiable manner. This shows that $K(v)$ varies differentiably in $v$. Similary we can show that $K(v)$ is smooth in $v$.  \n\n\n\n\n\n\\end{proof}\n\n\n\n\n\nIf there is $v_{0}\\in[0,1]$ such that $Ker\\big(\\mathcal{O}^{\\mathcal{S}}_{v}\\big)= 0$ then the Theorem \\ref{theorem} follows from Proposition \\ref{pert1prop}. In particular, in this case we have $dim\\,\\mathcal{W}^{\\mathcal{S}}=dim\\,\\mathcal{U}^{\\mathcal{S}}=0$. We assume that $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq 0$ for all $v\\in[0,1]$, that is $B_{0}=\\emptyset$, where $B_{0}$ is defined by \\eqref{tasinolab}. Then according to the above lemma there is a dense set of points $x$ which have an open neighborhood $V_{x}$ in which the kernels $K(v)$ vary smoothly.   In each of these intervals, we can find a smoothly varying in $v$ basis \n\\[\\mathcal{B}_{v}^{\\mathcal{S}}=\\left\\{ \\big(E_{1}^{\\mathcal{S}}\\big)_{v}, \\big(E_{2}^{\\mathcal{S}}\\big)_{v},\\cdots, \\big(E_{i}^{\\mathcal{S}}\\big)_{v}, \\ i=dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big) \\right\\}\\]\nof $K(v)=Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$. We  next localize in each of these intervals. \n\n\nLet $v_{0}\\in V_{x}$ for some $x$ as above. We can smoothly extend $\\psi\\left.\\right|_{S_{0}}$ and $\\psi\\left.\\right|_{S_{1}}$ in the cylinders $[0,v_{0}-\\epsilon]\\times\\mathbb{S}^{2}$ and $[v_{0}+\\epsilon,1]\\times \\mathbb{S}^{2}$, where $\\epsilon>0$ is sufficiently small such that $v_{0}-\\epsilon,v_{0}+\\epsilon\\in V_{x}$, such that $\\psi$ vanishes to infinite order  on $\\mathcal{H}$  at $\\left\\{v_{0}-\\epsilon\\right\\}\\times\\mathbb{S}^{2}$ and $\\left\\{v_{0}+\\epsilon\\right\\}\\times\\mathbb{S}^{2}$.  Then, $\\mathcal{O}^{\\mathcal{S}}_{v}\\psi$ is also known in the union of these two cylinders and hence, by \\eqref{wenull1}, $\\mathcal{F}^{\\mathcal{D}}\\psi$, given by \\eqref{defF}, is  known there. In fact $\\mathcal{F}^{\\mathcal{D}}\\psi$  vanishes to infinite order on $\\mathcal{H}$  at $\\left\\{v_{0}-\\epsilon\\right\\}\\times\\mathbb{S}^{2}$ and $\\left\\{v_{0}+\\epsilon\\right\\}\\times\\mathbb{S}^{2}$.  We wish to extend $\\psi$ smoothly in $[0,1]\\times\\mathbb{S}^{2}$ such that $\\psi$ solves \\eqref{wenull1} and the transversal derivatives $\\partial_{u}\\psi\\left.\\right|_{S_{0}}$ and $\\partial_{u}\\psi\\left.\\right|_{S_{1}}$ agree with the given characteristic data. \n\n\n We will do so by first smoothly extending $\\mathcal{F}^{\\mathcal{D}}\\psi$ everywhere in $[0,1]\\times\\mathbb{S}^{2}$ such that the above are satisfied and, using \\eqref{wenull1}, we can solve with respect to $\\psi$. In other words, gluing is possible if we can extend $\\mathcal{F}^{\\mathcal{D}}\\psi$, or equivalently $\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi$, in $[v_{0}-\\epsilon,v_{0}+\\epsilon]\\times\\mathbb{S}^{2}$ such that the following conditions hold:\n\n\\medskip\n\n\\noindent{\\underline{\\textbf{The conditions 1--3:}}}\n\n\\medskip\n\n\\begin{enumerate}\n\t\\item \\textbf{\\underline{Smoothness on $\\mathcal{H}$}:} The function $\\mathcal{F}^{\\mathcal{D}}\\psi$, or equivalently $\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi$, vanishes to all orders on $\\mathcal{H}$  at $\\left\\{v_{0}-\\epsilon\\right\\}\\times\\mathbb{S}^{2}$ and $\\left\\{v_{0}+\\epsilon\\right\\}\\times\\mathbb{S}^{2}$. \n\t\\item \\textbf{\\underline{Gluing for the transerval derivative $\\partial_{u}\\psi$}:} In view of \\eqref{defF} and \\eqref{wenull1}, $\\mathcal{F}^{\\mathcal{D}}\\psi$ satisfies on $\\mathcal{H}$:  \\[\\displaystyle\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}\\!\\!\\!\\big(\\mathcal{F}^{\\mathcal{D}}\\psi\\big)(v,\\theta^{1},\\theta^{2})\\cdot \\phi(v,\\theta^{1},\\theta^{2})\\, dv=\\rho(\\theta^{1},\\theta^{2}),\\] where $\\rho$ is a given (prescribed) function of the sphere (which depends only on the initial data at $S_{0}$ and $S_{1}$ and also the extension of $\\psi$ in the complement of the cylinder for which $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$).\n\t\n\t\\item \\textbf{\\underline{Integrability (orthogonality) condition}:} Given $\\mathcal{F}^{\\mathcal{D}}\\psi$ we can solve with respect to $\\psi$ (i.e. ``invert'' the operator $\\mathcal{F}^{\\mathcal{D}}$) if, using \\eqref{wenull1} and the comments of Section \\ref{sec:EllipticTheoryOnHh}, \n\t\\[\\mathcal{F}^{\\mathcal{D}}\\psi \\in  Im\\big(\\mathcal{Q}^{\\mathcal{S}}_{v}\\big)=\\left(Ker(\\mathcal{O}^{\\mathcal{S}}_{v})\\right)^{\\perp},   \\]\n\tor equivalently,\n\t\\begin{equation}\n\t\\displaystyle\\int_{S_{v}}\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\ d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=0\n\t\\label{1inte}\n\t\\end{equation} for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$ and $n=1,..., dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$.\n\t\\end{enumerate}\n\tNote that since\n\t\\begin{equation*}\n\\begin{split}\n\\int_{S_{v}}\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\ d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=&\\int_{S_{v}}2\\frac{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}}{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}\\cdot\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\, d\\theta^{1}\\, d\\theta^{2}\\\\=&\\int_{S_{v}}2\\frac{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}}\\cdot\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}\\, d\\theta^{1}\\, d\\theta^{2}\\\\=&\\int_{S_{v}}\\Big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\Big)\\cdot\\Big(\\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\cdot \\phi\\Big)\\, d\\mu_{_{\\mathbb{S}^{2}}},\n\\end{split}\n\\end{equation*}\nthe integrability condition \\eqref{1inte} is equivalent to the following:\n\\begin{equation}\n\\int_{S_{v}}\\Big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\Big)\\cdot\\Big(\\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\cdot \\phi\\Big)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0,\n\\label{integrability}\n\\end{equation}\nfor all $n=1,..., dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$. As we shall see, splitting $\\phi$ in both terms is very important as it also is the fact that the above integral is with respect to the standard unit metric on $S_{v}$. We define the functions\n\\begin{equation}\nG_{n}=\\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\cdot\\phi, \n\\label{g0}\n\\end{equation}\nfor $n=1,\\cdots, dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$ and thus \\eqref{1inte} is equivalent to \n\\begin{equation}\n\\int_{S_{v}}\\Big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\Big)\\cdot\\big(G_{n}\\big)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0,\n\\label{integrability}\n\\end{equation}\nfor $n=1,\\cdots, dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$. \n\\bigskip\n\n\\noindent\\underline{ \\textbf{A special case:}} $dim Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=1$ for all $v\\in[0,1]$\n\n\\medskip\n\nBefore we consider the  general case let us first consider the special case for which $dim Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=1$ for all $v\\in[0,1]$ (and hence $V_{x}=[0,1]$).  This will make our argument clear. We simplify the notation by denoting $G=G_{1}$. \n\n\n\nWe  decompose $\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi,\\,G$ in (standard) angular frequencies. Let\n\\begin{equation}\n\\big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\big)(v,\\theta^{1},\\theta^{2})=\\sum_{ml}F_{ml}(v)\\cdot Y^{ml}(\\theta^{1},\\theta^{2})\n\\label{f1}\n\\end{equation}\nand \n\\begin{equation}\nG(v,\\theta^{1},\\theta^{2})=\\sum_{ml}G_{ml}(v)\\cdot Y^{ml}(\\theta^{1},\\theta^{2}),\n\\label{g1}\n\\end{equation}\nwhere $Y^{ml}$ denote the standard spherical harmonics on $\\mathbb{S}^{2}$. \n\nWe can glue to first order characteristic data on $S_{0}$ and $S_{1}$ if there exist smooth functions  $F_{ml}(v)\\!:\\![v_{0}-\\epsilon,v_{0}+\\epsilon]\\rightarrow\\mathbb{R}$ such that the following conditions are satisfied\n\\begin{enumerate}\n\\item \\underline{Smoothness on $\\mathcal{H}$}:\nThe functions $F_{ml}(v)$ vanish to all orders  at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ for all $m,l$.\n\\item \n\\underline{Gluing for the transversal derivative $\\partial_{u}\\psi$}:\nThe integrals\n\\[\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{ml}(v)\\, dv\\]\n are all given. \n \\item \n\\underline{Orthogonality condition}: The integrability condition \\eqref{integrability} is satisfied:\n\\begin{equation*}\n\\begin{split}\n\\sum_{ml} F_{ml}(v)\\cdot G_{ml}(v)=0.\n\\end{split}\n\\end{equation*}\n\\end{enumerate}\n\n\n\n\\underline{\\textbf{Case I}}:\\medskip  \\\\ \nFor all $m,l$ we have $G_{ml}(v)=\\textbf{G}(v)\\cdot c_{ml}$, for all $v\\in[0,1]$, for some function $\\textbf{G}:[0,1]\\rightarrow\\mathbb{R}$ and some constant non-zero $l^{2}$ sequence $c_{ml}$.  That is to say, for all $m,l,m',l'$ we have $\\frac{G_{ml}(v)}{G_{m'l'}(v)}=c_{mlm'l'}$, where $c_{mlm'l'}$ is a constant. Then, the condition 3  becomes\n\\[\\sum_{ml}F_{ml}(v)\\cdot c_{ml}=0. \\]\n  The above equation implies that the functions $F_{ml}(v)$ are linearly dependent and hence condition  2 \\textbf{cannot} be satisfied \\textit{in general}. In fact, in this case we have that $G(v,\\theta^{1},\\theta^{2})=\\textbf{G}(v)\\cdot{\\Theta}(\\theta^{1},\\theta^{2}), $ for some function ${{\\Theta}}$ which is constant along the null generators, i.e.~$\\Theta\\in\\mathcal{V}_{\\hh}$. Then, by \\eqref{g0}, we have the splitting \n\\[  \\big(E^{\\mathcal{S}}_{v}\\cdot\\phi\\big)(v,\\theta^{1},\\theta^{2})= \\mathbf{G}(v)\\cdot\\Theta(\\theta^{1},\\theta^{2}) \\]\nwhich shows that \n\\[E_{v}^{\\mathcal{S}}(\\theta^{1},\\theta^{2})= \\mathbf{G}(v)\\cdot \\frac{1}{\\phi}\\cdot \\Theta(\\theta^{1},\\theta^{2}), \\]\nand since the function $\\mathbf{G}$ does not depends on the angular coordinates we have that \n\\[\\frac{1}{\\phi}\\cdot\\Theta\\in Ker(\\mathcal{O}_{v}^{\\mathcal{S}}),  \\]\nfor all $v\\in[0,1],$ as required. Furthermore,  by  \\eqref{integrability} and \\eqref{defF} we have \n\\[0=\\int_{S_{v}}\\big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{v}}\\Big(\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}} \\]\nand, since the function $\\Theta$ and the measure of integration do not depend on $v$, we obtain that the quantity\n\\begin{equation}\n\\int_{S_{v}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:conserved}\n\\end{equation}\nis conserved, i.e.~independent of $v$. Clearly the above conservation law is an obstruction to gluing of general initial data on $S_{0}$ to general initial data on $S_{1}$. \n\nWe will next show that this conservation law is the \\textbf{only} obstruction to gluing. Indeed suppose that  the initial data on $S_{0}$ and $S_{1}$ are such that \n\\[\\int_{S_{0}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{1}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}.\\]\nIn this case we need to construct functions $F_{ml}:[v_{0}-\\epsilon,v_{0}+\\epsilon]\\rightarrow\\mathbb{R}$ such that \n\\begin{enumerate}\n\\item $F_{ml}(v)$ vanishes to infinite order at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ for all $m,l$.\n\\item \\begin{equation}\n\\sum_{ml}F_{ml}(v)\\cdot c_{ml}=0\n\\label{in1}\n\\end{equation} for $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$, where $c_{ml}$ is a constant non-zero $l^{2}$ sequence. \n\n\\item The integrals\n\\[ \\mathbf{I}_{ml}=\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{ml}(v)\\, dv \\]are all prescribed such that    \n\\begin{equation}\n\\sum_{ml}\\mathbf{I}_{ml}\\cdot c_{ml} =0, \n\\label{in2}\n\\end{equation} \nsince\n\\[Char(S_{v_{0}-\\epsilon})=Char(S_{0})=Char(S_{1})=Char(S_{v_{0}+\\epsilon}),\\]\nand using \\eqref{defF} and \\eqref{malista}, where we denote \n\\[char(S_{v})=\\int_{S_{v}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}.\\] \\end{enumerate}\n\nFor simplicity we rename the sequences $F_{ml},G_{ml,}c_{ml}$ as $F_{i},G_{i},c_{i},i\\geq 0$. We assume without loss of generality that $c_{0}\\neq 0$. We construct the functions $F_{i}, i\\geq 1,$ such that condition 1 is satisfied and such that the integrals $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{i}(v)\\,dv$ agree with the prescribed values (condition 3).  We next construct $F_{0}$ by solving the equation \\eqref{in1} with respect to $F_{0}$. Clearly, condition 1 is satisfied for $F_{0}$. Condition 2 holds by construction. Moreover, the integral $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{0}(v)\\, dv$ agrees with its prescribed value in view of \\eqref{in2} and the linearity of the integrals. This finishes the construction of $F_{ml}$ which in turn allows us to extend $\\psi$ and hence to obtain the gluing of the characteristic data. \n\nWe remark that as along as gluing is possible then it can be achieved in a highly non-unique way.  \n\n\\bigskip\n\n\n\\underline{\\textbf{Case II}}:\\medskip\\\\ Using the above simplified notation, we can assume without loss of generality that \\begin{equation}\nG_{1}(v)=a(v)\\cdot G_{0}(v) \\text{ and } \\partial_{v} a\\neq 0,\\  G_{0}(v)\\neq 0\n\\label{gmalista}\n\\end{equation} in a (sufficiently) small interval $I$ of $v$ (and hence $G_{1}$ is not linearly dependent on $G_{0}$). We choose $v_{0}$ to be in $I$ and we take $\\epsilon>0$ small enough such that $[v_{0}-\\epsilon,v_{0}+\\epsilon]\\subset I$. By relabeling $m,l$, as before, we can rewrite condition 3 as follows \n\\[ F_{0}(v)\\cdot G_{0}(v)+F_{1}(v)\\cdot G_{1}(v)+\\sum_{i\\geq 2}F_{i}(v)\\cdot G_{i}(v)=0 \\]\nand hence \n\\begin{equation}\n F_{0}(v)=-\\frac{G_{1}(v)}{G_{0}(v)}F_{1}(v)-\\sum_{i\\geq 2}\\frac{G_{i}(v)}{G_{0}(v)}F_{i}(v)=-a(v)\\cdot F_{1}(v)-\\sum_{i\\geq 2}\\frac{G_{i}(v)}{G_{0}(v)}F_{i}(v). \n\\label{eq:pro3}\n\\end{equation}\nWe can then prescribe $F_{i}(v)$, $i\\geq 2$, such that condition 1 is satisfied and such that the integrals $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{i}(v)\\,dv$ agree with the prescribed values (condition 3). \n\nWe can then prescribe $F_{1}$ such that condition 1 is satisfied, the integral $\\int_{v_{0}-\\epsilon}^{v_{0}-\\epsilon}F_{1}(v)\\, dv$ agrees with its prescribed value \\textbf{and} the integral  $\\int_{v_{0}-\\epsilon}^{v_{0}-\\epsilon}a(v)\\cdot F_{1}(v)\\, dv$ is such that the integral $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{0}(v)\\, dv$, computed via  \\eqref{eq:pro3}, agrees with its prescribed value. Note that in view of \\eqref{gmalista}, the integrals $\\int_{v_{0}-\\epsilon}^{v_{0}-\\epsilon}a(v)\\cdot F_{1}(v)\\, dv$ and $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{0}(v)\\, dv$ are independent. In view of the fact that condition 1 is satisfied for all $F_{i},\\, i\\geq 1$, we have that it is automatically satisfied for $F_{0}$, again via \\eqref{eq:pro3}. The remaining conditions hold by construction.  This finishes the construction of $F_{ml}$'s for the case where $dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=1$ for all $v\\in [0,1]$.\n\n\n\\bigskip\n\n\\noindent\\underline{\\textbf{The general case}}\n\n\\medskip\n\nWe now return to the general case. We first derive the following lemmata.\n\n\\begin{lemma}\nLet $I$ be a closed interval of $\\mathbb{R}$. Given $n\\in\\mathbb{N}$ linearly independent functions $f_{1},f_{2},...,f_{n}\\in C^{\\infty}\\big(I\\big)$ and $\\lambda_{1},\\lambda_{2},\\cdots, \\lambda_{n}\\in \\mathbb{R}$ there is a function $\\alpha\\in C^{\\infty}\\big(I\\big)$ such that \n \\[\\int_{I}\\alpha(v)\\cdot f_{i}(v)\\, dv=\\lambda_{i}, \\text{ for all } i=1,2,...,n.\\]\n\\label{lemma1stoproof}\n\\end{lemma}\n\\begin{proof}\n\n\nLet $V_{n}=\\Big\\langle f_{1},f_{2},\\cdots, f_{n}\\Big\\rangle\\subset L^{2}(I)$\ndenote the $n$-dimensional span of the functions $f_{1},f_{2},\\cdots, f_{n}$. Using the Gram--Schmidt process we produce an orthonormal basis $\\left\\{e_{1},e_{2},\\cdots, e_{n}\\right\\}$ of $V_{n}$. We extend this basis to obtain an orthonormal basis $\\left\\{e_{1},\\cdots, e_{n},e_{n+1},\\cdots\\right\\}$ of $L^{2}(I)$. Clearly  for all $i=1,2,...,n$ we have that $e_{i}\\in C^{\\infty}\\big(I\\big)$ and  \n\\[f_{i}=\\sum_{k=1}^{i}\\left\\langle e_{k},f_{i}\\right\\rangle\\cdot e_{k} \\]\nwith $\\left\\langle e_{i},f_{i}\\right\\rangle>0$, where $\\left\\langle\\, \\cdot\\, ,\\, \\cdot\\, \\right\\rangle$ denotes the inner product of $L^{2}(I)$.\nWe want to construct a function $\\alpha\\in C^{\\infty}\\big(I\\big)$ such that $\\left\\langle a, f_{i}\\right\\rangle=\\lambda_{i}$ for $i=1,2,...,n$. The system\n\\[\\lambda_{i}= \\sum_{k=1}^{i}\\left\\langle e_{k},f_{i}\\right\\rangle\\cdot x_{i} \\]\nhas a unique solution with respect to $x_{1},x_{2},...,x_{n}$. We then define the function $\\alpha$ such that \n$\\left\\langle \\alpha, e_{i}\\right\\rangle=x_{i}$ for $i=1,2,...,n$ and $\\left\\langle \\alpha, e_{i}\\right\\rangle=0$ for $i\\geq n+1$, which clearly satisfies the required relations. \n\n\n\\end{proof}\n\n\\begin{lemma}\nLet $I$ be a compact interval of $\\mathbb{R}$. Let $G_{1},G_{2},...,G_{n}\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ be $n$ functions such that for each $v\\in I$ the functions $G_{1}(v,\\cdot), G_{2}(v,\\cdot),...,G_{n}(v,\\cdot)\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ are linearly independent and let  \\[\\Pi(v)=\\Big\\langle G_{1}(v,\\cdot), G_{2}(v,\\cdot),...,G_{n}(v,\\cdot)\\Big\\rangle\\subset L^{2}\\big(\\mathbb{S}^{2}\\big), \\] denote the ($v$-dependent) $n$-dimensional subspace of $L^{2}\\big(\\mathbb{S}^{2}\\big)$ spanned by them.  Then, given $\\rho\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ there is a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times\\mathbb{S}^{2}\\big)$  which vanishes to infinite order at $\\partial I\\times \\mathbb{S}^{2}$ and is such that \n\\begin{equation}\n\\int_{I}F_{\\rho}(v,\\cdot)\\, dv= \\rho(\\cdot)\n\\label{eq:rel1}\n\\end{equation}\nand \n\\begin{equation}\n\\int_{\\mathbb{S}^{2}}F_{\\rho}(v,\\cdot)\\cdot G_{i}(v,\\cdot)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0  \\text{ for all }   i=1,2,...,n,\n\\label{eq:rel2}\n\\end{equation}\nif and only if \n\\[\\rho\\in \\Bigg(\\bigcap_{v\\in I}\\Pi(v)\\Bigg)^{\\perp}\\subset L^{2}\\big(\\mathbb{S}^{2}\\big). \\]\n\\label{deuterolemma}\n\\end{lemma}\n\\begin{proof}\nWe denote \n\\begin{equation}\nV=\\bigcap_{v\\in I}\\Pi(v)\\subset L^{2}\\big(\\mathbb{S}^{2}\\big).\n\\label{sxesi1}\n\\end{equation}\n Clearly, $V$ is finite dimensional. \n\n If given a function $\\rho$ the function $F_{\\rho}$ exists then $F_{\\rho}\\in \\Big(\\Pi(v)\\Big)^{\\perp}$ for all $v$ and hence $F_{\\rho}\\in V^{\\perp}$. Therefore,  an immediate application of Fubini's theorem yields that $\\rho\\in V^{\\perp}$.\n\n\nLet us assume now that $\\rho \\in V^{\\perp}$. We will show that a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$  satisfying the above properties exists.  \n\nSince $V$ is finite dimensional, we have the decomposition\n\\[L^{2}\\big(\\mathbb{S}^{2} \\big)=V\\oplus V^{\\perp}.  \\]\nIf we define the spaces\n\\[\\Big.\\Pi(v)\\Big|_{V^{\\perp}} :=\\Big.\\text{proj}\\Big|_{V^{\\perp}}\\big(\\Pi(v)\\big).   \\]\nSince $V\\subset \\Pi(v)$ for all $v\\in I$ the space $\\Big.\\Pi(v)\\Big|_{V^{\\perp}} $ varies smoothly in $v$. Indeed, since $\\Pi(v)$ varies smoothly in $v\\in I$ we can write $\\Pi(v)=V\\oplus T(v)$ where $T(v)$ varies smoothly in $v\\in I$ and $\\text{dim}\\big(T(v)\\big)=n-\\text{dim}(V)$, where $\\text{dim}\\big(\\Pi(v)\\big)=n$ for all $v\\in I$. Then $\\Big.\\Pi(v)\\Big|_{V^{\\perp}}=\\Big.T(v)\\Big|_{V^{\\perp}}$. The projection\n\\[\\Big.\\text{proj}\\Big|_{V^{\\perp}}: T(v)\\rightarrow V^{\\perp} \\]\nhas full rank since otherwise we would have $T(v)\\cap V\\neq \\left\\{0\\right\\}$, contradiction. Therefore, since $T(v)$ varies smoothly in $v$ the space $\\Big.T(v)\\Big|_{V^{\\perp}}$ varies smoothly in $v$ and hence so does $\\Big.\\Pi(v)\\Big|_{V^{\\perp}}$. It also follows that \n\\begin{equation}\n\\text{dim}\\left(\\Big.\\Pi(v)\\Big|_{V^{\\perp}}\\right)=n-\\text{dim}(V).\n\\label{dimerelationw}\n\\end{equation}  \nFurthermore, we  obtain\n\\begin{equation}\n\\bigcap_{v\\in I}\\left( \\Big.\\Pi(v)\\Big|_{V^{\\perp}} \\right)=\\left\\{0\\right\\}.\n\\label{sxesi2}\n\\end{equation}\nIndeed, if the line $\\left\\langle l\\right\\rangle$ lies in the above intersection then for every $v\\in I$ there is $y_{v}\\in \\Pi(v)$ such that $y_{v}=x_{v}+l,$ where $x_{v}\\in V$ (and $l\\in V^{\\perp}$). However, by \\eqref{sxesi1}, $x_{v}\\in \\Pi(v)$ and hence by linearity $l\\in \\Pi(v)$. Since this holds of all $v$, it immediately contradicts \\eqref{sxesi1}.\n\n\n\nWe next show that there is a finite dimensional subspace $W\\subset V^{\\perp}$ such that if \n\\[\\widetilde{\\Pi}(v):= \\Big.\\text{proj}\\Big|_{W} \\Big( \\Big.\\Pi(v)\\Big|_{V^{\\perp}}\\Big)\\subset W,\\]\nthen \n\\begin{equation}\n\\bigcap_{v\\in[0,1]}\\widetilde{\\Pi}(v)=\\left\\{0\\right\\}.\n\\label{xrisimipi}\n\\end{equation}\nIndeed, in view of \\eqref{dimerelationw},\\eqref{sxesi2}, there are $v_{1},v_{2}\\in I$ such that $K_{1}= \\Big.\\Pi(v_{1})\\Big|_{V^{\\perp}}\\cap \\Big.\\Pi(v_{2})\\Big|_{V^{\\perp}}$ is at most $(n-1)$-dimensional (where  the dimension of $\\Pi(v)$ is $n$). In view of \\eqref{sxesi2}, there is $v_{3}\\in I$ such that $K_{2}=K_{1}\\cap \\Big.\\Pi(v_{3})\\Big|_{V^{\\perp}}$ is at most $(n-2)$-dimensional. Continuing inductively we deduce that there  are $v_{1},v_{2},...,v_{n}$ such that \n\\begin{equation}\n\\bigcap_{v_{i}}\\Big.\\Pi(v_{i})\\Big|_{V^{\\perp}}=\\left\\{0\\right\\}.\n\\label{finiteintersectionmiden}\n\\end{equation}\n We then define $W$ by\n\\begin{equation}\nW=\\Big\\langle \\Big.\\Pi(v_{1})\\Big|_{V^{\\perp}},\\Big.\\Pi(v_{2})\\Big|_{V^{\\perp}},...,\\Big.\\Pi(v_{n})\\Big|_{V^{\\perp}} \\Big\\rangle,\n\\label{eq:definitionofw}\n\\end{equation}\nwhich is clearly finite dimensional and satisfies \\eqref{xrisimipi}. Indeed, by the construction of $W$,  \\eqref{xrisimipi} is satisfied even if we restrict the intersection for the values of $v$ in the set $ \\left\\{v_{1},...,v_{n}\\right\\}$.\n\nWe next consider the projection\n\\[\\Big.\\text{proj}\\Big|_{W}(v): \\Big.\\Pi(v)\\Big|_{V^{\\perp}}\\rightarrow W. \\]\nBy virtue of \\eqref{eq:definitionofw} we have that $\\Big.\\text{proj}\\Big|_{W}(v)$ has full rank for $v=v_{1},...,v_{n}$. Since $\\Big.\\Pi(v)\\Big|_{V^{\\perp}}$ varies smoothly in $v$ we have that $\\Big.\\text{proj}\\Big|_{W}(v)$ has full rank in the union of sufficiently small intervals $J_{i}$ containing $v_{i}$. \n\n\nLet also $\\widetilde{O}(v)$ denote the orthogonal complement of $\\widetilde{\\Pi}(v)$ in $W$. It is important to note that the space $\\widetilde{O}(v)$ varies smoothly for $v\\in \\cup J_{i}$ since $\\widetilde{\\Pi}(v)$ varies smoothly for $v\\in \\cup J_{i}$. In view of \\eqref{finiteintersectionmiden} we have that \n\\begin{equation}\n\\Big\\langle \\widetilde{O}(v_{1}),...,\\widetilde{O}(v_{n}) \\Big\\rangle= W.\n\\label{finitespanofo}\n\\end{equation}\nIndeed, if there is a proper subspace $X\\subset W$ such that $\\widetilde{O}(v)\\subset X$ for all $v\\in \\left\\{v_{1},...v_{n}\\right\\}$ then $X^{\\perp}\\subset \\big(\\widetilde{O}(v)\\big)^{\\perp}\\subset W$ for all $v\\in \\left\\{v_{1},...v_{n}\\right\\}$. This is however contradiction since, by definition, $\\big(\\widetilde{O}(v)\\big)^{\\perp}=\\widetilde{\\Pi}(v)$ and by \\eqref{finiteintersectionmiden} these spaces cannot have non-trivial common intersection. Therefore, there are $u_{1},...,u_{\\text{dim}(W)}\\in\\left\\{v_{1},...,v_{n}\\right\\}\\subset \\cup J_{i}$ and $x_{i}\\in \\widetilde{O}(u_{i})$ with $i=1,...,\\text{dim}(W)$ such that the set $\\left\\{x_{i}, i=1,...,\\text{dim}(W)\\right\\}$ is a basis of $W$. We can assume that $u_{i}$ are pairwise distinct since otherwise we can consider small perturbations $u_{i}^{\\text{perp}}$ of them in the union $\\cup J_{i}$. Then, since $\\widetilde{O}(v)$ varies smoothly in $\\cup J_{i}$, the perturbed vectors $x_{i}^{\\text{perp}}=\\widetilde{O}\\big(u_{i}^{perp}\\big)$ still form a basis of $W$. Moreover, we can assume that $u_{i}^{\\text{perp}}$ lie in the interior of $I$ for all $i=1,...,\\text{dim}(W)$. We can thus define the following closed intervals $I_{i},i=1,...,\\text{dim}(W)$ as follows:\n\\begin{equation}\nI_{i}\\subset J_{i}\\cap \\text{int}I \\text{ such that } u_{i}^{\\text{perp}}\\in I_{i},\n\\label{orismosdiastimatos}\n\\end{equation}\nwhere $\\text{int}I$ denotes the interior of $I$.  \n\nLet now $W^{\\perp}$ denote the orthogonal complement of $W$ in $V^{\\perp}$ and hence \n\\[L^{2}\\big(\\mathbb{S}^{2}\\big)=V\\oplus W \\oplus W^{\\perp}.  \\] Since  $W=\\widetilde{\\Pi}(v)\\oplus \\widetilde{O}(v)$ we have\n\n\\begin{equation}\nL^{2}\\big(\\mathbb{S}^{2}\\big)= V\\oplus \\widetilde{\\Pi}(v)\\oplus \\widetilde{O}(v)\\oplus W^{\\perp}. \n\\label{sxesi3}\n\\end{equation}\nThen, for any $F\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ we have\n\\begin{equation}\nF(v,\\cdot)=\\Big.F\\Big|_{V}+\\Big.F\\Big|_{\\widetilde{\\Pi}(v)}+\\Big.F\\Big|_{\\widetilde{O}(v)}+\\Big.F\\Big|_{W^{\\perp}}\n\\label{sxesiF}\n\\end{equation}\nwhere we denote $\\Big.F\\Big|_{Z}=\\Big.\\text{proj}\\Big|_{Z}\\big(F\\big)$.\n\n\nGiven a function $\\rho\\in V^{\\perp}\\cap C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ we want to construct a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ such that \n\\begin{equation}\n\\int_{I}F_{\\rho}(v,\\cdot)dv=\\rho\\in V^{\\perp} \n\\label{edw1}\n\\end{equation}and\n\\begin{equation}\nF_{\\rho}(v,\\cdot)\\in \\Big(\\Pi(v)\\Big)^{\\perp}\\subset V^{\\perp}\\ \\text{ for all }\\ v\\in I.\n\\label{edw2}\n\\end{equation}\nThe first condition we impose on $F_{\\rho}$ is:\n\\begin{equation}\n\\Big.F_{\\rho}(v,\\cdot)\\Big|_{V}=0 \\text{ for all } v\\in I.\n\\label{conditionF1}\n\\end{equation}\nThen condition \\eqref{edw1} is equivalent to  \n\\begin{equation}\n\\int_{I}\\Big.F_{\\rho}(v,\\cdot)\\Big|_{W^{\\perp}}dv=\\Big.\\rho\\Big|_{W^{\\perp}} \n\\label{zitoumeno1}\n\\end{equation}\nand \n\\begin{equation}\n\\int_{I}\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{\\Pi}(v)}dv+\\int_{I}\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}dv=\\Big.\\rho\\Big|_{W}.\n\\label{zitoumeno2}\n\\end{equation}\nMoreover, condition \\eqref{edw2}, using \\eqref{conditionF1}, is equivalent to the following: \n\\begin{equation}\n\\Big\\langle \\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{\\Pi}(v)}, \\big. w\\big|_{\\widetilde{\\Pi}(v)} \\Big\\rangle =- \\Big\\langle \\Big. F_{\\rho}(v,\\cdot)\\Big|_{W^{\\perp}},  \\big. w\\big|_{W^{\\perp}}   \\Big\\rangle\\text{ for any }w\\in \\Pi(v). \n\\label{conditionF2}\n\\end{equation}\nIndeed, for any $w\\in \\Pi(v)$ we have\n\\begin{equation*}\n\\begin{split}\n0=&\\Big\\langle F_{\\rho}(v,\\cdot), w\\Big\\rangle=\\Big\\langle  \\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}+\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}+\\Big.F_{\\rho}\\Big|_{W^{\\perp}},  \\big.w\\big|_{V^{\\perp}} \\Big\\rangle \\\\\n& =\\Big\\langle \\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}+\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}+\\Big.F_{\\rho}\\Big|_{W^{\\perp}}, \\big. w\\big|_{\\widetilde{\\Pi}(v)} + \\big. w\\big|_{W^{\\perp}} \\Big\\rangle\\\\\n&=\\Big\\langle \\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)},\\big. w\\big|_{\\widetilde{\\Pi}(v)} \\Big\\rangle+\\Big\\langle \\Big.F_{\\rho}\\Big|_{W^{\\perp}},  \\big. w\\big|_{W^{\\perp}}  \\Big\\rangle,\n\\end{split}\n\\end{equation*}\nsince $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{V}=\\big.w\\big|_{\\widetilde{O}(v)}=0$. Note thus that  condition \\eqref{conditionF2} (and hence \\eqref{edw2}) is independent of  $\\Big. F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$. \n\n\\bigskip\n\nLet $\\mathcal{B}_{1}=\\left\\{ e_{1},...,e_{\\text{dim}(V)} \\right\\}$, $\\mathcal{B}_{2}=\\left\\{ e_{\\text{dim}(V)+1},...,e_{\\text{dim}(V)+\\text{dim}(W)} \\right\\}$, $\\mathcal{B}_{3}=\\left\\{ e_{\\text{dim}(V)+\\text{dim}(W)+1},... \\right\\}$ be orthonormal bases of the spaces $V,W, W^{\\perp}$, respectively. Clearly, $\\mathcal{B}_{1}\\cup\\mathcal{B}_{2}\\cup\\mathcal{B}_{3}$ is an orthonormal basis of $L^{2}\\big(\\mathbb{S}^{2}\\big)$. Since $\\Pi(v)$ are spanned by smooth functions, it is easy to see that we can take $e_{i}$ to be smooth functions on $\\mathbb{S}^{2}$.  Any function $F\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ can be written as\n\\[F(v,\\cdot)=\\sum_{i\\geq 1} F_{i}(v)\\cdot e_{i}. \\]\nWe impose $\\big(F_{\\rho}\\big)_{i}(v)$ such that  for  $1\\leq i\\leq \\text{dim}(V)$ as follows:\n\\begin{itemize}\n\t\\item  $\\big(F_{\\rho}\\big)_{i}(v)=0$ for all $v\\in I$,  in accordance with \\eqref{conditionF1}. \n\\end{itemize}\n Moreover, we  prescribe  $\\big(F_{\\rho}\\big)_{i}(v)$ for $i\\geq \\text{dim}(V)+\\text{dim}(W)+1$ such that: \n\\begin{itemize}\n\t\\item $\\left\\{\\big(F_{\\rho}\\big)_{i}(v) \\right\\}_{i}\\in \\ell^{2}$ for all $v\\in I,$\n\\item \t$\\big(F_{\\rho}\\big)_{i}(v)=0  $ for all $v\\in I\/ \\big(I_{1}\\cup ....\\cup I_{\\text{dim}(W)}\\big),$ where $I_{k}$ as defined in \\eqref{orismosdiastimatos},\n\\item $\\big(F_{\\rho}\\big)_{i}(v)$ for $v\\in I_{1}\\cup ....\\cup I_{\\text{dim}(W)}$ are such that   $\\int_{I}\\big(F_{\\rho}\\big)_{i}(v)dv$ are imposed by condition \\eqref{zitoumeno1}.\n\\end{itemize}\nClearly such functions exist and hence the functions $\\Big.F_{\\rho}\\Big|_{W^{\\perp}}(v)\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ are determined and, in particular are such that they vanish for $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$.  \n\nWe next construct $\\Big.F_{\\rho}\\Big|_{W}(v)=\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)+\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}(v)$ or, equivalently, to construct $\\big(F_{\\rho}\\big)_{i}(v)$ for $\\text{dim}(V)+1\\leq i\\leq \\text{dim}(V)+\\text{dim}(W)+1$. We first impose \n\\begin{itemize}\n\t\\item $\\Big.F_{\\rho}\\Big|_{W}(v)=0$ for all $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ and hence $\\big(F_{\\rho}\\big)_{i}(v)=0$ for $\\text{dim}(V)+1\\leq i\\leq \\text{dim}(V)+\\text{dim}(W)+1$ and $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\\end{itemize}\nClearly, condition \\eqref{conditionF2} is then satisfied for all $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$.\nIt remains to construct the functions  $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v),\\, \\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}(v)$ for $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$.\n\nWe note that condition \\eqref{conditionF2} uniquely determines a well-defined $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)$ for all $v\\in\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$.  Indeed, if $\\widetilde{\\Pi}(v)=\\left\\{0\\right\\}$ for some $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ then we have $\\widetilde{\\Pi}(v)=\\left\\{0\\right\\}$ for all $v\\in\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ and hence $\\Pi(v)=V$ for all $v\\in I$.  In this case we take $W=\\left\\{0\\right\\}$ and \\eqref{conditionF2} holds trivially. If, on the other hand, the spaces $\\widetilde{\\Pi}(v)$ are non-trivial and vary smoothly for all $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ then condition \\eqref{conditionF2} determines $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)$ for $v\\in\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. Since $\\Big.F_{\\rho}\\Big|_{W^{\\perp}}(v)$  a smooth function in $I$ which vanishes in $I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ we obtain that $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)$, as defined above,  is also a smooth function in $I$ which vanishes in $I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\nWe finally construct the function  $\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}(v)$ for $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ such that \\eqref{zitoumeno2} holds (recall that this function vanishes in the complement of this union in $I$).  We will indeed show that this is possible.\n\nRecall that  $\\widetilde{O}(v)$ varies smoothly in $v\\in I_{1}\\cup...\\cup I_{\\text{dim}(W)}$. Moreover,  by the definition of the intervals $I_{1},...,I_{\\text{dim}(W)}$, there are vectors $x_{i}\\in \\widetilde{O}(v_{i})$ with $v_{i}\\in I_{i}, i=1,...,\\text{dim}(W)$ such that $\\left\\{x_{i},i=1,...,\\text{dim}(W)\\right\\}$ is a basis of $W$. We can construct a smooth curve \n\\[\\gamma:I\\rightarrow W  \\]\nsuch that \n\\begin{equation}\n\\gamma(v_{i})=x_{i} \n\\label{gam1}\n\\end{equation}\nand \n\\begin{equation}\n\\gamma(v)=0 \\text{ for all } v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big).\n\\label{gam2}\n\\end{equation}\nThe smoothness of the curve $\\gamma$ can be guaranteed by the smoothness of $\\widetilde{O}(v)$ in $I_{1}\\cup...\\cup I_{\\text{dim}(W)}$. If $\\big(f_{1}(v),f_{2}(v),...,f_{\\text{dim}(W)}(v) \\big)$ are the coordinates of $\\gamma(v)$ with respect to the basis $\\mathcal{B}_{2}$ of $W$, then, in view of \\eqref{gam1}, the smooth functions $f_{i}:I\\rightarrow\\mathbb{R},i=1,...,\\text{dim}(W),$ are linearly independent.  We will find an appropriate smooth function $\\alpha:I\\rightarrow\\mathbb{R}$ such that \n\\begin{equation}\n\\Big. F_{\\rho}\\Big|_{\\widetilde{O}(v)}=\\alpha(v)\\cdot \\gamma(v)\\in \\widetilde{O}(v). \n\\label{ansatz1}\n\\end{equation}\nLet us assume that $\\Big(\\big(F_{\\rho}\\big)_{1}(v),...,\\big(F_{\\rho}\\big)_{\\text{dim}(W)}(v) \\Big)$ are the coordinates of $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$ with respect to the basis $\\mathcal{B}_{2}$ of $W$. Condition \\eqref{zitoumeno2} is satisfied if we choose these functions such that \n\\[ \\int_{I}\\big(F_{\\rho}\\big)_{i}(v)\\, dv=\\lambda_{i},\\]for all $i=1,2,...,\\text{dim}(W)$, \nwhere $\\lambda_{i}$ is completely determined by \\eqref{zitoumeno2} and our previous constructions. Equivalently, in view of our ansatz \\eqref{ansatz1}, it suffices to show the existence of a smooth function $\\alpha:I\\rightarrow\\mathbb{R}$ such that\\begin{equation}\n\\int_{I}\\alpha(v)\\cdot f_{i}(v)\\, dv=\\lambda_{i},\n\\label{eq:nai}\n\\end{equation}\nfor all $i=1,2,...,\\text{dim}(W)$. Since the functions $f_{i}$ are linearly independent, the existence of  $\\alpha$ follows from Lemma \\ref{lemma1stoproof}. This completes the construction of the function $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$. Note that in view of \\eqref{gam2} the function $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$ vanishes in $I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\nThis completes the construction of the function $F_{\\rho}$ with all the required properties.\n\\end{proof}\nLemma \\ref{lemma1stoproof} holds not only for intervals but also for union of intervals. Moreover, a modification of the above proof yields the following result\n\\begin{lemma}\nLet $I=\\bigcup_{k=1}^{n}I_{k}$, where $I_{k}$, with $k=1,2,...,n$, are compact intervals of $\\mathbb{R}$. For each $k\\in\\left\\{1,2,...,n\\right\\}$ consider  $\\Pi_{k}(v), v\\in I_{k}$, to be a smoothly varying $n_{k}$-dimensional subspace of $L^{2}\\big(\\mathbb{S}^{2} \\big)$ spanned by $k$ smooth functions on $\\mathbb{S}^{2}$. Define the subspaces $V_{k}\\subset L^{2}\\big(\\mathbb{S}^{2} \\big)$ as follows\n\\[ V_{k}=\\bigcap_{v\\in I_{k}}\\Pi_{k}(v).  \\]\nGiven a function $\\rho\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ there is a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times\\mathbb{S}^{2}\\big)$  which vanishes to infinite order at $\\partial I\\times \\mathbb{S}^{2}$ and is such that \n\\begin{equation}\n\\int_{I}F_{\\rho}(v,\\cdot)\\, dv= \\rho(\\cdot)\n\\label{eq:rel1}\n\\end{equation}\nand \n\\begin{equation}\nF_{\\rho}(v,\\cdot)\\in \\Big( \\Pi_{k}(v) \\Big)^{\\perp} \\text{ for all }v\\in I_{k} \\text{ and  }k=1,2...,n\n\\label{eq:arbitragelemma}\n\\end{equation}\nif and only if \n\\[\\rho\\in \\Big(V_{1}\\cap V_{2}\\cap...\\cap V_{n}\\Big)^{\\perp}\\subset L^{2}\\big(\\mathbb{S}^{2}\\big). \\]\n\\label{argitragelemma}\n\\end{lemma}\n\\begin{proof}\nThe above lemma is proved using the same arguments as in the proof of Lemma \\ref{deuterolemma} if we replace the space $V$ by $V_{1}\\cap V_{2}\\cap...\\cap V_{n}$.\n\\end{proof}\n\n\\bigskip\n\nWe now have all the tools needed for the proof of Theorem \\ref{theorem}.\n\n Let $x\\in\\mathcal{Z}$ (where $\\mathcal{Z}$ is given by Lemma \\ref{lemmagiaelliptickernel}) and $V_{x}$ is a closed interval containing $x$ and such that  $dimKer(\\mathcal{O}_{v}^{*})=n\\geq 1$. If we apply Lemma \\ref{deuterolemma} for $I=V_{x}$ and $\\Pi(v)=\\left\\langle G_{1}(v),...,G_{n}(v)\\right\\rangle$, where  the functions $G_{n}$ given by \\eqref{g0}, then the only obstruction to satisfying conditions 1--3  (and thus to gluing) is the existence of the following intersection\n\\begin{equation}\n\\mathcal{W}(V_{x})=\\bigcap_{v\\in V_{x}}\\Pi(v). \n\\label{subkernel}\n\\end{equation}\nIn other words, if there is $x\\in \\mathcal{Z}$ such that $\\mathcal{W}(V_{x})=\\left\\{0\\right\\}$ then we can perform gluing. Suppose now that for all $x\\in\\mathcal{Z}$ we have $\\mathcal{W}(V_{x})\\neq \\left\\{0\\right\\}$. Let us fix an $x\\in\\mathcal{Z}$ and let $\\Theta\\in \\mathcal{W}(V_{x})$ such that $\\Theta\\neq 0$.  By  \\eqref{defF} and \\eqref{integrability}, and since $\\partial_{v}\\Theta=0$ for $v\\in V_{x}$,   we have \n\\[0=\\int_{S_{v}}(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{v}}\\Big(\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\partial_{v}\\left(\\int_{S_{v}}\\Big(\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}\\right), \\]\nwhere we have also used that the measure of integration does not depend on $v$. Hence, the quantity\n\\begin{equation}\n\\int_{S_{v}}\\Big(\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:conserved}\n\\end{equation}\nis conserved in $V_{x}$, i.e.~independent of $v$ for all $v\\in V_{x}$. Therefore, we have conservation laws in each interval $V_{x}$ and the kernel of the conservation laws is precisely the  space\n $\\mathcal{W}(V_{x})$.  \n\nWe now consider two cases.\n\n\\medskip\n\n\\underline{ \\textbf{Case I:}} \n\n\\medskip\n\nSuppose that \n\\begin{equation}\n\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})=\\left\\{0\\right\\}.\n\\label{suppose1}\n\\end{equation}\nBy virtue of Lemma \\ref{lemmagiaelliptickernel}, we have that there is $n\\in\\mathbb{N}$ such that $\\text{dim}\\big(\\mathcal{W}(V_{x})\\big)\\leq n$ for all $x\\in\\mathcal{Z}$. In view of \\eqref{suppose1}, there are $x_{1},x_{2}\\in\\mathcal{Z}$ such that $\\mathcal{W}(V_{x_{1}})\\cap\\mathcal{W}(V_{x_{2}})$ is at most $(n-1)$-dimensional.  In view again of \\eqref{suppose1}, there is $x_{3}\\in\\mathcal{Z}$ such that $\\mathcal{W}(V_{x_{1}})\\cap\\mathcal{W}(V_{x_{2}})\\cap\\mathcal{W}(V_{x_{3}})$ is at most $(n-2)$-dimensional.  Continuing inductively we obtain $x_{1},...,x_{n+1}\\in\\mathcal{Z}$ such that $\\mathcal{W}(V_{x_{1}})\\cap\\mathcal{W}(V_{x_{2}})\\cap...\\cap\\mathcal{W}(V_{x_{n+1}})=\\left\\{0\\right\\}$. Thus, applying Lemma \\ref{argitragelemma} for $V_{x_{1}},...,V_{x_{n+1}}$ we can satisfy conditions 1--3 and thus perform gluing. The smoothness of the extension of $\\psi$ on $\\mathcal{H}$ follows from the results of Lemma \\ref{lemmagiaelliptickernel} and their analogue in higher Sobolev spaces.   \n\n\n\\medskip\n\n\\underline{ \\textbf{Case II:}} \n\n\\medskip\n\nSuppose that \n\\begin{equation}\n\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})\\neq\\left\\{0\\right\\}.\n\\label{suppose1}\n\\end{equation}\nLet $\\Theta \\in \\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})$ with $\\Theta\\neq 0$. Then the integrals \\eqref{eq:conserved} are conserved for $v\\in V_{x}$ for all $x\\in \\mathcal{Z}$. For any  $x,y\\in\\mathcal{Z}$ with $V_{x}\\cap V_{y}\\neq \\emptyset$ we have that the integrals \\eqref{eq:conserved} are conserved for $v\\in V_{x}\\cup V_{y}$. We will show that the integrals \\eqref{eq:conserved} are in fact conserved for all $v\\in[0,1]$. Let us denote the integral \\eqref{eq:conserved} over $S_{v}$ by $f(v)$. Then  $f$ is a smooth function on $[0,1]$ which is constant on the connected components of the open and dense subset $\\mathcal{Z}$ of $[0,1]$. If $f$ is not constant then there is $v_{0}\\in [0,1]$ for which $\\big.\\partial_{v}f\\big|_{v_{0}}\\neq 0$ and hence there is an interval containing $v_{0}$ where $f$ is strictly monotonic. This is however contradiction since, by assumption, for every interval there is a subinterval where $f$ is constant. \n\n\nTherefore, the integrals \\eqref{eq:conserved} are conserved for $v\\in[0,1]$ for all $\\Theta \\in \\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})$. Clearly, these conservation laws are legitimate obstructions to gluing since we cannot do gluing unless the corresponding integrals of the initial data at $S_{0}$ and $S_{1}$ are equal. Suppose, therefore, that the initial data at $S_{0}$ and $S_{1}$ are such that the corresponding integrals \\eqref{eq:conserved} are equal. Then we will show that we can perform gluing (and hence there are no additional obstructions to gluing apart from the above conservation laws). \n\nIn view of \\eqref{wenull1} and \\eqref{malista} and the second condition for the gluing of transversal derivative we need to construct a function $C^{\\infty}\\big([0,1]\\times\\mathbb{S}^{2}\\big)$ such that \n\\begin{equation}\n\\int_{0}^{1}F(v,\\cdot)\\, dv= 2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{1}}}-2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{0}}}.\n\\label{apopano}\n\\end{equation}\nIn view of our assumption on the initial data at $S_{0},S_{1}$ we have that the right hand side of \\eqref{apopano} is orthogonal to the space $\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})$ with respect to the inner product of $L^{2}\\big(\\mathbb{S}^{2}\\big)$. Using a similar argument as above, we can choose finitely many $x_{1},...,x_{m}\\in \\mathcal{Z}$ such that $V_{x_{i}}\\cap V_{x_{j}}=\\emptyset$ for $i\\neq j$ and  \n\\[\\mathcal{W}(V_{x_{1}})\\cap...\\cap\\mathcal{W}(V_{x_{m}})=\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x}). \\]\nIt now suffices to extend $\\psi$ in the complement of $V_{x_{1}}\\cup...\\cup V_{x_{m}}$ in $[0,1]$ and apply Lemma \\ref{argitragelemma} for the  intervals $V_{x_{1}}, ....,  V_{x_{m}}$ in order to complete the gluing construction.  This finishes the proof of Theorem \\ref{theorem}.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Change of foliation and conservation laws}\n\\label{sec:EffectOfGaugeOnConservationLaws}\n\nSo far we have consider characteristic gluing constructions and conservation laws on null hypersurfaces with respect to a \\textbf{fixed} foliation. In particular, it is clear the the conserved charges depend, at least a priori, on the choice of foliation\\footnote{After all, the conserved charges are appropriate integrals over the leaves of the foliation.}. In this subsection we address the issue of change of foliation and its effect on the conservation laws. \n\nLet $\\mathcal{S}=\\Big\\langle S_{0}, L_{geod}, \\Omega\\Big\\rangle$ and $\\mathcal{S}'=\\left\\langle S_{0}', L_{geod}', \\Omega'\\right\\rangle$ be two foliations of a regular null hypersurface $\\mathcal{H}$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Let $\\mathcal{V}_{\\hh}$ be the linear space defined by \\eqref{linearspace}. Consider the kernels $\\mathcal{W}^{\\mathcal{S}},\\mathcal{W}^{\\mathcal{S}'}\\subset \\mathcal{V}_{\\hh}$ of the conservation laws with respect to the foliations $\\mathcal{S},\\mathcal{S}'$, respectively, as defined in Section \\ref{sec:ConservationLawsForTheWaveEquations}. Given also the operators $\\mathcal{O}^{\\mathcal{S}},\\mathcal{O}^{\\mathcal{S}'}$ associated to the foliations $\\mathcal{S},\\mathcal{S}'$, respectively, and defined by \\eqref{adjoint} we consider the linear spaces $\\mathcal{U}^{\\mathcal{S}},\\mathcal{U}^{\\mathcal{S}'}\\subset\\mathcal{V}_{\\hh}$ which are (appropriately rescaled) subspaces of  $Ker\\big(\\mathcal{O}^{\\mathcal{S}}\\big),Ker\\big(\\mathcal{O}^{\\mathcal{S}'}\\big)$ defined as in Theorem \\ref{theorem}. Then, according to Theorem \\ref{theorem} we have\n\\begin{equation}\n\\mathcal{W}^{\\mathcal{S}}=\\mathcal{U}^{\\mathcal{S}}\\ \\text{ and }\\  \\mathcal{W}^{\\mathcal{S}'}=\\mathcal{U}^{\\mathcal{S}'}.\n\\label{eq:change1}\n\\end{equation}\nThe following proposition derives the relation between the spaces $\\mathcal{U}^{\\mathcal{S}}$ and $\\mathcal{U}^{\\mathcal{S}'}$. \n\\begin{proposition}\nLet $\\mathcal{S}=\\Big\\langle S_{0}, L_{geod}, \\Omega\\Big\\rangle$ and $\\mathcal{S}'=\\left\\langle S_{0}', L_{geod}', \\Omega'\\right\\rangle$ be two foliations of a (regular) null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Let $f\\in\\mathcal{V}_{\\hh}$ be such that   \n\\begin{equation}\nL_{geod}'=f^{2}\\cdot L_{geod}\\ : \\text{  on }\\, \\mathcal{H}.\n\\label{geodesicf}\n\\end{equation}\nThen, we have\n\\begin{equation}\n\\mathcal{U}^{\\mathcal{S}'}=f^{2}\\cdot\\mathcal{U}^{\\mathcal{S}}=\\Big\\{f^{2}\\cdot\\Theta\\ :\\ \\Theta\\in\\mathcal{U}^{\\mathcal{S}} \\Big\\}.\n\\label{changef12}\n\\end{equation}\n\\label{corm1}\n\\end{proposition}\n\\begin{proof}\nAccording to the main result of \\cite{aretakiselliptic}, the operators $\\mathcal{O}^{\\mathcal{S}}$ and $\\mathcal{O}^{\\mathcal{S}'}$ satisfy the following relation\n\\begin{equation}\n\\mathcal{O}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot\\Theta\\right)=\\left(\\frac{\\Omega}{\\Omega'}\\right)^{2}\\cdot\\frac{1}{f^{2}}\\cdot \\mathcal{O}^{\\mathcal{S}'}\\left(f^{2}\\cdot\\frac{1}{\\phi}\\cdot\\Theta \\right),\n\\label{fromotherpaper}\n\\end{equation}\non $\\mathcal{H}$, for all functions $\\Theta\\in\\mathcal{V}_{\\hh}$. Let now $\\Theta\\in\\mathcal{U}^{\\mathcal{S}}$ and $S'_{v'}$ be a section of the foliation $\\mathcal{S}'$. Then, $S'_{v'}$ can be sweeped by the sections $S_{v}$ of the foliation $\\mathcal{S}$ as depicted below. \n\\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.13]{nullsection2.png}\n\t\\label{fig:nullsectionxax}\n\\end{figure}\nFor each intersection point of $S'_{v'}$ with the sections of $\\mathcal{S}$ we apply the relation \\eqref{fromotherpaper} to deduce that \n\\[\\text{ If }\\ \\mathcal{O}_{v}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot \\Theta \\right)=0 \\ \\text{for all }v\\in\\mathbb{R}\\ \\text{ then }\\ \\mathcal{O}^{\\mathcal{S}'}_{v'}\\left(\\frac{1}{\\phi}\\cdot \\big(f^{2}\\cdot\\Theta \\big)\\right)=0. \\]\nHence, $ f^{2}\\cdot\\mathcal{U}^{\\mathcal{S}}  \\subseteq \\mathcal{U}^{\\mathcal{S}'}$. The inverse inclusion can be similarly shown by sweeping the section of the foliation $\\mathcal{S}$ with the sections of $\\mathcal{S}'$, yielding the required result. \n\\end{proof}\nWe remark that the equation \\eqref{changef12} holds only if we can sweep the sections of the foliation $\\mathcal{S}$ with those of $\\mathcal{S}'$ and vice versa. This shows that the existence of conservation laws (and hence the obstructions to gluing) are due to global properties of the sections of the null hypersurface (and not just pointwise or local; cf. Remark 1 in Section \\ref{sec:Remarks}). Specifically, they are related with the properties of the kernels of the elliptic operators $\\mathcal{O}^{\\mathcal{S}}_{v}, v\\in\\mathbb{R}$. The calculation of \\cite{aretakiselliptic} showed that these elliptic operators are covariant (in the sense of \\eqref{fromotherpaper}) under change of foliation. This covariance property holds also for the kernels of these operators as long as the sweeping property holds. Hence, although capturing the elliptic structure associated to the sections of $\\mathcal{S}$ is of fundamental importance in the present paper, it turns out this elliptic structure does not depend on the choice of the foliation of $\\mathcal{H}$.\n\nIn view of the results of Theorem \\ref{theorem}, an immediate corollary of Proposition \\ref{corm1} is the following\n\\begin{corollary}\nA null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$ admits conservation laws with respect to a foliation $\\mathcal{S}=\\Big\\langle S_{0}, L_{geod}, \\Omega\\Big\\rangle$ in the sense of Definition \\ref{definitionconservationlaw} if and only it admits conservation laws with respect to any other foliation $\\mathcal{S}'=\\Big\\langle S_{0}', L_{geod}', \\Omega'\\Big\\rangle$. Specifically, the kernels $\\mathcal{W}^{S},\\mathcal{W}^{\\mathcal{S}'}$ of the conservation laws satisfy\n\\begin{equation}\n\\mathcal{W}^{\\mathcal{S}'}=f^{2}\\cdot\\mathcal{W}^{\\mathcal{S}}=\\Big\\{f^{2}\\cdot\\Theta\\ :\\ \\Theta\\in\\mathcal{W}^{\\mathcal{S}} \\Big\\},\n\\label{corequchangeoffoliation}\n\\end{equation}\nwhere $f$ is given by \\eqref{geodesicf}. In other words, the integrals \n\\begin{equation}\nchar\\big(S_{v}\\big)[\\psi]=\\int_{S_{v}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta \\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{corollaryequation1}\n\\end{equation}\nwith $\\Theta\\in\\mathcal{V}_{\\hh}$ are conserved (i.e.~independent of $v$), for all solutions $\\psi$ to the wave equation, if and only if the integrals \n\\begin{equation}\nchar\\big(S'_{v'}\\big)[\\psi;\\Theta]=\\int_{S'_{v'}}f^{2}\\cdot Y^{\\mathcal{S}'}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta \\, d\\mu_{_{\\mathbb{S}^{2}}} \n\\label{corollaryequation2}\n\\end{equation}\nare conserved, i.e.~independent of $v$, for all solutions $\\psi$ to the wave equation. The vector fields  $Y^{\\mathcal{S}},\\, f^{2}\\cdot Y^{\\mathcal{S}'}$ are null and normal to the sections of $\\mathcal{S},\\mathcal{S}'$, respectively, conjugate to $\\mathcal{H}$ and normalized such that \n\\[g\\big(Y^{\\mathcal{S}},L_{geod}\\big)=g\\big(f^{2}\\cdot Y^{\\mathcal{S}'},L_{geod}\\big)=-1.  \\]\nMoreover, if $\\mathcal{H}$ admits conservation laws then we in fact have\n\\begin{equation}\nchar\\big(S_{v}\\big)[\\psi;\\Theta]=char\\big(S'_{v'}\\big)[\\psi;\\Theta]\n\\label{corollaryequation3}\n\\end{equation}\nand hence the value of the conserved charges is independent of the choice of foliation. \n\\label{corollarychangeofoliation}\n\\end{corollary}\n\n\n\n\n\n\n\\subsection{Perturbation analysis}\n\\label{sec:PerturbativeAnalysis}\n\nWe next investigate the stability\/genericity properties of the conservation laws on a null hypersurface $\\mathcal{H}$ of a Lorentzian manifold $(\\mathcal{M},g)$ under perturbations of the ambient metric $g$ (i.e.~under perturbations of the geometry of $\\mathcal{H}$). We have the following\n\n\n\\begin{proposition}\nLet $\\mathcal{H}$ be a null hypersurface as in Theorem \\ref{theoremmainintro} which admits conservation laws for the wave equation on a Lorentzian manifold $(\\mathcal{M},g)$. There are  arbitrarily small perturbations of the ambient metric $g$ for which $\\mathcal{H}$ is a null hypersurface without, however, admitting conservation laws (and hence gluing is possible on these perturbed null hypersurfaces).  In fact, the set of all ambient metrics for which $\\mathcal{H}$ admits consevation laws is of positive codimension. \n\nFurthermore, if the underlying Lorentzian manifold $(\\mathcal{M},g)$ satisfies the Einstein--vacuum equations then we can perturb the metric $g$ in the class of spacetimes satisfying the Einstein--vacuum equations such that the above conclusions still hold.\n\n\\label{prop2pert}\n\\end{proposition} \n\\begin{proof}\nIn view of the results of Section \\ref{sec:EffectOfGaugeOnConservationLaws} it suffices to consider a geodesic foliation $\\mathcal{S}=\\left\\langle S_{0}, \\,L_{geod}, \\, \\Omega=1\\right\\rangle$ of $\\mathcal{H}$. By assumption we have that $dim\\mathcal{W}^{\\mathcal{S}}=\\dim\\mathcal{U}^{\\mathcal{S}}\\geq 1$, which, in particular, implies that $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq \\left\\{0\\right\\}$ for all $v\\in\\mathbb{R}$. Recall that (for the unperturbed metric):\n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+\\Big[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+\\partial_{v}( tr\\underline{\\chi})+\\frac{1}{2}( tr\\underline{\\chi})\\cdot( tr\\chi) \\Big]\\cdot\\psi \n\\label{toxreiazomai}\n\\end{equation}\nWe will show that for any fixed $v$ there is an $\\epsilon_{1}>0$ such that for all $\\epsilon\\in(0,\\epsilon_{1})$ the operator\n\\[\\mathcal{O}_{v}^{\\mathcal{S},\\epsilon}\\psi=\\mathcal{O}_{v}^{\\mathcal{S}}\\psi-\\epsilon\\cdot \\psi\\]\nhas \\textit{trivial} kernel. Indeed, if we consider  the operator\n\\[\\mathcal{O}_{temp}\\psi=\\mathcal{O}_{v}^{\\mathcal{S}}\\psi-\\frac{1}{\\epsilon_{0}}\\cdot \\psi \\]\nthen we can show as before that if we take $\\epsilon_{0}$ sufficiently small (depending on $v$) then the operator $\\mathcal{O}_{temp}$ is invertible and hence has a discrete spectrum consisting only of eigenvalues. Therefore, the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ has a discrete set of eigenvalues in the spectrum (whose limit point is infinity). By assumption, one of these eigenvalues is zero. By the discreteness of the spectrum, if we take $\\epsilon_{1}$ sufficiently small, then the operator $\\mathcal{O}_{v}^{\\mathcal{S},\\epsilon}$ has trivial kernel. \n\nTherefore, if we consider a perturbed metric $g_{\\epsilon}$ for which $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\epsilon}\\!\\left.\\right|_{S_{0}}=\\mbox{$g \\mkern-8.8mu \/$\\,}\\!\\left.\\right|_{S_{0}}$, $\\zeta_{\\epsilon}\\!\\left.\\right|_{\\mathcal{H}}=\\zeta\\!\\left.\\right|_{\\mathcal{H}},$  $\\mbox{tr}\\chi_{\\epsilon}\\!\\left.\\right|_{\\mathcal{H}}=\\mbox{tr}\\chi\\!\\left.\\right|_{\\mathcal{H}},$ $tr\\underline{\\chi}_{\\epsilon}\\!\\left.\\right|_{S_{0}}=tr\\underline{\\chi}\\!\\left.\\right|_{S_{0}}$, $\\partial_{v}tr\\underline{\\chi}_{\\epsilon}\\!\\left.\\right|_{S_{0}}=\\partial_{v}tr\\underline{\\chi}\\!\\left.\\right|_{S_{0}}+\\epsilon$ for some $v$. Then, \n\\[\\widetilde{\\mathcal{O}}_{v}^{\\mathcal{S}}=\\mathcal{O}^{\\mathcal{S},\\epsilon}_{v},\\]\nwhere $\\widetilde{\\mathcal{O}}_{v}^{\\mathcal{S}}$ denotes the elliptic operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ with respect to the metric $g_{\\epsilon}$. By the above discussion and Proposition \\ref{pert1prop} we deduce that $\\mathcal{H}$ does not admit conservation laws as a null hypersurface embedded in the Lorentzian manifold  $\\big(\\mathcal{M},g_{\\epsilon}\\big)$. \n\n\n\nWe next consider perturbations in the class of spacetimes satisfying the Einstein--vacuum equations. The freely prescribable initial data of the geometry of $\\mathcal{H}$ in the context of the characteristic problem of the Einstein--vacuum equations is the following:\n\\begin{enumerate}\n\t\\item The conformal geometry of $\\mathcal{H}$,\n\\item The metric of $S_{0}$,\n\t\\item The expansions $tr\\chi$ and $tr\\underline{\\chi}$ at $S_{0}$ and\n\t\t\\item The torsion $\\zeta$ at $S_{0}$.\n\\end{enumerate}\nIn our perturbation argument, we will fix the conformal geometry of $\\mathcal{H}$, the metric of $S_{0}$ and the torsion $\\zeta$ at $S_{0}$ and we will perturb the initial expansions $tr\\chi,tr\\underline{\\chi}$ at $S_{0}$.\n\nThe propagation equation for the transversal null second fundamental form yields (see \\cite{DC09})\n\\[\\partial_{v}tr\\underline{\\chi}=\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+|\\zeta|^{2}+\\rho-(\\chi,\\underline{\\chi}). \\]\nFurthermore, the Gauss equation gives us\n\\[\\rho-(\\chi,\\underline{\\chi})=-K-tr\\chi tr\\underline{\\chi}. \\]\nTherefore, by eliminating the left hand side of the above equation we obtain a linear propagation equation for $tr\\underline{\\chi}$ along $\\mathcal{H}$:\n\\begin{equation}\n\\partial_{v}tr\\underline{\\chi}=\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+|\\zeta|^{2}-K-tr\\chi tr\\underline{\\chi}.\n\\label{propaforzeta}\n\\end{equation}\nHence the coefficient of the zeroth order term in \\eqref{toxreiazomai} becomes\n\\begin{equation}\nw=3\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+|\\zeta|^{2}-K-\\frac{1}{2}( tr\\underline{\\chi})\\cdot( tr\\chi). \n\\label{eq:w}\n\\end{equation}\nBy assumption the first three terms on the right hand side are fixed under our perturbations. We can then freely perturb $tr\\chi$ and $tr\\underline{\\chi}$ such that \n\\[w_{\\epsilon}=w+\\epsilon\\]\nat $S_{0}$. The result then follows from the above discussion that Proposition \\ref{pert1prop}.\n\n\n\\end{proof}\n\n\n\\section{Black hole spacetimes}\n\\label{sec:KillingHorizons}\n\nLet $(\\mathcal{M},g)$ be a stationary spacetime with a black hole region (for the relevant definitions see \\cite{haw}). For such a  spacetime the event horizon $\\mathcal{H}$ is  a Killing horizon, that is there is a Killing vector field $\\xi$ normal to $\\mathcal{H}$. In this case $\\xi$ satisfies\n\\begin{equation}\n\\nabla_{\\xi}\\xi=\\kappa\\cdot \\xi\\, : \\text{ on  }\\mathcal{H},\n\\label{surface}\n\\end{equation}\nwhere $\\kappa$ is constant along the null generators of $\\mathcal{H}$. In consistency with the zeroth law of black hole mechanics, we will assume that $\\kappa$ is globally constant on $\\mathcal{H}$ in which case $\\kappa$ is the so-called \\textit{surface gravity} of $\\mathcal{H}$. If $\\kappa=0$, then $\\mathcal{H}$ is called an \\textit{extremal horizon}. We will next investigate the existence of conservation laws on the event horizon $\\mathcal{H}$. \n\nThe following properties of Killing horizons were shown in \\cite{aretakiselliptic}:\n\\begin{lemma}\nLet $\\mathcal{H}$ be a Killing horizon and $\\mathcal{S}=\\left\\langle S_{0},L_{geod}, \\Omega=1\\right\\rangle$ be a geodesic foliation of $\\mathcal{H}$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Assume that $\\xi$ is a Killing vector field normal to $\\mathcal{H}$ and such that \\eqref{surface} is satisfied. Then, the following  relations hold on $\\mathcal{H}$: \n\\begin{enumerate}\n\t\\item $\\chi=0$, \n\\item $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\mbox{$g \\mkern-8.8mu \/$\\,}=0$,\n\t\\item $\\mbox{$d \\mkern-9.2mu \/$\\,}\\kappa=g(\\xi, \\underline{L})\\cdot \\beta$, where the curvature component $\\beta$ is given by \\eqref{curvcompdeflist},\n\t\\item $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\zeta=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{L}\\zeta=-\\beta$,\n\t\\item If, in addition, we take $\\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}}$ and $\\kappa$ is constant on $\\mathcal{H}$, then\n\t\n\t$\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\underline{\\chi}=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{L}\\underline{\\chi}=\\displaystyle\\frac{\\kappa}{f}\\cdot\\underline{\\chi}$, where $f$ is such that $\\xi=f\\cdot {L}_{geod}$ on $\\mathcal{H}$.\n\n\n\\end{enumerate}\n\\label{lemma}\n\\end{lemma}\n\n\n\n\n\\noindent Recall that since $\\Omega=1$ we have $L_{geod}=L=\\partial_{v}$. If we trace the last identity of the above lemma we obtain\n\\[L tr\\underline{\\chi}=\\frac{\\kappa}{f}\\cdot tr\\underline{\\chi}.  \\]\nSince $Lf=\\kappa$ we obtain\n\\begin{equation}\ntr\\underline{\\chi}=\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot f\n\\label{trchibarlambda}\n\\end{equation} and so \n\\begin{equation}\n\\partial_{v} tr\\underline{\\chi}=\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa.\n\\label{tracechibar}\n\\end{equation}\nTherefore, we obtain\n\\begin{equation}\n\t\\begin{split}\n\\mathcal{O}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+\\left[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\zeta^{\\sharp}+\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\right]\\cdot\\psi,\n\\end{split}\n\\label{killingoperator}\n\\end{equation}\nwith respect to the foliation \\begin{equation}\n\\mathcal{S}=\\left\\langle S_{0}, \\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\,  \\Omega=1\\right\\rangle.\n\\label{foliationkilling}\n\\end{equation}  \n\nSince $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\mbox{$g \\mkern-8.8mu \/$\\,}=0$ all the sections of $\\mathcal{H}$ are isometric. Moreover, since $\\kappa$ is constant on $\\mathcal{H}$, and hence $\\mbox{$d \\mkern-9.2mu \/$\\,} k=0$, by Lemma \\ref{lemma}, we obtain that  $\\beta=0$ on $\\mathcal{H}$ and hence $\\zeta$ is conserved on $\\mathcal{H}$, i.e.~$\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\zeta=0$. \tFrom \\eqref{tracechibar}, $\\partial_{v} tr\\underline{\\chi}$ does not depend on $v$. By virtue of the equation $L\\phi=\\frac{1}{2}tr\\chi\\cdot\\phi$, the conformal factor $\\phi$ also does not depend on $v$. Therefore, the operators $\\mathcal{O}_{v}^{\\mathcal{S}}$ do \\textbf{not} depend on $v$ (modulo identifying the sections $S_{v}$ with $S_{0}$ via the diffeomorphisms $\\Phi_{v}$). \n\nIf we now consider a general foliation ${\\mathcal{S}'}=\\left\\langle S_{0},\\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\Omega\\right\\rangle$ where $\\Omega$ is a smooth function on $\\mathcal{H}$, then by virtue of \\eqref{fromotherpaper} and using the fact that the geodesic vector field $L_{geod}$ is the same for both foliations $\\mathcal{S},\\mathcal{S}'$, we obtain\n\\[\\frac{1}{\\Omega^{2}}\\cdot \\mathcal{O}^{{\\mathcal{S}'}}\\Psi=\\mathcal{O}^{{\\mathcal{S}}}\\Psi\\ :\\ \\text{ for all }\\, \\Psi\\in\\mathcal{V}_{\\hh}. \\]\nand hence the operators $\\frac{1}{\\Omega^{2}}\\cdot \\mathcal{O}^{{\\mathcal{S}}'}_{v}$ do not depend on $v$ (again modulo identifying the sections $S_{v}'$ of $\\mathcal{S}'$ with $S_{0}$ via the diffeomorphisms $\\Phi_{v}$). \n\n\\begin{remark}\nGiven a foliation $\\mathcal{S}=\\left\\langle S_{0}, \\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\,  \\Omega=1\\right\\rangle$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions},  of a Killing horizon of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, we can rewrite the operator $\\mathcal{O}^{\\mathcal{S}}_{v}$ given by \\eqref{killingoperator} as follows\n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\underbrace{\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\zeta^{\\sharp}\\cdot \\psi}_{{\\mathcal{K}_{v}^{\\mathcal{S}}\\psi}}+\\underbrace{\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\cdot\\psi}_{\\mathcal{T}_{v}^{\\mathcal{S}}\\psi},\n\\label{killingoperator1}\n\\end{equation}\nThe section $S_{0}$ can be freely chosen for the foliation $\\mathcal{S}$. In view of the above discussion the operator $\\mathcal{K}_{v}^{\\mathcal{S}}$ does not depend on the choice of the section $S_{0}$ (again, modulo identifying all sections of $\\mathcal{H}$ via the flow of the null generators). However, clearly the operator $\\mathcal{T}_{v}^{\\mathcal{S}}$ depends on $S_{0}$. Specifically, if we consider another foliation $\\mathcal{S}'=\\left\\langle S_{0}', \\left.L_{geod}\\right|_{S_{0}'}=\\left.\\xi\\right|_{S_{0}'},\\,  \\Omega=1\\right\\rangle$ then in view of \\eqref{fromotherpaper}, and recalling that $L\\phi=0$, we have that\n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}(\\Psi)=\\frac{1}{f^{2}}\\cdot\\mathcal{O}_{v}^{\\mathcal{S}'}\\big(f^{2}\\cdot\\Psi\\big)\\ :\\ \\text{ for all }\\, \\Psi\\in\\mathcal{V}_{\\hh},\n\\label{changeofsection}\n\\end{equation}\nwhere $f$ is such that $L_{geod}'=f^{2}\\cdot L_{geod}$, where $L_{geod}',L_{geod}$ denote the geodesic vector fields of $\\mathcal{S}',\\mathcal{S}$, respectively.\n\\label{remarkgiazeta}\n\\end{remark}\nIn view of the above proposition, the main Theorem \\ref{theorem} and the fact that $L\\phi=0$, i.e.~$\\phi\\in\\mathcal{V}_{\\hh}$, the existence of conservation laws along Killing horizons is equivalent to the non-triviality of the kernel $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$ for some fixed $v$. Specifically, we have shown that \n\\begin{equation}\ndim\\, \\mathcal{W}^{\\mathcal{S}}= dim\\, \\mathcal{U}^{\\mathcal{S}}= \\phi\\cdot Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big).\n\\label{killingapotelesma}\n\\end{equation}\n\n\nSummarizing we have shown the following\n\\begin{proposition}\nLet $\\mathcal{H}$ be a Killing horizon with constant surface gravity $\\kappa$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. Let also $S=\\left\\langle S_{0},\\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\Omega\\right\\rangle$ be a foliation of $\\mathcal{H}$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Then the operators $\\frac{1}{\\Omega^{2}}\\cdot\\mathcal{O}^{\\mathcal{S}}_{v}$ do not depend on $v$ modulo identifying $S_{v}$ with $S_{0}$ via the diffeomorphism $\\Phi_{v}$, i.e.\n\\[ \\big(\\Phi_{v}\\big)^{*}\\left(\\frac{1}{\\Omega^{2}}\\cdot\\mathcal{O}_{v}^{\\mathcal{S}}\\right)=\\frac{1}{\\Omega^{2}}\\cdot\\mathcal{O}_{0}^{\\mathcal{S}}   .\\] \nUnder the same identification we have \n\\begin{equation}\nKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=Ker\\big(\\mathcal{O}_{0}^{\\mathcal{S}}\\big),\n\\label{kernelrelation}\n\\end{equation}\nfor all $v\\in\\mathbb{R}$.  Moreover, the kernel of the conservation laws along $\\mathcal{H}$ satisfies\n\\[dim\\, \\mathcal{W}^{\\mathcal{S}}= dim\\, \\mathcal{U}^{\\mathcal{S}}= \\phi\\cdot Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=\\left\\{\\phi\\cdot f\\, :\\, f\\in Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\right\\}.\\]\n\\label{lastprop}\n\\end{proposition}\n\n\n\n\n\n\n\n\\subsection{Conservation laws on extremal black holes}\n\\label{sec:ConservationLawsOnExtremalHorizons}\n\nBy definition,  extremal black holes satisfy $\\kappa=0$. In this case the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ takes the form \n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\big(2\\psi\\cdot\\zeta\\big)\n\\label{operatorextremal}\n\\end{equation}\nwith respect to the foliation $\\mathcal{S}$ given by \\eqref{foliationkilling}. \nFollowing the argument of Lucietti and Reall \\cite{hj2012} one obtains that $dim\\mathcal{U}^{\\mathcal{S}}=dimKer(\\mathcal{O}_{v}^{\\mathcal{S}})=1$ for all $v$. Indeed, since $\\int_{S_{v}}\\mathcal{O}^{\\mathcal{S}}_{v}\\psi=0$ for all $\\psi$, the unique positive principal eigenfunction $\\Psi$ of $\\mathcal{O}_{v}^{\\mathcal{S}}$ (see \\cite{andersson, evans}) must lie in the kernel of $\\mathcal{O}_{v}^{\\mathcal{S}}$. This shows the following\n\\begin{proposition}\nLet $\\mathcal{H}$ be an extremal horizon (i.e.~$\\kappa=0$) of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. If $\\mathcal{S}$ is the foliation of $\\mathcal{H}$ given by \\eqref{foliationkilling} then \n\\[dim\\,\\mathcal{W}^{\\mathcal{S}}=dim\\,\\mathcal{U}^{\\mathcal{S}}=dimKer(\\mathcal{O}_{v}^{\\mathcal{S}})=1.\\]\nTherefore, $\\mathcal{H}$ admits a unique conservation law with respect to the foliation $\\mathcal{S}$ (and thus with respect to any foliation). This law coincides with the \nconservation law found in \\cite{aretakis4, hj2012, murata2012}.\n\\label{propositionextremal}\n\\end{proposition}\n\n\nThe above conservation law coupled with dispersive estimates away from the event horizon forces higher order derivatives of generic solutions to the wave equation to blow up asymptotically along the event horizon (see \\cite{aretakis1,aretakis3}).  We remark that the previous result is in stark contrast with the subextremal case for which Dafermos and Rodnianski \\cite{lecturesMD,enadio,tria} have derived quantitative decay estimates for all higher order derivatives in the exterior region up to and including  the event horizon.  \n\n\n\n\\subsection{Gluing constructions for sub-extremal black holes}\n\\label{sec:GluingConstructionsForNonExtremalHorizons}\n\nWe next consider sub-extremal Killing horizons and specifically such that $\\kappa>0$. We also assume that there exists a section $S_{0}$ such that $\\left.tr\\underline{\\chi}\\right|_{S_{0}}<0$ on $S_{0}$.\\footnote{Note that under these assumptions we can use the calculations in \\cite{aretakiselliptic} and the method of the present subsection to deduce that there must exist a section $S$ such that $\\left.tr\\underline{\\chi}\\right|=c,$ where $c<0$ is constant on $S$. } Let $\\mathcal{S}$ be the foliation given by \\eqref{foliationkilling} such that its  `initial' section is the above one. In view of \\eqref{killingoperator} the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ can then be written as\n\\[\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\big(2\\psi\\cdot\\zeta\\big)+\\left[\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\right]\\cdot\\psi.\\]\nLet $\\Psi>0$ be the unique (up to rescaling) positive principal eigenfunction of $\\mathcal{O}_{v}^{\\mathcal{S}}$ and let $\\lambda$ be its principal (maximum) eigenvalue. Then, we immediately obtain\n\\[ \\int_{S_{v}}\\left(\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\right)\\cdot \\Psi \\, d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=\\lambda \\cdot\\int_{S_{v}}\\Psi\\, d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}. \\]\nThe left hand side is manifestly negative which forces the maximum eigenvalue $\\lambda$ to be strictly negative and hence $ Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=\\left\\{0\\right\\}$. We have thus shown the following\n\\begin{proposition}\nLet $\\mathcal{H}$ be a Killing horizon with positive surface gravity $\\kappa>0$. We additionally assume that there is a  spherical section $S_{0}$ of $\\mathcal{H}$ with negative transversal null expansion, i.e.~$\\big.tr\\underline{\\chi}\\big|_{S_{0}}<0$. Then $\\mathcal{H}$ does not admit any conservation laws (i.e.~$dim\\,\\mathcal{W}^{\\mathcal{S}}=0$) and hence gluing in the sense of Definition \\ref{firstordergluingdefinition} of characteristic data is always possible on $\\mathcal{H}$. \n\\label{gluingnonextremal}\n\\end{proposition}\nWe note that the non-existence of  conservation laws on the event horizon $\\mathcal{H}$ of a subextremal Kerr black hole $(|a|<M)$ does not follow from the previously mentioned decay results of Dafermos and Rodnianski. Indeed, according to Definition \\ref{definitionconservationlaw}, if $\\mathcal{H}$ admitted conservation laws then those would involve the $Y^{\\mathcal{S}}$ derivative of the scalar field $\\psi$. However, Dafermos and Rodnianski have shown decay results for the translation-invariant derivative $Y^{dec}$, where \n\\[Y^{\\mathcal{S}}=e^{\\kappa v}\\cdot Y^{dec}.   \\]\n\n\n\nWe also remark that the assumption on the negativity of the transversal null expansion is necessary. For a counterexample see the second example in Section \\ref{sec:AnExampleOfAMetricForWhichC1Neq0AndC20}. \n\n\n\n\n\n\n\n\n\n\n\\section{The Newman--Penrose constants}\n\\label{sec:TheNullInfinityMathcalI}\n\nIn this section we will show that the Newman--Penrose constants (see \\cite{np1}) can be recovered by Theorem \\ref{theorem}. We first start with a discussion about the geometry of the null infinity $\\mathcal{I}$ of general asymptotically flat four-dimensional Lorentzian manifolds. \n\n\\subsection{The null infinity $\\mathcal{I}$}\n\\label{sec:TheAsympoticGauge}\n\nLet $(\\mathcal{M},g)$ be a general asymptotically flat four-dimensional Lorentzian manifold. \nConsider an outgoing null hypersurface $\\mathcal{H}_{0}=\\left\\{u=0\\right\\}$ of $\\mathcal{M}$ with future complete null generators. Let $\\mathcal{S}=\\left\\langle S_{0},L_{geod},\\Omega=1\\right\\rangle$ be a foliation of $\\mathcal{H}_{0}$ such that  $S_{0}$ is a surface embedded in a Cauchy hypersurface $\\Sigma$ terminating at the spacelike infinity $i^{0}$ (see figure below). Let $\\tau$ be the affine parameter of $L_{geod}$ on $\\mathcal{H}_{0}$, i.e.~$\\left.\\tau\\right|_{S_{0}}=0$ and $L_{geod}(\\tau)=1$, and let $S_{\\tau}$ denote the corresponding sections on $\\mathcal{H}_{0}$. Let $\\underline{\\mathcal{H}}_{\\tau}$ denote the ingoing null hypersurface of $\\mathcal{M}$ generated by incoming null geodesics normal to  $S_{\\tau}$. We consider the  collection $\\left\\{\\mathcal{D}_{\\tau},\\, \\tau\\geq 0\\right\\}$ of the double null foliations generated by\n$\\mathcal{D}_{\\tau}=\\left\\langle S_{\\tau}, \\left.L_{geod}\\right|_{S_{\\tau}}, \\left.\\Omega\\right|_{\\mathcal{H}_{0}}=1, \\left.\\Omega\\right|_{\\underline{\\mathcal{H}}_{\\tau}}=1 \\right\\rangle$ \nwhere $\\left.L_{geod}\\right|_{S_{\\tau}}$ is normalized such that \n\\begin{equation}\ntr\\chi+tr\\underline{\\chi}=0\\ : \\ \\text{ on }\\, S_{\\tau}.\n\\label{normalizationnp}\n\\end{equation}\n\\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.115]{nullinf2}\n\t\t\\label{fig:nullinf2}\n\\end{figure}\nConsider now the induced foliation $\\underline{\\mathcal{S}}_{\\tau}=\\left\\langle S_{\\tau}, \\left.\\underline{L}_{geod}\\right|_{S_{\\tau}},  \\left.\\Omega\\right|_{\\underline{\\mathcal{H}}_{\\tau}}=1 \\right\\rangle$ on $\\underline{\\mathcal{H}}_{\\tau}$. Let $\\mbox{$g \\mkern-8.8mu \/$\\,}$ be the induced metric, $\\mbox{$\\nabla \\mkern-13mu \/$\\,}$ the induced covariant derivative and $\\mbox{$\\triangle \\mkern-13mu \/$\\,}$ the induced Laplacian. Let also $A$ be the area and $r=\\sqrt{A\/4\\pi}$ the radius function. \n\nSuppose that  as $\\tau\\rightarrow +\\infty$ we have $r(S_{\\tau})\\rightarrow +\\infty$.\nThen, the null infinity $\\mathcal{I}$ is defined to be the limit of the hypersurfaces $\\underline{\\mathcal{H}}_{\\tau}$ as $\\tau\\rightarrow +\\infty$.  In view of its limiting character, quantites associated to $\\mathcal{I}$ can only be understood in a limiting, appropriately rescaled, sense. The limit of the foliations $\\underline{\\mathcal{S}}_{\\tau}$, as $\\tau\\rightarrow+\\infty$, gives rise to a foliation of $\\mathcal{I}$. We have the following asymptotic behavior (see, for example, \\cite{memorychistodoulou}):\n\\begin{equation}\n\\begin{split}\n&\\frac{1}{r^{2}}\\mbox{$g \\mkern-8.8mu \/$\\,}\\rightarrow \\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}, \\ \\ \\ \\frac{1}{r^{2}}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\rightarrow \\sin\\theta,\\ \\ \\  {r^{2}}\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\rightarrow \\mbox{$\\nabla \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}, \\ \\ \\ r^{2}\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\rightarrow \\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}, \\ \\ \\ \\frac{1}{r}\\phi\\rightarrow 1,\\\\\n&\\partial_{v}r\\rightarrow \\frac{1}{\\sqrt{2}}, \\ \\ \\ \\partial_{u}r\\rightarrow- \\frac{1}{\\sqrt{2}}, \\ \\ \\ r\\zeta\\rightarrow Z,\n\\\\&  rtr\\chi\\rightarrow \\sqrt{2}, \\ \\ \\ rtr\\underline{\\chi}\\rightarrow -\\sqrt{2}, \\ \\ \\ r^{2}\\partial_{v}tr\\underline{\\chi}\\rightarrow 1, \\ \\ \\ r^{2}\\partial_{u}tr{\\chi}\\rightarrow 1,\\\\\n\\end{split}\n\\label{asym}\n\\end{equation}\n\\begin{figure}[H]\n   \\centering\n\t\t\\includegraphics[scale=0.125]{nullinfgaugepeelingpaper.png}\n\t\t\\label{fig:nullinfpeel}\n\\end{figure}\n\\noindent where $Z$ is a 1-form on $\\mathbb{S}^{2}$. The above limits should be understood in terms of  the pullback of the induced tensor fields to the standard sphere via tha diffeomorphism $\\Phi_{u,v}$ (see Section \\ref{sec:TheDoubleNullFoliation}). \n\n\n\n\n\n\\subsection{The Newman--Penrose constants}\n\\label{sec:TheNewmanPenroseConstants}\n\n\nLet us briefly recall the (first-order) Newman--Penrose constant. If we  use the coordinate system $(u,r,\\theta^{1},\\theta^{2})$ to make the dependence on the powers of $r$ more explicit then we obtain\n\\[\\psi(u,r,\\theta)=\\frac{\\alpha_{1}(u,\\theta^{1},\\theta^{2})}{r}+\\frac{\\alpha_{2}(u,\\theta^{1},\\theta^{2})}{r^{2}}+O\\left(\\frac{1}{r^{3}}\\right)\\]\nclose to null infinity $\\mathcal{I}$. Note that $\\partial_{u}$ is tangential to $\\mathcal{I}$ and $\\partial_{v}$ is transversal. \nHere, \\[ \\alpha_{1}(u,\\theta^{1},\\theta^{2})=\\lim_{r\\rightarrow+\\infty}(r\\psi)(u,r,\\theta^{1},\\theta^{2}) \\]\nis the radiation field of $\\psi$ on $\\mathcal{I}$. \n Furthermore,\n\\[ \\partial_{v}(r\\psi)= -\\sqrt{2}\\cdot \\alpha_{2}(u,\\theta^{1},\\theta^{2})\\cdot \\frac{1}{r^{2}}+O\\left(\\frac{1}{r^{3}}\\right).\\]\nTherefore, as $r\\rightarrow +\\infty$ we have $\\partial_{v}(r\\psi)\\rightarrow 0$ and $r\\cdot \\partial_{v}(r\\psi)\\rightarrow 0$. The higher order non-trivial quantity\n\\[r^{2}\\cdot \\partial_{v}(r\\psi)=-\\sqrt{2}\\cdot \\alpha_{2}(u,\\theta^{1},\\theta^{2})\\]  \ngives rise to the Newman--Penrose costant\n\\begin{equation}\n\\lim_{r\\rightarrow+\\infty}\\int_{S_{u}}r^{2}\\partial_{v}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}=-\\sqrt{2}\\cdot \\int_{S_{u}}\\alpha_{2}(u,\\theta) \\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{npconstant}\n\\end{equation}\nwhich turns out to be  conserved (i.e. independent of $u$) along $\\mathcal{I}$.\n\n\n\nWe will show that the (first-order) Newman--Penrose constant is a limiting case of Theorem \\ref{theorem}. The  wave equation with respect to the double null foliation $\\mathcal{D}_{\\tau}$ can be written on $\\underline{\\mathcal{H}}_{\\tau}$ as follows\n\\begin{equation}\n-2\\partial_{u}\\partial_{v}(\\phi\\cdot\\psi) +\\phi\\cdot \\mathcal{Q}^{\\underline{\\mathcal{S}}_{\\tau}}\\psi=0,\n\\label{wenullinf}\n\\end{equation}\nwhere \n\\begin{equation}\n\\mathcal{Q}^{\\underline{\\mathcal{S}}_{\\tau}}\\psi=  \\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi-2\\zeta^{\\sharp}\\cdot \\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+ \\left[\\partial_{u}( tr{{\\chi}})+\\frac{1}{2}(tr\\underline{\\chi})\\cdot(tr\\chi)\\right]\\cdot\\psi.\n\\label{ovoperatorinf}\n\\end{equation}\nSimilarly, the adjoint operator is given by\n \t\\begin{equation*}\n\t\\begin{split}\n\\mathcal{O}^{\\underline{\\mathcal{S}}_{\\tau}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot \\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+\\left[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\zeta^{\\sharp}+\\partial_{u}( tr{\\chi})+\\frac{1}{2}(tr\\underline{\\chi})\\cdot(tr\\chi)\\right]\\cdot\\psi\n\\end{split}\n\\end{equation*}\non $\\underline{\\mathcal{H}}_{\\tau}$. Let $\\underline{\\mathcal{S}}$ denote the limiting foliation  $\\lim_{\\tau\\rightarrow+\\infty}\\underline{\\mathcal{S}}_{\\tau}$. Clearly, in this limit $\\underline{\\mathcal{H}}_{\\tau}\\rightarrow\\mathcal{I}$. We define the following rescaled limiting operator\n\\begin{equation}\n\\mathcal{O}^{\\underline{\\mathcal{S}}}\\psi=\\lim_{r\\rightarrow+\\infty}\\mathcal{O}^{\\underline{\\mathcal{S}}_{\\tau}}\\big(r^{2}\\cdot\\psi\\big) \\ : \\ \\text{ on }\\, \\mathcal{I}.\n\\label{limitingoperator}\n\\end{equation}\nIn view of the  asymptotics \\eqref{asym}, we obtain\n\t\\begin{equation}\n\t\\begin{split}\n\\mathcal{O}^{\\underline{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{{\\mathbb{S}^{2}}}\\psi. \n\\end{split}\n\\label{operatorfori}\n\\end{equation}\n\\textbf{The key observation here is that, in view of the asymptotics \\eqref{asym}, $\\zeta$, and more importantly, $  r^{2}\\cdot\\bigg(\\partial_{u}tr{\\chi}+\\frac{1}{2}tr\\chi tr\\underline{\\chi}\\bigg)$  decay like $1\/r$} and hence all the terms but the Laplacian  vanish at the limit $r\\rightarrow +\\infty$.\n\nIn view of the presence of limiting quantities we need to be particularly careful in order to apply Theorem \\ref{theorem}. Indeed, the integrability condition \\eqref{1inte} does not hold on $\\mathcal{I}$ since the measure of integration has infinite area element. Using however that on $\\mathcal{I}$ we have $d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=r^{2}\\cdot d\\mu_{_{\\mathbb{S}^{2}}}$ and hence \\eqref{1inte}, using \\eqref{wenull1}, takes the following limiting form\n\\begin{equation}\n\\int_{S_{u}}\\Big(\\mathcal{Q}_{u}^{\\underline{\\mathcal{S}}}(\\psi)\\Big) \\cdot\\Big( \\big(E^{\\underline{S}}\\big)_{u}\\Big)\\, d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=\\int_{S_{u}}\\psi\\cdot \\mathcal{O}^{\\underline{\\mathcal{S}}}_{u}\\Big(\\big(E^{\\mathcal{S}}\\big)_{u}\\Big)\\, d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=\\int_{S_{u}}(r\\psi)\\cdot \\mathcal{O}^{\\underline{\\mathcal{S}}}_{u}\\Big(r\\cdot \\big(E^{\\underline{\\mathcal{S}}}\\big)_{u}\\Big) \\,d\\mu_{_{\\mathbb{S}^{2}}} =0\n\\label{limitingform}\n\\end{equation}\nwhere, for the last equation, we take $\\big(E^{\\underline{\\mathcal{S}}}\\big)_{u}$ such that \\begin{equation}\nr\\cdot\\big(E^{\\underline{\\mathcal{S}}}\\big)_{u}\\in Ker\\big(\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}\\big).\n\\label{eq:notethat}\n\\end{equation} Note that $r\\psi$, being the radiation field of $\\psi$ on $\\mathcal{I}$, is finite. \n\nRecall that the kernel $\\mathcal{W}^{\\underline{\\mathcal{S}}}_{\\mathcal{I}}$ of the conservation laws on $\\mathcal{I}$ consists of all functions $\\Theta^{\\underline{\\mathcal{S}}}$ which are ``constant'' along the null generators of $\\mathcal{I}$ such that \n\\begin{equation}\n\\Theta^{\\underline{\\mathcal{S}}}=\\phi\\cdot \\big(E^{\\underline{\\mathcal{S}}}\\big)_{u}=r\\cdot \\big(E^{\\underline{\\mathcal{S}}}\\big)_{u}.\n\\label{rescaledu}\n\\end{equation}\nIn view of \\eqref{limitingoperator}, \\eqref{operatorfori} and \\eqref{eq:notethat} we can (only) take $\\big(E^{\\underline{\\mathcal{S}}}\\big)_{u}=r$ and hence \n\\begin{equation}\n\\Theta^{\\underline{\\mathcal{S}}}=r^{2}.\n\\label{eq:thetanp}\n\\end{equation}\nModulo a trivial rescaling we have, $Y^{\\underline{\\mathcal{S}}}=\\partial_{u}$, where $Y^{\\underline{\\mathcal{S}}}$ is the vector field defined in Section \\ref{sec:ConservationLawsForTheWaveEquations},  and in view of the asymptotic behavior of the function $r$ we can in fact take\n\\begin{equation}\nY^{\\underline{\\mathcal{S}}}=\\partial_{r} \\ : \\ \\text{ on }\\, \\mathcal{I}{+}.\n\\label{ynp}\n\\end{equation}\nHence, the conserved charge \\eqref{eq:integrals} on $\\mathcal{I}$ is equal to\n\\begin{equation}\n\\lim_{r\\rightarrow+\\infty}\\int_{S_{u}}r^{2}\\cdot \\partial_{r}(r\\psi)\\,d\\mu_{_{\\mathbb{S}^{2}}},\n\\label{conpcha}\n\\end{equation}\nwhich coincides with the Newman--Penrose constant \\eqref{npconstant}.\n\\begin{remark}\nClearly, the function $r$ is not constant on the incoming null hypersurfaces $\\underline{\\mathcal{H}}_{\\tau}$. However, it can be considered constant on $\\mathcal{I}$ in a limiting sense . As far as the conservation laws are concerned we have the following: Since \n  $\\phi-r\\in O\\left(\\frac{1}{r^{a}}\\right),\\ a>0$, we obtain\n\\begin{equation*}\n\\begin{split}  r^{2}\\partial_{u}\\partial_{v}(\\phi\\cdot\\psi)&=\\partial_{u}\\big(r^{2}\\partial_{v}(\\phi\\cdot\\psi)\\big)-2r\\partial_{u}r\\partial_{v}(\\phi\\cdot\\psi)\\\\\n&=\\left[\\partial_{u}\\big(r^{2}\\partial_{v}(r\\psi)\\big)-2r\\partial_{u}r\\partial_{v}(r\\psi)+\\partial_{u}\\big(r^{2}\\partial_{v}(r^{-a}\\psi)\\big)-2r\\partial_{u}r\\partial_{v}(r^{-a}\\psi)\\right]\\\\\n&\\rightarrow \\partial_{\nu}\\big(r^{2}\\partial_{v}(r\\psi)\\big).\n\\end{split}\n\\end{equation*}\n\\label{remarknkconse}\n\\end{remark}\n\\noindent The restriction $\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}$ on $S_{u}$ of the operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}$ on $\\mathcal{I}$ is given by\n\\begin{equation}\n\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}=\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}\n\\label{eq:oni}\n\\end{equation}\nand hence is independent of $u$. This ``$u$-invariance''  of the operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}_{u}$ is due to the BMS symmetry group of $\\mathcal{I}$ (see \\cite{wald}). Recall from Section \\ref{sec:KillingHorizons} that a similar result holds for Killing horizon and in fact the kernel of $\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}$ is isomorphic to the kernel of the associated operator on extremal horizons. This reveals yet another common property of $\\mathcal{I}$ and extremal horizons. \n\nIf $\\mathcal{W}^{\\underline{\\mathcal{S}}}$ denotes the kernel of the (limiting) conservation laws on $\\mathcal{I}$ and $\\mathcal{U}^{\\underline{\\mathcal{S}}}$ denotes the appropriately rescaled kernel of the operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}$ (see \\eqref{rescaledu}) then, in view of \\eqref{operatorfori}, we have\n\\[dim\\,\\mathcal{W}^{\\underline{\\mathcal{S}}}=dim\\,\\mathcal{U}^{\\underline{\\mathcal{S}}}=1\\]\nand, therefore, there exists \\textbf{only} one (non-trivial limiting) conservation law on $\\mathcal{I}$, namely that given by the Newman--Penrose constants.  Moreover, the result of Section \\ref{sec:EffectOfGaugeOnConservationLaws} applies for this conservation law. Specifically, if ${\\underline{\\mathcal{S}}}'$ is a foliation of $\\mathcal{I}$ and $\\widetilde{Y}^{\\underline{\\mathcal{S}}'}$ denotes the unique null conjugate to ${\\underline{\\mathcal{S}}}'$ vector field normalized  such that $\\widetilde{Y}^{\\underline{\\mathcal{S}}'}r=1$ then the integrals\n\\[\\lim_{r\\rightarrow+\\infty}\\int_{S'_{u'}}r^{2}\\cdot \\widetilde{Y}^{\\underline{\\mathcal{S}}'}(r\\cdot\\psi) \\, d\\mu_{_{\\mathbb{S}^{2}}} \\]\nare conserved, i.e.~independent of $u$. \n\nSummarizing  we have shown the following\n\\begin{proposition}\nLet $\\underline{\\mathcal{S}}=\\big(S_{u}\\big)_{u\\in\\mathbb{R}}$ be a foliation of the null infinity $\\mathcal{I}$ of an asymptotically flat spacetime $(\\mathcal{M},g)$, as defined in Section \\ref{sec:TheAsympoticGauge}. Then, the appropriately rescaled operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}$ given by \\eqref{operatorfori} is $u$-invariant, i.e.~the operators $\\mathcal{O}^{\\underline{\\mathcal{S}}}_{u}$ do not depend on $u$. Moreover, $dim\\,\\mathcal{W}^{\\underline{\\mathcal{S}}}=dim\\,\\mathcal{U}^{\\underline{\\mathcal{S}}}=1$ and the unique associated conservation law on $\\mathcal{I}$ gives rise to the (first-order) Newman--Penrose constant. \n\\label{nullinfiprop}\n\\end{proposition}\n\n\n\n Note that all the higher order Newman--Penrose constants can be obtained by commuting the wave equation with $\\partial_{v}^{k}$ (see also Section \\ref{sec:Genericity}). \n\n\n\n\n\\section{Spherical symmetry}\n\\label{sec:Genericity}\n\nIn this section we investigate the existence of higher order conservation laws. Although our method applies for general spacetimes for the sake of simplicity we focus on spherically symmetric backgrounds. \n\n\n\nLet $\\mathcal{H}$ be a spherically symmetric null hypersurface and  $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ be a spherically symmetric  foliation on $\\mathcal{H}$ in a spherically symmetric four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. Then the wave equation restricted on $\\mathcal{H}$ can be written as\n\\begin{equation*}\n\\begin{split}\n\\Box_{g}\\psi\n=&-2\\partial_{u}\\partial_{v}(r\\psi)+ \\mathcal{O}^{\\mathcal{S}}(r\\psi)=0,  \n\\end{split}\n\\end{equation*}\nwhere \\[\\mathcal{O}^{\\mathcal{S}}\\psi=\\Omega^{2}\\frac{1}{r^{2}}\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}\\psi +2\\Omega^{2}\\cdot \\frac{(\\partial_{u}\\partial_{v}r)}{r}\\cdot \\psi.\\]\nWe next assume that  $\\Omega=1$ on $\\mathcal{H}$. Since all the expressions are spherically symmetric, in view of Theorem \\ref{theorem}, we have a (first order) conservation law if and only if \n\\[\\frac{2\\partial_{u}\\partial_{v} r}{r}=\\frac{l(l+1)}{r^{2}}, \\]\nfor some $l\\in\\mathbb{N}$. \n\n\n\\subsection{Higher order conservation laws}\n\\label{sec:HigherOrderConservationLaws}\n\nFor a given $n\\in\\mathbb{N}$, we want to find necessary and sufficient conditions under which we can glue general data \n\\begin{equation}\n\\big.\\psi\\big|_{S_{0}}, \\ \\ \\ \\left.\\partial_{u}^{k}\\psi\\right|_{S_{0}},\\ \\ \\  1\\leq k\\leq n\n\\label{data1}\n\\end{equation}\non $S_{0}$ to general data  \n\\begin{equation}\n\\big.\\psi\\big|_{S_{1}}, \\ \\ \\ \\left.\\partial_{u}^{k}\\psi\\right|_{S_{1}},\\  \\ 1\\leq k\\leq n, \n\\label{data2}\n\\end{equation}\non $S_{1}$ as explained in Section \\ref{sec:TheCharacteristicInitialValueProblem}. \n\n\n\nAs we shall see the only obstruction to such gluings is higher order conservation laws. By decomposing $\\psi$ in angular frequencies we can assume that it is supported on the $l$ angular frequency. Then, the wave equation on $\\mathcal{H}$ reads\n\\begin{equation*}\n\\begin{split}\n\\Box_{g}\\psi\n=&-2\\partial_{u}\\partial_{v}(r\\psi)+ \\mathcal{O}^{\\mathcal{S}}(r\\psi)=0,  \n\\end{split}\n\\end{equation*}\nwhere \\[\\mathcal{O}^{\\mathcal{S}}\\psi=\\Omega^{2}\\cdot\\left[2\\frac{\\partial_{u}\\partial_{v} r}{r}-\\frac{l(l+1)}{r^{2}} \\right]\\cdot\\psi.\\]\nSet \n\\begin{equation}\n\\Psi=r\\psi, \\ \\ \\ c_{1}=\\Omega^{2}\\cdot\\left[\\frac{\\partial_{u}\\partial_{v} r}{r}-\\frac{l(l+1)}{2r^{2}} \\right],\\ \\ \\ Y=\\partial_{u}.\n\\label{phi}\n\\end{equation}\nThe data for $\\Psi $ must satisfy the constraint equations\n\\begin{equation}\n\\partial_{v}\\big(Y\\Psi \\big)=c_{1}\\cdot\\Psi , \\ \\ \\ \\partial_{v}\\big(Y^{k+1}_{u}\\Psi \\big)=Y^{k}\\big(c_{1}\\cdot\\Psi \\big), \\ k\\geq 1,\n\\label{constraint}\n\\end{equation}\n By integrating along the null generators we obtain\n\\begin{equation}\n(Y\\Psi )(\\tau)=(Y\\Psi )(0)+\\int_{0}^{1}c_{1}\\cdot\\Psi ,\n\\label{con1}\n\\end{equation}\nand more generally\n\\begin{equation}\n(Y^{k+1}\\Psi )(\\tau)=(Y^{k+1}\\Psi )(0)+ \\int_{0}^{1}Y^{k}(c_{1}\\cdot\\Psi )\n\\label{conk}\n\\end{equation}\nWe consider first the case $k=1$. Clearly, we want to construct $\\Psi $ such that \n\\begin{equation*}\n\\begin{split}\n\\int_{0}^{1}c_{1}\\cdot\\Psi =\\alpha_{1}, \\ \\ \\ \n\\int_{0}^{1}\\Big((Yc_{1})\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )\\Big)d\\tilde{\\tau}=\\alpha_{2}, \n \\end{split}\n\\end{equation*}\nwhere $\\alpha_{1},\\alpha_{2}$ are arbitrary real numbers. Using \\eqref{con1}, with $\\tau\\mapsto\\tilde{\\tau}$, the above is equivalent to \n\\begin{equation*}\n\\begin{split}\n\\int_{0}^{1}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )(0)+ c_{1}\\cdot \\int_{0}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi   \\bigg]d\\tilde{\\tau}=\\alpha_{2}.\n\\end{split}\n\\end{equation*}\nIf $c_{1}=0$ but and $Yc_{1}=0$ then we have two conservation laws. If $c_{1}=0$ but $Yc_{1}\\neq 0$ then we have a first order conservation law and gluing for second order derivatives.  If $c_{1}\\neq 0$  for all  $v\\in[a,b]$ then we compute\n\\begin{equation*}\n\\begin{split}\n\\alpha_{2}=&\\int_{0}^{a}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )(0) + c_{1}\\cdot \\int_{0}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}+\\int_{b}^{1}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )(0) \\bigg]d\\tilde{\\tau}\\\\\n&+\\int_{b}^{1}\\bigg[ c_{1}\\cdot \\int_{0}^{a} c_{1}\\cdot \\Psi + c_{1}\\cdot \\int_{b}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}\\\\\n&+\\int_{a}^{b}\\bigg[ c_{1}\\cdot (Y\\Psi )(0)+ c_{1}\\cdot \\int_{0}^{a} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau} \\\\\n&+\\int_{a}^{b}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot \\int_{a}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi  \\bigg]d\\tilde{\\tau}+\n\\Bigg(\\int_{a}^{b} c_{1}\\cdot \\Psi \\Bigg)\\cdot\\Bigg(\\int_{b}^{1}c_{1}\\Bigg). \n \\end{split}\n\\end{equation*}\nNote that the quantities in the first three lines depend only of the values of $\\Psi $ in the region $\\mathcal{C}=[0,a]\\cup[b,1]$ and on $(Y\\Psi )(0)$. We want to extend $\\Psi$ everywhere in $[0,1]$ so we can do gluing. It suffices to construct $\\Psi $  in $[a,b]$ such that \n\\begin{equation*}\n\\begin{split}\n\\int_{a}^{b} c_{1}\\cdot \\Psi =\\beta_{1},\\\\\n\\int_{a}^{b}\\bigg[(Y c_{1} )\\cdot\\Psi  &+ c_{1}\\cdot \\int_{a}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi  \\bigg]d\\tilde{\\tau}=\\beta_{2},\n\\end{split}\n\\end{equation*}\nwhere $\\beta_{1},\\beta_{2}$ are given.\nDefine the function\n\\begin{equation}\n\\Phi_{1} :[a,b]\\rightarrow\\mathbb{R}: \\ \\ \\ \\Phi_{1} (t)=\\int_{a}^{t} c_{1}\\cdot \\Psi .\n\\label{phidef}\n\\end{equation}\nNote $\\Phi_{1} '(t)=c_{1}(t)\\cdot\\Psi (t)$ and hence, since $c_{1}\\neq 0$ in $[a,b]$, the function  $\\Phi_{1} $ determines $\\Psi $ in $[a,b]$. Therefore, it suffices to construct $\\Phi_{1}$ such that $\\Phi_{1} (b)=\\beta_{1}$ and all derivatives of $\\Phi$ at $b$ are prescribed and such that \n\\begin{equation}\n\\begin{split}\n\\int_{a}^{b}\\bigg[\\frac{Y c_{1} }{c_{1}}\\cdot \\Phi_{1} '+ c_{1}\\cdot \\Phi_{1}  \\bigg]d\\tilde{\\tau}=\\beta_{2}\n\\end{split}\n\\label{conditions1}\n\\end{equation}\nNow, by integration by parts we obtain\n\\[\\int_{a}^{b}\\frac{Y c_{1} }{c_{1}}\\cdot \\Phi_{1} '=\\frac{(Y c_{1} )(b)}{c_{1}(b)}\\cdot\\Phi_{1} (b)-\\int_{a}^{b}\\bigg(\\frac{Y c_{1} }{c_{1}}\\Bigg)'\\cdot\\Phi_{1} .   \\]\nTherefore, if $f=-\\bigg(\\frac{Y c_{1} }{c_{1}}\\Bigg)'$ then \\eqref{conditions1} can be rewritten\n\\begin{equation}\n\\begin{split}\n\\int_{a}^{b}\\bigg[(f+ c_{1}) \\cdot\\Phi_{1}  \\bigg]d\\tilde{\\tau}=\\beta_{3}.\n\\end{split}\n\\label{conditions2}\n\\end{equation}\nClearly, this has a solution if and only if $(f+ c_{1}) \\neq 0$ at a point $\\tau_{0}\\in[a,b]$. Since\n\\begin{equation*}\n\\begin{split}\nf+c_{1}=-\\frac{(Y c_{1} )'\\cdot c_{1}-c_{1}'\\cdot(Y c_{1} )-c_{1}^{3}}{c_{1}^{2}}\n\\end{split}\n\\end{equation*}\nwe obtain that we have glue data to second order if  $c_{1}\\neq 0$ and \n\\begin{equation}c_{2}=\nc_{1}^{3}+(\\partial_{v} c_{1}) (\\partial_{u} c_{1}) - c_{1}\\cdot (\\partial_{v}\\partial_{u} c_{1}) \\neq 0. \n\\label{h1condition}\n\\end{equation}\nWe will next show that if $c_{2}=0$ (and $c_{1}\\neq 0$) on $\\mathcal{H}$ then we have a second order conservation law. \n\n Recall the definition \\eqref{phidef} for the function $\\Phi_{1} $. Then, in view of \\eqref{con1} and \\eqref{conk} we obtain\n\\[\\Phi_{1} '= c_{1}\\cdot \\Psi ,\\]\n\\begin{equation*}\n\\begin{split}\n (Y\\Psi )(\\tau)&=(Y\\Psi )(0)+\\Phi_{1} (\\tau)\n \\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split} (YY\\Psi)(\\tau)&=(YY\\Psi)(0)+\\int_{0}^{\\tau}\\bigg[(Yc_{1})\\cdot\\Psi+c_{1}\\cdot (Y\\Psi)(0)+c_{1}\\cdot \\int_{0}^{\\tilde{\\tau}}c_{1}\\cdot\\Psi \\bigg]d\\tilde{\\tau}\\\\\n &=(YY\\Psi)(0)+\\int_{0}^{\\tau}\\bigg[\\frac{Yc_{1}}{c_{1}}\\cdot\\Phi_{1}' +c_{1}\\cdot(Y\\Psi)(0)+c_{1}\\cdot\\Phi_{1} \\bigg]d\\tilde{\\tau}\\\\\n  &=(YY\\Psi)(0)+(Y\\Psi)(0)\\cdot\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg) +\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\Phi_{1}(\\tau)+\\int_{0}^{\\tau}\\bigg[\\bigg(-\\left(\\frac{Yc_{1}}{c_{1}}\\right)'+c_{1}\\bigg)\\cdot\\Phi_{1}    \\bigg]d\\tilde{\\tau}\\\\\n   &=(YY\\Psi)(0)+(Y\\Psi)(0)\\cdot\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg) +\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\Phi_{1}(\\tau)\\\\\n      &=(YY\\Psi)(0)+(Y\\Psi)(0)\\cdot\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg) +\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\Big((Y\\Psi)(\\tau)-(Y\\Psi)(0)\\Big)\\\\\n&=(YY\\Psi)(0)+\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\big((Y\\Psi)(\\tau)\\big)+(Y\\Psi)(0)\\cdot\\bigg[\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg)-\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\bigg]\\\\\n&=(YY\\Psi)(0)+\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\big((Y\\Psi)(\\tau)\\big)-(Y\\Psi)(0)\\cdot\\frac{(Yc_{1})(0)}{c_{1}(0)},\\\\\n  \\end{split}\n\\end{equation*}\nwhere we repeatedly used that $c_{1}=\\left(\\frac{Y c_{1} }{c_{1}}\\right)'$. Therefore,  if $c_{1}\\neq 0$ and $c_{1}=\\left(\\frac{Y c_{1} }{c_{1}}\\right)'$ then the quantity\n\\begin{equation}\nY^{2}\\Psi -\\frac{Y c_{1} }{c_{1}}\\cdot Y\\Psi \n\\label{newcon1}\n\\end{equation}\nis conserved on $\\mathcal{H}$. \n\nThe above provides a scheme in order to find necessary and sufficient conditions for the existence of higher order conservations laws. Clearly, these conservation laws are the only obstruction to gluing.\n\n\nWe next consider third order gluing constructions.   We have\n\\begin{equation*}\n\\begin{split}\n(Y^{3}\\Psi )(\\tau)-(Y^{3}\\Psi )(0)=&\\int_{0}^{\\tau}\\bigg[c_{1}\\cdot(Y^{2}\\Psi )(0)+2\\cdot(Y c_{1} )\\cdot(Y\\Psi )(0)+(Y^{2} c_{1}) \\cdot\\Psi      \\bigg]d\\tau_{1}\\\\\n&+ \\int_{0}^{\\tau}2(Y c_{1} )\\Bigg[\\int_{0}^{\\tau_{1}} c_{1}\\cdot \\Psi     \\Bigg]d\\tau_{1}\\\\\n&+\\int_{0}^{\\tau}\\Bigg[c_{1}\\cdot\\int_{0}^{\\tau_{1}}\\bigg[\\bigg(\n(Y c_{1} )\\cdot\\Psi +c_{1}\\cdot(Y\\Psi )(0)+c_{1}\\cdot\\int_{0}^{\\tau_{2}}c_{1}\\cdot\\Psi  \\Bigg)\\bigg]d\\tau_{2}  \\Bigg]d\\tau_{1}.\n\\end{split}\n\\end{equation*}\nThen,\n\\begin{equation*}\n\\begin{split}\n&(Y^{3}\\Psi )(\\tau)-(Y^{3}\\Psi )(0)\n\\\\=& \\int_{0}^{\\tau}\\bigg[ c_{1}\\cdot (Y^{2}\\Psi )(0)+2\\cdot(Y c_{1} )\\cdot(Y\\Psi )(0)+\\frac{Y^{2}c_{1}}{c_{1}}\\cdot (\\Phi _{1})'+2(Y c_{1} )\\cdot\\Phi _{1} +(Y\\Psi )(0)\\cdot c_{1}\\cdot\\int_{0}^{\\tau_{1}}c_{1}\\bigg]d\\tau_{1}\n\\\\&+\\int_{0}^{\\tau}\\Bigg[(Y c_{1} )\\cdot\\Phi _{1}+c_{1}\\cdot\\int_{0}^{\\tau_{1}}\\bigg[ \\bigg(-\\left(\\frac{Y c_{1} }{c_{1}}\\right)'+c_{1}\\Bigg)\\cdot\\Phi _{1} \\bigg]d\\tau_{2}\\Bigg]d\\tau_{1}.\\\\\n\\end{split}\n\\end{equation*}\nWe now define the function\n\\[\\Phi _{2}(t)=\\int_{0}^{t}\\bigg(-\\left(\\frac{Y c_{1} }{c_{1}}\\right)'+c_{1}\\Bigg)\\cdot\\Phi _{1}\\]\nand the function\n\\[c_{2}=\\bigg(-\\left(\\frac{Y c_{1} }{c_{1}}\\right)'+c_{1}\\Bigg)\\neq 0.\\]\nWe thus obtain\n\\begin{equation*}\n\\begin{split}\n(Y^{3}\\Psi )(\\tau)-(Y^{3}\\Psi )(0)=&\\int_{0}^{\\tau}\\bigg[ c_{1}\\cdot (Y^{2}\\Psi )(0)+2\\cdot(Y c_{1} )\\cdot(Y\\Psi )(0)+(Y\\Psi )(0)\\cdot c_{1}\\cdot\\int_{0}^{\\tau_{1}}c_{1}\\bigg]d\\tau_{1}\n\\\\&+\\frac{(Y^{2} c_{1}) (\\tau)}{c_{1}(\\tau)\\cdot c_{2}(\\tau)}\\cdot\\Phi '_{2}(\\tau)\n+\\frac{1}{c_{2}(\\tau)}\\Bigg(-\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'+3(Y c_{1} )\\Bigg)(\\tau)\\cdot\\Phi _{2}(\\tau)\\\\&\n+\\int_{0}^{\\tau}\\Bigg[\\Bigg(\\Bigg(\\frac{1}{c_{2}}\\Bigg(\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'-3(Y c_{1} )\\Bigg)\\Bigg)'+c_{1}\\Bigg)\\cdot\\Phi _{2}\\Bigg]d\\tau_{1}.\\\\\n\\end{split}\n\\end{equation*}\nThe first line is completely determined by the initial data and the geometry of $\\mathcal{H}$. The second line is determined (by the geometry of $\\mathcal{H}$) and by $\\Phi _{2}(\\tau), \\Phi '_{2}(\\tau)$. \n\n \n Hence, we can arbitrarily prescribe $\\Psi (\\tau), Y\\Psi (\\tau),Y^{2}\\Psi (\\tau),Y^{3}\\Psi (\\tau)$ if and only if the function\n \\begin{equation}\nc_{3}=\\Bigg(\\Bigg(\\frac{1}{c_{2}}\\Bigg(\\left(\\frac{Y^{2}g}{g}\\right)'-3(Y c_{1} )\\Bigg) \\Bigg) '+g\\Bigg)\n\\label{eq:3}\n\\end{equation}\nis non-zero at at least a point on $\\mathcal{H}$. If, on the other hand, we have $c_{3}=0$ everywhere on $\\mathcal{H}$ then the quantity\n\\begin{equation*}\n Y^{3}\\Psi +\\frac{1}{c_{2}}\\cdot\\left(\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'-3(Y c_{1} )\\right)\\cdot Y^{2}\\Psi +\\Bigg[\\frac{Y c_{1} }{c_{1}}\\cdot\\left[\\frac{1}{c_{2}}\\left(\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'-3(Y c_{1} )\\right)\\right]+\\frac{Y^{2}c_{1}}{c_{1}}\\Bigg]  \\cdot Y\\Psi \n\\label{phi3conserved}\n\\end{equation*}\nis conserved, i.e.~independent of $\\tau$. Using the above scheme, Theorem \\ref{theo4} can be proved inductively.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Some examples}\n\\label{sec:AnExampleOfAMetricForWhichC1Neq0AndC20}\n\n\n\\paragraph{1. An example of a metric for which $c_{1}\\neq 0$ and $c_{2}=0$ for $l=0$\\medskip\\\\}\n\\label{sec:AnExampleOfAMetricForWhichC1Neq0AndC20}\n\n\n\nConsider a spherically symmetric metric such that \n\\begin{equation}\n\\Omega(u,v)=1, \\ \\ \\ r(u,v)=\\frac{1}{2}+\\frac{1}{2}(1+uv)^{2}>0. \n\\label{examplec1c2}\n\\end{equation}\nLet now $\\mathcal{H}=\\left\\{u=0\\right\\}$ along which $r=1, \\partial_{u}r=v, \\partial_{u}\\partial_{u}r=v^{2}$. Then along $\\mathcal{H}$ we obtain   $c_{1}=1$ and  $c_{2}=0$. Therefore, $\\mathcal{H}$ admits a second-order conservation law for all spherically symmetric solutions to the wave equation. This example shows that the order of the conservation law and the angular frequency associated to its kernel are independent.\n\n\\paragraph{2. Conservation law on the Cauchy horizon of Reissner--Nordstr\\\"{o}m\\medskip\\\\}\n\\label{sec:AnExampleOfdsaf}\n\n\nThe  Reissner--Norstr\\\"{o}m metric satisfies:\n\\[\\partial_{u}\\partial_{v}r=-\\frac{\\Omega^{2}}{4r}-\\frac{1}{r}\\partial_{v}r\\partial_{u}r+\\frac{1}{4}\\Omega^{2}r^{-3}e^{2}. \\]\nOn the (inner) horizon we have $\\partial_{v}r=0$ and in fact \n\\[r=M-\\sqrt{M^{2}-e^{2}}.\\]\nThen, we have a first-order conservation law if:\n\\begin{equation}\n\\partial_{v}\\partial_{u}r=\\Omega^{2}\\cdot l(l+1)\\cdot\\frac{1}{2r}. \n\\label{malistatwra}\n\\end{equation}\nEquation \\eqref{malistatwra} is satisfied for a discrete set of values for the charge $e$. Indeed, we need $e$ to satisfy:\n\\[\\left(\\frac{e}{r}\\right)^{2}=\\frac{2l(l+1)+1}{2}.\\]\nIn this case the kernel of the conservation law consists of all the eigenfunctions of the standard spherical Laplacian $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ which correspond to the eigenvalue $-l(l+1)$.\n\n\\paragraph{3. A hierarchy of conservation laws for some spacetimes\\medskip\\\\}\n\\label{sec:AnExampleOfdsaf}\n\nThe spherically symmetric null hypersurfaces of Minkowski spacetimes, the null infinity of asymptotically flat spacetimes and the event horizon of extremal Reissner--Norstr\\\"{o}m and extremal Kerr black holes admit the following hierarchy of conservation laws. Specifically, we have $R_{l+1,l}=0$ for all $l\\in\\mathbb{N}$. Hence, for all $l\\in\\mathbb{N}$ there is an $(l+1)$-order conservation law and its kernel consists of all the eigenfunctions of the standard spherical Laplacian $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ which correspond to the eigenvalue $-l(l+1)$.\n\n\n\n\n\\section{Acknowledgements}\n\\label{sec:Acknowledgements}\n\tI would like to thank Mihalis Dafermos, Georgios Moschidis, Willie Wong and Shiwu Yang for their help and insights. I would also like to thank Harvey Reall, Sergiu Klainerman, Jan Sbierski and Jeremy Szeftel for several very stimulating discussions and comments. I acknowledge support through NSF grant  DMS-1265538.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nDynamic systems have been investigated since the time of Newton, but the interest for numerical experiments grew considerably from the works realised by Lorenz \\citep{Lorenz} in 1960. These works were carried out with the purpose of understanding the behaviour of nonlinear dynamic systems \\citep{Grebogi}. Among the dynamic systems, those without an analytical solution are usually investigated by means of numerical computation, such as nonlinear systems with chaotic behaviour. Thus, the numerical computation plays a fundamental role in the analysis of nonlinear dynamic systems \\citep{Lozi,Nepomuceno}\n    \nOne of the most important way to study dynamic systems are the discrete maps.\nThese maps can also be seen as recursive functions, which allow the description and solution to a vast set of problems \\citep{Feigenbaum}. They are the basis for the resolution of most nonlinear dynamic systems \\citep{Lozi}. Using discrete maps it is possible to build bifurcation diagrams, which although much studied, do not have a well-defined set of rules for its application. As it has been already seen in \\citep{Paiva}, the initial condition can be crucial in the constructing bifurcation diagram of the logistic map. In this paper, we are interested in doing a similar investigation of \\citep{Paiva} for quadratic map \\citep{Lorenz,Galias}, which has an important application in encryption of images based on chaos \\citep{Ramadan,Kar}.\n\n\\section{PRELIMINARY CONCEPTS}\n\\label{sec:concepts}\n\n\\subsection{Recursive Functions}\nRecursive functions that take the form $x_{n+1}=f(x_n)$ \ndescribe a wide range of problems. Being $\\mathbb{I} \\subseteq \\mathbb{R}$ a metric space with $f:\\mathbb{I} \\rightarrow \\mathbb{R}$ recursive function can be described as \\citep{Nepomuceno}:\n\n\\begin{equation}\n\\centering\nx_n=f(x_{n-1}). \n\\label{eq01}\n\\end{equation}\n\n\\subsection{Fixed Point}\n\nThe fixed point $x^*$ of a map is the point such that $x_{n+1}=x_n=x^*$. That is, starting from $x^*$, \nremains in $x^*$ in the next iteration. Therefore, $x^*$ satisfies the following relation $x^*=f(x^*)$, \\citep{Monteiro}.\n\n\\subsection{Quadratic Map}\n The quadratic map \\citep{Lorenz} is described by the following function:\n \\begin{equation}\n  x_{n+1}=a-x_n^2, \n  \\label{eq02}\n\\end{equation}\n\n\\noindent\nwhere, \\textit{n} \nis the number of iterations and \\textit{a} is the bifurcation parameter.\n\\vspace{-3.3cm}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.5 ]{pontofixo1.pdf}\n\\vspace{-3.3cm}\n\\caption{Result of the values of $x_n$ for n=50 and $a=1,9$.}\n\\label{fig1}\n\\end{figure}\n\n\\subsection{Bifurcation Diagram}\n\nThe term bifurcation is generally used to refer to the qualitative transition from regular to chaotic behavior by changing the control parameter \\citep{Ramadan}, \ngiven any initial condition.The bifurcation diagram is used to study the system in function of its control parameter, allowing to know regions of the system that converge to bifurcation or even to chaos, depending on the parameter. \\citep{Monteiro}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.4 ]{diagrama_mapa_quadratico.pdf}\n\\caption{Diagram of bifurcation for the quadratic map.}\n\\label{fig2}\n\\end{figure}\n\n\\subsection{Floating-Point Arithmetic}\n\nNumerical computing is a critical part of dynamic systems analysis. Therefore, it is necessary to emphasize the IEEE 754-2008 standard \\citep{IEEE} for floating-point arithmetic, since a large majority of numerical computation makes use of this norm. It establishes the rounding behavior, error handling and degree of accuracy of the calculations performed, causing the results to approach the expected value. Even with satisfactory results, numerical computation is not yet able to present results that are totally in agreement with reality, due to the fact that it has a finite representation of real numbers. \\citep{Rodrigues,IEEE}.\n\n\\section{METHODOLOGY}\n\\label{sec:methodology}\n\nThe behaviour of the quadratic map is determined by the variation of the bifurcation parameter $a$, \nwhere $-2.5 < a <2.5$. Thus, given an initial condition belonging to the defined interval, the function will have its behaviour as shown in the bifurcation diagram of Fig. (1a).\n\nGiven the equation $x_{n+1}=a-x_n^2$ which describes the Quadratic Map, it is possible to obtain its fixed point  $x^*$ doing  $x_{n+1}=x_n=x^*$. Thus the map equation takes the following form:\n \\begin{equation}\n \\centering\n  x^2 + x - a =0. \n  \\label{eq03}\n\\end{equation}\nFrom Eq.(\\ref{eq03}), it is possible to  find a value for \\textit{x} depending on the parameter \\textit{a} which is a fixed point, shown by Eq. (\\ref{eq04}).\n\\begin{equation}\n\\centering\n    x_1=\\frac{\\sqrt{4a+1}-1}{2};\n    x_2=-\\left ( \\frac{1+\\sqrt{4a+1}}{2} \\right )\n    \\label{eq04}.\n\\end{equation}\n\nThe use of the inverse of the function \\eqref{eq03}, presented in Eq. (\\ref{eq05}), allows to  verify the existence of a set of points that converge to the fixed point $x^*$. \n\n\\begin{equation}\n\\centering\n   f^{-1}(x^*)=\\pm \\sqrt{a-x^*}.\n\\label{eq05}\n\\end{equation}\n\n\\thispagestyle{empty} \nFrom Eq. (\\ref{eq05}) the following result was obtained:\n\\begin{equation}\n\\centering\n    x = \\pm \\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}.\n    \\label{eq06}\n\\end{equation}\nEq. (\\ref{eq06}) is so used as the initial condition $x_0$ to build the bifurcation diagram. Thereby, \nit was possible to show that $\\pm \\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$ belongs to the set of points that converge to the fixed point $x^*$.\n\n\\begin{equation*}\n  x_1 = f(x) = a - \\left [\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}\\right ]^2 = \\frac{\\sqrt{4a+1}-1}{2};\n\\end{equation*}\n\n\\begin{equation}\n\\centering\n  x_2 = f(x_1) = a - \\left [\\frac{\\sqrt{4a+1}-1}{2}\\right ]^2 = a - \\left [\\frac{4a+1 -2\\sqrt{4a+1} +1}{4}\\right] = \\frac{\\sqrt{4a+1}-1}{2};\n \\label{eq07}\n\\end{equation}\n\\vspace{0.1cm}\n\n\\noindent\nand $x_1 = x_2 = ... = x_n$, then $d(f^p(x),x^*) \\to 0$ when $p \\to \\infty$. \n\n\n\\section{RESULTS}\n\nThe bifurcation diagrams are shown in Fig. (\\ref{fig5}) analysis of the behavior of the map given a value for \\textit{a}. When $a=1,9$ the map assumes an unstable fixed-point region, shown in Fig.(\\ref{fig1}). The effect of the application of $ x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$ as initial condition can be observed by Fig. (\\ref{fig3}) where to $a=1,9$, constraint to the initial condition, it was possible to find a convergence for $x^*$, showing a result that is considerably different from that found in Fig. (\\ref{fig1}), \nwhere no restrictions were made for the initial condition and only 50 iterations were performed.\n\n\\newpage\n\\begin{figure}[h!]\n\\centering\n\\vspace{-3.3cm}\n\\includegraphics[angle=0, scale=0.5 ]{pontofixo2.pdf}\n\\vspace{-3.3cm}\n\\caption{Result of the values of \\textit{x} for n=200 and $a=1,9$.}\n\\label{fig3}\n\\end{figure}\n\nFrom these results, it is necessary to obtain a new bifurcation diagram from the initial condition $ x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$. \nThis diagram is shown in Fig.(\\ref{fig4}). The diagrams shown in Figs.(\\ref{fig4}) and (\\ref{fig2}) were obtained in conventional manner in the literature using Matlab and double precision floating point arithmetic.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.4 ]{novo_diagrama_mq.pdf}\n\\caption{Bifurcation Diagram for Initial Condition $x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$.}\n\\label{fig4}\n\\end{figure}\n\nThe result shown in Fig.(\\ref{fig4}) is largely incorrect, although, due to its similarity with Fig.(\\ref{fig2}), could be easily understood as correct.  With initial condition given by Eq.(\\ref{eq06}), the correct answer is shown in Fig. (\\ref{fig5}), which the transient is not discharged, and on the contrary, the bifurcation diagram has been built using only the first result of the iteration process, that is, using only $x_1$.  Similar results were found by \\citep{Paiva}, for the logistic map.\n\\newpage\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.4 ]{diagrama_real.eps}\n\\caption{Bifurcation diagram for initial condition $x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$.}\n\n\\label{fig5}\n\\end{figure}\n\n\\section{CONCLUSIONS} \n\\label{sec:conclusions}\nAfter all the analyses, it is concluded that for the construction of the bifurcation diagram of the quadratic map the value of the initial condition has a fundamental role. The diagrams shown in Figs. (\\ref{fig2}), (\\ref{fig4}) and (\\ref{fig5}) show different results. For Fig. (\\ref{fig2}), the diagram was obtained following the methodology proposed in the literature for an equal initial condition $x_0 = 0.2$. While in the simulations of the diagrams shown in Figs. (\\ref{fig4}) and (\\ref{fig5}), a constraint associated with the parameter $ a $ has been applied. In Fig. (\\ref{fig4} the numerical simulation did not present the expected results because using the proposed initial condition, for any value of $ a $, convergence occurs at the first iteration. However, it was observed that after a number of iterations, the result diverges. This divergence can be understood as the result of a numerical error due to the finite-precision of the floating-point arithmetic. Therefore,  it is necessary more care on the constructing bifurcation diagram of the quadratic map.\n\n\\section{ACKNOWLEDGEMENTS}\n\nWe thank CAPES, Fapemig and UFSJ for the support.\n\\section{REFERENCES} \n\\bibliographystyle{abcm}\n\\renewcommand{\\refname}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\\label{sec:intr}\n\nA considerable amount of work has been done over the last few years to improve the simulations of MPGDs. A full understanding of the micro-physics associated with MPGDs operation is vital for the improvement of such detectors, whose some behaviours are still not completely understood.\\cite{COliveira2012JINST,Ozkan2010JINST,spark_simulation}.\n\n\nSome works report that there is a transient period during which the effective gain changes, after voltages are applied and the detector irradiated\\cite{Azmoun200611,THGEM_operation_Ne_CH4}. \nThe gain ends up stabilizing after minutes or even hours, depending on the MPGD and the rates of irradiation.\n\nMPGDs were developed to detect radiation, and their main applications are for high energy physics, astrophysics, rare-event searches and medical imaging\\cite{mpgd_progression,natal_luz}. \nThe \\mbox{\\textit{Gas Electron Multiplier}} (GEM)\\cite{Sauli1997531} has been largely used in many of those applications. \n\nThe device consists of a thin polyimide (insulator) foil typically 50 $\\mu$m thick. The foil is covered on both sides with 5 $\\mu$m thick layers of a conductor and etched with an hexagonal pattern of holes. The device operates inside a gas medium and suitable electric potentials are applied between the upper and the lower electrodes of the structure. In this way, a very high electric field is created inside the holes. Electrons created in the drift region by interaction of external radiation, travel towards the micro-structure, being focused into the holes and accelerated. They acquire enough energy to ionize atoms\/molecules of the gas, creating new ionizations. The secondary electrons undergo the same process while inside the hole and an avalanche ends up being produced, the Townsend avalanche.\n\nThe charges produced during multiplication have two possible destinations: they may be collected by conductor electrodes, both those of the GEM itself or of any other readout setup; and a fraction of them ends up accumulating in the insulator surfaces.\nThe number of simulated avalanches is correlated with the number of primary electrons that undergo to the hole, since we assume that  no charges will drift in the insulator. \n\nEffective gain is defined as the number of secondary electrons, for each primary electron, that are collected in an electrode plane located below the GEM (figure~\\ref{fig:plane_section}).\n \n The electronic affinity of the polyimide usually used in these devices is high, 1.4 eV \\cite{poly_affinity}. Once electrons are trapped, it is unlikely that they are able to leave the surface. The same process happens with ions, according to \\cite{sessler}. \n\nThe effective gain of the detector strongly depends on the intensity of the electric field produced in the multiplication region. Charges accumulated in the insulator surfaces change locally the electric field, changing the amplification gain. This is known as the charging-up effect in the insulator.\n\n\nDeposited charges can flow through the insulator surface and insulator bulk under the action of the electric field. \nPrevious studies propose that the positive ions are not captured in insulators surfaces, instead they transfer their charge to intrinsic carriers of the insulator, and the conduction should be made by electrons and holes\\cite{sessler}. \nThe time for charges evacuation is of the order of several hours to days, so we did not include this effect in our method, i.e. all deposited charges will remain in the same surface during the whole simulation time. \nThis approach should remain valid if the charging-up process is much faster than the draining of charges.\n\nSimulations of the charging-up influence in the GEM transparency were already reported\\cite{Alfonsi20126}. \nIn order to study the contribution of the charging-up for the effective gain variations, two methods to simulate the charge accumulation in the detector are presented.\nWe also compare our results with available experimental data. \n\n\n\\section*{Calculations}\\label{sec:gem_thgem}\n\n\n\\subsection*{Geometry}\\label{subsec:mpgds_geom}\n\nAn hexagonal hole pattern with a distance between holes of 140 $\\mu$m was considered, the insulator thickness is 50 $\\mu$m and the metal electrodes are 5 $\\mu$m  thick. Figure \\ref{fig:cross_section} shows the GEM geometry.\nThe corresponding electric field configuration is depicted in figure \\ref{fig:plane_section}. The hole has a bi-conical shape, the outer diameter is 70 $\\mu$m, while the narrower part has 50 $\\mu$m of diameter.\n\n\\begin{figure}[htp]\n\\centering\n\\subfloat[]{\\includegraphics[width=.45\\textwidth]{gem_cross_section_dimensoes.pdf}\n\\label{fig:cross_section}}\\quad\n\\subfloat[]{\\includegraphics[width=.45\\textwidth]{gem_cross_section_plano.pdf}\n\\label{fig:plane_section}}\n\\caption{a) Cross sections of a GEM, with the dimensions and voltages between electrodes. b) Simulated configuration and applied drift and induction fields.}\n\\end{figure}\n\n\\subsection*{Gases}\\label{subsec:gases}\n\nIn order to compare simulation results with our measurements, we simulated a gas mixture of $\\mathrm{Ar \\ 70\\% \\ \/ \\ CO_2 \\ 30\\%}$. This is a penning-mixture, due to the presence of the quencher molecule CO$_{2}$, that opens new de-excitation ways for the previously excited noble gas molecules. If this excess of energy is above the ionization threshold of the quencher molecule, an ionization may occur, whit a probability called Penning probability. We used a Penning probability of 0.7 in this simulations based on previous calculations for $\\mathrm{Ar \\ 70\\% \\ \/ \\ CO_2 \\ 30\\%}$ mixture\\cite{Ozkan2010JINST,sven,penning1,penning2}.\n\nThe drift and induction fields (electric fields applied up and above the GEM, respectively) were 0.2 and 0.3 kVcm$^{-1}$.\nTaking into account that the computational time strongly depends on the gain, because a higher number of electrons need to be tracked within the avalanche, a potential of 400 $V$ between electrodes was used for the firsts simulations tests, corresponding to gains of $\\sim$10$^{2}$.\nAll simulations were performed considering a temperature of 293 K and a pressure of 760 Torr.\n\n\\section*{Simulation details}\n\n\\subsection*{Software platforms}\n\nThe Monte Carlo calculations involved three programs. Due to the complex shape of the GEM structure, an analytic solution for the electric field in the interest region is not possible to obtain.\nTo overcome this problem, the electric field is computed with \\textit{Finit Element Methods} (FEM) software, that is used to calculate the electric potential in discrete nodes of a mesh, using boundary conditions. \nWe used ANSYS$^\\circledR$\\footnote{www.ansys.com} to produce potential maps, to which we call generally as field maps, selecting the curved tetrahedral elements as our mesh elements, because they easily fit in sharp curved surfaces present in GEMs.\n\nTo simulate the drift and transport properties of electrons and ions in the MPGD gas medium, we used Garfield++ \\cite{garfieldpp}. As input, this software requires the electric field configuration in the MPGD, the gas mixture, temperature, pressure and initial conditions of the primary charges (position, velocity and energy). \n\nIn what regards the electric field configuration, we used Garfield++ to read the potential maps calculated with ANSYS$^\\circledR$ and to calculate the electric field in any point of the space by interpolation between nodes.\n\nA microscopic approach is used to simulate the drift of the charges. This uses Monte-Carlo methods to calculate the probability to occur each type of collision during the drift (elastic, excitation or ionization). \nThe cross sections associated with each collision type are obtained from Magboltz\\cite{magboltz,Biagi1999234}.\n\nPrimary electrons starts with assigned $\\vec{r}_\\mathrm{start}=(x_\\mathrm{s},y_\\mathrm{s},z_\\mathrm{s})$, velocity $\\vec{v}_\\mathrm{start}=(v_\\mathrm{x,s},v_\\mathrm{y,s},v_\\mathrm{z,s})$ and kinetic energy $E_\\mathrm{start}$, drifting through the gas and producing secondary charges as it passes the multiplication region.\nThe final position of each secondary charge, $\\vec{r}_\\mathrm{end}$ and the effective gain are the observables of interested that are recorded for further analysis of the charging-up effect.\n\n\n\\subsection*{Initial attempts}\n \nTo start our simulations, we randomly distributed 10$^{4}$ primary electrons in the surface of a plane parallel to the GEM, located 100 $\\mu$m above the GEM, indicated as the start plane in figure \\ref{fig:cross_section}. \n\n\nIn order to determine the number of collected and deposited electrons and ions, the final position of each electron and ion from avalanches are analysed:\n\\begin{itemize}\n  \\item Electrons are collected if the final z is -100 $\\mu$m below the GEM (the collection plane represented in figure \\ref{fig:plane_section}.\n  \\item Ions are collected if the final z coordinate is in the top (positive) electrode of the GEM.\n  \\item Electrons and ions are deposited in the insulator surface if after the drift the z coordinate is between -25 $\\mu$m to  +25 $\\mu$m.\n\\end{itemize}  \n\nGEMs previous to charging-up, i.e. without deposited charges, are called uncharged GEMs, being the charged GEMs those who have already deposited charges due to charging-up. \nThe deposition distributions of charges (electrons and ions separately) in the insulator, for the case previous to charging-up (figure \\ref{fig:depo_hist_with_no_charg}) shows that the charges are not deposited uniformly on the hole surface. \nIn addition, the number of deposited electrons is higher than the number of deposited ions. The reason is related with the mass of each particle. Ions are heavier than electrons, and tend to follow very well the field lines, in the direction of the electrodes. Electrons has a much more chaotic movement, due to the lower mass, having more probability of ending in the insulator surfaces. This originates variations on the local electric field.\n\n\\begin{figure}[htp]\n  \\centering\n  \\subfloat[Uncharged GEM]{\\includegraphics[width=.45\\textwidth]{deposition_400V_gem_uncharged.eps}\\label{fig:depo_hist_with_no_charg}}\\quad\n  \\subfloat[Charged GEM]{\\includegraphics[width=.45\\textwidth]{deposition_400V_gem_charged.eps}\\label{fig:depo_hist_with_charg}}\\\\\n  \\caption{Spatial distribution of charges deposited in the insulator surface of the GEM detector, before (\\ref{fig:depo_hist_with_no_charg}) and after (\\ref{fig:depo_hist_with_charg}) simulation of primary avalanches, at $V_\\mathrm{GEM}=$ 400 V.}\n  \\label{depo_hist_with_charg}\n\\end{figure}\n\\\nAfter some avalanches, the distribution of new electrons and ions that reaches the insulator tend to compensate each other, due to Coulomb attraction between previous and future deposited charges (figure \\ref{fig:depo_hist_with_charg}). The local variation in the electric field will therefore vanish and a stable configuration will be achieved.\n\n\nIn order to simulate the effective gain variation as avalanches happen, we needed to iteratively include this charge deposition in the potential maps computed with ANSYS$^\\circledR$. The software does not provide the option to put single charges in their exact deposition position in the insulator surface. In addiction, this scenario would lead to discontinuities and numerical issues.\nInstead, we created small slice surfaces in the insulator foil and add the correspondent density charge to each surface. Due to the shape of the deposition, and to computational limitations of field maps files for very small finite elements, we used 24 different slices in the insulator, achieving in this way a good balance between the detail of the calculations and the needed computing power. The slices are not regularly distributed, as shown in figure \\ref{fig:gem_slice}, trying to match the z profile of the charge deposition histograms (figure \\ref{fig:depo_hist_with_no_charg}).\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=.3\\textwidth]{gem_pictures003-eps-converted-to.pdf}\n\\caption{Unity cell of a GEM, used to calculate the field maps with ANSYS$^\\circledR$. 24 slices of different sizes were non regularly distributed due to the charge deposition non uniformity in the insulator surface, shown in figure \\ref{fig:depo_hist_with_no_charg}.\\label{fig:gem_slice}}\n\\end{figure}\n\n\n \n\\subsection*{Constant Step Method}\\label{subsec:const_meth}\n\n\nThe flow-chart of the first iterative algorithm used to simulate charging-up iterations is in figure \\ref{fig:cont_Step_meth}.\n\n\\begin{figure}[htp]\n  \\centering\n  {\\includegraphics[width=.5\\textwidth]{charg_corrigido.pdf}\\caption{Constant step method flow-chart.\\label{fig:cont_Step_meth}}}\n\\end{figure}\n\nAt the first iteration, we compute the electric field map assuming no charges in the insulator surface. Then, we import that field map into Garfield++, simulate $10^{4}$ primary avalanches and determine the density charge deposited in each insulator slice surface.\nA new field map is created, with the contribution of previously deposited charges. The density charge in each slice is calculated taking into account the contribution of both the ions and the electrons ending up in the insulator surface. \nA new set of $10^{4}$ primary avalanches is simulated and the process is repeated iteratively.\n\nIt was found that statistical fluctuations in the calculated gain depends on the number of simulated avalanches per step, but the number of deposited charges per avalanche seems to be less sensitive to fluctuations.\nA small step-size of $10^{4}$ primary electrons\nwas chosen in order to obtain good detail in the time evolution of charging-up. However, this small step implies hundreds of iterations until stabilization, which lead to a very heavy computation.\n\nSince the number of deposited charges is the responsible for the local variation in the electric field, we use that observable as our control function for the iterative simulation, i.e. we stop our iterations when its value stabilizes over iterations (corresponding also to a gain stabilization).\n \n\n\\subsection*{Dynamic Step Method}\n\nIn order to accelerate the simulation process, we developed an extended method that uses a dynamic step-size in each iteration.\nThis step-size is smaller when the number of deposited charges per avalanche changes quickly, and is larger when this quantity is more constant, i.e. the deposition stabilizes.\n\nTo constrain the size of the step, we defined that the maximum total charge (sum of signs of ions and electrons) that can be added to the new field map should not be larger than 2$\\times$10$^{4}$ $q_{e}$, (where $q_{e}$ is the elemental charge 1.6$\\times10^{-19}$ C). This way, the maximum allowed number of avalanches per step is equal to $\\frac{2\\times10^{4}q_{e}}{G_{tot}}$, where $G_{tot}$ is the absolute gain in each iteration. The output of this calculation give us an upper limit for the step-size, considering a maximum charge that can be added to new potential maps in each iteration. \nOur attempts show that this upper limit value an acceptable value in terms of the convergence and speed of the method, but other limits can be defined.\nThe dynamic method is briefly described in the flow-chart in figure \\ref{fig:din_Step_meth}.\n\\begin{figure}[htb]\n  \\centering\n  {\\includegraphics[width=.65\\textwidth]{charg_din_corrigido.pdf}\\caption{Dynamic step method diagram.\\label{fig:din_Step_meth}}}\n\\end{figure}\n\n\nThe method starts with an uncharged ANSYS$^\\circledR$ field map of the GEM. In each iteration we simulate $10^{3}$ primary avalanches, which is a good compromise between statistical fluctuations and computational time. \n\n\nThe number of deposited charges per avalanche, in each slice of the insulator surface, is multiplied by the variable step. \nFor the firsts iterations, steps between $0.5\\times10^{3}$ and $10^{3}$ primary avalanches were used. \n\nAfter the first 5 iterations (sufficient number of iterations that allows a reasonable fit), we fit the number of deposited charges per avalanche to a first order polynomial, and calculate for a given step, what the value of that function should be for the new iteration. \n\nWe then simulate iteration number 6, and compare with the predicted value from the fit:\n\n\\begin{itemize}\n\\item If the difference between simulated and fitted value is larger than the maximum defined step, discard the iteration, the step is reduced to its half, and repeat the iteration.\n\n\\item If the difference between simulated and fitted value is smaller than the maximum defined step, the iteration is saved, we increase the step to the double. A new iteration is calculated and new fit considers only the last 5 valid iterations\n\\end{itemize} \n\n\\section*{Results}\\label{GEM_sub}\n\\subsection*{Comparison between methods}\nThe sum of all electric charges (the integral of the deposition histograms in figures \\ref{fig:depo_hist_with_no_charg} and \\ref{fig:depo_hist_with_charg}) deposited in the insulator surface, per primary avalanche, is shown in figure \\ref{fig:depo_2method} for both methods, as function of the charge produced by each avalanche, per hole (is simply the number of primary simulated electrons in each hole multiplied by the total gain).\n\n\\begin{figure}[htb]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.45\\textwidth]{deposition400V.eps}\\label{fig:depo_2method}}\\quad\n\\subfloat[]{\\includegraphics[width=0.45\\textwidth]{cons_vs_dyn_method_400V.eps}\\label{fig:gain400V}}\n\\caption{a) Total number of deposited charges per produced secondary charge per hole, for both constant and dynamic methods. b) Comparison of the total gain (total number of secondary electrons produced per avalanche), between the constant and dynamic method. Both plots obtained for V$_{\\mathrm{GEM}}=400$  V. \\label{fig:compar}}\n\\end{figure}\n\nThe agreement between both methods is clear. However, the dynamic-step method saves computational resources, using about one tenth of iterations.\n\n\nFigure \\ref{fig:gain400V} represents the total gain evolution for the two methods. We observe an increase in effective gain, followed by a stabilization plateau, reached in both methods. \nDue to previous results, from now on we will only consider the dynamic-step method for calculations.\n\n\n\\subsection*{Charging-up effect in the GEM transmission}\nPrimary electrons produced by incident radiation and drifting towards the GEM holes can be collected in the top electrodes, ending up not producing avalanches. \nThe ratio between the number of primary electrons that enter the holes, producing avalanches and the total number of primary electrons simulated is defined as the electron transmission, shown in figure \\ref{fig:trasmission}, for several voltages applied to the GEM electrodes. \n\n\nThe contribution of the charging-up effect in the electron transmission is more important when low voltages (<400 V) are used and negligible when higher electrical potentials are used.\n\n\\subsection*{Effective gain with and without charging-up}\nThe dependence of effective gain on the voltage applied between electrodes in the GEM detector, is shown in figure \\ref{fig:gain_vs_vgem}. \nThe gain, after charging-up stabilization, is 10-15\\% higher than the situation without charging-up.\n\n\n\\begin{figure}[htb]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.42\\textwidth]{transmission_vs_vgem.eps}\\label{fig:trasmission}}\\quad\n\\subfloat[]{\\includegraphics[width=0.43\\textwidth]{effgain_vs_vgem.eps}\\label{fig:gain_vs_vgem}}\n\\caption{a) Electron transmission as a function of the voltage applied bettween the GEM electrodes. b) Effective gain comparison between charged (red) and uncharged (green) GEM, for different voltage between electrodes.}\\label{fig:gain_and_transmission_with_charging}\n\\end{figure}\n\n\n\\subsection*{Electric field intensity variation}\nA 2D representation of the electric field in the GEM is shown in figure~\\ref{fig:efield_variation}. Each plot is obtained by calculation of the intensity of the electric field along a plane corresponding to a vertical cross section of the GEM hole, at four different stages of the charging-up process.\n\n\\begin{figure}[htp]\n\\centering\n\\subfloat[Without charging-up.]{\\includegraphics[width=0.45\\textwidth]{Efile0.jpg}\\label{fig:efield_a}}\\quad\n\\subfloat[3 $\\times 10^{6}$ avalanches.]{\\includegraphics[width=0.45\\textwidth]{Efile30.jpg}\\label{fig:efield_b}}\\quad\n\\subfloat[6 $\\times 10^{6}$ avalanches.]{\\includegraphics[width=0.45\\textwidth]{Efile60.jpg}\\label{fig:efield_c}}\\quad\n\\subfloat[10 $\\times 10^{6}$ avalanches.]{\\includegraphics[width=0.45\\textwidth]{Efile100.jpg}\\label{fig:efield_d}}\\quad\n\\caption{Evolution of the intensity of the electric field, in a GEM cross section. Computed with ANSYS$^\\circledR$. The colorbar refers to the logarithm of \\textbf{E}. Only intensities above 100 kVcm$^{-1}$ ($\\ln (100)\\simeq 4.6$) are colored. }\\label{fig:efield_variation}\n\\end{figure}\n\n We can observe that the biggest change in the electric field occurs near the electrodes. While the intensity of the electric field near the top (negative polarized) electrode decreases, it increases near the center of the hole and the bottom (positive polarized) electrode. \nThe development of an avalanche inside the hole follows a nearly exponential model. The bigger fraction of secondary electrons is produced at the exit of the hole, in the last stages of the avalanches. There, the electric field is higher due to the charging-up effect, and thus, the effective gain increases as a result of this process.\n \n \\subsection*{Comparison with experimental results}\n\nExperimental measurements were performed at CERN. Physical parameters of the GEM, and the gas mixture used for measurements correspond to the simulation settings. X-ray photons were used as ionizing radiation. $\\mathrm{K_{\\alpha}}$ and $\\mathrm{K_{\\beta}}$ photons corresponding to energies 8.0 $\\mathrm{keV}$ and 8.9 $\\mathrm{keV}$ respectively were emitted by the X-ray tube which employed a copper target. \n\nCollimators to control the photon flux were used in order to regulate the rate of charging up. The gain was measured over the time, for a constant irradiation flux. The GEM structure was housed inside an air-tight chamber, shown in figure \\ref{fig:Timing}, that had a constant circulation of mixture, with a gas flow rate of $\\mathrm{6\\ l.h^{-1}}$. Chamber pressure was maintained at $\\mathrm{760\\ Torr}$. Figure \\ref{fig:Flowchart} shows the schematics setup for gain calibration (left) and gain measurement (right). In order to compare with simulations, the experimental results were normalized from the time scale to charges per hole. This is done multiplying the number of primary electrons produced per second by the absolute gain, divided by the number of irradiated holes of the GEM.\n\nFor a detailed description of the measurement procedure refer to \\cite{mythra}.\n\n\n\\begin{figure}[ht]\n{\\centering \\resizebox*{3in}{3in}{\\includegraphics{TimingGEM.jpg}} \\par}\n\\caption{GEM foil being mounted on a timing-GEM chamber}\n\\label{fig:Timing}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{minipage}[b]{0.5\\linewidth}\n{\\centering \\resizebox*{3in}{3in}{\\includegraphics{Gain_calibration.png}} \\par}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n{\\centering \\resizebox*{3in}{3in}{\\includegraphics{Gain_measurement.png}} \\par}\n\\end{minipage}\n\\caption{Flow chart depicting the setup for Gain calibration (left) and Gain measurement (right).}\n\\label{fig:Flowchart}\n\\end{figure}\n \nA comparison between Monte Carlo calculations and measurements is shown in figure~\\ref{fig:mythra_gain_comp}.\nThe total gain increases as the GEM starts to be irradiated, in both cases, achieving then a plateau.\nA normalization to the plateau of both data was done to directly compare the time evolution of Monte Carlo simulations and experimental values, as shown in figure \\ref{fig:mythra_gain_comp_norm}. From this observation, Monte Carlo simulations reproduce the time evolution of the gain.\n\nOn the other hand, the value of the total gain still do not match. This can be related with the mobility of the charges in the insulator surfaces and bulk, numerical issues associated with the finite elements method, computed with ANSYS$^\\circledR$, impurities in gas and imperfections in GEMs dimensions introduced during the production.\n\n\\begin{figure}[ht]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.42\\textwidth]{comparation_mytra_380V.eps}\\label{fig:mythra_gain_comp}}\\quad\n\\subfloat[]{\\includegraphics[width=0.42\\textwidth]{comparation_mytra_380V_normalized.eps}\\label{fig:mythra_gain_comp_norm}}\\quad\n\\caption{a) Absolute gain comparison between measuremments (red) and simulated (green) results. Same situation as figure \\ref{fig:gain400V} but with $V_\\mathrm{GEM}$=380 V. b) Same plot as figure \\ref{fig:mythra_gain_comp} but with gain normalized. Experimental data taken by Mythra Varun Nemallapudi at RD51 facilities, CERN.\\label{fig:mythra_gain_comp_tot}}\n\\end{figure}\n\n\n\n\n\\section*{Conclusion and future work}\nIn this work we have presented two iterative methods for the simulation of the insulator surface charging-up in GEMs, allowing a better understanding of their response.\n\nBoth methods agree between each other. However, the dynamic-step method saves computational resources.\nThe Monte Carlo functional time behaviour of the gain as the GEM is irradiated reproduces that observed experimentally.  \nHowever, the absolute scale still not agree. \n\nPrimary electrons transmission should be affected by the charging-up at lower voltages between electrodes of the GEM, but for higher voltages, used in regular applications, it does not play an important role.\n\n\nFuture work will include the application of the presented charging-up simulation methods to other MPGDs (e.g. THGEM) and the study of new geometries and detectors that could take advantages of this effect or minimize it.\n\n\nThe simulation of the mobility of deposited charges in the insulator surfaces could contribute to obtain more precise values. Refine the calculations of the electric field calculations (with ANSYS$^\\circledR$ or other method) can also be important to get agreement between absolute simulated and measured gain values.\n\n\\acknowledgments\n\\noindent\nThis work was partially supported by projects CERN\/FP\/123604\/2011 and  PTDC\/FIS\/110925\/2009 through COMPETE, FEDER and FCT (Lisbon) programs.\n\n\\noindent P.M.M. Correia was supported by FCT (Lisbon) grant BIC\/UI96\/5496\/2011.\n\n\\noindent C.D.R. Azevedo was supported by FCT (Lisbon) grant SFHR\/BPD\/79163\/2011.\n\n\n\\bibliographystyle{PedroJINST}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is believed that supermassive black holes (SMBHs) reside in central\nnuclei of many galaxies, and they occasionally capture a stellar mass\ncompact object (SMCO) which surrounds them.\nGravitational waves from such binary systems with extreme mass\nratios bring us information on the orbits of SMCOs and the spacetime\nstructure near black holes.\nTherefore such systems are considered to be one of the most important\ntargets of the LISA space-based gravitational wave detector\n\\cite{LISA}.\nIn order to detect gravitational waves emitted by extreme mass\nratio inspirals (EMRIs) and to extract physical information from them \nefficiently, we need to predict accurate theoretical waveforms \nin advance.\nOur goal along the line of this paper \nis to precisely calculate theoretical waveforms from EMRIs.\n\nTo investigate gravitational waves from EMRIs,\nour strategy is to adopt the black hole perturbation method:\n\\cite{Mino:1997bx}\nwe consider metric perturbations induced by a SMCO in\na black hole spacetime governed by a SMBH. \nWe also assume that a SMCO is described by a point particle,  \nneglecting its internal structure. \nUnder the above approximations, we can calculate the metric perturbation\nevaluated at infinity to predict gravitational waveforms.\nAt the lowest order with respect to the mass ratio, \nwe may calculate the metric perturbation \nby approximating the particle's orbit by\na background geodesic.\nTo step further, \nwe consider the orbital shift from the background geodesic by \ntaking account of the self-force induced by the particle itself. \n\nIn a Schwarzschild background, we can assume that the \norbit is in the equatorial plane from the symmetry without loss \nof generality. \nHence, the orbital velocity can be specified solely by the energy \nand the azimuthal angular momentum of the particle.\nNamely, we can evaluate the orbital evolution from \nthe change rates of the energy and the angular momentum.\nTheir averaged change rates \ncan be evaluated by using the balance argument; \nthe energy and the angular momentum that a particle loses are\nequal to the ones that are radiated to the infinity or across the\nhorizon as gravitational waves because of the conservation laws.\nIn the limit of a large mass ratio, averaged change rates will \nbe sufficient to determine the leading order effects on the \norbital evolution due to the self-force. \nIn this sense the leading order effects can be read from \nthe asymptotic \nbehavior of the metric perturbation in the Schwarzschild case.\n\nOn the other hand, the third constant of motion, i.e., the Carter\nconstant, is necessary in addition to the energy and the azimuthal \nangular momentum to specify a geodesic in a Kerr background.\nHowever, there is no known conserved current composed of gravitational\nwaves that is associated with the Carter constant, \nand hence we cannot use the balance argument to evaluate\nthe change rate of the Carter constant. \nTherefore we have to calculate the self-force acting on a particle\n\\cite{Ori:1995ed,Ori:1997be}.\nWhen we calculate the self-force, we are faced with the regularization\nproblem.\nAlthough the formal expression for the regularized \nself-force had been derived\n\\cite{Mino:1996nk,Quinn:1996am,Detweiler:2002mi},\ndoing explicit calculation is not so straightforward.\n\nGal'tsov \\cite{Gal'tsov82} proposed a method of calculating\nthe loss rates of the energy and angular momentum of a particle\nby using the radiative part of metric perturbation, which was\nintroduced earlier by Dirac \\cite{Dirac:1938nz}.\nThe radiative field is defined by half retarded field minus half\nadvanced one, which is a homogeneous solution of the field equation.\nIt was shown that the time-averaged loss rates of the energy and\nangular momentum evaluated by using the radiative field \nare identical with the results obtained from the balance argument.\nRecently, Mino proved that the Gal'tsov's scheme also gives \nthe correct averaged change rate of the Carter constant\n\\cite{Mino:2003yg}.\nThe Gal'tsov-Mino method has a great advantage that \nwe do not need any regularization procedure because\nthe radiative field is free from divergence from the beginning. \nIn Ref.~\\citen{Sago:2005gd},\nwe briefly reported that the formula for the adiabatic evolution\nof the Carter constant based on Gal'tsov-Mino method\ncan be largely simplified. \nIn this paper, we explain the derivation of this new formula\nin detail.\nApplying our new formula, \nwe explicitly calculate\nthe change rate of the Carter constant for orbits with \nsmall eccentricities and inclinations. \n\nThis paper is organized as follows.\nIn Sec.~\\ref{sec:geodesic},\nwe give a brief review of the Kerr geometry and the geodesic motion.\nNext, we show a practical prescription to calculate the time-averaged\nchange rates of the constants of motion in Sec.~\\ref{sec:COMdot}.\nWe also derive a simplified expression for the change rate\nof the Carter constant.\nIn Sec.~\\ref{sec:example}, applying our prescription,\nwe calculate the change rates of the constants of motion\nand then show the analytic formulae of them\nfor slightly eccentric and inclined orbits.\nFinally we devote Sec.~\\ref{sec:summary} to summarize\nthis paper.\nIn Appendix~\\ref{sec:radiative}, we show the derivation of\nthe radiative part of metric perturbation.\nAnd we also give short reviews on analytic methods \nof solving the radial Teukolsky equation and \nobtaining the spheroidal harmonics in Appendices~\\ref{sec:MST}\nand \\ref{sec:spheroidal}.\n\n\n\\section{Geodesic motion in the Kerr spacetime} \\label{sec:geodesic}\nIn this section, we give a brief review on geodesics in the \nKerr geometry.\nThe metric of the Kerr spacetime in the Boyer-Lindquist\ncoordinates is \n\\begin{eqnarray}\nds^2 &=&\n-\\left(1-\\frac{2Mr}{\\Sigma}\\right)dt^2\n-\\frac{4Mar\\sin^2\\theta}{\\Sigma}dtd\\varphi\n+\\frac{\\Sigma}{\\Delta}dr^2\n\\nonumber \\\\ &&\n\\hspace*{2cm} +\\Sigma d\\theta^2\n+\\left(r^2+a^2+\\frac{2Ma^2r}{\\Sigma}\\sin^2\\theta\\right)\n\\sin^2\\theta d\\varphi^2, \\label{eq:Kerr}\n\\end{eqnarray}\nwhere\n\\[\n\\Sigma=r^2+a^2\\cos^2\\theta, \\quad\n\\Delta=r^2-2Mr+a^2.\n\\]\n$M$ and $aM$ are the mass and angular momentum of\nthe black hole, respectively.\nThere are two Killing vectors reflecting the stationary\nand axisymmetric properties of the Kerr geometry:\n\\begin{equation}\n\\xi_{(t)}^{\\mu}=(1,0,0,0), \\quad\n\\xi_{(\\varphi)}^{\\mu}=(0,0,0,1).\n\\end{equation}\nIn addition, the Kerr spacetime possesses a Killing tensor,\n\\begin{equation}\nK_{\\mu\\nu}=2\\Sigma l_{(\\mu}n_{\\nu)}+r^2g_{\\mu\\nu},\n\\end{equation}\nwhich satisfies $K_{(\\mu\\nu;\\rho)}=0$, where the parenthese\noperating on the indices is the notation for symmetric part of\ntensors. Here we have introduced null vectors,\n\\begin{eqnarray}\nl^{\\mu} &:=&\n\\left(\\frac{r^2+a^2}{\\Delta},1,0,\\frac{a}{\\Delta} \\right), \\quad\nn^{\\mu}:=\n\\left(\\frac{r^2+a^2}{2\\Sigma},-\\frac{\\Delta}{2\\Sigma},\n0,\\frac{a}{2\\Sigma}\\right),\n\\nonumber \\\\\nm^{\\mu} &:=&\n\\frac{1}{\\sqrt{2}(r+ia\\cos\\theta)}\n\\left(ia\\sin\\theta,0,1,\\frac{i}{\\sin\\theta}\\right). \n\\end{eqnarray}\nWe consider a point particle moving in the Kerr geometry:\n\\[\nz^{\\alpha}(\\tau) =\n\\left(t_z(\\tau),r_z(\\tau),\\theta_z(\\tau),\\varphi_z(\\tau)\\right),\n\\]\nwhere $\\tau$ is the proper time along the orbit.\nHere we introduce quantities defined by \n\\begin{eqnarray}\n\\hat{E} &:=&\n-u^{\\alpha}\\xi_{\\alpha}^{(t)}=\n\\left(1-\\frac{2Mr_z}{\\Sigma}\\right)u^t\n+\\frac{2Mar_z\\sin^2\\theta_z}{\\Sigma}u^{\\varphi},\n\\label{eq:Energy} \\\\\n\\hat{L} &:=&\nu^{\\alpha}\\xi_{\\alpha}^{(\\varphi)}=\n-\\frac{2Mar_z\\sin^2\\theta_z}{\\Sigma}u^t\n+\\frac{(r_z^2+a^2)^2-\\Delta a^2\\sin^2\\theta_z}{\\Sigma}\n\\sin^2\\theta_z u^\\varphi, \\label{eq:Momentum} \\\\\n\\hat{Q} &:=&\nK_{\\alpha\\beta}u^{\\alpha}u^{\\beta}\n=\\frac{(\\hat{L}-a\\hat{E}\\sin^2\\theta_z)^2}{\\sin^2\\theta_z}\n+a^2\\cos^2\\theta_z+\\Sigma^2 (u^\\theta)^2, \\label{eq:Carter}\n\\end{eqnarray}\nwhere $u^{\\alpha}=dz^{\\alpha}\/d\\tau$.\nThese quantities remain constant as long as the orbit is a geodesic.\n$\\hat{E}$ and $\\hat{L}$ represent the energy and the (azimuthal) angular\nmomentum per unit mass, respectively.\n$\\hat{Q}$ is called the Carter constant.\nDenoting the mass of a particle by $\\mu$, \nthe energy, the angular momentum and the Carter constant\nof a particle are \n$E\\equiv\\mu\\hat{E}$, $L\\equiv\\mu\\hat{L}$ and $Q\\equiv\\mu^2\\hat{Q}$,\nrespectively.\nAnother notation for the Carter constant defined by \n\\begin{equation}\nC \\equiv\nQ-(aE-L)^2, \\label{eq:Carter2}\n\\end{equation}\nis also convenient since $C$ vanishes for orbits in the \nequatorial plane.\nWe also use $\\hat{C}\\equiv C\/\\mu^2$.\n\nWe can specify an orbit of a particle \nby using three constants of motion,\nthe total energy, angular momentum and Carter constant.\nIntroducing a new parameter $\\lambda$ by \n$d\\lambda=d\\tau\/\\Sigma$,\nthe equations of motion are given as \n\\begin{eqnarray}\n\\frac{dt_z}{d\\lambda} &=&\n-a(a\\hat{E}\\sin^2\\theta_z-\\hat{L})\n+\\frac{r_z^2+a^2}{\\Delta}P(r_z), \\label{eq:eom_t}\\\\\n\\left(\\frac{dr_z}{d\\lambda}\\right)^2 &=&\nR(r_z), \\label{eq:eom_r} \\\\\n\\left(\\frac{d\\cos\\theta_z}{d\\lambda}\\right)^2 &=&\n\\Theta(\\cos\\theta_z), \\label{eq:eom_theta} \\\\\n\\frac{d\\varphi_z}{d\\lambda} &=&\n-\\left(a\\hat{E}-\\frac{\\hat{L}}{\\sin^2\\theta_z}\\right)\n+\\frac{a}{\\Delta}P(r_z), \\label{eq:eom_phi}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nP(r)&:=&\\hat{E}(r^2+a^2)-a\\hat{L}, \\\\\nR(r)&:=&[P(r)]^2-\\Delta[r^2+(a\\hat{E}-\\hat{L})^2+\\hat{C}], \\\\\n\\Theta(\\cos\\theta)&:=&\n\\hat{C} - (\\hat{C}+a^2(1-\\hat{E}^2)+\\hat{L}^2)\\cos^2\\theta\n+ a^2(1-\\hat{E}^2)\\cos^4\\theta.\n\\end{eqnarray}\nIt should be noted that the equations for $r_z$ and\n$\\theta_z$ are completely decoupled by using $\\lambda$. \nMoreover, $R(r)$ and $\\Theta(\\cos\\theta)$\nare quartic functions of $r$ and $\\cos\\theta$, respectively.\n\nWe first consider the radial component of the geodesic equations.\nWhen the radial motion is bounded by the \nminimal and the maximal radii \n$r_{{\\rm min}}$ and $ r_{{\\rm max}}$,\n$r_z(\\lambda)$ becomes a periodic function which satisfies \n$r_z(\\lambda+\\Lambda_r) = r_z(\\lambda)$  \nwith period \n\\begin{equation}\n\\Lambda_r =\n2 \\int_{r_{{\\rm min}}}^{r_{{\\rm max}}} \\frac{dr}{\\sqrt{R(r)}}.\n\\end{equation}\nTherefore, we can expand the radial motion in a Fourier series as\n\\begin{equation}\nr_z(\\lambda) = \\sum_n \\tilde{r}_n e^{-in\\Omega_r\\lambda}\n\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\Omega_r = {2\\pi\/\\Lambda_r}.\n\\end{equation}\n\nWe can deal with the motion in $\\theta$-direction \nin a similar manner.\nWhen the minimum of $\\theta$ is given by \n$\\theta_{{\\rm min}} (\\le \\pi\/2)$, the maximum is \n$\\theta_{{\\rm max}} = \\pi - \\theta_{{\\rm min}}$\nbecause of the symmetry with respect to the equatorial plane.\nAs in the case of the radial motion, \n$\\cos\\theta_z(\\lambda)$ becomes a periodic function which satisfies \n$\\cos\\theta_z(\\lambda+\\Lambda_\\theta) = \\cos\\theta_z(\\lambda)$ \nwith period \n\\begin{equation}\n\\Lambda_\\theta =\n4\\int_0^{\\cos\\theta_{{\\rm min}}}\n\\frac{d(\\cos\\theta)}{\\sqrt{\\Theta(\\cos\\theta)}}.\n\\end{equation}\nWe can expand $\\cos\\theta_z(\\lambda)$ in a Fourier series as\n\\begin{equation}\n\\cos\\theta_z(\\lambda) = \\sum_n \\tilde{z}_n e^{-in\\Omega_\\theta \\lambda}\n\\,,\n\\end{equation}\nwhere $\\Omega_\\theta = {2\\pi\/\\Lambda_\\theta}$.\n\n\nNext, we consider the $t$- and $\\varphi$-components of geodesic equations.\nEqs.~(\\ref{eq:eom_t}) and (\\ref{eq:eom_phi}) can be integrated as\n\\begin{eqnarray}\nt_z(\\lambda)&=&t^{(r)}(\\lambda)+\n   t^{(\\theta)}(\\lambda)+\n   \\left\\langle {dt_z\\over d\\lambda}\\right\\rangle \\lambda, \\\\\n\\varphi_z(\\lambda)&=&\\varphi^{(r)}(\\lambda)+\n   \\varphi^{(\\theta)}(\\lambda)+\n   \\left\\langle {d\\varphi_z\\over d\\lambda}\\right\\rangle \\lambda, \n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\nt^{(r)}(\\lambda) &:=&\n\\int d\\lambda \\left[ \\frac{(r_z^2+a^2)P(r_z)}{\\Delta(r_z)}\n- \\left\\langle \\frac{(r_z^2+a^2)P(r_z)}{\\Delta(r_z)}\n  \\right\\rangle \\right], \\\\\nt^{(\\theta)}(\\lambda) &:=&\n-\\int d\\lambda \\left[ a^2 \\hat{E}\\sin^2\\theta_z - a \\hat{L}\n- \\left\\langle a^2 \\hat{E}\\sin^2\\theta_z - a \\hat{L} \\right\\rangle\n\\right], \\\\\n\\varphi^{(r)}(\\lambda) &:=&\n\\int d\\lambda \\left[ \\frac{aP(r_z)}{\\Delta(r_z)}\n- \\left\\langle \\frac{aP(r_z)}{\\Delta(r_z)} \\right\\rangle\n\\right], \\\\\n\\varphi^{(\\theta)}(\\lambda) &:=&\n\\int d\\lambda \\left[ \\frac{\\hat{L}}{\\sin^2\\theta_z} -a\\hat{E}\n- \\left\\langle \\frac{\\hat{L}}{\\sin^2\\theta_z} -a\\hat{E} \\right\\rangle\n\\right]. \\label{eqs:tphi-motion}\n\\end{eqnarray*}\n$\\langle \\cdots \\rangle$ represents the time average along the  \ngeodesic:\n\\[\n\\left\\langle F(\\lambda) \\right\\rangle\n:=\n\\lim_{T\\to\\infty}\\frac{1}{2T}\\int_{-T}^{T}\nd\\lambda' \\, F(\\lambda').\n\\]\nHere, $t^{(r)}(\\lambda)$ and $\\varphi^{(r)}(\\lambda)$ are\nperiodic functions with period $\\Lambda_r$, while\n$t^{(\\theta)}(\\lambda)$ and $\\varphi^{(\\theta)}(\\lambda)$ are \nthose with period $\\Lambda_\\theta$.\n\n\n\\section{The Time Evolution of the Constants of motion}\n\\label{sec:COMdot}\nIf the timescale of the orbital evolution due to gravitational\nradiation reaction is much longer than the typical dynamical\ntimescale, we may be able to approximate the particle's motion by \nthe geodesic in the background spacetime that is momentarily \ntangential to the orbit \n(osculating geodesic approximation).\nUnder this assumption, we evaluate the change rates of the constants\nof motion at each moment. \nFor bound orbits \nwe can express the change rates of the constants of motion, \n$I^i=\\left\\{E,L,Q\\right\\}$, as \n\\begin{eqnarray}\n{d I^i\\over d\\lambda} \n=\n\\left\\langle \\frac{dI^i}{d\\lambda} \\right\\rangle \n+ \\sum_{(n_r,r_\\theta)\\not=(0,0)}\n      \\dot{I}^{i(n_r,n_\\theta)}\n   \\exp\\left[ - i(n_r\\Omega_r+n_\\theta\\Omega_\\theta) \\lambda\\right]. \n\\label{eq:bareQdot}\n\\end{eqnarray}\nThe first term on the right hand side is a time-independent dissipative\ncontribution due to radiation reaction, while \nthe others are oscillating. \nIntegrating over a long period, the first term \nbecomes dominant. \nIn the same spirit in Ref.~\\citen{Mino:2003yg},\nhere we define the 'adiabatic' evolution as an approximation \nwhich takes account of only the first term. \nNamely, the adiabatic evolution is solely determined by \nthe time averaged change rates of the constants\nof motion.\n\nOwing to the argument given in Ref.~\\citen{Mino:2003yg}, \nwe can evaluate the averaged change rates of the constants of \nmotion by using\nthe radiative field of the metric perturbation\n\\begin{equation}\n \\left\\langle \\frac{dI^i}{d\\lambda} \\right\\rangle =\n\\lim_{T\\to\\infty}\\frac{1}{2T}\\int_{-T}^{T}d\\lambda\\,\\Sigma \n {\\partial I^i\\over\\partial u^\\alpha}\n {f}^{\\alpha}[h_{\\mu\\nu}^{\\rm rad}],\n\\label{eq:meanQdot}\n\\end{equation}\nwhere $h_{\\mu\\nu}^{{\\rm rad}}$ is the radiative part of the\nmetric perturbation defined by half retarded field minus half \nadvanced field, i.e.,  \n$\nh_{\\mu\\nu}^{\\rm rad}:=\n(h_{\\mu\\nu}^{\\rm ret}-h_{\\mu\\nu}^{\\rm adv})\/2. \n$\nRadiative field is a solution of source-free vacuum \nEinstein equation. The singular parts contained in both retarded \nand advanced fields cancel out. \nTherefore we can avoid the tedious issue of regularizing \nthe self-force. ${f}^{\\alpha}$ is a differential operator,\n\\begin{equation}\nf^{\\alpha}[h_{\\mu\\nu}]:=\n-\\frac{1}{2}(g^{\\alpha\\beta}+u^{\\alpha}u^{\\beta})\n(h_{\\beta\\gamma;\\delta}+h_{\\beta\\delta;\\gamma}-h_{\\gamma\\delta;\\beta})\nu^{\\gamma}u^{\\delta}.\n\\end{equation}\nThis operator with its index lowered reduces to \n\\begin{eqnarray}\nf_\\alpha [h_{\\mu\\nu}]\n&=& g_{\\alpha\\beta} f^\\beta [h_{\\mu\\nu}] \\nonumber \\\\\n&=&\n\\frac{1}{2}\\left( \\partial_\\alpha h_{\\gamma\\delta} \\right)\nu^\\gamma u^\\delta\n-\\frac{d}{d\\tau}\\left( h_{\\alpha\\gamma}u^\\gamma \\right)\n-\\frac{1}{2}u_\\alpha \\frac{d}{d\\tau}\n\\left( h_{\\gamma\\delta} u^\\gamma u^\\delta \\right)\n+ O(\\mu^2),\n\\end{eqnarray}\nignoring the second order terms.\n\n\n\\subsection{Calculation of $dE\/dt$ and $dL\/dt$}\nFrom Eq.~(\\ref{eq:meanQdot}), we obtain\n\\begin{eqnarray}\n \\left\\langle \\frac{dE}{d\\lambda} \\right\\rangle &=&\n\\lim_{T\\to\\infty}\\frac{\\mu}{2T}\\int_{-T}^{T}d\\lambda\\,\\Sigma \n\\left( -\\xi_\\alpha^{(t)} \\right)\n {f}^{\\alpha}[h_{\\mu\\nu}^{\\rm rad}] \\nonumber \\\\\n&=&\n\\lim_{T\\to\\infty}\\frac{-\\mu}{2T}\\int_{-T}^{T}d\\lambda\n\\left[ \\frac{\\Sigma}{2}\\left( \\partial_t h_{\\gamma\\delta}^{{\\rm rad}} \\right)\n       u^\\gamma u^\\delta\n      -\\frac{d}{d\\lambda}\\left(\n        h_{t \\gamma}^{{\\rm rad}}u^\\gamma \\right)\n      +\\frac{\\hat{E}}{2}\\frac{d}{d\\lambda}\\left(\n        h_{\\gamma\\delta}^{{\\rm rad}}u^\\gamma u^\\delta \\right)\n\\right] \\nonumber \\\\\n&=&\n\\lim_{T\\to\\infty}\\frac{-\\mu}{2T}\\int_{-T}^{T}d\\lambda\n\\left[ \\frac{\\Sigma}{2}\\left( \\partial_t h_{\\gamma\\delta}^{{\\rm rad}} \\right)\n       u^\\gamma u^\\delta\n\\right]. \\label{eq:Edot-int}\n\\end{eqnarray}\nIn the last equality, \nthe total derivative terms are neglected.\n\nNext, we introduce a vector field\n$\\tilde{u}^\\mu (x)$ by~\\cite{Sago:2005gd}\n\\begin{equation}\n(\\tilde{u}_t, \\tilde{u}_r, \\tilde{u}_\\theta, \\tilde{u}_\\varphi)\n:=\n\\left( -\\hat{E}, \\pm\\frac{\\sqrt{R(r)}}{\\Delta(r)},\n       \\pm\\frac{\\Theta(\\cos\\theta)}{\\sin\\theta}, \\hat{L} \\right).\n\\end{equation}\nThis vector field is a natural extension of the \nfour-velocity of a particle. \nIn fact, it satisfies\n$\\tilde{u}_\\mu (z(\\lambda)) = u_\\mu(\\lambda)$.\n$\\tilde{u}_\\mu$ depends only on $r$ and $\\theta$.\nFurthermore, since $\\tilde{u}_r$ and $\\tilde{u}_\\theta$ depend\nonly on $r$ and $\\theta$, respectively, we have the relation\n$u_{\\mu;\\nu} = u_{\\nu;\\mu}$.\n\nUsing this vector field, we can rewrite Eq.~(\\ref{eq:Edot-int})\nas\n\\begin{equation}\n \\left\\langle \\frac{dE}{d\\lambda} \\right\\rangle =\n\\lim_{T\\to\\infty}\\frac{-\\mu}{2T}\\int_{-T}^{T}d\\lambda \\left[\n\\partial_t \\left(\n\\frac{\\Sigma}{2}h_{\\gamma\\delta}^{{\\rm rad}}\\tilde{u}^\\gamma\\tilde{u}^\\delta\n\\right) \\right]_{x\\to z(\\lambda)},\n\\label{eq:dEdl}\n\\end{equation}\nwhere we used the fact that $\\Sigma$ and $\\tilde{u}_\\mu$\nare independent of $t$ (and $\\varphi$).\n\nAs shown in Appendix~\\ref{sec:radiative}\n(Eq.~(\\ref{hrad})),\nthe radiative field of metric perturbation is given by\n\\begin{eqnarray}\nh_{\\mu\\nu}^{{\\rm rad}}(x) &=&\n\\mu \\int d\\omega \\sum_{\\ell m} \\frac{1}{2i\\omega^3} \\bigg\\{\n|N_{s}^{{\\rm out}}|^2 {}_s\\Pi_{\\Lambda,\\mu\\nu}^{{\\rm out}}(x)\n\\int d\\lambda \\left[\n\\Sigma {}_s\\bar{\\Pi}_{\\Lambda,\\alpha\\beta}^{{\\rm out}}(z(\\lambda))\nu^{\\alpha}u^{\\beta} \\right]\n\\cr &&\n+ \\frac{\\omega}{k}\n|N_{s}^{{\\rm down}}|^2 {}_s\\Pi_{\\Lambda,\\mu\\nu}^{{\\rm down}}(x)\n\\int d\\lambda \\left[\n\\Sigma {}_s\\bar{\\Pi}_{\\Lambda,\\alpha\\beta}^{{\\rm down}}(z(\\lambda))\nu^{\\alpha}u^{\\beta} \\right] \\bigg\\}\n+ ({\\rm c.c.}),\n\\end{eqnarray}\nwhere $\\Lambda=\\{\\ell m \\omega\\}$,\n$k=\\omega-ma\/2Mr_+$ and $r_+=M+\\sqrt{M^2-a^2}$.\n$ {}_s\\Pi_{\\Lambda,\\mu\\nu}^{{\\rm (out)}}(x) $ \nand $ {}_s\\Pi_{\\Lambda,\\mu\\nu}^{{\\rm (down)}}(x) $ \nare the out-going and down-going mode solutions \nfor $h_{\\mu\\nu}$, respectively.\n$N_s^{{\\rm out}}$ and $N_s^{{\\rm down}}$\nare normalization factors, given by\nEqs.~(\\ref{eq:Namp-out}) and (\\ref{eq:Namp-down}). \nA bar represents complex conjugation. \nUsing this formula, we obtain \n\\begin{eqnarray}\n\\psi^{{\\rm rad}}(x) &:=&\n\\frac{1}{2}\\Sigma h_{\\gamma\\delta}^{{\\rm rad}}\n\\tilde{u}^\\gamma \\tilde{u}^\\delta\n\\cr &=&\n\\mu \\int d\\omega \\sum_{\\ell m} \\frac{1}{4i\\omega^3} \\bigg[\n\\phi_\\Lambda^{{\\rm out}}(x)\n\\int d\\lambda' \\bar{\\phi}_\\Lambda^{{\\rm out}}(z(\\lambda'))\n\\cr && \\hspace*{2cm}\n+ \\frac{\\omega}{k} \\phi_\\Lambda^{{\\rm down}}(x)\n\\int d\\lambda' \\bar{\\phi}_\\Lambda^{{\\rm down}}(z(\\lambda'))\n\\bigg] + ({\\rm c.c.}),\n\\label{eq:huu}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\phi_\\Lambda^{{\\rm (out\/down)}}(x) :=\nN_s^{{\\rm (out\/down)}}\n\\Sigma(x) {}_s\\Pi_{\\Lambda,\\gamma\\delta}^{{\\rm (out\/down)}}(x)\n\\tilde{u}^\\gamma(x) \\tilde{u}^\\delta(x).\n\\label{eq:def-phi-out}\n\\end{equation}\nFor a bound orbit, we can expand $\\phi_\\Lambda^{{\\rm out}}$\nin a Fourier series as:\n\\begin{equation}\n\\phi_\\Lambda^{{\\rm (out\/down)}}(z(\\lambda)) =\n\\frac{1}{2\\pi} \\!\n\\left\\langle\\frac{dt_z}{d\\lambda}\\right\\rangle\n\\!\\! \\sum_{n_r, n_\\theta} \\!\n\\bar{\\tilde{Z}}_{\\ell m n_r n_\\theta}^{{\\rm (out\/down)}}(\\omega)\n\\exp\\left[ i\\left\\langle\\frac{dt_z}{d\\lambda}\\right\\rangle\n(\\omega - \\omega_{mn_r n_\\theta})\\lambda \\right],\n\\label{eq:phi-out-ft}\n\\end{equation}\nwhere\n\\begin{equation}\n\\omega_{m n_r n_\\theta} :=\n\\left\\langle\\frac{dt_z}{d\\lambda}\\right\\rangle^{-1}\n\\left( m \\left\\langle\\frac{d\\varphi_z}{d\\lambda}\\right\\rangle\n       + n_r \\Omega_r + n_\\theta \\Omega_\\theta \\right).\n\\label{eq:disc-omega}\n\\end{equation}\nSubstituting Eqs.~(\\ref{eq:huu}) and (\\ref{eq:phi-out-ft})\ninto (\\ref{eq:dEdl}), we obtain:\n\\begin{equation}\n\\left\\langle\n\\frac{dE}{dt}\\right\\rangle_{\\!\\!t} =\n- \\mu^2 \\sum_{\\ell m n_r n_\\theta}\n\\frac{1}{4\\pi\\omega_{m n_r n_\\theta}^2}\n\\left( \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm out}} \\right|^2\n+ \\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n  \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm down}} \\right|^2 \\right), \n\\label{eq:ad-Edot}\n\\end{equation}\nwhere $k_{m n_r n_\\theta}=\\omega_{m n_r n_\\theta}-ma\/2Mr_+$,\n$\n\\langle F(t) \\rangle_t :=\n\\lim_{T\\to\\infty}\\frac{1}{2T}\\int_{-T}^{T} dt \\, F(t)\n$, and\n\\begin{equation}\nZ_{\\ell m n_r n_\\theta}^{{\\rm (out\/down)}} \\equiv\n\\tilde{Z}_{\\ell m n_r n_\\theta}^{{\\rm (out\/down)}}(\\omega_{m n_r n_\\theta}).\n\\label{eq:disc-Z}\n\\end{equation}\nIn a similar manner, the formula for the loss rate of the angular\nmomentum is given by \n\\begin{equation}\n\\left\\langle\\frac{dL}{dt}\\right\\rangle_{\\!\\!t} =\n- \\mu^2 \\sum_{\\ell m n_r n_\\theta}\n\\frac{m}{4\\pi\\omega_{m n_r n_\\theta}^3} \\left(\n\\left|Z_{\\ell m n_r n_\\theta}^{{\\rm out}} \\right|^2\n+ \\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n  \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm down}} \\right|^2 \\right).\n\\label{eq:ad-Ldot}\n\\end{equation}\n\n\n\\subsection{Calculation of $dQ\/dt$}\nTo obtain the change rate of the Carter constant, \nwe need to evaluate \n\\begin{equation}\n \\left\\langle \\frac{dQ}{d\\lambda} \\right\\rangle =\n\\lim_{T\\to\\infty}\\frac{\\mu^2}{2T}\\int_{-T}^{T}d\\lambda\n2\\Sigma K_{\\beta}^{\\alpha}u^{\\beta}\n{f}_{\\alpha}[h_{\\mu\\nu}^{\\rm rad}].\n\\end{equation}\nUsing the vector field $\\tilde{u}^{\\alpha}(x)$,\nwhich was introduced in the previous subsection,\nwe obtain \n\\begin{eqnarray}\n2 K_\\beta^\\alpha u^\\beta f_\\alpha &=&\n\\lim_{x\\rightarrow z}\\left[\nK_\\beta^\\alpha \\tilde{u}^\\beta \\partial_\\alpha\n(h_{\\gamma\\delta}\\tilde{u}^\\gamma \\tilde{u}^\\delta)\n+2h_{\\gamma\\delta}\\tilde{u}^\\beta \\tilde{u}^\\gamma\n(K^\\delta_{\\beta;\\alpha}\\tilde{u}^\\alpha\n -K^\\alpha_\\beta \\tilde{u}^\\delta_{;\\alpha})\n\\right],\n\\end{eqnarray}\nto the first order in perturbation, excluding \ntotal derivative terms with respect to $\\tau$. \nThose total derivative terms do not contribute after\ntaking a long-time average. \nFurthermore, one can show that \nthe second term also vanishes by using \n$K_{(\\alpha\\beta;\\gamma)}=0$ and \n$\\tilde{u}_{\\alpha;\\beta}=\\tilde{u}_{\\beta;\\alpha}$.\nAfter all, we find \n\\begin{eqnarray}\n \\left\\langle \\frac{dQ}{d\\lambda} \\right\\rangle &=&\n\\lim_{T\\to\\infty}\\frac{\\mu^2}{2T}\\int_{-T}^{T}d\\lambda\n\\left[\n2 \\Sigma K_\\beta^\\alpha \\tilde{u}^\\beta \\partial_\\alpha\n\\left( \\frac{\\psi^{{\\rm rad}}(x)}{\\Sigma} \\right)\n\\right]_{x\\to z(\\lambda)} \\nonumber \\\\\n& = &\n\\lim_{T\\to\\infty}\\frac{-\\mu^2}{T}\\int_{-T}^{T}d\\lambda\n\\nonumber \\\\ && \\times\n\\left[\\left\\{\n {P(r)\\over \\Delta}\\left(\n  (r^2+a^2)\\partial_t+a\\partial_{\\varphi}\\right)\n + {dr_z\\over d\\lambda}\\partial_r\n\\right\\} \\psi^{{\\rm rad}}(x)\\right]_{x\\to z(\\lambda)}~~.\n\\label{eq:Qdot-form1}\n\\end{eqnarray}\nTo obtain the last term in the last line, \nthe term with $\\tilde u^\\mu\\partial_\\mu$ was rewritten \ninto $\\Sigma^{-1}{d\/d\\lambda}$, and integration by parts  \nwas applied. \n\nSubstituting Eqs.~(\\ref{eq:huu}) and (\\ref{eq:phi-out-ft})\ninto Eq.~(\\ref{eq:Qdot-form1}), we obtain:\n\\begin{eqnarray}\n \\left\\langle \\frac{dQ}{d\\lambda} \\right\\rangle &=&\n\\lim_{T\\to\\infty}\\frac{-\\mu^3}{2T} \\int_{-T}^{T}d\\lambda\n\\int d\\omega \\sum_{\\ell m n_r n_\\theta}\n\\frac{1}{2i\\omega^3}\\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\cr && \\hspace*{-5mm}\n\\times \\bigg[\nZ_{\\ell m n_r n_\\theta}^{{\\rm out}}\n\\bigg\\{ \\frac{P(r)}{\\Delta}\n\\big( (r^2+a^2)\\partial_t + a\\partial_\\varphi \\big)\n+\\frac{dr_z}{d\\lambda}\\partial_r \\bigg\\}\n\\phi_\\Lambda^{{\\rm out}}(x)\n\\cr && \\hspace*{-2mm}\n+ \\frac{\\omega}{k}\nZ_{\\ell m n_r n_\\theta}^{{\\rm down}}\n\\bigg\\{ \\frac{P(r)}{\\Delta}\n\\big( (r^2+a^2)\\partial_t + a\\partial_\\varphi \\big)\n+\\frac{dr_z}{d\\lambda}\\partial_r \\bigg\\}\n\\phi_\\Lambda^{{\\rm down}}(x) \\bigg]_{x\\to z(\\lambda)}\n\\nonumber \\\\ && \\hspace*{-2mm} + ({\\rm c.c.}).\n\\label{eq:Qdot-form2}\n\\end{eqnarray}\nNow we focus on the $r$-derivative term in the curly brackets.\nSince $\\phi_\\Lambda^{{\\rm out}}$ and $\\phi_\\Lambda^{{\\rm down}}$\ndepend on $t$ and $\\varphi$ only through an exponential function\n$e^{-i\\omega t + im\\varphi}$, we can write \n\\begin{eqnarray}\n&& \\hspace*{-1.5cm}\n\\phi_\\Lambda(z(\\lambda))\n\\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\nonumber \\\\ &&\n= f(r_z(\\lambda), \\cos\\theta_z(\\lambda))\n\\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\cr && \\hspace*{5mm} \\times\n\\exp\\bigg[-i\\omega_{m n_r n_\\theta} \\left(\n\\left\\langle \\frac{dt_z}{d\\lambda} \\right\\rangle \\lambda\n+ t^{(r)}(\\lambda) + t^{(\\theta)}(\\lambda) \\right)\n\\cr && \\hspace*{2.5cm}\n+ im\\left( \\left\\langle \\frac{d\\varphi_z}{d\\lambda} \\right\\rangle \\lambda\n+ \\varphi^{(r)}(\\lambda) + \\varphi^{(\\theta)}(\\lambda) \\right) \\bigg]\n\\nonumber \\\\ &&\n= f(r_z(\\lambda), \\cos\\theta_z(\\lambda))\n\\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\cr && \\hspace*{5mm} \\times\n\\exp\\Big[ - in_r\\Omega_r\\lambda\n          - i\\omega_{m n_r n_\\theta} t^{(r)}(\\lambda)\n\t  + im\\varphi^{(r)}(\\lambda)\n\\cr && \\hspace*{2.5cm}\n          - in_\\theta\\Omega_\\theta\\lambda\n          - i\\omega_{m n_r n_\\theta} t^{(\\theta)}(\\lambda)\n\t  + im\\varphi^{(\\theta)}(\\lambda)\n\\Big],\n\\end{eqnarray}\nwhere $f(r,\\cos\\theta)$ represents the \ndependence on $r$ and $\\cos\\theta$ in \n$\\phi_\\Lambda(x)$. \n$r_z(\\lambda)$, $t^{(r)}(\\lambda)$ and $\\varphi^{(r)}(\\lambda)$\nare periodic functions with period $\\Lambda_r$, while\n$\\theta_z(\\lambda)$, $t^{(\\theta)}(\\lambda)$ and\n$\\varphi^{(\\theta)}(\\lambda)$ are \nthose with period $\\Lambda_\\theta$.\nWe introduce two different time variables $\\lambda_r$\nand $\\lambda_\\theta$. We use them instead of $\\lambda$ for functions\nwith period $\\Lambda_r$ and $\\Lambda_\\theta$.\nThen, by using these new variables, we can replace \nthe infinitely long time average with a double integral\nover a finite region:\n\\begin{eqnarray}\n&& \\hspace*{-1cm}\n\\lim_{T\\to\\infty}\\frac{1}{2T}\\int_{-T}^{T} \\!\\! d\\lambda\n\\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\frac{dr_z}{d\\lambda}\n\\partial_r \\phi_\\Lambda(z(\\lambda))\n\\nonumber \\\\\n&&\n= \\frac{1}{\\Lambda_r\\Lambda_\\theta}\n\\int_0^{\\Lambda_r} \\!\\!\\! d\\lambda_r\n\\int_0^{\\Lambda_\\theta} \\!\\!\\! d\\lambda_\\theta\n\\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\frac{dr_z}{d\\lambda_r} \\partial_r\\bigg\\{\nf(r_z(\\lambda_r), \\cos\\theta_z(\\lambda_\\theta))\n\\cr && \\hspace*{2cm} \\times\n\\exp\\bigg[ - in_r\\Omega_r\\lambda_r\n          - i\\omega_{m n_r n_\\theta} t^{(r)}(\\lambda_r)\n          + im\\varphi^{(r)}(\\lambda_r) \n\\cr && \\hspace*{3.5cm}\n          - i n_\\theta\\Omega_\\theta\\lambda_\\theta\n          - i \\omega_{m n_r n_\\theta} t^{(\\theta)}(\\lambda_\\theta)\n          + i m \\varphi^{(\\theta)}(\\lambda_\\theta)\n\\bigg]\\bigg\\}.\n\\label{3.22}\n\\end{eqnarray}\nWe only need to integrate over one cycle for each of $\\lambda_r$ and \n$\\lambda_\\theta$. Using the relation\n\\begin{eqnarray*}\n&& \\hspace*{-1cm}\n\\frac{d}{d\\lambda_r}\\left\\{\nf(r_z(\\lambda_r), \\cos\\theta_z(\\lambda_\\theta))\n\\exp[ - in_r\\Omega_r\\lambda_r\n          - i\\omega_{m n_r n_\\theta} t^{(r)}\n          + im\\varphi^{(r)} ] \n\\right\\}\n\\nonumber \\\\ &&\n= \\bigg[ \\frac{dt^{(r)}}{d\\lambda_r}\\partial_t\n     + \\frac{dr_z}{d\\lambda_r}\\partial_r\n     + \\frac{d\\varphi^{(r)}}{d\\lambda_r}\\partial_\\varphi\n     + \\partial_{\\lambda_r} \\bigg]\n\\cr && \\hspace*{5mm} \\times\nf(r_z(\\lambda_r), \\cos\\theta_z(\\lambda_\\theta))\n\\exp\\big[ - in_r\\Omega_r\\lambda_r\n          - i\\omega_{m n_r n_\\theta} t^{(r)}\n          + im\\varphi^{(r)} \\big],\n\\end{eqnarray*}\n$\\lambda_r$-integral in (\\ref{3.22}) can be rewritten as \n\\begin{eqnarray}\n&& \\hspace*{-15mm} \n\\int_0^{\\Lambda_r} \\!\\!\\! d\\lambda_r\n\\frac{dr_z}{d\\lambda_r}\\partial_r\\big\\{\nf(r_z(\\lambda_r), \\cos\\theta_z(\\lambda_\\theta))\n\\cr && \\hspace*{1cm} \\times\n\\exp\\big[ - i n_r \\Omega_r \\lambda_r\n          - i \\omega_{m n_r n_\\theta} t^{(r)}\n           + i m \\varphi^{(r)}\n\\big] \\big\\}\n\\nonumber \\\\ &=&\n\\int_0^{\\Lambda_r} \\!\\!\\! d\\lambda_r \\bigg[\n- \\frac{dt^{(r)}}{d\\lambda_r}\\partial_t\n- \\frac{d\\varphi^{(r)}}{d\\lambda_r}\\partial_\\varphi\n+ i n_r \\Omega_r \\lambda_r\n\\bigg]\n\\nonumber \\\\\n&& \\times \\big\\{\nf(r_z(\\lambda_r), \\cos\\theta_z(\\lambda_r))\n\\exp\\big[ - i n_r \\Omega_r \\lambda_r\n           - i \\omega_{m n_r n_\\theta} t^{(r)}\n           + i m \\varphi^{(r)}\n\\big] \\big\\} .\n\\end{eqnarray}\nEliminating the $r$-derivative term from (\\ref{eq:Qdot-form2}) \nby using the above relations,\nwe obtain \n\\begin{eqnarray}\n\\hspace*{-5mm}\\left\\langle {dQ\\over d\\lambda}\\right\\rangle\n& = & \n\\lim_{T\\to\\infty}\\frac{-\\mu^3}{2T} \\int_{-T}^T \\!\\!\\! d\\lambda\n\\int d\\omega \\!\\!\\! \\sum_{\\ell m n_r n_\\theta}\n\\!\\! \\frac{1}{2i\\omega^3} \\delta(\\omega-\\omega_{m n_r n_\\theta})\n\\cr &&\n\\times \\bigg[\nZ_{\\ell m n_r n_\\theta}^{{\\rm out}} \\left\\{\n\\left\\langle {(r^2+a^2)P\\over \\Delta}\\right\\rangle \\partial_t\n+ \\left\\langle {a P\\over \\Delta}\\right\\rangle \\partial_\\varphi\n+ i n_r\\Omega_r \\right\\} \\phi_\\Lambda^{{\\rm out}}(x)\n\\cr &&\n\\quad + \\frac{\\omega}{k}\nZ_{\\ell m n_r n_\\theta}^{{\\rm down}} \\left\\{\n\\left\\langle {(r^2+a^2)P\\over \\Delta}\\right\\rangle \\partial_t\n+ \\left\\langle {a P\\over \\Delta}\\right\\rangle \\partial_\\varphi\n+ i n_r\\Omega_r \\right\\} \\phi_\\Lambda^{{\\rm down}}(x)\n\\bigg]_{x\\to z(\\lambda)}\n\\cr && \\quad+ ({\\rm c.c.})\n\\nonumber \\\\\n&=&\n- 2 \\mu^3 \\left\\langle \\frac{dt_z}{d\\lambda} \\right\\rangle\n\\!\\! \\sum_{\\ell m n_r n_\\theta} \\!\\!\\!\n\\frac{1}{4\\pi\\omega_{m n_r n_\\theta}^2}\n\\cr && \\hspace*{1cm} \\times\n\\bigg[ - \\left\\langle {(r^2+a^2)P\\over \\Delta}\\right\\rangle\n       + \\frac{m}{\\omega_{m n_r n_\\theta}}\n         \\left\\langle {a P\\over \\Delta}\\right\\rangle\n       + \\frac{n_r \\Omega_r}{\\omega_{m n_r n_\\theta}}\n\\bigg]\n\\cr && \\hspace*{1cm}\n\\times \\left(\n|Z_{\\ell m n_r n_\\theta}^{{\\rm out}}|^2\n+\\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n|Z_{\\ell m n_r n_\\theta}^{{\\rm down}}|^2\n\\right).\n\\end{eqnarray}\nHere we used Eqs.~(\\ref{eq:ad-Edot}) and (\\ref{eq:ad-Ldot})\nin the last equality. Finally, we obtain:\n\\begin{eqnarray}\n\\left\\langle {dQ\\over dt}\\right\\rangle_{\\!\\!t}\n& = & \n2\\mu\\left\\langle {(r^2+a^2)P\\over \\Delta}\\right\\rangle\n \\left\\langle{dE\\over dt}\\right\\rangle_{\\!\\!t}\n-2\\mu\\left\\langle {a P\\over \\Delta}\\right\\rangle\n  \\left\\langle{dL\\over dt}\\right\\rangle_{\\!\\!t}\n\\cr &&\n+ \\mu^3 \\!\\!\\! \\sum_{\\ell m n_r n_\\theta} \\!\\!\\!\n  \\frac{n_r \\Omega_r}{2 \\pi \\omega_{m n_r,n_\\theta}^3}\n\\bigg(\n|Z_{\\ell m n_r n_\\theta}^{{\\rm out}}|^2\n+\\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n|Z_{\\ell m n_r n_\\theta}^{{\\rm down}}|^2\n\\bigg).\n\\label{eq:ad-Qdot}\n\\end{eqnarray}\n\n\n\\subsection{Consistency of our formulae in simple cases}\nIn this subsection, we examine our formulae in a few simple cases.\nFirst, we consider circular orbits. \nWe know that a circular orbit remains circular\nunder radiation \nreaction\\cite{Kennefick:1995za}. \nThis condition fixes $dQ\/dt$ for circular orbits as \n\\begin{equation}\n\\frac{dQ}{dt} =\n\\frac{2\\mu(r^2+a^2)P}{\\Delta} \\frac{dE}{dt}\n- \\frac{2\\mu aP}{\\Delta} \\frac{dL}{dt}.\n\\label{eq:check-cir}\n\\end{equation}\nSince we have $Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}=0$\nfor $n_r\\ne 0$ in the case of a circular orbit,\nthe last term in Eq.~(\\ref{eq:ad-Qdot}) vanishes. \nThus Eq.~(\\ref{eq:ad-Qdot}) is consistent with the above condition\nthat a circular orbit remains circular. \n\nNext, we consider orbits in the equatorial plane.\nAn orbit in the equatorial plane should not \nleave the plane by symmetry. This can be \nconfirmed by rewriting the above formula in terms of $C$. \nFrom the definition of $\\omega_{m n_r n_\\theta}$\n(\\ref{eq:disc-omega}), we obtain the following identity:\n\\begin{eqnarray*}\n&& \\hspace*{-1cm}\n\\mu^2\\!\\!\\sum_{\\ell m n_r n_\\theta}\n\\!\\! \\frac{n_r \\Omega_r}{4\\pi\\omega_{m n_r n_\\theta}^3}\n\\left(\n\\left|Z_{\\ell m n_r n_\\theta}^{{\\rm out}} \\right|^2\n+ \\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n  \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm down}} \\right|^2 \\right) \\\\\n&=&\n\\mu^2\\!\\!\\sum_{\\ell m n_r n_\\theta}\n\\!\\! \\frac{1}{4\\pi\\omega_{m n_r n_\\theta}^2} \\left(\n\\left\\langle \\frac{dt_z}{d\\lambda} \\right\\rangle\n-\\frac{m}{\\omega_{m n_r n_\\theta}}\n\\left\\langle \\frac{d\\varphi_z}{d\\lambda} \\right\\rangle\n-\\frac{n_\\theta \\Omega_\\theta}{4\\pi\\omega_{m n_r n_\\theta}}\n\\right)\n\\cr && \\hspace*{2.5cm} \\times \\left(\n\\left|Z_{\\ell m n_r n_\\theta}^{{\\rm out}} \\right|^2\n+ \\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n  \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm down}} \\right|^2 \\right) \\\\\n&=&\n-\\left\\langle \\frac{dt_z}{d\\lambda} \\right\\rangle\n\\left\\langle \\frac{dE}{dt} \\right\\rangle_{\\!\\!t}\n+\\left\\langle \\frac{d\\varphi_z}{d\\lambda} \\right\\rangle\n\\left\\langle \\frac{dL}{dt} \\right\\rangle_{\\!\\!t}\n\\cr && \\hspace*{1cm}\n- \\mu^2 \\!\\! \\sum_{\\ell m n_r n_\\theta}\n\\!\\! \\frac{n_\\theta \\Omega_\\theta}{4\\pi\\omega_{m n_r n_\\theta}^3}\n\\left(\n\\left|Z_{\\ell m n_r n_\\theta}^{{\\rm out}} \\right|^2\n+ \\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n  \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm down}} \\right|^2 \\right),\n\\end{eqnarray*}\nwhere we used the the expressions of\n$\\langle{dE\/dt}\\rangle_t$ and $\\langle{dL\/dt}\\rangle_t$ \ngiven in Eqs.~(\\ref{eq:ad-Edot}) and (\\ref{eq:ad-Ldot}).\nUsing this identity, we have\n\\begin{eqnarray}\n\\left\\langle {dC\\over dt}\\right\\rangle_{\\!\\!t}\n& = &\n\\left\\langle {dQ\\over dt}\\right\\rangle_{\\!\\!t}\n-2(aE-L)\\left(\na\\left\\langle {dE\\over dt}\\right\\rangle_{\\!\\!t}\n-\\left\\langle {dL\\over dt}\\right\\rangle_{\\!\\!t}\n\\right)\n\\nonumber \\\\ &=&\n- 2\\left\\langle a^2 E \\cos^2\\theta_z\\right\\rangle\n   \\left\\langle{dE\\over dt}\\right\\rangle_{\\!\\!t}\n+ 2\\left\\langle {L \\cot^2\\theta_z}\\right\\rangle\n            \\left\\langle{dL\\over dt}\\right\\rangle_{\\!\\!t}\n\\cr &&\n-\\mu^3 \\!\\!\\!\\!\\!\\sum_{\\ell,m,n_r,n_\\theta}\\!\\!\\!\\!\\!\n      {n_\\theta\\Omega_\\theta\\over 2\\pi\\omega^3_{m n_r n_\\theta}} \n\\left(\n\\left|Z_{\\ell m n_r n_\\theta}^{{\\rm out}} \\right|^2\n+ \\frac{\\omega_{m n_r n_\\theta}}{k_{m n_r n_\\theta}}\n  \\left|Z_{\\ell m n_r n_\\theta}^{{\\rm down}} \\right|^2 \\right),\n\\end{eqnarray}\nwhere we have used the following relations:\n\\begin{eqnarray*}\n\\left\\langle \\frac{dt_z}{d\\lambda} \\right\\rangle &=&\n-a(a\\hat{E}-\\hat{L})\n+ \\left\\langle a^2 \\hat{E} \\cos^2\\theta_z \\right\\rangle\n+ \\left\\langle \\frac{r_z^2+a^2}{\\Delta}P \\right\\rangle, \\\\\n\\left\\langle \\frac{d\\varphi_z}{d\\lambda} \\right\\rangle &=&\n-a\\hat{E} + \\hat{L}\n+ \\left\\langle \\hat{L} \\cot^2\\theta_z \\right\\rangle\n+ \\left\\langle \\frac{aP}{\\Delta} \\right\\rangle.\n\\end{eqnarray*}\nFrom this equation, it is found\nthat $\\langle dC\/dt\\rangle_t=0$ when $\\theta=\\pi\/2$.\nNote that we have\n$Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}} \\ne 0$\nonly for $n_\\theta=0$ in  the case of equatorial orbits. \n\n\n\n\\section{Application of our formulation to orbits with\nsmall eccentricity and inclination} \\label{sec:example}\nIn this section, as an application of our formulation,\nwe consider a slightly eccentric orbit with\nsmall inclination from the equatorial plane.\nSince, in this case, we can expand an orbit with respect to\nthe eccentricity and inclination, we can analytically \ncalculate the change rates of the constants of motion.\n\n\\subsection{Orbits}\nHere we define $r_0$ so that the potential in $r$-direction \n$R(r)$ takes its minimum at $r=r_0$:\n\\begin{equation}\n\\left. \\frac{dR}{dr}\\right|_{r=r_0}=0. \\label{eq:min-condition}\n\\end{equation}\nWe denote the outer turning point by $r_0(1+e)$. \nNamely, \n\\begin{equation}\nR(r_0(1+e))=0, \\label{eq:turn-condition}\n\\end{equation}\nwhich gives the definition of the eccentricity $e$. \nWe also define a parameter $y=C\/L^2$, which is \nrelated to the inclination angle. For orbits in the equatorial \nplane, we have $y=0$.  \nFurther, we introduce a new parameter $v=\\sqrt{M\/r_0}$.\nFor circular orbits \n$v$ represents the orbital velocity at the Newtonian order. \nHence, we regard $v$ as a parameter whose power indicates \ntwice the post-Newtonian (PN) order.\n\nSolving (\\ref{eq:min-condition}) and\n(\\ref{eq:turn-condition}) for $\\hat{E}$ and $\\hat{L}$, \nthey are expressed in terms of \n$e$ and $y$ as \n\\begin{eqnarray}\n\\hat{E} &=&\n1-\\frac{1}{2} v^2 + \\frac{3}{8} v^4 - q v^5\n- \\left( \\frac{1}{2} v^2 - \\frac{1}{4} v^4 + 2 q v^5 \\right) e^2\n+\\frac{1}{2} q v^5 y + q v^{5} e^2 y,\n\\label{eq:E_exp} \\\\\n\\hat{L} &=& r_0 v \\bigg[\n1 + \\frac{3}{2} v^2 -3 q v^3\n+ \\frac{27}{8} v^4 + q^2 v^4\n- \\frac{15}{2} q v^5\n\\cr && \\hspace*{1cm}\n+ \\Big( - 1 + \\frac{3}{2} v^2 - 6 q v^3\n         + \\frac{81}{8} v^4 + \\frac{7}{2} q^2 v^4\n         - \\frac{63}{2} q v^5 \\Big) e^2\n\\cr && \\hspace*{1cm}\n+ \\Big( -\\frac{1}{2} - \\frac{3}{4} v^2 + 3 q v^3 \n         - \\frac{27}{16} v^4 - \\frac{3}{2} q^2 v^4\n         + \\frac{15}{2} q v^5 \\Big) y\n\\cr && \\hspace*{1cm}\n+ \\Big( \\frac{1}{2} - \\frac{3}{4} v^2 + 6 q v^3\n         - \\frac{81}{16} v^4 - \\frac{19}{4} q^2 v^4\n         + \\frac{63}{2} q v^5 \\Big) e^2 y\n\\bigg], \n\\label{eq:L_exp}\n\\end{eqnarray}\nwhere $q:=a\/M$.\nHereafter we keep terms up to $O(v^5 e^2 y)$\nrelative to the leading order.\n\nWith the initial condition set to $r_z(\\lambda=0)=r_0(1+e)$,\nthe solution for $r_z(\\lambda)$ is obtained in an expansion \nwith respect to $e$ as \n\\begin{eqnarray}\nr_z(\\lambda) &=&\nr_0[1+er^{(1)}+e^2r^{(2)}],\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nr^{(1)} &=& \\cos\\Omega_r\\lambda, \\nonumber \\\\\nr^{(2)} &=&\np^{(1)} (1-\\cos\\Omega_r\\lambda)\n+ p^{(2)} (1-\\cos 2\\Omega_r\\lambda), \\nonumber \\\\\n\\Omega_r &=& r_0 v \\bigg[\n1 - \\frac{3}{2} v^2 + 3 q v^3\n- \\frac{45}{8} v^4 - \\frac{3}{2} q^2 v^4\n+ \\frac{33}{2} q v^5\n\\cr && \\hspace*{5mm}\n- \\bigg\\{ 1 + \\frac{3}{2} v^2 - 6 q v^3\n          + \\Big( \\frac{165}{8} + \\frac{9}{2} q^2 \\Big) v^4\n          - \\frac{165}{2} q v^5 \\bigg\\} e^2\n\\cr && \\hspace*{5mm}\n- \\Big( \\frac{3}{2} q v^3 - 2 q^2 v^4\n        + \\frac{33}{4} q v^5 \\Big) y\n- \\Big( 3 q v^3 - \\frac{27}{4} q^2 v^4 \n         + \\frac{165}{4} q v^5 \\Big) e^2 y\n\\bigg], \\cr\np^{(1)} &=&\n - 1 - v^2 + 2 q v^3 -6 v^4 - q^2 v^4 + 20 q v^5 \n- \\left( q v^3 - 2 q^2 v^4 + 10 q v^5 \\right) y, \\cr\np^{(2)} &=&\n- \\frac{1}{2} - \\frac{1}{2} v^2 + q v^3 - 3 v^4\n- \\frac{1}{2} q^2 v^4 + 10 q v^5\n- \\Big( \\frac{1}{2} q v^3 - q^2 v^4 + 5 q v^5 \\Big) y.\n\\nonumber\n\\end{eqnarray}\n\nWe also compute $\\cos\\theta_z(\\lambda)$ in a series expansion \nin $y$ as\n\\begin{equation}\n\\cos\\theta_z(\\lambda) =\n\\sqrt{y}[ c_z^{(0)}(\\lambda) + y c_z^{(1)}(\\lambda)],\n\\end{equation}\nwhere\n\\begin{eqnarray*}\nc_z^{(0)} &=&\n\\Big( 1 - \\frac{1}{2} q^2 v^4 - \\frac{3}{2}q^2 v^4 e^2 \\Big)\n\\sin\\Omega_\\theta \\lambda, \\\\\nc_z^{(1)} &=&\n\\Big( - \\frac{1}{2} + \\frac{13}{16} q^2 v^4\n       + \\frac {39}{16} q^2 v^4 e^2 \\Big)\n\\sin\\Omega_\\theta \\lambda\n+ \\Big( \\frac{1}{16} q^2 v^4 + \\frac{3}{16} q^2 v^4 e^2 \\Big)\n\\sin 3\\Omega_\\theta \\lambda, \\\\\n\\Omega_\\theta &=& r_0 v \\bigg[\n1 + \\frac{3}{2} v^2 - 3 q v^3 + \\frac{27}{8} v^4\n+ \\frac{3}{2} q^2 v^4 - \\frac{15}{2} q v^5\n\\cr && \\hspace*{5mm}\n+ \\Big( - 1  + \\frac{3}{2} v^2 -6 q v^3\n         + \\frac{81}{8} v^4 + \\frac{9}{2} q^2 v^4\n         - \\frac{63}{2} q v^{5} \\Big) e^2\n\\cr && \\hspace*{5mm}\n+ \\Big( \\frac{3}{2} q v^3 - \\frac{7}{4} q^2 v^4 + \\frac{15}{4} q v^5 \\Big) y\n+ \\Big( 3 q v^3 - \\frac{9}{2} q^2 v^4 + \\frac {63}{4} q v^{5} \\Big) e^2 y\n\\bigg].\n\\end{eqnarray*}\nHere the solution satisfies the condition,\n$\\cos\\theta_z(\\lambda=0) = 0$.\n\nSubstituting $r_z$ and $\\cos\\theta_z$\ninto Eqs.(\\ref{eq:eom_t}), (\\ref{eq:eom_phi})\nand (\\ref{eqs:tphi-motion}), we obtain\n\\begin{eqnarray}\nt^{(r)} &=& \\frac{r_0 e}{v} \\bigg[\n\\Big\\{\n( 2 + 4 v^2 - 6 q v^3 + 17 v^4 + 3 q^2 v^4 - 54 q v^5)\n\\cr && \\hspace*{1cm}\n+ ( 2 + 6 v^2 - 10 q v^3\n  + 33 v^4 + 5 q^2 v^4 - 108 q v^5) e\n\\cr &&  \\hspace*{1cm}\n+ ( 3 q v^3 - 4 q^2 v^4 + 27 q v^5) y\n\\cr &&  \\hspace*{1cm}\n+ ( 5 q v^3 - 8 q^2 v^4 + 54 q v^5) e y\n\\Big\\} \\sin \\Omega_r \\lambda\n\\cr && \\hspace*{1cm}\n+ \\Big\\{\n\\Big( \\frac{3}{4} + \\frac{7}{4} v^2 - \\frac{13}{4} q v^3\n       + \\frac{81}{8} v^4 + \\frac{13}{8} q^2 v^4\n       - \\frac{135}{4} q v^5\n\\Big) e\n\\cr && \\hspace*{1.5cm}\n+ \\Big( \\frac{13}{8} q v^3 - \\frac{5}{2} q^2 v^4\n         + \\frac{135}{8} q v^5 \n\\Big) e y\n\\Big\\} \\sin 2\\Omega_r \\lambda\n\\bigg], \\\\\nt^{(\\theta)} &=& q^2 v^3 r_0 y \\bigg[\n\\Big\\{\n\\Big( -\\frac{1}{4} + \\frac{1}{2} v^2\n       - \\frac{3}{4} q v^3 + \\frac{5}{8} q^2 v^4 + q v^5\n\\Big)\n\\cr && \\hspace*{1.5cm}\n+ \\Big( - \\frac{1}{4} + \\frac{11}{8} v^2 -3 q v^3\n         + \\frac{1}{2} v^4 + \\frac{23}{8} q^2 v^4\n         + \\frac{9}{2} q v^5\n\\Big) e^2\n\\Big\\} \\sin 2\\Omega_\\theta \\lambda\n\\bigg], \\\\\n\\left\\langle {dt_z\\over d\\lambda}\\right\\rangle &=&\nr_0^2 \\Big[\n1 + \\frac{3}{2} v^2 + \\frac{27}{8} v^4 - 3 q v^5\n- \\Big( \\frac{5}{2} + \\frac{21}{4} v^2 - 6 q v^3\n         + \\frac{315}{16} v^4 + 3 q^2 v^4\n         - \\frac{123}{2} q v^5\n\\Big) e^2\n\\cr && \\hspace*{5mm}\n+ \\Big( \\frac{1}{2} q^2 v^4 + \\frac{3}{2} q v^5\n\\Big) y\n+ \\Big( -3 q v^3 + 6 q^2 v^4 - \\frac{123}{4} q v^5\n\\Big) e^2 y \\Big], \\\\\n\\varphi^{(r)} &=&\nq v^3 e \\bigg[ \\Big\\{\n( - 2 + 2 q v - 10 v^2 + 18 q v^{3})\n+ ( - 2 + 2 q v - 12 v^2 + 24 q v^3) e\n\\cr && \\hspace*{3cm}\n- ( q v + 9 q v^3) y \n- ( q v +12 q v^3 ) e y\n\\Big\\} \\sin\\Omega_r \\lambda\n\\cr && \\hspace*{1cm}\n+ \\Big\\{\n\\Big( - \\frac{1}{4} q v + \\frac{1}{2} v^2 - \\frac{3}{4} q v^3 \\Big) e\n+ \\Big( \\frac{1}{8} q v + \\frac{3}{8} q v^3 \\Big) e y\n\\Big\\} \\sin 2\\Omega_r \\lambda \\bigg], \\\\\n\\varphi^{(\\theta)} &=&\ny \\bigg[\n\\Big( - \\frac{1}{4} + \\frac{3}{8} q^2 v^4 \\Big)\n+ \\frac{9}{8} q^2 v^4 e^2 \\bigg]\n\\sin 2\\Omega_\\theta \\lambda, \\\\\n\\left\\langle {d\\varphi_z\\over d\\lambda}\\right\\rangle &=&\nr_0 v \\bigg[\n1 + \\frac{3}{2} v^2 - q v^3\n+ \\frac{27}{8} v^4 - \\frac{9}{2} q v^5\n- \\Big( 1 - \\frac{3}{2} v^2 + 2 q v^3\n       - \\frac{81}{8} v^4 + \\frac{27}{2} q v^5\n\\Big) e^{2}\n\\cr && \\hspace*{1cm}\n+ \\Big(\n\\frac{3}{2} q v^3 - q^2 v^4 + \\frac{15}{4} q v^5\n\\Big) y\n+ \\Big(\n3 q v^3 - \\frac{9}{4} q^2 v^4 + \\frac{63}{4} q v^5\n\\Big) e^{2} y \\bigg].\n\\end{eqnarray}\n\n\n\\subsection{Calculation of $Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}$}\nIn order to obtain the averaged change rates of the energy,\nangular momentum and Carter constant, we have to calculate\n$Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}$ defined by\nEq.~(\\ref{eq:disc-Z}) with Eq.~(\\ref{eq:phi-out-ft}). \nIntegrating Eq.~(\\ref{eq:def-phi-out}) with respect to\n$\\lambda$, we obtain\n\\begin{eqnarray}\n&&\\hspace*{-1cm}\n\\hat{Z}_{\\Lambda}^{{\\rm (out\/down)}}\n\\equiv\n\\int d\\lambda \\bar{\\phi}_\\Lambda^{{\\rm (out\/down)}} (z(\\lambda))\n\\nonumber \\\\\n&=&\nN_s^{{\\rm (out\/down)}} \\int d^4x \\sqrt{-g(x)} \n~{}_s\\bar{\\Pi}_{\\Lambda,\\alpha\\beta}^{{\\rm (out\/down)}}(x)\n\\int d\\tau\n\\frac{\\tilde{u}^\\alpha(x)\\tilde{u}^\\beta(x)}{\\sqrt{-g(x)}}\n\\delta^{(4)}(x-z(\\lambda))\n\\nonumber \\\\\n&=&\n{N_s^{\\rm (out\/down)}\\over \\mu} \\int d^4x \\sqrt{-g(x)}\n~{}_s\\bar{\\Pi}_{\\Lambda,\\alpha\\beta}^{{\\rm (out\/down)}}(x)\nT^{\\alpha\\beta}(x),\n\\end{eqnarray}\nwhere \n\\begin{equation}\nT^{\\alpha\\beta}(x) =\n\\mu \\int d\\tau \\frac{1}{\\sqrt{-g(x)}}\nu^\\alpha u^\\beta \\delta^{(4)}(x-z(\\tau)).\n\\label{ppEM}\n\\end{equation}\nis the energy momentum tensor of a mono-pole particle of mass \n$\\mu$. \nUsing the relation given in Eq.~(\\ref{TT}), \n$\\hat{Z}_{\\Lambda}^{{\\rm (out\/down)}}$ can be also expressed \nin the familiar form which appears as an integration over \nthe source term in the standard Teukolsky formalism as \n\\begin{eqnarray}\n\\hspace*{-5mm}\n\\hat{Z}_{\\Lambda}^{{\\rm (out\/down)}} & = &\n\\frac{N_s^{\\rm (out\/down)}}{\\mu}\\bar{\\zeta}_s \\int d^4x\\sqrt{-g(x)} \n~{}_s R_\\Lambda^{\\rm (in\/up)}(r){}_s\\bar{Z}_\\Lambda(\\theta,\\varphi)\ne^{i\\omega t} {}_s\\hat{T}(x), \n\\label{eq:Zout}\n\\end{eqnarray}\nwhere ${}_s\\hat{T}(x)$ is a projected \nenergy momentum tensor defined by \n${}_s \\hat{T}:={}_s\\tau_{\\mu\\nu} \nT^{\\mu\\nu}$ with (\\ref{taudef}), and  \n$~{}_{s} R_\\Lambda^{\\rm (in\/up)}(r)(={}_{-s}\\bar R_\\Lambda^{\\rm (out\/down)}(r))$  and \n${}_s Z_\\Lambda(\\theta,\\varphi)$\nare, respectively, the radial mode functions and the spheroidal\nharmonics introduced in Appendix~\\ref{sec:radiative}.2.\n\n\nIn the following discussion we concentrate on the case with $s=-2$. \nSubstituting the explicit forms of \nthe energy momentum tensor and the projection operator \n${}_{-2}\\tau_{\\mu\\nu} $, we obtain \n\\begin{eqnarray}\n\\hat{Z}_{\\Lambda}^{{\\rm (out\/down)}}\n= 2 N_s^{{\\rm (out\/down)}}\\bar \\zeta_s \n\\int_{-\\infty}^{\\infty} dt e^{i\\omega t - im\\varphi(t)}\n{\\cal I}_\\Lambda^{{\\rm (in\/up)}}(r(t),\\theta(t)),\n\\end{eqnarray}\nwith\n\\begin{eqnarray*}\n{\\cal I}_\\Lambda &=&\n\\bigg[ R_\\Lambda ( A_{nn0}+A_{\\bar{m}n0}+A_{\\bar{m}\\bar{m}0} )\n\\cr && \\hspace*{1cm}\n-\\frac{dR_\\Lambda}{dr}(A_{\\bar{m}n1}+A_{\\bar{m}\\bar{m}1})\n+\\frac{d^2R_\\Lambda}{dr^2}A_{\\bar{m}\\bar{m}2}\n\\bigg]_{r=r(t),\\theta=\\theta(t)}, \\\\\nA_{nn0}&=& \\frac{-2}{\\sqrt{2\\pi}\\Delta^2}\nC_{nn}\\bar z^{2}z\\mathcal{L}_{1}^{\\dag}\n\\left\\{\\bar z^{4}\\mathcal{L}_2^{\\dagger}(\\bar z^{-3} S_\\Lambda )\\right\\},\\\\\nA_{\\bar{m}n0}&=& \\frac{2}{\\sqrt{\\pi}\\Delta}\nC_{\\bar{m}n}\\bar z^{3}\\bigg[\n\\Big(\\frac{iK}{\\Delta}+z^{-1}+\\bar z^{-1}\\Big)\n\\mathcal{L}^{\\dag}_2 S_\\Lambda\n-\\frac{K}{\\Delta}(z^{-1}-\\bar z^{-1})a\\sin\\theta S_\\Lambda \\bigg],\\\\\nA_{\\bar{m}\\bar{m}0} &=& -\\frac{1}{\\sqrt{2\\pi}}\\bar z^{3}z^{-1}\nC_{\\bar{m}\\bar{m}}S_\\Lambda\\left[-i\\left(\\frac{K}{\\Delta}\\right)_{,r}\n-\\frac{K^2}{\\Delta^2}+{2i\\over\\bar z}\\frac{K}{\\Delta}\\right],\\\\\nA_{\\bar{m}n1}&=& \\frac{2}{\\sqrt{\\pi}\\Delta}\\bar z^{3}\nC_{\\bar{m}n}\\left[\\mathcal{L}^{\\dag}_2S_\\Lambda\n+ia\\sin\\theta(z^{-1}-\\bar z^{-1})S_\\Lambda\\right],\\\\\nA_{\\bar{m}\\bar{m}1}&=& -\\frac{2}{\\sqrt{2\\pi}}\\bar z^{3}z^{-1}\nC_{\\bar{m}\\bar{m}}S_\\Lambda\\left(i\\frac{K}{\\Delta}+\\bar z^{-1}\\right),\\\\\nA_{\\bar{m}\\bar{m}2}&=&-\\frac{1}{\\sqrt{2\\pi}}\\bar z^{3}z^{-1}\nC_{\\bar{m}\\bar{m}}S_\\Lambda, \\\\\nC^{\\mu\\nu}&=&\\frac{u^\\mu u^\\nu}{\\Sigma u^t},\n\\end{eqnarray*}\nwhere $S_\\Lambda$ represents ${}_{-2}S_{\\Lambda}(\\theta)$ defined \nin Appendix.~\\ref{sec:radiative}, and \n\\begin{eqnarray*}\nz &=& r+ia\\cos\\theta, \\\\\nK &=&\n(r^2+a^2)\\omega -ma, \\\\\n{\\cal L}_s &=&\n\\partial_\\theta + \\frac{m}{\\sin\\theta}\n-a\\omega\\sin\\theta + s\\cot\\theta.\n\\end{eqnarray*}\nHere dagger ($\\dag$) means \nan operation that transforms $(m, \\omega)$ to $(-m, -\\omega)$.\nThe radial functions and the spheroidal harmonics appearing\nin the above equations can be evaluated analytically,\nas shown in Appendices~\\ref{sec:MST} and \\ref{sec:spheroidal}.\nFor a bound orbit, since \n$e^{-im\\varphi(t)}{\\cal I}^{\\rm (in\/up)}_\\Lambda(r(t),\\theta(t))$ \nis a double periodic function, \n$\\hat{Z}_\\Lambda^{\\rm (out\/down)}$ has a discrete spectrum \nas \n\\begin{equation}\n\\hat{Z}_\\Lambda^{{\\rm out\/down}} =\n\\sum_{n_r, n_\\theta} Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}\n\\delta(\\omega - \\omega_{mn_r n_\\theta}),\n\\end{equation}\nwhere the coefficients $Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}$ are\nthose already introduced in Eq.~(\\ref{eq:disc-Z})\nwith Eq.~(\\ref{eq:phi-out-ft}).\nAlthough we cannot show all the processes explicitly here,\nit is straight forward to calculate \n$Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}$\nfor each $\\omega_{m n_r n_\\theta}$\nby substituting the analytic expansions of the orbits,\nthe radial functions and the spheroidal harmonics.\n\n\n\\subsection{Results}\nSubstituting $Z_{\\ell m n_r n_\\theta}^{{\\rm out\/down}}$\nobtained by following the scheme explained \nin the preceding subsection into\nEqs.~(\\ref{eq:ad-Edot}), (\\ref{eq:ad-Ldot}) and (\\ref{eq:ad-Qdot}),\nwe obtain:\n\\begin{eqnarray}\n\\left\\langle \\frac{dE}{dt} \\right\\rangle_{\\!\\!t}\n&=&\n- \\frac{32}{5}\\left(\\frac{\\mu}{M}\\right)^2 v^{10} \n\\cr && \\times\n\\bigg[\n1 - \\frac{1247}{336} v^2\n- \\bigg( \\frac{73}{12} q - 4 \\pi \\bigg) v^3\n\\cr && \\hspace*{5mm}\n- \\bigg( \\frac{44711}{9072} - \\frac{33}{16} q^2 \\bigg) v^4\n+ \\bigg( \\frac{3749}{336} q - \\frac{8191}{672} \\pi \\bigg) v^5\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ \\frac{277}{24} - \\frac{4001}{84} v^2\n         + \\bigg( \\frac{3583}{48} \\pi - \\frac{457}{4} q \\bigg) v^3\n\\cr && \\hspace*{1cm}\n         + \\bigg( 42 q^2 - \\frac{1091291}{9072} \\bigg) v^4\n         + \\bigg( \\frac{58487}{672} q - \\frac{364337}{1344} \\pi \\bigg) v^5\n\\bigg\\} e^2\n\\cr && \\hspace*{5mm}\n+ \\bigg( \\frac {73}{24} q v^3 - \\frac{527}{96} q^2 v^4\n         - \\frac{3749}{672} q v^5 \\bigg) y\n\\cr && \\hspace*{5mm}\n+ \\bigg( \\frac{457}{8} q v^3 - \\frac{5407}{48} q^2 v^4\n         - \\frac{58487}{1344} q v^5 \\bigg) e^2 y\n\\bigg], \\\\\n\\left\\langle \\frac{dL}{dt} \\right\\rangle_{\\!\\!t}\n&=&\n- \\frac{32}{5}\\left(\\frac{\\mu^2}{M}\\right)  v^{7}\n\\cr && \\times\n\\bigg[\n1 - \\frac{1247}{336} v^2\n- \\bigg( \\frac{61}{12} q - 4 \\pi \\bigg) v^3\n\\cr && \\hspace*{5mm}\n- \\bigg( \\frac{44711}{9072}\n- \\frac{33}{16} q^2 \\bigg) v^4\n+ \\bigg( \\frac{417}{56} q - \\frac{8191}{672} \\pi \\bigg) v^5\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ \\frac{51}{8} - \\frac{17203}{672} v^2\n+ \\bigg( - \\frac{781}{12} q + \\frac{369}{8} \\pi \\bigg) v^3\n\\cr && \\hspace*{1cm}\n+ \\bigg( \\frac{929}{32} q^2 - \\frac{1680185}{18144} \\bigg) v^4\n+ \\bigg( \\frac{1809}{224} q - \\frac{48373}{336} \\pi \\bigg) v^5\n\\bigg\\} e^2\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ - \\frac{1}{2} + \\frac{1247}{672} v^2\n+ \\bigg( \\frac{61}{8} q - 2 \\pi \\bigg) v^3\n\\cr && \\hspace*{1cm}\n- \\bigg( \\frac{213}{32} q^2 - \\frac{44711}{18144} \\bigg) v^4\n- \\bigg( \\frac{4301}{224} q - \\frac{8191}{1344} \\pi \\bigg) v^5\n\\bigg\\} y\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ - \\frac{51}{16} + \\frac{17203}{1344} v^2\n+ \\bigg( \\frac{1513}{16} q - \\frac{369}{16} \\pi \\bigg) v^3\n\\cr && \\hspace*{1cm}\n+ \\bigg( \\frac{1680185}{36288} - \\frac{5981}{64} q^2 \\bigg) v^4\n- \\bigg( 168 q - \\frac{48373}{672} \\pi \\bigg) v^5\n\\bigg\\} e^2 y\n\\bigg], \\\\\n\\left\\langle \\frac{dQ}{dt} \\right\\rangle_{\\!\\!t}\n&=&\n- \\frac{64}{5} \\mu^3 v^{6}\n\\cr && \\times\n\\bigg[\n1 - q v - \\frac{743}{336} v^2\n- \\bigg( \\frac{1637}{336} q - 4 \\pi \\bigg) v^3\n\\cr && \\hspace*{5mm}\n+ \\bigg( \\frac{439}{48} q^2 - \\frac{129193}{18144} - 4 \\pi q \\bigg) v^4\n+ \\bigg( \\frac{151765}{18144} q - \\frac{4159}{672} \\pi\n        - \\frac{33}{16} q^3 \\bigg) v^5\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ \\frac{43}{8} - \\frac{51}{8} q v - \\frac{2425}{224} v^2\n       - \\bigg( \\frac{14869}{224} q - \\frac{337}{8} \\pi \\bigg) v^3\n\\cr && \\hspace*{1cm}\n       - \\bigg( \\frac{453601}{4536} - \\frac{3631}{32} q^2\n                + \\frac{369}{8} \\pi q \\bigg) v^4\n\\cr && \\hspace*{1cm}\n       + \\bigg( \\frac{141049}{9072} q - \\frac{38029}{672} \\pi\n       - \\frac{929}{32} q^3 \\bigg) v^5 \\bigg\\} e^2\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ \\frac{1}{2} q v + \\frac{1637}{672} q v^3\n       - \\bigg( \\frac{1355}{96} q^2 - 2 \\pi q \\bigg) v^4\n\\cr && \\hspace*{1cm}\n       - \\bigg( \\frac{151765}{36288} q - \\frac{213}{32} q^3 \\bigg) v^5\n\\bigg\\} y\n\\cr && \\hspace*{5mm}\n+ \\bigg\\{ \\frac{51}{16} q v + \\frac{14869}{448} q v^3\n       + \\bigg( \\frac{369}{16} \\pi q - \\frac{33257}{192} q^2\n         \\bigg) v^4\n\\cr && \\hspace*{1cm}\n       + \\bigg( - \\frac{141049}{18144} q + \\frac{5981}{64} q^3 \\bigg) v^5\n  \\bigg\\} e^2 y\n\\bigg].\n\\end{eqnarray}\n\nFrom the above results we can compute \n\\begin{eqnarray}\n&& \\hspace*{-1cm}\n\\left\\langle \\frac{dQ}{dt} \\right\\rangle_{\\!\\!t}\n- \\left\\langle \\frac{2\\mu(r^2+a^2)P}{\\Delta}\\right\\rangle\n\\left\\langle \n   \\frac{dE}{dt} \\right\\rangle_{\\!\\!t}\n+ \\left\\langle \\frac{2\\mu aP}{\\Delta}\\right\\rangle\n\\left\\langle \n \\frac{dL}{dt} \\right\\rangle_{\\!\\!t}\n\\nonumber \\\\ &=&\n- \\frac{64}{5} \\mu^3 v^6 e^2 \\bigg[\n- \\frac{37}{6}\n+ \\frac{13435}{672} v^2\n- \\bigg( \\frac{1561}{48} \\pi - \\frac{335}{8} q \\bigg) v^3\n\\cr && \\hspace*{2.2cm}\n+ \\bigg( \\frac{625117}{12096} - \\frac{337}{32} q^2 \\bigg) v^4\n+ \\bigg( \\frac {46827}{448} \\pi - \\frac{1355}{672} q \\bigg) v^5\n\\cr && \\hspace*{2.2cm}\n- \\bigg( \\frac{335}{16} q v^3 - \\frac{7559}{192} q^2 v^4\n         - \\frac{1355}{1344} q v^5\n\\bigg) y\n\\bigg].\n\\end{eqnarray}\nThe left hand side of this equation vanishes for circular orbits\n\\cite{Kennefick:1995za}.\nIn fact, the right hand side vanishes for $e=0$.\nWhen we did not know how to compute $\\langle dQ\/dt\\rangle_t$, \nthe best guess that we could do for $\\langle dQ\/dt\\rangle_t$ \nwas to assume that the left hand side vanishes for general \norbits\\cite{Hughes:1999bq}. \nTherefore this combination represents the errors coming from \nthis hand-waving working hypothesis. \nWe can also compute \n\\begin{eqnarray}\n\\left\\langle \\frac{dC}{dt} \\right\\rangle_{\\!\\!t} &=&\n\\left\\langle \\frac{dQ}{dt} \\right\\rangle_{\\!\\!t}\n-  2(aE - L) \\bigg(\na \\left\\langle \\frac{dE}{dt} \\right\\rangle_{\\!\\!t}\n- \\left\\langle \\frac{dL}{dt} \\right\\rangle_{\\!\\!t}\n\\bigg)\n\\nonumber \\\\ &=&\n- \\frac{64}{5} \\mu^3 v^6 y \\bigg[\n1 - \\frac{743}{336} v^2\n- \\bigg( \\frac{85}{8} q - 4 \\pi \\bigg) v^3\n\\cr && \\hspace*{2.2cm}\n- \\bigg( \\frac{129193}{18144} - \\frac{307}{96} q^2 \\bigg) v^4\n+ \\bigg( \\frac{2553}{224} q - \\frac{4159}{672} \\pi \\bigg) v^5\n\\cr && \\hspace*{2.2cm}\n+ \\bigg\\{ \\frac{43}{8} - \\frac{2425}{224} v^2\n+ \\bigg( \\frac{337}{8} \\pi - \\frac{1793}{16} q \\bigg) v^3\n\\cr && \\hspace*{2.7cm}\n- \\bigg( \\frac{453601}{4536} - \\frac{7849}{192} q^2 \\bigg) v^4\n\\cr && \\hspace*{2.7cm}\n+ \\bigg( \\frac{3421}{224} q - \\frac{38029}{672} \\pi \\bigg) v^5\n\\bigg\\} e^2\n\\bigg].\n\\end{eqnarray}\nSince $y=0$ (i.e., $C=0$) corresponds to $\\theta=\\pi\/2$, \n$C$ does not \nevolve for equatorial orbits. This is consistent \nwith the requirement from the symmetry that an orbit in the\nequatorial plane stays in the equatorial plane.\n\nFinally, we consider \nthe evolution of an inclination angle $\\iota$ defined by \n\\cite{Hughes:1999bq},\n\\begin{equation}\n\\cos\\iota = \\frac{L}{\\sqrt{L^2+C}}, \n\\end{equation}\nwhich, roughly speaking, represents an angle between\nthe normal vector of an orbital plane and the rotational\naxis of the central black hole, \nbut this is not the unique definition. \nAlthough the definition of inclination angle \ncan be changed at will to some extent in Kerr case, \nthus defined inclination angle reduces correctly to the \nusual one in the $q=0$ Schwarzschild limit.  \nTaking the average of the time derivative of $\\cos\\iota$,\nwe obtain\n\\begin{eqnarray}\n\\left\\langle \\frac{d\\cos\\iota}{dt} \\right\\rangle_{\\!\\!t} &=&\n\\frac{1}{2(L^2+C)^{\\frac{3}{2}}}\n\\bigg( 2 \\left\\langle \\frac{dL}{dt} \\right\\rangle_{\\!\\!t} C\n       -L \\left\\langle \\frac{dC}{dt} \\right\\rangle_{\\!\\!t}\n\\bigg)\n\\nonumber \\\\ &=&\n\\frac{32\\mu^3 v^6}{5L^2(1+y)^{\\frac{3}{2}}}\nq y \\bigg[\n\\bigg( -\\frac{61}{24} v^3 + \\frac{13}{96} q v^4\n+ \\frac{1779}{224} v^5 \\bigg)\n\\cr && \\hspace*{2.7cm}\n- \\bigg( \\frac{431}{16} v^3 - \\frac{775}{192} q v^4\n- \\frac{22431}{224} v^5 \\bigg) e^2\n\\bigg].\n\\end{eqnarray}\nSubstituting $q=0$ into this equation, we can confirm that \n$\\iota$ does not change in the case of Schwarzschild limit,  \nwhich must be so \nbecause of the spherical symmetry of Schwarzschild spacetime.\n\n\n\\section{Summary} \\label{sec:summary}\nIn this paper, we have considered a scheme to evaluate the change\nrates of the orbital parameters of a particle orbiting Kerr black\nhole under the adiabatic approximation.\nWe have adopted the method proposed by Mino \\cite{Mino:2003yg},\nin which we use the radiative field instead of the retarded\nfield in order to compute the change rates for ``the \nconstants of motion'' due to radiation reaction approximately. \nBased on Mino's method, we have developed a simplified scheme\nto evaluate the long term average of the change rates. \nApplying our new scheme, we have performed explicit \ncalculations to present analytic formulas of change\nrates, $\\langle dE\/dt \\rangle_t$,\n$\\langle dL\/dt \\rangle_t$ and $\\langle dQ\/dt \\rangle_t$,\nfor orbits with small eccentricity and inclination angle.\n\nHere we used the expansions with respect to \nthe post-Newtonian order, the eccentricity and the inclination \nangle in evaluating $\\langle dE\/dt \\rangle_t$, $\\langle dL\/dt \\rangle_t$\nand $\\langle dQ\/dt \\rangle_t$. \nAs a next step therefore we need to examine how large parameter region \nis covered by our formulae with a sufficient accuracy.  \nAs for the inclination, we recently found a formulation to obtain\nthe analytic formulae for the change rates \nwithout assuming a small inclination angle~\\cite{Ganz:2005}.\nOn the other hand, it is almost certain that \nwe need numerical calculation for the cases with a \nlarge eccentricity.\nDrasco and Hughes \\cite{Drasco:2005kz} developed a numerical code to\ncalculate the gravitational wave fluxes of energy and azimuthal\nangular momentum evaluated at infinity and at the event horizon\nfor general geodesic orbits.\nFujita and Tagoshi also developed a numerical code based on\nan analytic method of solving the radial Teukolsky equation.\nBy applying such codes to our scheme, we can evaluate\nthe time-averaged change rate of the Carter constant\nfor general orbits, although computational cost will not \nbe small because we need to take into account a large number \nof frequency modes. \n\nOnce we obtain the change rates of ``the constants of motion'', \nas a next step, we want to use them to trace\nthe evolution of orbits.\nSome strategies to solve the orbital evolution \ntaking into account \nthe radiation reaction effects were proposed\nin Refs.~\\citen{Mino:2005an} and \\citen{Tanaka:2005ue}.\nHowever, it should be noted that the adiabatic approximation\nused here contains only the dissipative part of the self-force\non a particle, and it does not contain the conservative part.\nIn general, the conservative part also contributes\nto the secular evolution of orbits, thought it is not \nthe dominant part in the limit $\\mu\\to 0$. \nTherefore the adiabatic approximation may not be \nsufficient to evaluate the orbital evolution.\n\nRecently, Pound, Poisson and Nickel showed that\nthe conservative part of the self-force can produce\nsignificant shifts in orbital phases  \nin an analogous problem with a charged particle in \nelectromagnetism\\cite{Pound:2005fs}. \nThey suggested that the conservative contribution to\nthe phase shift is relatively large in weak field, slow motion cases,\nwhile it is suppressed in strong field, rapid motion cases.\nFurthermore, there are different types of effects \nhigher order in $\\mu$ which may produce significant shifts \nin phases.\nTherefore it is important to quantify the range of validity \nof the adiabatic approximation for appropriate \napplications of the results obtained in this paper. \nAlthough it requires computing second order perturbations in \n$\\mu$ in order to understand the whole effects which potentially\ngive phase shifts greater than $O(1)$, \nsome of effects can be evaluated by studying \nthe first order self-force at each moment without \naveraging over a long period. \nWe will come back to this issue in one of our forthcoming \npapers~\\cite{Hikida:2005}.\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank S.~Drasco, S.~Jhingan,\nY.~Mino, T.~Nakamura, M.~Sasaki and H.~Tagoshi\nfor invaluable discussions.\nNS, WH and HN would like to thank all participants of\nthe 8th Capra Meeting at the Rutherford Appleton Laboratory\nin UK for useful discussions.\nThis work was supported by Monbukagaku-sho Grant-in-Aid for\nScientific Research of the Japanese Ministry of Education,\nCulture, Sports, Science and Technology, Nos. 14047212 and 14047214.\nHN and WH are supported by a JSPS Research Fellowship \nfor Young Scientists, No.~5919 and No.~1756, respectively.\n\n\n\n\\begin{appendix}\n\\section{Radiative solution for the metric perturbation}\n\\label{sec:radiative}\nIn this Appendix, we give a brief review on Teukolsky formalism, \nfollowed by a derivation of the radiative Green function\nof the linearized Einstein equations.\nThis derivation is based on \nRefs.~\\citen{Chrzanowski:1975wv,Wald:1978vm} and \\citen{Gal'tsov82}.\n\n\n\\subsection{Teukolsky equation}\nAs a master variable we consider the Teukolsky functions defined by\n\\begin{eqnarray}\n{}_s\\Psi &:=& {}_sD^{\\mu\\nu}h_{\\mu\\nu} =\n\\left\\{\n \\begin{array}{ll}\n  -C_{\\alpha\\beta\\gamma\\delta}l^{\\alpha}m^{\\beta}l^{\\gamma}m^{\\delta},\n  & s=2, \\\\\n  -\\bar{z}^4C_{\\alpha\\beta\\gamma\\delta}\n   n^{\\alpha}\\bar{m}^{\\beta}n^{\\gamma}\\bar{m}^{\\delta}, & s=-2,\n \\end{array}\n\\right. \n\\label{defPsi}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n{}_2 D^{\\mu\\nu} &=& -{1\\over 2z}\\bigg[{1\\over 2}\n   {\\cal L}^{\\dag}_{-1}{\\cal L}^{\\dag}_{0}{1\\over z}l^\\mu l^\\nu\n   +{\\cal D}^2_0 zm^\\mu m^\\nu \n\\cr && \\hspace*{1cm}\n   -{1\\over 2\\sqrt{2}}\\left( {\\cal D}_0{1\\over z^2}\n    {\\cal L}^{\\dag}_{-1}z^2+{\\cal L}^{\\dag}_{-1}{1\\over z^2}\n    {\\cal D}_0 z^2\\right)(l^\\mu m^\\nu +m^\\mu l^\\nu)\\bigg],\n\\nonumber \\\\\n{}_{-2} D^{\\mu\\nu} &=& -{1\\over 2z}\\Biggl[{1\\over 2}\n   {\\cal L}_{-1}{\\cal L}_{0}z\\bar z^2 n^\\mu n^\\nu\n   +{1\\over 4}\\Delta^2 {\\cal D}^{\\dag2}_0{\\bar z^2\\over z} \n   \\bar m^\\mu \\bar m^\\nu \n\\cr && \\hspace*{5mm}\n   +{\\Delta^2 \\over 4\\sqrt{2}}\\bigg({\\cal D}^{\\dag}_0\n     {1\\over \\Delta z^2}\n    {\\cal L}_{-1}z^2\\bar z^2+{\\cal L}_{-1}\\frac{1}{z^2}\n    {\\cal D}^{\\dag}_0{z^2\\bar z^2\\over \\Delta}\\bigg)\n    (n^\\mu \\bar m^\\nu +\\bar m^\\mu n^\\nu)\\Biggr],\n \\label{Ddef}\n\\end{eqnarray}\n$z:=r+ia\\cos\\theta$,\n$\\Delta:=r^2-2Mr+a^2$, and $\\Sigma:=r^2+a^2\\cos^2\\theta$.\n${\\cal D}_n$ and ${\\cal L}_s$ are the differential operators defined by \n\\begin{eqnarray}\n{\\cal D}_n &:=&\n\\partial_r + \\frac{(r^2+a^2)}{\\Delta}\n \\partial_t+\\frac{a}{\\Delta}\\partial_\\varphi\n+ \\frac{2n(r-M)}{\\Delta} , \\\\\n{\\cal L}_s &:=&\n\\partial_{\\theta} - \\frac{i}{\\sin\\theta}\\partial_\\varphi\n-i a\\sin\\theta\\partial_t + s\\cot\\theta, \n\\end{eqnarray}\nand a dagger ($\\dag$) acting on an operator \nmeans transformation \nof $(\\partial_t, \\partial_\\varphi) \\to \n(-\\partial_t, -\\partial_\\varphi)$, \nwhich reduces to the one defined in the main text by \n$(\\omega, m) \\to (-\\omega, -m)$ \nunder the assumption of Fourier expansion.\nThe Teukolsky functions satisfy a separable partial differential\nequation~\\cite{Teukolsky:1973ha}\n\\begin{equation}\n{}_s{\\cal O}~{}_s\\Psi = 4\\pi\\Sigma ~_s\\hat{T}, \n \\label{eq:Teukolsky-eq}\n\\end{equation}\nwhere \n\\begin{equation}\n{}_s \\hat{T}:=~_s\\tau_{\\mu\\nu} T^{\\mu\\nu},\n\\end{equation}\nand\n${}_s{\\cal O}$ is the Teukolsky differential operator, \n\\begin{eqnarray}\n{}_s{\\cal O} &:=& {}_s{\\cal O}_r + {}_s{\\cal O}_\\theta,\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n{}_s{\\cal O}_r &:=& \n-\\frac{(r^2+a^2)^2}{\\Delta}\\partial_t^2\n+\\Delta^{-s}\\partial_r(\\Delta^{s+1}\\partial_r)\n-\\frac{a^2}{\\Delta}\\partial_{\\varphi}^2\n-\\frac{4Mar}{\\Delta}\\partial_t\\,\\partial_{\\varphi}\n+\\frac{2sa(r-M)}{\\Delta}\\partial_{\\varphi}\n\\cr &&\n+2s\\left(\\frac{M(r^2-a^2)}{\\Delta}-r\\right)\n\\partial_t + s, \\cr\n{}_s{\\cal O}_\\theta &:=& \na^2\\sin^2\\theta\\,\\partial_t^2\n+\\frac{1}{\\sin\\theta}\\,\\partial_{\\theta}(\\sin\\theta\\,\\partial_{\\theta})\n+\\frac{1}{\\sin^2\\theta}\\,\\partial_{\\varphi}^2\n+\\frac{2is\\cos\\theta}{\\sin^2\\theta}\\,\\partial_{\\varphi}\n\\cr\n&&-2isa\\cos\\theta\\,\\partial_t-s^2\\cot^2\\theta,\n\\label{calOs}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{}_2 \\tau_{\\mu\\nu} & := & {1\\over\\bar z^4 z}\\bigg[\n  {1\\over\\sqrt{2}}\\left({\\cal L}^{\\dag}_{-1}{\\bar z^4\\over z^2}\n  {\\cal D}_0+{\\cal D}_0{\\bar z^4\\over z^2}{\\cal L}^{\\dag}_{-1}\n  \\right)z^2 (l_\\mu m_\\nu +m_\\mu l_\\nu)\n\\cr && \\hspace*{2.5cm}\n  -{\\cal L}^{\\dag}_{-1}\\bar z^4{\\cal L}^{\\dag}_0 {1\\over z}l_\\mu l_\\nu\n  -2{\\cal D}_0\\bar z^4{\\cal D}_0 z m_\\mu m_\\nu\\bigg],\n \\nonumber \\\\\n _{-2}\\tau_{\\mu\\nu} & := & -{1\\over\\bar z^4 z}\\Biggl[\n  {1\\over2\\sqrt{2}}\\Delta\\left({\\cal L}_{-1}{\\bar z^4\\over z^2}\n  {\\cal D}^{\\dag}_{-1}+{\\cal D}^{\\dag}_{-1}{\\bar z^4\\over z^2}{\\cal L}_{-1}\n  \\right)\\Sigma^2 (n_\\mu \\bar m_\\nu +\\bar m_\\mu n_\\nu)\n\\cr && \\hspace*{2cm}\n  +{\\cal L}_{-1}\\bar z^4{\\cal L}_0 \\bar z \\Sigma n_\\mu n_\\nu\n  +{1\\over 2}\\Delta^2{\\cal D}^{\\dag}_0\\bar z^4{\\cal D}^{\\dag}_0 \n  {\\bar z^2\\over z} \\bar m_\\mu \\bar m_\\nu\\Biggr].\n \\label{taudef}\n\\end{eqnarray}\nWe consider the following forms of expansions for ${}_s\\Psi$\nand ${}_s\\hat{T}$:\n\\begin{eqnarray*}\n{}_s\\Psi &=&\n\\int_{-\\infty}^{\\infty}d\\omega\\sum_{\\ell m}\ne^{-i\\omega t}{}_sX_\\Lambda(r){}_sS_\\Lambda(\\theta)\n\\frac{e^{im\\varphi}}{\\sqrt{2\\pi}}, \\\\\n4\\pi\\Sigma~_s\\hat{T} &=&\n\\int_{-\\infty}^{\\infty}d\\omega\\sum_{\\ell m}\ne^{-i\\omega t}~_sT_\\Lambda(r)~_sS_\\Lambda(\\theta)\n\\frac{e^{im\\varphi}}{\\sqrt{2\\pi}},\n\\end{eqnarray*}\nwhere $\\Lambda:=\\{lm\\omega\\}$.\nSubstituting these expressions into the Teukolsky equation \n(\\ref{eq:Teukolsky-eq}),\nwe obtain equations separated for the radial and angular parts\nas\n\\begin{eqnarray}\n\\left[\\Delta^{-s}\\frac{d}{dr}\\left(\\Delta^{s+1}\\frac{d}{dr}\\right)\n+ \\frac{K^2-2is(r-M)K}{\\Delta}+4is\\omega r\n-\\lambda \\right]{}_sX_\\Lambda(r)\n&=& {}_sT_\\Lambda, \\label{eq:radial-Teuk}\\\\\n\\bigg[\\frac{1}{\\sin\\theta}\\frac{d}{d\\theta}\n\\left(\\sin\\theta\\frac{d}{d\\theta}\\right)\n- a^2\\omega^2\\sin^2\\theta\n- \\frac{(m+s\\cos^2\\theta)^2}{\\sin^2\\theta}\n\\hspace*{2cm} && \\cr \n- 2a\\omega s \\cos\\theta + \\lambda + s + 2am\\omega\n\\bigg]{}_sS_\\Lambda(\\theta)\n&=& 0,\n\\label{eq:spheroid-eq}\n\\end{eqnarray}\nwhere $K:=(r^2+a^2)\\omega -ma$ and\n$\\lambda := {}_sE_{\\ell m}(a\\omega) - s(s+1) + a^2\\omega^2 - 2am\\omega$. \nThe eigenvalue ${}_sE_{\\ell m}(a\\omega)$ is determined \nby solving Eq.(\\ref{eq:spheroid-eq}) as an eigenvalue \nproblem imposing regular boundary conditions on ${}_sS_\\Lambda(\\theta)$ \nat $\\theta=\\pm\\pi\/2$. Here $\\ell$ is an index that labels  \ndiffernt eigen values. We also give a brief review on \nhow to solve this equation analytically in Appendix C. \n\n\n\n\\subsection{Mode functions}\nWe write mode functions for the Teukolsky equation \n(\\ref{eq:Teukolsky-eq}) in the form \n\\begin{equation}\n{}_s \\Omega_\\Lambda :=\n{}_sR_\\Lambda(r)~_s Z_\\Lambda(\\theta,\\varphi) e^{-i\\omega t},\n \\label{eq:Teuk-mode-fnc}\n\\end{equation}\nwhere ${}_sR_\\Lambda(r)$ is a homogeneous solution of\nthe radial Teukolsky equation (\\ref{eq:radial-Teuk}),\nand $_s Z_\\Lambda(\\theta,\\varphi)$ is the \nspheroidal harmonics\n\\begin{equation}\n_sZ_\\Lambda(\\theta,\\varphi) =\n\\frac{1}{\\sqrt{2\\pi}}~_sS_\\Lambda(\\theta) e^{im\\varphi},\n\\end{equation}\nnormalized as\n\\begin{equation}\n\\int_0^\\pi d\\theta \\sin\\theta |_sS_\\Lambda(\\theta)|^2 =1.\n\\end{equation}\nUsing the symmetry of the radial equation \n(\\ref{eq:radial-Teuk}) under the \nsimultaneous operations of the \ncomplex conjugation and the transformation \nof $(m,\\omega)\\rightarrow(-m,-\\omega)$,\nwe impose \n\\begin{equation}\n{}_s R_\\Lambda=~_s\\bar R^{\\dag}_\\Lambda,\n \\label{Rdag}\n\\end{equation}\nwhere a dagger ($\\dag$) acting on a mode function \nmeans transformation\nof $(\\omega, m) \\to (-\\omega, -m)$. \nIn a similar manner, \nby virtue of the symmetries of\nEq.~(\\ref{eq:spheroid-eq}), we arrange \nthe spheroidal harmonics to satisfy \n\\begin{equation}\n{}_s Z_\\Lambda =(-1)^m~_{-s}\\bar Z_\\Lambda^{\\dag}. \n\\label{eq:symm-spheroid}\n\\end{equation}\nIn our later discussions, we also \nneed the well-known Teukolsky-Starobinsky identities:\n\\begin{equation}\n {}_{-s} R_\\Lambda =~ _s U _s R_\\Lambda,\n\\qquad\n(\\mbox{for}~~ |s|=2),  \n \\label{Utransform}\n\\end{equation}\nwith\n\\begin{equation}\n {}_{-2} U:={A \\over {\\cal C}}{\\cal D}^4_0,\\qquad \n {}_2 U:={1\\over A\\bar {\\cal C}}\\Delta^2\n{\\cal D}^{\\dag 4}_0 \\Delta^2,\n \\label{Udef}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n {\\cal C}& = &[((\\lambda+s(s+1))^2\n +4a\\omega m -4a^2\\omega^2)\n   \\{(\\lambda+s(s+1)-2\n    )^2+36a\\omega m-36 a^2\\omega^2\\}\\cr\n  &&\\quad +(2\\lambda+2s(s+1)-1)(96a^2\\omega^2-48a\\omega m)-\n   144 a^2\\omega^2]^{1\/2}+12i\\omega M, \n\\end{eqnarray}\nand $A$ is a factor which depends on how we normalize \nthe radial functions.\nIn this paper, we simply adopt $A=1$. \n(This convention is the one used in Ref.~\\citen{Mano:1996gn}). \n\nNow we discuss how to construct mode functions \nfor metric perturbations\nfrom mode functions of the Teukolsky equation. \nThe basic idea owes to \nChrzanowski~\\cite{Chrzanowski:1975wv}. \nHere we follow a more rigorous approach taken by \nWald~\\cite{Wald:1978vm}. \nUsing the relation (\\ref{defPsi}), \nthe Teukolsky equation (\\ref{eq:Teukolsky-eq}) \nis rewritten as \n\\begin{equation}\n {1\\over 4\\pi\\Sigma}{}_s{\\cal O}\\,_sD^{\\mu\\nu}\n    h_{\\mu\\nu}={}_s\\hat T.\n\\end{equation}\nOn the other hand, operating ${}_s\\tau_{\\alpha\\beta}$ on \nthe linearlized Einstein equation, which we schematically denote as \n$\n G^{\\alpha\\beta\\mu\\nu} h_{\\mu\\nu}=4\\pi T_{\\alpha\\beta}, \n$\nwe obtain \n\\begin{equation}\n{1\\over 4\\pi}{}_s\\tau_{\\alpha\\beta} G^{\\alpha\\beta\\mu\\nu} \n h_{\\mu\\nu}={}_s\\hat T.\n\\end{equation}\nFrom the comparison of these equations, we find an identity at the \noperator level:\n\\begin{equation}\n{1\\over \\Sigma}{}_s{\\cal O}\\,_sD^{\\mu\\nu}\n  ={}_s\\tau_{\\alpha\\beta} G^{\\alpha\\beta\\mu\\nu}.\n\\label{operatorID}\n\\end{equation}\n\nHere we define $O^{*\\mu\\nu}$, \nthe adjoint of an operator $O^{\\mu\\nu}$, so as \nto satisfy \n\\begin{equation}\n \\int\\sqrt{-g} \\bar X O^{\\mu\\nu} Y_{\\mu\\nu} d^4x\n = \\int\\sqrt{-g} Y_{\\mu\\nu}  \\overline{ O^{*\\mu\\nu} X} \n       d^4x, \n\\end{equation}\nfor arbitrary scalar field $X$ and tensor field $Y_{\\mu\\nu}$. \nThe definition of the adjoint operators for \ndifferent types of tensor operators is a straight forward \ngeneralization of this definition.  \nIt will be worth noting $\\sqrt{-g}\\,d^4x=\\sin\\theta\\Sigma \\,\ndt\\,dr\\,d\\theta\\,d\\varphi$, \nand \n\\begin{eqnarray}\n(AB)^*=B^* A^*,\\qquad \n{\\cal D}_n^*=-\\Sigma^{-1}{\\cal D}_{-n}^\\dag\\Sigma,\\qquad\n{\\cal L}_s^*=-\\Sigma^{-1}{\\cal L}_{1-s}^\\dag\\Sigma. \n\\end{eqnarray}\nBy taking adjoint of each side in \nEq.~(\\ref{operatorID}), we obtain \n\\begin{equation}\n{}_sD^{*\\mu\\nu}\n\\left(\\Sigma^{-1}{}_s{\\cal O}\\right)^*\n  =G^{\\alpha\\beta\\mu\\nu} {}_s\\tau^*_{\\alpha\\beta}. \n\\end{equation}\nHere we used the fact that the linearlized Einstein \noperator $G^{\\alpha\\beta\\mu\\nu}$ \nis self-adjoint, i.e., $G^{*\\alpha\\beta\\mu\\nu}\n=G^{\\alpha\\beta\\mu\\nu}$. \nThen, from the definition of ${}_s{\\cal O}_r$ and \n${}_s{\\cal O}_\\theta$ given in Eqs.~(\\ref{calOs}), \nit is easy to see that \n\\begin{equation}\n\\left(\\Sigma{}^{-1}~_s{\\cal O}_r\\right)^*=\n    \\Sigma{}^{-1} {}_{-s}{\\cal O}_r, \n\\qquad\n\\left(\\Sigma^{-1}~{}_s{\\cal O}_\\theta\\right)^*=\n    \\Sigma{}^{-1} {}_s{\\cal O}_\\theta.  \n\\end{equation}\nTherefore we have \n$(\\Sigma{}_s{\\cal O})^*{}_{-s} R_{\\Lambda}\\,_s Z_{\\Lambda}\\,\ne^{-i\\omega t}=0$,  \nwhihc means that \n\\begin{equation}\nG^{\\alpha\\beta\\mu\\nu} {}_s\\tau^*_{\\alpha\\beta}\\,\n {}_{-s}R_{\\Lambda}\\,{}_s Z_{\\Lambda}\\,e^{-i\\omega t}=0. \n\\end{equation}\nHere the explicit form of the \nadjoint operators ${}_s\\tau_{\\mu\\nu}^*$ are  \n\\begin{eqnarray}\n{}_2\\tau^*_{\\mu\\nu} & = &\n\\bigg[\n {1\\over\\sqrt{2}}(l_\\mu \\bar m_\\nu +\\bar m_\\mu l_\\nu) \n {\\bar z\\over z}\\left({\\cal D}_0{z^4\\over \\bar z^2}{\\cal L}_2+\n {\\cal L}_2 {z^4\\over \\bar z^2}{\\cal D}_0\\right)\n\\cr && \\hspace*{1.5cm}\n -l_\\mu l_\\nu{1\\over z\\bar z^2}\n {\\cal L}_1 z^4{\\cal L}_2 -2\\bar m_\\mu \\bar m_\\nu {1\\over z}\n {\\cal D}_0 z^4 {\\cal D}_0\\bigg]{1\\over z^3},\n\\\\\n{}_{-2} \\tau^*_{\\mu\\nu} & = &\n-\\bigg[\n {1\\over 2\\sqrt{2}}(n_\\mu m_\\nu +m_\\mu n_\\nu) \n  z\\bar z\\left({\\cal D}^{\\dag}_1{z^4\\over \\bar z^2}{\\cal L}^{\\dag}_2+\n {\\cal L}^{\\dag}_2 {z^4\\over \\bar z^2}{\\cal D}^{\\dag}_1\\right)\\Delta\n\\cr && \\hspace*{1.5cm}\n  +n_\\mu n_\\nu z{\\cal L}^{\\dag}_1 z^4{\\cal L}^{\\dag}_2 \n  +{1\\over 2}m_\\mu m_\\nu {z\\over\\bar z^2}\n {\\cal D}^{\\dag}_0 z^4 {\\cal D}^{\\dag}_0\\Delta^2\\bigg]{1\\over z^3}. \n\\label{taustar}\n\\end{eqnarray}\nHence, \n\\begin{equation}\n{}_s\\Pi_{\\Lambda,\\mu\\nu} :=\n\\zeta_s~_s\\tau_{\\mu\\nu}^{*}\\,\n{}_s\\tilde\\Omega_\\Lambda, \n \\label{Omgtopi}\n\\end{equation}\nwith \n\\begin{equation}\n{}_s\\tilde\\Omega_\\Lambda\n = {}_{-s}R_{\\Lambda}\\,_s Z_\\Lambda \\,e^{-i\\omega t}, \n\\end{equation}\nis a complex-valued homogeneous solution of the \nlinearized Einstein equations. \nHere $\\zeta_s$ is a numerical coefficient which we \ndetermine so as to satisfy \n\\begin{eqnarray}{}_sD^{\\mu\\nu}{}\n\\sum_\\Lambda ({\\cal A}_\\Lambda\\,_s \\Pi_{\\Lambda,\\mu\\nu}\n+\\overline{{\\cal A}_\\Lambda\\,_s\\Pi_{\\Lambda,\\mu\\nu}}) \n=\\sum_\\Lambda {\\cal A}_\\Lambda\\, {}_s\\Omega_{\\Lambda}, \n\\end{eqnarray}\nfor any complex-valued amplitude of each mode, ${\\cal A}_\\Lambda$. \nHere the coplex conjugate term in parentheses\nis necessary to make the metric perturbation real. \nUsing Eqs.~(\\ref{Ddef}) and (\\ref{taustar}), we can verify\n\\begin{eqnarray}\n{}_2 D^{\\mu\\nu}{}_{-2}\\tau^*_{\\mu\\nu} &= &{1\\over 4} \n   {\\cal L}^{\\dag}_{-1} {\\cal L}^{\\dag}_0 \n   {\\cal L}^{\\dag}_{1} {\\cal L}^{\\dag}_2, \\quad\n{}_{-2}D^{\\mu\\nu}{}_{-2}\\tau^*_{\\mu\\nu}={1\\over 16} \n   \\Delta^2{\\cal D}^{\\dag 4}_0\\Delta^2,\n\\nonumber \\\\\n{}_{-2}D^{\\mu\\nu}{}_{-2}\\bar\\tau^{*}_{\\mu\\nu} &=& 0,  \\quad\n{}_{2}D^{\\mu\\nu}{}_{2}\\tau^*_{\\mu\\nu}={\\cal D}^{4}_0,\n\\\\\n{}_{-2}D^{\\mu\\nu}{}_{2}\\tau^*_{\\mu\\nu} &=&\n{1\\over 4} {\\cal L}_{-1} {\\cal L}_0 {\\cal L}_{1} {\\cal L}_2, \\quad\n{}_{2}D^{\\mu\\nu}{}_{2}\\bar\\tau^{*}_{\\mu\\nu} = 0. \n\\label{Dtau}\n\\end{eqnarray}\nIn literature ${}_s\\bar\\tau^{*\\dag}_{\\mu\\nu}$ is \nused to represent what we denote here by ${}_s\\bar\\tau^{*}_{\\mu\\nu}$. \nThe difference arises because we use the notation for \nthe differential operators without assuming that \nthey always act on a single Fourier mode. \nNamely, instead of writing $(-i\\omega, im)$, we are using here \n$(\\partial_t,\\partial_\\varphi)$. The complex conjugation \nof the former gives rise a flip of signature, while that \nof the latter does not. \nWith the aid of the above relations (\\ref{Utransform}) and \n(\\ref{Dtau}), we find that the complex conjugate terms vanish\nto obtain \n\\begin{eqnarray}\n _2 D^{\\mu\\nu}{}_2\\Pi_{\\Lambda,\\mu\\nu} &=&\n \\zeta_2 \\,{\\cal C}\\, {}_2\\Omega_\\Lambda,\n \\nonumber \\\\\n {}_{-2} D^{\\mu\\nu}{}_{-2}\\Pi_{\\Lambda,\\mu\\nu} &=&\n {\\zeta_{-2}\\, \\bar {\\cal C}\\over 16}{}_{-2}\\Omega_\\Lambda.\n\\end{eqnarray}\nThus the normalization constants are fixed as  \n\\begin{equation}\n \\zeta_2={1\\over {\\cal C}}, \\qquad \n \\zeta_{-2}={16\\over \\bar {\\cal C}}.\\label{eq:zeta-norm}\n\\end{equation}\n\n\n\\subsection{Radiative field}\nHere we explain a method of constructing radiative field \nfor metric perturbations. Radiative field is a homogeneous solution \nof field equations. Hence, once we obtain the radiative field \nfor the Teukolsky function, it can be easily transformed into \nthat for metric perturbations by using the relations established \nin the preceding subsection. \nWe therefore first derive the radiative field \nfor the Teukolsky function. \n\nThe retarded Green function of the Teukolsky function \nis defined as a solution of\n\\begin{equation}\n{}_s{\\cal O} {}_sG(x,x') =\n\\frac{\\delta^{(4)}(x-x')}{\\Delta^s}, \n\\end{equation}\nwith the retarded boundary condition: ${}_sG(x,x') =0$ for \n$t<t'$. \nWe write the retarded Green function of the Teukolsky equation \nin the form of Fourier-harmonic expansion as \n\\begin{eqnarray}\n{}_sG(x,x') &=&\n\\int \\frac{d\\omega}{2\\pi} \\sum_{\\ell m}\n{}_s g_\\Lambda (r,r')\n{}_sZ_\\Lambda(\\theta,\\varphi){}_s\\bar{Z}_\\Lambda(\\theta',\\varphi')\ne^{-i\\omega(t-t')}. \n\\end{eqnarray}\nThen the radial part \nof the Green function \n${}_s g_\\Lambda(r,r')$ is given by \n\\begin{eqnarray}\n{}_s g_\\Lambda(r,r') &=&\n\\frac{1}{W({}_sR_\\Lambda^{{\\rm in}},{}_sR_\\Lambda^{{\\rm up}})}\n\\bigg[\n{}_sR_\\Lambda^{{\\rm up}}(r){}_sR_\\Lambda^{{\\rm in}}(r')\\theta(r-r')\n\\cr && \\hspace*{3cm}\n+{}_sR_\\Lambda^{{\\rm in}}(r){}_sR_\\Lambda^{{\\rm up}}(r')\\theta(r'-r)\n\\bigg], \n\\end{eqnarray}\nwith the Wronskian defined by \n\\begin{eqnarray}\nW({}_sR_\\Lambda^{{\\rm in}},{}_sR_\\Lambda^{{\\rm up}}) &:=&\n\\Delta^{s+1}\n\\bigg[\n{}_sR_\\Lambda^{{\\rm in}}(r)\\frac{d}{dr}{}_sR_\\Lambda^{{\\rm up}}(r)\n-{}_sR_\\Lambda^{{\\rm up}}(r)\\frac{d}{dr}{}_sR_\\Lambda^{{\\rm in}}(r)\n\\bigg].\n\\end{eqnarray}\n\nThe advanced Green function can be constructed in a similar manner \njust by replacing ``in'' and ``up'' with ``out'' and ``down'', \nrespectively. \nThen it is easy to show that \nthe radiative Green function has a simple structure which \ndoes not contain any step function $\\theta(r-r')$. \nTo show this, let us start with the following expression\nfor the radial part of the radiative Green function for $r>r'$:\n\\begin{equation}\n{}_sg^{\\rm rad}_{\\Lambda}(r,r') =\n\\frac{1}{2}\\left[\n \\frac{{}_s R^{\\rm up}_\\Lambda(r){}_s R^{\\rm in}_\\Lambda(r')}\n      {W({}_s R^{\\rm in}_\\Lambda,{}_s R^{\\rm up}_\\Lambda)}-\n \\frac{{}_s R^{\\rm down}_\\Lambda(r){}_s R^{\\rm out}_\\Lambda(r')}\n    {W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm down}_\\Lambda)}\n\\right]. \n\\label{eq1}\n\\end{equation}\nWe rewrite this expression in terms of the \ndown-field and the out-field, \neliminating ${}_s R^{\\rm up}_\\Lambda(r)$\nand ${}_s R^{\\rm out}_\\Lambda(r')(=\\Delta^{-s}\n   {}_{-s} \\bar R^{\\rm in}_\\Lambda(r'))$ \nin Eq.~(\\ref{eq1}). Hence we expand \n${}_s R^{\\rm up}_\\Lambda$ and ${}_{s}R^{\\rm out}_\\Lambda$ as  \n\\begin{eqnarray}\n {}_s R^{\\rm up}_\\Lambda \n & = & \\alpha~ {}_s R^{\\rm out}_\\Lambda\n +\\beta~ {}_s R^{\\rm down}_\\Lambda,\\cr\n {}_s R^{\\rm out}_\\Lambda & = & \\gamma~ {}_s R^{\\rm up}_\\Lambda\n  +\\delta~ {}_s R^{\\rm in}_\\Lambda~. \n\\label{expansion}\n\\end{eqnarray}\nTaking the Wronskians of both sides of Eqs.~(\\ref{expansion})\nwith appropriate radial functions, \none can easily obtain \n\\begin{eqnarray*}\n && W({}_s R^{\\rm up}_\\Lambda,{}_s R^{\\rm down}_\\Lambda)\n   =\\alpha\\, W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm down}_\\Lambda),\n\\qquad\n  W({}_s R^{\\rm up}_\\Lambda,{}_s R^{\\rm out}_\\Lambda)\n   =\\beta\\, W({}_s R^{\\rm down}_\\Lambda,{}_s R^{\\rm out}_\\Lambda), \n\\cr\n  &&   W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm in}_\\Lambda)\n   =\\gamma\\, W({}_s R^{\\rm up}_\\Lambda,{}_s R^{\\rm in}_\\Lambda),\n\\qquad\n  W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm up}_\\Lambda)\n   =\\delta\\, W({}_s R^{\\rm in}_\\Lambda,{}_s R^{\\rm up}_\\Lambda). \n\\end{eqnarray*}\nSubstituting these relations, the expression (\\ref{eq1}) \nreduces to\n\\begin{eqnarray}\n_sg^{\\rm rad}_\\Lambda(r,r')&=& {\\Delta^{-s}(r')\\over\n    2 W({}_s R^{\\rm in}_\\Lambda,{}_s R^{\\rm up}_\\Lambda)\n    W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm down}_\\Lambda) }\\cr\n&&\\quad   \\times \\Bigl[\n    W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm in}_\\Lambda) \n     \\,_sR^{\\rm down}_{\\Lambda}(r) \n     \\,_{-s}\\bar R^{\\rm down}_{\\Lambda}(r')\\cr\n  &&\\hspace*{2cm} \n     +W({}_s R^{\\rm up}_\\Lambda,{}_s R^{\\rm down}_\\Lambda) \n    \\,_s R^{\\rm out}_{\\Lambda}(r) \n    \\,_{-s}\\bar{R}^{\\rm out}_{\\Lambda}(r') \n    \\Bigr].  \\label{eq2}\n\\end{eqnarray}\nWe can do an analogous reduction for $r<r'$, and the result \nturns out to be the same as that \nfor $r>r'$. Namely, the step functions \nwhich was present in the retarded and the advanced Green functions \ndo not appear in the radiative Green function. \nThis is consistent with the fact \nthat the radiative field is a source-free \nhomogeneous solution. \n\nSince the radiative field is a homogeneous solution, \nwe can use the method for reconstruction of metric \nperturbation explained in the preceding subsection. \nWhen we consider the metric perturbation by a point mass,\nthe energy-momentum tensor is given by (\\ref{ppEM}). \nIn this case it is easy to verify that the radiative field \nof the metric perturbations is given by \n\\begin{eqnarray}\nh_{\\mu\\nu}^{{\\rm rad}}(x) &=&\n\\mu \\! \\int \\!\\! \nd\\omega \\sum_{\\ell m} \n\\bigg\\{\n{\\cal N}_{s}^{\\rm out} \n{}_s\\Pi_{\\Lambda,\\mu\\nu}^{{\\rm out}}(x)\n\\int \\!\\! d\\tau \\Big[\n{}_s\\bar{\\Pi}_{\\Lambda,\\alpha\\beta}^{{\\rm out}}(z(\\lambda))\nu^{\\alpha}u^{\\beta} \\Big]\n\\cr &&\n+ {\\cal N}_{s}^{\\rm down} {}_s\\Pi_{\\Lambda,\\mu\\nu}^{{\\rm down}}(x)\n\\int \\!\\! d\\tau \\Big[\n{}_s\\bar{\\Pi}_{\\Lambda,\\alpha\\beta}^{{\\rm down}}(z(\\lambda))\nu^{\\alpha}u^{\\beta}\n\\Big] \\bigg\\}\n+ ({\\rm c.c.}), \n\\label{eq:rad-field}\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n {\\cal N}_{s}^{\\rm out} \n  & = & { W({}_s R^{\\rm up}_\\Lambda,{}_s R^{\\rm down}_\\Lambda)\n    \\over \\bar\\zeta_s \n    W({}_s R^{\\rm in}_\\Lambda,{}_s R^{\\rm up}_\\Lambda)\n    W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm down}_\\Lambda) },\\cr\n {\\cal N}_{s}^{\\rm down} \n  & = & { W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm in}_\\Lambda)\n    \\over \\bar\\zeta_s \n    W({}_s R^{\\rm in}_\\Lambda,{}_s R^{\\rm up}_\\Lambda)\n    W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm down}_\\Lambda) }. \n\\end{eqnarray}\nIn fact, if we apply ${}_s{\\cal D}^{\\mu\\nu}$, we correctly \nrecover\n${}_s \\Psi^{\\rm rad}(x)=4\\pi\\int \nG^{\\rm rad}(x,x') \\Sigma(x')$\n$\n\\Delta^s(x') {}_s\\hat T(x') d^4 x$. \nTo show this, we also used\n\\begin{eqnarray}\n\\bar\\zeta_s \\int \\sqrt{-g}\n   {}_s\\bar{\\tilde{\\Omega}}_{\\Lambda} \n       \\,{}_s\\hat T d^4x\n&=&\\bar\\zeta_s\\int \\sqrt{-g} \n  \\overline{\\left({}_s\\tau^{*}_{\\mu\\nu}\\, \n   {}_s\\tilde\\Omega_{\\Lambda}\\right)} T^{\\mu\\nu} d^4x \\cr\n&=&\\mu\\int d\\tau\\,\n    u^{\\mu}  u^{\\nu} \n    \\bar\\Pi_{\\Lambda,\\mu\\nu}(z(\\tau)). \n\\label{TT}\n\\end{eqnarray}\n\nIt is more convenient to rewrite ${\\cal N}_s$ written \nin terms of Wronskians by using the coefficients \nin the asymptotic forms of radial functions. \nThe radial functions take the asymptotic forms,\n\\begin{eqnarray}\n{}_s R_\\Lambda^{{\\rm in}} &:=&\n\\left\\{\n\\begin{array}{ll}\n {}_sB_{\\Lambda}^{{\\rm inc}}\\displaystyle r^{-1} e^{-i\\omega r^{*}}\n +{}_sB_{\\Lambda}^{{\\rm ref}} r^{-2s-1} e^{i\\omega r^{*}}, \\hskip1cm\n&\n {\\rm for}~~r^{*}\\rightarrow\\infty,\n\\\\\n \\displaystyle\n {}_sB_{\\Lambda}^{{\\rm trans}}\\Delta^{-s}e^{-ik r^{*}},\n&\n {\\rm for}~~r^{*}\\rightarrow -\\infty,\n\\end{array}\n\\right. \\label{eq:asymptotic-in} \\\\\n{}_s R_\\Lambda^{{\\rm up}} &:=&\n\\left\\{\n\\begin{array}{ll}\n \\displaystyle\n  {}_sC_{\\Lambda}^{{\\rm trans}}r^{-2s-1} e^{i\\omega r^{*}}, \\hskip3cm\n\\hspace*{7mm}\n&\n  {\\rm for}~~r^{*}\\rightarrow\\infty, \n\\\\\n \\displaystyle\n {}_sC_{\\Lambda}^{{\\rm up}}e^{ik r^{*}}\n + {}_sC_{\\Lambda}^{{\\rm ref}}\\Delta^{-s}e^{-ik r^{*}},\n&\n {\\rm for}~~r^{*}\\rightarrow -\\infty, \n\\end{array}\n\\right.\n\\label{eq:asymptotic-up}\n\\end{eqnarray}\nwhere $r^*$ is the tortoise coordinate defined by\n$dr^*\/dr=(r^2+a^2)\/\\Delta$.\nUsing the relations \n${}_s R^{\\rm out}_\\Lambda=\n\\Delta^{-s}{}_{-s} \\bar R^{\\rm out}_\\Lambda$\nand\n${}_s R^{\\rm down}_\\Lambda=\n\\Delta^{-s}{}_{-s} \\bar R^{\\rm up}_\\Lambda$, \nwe can describe the asymptotic forms\nof out- and down- fields with the same coefficients\nthat appear in Eqs.~(\\ref{eq:asymptotic-in}) and\n(\\ref{eq:asymptotic-up}).\nThen, the Wronskians that we need to evaluate are \n\\begin{eqnarray}\n    W({}_s R^{\\rm in}_\\Lambda\n,{}_s R^{\\rm up}_\\Lambda) \n& = & \n  2i\\omega\\,  {}_{s}B_\\Lambda^{\\rm inc} \\,\n      {}_{s}C_\\Lambda^{\\rm trans}, \n\\cr \n    W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm down}_\\Lambda) \n& = & \n  -2i\\omega\\,  {}_{-s}\\bar C_\\Lambda^{\\rm trans} \\,\n      {}_{-s}\\bar B_\\Lambda^{\\rm inc}, \\cr \n    W({}_s R^{\\rm out}_\\Lambda,{}_s R^{\\rm in}_\\Lambda) \n& = & \n  -4ikM r_+\\kappa_s \\,  {}_{s} B_\\Lambda^{\\rm trans} \\,\n      {}_{-s}\\bar B_\\Lambda^{\\rm trans}, \\cr \n    W({}_s R^{\\rm up}_\\Lambda,{}_s R^{\\rm down}_\\Lambda) \n& = & \n  -2i\\omega\\,  {}_{-s}\\bar C_\\Lambda^{\\rm trans} \\,\n      {}_{s} C_\\Lambda^{\\rm trans}, \n\\end{eqnarray}\nwhere $\\displaystyle\\kappa_s:=1-{is(r_+-M)\/ 2kMr_{+}}$.\nThe coefficients with $(-s)$-spin can be erased by using \nthe Teukolsky-Starobinsky identities~(\\ref{Utransform}). \nSubstituting the asymptotic forms (\\ref{eq:asymptotic-in}) and\n(\\ref{eq:asymptotic-up}) into Eqs.~(\\ref{Utransform}), we obtain\n\\begin{eqnarray}\n {}_{-2}B_\\Lambda^{\\rm inc} &= \n  &{{\\cal C}\\over(2\\omega)^4}{}_{2}B_\\Lambda^{\\rm inc}, \\qquad\n {}_{-2}B_\\Lambda^{\\rm trans}\n   =  \\left({1\\over 4M r_{+} k}\\right)^4\n  {{\\cal C}\n   \\over \\kappa_{-2}\\kappa_{-1}\\kappa_1}{}_{2}B_\\Lambda^{\\rm trans}, \n \\nonumber \\\\\n  {}_{-2}C_\\Lambda^{\\rm trans} \n &= &{(2\\omega)^4\\over \\bar {\\cal C}}\n    {}_{2}C_\\Lambda^{\\rm trans} , \\qquad\n  {}_{-2}B_\\Lambda^{\\rm ref} \n  ={(2\\omega)^4 \\over \\bar {\\cal C}}  {}_{2}B_\\Lambda^{\\rm ref}.\n \\label{relcoeff}\n\\end{eqnarray}\nUsing the above relations, the coefficients \n${\\cal N}_s$ are rewritten as \n\\begin{equation}\n {\\cal N}^{\\rm out}_s={1\\over 2i\\omega^3}|N_s^{\\rm out}|^2 ,\n\\qquad\n  {\\cal N}^{\\rm down}_s={1\\over 2i\\omega^2 k}|N_s^{\\rm down}|^2 ,\n\\end{equation}\nwith\n\\begin{eqnarray}\n|N_s^{{\\rm out}}|^2 \n&\\equiv&\n{2^{3s-2}\\omega^{2s+2} \\over |{\\cal C}|^{s\/2-1}}\n{1\\over |{}_sB^{\\rm inc}_\\Lambda|^{2}}, \n\\label{eq:Namp-out} \\\\\n|N_s^{{\\rm down}}|^2 &\\equiv&\n{2^{-3s-2} k^{-2s+2} \n|{\\cal C}|^{s\/2+1}\\over |\\kappa_2|^{s\/2-1}|\\kappa_1|^{s}\n(2Mr_+)^{2s-1}}\n  {|{}_sB^{\\rm trans}_\\Lambda|^{2}\\over \n   |{}_sB^{\\rm inc}_\\Lambda|^{2}\\, \n   |{}_sC^{\\rm trans}_\\Lambda|^{2}}.\n\\label{eq:Namp-down}\n\\end{eqnarray}\nHence, we finally obtain  \n\\begin{eqnarray}\nh_{\\mu\\nu}^{\\rm rad} &= & \\mu\\int d\\omega \n  \\sum_{\\ell m}{1\\over 2i\\omega^3}\\Bigl(\n   N^{\\rm out}_s \n    {}_s\\Pi^{\\rm out}_{\\Lambda,\\mu\\nu}(x) \n    \\int {d\\tau\\over \\Sigma}\\, \\bar \\phi_{\\Lambda}^{out}(\\tau)\n\\cr &&\\qquad\\qquad\n   + {\\omega\\over k}N^{\\rm down}_s \n     {}_s\\Pi^{\\rm down}_{\\Lambda,\\mu\\nu}(x) \n    \\int {d\\tau\\over\\Sigma}\n      \\, \\bar \\phi_{\\Lambda}^{\\rm down}(\\tau)\n    \\Bigr)+({\\rm c.c.}),\n\\label{hrad}\n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\phi_\\Lambda^{{\\rm (out\/down)}}(\\tau) :=\nN_s^{{\\rm (out\/down)}} \\Sigma(z(\\tau))\n{}_s\\Pi_{\\Lambda,\\gamma\\delta}^{{\\rm (out\/down)}}(z(\\tau))\nu^\\gamma(\\tau) u^\\delta(\\tau), \n\\end{equation}\nwhose extension to a field is \n$ \\phi^{\\rm (out\/down)}_{\\Lambda}(x)$ \ndefined in Eq.~(\\ref{eq:def-phi-out}). \n\n\n\\section{Mano-Suzuki-Takasugi method} \\label{sec:MST}\nMano, Suzuki and Takasugi formulated a method of\nconstructing a homogeneous solution for the radial Teukolsky\nequation in two kinds of series by using\nthe Coulomb wave function and the hypergeometric functions\n\\cite{Mano:1996vt,Mano:1996gn,Sasaki:2003xr}.\nBy applying this method under slow motion approximation,\nwe can express homogeneous solutions in an analytic form.\nFurthermore, this method determines the asymptotic amplitudes \nof homogeneous solutions without numerical integration.\nThis allows us to compute the gravitational wave flux\nat infinity and on the horizon with a high accuracy\n\\cite{Fujita:2004rb}.\nWe summarize this method in this appendix. \n\n\\subsection{Outer solution of radial Teukolsky equation}\nAccording to \\citen{Mano:1996gn,Mano:1996vt,Sasaki:2003xr},\nwe can expand ${}_sR_C^{\\nu}$, a homogeneous solution\nof the radial Teukolsky equation (\\ref{eq:radial-Teuk}),\nin terms of the Coulomb wave functions as \n\\begin{eqnarray}\n{}_sR_C^{\\nu} \n&=&\n\\frac{\\Gamma(\\nu+1-s+i\\epsilon)}{\\Gamma(2\\nu+2)}\n\\hat{z}^{-s}(2\\hat{z})^{\\nu} e^{-i\\hat{z}}\n\\left(1-\\frac{\\epsilon\\kappa}{\\hat{z}}\\right)^{-s-i\\epsilon_+}\n\\cr && \\times\n\\sum_{n=-\\infty}^{\\infty} \\!\\! (-2i\\hat{z})^n\n\\frac{(\\nu+1+s-i\\epsilon)_n}{(2\\nu+2)_{2n}} a_n^{\\nu,s}\n\\cr && \\hspace*{1.5cm} \\times\n{}_1F_1(n+\\nu+1-s+i\\epsilon,2n+2\\nu+2 ; 2i\\hat{z}),\n\\label{eq:Coulomb-series}\n\\end{eqnarray}\nwhere $\\epsilon=2M\\omega$, $\\epsilon_+=\\epsilon+\\tau$,\n$\\tau=\\kappa^{-1}(\\epsilon-ma\/M)$,\n$\\kappa=\\sqrt{1-(a\/M)^2}$, $(x)_n:=\\Gamma(x+n)\/\\Gamma(x)$,\n$\\hat{z}:=\\omega(r-r_-)$, and $r_-=M-\\sqrt{M^2-a^2}$.\nThe coefficients $a_n^{\\nu,s}$ satisfies the following \nthree term recurrence relation, \n\\begin{equation}\n\\alpha_n^{\\nu} a_{n+1}^{\\nu,s}+\\beta_n^{\\nu} a_n^{\\nu,s}\n+\\gamma_n^{\\nu} a_{n-1}^{\\nu,s}=0,\n\\label{eq:recurrence}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\alpha_n^{\\nu} &=& \n\\frac{i\\epsilon \\kappa\n(n+\\nu+1+s+i\\epsilon)(n+\\nu+1+s-i\\epsilon)(n+\\nu+1+i\\tau)}\n{(n+\\nu+1)(2n+2\\nu+3)}, \\nonumber \\\\\n\\beta_n^{\\nu} &=& \n-\\lambda-s(s+1)+(n+\\nu)(n+\\nu+1)\n+\\epsilon^2+\\epsilon(\\epsilon-mq) \n +\\frac{\\epsilon (\\epsilon-mq)(s^2+\\epsilon^2)}{(n+\\nu)(n+\\nu+1)},\n\\nonumber \\\\\n\\gamma_n^{\\nu} &=&\n-\\frac{i\\epsilon \\kappa (n+\\nu-s+i\\epsilon)\n(n+\\nu-s-i\\epsilon)(n+\\nu-i\\tau)}\n{(n+\\nu)(2n+2\\nu-1)},\n\\label{eq:coeff-Cwave-fnc}\n\\end{eqnarray}\nand $q=a\/M$. The renormalized angular momentum $\\nu$ is determined\nby the conditions \n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\nn\\frac{a_n^{\\nu,s}}{a_{n-1}^{\\nu,s}} = \\frac{i\\epsilon\\kappa}{2}, \n\\qquad\n\\lim_{n\\rightarrow -\\infty}\nn\\frac{a_n^{\\nu,s}}{a_{n+1}^{\\nu,s}} = -\\frac{i\\epsilon\\kappa}{2}.\n\\end{equation}\nUnder this condition, the series of Coulomb wave functions\n(\\ref{eq:Coulomb-series}) converges for any $r>r_+$.\n\nFrom the equations in (\\ref{eq:coeff-Cwave-fnc}),\nwe can show that $\\alpha_{-n}^{-\\nu-1}=\\gamma_n^\\nu$ and\n$\\beta_{-n}^{-\\nu-1}=\\beta_n^\\nu$.\nBy using these relations, we can find that\n$a_n^{-\\nu-1,s}=a_{-n}^{\\nu,s}$ and\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\nn\\frac{a_n^{-\\nu-1,s}}{a_{n-1}^{-\\nu-1,s}} = \\frac{i\\epsilon\\kappa}{2}, \n\\qquad\n\\lim_{n\\rightarrow -\\infty}\nn\\frac{a_n^{-\\nu-1,s}}{a_{n+1}^{-\\nu-1,s}} = -\\frac{i\\epsilon\\kappa}{2}.\n\\end{equation}\nThis fact shows that ${}_sR_C^{-\\nu-1}$ is also a solution of the\nradial Teukolsky equation, which converges within the region $r>r_+$.\n\n\n\\subsection{In-going and up-going solutions}\nThe in-going solution of the radial Teukolsky equation is given\nin terms of the Coulomb type solutions (\\ref{eq:Coulomb-series}) \nas \n\\begin{equation}\n_sR_\\Lambda^{{\\rm in}} =\nA_s e^{i\\epsilon\\kappa}(\nK_{s,\\nu}~_sR_{C}^{\\nu} + K_{s,-\\nu-1}~_sR_{C}^{-\\nu-1}),\n\\label{eq:Rin}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nA_2 &=& \\bar{{\\cal C}}\\left(\\frac{\\omega}{\\epsilon\\kappa}\\right)^4\n\\frac{\\Gamma(3-2i\\epsilon_+)}{\\Gamma(-1-2i\\epsilon_+)}\n\\left| \\frac{\\Gamma(\\nu-1+i\\epsilon)}\n{\\Gamma(\\nu+3+i\\epsilon)}\\right|^2, \\quad A_{-2} = 1, \\\\\nK_{s,\\nu} &=&\n\\frac{(2\\epsilon\\kappa)^{s-\\nu-r}2^{-s} i^r}\n{(\\nu+1+i\\tau)_r(\\nu+1+s+i\\epsilon)_r}\n\\cr && \\times\n\\frac{\\Gamma(1-s-2i\\epsilon_+)\\Gamma(n+2\\nu+2)\\Gamma(n+2\\nu+1)}\n{\\Gamma(r+\\nu+1-s+i\\epsilon)\\Gamma(\\nu+1-s-i\\epsilon)\n\\Gamma(\\nu+1-i\\tau)} \\cr\n&& \\times \\left[\\sum_{n=r}^{\\infty} (-1)^n\n\\frac{(r+2\\nu+1)_n(\\nu+1+s+i\\epsilon)_n(\\nu+1+i\\tau)_n}\n{(n-r)!~(\\nu+1-s-i\\epsilon)_n(\\nu+1-i\\tau)_n}a_n^{\\nu,s}\\right] \\cr\n&& \\times \\left[\\sum_{n=-\\infty}^{r}\n\\frac{(-1)^n}{(r-n)!~(r+2\\nu+2)_n}\n\\frac{(\\nu+1+s-i\\epsilon)_n}{(\\nu+1-s+i\\epsilon)_n}\na_n^{\\nu,s}\\right]^{-1}.\n\\end{eqnarray}\nHere $r$ is an arbitrary integer and $K_{s,\\nu}$ is independent\nof the choice of $r$.\n\nNext, we consider the up-going solution.\n$_sR_C^{\\nu}$ can be divided into two parts as\n\\begin{equation}\n{}_sR_C^{\\nu} = {}_sR_+^{\\nu} + {}_sR_-^{\\nu},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n{}_sR_+^{\\nu} &=&\ne^{-\\pi\\epsilon}e^{i\\pi(\\nu+1-s)}e^{-i\\hat{z}}\n(2\\hat{z})^{\\nu}\\hat{z}^{-s}\n\\left(1-\\frac{\\epsilon\\kappa}{\\hat{z}}\\right)^{-s-i\\epsilon_+}\n\\frac{\\Gamma(\\nu+1-s+i\\epsilon)}{\\Gamma(\\nu+1+s-i\\epsilon)} \\cr\n&& \\times\n\\sum_{n=-\\infty}^{\\infty}(2i\\hat{z})^n a_n^{\\nu,s}\n\\Psi(n+\\nu+1-s+i\\epsilon,2n+2\\nu+2; 2i\\hat{z}), \\\\\n{}_sR_-^{\\nu} &=&\ne^{-\\pi\\epsilon}e^{-i\\pi(\\nu+1+s)}e^{i\\hat{z}}\n(2\\hat{z})^{\\nu}\\hat{z}^{-s}\n\\left(1-\\frac{\\epsilon\\kappa}{\\hat{z}}\\right)^{-s-i\\epsilon_+}\n\\cr && \\times\n\\sum_{n=-\\infty}^{\\infty} \\!\\! (2i\\hat{z})^n\n\\frac{(\\nu+1+s-i\\epsilon)_n}{(\\nu+1-s+i\\epsilon)_n}a_n^{\\nu,s}\n\\cr && \\hspace*{1.5cm} \\times\n\\Psi(n+\\nu+1+s-i\\epsilon,2n+2\\nu+2; -2i\\hat{z}), \n\\end{eqnarray}\nand $\\Psi(a,c;x)$ \nis the irregular confluent hypergeometric function. \nFrom the asymptotic form of $\\Psi(a,c;x)$,  \n\\begin{equation}\n\\Psi(a,c; x) \\rightarrow x^{-a}, \\qquad\n(|x|\\rightarrow\\infty),\n\\end{equation}\nthe asymptotic forms of $_sR_+^{\\nu}$ and $_sR_-^{\\nu}$ \nbecome \n\\begin{equation}\n_sR_+^{\\nu} = ~_sA_+^{\\nu}z^{-1}e^{-i(z+\\epsilon\\ln z)},\n\\qquad\n_sR_-^{\\nu} = ~_sA_-^{\\nu}z^{-1-2s}e^{i(z+\\epsilon\\ln z)},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n{}_sA_+^{\\nu} &=& \ne^{-\\pi\\epsilon\/2}e^{i\\pi(\\nu+1-s)\/2}2^{s-1-i\\epsilon}\n\\frac{\\Gamma(\\nu+1-s+i\\epsilon)}{\\Gamma(\\nu+1+s-i\\epsilon)}\n\\sum_{n=-\\infty}^{\\infty} \\!\\! a_n^{\\nu,s}, \\\\\n{}_sA_-^{\\nu} &=&\ne^{-\\pi\\epsilon\/2}e^{-i\\pi(\\nu+1+s)\/2}2^{-s-1+i\\epsilon}\n\\sum_{n=-\\infty}^{\\infty} \\!\\! (-1)^n\n\\frac{(\\nu+1+s-i\\epsilon)_n}{(\\nu+1-s+i\\epsilon)_n}\na_n^{\\nu,s}.\n\\end{eqnarray}\nThis shows that $_sR_-^{\\nu}$ ($_sR_+^{\\nu}$) satisfies the\nup-going (down-coming) boundary condition at infinity.\nSo we can take the up-going solution as\n\\begin{equation}\n{}_sR_\\Lambda^{{\\rm up}} = B_s {}_sR_-^{\\nu},\n\\label{eq:Rup}\n\\end{equation}\nwhere $B_2=\\bar{C}\\omega^{2s}$ and $B_{-2}=1$.\nTaking the limit $r^*\\to\\pm\\infty$ \nin Eqs.~(\\ref{eq:Rin}) and (\\ref{eq:Rup}) \nby means of the asymptotic form of $r^*$, \n\\begin{eqnarray}\n\\omega r^* &\\rightarrow&\n\\hat{z}+\\epsilon\\ln\\hat{z}-\\epsilon\\ln\\epsilon \\quad\n(r\\rightarrow\\infty), \\\\\nkr^* &\\rightarrow&\n\\epsilon_+\\ln(-x) + \\kappa\\epsilon_+\n+\\frac{2\\kappa\\epsilon_+}{1+\\kappa}\\ln\\kappa \\quad\n(r\\rightarrow r_+),\n\\end{eqnarray}\nwe find that the coefficients which appear in the asymptotic\nforms of Eqs.~(\\ref{eq:asymptotic-in}) and (\\ref{eq:asymptotic-up})\nare given by\n\\begin{eqnarray}\n{}_sB_\\Lambda^{{\\rm inc}} &=&\n\\frac{A_s e^{i\\epsilon\\kappa}}{\\omega}\n\\left[ K_{s,\\nu} - ie^{-i\\pi\\nu}\n\\frac{\\sin\\pi(\\nu-s+i\\epsilon)}{\\sin\\pi(\\nu+s-i\\epsilon)}\nK_{s,-\\nu-1} \\right] {}_sA_+^{\\nu}, \\\\\n{}_sB_\\Lambda^{{\\rm trans}} &=&\nA_s\\left(\\frac{\\epsilon\\kappa}{\\omega}\\right)^{2s}\n\\sum_{n=-\\infty}^{\\infty} a_n^{\\nu,s}, \\\\\n{}_sC_\\Lambda^{{\\rm trans}} &=&\n\\omega^{-1-2s}e^{i\\epsilon\\ln\\epsilon} {}_sA_-^{\\nu}. \n\\end{eqnarray}\n\n\n\\section{Spheroidal harmonics} \\label{sec:spheroidal}\nHere, we review the formalism to represent the spin-weighted spheroidal \nharmonics in a series of Jacobi polynomials based on\nRef.~\\citen{Fackerell}, which was slightly improved in \nRef.~\\citen{Fujita:2004rb}.\n\nWe first transform the angular part of the Teukolsky equation \n(\\ref{eq:spheroid-eq}) as \n\\begin{eqnarray}\n\\bigg[(1-x^{2})\\frac{d^2}{dx^{2}}-2x\\frac{d}{dx}+{\\xi}^{2} x^{2}\n\\hspace*{4cm} && \\cr\n-\\frac{m^{2}+s^{2}+2msx}{1-x^{2}}-2s\\xi x+{}_sE_{\\ell m}(\\xi)\n\\bigg]\n{}_sS_{\\ell m}^{\\xi}(x) &=& 0 \\,,\n\\label{eq:Sphe diff}\n\\end{eqnarray}\nwhere $\\xi = a\\,\\omega, x = \\cos\\theta$\nand ${}_sE_{\\ell m}(\\xi)=\\lambda+s(s+1)-\\xi^{2}+2m\\xi$.\nThe angular function $_{s}S_{\\ell m}^{\\xi}(x)$ is called the\nspin-weighted spheroidal harmonics. Equation (\\ref{eq:Sphe diff}) \nis a Sturm-Liouville type eigenvalue equation with\nregular boundary conditions at $x=\\pm 1$. \nSince there are a countable number of eigenvalues \nfor fixed parameters $s$, $m$ and $\\xi$, \nwe introduced an index $\\ell$ starting with max($|m|,|s|$) \nas such a label \nthat sorts the eigenvalues ${}_sE_{\\ell m}(\\xi)$ in an ascending \norder. \nWhen $\\xi=0$, $_{s}S_{\\ell m}^{\\xi}(x)$ \nis reduced to the spin-weighted spherical \nharmonics, and the eigenvalue ${}_sE_{\\ell m}(\\xi)$ becomes $\\ell(\\ell+1)$.\nWe normalize the amplitude of $_{s}S_{\\ell m}^{\\xi}(x)$ as \n\\begin{eqnarray}\n\\label{eq:normalSp}\n\\int _{0}^{\\pi}\\left |{}_{s}S_{\\ell m}^{\\xi}\\right |^2\\sin \\theta d\\theta=1 \\,.\n\\end{eqnarray}\n\nThe differential equation (\\ref{eq:Sphe diff}) has singularities at\n$x=\\pm 1$ and at $x=\\infty$. We transform the angular function as \n\\begin{eqnarray}\n_{s}S_{\\ell m}^{\\xi}(x) \\equiv\ne^{\\xi x}\\left(\\frac{1-x}{2}\\right)^{\\frac{\\alpha}{2}}\n\\left(\\frac{1+x}{2}\\right)^{\\frac{\\beta}{2}}\\, _{s}U_{\\ell m}(x) \\,,\n\\label{eq:SpheU}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\label{eq:SpheV}\n_{s}S_{\\ell m}^{\\xi}(x) \\equiv\ne^{-\\xi x}\\left(\\frac{1-x}{2}\\right)^{\\frac{\\alpha}{2}}\n\\left(\\frac{1+x}{2}\\right)^{\\frac{\\beta}{2}}\\, _{s}V_{\\ell m}(x) \\,,\n\\label{Vseries}\n\\end{eqnarray}\nwhere $\\alpha = |m+s|$ and $\\beta = |m-s|$. Then, \nEq.~(\\ref{eq:Sphe diff}) becomes\n\\begin{eqnarray}\n\\label{eq:proto-Jacobi}\n&&(1-x^{2})\\,_{s}U_{\\ell m}''(x)+\\left[\\beta-\\alpha-(2+\\alpha+\\beta)x\\right]\\,_{s}U_{\\ell m}'(x)\n\\nonumber\\\\\n&&\\quad\n+\\left[\\,_{s}E_{\\ell m}(\\xi)-\\frac{\\alpha+\\beta}{2}\\left(\\frac{\\alpha+\\beta}{2}+\n1\\right)\\right]\\,_{s}U_{\\ell m}(x)\\nonumber \\\\\n&&\\quad\n=\\xi\\left[-2(1-x^{2})\\,_{s}U_{\\ell m}'(x)+(\\alpha+\\beta+2s+2)x\\,_{s}U_{\\ell m}(x)\n\\right.\n\\nonumber\\\\\n&&\\quad\\quad\\left. \n-(\\xi+\\beta-\\alpha)\\,_{s}U_{\\ell m}(x)\\right] \\,,\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\label{eq:proto-JacobiV}\n&&(1-x^{2})\\,_{s}V_{\\ell m}''(x)+\\left[\\beta-\\alpha-(2+\\alpha+\\beta)x\\right]\\,\n_{s}V_{\\ell m}'(x)\n\\nonumber\\\\\n&&\\quad\n+\\left[\\,_{s}E_{\\ell m}(\\xi)-\\frac{\\alpha+\\beta}{2}\\left(\\frac{\\alpha+\\beta}{2}+\n1\\right)\\right]\\,_{s}V_{\\ell m}(x)\\nonumber \\\\\n&=&\\xi\\left[2(1-x^{2})\\,_{s}V_{\\ell m}'(x)-(\\alpha+\\beta-2s+2)x\\,_{s}V_{\\ell m}(x)\n\\right.\n\\nonumber\\\\\n&&\\quad\n\\left.\n-(\\xi-\\beta+\\alpha)\\,_{s}V_{\\ell m}(x)\\right] \\,.\n\\end{eqnarray}\nFrom Eqs.~(\\ref{eq:SpheU}) and (\\ref{eq:SpheV}), we find\n\\begin{eqnarray}\n\\label{eq:sphUtoV}\n\\,_{s}V_{\\ell m}(x)={\\rm exp}(2\\xi x)\\,_{s}U_{\\ell m}(x) \\,.\n\\end{eqnarray}\n\nWhen $\\xi=0$, the right-hand sides of Eqs.~(\\ref{eq:proto-Jacobi})\nand (\\ref{eq:proto-JacobiV}) are zero, and they reduce to\nthe differential equation satisfied by the Jacobi polynomials, \n\\begin{eqnarray}\n&&(1-x^{2})\\,P_{n}^{(\\alpha,\\beta)}{}^{''}(x)\n+\\left[\\beta-\\alpha-(2+\\alpha+\\beta)x\\right]\\,P_{n}^{(\\alpha,\\beta)}{}^{'}(x)\n\\nonumber\\\\\n&&\n\\quad +n(n+\\alpha+\\beta+1)\\,P_{n}^{(\\alpha,\\beta)}(x)=0.\n\\label{eq:Jacobi}\n\\end{eqnarray}\nIn this limit, the eigenvalue $_{s}E_{\\ell m}(\\xi)$ in the equation \n(\\ref{eq:proto-Jacobi}) becomes $\\ell(\\ell+1)$, \nwhere $n=\\ell-(\\alpha+\\beta)\/2=\\ell-{\\rm max}(\\mid m\\mid ,\\mid s\\mid )$.\nHere, the Jacobi polynomials are defined by the Rodrigue's formula by\n\\begin{eqnarray}\nP_{n}^{(\\alpha,\\beta)}(x) :=\n\\frac{(-1)^{n}}{2^{n}\\,n!}(1-x)^{-\\alpha}(1+x)^\n{-\\beta}\\left(\\frac{d}{dx}\\right)^{n}\\left[(1-x)^{\\alpha+n}(1+x)^{\\beta+n}\n\\right].\n\\end{eqnarray}\n\nNow, we expand $_{s}U_{\\ell m}(x)$ and $_{s}V_{\\ell m}(x)$ in a series of\nJacobi polynomials: \n\\begin{eqnarray}\n\\label{eq:Jacobi-series}\n_{s}U_{\\ell m}(x)&=&\\sum_{n=0}^{\\infty}\\,_{s}A_{\\ell m}^{(n)}(\\xi)\\,P_{n}^{(\\alpha,\\beta)}(x) \\,,\n\\\\ \n\\label{eq:Jacobi-series2}\n_{s}V_{\\ell m}(x)&=&\\sum_{n=0}^{\\infty}\\,_{s}B_{\\ell m}^{(n)} \\,P_{n}^{(\\alpha,\\beta)}(x) \\,.\n\\end{eqnarray}\nThe expansion coefficients $_{s}A_{\\ell m}^{(n)}(\\xi)$ \nand $_{s}B_{\\ell m}^{(n)}(\\xi)$  satisfy the recurrence \nrelations\n\\begin{eqnarray}\n\\alpha^{(0)}\\,_{s}A_{\\ell m}^{(1)}(\\xi)\n+\\beta^{(0)}\\,_{s}A_{\\ell m}^{(0)}(\\xi)&=&0, \n\\label{eq:3termElm0}\\\\\n\\alpha^{(n)}\\,_{s}A_{\\ell m}^{(n+1)}(\\xi)\n+\\beta^{(n)}\\,_{s}A_{\\ell m}^{(n)}(\\xi)\n+\\gamma^{(n)}\\,_{s}A_{\\ell m}^{(n-1)}(\\xi)&=&0, \\, (n\\ge 1) \\,,\n\\label{eq:3termElm}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\alpha^{(n)}&:=&\n\\frac{4\\xi(n+\\alpha+1)(n+\\beta+1)(n+(\\alpha+\\beta)\/2+1-s)}\n{(2n+\\alpha+\\beta+2)(2n+\\alpha+\\beta+3)},\\nonumber \\\\\n\\beta^{(n)}&:=&\n\\,_{s}E_{\\ell m}(\\xi)+\\xi ^2-\\left(n+\\frac{\\alpha+\\beta}{2}\\right)\n\\left(n+\\frac{\\alpha+\\beta}{2}+1\\right)\\nonumber \\\\\n&&+\\frac{2\\xi s(\\alpha-\\beta)(\\alpha+\\beta)}\n{(2n+\\alpha+\\beta)(2n+\\alpha+\\beta+2)},\\nonumber \\\\\n\\gamma^{(n)}&:=&\n-\\frac{4\\xi n(n+\\alpha+\\beta)(n+(\\alpha+\\beta)\/2+s)}\n{(2n+\\alpha+\\beta-1)(2n+\\alpha+\\beta)} \\,,\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\tilde{\\alpha}^{(0)}\\,_{s}B_{\\ell m}^{(1)}(\\xi)\n+\\tilde{\\beta}^{(0)}\\,_{s}B_{\\ell m}^{(0)}(\\xi)\n&=&0, \\nonumber \\\\\n\\tilde{\\alpha}^{(n)}\\,_{s}B_{\\ell m}^{(n+1)}(\\xi)\n+\\tilde{\\beta}^{(n)}\\,_{s}B_{\\ell m}^{(n)}(\\xi)\n+\\tilde{\\gamma}^{(n)}\\,_{s}B_{\\ell m}^{(n-1)}(\\xi)&=&0, \\quad (n\\ge 1) \\,,\n\\label{eq:3termElm2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\tilde{\\alpha}^{(n)}&:=&\n-\\frac{4\\xi(n+\\alpha+1)(n+\\beta+1)(n+(\\alpha+\\beta)\/2+1+s)}\n{(2n+\\alpha+\\beta+2)(2n+\\alpha+\\beta+3)},\\nonumber \\\\\n\\tilde{\\beta}^{(n)}&:=&\n\\,_{s}E_{\\ell m}(\\xi)+\\xi ^2-\\left(n+\\frac{\\alpha+\\beta}{2}\\right)\n\\left(n+\\frac{\\alpha+\\beta}{2}+1\\right)\\nonumber \\\\\n&&+\\frac{2\\xi s(\\alpha-\\beta)(\\alpha+\\beta)}{(2n+\\alpha+\\beta)(2n+\\alpha+\\beta+2)},\\nonumber \\\\\n\\tilde{\\gamma}^{(n)}&:=&\n\\frac{4\\xi n(n+\\alpha+\\beta)(n+(\\alpha+\\beta)\/2-s)}\n{(2n+\\alpha+\\beta-1)(2n+\\alpha+\\beta)} \\,.\n\\end{eqnarray}\n\nThe eigenvalues $\\,{_s}E_{\\ell m}(\\xi)$ are determined \nin a way similar to \nthe renormalized angular momentum $\\nu$. \nThe three-term recurrence relation Eq.~(\\ref{eq:3termElm}) \nhas two independent solutions, which respectively behave for large $n$ as\n\\begin{eqnarray}\n&&\nA_{(1)}^{(n)}\\sim \\frac{({\\rm const.})\\,(-\\xi)^n}\n{\\Gamma(n+(\\alpha+\\beta+3)\/2-s)} \\,, \\label{eq:AlmMin}\n\\\\ &&\nA_{(2)}^{(n)}\\sim ({\\rm const.})\\,\\xi^n \\Gamma(n+(\\alpha+\\beta+1)\/2+s)\\,.\n\\label{eq:AlmDom}\n\\end{eqnarray}\nThe first one, $A_{(1)}^{(n)}$, is the minimal solution, and the \nsecond one, $A_{(2)}^{(n)}$, is a dominant solution, since\n$\\displaystyle\\lim_{n\\rightarrow \\infty}A_{(1)}^{(n)}\/A_{(2)}^{(n)}=0$. \nIn the case of the dominant solution\nthese coefficients $A_{(2)}^{(n)}$ increase with $n$, \nand the series~(\\ref{eq:Jacobi-series}) diverges for \nall values of $x$. \nIn the case of the minimal solution \nthis series converges. \nThus, we have to choose $A_{(1)}^{(n)}$ in the \nseries expansion~(\\ref{eq:Jacobi-series}). \nFor a general ${_s}E_{\\ell m}(\\xi)$, \n$A_{(1)}^{n}$ \ndoes not satsify the relation (\\ref{eq:3termElm0}). Hence, \nthe requirement to satisfy this condition determines \nthe descrete eigen values ${_s}E_{\\ell m}(\\xi)$.\n\n{}As a practical way to obtain $A_{(1)}^{(n)}$ \nas well as ${_s}E_{\\ell m}(\\xi)$, \nwe introduce \n\\begin{equation}\nR_n\\equiv {A_{(1)}^{n}\\over A_{(1)}^{n-1}},\\qquad\nL_n\\equiv {A_{(1)}^{n}\\over A_{(1)}^{n+1}}.\n\\end{equation}\nThe ratio $R_n$ can be expressed as a continued fraction,\n\\begin{equation}\nR_n =-{\\gamma^{(n)}\\over {\\beta^{(n)}+\\alpha^{(n)} R_{n+1}}}\n=-{\\gamma^{(n)}\\over \\beta^{(n)}-}\n{\\alpha^{(n)}\\gamma^{(n+1)}\\over \\beta^{(n+1)}-}\n{\\alpha^{(n+1)}\\gamma^{(n+2)}\\over \\beta^{(n+2)}-}\\cdots . \n\\label{eq:RncontElm}\n\\end{equation}\nWe can also express $L_n$ in a similar way as\n\\begin{eqnarray}\nL_n&=&-{\\alpha^{(n)}\\over {\\beta^{(n)}+\\gamma^{(n)} L_{n-1}}}\n\\cr\n&=&-{\\alpha^{(n)}\\over \\beta^{(n)}-}\\,\n{\\alpha^{(n-1)}\\gamma^{(n)}\\over \\beta^{(n-1)}-}\\,\n{\\alpha^{(n-2)}\\gamma^{(n-1)}\\over \\beta^{(n-2)}-}\\cdots\n{\\alpha^{(1)}\\gamma^{(2)}\\over \\beta^{(1)}-}\\,\n{\\alpha^{(0)}\\gamma^{(1)}\\over \\beta^{(0)}}.\n\\label{eq:LncontElm}\n\\end{eqnarray}\nThis expressions for $R_n$ and $L_n$ are valid \nif the continued fraction~(\\ref{eq:RncontElm}) converge. \n(Notice that the last step of Eq.~(\\ref{eq:LncontElm})\nis not a continued fraction, but just a rational function.)\nBy using the properties of the three-term recurrence relations, \nit is proved that the continued fraction~(\\ref{eq:RncontElm})\nconverges as long as the eigenvalue $_{s}E_{\\ell m}(\\xi)$\nis finite. \n\nDividing Eq.~(\\ref{eq:3termElm}) by the expansion coefficients \n$_{s}A_{\\ell m}^{(n)}$, we obtain \n\\begin{eqnarray}\n\\beta^{(n)}+\\alpha^{(n)}R_{n+1}+\\gamma^{(n)}L_{n-1}=0 \\,.\n\\label{eq:determine_elm}\n\\end{eqnarray}\nWe replace $R_{n+1}$ and $L_{n-1}$ by Eqs.~(\\ref{eq:RncontElm})\nand (\\ref{eq:LncontElm}). \nThen we can determine the eigenvalue $\\,_{s}E_{\\ell m}$ as a root of \nEq.~(\\ref{eq:determine_elm}). There are many roots, and the above \nequations for all value of $n$ are equivalent. \nIn practice, however, \nwe truncate the continued fractions at finite lengths. \nIn this case the most efficient way is to choose the equation with \n$n=n_\\ell:=\\ell-(\\alpha+\\beta)\/2$. With this choice all terms in  \nEq.~(\\ref{eq:determine_elm}) become $O(\\xi^2)$, and \nthe length of the continued fractions that we must keep  \nto achieve a given accuracy goal is the shortest.\n\nAs was done in Fujita and Tagoshi's paper, in general, we can \nadopt {\\rm Brent's algorithm}\\cite{Recipes} \nin order to determine $_{s}E_{\\ell m}(\\xi)$. \nHowever, when $|\\xi|$ is not large, \nwe can derive an analytic expression\nfor $_{s}E_{\\ell m}(\\xi)$. The result is \n\\begin{eqnarray}\n_{s}E_{\\ell m}(\\xi) \n    = \\ell (\\ell +1) -\\frac{2 s^2 m}{\\ell (\\ell +1)} \\xi \n\t+ \\left[H(\\ell+1)-H(\\ell)-1\\right]\\xi^2 +O(\\xi^3),\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nH(\\ell)=\\frac{2(\\ell^2-m^2)(\\ell^2-s^2)^2}{(2\\ell-1)\\ell^3(2\\ell+1)}.\n\\end{eqnarray}\n\nAfter we obtain the eigenvalues $_{s}E_{\\ell m}(\\xi)$, \nwe can easily determine all the coefficients. \nThe coefficient with $n=n_\\ell$\nis usually the largest term. The ratio of the other terms \nto the dominant term, i.e.\n$A_{(1)}^{(n)}\/A_{(1)}^{(n_{\\ell})}$, \ncan be determined \nin the most efficient way with a minimal error due to truncation \nusing Eqs.~(\\ref{eq:RncontElm}) and (\\ref{eq:LncontElm})\nfor $0<n<n_\\ell$ and $n>n_\\ell$, respectively.  \n\nThe coefficient of the leading term \n$A_{(1)}^{(n_{\\ell})}$ $\\big({}_{s}A_{\\ell m}^{(n_{\\ell})}\\big)$ \nis determined by the normalization condition. \nSince (\\ref{Vseries}) represents the same eigen function, \nthe series~(\\ref{eq:Jacobi-series2}) should converge \nfor the same eigen values $_{s}E_{\\ell m}(\\xi)$,\nconstituting the minimal solution of the recurrence relation\nEq.~(\\ref{eq:3termElm2}).\nAs in the case of $\\{A_{(1)}^{(n)}\\}$,  \nwe have \n\\begin{eqnarray}\n\\frac{B_{(1)}^{(n)}}{B_{(1)}^{(n-1)}}&=&\n-{\\tilde{\\gamma}^{(n)}\\over \\tilde{\\beta}^{(n)}-}\\,\n{\\tilde{\\alpha}^{(n)}\\tilde{\\gamma}^{(n+1)}\\over \\tilde{\\beta}^{(n+1)}-}\\,\n{\\tilde{\\alpha}^{(n+1)}\\tilde{\\gamma}^{(n+2)}\\over \\tilde{\\beta}^{(n+2)}-}\\cdots , \n\\label{eq:RncontElm2}\n\\\\\n\\frac{B_{(1)}^{(n)}}{B_{(1)}^{(n+1)}}&=&\n-{\\tilde{\\alpha}^{(n)}\\over \\tilde{\\beta}^{(n)}-}\\,\n{\\tilde{\\alpha}^{(n-1)}\\tilde{\\gamma}^{(n)}\\over \\tilde{\\beta}^{(n-1)}-}\\,\n{\\tilde{\\alpha}^{(n-2)}\\tilde{\\gamma}^{(n-1)}\\over \\tilde{\\beta}^{(n-2)}-}\\cdots\n{\\tilde{\\alpha}^{(1)}\\tilde{\\gamma}^{(2)}\\over \\tilde{\\beta}^{(1)}-}\\,\n{\\tilde{\\alpha}^{(0)}\\tilde{\\gamma}^{(1)}\\over \\tilde{\\beta}^{(0)}}.\n\\label{eq:LncontElm2}\n\\end{eqnarray}\nFrom these equations, we can determine the ratios of \nall coefficients, $B_{(1)}^{(n)}\/B_{(1)}^{(n_{\\ell})}$. \n\nNow, we determine the values of the two coefficients\n$A_{(1)}^{(n_{\\ell})}$ and $B_{(1)}^{(n_{\\ell})}$ that \ndetermines the overall normalization.\nSince Eq.~(\\ref{eq:sphUtoV}) must hold for any value of $x$, \nwe can set $x=1$ in it to obtain\n\\begin{eqnarray}\n&& \\hspace*{-1cm}\n{}_{s}B_{\\ell m}^{(n_{\\ell})}(\\xi)\\sum_{n=0}^{\\infty}\n\\frac{\\,_{s}B_{\\ell m}^{(n)}(\\xi)}\n{\\,_{s}B_{\\ell m}^{(n_{\\ell})}(\\xi)}\n\\frac{\\Gamma(n+\\alpha+1)}{\\Gamma(n+1)\\,\\Gamma(\\alpha+1)}\n\\nonumber \\\\\n&& \\hspace*{5mm} =\n{\\rm exp}(2\\xi)\\,_{s}A_{\\ell m}^{(n_{\\ell})}(\\xi)\\sum_{n=0}^{\\infty}\n\\frac{\\,_{s}A_{\\ell m}^{(n)}(\\xi)}{\\,_{s}A_{\\ell m}^{(n_{\\ell})}(\\xi)}\n\\frac{\\Gamma(n+\\alpha+1)}{\\Gamma(n+1)\\,\\Gamma(\\alpha+1)}.\n\\label{eq:normalization1}\n\\end{eqnarray}\nOn the other hand, \nfrom the normalization condition~(\\ref{eq:normalSp}), we find\n\\begin{eqnarray}\n\\int_{-1}^{1} \\!\\! dx\n\\bigg(\\frac{1-x}{2}\\bigg)^{\\alpha}\n\\bigg(\\frac{1+x}{2}\\bigg)^{\\beta} \\!\n\\sum_{n_{1}=0}^{\\infty} \\! {}_{s}A_{\\ell m}^{(n_{1})}\nP_{n_{1}}^{(\\alpha,\\beta)}(x) \\!\n\\sum_{n_{2}=0}^{\\infty} \\! {}_{s}B_{\\ell m}^{(n_{2})}\nP_{n_{2}}^{(\\alpha,\\beta)}(x)=1.\n\\label{eq:AlmBlm}\n\\end{eqnarray}\nBecause the Jacobi polynomials are orthogonal, we have \n\\begin{eqnarray}\n&& \\hspace*{-1cm}\n\\int_{-1}^{1}{\\rm d}x\\left(\n\\frac{1-x}{2}\\right)^{\\alpha}\\left(\\frac{1+x}{2}\\right)^{\\beta}\nP_{n_{1}}^{(\\alpha,\\beta)}(x)P_{n_{2}}^{(\\alpha,\\beta)}(x)\n\\nonumber\\\\\n&&\\quad\n=\\frac{2\\, \\Gamma(n+\\alpha+1)\\Gamma(n+\\beta+1)\n\\delta_{n_{1},n_{2}}}{(2n+\\alpha+\\beta+1)\n\\Gamma(n+1)\\Gamma(n+\\alpha+\\beta+1)}.\n\\end{eqnarray}\nThen, Eq.~(\\ref{eq:AlmBlm}) reduces to \n\\begin{eqnarray}\n&&\\sum_{n=0}^{\\infty}\\left[\n\\frac{\\,_{s}A_{\\ell m}^{(n)}}{\\,_{s}A_{\\ell m}^{(n_{\\ell})}}\\right]\n\\left[\\frac{\\,_{s}B_{\\ell m}^{(n)}}{\\,_{s}B_{\\ell m}^{(n_{\\ell})}}\\right]\n\\frac{2\\, \\Gamma(n+\\alpha+1)\\Gamma(n+\\beta+1)}\n{(2n+\\alpha+\\beta+1)\\Gamma(n+1)\n\\Gamma(n+\\alpha+\\beta+1)}\n\\nonumber\\\\\n&&\\quad\n=\\frac{1}{\\,_{s}A_{\\ell m}^{(n_{\\ell})}\\,_{s}B_{\\ell m}^{(n_{\\ell})}}.\n\\label{eq:normalization2}\n\\end{eqnarray}\nCombining Eqs.~(\\ref{eq:normalization1}) and (\\ref{eq:normalization2}), \nwe can determine the squares of $\\,_{s}A_{\\ell m}^{(n_{\\ell})}$\nand $\\,_{s}B_{\\ell m}^{(n_{\\ell})}$.\nFinally, we fix the signatures of $\\,_{s}A_{\\ell m}^{(n_{\\ell})}$ and\n$\\,_{s}B_{\\ell m}^{(n_{\\ell})}$ so that \n$_{s}S_{\\ell m}^{\\xi}(x)$ reduces to \nthe spin-weighted spherical harmonics \nin the limit $\\xi\\rightarrow 0$. \n\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe ability to prepare and probe individual quantum systems in precisely controlled environments is a driving force in modern atomic, molecular and optical physics. Manipulating single atoms~\\cite{Meschede2006}, molecules~\\cite{Moerner2007} and ions~\\cite{Leibfried2003}, for example, is becoming a common practice. At the heart of these experiments are the powerful imaging techniques which have taken on great importance in diverse areas, such as chemical sensing and chemical reaction dynamics~\\cite{Betzig1993,*Xie1994}, probing superconducting materials~\\cite{Yazdani1997,*Pan2000}, and for quantum logic and quantum information processing~\\cite{Schrader2004,*Nelson2007,*Haffner2005}. More recently, new single atom and single site sensitive imaging techniques for optical lattices have opened the door to control and probe complex many-body quantum systems in strongly correlated regimes~\\cite{Gericke2008,*Bakr2009,*Weitenberg2011}.\n\nThe usual approach to detect atoms is to measure the fluorescence or absorption of light by driving a strong optical cycling-transition. Weak or open transitions present a difficulty since the maximum number of scattered photons per atom becomes greatly limited. In the case of long lived states of trapped ions, the technique of electron shelving has been used as an amplifying mechanism in order to directly observe quantum jumps~\\cite{Nagourney86}. Another approach involves the use of an optical cavity to enhance the interaction of the atoms with the light field~\\cite{Bochmann2010,*Gehr2010,*Brahms2011}. This makes it possible to reach single-atom sensitivity, but usually at the expense of greatly reduced spatial resolution.\n\n\n\\begin{figure}[t!]\n\\centering\\includegraphics[width=0.88\\columnwidth]{figures\/figure1v7.eps}\n\\caption{{\\bf Scheme for imaging individual impurity atoms within a dense atomic gas.} Impurity atoms (crosses) are embedded within a dense two-dimensional atomic gas of background atoms. The background atoms interact with two light fields (coupling and probe) via a two-photon resonance with an excited state $|r\\rangle$. This coupling produces an EIT resonance on the ground-state probe transition. However, strong interactions with an impurity atom lead to a frequency shift $U$ of the resonance within a critical radius $R_c$. The change in absorption properties of many surrounding atoms makes it possible to map the impurity atom distribution to the absorption profile of a probe laser for analysis.} \\label{fig:scheme}\n\\end{figure}\n\nHere we propose a new method to image individual atoms embedded within a dense atomic gas. The concept exploits strong interactions of the atoms with highly polarizable Rydberg states of the surrounding gas. The induced level shifts can then be transferred to a strong optical transition and to many surrounding atoms within a critical radius, thereby providing two mechanisms which greatly enhance the effect of a single impurity on the light field. The Rydberg states could act as non-destructive probes for individual trapped ions, nearby surface charges, dipolar molecules, or other Rydberg atoms. In our approach, the interaction-induced shifts are spatially resolved via an electromagnetically-induced-transparency (EIT) resonance involving a weak probe and a strong coupling laser in a ladder configuration~\\cite{Fleischhauer2005}. Even though the Rydberg state is barely populated, the EIT resonance is extremely sensitive to its properties~\\cite{Mohapatra07,*Weatherill2008,*Pritchard2010,*Schempp2010,Tauschinsky2010}, thereby providing the means to obtain a strong absorption signal and great sensitivity combined with high spatial resolution for detecting individual atoms.\n\nWe exemplify our imaging scheme for the specific case of probing many-body states of strongly-interacting Rydberg atoms in a quasi-two-dimensional atomic gas (depicted in Fig.~\\ref{fig:scheme}). Rydberg atoms are of great interest because their typical interaction ranges are comparable to, or larger than, the typical interatomic separations in trapped quantum gases. Traditionally Rydberg atoms are field ionized and the resulting ions are subsequently detected, which provides rather limited spatial resolution. As a result, much of the work done so far, such as the scaling laws for excitation~\\cite{Low2009}, excitation statistics~\\cite{Liebisch2005,*Amthor2010,*Viteau2011} and light-matter interactions~\\cite{Mohapatra07,*Weatherill2008,*Pritchard2010,*Schempp2010}, has been restricted to the study of cloud averaged properties.  M\\\"uller \\emph{et al.} proposed to use a single Rydberg atom to conditionally transfer an ensemble of atoms between two states~\\cite{Muller2009}. Our method exploits the strong Rydberg interactions with a background gas of atoms to realize non-destructive single-shot optical images of Rydberg atoms with high resolution and enhanced sensitivity. We anticipate this technique will complement the new optical lattice imaging techniques~\\cite{Gericke2008,Bakr2009,*Weitenberg2011}, but with the capability to directly image many-body systems of Rydberg atoms. We show in particular that this will provide immediate experimental access to spatial correlations in recently predicted crystalline states of highly excited Rydberg atoms~\\cite{Weimer2008,*Pohl2010,*Schachenmayer2010,*vanbijnen2011}.\n\nTo quantitatively describe the absorption of probe light by a background gas of atoms surrounding a Rydberg atom we follow an approach based on the optical Bloch equations~\\cite{Fleischhauer2005}. The Hamiltonian describing the atom-light coupling is\n\\begin{eqnarray}\n\\label{eq1}\nH_0 &=& \\frac{\\hbar}{2}\\big(\\Omega_p | e \\rangle \\langle g | + \\Omega_c | r \\rangle \\langle e |  \\nonumber \\\\\n&&+\\Delta_p | e \\rangle \\langle e | + (\\Delta_p+ \\Delta_c) | r \\rangle \\langle r | + \\textrm{h.c.}\\big).\n\\end{eqnarray}\nFor resonant driving $\\Delta_p=\\Delta_c=0$ a dark-state is formed $|dark\\rangle\\approx\\Omega_c|g\\rangle-\\Omega_p|r\\rangle$, which no longer couples to the light field. Consequently, the complex susceptibility $\\chi$ of the probe transition vanishes and the atoms become transparent.\n\nThe presence of a nearby Rydberg atom, however, causes an additional energy shift $U=\\hbar C_6\/|d|^6$ for the state $|r\\rangle$, where $d$ is the distance to the Rydberg atom and the interaction coefficient $C_6$ reflects the sign and strength of interactions on the $|r\\rangle$ state.\nOne should also account for interactions between atoms in state $|r\\rangle$, but these can be neglected for $\\Omega_p\\ll \\Omega_c$ when the population in $|r\\rangle$ becomes small. We also include spontaneous decay from the states $|e\\rangle$ and $|r\\rangle$ with rates $\\Gamma_p$, and $\\Gamma_c$ respectively. From the master equation for the density matrix $\\rho$ we calculate the steady-state absorption and solve for the complex susceptibility of the probe transition numerically.\n\nIn the weak probe limit ($\\Omega_p\\ll \\Omega_c, \\Gamma_{p}$) we assume the population stays mostly in the ground state ($\\rho_{gg}\\approx 1$). In this case we obtain for the susceptibility\n\\begin{eqnarray}\\label{eq:chi}\n\\chi_=\\frac{i\\Gamma_p}{(\\Gamma_p-2i\\Delta_p)+\\Omega_c^2(\\Gamma_c-2i\\Delta)^{-1}},\n\\end{eqnarray}\nwhere $\\Delta=\\Delta_p+\\Delta_c+C_6\/|d|^6$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[height=0.45\\columnwidth]{figures\/fig_susceptibility1.eps}\n\\includegraphics[height=0.45\\columnwidth]{figures\/fig_susceptibility2.eps}\n\\caption{{\\bf Probe absorption given by the imaginary part of the susceptibility.} (a) $\\mathrm{Im}[\\chi]$ as a function of probe detuning for $\\Omega_c=1$, $\\Gamma_c=0.05$ and $\\Delta_c=0$ (in units of $\\Gamma_p$) for various distances from the Rydberg atom. The solid line is for $d\\rightarrow\\infty$, dashed corresponds to $d=R_c$ and the dotted line is for $d=R_c\/2$. (b) Dependence of $\\mathrm{Im}[\\chi]$ as a function of distance from the Rydberg atom with $\\Delta_p=0$. } \\label{fig:susceptibility}\n\\end{figure}\n\nFig.~\\ref{fig:susceptibility} shows the probe absorption proportional to the imaginary part of $\\chi$ for different laser parameters and for different distances to a Rydberg atom. Far from the influence of the Rydberg atom ($d\\rightarrow\\infty$), the susceptibility takes on a characteristic shape with vanishing absorption on resonance. For shorter distances, interactions tend to shift the transparency window and the on-resonant susceptibility increases. At a critical distance $d=R_c$, $\\Chi=1\/2$. For $d<R_c$, the excited states $|r\\rangle$ become far detuned and the background atoms effectively act as two-level systems. In this case the usual Lorentzian lineshape is recovered with maximum absorption on resonance. Hence, in the presence of a Rydberg atom, $N=n_{2D}\\pi R_c^2\\gg 1$ atoms, each with an absorption cross-section $\\sim\\lambda^2$, can scatter many photons to produce a dark disk with radius $R_c$. From Eq.~\\ref{eq:chi} and taking $\\Delta_p=0$ and $\\Gamma_c\\approx0$ we find the critical distance $R_c=(2 C_6\\Gamma_p\/\\Omega_c^2)^{1\/6}$, and the distance dependent susceptibility $\\mathrm{Im}[\\chi]\\approx\\big(1+(d\/R_c)^{12}\\big)^{-1}$. Since for typical parameters $N\\approx 50$, and $R_c\\approx 1~\\mu$m comparable to the optical resolution, the spatially resolved probe absorption provides an excellent signature for the presence of a Rydberg atom within a dense gas.\n\nTo apply this scheme to realistic situations one also has to analyze the influence of noise. We identify two major sources, one can be attributed to photon shot noise of the probe while a second contribution is associated with intrinsic atomic density fluctuations. Both noise sources can be accurately described as Poissonian processes. This suggests that the signal-to-noise ratio can be made arbitrarily large for large intensity and large density. However, to neglect interactions between background atoms we require the probability to find more than one atom in the $|r\\rangle$ state within the range of background-background interactions $R_c'$ to be $\\lesssim 1$ (for the states we consider $R_c'\\approx R_c$). This constraint imposes a relationship between the maximum density of background atoms and the maximum probe intensity $n_{2D} \\lesssim \\Omega_c^2\/\\pi R_c'^2\\Omega_p^2$. Above this critical density the contrast of the image would decrease due to the blockade effect, where only one atom can contribute to the EIT signal, while the remainder act as two-level atoms and couple resonantly to the probe laser\\cite{Pritchard2010,Ates2011}. In general, the maximum signal-to-noise ratios (see supplementary material) are achieved for large coupling strengths $\\Omega_c$ and long exposure times $\\tau$, but in practice these will be limited by the available laser power and by the required time resolution, which should be compared to the typical lifetime of a Rydberg atom ($\\sim100~\\mu$s).\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.49\\columnwidth]{figures\/fig_absimage1.eps}\n\\includegraphics[width=0.49\\columnwidth]{figures\/fig_absimage2.eps}\n\\caption{{\\bf Simulated absorption images of atom distributions including photon shot noise and atomic density fluctuations.}\nIn (a), without the coupling beam, a regular absorption image of the background atoms is obtained. The color code indicates absorption. With the coupling on (b) the background atoms are rendered transparent, except for those in the vicinity of a Rydberg atom. Parameters of the simulation can be found in the text.} \\label{fig:EITimage}\n\\end{figure}\n\nTo show the potential of this imaging scheme we have carried out numerical calculations of the EIT imaging process on simulated distributions of Rydberg atoms excited from a quasi-2D ideal gas. This situation can be realized with an optical dipole trap made using cylindrically focused Gaussian beams. Of particular interest for current experiments is the possibility to directly observe strong spatial correlations between Rydberg atoms induced by interactions in an otherwise disordered gas~\\cite{Robicheaux2005, Amthor2010,Weimer2008,*Pohl2010,*Schachenmayer2010,*vanbijnen2011}.\n\nWe use a simple semi-classical model to simulate the excitation of Rydberg atoms during a chirped Rydberg excitation pulse. The model is closely related to those used to describe optical control of cold collisions~\\cite{Suominen1996,Wall2007}. We consider a thermally distributed gas of 25\\,000 background atoms with a peak density $n_{2D} = 40\\,\\rm{atoms}\/\\mu$m$^2$ and a cloud radius of $\\sigma = 10\\,\\mu m$. Each atom can be in either the electronic ground state or a Rydberg state. We take the strength of the Rydberg-Rydberg interactions as $2\\pi\\times50~$GHz$~\\mu m^6$, typical of the $55S$ state~\\cite{Singer2005}. The detuning of the coupling field is swept from 0 to $+200$~MHz within $6~\\mu$s with an effective Rabi frequency of $\\Omega= 2\\pi\\times2.0~$MHz. During each time step atoms can undergo a transition to the Rydberg state with a probability estimated from the Landau-Zener formula~\\cite{Wittig2005}. Successive excitation events are treated independently, however the previously excited atoms influence successive transitions through the Rydberg-Rydberg interactions. This model also includes the effects of mechanical forces between Rydberg atoms by simultaneously solving the classical equations of motion for each Rydberg excited atom.\n\n We then calculate absorption images of the simulated Rydberg distributions by numerically solving the optical Bloch equations for the background atoms at each spatial position, accounting for the level-shifts produced by all Rydberg atoms. From the mean intensity at each pixel we generate Poisson distributed photon-shot noise. Similarly, a reference image is generated with uncorrelated noise for background division. Fig.~\\ref{fig:EITimage} shows calculated absorption images without and with the coupling laser from a single run of the simulation. Each pixel corresponds to a region of $(0.5\\,\\mu\\rm{m})^2$ in the plane of the atoms and we assume a numerical aperture of $0.25$ and an exposure time of $10\\,\\mu$s. For the EIT ladder system, we take the $^{87}$Rb states, $|5S_{1\/2}, F=2, m_F = 2\\rangle $ for the ground state, $|5P_{3\/2}, F=3, m_F = 3\\rangle$ for the intermediate state, and $|r = 28S\\rangle$ for the excited state. The decay rates are $\\Gamma_p=2\\pi\\times 6.1$ MHz and $\\Gamma_c \\approx 2\\pi\\times 10\\,$kHz, and for the coupling laser we assume $\\Omega_c = 2 \\pi\\times 50\\,$MHz. Laser line widths of $2\\pi \\times 1\\,$MHz were assumed for both probe and coupling lasers. The interactions between $|55S\\rangle$ and $|28S\\rangle$ states were calculated to obtain a van der Waals coefficient of $C_6(28S-55S) = -2\\pi\\times 8.7\\,\\rm{MHz}\\,\\mu \\rm{m}^6$ giving $R_c = 0.59\\,\\mu$m. Interactions between background atoms are taken as $C_6(28S-28S) = 2\\pi\\times 10.1\\,\\rm{MHz}\\,\\mu \\rm{m}^6$ ($R_c'=0.61\\mu$m).  For these parameters the optimal signal-to-noise ratio is obtained for $n_{2D}\\approx 40\\,$atoms\/$\\mu$m$^2$. The probe intensity ($\\Omega_p =2\\pi \\times 5.8\\,$MHz) is chosen such that on average the $|r\\rangle$ state density remains below 1 per $\\pi R_c'^2$. Such parameters are readily achieved in current experiments with quasi-2D atomic gases~\\cite{Gorlitz01}.\n\nIn the background region of the image the signal is dominated by photon-shot noise, while at the center atom-shot noise dominates. With the coupling laser on the ground state atoms are rendered mostly transparent, except for regions of high absorption around each Rydberg atom (Fig.~\\ref{fig:EITimage}b). The locations of the individual Rydberg atoms are clearly resolved in the image as bright (absorbing) spots with a spatial extent of $2.3~\\mu$m FWHM comparable to the assumed optical resolution. One can easily envisage higher resolutions using state-of-the-art imaging systems~\\cite{Bakr2009,*Weitenberg2011}, with the fundamental limit given by the density of background atoms surrounding the impurities. The signal-to-noise ratio of our images is sufficiently high that we can fit the position of each Rydberg atom with subpixel precision.\n\n Despite the simplicity of the excitation model, the final distribution of Rydberg atoms appears highly-correlated, reproducing some of the features of a full quantum mechanical treatment~\\cite{Pohl2010}. To characterize the translational order of the simulated Rydberg distributions, we calculate the pair distribution function $g(r)$ from 15 simulated images (Fig.~\\ref{fig:correlation}). To account for the inhomogeneous density distribution we also normalize by the autocorrelation of the mean image (see supplementary material). For a random distribution of atoms $g(r) \\approx 1$. Larger correlation values indicate an enhanced probability to find two Rydberg atoms at a given separation, while lower values indicate the absence of pairs. Since there is no preferred orientation in our system $g(r)$ takes on cylindrical symmetry. We clearly observe a shell with $g(r)\\approx 0$ at a radius of $\\sim$2.5~$\\mu$m which reflects the strong blockade of excitation due to Rydberg-Rydberg interactions. At larger distances, we observe two positive-correlated shells (around 4 and 8~$\\mu$m), which indicate translational correlations between nearest- and next-nearest neighbours. The observed shell structure decays rapidly indicating the absence of true long-range order. We note very similar behaviour of $g(r)$ for the raw atom positions (shaded bars). From this we conclude that the information regarding density-density correlations can be reliably extracted from the images, even under realistic imaging conditions.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{figures\/fig4.eps}\n\\caption{\\label{fig:correlation}\n{\\bf Pair distribution function computed for simulated Rydberg images from 15 realisations.} The inset shows the averaged two-dimensional autocorrelation function as computed from the absorption images. Taking the radial average gives $g(r)-1$ as shown in the main figure (solid black line). The shaded bars show $g(r)-1$ with $0.5~\\mu$m bin size as obtained from the simulated Rydberg atom coordinates. The clear shell structure which reflects translational order between nearest and next-nearest neighbours is preserved by the images.\n}\n\\end{figure}\n\nWe have also computed the angular correlation function $\\Phi(\\theta)$ at the radius of the first shell (see supplementary material). This gives the probability, starting from an atom to find two neighbours forming an angle $\\theta$. We observe the presence of two peaks at $\\sim$1.1 and $\\sim$5.3 rad, corresponding closely to $\\pi\/ 3$ and $5 \\pi \/ 3$, reflecting the 6-fold symmetry present among most nearest-neighbours. Even more information could be obtained from these images by studying higher-order correlation functions, or by first estimating the Rydberg atom positions to fully characterize the many-body state.\n\nOur new imaging method provides the means to optically image individual atoms within a dense atomic gas using Rydberg state electromagnetically induced transparency. The conditions we find to optimise the signal closely match those of current cold atom experiments. As an example of the full potential of this new imaging scheme we have carried out numerical simulations of Rydberg atoms excited from a quasi-2D gas. Remarkably, this simple model already gives rise to the appearance of strong spatial correlations between Rydberg atoms. The EIT imaging scheme provides the means for single-shot, non-destructive and time resolved images of such many-body states. We can foresee numerous other applications of the EIT imaging method. For example, Rydberg state EIT has already been used to measure spatially inhomogeneous electric fields near a surface~\\cite{Tauschinsky2010}. Another exciting prospect would be to directly image single ions within an atomic gas which could be directly observed in current experiments~\\cite{Grier2009,*Zipkes2010,*Schmid2010}. Closely related ideas could be used to realise a single atom optical transistor~\\cite{Hwang2009} or to impose non-classical spatial correlations onto the light field~\\cite{Sevlinci2011}.\n\n\\acknowledgements{ We would like to thank J. Evers, B. Olmos and I. Lesanovsky for valuable discussions. This work is supported in part by the Heidelberg Center for Quantum Dynamics and the Deutsche Forschungsgemeinschaft under WE2661\/10.1. SW acknowledges support from the EU Marie-Curie program (grant number PERG08-GA-2010-277017).}\n\n\\emph{Note added.--} During preparation of this manuscript we became aware of related work~\\cite{Olmos2011}\n\n\n\\section{Supplementary material}\n\\subsection{Influence of noise}\n\nWe provide the criteria to optimize the quality of images in the presence of realistic noise sources. Suppose an impurity atom is situated in region $(A)$ and we wish to distinguish its position from another region $(B)$ by measuring the difference in transmission $ \\Delta T$. $N_{ph}^{(A\/B)}$ is the number of detected photons (proportional to CCD counts; we assume a quantum efficiency $\\approx1$) and $\\NR$ is a reference used to normalize probe intensity variations. We consider the resonant case ($\\Delta_p=\\Delta_c=0$) since this maximizes the contrast while keeping the density of atoms in state $|r\\rangle$ small. In region $(A)$ the absorption is greatly enhanced and $N_{ph}^{A}\\approx T_A\\NR$ with $T_A=\\exp{(-\\sigma_0 n_{2D} \\Chi)}$ and $\\sigma_0$ is the resonant absorption cross-section for the probe transition. In region $(B)$ $\\ChiB\\approx 0$ therefore $\\Delta T >0 $.\n\nThe precision with which a measurement of $\\Delta T$ can be made depends on the noise in both regions:\n\\begin{eqnarray}\\label{eq:varT1}\n\\var(\\Delta T)\\!\\approx\\! \\frac{\\var(\\NR)\\langle\\NA\\rangle^2}{\\langle\\NR\\rangle^4}\\!+\\!\\frac{\\var(\\NA)}{\\langle\\NR\\rangle^2}\\!+\\!\\frac{2 \\, \\var(\\NR) }{\\langle\\NR\\rangle^2}.\\nonumber\n\\end{eqnarray}\nWe assume Poisson distributed noise for the intensity and density fluctuations, so $\\var(\\NR)=\\nobreak\\langle\\NR\\rangle$ and $\\var(N_{ph})\\approx\\nobreak\\langle T_A\\rangle\\langle\\NR\\rangle+\\langle\\NR\\rangle^2\\var(T_A)$. Atom shot noise is accounted for by $\\var(T_A)=\\nobreak\\sigma_0^2 \\ChiA^2\\langle T_A\\rangle^2 n_{2D}\/ a$, with $a$ the area of each region (for example the area of a pixel),\n\\begin{eqnarray}\n\\var(\\Delta T)\\!=\\!\\frac{\\langle T_A\\rangle\\!+\\!\\langle T_A\\rangle^2}{\\langle\\NR\\rangle}\\!+\\!  \\frac{2}{\\langle\\NR\\rangle}\\!+\\!\\frac{\\sigma_0^2n_{2D}}{a} \\ChiA^2\\langle T_A\\rangle^2.\\nonumber\n\\end{eqnarray}\nThe first two terms can be attributed to photon shot noise while the last term is from density fluctuations. Including saturation, $\\ChiA=\\nobreak\\Gamma_p^2\/(\\Gamma_p^2+\\nobreak 2 \\Omega_p^2)$. This suggests that the signal-to-noise ratio (SNR) can be made arbitrarily high for large $\\langle \\NR\\rangle$ and large $n_{2D}$. However, to ensure that interactions between background atoms can be neglected, we require that the density of atoms in the $|r\\rangle$ state is kept low ($\\rho_{rr}n_{2D}\\pi R_c'^2\\lesssim 1$). For strong coupling $\\rho_{rr}\\approx \\Omega_p^2\/\\Omega_c^2$ and this implies $\\langle\\NR\\rangle\\lesssim\\nobreak a \\tau \\Omega_c^2\/\\sigma_0n_{2D} \\pi R_c'^2\\Gamma_p$, with exposure time $\\tau$. In the limit of strong absorption $\\langle T_A\\rangle\\ll 1$, and substituting for the maximum value of $\\langle\\NR\\rangle$:\n\\begin{eqnarray}\\label{eq:varfinal}\n\\var(\\Delta T)&=& \\frac{2 \\sigma_0 \\Gamma_p n_{2D}\\pi R_c'^2}{a \\Omega_c^2 \\tau}\n\\\\ \\nonumber\n&\\times &\n\\biggr(1+\\frac{\\Omega_c^2 \\tau \\sigma_0}{2 \\pi \\Gamma_p R_c'^2}\\ChiA^2\\exp{(-2 \\sigma_0 n_{2D} \\ChiA})\\biggr)\n\\end{eqnarray}\nwith $\\ChiA=\\big(1+2\\Omega_c^2\/\\Gamma_p^2 \\pi R_c'^2 n_{2D}  \\big)^{-1}$.\n\nIn general, the best SNR is obtained for large coupling strengths $\\Omega_c$ and long exposure times $\\tau$, but in practice these will be limited by the available laser power and by the required time resolution. To find the optimal values for $n_{2D}$ and $\\Omega_p$ given fixed values of $\\tau$ and $\\Omega_c$ we numerically maximize the SNR using Eq.~\\eqref{eq:varfinal}. The final parameters used in the paper include the additional effect of finite laser linewidths which tends to increase $\\rho_{rr}$ slightly for the same $\\Omega_p$. This shifts the optimum density to slightly lower values. Assuming $\\Omega_c=2\\pi\\times 50$~MHz and $\\tau=10~\\mu$s we find $n_{2D}^{opt}=40 \\mu$m$^{-2}$ (neglecting linewidth $n_{2D}^{opt}\\approx 50 \\mu$m$^{-2}$).\n\n\\subsection{Rydberg excitation model}\n\nTo simulate the excitation of Rydberg atoms by a chirped laser pulse we consider a randomly (thermally) distributed ensemble of atoms. Each atom is treated as a point-like classical particle which can be in either the electronic ground state or in a Rydberg state. As the coupling field is swept from low to high detuning, each atom can undergo a transition. The transition probability is estimated using the Landau-Zener (LZ) formula for a sweep through an avoided crossing \\cite{Wittig2005}. The effect of Rydberg-Rydberg interactions causes level shifts for the nearby atoms which subsequently alters their probability to be excited by the laser pulse, giving rise to strong spatial correlations.\n\nThe simulation starts with zero detuning for the excitation laser and one atom is chosen at random to start in the Rydberg state. In the next time step the laser frequency is varied according to a fixed sweep rate, and we calculate all level shifts due to Rydberg-Rydberg interactions. From the atoms which crossed the resonance condition in the previous timestep we randomly select newly excited atoms based on their LZ probabilities. Any successful excitation immediately influences all other surrounding atoms, and thus the simulation also reproduces the excitation blockade effect. For each time step we also solve the Newtonian equations of motion of the Rydberg atoms to account for the interparticle mechanical forces. We do not consider the motion of the ground state atoms for the simulation (frozen gas regime). The simulation returns a list of the final coordinates of all the ground-state and Rydberg atoms within the gas after the laser sweep. These coordinates are then used as inputs to calculate the corresponding absorption image.\n\n\\subsection{Correlation analysis}\n\nTo characterize the translational order of the simulated Rydberg distributions, we define a pair distribution function from the absorption images $n(\\vec r)$:\n\\begin{equation}\nG[n](\\vec r) = \\frac{ \\int d^2r_0  ~n(\\vec r_0) n(\\vec r_0 + \\vec r)} { \\left( \\int d^2r_0 ~ n(\\vec r_0)    \\right) ^2}.\n\\end{equation}\nTo account for the inhomogeneous density and finite size of the system we define the following rescaled pair distribution function :\n\\begin{equation}\ng(\\vec r)  = \\frac{\\langle G_2[n] \\rangle }{ G_2[\\langle n\\rangle] }\n\\label{eq:g2}\n\\end{equation}\nwhere the brackets reflect averages over independent realisations. For a random distribution of atoms $g(r) \\approx 1$. Larger correlation values indicate an enhanced probability to find two Rydberg atoms at a given separation, while lower values indicate the absence of pairs.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{figures\/fig-suppl.eps}\n\\caption{\\label{fig:correlation2}\n{\\bf Angular correlation function computed from 15 simulated images.} Angular correlation function $\\Phi(\\theta )$ taken at the radius of the first shell. We observe peaks at angles around $\\pm \\pi \/ 3$ indicating a six-fold symmetry among nearest neighbours.\n}\n\\end{figure}\n\nWe can also extract information about the angular correlations in the images. For this we define the angular correlation function :\n\\begin{equation}\n\\Phi(\\theta)\\!\\propto\\!\\left\\langle\\! \\frac{        \\int\\! d^2r_0~n(\\vec r_0)  \\int\\! d \\phi~n(\\vec r_0\\!+\\! R_{nn}\\vec{e}_{\\phi} ) n(\\vec r_0\\!+\\!R_{nn} \\vec{e}_{\\phi+\\theta} )                       }                  { \\left(\\int d^2r_0  ~n(\\vec r_0) \\right) ^3}  \\!\\right\\rangle\n\\label{eq:g3}\n\\end{equation}\nwhere $\\vec e_{\\phi}$ is defined as the unit vector with angle $\\phi$ with respect to a reference axis $\\vec e_x$, and $R_{nn}$ is the radius of the first positive shell of the pair distribution function. This gives the probability, starting from an atom and one of its nearest neighbours, to find a second nearest neighbour forming an angle $\\theta$ with the first.\n\nFigure~\\ref{fig:correlation2} shows the angular correlation function computed from 15 simulated images at the radius of the first shell. We observe two clear peaks at $\\sim \\pi\/3$ and $\\sim 5\\pi\/3$ reflecting the 6-fold symmetry present among nearest neighbours. The other peaks at $\\theta=2n\\pi\/6,n=2,3,4$ are washed out indicating the absence of true long range orientational order.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDifferential equations are used for an enormous variety of applications, including industrial design and weather prediction. In fact, many of the main applications of supercomputers are in the form of large systems of differential equations \\cite{super}. Therefore quantum algorithms for solving differential equations would be extraordinarily valuable. A quantum algorithm for differential equations was proposed in Ref.\\ \\cite{Leyton08}, but that algorithm had very poor scaling in the time. The complexity of the simulation scaled exponentially in the number of time-steps over which to perform the simulation.\n\nThe algorithm in Ref.\\ \\cite{Leyton08} may have been overly ambitious, because it aimed to solve nonlinear differential equations. A more natural application for quantum computers is \\emph{linear} differential equations. This is because quantum mechanics is described by linear differential equations. We find that, when we restrict to linear differential equations, it is possible to obtain an algorithm that is far more efficient than that proposed in Ref.\\ \\cite{Leyton08}.\n\nWe consider first-order linear differential equations. Using standard techniques, any linear differential equation with higher-order derivatives can be converted to a first-order linear differential equation with larger dimension. A first-order ordinary differential equation may be written as\n\\begin{equation}\n\\dot x(t) = A(t)x(t) + b(t),\n\\end{equation}\nwhere $x$ and $b$ are $N_x$-component vectors, and $A$ is an $N_x\\times N_x$ matrix. Classically, the complexity of solving the differential equation must be at least linear in $N_x$. The goal of the quantum algorithm is to solve the differential equation in time $O(\\poly\\log N_x)$.\n\nQuantum mechanics is described by differential equations of this form, except they are homogeneous ($b(t)=0$), and $A(t)=iH(t)$, where $H(t)$ is Hermitian. This means that the solutions in quantum mechanics only include oscillating terms, whereas more general differential equations have solutions that may grow or decay exponentially. Quantum algorithms for simulating quantum mechanical systems have been extensively studied \\cite{Lloyd96,Aharonov03,Childs04,Berry07,Childs08,Berry09,Wiebe10}.\n\nClassical physics is described by more general differential equations. Large systems of ordinary differential equations are produced by discreti{\\s}ation of partial differential equations. Many equations in physics are linear partial differential equations, where the time derivative depends linearly on spatial derivatives and the value of a quantity at some point in physical space. Examples include Stokes equations (for creeping fluid flow), the heat equation, and Maxwell's equations. Discreti{\\s}ation of the partial differential equation on a mesh of points results in an ordinary differential equation with a very large value of $N_x$.\n\nIn the case where $A$ and $b$ are time independent, then one can find the equilibrium solution of the differential equation by solving\n\\begin{equation}\nA x = -b.\n\\end{equation}\nA quantum algorithm for this problem was given by Harrow, Hassadim and Lloyd \\cite{Harrow09}, with runtime that is polynomial in $\\log(N_x)$ and the condition number of $A$. Ambainis has reported development of an improved algorithm \\cite{Ambainis10}, though this algorithm has not yet been released. We consider the more difficult case of solving the time evolution under linear differential equations, rather than just the equilibrium solution. We find that this case can also be solved using a modification of the method of Harrow, Hassadim and Lloyd.\n\n\\section{Trotter formula approach}\n\nBefore explaining that approach, we first describe an approach using Trotter formulae, and the drawback to that approach. This will not be described rigorously, because it is not our main proposal for solving differential equations.\n\nThe homogeneous case, where $b=0$, is analogous to Hamiltonian evolution. If $A$ is antiHermitian, then we can take $A=iH$, where $H$ is a Hermitian Hamiltonian. Evolution under this Hamiltonian can be solved by methods considered in previous work \\cite{Berry07,Berry09}. Another case that can be considered is where $A$ is Hermitian. In this case, the eigenvalues of $A$ are real, and $A$ can be diagonali{\\s}ed in the form $A=V D V^{-1}$, where $D$ is a real diagonal matrix and $V$ is unitary. The formal solution is then, for $A$ independent of time, $x(t)=V e^{D(t-t_0)} V^{-1} x(t_0)$.\n\nThe differential equation can be solved using a similar method to that used in Ref.\\ \\cite{Harrow09}. The value of $x$ is encoded in a quantum state as\n\\begin{equation}\n\\ket{x} = {\\cal N}_x \\sum_{j=1}^{N_x} x^{[j]} \\ket{j},\n\\end{equation}\nwhere $\\ket{j}$ are computational basis states of the quantum computer, $x^{[j]}$ are the components of the vector $x$, and ${\\cal N}_x$ is a normali{\\s}ation constant. The state can be written in a basis corresponding to the eigenvectors of $A$:\n\\begin{equation}\n\\ket{x} = \\sum_{j} \\lambda_j \\ket{\\lambda_j}.\n\\end{equation}\nUsing methods for Hamiltonian simulation, $iA$ can be simulated. By using phase estimation, if the state is an eigenstate $\\ket{\\lambda_j}$, then the eigenvalue $\\lambda_j$ can be determined. Given maximum eigenvalue $\\lambda_{\\rm max}$, we would change the amplitude by a factor of $e^{(t-t_0)(\\lambda_j-\\lambda_{\\rm max})}$. See Ref.\\ \\cite{Harrow09} for the method of changing the amplitude. If this is done coherently, then the final state will encode $x(t)$.\n\nFor more general differential equations, $A$ will be neither Hermitian nor antiHermitian. In this case, one can break $A$ up into Hermitian ($A_{H}$) and antiHermitian ($A_{aH}$) components. The evolution under each of these components can be simulated individually, and the overall evolution simulated by combining these evolutions via the Trotter formula. The drawback to this approach is that it appears to give a complexity that increases exponentially with the time interval $\\Delta t = t-t_0$ (though the complexity is still greatly improved over Ref.\\ \\cite{Leyton08}).\n\nIf $A$ were just Hermitian, then the eigenvector (or eigenspace) corresponding to the largest eigenvalue would not decay, and the system would end up in that state. Therefore the amplitude would not drop below the amplitude on the eigenspace corresponding to the largest eigenvalue. That is not the case when $A$ is a more general matrix, because usually the maximum real part of an eigenvalue of $A$ will be strictly less than the maximum eigenvalue of $A_H$. The amplitude must therefore decay exponentially, because we must use the maximum eigenvalue of $A_H$ in simulating evolution under $A_H$.\n\nThe result of this is that the complexity of the simulation will scale exponentially in the time that the differential equation needs to be simulated over, $\\Delta t$. The scaling will be considerably improved over that in Ref.\\ \\cite{Leyton08}, but it is desirable to obtain scaling that is polynomial in $\\Delta t$. Another drawback is that this approach does not enable simulation of inhomogeneous differential equations.\n\n\n\n\\section{Linear systems approach}\n\nTo avoid this problem we propose an approach based on the algorithm for solving linear systems from Ref.\\ \\cite{Harrow09}. The trick is to encode the solution of the differential equation at different times using the one state. That is, we wish to obtain the final state proportional to\n\\begin{equation}\n\\label{eq:fineq}\n\\ket{\\psi} := \\sum_{j=0}^{N_t} \\ket{t_j} \\ket{x_j}.\n\\end{equation}\nThe number $N_t$ is the number of time steps, $t_j$ is the time $t_0+j\\dt$, where $\\dt$ is the time interval in the discreti{\\s}ation of the differential equation, $x_j$ is the approximation of the value of $x$ at time $t_j$, and $\\Delta t$ is the total time interval over which the differential equation is to be solved. We use the subscript $j$ to index the vectors, and superscript for components of these vectors.\n\nOnce this state has been created, the state encoding the solution at the final time $t_0+\\Delta t$ can be approximated by measuring the register encoding the time and getting that time. Just using this method, the probability of obtaining the final time is small ($1\/(N_t+1)$). To obtain a significant probability of success, one can add times beyond $t_0+\\Delta t$ where $x$ is constant. We take $x$ to be constant for $t_0+\\Delta t$ to $t_0+2\\Delta t$, so $N_t=2\\Delta t\/\\dt$. Then any measurement result for the time in this interval will give the state corresponding to the solution. By this method, the probability of success can be boosted significantly, without changing the scaling for $N_t$.\n\nTo numerically solve differential equations, the simplest method is the Euler method, which discreti{\\s}es the differential equation as\n\\begin{equation}\n\\frac{x_{j+1}-x_j}{\\dt} = A(t_j) x_j +b(t_j).\n\\end{equation}\nFor times after $t_0+\\Delta t$, we set $x_{j+1}=x_j$ to ensure that $x$ is constant. The Euler method yields an error that scales as $O(\\dt^2)$ for a single time step. Therefore, we expect that the error in the total simulation is $O(N_t \\dt^2)=O(\\Delta t^2\/N_t)$. To achieve error bounded by $\\epsilon$, we can take $N_t = O(\\Delta t^2\/\\epsilon)$. To show these scalings rigorously requires additional constraints on the problem.\n\nIn particular, to rigorously bound the error it is necessary that the eigenvalues of $A(t_j)$ have no positive real part. Otherwise the error can grow exponentially. In cases where $A(t_j)$ does have an eigenvalue with positive real part, one can simply subtract a multiple of the identity, and rescale the solution. Note that $\\epsilon$ is the error in the solution of the differential equation, and is distinct from error in the solution of linear systems.\n\nMore generally, linear multistep methods have the form \\cite{Butcher,Hairer}\n\\begin{equation}\n\\label{eq:multi}\n\\sum_{\\ell=0}^{k} \\alpha_\\ell x_{j+\\ell} = \\dt \\sum_{\\ell=0}^{k}\\beta_\\ell [A(t_{j+\\ell})  x_{j+\\ell}+b(t_{j+\\ell})].\n\\end{equation}\nMultistep methods can be chosen such that the error is of higher order in $\\dt$, but there is the problem that the method may not be stable. That is, even if the exact solution of the differential equation is bounded, the solution of the difference equation may be unbounded.\n\nTo examine the stability, one defines the generating polynomials\n\\begin{equation}\n\\rho(\\zeta)=\\sum_{j=0}^k \\alpha_j \\zeta^j, \\qquad \\sigma(\\zeta) = \\sum_{j=0}^k \\beta_j \\zeta^j.\n\\end{equation}\nThe stability can be examined via the roots of the equation\n\\begin{equation}\n\\label{eq:stpol}\n\\rho(\\zeta)-\\mu \\sigma(\\zeta) = 0.\n\\end{equation}\nOne defines the set $S$ by\n\\begin{equation}\nS := \\left\\{ \\mu\\in {\\mathbb{C}}; \\begin{array}{*{20}l}\n{{\\rm all~roots~} \\zeta_j(\\mu) {\\rm~of~} \\eqref{eq:stpol} {\\rm~satisfy~} |\\zeta_j(\\mu)|} \\le 1 \\\\\n{{\\rm multiple~roots~satisfy~} |\\zeta_j(\\mu)|< 1} \\\\ \\end{array} \\right\\}.\n\\end{equation}\n$S$ is called the stability domain or stability region of the multistep method. In addition, if the roots of $\\sigma(\\zeta)$ all satisfy $|\\zeta|\\le 1$, and repeated roots satisfy $|\\zeta|<1$, then the method is said to be stable at infinity.\n\nA linear multistep method is said to be order $p$ if it introduces local errors $O(\\dt^{p+1})$. This means that, if it is applied with exact starting values to the problem $\\dot x = t^q$ ($0\\le q \\le p$), it integrates the problem without error. A linear multistep method has order $p$ if and only if \\cite{Butcher}\n\\begin{equation}\n\\rho(e^h)-h\\sigma(e^h) = O(h^{p+1}).\n\\end{equation}\n\nA useful property of linear multistep methods is for them to be $A$-stable \\cite{Hairer,Dahlquist}.\n\\begin{definition}\nA linear multistep method is called $A$-stable if $S \\supset \\mathbb{C}^-$, i.e., if\n\\begin{equation}\n{\\rm Re}\\, \\lambda \\le 0 \\implies \\text{numerical solution for } \\dot x = \\lambda x \\text{ is bounded.}\n\\end{equation}\n\\end{definition}\nThis definition means that, if the solution of the differential equation is bounded, then the approximation given by the multistep method is bounded as well. For a scalar differential equation, the multistep method is bounded whenever $\\lambda$ is in the left half of the complex plane. The Euler method is $A$-stable, but it is not possible to construct arbitrary order $A$-stable multistep methods. The second Dahlquist barrier is that an $A$-stable multistep method must be of order $p\\le 2$ \\cite{Hairer,Dahlquist}. As we wish to consider higher-order multistep methods, we relax the condition and require that the linear multistep method is $A(\\alpha)$-stable \\cite{Hairer,Widlund}.\n\n\\begin{definition}\nA linear multistep method is $A(\\alpha)$-stable, $0<\\alpha<\\pi\/2$, if\n\\begin{equation}\nS \\supset S_\\alpha = \\{\\mu ; |\\arg(-\\mu)| < \\alpha, \\mu \\ne 0 \\}.\n\\end{equation}\n\\end{definition}\nThis definition means that, in the case of a scalar differential equation, the multistep method is bounded whenever $\\lambda$ is within a wedge in the left half of the complex plane. For a vector differential equation, the eigenvalues of $A$ should be within this wedge. It is known that, for any $\\alpha<\\pi\/2$ and $k\\in \\mathbb{N}$, there is an $A(\\alpha)$-stable linear $k$-step method of order $p=k$ \\cite{Grigoreff,Butcher}.\n\nThe error in the total solution of the differential equation will be $O(N_t (\\Delta t)^{p+1})$. In order to obtain a rigorous result, we specialise to the case that $A$ and $b$ are independent of time. The relevant bound is given in Theorem 7.6 in Chapter V of Ref.\\ \\cite{Hairer}.\n\\begin{theorem}\n\\label{thm2}\nSuppose a linear multistep method is of order $p$, $A(\\alpha)$-stable and stable at infinity. If the matrix $A$ is diagonali{\\s}able (i.e.\\ there exists a matrix $V$ such that $V^{-1}AV=D=\\diag(\\lambda_1,\\ldots,\\lambda_n)$) with eigenvalues satisfying\n\\begin{equation}\n|\\arg(-\\lambda_i)|\\le \\alpha \\qquad for~i=1,\\ldots,N_x,\n\\end{equation} \nthen there exists a constant $M$ (depending only on the method) such that for all $\\dt>0$ the global error satisfies\n\\begin{equation}\n\\|x(t_m)-x_m\\| \\le M \\kappa_V \\left( \\max_{0\\le j< k} \\| x(t_j)-x_j \\| + \\dt^p \\int_{t_0}^{t_m} \\|x^{(p+1)}(\\xi)\\|d\\xi\\right),\n\\end{equation}\nwhere $\\kappa_V=\\|V\\| \\cdot \\|V^{-1}\\|$ is the condition number of $V$.\n\\end{theorem}\n\nHere the superscript with round brackets denotes repeated derivative. We can use this result to show a lemma on the scaling of the error.\n\n\\begin{lemma}\n\\label{lem:ersca}\nSuppose a linear multistep method is of order $p$, $A(\\alpha)$-stable and stable at infinity. If the matrix $A$ is diagonali{\\s}able (i.e.\\ there exists a matrix $V$ such that $V^{-1}AV=D=\\diag(\\lambda_1,\\ldots,\\lambda_n)$) with eigenvalues satisfying\n\\begin{equation}\n|\\arg(-\\lambda_i)|\\le \\alpha \\qquad for~i=1,\\ldots,N_x,\n\\end{equation} \nand $b$ is constant, then the global error satisfies\n\\begin{equation}\n\\|x(t_m)-x_m\\| = O\\left( \\kappa_V^2 (\\|x_{\\rm init}\\| + \\|b\\|\/\\|A\\|)\\left[ \\kappa_V (\\dt \\|A\\|)^2\n + m(\\dt \\|A\\|)^{p+1} \\right] \\right),\n\\end{equation}\nwhere $\\kappa_V=\\|V\\| \\cdot \\|V^{-1}\\|$ is the condition number of $V$.\n\\end{lemma}\n\n\\begin{proof}\nThe linear multistep method requires a starting method to obtain the values of $x_j$ for $0<j<k$. The term $\\max_{0\\le j< k} \\| x(t_j)-x_j \\|$ arises from the inaccuracy in this procedure. One can simply use the Euler method, in which case the error is $O(\\dt^2)$. It is also possible to use higher-order starting methods, but there is not a convenient rigorous result that can be used. To determine the error in the Euler method, one can simply use Theorem \\ref{thm2}. Because $k=1$ and $p=1$ for the Euler method, and for the initial point there is zero error ($x(t_0)=x_0$), Theorem \\ref{thm2} gives\n\\begin{equation}\n\\|x(t_j)-x_j\\| \\le M_E \\kappa_V \\dt\\int_{t_0}^{t_j}\\|x^{(2)}(\\xi)\\|d\\xi,\n\\end{equation}\nwhere $M_E$ is the constant for the Euler method. We consider this expression for $0\\le j < k$, and obtain\n\\begin{equation}\n\\max_{0\\le j< k}\\|x(t_j)-x_j\\| \\le M_E \\kappa_V \\dt^2 (k-1) \\max_{\\xi\\in[t_0,t_0+(k-1)\\dt]}\\|x^{(2)}(\\xi)\\|.\n\\end{equation}\n\nIn using these results, it is necessary to place upper bounds on the values of $\\|x^{(p+1)}(\\xi)\\|$ and $\\|x^{(2)}(\\xi)\\|$. In general these will depend on the value of $b(t)$, and its time-dependence. It is well-behaved if $b$ is a constant, in which case the exact solution is\n\\begin{equation}\nx(t) = e^{A(t-t_0)}(x_{\\rm init} + A^{-1} b) - A^{-1} b.\n\\end{equation}\nThen\n\\begin{equation}\nx^{(\\ell)}(t) = e^{A(t-t_0)}(A^\\ell x_{\\rm init} + A^{\\ell-1} b),\n\\end{equation}\nso\n\\begin{align}\n\\|x^{(\\ell)}(t)\\| &= \\|V e^{D(t-t_0)} V^{-1}(A^\\ell x_{\\rm init} + A^{\\ell-1} b)\\| \\nn\n&\\le \\kappa_V (\\|A\\|^\\ell \\|x_{\\rm init}\\| + \\|A\\|^{\\ell-1}\\|b\\|)\n\\end{align}\nIn the first line we have used the diagonali{\\s}ation of $A$, and in the second line we have used the condition that $|\\arg(-\\lambda_i)|\\le \\alpha$.\nUsing Theorem \\ref{thm2}, the error is bounded as\n\\begin{align}\n\\|x(t_m)-x_m\\| &\\le M \\kappa_V \\left( \\max_{0\\le j< k} \\| x(t_j)-x_j \\| + \\dt^p \\int_{t_0}^{t_m} \\|x^{(p+1)}(\\xi)\\|d\\xi\\right) \\nn\n&\\le M \\kappa_V \\left[ M_E \\kappa_V^2 \\dt^2 k (\\|x_{\\rm init}\\| + \\|b\\|\/\\|A\\|)\\|A\\|^2\n + \\dt^p (t_m-t_0) \\kappa_V (\\|x_{\\rm init}\\| + \\|b\\|\/\\|A\\|)\\|A\\|^{p+1}\\right] \\nn\n&= O\\left( \\kappa_V^2 (\\|x_{\\rm init}\\| + \\|b\\|\/\\|A\\|)\\left[ \\kappa_V (\\dt \\|A\\|)^2\n + m(\\dt \\|A\\|)^{p+1}  \\right] \\right) .\n\\end{align}\n\\end{proof}\n\nThis result means that, disregarding the dependence on many of the quantities, and omitting the error due to the starting method, the error scales as $O((\\|A\\|\\dt)^p \\|A\\| \\Delta t)$ for total time $\\Delta t$. To achieve error bounded by $\\epsilon$, we then use\n\\begin{equation}\n\\label{eq:nosteps}\nN_t = O\\left( \\frac{(\\|A\\|\\Delta t)^{1+1\/p}}{\\epsilon^{1\/p}} \\right).\n\\end{equation}\nThat is, the number of time steps required is close to linear in the time.\n\nGiven a linear multistep method, it is straightforward to encode this method as a linear system\n\\begin{equation}\n\\matr{\\vec x} = {\\vec b}.\n\\end{equation}\nHere $\\vec x$ is the vector of blocks $x_j$, $\\vec b$ is a vector of the blocks $b$, and ${\\cal A}$ is a matrix describing the initial condition and discretised differential equation. As an example, the equations for the Euler method and $A$ and $b$ independent of time may be expressed as\n\\begin{equation}\n\\label{eq:exmat}\n\\left[ {\\begin{array}{*{20}c}\n   \\openone & 0 & 0 & 0 & 0  \\\\\n   {-(\\openone + A\\dt)} & \\openone & 0 & 0 & 0  \\\\\n   0 & {-(\\openone + A\\dt)} & \\openone & 0 & 0  \\\\\n   0 & 0 & -\\openone & \\openone & 0  \\\\\n   0 & 0 & 0 & -\\openone & \\openone  \\\\\n\\end{array}} \\right]\\left[ {\\begin{array}{*{20}c}\n   {x_0 }  \\\\\n   {x_1 }  \\\\\n   {x_2 }  \\\\\n   {x_3 }  \\\\\n   {x_4 }  \\\\\n\\end{array}} \\right] = \\left[ {\\begin{array}{*{20}c}\n   {x_{\\rm in} }  \\\\\n   b\\dt  \\\\\n   b\\dt  \\\\\n   0  \\\\\n   0  \\\\\n\\end{array}} \\right].\n\\end{equation}\n\nEach entry of $\\matr$ is a block of the dimension of $A$, and each entry of $\\vec x$ and $\\vec b$ is a block of the dimension of $x$. \nThe first row sets the initial value, $x_0=x_{\\rm in}$. The next rows give $x_{j+1}-(x_j+A x_j \\dt)=b\\dt$, corresponding to the discreti{\\s}ation of the differential equation via the Euler method. The final rows indicate equations where $x_{j+1}-x_j=0$. This is for the times where $x$ is required to be constant. More generally, we use the Euler method as a starting method, then continue with a higher-order method, then for the final rows again have $x_{j+1}-x_j=0$ to ensure that all the final values are equal. Therefore, we set the blocks of $\\matr$ as, for $N_t\\ge 2k$,\n\\begin{equation}\n\\label{eq:explicit}\n\\begin{array}{*{20}l}\n\\matr_{j,j} = \\openone, & 0\\le j < k, \\quad N_t\/2 < j \\le N_t, \\\\\n\\matr_{j,j-1} = -(\\openone + A\\dt), & 1\\le j < k, \\\\\n\\matr_{j,j-k+\\ell} = \\alpha_l \\openone - \\beta_\\ell A \\dt, & k\\le j \\le N_t\/2, \\quad 0\\le \\ell\\le k. \\\\\n\\matr_{j,j-1} = -\\openone, & N_t\/2 < j \\le N_t. \\\\ \\end{array}\n\\end{equation}\nWe will always require $N_t\\ge 2k$ when using $\\matr$, because otherwise there are not enough time steps to start the linear multistep method.\nWe also set the blocks of $\\vec b$ as\n\\begin{equation}\n\\label{eq:explicitb}\n\\begin{array}{*{20}l}\nb_{0} = x_{\\rm in}, & \\\\\nb_{j} = b\\dt, & 1 \\le j < k, \\\\\nb_{j} = \\sum_{\\ell=0}^k \\beta_\\ell b\\dt, & k\\le j \\le N_t\/2, \\\\\nb_{j} = 0, & N_t\/2< j \\le N_t. \\\\\n\\end{array}\n\\end{equation}\n\nWe require $A$, $b$, and $x_{\\rm in}$ to be sparse, with no more than $s$ nonzero elements in any row or column. We assume that the oracles are of the same form as in Ref.\\ \\cite{Berry09}. That is, the oracle for $A$ is a unitary operator acting as\n\\begin{equation}\nO_A \\ket{j,\\ell}\\ket{z} = \\ket{j,\\ell} \\ket{z\\oplus A^{[j,\\ell]}}.\n\\end{equation}\nNote that we use superscript to denote indexing within the block $A$.\nWe also require an oracle for the sparseness, that locates the nonzero elements. Given a function $f(j,\\ell)$ that gives the row index of the $\\ell$th nonzero element in column $j$, we require a unitary oracle\n\\begin{equation}\nO_F \\ket{j,\\ell} = \\ket{j,f(j,\\ell)}.\n\\end{equation}\nBecause $A$ is not Hermitian, we require a similar oracle to give the positions of the nonzero elements in a given row.\nWe also require oracles to give the values and locations of non-zero elements for $b$ and $x_{\\rm in}$. These oracles ensure that the initial state corresponding to $\\vec b$ can be prepared efficiently. Alternatively, it is also possible to consider $b$ and $x_{\\rm in}$ such the efficient preparation procedure of Ref.\\ \\cite{Grover02} can be used.\n\nA linear system of equations can be solved using the algorithm of Ref.\\ \\cite{Harrow09} with complexity $\\tilde O(\\log(N)s^4\\kappa^2\/\\epsilon_L)$, where $\\kappa$ is the condition number of the matrix $\\matr$, and $\\epsilon_L$ is the allowable error. (Note that the power of $s$ should be 4, not 2 as given in Ref.\\ \\cite{Harrow09}.) We use the symbol $\\epsilon_L$ to indicate the allowable error for the solution of the linear systems, which is distinct from the allowable error for the solution of the differential equation. The scaling can be improved if the method reported in Ref.\\ \\cite{Ambainis10} is used (the method is not given there, but is reported as being in a manuscript in preparation). The scaling reported there is $O(\\kappa^{1+o(1)}\\log^cN)$, which does not include the scaling in $s$ or $\\epsilon_L$.\n\n\\section{Bounding the condition number}\n\nTo determine the complexity in either case it is necessary to determine the value of the condition number $\\kappa$. To bound this condition number we first determine bounds on the norms of $\\|\\matr\\|$ and $\\|\\matr^{-1}\\|$.\n\n\n\n\n\\begin{lemma}\n\\label{norm1}\nThe matrix $\\matr$, with blocks given by Eq.\\ \\eqref{eq:explicit}, satisfies $\\|\\matr\\|=O(1)$ provided $\\dt=O(1\/\\|A\\|)$.\n\\end{lemma}\n\n\\begin{proof}\nTo determine the upper bound on $\\|\\matr\\|$, we express $\\matr$ as a sum of block-diagonal matrices, and use the triangle inequality. Let us define $\\matr^{\\{\\ell\\}}$ to be the block diagonal matrix with all entries zero, except\n\\begin{equation}\n\\matr^{\\{\\ell\\}}_{j,j-\\ell} = \\matr_{j,j-\\ell}.\n\\end{equation}\nWe then have\n\\begin{equation}\n\\matr = \\sum_{\\ell=0}^k \\matr^{\\{\\ell\\}} ,\n\\end{equation}\nso, via the triangle inequality,\n\\begin{equation}\n\\|\\matr\\| \\le \\sum_{\\ell=0}^k \\|\\matr^{\\{\\ell\\}}\\|.\n\\end{equation}\nThe norm of a block-diagonal matrix is just the maximum norm of the blocks, so we find\n\\begin{align}\n\\|\\matr^{\\{0\\}}\\| &\\le \\max(1,|\\alpha_k|+|\\beta_k| h \\|A\\|), \\nn\n\\|\\matr^{\\{1\\}}\\| &\\le \\max(1 + h\\|A\\|,|\\alpha_{k-1}|+|\\beta_{k-1}| h \\|A\\|), \\nn\n\\|\\matr^{\\{\\ell\\}}\\| &\\le |\\alpha_{\\ell}|+|\\beta_{\\ell}| h \\|A\\|, \\qquad 1<\\ell\\le k.\n\\end{align}\nBecause we require that $\\dt=O(1\/\\|A\\|)$, each of these norms is $O(1)$, and hence the overall norm is $O(1)$.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\\label{norm2}\nSuppose that the multistep method is $A(\\alpha)$-stable, the matrix $A$ may be diagonalised as $A=VDV^{-1}$, and the eigenvalues of $A$ all satisfy $|\\arg(-\\lambda_i)|\\le \\alpha$. Then the matrix $\\matr$, with blocks given by Eq.\\ \\eqref{eq:explicit}, satisfies $\\|\\matr^{-1}\\|=O(N_t\\kappa_V)$, where $\\kappa_V$ is the condition number of $V$.\n\\end{lemma}\n\n\\begin{proof}\nTo upper bound $\\|\\matr^{-1}\\|$, we use a method analogous to that used to bound the error in Ref.\\ \\cite{Hairer}. As in the condition for Theorem \\ref{thm2}, we assume that $A$ may be diagonalised as\n\\begin{equation}\nA = V D V^{-1}.\n\\end{equation}\nNote that $A$ need not be Hermitian, so $V$ need not be unitary. If we define ${\\cal V}$ to be to the block matrix with $V$ on the diagonal, and ${\\cal D}$ to be the matrix corresponding to $\\matr$ except with $A$ replaced with $D$, then $\\matr={\\cal V}{\\cal D}{\\cal V}^{-1}$. We obtain\n\\begin{equation}\n\\|\\matr^{-1}\\|\\le \\|{\\cal V}\\|\\cdot\\|{\\cal D}^{-1}\\|\\cdot\\|{\\cal V}^{-1}\\| = \\kappa_V \\|{\\cal D}^{-1}\\|.\n\\end{equation}\nTo bound $\\|\\matr^{-1}\\|$ we therefore just need to bound $\\|{\\cal D}^{-1}\\|$.\n\nThe matrix ${\\cal D}$ corresponds to the linear multistep solution of decoupled scalar differential equations. That is, taking $z=V^{-1}x$, the differential equation becomes $N_x$ decoupled differential equations\n\\begin{equation}\n\\dot z^{[j]}(t) = \\lambda_j z^{[j]}(t) + [V^{-1}b]^{[j]}.\n\\end{equation}\nThe matrix ${\\cal D}$ gives decoupled linear multistep solutions of each of these differential equations. It may therefore be written in block-diagonal form, with each block corresponding to solution of each of these decoupled equations. The value of $\\|{\\cal D}^{-1}\\|$ can therefore be bounded by the maximum of the norm of the inverse of each of these blocks.\n\nTo bound the norm of the inverse, we can take ${\\cal D}\\vec z = \\vec y$, and determine a bound on the norm of $\\vec z$ for a given norm of $\\vec y$. We can determine this by separately examining the uncoupled blocks in ${\\cal D}$. For each of these blocks (labelled by $j$) we have the linear multistep equation, for $m=0,\\ldots,N_t\/2-k$,\n\\begin{equation}\n\\label{eq:multier}\n\\sum_{i=0}^k (\\alpha_i - \\dt \\lambda_j \\beta_i) z_{m+i}^{[j]} = y_{m+k}^{[j]}.\n\\end{equation}\nWe also have, for the initial condition, $z_{0}^{[j]} = y_{0}^{[j]}$,\nand for the Euler method as the starting method with $0\\le m<k-1$,\n\\begin{equation}\nz_{m+1}^{[j]} - (1 + \\dt \\lambda_j) z_{m}^{[j]} = y_{m+1}^{[j]}.\n\\end{equation}\nFor the end of the simulation, we have for $N_t\/2\\le m<N_t$,\n\\begin{equation}\nz_{m+1}^{[j]} - z_{m}^{[j]} = y_{m+1}^{[j]}.\n\\end{equation}\n\nWe can see that Eq.\\ \\eqref{eq:multier} is equivalent to Eq.\\ (7.11) in the method used to bound error in Ref.\\ \\cite{Hairer}. We identify $z_{m+i}^{[j]}$ as equivalent to $e_{m+i}$ in Ref.\\ \\cite{Hairer}, and $y_{m+k}^{[j]}$ as equivalent to $\\delta_h(x_m)$ in Ref.\\ \\cite{Hairer}.\nAs in that method, we can define\n\\begin{align}\nE_m &:= (z_{m+k-1}^{[j]},\\ldots,z_{m+1}^{[j]},z_{m}^{[j]})^T, \\nn\n\\Delta_m &:= (y_{m+k}^{[j]}\/(\\alpha_k-\\dt \\lambda_j\\beta_k),0,\\ldots,0)^T.\n\\end{align}\nAs the problem is equivalent to that considered in Ref.\\ \\cite{Hairer}, the result given in Eq.\\ (7.24) of that reference holds:\n\\begin{equation}\n\\|E_{m+1}\\| \\le M\\left(\\|E_0\\|+\\sum_{\\ell=0}^m \\|\\Delta_\\ell\\| \\right),\n\\end{equation}\nwhere $M$ is a constant depending only on the method. Using the definition of $\\Delta_\\ell$ gives\n\\begin{align}\n\\|E_{m+1}\\| &\\le M\\left(\\|E_0\\|+\\sum_{\\ell=0}^{m} |y_{\\ell+k}^{[j]}|\/|\\alpha_k-\\dt \\lambda_j\\beta_k| \\right) \\nn\n&\\le M\\left(\\|E_0\\|+\\sum_{\\ell=0}^{m} |y_{\\ell+k}^{[j]}|\/|\\alpha_k| \\right).\n\\end{align}\nIn the last line we have used the fact that the condition of $A(\\alpha)$ stability means $\\alpha_k\\cdot\\beta_k>0$, so $|\\alpha_k - \\dt \\lambda_j\\beta_k|^{-1}\\le |\\alpha_k|^{-1}$.\n\nFor the starting method, we have used the Euler method, and the result is simpler. For the Euler method, $E_m$ and $\\Delta_m$ are scalars, and are just $z_m^{[j]}$ and $y_{m+1}^{[j]}$. The corresponding result is therefore, for $0< m < k$,\n\\begin{align}\n|z_m^{[j]}| &\\le M_E\\left(|z_0^{[j]}|+\\sum_{\\ell=0}^{m-1} |y_{\\ell+1}^{[j]}| \\right) \\nn\n&= M_E\\sum_{\\ell=0}^{m} |y_{\\ell}^{[j]}|.\n\\end{align}\nHere $M_E$ is the corresponding constant for the Euler method.\nFor the end of the simulation, we have for $N_t\/2\\le m<N_t$, $z_{m+1}^{[j]} - z_{m}^{[j]} = y_{m+1}^{[j]}$, so\n\\begin{equation}\n|z_m^{[j]}| \\le |z_{N_t\/2}^{[j]}| + \\sum_{\\ell=N_t\/2+1}^m | y_\\ell^{[j]} |.\n\\end{equation}\n\nNow we can bound the norm of $E_0$ as\n\\begin{align}\n\\|E_0\\| &\\le \\sum_{m=0}^{k-1} |z_m^{[j]}| \\nn\n&\\le M_E k \\sum_{\\ell=0}^{k-1} |y_{\\ell}^{[j]}|.\n\\end{align}\nWe can use this result to bound $|z_{m}^{[j]}|$ as, for $k\\le m \\le N_t\/2$,\n\\begin{align}\n|z_{m}^{[j]}| &\\le \\| E_{m+1-k}\\| \\nn\n& \\le M\\left(\\|E_0\\|+\\sum_{\\ell=0}^{m-k} |y_{\\ell+k}^{[j]}|\/|\\alpha_k| \\right) \\nn\n& \\le M\\left(M_E k \\sum_{\\ell=0}^{k-1} |y_{\\ell}^{[j]}|+\\sum_{\\ell=k}^{m} |y_{\\ell}^{[j]}|\/|\\alpha_k| \\right) .\n\\end{align}\n\nFor convenience we define the quantity\n\\begin{equation}\nM_T := \\max(MM_Ek,M\/|\\alpha_k|,M_E,1).\n\\end{equation}\nThen we find that, for all $0\\le m\\le N_t$,\n\\begin{equation}\n|z_{m}^{[j]}| \\le M_T \\sum_{\\ell=0}^{m} |y_{\\ell}^{[j]}|.\n\\end{equation}\nHence we can determine an overall upper bound on the norm of $z^{[j]}$ as\n\\begin{align}\n\\|z^{[j]}\\|^2 &\\le M_T^2\\sum_{m=0}^{N_t} \\left(\\sum_{\\ell=0}^{m} |y_{\\ell}^{[j]}|\\right)^2 \\nn\n&\\le M_T^2 N_t^2 \\|y^{[j]}\\|^2. \n\\end{align}\nSumming over $j$, this then gives\n\\begin{align}\n\\|\\vec z\\| \\le M_T N_t \\|\\vec y\\|. \n\\end{align}\nThis result means that $\\|{\\cal D}^{-1}\\|\\le M_TN_t$, where $M_T$ depends only on the method. This then bounds the norm of $\\matr^{-1}$ as\n\\begin{equation}\n\\|\\matr^{-1}\\| = O(N_t\\kappa_V).\n\\end{equation}\n\\end{proof}\n\nWe can now use these results to bound the condition number of $\\matr$.\n\n\n\n\n\n\\begin{theorem}\n\\label{thm:conthm}\nSuppose that the multistep method is $A(\\alpha)$-stable, the matrix $A$ may be diagonalised as $A=VDV^{-1}$, the eigenvalues of $A$ all satisfy $|\\arg(-\\lambda_i)|\\le \\alpha$, and $\\dt=O(1\/\\|A\\|)$. Then the matrix $\\matr$, with blocks given by Eq.\\ \\eqref{eq:explicit}, has condition number $\\kappa=O(N_t\\kappa_V)$, where $\\kappa_V$ is the condition number of $V$.\n\\end{theorem}\n\n\\begin{proof}\nThe condition number is given by the formula\n\\begin{equation}\n\\kappa = \\left(\\max_{\\vec x} \\frac{\\| \\matr{\\vec x} \\|}{\\|\\vec x\\|}\\right)\n\\left(\\max_{\\vec x} \\frac{\\|\\vec x\\|}{\\| \\matr{\\vec x} \\|}\\right) = \\|\\matr\\| \\cdot \\|\\matr^{-1}\\|.\n\\end{equation}\nThe conditions of this theorem ensure that the conditions of Lemmas \\ref{norm1} and \\ref{norm2} hold. Therefore we can use the bounds on $\\|\\matr\\|$ and $\\|\\matr^{-1}\\|$ from those lemmas to obtain $\\kappa=O(N_t \\kappa_V)$.\n\\end{proof}\n\n\n\n\nThis result for the condition number can be explained in a more intuitive way. Each value of $y_i^{[j]}$ is equivalent to an excitation of the differential equation at a single time. Therefore $\\vec z$ is close to the solution of the differential equation with each of those individual excitations. An excitation can not cause a growing solution, because of the stability condition. This means that the worst that an excitation can do is cause a solution that is displaced by a proportional amount for the remaining time. Therefore the norm of $\\vec z$ can not be more than a factor of $N_t$ times the norm of $\\vec y$.\n\n\\section{Algorithm for solving linear systems}\n\nWe can use the bound on the condition number to estimate the complexity of the quantum algorithm. We will first explain the scaling in a simple way, then give a more rigorous result. Using Eq.\\ \\eqref{eq:nosteps} for the number of time steps, we have (ignoring dependence on many of the quantities)\n\\begin{equation}\n\\kappa = O\\left( \\frac{(\\|A\\|\\Delta t)^{1+1\/p}}{\\epsilon^{1\/p}} \\right).\n\\end{equation}\nUsing this expression in the result for the complexity of solving linear systems from Ref.\\ \\cite{Harrow09} gives\n\\begin{equation}\n\\tilde O(\\log(N)s^4(\\|A\\|\\Delta t)^{2+2\/p}\/(\\epsilon^{2\/p}\\epsilon_L)).\n\\end{equation}\nIf the technique of Ref.\\ \\cite{Ambainis10} can be used, then the scaling can be improved to\n\\begin{equation}\nO(\\log^c(N)(\\|A\\|\\Delta t)^{1+1\/p}\/\\epsilon^{1\/p}).\n\\end{equation}\n\nThere is a question of whether the scaling in Ref.\\ \\cite{Harrow09} can be improved because we consider a specific application of the solution of linear systems. The scaling obtained in Ref.\\ \\cite{Harrow09} is based on a worst-case scenario, that $\\|\\vec b\\|$ scales as $\\Lambda_{\\rm max} \\|\\vec x\\|$. It is easily shown that $\\|\\vec x\\|$ scales as $\\sqrt{N_t}$, provided the magnitude of the solution of the differential equation does not vary greatly in time. In contrast, the magnitude of $\\vec b$ is given by\n\\begin{align}\n\\|\\vec b\\|^2 &= \\|x_{\\rm in}\\|^2 + (k-1)\\|b\\|^2h^2 + (N_t\/2-k+1) \\|b\\|^2 \\dt^2 \\left( \\sum_{\\ell=0}^k \\beta_\\ell \\right)^2 \\nn\n&\\le \\|x_{\\rm in}\\|^2 +(k-1)\\|b\\|^2h^2 + \\dt \\Delta t \\|b\\|^2 \\left( \\sum_{\\ell=0}^k \\beta_\\ell \\right)^2.\n\\end{align}\nIn the case that we use the Euler method, then $\\dt\\propto 1\/\\Delta t$, so $\\|\\vec b\\|=O(1)$. In that case it is possible to solve the linear systems more efficiently than the worst-case scaling given by Ref.\\ \\cite{Harrow09}. However, here we are concerned with using higher-order linear multistep methods. For these methods the scaling of $h\\Delta t$ is close to $N_t$. In that case $\\|\\vec b\\|$ has similar scaling to $\\Lambda_{\\rm max} \\|\\vec x\\|$, so there is not a significant advantage to using this approach to improve on the result in \\cite{Harrow09}. Therefore we will just use the result stated in Ref.\\ \\cite{Harrow09}.\n\n\n\n\n\n\nBefore obtaining the overall scaling, another factor that needs to be considered is the scaling for creating the state encoding $\\vec b$. This is because the algorithm of Ref.\\ \\cite{Harrow09} uses an amplitude amplification approach, which requires the preparation of this state at each step. Therefore, the overall complexity is multiplied by the complexity of performing this state preparation. We assume that we have oracles that give the elements of $x_{\\rm in}$ and $\\vec b$ in the computational basis, and that these vectors are $s$-sparse. We find the following result.\n\n\\begin{lemma}\n\\label{lem:prep}\nThe state encoding $\\vec b$,\n\\begin{equation}\n\\ket{\\vec b} \\propto \\sum_{j,\\ell} \\vec b_j^{[\\ell]} \\ket{j,\\ell},\n\\end{equation}\ncan be prepared using $O(\\sqrt{s}+\\log(N_t))$ calls to the oracles for $x_{\\rm in}$ and $b$, provided the normali{\\s}ations of $x_{\\rm in}$ and $b$ are known.\n\\end{lemma}\n\n\\begin{proof}\nFor this preparation we can assume that the normali{\\s}ations of $x_{\\rm in}$ and $b$ are known. That is because this normali{\\s}ation can be determined with complexity $O(s)$, and need only be determined once. Because the overall complexity of the simulation is greater than this, it can just be assumed that the normali{\\s}ation is known.\n\nA state of dimension $s$ can be prepared with complexity $O(\\sqrt{s})$ using the method of Ref.\\ \\cite{Grover00}. As discussed above, we assume an oracle of the form used in Ref.\\ \\cite{Berry09} for the sparseness. This means that it only requires one oracle call to prepare an $s$-sparse state from a dimension $s$ state with the same coefficients \\cite{Berry09}. Therefore the complexity of preparing an $s$-sparse state is also $O(\\sqrt{s})$.\n\nLet us encode the state in three registers (which may themselves be composed of multiple qubits). The first is one qubit encoding $\\ket{0}$ for the times $t_1,\\ldots,t_{N_t\/2}$, and $\\ket{1}$ for the times $t_0,t_{N_t\/2+1},\\ldots,t_{N_t}$. The second register provides the remainder of the encoding of the time. The third register is of dimension $N_x$, and encodes $b$ or $x_{\\rm in}$.\n\nBy performing a rotation on the first qubit, based on the normali{\\s}ations of $x_{\\rm in}$ and $b$, we obtain the correct relative weighting of the two time ranges. Then, conditional on the qubit being in the state $\\ket{0}$, we can prepare a superposition of times $t_1,\\ldots,t_{N_t\/2}$ in the second register, as well as a state encoding $b$ in the third register. Conditional on the qubit being in the state $\\ket{1}$, we can prepare the time $t_0$ in the second register, and $x_{\\rm in}$ in the third register. The complexity of these controlled state preparations will be $O(\\sqrt{s})$. The complexity of preparing the superposition of times can be made $O(\\log(N_t))$ simply by choosing $N_t$ to be a power of two (which does not change the scaling).\n\\end{proof}\n\nWe now translate the complexity of solving linear systems into the complexity of obtaining a state corresponding to the solution of the differential equation.\n\n\\begin{theorem}\n\\label{thm:final}\nSuppose that the multistep method is order $p$ and $A(\\alpha)$-stable, the matrix $A$ may be diagonalised as $A=VDV^{-1}$, the eigenvalues of $A$ all satisfy $|\\arg(-\\lambda_i)|\\le\\alpha$, and\n\\begin{align}\n\\label{eq:varcon}\n\\max_{t\\in[t_0,t_0+\\Delta t]}\\|x(t)\\| &= O(\\|x(t_0+\\Delta t)\\|), \\\\\n\\label{eq:xfbig}\n\\epsilon &= o(\\|x_{\\rm in}\\|).\n\\end{align}\nThen a state encoding the solution of the differential equation at time $t_0+\\Delta t$ may be obtained to within trace distance $\\epsilon$ using\n\\begin{equation}\n\\tilde O\\left(\\log(N_x)s^{9\/2} (\\|A\\|\\Delta t)^{5\/2}\\kappa_V^{23\/4}(\\|x_{\\rm in}\\|+\\|b\\|\/\\|A\\|)^{5\/4}\\|x(t_0+\\Delta t\\|\/\\epsilon^{9\/4}\\right)\n\\end{equation}\ncalls to the oracles for $A$, $b$, and $x_{\\rm in}$.\n\\end{theorem}\n\n\\begin{proof}\nThere are two main issues that we need to take into account in determining the complexity of obtaining the solution of the differential equation. The first is the probability of obtaining the correct time, and the second is the relation of the allowable error for solving the differential equation to the allowable error for the solution of the linear systems.\n\nTo obtain the correct final state, one would need to measure the time in the ancilla register, and obtain a value in the range $t_0+\\Delta t$ to $t_0+2\\Delta t$. This probability can, in principle, be small or even zero. However, the conditions of the theorem ensure that it is not. The probability of success is given by\n\\begin{equation}\np_{\\rm time} = \\frac{\\sum_{j=N_t\/2}^{N_t} \\|x_j\\|^2}{\\braket{\\psi}{\\psi}}.\n\\end{equation}\nThe normali{\\s}ation of the state is given by\n\\begin{align}\n\\label{eq:norm}\n\\braket{\\psi}{\\psi} &= \\sum_{j=0}^{N_t} \\|x_j\\|^2 \\nn\n&= \\sum_{j=0}^{N_t} [|x(t_0+\\dt j)\\|+O(\\epsilon)]^2 \\nn\n&= O\\left(N_t\\max_{t\\in[t_0,t_0+\\Delta t]}\\|x(t)\\|^2\\right) \\nn\n&= O\\left(N_t\\|x(t_0+\\Delta t)\\|^2\\right).\n\\end{align}\nHere we have bounded the error in the state at all times by $\\epsilon$. This is because we choose parameters that bound the error at time $t_0+\\Delta t$ by $\\epsilon$. The bound on the error increases monotonically with time, so the error at earlier times will also be bounded by $\\epsilon$. We have also used the conditions \\eqref{eq:varcon} and \\eqref{eq:xfbig} to obtain that $\\epsilon=o(\\|x(t_0+\\Delta t)\\|)$.\nUsing this result for $\\braket{\\psi}{\\psi}$ we obtain\n\\begin{align}\np_{\\rm time} &= \\Omega\\left( \\frac{\\sum_{j=N_t\/2}^{N_t} [\\|x(t_0+\\Delta t)\\|+O(\\epsilon)]^2}{N_t\\|x(t_0+\\Delta t)\\|^2} \\right) \\nn\n&= \\Omega(1).\n\\end{align}\nTherefore the probability of obtaining the correct time does not change the scaling of the complexity.\n\nThe other main issue that needs to be addressed to obtain the overall scaling is the scaling in the error. The $\\epsilon_L$ used in the expression for the solution of the linear systems is the error in the state encoding all times, not the error in the estimate of $x(t_0+\\Delta t)$ obtained (for which we use $\\epsilon$).\nTo determine the relationship between $\\epsilon$ and $\\epsilon_L$, we use the bound on the normali{\\s}ation of the state. \nIn solving for the state $\\ket{\\psi}$, we obtain the state coefficients approximating\n\\begin{equation}\nx(t_0+\\Delta t)\/\\sqrt{\\braket{\\psi}{\\psi}}.\n\\end{equation}\nUsing the bound on the normali{\\s}ation in Eq.\\ \\eqref{eq:norm}, if the error in the coefficients is no more than $\\epsilon_L$, the error in the solution of the differential equation will be $O(\\epsilon_L\\sqrt{N_t}\\|x(t_0+\\Delta t\\|)$. Therefore, the error in the solution of the differential equation will be bounded by $\\epsilon$ if we take\n\\begin{equation}\n\\epsilon_L = \\Theta\\left(\\epsilon\/[\\sqrt{N_t}\\|x(t_0+\\Delta t)\\|]\\right).\n\\end{equation}\n\nUsing this value of $\\epsilon_L$ in the scaling from Ref.\\ \\cite{Harrow09}, together with $\\kappa=O(N_t \\kappa_V)$ from Theorem \\ref{thm:conthm}, and the complexity of state preparation from Lemma \\ref{lem:prep}, the number of oracle calls is\n\\begin{equation}\n\\tilde O(\\log(N_x)s^{9\/2} N_t^{5\/2}\\kappa_V^2\\|x(t_0+\\Delta t\\|\/\\epsilon).\n\\end{equation}\nNote that we can omit $\\log(N_t)$ from Lemma \\ref{lem:prep}, because the $\\tilde O$ notation omits logarithmic factors. For the same reason, we have replaced $N=N_x N_t$ with $N_x$.\nWe use a value of $N_t$ that is sufficient to ensure that the error is no greater than $\\epsilon$, and that $\\dt=O(1\/\\|A\\|)$, which is a condition needed to use Theorem \\ref{thm:conthm}. If we take\n\\begin{equation}\nN_t = \\Theta \\left( \\|A\\|\\Delta t \\sqrt{\\frac{\\kappa_V^3(\\|x_{\\rm in}\\|+\\|b\\|\/\\|A\\|)}{\\epsilon}}\n+ (\\|A\\|\\Delta t)^{1+1\/p} \\left( \\frac{\\kappa_V^2(\\|x_{\\rm in}\\|+\\|b\\|\/\\|A\\|)}{\\epsilon} \\right)^{1\/p} \\right),\n\\end{equation}\nthen using Lemma \\ref{lem:ersca}, the error will be bounded by $\\epsilon$. In addition, because $\\epsilon=o(\\|x_{\\rm in}\\|)$, we obtain $N_t = \\Omega (\\|A\\|\\Delta t)$, which ensures that $\\dt=O(1\/\\|A\\|)$.\n\nWe can simplify the result by taking\n\\begin{equation}\nN_t = \\Theta \\left( (\\|A\\|\\Delta t)^{1+1\/p} \\sqrt{\\frac{\\kappa_V^3(\\|x_{\\rm in}\\|+\\|b\\|\/\\|A\\|)}{\\epsilon}}\n\\right).\n\\end{equation}\nThen the overall scaling of the number of black-box calls is\n\\begin{equation}\n\\tilde O\\left(\\log(N_x)s^{9\/2} (\\|A\\|\\Delta t)^{(5\/2)(1+1\/p)}\\kappa_V^{23\/4}(\\|x_{\\rm in}\\|+\\|b\\|\/\\|A\\|)^{5\/4}\\|x(t_0+\\Delta t\\|\/\\epsilon^{9\/4}\\right).\n\\end{equation}\nBecause we are using the $\\tilde O$ notation, which omits sublinear terms, we can omit $1\/p$ in giving the result.\n\\end{proof}\n\nThis result is somewhat conservative, because we have included the term due to the error in starting the linear multistep method. If we assume that this error is negligible, then we obtain\n\\begin{equation}\n\\tilde O\\left(\\log(N_x)s^{9\/2} (\\|A\\|\\Delta t)^{(5\/2)(1+1\/p)}\\kappa_V^{2+5\/p}(\\|x_{\\rm in}\\|+\\|b\\|\/\\|A\\|)^{5\/2p}\\|x(t_0+\\Delta t\\|\/\\epsilon^{1+5\/2p}\\right).\n\\end{equation}\nThis has the same scaling in $\\|A\\|\\Delta t$, but improved scaling in other quantities.\n\n\\section{Conclusions}\nA quantum computer may be used to solve sparse systems of linear differential equations, provided the result may be encoded in a quantum state, rather than given explicitly. By encoding the differential equation as a linear system, and using the algorithm of Ref.\\ \\cite{Harrow09} for solving linear systems, the complexity is (including only scaling in $\\|A\\|$ and $\\Delta t$),\n\\begin{equation}\n\\tilde O\\left((\\|A\\|\\Delta t)^{5\/2}\\right).\n\\end{equation}\nThis improves upon previous results for nonlinear differential equations, which were strongly exponential in $\\Delta t$ \\cite{Leyton08}. This algorithm has an enormous range of possible applications, because large systems of differential equations are ubiquitous in science and engineering.  In particular they arise from the discreti{\\s}ation of partial differential equations.\n\nThese results are for constant coefficients, because that enables an analytic error analysis. This approach can also be used to solve linear differential equations with time-dependent coefficients, though the error analysis will be more difficult.\n\nAn interesting question is how the complexity will scale if the method referred to in Ref.\\ \\cite{Ambainis10} can be used. That reference refers to work in preparation providing an improved scaling for solving linear systems. The scaling is close to linear in the condition number, which would improve the scaling in $\\|A\\|\\Delta t$ for the solution of linear differential equations.\n\nAnother interesting direction for future research is the problem of nonlinear differential equations. It is likely that the exponential scaling obtained in Ref.\\ \\cite{Leyton08} is fundamental, because quantum mechanics is linear. However, it may potentially be possible to improve the constant of the exponential scaling.\n\n\\acknowledgements\nThe author is grateful for enlightening discussions with Andrew Childs and Jon Tyson.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Keywords}\n      Stochastic Differential Equations,\n      Fokker--Planck Equation,\n      Exponential Families,\n      Stock Price Models,\n      Black and Scholes model, Option Pricing,\n      Trading Time Grid, $\\Delta$-Markovianity,\n      Market Incompleteness, Option replication error.\n\\section{Introduction}\nSince the seminal papers by Black and Scholes (1973) and Merton (1973),\nmany researchers have tackled the problem of option pricing and hedging\nby resorting to continuous-time mathematics to fully exploit the richness\nof its theoretical results. In particular, the underlying stock price dynamics\nhas often been modelled through a geometric Brownian motion. This assumption is still\nquite popular, especially among practitioners, even though the empirical distribution of\nstock returns is often found more leptokurtic than the normal one.\nIndeed, the tradeoff between analytical tractability and empirical\nfindings still makes the geometric Brownian motion a quite acceptable choice.\n\nHowever, as we shall prove in this paper, the geometric Brownian motion\nis not the only continuous-time process that possesses a lognormal marginal distribution and\nnormal log-returns. In fact, we shall construct a family of processes with these\ncharacteristics and then we shall also analyse the main implications of the existence of\nsuch a family as far as option pricing is concerned.\n\nIn order to achieve our goal, we consider scalar\nnonlinear diffusion processes whose densities evolve in\ngiven finite--dimensional exponential families, as from Chapter 7 of Brigo (1996).\nWe begin by a stochastic differential equation (SDE) a la Black and\nScholes for the stock price, whose solution is the traditional\ngeometric Brownian motion. The probability density of such a stock\nprice model evolves in the exponential family of log-normal densities.\nIt is then possible to define a second SDE with a different (and arbitrarily given)\ndiffusion coefficient such that its solution has the same marginal\nlognormal density as that of the Black and Scholes (1973) geometric Brownian motion.\n\nFurthermore, one can choose the drift of this second SDE in such a\nway that its solution has transition densities, between two any instants\nof a prescribed finite subset of the time interval, that match the\ntransition densities of the Black and Scholes process.\nWe thus derive a family of stock price processes that behave almost\nequivalently to a geometric Brownian motion in that they\nare probabilistically identical along the finite subset of the time interval.\n\nWe then consider the problem of option pricing in a\ncontinuous-time framework. The interesting result we obtain is\nthat notwithstanding the above mentioned affinity with the Black and Scholes model,\nthe stock price processes of our family\nlead to completely arbitrary option prices, ranging the whole interval of prices\ninbetween the no--arbitrage lower and upper bounds.\nSuch a seeming paradox can be explained in terms of the differences between the dynamics of\nthese stock price processes at an infinitesimal level.\n\nOur result is similar in spirit to that of Rogers and Satchell (1996). However,\nour approach is much more constructive in that we are able to provide, under the\noriginal measure, easy and explicit dynamics for the stock price that are\ntheoretically consistent with whatever option price is observed in reality.\n\nThe practical implications of our result can be analysed by considering the case of\na practitioner who has to price an option. Although the physical time he works in is\ndiscrete, he usually resorts to continuous-time mathematics modelling the stock price\nunderlying the option through a Black and Scholes (1973) process. However,\nas the previous theoretical result implies, the geometric Brownian motion\nis only one of infinitely many processes that possess the discrete-time properties\nrequired by the practitioner. Therefore, as far as the stock price dynamics is concerned,\nthe practitioner must view all these processes as equivalent to each other. Yet, when\noption pricing is taken into account, he must also be aware that they all imply different,\nand arbitrary, option prices. Since this paradoxical situation reveals a clear limit of\nthe approximation of discrete time with continuous time, our practitioner can react as\nfollows. He simply avoids the theory of market completeness and just chooses a process\nwhich is consistent with the option price he believes in.\nSuch an option price can be either exogenously given by the market or\nendogenously produced by a model for pricing and hedging in incomplete\nmarkets. This is a further contribution of the paper. We do not derive\noption prices and hedging strategies assuming a specific evolution of the\nunderlying stock price. Instead, we define a family of processes with equivalent\ncharacteristics among which one can select a single process according to his pricing\nand hedging purposes.\n\nThe paper is structured as follows. In Section 2, the theoretical\nresult on the construction of some suitable diffusion processes is briefly\nreviewed.\nSection 3 deals with the application of this result to the case of the\nBlack and Scholes (1973) model. Section 4 analyses the problem of option\npricing in continuous time and considers a natural particular case.\nSection 5 analyses some practical implications of using any of our alternative\nprocesses in describing the asset price dynamics.\nSection 6 concludes the paper.\n\nA reduced version of this paper can be found in Brigo and Mercurio (2000).\n\n\\section{Stochastic differential equations and exponential families}\nIn this section we consider the technical result the paper is based on.\nThis result solves the following problem\nfor scalar diffusion processes: Given a diffusion coefficient\nand a curve in an exponential family of densities, find a drift such that\nthe solution of the resulting stochastic differential equation (SDE)\nhas a density evolving in the prescribed exponential family according to the\ngiven curve.\nThis problem has a straightforward solution.\nIn this section, we shall present a short summary of the steps leading to the problem\nformulation and to its solution.\nThis summary is based on Chapter 7 of Brigo (1996). A related result appeared in Brigo (1997),\nand the general result with its applications both to mathematical finance and\nto stochastic nonlinear filtering is also reported in Brigo (2000).\n\nLet us consider the SDE\n\\begin{eqnarray} \\label{sde1}\n   dX_t =  f_t(X_t) dt + \\sigma_t(X_t) d W_t,\n\\end{eqnarray}\nwhere $\\{W_t, t\\ge 0\\}$ is a standard Brownian motion independent\nof the initial condition $X_0$.\nIn order to make sure that we are dealing with an equation whose\nsolution exists unique, we formulate the following assumption.\n\n\\begin{itemize}\n   \\item[(A1)] The initial condition $X_0$ is\n      a continuous random variable with density $p_0(x) >0$\n      for all $x \\in {\\bf R}$\n      w.r.t.\\ the Lebesgue measure on ${\\bf R}$ whose moments\n      of any order are finite.\n      Moreover, the stochastic differential equation~(\\ref{sde1})\n      characterized by the coefficients $f$, $\\sigma$,\n      and by the initial condition $X_0$ admits a unique strong solution.\n\\end{itemize}\n\nExplicit conditions ensuring (A1) are, for example, local Lipschitz continuity\nand linear growth, or the Yamada-Watanabe\ncondition (see e.g. Rogers and Williams (1987), Section V-40).\n\nOnce existence and uniqueness of the solution\nof a SDE have been established, we can analyse\nthe distribution of its solution at all time instants.\nIn describing the evolution of the distribution of a diffusion process,\nthe Fokker--Planck partial differential equation is a fundamental tool.\nWe therefore introduce the following assumption.\n\\begin{itemize}\n \\item[(A2)]\nThe unique solution $X_t$ of ~(\\ref{sde1}) admits a density $p_t$ that is absolutely\ncontinuous with respect to the Lebesgue measure, i.e.,\n\\begin{eqnarray*}\n\\mbox{Prob}\\{X_t\\in A\\} = \\int_A p_t(x) dx, \\ \\ \\mbox{for all Borel sets} \\ \\ A,\n\\end{eqnarray*}\nand that satisfies the Fokker--Planck equation:\n\\begin{eqnarray} \\label{FPESQRT}\n\\frac{\\partial p_t}{\\partial t} =  -\\frac{\\partial}{\\partial x} (f_t p_t) + \\inverse{2}\n   \\frac{\\partial^2}{\\partial x^2} (a_t p_t),  \\ \\    a_t(\\cdot) = \\sigma_t^2(\\cdot) \\ .\n\\end{eqnarray}\n\\end{itemize}\nExamples of assumptions on the coefficients $f$, $a$ and on their\npartial derivatives\nunder which (A2) holds are given in the literature. See for example\nStroock and Varadhan (1979)\nor Friedman (1975).\n\nIn order to  appropriately introduce the problem we mentioned at the beginning\nof the section, we now present a definition of exponential family.\n\\begin{definition}\n   Let $\\{c_1,\\cdots,c_m\\}$ be scalar functions defined on ${\\bf R}$,\n   such that $\\{1,c_1,\\cdots,c_m\\}$ are {\\em linearly independent},\n   have at most polynomial growth, are\n   twice continuously differentiable and the convex set\n\\begin{displaymath}\n   \\Theta_0 := \\left\\{\\theta=\\{\\theta^1,\\ldots,\\theta^m\\}'\\in {\\bf R}^m\\,:\\,\n   \\psi(\\theta) = \\log\\; \\int \\exp[ \\theta' c(x) ]\\, d x\n   < \\infty \\right\\}\\ ,\n\\end{displaymath}\n   has {\\em non--empty interior}, where $c(x)=\\{c_1(x),\\cdots,c_m(x)\\}'$ and\n   `` $'$ '' denotes transposition.\n   Then\n\\begin{displaymath}\n   EM(c) = \\{ p(\\cdot,\\theta)\\,,\\, \\theta \\in \\Theta \\},\n   \\hspace{1cm} p(x,\\theta):= \\exp[\\theta' c(x) - \\psi(\\theta)]\\ ,\n\\end{displaymath}\n   where $\\Theta \\subseteq \\Theta_0$ is open,\n   is called an exponential family of probability densities.\n\\end{definition}\n\nOur problem consists in finding a SDE whose solution $X_t$\nhas a density $p_t$ that follows a prescribed curve in a given exponential family.\nMore precisely, we require the curve $t \\mapsto p_t$, in the space\nof all densities, to coincide with a given curve\n$t \\mapsto p(\\cdot,\\theta_t)$ in a given $EM(c)$.\\footnote{In order to contain space\nand notation the underlying geometric setup\nis not fully developed here. We just say that the problem originated from the\nuse of differential geometry and statistics for the nonlinear filtering\nproblem. The reader interested in geometric aspects and other details is referred to\nBrigo (1996),  Brigo, Hanzon and Le Gland (1999), or to the tutorial in Brigo (1999).}\n\nThis is formalized in the following.\n\\begin{problem} \\label{fin-dim:pro}\nLet be given an exponential family $EM(c)$, an initial density $p_0$\ncontained in $EM(c)$, and\na diffusion coefficient $a_t(\\cdot) := \\sigma^2_t(\\cdot)$,\n$t\\ge 0$.\nLet ${\\cal U}(p_0,\\sigma)$ denote the set of all drifts\n$f$ such that $p_0$, $f$ and $\\sigma$ and its related SDE~(\\ref{sde1}) satisfy assumptions\n(A1) and (A2). Assume ${\\cal U}(p_0,\\sigma)$ to be non-empty.\n\nThen, given the curve $t \\mapsto p(\\cdot,\\theta_t)$ in $EM(c)$\n(where $t \\mapsto \\theta_t$ is a $C^1$--curve in the parameter space\n$\\Theta$), find a drift in ${\\cal U}(p_0,\\sigma)$ whose related SDE has a solution\nwith density $p_t = p(\\cdot,\\theta_t)$.\n\n\\end{problem}\nThe solution of this problem is given by the following.\n\\begin{theorem} \\label{sol-prob1}\n{\\bf (Solution of Problem~\\ref{fin-dim:pro})}\nAssumptions and notation of Problem~\\ref{fin-dim:pro} in force.\nConsider the stochastic differential equation\n\\begin{eqnarray} \\label{sol:prob1}\nd Y_t &=& u^\\sigma_t(Y_t) dt + \\sigma_t(Y_t) dW_t, \\ \\ Y_0 \\sim p_0, \\nonumber \\\\  \\\\\n u^\\sigma_t(x) &:=& \\inverse{2} \\frac{\\partial a_t}{\\partial x}(x) +\n      \\inverse{2} a_t(x) \\theta_t' \\frac{\\partial c}{\\partial x}(x)\n      \\nonumber \\\\ \\nonumber \\\\ \\nonumber\n&& - \\left(\\frac{d}{dt}\\theta_t' \\right)\n         \\int_{-\\infty}^x \\left(c(\\xi) - \\nabla_{\\theta}\\psi(\\theta_t)\\right)\n         \\ \\exp[\\theta_t' (c(\\xi) - c(x))] d\\xi,\n\\end{eqnarray}\nwhere $\\nabla_{\\theta}\\psi(\\theta_t)=\\{\\partial \\psi \/ \\partial\n\\theta^1(\\theta_t),\\ldots,\\partial \\psi \/\\partial \\theta^m(\\theta_t) \\}'$,\nwith the symbol ``$\\sim$'' to be read as ``distributed as''.\n\nIf  $u^\\sigma \\in {\\cal U}(p_0,\\sigma)$,\nthen the SDE (\\ref{sol:prob1}) solves Problem~\\ref{fin-dim:pro}, in that\n\\begin{displaymath}\np_{Y_t}(x) = \\exp\\left[\\theta_t' \\ c(x) - \\psi(\\theta_t)\\right],\n\\ \\ t \\ge 0.\n\\end{displaymath}\n\\end{theorem}\nThe proof of the theorem is rather straightforward.\nIt is sufficient to write the\nFokker--Planck equation for the SDE (\\ref{sol:prob1}) and, after\nlengthy computations, verify that indeed\n\\begin{displaymath}\n \\frac{\\partial}{\\partial t} \\exp[\\theta_t' c(x) - \\psi(\\theta_t)]\\\n= - \\frac{\\partial}{\\partial x}\n\\left(u^\\sigma_t(x) \\exp[\\theta_t' c(x) - \\psi(\\theta_t)]\\right)\n+ \\inverse{2}  \\frac{\\partial^2}{\\partial x^2}\n\\left( a_t(x) \\exp[\\theta_t' c(x) - \\psi(\\theta_t)] \\right)  \\\n\\end{displaymath}\nby substituting the expression for $u$ given in the theorem.\nA different proof can be found in Chapter 7 of Brigo (1996) or in Brigo (2000),\nwhere in deriving the expression for $u$ it was tacitly assumed, as is done here, that\n\\begin{displaymath}\n\\lim_{x\\rightarrow -\\infty}u^\\sigma_t(x)p_t(x) = 0 \\ \\ \\mbox{for all } \\ t \\ge 0.\n\\end{displaymath}\n\nIn the next section, we shall consider an interesting application of this theorem to\nthe option pricing problem. Indeed, we shall use such result\nmore as a  ``guiding tool'' rather than applying it immediately\nas it stands. In particular, assumptions (A1) and (A2) will be checked\ndirectly and not via the sufficient conditions usually considered in the literature.\n\n\\section{Alternatives to the Black and Scholes model}\nLet us consider the Black and Scholes (1973) stock price model,\n\\begin{eqnarray} \\label{BeS}\n d S_t = \\mu S_t dt + \\bar{\\sigma}  S_t \\ dW_t, \\ \\  S_0 = s_0 , \\ \\\nt \\in [0,T],\n\\end{eqnarray}\nwhere $s_0$ is a positive deterministic initial condition, and $\\mu$,\n$\\bar{\\sigma}$ and $T$ are positive real constants.\n\nThe probability density $p_{S_t}$ of $S_t$, at any time $t>0$,\nis given by\n\\begin{eqnarray} \\label{BeSlnd}\np_{S_t}(x) &=& \\exp\\left\\{\\zeta \\ln\\frac{x}{s_0} + \\rho(t)\n  \\ln^2\\frac{x}{s_0} - \\psi(\\zeta,\\rho(t))  \\right\\}, \\ \\ x > 0, \\\\ \\nonumber \\\\ \\nonumber\n\\zeta &=& \\frac{\\mu}{\\bar{\\sigma}^2} - \\frac{3}{2},\n\\ \\ \\rho(t) = - \\frac{1}{2 \\bar{\\sigma}^2 t}, \\\\ \\nonumber \\\\ \\nonumber\n\\psi(\\zeta,\\rho(t)) &=& - \\frac{(\\zeta+1)^2}{4 \\rho(t)}\n+ \\inverse{2} \\ln\\left(\\frac{-\\pi}{\\rho(t)}\\right) + \\ln(s_0).\n\\end{eqnarray}\nWith the notation for exponential families introduced in the previous section,\none writes\n\\begin{eqnarray*}\nc_1(x) =  \\ln\\frac{x}{s_0}, \\ \\ c_2(x) =  \\ln^2\\frac{x}{s_0} , \\ \\\n\\theta_t = \\{\\zeta, \\ \\rho(t)\\}' \\ .\n\\end{eqnarray*}\n\nOne might wish to model the stock price process by considering a different volatility\nfunction $\\sigma$, instead of $\\bar{\\sigma} S_t$ in (\\ref{BeS})\\footnote{We use\nthe term ``volatility'' to denote the whole diffusion coefficient $\\sigma_t(\\cdot)$ rather\nthan the standard deviation rate of the instantaneous return as usually done in practice.},\nwhile preserving major properties of the original process\n(\\ref{BeS}). The purpose of this section is then the construction of\nalternative stock price dynamics that differs from (\\ref{BeS}), yet sharing\nsimilar features from a probabilistic point of view.\n\nLet us approach this problem by applying Theorem~\\ref{sol-prob1}\nto find a SDE with a given diffusion\ncoefficient $\\sigma_t(\\cdot)$ and whose marginal density is\nequal to the marginal density of $S$ in all time instants\nof the time interval ${\\cal T} = [\\epsilon,T]$, where $0<\\epsilon<T$ and $\\epsilon$\ncan be chosen arbitrarily close to $0$.\nWe remark that the density of $S$ is not concentrated in the whole\nreal line, but in $(0, +\\infty)$.\nHowever, the procedure resulting in Theorem~\\ref{sol-prob1}\ncan be adapted in a straightforward way to the latter case\nonce $-\\infty$ is replaced by $0$.\n\nIt easily follows from (\\ref{sol:prob1}) that the equation sought for is\n\\begin{eqnarray} \\label{sol:bes1}\nd Y_t &=& u^\\sigma_t(Y_t,s_0,0) dt + \\sigma_t(Y_t) dW_t,\n\\ \\ Y_\\epsilon = S_\\epsilon,\n\\ \\  \\epsilon \\le t \\le T, \\nonumber \\\\  \\\\\n\\nonumber\n u^\\sigma_t(x,y,\\alpha) &:=& \\inverse{2} \\frac{\\partial a_t}{\\partial x}(x) +\n\\inverse{2} \\frac{a_t(x)}{x} \\left[ \\zeta + 2 \\rho(t-\\alpha)\n\\ln\\frac{x}{y}\\right] +\\frac{x}{2(t-\\alpha)}\n\\left[\\ln\\frac{x}{y} - \\frac{\\zeta+1}{2 \\rho(t-\\alpha)}\\right].\n\\end{eqnarray}\nThe definition of $Y$ is then extended to the whole interval\n$[0, T]$ by setting\n\\[ d Y_t = \\mu Y_t \\ dt + \\bar{\\sigma} Y_t dW_t, \\ 0<t<\\epsilon, \\  Y_0 = s_0 \\ . \\]\nIn other terms, $Y$ is assumed to follow the same dynamics as the original\nprocess $S$ in $[0,\\epsilon)$, and to start from the same initial condition.\n\nThe reason for splitting the interval $[0,T]$ into two subintervals\nis simply to avoid problems concerning the definition of the drift at time $t=0$.\nThe variable $\\alpha$ is here introduced to allow for subsequent generalizations.\n\nIn general, we cannot prove that (\\ref{sol:bes1}) satisfies assumptions\n(A1) and (A2) for a prescribed $\\sigma$, i.e. that $u^\\sigma \\in {\\cal U}(p_0,\\sigma)$\nso that ${\\cal U}(p_0,\\sigma)$ is nonempty. Therefore, at this stage,\n(\\ref{sol:bes1}) is simply providing us with a {\\em candidate} solution\nof our problem for a generic $\\sigma$.\n\nIn the following table we report some possible choices for $\\sigma$\nand the corresponding candidate $u^\\sigma$.\nIn particular, we can notice that for $\\sigma_t(x) = \\bar{\\sigma} x$, $u^\\sigma_t(x,s_0,0)$ equals,\nas it must be, the original drift $\\mu x$ of the Black and Scholes model (\\ref{BeS}).\n\n\\begin{table}[ht]\n\\begin{center}\n{\\bf Table 1}\\\\\nExamples of volatility functions and corresponding candidate drift\n\\mbox{} \\\\\n\\begin{tabular}{|c|c|}\n\\hline $\\mbox{Volatility function} \\ \\ \\sigma_t(x)$ & $ \\mbox{Drift} \\ \\ u^\\sigma_t(x,s_0,0)$\n\\\\ \\hline\n$\\nu$ & $ \\inverse{2} \\frac{\\nu^2}{x}\n\\left[ \\zeta + 2 \\rho(t) \\ln\\frac{x}{s_0}\\right] +\n\\frac{x}{2t} \\left[\\ln\\frac{x}{s_0} - \\frac{\\zeta+1}{2 \\rho(t)}\\right]$\n\\\\\n\\hline $\\nu \\sqrt{x}$  & $\\frac{\\nu^2}{2}\n(\\frac{\\mu}{\\bar{\\sigma}^2}-\\inverse{2}) -\n\\inverse{2t} \\frac{\\nu^2}{\\bar{\\sigma}^2} \\ln\\frac{x}{s_0} +\n\\frac{x}{2}(\\mu - \\frac{\\bar{\\sigma}^2}{2}) +  \\frac{x}{2t} \\ln\\frac{x}{s_0}$\n\\\\\n\\hline $\\nu x$  & $ x \\left[ \\inverse{4}(\\nu^2 - \\bar{\\sigma}^2) +\n\\frac{\\mu}{2}(\\frac{\\nu^2}{\\bar{\\sigma}^2} + 1) \\right]\n+ \\frac{x}{2t}\\ln\\frac{x}{s_0}(1-\\frac{\\nu^2}{\\bar{\\sigma}^2} )$  \\\\\n\\hline $\\bar{\\sigma} x$ & $\\mu x$ \\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n\\end{table}\nLet us now assume that the\nparticular $\\sigma$ we are working with is such that\n$u^\\sigma \\in {\\cal U}(p_0,\\sigma)$.\n\nWe have seen so far that, by applying the results of the previous section,\nit is rather straightforward to produce processes with the same marginal\ndistribution as that of (\\ref{BeS}). However, a further fundamental\nproperty of process (\\ref{BeS}) is that its log-returns\nbetween any two time instants are normally distributed,\nindependently of the prices at the considered instants, i.e.,\n\\begin{eqnarray} \\label{BeSret}\n \\ln \\frac{S_{t+\\delta}}{S_t}\n\\sim {\\cal N}\\left((\\mu - \\inverse{2} \\bar{\\sigma}^2)\\delta ,\\ \\bar{\\sigma}^2 \\delta\\right),\n\\ \\ \\delta > 0, \\ \\ t\\in [0,T-\\delta].\n\\end{eqnarray}\n\nAlternative models such as (\\ref{sol:bes1}) do not share this\nproperty in general. In fact, identity of the marginal laws alone\ndoes not suffice to ensure (\\ref{BeSret}), for which equality of\nsecond order laws or of transition densities would be sufficient instead.\nHow can we obtain  alternative models whose properties concerning log-returns\nare as close as possible to property~(\\ref{BeSret})?\n\nTo tackle this issue, we have to find a compromise between our alternative\nmodel (\\ref{sol:bes1}) and model (\\ref{BeS}). To this end, we consider a\nweaker version of (\\ref{BeSret}) in that we restrict the set of dates for\nwhich the property holds true. Precisely, we modify the\ndefinition of $Y$ so that, given the time instants\n${\\cal T}^\\Delta:=\\{0,\\Delta, 2\\Delta, \\ldots, N\\Delta\\}$,\n$\\Delta = T\/N$, $\\Delta > \\epsilon$,\nproperty (\\ref{BeSret}) is satisfied by $Y$ in ${\\cal T}^\\Delta$, i.e.\n\\begin{eqnarray} \\label{yret}\n \\ln \\frac{Y_{i \\Delta}}{Y_{j \\Delta}}\n\\sim {\\cal N}((\\mu - \\inverse{2} \\bar{\\sigma}^2)(i-j) \\Delta ,\\ \\bar{\\sigma}^2(i-j) \\Delta),\n\\ \\ i > j, \\ \\ i=1,\\ldots,N, \\ \\ j=0,\\ldots,N-1.\n\\end{eqnarray}\nLimiting such key property to a finite set of times is not so\ndramatic. Indeed, only discrete time samples are observed in practice,\nso that once the time instants are fixed, our  process $Y$ can not be\ndistinguished from Black and Scholes process'. The issue of discrete versus\ncontinuous time will be further developed in Section 5.\n\nThe new definition of $Y$ is still based on Theorem~\\ref{sol-prob1}.\nHowever, we use this theorem ``locally'' in each time interval $[(i-1)\\Delta, \\ i\\Delta)$.\nThis means that in such interval we define iteratively the drift $u^\\sigma$ as in the\ntheorem but\n\\begin{itemize}\n\\item we translate back the time--dependence of a time amount $(i-1)\\Delta$\n     (thus locally restoring the dynamics of the original result) and\n\\item we replace the distribution $p_0$ for the initial condition with\n      the distribution of the final value of $Y$ relative to the previous interval.\n\\end{itemize}\nWe obtain:\n\\begin{eqnarray}\n\\label{sol:bes2}\nd Y_t &=& u^\\sigma_t(Y_t,Y_{\\alpha(t)},\\alpha(t)) dt +  \\sigma_t(Y_t) dW_t, \\ \\\nt \\in [i\\Delta + \\epsilon, (i+1)\\Delta),  \\\\ \\nonumber \\\\ \\nonumber\ndY_t &=& \\mu Y_t dt + \\bar{\\sigma} Y_t dW_t,  \\ \\ \\mbox{for} \\ \\  t \\in [i\\Delta,i\\Delta+ \\epsilon),\\ \\\n\\alpha(t) = i \\Delta \\ \\ \\mbox{for} \\ \\  t \\in [i\\Delta, \\ (i+1)\\Delta) \\ ,\n\\end{eqnarray}\nwhere $u^\\sigma_t(x,y,\\alpha)$ was defined in (\\ref{sol:bes1}).\n\nIt is clear by construction that the transition densities\nof $S$ and $Y$ satisfy\n$p_{Y_{(i+1) \\Delta}|Y_{i \\Delta}}(x;y) = p_{S_{(i+1) \\Delta}|S_{i\\Delta}}(x;y)$.\nThen, starting from the equality of the marginal laws of $S$ and $Y$ in the first\ninterval that holds by construction, we inductively obtain the equality of the\nmarginal laws also in each other interval.\nAs a consequence, the second order densities are also equal\namong consecutive instants $(i-1)\\Delta, \\ i\\Delta$, i.e.,\n\\begin{displaymath}\np_{[Y_{(i+1) \\Delta},Y_{i\\Delta}]}(x,y) =\np_{[S_{(i+1) \\Delta},S_{i\\Delta}]}(x,y).\n\\end{displaymath}\nIt follows that\n\\begin{eqnarray} \\label{yret1}\n \\ln \\frac{Y_{(i+1) \\Delta}}{Y_{i \\Delta}}\n\\sim {\\cal N}((\\mu - \\inverse{2} \\bar{\\sigma}^2) \\Delta ,\\ \\bar{\\sigma}^2 \\Delta),\n\\ \\ i=0,\\ldots,N-1.\n\\end{eqnarray}\nAt this point we remark that the process $Y$ in (\\ref{sol:bes2}) is not a Markov process\nin $[0, T]$. However, it is Markov in all time instants of ${\\cal T}^\\Delta$.\nFormally,\n\\begin{displaymath}\np_{Y_{m \\Delta}|Y_{(m-1)\\Delta},Y_{(m-2)\\Delta},\\ldots,Y_0} =\np_{Y_{m \\Delta}|Y_{(m-1)\\Delta}}.\n\\end{displaymath}\nThis property follows from the fact that in $[(m-1)\\Delta, m \\Delta)$\nthe dynamics of the SDE defining $Y$  does not depend on\n$Y_{(m-2)\\Delta},\\ldots,Y_0$, and that when such equation\nis considered for $t\\in [(m-1)\\Delta, m\\Delta)$, in its drift $u^\\sigma$\nthe local initial condition for the entry $Y$ is set to $Y_{(m-1)\\Delta}$.\n\nFrom now on, we refer to markovianity in  ${\\cal T}^\\Delta$ as to\n$\\Delta${\\em --Markovianity}.\n\nWe finally notice that, through the $\\Delta$--Markovianity,\nproperty (\\ref{yret1}) extends to any pair of instants in\n${\\cal T}^\\Delta$, so as to yield (\\ref{yret}).\nMoreover, the inductive application of the $\\Delta$--Markovianity\nand the identity of transition densities in the grid leads to the\nidentity of the finite dimentional distributions of $S$ and $Y$ in the grid.\n\n\\section{Option pricing in continuous-time} \\label{ophct}\nLet us now consider the process $\\{B_t:t\\ge 0\\}$ whose value evolves\naccording to\n\\begin{equation}\ndB_t=B_t r dt,\n\\end{equation}\nwith $B_0=1$ and where $r$ is a positive real number, so that\n$B_t = \\exp(r t)$. The process $B$ is assumed to describe the evolution of\na money market account in a given financial market. The process $Y_t$ in\n(\\ref{sol:bes2}) is instead assumed to model the evolution of some traded\nfinancial (risky) asset, typically a stock.\n\nThe financial market thus defined might admit arbitrage opportunities. As\nis well known, a sufficient condition which ensures\narbitrage-free dynamics is the existence of an equivalent martingale measure\nwith respect to the initially chosen numeraire. In this paper, we use the\nprocess $B$ as a numeraire, so that an equivalent martingale measure is a\nprobability measure that is equivalent to the initial one, $P$, and under\nwhich the process $\\{Y_t\/B_t:t\\ge 0\\}$ is a martingale. A necessary condition for\nthe existence of an equivalent martingale measure is the semimartingale\nproperty for the process $Y$. The process $Y$ is indeed a semimartingale\nunder $P$ for sufficiently well behaved volatility functions $\\sigma_t(\\cdot)$.\n\nWe denote by $\\cal S$ the set of all volatility functions $\\sigma_t(\\cdot)$\nsuch that $u^\\sigma \\in {\\cal U}(p_0,\\sigma)$ and for which there exists a unique\nequivalent martingale measure.\n\nThe set $\\cal S$ is obviously non-empty, since it contains at least the\nBlack and Scholes volatility function $\\sigma_t(x)=\\bar{\\sigma} x$. Moreover, as we will\nprove in the sequel, all volatilities functions of type $\\nu I$, $\\nu >0$, belong\nto $\\cal S$, with $I$ denoting the identity map. An interesting example of\nvolatility functions which do not belong to ${\\cal S}$ is instead provided in the appendix.\n\nWe now assume that we have chosen $\\sigma\\in {\\cal S}$ and the\ncorresponding equivalent martingale measure $Q^\\sigma$.\nSince $\\{Y_t\/B_t:t\\ge 0\\}$ is a martingale under such a measure, it easily\nfollows that under $Q^\\sigma$ the process $Y$ satisfies the SDE\n\\begin{eqnarray*}\nd Y_t &=& r Y_t \\ dt + \\bar{\\sigma} Y_t \\ d \\widetilde{W}_t,  \\ \\  t \\in [i\\Delta, i\\Delta+\\epsilon), \\\\\nd Y_t &=& r Y_t \\ dt + \\sigma_t(Y_t) \\ d \\widetilde{W}_t,  \\ \\ t \\in [i\\Delta+\\epsilon, (i+1)\\Delta),\n\\end{eqnarray*}\nwhere  $\\widetilde{W}$ is a standard Brownian motion under\n$Q^\\sigma$.\n\nFurthermore, under the assumption that\ni) there are no-transaction costs, ii) the borrowing and lending rates are both equal to $r$, iii)\nshort selling is allowed with no restriction or penalty, and iv)\nthe stock is infinitely divisible and pays no dividends,\nthe unique no-arbitrage price for a given contingent claim\n$H\\in L^2(Q^\\sigma)$ is (see Harrison and Pliska (1981, 1983))\n\\begin{equation}\n\\label{optpr}\nV_t=\\frac{B_t}{B_T}E^{Q^\\sigma}\\left\\{\\left.H\\right| {\\cal F}_t \\right\\},\n\\end{equation}\nwhere $\\{{\\cal F}_t:t\\ge 0\\}$ denotes the filtration associated to the\nprocess $Y$.\n\nIn the special case of a European call option, the following are\ninteresting problems to solve.\n\\begin{problems} \\label{infsup}\nLet us assume that the given claim is a European call option, written on\nthe stock, with maturity $T$ and strike $K$. Find:\n\\begin{equation}\n\\label{optsiginf}\n\\inf_{\\epsilon>0,\\sigma\\in {\\cal S}} B_T^{-1}\nE^{Q^\\sigma}\\left\\{\\left.(Y_T-K)^+\\right| {\\cal F}_0 \\right\\},\n\\end{equation}\n\\begin{equation}\n\\label{optsigsup}\n\\sup_{\\epsilon>0,\\sigma\\in {\\cal S}} B_T^{-1}\nE^{Q^\\sigma}\\left\\{\\left.(Y_T-K)^+\\right| {\\cal F}_0 \\right\\}.\n\\end{equation}\n\\end{problems}\nSolving these problems is equivalent to finding the lowest and highest\ntheoretical price of the option for which the underlying stock price has lognormal\nmarginal distribution and normal log-returns on the grid ${\\cal T}^\\Delta$, with\nstandard deviations proportional to $\\bar{\\sigma}$.\n\nIf we denote by $V_{*}$ the value of the infimum in\n(\\ref{optsiginf}),\nand by $V^{*}$ the value of the supremum in (\\ref{optsigsup}),\nthe following inequalities obviously hold:\n\\begin{equation}\n\\label{ineqbs}\n (s_0 - Ke^{-rT})^+ \\le V_{*}\\leq V_{BS}(\\bar{\\sigma})\n \\le V^{*} \\le s_0,\n\\end{equation}\nwhere $V_{BS}(\\bar{\\sigma})$ denotes the option Black and Scholes price at time 0\nas determined by $\\bar{\\sigma}$, the volatility parameter in (\\ref{BeS}).\nIndeed, since $\\bar{\\sigma} I\\in {\\cal S}$, the central inequalities hold by\ndefinition of $V_{*}$ and $V^{*}$, whereas the first and\nthe last ones feature respectively the well known no-arbitrage lower and upper\nbounds for option prices.\n\nIn the next subsection we will show that the first and last inequalities in\n(\\ref{ineqbs}) are actually equalities. To prove this statement, it will be\nsufficient to restrict our analysis to the class of volatilities\n$\\{\\sigma_t(x) = \\nu x$, $\\nu > 0\\}$. This result is at first sight surprising.\nIndeed, one would naively expect that the difference\nbetween prices implied by models which are equivalent in the $\\Delta$--grid\nis bounded by a quantity that is somehow related to $\\Delta$, typically\n${\\cal O}(\\Delta^\\lambda)$ for some positive real $\\lambda$. In fact, by\nhalving the size of $\\Delta$, we double the discrete--time instants where\nthe models in our family are equivalent. Accordingly, we would expect the prices\nimplied by these now ``closer'' models to range in a narrower interval. However, as\nwe shall soon see, this is not the case.\n\n\\subsection{A fundamental case} \\label{fundamental}\nWe begin by stating the following.\n\\begin{lemma}\nIn the fundamental case where $\\sigma_t(x) = \\nu x$, $\\nu>0$, the\nprocess $Y^\\nu$ given by\n\\begin{eqnarray} \\label{sol:bes3}\nd Y_t^\\nu &=& u^{\\nu I}_t(Y_t,Y_{\\alpha(t)},\\alpha(t)) dt + \\nu \\ Y_t \\ dW_t,\n\\ \\ t \\in [i\\Delta + \\epsilon, (i+1)\\Delta), \\nonumber \\\\ \\\\ \\nonumber\nu^{\\nu I}_t(y,y_\\alpha,\\alpha) &=&\ny \\left[ \\inverse{4}(\\nu^2 - \\bar{\\sigma}^2) +\n\\frac{\\mu}{2}(\\frac{\\nu^2}{\\bar{\\sigma}^2} + 1) \\right]\n+ \\frac{y}{2(t-\\alpha)} (1-\\frac{\\nu^2}{\\bar{\\sigma}^2} ) \\ln\\frac{y}{y_\\alpha},\n\\\\ \\nonumber \\\\ \\nonumber\n d Y_t^\\nu &=& \\mu Y_t^\\nu dt + \\bar{\\sigma} Y_t^\\nu dW_t \\ \\ \\mbox{for} \\ \\  t \\in [i\\Delta,i\\Delta+ \\epsilon),\\ \\\n\\alpha(t) = i \\Delta \\ \\ \\mbox{for} \\ \\  t \\in [i\\Delta, \\ (i+1)\\Delta) \\\n\\end{eqnarray}\nsolves Problem~\\ref{fin-dim:pro} when $p_t$ is given by (\\ref{BeSlnd}).\nMoreover, the volatility function $\\sigma_t(x) = \\nu x$ belongs to $\\cal S$,\nfor any $\\nu>0$.\n\\end{lemma}\n{\\em Proof}.\nSince $Y^\\nu$ has the same marginal distribution\nas $S$ under $P$, it follows that $Y_t^\\nu>0$. Then, the process $Z_t = \\ln Y_t^\\nu$ is\nwell defined and, by It\\^o's formula, the SDE for $Z_t$ is piecewise linear in a narrow\nsense (in that its diffusion coefficient is purely deterministic), and hence\nadmits a unique strong solution which, for each $t\\in [j\\Delta,(j+1)\\Delta)$,\nis explicitly given by\n\\begin{eqnarray} \\label{intediff}\nZ_t &=& Z_{j\\Delta} + (\\mu -\\inverse{2}\\bar{\\sigma}^2)(t-j\\Delta)  \\\\ \\nonumber\n& &\n+\\left\\{\n\\begin{array}{ll}\n\\bar{\\sigma}(W_t-W_{j\\Delta})  & t\\in [j\\Delta,j\\Delta+\\epsilon), \\\\ \\nonumber\n\\left(\\frac{t-j\\Delta}{\\epsilon}\\right)^{\\beta\/2}\n\\left[\\bar{\\sigma}(W_{j\\Delta+\\epsilon}-W_{j\\Delta})+ \\nu\\int_{j\\Delta+\\epsilon}^t\n\\left(\\frac{u-j\\Delta}{\\epsilon}\\right)^{-\\beta\/2} dW_u\\right]\n& t\\in [j\\Delta+\\epsilon,(j+1)\\Delta),\n\\end{array}\n\\right.\n\\end{eqnarray}\nwhere $\\beta = 1-\\frac{\\nu^2}{\\bar{\\sigma}^2}$.\n\nAs a consequence, the assumptions of Problem~\\ref{fin-dim:pro} are satisfied\nso that $u^{\\nu I}$ solves Problem~\\ref{fin-dim:pro} when $p_t$ is given by\n(\\ref{BeSlnd}). Moreover, the Girsanov change of measure from $P$ to $Q^{\\nu I}$\nis well defined since one can show that the Novikov condition is satisfied through\napplication of the ``tower property'' of conditional expectations.\nHence, the measure $Q^{\\nu I}$ exists unique and $\\nu I$ belongs to\n$\\cal S$ for each $\\nu > 0$. For the uniqueness of the measure $Q^{\\nu I}$,\nwe refer for instance to Duffie (1996).\n\n\\begin{theorem}\nIn the fundamental case where $\\sigma_t(x) = \\nu x$, $\\nu>0$,\nthe unique no-arbitrage option price at time $t$ is the Black and Scholes price\n\\begin{equation}\n\\label{bslem}\nU^\\epsilon(t,\\nu)=Y_t^\\nu \\Phi(d_1) - K e^{-r(T-t)}\\Phi(d_2),\n\\end{equation}\nwhere\n\\begin{eqnarray*}\n&& d_1=\\frac{\\ln(Y_t^\\nu\/K)+(r+\\bar{\\sbar}^\\epsilon (t)^2\/2)(T-t)}{\\bar{\\sbar}^\\epsilon (t) \\sqrt{T-t}},\\\\ \\\\\n&& d_2=d_1(t) - \\bar{\\sbar}^\\epsilon (t) \\sqrt{T-t},\n\\end{eqnarray*}\n\\begin{equation}\n\\label{nubar}\n\\bar{\\sbar}^\\epsilon (t)=\n\\left\\{\n\\begin{array}{ll}\n\\sqrt{\\frac{\n\\frac{\\epsilon}{\\Delta}(\\bar{\\sigma}^2-\\nu^2)(T-\\alpha(t))+\\nu^2(T-\\alpha(t))\n+\\bar{\\sigma}^2(\\alpha(t)-t)}{T-t}} & t \\in [\\alpha(t),\\alpha(t)+\\epsilon)\n\\\\ \\sqrt{\\frac{\n\\frac{\\epsilon}{\\Delta}(\\bar{\\sigma}^2-\\nu^2)(T-\\alpha(t)-\\Delta)+\n\\nu^2(T-t)}{T-t}} & t \\in [\\alpha(t)+\\epsilon,\\alpha(t)+\\Delta)\n\\end{array}\n\\right.\n\\end{equation}\n\\end{theorem}\n{\\em Proof}.\nFrom the previous lemma, we infer the existence of a unique\nno-arbitrage option price that can be calculated through (\\ref{optpr}).\nThen (\\ref{bslem}) is obtained by noticing that under the equivalent martingale measure\n\\[ \\ln \\frac{Y_T^\\nu}{Y_t^\\nu}\n\\sim {\\cal N}\\left((r - \\inverse{2} \\bar{\\sbar}^\\epsilon (t)^2)(T-t) ,\\ \\bar{\\sbar}^\\epsilon\n(t)^2 (T-t)\\right), \\ t\\in [0,T],\\]\nwith $\\bar{\\sbar}^\\epsilon$ given by (\\ref{nubar}). This implies that the option price\nat time $t$ corresponding to $Y^\\nu$ is the Black and Scholes price with\nvolatility coefficient $\\bar{\\sbar}^\\epsilon(t)$, i.e., that (\\ref{bslem}) holds.\n\n\\mbox{}\\newline\nThe key point of our result is that, for any given volatility\ncoefficient $\\nu$, we are free to adjust the drift of the SDE defining the dynamics of\nthe stock--price process under the objective measure,\nin such a way that the resulting $Y^\\nu$ has the same distributional properties of\nthe Black and Scholes process on discrete-time dates. As opposed to this,\nthe risk--neutral valuation for pricing options {\\em imposes the drift} $r Y^\\nu$ to\nthe SDE followed by $Y^\\nu$ under the equivalent martingale measure.\nThis causes the option price implied by the alternative model to coincide, at first order\nin $\\epsilon$, with the Black and Scholes price with volatility parameter $\\nu$. In fact,\nimposing the drift $r Y^\\nu$ to $Y^\\nu$ leads to the same risk neutral process as that\nof Black and Scholes' (obtained by imposing the drift $rS$ to $S$), with the only difference\nthat $\\bar{\\sigma}$ is replaced by $\\nu$.\n\n\\begin{remark} {\\bf (Historical versus Implied volatility).}\nA first interesting property that can be deduced from this theorem\napplies in case one believes that option prices trade independently of the\nunderlying stock price.  We have in fact been able to\nconstruct a stock price process, the process (\\ref{sol:bes3}), whose marginal\ndistribution and transition density depend on the volatility coefficient $\\bar{\\sigma}$,\nwhereas the corresponding option price, in the limit for $\\epsilon \\rightarrow 0$,\nonly depends on the volatility coefficient $\\bar{\\sbar}$. As a consequence, we can  provide a\nconsistent theoretical framework which justifies the differences between\nhistorical and implied volatility that are commonly observed in real markets.\n\\end{remark}\n\nStraightforward application of the previous theorem leads to the main result\nof this section which is summarized in the following.\n\\begin{corollary} \\label{inf}\nThe solutions of problems\n(\\ref{optsiginf}) and (\\ref{optsigsup}) are\n\\begin{eqnarray}\n\\label{solinf}\n&V_{*}=(s_0-Ke^{-rT})^+\\nonumber \\\\ \\\\\n&V^{*}=s_0.\\nonumber\n\\end{eqnarray}\nMoreover, for any other candidate price $\\bar{V} \\in (V_{*},V^{*})$\nthere exist a volatility $\\nu$ and an $\\epsilon > 0$ such that\n\\begin{eqnarray*}\nU^\\epsilon(0,\\nu) = \\bar{V} .\n\\end{eqnarray*}\n\\end{corollary}\n{\\em Proof}.\nTo prove (\\ref{solinf}), we simply have to take the limit of expression\n(\\ref{bslem}) (with $t=0$) for $\\epsilon$ going to zero and $\\nu$ either going to zero\nor going to infinity, since\n\\begin{eqnarray*}\n&&\\lim_{v \\rightarrow 0} V_{BS}(v) = (s_0 - Ke^{-rT})^+, \\\\\n&&\\lim_{v \\rightarrow +\\infty} V_{BS}(v) = s_0,\n\\end{eqnarray*}\nand\n\\[ \\lim_{\\epsilon\\rightarrow 0} \\bar{\\sbar}^\\epsilon (0)=\\nu.\\]\nFinally, we remember that, ceteris paribus, $V_{BS}(v)$ is a strictly increasing\nfunction of $v$, so that $V_{BS}(v)=\\hat{V}$ has a unique solution for\n$V \\in (V_{*},V^{*})$, hence $U^\\epsilon(0,\\nu) = \\bar{V}$ has a solution in\n$(0,+\\infty)\\times (V_{*},V^{*})$.\n\n\\begin{remark} {\\bf (Taking $\\epsilon \\rightarrow 0$ ).}\nIt is possible to consider the limit for $\\epsilon \\rightarrow 0$ in the above\nexpressions so as to present our result in a simpler and more elegant way.\nHowever, the treatment with $\\epsilon$ does not involve limit considerations\nand permits to contain analytical effort, so that we decided to keep $\\epsilon > 0$.\nThis can be also useful in numerical implementations.\nWe just observe that, for $\\epsilon \\rightarrow 0$, since $\\beta<1$,\n(\\ref{intediff}) becomes\n\\begin{eqnarray*}\nZ_t &=& Z_{j\\Delta} + (\\mu -\\inverse{2}\\bar{\\sigma}^2)(t-j\\Delta)  \\\\ \\nonumber\n& & +\n (t-j\\Delta)^{\\beta\/2}  \\nu\\int_{j\\Delta}^t\n (u-j\\Delta)^{-\\beta\/2} dW_u\n\\ \\ \\  t\\in [j\\Delta ,(j+1)\\Delta).\n\\end{eqnarray*}\nThis process is well defined since the integral in the right-hand side exists finite a.s.\neven though its integrand diverges when $u \\rightarrow j \\Delta^+$.\n\nThe above equation can be better compared to the Black and Scholes\nprocess when written in differential form:\n\\begin{eqnarray*}\nd Z_t = (\\mu -\\inverse{2}\\bar{\\sigma}^2)\\ dt +\n\\frac{\\beta}{2}(t-j\\Delta)^{\\beta\/2-1}  \\nu\\int_{j\\Delta}^t (u-j\\Delta)^{-\\beta\/2} dW_u\\ dt\n  + \\nu \\ dW_t  \\ \\  \\  t\\in [j\\Delta ,(j+1)\\Delta).\n\\end{eqnarray*}\nBy observing this last equation we can isolate three terms in the right-hand side.\nThe first term is the same drift as in the log--returns of the Black and Scholes\nprocess. The third term is the same as in the log--returns of the Black and Scholes\nprocess, but the volatility parameter $\\bar{\\sigma}$ is replaced by our $\\nu$.\nFinally, the central term is the term which is needed to have returns equal to\nthe returns in the Black and Scholes process even after changing the volatility\nfrom $\\bar{\\sigma}$ to $\\nu$. Note that this term goes to zero for $\\bar{\\sigma} = \\nu$.\nIt is this term that makes our process non-Markov outside the trading time grid.\n\\end{remark}\n\nThe interpretation of the previous theorem and corollary is as\nfollows. If we are given a discretely observed stock price, the particular\nway we use to ``complete'' the model with any of our continuous-time processes $Y$ has a\nheavy impact on the associated option price. The influence is in fact so\nrelevant that such a price can be arbitrarily close to either no-arbitrage bounds\nfor option prices.\n\nFrom an intuitive point of view, the reason why option prices can be so\ndifferent is because the time step in the grid ${\\cal T}^\\Delta$\nis never infinitesimal. In other words, continuous-time option prices\nwould simply reveal the differences existing at an infinitesimal level\namong all the stock price processes $Y^\\nu$.\n\n\\section{Option pricing and hedging in the real world}\nLet us now consider a trader who needs to price an option on a given stock.\nHis usual practice is to resort to continuous-time mathematics to fully\nexploit the richness of its theoretical results, and to model stock returns with\na normal distribution.\n\nLet us denote by $\\delta t$ the length of the smallest time interval\nwhen an actual transaction can occur. Such $\\delta t$ is the best realistic\napproximation of the infinitesimal time distance ``$dt$''.\n\nThe results of the previous section imply that\nthe geometric Brownian motion (\\ref{BeS}) is\njust one of the infinitely many processes $Y$ that possess the\nproperties required by the practitioner along intervals of equal length $\\delta t$.\nHowever, the basic equivalence in the description of\nthe stock price dynamics can not be extended to the corresponding option\nprices. Indeed, any real number in the interval $(V_{*},V^{*})$ can be\nviewed as the unique no-arbitrage option price for some process $Y$.\n\nWhich process $Y$ should then be chosen by the practitioner?\n\nA possible answer to this question can be provided,\nfor example, through the estimation of the option replication\nerror in discrete time.\nTo this end, for any $\\nu >0$, let us denote by $(\\xi^\\nu,\\eta^\\nu)$\nthe self-financing strategy that replicates in continuos time the option payoff for\nthe process $Y^\\nu$, where\n$\\xi_t^\\nu$ and $\\eta_t^\\nu$ are respectively interpreted as the number of stock\nshares and money units held at time $t$.\n\n\nThen, fixing a set of dates $\\tau_0=0<\\tau_1<\\cdots<\\tau_n=T$ such that\n$\\tau_i=i \\delta t$,\\footnote{\nAlthough fixed a priori, the set ${\\cal T}^\\Delta$\nintroduced earlier is arbitrarily chosen, so that we can set\n$\\Delta=\\delta t\/k$, with $k$ any positive integer implying that\n$\\{\\tau_1,...,\\tau_n\\}  \\subset {\\cal T}^\\Delta$. }\nand, denoting the observed stock price at time $\\tau_j$ by $\\bar{S}_{\\tau_j}$,\n$j=0,\\ldots,n$, the replication error when hedging according to the\nstrategy $(\\xi^\\nu,\\eta^\\nu)$, starting from the endowment $U^\\epsilon(0,\\nu)$, is\n\\begin{eqnarray*}\n\\varepsilon(\\nu):=(\\bar{S}_{T}-K)^+ -U^\\epsilon(0,\\nu)-\\sum_{j=0}^{n-1}\n\\xi^\\nu_{\\tau_j}(\\bar{S}_{\\tau_{j+1}}-\\bar{S}_{\\tau_j})\n-\\sum_{j=0}^{n-1}\\eta^\\nu_{\\tau_j}(e^{r \\tau_{j+1}}-e^{r \\tau_j}).\n\\end{eqnarray*}\nAt this stage, we can solve, for example, an\noptimization problem where $\\varepsilon(\\nu)$ is minimized,\naccording to some criterion, over all $\\nu>0$.\nSuch procedure, however, is justified only to\nmeasure the performance of our continuous-time prices and strategies on the fixed set\nof discrete-time instants.\nMore generally, the issue of deriving a fair $\\delta t$-time option price and hedging\nstrategy should be tackled by resorting to the existing literature on incomplete markets.\n\nMany are the criterions one can choose from for pricing and hedging in\nincomplete markets. We mention for instance those of\nF\\\"{o}llmer and Sondermann (1986), F\\\"{o}llmer and Schweizer (1991),\nSchweizer (1988, 1991, 1993, 1994, 1995, 1996), Sch\\\"{a}l (1994), Bouleau and\nLamberton (1989), Barron and Jensen (1990), El Karoui, Jeanblanc-Picqu\\'{e}\nand Viswanathan (1991), Davis (1994), Frasson and Runggaldier (1997),\nEl Karoui and Quenez (1995), Frittelli (1996), Mercurio (1996), Mercurio\nand Vorst (1997), Bellini and Frittelli (1997), Frey (1998) and F\\\"ollmer and Leukert\n(1998).\n\nHowever, the purpose of this section is not to favor any particular approach.\nWe want, instead, to stress the following innovative feature in the theoretical\nproblem of option pricing. Instead of fixing a stock price process and then deriving\na fair option price and an ``optimal'' hedging strategy, we can in fact consider\na family of processes, that are somehow equivalent in the description of the\nstock price evolution, among which we can select a convenient one by means of our\nfavorite incomplete markets criterion.\n\n\nA comparison between the performances of the strategy $(\\xi^\\nu,\\eta^\\nu)$ and the hedging\nstrategy associated to any incomplete-market criterion is beyond the scope of the paper and is\nleft to future research.\n\n\\section{Conclusions}\nIn the present paper we consider an option-pricing application of a result which\nis based on the construction of nonlinear SDE's with densities evolving\nin a given finite-dimensional exponential family.\nPrecisely, we derive a family of stock price models that behave almost\nequivalently to that of Black and Scholes.\nAll such models share the same distributions for the stock price process and\nits log-returns along any previously fixed `trading time-grid'.\nTherefore, all these models can be viewed as equivalent in the description of the stock\nprice evolution.\n\nHowever, the continuous-time dynamics chosen to `complete' the model\nin between the instants of the trading time-grid,\nreflects heavily on the option price. The option price in fact can assume any value\nbetween the option intrinsic value and the underlying stock price.\n\nAs a conclusion, our result points out that no dynamics is the {\\em right}\none a priori, and that an incomplete-market criterion is needed to choose\namong all the different models.\nPractitioners with different criteria can still pick up a model\nfrom our family, so as to match their expectations or to minimize their exposures.\n\nEven though our results are based on the assumption of a lognormal distribution\nfor the stock price and a normal distribution for its log-returns, the generalization\nto many other distributions is possible. Notice, indeed, that\nthe results of Section 2 can be applied to any curve of densities in a given\nexponential family. However, the generalization is not straightforward and heavily relies\non the particular distributions which are considered.\n\n\n\\section*{Appendix}\nIn this appendix we consider the case $\\sigma_t(\\cdot) \\equiv \\nu \\neq 0$\nas an interesting example of a class of volatilities which do not belong to ${\\cal S}$.\nSuppose, for the sake of simplicity,\nthat we require only the marginal law of $Y$ to coincide with\nthe marginal law of $S$, ignoring the returns distributions.\nNow assume by contradiction that  $\\nu \\in {\\cal S}$.\nThen by applying~(\\ref{sol:bes1}) we obtain a Markov process~$Y$\n\\begin{eqnarray} \\label{gaussy}\ndY_t =  \\inverse{2} \\frac{\\nu^2}{Y_t}\n\\left[ \\zeta + 2 \\rho(t) \\log\\frac{Y_t}{s_0}\\right] \\ dt +\n\\frac{Y_t}{2t} \\left[\\log\\frac{Y_t}{s_0} - \\frac{\\zeta+1}{2 \\rho(t)}\\right]\n\\ dt + \\nu \\ dW_t \\ , \\ \\ \\epsilon \\le t \\le T \\ ,\n\\end{eqnarray}\nwith $Y_t=S_t$ for each $t \\in [0,\\epsilon]$.\n\nLet us focus on $t \\ge \\epsilon$.\nSince we now know that $Y_t$ and $S_t$ have the same\ndistribution under the objective probability measure, $Y_t$ is lognormally\ndistributed, and in particular $Y_t > 0$ for all $t$.\nSince $\\nu \\in {\\cal S}$, there exists an equivalent martingale measure,\nso that, under such a measure,\n\\begin{eqnarray*}\ndY_t = r Y_t \\ dt + \\nu \\ d\\widetilde{W}_t \\ ,\n\\end{eqnarray*}\nwhere $\\widetilde{W}$ is a standard Brownian motion under\nthe martingale measure. Now notice that this last SDE is a linear\nequation, so that its solution has a density whose\nsupport is the whole real line. In other terms, such solution\ncan be negative with positive probability at any fixed\ntime instant, as opposed to what we have seen\nfor $Y$ under the objective measure. However, there cannot be an equivalent measure\nthat transforms a process whose support is the positive halfline\ninto another one whose support is the whole real line. Therefore, we\nhave contradicted our assumption that $\\nu \\in {\\cal S}$.\n\nOur example is similar in spirit to that of Delbaen and Schachermayer (1995)\nwho consider a Bessel process (taking positive values at all times)\nwhich cannot be transformed into a Brownian motion. Some conditions\nfor the existence of an equivalent martingale measure are given\nfor example in Rydberg (1997).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nOne of the most controversial questions regarding the formation and evolution \nof galaxies is when and how today's most massive galaxies formed. In the \nstandard paradigm of structure formation, dark matter halos build in a \nhierarchical fashion through the dissipationless mechanism of gravitational \ninstability, while galaxies form inside these structures following the \nradiative cooling of baryons, with stars forming out of gas that cools within \nthe halos. In this picture, the most massive galaxies are naturally at the \nendpoint of the hierarchical merging process, and offer an opportunity to \nconstrain models of galaxy formation and evolution, since their predicted \ngalaxy properties are sensitive to various model assumptions, such as gas \ncooling, star formation, stellar and AGN feedback processes, chemical \nevolution, and mergers (e.g., \\citealt{baugh06}; \\citealt{delucia06}; \n\\citealt{henriques12}; and references therein).\n\nIn the local universe ($z\\sim0$), the most massive galaxies (i.e., stellar mass \n$M_{\\rm star}>5\\times10^{11}$~M$_{\\odot}$) constitute a homogeneous population. \nThese local ultra-massive galaxies tend to live in high-density \nenvironments (typically rich groups or clusters of galaxies; e.g., \n\\citealt{blanton09} for a review) and are characterized by early-type \nmorphologies, red rest-frame optical colors, extremely quiescent stellar \npopulations, old mean stellar ages, significant enhancements of the $\\alpha\/Fe$ \nelement ratio, and little dust attenuation, and populate the extreme massive \nend of the very tight red sequence (e.g., \\citealt{bower92}; \n\\citealt{kauffmann03}; \\citealt{kauffmann04}; \\citealt{gallazzi05}; \n\\citealt{nelan05}; \\citealt{thomas05}; \\citealt{baldry06}; \n\\citealt{gallazzi06}; \\citealt{thomas10}). In particular, two decades of \narcheological studies (i.e., the detailed investigation of the stellar \npopulations in local galaxies to constrain their formations and evolutions \nwith cosmic time) have robustly shown that the more massive the galaxy is \ntoday, the earlier its star formation must have started and more promptly must \nhave subsided, in an apparently ``anti-hierarchical'' fashion of the star \nformation. Quantitatively, most of the stars in local ultra-massive galaxies \nmust have formed in the first $\\sim3$~Gyr of cosmic history (i.e., $z>2$) \nthrough short ($<$1~Gyr, as low as $\\sim0.2$~Gyr for the most massive \ngalaxies), hence intense, bursts of star formation (e.g., \\citealt{renzini06}; \n\\citealt{vandokkum07}; \\citealt{thomas10}). \n\nIncreasingly more sophisticated models of galaxy formation and evolution have \nshown that the apparent down-sizing in the star formation of ultra-massive \ngalaxies is not in contradiction with the hierarchical paradigm. Recent models \ndo predict `anti-hierarchical' star-formation histories for the most massive \ngalaxies in a $\\Lambda$CDM universe (with most stars formed by $z\\sim3$), \nwhereas the assembly of these galaxies is indeed hierarchical, with $\\sim$80\\% \nof the final mass typically locked up in a single galaxy only at $z<0.4$ and \nwith a number of effective progenitors as high as $\\sim$5 for the most massive \ngalaxies (e.g., \\citealt{delucia06}; \\citealt{delucia07}). \n\nThese predictions are qualitatively in agreement with the nearly constant \nrest-frame optical luminosity density of red-sequence galaxies since $z\\sim1$ \nfound by the COMBO-17 and the DEEP2 surveys (\\citealt{bell04}; \n\\citealt{faber07}). This result in fact implies that the stellar mass density \nin red-sequence galaxies has roughly doubled since $z\\sim1$ driven by a \ncombination of galaxy merging and truncation of star formation of the blue \nstar-forming population. In particular, \\citet{faber07} have used these \nresults in support of a scenario for the formation of massive red-sequence \n(spheroidal) galaxies involving early mass assembly and star formation (with \nsuch progenitors living on the blue cloud), followed by quenching (which \nmigrates the progenitors onto the red sequence) and further dry merging \n(allowing for additional growth along the red sequence to explain the assembly \nof the most massive galaxies in the local universe). \n\nHowever, an independent analysis by \\citet{cimatti06} of the evolution of the \nrest-frame optical luminosity function of red-sequence galaxies since \n$z\\sim0.8$ from COMBO-17 and DEEP2 has shown that the massive \n($M_{\\rm star}>10^{11}$~M$_{\\odot}$) red-sequence galaxies have actually not \ngrown in mass over the past $\\sim$7 billion years, while the inferred growth \nby a factor of $\\sim$2 of the stellar mass density of red-sequence galaxies \nsince $z\\sim0.8$ appears due to the build-up of the less-massive red-sequence \npopulation. In addition to the anti-hierarchical formation of the stars, this \nresult implies also an anti-hierarchical assembly of the stellar content of \nthe most massive galaxies, in contrast to the predictions from theoretical \nmodels. \n\nRecent measurements of the stellar mass function of galaxies out to redshift \n$z=4$ have intensified the tension between observations and theoretical \npredictions. Indeed, increasingly wider and deeper near-infrared surveys have \nprovided evidence for the existence of very massive galaxies \n(i.e., $\\log{(M_{\\rm star}\/M_{\\odot})} \\gtrsim 11.5$) when the universe was only \n1.5 Gyr old (i.e., $z \\sim 4$), and of their number densities evolving very \nlittle in the following $\\sim$3--4~Gyr from $z=4$ to $z\\sim$1--1.5 (e.g., \n\\citealt{perez08}; \\citealt{marchesini09}; \\citealt{marchesini10}; \n\\citealt{brammer11}; \\citealt{muzzin13b}; and references therein). \n\nWhile the stellar mass function of galaxies provides a mean to measure the \nabundance of a galaxy population at a given time and the overall growth as a \nfunction of time of its stellar content, it does not tell us how individual \ngalaxies have evolved and assembled their mass. Ultimately, we would like to \nbe able to connect local ultra-massive galaxies with their progenitors in the \nearly universe to understand how their stellar populations have actually \nchanged over cosmic time and to answer the following outstanding questions. Do \nlocal ultra-massive galaxies really form in short and intense bursts of star \nformation at high redshift? When do they actually stop forming stars? How does \ntheir stellar population age? How much mass is assembled over time? Do they \nreally evolve from the blue cloud to the red sequence as suggested by, e.g., \n\\citet{faber07}? \n\nNeedless to say, answering these questions have proved difficult, as it \nrequires the non-trivial task of linking galaxies and their \ndescendants\/progenitors through cosmic time, which in turn requires \nassumptions for how galaxies evolve. However, in recent years, several \napproaches have been developed to link galaxies across cosmic time. A popular \napproach easy to implement is to compare galaxy properties at fixed cumulative \nnumber density (e.g., \\citealt{wake06}; \\citealt{tojeiro10}; \n\\citealt{brammer11}; \\citealt{papovich11}; \\citealt{tojeiro12}; \n\\citealt{patel13}; \\citealt{vandokkum13}; \\citealt{leja13}; \n\\citealt{muzzin13b}). While this approach has the advantage of relying solely \non observations, it does not take into account galaxy-galaxy mergers and \nscatter in mass accretion histories, both affecting the median cumulative \nnumber density of a galaxy population (\\citealt{leja13}; \\citealt{lin13}; \n\\citealt{behroozi13a}). More advanced comparisons have adopted either \nsemi-analytical or semi-empirical galaxy-halo connections to deduce galaxy \nevolution from simulated dark matter merger histories (e.g., \n\\citealt{conroy09}; \\citealt{leitner12}; \\citealt{behroozi13a}; \n\\citealt{moster13}; \\citealt{wang13}; \\citealt{lu12}; \\citealt{leja13}; \n\\citealt{lin13}; \\citealt{mutch13}; \\citealt{lu14}; \\citealt{behroozi13b}; and \nreferences therein). \n\nIn this paper, we investigate the evolution with cosmic time of the \nprogenitors of local ultra-massive ($M_{\\rm star} \\approx 6 \\times 10^{11}$ \nM$_{\\odot}$) galaxies from $z=3$. The progenitors are selected using both the \nfixed cumulative number density technique and a semi-empirical approach using \nabundance matching in the $\\Lambda$CDM paradigm. The stellar population \nproperties of the progenitors are then studied as a function of redshift. This \nanalysis provides a direct test of the star-formation histories predicted from \nthe archeology studies of local ultra-massive galaxies and returns a clear and \nconsistent picture of their evolution in the past 11.4 billion years of cosmic \nhistory. \n\nOur paper is structured as follows: The adopted dataset and the selection of \nthe sample of progenitors are described in \\S\\ref{sec-dataset} and \n\\S\\ref{sec-sample}, respectively. The evolution of the properties of the \nprogenitors of local ultra-massive galaxies is presented in \n\\S\\ref{sec-results}, while the results are summarized and discussed in \n\\S\\ref{sec-discussion}. We assume $\\Omega_{\\rm M}=0.3$, \n$\\Omega_{\\rm \\Lambda}=0.7$, and $H_{\\rm 0}=70$~km~s$^{-1}$Mpc$^{-1}$. A \n\\citet{kroupa01} initial mass function (IMF) is assumed throughout the paper. \nAll magnitudes are in the AB system.\n\n\n\\section{The Dataset}\\label{sec-dataset}\n\nThis study uses the K$_{\\rm S}$-selected catalog of the COSMOS\/UltraVISTA field \nfrom \\citet{muzzin13a}. The catalog includes PSF-matched photometry in 30 \nphotometric bands over the wavelength range 0.15$\\mu$m $\\rightarrow$ 24$\\mu$m \nfrom the available GALEX \\citep{martin05}, CFHT\/Subaru \\citep{capak07}, \nUltraVISTA \\citep{mccracken12}, and S-COSMOS \\citep{sanders07} datasets. \nSources are selected from the DR1 \nUltraVISTA K$_{\\rm S}$-band imaging \\citep{mccracken12} which reaches a depth \nof K$_{\\rm S,tot}<23.4$ at 90\\% completeness. A detailed description of the \nphotometric catalog construction, photometric redshift measurements, and \nstellar population properties' estimates (e.g., stellar mass) is presented in \n\\citet{muzzin13a}, which accompanies the public release of all data products \nfrom the catalog. A brief description of the relevant aspects of the catalog \nwith respect to our analysis is also provided in \\citet{muzzin13b}, presenting \nthe measurements of the stellar mass function of galaxies, including quiescent \nand star-forming galaxies, from $z=4$ to $z=0.2$. \n\nBriefly, stellar population properties were derived by fitting the observed \nspectral energy distributions (SEDs) from the {\\it GALEX} UV to the \n{\\it Spitzer}-IRAC 8$\\mu$m photometry with \\citet{bruzual03} models assuming \nexponentially-declining star-formation histories (SFHs) of the form \nSFR$\\propto e^{\\rm -t\/\\tau}$, where SFR is the star formation rate, $t$ is the \ntime since the onset of star formation, and $\\tau$ sets the timescale of the \ndecline in the SFR, solar metallicity, a \\citet{calzetti00} dust law, and a \n\\citet{kroupa01} IMF. We allow $\\log{(\\tau\/Gyr)}$ to range between 7.0 and \n10.0 Gyr, $\\log{(t\/Gyr)}$ between 7.0 and 10.1 Gyr, and $A_{\\rm V}$ between 0 \nand 4 mag. Median random errors on stellar mass, SFR, stellar age, and dust \nextinction are $^{+0.05}_{-0.09}$~dex, $\\pm$0.2~dex, $^{+0.1}_{-0.2}$~dex, and \n$^{+0.3}_{-0.2}$~mag, respectively, for the quiescent progenitors, and \n$^{+0.09}_{-0.18}$~dex, $\\pm$0.6~dex, $^{+0.2}_{-0.4}$~dex, and $^{+0.6}_{-0.4}$ \nmag, respectively, for the star-forming progenitors, similar to the typical \nuncertainties from low-resolution spectroscopy \\citep{muzzin09}. To assess the \nsystematic effects of the adopted SED-modeling assumptions, we have also \ninvestigated different SFHs, metallicities, and extinction curves (see \n\\S\\ref{s-sedmodel}).\n\nThe SFRs from SED modeling can be strongly influenced by the assumption of the \nSFH. A more robust estimate of the instantaneous SFR is obtained from the \ncombination of the rest-frame UV flux ($L_{\\rm 2800}$) and the rest-frame total \ninfrared luminosity ($L_{\\rm IR}$). To determine $L_{\\rm 2800}$ we used EAZY \n\\citep{brammer08} to integrate the best-fit template over the wavelength range \n2600-2950\\AA. The inclusion of the {\\it GALEX} photometry in the UltraVISTA \nmultiwavelength catalogs of \\citet{muzzin13a} guarantees that $L_{\\rm 2800}$ is \nconstrained by the data over the full targeted redshift range. The approach \npresented in \\citet{wuyts08} was followed to determine $L_{\\rm IR}$ from the \n24$\\mu$m emission. Specifically, as already described in detail in \n\\citet{marchesini10}, we used the infrared SEDs of star-forming galaxies from \n\\citet{dale02}, which allow us to derive the IR\/MIR flux ratio for different \nheating levels of the interstellar environment, parameterized by \n$dM(U) \\sim U^{-\\alpha} dU$, where $M(U)$ is the dust mass heated by an \nintensity $U$ of the interstellar field. The total infrared luminosity \n$L_{\\rm IR,\\alpha}$ was computed for each template in \\citet{dale02} within the \nrange $1 \\leq \\alpha \\leq 2.5$. We adopted the log-average of the resulting \n$L_{\\rm IR,\\alpha=1,...,2.5}$ as the best estimate for the IR luminosity. This \napproach returns SFRs in better agreement with the SFRs estimated from SED \nmodeling (\\citealt{franx08}; \\citealt{wuyts08}) and from dust-corrected \nH$\\alpha$ line fluxes (\\citealt{muzzin10}), compared to the often adopted local \nluminosity-dependent approaches, which systematically overestimate SFRs by a \nfactor of 4-6 (e.g., \\citealt{papovich07}; \\citealt{murphy09}; \n\\citealt{elbaz10}; \\citealt{muzzin10}). We convert the $L_{\\rm 2800}$ into \nSFR$_{\\rm UV,uncorr}$ using the conversion factor \nSFR$_{\\rm UV,uncorr}=3.234\\times10^{-10}~L_{\\rm 2800}$ from \\citet{kennicutt98}, \nadapted to a \\citet{kroupa01} IMF by \\citet{bell05}. SFR$_{\\rm UV,uncorr}$ is \nthe observed SFR, and is not corrected for dust extinction. The $L_{\\rm IR}$ is \nconverted into a SFR$_{\\rm IR}$ using \nSFR$_{\\rm IR}=0.98\\times10^{-10}~L_{\\rm IR}$ from \\citet{kennicutt98}, adapted \nto a \\citet{kroupa01} IMF. The total SFR of the galaxies is then derived via \nSFR$_{\\rm tot}$=SFR$_{\\rm UV,uncorr}+$SFR$_{\\rm IR}$.\n\n\\section{Selection of the Progenitors of Local Ultra-Massive Galaxies}\\label{sec-sample}\n\n\\subsection{Fixed Cumulative Number Density}\\label{fcnda}\n\n\\begin{figure*}\n\\epsscale{1.05}\n\\plotone{fig1.eps}\n\\caption{{\\bf Left panel:} Cumulative number density, $n(>M_{\\rm star})$, as a \nfunction of stellar mass obtained by integrating the stellar mass functions \nfrom the UltraVISTA survey \\citep{muzzin13b} at $0.2<z<3.0$. The different \ncolors correspond to the different targeted redshift intervals. Filled \ntriangles show the evolution in stellar mass for galaxies at fixed cumulative \nnumber densities of 3$\\times10^{-6}$ Mpc$^{-3}$, shown by the dashed horizontal \ngray line. Filled circles show evolution in stellar mass after inclusion of the \nevolving number density for galaxy progenitors using an abundance matching \napproach (\\S\\ref{seaam}). The hatched regions show the 1$\\sigma$ range of \n$n(>M_{\\rm star})$ including Poisson errors, cosmic variance, and photometric \nredshift uncertainties. The 1$\\sigma$ range of $n(>M_{\\rm star})$ is plotted \nonly for the lowest and highest targeted redshift intervals for clarity. \n{\\bf Right panel:} Evolution of the stellar mass of the progenitors, \n$M_{\\rm progen}$, as a function of redshift. Filled triangles are obtained \nusing the fixed cumulative number density approach, whereas filled circles are \nderived using the semi-empirical approach using abundance matching. The error \nbars show the width of the redshift bin and the formal 1$\\sigma$ uncertainty \nin stellar mass at a given cumulative number density and includes Poisson \nerrors, cosmic variance, and photometric redshift uncertainties. The black \ndashed and solid curves show the evolutions of the stellar mass as a function \nof redshift (parameterized as $\\log{(M_{\\rm star})}=A+Bz+Cz^{2}$) derived from \nthe constant cumulative number density and abundance matching approaches, \nrespectively. The progenitors of $z\\sim0$ UMGs (i.e., \n$\\log{(M_{\\rm star}\/M_{\\odot})}=11.8$) are selected in a narrow range of stellar \nmass around these curves. The dark gray filled region shows the 1$\\sigma$ \nuncertainty on the stellar mass growth of the progenitors as selected using \nthe abundance matching approach when only the errors on the observed stellar \nmass functions are considered. The light gray filled region shows instead the \n68th-percentile range in stellar mass including both the errors on the \nobserved stellar mass functions and the spread from individual galaxy growth \nhistories. Specifically, the 68-th percentile range was estimated by \ngenerating 1000 realizations of the stellar mass tracks from 1000 realizations \nof the stellar mass functions and of the galaxy accretion histories \n\\citep{behroozi13b}. The color curve and hatched region represent the median \nand the 15$^{\\rm th}$\/85$^{\\rm th}$ percentile range, respectively, of the \nassembly history (light blue) and the star-formation history (light green) of \nBCGs as simulated in \\citet{delucia07}.\\label{fig1}}\n\\end{figure*}\n\nFollowing previous studies (e.g., \\citealt{wake06}; \\citealt{tojeiro10}; \n\\citealt{brammer11}; \\citealt{papovich11}; \\citealt{tojeiro12}; \n\\citealt{patel13}; \\citealt{vandokkum13}; \\citealt{leja13}; \n\\citealt{muzzin13b}), we connect progenitor galaxies by requiring that they \nhave the same cumulative co-moving number density of the low-redshift galaxy \npopulation, i.e., galaxies at $0.2<z<0.5$ with $M_{\\rm star} \\approx 5 \\times \n10^{11}$ M$_{\\odot}$. Effectively, galaxies are ranked according to their \nstellar mass and we select galaxies at different redshifts that have the same \nrank order as the low-redshift population. The implicit assumption is that rank \norder is conserved through cosmic time, or that processes that break the rank \norder do not have a strong effect on the average measured properties.\n\nIn order to apply the fixed cumulative number density selection, we derive \ncumulative number densities, $n(>M_{\\rm star})$, as a function of stellar mass \nusing the recently published and publicly available stellar mass functions \nfrom the UltraVISTA survey \\citep{muzzin13b}. The contributions from Poisson \nerrors, cosmic variance, and photometric redshift uncertainties to the total \nerror budget of the stellar mass functions were quantified and presented in \n\\citet{muzzin13b}. We refer to this paper for a detailed description of the \nderivations of the errors of the stellar mass functions. The left panel of \nFigure~\\ref{fig1} shows $n(>M_{\\rm star})$ in the six targeted redshift ranges \nfrom $z=0.2$ to $z=3.0$, namely $0.2 \\le z<0.5$, $0.5 \\le z<1.0$, \n$1.0 \\le z<1.5$, $1.5 \\le z<2.0$, $2.0 \\le z<2.5$, $2.5 \\le z<3.0$. Also shown \nis the total 1$\\sigma$ error (including Poisson errors, cosmic variance, and \nphotometric redshift uncertainties) on $n(>M_{\\rm star})$ for the lowest and \nhighest targeted redshift bins. Galaxies with stellar masses \n$M_{\\rm star}\\approx5 \\times 10^{11}$~M$_{\\odot}=10^{11.7}$ M$_{\\odot}$ at \n$z=0.35$ have a cumulative number density of $3\\times10^{-6}$ Mpc$^{-3}$ (shown \nby the horizontal dashed line in the left panel of Fig.~\\ref{fig1}). We then \ntrace the progenitors of these galaxies by identifying, at each redshift, the \nstellar mass for which the cumulative number density is $3\\times10^{-6}$ \nMpc$^{-3}$ (indicated in the left panel of Fig.~\\ref{fig1} by the filled \ntriangles). \n\nThe stellar mass evolution for galaxies with the rank order of the low-$z$ \npopulation is shown in the right panel of Figure~\\ref{fig1} with colored filled \ntriangles. Also shown is the formal 1$\\sigma$ uncertainty on the stellar mass \nof the progenitors at the given cumulative number density and including the \ncontributions from Poisson errors, cosmic variance, and photometric redshift \nuncertainties. As previously done in the literature (e.g., \n\\citealt{vandokkum13}), we parameterize the evolution of the stellar mass of \nthe progenitors with a quadratic function of the form \n$\\log{(M_{\\rm progen}(z)\/M_{\\odot})}=A+Bz+Cz^{2}$, with $A=11.796\\pm0.043$, \n$B=-0.291\\pm0.076$, and $C=0.060\\pm0.028$ (dashed curve in the right panel of \nFig.~\\ref{fig1}), corresponding to a value of the stellar mass in the local \nuniverse of $\\log{(M_{\\rm progen}(z=0)\/M_{\\odot})}=11.8$. This value is \ncomparable to the typical stellar masses of local brightest cluster galaxies \n(BCGs; e.g., \\citealt{lidman12}; \\citealt{lin13}). \n\n\\subsection{Semi-empirical Approach Using Abundance Matching}\\label{seaam}\n\nThe fixed cumulative number density approach does not take into account \ngalaxy-galaxy mergers and scatter in mass accretion histories, both altering \nthe median cumulative number density of a galaxy population (\\citealt{leja13}; \n\\citealt{lin13}; \\citealt{behroozi13a}). Therefore, to select the progenitors \nwe also adopt a semi-empirical approach using abundance matching in the \n$\\Lambda$CDM paradigm, accounting for mergers and scatter in mass accretion \nhistories. A detailed description of this technique is presented in \n\\citet{behroozi13b} (and references therein), who also made publicly available \nthe code implementing this \ntechnique.\\footnote{http:\/\/code.google.com\/p\/nd-redshift\/} Briefly, the galaxy \ncumulative number density at redshift $z_{\\rm 1}$ is converted to a halo mass \nwith equal cumulative number density using peak halo mass functions. Then, for \nhalos at that mass at $z_{\\rm 1}$, the masses of the most-massive progenitor \nhalos at $z_{\\rm 2}>z_{\\rm 1}$ are recorded using to the halos' mass accretion \nhistories. Finally, the median halo progenitor mass at $z_{\\rm 2}$ is converted \nback into cumulative number densities using the halo mass function at \n$z_{\\rm 2}$. \n\nSimilarly to the fixed cumulative number density approach, the progenitors of \nthe low-$z$ population of very massive galaxies are traced by identifying, at \neach redshift, the stellar mass for which the evolving cumulative number \ndensity (determined with the semi-empirical approach using abundance matching) \nintersects the cumulative number density curves derived from the UltraVISTA \nstellar mass functions (plotted as filled circles in the left panel \nFig.~\\ref{fig1}).\n\nThe resulting stellar mass evolution with the corresponding 1$\\sigma$ \nuncertainty is shown in the right panel of Figure~\\ref{fig1} with colored \nfilled circles. Errors on the stellar mass of the progenitors were derived in \nthe same way as described in \\S\\ref{fcnda}. Adopting the same parameterization \nas for the fixed cumulative number density approach, we find \n$A=11.801\\pm0.038$, $B=-0.304\\pm0.054$, and $C=0.039\\pm0.014$, also implying a \nvalue of the stellar mass in the local universe of \n$\\log{(M_{\\rm progen}(z=0)\/M_{\\odot})}=11.8$ (continuous curve in the right \npanel of Fig.~\\ref{fig1}).\n\n\\section{Evolution of the Progenitors' Properties}\\label{sec-results}\n\nHere we study and present the evolution as a function of cosmic time of the \nproperties of the progenitors of local ultra-massive \n($\\log{(M_{\\rm star}\/M_{\\odot})}=11.8$) galaxies (UMGs, hereafter) since $z=3$. \nFor a given redshift, we select the progenitors of $z\\sim0$ UMGs within a bin \nof size $\\sim0.2$~dex in stellar mass centered on $M_{\\rm progen}(z)$, following \nthe approach adopted by \\citet{patel13}. The actual boundaries of the bins are \nadjusted such that the median mass is close to the value given by \n$M_{\\rm progen}(z)$. Given the steepness of the stellar mass function, in \npractice this results in selecting galaxies at \n$(\\log{M_{\\rm progen}(z)\/M_{\\odot}})^{+0.15}_{-0.07}$. \n\n\\subsection{Stellar Mass Evolution}\n\nAs shown by the dashed curve in the right panel of Figure~\\ref{fig1}, the \nevolution in stellar mass of the progenitors of $z\\sim0$ UMGs is quite small \nwhen the progenitors are selected using the fixed cumulative number density \nmethod, i.e., $0.33\\pm0.15$~dex (i.e., a factor of $\\sim$2.1$^{+1.0}_{-0.6}$) \nfrom $z=3.0$ to $z=0$. Therefore, about half of the stellar mass in local UMGs \nwas assembled at $z>3$, i.e., in the first 2~Gyr of cosmic history. If the \nprogenitors are instead selected with the semi-empirical approach using \nabundance matching, the evolution in stellar mass is much more significant, \n$0.56\\pm0.21$~dex, $0.45\\pm0.13$~dex, and $0.27\\pm0.08$~dex from $z=3.0$, \n$z=2.0$, and $z=1.0$, respectively, to $z=0$. Therefore, the progenitors have \ngrown by a factor of $\\sim$3.6$^{+2.3}_{-1.4}$ over the last 11.4~Gyr from \n$z=3.0$ to $z=0$, with only about a fourth of the stellar mass in local UMGs \nhaving been assembled by $z\\sim3$. If the 68th-percentile range in stellar \nmass (shown in Figure~\\ref{fig1} as a light gray filled region) resulting from \nthe scatter in the progenitors' number density is included in the error \nbudget, the uncertainty on the inferred growth would be larger by a factor up \nto $\\sim$1.7, i.e., $0.56^{+0.35}_{-0.25}$~dex, $0.45^{+0.16}_{-0.20}$ dex, and \n$0.27^{+0.08}_{-0.12}$~dex from $z=3.0$, $z=2.0$, and $z=1.0$, respectively, to \n$z=0$.\n\nThe right panel of Figure~\\ref{fig1} also shows the predicted median and \n15$^{\\rm th}$\/85$^{\\rm th}$ percentiles of the `formation' (light blue) and \n`assembly' (light green) histories of BCGs from the theoretical study using \nsemi-analytic techniques presented in \\citet{delucia07}. The formation history \nof BCGs is defined as the sum of the \nstellar masses in all progenitors at each time, whereas their assembly history \ncorresponds to the stellar mass of the main branch. As highlighted in \n\\citet{delucia07}, there is a very small scatter in the formation histories of \nBCGs, with most of their stars having formed at $z>$3 ($\\sim$90\\% of the stars \nformed at $z>2.5$). The assembly histories exhibit a much larger scatter with \na much later assembly time. Specifically, the fraction of mass in the main \nprogenitor varies between 15\\% and 40\\% at $z=1$, and between 40\\% and 70\\% at \n$z=0.5$ \\citep{delucia07}. Figure~\\ref{fig1} shows that the inferred \nevolutions in stellar mass of the progenitors of $z\\sim0$ UMGs (solid or \ndashed curves) are faster than the model-predicted assembly histories, but \nmuch slower than the predicted formation histories of BCGs. In other words, in \nsemi-analytic models most of the stars were not formed in the main branch, but \nstar formation happened very early in low- to intermediate-mass halos (with \nstellar masses $\\sim$a few $\\times$10$^{10}$~M$_{\\odot}$), which were accreted \nsteadily over time following the assembly history of the dark matter. We note \nthat, as the median cumulative number density of the progenitors increases \nwith redshift as $\\sim(0.2 \\Delta z)$ \\citep{behroozi13b}, using the fixed \ncumulative number density approach would preferentially select progenitors \nwhich potentially had an unusually rapid assembly at earlier times. This \nshould be kept in mind in the comparison between the model-predicted assembly \nhistory of BCGs with the inferred growth in stellar mass of the progenitors of \nlocal UMGs selected with the fixed cumulative number density approach. At \n$z=3$, the progenitors of UMGs have a typical observed stellar mass of \n$\\log{(M_{\\rm star}\/M_{\\odot})}\\approx11.2$, a factor of $\\sim5$ larger than the \nmodel-predicted stellar mass assembled by $z=3$, implying a predicted \nabundance of BCGs' progenitors at $z=3$ significantly larger than observed. \nWe note that the dominant source of uncertainty in the stellar mass evolution \nof the progenitors is due to the scatter in the progenitors' cumulative number \ndensity. When this source of error is included, as shown by the light gray \nfilled region in the right panel of Figure~\\ref{fig1}, the disagreement \nbetween the observed stellar growth of local UMGs and the model-predicted \nassembly history of BCGs becomes less significant, and the two marginally \nagree at the $\\sim$2$\\sigma$ level.\n\nThe disagreement between observed and model-predicted evolutions of the \nstellar mass content of the progenitors of local UMGs is another manifestation \nof the previously highlighted tensions between observed and model-predicted \nstellar mass functions. Specifically, semi-analytic models were found to \nsignificantly overpredict the observed number density of galaxies below \n$10^{11}$~M$_{\\odot}$ at $z\\gtrsim1$-1.5 and to underpredict the number density \nof massive galaxies at $z>2$ (e.g., \\citealt{marchesini09}), with low-mass \ngalaxies in the stellar mass range $10^{9}-10^{11}$~M$_{\\odot}$ forming too \nearly in the models and being too passive at later times (e.g., \n\\citealt{fontanot09}; see also \\citealt{henriques12}). We note that, whereas \nthe typical errors on stellar masses are not large enough to explain the \ndisagreement at the high-mass end between the observed and the model-predicted \nstellar mass functions at $z \\gtrsim 1$ (e.g., \\citealt{ilbert13}), the \ndisagreement between the inferred evolution in stellar mass of the progenitors \nof local UMGs and the model-predicted assembly history of BCGs shown in \nFigure~\\ref{fig1} could potentially be further reduced (although at most only \npartly) if the model predictions were to be convolved by the typical \nstatistical errors in the stellar masses of UltraVISTA.\n\nWhereas the inferred growth in stellar mass from $z=3$ is quite different \ndepending on the adopted method to select the progenitors, the derived \nevolution in the stellar population parameters (e.g., rest-frame colors, level \nof star formation activity, stellar ages, extinction, rest-frame optical \nabsolute magnitudes, and mass-to-light ratios) is qualitatively similar. \nMoreover, observationally, recent works have robustly revealed that most \nearly-type galaxies show tidally disrupted features indicative of their past \nmerging histories (e.g., \\citealt{malin83}; \\citealt{vandokkum05}; \n\\citealt{tal09}; \\citealt{janowiecki10}; \\citealt{sheen12}), with evidence of \nmajor dry (i.e., dissipationless) merging being an increasingly important \nmechanism for the evolution of galaxies with $M_{\\rm star}>10^{11.4}$~M$_{\\odot}$ \n(e.g., \\citealt{kim13}). Contrary to the fixed cumulative number density \nmethod, the semi-empirical approach using abundance matching accounts for \nmergers, making the latter a more desirable approach to constrain the evolution \nof the median cumulative number density for a galaxy population \n\\citep{behroozi13b}. Therefore, in the sections below, we present only the \nevolution of the stellar population properties of the progenitors of $z\\sim0$ \nUMGs when selected with the semi-empirical approach using abundance matching. \nFor completeness, Appendix~\\ref{appendixA} presents the evolution of the \nstellar population properties of the progenitors of local UMGs when selected \nwith the fixed cumulative number density method.\n\n\\subsection{Evolution in Rest-frame $U-V$ and $V-J$ Colors}\n\n\\begin{figure*}\n\\epsscale{1.1}\n\\plotone{fig2.eps}\n\\caption{Rest-frame $U-V$ versus $V-J$ diagram in the targeted redshift \nintervals between $z=0.2$ and $z=3.0$ for the progenitors of local UMGs \nselected with the semi-empirical approach using abundance matching. The cuts \nused to separate star forming from quiescent galaxies from \\citet{muzzin13b} \nare shown as the solid dark gray lines. Red and blue filled circles highlight \nthe quiescent and star-forming galaxies among the progenitors' sample. Listed \nare the numbers of quiescent (red) and star-forming (blue) progenitors in each \nredshift interval. The grayscale representation of the overall galaxy \npopulation above the 95\\% mass-completeness limit from the UltraVISTA survey \nis also plotted in each redshift interval. Small green circles show the \nprogenitors detected in the {\\it Spitzer}-MIPS 24$\\mu$m data with \nsignal-to-noise $S\/N>5$. The numbers of MIPS-detected galaxies among the \nquiescent and star-forming progenitors are listed in green. Most of the \nstar-forming galaxies are robustly detected in MIPS, whereas only a small \nfraction of quiescent galaxies is detected at 24$\\mu$m. Color evolution tracks \nof \\citet{bruzual03} models are also shown: an exponentially declining SFH \nwith no dust ($\\tau$=100~Myr; dark gray), a constant SFH with no dust (CSF; \nlight blue), and the same CSF model with $A_{\\rm V}=2$~mag of extinction \n(light brown). The evolution tracks are plotted up to the maximum allowed age \nof the universe corresponding to the lower limit of each redshift interval. \nThe empty circles represent the model colors at the specified ages (in Gyr). \nThe dust vector indicates an extinction of $A_{\\rm V}=1$~mag for a \n\\citet{calzetti00} extinction curve. The population of progenitors of local \nUMGs shows a range in properties at $2.5<z<3.0$, typical of a heterogeneous \npopulation mostly dominated by dusty ($A_{\\rm V}=1.5-2$~mag) star-forming \ngalaxies, but also including post-starburst quiescent galaxies. By $z=0.35$, \nsuch heterogeneous population has turned into a homogeneous population of \nmaximally old, quiescent galaxies with very similar colors. \n\\label{fig2}}\n\\end{figure*}\n\nThe rest-frame $U-V$ versus $V-J$ color-color diagram (hereafter the $UVJ$ \ndiagram) has become, in recent years, a common tool to separate quiescent from \nstar-forming galaxies for its ability to separate red galaxies that are \nquiescent from reddened (i.e., dust obscured) star-forming galaxies (see, \ne.g., \\citealt{labbe06}; \\citealt{wuyts07}; \\citealt{williams09}; \n\\citealt{brammer11}; \\citealt{patel11}; \\citealt{whitaker11}; \n\\citealt{muzzin13b}). Figure~\\ref{fig2} shows the evolution of the rest-frame \n$U-V$ and $V-J$ colors of the progenitors of UMGs since $z=3$ (colored filled \ncircles), along with the evolution of the rest-frame $U-V$ and $V-J$ colors of \nthe overall galaxy population (grayscale representation) above the 95\\% \nmass-completeness limit from the UltraVISTA survey. To distinguish between \nstar-forming and quiescent galaxies we use the same box adopted in \n\\citet{muzzin13b} for the study of the evolution of the stellar mass functions \nof star-forming and quiescent galaxies since $z=4$. The adopted box is plotted \nin Figure~\\ref{fig2}, with the quiescent and star-forming progenitors \nhighlighted in red and blue, respectively. Color evolution tracks from the \nstellar population synthesis models of \\citet{bruzual03} are also plotted in \nFigure~\\ref{fig2} for different assumed star-formation histories (SFHs) and \namount of dust extinction: an exponentially declining SFH with an e-folding \ntimescale of $\\tau$=100~Myr and no dust extinction ($\\tau_{\\rm 100}$; dark gray \ntrack), a constant SFH with no dust (CSF; light blue), and the same CSF model \nwith $A_{\\rm V}=2$~mag of extinction (light brown). The color tracks are \nplotted up to the maximum allowed age of the universe corresponding to the \nlower limit of each redshift interval. We note that the $\\tau_{\\rm 100}$ color \ntrack is very similar to the color track of a single stellar population, with \nthe latter reaching the same $U-V$ and $V-J$ colors of the $\\tau_{\\rm 100}$ \nmodel at slightly younger (by $\\sim0.2$~Gyr) ages. \n\nAt $0.2<z<0.5$, the progenitors of UMGs constitute a homogeneous population \nwith similar rest-frame $U-V$ and $V-J$ colors, typical of \nquiescent galaxies with old (age$>8$~Gyr) stellar populations (from the \ncomparison with the $\\tau_{\\rm 100}$ color track). As the progenitors of UMGs \nare followed to higher redshifts, such population becomes increasingly \ndiversified. Out to $z=1$, the progenitors are still all quiescent galaxies \nand constitute a homogeneous population, while at $z>1$ the contribution \nfrom star-forming galaxies progressively increases, with the population of the \nprogenitors being dominated by star-forming galaxies at $2<z<3$. \nQuantitatively, the fraction of quiescent galaxies in the progenitors of local \nUMGs is seen to decrease from 100\\% at $z \\le 1$, to $\\sim$86\\% at $z=1.25$, \nand down to $\\sim$17\\% by $z=2.75$. \n\nThe location of the star-forming progenitors in the $UVJ$ diagram clearly \nindicates that these galaxies are quite dusty, with typical extinction in the \nvisual band of $A_{\\rm V}\\sim1.5-2$~mag (inferred from the comparison with the \ncolor track of a CSF model with $A_{\\rm V}=2$~mag). The dusty nature of \nthe star-forming progenitors can be further tested with {\\it Spitzer}-MIPS \n24$\\mu$m data. For the targeted redshift range, the MIPS~24$\\mu$m probes the \nrest-frame mid-infrared (MIR; $\\sim$6 and 20$\\mu$m at $z=3$ and $z=0.2$, \nrespectively), offering a powerful method of separating two basic SED types: \nevolved quiescent systems (whose stellar emission drops drastically at \n$\\lambda_{\\rm rest} >$2$\\mu$m), and active dusty galaxies, i.e., those that are \npowered by either star formation or active galactic nuclei (AGNs), both of \nwhich produce substantial MIR emission (e.g., \\citealt{webb06}). Specifically, \nin pure starburst galaxies, the MIR emission is dominated by features from \npolycyclic aromatic hydrocarbons (PAH) that are strong relative to the \nunderlying dust continuum, which only begins to rise above $\\sim10$$\\mu$m \n\\citep{draine07}. The hard radiation field of an AGN destroys PAH carriers, \nand the continuum emission from hot small dust grains is strong throughout the \nMIR \\citep{genzel00}. \n\nThe progenitors of UMGs detected with a signal-to-noise ratio $S\/N>5$ in the \nMIPS 24$\\mu$m data are shown in Figure~\\ref{fig2} by green filled circles. On \naverage, $\\sim$84\\% of the star-forming progenitors are robustly detected with \n$S\/N>5$ in MIPS, with a fraction of 24$\\mu$m-detected sources as high as 90\\% \nat $z<2$, supporting the dusty nature of the star-forming progenitors. We note \nthat the adopted MIPS flux limit (35$\\mu$Jy, 5$\\sigma$) translates into a \nlarge range of MIR luminosities sampled at a broad range of rest-frame \nwavelengths over the large considered redshift range, preventing the detection \nrate alone to be an ideal proxy of the level of obscured star-formation \nactivity as a function of redshift, since larger SFRs are needed in order to \nbe detected at higher redshifts. Therefore, the roughly constant detection \nrate of the star-forming progenitors over the targeted redshift interval is a \nstrong evidence for increasing levels of the dust-enshrouded activity with \nredshift in the star-forming progenitors of UMGs (see a more quantitative \nanalysis of the evolution of the SFR and specific-SFR in \\S\\ref{ssfr}). \n\n\\begin{figure*}\n\\epsscale{1.1}\n\\plotone{fig3.eps}\n\\caption{Rest-frame $U-V$ color versus stellar mass diagram. The grayscale \nrepresentation shows the $U-V$ versus $M_{\\rm star}$ diagram for the overall \ngalaxy population from the UltraVISTA survey in each redshift interval, with \nthe vertical dashed yellow line representing the 95\\% completeness in stellar \nmass. The $U-V$ versus $M_{\\rm star}$ diagram at $0.2 \\leq z<0.5$ is overplotted \nin all panels at $z>0.5$ as gray contours to highlight the evolution as a \nfunction of redshift of the overall galaxy population in the $U-V$ versus \n$M_{\\rm star}$ diagram. Orange and cyan small filled circles represent the \nquiescent and star-forming progenitors of UMGs. The open red circle and blue \nstar represent the median value of the quiescent and star-forming progenitors, \nrespectively. In the panel corresponding to the $2.5 \\le z < 3.0$ redshift \ninterval, the small filled red circles and blue stars represent the lower \nredshift median values of the quiescent and star-forming progenitors, \nrespectively, to highlight their evolution with redshift in the color-stellar \nmass diagram. The star-forming progenitors have never lived in the blue cloud \nsince $z=3$, and they are as red as the quiescent red-sequence galaxies at all \nredshifts.\\label{fig3}}\n\\end{figure*}\n\nIn contrast to the star-forming progenitors, only $\\sim$21\\% of the quiescent \nprogenitors are on average detected at 24$\\mu$m with $S\/N>5$, with this \nfraction ranging from $\\sim$36\\% at $z=0.35$ to $\\sim$12\\% at $z=2.75$. This \ndetection rate is quantitatively consistent with the MIPS detection rate found \nin a sample of quiescent galaxies at $0.3<z<2.5$ with \n$\\log{(M_{\\rm star})\/M_{\\odot}}>10.3$ selected from the 3D-HST survey \n\\citep{fumagalli13}. While the MIPS detection rate in the quiescent \nprogenitors is a factor of $\\sim$4 smaller than in the star-forming \nprogenitors, its not-zero value is potentially indicative of low levels of \nobscured star formation or AGN contamination. We see from Figure~\\ref{fig2} \nthat the MIPS detection in the quiescent progenitors have preferentially \nredder $V-J$ colors, lying closer to the line separating quiescent from \ndusty star-forming galaxies, suggesting the possibility of these galaxies \nbeing in their terminal phases of star formation and on their ways to become \nfully quenched systems. If these systems are indeed transitioning systems in \ntheir final approaches to the quiescent population, it is tempting to \ninterpret their MIPS detections as potentially due to obscured AGNs \nresponsible for the quenching of their star-formation activities, given the \nincreasing fraction of AGN activity with increasing stellar mass found both in \nthe local universe (e.g., \\citealt{kauffmann03b}) and out to $z\\sim2.5$ (e.g., \n\\citealt{papovich06}; \\citealp{kriek07}). We note however that the \ninvestigation of the origin of the MIPS fluxes in the sample of quiescent, \nlower-mass galaxies in \\citet{fumagalli13} appears to indicate that the low \nMIPS fluxes can be fully accounted by processes unrelated to obscured AGNs and \non-going star formation, such as cirrus dust heated by old stellar populations \nand circumstellar dust embedding asymptotic giant branch stars. Related to the \ndust content, we finally note that the comparison in the $UVJ$ diagram of the \nquiescent progenitors and the evolution tracks indicates an increasing amount \nof dust extinction with increasing redshift also in the quiescent progenitors \nof UMGs, especially at $z>1-1.5$ (see also \\S\\ref{sec-dust}). \n\n\\subsection{Evolution in the $U-V$ versus $M_{\\rm star}$ diagram}\\label{cmd}\n\nAs highlighted in the previous section, Figure~\\ref{fig2} shows that the \nprogenitors of UMGs at $z<1$ constitute a homogeneous population, with \nvery similar rest-frame $U-V$ and $V-J$ colors, typical of quiescent and old \nstellar populations. Two additional remarkable results are evident from \nFigure~\\ref{fig2}. First, the scatter in the rest-frame $U-V$ color of the \nquiescent progenitors becomes progressively larger with increasing redshift, \nespecially at $z>1.5$. Second, the star-forming progenitors show similar \nrest-frame $U-V$ colors compared to the quiescent progenitors, indicative that \nthe star-forming progenitors never lived on the blue cloud since $z=3$. Even \nmore remarkable is that a significant fraction of star-forming progenitors in \nthe highest redshift bin show rest-frame $U-V$ colors redder than the \nquiescent progenitors, in addition to significantly redder rest-frame $V-J$ \ncolors, suggesting that the star-forming progenitors at $z\\sim3$ of local UMGs \nmay be actually {\\it redder} than the quiescent progenitors. \n\nThese results can be better seen in Figure~\\ref{fig3}, which shows the \nrest-frame $U-V$ versus $M_{\\rm star}$ diagram (or color-mass diagram; CMD, \nhereafter) of the quiescent and star-forming progenitors of UMGs. \nFigure~\\ref{fig3} also shows the evolution in the CMD of the overall galaxy \npopulation (grayscale representation) from the UltraVISTA survey. \n\nAt $0.2 \\le z<0.5$, the progenitors of local UMGs are the reddest objects and \nhave similar rest-frame $U-V$ colors, with a very small intrinsic scatter \n($\\sim0.04$~mag). To measure the intrinsic scatter ($\\sigma_{\\rm int}$), we \ncorrected the observed scatter ($\\sigma_{\\rm obs}$) for the scatter introduced \nby photometric errors ($\\sigma_{\\rm phot}$) using the formula \n$(\\sigma_{\\rm obs})^{2}=(\\sigma_{\\rm int})^{2}+(\\sigma_{\\rm phot})^{2}$. Following \n\\citet{whitaker10}, for each progenitor, the observed SED fluxes were \nperturbed by a normally distributed, pseudo-random number from a Gaussian \ndistribution with a mean of zero and a standard deviation of the photometric \nerror for each respective filter. From these perturbed flux values, we \ngenerated 100 simulated catalogs, and re-determined $U-V$, refitting the \nphotometric redshifts. The uncertainty in $U-V$ color for each progenitor is \nestimated from the biweight sigma of the its $U-V$ distribution. At each \nredshift interval, we estimated the values of $\\sigma_{\\rm phot}$ for each \nsample of quiescent and star-forming progenitors by taking the average of the \nbiweight sigma for the progenitors in the considered sample. The average \nuncertainty in $U-V$ colors for the sample of quiescent progenitors ranges \nfrom 0.02~mag at $z=0.35$ to 0.06~mag at $z=2.75$, while for the sample of \nstar-forming progenitors it ranges from 0.03~mag at $z=1.25$ to 0.12~mag at \n$z=2.75$.\n\nThe derived scatter of the progenitors at $z=0.35$ is quantitative consistent \nwith the tightness of the red sequence in the local universe \n($\\sim$0.03$\\pm$0.02 mag; e.g., \\citealt{bower92}). While at \n$0.5 \\le z < 1.0$ the intrinsic scatter in the $U-V$ color is still quite small \n($\\sim0.07$~mag), consistent with the scatter of $\\sim$0.08$\\pm$0.03 mag from \nseveral studies done in galaxy clusters out to $z\\sim0.8$ \n(\\citealt{vandokkum00}; \\citealt{holden04}; \\citealt{mcintosh05}; \n\\citealt{mei09}; \\citealt{ruhland09}; \\citealt{vandokkum08}), the scatter in \nthe rest-frame $U-V$ color of the quiescent progenitors becomes progressively \nlarger with increasing redshift, i.e., $\\sim0.11$~mag at $1.0 \\le z < 1.5$ and \n$\\sim0.17$~mag at $z>1.5$. The scatter and its evolution is in good \nagreement with the evolution of the rest-frame $U-V$ scatter derived from the \nNEWFIRM Medium-Band Survey (NMBS) in a mass-complete sample of quiescent \ngalaxies at $0.2<z<2.2$ with $\\log{(M_{\\rm star}\/M_{\\odot})}>11$ \n\\citep{whitaker10}. Our work shows, for the first time, the increasing scatter \nwith redshift of the rest-frame $U-V$ color for a sample of quiescent \nprogenitors of local UMGs rather than for samples selected above a fixed \nstellar mass at all redshifts and which are likely not evolutionarily \nconnected. \n\nFigure~\\ref{fig3} clearly demonstrates that the star-forming progenitors have \nnever lived in the blue star-forming cloud in the last 11.4~Gyr, i.e., since \n$z=3$. Indeed, the star-forming progenitors appear to be as red as the \nquiescent galaxies at all redshifts. In the highest redshift bin, i.e., at \n$z=2.75$, quiescent and star-forming progenitors have similar median \n$U-V$ colors, although there is a significant fraction of star-forming \nprogenitors with redder colors compared to most of the quiescent progenitors. \nThe intrinsic scatter in $U-V$ color of the star-forming progenitors is much \nlarger than the quiescent progenitors, i.e, $\\sim0.28$~mag at $z=2.75$ and \n$\\sim0.22$~mag at $z<2.5$. As shown in the bottom-right panel of \nFigure~\\ref{fig3}, the median $U-V$ color of the star-forming progenitors \ndoes not seem to evolve much in the $\\sim1.3$~Gyr from $z=2.75$ to $z=1.75$, \nwhereas the median $U-V$ color of the population of quiescent progenitors \nbecomes redder as it matures from a young, post-starburst to a more evolved \npopulation (see also \\S\\ref{ages} and \\ref{seds}). \n\nThe large scatter in $U-V$ of the quiescent progenitors at $z>1.5$ is evidence \nthat the massive end of the local red sequence is in the process of assembly \nbetween $z=3$ and $z=1.5$. At $z=2.75$, $\\sim$15\\% of the progenitors of UMGs \nhave already quenched, most of which only recently (from Fig.~\\ref{fig2}; see \nalso \\S\\ref{ages}, \\ref{ssfr}, and \\ref{seds}). The majority of the \nprogenitors at $z=2.75$ are star bursting. Then, the quiescent progenitors age \nwith cosmic time and redden, while most of the star-forming progenitors quench \nin the $\\sim2.6$~Gyr from $z=2.75$ to $z=1.25$. The few remaining star-forming \nprogenitors at $z=1.25$ complete their migration to the quiescent population \nbefore $z=1$, i.e., in less than 1~Gyr. Therefore, the massive end of the \nquiescent red sequence is fully formed by $z=1$, and it ages thereafter in the \nfollowing $\\sim7.7$~Gyr from $z=1$ to $z\\sim0$.\n\n\\subsection{Evolution in Stellar Population Properties}\\label{sec-prop}\n\nWe now focus on the evolution with redshift of the stellar population \nproperties of the progenitors of UMGs as derived from the modeling of the \nUV-to-8$\\mu$m SEDs. \n\n\\subsubsection{Stellar ages}\\label{ages}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig4.eps}\n\\caption{Evolution with redshift of the stellar age \n$\\langle t \\rangle_{\\rm SFR}$ derived from the modeling of the UV-to-8$\\mu$m \nSEDs of the progenitors of local UMGs selected with the semi-empirical \napproach using abundance matching (black filled circles). Red filled circles \nand blue empty stars represent the quiescent and star-forming galaxies among \nthe progenitors' sample, respectively. The error bars show the upper and lower \nranges including 68\\% of the populations, while the gray, orange, and light \nblue filled regions represent the mean 1$\\sigma$ errors in the estimated \nstellar ages for the whole progenitors' sample, the quiescent, and \nstar-forming progenitors, respectively. The continuous curve represents the \nage of the universe at the corresponding redshift, while the dotted curve is \nthe continuous curve shifted down by 1.8~Gyr, and going through the \nmeasurements at $z\\sim0$. The red and blue symbols are slightly offset along \nthe x-axis (in opposite directions) for clarity. Also plotted are the average \nage of quiescent galaxies at $1.4<z<2.2$ with \n$\\log{(M_{\\rm star}\/M_{\\odot})}>10.5$ selected from the 3D-HST survey (hatched \ngray box; \\citealt{whitaker13}); the ages of early-type galaxies at $z\\sim 0$ \nwith $\\log{(M_{\\rm star}\/M_{\\odot})}\\approx11.8$ in high- and low-density \nenvironments (orange and green filled upward triangles; \\citealt{thomas05}); \nthe age of early-type galaxies at $0.05<z<0.06$ with \n$\\log{(M_{\\rm dyn}\/M_{\\odot})}\\approx11.8$ from the SDSS (purple filled downward \ntriangle; \\citealt{thomas10}); the typical age of all galaxies at $z\\sim 0.13$ \nwith $\\log{(M_{\\rm star}\/M_{\\odot})}\\approx11.8$ from the SDSS (black filled \nsquare; \\citealt{gallazzi05}); and the typical age of early-type galaxies at \n$z\\sim 0.13$ with $\\log{(M_{\\rm star}\/M_{\\odot})}\\approx11.8$ from the SDSS \n(filled red square; \\citealt{gallazzi06}). Whereas the median age of the \nstar-forming progenitors remains roughly constant from $z=3$ to $z=1$ with \nages$\\sim$0.8-1.6~Gyr (although with large uncertainties), the median stellar \nage of the quiescent progenitors increases steadily from \n$\\langle t \\rangle_{\\rm SFR}\\sim0.7$~Gyr at $z=2.75$ to \n$\\langle t \\rangle_{\\rm SFR}\\sim7$~Gyr at $z=0.35$, reaching an age of \n$\\sim$9-10~Gyr by $z\\sim0$.\\label{fig4}}\n\\end{figure}\n\nFigure~\\ref{fig4} shows the evolution with redshift of the stellar age derived \nfrom SED modeling of the progenitors of UMGs. As stellar age of the galaxy, we \nadopt the SFR-weighted mean age of the stellar population, \n$\\langle t \\rangle_{\\rm SFR}$, as defined in \\citet{natascha04}. When modeling \nSEDs, the best-fit age is strictly speaking the time elapsed since the onset \nof star formation. However, in exponentially declining SFHs, galaxies are \ncontinually forming stars, and therefore $\\langle t \\rangle_{\\rm SFR}$ \ncorresponds more closely to the age of the stars contributing the bulk of the \nstellar mass (and dominating the light of the galaxy). \n\nThe typical age of the star-forming progenitors is roughly constant at \n$\\langle t \\rangle_{\\rm SFR} \\approx$0.8-1~Gyr from $z=3$ to $z=1.5$, although \nwith a large range of ages spanned at each redshift, and it increases to \n$\\sim$1.6~Gyr by $z=1.25$. The 68\\% of the population of star-forming \nprogenitors at $z>1.5$ is bracketed by a lower limit in age of \n$\\sim$0.2-0.3~Gyr and an upper limit in age of $\\sim$1.6-1.8~Gyr. On the \ncontrary, the stellar age of the quiescent progenitors increases steadily from \n$z=2.75$ (with a median age of $\\sim$0.7 Gyr) to $z=0.35$ (with a median age of \n$\\sim$7~Gyr). The quiescent progenitors also show a smaller spread in stellar \nage at each redshift compared to the star-forming progenitors. Specifically, \nat $z>1$, the 68\\% of the quiescent population is bracketed by a lower limit \nin age $\\sim$20\\% smaller than the median value (compared to 60\\%-80\\% for \nthe star-forming progenitors). We note that the stellar ages derived for the \nstar-forming progenitors at $z>2$ are significantly more uncertain compared to \nthe estimated ages of the quiescent progenitors. As indicated by the orange and \nlight blue filled regions in Figure~\\ref{fig4}, the formal error on the \nstellar ages of the star-forming progenitors is 0.7-1.1~Gyr at $z>2$, a factor \nof $\\sim2$-2.5 larger than for quiescent progenitors at the same redshifts. \n\nFigure~\\ref{fig4} clearly shows the aging of the quiescent progenitors of \nlocal UMGs with decreasing redshift. This result is also evident from the \n$UVJ$ diagram shown in Figure~\\ref{fig2}. Specifically, at $2.5<z<3.0$, the \nquiescent progenitors mostly populate the lower-left corner of the quiescent \nbox. From the comparison with the $\\tau_{\\rm 100}$ color evolution track (dark \ngray curve in Fig.~\\ref{fig2}), this region in the quiescent box of the $UVJ$ \ndiagram corresponds to ages of 0.8-1.2~Gyr, and even younger ages for \nnon-negligible amounts of dust extinction (see \\S\\ref{sec-dust}), in good \nagreement with the ages derived from SED modeling. With decreasing redshift, \nwe see from Figure~\\ref{fig2} that the population of quiescent progenitors of \nUMGs progressively migrates toward the upper-right corner of the quiescent \nbox, clearly showing the aging of the stellar population of the quiescent \nprogenitors as a function of cosmic time. \n\nWhereas the above results are photometrically derived, they agree well with \nrecent results showing the spread in spectroscopically-derived ages of \nquiescent galaxies at $1.4<z<2.2$ from the 3D-HST dataset \\citep{whitaker13}. \nSpecifically, in \\citet{whitaker13}, massive \n($\\log{(M_{\\rm star}\/M_{\\odot})}>10.5$), quiescent galaxies at $1.4<z<2.2$ were \nselected using the $UVJ$ diagram and the 3D-HST grism spectra were used to \nunambiguously identify metal absorption lines in the stacked spectra, \nincluding the G band ($\\lambda$4304\\AA), MgI ($\\lambda$5175\\AA), and NaI \n($\\lambda$5894\\AA), and to spectroscopically derive an average age of \n$1.3^{+0.1}_{-0.3}$~Gyr, in good agreement with our photometrically-derived \nages in the overlapping redshift interval (see hatched gray box in \nFigure~\\ref{fig4}). \\citet{whitaker13} also showed that the reddest 80\\% of \nthe quiescent sample is dominated by metal lines and have older mean stellar \nages ($1.6^{+0.5}_{-0.4}$ Gyr), whereas the bluest galaxies have strong Balmer \nlines and a spectroscopic age of $0.9^{+0.2}_{-0.1}$~Gyr. This range in stellar \nages for the population of massive quiescent galaxies at $1.4<z<2.2$ is in \ngood agreement with the spread in ages we derived photometrically for the \nsample of quiescent progenitors of UMGs. \n\nFigure~\\ref{fig4} also shows the spectroscopically measured light-weighted ages \nof $z\\sim0$ UMGs. \nSpecifically, the local measurements plotted in Figure~\\ref{fig4} correspond \nto the typical ages of early-type galaxies at $z\\sim0$ with \n$\\log{(M_{\\rm star}\/M_{\\odot})}\\approx 11.8$ in high- and low-density \nenvironments \\citep{thomas05}; the age of early-type galaxies at \n$0.05 < z < 0.06$ with $\\log{(M_{\\rm star}\/M_{\\odot})}\\approx 11.8$ from the \nSloan Digital Sky Survey (SDSS; \\citealt{thomas10}); the typical age of all \ngalaxies at $z\\sim0.13$ with $\\log{(M_{\\rm star}\/M_{\\odot})}\\approx 11.8$ from \nthe SDSS \\citep{gallazzi05}; and the typical age of early-type galaxies at \n$z\\sim0.13$ with $\\log{(M_{\\rm star}\/M_{\\odot})}\\approx 11.8$ from the SDSS \n\\citep{gallazzi06}. \n\nOur study provides the first direct proof in the early universe of the results \nand implications of the archeological studies of UMGs in the local universe. \nSpecifically, from the study of the local stellar population scaling relations \npresented in \\citet{gallazzi06}, $z\\sim0$ UMGs appear to have a typical \nlight-weighted ages of $\\sim$10~Gyr, implying a median formation redshift of \n$z \\approx 1.9$. The inferred scatter in age at the very high-mass end \n($\\sim0.078$-0.088~dex, or $\\approx$1.8-2~Gyr; \\citealt{gallazzi06}) implies a \nformation redshift for UMGs as low as $z\\sim$1.1, and as high as \n$z\\sim$3.9-4.2. The formation redshift and spread in age of local UMGs agree \nremarkably well with our findings and our estimated ages for the progenitors \nof UMGs. Indeed, at $2<z<3$, most of the progenitors of $z\\sim0$ UMGs are \nstar-forming and the massive end of the local quiescent red sequence is being \nvigorously assembled, in agreement with a median formation redshift of UMGs of \n$z\\sim2$ from archeology studies. As quiescent progenitors are already present \nat $z=2.75$, their formation must have started at $z>3$ for at least some of \nthem, in good agreement with archeological studies in the local universe. \nThis is also quantitatively consistent with recent results showing the \nexistence of very massive galaxies at $3<z<4$, and the dominance of \nstar-bursting galaxies at the massive end (\\citealt{marchesini10}; \n\\citealt{muzzin13b}). By $1.0<z<1.5$, most star-forming progenitors have \nquenched, in remarkably good agreement with the low formation redshift of \n$z\\sim$1.1 from the archeological studies of local UMGs. Moreover, assuming \nthat the progenitors of local UMGs start quenching at $2.5<z<3.0$ and that \ntheir quenching is completed at $1.0<z<1.5$ (since no star-forming progenitors \nare seen at $z<1$), and assuming passive evolution since $z=1$, then the mean \nage difference between the quiescent progenitors of local UMGs should be \n$\\sim2.6$~Gyr, i.e., the age difference of the universe between $z=2.75$ and \n$z=1.25$. This is in good agreement with the age dispersion of local UMGs as \nderived from their fossil records (\\citealt{gallazzi05}; \\citealt{thomas05}; \n\\citealt{gallazzi06}; \\citealt{thomas10}).\n\n\\subsubsection{Star-formation activity}\\label{ssfr}\n\nIn this section we present a more quantitative analysis of the evolution with \nredshift of the SFRs and the specific star-formation rates \n(sSFR=SFR\/$M_{\\rm star}$) of the progenitors of UMGs. \n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig5.eps}\n\\caption{Evolution with redshift of the sSFR (top panels) and the SFR (bottom \npanels) derived from SED modeling (left panels) and from the combination of the \nUV and IR luminosities (right panels) for the progenitors of UMGs selected \nwith the semi-empirical approach using abundance matching. Symbols as in \nFig.~\\ref{fig4}. The continuous line in the top panels represents the value \nof sSFR$=10^{-11}$~yr$^{-1}$ typically used in the literature to separate \nstar-forming from quiescent galaxies.\\label{fig5}}\n\\end{figure}\n\nFigure~\\ref{fig5} shows the evolution with redshift of the SFRs and sSFR \nderived from SED modeling (left-hand panels) and from the combination of the \nUV and IR luminosities (right-hand panels) for the progenitors of UMGs.\nAt $z \\lesssim 1.5$, the progenitors of UMGs are predominantly quiescent, with \nsSFR$<10^{\\rm -11}$~yr, whereas at $z \\gtrsim 1.5$ they become increasingly \ndominated by star-forming galaxies. The median SFRs of the star-forming \nprogenitors from the combination of the UV and IR luminosities are \n$\\sim$170-330~M$_{\\odot}$ yr$^{-1}$ at $1.5<z<3$, i.e., the star-forming \nprogenitors are intensely star-bursting galaxies, with \nsSFR$\\approx$0.8-2~Gyr$^{-1}$, hence doubling their stellar mass in \n$\\sim$0.5-1.2~Gyr if these SFRs were to be sustained over this period of time. \nThe evolutions with redshift of the sSFR and SFR derived from SED modeling are \nqualitatively similar to the evolutions derived from the combination of the UV \nand IR luminosities, although they differ quantitatively. Specifically, SFRs \nfrom the combination of the UV and IR luminosities are typically larger than \nthe SFRs from SED modeling, resulting in larger sSFRs. This is especially true \nfor the star-forming progenitors, most of which are robustly detected in the \nMIPS 24$\\mu$m, with SFRs from the combination of the UV and IR luminosities an \norder of magnitude larger than the SFRs from SED modeling. \n\n\\begin{figure*}\n\\epsscale{1.}\n\\plotone{fig6.eps}\n\\caption{Evolution with redshift of the dust extinction (left), the \nrest-frame $V$-band absolute magnitude (middle), and the mass-to-light ratio \nin the rest-frame $V$-band, $M_{\\rm star}\/L_{\\rm V}$, (right) for the \nprogenitors of UMGs selected with the semi-empirical approach using abundance \nmatching. Symbols as in Fig.~\\ref{fig4}. Dust extinction of the \nprogenitors increases with redshift, driven by the growing importance of dusty \nstar-bursting galaxies among the progenitors at $z>1.5$. A mild increase of \nthe dust extinction with redshift in the quiescent progenitors is also \ndetected. The $M_{\\rm star}\/L_{\\rm V}$ of the star-forming progenitors remains \nroughly constant with redshift, but larger than the $M_{\\rm star}\/L_{\\rm V}$ of \nthe quiescent progenitors at $z>1.5$ due to larger dust obscuration. The \n$M_{\\rm star}\/L_{\\rm V}$ of the quiescent progenitors increases steadily with \ncosmic time due to aging of the overall stellar population.\\label{fig6}}\n\\end{figure*}\n\nIt is worth noting that the progenitors selected as quiescent using the \n$UVJ$ diagram become increasingly more actively star forming at high redshift. \nThe reason for this is twofold. On one hand, the quiescent galaxies at $z>2$ \nmay indeed have non-negligible levels of star-formation activity. As the early \nuniverse is investigated, the quiescent population will be preferentially \ndominated by post-starburst, relatively younger galaxies whose star-formation \nactivity is still being quenched or has only recently been quenched. On the \nother hand, contamination of the $UVJ$ quiescent population due to \nmisclassified dusty star-bursting galaxies may increase with redshift and be \nsignificant at $z>2$. From the $UVJ$ diagram, there is a non-negligible \nfraction of galaxies at $z>2$ straddling the quiescent and star-forming \nregions. Interestingly, most of these galaxies have MIPS 24$\\mu$m detections at \n$>5$~$\\sigma$, indicative of dusty star-bursting galaxies. The SFRs of \n$\\sim$98\\% of the quiescent progenitors with robust MIPS detections overlap \nwith the low-SFR tail of the distribution of SFRs of the star-forming \nprogenitors with robust MIPS detections, suggesting that most of the quiescent \nprogenitors with 24$\\mu$m detection may be either transitioning objects from \nthe star-bursting to the quiescent population, or objects in a phase of \nrejuvenated star formation characterized by relatively low levels of \nstar-formation activity.\n\nIt should also be kept in mind that the observed MIPS flux could be \npotentially contaminated by MIR emission related to obscured AGNs, whose \nfraction has been shown to increase with stellar mass, especially at higher \nredshifts (e.g., \\citealt{kriek07}). While the relative contributions of star \nformation and AGN to the 24$\\mu$m flux are difficult to assess here, AGN \ncontamination can be partly responsible for the discrepancy between the SFRs \nderived from SED modeling and UV+IR, and it may contribute in making some of \nthe $UVJ$ quiescent progenitors appear star-forming in Figure~\\ref{fig5}. \nWe finally note that low MIPS fluxes can also be fully accounted by \nprocesses unrelated to obscured AGNs and on-going star formation, such as \ncirrus dust heated by old stellar populations and circumstellar dust embedding \nasymptotic giant branch stars \\citep{fumagalli13}.\n\nWe therefore caution against taking the values of the SFRs and sSFRs of the \nquiescent galaxies at $z>1.5$ at face value, and argue that these should be \ninterpreted as upper limits due to the potential contamination from dusty \nstar-bursting galaxies and\/or obscured AGNs. Even with these caveats, the \nresult that the progenitors of UMGs are increasingly dominated by dusty \nstar-bursting galaxies at higher redshifts (i.e., $z>1.5-2$) is robust. \n\n\\subsubsection{Dust extinction}\\label{sec-dust}\n\nThe left panel of Figure~\\ref{fig6} shows the evolution with redshift of the \ndust extinction ($A_{\\rm V}$) derived from the modeling of the UV-to-8$\\mu$m \nSEDs. The progenitors of UMGs are progressively dustier with increasing \nredshift, with the median dust extinction increasing from \n$A_{\\rm V}\\sim0.3$~mag at $z=0.35$ to $A_{\\rm V}\\sim1.3$~mag at $z=2.75$. The \nobserved growth is driven by the growing importance with redshift of the \npopulation of dusty starburst galaxies, having a median extinction of \n$A_{\\rm V}\\sim1.6$~mag and a 68\\% range of $A_{\\rm V}\\sim1-2.2$~mag at $z=2.75$. \nOn the contrary the quiescent population is characterized by dust extinction \n$A_{\\rm V} < 1$~mag over the full targeted redshift interval $0.2<z<3.0$.\n\nWe note that the quiescent progenitors also show a mild increase with \nredshift of the dust extinction. Evidence for an increase with redshift of the \ndust extinction in both the overall and quiescent populations of massive \ngalaxies were previously presented by \\citet{whitaker10} and \n\\citet{marchesini10}. \\citet{whitaker10} found a median extinction of \n$A_{\\rm V}=0.2-0.3$~mag in massive ($M_{\\rm star}>10^{11}$~M$_{\\odot}$) quiescent \ngalaxies at $z\\sim1-2$, in qualitative agreement with the range of $A_{\\rm V}$ \nfound in our sample of quiescent progenitors in the overlapping redshift \nrange. \\citet{marchesini10} studied a complete sample of galaxies at $3<z<4$ \nwith $M_{\\rm star}>4\\times10^{11}$~M$_{\\odot}$, finding a median dust extinction \nof $A_{\\rm V}=1.4$~mag, in good agreement with the median extinction in \nour sample of progenitors at $2.5<z<3.0$, and evidence of increasing amount of \ndust in the population of very massive galaxies in the early universe.\n\n\\subsubsection{Rest-frame absolute $V$-band magnitude}\n\nThe central panel of Figure~\\ref{fig6} shows the evolution with redshift of the \nrest-frame $V$-band absolute magnitude ($M_{\\rm V}$) derived by integrating the \nredshifted rest-frame filter bandpass from the best-fit EAZY template, as \ndescribed in \\citet{brammer11}. This method produces similar rest-frame \nluminosities as other methods interpolating between the observed bands that \nbracket the rest-frame band at a given redshift (e.g., \\citealt{rudnick03}).\n\nThe progenitors of UMGs have median rest-frame $V$-band absolute magnitude in \nthe range from $M_{\\rm V}=-23.0$ to $M_{\\rm V}=-23.5$. When the progenitors are \nseparated in quiescent and star-forming galaxies, a clear trend with redshift \nbecomes evident. The star-forming progenitors span a large range in \n$M_{\\rm V}$ at each redshift (0.7-1.4 mag, depending on the redshift interval). \nThe median rest-frame $V$-band absolute magnitude of the star-forming \nprogenitors increases with cosmic time from $M_{\\rm V}=-22.9$ at $2<z<3$ to \n$M_{\\rm V}=-23.3$ at $1.0<z<1.5$, which correspond to \n$1.3\\times L^{\\star}_{\\rm V}$ and 2.5$\\times L^{\\star}_{\\rm V}$, respectively \n(with $L^{\\star}_{\\rm V}$ the characteristic luminosity of the rest-frame \n$V$-band luminosity function of galaxies at the corresponding redshift; \n\\citealt{marchesini12}). On the contrary, the rest-frame $V$-band luminosity of \nthe quiescent progenitors decreases with cosmic time from $M_{\\rm V}=-24.0$ \n(corresponding to $\\sim 3.3 \\times L^{\\star}_{\\rm V}$) at $2.5<z<3.0$ to \n$M_{\\rm V}=-23.1$ (corresponding to $\\sim 4 \\times L^{\\star}_{\\rm V}$) at \n$0.2<z<0.5$. \n\n\\begin{figure*}\n\\epsscale{1.}\n\\plotone{fig7.eps}\n\\caption{Evolution with redshift of the median rest-frame SEDs of the \nprogenitors of UMGs selected with the semi-empirical approach using abundance \nmatching. The left panel shows all progenitors at different redshifts; the \nmiddle and right panels show the evolution of the rest-frame SEDs of the \nquiescent and star-forming progenitors, respectively. Filled colored circled \nrepresent the median rest-frame SEDs derived from the observed photometry, \nwhereas the colored curves represent the median best-fits from SED modeling. \nThe SEDs derived from these two methods are in good agreement. The \nprogenitors of UMGs have rest-frame SEDs typical of dusty star-bursting \ngalaxies at high redshift (i.e., $z\\gtrsim$2), whereas their rest-frame SEDs \nare dominated at low redshifts by the SEDs typical of quiescent galaxies with \nlittle or no dust extinction, and old ages. The quiescent progenitors at high \nredshift have rest-frame SEDs typical of post-starburst galaxies, \ncharacterized by strong Balmer breaks and ``peaky'' rest-frame optical SEDs. \n\\label{fig7}}\n\\end{figure*}\n\n\\subsubsection{Mass-to-light ratio}\n\nThe right panel of Figure~\\ref{fig6} shows the evolution with redshift of the \nmass-to-light ratio $M_{\\rm star}\/L_{\\rm V}$. The $M_{\\rm star}\/L_{\\rm V}$ of the \nprogenitors of UMGs is roughly constant at $z>1$ \n($M_{\\rm star}\/L_{\\rm V} \\sim 1.1$~M$_{\\odot}$~L$^{-1}_{\\odot}$) and increases with \ncosmic time to $M_{\\rm star}\/L_{\\rm V} \\sim 3.3$~M$_{\\odot}$~L$^{-1}_{\\odot}$ at \n$z=0.35$. The star-forming and quiescent progenitors show once again different \ndependency of their mass-to-light ratios as a function of redshift. \nThe star-forming progenitors are characterized by a constant mass-to-light \nratio, $M_{\\rm star}\/L_{\\rm V} \\sim 1.7$~M$_{\\odot}$~L$^{-1}_{\\odot}$, over the \nentire $1<z<3$ redshift range. On the contrary, the mass-to-light ratio of the \nquiescent progenitors increases steadily with cosmic time, from \n$M_{\\rm star}\/L_{\\rm V} \\sim 0.7$~M$_{\\odot}$~L$^{-1}_{\\odot}$ at $z=2.75$ to \n$M_{\\rm star}\/L_{\\rm V} \\sim 3.3$~M$_{\\odot}$~L$^{-1}_{\\odot}$ at $z=0.35$. \nInterestingly, at $z>1$, the mass-to-light ratio of the star-forming \nprogenitors is larger than the mass-to-light ratio of the quiescent progenitors \nby a factor of $\\sim 1.5-2.3$ due to the significantly larger amount of dust \nextinction in the star-forming population. At $z<1$, the mass-to-light ratio \nof the quiescent progenitors increases due to the aging (and the fainting) of \nthe overall stellar population.\n\n\\subsection{Evolution of the SEDs}\\label{seds}\n\nFigure~\\ref{fig7} shows the evolution of the rest-frame SEDs of the \nprogenitors of UMGs from $z=3$ to $z=0.2$, along with the evolution of the \nrest-frame SEDs of the quiescent and star-forming progenitors (middle and \nright panels, respectively). The SEDs have been normalized using the flux in \nthe rest-frame $J$-band (i.e., $\\lambda_{\\rm rest}=1.24$~$\\mu$m). This figure \nsummarizes the results for the evolution of the stellar population properties \nfound in previous \\S\\ref{sec-prop}. Whereas little evolution is seen in the \nstar-forming progenitors (which dominate at $z>2$), consistently with the \nderived evolution of their stellar population properties, significant \nevolution is observed for the quiescent progenitors, which dominate the \noverall progenitors' population at $z<2$. Specifically, the SEDs of the \nquiescent progenitors evolve from an SED typical of a young post-starburst \ngalaxy with moderate dust extinction at $z=2.75$ into an SED typical of a \nquiescent, maximally old stellar population with little or no dust extinction \nat $z=0.35$. As Figure~\\ref{fig7} shows, the population of progenitors of UMGs \nwas dominated by dusty star-bursting galaxies in the early universe. Such \npopulation has mostly quenched by $z\\sim1.5$, with all progenitors having \nbecome quiescent by $z=1$, and has aged thereafter with cosmic time, evolving \ninto a homogeneous population of quiescent, maximally old progenitors by \n$z=0.35$. \n\n\\subsection{Possible systematics effects}\\label{caveats}\n\nIn this section we address three possible systematic effects which can \npotentially bias the results presented in the previous sections, namely \nsystematic errors in photometric redshift estimates, contamination from \nemission lines, and systematic uncertainties in the modeling of the observed \nSEDs. \n\n\\subsubsection{Systematic errors in photometric redshifts}\n\nPrevious searches for old and massive galaxies at $z>4$ highlighted the \ndifficulty in unambiguously identifying old and massive objects at extreme \nredshifts on the basis of spectral fitting, with equally acceptable solutions \nbetween high-redshift very massive galaxies with low-extinction and massive \n($10^{11}$~M$_{\\odot}$) galaxies with high extinction ($A_{\\rm V}\\sim4$~mag) at \nintermediate redshifts ($z\\sim2$) (e.g., \\citealt{dunlop07}). Moreover, using \nNIR medium band data from the NMBS, \\citet{marchesini10} found that the \ninclusion of an ``old and dusty'' template in the photometric redshift \nestimate caused approximately half of the massive galaxies population at $z>3$ \nto be consistent with a somewhat lower photometric redshift in the range \n$2<z<3$. Whereas, at present, there is no strong evidence from spectroscopic \nstudies of the existence of such an old and dusty population, \n\\citet{muzzin13b} showed that the inclusion of such ``old and dusty'' template \nhas significant effects on the high-mass end of the stellar mass function of \nstar-forming galaxies at $z>1.5$, especially at $3<z<4$. \n\nGiven that at $z>2$ our sample of progenitors of UMGs are dominated by a \npopulation of massive galaxies with very red $U-V$ and $V-J$ colors and \nsignificant amount of dust extinction (see Fig.~\\ref{fig8} for four \nrepresentative examples of SEDs), it is worth examining the potential \nsystematic effect of including this ``old and dusty'' template on the inferred \nevolution of the progenitors of local UMGs. We have repeated the entire \nanalysis after inclusion of the same ``old and dusty'' template adopted in \n\\citet{muzzin13b}. Specifically, we used the corresponding stellar mass \nfunctions from \\citet{muzzin13b} to select progenitors and we studied the \nevolution of their properties. \n\nConsistently with what found by \\citet{marchesini10} and \\citet{muzzin13b}, we \nfind that quiescent galaxies are not affected by the inclusion of this new \ntemplate, whereas the star-forming galaxies with extreme colors tend to \nsystematically shift to lower redshifts. Whereas individual star-forming \ngalaxies at $z>2$ may change their redshifts substantially, the results agree \nwell qualitatively and quantitatively at $z<2.5$, with relevant differences \nonly in the highest targeted redshift bin. Specifically, the progenitors at \n$2.5<z<3.0$ tend to be slightly less massive (by $\\sim$0.15 dex), less dust \nextincted ($\\sim$0.3~mag), and with a slightly bluer median $U-V$ color (by \n$\\sim$0.2 mag, although with the same spanned range in $U-V$ color) when the \n``old and dusty'' template is included. Worth noting is also a significant \nincrease in the spanned ranges of rest-frame $U-V$ and $V-J$ colors toward \nredder values, especially at $2<z<2.5$, when the new template is included. \n\nDespite these differences, the overall evolutionary picture of the progenitors \nof local UMGs does not change at all in, e.g., the $U-V$ versus $M_{\\rm star}$ \ndiagram, hence our main results and conclusions are robust against the \ninclusion of the ``old and dusty'' template, whose relevance has yet to be \nconfirmed and assessed spectroscopically.\n\n\\subsubsection{Emission line contamination}\n\nContamination of the photometry from emission lines is another source of \nsystematic effects, potentially resulting in significant overestimated stellar \nmasses. Several recent spectroscopic investigations have shown serious \nsystematic effects due to emission lines on the derived stellar masses of \nrest-frame UV selected, star-forming galaxies at $z>4$ (e.g., \n\\citealt{bowler12}; \\citealt{ellis13}; \\citealt{labbe13}; \\citealt{stark13}). \n\nWe have used the best-fit EAZY template to quantify the impact of emission \nlines on the estimated stellar masses. Specifically, the EAZY templates used \nto derive photometric redshifts include emission lines, and we have measured \nemission lines equivalent widths on the best-fit EAZY templates of the \nprogenitors' sample. Then, we used the measured equivalent widths to correct \nthe observed SEDs, and we re-estimated stellar masses using the emission lines \ncorrected SEDs. This modeling leads to conclude that the majority of the \nprogenitors' sample at $0.2<z<3.0$ is not affected significantly by emission \nline contamination, with a mean difference of 0.03$\\pm$0.05~dex in the stellar \nmass estimated before and after emission line correction. We estimate that \n$\\sim$5\\% of the sample appears to be affected significantly by emission line \ncontamination, with stellar masses overestimated on average by a factor of \n$\\sim$2.5. Since these galaxies represent less than 10\\% of the sample of \nstar-forming progenitors at $z>1.5$, we conclude that, based on the emission \nline ratios and simplistic modeling adopted in EAZY, our results appears to be \nrelatively robust against emission line contamination. \n\nHowever, we caution that unusually high line ratios, especially \n[OIII]\/H$\\beta$, are being consistently found in star-forming galaxies at \n$z>3$, significantly biasing estimated stellar masses \\citep{holden14}. While \nthese studies have been currently targeting only rest-frame UV selected \ngalaxies with smaller stellar masses ($\\log{(M_{\\rm star}\/M_{\\odot})}<10.7$) \ncompared to our progenitors' sample, we stress that a robust quantitative \nassessment of emission line contamination in the $z>2$ galaxy population at \nthe high-mass end will necessarily require spectroscopic studies for large \nsamples of massive star-forming galaxies, currently missing.\n\n\\subsubsection{SED modeling}\\label{s-sedmodel}\n\nThe derived stellar population properties depend on the adopted choice of \nSED-modeling assumptions. We have investigated the effects of different \nSED-modeling assumptions by adopting different SFHs, extinction curves, and \nmetallicities. Specifically, in place of the exponentially-declining SFH we \nhave used a delayed-$\\tau$ SFH of the form $SFR \\propto t~exp(-t\/\\tau)$, which \nallows for increasing SFR at earlier times. In place of the \\citet{calzetti00} \nextinction curve, we have adopted both the Milky Way extinction law \n\\citep{cardelli89} and the extinction curve proposed by \\citet{kriek13}, which \nassumes a dust attenuation curve with $R_{\\rm V}=4.05$ and a UV dust bump \nwhich is 20\\% of the strength of the Milky Way bump, following the \nprescription by \\citet{noll09}. Finally, we have relaxed the assumption on \nmetallicity by leaving it as a free parameter in the SED modeling. \n\nThe systematic effects on the stellar population properties of the progenitor \ngalaxies are found to be significantly smaller than the corresponding typical \nrandom uncertainties for most of the different SED-modeling assumptions. For \nthe stellar mass, the systematic effect is found to be relevant only when \nadopting the Milky Way extinction curve, although still comparable to the \ntypical random uncertainty, with $M_{\\rm star}$ smaller by 0.05(0.09) dex for \nthe quiescent (star-forming) progenitors. The adoption of the Milky Way \nextinction curve also results in the largest systematic effects for the dust \nextinction, although significantly smaller compared to its typical random \nuncertainty, with $A_{\\rm V}$ smaller by 0.05(0.2)~mag for the quiescent \n(star-forming) progenitors compared to the case when the \\citet{calzetti00} \nlaw is adopted. The largest effect on the star-formation rate for the \nquiescent progenitors is found when the delayed-$\\tau$ model is adopted, with \nSFR smaller by 0.1~dex, whereas the largest effect on the star-formation rate \nfor the star-forming progenitors is found when the Milky Way extinction curve \nis adopted, with SFR smaller by 0.13~dex. Finally, the largest effects on the \nstellar ages of the star-forming progenitors are found when the delayed-$\\tau$ \nmodel is adopted, with $\\langle t \\rangle_{\\rm SFR}$ larger by $\\sim$0.1~dex \ncompared to the default SED-modeling assumptions. The stellar ages of the \nquiescent progenitors changes at most by 0.02~dex depending on the different \nSED-modeling assumptions. We therefore conclude that the inferred evolution \nwith redshift of the stellar population properties of the quiescent and \nstar-forming progenitors of local UMGs are robust and not very sensitive to \nreasonable choices of the SED-modeling assumptions.\n\nHowever, systematic effects in the modeling of the observed SEDs, and hence in \nthe estimated stellar masses, can be relevant in galaxies characterized by \ncomposite stellar populations. Specifically, the stellar masses of the dusty, \nmassive star-forming progenitors at $z>1.5$ could be potentially overestimated \nwhen modeled with a single stellar population, especially if characterized by \nmultiple stellar population components with differential dust extinctions, \nsuch as a quiescent un-obscured population and a young, dusty star-bursting \ncomponent. \n\nTo investigate potential biases in the stellar masses of the star-forming \nprogenitors at $2<z<3$, we have derived stellar masses by allowing for \nmultiple stellar components to contribute to the observed SEDs of the dusty \nstar-forming progenitors. No systematic differences are found, with a median \ndifference smaller than 0.1~dex and a scatter of $\\sim$0.5~dex (Brammer et al. \n2014, in preparation). We also note that the observed SEDs of most of the \ndusty star-forming progenitors of UMGs at $z>1.5$ cannot be well modeled by an \nad hoc combination of a quiescent un-obscured population and a young, dusty \nstar-bursting component, which would arguably cause the largest biases in the \nstellar mass estimates (Brammer et al. 2014, in preparation). \n\nWe therefore conclude that, while the stellar masses of individual \nstar-forming progenitors may be systematically different by a factor of a \nfew-to-several due to simplified SED-modeling assumptions, the stellar mass \ndistribution of the overall population appears to be largely unaffected by the \nadopted SED-modeling assumptions. Nonetheless, we stress that the \nmulti-component modeling in Brammer et al. (2014) is still over simplified \n(e.g., a dust screen with an adopted \\citealt{calzetti00} extinction curve), \nand systematic errors up to a factor of several in stellar mass cannot be \nexcluded for a subset of the most extreme star-forming progenitors. \n\n\\subsubsection{The reddest star-forming progenitors: a new population?}\n\n\\begin{figure}\n\\epsscale{1.15}\n\\plotone{fig8.eps}\n\\caption{Examples of SEDs of reddest star-forming progenitors with $V-J>1.9$. \nBlack filled circles and error bars represent the observed photometry, whereas \nthe continuous gray curve shows the best-fit model. The top and bottom rows \nshow SEDs at $2.0<z<2.5$ and $2.5<z<3.0$, respectively. \\label{fig8}}\n\\end{figure}\n\nIn this study we have demonstrated that a significant fraction of the \nprogenitors of UMGs at $z>2$ are dusty star-forming galaxies. Amongst that \npopulation, $\\sim$19\\% have extraordinary SEDs with very red $V-J$ colors \n($V-J>1.9$).  As shown in Figure~\\ref{fig2}, effectively zero galaxies  with \nsuch red $V-J$ colors exist at $z<1$ (down to the UltraVISTA mass limit), and \ntherefore it is tempting to classify them as a ``new'' population of galaxies.\n\nThese extremely red galaxies are intriguing, and in Figure~\\ref{fig8} we show \nfour representative examples of their SEDs. The solid lines are the best-fit \nFAST SEDs using the default SED modeling assumptions.  The fits are reasonable \nand hence with the default assumptions we naturally interpret this population \nas massive, extremely red, star-forming progenitors.  However, the SEDs are \nquite unlike local galaxies, and given that they only appear at $z>1$, it is \nimportant to consider carefully what else this unusual population might be and \nhow accurate their stellar masses are.  As already discussed, systematic \neffects likely dominate the uncertainties in both the photometric redshifts \nand SED modeling of individual galaxies among this extreme population. \n\nSpecifically, strong emission lines (or unusual emission-line ratios) could \nresult in significantly overestimated stellar masses, while the presence of \noptically thick regions could even cause the underestimating of their stellar \nmasses. Perhaps of most concern are composite stellar populations and\/or \npotential blending\/confusion of sources in ground-based images which could \nresult in the unusual SEDs and therefore biased stellar masses. \n\nWithout additional data it is difficult to say how reasonable the SED modeling \nassumptions are for this population, and how reliable their stellar masses are. \nTherefore, currently we advise caution in interpreting this extreme population. \nWhile they are only $\\sim$19\\% of the overall population, and unlikely to \nchange our overall picture of UMG formation, they are intriguing, and detailed \ninvestigations, beyond the capabilities of current photometric catalogs, will \nbe needed to confirm the existence of this population, and to assess its \nrelevance in the formation of massive galaxies in the early universe. \n\nWe note that a better understanding of this population could be made with \ncurrent instrumentation. Spectroscopy from ground-based near-infrared \nspectrographs and ALMA could measure spectroscopic redshifts and clearly \nidentify the level of contamination of the broad band colors from emission \nlines (if any).  Likewise, {\\it Hubble Space Telescope} near-infrared \nobservations could resolve them spatially and be used to test for \nconfusion\/blending, as well as potential systematics from resolved composite \nstellar populations. Lastly, deep spectroscopy could be used to determine \nvelocity dispersions, which will be the key for understanding how reliable the \nphotometrically-derived masses are. Followup of this extreme population will \nbe key for a more complete understanding of the evolutionary history of UMGs.\n\n\\section{Summary and Conclusions}\\label{sec-discussion}\n\nIn this paper we have investigated the evolution since $z=3$ of the properties \nof the progenitors of ultra-massive galaxies \n($\\log{(M_{\\rm star}\/M_{\\odot})}\\approx11.8$) at $z=0$ (UMGs). The progenitors \nhave been selected using the fixed cumulative number density approach and a \nsemi-empirical method using abundance matching in the $\\Lambda$CDM paradigm. \nWhen the progenitors are selected with the fixed cumulative number density \napproach, the stellar mass of the progenitors is seen to grow by a factor of \n$\\sim2.2$ from $z=3$ to $z=0$, implying that about half of the stellar mass in \nlocal UMGs was assembled at $z>3$. If the progenitors are selected with the \nsemi-empirical method using abundance matching, the evolution in stellar mass \nis much more significant ($0.56^{+0.35}_{-0.25}$~dex), with only about a fourth \nof the stellar mass in local UMGs having been assembled by $z=3$. Since the \nsemi-empirical approach using abundance matching accounts for merging, whereas \nthe fixed cumulative number density method does not, the former is a more \ndesirable and robust approach to constrain the evolution of the median \ncumulative number density and therefore to select progenitors. The comparison \nbetween the assembly history of local brightest cluster galaxies (BCGs) as \npredicted from semi-analytic models with the inferred evolution in stellar \nmass since $z=3$ of the progenitors of local UMGs provides yet another \nmanifestation of the previously highlighted tensions between the observed and \nmodel-predicted evolutions of the stellar mass function since $z=4$ (e.g., \n\\citealt{marchesini09}). Specifically, low-mass galaxies in the stellar mass \nrange $10^{9}-10^{11}$~M$_{\\odot}$ form too early in the models and are too \npassive at later times \\citep{fontanot09}, with the mass growth of simulated \nBCGs being dominated by accretion since $z\\sim4$ \\citep{delucia07}, in \ncontrast to what is measured in our study. \n\nThe progenitors of UMGs have been separated in quiescent and star-forming \ngalaxies using the rest-frame $U-V$ versus $V-J$ diagram ($UVJ$ diagram). At \n$z=0.35$, the progenitors of UMGs constitute a homogeneous population \nwith similar rest-frame $U-V$ and $V-J$ colors, typical of quiescent \ngalaxies with old stellar populations (age$\\sim$7~Gyr). As the progenitors are \nfollowed to higher redshifts, this population becomes increasingly \ndiversified. At $z<1$, the progenitors are all quiescent, while at $z>1$ the \ncontribution from star-forming galaxies progressively increases, with the \nprogenitors' population being dominated by dusty, star-bursting galaxies at \n$2<z<3$. \n\nThe stellar population properties (i.e., stellar age, dust extinction, \nstar-formation rate, specific star-formation rate, rest-frame absolute \n$V$-band magnitude, and mass-to-light ratio in the rest-frame $V$ band) have \nbeen estimated from the modeling of the UV-to-8$\\mu$m spectral energy \ndistributions and 24$\\mu$m data, and their evolutions with redshift have been \ninvestigated. At $z=2.75$, most of the progenitors of UMGs are dusty \n($A_{\\rm V}\\sim$1-2.2~mag) star-bursting (SFR$\\sim$100-700~M$_{\\odot}$~yr$^{-1}$; \nsSFR$\\sim$0.8-4~Gyr$^{-1}$) galaxies with a large range in stellar ages, i.e., \n0.3-1.6~Gyr. At the same redshift, the quiescent progenitors have properties \ntypical of young (age $\\sim$0.6-1 Gyr) post-starburst galaxies with little \ndust extinction ($A_{\\rm V}\\sim$0.4 mag) and strong Balmer breaks, showing a \nlarge scatter in rest-frame $U-V$ color ($\\sim$0.2 mag). With decreasing \nredshift, the fraction of quiescent progenitors increases as the star-forming \nprogenitors quench to build the quiescent population. At the same time, the \nstellar population of the already quiescent progenitors is clearly seen to age \nwith cosmic time, with their properties changing accordingly. Specifically, \ntheir stellar ages and mass-to-light ratio increase with decreasing redshift, \nand their median SEDs evolve from being characterized by a strong Balmer break \nto being shaped by the 4000\\AA~break in the rest-frame optical. Moreover, the \nquiescent progenitors are seen to migrate from the bottom-left to the \nupper-right corner of the $UVJ$ quiescent box and their rest-frame $U-V$ color \nto redden, as expected from an aging quiescent stellar population. \n\nIt is interesting to note that the stellar properties of the star-forming \nprogenitors do not appear to be changing substantially over the redshift \ninterval $1.5<z<3.0$, and that over this same interval the scatter in the \n$U-V$ color of the quiescent progenitors remains constant and large \n($\\sim$0.17~mag) and it starts to decrease only at $z<1.5$. It appears indeed \nthat $z=1.5$ marks the time by which most of the star-forming progenitors have \nquenched, as indicated by the very few star-forming progenitors at \n$1.0<z<1.5$. Consistent with this interpretation, the last star-forming \nprogenitors to quench at $z=1.25$ are characterized by lower levels of star \nformation and older ages on average compared to the star-forming progenitors \nat $z>1.5$. By $z=1$, all star-forming progenitors have quenched, and the \nquiescent progenitors are simply left aging thereafter in the following \n$\\sim$7.7~Gyr to $z=0$, and some additional growing due to, e.g., minor dry \nmergers. \n\nWhereas the existence at $z=2.75$ of a small fraction ($\\sim$15\\%) of \nquiescent progenitors implies that the assembly of the very massive end of the \nlocal quiescent red-sequence population must have started at $z>3$, our \nresults suggest it had been mostly assembled between $z=3$ and $z=1.5$, with \nall the growth by in-situ star formation ceased by $z=1$. Most of the \nquenching of the star-forming progenitors is seen to happen between $z=2.75$ \nand $z=1.25$, implying a mean age difference of $\\sim$2.6~Gyr among the UMGs. \nOur results are in remarkably good agreement with the typical formation \nredshift of quiescent, red-sequence UMGs in the local universe ($z\\sim1.9$) \nand their rms scatter in age ($\\approx$1.8-2~Gyr) as derived from \narcheological studies (e.g., \\citealt{gallazzi06}), and provide a direct probe \nin the early universe of the results and implications of the fossil records in \nthe population of UMGs today.\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig9.eps}\n\\caption{Schematic evolutionary tracks plotted in the color-mass diagram \nshowing possible paths for the formation of massive red-sequence galaxies in \nthe local universe. Track C, proposed by \\citet{faber07}, is a ``mixed'' \nscenario involving both early and late quenching of blue, star-forming \ngalaxies. The red sequence galaxy arise from the blue galaxy when star \nformation is quenched (upward black arrows from the blue cloud to the red \nsequence). Once the galaxy arrives on the red sequence, it may evolve more \nslowly along it through a series of dry mergers (white arrows). The green \narrow represents our newly proposed path for the formation of UMGs in the \nlocal universe based on our findings. This new scenario involves in-situ early \nstellar growth during a short and intense dusty burst of star formation at \nwhich point the progenitor manifest as a red, massive, heavily dust-extincted \nstar-bursting galaxy. After quenching, the progenitor continues reddening due \nto aging and potentially growing in mass via dry mergers, reaching its final \nposition on the color-mass diagram. In this new scenario, the progenitor of \nlocal UMGs have never lived on the blue star-forming cloud since $z=3$. The \npopulation of red galaxies is seen to change with cosmic time, dominated by \ndusty star-bursting galaxies at $z>2$ and by quiescent galaxies at lower \nredshift. \\label{fig9}}\n\\end{figure}\n\nFinally, we have shown that the progenitors of UMGs have never lived on the \nblue star-forming cloud in the last 11.4~Gyr of cosmic history, i.e., since \n$z=3$. This is true also for the star-forming progenitors, which have a median \nrest-frame $U-V$ color at $z=2.75$ similar to the quiescent progenitors, with \na significant fraction of them having $U-V$ colors much redder than most of \nthe quiescent progenitors at the same redshift. In light of our findings, \nthe picture in which early-type UMGs at $z\\sim0$ have formed from star-forming \ngalaxies living on the blue cloud that reddened after quenching and further \ngrew through moderate dry merging has to be refined. Such scenario is pictured \nin Figure~\\ref{fig9} as taken from the ``Evolution Track C'' in \n\\citet{faber07}. This scenario involves early mass assembly and star formation \nin galaxies on the blue cloud of star-forming galaxies. When the \nstar-formation activity in the blue galaxies gets quenched, they rapidly move \nonto the red sequence. Once on the red sequence of quiescent galaxies, any \nfurther growth in stellar mass in these progenitors mostly proceeds via dry \nmerging among galaxies on the red sequence. Whereas such path is still \nviable to explain the formation of less massive (e.g., $10^{11}$~M$_{\\odot}$) \nlocal red-sequence galaxies which may have gone through mostly unobscured \n(hence blue) episodes of early star formation, our study suggest an \nalternative path for the formation of UMGs at $z\\sim0$. This new scenario, \npictured as a green arrow in Figure~\\ref{fig9}, involves early mass assembly \nand stellar growth in a short and intense dusty burst of star formation during \nwhich the progenitors manifest as a red, heavily dust-obscured star-bursting \ngalaxies. After quenching, the progenitors continue reddening due to the aging \nof their stellar populations, and eventually grow in mass through (mostly \nminor) dry merging, finally reaching their current positions on the color-mass \ndiagram. This scenario qualitatively agrees with the predictions from the \ntheoretical model of the co-evolution of massive galaxies and super-massive \nblack holes originally proposed in \\citet{granato04} and further developed in \n\\citet{cai13} and \\citet{lapi14}, which indicates that the star formation in \nthe progenitors of local very massive spheroids proceeded within a heavily \ndust-enshrouded medium at $z>1.5$ over a timescale $\\lesssim 0.5-1$~Gyr.\n\nOur study provides a complete and consistent picture of how the most massive \ngalaxies in the local universe have formed and assembled since $z=3$. While \nwe see the star-forming progenitors disappearing and turning into the quiescent \nprogenitors, we do not know what physical mechanisms are responsible for \nhalting the star formation. Is the AGN, if present, playing a major role? What \nabout feedback from the starburst itself? For the first time, our analysis \nprovides a sample of star-forming progenitors that must quench by $z=1$. \nDetailed studies of the AGN content, physical conditions of the gas (e.g., gas \nfraction and gas-phase metallicity), stellar population properties (e.g., \nstellar ages and metallicities, SFRs), structural and dynamical properties, \nand environment of these star-forming progenitors can shed light into the \nphysical processes responsible for the inevitable quenching of their \nstar-formation activity. High $S\/N$ optical and near-infrared spectroscopic \nstudies, as well as deep sub-millimeter studies with ALMA will provide the \ndatasets to answer the remaining outstanding questions. \n\nNeedless to say, our study needs to be extended to earlier times to \ninvestigate the properties of the progenitors at $z>3$, i.e., in the first 2 \nbillion years of cosmic history. Have the progenitors ever lived on the blue \nstar-forming cloud? Which are the progenitors of the quiescent, massive, \npost-starburst galaxies we see at $2.5<z<3$? Much deeper near-infrared \nselected catalogs over a large surveyed area are required to push this \ninvestigation to higher redshift. The second data release (DR2) of the \nUltraVISTA survey will provide much deeper near-infrared coverage over \n$\\sim$70\\% of the full COSMOS field, reaching $\\sim1.2$~mag deeper in the \n$K_{\\rm S}$ band, and allowing to probe the stellar population 0.5~dex deeper \nin stellar mass. \n\nThe observed assembly since $z=3$ of the most massive galaxies need \nto be investigated in more detail in the context of theoretical models of \ngalaxy formation and evolution. The new generation of semi-analytic models \naccounting for a better treatment of satellite galaxies, for the formation of \nthe intra-cluster component, including detailed chemical enrichment models, \nand coupled with radiative transfer codes or templates to predict the infrared \nluminosities of model galaxies (e.g., \\citealt{vega05}; \\citealt{lagos12}) \nwill provide valuable tools to further test the physical mechanisms driving \nthe formation and evolution of today's most massive galaxies. \n\nFinally, we note that $\\sim$19\\% of the star-forming progenitors at $z>2$ have \nextraordinary SEDs with very red $V-J$ colors ($V-J>1.9$), potentially \nrepresenting a ``new'' population of galaxies in the early universe that \neffectively disappear by $z\\sim1$. While unlikely to change our overall \npicture of the formation of UMGs, systematic effects dominate the \nuncertainties in both the photometric redshifts and SED modeling of individual \ngalaxies among this extreme population. Detailed investigations, including \nspectroscopy with near-infrared spectrographs and ALMA, and near-infrared \nimaging with the {\\it Hubble Space Telescope}, will be needed to confirm the \nexistence of this intriguing population and for a more complete understanding \nof the evolutionary history of UMGs.\n\n\n\\acknowledgments\nWe are grateful to the anonymous referee whose comments and suggestions helped \nimproving significantly this paper. We gratefully acknowledge Peter Behroozi \nfor providing the code implementing the semi-empirical approach using \nabundance matching to select progenitors. DM acknowledges the support of the \nResearch Corporation for Science Advancement's Cottrell Scholarship. DM also \nacknowledges support from Tufts University Mellon Research Fellowship in Arts \nand Sciences. AM and MF acknowledge support from the Netherlands Foundation \nfor Research (NWO) Spinoza grant. MS acknowledges support from the programs \nHST-GO-12286.11 and HST-GO-12060.95, provided by NASA through a grant from the \nSpace Telescope Science Institute, which is operated by the Association of \nUniversities for Research in Astronomy, Incorporated, under the NASA contract \nNAS5-26555. BMJ and JPUF acknowledge support from the ERC-StG grant \nEGGS-278202. JSD acknowledges the support of the European Research Council \nthrough the award of an Advanced Grant, and the support of the Royal Society \nvia a Wolfson Research Merit Award. JSD also acknowledges the contribution of \nthe EC FP7 SPACE project ASTRODEEP (Ref.No: 312725). The Dark Cosmology Centre \nis funded by the DNRF. This study is based on data products from observations \nmade with ESO Telescopes at the La Silla Paranal Observatory under ESO \nprogramme ID 179.A-2005 and on data products produced by TERAPIX and the \nCambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider the system of nonlinear Volterra difference equations with infinite\ndela\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{c\n\\Delta x_{n}=h_{n}x_{n}+\\sum\\limits_{i=-\\infty}^{n}a_{n,i}f(y_{i})\\\\\n\\Delta y_{n}=p_{n}y_{n}+\\sum\\limits_{i=-\\infty}^{n}b_{n,i}g(x_{i})\\text{\\ \n\\end{array}\n\\right.  , \\label{1.1\n\\end{equation}\nwhere $f$ and $g$ are real valued and continuous functions, and $\\left\\{\na_{n,i}\\right\\}  $, $\\left\\{  b_{n,i}\\right\\}  $, $\\left\\{  h_{n}\\right\\}  $,\nand $\\left\\{  p_{n}\\right\\}  $ are real sequences. In this study, we use\n\\emph{Schauder's fixed point theorem }to provide sufficient conditions\nguaranteeing the existence of periodic and asymptotically periodic solutions\nof the system (\\ref{1.1}). Since we are seeking the existence of periodic\nsolutions it is natural to ask that there exists a least positive integer $T$\nsuch tha\n\\begin{equation}\nh_{n+T}=h_{n}\\text{,\\ }p_{n+T}=p_{n}\\text{,} \\label{1.2\n\\end{equation\n\\begin{equation}\na_{n+T,i+T}=a_{n,i}, \\label{1.2.1\n\\end{equation}\nan\n\\begin{equation}\nb_{n+T,i+T}=b_{n,i} \\label{1.2.2\n\\end{equation}\nhold for all $n\\in\\mathbb{N}$, where $\\mathbb{N}$ indicates the set of all\nnonnegative integers.\n\nRecently, there has been a remarkable interest in the study of Volterra\nequations due to their applications in numerical analysis and biological\nsystems, see e.g. \\cite{ab}, \\cite{hb}, and \\cite{cs}. There is a vast\nliterature on this subject in the continuous and discrete cases. For\ninstance,\\ in \\cite{[2]} the authors considered the two dimensional system of\nnonlinear Volterra difference equation\n\\[\n\\left\\{\n\\begin{array}\n[c]{c\n\\Delta x_{n}=h_{n}x_{n}+\\sum\\limits_{i=1}^{n}a_{n,i}f(y_{i})\\\\\n\\Delta y_{n}=p_{n}y_{n}+\\sum\\limits_{i=1}^{n}b_{n,i}g(x_{i})\n\\end{array}\n\\right.  ,\\ \\ \\ \\ n=1,2,...\n\\]\nand classified the limiting behavior and the existence of its positive\nsolutions with the help of fixed point theory. Also, the authors of\n\\cite{[xy]} analyzed the asymptotic behavior of positive solutions of second\norder nonlinear difference systems, while the authors of \\cite{[wy]} studied\nthe classification and the existence of positive solutions of the system of\nVolterra nonlinear difference equations. Periodicity of the solutions of\ndifference equations has been handled by \\cite{[1]}, \\cite{[3]}-\\cite{[5]}. In\n\\cite{[6]} and \\cite{[7]}, the authors focused on a system of Volterra\ndifference equations of the form\n\\[\nx_{s}(n)=a_{s}(n)+b_{s}(n)x_{s}(n)+\\sum_{p=1}^{r}\\sum_{i=0}^{n}K_{sp\n(n,i)x_{p}(i),\\ \\ n\\in\\mathbb{N},\n\\]\nwhere $a_{s}$, $b_{s}$, $x_{s}:\\mathbb{N}\\rightarrow\\mathbb{R}$ and\n$K_{sp}:\\mathbb{N}\\times\\mathbb{N}\\rightarrow\\mathbb{R}$, $s=1,2,...,r$, and\n$\\mathbb{R}$ denotes the set of all real numbers and obtained sufficient\nconditions for the existence of asymptotically periodic solutions. They had to\nconstruct a mapping on an appropriate space and then obtain a fixed point.\nFurthermore, in \\cite{[8]} \\ the authors investigated the existence of\nperiodic and positive periodic solutions of system of nonlinear Volterra\nintegro-differential equations. The paper \\cite{el} of Elaydi, was one of the\nfirst to address the existence of periodic solutions and the stability\nanalysis of Volterra difference equations. Since then, the study of Volterra\ndifference equations has been vastly increasing. For instance, we mention the\npapers \\cite{kr}, \\cite{md}, and the references therein. In addition to\nperiodicity we refer to \\cite{kc} and \\cite{mdi} for results regarding\nboundedness. \\newline The main purpose of this paper is to extend the results\nof the above mentioned literature by investigating the possibility of\nexistence of periodic and the asymptotic periodic solutions for systems of\nnonlinear Volterra difference equations with infinite delay.\n\nDenote by $\\mathbb{Z}$ and $\\mathbb{Z}^{-}$ the set of integers and the set of\nnonpositive integers, respectively. By a solution of the system (\\ref{1.1}) we\nmean a pair of sequences $\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}\n_{n\\in\\mathbb{Z}}$ of real numbers which satisfies (\\ref{1.1}) for all\n$n\\in\\mathbb{N}$. The initial sequence space for the solutions of the system\n(\\ref{1.1}) can be constructed as follows. Let $S$ denote the nonempty set of\npairs of all sequences $\\left(  \\eta,\\zeta\\right)  =\\left\\{  \\left(  \\eta\n_{n},\\zeta_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}^{-}}$ of real numbers such\nthat\n\\[\n\\max\\left\\{  \\sup_{n\\in\\mathbb{Z}^{-}}\\left\\vert \\eta_{n}\\right\\vert\n,\\sup_{n\\in\\mathbb{Z}^{-}}\\left\\vert \\zeta_{n}\\right\\vert \\right\\}  <\\infty,\n\\]\nand for each $n\\in\\mathbb{N}$ the serie\n\\\n{\\displaystyle\\sum\\limits_{i=-\\infty}^{0}}\na_{n,i}f(\\eta_{i})\\text{ and \n{\\displaystyle\\sum\\limits_{i=-\\infty}^{0}}\nb_{n,i}g(\\zeta_{i})\n\\]\nconverge. It is clear that for any given pair of initial sequences $\\left\\{\n\\left(  \\eta_{n},\\zeta_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}^{-}}$ in $S$\nthere exists a unique solution $\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}\n_{n\\in\\mathbb{Z}}$ of the system (\\ref{1.1}) which satisfies the initial\nconditio\n\\begin{equation}\n\\left(\n\\begin{array}\n[c]{c\nx_{n}\\\\\ny_{n\n\\end{array}\n\\right)  =\\left(\n\\begin{array}\n[c]{c\n\\eta_{n}\\\\\n\\zeta_{n\n\\end{array}\n\\right)  \\text{ for }n\\in\\mathbb{Z}^{-}, \\label{initial condition\n\\end{equation}\nsuch solution $\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}  _{n\\in\n\\mathbb{Z}}$ is said to be the solution of the initial problem (\\ref{1.1\n-\\ref{initial condition}). For any pair $\\left(  \\eta,\\zeta\\right)  \\in S$,\none can specify a solution of (\\ref{1.1}-\\ref{initial condition}) by denoting\nit by $\\left\\{  \\left(  x_{n}\\left(  \\eta\\right)  ,y_{n}\\left(  \\zeta\\right)\n\\right)  \\right\\}  _{n\\in\\mathbb{Z}}$, wher\n\\[\n\\left(  x_{n}\\left(  \\eta\\right)  ,y_{n}\\left(  \\zeta\\right)  \\right)\n=\\left\\{\n\\begin{array}\n[c]{ll\n\\left(  \\eta_{n},\\zeta_{n}\\right)  & \\text{for }n\\in\\mathbb{Z}^{-}\\\\\n\\left(  x_{n},y_{n}\\right)  & \\text{for }n\\in\\mathbb{N\n\\end{array}\n\\right.  .\n\\]\nIn our analysis, we apply a fixed point theorem to general operators over a\nBanach space of bounded sequences defined on the whole set of integers. Unlike\nthe above mentioned literature that dealt with stability of delayed difference\nsystems, in the construction of our existence type theorems we neglect the\nconsideration of phase space, for simplicity. For a similar approach we refer\nto \\cite{burton}.\n\nWe end this section by recalling the fixed point theorem that we use in our\nfurther analysis.\n\n\\begin{theorem}\n\\label{thm1}\\emph{(Schauder's fixed point theorem)}Let $X$ be a Banach Space.\nAssume that $K$ is a closed, bounded and convex subset of $X$. If\n$T:K\\rightarrow K$ is a compact operator, then it has a fixed point in$\\ K.$\n\\end{theorem}\n\n\\section{Periodicity}\n\nIn this section, we use Schauder's fixed point theorem to show that system\n(\\ref{1.1}) has a periodic solution.\n\nLet $P_{T}$ be the set of all pairs of sequences $\\left(  x,y\\right)\n=\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}}$ satisfying\n$x_{n+T}=x_{n}$ and $y_{n+T}=y_{n}$ for all $n\\in\\mathbb{N}$. Then $P_{T}$ is\na Banach space when it is endowed with the maximum nor\n\\[\n\\left\\Vert (x,y)\\right\\Vert :=\\max\\left\\{  \\max_{n\\in\\left[  1,T\\right]\n_{\\mathbb{Z}}}\\left\\vert x_{n}\\right\\vert ,\\max_{n\\in\\left[  1,T\\right]\n_{\\mathbb{Z}}}\\left\\vert y_{n}\\right\\vert \\right\\}  ,\n\\]\nwhere $[1,T]_{\\mathbb{Z}}:=\\left[  1,T\\right]  \\cap\\mathbb{Z}$. Let us define\nthe subset $\\Omega\\left(  W\\right)  $ of $P_{T}$ by\n\\[\n\\Omega\\left(  W\\right)  :=\\{(x,y)\\in P_{T}:\\left\\Vert (x,y)\\right\\Vert \\leq\nW\\},\n\\]\nwhere $W>0$ is a constant. Then $\\Omega\\left(  W\\right)  $ is a bounded,\nclosed and convex subset of $P_{T}$.\n\nFor any pair $\\left(  x,y\\right)  =\\left\\{  \\left(  x_{n}\\left(  \\eta\\right)\n,y_{n}\\left(  \\zeta\\right)  \\right)  \\right\\}  _{n\\in\\mathbb{Z}}\\in\n\\Omega\\left(  W\\right)  $ with an initial sequence $\\left\\{  \\left(  \\eta\n_{n},\\zeta_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}^{-}}$ in $S$, define the\nmapping $E$ on $\\Omega\\left(  W\\right)  $ b\n\\[\nE\\left(  x,y\\right)  :=\\left\\{  E(x,y)_{n}\\right\\}  _{n\\in\\mathbb{Z\n}:=\\left\\{  \\left(\n\\begin{array}\n[c]{c\nE_{1}(x,y)_{n}\\\\\nE_{2}(x,y)_{n\n\\end{array}\n\\right)  \\right\\}  _{n\\in\\mathbb{Z}},\n\\]\nwhere\n\\begin{equation}\nE_{1}(x,y)_{n}:=\\left\\{\n\\begin{array}\n[c]{ll\n\\eta_{n} & \\text{for }n\\in\\mathbb{Z}^{-}\\\\\n\\alpha_{h}\\sum\\limits_{i=n}^{n+T-1}\\left(  \\prod\\limits_{l=i+1}^{n+T-1\n(1+h_{l})\\right)  \\sum\\limits_{m=-\\infty}^{i}a_{i,m}f(y_{m}) & \\text{for \nn\\in\\mathbb{N\n\\end{array}\n\\right.  , \\label{1.5\n\\end{equation\n\\begin{equation}\nE_{2}(x,y)_{n}:=\\left\\{\n\\begin{array}\n[c]{ll\n\\zeta_{n} & \\text{for }n\\in\\mathbb{Z}^{-}\\\\\n\\alpha_{p\n{\\textstyle\\sum\\limits_{i=n}^{n+T-1}}\n\\left(  \\prod\\limits_{l=i+1}^{n+T-1}(1+p_{l})\\right)  \\sum\\limits_{m=-\\infty\n}^{i}b_{i,m}g(x_{m}) & \\text{for }n\\in\\mathbb{N\n\\end{array}\n\\right.  , \\label{1.5.1\n\\end{equation}\nan\n\\begin{align*}\n\\alpha_{h}  &  :=\\left[  1-\\prod\\limits_{l=0}^{T-1}(1+h_{l})\\right]  ^{-1},\\\\\n\\alpha_{p}  &  :=\\left[  1-\\prod\\limits_{l=0}^{T-1}(1+p_{l})\\right]  ^{-1}.\n\\end{align*}\nWe shall use the following result on several occasions in our further analysis.\n\n\\begin{lemma}\n\\label{lem1}Assume that (\\ref{1.2}-\\ref{1.2.2}) hold. Suppose that\n$1+h_{n}\\neq0$, $1+p_{n}\\neq0$ for all $n\\in\\lbrack1,T]_{\\mathbb{Z}}$, and\nthat\n\\begin{equation}\n\\prod\\limits_{l=0}^{T-1}(1+h_{l})\\neq1\\text{ and }\\prod\\limits_{l=0\n^{T-1}(1+p_{l})\\neq1. \\label{1.3.\n\\end{equation}\nThe pair$\\ \\left(  x,y\\right)  =\\left\\{  \\left(  x_{n}\\left(  \\eta\\right)\n,y_{n}\\left(  \\zeta\\right)  \\right)  \\right\\}  _{n\\in\\mathbb{Z}}\\in\n\\Omega\\left(  W\\right)  $ with an initial sequence $\\left\\{  \\left(  \\eta\n_{n},\\zeta_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}^{-}}$ in $S$ satisfie\n\\[\nE(x,y)_{n}=(x_{n},y_{n})\n\\]\nfor all $n\\in\\mathbb{N}$ if and only if it is a $T$-periodic solution of\n(\\ref{1.1}).\n\\end{lemma}\n\n\\begin{proof}\nOne may easily verify that the pair $\\left(  x,y\\right)  =\\left\\{  \\left(\nx_{n}\\left(  \\eta\\right)  ,y_{n}\\left(  \\zeta\\right)  \\right)  \\right\\}\n_{n\\in\\mathbb{Z}}\\in\\Omega\\left(  W\\right)  $ satisfying $(x_{n\n,y_{n})=E(x,y)_{n}$ for all $n\\in\\mathbb{N}$ is a $T$-periodic solution of the\nsystem (\\ref{1.1}). Conversely, suppose that the pair $\\left(  x,y\\right)\n=\\left\\{  \\left(  x_{n}\\left(  \\eta\\right)  ,y_{n}\\left(  \\zeta\\right)\n\\right)  \\right\\}  _{n\\in\\mathbb{Z}}\\in\\Omega\\left(  W\\right)  $ is a\n$T$--periodic solution of (\\ref{1.1}). Multiplying both sides of the first\nequation in (\\ref{1.1}) with $\\left(  \\prod\\limits_{l=0}^{n}(1+h_{l})\\right)\n^{-1}$ and taking the summation from $n$ to $n+T-1,$ we obtai\n\\[\n\\sum\\limits_{i=n}^{n+T-1}\\Delta\\left[  x_{i}\\left(  \\prod\\limits_{l=0\n^{i-1}(1+h_{l})\\right)  ^{-1}\\right]  =\\sum\\limits_{i=n}^{n+T-1}\\left(\n\\prod\\limits_{l=0}^{i}(1+h_{l})\\right)  ^{-1}\\sum\\limits_{m=-\\infty\n^{i}a_{i,m}f(y_{m}).\n\\]\nThis implies tha\n\\begin{align*}\n&  x_{n+T}\\left(  \\prod\\limits_{l=0}^{n+T-1}(1+h_{l})\\right)  ^{-1\n-x_{n}\\left(  \\prod\\limits_{l=0}^{n-1}(1+h_{l})\\right)  ^{-1}\\\\\n&  =\\sum\\limits_{i=n}^{n+T-1}\\left(  \\prod\\limits_{l=0}^{i}(1+h_{l})\\right)\n^{-1}\\sum\\limits_{m=-\\infty}^{i}a_{i,m}f(y_{m}).\n\\end{align*}\nUsing the equalities $x_{n+T}=$ $x_{n}$ and $\\prod\\limits_{l=n}^{n+T-1\n(1+h_{l})=\\prod\\limits_{l=0}^{T-1}(1+h_{l})$, $n\\in\\mathbb{N}$, we have\n$E_{1}(x,y)_{n}=(x_{n},y_{n})$ for all $n\\in\\mathbb{N}$. The equality\n$E_{2}(x,y)_{n}=(x_{n},y_{n})$ for $n\\in\\mathbb{N}$ can be obtained by using a\nsimilar procedure. The proof is complete.\n\\end{proof}\n\nIn preparation for the next result we assume that there exist positive\nconstants $W_{1},$ $W_{2},$ $K_{1}$, and $K_{2}$ such tha\n\\begin{align}\n\\left\\vert f(x)\\right\\vert  &  \\leq W_{1}\\label{2.1}\\\\\n\\left\\vert g(y)\\right\\vert  &  \\leq W_{2} \\label{2.2\n\\end{align\n\\begin{equation}\n\\left\\vert \\alpha_{h}\\right\\vert \\sum\\limits_{i=n}^{n+T-1}\\left\\vert\n\\prod\\limits_{l=i+1}^{n+T-1}(1+h_{l})\\right\\vert \\sum\\limits_{m=-\\infty\n^{i}\\left\\vert a_{i,m}\\right\\vert \\leq K_{1} \\label{2.3\n\\end{equation\n\\begin{equation}\n\\left\\vert \\alpha_{p}\\right\\vert \\sum\\limits_{i=n}^{n+T-1}\\left\\vert\n\\prod\\limits_{l=i+1}^{n+T-1}(1+p_{l})\\right\\vert \\sum\\limits_{m=-\\infty\n^{i}\\left\\vert b_{i,m}\\right\\vert \\leq K_{2} \\label{2.4\n\\end{equation}\nfor all $n\\in\\mathbb{Z}$ and all $\\left(  x,y\\right)  \\in\\Omega\\left(\nW\\right)  $.\n\n\\begin{theorem}\n\\label{thm2} In addition to the assumptions of Lemma \\ref{lem1} suppose that\n(\\ref{2.1}-\\ref{2.4}) hold. Then (\\ref{1.1}) has a $T$-periodic solution.\n\\end{theorem}\n\n\\begin{proof}\nFrom Lemma \\ref{lem1}, we can deduce that $E(x,y)_{n+T}=E(x,y)_{n}$ for all\n$n\\in\\mathbb{N}$ and any $(x,y)\\in\\Omega\\left(  W\\right)  $. Moreover, if\n$(x,y)\\in\\Omega\\left(  W\\right)  $ the\n\\begin{equation}\n\\left\\vert E_{1}(x,y)_{n}\\right\\vert \\leq\\left\\vert \\alpha_{h}\\right\\vert\n\\sum\\limits_{i=n}^{n+T-1}\\left\\vert \\prod\\limits_{l=i+1}^{n+T-1\n(1+h_{l})\\right\\vert \\sum\\limits_{m=-\\infty}^{i}\\left\\vert a_{i,m}\\right\\vert\n\\left\\vert f(y_{m})\\right\\vert \\leq W_{1}K_{1}, \\label{2.4*\n\\end{equation}\nan\n\\begin{equation}\n\\left\\vert E_{2}(x,y)_{n}\\right\\vert \\leq\\left\\vert \\alpha_{p}\\right\\vert\n\\sum\\limits_{i=n}^{n+T-1}\\left\\vert \\alpha_{p}\\prod\\limits_{l=i+1\n^{n+T-1}(1+p_{l})\\right\\vert \\sum\\limits_{m=-\\infty}^{i}\\left\\vert\nb_{i,m}\\right\\vert \\left\\vert g(x_{m})\\right\\vert \\leq W_{2}K_{2} \\label{2.5*\n\\end{equation}\nfor all $n\\in\\mathbb{N}$. If we set $W=\\max\\{W_{1}K_{1},W_{2}K_{2}\\}$ then $E$\nmaps $\\Omega\\left(  W\\right)  $ into itself. Now we show that $E$ is\ncontinuous. Let $\\{(x^{l},y^{l})\\}$, $l\\in\\mathbb{N}$, be a sequence in\n$\\Omega\\left(  W\\right)  $ such tha\n\\begin{align*}\n\\lim_{l\\rightarrow\\infty}\\left\\Vert (x^{l},y^{l})-(x,y)\\right\\Vert  &\n=\\lim_{l\\rightarrow\\infty}\\left(  \\max_{n\\in\\left[  1,T\\right]  _{\\mathbb{Z}\n}\\left\\{  \\left\\vert x_{n}^{l}-x_{n}\\right\\vert ,\\left\\vert y_{n}^{l\n-y_{n}\\right\\vert \\right\\}  \\right) \\\\\n&  =0.\n\\end{align*}\nSince $\\Omega\\left(  W\\right)  $ is closed, we must have $(x,y)\\in\n\\Omega\\left(  W\\right)  $. Then by definition of $E$ we have\n\\begin{align*}\n\\left\\Vert E(x^{l},y^{l})-E(x,y)\\right\\Vert  &  =\\max\\left\\{  \\max\n_{n\\in\\left[  1,T\\right]  _{\\mathbb{Z}}}\\left\\vert E_{1}(x^{l},y^{l\n)_{n}-E_{1}(x,y)_{n}\\right\\vert ,\\right. \\\\\n&  \\left.  \\max_{n\\in\\left[  1,T\\right]  _{\\mathbb{Z}}}\\left\\vert E_{2\n(x^{l},y^{l})_{n}-E_{2}(x,y)_{n}\\right\\vert \\right\\}  ,\n\\end{align*}\nin whic\n\\begin{align*}\n\\left\\vert E_{1}(x^{l},y^{l})_{n}-E_{1}(x,y)_{n}\\right\\vert  &  =\\left\\vert\n\\alpha_{h}\\right\\vert \\left\\vert \\sum\\limits_{i=n}^{n+T-1}\\left(\n\\prod\\limits_{l=i+1}^{n+T-1}(1+h_{l})\\right)  \\sum\\limits_{m=-\\infty\n^{i}a_{i,m}f(y_{m}^{l})-\\right. \\\\\n&  \\left.  \\sum\\limits_{i=n}^{n+T-1}\\left(  \\prod\\limits_{l=i+1\n^{n+T-1}(1+h_{l})\\right)  \\sum\\limits_{m=-\\infty}^{i}a_{i,m}f(y_{m\n)\\right\\vert \\\\\n&  \\leq\\left\\vert \\alpha_{h}\\right\\vert \\sum\\limits_{i=n}^{n+T-1}\\left\\vert\n\\prod\\limits_{l=i+1}^{n+T-1}(1+h_{l})\\right\\vert \\sum\\limits_{m=-\\infty\n^{i}\\left\\vert a_{i,m}\\right\\vert \\left\\vert f(y_{m}^{l})-f(y_{m})\\right\\vert\n.\n\\end{align*}\nSimilarly\n\\[\n\\left\\vert E_{2}(x^{l},y^{l})_{n}-E_{2}(x,y)_{n}\\right\\vert \\leq\\left\\vert\n\\alpha_{p}\\right\\vert \\sum\\limits_{i=n}^{n+T-1}\\left\\vert \\prod\\limits_{l=i+1\n^{n+T-1}(1+p_{l})\\right\\vert \\sum\\limits_{m=-\\infty}^{i}\\left\\vert\nb_{i,m}\\right\\vert \\left\\vert g(x_{m}^{l})-g(x_{m})\\right\\vert .\n\\]\nThe continuity of $f$ and $g$ along with the Lebesgue dominated convergence\ntheorem imply tha\n\\[\n\\lim_{l\\rightarrow\\infty}\\left\\Vert E(x^{l},y^{l})-E(x,y)\\right\\Vert =0.\n\\]\nThis shows that $E$ is continuous. Finally, we have to show that\n$E\\Omega\\left(  W\\right)  $ is precompact. Let $\\{(x^{l},y^{l})\\}_{l\\in\n\\mathbb{N}}$ be a sequence in $\\Omega\\left(  W\\right)  $. For each fixed\n$l\\in\\mathbb{N}$, $\\{(x_{n}^{l},y_{n}^{l})\\}_{n\\in\\mathbb{Z}}$ is a bounded\nsequence of real pairs. Then by \\emph{Bolzano-Weierstrass Theorem,}\n$\\{(x_{n}^{l},y_{n}^{l})\\}_{n\\in\\mathbb{Z}}$ has a convergent subsequence\n$\\{(x_{n_{k}}^{l},y_{n_{k}}^{l})\\}$. By repeating the diagonalization process\nfor each $l\\in\\mathbb{N},$ we can construct a convergent subsequence\n$\\{(x^{l_{k}},y^{l_{k}})\\}_{l_{k}\\in\\mathbb{N}}$ of $\\{(x^{l},y^{l\n)\\}_{l\\in\\mathbb{N}}$ in $\\Omega\\left(  W\\right)  .$ Since $E$ is continuous,\nwe deduce that $\\{E(x^{l},y^{l})\\}_{l\\in\\mathbb{N}}$ has a convergent\nsubsequence in $E\\Omega\\left(  W\\right)  $. This means, $E\\Omega\\left(\nW\\right)  $ is precompact. By Schauder's fixed point theorem we conclude that\nthere exists a pair $(x,y)\\in\\Omega\\left(  W\\right)  $ such that\n$E(x,y)=(x,y)$.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm3}In addition to the assumptions of Lemma \\ref{lem1}, we assume that\n(\\ref{2.1}), (\\ref{2.3}) and (\\ref{2.4}) hold. If $g$ is a non-decreasing\nfunction satisfyin\n\\begin{equation}\n\\left\\vert g(x)\\right\\vert \\leq g(\\left\\vert x\\right\\vert ), \\label{2.6*\n\\end{equation}\nthen (\\ref{1.1}) has a $T$-periodic solution.\n\\end{theorem}\n\n\\begin{proof}\nBy (\\ref{2.3}) and (\\ref{2.4*}) we already hav\n\\[\n\\left\\vert E_{1}(x,y)\\right\\vert \\leq W_{1}K_{1}\\text{ for all }(x,y)\\in\n\\Omega\\left(  W\\right)  .\n\\]\nThis along with (\\ref{2.6*}) impl\n\\begin{align*}\n\\left\\vert E_{2}(x,y)_{n}\\right\\vert  &  \\leq\\sum\\limits_{i=n}^{n+T-1\n\\left\\vert \\alpha_{p}\\prod\\limits_{l=i+1}^{n+T-1}(1+p_{l})\\right\\vert\n\\sum\\limits_{m=-\\infty}^{i}\\left\\vert b_{i,m}\\right\\vert \\left\\vert\ng(x_{m})\\right\\vert \\\\\n&  \\leq\\sum\\limits_{i=n}^{n+T-1}\\left\\vert \\alpha_{p}\\prod\\limits_{l=i+1\n^{n+T-1}(1+p_{l})\\right\\vert \\sum\\limits_{m=-\\infty}^{i}\\left\\vert\nb_{i,m}\\right\\vert g(\\left\\vert E_{1}(x,y)\\right\\vert )\\\\\n&  \\leq K_{2}g(W_{1}K_{1}).\n\\end{align*}\nIf we set $W=\\max\\{W_{1}K_{1},K_{2}g(W_{1}K_{1})\\}$, then the rest of the\nproof is similar to the proof of Theorem \\ref{thm2} and hence we omit it.\n\\end{proof}\n\nSimilarly, we can give the following result.\n\n\\begin{theorem}\n\\label{thm4}In addition to the assumptions of Lemma \\ref{lem1}, we assume\n(\\ref{2.2}), (\\ref{2.3}) and (\\ref{2.4}) hold. If $f$ is a non-decreasing\nfunction satisfyin\n\\[\n\\left\\vert f(y)\\right\\vert \\leq f(\\left\\vert y\\right\\vert ),\n\\]\nthen (\\ref{1.1}) has a $T$-periodic solution.\n\\end{theorem}\n\n\\begin{example}\n\\label{Examp1}Let\n\\begin{align*}\nh_{n}  &  =1+\\cos n\\pi,\\\\\np_{n}  &  =1-\\cos n\\pi,\\\\\na_{n,i}  &  =b_{n,i}=e^{i-n},\n\\end{align*}\nan\n\\[\nf(x)=\\sin x\\text{ and }g(x)=\\sin2x.\n\\]\nThen (\\ref{1.1}) turns into the following syste\n\\[\n\\left\\{\n\\begin{array}\n[c]{c\n\\Delta x_{n}=(1+\\cos n\\pi)x_{n}+\\sum\\limits_{i=-\\infty}^{n}e^{i-n}\\sin\n(y_{i}),\\\\\n\\Delta y_{n}=(1-\\cos n\\pi)y_{n}+\\sum\\limits_{i=-\\infty}^{n}e^{i-n}\\sin\n(2x_{i})\\text{\\ \\ \n\\end{array}\n\\right.  .\n\\]\nIt can be easily verified that conditions (\\ref{1.2}-\\ref{1.3.}) and\n(\\ref{2.1}-\\ref{2.4}) hold. By Theorem \\ref{thm2}, there exists a $2$-periodic\nsolution $(x,y)=\\left\\{  (x_{n},y_{n})\\right\\}  _{ne\\mathbb{Z}}$ of system\n(\\ref{1.1}) satisfyin\n\\begin{align*}\nx_{n}  &  =-\\frac{1}{2}\\sum\\limits_{i=n}^{n+1}\\prod\\limits_{l=i+1\n^{n+1}(2+\\cos(l\\pi))\\sum\\limits_{m=-\\infty}^{i}e^{m-i}\\sin(y_{m})\\text{,}\\\\\ny_{n}  &  =-\\frac{1}{2}\\sum\\limits_{i=n}^{n+1}\\prod\\limits_{l=i+1\n^{n+1}(2-\\cos(l\\pi))\\sum\\limits_{m=-\\infty}^{i}e^{m-i}\\sin(2x_{m})\\text{,\n\\end{align*}\nfor all $n\\in\\mathbb{N}$.\n\\end{example}\n\n\\section{Asymptotic Periodicity}\n\nIn this section, we are going to show the existence of an asymptotically\n$T$-periodic solution of system (\\ref{1.1}) by using Schauder's fixed point\ntheorem. First we state the following definition.\n\n\\begin{definition}\n\\label{def2}A sequence $\\left\\{  x_{n}\\right\\}  _{n\\in\\mathbb{Z}}$ is called\nasymptotically $T$-periodic if there exist two sequences $u_{n}$ and $v_{n}$\nsuch that $u_{n}$ is $T$-periodic, $\\lim_{n\\rightarrow\\infty}v_{n}=0$, and\n$x_{n}=u_{n}+v_{n}$ for all $n\\in\\mathbb{Z}.$\n\\end{definition}\n\nFirst, we suppose that\n\\begin{equation}\n\\prod\\limits_{j=0}^{T-1}(1+h_{j})=1\\text{ and }\\prod\\limits_{j=0\n^{T-1}(1+p_{j})=1. \\label{3.2.1\n\\end{equation}\nThen we define the sequences $\\varphi:=\\{\\varphi_{n}\\}_{n\\in\\mathbb{N}}$ and\n$\\psi:=\\{\\psi_{n}\\}_{n\\in\\mathbb{N}}$ as follow\n\\begin{equation}\n\\varphi_{n}:=\\prod\\limits_{j=0}^{n-1}\\frac{1}{1+h_{j}}\\text{ and }\\psi\n_{n}:=\\prod\\limits_{j=0}^{n-1}\\frac{1}{1+p_{j}}. \\label{3.2\n\\end{equation}\nFurthermore, we define the constants $m_{k}$, $M_{k}$, $k=1,2$, b\n\\[\nm_{1}:=\\min_{i\\in\\lbrack1,T]_{\\mathbb{Z}}}\\left\\vert \\varphi_{i}\\right\\vert\n,\\ \\ M_{1}:=\\max_{i\\in\\lbrack1,T]_{\\mathbb{Z}}}\\left\\vert \\varphi\n_{i}\\right\\vert ,\\ m_{2}:=\\min_{i\\in\\lbrack1,T]_{\\mathbb{Z}}}\\left\\vert\n\\psi_{i}\\right\\vert ,\\ \\ M_{2}:=\\max_{i\\in\\lbrack1,T]_{\\mathbb{Z}}}\\left\\vert\n\\psi_{i}\\right\\vert .\n\\]\n\n\nWe note that in this section, we do not assume (\\ref{1.2.1}-\\ref{1.2.2}) but\ninstead we ask that the serie\n\\begin{equation}\n\\sum\\limits_{i=0}^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\left\\vert a_{i,m\n\\right\\vert <\\infty\\text{ and }\\sum\\limits_{i=0}^{\\infty}\\sum\n\\limits_{m=-\\infty}^{i}\\left\\vert b_{i,m}\\right\\vert <\\infty\\label{3.3\n\\end{equation}\nconverge to $a$ and $b$, respectively. Observe that (\\ref{3.3}) implie\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n^{i}\\left\\vert a_{i,m}\\right\\vert =\\lim_{n\\rightarrow\\infty}\\sum\n\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\left\\vert b_{i,m}\\right\\vert\n=0\\text{.} \\label{3.3.1\n\\end{equation}\n\n\n\\begin{theorem}\n\\label{thm6}Suppose that (\\ref{2.1}-\\ref{2.2}), (\\ref{3.2.1}), and\n(\\ref{3.3}-\\ref{3.3.1}) hold. Then system (\\ref{1.1}) has an asymptotically\n$T$-periodic solution $(x,y)=\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}\n_{n\\in\\mathbb{Z}}$ satisfyin\n\\begin{align*}\nx_{n}  &  :=u_{n}^{(1)}+v_{n}^{(1)}\\\\\ny_{n}  &  :=u_{n}^{(2)}+v_{n}^{(2)\n\\end{align*}\nfor $n\\in\\mathbb{N}$, where\n\\[\nu_{n}^{(1)}=c_{1}\\prod\\limits_{j=0}^{n-1}(1+h_{j}),\\ \\ \\ \\ u_{n}^{(2)\n=c_{2}\\prod\\limits_{j=0}^{n-1}(1+p_{j})\\text{, }n\\in\\mathbb{Z}^{+\n\\]\n$c_{1}$ and $c_{2}$ are positive constants, and\n\\[\n\\lim_{n\\rightarrow\\infty}v_{n}^{(1)}=\\lim_{n\\rightarrow\\infty}v_{n}^{(2)}=0.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nDue to the $T$-periodicity of the sequences $\\left\\{  h_{n}\\right\\}\n_{n\\in\\mathbb{Z}}$ and $\\left\\{  p_{n}\\right\\}  _{n\\in\\mathbb{Z}}$ and by\n(\\ref{3.2.1}-\\ref{3.2}) we have\n\\[\n\\varphi_{n}\\in\\{\\varphi_{1},\\varphi_{2},...,\\varphi_{T}\\}\\text{ and }\\psi\n_{n}\\in\\{\\psi_{1},\\psi_{2},...,\\psi_{T}\\}\n\\]\nfor all $n\\in\\mathbb{N}.$ This mean\n\\begin{align}\nm_{1}  &  \\leq\\left\\vert \\varphi_{n}\\right\\vert \\leq M_{1}\\label{3.4}\\\\\nm_{2}  &  \\leq\\left\\vert \\psi_{n}\\right\\vert \\leq M_{2} \\label{3.5\n\\end{align}\nfor all $n\\in\\mathbb{N}$. Let $B$ be the set of all real bounded sequences\n$x=\\left\\{  x_{n}\\right\\}  _{ne\\mathbb{Z}}$. Denote by $\\mathbb{B}$ the Banach\nspace of all pairs of real bounded sequences $\\left(  x,y\\right)\n=\\{(x_{n},y_{n})\\}_{n\\in\\mathbb{Z}}$, $x,y\\in B$, endowed with the maximum\nnorm\n\\[\n\\left\\Vert \\left(  x,y\\right)  \\right\\Vert =\\max\\{\\sup_{n\\in\\mathbb{Z\n}\\left\\vert x_{n}\\right\\vert ,\\sup_{n\\in\\mathbb{Z}}\\left\\vert y_{n}\\right\\vert\n\\}.\n\\]\n\n\nFor a positive constant $W^{\\ast}$ we define\n\\[\n\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  :=\\{(x,y)\\in\\mathbb{B}:\\left\\Vert\n(x,y)\\right\\Vert \\leq W^{\\ast}\\}.\n\\]\nThen, $\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $ is a nonempty bounded convex,\nand closed subset of $\\mathbb{B}.$ For any pair \\newline$\\left(  x,y\\right)\n=\\left\\{  \\left(  x_{n}\\left(  \\eta\\right)  ,y_{n}\\left(  \\zeta\\right)\n\\right)  \\right\\}  _{n\\in\\mathbb{Z}}\\in\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $\nwith an initial sequence $\\left\\{  \\left(  \\eta_{n},\\zeta_{n}\\right)\n\\right\\}  _{n\\in\\mathbb{Z}^{-}}$ in $S$ define the mapping $E^{\\ast}$ on\n$\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $ b\n\\[\nE^{\\ast}\\left(  x,y\\right)  =\\left\\{  E^{\\ast}\\left(  x,y\\right)\n_{n}\\right\\}  _{n\\in\\mathbb{Z}}=\\left\\{  \\left(\n\\begin{array}\n[c]{c\nE_{1}^{\\ast}(x,y)_{n}\\\\\nE_{2}^{\\ast}(x,y)_{n\n\\end{array}\n\\right)  \\right\\}  _{n\\in\\mathbb{Z}},\n\\]\nwher\n\\begin{equation}\nE_{1}^{\\ast}(x,y)_{n}:=\\left\\{\n\\begin{array}\n[c]{ll\n\\eta_{n} & \\text{for }n\\in\\mathbb{Z}^{-}\\\\\nc_{1}\\frac{1}{\\varphi_{n}}-\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n}^{i}\\frac{\\varphi_{i+1}}{\\varphi_{n}}a_{i,m}f(y_{m}) & \\text{for \nn\\in\\mathbb{N\n\\end{array}\n\\right.  \\text{,} \\label{3.8\n\\end{equation}\nan\n\\begin{equation}\nE_{2}^{\\ast}(x,y)_{n}:=\\left\\{\n\\begin{array}\n[c]{ll\n\\zeta_{n} & \\text{for }n\\in\\mathbb{Z}^{-}\\\\\nc_{2}\\frac{1}{\\psi_{n}}-\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n^{i}\\frac{\\psi_{i+1}}{\\psi_{n}}b_{i,m}g(x_{m}) & \\text{for }n\\in\\mathbb{N\n\\end{array}\n\\right.  . \\label{3.9\n\\end{equation}\nWe will show that the mapping $E^{\\ast}$ has a fixed point in $\\mathbb{B}$.\nFirst, we demonstrate that $E^{\\ast}\\Omega^{\\ast}\\left(  W^{\\ast}\\right)\n\\subset\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $. If $(x,y)\\in\\Omega^{\\ast\n}\\left(  W^{\\ast}\\right)  $, the\n\\begin{align}\n\\left\\vert E_{1}^{\\ast}(x,y)_{n}-c_{1}\\frac{1}{\\varphi_{n}}\\right\\vert  &\n\\leq M_{1}m_{1}^{-1}W_{1}\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n^{i}\\left\\vert a_{i,m}\\right\\vert \\label{3.10}\\\\\n&  \\leq M_{1}m_{1}^{-1}W_{1}\\sum\\limits_{i=0}^{\\infty}\\sum\\limits_{m=-\\infty\n}^{i}\\left\\vert a_{i,m}\\right\\vert \\nonumber\\\\\n&  =M_{1}m_{1}^{-1}W_{1}a,\n\\end{align}\nan\n\\begin{align}\n\\left\\vert E_{2}^{\\ast}(x,y)_{n}-c_{2}\\frac{1}{\\psi_{n}}\\right\\vert  &  \\leq\nM_{2}m_{2}^{-1}W_{2}\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n^{i}\\left\\vert b_{i,m}\\right\\vert \\label{3.11}\\\\\n&  \\leq M_{2}m_{2}^{-1}W_{2}\\sum\\limits_{i=0}^{\\infty}\\sum\\limits_{m=-\\infty\n}^{i}\\left\\vert b_{i,m}\\right\\vert \\nonumber\\\\\n&  =M_{2}m_{2}^{-1}W_{2}b\n\\end{align}\nfor all $n\\in\\mathbb{N}$. This implies tha\n\\[\n\\left\\vert E_{1}^{\\ast}(x,y)_{n}\\right\\vert \\leq M_{1}m_{1}^{-1}W_{1\na+\\frac{c_{1}}{m_{1}\n\\]\nan\n\\[\n\\left\\vert E_{2}^{\\ast}(x,y)_{n}\\right\\vert \\leq M_{2}m_{2}^{-1}W_{2\nb+\\frac{c_{2}}{m_{2}\n\\]\nfor all $n\\in\\mathbb{N}$. If we set\n\\[\nW^{\\ast}=\\max\\{M_{1}m_{1}^{-1}W_{1}a+\\frac{c_{1}}{m_{1}},M_{2}m_{2}^{-1\nW_{2}b+\\frac{c_{2}}{m_{2}}\\},\n\\]\nthen \\ we have $E^{\\ast}\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  \\subset\n\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $ as desired. \\newline Next, we show\nthat $E^{\\ast}$ is continuous. Let $\\{(x^{q},y^{q})\\}_{q\\in\\mathbb{N}}$ be a\nsequence in $\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $ such that \\newlin\n$\\lim_{q\\rightarrow\\infty}\\left\\Vert (x^{q},y^{q})-(x,y)\\right\\Vert =0$, where\n$\\left(  x,y\\right)  =\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}\n_{n\\in\\mathbb{Z}}$. Since $\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $ is closed,\nwe must have $(x,y)\\in\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $. From\n(\\ref{3.8}) and (\\ref{3.9}), we hav\n\\[\n\\left\\vert E_{1}^{\\ast}(x^{q},y^{q})_{n}-E_{1}^{\\ast}(x,y)_{n}\\right\\vert\n\\leq\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\left\\vert\n\\frac{\\varphi_{i+1}}{\\varphi_{n}}\\right\\vert \\left\\vert a_{i,m}\\right\\vert\n\\left\\vert f(y_{m}^{q})-f(y_{m})\\right\\vert\n\\]\nan\n\\[\n\\left\\vert E_{2}^{\\ast}(x^{q},y^{q})_{n}-E_{2}^{\\ast}(x,y)_{n}\\right\\vert\n\\leq\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\left\\vert \\frac\n{\\psi_{i+1}}{\\psi_{n}}\\right\\vert \\left\\vert b_{i,m}\\right\\vert \\left\\vert\ng(x_{m}^{q})-g(x_{m})\\right\\vert\n\\]\nfor all $n\\in\\mathbb{N}$. Since $f$ and $g$ are continuous, we have by the\nLebesgue dominated convergence theorem tha\n\\[\n\\lim_{q\\rightarrow\\infty}\\left\\Vert E^{\\ast}(x^{q},y^{q})-E^{\\ast\n}(x,y)\\right\\Vert =0.\n\\]\nAs we did in the proof of Theorem \\ref{thm2} we can show that $E^{\\ast}$ has a\nfixed point in $\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $. On the other hand,\nusing a similar procedure that we have employed in the proof of Lemma\n\\ref{lem1}, we can deduce that any solution $(x,y)=\\left\\{  \\left(\nx_{n},y_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}}$ of the system (\\ref{1.1}) is\na fixed point for the operator $E^{\\ast}$. This means $E^{\\ast}\\left(\nx,y\\right)  =\\left(  x,y\\right)  $ or equivalently\n\\begin{equation}\nx_{n}=c_{1}\\frac{1}{\\varphi_{n}}-\\sum\\limits_{i=n}^{\\infty}\\sum\n\\limits_{m=-\\infty}^{i}\\frac{\\varphi_{i+1}}{\\varphi_{n}}a_{i,m}f(y_{m})\n\\label{3.12\n\\end{equation}\nan\n\\begin{equation}\ny_{n}=c_{2}\\frac{1}{\\psi_{n}}-\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n}^{i}\\frac{\\psi_{i+1}}{\\psi_{n}}b_{i,m}g(x_{m}). \\label{3.13\n\\end{equation}\nConversely, any pair $(x,y)=\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}\n_{n\\in\\mathbb{Z}}$ satisfying (\\ref{3.12}) and (\\ref{3.13}) will also satisf\n\\begin{align*}\nx_{n+1}-x_{n}(1+h_{n})  &  =c_{1}(\\prod\\limits_{j=0}^{n}(1+h_{j\n)-(1+h_{n})\\prod\\limits_{j=0}^{n-1}(1+h_{j}))\\\\\n&  +(1+h_{n})\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty}^{i\n\\frac{\\varphi_{i+1}}{\\varphi_{n}}a_{i,m}f(y_{m})\\\\\n&  -\\sum\\limits_{i=n+1}^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\frac{\\varphi\n_{i+1}}{\\varphi_{n+1}}a_{i,m}f(y_{m}),\n\\end{align*}\nand hence\n\\begin{align*}\nx_{n+1}-x_{n}(1+h_{n})  &  =\\sum\\limits_{i=n}^{\\infty}\\sum\\limits_{m=-\\infty\n}^{i}\\frac{(1+h_{n})\\prod\\limits_{j=0}^{n-1}(1+h_{j})}{\\prod\\limits_{j=0\n^{i}(1+h_{j})}a_{i,m}f(y_{m})\\\\\n&  -\\sum\\limits_{i=n+1}^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\frac\n{\\prod\\limits_{j=0}^{n}(1+h_{j})}{\\prod\\limits_{j=0}^{i}(1+h_{j})\na_{i,m}f(y_{m})\\\\\n&  =\\sum\\limits_{m=-\\infty}^{n}a_{n,m}f(y_{m}).\n\\end{align*}\nThat is, any fixed point $\\left(  x,y\\right)  =\\left\\{  (x_{n},y_{n})\\right\\}\n_{n\\in\\mathbb{Z}}$ of the operator $E^{\\ast}$ satisfies the first equation in\n(\\ref{1.1}). Similarly, one may show that the second equation holds. \\newline\nFor an arbitrary fixed point $(x,y)\\in\\Omega^{\\ast}\\left(  W^{\\ast}\\right)  $\nof $E^{\\ast}$, we hav\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\\left\\vert x_{n}-c_{1}\\frac{1}{\\varphi_{n\n}\\right\\vert =\\lim_{n\\rightarrow\\infty}\\left\\vert E_{1}^{\\ast}(x,y)_{n\n-c_{1}\\frac{1}{\\varphi_{n}}\\right\\vert =0 \\label{3.13*\n\\end{equation}\nan\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}\\left\\vert y_{n}-c_{2}\\frac{1}{\\psi_{n}}\\right\\vert\n=\\lim_{n\\rightarrow\\infty}\\left\\vert E_{2}(x,y)_{n}-c_{2}\\frac{1}{\\psi_{n\n}\\right\\vert =0. \\label{3.13**\n\\end{equation}\nChoosin\n\\begin{equation}\nu_{n}^{(1)}=c_{1}\\frac{1}{\\varphi_{n}},\\ \\ v_{n}^{(1)}=-\\sum\\limits_{i=n\n^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\frac{\\varphi_{i+1}}{\\varphi_{n}\na_{i,m}f(y_{m}) \\label{3.13.1\n\\end{equation}\nan\n\\begin{equation}\nu_{n}^{(2)}=c_{2}\\frac{1}{\\psi_{n}},\\ \\ v_{n}^{(2)}=-\\sum\\limits_{i=n\n^{\\infty}\\sum\\limits_{m=-\\infty}^{i}\\frac{\\psi_{i+1}}{\\psi_{n}}b_{i,m\ng(x_{m}), \\label{3.13.2\n\\end{equation}\nwe have $x_{n}=u_{n}^{(1)}+v_{n}^{(1)}$ and $y_{n}=u_{n}^{(2)}+v_{n}^{(2)}$.\nBy (\\ref{3.13*}) and (\\ref{3.13**}), $v_{n}^{(1)}$ and $v_{n}^{(2)}$ tend to\n$0$ when $n\\rightarrow\\infty.$ Left to show that $u_{n}^{(1)}$ and\n$u_{n}^{(2)}$ are $T$-periodic.\n\\begin{align*}\nu_{n+T}^{(1)}  &  =c_{1}\\prod\\limits_{j=0}^{n+T-1}(1+h_{j})=c_{1\n\\prod\\limits_{j=0}^{n-1}(1+h_{j})\\prod\\limits_{j=n}^{n+T-1}(1+h_{j})\\\\\n&  =c_{1}\\prod\\limits_{j=0}^{n-1}(1+h_{j})\\prod\\limits_{j=0}^{T-1}(1+h_{j})\\\\\n&  =c_{1}\\prod\\limits_{j=0}^{n-1}(1+h_{j}),\\;\\mbox{by}\\;\\eqref{3.2.1}.\n\\end{align*}\nProof for $u_{n}^{(2)}$ is identical.\n\\end{proof}\n\n\\begin{example}\n\\label{examp2}Consider the system (\\ref{1.1}) with the following entrie\n\\begin{align*}\nh_{n}  &  =p_{n}=\\left\\{\n\\begin{array}\n[c]{cc\n1, & \\text{if }n=2k+1\\text{ for }k\\in\\mathbb{Z}\\text{ }\\\\\n-\\frac{1}{2}, & \\text{if }n=2k\\text{ for }k\\in\\mathbb{Z\n\\end{array}\n\\right.  ,\\\\\na_{n,i}  &  =e^{i-2n}\\text{, for }n,i\\in\\mathbb{Z}\\\\\nb_{n,i}  &  =e^{2i-3n}\\text{,}\\ \\ \\text{for }n,i\\in\\mathbb{Z}\\\\\nf(x)  &  =\\cos x\\text{ and }g(x)=\\cos2x.\n\\end{align*}\nThen (\\ref{1.1}) turns into the following system\n\\[\n\\left\\{\n\\begin{array}\n[c]{c\n\\Delta x_{n}=h_{n}x_{n}+\\sum\\limits_{i=-\\infty}^{n}e^{i-2n}\\cos(y_{i}),\\\\\n\\Delta y_{n}=p_{n}y_{n}+\\sum\\limits_{i=-\\infty}^{n}e^{2i-3n}\\cos\n(2x_{i})\\text{\\ \\ \n\\end{array}\n\\right.  .\n\\]\nObviously, the sequences $\\left\\{  h_{n}\\right\\}  _{n\\in\\mathbb{Z}}$ and\n$\\left\\{  p_{n}\\right\\}  _{n\\in\\mathbb{Z}}$ are $2-$periodic and all\nconditions of Theorem \\ref{thm6} are satisfied. Hence, we conclude by Theorem\n\\ref{thm6} the existence of an asymptotically $2-$periodic solution $\\left(\nx,y\\right)  =\\left\\{  \\left(  x_{n},y_{n}\\right)  \\right\\}  _{n\\in\\mathbb{Z}}$\nsatisfyin\n\\begin{align*}\nx_{n}  &  =c_{1}\\frac{1}{\\varphi_{n}}-\\sum\\limits_{i=n}^{\\infty\n\\sum\\limits_{m=-\\infty}^{i}\\frac{\\varphi_{i+1}}{\\varphi_{n}}e^{m-2i}\\cos\n(y_{m})\\\\\ny_{n}  &  =c_{2}\\frac{1}{\\psi_{n}}-\\sum\\limits_{i=n}^{\\infty}\\sum\n\\limits_{m=-\\infty}^{i}\\frac{\\psi_{i+1}}{\\psi_{n}}e^{2m-3i}\\cos(2x_{m}),\n\\end{align*}\nfor all $n\\in\\mathbb{N}$, where $c_{1}$ and $c_{2}$ are positive constants,\n$\\varphi:=\\{\\varphi_{n}\\}_{n\\in\\mathbb{N}}$ and $\\psi:=\\{\\psi_{n\n\\}_{n\\in\\mathbb{N}}$ are as in (\\ref{3.2}).\n\\end{example}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpin liquids are commonly understood as systems where quantum fluctuations prevent magnetic ordering down to the lowest temperatures giving rise to a long-range entangled quantum phase \\cite{Savary2016,knolle2018field}. An archetypical example of a spin liquid is Kitaev's honeycomb lattice model (HLM) \\cite{Kitaev2006}. \nIt can support Abelian anyonic phases as well as phases with non-Abelian anyons. \nIts physics is very rich and at the same time the model is analytically and numerically tractable. This tractability allowed a rigorous investigation of its dynamical correlations~\\cite{Knolle2014,Knolle2015a} and thermodynamic properties~\\cite{Nasu2014,Nasu2015b,Nasu2015a}.\nThe HLM is experimentally relevant too ~\\cite{Jackeli2009}. It has been recently related to various materials such as $\\text{A}_2\\text{IrO}_3$ ($\\text{A}=\\text{Na,Li}$)~\\cite{Singh2010} or $\\alpha-\\text{RuCl}_3$~\\cite{Plumb2014,banerjee2016proximate,winter2017models,hermanns2017physics}, where the heat capacity and thermal conductivities have been measured~\\cite{Hentrich2018unusual,Kasahara2018unusual}. Excitingly, a recent experimental observation~\\cite{kasahara2018majorana} of a quantised thermal Hall effect in the magnetic field induced~\\cite{Sears2017phase,Baek2017evidence,banerjee2018excitations} spin liquid state of $\\alpha-\\text{RuCl}_3$ opens the possibility for an experimental investigation of exotic phenomena related to anyonic excitations in a magnetic material. \n\nIn the non-Abelian coupling regime the spectrum of the HLM is given in terms of fermions and vortices that bind Majorana zero modes~\\cite{Kitaev2006}. While these bound Majoranas have zero energy when the vortices are well separated, they energetically split when they are brought into close proximity due to vortex-vortex couplings. Interestingly, these couplings alternate in sign depending on the distance between the vortices~\\cite{Lahtinen:2011gm}. It has been demonstrated that ensembles of two-dimensional lattices of Majorana zero modes paired with random sign couplings give rise to a so called `thermal metal phase'~\\cite{Altland1997,Chalker2001,Mildenberger2007,Laumann2012}. This phase, that we will refer to as `metallic', is characterised by electrically neutral energy currents traversing its bulk~\\cite{Beenakker2013}. A priori, the HLM is a translationally invariant spin lattice model. Nevertheless, due to its extensive number of conserved plaquette operators, so called fluxes, it is equivalent to a sum of inhomogeneous Hamiltonians of Majorana modes that describe different sectors of the model ~\\cite{Kitaev2006}. When the temperature is higher than the typical flux gap, it is natural to expect a significant contribution from Majorana Hamiltonians with a random sign of couplings from the random flux configurations. Here we address the natural question: can the HLM support metallic behaviour with extended states purely by increasing its temperature? \nWe can answer this question affirmatively via our new Monte Carlo method, which is not only able to reproduce the thermodynamic properties of the model~\\cite{Nasu2015a,Nasu2015b,Metavitsiadis2017}, but which also recovers the more subtle topological properties by an explicit projection onto the physical parity sectors.  \n\nInsulating and metallic behaviours are manifestations of macroscopic quantum effects. Typically, during the transition from an insulator to a metal a redistribution of the energy eigenvalues and delocalisation of eigenstates occurs via a change of external parameters. Here we demonstrate that increasing the temperature of the translationally invariant HLM leads to a behaviour similar to a thermal metal, even if it has an insulating gap at zero temperatures. The metallic behaviour arises from Majorana fermions moving in a disordered medium of fluxes effectively created by the finite temperature.\n\nTo discriminate the characteristics of the thermal metal emerging in the non-Abelian phase we compare it to the thermal behaviour of the Abelian Toric Code phase. We begin by studying the average fermionic energy gap. The average fermion gap in the non-Abelian phase vanishes at high temperature, whereas it does not in the Abelian phase. Probing deeper, we further show that in the non-Abelian phase the distribution of fermionic energy levels diverges logarithmically at low energy and that the wave functions of these low energy states acquire a fractal character. \nThese behaviours of the non-Abelian phase are similar to those of systems at the critical point of an Anderson transition between a metal and an insulator~\\cite{Evers2008}. \nOur study shows that energy transport mediated by non-localised states, as in metallic systems, can be thermally activated in a spin liquid.\n\n\\section{Kitaev's honeycomb lattice model at finite temperatures} \n\\label{sec:HLM}\n\nKitaev's honeycomb lattice model comprises spin-$1\/2$ particles arranged on the vertices of a honeycomb lattice. The spins interact via the Hamiltonian\n\\begin{equation}\n{H} = - \\sum_{(i,j):\\alpha} J_\\alpha \\sigma_\\alpha^i \\sigma_\\alpha^j +K \\sum_{(i,j,k)} \\sigma_{\\alpha}^i \\sigma_{\\beta}^j \\sigma_{\\gamma}^k ,\n\\label{eqn:Hamiltonian}\n\\end{equation}\nwhere $\\alpha, \\beta,\\gamma=x,y,z$ and $\\sigma^\\alpha$ are the Pauli matrices. The two-spin interaction terms are anisotropic bond-dependent Ising couplings. The three-spin terms break time-reversal invariance, with the triplets $(i,j,k)$ being consecutive indices around a plaquette.\n\nThe Hamiltonian (\\ref{eqn:Hamiltonian}) has an extensive number of conserved quantities, which makes it exactly solvable. It is solved by rewriting it in terms of a $\\mathbb{Z}_2$ gauge field associated with the links of the lattice, described by operators $\\hat u_{ij}$, and Majorana fermions living on the vertices \\cite{Kitaev2006}. The gauge field operators commute with the Hamiltonian, so we can consider fixed static choices of gauge $\\gras{u} = \\{{u}_{ij} =\\pm 1\\}$. In a given sector $\\gras{u}$, the Hamiltonian is\n\\begin{equation}\nH_{\\gras{u}}= \\frac{i}{4} \\, \\sum_{j,k=1}^{2 L^2} \\, A_{jk} \\, c_j c_k,\n\\label{eqn:ferm-Hamiltonian}\n\\end{equation} \nwhere $L$ is the linear size of the system (the number of hexagonal plaquettes), $c_j$ is a Majorana fermion living on site $j$. The couplings are given by $A_{jk}= 2 J_{jk} {u}_{jk}$ if $(j,k)$ is a nearest-neighbour pair, $A_{j,k}=2 K \\sum_l {u}_{jl} \\, {u}_{kl}$ if $(j,k)$ is a next-nearest-neighbour pair, otherwise $A_{j,k}=0$. As a quadratic fermionic Hamiltonian, $H_{\\gras{u}}$ can be efficiently diagonalised. \n(Viewed in this way, $H_{\\gras{u}}$ describes a spinless p-wave superconductor.)\nThe Wilson loops of the gauge field for configuration $\\gras{u}$, obtained by multiplying the values of ${u}_{ij}$ around a hexagonal lattice plaquette, corresponds to a quenched configuration of vortex defects in the superconductor.\nHowever, it is important to remember that in the original spin language these vortices will arise as quasiparticle excitations of eqn.~\\eqref{eqn:Hamiltonian}. \nIn the strongly dimerised limit where $J_x,J_y\\ll J_z$, a perturbative expansion shows that the model behaves as the toric code (Abelian phase) \\cite{Kitaev2006}, whereas for $J_x\\approx J_y \\approx J_z$ and $K \\ne 0$, it supports Ising anyons (non-Abelian phase).\n\nTo contrast different thermal behaviours, we concentrate on the two distinct gapped phases of the model. First, we consider the Abelian (toric code) phase associated with the strongly dimerised limit, $J_x,J_y\\ll J_z$ and $K=0$~\\cite{Kitaev2006}. Second, we consider the non-Abelian (Ising anyon) phase, where $J_x\\approx J_y \\approx J_z$ and $K \\ne 0$. The particle types of the toric code are the vacuum 1, the anyons $e$ and $m$ and the fermion $\\epsilon= e \\times m$. In the Ising anyon regime, the quasiparticle types are the vacuum $1$, the fermion $\\psi$ and the Ising anyon $\\sigma$. These anyons satisfy the non-Abelian fusion rule $\\sigma\\times\\sigma=1+\\psi$. Namely, if two $\\sigma$ are brought together they combine to either give the vacuum, 1, or a fermion, $\\psi$. In the non-Abelian regime of the HLM, vortices essentially behave as Ising anyons~\\cite{Lahtinen2009}. In our Abelian case, vortices on alternating rows of plaquettes are identified with $e$ or $m$ toric code anyons~\\cite{Kitaev2006}.\n\n\\subsection{Fermion parity projection}\n\nIn order to access all topologically distinct gauge field configurations we must consider the most general cases, where every gauge operator $\\hat u_{ij}$ can be fixed independently to its eigenvalues $u_{ij} =\\pm 1$.\nHowever, if we allow these completely general gauge configurations we must follow the Kitaev fermionisation~\\cite{Kitaev2006}. \nThis approach has the complication that the fermionic Hamiltonian eqn.~\\eqref{eqn:ferm-Hamiltonian} lives in an extended Hilbert space.\nGiven an eigenstate $|\\Psi_u\\rangle$ of some $H_{\\gras{u}}$, it is necessary to apply a projection operator $\\mathcal{P}$ in order to get an eigenstate of $H$, eqn.~\\eqref{eqn:Hamiltonian}. This projector is made of two parts: $\\mathcal{P}=\\mathcal{S} \\mathcal{P}_0$~\\cite{Pedrocchi2011}. The part $\\mathcal{S}$ acts purely on the gauge degrees of freedom $u_{ij}$, in such a way that it does not mix topologically distinct gauge configurations. Whereas $\\mathcal{P}_0$ acts on the fermions, in each sector $\\gras{u}$ it is given by\n\\begin{equation*}\n\\mathcal{P}_0 = \\frac{1}{2} \\left( 1 + \\text{det}(Q) \\bigg( \\prod_{(i, j)} u_{ij} \\bigg) \\mathcal{F} \\right) \\, .\n\\label{eqn:ferm-proj-op}\n\\end{equation*}\nThe product over pairs $(i, j)$ runs over every lattice link and must be consistent with the chosen orientations of the links~\\cite{Pedrocchi2011}. The orthogonal matrix $Q$ is the matrix that brings $H_{\\gras{u}}$ into Majorana normal form (see section 3 of Kitaev 2006~\\cite{Kitaev2006}). The operator $\\mathcal{F}$ counts the parity of the occupancies of the diagonal fermions $f_p$, $f^\\dagger_p$, given by\n\\begin{equation*}\n\\mathcal{F} = \\prod_p (1 - 2 f^\\dagger_p f_p)\n\\end{equation*} \nThe eigenvalues of $\\mathcal{F}$ are +1 (when an even number of modes are excited) and -1 (when the number is odd). The operator $\\mathcal{P}_0$ therefore projects into a particular parity of the diagonal fermions, fixed by the configuration $\\gras{u}$.\n\nNote that previous studies of the HLM at non-zero temperature employed alternative fermionisation approaches~\\cite{Nasu2015a,Nasu2015b,Metavitsiadis2017}. These have the advantage that the states obtained in that way do not need to be projected. However, these fermionisations correspond to only considering certain restricted sets of gauge configurations. Under these restrictions it is not possible to properly probe the topological degeneracy of the ground state.\nBy implementing the projection, we can study the finite temperature behaviour of the fermion parity and probe the contribution of the topological ground state degeneracy to the low temperature thermal entropy, as seen in the following.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{thermo_and_ferm_parity.pdf}\n\\caption{\nThermodynamic quantities in the (left) Abelian and (right) non-Abelian regime. \nIn the top panel we plot the entropy $S$ (red circles) and the specific heat $C\/N$ (purple rhombus).\nThe distinctive two peak structure of $C\/N$ is clearly visible, with the lower temperature peak associated with the activation of the vortex excitations and the higher temperature peak with the activation of the fermionic excitations. At low temperatures $S$ identifies the different topological degeneracies of the ground states, which is fourfold in the Abelian case ($S =2\\ln2$) and threefold in the non-Abelian case ($S=\\ln 3$). At higher temperatures the entropic signature of topological order is lost due to the proliferation of vortices. The lower panel plots vortex density $\\rho_V$ (green rhombus), internal energy per particle $U\/N$ (orange squares) and average fermion parity $\\langle \\mathcal{F} \\rangle$ (blue circles). The vortex density $\\rho_V$ is zero at low temperature, and increases to $1\/2$ when the first peak of $C\/N$ occurs, while the internal energy becomes non-zero at the second peak. The average fermion parity $\\langle \\mathcal{F} \\rangle$ shows distinctly different behaviours in the Abelian and non-Abelian regimes for intermediate temperatures between the two peaks.\nData is shown for $L=10$ system size, with couplings $J_x = J_z = 0.25$, $J_z=1$, $K=0$ in the Abelian case and $J_x = J_y = J_z=1$, $K=0.1$ in the non-Abelian case.\n}\n\\label{fig:thermo_and_ferm_parity}\n\\end{figure*}\n\n\\subsection{Finite temperature expectation values}\n\\label{sec:HLM:expt-vals}\n\nThe finite temperature physics of Hamiltonian (\\ref{eqn:Hamiltonian}) can be numerically studied by randomly sampling different gauge configurations $\\gras{u}$, in each case diagonalising the fermionic Hamiltonian $H_{\\gras{u}}$ fully. This is possible since the partition function at temperature $T = 1\/\\beta$ can be written\n\\begin{gather*}\n\\mathcal{Z}(\\beta) = \\textrm{tr}(e^{-\\beta H}) = \\sum_\\gras{u} {\\mathcal{Z}}'(\\beta;\\gras{u})  \\, ,\n\\label{eqn:full-part-func}\n\\end{gather*}\nwhere ${\\mathcal{Z}}'(\\beta;\\gras{u}) \\equiv \\textrm{tr}'( \\mathcal{P} \\,  e^{-\\beta H_\\gras{u}} )$ with $\\textrm{tr}'$ is over the fermionic degrees of freedom only. The projected fermionic partition functions ${\\mathcal{Z}}'(\\beta;\\gras{u})$ can be calculated exactly~\\cite{chrisUpcoming}. The weighting of each $\\gras{u}$ in the thermal state is thus given by $p_\\beta(\\gras{u}) = {{\\mathcal{Z}}'(\\beta;\\gras{u})}\/{\\mathcal{Z}(\\beta)}$, and finite temperature expectation values for observable $\\mathcal{O}$ can be estimated by approximating\n\\begin{align*}\n\\langle \\mathcal{O} (\\beta) \\rangle &= \\sum_{\\gras{u}} p_\\beta(\\gras{u}) \\, \\frac{\\textrm{tr}'( \\mathcal{P} \\, e^{-\\beta H_\\gras{u}} \\mathcal{O}_\\gras{u} )}{\\overline{\\mathcal{Z}}(\\beta;\\gras{u})} \\nonumber \\\\ \n&\\equiv \\sum_{\\gras{u}} p_\\beta(\\gras{u}) \\, \\langle \\mathcal{O}_\\gras{u} (\\beta) \\rangle' \\, ,\n\\end{align*}\nusing Monte Carlo methods.\nFor all quantities of interest the projected fermionic expectation values $\\langle \\mathcal{O}_\\gras{u} (\\beta) \\rangle'$ can again be computed exactly~\\cite{chrisUpcoming}. The heat capacity of the full model is given by\n\\begin{equation*}\nC = \\beta^2 \\, \\text{Var}_\\gras{u} \\Big( \\langle H_\\gras{u} \\rangle' \\Big) + \\text{Mean}_\\gras{u}\\Big( \\beta^2 \\big\\langle \\, (H_\\gras{u} - \\langle H_\\gras{u} \\rangle')^2 \\, \\big\\rangle' \\Big) \\, ,\n\\end{equation*}\nwhere $\\text{Var}_\\gras{u}$ and $\\text{Mean}_\\gras{u}$ are calculated with respect to the probability distribution $p_\\beta(\\gras{u})$. Numerically these variances, $\\text{Var}_\\gras{u}$, are estimated along with their uncertainties using bootstrap resampling~\\cite{Efron1979}. At infinite temperature the state consists of $N$ decoupled spin-1\/2 particles, which has thermal entropy $S = N \\ln(2)$. The thermal entropy can then be obtained by computing\n\\begin{equation*}\nS(T) = N \\ln(2) - \\int_T^\\infty \\text{d}T' \\, \\frac{C(T')}{T'} \\, .\n\\end{equation*}\n\n\\subsection{Numerical results} \n\nTo analyse the thermal behaviour of the HLM we calculate the specific heat, \\emph{i.e.} the heat capacity per spin, $C \/ N$, thermal entropy $S$, vortex density, $\\rho_V$, internal energy per spin, $U\/N$ ($U = \\langle H \\rangle$) and average fermion parity $\\langle {\\cal F}\\rangle$ of the model. These are plotted as functions of temperature for both the toric code and Ising coupling regimes in Fig.~\\ref{fig:thermo_and_ferm_parity}. \n\nFirst, we see that the splitting of the spins into gauge and fermionic degrees of freedom is not only a mathematical trick for solving Kitaev's honeycomb lattice model. The model exhibits a physical fractionalisation of its spins into gauge vortices and fermions. We observe the distinctive two peak structure of the specific heat, indicating the fractionalisation of the spins, seen in previous studies~\\cite{Nasu2015b}. For both the toric code and Ising regime, the low $T$ peak is associated with a change of the vortex density, in which it rises from its ground state value $\\rho_V = 0$ towards a completely disordered vortex configuration $\\rho_V = 1\/2$. The higher temperature peak is linked to a large change in internal energy $U$, corresponding to an unfreezing of the fermionic modes. These peaks are well separated by several orders of magnitude of $T$ in the toric code limit, whereas they are closer and blur together to a certain extent in the Ising regime. Nevertheless, the specific heat in both regimes exhibit similar qualitative behaviour. \n\nSecond, in Fig.~\\ref{fig:thermo_and_ferm_parity} (upper panel) we plot the behaviour of the thermal entropy, $S$, obtained by integrating the heat capacity down from infinite temperature. At small temperatures the thermal entropy is able to identify the topological degeneracy of the ground state. Specifically, the Boltzmann formula gives $S = \\ln(\\Omega) \\,$, for $\\Omega$ the number of microstates (with the Boltzmann constant set equal to one). As $T \\rightarrow 0$, $\\Omega$ gives the degeneracy of the ground state whose correct value can only be obtained by employing the projection onto physical states. Hence, the thermal entropy is in excellent agreement with the topological degeneracies expected for the toric code anyons $S=2\\ln2$ and for the Ising anyons $S=\\ln 3$. However, for larger temperatures when the vortices start to proliferate the entropic signatures of topological order are washed out~\\cite{Iblisdir2009scaling,iblisdir2010thermal}. \n\nIn contrast to the thermal entropy, the fermionic parity $\\langle \\mathcal{F} \\rangle$ shows interesting differences between the two regimes at higher temperatures and in fact hints at even more complex differences between the two. We begin by describing the behaviour we would expect $\\langle \\mathcal{F} \\rangle$ to have based purely on the anyon models.\nThe $T \\rightarrow 0$ state always has no vortices. In the toric code case this already rules out the possibility of odd fermion parity states because of the fusion rules. States containing two fermionic quasiparticles would be allowed, however these have very high energy. Whereas in the Ising regime odd fermion parity states that do not contain any vortices are allowed by the fusion rules (they correspond to states with anyonic flux around both handles of the torus). However, since they contain a fermion they have higher energy, so these states do not occur at low temperature. Hence at low $T$ we expect to find $\\langle \\mathcal{F} \\rangle=1$ in both regimes. In contrast, at high temperatures we would expect different values of $\\langle \\mathcal{F} \\rangle$ in the two cases. As $T \\rightarrow \\infty$ the vortices becomes completely disordered, $\\rho_V = 1\/2$. In the toric code case half of these configurations will contain odd numbers of $e$ (and $m$) anyons and half even numbers. From the fusion rules these correspond to odd and even fermion parities respectively, and averaging over them gives $\\langle \\mathcal{F} \\rangle = 0$. Whereas in the Ising theory the only way to generate a state with total odd fermion parity is to add non-trivial anyon flux around both torus handles. This is one of four flux configurations and so we would instead expect to find $\\langle \\mathcal{F} \\rangle = \\ilfrac{3}{4}(+1) + \\ilfrac{1}{4}(-1) = 0.5$. Fig.~\\ref{fig:thermo_and_ferm_parity} (lower panel) plots the fermion parity as a function of temperature. We see that at low temperature both regimes agree with the predictions of the anyon models. At high temperature, however, only the toric code case agrees with our predictions. The fermion parity also becomes zero in the Ising case at high temperatures. This is a first indication that the thermal proliferation of Ising vortices has driven a transition out of the topologically ordered phase (described by the anyon model) to a new kind of phase. We will demonstrate in the following that this phase is a thermal metal.\n\n\n\\section{Thermal metal behaviour} \n\nOf central interest to us is the behaviour of a two-dimensional Majorana lattice with random sign couplings. Such random configurations are renormalisation fixed points. Indeed, if block renormalisation is employed the resulting model is again a Majorana lattice with random sign of couplings~\\cite{Evers2008}. These signs can be interpreted as a $\\mathbb{Z}_2$ gauge field that gives rise to Majorana binding vortices. It has been shown that a Majorana lattice with $\\mathbb{Z}_2$ field couplings can be faithfully modelled by the same system with homogeneous couplings, superposed by an independent lattice of Majoranas at the position of the vortices created by the random signs~\\cite{Lahtinen2012}. As the vortices are farther apart than the original lattice spacing they interact more weakly. So this vortex lattice contributes states with energies below the band gap of the original lattice model. Moreover, the sign of the vortex couplings is random due to the random position of the vortices. This picture holds recursively providing subgap states all the way to zero energy. As a result we expect the randomisation of the coupling signs to give rise to a gapless phase. Moreover, as randomness persists with renormalisation, this critical phase should be described by an Anderson transition with wave functions exhibiting multifractality~\\cite{Evers2008}. The divergence of the number of fermionic states with close-to zero energy and the critical multifractal behaviour of the wave functions are the defining characteristic of a thermal metal phase.\n\nTwo-dimensional Majorana lattice with random sign couplings have been numerically proven to support  thermal metal phase~\\cite{Laumann2012,Lahtinen2014}. Thermal metal behaviour has also been seen in topological superconductors at temperatures above the Kosterlitz-Thouless transition point where vortices that bind Majorana zero-modes proliferate ~\\cite{Bauer2013}. However, in both of these examples the vortices are introduced by hand as they are not part of the thermal states of the system. Hence, ensembles of different realisations of the system have to be involved.\n\nIn the HLM case, in contrast, the vortices naturally occur as finite temperature excitations, as shown in Fig.~\\ref{fig:thermo_and_ferm_parity}. To intuitively understand why such a thermal metal phase can emerge in this situation, consider the system at $T=0$ in the presence of various patterns of vortices. When vortex quasiparticles of the non-Abelian regime are brought into proximity, they couple with a sign that depends on their relative position~\\cite{Lahtinen2012}. As temperature increases, in the HLM the average vortex density increases to the point that the vortex-vortex couplings become important. Moreover, the random distances between vortices mean the pairing energies between the vortex bound states are positive or negative at random. Thus we also expect a thermal metal phase to arise in the HLM at finite temperature. \n\n\\subsection{Fermionic gap}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{ferm_gaps.pdf}\n\\caption{\nAverage fermion gap for the (left) Abelian and (right) non-Abelian cases, with the specific heat profile superimposed.\nThe fermions in the Abelian limit are gapped at all temperatures, whereas in the non-Abelian case they become gapless when vortices start proliferating.\nThe large system size limits of the fermion gap for $T\\rightarrow0$ and $T\\rightarrow\\infty$ (obtained in appendix~\\ref{app:finite-size-gap}) are indicated with markers on the axes. Data is shown for $L=10$ system size with couplings set to $J_x = J_z = 0.25$, $J_z=1$, $K=0$ in the Abelian case and $J_x = J_y = J_z=1$, $K=0.1$ in the non-Abelian case.\n}\n\\label{fig:fermion-gap}\n\\end{figure}\n\nThermal metals conduct heat through their fermionic degrees of freedom. This metallic conduction property requires a vanishing fermion gap. The behaviours of the average fermion gap, $\\langle \\Delta \\rangle$, in the Abelian and non-Abelian regimes are shown in Fig.~\\ref{fig:fermion-gap}. In these plots the fermionic gap is averaged over the vortex sectors that are thermally excited for a given temperature $T$. We see that the Abelian regime remains gapped at all temperatures. \nWhereas, in the non-Abelian case the average fermionic gap vanishes above the temperature at which vortices begin to proliferate. In appendix~\\ref{app:finite-size-gap} we address the finite size scaling of the fermion gap in the zero temperature and infinite temperature limit. We give evidence that in the non-Abelian phase the infinite temperature limit $\\langle \\Delta \\rangle$ tends to zero as $L \\rightarrow \\infty$.\n\n\n\nA vanishing fermion gap is a necessary condition for a thermal metal. However, since the non-Abelian vortices bind Majorana modes at zero-energy the vanishing of the gap is not sufficient. It cannot distinguish between a metallic phase (which has fermionic energy levels occuring continuously down to zero energy) and an insulating phase with additional zero energy localised states associated with the vortices (isolated subgap fermionic states at zero energy). \nFurther, a thermal metal phase is expected to display its own distinct behaviours in both the distribution of energy levels at low energy and the spatial properties of the wave functions~\\cite{Altland1997,Mildenberger2007}. In the following we probe both of these properties in detail.\n\n\\subsection{Density of states}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{NV_dos.pdf}\n\\caption[DOS]{Distribution of fermion energies $\\rho(E)$ for low $E$ in the non-Abelian case. Computed for system sizes $L=10$ and 20 at fixed temperature $T \\approx 0.4$.\nAt low energies $\\rho(E)$ diverges logarithmically with decreasing $E$. This is indicated by the black dashed line, which is a fit to the $L=20$ data.\nFor the smallest values of $E$ oscillations are visible. (Inset) The oscillations are collapsed onto curves of the form Eqn.~\\eqref{eqn:rmt-osc}.\nThe system couplings are set to $J_x = J_y = J_z=1$, $K=0.1$.}\n\\label{fig:DOS}\n\\end{figure}\n\nThe fermionic distribution of energies (DOE) is given by $\\rho(E) = \\langle \\, \\sum_n \\delta(E - \\varepsilon_n) \\, \\rangle$, where the $\\varepsilon_n$ are the fermionic energies in a single vortex sector and the average is taken over the vortex sectors visited at temperature $T$~\\endnote{Viewing the system as an ensemble of disordered fermionic Hamiltonians with quenched vortex disorder this would be the ensemble averaged density of states. This is the situation in which thermal metals are usually discussed. However, since the vortex disorder we consider arises from the thermal state of the Kitaev spin Hamiltonian, we avoid that term so as not to confuse $\\rho(E)$ with the density of states of that underlying Hamiltonian.}. In a thermal metal, $\\rho(E)$ diverges logarithmically with decreasing energy at low energies~\\cite{Mildenberger2007,Bocquet:1999bb}. To clearly see this feature of $\\rho(E)$ we find it necessary to go to larger systems than are accessible when using the algorithm that properly samples from the thermal state, e.g. our $L=10$ data. For system size $L = 20$ data is obtained by randomly sampling over vortex sectors, matching the average vortex density to the value obtained from smaller system sizes (see Appendix~\\ref{app:sim:vortices}). The logarithmic divergence is extracted by a fit to $L=20$, and is in agreement with the behaviour of $L=10$, as shown in Fig.~\\ref{fig:DOS}. For the very lowest energies, $\\rho(E)$ additionally exhibits characteristic oscillations predicted by random matrix theory~\\cite{Bocquet:1999bb,wigner1993class}. In this limit the DOE is given by\n\\begin{equation}\n\\rho(E) = \\alpha \\left( 1 + \\frac{\\sin(2\\pi E \\alpha L^2)}{2\\pi E \\alpha L^2} \\right) \\, .\n\\label{eqn:rmt-osc}\n\\end{equation}\nThe parameter $\\alpha$ can be obtained by numerically solving the consistency  equation $\\rho(\\ilfrac{1}{\\alpha L^2}) = \\alpha$ and $\\alpha$ is expected to diverge logarithmically with $L$~\\cite{Mildenberger2007}. The first few oscillation periods of the random matrix theory are shown in Fig.~\\ref{fig:DOS} (Inset). Hence, the energy distribution is in agreement with the thermal metal phase.\n\n\\subsection{Weak multifractality of wavefunctions}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{NV_IPR.pdf}\n\\caption[IPR]{Weak multifractality of the fermionic wave functions in the non-Abelian case at high temperatures. (Left) Inverse participation ratios $I_q$ of the lowest-lying fermionic eigenmode as a function of system size $L$ (on a log-scale). The $I_q$ are averaged over the thermal distribution of vortices at $T \\approx 0.4$. (Right) Exponents $\\tau(q)$ (defined in Eq.(\\ref{eq:multifractality})) as a function of $q$ demonstrating multifractal critical behaviour, in-between the metallic $\\tau(q)=2(q-1)$ and the insulating $\\tau(q)=0$ behaviours.}\n\\label{fig:IPR}\n\\end{figure}\n\nBeyond the energy eigenvalues, the wave functions of thermal metals exhibit \\emph{weak multifractality}, which means they are spatially extended in a particular way~\\cite{Falko:2007bl}. This behaviour is shared with systems that exhibit Anderson localisation, where weak multifractality occurs at the critical point between insulating and conducting phases~\\cite{Evers2008}.\nWhether states are localised or extended can be studied via the inverse participation ratios of the fermionic eigenfunctions $\\psi_n(\\mathbf{r})$~\\cite{Evers2008}, given by\n\\begin{equation}\\label{eq:multifractality}\nI_q = \\int \\textrm{d}^2 r \\, |\\psi_n(\\mathbf{r})|^{2q} \\sim \\frac{1}{L^{\\tau(q)}} .\n\\end{equation}\nThe exponent is often written $\\tau(q) = (q-1) D_q$ where $D_q$ is called the \\emph{fractal dimension}.\nFor pure metallic states $D_q$ is equal to the spatial dimension, $D_q = 2$, and for entirely localised states $D_q = 0$. \nHowever, weak multifractality implies $D_q$ becomes dependent on $q$, and behaves as $D_q = 2 - \\gamma q$~\\cite{Falko:2007bl}. Fig.~\\ref{fig:IPR} shows the exponent $\\tau(q)$ plotted as a function of $q$ for the lowest-lying fermionic state, averaged over vortex sectors at temperature $T \\approx 0.4$. Fitting to the data we obtain the fractal dimension $D_q = 1.7(8) - 0.12(3) q$. Hence, the eigenstates also exhibit the critical Anderson behaviour expected from a thermal metal phase. \n\n\\section{Conclusions} \n\nWe have analysed the thermal behaviour of the Abelian and non-Abelian quantum spin liquid phases of the Kitaev honeycomb model. Using Monte Carlo simulations, we have studied the thermal distributions of vortices, finding thermodynamic behaviours that agree with past studies~\\cite{Nasu2015a,Nasu2015b}. Importantly, we find that the HLM at $T>0$ enters a thermal metal phase, with logarithmic divergent distribution of energies and with wave functions that exhibit multifractality. \n\nCompared to the study of thermal metals in disordered integer quantum Hall effect~\\cite{Laumann2012} that required fine tuning to zero chemical potential, the metallic phase obtained here is stable against changes in the coupling constants that do not move the system out of its non-Abelian phase. Compared to the p-wave superconductor~\\cite{Bauer2013} the metallic phase appears without having to introduce vortices by hand; once in the non-Abelian phase, the only necessary knob is the temperature. Also, no ensemble averaging is necessary to get different patterns of vortices; throughout, we have only needed to deal  with the Gibbs state of a single translationally invariant Hamiltonian. We believe this distinction is important and could eventually contribute to the experimental observation of a thermal metallic phase, \\emph{e.g.} in magnetic field-tuned $\\alpha$-RuCl$_3$ above the temperature window which has recently shown signatures of a quantised thermal Hall effect~\\cite{kasahara2018majorana}.\n\nOur work paves the way for a number of future investigations: It will be important to analyse how the signatures of the thermal metal become manifest in experimental observables, e.g. the temperature or system size scaling of the thermal conductivity or the frequency dependence of the low energy Raman response \\cite{Sandilands2015scattering,Brent2016}. A challenging question will be to investigate whether the metallic phase survives the addition of integrability breaking terms that render static fluxes dynamical or the coupling to acoustic phonons inevitably present in real materials. Finally, beyond the quantum spin liquid context we expect that temperature induced localisation-delocalisation transitions could also emerge in certain lattice gauge theories \\cite{Smith2017,Prosko2017,smith2018dynamical} recently discussed in the context of non-ergodic phases. \n\n\\begin{acknowledgments}\n\nWe thank Chris Turner for helpful discussions while carrying out this work. This work was undertaken on ARC2, part of the High Performance Computing facilities at the University of Leeds, UK. SI acknowledges support from MINECO (grant MTM2014-54240-P), Comunidad de Madrid (grant QUITEMAD+CM, ref. S2013\/ICE-2801), and Severo Ochoa project SEV-2015-556. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 648913). CS and JKP were supported by the EPSRC grant EP\/R020612\/1.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDirect simulation Monte Carlo method (DSMC)\\cite{Bird94} is a\nstandard method to solve the Boltzmann equation numerically. In this method\none divides space into cells of volume $V_{k}$ $(k=1,2,3,...)$ and takes a\nlarge number ($N$) of simulated particles ($10^{3}-10^{6}$) to represent\nreal gas molecules. The time evolution of the gas for a short time period $%\n\\Delta t$ is calculated in two steps. In the first step some pairs of\nparticles in the same cell are chosen randomly and are allowed to collide\nwithout changing their positions. A collision is allowed with a probability\nproportional to $u\\Sigma $ where $u$ is the relative velocity and $\\Sigma $\nis the total cross section. In the second step all particles are propagated\nwithout collisions for a time $\\Delta t$.\n\nThe method is invented by Bird and Bird introduced the method based on\nphysical arguments. A seminal paper of Bird\\cite{Bird70} gives somewhat\nheuristic arguments to justify its use to solve the Boltzmann equation. One\nvariant of the method was derived by Nanbu\\cite{Nanbu80} starting from the\nBoltzmann equation. Also it appears that essentially the same stochastic\nalgorithms for a homogenous gas were invented independently by people\ninterested in using them as a pedagogical tool to demonstrate evolution of a\ngas toward Maxwell-Boltzmann(MB) distribution.\\cite{Novak70} \\cite{Eger82}\n\\cite{Bonomo84}. In order to represent time evolution of the real gas such\nmethods should converge to the true solution of the Boltzmann equation in\nthe limit of $N\\rightarrow \\infty ,$ $V_{k}\\rightarrow 0,$ $\\Delta\nt\\rightarrow 0.$ Convergence proofs were given by Babovsky\\cite{Babovsky1}\nand Babovsky and Illner\\cite{Babovsky2} for Nanbu's method and by Wagner\\cite\n{Wagner92} for Bird's method.\n\nThe cited convergence proofs are very formal and they appear to be written\nfor mathematicians. In this paper we give a simple derivation of Birds no\ntime counter algorithm. We also show that, in DSMC, appropriately normalized\nsingle particle probability distribution satisfies Boltzmann equation for\nsimple gases and Wang Chang-Uhlenbeck equation for molecular gases and their\nmixtures. The language of this development is familiar to the physicist from\nthe well known BBGKY hierarchy.\n\nIn the next section we develop a general formalism for direct simulation. In\norder to demonstrate usefulness of the formalism we apply it to some simple\nmoney games. In the third section we apply the formalism to homogenous gases\nand show that, if appropriate collision kernels are chosen, the one particle\nprobability distribution obeys the Boltzmann equation for simple gases and\nthe Wang Chang-Uhlenbeck equation for molecular gases and their mixtures. In\nthe fourth section we derive DSMC\\ algorithm for inhomogeneous gases.\nFinally in the last section we give a summary and discussion.\n\n\\section{Direct simulation as a Markov process}\n\n\\subsection{The Master Equation}\n\nAssume that we have an assembly of things we call 'particles'. Particles can\nbe real particles in a gas or humans or anything you can imagine. There are $%\nN$ particles in the assembly where $N$ is a very large number. Each member\nof the assembly can be in any one of the 'states' where states are labeled\nby the parameter $\\mu $. For a real gas $\\mu $ can be velocity vectors and\nfor an assembly of people $\\mu $ can be the money in their pocket on bank\naccount. The $\\mu $ can be discrete or continuous and it can stand for a\ncollection of indices that can be both continuous and discrete. For the rest\nof this section we will treat $\\mu $ as a continuous index. Integration over\n$\\mu $ is actually integration over the continuous indices and summation\nover the discrete indices that $\\mu $ stands for.\n\nWe play a stochastic game with this assembly. We randomly pick pairs of\nparticles and force them to 'collide'. A collision is an event that the\nparticles change their states with a prescribed probability. Suppose we\npicked particles with states $\\mu _{A}$ and $\\mu _{B}.$ The probability that\nthey will end up with state labels $\\mu _{C}$ and $\\mu _{D}$ in the volume $%\nd\\mu _{C}d\\mu _{D}$ is $T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu _{D})d\\mu _{C}d\\mu\n_{D}$ where $T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu _{D})$ is the collision\nkernel. Collision kernel is assumed to be symmetric\n\\begin{eqnarray}\nT(\\mu _{A},\\mu _{B};\\mu _{C},\\mu _{D}) &=&T(\\mu _{C},\\mu _{D};\\mu _{A},\\mu\n_{B}),  \\label{a10} \\\\\nT(\\mu _{A},\\mu _{B};\\mu _{C},\\mu _{D}) &=&T(\\mu _{B},\\mu _{A};\\mu _{D},\\mu\n_{C}).  \\label{a20}\n\\end{eqnarray}\nAlso the probabilities are normalized\n\\begin{equation}\n\\int T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu _{D})\\,d\\mu _{C}\\,d\\mu _{D}=\\int T(\\mu\n_{A},\\mu _{B};\\mu _{C},\\mu _{D})\\,d\\mu _{A}\\,d\\mu _{B}=1.  \\label{a30}\n\\end{equation}\n\nWe define N-particle probability distribution $f^{(N)}(\\mu _{1},\\mu\n_{2},...,\\mu _{N};n)$ such that $f^{(N)}(\\mu _{1},\\mu _{2},...,\\mu\n_{N};n)d\\mu _{1}d\\mu _{2},...,d\\mu _{N}$ is the probability of finding the\nparticles $1,2,...,N$ in the $d\\mu _{1}d\\mu _{2},...,d\\mu _{N}$ phase space\nvolume after the $n^{th}$ collision. Since the particles are identical the $%\nf^{(N)}(\\mu _{1},\\mu _{2},...,\\mu _{N};n)$ is assumed to be completely\nsymmetric\n\\begin{equation}\nf^{(N)}(\\mu _{1},...,\\mu _{j},...,\\mu _{i},...,\\mu _{N};n)=f^{(N)}(\\mu\n_{1},...,\\mu _{i},...,\\mu _{j},...,\\mu _{N};n).  \\label{a40}\n\\end{equation}\n\nWe define reduced M-particle distribution as\n\\begin{equation}\nf^{(M)}(\\mu _{1},...,\\mu _{N};n)=\\int f^{(N)}(\\mu _{1},...,\\mu _{N};n)\\,d\\mu\n_{M+1}\\,d\\mu _{M+2},...,d\\mu _{N}.  \\label{a44}\n\\end{equation}\nWe will denote $f^{(M)}(\\mu _{1},....,\\mu _{M};n)$ $(M=1,2,...,N)$ as $%\nf^{(M)}(\\mu ;n)$ shortly. As a convenient notation we also define $%\nf_{ij}^{(M)}(\\mu _{A},\\mu _{B};n)$ as\n\\begin{equation}\nf_{ij}^{(M)}(\\mu _{A},\\mu _{B};n)=f^{(M)}(\\mu _{1},...,\\mu _{i}=\\mu\n_{A},...,\\mu _{j=}\\mu _{B},...,\\mu _{M};n),  \\label{a45}\n\\end{equation}\nwhere $\\mu _{i}$ and $\\mu _{j}$ are replaced with $\\mu _{A}$ and $\\mu _{B}$\nin $f^{(M)}(\\mu _{1},...,\\mu _{M};n)$. Examples are\n\\begin{eqnarray}\nf_{31}^{(N)}(\\mu _{A},\\mu _{B};n) &=&f(\\mu _{B},\\mu _{2},\\mu _{A},\\mu\n_{4},...,\\mu _{N};n)  \\label{a48} \\\\\nf_{24}^{(N)}(\\mu _{A},\\mu _{B};n) &=&f(\\mu _{1},\\mu _{A},\\mu _{3},\\mu\n_{B},\\mu _{5},...,\\mu _{N};n)  \\label{a49}\n\\end{eqnarray}\n\nWe are ready to start now. The equation satisfied by the $f^{(N)}(\\mu ;n)$\nis given by\n\\begin{equation}\nf^{(N)}(\\mu ;n+1)=\\frac{1}{N(N-1)}\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}\\int\nf_{ij}^{(N)}(\\mu _{A},\\mu _{B};n)\\,T(\\mu _{A},\\mu _{B};\\mu _{i},\\mu\n_{j})\\,d\\mu _{A}\\,d\\mu _{B}.  \\label{a66}\n\\end{equation}\nThe meaning of this equation is clear. If the last pair we collided is $i,j$\nmolecules, the probability of having $\\mu _{i},\\mu _{j}$ pairs at the end of\ncollision is the probability of having initial states $\\mu _{A},\\mu _{B}$\n(represented by $f_{ij}^{(N)}(\\mu _{A},\\mu _{B};n)d\\mu _{A}d\\mu _{B}$)\nmultiplied by the probability of ending with $\\mu _{i},\\mu _{j}$\n(represented by $T(\\mu _{A},\\mu _{B};\\mu _{i},\\mu _{j})$). The sum over $i,j$\nand the factor $1\/N(N-1)$ takes care of the fact that all pairs (respecting\norder of the molecules) are possible with the probability $1\/N(N-1).$ The\nstate of the system after $n+1$ collisions depends on the state of system\nafter $n$ collisions and the direct simulation game is a Markov process\nactually. The eq.(\\ref{a66}) is the master equation for this stochastic\nprocess.\n\nIn order to see clearly how this equation is derived let us multiply this\nwith $d\\mu _{1}d\\mu _{2}...d\\mu _{N}$. The left hand side is\n\\begin{equation}\nf^{(N)}(\\mu ;n+1)d\\mu _{1}d\\mu _{2}...d\\mu _{N}  \\label{ad10}\n\\end{equation}\nand it is the probability of the system being in the phase space volume $%\nd\\mu _{1}d\\mu _{2}...d\\mu _{N}$ after the $(n+1)^{th}$ collision. On the\nright side we have\n\\begin{equation}\n\\frac{1}{N(N-1)}\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}\\int f_{ij}^{(N)}(\\mu\n_{A},\\mu _{B};n)T(\\mu _{A},\\mu _{B};\\mu _{i},\\mu _{j})d\\mu _{A}d\\mu _{B}d\\mu\n_{1}d\\mu _{2}...d\\mu _{N}.  \\label{ad20}\n\\end{equation}\n(Here the integration is over $\\mu _{A}$ and $\\mu _{B}$ only) In order to\ninterpret this lets us look at $i=1$ and $j=2$ term. It is the following\nterm\n\\begin{eqnarray}\n&&\\left[ \\frac{1}{N(N-1)}\\right] \\left[ f^{(N)}(\\mu _{A},\\mu _{B},\\mu\n_{3},\\mu _{4,}...,\\mu _{N})d\\mu _{A}d\\mu _{B}d\\mu _{3}d\\mu _{4}...d\\mu\n_{N}\\right]  \\nonumber \\\\\n&&\\times \\left[ T(\\mu _{A},\\mu _{B};\\mu _{1},\\mu _{2})d\\mu _{1}d\\mu\n_{2}\\right]  \\label{ad30}\n\\end{eqnarray}\nintegrated over $\\mu _{A},$ $\\mu _{B}.$ In this form the terms under the\nintegration are product of three probabilities. $1\/N(N-1)$ is the\nprobability of choosing $i=1,j=2\\,$pair. The second parenthesis is the\nprobability of finding the system in $d\\mu _{A}d\\mu _{B}d\\mu _{3}d\\mu\n_{4}...d\\mu _{N}$ phase space volume before the collision. The last\nparenthesis is the probability of taking particles one and two from $d\\mu\n_{A}d\\mu _{B}$ to $d\\mu _{1}d\\mu _{2}$ interval after the collision. When\nintegrated over $\\mu _{A},$ $\\mu _{B}$ this term becomes the probability of\narriving in $d\\mu _{1}d\\mu _{2}...d\\mu _{N}$ phase space volume after $%\n(n+1)^{th}$ collision via a collision between particles one and two. If all\nsuch term are summed over $i$ and $j$ we find the probability of probability\nof arriving in $d\\mu _{1}d\\mu _{2}...d\\mu _{N}$ phase space volume after $%\n(n+1)^{th}$ collision which is the same as eq.(\\ref{ad10}).\n\n\\subsection{Asymptotic Behavior of the Master Equation}\n\nLet us introduce a short notation for state variables:\n\\begin{equation}\n\\begin{array}{ll}\nX=(x_{1},x_{2,}...,x_{N}) & dX=dx_{1}dx_{2}...dx_{N} \\\\\nY=(y_{1},y_{2,}...,y_{N}) & dY=dy_{1}dy_{2}...dy_{N} \\\\\nZ=(z_{1},z_{2,}...,z_{N}) & dZ=dz_{1}dz_{2}...dz_{N}\n\\end{array}\n.  \\label{ab10}\n\\end{equation}\nThen the Master equation can be written in the form\n\\begin{equation}\nf(X;n+1)=\\int P(X,Y)f(Y;n)dY.  \\label{ab20}\n\\end{equation}\nThe $P(X,Y)$ has $N(N-1)$ terms and each one of the terms contains $N-2$\ndelta functions. For example $i=1$, $j=2$ term reads as\n\\begin{equation}\n\\frac{1}{N(N-1)}T(x_{1},x_{2};y_{1},y_{2})\\delta (x_{3}-y_{3})...\\delta\n(x_{N}-y_{N}).  \\label{ab30}\n\\end{equation}\nThe general expression for $P(X,Y)$ is\n\\begin{equation}\nP(X,Y)=\\frac{1}{N(N-1)}\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}\\left(\nT(x_{i},x_{j};y_{i},y_{j})\\prod_{k\\neq i,j}^{N}\\delta (x_{k}-y_{k})\\right)\n\\label{ab40}\n\\end{equation}\nThe $P(X,Y)dX$ is the probability that the system jumps from $Y$ to $dX$\nphase space volume after a collision. As can be seen directly from eq.(\\ref\n{ab40}) it is also symmetric: $P(X,Y)=P(Y,X).$ As a probability density it\nsatisfies the normalization condition\n\\begin{equation}\n\\int P(X,Y)dX=\\int P(X,Y)dY=1.  \\label{ab50}\n\\end{equation}\nWe will need convolution of $P(X,Y)$ shortly. Let us define $W(X,Y)$ as\n\\begin{equation}\nW(X,Y)=\\int P(X,Z)P(Y,Z)dZ  \\label{ab60}\n\\end{equation}\nIt is easily seen that $W(X,Y)$ is symmetric ($W(X,Y)=W(Y,X)$) and it also\nsatisfies a normalization condition\n\\begin{equation}\n\\int W(X,Y)dX=\\int W(X,Y)dY=1.  \\label{ab65}\n\\end{equation}\n\nNow we are ready to discuss asymptotic behavior or the master equation. Let\nus form $\\int f^{2}(X;n+1)dX$ as\n\\begin{eqnarray}\n\\int f^{2}(X;n+1)dX &=&\\int dX\\left( \\int P(X,Y)f(Y;n)dY\\right) \\left( \\int\nP(X,Z)f(Z;n)dZ\\right)  \\label{ab70} \\\\\n&=&\\int W(Y,Z)f(Y;n)f(Z;n)dYdZ  \\label{ab80}\n\\end{eqnarray}\nWe can also write $\\int f^{2}(X;n)dX$ as\n\\begin{equation}\n\\int f^{2}(X;n)dX=\\int W(Y,Z)f^{2}(Y)dYdZ=\\int W(Y,Z)f^{2}(Z)dYdZ\n\\label{ab90}\n\\end{equation}\nwhich follows from eq.(\\ref{ab65}). Using these two relations we can write\nthe following\n\\begin{eqnarray}\n\\int f^{2}(X;n+1)dX-\\int f^{2}(X;n)dX &=&\\int W(Y,Z)f(Y;n)f(Z;n)dYdZ\n\\label{ab100} \\\\\n&&-\\frac{1}{2}\\int W(Y,Z)f^{2}(Y)dYdZ  \\nonumber \\\\\n&&-\\frac{1}{2}\\int W(Y,Z)f^{2}(Z)dYdZ  \\nonumber\n\\end{eqnarray}\nThe right side can be written as\n\\begin{equation}\n\\int f^{2}(X;n+1)dX-\\int f^{2}(X;n)dX=-\\frac{1}{2}\\int W(Y,Z)\\left(\nf(Y;n)-f(Z;n)\\right) ^{2}dYdZ.  \\label{ab110}\n\\end{equation}\nSince $W(Y,Z)$ is always nonnegative the expression on the right is always\nnegative or zero. This means $\\int f^{2}(X;n)dX$ decreases after each\ncollision. The decrease stops when $f(Y;n)-f(Z;n)=0$ for all $Y$ and $Z$ and\nthis means $f(X;n)$ must be a constant. The equilibrium is reached when $%\nf(X;n)$ is microcanonical distribution.\n\nThere is a final point to be discussed here. The above argument proves that\nthe probability density in the direct simulation always converges towards\nmicrocanonical distribution. If the phase space is divided in separate\nregions such that collisions cannot take the system from one region to\nanother then the above argument must be modified. If $Y$ and $Z$ belong to\ndifferent regions then $W(Y,Z)=0$ and $f(Y;n)-f(Z;n)=0$ is not required. But\nif $Y$ and $Z$ belong to the same region then $W(Y,Z)\\neq 0$ and $%\nf(Y;n)-f(Z;n)=0$ is required. This means that $f(X;n)$ must be a constant in\neach region asymptotically but they can be different constants. For direct\nsimulation of a gas total energy and total momentum are conserved and the\nsystem stays on a constant total energy-total momentum shell. Asymptotically\nthe $f(X;n)$ will be constant on each shell but they will be different\nconstant for different shells.\n\n\\subsection{The hierarchy of Reduced probability distributions}\n\nIf we integrate the master equation over $d\\mu _{M+1},\\mu _{M+2},...,\\mu\n_{N} $ we obtain the equation\n\\begin{eqnarray}\nf^{(M)}(\\mu \\mathbf{;}n+1) &=&\\frac{(N-M)(N-M-1)}{N(N-1)}\\,\\,f^{(M)}(\\mu\n\\mathbf{;}n)  \\label{a60} \\\\\n&&+\\frac{2(N-M)}{N(N-1)}\\sum_{i=1}^{M}\\int f_{i,M+1}^{(M+1)}(\\mu _{A},\\mu\n_{B};n)\\,T(\\mu _{A},\\mu _{B};\\mu _{i},\\mu _{C})\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu\n_{C}  \\nonumber \\\\\n&&+\\frac{M(M-1)}{N(N-1)}\\sum_{i=1}^{M}\\sum_{j\\neq i}^{M}\\int\nf_{i,j}^{(M)}(\\mu _{A},\\mu _{B};n)\\,T(\\mu _{A},\\mu _{B};\\mu _{i},\\mu\n_{j})\\,d\\mu _{A}\\,d\\mu _{B}.  \\nonumber\n\\end{eqnarray}\nThe $f^{(M)}(\\mu \\mathbf{;}n+1)$ depends on $f^{(M+1)}(\\mu ;n)$ and this\nrepresents a hierarchy of equations similar to the well-known BBGKY hierarchy%\n\\cite{Huang}.\n\nThe first equation in the hierarchy is\n\\begin{eqnarray}\nf^{(1)}(\\mu \\mathbf{;}n+1) &=&(1-2\/N)\\,f^{(1)}(\\mu \\mathbf{;}n)  \\label{a70}\n\\\\\n&&+\\frac{2}{N}\\int f^{(2)}(\\mu _{A},\\mu _{B};n)\\,T(\\mu _{A},\\mu _{B};\\mu\n_{C},\\mu )\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C}.  \\nonumber\n\\end{eqnarray}\nIf we make the assumption of molecular chaos (AMC)\n\\begin{equation}\nf^{(2)}(\\mu _{A},\\mu _{B};n)=f^{(1)}(\\mu _{A};n)\\,f^{(1)}(\\mu _{B};n),\n\\label{a80}\n\\end{equation}\nwe obtain a nonlinear equation for $f^{(1)}(\\mu ;n)$ similar to the\nBoltzmann equation.\n\nFrom now on we will suppress the superscript $(1)$ in $f^{(1)}(\\mu \\mathbf{;}%\n\\tau )$ wherever it does not cause confusion. Using the relation\n\\begin{equation}\nf(\\mu ,n)=\\int f(\\mu ,n)\\,f(\\mu _{C},n)\\,T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu\n)\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C},  \\label{a90}\n\\end{equation}\nwhich follows from Eq.(\\ref{a30}) and the normalization of $f(\\mu _{C})$ and\nimposing the assumption of molecular chaos we can write eq.(\\ref{a70}) as\n\\begin{equation}\nf(\\mu \\mathbf{;}n+1)=f(\\mu \\mathbf{;}n)+\\frac{2}{N}\\int [f,f]\\,T(\\mu\n_{A},\\mu _{B};\\mu _{C},\\mu )\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C}  \\label{a101}\n\\end{equation}\n\\begin{equation}\n\\lbrack f,f]=f(\\mu _{A},n)\\,f(\\mu _{B},n)-f(\\mu _{C},n)\\,f(\\mu ,n)\n\\label{a103}\n\\end{equation}\n\nA\\ second simplification occurs for large $N.$ The $2\/N$ appearing in eq.(%\n\\ref{a101}) is a small number and we can take $\\tau =2n\/N$ as a continuous\nparameter which we call the collision time. Then $\\Delta \\tau =2\/N$ and $%\n\\left[ f(\\mu \\mathbf{;}n+1)-f(\\mu \\mathbf{;}n)\\right] \/\\Delta \\tau $ can be\ntaken as $\\partial f(\\mu \\mathbf{,}\\tau )\/\\partial \\tau $. The eq.(\\ref{a101}%\n) can be written in either of the following forms:\n\\begin{eqnarray}\n\\frac{\\partial f(\\mu ,\\tau \\mathbf{)}}{\\partial \\tau } &=&\\int [f,f]\\,T(\\mu\n_{A},\\mu _{B};\\mu _{C},\\mu )\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C}.  \\label{a130}\n\\\\\n\\frac{\\partial f(\\mu ,\\tau \\mathbf{)}}{\\partial \\tau } &=&-f(\\mu )+\\int\nf(\\mu _{A})f\\,(\\mu _{B})T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )\\,d\\mu _{A}\\,d\\mu\n_{B}\\,d\\mu _{C}.  \\label{a131}\n\\end{eqnarray}\n\nWe will call the first equation in the hierarchy 'the first equation'\nbriefly for the rest of the paper. In latter parts of this paper we will\ncall the integral on the right side of eq.(\\ref{a130}) 'the collision\nintegral'. From now on we will also suppress the collision time $\\tau $ in $%\nf(\\mu \\mathbf{,}\\tau )$ wherever it is convenient$.$\n\n\\subsection{Justification of assumption of molecular chaos}\n\nThe only thing in this paper that is not fully rigorous is the assumption of\nmolecular chaos. In order to have assumption of molecular chaos valid from\nthe beginning we must start from an uncorrelated state\n\\begin{equation}\nf^{(N)}(\\mu _{1},\\mu _{2},...,\\mu _{N};n=0)=h(\\mu _{1})\\,h(\\mu\n_{2})....h(\\mu _{N}),  \\label{a83}\n\\end{equation}\nwhich is what is done in direct simulations mostly. The master equation eq.(%\n\\ref{a66}) should be used to justify AMC. For finite $N$ the AMC\\ is not\nstrictly valid and the AMC\\ should get better and better as $N\\rightarrow\n\\infty $. For $M\/N<<1$ the eq. (\\ref{a60}) is written as\n\\begin{eqnarray}\nf^{(M)}(\\mu \\mathbf{;}n+1) &=&(1-2M\/N)\\,\\,f^{(M)}(\\mu \\mathbf{;}n)+O(1\/N^{2})\n\\label{a133} \\\\\n&&+\\frac{2}{N}\\sum_{i=1}^{M}\\int f_{i,M+1}^{(M+1)}(\\mu _{A},\\mu _{B};n)\n\\nonumber \\\\\n&&\\times \\,T(\\mu _{A},\\mu _{B};\\mu _{i},\\mu _{C})\\,d\\mu _{A}\\,d\\mu\n_{B}\\,d\\mu _{C}  \\nonumber\n\\end{eqnarray}\nwhere $O(1\/N^{2})$ are the terms of order $1\/N^{2}$. If we invoke collision\ntime $\\tau =2n\/N$ again and write $\\left[ f^{(M)}(\\mu \\mathbf{;}%\nn+1)-f^{(M)}(\\mu \\mathbf{;}n)\\right] \/\\Delta \\tau =\\partial f^{(M)}(\\mu\n\\mathbf{;}\\tau )\/\\partial \\tau $ and we take the limit $N\\rightarrow \\infty $\nwe obtain\n\\begin{eqnarray}\n\\frac{\\partial f^{(M)}(\\mu \\mathbf{;}\\tau )}{\\partial \\tau }\n&=&-Mf^{(M)}(\\mu \\mathbf{;}\\tau )  \\label{a134} \\\\\n&&+\\sum_{i=1}^{M}\\int f_{i,M+1}^{(M+1)}(\\mu _{A},\\mu _{B};\\tau )\\,T(\\mu\n_{A},\\mu _{B};\\mu _{i},\\mu _{C})\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C}  \\nonumber\n\\end{eqnarray}\nwhere $M=1,2,...,\\infty $. This is an infinite chain of coupled differential\nequations. If we invoke\n\\begin{equation}\nf^{(M)}(\\mu _{1},\\mu _{2},...,\\mu _{M};\\tau )=f^{(1)}(\\mu _{1}\\mathbf{;}\\tau\n)\\,f^{(1)}(\\mu _{2}\\mathbf{;}\\tau )....f^{(1)}(\\mu _{M}\\mathbf{;}\\tau ).\n\\label{a135}\n\\end{equation}\nin the eq.(\\ref{a134}) all the equations in the infinite chain are satisfies\nprovided $f^{(1)}(\\mu \\mathbf{;}\\tau )$ satisfies eq. (\\ref{a130}). This\nproves that in the limit $N\\rightarrow \\infty $ the AMC remains valid for\nall $\\tau $ if we start from an uncorrelated initial state.\n\nWhat happens if we start from a correlated state that does not satisfy AMC?\nFor finite $N$ there are always some correlations to any order. We know that\nthe system evolves towards microcanonical distribution. In the limit $%\nN\\rightarrow \\infty $ microcanonical distribution obeys AMC. This means even\nif we start from a correlated state the system will satisfy AMC better and\nbetter as the system evolves towards equilibrium for large $N.$ Collisions\ndestroys correlations and It should take only a few collisions per particle\nto destroy initial correlations. Moreover in the practical applications of\nDSMC in gas dynamics the $N$ is almost always large and initial state is\nchosen as almost uncorrelated from the beginning. Therefore using the first\nequation to determine the single particle probability density is a\njustifiable process.\n\n\\subsection{Collision invariants and the H-theorem}\n\nWe now show that expectation value $\\left\\langle g(\\mu \\mathbf{)}%\n\\right\\rangle $ of a collision invariant $g(\\mu )$ is conserved. The $g(\\mu\n) $ is a collision invariant if\n\\begin{equation}\n\\Delta g=g(\\mu )+g(\\mu _{C})-g(\\mu _{A})-g(\\mu _{B})=0.  \\label{a140}\n\\end{equation}\nMultiplying eq.(\\ref{a130}) and integrating over $\\mu $ we obtain\n\\begin{equation}\n\\frac{d}{d\\tau }\\int f(\\mu \\mathbf{)\\,}g(\\mu \\mathbf{)\\,}d\\mu =\\int\n[f,f]\\,\\,g(\\mu \\mathbf{)}T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )\\,\\,d\\mu\n_{A}\\,d\\mu _{B}\\,d\\mu _{C}\\,d\\mu .  \\label{a150}\n\\end{equation}\nUsing symmetries of $T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )$ and relabeling\nintegration variables among themselves we can write this as\n\\begin{equation}\n\\frac{d}{d\\tau }\\left\\langle g(\\mu \\mathbf{)}\\right\\rangle =\\frac{1}{4}\\int\n[f,f]\\,\\Delta g\\,T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )\\,d\\mu _{A}\\,d\\mu\n_{B}\\,d\\mu _{C}\\,d\\mu .  \\label{a160}\n\\end{equation}\nThe integral is zero because of eq.(\\ref{a140})$.$\n\nWe can derive an H-theorem for the first equation. Defining $H(\\tau )$ a\n\\begin{equation}\nH(\\tau )=\\int f(\\mu )\\,\\ln (f(\\mu ))\\,d\\mu \\mathbf{,}  \\label{b10}\n\\end{equation}\nand using the eqs. (\\ref{a10},\\ref{a20}) and eq.(\\ref{a130}) we can express $%\ndH\/d\\tau $ as\n\\begin{equation}\n\\frac{dH}{d\\tau }=-\\frac{1}{4}\\int \\Phi [f]\\,T(\\mu _{A},\\mu _{B};\\mu\n_{C},\\mu )\\,\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C}\\,d\\mu \\mathbf{,}  \\label{b20}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Phi [f]=\\left[ f(\\mu _{A})\\,\\,f(\\mu _{B})-f(\\mu )\\,\\,f(\\mu _{C})\\right]\n\\left[ \\ln f(\\mu _{A})\\,\\,f(\\mu _{B})-\\ln f(\\mu )\\,\\,f(\\mu _{C})\\right] .\n\\label{b31}\n\\end{equation}\nThe $\\Phi [f]$ can be shown to be always nonnegative as done in all kinetic\ntheory books and $T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )$ is intrinsically\npositive. Therefore $dH\/d\\tau $ is nonpositive. There are two possibilities\nhere. The $H$ keeps decreasing toward negative infinity or it approaches an\nabsolute minimum asymptotically and the system approaches toward an\nequilibrium distribution. Following the usual arguments of the H-theorem,\nthe decrease of $H$ stops only when\n\\begin{equation}\n\\ln f(\\mu _{A})+\\ln f(\\mu _{B})=\\ln f(\\mu _{C})+\\ln f(\\mu ),  \\label{b40}\n\\end{equation}\nis satisfied which implies that $\\ln f(\\mu )$ is a collision invariant. If\nwe choose the $T(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )$ such that there are\ncollision invariants $g_{i}(\\mu )$ $(i=1,2,...,L)$ then $\\ln f(\\mu )$ must\nbe expressible as a linear combinations of these collision invariants as\n\\begin{equation}\n\\ln f(\\mu )=c_{1}g_{1}(\\mu )+c_{2}g_{2}(\\mu )+...+c_{L}g_{L}(\\mu ),\n\\label{b50}\n\\end{equation}\nwhere $c_{1},...,c_{L}$ are parameters describing the equilibrium.\n\nThere is at least one trivial collision invariant. It is the number of\nparticles entering and exiting the collision which corresponds to $g_{1}(\\mu\n)=1$. When there are additional collision invariants the $H$ has a lower\nbound usually. For the case of real gases momentum and energy are collision\ninvariants and this makes $H$ bounded from below.\n\n\\subsection{Example: A game of discrete money gambling}\n\nHere we give a simple example of a direct simulation money game with finite\nnumber of discrete states. Suppose everybody is given some random amount of\nmoney at the beginning. Everybody in the assembly has one, two or three\ndollars in their pocket. The random assignment of initial money ensures\nassumption of molecular chaos from the beginning. The collisions takes place\nas follows: Player 1 and player 2 share their total money such that nobody\ngets more than three dollars and both players get at least one dollar. All\nthe possibilities satisfying these conditions have equal probabilities. If\nthey have total two dollars (one dollar each) then the only possibility is\nthat they will have one dollar each at the end with unity probability. If\nthey have total three dollars then the possible outcomes are (1,2) and (2,1)\nwith equal 1\/2 probabilities. If they have total four dollars then possible\noutcomes are (1,3), (3,1), (2,2) with 1\/3 probability each. If they have\ntotal five dollars then possible outcomes are (2,3) and (3,2) with 1\/2\nprobability each. Finally if they have total six dollars (three dollars\neach) then the only possibility is (3,3) with unity probability.\n\nFor this game the money is conserved in collisions and transitions between\nstates with equal amount of total money is possible only. For $N$ particles\nthe total money can have values between $N$ to $3N$ and there are a total of\n$2N+1$ separate regions in phase space. One cannot cross from one to another\nof these regions by making collisions.\n\nNow that we defined the game, how does single particle distribution evolves\nas we make collisions? The state variable $\\mu $ is the amount of the money\nin the persons pocket and it takes the values 1,2,3. Let $P_{\\mu }(\\tau )$\nbe the probability that a chosen person will have the money $\\mu $ at the\ncollision time $\\tau .$ From eq.(\\ref{a131}) the $P_{\\mu }(\\tau )$ satisfies\n\\begin{eqnarray}\n\\frac{dP_{1}}{d\\tau } &=&-P_{1}+P_{1}^{2}\\,\\,T(1,1,1,1)+P_{1}P_{2}\\,%\n\\,T(1,2;2,1)  \\label{c10} \\\\\n&&+P_{2}P_{1}\\,\\,T(2,1;2,1)+P_{1}P_{3}\\,\\,T(1,3;3,1)  \\nonumber \\\\\n&&+P_{3}P_{1}\\,\\,T(3,1;3,1)+P_{2}P_{2}\\,\\,T(2,2;2,1),  \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{dP_{2}}{d\\tau } &=&-P_{2}+P_{2}^{2}\\,\\,T(2,2,2,2)  \\label{c20} \\\\\n&&+P_{1}P_{2}\\,\\,T(1,2;1,2)+P_{2}P_{1}\\,\\,T(2,1;1,2)  \\nonumber \\\\\n&&+P_{1}P_{3}\\,\\,T(1,3;2,2)+P_{3}P_{1}\\,\\,T(3,1;2,2)  \\nonumber \\\\\n&&+P_{2}P_{3}\\,\\,T(2,3;3,2)+P_{3}P_{2}\\,\\,T(3,2;3,2),  \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\frac{dP_{3}}{d\\tau } &=&-P_{3}+P_{1}P_{3}\\,\\,T(1,3;1,3)+P_{3}P_{1}\\,%\n\\,T(3,1;1,3)  \\label{c30} \\\\\n&&+P_{3}^{2}\\,\\,T(3,3;3,3)+P_{2}P_{3}\\,\\,T(2,3;2,3)  \\nonumber \\\\\n&&+P_{3}P_{2}\\,\\,T(3,2;2,3)+P_{2}^{2}\\,\\,T(2,2,1,3).  \\nonumber\n\\end{eqnarray}\nInserting the $T$ values this can be written as\n\\begin{eqnarray}\n\\frac{dP_{1}}{d\\tau } &=&-P_{1}+P_{1}^{2}+P_{1}P_{2}+\\frac{2}{3}P_{1}P_{3}+%\n\\frac{1}{3}P_{2}^{2},  \\label{c40} \\\\\n\\frac{dP_{2}}{d\\tau } &=&-P_{2}+\\frac{1}{3}P_{2}^{2}+P_{1}P_{2}+\\frac{2}{3}%\nP_{1}P_{3}+P_{2}P_{3},  \\label{c41} \\\\\n\\frac{dP_{3}}{d\\tau } &=&-P_{3}+\\frac{2}{3}P_{1}P_{3}+P_{2}P_{3}+P_{3}^{2}+%\n\\frac{1}{3}P_{2}^{2}.  \\label{c42}\n\\end{eqnarray}\n\nThis is a complicated set of nonlinear differential equations. But there are\nsimplifying features because we know the collision invariants $g_{1}(\\mu )=1$\nand $g_{2}(\\mu )=\\mu $. Summing the eqs.(\\ref{c40},\\ref{c41},\\ref{c42}) we\nobtain\n\\begin{equation}\n\\frac{d}{d\\tau }\\left( P_{1}+P_{2}+P_{3}\\right) =\\left(\nP_{1}+P_{2}+P_{3}-1\\right) \\left( P_{1}+P_{2}+P_{3}\\right) ,  \\label{c46}\n\\end{equation}\nand\n\\begin{equation}\n\\frac{d}{d\\tau }\\left( P_{1}+2P_{2}+3P_{3}\\right) =\\left(\nP_{1}+P_{2}+P_{3}-1\\right) \\left( P_{1}+2P_{2}+3P_{3}\\right) .  \\label{c46b}\n\\end{equation}\nThe first equation tells us that since $P_{1}+P_{2}+P_{3}=1$ at the\nbeginning it always remains unity and probability is conserved. The second\nequation tells us that since $P_{1}+P_{2}+P_{3}-1=0$ always the expectation\nvalue $\\left\\langle \\mu \\right\\rangle =P_{1}+2P_{2}+3P_{3}$ is conserved.\n\nWe denote expected money in the pocket with $m$. We have two equations\n\\begin{eqnarray}\nP_{1}+P_{2}+P_{3} &=&1,  \\label{c47} \\\\\nP_{1}+2P_{2}+3P_{3} &=&m,  \\label{c48}\n\\end{eqnarray}\nfrom which we solve $P_{2}$ and $P_{3}$ as\n\\begin{eqnarray}\nP_{2} &=&-2P_{1}+3-m,  \\label{c49} \\\\\nP_{3} &=&P_{1}+m-2.  \\label{c50}\n\\end{eqnarray}\nInserting $P_{2}$ and $P_{3}$ in the eq.(\\ref{c40}) we obtain\n\\begin{equation}\n\\frac{dP_{1}}{d\\tau }=P_{1}^{2}+(m-\\frac{10}{3})P_{1}+\\frac{1}{3}(3-m)^{2}.\n\\label{ek10}\n\\end{equation}\nCalculating roots of the quadratic term on the right we write this as\n\\begin{equation}\n\\frac{dP_{1}}{d\\tau }=\\left( P_{1}-r_{1}\\right) \\left( P_{1}-r_{2}\\right) ,\n\\label{ek20}\n\\end{equation}\nwhere $r_{1}$ and $r_{2}$ are\n\\begin{eqnarray}\nr_{1} &=&\\frac{1}{6}\\left( 10-3m+\\sqrt{1+3(m-1)(3-m)}\\right) ,  \\label{ek30}\n\\\\\nr_{2} &=&\\frac{1}{6}\\left( 10-3m-\\sqrt{1+3(m-1)(3-m)}\\right) .  \\label{ek40}\n\\end{eqnarray}\nNotice that since $1\\leq m\\leq 3$ the term under the square root is always\ngreater than or equal to unity.\n\nSolving eq.(\\ref{ek20}) is straightforward and we obtain\n\\begin{equation}\nP_{1}(\\tau )=\\frac{r_{2}(p_{0}-r_{1})-r_{1}(p_{0}-r_{2})\\mathrm{e}^{-\\lambda\n\\tau }}{(p_{0}-r_{1})-(p_{0}-r_{2})\\mathrm{e}^{-\\lambda \\tau }},\n\\label{ek50}\n\\end{equation}\nwhere $p_{0}=P_{1}(\\tau =0)$ and $\\lambda =r_{1}-r_{2}$. It is easy to\nverify that $P_{1}(\\infty )=r_{2}$ and $P_{1}(\\tau )\\,$approaches this limit\nexponentially fast. One can check from eq.(\\ref{ek40}) that $r_{2}=1$ at $%\nm=1 $ and $r_{2}=0$ at $m=3$ and it behaves as it is expected.\n\nThe conditions $0\\leq P_{2}\\leq 1$ and $0\\leq P_{3}\\leq 1$ together with\neqs.(\\ref{c49},\\ref{c50}) gives conditions that $P_{1}(\\tau )$ must satisfy.\nThese conditions are expressed as $2-m\\leq P_{1}\\leq (3-m)\/2$ when $m\\leq 2$\nand $0\\leq P_{1}\\leq (3-m)\/2$ when $m>2$. Therefore $P_{1}(\\tau =0)\\,$%\ninitial value should obey these limitations.\n\nTo find the equilibrium distribution directly without solving the\ndifferential equation we set $dP_{\\mu }\/d\\tau =0$ for $\\mu =1,2,3$ in eqs.(%\n\\ref{c40},\\ref{c41},\\ref{c42}) and we obtain a set of algebraic nonlinear\nequations. Setting $P_{1}=a$, $P_{2}=ab$, $P_{3}=ab^{2}$ all three equations\nare satisfied provided the normalization condition\n\\begin{equation}\na(1+b+b^{2})=1,  \\label{c59}\n\\end{equation}\nholds. We were able to guess this solution from the H-theorem. There are two\ncollision invariants $g_{1}=1$ and $g_{2}(\\mu )=\\mu $. The second one is a\nresult of conservation of money in the collisions. Therefore according to\nthe H-theorem we must have $\\ln P_{\\mu }=C_{1}+C_{2}\\mu $ and this gives the\nsolution $P_{\\mu }=ab^{\\mu -1}$. We need one more relation to determine both\n$a$ and $b$. This comes from expected money in the pocket:\n\\begin{equation}\nm=a\\left( 1+2b+3b^{2}\\right) ,  \\label{c60}\n\\end{equation}\nwhich is a conserved quantity during the 'time' evolution and it is set by\nthe initial conditions. Solving these two equation we obtain\n\\begin{eqnarray}\na &=&\\frac{1}{6}\\left( 10-3m-\\sqrt{1+3(m-1)(3-m)}\\right) ,  \\label{c65} \\\\\nb &=&\\left( m-2+\\sqrt{1+3(m-1)(3-m)}\\right) \/2(3-m).  \\nonumber\n\\end{eqnarray}\nNotice that $a=r_{2}$ and this agrees with solution of the differential\nequation.\n\nThe H-function\n\\begin{equation}\nH=P_{1}\\ln P_{1}+P_{2}\\ln P_{2}+P_{3}\\ln P_{3},  \\label{c70}\n\\end{equation}\nis bounded from below for this problem since the function $x\\ln x$ is\nbounded from below and $0\\leq P_{\\mu }\\leq 1$. We minimize $H$ with the\nconstraint that the expected money is fixed and probabilities are\nnormalized. The constraints can be adopted with Lagrange multipliers. Taking\nthe auxiliary function\n\\begin{eqnarray}\n\\Psi &=&P_{1}\\ln P_{1}+P_{2}\\ln P_{2}+P_{3}\\ln P_{3}  \\label{c80} \\\\\n&&-\\lambda _{2}(P_{1}+P_{2}+P_{3}-1)-\\lambda _{2}(P_{1}+2P_{2}+3P_{3}-m),\n\\nonumber\n\\end{eqnarray}\nand setting $\\partial \\Psi \/\\partial P_{1}=\\partial \\Psi \/\\partial\nP_{2}=\\partial \\Psi \/\\partial P_{3}=0$ we obtain the same solution $P_{\\mu\n}=ab^{\\mu -1}$ where $a$ and $b$ satisfies the eqs.(\\ref{c59},\\ref{c60}).\nThe minimum value of H becomes\n\\begin{equation}\nH=a\\ln a+ab\\ln ab+ab^{2}\\ln ab^{2}=\\ln (ab^{m-1}).  \\label{c90}\n\\end{equation}\n\n\\subsection{Example2: A game of continuous money gambling}\n\nHere we give another example of direct simulation money games with\ncontinuous states. In this case we were not even able to solve one particle\nprobability distribution. We just find the equation for one particle\ndistribution and guess the stationary one particle distribution from the\nH-theorem. We then show that it satisfies the equation for single particle\nprobability equation.\n\nThis time initially we give players a random amount of money between zero\nand, say, ten dollars. Suppose we pick a pair to collide. player1 has $\\mu\n_{1}$ and player2 has $\\mu _{2}$ amount of money. A computer produces a\nrandom number $p$ between zero and one. Player1 takes $p(\\mu _{1}+\\mu _{2})$\nand player2 takes $(1-p)(\\mu _{1}+\\mu _{2})$ amounts of money and we pick\nanother pair to collide. What is the final distribution when the system\ncomes to equilibrium?\n\nThe probability distribution that a person will have money $\\mu $ satisfies\nthe eq.(\\ref{a131})\n\\begin{equation}\n\\frac{\\partial f(\\mu \\mathbf{)}}{\\partial \\tau }=-f(\\mu )+\\int_{0}^{\\infty\n}da\\int_{0}^{\\infty }db\\,f(a)\\,f(b)\\,T(a,b,\\mu ,\\nu )\\,da\\,db\\,d\\nu ,\n\\label{d10}\n\\end{equation}\nwhere the collision kernel is\n\\begin{equation}\nT(a,b,\\mu ,\\nu )=\\frac{1}{a+b}\\delta (a+b-\\mu -\\nu )\\Theta (a)\\,\\Theta\n(b)\\,\\Theta (\\mu )\\,\\Theta (\\nu ).  \\label{d21}\n\\end{equation}\nHere $\\Theta (x)$ is the standard step function\n\\begin{equation}\n\\Theta (x)=\\left\\{\n\\begin{array}{ll}\n0\\quad & {}x<0 \\\\\n1\\quad & x\\geq 0\n\\end{array}\n\\right. .  \\label{d30}\n\\end{equation}\nIf we insert the $T(a,b,\\mu ,\\nu )$ given in the eq.(\\ref{d21}) into the eq.(%\n\\ref{d10}) and perform the $\\nu $ integral we obtain\n\\begin{equation}\n\\frac{\\partial f(\\mu \\mathbf{)}}{\\partial \\tau }=-f(\\mu )+\\int_{0}^{\\infty\n}da\\int_{0}^{\\infty }db\\,\\Theta (a+b-\\mu )\\,\\frac{f(b)\\,f(a)}{a+b}.\n\\label{d40}\n\\end{equation}\nThis can be further simplified by changing variables $x=a+b$ and $y=a$ which\nyields\n\\begin{equation}\n\\frac{\\partial f(\\mu \\mathbf{)}}{\\partial \\tau }=-f(\\mu )+\\int_{\\mu\n}^{\\infty }dx\\int_{0}^{x}dy\\,\\frac{f(y)\\,f(x-y)}{x}.  \\label{d50}\n\\end{equation}\n\nThe H-theorem insures that this equation will converge to an equilibrium\ndistribution as $\\tau \\rightarrow \\infty $. Since we have money conservation\nin the collisions there are two collision invariants $g_{1}(\\mu )=1$ and $%\ng_{2}(\\mu )=\\mu $. Then the equilibrium distribution is\n\\begin{equation}\nf_{eq}(\\mu )=A\\,e^{-B\\mu }.  \\label{d60}\n\\end{equation}\nIf the average money initially given to each person is $m$, the $f(\\mu )$\nshould satisfy two conditions\n\\begin{eqnarray}\n\\int_{0}^{\\infty }f(\\mu )\\,d\\mu &=&1,  \\label{d80} \\\\\n\\int_{0}^{\\infty }\\mu \\,f(\\mu )\\,d\\mu &=&m,  \\label{d81}\n\\end{eqnarray}\nand they fix the values of $A$ and $B$ in the eq.(\\ref{d60}). The solution\nis\n\\begin{equation}\nf_{eq}(\\mu )=\\frac{1}{m}\\,e^{-\\mu \/m}.  \\label{d90}\n\\end{equation}\nIf we insert this solution into eq.(\\ref{d50}) we can easily check that\nright side of the equation becomes zero which confirms that $f_{eq}(\\mu )$\nis the equilibrium distribution.\n\n\\section{ Application of the direct simulation formalism to homogenous gases}\n\n\\subsection{Center of mass frame}\n\nIn the following sections we will need some results from studying the\ncollision in the center of mass frame. Instead of deriving them for each\ncase separately we derive the relevant results once for the most general\ncase in this subsection and refer to formulae derived here as needed in the\nfollowing subsections. In the rest of the paper bold letters denote vector\nquantities.\n\nParticles with states $\\mu _{A}=\\mathbf{v}_{A}\\mathbf{\\ }$and $\\mu _{B}=%\n\\mathbf{v}_{B}$ and enter the collision and particles with states $\\mu _{C}=%\n\\mathbf{v}_{C}$ and $\\mu _{D}=\\mathbf{v}$ exit the collision. We define the\ncenter of mass (CM) coordinates as\n\\begin{eqnarray}\n\\mathbf{H} &=&(m_{A}\\mathbf{v}_{A}+m_{B}\\mathbf{v}_{B})\/(m_{A}+m_{B})\n\\label{e3} \\\\\n\\mathbf{H}^{\\prime } &=&(m_{A}\\mathbf{v}_{C}+m_{B}\\mathbf{v})\/(m_{A}+m_{B}),\n\\label{e3b}\n\\end{eqnarray}\nand\n\\begin{equation}\n\\begin{array}{lll}\n\\mathbf{u=v}_{A}-\\mathbf{v}_{B},\\;\\;\\; & u=\\left| \\mathbf{u}\\right| , &\n\\mathbf{n}=\\mathbf{u}\/u \\\\\n\\mathbf{u}^{\\prime }\\mathbf{=v}_{C}-\\mathbf{v}, & u^{\\prime }=\\left| \\mathbf{%\nu}^{\\prime }\\right| ,\\;\\;\\; & \\mathbf{n}^{\\prime }=\\mathbf{u}^{\\prime\n}\/u^{\\prime }\n\\end{array}\n\\label{e4}\n\\end{equation}\nwhere $m_{A}$ is the mass of particles $A$ and $C$ and $m_{B}$ is the mass\nof particles $B$ and $D$. For one kind of gas all masses are equal and\nformulae for CM velocities $\\mathbf{H}$ and $\\mathbf{H}^{\\prime }$ reduce to\n\\begin{equation}\n\\begin{array}{ll}\n\\mathbf{H}=(\\mathbf{v}_{A}+\\mathbf{v}_{B})\/2,\\;\\;\\; & \\mathbf{H}^{\\prime }=(%\n\\mathbf{v}_{C}+\\mathbf{v})\/2.\n\\end{array}\n\\label{e5}\n\\end{equation}\nIntegrations over $\\mathbf{v}_{A}$ and $\\mathbf{v}_{B}$ can be carried over\nin the variables $\\mathbf{H}$ and $\\mathbf{u}$. The transformation between\nthese two sets of variables are linear and the Jacobian is unity. Therefore\n\\begin{equation}\n\\int f(\\mathbf{v}_{A},\\mathbf{v}_{B})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}%\n_{B}=\\int f(\\mathbf{H},\\mathbf{u})d^{3}\\mathbf{H}d^{3}\\mathbf{u.}\n\\label{e34}\n\\end{equation}\nIn the following subsections we will deal with integrations over $\\mathbf{v}%\n_{A}$, $\\mathbf{v}_{B}$, $\\mathbf{v}_{C}$. Integrations over $\\mathbf{v}_{A}$%\n, $\\mathbf{v}_{B}$ will be converted to integration over $\\mathbf{H}$ and $%\n\\mathbf{u}$ in the CM\\ frame. In each case there will be a Dirac delta\nfunction removing the integral over $\\mathbf{H}$. Integration over $\\mathbf{v%\n}_{C}$ will be converted to integration over $\\mathbf{u}^{\\prime }$ since $%\n\\mathbf{v}_{C}=\\mathbf{u}^{\\prime }\\mathbf{+v}$ and there is no integration\nover $\\mathbf{v.}$ Furthermore integrations over $\\mathbf{u}^{\\prime }$ will\nbe carried in spherical coordinates as\n\\begin{equation}\n\\int f(\\mathbf{u}^{\\prime })d^{3}\\mathbf{u}^{\\prime }=\\int f(\\mathbf{u}%\n^{\\prime })(u^{\\prime })^{2}du^{\\prime }d\\mathbf{n}^{\\prime }  \\label{e36}\n\\end{equation}\nand in each case there will be a Dirac delta function removing the\nintegration over $u^{\\prime }.$ In the final expressions the integration\nover solid angle $\\mathbf{n}^{\\prime }$ and $\\mathbf{u}$ remain at the end.\n\nIn order to evaluate the integrals we will encounter in the following\nsubsections we must express $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$\nin terms of the variables $\\mathbf{v,u,n}^{\\prime }.$ This is a simple\nexercise in collision kinetics. We will do this for the inelastic collisions\nwith unequal masses. This is the most general case we will deal in this\npaper. We will assume that molecules have internal energies $\\epsilon (A),$ $%\n\\epsilon (B)$ and $\\epsilon (C),$ $\\epsilon (D)$. Let $\\epsilon =\\epsilon\n(A)+\\epsilon (B)$ and $\\epsilon ^{\\prime }=\\epsilon (C)+\\epsilon (D)$. From\nenergy conservation we have $u^{\\prime }(u)=\\sqrt{u^{2}+2(\\epsilon -\\epsilon\n^{\\prime })\/m_{r}}$ where $m_{r}=m_{A}m_{B}\/(m_{A}+m_{B})$ is the reduced\nmass and $m_{A}$, $m_{B}$ are masses of the colliding particles. We can\nwrite $\\mathbf{u}^{\\prime }=u^{\\prime }(u)\\mathbf{n}^{\\prime }$ and $\\mathbf{%\nv}_{C}=\\mathbf{v+}u^{\\prime }(u)\\mathbf{n}^{\\prime }$. From CM velocity\nconservation we have\n\\begin{equation}\nm_{A}\\mathbf{v}_{A}+m_{B}\\mathbf{v}_{B}=m_{A}\\mathbf{v}_{C}+m_{B}\\mathbf{v=}%\n(m_{A}+m_{B}\\mathbf{)v}+m_{A}u^{\\prime }(u)\\mathbf{n}^{\\prime }  \\label{e93}\n\\end{equation}\nand we also have $\\mathbf{v}_{A}-\\mathbf{v}_{B}=\\mathbf{u}.$ We solve $%\n\\mathbf{v}_{A}$, $\\mathbf{v}_{B}$, $\\mathbf{v}_{C}$ from these as\n\\begin{eqnarray}\n\\mathbf{v}_{A} &=&\\mathbf{v}+\\frac{m_{A}}{m_{A}+m_{B}}u^{\\prime }(u)\\mathbf{n%\n}^{\\prime }+\\frac{m_{B}}{m_{A}+m_{B}}\\mathbf{u}  \\label{e100} \\\\\n\\mathbf{v}_{B} &=&\\mathbf{v}+\\frac{m_{A}}{m_{A}+m_{B}}u^{\\prime }(u)\\mathbf{n%\n}^{\\prime }-\\frac{m_{A}}{m_{A}+m_{B}}\\mathbf{u}  \\label{e101} \\\\\n\\mathbf{v}_{C} &=&\\mathbf{v+}u^{\\prime }(u)\\mathbf{n}^{\\prime }  \\label{e102}\n\\\\\nu^{\\prime }(u) &=&\\sqrt{u^{2}+2(\\epsilon -\\epsilon ^{\\prime })\/m_{r}}\n\\label{e103}\n\\end{eqnarray}\n\nFor one kind of gas ($m_{A}=m_{B}=m$ ) without internal states ($\\epsilon\n(A)=\\epsilon (B)=\\epsilon (C)=\\epsilon (D)=0$) these equations reduce to\n\\begin{eqnarray}\n\\mathbf{v}_{A} &=&\\mathbf{v}+(u\\mathbf{n}^{\\prime }+\\mathbf{u)\/2}\n\\label{e105} \\\\\n\\mathbf{v}_{B} &=&\\mathbf{v}+(u\\mathbf{n}^{\\prime }-\\mathbf{u)\/2}\n\\label{e106} \\\\\n\\mathbf{v}_{C} &=&\\mathbf{v+}u\\mathbf{n}^{\\prime }  \\label{e107}\n\\end{eqnarray}\nAgain for one kind of gas ($m_{A}=m_{B}=m$ and $m_{r}=m\/2$) with internal\nstates eqs.(\\ref{e100},\\ref{e101},\\ref{e102},\\ref{e103}) reduce to\n\\begin{eqnarray}\n\\mathbf{v}_{A} &=&\\mathbf{v}+\\left[ u^{\\prime }(u)\\mathbf{n}^{\\prime }+%\n\\mathbf{u}\\right] \\mathbf{\/}2  \\label{g75} \\\\\n\\mathbf{v}_{B} &=&\\mathbf{v}+\\left[ u^{\\prime }(u)\\mathbf{n}^{\\prime }-%\n\\mathbf{u}\\right] \/2  \\label{g76} \\\\\n\\mathbf{v}_{C} &=&\\mathbf{v+}u^{\\prime }(u)\\mathbf{n}^{\\prime }  \\label{g77}\n\\\\\nu^{\\prime }(u) &=&\\sqrt{u^{2}+4(\\epsilon -\\epsilon ^{\\prime })\/m}\n\\label{g78}\n\\end{eqnarray}\nFor a mixture of gases without internal states eqs.(\\ref{e100},\\ref{e101},%\n\\ref{e102},\\ref{e103}) reduce to\n\\begin{eqnarray}\n\\mathbf{v}_{A} &=&\\mathbf{v}+\\frac{m_{A}}{m_{A}+m_{B}}\\,u\\mathbf{n}^{\\prime\n}+\\frac{m_{B}}{m_{A}+m_{B}}\\,\\mathbf{u,}  \\label{f80} \\\\\n\\mathbf{v}_{B} &=&\\mathbf{v}+\\frac{m_{A}}{m_{A}+m_{B}}\\,u\\mathbf{n}^{\\prime\n}-\\frac{m_{A}}{m_{A}+m_{B}}\\,\\mathbf{u,}  \\label{f81} \\\\\n\\mathbf{v}_{C} &=&\\mathbf{v+}u\\mathbf{n}^{\\prime }.  \\label{f82}\n\\end{eqnarray}\nAnd for a mixture of gases with internal states eqs.(\\ref{e100},\\ref{e101},%\n\\ref{e102},\\ref{e103}) are the formule.\n\n\\subsection{One kind of gas without internal degrees of freedom}\n\nThe state of particles are defined by three components of the velocity\nvector $\\mathbf{v}$. (We use bold letters for vectors throughout this paper)\nBird's original algorithm to keep track of time in the simulation was the\n'time counter method'. Later Bird introduced 'No time counter method' (NTC)\nand declared time counter method 'obsolete' in his book.\\cite{Bird94} Time\ncounter method is more difficult (if not impossible) to formulate in the\ndirect simulation formalism given in this paper and since NTC is the\nalgorithm currently used we will derive NTC algorithms only in this paper.\n\nHere the state index $\\mu $ refer the velocity vectors and the integration\nover $\\mu $ stands for three integrations over components of velocities. The\nNTC\\ kernel $S(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}%\n)=S_{1}+S_{2}$ is given by\n\\begin{eqnarray}\nS_{1} &=&\\frac{2}{R}\\delta \\left( \\mathbf{H}-\\mathbf{H}^{\\prime }\\right)\n\\,\\delta \\left( u^{2}-(u^{\\prime })^{2}\\right) \\,\\sigma (\\mathbf{n},\\mathbf{n%\n}^{\\prime })  \\label{e20} \\\\\nS_{2} &=&\\left( 1-\\frac{u\\Sigma }{R}\\right) \\,\\delta \\left( \\mathbf{v}_{C}-%\n\\mathbf{v}_{A}\\right) \\,\\delta \\left( \\mathbf{v}-\\mathbf{v}_{B}\\right)\n\\label{e30}\n\\end{eqnarray}\nHere $\\sigma (\\mathbf{n},\\mathbf{n}^{\\prime })$ is the differential cross\nsection and $\\Sigma \\,$is the total cross section which is given by\n\\begin{equation}\n\\Sigma =\\int \\sigma (\\mathbf{n},\\mathbf{n}^{\\prime })\\,d\\mathbf{n}^{\\prime },\n\\label{e40}\n\\end{equation}\nwhere $d\\mathbf{n}^{\\prime }$ is the solid angle in the direction of $%\n\\mathbf{n}^{\\prime }$. The $\\sigma (\\mathbf{n},\\mathbf{n}^{\\prime })$\ndepends on the angle $\\theta $ between $\\mathbf{n}$ and $\\mathbf{n}^{\\prime\n} $ ($\\mathbf{n}^{\\prime }\\cdot \\mathbf{n}=\\cos \\theta $). Hence $\\sigma (%\n\\mathbf{n},\\mathbf{n}^{\\prime })=\\sigma (\\mathbf{n}^{\\prime },\\mathbf{n})$\nand the kernel is obviously symmetric. The term $\\delta (u^{2}-(u^{\\prime\n})^{2})=\\delta (u-u^{\\prime })\/2u$ represents energy conservation and $%\n\\delta \\left( \\mathbf{H}-\\mathbf{H}^{\\prime }\\right) $ represents\nconservation of center of mass (CM) velocity which is the same thing as the\nconservation of momentum. The kernel satisfies the normalization condition\n\\begin{equation}\n\\int S(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}%\n\\mathbf{v}_{C}\\,d^{3}\\mathbf{v=\\int }S(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{%\nv}_{C},\\mathbf{v})\\,d^{3}\\mathbf{H}^{\\prime }\\,d^{3}\\mathbf{u}^{\\prime }=1.\n\\label{e50}\n\\end{equation}\nHere the integral is taken in the CM coordinates. The Jacobian of the CM\ntransformation is unity and $d^{3}\\mathbf{u}^{\\prime }=(u^{\\prime\n})^{2}du^{\\prime }d\\mathbf{n}^{\\prime }$.\n\nThe $S_{2}$ part of the kernel directly transfer initial velocities to the\nfinal velocities with a probability $\\left( 1-u\\Sigma \/R\\right) $ and hence\ncauses a null collision. A null collision is a collision that particles do\nnot change their states. The probability of making a real collision is\n\\begin{equation}\n\\int S_{1}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}%\n\\mathbf{v}_{C}\\,d^{3}\\mathbf{v=}\\frac{u\\Sigma }{R}  \\label{e60}\n\\end{equation}\nwhere integral is calculated in the CM coordinates.\n\nInserting $S(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})$ in\neq.(\\ref{a130}) we obtain\n\\begin{equation}\n\\frac{\\partial f(\\mathbf{v})}{\\partial \\tau }=\\int [f,f]\\,\\,S_{1}(\\mathbf{v}%\n_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}\\mathbf{v}_{A}\\,d^{3}%\n\\mathbf{v}_{B}\\,d^{3}\\mathbf{v}_{C}.  \\label{e70}\n\\end{equation}\nwhere\n\\begin{equation}\n\\lbrack f,f]=f(\\mathbf{v}_{A})\\,f(\\mathbf{v}_{B})-f(\\mathbf{v}_{C})\\,f(%\n\\mathbf{v}),  \\label{e72}\n\\end{equation}\nThe $S_{2}\\,$part of the kernel gives zero contribution in the collision\nintegral\n\\begin{equation}\n\\int [f,f]\\,\\,\\delta \\left( \\mathbf{v}_{C}-\\mathbf{v}_{A}\\right) \\,\\delta\n\\left( \\mathbf{v}-\\mathbf{v}_{B}\\right) \\,d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v%\n}_{B}\\,d^{3}\\mathbf{v}_{C}=0.  \\label{e80}\n\\end{equation}\n\nWe evaluate the integral in eq.(\\ref{e70}) in the CM coordinates. We write $%\nd^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}_{B}=d^{3}\\mathbf{H}\\,d^{3}\\mathbf{u}$\nand $d^{3}\\mathbf{v}_{C}=d^{3}\\mathbf{u}^{\\prime }=(u^{\\prime\n})^{2}\\,du^{\\prime }\\,d\\mathbf{n}^{\\prime }$. When we do the integral we\nobtain\n\\begin{equation}\n\\frac{\\partial f(\\mathbf{v})}{\\partial \\tau }=\\frac{1}{R}\\int\n[f,f]\\,u\\,\\sigma (\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u}\\,d%\n\\mathbf{n}^{\\prime },  \\label{e90}\n\\end{equation}\nwhere $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$ are expressed in terms\nof the variables $\\mathbf{v,u,n}^{\\prime }$ in eqs.(\\ref{e105},\\ref{e106},%\n\\ref{e107}).\n\nThe equation(\\ref{e90}) is essentially the Boltzmann equation with the\ndifference that the Boltzmann equation is written for density in physical\nspace. To obtain the Boltzmann equation we write this equation for $F(%\n\\mathbf{v})=\\left( N\/V\\right) f(\\mathbf{v})$ where $V\\,$is the volume of the\ngas. Then we obtain\n\\begin{equation}\n\\frac{\\partial F(\\mathbf{v})}{\\partial \\tau }=\\frac{1}{R}\\left( \\frac{V}{N}%\n\\right) \\int \\left[ F(\\mathbf{v}_{A})F(\\mathbf{v}_{B})-F(\\mathbf{v}_{C})F(%\n\\mathbf{v})\\right] \\,u\\,\\sigma (\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}%\n\\mathbf{u}\\,d\\mathbf{n}^{\\prime }  \\label{e110}\n\\end{equation}\nNow, if we change to the variable $t=\\tau V\/RN=2nV\/RN^{2}$ we obtain the\nBoltzmann equation for a homogenous gas\n\\begin{equation}\n\\frac{\\partial F(\\mathbf{v})}{\\partial t}=\\int \\left[ F(\\mathbf{v}_{A})\\,F(%\n\\mathbf{v}_{B})-F(\\mathbf{v}_{C})\\,F(\\mathbf{v})\\right] \\,u\\,\\sigma (\\mathbf{%\nn},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime }\n\\label{e120}\n\\end{equation}\nHere $t$ must be interpreted as the physical time and $t=2nV\/RN^{2}$ formula\nconnects the physical time $t$ and number of collision attempts $n$.\n\nLet us state the algorithm for a homogenous gas. We choose a number $R$ big\nenough such that for only very few (say less than one in thousand) pairs $%\nu\\Sigma \/R$ will exceed unity. We make $n=RN^{2}t\/2V$ collision attempts to\nreach the desired time. For each pair we take a random number $r$ and we\nallow the collision to happen if $r<u\\Sigma \/R$. If the collision is\nallowed, we choose the direction of scattering $\\mathbf{n}^{\\prime }$\naccording to the probability density $\\sigma (\\mathbf{n},\\mathbf{n}^{\\prime\n})\/\\Sigma $ and a few more random numbers are used for that. Then we\ncalculate and store final velocities for the colliding pairs and pick\nanother pair. We keep taking and colliding pairs until we reach the desired\ntime.\n\nSuppose the formula $n=RN^{2}t\/2V$ yields 234.783 collisions. How do you\nmake 0.783 collisions? The way to do this in practise is to make 234\ncollisions first. Then throw a random number $r$ and if $r<0.783$ then go on\nto make a collision attempt. This can be justified from the formula\n\\begin{equation}\nf(\\mu \\mathbf{;}n+1)=f(\\mu \\mathbf{;}n)+\\frac{2}{N}\\int [f,f]\\,T(\\mu\n_{A},\\mu _{B};\\mu _{C},\\mu )\\,d\\mu _{A}\\,d\\mu _{B}\\,d\\mu _{C}.  \\label{e131}\n\\end{equation}\nAfter making $n$ collision attempts with the NTC\\ kernel $S(\\mathbf{v}_{A},%\n\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})\\,$we can change the kernel to\n\\begin{equation}\nP(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})=q\\,S(\\mathbf{v}%\n_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})+(1-q)\\,\\delta \\left( \\mathbf{v%\n}_{C}-\\mathbf{v}_{A}\\right) \\,\\delta \\left( \\mathbf{v}-\\mathbf{v}_{B}\\right)\n.  \\label{e140}\n\\end{equation}\nThis kernel makes a NTC\\ collision attempt with a probability $q\\,$(which\nwas $0.783$ in the above example) and a null collision happens with the\nprobability $1-q$. We use this kernel for the $(n+1)^{th}$ collision attempt\n(it is permissible to change the kernel) and this causes another $\\Delta\n\\tau =2q\/N\\,$collision time and $\\Delta t=q(2V\/RN^{2})$ real time increase.\n\n\\subsection{Mixture of gases without internal degrees of freedom}\n\nThe state of particles are defined by three components of the velocity\nvector $\\mathbf{v}$ and one kind index for which we will use $p,q,r,s$\ncharacters. We have $M$ kind of gas without internal states in the mixture\nand there are $N_{p}$ number of molecules of the $p^{th}$ kind. The mass of $%\np^{th}$ kind molecule is $m_{p}$. The probability density $f(\\mu )=f(\\mathbf{%\nv,}p)$ will be written as $f^{p}(\\mathbf{v})$.\n\nParticles with states $\\mu _{A}=(\\mathbf{v}_{A},s),\\mathbf{\\ }$and $\\mu\n_{B}=(\\mathbf{v}_{B},r)$ enter the collision and particles with states $\\mu\n_{C}=(\\mathbf{v}_{C},q)$ and $\\mu _{D}=(\\mathbf{v,}p\\mathbf{)}$ exits the\ncollision. The integration over $\\mu $ such as $\\int f^{p}(\\mathbf{v})d\\mu $\nstands for three integrations over $\\mathbf{v}$ and summation over $p$. The\ncenter of mass (CM) coordinates are defined in eqs.(\\ref{e3},\\ref{e3b},\\ref\n{e4}).\n\nThe NTC\\ kernel $G_{pq}^{rs}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},%\n\\mathbf{v})=G_{1}+G_{2}$ is given by\n\\begin{eqnarray}\nG_{1} &=&\\frac{2}{R}\\,\\delta \\left( \\mathbf{H}-\\mathbf{H}^{\\prime }\\right)\n\\,\\,\\delta \\left( u^{2}-(u^{\\prime })^{2}\\right) \\,\\sigma _{pq}(\\mathbf{n},%\n\\mathbf{n}^{\\prime })\\,\\delta _{pr}\\,\\delta _{qs},  \\label{f20} \\\\\nG_{2} &=&\\left( 1-\\frac{u\\Sigma _{pq}}{R}\\right) \\,\\delta \\left( \\mathbf{v}%\n_{C}-\\mathbf{v}_{A}\\right) \\,\\delta \\left( \\mathbf{v}-\\mathbf{v}_{B}\\right)\n\\,\\delta _{pr}\\,\\delta _{qs}.  \\label{f30}\n\\end{eqnarray}\nHere $\\sigma _{pq}(\\mathbf{n},\\mathbf{n}^{\\prime })$ is the differential\ncross section between gases of the $p^{th}$ and $q^{th}$ kind and $\\Sigma\n_{pq}\\,$is the total cross section which is given by\n\\begin{equation}\n\\Sigma _{pq}=\\int \\sigma _{pq}(\\mathbf{n},\\mathbf{n}^{\\prime })\\,d\\mathbf{n}%\n^{\\prime },  \\label{f40}\n\\end{equation}\nwhere $d\\mathbf{n}^{\\prime }$ is the solid angle in the direction of $%\n\\mathbf{n}^{\\prime }$. The $\\delta _{pr}\\delta _{qs}$ term in the kernel\ninsures that particles do not loose their identities during the collisions.\nAgain $\\sigma _{pq}(\\mathbf{n},\\mathbf{n}^{\\prime })=\\sigma _{rs}(\\mathbf{n,n%\n}^{\\prime })\\,$due to the $\\delta _{pr}\\delta _{qs}$ term and we also have\nthe symmetry $\\sigma _{pq}(\\mathbf{n},\\mathbf{n}^{\\prime })=\\sigma _{qp}(%\n\\mathbf{n}^{\\prime },\\mathbf{n})$. The kernel is obviously symmetric. The\nterm $\\delta (u^{2}-(u^{\\prime })^{2})\\ $ and $\\delta \\left( \\mathbf{H}-%\n\\mathbf{H}^{\\prime }\\right) $ have the same meanings as before and the\nkernel satisfies the normalization condition\n\\begin{equation}\n\\sum_{p=1}^{M}\\sum_{q=1}^{M}\\int G_{pq}^{rs}(\\mathbf{v}_{A},\\mathbf{v}_{B};%\n\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v}=1.\n\\label{f50}\n\\end{equation}\n\nAgain $G_{2}$ part of the kernel directly transfer initial velocities to the\nfinal velocities with a probability $1-(u\\Sigma _{rs})\/R$ and hence causes a\nnull collision. The probability of making a real collision is\n\\begin{equation}\n\\sum_{p=1}^{M}\\sum_{q=1}^{M}\\int (G_{1})_{pq}^{rs}(\\mathbf{v}_{A},\\mathbf{v}%\n_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v=}\\frac{%\nu\\Sigma _{rs}}{R},  \\label{f60}\n\\end{equation}\nwhere integral is calculated in the CM coordinates.\n\nInserting $G_{pq}^{rs}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v%\n})$ in eq.(\\ref{a130}) and doing the summations over $r,s$ and doing the\nintegrals in the CM coordinates we obtain\n\\begin{eqnarray}\n\\frac{\\partial f^{p}(\\mathbf{v})}{\\partial \\tau } &=&\\sum_{q=1}^{M}\\int\nG_{pq}^{pq}(\\mu _{A},\\mu _{B};\\mu _{C},\\mu )\\,[f^{q},f^{p}]\\,\\,d^{3}\\mathbf{v%\n}_{A}\\,d^{3}\\mathbf{v}_{B}\\,d^{3}\\mathbf{v}_{C},  \\label{f70} \\\\\n&=&\\frac{1}{R}\\sum_{q=1}^{M}\\int [f^{q},f^{p}]\\,\\,u\\,\\sigma _{pq}(\\mathbf{n},%\n\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u\\,}d\\mathbf{n}^{\\prime },  \\label{f71}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\lbrack f^{q},f^{p}]=f^{q}(\\mathbf{v}_{A})\\,f^{p}(\\mathbf{v}_{B})-f^{q}(%\n\\mathbf{v}_{C})\\,f^{p}(\\mathbf{v})  \\label{f74}\n\\end{equation}\n\nAgain we write this equation for $F^{p}(\\mathbf{v})=\\left( N\/V\\right) f^{p}(%\n\\mathbf{v})$ and take $t=2nV\/RN^{2}$ to obtain Boltzmann equation for a\nmixture of homogenous gases without internal states\n\\begin{equation}\n\\frac{\\partial F^{p}(\\mathbf{v})}{\\partial t}=\\sum_{q=1}^{M}\\int \\left[\nF^{q}(\\mathbf{v}_{A})\\,F^{p}(\\mathbf{v}_{B})-F^{q}(\\mathbf{v}_{C})\\,F^{p}(%\n\\mathbf{v})\\right] \\,u\\,\\sigma _{pq}(\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}%\n\\mathbf{u\\,}d\\mathbf{n}^{\\prime }.  \\label{f90}\n\\end{equation}\nHere $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$ are expressed in terms\nof the variables $\\mathbf{v,u,n}^{\\prime }$ in eqs.(\\ref{f80},\\ref{f81},\\ref\n{f82}).\n\nThe algorithm is the same. We take $n=RN^{2}t\/2V$ pairs and allow each\ncollision with a probability $(u\\Sigma _{rs})\/R.$ If the collision is\nallowed we choose the scattering angle according to the $\\sigma _{rs}(%\n\\mathbf{n},\\mathbf{n}^{\\prime })\/\\Sigma _{rs}$ probability distribution.\n\nNote that the normalization of $f^{p}(\\mathbf{v})\\,$is given by\n\\begin{equation}\n\\sum_{p=1}^{M}\\int f^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v}=1.  \\label{f100}\n\\end{equation}\nThe integral $\\int f^{p}(\\mathbf{v})d^{3}\\mathbf{v}$ is conserved during the\nsimulation. From eq.(\\ref{f70}) its rate of change is\n\\begin{eqnarray}\n\\frac{d}{d\\tau }\\int f^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v} &=&\\int \\frac{%\n\\partial f^{p}(\\mathbf{v})}{\\partial \\tau }\\,d^{3}\\mathbf{v=}%\n\\sum_{q=1}^{M}\\int G_{pq}^{pq}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},%\n\\mathbf{v})  \\label{f105} \\\\\n&&\\times \\ \\left[ f^{q}(\\mathbf{v}_{A})\\,f^{p}(\\mathbf{v}_{B})-f^{q}(\\mathbf{%\nv}_{C})\\,f^{p}(\\mathbf{v})\\right] \\,d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}%\n_{B}\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v.}  \\nonumber\n\\end{eqnarray}\n\\newline\nFrom normalization of probabilities in eqs.(\\ref{a30},\\ref{f50}) we have\n\\begin{eqnarray}\n\\int G_{pq}^{pq}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}%\n)\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v} &=&1  \\label{f111} \\\\\n\\int G_{pq}^{pq}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}%\n)\\,d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}_{B} &=&1.  \\label{f111a}\n\\end{eqnarray}\nUsing these relations the integral on the right side of eq.(\\ref{f105}) can\nbe written as\n\\begin{eqnarray}\n\\frac{d}{d\\tau }\\int f^{p}(\\mathbf{v})d^{3}\\mathbf{v} &=&\\sum_{q=1}^{M}\\int\nf^{q}(\\mathbf{v}_{A})\\,\\,f^{p}(\\mathbf{v}_{B})\\,d^{3}\\mathbf{v}_{A}\\,d^{3}%\n\\mathbf{v}_{B}  \\label{f112} \\\\\n&&-\\sum_{q=1}^{M}\\int f^{q}(\\mathbf{v}_{C})\\,\\,f^{p}(\\mathbf{v})d^{3}\\mathbf{%\nv}_{C}\\,d^{3}\\mathbf{v.}  \\nonumber\n\\end{eqnarray}\nThese two terms are equal and they cancel each other yielding constancy of $%\n\\int f^{p}(\\mathbf{v})d^{3}\\mathbf{v}$.\n\nThe number of molecules of the $p^{th}\\,$kind is\n\\begin{equation}\nN_{p}=N\\int f^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v,}  \\label{f120}\n\\end{equation}\nand it remains constant as it should. Hence the $F^{p}(\\mathbf{v})$ is\nnormalized as\n\\begin{equation}\n\\int F^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v\\,}d^{3}\\mathbf{x=}N_{p},  \\label{f125}\n\\end{equation}\nwhere $\\mathbf{x}$ is position of the molecule.\n\n\\subsection{One kind of gas with internal degrees of freedom}\n\nFor a homogeneous gas with internal states the $\\mu $ stands for velocity $%\n\\mathbf{v}$ and a discrete index (for which we use $\\alpha ,\\beta ,i,j$)\ndefining the internal quantum state of the molecule. The mass of the\nmolecules is $m$. Particles with states $\\mu _{A}=(\\mathbf{v}_{A}\\mathbf{,}%\n\\beta )$ and $\\mu _{B}=(\\mathbf{v}_{B}\\mathbf{,}\\alpha )$ enter the\ncollision and particles with states $\\mu _{C}=(\\mathbf{v}_{C}\\mathbf{,}j)$\nand $\\mu _{D}=(\\mathbf{v,}i)$ exits the collision. The integral over $\\mu $\nstands for integration over $\\mathbf{v}$ and summation over the internal\nstate index. The internal energy of molecule in the state $\\gamma $ is $%\nE_{\\gamma }$ and $\\epsilon =E_{\\alpha }+E_{\\beta }$ and $\\epsilon ^{\\prime\n}=E_{i}+E_{j}$. The center of mass (CM) coordinates are defined in eqs.(\\ref\n{e4},\\ref{e5}).\n\nLet us define the no time counter (NTC) kernel $K_{ij}^{\\alpha \\beta }(%\n\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})=K_{1}+K_{2}$ where\n\\begin{equation}\nK_{1}=\\frac{1}{R}\\delta (\\mathbf{H}-\\mathbf{H}^{\\prime })\\,\\delta \\left[\n\\frac{2}{m_{r}}\\epsilon +u^{2}-\\frac{2}{m_{r}}\\epsilon ^{\\prime }-(u^{\\prime\n})^{2}\\right] \\,\\frac{2u}{u^{\\prime }}\\,\\,\\sigma _{ij}^{\\alpha \\beta }(%\n\\mathbf{n},\\mathbf{n}^{\\prime }),  \\label{g21}\n\\end{equation}\nand\n\\begin{equation}\nK_{2}=\\left( 1-\\frac{1}{R}\\sum_{i}\\sum_{j}u\\Sigma _{ij}^{\\alpha \\beta\n}\\right) \\delta (\\mathbf{v}_{C}-\\mathbf{v}_{A})\\,\\delta (\\mathbf{v}-\\mathbf{v%\n}_{B})\\,\\delta _{i\\alpha }\\,\\delta _{j\\beta }.  \\label{g30}\n\\end{equation}\nHere $m_{r}=m\/2$ is the reduced mass where $m$ is the mass of the molecules\nand $R$ is a chosen parameter. The $\\sigma _{ij}^{\\alpha \\beta }(\\mathbf{n},%\n\\mathbf{n}^{\\prime })$ is differential and the $\\Sigma _{ij}^{\\alpha \\beta }$\nis the total cross section into the internal states $i,j$\n\\begin{equation}\n\\Sigma _{ij}^{\\alpha \\beta }=\\int \\sigma _{ij}^{\\alpha \\beta }(\\mathbf{n},%\n\\mathbf{n}^{\\prime })\\,d\\mathbf{n}^{\\prime },  \\label{g40}\n\\end{equation}\nwhere $d\\mathbf{n}^{\\prime }$ is the solid angle in the direction of $%\n\\mathbf{n}^{\\prime }$. This kernel is symmetric due to the reciprocity\nrelation of the inelastic scattering cross sections\\cite{Reciprocity}\n\\begin{equation}\nu^{2}\\,\\sigma _{ij}^{\\alpha \\beta }(\\mathbf{n},\\mathbf{n}^{\\prime\n})=(u^{\\prime })^{2}\\,\\sigma _{\\alpha \\beta }^{ij}(\\mathbf{n}^{\\prime },%\n\\mathbf{n}),  \\label{g50}\n\\end{equation}\nbecause $(u\/u^{\\prime })\\,\\sigma _{ij}^{\\alpha \\beta }=(u^{\\prime\n}\/u)\\,\\sigma _{\\alpha \\beta }^{ij}$.\n\nThe $K_{2}$ part of $K_{ij}^{\\alpha \\beta }(\\mathbf{v}_{A},\\mathbf{v}_{B};%\n\\mathbf{v}_{C},\\mathbf{v})$ directly transfers initial state to the final\nstate and causes a null collision. The probability of making a real\ncollision into the states $(i,j)$ is\n\\begin{equation}\nP_{ij}=\\int K_{1}\\,d\\mathbf{v}_{C}\\,d\\mathbf{v=}\\frac{u\\,\\Sigma\n_{ij}^{\\alpha \\beta }}{R}.  \\label{g60}\n\\end{equation}\nTherefore total probability of making a real collision is $%\n(\\sum_{i}\\sum_{j}\\,u\\Sigma _{ij}^{\\alpha \\beta })\/R$.\n\nInserting the $K_{ij}^{\\alpha \\beta }(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v%\n}_{C},\\mathbf{v})$ into the eq.(\\ref{a130}) and doing the integrals in the\nCM\\ coordinates we obtain\n\\begin{equation}\n\\frac{\\partial f_{i}(\\mathbf{v})}{\\partial \\tau }=\\frac{1}{R}\\sum_{\\alpha\n}\\sum_{\\beta }\\sum_{j}\\int \\left[ f_{\\beta }(\\mathbf{v}_{A})\\,f_{\\alpha }(%\n\\mathbf{v}_{B})-f_{j}(\\mathbf{v}_{C})\\,f_{i}(\\mathbf{v})\\right] \\,u\\,\\sigma\n_{ij}^{\\alpha \\beta }(\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u}\\,d%\n\\mathbf{n}^{\\prime }.  \\label{g71}\n\\end{equation}\nHere the $K_{2}$ part does not contribute to the collision integral as\nbefore.\n\nAgain defining time as $t=\\tau V\/RN=2nV\/RN^{2}$ and defining the new\nfunctions $F_{i}(\\mathbf{v})=(N\/V)f_{i}(\\mathbf{v})$ this is expressed as\n\\begin{equation}\n\\frac{\\partial F_{i}}{\\partial t}=\\sum_{\\alpha }\\sum_{\\beta }\\sum_{j}\\int\n\\left[ F_{\\beta }(\\mathbf{v}_{A})\\,F_{\\alpha }(\\mathbf{v}_{B})-F_{j}(\\mathbf{%\nv}_{C})\\,F_{i}(\\mathbf{v})\\right] \\,\\,u\\,\\sigma _{ij}^{\\alpha \\beta }(%\n\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime },\n\\label{g80}\n\\end{equation}\nwhere $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$ are expressed in terms\nof the variables $\\mathbf{v,u,n}^{\\prime }$ in eqs.(\\ref{g75},\\ref{g76},\\ref\n{g77},\\ref{g78}). These equations are the Wang Chang-Uhlenbeck equations for\na gas with internal degrees of freedom. Here the states are assumed\nnondegenerate for simplicity. Generalization to degenerate states is also\nvery straightforward.\n\nAgain we choose a number $R$ big enough such that for only very few (say\nless than one in thousand) pairs $(\\sum_{i}\\sum_{j}u\\Sigma _{ij}^{\\alpha\n\\beta })\/R$ will exceed unity. We chose $n=RN^{2}t\/2V$ random pairs. For\neach pair we take a random number $r$ and we allow the collision to happen\nif $r<(\\sum_{i}\\sum_{j}u\\Sigma _{ij}^{\\alpha \\beta })\/R$. If collision is\nallowed we choose the final state $(i,j)$ with the probability $\\Sigma\n_{ij}^{\\alpha \\beta }\/(\\sum_{i}\\sum_{j}\\Sigma _{ij}^{\\alpha \\beta })$ and\nanother random number is used to choose the final state. Finally we choose\nthe direction of scattering $\\mathbf{n}^{\\prime }$ according to the\nprobability density $\\sigma _{ij}^{\\alpha \\beta }(\\mathbf{n},\\mathbf{n}%\n^{\\prime })\/\\Sigma _{ij}^{\\alpha \\beta }$ and a few more random numbers are\nused for that. Then we calculate and store final velocities and state\nindices for the colliding pair and go on to choose next pair.\n\n\\subsection{Mixture of gases with internal degrees of freedom}\n\nThis case is a combination of previous two cases and it is very\nstraightforward but unfortunately there are too many indices. The state of\nparticles are defined by three components of the velocity vector $\\mathbf{v}$\nand one kind index for which we use $p,q,r,s$ and one internal state index\nfor which we use $i,j,\\alpha ,\\beta $. We have $M$ kind of gas with internal\nstates in the mixture and there are $N_{p}$ number of molecules of the $%\np^{th}$ kind. The internal energy of $i^{th}$ internal state of $p^{th}$\nkind molecule is $E_{i}^{p}$. The probability density $f(\\mu )=f(\\mathbf{v,}%\ni,p)$ will be written as $f_{i}^{p}(\\mathbf{v})$.\n\nParticles with states $\\mu _{A}=(\\mathbf{v}_{A},\\beta ,s),\\mathbf{\\ }$and $%\n\\mu _{B}=(\\mathbf{v}_{B},\\alpha ,r)$ enter the collision and particles with\nstates $\\mu _{C}=(\\mathbf{v}_{C},j,q)$ and $\\mu _{D}=(\\mathbf{v,}i,p\\mathbf{)%\n}$ exits the collision. We also define $\\epsilon =E_{\\beta }^{s}+E_{\\alpha\n}^{r}$ and $\\epsilon ^{\\prime }=E_{j}^{q}+E_{i}^{p}$. The integration over $%\n\\mu $ such as $\\int f_{i}^{p}(\\mathbf{v})d\\mu $ stands for three\nintegrations over $\\mathbf{v}$ and summations over $i$ and $p$. The center\nof mass (CM) coordinates are defined in eqs.(\\ref{e3},\\ref{e3b},\\ref{e4}).\n\nThe NTC kernel is $Q_{ij,pq}^{\\alpha \\beta ,rs}(\\mathbf{v}_{A},\\mathbf{v}%\n_{B};\\mathbf{v}_{C},\\mathbf{v})=Q_{1}+Q_{2}$ where $Q_{1}$ and $Q_{2}$ are\ndefined as\n\\begin{equation}\nQ_{1}=\\frac{1}{R}\\delta (\\mathbf{H}-\\mathbf{H}^{\\prime })\\,\\delta \\left[\n\\frac{2}{m_{r}}\\epsilon +u^{2}-\\frac{2}{m_{r}}\\epsilon ^{\\prime }-(u^{\\prime\n})^{2}\\right] \\frac{2u}{u^{\\prime }}\\,\\,\\sigma _{ij,pq}^{\\alpha \\beta ,pq}(%\n\\mathbf{n},\\mathbf{n}^{\\prime })\\,\\delta _{pr}\\,\\delta _{qs}.  \\label{h21}\n\\end{equation}\nand\n\\begin{equation}\nQ_{2}=\\left( 1-\\frac{1}{R}\\sum_{i}\\sum_{j}u\\Sigma _{ij,pq}^{\\alpha \\beta\n,pq}\\right) \\delta (\\mathbf{v}_{C}-\\mathbf{v}_{A})\\,\\delta (\\mathbf{v}-%\n\\mathbf{v}_{B})\\,\\delta _{i\\alpha }\\,\\delta _{j\\beta }\\,\\delta _{pr}\\,\\delta\n_{qs}.  \\label{h30}\n\\end{equation}\nThe delta functions $\\delta _{pr}\\delta _{qs}$ insures that the molecules do\nno change identities during the collision. Here $%\nm_{r}=m_{A}m_{B}\/(m_{A}+m_{B})$ is the reduced mass, $R$ is a chosen\nparameter. The $\\sigma _{ij,pq}^{\\alpha \\beta ,pq}(\\mathbf{n},\\mathbf{n}%\n^{\\prime })$ is the differential cross section between species of the $%\np^{th} $ kind in the state $\\alpha $ and $q^{th}$ kind in the state $\\beta $\nand $\\Sigma _{ij,pq}^{\\alpha \\beta ,pq}$ is the total cross section into the\nchannel $(i,j)$\n\\begin{equation}\n\\Sigma _{ij,pq}^{\\alpha \\beta ,pq}=\\int \\sigma _{ij,pq}^{\\alpha \\beta ,pq}(%\n\\mathbf{n},\\mathbf{n}^{\\prime })\\,d\\mathbf{n}^{\\prime }  \\label{h40}\n\\end{equation}\nwhere $d\\mathbf{n}^{\\prime }$ is the solid angle in the direction of $%\n\\mathbf{n}^{\\prime }$. The $Q_{ij,pq}^{\\alpha \\beta ,rs}(\\mathbf{v}_{A},%\n\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})$ is also symmetric due to eq.(\\ref\n{g50}). The $Q_{2}$ directly transfers initial states to the final states\nand causes a null collision. The probability of making a real collision into\nthe states $(i,j)$ is\n\\begin{equation}\nP_{ij}=\\int Q_{1}\\,d\\mathbf{v}_{C}\\,d\\mathbf{v=}\\frac{u\\,\\Sigma\n_{ij,pq}^{\\alpha \\beta ,pq}}{R}  \\label{h50}\n\\end{equation}\nTherefore total probability of making a real collision is $%\n(\\sum_{i}\\sum_{j}u\\Sigma _{ij,pq}^{\\alpha \\beta ,pq})\/R$.\n\nInserting the $Q_{ij,pq}^{\\alpha \\beta ,rs}(\\mathbf{v}_{A},\\mathbf{v}_{B};%\n\\mathbf{v}_{C},\\mathbf{v})$ into the eq.(\\ref{a130}) and doing the integrals\nin the CM\\ coordinates we obtain\n\\begin{equation}\n\\frac{\\partial f_{i}^{p}(\\mathbf{v})}{\\partial \\tau }=\\sum_{q=1}^{M}\\sum_{%\n\\alpha }\\sum_{\\beta }\\sum_{j}\\int [f^{q},f^{p}]_{ij}^{\\alpha \\beta\n}\\,Q_{ij,pq}^{\\alpha \\beta ,pq}(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},%\n\\mathbf{v})\\,d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}_{B}\\,d^{3}\\mathbf{v}_{C},\n\\label{h65}\n\\end{equation}\nwhere\n\\begin{equation}\n\\lbrack f^{q},f^{p}]_{ij}^{\\alpha \\beta }=f_{\\beta }^{q}(\\mathbf{v}%\n_{A})\\,f_{\\alpha }^{p}(\\mathbf{v}_{B})-f_{j}^{q}(\\mathbf{v}_{C})\\,f_{i}^{p}(%\n\\mathbf{v}).  \\label{h66}\n\\end{equation}\nAfter inserting $Q_{ij,pq}^{\\alpha \\beta ,pq}$ we obtain\n\\begin{equation}\n\\frac{\\partial f_{i}^{p}(\\mathbf{v})}{\\partial \\tau }=\\frac{1}{R}\n\\sum_{q=1}^{M}\\sum_{\\alpha }\\sum_{\\beta }\\sum_{j}\\int\n[f^{q},f^{p}]_{ij}^{\\alpha \\beta }\\,u\\,\\sigma _{ij,pq}^{\\alpha \\beta ,pq}(%\n\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u\\,}d\\mathbf{n}^{\\prime }.\n\\label{h67}\n\\end{equation}\nThe $Q_{2}$ part does not contribute to the collision integral as before.\nExpressions of $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$ in terms of $%\n\\mathbf{v},\\mathbf{u,}\\mathbf{n}^{\\prime }$ are given in eqs.(\\ref{e100},\\ref\n{e101},\\ref{e102},\\ref{e103})\n\nAgain defining time as $t=\\tau V\/RN=2nV\/RN^{2}$ and defining the new\nfunctions $F_{i}^{p}(\\mathbf{v})=(N\/V)\\,f_{i}^{p}(\\mathbf{v})$ this is\nexpressed as\n\\begin{eqnarray}\n\\frac{\\partial F_{i}^{p}(\\mathbf{v})}{\\partial t} &=&\\sum_{q=1}^{M}\\sum_{%\n\\alpha }\\sum_{\\beta }\\sum_{j}\\int \\left( F_{\\beta }^{q}(\\mathbf{v}%\n_{A})\\,F_{\\alpha }^{p}(\\mathbf{v}_{B})-F_{j}^{q}(\\mathbf{v}_{C})\\,F_{i}^{p}(%\n\\mathbf{v})\\right)  \\label{h70} \\\\\n&&\\times u\\,\\sigma _{ij,pq}^{\\alpha \\beta ,pq}(\\mathbf{n},\\mathbf{n}^{\\prime\n})\\,d^{3}\\mathbf{u\\,}d\\mathbf{n}^{\\prime }.  \\nonumber\n\\end{eqnarray}\nThese equations are the Wang Chang-Uhlenbeck equations for a mixture of\ngases with internal degrees of freedom. Here the states are assumed\nnondegenerate for simplicity again.\n\nAgain we choose a number $R$ big enough such that for only very few (say\nless than one in thousand) pairs $(\\sum_{i}\\sum_{j}u\\Sigma _{ij,pq}^{\\alpha\n\\beta ,pq})\/R$ will exceed unity. We chose $n=RN^{2}t\/2V$ random pairs. For\neach pair we take a random number $r$ and we allow the collision to happen\nif $r<(\\sum_{i}\\sum_{j}u\\Sigma _{ij,pq}^{\\alpha \\beta ,pq})\/R$. If collision\nis allowed we choose the final state $(i,j)$ with the probability $\\Sigma\n_{ij,pq}^{\\alpha \\beta ,pq}\/(\\sum_{i}\\sum_{j}\\Sigma _{ij,pq}^{\\alpha \\beta\n,pq})$ and another random number is used to choose the final state. Finally\nwe choose the direction of scattering $\\mathbf{n}^{\\prime }$ according to\nthe probability density $\\sigma _{ij,pq}^{\\alpha \\beta ,pq}(\\mathbf{n},%\n\\mathbf{n}^{\\prime })\/\\Sigma _{ij,pq}^{\\alpha \\beta ,pq}$ and a few more\nrandom numbers are used for that. Then we calculate and store final\nvelocities and state indices for the colliding pair and go on to choose next\npair.\n\nNote that the normalization of $f_{i}^{p}(\\mathbf{v})\\,$is given by\n\\begin{equation}\n\\sum_{p}\\sum_{i}\\int f_{i}^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v}=1.  \\label{h80}\n\\end{equation}\nThe expression $\\sum_{i}\\int f_{i}^{p}(\\mathbf{v})d^{3}\\mathbf{v}$ is\nconserved during the simulation. From eq.(\\ref{h65}) its rate of change is\n\\begin{eqnarray}\n\\frac{d}{d\\tau }\\sum_{i}\\int f_{i}^{p}(\\mathbf{v})d^{3}\\mathbf{v}\n&=&\\sum_{i}\\int \\frac{\\partial f_{i}^{p}(\\mathbf{v})}{\\partial \\tau }\\,d^{3}%\n\\mathbf{v}=\\sum_{q=1}^{M}\\sum_{\\alpha }\\sum_{\\beta }\\sum_{i}\\sum_{j}\n\\label{h91} \\\\\n&&\\int [f^{q},f^{p}]_{ij}^{\\alpha \\beta }\\,Q_{ij,pq}^{\\alpha \\beta ,pq}(%\n\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}\\mathbf{v}%\n_{A}\\,d^{3}\\mathbf{v}_{B}\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v}  \\nonumber\n\\end{eqnarray}\n\\newline\nFrom symmetry and normalization of the kernel given in eqs.(\\ref{a10},\\ref\n{a20},\\ref{a30}) we have\n\\begin{eqnarray}\n\\sum_{i}\\sum_{j}\\int Q_{ij,pq}^{\\alpha \\beta ,pq}(\\mathbf{v}_{A},\\mathbf{v}%\n_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v} &=&1\n\\label{h120} \\\\\n\\sum_{\\alpha }\\sum_{\\beta }\\int Q_{ij,pq}^{\\alpha \\beta ,pq}(\\mathbf{v}_{A},%\n\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})\\,d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{%\nv}_{B} &=&1  \\label{h121}\n\\end{eqnarray}\nUsing this, we express eq.(\\ref{h91}) as\n\\begin{eqnarray}\n\\frac{d}{d\\tau }\\sum_{i}\\int f_{i}^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v}\n&=&\\sum_{q=1}^{M}\\sum_{\\alpha }\\sum_{\\beta }\\int f_{\\beta }^{q}(\\mathbf{v}%\n_{A})\\,f_{\\alpha }^{p}(\\mathbf{v}_{B})\\,d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}%\n_{B}  \\label{h130} \\\\\n&&-\\sum_{q=1}^{M}\\sum_{i}\\sum_{j}\\int f_{j}^{q}(\\mathbf{v}_{C})\\,f_{i}^{p}(%\n\\mathbf{v})\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{v}  \\nonumber\n\\end{eqnarray}\nThese two terms are equal and they cancel each other yielding constancy of $%\n\\sum_{i}f_{i}^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v}$. The number of molecules of\nthe $p^{th}\\,$kind is\n\\begin{equation}\nN_{p}=N\\sum_{i}\\int f_{i}^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v}  \\label{h140}\n\\end{equation}\nand as the above argument shows, it remains constant as it should. Hence the\n$F_{i}^{p}(\\mathbf{v})$ is normalized as\n\\begin{equation}\n\\sum_{i}\\int F_{i}^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v\\,}d^{3}\\mathbf{x=}N_{p},\n\\label{h150}\n\\end{equation}\nwhere $\\mathbf{x}$ is position of the molecule.\n\n\\subsection{Relation to Kac's work}\n\nFifty years ago M. Kac\\cite{Kac} introduced a master equation similar to\nours and derived the Boltzmann equation for a homogenous gas from it. Here\nwe summarize his work and point out similarities. We will use a different\nnotation than his.\n\nSuppose we have $N$ particles in a gas contained in volume $V$. Collisions\nare assumed to take place randomly within the gas. Again we have a\nprobability distribution $f^{(N)}(\\mathbf{v}_{1},\\mathbf{v}_{2},...,\\mathbf{v%\n}_{N};t)$ for their velocities. For brevity we will show this as $f^{(N)}(%\n\\mathbf{v};t)$ wherever convenient. Probability that the $i^{th}$ and $%\nj^{th} $ particles having velocities $\\mathbf{v}_{A}$ and $\\mathbf{v}_{B}$\nwill collide and emerge with velocities $\\mathbf{v}_{C}\\,$and $\\mathbf{v}%\n_{D} $ in the phase space $d^{3}\\mathbf{v}_{C}d^{3}\\mathbf{v}_{D}$ in a time\ninterval $dt$ is $R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}%\n_{D})d^{3}\\mathbf{v}_{C}d^{3}\\mathbf{v}_{D}dt.$ Here the $R(\\mathbf{v}_{A},%\n\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}_{D})$ is a function connected to\ndifferential cross section but we will not need the precise relation until\nlater. The total collision probability in $dt$ time interval is $S(\\mathbf{v}%\n_{A},\\mathbf{v}_{B})dt$ where $S(\\mathbf{v}_{A},\\mathbf{v}_{B})\\,$is given\nby\n\\begin{equation}\nS(\\mathbf{v}_{A},\\mathbf{v}_{B})=\\int R(\\mathbf{v}_{A},\\mathbf{v}_{B};%\n\\mathbf{v}_{C},\\mathbf{v}_{D})d^{3}\\mathbf{v}_{C}d^{3}\\mathbf{v}_{D}.\n\\label{kac10}\n\\end{equation}\nAs usual we assume some symmetries for the $R(\\mathbf{v}_{A},\\mathbf{v}_{B},%\n\\mathbf{v}_{C},\\mathbf{v}_{D})$ function:\n\\begin{eqnarray}\nR(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}_{D}) &=&R(\\mathbf{v%\n}_{C},\\mathbf{v}_{D};\\mathbf{v}_{A},\\mathbf{v}_{B}),  \\label{kac20} \\\\\nR(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v}_{D}) &=&R(\\mathbf{v%\n}_{B},\\mathbf{v}_{A};\\mathbf{v}_{D},\\mathbf{v}_{C}).  \\label{kac30}\n\\end{eqnarray}\n\nThe $f^{(N)}(\\mathbf{v}_{1},\\mathbf{v}_{2},...,\\mathbf{v}_{N};t)$ satisfies\nthe master equation\n\\begin{equation}\n\\frac{\\partial f^{(N)}(\\mathbf{v})}{\\partial t}=-f^{(N)}(\\mathbf{v}%\n)\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}S(\\mathbf{v}_{i},\\mathbf{v}%\n_{j})+\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}\\int f_{ij}^{(N)}(\\mathbf{v}_{A},%\n\\mathbf{v}_{B})R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{i},\\mathbf{v}%\n_{j})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}_{B}  \\label{kac40}\n\\end{equation}\nIn order to see where this comes from we write it for infinitesimal time\ninterval $dt$:\n\\begin{eqnarray}\nf^{(N)}(\\mathbf{v;}t+dt) &=&f^{(N)}(\\mathbf{v;}t)\\left(\n1-dt\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}S(\\mathbf{v}_{i},\\mathbf{v}_{j})\\right)\n\\label{kac50} \\\\\n&&+dt\\left( \\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}\\int f_{ij}^{(N)}(\\mathbf{v}_{A},%\n\\mathbf{v}_{B})R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{i},\\mathbf{v}%\n_{j})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}_{B}\\right) .  \\nonumber\n\\end{eqnarray}\nLet us multiply both sides with $d^{3}\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}.$\nThen $f^{(N)}(\\mathbf{v;}t+dt)d^{3}\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}$ is\nthe probability that the velocities are in the phase space volume $d^{3}%\n\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}$ at time $t+dt$. The first term on the\nright is\n\\begin{equation}\n\\left( f^{(N)}(\\mathbf{v;}t)d^{3}\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}\\right)\n\\left( 1-dt\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}S(\\mathbf{v}_{i},\\mathbf{v}%\n_{j})\\right) .  \\label{kac60}\n\\end{equation}\nThe first parenthesis is the probability that the system was in $d^{3}%\n\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}$ phase space volume at time $t$ and the\nsecond parenthesis is the probability that no collisions occurred in $dt$\ntime interval. Their product is the probability of arriving $d^{3}\\mathbf{v}%\n_{1}...d^{3}\\mathbf{v}_{N}$ phase space volume at $t+dt$ without making a\ncollision. The second term in the right side are probabilities of arriving\nin $d^{3}\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}$ by making collisions with\ndifferent pairs. For example let us write $i=1,$ $j=2$ term:\n\\begin{equation}\n\\int \\left( f^{(N)}(\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{3},...,\\mathbf{%\nv}_{N})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}_{B}d^{3}\\mathbf{v}_{3}...d^{3}%\n\\mathbf{v}_{N}\\right) \\left( R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{1},%\n\\mathbf{v}_{2})d^{3}\\mathbf{v}_{1}d^{3}\\mathbf{v}_{2}dt\\right) .\n\\label{kac70}\n\\end{equation}\nThe first parenthesis under the integral is the probability that the system\nwas in the phase space volume $d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}_{B}d^{3}%\n\\mathbf{v}_{3}...d^{3}\\mathbf{v}_{N}$ at time $t$ and the second parenthesis\nis the probability that the collision between particles one and two took\nthem to $d^{3}\\mathbf{v}_{1}d^{3}\\mathbf{v}_{2}$ phase space volume. If we\nintegrate this product over $\\mathbf{v}_{A},\\mathbf{v}_{B}$ we obtain\nprobability of arriving in $d^{3}\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}$ at\ntime $t+dt$ via a collision between particles one and two. To obtain total\nprobability of arriving in $d^{3}\\mathbf{v}_{1}...d^{3}\\mathbf{v}_{N}$ at\ntime $t+dt$ via a collision we sum such terms over all possible pairs. This\nargument clearly shows how the master equation is derived.\n\nWriting $S(\\mathbf{v}_{i},\\mathbf{v}_{j})$ as\n\\begin{equation}\nS(\\mathbf{v}_{i},\\mathbf{v}_{j})=\\int R(\\mathbf{v}_{A},\\mathbf{v}_{B};%\n\\mathbf{v}_{j},\\mathbf{v}_{i})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}_{B}\n\\label{kac80}\n\\end{equation}\nthe master equation can be written in a more symmetric form\n\\begin{equation}\n\\frac{\\partial f^{(N)}(\\mathbf{v})}{\\partial t}=\\sum_{i=1}^{N}\\sum_{j\\neq\ni}^{N}\\int \\left( f_{ij}^{(N)}(\\mathbf{v}_{A},\\mathbf{v}_{B})-f^{(N)}(%\n\\mathbf{v})\\right) R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{i},\\mathbf{v}%\n_{j})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}_{B}  \\label{kac90}\n\\end{equation}\n\nAll of the results we obtained from our master equation can be obtained for\nthis master equation too. Kac\\cite{Kac} showed that the distribution goes to\nmicrocanonical distribution as $t\\rightarrow \\infty $. A hierarchy of\nreduced probability equations can be obtained for this master equation too.\nKac\\cite{Kac} showed that in the limit $N\\rightarrow \\infty $ if one starts\nfrom uncorrelated state at $t=0$ the system always remains uncorrelated. His\narguments was different than ours.\n\nThe first equation in the hierarchy (obtained by integrating over $\\mathbf{v}%\n_{2},\\mathbf{v}_{3},...,\\mathbf{v}_{N}$ ) is\n\\begin{equation}\n\\frac{\\partial f^{(1)}(\\mathbf{v})}{\\partial t}=2N\\int \\left( f^{(2)}(%\n\\mathbf{v}_{A},\\mathbf{v}_{B})-f^{(2)}(\\mathbf{v,v}_{C})\\right) R(\\mathbf{v}%\n_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})d^{3}\\mathbf{v}_{A}d^{3}%\n\\mathbf{v}_{B}d^{3}\\mathbf{v}_{C}  \\label{kac100}\n\\end{equation}\nIf we introduce AMC this equation becomes\n\\begin{equation}\n\\frac{\\partial f(\\mathbf{v})}{\\partial t}=2N\\int \\left( f(\\mathbf{v}_{A})f(%\n\\mathbf{v}_{B})-f(\\mathbf{v})f(\\mathbf{v}_{C})\\right) R(\\mathbf{v}_{A},%\n\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})d^{3}\\mathbf{v}_{A}d^{3}\\mathbf{v}%\n_{B}d^{3}\\mathbf{v}_{C}.  \\label{kac110}\n\\end{equation}\nHere the superscript (1) is dropped and time $t$ is suppressed in $f^{(1)}(%\n\\mathbf{v;}t)$.\n\nNow we go to center of mass frame (Equations \\ref{e4},\\ref{e5}). In the CM\ncoordinates the $R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})$\nis expressed as\n\\begin{equation}\nR(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v})=\\frac{1}{V}\\delta\n\\left( \\mathbf{H}-\\mathbf{H}^{\\prime }\\right) \\,\\delta \\left(\nu^{2}-(u^{\\prime })^{2}\\right) \\,\\sigma (\\mathbf{n},\\mathbf{n}^{\\prime })\n\\label{kac140}\n\\end{equation}\nwhere $V$ is the volume of the gas and $\\sigma (\\mathbf{n},\\mathbf{n}%\n^{\\prime })$ is the differential cross section. Inserting this into eq.(\\ref\n{kac110}) and doing the integrals over the center of mass frame we obtain\n\\begin{equation}\n\\frac{\\partial f(\\mathbf{v})}{\\partial t}=\\frac{N}{V}\\int \\left[ f(\\mathbf{v}%\n_{A})\\,f(\\mathbf{v}_{B})-f(\\mathbf{v}_{C})\\,f(\\mathbf{v})\\right] \\,u\\,\\sigma\n(\\mathbf{n},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime },\n\\label{kac150}\n\\end{equation}\nwhere $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$ are expressed in terms\nof the variables $\\mathbf{v,u,n}^{\\prime }$ in eqs.(\\ref{e105},\\ref{e106},%\n\\ref{e107}). If we write this equation for $F(\\mathbf{v})=(N\/V)f(\\mathbf{v})$\nwhich is velocity distribution normalized to the number density per unit\nvolume, we obtain the Boltzmann equation for a homogenous gas\n\\begin{equation}\n\\frac{\\partial F(\\mathbf{v})}{\\partial t}=\\int \\left[ F(\\mathbf{v}_{A})\\,F(%\n\\mathbf{v}_{B})-F(\\mathbf{v}_{C})\\,F(\\mathbf{v})\\right] \\,u\\,\\sigma (\\mathbf{%\nn},\\mathbf{n}^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime }.\n\\label{kac160}\n\\end{equation}\n\nAlthough both master equations have similar structures their philosophies\nare different. In Kac's work the collisions happens randomly and\nspontaneously in the gas whereas in direct simulation we take pairs and\nforce them to collide. Direct simulation has applications to systems other\nthan gases as we showed in the money games examples. In these systems there\nare not physical processes driving the collisions and instead we make the\ncollisions. In Kac's work his motivation was to describe Boltzmann equation\nfor gases as a stochastic equation and the DSMC method had not been invented\nyet. Just as in our work, Kac's method can be generalized to molecular gases\nand gas mixtures and one can obtain Boltzmann equations for these cases by\ndefining a suitable $R(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C},\\mathbf{v%\n})$ for each case.\n\n\\section{Direct simulation for an inhomogeneous gas}\n\nIn this section we study NTC algorithm of DSMC method for inhomogeneous gas.\nWe will not actually derive Bird's algorithm but we will define a similar\nalgorithm to simulate inhomogeneous gas. We will show that single particle\nprobability distribution of our algorithm satisfies the Boltzmann equation\nfor an inhomogeneous gas. Then we will argue that both algorithms give the\nsame results in the limit $N\\rightarrow \\infty $.\n\nWe divide the physical space into cells as in the Bird's method. In our\nalgorithm we take pairs not from the same cell but from all of the volume\nand we let each pair to make a collision attempt if both of them are in the\nsame cell.\n\nWe divide the physical space into cells and the $k^{th}$ cell has the volume\n$V_{k}$. Now let us define the functions\n\\begin{equation}\n\\Delta _{k}(\\mathbf{x})=\\left\\{\n\\begin{array}{ll}\n1\\quad & \\mathbf{{}x}\\in V_{k} \\\\\n0\\quad & \\mathbf{x}\\notin V_{k}\n\\end{array}\n\\right. .  \\label{k10}\n\\end{equation}\nWe will also need the function\n\\begin{equation}\n\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })=\\sum_{k}\\frac{\\Delta _{k}(\\mathbf{x}%\n)\\Delta _{k}(\\mathbf{x}^{\\prime })}{V_{k}}.  \\label{k20}\n\\end{equation}\nThis function is zero when $\\mathbf{x}$ and $\\mathbf{x}^{\\prime }$ are not\nin the same cell and $1\/V_{k}$ when they are in the same cell. Its integral\nover $\\mathbf{x}$ or $\\mathbf{x}^{\\prime }$ is unity\n\\begin{equation}\n\\int \\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })d^{3}\\mathbf{x}^{\\prime }=\\int\n\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })d^{3}\\mathbf{x}=1.  \\label{k30}\n\\end{equation}\nAt the end of this section we will take the limit $V_{k}\\rightarrow 0$. In\nthis limit $\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })=0$ for $\\mathbf{x}\\neq\n\\mathbf{x}^{\\prime }$ and $\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })=\\infty $\nfor $\\mathbf{x}=\\mathbf{x}^{\\prime }$ and eq.(\\ref{k30}) is still satisfied.\nThese are properties of the Dirac delta function and we have the limit\n\\begin{equation}\n\\lim_{V_{k}\\rightarrow 0}\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })=\\delta (%\n\\mathbf{x}-\\mathbf{x}^{\\prime })  \\label{k40}\n\\end{equation}\n\nNow we can start the discussion. We will treat the simplest case for\nclarity. We develop our arguments for one kind of gas without internal\ndegrees of freedom. The generalization to the other cases is very\nstraightforward and will be summarized at the end of the section.\n\nThe state index $\\mu $ represent position of the particle $\\mathbf{x}$ and\nthe velocity $\\mathbf{v.}$ The collision kernel is $Z=Z_{1}+Z_{2}$ where $%\nZ_{1}$ and $Z_{2}$ are\n\\begin{eqnarray}\nZ_{1}(\\mathbf{x}_{A}\\mathbf{v}_{A},\\mathbf{x}_{B}\\mathbf{v}_{B};\\mathbf{x}%\n_{C}\\mathbf{v}_{C},\\mathbf{x}_{D}\\mathbf{v}_{D}) &=&S(\\mathbf{v}_{A},\\mathbf{%\nv}_{B};\\mathbf{v}_{C}\\mathbf{,v}_{D})\\,  \\label{k50} \\\\\n&&\\times \\Gamma (\\mathbf{x}_{A},\\mathbf{x}_{B})\\,\\Omega \\,\\delta (\\mathbf{x}%\n_{C}-\\mathbf{x}_{A})\\,\\delta (\\mathbf{x}_{D}-\\mathbf{x}_{B})\\,,  \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nZ_{2}(\\mathbf{x}_{A}\\mathbf{v}_{A},\\mathbf{x}_{B}\\mathbf{v}_{B};\\mathbf{x}%\n_{C}\\mathbf{v}_{C},\\mathbf{x}_{D}\\mathbf{v}_{D}) &=&\\,\\,\\left( 1-\\Omega\n\\,\\Gamma (\\mathbf{x}_{A},\\mathbf{x}_{B})\\right) \\delta (\\mathbf{x}_{C}-%\n\\mathbf{x}_{A})\\,\\,\\,  \\label{k61} \\\\\n&&\\,\\,\\times \\delta (\\mathbf{x}_{D}-\\mathbf{x}_{B})\\,\\delta (\\mathbf{v}_{C}-%\n\\mathbf{v}_{A})\\,\\delta (\\mathbf{v}_{D}-\\mathbf{v}_{B}).  \\nonumber\n\\end{eqnarray}\nHere\n\\begin{equation}\n\\Omega \\,=\\left( \\sum_{k}\\frac{1}{V_{k}}\\right) ^{-1},  \\label{k70}\n\\end{equation}\nis a constant chosen to insure that probability of making a collision in any\ncell is less than unity. The $S(\\mathbf{v}_{A},\\mathbf{v}_{B};\\mathbf{v}_{C}%\n\\mathbf{,v}_{D})$ is given in eqs.(\\ref{e20},\\ref{e30}). The $Z_{2}$ does\nnot change states of the of the particles and the pair will not be allowed\nto make a collision attempt with a probability $\\left( 1-\\Omega \\,\\Gamma (%\n\\mathbf{x}_{A},\\mathbf{x}_{B})\\right) .$ The probability of a collision\nattempt is $\\Omega \\Gamma (\\mathbf{x}_{A},\\mathbf{x}_{B})$ and in a real\ncollision positions of particles do not change because of the $\\delta (%\n\\mathbf{x}_{C}-\\mathbf{x}_{A})\\,\\delta (\\mathbf{x}_{D}-\\mathbf{x}_{B})$ term\nin the $Z$. The $Z(\\mathbf{x}_{A}\\mathbf{v}_{A},\\mathbf{x}_{B}\\mathbf{v}_{B};%\n\\mathbf{x}_{C}\\mathbf{v}_{C},\\mathbf{x}_{D}\\mathbf{v}_{D})$ is symmetric and\nsatisfies the normalization condition\n\\begin{eqnarray}\n\\int Z(\\mathbf{x}_{A}\\mathbf{v}_{A},\\mathbf{x}_{B}\\mathbf{v}_{B};\\mathbf{x}%\n_{C}\\mathbf{v}_{C},\\mathbf{x}_{D}\\mathbf{v}_{D})d^{3}\\mathbf{v}_{A}d^{3}%\n\\mathbf{v}_{B}d^{3}\\mathbf{x}_{A}d^{3}\\mathbf{x}_{B} &=&1,  \\label{k81} \\\\\n\\int Z(\\mathbf{x}_{A}\\mathbf{v}_{A},\\mathbf{x}_{B}\\mathbf{v}_{B};\\mathbf{x}%\n_{C}\\mathbf{v}_{C},\\mathbf{x}_{D}\\mathbf{v}_{D})d^{3}\\mathbf{v}_{C}d^{3}%\n\\mathbf{v}_{D}d^{3}\\mathbf{x}_{C}d^{3}\\mathbf{x}_{D} &=&1.  \\label{k82}\n\\end{eqnarray}\n\nThe $\\Omega \\Gamma (\\mathbf{x}_{A},\\mathbf{x}_{B})$ vanishes unless $\\mathbf{%\nx}_{A}$ and $\\mathbf{x}_{B}$ are in the same cell and $\\Omega \\Gamma (%\n\\mathbf{x}_{A},\\mathbf{x}_{B})=\\Omega \/V_{k}$ when $\\mathbf{x}_{A}$ and $%\n\\mathbf{x}_{B}$ are in the cell $V_{k}$. The probability of having both\nparticles in the cell $V_{k}$ is $(N_{k}\/N)^{2}$ where $N_{k}$ is the number\nof particles in the cell $V_{k}$ during the collisions part of the\nsimulation. Therefore the probability of a pair making a collision attempt\nin the $k^{th}$ cell is $p_{k}=(\\Omega \/V_{k})(N_{k}\/N)^{2}$. The $1\/V_{k}\\,$%\nterm looks awkward in this probability but it is absolutely necessary as the\nfollowing argument shows. Suppose the physical density is uniform and\ntherefore $N_{k}\/N=V_{k}\/V$ where $V$ is the total volume. When density is\nuniform we expect that the probability of having a collision in $V_{k}$ is\nproportional to $V_{k}.$ When $N_{k}\/N=V_{k}\/V$ in inserted in $p_{k}\\,$we\nfind $p_{k}=\\Omega V_{k}\/V^{2}$ which is proportional to $V_{k}$ as expected.\n\nNow we insert the kernel $Z\\,$in the eq.(\\ref{a130}) to obtain\n\\begin{eqnarray}\n\\frac{\\partial f(\\mathbf{xv},\\tau \\mathbf{)}}{\\partial \\tau } &=&\\int\n[f,f]\\,Z(\\mathbf{x}_{A}\\mathbf{v}_{A},\\mathbf{x}_{B}\\mathbf{v}_{B};\\mathbf{x}%\n_{C}\\mathbf{v}_{C},\\mathbf{xv})  \\label{k90} \\\\\n&&\\times d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}_{B}\\,d^{3}\\mathbf{v}_{C}\\,d^{3}%\n\\mathbf{x}_{A}\\,d^{3}\\mathbf{x}_{B}\\,d^{3}\\mathbf{x}_{C}  \\nonumber\n\\end{eqnarray}\nwhere $[f,f]\\,$is\n\\begin{equation}\n\\lbrack f,f]\\,=f(\\mathbf{x}_{A}\\mathbf{v}_{A},\\tau )f(\\mathbf{x}_{B}\\mathbf{v%\n}_{B},\\tau )-f(\\mathbf{x}_{C}\\mathbf{v}_{C},\\tau )f(\\mathbf{xv},\\tau ).\n\\label{k100}\n\\end{equation}\nThe $Z_{2}$ part of the collision kernel does not contribute to the\ncollision integral. After doing the delta function integrals over positions $%\n\\mathbf{x}_{A}\\,,\\mathbf{x}_{B}$ we obtain\n\\begin{eqnarray}\n\\frac{\\partial f(\\mathbf{xv},\\tau \\mathbf{)}}{\\partial \\tau } &=&\\Omega \\int\n\\left[ f(\\mathbf{x}^{\\prime }\\mathbf{v}_{A},\\tau )f(\\mathbf{xv}_{B},\\tau )-f(%\n\\mathbf{x}^{\\prime }\\mathbf{v}_{C},\\tau )f(\\mathbf{xv},\\tau )\\right]\n\\label{k110} \\\\\n&&\\times \\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })\\,S(\\mathbf{v}_{A},\\mathbf{v%\n}_{B};\\mathbf{v}_{C}\\mathbf{,v})d^{3}\\mathbf{v}_{A}\\,d^{3}\\mathbf{v}%\n_{B}\\,d^{3}\\mathbf{v}_{C}\\,d^{3}\\mathbf{x}^{\\prime }.  \\nonumber\n\\end{eqnarray}\nNow we insert $S=S_{1}+S_{2}$ from eqs.(\\ref{e20},\\ref{e30}) in this\nequation. The $S_{2}$ part gives no contribution to the integral as before.\nDoing the integrals over $\\mathbf{v}_{A}\\,,\\mathbf{v}_{B}\\,,\\mathbf{v}_{C}\\,$\nin the center of mass coordinates we obtain\n\\begin{eqnarray}\n\\frac{\\partial f(\\mathbf{xv},\\tau \\mathbf{)}}{\\partial \\tau } &=&\\frac{%\n\\Omega }{R}\\int \\left[ f(\\mathbf{x}^{\\prime }\\mathbf{v}_{A},\\tau )f(\\mathbf{%\nxv}_{B},\\tau )-f(\\mathbf{x}^{\\prime }\\mathbf{v}_{C},\\tau )f(\\mathbf{xv},\\tau\n)\\right]  \\label{k113} \\\\\n&&\\times \\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })\\,\\sigma (\\mathbf{n,n}%\n^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime }\\,d^{3}\\mathbf{x}^{\\prime\n}.  \\nonumber\n\\end{eqnarray}\nwhere $\\mathbf{v}_{A},\\mathbf{v}_{B},\\mathbf{v}_{C}$ are given in eqs.(\\ref\n{e105},\\ref{e106},\\ref{e107}). In order to have complete correspondence with\nthe Boltzmann equation we define the new function $F(\\mathbf{xv},\\tau )=Nf(%\n\\mathbf{xv},\\tau \\mathbf{)}$ and we also define the new variable $t=\\Omega\n\\tau \/RN=2\\Omega n\/RN^{2}$ to obtain\n\\begin{equation}\n\\frac{\\partial F(\\mathbf{xv},t\\mathbf{)}}{\\partial t}=\\widehat{L}_{C}F(%\n\\mathbf{xv,}t\\mathbf{)}  \\label{k122}\n\\end{equation}\n\nwhere the operator $\\widehat{L}_{C}$ is defined as\n\\begin{eqnarray}\n\\widehat{L}_{C}F(\\mathbf{xv,}t\\mathbf{)} &=&\\ \\int \\left[ F(\\mathbf{x}%\n^{\\prime }\\mathbf{v}_{A},t)F(\\mathbf{xv}_{B},t)-F(\\mathbf{x}^{\\prime }%\n\\mathbf{v}_{C},t)F(\\mathbf{xv},t)\\right]  \\label{k150} \\\\\n&&\\times \\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })\\,\\sigma (\\mathbf{n,n}%\n^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime }\\,d^{3}\\mathbf{x}^{\\prime\n}.  \\nonumber\n\\end{eqnarray}\nHere $t$ is interpreted as the physical time.\n\nIn the collisions part of the DSMC\\ method we make collision attempts for a\ntime $\\Delta t$ where $\\Delta t$ is a small time interval. This corresponds\nto $\\Delta \\tau =RN\\Delta t\/\\Omega $ collision time passage or $\\Delta\nn=RN^{2}\\Delta t\/2\\Omega $ pairs chosen. From eq.(\\ref{k122}), after making $%\n\\Delta n$ collisions attempt $F(\\mathbf{xv},t\\mathbf{)}$ becomes $F^{*}(%\n\\mathbf{xv},t\\mathbf{)}$%\n\\begin{equation}\nF^{*}(\\mathbf{xv},t\\mathbf{)=}(1+\\Delta t\\widehat{L}_{C})F(\\mathbf{xv},t%\n\\mathbf{)+}O((\\Delta t)^{2})  \\label{m10}\n\\end{equation}\nwhere $O((\\Delta t)^{2})$ is an error term of order $(\\Delta t)^{2}$.\n\nNext we perform free propagation step where $\\mathbf{x\\rightarrow x+}\\Delta t%\n\\mathbf{v}$ and $\\mathbf{v\\rightarrow v+}\\Delta t\\mathbf{a}$ transformation\nis made for each particle. Here $\\mathbf{a=F\/}m$ is the acceleration of the\nparticle due to the force $\\mathbf{F}$ and it can depend on both position\nand velocity of the particle. This changes the $N$ particle distribution\nfunction $f^{(N)}(\\mathbf{x}_{1},\\mathbf{v}_{1};\\mathbf{x}_{2},\\mathbf{v}%\n_{2};...,\\mathbf{x}_{N},\\mathbf{v}_{N})$ to\n\\begin{equation}\nf^{(N)}(\\mathbf{x}_{1}-\\Delta t\\mathbf{v}_{1},\\mathbf{v}_{1}-\\Delta t\\mathbf{%\na}_{1};...;\\mathbf{x}_{N}-\\Delta t\\mathbf{v}_{N},\\mathbf{v}_{N}-\\Delta t%\n\\mathbf{a}_{N}).  \\label{m20}\n\\end{equation}\nThe jacobian of the transformation is unity with a correction of order $%\n(\\Delta t)^{2}$ and therefore this expression is correct with an error of\nthe same order. Integrating this over $\\mathbf{x}_{2},\\mathbf{v}_{2};...,%\n\\mathbf{x}_{N},\\mathbf{v}_{N}$ we find that the single particle probability\ndistribution $f^{(1)}(\\mathbf{x},\\mathbf{v)}$ changes to $f^{(1)}(\\mathbf{x-}%\n\\Delta t\\mathbf{v},\\mathbf{v-}\\Delta t\\mathbf{a)}$ with an error term of\norder $(\\Delta t)^{2}$. Therefore $F^{*}(\\mathbf{x},\\mathbf{v},t\\mathbf{)}$\nbecomes $F^{*}(\\mathbf{x-}\\Delta t\\mathbf{v},\\mathbf{v-}\\Delta t\\mathbf{a,}t%\n\\mathbf{)}$ which is taken as $F(\\mathbf{x},\\mathbf{v},t+\\Delta t\\mathbf{)}$%\n. Hence\n\\begin{equation}\nF(\\mathbf{x},\\mathbf{v},t+\\Delta t\\mathbf{)}=F^{*}(\\mathbf{x-}\\Delta t%\n\\mathbf{v},\\mathbf{v-}\\Delta t\\mathbf{a,}t\\mathbf{).}  \\label{m31}\n\\end{equation}\nUsing eq.(\\ref{m10}) and expanding $F(\\mathbf{x-}\\Delta t\\mathbf{v},\\mathbf{%\nv-}\\Delta t\\mathbf{a,}t\\mathbf{)}$ up to first order terms in $\\Delta t$ we\nobtain\n\\begin{equation}\nF(\\mathbf{x},\\mathbf{v},t+\\Delta t\\mathbf{)}=\\left( 1-\\Delta t\\mathbf{v}%\n\\frac{\\partial }{\\partial \\mathbf{x}}-\\Delta t\\mathbf{a}\\frac{\\partial }{%\n\\partial \\mathbf{v}}+\\Delta t\\widehat{L}_{C}\\right) F(\\mathbf{x},\\mathbf{v}%\n,t)+O((\\Delta t)^{2})  \\label{m35}\n\\end{equation}\nwhere $O((\\Delta t)^{2})$ is the error terms of order $(\\Delta t)^{2}$.\nTaking the limit $\\Delta t\\rightarrow 0$ we obtain\n\\begin{equation}\n\\frac{\\partial F(\\mathbf{x},\\mathbf{v},t\\mathbf{)}}{\\partial t}+\\mathbf{%\nv\\cdot }\\frac{\\partial F(\\mathbf{x},\\mathbf{v},t\\mathbf{)}}{\\partial \\mathbf{%\nx}}+\\frac{\\mathbf{F}}{m}\\mathbf{\\cdot }\\frac{\\partial F(\\mathbf{x},\\mathbf{v}%\n,t\\mathbf{)}}{\\partial \\mathbf{v}}=\\widehat{L}_{C}F(\\mathbf{x},\\mathbf{v},t%\n\\mathbf{)}  \\label{m40}\n\\end{equation}\n\nThis equation is similar to the Boltzmann equation but it is not the same.\nAlready when treating $\\tau =2n\/N$ as a continuous parameter we took $%\nN\\rightarrow \\infty $ limit implicitly. The remaining limit is $%\nV_{k}\\rightarrow 0$ and we know that $\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime\n})\\,\\rightarrow \\delta (\\mathbf{x}-\\mathbf{x}^{\\prime })$ in this limit.\nAfter setting $\\Gamma (\\mathbf{x},\\mathbf{x}^{\\prime })\\,=\\delta (\\mathbf{x}-%\n\\mathbf{x}^{\\prime })$ performing the $\\mathbf{x}^{\\prime }$ integral the\noperator $\\widehat{L}_{C}$ reduces to\n\\begin{equation}\n\\widehat{L}_{C}F(\\mathbf{xv)}=\\int \\left[ F(\\mathbf{xv}_{A},t)F(\\mathbf{xv}%\n_{B},t)-F(\\mathbf{xv}_{C},t)F(\\mathbf{xv},t)\\right] \\sigma (\\mathbf{n,n}%\n^{\\prime })\\,d^{3}\\mathbf{u}\\,d\\mathbf{n}^{\\prime }\\,.  \\label{m51}\n\\end{equation}\nWith this form of the $\\widehat{L}_{C}$ the eq. (\\ref{m40}) is the Boltzmann\nequation.\n\nHence we have shown that in direct simulation algorithm for inhomogeneous\ngas the one particle probability distribution satisfies the Boltzmann\nequation. Now, how do we connect this to the Bird's NTC\\ algorithm? Clearly\nthey are not the same. In fact our algorithm is not practical since great\nmajority of chosen pairs will not be in the same cell and therefore will not\nmake collisions.\n\nIn the time interval $\\Delta t\\,$we choose $\\ \\Delta n=RN^{2}\\Delta\nt\/2\\Omega $ pairs. The probability that each pair will make a collision\nattempt in the $k^{th}$ cell is $p_{k}=(\\Omega \/V_{k})(N_{k}\/N)^{2}.$ Let $%\nn_{k}$ be the number of collision attempts that take place in $V_{k}$. The\nexpected value of $n_{k}$ is\n\\begin{equation}\n\\overline{n}_{k}=\\Delta n\\cdot p_{k}=\\frac{RN_{k}^{2}}{2V_{k}}\\Delta t.\n\\label{m60}\n\\end{equation}\nThis is the same as number of collision attempts in $V_{k}$ in Birds\nalgorithm. The difference is that in Birds algorithm the number of collision\nattempts in each cell is fixed as $n_{k}=RN_{k}^{2}\\Delta t\/2V_{k}$ whereas\nin our algorithm the $n_{k}$ has a probability distribution with a mean\nvalue $RN_{k}^{2}\\Delta t\/2V_{k}$. The probability distribution for $n_{k}$\nis given as\n\\begin{equation}\nP(n_{k})=\\frac{(\\Delta n)!}{(\\Delta n-n_{k})!\\,(n_{k})!}%\n(p_{k})^{n_{k}}(1-p_{k})^{\\Delta n-n_{k}}.  \\label{m71}\n\\end{equation}\nIn the limit of $V_{k}\\rightarrow 0$ we have $p_{k}\\rightarrow 0$ and the $%\nP(n_{k})$ becomes the Poisson probability distribution\n\\begin{equation}\nP(n_{k})=\\frac{(\\overline{n}_{k})^{n_{k}}}{(n_{k})!}\\exp (-\\overline{n}_{k}).\n\\label{m80}\n\\end{equation}\nThe width of distributions in eqs.(\\ref{m71},\\ref{m80}) is of order $\\sqrt{%\n\\overline{n}_{k}}$. For large values of $\\overline{n}_{k}$ we have $n_{k}\/%\n\\overline{n}_{k}=1+O(1\/\\sqrt{\\overline{n}_{k}})$ where $O(1\/\\sqrt{\\overline{n%\n}_{k}})$ is a term of order $1\/\\sqrt{\\overline{n}_{k}}$.\n\nNow we take the limit $N_{k}\\rightarrow \\infty $ and $O(1\/\\sqrt{\\overline{n}%\n_{k}})$ error term vanishes. In a more mathematical language, probability\nthat $n_{k}\/\\overline{n}_{k}=1$ is unity. Hence both methods approach each\nother in the limit $N_{k}\\rightarrow \\infty $ and single particle\nprobability distribution in Bird's method too should satisfy the Boltzmann\nequation (eq.(\\ref{m40})) in this limit.\n\nThere is an important distinction in the limits taken for both method to\nsatisfy the Boltzmann equation. In our algorithm we take $N\\rightarrow\n\\infty ,$ $\\Delta t\\rightarrow 0$ and $V_{k}\\rightarrow 0$ limits. This does\nnot mean that number of particles in each cell ($N_{k}$) will go to\ninfinity. For example for a uniform density we have $N_{k}=(N\/V)V_{k}.$ Here\n$N\\rightarrow \\infty $ and $V_{k}\\rightarrow 0$ limits does not imply\nanything about $N_{k}$. $NV_{k}$ can remain finite and even can go to zero\nand still our algorithm satisfies the Boltzmann equation. The Bird's\nalgorithm requires $N_{k}\\rightarrow \\infty $ to satisfy the Boltzmann\nequation however and this is a more stringent requirement.\n\nWe did this analysis for the simplest case of one kind of gas without\ninternal degrees of freedom for clarity. It is very simple to generalize\nthis to the other cases by replacing the kernel $S$ in eq.(\\ref{k50}) with $%\nG_{pq}^{rs}$ in eq.(\\ref{f20}) or with $K_{ij}^{\\alpha \\beta }$ in eq.(\\ref\n{g21}) or with $Q_{ij,pq}^{\\alpha \\beta ,rs}$ in eq.(\\ref{h21}). Then the\nBoltzmann equation will be replaced by the Wang Chang-Uhlenbeck equation but\nall of the arguments will remain the same.\n\n\\section{Conclusions}\n\nLet us list our contributions in this paper.\n\n\\begin{itemize}\n\\item  In this paper we introduced a general formalism for direct simulation\nprocesses. We defined the direct simulation as a markov process with a\nmaster equation and we found the master equation given in eq.(\\ref{a51}).\nDefinition the DSMC\\ algorithm as a stochastic process governed by a master\nequation does not exist in the literature of the DSMC\\ method to our\nknowledge.\n\n\\item  Starting from the master equation we showed that the N-particle\nprobability density evolves towards microcanonical distribution as the\nnumber of collisions go to infinity.\n\n\\item  We derived a hierarchy of equations similar to the BBGKY hierarchy\nfor the reduced probability densities given in eq.(\\ref{a60})\n\n\\item  We showed that if AMC\\ approximation is employed the single particle\nprobability distribution satisfies an equation given in eq.(\\ref{a101}). In\nthe limit $N\\rightarrow \\infty $ this reduces to eq.(\\ref{a130}) which is an\nequation similar to the Boltzmann equation.\n\n\\item  We found the equations of the hierarchy in the limit $N\\rightarrow\n\\infty $ (the eq.(\\ref{a134}) )and showed that the ansatz $f^{(M)}(\\mu\n_{1},\\mu _{2},...,\\mu _{M};\\tau )=f^{(1)}(\\mu _{1}\\mathbf{;}\\tau\n)\\,f^{(1)}(\\mu _{2}\\mathbf{;}\\tau )....f^{(1)}(\\mu _{M}\\mathbf{;}\\tau )$\nsatisfies all the equations in the hierarchy provided the $f^{(1)}(\\mu\n\\mathbf{;}\\tau )$ satisfies the eq.(\\ref{a130}). This ensures that in the\nlimit $N\\rightarrow \\infty $ the AMC is satisfied for all times if one\nstarts from an uncorrelated initial state.\n\n\\item  We gave two simple examples from direct simulation money games. The\ndiscrete money game example has the nice feature that it is exactly solvable\nand we observe from the solution that the approach to the equilibrium is\nexponentially fast.\n\n\\item  We obtained the H-theorem and conservation of expectation values of\ncollision invariants. These results are familiar to most readers from the\nstandard treatments of the Boltzmann equation. But it is worth repeating\nthem here because although the equations are similar they are applied to\nwide variety of different problems in the direct simulation setting, not\njust to gases.\n\n\\item  We applied the formalism to the direct simulation Monte Carlo method\nfor real homogenous gases which is a standard method to solve the Boltzmann\nequation. Introducing appropriate kernels we obtained NTC\\ algorithm for a\nhomogenous gas and we showed that the appropriately normalized single\nparticle probability distribution satisfies Boltzmann equation for simple\nhomogenous gases and Wang Chang-Uhlenbeck equations for homogenous molecular\ngases and their mixtures. The derivation of conservation of $\\int f^{p}(%\n\\mathbf{v})\\,d^{3}\\mathbf{v}$ for mixture of gases without internal degrees\nof freedom and $\\sum_{i}\\int f_{i}^{p}(\\mathbf{v})\\,d^{3}\\mathbf{v}$ for\nmixture of gases with internal degrees of freedom should be also familiar to\nthe reader from the standard treatments of the Boltzmann equation. The novel\nfeature of our derivation is the significant simplification that the\nnormalization of $T(\\mu _{A},\\mu _{B},\\mu _{C},\\mu _{D})$ given in the\nequations (\\ref{a30},\\ref{f111},\\ref{f111a},\\ref{h120},\\ref{h121}) provide\nto obtain the result. If we try to obtain the same result from the Boltzmann\nequation we would have to use the argument that the integrals in (\\ref{f111},%\n\\ref{f111a},\\ref{h120},\\ref{h121}) are functions of the collision invariants.\n\n\\item  We introduced a new algorithm to do the DSMC\\ calculations for an\ninhomogeneous gas. Our algorithm is not practical for the actual practice of\nthe art because of wasting the great majority of the chosen pairs. We showed\nthat the single particle probability distribution satisfies the Boltzmann\nequation in our algorithm in the limits $N\\rightarrow \\infty ,$ $\\Delta\nt\\rightarrow 0$ and $V_{k}\\rightarrow 0$. We also showed that Bird's\nalgorithm for DSMC converges to our algorithm if $N_{k}\\rightarrow \\infty $\nis taken in addition to the limits $\\Delta t\\rightarrow 0$ and $%\nV_{k}\\rightarrow 0$. Birds algorithm requires more stringent requirements to\nsatisfy the Boltzmann equation. To prevent any misunderstanding we stress\nhere that our algorithm is not intended as a practical scheme to implement\nDSMC calculations. The Bird's algorithm does not easily fit in the direct\nsimulation formalism presented in this paper whereas the algorithm we\npresented does. We showed that our algorithm gives the Boltzmann equation in\nthe limits $N\\rightarrow \\infty ,$ $\\Delta t\\rightarrow 0$ and $%\nV_{k}\\rightarrow 0$ and we also showed that our algorithm and Bird's\nalgorithm converges to each other if we go to more stringent limit of $%\nN_{k}\\rightarrow \\infty $. Therefore we proved indirectly that Birds\nalgorithm satisfies Boltzmann equation in the limit $N_{k}\\rightarrow \\infty\n$, $\\Delta t\\rightarrow 0$ and $V_{k}\\rightarrow 0$. Therefore we introduced\nour algorithm as a tool to study convergence of Bird's method and not as a\npractical way of doing DSMC calculations.\n\\end{itemize}\n\nMeaning of the convergence here should be interpreted according to the\nensemble theory of statistical mechanics. We imagine practically infinite\nnumber of identical systems (computers with human operators) doing the same\ndirect simulation and call this the ensemble. The $f^{(1)}(\\mu ;\\tau )d\\mu $\nrepresents ratio of number of particles in $d\\mu $ to the total number of\nparticles averaged over all the ensemble. When you perform a direct\nsimulation on a computer you are just one member of the ensemble. Your\nresults will show statistical fluctuations. But when you do the same\nsimulation many times with different initial states chosen according to a\nuncorrelated probability distribution $f^{(N)}(\\mu _{1},\\mu _{2},...,\\mu\n_{N};n=0)=\\,h(\\mu _{1})h(\\mu _{2})....h(\\mu _{N})$ you form your own\nensemble and averages over them will nicely follow $f^{(1)}(\\mu ;\\tau )$\nobtained by solving eq.(\\ref{a130}) with the initial value $f^{(1)}(\\mu\n;\\tau =0)=h(\\mu ) $.\n\nThis work can generalize to chemical reactions and radiative processes in a\nmore or less straightforward fashion. But there are enough number of\nsubtleties such that we leave them to future publications.\n\nA simplified version of this paper\\cite{AJP} containing only one kind of\nhomogenous gas without internal degrees of freedom is published in American\nJournal of Physics. The material in that paper makes a small fraction of the\nmaterial in this paper. The present paper contains much new material and\noverlap between the two papers is small.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\\PARstart{T}{his} document is a template for \\LaTeX. If you are \nreading a paper or PDF version of this document, please download the \nelectronic file, trans\\_jour.tex, from the IEEE Web site at \\underline\n{http:\/\/ieeeauthorcenter.ieee.org\/create-your-ieee-article\/}\\break\\underline{use-authoring-tools-and-ieee-article-templates\/ieee-article-}\\break\\underline{templates\/} so you can use it to prepare your manuscript. If \nyou would prefer to use LaTeX, download IEEE's LaTeX style and sample files \nfrom the same Web page. You can also explore using the Overleaf editor at \n\\underline\n{https:\/\/www.overleaf.com\/blog\/278-how-to-use-overleaf-}\\break\\underline{with-ieee-collabratec-your-quick-guide-to-getting-started}\\break\\underline{\\#.xsVp6tpPkrKM9}\n\nIf your paper is intended for a conference, please contact your conference \neditor concerning acceptable word processor formats for your particular \nconference. \n\nIEEE will do the final formatting of your paper. If your paper is intended \nfor a conference, please observe the conference page limits. \n\n\\subsection{Abbreviations and Acronyms}\nDefine abbreviations and acronyms the first time they are used in the text, \neven after they have already been defined in the abstract. Abbreviations \nsuch as IEEE, SI, ac, and dc do not have to be defined. Abbreviations that \nincorporate periods should not have spaces: write ``C.N.R.S.,'' not ``C. N. \nR. S.'' Do not use abbreviations in the title unless they are unavoidable \n(for example, ``IEEE'' in the title of this article).\n\n\\subsection{Other Recommendations}\nUse one space after periods and colons. Hyphenate complex modifiers: \n``zero-field-cooled magnetization.'' Avoid dangling participles, such as, \n``Using \\eqref{eq}, the potential was calculated.'' [It is not clear who or what \nused \\eqref{eq}.] Write instead, ``The potential was calculated by using \\eqref{eq},'' or \n``Using \\eqref{eq}, we calculated the potential.''\n\nUse a zero before decimal points: ``0.25,'' not ``.25.'' Use \n``cm$^{3}$,'' not ``cc.'' Indicate sample dimensions as ``0.1 cm \n$\\times $ 0.2 cm,'' not ``0.1 $\\times $ 0.2 cm$^{2}$.'' The \nabbreviation for ``seconds'' is ``s,'' not ``sec.'' Use \n``Wb\/m$^{2}$'' or ``webers per square meter,'' not \n``webers\/m$^{2}$.'' When expressing a range of values, write ``7 to \n9'' or ``7--9,'' not ``7$\\sim $9.''\n\nA parenthetical statement at the end of a sentence is punctuated outside of \nthe closing parenthesis (like this). (A parenthetical sentence is punctuated \nwithin the parentheses.) In American English, periods and commas are within \nquotation marks, like ``this period.'' Other punctuation is ``outside''! \nAvoid contractions; for example, write ``do not'' instead of ``don't.'' The \nserial comma is preferred: ``A, B, and C'' instead of ``A, B and C.''\n\nIf you wish, you may write in the first person singular or plural and use \nthe active voice (``I observed that $\\ldots$'' or ``We observed that $\\ldots$'' \ninstead of ``It was observed that $\\ldots$''). Remember to check spelling. If \nyour native language is not English, please get a native English-speaking \ncolleague to carefully proofread your paper.\n\nTry not to use too many typefaces in the same article. You're writing\nscholarly papers, not ransom notes. Also please remember that MathJax\ncan't handle really weird typefaces.\n\n\\subsection{Equations}\nNumber equations consecutively with equation numbers in parentheses flush \nwith the right margin, as in \\eqref{eq}. To make your equations more \ncompact, you may use the solidus (~\/~), the exp function, or appropriate \nexponents. Use parentheses to avoid ambiguities in denominators. Punctuate \nequations when they are part of a sentence, as in\n\\begin{equation}E=mc^2.\\label{eq}\\end{equation}\n\nBe sure that the symbols in your equation have been defined before the \nequation appears or immediately following. Italicize symbols ($T$ might refer \nto temperature, but T is the unit tesla). Refer to ``\\eqref{eq},'' not ``Eq. \\eqref{eq}'' \nor ``equation \\eqref{eq},'' except at the beginning of a sentence: ``Equation \\eqref{eq} \nis $\\ldots$ .''\n\n\\subsection{\\LaTeX-Specific Advice}\n\nPlease use ``soft'' (e.g., \\verb|\\eqref{Eq}|) cross references instead\nof ``hard'' references (e.g., \\verb|(1)|). That will make it possible\nto combine sections, add equations, or change the order of figures or\ncitations without having to go through the file line by line.\n\nPlease don't use the \\verb|{eqnarray}| equation environment. Use\n\\verb|{align}| or \\verb|{IEEEeqnarray}| instead. The \\verb|{eqnarray}|\nenvironment leaves unsightly spaces around relation symbols.\n\nPlease note that the \\verb|{subequations}| environment in {\\LaTeX}\nwill increment the main equation counter even when there are no\nequation numbers displayed. 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It\nwill not stop equation numbers inside \\verb|{array}| (there won't be\nany anyway) and it might stop a wanted equation number in the\nsurrounding equation.\n\n\\section{Units}\nUse either SI (MKS) or CGS as primary units. (SI units are strongly \nencouraged.) English units may be used as secondary units (in parentheses). \nThis applies to papers in data storage. For example, write ``15 \nGb\/cm$^{2}$ (100 Gb\/in$^{2})$.'' An exception is when \nEnglish units are used as identifiers in trade, such as ``3\\textonehalf-in \ndisk drive.'' Avoid combining SI and CGS units, such as current in amperes \nand magnetic field in oersteds. This often leads to confusion because \nequations do not balance dimensionally. If you must use mixed units, clearly \nstate the units for each quantity in an equation.\n\nThe SI unit for magnetic field strength $H$ is A\/m. However, if you wish to use \nunits of T, either refer to magnetic flux density $B$ or magnetic field \nstrength symbolized as $\\mu _{0}H$. Use the center dot to separate \ncompound units, e.g., ``A$\\cdot $m$^{2}$.''\n\n\\section{Some Common Mistakes}\nThe word ``data'' is plural, not singular. The subscript for the \npermeability of vacuum $\\mu _{0}$ is zero, not a lowercase letter \n``o.'' The term for residual magnetization is ``remanence''; the adjective \nis ``remanent''; do not write ``remnance'' or ``remnant.'' Use the word \n``micrometer'' instead of ``micron.'' A graph within a graph is an \n``inset,'' not an ``insert.'' The word ``alternatively'' is preferred to the \nword ``alternately'' (unless you really mean something that alternates). Use \nthe word ``whereas'' instead of ``while'' (unless you are referring to \nsimultaneous events). Do not use the word ``essentially'' to mean \n``approximately'' or ``effectively.'' Do not use the word ``issue'' as a \neuphemism for ``problem.'' When compositions are not specified, separate \nchemical symbols by en-dashes; for example, ``NiMn'' indicates the \nintermetallic compound Ni$_{0.5}$Mn$_{0.5}$ whereas \n``Ni--Mn'' indicates an alloy of some composition \nNi$_{x}$Mn$_{1-x}$.\n\n\\Figure[t!](topskip=0pt, botskip=0pt, midskip=0pt){fig1.png}\n{Magnetization as a function of applied field.\nIt is good practice to explain the significance of the figure in the caption.\\label{fig1}}\n\nBe aware of the different meanings of the homophones ``affect'' (usually a \nverb) and ``effect'' (usually a noun), ``complement'' and ``compliment,'' \n``discreet'' and ``discrete,'' ``principal'' (e.g., ``principal \ninvestigator'') and ``principle'' (e.g., ``principle of measurement''). Do \nnot confuse ``imply'' and ``infer.'' \n\nPrefixes such as ``non,'' ``sub,'' ``micro,'' ``multi,'' and ``ultra'' are \nnot independent words; they should be joined to the words they modify, \nusually without a hyphen. There is no period after the ``et'' in the Latin \nabbreviation ``\\emph{et al.}'' (it is also italicized). The abbreviation ``i.e.,'' means \n``that is,'' and the abbreviation ``e.g.,'' means ``for example'' (these \nabbreviations are not italicized).\n\nA general IEEE styleguide is available at \\underline{http:\/\/www.ieee.org\/}\\break\\underline{authortools}.\n\n\\section{Guidelines for Graphics Preparation and Submission}\n\\label{sec:guidelines}\n\n\\subsection{Types of Graphics}\nThe following list outlines the different types of graphics published in \nIEEE journals. They are categorized based on their construction, and use of \ncolor\/shades of gray:\n\n\\subsubsection{Color\/Grayscale figures}\n{Figures that are meant to appear in color, or shades of black\/gray. Such \nfigures may include photographs, illustrations, multicolor graphs, and \nflowcharts.}\n\n\\subsubsection{Line Art figures}\n{Figures that are composed of only black lines and shapes. These figures \nshould have no shades or half-tones of gray, only black and white.}\n\n\\subsubsection{Author photos}\n{Head and shoulders shots of authors that appear at the end of our papers. }\n\n\\subsubsection{Tables}\n{Data charts which are typically black and white, but sometimes include \ncolor.}\n\n\\begin{table}\n\\caption{Units for Magnetic Properties}\n\\label{table}\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{|p{25pt}|p{75pt}|p{115pt}|}\n\\hline\nSymbol& \nQuantity& \nConversion from Gaussian and \\par CGS EMU to SI $^{\\mathrm{a}}$ \\\\\n\\hline\n$\\Phi $& \nmagnetic flux& \n1 Mx $\\to  10^{-8}$ Wb $= 10^{-8}$ V$\\cdot $s \\\\\n$B$& \nmagnetic flux density, \\par magnetic induction& \n1 G $\\to  10^{-4}$ T $= 10^{-4}$ Wb\/m$^{2}$ \\\\\n$H$& \nmagnetic field strength& \n1 Oe $\\to  10^{3}\/(4\\pi )$ A\/m \\\\\n$m$& \nmagnetic moment& \n1 erg\/G $=$ 1 emu \\par $\\to 10^{-3}$ A$\\cdot $m$^{2} = 10^{-3}$ J\/T \\\\\n$M$& \nmagnetization& \n1 erg\/(G$\\cdot $cm$^{3}) =$ 1 emu\/cm$^{3}$ \\par $\\to 10^{3}$ A\/m \\\\\n4$\\pi M$& \nmagnetization& \n1 G $\\to  10^{3}\/(4\\pi )$ A\/m \\\\\n$\\sigma $& \nspecific magnetization& \n1 erg\/(G$\\cdot $g) $=$ 1 emu\/g $\\to $ 1 A$\\cdot $m$^{2}$\/kg \\\\\n$j$& \nmagnetic dipole \\par moment& \n1 erg\/G $=$ 1 emu \\par $\\to 4\\pi \\times  10^{-10}$ Wb$\\cdot $m \\\\\n$J$& \nmagnetic polarization& \n1 erg\/(G$\\cdot $cm$^{3}) =$ 1 emu\/cm$^{3}$ \\par $\\to 4\\pi \\times  10^{-4}$ T \\\\\n$\\chi , \\kappa $& \nsusceptibility& \n1 $\\to  4\\pi $ \\\\\n$\\chi_{\\rho }$& \nmass susceptibility& \n1 cm$^{3}$\/g $\\to  4\\pi \\times  10^{-3}$ m$^{3}$\/kg \\\\\n$\\mu $& \npermeability& \n1 $\\to  4\\pi \\times  10^{-7}$ H\/m \\par $= 4\\pi \\times  10^{-7}$ Wb\/(A$\\cdot $m) \\\\\n$\\mu_{r}$& \nrelative permeability& \n$\\mu \\to \\mu_{r}$ \\\\\n$w, W$& \nenergy density& \n1 erg\/cm$^{3} \\to  10^{-1}$ J\/m$^{3}$ \\\\\n$N, D$& \ndemagnetizing factor& \n1 $\\to  1\/(4\\pi )$ \\\\\n\\hline\n\\multicolumn{3}{p{251pt}}{Vertical lines are optional in tables. Statements that serve as captions for \nthe entire table do not need footnote letters. }\\\\\n\\multicolumn{3}{p{251pt}}{$^{\\mathrm{a}}$Gaussian units are the same as cg emu for magnetostatics; Mx \n$=$ maxwell, G $=$ gauss, Oe $=$ oersted; Wb $=$ weber, V $=$ volt, s $=$ \nsecond, T $=$ tesla, m $=$ meter, A $=$ ampere, J $=$ joule, kg $=$ \nkilogram, H $=$ henry.}\n\\end{tabular}\n\\label{tab1}\n\\end{table}\n\n\\subsection{Multipart figures}\nFigures compiled of more than one sub-figure presented side-by-side, or \nstacked. If a multipart figure is made up of multiple figure\ntypes (one part is lineart, and another is grayscale or color) the figure \nshould meet the stricter guidelines.\n\n\\subsection{File Formats For Graphics}\\label{formats}\nFormat and save your graphics using a suitable graphics processing program \nthat will allow you to create the images as PostScript (PS), Encapsulated \nPostScript (.EPS), Tagged Image File Format (.TIFF), Portable Document \nFormat (.PDF), Portable Network Graphics (.PNG), or Metapost (.MPS), sizes them, and adjusts \nthe resolution settings. When \nsubmitting your final paper, your graphics should all be submitted \nindividually in one of these formats along with the manuscript.\n\n\\subsection{Sizing of Graphics}\nMost charts, graphs, and tables are one column wide (3.5 inches\/88 \nmillimeters\/21 picas) or page wide (7.16 inches\/181 millimeters\/43 \npicas). The maximum depth a graphic can be is 8.5 inches (216 millimeters\/54\npicas). When choosing the depth of a graphic, please allow space for a \ncaption. Figures can be sized between column and page widths if the author \nchooses, however it is recommended that figures are not sized less than \ncolumn width unless when necessary. \n\nThere is currently one publication with column measurements that do not \ncoincide with those listed above. Proceedings of the IEEE has a column \nmeasurement of 3.25 inches (82.5 millimeters\/19.5 picas). \n\nThe final printed size of author photographs is exactly\n1 inch wide by 1.25 inches tall (25.4 millimeters$\\,\\times\\,$31.75 millimeters\/6 \npicas$\\,\\times\\,$7.5 picas). Author photos printed in editorials measure 1.59 inches \nwide by 2 inches tall (40 millimeters$\\,\\times\\,$50 millimeters\/9.5 picas$\\,\\times\\,$12 \npicas).\n\n\\subsection{Resolution }\nThe proper resolution of your figures will depend on the type of figure it \nis as defined in the ``Types of Figures'' section. Author photographs, \ncolor, and grayscale figures should be at least 300dpi. Line art, including \ntables should be a minimum of 600dpi.\n\n\\subsection{Vector Art}\nIn order to preserve the figures' integrity across multiple computer \nplatforms, we accept files in the following formats: .EPS\/.PDF\/.PS. All \nfonts must be embedded or text converted to outlines in order to achieve the \nbest-quality results.\n\n\\subsection{Color Space}\nThe term color space refers to the entire sum of colors that can be \nrepresented within the said medium. For our purposes, the three main color \nspaces are Grayscale, RGB (red\/green\/blue) and CMYK \n(cyan\/magenta\/yellow\/black). RGB is generally used with on-screen graphics, \nwhereas CMYK is used for printing purposes.\n\nAll color figures should be generated in RGB or CMYK color space. Grayscale \nimages should be submitted in Grayscale color space. Line art may be \nprovided in grayscale OR bitmap colorspace. Note that ``bitmap colorspace'' \nand ``bitmap file format'' are not the same thing. When bitmap color space \nis selected, .TIF\/.TIFF\/.PNG are the recommended file formats.\n\n\\subsection{Accepted Fonts Within Figures}\nWhen preparing your graphics IEEE suggests that you use of one of the \nfollowing Open Type fonts: Times New Roman, Helvetica, Arial, Cambria, and \nSymbol. If you are supplying EPS, PS, or PDF files all fonts must be \nembedded. Some fonts may only be native to your operating system; without \nthe fonts embedded, parts of the graphic may be distorted or missing.\n\nA safe option when finalizing your figures is to strip out the fonts before \nyou save the files, creating ``outline'' type. This converts fonts to \nartwork what will appear uniformly on any screen.\n\n\\subsection{Using Labels Within Figures}\n\n\\subsubsection{Figure Axis labels }\nFigure axis labels are often a source of confusion. Use words rather than \nsymbols. As an example, write the quantity ``Magnetization,'' or \n``Magnetization M,'' not just ``M.'' Put units in parentheses. Do not label \naxes only with units. As in Fig. 1, for example, write ``Magnetization \n(A\/m)'' or ``Magnetization (A$\\cdot$m$^{-1}$),'' not just ``A\/m.'' Do not label axes with a ratio of quantities and \nunits. For example, write ``Temperature (K),'' not ``Temperature\/K.'' \n\nMultipliers can be especially confusing. Write ``Magnetization (kA\/m)'' or \n``Magnetization (10$^{3}$ A\/m).'' Do not write ``Magnetization \n(A\/m)$\\,\\times\\,$1000'' because the reader would not know whether the top \naxis label in Fig. 1 meant 16000 A\/m or 0.016 A\/m. Figure labels should be \nlegible, approximately 8 to 10 point type.\n\n\\subsubsection{Subfigure Labels in Multipart Figures and Tables}\nMultipart figures should be combined and labeled before final submission. \nLabels should appear centered below each subfigure in 8 point Times New \nRoman font in the format of (a) (b) (c). \n\n\\subsection{File Naming}\nFigures (line artwork or photographs) should be named starting with the \nfirst 5 letters of the author's last name. The next characters in the \nfilename should be the number that represents the sequential \nlocation of this image in your article. For example, in author \n``Anderson's'' paper, the first three figures would be named ander1.tif, \nander2.tif, and ander3.ps.\n\nTables should contain only the body of the table (not the caption) and \nshould be named similarly to figures, except that `.t' is inserted \nin-between the author's name and the table number. For example, author \nAnderson's first three tables would be named ander.t1.tif, ander.t2.ps, \nander.t3.eps.\n\nAuthor photographs should be named using the first five characters of the \npictured author's last name. For example, four author photographs for a \npaper may be named: oppen.ps, moshc.tif, chen.eps, and duran.pdf.\n\nIf two authors or more have the same last name, their first initial(s) can \nbe substituted for the fifth, fourth, third$\\ldots$ letters of their surname \nuntil the degree where there is differentiation. For example, two authors \nMichael and Monica Oppenheimer's photos would be named oppmi.tif, and \noppmo.eps.\n\n\\subsection{Referencing a Figure or Table Within Your Paper}\nWhen referencing your figures and tables within your paper, use the \nabbreviation ``Fig.'' even at the beginning of a sentence. Do not abbreviate \n``Table.'' Tables should be numbered with Roman Numerals.\n\n\\subsection{Checking Your Figures: The IEEE Graphics Analyzer}\nThe IEEE Graphics Analyzer enables authors to pre-screen their graphics for \ncompliance with IEEE Access standards before submission. \nThe online tool, located at\n\\underline{http:\/\/graphicsqc.ieee.org\/}, allows authors to \nupload their graphics in order to check that each file is the correct file \nformat, resolution, size and colorspace; that no fonts are missing or \ncorrupt; that figures are not compiled in layers or have transparency, and \nthat they are named according to the IEEE Access naming \nconvention. At the end of this automated process, authors are provided with \na detailed report on each graphic within the web applet, as well as by \nemail.\n\nFor more information on using the Graphics Analyzer or any other graphics \nrelated topic, contact the IEEE Graphics Help Desk by e-mail at \ngraphics@ieee.org.\n\n\\subsection{Submitting Your Graphics}\nBecause IEEE will do the final formatting of your paper,\nyou do not need to position figures and tables at the top and bottom of each \ncolumn. In fact, all figures, figure captions, and tables can be placed at \nthe end of your paper. In addition to, or even in lieu of submitting figures \nwithin your final manuscript, figures should be submitted individually, \nseparate from the manuscript in one of the file formats listed above in \nSection \\ref{formats}. Place figure captions below the figures; place table titles \nabove the tables. Please do not include captions as part of the figures, or \nput them in ``text boxes'' linked to the figures. Also, do not place borders \naround the outside of your figures.\n\n\\subsection{Color Processing\/Printing in IEEE Journals}\nAll IEEE Transactions, Journals, and Letters allow an author to publish \ncolor figures on IEEE Xplore\\textregistered\\ at no charge, and automatically \nconvert them to grayscale for print versions. In most journals, figures and \ntables may alternatively be printed in color if an author chooses to do so. \nPlease note that this service comes at an extra expense to the author. If \nyou intend to have print color graphics, include a note with your final \npaper indicating which figures or tables you would like to be handled that \nway, and stating that you are willing to pay the additional fee.\n\n\\section{Conclusion}\nA conclusion section is not required. Although a conclusion may review the \nmain points of the paper, do not replicate the abstract as the conclusion. A \nconclusion might elaborate on the importance of the work or suggest \napplications and extensions. \n\n\\appendices\n\nAppendixes, if needed, appear before the acknowledgment.\n\n\\section*{Acknowledgment}\n\nThe preferred spelling of the word ``acknowledgment'' in American English is \nwithout an ``e'' after the ``g.'' Use the singular heading even if you have \nmany acknowledgments. Avoid expressions such as ``One of us (S.B.A.) would \nlike to thank $\\ldots$ .'' Instead, write ``F. A. Author thanks $\\ldots$ .'' In most \ncases, sponsor and financial support acknowledgments are placed in the \nunnumbered footnote on the first page, not here.\n\n\\section*{References and Footnotes}\n\n\\subsection{References}\nReferences need not be cited in text. When they are, they appear on the \nline, in square brackets, inside the punctuation. Multiple references are \neach numbered with separate brackets. When citing a section in a book, \nplease give the relevant page numbers. In text, refer simply to the \nreference number. Do not use ``Ref.'' or ``reference'' except at the \nbeginning of a sentence: ``Reference \\cite{b3} shows $\\ldots$ .'' Please do not use \nautomatic endnotes in \\emph{Word}, rather, type the reference list at the end of the \npaper using the ``References'' style.\n\nReference numbers are set flush left and form a column of their own, hanging \nout beyond the body of the reference. The reference numbers are on the line, \nenclosed in square brackets. In all references, the given name of the author \nor editor is abbreviated to the initial only and precedes the last name. Use \nthem all; use \\emph{et al.} only if names are not given. Use commas around Jr., \nSr., and III in names. Abbreviate conference titles. When citing IEEE \ntransactions, provide the issue number, page range, volume number, year, \nand\/or month if available. When referencing a patent, provide the day and \nthe month of issue, or application. References may not include all \ninformation; please obtain and include relevant information. Do not combine \nreferences. There must be only one reference with each number. If there is a \nURL included with the print reference, it can be included at the end of the \nreference. \n\nOther than books, capitalize only the first word in a paper title, except \nfor proper nouns and element symbols. For papers published in translation \njournals, please give the English citation first, followed by the original \nforeign-language citation See the end of this document for formats and \nexamples of common references. For a complete discussion of references and \ntheir formats, see the IEEE style manual at\n\\underline{http:\/\/www.ieee.org\/authortools}.\n\n\\subsection{Footnotes}\nNumber footnotes separately in superscript numbers.\\footnote{It is recommended that footnotes be avoided (except for \nthe unnumbered footnote with the receipt date on the first page). Instead, \ntry to integrate the footnote information into the text.} Place the actual \nfootnote at the bottom of the column in which it is cited; do not put \nfootnotes in the reference list (endnotes). Use letters for table footnotes \n(see Table \\ref{table}).\n\n\\section{Submitting Your Paper for Review}\n\n\\subsection{Final Stage}\nWhen you submit your final version (after your paper has been accepted), \nprint it in two-column format, including figures and tables. You must also \nsend your final manuscript on a disk, via e-mail, or through a Web \nmanuscript submission system as directed by the society contact. You may use \n\\emph{Zip} for large files, or compress files using \\emph{Compress, Pkzip, Stuffit,} or \\emph{Gzip.} \n\nAlso, send a sheet of paper or PDF with complete contact information for all \nauthors. Include full mailing addresses, telephone numbers, fax numbers, and \ne-mail addresses. This information will be used to send each author a \ncomplimentary copy of the journal in which the paper appears. In addition, \ndesignate one author as the ``corresponding author.'' This is the author to \nwhom proofs of the paper will be sent. Proofs are sent to the corresponding \nauthor only.\n\n\\subsection{Review Stage Using ScholarOne\\textregistered\\ Manuscripts}\nContributions to the Transactions, Journals, and Letters may be submitted \nelectronically on IEEE's on-line manuscript submission and peer-review \nsystem, ScholarOne\\textregistered\\ Manuscripts. You can get a listing of the \npublications that participate in ScholarOne at \n\\underline{http:\/\/www.ieee.org\/publications\\_standards\/publications\/}\\break\\underline{authors\/authors\\_submission.html}.\nFirst check if you have an existing account. If there is none, please create \na new account. After logging in, go to your Author Center and click ``Submit \nFirst Draft of a New Manuscript.'' \n\nAlong with other information, you will be asked to select the subject from a \npull-down list. Depending on the journal, there are various steps to the \nsubmission process; you must complete all steps for a complete submission. \nAt the end of each step you must click ``Save and Continue''; just uploading \nthe paper is not sufficient. After the last step, you should see a \nconfirmation that the submission is complete. You should also receive an \ne-mail confirmation. For inquiries regarding the submission of your paper on \nScholarOne Manuscripts, please contact oprs-support@ieee.org or call +1 732 \n465 5861.\n\nScholarOne Manuscripts will accept files for review in various formats. \nPlease check the guidelines of the specific journal for which you plan to \nsubmit.\n\nYou will be asked to file an electronic copyright form immediately upon \ncompleting the submission process (authors are responsible for obtaining any \nsecurity clearances). Failure to submit the electronic copyright could \nresult in publishing delays later. You will also have the opportunity to \ndesignate your article as ``open access'' if you agree to pay the IEEE open \naccess fee. \n\n\\subsection{Final Stage Using ScholarOne Manuscripts}\nUpon acceptance, you will receive an email with specific instructions \nregarding the submission of your final files. To avoid any delays in \npublication, please be sure to follow these instructions. Most journals \nrequire that final submissions be uploaded through ScholarOne Manuscripts, \nalthough some may still accept final submissions via email. Final \nsubmissions should include source files of your accepted manuscript, high \nquality graphic files, and a formatted pdf file. If you have any questions \nregarding the final submission process, please contact the administrative \ncontact for the journal. \n\nIn addition to this, upload a file with complete contact information for all \nauthors. Include full mailing addresses, telephone numbers, fax numbers, and \ne-mail addresses. Designate the author who submitted the manuscript on \nScholarOne Manuscripts as the ``corresponding author.'' This is the only \nauthor to whom proofs of the paper will be sent. \n\n\\subsection{Copyright Form}\nAuthors must submit an electronic IEEE Copyright Form (eCF) upon submitting \ntheir final manuscript files. You can access the eCF system through your \nmanuscript submission system or through the Author Gateway. You are \nresponsible for obtaining any necessary approvals and\/or security \nclearances. For additional information on intellectual property rights, \nvisit the IEEE Intellectual Property Rights department web page at \n\\underline{http:\/\/www.ieee.org\/publications\\_standards\/publications\/}\\break\\underline{rights\/index.html}. \n\n\\section{IEEE Publishing Policy}\nThe general IEEE policy requires that authors should only submit original \nwork that has neither appeared elsewhere for publication, nor is under \nreview for another refereed publication. The submitting author must disclose \nall prior publication(s) and current submissions when submitting a \nmanuscript. Do not publish ``preliminary'' data or results. The submitting \nauthor is responsible for obtaining agreement of all coauthors and any \nconsent required from employers or sponsors before submitting an article. \nThe IEEE Access Department strongly discourages courtesy \nauthorship; it is the obligation of the authors to cite only relevant prior \nwork.\n\nThe IEEE Access Department does not publish conference \nrecords or proceedings, but can publish articles related to conferences that \nhave undergone rigorous peer review. Minimally, two reviews are required for \nevery article submitted for peer review.\n\n\\section{Publication Principles}\nThe two types of contents of that are published are; 1) peer-reviewed and 2) \narchival. The Access Department publishes scholarly \narticles of archival value as well as tutorial expositions and critical \nreviews of classical subjects and topics of current interest. \n\nAuthors should consider the following points:\n\n\\begin{enumerate}\n\\item Technical papers submitted for publication must advance the state of knowledge and must cite relevant prior work. \n\\item The length of a submitted paper should be commensurate with the importance, or appropriate to the complexity, of the work. For example, an obvious extension of previously published work might not be appropriate for publication or might be adequately treated in just a few pages.\n\\item Authors must convince both peer reviewers and the editors of the scientific and technical merit of a paper; the standards of proof are higher when extraordinary or unexpected results are reported. \n\\item Because replication is required for scientific progress, papers submitted for publication must provide sufficient information to allow readers to perform similar experiments or calculations and \nuse the reported results. Although not everything need be disclosed, a paper \nmust contain new, useable, and fully described information. For example, a \nspecimen's chemical composition need not be reported if the main purpose of \na paper is to introduce a new measurement technique. Authors should expect \nto be challenged by reviewers if the results are not supported by adequate \ndata and critical details.\n\\item Papers that describe ongoing work or announce the latest technical achievement, which are suitable for presentation at a professional conference, may not be appropriate for publication.\n\\end{enumerate}\n\n\\section{Reference Examples}\n\n\\begin{itemize}\n\n\\item \\emph{Basic format for books:}\\\\\nJ. K. Author, ``Title of chapter in the book,'' in \\emph{Title of His Published Book, x}th ed. City of Publisher, (only U.S. State), Country: Abbrev. of Publisher, year, ch. $x$, sec. $x$, pp. \\emph{xxx--xxx.}\\\\\nSee \\cite{b1,b2}.\n\n\\item \\emph{Basic format for periodicals:}\\\\\nJ. K. Author, ``Name of paper,'' \\emph{Abbrev. Title of Periodical}, vol. \\emph{x, no}. $x, $pp\\emph{. xxx--xxx, }Abbrev. Month, year, DOI. 10.1109.\\emph{XXX}.123456.\\\\\nSee \\cite{b3}--\\cite{b5}.\n\n\\item \\emph{Basic format for reports:}\\\\\nJ. K. Author, ``Title of report,'' Abbrev. Name of Co., City of Co., Abbrev. State, Country, Rep. \\emph{xxx}, year.\\\\\nSee \\cite{b6,b7}.\n\n\\item \\emph{Basic format for handbooks:}\\\\\n\\emph{Name of Manual\/Handbook, x} ed., Abbrev. Name of Co., City of Co., Abbrev. State, Country, year, pp. \\emph{xxx--xxx.}\\\\\nSee \\cite{b8,b9}.\n\n\\item \\emph{Basic format for books (when available online):}\\\\\nJ. K. Author, ``Title of chapter in the book,'' in \\emph{Title of\nPublished Book}, $x$th ed. City of Publisher, State, Country: Abbrev.\nof Publisher, year, ch. $x$, sec. $x$, pp. \\emph{xxx--xxx}. [Online].\nAvailable: \\underline{http:\/\/www.web.com}\\\\\nSee \\cite{b10}--\\cite{b13}.\n\n\\item \\emph{Basic format for journals (when available online):}\\\\\nJ. K. Author, ``Name of paper,'' \\emph{Abbrev. Title of Periodical}, vol. $x$, no. $x$, pp. \\emph{xxx--xxx}, Abbrev. Month, year. Accessed on: Month, Day, year, DOI: 10.1109.\\emph{XXX}.123456, [Online].\\\\\nSee \\cite{b14}--\\cite{b16}.\n\n\\item \\emph{Basic format for papers presented at conferences (when available online): }\\\\\nJ.K. Author. (year, month). Title. presented at abbrev. conference title. [Type of Medium]. Available: site\/path\/file\\\\\nSee \\cite{b17}.\n\n\\item \\emph{Basic format for reports and handbooks (when available online):}\\\\\nJ. K. Author. ``Title of report,'' Company. City, State, Country. Rep. no., (optional: vol.\/issue), Date. [Online] Available: site\/path\/file\\\\\nSee \\cite{b18,b19}.\n\n\\item \\emph{Basic format for computer programs and electronic documents (when available online): }\\\\\nLegislative body. Number of Congress, Session. (year, month day). \\emph{Number of bill or resolution}, \\emph{Title}. [Type of medium]. Available: site\/path\/file\\\\\n\\textbf{\\emph{NOTE: }ISO recommends that capitalization follow the accepted practice for the language or script in which the information is given.}\\\\\nSee \\cite{b20}.\n\n\\item \\emph{Basic format for patents (when available online):}\\\\\nName of the invention, by inventor's name. (year, month day). Patent Number [Type of medium]. Available: site\/path\/file\\\\\nSee \\cite{b21}.\n\n\\item \\emph{Basic format}\\emph{for conference proceedings (published):}\\\\\nJ. K. Author, ``Title of paper,'' in \\emph{Abbreviated Name of Conf.}, City of Conf., Abbrev. State (if given), Country, year, pp. \\emph{xxxxxx.}\\\\\nSee \\cite{b22}.\n\n\\item \\emph{Example for papers presented at conferences (unpublished):}\\\\\nSee \\cite{b23}.\n\n\\item \\emph{Basic format for patents}$:$\\\\\nJ. K. Author, ``Title of patent,'' U.S. Patent \\emph{x xxx xxx}, Abbrev. Month, day, year.\\\\\nSee \\cite{b24}.\n\n\\item \\emph{Basic format for theses (M.S.) and dissertations (Ph.D.):}\n\\begin{enumerate}\n\\item J. K. Author, ``Title of thesis,'' M.S. thesis, Abbrev. Dept., Abbrev. Univ., City of Univ., Abbrev. State, year.\n\\item J. K. Author, ``Title of dissertation,'' Ph.D. dissertation, Abbrev. Dept., Abbrev. Univ., City of Univ., Abbrev. State, year.\n\\end{enumerate}\nSee \\cite{b25,b26}.\n\n\\item \\emph{Basic format for the most common types of unpublished references:}\n\\begin{enumerate}\n\\item J. K. Author, private communication, Abbrev. Month, year.\n\\item J. K. Author, ``Title of paper,'' unpublished.\n\\item J. K. Author, ``Title of paper,'' to be published.\n\\end{enumerate}\nSee \\cite{b27}--\\cite{b29}.\n\n\\item \\emph{Basic formats for standards:}\n\\begin{enumerate}\n\\item \\emph{Title of Standard}, Standard number, date.\n\\item \\emph{Title of Standard}, Standard number, Corporate author, location, date.\n\\end{enumerate}\nSee \\cite{b30,b31}.\n\n\\item \\emph{Article number in~reference examples:}\\\\\nSee \\cite{b32,b33}.\n\n\\item \\emph{Example when using et al.:}\\\\\nSee \\cite{b34}.\n\n\\end{itemize}\n\n\n\\section{Introduction}\n\n\nColorectal polyps, especially adenomas with high-grade dysplasia, carry high risks of progressing into colorectal cancer (CRC) \\cite{gschwantler2002high}, which claims over 640,000 lives each year \\cite{bernal2017comparative}. There are available procedures to screen and detect high-risk polyps in an early stage, increasing chances of successful treatment. Polyp detection and removal in colonoscopy are the most effective method to prevent colorectal cancer \\cite{issa2017colorectal}.\n\nAt the same time, practical factors such as overloading healthcare systems, low-quality endoscopy equipment, or personnel's lack of experience \\cite{armin2015visibility,lee2008adequate} can severely limit the effectiveness of colonoscopy. A review by Leufkens et al. \\cite{leufkens2012factors} pointed out that $20-47\\%$ of polyps might have been missed during colonoscopies. Several types of image-enhanced endoscopies and accessories have been proposed to alleviate these, yet they can be prohibitively expensive for practical applications, especially in poor medical facilities. On the other hand, computer-aided systems for colonoscopy have shown a lot of promise and have attracted many researchers in recent years. Several works have achieved very good performance on benchmark datasets \\cite{fan2020pranet,huang2021hardnet,tang2019cu}.\n\nPolyp segmentation is a subset of medical image analysis that has gained much attention recently. Traditional machine learning methods for solving the problem are mostly based on hand-crafted features \\cite{iwahori2013automatic,silva2014toward} to extract image information such as color, shape, and textures. Since polyps have very high intra-class diversity and low inter-class variation, such approaches are often limited in representing and detecting polyps. Deep neural networks, and especially U-Net \\cite{ronneberger2015u} have been the state-of-the-art methods for polyp segmentation in the last few years. These networks can learn highly abstract and complex features, allowing them to achieve good performances. At the same time, deep neural networks also come with a complexity trade-off, as models can be very large (up to several hundred million parameters) and cause high latency during inference.\n\n\nMost of the existing research in endoscopy image analysis has focused on the polyp segmentation problem, in which lesion regions or polyps are segmented from the background pixels. Those works attempted to improve the learning models to provide accurate segmentation of polyps. However, the segmentation of polyps only does not provide information about the type of polyps, i.e., benign (non-cancerous) or malignancy (presence of cancerous cells).\nIn our seminal work \\cite{lan2021neounet}, we have defined the problem of Polyp Segmentation and Neoplasm Detection (PSND), aiming at fine-grained segmentation of polyps. This can be considered as an extension of the polyp segmentation problem, providing richer semantic information for the segmented regions. Particularly, besides classifying image pixels as polyp or background, the proposed formulation further identifies each polyp pixel as non-neoplastic, neoplastic, or undefined. In general, non-neoplastic polyps are typically benign, while neoplastic polyps have a risk of developing cancer.\nOur newly developed UNet-based neural network architecture, NeoUNet, has obtained state-of-the-art performance in solving this problem in terms of accuracy. However, attention has not yet been paid to the model size and speed, which is challenging for practical deployment.\n\nIn this paper, we further improve on our previous works on Polyp Segmentation and Neoplasm Detection with the proposal of BlazeNeo\\xspace{}, a novel deep neural network architecture with an efficient learning mechanism. Our main contributions are:\n\\begin{itemize}\n   \n   \n   \n    \\item A new deep neural network architecture called BlazeNeo\\xspace{} for polyp segmentation and neoplasm detection, aiming at reducing the model size and therefore improving inference speed;\n    \\item An auxiliary training strategy to fully exploit training data for maintaining high accuracy while reducing model size;\n    \\item Extensive experiments on the newly collected NeoPolyp\\xspace dataset and comparisons to existing models. Moreover, we measure model latency and throughput on dedicated hardware in a setting similar to real-life deployments of polyp segmentation and neoplasm detection.\n\\end{itemize}\n\nThe rest of the paper is organized as follows. We provide a brief review of related works in Section \\ref{sec:related}. Section \\ref{sec:problem} briefly describes the polyp segmentation and neoplasm detection problem as formulated in \\cite{lan2021neounet}. The BlazeNeo\\xspace architecture is presented in Section \\ref{sec:propose}. Section \\ref{sec:experiment} showcases our experimental studies. Finally, we conclude the paper and highlight future works in Section \\ref{sec:conclude}.\n\n\\section{Related work}\n\\label{sec:related}\nIn recent years, many computer vision tasks have seen massive improvements through the advancements of convolutional neural networks (CNNs). AlexNet \\cite{krizhevsky2012imagenet} and VGG \\cite{simonyan2014very} are among the first successful CNNs for the image classification problem. However, these early models still suffer from degradation when increasing network depth. Many works have attempted to modify the network architectures to improve learning capability aiming at improving network performance. Skip connections, first introduced in ResNet \\cite{he2016deep} in 2016, helped alleviate the degradation and smoothed out the loss landscape. ResNeXt \\cite{xie2017aggregated} combined the idea of skip connections with a multi-branch design first proposed by the authors of GoogLeNet \\cite{szegedy2015going}. More recently, Tan et al. \\cite{tan2019efficientnet} employed neural architecture search to produce EfficientNet, a family of neural networks with varying levels of the trade-off between accuracy and latency. Meanwhile, HarDNet \\cite{chao2019hardnet} is a model highly focused on optimizing inference latency and memory traffic.\n\nMany CNN architectures have been designed for the semantic image segmentation task, especially in medical images. Among the earliest was the work by Long et al. \\cite{long2015fully}, who adopted several well-known architectures using transfer learning. In the same year, U-Net \\cite{ronneberger2015u} became a breakthrough model in medical imaging, achieving highly promising results for medical image segmentation. Later works such as UNet++ \\cite{zhou2019unet++}, DoubleUNet \\cite{jha2020doubleu} and Coupled U-Net \\cite{tang2019cu} further improved and alleviated limitations in U-Net. ColonSegNet \\cite{jha2021real} was a lightweight encoder-decoder architecture that uses residual connections with squeeze and excitation network as the main component. ColonSegNet achieved a high inference speed but with sacrificing accuracy a lot. DDANet \\cite{tomar2020ddanet} was another encoder-decoder design that leverages the strength of residual connection and squeeze-and-excitation modules. DDANet incorporated a single encoder followed by two dual decoders. The first decoder is used for the segmentation mask, while the second one acts as an autoencoder model that reconstructs the grayscale image and helps strengthen the feature representation of the encoder. Attention-UNet \\cite{oktay2018attention} proposed attention gates as a filter mechanism for selecting useful salient features. Development on non-UNet models in medical segmentation has also remained active. DeepLabV3 \\cite{chen2017rethinking} is a prominent architecture that utilizes atrous convolutions for dense feature extraction. Fan et al. \\cite{fan2020pranet} enhanced an FCN-like model with parallel partial decoder and reverse attention to form PraNet, a network that achieved state-of-the-art performance on many benchmark datasets. HarDNet-MSEG \\cite{huang2021hardnet} employed an encoder-decoder structure with HarDNet as the encoder backbone, achieving good performance on the Kvasir-SEG dataset and very high inference speed. Meanwhile, TransFuse \\cite{zhang2021transfuse} combined a CNN with the Transformer architecture using a fusion module called BiFusion.\n\nMany deep learning methods are also specially designed for the polyp segmentation and detection problem. Qadir et al. \\cite{qadir2019framework} proposed a framework that incorporates a CNN architecture for labeling segmentation masks for polyps. The framework allows doctors to receive pre-annotations from a model trained in a semi-supervised manner. In \\cite{shin2018automatic}, the authors proposed a model with an Inception-ResNet backbone combined with several post-learning methods to enhance polyp detection accuracy. Shin et al. \\cite{shin2018abnormal} used conditional adversarial networks to generate abnormal samples for training polyp detection models. Liu et al. \\cite{liu2019colonic} applied different CNN backbones including InceptionV3 \\cite{szegedy2015going}, ResNet50 \\cite{he2016deep} and VGG16 \\cite{simonyan2014very} to the SSD  framework, whose accuracy was much higher than other one-stage object detectors and comparable to the Faster-RCNN two-stage one. In \\cite{li2021colonoscopy}, Li et al. compared the performance of eight state-of-the-art deep learning object detectors and demonstrated promising results in colonoscopy.\n\nWhile there have been few works in the past several years concerning polyp neoplasm detection (such as that of Ribeiro et al. \\cite{ribeiro2016exploring}), it was only very recently that neoplasm detection has been incorporated into polyp segmentation. Phan et al. \\cite{lan2021neounet} formally described the problem (denoted as PSND) and proposed NeoUNet, a UNet-based architecture that established the baseline for PSND. While NeoUNet strikes a balance between accuracy and inference speed, typical deployment scenarios for polyp segmentation systems require relatively low latency to ensure smooth operation. Moreover, such scenarios would have the neural network run on lightweight systems embedded within the colonoscopy device. Therefore, this paper focuses on building a network architecture optimized for speed and small in size.\n\n\n\\begin{figure*}[ht!]\n    \\begin{center}\n        \\includegraphics[width=0.9\\textwidth]{images\/PSND.png}\n    \\end{center}\n    \\caption{PSND is an extension of polyp segmentation, which further discriminates whether a polyp is neoplastic or non-neoplastic. For a given input image shown on the top right corner, expected outputs for polyp segmentation and PSND are depicted on the middle and the bottom images on the right, respectively. The black color denotes background pixels, white color denotes polyp regions; green and red colors denote non-neoplastic and neoplastic polyps, respectively.}\n    \\label{fig:psnd}\n\\end{figure*}\n\n\\section{Polyp Segmentation and Neoplasm Detection}\n\\label{sec:problem}\n\nPolyp Segmentation and Neoplasm Detection (PSND) has been formulated as a type of fine-grained polyp segmentation problem \\cite{lan2021neounet}. Besides segmentation of polyps, this formulation further classifies a polyp pixel into two classes: non-neoplastic or neoplastic. In medical image analysis, non-neoplastic polyps are considered benign, while neoplastic polyps may further progress with a higher risk of cancer. During a colonoscopy, the operator must decide the types of polyps, neoplasm or non-neoplastic, to consider an optimal management strategy, i.e., removal or resection in endoscopy procedure or biopsy and operation. Figures \\ref{fig:psnd} depicts the PSND problem as a semantic extension of the polyp segmentation problem.\n\nIn practice, an ``undefined'' class is labeled if there is not enough information from the endoscopic image to categorize the risk of neoplasm. This undefined subgroup is not a specific class we want to predict since such predictions would not bring any insight for the endoscopist. Therefore, the model only needs to learn to discriminate between neoplastic and non-neoplastic polyps. However, the undefined class still gives supplementary information about polyp regions that can be exploited to help the model gain more representation power in the training phase.\n\n\n\n\\section{Proposed Method}\n\\label{sec:propose}\n\n\\begin{figure*}[ht!]\n    \\begin{center}\n        \\includegraphics[width=\\textwidth]{images\/NeoSeg-Edit.png}\n    \\end{center}\n    \\caption{Proposed architectures of our BlazeNeo\\xspace: (a) Single-headed Binary BlazeNeo (SB-BlazeNeo) has one output branch that produces two binary segmentation maps corresponding to neoplastic and non-neoplastic classes; (b) Single-headed Trinary BlazeNeo (ST-BlazeNeo) also has one output branch that directly predicts a trinary segmentation map; (c) Multi-headed BlazeNeo (or BlazeNeo for short) contains two output branches that are responsible for the two tasks: neoplasm segmentation treated as the main task, and polyp segmentation treated as the auxiliary task. Both branches share the same architecture of the feature aggregation module, but they are trained separately without sharing their parameters.}\n    \\label{fig:model}\n\\end{figure*}\n\nFigure~\\ref{fig:model} depicts three versions of the proposed BlazeNeo\\xspace model. The models are developed based on HarDNet-MSEG~\\cite{huang2021hardnet}, with several improvements tailored for the PSND problem. First, we use a simplified version of RFB~\\cite{liu2018receptive} with smaller kernel sizes and apply different feature aggregation schemes. For the encoder layer, we keep the HarDNet-68~\\cite{chao2019hardnet} backbone as in HarDNet-MSEG.\n\nThe three BlazeNeo\\xspace variants differ in how outputs from the model are generated and processed. Inspired by NeoUNet \\cite{lan2021neounet}, Single-headed Binary BlazeNeo\\xspace (SB-BlazeNeo\\xspace) solves two binary segmentation tasks corresponding to neoplastic and non-neoplastic classes.\nThe second variant, Single-headed Trinary BlazeNeo\\xspace (ST-BlazeNeo\\xspace), predicts a trinary map for three classes in the neoplasm segmentation task. Finally, Multi-headed BlazeNeo\\xspace (or BlazeNeo\\xspace for short) uses two output branches. The main output branch is used for the neoplasm segmentation task, and the auxiliary branch is for the polyp segmentation task. The following subsections describe our design and motivations in detail.\n\n\\subsection{Lightweight Encoder: HarDNet}\n\n\\begin{center}\n    \\begin{figure}[th]\n        \\begin{center}\n            \\includegraphics[width=0.48\\textwidth]{images\/HarDNet.png}\n        \\end{center}\n        \\caption{Illustrations for DenseNet block and Harmonic DenseNet (HarDNet) block. Each of the layers is a $3\\times3$ convolution. The value on each layer denotes the number of output channels.}\n        \\label{fig:hardnet}\n    \\end{figure}\n\\end{center}\n\nHarDNet (Harmonic Densely Connected Network)~\\cite{chao2019hardnet} is an improvement over DenseNet~\\cite{huang2017densely}. The primary goal in HarDNet's design is to lower latency by reducing memory traffic. The authors argued that the connection pattern of Dense Blocks, in which each layer has skip connections toward every proceeding layer in the block, causes ineffective memory access during runtime that severely hinders performance. HarDNet reduces the number of skip connections to form a pattern similar to the harmonic wave function, as well as scaling channel width according to a layer's influence level. We illustrate the difference between the two architectures in Figure~\\ref{fig:hardnet}.\n\nInside a HarDNet block, each layer is indexed from the input layer 0. A layer $l$ receives a skip connection from layer $l-2^n$ if $2^n$ divides $l$ ($n \\geq 0$, $l-2^n \\geq 0$). Given the initial growth rate $k$ and compression factor $m$, layer $l$'s channel width is equal to $k \\times m^x$, where $x = max \\{\\nu \\ |\\ l\\ \\vdots\\ 2^{\\nu} \\}$.\n\nChao et al. \\cite{chao2019hardnet} further proposed HarDNet-68 for small-object detection problems. While most CNNs focus on stride-16 to enhance classification ability, HarDNet-68 distributes most of the layers on stride-8 to aid small-scale object detection, as shown in Figure~\\ref{fig:hardnet-68}. Experiments in~\\cite{chao2019hardnet} show that HarDNet-68 is not only 30\\% faster than ResNet-50~\\cite{he2016deep} but also more accurate than ResNet-101~\\cite{he2016deep} when used as backbone for SSD~\\cite{liu2016ssd} in the object detection problem.\n\nOur BlazeNeo\\xspace also uses HarDNet-68 as the encoder backbone. For an input colonoscopy image $I$ with size  ${h \\times w}$, five levels of features ${\\{f_{i}, i = 1, 2, 3, 4, 5\\}}$ with resolution ${[h\/2^{i-1}, w\/2^{i-1}]}$ are produced from the encoder. Wu et al. \\cite{wu2019cascaded} showed that low-level features (corresponding to $f_1$ and $f_2$) contribute less to performance while being computationally expensive due to their size. Therefore, our BlazeNeo\\xspace discards $f_1$ and $f_2$, using only the final three feature maps for the decoder module to accelerate its inference speed.\n\n\\begin{center}\n    \\begin{figure}[th]\n        \\begin{center}\n            \\includegraphics[width=0.45\\textwidth]{images\/HarDNet-68.png}\n        \\end{center}\n        \\caption{An overview of HarDNet-68 architecture. Following each HarDNet block is a transitional Conv 1x1 layer.}\n        \\label{fig:hardnet-68}\n    \\end{figure}\n\\end{center}\n\n\\subsection{Parallel Partial Decoder}\nIn all three variants of our BlazeNeo\\xspace, the decoder module consists of a Receptive Field Block (RFB) series. Three last feature maps of the encoder are independently passed through the RFB blocks and then fused by the feature aggregation blocks.\n\n\\subsubsection{Receptive Field Block}\n\nPolyps can appear in various scales on endoscopic images depending on their actual size, their distance to the colonoscopy camera, or the angle between them and the camera. This is a challenge for CNN architectures, in which the receptive field size is often fixed. Several studies have suggested different mechanisms to create more robust receptive fields, including the Inception block~\\cite{szegedy2015going}, ASPP block~\\cite{chen2017deeplab}, and Deformable Convolution block~\\cite{dai2017deformable}. Conceptually, these proposals are similar in that they all use multiple convolutional branches with different kernel sizes, merging the outputs to form adaptive receptive fields. However, they also have their own limitations. The Inception block samples all the kernels of each branch at their center, ignoring crucial edge details due to small sampling coverage. Meanwhile, the ASPP and Deformable Convolution blocks do not differentiate between different pixel positions, making it difficult for the model to focus on segmentation targets.\n\nThe Receptive Field Block (RFB) \\cite{liu2018receptive} has been proposed to address these limitations. RFB also uses the multi-branch convolution approach, with improvements inspired by the human visual cortex. In addition, it highlights the relationship between receptive field size and eccentricity, where the distance to the center is proportional to the weight specified or the importance level.\n\nThe RFB module used in PraNet \\cite{fan2020pranet} and HarDNet-MSEG \\cite{huang2021hardnet} is modified with a larger kernel size and larger dilation rate. As this version of RFB is more computationally expensive, our BlazeNeo\\xspace instead uses a simplified version of RFB (see Figure~\\ref{fig:rfb}) as proposed in \\cite{liu2018receptive} for faster inference.\n\n\\begin{center}\n    \\begin{figure}[th]\n        \\begin{center}\n            \\includegraphics[scale=.3]{images\/RFB.png}\n        \\end{center}\n        \\caption{Structure of the Receptive Field Block (RFB).}\n        \\label{fig:rfb}\n    \\end{figure}\n\\end{center}\n\n\\subsubsection{Feature Aggregation}\nA high polyp miss rate is often associated with small and flat polyps (whose perimeters are below $10 mm$)~\\cite{kim2017miss}. In order to detect these polyps, it is important to obtain high-resolution features from multiple image scales. Feature fusion (or aggregation) is a well-studied technique to achieve this for CNNs, in which feature maps from different scales are fused to form a multi-scale feature map. Figure \\ref{fig:fa} demonstrates four feature aggregation techniques, in order of increasing complexity \\cite{zhang2020multi}.\n\n\\begin{center}\n    \\begin{figure}[th]\n        \\begin{center}\n            \\includegraphics[scale=.25]{images\/FA.png}\n        \\end{center}\n        \\caption{Different feature aggregation schemes as shown in \\cite{zhang2020multi}. \\textbf{(a)} Long skip connection (LSC); \\textbf{(b)} Iterative Deep Aggregation (IDA); \\textbf{(c)} Dense Iterative Aggregation (DIA); \\textbf{(d)} Dense Hierarchical Aggregation (DHA).}\n        \\label{fig:fa}\n    \\end{figure}\n\\end{center}\n\nLong Skip Connection (LSC) illustrated in Figure~\\ref{fig:fa}a is an early aggregation scheme used by segmentation networks such as UNet~\\cite{ronneberger2015u}, UNet++~\\cite{zhou2019unet++}, and Attention UNet~\\cite{oktay2018attention}. Given feature maps $X_1$, $X_2$, $X_3$, higher-level feature maps are upsampled and fused with their adjacent low-level features by a long skip connection, gradually restoring the spatial information. Each fusion module includes a concatenation layer and a convolutional layer with kernel size $3\\times3$.\n\nLSC is a well-tested and straightforward technique, but it is not without limitations. Zhang et al. \\cite{zhang2020multi} proposed three alterations to produce higher-quality features, as described below.\n\nIterative Deep Aggregation (IDA) depicted in Figure~\\ref{fig:fa}b produces finer feature maps by using multiple iterative convolutions for a single scale.\n\nInspired by DenseNet, Dense Iterative Aggregation (DIA) introduces dense skip connections to the iterative convolutions in IDA (see Figure~\\ref{fig:fa}c). This addition ensures maximum information flow and reduces overfitting.\n\nDense Hierarchical Aggregation (DHA) further enhances semantic information by re-combining the output with high-level feature maps as shown in Figure~\\ref{fig:fa}d.\n\nFrom these observations, in our BlazeNeo\\xspace, we propose to apply feature aggregation to the outputs of RFB modules at different scales. We examine variants with each of the aforementioned aggregation schemes, namely BlazeNeo-LSC\\xspace, BlazeNeo-IDA\\xspace, BlazeNeo-DIA\\xspace, and BlazeNeo-DHA\\xspace, in section \\ref{sec:experiment}.\n\n\\subsection{Loss Function and Auxiliary Training}\nWe aim to exploit the information from data with the undefined labels for training in a way similar to \\cite{lan2021neounet}. The intuition is that while these data do not provide\ninformation for deciding on neoplasm class, they still can provide some semantic meaning from the data for the segmentation.\n\nTo training BlazeNeo\\xspace, we propose the loss function $\\mathcal{L}_{total}$ consisting of two components: a main loss $\\mathcal{L}_{main}$ associated with the main task of neoplasm segmentation, and an auxiliary loss $\\mathcal{L}_{aux}$ associated with the auxiliary task of polyp segmentation. The total loss can be expressed as follows:\n\n\\begin{equation}\n    \\label{eq:total_loss}\n    \\mathcal{L}_{total} =  \\mathcal{L}_{main} + \\mathcal{L}_{aux}\n\\end{equation}\n\nThe main loss $\\mathcal{L}_{main}$ drives the model toward making accurate class-specific segmentation. The pixels with undefined labels are excluded when calculating $\\mathcal{L}_{main}$. The auxiliary loss $\\mathcal{L}_{aux}$ drives the model toward making accurate foreground-background segmentation, in which all pixels are used for training.\n\nTo investigate this training strategy, we iterate through three different ways to incorporate it into BlazeNeo\\xspace, which creates three different variants: Single-headed Binary BlazeNeo\\xspace (SB-BlazeNeo\\xspace), Single-headed Trinary BlazeNeo\\xspace (ST-BlazeNeo\\xspace), and Multi-headed BlazeNeo\\xspace (the final version of BlazeNeo\\xspace).\n\n\\subsubsection{Single-headed Binary BlazeNeo\\xspace}\nOur first variant, SB-BlazeNeo\\xspace (Figure~\\ref{fig:model}a), uses the same loss formulation as the NeoUNet model proposed in \\cite{lan2021neounet}. The final output layer produces two binary segmentation maps, one for the neoplastic class and another for the non-neoplastic class. Losses are calculated separately for each map and then averaged. The main loss is a combination of Binary Cross Entropy and Focal Tversky loss \\cite{abraham2019novel} as follows:\n\\begin{align}\n     & \\mathcal{L}_{main} = \\mathcal{L}_{main}^{neo} + \\mathcal{L}_{main}^{non} \\nonumber                               \\\\\n     & = \\mathcal{L}_{BCE}(P_{main}^{neo}, G_{main}^{neo}) + \\mathcal{L}_{FT}(P_{main}^{neo}, G_{main}^{neo}) \\nonumber \\\\\n     & +  \\mathcal{L}_{BCE}(P_{main}^{non}, G_{main}^{non}) + \\mathcal{L}_{FT}(P_{main}^{non}, G_{main}^{non})\n    \\label{eq:main_loss_sb}\n\\end{align}\nwhere $P_{main}^{neo}$ and $P_{main}^{non}$ denote the prediction maps for the neoplastic and non-neoplastic classes, respectively; $G_{main}^{neo}$ and $G_{main}^{non}$ are ground truths. We choose the Focal Tversky loss for $\\mathcal{L}_{main}$ to alleviate class imbalance due to small amount of non-neoplastic polyp pixels as shown later in Figure~\\ref{fig:data_dist}.\n\nThe auxiliary loss is a combination of Binary Cross Entropy and Tversky loss, which uses an auxiliary polyp segmentation map inferred from the two binary class-specific maps:\n\\begin{equation}\n    \\label{eq:aux_loss_sb}\n    \\mathcal{L}_{aux} =  \\mathcal{L}_{BCE}(P_{aux}^{polyp}, G_{aux}^{polyp}) + \\mathcal{L}_{T}(P_{aux}^{polyp}, G_{aux}^{polyp})\n\\end{equation}\nwhere $P_{aux}^{polyp}$ and $G_{aux}^{polyp}$ denote the auxiliary polyp prediction map and the corresponding ground truth.\n\nThe auxiliary polyp prediction map $P_{aux}^{polyp}$ is inferred using element-wise max:\n\\begin{equation}\n    P_{aux}^{polyp} = \\max(P_{main}^{neo}, P_{main}^{non})\n    \\label{eq:mapconvert}\n\\end{equation}\n\n\n\\subsubsection{Single-headed Trinary BlazeNeo\\xspace}\nWhile the method using binary map \\cite{lan2021neounet} yielded promising classification results, we found that a lighter model can benefit from imposing an additional constraint on the outputs.\nSpecifically, when the model outputs two separate class-specific maps for one image, it may make both maps have high prediction values for the same pixel. Therefore, a class constraint is needed to ensure that one class is chosen for each pixel. In our second variant, ST-BlazeNeo\\xspace (Figure~\\ref{fig:model}b), we use a 3-channel output map $P_{main}^{trinary}$, denoting the probabilities for the neoplastic, non-neoplastic, and background class, respectively. A softmax activation is used on the channel dimension, meaning each pixel may only belong to one class.\n\nThe main loss is a combination of Categorical Cross Entropy and Focal Tversky loss as follows:\n\n\\begin{align}\n    \\label{eq:main_loss}\n    \\mathcal{L}_{main} & = \\mathcal{L}_{CCE}(P_{main}^{trinary}, G_{main}^{trinary}) \\nonumber \\\\\n                       & + \\mathcal{L}_{FT}(P_{main}^{trinary}, G_{main}^{trinary})\n\\end{align}\nwhere $G_{main}^{trinary}$ is the trinary ground truth.\n\nSimilarly, the auxiliary loss is a combination of Categorical Cross Entropy and Tversky loss:\n\n\\begin{equation}\n    \\label{eq:aux_loss}\n    \\mathcal{L}_{aux} =  \\mathcal{L}_{CCE}(P_{aux}^{polyp}, G_{aux}^{polyp}) + \\mathcal{L}_{T}(P_{aux}^{polyp}, G_{aux}^{polyp})\n\\end{equation}\nHere we apply element-wise max to the two channels of the map $P_{main}^{trinary}$, which correspond to the neoplastic and non-neoplastic classes, to produce the auxiliary polyp segmentation map $P_{aux}^{polyp}$.\n\n\n\\subsubsection{Multi-headed BlazeNeo\\xspace}\nFor our final version of BlazeNeo\\xspace, we further evolve the use of auxiliary loss by adding an auxiliary segmentation branch.\nAuxiliary training is the process of jointly learning a \\textit{side} or \\textit{auxiliary} task to enhance the main task's performance \\cite{chennupati2019auxnet,zhang2020auxiliary}. This idea is similar to multi-task learning, except the auxiliary branch does not activate in the inference phase.\n\nThe auxiliary branch uses an identical (with separate weights) feature aggregation module but outputs a binary segmentation map instead of a 3-channel multi-class map. The main loss and auxiliary loss are calculated exactly as in Eq.~(\\ref{eq:main_loss}) and Eq.~(\\ref{eq:aux_loss}), respectively. However, there is no conversion between multi-class and polyp segmentation maps as in Eq.~(\\ref{eq:mapconvert}). Instead, the main loss is calculated with the 3-channel multi-class map, and the auxiliary loss is calculated with the output map from the auxiliary branch. Intuitively, we believe the model will benefit from adding an auxiliary network branch since it strengthens the supervised signal and betters the optimization process during training the model. Figure~\\ref{fig:model}c describes how BlazeNeo\\xspace incorporates the auxiliary branch for training in our BlazeNeo.\n\n\n\n\\section{Experiments}\n\\label{sec:experiment}\n\n\\subsection{Benchmark dataset}\n\n\\begin{figure*}[ht!]\n    \\begin{center}\n        \\includegraphics[width=\\textwidth]{images\/examples\/polyp_examples.png}\n    \\end{center}\n    \\caption{Some examples from the NeoPolyp dataset. The first row displays original images from the dataset. The second row shows the ground truths for polyp segmentation. The last row shows the ground truths for neoplasm segmentation, where some polyps are undefined and marked by yellow color. From left to right, the color modes are WLI, BLI, LCI, FICE, and FICE, respectively.}\n    \\label{fig:polyp:examples}\n\\end{figure*}\n\n\\begin{center}\n    \\begin{figure*}[ht!]\n        \\subcaptionbox{Polyp-wise distribution}{\n            \\includegraphics[width=0.48\\textwidth]{images\/polyp-wise-distribution.png}\n        }\n        \\subcaptionbox{Pixel-wise distribution}{\n            \\includegraphics[width=0.48\\textwidth]{images\/pixel-wise-distribution.png}\n        }\n        \\caption{Data distribution of polyp class labels in the NeoPolyp\\xspace dataset. In the pixel-wise distribution on the right, percentages are calculated on polyp pixels only (not including background pixels.)}\n        \\label{fig:data_dist}\n    \\end{figure*}\n\\end{center}\n\nWe use the NeoPolyp\\xspace dataset as introduced in \\cite{lan2021neounet} to train and benchmark the proposed BlazeNeo\\xspace{}. The dataset consists of 7,466 annotated endoscopic images captured directly during endoscopic recording and includes all four lighting modes: WLI (White Light Imaging), FICE (Flexible spectral Imaging Color Enhancement), BLI (Blue Light Imaging), and LCI (Linked Color Imaging). NeoPolyp\\xspace is split into a training set of 5,966 images and a test set of 1,500 images. Some examples of the NeoPolyp\\xspace dataset are shown in Figure~\\ref{fig:polyp:examples}. For comparison with baseline models, we also use the NeoPolyp-Clean\\xspace dataset, which does not contain any polyps with undefined class labels. This dataset consists of 5,277 training images and 1,353 test images.\n\nIn practice, most non-neoplastic polyps are small, and the endoscopists can immediately remove them without the need to capture images or take a biopsy for lesions less than 5mm for post-checking. Due to that reason, neoplastic polyps take up a majority of the polyps present in NeoPolyp\\xspace (see Figure~\\ref{fig:data_dist}). The number of neoplastic, non-neoplastic, and undefined polyps are 5113, 3185, and 1031, respectively. However, if we look at the pixel-wise level as shown in Figure~\\ref{fig:data_dist}b, we can observe a strong data imbalance between the three classes. The number of neoplastic polyp pixels takes up to 80\\% among all polyp pixels in the dataset. Meanwhile, these numbers for non-neoplastic and undefined classes are 13\\% and 7\\%, respectively. This data imbalance, combined with the inherent challenges of PSND, creates a difficult benchmark for models to overcome.\n\n\n\\subsection{Experiment setup}\n\\begin{center}\n    \\begin{figure*}[ht]\n       \n       \n       \n        \\subcaptionbox{Non-neoplastic class}{\n            \\includegraphics[width=0.5\\textwidth]{images\/aux-effectiveness-non.png}\n        }\n        \\subcaptionbox{Neoplastic class}{\n            \\includegraphics[width=0.5\\textwidth]{images\/aux-effectiveness-neo.png}\n        }\n        \\caption{Dice scores and FPS for different BlazeNeo\\xspace variations, with and without auxiliary training. Red triangles denote results with auxiliary training, while blue circles are those without auxiliary training.}\n        \\label{fig:aux-effectiveness}\n    \\end{figure*}\n\\end{center}\n\n\nOur experiments include several ablation studies to verify the effectiveness of each component in BlazeNeo\\xspace, a comparison against several polyp segmentation methods, and benchmarks on the NVIDIA Jetson AGX Xavier developer kit \\footnote{https:\/\/developer.nvidia.com\/embedded\/jetson-agx-xavier-developer-kit}, which closely resembles real deployments. The Jetson device is configured to run at MAXN power mode.\n\nWe oversample non-neoplastic polyps to account for class imbalance in the NeoPolyp\\xspace dataset, as addressed by \\cite{lan2021neounet}. Images containing non-neoplastic polyps are duplicated such that $P_{non} \\approx P_{neo}$, where $P_{non}$ and $P_{neo}$ are the number of pixels containing in  non-neoplastic and neoplastic polyps, respectively.\n\nThe models are trained using Stochastic Gradient Descent (SGD) with Nesterov momentum and an initial learning rate of $0.001$. The learning rate is adjusted according to a combination of linear warmup and a cosine annealing schedule.\n\nImages used for training are at 3 different scales: $256 \\times 256$, $352 \\times 352$ and $512 \\times 512$.\n\nDuring training, augmentations including random scaling, rotation, horizontal\/vertical flip, motion blur, and color jittering are added to improve generality. These augmentations are performed on-the-fly with a probability of $0.5$.\n\nBlazeNeo\\xspace and other baseline models are implemented in Python 3.7 using the PyTorch framework.\n\n\\subsection{Evaluation metrics}\nCommonly used metrics Dice score and IoU score are employed to measure the model's output quality. They are evaluated on three classes: neoplastic polyp, non-neoplastic polyp, and generic polyp (same as polyp segmentation). Dice and IoU are calculated pixel-wise on the entire test set (micro-averaged). Equations (\\ref{eq:dice_score}) and (\\ref{eq:iou_score}) describe how these metrics are calculated.\n\n\\begin{equation}\n    \\label{eq:dice_score}\n    Dice_{c} = \\frac{ 2\\times \\sum_{i\\in{I}}^{} u_{i}^{c} v_{i}^{c} }{ \\sum_{i\\in{I}}^{} u_{i}^{c} + \\sum_{i\\in{I}}^{} v_{i}^{c}}\n\\end{equation}\n\n\\begin{equation}\n    \\label{eq:iou_score}\n    IoU_{c} = \\frac{ \\sum_{i\\in{I}}^{} u_{i}^{c} v_{i}^{c} }{ \\sum_{i\\in{I}}^{} u_{i}^{c} + \\sum_{i\\in{I}}^{} v_{i}^{c} - \\sum_{i\\in{I}}^{} u_{i}^{c} v_{i}^{c} }\n\\end{equation}\nwhere $i \\in I$ denotes a prediction pixel within the entire test set. $u_{i}^{c} = 1$ if the model predicts pixel $i$ to have class $c$, and $0$ otherwise. Similarly, $v_{i}^{c} = 1$ if the ground truth map states that pixel $i$ has class $c$, and $0$ otherwise. For neoplastic and non-neoplastic class evaluation, $I$ does not include undefined neoplasm pixels.\n\nWe also evaluate each model's inference speed using the number of processed frames per second (FPS). This metric is measured by running each model with a batch size of 1 on 100 colonoscopy images. FPS is measured on a Google Colaboratory instance with an NVIDIA Tesla V100 GPU when not specified otherwise.\n\nFinally, we log each model's number of parameters and floating-point operations (measured in GFLOPs) to evaluate their size and complexity.\n\n\\section{Results and Discussion}\n\\subsection{Ablation Study}\n\\subsubsection{The effectiveness of different output architectures}\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Performance metrics on the NeoPolyp\\xspace test set for the three variants of BlazeNeo using the same DHA feature aggregation scheme}\n    \\label{tab:exp3_2}\n    \\begin{tabular}{@{} c| c c c c c c @{}}\n        \\toprule\n        \n        \\multicolumn{1}{c|}{Model} & \\multicolumn{1}{c}{$\\text{Dice}_{seg}$} & \\multicolumn{1}{c}{$\\text{IoU}_{seg}$} & \\multicolumn{1}{c}{$\\text{Dice}_{non}$} & \\multicolumn{1}{c}{$\\text{IoU}_{non}$} & \\multicolumn{1}{c}{$\\text{Dice}_{neo}$} & \\multicolumn{1}{c}{$\\text{IoU}_{neo}$} \\\\ \\midrule \\midrule\n        SB-BlazeNeo\\xspace              & 0.866                                   & 0.764                                  & 0.683                                   & 0.518                                  & 0.862                                   & 0.758                                  \\\\\n        ST-BlazeNeo\\xspace              & 0.874                                   & 0.777                                  & 0.671                                   & 0.505                                  & 0.865                                   & 0.762                                  \\\\\n        BlazeNeo\\xspace                 & \\textbf{0.901}                          & \\textbf{0.820}                         & \\textbf{0.728}                          & \\textbf{0.572}                         & \\textbf{0.888}                          & \\textbf{0.800}                         \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\nFirstly, we compare our three BlazeNeo\\xspace variants shown in Figure \\ref{fig:model} to evaluate the effectiveness of different output architectures. Table \\ref{tab:exp3_2} shows performance metrics for each variant on the NeoPolyp\\xspace test set. Here we use the same DHA feature aggregation scheme for all three of BlazeNeo\\xspace. The final BlazeNeo\\xspace model with multi heads achieves the best results on all metrics. Notably, this final variant outperforms the other two variants by over $6\\%$ in IoU score for the non-neoplastic class. This shows the effectiveness of the auxiliary branch, which helps anchor segmentation performance and makes use of undefined labels. We also see that ST-BlazeNeo\\xspace outperforms SB-BlazeNeo\\xspace, which justifies our use of trinary output in the final variant.\n\n\\subsubsection{The effectiveness of different feature aggregation schemes}\nThis experiment investigates the use of four different feature aggregation schemes in BlazeNeo\\xspace{}: Long Skip Connection (LSC), Iterative Deep Aggregation (IDA), Dense Iterative Aggregation (DIA), and Dense Hierarchical Aggregation (DHA).\n\nTable~\\ref{tab:exp1} shows performance metrics for each model variation on the NeoPolyp\\xspace dataset. We can see that BlazeNeo-DHA\\xspace produces the highest values of Dice and IoU on all classes. This is what we expected, as DHA is the most complex aggregation mechanism that preserves a lot of high-resolution features. However, this improvement comes at a cost as BlazeNeo-DHA\\xspace is also the slowest variation at only 81.5 FPS, compared to the fastest BlazeNeo-LSC\\xspace at 88.6 FPS. Interestingly, the nested IDA aggregation scheme performs worse than basic LSC in the segmentation task. This is alleviated in DIA and DHA with the use of skip connections.\n\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Performance metrics on the NeoPolyp\\xspace test set for BlazeNeo\\xspace with different feature aggregation schemes}\n    \\label{tab:exp1}\n    \\begin{tabular}{@{} c | c c c c c c | c@{}}\n        \\toprule\n        \n        \\multicolumn{1}{c | }{Method} & \\multicolumn{1}{c}{$\\text{Dice}_{seg}$} & \\multicolumn{1}{c}{$\\text{IoU}_{seg}$} & \\multicolumn{1}{c}{$\\text{Dice}_{non}$} & \\multicolumn{1}{c}{$\\text{IoU}_{non}$} & \\multicolumn{1}{c}{$\\text{Dice}_{neo}$} & \\multicolumn{1}{c|}{$\\text{IoU}_{neo}$} & \\multicolumn{1}{c}{FPS} \\\\ \\midrule \\midrule\n        BlazeNeo-LSC\\xspace                          & 0.897                                   & 0.814                                  & 0.709                                   & 0.550                                  & 0.884                                   & 0.792                                   & \\textbf{88.6}           \\\\\n        BlazeNeo-IDA\\xspace                          & 0.890                                   & 0.803                                  & 0.715                                   & 0.557                                  & 0.887                                   & 0.798                                   & 84.6                    \\\\\n        BlazeNeo-DIA\\xspace                          & 0.897                                   & 0.814                                  & 0.719                                   & 0.562                                  & \\textbf{0.888}                          & \\textbf{0.800}                          & 84.0                    \\\\\n        BlazeNeo-DHA\\xspace                          & \\textbf{0.901}                          & \\textbf{0.820}                         & \\textbf{0.728}                          & \\textbf{0.572}                         & \\textbf{0.888}                          & \\textbf{0.800}                          & 81.5                    \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\\subsubsection{The effectiveness of auxiliary training}\n\nThis experiment looks into the effectiveness of enabling and disabling auxiliary training for each variation of BlazeNeo\\xspace, namely BlazeNeo-LSC\\xspace, BlazeNeo-IDA\\xspace, BlazeNeo-DIA\\xspace, and BlazeNeo-DHA\\xspace.\n\nFigure~\\ref{fig:aux-effectiveness} shows that auxiliary training generally improves output quality. In fact, without auxiliary segmentation learning, BlazeNeo-DIA\\xspace and BlazeNeo-DHA\\xspace achieve even lower accuracy than BlazeNeo-IDA\\xspace. The non-neoplastic class benefits the most from auxiliary training, improving BlazeNeo-LSC\\xspace, BlazeNeo-IDA\\xspace, BlazeNeo-DIA\\xspace, and BlazeNeo-DHA\\xspace by 3\\%, 1.9\\%, 4\\%, and 4.9\\%, respectively.\n\n\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Performance metrics for BlazeNeo-DHA\\xspace when training on NeoPolyp\\xspace and NeoPolyp-Clean\\xspace, measured on the NeoPolyp\\xspace test set}\n    \\label{tab:exp3}\n    \\begin{tabular}{@{} c| c c c c c c @{}}\n        \\toprule\n        \n        \\multicolumn{1}{c|}{Training dataset} & \\multicolumn{1}{c}{$\\text{Dice}_{seg}$} & \\multicolumn{1}{c}{$\\text{IoU}_{seg}$} & \\multicolumn{1}{c}{$\\text{Dice}_{non}$} & \\multicolumn{1}{c}{$\\text{IoU}_{non}$} & \\multicolumn{1}{c}{$\\text{Dice}_{neo}$} & \\multicolumn{1}{c}{$\\text{IoU}_{neo}$} \\\\ \\midrule \\midrule\n        NeoPolyp-Clean\\xspace                     & 0.900                                   & 0.818                                  & 0.714                                   & 0.555                                  & 0.884                                   & 0.792                                  \\\\\n        NeoPolyp\\xspace                          & \\textbf{0.901}                          & \\textbf{0.820}                         & \\textbf{0.728}                          & \\textbf{0.572}                         & \\textbf{0.888}                          & \\textbf{0.799}                         \\\\\n        \n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\\subsubsection{The effectiveness of including undefined polyps}\nThis experiment examines the effectiveness of using undefined neoplasm pixels via the auxiliary module. Table~\\ref{tab:exp3} shows performance metrics for BlazeNeo-DHA\\xspace, the best-performing BlazeNeo\\xspace model, when trained on the NeoPolyp\\xspace dataset (which contains undefined polyps) and the NeoPolyp-Clean\\xspace dataset (which does not contain undefined polyps). Results show that the addition of these undefined pixels leads to improvements across the board, especially for the non-neoplastic class.\n\n\\subsection{Comparison with state-of-the-art models}\n\\subsubsection{Quantitative comparison}\nWe compare the performance of BlazeNeo-DHA\\xspace, the best-performing BlazeNeo\\xspace model, with seven state-of-the-art models for the polyp segmentation and PSND problem: U-Net \\cite{ronneberger2015u}, ColonSegNet \\cite{jha2021real}, DDANet \\cite{tomar2020ddanet}, DoubleUNet \\cite{jha2020doubleu},  HarDNet-MSEG \\cite{huang2021hardnet}, PraNet \\cite{fan2020pranet}, and NeoUNet \\cite{lan2021neounet}. Except for NeoUNet, the models mentioned above do not handle undefined neoplasm pixels during training. Thus, in the interest of a fair comparison, we use the NeoPolyp-Clean\\xspace dataset for this experiment, which does not contain undefined polyps. Results are shown in Table \\ref{tab:compare_sota}.\n\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Performance metrics of different models on the NeoPolyp-Clean\\xspace test set}\n    \\label{tab:compare_sota}\n    \\begin{tabular}{@{} c | c c c c c c | c c c@{}}\n        \\toprule\n        \n        \\multicolumn{1}{c | }{Method}        & \\multicolumn{1}{c}{$\\text{Dice}_{seg}$} & \\multicolumn{1}{c}{$\\text{IoU}_{seg}$} & \\multicolumn{1}{c}{$\\text{Dice}_{non}$} & \\multicolumn{1}{c}{$\\text{IoU}_{non}$} & \\multicolumn{1}{c}{$\\text{Dice}_{neo}$} & \\multicolumn{1}{c | }{$\\text{IoU}_{neo}$} & \\multicolumn{1}{c}{FPS} & \\multicolumn{1}{c}{Parameters} & \\multicolumn{1}{c}{GFLOPs} \\\\ \\midrule\n        \\midrule\n        ColonSegNet \\cite{jha2021real}       & 0.738                                   & 0.585                                  & 0.505                                   & 0.338                                  & 0.732                                   & 0.577                                     & 44.9                    & 5,010,000                      & 64.84                      \\\\\n        U-Net \\cite{ronneberger2015u}        & 0.785                                   & 0.646                                  & 0.525                                   & 0.356                                  & 0.773                                   & 0.631                                     & 69.6                    & 31,043,651                     & 103.59                     \\\\\n        DDANet \\cite{tomar2020ddanet}        & 0.813                                   & 0.684                                  & 0.578                                   & 0.406                                  & 0.802                                   & 0.670                                     & 46.2                    & 6,840,000                      & 31.45                      \\\\\n        DoubleU-Net \\cite{jha2020doubleu}    & 0.837                                   & 0.720                                  & 0.621                                   & 0.450                                  & 0.832                                   & 0.712                                     & 43.2                    & 18,836,804                     & 83.62                      \\\\\n        HarDNet-MSEG \\cite{huang2021hardnet} & 0.883                                   & 0.791                                  & 0.659                                   & 0.492                                  & 0.869                                   & 0.769                                     & \\underline{77.1}        & \\underline{17,424,031}         & \\underline{11.38}          \\\\\n        PraNet \\cite{fan2020pranet}          & 0.895                                   & 0.811                                  & 0.705                                   & 0.544                                  & 0.873                                   & 0.775                                     & 55.6                    & 30,501,341                     & 13.11                      \\\\\n        NeoUNet \\cite{lan2021neounet}        & \\textbf{0.911}                          & \\textbf{0.837}                         & \\textbf{0.720}                          & \\textbf{0.563}                         & \\textbf{0.889}                          & \\textbf{0.800}                            & 68.3                    & 38,288,397                     & 39.88                      \\\\\n        BlazeNeo-DHA\\xspace (Ours)                          & \\underline{0.904}                       & \\underline{0.825}                      & \\underline{0.717}                       & \\underline{0.559}                      & \\underline{0.885}                       & \\underline{0.792}                         & \\textbf{81.5}           & \\textbf{17,143,324}            & \\textbf{11.06}             \\\\\n        \n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\nWe can see that while NeoUNet remains the most accurate model, BlazeNeo\\xspace is a close second, with a difference of less than $1\\%$ on most accuracy metrics. At the same time, BlazeNeo\\xspace is a much more lightweight and faster model. Compared to NeoUNet, BlazeNeo\\xspace achieves higher FPS ($81.5$ versus $68.3$), has half as many parameters ($17,143,324$ versus $38,288,397$), and lower GFLOPs ($11.06$ versus $39.88$). Notably, BlazeNeo-DHA\\xspace is faster and more accurate than HarDNet-MSEG, while NeoUNet is slower than HarDNet-MSEG and U-Net. We attribute the speed improvement of BlazeNeo\\xspace over HarDNet-MSEG to the different decoder designs, especially the use of the small RFB module.\n\nAlthough achieving a high overall dice score, our BlazeNeo\\xspace is not without limitations. The dice and IoU scores of BlazeNeo\\xspace are still low for non-neoplastic polyps, and both of them are significantly reduced when the model is converted into the INT8 precision. The reason may be due to the small size of non-neoplastic polyps. When the weights are converted into integers ranging from -128 to 127 in the INT8 precision, the model seems to lose the capacity of representing detailed information and, therefore, is easier to miss small objects in the input images. \n\n\\subsubsection{Qualitative comparison}\nFigure~\\ref{fig:good} shows examples outputs of BlazeNeo-DHA\\xspace and other baseline models. Overall, BlazeNeo-DHA\\xspace produces the most accurate segmentation and classification results for different types of polyps.\n\nThe first five rows of Figure~\\ref{fig:good} contain ``easier'' polyp examples. BlazeNeo-DHA\\xspace and NeoUNet perform quite well in these examples, with similarly high accuracy. Meanwhile, PraNet, HarDNet-MSEG, DoubleU-Net, DDANet, U-Net and ColonSegNet produce predictions with less uniformity, with some polyps containing both neoplastic and non-neoplastic regions. For larger polyps such as the 5th row, PraNet is unable to fully segment the area.\n\nThe final five rows in Figure~\\ref{fig:good} represent more challenging examples, in which all models struggle to provide accurate segmentation masks. This is because non-neoplastic polyps are usually small in size and easier to be miss detected. BlazeNeo-DHA\\xspace and NeoUNet produce fewer false positives in these situations, while U-Net and ColonSegNet create the most inaccurate masks.\n\nThe last two rows show two non-neoplastic polyps in BLI and FICE modes. Contributing factors for these struggles in enhanced color modes include their smaller proportions in the training dataset. Furthermore, the endoscopist put the camera scope close to the mucosa in these cases, making the polyps' surface look very clear and sharp such that one can even observe small dots that are glandular holes on the surface. Therefore, these polyps are easily confused with angiogenesis in neoplastic lesions, which is why all models failed in classifying them.\n\n\n\\begin{center}\n    \\begin{figure*}[th!]\n        \\begin{center}\n            \\includegraphics[scale=.21]{images\/compare_models.png}\n        \\end{center}\n        \\caption{Qualitative comparison of the proposed method with other baseline methods: (a) image, (b) ground truth, (c) BlazeNeo (Ours), (d) NeoUNet, (e) PraNet, (f) HarDNet-MSEG, (g) UNet, (h) DoubleU-Net, (i) DDANet, and (j) ColonSegNet.}\n        \\label{fig:good}\n    \\end{figure*}\n\\end{center}\n\n\n\\subsection{Benchmark for embedded device}\nIn this experiment, we apply model compression techniques via the NVIDIA TensorRT 7.1 toolkit \\cite{tensorrt} to BlazeNeo\\xspace and the baseline models. The compressed models are then benchmarked on the NVIDIA Jetson AGX Xavier developer kit, an embedded computation unit with an NVIDIA GPU specialized for edge AI deployments. This setup is more in-line with deployment scenarios for polyp segmentation and PSND models, i.e., embedded on-site into colonoscopy devices.\n\nThree available precision modes are tested for compressing each model: FP32, FP16, and INT8. Each mode can be seen as a different trade-off level between accuracy and speed. FP32 precision applies techniques such as layer\/tensor fusion while keeping parameters as 32-bit floating-point numbers. Hence, this mode provides some speed-up while minimizing accuracy degradation. FP16 precision converts suitable parameters to 16-bit floating-point numbers, greatly reducing model size and latency but is subject to more degradation. Finally, INT8 precision mode quantizes model parameters to 8-bit integers. This precision mode requires an additional calibration procedure to maintain model integrity, which attempts to replicate the original model's output on a small calibration dataset. Despite calibration, INT8 precision is susceptible to a lot more degradation. For this experiment, INT8 calibration is done on a randomized set of images.\n\nTable \\ref{tab:embedded} shows performance metrics in different precisions for U-Net, PraNet, HarDNet-MSEG, NeoUNet, and BlazeNeo-DHA\\xspace. We can see a stark difference in FPS when running models on the Jetson AGX Xavier compared to the Google Colab environment. At FP32 precision, the fastest model (BlazeNeo\\xspace) achieves only 53.3 FPS on the device, while the slowest model in the Colab environment (PraNet) without any compression still achieves 55.6 FPS. On the other hand, Dice and IoU measures are hardly affected at this precision level. In fact, U-Net and HarDNet-MSEG see improvements up to $1.5\\%$ across all metrics after compression. DoubleU-Net and ColonSegNet achieve a slight improvement of about $0.5-1\\%$ after compression. DDANet suffers a little from compression, dropping by about $1.5\\%$ in segmentation metrics. Finally, PraNet is most affected after compression, dropping by $4.6\\%$ in segmentation metrics.\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Performance metrics of different models with FP16, FP32 and INT8 precision levels on NVIDIA Jetson AGX Xavier}\n    \\label{tab:embedded}\n    \\begin{tabular}{@{} c | c | c c c c c c | c @{}}\n        \\toprule\n        \n        \\multicolumn{1}{c |}{Model} & \\multicolumn{1}{c | }{Precision} & \\multicolumn{1}{c}{$\\text{Dice}_{seg}$} & \\multicolumn{1}{c}{$\\text{IoU}_{seg}$} & \\multicolumn{1}{c}{$\\text{Dice}_{non}$} & \\multicolumn{1}{c}{$\\text{IoU}_{non}$} & \\multicolumn{1}{c}{$\\text{Dice}_{neo}$} & \\multicolumn{1}{c|}{$\\text{IoU}_{neo}$} & \\multicolumn{1}{c}{FPS} \\\\ \\midrule \\midrule\n        ColonSegNet@FP32            & FP32                             & 0.745                                   & 0.594                                  & 0.528                                   & 0.359                                  & 0.728                                   & 0.572                                   & 9.4                     \\\\\n        UNet@FP32                   & FP32                             & 0.800                                   & 0.667                                  & 0.537                                   & 0.367                                  & 0.781                                   & 0.641                                   & 10.7                    \\\\\n        DDANet@FP32                 & FP32                             & 0.799                                   & 0.665                                  & 0.557                                   & 0.386                                  & 0.776                                   & 0.635                                   & 16.6                    \\\\\n        DoubleU-Net@FP32            & FP32                             & 0.840                                   & 0.725                                  & 0.627                                   & 0.456                                  & 0.835                                   & 0.717                                   & 10.6                    \\\\\n        HarDNet-MSEG@FP32           & FP32                             & 0.891                                   & 0.804                                  & 0.685                                   & 0.521                                  & 0.871                                   & 0.771                                   & \\underline{52.2}        \\\\\n        PraNet@FP32                 & FP32                             & 0.849                                   & 0.738                                  & 0.571                                   & 0.400                                  & 0.844                                   & 0.730                                   & 37.0                    \\\\\n        NeoUNet@FP32                & FP32                             & \\textbf{0.909}                          & \\textbf{0.832}                         & \\textbf{0.725}                          & \\textbf{0.568}                         & \\textbf{0.893}                          & \\textbf{0.806}                          & 25.4                    \\\\\n        BlazeNeo\\xspace{}@FP32 (Ours)    & FP32                             & \\underline{0.906}                       & \\underline{0.828}                      & \\underline{0.721}                       & \\underline{0.563}                      & \\underline{0.887}                       & \\underline{0.796}                       & \\textbf{53.3}           \\\\\n        \\midrule\n        ColonSegNet@FP16            & FP16                             & 0.746                                   & 0.594                                  & 0.528                                   & 0.359                                  & 0.728                                   & 0.572                                   & 26.3                    \\\\\n        UNet@FP16                   & FP16                             & 0.800                                   & 0.666                                  & 0.537                                   & 0.367                                  & 0.781                                   & 0.641                                   & 35.5                    \\\\\n        DDANet@FP16                 & FP16                             & 0.799                                   & 0.665                                  & 0.557                                   & 0.386                                  & 0.776                                   & 0.635                                   & 31.8                    \\\\\n        DoubleU-Net@FP16            & FP16                             & 0.840                                   & 0.725                                  & 0.627                                   & 0.456                                  & 0.835                                   & 0.717                                   & 22.9                    \\\\\n        HarDNet-MSEG@FP16           & FP16                             & 0.891                                   & 0.804                                  & 0.685                                   & 0.521                                  & 0.871                                   & 0.771                                   & \\underline{118.7}       \\\\\n        PraNet@FP16                 & FP16                             & 0.850                                   & 0.740                                  & 0.572                                   & 0.401                                  & 0.845                                   & 0.731                                   & 95.1                    \\\\\n        NeoUNet@FP16                & FP16                             & \\textbf{0.908}                          & \\textbf{0.832}                         & \\textbf{0.724}                          & \\textbf{0.568}                         & \\textbf{0.893}                          & \\textbf{0.806}                          & 73.4                    \\\\\n        BlazeNeo\\xspace{}@FP16 (Ours)    & FP16                             & \\underline{0.906}                       & \\underline{0.828}                      & \\underline{0.721}                       & \\underline{0.563}                      & \\underline{0.887}                       & \\underline{0.796}                       & \\textbf{121.9}          \\\\\n        \\midrule\n        ColonSegNet@INT8            & INT8                             & 0.672                                   & 0.507                                  & 0.456                                   & 0.295                                  & 0.633                                   & 0.463                                   & 41.6                    \\\\\n        UNet@INT8                   & INT8                             & 0.764                                   & 0.618                                  & 0.501                                   & 0.334                                  & 0.753                                   & 0.604                                   & 44.8                    \\\\\n        DDANet@INT8                 & INT8                             & 0.770                                   & 0.626                                  & 0.492                                   & 0.326                                  & 0.748                                   & 0.598                                   & 46.4                    \\\\\n        DoubleU-Net@INT8            & INT8                             & 0.835                                   & 0.717                                  & 0.562                                   & 0.391                                  & 0.830                                   & 0.709                                   & 46.7                    \\\\\n        HarDNet-MSEG@INT8           & INT8                             & 0.810                                   & 0.680                                  & 0.594                                   & 0.422                                  & 0.799                                   & 0.665                                   & \\underline{143.7}       \\\\\n        PraNet@INT8                 & INT8                             & 0.817                                   & 0.691                                  & 0.575                                   & 0.404                                  & 0.815                                   & 0.688                                   & 116.7                   \\\\\n        NeoUNet@INT8                & INT8                             & \\underline{0.848}                       & \\underline{0.736}                      & \\underline{0.638}                       & \\underline{0.468}                      & \\underline{0.848}                       & \\underline{0.737}                       & 100.3                   \\\\\n        BlazeNeo\\xspace{}@INT8 (Ours)    & INT8                             & \\textbf{0.870}                          & \\textbf{0.770}                         & \\textbf{0.678}                          & \\textbf{0.513}                         & \\textbf{0.857}                          & \\textbf{0.750}                          & \\textbf{155.6}          \\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\\begin{figure*}[ht!]\n    \\begin{center}\n        \\includegraphics[width=0.9\\textwidth]{images\/Comparision.png}\n    \\end{center}\n    \\caption{Comparison between different models in different precisions. Red, green and blue markers denote INT8, FP16 and FP32 precisions, respectively.}\n    \\label{fig:perf_comparision}\n\\end{figure*}\n\nAt FP16 precision, latency for all models are vastly improved, the fastest being BlazeNeo\\xspace (121.9 FPS) and HarDNet-MSEG (118.7 FPS). In addition, accuracy metrics for all models are within $0.01\\%$ of their FP32 counterparts. This further shows that large neural networks do not require high float precision to remain effective.\n\nINT8 precision gives the most speed gain out of the three modes, but at great expense in terms of accuracy. BlazeNeo\\xspace runs at 155.6 FPS in this mode, whereas HarDNet-MSEG is the second fastest model at 143.7 FPS. However, HarDNet-MSEG also suffers the largest drop in accuracy at about $8.1\\%$ on all metrics. This drop is a bit lower for PraNet ($\\approx 3.3\\%$), NeoUNet ($\\approx 6\\%$) and BlazeNeo\\xspace ($\\approx 3.6\\%$).\n\nIn every precision mode, BlazeNeo\\xspace is consistently the fastest model while being a close second in terms of accuracy behind NeoUNet. BlazeNeo\\xspace also displays its robustness to compression techniques, incurring significantly less degradation compared to models such as PraNet. Its small size also gives the proposed model advantage in long-term deployment, as energy usage and equipment wear become factors. In Figure~\\ref{fig:perf_comparision}, we visualize the performance of BlazeNeo in terms of speed (FPS) and accuracy (dice score on polyp segmentation task) compared to other existing methods.\n\nIt should be emphasized that high FPS is essential in real scenarios because it helps endoscopists operate smoother and easier to monitor and detect lesions. According to endoscopists' experience, the idea FPS for colonoscopy should be at least 60. In addition, the fast inference speed gives us the potential to deploy the model on even more low-cost devices with less computational power, such as NVIDIA Jetson TX1\/TX2, while still satisfying the minimal required FPS. This is important because it allows us to deploy the application on a large scale to many medical facilities in remote areas where the economic conditions are difficult.\n\n\n\nFor further discussion, we compare the performance of the models quantitatively and qualitatively in different precisions.\n\nTable~\\ref{tab:compress_acc} shows the accuracy of our BlazeNeo-DHA\\xspace models in different precisions. One can observe that the models with FP32 and FP16 precisions give the same result, which is lower than the original model by a mean margin of 2.32\\%. The TensorRT INT8 model yields the worst accuracy, which is lower than the original model by a mean margin of 5.75\\%.\n\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Accuracy metrics on the NeoPolyp-Clean\\xspace test set for BlazeNeo-DHA\\xspace models in different precisions}\n    \\label{tab:compress_acc}\n    \\begin{tabular}{@{} c | r r r r r r r@{}}\n        \\toprule\n        \n        \\multicolumn{1}{c|}{Model} & \\multicolumn{1}{c}{$\\text{Dice}_{seg}$} & \\multicolumn{1}{c}{$\\text{IoU}_{seg}$} & \\multicolumn{1}{c}{$\\text{Dice}_{non}$} & \\multicolumn{1}{c}{$\\text{IoU}_{non}$} & \\multicolumn{1}{c}{$\\text{Dice}_{neo}$} & \\multicolumn{1}{c}{$\\text{IoU}_{neo}$} \\\\\n        \\midrule\n        \\midrule\n        PyTorch (w\/o compression)                   & 0.904                                   & 0.825                                  & 0.717                                   & 0.559                                  & 0.885                                   & 0.792                                  \\\\\n        TensorRT FP32              & 0.906                                   & 0.828                                  & 0.721                                   & 0.563                                  & 0.887                                   & 0.796                                  \\\\\n        TensorRT FP16              & 0.906                                   & 0.828                                  & 0.721                                   & 0.563                                  & 0.887                                   & 0.796                                  \\\\\n        TensorRT INT8              & 0.870                                   & 0.770                                  & 0.678                                   & 0.513                                  & 0.857                                   & 0.750                                  \\\\\n        \n        \\bottomrule\n    \\end{tabular}\n\\end{table*}\n\nWith lower precision, a model can reduce latency, throughput and increase power efficiency. To compare the speed performance of the models in different precisions, we present detailed benchmarking results of them for 100 iterations on the Jetson AGX Xavier with two metrics: host latency and GPU compute time. Host latency is measured as the end-to-end execution time from the CPU point of view, while GPU computing time is the actual working time for GPU calculation.\n\\begin{figure*}[ht!]\n    \\begin{center}\n        \\includegraphics[scale=.33]{images\/compress.png}\n    \\end{center}\n    \\caption{Qualitative comparison of BlazeNeo-DHA\\xspace models in different precisions: (a) image, (b) ground truth, (c) Pytorch, (d) TensorRT FP32, (e) TensorRT FP16, (f) TensorRT INT8.}\n    \\label{fig:compress}\n\\end{figure*}\n\nAs shown in Table~\\ref{tab:compress_latency}, the model in INT8 precision has the shortest latency compared to all other models, less than one-third of the model in FP32 precision. However, for deep learning inference, we must consider the trade-off between accuracy and speed. Therefore, the model in FP16 precision is the best choice, which gives the best result in all precisions and has a median latency.\n\n\\begin{table*}[ht!]\n    \\centering\n    \\caption{Latency metrics for BlazeNeo-DHA\\xspace models in different precisions.  The latency is measured on Jetson Xavier AGX with power mode MAXN}\n    \\label{tab:compress_latency}\n    \\begin{tabular}{l|c c c c|c c c c}\n        \\hline\n        \\multicolumn{1}{c|}{\\multirow{2}{*}{Precision}} & \\multicolumn{4}{c|}{\\begin{tabular}[c]{@{}c@{}}Host Latency (ms)\\end{tabular}} & \\multicolumn{4}{c}{ \\begin{tabular}[c]{@{}c@{}}GPU Compute (ms)\\end{tabular}}                                                                                                 \\\\ \\cline{2-9}\n        \\multicolumn{1}{c|}{}                           & min                                                 & max                                                 & mean          & median        & min           & max           & mean          & median        \\\\ \\midrule\n        \\midrule\n        TensorRT FP32                                   & 18.50                                               & 22.04                                               & 18.67         & 18.55         & 18.40         & 21.94         & 18.57         & 18.45         \\\\\n        TensorRT FP16                                   & 8.10                                                & 10.24                                               & 8.20          & 8.15          & 7.99          & 10.13         & 8.09          & 8.04          \\\\\n        TensorRT INT8                                   & \\textbf{6.35}                                       & \\textbf{6.51}                                       & \\textbf{6.42} & \\textbf{6.42} & \\textbf{6.25} & \\textbf{6.42} & \\textbf{6.32} & \\textbf{6.32} \\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\n\nFor a more intuitive understanding of the loss in accuracy of each compressed model, Figure~\\ref{fig:compress} illustrates the sample results of BlazeNeo-DHA\\xspace models in different precisions. As we can observe, the prediction result of models in FP32 and FP16 precisions are almost the same in every example. In general, the INT8 compressed model loses the ability to segment smaller polyps compared to other modes. For segmented polyps, the neoplasm prediction remains constant for every compression mode in all examples. This demonstrates that BlazeNeo\\xspace can be quite robust to model compression and can be deployed in high compression modes such as FP16 with confidence.\n\n\\section{Conclusion}\n\\label{sec:conclude}\nThis paper has proposed BlazeNeo\\xspace, a novel neural network architecture for the polyp segmentation and neoplasm detection problem, with an emphasis on speed and deployability. BlazeNeo\\xspace is an extremely lightweight and fast neural network, thanks to the use of an efficient HarDNet backbone, a multi-level feature aggregation structure, and an auxiliary training module to take advantage of undefined labels. Our experiments show that BlazeNeo\\xspace outperforms all other state-of-the-art models in terms of inference latency while providing competitive accuracy. We also show that BlazeNeo\\xspace can be very robust against degradation caused by compression techniques. In general, the proposed model is highly suitable for lightweight deployments with real-time requirements. Our source code is available at \\url{https:\/\/github.com\/tofuai\/neoplasm-segmentation}.\n\nIn future works, we would like to exploit recent advancements in Transformer-based architectures to improve the performance of the models. Especially, lightweight Transformers such as SegFormer \\cite{xie2021segformer} will be the primary focus since they ensure both high accuracy and high inference speed which are crucial factors for a real computer-aided system in colonoscopy.  \n\n\\section{Acknowledgment}\nThis work was funded by Vingroup Innovation Foundation (VINIF) under project code VINIF.2020.DA17.\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAlternating direction method of multipliers (ADMM \\cite{gabay1976dual,boyd2011distributed}) is a popular optimization tool\nfor solving the composite and constrained problems in machine learning.\nIn particular, ADMM can efficiently optimize some problems with complicated structure regularization\nsuch as the graph-guided fused lasso \\cite{kim2009multivariate},\nwhich is too complicated for the other popular optimization methods\nsuch as proximal gradient methods \\cite{beck2009fast}.\nFor the large-scale optimization,\nthe stochastic ADMM method \\cite{ouyang2013stochastic} has been proposed.\nRecently, some faster stochastic ADMM methods \\cite{suzuki2014stochastic,zheng2016fast}\nhave been proposed by using the variance reduced (VR) techniques such as the SVRG \\cite{johnson2013accelerating}.\nIn fact, ADMM is also highly successful in solving various nonconvex problems\nsuch as training deep neural networks \\cite{Taylor2016Training}.\nThus, some fast nonconvex stochastic ADMM methods have been developed in \\cite{huang2016stochastic,Huang2019fasterst}.\n\nCurrently, most of the ADMM methods\nneed to  compute the gradients\nof objective functions over each iteration.\nHowever, in many machine learning problems,\nthe explicit expression of gradient for objective function\nis difficult or infeasible to obtain.\nFor example, in black-box situations, only prediction results (\\emph{i.e.}, function\nvalues) are provided \\cite{chen2017zoo,liu2018zeroth}.\nIn bandit settings \\cite{agarwal2010optimal},\nplayer only receives the partial feedback in terms of loss function values,\nso it is impossible to obtain expressive gradient of the loss function.\nClearly, the classic optimization methods, based on the first-order gradient or second-order information, are not competent to these problems.\nRecently, the zeroth-order optimization methods \\cite{duchi2015optimal,Nesterov2017RandomGM}\nare developed by only using the function values in the optimization.\n\\begin{table*}\n  \\centering\n  \\begin{tabular}{c|c|c|c|c}\n  \\hline\n \n \\textbf{Algorithm} & \\textbf{Reference} & \\textbf{Gradient Estimator} &  \\textbf{Problem}  & \\textbf{Convergence Rate} \\\\ \\hline\n  ZOO-ADMM & \\cite{liu2018admm} &GauSGE &  C(S) + C(NS) & $O(\\sqrt{1\/T})$  \\\\ \\hline\n  ZO-GADM & \\cite{gao2018information} & UniSGE & C(S) + C(NS)&  $O(\\sqrt{1\/T})$  \\\\ \\hline\n  RSPGF  &\\cite{ghadimi2016mini} & GauSGE &NC(S) + C(NS)  & $O(\\sqrt{1\/T})$ \\\\ \\hline\n  ZO-ProxSVRG & \\multirow{2}*{\\cite{Huang2019faster}} & \\multirow{2}*{CooSGE} & \\multirow{2}*{NC(S) + C(NS)} & \\multirow{2}*{$O(1\/T)$} \\\\\n  ZO-ProxSAGA &  & & &   \\\\ \\hline\n  ZO-SVRG-ADMM & \\multirow{2}*{Ours} & \\multirow{2}*{CooSGE} & \\multirow{2}*{NC(S) + C(mNS)} & \\multirow{2}*{$O(1\/T)$} \\\\\n  ZO-SAGA-ADMM &  & & &   \\\\ \\hline\n  \\end{tabular}\n  \\caption{  Convergence properties comparison of the zeroth-order ADMM algorithms and other ones.\n  C, NC, S, NS and mNS are the abbreviations of convex,\n    non-convex, smooth, non-smooth and the sum of multiple non-smooth functions, respectively.\n    $T$ is the whole iteration number.\n    Gaussian Smoothing Gradient Estimator (GauSGE),\n    Uniform Smoothing Gradient Estimator (UniSGE) and Coordinate Smoothing Gradient Estimator (CooSGE). }\\label{tab:1}\n\\end{table*}\n\nIn the paper, we focus on using the zeroth-order methods to solve the following nonconvex nonsmooth problem:\n\\begin{align} \\label{eq:1}\n \\min_{x,\\{y_j\\}_{j=1}^k} & F(x,y_{[k]}) := \\frac{1}{n}\\sum_{i=1}^n f_i(x) + \\sum_{j=1}^{k}\\psi_j(y_j)  \\\\\n \\mbox{s.t.} \\ & Ax + \\sum_{j=1}^kB_jy_j =c, \\nonumber\n\\end{align}\nwhere $A\\in \\mathbb{R}^{p\\times d}$, $B_j\\in \\mathbb{R}^{p\\times q}$ for all $j\\in [k],\\ k \\geq 1$,\n$f(x)=\\frac{1}{n}\\sum_{i=1}^n f_i(x): \\mathbb{R}^d\\rightarrow \\mathbb{R}$\nis a \\emph{nonconvex} and black-box function, and each $\\psi_j(y_j): \\mathbb{R}^q\\rightarrow \\mathbb{R}$ is a convex and\n\\emph{nonsmooth} function.\nIn machine learning, function $f(x)$ can be used for the empirical loss,\n$\\sum_{j=1}^{k}\\psi_j(y_j)$ for multiple structure penalties (\\emph{e.g.}, sparse + group sparse), and the constraint for encoding\nthe structure pattern of model parameters such as graph structure.\nDue to the flexibility in splitting the objective function into loss $f(x)$ and each penalty $\\psi_j(y_j)$,\nADMM is an efficient method to solve the obove problem. However,\nin the problem \\eqref{eq:1}, we only access the objective values\nrather than the whole explicit function $F(x,y_{[k]})$,\nthus the classic ADMM methods are unsuitable for the problem \\eqref{eq:1}.\n\nRecently, \\cite{gao2018information,liu2018admm} proposed the zeroth-order stochastic ADMM methods,\nwhich only use the objective values to optimize. However,\nthese zeroth-order ADMM-based methods build on the convexity of objective function. Clearly, these methods\nare limited in many nonconvex problems such as adversarial attack on black-box deep neural network (DNN).\nAt the same time, due to that the problem \\eqref{eq:1} includes multiple nonsmooth regularization functions and an equality constraint,\nthe existing zeroth-order algorithms \\cite{liu2018zeroth,Huang2019faster}\nare not suitable for solving this problem.\n\nIn the paper, thus, we propose a class of\nfast zeroth-order stochastic ADMM methods (\\emph{i.e.}, ZO-SVRG-ADMM and ZO-SAGA-ADMM) to solve the problem \\eqref{eq:1} based on the coordinate\nsmoothing gradient estimator \\cite{liu2018zeroth}.\nIn particular, the ZO-SVRG-ADMM and ZO-SAGA-ADMM methods build on\nthe SVRG \\cite{johnson2013accelerating} and SAGA \\cite{defazio2014saga}, respectively.\nMoreover, we study the convergence properties of the proposed methods.\nTable \\ref{tab:1} shows the convergence properties\nof the proposed methods and other related ones.\n\\subsection{Challenges and Contributions}\nAlthough both SVRG and SAGA show good performances in the first-order and second-order methods,\napplying these techniques to the nonconvex zeroth-order ADMM method\nis \\emph{not trivial}. There exists at least two main \\textbf{challenges}:\n\\begin{itemize}\n \\item Due to failure of the Fej\\'{e}r monotonicity of iteration, the convergence analysis of the nonconvex ADMM is generally quite difficult \\cite{wang2015convergence}.\n       With using the inexact zeroth-order estimated gradient, this difficulty becomes greater in the nonconvex ADMM methods.\n \\item To guarantee convergence of our zeroth-order ADMM methods, we need to design a new effective \\emph{Lyapunov} function, which can not follow\n       the existing nonconvex (stochastic) ADMM methods \\cite{jiang2019structured,huang2016stochastic}.\n\\end{itemize}\nThus, we carefully establish the \\emph{Lyapunov} functions in the following theoretical analysis to\nensure convergence of the proposed methods.\nIn summary, our major \\textbf{contributions} are given below:\n\\begin{itemize}\n\\item[1)] We propose a class of fast zeroth-order stochastic ADMM methods (\\emph{i.e.}, ZO-SVRG-ADMM and ZO-SAGA-ADMM) to solve the problem \\eqref{eq:1}.\n\\item[2)] We prove that both the ZO-SVRG-ADMM and ZO-SAGA-ADMM have convergence rate of $O(\\frac{1}{T})$ for nonconvex nonsmooth optimization.\n          In particular, our methods not only reach the existing best convergence rate $O(\\frac{1}{T})$ for the nonconvex optimization,\n          but also are able to effectively solve many machine learning problems with multiple complex regularized penalties.\n\\item[3)] Extensive experiments conducted on black-box classification and structured adversarial attack on black-box DNNs\n          validate efficiency of the proposed algorithms.\n\\end{itemize}\n\\section{Related Works}\nZeroth-order (gradient-free) optimization is a powerful optimization tool for solving many\nmachine learning problems, where the gradient\nof objective function is not available or computationally prohibitive.\nRecently, the zeroth-order optimization methods are widely applied and studied.\nFor example, zeroth-order optimization methods have been applied to bandit feedback analysis \\cite{agarwal2010optimal} and\nblack-box attacks on DNNs \\cite{chen2017zoo,liu2018zeroth}.\n\\cite{Nesterov2017RandomGM} have proposed several random zeroth-order\nmethods based on the Gaussian smoothing gradient estimator.\nTo deal with the nonsmooth regularization, \\cite{gao2018information,liu2018admm} have proposed\nthe zeroth-order online\/stochastic ADMM-based methods.\n\nSo far, the above algorithms mainly build on the convexity of problems.\nIn fact, the zeroth-order methods are also highly successful in solving various nonconvex problems such as\nadversarial attack to black-box DNNs \\cite{liu2018zeroth}.\nThus, \\cite{ghadimi2013stochastic,liu2018zeroth,Gu2018faster}\nhave begun to study the zeroth-order stochastic methods for the nonconvex optimization.\nTo deal with the nonsmooth regularization, \\cite{ghadimi2016mini,Huang2019faster} have proposed\nsome non-convex zeroth-order proximal stochastic gradient methods.\nHowever, these methods still are not well competent to some complex machine learning problems\nsuch as a task of structured adversarial attack to the black-box DNNs, which is described in the following experiment.\n\\subsection{Notations}\nLet $y_{[k]}= \\{y_1,\\cdots,y_k\\}$ and $y_{[j:k]}= \\{y_{j},\\cdots,y_k\\}$ for $j\\in[k]$.\nGiven a positive definite matrix $G$, $\\|x\\|^2_G = x^TGx$;\n$\\sigma_{\\max}(G)$ and $\\sigma_{\\min}(G)$\ndenote the largest and smallest eigenvalues of $G$, respectively,\nand $\\kappa_G = \\frac{\\sigma_{\\max}(G)}{\\sigma_{\\min}(G)}$.\n$\\sigma^A_{\\max}$ and $\\sigma^A_{\\min}$ denote the largest and smallest eigenvalues of matrix $A^TA$.\n\\section{Preliminaries}\nIn the section, we begin with restating a standard $\\epsilon$-approximate stationary point of the problem \\eqref{eq:1},\nas in \\cite{jiang2019structured,Huang2019fasterst}.\n\\begin{definition} \\label{def:1}\nGiven $\\epsilon>0$, the point $(x^*,y_{[k]}^*,\\lambda^*)$ is said to be an\n$\\epsilon$-approximate stationary point of the problems \\eqref{eq:1},\nif it holds that\n\\begin{align}\n \\mathbb{E}\\big[ \\mbox{dist}(0,\\partial L(x^*,y_{[k]}^*,\\lambda^*))^2 \\big] \\leq \\epsilon,\n\\end{align}\nwhere $L(x,y_{[k]},\\lambda)=f(x) + \\sum_{j=1}^k\\psi_j(y_j) - \\langle \\lambda, Ax+\\sum_{j=1}^kB_jy_j-c\\rangle$,\n\\begin{align}\n   \\partial L(x,y_{[k]},\\lambda) = \\left [ \\begin{matrix}\n     \\nabla_x L(x,y_{[k]},\\lambda) \\\\\n     \\partial_{y_1} L(x,y_{[k]},\\lambda) \\\\\n     \\cdots \\\\\n     \\partial_{y_k} L(x,y_{[k]},\\lambda) \\\\\n     -Ax - \\sum_{j=1}^kB_jy_j+c\n \\end{matrix}\n \\right ], \\nonumber\n\\end{align}\n$\\mbox{dist}(0,\\partial L)=\\inf_{L'\\in \\partial L} \\|0-L'\\|.$\n\\end{definition}\n\nNext, we make some mild assumptions regarding problem \\eqref{eq:1} as follows:\n\\begin{assumption}\nEach function $f_i(x)$ is $L$-smooth for $\\forall i \\in \\{1,2,\\cdots,n\\}$ such that\n\\begin{align}\n\\|\\nabla f_i(x)-\\nabla f_i(y)\\| \\leq L \\|x - y\\|, \\ \\forall x,y \\in \\mathbb{R}^d, \\nonumber\n\\end{align}\nwhich is equivalent to\n\\begin{align}\nf_i(x) \\leq f_i(y) + \\nabla f_i(y)^T(x-y) + \\frac{L}{2}\\|x-y\\|^2.  \\nonumber\n\\end{align}\n\\end{assumption}\n\\begin{assumption}\nFull gradient of loss function $f(x)$ is bounded, i.e., there exists a constant $\\delta >0$ such that for all $x$,\nit follows that $\\|\\nabla f(x)\\|^2 \\leq \\delta^2$.\n\\end{assumption}\n\\begin{assumption}\n$f(x) $ and $\\psi_j(y_j)$ for all $j\\in [k]$ are all lower bounded, and denote\n$f^* = \\inf_x f(x)$ and $\\psi_j^* = \\inf_y \\psi_j(y)$ for $j\\in [k]$.\n\\end{assumption}\n\\begin{assumption}\n$A$ is a full row or column rank matrix.\n\\end{assumption}\nAssumption 1 has been commonly used in the convergence analysis of\nnonconvex algorithms \\cite{ghadimi2016mini}.\nAssumption 2 is widely used for\nstochastic gradient-based and ADMM-type methods \\cite{boyd2011distributed}.\nAssumptions 3 and 4 are usually used in the convergence analysis of ADMM methods \\cite{jiang2019structured,huang2016stochastic,Huang2019fasterst}.\nWithout loss of generality, we will use the full column rank of matrix $A$ in the rest of this paper.\n\\section{ Fast Zeroth-Order Stochastic ADMMs }\n\\begin{algorithm}[htb]\n   \\caption{ Nonconvex ZO-SVRG-ADMM Algorithm }\n   \\label{alg:1}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} $b$, $m$, $T$, $S=\\lceil T\/m\\rceil$, $\\eta>0$ and $\\rho>0$;\n   \\STATE {\\bfseries Initialize:} $x_0^1$, $y^{0,1}_j$ for $j\\in [k]$ and $\\lambda_0^1$;\n   \\FOR {$s=1,2,\\cdots,S$}\n   \\STATE{} $\\tilde{x}^{s+1}=x_0^{s+1}$, $\\hat{\\nabla} f(\\tilde{x}^{s})=\\frac{1}{n}\\sum_{i=1}^n\\hat{\\nabla} f_i(\\tilde{x}^{s})$;\n   \\FOR {$t=0,1,\\cdots,m-1$}\n   \\STATE{} Uniformly randomly pick a mini-batch $\\mathcal{I}_t$ (with replacement) from $\\{1,2,\\cdots,n\\}$, and $|\\mathcal{I}_t|=b$ ;\n   \\STATE{}  Using \\eqref{eq:5} to estimate stochastic gradient\n             $\\hat{g}_{t}^{s} = \\hat{\\nabla} f_{\\mathcal{I}_t}(x_{t}^{s})-\\hat{\\nabla} f_{\\mathcal{I}_t}(\\tilde{x}^s)+\\hat{\\nabla} f(\\tilde{x}^s)$;\n   \\STATE{}  $y_j^{s,t+1} \\!= \\! \\arg\\min_{y_j} \\! \\big\\{ \\mathcal {L}_{\\rho}(x_t^s,y_{[j-1]}^{s,t+1},y_j,y_{[j+1:k]}^{s,t},\\lambda_t^s)$\n             $ \\quad + \\frac{1}{2}\\|y_j-y_j^{s,t}\\|^2_{H_j} \\big\\} $, for all $j\\in [k]$;\n   \\STATE{}  $x^s_{t+1}= \\arg\\min_x \\hat{\\mathcal {L}}_{\\rho}\\big( x,y_{[k]}^{s,t+1},\\lambda^s_t, \\hat{g}_{t}^{s} \\big)$;\n   \\STATE{}  $\\lambda^s_{t+1} = \\lambda^s_t-\\rho(Ax^s_{t+1} + \\sum_{j=1}^kB_jy_j^{s,t+1}-c)$;\n   \\ENDFOR\n   \\STATE{} $x_0^{s+1}=x_{m}^{s}$, $y_j^{s+1,0}=y_j^{s,m}$ for $j\\in [k]$, $\\lambda_0^{s+1}=\\lambda_{m}^{s}$;\n   \\ENDFOR\n   \\STATE {\\bfseries Output:} $\\{x,y_{[k]},\\lambda\\}$ chosen at random uniformly from $\\{(x_{t}^s,y_{[k]}^{s,t},\\lambda_t^s)_{t=1}^{m}\\}_{s=1}^S$.\n\\end{algorithmic}\n\\end{algorithm}\n\nIn this section, we propose a class of zeroth-order stochastic ADMM methods to solve\nthe problem \\eqref{eq:1}. First, we define an augmented\nLagrangian function of the problem \\eqref{eq:1}:\n\\begin{align}\n \\mathcal {L}_{\\rho}(x,y_{[k]},\\lambda) & \\!= f(x) + \\sum_{j=1}^k\\psi_j(y_j) \\!-\\! \\langle\\lambda, Ax \\!+\\! \\sum_{j=1}^kB_jy_j-c\\rangle \\nonumber \\\\\n &  + \\frac{\\rho}{2} \\|Ax + \\sum_{j=1}^kB_jy_j - c\\|^2,\n\\end{align}\nwhere $\\lambda\\in \\mathbb{R}^{p}$ and $\\rho >0$ denotes the dual variable and penalty parameter, respectively.\n\n\\begin{algorithm}[htb]\n   \\caption{ Nonconvex ZO-SAGA-ADMM Algorithm }\n   \\label{alg:2}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} $b$, $T$, $\\eta>0$ and $\\rho>0$;\n   \\STATE {\\bfseries Initialize:} $z_i^0=x_0$ for $i\\in \\{1,2,\\cdots,n\\}$, $\\hat{\\phi}_0=\\frac{1}{n}\\sum_{i=1}^n\\nabla f_i(z^0_i)$, $y^{0}_j$ for $j\\in [k]$ and $\\lambda_0$;\n   \\FOR {$t=0,1,\\cdots,T-1$}\n   \\STATE{} Uniformly randomly pick a mini-batch $\\mathcal{I}_t$ (with replacement) from $\\{1,2,\\cdots,n\\}$, and $|\\mathcal{I}_t|=b$ ;\n   \\STATE{} Using \\eqref{eq:5} to estimate stochastic gradient $\\hat{g}_{t} \\!=\\! \\frac{1}{b}\\sum_{i_t\\in \\mathcal{I}_t}\n             \\big(\\hat{\\nabla} f_{i_t}(x_{t})\\!-\\!\\hat{\\nabla} f_{i_t}(z^t_{i_t}) \\big)\\!+\\!\\hat{\\phi}_t$\n            with $\\hat{\\phi}_t\\!=\\!\\frac{1}{n}\\sum_{i=1}^n\\hat{\\nabla} f_i(z^t_i)$;\n   \\STATE{}  $y_j^{t+1} = \\arg\\min_{y_j} \\big\\{ \\mathcal {L}_{\\rho}(x_t,y_{[j-1]}^{t+1},y_j,y_{[j+1:k]}^{t},\\lambda_t)\n              + \\frac{1}{2}\\|y_j-y_j^t\\|^2_{H_j} \\big\\}$, for all $j\\in [k]$;\n   \\STATE{}  $x_{t+1}= \\arg\\min_x \\hat{\\mathcal {L}}_{\\rho}\\big( x,y_{[k]}^{t+1},\\lambda_t, \\hat{g}_{t} \\big)$;\n   \\STATE{}  $\\lambda_{t+1} = \\lambda_t-\\rho(Ax_{t+1} + \\sum_{j=1}^kB_jy_j^{t+1}-c)$;\n   \\STATE{} $z^{t+1}_{i_t}= x_{t}$ for $i_t \\in \\mathcal{I}_t$ and $z_{i_t}^{t+1}=z^t_{i_t}$ for $i_t \\not\\in \\mathcal{I}_t$;\n   \\STATE{} $\\hat{\\phi}_{t+1}=\\hat{\\phi}_t-\\frac{1}{n}\\sum_{i_t\\in \\mathcal{I}_t} \\big(\\hat{\\nabla} f_{i_t}(z^t_{i_t})-\\hat{\\nabla} f_{i_t}(z^{t+1}_{i_t})\\big)$;\n   \\ENDFOR\n   \\STATE {\\bfseries Output:} $\\{x,y_{[k]},\\lambda\\}$ chosen at random uniformly from $\\{x_{t},y_{[k]}^{t},\\lambda_t\\}_{t=1}^{T}$.\n\\end{algorithmic}\n\\end{algorithm}\nIn the problem \\eqref{eq:1}, the explicit expression of objective function $f_i(x)$ is not available, and\nonly the function value of $f_i(x)$ is available. To avoid\ncomputing explicit gradient, thus, we use the coordinate smoothing gradient estimator\n\\cite{liu2018zeroth} to estimate gradients: for $i\\in [n]$,\n\\begin{align} \\label{eq:5}\n \\hat{\\nabla} f_i(x) = \\sum_{j=1}^d \\frac{1}{2\\mu_j}\\big(\n f_i(x+\\mu_j e_j)-f_i(x-\\mu_j e_j)\\big)e_j,\n\\end{align}\nwhere $\\mu_j$ is a coordinate-wise smoothing parameter, and $e_j$ is\na standard basis vector with 1 at its $j$-th coordinate, and 0\notherwise.\n\nBased on the above estimated gradients, we propose a zeroth-order\nADMM (ZO-ADMM) method to solve the problem \\eqref{eq:1} by executing the following iterations,\nfor $t \\!=\\! 1,2,\\cdots$\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n& y_j^{t+1} = \\arg\\min_{y_j} \\big\\{ \\mathcal {L}_{\\rho}(x_t,y_{[j-1]}^{t+1},y_j,y_{[j+1:k]}^{t},\\lambda_t) \\\\\n& \\quad \\qquad \\qquad \\qquad + \\frac{1}{2}\\|y_j-y_j^t\\|^2_{H_j} \\big\\}, \\ \\forall j\\in [k] \\\\\n& x_{t+1} = \\arg\\min_x \\hat{\\mathcal {L}}_{\\rho}(x,y_{t+1},\\lambda_t,\\hat{\\nabla} f(x)) \\\\\n& \\lambda_{t+1} = \\lambda_t - \\rho(Ax_{t+1}+By_{t+1}-c),\n\\end{aligned} \\right.\n\\end{equation*}\nwhere the term $\\frac{1}{2}\\|y_j-y_j^t\\|^2_{H_j}$ with $H_j\\succ 0$ to linearize the term $\\|Ax + \\sum_{j=1}^kB_jy_j-c\\|^2$.\nHere, due to using the inexact zeroth-order gradient to update $x$, we define an approximate function over $x_t$ as follows:\n\\begin{align}\n\\small\n & \\hat{\\mathcal {L}}_{\\rho}\\big(x,y_{[k]}^{t+1},\\lambda_t,\\hat{\\nabla} f(x)\\big) \\!=\\!\n f(x_t) \\!+\\! \\hat{\\nabla}f(x)^T(x-x_t) \\nonumber \\\\\n & \\!+\\! \\frac{1}{2\\eta}\\|x\\!-\\!x_t\\|^2_G \\!+\\! \\sum_{j=1}^k \\psi_j(y_j^{t+1}) \\!-\\! \\lambda_t^T(Ax \\!+\\! \\sum_{j=1}^k B_jy_j^{t+1}\\!-\\!c) \\nonumber \\\\\n & \\!+\\! \\frac{\\rho}{2}\\|Ax \\!+\\! \\sum_{j=1}^kB_jy_j^{t+1}\\!-\\!c\\|^2,\n\\end{align}\nwhere $G\\succ0$, $\\hat{\\nabla} f(x)$ is the zeroth-order gradient and $\\eta>0$ is a step size.\nConsidering the matrix $A^TA$ is large, set $G = r I - \\rho \\eta A^TA \\succ I$ with $r > \\rho \\eta\n\\sigma_{\\max}(A^TA) + 1 $ to linearize the term $\\|Ax + \\sum_{j=1}^kB_jy_j^{t+1}-c\\|^2$.\nIn the problem \\eqref{eq:1}, not only the noisy\ngradient of $f_i(x)$ is not available, but also the sample size $n$\nis very large. Thus, we propose fast ZO-SVRG-ADMM and ZO-SAGA-ADMM to solve the problem \\eqref{eq:1},\nbased on the SVRG and SAGA, respectively.\n\nAlgorithm \\ref{alg:1} shows the algorithmic framework of ZO-SVRG-ADMM.\nIn Algorithm \\ref{alg:1}, we use the estimated stochastic gradient\n$\\hat{g}_{t}^{s} = \\hat{\\nabla} f_{\\mathcal{I}_t}(x_{t}^{s})-\\hat{\\nabla} f_{\\mathcal{I}_t}(\\tilde{x}^s)+\\hat{\\nabla} f(\\tilde{x}^s)$\nwith $\\hat{\\nabla} f_{\\mathcal{I}_t}(x^s_t) = \\frac{1}{b}\\sum_{i_t\\in \\mathcal{I}_t} \\hat{\\nabla} f_{i_t}(x^s_t)$.\nWe have $\\mathbb{E}_{\\mathcal{I}_t}[\\hat{g}_{t}^{s}] = \\hat{\\nabla} f(x_{t}^{s}) \\neq \\nabla f(x_{t}^{s})$, \\emph{i.e.}, this stochastic gradient\nis a \\textbf{biased} estimate of the true full gradient.\nAlthough the SVRG has shown a great promise,\nit relies upon the assumption that the stochastic\ngradient is an \\textbf{unbiased }estimate of true full gradient.\nThus, adapting the similar ideas of SVRG to zeroth-order ADMM optimization\nis not a trivial task. To handle this challenge, we choose the appropriate step size\n$\\eta$, penalty parameter $\\rho$ and smoothing parameter $\\mu$ to guarantee the convergence of our algorithms,\nwhich will be discussed in the following convergence analysis.\n\nAlgorithm \\ref{alg:2} shows the algorithmic framework of ZO-SAGA-ADMM.\nIn Algorithm \\ref{alg:2}, we use the estimated stochastic gradient $\\hat{g}_{t} = \\frac{1}{b} \\sum_{i_t\\in \\mathcal{I}_t}\n\\big(\\hat{\\nabla} f_{i_t}(x_{t})-\\hat{\\nabla} f_{i_t}(z^t_{i_t}) \\big) + \\hat{\\phi}_t$\nwith $\\hat{\\phi}_t=\\frac{1}{n}\\sum_{i=1}^n \\hat{\\nabla} f_i(z^t_i)$.\nSimilarly, we have $\\mathbb{E}_{\\mathcal{I}_t}[\\hat{g}_{t}] = \\hat{\\nabla} f(x_t) \\neq \\nabla f(x_t)$.\n\\section{Convergence Analysis}\nIn this section, we will study the convergence properties of\nthe proposed algorithms (ZO-SVRG-ADMM and ZO-SAGA-ADMM).\n\\subsection{Convergence Analysis of ZO-SVRG-ADMM }\nIn this subsection, we analyze convergence properties of the ZO-SVRG-ADMM.\n\nGiven the sequence $\\{(x^{s}_t,y_{[k]}^{s,t},\\lambda^{s}_t)_{t=1}^m\\}_{s=1}^S$ generated from Algorithm \\ref{alg:1},\nwe define a \\emph{Lyapunov} function:\n \\begin{align}\n R^s_t \\! = & \\mathbb{E}\\big[\\mathcal{L}_{\\rho} (x^s_t,y_{[k]}^{s,t},\\lambda^s_t) \\!+\\! (\\frac{3\\sigma^2_{\\max}(G)}{\\sigma^A_{\\min}\\eta^2\\rho} \\!+\\! \\frac{9L^2}{\\sigma^A_{\\min}\\rho})\\|x^s_{t}\\!-\\!x^s_{t-1}\\|^2 \\nonumber \\\\\n & \\quad + \\frac{18 L^2d }{\\sigma^A_{\\min}\\rho b}\\|x^s_{t-1}-\\tilde{x}^s\\|^2 + c_t\\|x^s_{t}-\\tilde{x}^s\\|^2\\big], \\nonumber\n \\end{align}\n where the positive sequence $\\{c_t\\}$ satisfies\n \\begin{equation*}\n  c_t= \\left\\{\n  \\begin{aligned}\n  & \\frac{36 L^2d }{\\sigma^A_{\\min}\\rho b} +\n     \\frac{2Ld}{b} + (1+\\beta)c_{t+1}, \\ 1 \\leq t \\leq m, \\\\\n  & 0, \\ t \\geq m+1.\n  \\end{aligned}\n  \\right.\\end{equation*}\nNext, we definite a useful variable $\\theta^s_{t} \\!=\\! \\mathbb{E} \\big[ \\|x^s_{t+1}-x^s_{t}\\|^2 + \\|x^s_{t}-x^s_{t-1}\\|^2 + \\frac{d}{b}(\\|x^s_{t}-\\tilde{x}^s\\|^2 + \\|x^s_{t-1}-\\tilde{x}^s\\|^2 )\n   + \\sum_{j=1}^k \\|y_j^{s,t}-y_j^{s,t+1}\\|^2\\big]$.\n\\begin{theorem} \\label{th:1}\n Suppose the sequence $\\{(x^{s}_t,y_{[k]}^{s,t},\\lambda^{s}_t)_{t=1}^m\\}_{s=1}^S$ is generated from Algorithm \\ref{alg:1}. Let $m=n^{\\frac{1}{3}}$,\n $b=d^{1-l} n^{\\frac{2}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$, $\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{9d^lL} \\ (0 < \\alpha \\leq 1 )$ and\n $\\rho = \\frac{6\\sqrt{71}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$,\n then we have\n \\begin{align}\n\\min_{s,t} \\mathbb{E}\\big[ \\mbox{dist}(0,\\partial L(x^s_t,y_{[k]}^{s,t},\\lambda^s_t))^2\\big] \\leq O(\\frac{\\tilde{\\nu} d^{2l}}{T}) + O(d^{2+2l}\\mu^2), \\nonumber\n\\end{align}\nwhere $\\tilde{\\nu}=R^1_0 - R^*$, and $R^*$ is a lower bound of function $R^s_t$.\nIt follows that suppose the smoothing parameter $\\mu$ and the whole iteration number $T=mS$ satisfy\n \\begin{align}\n  \\frac{1}{\\mu} = O \\big( \\frac{d^{1+l}}{\\sqrt{\\epsilon}} \\big), \\quad\n   T = O\\big( \\frac{\\tilde{\\nu} d^{2l}}{\\epsilon} \\big), \\nonumber\n \\end{align}\n then $(x^{s^*}_{t^*},y_{[k]}^{s^*,t^*},\\lambda^{s^*}_{t^*})$\n is an $\\epsilon$-approximate stationary point of the problems \\eqref{eq:1}, where $(t^*,s^*) = \\mathop{\\arg\\min}_{t,s}\\theta^s_{t}$.\n\\end{theorem}\n\\begin{remark}\nTheorem \\ref{th:1} shows that given $m=n^{\\frac{1}{3}}$,\n $b=d^{1-l} n^{\\frac{2}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$, $\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{9d^lL} \\ (0 < \\alpha \\leq 1 )$,\n $\\rho = \\frac{6\\sqrt{71}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$ and $\\mu= O(\\frac{1}{d\\sqrt{T}})$,\n the ZO-SVRG-ADMM has convergence rate of $O(\\frac{d^{2l}}{T})$. Specifically, when $1\\leq d < n^{\\frac{1}{3}}$, given $l=0$,\n the ZO-SVRG-ADMM has convergence rate of $O(\\frac{1}{T})$; when $n^{\\frac{1}{3}} \\leq d < n^{\\frac{2}{3}}$, given $l=\\frac{1}{2}$,\n it has convergence rate of $O(\\frac{\\sqrt{d}}{T})$; when $n^{\\frac{2}{3}} \\leq d$, given $l=1$,\n it has convergence rate of $O(\\frac{d}{T})$.\n\\end{remark}\n\\subsection{Convergence Analysis of ZO-SAGA-ADMM }\nIn this subsection, we provide the convergence analysis of\nthe ZO-SAGA-ADMM.\n\nGiven the sequence $\\{x_t,y_{[k]}^{t},\\lambda_t\\}_{t=1}^T$ generated from Algorithm \\ref{alg:2},\nwe define a \\emph{Lyapunov} function\n \\begin{align}\n & \\Omega_t \\! = \\mathbb{E}\\big[ \\mathcal{L}_{\\rho} (x_t,y_{[k]}^{t},\\lambda_t) \\!+\\! (\\frac{3\\sigma^2_{\\max}(G)}{\\sigma^A_{\\min}\\rho\\eta^2}\n  \\!+\\! \\frac{9L^2}{\\sigma^A_{\\min}\\rho}) \\|x_{t}\\!-\\!x_{t-1}\\|^2  \\nonumber \\\\\n &\\quad + \\frac{18 L^2d }{\\sigma^A_{\\min}\\rho b}\\frac{1}{n} \\sum_{i=1}^n\\|x_{t-1}-z^{t-1}_i\\|^2 + c_t\\frac{1}{n} \\sum_{i=1}^n\\|x_{t}-z^t_i\\|^2 \\big]. \\nonumber\n\\end{align}\nHere the positive sequence $\\{c_t\\}$ satisfies\n \\begin{equation*}\n c_t \\!= \\left\\{\n  \\begin{aligned}\n  & \\frac{36L^2d }{\\sigma^A_{\\min}\\rho b} \\!+ \\frac{2Ld}{b} \\!+ (1-\\hat{p})(1+\\beta)c_{t+1}, \\ 0 \\leq t \\leq T-1, \\\\\n  & 0, \\ t \\geq T,\n  \\end{aligned}\n  \\right.    \\end{equation*}\n where $\\hat{p}$ denotes probability of an index $i$ in $\\mathcal{I}_t$.\nNext, we definite a useful variable $\\theta_{t} \\!=\\! \\mathbb{E} \\big[\\|x_{t+1}\\!-\\!x_t\\|^2 \\!+\\! \\|x_t\\!-\\!x_{t-1}\\|^2\n\\!+\\! \\frac{d}{bn}\\sum^n_{i=1} (\\|x_t\\!-\\!z^t_i\\|^2 \\!+\\! \\|x_{t-1}\\!-\\!z^{t-1}_i\\|^2)\\!+\\! \\sum_{j=1}^k \\|y_j^{t}\\!-\\!y_j^{t+1}\\|^2 \\big]$.\n\\begin{theorem} \\label{th:2}\nSuppose the sequence $\\{x_t,y_{[k]}^{t},\\lambda_t\\}_{t=1}^T$ is generated from Algorithm \\ref{alg:2}. Let $b = n^{\\frac{2}{3}}d^{\\frac{1-l}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$,\n$\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{33d^lL} \\ (0 < \\alpha \\leq 1)$ and $\\rho = \\frac{6\\sqrt{791}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$\nthen we have\n\\begin{align}\n\\min_{1\\leq t \\leq T} \\mathbb{E}\\big[ \\mbox{dist}(0,\\partial L(x_t,y_{[k]}^{t},\\lambda_t))^2\\big] \\leq O(\\frac{\\tilde{\\nu}d^{2l}}{T}) \\!+ O(d^{2+2l}\\mu^2), \\nonumber\n\\end{align}\nwhere $\\tilde{\\nu}=\\Omega_0- \\Omega^*$, and $\\Omega^*$ is a lower bound of function $\\Omega_t$.\nIt follows that suppose the parameters $\\mu$ and $T$ satisfy\n\\begin{align}\n \\frac{1}{\\mu} = O \\big( \\frac{d^{1+l} }{\\sqrt{\\epsilon}} \\big), \\quad  T = O\\big(\\frac{\\tilde{\\nu}d^{2l}}{\\epsilon} \\big), \\nonumber\n\\end{align}\nthen $(x_{t^*},y_{[k]}^{t^*},\\lambda_{t^*})$ is an $\\epsilon$-approximate stationary point of the problems \\eqref{eq:1},\nwhere $t^* = \\mathop{\\arg\\min}_{ 1\\leq t\\leq T}\\theta_{t}$.\n\\end{theorem}\n\\begin{remark}\nTheorem \\ref{th:2} shows that $b = n^{\\frac{2}{3}}d^{\\frac{1-l}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$,\n$\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{33d^lL} \\ (0 < \\alpha \\leq 1)$, $\\rho = \\frac{6\\sqrt{791}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$\nand $\\mu= O(\\frac{1}{d\\sqrt{T}})$, the ZO-SAGA-ADMM has the $O(\\frac{d^{2l}}{T})$ of convergence rate. Specifically, when $1\\leq d < n$, given $l=0$,\n the ZO-SAGA-ADMM has convergence rate of $O(\\frac{1}{T})$; when $n \\leq d < n^2$, given $l=\\frac{1}{2}$,\n it has convergence rate of $O(\\frac{d}{T})$; when $n^2 \\leq d$, given $l=1$,\n it has convergence rate of $O(\\frac{d^2}{T})$.\n\\end{remark}\n\\begin{figure*}[!t]\n\\centering\n\\subfigure[20news]{\\includegraphics[width=0.235\\textwidth]{20news_obj}}\n\\subfigure[a9a]{\\includegraphics[width=0.235\\textwidth]{a9a_obj}}\n\\subfigure[w8a]{\\includegraphics[width=0.235\\textwidth]{w8a_obj}}\n\\subfigure[covtype.binary]{\\includegraphics[width=0.235\\textwidth]{covtype_obj}}\n\\caption{Objective value gaps \\emph{versus} CPU time on benchmark datasets.}\n\\label{fig:1}\n\\end{figure*}\n\\section{Experiments}\nIn this section, we compare our algorithms\n(ZO-SVRG-ADMM, ZO-SAGA-ADMM) with the ZO-ProxSVRG, ZO-ProxSAGA \\cite{Huang2019faster}, the deterministic zeroth-order ADMM (ZO-ADMM),\nand zeroth-order stochastic ADMM (ZO-SGD-ADMM) without variance reduction\non two applications: 1) robust black-box binary classification, and 2) structured adversarial\nattacks on black-box DNNs.\n\\begin{table}[!h]\n  \\centering\n  \\begin{tabular}{c|c|c|c}\n  \\hline\n  datasets & \\#samples & \\#features & \\#classes \\\\ \\hline\n  \\emph{20news} & 16,242 &  100 & 2 \\\\\n  \\emph{a9a}   & 32,561 & 123 & 2 \\\\\n  \\emph{w8a} & 64,700 & 300 & 2 \\\\\n  \\emph{covtype.binary} & 581,012 & 54 & 2\\\\\n  \\hline\n  \\end{tabular}\n  \\caption{Real Datasets for Black-Box Binary Classification }\\label{tab:2}\n\\end{table}\n\n\\begin{figure*}[!t]\n  \\centering\n \n  \\includegraphics[width=0.75\\textwidth]{Adversarial_attacks}\\\\\n  \\caption{Group-sparsity perturbations are learned from MNIST and CIFAR-10 datasets. Blue and red labels denote the initial label,\n  and the label after attack, respectively. }\\label{fig:3}\n\\end{figure*}\n\\subsection{ Robust Black-Box Binary Classification }\nIn this subsection, we focus on a robust black-box binary classification task\nwith graph-guided fused lasso.\nGiven a set of training samples $(a_i,l_i)_{i=1}^n$,\nwhere $a_i\\in \\mathbb{R}^d$ and $l_i \\in \\{-1,+1\\}$,\nwe find the optimal parameter $x\\in \\mathbb{R}^d$ by solving the problem:\n\\begin{align} \\label{eq:8}\n \\min_{x\\in \\mathbb{R}^d} \\frac{1}{n}\\sum_{i=1}^n f_i(x) + \\tau_1\\|x\\|_1 + \\tau_2\\|\\hat{G}x\\|_1,\n\\end{align}\nwhere $f_i(x)$ is the black-box loss function,\nthat only returns the function value given an input.\nHere, we specify the loss function\n$f_i(x)= \\frac{\\sigma^2}{2}\\big(1-\\exp(-\\frac{(l_i-a_i^Tx)^2}{\\sigma^2})\\big)$,\nwhich is the \\emph{nonconvex} robust correntropy\ninduced loss \\cite{he2011maximum}.\nMatrix $\\hat{G}$ decodes the sparsity pattern of graph obtained by learning sparse Gaussian graphical model \\cite{huang2015joint}.\nIn the experiment, we give mini-batch size $b=20$,\nsmoothing parameter $\\mu=\\frac{1}{d\\sqrt{t}}$ and penalty parameters $\\tau_1=\\tau_2=10^{-5}$.\n\nIn the experiment, we use\nsome public real datasets\\footnote{\\emph{20news} is from \\url{https:\/\/cs.nyu.edu\/~roweis\/data.html};\nothers are from \\url{www.csie.ntu.edu.tw\/~cjlin\/libsvmtools\/datasets\/}.},\nwhich are summarized in Table \\ref{tab:2}. For each dataset, we use half of the samples as training data\nand the rest as testing data. Figure \\ref{fig:1} shows that the objective values\nof our algorithms faster decrease than the\nother algorithms, as the CPU time increases. In particular, our algorithms show better performances than\nthe zeroth-order proximal algorithms. It is relatively difficult that these zeroth-order proximal methods deal with\nthe nonsmooth penalties in the problem \\eqref{eq:8}.\nThus, we have to use some iterative methods\nto solve the proximal operator in these proximal methods.\n\\subsection{Structured Attacks on Black-Box DNNs }\nIn this subsection, we use our algorithms to generate adversarial examples to attack the pre-trained DNN models,\nwhose parameters are hidden from us and only its outputs are accessible.\nMoreover, we consider an interesting problem: ``What possible\nstructures could adversarial perturbations have to fool black-box DNNs ?\"\nThus, we use the zeroth-order algorithms to find an universal structured adversarial perturbation $x \\in \\mathbb{R}^d$\nthat could fool the samples $\\{a_i\\in \\mathbb{R}^d , \\ l_i \\in \\mathbb{N} \\}_{i=1}^n$,\nwhich can be regarded as the following problem:\n\\begin{align} \\label{eq:9}\n\\min_{x \\in \\mathbb{R}^d} &\\frac{1}{n}\\sum\\limits_{i=1}^n \\max\\big\\{F_{l_i}(a_i+x) - \\max\\limits_{j\\neq l_i} F_j(a_{i}+x), 0 \\big\\} \\nonumber \\\\\n &  + \\tau_1 \\sum_{p=1}^P\\sum_{q=1}^Q \\|x_{\\mathcal{G}_{p,q}}\\|_2 + \\tau_2 \\|x \\|_2^2 + \\tau_3 h(x),\n\\end{align}\nwhere $F(a)$ represents the final layer output before softmax of neural network, and $h(x)$ ensures the validness of created adversarial examples.\nSpecifically, $h(x) = 0$ if $a_i+x \\in [0,1]^d$ for all $i\\in [n]$ and $\\|x\\|_\\infty \\leq \\epsilon$, otherwise $h(x) = \\infty$.\nFollowing \\cite{xu2018structured}, we use the overlapping lasso to obtain structured perturbations.\nHere, the overlapping groups $\\{\\mathcal{G}_{p,q}\\},\\ p=1,\\cdots,P, \\ q = 1,\\cdots,Q$\ngenerate from dividing an image into\nsub-groups of pixels.\n\nIn the experiment, we use the pre-trained DNN models on MNIST and CIFAR-10 as the target black-box models, which can attain $99.4\\%$ and $80.8\\%$ test accuracy, respectively.\nFor MNIST, we select 20 samples from a target class and set batch size $b=4$; For CIFAR-10, we select 30 samples and set $b=5$.\nIn the experiment, we set $\\mu=\\frac{1}{d\\sqrt{t}}$, where $d=28 \\times 28$ and $d=3\\times 32\\times 32$ for MNIST and CIFAR-10, respectively.\nAt the same time, we set the parameters $\\epsilon=0.4$, $\\tau_1 = 1$, $\\tau_2=2$ and $\\tau_3=1$.\nFor both datasets, the kernel size for overlapping group lasso is set to $3 \\times 3$ and the stride is one.\n\nFigure \\ref{fig:2} shows that attack losses (\\emph{i.e.} the first term of the problem \\eqref{eq:9}) of\nour methods faster decrease than the other methods, as the number of iteration increases.\nFigure \\ref{fig:3} shows that our algorithms can learn some structure perturbations,\nand can successfully attack the corresponding DNNs.\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[\\emph{MNIST}]{\\includegraphics[width=0.236\\textwidth]{Mnist_attack_loss}}\n\\subfigure[\\emph{CIFAR-10}]{\\includegraphics[width=0.236\\textwidth]{CIFAR_attack_loss}}\n\\caption{Attack loss on adversarial attacks black-box DNNs. }\n\\label{fig:2}\n\\end{figure}\n\\section{Conclusions}\nIn the paper, we proposed fast ZO-SVRG-ADMM and ZO-SAGA-ADMM methods based on\nthe coordinate smoothing gradient estimator, which only uses the objective function\nvalues to optimize. Moreover, we prove that the proposed methods have a convergence rate of $O(\\frac{1}{T})$.\nIn particular, our methods not only reach the existing best convergence rate $O(\\frac{1}{T})$ for the nonconvex optimization,\nbut also are able to effectively solve many machine learning problems with the complex nonsmooth regularizations.\n\\section*{Acknowledgments}\nF.H., S.G., H.H. were partially supported by\nU.S. NSF IIS 1836945, IIS 1836938, DBI 1836866, IIS 1845666, IIS 1852606, IIS 1838627, IIS 1837956.\nS.C. was partially supported by the NSFC under Grant No. 61806093 and No. 61682281, and the Key Program of NSFC under Grant No. 61732006.\n\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nThis document is a model and instructions for \\LaTeX.\nPlease observe the conference page limits. \n\n\\section{Ease of Use}\n\n\\subsection{Maintaining the Integrity of the Specifications}\n\nThe IEEEtran class file is used to format your paper and style the text. All margins, \ncolumn widths, line spaces, and text fonts are prescribed; please do not \nalter them. You may note peculiarities. For example, the head margin\nmeasures proportionately more than is customary. This measurement \nand others are deliberate, using specifications that anticipate your paper \nas one part of the entire proceedings, and not as an independent document. \nPlease do not revise any of the current designations.\n\n\\section{Prepare Your Paper Before Styling}\nBefore you begin to format your paper, first write and save the content as a \nseparate text file. Complete all content and organizational editing before \nformatting. 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Use the abbreviation \n``Fig.~\\ref{fig}'', even at the beginning of a sentence.\n\n\\begin{table}[htbp]\n\\caption{Table Type Styles}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{Table}&\\multicolumn{3}{|c|}{\\textbf{Table Column Head}} \\\\\n\\cline{2-4} \n\\textbf{Head} & \\textbf{\\textit{Table column subhead}}& \\textbf{\\textit{Subhead}}& \\textbf{\\textit{Subhead}} \\\\\n\\hline\ncopy& More table copy$^{\\mathrm{a}}$& &  \\\\\n\\hline\n\\multicolumn{4}{l}{$^{\\mathrm{a}}$Sample of a Table footnote.}\n\\end{tabular}\n\\label{tab1}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics{fig1.png}}\n\\caption{Example of a figure caption.}\n\\label{fig}\n\\end{figure}\n\nFigure Labels: Use 8 point Times New Roman for Figure labels. Use words \nrather than symbols or abbreviations when writing Figure axis labels to \navoid confusing the reader. As an example, write the quantity \n``Magnetization'', or ``Magnetization, M'', not just ``M''. If including \nunits in the label, present them within parentheses. Do not label axes only \nwith units. In the example, write ``Magnetization (A\/m)'' or ``Magnetization \n\\{A[m(1)]\\}'', not just ``A\/m''. Do not label axes with a ratio of \nquantities and units. For example, write ``Temperature (K)'', not \n``Temperature\/K''.\n\n\\section*{Acknowledgment}\n\nThe preferred spelling of the word ``acknowledgment'' in America is without \nan ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B. \nG.) thanks $\\ldots$''. Instead, try ``R. B. G. thanks$\\ldots$''. Put sponsor \nacknowledgments in the unnumbered footnote on the first page.\n\n\\section*{References}\n\nPlease number citations consecutively within brackets \\cite{b1}. The \nsentence punctuation follows the bracket \\cite{b2}. Refer simply to the reference \nnumber, as in \\cite{b3}---do not use ``Ref. \\cite{b3}'' or ``reference \\cite{b3}'' except at \nthe beginning of a sentence: ``Reference \\cite{b3} was the first $\\ldots$''\n\nNumber footnotes separately in superscripts. Place the actual footnote at \nthe bottom of the column in which it was cited. Do not put footnotes in the \nabstract or reference list. Use letters for table footnotes.\n\nUnless there are six authors or more give all authors' names; do not use \n``et al.''. Papers that have not been published, even if they have been \nsubmitted for publication, should be cited as ``unpublished'' \\cite{b4}. Papers \nthat have been accepted for publication should be cited as ``in press'' \\cite{b5}. \nCapitalize only the first word in a paper title, except for proper nouns and \nelement symbols.\n\nFor papers published in translation journals, please give the English \ncitation first, followed by the original foreign-language citation \\cite{b6}.\n\n\n\\section{Introduction}\n\nCritically ill patients in the intensive care unit (ICU) commonly experience pain because of a multitude of factors including surgery, trauma, burns, and cancer, as well as enduring procedural pain \\cite{Barr2013}. Regardless of source, poorly managed pain adversely affects multiple organ systems and has a direct impact on outcomes, as the negative physiologic and psychologic consequences of pain are significant, long lasting, and can  preclude patients from participating in their care.\n\nIn the ICU, opioids are the primary systemic medications for managing pain. The optimal choice of opioid and the dosing regimen used for an individual patient depend on many factors, including the intensity of pain. To assess the latter as well as to measure  response to treatment, multiple pain intensity scales have been developed. These are unidimensional scales that measure intensity from 0 (no pain) to 10 (worst possible pain).\nUnfortunately, pain assessment scales fail in multiple patient populations (e.g. non-verbal children) and clinical situations (e.g. during paralysis). Hence, in recent years, an increasing body of work is addressing the problem of pain measurement using machine leaning with autonomic  \\cite{dlmEMBC2018}, brain activity \\cite{dlm_ICPR_2018} and facial expression \\cite{LopezMartinez2017c} data, and by stratifying patients based on their pain responses  to allow for personalized assessments \\cite{dlmNIPS2017}. However, machine learning for automatic pain management interventions remains an unexplored area.\n\n\nIn this work, we propose a novel approach for the automatic dosing of analgesics based on deep reinforcement learning. Specifically, we focus on morphine, one of the most commonly prescribed opioids in the ICU and a first line for the treatment of pain. Both under-treatment and over-treatment with morphine lead to severe adverse effects, such as those related to unrelieved pain and respiratory depression respectively. Our model aims to learn clinically interpretable optimal treatment policies and recommend individualized morphine interventions adapted to each patient's evolving state, to minimize pain while avoiding adverse effects.\nTo our knowledge, our work is the first to apply reinforcement learning for the recommendation of pain management regimes and the automatic dosing of analgesics.\n\n\\section{Background and related work}\n\nReinforcement learning (RL) algorithms model time-varying state and action spaces, and learn actionable policies $\\pi$ (mappings from states to actions) from training examples that do not represent optimal behaviour. At each time step $t$, the RL agent observes the current environment state $s_t$ and takes an action $a_t \\in \\mathcal{A}$ from a discrete set of actions $\\mathcal{A}$, subsequently receiving a reward $r_t$ and transitioning to a new state $s_{t+1}$. The action $a_t$ is chosen following policy $\\pi$ so that the return or expected sum of future rewards $R_t = r_t + \\gamma r_{t+1} + \\gamma^2 t_{t+2}+...$ when taking that action and following policy $\\pi$ thereafter is maximized. The future rewards are discounted by a factor $\\gamma$ that represents the trade off between  immediate and later rewards.\n\nCombining RL with deep neural networks has achieved tremendous success in complex problems such as the Atari, Go, and chess games \\cite{Mnih2015, Silver2018}. In medicine, deep reinforcement learning has been applied, for example, to unfractioned heparin dosing in the ICU \\cite{Nemati2016}, intravenous fluid and vasopressor dosing in sepsis patients \\cite{Raghu2017,Raghu2017a}, and chemotherapy dosing in cancer patients enrolled in clinical trials \\cite{Yauney2018}.\n\n\n\n\n\\section{Methods}\nMorphine  analgesia is characterized by tremendous interindividual variation and is influenced by multiple factors \\cite{Fillingim2005}. Hence, in the ICU, physicians continuously monitor patients to personalize morphine pain management. Our goal is to deduce optimal treatment policies, similar to those of physicians, that account for pain level as well as physiological state as captured by vital signs. \nWe chose RL over supervised learning because no ground truth for treatment strategies exists and RL is able to infer optimal strategies from suboptimal training examples.\n\n\\subsection{Data and preprocessing}\n\nThis work was  conducted on the publicly available Multiparameter Intelligent Monitoring in Intensive Care III (MIMIC-3) database \\cite{Johnson2016}, which contains hospital admissions from approximately 38600 individuals aged $\\geq 15$ years. From MIMIC-3, we extracted a cohort of ICU patients fulfilling 2 criteria: (1) at least one pain intensity score  had been collected, and (2) at least one dose of intravenous morphine  had been administered.\nA total of 6843 admissions fulfilled these criteria. We chose to focus on morphine since this was the most commonly prescribed opioid in the MIMIC-3 database, followed by hydromorphone and oxycodone.\n\n\\subsection{Feature extraction}\n\nFor each patient, we extracted pain scores, analgesic administration events and vital signs. Data were aggregated into non-overlapping one-hour windows. When several data points were available within a window, we replaced them by either the mean (pain scores, vital signs) or the sum (analgesic administrations).  To account for missing data, we utilized sample-and-hold interpolation as in \\cite{Nemati2016}. In total, we extracted 20 hourly features.\n\n\n\\subsubsection{Pain scores}  MIMIC-III  contains scores in both numerical and text format. The latter were converted to a numeric score by using the scores referenced in the text (e.g. \\textit{\"3-Mild to Mod\"} becomes 3). All scores were reported in a 0 (no pain) to 10 (maximum pain) scale.\n\n\n\n\\subsubsection{Morphine interventions}\nMorphine sulfate can be administered intravenously either as a bolus  or as a continuous infusion with a steady rate for an extended period of time. In this work, we considered intravenous boluses only, which represented a majority of administrations for the studied admissions (59.4\\%).\nThese typically have an onset of 5-10 min and a half life of 3-4 hr.  \\cite{Narayanan2016a}. The dose, in addition to time of administration, was recorded.\n\n\n\\subsubsection{Non-morphine analgesic interventions} \nIn order to  lower opioid consumption and to improve the quality of analgesia, morphine is commmonly administered with adjuvant analgesics \\cite{Narayanan2016a}. This includes nonopiod coanalgesics such as nonsteroidal anti-inflammatory agents. Therefore, we also extracted administrations of 16 analgesics in addition to morphine.\n\n\n\n\n\\subsubsection{Physiological signals}\nMorphine induces respiratory depression \\cite{Pattinson2008} and impacts the cardiovascular system \\cite{Chen2015} with potentially fatal consequences.\nHence, we extracted two relevant physiological signals:  respiration rate (RR) and  heart rate (HR).\n\n\n\\subsection{Continuous State-space and Discretized Action-space }\n\\label{sec:spaces}\nThe action space was discretized to 14 actions: $0$, $(0,1]$, $(1,2]$, ..., $(9,10]$, $(10,15]$, $(15,20]$ and $(20,\\infty)$ mg of morphine intravenously. The state space, on the other hand, was continuous and directly  captured the patient's self-reported pain intensity and physiological state, as well as non-morphine analgesic interventions.\n\n\n\\subsection{Reward function}\n\nAt each time stamp $t$, the RL agent interacts with the environment and observes the state $s_t$ to  choose whether to deliver a morphine IV bolus (actions 1-13) or to withhold morphine (action 0). The action $a_t$ is chosen to  maximize the return, that is, the expected discounted future reward, defined as $R_t = \\sum_{t'=t}^{T} \\gamma^{t'-t} r_{t'}$, where $T$ is the terminal timestep.\n\n\nWe defined a reward function $r_t(hr,rr,p)$ that incentivizes \npain reduction while seeking to maintain heart rate and respiration rate within adequate physiological ranges. For healthy adults at rest, heart rates between 60 and 100 beats per minute and respiration rates between 12 and 20 breaths per minute are considered acceptable. \n\\begin{equation}\nr_t(hr,rr,p) = C_\\text{P} \\cdot r^{\\text{P}}_t (p_t)\n    + C_\\text{HR} \\cdot  r^{\\text{HR}}_t (hr_t)\n    + C_\\text{RR} \\cdot   r^{\\text{RR}}_t (rr_t)\n\\end{equation}\nwhere $C_\\text{P} = C_\\text{HR} =  C_\\text{RR} = \\frac{1}{3}$ and \n{\\small\n\\begin{flalign}\n    & r^{\\text{HR}}_t (hr_t) =     \\frac{2}{1 + e^{-({hr}_t- 60)} } \n                        -  \\frac{2}{1 + e^{-({hr}_t-100)} } \n                        -  1  &\\\\\n    & r^{\\text{RR}}_t (rr_t) =    \\frac{2}{1 + e^{-({rr}_t-12)} } \n                       -  \\frac{2}{1 + e^{-({rr}_t-20)} } \n                       -  1    &\\\\ \n    & r^{\\text{P}}_t (p_t) = 1 - \\frac{2 p_t}{10}  & \n\\end{flalign}\n}\nThis function assigns the maximum reward when pain is absent and both heart rate and respiration rate are within the acceptable range, while diminishing as they move away from the ideal range.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{figures\/Example_v3.png}\n\t\\caption{Example of a MIMIC-III patient's time series of physiological data and pain scores, together with the morphine administrations (discretized to 14  actions as described in Sec.\\ref{sec:spaces}).}\n\t\\label{fig:Example}\n\\end{figure}\n\n\\subsection{Reinforcement learning model architecture}\n\n\\begin{comment}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{figures\/Model.png}\n\t\\caption{DQN model used in this research.}\n\t\\label{fig:Model}\n\\end{figure}\n\\end{comment}\n\n\nTo learn the optimal treatment policies, we use Q-learning \\cite{Watkins92}, which works by learning an action-value $Q(s_t,a_t)$ function that provides the expected return $R_t$ of taking a given action at each state, that is, \n$\n    Q(s_t,a_t) = \\max_\\pi \\mathbb{E}[R_t]\n$\nwhere the maximum is taken over all possible policies $\\pi$. In Q-learning, the optimal action-value function is estimated using the Bellman equation: \n{ \\small\n\\begin{equation}\nQ(s_t, a_t) = \\mathbb{E}_{s_{t+1} \\sim T(s_{t+1}|s_t,a_t)}  [r + \\gamma \\max_{a_{t+1}} Q(s_{t+1},a_{t+1}) | s_t, a_t ]\n\\end{equation}\n}\nwhere $T(s_{t+1}|s_t,a_t)$ is the state transition distribution. Deep Q-learning uses neural networks to approximate $Q(s_t,a_t)$ \\cite{Mnih2015} by sampling tuples $<s_t,a_t,r_t,s_{t+1}>$ and \nminimizing the squared error loss between the output and  target $Q$ values, where $Q_{\\mbox{target}} = r_t+ \\gamma \\max_{a_{t+1}} Q (s_{t+1}, a_{t+1}; \\theta)$ and $\\theta$  are the weights used to parametrize the target network, using stochastic gradient descent.\n\n\nIn this work, following \\cite{Raghu2017,Raghu2017a}, we combined two common extensions of the deep Q network (DQN) model: double-deep Q networks (DDQN) \\cite{Hasselt2016} and dueling Q networks (dueling DQN) \\cite{Wang2015}. These address several shortcomings of simple DQNs, such as their tendency to overestimate Q values.\n\\begin{comment}\nDouble DQNs overcome the tendency of DQNs to overestimate Q values by modifying the Bellman update to:\n\\begin{equation}\n    Q(s_t, a_t) = r_t + \\gamma \\max_a Q'(s_{t+1}, \\arg\\max_a Q(s_{t+1},a) )\n\\end{equation}\nDueling DQNs, on the other hand, into separate \\textit{value} and \\textit{advantage} streams, where the \\textit{value} represents the quality of the current state, and the \\textit{advantage} represents the quality of the chosen action, such that:\n\\begin{equation}\nQ(s,a) = V(s) + A(s,a)\n\\end{equation}\n\\end{comment}\n\nOur final network architecture was therefore a Dueling Double-Deep Q Network (Dueling DDQN),\nIt contained two hidden layers with 64 nodes each, with Leaky-ReLu activation functions and equally sized advantage and value streams. As in \\cite{Raghu2017, Raghu2017a}, we used prioritized experience replay to sample transitions from the training set with probability proportional to the previous error.  The network was implemented in TensorFlow.\n\n\n\n\\section{Results}\n\nWe separated  6843 hospital admissions into 70\\%\/20\\%\/10\\% training, validation and test sets respectively. The action space was discretized to 14 actions as described in Sec.\\ref{sec:spaces}, based on visual inspection of the histogram of morphine doses in MIMIC-III.\nThe dueling DDQN model was then trained using Adam optimization \\cite{Kingma2015} with batch size 32. After training, we obtained the optimal policy $\\pi$ for a given patient state $s$, such that $\\pi(s)= \\arg \\max _a Q(s,a)$. Unfortunately, quantitative evaluation of off-policy models is challenging because it is difficult to assess the impact of deploying the learned policy $\\pi$ in patient outcomes. Hence, we evaluated $\\pi$ by qualitatively examining the choices of treatments proposed by the model, and comparing these with those made by physicians and our understanding of accepted clinical guidelines.\n\nFig.\\ref{fig:histogram1} depicts the comparison of actions taken by the physicians with those recommended by the model, for every hour in the test set. Action 0 refers to no morphine administered, whereas actions 1-13 represent increasing morphine doses. Physicians do not often prescribe morphine to patients (only 5.80\\% hourly timestamps contained morphine administration), but when they do, in 99\\% of instances the model also recommended morphine administration. The agreement between physician actions and model recommendations in these instances is shown in Fig.\\ref{fig:2DHistogram}. The actions recommended for the model for the 94.2\\% instances in which physicians chose to withhold morphine are shown in Fig.\\ref{fig:histogram1}.\n\nWhile physicians administer morphine sparsely, when they do, a second morphine dose is administered in the next hour in more than 40\\% of occasions, as shown in Fig.\\ref{fig:1DHistograms}. The model, on the other hand, recommended more continuous administration of morphine. This result is consistent with previous studies comparing intermittent bolus versus continuous infusion dosing of morphine, which have shown that continuous dosing provides similar or even better pain relief with no increase in acute adverse effects \\cite{Yu2014,Rutter1980}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/combined_v3_tight.png}\n\t\\caption{\\textit{(Left)} Normalized histogram of physician and dueling DDQN actions, where all actions that involved administration of morphine (actions 1-13) were grouped together. \\textit{(Right)} Histogram of dueling DDQN recommended actions for those instances in which physicians chose to withhold morphine (action 0).}\n\t\\label{fig:histogram1}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{figures\/2DHistogram_v3.png}\n\t\\caption{Histogram of physician against dueling DDQN actions  for those instances in which both physician and dueling DDQN recommended  administration of morphine (actions 1-13).}\n\t\\label{fig:2DHistogram}\n\\end{figure}\n\n\n\\section{Conclusion}\nThis work builds on recent research using  reinforcement learning to automatically find optimal treatment policies to improve patient outcome from training examples that do not represent optimal behavior  \\cite{Nemati2016,Raghu2017,Raghu2017a}. Here, we focus on the problem of dosing opioid analgesics in the ICU. While machine learning has been applied extensively to the problem of pain recognition, to our knowledge this is the first application of reinforcement learning\nfor deducing optimal intervention regimens to manage pain. We aimed to demonstrate how deep reinforcement learning could be used to suggest clinically interpretable treatment strategies for pain using real-time patient data that are readily accessible.  \nThis work has many limitations and should be taken as an illustrative example. Morphine analgesia in the ICU is a complex decision problem influenced by multiple factors. However, our model simplified these clinical factors by constraining  the amount of variables considered \nand defining a reward function that only aims to reduce pain while keeping heart and respiration rates within a pre-defined range. Therefore, possible directions for future work include improving the state space by adding more data modalities, extending the action space to include more dosing regimes (e.g. co-analgesics), and increasing the sophistication of the reward function to reflect the broader range of therapeutic goals. Another limitation is that we used data from a single center, which raises concerns about the generalizability of our findings. Hence, future studies should include multiple centers.\nDespite the limitations of this work, our learned policy is consistent with research that shows that continuous dosing is recommended over intermittent dosing. \n\nFinally, deployment of this approach to opioid dosing in clinical settings would first require extensive prospective evaluation of the learned policies. The goal would not be to replace physicians'  clinical judgments about treatment, but to aid clinical decision making with insights about optimal decisions and automatically guide therapy.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/fourfigs_v3_tight.png}\n\t\\caption{Histogram of morphine administration doses \\textit{(top)} and time between administrations \\textit{(bottom)} for both physicians \\textit{(left)} and the dueling DDQN algorithm \\textit{(right)}.}\n\t\\label{fig:1DHistograms}\n\\end{figure}\n\n{\\small\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nAfter a recent discovery of  topological insulators (TIs) \\cite{HasanRMP2010,QiRMP2011}, the quest for numerous fascinating effects in response to external perturbations in TIs has started. One of them is the magnetoelectric (ME) effect, which is a long standing issue in multiferroics \\cite{Tokura2007} and has been recently introduced in TIs \\cite{HasanRMP2010,QiRMP2011,Qi2008,Essin2009,Vazifeh2010,Nomura2011,Tse2010}. The essence of the ME effect is that the external electric or magnetic fields induce magnetization or polarization in a TI, respectively. From the effective topological field theory predictions \\cite{Qi2008}, the ME effect in three-dimensional (3D) TIs can be described by introducing a new term $U_{\\theta}$ in the energy density for the axion electrodynamics \\cite{HasanRMP2010,QiRMP2011,Qi2008,Essin2009,Chen2011,Nomura2011,Tse2010,Wilczek1987, Vazifeh2010,Coh2011,Sitte2012}:\n\\begin{equation}\n\\label{Eq:Enden}\nU_{\\theta}=-\\frac{e^2}{4\\pi^2\\hbar c}\\theta\\mathbf{E}\\cdot\\mathbf{B},\n\\end{equation}\nwhere $\\mathbf{E}$ and $\\mathbf{B}$ are the electric and magnetic fields respectively. The $\\theta$ term, $U_{\\theta}$ \\cite{Wilczek1987}, characterizes the 3D TIs. In Eq.~(\\ref{Eq:Enden}), parameter $\\theta$ takes values $\\pm\\pi$  for TIs, whereas it is $0$ for ordinary insulators.\nThe magnetization $\\mathbf{M}$ and electric polarization $\\mathbf{P}$ are obtained from the energy density as follows:\n\\begin{eqnarray}\n\\mathbf{M}&=&-\\frac{\\partial U_{\\theta}}{\\partial \\mathbf{B}}=\\frac{e^2 }{4\\pi^2\\hbar c}\\theta \\mathbf{E},\\\\\n\\qquad \\mathbf{P}&=&-\\frac{\\partial U_{\\theta}}{\\partial\\mathbf{E}}=\\frac{e^2}{4\\pi^2\\hbar c}\\theta\\mathbf{B}, \n\\end{eqnarray}\nand clearly show the cross-correlated responses \\cite{Tokura2007,Qi2008,Essin2009,Essin2010,Malashevich2010,Malashevich2011}.\n\nIn a strong magnetic field, which breaks time reversal symmetry, however, a controversy still remains in the literature regarding the quantization of ME responses. According to Ref.~\\onlinecite{Tse2010}, which considers only TI surface states, the response should be quantized: $\\theta\/\\pi=2N+1$ ($N$ being an integer) when the electron densities on the top and bottom surfaces are balanced, i.e., no gate voltage is applied. On the other hand, Ref.~\\onlinecite{Coh2011}, considering only bulk states in a magnetic field, concludes that $\\theta$ is not quantized but rather arbitrary. \n\nIn this paper, to resolve this controversy, we study the effect of an ultrahigh magnetic field on TIs. To fully treat both bulk and surfaces, we consider the lattice Hamiltonian \\cite{Zhang2009,Liu2010} for a 3D TI in a slab geometry, sandwiched between two thin ferromagnets with magnetizations pointing in the opposite directions along $z$ axis, see Fig.~\\ref{fig:EM}. The role of ferromagnetic films can also be played by an appropriate doping of the TI with magnetic impurities. \n\\begin{figure}\n\\includegraphics[width=0.99\\columnwidth]{me_TI}\n \\caption{(Color online) The ME effect in a 3D TI of slab geometry sandwiched between two ferromagnets (FMs) with opposite magnetizations.}\n \\label{fig:EM}\n\\end{figure}\nWe first reproduce the $\\theta$ term (\\ref{Eq:Enden}) with $\\theta=\\pm\\pi$, in the balanced case, from the total energy of electrons. This means that the axion electrodynamics remains unchanged as long as the bulk gap is not destroyed, even though time-reversal symmetry, which plays essential role in TIs, is broken by the magnetic field. However, as the TI thickness is decreased, we observe that the ME effect starts to vanish. Based on this result, we estimate the critical thickness of TI film capable of displaying the robust ME effect, which can be useful for future experimental studies, especially in the view of recent experimental progress on TI thin films \\cite{Checkelsky2012,Chang2013}. Moreover, we also study the case of both strong electric and magnetic fields. We show that the response to a small change in the electric field can be characterized in this regime by different $\\theta$, which are \\textit{still quantized} as long as the Fermi level resides within the bulk gap. We show that the quantization rule of $\\theta$ is determined by Landau level structures on the top and bottom surfaces, while the bulk states do not play an important role in thick TI films.\n\n\\section{Model}\n\nTo study the ME effect in TIs, we employ the Wilson-Dirac tight-binding Hamiltonian \nfor 3D lattice TI model \\cite{Zhang2009,Liu2010}, which is a simplified version of the effective four-band Hamiltonian for newly discovered 3D TIs such as $\\mathrm{Bi}_2\\mathrm{Se}_3,\\mathrm{Bi}_2\\mathrm{Te}_3$, and $\\mathrm{Sb}_2\\mathrm{Te}_3$ \\cite{Xia2009,Chen2009,Hsieh2009}. We introduce external magnetic field $\\mathbf{B}$, applied in $z$ direction, through the Peierls substitution and set its magnitude $B$ so that the magnetic flux going through a single plaquette is a rational fraction of the magnetic flux quantum. In other words, $B=(p\/q)\\Phi_0\/a^2$, where $p$ and $q$ are natural numbers with $p<q$, $\\Phi_0=hc\/e$ is the magnetic flux quantum, and $a$ is the lattice constant. For convenience, we set $e=c=\\hbar=a=1$ in the following so the magnetic field becomes $B=2\\pi p\/q$. Using Landau gauge $\\mathbf{A}=(0,Bx,0)$, the Hamiltonian is defined as \n\\begin{equation}\t\t\n\\label{eq:Wlsn}\t\nH_0=-\\frac{1}{2}\\sum_{\\mu=x,y,z}(T_{\\mu}^{}+T_{\\mu}^{\\dag})+(m+3r)\\sum_{\\mathbf{R}} c^\\dag_{\\mathbf{R}}\\beta c_{\\mathbf{R}}. \n\\end{equation}\nHere, the translation operators in $\\mu=x,y,z$ directions are\n\\begin{equation}\n\\label{eq:Wlsn2}\nT_{\\mu}=\\sum_{\\mathbf{R}}c^\\dag_{\\mathbf{R}+\\mathbf{a}_{\\mu}}e^{i A_{\\mu}}(r\\beta-it\\alpha_{\\mu})c_{\\mathbf{R}},\n\\end{equation}\nwhere $t$ and $r$ are the hopping parameters with and without spin flip, respectively, $m$ is the mass characterizing the spin-orbit interaction, $\\mathbf{R}=(n_x,n_y,n_z)$ is the lattice coordinate, $\\mathbf{a}_{\\mu}$ is the unit lattice vector in $\\mu$ direction, and $\\alpha_{\\mu}$, $\\beta$ are the standard Dirac matrices:\n\\begin{equation*}\n  \\alpha_\\mu=\n  \\begin{pmatrix}\n    0&\\mathbf{\\sigma_\\mu}\\\\\n    \\mathbf{\\sigma_\\mu}&0\n  \\end{pmatrix},\\quad\\beta=\n  \\begin{pmatrix}\n    I&0\\\\\n    0&-I\n  \\end{pmatrix},\n\\end{equation*}\nwhere $\\sigma_{\\mu}$ are the Pauli matrices. \n\nTo take into account the effect of an external electric field and the interaction of electron spins on TI surfaces with magnetization, we introduce additional terms $H_E$ and $H_s$, respectively. Then, the total Hamiltonian is\n\\begin{eqnarray}\n \\label{eq:Ham}\n  H&=&H_0+H_E+H_s,\\\\\n  H_E&=&\\sum_{\\mathbf{R}}c^\\dag_{\\mathbf{R}}U(n_z)\n  \\begin{pmatrix}\n    I&0\\\\\n    0&I\n  \\end{pmatrix}\n  c_{\\mathbf{R}}\n\\end{eqnarray}\nwith the potential energy $U(n_z)=-E(n_z-L_z\/2)$, where $L_z$ is the TI film thickness. The surface spin contribution \\cite{Qi2008, Nomura2011, LiuPRL2009, Abanin2011}, see Fig.~\\ref{fig:EM}, is given by\n\\begin{equation}\nH_s=\\sum_{\\mathbf{R}}c^\\dag_{\\mathbf{R}} b(n_z) \\Sigma_s^+c_{\\mathbf{R}}, \\,\nb(n_z)=\n\\begin{cases}\nb_s,&n_z=L_z,\\\\\n-b_s,&n_z=0.\n\\end{cases}\n\\end{equation}\nHere, $\\Sigma_s^+=\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ and $b_s$ is the surface spin constant. \n    \n\\section{Results and Discussion} \nWe investigate the magnetoelectric effect by calculating the energy eigenvalues of electrons. Since our model has translational invariance in $x$ and $y$ directions, Eq.~(\\ref{eq:Ham}) can be written as \n\\begin{equation}\n H=\\sum_{k_x,k_y,n_z}c^{\\dag}_{n_z}(k_x,k_y){\\cal H}_{n_z,n_z'}(k_x,k_y)c_{n_z'}(k_x,k_y),\n\\end{equation}\nwhere ${\\cal H}_{n_z,n_z'}(k_x,k_y)$ is $4q(L_z+1)\\times 4q(L_z+1)$ matrix, see Appendix for details. By numerically performing an exact diagonalization of this Hamiltonian matrix, we solve the eigenvalue equation \n${\\cal H}(k_x,k_y)|\\lambda,k_x,k_y\\rangle=\\epsilon_{\\lambda}(k_x,k_y)|\\lambda,k_x,k_y\\rangle$.\nThe ME effect is described by the energy density of electrons:\n\\begin{equation}\n U_{\\rm tot}=\\frac{1}{\\mathcal{V}}\\sum_{k_x,k_y}{\\sum_{\\lambda}}'\\epsilon_{\\lambda}(k_x,k_y),\n\\end{equation}\nwhere $\\mathcal{V}$ is the TI volume and $\\sum_{\\lambda}'$ is the sum of eigenvalues below the Fermi level, which is fixed at $\\epsilon_F=0$ in the following.\nIn the presence of both electric $\\mathbf{E}$ and magnetic $\\mathbf{B}$ fields, $U_{\\rm tot}$ has a term $\\propto\\mathbf{E}\\cdot\\mathbf{B}$ as expected from the axion electrodynamics, Eq.~(\\ref{Eq:Enden}).\n\nThe energy density for a typical case is plotted in the inset of Fig.~\\ref{fig:thetaLz}. For fixed magnetic field, we change the electric field and plot the energy density as a function of $\\mathbf{E}\\cdot\\mathbf{B}$.\nFrom Eq.~(\\ref{Eq:Enden}), we estimate the parameter $\\theta$ by $\\theta=-4\\pi^2\\tan\\gamma$, where $\\tan\\gamma$ is the slope of the linear dependence in the inset of Fig.~\\ref{fig:thetaLz}. In all our calculations, we set the hopping parameters $t=r=1$, the mass $m=-1$, and choose the number of layers in $x$ direction $L_x$, so that the number of points in $k_x$ direction of the magnetic Brillouin zone is $L_x\/q=32$.\n\n\\begin{figure}\n\\includegraphics[width=0.97\\columnwidth]{thetaLzcom}\n\\caption{(Color online) Parameter $\\theta$ as a function of TI film thickness $L_z$. The inset shows the energy density of electrons in a weak electric field and its linear relationship to the product of electric and magnetic fields for $L_z=30$.}\n\\label{fig:thetaLz}\n\\end{figure}\n\nFirst, to verify our method, we study $\\theta$ as a function of TI thickness $L_z$ for the fixed magnetic field $B=\\pi\/5$ corresponding to $p=1$ and $q=10$, see the main panel of Fig.~\\ref{fig:thetaLz}. The data show that $\\theta$ is close to $\\pi$ at large $L_z$, whereas it decreases notably below $L_z\\approx10$. As $L_z$ approaches 1, the parameter $\\theta$ almost vanishes, which means that in very thin TI films the ME effect disappears. When the TI thickness is very small, it is expected that its surface wave functions hybridize with each other \\cite{Linder09,Essin2009}. To confirm it, we calculate the spacial profiles of the wave functions in $z$ direction for the surface states.  At the $\\Gamma$ point $(k_x=k_y=0)$, they are calculated for several values of $L_z$ and shown in Fig.~\\ref{fig:LzComb}. We observe that for $L_z=30$ the wave functions are strongly localized near the TI surfaces, whereas overlap of these wave functions increases with decreasing TI thickness. At $L_z=4$ the top and bottom wave functions almost completely overlap. This clearly indicates that the reduction of the ME effect is associated with the hybridization of the surface wave functions. From Fig.~\\ref{fig:thetaLz}, the value of $L_z$ below which the ME effect starts to significantly diminish can be estimated as $\\sim 10$. For typical TIs \\cite{Cui2004,Hsieh2008,Chen2009,Xia2009,Hsieh2009}, this value corresponds to $\\sim 10$ nm. \n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1\\columnwidth]{hybridization_combined}\n\\caption{(Color online) The hybridization of surface wave functions $\\Psi$ for TI thicknesses $L_z=4,7,10$, and $30$. The red dashed line corresponds to the bottom TI surface wave function and the blue solid line corresponds to the top surface wave function.}\n\\label{fig:LzComb}\n\\end{figure}\n\nSo far, we have considered a TI in a weak electric field. Experimentally, however, ultrahigh electric fields can be achieved by ionic liquid gating \\cite{Checkelsky2012}, and it is therefore interesting to study a nontrivial response of a TI to both strong electric and magnetic fields.  In the presence of a strong magnetic field, Landau levels form on top and bottom TI surfaces. The existence of well-defined Landau levels in TIs has been confirmed by recent experimental observations of the quantum Hall effect \\cite{Brune2011,Chang2013}. In our simulation, we fix for simplicity the Landau levels on the bottom surface, whereas the gating induced electric field, $E_g$, shifts the energy levels on the top surface. As the gate voltage changes from $V_g\\equiv E_gL_z=0$, the Landau level energies on the top surface move. Figure~\\ref{fig:EgLz}(a) shows that Landau levels cross the Fermi level\nat the following gate voltages: $V_{-5},\\,V_{-4},\\,V_{-3},\\ldots,V_{5}$. In the nonlinear regime of strong electric or magnetic fields, the axion electrodynamics corresponding to $\\theta=\\pm\\pi$ in general is not expected to be applicable. To probe this regime, we vary the electric field by small $\\delta \\mathbf{E}$ near $\\mathbf{E}_g$, i. e. the total electric field is $\\mathbf{E}=\\mathbf{E}_g+\\delta\\mathbf{E}$, and study the response to $\\delta\\mathbf{E}\\cdot\\mathbf{B}$. \nThen $\\delta\\mathbf{E}$ has a role of $\\mathbf{E}$ in Eq.~(\\ref{Eq:Enden}) and $\\theta(E_g,B)$ is defined as the overall coefficient describing the ME effect response.\n\nFigure~\\ref{fig:EgLz}(b) shows $\\theta\/\\pi$ as a function of $V_g$. One can see that there are several plateaus where $\\theta\/\\pi$ is approximately quantized in integers $\\{-4,-3,-2,-1,0,1,2,3,4,5\\}$. Every new quantization plateau in Fig.~\\ref{fig:EgLz}(b) occurs at the points where Landau levels cross the Fermi level.\nThe exact quantization would be in agreement with the prediction of Ref.~\\onlinecite{Tse2010} for weak electric fields. \nAs we show below, the deviation from the exact quantization is mostly attributed to the asymmetry of TI surface wave functions due to the application of the large gate voltage. Thus, Fig.~\\ref{fig:EgLz}(b) shows an important fact that although time-reversal symmetry is broken in a 3D TI by a strong magnetic field, $\\theta$ is not arbitrary as it has been claimed in Refs.~\\onlinecite{Coh2011,Sitte2012}, but rather quantized as long as the Fermi level is still within the bulk gap. In accord with this, the two continuous regions of the dependence in Fig.~\\ref{fig:EgLz}(b) at the left and right edges correspond to the cases when the Fermi level enters continuous (valence or conduction) bands.  In the two continuous regions, however, our results only qualitatively show the behavior of the parameter $\\theta$, because we do not consider any screening effects coming from conduction and valence bands that may suppress the $\\theta$-term, and also because beyond the range bounded by $V_{-5}$ and $V_{5}$, see Fig.~\\ref{fig:EgLz}(a), there are multiple crossings of Landau levels very close to each other. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{thetacom}\n\\caption{(Color online) (a) The TI energy levels (at $\\Gamma$ point) as a function of gate voltage $V_g$ measured in units of $E_{g}L_z$. The surface Landau levels cross the Fermi level. The red lines represent the surface Landau levels and the blue line is the Fermi level. (b) The ME response quantization in a TI vs $V_g$. }\n\\label{fig:EgLz}\n\\end{figure}\n\nTo investigate the hypothesis that the lack of exact quantization in $\\theta$ is attributed to the asymmetry of TI surface wave functions due to gating, we repeat the same calculation for a thicker TI film. The details of the quantization of $\\theta$ as a function of gate voltage $V_g$ are shown in Fig.~\\ref{fig:Lz10vsLz20} for two different TI film thicknesses. The results for $L_z=10$ are shown by blue circles and for $L_z=20$ by red triangles. At a fixed gate voltage $V_g$ ($=E_{g}L_{z}$), applied electric field for $L_z=10$ case is two times stronger than that for $L_z=20$. The comparison shows that when $L_z$ increases two times the slope of each quantization plateau becomes two times smaller. This indicates that for thick enough TI films the plateaus should become horizontal, because in this case the asymmetry of surface wave functions can be disregarded since their overlap is too small. As a result, for the thick films, $\\theta$ depends only on the surface states, but not the bulk states. When the bulk gap is not destroyed, $\\theta$ is determined by the surfaces and thus quantized as the Hall conductivity is quantized, i.e.,\n\\begin{equation}\n\\frac{\\theta}{\\pi} = \\nu_t -\\nu_b,\n\\end{equation}\nwhere $\\nu_{t(b)}$ are the Landau level filling factors for the top (bottom) TI surfaces. In the case of zero applied gate voltage,  $\\nu_{t}=-1\/2$, $\\nu_{b}=1\/2$ and therefore $|\\theta\/\\pi|=1$, c.f. Fig.~\\ref{fig:thetaLz}. \n\nBy applying a magnetic field to the TI model, Eq.~(\\ref{eq:Wlsn}), we break time-reversal symmetry and as a result the system crosses from symmetry class AII, where $\\theta$ is quantized $0$ or $\\pi$, to class A where in general $\\theta$ can be arbitrary. Thus we show that depending on the perturbations (orbital or Zeeman magnetic fields or electric field), $\\theta$ is still quantized in class A for our model. The quantization rules can be rather nontrivial depending on a combination of perturbations, however, we can successfully explain these rules based on the Landau level physics.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.97\\columnwidth]{thetaLz10vsLz20}\n\\caption{(Color online) The quantization of $\\theta$ as a function of gate voltage $V_g$ for $L_z=10$ (blue circles) and $L_z=20$ (red triangles). The comparison shows that when the TI thickness $L_z$ increases two times the slope of each quantization plateau becomes two times smaller.}\n\\label{fig:Lz10vsLz20}\n\\end{figure}\n\nFinally, we study the effects of bulk magnetic interaction on the ME effect. The Zeeman type interaction between the electron spin and applied magnetic field is described by \n\\begin{eqnarray}\n\\label{minus}\n\\mathcal{H}'&=&\\mu_BB_z\n\\begin{pmatrix}\ng_1\\sigma_z&0\\\\\n0&g_2\\sigma_z\n\\end{pmatrix},\n\\end{eqnarray}\nwhere $g_1$ and $g_2$ are the effective $g$ factors depending on the TI materials. In most of cases, for materials such as Bi$_2$Se$_3$ and Bi$_2$Te$_3$, $g_1$ and $g_2$ have opposite signs \\cite{Liu2010}. Therefore, as a simplified model, we consider the case of $g_2=-g_1$. In addition, we study the influence of the exchange interaction between the electron spins and magnetic impurities that can be introduced by the appropriate doping of the TI bulk. Within the virtual crystal approximation, this interaction has the form\n\\begin{eqnarray}\n\\label{plus}\n\\mathcal{H}'&=&J\\overline{M}_z\n\\begin{pmatrix}\n\\sigma_z&0\\\\\n0&\\sigma_z\n\\end{pmatrix},\n\\end{eqnarray}\nwhere $J$ is the exchange interaction constant and $\\overline{M}_z$ is the mean value of magnetic moments in the bulk. These two interactions, Eqs.~(\\ref{minus}) and (\\ref{plus}), can be covered by two simple model Hamiltonians: $\\mathcal{H}'=b_b\\Sigma_z^{\\pm}$ with $\\Sigma_z^{\\pm}=\\mathrm{diag}(\\mathbf{\\sigma}_z, \\pm\\mathbf{\\sigma}_z)$ and $b_b$ being the bulk spin constant. The sign ``$+$'' in $\\Sigma_z^+$ indicates bulk electrons coupling with magnetization, while ``$-$'' sign indicates coupling with the external magnetic field.  \n\nThe calculation results for magnetic field $B=\\pi\/10$ are shown in Fig.~\\ref{fig:bulkspin}, where the red and blue curves correspond to $\\Sigma_z^-=\\mathrm{diag}(\\mathbf{\\sigma}_z,-\\mathbf{\\sigma}_z)$, red is for $L_z=15$ and blue is for $L_z=10$.  The black and green curves correspond to $\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ case, with the black for $L_z=15$ and green for $L_z=10$. One can see from Fig.~\\ref{fig:bulkspin} that $\\theta\/\\pi=1$ persists approximately up to $b_b=1$, which corresponds to the TI regime. At $b_b=|m|=1$, the system crosses to the Weyl semimetal regime \\cite{Wan2011,Burkov2011} for $\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ case and therefore no quantization of $\\theta$ beyond $b_b=1$ is expected \\cite{Zyuzin2012}. These results show remarkable robustness of the ME effect in TI phase even for very strong magnetic fields or high concentration of magnetic impurities that affect the bulk structure of the TI. It seems to be irrespective of microscopic details of coupling with the magnetic impurities or field, since it works equally well in the TI regime for the toy models with both $\\pm$ signs.\n \n\\begin{figure}\n\\centering \n\\includegraphics[width=0.97\\columnwidth]{bulkspin}\n\\caption{(Color online) The bulk spin interaction influence on the ME effect. The triangles and circles represent $\\Sigma_z^+=\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ case, whereas ``$\\times$'' and ``$+$'' symbols represent $\\Sigma_z^-=\\mathrm{diag}(\\mathbf{\\sigma}_z,-\\mathbf{\\sigma}_z)$ case.}\n\\label{fig:bulkspin}\n\\end{figure}\n\n\\section{Summary}\n\nWe have shown that the ME effect in TIs survives even in both strong magnetic and electric fields even though time-reversal symmetry is broken. We have described the ME effect  beyond the linear response regime and have confirmed that it is robust but takes a fascinating form of multiple quantization related to Landau levels crossing of the Fermi level. In addition, the influence of the TI surface hybridization has been studied in very thin TI films, describing how the ME effect vanishes in the limit of very thin TIs.  Furthermore, the ME effect is shown to persist even for strong bulk interactions with magnetic field or magnetic impurities.  \n\n\\acknowledgments \n\nWe would like to thank the staff of the Center for Computational Materials Science at the IMR, Tohoku University for their support of the supercomputing facilities. K.N. was supported by Grant-in-Aid for Scientific Research (Nos. 24740211 and 25103703) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. O.A.T. acknowledges support  by the Grants-in-Aid for Scientific Research (Nos. 25800184 and 25247056) from the MEXT, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Nodal components of random functions}\n\nGiven a ``nice\" function\n$f:{\\mathbb R}^{d}\\rightarrow{\\mathbb R}$, $d\\ge 2$, or $f:\\M\\rightarrow{\\mathbb R}$ with $\\M$ a compact Riemannian $d$-manifold, the nodal set\nof $f$ is its zero set $f^{-1}(0)$; if $f$ is Morse, then its nodal set is a smooth hypersurface in both the ``Euclidean\" and\n``Riemannian\" contexts.\nIn either scenario, the {\\em nodal components} of $f$ are the connected components of the nodal set, and\nthe {\\em nodal domains} are the (positive or negative) connected components of the {\\em complement} of the nodal set.\nThe most basic question one is interested is in the {\\em nodal count} of $f$, i.e.\nthe total number of the nodal components of $f$, also yielding the total number of nodal domains of $f$ via Euler's identity,\nat least, in the Riemannian context $f:\\M\\rightarrow{\\mathbb R}$ (where the nodal set is of finite\nhypersurface volume, thanks to the assumed compactness of $\\M$).\n\nTo make sense of the analogous question in the Euclidean context $f:{\\mathbb R}^{d}\\rightarrow{\\mathbb R}$, one usually takes a large parameter $R>0$\nand studies the asymptotics of the nodal count of the restriction $f|_{B(R)}$ to the centred radius-$R$ ball $B(R)\\subseteq {\\mathbb R}^{d}$, as\n$R\\rightarrow\\infty$. Other than the nodal count, one might refine the said question by separately counting the nodal components of $f$ belonging to\ne.g. a given diffeomorphism type or otherwise, study the mutual positions of the components (``nestings\") etc.\n\n\\vspace{2mm}\n\nUnderstanding the ``typical\" nature of the nodal structures of {\\em Gaussian random fields}, rather than individual functions,\nis an actively pursued subject within several disciplines, in the last few years in particular.\nLet $F:{\\mathbb R}^{2}\\rightarrow {\\mathbb R}$ (more generally, $F:{\\mathbb R}^{d}\\rightarrow{\\mathbb R}$) be a Gaussian random field, that will be assumed {\\em stationary},\nand $R>0$ be a large parameter.\nThen the number $\\pazocal{N}(F,R)$ of nodal components of $F$ {\\em fully contained} inside $B(R)$ is a random variable; alternatively, one\ncould count those components merely {\\em intersecting} $B(R)$. It was shown by Nazarov-Sodin ~\\cite{NS09,So12,NS15} that, under very\nmild smoothness and non-degeneracy assumptions on the law of $F$, there exists a constant $c_{NS}\\ge 0$ so that\n$\\frac{\\pazocal{N}(F,R)}{\\operatorname{Vol}(B(R))}$ converges to $c_{NS}$ both a.s. and in mean, i.e.\n\\begin{equation}\n\\label{eq:cNS conv mean}\n{\\mathbf E}\\left[\\left| \\frac{\\pazocal{N}(F,R)}{\\operatorname{Vol}(B(R))} - c_{NS}  \\right|\\right] \\rightarrow 0\n\\end{equation}\nas $R\\rightarrow\\infty$. Sarnak-Wigman ~\\cite{SW18}, and Beliaev-Wigman ~\\cite{BW17} further developed the techniques\ndue to Nazarov-Sodin by considering the more refined questions of separate nodal counts belonging to particular diffeomorphism types or given hypersurface measure.\n\n\n\\subsection{Random band-limited functions on smooth manifolds} \\label{s:band_lim}\n\nRather than for its own intrinsic interest, this Euclidean scenario above serves as a ground state for the Riemannian one, namely,\nas its scaling limit. Here we consider\na sequence (``ensemble\") $\\{f_{l}\\}_{l\\in\\Lc}$, of smooth random Gaussian fields $f_{l}:\\M\\rightarrow{\\mathbb R}$, satisfying a natural scaling\nproperty, with the scaling parameter $l$ lying in some countable set $\\Lc$, and our objective is to study the distribution\nof the {\\em total} number of nodal components of $f_{l}$ as $l\\rightarrow\\infty$, their typical topology, geometry, relative positions,\nand other important properties. A particularly important such ensemble,\nmotivating the work ~\\cite{SW18}, is\nthe ensemble of band-limited functions, depending on a fixed number $\\alpha\\in [0,1]$. This includes the important ensembles of\nrandom degree-$l$ spherical harmonics (see \\S\\ref{sec:spher harm RMW} below), and Arithmetic Random Waves (\\S\\ref{sec:ARW} below).\n\n\nIt is well-known that, since we assumed $\\M$ to be a smooth, compact, Riemannian $d$-manifold, the space $L^{2}(\\M)$ of square-summable functions on $\\M$\nhas an orthonormal basis $\\{\\varphi_{j}\\}_{j=1}^{\\infty}$ consisting of Laplace eigenfunctions, i.e.\n\\begin{equation}\n\\label{eq:Helmholts}\n\\Delta \\phi_{j}+t_{j}^{2}\\phi_{j} = 0,\n\\end{equation}\nwhere $\\Delta$ is the Laplace-Beltrami operator on $\\M$ acting on $L^{2}$, and $\\{t_{j}\\}_{j\\ge 0}$ is its purely discrete spectrum\n$$0=t_{0}\\le t_{2}\\le\\ldots,$$ satisfying $t_{j}\\rightarrow\\infty$. For a ``band\" $\\alpha\\in [0,1)$ and spectral\nparameter $T>0$ (with the intention of taking the limit $T\\rightarrow\\infty$), we define ~\\cite{SW18} the random\nband-limited functions to be\n\\begin{equation}\n\\label{eq:fT band lim alpha<1}\nf_{T}(x)=f_{\\alpha;T}(x)=\\frac{1}{\\left|\\{j:\\alpha\\cdot T<t_{j}<T\\}\\right|^{1\/2}}\\sum\\limits_{\\alpha\\cdot T<t_{j}<T}c_{j}\\varphi_{j}(x),\n\\end{equation}\nwhere the $c_{j}$ are i.i.d. standard Gaussian. For $\\alpha=1$ the above definition usually makes no sense, as the summation on the r.h.s.\nof \\eqref{eq:fT band lim alpha<1} is typically one summand (or empty), so in this case we interpret the summation as\n\\begin{equation}\n\\label{eq:fT band lim alpha=1}\nf_{1;T}(x)=\\frac{1}{\\left|\\{j:T-\\eta(T)<t_{j}<T\\}\\right|^{1\/2}}\\sum\\limits_{T-\\eta(T)<t_{j}<T}c_{j}\\varphi_{j}(x),\n\\end{equation}\nwith the convention $\\eta(T)=o_{T\\rightarrow\\infty}(T)$ but $\\eta(T)\\rightarrow\\infty$.\n\nAlternatively, $f_{T}(\\cdot)$ is the (uniquely defined) centred Gaussian random field of covariance\n\\begin{equation}\n\\label{eq:KT covar func}\nK_{T}(x,y)=K_{\\alpha;T}:={\\mathbf E}[f_{\\alpha;T}(x)\\cdot f_{\\alpha;T}(y)] = \\sum \\phi_{j}(x)\\cdot\\phi_{j}(y),\n\\end{equation}\n$x,y\\in\\M$, where the summation is over the same energy window as \\eqref{eq:fT band lim alpha<1} for $\\alpha<1$\n(resp. \\eqref{eq:fT band lim alpha=1} for $\\alpha=1$); we identify $K_{T}(\\cdot,\\cdot)$ as the {\\em spectral projector}\nfor the corresponding energy window. The kernel $K_{T}(\\cdot, \\cdot)$ and its derivatives possess scaling limits once appropriately scaled by $T$,\nsee \\S\\ref{sec:proofs outline} for more details, and in particular \\eqref{eq:covar func band lim der conv}.\n\n\\subsubsection{Random spherical harmonics and Berry's Random Wave Model}\n\\label{sec:spher harm RMW}\n\nIt is well-known that the Laplace eigenfunctions on the $2$-sphere $S^{2}\\subseteq {\\mathbb R}^{3}$, that is the spherical harmonics, are restrictions of harmonic polynomials of some degree $l\\ge 1$. The space of degree-$l$ spherical harmonics is of\ndimension $2l+1$, so given a number $l$ we may obtain an $L^{2}$-orthonormal basis $\\Phi_{l}:=\\{\\eta_{l;1},\\ldots,\\eta_{l;2l+1}\\}$,\nand define\n\\begin{equation}\n\\label{eq:Tl spher harm}\nT_{l}(x):=\\frac{\\sqrt{4\\pi}}{\\sqrt{2l+1}}\\sum\\limits_{m=1}^{2l+1}a_{m}\\eta_{l;m}(x)\n\\end{equation}\nwith $\\{a_{m}\\}_{m=1}^{2l+1}$ standard Gaussian i.i.d.; the law of the random spherical harmonics $T_{l}(\\cdot)$ is\ninvariant w.r.t. the choice of $\\Phi_{l}$. The random fields $T_{l}(\\cdot)$ are the Fourier components of every\nrotation invariant random field on $S^{2}$, hence its importance in a variety of disciplines within mathematics,\nphysics and cosmology.\n\n\nEquivalently, the random spherical harmonic $T_{l}(\\cdot)$ is a centred\nGaussian random field defined via the covariance function $$r_{l}(x,y)=P_{l}(\\cos d(x,y)),$$ $x,y\\inS^{2}$,\n$d(\\cdot,\\cdot)$ is the spherical geodesic distance, and $P_{l}(\\cdot)$ is the degree-$l$ Legendre polynomial.\nSince, by the standard Hilb's asymptotics ~\\cite{Szego}, $$P_{l}(\\cos\\theta)\\approx J_{0}((l+1)\\theta),$$\nthe geometry of the nodal line of $T_{l}$ could be compared to the geometry of the nodal line of the stationary isotropic\nrandom field on ${\\mathbb R}^{2}$ defined by the covariance function $J_{0}(\\|x\\|)$, usually referred to ``Berry's Random Wave Model\"\n(RWM), for it is postulated ~\\cite{Berry 1977} to be a stand-in for deterministic Laplace eigenfunctions on generic chaotic surfaces.\n\n\\subsubsection{Arithmetic random waves}\n\\label{sec:ARW}\n\nThe Arithmetic Random Waves are random Gaussian toral Laplace eigenfunctions $\\{f_{n}:\\Tb^{2}\\rightarrow{\\mathbb R}\\}_{n\\in S}$,\nwhere $\\Tb^{2}={\\mathbb R}^{2}\/{\\mathbb Z}^{2}$ is the $2$-dimensional standard torus, and $S=\\{a^{2}+b^{2}:\\:a,b\\in{\\mathbb Z}\\}$ is the set of all numbers\nexpressible as sum of two integer squares. For $n\\in S$ let $$\\Lambda_{n}:=\\{(a,b)\\in {\\mathbb Z}^{2}:\\: a^{2}+b^{2}=n\\}$$ be the set\nof all lattice points lying on the centred circle of radius $\\sqrt{n}$. We may define\n\\begin{equation}\n\\label{eq:fn ARW def}\nf_{n}(x)=\\sum\\limits_{\\lambda\\in\\Lambda_{n}} a_{\\lambda}e(\\langle \\lambda,x  \\rangle),\n\\end{equation}\nwhere $e(t)=e^{2\\pi i t}$, and the $\\{a_{\\lambda}\\}_{\\lambda\\in\\Lambda_{n}}$ are complex standard Gaussian i.i.d., save to the condition\n$a_{-\\lambda}=\\overline{a_{\\lambda}}$, so that $f_{n}$ are real valued.\n\nThe random fields $f_{n}$ are the ``Arithmetic Random Waves\"\nthat serve as a motivation to our research, following the work ~\\cite{RudWig2018}; $f_{n}$ should be compared\nto the band-limited functions \\eqref{eq:fT band lim alpha=1}, with $n\\approx T^{2}$\n(and, thanks to the spectral multiplicity, we may take an infinitesimal energy window).\nEquivalently to the explicit definition \\eqref{eq:fn ARW def},\n$f_{n}$ could be defined as the stationary centred Gaussian random field on $\\Tb^{2}$ with the covariance function\n\\begin{equation}\n\\label{eq:ARW covar def}\nr_{n}(x)={\\mathbf E}[f_{n}(y)\\cdot f_{n}(y+x)] = \\frac{1}{|\\Lambda_{n}|}\\sum\\limits_{\\lambda\\in\\Lambda_{n}}\ne(\\langle \\lambda,x \\rangle),\n\\end{equation}\nor via the spectral measure of $f_{n}$, i.e. the atomic measure\n\\begin{equation}\n\\label{eq:ARW spec measure rho}\n\\rho_{n}:=\\frac{1}{|\\Lambda_{n}|}\\sum\\limits_{\\lambda\\in\\Lambda_{n}}\\delta_{\\lambda\/\\sqrt{n}}\n\\end{equation}\non $S^{1}\\subseteq{\\mathbb R}^{2}$.\n\n\\subsection{Direction distribution and statement of the principal result}\n\n\\subsubsection{Direction distribution on $\\mathbb{T}^d$}\n\nIn $2$d Rudnick and Wigman ~\\cite{RudWig2018} proposed to study the {\\em ``direction distribution\"} of the nodal line, a different quantity they\nintroduced for functions defined on the $2$d standard torus $\\Tb^{2}={\\mathbb R}^{2}\/{\\mathbb Z}^{2}$, also related to ~\\cite{Swerling}.\nLet $\\zeta\\inS^{1}$ be\na direction, and $f:\\Tb^{2}\\rightarrow{\\mathbb R}$ any ``nice\" function. The number\n\\begin{equation}\n\\label{eq:Nzeta torus}\n\\mathcal{N}_{\\zeta}(f):= \\left|\\left\\{x\\in \\Tb^{2}:f(x)=0,\\, \\frac{\\nabla f(x)}{\\|\\nabla f(x)\\|}=\\pm \\zeta  \\right\\} \\right|\n\\end{equation}\nis the number of\npoints on the nodal line of $f$ normal to $\\zeta$; equivalently, $\\mathcal{N}_{\\zeta}(f)$ is the number of nodal points of $f$ in direction\n$\\pm\\xi:=\\zeta^{\\perp}$. The direction distribution carries a lot of information on the nodal line of $f$; for example, the number of\nnodal components of $f$ on $\\Tb^{2}$ essentially majorizes ~\\cite{KuWiAdv} the nodal count of $f$ for every $\\zeta$,\nas every nodal component of $f$ of trivial\nhomology contains at least two points tangent to $\\xi$, and it is usually easy to control the contribution of all the other\ncomponents.\n\nFor the $\\mathcal{N}_{\\zeta}(\\cdot)$ corresponding to\ntoral Laplace eigenfunctions Rudnick and Wigman ~\\cite{RudWig2018} gave the optimal upper bounds, as well as evaluated their total expected number for the associated Gaussian random model, ``Arithmetic Random Waves\" \\eqref{eq:fn ARW def}, precisely, appealing to the Kac-Rice method. Their result, however, does not allow for the separation of the $\\xi$-tangencies into nodal components of $f$, i.e. how many points on a\n{\\em given component} represent $\\xi$, or how many nodal components of $f$ represent the direction $\\xi$ {\\em precisely} $k$ times.\n\n\\subsubsection{Direction distribution on curved surfaces}\n\nThe purpose of this manuscript is two-fold. First, we propose, what it seems, a natural generalisation of the direction distribution concept for smooth manifolds with curvature, and study its properties for the random band-limited functions \\eqref{eq:fT band lim alpha<1} (and\n\\eqref{eq:fT band lim alpha=1}).\nSecond, in this random setting (random band-limited functions, including, in particular Arithmetic Random Waves) we aim at refining\nthe said results due to Rudnick-Wigman by keeping separate accounts for the number of $\\xi$-tangencies on individual components rather than merely the total number of $\\xi$-tangencies (though the latter is useful for our purposes).\n\nLet us now introduce a space of vector fields that play a central role in our article. We let $\\mathcal{V}(\\M)$ be the class of all $C^{\\infty}$-smooth vector fields on $\\M$, with finitely many zeros; the class $\\mathcal{V}(\\M)$ is non-empty for\nevery smooth $\\M$ by Lemma \\ref{l:vfs_finite_zeroes}. First, given $f:\\M\\rightarrow{\\mathbb R}$ a nice function, and $V\\in\\mathcal{V}(\\M)$, we introduce\n\\begin{equation*}\n\\mathcal{N}_{V}(f)= \\#\\{x:\\:V(x)\\ne 0,\\, f(x)=Vf(x)=0\\},\n\\end{equation*}\nthe announced generalisation of the ``flat\" direction distribution, with the variable direction $V(x)$ in place of\n$\\xi=\\zeta^{\\perp}$ in \\eqref{eq:Nzeta torus}. Other than for the torus, a precise asymptotic expression for the total number of\n$V$-tangencies for the random spherical harmonics \\eqref{eq:Tl spher harm} was recently obtained \\cite{E18} by the first author.  Related results were obtained by Dang and Rivi\\`ere \\cite{DaRiJEMS} following some substantial and general results of Gayet and Welschinger \\cite{GaWeJussieu}.\n\n\n\nNow we separate the tangency counts for individual nodal components of $f$: for a nodal component $\\gamma\\subseteq f^{-1}(0)$ define\nthe number of tangencies w.r.t. $V$,\n\\begin{equation*}\nT_{V}(\\gamma)= \\#\\{x\\in \\gamma:\\: V(x)\\ne 0,\\, f(x)=Vf(x)=0\\},\n\\end{equation*}\nand\n\\begin{equation*}\n\\pazocal{N}_{V}(f,k) :=\\#\\{\\gamma  \\subseteq f^{-1}(0):\\: T_{V}(\\gamma)=k\\}\n\\end{equation*}\nis the total number of nodal components of $f$ with precisely $k$ tangencies w.r.t. $V$,\nso that we have the\nidentity\n\\begin{equation}\n\\label{eq:tot numb tang=sum mean}\n\\mathcal{N}_{V}(f) = \\sum\\limits_{k=1}^{\\infty}k\\cdot \\pazocal{N}_{V}(f,k).\n\\end{equation}\nFinally, we let\n\\begin{equation*}\n\\pazocal{N} (f) = \\sum\\limits_{k=0}^{\\infty}\\pazocal{N}_{V}(f,k)\n\\end{equation*}\nbe the total number of the nodal components of $f$.\n\n\\subsubsection{Statement of the principal result}\n\nOur main result concerns the asymptotic law for the direction distribution measure corresponding to the band-limited functions\n$f_{\\alpha;T}(\\cdot)$ in \\eqref{eq:fT band lim alpha<1}-\\eqref{eq:fT band lim alpha=1}.\nFollowing the approach of Sarnak-Wigman ~\\cite{SW18},\nwe may incorporate, or ``encapsulate\", all the individual counts $\\pazocal{N}_{V}(f,\\cdot)$ into a single (random) probability measure,\nthe ``direction distribution measure\",\n\\begin{equation}\n\\label{eq:mu(f) ddm def}\n\\mu_{f}(V) =\\frac{1}{\\pazocal{N}(f)}\\sum\\limits_{\\gamma\\subseteq f^{-1}(0)}\\delta_{T_{V}(\\gamma)} =\\frac{1}{\\pazocal{N}(f)}\\sum\\limits_{k=0}^{\\infty}\\pazocal{N}_{V}(f,k)\\cdot\\delta_{k},\n\\end{equation}\non ${\\mathbb Z}_{\\ge 0}$.\nGiven two probability measures $\\mu_{1},\\mu_{2}$ on ${\\mathbb Z}$ we will use the total variation distance function\n\\begin{equation}\n\\label{eq:D tot var dist def}\n\\mathcal{D}(\\mu_{1},\\mu_{2}) = \\sup\\limits_{F\\subseteq {\\mathbb Z}_{\\ge 0}} |\\mu_{1}(F)-\\mu_{2}(F)|.\n\\end{equation}\n\n\n\\begin{theo}\n\\label{thm:main meas conv}\n\nGiven $\\alpha\\in [0,1]$, there exists a (deterministic) probability measure $\\mu_{\\alpha}$ on ${\\mathbb Z}_{\\ge 0}$,\nsupported on the positive even integers $2{\\mathbb Z}_{>0}$, so that for all $V\\in\\mathcal{V} (\\M)$ and every $\\epsilon>0$,\n\\begin{equation}\n\\label{eq:main meas conv}\n\\lim\\limits_{T\\rightarrow\\infty}\\pazocal{P}r\\left(\\mathcal{D}(\\mu_{f_{\\alpha;T}}(V),\\mu_{\\alpha})>\\epsilon\\right)=0,\n\\end{equation}\nwhere $\\mathcal{D}(\\cdot,\\cdot)$ is the total variation distance \\eqref{eq:D tot var dist def}.\n\n\\end{theo}\n\n\\subsubsection{On the principal result}\n\nThe band-limited functions \\eqref{eq:fT band lim alpha<1}-\\eqref{eq:fT band lim alpha=1} is a particular case of an aforementioned ensemble of Gaussian random fields possessing a natural scaling by the wave number $T$, see \\S\\ref{sec:proofs discussions} below, and, in particular \\eqref{eq:covar func band lim der conv}.\nOur proofs are applicable for such ensembles with scaling,\nin a more general scenario than merely the band-limited functions, as long as an analogue of \\eqref{eq:covar func band lim der conv} holds,\nwith $T$ replaced by the scaling parameter (see e.g. the important application on Arithmetic Random Waves from \\S\\ref{sec:ARW},\nin \\S\\ref{sec:appl ARW} below).\nIn general, in a scenario like \\eqref{eq:main meas conv}, when probability measures (on ${\\mathbb Z}_{\\ge 0}$) weak-$*$\nconverge to a limit measure $\\mu_{\\alpha}$, it forces the limit measure to have $\\mu_{\\alpha}({\\mathbb Z}_{\\ge 0}) \\le 1$\nby Fatou's Lemma, and $\\mu_{\\alpha}({\\mathbb Z}_{\\ge 0}) < 1$ would mean escape of probability to infinity. As part\nof Theorem \\ref{thm:main meas conv}, we will be able to rule this kind of losses of mass, cf. \\cite[Theorem 1.1]{SW18}.\n\nWe stress that, despite the fact that the measures $\\mu_{f_{\\alpha;T}}$ are {\\em random}, the limit measure $\\mu_{\\alpha}$\nis {\\em deterministic}, and, notably, in addition, $\\mu_{\\alpha}$ is {\\em independent} of $V$.\nThanks to the asymptotic scaling \\eqref{eq:covar func band lim der conv}, we will be able to compare the nodal counts $\\pazocal{N}_{V}(f,k)$ to\nthe nodal counts of some Gaussian isotropic random field $\\mathfrak{g}_{\\alpha}(\\cdot)$ on ${\\mathbb R}^{2}$\n(to be defined in \\S\\ref{sec:proofs outline}),\nwith $V\\equiv \\xi$ a constant vector field,\nup to an admissible error term. The measures $\\mu_{\\alpha}$ are $V$-independent, since, by the isotropic property of $\\mathfrak{g}_{\\alpha}(\\cdot)$,\nthe corresponding distribution of $\\xi$-tangencies counts is independent of $\\xi$.\nThe support of $\\mu_{\\alpha}$ is a manifestation of the fact that the number of $\\xi$-tangencies\nfor a simple smooth planar curve is necessarily {\\em even}, unless a degeneracy occurs, with probability $0$ (though,\nstrictly speaking, it is possible to construct simple curve with an odd number of $\\xi$-tangencies). However,\nfor $k>0$ even it is possible to construct a simple curve with number of $\\xi$-tangencies being precisely $k$, that occurs\nas a nodal component of the scaled random field with positive density (the most subtle or delicate case being $\\alpha=1$, cf.\n~\\cite[Proposition $5.3$]{SW18} and ~\\cite{CanSar18}).\nWe refer the reader to \\S\\ref{sec:proofs outline} for more details on proofs and intuitions as of why these peculiarities hold.\n\n\\subsubsection{$T^*\\M$ and conormal cycles}\nWe conclude this section by emphasizing that the microlocal geometry of nodal sets, or more specifically the interaction between ``cotangent\/conormal spaces\" to nodal sets and various other geometric quantities of these submanifolds, is a natural topic of study. The work of Dang and Rivi\\`ere gives asymptotics pertaining to the equidistribution (in $T^* \\M$) of ``conormal cycles\" for $f_T$ on general compact manifolds \\cite{DaRiJEMS}, with conormal cycles being phase-space quantities related to the conormal bundle of $f^{-1}_T(0)$, namely $N^*(\\{ f_T = 0 \\}) \\subset T^* \\M$.  In other words, for example in odd dimensions, the authors show that the expected value of a natural and re-scaled volume measure on the conormal bundle to the nodal set $N^*(\\{ f_T = 0 \\})$ converges to a uniform volume measure on $T^*\\M$ as $T \\rightarrow \\infty$.\nIt is worth noting that Dang-Rivi\\`ere were themselves motivated by the work \\cite{GaWeJussieu} on expected Betti numbers for nodal sets associated to elliptic pseudodifferential operators; a local refinement of their lower bound was very recently obtained by Wigman \\cite{Wig19}.\n\n\n\\subsection{Applications to arithmetic random waves}\n\n\\label{sec:appl ARW}\n\n\nRecall the Arithmetic Random Waves $f_{n}$ in \\eqref{eq:fn ARW def} for $n\\in S$, the set of numbers expressible as sum of two squares, the corresponding covariance function \\eqref{eq:ARW covar def},\nand their spectral measure \\eqref{eq:ARW spec measure rho}.\nIt is well-known ~\\cite{KaKo,ErHa},\nthat for a ``generic\" (density-$1$) sequence $\\{n\\}\\subseteq S$, the measures $\\rho_{n}$ equidistribute\non $S^{1}$, i.e.\n\\begin{equation}\n\\label{eq:rhon equidistr}\n\\rho_{n}\\Rightarrow \\frac{d\\theta}{2\\pi}.\n\\end{equation}\nHowever, there exist ~\\cite{Cil,KuWiMAnn} other weak-$*$ partial limits of the sequence $\\{\\rho_{n}\\}_{n\\in S}$.\nIt was established ~\\cite{KKW,So12,KuWiAdv}, that for the nodal structures of $f_{n}$ to exhibit a limit law,\nit is essential to divide $S$ into\nsubsequences whose corresponding $\\rho_{n}$ obey a limiting distribution, i.e. take a subsequence $\\{n\\}\\subseteq S$\nso that $\\rho_{n}\\Rightarrow\\tau$ for some probability measure $\\tau$ on $S^{1}$. In this case\nthe corresponding covariance functions $r_{n}$ converge uniformly locally to $r$, the (inverse) Fourier transform of\n$\\tau$, in the sense of \\eqref{eq:covar func band lim der conv}.\n\n\nLet $\\pazocal{N}(f_{n})$ be the total number of nodal components of $f_{n}$. A combination of the techniques by Nazarov-Sodin\n~\\cite{So12,NS15} and ~\\cite{KuWiAdv} implies that for every sequence $\\{n\\}\\subseteq S$ so that\n\\begin{equation*}\n\\rho_{n}\\Rightarrow \\tau\n\\end{equation*}\nfor some probability measure $\\tau$ on $S^{1}$, there exists a constant $c_{NS}(\\tau)\\ge 0$ so that\n\\begin{equation*}\n{\\mathbf E}[\\pazocal{N}(f_{n})]= c_{NS}(\\tau)\\cdot n +o_{n\\rightarrow\\infty}(1).\n\\end{equation*}\nMoreover, $c_{NS}(\\tau)=0$, if and only if $\\tau$ is either the ``Cilleruelo\" measure\n\\begin{equation*}\n\\tau_{0}=\\frac{1}{4}\\left(\\delta_{\\pm 1}+\\delta_{\\pm i}\\right),\n\\end{equation*}\ni.e. the atomic probability measure supported on the $4$ antipodal points $\\pm 1$ and $\\pm i$ (viewing ${\\mathbb R}^{2}\\cong {\\mathbb C}$),\nor its tilted by $\\pi\/4$ variant\n\\begin{equation*}\n\\widetilde{\\tau_{0}}=\\frac{1}{4}\\left(\\delta_{\\pm \\pi\/4}+\\delta_{3\\pi\/4}\\right).\n\\end{equation*}\nThe probability measures $\\tau_{0}$ and $\\widetilde{\\tau_{0}}$ on $S^{1}$ are the only measures\nsatisfying all the underlying symmetries (invariance w.r.t. rotation by $\\pi\/2$ and complex conjugation) that\nare supported on $4$ points only.\n\n\\vspace{2mm}\n\nThough not applicable as a black box, our Theorem \\ref{thm:main meas conv} (or, rather, the associated techniques)\nyields the following extension of the said results\nconcerning the $V$-direction distribution measures $\\mu_{f_{n}}(V)$ of $f_{n}$\nfor some $V\\in\\mathcal{V}(\\Tb^{2})$ (for example, $V$ could be the constant vector field\n$V\\equiv \\xi$), defined as in \\eqref{eq:mu(f) ddm def}.\nLet $V\\in\\mathcal{V}(\\Tb^{2})$, and $\\{n\\}\\subseteq S$ be a subsequence so that\n\\begin{equation*}\n\\rho_{n}\\Rightarrow \\tau\n\\end{equation*}\nfor some $\\tau\\ne \\tau_{0},\\widetilde{\\tau_{0}}$.\nThen there exists a deterministic probability measure $\\mu_{\\tau;V}$ on ${\\mathbb Z}_{\\ge 0}$, so that\n\\begin{equation*}\n\\mathcal{D}\\left( \\mu_{f_{n}}(V), \\mu_{\\tau;V}  \\right) \\rightarrow 0,\n\\end{equation*}\nas $n\\rightarrow\\infty$. In addition, the support of $\\mu_{\\tau;V}$ is contained in the set of positive even\nintegers $2{\\mathbb Z}_{>0}$.\n\nWe emphasize that for the Arithmetic Random Waves, unlike the band-limited functions in the statement of Theorem \\ref{thm:main meas conv},\nthe limit direction distribution\nmeasure $\\mu_{\\tau;V}$ does depend on $V$, and we are not aware whether it is possible to explicate the dependency of $\\mu_{\\tau;V}$ on\n$V$; this is a by-product\nof the fact that the spectral measure $\\tau$ is not invariant w.r.t. rotations. Moreover, the support might fail to attain the whole of $2\\cdot{\\mathbb Z}_{>0}$,\ndepending on $\\tau$ (and possibly $V$). However, for the generic case \\eqref{eq:rhon equidistr},\nthe conclusions of Theorem \\ref{thm:main meas conv} hold true in their full strength with $\\mu_{d\\theta\/2\\pi}=\\mu_{1}$ same\nas in \\eqref{eq:main meas conv}, and, in particular,\n$\\mu_{d\\theta\/2\\pi}=\\mu_{d\\theta\/2\\pi;V}$ does not depend on $V$, and its support equals to precisely $2{\\mathbb Z}_{>0}$.\n\n\\subsection*{Acknowledgements}\n\nWe warmly thank Ze\\'{e}v Rudnick for suggesting this collaboration, his interest in our work and his support,\nand to Manjunath Krishnapur and Lakshmi Priya for a very enlightening discussion during the conference\n``Random Waves in Oxford\" that took place in June $2018$. Thanks also to Dmitry Beliaev, Matthew de Courcy-Ireland, Gabriel Rivi\\`ere, Nguyen Viet Dang, and Damien Gayet for various discussions on their own work which themselves gave us insights.  A part of the presented research was\nconducted during both authors' visit at the Banff International Research Station (BIRS) in July $2019$,\nand we are grateful to them for providing us with a unique opportunity to use the outstanding facilities of BIRS as part of the\nprogramme ``Research in Pairs\". SE was in residence at McGill University as a Visiting Professor for a majority of the writing of this article and warmly thanks the Department of Mathematics for their hospitality. However, the discussions that lead to the writing of this article began while SE was Lecturer in Analysis at Cardiff University; he is grateful for the opportunity to work in a such stimulating environment. The research leading to these results has received funding from the European Research\nCouncil under the European Union's Seventh Framework Programme (FP7\/2007-2013), ERC grant agreement n$^{\\text{o}}$ 335141 (I.W.).\n\n\n\n\\section{Outline of the proofs and discussion}\n\n\\label{sec:proofs discussions}\n\n\\subsection{Outline of the proofs}\n\n\\label{sec:proofs outline}\n\nIt is known\nthat ~\\cite{Lax,Horm,CH1,CH2, S88, SV96}, under the assumptions above, the covariance functions $K_{T}(\\cdot, \\cdot)$ as in \\eqref{eq:KT covar func}\nscales by $T$ around every point of $\\M$, in the following sense.\nFor $x\\in \\M$, and $u,v\\in {\\mathbb R}^{2}$ so that both $\\frac{\\|u\\|}{T},\\frac{\\|v\\|}{T}$ are less than the injectivity\nradius of $x$ define the {\\em scaled} covariance function using the identification $I_{x}:{\\mathbb R}^{2}\\rightarrow T_{x}\\M$:\n\\begin{equation*}\nK_{T;x}(u,v):= K_{T}(\\exp_{x}(I_{x}(u\/T)),\\exp_{x}(I_{x}(v\/T)))\n\\end{equation*}\non $u,v\\in{\\mathbb R}^{2}$, corresponding to the {\\em scaled} random fields\n\\begin{equation}\n\\label{eq:fxT scaled def}\nf_{x,T}(u)=f_{\\alpha;x,T}(u)=f_{T}\\left(\\exp_{x}(I_{x}(u\/T))\\right),\n\\end{equation}\n$u\\in{\\mathbb R}$ ($\\|u\\|$ smaller than the injectivity radius of $x$).\n\nThen, for all $u,v\\in {\\mathbb R}^{2}$ {\\em fixed},\n\\begin{equation}\n\\label{eq:KTx(u,v)->B(|u-v|)}\nK_{T;x}(u,v)\\rightarrow B(\\|u-v\\|),\n\\end{equation}\nlocally uniformly, where\n\\begin{equation}\n\\label{eq:Bu Fourier annulus}\nB(u):= \\frac{1}{|A_{\\alpha}|}\\int\\limits_{A_{\\alpha}}e^{-2\\pi i \\langle u,v\\rangle} dv\n\\end{equation}\nis the Fourier transform of the characteristic function of the annulus $$A_{\\alpha}=\\{v\\in{\\mathbb R}^{2}:\\: \\alpha<\\|v\\|<1\\}$$\n(the unit circle $S^{1}\\subseteq{\\mathbb R}^{2}$ for $\\alpha=1$),\nand $|A_{\\alpha}|$ is the volume of $A_{\\alpha}$ (resp. the length $\\pi$ of $S^{1}$ for $\\alpha=1$).\nExplicitly, \\eqref{eq:KTx(u,v)->B(|u-v|)} means that for every $R>0$,\n\\begin{equation}\n\\label{eq:covar func band lim der conv}\n\\sup\\limits_{\\|u\\|,\\|v\\|<R} \\left| K_{T;x}(u,v)- B_{\\alpha}(\\|u-v\\|)\\right|\\rightarrow 0,\n\\end{equation}\nand the same holds for all derivatives of $K$, where the rate of convergence in \\eqref{eq:covar func band lim der conv}\ndepends on the order of the derivative only.\n\nThe rotation invariant function $B_{\\alpha}(\\cdot)$ defines a stationary isotropic random field\n$\\mathfrak{g}_{\\alpha}:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$, i.e. the covariance\nfunction of $\\mathfrak{g}_{\\alpha}(\\cdot)$ is $${\\mathbf E}[\\mathfrak{g}_{\\alpha}(x)\\cdot \\mathfrak{g}_{\\alpha}(y)]=B_{\\alpha}(\\|x-y\\|).$$\nBy \\eqref{eq:Bu Fourier annulus}, the spectral measure of $\\mathfrak{g}_{\\alpha}$ is the volume measure of $A_{\\alpha}$ for $\\alpha<1$\n(resp. arc length of $S^{1}$ for $\\alpha=1$).\nIntuitively, \\eqref{eq:covar func band lim der conv} means that there exists a {\\em coupling} of the relevant fields, so that\nthe random fields \\eqref{eq:fxT scaled def} converge, in a sense to be made precise below, to $\\mathfrak{g}_{\\alpha}$.\nNote that the random field $\\mathfrak{g}_{1}$ coincides with Berry's\nRandom Waves Model in \\S\\ref{sec:spher harm RMW}.\n\n\\vspace{2mm}\n\n\nWe follow the general strategy of Nazarov-Sodin ~\\cite{So12,NS15} by first establishing the analogous\nresults for the limit random field $\\mathfrak{g}_{\\alpha}$ (while separating the individual $k$) on ${\\mathbb R}^{2}$,\nthen prove that these extend to $\\M$, locally around every point $x\\in \\M$ (restricted to geodesic balls of\nradius commensurable with $1\/T$), and finally glue all the local results on $\\M$ into a global one.\nCounting tangencies poses a number of marked significant challenges as compared to the original setting\nof Nazarov-Sodin, and also ~\\cite{SW18}.\n\nTo pass from the Euclidean setting into the Riemannian one, a high probability ``stable\" event is created\n(``few\" low lying critical points),\nso that, if it occurs, ``most\" of the nodal components cannot disappear or merge while perturbing the sample function,\nand only few new ones can appear. It was noticed in ~\\cite{SW18} that, on the stable event, not only the number of nodal components is (almost) preserved, but also their topologies. However, the number of $V$-tangencies of a component is {\\em not} a topological\nproperty, and so can vary upon an arbitrarily small perturbation of the sample function. We then have to redefine the stable\nevent to account for possible ``non-transversality\" of the tangencies by translating our problem into one surrounding the quantitative transversal intersection between curves. We also have to use a somewhat\ndifferent analysis for proving that the new stable event is of high probability.\n\nThe reason why the support of $\\mu_{\\alpha}$ does not contain odd integers nor $0$ is that, in the Euclidean setup, if a direction $\\zeta$ is fixed, for a closed simple curve to have $0$ or odd number of $\\zeta$-tangencies would force some of the tangencies to be non-transversal, occurring with\nprobability $0$. Since the Riemannian setup will inherit the direction distribution measures from the Euclidean one, it will also induce the analogous results for the band-limited functions (though having a few nodal components with $0$ or odd number of $V$-tangencies is not dismissed a.s.). The proof for the support being exactly $2 {\\mathbb Z}_+$ follows essentially from a $C^2$ analogue of the barrier method as utilized in \\cite{NS15, SW18, CanSar18} in combination with a necessary $C^2$-closeness estimate for transversal intersections established in \\S\\ref{s:local_results}.\n\nAnother new challenge we encountered is that no analysis of the type explained above is possible around the\n(finitely many) points $x\\in \\M$ where $V(x)=0$.\nTo deal with this situation we will excise small radius-$\\rho$ balls around these problematic points, and bound\nthe contribution of their neighbourhoods to our counts, taking $\\rho\\rightarrow 0$ at the very end. Fortunately,\nit is possible to tune all the parameters encountered and use their relations in our favour. Finally, the local computations with Kac-Rice are useful in order to establish that\nthe limit measure $\\mu_{\\alpha}$ is probability, as their means \\eqref{eq:tot numb tang=sum mean} (or, rather, their Euclidean counterparts in\n\\eqref{eq:sum k*nod(k)<=Ncc} below) stay bounded.\n\n\n\n\n\\subsection{Discussion: further questions}\n\n\n\\subsubsection{Deterministic bounds}\n\nYau's conjecture ~\\cite{Yau1,Yau2} states that if a function $\\phi_{j}:\\M\\rightarrow{\\mathbb R}$ satisfies \\eqref{eq:Helmholts}\non a $d$-manifold $\\M$,\nthen its nodal volume, i.e. the hypersurface volume of the nodal set $\\phi_{j}^{-1}(0)$ is commensurable with $t_{j}$, i.e.\n\\begin{equation*}\nt_{j}\\ll_{\\M}\\operatorname{Vol}(\\phi_{j}^{-1}(0)) \\ll_{\\M} t_{j}.\n\\end{equation*}\nYau's conjecture was settled by Br\\\"{u}ning ~\\cite{Br}, Br\\\"{u}ning-Gromes ~\\cite{BrGr}, and Donnelly-Fefferman ~\\cite{DF}\nfor real analytic manifolds (lower and upper bounds), and, more recently,\na breakthrough progress was made towards the general smooth case ~\\cite{Log Mal,Log Lower, Log Upper}. It would be desirable to\nestablish the analogous deterministic bounds for the direction distribution of the type\n\\begin{equation*}\nt_{j}^{2}\\ll_{\\M} \\mathcal{N}_{V}(\\phi_{j}) \\ll_{\\M} t_{j}^{2}\n\\end{equation*}\nfor a $2$-manifold $\\M$, and $V\\in \\mathcal{V}(\\M)$, in some ``generic\" scenario. Other than\nthe aforementioned bounds for the toral Laplace eigenfunctions \\eqref{eq:fn ARW def}\ndue to Rudnick-Wigman ~\\cite{RudWig2018}, appealing to B\\'ezout's Theorem,\nand the case of spherical harmonics\nwhere one may exploit the fact that these are restrictions of polynomials and can therefore also employ B\\'ezout, this problem is entirely open, at least to our knowledge.\n\n\n\\subsubsection{Uniform convergence w.r.t. $V$}\n\nThe convergence \\eqref{eq:main meas conv} of the direction distribution measure to the limit measure is a priori dependent on $V$.\nHowever, one could explicate the proofs to control the corresponding constants uniformly on compact sets of $\\mathcal{V}(\\M)$ with respect to some reasonable topology that takes into account $V$ and finitely many of its derivatives. One would be interested in characterising the geometry of such vector fields.\n\n\n\\subsubsection{Nonconstant vector fields on ${\\mathbb R}^{2}$}\n\nOur proofs below that $\\mu_{\\alpha}$ is independent of $V\\in\\mathcal{V}(\\M)$ are based\non the fact that if $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ is a stationary isotropic Gaussian random field, then\nthe law of $\\pazocal{N}_{\\xi}(F|_{B(R)})$ (and $\\mu_{F|_{B(R)}}(\\xi)$), corresponding to the restriction of $F$ on a (large)\nball $B(R)$, is independent of the direction $\\xi$. Here $\\xi$ should be\nthought as the direction corresponding to $V(x)$, $x\\in\\M$ is a fixed point on a Riemannian manifold,\nand $F$ should be thought of a scaled version of a random field defined on $\\M$, in the vicinity of $x$.\nThis naturally raises the question whether the same is true for $W\\in\\mathcal{V}({\\mathbb R}^{2})$ in place of $\\xi$,\ni.e. the distribution of $\\pazocal{N}_{W}(F|_{B(R)})$ is independent of $W$, for $W$ that is no longer assumed to be constant,\nat least, asymptotically as $R\\rightarrow\\infty$.\n\n\n\\subsection{Counting version of Theorem \\ref{thm:main meas conv}}\n\nUnlike in Theorem \\ref{thm:main meas conv}, the following Theorem \\ref{t:main_thm_less_equal_1} will separate the counts for different $k$.\nA short argument will then allow to deduce Theorem \\ref{thm:main meas conv} from Theorem \\ref{t:main_thm_less_equal_1}\n(see \\S\\ref{sec:proof main thm meas}).\n\n\\begin{theo} \\label{t:main_thm_less_equal_1}\nLet $V \\in \\mathcal{V}(\\M)$ and $\\alpha \\in [0,1]$. Let $f_{\\alpha, T}$ be the random $\\alpha$-band limited functions \\eqref{eq:fT band lim alpha<1} (or \\eqref{eq:fT band lim alpha=1}) of degree $T$,\nand $\\pazocal{N}_{V}(f_T,k)$ be the number of components whose number of $V$-tangencies is precisely $k$, $k\\ge 0$. Then the following hold:\n\\begin{enumerate}\n\\item There exists $C_{\\alpha, k} \\geq 0$ such that\n\\begin{equation*}\n{\\mathbf E} \\left[ \\left| \\frac{\\pazocal{N}_{V}(f_{\\alpha, T},k)}{\\pazocal{N}(f_{\\alpha, T})} - C_{\\alpha, k} \\right| \\right]\n\\xrightarrow[T \\rightarrow \\infty]{} 0 .\n\\end{equation*}\n\n\\item\nFurthermore, $C_{\\alpha, k} > 0$ for $k \\in 2 {\\mathbb Z}_{> 0}$ and $C_{\\alpha, k} = 0$ otherwise.\n\\end{enumerate}\n\\end{theo}\n\n\n\\section{Direction distribution for Euclidean random fields}\n\n\\subsection{Euclidean random fields: scale invariant model}\n\nThroughout this article we will be interested in centered Gaussian random fields $F: \\mathbb{R}^2 \\rightarrow {\\mathbb R}$.\nBy Kolmogorov's Theorem we know that the law of a centered Gaussian field is determined by its covariance kernel\n\\begin{equation*}\nK(x,y) = {\\mathbf E} \\left[ F(x) \\cdot F(y) \\right].\n\\end{equation*}\nIn particular, we concern ourselves with those fields which are isotropic, that is,\n\\begin{equation*}\nK(x,y) = K(|x-y|),\n\\end{equation*}\nimplying invariance under all rotations and translations. Furthermore, we assume for convenience that $K(0) = 1$.\n\nIt is known that such covariance kernels $K$ can be expressed as the Fourier transform of a measure $\\rho$, called the \\textit{spectral measure} of $F$. In many cases, it is more convenient to describe the field $F$ in terms of $\\rho$ instead of the covariance kernel.  This alternative way of writing $F$ is through the Hilbert space $\\mathcal{H}(\\rho) = \\mathcal{F} \\left( L^2_{sym}(\\rho) \\right)$, that is the space of functions $h$ which are $L^2$-orthonormal with respect to the measure $\\rho$ and satisfy the symmetry rule $\\overline{h}(-Y) = h(Y)$.  In particular, if $\\{\\phi_k\\}_k$ is an orthonormal basis for $\\mathcal{H}(\\rho)$, we can formally write\n\\begin{equation*}\nF = \\sum \\zeta_k \\phi_k\n\\end{equation*}\nwhere $\\zeta_k$  are i.i.d Gaussian random variables. This series diverges a.s. in $\\mathcal{H}(\\rho)$ but under suitable assumptions on $\\rho$, converges a.s. pointwise or in some other sense to a well-defined function, the nature of the convergence depending on the properties of $\\rho$.\n\nFor our purposes, the series of interest converges locally uniformly in $C^k$ for all $k \\in {\\mathbb N}$.\nAs a concrete example, the spectral measure $\\rho$ corresponding to Berry's RWM mentioned in \\S\\ref{sec:spher harm RMW} as the scaling limit\nof $T_{l}$ is the 1-dimensional arc-length measure $ds$ on the circle $S^1$, and we can explicitly compute an orthonormal basis for $\\mathcal{H}(ds)$, so we can express it as\n\\begin{equation*}\n\\sum_{n \\in {\\mathbb Z}} c_n J_{|n|}(r) e^{in \\theta}\n\\end{equation*}\nwhere the $c_k$ are i.i.d. standard complex Gaussians.\n\n\n\\subsection{Kac-Rice method}\n\n\nThe Kac-Rice formula is a standard tool or a meta-theorem for expressing all the moments of {\\em local} quantities, an example being the nodal volume, and the number of critical points of a random field $F:\\mathcal{D}\\rightarrow{\\mathbb R}$ where\n$\\mathcal{D}$ might be a compact subdomain of the Euclidean space or a subdomain of $\\M$. First, for our purposes\nwe need the following two upper bounds on the nodal count, which are direct conclusions from the application of Kac-Rice formula on appropriately defined critical points (either of a function $F$ or its restriction on the boundary of a disc or geodesic ball in $\\M$)\nappearing in ~\\cite{BW17}, which were themselves borrowed from ~\\cite{So12}. These results are valid for all random fields of our interest,\nlying in the $2$ dimensional case only.\n\n\n\\begin{lemma}[Cf. {~\\cite[Lemma 3]{BW17}}, {~\\cite[Corollary 2.3]{So12}}]\n\\label{lem:Kac Rice Euclid}\n\nLet $R>0$ and $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ be a stationary Gaussian random field, so that $F(\\cdot)$ is a.s. $C^{2}$-smooth and so that for every $x\\in {\\mathbb R}^{2}$\nthe vector $\\nabla F(x)$ is non-degenerate Gaussian.\n\n\\begin{enumerate}\n\n\\item Let $\\pazocal{N}(F,R)$ be the number of nodal components of $F$ lying entirely in $B(R)$. Then\n\\begin{equation*}\n{\\mathbf E}[\\pazocal{N}(F,R)] = \\mathcal{O}(R^{2}),\n\\end{equation*}\nwith the constant involved in the $\\mathcal{O}$-notation depending on the law of $F$ only.\n\n\\item Let $\\widetilde{\\pazocal{N}}(F,R)$ be the number of nodal components of $F$ intersecting the circle $\\partial B(R)$. Then\n\\begin{equation*}\n{\\mathbf E}[\\widetilde{\\pazocal{N}}(F,R)] = \\mathcal{O}(R),\n\\end{equation*}\nwith the constant involved in the $\\mathcal{O}$-notation depending on the law of $F$ only.\n\n\n\\end{enumerate}\n\n\n\\end{lemma}\n\nNext, the following result will be useful for ruling out mass leaking for the direction distribution measures (see the proof of\nTheorem \\ref{thm:euclid_count} (4) in \\S\\ref{sec:proof thm euclid no leak} below). It is a direct consequence of ~\\cite[Theorem 6.3]{AW}.\n\n\\begin{lemma}\n\\label{lem:Kac-Rice Euclid tangencies}\nLet $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ be a stationary Gaussian random field, so that the random vector $(F(0),\\nabla F(0))$\nhas a non-degenerate $3$-variate Gaussian distribution, $\\zeta\\in S^{1}$,\nand denote $\\mathcal{N}_{\\zeta}(F,R)$ to be the number of $\\zeta$-tangencies of $F$. Then\n\\begin{equation*}\n{\\mathbf E}[\\mathcal{N}_{\\zeta}(F,R)] = C_{0}\\cdot \\operatorname{Vol}(B(R)),\n\\end{equation*}\nfor some $C_{0}>0$ depending on the law of $F$ only.\n\\end{lemma}\n\nFinally, the following lemma yields an upper bound for the number of nodal components lying in a small geodesic ball\nfor the band limited functions only.\n\n\n\\begin{lemma}[{~\\cite[Lemma 2]{So12}}]\n\\label{lem:Kac Rice manifold}\nLet $\\alpha \\in [0,1]$ and $f_{T}=f_{\\alpha;T}$ be the random band-limited function \\eqref{eq:fT band lim alpha<1}\n(or \\eqref{eq:fT band lim alpha=1}).\nFor $x\\in \\M$ and $r>0$ sufficiently small, denote $\\pazocal{N} (f_{x,T},r)$ to be the number of nodal components of $f_{T}$ lying inside\nthe geodesic ball $B(x,r)\\subseteq\\M$ centred at $x$ with radius $r$. Then their expected number satisfies the\nfollowing estimate:\n\\begin{equation*}\n{\\mathbf E}[\\pazocal{N} (f_{x,T},r)] = \\mathcal{O}(T^{2}\\cdot r^{2}),\n\\end{equation*}\nwith constant involved in the $\\mathcal{O}$-notation depending on $\\M$ and $\\alpha$ only.\n\\end{lemma}\n\n\n\\subsection{Direction distribution for Euclidean random fields}\n\nLet $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ be a smooth Gaussian random field with spectral measure $\\rho$,\n$\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ be non-vanishing, and $k \\in {\\mathbb Z}_{\\geq 0}$. We denote by\n\\begin{equation*}\n\\pazocal{N}_{\\tilde{V}}(F,k,R)\n\\end{equation*}\n(resp. $\\pazocal{N}(F,R)$) the number of connected components of $F^{-1}(0)$ (which are all smooth thanks to Bulinskaya's Lemma \\cite[Proposition 6.11]{AW}) that are both completely contained in $B(R)$ and whose number of $\\tilde{V}$-tangencies is precisely $k$\n(resp. the total number of connected components of $F^{-1}$ contained in $B(R)$).\n\n\nAs we would like to study the distribution of $\\pazocal{N}_{\\tilde{V}}(F, \\cdot, \\cdot)$, it is essential to show that this function is in fact a random variable (i.e. that it is measurable on the given sample space of Gaussian fields $F$). A detailed verification of this fact for the related random variable $N(F, R, \\mathcal{T})$, that is the number of connected components contained in the ball $B(R)$ with topological type $\\mathcal{T}$, was carried out in the Appendix of \\cite{SW18} and is sufficiently robust to prove measurability in our case. We leave the details to the interested reader. We now state some axioms on $F$ (rather, its spectral measure $\\rho$) so that to be able to formulate the main result (Theorem \\ref{thm:euclid_count} immediately below)\non Euclidean random field.\n\n\\begin{definition}[Axioms on $\\rho$]\n\\label{def:axiom spec meas}\nFor a given Gaussian stationary field $F$, we will sometimes impose the following restrictions on the corresponding spectral measure $\\rho=\\rho_{F}$:\n\\begin{itemize}\n\\item $(\\rho 1)$ The measure $\\rho$ has no atoms (if and only if the action of the translations is ergodic by Theorem \\ref{thm:GFM}).\n\\item $(\\rho 2)$ For some $p > 4$,\n\\begin{equation*}\n\\int\\limits_{{\\mathbb R}^2} |\\lambda|^{p} \\, d \\rho(\\lambda) < \\infty\n\\end{equation*}\n(this ensures that a.s. $C^2$-smoothness of $F$).\n\\item $(\\rho 3)$ The support of $\\rho$ does not lie in a linear hyperplane (this ensures that the Gaussian field, together with its gradient, is not degenerate).\n\n\\item $(\\rho 4^*)$ The support of $\\rho$ has non-empty interior.\n\n\\end{itemize}\n\\end{definition}\n\n\nNazarov and Sodin ~\\cite{So12,NS15}\nproved that if $F$ satisfies the axioms $(\\rho 1)-(\\rho 3)$, then there exists a constant $c(\\rho)=c_{NS}(\\rho)\\ge 0$\n(``Nazarov-Sodin constant \"), so that\n\\begin{equation}\n\\label{eq:NS const def prop}\n{\\mathbf E}\\left[\\left|\\frac{\\pazocal{N}(F,R)}{\\operatorname{Vol}(B(R))} - c(\\rho) \\right|\\right] \\rightarrow 0.\n\\end{equation}\nIf, further, $(\\rho 4^{*})$ is satisfied, then $c(\\rho)>0$. In fact, in most of what follows, we will work with the following weaker version\nof $(\\rho 4^{*})$.\n\n\n\\begin{definition}[Axiom $(\\rho 4)$]\n\n$(\\rho 4)$ The Nazarov-Sodin constant $c(\\rho)$ is positive.\n\n\\end{definition}\n\n\n\n\\begin{theo} \\label{thm:euclid_count}\nLet $F: {\\mathbb R}^n \\rightarrow {\\mathbb R}$ be a stationary random field whose spectral measure $\\rho$ satisfies axioms $(\\rho 1) - (\\rho 4)$,\n$k \\ge 0$, $\\zeta\\in{\\mathbb R}^{2}$ be a fixed direction, and recall the Nazarov-Sodin constant $c(\\rho)>0$ satisfying the defining\nproperty \\eqref{eq:NS const def prop}. Then:\n\\begin{enumerate}\n\\item There exists $C_{k, \\zeta}=C_{k, \\zeta}(\\rho) \\geq 0$\n\\begin{equation}\n\\label{eq:conv L1 Euclid}\n{\\mathbf E} \\left[ \\left| \\frac{\\pazocal{N}_{\\zeta}(F,k,R)}{c(\\rho)\\cdot \\operatorname{Vol} (B(R))} - C_{k, \\zeta} \\right| \\right] \\rightarrow 0\n\\end{equation}\nas $R \\rightarrow \\infty$. If $k=0$ or $k$ is odd, then $C_{k,\\zeta}=0$.\n\n\\item Assuming further that $F$ is isotropic, we have that $C_k=C_{k, \\zeta}$ is independent of $\\zeta$.\n\n\\item If $k\\in 2{\\mathbb Z}_{>0}$, and either $F$ satisfies the axiom $(\\rho 4^*)$ (without assuming that $F$ is isotropic),\nor $\\rho = \\sigma_{S^1}$ (i.e. $F$ is Berry's monochromatic isotropic random waves),\nthe normalized arc-length measure on $S^1$, then $C_{k,\\zeta} > 0$.\n\n\\item We have\n\\begin{equation*}\n\\sum\\limits_{k=0}^{\\infty}C_{k,\\zeta} = 1.\n\\end{equation*}\n\n\\end{enumerate}\n\\end{theo}\n\n\\section{Various facts from plane geometry}\n\n\\subsection{Vector fields and exponential maps}\n\n\\begin{defn} \\label{d:blown_up_V}\nLet $F  \\in C^{\\infty}(T_x\\M)$ and $T>0$. For $V \\in \\mathcal{V}(\\M)$ and $Y \\in T_x\\M$, we define the \\textit{blown-up (at scale $T$) vector field} $\\tilde{V}_{x,T} \\in \\mathcal{V}(T_x\\M)$ at $x \\in \\M$, where $exp_x(0) = x$ and $exp_x(Y)=y$, via the local coordinate formula\n\\begin{equation}\n\\label{eq:tild(V) def}\n(\\tilde{V}_{x,T} F) =\n a_1 \\left( \\exp_x \\left( \\frac{Y_1}{T}, \\frac{Y_2}{T} \\right) \\right) \\cdot \\frac{\\partial F}{\\partial Y_1}\\left( Y  \\right)\n+ a_2 \\left( \\exp_x \\left( \\frac{Y_1}{T}, \\frac{Y_2}{T} \\right) \\right) \\cdot \\frac{\\partial F}{\\partial Y_2} \\left( Y \\right).\n\\end{equation}\n\\end{defn}\n\nWe will shortly consider the random fields\n$F(Y_1, Y_2) = f_{\\alpha, T}(\\exp_x(\\frac{Y}{T}))$ as in \\eqref{eq:fxT scaled def}. Notice that we are pushing forward the vector field $V$, extracting the ``top order in $T^{-1}$\" differential operator, and then applying that to $F$. We {\\em are not} pushing forward $V$ by $M_{1\/T} \\circ (\\exp_x)^{-1}_*$, where $M_{1\/T}$ is simply multiplication in the fibers of $T_xM$ by $1\/T$. The reason for this particular definition is so that the coefficients of $V$ are localizing at the same scale as $f_{x, T}$.\nThus in the regime $|Y| \\leq R$ with $R\/T = o(1)$, for a large but fixed $R$ with $T \\rightarrow \\infty$, we find that the coefficients $a_1,a_2$ become constants up to first order in $T^{-1}$. More specifically, we have that, for instance, $a_1 \\left( \\exp_x(\\frac{Y_1}{T}, \\frac{Y_2}{T}) \\right) = a_1(x) + \\mathcal{O} \\left( \\frac{|Y|}{T} \\right)$ with $\\frac{|Y|}{T} = o(1)$ as seen in upcoming sections discussing limiting regimes.\n\nFrom here onward, we reserve the notation of $\\tilde{V}$ for vector fields on ${\\mathbb R}^2$. Now, in order to show that our main result Theorem \\ref{t:main_thm_less_equal_1} is not vacuous, we give the following result on the existence of vector fields with finitely many zeroes.\n\n\\begin{lem} \\label{l:vfs_finite_zeroes}\nGiven a smooth and compact surface $(\\M,g)$, there exists smooth vector fields $V$ that have finitely many zeroes. That is, our class $\\mathcal{V}(\\M) \\neq \\emptyset$ for compact Riemannian surfaces $\\M$.\n\\end{lem}\n\nThis result holds in $d\\ge 2$ dimensions, but we state it for surfaces given our context.\n\n\\begin{proof}\nConsider any Morse function $f$ on $\\M$; we know there exists such a function, for example one being the height function on $\\M$ after applying an appropriate embedding into ${\\mathbb R}^N$. The metric $g$ allows us to associate the differential $df$ to its gradient field $\\nabla_g f$, each of whose zeroes is isolated by the non-degenerate critical point condition. Compactness of $\\M$ yields that the number of such zeroes must be finite.\n\\end{proof}\n\n\\subsection{Global plane geometry}\n\nIn this section we begin our study of curves in ${\\mathbb R}^2$ that have a fixed number of tangencies to a given vector field $\\tilde{V}$ and its relation to the study of transversality between the sets $f^{-1}(0)$ and $(Vf)^{-1}(0)$. These elementary results, although simple to state and are intuitively clear, prove very useful in the pursuit of Theorem \\ref{t:main_thm_less_equal_1}. We begin with a simple but crucial lemma that will play a key role in showing that the limiting constants $C_k$, for $k=0,1$, in Theorem \\ref{t:main_thm_less_equal_1} are $0$:\n\n\\begin{lem} \\label{l:at_least_two_tangencies}\nSuppose $\\zeta$ is a fixed direction on ${\\mathbb R}^2$. Then any closed simple curve in the plane must have at least two tangencies $\\zeta$.\n\\end{lem}\n\nBefore proving this, we remind ourselves of a simple form of the Gauss-Bonnet Index Theorem, which goes by the name of the Theorem of Turning Tangents in elementary differential geometry, due to Hopf:\n\n\\begin{theo}[Theorem of Turning Tangents] \\label{t:turn_tang}\nThe rotation number of an embedded $S^1$ in ${\\mathbb R}^2$ (i.e. the oriented number of times the unit tangent vector $\\overrightarrow{T}$ to this embedding transverses $S^1$) is equal to $\\pm 1$.\n\nIn other words, the map induced by $\\overrightarrow{T}$ from $S^1$ to $S^1$ must be surjective and in terms of topological language, $\\overrightarrow{T}$ induces a map of degree $\\pm 1$.\n\\end{theo}\n\n\\begin{proof}[Proof of Lemma \\ref{l:at_least_two_tangencies}]\nBy Theorem \\ref{t:turn_tang}, $\\overrightarrow{T}$ must be parallel to $\\zeta$ and -$\\zeta$ at at least two distinct points.  Since $\\overrightarrow{T}$ parallel to -$\\zeta$ is counted as a $\\zeta$-tangency, this proves our claim.\n\\end{proof}\n\nIt is possible to construct closed curves in ${\\mathbb R}^2$ with an odd number of tangencies:\nall we must do is properly glue together neighbourhoods of the singular parts of a cubic to the neighbourhoods of singular parts of the quadratics. However, we will see that this situation is unstable in an appropriate sense, and therefore will not contribute towards our final result.\n\n\\begin{lem} \\label{l:even_int}\nConsider two $C^2$ curves $\\gamma_1, \\gamma_2 \\subsetneq {\\mathbb R}^2$ that arise as components of the zero sets of respectively two different functions, both of which has 0 as a regular value. Assume that $\\# ( \\gamma_1 \\cap \\gamma_2)  < \\infty$.  If $\\gamma_1$ is both simple and closed or $\\gamma_2$ is both simple and closed, and each point in $\\gamma_1 \\cap \\gamma_2$ is transversal, then $\\# \\gamma_1 \\cap \\gamma_2$ is even.\n\\end{lem}\n\n\\begin{proof}\nWithout loss of generality, let $\\gamma_1$ be simple and closed.  By the Jordan Curve Theorem, $\\gamma_1$ encloses a bounded region $\\Omega_1$.  By the transversality assumption, every point of intersection is uniquely connected to another intersection  by an arc  of $\\gamma_2$ that is completely contained in $\\Omega_1$ (for otherwise, this contradicts the regularity assumption or the transversality assumption).  Hence, we can associate two unique points of intersection to each interior arc.  Counting the number of such arcs, we arrive at our conclusion.\n\\end{proof}\n\n\\begin{cor}\n\\label{cor:tangencies even}\nFor a fixed direction $\\zeta$ on ${\\mathbb R}^2$, any simple closed curve in the plane without inflection points that are also $\\zeta$-tangent must have an even positive number of $\\zeta$-tangencies.\n\\end{cor}\n\nWe close this section by emphasizing that by the qualitative version of Bulinskaya's Lemma (see \\cite[Proposition 6.11]{AW}),\nwith probability 1 any nodal component of a Gaussian field $F$ on $B(0,R)$ is simple and regular (that is 0 is regular value for $F$) although there may exists those that are not closed.\n\n\\subsection{Intersections in the plane: local results} \\label{s:local_results}\n\nThe terminology established in the section is important from both analytic and geometric points of view. It is here where we establish our main observation: \\textit{the study of nodal curves with a fixed number of $V$-tangencies is equivalent to the study of the transversal and tangential intersections of the sets $f_{T, \\alpha}^{-1}(0)$ and $(Vf_{T, \\alpha})^{-1}(0)$.}\n\nOur aim is to interchange the words of ``intersection\" and ``$V$-tangency\" with the latter being a special case of the former in the upcoming dictionary. Although this may appear counter-intuitive at first, it is crucial that we identify our geometric property with a suitable analytic characterization.\n\n\\subsubsection{Deterministic intersection results: $\\beta$-transversal tangencies} \\label{s:deterministic}\n\n\\begin{defn} \\label{d:beta_trans}\n\\begin{enumerate}\n\\item For a pair of functions $G=(g_1, g_2): {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ such that $g_1, g_2 \\in C^1(B(2R))$, $0$\nis a regular value for both $g_1$ and\n$g_2$, and $\\beta>0$ (possibly small), we say that $u_0$ is a point of \\textit{$\\beta$-transverse intersection} if $g_1(u_0)=g_2(u_0) = 0$ and $\\left| \\det DG(u_0) \\right| >  \\beta$.\n\n\\item Let $\\beta>0$ be a small parameter.\nGiven $g \\in C^2({\\mathbb R}^2)$ such that $0$ is a regular value with $\\tilde{V}$ a smooth vector field on ${\\mathbb R}^2$, we say that $u_0$ is a \\textit{$\\beta$-transverse $\\tilde{V}$-tangency} (or simply a ``non-degenerate\" tangency) if $\n\\tilde{V}(u_0) \\neq 0 \\in {\\mathbb R}^2$, $g(u_0)=\\tilde{V}g(u_0)=0$, and $\\left| \\det \\left( D_{u=u_0} \\left[ \\nabla g, \\nabla \\tilde{V}g \\right] \\right) \\right| > \\beta$.\n\\end{enumerate}\n\\end{defn}\n\nNow that we have fixed some useful terminology, we proceed to establishing some key yet general estimates on the intersections of curves in the planes and their behaviors under perturbations.  In this section, we analyse a situation of quantitative transversality:\n\n\\begin{prop}[Deterministic stability of transversal intersections, cf. {\\cite[Lemma 10]{BW17}}] \\label{p:stab_trans}\nLet $R>0$ and $k \\ge 0$ be given. Let $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ be non-vanishing, $g,h \\in C^3(\\overline{B}(2R))$, and denote $G:=(g, \\tilde{V}g)$. Assume that the following restrictions hold:\n\\begin{enumerate}\n\\item There exists $\\beta_1 >0$ such that\n\\begin{equation*}\n\\min_{\\overline{B}(2R)} \\max \\{ |g(u)|, |\\nabla g(u)| \\}, \\,  \\min_{\\overline{B}(2R)} \\max \\{ |\\tilde{V}g(u)|, |\\nabla (\\tilde{V} g)(u)| \\} > \\beta_1.\n\\end{equation*}\nThis immediately implies that\n\\begin{equation*}\n\\min_{\\overline{B}(2R)} \\max \\{ \\|G\\|, \\|DG\\|_{\\mathcal{L}({\\mathbb R}^2 \\rightarrow {\\mathbb R}^2)} \\} > \\frac{\\beta_1}{\\sqrt{2}},\n\\end{equation*}\nand in particular that $\\|g\\|_{C^2} > c_2\\beta_1$ where $c_2>0$ is a constant depending only on the ambient dimension.\n\\item There exists $\\beta_2$ such that\n\\begin{equation*}\n\\left| \\det DG(u) \\right| >  \\beta_2\n\\end{equation*}\nfor $u \\in \\{ (g, \\tilde{V}g) = 0 \\}$ in $B(2R)$.\n\\item The $C^3$ norms of $g,h$ are bounded above by $M_0$ for some $M_{0}>0$.\n\\item We have\n\\begin{equation*}\n\\| g - h \\|_{C^2(B(R+1))} < b_0\n\\end{equation*}\nfor some $b_{0}=b_{0}(M_{0},\\beta_{0},\\beta_{1})>0$ sufficiently small.\n\n\\end{enumerate}\n\nThen there exists an injective map $\\gamma \\rightarrow \\gamma^h$ between the components of $g^{-1}(0)$ and $h^{-1}(0)$ with the property that:\n\\begin{enumerate}\n\\item The connected components $\\gamma \\subset B(R-1)$ of $g^{-1}(0)$ whose number of $\\beta$-transversal $\\tilde{V}$-tangencies are precisely\n$k$ is in one-to-one correspondence with a subset of the connected components $\\gamma^{h} \\subset B(R)$ of $h^{-1}(0)$ whose number of $(\\beta-b)$-transversal $\\tilde{V}$-tangencies is precisely $k$.\n\n\\item For every $\\gamma$ as above, the components $\\gamma$ and $\\gamma^h$ are uniformly close in the following sense: there exists a smooth bijective map $\\psi_{\\gamma}: \\gamma \\rightarrow \\gamma^h$ such that for all $u \\in \\gamma$, we have the estimate\n\\begin{equation} \\label{e:components_bouge}\n\\| \\psi_{\\gamma}(u) - u \\|_{L^{\\infty}(B(R))} = \\mathcal{O}(b)\n\\end{equation}\nwhere the implicit constant depends on the quantities $\\|G\\|_{C^2(B(0,2R))}$ and $\\beta_1$.\n\\end{enumerate}\n\\end{prop}\n\nThe dependence of the implicit constant in the estimate \\eqref{e:components_bouge} for the new zero's positioning \\textit{linear} in $\\beta_1$ but of order $-1$ in $\\beta_2$.\n\n\n\\subsubsection{Some technical lemmas}\n\nIn this section we give a key lemma following from multivariable calculus, which is essentially a restatement of the above proposition, and is analogous to \\cite[Lemma 7]{So12}.\n\n\\begin{lem} \\label{l:Sodin_type_calc_lem}\nConsider $G(u) = (g_1(u), g_2(u))$ for $g_1, g_2 \\in C^2(B(2R)$ for some given $R>0$ possibly large. Suppose $u_0 \\in G^{-1}(0) \\cap B(2R)$. We make the following assumptions:\n\\begin{itemize}\n\\item There exists $1 > a>0$ such that for all $u \\in B(2R)$, we have\n\\begin{equation} \\label{e:no_low_lying_cp}\n\\min_{\\overline{B}(2R)} \\max \\{ |g_1(u)|, |\\nabla g_1(u)| \\}, \\,  \\min_{\\overline{B}(2R)} \\max \\{ |g_2(u)|, |g_2(u)| \\} > a.\n\\end{equation}\n\\item There exists $\\alpha> 0$ such that $\\alpha < |\\det DG(u)|$ for all $u \\in B(2 R) \\cap G^{-1}(0)$.\n\\item Set\n\\begin{equation} \\label{e:perturb_size}\n b =\\frac{1}{4} \\min \\left\\{ a , \\frac{1}{ \\|G\\|_{C^2(B(u_0, R_0+1))}}, \\alpha \\right\\}.\n\\end{equation}\n\n\\end{itemize}\nLet $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ be non-vanishing.  Consider a perturbation $\\Psi = (\\psi, \\tilde{V} \\psi)$ where $\\psi \\in C^2(B(2R))$.  Set $G_{\\Psi}(u) = G(u) + \\Psi(u)$ and consider only zeroes $u_0 \\in G^{-1}(0)$ such that $B(u_0, b) \\subset B(0, R_0)$.  Then for $\\| \\psi \\|_{C^2(B(2R))} < b$ we have that\n\\begin{enumerate}\n\\item $u_0$ is a isolated zero of $G$,\n\\item $G_{\\Psi}$ is a global diffeomorphism on $B ( u_0,  b )$,\n\\item $G_{\\Psi}$ also has a unique isolated zero $\\tilde{u_0} \\in B(u_0, b) $ with\n\\begin{equation*} \\label{e:new_zero_pos}\n\\|u_0 - \\tilde{u_0} \\| =  \\mathcal{O}(b)\n\\end{equation*}\nwhere the implicit constant depends on the quantities $\\|G\\|_{C^2}$ and $a^{-1}$.\nMoreover, the new zero $\\tilde{u_0}$ has the property that $|det DG_{\\psi}(\\tilde{u_0})| > \\alpha - b.$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nEach of these claims follows almost directly from the Inverse Function theorem and its proof. Part (1) is an immediate implication of the inverse function theorem.\n\nParts (2) and (3) use elements from the proof of the Inverse Function theorem. Here, we closely follow \\cite[Theorem 22.26]{B03}. If we can show that the so-called \\textit{Newton's function} $\\epsilon_{\\Psi}(u) = DG_{\\Psi}^{-1}(u_0)G_{\\Psi}(u+u_0) - u$ has the property $\\|D\\epsilon_{\\Psi}(u)\\|  < 1$ for all $u \\in B(0, b)$ with $b$ satisfying the stated inequality, the referenced proof of the Inverse Function theorem shows that $\\epsilon_{\\Psi}$ is invertible on $B(0, b)$ and therefore $G_{\\Psi}$ is invertible on $B(u_0, b)$. Thus, we estimate the operator norm for the matrix $D \\epsilon_{\\Psi}$:\n\\begin{align*}\n& \\left( DG(u_0) + D\\Psi(u_0) \\right)^{-1} \\left( DG(u+u_0) + D\\Psi(u+u_0)  \\right) - I \\\\\n& = \\left( DG(u_0) + D\\Psi(u_0) \\right)^{-1} \\left( DG(u_0) + D\\Psi(u+u_0) + \\mathcal{O}_{\\|G\\|_{C^2(B(2R))}}(\\|u\\|)  \\right) - I \\\\\n&  = \\left(I + DG(u_0)^{-1} D\\Psi(u_0) \\right)^{-1} (DG(u_0))^{-1} \\left( DG(u_0) + D\\Psi(u+u_0) +  \\mathcal{O}_{\\|G\\|_{C^2(B(2R))}}(\\|u\\|) \\right) - I  \\\\\n& = \\left( I + DG(u_0)^{-1} D\\Psi(u_0) \\right)^{-1} \\left( I + DG(u_0)^{-1} D\\Psi(u+u_0) + \\mathcal{O}_{ \\|G\\|_{C^2(B(2R))}}(a^{-1} b )  \\right) - I \\\\\n&= \\mathcal{O}\\left(a^{-1} \\cdot b \\right) +  \\mathcal{O}\\left( \\|G\\|_{C^2(B(u_0, b))} \\cdot a^{-1} \\cdot b \\right) +  \\mathcal{O} \\left((a^{-1}b)^2 \\right) \\\\\n& = \\mathcal{O}_{a, \\|G\\|_{C^2}}(b),\n\\end{align*}\nprovided we have the restriction \\eqref{e:perturb_size} and $a < 1$, where we have used some well-known formulas for inverses of small perturbations of the identity matrix. This in turn implies that the inverse function $G_{\\psi}^{-1}$ exists on $B(u_0, b)$, therefore establishing claim (2).  Hence any zero, if it exists, lying inside of $B(u_0, b)$ must be unique and isolated.  We have now reduced ourselves to proving the existence of a new zero $\\widetilde{u_0}$.\n\nThe proof of the existence and exact positioning of the isolated zero is motivated by the proof of \\cite[Lemma 10]{BW17}.\nAs $N_i(u_0) = \\frac{\\nabla g_i(u_0)}{\\| \\nabla g_i(u_0)\\|}$ is well-defined thanks to $|\\det DG(u_0) | > \\alpha$, we consider the function\n\\begin{equation*}\n\\zeta(s_1, s_2) := G_{\\psi}\\left(u_0 + s_1 N_1(u_0) + s_2 N_2(u_0) \\right);\n\\end{equation*}\nwe would like to show that $\\zeta$ does obtain a zero $\\tilde{s}_0$ and give an estimate on $\\|\\tilde{s}_0\\|$. This would establish claim (3).\nTo begin, consider the auxiliary function\n\\begin{equation*}\n\\mathfrak{S}_{(t_1, t_2)}(s_1, s_2) := (t_1, t_2) - \\epsilon_{\\zeta}(s_1,s_2)\n\\end{equation*}\nwhere $\\epsilon_{\\zeta}(s_1, s_2) = D\\zeta^{-1}(0,0) \\zeta(s_1,s_2) - (s_1, s_2)$ and $(t_1,t_2)$ are small numbers; we will employ an effective version of the Contraction Mapping Theorem in order to determine the existence and size of $\\tilde{s}_0$.\n\nLet us take a pause to understand the purposes of our various newly-defined auxiliary functions.  Supposing that $\\epsilon_{\\zeta}(s_1, s_2)$ were well-defined and that we were to able solve the fixed point equation\n\\begin{equation} \\label{e:fixed_point_H}\n\\mathfrak{S}_{(t_1,t_2)} (H(t_1,t_2)) = H(t_1,t_2)\n\\end{equation}\nfor $(t_1, t_2)$ varying over some small neighborhood around the origin. This would imply that $H(0,0) = - \\epsilon_{\\zeta}(H(0,0))$. As $D\\zeta^{-1}(0,0)$ is invertible our assumption $\\epsilon_{\\zeta}$ being well-defined, then this along with the equation $H(0,0) = - \\epsilon_{\\zeta}(H(0,0))$ would show that $ D\\zeta^{-1}(0,0) \\zeta(H(0,0))=(0,0)$.  Thus $ \\zeta(H(0,0))=(0,0)$ and moreover $\\zeta$ obtains a zero.  If we can show that  $$\\|D_{s_1,s_2} \\mathfrak{S}_{(t_1,t_2)}(s_1, s_2) \\| < 1$$ for suitably small values of $(t_1,t_2)$ and $(s_1,s_2)$, then the Contraction Mapping Theorem as used in the proof of the Inverse Function Theorem gives us such an $H(t_1,t_2)$ as in (\\ref{e:fixed_point_H}) and therefore $\\tilde{s}_0:=H(0,0)$ is achieved.\n\nTo sum up our reasoning, first we must estimate the operator norm of $D_{s_1, s_2}\\mathfrak{S}_{(t_1, t_2)}$ for $\\|(t_1,t_2)\\|, \\|(s_1,s_2)\\| \\leq Cb$ to show the existence of $H(s_1,s_2)$ which in turn gives the existence of a zero $\\tilde{s}_0 = H(0,0)$. Second, an estimate on $\\|\\epsilon_{\\zeta}(s_1,s_2)\\|$ for $\\|(s_1, s_2)\\| \\leq C b$ where $C>0$ is some uniform constant gives us the positioning of $\\tilde{s}_0$; this follows from the equation $H(0,0) = - \\epsilon_{\\zeta}(H(0,0))$. We now proceed to carrying out these steps.\n\nA direct calculation will show that\n\\begin{equation*}\nD_{s_1,s_2} \\zeta_{|(0,0)} = D_{s_1,s_2} G_{\\Psi} \\left( u_0 + s_1 N_1(u_0) + s_2 N_2(u_0)\\right)_{|(0,0)}\n\\end{equation*}\nis invertible. Thanks to $G$ being a local diffeomorphism in $B(u_0, b)$ (and therefore $\\{N_1, N_2\\}$ forming a linearly independent set over $B(u_0, b)$), this combined with  $|\\det (DG + D\\psi)(u)| \\geq \\alpha - 2b > 0$, which follows by (\\ref{e:no_low_lying_cp}) and (\\ref{e:perturb_size}), for all $u \\in B(u_0, b)$ finally shows that $D\\zeta(0,0)$ is invertible. Thus, we have an expression\n\\begin{align*}\n&  D_{s_1,s_2} \\mathfrak{S}_{(t_1,t_2)}(s_1, s_2) = -D \\epsilon_{\\zeta}(s_1,s_2) \\\\\n& = \\left[N_1(u_0), N_2(u_0) \\right]^{-1} \\circ DG^{-1}_{\\psi}(u_0) \\circ DG_{\\psi}(u_0 + sN_1 + sN_2) \\circ \\left[N_1(u_0), N_2(u_0) \\right] - I\n\\end{align*}\nwhere we arrive at a similar Taylor expansion analysis as carried out in the first half of our proof. We leave it to the interested reader to see that for $\\|D_{s_1,s_2} \\mathfrak{S}_{(s_1,s_2)}(s_1, s_2)\\| = \\mathcal{O}(b) < 1$, where the implicit constant again depends on $\\|G\\|_{C^2}$ and $a^{-1}$.  Another application of the Contraction Mapping Theorem, thanks to the operator norm of $D_{s_1,s_2} \\mathfrak{S}_{(t_1,t_2)}(s_1, s_2)$ being smaller than 1, brings us to the conclusion that such a solution $H$ exists.\n\nWe know that $\\epsilon_{\\zeta}(s_1, s_2) = \\epsilon_{\\zeta}(0, 0) + \\mathcal{O}_{\\|D\\epsilon_{\\zeta}(0,0)\\|}(b)$. Furthermore,\n$\\epsilon_{\\zeta}(0,0) = D \\zeta^{-1}(0,0) \\Psi(u_0)$ since $u_0$ is a zero of $G$. Finally as $\\|\\psi\\|_{C^2} < b$, we conclude that $\\tilde{s}_0 = H(0,0) =  \\mathcal{O}(b)$ with the corresponding implicit constants.\n\\end{proof}\n\nNote that our application does require $C^2$ regularity due to setting $g_2 = \\tilde{V}_{x,T}g_1$. Our proof demonstrates that the amount in which our zeroes move after perturbation is controlled by the $C^1$ norm of the perturbation $\\Psi$ (thanks to our perturbation being simply additive in nature), emphasizing that from the point of view of analysing zeroes the $C^1$ approximation of the scaling limit $\\mathfrak{g}_{\\alpha}$ is sufficient.\n\n\\begin{proof}[Proof of Proposition \\ref{p:stab_trans}]\nSet $g = g_1$ and $\\tilde{V}g = g_2$. This demonstrates why $C^2$ estimates are imposed. Now, we simply set $\\beta_1 = a$, $\\beta_2 = \\alpha$, and take $b$ equal to the strength of the perturbation. The final step involves applying in tandem \\cite[Lemma 10]{BW17} (although we can easily repeat the steps of the proof of Lemma \\ref{l:Sodin_type_calc_lem} to recover this result) and conclusion (3) of Lemma \\ref{l:Sodin_type_calc_lem} to show that non-degenerate components map to a subset of non-degenerate components and stay $\\mathcal{O}(b)$ close in $L^{\\infty}$.\n\\end{proof}\n\nA combination the Proposition \\ref{p:stab_trans} and \\cite[Lemma 7]{So12} immediately gives us that:\n\n\\begin{cor} \\label{l:Sodin_annuli}\nUnder the hypotheses of Lemma \\ref{l:Sodin_type_calc_lem}, let $C>0$ be the implicit constant that appears in the estimate (\\ref{e:new_zero_pos}).  Take $\\Gamma_g$ a regular component of $g^{-1}(0)$. Then, if given a tubular neighborhood $A_{\\Gamma_g}(Cb)$ that uniquely contains the component $\\Gamma_g \\subset g^{-1}(0)$ and has width $2Cb$, it follows that the points of $\\{ (g, \\tilde{V}g) = 0 \\}$ which meet $\\Gamma_g$ persist inside of $A_{\\Gamma_g}(Cb)$ after perturbation by $h-g$.\n\\end{cor}\n\n\n\\subsubsection{Deterministic intersection results: sub-$\\beta$ tangencies}\n\nThe main purpose of this section is to provide the reader with the intuition that components with ``near-tangential\" intersections can be unstable in terms of intersection counts being constant when subjected to small perturbations. The following lemma will not be employed in the proofs of our main results, however, it serves as an intuition what might happen to the tangency points if we do not exclude the unstable event $\\Delta_7$, as\ndefined in \\S\\ref{s:exceptional_events} below.\n\n\\begin{defn} \\label{d:sub_beta}\n\\begin{enumerate}\n\\item For a function $G=(g_1, g_2)$ such that $g_1, g_2 \\in C^1(B(2R))$ and $\\beta>0$ (possibly small), we say that $u_0$ is a point of \\textit{sub-$\\beta$ intersection} if $G(u_0) =  (g_1(u_0), g_2(u_0)) = 0$ and $\\left| \\det DG(u_0) \\right| \\leq  \\beta$.\n\n\\item Let $\\beta>0$ be a small parameter. Given $g \\in C^2({\\mathbb R}^2)$ with $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ non-vanishing, we say that $u_0$ is a point of \\textit{sub-$\\beta$ $\\tilde{V}$-tangency} (or simply a ``near-degenerate\" tangency) if $( g(u_0), \\tilde{V}g(u_0)) = 0$ and $|\\det \\left( D_{u=u_0} \\left[ \\nabla g, \\nabla \\tilde{V}g \\right] \\right) | \\leq \\beta$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{lem}[Classification of perturbations of sub-$\\beta$ intersections] \\label{l:class_tang}\nLet $r>0$ and $\\beta>0$ be given. Let $0$ be a regular value of $g_1, g_2 \\in C^{\\infty}({\\mathbb R}^2)$. Furthermore, we assume that $g_1^{-1}(0) \\cap g_2^{-1}(0)$ is regular, i.e. that $|\\det(\\nabla g_1(u), \\nabla g_2(u))| \\neq 0$ for each point $u \\in g_1^{-1}(0) \\cap g_2^{-1}(0)$.  Suppose $u_0 \\in B(u_0, r) \\cap g_1^{-1}(0) \\cap g_2^{-1}(0) \\subset B(R)$ is an isolated point of ``near\"-tangential intersection, that is\n\\begin{equation*}\n|\\det(\\nabla g_1(u_0), \\nabla g_2(u_0))| \\leq \\beta.\n\\end{equation*}\n\nThen there exists $b(r)>0$ with $b < r$, such that for perturbations $\\psi_1, \\psi_2$ with $\\|\\psi_i\\|_{C^2(B(R)} < b$ for $i=1,2$, the only possibilities of zeroes of $(g_1 + \\psi_1)^{-1}(0)  \\cap (g_2 + \\psi_2)^{-1}(0)$ appearing inside of $B(u_0, r + b)$ are: 1) two sub-$(\\beta+b)$ zeroes, that is we have $|\\det(\\nabla (g_1 + \\psi_1), \\nabla (g_2 + \\psi_2))| < \\beta + b$ on $B(u_i, r + b)$ for $i=1,2$ being the new zeroes, or 2) a single sub-$(\\beta + b)$ zero in $B(u_0, r + b)$.\n\\end{lem}\n\n\\begin{proof}\nBy an elementary notion from differential topology \\cite[Chapter 2]{GP}, if $K$ is a compact set and $f: K \\rightarrow {\\mathbb R}^2$ is transverse to a closed submanifold $L \\subsetneq {\\mathbb R}^2$, then the transversality is stable under suitable perturbations (i.e. of class $C^1$) of $f$.\nConsidering this fact, let $f$ be the parametrization of the portion of the nodal component of $g_1^{-1}(0)$ inside of $B(u_0, r)$ and $L$ the nodal component of $g_2^{-1}(0)$ inside of $B(u_0,r)$. If our initial choice of $b$ is small enough, then the transversality will persist. For a possibly smaller choice of $b$ depending on $t$, given value of $r$, this then yields 1) or 2).\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{thm:euclid_count}}\n\n\\subsection{Some preliminaries}\n\n\\begin{theo}[Grenander-Fomin-Maruyama] \\label{thm:GFM}\nLet $F$ be a random stationary Gaussian field with spectral measure $\\rho$. Then, if $\\rho$ contains no atoms, the action of the translations group\n\\begin{equation*}\n(\\tau_uF)(x) = F(x - u)\n\\end{equation*}\nis ergodic ($F$ is ``ergodic\").\n\\end{theo}\n\n\\begin{theo}[Wiener's Ergodic Theorem] \\label{thm:weiner}\nSuppose that $F$ is ergodic and the translation map ${\\mathbb R}^n \\times C({\\mathbb R}^n) \\rightarrow C({\\mathbb R}^n)$\n\\begin{equation*}\n(u,F) \\mapsto \\tau_u F\n\\end{equation*}\nis measurable with respect to the product $\\sigma$-algebra $\\mathcal{B}({\\mathbb R}^n) \\times \\mathcal{A}$ and $\\mathcal{A}$.  Then every random variable $\\Phi(F)$ with finite expectation ${\\mathbf E} \\left[ |\\Phi(F)| \\right] < \\infty$ satisfies\n\\begin{equation*}\n\\lim\\limits_{R \\rightarrow \\infty} \\frac{1}{\\operatorname{Vol} (B(R))} \\int\\limits_{B(R)} \\Phi(\\tau_u F) \\, du \\rightarrow {\\mathbf E} \\left[ \\Phi(F) \\right],\n\\end{equation*}\nboth almost surely and in $L^1$.\n\\end{theo}\n\n\n\\begin{defn}\n\\label{def:counts}\nLet $\\Gamma \\subset {\\mathbb R}^2$ be a smooth curve (connected or not) and $\\zeta$ be a fixed direction. For $u \\in {\\mathbb R}^2$, $r >0$, and $k \\ge 0$, we denote by\n\\begin{equation*}\n\\pazocal{N}_{\\zeta}(\\Gamma, k, u, r)\n\\end{equation*}\nthe number of connected components of $\\Gamma$ whose number of $\\zeta$-tangencies, that are also lying entirely inside of the ball $B(u,r)$, is precisely $k$.  In the case of $u=0$, we simply write $\\pazocal{N}_{\\zeta}(\\Gamma, k, r)$.  Similarly, we define\n\\begin{equation*}\n\\pazocal{N}^*_{\\zeta}(\\Gamma, k, u, r)\n\\end{equation*}\nby relaxing the condition that the components of $\\Gamma$ merely intersect $B(u,r)$.\n\\end{defn}\n\n\\begin{lem}[Integral-Geometric Sandwich] \\label{l:integro_geom_sandwich}\nLet $\\Gamma$ be a smooth, closed curve and $\\zeta$ be a fixed direction. Then for $0 < r < R$, $k \\ge 0$, we have that\n\\begin{align*}\n\\frac{1}{\\operatorname{Vol}(B(r))} \\int\\limits_{B(R-r)} \\pazocal{N}_{\\zeta}(\\Gamma, k, u, r) \\, du &\\leq \\pazocal{N}_{\\zeta}(\\Gamma, k, R) \\\\\n& \\leq \\frac{1}{\\operatorname{Vol}(B(r))} \\int\\limits_{B(R+r)} \\pazocal{N}^*_{\\zeta}(\\Gamma, k, u, r) \\, du\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nOur proof follows almost exactly the steps of \\cite[Lemma 1]{So12} and \\cite[Lemma 5]{BW17}. Here, we present only the proof for the upper bound of $\\pazocal{N}_{\\zeta}(\\Gamma, k, R)$.\n\nLet $\\gamma \\subset \\Gamma$ be a connected component.  Define\n\\begin{equation*}\nG_*(\\gamma) = \\bigcap_{v \\in \\gamma} B(v,r) = \\{ u \\, : \\, \\gamma \\subset B(u,r) \\}\n\\end{equation*}\nand\n\\begin{equation*}\nG^*(\\gamma) = \\bigcup_{v \\in \\gamma} B(v,r) = \\{ u \\, : \\, \\gamma \\cap B(u,r) \\neq \\emptyset \\}.\n\\end{equation*}\nIt follows by the above definition that for all $v \\in \\gamma$,\n\\begin{equation*}\nG_*(\\gamma) \\subset B(v,r) \\subset G^*(\\gamma),\n\\end{equation*}\nwhich in turn immediately yields the inequality\n\\begin{equation*}\n\\operatorname{Vol}(G_*(\\gamma)) \\leq \\operatorname{Vol}(B(v,r)) \\leq \\operatorname{Vol}(G^*(\\gamma))\n\\end{equation*}\nFinally, a sum over the components $\\gamma \\subset B(R)$ whose number of $\\zeta$-tangencies is precisely k gives us\n\\begin{align*}\n\\sum_{\\left\\{\\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}} \\operatorname{Vol}(G_*(\\gamma)) & \\leq \\operatorname{Vol}(B(r)) \\, \\pazocal{N}_{\\zeta}(\\Gamma,k, R) \\\\\n& \\leq \\sum_{\\left\\{ \\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}} \\operatorname{Vol}(G^*(\\gamma))\n\\end{align*}\n\n\nUsing the definition of $G_*(\\gamma)$, we know that\n\\begin{equation*}\n\\sum_{\\left\\{\\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}} \\operatorname{Vol}(G_*(\\gamma)) = \\sum_{\\left\\{ \\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}}\\left( \\int\\limits_{\\{u: \\gamma \\subset B(u,r)\\}} \\, du \\right).\n\\end{equation*}\n\nNote that if $u \\in B(R-r)$ then we have the inclusion $B(u,r) \\subset B(R)$. Hence we have\n\\begin{align*}\n&\\sum_{\\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}} \\operatorname{Vol}(G_*(\\gamma)) \\geq \\\\\n&\\int\\limits_{B(R-r)} \\left( \\sum_{\\substack{ \\gamma \\subset B(u,r)\\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies} }} \\right) \\, du = \\int\\limits_{B(R-r)} \\pazocal{N}_{\\zeta}(\\Gamma, k, u, r) \\, du.\n\\end{align*}\nA division by $\\operatorname{Vol}(B(r))$ gives us the desired lower bound. A somewhat similar argument gives the proposed upper bound but we leave this to the reader.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:euclid_count} (1)-(2): existence of an asymptotic law}\n\n\n\\begin{proof}\n\nWe start with $r>0$ fixed and $k \\ge 0$ given.\nDenote $\\pazocal{N}_{\\zeta}(F, k, u, r)$ to be the number of nodal components of $F$ contained in $B(u,r)$ with $k$ $\\zeta$-tangencies,\nand $\\widetilde{\\pazocal{N}}(F,u, r)$ the (total) number of nodal components of $F$ intersecting $\\partial B(u,r)$ (cf. Definition \\ref{def:counts}).\nAn application of Lemma \\ref{l:integro_geom_sandwich} with $\\Gamma=F^{-1}(0)$ and $R$ sufficiently large so that $(1\\pm r\/R)^{2}$ is $\\epsilon$-close to $1$, and bearing in mind the obvious estimate $$\\pazocal{N}^{*}_{\\zeta}(F, k, u, r)\\le \\pazocal{N}_{\\zeta}(F, k, u, r) + \\widetilde{\\pazocal{N}}(F,u, r),$$\nyields the inequality\n\\begin{equation}\n\\label{eq:IGS apply}\n\\begin{split}\n&(1-\\epsilon)\\cdot\\frac{1}{\\operatorname{Vol}(B(R-r))}\n\\int\\limits_{B(R-r)} \\frac{\\pazocal{N}_{\\zeta}(F, k, u, r)}{\\operatorname{Vol}(B(r))} \\, du \\leq  \\frac{\\pazocal{N}_{\\zeta}(F, k, R)}{\\operatorname{Vol}(B(R))} \\\\\n&\\le  (1+\\epsilon)\\cdot\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}_{\\zeta}(F, k, u, r) + \\widetilde{\\pazocal{N}}(F,u, r)}{\\operatorname{Vol}(B(r))}.\n\\end{split}\n\\end{equation}\nNote also that, under the notation of the ergodic Theorem \\ref{thm:weiner}, we have the identity\n$$\\pazocal{N}_{\\zeta}(F, k, u, r)=\\pazocal{N}_{\\zeta}(\\tau_{u}F, k,r)$$ and $\\widetilde{\\pazocal{N}}(F,u, r) = \\widetilde{\\pazocal{N}}(\\tau_{u}F,r)$,\nso that \\eqref{eq:IGS apply} reads\n\\begin{equation}\n\\label{eq:IGS apply trans}\n\\begin{split}\n&(1-\\epsilon)\\cdot \\int\\limits_{B(R-r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, r)}{\\operatorname{Vol}(B(r))} \\, du \\leq  \\frac{\\pazocal{N}_{\\zeta}(F, k, R)}{\\operatorname{Vol}(B(R))} \\\\\n& \\le (1+\\epsilon)\\cdot\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, r) + \\widetilde{\\pazocal{N}}(\\tau_{u}F, r)}{\\operatorname{Vol}(B(r))}.\n\\end{split}\n\\end{equation}\n\nAn application of Theorem \\ref{thm:weiner} on the random variable $$\\Phi_{r;\\zeta,k}(F):=\\frac{\\pazocal{N}_{\\zeta}(F, k, r)}{\\operatorname{Vol}(B(R))},$$ of finite expectation in light of Lemma \\ref{lem:Kac Rice Euclid} (1), yields that both\n\\begin{equation}\n\\label{eq:int mean -> L1 Crzk}\n\\frac{1}{\\operatorname{Vol}(B(R-r))}\\int\\limits_{B(R-r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, r)}{\\operatorname{Vol}(B(r))} ,\\,\n\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, u, r)}{\\operatorname{Vol}(B(r))} \\xrightarrow[L^{1}]{} \\widetilde{C}_{r;\\zeta,k}\n\\end{equation}\nconverge {\\em in mean} to $$\\widetilde{C}_{r;\\zeta,k}=\\widetilde{C}_{F,r;\\zeta,k}:= \\frac{{\\mathbf E}[\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, u, r)]}{\\operatorname{Vol}(B(r))} \\ge 0 .$$\nSame argument, now employing\nLemma \\ref{lem:Kac Rice Euclid} (2) yields that\n\\begin{equation}\n\\label{eq:intrsct->a}\n\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}(\\tau_{u} F, r)}{\\operatorname{Vol}(B(r))} \\xrightarrow[L^{1}]{} a_{F,r},\n\\end{equation}\nwith\n\\begin{equation}\n\\label{eq:a const=O(1\/r)}\na_{F,r} := \\frac{{\\mathbf E}[\\widetilde{\\pazocal{N}}(\\tau_{u}F,r)]}{\\operatorname{Vol}(B(r))} = \\mathcal{O}_{r\\rightarrow\\infty}\\left(\\frac{1}{r} \\right).\n\\end{equation}\n\nSubstituting \\eqref{eq:a const=O(1\/r)} into \\eqref{eq:intrsct->a}, and then together with \\eqref{eq:int mean -> L1 Crzk} into \\eqref{eq:IGS apply trans} implies that\n\\begin{equation}\n\\label{eq:lim const Cauchy}\n{\\mathbf E}\\left[\\left| \\frac{\\pazocal{N}_{\\zeta}(F, k, R)}{\\operatorname{Vol}(B(R))} -  \\widetilde{C}_{r;\\zeta,k}  \\right|  \\right]= \\mathcal{O}\\left( \\epsilon+\\frac{1}{r}  \\right).\n\\end{equation}\nFinally, taking $r\\rightarrow\\infty$ in \\eqref{eq:lim const Cauchy} implies the existence of the limit\n$$\\widetilde{C}_{k,\\zeta}:=\\lim\\limits_{r\\rightarrow\\infty}\\widetilde{C}_{r;\\zeta,k}.$$\nNow, recalling the Nazarov-Sodin constant $c(\\rho)$ satisfying the defining property \\eqref{eq:NS const def prop},\nyields the convergence \\eqref{eq:conv L1 Euclid}\nto the constants $C_{k,\\zeta}:=\\frac{\\widetilde{C}_{k,\\zeta}}{c(\\rho)}$, at the prescribed rate of $\\mathcal{O}(1\/r)$ of convergence.\nFinally, if $k=0$ or $k$ is odd, then a.s. $\\pazocal{N}_{\\zeta}(F, k, R)=0$ by Lemma \\ref{l:at_least_two_tangencies}, Corollary \\ref{cor:tangencies even} and the classical Bulinskaya Lemma \\cite[Proposition 6.11]{AW}, stronger than merely $C_{k,\\zeta}=0$.\nThis concludes the proof of (1) of Theorem \\ref{thm:euclid_count}.\n\nIn case $F$ is isotropic, that $C_{k,\\zeta}$ are independent of $\\zeta$ follows directly\nfrom the invariance of the law of $F$ w.r.t. the rotations. Alternatively, one can note that, in this case,\nthe constants $C_{r;\\zeta,k}$ as in \\eqref{eq:int mean -> L1 Crzk}, a priori\ndependent on direction $\\zeta$ are, in fact, independent of $\\zeta$ thanks to the invariance of $F$ w.r.t. the rotations of ${\\mathbb R}^{2}$,\nand so are the constants $C_{k,\\zeta}$, which is (2) of Theorem \\ref{thm:euclid_count}.\n\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:euclid_count} (3): support of limit direction distribution}\n\nThe deterministic intersection results in \\S\\ref{s:deterministic} play a crucial role in this section. We remind ourselves that for $k$ odd, $C_{k,\\zeta}=0$ follows immediately from the stronger property that $\\pazocal{N}_{\\zeta}(F, k, R)=0$ almost surely.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:euclid_count} (3)]\n\nWe distinguish between two main cases: when $(\\rho 4^{*})$ is satisfied (i.e. the interior of the support of $\\rho$ is non-empty), and when $\\rho=\\sigma_{S^{1}}$.\n\n{\\bf Case $(\\rho 4^{*})$ is satisfied.}\nThe proof follows along similar lines to those of \\cite[Theorem 6]{BW17} and \\cite[Proposition 5.2]{SW18}, namely the ``barrier method\" as initiated in \\cite{NS09} and later generalized in \\cite{So12}.\nThe three main steps are standard, and we summarize them as follows: 1) find a deterministic function on $B(r_0)$, for some $r_0>0$ sufficiently large, so that it contains at least one component with the desired number of tangencies, 2) show that the probability of selecting a random function with almost the same nodal component is positive by establishing that $\\mathcal{H}(\\rho) \\subset C^m(B(R))$ is dense and then using the structure of the Gaussian measure on our probability space $\\Omega$, and 3) assuming the previous two points, show that the expected density of such nodal components is positive via a simple monotonicity argument originating from \\cite{NS09}. Here we give only the details that deviate from referenced article at critical junctures and leave the remaining details to the interested reader.\n\nRecall that by the virtue of axiom $(\\rho 4^{*})$, the interior of supp $\\rho$ is non-empty. This implies, via the same arguments using the existence of approximations to finite sums of Dirac $\\delta$-functions to generate the set of all monomials as in \\cite[Section 3.4]{BW17} that the Hilbert space $\\mathcal{H}(\\rho)$ is dense in $C^m(Q)$, for a given compact set $Q \\subset B(R)$ and for all $m$.\n\nLet $\\zeta$ be a given direction, and $g_k$ be a $C^2({\\mathbb R}^{2})$ function with a regular nodal component $\\Gamma$ that contains the origin, has precisely $k$ tangencies to with respect to the constant vector field $\\zeta$ with each tangency being quantitatively transverse, and is contained in $B(r_0)$. Moreover, we know by the deterministic analysis in \\S\\ref{s:deterministic}, we know there exists a $\\beta_0>0$ such that each point of tangency is $\\beta_0$-transverse for some $\\beta_0>0$ and that each other point of $\\Gamma$ is not sub-$\\beta_0$. By the density of $\\mathcal{H}(\\rho) \\subset C^k(B(R))$, there exists $\\mathfrak{g}_k \\in \\mathcal{H}(\\rho)$ such that $\\|\\mathfrak{g}_k - g_k \\|_{C^2(B(R))} < \\beta_0\/4$.  Furthermore, we find that the neighborhood defined by $\\|\\mathfrak{g} - \\mathfrak{g}_k \\|_{C^2(B(R))} < \\beta_0\/4$ has only random functions $\\mathfrak{g}$ with a nodal component encompassing the origin and with exactly $k$ $\\zeta$-tangencies.  The positivity of the quantity $\\mathbb{P} \\left\\{ \\|\\mathfrak{g} - \\mathfrak{g}_k \\|_{C^2(B(R))} < \\beta_0\/4 \\right\\}$ and consequently of $C_{k, \\zeta}$ are standard; see \\cite{NS15, SW18, BW17}.\n\n{\\bf Case $\\rho=\\sigma_{S^{1}}$.}\nOnce again, our proof follows similar lines as that of \\cite[Theorem 7]{BW17} and \\cite[Proposition 5.2]{SW18} in the case of $\\alpha =1$ except that now we utilize the stronger $C^k(B(R))$ versions of both Whitney's approximation\/extension theorem and the Lax-Malgrange extenstion theorem; see \\cite[Theorem 2.6.3]{Horm} and references therein, and respectively \\cite[Lemma 7.2]{EPS}, for the exact statements. For our purposes, $k=2$ is sufficient.\n\nLet $\\gamma$ be a smooth closed curve in ${\\mathbb R}^2$ that has precisely $k$ (even) tangencies, each of which is quantitatively transverse, with respect to $\\zeta$: that is, we know there exists a $\\beta_0 >0$ such that each of its tangencies is $\\beta_0$-transverse and each other point of $\\gamma$ is not sub-$\\beta_0$. Via an application of the Implicit Function Theorem, we can find a tubular neighborhood $V_{\\gamma}$ of $\\gamma$ and a smooth function $H_{\\gamma}: V_{\\gamma} \\rightarrow {\\mathbb R}$ with $\\gamma = H^{-1}(0)$; furthermore, we can assume that $\\inf_{u \\in V_{\\gamma}} \\| \\nabla H_{\\gamma}(u) \\| > 0$.  Now, the $C^2(B(R))$ version of Whitney's theorem states that for a given $\\epsilon >0$, there exists a real analytic function $G$ such that $\\| G - H_{\\gamma} \\|_{C^2(V_{\\gamma})} < \\epsilon$.\n\nConsider the range $\\beta_0 > \\epsilon > 0$.  By Proposition \\ref{p:stab_trans}, for $\\epsilon$ sufficiently small, we know that $\\widetilde{\\gamma} := G^{-1}(0) \\cap V_{\\gamma}$ has the property that $\\mbox{dist}(\\widetilde{\\gamma}, \\gamma) < \\epsilon$ (this can also be seen from $C^2(B(R))$ version of Thom's Isotopy Theorem; see \\cite{AR}).  Moreover we have that $\\widetilde{\\gamma}$ also has precisely $k$ tangencies with respect to $\\zeta$ each of which is $\\beta_0 - \\epsilon$-transverse. Now, it remains to find a solution of $\\Delta + 1$ that captures this analytic curve $\\widetilde{\\gamma}$ as part of its nodal set.\n\nRepeating the argument as described in the proof of \\cite[Theorem 1.1]{CanSar18}, we use that $\\widetilde{\\gamma}$ separates ${\\mathbb R}^2$ into two components, a bounded component $A_{\\widetilde{\\gamma}}$ and an unbounded component. Let $\\lambda^2$ be the first Dirichlet eigenvalue for the domain $A_{\\widetilde{\\gamma}}$ and let $h_{\\lambda}$ be the corresponding eigenfunction: that is, $(\\nabla + \\lambda^2) h_{\\lambda}(u) = 0$ for $x \\in \\overline{A_{\\widetilde{\\gamma}}}$ and $h_{\\lambda \\, | \\widetilde{\\gamma}} = 0$.\n\nLet $\\overline{\\lambda A_{\\widetilde{\\gamma}}} := \\{ x \\in {\\mathbb R}^2: u\/\\lambda \\in A_{\\widetilde{\\gamma}} \\}$ It follows that $h(u) := h_{\\lambda}(u\/\\lambda)$ solves the Dirichlet problem for $(\\Delta + 1)$ on $\\overline{\\lambda A_{\\widetilde{\\gamma}}}$ and can be extended (thanks to analyticity) to an open set $B_{\\widetilde{\\gamma}}$ such that $\\overline{\\lambda A_{\\widetilde{\\gamma}}} \\subset B_{\\widetilde{\\gamma}}$ as well as solving the following boundary value-type problem:\n\\begin{equation*}\n\\begin{cases}\n(\\Delta + 1)h(u) = 0, & x \\in B_{\\widetilde{\\gamma}} \\\\\nh(u) = 0, & x \\in \\lambda \\widetilde{\\gamma}\n\\end{cases}\n\\end{equation*}\nwhere $\\lambda \\widetilde{\\gamma} := \\{ x \\in {\\mathbb R}^2: u\/\\lambda \\in \\widetilde{\\gamma} \\}$. It is important to notice that $\\lambda \\widetilde{\\gamma}$ continues to have precisely $k$ $\\frac{(\\beta_0-\\epsilon)}{\\lambda^3}$-transverse tangencies with respect to $\\zeta$ thanks to the fact that dilation by $\\lambda$ commutes with the action generated by the constant vector field $\\zeta$ which is translation in the direction $\\zeta$. Added: Now, let $\\epsilon' < \\frac{\\beta_0 - \\epsilon}{10\\lambda^3}$.\n\nOur next-to-penultimate step is an application of a $C^2(B(R))$ version of Lax-Malgrange \\cite[Lemma 7.2]{EPS} which states if $B_{\\widetilde{\\gamma}}^{\\complement}$ has no compact components, then there exists a global solution on ${\\mathbb R}^2$ to the equation $(\\Delta + 1)g(u) = 0$ with the property that $\\|g - h \\|_{C^2(B_{\\widetilde{\\gamma}})} < \\epsilon'$. Another application of Proposition \\ref{p:stab_trans} implies that $g^{-1}(0) \\cap V_{\\widetilde{\\gamma}}$ has precisely $k$ (transverse) tangencies with respect to $\\zeta$. The final step involving disc packing is exactly the same as that for disc packing in the case of $\\alpha < 1$.\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:euclid_count} (4): direction distributions not leaking mass}\n\n\\label{sec:proof thm euclid no leak}\n\n\\begin{proof}\n\nGiven a direction $\\zeta\\in S^{1}$ and $R>0$, denote $\\mathcal{N}_{\\zeta}(R)$ to be the number of\n$\\zeta$-tangency points of $F^{-1}(0)$ lying inside $B(R)$ (where the corresponding nodal component might and might not\nbe contained in $B(R)$).\nThe Euclidean analogue of \\eqref{eq:tot numb tang=sum mean} is the inequality\n\\begin{equation}\n\\label{eq:sum k*nod(k)<=Ncc}\n\\sum\\limits_{k=1}^{\\infty}k\\cdot \\pazocal{N}_{\\zeta}(F,k,R) \\le \\mathcal{N}_{\\zeta}(F,R),\n\\end{equation}\nas we have to discount for those tangency points that lye inside $B(R)$, but belong to the nodal components of $F$\nmerely intersecting $\\partial B(R)$. Since all the involved quantities are nonnegative, we may take the expectation\nof both sides of \\eqref{eq:sum k*nod(k)<=Ncc} and choose the order of summation and expectation as we please (e.g. by Monotone Convergence Theorem), so that, upon invoking Lemma \\ref{lem:Kac-Rice Euclid tangencies}, we may write\n\\begin{equation*}\n\\sum\\limits_{k=1}^{\\infty}k\\cdot {\\mathbf E}[\\pazocal{N}_{\\zeta}(F,k,R)] \\le {\\mathbf E}[\\mathcal{N}_{\\zeta}(F,R)]=C_{0}\\cdot \\operatorname{Vol}(B(R)),\n\\end{equation*}\nfor some $C_{0}>0$ depending on the law of $F$ only.\nEquivalently,\n\\begin{equation}\n\\label{eq:sum k*E[k]\/E<=C0}\n\\sum\\limits_{k=1}^{\\infty}k\\cdot \\frac{{\\mathbf E}[\\pazocal{N}_{\\zeta}(F,k,R)]}{{\\mathbf E}[\\pazocal{N}(F,R)]} \\le C_{0}\\cdot \\operatorname{Vol}(B(R))\/{\\mathbf E}[\\pazocal{N}(F,R)] \\le C_{1},\n\\end{equation}\nwith some $C_{1}>0$, holding for $R$ sufficiently large, by \\eqref{eq:NS const def prop}.\n\n\\vspace{2mm}\n\nWe interpret the l.h.s. of \\eqref{eq:sum k*E[k]\/E<=C0} as the average number of $\\zeta$-tangent points per nodal component of $F$,\nimportantly, bounded by the r.h.s. of \\eqref{eq:sum k*E[k]\/E<=C0} for every $R>0$ sufficiently large.\nThe sequence\n\\begin{equation}\n\\label{eq:aRk def}\na_{R;k}:=\\frac{{\\mathbf E}[\\pazocal{N}_{\\zeta}(F,k,R)]}{{\\mathbf E}[\\pazocal{N}(F,R)]}\n\\end{equation}\nsatisfies for every $R>0$ the equality\n\\begin{equation}\n\\label{eq:sum(aRk)=1}\n\\sum\\limits_{k= 0}^{\\infty}a_{R;k}=1,\n\\end{equation}\nand also, for every $k\\ge 0$,\n\\begin{equation}\n\\label{eq:lim(aRk)=Ckz}\n\\lim\\limits_{R\\rightarrow\\infty}a_{R;k}=C_{k,\\zeta},\n\\end{equation}\nthanks to Theorem \\ref{thm:euclid_count} (1), and \\eqref{eq:NS const def prop}. We claim that \\eqref{eq:sum(aRk)=1}\ntogether with \\eqref{eq:lim(aRk)=Ckz} yield\n\\begin{equation}\n\\label{eq:sum(Ck=1)}\n\\sum\\limits_{k\\ge 0}C_{k,\\zeta}=1,\n\\end{equation}\nwhich is the statement of Theorem \\ref{thm:euclid_count} (4). To this end, it is sufficient to prove the {\\em tightness}\nof $\\{a_{R;k}\\}$ w.r.t. $k$, i.e. that for every $\\epsilon>0$ there exists $K_{0}=K_{0}(\\epsilon)$ and $R_{0}=R_{0}(\\epsilon)>0$ sufficiently large,\nso that\n\\begin{equation*}\n\\sum\\limits_{k=K_{0}}^{\\infty}a_{R;k}<\\epsilon\n\\end{equation*}\nholds for all $R>R_{0}$.\nHowever the said tightness condition follows directly from \\eqref{eq:sum k*E[k]\/E<=C0} (on recalling that $a_{R;k}$ are given by\n\\eqref{eq:aRk def}).\nAs mentioned above, this is sufficient for \\eqref{eq:sum(Ck=1)}, which, in turn, is the statement of Theorem \\ref{thm:euclid_count} (4).\n\n\n\\end{proof}\n\n\n\\section{Riemannian scenario: Local results}\n\n\\subsection{Local setting and statement of the main local result}\n\nLet $V \\in \\mathcal{V}(\\M)$, and define\n\\begin{equation*}\n\\pazocal{N}_{V}(f_{T, \\alpha}, k, x, r)\n\\end{equation*}\nto be the number of components of $f_{T, \\alpha}^{-1}(0)$ completely contained in the geodesic ball $B(x, r)$ whose number of $V$-tangencies (not counting the zeroes of $V$ that might touch $f_{T, \\alpha}^{-1}(0)$) is precisely $k$. That is to say, we are counting the number of  components of $f_{T, \\alpha}^{-1}(0)$ completely contained in the geodesic ball $B(x, r)$ with precisely $k$ zeroes of $V f_T$ each.\n\nRecalling the scaled random fields \\eqref{eq:fxT scaled def} and Definition \\ref{d:blown_up_V} of $\\tilde{V}_{x,T}$, the analogous local quantity for $x \\in \\M \\setminus \\{V=0\\}$ is\n\\begin{equation*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R),\n\\end{equation*}\nwhich, by definition, is equal to the number of components of $f_{x,T}^{-1}(0)$ completely contained in the Euclidean ball $B(0, R)$ whose number of $\\tilde{V}_{x,T}$-tangencies is precisely $k$. Equivalently, $\\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R)$ is the number of connected components of\n$f_{x, T}^{-1}(0)$ completely contained in $B(0, R)$ with precisely $k$ zeroes of $\\tilde{V}_{x,T} f_{x,T}$ each.\n\n\\begin{theo} \\label{t:conv_prob}\nLet $f_{T, \\alpha}$ be the random band-limited functions \\eqref{eq:fT band lim alpha<1} (or \\eqref{eq:fT band lim alpha=1}),\nand $V \\in \\mathcal{V}(\\M)$. Then for all $x \\in \\M \\setminus \\{ V=0 \\}$, $k \\ge 0$, and $\\epsilon>0$, we have\n\\begin{equation}\n\\label{eq:double lim R,T conv loc}\n\\lim\\limits_{R \\rightarrow \\infty} \\limsup\\limits_{T \\rightarrow \\infty} \\, \\mathbb{P} \\left[ \\left| \\frac{\\pazocal{N}_{V}(f_T, k, x, R\/T)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R))}  - C_{k} \\right| > \\epsilon \\right] = 0\n\\end{equation}\nwhere the constants $C_{k}$ are same as in Theorem \\ref{thm:euclid_count} (1)\napplied on the random field $F=\\mathfrak{g}_{\\alpha}$ with $\\zeta$ arbitrary\n($C_{k}$ are independent of $\\zeta$ by Theorem \\ref{thm:euclid_count} (2)), and $c_{2,\\alpha}>0$ is the Nazarov-Sodin constant\nof $\\mathfrak{g}_{\\alpha}$, i.e. satisfying the defining property \\eqref{eq:NS const def prop} with $F=\\mathfrak{g}_{\\alpha}$.\n\\end{theo}\n\n\\subsection{Exceptional events} \\label{s:exceptional_events}\n\nIn this section, we introduce the following parameters and their purposes: a small parameter $\\delta>0$ to control probabilities, $b>0$ to control the quality of coupling approximation, $\\beta_{1,i} > 0$ to control the regularity of our components of $\\mathfrak{g}_{\\alpha}$ for $i=1,2$, $\\beta_{2,i} >0$ to control the nature of our tangencies for $i=1,2$, $M_i$ to control the $C^k$ norms of our random functions for $i=0,1$, large spectral parameter $T>0$, and related $R>0$ controlling our convergence to the scaling limit.\n\n\\begin{defn}[Exceptional events] \\label{d:standard_events}\nLet $R,T, b, M_0, M_1, \\beta_1, \\beta_2 >0$ and $x \\in \\M \\setminus \\{ V=0 \\}$. We define the events as follows:\n\\begin{align*}\n& \\Delta_1 = \\Delta_1(x, R,T;b) = \\{ \\|f_{x,T} - \\mathfrak{g}_{\\alpha} \\|_{C^2(B(2R))} \\geq b \\} \\\\\n& \\Delta_2 = \\Delta_2(R, M_0) = \\{ \\| \\mathfrak{g}_{\\alpha} \\|_{C^2(B(2R))} \\geq M_0 \\} \\\\\n& \\Delta_3 = \\Delta_3(x, R, T, M_1) = \\{ \\| f_{x,T} \\|_{C^2(B(2R))} \\geq M_1 \\} \\\\\n& \\Delta_4 = \\Delta_4(R, \\beta_{1,1}) = \\{ \\min_{u \\in B(2R)} \\max \\{ |\\mathfrak{g}_{\\alpha}(u)|, \\|\\nabla \\mathfrak{g}_{\\alpha}(u)\\|_{2}  \\} \\leq \\beta_{1,1} \\} \\\\\n& \\Delta_5 = \\Delta_5(x, R, T, \\beta_{1,2}) = \\{ \\min_{u \\in B(2R)} \\max \\{ |\\tilde{V}_{x,T}(\\mathfrak{g}_{\\alpha})(u)|, \\|\\nabla (\\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha})(u)\\|_{2}  \\} \\leq \\beta_{1,2} \\}.\n\\end{align*}\n\\end{defn}\n\nWe would like to emphasize that for the event $\\Delta_1$, its analogue in \\cite[Lemma 4]{So12}  only involves the $C^1$ norm and is stated to have a quantitatively low probability. It is however clear, thanks to the flexibility of the Hadamard-Landau inequality, that we can immediately replace $C^1$ by $C^2$ both in the statement of \\cite[Lemma 4]{So12} and its proof. The parameter $\\beta_{1,2}$ in some sense measures our distance away from the set $\\{ V = 0\\}$ on $\\M$.\n\n\\begin{defn} \\label{d:unif_stability_fields}\n For $R>0$, and $\\eta, \\beta_{1,3}>0$ small, we define the \\textit{unstable components} event $\\Delta_6(\\eta, R, \\beta_{1,3})$ as follows:\n\\begin{align*}\n\\Delta_6(\\eta, R,\\beta_{1,3}) := &\\{ \\mbox{the number of components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ with a point $u_0$} \\\\\n& \\mbox{such that } | \\nabla \\mathfrak{g}_{\\alpha}(u_0)| \\leq  \\beta_{1,3} \\mbox{ is at least $\\eta R^2$} \\}\n\\end{align*}\n\\end{defn}\n\n\\begin{defn}  \\label{d:unstable_event}\nFor $x \\in \\M\\setminus \\{V = 0\\}$, $R, T>0$, and $\\eta, \\beta_2>0$ small, we define the \\textit{tangentially unstable components} event $\\Delta_7(V, x, R, T, \\beta_2, \\eta)$ as follows:\n\\begin{align*}\n\\Delta_7(V, x, R, T, \\beta_2, \\eta) := &\\{ \\mbox{the number of components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$, with point $u_0$ such that} \\\\\n& \\det [\\nabla \\mathfrak{g}_{\\alpha}(u_0),  \\nabla \\left( \\tilde{V}_{x,T} \\, \\mathfrak{g}_{\\alpha} \\right) (u_0) ] | \\leq  \\beta_2 \\mbox{, is at least $\\eta R^2$} \\}\n\\end{align*}\nwhere $\\tilde{V}_{x,T}$ is the blown-up vector field at $x$, at scale $T$, as given in Definition \\ref{d:blown_up_V}.\n\\end{defn}\n\nA combination of the Borel-TIS and Sudakov-Fernique inequalities along with the argument in \\cite[Section 4.3]{BW17} using Chebyshev's Inequality give us a probability of less than $\\delta$ for the events $\\Delta_2$ and $\\Delta_3$.\n\n\\begin{lem}[Uniform stability for components of smooth Gaussian fields, Cf. {\\cite[Proposition 4.3]{SW18}}] \\label{p:unstable_comp}\nGiven $\\delta>0$, $\\eta >0$, there exists $\\beta_{1,3}(\\delta,\\eta)>0$ (possibly small) and $R_0(\\delta, \\eta)$ such that for all $R \\geq R_0$, we have that $\\mathbb{P}[\\Delta_6(\\eta, R, \\beta_{1,3})] < \\delta$.\n\\end{lem}\n\n\\begin{prop}[Uniform stability of $\\beta_2$-transverse tangencies for smooth Gaussian fields] \\label{p:unstable_tangencies}\nGiven $\\eta, \\delta > 0$ and $x \\in \\M\\setminus\\{V = 0\\}$, there exists $\\beta_{2}=\\beta_{2}(\\eta,\\delta, V)>0$, and in turn\n$R_0(\\delta, \\eta, \\beta_{2},x)$ such that for all $R \\geq R_0$, there exists a $T_0(R, \\beta_{2}, V)>0$ such that for all $T \\geq T_0$, we have $\\mathbb{P}[\\Delta_7(V,x,R, T, \\beta_2, \\eta)] < \\delta$.\n\\end{prop}\n\n\nWe would like to remind the reader of the following standard fact:\n\\begin{lem} \\label{l:qual_bulinskaya}\nFor all $x \\in \\M \\setminus \\{V=0\\}$, we have that the event\n\\begin{align*}\n\\Delta_8(V, x, T, R) &= \\bigg\\{ \\mbox{there exists } u_0 \\in B(R) \\mbox{ such that } \\mathfrak{g}_{\\alpha}(u_0)=\\left(\\tilde{V}_{x,T} \\, \\mathfrak{g}_{\\alpha} \\right) (u_0) \\\\\n&=  \\det \\left[\\nabla \\mathfrak{g}_{\\alpha}(u_0),  \\nabla \\left( \\tilde{V}_{x,T} \\, \\mathfrak{g}_{\\alpha} \\right)(u_0) \\right] = 0 \\bigg\\}\n\\end{align*}\nhas probability 0.\n\\end{lem}\n\n\n\\begin{proof}[Proof of Lemma \\ref{l:qual_bulinskaya}]\nThis is an immediate application of Bulinskaya's lemma \\cite[Proposition 6.11]{AW} on the random field $(\\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T} \\mathfrak{g}_{\\alpha})$.\n\\end{proof}\n\nThanks to having both $\\mathfrak{g}_{\\alpha}^{-1}(0)$ and $( \\tilde{V}_{x,T} \\mathfrak{g}_{\\alpha})^{-1}(0)$ (and the zero sets for the corresponding perturbations) being regular with probability 1, that the intersection between these two sets is always transversal by Lemma \\ref{l:qual_bulinskaya}, and finally Lemma \\ref{l:even_int}, we are able to focus only on events where the nodal components have an even number of tangencies to $\\tilde{V}_{x,T}$ (although $k=0$ may occur and is addressed in the proof of Proposition \\ref{p:stab_k_even}). Thus, we have reduce our analysis to that of $\\beta$-transverse tangencies and sub-$\\beta$ tangencies.\n\nIn closing this, we set\n\\begin{equation}\n\\label{eq:E good event def}\nE = \\cap_{i=1}^8 \\Delta_i^{\\complement}\n\\end{equation}\nwith the corresponding range of parameters so that $\\mathbb{P}[E] > 1 - \\delta$, where $\\delta > 0$ is given. We will refer to this $E$ numerous times in upcoming sections.\n\n\n\\subsection{High probability stability estimates for local counts} \\label{s:stab_est}\n\nA $\\mbox{sub}-\\frac{3}{2} \\beta$ tangency of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ is mapped, under a small $C^{2}$ perturbation, to $\\mbox{sub}-\\left( \\frac{3}{2}\\beta + b \\right)$ tangency of $f_{x,T}^{-1}(0)$, where $b$ is small (see Definition \\ref{d:standard_events}).  We will later prove that components containing these near-degenerate tangencies are {\\em few} in number, with high probability (see Proposition \\ref{p:unstable_tangencies}).\n\n\n\\begin{prop}[Stability for the number of tangencies] \\label{p:stab_k_even}\nLet $V \\in \\mathcal{V}(\\M)$ be given, and let $\\tilde{V}_{x,T}$ denote the blow-up of $V$ at $x$. We have that for all $x \\in \\M \\setminus \\{V = 0\\}$, $\\delta>0$, $\\eta >0$, and $k \\ge 0$, there exists a $R_0(x, \\delta, \\eta, \\|V(x)\\|)$ such that for all $R \\geq R_0$, there exists a $T_0(R, \\|V(x)\\|, \\delta, k)$ such that for all $T \\geq T_0$, we have the estimate\n\\begin{align}\n\\label{eq:perturb count}\n \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R-1) - \\eta R^2 & \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k , R)\n \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R+1) +  \\eta R^2,\n\\end{align}\nwith probability $> 1 - \\delta.$\n\\end{prop}\n\nNote that in Proposition \\ref{p:stab_k_even} we are counting all $k$ tangencies, whether they are $\\beta$-transversal or sub-$\\beta$. This is the source of the extra terms of $\\pm\\eta R^{2}$ on both l.h.s. and r.h.s. of \\eqref{eq:perturb count}, accounting for the sub-$\\beta$, or rather near-degenerate, tangencies.\n\n\n\nLet $V \\in \\mathcal{V}$. We will apply our deterministic intersection results from \\S\\ref{s:deterministic} to smooth Gaussian fields $G = (\\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha})$, where $\\tilde{V}_{x,T}$ is the blow-up of $V$ at $x \\in \\M$,  and $(f_{x,T}, \\frac{1}{T}(Vf_T)_{x,T})$. It is important to note that there two advantages to considering $\\frac{1}{T}(Vf_T)_{x,T}$. First $(\\frac{1}{T} V)f_T$ and $ V(f_T)$ share the same zero sets and second, the zeroes of $\\frac{1}{T}(Vf_T)_{x,T}$ and $\\tilde{V}_{x,T}(f_{x,T})$ are close with respect to a natural perturbation parameter that appears when summoning certain coupling results (cf. \\cite{SW18, So12} and Definition \\ref{d:standard_events}).\n\n\\begin{lem}[Blowing up ``almost\" commutes with differentiation] \\label{l:non_commuting}\nGiven $x \\in \\M\\setminus\\{V=0\\}$, and $R>0$, there exists $T_0(x,R,b)$ such that for all $T \\geq T_0$, $\\tilde{V}_{x,T}$ is non-vanishing in $B(R) \\subset T_x\\M$ and we have that\n\\begin{align*}\n T \\tilde{V}_{x,T} \\left( f_{x,T} \\right)  = (Vf_T)_{x,T}+ \\mathcal{O}_{\\|f_{x,T}\\|_{C^1}}\\left( \\frac{R}{T}\\right).\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nThis is an immediate consequence of Taylor's theorem and the fact that for $T$ large enough, $R\/T < \\min \\{inj(\\M), 1 \\}$ and that $D(\\exp_x)_{Z = \\frac{Y}{T}} = Id + \\mathcal{O} \\left( \\frac{R}{T} \\right)$ as $|Y|\/T \\leq R\/T$.\n\\end{proof}\n\nWe need an intermediate estimate on the plane before proceeding to the proof of Theorem \\ref{t:conv_prob}:\n\\begin{lem}[Close vector fields generate the same count] \\label{l:close_vector_fields}\nLet $\\delta > 0$ be given.  Consider the parameter $\\beta_2$ that controls degeneracy of our tangencies as described in \\S\\ref{s:exceptional_events}.  Let $0 < \\epsilon \\leq \\beta_2$ be given. Suppose $\\tilde{V}_1, \\tilde{V}_2 \\in \\mathcal{V}({\\mathbb R}^2)$ are non-vanishing and that $\\| \\tilde{V}_1 - \\tilde{V}_2 \\|_{C^1(B(R))} < \\epsilon$; that is, their coefficients with respect to the standard basis are $C^1$ close.\nThen, for all $k \\ge 0$, for all $\\eta >0$, on the event $E$ as in \\eqref{eq:E good event def} (of probability $>1-\\delta$), there exists $R_0(\\eta, \\epsilon)$ such that for all $R \\geq R_0$, the following inequality holds:\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_1}(\\mathfrak{g}_{\\alpha}, k , R-1) - \\eta R^2 & \\leq \\pazocal{N}_{\\tilde{V}_2}(\\mathfrak{g}_{\\alpha}, k , R) \\leq  \\pazocal{N}_{\\tilde{V}_1}(\\mathfrak{g}_{\\alpha}, k , R+1) + \\eta R^2.\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}[Proof of Lemma \\ref{l:close_vector_fields}]\nThe proof follows almost every step of the proof of our main Proposition \\ref{p:stab_k_even} given in \\S\\ref{sect:proof_main_stab_prop} except that we do not perturb by the vector function $( f_{x,T} - \\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T}(f_{x, T}  - \\mathfrak{g}_{\\alpha}) )$ but by the simpler function $(0, (V_2-V_1) \\mathfrak{g}_{\\alpha})$. This type of perturbation is under the scope of the hypotheses of Lemma \\ref{l:Sodin_type_calc_lem} where we set $b < \\epsilon$ and $\\Psi = (0, (V_2-V_1) \\mathfrak{g}_{\\alpha})$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:conv_prob} assuming Proposition \\ref{p:stab_k_even}, Lemma \\ref{l:non_commuting} and Lemma \\ref{l:close_vector_fields}]\n\nConsidering the listed proposition and lemmas, we let $\\delta, \\eta>0$ be given and $k\\ge 0$. Now take the parameters as prescribed in \\S\\ref{s:exceptional_events}: that is, take $R_0, M_0, M_1, \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2$, and $T_0$.\n\nWe assume that\n\\begin{equation} \\label{e:final_freq_stab}\nR\/T < \\min \\left\\{1, \\mbox{inj}(\\M), \\frac{1}{10 M_0}, \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2 \\right\\}\n\\end{equation}\nafter possibly taking the initial $T_0$ larger than is posed in \\S\\ref{s:exceptional_events}.  We know that $(Vf_T)_{x,T} = T (\\tilde{V}_{x,T})(f_{x,T}) + \\mathcal{O}_{\\|f_{x,T}\\|_{C^1}}\\left( \\frac{R}{T} \\right)$ in the local coordinates w.r.t. $\\exp_x$ by Lemma \\ref{l:non_commuting}.  Let us define a new quantity $\\pazocal{N} \\left(f_{x,T}, (Vf_T)_{x,T}, k,  R  \\right)$ and set it equal to the number of components of $f_{x,T}^{-1}(0)$ contained inside of $B(R)$ whose number of points of intersection with $(Vf_T)_{x,T}^{-1}(0)$ is exactly $k$.  Recall that in Section 4, we established that understanding the number of tangencies was equivalent to understanding the number of intersections between two sets of curves.  From this, we see immediately that $\\pazocal{N}_{V}(f_T, x, k, \\frac{R}{T})$ is bounded above and below by\n\\begin{equation*}\n\\pazocal{N} \\left(f_{x,T}, (Vf_T)_{x,T}, k,  R \\pm 1  \\right).\n\\end{equation*}\n\nNow by our choice of $T_0$ as indicated after \\eqref{e:final_freq_stab}, Lemma \\ref{l:close_vector_fields}\n(in particular, its proof which uses the equivalence between counting tangencies and counting intersections), and Corollary \\ref{l:Sodin_annuli}, we have that $\\pazocal{N} \\left(f_{x,T}, (Vf_T)_{x,T}, k,  R \\pm 1  \\right)$ itself is bounded above and below\nby $\\pazocal{N}_{T\\tilde{V}_{x,T}}\\left( f_{x,T}, k,  R \\pm 2  \\right) \\pm \\eta R^2.$ Thus,\n\\begin{equation*}\n \\pazocal{N}_{T \\, \\tilde{V}_{x,T}}(f_{x,T}, k, R-2)  - \\eta R^2 \\leq \\pazocal{N}_{V}\\left( f_T,x, k,  \\frac{R}{T}  \\right) \\leq \\pazocal{N}_{T \\, \\tilde{V}_{x,T}}(f_{x,T}, k, R+2)   + \\eta R^2.\n\\end{equation*}\nNow, we emphasis that $\\pazocal{N}_{T \\, \\tilde{V}_{x,T}}(f_{x,T}, k, R \\pm 2) =  \\pazocal{N}_{ \\tilde{V}_{x,T}}(f_{x,T}, k, R \\pm 2)$ since joint zeroes of $(f_{x,T}, T \\, \\tilde{V}_{x,T}(f_{x,T}))$ are the same as those of $(f_{x,T}, \\tilde{V}_{x,T}(f_{x,T}))$, therefore giving us a one-to-one correspondence between components of $f_{x,T}^{-1}(0)$ with precisely $k$ zeroes of $\\tilde{V}_{x,T}(f_{x,T})$ and those with precisely $k$ zeroes of $T\\tilde{V}_{x,T}(f_{x,T})$.  This leads us to\n\\begin{equation*}\n \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R-2)  - \\eta R^2 \\leq \\pazocal{N}_{V}\\left( f_T,x, k,  \\frac{R}{T}  \\right) \\leq \\pazocal{N}_{ \\tilde{V}_{x,T}}(f_{x,T}, k, R+2)   + \\eta R^2.\n\\end{equation*}\n\nIn the local coordinates around $x$ given by $y=\\exp_x(Y)$, we set $\\tilde{V}(x)$ to be the constant vector field given by the trivial extension of $(\\tilde{V}_{x,T})_{|Y=0}$ to ${\\mathbb R}^2$. Proposition \\ref{p:stab_k_even} yields us with probability $1 - \\delta$ that there exists $R_1$ such that $R \\geq R_1(x)$ and $T_1(R)$ such that $T \\geq T_1$, along with an application of the vector field comparison statement Lemma \\ref{l:close_vector_fields} in comparing $\\tilde{V}_{x,T}$ to the constant field $\\tilde{V}(x)$ by taking $T$ as in (\\ref{e:final_freq_stab}), we have\n\\begin{align*}\n \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R-1) - 2\\eta R^2 & \\leq \\pazocal{N}_{\\tilde{V}(x)}(f_{x,T},  k , R)  \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R+1) + 2\\eta R^2.\n\\end{align*}\nNow, apply Lemma \\ref{l:close_vector_fields} for the second time in the case of $f_{x,T}$, $V_1 = \\tilde{V}_{x,T}$, $V_2 = \\tilde{V}(x)$, and the same values of $R,T$ as in \\eqref{e:final_freq_stab}, to obtain\n\\begin{equation*}\n \\pazocal{N}_{\\tilde{V}(x)}(f_{x,T}, k, R-3)  - \\eta R^2 \\leq \\pazocal{N}_{V}\\left( f_T,x, k,  \\frac{R}{T}  \\right) \\leq \\pazocal{N}_{\\tilde{V}(x)}(f_{x,T}, k, R+3)   + \\eta R^2.\n\\end{equation*}\n\nLet $\\epsilon_1>0$ be given. By the above estimates, occurring with probability $> 1 - \\delta$, this implies\n\\begin{align*}\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{V}\\left( f_T, x,  k , \\frac{R}{T} \\right)}{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 \\right] & \\leq\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{\\tilde{V}(x)}(\\mathfrak{g}_{\\alpha}, k , R-4) }{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 - \\frac{4\\eta}{c_{2, \\alpha} \\pi} \\right] + \\\\\n& \\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{\\tilde{V}(x)}(\\mathfrak{g}_{\\alpha}, k , R+4) }{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 - \\frac{4\\eta}{c_{2, \\alpha} \\pi} \\right] +  \\mathcal{O}(\\delta); \\\\\n\\end{align*}\nnote we have used that $R^2 = (R - 2)^2 + \\mathcal{O}(R)$ in $R$ along with the Kac-Rice formula applied to critical points for $\\mathfrak{g}_{\\alpha}$ as in \\cite[Corollary 2.3]{SW18} to show that\n\\begin{equation*}\n\\frac{ \\pazocal{N}_{\\frac{\\partial}{\\partial Y_1}}(\\mathfrak{g}_{\\alpha}, k , R-4) }{c_{2, \\alpha} \\pi R^2}  = \\frac{ \\pazocal{N}_{\\frac{\\partial}{\\partial Y_1}}(\\mathfrak{g}_{\\alpha}, k , R-4) }{c_{2, \\alpha} \\pi (R-4)^2} + o(1)\n\\end{equation*}\nfor $R \\geq R_3$.  To guarantee that $ \\epsilon_1 - \\frac{4\\eta}{c_{2, \\alpha} \\pi} > \\frac{\\epsilon_1}{2}$, we choose $\\eta$ possibly smaller.\n\nThanks to all of this, we have that for any radius parameter $R \\geq R_4$, we can take $T \\geq T_4$ (for $R_4$ and $T_4$ chosen possibly larger once again after invoking our Euclidean nodal count result Theorem \\ref{thm:euclid_count}) to obtain the estimate\n\\begin{equation*}\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{\\tilde{V}(x)}(\\mathfrak{g}_{\\alpha}, k , R \\pm 4) }{c_{2, \\alpha} \\pi (R \\pm 4)^2} - C_k \\right| > \\frac{\\epsilon_1}{2} \\right] = o(1)\n\\end{equation*}\nwhich in turn implies that\n\\begin{equation*}\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{V}\\left( f_T, x,  k , \\frac{R}{T} \\right)}{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 \\right] = o(1)\n\\end{equation*}\nThis concludes our proof.\n\\end{proof}\n\n\\subsection{Proof of Proposition \\ref{p:stab_k_even}: stability estimate} \\label{sect:proof_main_stab_prop}\n\nIn this section we always take $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ to be nowhere vanishing.\n\n\\begin{defn}\n\n\\begin{enumerate}\n\n\\item We denote by $\\pazocal{N}_{\\tilde{V}}^{\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R)$ the number of connected components of\n$\\mathfrak{g}_{\\alpha}^{-1}(0)$, strictly contained in $B(R)$, whose number of $\\tilde{V}$-tangencies is precisely $k$ with each point of tangency being $\\beta$-transverse (cf. Definition \\ref{d:beta_trans}).\n\n\\item We denote by $\\pazocal{N}_{\\tilde{V}}^{\\mbox{sub}-\\beta}(\\mathfrak{g}_{\\alpha}, k, R)$ the number of connected components of\n$\\mathfrak{g}_{\\alpha}^{-1}(0)$, strictly contained in $B(R)$,  whose number of $\\tilde{V}$-tangencies is precisely $k$ with at least one point of tangency being sub-$\\beta$ (cf. Definition \\ref{d:sub_beta}).\n\n\\end{enumerate}\n\n\\end{defn}\n\n\\begin{prop}[Stability of $\\beta$-transversality] \\label{p:b_trans_est}\nLet $k \\ge 0, \\eta, \\delta > 0$, and $x \\in \\M \\setminus \\{V = 0\\}$ be given. Then there exists $\\beta_0(x)$ and $R_0(V,x, \\eta)>0$ such that for all $\\beta < \\beta_0$ and $R \\geq R_0$, there exists $T_0(V,x, R, \\beta, \\eta)>0$ and $b(\\beta, V)>0$ such that for all $T \\geq T_0$, we have the estimate\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{3}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) & \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\beta-\\mbox{trans}}(f_{x,T}, k, R)\n \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{1}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R+1)\n\\end{align*}\nfor an event $E$, where $\\mathbb{P} \\left[ E \\right] > 1 - \\delta$ and $\\|f_{x,T} - \\mathfrak{g}_{\\alpha} \\|_{C^2(B(R))} < b$.\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition \\ref{p:stab_k_even} assuming Proposition \\ref{p:b_trans_est} and Proposition \\ref{p:unstable_tangencies}]\nWe prove the first inequality as the second inequality will be proved in an identical fashion. Let $\\eta>0$ be given as in the referenced propositions, and assume that $E$ as in \\eqref{eq:E good event def} occurs.\nSince we have\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k, R-1) &= \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{3}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) + \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\mbox{sub}-\\frac{3}{2}\\beta}(\\mathfrak{g}_{\\alpha}, k, R-1),\n\\end{align*}\nProposition \\ref{p:unstable_tangencies} implies that for $\\beta$ possibly smaller than $\\beta_0(\\eta) = \\beta_2$ as initally given in \\S\\ref{s:exceptional_events}, that\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k, R-1) \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{3}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) + \\eta R^2\n\\end{align*}\nfor all $R \\geq R_0$ large enough and $T \\geq T_0$ with $T_0$ being large enough and depending on $R$ amongst other parameters.  We are now in a position to apply Proposition \\ref{p:b_trans_est} and repeat this argument for $\\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R)$; that is, we write\n\\begin{align*}\n& \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k, R-1) \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\beta-\\mbox{trans}}(f_{x,T}, k, R) + \\eta R^2 \\\\\n& = \\left( \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R) - \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\mbox{sub}-\\beta}(f_{x,T}, k, R) \\right) + \\eta R^2 \\\\\n& \\leq  \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R) + \\eta R^2\n\\end{align*}\nsince $\\pazocal{N}_{\\tilde{V}_{x,T}}^{\\mbox{sub}-\\beta}(f_{x,T}, k, R) \\geq 0$. Note that for the upper bound of $ \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R)$, a similar argument follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{p:b_trans_est} assuming the results in \\S\\ref{s:exceptional_events} and \\S\\ref{s:local_results}]\n\nLet $\\eta, \\delta>0$ be given and $\\mathfrak{G} = (\\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha})$. Now, consider the multi-parameter-dependent event $$E(x,R,T, \\eta, \\delta, \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2, b),$$ as in \\eqref{eq:E good event def}.\nLet $\\beta_0 := \\min \\{  \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2\\}$ and set $b < \\frac{\\beta_0}{4}$. We now satisfy the hypotheses of Proposition \\ref{p:stab_trans}.  On the corresponding event $E$, we obtain some mapping properties of the components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ to the components of $f_{x,T}$ after perturbations of $C^1$ size $b$. More specifically, we know that after adding the perturbation $f_{x,T} - \\mathfrak{g}_{\\alpha}$ to $\\mathfrak{g}_{\\alpha}$, we have that each of the $\\frac{3}{2} \\beta_2$-transverse tangencies of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ goes to a $ \\left( \\frac{3}{2}\\beta_2 - b \\right)$-transverse tangency of $f_{x,T}^{-1}(0)$. Thus, by Corollary \\ref{l:Sodin_annuli} a component of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ whose number of tangencies is precisely $k$ and each of which is $\\frac{3}{2} \\beta_2$-transverse is mapped to a component of $f_{x,T}^{-1}(0)$ whose number of tangencies is precisely $k$ and each of which is $(\\frac{3}{2} \\beta_2 - b)$- transverse. Thus, we have some form of injectivity and it follows that\n\\begin{equation*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}^{(\\frac{3}{2}\\beta_2 )-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{(\\frac{3}{2}\\beta_2 - b)-\\mbox{trans}}(f_{x,T}, k, R).\n\\end{equation*}\nThe second inequality follows similarly.\n\\end{proof}\n\n\n\\subsection{Proof of Proposition \\ref{p:unstable_tangencies}: high probability of few components with near-degenerate tangencies}\nIn this section, let $T>0$ and $\\mathfrak{G}: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be the smooth Gaussian field\n\\begin{equation}\n\\label{eq:frakG def}\n\\mathfrak{G}(u) := \\left( \\mathfrak{g}_{\\alpha}(u), \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha}(u)\\right).\n\\end{equation}\nIt is important to note that all the results in this section also hold for the Gaussian field $\\mathfrak{F}(u) := \\left( f_{x,T}(u), \\tilde{V}_{x,T}f_{x,T}(u)\\right)$ thanks to the exclusion of the $C^2$ decoupling event $\\Delta_1$ as in \\S\\ref{s:exceptional_events}.\n\n\\begin{defn}\nLet $\\eta>0$, and $\\beta_{2,2}>0$ be small, $R, R_1>0$ be large parameters such that $0< R_1 < R$ and $R\/R_1$ is itself large. Let us cover $B(R)$ with approximately $(R\/R_1)^2$ balls $\\mathcal{D}_i$ of radius $R_1$ such that the multiplicity of the covering is bounded by a constant $\\kappa>0$.  Denote by $\\mathcal{G}_i$ the balls centred at the same points as $\\mathcal{D}_i$ with radii $3R_1$. Note that the multiplicity of the covering $\\{ \\mathcal{G}_i \\}_i$ is bounded by $c_0(2) \\kappa$.\n\n\\begin{enumerate}\n\\item We say that the smooth random field $G=(g_1,g_2)$ is $( \\beta_{2,2}, 3R_1)$-stable on a ball $\\mathcal{G}_i$ if for all $u \\in \\mathcal{G}_i$, we\nhave $|g_1(u)| > \\beta_{2,2}$ or\n$|\\det DG(u)| > \\beta_{2,2}$; otherwise we say that\n$G$ is $(\\beta_{2,2})$-unstable on $\\mathcal{G}_i$.\n\\item We say that $G$ is $(\\eta, \\beta_{2,2}, 3R_1)$-stable if $G$ is $(\\beta_{2,2}, 3R_1)$-stable on all $\\mathcal{G}_i$ except\nfor up to $\\eta R^2$ ones.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{prop} \\label{p:abc_stability}\nLet $x \\in \\M \\setminus \\{V=0\\}$, and $R_{1}>0$. Given $\\delta>0$, $\\eta>0$, there exists a positive number $\\beta_{2,2}(\\delta,R_{1},\\eta)$ so that for all $R>0$ with $R\/R_1 > 100$, there exists a spectral parameter value $T_0(V, R, \\beta_{2,2})$ such that for all $T \\geq T_0$, any random field $\\mathfrak{G}_{|B(R)}$ as in \\eqref{eq:frakG def} is also $(\\eta,\\beta_{2,2}, 3R_1)$-stable\nwith probability $1 - \\delta$.\n\\end{prop}\n\nNote that there is an implicit dependence of all the parameters above on the radius $R_1$ for the elements in our covering. We will track this throughout our calculations. To prove this proposition, we quote the following two lemmas from \\cite{SW18}:\n\n\\begin{lem}[Cf. {\\cite[Lemma 4.9]{SW18}}] \\label{l:4_separation}\nThere exists a constant $c_0 = c_0(\\kappa) > 0$ depending only on $\\kappa$ with the following property: for $\\mathcal{G} = \\{ \\mathcal{G}_i \\}_{i \\leq K}$ a collection of radius-$3R_1$ balls lying in $B(R)$ such that each point $x \\in B(R)$ lies in at most $\\kappa$ elements of $\\mathcal{G}$, we have that $\\mathcal{G}$ contains at most $c_0 \\, K$ balls that are in addition $4$-separated.\n\\end{lem}\n\n\n\\begin{lem}[Cf. {\\cite[Lemma 4.10]{SW18}}] \\label{l:derivative_bounds}\nLet $F$ be a stationary Gaussian random field. For all $\\delta>0$ and for all $K, m \\in \\mathbb{N}$, there exists $C_0(\\epsilon) >0$ such that for any (possibly random) collection of centers $\\{ u_i \\}_{i \\leq K}$ satisfying $d(u_i, u_j) > 4$ for $i \\neq j$, there exist $\\lfloor K\/2 \\rfloor$ points $\\{ u_{i_j} \\}_{i_j \\in I}$ with $|I| = \\lfloor K\/2 \\rfloor$ such that\n\\begin{equation} \\label{e:small_derivatives}\n\\sup\\limits_{| \\nu | \\leq m, B(u_{i_j}, 1)} | \\partial^{\\nu} F(u) | \\leq C_0 \\frac{R}{\\sqrt{K}}\n\\end{equation}\nholds with probability $> 1 - \\delta$.\n\\end{lem}\n\nFor the field $F = \\mathfrak{g}_{\\alpha}$, the constant $C_0$ in (\\ref{e:small_derivatives}) depends on the quantity $\\sum_{|\\nu| \\leq s_0} {\\mathbf E} \\left[  | \\partial^{\\nu} F(0) |  \\right]$, thanks to our Gaussian process being stationary. For $F= \\tilde{V}_{x,T} \\mathfrak{g}_{\\alpha}$,\nLemma \\ref{l:derivative_bounds} is not directly applicable, since $F$ is no longer stationary. However, in what follows, we argue that the statement\n\\eqref{e:small_derivatives} of Lemma \\ref{l:derivative_bounds} does hold with $F$, possibly increasing the $C_{0}$ in \\eqref{e:small_derivatives}.\nFirst, the derivatives of $F$\ncould be expressed in terms of the derivatives of $\\mathfrak{g}_{\\alpha}$ and the derivatives of $\\tilde{V}_{x,T}$.\nAlso, the vector field $\\tilde{V}_{x,T}$ is asymptotically constant, as can be seen from Taylor expanding the coefficients $(a_1)_{x,T}$ and $(a_2)_{x,T}$ as in \\eqref{eq:tild(V) def},\nand using $$D(\\exp_x)_{Z = \\frac{Y}{T}} = Id + \\mathcal{O} \\left( \\frac{R}{T} \\right)$$ for $|Y|\/T \\leq R\/T$.\nHence it follows that \\eqref{e:small_derivatives} holds, with the constant $C_0$ in the r.h.s. of \\eqref{e:small_derivatives} now depending on the derivatives of these coefficients of $V$, in Riemannian normal coordinates, evaluated at $0$.\n\n\\vspace{2mm}\n\nIn preparation for the proof of Proposition \\ref{p:abc_stability}, we need an estimate on some volume quantities.\n\n\\begin{lem}[Volumes of neighborhoods of sub-$\\beta$ tangencies] \\label{l:det_cov}\nLet $x \\in \\M \\setminus \\{V=0\\}$, $A, B >0$ be small and $u \\in B(R)$ be fixed.\nThen there exists $C, \\sigma_1, \\sigma_2>0$ and $T_0(V,x)>0$ such that for all $T \\geq T_0$,\n\\begin{align*}\n& \\mathbb{P} \\left[ \\mathfrak{g}_{\\alpha} : |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right] \\leq \\\\\n& C A^2 \\cdot \\sqrt{\\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1 - \\sigma_1} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-2\\sigma_1} + A^{1 - \\sigma_2} + B },\n\\end{align*}\nwith $\\mathfrak{G}(\\cdot)$ as in \\eqref{eq:frakG def} (implicitly depending on $x$ and $T$),\nand constants involved in the $``\\mathcal{O}\"$-notation depending on $\\alpha$ and $V$ only.\n\\end{lem}\n\\begin{proof}\nConsider the smooth Gaussian field\n\\begin{equation*}\nH:=\\left( \\mathfrak{g}_{\\alpha}, \\nabla \\mathfrak{g}_{\\alpha}, \\partial_{1,2} \\mathfrak{g}_{\\alpha}, \\partial_{1,1} \\mathfrak{g}_{\\alpha} \\right).\n\\end{equation*}\nThe distribution of $H(u)$ is non-degenerate Gaussian for every $u\\in{\\mathbb R}^{2}$ and $\\alpha\\in [0,1]$ (for $\\alpha<1$ it follows from the axiom\n$(\\rho 4^{*})$, whereas for the most subtle $\\alpha=1$ the non-degeneracy of the distribution was shown in \\cite[Appendix 1]{CMW16} via an explicit computation; it is no longer non-degenerate if $\\partial_{2,2}\\mathfrak{g}_{\\alpha}$ is added to the vector). The field $H$ proves useful for the following reasons. By rotating the plane ${\\mathbb R}^{2} \\simeq T_x \\M$\nif necessary (and using the rotation invariance of the law of $\\mathfrak{g}_{\\alpha}(\\cdot) $), we may assume that\n\\begin{equation*}\n\\tilde{V}_{x,T} = (1 + a_1(u)) \\partial_1 + a_2(u) \\partial_2\n\\end{equation*}\nwhere $a_1=a_2= \\mathcal{O}(R\/T)$ and have the additional property of $a_1(0)=a_2(0)=0$. Thus, thanks to the stationarity of $\\mathfrak{g}_{\\alpha}$ (and hence of $\\mathfrak{G}$), we have\n\\begin{align*}\n& \\mathbb{P} \\left[ \\{ |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right] =\n\\mathbb{P} \\left[ \\{ |\\mathfrak{G}(0)| < A, | \\det D \\mathfrak{G}(0)| < B \\} \\right] \\\\&=\n \\mathbb{P} \\left[ \\{ |(\\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0))| < A, |\\det D \\mathfrak{G}(0)| < B \\} \\right]\n\\end{align*}\nwhere\n\\begin{align*}\n\\det D \\mathfrak{G}(0) & = \\partial_2 \\mathfrak{g}_{\\alpha} \\cdot \\partial_{1,1} \\mathfrak{g}_{\\alpha} - \\partial_{1} \\mathfrak{g}_{\\alpha} \\cdot \\partial_{2,1} \\mathfrak{g}_{\\alpha} + \\left( \\partial_1 a_1 - \\partial_2 a_2 \\right) \\cdot (\\partial_1 \\mathfrak{g}_{\\alpha}) \\cdot (\\partial_2 \\mathfrak{g}_{\\alpha}) \\\\ & + \\partial_1 a_2 \\cdot (\\partial_2 \\mathfrak{g}_{\\alpha})^2 - \\partial_2 a_1 \\cdot (\\partial_1 \\mathfrak{g}_{\\alpha})^2_{|u=0}.\n\\end{align*}\n\nIt follows that,\nfor some absolute constant $c>0$,\n\\begin{align} \\label{e:ub_vol_bound}\n\\nonumber & \\mathbb{P} \\left[ \\{  |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right]\n \\leq cA^2 \\cdot \\mathbb{P} \\left[ \\{ | \\det D \\mathfrak{G}(0)| < B \\, \\bigg| \\, \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A  \\} \\right] \\\\\n& \\leq cA^2 \\cdot \\mathbb{P} \\left[ \\{ | \\partial_2 \\mathfrak{g}_{\\alpha} \\cdot \\partial_{1,1} \\mathfrak{g}_{\\alpha} - \\partial_{1} \\mathfrak{g}_{\\alpha} \\cdot \\partial_{2,1} \\mathfrak{g}_{\\alpha} | < B + F(u) \\, \\bigg| \\, \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A  \\} \\right],\n\\end{align}\nwhere $F(u) = \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) \\cdot A \\cdot \\partial_2 \\mathfrak{g}_{\\alpha} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) \\cdot (\\partial_2 \\mathfrak{g}_{\\alpha})^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) \\cdot A^2$ is an estimate on the remaining terms appearing the expression for $\\det D \\mathfrak{G}(u)$. Note that given $\\sigma_1, \\sigma_2 > 0$, there exist constants $C_1,C_2,c_1,c_2 >0$, independent of $A$, such that $\\mathbb{P} \\left[ \\bigg| \\partial_2 \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_1} \\bigg| |\\mathfrak{G}(0)| < A \\right] \\leq C_1 e^{-\\frac{c_1}{A^{2\\sigma_1}}}$ and $\\mathbb{P} \\left[ | \\partial_{2,1} \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_2} \\bigg| |\\mathfrak{G}(0)| < A\\right] \\leq C_2 e^{-\\frac{c_2}{A^{2\\sigma_2}}}$ thanks to the non-degeneracy of our conditional covariance matrices which follows directly from the non-degeneracy of that of $\\tilde{\\mathfrak{G}}$.\n\nLet us combine the right-hand side of \\eqref{e:ub_vol_bound} with our estimate on $F(u)$, the estimates for the probabilities $\\mathbb{P} \\left[ \\bigg| \\partial_2 \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_1} \\bigg| |\\mathfrak{G}(0)| < A \\right]$ and $\\mathbb{P} \\left[ | \\partial_{2,1} \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_2} \\bigg| |\\mathfrak{G}(0)| < A\\right]$, and the explicit quantity of $\\mathbb{P} \\bigg[ \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A \\bigg]$.  Thus we have reduced to calculating\n\\begin{align*}\n\\mathbb{P} \\bigg[ | \\partial_2 \\mathfrak{g}_{\\alpha} \\cdot \\partial_{1,1} \\mathfrak{g}_{\\alpha} | & \\leq \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) A^{1 - \\sigma_1} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-2\\sigma_1} + A^{1 - \\sigma_2} + B  \\\\\n& \\bigg| \\, \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A \\bigg].\n\\end{align*}\nThe fact that at least one of the terms $\\partial_2 \\mathfrak{g}_{\\alpha}, \\partial_{1,1} \\mathfrak{g}_{\\alpha}$ must be bounded above by the squareroot of the right-hand side along with the non-degeneracy of the corresponding conditional covariance matrix gives the desired upperbound of\n\\begin{equation*}\n2 \\sqrt{\\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) A^{1 - \\sigma_1} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-2\\sigma_1} + A^{1 - \\sigma_2} + B }.\n\\end{equation*}\nNow, set $\\sigma_1=\\sigma_2=1\/2$ (although this may not be an optimal choice of parameters).\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{p:abc_stability}]\n\nOur proof follows the proof of \\cite[Proposition 4.7]{SW18} with some important modifications of the argument. The main idea is to show that if the count for critical points of $\\mathfrak{G}$, that are also low-lying in the sense that the first component of $\\mathfrak{G}$ is small, is too large, then for sufficiently small parameters $\\beta_{2,2}$, the smoothness of the field $\\mathfrak{G}$ allows us to derive contradictions in volume comparisons for neighborhoods of such critical points.\n\nFirst, we restrict ourselves to the event $\\left( \\cap_{i=1}^6 \\Delta_i^{\\complement}\\right)$ where $R \\geq R_0(\\eta,\\delta)$, $\\beta_{1,3}=\\beta_{1,3}(\\eta,\\delta)$; the parameters governing the events $\\Delta_i$, $i=1,...,5$, have been chosen such that $\\cap_{i=1}^5 \\Delta_i^{\\complement}$ has probability $> 1 - \\frac{\\delta}{8}$. Note that $\\mathbb{P}[\\Delta_6(\\eta, R, \\beta_{1,3})^{\\complement}] > 1 - \\frac{\\delta}{8}$. We are now in the situation where the nodal components of $\\mathfrak{g}_{\\alpha}$ are stable in the sense of Definition \\ref{d:unif_stability_fields}.\nFor a parameter $\\beta_{2,2}$, we let $K$ be the number of $\\beta_{2,2}$-unstable balls for $\\mathfrak{G}$. We will eventually be able to find a number $\\beta_{2,2}$ that satisfies the conclusion of Proposition \\ref{p:unstable_tangencies}.\n\nNext, we introduce a small parameter $\\gamma = \\gamma(R_1)$ that we will specify towards the end of our proof. We set\n\\begin{equation}\n\\label{eq:Ktild=K}\n\\tilde{K} = c_1 K\n\\end{equation}\nafter invoking the separation statement Lemma \\ref{l:4_separation}, and consider the new associated covering $\\{ \\mathcal{G}_{i_j} \\}_{i_j \\leq \\tilde{K}}$, with centers $u_{i_j}$, where we have possibly reordered the indices $i_j \\leq \\tilde{K}$. Now Taylor expand $\\det D \\mathfrak{G}$ to obtain the bound\n\\begin{equation} \\label{e:DG_expansion}\n|\\det D \\mathfrak{G}(u)| < \\beta_{2,2} + C_2 \\bigg( \\sup\\limits_{|\\nu| = 1, |\\tau|=1 ,\\, x \\in B(u_{i_j}, \\gamma)} |\\partial^{\\nu}  \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha}(u) | \\, |\\partial^{\\tau}  \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha}(u)| \\bigg) \\cdot \\gamma\n\\end{equation}\non each ball $B(u_{i_j}, \\gamma)$.\nAs a result of an application of Lemma \\ref{l:derivative_bounds} on \\eqref{e:DG_expansion}, we obtain the numbers\n\\begin{align*}\nA &:=  \\beta_{2,2} + c_2 \\frac{R}{\\tilde{\\sqrt{K}}}\\cdot \\gamma \\\\\nB &:= \\beta_{2,2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma\n\\end{align*}\nwith $c_{2}, c_3$ absolute, so that with probability $1 - \\delta\/4$ for $R \\geq R_1$, the bounds $|\\mathfrak{G}(u)| < A$ and $|\\det D \\mathfrak{G}(u)| < B$ hold on at least half of the $B(u_j, \\gamma)$, assuming $R \\geq R_1$\n(though we may neglect the distinction between ``all of the balls\" and ``half of them\" by appropriately modifying\nthe constant in \\eqref{eq:Ktild=K}).\n\n\nOur third, and most important, step is to manipulate the random variable\n\\begin{equation*}\n\\mathcal{A}_{A,B}  = \\mbox{Area} \\left( \\{ u:  |\\mathfrak{G}(u)|<A, |\\det D\\mathfrak{G}(u)| < B \\} \\right)\n\\end{equation*}\nin order to obtain a sufficient estimate on $K$ and deduce some necessary restrictions on $\\beta_{2,2}$.\nWe now have with probability $> 1 - \\delta\/4$, following immediately from the assumptions on the balls $B(u_j, \\gamma)$ and that $\\tilde{K} = c_1K$, that there exists $c_3 > 0$ uniform such that the following bound holds:\n\\begin{equation}\n\\label{eq:Ac(A,B)>=cgamma^2}\n\\mathcal{A}_{A,B} \\geq c_3 \\gamma^2 \\cdot K.\n\\end{equation}\n\nWe have\n\\begin{equation*}\n\\mathcal{A}_{A,B} = \\int\\limits_{B(R)} \\chi_{A,B} \\left( g_1,g_2, \\frac{\\partial g_1}{\\partial x_1}  \\frac{\\partial g_2}{\\partial x_2} -  \\frac{\\partial g_1}{\\partial x_2} \\frac{\\partial g_2}{\\partial x_1} \\right) \\, dx\n\\end{equation*}\nwhere $\\chi_{A,B}(\\cdot, \\cdot, \\cdot)$ is the indicator function for the set $\\{|(v_1, v_2)| < A, \\, |v_3| < B \\}$. If we can obtain a sufficient bound on ${\\mathbf E}[\\mathcal{A}_{A,B}]$ then, after an application of Chebyshev's Inequality, we have a sufficient estimate on $\\mathcal{A}_{A,B}$ with high probability. This is where Lemma \\ref{l:det_cov} is invoked, since\n\\begin{equation*}\n{\\mathbf E} \\left[\\chi_{A,B} \\left( g_1,g_2, \\frac{\\partial g_1}{\\partial x_1}  \\frac{\\partial g_2}{\\partial x_2} -  \\frac{\\partial g_1}{\\partial x_2} \\frac{\\partial g_2}{\\partial x_1} \\right) \\right] =  \\mathbb{P} \\left[ \\mathfrak{g}_{\\alpha} : |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right].\n\\end{equation*}\n\nWe remind ourselves (as already done in the proof of Theorem \\ref{t:conv_prob}) that in the local coordinates around $x$ given by $y=\\exp_x(Y)$, which allows us to identify ${\\mathbb R}^2$ with $T_xM$, we set $\\tilde{V}(x)$ to be the constant vector field given by the trivial extension of $(\\tilde{V}_{x,T})_{|Y=0}$ to ${\\mathbb R}^2$.  We can choose $T \\geq T_1(V(x))$ such that $\\tilde{V}_{x,T} = \\tilde{V}(x) + o(1)$; we remind ourselves that we have also assumed $\\tilde{V}(x) \\neq 0$. Note our $T_1$ can be taken to be greater than the $T_0$ appearing in the statement of Lemma \\ref{l:det_cov}. Let $s_{1},s_{2}$ be any positive numbers satisfying $2s_{1}+s_{2}=2$, so that $\\gamma^{-2} = \\gamma^{-2s_1 - s_2}$.\nThe estimate given by Lemma \\ref{l:det_cov} together with \\eqref{eq:Ac(A,B)>=cgamma^2}, yields that\n\\begin{align} \\label{e:penult_step_small_number_K}\n\\nonumber K & \\leq C_3 \\gamma^{-2}  \\,  \\left(  \\beta_{2,2} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma  \\right)^2 \\\\\n\\nonumber & \\times \\sqrt{\\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1\/2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} + A^{1\/2} +  \\left( \\beta_{2,2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma \\right)  } \\times R^2 \\\\\n\\nonumber & = C_3 \\,  \\left(\\beta_{2,2}\\gamma^{-s_1} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma^{1-s_1}   \\right)^2 \\cdot \\bigg( \\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1\/2} \\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2\\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} \\gamma^{-2s_2} \\\\\n& + A^{1\/2}  \\gamma^{-2s_2}+  \\left( \\beta_{2,2}\\gamma^{-2s_2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma^{1-2s_2} \\right)  \\bigg)^{\\frac{1}{2}} \\times R^2.\n\\end{align}\nIn preparation for the final steps of our proof, let us define\n\\begin{align} \\label{e:xi_value}\n\\nonumber & \\xi := C_3 \\left(\\beta_{2,2}\\gamma^{-s_1} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma^{1-s_1}   \\right)^2 \\cdot \\bigg( \\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1\/2} \\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2\\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} \\gamma^{-2s_2} \\\\\n& + A^{1\/2}  \\gamma^{-2s_2}+  \\left( \\beta_{2,2}\\gamma^{-2s_2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma^{1-2s_2} \\right)  \\bigg)^{\\frac{1}{2}}\n\\end{align}\nso that we have $K \\leq \\xi R^2$; without loss of generality, if necessary, we increase $C_3$ to be greater than $1$.\n\nLet us now proceed by contradiction. To wit, let us assume that $K > \\eta R^2$ from which it follows that $\\frac{R}{\\sqrt{\\tilde{K}}} < \\frac{1}{c_1 \\sqrt{\\eta}}$, on recalling \\eqref{eq:Ktild=K}. In order to make our choice of parameters $\\gamma, \\beta_{2,2},$ and $T_0$ properly so that we can eventually make $\\xi < \\eta$, we proceed as follows.\n\\begin{enumerate}\n\\item To start, we let $\\gamma <1$ and $\\beta_{2,2} < 1$.\nWe would like to make $A^{1\/2} \\gamma^{-2s_2} < \\frac{1}{100}$. Therefore when given $\\eta$, this leads us the first restriction of\n\\begin{equation*}\n\\gamma^{1-4s_2} < \\frac{1}{100} \\frac{c_1 \\sqrt{\\eta}}{c_2},\n\\end{equation*}\ntherefore requiring that $1 - 4s_2 > 0 $.\n\\item We aim to have that $A^2 \\gamma^{-2s_1} =  \\left(\\beta_{2,2}\\gamma^{-s_1} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma^{1-s_1}   \\right)^2 $ in the r.h.s of (\\ref{e:xi_value}) is less than $\\frac{\\eta}{100C_3}$. This requires that\n\\begin{equation*}\n\\gamma^{1-s_1} < \\frac{c_1 \\,\\eta}{100 \\, c_2 \\, C_3}\n\\end{equation*}\ntherefore requiring that $1 - s_1 > 0 $. As we have the initial restriction of $2s_1 + s_2 =2$, as a concrete working set of parameters, we make the choice of $s_1 = 15\/16$ and $s_2 = 1\/8$. Hence, we take\n\\begin{equation*}\n\\gamma  < \\min \\left\\{ \\left(\\frac{c_1 \\, \\sqrt{\\eta}}{100 \\, c_2 \\, C_3} \\right)^{2}, \\left(\\frac{c_1 \\eta}{100 \\, c_2 \\, C_3} \\right)^{16} \\right\\} = \\left( \\frac{c_1}{100 c_2 C_3} \\right)^{16} \\eta^{16}.\n\\end{equation*}\n\\item With the goals of eventually having $A^2 \\gamma^{-2s_1} < \\frac{\\eta}{100C_3}$ in the r.h.s of (\\ref{e:xi_value}) and the second (non-constant) factor that is raised to the power of $1\/2$ be $< 1$, we make another restriction of\n\\begin{align*}\n& \\beta_{2,2} < \\min \\left\\{ \\frac{\\eta^{1\/2}}{\\sqrt{100C_3}} \\gamma^{s_1}, \\frac{1}{10} \\gamma^{2s_2} \\right\\} =  \\min \\left\\{ \\frac{\\eta^{1\/2}}{100C_3} \\gamma^{15\/16}, \\frac{1}{100} \\gamma^{1\/4} \\right\\} \\\\\n& = \\frac{\\eta^{1\/2}}{100C_3} \\gamma^{15\/16} < \\frac{1}{100C_3} \\,  \\left( \\frac{c_1}{100 c_2 C_3} \\right)^{15} \\eta^{31\/2}.\n\\end{align*}\nafter using the restriction on $\\gamma$.  Note that our restrictions on $\\gamma$ and $\\beta$ make all terms which do not depend on $T$ sufficiently small.\n\\item Now, considering all the terms in (\\ref{e:xi_value}) involving factors of $T^{-1}$, appropriately bounding the term $\\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} \\gamma^{-2s_2}$ above by $\\frac{1}{100}$ will place similar bounds on the aforementioned remaining terms.  Letting $C$ be the supremum amongst all the implicit constants depending on a derivative of $V$ (that is, the $C^1$ norm of $V$'s coefficients) and appearing in the $\\mathcal{O}$ notations, we make the final restriction of\n\\begin{equation*}\n\\frac{C}{T} < \\frac{\\sqrt{A}}{10R} \\gamma^{2s_2}= \\frac{\\sqrt{A}}{10R} \\gamma^{1\/4} < \\frac{1}{10R} \\left(\\frac{c_1}{c_2 C_3} \\right)^4 \\eta^{15\/4}.\n\\end{equation*}\n\\end{enumerate}\nHence, we enlarge $T_0$ to satisfy our final constraint (and therefore inheriting a possible dependence on $R$). With all these choices made, in the order described above, we have arrive at $\\xi < \\frac{\\eta}{2500}$ and therefore contradict the hypothesis of $K > \\eta R^2$. This concludes the proof of Proposition \\ref{p:abc_stability}.\n\\end{proof}\n\nWe now collect some facts on nodal components that fall outside of our local approximation and are exceptional in terms of the volumes of the nodal domains for which they form the boundary. But first, we record some necessary definitions:\n\n\\begin{defn}\n\\begin{enumerate}\n\\item We say that a nodal component of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ is $\\xi$-small if it is adjacent to a domain of volume $< \\xi$.\n\n\\item For $R >0$, let $\\pazocal{N}_{\\xi-sm}(\\mathfrak{g}_{\\alpha}, R)$ be the number of $\\xi$-small components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ lying entirely inside of $B(R)$.\n\n\\item We say that a nodal component of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ is $R_1$-long if its diameter is $> R_1$.\n\n\\item For $R > R_1 >0$, let $\\pazocal{N}_{diam > R_1}(\\mathfrak{g}_{\\alpha};R)$ be the number of $R_1$-long components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ lying entirely inside of $B(R)$.\n\\end{enumerate}\n\\end{defn}\n\nThe following results are taken directly from \\cite{So12}:\n\n\\begin{lem}[Cf. {\\cite[Lemma 8]{So12}}] \\label{l:long_components}\nConsider $\\mathfrak{g}_{\\alpha}$ on $B(R)$. Let $\\delta>0$ be given. Then given $R_1 >0$, there exists $R_0(R_1)$ such that for all $R \\geq R_0 \\geq R_1$ and a uniform constant $C>0$, such that the number of components $\\Gamma \\subset B(R)$ of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ of diameter $> R_1$ is\n\\begin{equation*}\n\\pazocal{N}_{diam > R_1}(\\mathfrak{g}_{\\alpha};R) < \\frac{C}{R_1} R^2\n\\end{equation*}\nwith probability $> 1 - \\delta$.\n\\end{lem}\n\nIn our $2$-dimensional case Lemma \\ref{l:long_components} follows from the isoperimetric inequality; in the more general setting a slightly heavier\nmachinery involving the restriction of $\\mathfrak{g}_{\\alpha}$ to the boundary of $B(R)$ is employed. We note that it, thanks to discarding the long components, coverings of $B(R)$ with balls of radius $R_1 < R$ as in the previous sequence of statements will give us an advantage.\n\n\n\\begin{lem}[Cf. {\\cite[Lemma 9]{So12}}] \\label{l:small_vol}\nLet $\\xi>0$ be given.  There exist constants $c_0, C_0>0$ such that\n\\begin{equation*}\n\\limsup\\limits_{R \\rightarrow \\infty} \\frac{{\\mathbf E} \\left[ \\pazocal{N}_{\\xi-sm}(\\mathfrak{g}_{ \\alpha}, R) \\right]}{R^2} \\leq C_0 \\, \\xi^{c_0}.\n\\end{equation*}\n\\end{lem}\n\nGiven our facts about $\\xi$-small and $R_1$-long components, we can now state the following important lemma:\n\n\\begin{lem} \\label{l:sub_tang_low_count}\nLet $x \\in \\M\\setminus\\{V=0\\}$. Given the parameters $R, \\delta, \\eta_0>0$, there exists a covering radius parameter $R_{1,0} < R$, an auxiliary density parameter $\\eta_1(R, \\eta_0) < \\eta_0$, an event $E$ with $\\mathbb{P}[E] < \\delta$ and a stability parameter $\\beta_{2,2}$ such that outside $E$, if $F$ is $(\\eta_1, \\beta_{2,2}, 3R_{1,0})$-stable for all $T \\geq T_0(R)$, then $|\\det D \\mathfrak{G}| > \\beta_{2,2}$ on all but $\\eta_0 R^2$ components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ that are also uniformly stable as in Definition \\ref{d:unif_stability_fields}.\n\\end{lem}\n\n\\begin{proof}\nOur proof is almost verbatim that of \\cite[Lemma 4.8]{SW18} which is similar in spirit to that in \\cite[Section 4.2]{NS09}. We refer the interested reader to this paper but highlight the main details as well as an explicit formula which gives further insight into the dependences between our variety of parameters. To ease the notation and intuitively connect the parameters in this lemma with those in our previous statements, we replace $R_{1,0}$ by $R_1$ in our proof.\n\nThanks to lemmas \\ref{l:long_components} and \\ref{l:small_vol}, we know that with probability $1  - \\frac{\\delta}{2}$ there exists uniform constants $C_1,C_2, c_0>0$ such that we have that\n\\begin{equation} \\label{e:long_toosmall_comps}\n\\pazocal{N}_{diam>R_1}(\\mathfrak{g}_{\\alpha}, R) \\leq C_1 \\, \\frac{R^2}{R_1} \\, \\mbox{ and } \\, \\pazocal{N}_{\\xi_0-sm}(\\mathfrak{g}_{\\alpha}, R) \\leq C_2 \\, \\xi^{c_0} \\, R^2\n\\end{equation}\nfor some $R_1 < R$, where $R \\geq R_0$. This reduces us to now having to count components that are not $R_1$-long or are not $\\xi_0$-small.  This particular $R_1$ is what we will take as our covering radius parameter, for at least the time being.\n\nProposition \\ref{p:unstable_comp} states that with probability $1 - \\frac{\\delta}{2}$ and $R_0$ possibly larger, the total number of components that are not long or small in the above senses and are stable in terms of having low-lying zeroes of $\\nabla \\mathfrak{g}_{\\alpha}$ is $\\leq \\eta_1 R^2$.  Now, we apply Proposition \\ref{p:abc_stability} and the uniform lower bound we have on the volume encompassed by the individual components to see that the number of components that are neither long nor small with no sub-$\\beta_{2,2}$ $\\tilde{V}_{x,T}$- tangencies is $\\leq C_3 \\frac{R_1^2}{\\xi_0} \\frac{\\eta_1}{2} R^2$ for some uniform $C_3 >0$. Summing up all these counts gives an upper bound for the number of components that are either unstable in the sense of Definitions \\ref{d:unif_stability_fields} and \\ref{d:unstable_event} and those that are not amenable to our local methods.\n\nBy summing all of our declared estimates appearing in (\\ref{e:long_toosmall_comps}) and the previous paragraph, and therefore collating all the parameters we can adjust, we aim to establish the following inequality:\n\\begin{equation*}\nC \\left( \\frac{1}{R_1} + \\xi_0^{c_1} + \\frac{R_1^2}{2 \\xi_0}\\eta_{1}  + \\eta_1 \\right) < \\eta_0\n\\end{equation*}\nfor some uniform constants $C, c_1>0$. Motivated by this goal, we can choose the range for our medley of parameters in the following order: our radius parameter $R_1 > \\frac{4C}{ \\eta_0}$, $\\xi_0 < (\\frac{\\eta_0}{4C})^{1\/c_1}$, and finally our auxiliary density parameter $\\eta_1 < \\min \\{\\frac{\\eta_0}{4C}, \\frac{2 \\xi_0 }{R_1^2} \\frac{\\eta_0}{4C} \\}$. Our choice of $T_0$ comes from Proposition \\ref{p:abc_stability}. This gives our desired result.\n\\end{proof}\n\n\\begin{rem}\nNote that in the above proof, our choice of parameters $R_1 = R_{1,0}, \\xi_0, \\eta_1$ is not dictated by our choice of stability parameter $\\beta_{2,2}$ and hence not making our reductions circular. However, these choices of parameters could possibly make $\\beta_{2.2}$ smaller. Given the restrictions we derived in the proof of Proposition \\ref{p:abc_stability}, this does not harm our overall argument.\n\\end{rem}\n\nFinally, we can deduce Proposition \\ref{p:unstable_tangencies} from Lemma \\ref{l:sub_tang_low_count}.\n\n\\begin{proof}[Proof of Proposition \\ref{p:unstable_tangencies}]\nWe apply Lemma \\ref{l:sub_tang_low_count} directly. Considering that we are given $\\eta, \\delta>0$ and $x$, we take any $R_0 > R_{1,0}$ such that $\\frac{R_0}{R_{1,0}}$ is large, take $\\eta_0 < \\eta$, set $\\beta_2 = \\beta_{2,2}$, and take the same $T_0$ as provided.\n\\end{proof}\n\nWe close this section by emphasizing that our count of $\\eta R^2$ is for the total number of components with at least one near-degenerate tangency.  Thus, there is still room to make more precise this upper bound for those with precisely $k$ tangencies with at least one being near-degenerate.\n\n\n\n\\section{Proof of Theorem \\ref{thm:main meas conv}: global results in the Riemannnian scenario}\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:main meas conv} assuming Theorem \\ref{t:main_thm_less_equal_1}: consolidating the individual counts:}\n\n\\label{sec:proof main thm meas}\n\n\\begin{proof}\n\nUsing the constants $C_{\\alpha,k}$ prescribed by Theorem \\ref{t:main_thm_less_equal_1},\nwe define the measure\n\\begin{equation}\n\\label{eq:mualpha def}\n\\mu_{\\alpha}:=\\sum\\limits_{k=0}^{\\infty}C_{\\alpha,k}\\cdot\\delta_{k},\n\\end{equation}\nand Theorem \\ref{t:main_thm_less_equal_1} part (1) reads for every $k\\ge 0$,\n\\begin{equation*}\n{\\mathbf E}[|\\left(\\mu_{f_{\\alpha,T}}(V)\\right)(k) - \\mu_{\\alpha}(k) |] \\rightarrow 0\n\\end{equation*}\nas $T\\rightarrow\\infty$.\nWe claim that Theorem \\ref{thm:main meas conv} holds true with $\\mu_{\\alpha}$ as in \\eqref{eq:mualpha def}.\nFirst, since by the proof of Theorem \\ref{t:main_thm_less_equal_1} below, the constants $C_{\\alpha}$ are\nsame as in Theorem \\ref{thm:euclid_count} (1)\napplied on the random field $F=\\mathfrak{g}_{\\alpha}$ with $\\zeta$ arbitrary (cf. Proposition \\ref{p:almost_main_theorem}), where\nthe $C_{k}$ are independent of $\\zeta$ by Theorem \\ref{thm:euclid_count} (2), the measure\n$\\mu_{\\alpha}$, as defined in \\eqref{eq:mualpha def}, is a probability measure, thanks to Theorem \\ref{thm:euclid_count} (4).\n\nTherefore $\\{\\mu_{f_{\\alpha,T}}(V)\\}_{T}$ on ${\\mathbb Z}_{\\ge 0}$ is a collection of random probability measures,\nand $\\mu_{\\alpha}$ a deterministic probability measure, so that,\nfor every $k\\ge 0$, as $T\\rightarrow\\infty$, $(\\mu_{f_{\\alpha,T}}(V))(k)$ converges in mean (and hence in probability) to $\\mu_{\\alpha}(k)$.\nThis is precisely the scenario considered by ~\\cite[Lemma 6.3]{ALL}, proof (and same result)\nalready contained within ~\\cite[Proof of Theorem 1.1, \\S 7.1 on p. 57]{SW18},\nwhose conclusion yields the convergence in probability of\n$\\{\\mu_{f_{\\alpha,T}}(V)\\}_{T}$ as $T\\rightarrow\\infty$ to $\\mu_{\\alpha}$ w.r.t. the total variation distance \\eqref{eq:D tot var dist def},\ni.e. it gives \\eqref{eq:main meas conv}.\nFinally, the support statement of Theorem \\ref{thm:main meas conv} for $\\mu_{\\alpha}$ is an immediate corollary from\nthe definition \\eqref{eq:mualpha def} of $\\mu_{\\alpha}$ and Theorem \\ref{t:main_thm_less_equal_1} part (2).\nTheorem \\ref{thm:main meas conv} is now proved.\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{t:main_thm_less_equal_1}}\n\n\\begin{defn}\\label{def:plus_minus}\nFor $y \\in {\\mathbb R}$, we denote\n\\begin{equation*}\n|y|_+ = \\max \\{0,y\\}\n\\end{equation*}\nand\n\\begin{equation*}\n|y|_-= \\min \\{0,y\\},\n\\end{equation*}\nso that\n\\begin{equation*}\n| \\cdot | = | \\cdot |_+ + | \\cdot |_-.\n\\end{equation*}\n\\end{defn}\n\nLet $V \\in \\mathcal{V}(\\M)$ throughout this section. We remind ourselves that\n\\begin{equation*}\n\\pazocal{N}_{V}(f,k) :=\\#\\{\\gamma  \\subseteq f^{-1}(0):\\: \\pazocal{N}_{V}(\\gamma)=k\\}\n\\end{equation*}\nis the total number of nodal components of $f$ with precisely $k$ tangencies w.r.t. $V$. We now present a key proposition that builds upon on our previous (and crucial) local results and essentially gives the conclusions of our main theorems:\n\n\\begin{prop} \\label{p:almost_main_theorem}\nLet $k \\ge 0$ be given and $c_{2, \\alpha}$ be the Nazarov-Sodin constant of $\\mathfrak{g}_{\\alpha}$,\nand the constants $C_{\\alpha,k}$ prescribed by an application of Theorem \\ref{t:main_thm_less_equal_1} (1) on the\nfield $\\mathfrak{g}_{\\alpha}$ (with $\\zeta$ arbitrary). Then, as $T \\rightarrow \\infty$, we have that\n\\begin{equation*}\n{\\mathbf E} \\left[  \\left| \\frac{ \\pazocal{N}_{V}\\left( f_T,  k  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(\\M) \\cdot T^2} - C_{\\alpha,k} \\right|_{\\pm}  \\right] \\rightarrow 0.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}[Proof of Theorem \\ref{t:main_thm_less_equal_1} assuming Proposition \\ref{p:almost_main_theorem}]\n\nTheorem \\ref{t:main_thm_less_equal_1} follows immediately from Proposition \\ref{p:almost_main_theorem}.\n\n\\end{proof}\n\n\\subsection{Excising small volume and long components}\n\n\\begin{defn}\nLet $\\xi, D> 0$ be parameters and $f_T$ the band-limited functions \\eqref{eq:fT band lim alpha<1} (or \\eqref{eq:fT band lim alpha=1}).\n\n\\begin{enumerate}\n\\item A component of $f^{-1}_T(0)$ is \\textit{$\\xi$-small} if it is a boundary of nodal domain whose volume in $\\M$ is less than $\\xi T^{-2}$.  Let $\\pazocal{N}_{\\xi-sm}(f_T)$ be the total number of $\\xi$-small components of $f_T$ on $\\M$.\n\\item For $D>0$, a component of $f^{-1}_T(0)$ is \\textit{$D$-long} if its diameter is greater than $D\/T$. Let $\\pazocal{N}_{D-long}(f_T)$ be their total number.\n\\item Given the parameters $D, \\xi>0$, a component of $f^{-1}_T(0)$ is \\textit{$(D, \\xi)$-normal}, if it is neither $\\xi$-small nor $D$-long.\n\\item Let $\\pazocal{N}_{norm}(f_T)$ be the total number of $(D,\\xi)$-normal components of $f^{-1}_T(0)$ (analogously, define $\\pazocal{N}_{V, norm}(f_T, k)$ as the count for the subset of $(D,\\xi)$-normal components whose number of tangencies is precisely $k$ with respect to $V$).\n\\item  Let $\\pazocal{N}_{V, norm}(f_T, x, r)$ be the total number of $(D,\\xi)$-normal components of $f^{-1}_T(0)$ that are completely contained inside the geodesic ball $B(x;r)$ (respectively, $\\pazocal{N}_{V,norm}(f_T, k, x, r)).$\n\\item  Let $\\pazocal{N}^*_{norm}(f_T, x, r)$ be the total number of $(D,\\xi)$-normal components of $f^{-1}_T(0)$ that intersect the geodesic ball $B(x;r)$ (respectively, $\\pazocal{N}^*_{V,norm}(f_T, k, x, r)).$\n\\end{enumerate}\n\\end{defn}\n\nThe following two lemmas are taken directly out of \\cite{So12}:\n\n\\begin{lem}[Cf. {\\cite[Lemma 9]{So12}}]\nThere exists constants $c_0, C_0 >0$ so that the following estimate on the number of $\\xi$-small components holds:\n\\begin{equation*}\n\\limsup\\limits_{T \\rightarrow \\infty} \\frac{{\\mathbf E} \\left[  \\pazocal{N}_{\\xi-sm}(f_T) \\right]}{T^2} \\leq C_0 \\, \\xi^{c_0}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{lem}[Cf. {\\cite[Lemma 8]{So12}}]\nThere exists a constant $C_0 >0$ such that the following bound holds on the number of $D$-long components:\n\\begin{equation*}\n\\limsup\\limits_{T \\rightarrow \\infty} \\frac{{\\mathbf E} \\left[  \\pazocal{N}_{D-long}(f_T) \\right]}{T^2} \\leq C_0 \\, \\frac{1}{D}.\n\\end{equation*}\n\\end{lem}\n\n\nWe can now state the main proposition of this section:\n\n\\begin{prop} \\label{p:conv_in_mean_normal}\nLet $\\xi, D>0$ be given and $k \\ge 0$. Then as $T \\rightarrow \\infty$, we have that\n\\begin{equation*}\n{\\mathbf E} \\left[  \\left| \\frac{ \\pazocal{N}_{V,norm}\\left( f_T,  k  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(\\M) \\cdot T^2} - C_k \\right|_{\\pm}  \\right] \\rightarrow 0,\n\\end{equation*}\nfor $| \\cdot |_{\\pm}$ as in Definition \\ref{def:plus_minus}, and $C_{k}=C_{\\alpha,k}$ is in Proposition \\ref{p:almost_main_theorem}.\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition \\ref{p:almost_main_theorem} assuming Proposition \\ref{p:conv_in_mean_normal}]\nThis is immediate given lemmas \\ref{l:small_vol} and \\ref{l:long_components}.\n\\end{proof}\n\n\\subsection{Proof of Proposition \\ref{p:conv_in_mean_normal}}\n\nWe begin with a global analogue of Lemma \\ref{l:integro_geom_sandwich}.\n\n\\begin{lem} \\label{l:integro_geom_sandwich_global}\nGiven $\\epsilon > 0$ and $k \\ge 0$, there exists $r_0$ such that for all $r < r_0$, we have\n\\begin{align*}\n& (1 - \\epsilon) \\int\\limits_{\\M} \\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, r  \\right)}{\\operatorname{Vol}(B(r))} \\, dx \\leq \\pazocal{N}_{V,norm}\\left( f_T,  k \\right)\\leq (1 + \\epsilon) \\int\\limits_{\\M} \\frac{\\pazocal{N}^*_{V,norm}\\left( f_T,  k, x, r  \\right)}{\\operatorname{Vol}(B(r))} \\, dx.\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nAs the proof is almost exactly the same as that of Lemma \\ref{l:integro_geom_sandwich}, we omit it and leave the details to the reader.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{p:conv_in_mean_normal}]\nOnce again, our proof mirrors that of \\cite[Proposition 3]{BW17} so we give the major steps of the proof, with particular details given at the critical junctures, and leave the remaining similar details to the interested reader.  For convenience, we assume that $\\operatorname{Vol}(\\M)=1$.\nLet $\\epsilon > 0$ and $k \\ge 0$ be given. Consider a regime of $R,T$ such that $R\/T < r_0 < 1$ as in Lemma \\ref{l:integro_geom_sandwich_global} and set $D = \\sqrt{R}$ so that\n\\begin{equation*}\n\\frac{\\operatorname{Vol}(B(R+D))}{\\operatorname{Vol}(B(R))} < 1 + \\epsilon.\n\\end{equation*}\n\nWe focus on the case for $| \\cdot |_+$ as the proof for $| \\cdot |_{-}$ is similar. We begin by applying Lemma \\ref{l:integro_geom_sandwich_global} with $r = R\/T$. It follows that\n\\begin{align*}\n {\\mathbf E} \\left[  \\left| \\frac{ \\pazocal{N}_{V,norm}\\left( f_T,  k  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(\\M) \\cdot T^2} - C_k \\right|_{+}  \\right] &\\leq {\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|(1 + 2\\epsilon) \\frac{\\pazocal{N}^*_{V,norm}\\left( f_T,  k, x, R\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} \\, dx - C_k  \\right|_{+} \\, dx  \\right] \\\\\n& \\leq {\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx  \\right] \\\\\n& + \\mathcal{O} \\left( \\epsilon \\cdot \\int\\limits_{\\M} \\frac{{\\mathbf E} \\left[\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right) \\right]}{ c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} \\, dx \\right).\n\\end{align*}\nNote that\n\\begin{equation*}\n\\int\\limits_{\\M} \\frac{{\\mathbf E} \\left[\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right) \\right]}{ c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} \\, dx  =  \\mathcal{O}(1)\n\\end{equation*}\nby \\cite[Lemma 2]{So12} on using Lemma \\ref{lem:Kac Rice manifold} to obtain\nan upper bound for the total number of components. Thus, our focus is on proving that\n\\begin{equation*}\n{\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx  \\right]  \\rightarrow 0.\n\\end{equation*}\n\nLet $\\rho < \\frac{\\epsilon}{5\\sqrt{\\pi} N}$, where $N$ is the cardinality of $\\{ V = 0 \\}$, be a parameter that will control how close we are allowed to approach the zero set $\\{V=0\\}$. Let $\\xi >0$ be a small number, to be specified shortly, that controls how small a volume a nodal component can encompass and will also be related to our given $\\epsilon$ later.\nWe now consider the event\n\\begin{equation*}\n\\Delta_{T,k,x,R} := \\left\\{ \\left| \\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))}  - C_k  \\right| > \\frac{\\epsilon}{5} \\right\\}.\n\\end{equation*}\nLet $\\{ V= 0\\}^{\\sqrt{\\rho\/2}}$ be a neighborhood consisting of balls of radius $\\sqrt{\\rho\/2} < inj(\\M)$ centered at the finitely many points of $\\{V=0\\}$. An application of an Egorov-type Theorem w.r.t. the double limit $\\lim\\limits_{R\\rightarrow\\infty}\\lim\\limits_{T\\rightarrow\\infty}$\n(as carried out in \\cite[Proposition 3]{BW17}), along with Theorem \\ref{t:conv_prob} allows us to conclude that on $\\M \\setminus (\\{ V= 0\\}^{\\sqrt{\\rho\/2}})$, there exists a set $\\M_{\\epsilon} \\subset \\M$ with $\\operatorname{Vol}(\\M_{\\epsilon}) > 1 - \\sqrt{\\pi} \\frac{\\epsilon}{5}$ (with the additional property that $\\M_{\\epsilon} \\cap \\{ V= 0\\}^{\\sqrt{\\rho\/2}} = \\emptyset$ thanks to taking a tubular neighborhood of radius $\\sqrt{\\rho\/2}$), such that\n\\begin{equation}\n\\label{eq:double limit liminf=0}\n\\liminf_{R \\rightarrow \\infty} \\limsup_{T \\rightarrow \\infty} \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}\\left[  \\Delta_{T,k,x,R} \\right] = 0,\n\\end{equation}\ni.e. the limit \\eqref{eq:double lim R,T conv loc} is {\\em almost} uniform along some sequence $R_{j}\\rightarrow\\infty$ attaining the $\\liminf\\limits_{R\\rightarrow\\infty}$ in \\eqref{eq:double limit liminf=0}, so that\n\\begin{equation}\n\\label{eq:double limit R attain liminf}\n\\lim\\limits_{j \\rightarrow \\infty} \\limsup_{T \\rightarrow \\infty} \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}\\left[  \\Delta_{T,k,x,R_{j}} \\right] = 0.\n\\end{equation}\nWithout loss of generality, we can restrict ourselves to a regime of $R,T$ where there exists a large enough $C=C_{\\M} >0$ such that for all ratios $R\/T \\leq \\frac{inj(\\M)}{C}$, we have that $\\mbox{vol}_g B(x, R\/T) = \\pi \\frac{R^2}{T^2} + o(\\frac{R^2}{T^2})$.\n\nWe re-express\n\\begin{align} \\label{e:expect_int_geom}\n& {\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx  \\right] \\\\\n\\nonumber & = \\int\\limits_{\\Omega} \\int\\limits_{\\M}  \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx \\, d \\mathbb{P}(\\omega).\n\\end{align}\nAn application of Fubini's Theorem along with the measure bounds for the sets $\\Delta_{T,k,x,R}$ and $\\M_{\\epsilon}$ reduces bounding (\\ref{e:expect_int_geom}) to understanding the sum\n\\begin{equation*}\n\\left( \\int\\limits_{\\M} \\int\\limits_{\\Delta_{T,k,x,R}} + \\int\\limits_{\\M} \\int\\limits_{\\Omega \\cap \\Delta_{T,k,x,R}^{\\complement}} \\right) \\underbrace{\\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+}\\, d \\mathbb{P}(\\omega)   \\, dx }_{=: F \\, d \\mu} .\n\\end{equation*}\nThis the sum of integrals breaks up into a further $5$ integrals in total and we estimate each accordingly:\n\\begin{equation*}\n\\int\\limits_{\\M_{\\epsilon}} \\int\\limits_{\\Delta_{T,k,x,R}} F  \\, d \\mu \\leq \\xi^{-c_0} \\cdot \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}[\\Delta_{T,k,x,R}],\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{\\M\\setminus\\M_{\\epsilon}} \\int\\limits_{\\Delta_{T,k,x,R}} F  \\, d \\mu \\leq \\frac{\\epsilon}{5} \\cdot \\xi^{-c_0} \\cdot \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}[\\Delta_{T,k,x,R}],\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{\\M_{\\rho}} \\int\\limits_{\\Delta_{T,k,x,R}^{\\complement}} F  \\, d \\mu \\leq \\frac{\\epsilon}{5},\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{  \\{V=0\\}^{\\sqrt{\\rho\/2}} } \\int\\limits_{\\Delta_{T,k,x,R}^{\\complement}} F  \\, d \\mu \\leq C_{\\M} \\cdot \\frac{\\epsilon^2}{50}, \\mbox{ thanks to the finiteness of } \\{V=0\\} \\mbox {, and }\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{\\M\\setminus \\left( \\M_{\\rho} \\cup (\\{V=0\\}^{\\sqrt{\\rho\/2}} \\right)} \\int\\limits_{\\Delta_{T,k,x,R}^{\\complement}} F  \\, d \\mu \\leq C_{\\M} \\cdot \\frac{\\epsilon^2}{25}, \\mbox{ as }  \\left( \\M\\setminus  \\M_{\\rho} \\right) \\cap (\\{V=0\\}^{\\sqrt{\\rho\/2}}  \\neq \\emptyset \\mbox{ is possible}\n\\end{equation*}\nwhere $C_{\\M} > 0$ is an absolute constant depending only on the volume of $\\M$, which we have normalized to $1$.\n\nKeeping in mind that $\\epsilon$ was given to us, we restrict the auxiliary parameters introduced in our proof as follows: 1) $\\xi < (\\frac{\\epsilon}{5})^{1\/c_0}$, 2) take $R_0$ such that for all $R_{j}  \\geq R_0$ with $\\{R_{j}\\}_j$ as in \\eqref{eq:double limit R attain liminf} (which in turn, as in the proof of Theorem \\ref{t:conv_prob}, requires us to carefully select our parameters $\\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2$ each of which themselves depends on $\\eta$), we know there exists $T_0(R_{j})$ (implicitly depending upon our string of stability parameters $\\beta$) such that for all $T \\geq T_0$ we have $ \\sup\\limits_{x \\in \\M_{\\rho}} \\mathbb{P}[\\Delta_{T,k,x,R_{j}}] < \\frac{\\epsilon \\xi^{c_0}}{5}$. We conclude that \\eqref{e:expect_int_geom} is bounded above by $\\epsilon$. This completes the proof of our proposition.\n\\end{proof}\n\nWe conclude by noting that our spectral parameter $T_0$ being dependent on the previous parameters $\\beta_{1,2}$ and $\\epsilon$ is a reflection of how our covering of $\\M \\setminus \\M_{\\epsilon}$, by geodesic disks of radius $R\/T$, is governed by the maximal order of vanishing amongst all zeroes of $V$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:Introduction}Introduction}\n\n\\section{Introduction} \n\nUltracold atomic gases confined in optical lattices with their unique controllability allow for artificial quantum engineering of lattice Hamiltonians with large system sizes beyond the computational reachability of classical simulations~\\cite{1998_Zoller_Jaksch_PRL,2002_Hofstetter_Cirac_PRL,2007_Lewenstein_AP,2008_Bloch_Dalibard_RMP,2010_Esslinger_CMP,2015_Lewenstein_RPP}. In the experiments, Bose and Fermi Hubbard models as well as spin Hamiltonians have  all been emulated to study both equilibrium quantum phase transitions of ground states and also  out-of-equilibrium many-body dynamics. Mott transitions of both bosons and fermions have been found in experiments~\\cite{2002_Greiner_Mandel_Nature,2008_Jordens_Strohmaier_Nature,2008_Schneider_Hackermuller_Science}, and even super-change mediated  magnetic phases~\\cite{2013_Greif_Uehlinger_Science,2015_Hart_Duarte_Nature,2016_Mazurenko_Chiu_arXiv} have recently been accomplished through the state-of-the-art quantum microscope techniques~\\cite{2015_Haller_Hudson_NatPhys,2015_Cheuk_Nichols_PRL,2015_Parsons_Huber_PRL,2015_Edge_Anderson_PRA,2015_Omran_Boll_PRL,2016_Greif_Parsons_Science,2016_Cheuk_Nichols_PRL,2016_Parsons_Mazurenko_Science,2016_Boll_Hilker_Science,2016_Cheuk_Nichols_Science,2016_Brown_Mitra_arXiv}.  Observing novel quantum dynamics in a strongly correlated setting is presently attracting growing experimental research interests in cold atoms and other synthetic quantum systems~\\cite{2012_Bloch_Light_Nature,2014_Zwierlein_PRL,2014_Monroe_Nature,2016_Greiner_Quantum_Science,2017_Nagerl_Bloch_Science,2017_Lukin_Observation_Nature,2017_Monroe_Observation_Nature,2017_Hulet_Formation_Science,2017_Everitt_observation}. \n\nThermalization---a most common phenomenon for interacting particles---could break down in isolated quantum systems subjected to random disorder potentials. \nStarting from Anderson's seminal work on localization~\\cite{1958_Anderson_Absence_PRL}, it has now been well-established that non-interacting particles moving in a disordered medium will be localized. \nThe robustness against interaction effects yet remained controversial until the recent studies of many-body localization~\\cite{Basko06,huse2015many,2015_Altman_review}. Through recent studies, the persistence of  localization in interacting systems has been established through a perturbative field theory calculation~\\cite{Basko06} and a rigorous mathematical proof~\\cite{imbrie2014many} together with extensive numerical works~\\cite{oganesyan2007,pal2010,moore2012,2013_Bauer_JSM,vadim,bardarson2014,2015_Singh_MBL_PRL,2015_Khemani_Pollmann_PRL,2015_Yu_Pekker_arXiv,2015_Lim_Sheng_PRB,2016_Kennes_Karrasch_PRB,2015_Bera_Many_PRL}. To describe the MBL phase, a phenomenological theory of local integrals of motion has been proposed, which provides a physical picture of highly constrained dynamics in the localized phase~\\cite{2014_Huse_MBL_PRB,2013_Serbyn_PRL,chandran2015,Ros2015420}. The consequent dynamical phenomena have been observed in experiments of ultracold atoms and trapped ions~\\cite{blochmbl,DemarcoDisorder,2015_Monroe_MBL_NatPhys,2016_Bordia_Bloch_MBL_PRL,2017_Bloch_Periodically_NatPhys,2016_Choi_Bloch_MBL_Science,2017_Bloch_Probing_arXiv,2017_Bloch_Signatures_PRX}. \n\nMore recent theoretical efforts in the MBL context are devoted to understanding the delocalization transition~\\cite{2015_Huse_Theory_PRX,2015_Potter_Universal_PRX,2017_Huse_Rare_AdP,2017_Huse_Critical_PRX} to the quantum thermal phase where eigenstate thermalization hypothesis~\\cite{1991_Deutsch_Quantum_PRA,1994_Srednicki_Chaos_PRE} is expected to hold, for that this type of transition does not have an analogue in the non-interacting problem of Anderson localization. To characterize the transition, various diagnostics such as entanglement entropy scaling and many-body energy level statistics have thoroughly  been investigated in theory, and scaling functions based on these quantities have been proposed to describe the MBL-to-ergodic criticality. Across the transition the quantum entanglement entropy scaling would switch from area- to volume-law~\\cite{2013_Bauer_JSM}. The level statistics exhibits a transition from Poisson type to Wigner-Dyson~\\cite{oganesyan2007}. These quantum entanglement and statistical quantities offer concrete measures to describe the criticality,  but these theoretical ``observables\"  turn out to be extremely challenging to measure, and it remains unclear how to accurately probe MBL localization length in experiments. The present theoretical debate about the validity of Harris-Chayes bound~\\cite{1974_Harris_Effect_JPC,1986_Chayes_Finite_PRL} for MBL transition~\\cite{2014_Kjall_MBL_PRL,2015_Huse_Theory_PRX,2015_Potter_Universal_PRX,2015_Luitz_MBL_PRB}  makes the task to probe localization length exceedingly desirable. \n\n\nHere we propose to use quench dynamics to probe the localization length of interacting atoms in disorder potential and the corresponding MBL criticality. \nWe find that an added hole in the MBL phase shows an exponentially decaying density profile after long-time dynamical evolution, whereas in the ergodic phase its density distribution completely spreads over the whole lattice system. The size of the density profile in the long-time-evolved final state defines a localization length, whose critical behavior can be directly probed in experiments. \nThe transition point determined by our defined localization length agrees with  that by other diagnostics. \nIn our numerical results, the critical exponent is found to violate Harris-Chayes bound~\\cite{1974_Harris_Effect_JPC,1986_Chayes_Finite_PRL}, \nwhich implies many-body localization criticality is beyond the description of conventional field theory or renormalization group study for disorder systems.\n It is worth remarking here that this work is rather to propose an experimental scheme to detect the MBL localization length and criticality than to calculate a precise critical exponent. Whether the Harris-Chayes bound holds or not at the MBL transition would rely on future experiments. \nThe proposed schemes are justified by benchmarking to  the extensively studied 3D Anderson localization.\n\n\n\n\n\n\n\n\n\n\\section{Model and Method}\n\nTo be concrete we consider a model Hamiltonian of spinless fermions with nearest neighbor interactions, \n\\begin{equation} \\label{Hamiltonian}\n\\hat{H} = \\sum_{<jj'>}\\left[-t\\left(\\hat{c}_{j}^{\\dagger}\\hat{c}_{j'}+h.c.\\right)+V\\hat{n}_{j}\\hat{n}_{j'}\\right] + \\sum_{j}h_{j} \\hat{n}_{j}.  \n\\end{equation}\nHere $<jj'>$ denotes nearest-neighboring sites, $t$ is the tunneling matrix element ($t$ is set to be the energy unit throughout), $\\hat{c}_{j}^{\\dagger}$ ($\\hat{c}_{j}$)  is a  fermionic creation (annihilation) operator. The second term describes the interaction between nearest-neighboring sites with $V$  the interaction strength and $\\hat{n}_{j} = \\hat{c}_{j}^{\\dagger}\\hat{c}_{j}$ is the number operator. The last term corresponds to the disorder potential, where $h_{i}$ is drawn from uniform distribution $\\left[-W,W\\right]$. This model is equivalent to the spin-1\/2 XXZ model with random field via Jordan-Wigner transformation.\nWe consider this model instead of the experimental system of spinful fermions for the interests of performing numerical calculations of large system sizes. The proposed method and the findings based on the spinless fermion model to present below are also expected to hold for spinful fermions as well due to the robust universality of the MBL transition. We focus on half filling in this study. \n\n\n We propose to use quench dynamics to probe MBL criticality, which is experimentally accessible with quantum microscope techniques in ultra-cold atomic gases. As elaborated in previous studies, interacting fermions will display many-body localization with strong disorder potential. In analogy to Anderson localization, the response to a local quench is expected to be bounded within a local region restricted by the localization length~\\cite{2017_Deng_Log_PRB}.  We thus propose to measure the localization length by monitoring the perturbed density profile  after a local quantum quench. \n\nMore precisely, the initial state we choose is a ``charge density wave (CDW) state\" with atoms occupying every other lattice site, which is the same as used in the experiment~\\cite{blochmbl}. We average over two different types of CDW states, atoms occupying either all even sites or odd sites. \nThen we let the state evolve for long enough time say $\\tau_1$ such that the degrees of freedom would ``locally equilibrate\" with each other. \n \nThen we introduce a sudden local quench. We provide two quench protocols for comparison---(I) lowering the potential of the quench site to zero, that had much higher energy than other sites and was initially empty before the quantum quench; and (II) performing a measurement on the quench site and remove the atom at this site. The details of the quench protocols are to be specified in Section~\\ref{sec:quenchresults}. \nThe dynamics following quench protocol-(I) is completely unitary and is thus more convenient for theoretical analysis, whereas the dynamics in protocol-(II) is non-unitary because of the involved measurement, yet has the advantage of being more experimentally feasible with quantum microscope techniques~\\cite{2015_Haller_Hudson_NatPhys,2015_Cheuk_Nichols_PRL,2015_Parsons_Huber_PRL,2015_Edge_Anderson_PRA,2015_Omran_Boll_PRL,2016_Greif_Parsons_Science,2016_Cheuk_Nichols_PRL,2016_Parsons_Mazurenko_Science,2016_Boll_Hilker_Science,2016_Cheuk_Nichols_Science,2016_Brown_Mitra_arXiv}. We will provide theoretical analysis based on the unitary evolution following quench protocol-(I) in this section, and provide simulated numerical results for both protocols in Section~\\ref{sec:quenchresults}. \n\n\n\n\nDespite the difference, both quench protocols lead to a hole on the quench site in the density profile. \nThen the density profile of this added hole is monitored. This density profile is expected to be localized (delocalized) in the MBL (ergodic) phase. The localization length can be correspondingly extracted from the time-evolved density profile at long-time limit say at $\\tau_2$. It is worth emphasizing here that our proposed scheme to probe MBL criticality is rather easily adaptable to specific experimental setups instead of being restricted to the particular local quench protocol considered here. \n\n\n\n\n\n To make it quantitative, here we analyze the quench dynamics following protocol-(I). We have total number of $L+1$ sites labeled from $0$ to $L$. The potential energy at site-$0$ is set to be  much higher  than other sites before quench, so is initially empty. After the quench this site gets filled in dynamics because its potential is then lowered down. Denoting the pre- and post- quench Hamiltonian as $H_0$ and $H$,  the perturbed density profile measures\n$\n\\delta n_j = \\langle \\psi(\\tau_1) |  {\\cal O}_j  |\\psi(\\tau_1) \\rangle \n$   \nwhere we have \n$$\n{ {\\cal O}_j} = e^{iH_0 \\Delta \\tau} \\hat{n}_j e^{-iH_0 \\Delta \\tau } - e^{iH \\Delta \\tau} \\hat{n}_j e^{-iH \\Delta \\tau }, \n$$\nwith $\\Delta \\tau = \\tau_2 - \\tau_1$. \nThe difference between $H$ and $H_0$ is strictly local near site-$0$. In the MBL phase, the phenomenological theory of local integrals of motion~\\cite{2014_Huse_MBL_PRB,2013_Serbyn_PRL,chandran2015,Ros2015420} implies that \nthe operator norm of ${\\cal O}_j$ decays exponentially  \n\\begin{equation} \n\\overline{ || {\\cal O} _j ||}  \\propto e^{-d_j \/\\xi}  \n\\end{equation} \nwith $\\xi$ a localization length, and $d_j $ the distance of the $j$-th lattice site to site-$0$, which is given by  $d_j = {\\rm min}(j, L+1-j)$ in a periodic boundary system with  size $L+1$. In the thermal phase, the support of ${\\cal O} _j$ would spread over the whole system through a linear light-cone dynamics. ~\\cite{2012_Bloch_Light_Nature, 2017_Deng_Log_PRB, Lieb1972}\nWe can thus extract the localization length from the perturbed density profile $\\delta n_j$ according to  \n\\begin{equation} \n\\xi = \\frac{\\sum_j  d_j \\overline{ \\delta  n_j} }{\\sum_j \\overline{ \\delta n_j}}. \n\\end{equation}\nIn our numerical results shown in Fig.\\ref{fig:Figure1}, we find stronger disorder strength makes the density profile more  localized and thus  $\\xi$ smaller.  Increasing interaction strength makes the density profile more extended, and  $\\xi$ becomes larger.\n\n\n\n\n\n\n\n\n\nThe system is completely localized at the strong disorder limit, whose eigenstates are simply product states. In the deep localized phase, the localization length is vanishing in our definition. Upon decreasing disorder, the localization length becomes larger but still remains to be a finite number in the MBL phase, i.e., with the scaling form $L^0$ as we change the system size. Further decreasing disorder, the scaling form of the localization length switch to $L\/4$ immediately after the ergodic transition happens. In the one dimensional system, the scaling behavior of $\\xi$ would resemble the entanglement entropy in both the localized and the ergodic phases. The difference is that $\\xi$ can be measured in experiments whereas to measure the entanglement entropy is difficult. \n \nAssuming a unique diverging length scale $\\delta W^{-\\nu}$ at  the MBL transition leads to a natural finite-size scaling ansatz for the localization length\n\\begin{equation} \\label{eq:scaling}\n\\xi\/L \\sim  g(L^{1\/\\nu} \\delta W)\n\\end{equation} \nwith $\\delta W = W - W_c$. \n \n This scaling ansatz is consistent with the fact that $\\xi\/L$ has a jump across the MBL-to-ergodic transition in the thermodynamic limit. \n\nGiven the scaling form of $\\xi\/L$, the transition point can be extracted from the crossing of the curves for $\\xi\/L $ versus the disorder strength $W$ with different system sizes. \nThe critical exponent $\\nu$ as well as critical value $W_c$ are determined by finite-size scaling analysis~\\cite{2014_Kjall_MBL_PRL,2015_Luitz_MBL_PRB}.  \n\n\n\\begin{figure}[htp]\n\t\\includegraphics[width=0.8\\linewidth]{Figure1_a}\n\t\\includegraphics[width=0.8\\linewidth]{Figure1_b}\n\t\\caption{(Color online)\n          The perturbed density profile after a local quench following protocol-(I) (see main text). \n\tHere we choose the system size $L = 16$, and average over $1000$ disorder realizations in this figure. \n\t(a), The density profile distribution with varying interaction strengths with disorder strength fixed at $W\/t = 9$. \n\t(b), The distribution for different disorder strengths with interaction fixed at $V\/t =1 $.}\n\\label{fig:Figure1} \n\\end{figure}\t\n\n\n\\begin{figure}[htp]\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure2_a}\n\t\\quad\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure2_b}\n\t\\quad\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure2_c}\n\t\\caption{\\label{fig:Figure2} (Color online) \nMany-body localization criticality from our proposed quench dynamics with protocol-(I) (see main text). (a) and (b) MBL length as a function of disorder strength for interaction strength $V\/t = 1$ and $V\/t = 1.5$ respectively. The curve intersects at $W_{c}\/t = 4.90 \\pm 0.03$ for $V\/t = 1$ and at $W_{c}\/t = 5.57 \\pm 0.04$ for $V\/t =1.5$. (c) Finite size scaling with scaling function $g\\left( L^{1\/\\nu}\\delta W\\right )$ where $\\delta W = W - W_{c}$. Critical exponent is estimated to be $\\nu = 1.02 \\pm 0.09$.}\n\\end{figure}\n\n\n\n\\section{Simulated results for post-quench dynamics} \n\\label{sec:quenchresults} \n\n \nThe proposed dynamical experiment is simulated by performing exact diagonalization calculation. We use periodic boundary condition to minimize finite-size effects in the numerical calculation, and the system sizes are $L = 8, 10, \\ldots 16$. For the experimental case, the finite-size effects are not expected to be too significant for that the number of sites in optical lattice experiments could reach the order of one hundred, much larger than that can be numerically simulated. \n\n\n\\subsection{Quench Protocol-(I) } \n\nIn the quench protocol-(I), we have an empty site labeled by an index $0$, that has much higher potential energy than other lattice sites in the system. \nThe pre-quench lattice sites are labeled as $1, 2, \\ldots L$. In the numerical simulation of pre-quench dynamics, we restrict to the Hilbert space setting the particle number at site $0$ to be explicitly $0$. \nAt $\\tau_1$ the potential at site-$0$ is lowered down to the same level as other lattice sites, which instantaneously creates a hole in the system at site $0$. The system is then evolved with a long time $\\tau_2$, where the created hole eventually stabilizes by interacting with other particles. \nFor $L \\le 12$, we perform the full matrix diagonalization and average over $10^4$ disorder realizations. For $L \\ge 14$, a  Krylov space expansion is implemented for the unitary $e^{-iHt}$ in order to save cost on memory, and we average over $10^3$ disorder realizations. \nThe dynamical evolution of local density $\\delta n_{j} = n_{j}\\left(\\tau_1\\right) - n_{j}\\left(\\tau_2\\right)$  is calculated, $n_j (\\tau) = \\langle \\psi (\\tau) | \\hat{n}_j |\\psi(\\tau)\\rangle$, with $|\\psi(\\tau)\\rangle$ the time-dependent quantum state. \n\n\nHere we would like to describe a special treatment on the perturbed density profile at the quenched site, i.e. $\\delta n_0$.  If we are in the thermal phase, the final state in our proposed dynamical procedure is ergodic, and the density distribution is uniform on every lattice site. \n\nSince at time $\\tau_1$ the occupation number at site-$0$ is different from other sites, \n$\\delta  n_0$ is compensated by adding $1\/2$.  This compensation makes $\\delta n_j$ a flat profile in the thermal phase. Meantime, the so-defined $\\delta n_j$ then obeys a normalization condition  $\\sum_j \\delta n_j = 1\/2$. \n \n\n\n\nFrom Fig.~\\ref{fig:Figure1}, averaged over different disorder realizations, $\\overline {\\delta n_i}$ indeed shows exponential decay in the MBL phase with strong disorder, and becomes flat in the ergodic phase at weak disorder. Note that the notation $\\overline{\\ldots}$ implies averaging over disorder throughout. The interaction dependence is also studied. Increasing interaction strength in the localized phase only makes the density profile of the hole a bit more delocalized, and it still shows an exponential decay, showing that the signature of many-body localization in quench dynamics is stable against interaction effects. \n\n\nFig.~\\ref{fig:Figure2} (a) shows the many-body localization criticality exhibited by the hole-profile localization length in our proposed quench dynamics. \nWith an interaction strength $V\/t = 1$,  the extracted critical disorder strength from the localization length is found to locate at $W_{c}\/t = 4.90 \\pm 0.03$.  As we increase the interaction strength, the localization becomes less robust and the required disorder strength to stabilize MBL gets larger (compare Figs.~\\ref{fig:Figure2} (a, b)). To find out the critical exponent $\\nu$, we collapse the data to the  scaling function $g\\left(L^{1\/\\nu}\\delta W\\right)$, which gives an estimate that $\\nu = 1.02 \\pm 0.09$. This value breaks the Harris-Chayes criterion $\\nu \\ge 2\/d$ with $d$ the spatial dimension~\\cite{1974_Harris_Effect_JPC,1986_Chayes_Finite_PRL}.  \n This violation would imply many-body localization criticality is beyond the description of conventional field theory or renormalization group study for disorder systems. \n\nWith the numerical results, we explicitly confirm that the proposed quench dynamics can be used to study localization length and MBL criticality. \n\n\n\n\n\\subsection{Quench Protocol---(II)} \n\n\n \nIn the quench protocol-(II), we still let the initial state evolve for long enough time $\\tau_1$. The quench is performed in a different way from protocol-(I). At $\\tau_1$, we  perform a  conditional measurement on a given site, here labeled as site-$1$. Then we do a post-selection, where measurement-outcome state is discarded if there is no particle on the measured lattice site, and the state is kept otherwise. For the kept state, we remove the particle at this site to create a hole in the density profile. This process can be carried out by quantum microscope techniques in a straightforward way. For the initial state, we choose both CDW and random states---the random state case is used as a comparison to determine whether there is any artifact due to special choice of CDW states. CDW initial states have been used in cold atom experiments to study many-body localization~\\cite{blochmbl}. \n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure3}\n\t\\caption{(Color online)\n\t\tThe density profile after removing a particle on site-$1$ for different disorder strengths following quench protocol-(II) (see main text). We choose  a system size $L = 16$, and an interaction strength $V\/t = 1$, and average over 1000 disorder realizations in this figure. The density profile in this plot is normalized to $1$. }\n\t\\label{fig:Figure3} \n\\end{figure}\n\nWe monitor the post-quench density profile of the hole $\\delta n_j$ at a long time $\\tau_{2}$. \nFrom Fig.~\\ref{fig:Figure3}, we confirm that the hole-profile is localized (extended) in MBL (ergodic) phase. \nThe local hole density is defined as $\\delta n_{j} = 1\/2 - n_{j}\\left(\\tau_{2}\\right)$, and it obeys a normalization condition $\\sum_j \\delta n_j = 1$. \nHere, we remark that removing one particle is necessary in this quench protocol to study MBL localization length---a conditional measurement  without removing the particle could not create a well-defined hole in the density profile. \n\nThe localization length extracted from this protocol is shown in Fig.~\\ref{fig:Figure4}.  We find its behavior is similar to the results for quench protocol-(I) (see Fig.~\\ref{fig:Figure2}). \nIn comparison of  CDW with random initial states,  we find no qualitative difference.  For  these two different choices of inital states, the crossing point of MBL localization length  versus disorder strength with different system sizes is consistent with each other. We find systematic data-collapse using the scaling form in Eq.~\\eqref{eq:scaling} for both of them.    \nThis implies the MBL criticality can be studied by using CDW initial states which is relatively simpler to implement in optical lattice experiments. \n\n\n\nComparing the results in  Fig.~\\ref{fig:Figure2} and Fig.~\\ref{fig:Figure4}, it is evident that quench protocol-(II) works as well as quench protocol-(I), although the dynamics is non-unitary for (II) but unitary for (I). The similarity suggests that the details of the quench protocol are not crucial for  the study of localization length and MBL criticality, provided that the quench creates a well-defined hole in the density profile. \n\n\n\n \n \nIn our proposal, it is crucial to know the required dynamical time scales before and after the quench, i.e., $\\tau_1$ and $\\tau_2$.  \nFor the dynamics to reveal intrinsic MBL physics, it is necessary that the system stabilizes for both before and after the quantum quench. \nIn Fig.~\\ref{fig:Figure5},  we show details of the dynamical evolution. \nFig.~\\ref{fig:Figure5} (a, c) show the pre-quench dynamics with $V\/t = 1$ and $1.5$. We take one type of CDW state with atoms occupying all odd sites and monitor the atom number imbalance  as $I = \\frac{N_{o}-N_{e}}{N_{o}+N_{e}}$, with $N_o$ and $N_e$ the total particle numbers on odd and even lattice sites.\n It can be seen that the system stabilizes after about $20$ times of tunneling time. \nFig.~\\ref{fig:Figure5} (b, d) show the post-quench dynamics in the hole density profile $\\delta n_j$. \nThe system is found to stabilize after about $10$ times tunneling time. \nTaking the two steps into account, the required time scale to perform the quench experiment is around $30$ times tunneling time, which is about $30-100$ milliseconds for a typical optical lattice with  tunneling time around one millisecond~\\cite{2008_Bloch_Dalibard_RMP}. This is reasonably within the lifetime of cold atom experiments. \n \n \\begin{figure}[H]\n \t\\centering\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_a}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_b}\n \t\\end{minipage}}\n \t\\quad\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_c}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_d}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_e}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_f}\n \t\\end{minipage}}\n \t\n \t\\caption{ (Color online) \n \t\tMany-body localization criticality out of the quench protocol-(II) (see main text). (a, b, c, d) show the localization length versus disorder strength with the interaction fixed at $V\/t = 1$. The transition point and criticality is estimated to be $W_{c}\/t = 4.08 \\pm 0.04$ and $\\nu = 0.86 \\pm 0.06$ for CDW initial states and $W_{c}\/t = 3.82 \\pm 0.04$ and $\\nu = 0.73 \\pm 0.06$ for random initial state. In (e) and (f), interaction strength is $V\/t = 1.5$. The transition point and criticality is estimated to be $W_{c}\/t = 4.69 \\pm 0.05$ and $\\nu = 0.92 \\pm 0.08$. In the calculation, for $L \\le 12$, the full matrix diagonalization is performed and the results are averaged over $10^4$ disorder realizations. For $L \\ge 14$, a Krylov space expansion is implemented for the unitary $e^{-iHt}$, and we average over $10^3$ disorder realizations.  }\n \t\\label{fig:Figure4}\n \\end{figure}\n\n\\begin{figure}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_a}\n\t\\end{minipage}}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_b}\n\t\\end{minipage}}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_c}\n\t\\end{minipage}}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_d}\n\t\\end{minipage}}\n\t\\caption{\\label{fig:Figure5} \nPre- and post-quench dynamics following the quench protocol-(II). Here we take CDW initial states, and choose $L = 12$. (a) and (c) correspond to pre-quench dynamics in number imbalance $I$ (see main text) with $V\/t = 1$ and $1.5$ respectively. \n(b) and (d) show the post-quench dynamics in the density profile $\\delta n_2$ for interaction strengths $V\/t = 1$ and $1.5$.}\n\\end{figure}\n\n\n\\subsection{Interaction dependence of the critical disorder strength from local quench dynamics} \nFig.~\\ref{fig:Figure7} shows a systematic study of interaction effects on MBL transition for both quench protocols.  At the interaction strength $V\/t = 2$, the fermion lattice model maps onto  random field Heisenberg model which has been extensively studied in the literature. The transition point determined from our proposed dynamical experiment agrees with previous studies using other diagnostics~\\cite{oganesyan2007,pal2010,moore2012,2013_Bauer_JSM,vadim,bardarson2014,2015_Singh_MBL_PRL,2015_Khemani_Pollmann_PRL,2015_Yu_Pekker_arXiv,2015_Lim_Sheng_PRB,2016_Kennes_Karrasch_PRB}. Our approach has an advantage in that all the quantities required to extract the localization transition and criticality can be probed directly in optical lattice experiments. \n\n\nIt is worth noting that for large system size (see Fig.\\ref{fig:Figure2}), the data crossing to determine the localization transition point has a slight rightward drift as a result of finite-size effect. To reach a conclusive answer for MBL criticality would rely on the experiments which can go to large system sizes beyond the computation capability of numerical simulations with classical resources. The advantage of our proposed strategy to probe MBL criticality is that the required ingredients are all presently accessible with cold atoms in optical lattices. \n\n\\begin{figure} [htp]\n\t\\includegraphics[angle=0,width=.7\\linewidth]{Figure6}\n\t\\caption{\\label{fig:Figure7} {{Critical disorder strength as a function of interaction strength as extracted from our proposed local quench dynamics.}}}\n\\end{figure}\n\n\n\\section{Benchmarking the dynamical protocol with Anderson localization}\n\nTo benchmark the method of probing the many-body localization length in our study,  we carry out a simulation for the well-understood three-dimensional Anderson localization whose Hamiltonian reads, \n\\begin{equation} \\label{Hamiltonian2}\n\\hat{H} = \\sum_{<jj'>}-t\\left(\\hat{c}_{j}^{\\dagger}\\hat{c}_{j'}+h.c.\\right) + \\sum_{j}h_{j}\\hat{n}_{j}. \n\\end{equation} \nFor this non-interacting model, we extract the localization length from the long-time-evolved density profile of a single particle initialized at one lattice site. This single-particle  density profile indeed takes an exponential decay form. For $L = 16$ ($20$, $24$ and $32$), the density profile is averaged over $1000$ ($100$) realizations and in the vicinity of expected transition interval, we average over 1000 realizations for all sizes. \nBy calculating the localization length from the density profile, we find $W_c\/t = 15.88 \\pm 0.14$ (see Fig.~\\ref{fig:Figure8}) and the critical exponent $\\nu = 1.6 \\pm 0.2$, which are consistent with well-known results for the Anderson model~\\cite{1999_Slevin_Corrections_PRL}.  This justifies the validity of our proposed dynamical protocols to extract localization criticality. \n\n\n\n\\begin{figure} [H]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure7_a}\n\t\\end{minipage}}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure7_b}\n\t\\end{minipage}}\n\t\\caption{\\label{fig:Figure8} (Color online) (a) Localization length as a function of disorder strength for the three dimensional Anderson model to benchmark our method. The curves for different system sizes intersects at $W_c\/t = 15.88 \\pm 0.14$ . (b) Data collapse to a  scaling function $g_{\\rm AL} \\left[L^{1\/\\nu}\\delta\\right]$ where $\\delta = W - W_{c}$, which leads to  $\\nu = 1.6 \\pm 0.2$.}\n\\end{figure}\t\n\n\n\n\\section{Conclusion} \n\nIn summary, we propose to use local quench dynamics to probe the MBL-to-ergodic criticality. Its validity is confirmed by benchmarking towards  the well-known 3D Anderson localization. In our proposed schemes, the localization length could be extracted from the exponential decay of a perturbed density profile after a local quantum quench. This proposal is expected to guide future experiments in probing MBL criticality with ultracold atoms in disordered optical lattices. Moreover, we provide a finite-size scaling form of the localization length across the transition, which is directly applicable in analyzing the quench-dynamics data out of the proposed experiments. It is worth future study to sort out the finite-size effects in the quench dynamics of many-body localization, in particular about violation of Harris-Chayes bound, for example by developing dynamical renormalization group techniques.\n\n\\section{Acknowledgments}\n{We thank I. Bloch, Y. Takahashi, D.A. Huse, E. Altman, G. Refael, X. Chen, J. Alicea, L. Jiang and S. M. Girvin for helpful discussions. \nThis work is supported by National Program on Key Basic Research Project of China under Grant No. 2017YFA0304204, National Natural Science Foundation of China under Grants No. 117740067,  and the Thousand-Youth-Talent Program of China. \n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Acknowledgements}\nThis research is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG2-RP-2020-018). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of National Research Foundation, Singapore. The authors thank Jianxin Wu, Chaoyang He, Shixuan Sun, Yaqi Xie and Yuhang Chen for their feedback. The authors also thank Yuzhi Zhao, Wei Wang, and Mo Sha for their supports of computing resources.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{More Details of the Datasets}\nThe statistics of each dataset are shown in Table \\ref{tbl:data} when setting $\\beta=0.5$. All datasets provide a training dataset and test dataset. All the reported accuracies are computed on the test dataset.\n\n\\begin{table}[h]\n\\centering\n\\caption{The statistics of datasets.}\n\\label{tbl:data}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{dataset} & \\multicolumn{2}{c|}{\\#training samples\/party} & \\multirow{2}{*}{\\#test samples} \\\\ \\cline{2-3}\n & mean & std &  \\\\ \\hline \\hline\nCIFAR10 & 5,000 & 1,165 & 10,000 \\\\ \\hline\nCIFAR100 & 5,000 & 181 & 10,000 \\\\ \\hline\nTiny-Imagenet & 10,000 & 99 & 10,000 \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\nFigure \\ref{fig:datadis_beta01} and Figure \\ref{fig:datadis_beta5} show the data distribution of $\\beta=0.1$ and $\\beta=5$ (used in Section 4.6 of the main paper), respectively.\n\n\n\n\\begin{figure*}\n\\centering\n\\subfloat[CIFAR-10]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar10_data_beta01.pdf}%\n}\n\\hfill\n\\subfloat[CIFAR-100]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar100_data_beta01.pdf}%\n}\n\\hfill\n\\subfloat[Tiny-Imagenet]{\\includegraphics[width=0.32\\textwidth]{figures\/tinyimg_data_beta01.pdf}}\n\\caption{The data distribution of each party using non-IID data partition with $\\beta=0.1$. }\n\\label{fig:datadis_beta01}\n\\vspace{-10pt}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\subfloat[CIFAR-10]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar10_data_beta5.pdf}%\n}\n\\hfill\n\\subfloat[CIFAR-100]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar100_data_beta5.pdf}%\n}\n\\hfill\n\\subfloat[Tiny-Imagenet]{\\includegraphics[width=0.32\\textwidth]{figures\/tinyimg_data_beta5.pdf}}\n\\caption{The data distribution of each party using non-IID data partition with $\\beta=5$.}\n\\label{fig:datadis_beta5}\n\\vspace{-10pt}\n\\end{figure*}\n\n\n\\section{Projection Head}\nWe use a projection head to map the representation like \\cite{simclr}. Here we study the effect of the projection head. We remove the projection head and conduct experiments on CIFAR-10 and CIFAR-100 (Note that the network architecture changes for all approaches). The results are shown in Table \\ref{tbl:projhead}. We can observe that MOON can benefit a lot from the projection head. The accuracy of MOON can be improved by about 2\\% on average with a projection head.\n\n\n\\begin{table}[h]\n\\centering\n\\caption{The top1-accuracy with\/without projection head.}\n\\label{tbl:projhead}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Method} & CIFAR-10 & CIFAR-100 \\\\ \\hline\\hline\n\\multirow{5}{*}{\\begin{tabular}[c]{@{}c@{}}without\\\\ projection \\\\ head\\end{tabular}} & MOON & 66.8\\% & 66.1\\% \\\\ \\cline{2-4} \n & FedAvg & 66.7\\% & 65.0\\% \\\\ \\cline{2-4} \n & FedProx & 67.5\\% & 65.4\\% \\\\ \\cline{2-4} \n & SCAFFOLD & 67.1\\% & 49.5\\% \\\\ \\cline{2-4} \n & SOLO & 39.8\\%$\\pm$3.9\\% & 22.5\\%$\\pm$1.1\\% \\\\ \\hline\n\\multirow{5}{*}{\\begin{tabular}[c]{@{}c@{}}with\\\\  projection \\\\ head\\end{tabular}} & MOON & \\tb{69.1\\%} & \\tb{67.5\\%} \\\\ \\cline{2-4} \n & FedAvg & 66.3\\% & 64.5\\% \\\\ \\cline{2-4} \n & FedProx & 66.9\\% & 64.6\\% \\\\ \\cline{2-4} \n & SCAFFOLD & 66.6\\% & 52.5\\% \\\\ \\cline{2-4} \n & SOLO & 46.3\\%$\\pm$5.1\\% & 22.3\\%$\\pm$1.0\\% \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\\section{IID Partition}\nTo further show the effect of our model-contrastive loss, we compare MOON and FedAvg when there is no heterogeneity among local datasets. The dataset is randomly and equally partitioned into the parties. The results are shown in Table \\ref{tbl:homo}. We can observe that the model-contrastive loss has little influence on the training when the local datasets are IID. The accuracy of MOON is still very close to FedAvg even though with a large $\\mu$. MOON is still applicable when there is no heterogeneity issue in data distributions across parties.\n\n\n\\begin{table}[h]\n\\centering\n\\caption{The top-1 accuracy of MOON and FedAvg with IID data partition on CIFAR-10.}\n\\label{tbl:homo}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Method} & Top-1 accuracy\\\\ \\hline\\hline\n\\multirow{4}{*}{MOON} & $\\mu=0.1$ & 73.6\\%   \\\\ \\cline{2-3} \n & $\\mu=1$ & 73.6\\%   \\\\ \\cline{2-3} \n & $\\mu=5$ & 73.0\\%   \\\\ \\cline{2-3} \n & $\\mu=10$ & 72.8\\%   \\\\ \\hline\n\\multicolumn{2}{|c|}{FedAvg ($\\mu=0$)} & 73.4\\%  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\section{Hyper-Parameters Study}\n\n\\subsection{Effect of $\\mu$}\nWe show the accuracy of MOON with different $\\mu$ in Table \\ref{tbl:mu}. The best $\\mu$ for CIFAR-10, CIFAR-100, and Tiny-Imagenet are 5, 1, and 1, respectively. When $\\mu$ is set to a small value (i.e., $\\mu=0.1$), the accuracy of MOON is very close to FedAvg (i.e., $\\mu=0$) since the impact of model-contrastive loss is small. As long as we set $\\mu \\geq 1$, MOON can benefit a lot from the model-contrastive loss. Overall, we find that $\\mu=1$ is a reasonable good choice if they do not want to tune the parameter, where MOON achieves at least 2\\% higher accuracy than FedAvg.\n\n\\begin{table}[!]\n\\centering\n\\caption{The test accuracy of MOON with $\\mu$ from \\{0, 0.1, 1, 5, 10\\}. Note that MOON is actually FedAvg when $\\mu=0$.}\n\\label{tbl:mu}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$\\mu$ & CIFAR-10 & CIFAR-100 & Tiny-Imagenet \\\\ \\hline\\hline\n0 & 66.3\\% & 64.5\\% & 23.0\\% \\\\ \\hline\n0.1 & 66.5\\% & 65.1\\% & 23.4\\% \\\\ \\hline\n1 & 68.4\\% & \\tb{67.5\\%} & \\tb{25.1\\%} \\\\ \\hline\n5 & \\tb{69.1\\%} & 67.1\\% & 24.4\\% \\\\ \\hline\n10 & 68.3\\% & 67.3\\% & 25.0\\% \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Effect of temperature and output dimension} We tune $\\tau$ from \\{0.1, 0.5, 1.0\\} and tune the output dimension of projection head from \\{64, 128, 256\\}. The results are shown in Figure \\ref{fig:tau_outdim}. The best $\\tau$ for CIFAR-10, CIFAR-100, and Tiny-Imagenet are 0.5, 1.0, and 0.5, respectively. The best output dimension for CIFAR-10, CIFAR-100, and Tiny-Imagenet are 128, 256, and 128, respectively. Generally, MOON is stable regarding the change of temperature and output dimension. As we have shown in the main paper, MOON already improves FedAvg a lot with a default setting of temperature (i.e., 0.5) and output dimension (i.e., 256). Users may tune these two hyper-parameters to achieve even better accuracy.\n\n\\begin{figure}[h]\n\\centering\n\\subfloat[The effect of $\\tau$\\label{fig:tau}]{\\includegraphics[width=0.48\\linewidth]{figures\/tunetempe.pdf}\n}\n\\subfloat[The effect of output dimension\\label{fig:outdim}]{\\includegraphics[width=0.48\\linewidth]{figures\/tunedim.pdf}}\n\\caption{The top-1 accuracy of MOON trained with different temperatures and output dimensions.}\n\\label{fig:tau_outdim}\n\\vspace{-10pt}\n\\end{figure}\n\n\n\n\n\n\\section{Combining with FedAvgM}\nAs we have mentioned in the fourth paragraph of Section 2.1, MOON can be combined with the approaches working on improving the aggregation phase. Here we combine MOON and FedAvgM \\cite{hsu2019measuring}. We tune the server momentum $\\beta \\in \\{0.1, 0.7, 0.9\\}$. With the default experimental setting in Section 4.1, the results are shown in Table \\ref{tbl:fedavgm}. While FedAvgM is better than FedAvg, MOON can further improve FedAvgM by 2-3\\%.\n\n\\begin{table}[!]\n\\centering\n\\caption{The combining of MOON and FedAvgM.}\n\\label{tbl:fedavgm}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nDatasets & FedAvg & MOON & FedAvgM & MOON+FedAvgM \\\\ \\hline\\hline\nCIFAR-10 & 66.3\\% & 69.1\\% & 67.1\\% & 69.6\\% \\\\ \\hline\nCIFAR-100 & 64.5\\% & 67.5\\% &65.1\\%  &67.8\\%  \\\\ \\hline\nTiny-Imagenet & 23.0\\% & 25.1\\% &23.4\\%  &25.5\\%  \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\\section{Computation Cost}\nSince MOON introduces an additional loss term in the local training phase, the training of MOON will be slower than FedAvg. For the experiments in Table 1, the average training time per round with a NVIDIA Tesla V100 GPU and four Intel Xeon E5-2684 20-core CPUs are shown in Table \\ref{tbl:local_time}. Compared with FedAvg, the computation overhead of MOON is acceptable especially on CIFAR-10 and CIFAR-100.\n\n\\begin{table}[!]\n\\centering\n\\caption{The average training time per round.}\n\\label{tbl:local_time}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nMethod & CIFAR-10 & CIFAR-100 & Tiny-Imagenet \\\\ \\hline\\hline\nFedAvg & 330s & 20min & 103min \\\\ \\hline\nFedProx & 340s & 24min & 135min \\\\ \\hline\nSCAFFOLD & 332s & 20min & 112min \\\\ \\hline\nMOON & 337s & 31min & 197min \\\\ \\hline\n\\end{tabular}\n}\n\\vspace{-10pt}\n\\end{table}\n\n\\section{Number of Negative Pairs}\nIn typical contrastive learning, the performance usually can be improved by increasing the number of negative pairs (i.e., views of different images). In MOON, the negative pair is the local model being updated and the local model from the previous round. We consider using a single negative pair during training in the main paper. It is possible to consider multiple negative pairs if we include multiple local models from the previous rounds. Suppose the current round is $t$. Let $k$ denotes the maximum number of negative pairs. Let $z_{prev}^i = R_{w_i^{t-i}}(x)$ (i.e., $z_{prev}^i$ is the representation learned by the local model from $(t-i)$ round). Then, our local objective is\n\n{\\small\n\\begin{equation}\n    \\ell_{con}  = -\\log{\\frac{\\exp(\\textrm{sim}(z, z_{glob})\/\\tau)}{\\exp(\\textrm{sim}(z, z_{glob})\/\\tau) + \\sum_{i=1}^k\\exp(\\textrm{sim}(z, z_{prev}^i)\/\\tau)}}\n\\end{equation}\n}\n\nIf $k=1$, then the objective is the same as MOON presented in the main paper. If $k>t$, since there are at most $t$ local models from previous rounds, we only consider the previous $t$ local models (i.e., only $t$ negative pairs). There is no model-contrastive loss if $t=0$ (i.e., the first round). Here we study the effect of the maximum number of negative pairs on CIFAR-10. The results are shown in Table \\ref{tbl:n_pair}. Unlike typical contrastive learning, the accuracy of MOON cannot be increased by increasing the number of negative pairs. MOON can achieve the best accuracy when $k=1$, which is presented in our main paper.\n\n\\begin{table}[h]\n\\centering\n\\caption{The effect of maximum number of negative pairs. We tune $\\mu$ from \\{0.1, 1, 5, 10\\} for all approaches and report the best accuracy.}\n\\label{tbl:n_pair}\n\\begin{tabular}{|c|c|}\n\\hline\nmaximum number of negative pairs & top-1 accuracy \\\\ \\hline \\hline\n$k=1$ & \\tb{69.1\\%} \\\\ \\hline\n$k=2$ & 67.2\\% \\\\ \\hline\n$k=5$ & 67.7\\% \\\\ \\hline\n$k=100$ & 63.5\\% \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\section{Background and Related Work}\n\n\\subsection{Federated Learning}\nFedAvg \\cite{mcmahan2016communication} has been a de facto approach for federated learning. The framework of FedAvg is shown in Figure \\ref{fig:fedavg_fram}. There are four steps in each round of FedAvg. First, the server sends a global model to the parties. Second, the parties perform stochastic gradient descent (SGD) to update their models locally. Third, the local models are sent to a central server. Last, the server averages the model weights to produce a global model for the training of the next round.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/fl_fram_print.pdf}\n    \\caption{The FedAvg framework. In this paper, we focus on the second step, i.e., the local training phase.}\n    \\label{fig:fedavg_fram}\n    \\vspace{-5pt}\n\\end{figure}\n\n\nThere have been quite some studies trying to improve FedAvg on non-IID data. Those studies can be divided into two categories: improvement on local training (i.e., step 2 of Figure \\ref{fig:fedavg_fram}) and on aggregation (i.e., step 4 of Figure \\ref{fig:fedavg_fram}). This study belongs to the first category.\n\nAs for studies on improving local training, FedProx \\cite{lifedprox} introduces a proximal term into the objective during local training. The proximal term is computed based on the $\\ell_2$-norm distance between the current global model and the local model. Thus, the local model update is limited by the proximal term during the local training. SCAFFOLD \\cite{karimireddy2019scaffold} corrects the local updates by introducing control variates. Like the training model, the control variates are also updated by each party during local training. The difference between the local control variate and the global control variate is used to correct the gradients in local training. However, FedProx shows experiments on MNIST and EMNIST only with multinomial logistic regression, while SCAFFOLD only shows experiments on EMNIST with logistic regression and 2-layer fully connected layer. The effectiveness of FedProx and SCAFFOLD on image datasets with deep learning models has not been well explored. As shown in our experiments, those studies have little or even no advantage over FedAvg, which motivates this study for a new approach of handling non-IID image datasets with deep learning models. We also notice that there are other related contemporary work \\cite{acar2021federated,li2021fedbn,wang2020addressing} when preparing this paper. We leave the comparison between MOON and these contemporary work as future studies.\n\nAs for studies on improving the aggregation phase, FedMA \\cite{Wang2020Federated} utilizes Bayesian non-parametric methods to match and average weights in a layer-wise manner. FedAvgM \\cite{hsu2019measuring} applies momentum when updating the global model on the server. Another recent study, FedNova \\cite{wang2020tackling}, normalizes the local updates before averaging.\nOur study is orthogonal to them and potentially can be combined with these techniques as we work on the local training phase.\n\nAnother research direction is personalized federated learning \\cite{fallah2020personalized,dinh2020personalized,hanzely2020lower,zhang2021personalized,huang2021personalized}, which tries to learn personalized local models for each party. In this paper, we study the typical federated learning, which tries to learn a single global model for all parties.\n\n\n\\subsection{Contrastive Learning}\n\nSelf-supervised learning \\cite{jing2020self,grill2020bootstrap,simclr,simclr2,he2020momentum,misra2020self} is a recent hot research direction, which tries to learn good data representations from unlabeled data. Among those studies, contrastive learning approaches \\cite{simclr,simclr2,he2020momentum,misra2020self} achieve state-of-the-art results on learning visual representations. The key idea of contrastive learning is to reduce the distance between the representations of different augmented views of the same image (i.e., \\emph{positive pairs}), and increase the distance between the representations of augmented views of different images (i.e., \\emph{negative pairs}).\n\nA typical contrastive learning framework is SimCLR \\cite{simclr}. Given an image $x$, SimCLR first creates two correlated views of this image using different data augmentation operators, denoted $x_i$ and $x_j$. A base encoder $f(\\cdot)$ and a projection head $g(\\cdot)$ are trained to extract the representation vectors and map the representations to a latent space, respectively. Then, a contrastive loss (i.e., NT-Xent \\cite{sohn2016improved}) is applied on the projected vector $g(f(\\cdot))$, which tries to maximize agreement between differently augmented views of the same image. Specifically, given $2N$ augmented views and a pair of view $x_i$ and $x_j$ of same image, the contrastive loss for this pair is defined as \n\n\\begin{equation}\nl_{i, j} = -\\log{\\frac{\\exp(\\textrm{sim}(x_i, x_j)\/\\tau)}{\\sum_{k=1}^{2N} \\mathbb{I}_{[k\\neq i]}\\exp(\\textrm{sim}(x_i, x_k)\/\\tau)}}\n\\end{equation}\n\nwhere $\\textrm{sim}(\\cdot,\\cdot)$ is a cosine similarity function and $\\tau$ denotes a temperature parameter. The final loss is computed by summing the contrastive loss of all pairs of the same image in a mini-batch. \n\nBesides SimCLR, there are also other contrastive learning frameworks such as CPC \\cite{oord2018representation}, CMC \\cite{tian2019contrastive} and MoCo \\cite{he2020momentum}. We choose SimCLR for its simplicity and effectiveness in many computer vision tasks. Still, the basic idea of contrastive learning is similar among these studies: the representations obtained from different images should be far from each other and the representations obtained from the same image should be related to each other. The idea is intuitive and has been shown to be effective. \n\nThere is one recent study \\cite{zhang2020federated} that combines federated learning with contrastive learning. They focus on the unsupervised learning setting. Like SimCLR, they use contrastive loss to compare the representations of different images. In this paper, we focus on the supervised learning setting and propose model-contrastive learning to compare representations learned by different models.\n\\section{Conclusion}\nFederated learning has become a promising approach to resolve the pain of data silos in many domains such as medical imaging, object detection, and landmark classification. Non-IID is a key challenge for the effectiveness of federated learning. To improve the performance of federated deep learning models on non-IID datasets, we propose model-contrastive learning (MOON), a simple and effective approach for federated learning. MOON introduces a new learning concept, i.e., contrastive learning in model-level. Our extensive experiments show that MOON achieves significant improvement over state-of-the-art approaches on various image classification tasks. As MOON does not require the inputs to be images, it potentially can be applied to non-vision problems.\n\n\\section{Experiments}\n\\label{sec:exp}\n\\subsection{Experimental Setup}\n\\label{sec:exp_setup}\nWe compare MOON with three state-of-the-art approaches including (1) FedAvg \\cite{mcmahan2016communication},  (2) FedProx \\cite{lifedprox}, and (3) SCAFFOLD \\cite{karimireddy2019scaffold}.  We also compare a baseline approach named SOLO, where each party trains a model with its local data without federated learning. We conduct experiments on three datasets including CIFAR-10, CIFAR-100, and Tiny-Imagenet\\footnote{\\url{https:\/\/www.kaggle.com\/c\/tiny-imagenet}} (100,000 images with 200 classes). Moreover, we try two different network architectures. For CIFAR-10, we use a CNN network as the base encoder, which has two 5x5 convolution layers followed by 2x2 max pooling (the first with 6 channels and the second with 16 channels) and two fully connected layers with ReLU activation (the first with 120 units and the second with 84 units). For CIFAR-100 and Tiny-Imagenet, we use ResNet-50 \\cite{he2016deep} as the base encoder. For all datasets, like \\cite{simclr}, we use a 2-layer MLP as the projection head. The output dimension of the projection head is set to 256 by default. Note that all baselines use the same network architecture as MOON (including the projection head) for fair comparison.\n\nWe use PyTorch \\cite{paszke2019pytorch} to implement MOON and the other baselines. The code is publicly available\\footnote{\\url{https:\/\/github.com\/QinbinLi\/MOON}}. We use the SGD optimizer with a learning rate 0.01 for all approaches. The SGD weight decay is set to 0.00001 and the SGD momentum is set to 0.9. The batch size is set to 64. The number of local epochs is set to 300 for SOLO. The number of local epochs is set to 10 for all federated learning approaches unless explicitly specified. The number of communication rounds is set to 100 for CIFAR-10\/100 and 20 for Tiny-ImageNet, where all federated learning approaches have little or no accuracy gain with more communications. For MOON, we set the temperature parameter to 0.5 by default like \\cite{simclr}.\n\n\n\nLike previous studies \\cite{pmlr-v97-yurochkin19a,Wang2020Federated}, we use Dirichlet distribution to generate the non-IID data partition among parties. Specifically, we sample $p_k \\sim Dir_N(\\beta)$ and allocate a $p_{k,j}$ proportion of the instances of class $k$ to party $j$, where $Dir(\\beta)$ is the Dirichlet distribution with a concentration parameter $\\beta$ (0.5 by default). With the above partitioning strategy, each party can have relatively few (even no) data samples in some classes. We set the number of parties to 10 by default. The data distributions among parties in default settings are shown in Figure \\ref{fig:datadis}. For more experimental results, please refer to Appendix.\n\n\\begin{figure*}[!]\n\\centering\n\\subfloat[CIFAR-10]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar10_data.pdf}%\n}\n\\hfill\n\\subfloat[CIFAR-100]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar100_data.pdf}%\n}\n\\hfill\n\\subfloat[Tiny-Imagenet]{\\includegraphics[width=0.32\\textwidth]{figures\/tinyimg_data.pdf}}\n\\caption{The data distribution of each party using non-IID data partition. The color bar denotes the number of data samples. Each rectangle represents the number of data samples of a specific class in a party. }\n\\label{fig:datadis}\n\\end{figure*}\n\n\n\\subsection{Accuracy Comparison}\n\\label{sec:overall_acc}\nFor MOON, we tune $\\mu$ from $\\{0.1, 1, 5, 10\\}$ and report the best result. The best $\\mu$ of MOON for CIFAR-10, CIFAR-100, and Tiny-Imagenet are 5, 1, and 1, respectively. Note that FedProx also has a hyper-parameter $\\mu$ to control the weight of its proximal term (i.e., $L_{FedProx} = \\ell_{FedAvg} + \\mu \\ell_{prox}$). For FedProx, we tune $\\mu$ from $\\{0.001, 0.01, 0.1, 1\\}$ (the range is also used in the previous paper \\cite{lifedprox}) and report the best result. The best $\\mu$ of FedProx for CIFAR-10, CIFAR-100, and Tiny-Imagenet are $0.01$, $0.001$, and $0.001$, respectively. Unless explicitly specified, we use these $\\mu$ settings for all the remaining experiments.\n\nTable \\ref{tbl:allacc} shows the top-1 test accuracy of all approaches with the above default setting. Under non-IID settings, SOLO demonstrates much worse accuracy than other federated learning approaches. This demonstrates the benefits of federated learning. Comparing different federated learning approaches, we can observe that MOON is consistently the best approach among all tasks. It can outperform FedAvg by 2.6\\% accuracy on average of all tasks. For FedProx, its accuracy is very close to FedAvg. The proximal term in FedProx has little influence in the training since $\\mu$ is small. However, when $\\mu$ is not set to a very small value, the convergence of FedProx is quite slow (see Section \\ref{sec:comm_effi}) and the accuracy of FedProx is bad. For SCAFFOLD, it has much worse accuracy on CIFAR-100 and Tiny-Imagenet than other federated learning approaches.\n\n\n\\begin{table}[]\n\\centering\n\\caption{The top-1 accuracy of MOON and the other baselines on test datasets. For MOON, FedAvg, FedProx, and SCAFFOLD, we run three trials and report the mean and standard derivation. For SOLO, we report the mean and standard derivation among all parties.}\n\\label{tbl:allacc}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nMethod & CIFAR-10 & CIFAR-100 & Tiny-Imagenet \\\\ \\hline \\hline\nMOON & \\tb{69.1\\%}$\\pm$0.4\\% & \\tb{67.5\\%}$\\pm$0.4\\% & \\tb{25.1\\%}$\\pm$0.1\\%  \\\\ \\hline\nFedAvg & 66.3\\%$\\pm$0.5\\% & 64.5\\% $\\pm$0.4\\% & 23.0\\%$\\pm$0.1\\%   \\\\ \\hline\nFedProx & 66.9\\%$\\pm$0.2\\% & 64.6\\%$\\pm$0.2\\% & 23.2\\%$\\pm$0.2\\% \\\\ \\hline\nSCAFFOLD & 66.6\\%$\\pm$0.2\\% & 52.5\\% $\\pm$0.3\\% & 16.0\\%$\\pm$0.2\\%  \\\\ \\hline\nSOLO &46.3\\% $\\pm$5.1\\% &22.3\\%$\\pm$1.0\\% &8.6\\%$\\pm$0.4\\%\\\\\\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\\subsection{Communication Efficiency}\n\\label{sec:comm_effi}\nFigure \\ref{fig:comm} shows the accuracy in each round during training. As we can see, the model-contrastive loss term has little influence on the convergence rate with best $\\mu$. The speed of accuracy improvement in MOON is almost the same as FedAvg at the beginning, while it can achieve a better accuracy later benefit from the model-contrastive loss. Since the best $\\mu$ values are generally small in FedProx, FedProx with best $\\mu$ is very close to FedAvg, especially on CIFAR-10 and CIFAR-100. However, when setting $\\mu=1$, FedProx becomes very slow due to the additional proximal term. This implies that limiting the $\\ell_2$-norm distance between the local model and the global model is not an effective solution. Our model-contrastive loss can effectively increase the accuracy without slowing down the convergence.\n\n\n\\begin{figure*}[!]\n\\centering\n\\subfloat[CIFAR-10]{\\includegraphics[width=0.32\\textwidth]{figures\/newcomm_cifar10.pdf}%\n}\n\\hfill\n\\subfloat[CIFAR-100]{\\includegraphics[width=0.32\\textwidth]{figures\/newcomm_cifar100.pdf}%\n}\n\\hfill\n\\subfloat[Tiny-Imagenet]{\\includegraphics[width=0.32\\textwidth]{figures\/newcomm_tinyimg.pdf}}\n\\caption{The top-1 test accuracy in different number of communication rounds. For FedProx, we report both the accuracy with best $\\mu$ and the accuracy with $\\mu=1$.}\n\\label{fig:comm}\n\\vspace{-15pt}\n\\end{figure*}\n\nWe show the number of communication rounds to achieve the same accuracy as running FedAvg for 100 rounds on CIFAR-10\/100 or 20 rounds on Tiny-Imagenet in Table \\ref{tbl:comm}. We can observe that the number of communication rounds is significantly reduced in MOON. MOON needs about half the number of communication rounds on CIFAR-100 and Tiny-Imagenet compared with FedAvg. On CIFAR-10, the speedup of MOON is even close to 4. MOON is much more communication-efficient than the other approaches.\n\n\\begin{table}[]\n\\centering\n\\caption{The number of rounds of different approaches to achieve the same accuracy as running FedAvg for 100 rounds (CIFAR-10\/100) or 20 rounds (Tiny-Imagenet). The speedup of an approach is computed against FedAvg.}\n\\label{tbl:comm}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n \\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{CIFAR-10} & \\multicolumn{2}{c|}{CIFAR-100} & \\multicolumn{2}{c|}{Tiny-Imagenet} \\\\ \\cline{2-7} \n & \\#rounds & speedup & \\#rounds & speedup & \\#rounds & speedup \\\\ \\hline \\hline\nFedAvg & 100 & 1$\\times$ & 100 & 1$\\times$ &20  & 1$\\times$ \\\\ \\hline\nFedProx & 52 & 1.9$\\times$ & 75 & 1.3$\\times$ & 17 & 1.2$\\times$ \\\\ \\hline\nSCAFFOLD & 80 & 1.3$\\times$ & \\backslashbox{}{} & \\textless{}1$\\times$  & \\backslashbox{}{} & \\textless{}1$\\times$  \\\\ \\hline\nMOON & \\tb{27} & \\tb{3.7}$\\times$ & \\tb{43} & \\tb{2.3}$\\times$ & \\tb{11} & \\tb{1.8}$\\times$ \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\\begin{figure*}\n\\centering\n\\subfloat[CIFAR-10]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar10tunep.pdf}%\n}\n\\hfill\n\\subfloat[CIFAR-100]{\\includegraphics[width=0.32\\textwidth]{figures\/cifar100tunep.pdf}%\n}\n\\hfill\n\\subfloat[Tiny-Imagenet]{\\includegraphics[width=0.32\\textwidth]{figures\/tinyimgtunep.pdf}}\n\\caption{The top-1 test accuracy with different number of local epochs. For MOON and FedProx, $\\mu$ is set to the best $\\mu$ from Section \\ref{sec:overall_acc} for all numbers of local epochs. The accuracy of SCAFFOLD is quite bad when number of local epochs is set to 1 (45.3\\% on CIFAR10, 20.4\\% on CIFAR-100, 2.6\\% on Tiny-Imagenet). The accuracy of FedProx on Tiny-Imagenet with one local epoch is 1.2\\%.}\n\\label{fig:epoch}\n\\vspace{-5pt}\n\\end{figure*}\n\n\\subsection{Number of Local Epochs}\nWe study the effect of number of local epochs on the accuracy of final model. The results are shown in Figure \\ref{fig:epoch}. When the number of local epochs is 1, the local update is very small. Thus, the training is slow and the accuracy is relatively low given the same number of communication rounds. All approaches have a close accuracy (MOON is still the best). When the number of local epochs becomes too large, the accuracy of all approaches drops, which is due to the drift of local updates, i.e., the local optima are not consistent with the global optima. Nevertheless, MOON clearly outperforms the other approaches. This further verifies that MOON can effectively mitigate the negative effects of the drift by too many local updates.\n\n\n\n\n\\subsection{Scalability}\n\nTo show the scalability of MOON, we try a larger number of parties on CIFAR-100. Specifically, we try two settings: (1) We partition the dataset into 50 parties and all parties participate in federated learning in each round. (2) We partition the dataset into 100 parties and randomly sample 20 parties to participate in federated learning in each round (client sampling technique introduced in FedAvg \\cite{mcmahan2016communication}). The results are shown in Table \\ref{tbl:scala} and Figure \\ref{fig:scala}. For MOON, we show the results with $\\mu=1$ (best $\\mu$ from Section \\ref{sec:overall_acc}) and $\\mu=10$. For MOON ($\\mu=1$), it outperforms the FedAvg and FedProx over 2\\% accuracy at 200 rounds with 50 parties and 3\\% accuracy at 500 rounds with 100 parties. Moreover, for MOON ($\\mu=10$), although the large model-contrastive loss slows down the training at the beginning as shown in Figure \\ref{fig:scala}, MOON can outperform the other approaches a lot with more communication rounds. Compared with FedAvg and FedProx, MOON achieves about about 7\\% higher accuracy at 200 rounds with 50 parties and at 500 rounds with 100 parties. SCAFFOLD has a low accuracy with a relatively large number of parties.\n\n\n\n\\begin{table}[]\n\\centering\n\\caption{The accuracy with 50 parties and 100 parties (sample fraction=0.2) on CIFAR-100.}\n\\label{tbl:scala}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{\\#parties=50} & \\multicolumn{2}{c|}{\\#parties=100} \\\\ \\cline{2-5} \n  & 100 rounds & 200 rounds & 250 rounds & 500 rounds \\\\ \\hline \\hline\nMOON ($\\mu$=1) & 54.7\\% &58.8\\% & 54.5\\% & 58.2\\% \\\\ \\hline\nMOON ($\\mu$=10)& \\tb{58.2\\%} &\\tb{63.2\\%} & \\tb{56.9\\%} & \\tb{61.8\\%} \\\\ \\hline\nFedAvg & 51.9\\% &56.4\\% & 51.0\\% & 55.0\\% \\\\ \\hline\nFedProx  & 52.7\\% & 56.6\\% & 51.3\\% & 54.6\\% \\\\ \\hline\nSCAFFOLD & 35.8\\% & 44.9\\% & 37.4\\% & 44.5\\% \\\\ \\hline\nSOLO & \\multicolumn{2}{c|}{10\\%$\\pm$0.9\\%} & \\multicolumn{2}{c|}{7.3\\%$\\pm$0.6\\%} \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\\begin{figure*}\n\\centering\n\\subfloat[50 parties]{\\includegraphics[width=0.45\\linewidth]{figures\/200rounds_comm_cifar100_p50.pdf}\n}\n\\hfil\n\\subfloat[100 parties (sample fraction=0.2)]{\\includegraphics[width=0.45\\linewidth]{figures\/500rounds_newcomm_cifar100_p100.pdf}%\n}\n\\hfil\n\\caption{The top-1 test accuracy on CIFAR-100 with 50\/100 parties.}\n\\label{fig:scala}\n\\end{figure*}\n\n\\subsection{Heterogeneity}\nWe study the effect of data heterogeneity by varying the concentration parameter $\\beta$ of Dirichlet distribution on CIFAR-100. For a smaller $\\beta$, the partition will be more unbalanced. The results are shown in Table \\ref{tbl:beta}. MOON always achieves the best accuracy among three unbalanced levels. When the unbalanced level decreases (i.e., $\\beta=5$), FedProx is worse than FedAvg, while MOON still outperforms FedAvg with more than 2\\% accuracy. The experiments demonstrate the effectiveness and robustness of MOON. \n\n\n\\begin{table}[]\n\\centering\n\\caption{The test accuracy with $\\beta$ from \\{0.1, 0.5, 5\\}. }\n\\label{tbl:beta}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nMethod & $\\beta=0.1$ & $\\beta=0.5$ & $\\beta=5$ \\\\ \\hline \\hline\nMOON & \\tb{64.0\\%} & \\tb{67.5\\%} & \\tb{68.0\\%} \\\\ \\hline\nFedAvg & 62.5\\% & 64.5\\% & 65.7\\% \\\\ \\hline\nFedProx & 62.9\\% & 64.6\\% & 64.9\\% \\\\ \\hline\nSCAFFOLD & 47.3\\% & 52.5\\% & 55.0\\% \\\\ \\hline\nSOLO & 15.9\\%$\\pm$1.5\\% & 22.3\\%$\\pm$1\\% & 26.6\\%$\\pm$1.4\\% \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\n\n\\subsection{Loss Function}\n\nTo maximize the agreement between the representation learned by the global model and the representation learned by the local model, our model-contrastive loss $\\ell_{con}$ is proposed inspired by NT-Xent loss \\cite{simclr}. Another intuitive option is to use $\\ell_2$ regularization, and the local loss is\n\\begin{equation}\n    \\ell = \\ell_{sup} + \\mu \\norm{z-z_{glob}}_2\n\\end{equation}\n\nHere we compare the approaches using different kinds of loss functions to limit the representation: no additional term (i.e., FedAvg: $L = \\ell_{sup}$), $\\ell_2$ norm, and our model-contrastive loss. The results are shown in Table \\ref{tbl:loss}. We can observe that simply using $\\ell_2$ norm even cannot improve the accuracy compared with FedAvg on CIFAR-10. While using $\\ell_2$ norm can improve the accuracy on CIFAR-100 and Tiny-Imagenet, the accuracy is still lower than MOON. Our model-contrastive loss is an effective way to constrain the representations. \n\nOur model-contrastive loss influences the local model from two aspects. Firstly, the local model learns a close representation to the global model. Secondly, the local model also learns a better representation than the previous one until the local model is good enough (i.e., $z=z_{glob}$ and $\\ell_{con}$ becomes a constant).\n\n\n\\begin{table}[]\n\\centering\n\\caption{The top-1 accuracy with different kinds of loss for the second term of local objective. We tune $\\mu$ from \\{0.001, 0.01 , 0.1, 1, 5, 10\\} for the $\\ell_2$ norm approach and report the best accuracy.}\n\\label{tbl:loss}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nsecond term & CIFAR-10 & CIFAR-100 & Tiny-Imagenet \\\\ \\hline\\hline\nnone (FedAvg) & 66.3\\% & 64.5\\% & 23.0\\% \\\\ \\hline\n$\\ell_2$ norm & 65.8\\% & 66.9\\% & 24.0\\% \\\\ \\hline\nMOON & \\tb{69.1\\%} & \\tb{67.5\\%} & \\tb{25.1\\%} \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\\section{Introduction}\nDeep learning is data hungry. Model training can benefit a lot from a large and representative dataset (\\eg, ImageNet \\cite{deng2009imagenet} and COCO~\\cite{lin2014microsoft}). However, data are usually dispersed among different parties in practice (\\eg, mobile devices and companies). Due to the increasing privacy concerns and data protection regulations \\cite{voigt2017eu}, parties cannot send their private data to a centralized server to train a model. \n\nTo address the above challenge, federated learning \\cite{kairouz2019advances,yang2019federated,litian2019survey,li2019flsurvey} enables multiple parties to jointly learn a machine learning model without exchanging their local data. A popular federated learning algorithm is FedAvg~\\cite{mcmahan2016communication}. In each round of FedAvg, the updated local models of the parties are transferred to the server, which further aggregates the local models to update the global model. The raw data is not exchanged during the learning process. Federated learning has emerged as an important machine learning area and attracted many research interests \\cite{mcmahan2016communication,lifedprox,karimireddy2019scaffold,li2020practical,Wang2020Federated,dai2020federated,hu2020oarf,caldas2018leaf,chaoyanghe2020fedml}. Moreover, it has been applied in many applications such as medical imaging \\cite{kaissis2020secure,kumar2020blockchain}, object detection \\cite{liu2020fedvision}, and landmark classification \\cite{hsu2020federated}.\n\nA key challenge in federated learning is the heterogeneity of data distribution on different parties \\cite{kairouz2019advances}. The data can be non-identically distributed among the parties in many real-world applications, which can degrade the performance of federated learning~\\cite{karimireddy2019scaffold,Li2020On,niidbench}. When each party updates its local model, its local objective may be far from the global objective. Thus, the averaged global model is away from the global optima. There have been some studies trying to address the non-IID issue in the local training phase~\\cite{lifedprox,karimireddy2019scaffold}. FedProx \\cite{lifedprox} directly limits the local updates by $\\ell_2$-norm distance, while SCAFFOLD \\cite{karimireddy2019scaffold} corrects the local updates via variance reduction \\cite{johnson2013accelerating}. However, as we show in the experiments (see Section \\ref{sec:exp}), these approaches fail to achieve good performance on image datasets with deep learning models, which can be as bad as FedAvg.\n\nIn this work, we address the non-IID issue from a novel perspective based on an intuitive observation: \\emph{the global model trained on a whole dataset is able to learn a better representation than the local model trained on a skewed subset}. Specifically, we propose \\tb{mo}del-c\\tb{on}trastive learning (MOON), which corrects the local updates by maximizing the agreement of representation learned by the current local model and the representation learned by the global model. Unlike the traditional contrastive learning approaches \\cite{simclr,simclr2,he2020momentum,misra2020self}, which achieve state-of-the-art results on learning visual representations by comparing the representations of different images, MOON conducts contrastive learning in model-level by comparing the representations learned by different models. Overall, MOON is a simple and effective federated learning framework, and addresses the non-IID data issue with the novel design of model-based contrastive learning.\n\n\n\nWe conduct extensive experiments to evaluate the effectiveness of MOON. MOON significantly outperforms the other state-of-the-art federated learning algorithms~\\cite{mcmahan2016communication,lifedprox,karimireddy2019scaffold} on various image classification datasets including CIFAR-10, CIFAR-100, and Tiny-Imagenet. With only lightweight modifications to FedAvg, MOON outperforms existing approaches by at least 2\\% accuracy in most cases. Moreover, the improvement of MOON is very significant on some settings. For example, on CIFAR-100 dataset with 100 parties, MOON achieves 61.8\\% top-1 accuracy, while the best top-1 accuracy of existing studies is 55\\%. \n\n\\section{Model-Contrastive Federated Learning}\n\n\n\\subsection{Problem Statement}\nSuppose there are $N$ parties, denoted $P_1, ..., P_N$. Party $P_i$ has a local dataset $\\mathcal{D}^i$. Our goal is to learn a machine learning model $w$ over the dataset $\\mathcal{D}\\triangleq \\bigcup_{i\\in[N]}\\mathcal{D}^i$ with the help of a central server, while the raw data are not exchanged. The objective is to solve \n\\begin{equation}\n    \\argmin_{w} \\mathcal{L}(w) = \\sum_{i=1}^N \\frac{|\\mathcal{D}^i|}{|\\mathcal{D}|}L_i(w),\n\\end{equation}\nwhere $L_i(w) = \\mathbb{E}_{(x,y)\\sim \\mathcal{D}^i} [\\ell_i(w; (x, y))]$ is the empirical loss of $P_i$.\n\n\n\\subsection{Motivation}\n\nMOON is based on an intuitive idea: the model trained on the whole dataset is able to extract a better feature representation than the model trained on a skewed subset. For example, given a model trained on dog and cat images, we cannot expect the features learned by the model to distinguish birds and frogs which never exist during training. \n\nTo further verify this intuition, we conduct a simple experiment on CIFAR-10. Specifically, we first train a CNN model (see Section \\ref{sec:exp_setup} for the detailed structure) on CIFAR-10. We use t-SNE \\cite{maaten2008visualizing} to visualize the hidden vectors of images from the test dataset as shown in Figure \\ref{fig:global}. Then, we partition the dataset into 10 subsets in an unbalanced way (see Section \\ref{sec:exp_setup} for the partition strategy) and train a CNN model on each subset. Figure \\ref{fig:local} shows the t-SNE visualization of a randomly selected model. Apparently, the model trained on the subset learns poor features. The feature representations of most classes are even mixed and cannot be distinguished. Then, we run FedAvg algorithm on 10 subsets and show the representation learned by the global model in Figure \\ref{fig:fedavg_global} and the representation learned by a selected local model (trained based on the global model) in Figure \\ref{fig:fedavg_local}. We can observe that the points with the same class are more divergent in Figure \\ref{fig:fedavg_local} compared with Figure \\ref{fig:fedavg_global} (\\eg, see class 9). The local training phase even leads the model to learn a worse representation due to the skewed local data distribution. This further verifies that the global model should be able to learn a better feature representation than the local model, and there is a \\emph{drift} in the local updates. Therefore, under non-IID data scenarios, we should control the drift and bridge the gap between the representations learned by the local model and the global model.\n\n\n\\begin{figure}\n\\centering\n\\subfloat[global model\\label{fig:global}]{\\includegraphics[width=0.5\\linewidth]{figures\/cifar_allin_print.pdf}%\n}\n\\subfloat[local model\\label{fig:local}]{\\includegraphics[width=0.5\\linewidth]{figures\/cifar_solo_print.pdf}%\n}\n\\hfil\n\\subfloat[FedAvg global model\\label{fig:fedavg_global}]{\\includegraphics[width=0.5\\linewidth]{figures\/cifar_global_35_print.pdf}%\n}\n\\subfloat[FedAvg local model\\label{fig:fedavg_local}]{\\includegraphics[width=0.5\\linewidth]{figures\/cifar_local_35_print.pdf}%\n}\n\\caption{T-SNE visualizations of hidden vectors on CIFAR-10.}\n\\label{fig:nparty}\n\\vspace{-5pt}\n\\end{figure}\n\n\\subsection{Method}\nBased on the above intuition, we propose MOON. MOON is designed as a simple and effective approach based on FedAvg, only introducing lightweight but novel modifications in the local training phase. Since there is always drift in local training and the global model learns a better representation than the local model, MOON aims to decrease the distance between the representation learned by the local model and the representation learned by the global model, and increase the distance between the representation learned by the local model and the representation learned by the previous local model. We achieve this from the inspiration of contrastive learning, which is now mainly used to learn visual representations. In the following, we present the network architecture, the local learning objective and the learning procedure. At last, we discuss the relation to contrastive learning.\n\n\n\\subsubsection{Network Architecture}\nThe network has three components: a base encoder, a projection head, and an output layer. The base encoder is used to extract representation vectors from inputs. Like \\cite{simclr}, an additional projection head is introduced to map the representation to a space with a fixed dimension. Last, as we study on the supervised setting, the output layer is used to produce predicted values for each class. For ease of presentation, with model weight $w$, we use $F_w(\\cdot)$ to denote the whole network and $R_w(\\cdot)$ to denote the network before the output layer (i.e., $R_w(x)$ is the mapped representation vector of input $x$).\n\n\\subsubsection{Local Objective}\nAs shown in Figure \\ref{fig:local_fram}, our local loss consists two parts. The first part is a typical loss term (\\eg, cross-entropy loss) in supervised learning denoted as $\\ell_{sup}$. The second part is our proposed model-contrastive loss term denoted as $\\ell_{con}$.\n\nSuppose party $P_i$ is conducting the local training. It receives the global model $w^t$ from the server and updates the model to $w_i^t$ in the local training phase. For every input $x$, we extract the representation of $x$ from the global model $w^t$ (i.e., $z_{glob} = R_{w^t}(x)$), the representation of $x$ from the local model of last round $w_i^{t-1}$ (i.e., $z_{prev} = R_{w_i^{t-1}}(x)$), and the representation of $x$ from the local model being updated $w_i^t$ (i.e., $z = R_{w_i^t}(x)$). Since the global model should be able to extract better representations, our goal is to decrease the distance between $z$ and $z_{glob}$, and increase the distance between $z$ and $z_{prev}$. Similar to NT-Xent loss \\cite{sohn2016improved}, we define model-contrastive loss as\n\n\\begin{equation}\n\\begin{aligned}\n\\small\n    &\\ell_{con}  = -\\log{\\frac{\\exp(\\textrm{sim}(z, z_{glob})\/\\tau)}{\\exp(\\textrm{sim}(z, z_{glob})\/\\tau) + \\exp(\\textrm{sim}(z, z_{prev})\/\\tau)}} \n\\end{aligned}\n\\end{equation}\n\nwhere $\\tau$ denotes a temperature parameter. The loss of an input $(x,y)$ is computed by \n\n\\begin{equation}\\label{eq:L}\n    \\ell = \\ell_{sup}(w_i^t; (x,y)) + \\mu\\ell_{con}(w_i^t; w_i^{t-1}; w^t; x),\n\\end{equation}\n\nwhere $\\mu$ is a hyper-parameter to control the weight of model-contrastive loss. The local objective is to minimize \n\\begin{equation}\\label{eq:obj}\n\\min_{w_i^t} \\mathbb{E}_{(x,y)\\sim D^i} [\\ell_{sup}(w_i^t; (x,y)) + \\mu \\ell_{con}(w_i^t; w_i^{t-1}; w^t; x)].\n\\end{equation}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/local_fram_print.pdf}\n    \\caption{The local loss in MOON.}\n    \\label{fig:local_fram}\n   \n\\end{figure}\n\nThe overall federated learning algorithm is shown in Algorithm \\ref{alg:moon}. In each round, the server sends the global model to the parties, receives the local model from the parties, and updates the global model using weighted averaging. In local training, each party uses stochastic gradient descent to update the global model with its local data, while the objective is defined in Eq.~\\eqref{eq:obj}.\n\n\\begin{algorithm}[t]\n\\SetNoFillComment\n\\LinesNumbered\n\\SetArgSty{textnormal}\n\\KwIn{number of communication rounds $T$, number of parties $N$, number of local epochs $E$, temperature $\\tau$, learning rate $\\eta$, hyper-parameter $\\mu$}\n\\KwOut{The final model $w^T$}\n\n\\BlankLine\n\\tb{Server executes}:\n\ninitialize $w^0$\\\\\n\\For {$t=0, 1, ..., T-1$}{\n\n    \\For {$i=1, 2, ..., N$ \\tb{in parallel}}{\n   \n        send the global model $w^t$ to $P_i$\n        \n        $w_i^t \\leftarrow$ \\tb{PartyLocalTraining}($i$, $w^t$)\n    }\n    $w^{t+1} \\leftarrow \\sum_{k=1}^N \\frac{|\\mathcal{D}^i|}{|\\mathcal{D}|} w_k^{t}$\n}\n\nreturn $w^T$\n\n\\BlankLine\n\\tb{PartyLocalTraining}($i$, $w^t$):\n\n$w_i^t \\leftarrow w^t$ \n\n\\For{epoch $i = 1, 2, ..., E$}{\n    \\For{each batch $\\tb{b} = \\{x, y\\}$ of $\\mathcal{D}^i$}{\n        $\\ell_{sup} \\leftarrow CrossEntropyLoss(F_{w_i^t}(x), y)$\n        \n        $z \\leftarrow R_{w_i^t}(x)$\n        \n        $z_{glob} \\leftarrow R_{w^t}(x)$\n        \n        $z_{prev} \\leftarrow R_{w_i^{t-1}}(x)$\n        \n        $\\ell_{con} \\leftarrow -\\log{\\frac{\\exp(\\textrm{sim}(z, z_{glob})\/\\tau)}{\\exp(\\textrm{sim}(z, z_{glob})\/\\tau) + \\exp(\\textrm{sim}(z, z_{prev})\/\\tau)}}$\n        \n        $\\ell \\leftarrow \\ell_{sup} + \\mu\\ell_{con}$\n        \n        $w_i^t \\leftarrow w_i^t - \\eta \\nabla \\ell$\n    }\n}\n\nreturn $w_i^t$ to server\n\\caption{The MOON framework}\n\\label{alg:moon}\n\\end{algorithm}\n\nFor simplicity, we describe MOON without applying sampling technique in Algorithm \\ref{alg:moon}. MOON is still applicable when only a sample set of parties participate in federated learning each round. Like FedAvg, each party maintains its local model, which will be replaced\nby the global model and updated only if the party is selected\nto participate in a round. MOON only needs the latest local\nmodel that the party has, even though it may not be updated\nin round $(t-1)$ (e.g., $w_i^{t-1} = w_i^{t-2}$).\n\nAn notable thing is that considering an ideal case where the local model is good enough and learns (almost) the same representation as the global model (i.e., $z_{glob} = z_{prev}$), the model-contrastive loss will be a constant (i.e., $-\\log\\frac{1}{2}$). Thus, MOON will produce the same result as FedAvg, since there is no heterogeneity issue. In this sense, our approach is robust regardless of different amount of drifts.\n\n\n\\subsection{Comparisons with Contrastive Learning}\nA comparison between MOON and SimCLR is shown in Figure \\ref{fig:comp}. The model-contrastive loss compares representations learned by different models, while the contrastive loss compares representations of different images. We also highlight the key difference between MOON and traditional contrastive learning: MOON is currently for supervised learning in a federated setting while contrastive learning is for unsupervised learning in a centralized setting. Drawing the inspirations from contrastive learning, MOON is a new learning methodology in handling non-IID data distributions among parties in federated learning. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/compare_loss_print.pdf}\n\\caption{The comparison between SimCLR and MOON. Here $x$ denotes an image, $w$ denotes a model, and $R$ denotes the function to compute representation. SimCLR maximizes the agreement between representations of different views of the same image, while MOON maximizes the agreement between representations of the local model and the global model {on the mini-batches.}}\n\\label{fig:comp}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of fault-tolerant quantum error correction (FT-QEC) is one of the\npillars that the field of quantum information rests on. Starting with the\ndiscovery of quantum error correcting codes \\cite{Shor:95,Steane:96a}, and\nthe subsequent introduction of fault tolerance \\cite{Shor:96}, this theory\nhas been the subject of many improvements and important progress \\cite%\n{Knill:96,Knill:98,Aharonov:96b,Aharonov:99,Zalka:96,Gottesman:97b,Gottesman:99,Gottesman:99a,Preskill:97a,Steane:99a,Steane:03,Knill:04,Reichardt:04,Knill:05}%\n, leading to the well-known error correction threshold condition. Most\nrecently, work by Steane \\cite{Steane:03} and Knill \\cite{Knill:05} (see\nalso Reichardt \\cite{Reichardt:04}) has pushed the threshold down to values\nthat are claimed to be very close to being within experimental reach. A\nnotable feature of much of the work on FT-QEC is that the error models are \n\\emph{phenomenological}. By this we mean that the underlying models often do\nnot start from a Hamiltonian, microscopic description of the system-bath\ninteraction, but rather from a higher level, effective description, most\nnotably that of Markovian dynamics. E.g., Knill writes: \\textquotedblleft We\nassume that a gate's error consists of \\emph{random, independent}\napplications of products of Pauli operators with probabilities determined by\nthe gate\\textquotedblright\\ (our emphasis) \\cite{Knill:05}. This approach is\nnatural given the considerable difficulty of obtaining error thresholds\nstarting from a purely Hamiltonian description. Nevertheless, Hamiltonian\napproaches to decoherence management in a fault-tolerant setting have been\npursued, e.g., a mixed phenomenological-Hamiltonian treatment of FT-QEC \\cite%\n{MohseniLidar:05,Terhal:04,Aliferis:05,Aharonov:05}, and a fully Hamiltonian study of\nfault tolerance in dynamical decoupling \\cite{KhodjastehLidar:04}. Also\nnoteworthy are recent mixed phenomenological-Hamiltonian \\emph{continuous\ntime} treatments of QEC \\cite{Ahn:01,Sarovar:04,Sarovar:05}.\n\nHere we are concerned with a critical re-evaluation of the physical\nassumptions entering the theory of FT-QEC. We scrutinize, in particular, the\nconsistency of the assumption of Markovian dynamics within the larger\nframework of FT-QEC. We point out that there may be an inherent\ninconsistency in the theory of Markovian FT-QEC, when viewed from the\nperspective of the validity of the Markovian approximation. We begin by\nbriefly reviewing, in Section~\\ref{FTQEC-review}, a set of minimal and\nstandard, universally agreed upon assumptions made in Markovian FT-QEC theory. We then\nreview, in Section~\\ref{MME-review}, the derivation of Markovian master\nequations, emphasizing the physical assumptions entering the Markovian\napproximation, in particular the requirement of consistency with\nthermodynamics. Having delineated the set of assumptions entering FT-QEC and\nthe quantum Markov approximation, we discuss in Section~\\ref{consistency}\nthe internal consistency of Markovian FT-QEC theory. We point out where\naccording to our analysis there is an inconsistency, and discuss possible\nobjections. In Section~\\ref{alternatives} we then discuss how one may\novercome the inconsistency using a variety of alternative approaches,\nincluding adiabatic quantum computing (QC), holonomic QC, topological QC,\nand recent work on FT-QEC in a non-Markovian setting \\cite%\n{Terhal:04,Aliferis:05,Aharonov:05}. We conclude in Section~\\ref{conc}.\n\n\\section{Review of Standard Assumptions of FT-QEC}\n\n\\label{FTQEC-review}\n\nThe following are a set of minimal assumptions made in the theory of FT-QEC \n\\cite%\n{Shor:96,Knill:96,Knill:98,Aharonov:96b,Aharonov:99,Zalka:96,Gottesman:97b,Gottesman:99,Gottesman:99a,Preskill:97a,Steane:99a,Steane:03,Knill:04,Reichardt:04,Knill:05}%\n:\n\n\\begin{enumerate}\n\\item \\textbf{A1} --- \\emph{Gates can be executed in a time} $\\tau _{g}$ \n\\emph{such that} $\\tau _{g}\\omega =O(\\pi )$, \\emph{where} $\\omega $ \\emph{is\na typical Bohr or Rabi frequency}.\\footnote{%\nOne might object that slower (even adiabatic) gates could be used instead.\nWe analyze this possibility in detail in Section~\\ref{adiabatic-gates}, and\nshow that it does not lead to an improvement.}\n\n\\item \\textbf{A2} --- \\emph{A constant supply of fresh and nearly pure\nancillas}: at every time step we are given a supply of many qubits in the\nstate $|0\\rangle $, each of which can be faulty with some error parameter $%\n\\eta \\ll 1$.\n\n\\item \\textbf{A3} --- \\emph{Error correlations decay exponentially in time\nand space}.\n\\end{enumerate}\n\nSome remarks:\n\n\\noindent (i) \\textbf{A1} is not typically stated explicitly in the\nFT-QEC\\ literature, but can be understood as resulting from the definition\nof a quantum gate, which is a unitary transformation $U=\\exp (iA)$; when $%\nA=\\tau _{g}H$, where $H$ is a Hamiltonian generating the gate, \\textbf{A1}\nfollows from the absence of a free parameter:\\ when $\\tau _{g}$ is scaled up \n$H$ (and hence its eigenvalues) must be scaled down, and \\textit{vice versa}.\n\n\\noindent (ii) The distinction between Bohr and Rabi frequencies in \\textbf{%\nA1} is related to the application of constant \\textit{vs} periodic control\nfields, respectively. In the case of a constant control field \\textbf{A1}\ncan be understood as the condition that saturates the \\textquotedblleft\nMargolus-Levitin theorem\\textquotedblright\\ \\cite{Margolus:98}, which states\nthat the time required to transform an initial state $|\\psi \\rangle $ to an\northogonal state $|\\psi ^{\\bot }\\rangle $ using a constant Hamiltonian $H$\nis lower-bounded by $\\tau _{\\min }=\\pi \\hbar \/(2E)$, where $E=\\langle \\psi\n|H|\\psi \\rangle $; when $|\\psi \\rangle $ is an eigenstate of $H$ we have $%\n\\tau _{g}\\geq \\pi \/(2\\omega )$, where $\\omega =E\/\\hbar $ is the Bohr\nfrequency. See also \\cite{Andrecut:04} for the adiabatic version of the\nMargolus-Levitin theorem, and \\cite{Gea:02} for a lower bound on the amount\nof energy needed to carry out an elementary logical operation on a quantum\ncomputer, with a given accuracy and in a given time. In the case of periodic\ncontrol fields one can understand \\textbf{A1} as the result of the standard\nsolution to the driven two-level atom problem, where the probability of a\ntransition between ground and excited state is given by $(\\Omega _{\\mathrm{R}%\n}\/\\Omega _{\\mathrm{R}}^{\\prime })\\sin ^{2}(\\Omega _{\\mathrm{R}}^{\\prime\n}t\/2) $, where $\\Omega _{\\mathrm{R}}$ is the Rabi frequency and $\\Omega _{%\n\\mathrm{R}}^{\\prime }=(\\Omega _{\\mathrm{R}}+\\delta ^{2})^{1\/2}$, where $%\n\\delta $ is the detuning. This expression for the transition probability\nyields $\\tau _{g}\\Omega _{\\mathrm{R}}^{\\prime }=O(\\pi )$.\n\n\\noindent (iii) \\textbf{A2} is shown to be necessary in \\cite%\n{Aharonov:96b}. \\textbf{A3} is stated clearly in \\cite{Aharonov:99} (see the\ndiscussion in Sections 2.10 and 10 there). These, and additional assumptions\n[such as constant fault rate (independent of number of qubits) and\nparallelism (to correct errors in all blocks simultaneously)] are explicitly\nlisted, e.g., also in \\cite{Dennis:02}, Section II.\n\n\\noindent (iv) \\textbf{A3} is usually related to the\nMarkovian assumption, however both notions, the space-time correlations of\nerrors and the Markovian property, need some comments and explanations.\nUsing the convolutionless formalism in the theory of open systems (see, e.g.,\n\\cite{Breuer:02}) it is always possible to resolve the total superoperator $%\n\\Lambda (t)$ as \n\\begin{equation}\n\\Lambda (t)=\\prod_{i=1}^{n}\\Lambda _{i}U_{i}  \\label{eq:Lambda}\n\\end{equation}%\nwhere $U_{i}$ are ideal unitary superoperators (corresponding to quantum\nlogic gates), and $\\Lambda _{i}$ are linear maps, not necessarily completely\npositive (CP) or even positive. If $\\Lambda _{i}$ are CP then we can always\nrealize them by coupling to an evironment which is \\textquotedblleft renewed\neach time step\\textquotedblright . This is the \\textquotedblleft Markovian\ncondition\\textquotedblright\\ as formulated in \\cite{Aharonov:99} (section\n2.10). However, complete positivity is not a necessary condition for QEC,\nwhich only requires a linear structure \\cite{Knill:97b,ShabaniLidar:06}. To obtain the\nThreshold Theorem one needs the following bound on the probability \\cite%\n{Aharonov:99}[Eq.~(2.6)]: \n\\begin{equation}\n\\text{Pr}(\\text{fault\\ path\\ with\\ }k\\text{\\ errors})\\leq c\\eta ^{k}(1-\\eta\n)^{v-k},  \\label{Pr}\n\\end{equation}%\nwhere $\\eta $ is the probability of a single error, $c$ is a certain\nconstant independent of $\\eta $, and $v$ is the number of error locations in\nthe circuit. This bound implies that the $k$-qubit errors should scale as $%\n\\sim \\eta ^{k}$, i.e., that in the decomposition of $\\Lambda _{j}$ into $k$%\n-qubit superoperators $L_{j}(k)$ \n\\begin{equation}\n\\Vert L_{j}(k)\\Vert \\leq c\\eta ^{k}\\ .  \\label{k-qub}\n\\end{equation}%\nAs discussed in \\cite{Alicki:02} (within the Born approximation), the\ncondition (\\ref{k-qub}) can strictly be satisfied only for temporally \\emph{%\nexponentially}{\\ decaying reservoir correlation functions,} while for\nrealistic reservoir models the temporal decay is generically powerlike. The\ndecay of reservoir correlation functions (i.e., localization in time)\ntranslates into localization of errors in space due to the finite speed of\nerror propagation. On the other hand it is widely believed that the\nMarkovian model can be understood as arising, to an excellent approximation,\nfrom coupling to a reservoir which is not only renewed at each time step,\nbut whose influence is independent of the actual Hamiltonian dynamics of the\nopen system, and is localized in space (independent errors model) \\cite%\n{Giulini:book}. A large part of the present paper is devoted to a critical\ndiscussion of this claim.\n\n\\noindent (v) We note that the recent papers on FT-QEC theory \\cite%\n{Terhal:04,Aliferis:05,Aharonov:05} relax the (Markovian) assumption \\textbf{A3}, but do\nmake \\textbf{A1} (implicitly) and \\textbf{A2}. We comment on these\npapers in Section~\\ref{nonM-FTQEC}.\n\n\\section{Review of Markovian Master Equations}\n\n\\label{MME-review}\n\nThe field of derivations of the quantum Markovian master equation (MME) is\nstrewn with pitfalls: it is in fact non-trivial to derive the MME in a fully\nconsistent manner. There are essentially two types of fully rigorous\napproaches, known as the \\emph{singular coupling limit} (SCL) and the \\emph{%\nweak coupling limit} (WCL), both of which we consider below. See, e.g., the books \n\\cite{Alicki:87,Breuer:book} for more details, as well as the derivation in \n\\cite{Alicki:89}.\n\nConsider a system and a reservoir (bath), with self Hamiltonians $H_{S}^{0}$\nand $H_{R}$, interacting via the Hamiltonian $H_{SR}=\\lambda S\\otimes R$,\nwhere $S$ ($R$) is a Hermitian system (reservoir) operator and $\\lambda $ is\nthe coupling strength. A more general model of the form $H_{SR}=\\sum_{\\alpha\n}\\lambda _{\\alpha }S_{\\alpha }\\otimes R_{\\alpha }$ can of course also be\nconsidered and results in the same qualitative conclusions. Thus the total\nHamiltonian is \n\\begin{equation}\nH=(H_{S}^{0}+H_{C}(t))\\otimes I_{R}+I_{S}\\otimes H_{R}+H_{SR},  \\label{ham}\n\\end{equation}%\nwhere $H_{C}(t)$ describes control over the quantum device (system), and $I$\nis the identity operator.\n\nThe SCL and WCL derivations start from the expansion of the propagator $%\n\\Lambda $ for the reduced, system-only dynamics, \n\\begin{equation}\n\\rho _{S}(t)=\\Lambda (t,0)\\rho _{S}(0),\n\\end{equation}%\ncomputed in the interaction picture with respect to the renormalized, \\emph{%\nphysical}, time-dependent Hamiltonian $H_{S}(t)=H_{S}+H_{C}(t)$, where \n\\begin{equation}\nH_{S}=H_{S}^{0}+\\lambda ^{2}H_{1}^{\\mathrm{corr}}(t)+\\cdots .  \\label{eq:H_S}\n\\end{equation}%\nThe renormalizing terms containing powers of $\\lambda $ are\n\\textquotedblleft Lamb-shift\\textquotedblright\\ corrections due to the\ninteraction with the bath (see, e.g., \\cite{Lidar:CP01}). The lowest order\n(Born) approximation with respect to the coupling constant $\\lambda $ yields \n$H_{1}^{\\mathrm{corr}}$, while the higher order terms ($\\cdots $) require\ngoing beyond the Born approximation. Introducing a cumulant expansion for\nthe propagator, \n\\begin{equation}\n\\Lambda (t,0)=\\exp \\sum_{n=1}^{\\infty }[\\lambda ^{n}K^{(n)}(t)],\n\\end{equation}%\none finds that $K^{(1)}=0$. The Born approximation consists of terminating\nthe cumulant expansion at $n=2$, whence we denote $K^{(2)}\\equiv K$: \n\\begin{equation}\n\\Lambda (t,0)=\\exp [\\lambda ^{2}K(t)+O(\\lambda ^{3})].\n\\end{equation}%\nOne obtains%\n\\begin{eqnarray}\nK(t)\\rho _{S}=\\int_{0}^{t}ds\\int_{0}^{t}duF(s-u)S(s)\\rho\n_{S}S(u)^{\\dag } \\nonumber \\\\\n+(\\mathrm{similar \\ terms})\n\\label{eq:K(t)}\n\\end{eqnarray}\nas the first term in a cumulant expansion \\cite{Alicki:89}. Here $F(s)=%\n\\mathrm{Tr}(\\rho _{R}R(s)R)$ is the autocorrelation function, where $\\rho\n_{R}$ is the reservoir state and $R(s)$ is $R$ in the $H_{R}$-interaction\npicture, and $S(u)$ is $S$ in the interaction picture with respect to the\nphysical Hamiltonian $H_{S}(t)$. The \\textquotedblleft similar\nterms\\textquotedblright\\ in Eq.~(\\ref{eq:K(t)}) are of the form $\\rho\n_{S}S(s)S(u)^{\\dag }$ and $S(s)S(u)^{\\dag }\\rho _{S}$.\n\nAt first sight $K(t)\\sim t^{2}$, and this is true for small times (Zeno\neffect \\cite{Facchi:PRL02}). The Markov approximation means that we can\nreplace $K(t)$ by an expression that is linear in $t$, i.e. \n\\begin{equation}\nK(t)\\simeq \\int_{0}^{t}\\mathcal{L}(s)ds  \\label{eq:L}\n\\end{equation}%\nwhere $\\mathcal{L}(t)$ is a time-dependent Lindblad generator. That the\nLindblad generator can be time-dependent even after transforming back to the\nSchr\\\"{o}dinger picture is important for our considerations below.\n\n\\subsection{Singular Coupling Limit}\n\n\\label{SCL}\n\nThe SCL approach we present in this subsection underlies the standard\nderivation of the MME that can be found in almost any text concerning the\nMarkov approximation, though not always under the heading \\textquotedblleft\nSCL\\textquotedblright\\ (e.g., \\cite{Carmichael:book}, p.8, Eq.~(1.36)). The\nrigorous derivation of the SCL is briefly discussed (with references) in \n\\cite{Alicki:87}, pp.36-38. It is based on a rescaling of the bath and\nsystem-bath Hamiltonians, which physically makes sense in the\nhigh-temperature limit only. We will shortly see the emergence of this limit.\n\nIn essence, the \\textquotedblleft naive SCL-Markov\napproximation\\textquotedblright\\ is obtained by the ansatz $F(s)=a\\delta (s)$\nfor the autocorrelation function, whence \n\\begin{equation}\nL(s)\\rho _{S}=aS(s)\\rho _{S}S(s)^{\\dag }+(\\mathrm{similar\\ terms}).\n\\end{equation}%\n\\textbf{\\ }As a consequence, return to the Schr\\\"{o}dinger picture gives a\nMME with the dissipative part independent of the Hamiltonian: \n\\begin{eqnarray}\n\\frac{d\\rho _{S}}{dt} &=&-i[H_{S}(t),\\rho _{S}]+\\mathcal{L}\\rho _{S},  \\notag\n\\\\\n\\mathcal{L}\\rho _{S} &\\equiv &-\\frac{1}{2}\\lambda ^{2}a[S,[S,\\rho _{S}]]\n\\label{eq:SCL}\n\\end{eqnarray}\n\nMore precisely, we must consider the multi-time bath correlation functions $%\nF(t_{1},...,t_{n}):=\\mathrm{Tr}[\\rho _{R}R(t_{1})...R(t_{n})]:=\\langle\nR(t_{1})...R(t_{n})\\rangle $. Here $R(t):=\\exp (iH_{R}t)R\\exp (-iH_{R}t)$\nare the bath operators in the interaction picture, $\\rho _{R}=\\exp (-\\beta\nH_{R})\/Z$ (where $\\beta =1\/kT$, $Z=\\mathrm{Tr}[\\exp (-H_{R}\/kT)]$) is\nthe bath thermal equilibrium state at temperature $T$, which is a stationary\nstate of the reservoir, i.e., $[H_{R},\\rho _{R}]=0$. The influence of the\nenvironment on the system is entirely encoded into the $\\{F(t_{1},...,t_{n})%\n\\}_{n=2}^{\\infty }$.\\footnote{$F(t_{1})$ is constant by stationarity. We\nreserve the notation $F(t)$ for $F(t_{1},t_{2})\\equiv F(t_{2}-t_{1})$ below.}\nHeuristically, the Markov approximation can be justified under the following\nconditions:\n\n\\begin{enumerate}\n\\item The lowest order correlation function, \n\\begin{equation}\nF(t)=\\langle R(s+t)R(s)\\rangle =\\int_{-\\infty }^{\\infty }G(\\omega\n)e^{-i\\omega t}d\\omega ,  \\label{2corr}\n\\end{equation}%\ncan be approximated by a Dirac delta function:\\footnote{%\nNote that stationarity implies that $F(t)$ does not depend on $s$.} \n\\begin{equation}\nF(t)\\simeq \\left( \\int_{-\\infty }^{\\infty }F(s)ds\\right) \\delta\n(t)=G(0)\\delta (t)  \\label{F-delta}\n\\end{equation}%\n(white-noise approximation). Eq.~(\\ref{2corr}) defines the \\emph{spectral\ndensity} $G(\\omega )$, which is a key object in the theory.\n\n\\item Higher order correlation functions exhibit a Gaussian-type behavior,\ni.e., can be estimated by sums of products of the lowest order ones, and\nthen, by condition (\\ref{F-delta}), decay sufficiently rapidly.\n\\end{enumerate}\n\nLet us now comment on the physical relevance of the white-noise\napproximation.\n\nFirst, the condition (\\ref{F-delta}) cannot be satisfied in general. For\nexample, in the important case of linear coupling to a bosonic field (e.g.,\nelectromagnetic field, phonons in solid state), we have $G(0)=0$, which\nmeans (by inverse Fourier transform) that $\\int_{-\\infty }^{+\\infty\n}F(t)dt=0 $, and therefore \\emph{$F(t)$ cannot be well approximated by} $%\n\\delta (t)$.\n\nSecond, even for models with $G(0)>0$ there exists a universal relation, the\nso-called Kubo-Martin-Schwinger (KMS) condition, $\\langle R(t)R(0)\\rangle\n=\\langle R(0)R(t+i\\beta )\\rangle $, which is valid for all quantum systems\nat thermal equilibrium. This implies: \n\\begin{equation}\nG(-\\omega )=e^{-\\beta \\omega }G(\\omega )\\ .  \\label{KMS}\n\\end{equation}%\n(See, e.g., \\cite{Alicki:87}[pp.90-91], \\cite{Thirring:book}[pp.176-177], or \n\\cite{Breuer:book}[p.137].) The fundamental importance of the KMS condition\nis captured by the fact that it is necessary in order for thermodynamics to\nhold. The KMS condition implies a strong asymmetry of the spectral density $%\nG(\\omega )$ for low $T$, where $T$ is measured relative to the presence of $%\nkT$ energy scales in the bath, i.e., relative to the range where $G(\\omega )$\nis non-vanishing. The KMS condition is relevant to our discussion since we\nmake the reasonably minimalistic assumption that the reservoir (not the QC)\nis in thermal equilibrium.\\footnote{%\nOne may challenge the notion that the bath must always be in thermal\nequilibrium. E.g., consider an atom in a microwave cavity, with the cavity\nelectromagnetic field initially in thermal equilibrium. Now suppose the atom\nis driven and is coupled to the cavity electromagnetic field, which\ntherefore is no longer in equilibrium. However, is the internal\nelectromagnetic field the relevant environment, or is it the external one?\nClearly, the electromagnetic field inside the cavity is not a reservoir but\nitself a part of the system. This is because: a) its spectrum is discrete,\nb) its coupling to the atom (close to resonance) is enhanced. The reason\nthese considerations matter is because b) implies the strong coupling\nregime, hence failure of the initial state tensor product structure\nassumption, hence difficulties with the separation of the system from the\nreservoir (dressed atom picture); a) implies that $F(t)$ is (quasi)-periodic\nwith short Poincar\\'{e} recurrences, hence a strongly non-Markovian regime,\nand thus associated difficulties for Markovian FT-QEC. On the other hand the\nexternal electromagnetic field has a continuous spectrum and the state\nproduct structure is easily satisfied, hence qualifies as a reservoir. This\nexample merely serves to illustrate accepted notions regarding the division\ninto well defined system and bath; for most practical purposes a thermal\nequlibrium is the simplest and most relevant model of an environment, and\nFT-QEC\\ theory must be applicable to this setting.}\n\nThird, $G(\\omega )$ need not be flat even at high $T$ (indeed, the KMS\ncondition only implies that $G(\\omega )$ is symmetric at high $T$). For\nexample, this is the case for the electromagnetic field and for phonons, for\nwhich at $T>0$ one has $G(\\omega )\\sim \\omega ^{3}\/(1-e^{-\\hbar \\omega \/kT})$\nfor $|\\omega |\\leq \\omega _{\\mathrm{cut}}$, and $G(\\omega )=0$ for $|\\omega\n|>\\omega _{\\mathrm{cut}}$. One can see that for high $T$ ($kT\\gg \\hbar\n\\omega _{\\mathrm{cut}}$), $G(\\omega )\\sim kT\\omega ^{2}$ is symmetric. Here $%\n\\omega _{\\mathrm{cut}}$ is the Debye frequency in the case of phonons, while\nfor the electromagnetic field $\\omega _{\\mathrm{cut}}$ should tend to\ninfinity in the renormalization procedure. A flat $G(\\omega )$ means a\nstructureless bath, while physical systems always have a nontrivial\nstructure depending on relevant energy scales.\\footnote{%\nIt is interesting to note that even if we try to enforce a flat $G(\\omega )$\nby, e.g., choosing an appropriate form factor for the spin-boson system, the\nobtained model -- the so-called \\textquotedblleft Ohmic\ncase\\textquotedblright\\ -- is mathematically and physically ill defined (see \n\\cite{Alicki:02a}).}\n\nNow let us return to the implications of the SCL assumptions for the problem\nof FT-QEC. In order to derive the SCL from first principles, one rescales $%\nH_{R}\\rightarrow H_{R}\/\\epsilon ^{2}$, rescales $H_{SR}\\rightarrow\nH_{SR}\/\\epsilon $, but keeps $H_{S}$ and $\\rho _{R}$ fixed.\\footnote{%\nNote that because different Hamiltonians are rescaled differently, this\nrescaling procedure is \\emph{not} equivalent to a direct rescaling of the\ntime variable (which is what is done in the WCL, below).} The idea of this\nrescaling is that it accelerates the reservoir's evolution (via $%\nH_{R}\\rightarrow H_{R}\/\\epsilon ^{2}$) and hence produces faster decay of\nthe reservoir correlations, $F(t)$. To see this, note that the rescaling $%\nH_{SR}\\rightarrow H_{SR}\/\\epsilon $ increases the amplitude $F(0)$ to $%\nF(0)\/\\epsilon ^{2}$ (proportional to $H_{SR}^{2}$), while keeping the\nstrength of the noise $\\int_{-\\infty }^{+\\infty }F(t)dt=G(0)$ fixed (as can\nbe seen via a change of variables $t\\rightarrow t\/\\epsilon ^{2}$ in the\nintegral). This implies a faster decay of $F(t)$. The rescaling procedure is\nspecifically designed to yield the delta correlation [Eq.~(\\ref{F-delta})]\nin the limit as $\\epsilon \\rightarrow 0$. Note that if $\\rho _{R}$ is at\nthermal equilibrium at temperature $T$ with respect to $H_{R}$, then, since $%\n\\rho _{R}=\\exp (-\\beta H_{R})\/Z$ is fixed, it must be at thermal equilibrium\nwith respect to $H_{R}\/\\epsilon ^{2}$ at the temperature $T\/\\epsilon\n^{2}\\rightarrow \\infty $, whence our mention of the high temperature limit,\nabove. Further note that $H_{S}$ is not rescaled since the SCL is\n(artificially) designed to produce \\textquotedblleft white\nnoise\\textquotedblright\\ on the natural time scale of system's evolution,\nwhich is given by $H_{S}$.\n\nAnother, equivalent way to understand the emergence of the high-$T$ limit is\nthe following: For the Markovian condition $F(t)\\simeq a\\delta (t)$ to hold\nthe spectral density must be flat: $G(\\omega )=\\mathrm{const}$. However,\nthis is possible only in the limit $T\\rightarrow \\infty $ of the KMS\ncondition. More precisely, \\emph{the Markovian condition can hold only if }$%\nkT\\gg \\omega $ \\emph{over the entire spectrum of the system's Bohr\nfrequencies}. Strictly speaking, $G(\\omega )$ is never constant. The\nvariation of $G(\\omega )$ happens over the \\textquotedblleft thermal\nmemory\\textquotedblright\\ time $\\tau _{\\mathrm{th}}:=1\/kT$. In the infinite $%\nT$ limit we then recover the case of zero memory-time, i.e., Markovian\ndynamics. Physically, it is enough to assume that $G(\\omega )$ is\nessentially constant over the interval $[-\\omega _{0},\\omega _{0}]$ where $%\nkT>\\omega _{0}\\gg $ system's Bohr frequencies. I.e., system energy scales\nmust be compared to $1\/\\tau _{\\mathrm{th}}$ and this leads to the important\nrealization that \\emph{the Markovian approximation can be consistent with\nthe KMS condition only in the high temperature regime }$kT\\gg E$, \\emph{where%\n} $E$ \\emph{is the system energy scale}. As we argue below, this fact\npresents a serious difficulty in the context of FT-QEC, the issue being\nessentially that the requirement of a constant supply of nearly pure and\ncold ancillas contradicts the high-$T$ limit needed for the Markov\napproximation to hold.\n\n\\subsection{Weak Coupling Limit}\n\n\\label{WCL}\n\nIn the SCL approach above there was no restriction on the time-dependence of\nthe system Hamiltonian. However, the price paid is the high-$T$ limit.\nMoreover, while mathematically the SCL is rigorous in the scaling limit, it\nis inconsistent with thermodynamics except in the $T\\rightarrow \\infty $\nlimit. On the other hand, the derivation by Davies, in his seminal 1974\npaper \\cite{Davies:74}, is perhaps the only derivation of the MME that is\nentirely consistent from \\emph{both the mathematical and physical} points of\nview. The Davies approach is based not on the high-$T$ limit, but rather on\nthe physically plausible idea of weak coupling. This is natural and\nconsistent with thermodynamics at all temperatures.\n\nMore specifically, Davies' derivation does not invoke a flatness condition\non $G(\\omega )$ but is, of course, still subject to the KMS condition. In\nthe Davies approach the Markov approximation is a consequence of weak\ncoupling (and hence slow dynamics of the system in the interaction picture),\nand time coarse-graining, which leads to cancellation of the non-Markovian\noscillating terms. The price we pay is the invalidity of this approach for\ntime-dependent Hamiltonians, except in the adiabatic case. We explain this\nimportant comment below. Hence, while the Davies approach does not require\nthe high-$T$ limit, it imposes severe restrictions on the speed of quantum\ngates.\n\nIn his rigorous derivation Davies replaced the heuristic condition (\\ref%\n{F-delta}) by the weaker \n\\begin{equation}\n\\int |F(t)|dt<\\infty .  \\label{F-cond}\n\\end{equation}%\nThis condition avoids the difficulties originating from the singularity of\nthe SCL condition (\\ref{F-delta}), and preempts the corresponding problems\nwith the high-$T$ limit.\\footnote{%\nIn some sense the weak coupling limit is similar to the Central Limit\nTheorem (CLT) in probability, and condition~(\\ref{F-cond}) is analogous to a\nrough upper bound on the second moment in the CLT. If it is not satisfied\nthen the noise may be not Gaussian in the weak coupling limit. The value of $%\n\\int |F(t)|dt$ itself does not provide any meaningful physical parameter and\ncan depend on some regularization\/cut-off parameters.}$^{,}$\\footnote{%\nOne can go further and ask how generic the Markovian case is, in the sense\nthat Eq.~(\\ref{F-cond}) is satisfied. In fact, typically $F(t)$ decays as $%\n1\/t^{\\alpha }$ (e.g., for the vacuum bath $\\alpha =4$ \\cite{Alicki:02}%\n\\textbf{)}, which means that in some cases ($\\alpha \\leq 1$) Eq.~(\\ref%\n{F-cond}) can be violated. For a systematic treatment of these non-Markovian\neffects see, e.g., \\cite{Breuer:02}.} We now consider the cases of a\nconstant, periodic, and arbitrarily time-dependent control Hamiltonian. The\nconstant case is the one originally treated by Davies \\cite{Davies:74}, and\nextended in \\cite{Davies:78} to time-dependent Hamiltonians assuming a slow\n(\\textquotedblleft adiabatic\\textquotedblright ) change on the dissipation\ntime scale $\\lambda ^{2}t$. The non-constant cases we study here have, as\nfar as we know, not been published before in the general scientific\nliterature.\n\n\\subsubsection{WCL for Constant $H_{C}$: Summary of the Original Davies\nDerivation}\n\nWe present a simplified version of the discussion of the Markov\napproximation in \\cite{Alicki:89}. Denote by ${E_{k}}$ the Bohr energies\n(eigenvalues of $H_{S}$), let $\\omega \\in \\{\\omega _{kl}=E_{k}-E_{l}\\}_{k,l}$%\n, and let $S_{\\omega }$ be the discrete Fourier components of the\ninteraction picture $S$, i.e., \n\\begin{equation}\nS(t)=\\exp (iH_{S}t)S\\exp (-iH_{S}t)=\\sum_{\\omega }S_{\\omega }\\exp (i\\omega\nt),  \\label{eq:S}\n\\end{equation}%\nwhere $H_{S}$ is the renormalized (physical) system Hamiltonian: the sum of\nthe \\textquotedblleft bare\\textquotedblright\\ $H_{S}^{0}$ and a Lamb shift\nterm (bath induced), as in Eq.~(\\ref{eq:H_S}). Equivalently, \n\\begin{equation}\n\\lbrack H_{S},S_{\\omega }]=\\omega S_{\\omega }.  \\label{Dav1}\n\\end{equation}%\nWe remark that in the original Davies paper the Bohr energies and Eq.~(\\ref%\n{Dav1}) are computed with respect to the bare Hamiltonian $H_{S}^{0}$. Here\nwe use the physical Hamiltonian $H_{S}$ in order to take into account the\nfact that the Lamb shift term, although formally proportional to $\\lambda\n^{2}$, can be large or even infinite after cut-off removal.\n\nThen, it follows from Eq.~(\\ref{eq:K(t)}) that \n\\begin{eqnarray}\nK(t)\\rho _{S}&=&\\sum_{\\omega ,\\omega ^{\\prime }}S_{\\omega }\\rho _{S}S_{\\omega\n^{\\prime }}^{\\dag }\\int_{0}^{t}e^{i(\\omega -\\omega ^{\\prime\n  })u}du\\int_{-u}^{t-u}F(\\tau )e^{i\\omega \\tau }d\\tau \\notag \\\\\n&+&(\\mathrm{similar}\\text{ \n}\\mathrm{terms}).  \\label{eq:K2}\n\\end{eqnarray}\nThe weak coupling limit is next formally introduced by rescaling the time $t$\nto $t\/\\lambda ^{2}$ (van Hove limit). This enables two crucial\napproximations, which are valid in the resulting large-$t$ limit:\n\n\\begin{enumerate}\n\\item We replace\\footnote{%\nIn a more rigorous treatment the Cauchy principal value must be used, but\nthe result is essentially the same \\cite{Alicki:89}.}%\n\\begin{equation}\n\\int_{0}^{t}e^{i(\\omega -\\omega ^{\\prime })u}du\\approx t\\delta _{\\omega\n\\omega ^{\\prime }}.  \\label{eq:rep1}\n\\end{equation}%\nThis makes sense for \n\\begin{equation}\nt\\gg \\max \\{1\/(\\omega -\\omega ^{\\prime })\\}.  \\label{eq:tbig}\n\\end{equation}%\n\\emph{This violates \\textbf{A1}, expressed in terms of the Bohr frequencies}%\n. We see here already the emergence of an adiabatic criterion for the\nvalidity of the Markov approximation.\n\n\\item We replace $\\int_{-u}^{t-u}F(\\tau )e^{i\\omega \\tau }d\\tau $ by the\nFourier transform:%\n\\begin{equation}\n\\int_{-u}^{t-u}F(\\tau )e^{i\\omega \\tau }d\\tau \\approx {G}(\\omega\n)=\\int_{-\\infty }^{\\infty }F(\\tau )e^{i\\omega \\tau }d\\tau .  \\label{eq:rep2}\n\\end{equation}%\n\\end{enumerate}\n\nThe physical validity of the last approximation is usually ignored, though one\ncan make the following argument: On the LHS of Eq.~(\\ref{eq:rep2}), for a\ngiven Bohr frequency $\\omega $ the Fourier-like integral must sample the\nfunction $F(\\tau )$ with sufficiently high accuracy so that the Fourier\ntransform approximation will be valid. To this end one needs a time $t$ such\nthat: (i) $t\\gg 1\/\\omega $. This is a weaker condition than the previous one\n[$t\\gg \\max \\{1\/(\\omega -\\omega ^{\\prime })\\}$] which involves differences\nof Bohr frequencies. (ii) The time $t$ must be also much longer than the\ntime scale of the wildest variations of $F(\\tau )$, which is typically [as\nmay be checked for simple models of spectral densities $G(\\omega )$] given\nby $1\/\\omega _{\\mathrm{cut}}$, where $\\omega _{\\mathrm{cut}}$ is a\nhigh-frequency cutoff. When $\\omega <\\omega _{\\mathrm{cut}}$ (i) implies\n(ii). Therefore typically Eq.~(\\ref{eq:rep1}) is a stronger assumption than\nEq.~(\\ref{eq:rep2}).\n\n\nApplying the approximations (\\ref{eq:rep1}) and (\\ref{eq:rep2}), we obtain $K(t)\\rho _{S}=t\\sum_{\\omega\n}S_{\\omega }\\rho _{S}S_{\\omega }^{\\dag }{G}(\\omega )+(\\mathrm{similar}$ $%\n\\mathrm{terms})$, and hence it follows from Eq.~(\\ref{eq:L}) that $\\mathcal{L%\n}(s)=\\mathcal{L}$ is the Davies generator in the familiar Lindblad form: \n\\begin{eqnarray}\n\\frac{d\\rho _{S}}{dt} &=&-i[H_{S},\\rho _{S}]+\\mathcal{L}\\rho _{S},  \\notag \\\\\n\\mathcal{L}\\rho _{S} &\\equiv &\\frac{1}{2}\\lambda ^{2}\\sum_{\\omega }G(\\omega\n)([S_{\\omega },\\rho S_{\\omega }^{\\dagger }]+[S_{\\omega }\\rho ,S_{\\omega\n}^{\\dagger }])  \\label{Dav}\n\\end{eqnarray}%\nSeveral remarks are in order:\n\n\\noindent (i) The absence of off-diagonal terms in Eq.~(\\ref{Dav}), compared\nto Eq.~(\\ref{eq:K2}), is the hallmark of the Markovian limit. Namely, the\nDavies derivation relies on the cancellation of the non-Markovian\noff-diagonal terms $\\sum_{\\omega \\neq \\omega ^{\\prime }}S_{\\omega }\\rho\n_{S}S_{\\omega ^{\\prime }}^{\\dag }\\int_{0}^{t}e^{i(\\omega -\\omega ^{\\prime\n})u}du$. This time coarse-graining is possible due to integration over the\nfast-oscillating $\\int_{0}^{t}e^{i(\\omega -\\omega ^{\\prime })u}$ terms over\na long timescale, i.e., over $t\\gg \\max \\{1\/(\\omega -\\omega ^{\\prime })\\}$\n(see also \\cite{Lidar:CP01}). As remarked above, this violates \\textbf{A1},\nexpressed in terms of the Bohr frequencies.\n\n\\noindent (ii) It follows from Bochner's theorem applied to the Fourier\ntransform definition of ${G}(\\omega )$ that $G(\\omega )\\geq 0$ \\cite%\n{Alicki:87}[p.90], \\cite{Breuer:book}[p.136]; this result is essential for\nthe complete positivity of the Markovian master equation in the WCL.\n\n\\noindent (iii) Davies' derivation showed implicitly that the notion of\n\\textquotedblleft bath's correlation time\\textquotedblright\\ is not\nwell-defined -- Markovian behavior involves a rather complicated cooperation\nbetween system and bath dynamics. More specifically, the relations~(\\ref{Dav}%\n) and~(\\ref{Dav1}) together imply that the noise and $H_{S}$ are strongly\ncorrelated. In other words, contrary to what is often done in\nphenomenological treatments, \\emph{one cannot combine arbitrary }$H_{S}$%\n\\emph{'s with given }$S_{\\omega }$\\emph{'s.} This point is particularly\nrelevant in the context of FT-QEC, where it is common to assume Markovian\ndynamics and apply arbitrary control Hamiltonians.\n\nDavies did not consider time-dependent system Hamiltonians in \\cite%\n{Davies:74}, but it is possible to generalize his derivation to allow for\nslowly varying system Hamiltonians \\cite{Davies:78,Alicki:79,Alicki:89}.\nThat is, whenever the time scale of the variation of $H_{C}(t)$ is much\nlonger than the inverse of the typical Bohr frequency (of $H_{S}$), it is\npossible to add $H_{C}(t)$ to the system Hamiltonian in Eq.~(\\ref{Dav}),\nnecessitating at the same time this change also in Eq.~(\\ref{Dav1}). This is\na type of adiabatic limit (indeed, the $S_{\\omega }$ in Eq.~(\\ref{Dav1}) can\nbe interpreted, with $H_{S}$ replaced by $H_{S}+H_{C}(t)$, as being\nadiabatic eigenvectors of the superoperator $[H_{S}+H_{C}(t),\\cdot\n]$). We note that an alternative approach to adiabaticity in open\nquantum systems was recently developed in\nRef.~\\cite{SarandyLidar:04}. This approach, while being very general, is more\nphenomenological in that it postulates a\nconvolutionless master equation, and then derives corresponding\nadiabaticity conditions. Closer in spirit to the Davies derivation is\nanother recent approach to adiabaticity in open systems, which assumes\nslow system variation together with weak system-bath coupling \\cite{Thunstrom:05}.\n\n\\subsubsection{WCL for Periodic Driving:\\ Floquet Analysis}\n\n\\label{Floquet}\n\nBefore considering the case of periodic $H_{C}$ let us consider briefly once\nmore the case of a constant Hamiltonian in the so-called covariant\ndissipation setting. Covariance is the commutation condition $\\mathcal{H}%\n\\mathcal{L}=\\mathcal{L}\\mathcal{H}$ where $\\mathcal{H}=[H_{S},\\cdot ]$ is\nthe super-operator constant Hamiltonian, and $\\mathcal{L}$ is the Davies\ngenerator. Covariance is an abstract property which is automatically\nfulfilled for the Davies generator.\\footnote{%\nThis can be verified by directly computing $\\mathcal{HL}$ and making use of\nEq.~(\\ref{Dav1}) and the relation $[A,BC]=[A,B]C+B[A,C]$ (for operators $A,B$\nand $C$). A more elegant way to see this is to consider $\\mathcal{L}(t)=\\exp\n(-it\\mathcal{H})\\mathcal{L}\\exp (it\\mathcal{H})$ and note that Eq.~(\\ref%\n{Dav1}) implies that $S(t)$ and $S^{\\dag }(t)$ rotate in opposite\ndirections. Hence $\\mathcal{L}(t)=\\mathcal{L}$, whence $d\\mathcal{L}(t)\/dt=0$\ngives the result.} It is convenient since it implies factorization of the\nfull propagator into Hamiltonian and dissipative parts. Markovian dynamics\nobtained in the WCL as discussed above takes the form\n\n\\begin{equation}\n{\\frac{d\\rho }{dt}}=(-i\\mathcal{H}+\\mathcal{L})\\rho \\ ,\\ t\\geq 0,  \\label{1}\n\\end{equation}%\nwhere the most general form of the Lindblad (or Davies) $\\mathcal{L}$\nsatisfying Eq.~(\\ref{1}) is \n\\begin{equation}\n\\mathcal{L}\\rho ={\\frac{1}{2}}\\sum_{\\{\\omega \\},j}\\left( [V_{j}(\\omega\n),\\rho V_{j}(\\omega )^{\\dagger }]+[V_{j}(\\omega )\\rho ,V_{j}(\\omega\n)^{\\dagger }]\\right) .  \\label{2}\n\\end{equation}%\nHere $\\{\\omega \\}\\equiv \\mathrm{Spectrum}(\\mathcal{H})$, i.e., the Bohr\nfrequencies (differences of eigenvalues of $H$), and \n\\begin{equation}\n\\mathcal{H}V_{j}(\\omega )=\\omega V_{j}(\\omega )  \\label{3}\n\\end{equation}%\n[i.e., Eq.~(\\ref{Dav1})]. The solution, i.e., the dynamical semigroup is%\n\\begin{equation}\n\\rho (t)=\\Lambda (t)\\rho (0),\\ \\Lambda (t)=e^{-it\\mathcal{H}}e^{t\\mathcal{L}%\n}.  \\label{4}\n\\end{equation}%\nNow consider a periodic control Hamiltonian with period $\\Theta $ \n\\begin{equation}\nH_{C}(t)=H_{C}(t+\\Theta ),\\ \\Omega =2\\pi \/\\Theta .  \\label{5}\n\\end{equation}%\n(Note that $\\Omega $ is \\emph{not} the Rabi frequency, which throughout this\npaper we denote by $\\Omega _{\\mathrm{R}}$.) The situation is then very\nsimilar to the standard (time-independent $H_{C}$) WCL, but the set of\n\\textquotedblleft effective Bohr frequencies\\textquotedblright\\ (Floquet\nspectrum)\\ ${\\omega }$ is now larger and is of the form $\\{\\omega +q\\Omega\n\\} $, $q=0,\\pm 1,...$. Here $\\omega $ are Bohr frequencies for the Floquet\nunitary [defined in Eq.~(\\ref{eq:floquet}) below], i.e., differences of\neigenvalues $\\epsilon _{\\alpha }$ of the Floquet unitary, rather than $%\n\\{\\omega \\}=\\mathrm{Spectrum}(\\mathcal{H})$ as above. As this set of\n\\textquotedblleft effective Bohr frequencies\\textquotedblright\\ is discrete\nthe WCL still works, but the final Davies generator is more complicated, as\nwe now show.\n\nDefine the time-ordered unitary propagator \n\\begin{equation}\nU(t,s)\\equiv \\mathcal{T}\\exp \\left( -i\\int_{s}^{t}H_{S}(u)du\\right) ,\\ t\\geq\ns  \\label{6}\n\\end{equation}%\nwhich satisfies the properties $U(s,t)\\equiv U(t,s)^{-1}=U(t,s)^{\\dagger }$, \n$U(t,s)U(s,u)=U(t,u)$, $U(t+\\Theta ,s+\\Theta )=U(t,s)$, and ${\\frac{d}{dt}}%\nU(t,s)=-iH_{S}(t)U(t,s)$, ${\\frac{d}{dt}}U(t,s)^{\\dagger }=iU(t,s)^{\\dagger\n}H_{S}(t)$. The \\emph{Floquet unitary operator} is%\n\\begin{equation}\nF(s)\\equiv U(s+\\Theta ,s)\\bigskip ,  \\label{eq:floquet}\n\\end{equation}%\nwith corresponding super-operator action%\n\\begin{equation}\n\\mathcal{F}(s)\\rho \\equiv F(s)\\rho F(s)^{\\dagger },\n\\end{equation}%\nand Floquet eigenvectors $|\\phi _{\\alpha }\\rangle $ and eigenvalues\n(quasi-energies) $\\epsilon _{\\alpha }$ satisfying\\footnote{%\nNote that the Floquet Hamiltonian $H_{S}(t)-id\/dt$ operates on a different\nHilbert space than $F(0)$ (the space of periodic functions with values in\nthe system's Hilbert space). But its eigenvalues coincide with $\\epsilon\n_{\\alpha }$ from Eq.~(\\ref{19}).} \n\\begin{equation}\nF(0)|\\phi _{\\alpha }\\rangle =e^{-i\\epsilon _{\\alpha }\\Theta }|\\phi _{\\alpha\n}\\rangle .  \\label{19}\n\\end{equation}%\nIt follows from standard Floquet theory that%\n\\begin{equation}\nU(t,0)|\\phi _{\\alpha }\\rangle =e^{-it\\epsilon _{\\alpha }}\\sum_{q\\in \\mathbf{Z%\n}}e^{-itq\\Omega }|\\phi _{\\alpha }(q)\\rangle ,  \\label{20}\n\\end{equation}%\ni.e., the set $\\{|\\phi _{\\alpha }(q)\\rangle \\}$ is a complete basis.\nTherefore we have at most as many $q$'s as the dimension of the Hilbert\nspace. That the number of $q$'s is finite is important for our\nconsiderations below.\n\nWe call a Lindblad generator $\\mathcal{L}$ a \\textquotedblleft covariant\ndissipative perturbation of $H_{S}(t)$\\textquotedblright\\ if \n\\begin{equation}\n\\mathcal{F}(0)\\mathcal{L}=\\mathcal{L}\\mathcal{F}(0)  \\label{9}\n\\end{equation}\n\nWe will assume this property, similarly to the case of a constant\nHamiltonian described above. In fact, covariance holds for a periodic $%\nH_{S}(t)$ and also for the corresponding WCL Davies generator. One can then\nderive the \\emph{covariant master equation} (we sketch the\nderivation below):%\n\\begin{equation}\n{\\frac{d\\rho }{dt}}=\\left( -i\\mathcal{H}(t)+\\mathcal{L}(t)\\right) \\rho \\ ,\\\nt\\geq 0,  \\label{10}\n\\end{equation}%\n[compare to Eq.~(\\ref{1})] where \n\\begin{eqnarray}\n\\mathcal{L}(t) &=&\\mathcal{U}(t,0)\\mathcal{L}\\mathcal{U}(t,0)^{\\dagger },\n\\label{11} \\\\\n{\\frac{d}{dt}}\\mathcal{U}(t,s) &=&-i\\mathcal{H}(t)\\mathcal{U}(t,s),\n\\end{eqnarray}%\nand where the general form of $\\mathcal{L}$ appearing in Eq.~(\\ref{11}) is\ngiven by Eq.~(\\ref{2}), with $V_{j}(\\omega )$ now being eigenvectors of $%\n\\mathcal{F}(0)$, \n\\begin{equation}\n\\mathcal{F}(0)V_{j}(\\omega )=e^{-i\\omega \\Theta }V_{j}(\\omega ),  \\label{14}\n\\end{equation}%\nrather than of $\\mathcal{H}$, as in Eq.~(\\ref{3}). Moreover, here $\\{\\omega\n\\}\\equiv \\{\\epsilon _{\\alpha }-\\epsilon _{\\beta }\\}$, where $\\epsilon\n_{\\alpha }$ are quasi-energies (effective Bohr frequencies) of the Floquet\noperator, rather $\\{\\omega \\}\\equiv \\mathrm{Spectrum}(\\mathcal{H})$ as we\nsaw in the case of constant $H_{C}$.\n\nThe solution replacing Eq.~(\\ref{4}) is \n\\begin{eqnarray}\n\\rho (t) &=&\\Lambda (t,s)\\rho (s),\\text{ \\ }t\\geq s  \\notag \\\\\n\\Lambda (t,s) &=&\\mathcal{T}\\exp \\left\\{ \\int_{s}^{t}\\left( -i\\mathcal{H}(u)+%\n\\mathcal{L}(u)\\right) du\\right\\}  \\label{12}\n\\end{eqnarray}%\nBy direct computation one can prove the following properties: \n\\begin{eqnarray}\n\\mathcal{L}(t+\\Theta ) &=&\\mathcal{L}(t), \\\\\n\\mathcal{F}(s)\\mathcal{L}(s)\\mathcal{F}(s)^{\\dagger } &=&\\mathcal{L}(s), \\\\\n\\Lambda (t,s)\\Lambda (s,u) &=&\\Lambda (t,u)\\text{ for }t\\geq s\\geq u, \\\\\n\\Lambda (t+\\Theta ,s+\\Theta ) &=&\\Lambda (t,s), \\\\\n\\Lambda (t,s) &=&\\mathcal{U}(t,s)e^{-(t-s)\\mathcal{L}(s)}.  \\label{13}\n\\end{eqnarray}\n\nTo derive the covariant master equation (\\ref{10}) one considers the\nstandard picture of an open system $S+R$ with the total Hamiltonian \n\\begin{equation}\nH_{SR}(t)=H_{S}^{0}(t)+H_{R}+\\sum_{k}S_{k}\\otimes R_{k},  \\label{15}\n\\end{equation}%\n(we neglect the Lamb shift correction here; it can be included, changing $%\nH_{S}^{0}(t)$ into the physical Hamiltonian $H_{S}(t)$, by a suitable\nrenormalization procedure), stationary reservoir state $\\rho _{R}$, $%\n[H_{R},\\rho _{R}]=0$, $\\mathrm{Tr}(\\rho _{R}\\cdot )\\equiv \\langle \\cdot\n\\rangle _{R}$, $\\langle R_{k}\\rangle _{R}=0$. Then, exactly following a\nDavies-like calculation using a Fourier decomposition of $S(t)$, now\ngoverned by a periodic Hamiltonian, and making in particular again the\ncrucial assumption Eq.~(\\ref{eq:tbig}), which now reads \n\\begin{equation}\nt\\gg \\max \\{1\/(\\omega -\\omega ^{\\prime }+m\\Omega )\\},~m=0,\\pm 1,\\pm 2,...\n\\label{MA-per}\n\\end{equation}%\nwith $|m|$ upper-bounded by the dimension of the Hilbert space [see remark\nafter Eq.~(\\ref{20})], one obtains Eq.~(\\ref{10}) in the Davies WCL. The\nexplicit form of the generator is: \n\\begin{eqnarray}\n\\mathcal{L}\\rho &=& {\\frac{1}{2}}\\sum_{k,l}\\sum_{q\\in \\mathbf{Z}}\\sum_{\\{\\omega\n\\}}{\\hat{R}}_{kl}(\\omega +q\\Omega )\\{ [S_{l}(q,\\omega )\\rho\n  ,S_{k}(q,\\omega )^{\\dagger }] \\nonumber \\\\\n&+& [S_{l}(q,\\omega ),\\rho S_{k}(q,\\omega\n)^{\\dagger }] \\} .  \\label{16}\n\\end{eqnarray}\nHere $\\{\\omega \\}\\equiv \\{\\epsilon _{\\alpha }-\\epsilon _{\\beta }\\}$, the\nFloquet spectrum, and \n\\begin{equation}\n{\\hat{R}}_{kl}(x)=\\int_{-\\infty }^{\\infty }e^{-itx}\\langle\nR_{k}(t)R_{l}\\rangle _{R}dt  \\label{17}\n\\end{equation}%\n\\begin{equation}\nS_{k}(q,\\omega )=\\sum_{p\\in \\mathbf{Z}}\\sum_{\\{\\epsilon _{\\alpha }-\\epsilon\n_{\\alpha ^{\\prime }}=\\omega \\}}\\langle \\phi _{\\alpha }(p+q)|S_{k}|\\phi\n_{\\alpha ^{\\prime }}(p)\\rangle |\\phi _{\\alpha }\\rangle \\langle \\phi _{\\alpha\n^{\\prime }}|.  \\label{18}\n\\end{equation}%\n$S_{k}(q,\\omega )$ is the part of $S(t)$ which rotates with frequency $%\n\\omega +q\\Omega $ and can be computed using Eq.~(\\ref{20}). Note that by\ndiagonalizing the matrices ${\\hat{R}}_{kl}$ one can transform the generator $%\n\\mathcal{L}$ of Eq.~(\\ref{16}) into the form of Eq.~(\\ref{2}), which allows\none to read off the operators $V_{j}(\\omega )\\ $appearing there.\n\nNow to some important comments:\n\n\\noindent -- \\emph{Timescale analysis}: Note that for the periodic case the\ndifferences of \\textquotedblleft Bohr frequencies\\textquotedblright\\ may be\nof the order of $1\/\\Theta $. Hence we conclude from Eq.~(\\ref{MA-per})\\ that\none must average over many periods $\\Theta $, i.e., require $t\\gg \\Theta $.\nThis can be interpreted as a condition that \\textquotedblleft the\nenvironment must learn that the Hamiltonian is periodic\\textquotedblright .\nThis is exactly analogous to the adiabaticity condition in the adiabatic\ncase: $H(t)$ must be constant over many inverse Bohr frequencies to\n\\textquotedblleft be recognised\\textquotedblright\\ by the environment. The\nperiodic WCL is also a coarse-grained time description with the additional\ntime scale $\\Theta $.\n\\emph{Note that arbitrarily fast\nperiodic driving (small $\\Theta $) is incompatible even with the kind\nof generalized, finitely localized MME derived here, since then differences\nof Bohr frequencies matter in Eq.~(\\ref{MA-per})} (recall that $\\max\n|m|$ is bounded by the -- typically small -- dimension of the system\nHilbert space).\\\\\n\n\\noindent -- \\emph{Where is the Rabi frequency?} Note the dependence of the\noperators $S_{k}(q,\\omega )$ on the Floquet eigenvalue differences $\\epsilon\n_{\\alpha }-\\epsilon _{\\alpha ^{\\prime }}$. The usual Rabi frequency, $\\Omega\n_{\\mathrm{R}}=2dE\/\\hbar $ ($d$ is the dipole moment, $E$ is the electric\nfield amplitude) arises in the dipole approximation, which we have not made\nhere. The usual Rabi frequency is replaced in our non-perturbative treatment\n(in the sense of no multipole expansion) by the difference of Floquet\neigenvalues $\\epsilon _{\\alpha }-\\epsilon _{\\alpha ^{\\prime }}$ in Eq.~(\\ref%\n{18}).\\footnote{%\nOne can see that such a term also arises in the usual dipole approximation\nby considering, e.g., Eq. (2.94) in \\cite{Carmichael:book}. The interaction\npicture raising and lowering operators $\\sigma _{\\pm }(t)$ (for a two-level\natom driven by a classical field) there oscillate with three\n\\textquotedblleft Bohr frequencies\\textquotedblright\\ $\\omega _{A},\\omega\n_{A}\\pm \\Omega $\\thinspace , where $\\Omega =2dE\/\\hbar $ denotes the usual\nRabi frequency. Hence the Rabi frequency is a difference of two Bohr\nfrequencies.}\n\n\\noindent -- \\emph{More on the Rabi frequency}:\\ As we saw, the non-Markovian terms\nvanish because of the time coarse-grained description. To attain this, we\nmust average over times $t$ $\\gg \\max_{\\omega ,\\omega ^{\\prime }}\\{1\/(\\omega\n-\\omega ^{\\prime })\\}$, but must also keep in mind that the longest relevant\nscale for coarse-graining is given by the exponential decay time $\\tau $ (a \n\\emph{derived} quantity), i.e., we must have $t<\\tau $. The Rabi frequency $%\n\\Omega _{\\mathrm{R}}$ is a difference of two Bohr frequencies $\\omega\n,\\omega ^{\\prime }$:\\ $\\Omega _{\\mathrm{R}}=\\omega -\\omega ^{\\prime }$. This\nimplies that coarse-graining does not makes sense if $\\Omega _{\\mathrm{R}%\n}\\tau \\ll 1$ [since then $t<\\tau \\ll 1\/\\Omega _{\\mathrm{R}}=1\/(\\omega\n-\\omega ^{\\prime })$, in contradiction to the fundamental requirement on $t$%\n]. In physical terms this means that the width of the spectral line ($\\gamma\n=1\/\\tau $) is larger than the level splitting $\\Omega _{\\mathrm{R}}$ (see,\ne.g., Fig. 2.5 (i),(ii) in \\cite{Carmichael:book} for an illustration in the\ncase of the incoherent fluorescence spectrum) and therefore\n\\textquotedblleft the environment has no time to recognize the details of\nthe spectrum\\textquotedblright . On the other hand, when $\\Omega _{\\mathrm{R}%\n}\\tau \\gg 1$ (not inconsistent with the WCL), $\\Omega _{\\mathrm{R}}$ must\nappear in the generator, as appears from our treatment of the case of\nperiodic driving in the WCL, above. Unfortunately there are examples in the\nliterature where an MME\\ is written down subject to $\\Omega _{\\mathrm{R}%\n}\\tau \\gg 1$ but $\\Omega _{\\mathrm{R}}$ does not appear in the generator\n[e.g., Eq.~(2.96) in \\cite{Carmichael:book}, where $\\Omega _{\\mathrm{R}}\\sim\n10^{10}\\mathrm{Hz}$ and $\\tau \\sim 10^{-8}\\mathrm{s}$].\n\n\\noindent -- \\emph{Quantum optics considerations}:\\ The Markov approximation is\ncommonly accepted as an excellent approximation in quantum optics; see,\ne.g., the discussion of resonance fluorescence in \\cite{Carmichael:book}%\n[Ch.2]. This is also the basis for substantial confidence in the possibility\nof FT-QEC in quantum optical systems, such as trapped ions \\cite{Cirac:95}\nand atoms trapped in microwave cavities \\cite{Turchette:95}. Such arguments\nare based on the relative flatness of the damping constants $\\gamma (\\omega\n) $ as a function of frequency. This argument is closely related to the\nnotion of the flatness of the spectral density $G(\\omega )$ in the SCL,\nsince the damping constants are proportional to $G(\\omega )$ [see Eq.~(\\ref%\n{Dav})]. For example, below Eq. (2.95) in \\cite{Carmichael:book} the author\nargues that one can write down a Rabi frequency-independent MME for\nresonance fluorescence since $\\gamma (\\omega _{A})$ and $\\gamma (\\omega\n_{A}\\pm \\Omega _{\\mathrm{R}})$ (where $\\omega _{A}$ is the Bohr frequency)\ndiffer by less than 0.01$\\%$ at optical frequencies and reasonable laser\nintensities. However, this ignores the corrections due to the Rabi frequency\nto the operators $S_{k}(q,\\omega )$ [Eq.~(\\ref{18})]. This disagreement can\nbe traced to the question of at which point in the derivation it is safe to\nneglect $\\Omega _{\\mathrm{R}}$; in \\cite{Carmichael:book} this is done on\nthe basis of the flatness of $\\gamma (\\omega )$ before \\textquotedblleft a\nlot of tedious algebra\\textquotedblright\\ \\cite{Carmichael:book}[p.48], but\nour Floquet analysis shows that, in fact, one cannot neglect the Rabi\nfrequency relative to the Bohr frequency. This is relevant for our general\ndiscussion since the \\textquotedblleft \\emph{finitely localized}\nMME\\textquotedblright\\ which is the outcome of the Floquet analysis (see\nnext comment) actually exhibits a weak non-Markovian character. Such\ndeviations are, of course, important for FT-QEC, even if the effects are\nsmall. We revisit this point in Section \\ref{objections} below.\n\n\\noindent -- \\emph{Are there any non-Markovian effects at work here?} It seems that\none should accept the \\emph{generalized notion} of a quantum Markovian\nmaster equation as the one given by Eqs.~(\\ref{10}), (\\ref{11}) and (\\ref{2}%\n), i.e., a master equation with a possibly time-dependent Lindblad\ngenerator. In Davies' generalization to the time-dependent case \\cite%\n{Davies:78} (\\textquotedblleft adiabatic WCL\\textquotedblright ) the\ndissipative generator $\\mathcal{L}(t)$ depends on the Hamiltonian at the \n\\emph{same time} $t$. This is a type of \\textquotedblleft \\emph{local}\ngeneralized MME\\textquotedblright . On the other hand, in the periodic WCL\ntreated here, the dissipative generator $\\mathcal{L}(t)$ depends on the\nHamiltonians $H_{S}(u)$ from an interval, say $[0,t]$ ($t<\\Theta $), as can\nbe seen from Eq.~(\\ref{11}), which involves $\\mathcal{U}(t,0)$. This is\ntherefore a type of \\textquotedblleft \\emph{finitely localized}\nMME\\textquotedblright , though one could argue that it exhibits a weakly\nnon-Markovian character because of this dependence of the dissipative\ngenerator on the past. On the other hand, a non-Markovian master equation\n(in the convolutionless formalism \\cite{Breuer:book}) is also given by Eq.~(%\n\\ref{10}), but the generator is \\emph{not} of Lindblad form [in particular,\nit is not of the form (\\ref{19})], and may depend on the Hamiltonian in the \n\\emph{distant} past. The weight of distant past contributions depends on the\ndecay properties of $F(t)$ which are, generically, not exponential but\nrather powerlike. In the WCL the non-Lindbladian terms vanish due to the\noscillating character of the $e^{i(\\omega -\\omega ^{\\prime })u}$ terms in\nEq.~(\\ref{eq:K2}).\n\n\\noindent -- \\emph{The original Davies derivation}:\\ We note that the Davies result\nis a limit theorem which states that for a sufficiently small coupling\nconstant the WCL semigroup is a good approximation to the real dynamics.\nHowever, Davies' theorem itself does not provide the conditions under which\na given physical coupling is \\textquotedblleft small\nenough\\textquotedblright . In particular, one cannot extract from Davies'\ntheorem under what conditions the fast oscillating terms vanish. This can,\nhowever, be done by a more heuristic analysis, as done above.\n\n\n\\subsubsection{WCL for an Arbitrary Pulse}\n\nWe now consider the case \n\\begin{equation}\nH_{C}(t)=H_{0}+f(t)H_{1},\n\\end{equation}%\ni.e., an arbitrary driving field. This is, of course, the case of most\ninterest in FT-QEC. It follows from Fourier analysis that this case can be\ntreated qualitatively as a \\textquotedblleft\nsuperposition\\textquotedblright\\ of periodic perturbations discussed above.\nFor a single frequency $\\Omega $ the validity of the Markovian approximation\nis restricted by the condition (\\ref{MA-per}):\\ $t\\gg \\max \\{1\/(\\omega\n-\\omega ^{\\prime }+m\\Omega )\\}$. The discreteness of the frequencies $%\n\\{\\omega \\}$ and $\\{m\\Omega \\}$ is key:\\ it allows for condition (\\ref%\n{MA-per}) to be satisfied with finite $t$. A pulse $f(t)$ has a continuous\nband of frequencies of width $\\Gamma \\simeq 1\/\\tau _{g}$ (where $\\tau _{g}$\nis the gate duration), with amplitudes (Fourier transform) $\\hat{f}(\\Omega )$%\n, which add to and smear the effective Bohr spectrum $\\{\\omega \\}$. If the\npulse is long (a slow gate) then only a narrow band appears, and the\nsmearing effect is unimportant. More precisely, if $1\/\\tau _{g}$ is much\nsmaller than the typical difference of the Bohr frequencies, the\n\\textquotedblleft energy quanta\\textquotedblright\\ $m\\Omega $ [with $|m|$\nrestricted by the (typically small) dimension of the system Hilbert space]\ncannot fill the gap between $\\omega $ and $\\omega ^{\\prime }$ and the\ncondition (\\ref{MA-per}) can be satisfied. This is our adiabatic\napproximation. For fast pulses, when $1\/\\tau _{g}$ is comparable to $|\\omega\n-\\omega ^{\\prime }|$, the condition (\\ref{MA-per}) cannot be fulfilled: the\neffective Bohr spectrum becomes quasi-continuous and the denominator in\ncondition (\\ref{MA-per}) becomes abitrarily small. The result is that the\nWCL\\ analysis breaks down and non-Markovian effects dominate.\n\nThus, the condition for the adiabatic limit (Markov approximation valid) is:\n\\textquotedblleft the width of the band is much smaller than the minimal\ndifference of the effective Bohr frequencies\\textquotedblright . This is in\ncontradiction with the fast gate assumption, \\textbf{A1}.\n\n\\subsection{Section Summary}\n\nThe main advantage of the MME (\\ref{Dav}) is its consistency with\nthermodynamics. Namely, as a consequence of the KMS condition (\\ref{KMS})\nand the condition (\\ref{Dav1}), for a generic \ninitial state the system tends\nto its thermal equilibrium (Gibbs) state at the temperature of the heat\nbath \\cite{Alicki:87}.\n(An important exception to this rule are states within a\ndecoherence-free subspace \\cite{Zanardi:97c,LidarWhaley:03}, \nbut these states are not generic due\nto required symmetry properties of the\nsystem-bath interaction.) Therefore the dissipative part of the generator\nmust depend strongly on the Hamiltonian dynamics. This is consistent with\nthe notion of a coarse-grained description familiar from the study of MMEs:\nthe bath needs a time much longer than $\\max_{\\omega _{kl}}1\/\\omega _{kl}$\nto \\textquotedblleft learn\\textquotedblright\\ the system's Hamiltonian in\norder to drive it to a proper Gibbs state. In other words, \\emph{the Markov\napproximation is, equivalently, a long-time limit} (compared to $%\n\\max_{\\omega _{kl}}1\/\\omega _{kl}$ -- the system's Bohr frequencies), and\none cannot expect this approximation to be valid at short times. However,\nFT-QEC assumes operations on a time-scale that is short on the scale set by $%\n\\max_{\\omega _{kl}}1\/\\omega _{kl}$.\n\nStrictly speaking the MME (\\ref{Dav}) is valid only when $H_{S}$ is not time\ndependent. As we have shown, one can relax this by assuming slowly varying $%\nH_{S}$, giving rise to an \\textquotedblleft adiabatic MME\\textquotedblright\n, Eqs.~(\\ref{10}), (\\ref{11}) and (\\ref{2}). However, to accept Eqs.~(\\ref%\n{10}), (\\ref{11}) and (\\ref{2}) as a genuine Markovian description is\nsomewhat of a stretch, since the real question is not whether one obtains\nthe Lindblad form, but rather \\emph{how }$\\mathcal{L}(t)$\\emph{\\ depends on\nthe Hamiltonians }$H_{S}(u)$\\emph{, locally (i.e. }$u\\simeq t$\\emph{) or\nnonlocally. For fast gates and generic environments the dependence is\nnon-local, involving memory effects. }In any case, the crucial condition\nthat must be satisfied for a (generalized) MME is Eq.~(\\ref{MA-per}), which\nimplies that the average Bohr spectrum must be discrete. In essence, as long\nas the applied control does not spoil this discreteness a (generalized) MME\\\ncan be derived. On the other hand, \\emph{this means that fast gates are\nincompatible with the MME}, in violation of \\textbf{A1} of FT-QEC theory.\nThe corollary: \\emph{finite speed of gates implies non-Markovian effects}.\n\n\\section{Are the Standard FT-QEC Assumptions Internally Consistent?}\n\n\\label{consistency}\n\nWe now briefly summarize our examination of the assumptions of FT-QEC, in\nlight of the considerations above, and highlight where there may be internal\ninconsistencies in FT-QEC. As discussed above, there are essentially two\nrigorous approaches to the derivation of the MME: (i) the SCL, which is\ncompatible with arbitrarily fast Hamiltonian manipulations, but requires the\nhigh-$T$ limit; (ii) the WCL, which is compatible with thermodynamics at\narbitrary $T$, but requires adiabatic Hamiltonian manipulations.\n\nThe standard theory of FT-QEC (excluding Refs.~\\cite%\n{Terhal:04,Aliferis:05,Aharonov:05}) requires a quantum computer (QC)\nundergoing Markovian dynamics, supplemented with a constant supply of cold\nand fresh ancillas. These assumptions are contradictory under the SCL, since\nthe QC would have to be at high-$T$, while the ancillas require low-$T$ on\nthe same energy scale $E$ (set by the Bohr energies of the system = computer\n+ ancillas). Specifically, if we were to assume that for the ancillas too $%\nkT\\gg E$, they would quickly become highly mixed. If we insist that $kT\\ll E$\nfor the ancillas, then by coupling them to the QC we can no longer assume,\nin the SCL, that the total system = QC + ancillas is described by Markovian\ndynamics.\n\nIf, on the other hand, we approach the problem from the (physically more\nconsistent) WCL, then \\textbf{A3} is incompatible with \\textbf{A1} (the\nassumption of fast gates). Namely, in the WCL only adiabatic Hamiltonian\nmanipulations are allowed. Specifically, the Markov approximation in the WCL\nrequires a \\emph{discrete} system (effective) Bohr frequency spectrum, such\nthat the condition$\\ \\tau _{g}\\gg \\max_{\\omega _{kl}}1\/\\omega _{kl}$ can be\nsatisfied, hence violating the $\\tau _{g}\\omega _{B}=O(\\pi )$ condition of \n\\textbf{A1}. These conclusions are unavoidable if one accepts\nthermodynamics, since they follow from seeking a Markovian master equation\nthat satisfies the KMS condition -- a necessary condition for return to\nthermodynamic equilibrium in the absence of external driving. We take here\nthe reasonable position that a fault tolerant QC cannot be in violation of\nthermodynamics.\n\n\\section{Possible objections to the inconsistency}\n\n\\label{objections}\n\nIn this section we analyze a list of possible objections to the\ninconsistency we have pointed out.\n\n\\subsection{Is thermodynamics relevant?}\n\nWith respect to the SCL: \\textquotedblleft \\emph{Thermodynamics is\nirrelevant (since a QC need not ever be in thermal equilibrium)}%\n.\\textquotedblright\\ \n\nNote that we never claim that the QC is in thermal equilibrium; only the\nbath is. This assumption is a simplification which allows us to use a single\nparameter $T$ and therefore a single \\textquotedblleft thermal memory\ntime\\textquotedblright\\ $\\hbar \/kT$. There is no reason to use a nonthermal\nbath or many heat baths with different temperatures:\\ this does not make the\nspectral density flat and can only introduce more parameters.\n\n\\subsection{Doesn't the interaction picture save the day?}\n\nWith respect to the WCL: \\textquotedblleft \\emph{Suppose we have the\nfollowing Hamiltonian in the Schrodinger picture: }$%\nH=H_{S}+H_{C}(t)+H_{SR}+H_{R}$\\emph{\\ where }$||H_{S}||\\gg ||H_{C}||=$\\emph{%\ncontrol Hamiltonian }$\\gg ||H_{SR}||$\\emph{. Then in the interaction picture\nwith respect to} $H_{S}$\\emph{\\ the term }$H_{C}$\\emph{\\ is dominant and hence can\nimplement fast gates. However, in the original Schr\\\"{o}dinger picture }$%\nH_{C}$ \\emph{is small and hence the adiabatic limit for the derivation of\nthe MME is satisfied. Thus we have an example where we can have fast gates\n(in the interaction picture) and still the WCL can be satisfied so that the\nMarkovian limit can be reached. Moreover, this is the relevant limit relevant\nfor quantum optics, e.g., trapped ions.}\\textquotedblright\n\nThere are a number of problems with this argument. First, one should be more\ncareful about the formulation of the condition for adiabaticity. It can be\nstated as $|d\\omega (t)\/dt|\\ll \\omega (t)^{2}$, where $\\omega (t)$ is a\n\\textquotedblleft relevant\\textquotedblright\\ Bohr frequency. Merely\ncomparing norms as above does not guarantee adiabaticity. Second, in the\nquantum optics context we note the following. For three-level trapped ions\nwe have two Bohr frequencies: a large, time-independent $\\omega _{1}$, and a\nsmall, time-dependent $\\omega _{2}(t)$ (degenerate levels splitting). Only $%\n\\omega _{2}$ is \\textquotedblleft relevant\\textquotedblright\\ because it is\nrelated to gates, and then the adiabatic condition implies that $|d\\omega\n_{2}(t)\/dt|$ is correspondingly small, which contradicts the fast gate\ncondition \\textbf{A1}. Third, the inequality $||H_{C}||\\gg ||H_{SR}||$ is in\nfact not satisfied in the Markovian WCL, where $||H_{SR}||$ diverges (one\nshould not confuse the small system-reservoir coupling parameter involved in\nthe van-Hove limit with the operator norm, which can be infinite).\n\n\\subsection{Doesn't quantum optics provide a counterexample?}\n\nWith respect to the WCL: \\textquotedblleft \\emph{Trapped ions and other\nquantum optics systems provide a counter-example: a system experimentally\nsatisfying Markovian dynamics and allowing fast Rabi operations}%\n.\\textquotedblright\\ \n\nWe have already addressed quantum optical systems in Section \\ref{Floquet}.\nLet us add a few comments. We do not know of any quantum optics experiment\ntesting the Markov approximation with the accuracy relevant for FT-QEC (for\nquantum dots, on the other hand, non-Markovian effects are very visible). We\nknow that for constant, and also for strictly periodic Hamiltonians (which\ncorresponds in quantum optics to a constant external laser field), the\nDavies derivation can be applied (or extended, as in Section \\ref{WCL}) and\nthe Markov approximation is applicable. The problem appears for fast gates.\nIt would be difficult to test the Markov approximation in this case with the\nrequired accuracy, because, e.g., the results depend on the shape of the\npulse. A relevant example is resonance fluorescence, as described in \\cite%\n{Carmichael:book}[pp.43-61], and as discussed in Section \\ref{Floquet}. The\ndamping effects are only present in the widths of spectral lines -- see \\cite%\n{Carmichael:book}[p.61, Fig. 2.5]. The Markov approximation gives\nLorentzians while non-Markovian dynamics may give rise to more complicated\nlineshapes. Consider a 2-level atom like in \\cite{Carmichael:book} Section\n2.3.2., and in particular the final formula Eq. (2.96), which describes\nresonance fluorescence via a MME. The author claims that for typical\nparameters in quantum optics the dissipative part does not depend on the\nRabi frequency $\\Omega _{\\mathrm{R}}$ [recall our discussion in Section \\ref%\n{Floquet}]. Hence, as the gates are entirely related to $\\Omega _{\\mathrm{R}}\n$, it appears that either fast or slow gates are possible. The argument is\nbased on the small ratio $\\Omega _{\\mathrm{R}}\/\\omega _{A}<10^{10}\/10^{15}$\n(where $\\omega _{A}$ is the Bohr frequency). This is fine for replacing the\nspectral density at $\\omega _{A}\\pm \\Omega _{\\mathrm{R}}$ by the density at $%\n\\omega _{A}$, but the subsequent argument that we can replace [in Eq.\n(2.94)] $\\Omega _{\\mathrm{R}}$ by $0$ is inaccurate. This would be correct\nonly if the decay time $\\tau =1\/\\gamma $ is short enough such that $\\Omega _{%\n\\mathrm{R}}\\tau \\ll 1$. However, as explained in Section \\ref{Floquet}, in\nthis case the Davies type averaging makes no sense physically. In fact,\ntypically for radiation damping $\\tau =10^{-8}$s, and then $\\Omega _{\\mathrm{%\nR}}\\tau <100$ only. Hence for a fixed $\\Omega _{\\mathrm{R}}$ we do in fact\nnot have a simple Lindblad generator (of the type (2.96) in\n\\cite{Carmichael:book}), but rather a more complicated generator with\nLindblad \noperators depending on the Rabi frequency, as in Eq.~(\\ref{16}). Again, in\nthe derivation of a proper generator an averaging over terms of the form $%\n\\exp (-i\\Omega _{\\mathrm{R}}t)$ must be performed. Therefore the condition\nfor the adiabatic approximation involves the Rabi frequency $\\Omega _{%\n\\mathrm{R}}$ and cannot be satisfied for fast gates. For experiments based\non \\emph{spectral measurements} the difference between the two types of\ngenerators we have just discussed is probably irrelevant for many reasons;\nhowever, the quantum state of the atom at a given moment is sensitive to a\nsmall change in the Lindblad operators, and this is important in a fault\ntolerant implementation of quantum logic gates.\n\n\\subsection{Is \\textbf{A1} truly an assumption of FT-QEC?}\n\n\\label{adiabatic-gates}\n\nWith respect to the WCL: ``\\emph{Doesn't \\textbf{A1} impose an unnecessary\nconstraint on FT-QEC, in that gates are not required to satisfy the\ncondition }$\\tau _{g}\\omega =O(\\pi )$\\emph{?}''\n\nIn other words, one might argue in favor of slow gates, where instead the\ncondition is $\\tau _{g}\\omega \\gg O(\\pi )$. Such gates are certainly\nrelevant in the context of the adiabatic quantum computing (AQC) paradigm \n\\cite{Farhi:01}, holonomic QC \\cite{ZanardiRasetti:99,ZanardiRasetti:2000},\nor topological quantum computing (TQC) \\cite%\n{Kitaev:97,Freedman:01,DasSarma:05}. We comment in more detail on AQC, HQC,\\\nand TQC in Section \\ref{alternatives}. The question of interest to us is\nwhether an adiabatic gate satisfying $\\tau _{g}\\omega \\gg O(\\pi )$ is\napplicable to the standard FT-QEC paradigm we are considering here, and\nwhich is very different from AQC, HQC,\\ and TQC.\n\nFirst, let us clarify that by gates we mean one and two-qubit unitaries\npicked from well-known discrete and small sets of universal gates\\ \\cite%\n{Nielsen:book}. An algorithm is constructed via a sequence of such gates,\nand computational complexity is measured in terms of the minimal number of\nrequired gates. Of course one can instead join all gates used in a given\nalgorithm into a single unitary and call this a gate, but then one runs into\nthe problem of finding a relevant (physical) Hamiltonian and quantifying\ncomputational complexity. For a given gate there are infinitely many\nHamiltonian realizations. Among these are fast ones (optimal) which satisfy $%\n\\tau _{g}\\omega =O(\\pi )$ and slow ones (adiabatic) satisfying $\\tau\n_{g}\\omega \\gg O(\\pi )$ (all inequalities here are in the sense of orders of\nmagnitude). For example, consider a $\\pi $-rotation. The fast (optimal)\nrealization satisfies $\\tau _{g}\\omega =\\pi $ (compatible with \\textbf{A1}),\nwhile the slow (adiabatic) one satisfies $\\tau _{g}\\omega =\\pi +2\\pi n$ with \n$n\\gg 1$ (contradicts \\textbf{A1}).\n\nNow, one may ask whether a slow realization of gates can prevent the\ninconsistency with the WCL. We argue, based on computational complexity\nconsiderations, that the answer to this question is negative. To see this,\nnote first that non-Markovian errors are uncorrectable in standard FT-QEC.\nTherefore such non-Markovian, uncorrectable errors accumulate during the\ncomputation (by definition, they are not corrected by \\textquotedblleft\nMarkovian FT-QEC\\textquotedblright ), and in order to keep them under\ncontrol, the probability of such errors per gate, $p_{\\mathrm{non-M}}$,\nshould scale as \n\\begin{eqnarray}\n  p_{\\mathrm{non-M}} &\\sim& O[1\/(\\text{volume of algorithm})]\n  \\nonumber \\\\\n  &=& O[1\/(\\text{input\nsize})^{\\alpha }],\n\\end{eqnarray}\nwhere $\\alpha $ is some fixed power. Now, it follows from our discussion in\nSection \\ref{WCL} that the more adiabatic the evolution, the smaller is the\nprobability of the non-Markovian errors per gate. Therefore, if one writes\nthe adiabaticity condition as $\\tau _{g}\\omega >M$, where $M\\gg 1$ is the\n\\textquotedblleft adiabatic slowness parameter\\textquotedblright , then the\nprobability of non-Markovian errors should satisfy \n\\begin{equation}\np_{\\mathrm{non-M}}\\sim O(1\/M^{\\beta }),\n\\end{equation}%\nwhere $\\beta $ is another fixed power [$\\omega $ (the Bohr or Rabi\nfrequency) is limited essentially by the choice of physical system].\nComparing the two expressions for $p_{\\mathrm{non-M}}$, we see that $M$ must\ngrow with input size. This means that if one works with adiabatic gates in\norder to keep the dynamics (approximately) Markovian, the result is that one\nmust slow the gates in proportion to the input size (to some power). This,\nhowever, violates the threshold condition of FT-QEC, in which the input size\nand gate times are independent parameters (see, e.g., Theorem 12 in \\cite%\n{Aharonov:99}).\n\n\\subsection{Measurements}\n\nWith respect to both the WCL and the SCL:\\ \\textquotedblleft \\emph{Recent\nresults on fault-tolerant QC using measurements only (e.g., \\cite%\n{Nielsen:04,Raussendorf:05}) render all the claimed problems irrelevant}%\n.\\textquotedblright\\ \n\nIndeed, we have so far discussed only the problems with quantum logic gates.\nMoreover, measurements are an integral part of FT-QEC theory as well, in\nparticular to reset and disentangle ancillas before they are introduced into\nan error-correction circuit. Therefore some remarks on the use of\nmeasurements are in order.\n\nIn the most advanced FT-QEC scheme of \\cite{Aharonov:99}, measurements are\nperformed at the end of the computation. However, this approach demands a\nhigh resource overhead, which may make it impractical. Therefore, more\nrecent proposals (e.g., \\cite{Knill:05,Steane:04}) rely on feedback\nmechanisms employing the results of quantum measurements. Those\n\\textquotedblleft measurements in the middle of\ncomputation\\textquotedblright\\ are treated for simplicity as certain\nvon-Neumann projective measurements (but with efficiency $\\ll 1$) satisfying\na \\emph{repeatability condition}. The latter implies that the subsequent\nmeasurements reduce the measurement error exponentially as their number\nincreases. This assumption should be carefully scrutinized, within realistic\nHamiltonian models of quantum measurement treated as a dynamical process.\nHere, again one can expect that the tacit assumption of statistical\nindependence of repeated measurements is in conflict with the non-Markovian\ncharacter of the dynamics of open quantum systems.\n\nAs all proposed measurement schemes are based on electromagnetic\ninteractions, it should be possible to construct a rather general\nHamiltonian framework and apply it to various particular implementations.\nIndeed, this has been done, e.g., for a single-electron tunneling (SET)\ntransistor coupled capacitively to a Josephson junction qubit \\cite%\n{Shnirman:98}. Rather than assuming that the measurement apparatus is\ncoupled to the system whenever measurements must be performed -- an option\nwhich is hard to achieve in mesoscopic systems -- Ref.~\\cite{Shnirman:98}\nmakes the reasonable assumption that the measurement apparatus is always\ncoupled to the system, but is in a state of equilibrium when it is not\nneeded. A measurement is then performed by driving the measuring device out\nof equilibrium, in a manner that dephases the qubit to be measured. Generic\nfeatures emerging from this analysis are the existence of three different\ntime-scales characterizing the measurement: the dephasing time, the\nmeasurement time (which may be longer than the dephasing time), and the\nmixing time (the time after which all the information about the initial\nquantum state is lost due to the transitions induced by the measurement).\nRef.~\\cite{Shnirman:98} thus arrives at a criterion for a ``good'' quantum\nmeasurement: the mixing time should be longer than the measurement time. A\ntime-scale analysis of measurements in optical systems, accounting for\nspontaneous emission, can be found, e.g., in Ref.~\\cite{Teich:89}. A fully\nconsistent analysis of FT-QEC should account for the existence of such\ntime-scales in a dynamic description of the measurement process. In\nparticular, it is important to set appropriate bounds on these time-scales,\nso that they may be taken into account in a threshold calculation (an\nanalysis based on a stochastic error model was reported in Ref.~\\cite%\n{Steane:03}).\n\n\\subsection{Degenerate Qubits}\n\nWith respect to the SCL: \\textquotedblleft \\emph{Degenerate qubits\nautomatically satisfy the high }$T$\\emph{\\ limit since their intrinsic\nenergy scale vanishes}.\\textquotedblright\\ \n\nExamples of degenerate qubits are common, e.g., in trapped ion quantum\ncomputing implementations where a pair of degenerate hyperfine states can\nserve as a qubit, with an auxiliary third level used to implement quantum\nlogic gates via Raman transitions \\cite{Wineland:98}. The case of degenerate\nqubits is somewhat more subtle to analyze within the context we have\nexplained above. Naively, in such a case the high-$T$ limit is indeed\nautomatically satisfied, since the system energy scale is zero. Therefore it\nappears that one could claim that the SCL version of the Markov\napproximation is attainable. However, upon closer examination this still\nseems problematic. Indeed, the vanishing of an energy scale for degenerate\nqubits holds, strictly speaking, only for fully adiabatic techniques, e.g.,\nHQC \\cite{ZanardiRasetti:99,ZanardiRasetti:2000}. Otherwise transformations\nbetween logical states are achieved by resorting to effective Hamiltonians\nwhich involve \\emph{virtual} transitions. For instance, if $|0\\rangle $ and $%\n|1\\rangle $ denote degenerate qubit levels (e.g., hyperfine levels of an\nion), one can introduce far-detuned (e.g., laser) couplings of $|0\\rangle $\nand $|1\\rangle $ with a third auxiliary level. Second order perturbation\ntheory then yields the effective Hamiltonian $H_{\\mathrm{eff}}=-(\\Omega _{%\n\\mathrm{R}}^{2}\/\\Delta )|1\\rangle \\langle 0|+\\mathrm{h.c.}$, where $\\Omega _{%\n\\mathrm{R}}$ and $\\Delta $ are the laser Rabi coupling and detuning,\nrespectively. Therefore we see that an effective, small but non-vanishing,\nenergy scale $E_{1}:=\\Omega _{\\mathrm{R}}^{2}\/\\Delta $ is introduced. (Note\nthat in order for perturbation theory to be valid one must have $\\Omega _{%\n\\mathrm{R}}\\ll \\Delta $, which in turn implies $E_{1}\\ll \\Delta $.) Yet\nanother energy scale is provided by the spectral width $E_{2}$ of the laser\npulse shape $\\Omega _{\\mathrm{R}}(t)$; in order to suppress unwanted \\emph{%\nreal} transitions, one must impose in addition that $E_{2}\\ll \\Delta $. At\nany rate, the appearance of these new system-energy scales implies that once\nagain the SCL-type contradiction applies. On the other hand, we can make\nboth $E_{1}$ and $E_{2}$ small at the price of lengthening the gating time ($%\n\\tau _{g}\\simeq \\max \\{1\/E_{1},\\,1\/E_{2}\\}$). This implies, once again, an\nadiabatic limit and the applicability of the WCL. Therefore it appears that\nas long as one restricts manipulations to adiabatic ones (thus contradicting \n\\textbf{A1}), quantum computing with degenerate qubits is possible even in\nthe Markovian limit. We expand on this viewpoint below.\n\n\\subsection{Impure Ancillas}\n\nWith respect to the SCL:\\ \\textquotedblleft \\emph{Do ancillas really need to\nbe pure?}\\textquotedblright\\ \n\nWhat precisely is the role of the ancillas in QEC? A popular answer is that\nthey serve as an \\textquotedblleft entropy sink\\textquotedblright\\ for the\nerrors accumulated during the quantum computation. This entropy in the\nsystem arises from the entanglement between system and bath, and the role of\nthe ancillas is to remove this entanglement. I.e., in a perfect quantum\nerror correction step the entanglement between system and bath is\ntransferred to the ancillas and bath. A natural objection to our SCL-based\ninconsistency is to claim that, in fact, ancillas need not be pure, or could\nperhaps even be highly mixed. However, this is not supported by the\n(current) standard theory of FT-QEC. Consider, e.g., an error correction\ncircuit based on the Steane 7-qubit code. It takes as input ancillas\nprepared in the $|\\psi \\rangle _{a}=(|0_{L}\\rangle +|1_{L}\\rangle )\/\\sqrt{2}$\nstate, where $|0_{L}\\rangle $ and $|1_{L}\\rangle $ are codewords. The\nphysical qubits which comprise such ancillas, are coupled bitwise via CNOT\ngates to the physical qubits making up the encoded data qubits in the\ncircuit. If instead we input an ancilla in a mixed state, this is equivalent\nto inputting a classical mixture with erred codewords, e.g., $(1-p)|\\psi\n\\rangle _{a}\\langle \\psi |+p|\\phi \\rangle _{a}\\langle \\phi |$, where $|\\phi\n\\rangle _{a}$ is an erred codeword. If one of these errors is a phase-flip,\nit feeds back (via the CNOT gates) into the data qubits, producing an error \n\\cite{Gottesman:97a}. Without fault-tolerance this means that there are now\ntwo errors (in the ancillas block and the data block), which may be more\nthan the code can handle. In FT-QEC theory such errors are accounted for,\nbut their magnitude is bounded from above (e.g., $p$ in the above example\nmust be small). We note that an ancilla which is initially entangled with\nthe data qubits (violating the assumption of being introduced into the\ncircuit in a tensor-product state) is essentially equivalent to the case of\nan impure ancilla just described (tracing over the data qubits yields an\nimpure ancilla state).\n\nA more general approach showing the importance of the assumption of pure\nancillas is the following (fairly standard account of QEC).\n\n\\noindent i) \\textit{Preparation}.--\n\nLet the initial state of system + reservoir + ancillas, with respective\nHilbert spaces $\\mathcal{H}_{S},\\mathcal{H}_{R},\\mathcal{H}_{A}$, be: $\\rho\n_{SRA}^{0}=|\\psi _{S}\\rangle \\langle \\psi _{S}|\\otimes |0_{R}\\rangle \\langle\n0_{R}|\\otimes \\rho _{A}$, where we have allowed for ancillas in a mixed\nstate $\\rho _{A}$.\n\n\\noindent ii) \\textit{System-reservoir interaction (decoherence)}.-- \n\\begin{equation}\n\\rho _{SRA}^{0}\\overset{U_{SR}}{\\longrightarrow }\\rho\n_{SRA}^{1}=\\sum_{e,e^{\\prime }\\in \\mathcal{E}}U_{e}|\\psi _{S}\\rangle \\langle\n\\psi _{S}|U_{e}^{\\dag }\\otimes |e_{R}\\rangle \\langle e_{R}^{\\prime }|\\otimes\n\\rho _{A},\n\\end{equation}%\nwhere $e$'s denote the \\emph{errors} belonging to the set $\\mathcal{E}$ that\nthe code $\\mathcal{C}$ can correct, and where $|e_{R}\\rangle $ are the\ncorresponding states of the reservoir. The error operators $U_{e}$ are\nassumed to be unitary and with linear span of dimension $|\\mathcal{E}|$.\n\n\\noindent iii) \\textit{System-ancilla interaction (syndrome extraction)}.--\n\nThis interaction takes the form $U_{SA}=\\sum_{e\\in \\mathcal{E}}\\Pi\n_{e}\\otimes T_{e}$ where the $T_{e}$'s are unitaries over $\\mathcal{H}_{A}$\nsuch that $T_{e}|0_{A}\\rangle =|e_{A}\\rangle $ and $\\Pi _{e}\\cong I_{%\n\\mathcal{C}}\\otimes |e\\rangle \\langle e|$ \\footnote{%\nWe know that $\\mathcal{H}_{S}\\cong \\mathcal{C}\\otimes \\mathcal{S}\\oplus \n\\mathcal{D}$ [$\\mathcal{S}$=syndrome subsystem, dim$\\mathcal{D}=|\\mathcal{E}|\n$; $\\mathcal{D}$=remainder (=$0$ for subspace-based codes)] \\cite%\n{Knill:97b,Knill:99a,Zanardi:99d}.}: \n\\begin{eqnarray}\n\\rho _{SRA}^{1} &\\overset{U_{SA}}{\\longrightarrow }&\\rho _{SRA}^{2}  \\notag\n\\\\\n&=&\\sum_{e,e^{\\prime }\\in \\mathcal{E}}U_{e}|\\psi _{S}\\rangle \\langle \\psi\n_{S}|U_{e^{\\prime }}^{\\dagger }\\otimes |e_{R}\\rangle \\langle e_{R}^{\\prime\n}|\\otimes T_{e}\\rho _{A}T_{e^{\\prime }}.  \\notag \\\\\n&&\n\\end{eqnarray}\n\n\\noindent iv) \\textit{Error recovery}.---\n\nUnitary recovery is implemented via $\\tilde{U}_{SA}=|\\mathcal{E}%\n|^{-1\/2}\\sum_{e\\in \\mathcal{E}}U_{e}^{\\dagger }\\otimes I_{R}\\otimes\n|e_{A}\\rangle \\langle e_{A}|$, where for unitarity we need $\\langle\ne_{A}|e_{A}^{\\prime }\\rangle =\\delta _{e,e^{\\prime }}$. By applying $\\tilde{U%\n}_{SA}$ and tracing over both $R$ and $A$ (assuming the $|e_{R}\\rangle $'s\ntoo are orthonormal) one obtains \n\\begin{equation}\n\\rho _{S}^{\\mathrm{out}}=\\frac{1}{|\\mathcal{E}|}\\sum_{e,f\\in {E}%\n}U_{f}^{\\dagger }U_{e}|\\psi _{S}\\rangle \\langle \\psi _{S}|U_{e}^{\\dagger\n}U_{f}\\,\\langle f_{A}|T_{e}\\rho _{A}T_{e}^{\\dagger }|f_{A}\\rangle .\n\\end{equation}%\nIn the case of a pure ancillas $\\rho _{A}=|0_{A}\\rangle \\langle 0_{A}|$ one\nhas $\\langle f_{A}|T_{e}\\rho _{A}T_{e}^{\\dagger }|f_{A}\\rangle =|\\langle\nf_{A}|e_{A}\\rangle |^{2}=\\delta _{f,e}$ and therefore the ideal case $\\rho\n_{A}^{\\mathrm{out}}=|\\psi _{S}\\rangle \\langle \\psi _{S}|$ is recovered. One\ncan also consider the fidelity \n\\begin{eqnarray}\nF &:=&\\langle \\psi _{S}|\\rho _{S}^{\\mathrm{out}}|\\psi _{S}\\rangle   \\notag \\\\\n&=&|\\mathcal{E}|^{-1}\\sum_{e,f\\in {E}}|\\langle \\psi _{S}|U_{f}^{\\dagger\n}U_{e}|\\psi _{S}\\rangle |^{2}\\,\\langle f_{A}|T_{e}\\rho _{A}T_{e}^{\\dagger\n}|f_{A}\\rangle .  \\notag \\\\\n&&\n\\end{eqnarray}%\nProvided the error operators $U_{f}$ satisfy the condition for a\nnon-degenerate code $\\langle \\psi _{S}|U_{f}^{\\dagger }U_{e}|\\psi\n_{S}\\rangle =\\delta _{f,e}$ \\cite{Knill:97b}, one obtains $F=|\\mathcal{E}%\n|^{-1}\\sum_{e\\in \\mathcal{E}}\\langle e_{A}|T_{e}\\rho _{A}T_{e}^{\\dagger\n}|e_{A}\\rangle =\\langle 0_{A}|\\rho _{A}|0_{A}\\rangle .$ Clearly, $F=1$ iff $\\rho _{A}=|0_{A}\\rangle \\langle 0_{A}|,$ i.e., \\emph{%\nthe ancillas are pure}. One can also consider non-unitary recovery via\nancilla measurements and conditional unitaries, with Kraus operators given\nby $A_{e}=|\\mathcal{E}|^{-1\/2}U_{e}^{\\dagger }\\otimes I_{R}\\otimes\n|e_{A}\\rangle \\langle e_{A}|$. The conclusion that the ancillas' state must\nbe pure is unchanged.\n\nWe note that FT is obtained by adding concatenation and, in steps iii) and\niv), preparing and coupling encoded ancillas with the system in a suitable\nway, e.g., as in the Steane-code example above. In this case it is\npermissible to allow slightly impure ancillas, and relax the assumptions\nthat, in step ii) the environment couples only to the system, and in steps\niii,iv), the environment does not act. This formulation, however, does not\nallow arbitrarily mixed-state ancillas, as argued in the Steane-code\nexample. While such a formulation of FT-QEC theory might still emerge (for\nexample, by using algorithmic cooling techniques \\cite%\n{Schulman:98,Schulman:05}, which, however, at present assume perfect gates),\nit does not appear possible at present to relax the assumption of cold\nancillas.\n\n\\subsection{Hot QC, cold ancillas, and fast QC-ancilla interactions in the\nSCL}\n\n\\label{Strong}\n\nWith respect to the SCL: \\emph{\\textquotedblleft One can keep the ancillas\ncoupled to a separate cold bath and then couple them for only a short time\nto the QC: what matters then is the }$\\emph{T}_{1}$\\emph{\\ timescale and\nthat one can be very long compared to the required ancilla-QC coupling\ntime\\textquotedblright }.\n\nLet us paraphrase this objection. If one can make $H_{SA}$ (system-ancillas)\nvery large then one could beat the rate of ancilla heating by strongly\ncoupling the QC and ancillas. I.e., suppose one would like to bring the\nancillas in from their cold reservoir to couple to the system, which is\ncoupled to a hot reservoir as required for the SCL. The ancillas then heat\nup fast, but there is a timescale associated with this heating\n(\\textquotedblleft $T_{1}$\\textquotedblright ), which one wishes to beat.\nNow if one could make the system-ancilla coupling very strong then one\ncould, presumably, use the ancillas (e.g. for syndrome extraction) faster\nthan their heating rate, while they are still sufficiently pure for fault\ntolerance purposes.\n\nThe simplest argument against this objection is the following. In the\nsetting of the objection, the QC is described by the SCL (high $T$) while\nthe ancillas are described by the WCL (low $T$). Strong and fast coupling\nbetween the QC and the ancillas is unacceptable according to the WCL because\nit is fast\n(only adiabatic manipulations are allowed),\nand according to the  SCL because it is strong\n(``strong'' refers to the system's Hamiltonian part, while in the SCL this Hamiltonian is weak\nin comparison with the system-bath coupling).\n\nHowever, one could go on to argue that the ancillas are a different species\nthan the QC qubits, and in particular have a different intrinsic\n(less\ndense) energy scale, so that they are at low $T$ on the scale set by the QC\\\nqubits. In this case both ancillas and QC are described by the SCL. Then the\nproblem with the objection is the following: recall that in the SCL (see\nSection \\ref{SCL}) one must rescale $H_{SR}$ and $H_{AR}$ as $%\nH_{SR}\/\\epsilon $ and $H_{AR}\/\\epsilon $ respectively, where here $R$\ndenotes the common reservoir the system and the ancillas are coupled to.\nThe heating rate is proportional to the square of the coupling\nstrength to the reservoir, i.e., to $1\/\\epsilon ^{2}$, and hence diverges in\nthe SCL. Therefore to beat the ancilla heating process via fast manipulation\nof the system-ancilla coupling one would have to rescale $H_{SA}$ at least\nby $1\/\\epsilon ^{2}$, but this contradicts the SCL derivation, where in fact\none must keep $H_{SA}$ fixed while rescaling $H_{SR}$. The reason for this\nis that, in the SCL derivation, it is the system (now including the\nancillas) that sets the timescale against which reservoir correlations must\nbe accelerated.\\footnote{%\nLet us also consider the issue from the perspective of thermodynamics. This\nis not really necessary, since the arguments above about the SCL are\nrigorous, but is interesting in its own right. \nFirst, we remark that error correction should really be made to work at the\ncommon lower (initial ancillas') temperature. Heating a part of a QC only to\nbe closer to the Markovian limit is a suboptimal strategy, because it increases\nthe strength of the noise and stimulates entropy production. \nSecond, in standard FT-QEC heat (entropy) flows from the QC to the ancillas\nonly, while in reality one should expect a flow in both directions and\nadditionally an entropy production. To see this let us ignore for the moment\nthe coupling of the QC to the bath, and consider ancillas coupled to a heat\nbath at temperature $T$. The ancillas can be kept pure by maintaining an\nenergy gap $\\gg $ $kT$. Assume that the initial state of QC ($C$) and\nancillas ($A$) is a product state $|\\psi _{C}\\rangle \\otimes |\\psi\n_{A}\\rangle $. Switching on the interaction $H_{CA}$ we induce an\nequilibration process (because the dynamics is Markovian) of $C+A$ towards\nthe Gibbs state $\\rho _{CA}=\\exp (-H_{CA}\/kT)\/Z$, which is \\emph{entangled}\n(here for simplicity $H_{CA}$ contains not only the interaction but is the\ntotal Hamiltonian of $C+A$). After a single step of error correction the\ntotal state of $C+A$ can be modeled by $(1-p)U|\\psi _{C}\\rangle \\otimes\n|\\psi _{A}\\rangle U^{\\dag }+p\\rho _{CA}$, where $U$ is unitary and $0<p\\ll 1$%\n. Then we switch off the interaction with the ancillas. Whatever we do next\nseparately with $C$ and $A$, we cannot eliminate the error due to the\nentanglement present in the term $p\\rho _{CA}$. This type of incorrectable\nerror accumulates and destroys FT-QEC. This is the back flow of entropy from\nthe ancillas bath to the QC, mentioned above.}\n\n\\section{Alternatives to Markovian FT-QEC}\n\n\\label{alternatives}\n\n\\subsection{Nature of the non-Markovian errors in the WCL}\n\nWhile we have pointed out that, in the WCL, the application of fast gates is\nlikely to violate the conditions required for Markovian dynamics to persist,\nwe have not been specific about the type of non-Markovian effects that will\nemerge. It is well known that FT-QEC is capable of dealing with errors that\nchange due to the application of gates. Namely, assume (slightly) faulty\ngates correcting a specific error model described by a CP map $\\Lambda$\n[recall Eq.~(\\ref{eq:Lambda})], are applied in sequence, $\\Lambda\nU_{N}^{\\prime }\\Lambda U_{N-1}^{\\prime }\\cdots \\Lambda U_{1}^{\\prime }$, and\nthese gates are (in some appropriate norm) close to the ideal gates $%\n\\{U_{i}\\}_{i=1}^{N}$, i.e., for all $i$, $||U_{i}^{\\prime }U_{i}^{\\dag\n}-I||\\ll 1$. Then by inserting $U_{i}^{\\dag }U_{i}$'s everywhere one obtains\nthe new sequence $\\Lambda_{N}U_{N}\\Lambda_{N-1}U_{N-1}\\cdots\n\\Lambda_{1}U_{1} $, where $\\Lambda_{i}:=\\Lambda U_{i}^{\\prime }U_{i}^{\\dag }$%\n, and FT-QEC is capable of dealing with such a (gate-modified) error model.\nHowever, the non-Markovian effects that arise due to the application of fast\ngates in the WCL, will in general \\emph{not} be describable by a simple\ntime-local modification such as $\\Lambda\\rightarrow \\Lambda U_{i}^{\\prime\n}U_{i}^{\\dag } $. This can be worked out, e.g., using the methods of Ref.~%\n\\cite{Breuer:02}.\n\nIn order to formulate consistent alternatives to standard, Markovian FT-QEC\ntheory, it seems useful to start with a Hamiltonian formulation. As the\ndiscussion below will illustrate, it appears that a hybrid approach will be\nnecessary, which combines alternatives to standard QC with a new version of\nFT-QEC.\n\n\\subsection{Adiabatic Quantum Computing (AQC)}\n\nWe keep \\textbf{A2} and \\textbf{A3}, discard \\textbf{A1} (fast gates), and\nwork in a purely adiabatic mode, thus permitting a consistent WCL. This may\nindeed be possible using the adiabatic quantum computing (AQC) approach of\nFarhi et al. \\cite{Farhi:01}. At present there is little understanding of\nthe fault-tolerance of AQC. Some recent works explore AQC in the presence of\ndecoherence and\/or control errors \\cite%\n{Childs:02,Shenvi:03,Roland:04,Aberg:04,SarandyLidar:05,aberg-2005-72}.\nIndeed, the subject of the adiabatic approximation in open quantum systems\nhas only very recently been addressed \\cite{SarandyLidar:04}, and used to\nstudy AQC in open systems \\cite{SarandyLidar:05}. Error correcting codes for\nAQC\\ were introduced very recently in \\cite{Jordan:05}, but the\ncorresponding universal Hamiltonians involve many-body interactions (four\nand six-body for 1-local and 2-local errors, respectively).\n\n\\subsection{Holonomic Quantum Computing (HQC)}\n\nAnother possibility for keeping \\textbf{A2} and \\textbf{A3}, and discarding \n\\textbf{A1}, is provided by HQC \\cite{ZanardiRasetti:99,ZanardiRasetti:2000}%\n. HQC is an adiabatic scheme which relies on Abelian or non-Abelian\ngeometric phases to implement quantum logic gates. Quantum information is\nencoded in a \\emph{degenerate} set of eigenstates of a Hamiltonian depending\non a set of controllable parameters, e.g., external laser fields (recall our\ndiscussion of degenerate states above). When these are adiabatically driven\nalong a suitable closed path, the initial quantum state is transformed by a\nnon-trivial unitary transformation (holonomy) that is geometrical in nature.\nThe key point is that the geometrical nature of the quantum holonomies is\nbelieved to render HQC inherently robust against certain kinds of errors.\nThis alleged fault-tolerance has only recently been seriously begun to be\nexamined \\cite{Solinas:04}; the emerging picture is that, while stability\nagainst decoherence must still be assessed, HQC seems to exhibit a strong\nrobustness against stochastic errors in the control process generating the\nrequired adiabatic loops \\cite{Zhu:04}. Moreover, in the adiabatic\nlimit of Markovian dynamics it has been show that the geometric phase of a\nsingle qubit coupled to a magnetic field is robust against both\ndephasing and spontaneous emission (but not against bit\nflips) \\cite{SarandyLidar:05a}. Nevertheless, since deviations from strict\nadiabaticity are inevitable, and adiabaticity is particularly challenging to\nsatisfy in open quantum systems \\cite{SarandyLidar:04}, it is tempting to\ncombine HQC with FT-QEC in order to address the performance of HQC in the\npresence of decoherence errors. Alternatively, we note that a hybrid\napproach that seems to be rather promising is the embedding of HQC within a\nDFS \\cite{WuZanardiLidar:05}. This amounts to realizing a set of universal\nquantum gates, acting on a DFS, by means of non-abelian quantum holonomies.\nThis strategy brings together the \\textquotedblleft best of two\nworlds\\textquotedblright : the quantum decoherence avoidance virtues of DFSs\nand the fault-tolerance of the all-geometric holonomic control. It is\npossible that such an approach can be implemented for quantum information\nprocessing in, e.g., trapped ions and quantum dots.\n\n\\subsection{Topological Quantum Computing (TQC)}\n\nA robust way of performing quantum computations is based on excitations with\nfractional statistics, since they have several fault-tolerant properties\nbuilt in. This idea is known as topological quantum computing (TQC) \\cite%\n{Kitaev:97,Freedman:01,DasSarma:05}. Physical realizations of the simplest\nversions of TQC have been considered in the literature, using, e.g.,\nrotating Bose-Einstein condensates \\cite{Paredes:01} and superconducting\ncircuits \\cite{Ioffe:02}. Let $\\mathcal{C}$ denote the manifold of quantum\ncodewords. Strikingly, in TQC, one can have a trivial Hamiltonian, e.g., $%\nH|_{\\mathcal{C}}=0$, but nevertheless obtain non-trivial quantum evolution\ndue to the existence of an underlying topological \\emph{global} structure\n(boundary conditions). Quantum encoding is typically performed in a properly\ndesigned degenerate ground state $\\mathcal{C}$. This fact implies, for low\nenough temperature, an exponential suppression of errors on encoded quantum\ninformation due to thermal fluctuations. More importantly, topological\nfeatures can render such a ground state stable against errors represented by \n\\emph{local} operators, namely error operators that do not involve a number\nof qubits of the order of the size of the system. For example, in the\nso-called toric codes \\cite{Kitaev:97,Kitaev:book}, qubits are encoded in\nthe ground-state manifold of a lattice of interacting spins in such a way\nthat degenerate ground states are mutually connected only via high powers\n(scaling linearly with lattice size) of local operators. Thus, here the\nfault-tolerance properties are already built-in at the \\emph{physical level}%\n. However, while one can argue that topological encoding provides a stable\nand passive quantum memory, it is not self-correcting as in today's\n\\textquotedblleft effectively naturally fault-tolerant\\textquotedblright\\\nclassical architectures (see Ref.~\\cite{Bacon:05} for an eloquent exposition\nof this point). Moreover, it is important to realize that as far as we know,\nin its present state TQC still requires active intervention, in the form of\nFT-QEC, when one tries to compute fault-tolerantly. Indeed, Preskill writes\nin Ref.~\\cite{Preskill:TQC-notes}[p.62], \\textquotedblleft It is therefore\nimplicit that the temperature is small enough compared to the energy gap of\nthe model that thermally excited anyons are too rare to cause trouble, that\nthe anyons are kept far enough apart from one another that uncontrolled\nexchange of charge can be neglected, and in general that errors in the\ntopological quantum computation are unimportant. If the error rate is small\nbut not completely negligible, then the standard theory of quantum fault\ntolerance can be invoked to boost the accuracy of the simulation as\nneeded\\textquotedblright . Ref.~\\cite{Dennis:02} takes this approach and\nexplicitly lists \\textbf{A2} and \\textbf{A3} as necessary requirements for\nfault-tolerant TQC. In contrast, \\textbf{A1} is definitely \\emph{not}\nrequired in TQC:\\ one performs computations by adiabatically dragging\nquasiparticles around one another, and these operations must be slow\nrelative to the gap between the ground state and the first excited state.\nThe larger the gap the easier it is to satisfy this adiabaticity condition,\nso this requirements is compatible with the thermal suppression of errors\nmentioned above. In addition, TQC requires time-dependent controls to read\nout the encoded data (Ref.~\\cite{Freedman:00a} shows that all measurements\ncan be postponed until the readout of the final result of the computation).\nHowever, a fully Hamiltonian analysis of the fault-tolerance of such\nmeasurements is still lacking. Nevertheless, one could argue that the error\nrate in a topological quantum computer could be made arbitrarily small by\nincreasing the system size and careful engineering, so that (similarly to\ntoday's self-correcting, fault-tolerant classical computers), one could\nultimately perform TQC without any active intervention other than read-out\nof the encoded data. An interesting, recent development in this direction\nwas reported in Ref.~\\cite{Bacon:05}, which suggests that certain\nthree-dimensional quantum spin-lattices might be self-correcting.\n\n\\subsection{Non-Markovian Quantum Computing}\n\\label{nonM-FTQEC}\n\nWe keep \\textbf{A1} and \\textbf{A2} but discard \\textbf{A3} (the Markov\napproximation) at least in part. This appears to be a reasonable approach in\nmany cases, since the Markov approximation is clearly a highly idealized\nlimit (though it does hold remarkably well in some optical systems and in\nliquid state NMR). Indeed, the degree of accuracy to which the Markov\napproximation must be satisfied has been quantified, e.g., by Steane in \\cite%\n{Steane:04}: the probability of an uncorrectable (i.e., non-Markovian) error\nper gate must be $<$ $10^{-10}$ for a computation involving $10^{9}$ gates\n(this probability must scale with the input size, as explained in Section %\n\\ref{adiabatic-gates}). Alternative approaches to dealing with non-Markovian\nbaths are therefore of interest. For example, the papers \\cite%\n{Terhal:04,Aliferis:05,Aharonov:05} present an extension of FT-QEC theory to a\nnon-Markovian setting. We offer in this context the following observations:\n\n\\noindent 1. An important ingredient carried over directly and without\nchange from Markovian FT-QEC theory, is the crucial role of the fresh and\nnearly pure ancillas \\cite{Terhal:04,Aliferis:05,Aharonov:05}. We believe that the detailed mechanism\nfor introducing and discarding ancillas at specific times should be\nreconsidered within a fully Hamiltonian framework.\n\n\n\\noindent 2. As recognized and discussed in \\cite{Terhal:04}, the important\nassumption of a small norm of the system-bath interaction Hamiltonian (e.g.,\nEq.~(58) in Ref.~\\cite{Aliferis:05}) is not satisfied for some standard\nmodels of open systems. For example, a linear coupling to a bosonic heat\nbath involves unbounded interaction Hamiltonians and a high-frequency\ncutoff. In general, the assumption of a small norm of the system-bath\ninteraction Hamiltonian is much stricter than the WCL and is not satisfied\nfor most standard models of reservoirs.\n\nAnother approach to fault-tolerance in a non-Markovian setting is the\nrecently developed time-concatenated dynamical decoupling method \\cite%\n{KhodjastehLidar:04} (see also \\cite{Viola:02} for a version of dynamical\ndecoupling with bounded-strength controls). However, comment 2. above about\nthe small norm of the system-bath interaction Hamiltonian applies here as\nwell. Therefore more general methods are required to deal with the full\nscope of baths one can expect in quantum computing implementations. A\npromising possibility in this direction is to incorporate fault-tolerant\ndynamical decoupling in a feedback loop.\n\n\\section{Conclusions}\n\n\\label{conc}\n\nWe have listed a set of minimal assumptions made in the theory of\nfault-tolerant quantum error correction (FT-QEC): 1) fast gates (on the\ntimescale set by the inverse of the relevant Bohr or Rabi frequency), 2) a\nsupply of fresh and nearly pure ancillas, 3) a Markovian bath.\n\nWe have also reviewed the only two known rigorous general limits leading to\nMarkovian dynamics: the singular coupling limit (SCL), which involves taking\na high temperature limit, and the weak coupling limit (WCL), which requires\neither a constant or an adiabatic system Hamiltonian, and averaging over\nlong times in comparison with the inverse of the relevant Bohr frequency.\nThese two limits allow one to replace the reservoir autocorrelation function\nby a Dirac delta, which leads to the Markovian limit.\n\nA close examination of the assumptions of FT-QEC has led us to conclude that\nassumption 3 can be sustained together with assumption 1 in the SCL, and\ntogether with assumption 2 in the WCL. However, it is not possible to\nmaintain all three assumptions in either the SCL or the WCL. We therefore\nconclude that, at present, there exists an inconsistency in the formulation\nof the theory of FT-QEC for Markovian baths. We have also listed a number of\nalternatives to Markovian FT-QEC which, from the point of view adopted here,\nare free of inconsistencies. However, none of these alternatives is so\ncomprehensive as to include the full range of errors one might expect in a\nfull-scale implementation of quantum computing. In particular, recent\nresults on fault tolerance in non-Markovian settings \\cite%\n{Terhal:04,Aliferis:05,Aharonov:05,KhodjastehLidar:04}, while representing a\nsignificant step forward, make a crucial assumption about the smallness of\nthe norm of the system-bath interaction Hamiltonian, which severely\nrestricts the class of physical reservoirs.\n\n\\acknowledgments We thank Dave Bacon, Andrew Doherty, Daniel Gottesman, Hideo Mabuchi, John\nPreskill, Alireza Shabani, and Barbara Terhal for very useful discussions\n(though this does not imply their agreement with our conclusions). Their\ninsightful comments helped us sharpen our critique and formulate the\nquestions in Section~\\ref{objections}. \n\nR.A. thanks for the support from the Polish Ministry of Science and\nInformation Technology- grant PBZ-MIN-008\/P03\/2003 and the EC grant RESQ IST-2001-37559, D.A.L. thanks the Sloan\nFoundation for a Research Fellowship and the DARPA-QuIST\\ program for\nsupport. P.Z. acknowledges support by the European Union FET project TOPQIP\n(Contract No. IST-2001-39215).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe primary goal of this work is to compare radio sources in and out\nof the rich cluster environment to look for differences in either the\nhost galaxies or the distribution of sources in radio luminosity,\nsize, or morphology.  The surprising result that the amplitude and\nshape of the radio luminosity function (RLF) is identical inside and\noutside the cores of rich clusters \\cite{ledl96,fanti84}, is puzzling\ngiven what we know about the X-ray properties of galaxy clusters and\ngroups.  While poor clusters may simply be a lower-mass extension of\nrich clusters, given the differences in both ambient external density\nand local velocity dispersion, one might expect a significantly\ndifferent rate of galaxy interactions, a higher frequency of gas-rich\nneighbors, and a lower density medium to confine the radio sources.\nThe similarity of the RLF in and out of clusters suggests that the\nsame frequency of elliptical galaxies are radio emitting (also with a\nsimilar distribution in radio powers).  It does not, however, rule out\na different relation in either the radio power-size ($P,D$) plane or\nthe relationship between radio and optical luminosity and the FR I\/II\ndivision \\cite{deruiter90,ledl95b}.\n\nIn order to study these effects, we have put together the largest\navailable samples of radio sources to date, including 259 rich-cluster\nradio galaxies \\cite{ledl95b,ledl96} and 124 radio sources not found\nwithin 1 Abell radius ($\\sim 2h_{75}^{-1}$ Mpc) of an Abell cluster.\nThe non-cluster samples are based on the Bologna B2 Survey\n\\cite{fanti87} and a subset of the all-sky 2.7GHz survey from\n\\citeasnoun{wp}(WP).  We include here analysis of new optical (R-band)\nobservations of all 124 non-cluster radio sources, which were analyzed\nidentically to our rich cluster survey \\cite{ledl95b}.  We have also\nused the NRAO VLA D-Array Sky-Survey (NVSS) to measure source fluxes\n(and in some cases sizes) for the B2 and WP samples.  Radio sizes for\nthe rich-cluster radio galaxies are from \\citeasnoun{eilek}.\n\n\\section{Selection Effects}\n\\label{selection}\n\nBefore we can compare our two samples (cluster \\& non-cluster), we\nneed to explore the effects of the selection function on the various\nsamples.  The B2 sample has traditionally been divided into B2-Near\n\\cite{b2n} and B2-Far \\cite{b2f} subsamples, each with a different\nselection function.  The near-sample has a magnitude-limit of $m_{pg}<\n15.7$ ($m_{24.5}^R \\lesssim 14$).  The far-sample has a magnitude\ncutoff of $m_v=16.5$ ($m_{24.5}^R \\lesssim 15.9$).  Both subsamples\nessentially have the same radio flux-limit of $P_{408MHz}=200$ mJy\n($\\approx 100$ mJy assuming $F_{\\nu} \\propto \\nu^{-0.7}$).  The WP\nsample has no magnitude cutoff, but has a high flux-limit of\n$P_{2.7GHz}=2$ Jy ($P_{1400MHz} \\geq 3.2$ Jy, also for $\\alpha=0.7$).\n\nThe effect of these selection functions is illustrated in Fig.\n\\ref{selectioneffects} where we have shown the 75th percentile values\nof $M_{24.5}^R$ and $P_{1400}$ for each of the samples (75\\% of the\nobjects are to the right of the vertical lines and above the\nhorizontal lines).  Because of the magnitude-limits of the B2 sample,\nthe part of the optical\/radio luminosity plane fainter than $M^{*}_R =\n-22$ and $P_{1400}<10^{24}$ is essentially unsampled.  This net effect\nof these selection functions is to preferentially pick out FR I radio\nsources with optical luminosities $\\gg L^*$.  {\\bf It is for these\n  reasons that FR I's have mistakingly been identified with only the\n  brighter ellipticals, consistent with brightest cluster galaxies\n  (BCG)}.  It is clear from Fig. \\ref{selectioneffects} that FR I's\nactually span a range of more than a factor of 40 in optical\nluminosity, and trace the normal optical luminosity function for\nelliptical galaxies \\cite{ledl95b}.  Additionally, these same\nselections explain why FR II's have always been associated with $M^*$\nellipticals.  Given the steep slope of both the optical and radio\nluminosity functions for $M<-23$ and $P_{1400}>10^{25}$ and the small\nsearch volumes, all detected FR II's are likely to cluster near the\nsurvey-limits in optical luminosity (near $M^*$). It should be noted,\nhowever, that it is difficult to separate out the effects of redshift\nevolution with a limited search volume.\n\n\\begin{figure}\n\\begin{center}\n  \\includegraphics*[width=10cm]{lcfig1.eps}\n\\end{center}\n\\caption{FR I\/II diagram in the optical\/radio luminosity plane. The dotted\n  lines show the 75th percentile values of $M_{24.5}$ and $P_{20cm}$\n  for the various samples. The symbols are: FR I=1's, FR II=2's, and\n  Fat-Double=F. The solid-line indicates the fiducial FR I\/II break.}\n\\label{selectioneffects}\n\\end{figure}\n\n\\section{Rich Cluster vs. Non Rich-Cluster Radio Galaxies}\n\nIn order to compare our rich cluster radio sources to these other\nsamples, we have therefore applied the B2 selection function (both\n$M_{opt}$ and $P_{20cm}$ constraints) to our rich cluster survey,\nincluding an upper-redshift cutoff of z=0.24 to better match the two\nsamples.  We have compared a number of parameters from both the\noptical analysis as well as the radio properties between these two\nsamples.  See \\citeasnoun{ledl2000} for a complete presentation of\nthese results.\n\nFor the optical properties, we compared the distribution of\nellipticities, surface-brightness profile shape (via a power-law\nexponent and $r^{1\/4}$-law fits), departure from elliptical isophotes\n(A4,B4 parameters), the Kormendy relation ($\\mu_e$ vs. $r_e$ slope),\nand the luminosity-size ($M_{24.5}$ vs.  $r_{24.5}$) relationships for\nboth the cluster and non-cluster samples.  Measured at the same\noptical luminosity ($M_{24.5}$), we find no discernable differences\nbetween the two samples.\n\nWe also compared the optical properties of the FR I and FR II host\ngalaxies (combining cluster and non-cluster sources together). From\nthe median of the optical magnitude distributions, we find:\n$M_{24.5}(FR I)=-22.85\\pm0.04$ and $M_{24.5}(FR II)=-22.58 \\pm0.10$.\nThus, the FR I's are found in only slightly brighter galaxies, and\nsignificantly less luminous on average than BCG's ($M_{24.5} <\n-23.5$).  Additionally, we found no differences in the host galaxy\nproperties listed above between the FR I and FR II hosts, at least in\ntheir broadband photometric properties.\n\nOur radio comparison between cluster and non-cluster sources included\nthe size-distributions (at the same radio power), the distribution in\nthe radio\/optical luminosity plane, and the FR I\/II division.  We find\nno evidence for differences in either of these properties between the\ntwo samples. We do however note the following observations: 1) The\nsize-distribution is somewhat broader outside of clusters, and 2) We\ndo not see a population of FR II's with $P_{20cm} \\gtrsim 25.5$ in the\nrich cluster sample. However, these differences are minor, and argue\nfor very similar envirnoments (at least the aspect of the environment\nwhich most influences the evolution of radio sources) inside\/outside\nthe cores of rich clusters.  All three samples are plotted together in\nFig.\\ref{selectioneffects} and Fig.  \\ref{pdplots}.\n\n\\section{The (P,D) Diagram} \n\nGiven that we find no significant differences in our\ncluster\/non-cluster samples, we have combined them in order to improve\nthe statistics.  In Fig. \\ref{pdplots}, we show the radio-power\/size\n($P,D$) distributions in several ways.  In the bottom diagram we\nseparate the three samples by point-type to show their intrinsic\ndistributions.  In the middle plot we show the sample divided into FR\nI, FR II, and Fat-double classes.  For the FR I's, we find a fit\n(log-log) of the form: $Size (FR I) \\propto P_{20cm}^{0.31\\pm0.03}$\n(ignoring unresolved sources with only size upper-limits).  For the FR\nII's, we find essentially no dependence between power and size ($Size\n(FR II) \\propto P_{20cm}^{0.08\\pm0.09}$).  Note, however, that at the\nsame radio power, both FR I and FR II sources have the same\ndistribution in source-sizes.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[height=12cm]{lcfig2.eps}\n\\end{center}\n\\caption{(middle) ($P,D$) diagram for combined cluster\/non-cluster samples with a \n  cutoff of z=0.24.  The solid-lines show the fit for the FR I's with\n  and without the size upper- limits for the unresolved sources.  The\n  dashed-line is the fit for the FR II's. (bottom) Same as middle,\n  where the point types reflect the sample from which the sources were\n  taken (see legend on plot). (top) Median values of the radio size in\n  radio power bins as a function of optical magnitude ($\\Delta M$=0.75\n  mag); centered on 1=-21.38, 2=-22.13, 3=-22.88, 4=-23.63, 5=-24.38.}\n\\label{pdplots}\n\\end{figure}\n\nThe top plot in Fig. \\ref{pdplots} shows the median size as a function\nof radio power and optical luminosity (lower numbers represent bins of\nlower optical luminosity).  One sees from the plot that the trend is\nfor optically fainter galaxies to have larger radio sources. In bins\nof constant optical magnitude, we find that for fits of the form:\n$Size \\propto P_{20cm}^x$, x decreases with increasing $L_{opt}$ (from\n0.36-0.29).  We also find that the size distributions for $M\\gtrsim\nM^*$ and $M\\lesssim -23$ are different at the 98\\% level.  It thus\nappears that optically brighter galaxies tend to have smaller sizes\nwhen measured at the same radio power.  Is this result a consequence\nof environment (the brighter galaxies being in a higher-density\nenvironment or near the bottom of the local gravitational potential)\nor initial conditions (the physics near the black hole and the jet\ninitial conditions)?  We plan to address these issues by comparing our\nobserved distributions in ($M_{opt},P_{20cm},Size$) with model\npredictions.  It is clear, however, that we may need a\nnew\/more-detailed model for FR I sources (see \\citeasnoun{eileklc}\nthese proceedings) in order to understand these results.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe study of cold quark matter has been an interesting topic of research in recent years. In an astrophysical\ncontext,  quark matter may be used to explain the measureable properties of pulsar-like compact stars.\n\nIt has recently been proposed that realistic quark matter in compact stars could exist in a solid state\n~\\cite{xu03,Horvath05}, either as a super-solid or as a normal solid~\\cite{xu09}. The basic conjecture of normal\nsolid quark matter is that de-confined quarks tend to form quark-clusters when the temperature and density are\nrelatively low. Below some critical temperature, these clusters could be in periodic lattices immersed in a\ndegenerate, extremely relativistic, electron gas. Note that even though quark matter is usually described as\nweakly coupled, the interaction between quarks and gluons in a quark-gluon plasma is still very\nstrong~\\cite{Shuryak}. It is this strong coupling that could cause the quarks to cluster and form a solid-like\nmaterial.\n\nIn this paper, we argue that comtemporary observations of thermal X-ray emitting pulsars are consistent with the\nassumption that these sources are in fact Solid Quark Stars (SQSs).\nIn the successive Sect., the thermal X-ray observations on isolated pulsars are exhibited according to the\ncollation in this work.\nIn Sect. 3, the SQS pulsar model is explored to interpret these observations.\nConclusions and corresponding discussions are presented in Sect. 4\n\n\\section{The observations}\n\\begin{figure\n\\begin{center}\n\\epsfig{file=f5.eps,height=4.8in} \\caption{Functions $L_{\\rm bol}^{\\infty}(\\dot{E})$ ({\\it top} panel) and\n$t(\\dot{E})$ ({\\it bottom} panel) of active pulsar candidates. In the {\\it top} panel, the fits are carried out\nboth for Group A (including top 17 pulsars listed in Table 1 in \\cite{yx09}; they are marked by dark points and\ntheir numbers) and Group B (including the members in Group A and the other 10 sources whose upper limits on the\nbolometric luminosities could be defined observationally; the 10 sources are marked by grey points and letters).\nRed lines are the fitted results for Group A, while blue lines are those for Group B. In both groups, solid\nlines provide the best fits to the data, while the dashed lines give the fits by freezing $p1$ at 1. We note\nthat the 10 sources with upper limits on their bolometric luminosities defined are taken from ~\\cite{BA02}, and\nthey are a. B1509-58, b. B1951+32, c. B1046-58, d. B1259-63, e. B1800-21, f. B1929+10, g. B0540-69, h. B0950+08,\ni. B0355+54, j. B0823+26.}\\label{fig:edot}\n\\end{center}\n\\end{figure}\n\\begin{table}\n{\\scriptsize\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline \\\\\nGroup & $p1$$^a$ & $p2$ & Corr. Coef.$^b$ & $\\chi_r^2$ (d.o.f)$^c$ \\\\\n\\hline \\\\\nA. & $0.4561\\pm0.2315$ & $16.20\\pm8.30$ & -- & 0.3277(14) \\\\\n & (1) & $-3.280^{+0.495}_{-0.494}$ & 0.7487 & 0.8607(15) \\\\\nB. & $0.5918^{+0.2045}_{-0.2046}$ & $11.66^{+7.3100}_{-7.2990}$ &\n-- & 0.6081(24) \\\\\n & (1) & $-2.896\\pm0.403$ & 0.7730 & 0.9964(25) \\\\\n\\hline \\\\\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\n$^a$ Values in the parenthesis are frozen during the fits. The errors are in 95\\% confidence level.\n\\newline\n$^b$ Correlation coefficient of the data between bolometric luminosity and $\\dot{E}$.\\newline\n$^c$ Reduced $\\chi^2$, or $\\chi^2$ per degree of freedom (d.o.f).\n\\caption{Fitting parameters for $L_{\\rm bol}^{\\infty}$---$\\dot{E}$ data, which has been shown in Fig.\n\\ref{fig:edot} ({\\it top} panel).}\\label{tab:fit}\n\\end{flushleft}}\n\\end{table}\n\nWe collate the thermal X-ray observations on 29 isolated pulsars (see Table 1 in~\\cite{yx09}). The black body\nparameters summarized comprise of pulsar surface temperature components $T_{\\rm\ns,1\/2}^{\\infty}$\\footnote{``$\\infty$''---values measured at distant.}, emission size components\n$R_{1\/2}^{\\infty}$, ages and bolometric X-ray luminosities $L_{\\rm bol}^{\\infty}$.\nSpin would be a significant source of energy for pulsars. Potential relation between pulsar X-ray\nbolometric luminosity $L_{\\rm bol}^{\\infty}$ and its spin energy loss rate $\\dot{E}$ could exist. Such a study\nwas done and the results are exhibited in Fig. \\ref{fig:edot} and Table \\ref{tab:fit}.\nUsing the function of\n\\begin{equation}\n{\\rm log}L_{\\rm bol}^{\\infty}=p1\\cdot {\\rm log}\\dot{E}+p2, \\label{eq:fit}\n\\end{equation}\n$L_{\\rm bol}^{\\infty}-\\dot{E}$ data were fitted and the results show that the best fit demonstrates a possible 1\/2-law between\nthese two quantities, or\n\\begin{equation}\nL_{\\rm bol}^{\\infty}(\\dot{E})=C\\dot{E}^{1\/2}, \\label{eq:onesecond}\n\\end{equation}\nwhere the coefficient $C=10^{p2}$ is in the unit of ${\\rm erg^{1\/2} s^{-1}}$.\nHowever, when the parameter $p1$ is set to 1, a linear-law of\n\\begin{equation}\nL_{\\rm bol}^{\\infty}(\\dot{E})=\\eta\\cdot\\dot{E}, \\label{eq:linear}\n\\end{equation}\nis presented, and this coefficient $\\eta$ is $10^{-3}$ approximately.\n\n\\section{The SQS pulsar model}\n\\subsection{X-ray cooling and accreting pulsars}\nThe state of a pulsar would be manifested in multiple wave bands and by different photon components. Top 17\npulsars have significant observable X-ray non-thermal components, which could originate from energetic\nmagnetospheres. Such pulsars are considered to be undergoing cooling processes as proposing by SQSs.\nThe rest sources are dominated by thermal X-ray components observationally, and show little magnetospheric\nactivities. These pulsars may be accreting to the surrounding interstellar medium to generate visible X-ray\nluminosities, if they are in fact SQSs.\n\\subsection{Cooling of SQSs}\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=f3.eps,height=2.0in} \\caption{{\\it Left} panel: Cooling curves for SQSs, if the 1\/2-law between the\nbolometric luminosity and the spin energy loss rate holds. {\\it Right} panel: Corresponding temperature\ndifferences between the hot and warm components of SQSs, or $\\Delta T=T^{\\infty}_{\\rm p}-T^{\\infty}_{\\rm s}$.\nThe parameters in both panels: $M=0.1{\\rm M_\\odot}$, $C=10^{16}$ (solid lines); $M=1.0{\\rm M_\\odot}$,\n$C=10^{16}$ (dashed lines); $M=1.0{\\rm M_\\odot}$, $C=10^{15}$ (dotted lines); $M=0.01{\\rm M_\\odot}$, $C=10^{15}$\n(dash-dot lines) ($C$ is in the unit of erg$^{1\/2}$ s$^{-1\/2}$). For two curves with same $M$ and $C$, the upper\none corresponds to an initial spin of 10 ms, while that of the lower one is 100 ms. Noting that the errors on\nthe surface temperatures of PSRs J0205+6449 (No. 3) and J2043+2740 (No. 17) are not provided by the authors of\nthe references, we then conservatively adopt them as deviating from the central values by a factor of\n2.}\\label{fig:ccos}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=f4.eps,height=2.0in} \\caption{{\\it Left} panel: Cooling curves for SQSs, if the linear-law holds.\n{\\it Right} panel: Temperature differences for this case. The parameters: $M=1.0{\\rm M_\\odot}$, $\\eta=0.01$\n(solid lines); $M=1.0{\\rm M_\\odot}$, $\\eta=0.001$ (dashed lines); $M=0.1{\\rm M_\\odot}$, $\\eta=0.1$ (dash-dot\nlines); $M=0.01{\\rm M_\\odot}$, $\\eta=0.1$ (dotted lines). As in Fig. \\ref{fig:ccos}, for two curves with same\n$M$ and $\\eta$, the upper one corresponds to an initial spin of 10 ms, while the lower one 100\nms.}\\label{fig:cclr}\n\\end{center}\n\\end{figure}\nCooling of a quark star, from the initial birth, can be quite complicated to model.\nTwo facts induce this result. Firstly, recent experiments on collisions between heavy relativistic ions (e.g.\nRHIC) manifested that the potential quark-gluon plasma in the fire ball would be strongly coupled even in the\ntemperatures of a few factors of the critical temperature $\\sim$170 MeV~\\cite{Shuryak}. However, and secondly,\nthe initial temperature at the birth of a pulsar could only be a few factors of $\\sim$ 10 MeV. Hence, if the\nnatal pulsar is a quark star, the stellar interior quark-gluon plasma would be coupled strongly, so that current\nphysics could not model either in an analytic or a\nnumerical way.\n\nNevertheless, a phenomenological study on this problem could be carried out, and Fig. 1 in~\\cite{yx09}\nillustrates proposed cooling processes for a quark star. It could be quite uncertain that at which temperatures\nthe phase transitions would occur, and how long the star would stay in Stages 1 and 2.\nHowever, in consequence of the strong coupling in quark-gluon plasma, Stages 1 and 2 would not be long, and the\nobserved pulsars (if they are quark stars) would be in the solid phase (Stage 3), because of the measured low\ntemperatures.\nThe calculations below concentrate on the late Stage 3, where the surface temperature is in the range of the\nobserved $10^7$---$10^5$ K. The loss of the thermal energy of a SQS in the late 3rd stage could dominantly induced by the emission of\nphotons rather than neutrinos, as a result of the low neutrino emissivities in such low temperatures.\n\nIn general, the cooling process would be described by the equation of\n\\begin{equation}\n-C_v\\frac{{\\rm d}T_{\\rm s}}{{\\rm d}t}+L_{\\rm SH}= L_{\\rm bol}, \\label{eq:coolinggeneral}\n\\end{equation}\nwith assuming that the stellar volume is invariant and the internal energy of the star is only the function of\nthe stellar temperature $T_{\\rm s}$. $C_v$ is the stellar heat capacity. $L_{\\rm SH}$ is the luminosity of the\nprobable heating process, and $L_{\\rm bol}$ is the photon luminosity of the star.\nThe stellar heat capacity was found to be tiny, so that the internal energy would be far from sustaining\nlong-term X-ray emission. We note here that for a $1.4{\\rm M_{\\odot}}$-SQS, it would cool down to 1 K from\n$10^{11}$ K in $\\sim$0.2 s. The detailed exhibition of the calculation of this stellar residual internal energy\nas well as heat capacity can be found in Appendix A.\n\nHeating mechanism would take important effect during the cooling of SQSs. In such a way, the thermal evolution\nof a SQS could not be a self-sustaining cooling process, but an external-supplying heating process.\nThe bombardment by backflowing particles at the poles could be the prime heating mechanism for SQSs, which would\nintrinsically be spin-powered.\nCombining the negligible residual internal energy of a SQS and the observed $L_{\\rm bol}^{\\infty}$---$\\dot{E}$\nrelations. The heating luminosity $L_{\\rm SH}$ could also follow that 1\/2-law or linear-law. The equation\n(\\ref{eq:coolinggeneral}) would hence yields\n\\begin{equation}\nC\\dot{E}^{1\/2}=4\\pi R^2\\sigma T_{\\rm s}^{4}+4\\pi r_{\\rm p}^2\\sigma T_{\\rm p}^4, \\label{eq:coolingactiveos}\n\\end{equation}\nor\n\\begin{equation}\n\\eta\\dot{E}=4\\pi R^2\\sigma T_{\\rm s}^{4}+4\\pi r_{\\rm p}^2\\sigma T_{\\rm p}^4. \\label{eq:coolingactivelr}\n\\end{equation}\nThe two terms on the right side of the equal sign are the stellar primary black body emission component and that\nfor a probable hot pole component respectively. The relation between $T_{\\rm p}$ and $T_{\\rm s}$ could be found\nby\n\\begin{equation}\nH\\sim \\frac{L_{\\rm SH}-4\\pi r_{\\rm p}^{2}\\sigma T_{\\rm p}^{4}}{\\pi r_{\\rm p}^{2}}\\sim \\kappa_{\\rm e}\\frac{T_{\\rm\np}-T_{\\rm s}}{R}, \\label{eq:H}\n\\end{equation}\nwhere $H$ is the heat current flowing from the poles to the bulk of the star. The stellar heat conductivity\n$\\kappa_{\\rm e}$ mainly contributed by free electrons refers to the corresponding results for solid\nmetal~\\cite{FI81}.\nFigs. \\ref{fig:ccos} and \\ref{fig:cclr} exhibit the computed cooling curves for the above two cases\nrespectively. In these calculations, SQSs were assumed to slow down as orthogonal rotators that lose their spin\nenergy by magnetic dipole radiation. The average densities of the SQSs were adopted to be 3 times nuclear\nsaturation density. Note that the results will not vary significantly when the density values are in the range of 3--5 nuclear saturation density, which is thought to be representative values for the solid quark matter.\nThe spin-powered thermal emission of SQSs enable an estimate on pulsar moments of inertia~\\cite{yx09}.\n\\subsection{Accreting SQSs}\nDim X-ray isolated neutron stars (XDINs) and compact central objects (CCOs) (Sources No. 18-29 and 4) are\nthought to be inactive pulsar candidates with inert magnetospheric manifestations. These sources may have\nexcessive X-ray luminosities to their spin energy loss rates.\nTheir energy origin has been keeping disputable, but accretion origin would be a candidate. The relative high\nmagnetic fields of these sources may drive their surrounding shells of matter (in large radii, usually larger\nthan the corotation radii) to rotate together with them. Under this {\\it propeller} regime, nearly all of the\nmatter would be expelled outward, but a small fraction may diffuse starward and fall onto the stellar surfaces\neventually. The release of the gravitational energy as well as the latent heat inducing by the burning from the\nbaryonic matter phase to the three-flavor quark matter phase could sustain long-term soft X-ray radiations of\nsuch pulsars.\nIn this scenario, the stellar luminosity would then be\n\\begin{equation}\nL_{\\rm bol}=\\frac{GM\\cdot \\eta_{\\rm acc}\\cdot\\dot{M}}{R}+\\Delta\\varepsilon\\frac{\\eta_{\\rm\nacc}\\cdot\\dot{M}}{m_{\\rm p}}. \\label{eq:coolingdead}\n\\end{equation}\nFor the calculated accreting parameters for SQSs in this {\\it propeller} regime and the resulting luminosities,\nplease see \\cite{yx09}.\nNote here that for No. 4 RX J0822-4300, a CCO, we treated it both as a cooling SQS and a accreting SQS, as the\nresult of the disputable structure around it~\\cite{ZTP99,HB06}.\n\n\\section{Conclusions}\nWe collate the thermal observations of 29 X-ray isolated pulsars and, in the SQS regime, for the\nmagnetospherically active pulsar candidates, establish their cooling processes (Figs. \\ref{fig:ccos} and\n\\ref{fig:cclr}), while for the magnetospherically inactive or dead pulsar candidates, interpret the X-ray\nluminosities under the accretion scenario.\nSQSs, because of the possibility of being low-mass, could provide an approach to understand the small black body\nemission sizes. We note that for SQSs with mass of $0.01{\\rm M_\\odot}$, ${\\rm 0.1{\\rm M_\\odot}}$ and $1{\\rm\nM_\\odot}$, their radii are $\\sim$1.8, $\\sim$3.8 and $\\sim$8.3 km respectively.\nOn the other hand, a linkage between pulsar rotational kinetic energy loss rates and bolometric X-ray\nluminosities is explored by SQSs.\n\nWe hence conclude that the phenomenological SQS pulsar model could not be ruled out by the thermal observations\non X-ray isolated pulsars, though a full depiction on the formation and the thermal evolution for a quark star\nin all stages could hardly be given nowadays, as the lack of the physics in some extreme conditions (as has been\ndiscussed in Sect. 3.2).\nSQSs have significant distinguishable interiors with neutron stars, but the structures of the magnetosphere\nbetween these two pulsar models could be similar. Therefore, SQSs and neutron stars would have similar heating\nmechanisms, such as the bombardment by backflowing particles and the accretion to surrounding medium. However, a\nfull comparison between SQSs and neutron stars including their cooling processes as well as heating mechanisms\nhas beyond the scope of this paper, and should be interesting and necessary in the future study.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAlthough General Relativity (GR) provides very successful solutions, observations and predictions at the intermediate regimes, it fails to be a complete theory at both large (IR) and small (UV) scales. In the IR regime, GR does not give explanations to the accelerating expansion of the universe and rotational speed of galaxies without assuming a tremendous amount of dark energy and dark matter compared to the ordinary matter. As for small distances at the quantum level, it is a non-renormalizable theory according to perturbative quantum field theory perspective because of the infinities appearing in a renormalization procedure. These infinities coming from the self-interactions of gravitons (in the pure gravity case) cannot be regulated by a redefinition of finite numbers of parameters. GR has also black hole or cosmological type singularities at the classical level. The GR is expected to be modified at both regimes in order to have a complete theory. Here, the main question is what kind of modification in the UV will provide a complete model which may also solve cosmological or black hole singularity problems. In this respect, a possible way out of this problem was to add scalar higher order curvature terms to Einstein's theory such as the quadratic theory\n\\begin{equation}\nI=\\int d^4x(\\sigma R+\\alpha R^2+\\beta R_{\\mu\\nu}^2),\n\\end{equation}\nwhich describes massive and massless spin-$2$ gravitons together with a massless spin-$0$ particle \\cite{stelle}. By adding higher curvature terms, renormalizability is gained, but the unitarity (ghost and tachyon-free) of the theory is lost due to a conflict between the massless and massive spin-$2$ excitations. In other words, the theory has Ostragradsky type instabilities at the classical level which become ghosts at the quantum theory. Theory has an unbounded Hamiltonian density from below. That is to say, the addition of higher powers of curvature causes a conflict between the unitarity and the renormalizability. \n\nOn the other hand, it has been recently demonstrated that infinite derivative gravity (IDG) \\cite{Biswas1,Biswas2} has the potential to have a complete theory in the UV scale\\footnote{For recent developments on IDG, see \\cite{Tomboulis,Biswas4, Modesto,Biswas5,Biswas6,Buoninfante:2018xiw,Modesto1,Modesto2,Modesto22,Modesto3, Talaganis:2016ovm,Edholm:2016hbt,Modesto4,Boos:2018bxf,Moffat1,Moffat2,Briscese:2012ys}.}. IDG is described by an action constructed from non-local functions $F_i(\\Box)$ [given in Eq.(\\ref{idfunc})], where $\\Box$ is the d'Alembartian operator ($\\Box=g^{\\mu\\nu}\\nabla_\\mu\\nabla_\\nu$). The propagator of the IDG in a flat background in $3+1$ dimensions \n\\begin{equation}\n\\Pi_{IDG}=\\frac{P^2}{a(k^2)}-\\frac{P_s^0}{2a(k^2)}=\\frac{\\Pi_{GR}}{a(k^2)},\n\\end{equation}\nis given in terms of Barnes-Rivers spin projection operators ($P^2,P_s^0$) \\cite{Biswas1}. Here $a$ is given in terms of $F_i(\\Box)$ [see Eq.(\\ref{relations})] and $\\Pi_{GR}$ is the pure GR graviton propagator. One of the important points is to avoid introducing ghost-like instabilities and having additional scalar degrees of freedom other than the massless spin-$2$ graviton. To do this, $a(k^2)$ can be chosen to be an exponential of an entire function as $a(k^2)=e^{\\gamma(\\frac{k^2}{M^2})}$, where $\\gamma(\\frac{k^2}{M^2})$ is an entire function. This choice guarantees that the propagator has no additional poles other than massless graviton, in other words, $a(k^2)$ has no roots. In the $a(k^2)\\to 0$ or $k\\ll M$ limit, the propagator takes the usual Einsteinian form. Furthermore, as the propagator does not have any extra degrees of freedom, the modified propagator is free from ghost-like instabilities. The Hamiltonian density is bounded from below. Moreover, in \\cite{Talaganis}, it has been recently shown that loop divergences beyond one-loop may be handled by introducing some form factors. Furthermore, infinite derivative extension of GR may resolve the problem of singularities in black holes and cosmology \\cite{Biswas1, Biswas2,Tomboulis,Biswas4, Modesto,Biswas5,Biswas6,Buoninfante:2018xiw}.  \n\nIn this work, we would like to explore the weak field limit of the IDG and compare it with the result of GR. In \\cite{Biswas1}, the Newtonian potential for the point source was calculated for the IDG, here we extend this discussion to include the spin, velocities and orbital motion of the sources. By spin, we mean the rotation of the sources about their own axes. Therefore we calculate the spin-spin and spin-orbit interactions between two massive sources in IDG and show that the mass-mass interaction, the spin-spin interaction and the spin-orbit interaction part become non-singular as $r\\to 0$. These non-singular results in IDG show that the theory has improved behavior in the small scale compared to GR. \n\nThe layout of the paper is as follows: In Sec. II, we investigate the spin-spin interactions of localized point-like spinning massive objects in IDG and consider the large and small distance limits of potential energy. Section III is devoted to extend the calculations in the previous section to the case that the massive spinning  sources are also moving. In that section, in addition to mass-mass and spin-spin interactions, we studied the spin-orbit interactions in IDG. In conclusions and further discussions, we give the final result for a gravitational memory effect in IDG and discuss the effects of mass scale of non-locality on memory. In the Appendix, we give some of the details of calculations for Sec. III. \n\n\\section{Scattering Amplitude in IDG}\nThe matter coupled Lagrangian density of IDG is \\cite{Biswas1}\n\\begin{equation}\n\\mathcal{L}= \\sqrt{-g}\\bigg[\\frac{M^2_P}{2}  R\\ +\\frac{1}{2} R F_1 (\\Box) R + \\frac{1}{2} R_{\\mu\\nu} F_2(\\Box) R^{\\mu\\nu}\n       + \\frac{1}{2}C_{\\mu\\nu\\rho\\sigma} F_3(\\Box) C^{\\mu\\nu\\rho\\sigma}+\\mathcal{L}_{matter}\\bigg],\n\\end{equation}\nwhere $M_P$ is the Planck mass, $R$ is the scalar curvature, $R_{\\mu\\nu}$ is the Ricci tensor and $C_{\\mu\\nu\\rho\\sigma}$ is the Weyl tensor. The infinite derivative functions $F_i(\\Box)$ are given as\n\\begin{equation}\nF_i(\\Box)=\\sum_{n=1}^{\\infty}f_{i_n}\\frac{\\Box^n}{M^{2n}}, \n\\label{idfunc}\n\\end{equation} \nwhich are functions of the d'Alembartian operator. Here, $f_{i_n}$ are dimensionless coefficients and $M$ is the mass scale of non-locality. The linearized field equations around a Minkowski background of $g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$ reads\\footnote{We will work with the mostly plus signature $\\eta_{\\mu\\nu}={\\it{diag}}(-1,1,1,1)$.} \\cite{Biswas1}\n\\begin{equation}\na(\\Box)R^L_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}c(\\Box)R^L-\\frac{1}{2}f(\\Box)\\partial_\\mu\\partial_\\nu R^L=\\kappa T_{\\mu\\nu},\n\\label{FeqIDG}\n\\end{equation}\nwhere $L$ refers to linearization and non-linear functions are defined as\n\\begin{equation}\n\\begin{aligned}\n &a(\\Box) =1 + M^{-2}_P \\left(F_2(\\Box) \n        + 2  F_3(\\Box)\\right) \\Box, \\\\\n&        c(\\Box) = 1 - M^{-2}_P\\left(4 F_1(\\Box) +  F_2(\\Box)  \n        - \\frac{2}{3}F_3(\\Box)\\right)\\Box,\\\\\n&        f(\\Box) =M^{-2}_P \\left(4F_1(\\Box) + 2F_2(\\Box) +\\frac{4}{3} F_3(\\Box)\\right),\n        \\end{aligned}\n        \\label{relations}\n\\end{equation}\nwhich give the constraint $a(\\Box)-c(\\Box) = f(\\Box)\\Box$. After plugging the relevant linearized curvature tensors \\cite{deser_tekin} into (\\ref{FeqIDG}), one arrives at the linearized field equations\n\\begin{equation}\n\\begin{aligned}\n  &\\frac{1}{2} \\bigg[a(\\Box)\\left(\\Box h_{\\mu\\nu} \n        -\\partial_\\sigma\n        \\left(\\partial_\\mu h^\\sigma_\\nu + \\partial_\\nu h^\\sigma_\\mu \\right)\\right)\n        + c(\\Box) \\left(\\partial_\\mu \\partial_\\nu h + \\eta_{\\mu\\nu} \\partial_\\sigma \\partial_\\rho \n        h^{\\sigma\\rho}-\\eta_{\\mu\\nu} \\Box h\\right)\\\\&+ f(\\Box) \\partial_\\mu \\partial_\\nu \n \\partial_\\sigma \\partial_\\rho \nh^{\\sigma\\rho}\\bigg]=-\\kappa T_{\\mu\\nu} .\n\\label{IDGfeq1}\n\\end{aligned}\n\\end{equation}\n If we set $a(\\Box)=c(\\Box)$, we recover the pure GR propagator in the large distance limit without introducing additional degrees of freedom. Then, in the de Donder gauge $\\partial_\\mu h^{\\mu\\nu} = \\frac{1}{2} \\partial^\\nu h$, the linearized field equations (\\ref{IDGfeq1}) take the following compact form\n\\begin{equation}\n  a(\\Box){\\cal{G}}^L_{\\mu\\nu}=\\kappa T_{\\mu\\nu} , \n  \\label{IDGfeq2}\n\\end{equation}\nwhere ${\\cal{G}}^L_{\\mu\\nu}$ is the linearized Einstein tensor defined as ${\\cal{G}}^L_{\\mu\\nu}=-\\frac{1}{2}(\\Box h_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}\\Box h)$. Manipulation of (\\ref{IDGfeq2}) yields\n \\begin{equation}\n  a(\\Box) \\Box h_{\\mu\\nu}=-2\\kappa(T_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}T) ,\n  \\label{IDGfeq3}\n\\end{equation}\nwhich is the equation that we shall work with.\n\nFrom now on, we consider the tree-level scattering amplitude between two spinning conserved point-like sources and find the corresponding weak field potential energy. To do that, one needs to first eliminate the non-physical degrees of freedom from the theory. For this purpose, let us consider the following decomposition of the spin-$2$ field\n\\begin{equation}\n  h_{\\mu\\nu} \\equiv  h^{TT}_{\\mu\\nu}+\\bar{\\nabla}_{(\\mu}V_{\\nu)}+\\bar{\\nabla}_\\mu \\bar{\\nabla}_\\nu \\phi+\\bar{g}_{\\mu\\nu} \\psi,\n  \\label{dectth}\n \\end{equation}\n where $ h^{TT}_{\\mu\\nu} $ is the transverse-traceless part of the field, $ V_\\mu $ is the transverse helicity-$1$ mode and $ \\phi $ and $ \\psi $ are scalar helicity-$0$ components of the field. To obtain  $\\psi $ in terms of  field $h$, one needs to take the trace  and double divergence of (\\ref{dectth}) to arrive at\n\\begin{equation}\n h=\\partial^2\\phi+4 \\psi, \\hskip 1 cm \\frac{1}{2}\\partial^2h=\\partial^4\\phi+\\partial^2\\psi,\n \\label{scaleinsstre}\n\\end{equation}\nwhere we used $\\partial^\\mu\\partial^\\nu h_{\\mu\\nu}=\\frac{1}{2}\\partial^2h$. Then, by using (\\ref{scaleinsstre}) and (\\ref{IDGfeq2}), one obtains\n \\begin{equation}\n  \\psi = \\frac{\\kappa}{3}(a(\\Box)\\partial^2)^{-1}T.\n  \\label{psittt}\n \\end{equation}\nOn the other hand, inserting (\\ref{dectth}) into (\\ref{IDGfeq2}) yields the wave-type equation\n \\begin{equation}\n    h^{TT}_{\\rho\\nu}= -2\\kappa\\,{\\cal O}^{-1} T^{TT}_{\\rho\\nu},\n \\label{htt}\n \\end{equation}\nwhere the corresponding scalar Green's function is\n\\begin{equation}\n  G ({\\bf x},{\\bf x}^{'},t,t^{'})={\\cal O}^{-1} \\equiv (a(\\Box)\\partial^2)^{-1}.\n\\end{equation}\nAccordingly, the tensor decomposition of energy momentum tensor  $T_{\\rho\\nu} $ can be given as \\cite{Gullu:2009vy}\n\\begin{equation}\n T^{TT}_{\\rho\\nu}=T_{\\rho\\nu}-\\frac{1}{3} \\bar{g}_{\\rho\\nu}T+\\frac{1}{3} \\Big (\\bar{\\nabla}_\\rho \\bar{\\nabla}_\\nu  \\Big ) \\times (\\bar{\\square} )^{-1} T.\n\\label{ttstrener}\n \\end{equation}\nRecall that the tree-level scattering amplitude between two  sources via one graviton exchange is given by\n\\begin{equation}\n\\begin{aligned}\n {\\cal A}&=\\frac{1}{4} \\int d^4 x \\sqrt{-\\bar{g}} T^{'}_{\\rho\\nu}(x) h^{\\rho\\nu}(x) \\\\\n &=\\frac{1}{4} \\int d^4 x \\sqrt{-\\bar{g}} (T^{'}_{\\rho\\nu} h^{TT\\rho\\nu}+T^{'} \\psi ).\n \\label{scatdef}\n\\end{aligned}\n \\end{equation}\nConsequently, by plugging (\\ref{psittt}),(\\ref{htt}) and (\\ref{ttstrener}) into (\\ref{scatdef}), the scattering amplitude in a flat background can be obtained as follows\n\\begin{equation}\n \\begin{aligned}\n  4{\\cal A}=-2\\kappa T^{'}_{\\rho\\nu} {\\cal O}^{-1}  T^{\\rho\\nu}+\\kappa T^{'}{\\cal O}^{-1}T,\n\\label{mainressct}\n\\end{aligned}\n\\end{equation}\nwhere the integral signs are suppressed for notational simplicity. Now, we are ready to compute the tree-level scattering amplitude for IDG  between two covariantly conserved point-like spinning sources. For this purpose, let us consider the following localized spinning energy-momentum tensors \n\\begin{equation}\nT_{00}= m_a \\delta^{(3)}({\\bf x}-{\\bf x}_a),\\qquad T^{i}{_0}=-\\frac{1}{2} J^k_a \\epsilon^{ikj} \\partial_j \\delta^{(3)}({\\bf x}-{\\bf x}_a),\n\\label{sources}\n\\end{equation}\nwhere $m_a $ are the mass and $ J_a $ are the spin of the sources which have no dimension in our limits; here $a=1,2$.\nIn this respect,  we want to solve the linearized IDG equations for the sources given in (\\ref{sources}). The scattering amplitude (\\ref{mainressct}) can be explicitly recast as\n\\begin{equation}\n4A \n  =-2\\kappa T{}_{00}^{\\prime}\\left\\{ \\frac{1}{a(\\Box)\\partial^2}\\right\\} T^{00}\n+\\kappa T^{\\prime}\\left\\{ \\frac{1}{a(\\Box)\\partial^2}\\right\\} T \n+4\\kappa T{}_{0i}^{\\prime}\\left\\{ \\frac{1}{a(\\Box)\\partial^2}\\right\\} T^i\\,_{0}.\n\\label{scatflat2}\n\\end{equation}\nOn the other hand one must keep in mind that, to avoid ghosts, $a(\\Box)$ must be an entire function. For simplicity, let us choose  $a(\\Box)=e^{-\\frac{\\Box}{M^2}}$ with which the main propagator can be computed as \n\\begin{equation}\nG({\\bf x}, {\\bf x}^{'}, t, t^{'})=\\frac{1}{4\\pi r} \\mbox{erf} (\\frac{Mr}{2})\\delta({\\bf x}-{\\bf x}'-(t-t')),\n\\label{decgreen1}\n  \\end{equation}\nwhere $ r = \\lvert {\\bf x}_1-{\\bf x}_2 \\rvert$ and $\\mbox{erf} (r)$ is the error function defined by the integral\n\\begin{equation}\n\\mbox{erf} (r)=\\frac{2}{\\sqrt{\\pi}}\\int_0^r e^{-k^2}dk.\n\\end{equation}\nThus, by substituting (\\ref{decgreen1}) into (\\ref{scatflat2}) and carrying out the time integrals, one gets\n\\begin{equation}\n \\begin{aligned}\n4\\,{\\cal U}=&-2\\kappa m_1 m_2\\int d^3 x \\int d^3 x^{'}\\,\\,\\,\\delta^{(3)}({\\bf x}^{'}-{\\bf x}_2)  \\hat{G}({\\bf x}, {\\bf x}^{'}) \\delta^{(3)}({\\bf x}-{\\bf x}_1)\\\\& +  \\kappa m_1 m_2 \\int d^3 x \\int d^3 x^{'}\\,\\,\\, \\delta^{(3)}({\\bf x}^{'}-{\\bf x}_2) \\hat{G}({\\bf x}, {\\bf x}^{'}) \\delta^{(3)}({\\bf x}-{\\bf x}_1) \\\\&\n+\\kappa\\int d^3 x \\int d^3 x^{'}\\,\\,J_1^k\\,\\epsilon^{ikj} \\partial'_j \\delta^{(3)}({\\bf x}^{'}-{\\bf x}_2)\\hat{G}({\\bf x}, {\\bf x}^{'}) \\,J_2^l\\,\\epsilon^{ilm}\\partial_m\\delta^{(3)}({\\bf x}-{\\bf x}_1).\n  \\end{aligned}\n\\end{equation}\nHere, the potential energy is $ {\\cal U}={\\cal A}\/t $ \\cite{Gullu-Tekin, Dengiz:2013hka} and $\\hat{G}({\\bf x}, {\\bf x}^{'})$ denotes the time-integrated scalar Green's function defined as  \n\\begin{equation}\n  \\hat{G}({\\bf x},{\\bf x}^{'})=\\int d t^{'} \\, G ({\\bf x},{\\bf x}^{'},t,t^{'})=\\frac{1}{4\\pi r} \\mbox{erf} (\\frac{Mr}{2}).\n\\end{equation}\nFinally, the Newtonian potential energy can be obtained as\n\\begin{equation}\n \\begin{aligned}\n{\\cal U}=&-\\frac{Gm_1m_2}{r}\\mbox{erf} (\\frac{Mr}{2})+\\frac{M^3}{2\\sqrt{\\pi}}e^{-\\frac{M^2r^2}{4}}G[J_1.J_2-(J_1.\\hat{r})(J_2.\\hat{r})]\\\\&-G[J_1.J_2-3(J_1.\\hat{r})(J_2.\\hat{r})]\\times\\bigg[\\frac{1}{r^3}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi}r^2}e^{-\\frac{M^2r^2}{4}}\\bigg].\n  \\label{newtpot1}\n  \\end{aligned}\n\\end{equation}\nObserve that the first term is the ordinary potential energy in IDG which was found in \\cite{Biswas1}, and the last two terms are the spin-spin part which could be attractive or repulsive depending on the choice of spin alignments. Let us now turn our attention to the small and large distance behaviors of potential energy. For the large separations as $r\\to \\infty$, $\\mbox{erf}(r)\\to 1$, $e^{-r^2}\\to 0$, then potential energy takes the form\n \\begin{equation}\n \\begin{aligned}\n{\\cal U}=-\\frac{Gm_1m_2}{r}-\\frac{G}{r^3}\\bigg( J_1.J_2-3(J_1.\\hat{r})(J_2.\\hat{r})\\bigg),\n  \\label{newtpot11}\n  \\end{aligned}\n\\end{equation}\nwhich reproduces the pure GR result \\cite{Gullu-Tekin} as expected. That is, the first term is the usual Newtonian potential energy, and the second one is the spin-spin interactions in GR. On the other side, for the small distances, as expanding the error and the exponential functions into series around $r=0$ give \n \\begin{equation}\n\\mbox{erf}(r)= \\frac{2r}{\\sqrt{\\pi}}-\\frac{2r^3}{3\\sqrt{\\pi}}+ {\\cal{O}}(r^5),\\hskip .6 cm e^{-r^2}=1-r^2+{\\cal{O}}(r^4),\n \\end{equation}\n the potential energy reads\n\\begin{equation}\n \\begin{aligned}\n{\\cal U}=&-\\frac{ G m_1m_2 M}{\\sqrt{\\pi}}+\\frac{GM^3}{3\\sqrt{\\pi}}J_1.J_2+{\\cal{O}}(r^2).\n  \\label{newtpot12}\n  \\end{aligned}\n\\end{equation}\nHere, the ordinary Newtonian potential term and the spin-spin interaction term  in (\\ref{newtpot12}) are constant and hence the potential is not singular at the origin. In GR, the spin-spin part diverges according to $\\sim -\\frac{1}{r^3}$ \\cite{Gullu-Tekin}, whereas in the IDG, this part is non-singular. Though the potential energy is generated by matter sources which have dirac delta function singularities, it is regular due to the non-locality. Thus, in the IDG, not only the usual Newtonian potential but also the spin-spin part become regular as one approaches $r\\to 0$. Therefore, the theory has improved behavior in the small scale behavior.\n\n\\section{Further Gravitomagnetism effects in IDG}\nIn the previous part, we have shown that both usual Newtonian potential and spin-spin terms are finite at the origin. This is a remarkable result, but one can ask whether further gravitomagnetic effects such as spin-orbit interactions also have non-singular behavior or not. To answer this question, let us turn our attention to the tree-level scattering amplitude between two spinning sources that also have velocities and orbital motion. For this purpose, let us consider the following energy-momentum tensors \\cite{Weinberg}:\n\\begin{eqnarray} \n\tT_{00}=T^{\\left(0\\right)}_{00}+T^{\\left(2\\right)}_{00}, \\hspace{10 mm}\n\tT_{i0}=T^{\\left(1\\right)}_{i0}, \\hspace{10 mm}\n\tT_{ij}=T^{\\left(2\\right)}_{ij} , \n\t\\label{Source_mom}\n\\end{eqnarray}\nwhere the relevant tensors are\n\\begin{eqnarray} \n\tT^{\\left(0\\right)}_{00}&=&m_a\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{a}\\right),\\nonumber \\\\\n\tT^{\\left(2\\right)}_{00}&=&\\frac{1}{2}m_a\\vec{v}_a^{2} \\delta^{(3)} \\left(\\vec{x}-\\vec{x}_{a}\\right)-\\frac{1}{2}J_a^{k}\\,v_a^{i}\\epsilon^{ikj}\\partial_{j}\\delta^{(3)} \\left(\\vec{x}\n\t-\\vec{x}_{a}\\right),\\nonumber \\\\\n\tT^{\\left(1\\right)}_{i0}&=&-m_a v_a^{i} \\delta^{(3)} \\left(\\vec{x}-\\vec{x}_{a}\\right)+\\frac{1}{2}J_a^{k}\\,\\epsilon^{ikj}\\partial_{j}\\delta^{(3)}\\left(\\vec{x}\n\t-\\vec{x}_{a}\\right),\\nonumber \\\\\n\tT^{\\left(2\\right)}_{ij}&=&m_av_a^{i}v_a^{j}\\delta^{(3)} \\left(\\vec{x}-\\vec{x}_{a}\\right)+J_a^{l}v_a^{(i}\\epsilon^{j)kl}\\partial_{k}\\delta^{(3)} \\left(\\vec{x}\n\t-\\vec{x}_{a}\\right). \n\t\\label{en_mom}\n\\end{eqnarray}\nHere, $\\vec{v}_{i}$ are the velocities of the particles as defined in a rest frame, and $v^{(i}\\epsilon^{j)kl}$ denotes symmetrization.  We shall work in the small velocity and spin limits, in other words up to $O(v^{2})$ and $O(vJ)$. In this respect, the scattering amplitude (\\ref{mainressct}) turns into\n\n\\begin{equation}\n\t4A=-2\\kappa T{}_{00}^{\\prime}( a(\\Box)\\partial^{2})^{-1}T^{00} -4\\kappa T{}_{0i}^{\\prime}( a(\\Box)\\partial^{2})^{-1}T^{0i} \n\t-2\\kappa T{}_{ij}^{\\prime}( a(\\Box)\\partial^{2})^{-1}T^{ij}+\\kappa  T^{\\prime}( a(\\Box)\\partial^{2})^{-1}T,\n\t\\label{sctIDG}\n\\end{equation}\nwhere integral signs are suppressed and $( a(\\Box)\\partial^{2})^{-1}$ is the scalar Green's function as was given in (\\ref{decgreen1}). To find the weak field potential energy for the sources given in (\\ref{Source_mom}), let us calculate the amplitude by working each term in (\\ref{sctIDG}), separately. After evaluating the relevant integrals, the energy density interaction term takes the form\n\\begin{equation}\n\\begin{aligned}\n\t-2\\kappa T{}_{00}( a(\\Box)\\partial^{2})^{-1}T^{\\prime 00}&=-2\\kappa \\bigg [ \\frac{m_{1}m_{2}}{4\\pi r}\\bigg (1+\\frac{\\vec{v}^{2}_{1}+\\vec{v}^{2}_{2}}{2}\\bigg )\\mbox{erf} (\\frac{Mr}{2})\\\\&+\\frac{1}{4\\pi}\\bigg(\\frac{1}{ r^{2}}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi} r}e^{-\\frac{M^2r^2}{4}}\\bigg)\n\t\\bigg (\\frac{m_{1}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{2}}}{2}-\\frac{m_{2}(\\hat{r}\\times \\vec{v}_{1} )\\cdot\\vec{J_{1}}}{2} \\bigg )\\bigg ]t.\n\\end{aligned}\n\\end{equation}\n\nHere, we have dropped the term which includes higher order contributions $O(J^{2}v^{2})$. On the other hand, the trace-trace interaction term yields \n\\begin{equation}\n\\begin{aligned}\n\t\\kappa T^{\\prime}( a(\\Box)\\partial^{2})^{-1}T=&\t\\kappa\\bigg [\\frac{m_{1}m_{2}}{4\\pi r}\\bigg(1+\\frac{-\\vec{v}^{2}_{1}-\\vec{v}^{2}_{2}}{2}\\bigg)\\mbox{erf} (\\frac{Mr}{2})\\\\&\n\t+\\frac{1}{4\\pi}\\bigg(\\frac{1}{ r^{2}}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi} r}e^{-\\frac{M^2r^2}{4}}\\bigg)\\bigg (\n\t-\\frac{m_{1}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{2}}}{2}+\n\t\\frac{m_{2}(\\hat{r}\\times \\vec{v}_{1})\\cdot\\vec{J_{1}}}{2} \\bigg )\\bigg ] t.\n\\end{aligned}\n\\end{equation}\nSimilarly the $T{}_{0i}^{\\prime}(\\partial^{2})^{-1}T^{0i}$ term becomes \n\\begin{equation}\n\\begin{aligned}\n-4\\kappa T{}_{0i}^{\\prime}( a(\\Box)\\partial^{2})^{-1}T^{0i}&=-4\\kappa \\bigg [ -\\dfrac{m_{1}m_{2}\\vec{v}_{1}\\cdot\\vec{v}_{2} }{4\\pi r}\\mbox{erf} (\\frac{Mr}{2}) \\\\&+\\frac{1}{8\\pi}\\bigg(\\dfrac{1}{r^{2}} \\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi}r}\ne^{-\\frac{M^2r^2}{4}}\\bigg)   \\bigg ( -m_{1}(\\hat{r}\\times \\vec{v}_{1}  )\\cdot \\vec{J}_{2}+m_{2}(\\hat{r}\\times \\vec{v}_{2}  )\\cdot \\vec{J}_{1}\\bigg)\\\\&  - \\dfrac{1}{16 \\pi }\\bigg( \\frac{M^3}{2\\sqrt{\\pi}}e^{-\\frac{M^2r^2}{4}}[J_1.J_2-(J_1.\\hat{r})(J_2.\\hat{r})]\\\\&-[J_1.J_2-3(J_1.\\hat{r})(J_2.\\hat{r})]\\times[\\frac{1}{r^3}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi}r^2}e^{-\\frac{M^2r^2}{4}}]\\bigg)\\bigg]t.\n\\end{aligned}\n\\end{equation}\nNote that as the $T{}_{ij}^{\\prime}(\\partial^{2})^{-1}T{}^{ij}$  term in (\\ref{sctIDG}) contributes only at the higher order, it has been dropped.\nConsequently, by using all these results, the potential energy in IDG takes the form\n\n\\begin{equation}\n\\begin{aligned}\nU_{IDG} =& -\\frac{G}{r}  m_{1}m_{2} \\left [ 1+\\frac{3}{2}\\vec{v}^{2}_{1}+\\frac{3}{2}\\vec{v}^{2}_{2}-4\\vec{v}_{1}\\cdot \\vec{v}_{2} \\right ] \\mbox{erf} (\\frac{Mr}{2})\n+\\frac{M^3}{2\\sqrt{\\pi}}e^{-\\frac{M^2r^2}{4}}G[J_1.J_2-(J_1.\\hat{r})(J_2.\\hat{r})]\\\\&-G[J_1.J_2-3(J_1.\\hat{r})(J_2.\\hat{r})]\\times\\bigg[\\frac{1}{r^3}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi}r^2}e^{-\\frac{M^2r^2}{4}}\\bigg]\\\\\n&-G\\bigg(\\frac{1}{r^{2}}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi}r} e^{-\\frac{M^2r^2}{4}}\\bigg) \\bigg [ \\frac{3m_{1}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{2}}}{2}-\\frac{3m_{2}(\\hat{r}\\times \\vec{v}_{1})\\cdot\\vec{J_{1}}}{2}\\\\&-2m_{1}(\\hat{r}\\times \\vec{v}_{1})\\cdot\\vec{J_{2}}+2m_{2}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{1}}\n\t\\bigg ]. \\label{IDGgrpe}\n\t\\end{aligned}\n\\end{equation}\nObserve that potential energy has the ordinary Newtonian potential energy, spin-spin and spin-orbit interactions. \nFor large separations as $r\\to \\infty$, the potential energy becomes\n \\begin{equation}\n \\begin{aligned}\nU =& -\\frac{G}{r}  m_{1}m_{2} \\left [ 1+\\frac{3}{2}\\vec{v}^{2}_{1}+\\frac{3}{2}\\vec{v}^{2}_{2}-4\\vec{v}_{1}\\cdot \\vec{v}_{2} \\right ]\n-\\frac{G}{r^{3}}\\left[\\vec{J_{1}}\\centerdot\\vec{J_{2}}-3\\vec{J_{1}}\\centerdot\\hat{r}\\,\n\\vec{J_{2}}\\centerdot\\hat{r}\\right] \\\\\n&-\\frac{G}{r^{2}}  \\left [ \\frac{3m_{1}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{2}}}{2}-\\frac{3m_{2}(\\hat{r}\\times \\vec{v}_{1})\\cdot\\vec{J_{1}}}{2}-2m_{1}(\\hat{r}\\times \\vec{v}_{1})\\cdot\\vec{J_{2}}+2m_{2}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{1}}\n\t\\right ], \n\\end{aligned}\t\n \\end{equation}\nwhich matches with the pure GR result \\cite{Tasseten} as expected. That is, the potential energy contains the usual Newtonian potential energy and relativistic corrections. On the other hand, for small distances, the potential energy reduces to\n\\begin{equation}\n \\begin{aligned}\n{\\cal U}=&-\\frac{ G m_1m_2 M}{\\sqrt{\\pi}}\\bigg[ 1+\\frac{3}{2}\\vec{v}^{2}_{1}+\\frac{3}{2}\\vec{v}^{2}_{2}-4\\vec{v}_{1}\\cdot \\vec{v}_{2} \\bigg]+\\frac{GM^3}{3\\sqrt{\\pi}}J_1.J_2+{\\cal{O}}(r).\n  \\label{newtpot121}\n  \\end{aligned}\n\\end{equation}\nHere, the ordinary Newtonian potential term and the spin-spin interaction term  in (\\ref{newtpot121}) are constant and the spin-orbit interaction terms contribute at the order ${\\cal{O}}(r)$. Therefore the potential is regular at the origin. Thus, in the IDG, not only the usual Newtonian potential but also the spin-spin and spin-orbit interactions become regular as one approaches $r\\to 0$. These non-singular results in IDG show that the theory is very well-behaved in the UV region compared to GR.\n\n\\section{Conclusions and further discussions}\nWe have considered the IDG in $3+1$ dimensional flat backgrounds. We computed the tree-level scattering amplitude in IDG and accordingly found weak field potential energy between two point-like spinning sources interacting via one-graviton exchange. We have demonstrated that at large distances potential energy is the same as the GR result, whereas at small distances, it is discreetly different from GR. We have also shown that both the ordinary Newtonian potential energy  and the spin-spin term remain finite at the small distance limit ($r\\to 0$). Furthermore, in addition to spin-spin interactions, we studied the spin-orbit interactions in IDG by considering that the sources are also moving. We found that not only mass-mass but also spin-spin and spin-orbit interactions are non-singular and finite at the origin. That is, gravitational potential energy of spinning sources that also have velocities becomes non-singular for IDG. Consequently, the theory is a very well-behaved feature in the UV regime as compared to GR. \n\nNow, we would like to discuss the effects of mass scale of non-locality ($M$) on gravitational memory effect. Gravitational waves, induced by merger of neutron stars or black holes etc, create a  permanent effect on a system composed of inertial test particles. In other words, a pulse of gravitational wave changes the relative displacements of test particles. This effect is called gravitational memory effect and comes in two forms: ordinary (or linear) \\cite{zeldovich} and null  (or\nnon-linear) \\cite{Christodoulou}. The studies on gravitational memory effect have recently received more attention in various aspects \\cite{Garfinkle,Satishchandran:2017pek,Tolish1,Tolish2,Bieri2,Gibbons,Kilicarslan} because there is a hope that it could be measured by advanced LIGO.  To calculate gravitational memory effect in IDG in a flat spacetime, we can follow the method of \\cite{Satishchandran:2017pek, Garfinkle}: we first solved the geodesic deviation equation and then integrated it two times to find relative separation of the test particles.   Without giving the details, we shall  give the final result:\n\\begin{equation}\n\\begin{aligned}\n\\Delta\\xi^{i}&\n=\\frac{1}{r} \\mbox{erf}(\\frac{Mr}{2})\\Delta_j^i\\Theta(U)\\xi^{j},\n\\label{memoryeffect}\n\\end{aligned}\n\\end{equation} \nwhere $\\Theta$ is the step function, $\\xi$ is a spatial separation vector and $\\Delta_j^i$ are spatial components of the memory tensor (See Eq.(45) in \\cite{Garfinkle} for memory tensor). This result shows that the test particles have non-trivial change in their separations which is described by the memory tensor. Observe that the memory is dependent of the mass scale of non-locality and different from GR. In the large distance limits, memory is the same as the usual Einsteinian form as expected. Furthermore, for a lower bound on mass scale of non-locality ($M>4keV$) \\cite{Edholm:2018qkc}, the memory reduces to GR prediction above at very small distances.\n\n\\section{\\label{ackno} Acknowledgements}\nWe would like to thank B. Tekin for useful discussions, suggestions and comments. We would also like to thank S.Dengiz and J. Edholm  for suggestions and critical readings of the manuscript.\n\n\\section{Appendix: Details of the Calculations}\nIn this part, we would like to give the details of scattering amplitude calculations for the Sec. III. Before going into further details,  let us give the following identities:\n\\begin{align}\n\\partial_{k}r= \\dfrac{(x^{k}-x^{\\prime k})}{r}=\\hat{r}^{k}, \\hspace{1cm} \\partial_{k}\\dfrac{1}{r}=\\dfrac{-(x^{k}-x^{\\prime k})}{r^{3}}=\\dfrac{-\\hat{r}^{k}}{r^{2}}, \\nonumber \\\\\n\\partial_{k^{\\prime}}r= \\dfrac{-(x^{k}-x^{\\prime k})}{r}=-\\hat{r}^{k}, \\hspace{1cm} \\partial_{k^{\\prime}}\\dfrac{1}{r}=\\dfrac{(x^{k}-x^{\\prime k})}{r^{3}}=\\dfrac{\\hat{r}^{k}}{r^{2}}, \\nonumber \\\\\n\\partial_{k}\\partial_{n^{\\prime}}r=\\dfrac{1}{r}\\left( -\\delta^{kn}+\\hat{r}^{k}\\hat{r}^{n}\\right) , \\hspace{1cm}\n\\partial_{k}\\partial_{n^{\\prime}}\\dfrac{1}{r}= \\dfrac{1}{r^{3}}\\left( \\delta^{kn}-3\\hat{r}^{k}\\hat{r}^{n}\\right),\\nonumber \\\\\n\\partial_{k}\\mbox{erf} (r)=\\frac{2}{\\sqrt{\\pi}}e^{-r^2}\\hat{r}^{k} , \\hspace{1cm}\n\\partial_{k^{\\prime}}\\mbox{erf} (r)=-\\frac{2}{\\sqrt{\\pi}}e^{-r^2}\\hat{r}^{k},\n\\end{align}\nwhich are needed for computations. Let us now calculate the amplitude by working each term in (\\ref{sctIDG}), separately. The energy density interaction term becomes\n\n\\begin{equation}\n\\begin{aligned}\n\tT{}_{00}(a(\\Box)\\partial^{2})^{-1}T^{\\prime 00}=&\\bigg [  m_{1}\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)+\\frac{1}{2}m_{1}\\vec{v}^{2}_{1} \\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right) -\\frac{1}{2}J_{1}^{l}\\,v^{i}_{1}\\epsilon^{ilk}\\partial_{k}\\delta\\left(\\vec{x}\n\t-\\vec{x}_{1}\\right)\\bigg ]  (a(\\Box)\\partial^{2})^{-1} \\\\\n\t&\n\t\\bigg [ m_{2}\\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)+\\frac{1}{2}m_{2}\\vec{v}^{2}_{2} \\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)  -\\frac{1}{2}J_{2}^{m}\\,v^{j}_{2}\\epsilon^{jmn}\\partial_{n}^{\\prime}\\delta^{(3)}\\left(\\vec{x^{\\prime}}\n\t-\\vec{x}_{2}\\right)\\bigg ],\n\t\\end{aligned}\n\\end{equation}\nwhose each distinct term reads\n\\begin{equation}\n\tm_{1}\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)( a(\\Box)\\partial^{2})^{-1}m_{2}\\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)=\\frac{m_{1}m_{2}}{4\\pi r}\\mbox{erf} (\\frac{Mr}{2})t,\n\\end{equation}\n\n\n\\begin{equation}\n\tm_{1}\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)( a(\\Box)\\partial^{2})^{-1}\\frac{1}{2}m_{2}\\vec{v}^{2}_{2} \\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)=\\frac{1}{2}\\frac{m_{1}m_{2}\\vec{v}^{2}_{2}}{4\\pi r}\\mbox{erf} (\\frac{Mr}{2})t,\n\\end{equation}\n\n\\begin{equation}\n\\begin{aligned}\n\t-\\frac{1}{2}m_{1}\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)( a(\\Box)\\partial^{2})^{-1}J_{2}^{m}\\,v^{j}_{2}\\epsilon^{jmn}\\partial_{n}^{\\prime}\\delta^{(3)}\\left(\\vec{x^{\\prime}}\n\t-\\vec{x}_{2}\\right) \n\t=&\\frac{1}{2}\n\t\\frac{m_{1}(\\hat{r}\\times \\vec{v}_{2}).\\vec{J_{2}}}{4\\pi r^{2}}\\mbox{erf} (\\frac{Mr}{2})t\\\\&-\\frac{M}{2\\sqrt{\\pi}}e^{-\\frac{M^2r^2}{4}}\n\t\\frac{m_{1}(\\hat{r}\\times \\vec{v}_{2}).\\vec{J_{2}}}{4\\pi r}t,\t\n\t\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\n\\frac{1}{2}m_{1}\\vec{v}^{2}_{1} \\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)( a(\\Box)\\partial^{2})^{-1}m_{2}\\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)=\\frac{1}{2}\\frac{m_{1}m_{2}\\vec{v}^{2}_{1}}{4\\pi r}\\mbox{erf} (\\frac{Mr}{2})t,\n\\end{equation}\n\n\\begin{equation}\n\\begin{aligned}\n\t-\\frac{1}{2}J_{1}^{l}\\,v^{i}_{1}\\epsilon^{ilk}\\partial_{k}\\delta^{(3)}\\left(\\vec{x}\n\t-\\vec{x}_{1}\\right) ( a(\\Box)\\partial^{2})^{-1} m_{2}\\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)\n\t=&-\\frac{1}{2}\n\t\\frac{m_{2}(\\hat{r}\\times \\vec{v}_{1}).\\vec{J_{1}}}{4\\pi r^{2}}\\mbox{erf} (\\frac{Mr}{2})t\\\\&+\\frac{M}{2\\sqrt{\\pi}}\n\t\\frac{m_{2}(\\hat{r}\\times \\vec{v}_{1}).\\vec{J_{1}}}{4\\pi r}e^{-\\frac{M^2r^2}{4}}t,\n\t\\end{aligned}\n\\end{equation}\nwith these terms, one ultimately gets\n\\begin{equation}\n\\begin{aligned}\n\t-2\\kappa T{}_{00}( a(\\Box)\\partial^{2})^{-1}T^{\\prime 00}&=-2\\kappa \\bigg [ \\frac{m_{1}m_{2}}{4\\pi r}\\mbox{erf} (\\frac{Mr}{2})\\bigg (1+\\frac{\\vec{v}^{2}_{1}+\\vec{v}^{2}_{2}}{2}\\bigg )\\\\&+\\frac{1}{4\\pi}\\bigg(\\frac{1}{r^{2}}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi} r}e^{-\\frac{M^2r^2}{4}}\\bigg)\n\t\\bigg (\\frac{m_{1}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{2}}}{2}-\\frac{m_{2}(\\hat{r}\\times \\vec{v}_{1} )\\cdot\\vec{J_{1}}}{2} \\bigg )\\bigg ]t.\n\\end{aligned}\n\\end{equation}\n\nOn the other side, the trace-trace interaction term yields \n\\begin{equation}\n\\begin{aligned}\n\tT^{\\prime}( a(\\Box)\\partial^{2})^{-1}T= &\\bigg [ -m_{1}\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)+\\frac{1}{2}m_{1}\\vec{v}^{2}_{1} \\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right) -\\frac{1}{2}J_{1}^{l}\\,v^{i}_{1}\\epsilon^{ilk}\\partial_{k}\\delta^{(3)}\\left(\\vec{x}\n\t-\\vec{x}_{1}\\right)\\bigg ] (\\partial^{2})^{-1} \\\\\n\t&\\bigg [ -m_{2}\\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)+\\frac{1}{2}m_{2}\\vec{v}^{2}_{2} \\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)  -\\frac{1}{2}J_{2}^{m}\\,v^{j}_{2}\\epsilon^{jmn}\\partial_{n}^{\\prime}\\delta^{(3)}\\left(\\vec{x^{\\prime}}\n\t-\\vec{x}_{2}\\right)\\bigg ].\n\\end{aligned}\n\\end{equation}\nThen, by evaluating the relevant integrals, one eventually obtains \n\\begin{equation}\n\\begin{aligned}\n\t\\kappa T^{\\prime}( a(\\Box)\\partial^{2})^{-1}T=&\t\\kappa\\bigg [\\frac{m_{1}m_{2}}{4\\pi r}\\bigg(1+\\frac{-\\vec{v}^{2}_{1}-\\vec{v}^{2}_{2}}{2}\\bigg)\\mbox{erf} (\\frac{Mr}{2})\\\\&\n\t+\\bigg(\\frac{1}{4\\pi r^{2}}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{4\\pi^{\\frac{3}{2}} r}e^{-\\frac{M^2r^2}{4}}\\bigg)\\bigg (\n\t-\\frac{m_{1}(\\hat{r}\\times \\vec{v}_{2})\\cdot\\vec{J_{2}}}{2}+\n\t\\frac{m_{2}(\\hat{r}\\times \\vec{v}_{1})\\cdot\\vec{J_{1}}}{2} \\bigg )\\bigg ] t.\n\\end{aligned}\n\\end{equation}\nSimilarly, the $T{}_{0i}^{\\prime}(\\partial^{2})^{-1}T^{0i}$ term can be written as\n\\begin{equation}\n\\begin{aligned}\n\tT{}_{0i}^{\\prime}( a(\\Box)\\partial^{2})^{-1}T^{0i}=&\\bigg [ -m_{1}v^{i}_{1}\\delta^{(3)}\\left(\\vec{x}-\\vec{x}_{1}\\right)+\\frac{1}{2}J_{1}^{k}\\,\\epsilon^{ikj}\\partial_{j}\\delta^{(3)}\\left(\\vec{x}\n\t-\\vec{x}_{1}\\right)\\bigg ](a(\\Box)\\partial^{2})^{-1}\\\\&\\times\\bigg [ m_{2}v^{i}_{2}\\delta^{(3)}\\left(\\vec{x^{\\prime}}-\\vec{x}_{2}\\right)-\\frac{1}{2}J_{2}^{l}\\,\\epsilon^{ilm}\\partial^{\\prime}_{m}\\delta^{(3)}\\left(\\vec{x^{\\prime}}\n\t-\\vec{x}_{2}\\right)\\bigg ],\n\\end{aligned}\n\\end{equation}\nwhich after lengthy and tedious calculations becomes\n\\begin{equation}\n\\begin{aligned}\n-4\\kappa T{}_{0i}^{\\prime}( a(\\Box)\\partial^{2})^{-1}T^{0i}&=-4\\kappa \\bigg [ -\\dfrac{m_{1}m_{2}\\vec{v}_{1}\\cdot\\vec{v}_{2} }{4\\pi r}\\mbox{erf} (\\frac{Mr}{2}) \\\\&+\\frac{1}{8\\pi}\\bigg(\\dfrac{1}{ r^{2}} \\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi} r}\ne^{-\\frac{M^2r^2}{4}}\\bigg)  \\bigg ( -m_{1}(\\hat{r}\\times \\vec{v}_{1}  )\\cdot \\vec{J}_{2}+m_{2}(\\hat{r}\\times \\vec{v}_{2}  )\\cdot \\vec{J}_{1}\\bigg)\\\\&  - \\dfrac{1}{16 \\pi }\\bigg( \\frac{M^3}{2\\sqrt{\\pi}}e^{-\\frac{M^2r^2}{4}}[J_1.J_2-(J_1.\\hat{r})(J_2.\\hat{r})]\\\\&-[J_1.J_2-3(J_1.\\hat{r})(J_2.\\hat{r})]\\times(\\frac{1}{r^3}\\mbox{erf} (\\frac{Mr}{2})-\\frac{M}{\\sqrt{\\pi}r^2}e^{-\\frac{M^2r^2}{4}})\\bigg)\\bigg]t.\n\\end{aligned}\n\\end{equation}\nRecall that the $T{}_{ij}^{\\prime}(\\partial^{2})^{-1}T{}^{ij}$  term contributes in higher order corrections.\nConsequently, by using the results above obtained, the potential energy in IDG is obtained in the form as given in (\\ref{IDGgrpe}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nSuppose $M$ is a compact metric space and let ${\\rm Homeo}(M)$ be its group of homeomorphisms. We equip ${\\rm Homeo}(M)$ with the topology of uniform convergence or what is equivalent, since $M$ is compact metric, the compact-open topology. Thus, in this way, a neighbourhood basis at the identity $1$ consists of the sets\n$$\n\\{h\\in {\\rm Homeo}(M)\\del h(C_1)\\subseteq V_1\\;\\&\\;\\ldots\\;\\&\\; h(C_n)\\subseteq V_n\\},\n$$\nwhere $V_i\\subseteq M$ is open and $C_i\\subseteq V_i$ compact. Under this topology the group operations are continuous and thus ${\\rm Homeo}(M)$ is a topological group. Moreover, the topology is {\\em Polish}, that is, ${\\rm Homeo}(M)$ is separable and its topology can be induced by a complete metric.\n\nNow consider the case when $M$ is Cantor space $\\ca$. Then, as any two disjoint closed sets in $\\ca$ can be separated by a clopen set, we get a neighbourhood basis at the identity consisting of sets of the form\n$$\n\\{h\\in {\\rm Homeo}(\\ca)\\del h(C_1)=C_1\\;\\&\\;\\ldots\\;\\&\\; h(C_n)=C_n\\},\n$$\nwhere $C_1,\\ldots,C_n\\subseteq \\ca$ is a partition of $\\ca$ into clopen sets.\n\nBy Stone duality, the homeomorphisms of Cantor space are just the automorphisms of the boolean algebra of clopen subsets of $\\ca$, which we denote by $\\B_\\infty$. Thus, viewed in this way, the neighbourhood basis at the identity has the form\n$$\n\\{h\\in {\\rm Homeo}(\\ca)\\del h|_\\C={\\rm id}_\\C\\}\n$$\nwhere $\\C$ is a finite subalgebra of $\\B_\\infty$.\n\nCantor space $\\ca$ is of course naturally homeomorphic to the Cantor group $(\\Z_2)^\\N$ and therefore comes equipped with Haar measure $\\mu$. Up to a homeomorphism of Cantor space $\\mu$ is the unique atomless Borel probability measure on $\\ca$ such that\n\\begin{itemize}\n\\item if $C\\in \\B_\\infty$, then $\\mu(C)$ is a {\\em dyadic rational}, i.e., on the form $\\frac n{2^k}$,\n\\item if $C\\in \\B_\\infty$ and $\\mu(C)=\\frac n{2^k}$, then for all $l\\geq k$, there is some clopen $B\\subseteq C$ such that $\\mu(B)=\\frac 1{2^l}$,\n\\item if $\\tom\\neq C\\in \\B_\\infty$, then $\\mu (C)> 0$.\n\\end{itemize}\nThe measure $\\mu$ is of course the product probability measure of the coinflipping measure on each factor $2=\\{0,1\\}$. For simplicity, we call $\\mu$ Haar measure on $\\ca$.\n\nOne easily sees that the group of Haar measure preserving homeomorphisms ${\\rm Homeo}(\\ca,\\mu)$ of $\\ca$ is a closed subgroup of ${\\rm Homeo}(\\ca)$ and therefore a Polish group in its own right.\nIt was proved by Kechris and Rosendal in \\cite{turbulence} that there are comeagre conjugacy classes in both ${\\rm Homeo}(\\ca)$ and ${\\rm Homeo}(\\ca,\\mu)$. In fact, the result for ${\\rm Homeo}(\\ca,\\mu)$ is rather simple and also holds for many other sufficiently homogeneous measures on $\\ca$ (see Akin \\cite{akin}). This result allows us to refer to the {\\em generic} measure preserving homeomorphism of Cantor space (with Haar measure) knowing that they are all mutually conjugate. One of the aims of this paper is to show that they are all conjugate to their non-zero powers, which will in turn show that they  all are part of an action of the additive group $\\Q$ by measure preserving homeomorphisms of $\\ca$. Notice that this is to some extent an optimal result, for as ${\\rm Homeo}(\\ca,\\mu)$ is totally disconnected there are no non-trivial continuous homomorphism (or even measurable homomorphisms) from $\\R$ into ${\\rm Homeo}(\\ca,\\mu)$ and thus $\\R$ cannot act non-trivially by measure preserving homeomorphisms on $\\ca$.\n\n\n\\\n\n\n\n\nThe {\\em Urysohn metric space} $\\U$ is a universal separable metric space first constructed by Urysohn in the posthomously published\n\\cite{ury}. It soon went out of fashion following the discovery that many separable Banach spaces are already universal separable metric spaces, but has come to the forefront over the last twenty years as an analogue of  Fra\\\"iss\\'e theory in the case of metric spaces.\n\nThe Urysohn space $\\U$ is characterised up to isometry by being separable and complete together with the  following extension property.\n\\begin{quotation}\nIf $\\phi\\colon\\A\\rightarrow \\U$ is an isometric embedding of a\nfinite metric space $\\A$ into $\\U$ and $\\B=\\A\\cap \\{y\\}$ is a one\npoint metric extension of $\\A$, then $\\phi$ extends to an isometric\nembedding of $\\B$ into $\\U$.\n\\end{quotation}\nThere is also a rational variant of $\\U$ called the {\\em rational Urysohn metric space}, which we denote by $\\Q\\U$. This is up to isometry the unique countable metric space with only rational distances such that the following variant of the above extension property holds.\n\\begin{quotation}\nIf $\\phi\\colon\\A\\rightarrow \\Q\\U$ is an isometric embedding of a\nfinite  metric space $\\A$ into $\\Q\\U$ and $\\B=\\A\\cap \\{y\\}$ is a one\npoint metric extension of $\\A$ whose metric only takes rational distances, then $\\phi$ extends to an isometric\nembedding of $\\B$ into $\\Q\\U$.\n\\end{quotation}\nWe denote by ${\\rm Iso}(\\Q\\U)$ and ${\\rm Iso}(\\U)$ the isometry groups of $\\Q\\U$ and $\\U$ respectively. These are Polish groups when equipped with the topology of pointwise convergence on $\\Q\\U$ seen as a discrete set and $\\U$ seen as a metric space respectively. Thus, the basic neighbourhoods of the identity in ${\\rm Iso}(\\Q\\U)$ are of the form\n$$\n\\{h\\in {\\rm Iso}(\\Q\\U)\\del h|_\\A={\\rm id}_\\A\\},\n$$\nwhere $\\A$ is a finite subset of $\\Q\\U$. On the other hand, the basic open neighbourhoods of the identity in ${\\rm Iso}(\\U)$ are of the form\n$$\n\\{h\\in {\\rm Iso}(\\U)\\del \\a x\\in \\A\\; d(hx,x)<\\epsilon\\},\n$$\nwhere $\\A$ is a finite subset of $\\U$ and $\\epsilon>0$.\n\nIn \\cite{solecki} S. Solecki proved, building on work of Herwig and Lascar \\cite{herwig}, the following result.\n\\begin{thm}\\label{solecki}\nLet $\\A$ be a finite rational metric space. Then there is a finite rational metric space $\\B$ containing $\\A$ and such that any partial isometry of $\\A$ extends to a full isometry of $\\B$.\n\\end{thm}\nThis is turn has the consequence that ${\\rm Iso}(\\Q\\U)$ has a comegre conjugacy class and we can therefore refer to its elements as {\\em generic} isometries of $\\Q\\U$. The second aim of our paper is to prove that these are all conjugate to their non-zero powers, which again suffices to show that they  all are part of an action of the additive group $\\Q$ by isometries of $\\Q\\U$.\n\n\n\n\n\n\n\n\\\n\n\nIn the last section we briefly consider a coarse notion of conjugacy in Polish groups. We say that $f$ and $g$ belonging to a Polish group $G$ are {\\em topologically similar} if for all increasing sequences $(s_n)$ we have $f^{s_n}\\Lim{n\\rightarrow \\infty} 1$ if and only if $g^{s_n}\\Lim{n\\rightarrow \\infty}1$. As opposed to automorphism groups of countable structures there tend not to be comeagre conjugacy classes in large connected Polish groups and we shall provide new simple proofs of this for ${\\rm Aut}([0,1],\\lambda)$ and ${\\rm Iso}(\\U)$ by showing that in fact their topological similarity classes are meagre.\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{Powers of generic measure preserving homeomorphisms}\n\n\n\n\\subsection{Free amalgams of measured boolean algebras}\nSuppose $\\B_1, \\B_2,\\ldots, \\B_n$ are finite boolean algebras\ncontaining a common subalgebra $\\A$. We define the {\\em free\namalgam}\n$$\n\\otimes_\\A^l\\B_l=\\B_1\\otimes_\\A\\B_2\\otimes_\\A\\ldots\\otimes_\\A\\B_n\n$$ of $\\B_1,\\ldots, \\B_n$ over $\\A$ as follows.\n\nBy renaming, we can suppose that $\\B_i\\cap \\B_j=\\A$ for all $i\\neq\nj$. We then take as our atoms the set of formal products\n$$\nb_1\\otimes\\ldots\\otimes b_n,\n$$\nwhere each $b_i$ is an atom in $\\B_i$ and such that for some atom\n$a$ of $\\A$ we have $b_i\\leq a$ for all $i$. Also, for simplicity,\nif $c_i\\in \\B_i$ is not necessarily an atom, but nevertheless we\nhave some atom $a$ of $\\A$ such that $c_i\\leq a$ for all $i$, we\nwrite\n$$\nc_1\\otimes\\ldots\\otimes c_n=\\bigvee\\{b_1\\otimes\\ldots\\otimes b_n\\del\nb_i\\text{ is an atom in }\\B_i \\text { and } b_i\\leq c_i\\}.\n$$\nWe can now embed each $\\B_i$ into $\\otimes_\\A^l\\B_l$ by defining for\neach  $b\\in \\B_i$ minorising an atom $a\\in \\A$\n$$\n\\pi_i(b)=a\\otimes\\ldots\\otimes a\\otimes b\\otimes\na\\otimes\\ldots\\otimes a,\n$$\nwhere the $b$ appears in the $i$'th position. In particular,\n$$\n\\pi_i(a)=a\\otimes\\ldots \\otimes a\n$$\nfor all atoms $a$ of $\\A$. Thus, for each $i$, $\\pi_i\\colon \\B_i\\hookrightarrow\n\\otimes_\\A^l\\B_l$ is an embedding of boolean algebras and if\n$\\iota_i\\colon\\A\\hookrightarrow \\B_i$ denotes the inclusion mapping, then the\nfollowing diagram commutes\n$$\n\\begin{CD}\n\\A @>\\iota_i>> \\B_i\\\\\n@V\\iota_jVV   @VV\\pi_iV\\\\\n\\B_j@>>\\pi_j> \\otimes_\\A^l\\B_l\n\\end{CD}\n$$\n\n\n\n\nNow, if $\\mu_i$ are  measures on $\\B_i$ agreeing on $\\A$, then we\ncan define a new measure $\\mu$ on $\\otimes_\\A^l\\B_l$ by setting for\nall $b_i\\in \\B_i$ minorising the same atom $a\\in \\A$\n$$\n\\mu(b_1\\otimes\\ldots\\otimes\nb_n)=\\frac{\\mu_1(b_1)\\cdots\\mu_n(b_n)}{\\mu_1(a)^{n-1}}.\n$$\nThus,\n\\begin{align*}\n\\mu(\\pi_i(b))&=\\mu(a\\otimes\\ldots\\otimes a\\otimes b\\otimes a\\otimes\n\\ldots\\otimes\na)\\\\\n&=\\frac{\\mu_1(a)\\cdots\\mu_{i-1}(a)\\mu_i(b)\\mu_{i+1}(a)\\cdots\\mu_n(a)}{\\mu_1(a)^{n-1}}\\\\\n&=\\frac{\\mu_1(a)\\cdots\\mu_{1}(a)\\mu_i(b)\\mu_{1}(a)\\cdots\\mu_1(a)}{\\mu_1(a)^{n-1}}\\\\\n&=\\mu_i(b).\n\\end{align*}\nSo $\\pi_i\\colon (\\B_i,\\mu_i)\\rightarrow (\\otimes_\\A^l\\B_l,\\mu)$ is an\nembedding of measured boolean algebras.\n\nA special case is when $\\A$ and each $\\B_i$ are {\\em equidistributed\ndyadic} algebras, i.e., have $2^k$ atoms  each of measure $2^{-k}$\nfor some $k\\geq 0$. Then this implies that for each $i$, all atoms\nof $\\A$ are the join of the same number of atoms of $\\B_i$, namely,\n$2^{k_i-m}$, where $\\A$ has $2^m$ atoms and $\\B_i$ has $2^{k_i}$\natoms. In this case, one can verify that $\\otimes_\\A^l\\B_l$ has\n$2^{k_1+\\ldots+k_n-(n-1)m}$ atoms each of measure\n$2^{(n-1)m-k_1-\\ldots-k_n}$. So again this is an equidistributed\ndyadic algebra.\n\nA similar construction works for {\\em equidistributed } algebras,\ni.e., those having a finite number of atoms of the same (necessarily\nrational) measure. In this case, the amalgam is also\nequidistributed.\n\n\nIn general, an automorphism of a finite boolean algebra arises from\na permutation of the atoms, but in the case of equidistributed\n(dyadic) algebras, any permutation of the atoms conversely gives rise to a\nmeasure preserving automorphism. Thus, for equidistributed algebras\nan automorphism is necessarily a measure preserving automorphism and\nwe can therefore be a bit forgetful about the measure.\n\n\n\\begin{lemme}\nLet $\\A$ be an equidistributed (dyadic) finite boolean algebra. Then\nany partial measure preserving automorphism of $\\A$ extends to an\nautomorphism of $\\A$.\n\\end{lemme}\n\n\n\\begin{proof}\nSuppose that $\\B$ and $\\C$ are subalgebras of $\\A$ and $g\\colon\n\\B\\rightarrow \\C$ a measure preserving isomorphism. Then if $b$ is an atom\nof $\\B$, we have, as $g$ is measure preserving, that $b$ and $g(b)$\nare composed of the same number of atoms of $\\A$. Therefore, we can\nextend $g$ to an automorphism of $\\A$ by choosing a bijection\nbetween the constituents of $b$ and $g(b)$ for each atom $b$ of\n$\\B$.\n\\end{proof}\n\n\n\n\n\n\\subsection{Roots of measure preserving homeomorphisms}\n\n\\begin{prop}\\label{divisibility}\nSuppose $\\A\\subseteq \\B$ are equidistributed (dyadic) boolean\nalgebras, $g$ an automorphism of $\\A$ and $f$ an automorphism of\n$\\B$ such that $f|\\A=g^n$. Then there is an equidistributed (dyadic)\nalgebra $\\C\\supseteq \\B$ and an automorphism $h$ of $\\C$ extending\n$g$ and such that $h^n|\\B=f$.\n\\end{prop}\n\n\n\n\n\n\\begin{proof}\nEnumerate the atoms of $\\A$ as $a_1,\\ldots, a_m$ and the atoms of\n$\\B$ as\n$$\nb_1^1,b_1^2,\\ldots, b_1^k,b_2^1,b_2^2,\\ldots, b_2^k,\\ldots,\nb_m^1,b_m^2,\\ldots,b_m^k,\n$$\nwhere\n$$\na_i=b_i^1\\vee b_i^2\\vee \\ldots \\vee b_i^k.\n$$\nSince $g$ is an automorphism of $\\A$ we can find a permutation\n$\\phi$ of $\\{1,\\ldots,m\\}$ such that\n$$\ng(a_i)=a_{\\phi(i)}\n$$\nfor all $i$. Similarly, we can find a function $\\psi\\colon\n\\{1,\\ldots,m\\}\\times \\{1,\\ldots, k\\}\\rightarrow \\{1,\\ldots,k\\}$ such that\nfor all $i$ and $j$\n$$\nf(b_i^j)=b_{\\phi^n(i)}^{\\psi(i,j)}.\n$$\nFor $f(a_i)=g^n(a_i)=a_{\\phi^n(i)}$ and thus $f(b_i^j)\\leq f(a_i)=\na_{\\phi^n(i)}$, whence $f(b_i^j)=b_{\\phi^n(i)}^{\\psi(i,j)}$ for some\n$\\psi(i,j)\\in\\{1,\\ldots, k\\}$. Also, since\n\\begin{align*}\nb_{\\phi^n(i)}^{\\psi(i,1)}\\vee b_{\\phi^n(i)}^{\\psi(i,2)}\n\\vee\\ldots\\vee b_{\\phi^n(i)}^{\\psi(i,k)}\n&=f(b_i^1\\vee b_i^2\\vee \\ldots \\vee b_i^k)\\\\\n&=f(a_i)\\\\\n&=a_{\\phi^n(i)}\\\\\n&=b_{\\phi^n(i)}^1\\vee b_{\\phi^n(i)}^2\\vee \\ldots \\vee\nb_{\\phi^n(i)}^k,\n\\end{align*}\nwe see that $\\psi(i,\\cdot)\\colon \\{1,\\ldots,k\\}\\rightarrow \\{1,\\ldots,k\\}$\nis a bijection for each $i$.\n\n\nLet $\\B_1=\\B_2=\\ldots=\\B_n=\\B$ and consider the free amalgam\n$\\otimes_\\A^l\\B_l$. We can now define the automorphism $h$ of\n$\\otimes_\\A^l\\B_l$ as follows.\n\\begin{align*}\nh(b_{i}^{j_1}\\otimes b_{i}^{j_2}\\otimes\\ldots\\otimes b_{i}^{j_n})\n&=b_{\\phi(i)}^{\\psi(i,j_n)}\\otimes b_{\\phi(i)}^{j_1}\\otimes\\ldots\n\\otimes b_{\\phi(i)}^{j_{n-1}}.\n\\end{align*}\nIt follows from the fact that $\\psi(i,\\cdot)$ is a bijection that\n$h$ also is a bijection of the atoms of $\\otimes_\\A^l\\B_l$ and thus\ndefines an automorphism of $\\otimes_\\A^l\\B_l$. Consider now\n\\begin{align*}\nh^n(b_{i}^{j_1}\\otimes b_{i}^{j_2}\\otimes\\ldots\\otimes b_{i}^{j_n})\n&=h^{n-1}(b_{\\phi(i)}^{\\psi(i,j_n)}\\otimes\nb_{\\phi(i)}^{j_1}\\otimes\\ldots \\otimes b_{\\phi(i)}^{j_{n-1}})\\\\\n&=h^{n-2}(b_{\\phi^2(i)}^{\\psi(\\phi(i),j_{n-1})}\\otimes\nb_{\\phi^2(i)}^{\\psi(i,j_n)}\\otimes b_{\\phi^2(i)}^{j_1}\\otimes\\ldots\n\\otimes b_{\\phi^2(i)}^{j_{n-2}})\\\\\n&=\\ldots\\\\\n&=b_{\\phi^n(i)}^{\\psi(\\phi^{n-1}(i),j_{1})}\\otimes\nb_{\\phi^n(i)}^{\\psi(\\phi^{n-2}(i),j_2)}\\otimes \\ldots \\otimes\nb_{\\phi^n(i)}^{\\psi(i,j_n)}.\n\\end{align*}\nThus,\n\\begin{align*}\nh^n(a_i\\otimes a_i\\otimes\\ldots&\\otimes a_i\\otimes b_{i}^{j_n})\\\\\n=&h^n\\big(\\bigvee_{j_1=1}^k \\bigvee_{j_2=1}^k \\ldots\n\\bigvee_{j_{n-1}=1}^k b_{i}^{j_1}\\otimes\nb_{i}^{j_2}\\otimes\\ldots\\otimes\nb_{i}^{j_n}\\big) \\\\\n=&\\bigvee_{j_1=1}^k \\bigvee_{j_2=1}^k \\ldots \\bigvee_{j_{n-1}=1}^k\nh^n\\big(b_{i}^{j_1}\\otimes\nb_{i}^{j_2}\\otimes\\ldots\\otimes b_{i}^{j_n}\\big)\\\\\n=&\\bigvee_{j_1=1}^k \\bigvee_{j_2=1}^k \\ldots \\bigvee_{j_{n-1}=1}^k\nb_{\\phi^n(i)}^{\\psi(\\phi^{n-1}(i),j_1)}\\otimes\nb_{\\phi^n(i)}^{\\psi(\\phi^{n-2}(i),j_2)}\\otimes \\ldots \\otimes\nb_{\\phi^n(i)}^{\\psi(i,j_n)}\\\\\n=&a_{\\phi^n(i)}\\otimes a_{\\phi^n(i)}\\otimes \\ldots \\otimes\na_{\\phi^n(i)}\\otimes b_{\\phi^n(i)}^{\\psi(i,j_n)}.\n\\end{align*}\nSimilarly,\n\\begin{align*}\nh(a_i\\otimes a_i\\otimes\\ldots\\otimes a_i)\n=&h\\big(\\bigvee_{j_1=1}^k \\bigvee_{j_2=1}^k \\ldots\n\\bigvee_{j_{n}=1}^k b_{i}^{j_1}\\otimes\nb_{i}^{j_2}\\otimes\\ldots\\otimes\nb_{i}^{j_n}\\big) \\\\\n=&\\bigvee_{j_1=1}^k \\bigvee_{j_2=1}^k \\ldots \\bigvee_{j_{n}=1}^k\nh\\big(b_{i}^{j_1}\\otimes\nb_{i}^{j_2}\\otimes\\ldots\\otimes b_{i}^{j_n}\\big)\\\\\n=&\\bigvee_{j_1=1}^k \\bigvee_{j_2=1}^k \\ldots \\bigvee_{j_{n}=1}^k\nb_{\\phi(i)}^{\\psi(i,j_n)}\\otimes b_{\\phi(i)}^{j_1}\\otimes\\ldots\n\\otimes b_{\\phi(i)}^{j_{n-1}}\\\\\n=&a_{\\phi(i)}\\otimes a_{\\phi(i)}\\otimes \\ldots \\otimes a_{\\phi(i)}.\n\\end{align*}\n\nWe now identify $\\B$ with the image of $\\B_n$ by the embedding\n$\\pi_n$ of $\\B_n$ into $\\otimes_\\A^l\\B_l$. Thus, the atoms of $\\B$\nare of the form\n$$\na_i\\otimes a_i\\otimes\\ldots\\otimes a_i\\otimes b_{i}^{j}\n$$\nand the atoms of $\\A$ are\n$$\na_i\\otimes a_i\\otimes\\ldots\\otimes a_i.\n$$\nMoreover, $g$ acts by\n\\begin{align*}\ng(a_i\\otimes a_i\\otimes\\ldots\\otimes a_i) &=g(a_i)\\otimes\ng(a_i)\\otimes\\ldots \\otimes g(a_i)\\\\\n&=a_{\\phi(i)}\\otimes a_{\\phi(i)}\\otimes \\ldots\\otimes a_{\\phi(i)},\n\\end{align*}\nwhile $f$ acts by\n\\begin{align*}\nf(a_i\\otimes a_i\\otimes\\ldots\\otimes a_i\\otimes b_{i}^{j})\n&=a_{\\phi^n(i)}\\otimes a_{\\phi^n(i)}\\otimes\\ldots\\otimes\na_{\\phi^n(i)}\\otimes b_{\\phi^n(i)}^{\\psi(i,j)}.\n\\end{align*}\nTherefore, $h$ extends $g$, while $h^n$ extends $f$, which was what\nwe wanted.\n\\end{proof}\n\n\\begin{prop}\\label{power}\nLet $n\\geq 1$. Then the generic measure preserving homeomorphism of\nCantor space is conjugate with its $n$'th power.\n\\end{prop}\n\n\\begin{proof}\nNotice first that a basic open set in ${\\rm Homeo}(2^\\N,\\mu)$ is of\nthe form\n$$\nU(h,\\A)=\\{g\\in {\\rm Homeo}(\\ca,\\mu)\\del g|_\\A=h|_\\A\\},\n$$\nwhere $\\A$ is a finite equidistributed subalgebra of $\\B_\\infty$ and\n$h\\in {\\rm Homeo}(\\ca,\\mu)$. We claim that for any $U(h,\\A)$ there\nis some finite equidistributed $\\B\\subseteq \\B_\\infty$ containing\n$\\A$ and some measure preserving homeomorphism $k$ leaving $\\B$\ninvariant, such that $U(k,\\B)\\subseteq U(h,\\A)$. To see this, suppose  $h$ and\n$\\A$ are given. Choose an equidistributed $\\B$ containing both $\\A$\nand $h(\\A)$ and notice that the partial automorphism $h\\colon \\A\\rightarrow\nh(\\A)$ of $\\B$ extends to an automorphism $\\hat h$ of $\\B$. So let\n$k$ be any measure preserving homeomorphism of $\\ca$ that extends\n$\\hat h$. Then $\\B$ is $k$-invariant while $U(k,\\B)\\subseteq\nU(h,\\A)$.\n\nFor simplicity, if $k$ is an automorphism of a finite equidistributed algebra $\\B$, we also write $U(k,\\B)$ to denote the set $\\{g\\in {\\rm Homeo}(\\ca,\\mu)\\del g|_\\B=k\\}$.\n\nLet now $C$ be the comeagre conjugacy class of ${\\rm\nHomeo}(\\ca,\\mu)$  and find dense open sets $V_i\\subseteq {\\rm\nHomeo}(\\ca,\\mu)$ such that $C=\\bigcap_iV_i$.  Enumerate the clopen\nsubsets of $\\ca$ as $a_0,a_1,a_2,\\ldots$. We shall define a sequence\nof finite equidistributed algebras $\\A_0\\subseteq \\A_1\\subseteq\n\\A_2\\subseteq \\ldots$ of clopen sets and automorphisms $g_i$ and\n$f_i$ of $\\A_i$ such that\n\\begin{itemize}\n  \\item[(1)] $a_i\\in \\A_{i+1}$,\n  \\item[(2)] $g_{i+1}$ extends $g_i$,\n  \\item[(3)] $f_{i+1}$ extends $f_i$,\n  \\item[(4)] $g_i^n=f_i$,\n  \\item[(5)] $U(g_{i+1},\\A_{i+1})\\subseteq V_{i}$,\n  \\item[(6)] $U(f_{i+1},\\A_{i+1})\\subseteq V_{i}$.\n\\end{itemize}\nTo begin, let $\\A_0$ be the trivial algebra with automorphism\n$g_0=f_0$. So suppose $\\A_i$, $g_i$, and $f_i$ are defined. We let\n$\\B$ be an equidistributed algebra containing both $a_i$  and $\\A_i$\nand let $h$ be any automorphism of $\\B$ extending $g_i$. As $V_i$ is\ndense open we can find some $U(k,\\C)\\subseteq V_i$, where $\\C$ is a\n$k$-invariant equidistributed algebra containing $\\B$ and $k$\nextends $h$. Again, as $V_i$ is dense open, we can  find some $U(p,\n\\D)\\subseteq V_i$, where $\\D$ is a equidistributed algebra\ncontaining $\\C$, $p$ a measure preserving homeomorphism leaving $\\D$\ninvariant and extending $k^n|_\\C$.\n\nNow, by Proposition \\ref{divisibility}, we can find an\nequidistributed algebra $\\E$ containing $\\D$ and an automorphism $q$\nof $\\E$  extending $k|_\\C$ such that $q^n$ extends $p|_\\D$. Finally,\nset $\\A_{i+1}=\\E$,\n$$\ng_{i+1}=q\\supseteq k|_\\C\\supseteq h\\supseteq g_i,\n$$\nand\n$$\nf_{i+1}=q^n\\supseteq p|_\\D\\supseteq k^n|_\\C\\supseteq g_i^n=f_i.\n$$\nThen $U(g_{i+1},\\A_{i+1})\\subseteq U(k,\\C)\\subseteq V_n$ and\n$U(f_{i+1},\\A_{i+1})\\subseteq U(p,\\D)\\subseteq V_n$.\n\nSet now $g=\\bigcup_ig_i$ and $f=\\bigcup_if_i$. By (1),(2), and (3),\n$f$ and $g$ are measure preserving automorphisms of $\\B_\\infty$ and,\nthus by Stone duality, measure preserving homeomorphisms of $\\ca$.\nAnd by (4), $g^n=f$, while by (5) and (6), $f, g\\in \\bigcap_iV_i=C$.\nThus, $f$ and $g$ belong to the comeagre conjugacy class and are\ntherefore mutually conjugate.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{prop}\nLet $G$ be  a Polish group with a comeagre conjugacy class. Then the\ngeneric element of $G$ is conjugate to its inverse.\n\\end{prop}\n\n\\begin{proof}\nLet $C$ be the comeagre conjugacy class of $G$. Then also $C^{-1}$ is\ncomeagre, so must intersect $C$ in some point $g$. Thus both $g$ and\n$g^{-1}$ are generic and hence conjugate.  Now, being conjugate with\nyour inverse is a conjugacy invariant property and thus holds\ngenerically in $G$.\n\\end{proof}\n\n\n\n\\begin{thm}\\label{homeomorphisms}\nLet $n\\neq 0$. Then the generic measure preserving homeomorphism of\nCantor space is conjugate with its $n$'th power and hence has roots\nof all orders.\n\nThus, for the generic measure preserving homeomorphism $g$, there is\nan action of $(\\Q,+)$ by measure preserving homeomorphisms of $\\ca$\nsuch that $g$ is the action by $1\\in \\Q$.\n\\end{thm}\n\n\\begin{proof}\nWe know that the generic $g$ is conjugate to all its positive powers\nand to $g^{-1}$. But then $g^{-1}$ is generic and thus conjugate to\n$(g^{-1})^n=g^{-n}$, whence $g$ is conjugate with $g^{-n}$, $n\\geq\n1$.\n\nSo suppose $g$ is generic and $n\\geq 1$. Then,  there is some $f$\nsuch that $(fgf^{-1} )^n=fg^nf^{-1} =g$, and hence $g$ has a generic\n$n$'th root, namely, $fgf^{-1}$. This means that we can define a\nsequence $g=g_1,g_2,\\ldots$ of generic elements such that $g_{n+1}$\nis an $n+1$'th root of $g_n$, $g_{n+1}^{n+1}=g_n$. The following\ntherefore defines an embedding of $(\\Q,+)$ into ${\\rm\nHomeo}(\\ca,\\mu)$ with $1=\\frac1{1!}\\mapsto g_1$,\n$$\n\\frac{k}{n!}\\mapsto g_{n}^k,\n$$\n$k\\in \\Z$, $n\\geq 1$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Powers of generic isometries}\n\n\n\\subsection{Free amalgams of metric spaces}Suppose $\\A$ and $\\B_1,\\ldots,\\B_n$ are non-empty finite metric spaces\nand $\\iota_i\\colon \\A\\hookrightarrow\\B_i$ is an isometric embedding for each $i$.  We define the {\\em free amalgam}\n$\\bigsqcup_{\\A}\\B_l$ of $\\B_1,\\ldots,\\B_n$ over $\\A$ and the embeddings $\\iota_1,\\ldots,\\iota_n$ as follows.\n\nDenote by $d_i$ the metric on $\\B_i$ for each $i$ and let $\\C_i=\\B_i\\setminus \\iota_i[\\A]$. By renaming elements, we can suppose that $\\C_1,\\ldots,\\C_n$ and $\\A$ are pairwise disjoint.\n\n\nWe then let the universe of $\\bigsqcup_{\\A}\\B_l$ be\n$\\A\\cup \\bigcup_{i=1}^n\\C_i$ and define the metric $\\partial$ by the\nfollowing conditions\n\\begin{itemize}\n\\item[(1)] $\\partial(x,y)=d_i(\\iota_ix,\\iota_iy)$ for $x,y\\in \\A$,\n\\item[(2)] $\\partial(x,y)=d_i(\\iota_ix,y)$ for $x\\in \\A$ and $y\\in \\C_i$,\n\\item[(3)] $\\partial(x,y)=d_i(x,y)$ for $x,y\\in \\C_i$,\n\\item[(4)] $\\partial(x,y)=\\min_{z\\in\\A}d_i(x,\\iota_iz)+d_j(\\iota_jz,y)$ for $x\\in \\C_i$ and $y\\in \\C_j$, $i\\neq j$.\n\\end{itemize}\nWe notice first that in (1) the definition is independent of $i$ since each $\\iota_i$ is an isometry. Also, a careful checking of the triangle inequality shows that this indeed defines a metric $\\partial$ on $\\A\\cup \\bigcup_{i=1}^n\\C_i$.\n\nWe define for each $i$ an isometric embedding $\\pi_i\\colon \\B_i\\hookrightarrow \\bigsqcup_{\\A}\\B_l$ by\n\\begin{itemize}\n\\item $\\pi_i(x)=x$ for $x\\in \\C_i$,\n\\item $\\pi_i(\\iota_ix)=x$ for $x\\in \\A$.\n\\end{itemize}\nNotice that in this way the following diagram commutes\n$$\n\\begin{CD}\n\\A @>\\iota_i>> \\B_i\\\\\n@V\\iota_jVV   @VV\\pi_iV\\\\\n\\B_j@>>\\pi_j> \\bigsqcup_\\A\\B_l\n\\end{CD}\n$$\n\n\n\n\n\n\n\n\n\\subsection{Roots of isometries}\n\n\\begin{prop}\\label{divisibility isometry}\nLet $\\bf A\\subseteq B$ be finite rational metric spaces, $f$ and isometry of $\\A$ and $g$ an isometry of $\\B$ leaving $\\A$ invariant and such that $f^n=g|_\\A$ for some $n\\geq 1$. Then there is a finite rational metric\nspace $\\bf D\\supseteq B$ and an isometry $h$ of $\\D$ such that $h^n$ leaves $\\B$ invariant and  $h^n|_\\B=g$.\n\\end{prop}\n\n\n\n\\begin{proof}\nLet $\\B_1=\\ldots=\\B_{n}=\\B$ and define isometric embeddings $\\iota_i\\colon \\A\\hookrightarrow\\B_i$ by\n$$\n\\iota_i(x)=f^{-i}(x).\n$$\nTo distinguish between the different copies of $\\B$, we let for $x\\in \\B\\setminus \\A$, $x^i$ denote the copy of $x$ in $\\C_i=\\B_i\\setminus \\iota_i[\\A]=\\B_i\\setminus\\A$. Note also that $\\B=\\B_1=\\ldots=\\B_n$ all have the same metric, which we denote by $d$.\nWe now define $h$ on $\\bigsqcup_\\A\\B_l$ as follows.\n\\begin{itemize}\n\\item $h(x)=f(x)$ for $x\\in \\A$,\n\\item $h(x^i)=x^{i+1}$ for $x\\in \\B\\setminus \\A$ and $1\\leq i<n$,\n\\item $h(x^n)=(gx)^1$ for $x\\in \\B\\setminus \\A$.\n\\end{itemize}\nNow, obviously, $h$ is a permutation of $\\A$ and for $1\\leq i<n$, $h$ is a bijection between $\\C_i$ and $\\C_{i+1}$. Moreover, $h$ is a bijection between $\\C_n$ and $\\C_1$. Therefore, $h$ is a permutation of $\\bigsqcup_\\A\\B_l$. We check that $h$ is $1$-Lipschitz.\n\nSuppose first that $x,y\\in \\A$. Then\n\\begin{align*}\n\\partial(hx,hy)&=\\partial(fx,fy)\\\\\n&=d(\\iota_ifx,\\iota_ify)\\\\\n&=d(f^{-i}fx,f^{-i}fy)\\\\\n&=d(f^{1-i}x,f^{1-i}y)\\\\\n&=d(f^{-i}x,f^{-i}y)\\\\\n&=d(\\iota_ix,\\iota_iy)\\\\\n&=\\partial(x,y).\n\\end{align*}\nAlso, $h$ is clearly an isometry between $\\C_i$ and $\\C_{i+1}$ for $1\\leq i< n$. So consider the case $\\C_n$. Fix $x,y\\in \\B\\setminus \\A$. Then\n\\begin{align*}\n\\partial(h(x^n),h(y^n))&=\\partial((gx)^1,(gy)^1)\\\\\n&=d(gx,gy)\\\\\n&=d(x,y)\\\\\n&=\\partial(x^n,y^n).\n\\end{align*}\nNow, if $x\\in \\A$, $y\\in \\B\\setminus \\A$, and $1\\leq i<n$, then\n\\begin{align*}\n\\partial(h(x),h(y^i))&=\\partial(fx,y^{i+1})\\\\\n&=d(\\iota_{i+1}fx,y)\\\\\n&=d(f^{-(i+1)} fx,y)\\\\\n&=d(f^{-i}x,y)\\\\\n&=d(\\iota_ix,y)\\\\\n&=\\partial(x,y^i).\n\\end{align*}\nAlso, if $x\\in \\A$, $y\\in \\B\\setminus \\A$, then\n\\begin{align*}\n\\partial(h(x),h(y^n))&=\\partial(fx,(gy)^1)\\\\\n&=d(\\iota_{1}fx,gy)\\\\\n&=d(f^{-1}fx,gy)\\\\\n&=d(x,gy)\\\\\n&=d(g^{-1} x,y)\\\\\n&=d(f^{-n}x,y)\\\\\n&=d(\\iota_nx,y)\\\\\n&=\\partial(x,y^n).\n\\end{align*}\nAnd finally, if $x,y\\in \\B\\setminus \\A$ and $1\\leq i< j\\leq n$, we pick $z\\in \\A$ such that the distance $\\partial(x^i,y^j)$ is witnessed by $z$, i.e.,\n\\begin{align*}\n\\partial(x^i,y^j)=d(x,\\iota_iz)+d(\\iota_jz,y)=d(x,f^{-i}z)+d(f^{-j}z,y).\n\\end{align*}\nAssume first that $j<n$. Then\n\\begin{align*}\n\\partial(h(x^i),h(y^j))&=\\partial(x^{i+1},y^{j+1})\\\\\n&\\leq d(x,\\iota_{i+1}fz)+d(\\iota_{j+1} fz,y)\\\\\n&= d(x,f^{-i}z)+d(f^{-j}z,y)\\\\\n&=\\partial(x^i,y^j).\n\\end{align*}\nAnd if $j=n$, we have\n\\begin{align*}\n\\partial(h(x^i),h(y^n))&=\\partial(x^{i+1},(gy)^1)\\\\\n&\\leq d(x,\\iota_{i+1}fz)+d(\\iota_{1} fz,gy)\\\\\n&= d(x,f^{-i}z)+d(z,f^ny)\\\\\n&= d(x,f^{-i}z)+d(f^{-n}z,y)\\\\\n&=\\partial(x^i,y^n).\n\\end{align*}\nThus, $h$ is an isometry of $\\bigsqcup_\\A\\B_l$.\n\nNow see $g$ and $f$ as isometries of the first copy $\\B_1$ of $\\B$, i.e., $g(x^1)=(gx)^1$ for $x^1\\in \\C_1$. Let $\\pi_1\\colon \\B_1\\hookrightarrow \\bigsqcup_{\\A}\\B_l$ be the canonical isometric embedding defined by\n\\begin{itemize}\n\\item $\\pi_1(x^1)=x^1$ for $x^1\\in \\C_1$,\n\\item $\\pi_1(\\iota_1x)=x$ for $x\\in \\A$.\n\\end{itemize}\nTo finish the proof, we need to show that the following diagram commutes\n$$\n\\begin{CD}\n\\B_1 @>g>> \\B_1\\\\\n@V\\pi_1VV   @VV\\pi_1V\\\\\n\\bigsqcup_{\\A}\\B_l@>>h^n> \\bigsqcup_{\\A}\\B_l\n\\end{CD}\n$$\nFirst, suppose $y=\\iota_1x\\in \\A$. Then\n\\begin{align*}\nh^n\\pi_1y=&h^n\\pi_1\\iota_1x=h^nx=f^nx\\\\\n=&\\pi_1\\iota_1f^nx\n=\\pi_1f^{-1} f^nx\n=\\pi_1f^nf^{-1} x\\\\\n=&\\pi_1f^n\\iota_1x\n=\\pi_1f^ny\n=\\pi_1gy.\n\\end{align*}\nNow suppose that $x\\in \\B\\setminus \\A$. Then\n\\begin{align*}\nh^n\\pi_1(x^1)=&h^n(x^1)=h(x^n)=(gx)^1\n=\\pi_1(gx)^1=\\pi_1g(x^1).\n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{prop}\\label{power isometry}\nLet $n\\geq 1$. Then the generic isometry of\nthe rational Urysohn metric space is conjugate with its $n$'th power.\n\\end{prop}\n\n\\begin{proof}A basic open set in ${\\rm Iso}(\\Q\\U)$ is on\nthe form\n$$\nU(h,\\A)=\\{g\\in {\\rm Iso}(\\Q\\U)\\del g|_\\A=h|_\\A\\},\n$$\nwhere $\\A$ is a finite subspace  of $\\Q\\U$ and\n$h\\in {\\rm Iso}(\\Q\\U)$. We claim that for any $U(h,\\A)$ there\nis some finite $\\B\\subseteq \\Q\\U$ containing\n$\\A$ and some isometry $k$ leaving $\\B$\ninvariant, such that $U(k,\\B)\\subseteq U(h,\\A)$.\nFor if $h$ and $\\A$ are given, choose by Theorem \\ref{solecki} some finite $\\B\\subseteq \\Q\\U$ containing both $\\A$\nand $h(\\A)$ such that the partial isometry $h\\colon \\A\\rightarrow h(\\A)$ of $\\A\\cup h(\\A)$ extends to an isometry $\\hat h$ of $\\B$. Let\n$k$ be any isometry of $\\Q\\U$ that extends $\\hat h$. Then $\\B$ is $k$-invariant while $U(k,\\B)\\subseteq\nU(h,\\A)$.\n\nAgain, if $k$ is an isometry of some finite $\\B\\subseteq \\Q\\U$, we let $U(k,\\B)=\\{g\\in {\\rm Iso}(\\Q\\U)\\del g|_\\B=k\\}$.\n\n\n\nLet now $C$ be the comeagre conjugacy class of ${\\rm Iso}(\\Q\\U)$  and find dense open sets $V_i\\subseteq {\\rm Iso}(\\Q\\U)$ such that $C=\\bigcap_iV_i$.  Enumerate the points of $\\Q\\U$ as $a_0,a_1,a_2,\\ldots$. We shall define a sequence\nof finite subsets $\\A_0\\subseteq \\A_1\\subseteq\n\\A_2\\subseteq \\ldots\\subseteq \\Q\\U$ and isometries $g_i$ and\n$f_i$ of $\\A_i$ such that\n\\begin{itemize}\n  \\item[(1)] $a_i\\in \\A_{i+1}$,\n  \\item[(2)] $g_{i+1}$ extends $g_i$,\n  \\item[(3)] $f_{i+1}$ extends $f_i$,\n  \\item[(4)] $g_i^n=f_i$,\n  \\item[(5)] $U(g_{i+1},\\A_{i+1})\\subseteq V_{i}$,\n  \\item[(6)] $U(f_{i+1},\\A_{i+1})\\subseteq V_{i}$.\n\\end{itemize}\nTo begin, let $\\A_0=\\tom$ with trivial isometries $g_0=f_0$. So suppose $\\A_i$, $g_i$, and $f_i$ are defined. We let\n$\\B\\subseteq \\Q\\U$ be a finite subset containing both $a_i$  and $\\A_i$\nand such that there is some isometry $h$ of $\\B$ extending $g_i$. As $V_i$ is\ndense open we can find some $U(k,\\C)\\subseteq V_i$, where $\\C\\subseteq \\Q\\U$ is a\n$k$-invariant finite set containing $\\B$ and $k$\nextends $h$. Again, as $V_i$ is dense open, we can  find some $U(p,\n\\D)\\subseteq V_i$, where $\\D\\subseteq \\Q\\U$ is a finite set\ncontaining $\\C$, $p$ an isometry of $\\Q\\U$ leaving $\\D$\ninvariant and extending $k^n|_\\C$.\n\nNow, by Proposition \\ref{divisibility isometry}, we can find a finite subset $\\E\\subseteq \\Q\\U$ containing $\\D$ and an isometry $q$\nof $\\E$  extending $k|_\\C$ such that $q^n$ extends $p|_\\D$. Finally,\nset $\\A_{i+1}=\\E$,\n$$\ng_{i+1}=q\\supseteq k|_\\C\\supseteq h\\supseteq g_i,\n$$\nand\n$$\nf_{i+1}=q^n\\supseteq p|_\\D\\supseteq k^n|_\\C\\supseteq g_i^n=f_i.\n$$\nThen $U(g_{i+1},\\A_{i+1})\\subseteq U(k,\\C)\\subseteq V_n$ and\n$U(f_{i+1},\\A_{i+1})\\subseteq U(p,\\D)\\subseteq V_n$.\n\nSet now $g=\\bigcup_ig_i$ and $f=\\bigcup_if_i$. By (1),(2), and (3),\n$f$ and $g$ are isometries of $\\Q\\U$.\nAnd by (4), $g^n=f$, while by (5) and (6), $f, g\\in \\bigcap_iV_i=C$.\nThus, $f$ and $g$ belong to the comeagre conjugacy class and are\ntherefore mutually conjugate.\n\\end{proof}\n\nNow in exactly the same way as for measure preserving homeomorphisms, we can prove\n\n\\begin{thm}\\label{isometries}\nLet $n\\neq 0$. Then the generic isometry of\nthe rational Urysohn metric space is conjugate with its $n$'th power and hence has roots\nof all orders.\n\nThus, for the generic isometry $g$, there is\nan action of $(\\Q,+)$ by isometries of $\\Q\\U$\nsuch that $g$ is the action by $1\\in \\Q$.\n\\end{thm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Comeagre conjugacy classes and Baire category}\nWe now set up the framework allowing us to pass between a Polish group and a dense subgroup with a comeagre conjugacy class.\n\n\\begin{defi}\nLet $\\pi\\colon X\\rightarrow Y$ be a Borel map between Polish spaces. We say that $\\pi$ is {\\em categorical} if the following conditions hold.\n\\begin{itemize}\n\\item[(1)] If $A\\subseteq X$ is a nonmeagre analytic set, then $\\pi(A)\\subseteq Y$ is nonmeagre,\n\\item[(2)] if $A\\subseteq X$ is a comeagre analytic set, then $\\pi(A)\\subseteq Y$ is comeagre,\n\\item[(3)] if $B\\subseteq Y$ is a meagre analytic set, then $\\pi^{-1}(B)\\subseteq X$ is meagre,\n\\item[(4)] if $B\\subseteq Y$ is a nonmeagre analytic set, then $\\pi^{-1}(B)\\subseteq X$ is nonmeagre,\n\\item[(5)] and if $B\\subseteq Y$ is a comeagre analytic set, then $\\pi^{-1}(B)\\subseteq X$ is comeagre.\n\\end{itemize}\nWe say that $\\pi$ is {\\em strongly categorical} if, moreover,\n\\begin{itemize}\n\\item[(6)] If $A\\subseteq X$ is a meagre analytic set, then $\\pi(A)\\subseteq Y$ is meagre.\n\\end{itemize}\n\\end{defi}\nWe notice that (1)$\\Leftrightarrow$(3)$\\Leftrightarrow$(5) and  (2)$\\Leftrightarrow$(4). But (2)$\\not\\Rightarrow$(1)$\\not\\Rightarrow$(2). However, if $\\pi(X)$ is comeagre in $Y$, then (6)$\\Rightarrow$(2).\n\\begin{lemme}\nLet $\\pi\\colon X\\rightarrow Y$ be a surjective, continuous, and open map between Polish spaces. Then $\\pi$ is categorical.\n\\end{lemme}\n\n\\begin{proof}\n(5) follows from exercise (8.45) in Kechris \\cite{kechris}. So to see (2), assume towards a contradiction that $A\\subseteq X$ is comeagre, but $\\pi(A)$ is not comeagre. Then, pick some non-empty open $U\\subseteq Y$ in which $\\pi(A)$ is meagre and let $X_0=\\pi^{-1}(U)$, which is non-empty and  open.  The map $\\pi\\colon X_0\\rightarrow U$ is then surjective, continuous, and open, whereby (5) holds for this map. But then $\\pi(A\\cap X_0)$ is nonmeagre in $U$ being the image of the nonmeagre set $A\\cap X_0$ and using the equivalence of (1) and (5). This contradicts that $\\pi(A)$ is meagre in $U$.\n\\end{proof}\n\n\n\n\nBefore we state the next result, recall that, by a theorem of Marker and Sami \\cite{becker}, if a Polish group $G$ has a comeagre conjugacy class $C$, then $C$ is $G_\\delta$ in $G$ and thus a Polish space in its induced topology. Moreover, $G$ acts continuously and transitively on $C$ by conjugation.\n\n\\begin{prop}Suppose $G$ is a Polish group with a comeagre conjugacy class $C$. Suppose also that an element of $C$ is conjugate with all its non-zero powers.\nThen, for every $n\\neq 0$, the map $\\pi\\colon C\\rightarrow C$ defined by\n$$\n\\pi(g)=g^n\n$$\nis a surjective, continuous, and open $G$-map. In particular, $\\pi$ is categorical.\n\\end{prop}\n\n\\begin{proof}\nFirst notice that $\\pi$ is a continuous $G$-map, where $G$ acts on $C$ by conjugation,\n$$\n\\pi(ghg^{-1})=(ghg^{-1})^n=gh^ng^{-1}=g\\pi(h)g^{-1}.\n$$\nAlso, $\\pi$ is surjective. For if $f\\in C$, we know that $f$ and $f^n$ are conjugate and thus for some $g\\in G$, $gf^ng^{-1}=f$, whence $\\pi(gfg^{-1})=gf^ng^{-1}=f$.\nFinally, to see that $\\pi$ is open, suppose that $U\\subseteq C$ is open and let $g\\in U$. As $G$ acts continuously on $C$ by the conjugacy action, which we write as $h.g=hgh^{-1}$, we can find some open neighbourhood $V$ of $1$ in $G$ such that $V.g\\subseteq U$. But then by Effros' Theorem (see Becker and Kechris \\cite{becker}) the set $V.\\pi(g)$ is open in $C$ and $\\pi(g)\\in V.\\pi(g)=\\pi(V.g)\\subseteq \\pi(U)$. Thus $\\pi(U)$ is open.\n\\end{proof}\n\n\n\n\\begin{prop}\\label{transfer}\nSuppose $H$ is a Polish group with a dense subgroup $G$, which is Polish in a finer topology. Assume that $G$ has a comeagre conjugacy class $C$ whose elements are conjugate with their non-zero powers.\nAssume $n\\neq 0$ and let $\\pi\\colon H\\rightarrow H$ be the continuous $H$-map\n$$\n\\pi(g)=g^n.\n$$\nThen $\\pi$ is categorical.\n\\end{prop}\n\n\n\\begin{proof}\nSuppose towards a contradiction that $A\\subseteq H$ is nonmeagre, while $\\pi(A)$ is meagre. Find closed nowhere dense sets $F_n\\subseteq H$ such that $\\pi(A)\\subseteq \\bigcup_nF_n$. Then $\\pi^{-1}(F_n)$ is closed and $A\\subseteq \\bigcup_n\\pi^{-1}(F_n)$ is nonmeagre, whereby some $\\pi^{-1}(F_n)$ must be nonmeagre and thus have non-empty interior $U$. Now, $C$ is dense in $H$ and thus $U\\cap C$ is open in the topology of $C$, whereby $\\pi(U\\cap C)$ is nonmeagre and hence somewhere dense in $C$. Therefore,  $\\pi(U\\cap C)\\subseteq F_n$ is also somewhere dense in $H$, which is a contradiction. Thus, images of nonmeagre sets are nonmeagre.\n\nWe now need to show that also images of comeagre sets are comeagre. So suppose $A\\subseteq H$ is comeagre and let $H_0\\leq H$ be a countable dense subgroup. Set\n$$\nB=\\bigcap_{g\\in H_0}gAg^{-1},\n$$\nwhich is comeagre and $H_0$-invariant. But then $\\pi(B)$ is nonmeagre and $H_0$-invariant, whence, as the action of $H_0$ on $H$ is topologically transitive, $\\pi(B)$ must be comeagre. Thus also $\\pi(A)$ is comeagre.\n\\end{proof}\n\n\n\\section{Measure preserving automorphisms and isometries}\nWe can now apply our Theorems \\ref{homeomorphisms} and \\ref{isometries} in combination with Proposition \\ref{transfer} to deduce  results, the first of which is due to J.L.F. King, about respectively the automorphism group of a standard probability space and the isometry group of the Urysohn metric space. \n\nBefore stating the next result we recall the so called {\\em weak topology} on the group ${\\rm Aut}([0,1],\\lambda)$ of Lebesgue measure preserving automorphisms of the unit interval: It is the weakest topology such that for all Borel sets $A,B\\subseteq [0,1]$ the map $g\\mapsto\\lambda(gA\\triangle B)$ is continuous.\n\n\\begin{thm}[J.L.F. King \\cite{king}]\\label{king}\nThe generic measure preserving automorphism of the unit interval has roots of all orders.\n\\end{thm}\n\n\\begin{proof}\nIt is well-known that the group of Haar measure preserving homeomorphisms embeds continuously as a dense subgroup of the group $H$ of measure preserving automorphisms of the unit interval (see, e.g., Halmos \\cite{halmos}). Thus, by Theorem \\ref{isometries} and Proposition \\ref{transfer}, we know that for each $n\\neq 0$, the map $\\pi\\colon H\\rightarrow H$ given by $\\pi(g)=g^n$ is categorical. In particular, $\\pi(H)$ is comeagre in $H$ and thus the generic element of $H$ has an $n$'th root. As $n\\geq 1$ is arbitrary, this thus holds for all $n\\geq 1$ simultaneously.\n\\end{proof}\n\nWe should mention that there are now significatly better results known for the generic automorphism. Thus T. de la Rue and J. de Sam Lazaro \\cite{rue} prove, in response to a question of J.L.F. King \\cite{king}, that the generic automorphism is the time $1$ map of a measure preserving $\\R$-flow and hence of course has roots of all orders.\n\n\\begin{thm}\nThe generic isometry of the Urysohn metric space has roots of all orders.\n\\end{thm}\n\n\n\\begin{proof}\nWe repeat the proof of Theorem \\ref{king} using that the isometry group of the rational Urysohn metric space embeds continuously and densely into the isometry group of the Urysohn metric space.\n\\end{proof}\nIt is important to notice that there are in fact isometries of the Urysohn space without square roots. This is proved by J. Melleray in his thesis \\cite{melleray}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Topological similarity and Rohlin's Lemma for isometries}\n\n\nSuppose $G$ is a Polish group and $f,g\\in G$. We say that $f$ and $g$ are {\\em topologically similar} if the topological groups $\\langle f\\rangle\\leq G$ and $\\langle g\\rangle\\leq G$ are isomorphic.\n\nNotice first that any $f$ is topologically similar to $f^{-1}$. For if $\\psi(f^n)=f^{-n}$, then $\\psi$ is an involution homeomorphism, since inversion is continuous in $G$.\nOf course, if $f$ and $g$ have infinite order, then any isomorphic homeomorphism $\\phi$ between $\\langle f\\rangle$ and $\\langle g\\rangle$ must send the generators to the generators and so either $\\phi(f)=g$ or $\\phi(f)=g^{-1}$. But then composing with $\\psi$ we can always suppose that $\\phi(f)=g$.\n\nNotice also that as each $\\langle f\\rangle$ is metrisable, $f$ and $g$ are topologically similar if and only if for all increasing sequences $(s_n)\\subseteq \\N$, $f^{s_n}\\Lim{n}1$ if and only if $g^{s_n}\\Lim{n}1$. Thus, in particular, topological similarity is a coanalytic equivalence relation. We notice also that topological similarity is really independent of the ambient group $G$. For example, if $G$ is topologically embedded into another Polish group $H$, then $f$ and $g$ are topologically similar in $G$ if and only if they are topologically similar in $H$.\n\nTopological similarity is an obvious invariant for conjugacy, that is, if there is any way to make $f$ and $g$ conjugate in some Polish group, then they have to be topologically similar.\n\n\nOf particular interest are the cases $G={\\rm Aut}([0,1],\\lambda)$, $G=U(\\ell_2)$, and $G={\\rm Iso}(\\U)$. Here of course ${\\rm Aut}([0,1],\\lambda)$ sits inside of $U(\\ell_2)$ via the Koopman representation and two measure preserving transformations $f$ and $g$ are said to be {\\em spectrally equivalent} if they are conjugate in $U(\\ell_2)$. By the spectral theorem, spectral equivalence is Borel. Also, topological similarity is coarser than spectral equivalence. To see this, we notice that mixing is not a topological similarity invariant, whereas it is a spectral invariant. For if $f$ is mixing, then the automorphism $f\\oplus {\\rm id}$ is a non-mixing transformation of $[0,1]\\oplus[0,1]$ but generates a discrete subgroup of ${\\rm Aut}([0,1]\\oplus [0,1],\\lambda\\oplus \\lambda)$. So taking a transformation $h\\in {\\rm Aut}([0,1],\\lambda)$ conjugate with $f\\oplus {\\rm id}$, we see that $f$ and $h$ are topologically similar, since they both generate discrete groups.  A survey of the closely related topic of topological torsion elements in topological groups is given by Dikranjan in \\cite{dikranjan}.\n\n\n\n\\begin{prop}Let $G$ be a non-trivial Polish group  such that for all infinite $S\\subseteq \\N$ the set\n$\\mathbb A(S)=\\{g\\in G\\del \\e s\\in S\\; g^s=1\\}$ is dense. Then every topological similarity class of $G$ is meagre.\n\nMoreover, for every infinite  $S\\subseteq \\N$ the set\n$$\n\\mathbb C(S)=\\{g\\in G\\del \\e (s_n)\\subseteq S\\;\\;  g^{s_n}\\Lim{n} 1\\}.\n$$\nis dense $G_\\delta$ and invariant under topological similarity.\n\\end{prop}\n\n\n\\begin{proof}\nLet $V_0\\supseteq V_1\\supseteq\\ldots$ be a basis of open neighbourhoods of the identity and set\n$$\n\\mathbb B(S,k)=\\{g\\in G\\del \\e n\\in S\\setminus[1,k]\\;\\;g^n\\in V_k\\},\n$$\nwhich is open and dense since it contains $\\mathbb A(S\\setminus[1,k])$. Now set\n\\begin{align*}\n\\mathbb C(S)&=\\{g\\in G\\del \\e (s_n)\\subseteq S\\;\\;  g^{s_n}\\Lim{n}1\\}\\\\\n&=\\{g\\in G\\del \\a k\\; \\e n\\in S\\setminus [1,k]\\;\\; g^n\\in V_k\\}\\\\\n&=\\bigcap_k\\mathbb B(S,k).\n\\end{align*}\nThen $\\mathbb C(S)$ is invariant under topological similarity and dense $G_\\delta$.\n\nNow if some topological similarity class $C$ was nonmeagre, then\n$$\nC\\subseteq \\bigcap_{\\substack{S\\subseteq \\N\\\\\\text{ infinite}}}\\mathbb C(S)\n$$\nand hence for all $g\\in C$, $g^n\\Lim{n}1$, implying that $g=1$, which is impossible.\n\\end{proof}\n\n\nSince by Rohlin's Lemma the sets $\\{g\\in {\\rm Aut}([0,1],\\lambda)\\del \\e s\\in S\\; g^s=1\\}$ are dense in ${\\rm Aut}([0,1],\\lambda)$ for all infinite $S\\subseteq \\N$, we have\n\\begin{cor}\nEvery topological similarity class is meagre in ${\\rm Aut}([0,1],\\lambda)$.\n\\end{cor}\nThis improves a result of Rohlin saying that all conjugacy classes are meagre in ${\\rm Aut}([0,1],\\lambda)$.\nWe clearly see the importance of Rohlin's Lemma in these matters. However, interestingly, Rohlin's Lemma can also be used to prove the existence of {\\em dense} conjugacy classes in ${\\rm Aut}([0,1],\\lambda)$.\n\nWe now have the following analogue of Rohlin's Lemma for isometries of the Urysohn metric space.\n\\begin{prop}[Rohlin's Lemma for isometries]\\label{rohlin}\nSuppose $S\\subseteq \\N$ is infinite. Then the set\n$$\n\\{g\\in {\\rm Iso}(\\U)\\del \\e n\\in S\\; g^n=1\\}\n$$\nis dense in ${\\rm Iso}(\\U)$.\n\\end{prop}\n\n\nA {\\em finite circular order} is a finite subset $\\mathbb F$ of the unit circle $S^1$. If $x\\in \\mathbb F$, we denote by $x^+$ the first $y\\in \\mathbb F$ encountered by moving counterclockwise around $S^1$ beginning at $x$. We then denote $x$ by $y^-$, i.e., $x^+=y$ if and only if $y^-=x$.\n\n\n\n\\begin{lemme}\nSuppose $h$ is an isometry of $\\U$ and $\\delta>0$. Then for all finite $\\A\\subseteq \\U$ there is an isometry $f$ of $\\U$ such that $d(f(a),h(a))\\leq \\delta$ for all $a\\in \\A$ while $d(a,f(b))\\geq \\delta$ for all $a,b\\in \\A$.\n\\end{lemme}\n\n\n\\begin{proof}\nLet $\\B=\\A\\cup h[\\A]$ and let $\\C=\\B\\times \\{0,\\delta\\}$ be equipped with the $\\ell_1$-metric $d_1((b,x),(b',y))=d(b,b')+|x-y|$. Clearly, $\\B$ is isometric with $\\B\\times \\{0\\}$ and $\\B\\times \\{\\delta\\}$, so we can assume that $\\B$ is actually $\\B\\times \\{0\\}\\subseteq \\C\\subseteq \\U$. Now, let $f$ be any isometry of $\\U$ such that $f(a,0)=(h(a),\\delta)$ for $a\\in \\A$.\n\\end{proof}\n\nNow for the proof of Proposition \\ref{rohlin}.\n\\begin{proof}\nSuppose $\\A\\subseteq \\U$ is finite, $h$ an isometry of $\\U$, and $\\epsilon>0$. We wish to find some isometry $g$ such that $d(g(a),h(a))<\\epsilon$ for all $a\\in \\A$ and such that for some $s\\in S$, $g^s=1$. Find first some $f$ such that $d(f(a),h(a))<\\epsilon$ for all $a\\in \\A$ while $d(a,f(b))>\\epsilon\/2$ for all $a,b\\in \\A$. It is therefore enough to find some $g$ that agrees with $f$ on $\\A$ while $g^s=1$ for some $s\\in S$.\n\nWe let $\\Delta={\\rm diam}(\\A\\cup f[\\A])$ and $\\delta=\\min(d(x,f(y))\\del x,y\\in \\A)$. Fix a  number $s\\in S$ such that $\\delta\\cdot (s-2)\\geq\\Delta$ and take a finite circular order $\\mathbb F$ of cardinality $s$. Now let\n$$\n\\B=\\{a\\!\\bullet\\! x\\del a\\in \\A\\;\\&\\; x\\in \\mathbb F\\},\n$$\nwhere $a\\!\\bullet\\! x$ are formally new points.\n\nA {\\em path} in $\\B$ is a sequence $p=(a_0\\!\\bullet\\! x_0,a_1\\!\\bullet\\! x_1,\\ldots,a_n\\!\\bullet\\! x_n)$ where $n\\geq 1$ and such that for each $i$, $x_{i+1}$ is either $x_i^-$,  $x_i$, or $x_i^+$. We define the {\\em length} of a path by\n$$\n\\ell(p)=\\sum_{i=0}^{n-1}\\rho(a_i\\!\\bullet\\! x_i,a_{i+1}\\!\\bullet\\! x_{i+1}),\n$$\nwhere\n$$\n\\rho(a\\!\\bullet\\! x,b\\!\\bullet\\! y)=\\left\\{\n                              \\begin{array}{ll}\n                                d(a,b), & \\hbox{if $y=x$;} \\\\\n                                d(a,f(b)), & \\hbox{if $y=x^+$;} \\\\\n                                d(f(a),b), & \\hbox{if $y=x^-$,}\n                              \\end{array}\n                            \\right.\n$$\nand put $|p|=n+1$.\n\nTherefore, if $\\breve{p}$ denotes the reverse path of $p$ and $p\\centerdot q$ the concatenation of two paths (whenever it is defined), then $\\ell(\\breve{p})=\\ell(p)$ and $\\ell(p\\centerdot q)=\\ell(p)+\\ell(q)$. Thus, $\\ell$ is the distance function in a finite graph with weighted edges and hence the following defines a  a metric on $\\B$\n$$\nD(a\\!\\bullet\\! x,b\\!\\bullet\\! y)=\\inf\\big(\\ell(p)\\del p \\text{ is  a path with initial point } a\\!\\bullet\\! x \\text{ and end point } b\\!\\bullet\\! y\\big).\n$$\n\nWe say that two paths are {\\em equivalent} if they have the same initial point and the same end point. We also say that a path $p$ is {\\em positive} if either $p=(a\\!\\bullet\\! x,b\\!\\bullet\\! x)$ for some $x\\in \\mathbb F$ or $p=(a_0\\!\\bullet\\! x_0,a_1\\!\\bullet\\! x_1,\\ldots,a_n\\!\\bullet\\! x_n)$, where $x_{i+1}=x_i^+$ for all $i$. Similarly, $p$ is {\\em negative} if either $p=(a\\!\\bullet\\! x,b\\!\\bullet\\! x)$ for some $x\\in \\mathbb F$ or $p=(a_0\\!\\bullet\\! x_0,a_1\\!\\bullet\\! x_1,\\ldots,a_n\\!\\bullet\\! x_n)$, where $x_{i+1}=x_i^-$ for all $i$. So $p$ is positive if and only if $\\breve{p}$ is negative. Notice also that if $p$ is positive, then $\\ell(p)\\geq \\delta\\cdot (|p|-2)$.\n\n\n\\begin{lemme}\nFor every path $p$ there is an equivalent path $q$, with $\\ell(q)\\leq \\ell(p)$, which is either positive or negative.\n\\end{lemme}\n\n\\begin{proof}\nIf $p$ is not either positive or negative, then there is a segment of $p$ of one of the following forms\n\\begin{equation*}\n\\begin{aligned}\n&(1)\\quad (a\\!\\bullet\\! x,b\\!\\bullet\\! x,c\\!\\bullet\\! x),\\\\\n&(2)\\quad (a\\!\\bullet\\! x^+,b\\!\\bullet\\! x,c\\!\\bullet\\! x),\\\\\n&(3)\\quad (a\\!\\bullet\\! x^-,b\\!\\bullet\\! x,c\\!\\bullet\\! x),\\\\\n&(4)\\quad (a\\!\\bullet\\! x,b\\!\\bullet\\! x^+,c\\!\\bullet\\! x),\n\\end{aligned}\\qquad\\qquad\n\\begin{aligned}\n&(5)\\quad (a\\!\\bullet\\! x,b\\!\\bullet\\! x^-,c\\!\\bullet\\! x),\\\\\n&(6)\\quad (a\\!\\bullet\\! x,b\\!\\bullet\\! x,c\\!\\bullet\\! x^+),\\\\\n&(7)\\quad (a\\!\\bullet\\! x,b\\!\\bullet\\! x,c\\!\\bullet\\! x^-).\\\\\n\\,\n\\end{aligned}\n\\end{equation*}\nWe replace these by respectively\n\\begin{equation*}\n\\begin{aligned}\n&(1')\\quad (a\\!\\bullet\\! x,c\\!\\bullet\\! x),\\\\\n&(2')\\quad (a\\!\\bullet\\! x^+,c\\!\\bullet\\! x),\\\\\n&(3')\\quad (a\\!\\bullet\\! x^-,c\\!\\bullet\\! x),\\\\\n&(4')\\quad (a\\!\\bullet\\! x,c\\!\\bullet\\! x),\n\\end{aligned}\\qquad\\qquad\n\\begin{aligned}\n&(5')\\quad (a\\!\\bullet\\! x,c\\!\\bullet\\! x),\\\\\n&(6')\\quad (a\\!\\bullet\\! x,c\\!\\bullet\\! x^+),\\\\\n&(7')\\quad (a\\!\\bullet\\! x,c\\!\\bullet\\! x^-),\\\\\n\\,\n\\end{aligned}\n\\end{equation*}\nand see that by the triangle inequality for $d$ we can only decrease the value of $\\ell$. For example, in case (3), we see that\n\\begin{align*}\n \\rho(a\\!\\bullet\\! x^-,b\\!\\bullet\\! x)+\\rho(b\\!\\bullet\\! x,c\\!\\bullet\\! x)=&d(a,f(b))+d(b,c)\\\\\n =&d(a,f(b))+d(f(b),f(c))\\\\\n \\geq&d(a,f(c))\\\\\n =&\\rho(a\\!\\bullet\\! x^-,c\\!\\bullet\\! x).\n \\end{align*}\n We can then finish the proof by induction on $|p|$.\n\\end{proof}\nWe now claim that $D(a\\!\\bullet\\! x, b\\!\\bullet\\! x)=d(a,b)$. To see this, notice first that $D(a\\!\\bullet\\! x, b\\!\\bullet\\! x)\\leq d(a,b)$. For the other inequality, let $p$ be an either positive or negative path from $a\\!\\bullet\\! x$ to $b\\!\\bullet\\! x$. By symmetry, we can suppose $p$ is positive. But then, unless $p=(a\\!\\bullet\\! x, b\\!\\bullet\\! x)$, we must have $|p|\\geq s+1$, whence also $\\ell(p)\\geq \\delta\\cdot (|p|-2)\\geq \\delta\\cdot (s-1)\\geq\\Delta\\geq d(a,b)$. \nA similar argument shows that $D(a\\!\\bullet\\!  x,b\\!\\bullet\\!  x^+)=d(a,f(b))$.\n\nThis shows that for any $x_0\\in \\mathbb F$, $\\A\\cup f[\\A]$ is isometric with $\\A\\times \\{x_0,x_0^+\\}$ by the function $a\\mapsto a\\!\\bullet\\! x_0$ and $f(a)\\mapsto a\\!\\bullet\\! x_0^+$. So we can just identify $\\A\\cup f[\\A]$  with $\\A\\times \\{x_0,x_0^+\\}$. Notice also that the following mapping $g$ is an isometry of $\\B$:\n$$\na\\!\\bullet\\! x\\mapsto a\\!\\bullet\\! x^+.\n$$\nMoreover, it agrees with $f$ on their common domain $\\A\\times \\{x_0\\}$. Realising $\\B$ as a subset of $\\U$ containing $\\A$, we see that $g$ acts by isometries on $\\B$ with  $g^s=1$. It then follows that $g$ extends to a full isometry of $\\U$ still satisfying $g^s=1$.\n\\end{proof}\n\n\n\n\\begin{cor}\nEvery topological similarity class is meagre in ${\\rm Iso}(\\U)$.\n\\end{cor}\nAgain this strengthens a result of Kechris \\cite{glasner} saying that all conjugacy classes are meagre.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The Lang-Trotter conjectures}\n\nFix a non-CM elliptic curve $E$ defined over $\\QQ$ and let $N_E$ be its conductor.   Take any prime $p\\nmid N_E$.   Let $E_p$ be the reduction of $E$ modulo $p$; it is an elliptic curve over ${\\mathcal F}_p$.   Let $\\pi_p$ be the Frobenius endomorphism of $E_p$.    We have $\\pi_p^2-a_p(E) \\pi_p + p=0$ for a unique integer $a_p(E)$.   We can also define $a_p(E)$ by the formula $a_p(E) =  |E_p({\\mathcal F}_p)| - (p+1)$.  From Hasse, we know that $|a_p(E)|< 2\\sqrt{p}$ and hence $\\QQ(\\pi_p)$ in $\\End(E_p)\\otimes_{\\mathbb Z} \\QQ$ is an imaginary quadratic field. \n\nFix an integer $a$ and an imaginary quadratic field $k$.  We define the following functions of $x\\geq 2$:\n\\begin{align*}\nP_{E,a}(x) &:= \\#\\{ p\\leq x : p\\nmid N_E,\\, a_p(E)=a \\},\\\\\nP_{E,k}(x) &:= \\#\\{ p\\leq x : p\\nmid N_E,\\, \\QQ(\\pi_p) \\cong k \\}.\n\\end{align*}\nLang and Trotter made the following two conjectures concerning the asymptotics of $P_{E,a}(x)$ and $P_{E,k}(x)$, cf.~\\cite{MR0568299}.\n\n\\begin{conjecture}[Lang-Trotter] \\label{C:LT conj}\n\\begin{alphenum}\n\\item\nThere is an explicit constant $C_{E,a} \\geq 0$ such that \n\\[\nP_{E,a}(x) \\sim C_{E,a}\\cdot \\frac{x^{1\/2}}{\\log x}\n\\]\nas $x\\to \\infty$.   When $C_{E,a}=0$, we interpret this asymptotic as meaning that $P_{E,a}(x)$ is a bounded function of $x$.\n\\item\nThere is an explicit constant $C_{E,k} > 0$ such that \n\\[\nP_{E,k}(x) \\sim C_{E,k} \\cdot \\frac{x^{1\/2}}{\\log x}\n\\]\nas $x\\to \\infty$.   \n\\end{alphenum}\n\\end{conjecture}\n\n\\subsection{Upper bounds}\n\nIn this paper, we are interested in improving the best known upper bounds on $P_{E,a}(x)$ and $P_{E,k}(x)$ as functions of $x$; we will summarize previous results in \\S\\ref{SS:previous results}.   Some bounds are conditional on the Generalized Riemann Hypothesis (GRH) for number fields.\n\n\\begin{theorem} \\label{T:main a}\nLet $E$ be a non-CM elliptic curve defined over $\\QQ$ and let $a$ be an integer.   Assuming GRH, we have\n\\[\nP_{E,a}(x) \\ll_E \\frac{x^{4\/5}}{(\\log x)^{3\/5}} \\quad \\text{and}\\quad P_{E,0}(x) \\ll_E \\frac{x^{3\/4}}{(\\log x)^{1\/2}}.\n\\]\n\\end{theorem}\n\nThe best known unconditional bounds for $P_{E,a}(x)$ can be found in \\S\\ref{SS:previous results}.  We now give bounds for $P_{E,k}(x)$.\n\n\\begin{theorem} \\label{T:main k}\nLet $E$ be a non-CM elliptic curve  over $\\QQ$ and let $k$ be an imaginary quadratic field.  \n\\begin{romanenum}\n\\item \\label{T:main k i}\nAssume GRH.  Then\n\\[\nP_{E,k}(x) \\ll_E  \\,\\, \\frac{1}{h_k^{3\/5}}  \\frac{x^{4\/5}}{ (\\log x)^{3\/5}} + x^{1\/2}(\\log x)^3,\n\\]\nwhere $h_k$ is the class number of $k$.  In particular, $P_{E,k}(x) \\ll_E  {x^{4\/5}}\/{(\\log x)^{3\/5}}$.\n\\item \\label{T:main k ii}\nThere is a constant $c>0$, depending only on $E$ and $k$, such that \n\\[\nP_{E,k}(x) \\ll_E \\frac{x(\\log\\log x)^2}{(\\log x)^2}\n\\]\nwhenever $x\\geq c$.  In particular, $P_{E,k}(x) \\ll_{E,k} {x(\\log\\log x)^2}\/{(\\log x)^2}$.\n\\end{romanenum}\n\\end{theorem}\n\nLet $D_E(x)$ be the number of imaginary quadratic extensions $k$ of $\\QQ$, in some fixed algebraic closure of $\\QQ$, for which there exists a prime $p\\leq x$ with $\\QQ(\\pi_p) \\cong k$.   The following, which will be proved in \\S\\ref{S:D}, is an easy consequence of Theorem~\\ref{T:main k}(\\ref{T:main k i}).\n\n\\begin{corollary} \\label{C:D theorem}\nLet $E$ be a non-CM elliptic curve defined over $\\QQ$.  Assuming GRH, we have\n\\[ \nD_E(x) \\gg_E   \\frac{x^{2\/7}}{(\\log x)^{10\/7}} .\n\\]\n\\end{corollary}\n\nThis improves on the bound $D_E(x) \\gg_E x^{1\/14}\/(\\log x)^2$ from \\cite{MR2464027}.   The explicit dependence of $k$ in Theorem~\\ref{T:main k}(\\ref{T:main k i}) is very important here.\n\n\n\\subsection{Some earlier results}  \\label{SS:previous results}\nWe first describe bounds for $P_{E,a}(x)$.  Under GRH, Serre proved that $P_{E,a}(x) \\ll_E x^{7\/8}(\\log x)^{1\/2}$ and $P_{E,0}(x) \\ll_E x^{3\/4}$, cf.~\\cite{MR644559}.    Under GRH, Murty, Murty and Saradha obtained the improved bound \n\\[\nP_{E,a}(x) \\ll_E \\frac{x^{4\/5}}{(\\log x)^{1\/5}},\n\\] \ncf.~\\cite{MR935007}.    In \\cite{MR644559}, Serre proved (unconditionally) that $P_{E,a}(x)\\ll_{E,\\varepsilon}\\,  x\/(\\log x)^{5\/4-\\varepsilon}$ for any $\\varepsilon>0$.   The exponent $5\/4$ was improved to $2$ by D.~Wan \\cite{MR1062334}.  The best general unconditional bound for $P_{E,a}(x)$ is the bound \n\\begin{align} \\label{E:unconditional VKMurty}\nP_{E,a}(x)\\ll_E x \\frac{(\\log\\log x)^2}{(\\log x)^2}\n\\end{align}\nof V.~K.~Murty \\cite{MR1694997}.   For $a=0$, there is also the superior bound  $P_{E,0}(x) \\ll_E x^{3\/4}$ of Elkies, Kaneko and Murty, cf.~\\cite{MR1144318}.\n\\\\\n\nWe now describe bounds for $P_{E,k}(x)$.   In \\cite{MR644559}*{p.~191}, Serre claimed without proof that $P_{E,k}(x) \\ll_{E,k}\\, x^\\theta$ (under GRH) and $P_{E,k}(x) \\ll_{E,k}\\, x\/(\\log x)^{\\gamma+1}$ (unconditionally) for some positive constants $\\theta$ and $\\gamma$.    Under GRH, Cojocaru, Fouvry and Murty \\cite{MR2178556} showed that one could take $\\theta=7\/8$ and take any $\\gamma>1\/24$.  Under GRH, Cojocaru and David \\cite{MR2464027} obtained the bound\n\\[\nP_{E,k}(x) \\ll_{E,k} \\frac{x^{4\/5}}{(\\log x)^{1\/5}}.\\\\\n\\]\n\n\nUpper bounds for $P_{E,a}(x)$ and $P_{E,k}(x)$ are in general hard to improve.  The function of $x$ obtained indicates the strength of the methods used and often different methods will give the exact same bound.    For example, assume $E$ is semistable and  that the $L$-function for $E$ and its symmetric powers has analytic continuation and satisfies the appropriate analogue of the Riemann hypothesis, then Rouse and Thorner proved that $P_{E,0}(x) \\ll_E  x^{3\/4}\/(\\log x)^{1\/2}$, cf.~\\cite{Rouse-Thorner}.   This is the same bound as Theorem~\\ref{T:main a} under GRH!\n\nThe goal of this paper is to push the upper bounds obtained using Chebotarev to the limit.   It is not clear to the author how to improve them without completely new ideas.\n\n\n\\subsection{Overview}\n\nIn \\S\\ref{S:Chebotarev}, we recall several effective versions of the Chebotarev density theorem.   Under GRH and Artin's holomorphy conjecture, we also give some improved Chebotarev upper bounds; the key point being that we can obtain superior error terms if we count primes using a smoothed weighting.\n\nOne can also prove a smoothed and \\emph{unconditional} version of Chebotarev.   We have not done so because they did seem to lead to stronger Lang-Trotter bounds (in particular, we could not improve on the bound (\\ref{E:unconditional VKMurty}) of V.K.~Murty).\n\n\nIn \\S\\ref{S:Galois representations}, we review some of the Galois representations associated to $E$ and $k$.   These representations play a role in the heuristics of Lang and Trotter in \\cite{MR0568299}.    To understand their images we will need Serre's open image theorem and some class field theory.\n\nWe prove Theorem~\\ref{T:main a} in \\S\\ref{S:main a}.   We follow the proof of Murty, Murty and Saradha in \\cite{MR935007} and use our stronger Chebotarev bound.   We prove Theorem~\\ref{T:main k} in \\S\\ref{S:main k}.   We again follow the general strategy of \\cite{MR935007} though the groups are more complicated.\n\n\\begin{remark}\nLet us remark how Theorem~\\ref{T:main a} can be naturally generalized.   Suppose that $f(z) = \\sum_{n\\geq 1} a_n(f) e^{2\\pi i n z}$ is a non-CM newform of integral weight $k \\geq 2$ on $\\Gamma_0(N)$ whose Fourier coefficients are rational numbers.  For a fixed integer $a$, we define $P_{f,a}(x)$ to be the number of primes $p\\leq x$ for which $a_p(f)=a$.   \n\nUsing the methods in this paper, one can prove the Lang-Trotter bounds $P_{f,a}(x) \\ll_{f} x^{4\/5} (\\log x)^{-3\/5}$ and $P_{f,0}(x) \\ll_f  x^{3\/4} (\\log x)^{-1\/2}$ assuming GRH.   This is a generalization of Theorem~\\ref{T:main a} since by modularity every elliptic curve $E\/\\QQ$ gives rise to a newform $f$ of weight $2$ on $\\Gamma_0(N_E)$ satisfying $a_p(f) = a_p(E)$ for all primes $p\\nmid N_E$.   Note that many of the bounds mentioned in \\S\\ref{SS:previous results} are proved for such general $f$.    (For $k\\geq 3$, one should not expect the precise analog of Conjecture~\\ref{C:LT conj} to hold.)\n\nOur proof depends only on Galois representations and effective Chebotarev density theorems; we now make a few comments that describe what one needs to know about the Galois representations arising from $f$.  For each rational prime $\\ell$, we know from Deligne that there is a Galois representation $\\rho_{f,\\ell}\\colon \\Gal_\\QQ:=\\Gal({\\overline{\\Q}}\/\\QQ) \\to \\operatorname{GL}_2({\\mathcal F}_\\ell)$ such that $\\tr(\\rho_{f,\\ell}(\\Frob_p)) \\equiv a_p(f) \\pmod{\\ell}$ for all primes $p\\nmid N\\ell$.   For all but finitely many $\\ell$, we will  have $\\rho_{f,\\ell}(\\Gal_\\QQ)\\supseteq \\operatorname{SL}_2({\\mathcal F}_\\ell)$, cf.~\\cite{MR819838}.   \n\\end{remark}\n\n\n\n\\subsection*{Notation}\n\nFor two functions $f(x)$ and $g(x)$ of a real variable $x\\geq 2$, we say that $f \\ll g$ (or $g \\gg f$) if there is a positive constant $C$ such that $|f(x)| \\leq C|g(x)|$ for all $x\\geq 2$.  We shall use $O(f)$ to denote an unspecified function $g$ with $g\\ll f$.  We will always indicate the dependence of the implied\nconstant $C$ with subscripts on $\\ll$ or $O$ (in particular, no subscripts indicates that the constant is absolute).  The logarithm integral is $\\operatorname{Li}(x) := \\int^x_2 (\\log t)^{-1} \\, dt$; it satisfies $\\operatorname{Li}(x) \\ll x\/\\log x$.\n\nFor a number field $K$, let ${\\mathcal O}_K$ be its ring of integers.   Let $\\Sigma_K$ be the set of non-zero prime ideals of ${\\mathcal O}_K$.   For each $\\mathfrak{p} \\in \\Sigma_K$, let $N(\\mathfrak{p})$ be the cardinality of the residue field ${\\mathcal O}_K\/\\mathfrak{p}$.\n\nFor a number field $K$, let ${\\overline{K}}$ be a fixed algebraic closure of $K$.  Define the absolute Galois group $\\Gal_K:=\\Gal({\\overline{K}}\/K)$.   For each $\\mathfrak{p} \\in \\Sigma_K$, let $\\Frob_\\mathfrak{p} \\in \\Gal_K$ be a Frobenius automorphism for the prime $\\mathfrak{p}$.  If $\\rho\\colon \\Gal_K \\to G$ is a representation unramified at $\\mathfrak{p}$, then $\\rho(\\Frob_\\mathfrak{p})$ is a well-defined conjugacy class.\n\n\n\\subsection*{Acknowledgements}\nThis paper uses parts of an unpublished preprint that benefited from helpful comments from Bjorn Poonen.  Thanks also to Theodore Hui and the referee for making several corrections.\n\n\n\\section{Chebotarev bounds} \\label{S:Chebotarev}\n\nFix a Galois extension of number fields $L\/K$ with Galois group $G$.    Define\n\\[\nM(L\/K):= 2 [L:K] \\cdot d_K^{1\/[K:\\QQ]} \\cdot {\\prod}_{p\\in {\\mathcal P}(L\/K)}\\,p,\n\\]\nwhere $d_K$ is the absolute discriminant of $K$ and ${\\mathcal P}(L\/K)$ is the set of rational primes $p$ that are divisible by some $\\mathfrak{p} \\in \\Sigma_K$ that ramifies in $L$.\n\nWe say that $L\/K$ satisfies \\defi{Artin's Holomorphy Conjecture} (AHC) if for each irreducible character $\\chi \\colon G \\to \\mathbb C$, the Artin $L$-function $L(s,\\chi)$ extends to a function analytic on the whole complex plane except at $s=1$ when $\\chi=1$.   If $G=\\Gal(L\/K)$ is abelian, then AHC is known to hold for $L\/K$; the Artin $L$-function then agrees with a Hecke $L$-function that has the required properties.  The \\defi{Generalized Riemann Hypothesis} (GRH) for the field $L$ asserts that any zero $\\rho$ of the Dedekind $L$-function of the field $L$ with $0\\leq \\operatorname{Re}(s) \\leq 1$ satisfies $\\operatorname{Re}(\\rho)=1\/2$.   We say that GRH holds if it holds for all number fields.\n\n\\subsection{Chebotarev density theorem} \\label{SS:basic Cheb}\n\nLet $\\varphi\\colon G \\to \\mathbb C$ be a class function.  For each prime $\\mathfrak{p}\\in \\Sigma_K$, choose any $\\P\\in \\Sigma_L$ dividing $\\mathfrak{p}$.   We then have a distinguished (arithmetic) Frobenius element $\\sigma_{\\P} \\in D_\\P\/I_\\P$, where $D_\\P$ and $I_\\P$ are the decomposition and inertia subgroups of $G$, respectively, at $\\P$.    For each integer $m\\geq 1$, we define\n\\[\n\\varphi(\\Frob_\\mathfrak{p}^m) := \\frac{1}{|I_\\P|} \\sum_{\\substack{g\\in D_\\P,\\\\ gI_\\P = \\sigma_\\P^m \\in D_\\P\/I_\\P}} \\varphi(g).\n\\]\nAs the notation suggests, $\\varphi(\\Frob_\\mathfrak{p}^m)$ is independent of the choice of $\\P$.  For $\\mathfrak{p}$ unramified in $L$, this definition agrees with the value of $\\varphi$ on the conjugacy class $\\Frob_\\mathfrak{p}^m$ of $G$.  For $x\\geq 2$, define\n\\[\n\\pi_\\varphi(x) := \\sum_{\\substack{\\mathfrak{p}\\in\\Sigma_K \\text{ unramified in $L$}\\\\N(\\mathfrak{p})\\leq x}} \\varphi(\\Frob_\\mathfrak{p}) \\text{ \\quad and \\quad }\n\\widetilde{\\pi}_\\varphi(x) := \\sum_{\\substack{\\mathfrak{p}\\in\\Sigma_K, m\\geq 1\\\\ N(\\mathfrak{p}^m)\\leq x}} \\frac{1}{m} \\varphi(\\Frob_\\mathfrak{p}^m).\n\\]\n\nNow let $C$ be a subset of $G$ that is stable under conjugation and let $\\delta_C \\colon G \\to \\{0,1\\}$ be the class function such that $\\delta_C(g)=1$ if and only if $g\\in C$.    We define \n\\[\n\\pi_C(x,L\/K) := \\pi_{\\delta_C}(x)\\quad \\text{ and }\\quad \\widetilde\\pi_C(x,L\/K) := \\widetilde{\\pi}_{\\delta_C}(x).\n\\]\nIt is often more convenient to use $\\widetilde\\pi_C(x,L\/K)$ since it has better functorial properties, cf.~\\S\\ref{SS:functorial}.\n\nThe \\defi{Chebotarev density theorem} says that\n\\begin{equation} \\label{E:Chebotarev first}\n\\pi_C(x,L\/K) \\sim \\frac{|C|}{|G|} \\operatorname{Li}(x) \n\\end{equation}\nas $x\\to +\\infty$.   An effective form of Chebotarev is a version with an explicit error term.   The following theorems of Murty, Murty and Saradha give effective versions.\n\n\\begin{theorem}  \\label{T:effective}\n\\begin{romanenum}\n\\item \\label{T:effective i}\nSuppose that AHC holds for $L\/K$ and that GRH holds for $L$.  Then\n\\[\n\\pi_C(x,L\/K) = \\frac{|C|}{|G|}\\operatorname{Li}(x) + O\\Big( |C|^{1\/2}\\, [K:\\QQ] \\, x^{1\/2}  \\log  (M(L\/K)x) \\Big).\n\\]\n\\item \\label{T:effective ii}\nAssume that the group $G$ is abelian.   There are absolute constants $b,c>0$ such that if $\\log x \\geq b [K:\\QQ] \\log^2 M(L\/K)$, then\n\\begin{align*}\n&\\Big| \\pi_C(x,L\/K) - \\frac{|C|}{|G|} \\operatorname{Li}(x) \\Big| \\\\\n\\leq & \\frac{|C|}{|G|} \\operatorname{Li}(x^{\\beta_L}) + O\\bigg( |C|^{1\/2} [K:\\QQ] x \\exp\\bigg(-\\frac{c(\\log x)^{1\/2}}{[K:\\QQ]^{1\/2}} \\bigg) \\cdot\\log^2(M(L\/K)x) \\bigg),\n\\end{align*}\nwhere $\\beta_L$ is the possible exceptional zero of the Dedekind $L$-function of the field $L$ (it would be real and satisfy $1\/2< \\beta_L <1$).   The term $\\frac{|C|}{|G|} \\operatorname{Li}(x^{\\beta_L})$ is present only when $\\beta_L$ exists. \t\n\\end{romanenum}\n\\end{theorem}\n\\begin{proof}\nPart (\\ref{T:effective i}) is Corollary~3.7 of \\cite{MR935007}.     Part (\\ref{T:effective ii}) is a special case of Theorem~4.6 of \\cite{MR1694997} where the group is abelian.  In the notation of \\cite{MR1694997},  we have $H=1$ and $G\/H$  abelian, so $d_{G\/H}=1$ and $|\\chi_{G\/H}(\\bbar{C})|\\leq |C|$.\n\\end{proof}\n\n\n\\subsection{Smoothed Chebotarev}\n\nWe will prove the following smoothed analogue of Chebotarev in \\S\\ref{S:smoothed proof}.\n\n\\begin{theorem} \\label{T:smoothed}\nLet $L\/K$ be a finite Galois extension of number fields with Galois group $G$.  Assume that AHC holds for the extension $L\/K$ and that GRH holds for $L$.    Let $C$ be a subset of $G$ stable under conjugation.    \n\nTake any smooth function $f\\colon (0,\\infty) \\to {\\mathbb R}$ with compact support.  Then for $x\\geq 2$, we have\n\\begin{align*}\n&\\sum_{\\mathfrak{p}} \\delta_C(\\Frob_\\mathfrak{p}) \\log N(\\mathfrak{p}) \\cdot f(N(\\mathfrak{p})\/x) \n=  \\frac{|C|}{|G|} \\int^\\infty_0 f(t) dt\\, \\cdot x + O_f\\Big(|C|^{1\/2}  [K:\\QQ]  x^{1\/2} \\log M(L\/K) \\Big),\n\\end{align*}\nwhere the sum is over all the primes $\\mathfrak{p} \\in \\Sigma_K$ that are unramified in $L$.\n\\end{theorem}\n\nIn our application, we are only interested in asymptotic upper bounds for $\\pi_C(x,L\/K)$, so there is no harm in counting using a smoothed weight.  Under GRH, the following gives an upper bound that cannot be deduced from Theorem~\\ref{T:effective}(\\ref{T:effective i}).\n\n\\begin{theorem} \\label{T:Chebotarev upper bound}\nLet $L\/K$ be a Galois extension of number fields with Galois group $G$.  Assume that AHC holds for $L\/K$ and that GRH holds for $L$.  Let $C$ be a subset of $G$ that is stable under conjugation. Then\n\\[\n\\pi_C(x,L\/K) \\ll \\frac{|C|}{|G|}\\,  \\frac{x}{\\log x} + |C|^{1\/2}\\, [K:\\QQ] \\frac{x^{1\/2}}{\\log x}  \\log  M(L\/K).\n\\]\n\\end{theorem}\n\\begin{proof}\nFirst fix a smooth function $f\\colon (0,\\infty)\\to {\\mathbb R}$ with compact support that is non-negative and satisfies $f(t) \\geq 1$ for all $1\/2\\leq t \\leq 1$.   For every $x\\geq 2$, define\n\\[\nA(x):= \\sum_{\\substack{\\mathfrak{p} \\in \\Sigma_K,\\, \\sqrt{x}\\leq N(\\mathfrak{p}) \\leq x \\\\ \\mathfrak{p} \\text{ unramified in $L$}}} \\delta_C(\\Frob_\\mathfrak{p}) \\log N(\\mathfrak{p}) \\quad \\text{and}\\quad \\Pi(x) := \\sum_{\\mathfrak{p} \\in \\Sigma_K, \\, x\/2\\leq N(\\mathfrak{p}) \\leq x} \\delta_C(\\Frob_\\mathfrak{p}) \\log N(\\mathfrak{p}).\n\\]\nBy our choice of $f$, we have $\\Pi(x) \\leq \\sum_{\\mathfrak{p} \\in \\Sigma_K} \\delta_C(\\Frob_\\mathfrak{p}) \\log N(\\mathfrak{p}) \\cdot f(N(\\mathfrak{p})\/x)$.  By Theorem~\\ref{T:smoothed}, we have $\\Pi(x)\\ll_f \\tfrac{|C|}{|G|} x + |C|^{1\/2}  [K:\\QQ]  x^{1\/2} \\log M(L\/K)$.\n\nLet $m\\geq 1$ be the largest integer for which $x\/2^m \\geq \\sqrt{x}$.  We have $A(x)\\leq \\sum_{i=0}^m \\Pi(x\/2^i)$, so our bound for $\\Pi(x)$ gives\n\\begin{align*}\nA(x) & \\ll_f  \\frac{|C|}{|G|} x \\sum_{i=0}^m \\frac{1}{2^i} + |C|^{1\/2}  [K:\\QQ]  x^{1\/2} \\log M(L\/K) \\sum_{i=0}^m \\frac{1}{2^{i\/2}} \\\\\n& \\ll  \\frac{|C|}{|G|} x + |C|^{1\/2}  [K:\\QQ]  x^{1\/2} \\log M(L\/K).\n\\end{align*}\nThere are at most $[K:\\QQ]$ primes $\\mathfrak{p}$ dividing any rational prime $p$, so \n\\[\n|\\{\\mathfrak{p} \\in \\Sigma_K : N(\\mathfrak{p})\\leq \\sqrt{x}\\}| \\leq [K:\\QQ] \\cdot |\\{p: p\\leq \\sqrt{x}\\}| \\ll [K:\\QQ] \\tfrac{\\sqrt{x}}{\\log \\sqrt{x}}.\n\\]\n Therefore,\n\\begin{align*}\n\\pi_C(x,L\/K) &\\ll  A(x)\/\\log(\\sqrt{x}) + [K:\\QQ] x^{1\/2}\/\\log(\\sqrt{x})\\\\\n& \\ll_f   \\frac{|C|}{|G|} \\frac{x}{\\log x} + |C|^{1\/2}  [K:\\QQ]  \\frac{x^{1\/2}}{\\log x} \\log M(L\/K) + [K:\\QQ] \\frac{x^{1\/2}}{\\log x}.\n\\end{align*}\nThe theorem now follows if $|C|\\neq 0$.  If $|C|=0$, then the theorem is trivial.\n\\end{proof}\n\nFor future use, we also give the following consequence of Theorem~\\ref{T:smoothed}.\n\n\\begin{corollary} \\label{C:cheb existence}\nFix notation and assumptions as in Theorem~\\ref{T:smoothed}.  Assume that $C\\neq \\emptyset$.  There is an absolute constant $c>0$ such that if \n\\[\nx \\geq c \\frac{\\;|G|^2}{|C|} [K:\\QQ]^2 \\log^2 M(L\/K), \n\\]\nthen there is a prime $\\mathfrak{p} \\in \\Sigma_K$ unramified in $L$ with $x\/2 \\leq N(\\mathfrak{p}) \\leq x$ such that $\\delta_C(\\Frob_\\mathfrak{p})=1$.  \n\\end{corollary}\n\\begin{proof}\nLet $f\\colon (0,\\infty)\\to {\\mathbb R}$ be a non-negative smooth function with compact support whose support is in the interval $[1\/2,1]$ and is non-zero.   Suppose that there are no primes $\\mathfrak{p} \\in \\Sigma_K$ with $x\/2\\leq N(\\mathfrak{p}) \\leq x$ such that $\\mathfrak{p}$ is unramified in $L$ and $\\delta_C(\\Frob_\\mathfrak{p})=1$.   The sum in Theorem~\\ref{T:smoothed} is then zero, so\n\\[\n \\frac{|C|}{|G|} x \\ll_f |C|^{1\/2}  [K:\\QQ]  x^{1\/2} \\log M(L\/K);\n\\]\nnote that the integral $\\int^\\infty_0 f(t) dt$ is positive by our choice of $f$.  Rearranging, we deduce that $x \\ll_f |G|^2\/|C|\\cdot [K:\\QQ]^2 \\log^2 M(L\/K)$.   We obtain the desired contradiction by choosing the constant $c$ in the statement of the corollary sufficiently large.\n \\end{proof}\n\n\n\\subsection{Functorial properties} \\label{SS:functorial}\n\nLet $H$ be a subgroup of $G$.  Take any class function $\\varphi\\colon H \\to \\mathbb C$.  As above, we can define $\\widetilde{\\pi}_\\varphi(x)$; note that $H$ is the Galois group of the extension $L\/L^H$.  Define the induced function\n\\[\n\\Ind_H^G\\varphi \\colon G \\to \\mathbb C,\\quad g\\mapsto \\frac{1}{|H|} \\sum_{{t\\in G},\\,{t^{-1}gt\\in H}} \\varphi(t^{-1}gt);\n\\]\nit is a class function of $G$.\n\n\\begin{lemma} \\label{L:invariance under induction}\n\n\\begin{romanenum}\n\\item \\label{L:invariance under induction i}\nLet $H$ be a subgroup of $G$ and let $\\varphi$ be a class function of $H$.  Then $\\widetilde{\\pi}_{\\Ind^G_H \\varphi}(x) = \\widetilde{\\pi}_\\varphi(x)$.\n\\item \\label{L:invariance under induction ii}\nLet $N$ be a normal subgroup of $G$ and let $\\varphi'$ be a class function of $G\/N$.   Then $\\widetilde{\\pi}_{\\varphi'}(x)=\\widetilde{\\pi}_{\\varphi}(x)$,\nwhere $\\varphi$ is the function obtained by composing the projection $G\\to G\/N$ with $\\varphi'$.\n\\end{romanenum}\n\\end{lemma}\n\\begin{proof}\nSee Proposition~8 of \\cite{MR644559}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{L:invariance C}\n\\begin{romanenum}\n\\item \\label{L:invariance C i}\nLet $H$ be a subgroup of $G$ and let $C$ be a subset of $G$ stable under conjugation.   Suppose that every element of $C$ is conjugate to some element of $H$. Then\n\\[\n\\widetilde{\\pi}_C(x,L\/K) \\leq \\widetilde{\\pi}_{C\\cap H}(x,L\/L^H).\n\\]\n\\item \\label{L:invariance C ii}\nLet $N$ be a normal subgroup of $G$ and let $C$ be a subset of $G$ stable under conjugation that satisfies $NC \\subseteq C$.   Then\n\\[\n\\widetilde{\\pi}_C(x,L\/K) = \\widetilde{\\pi}_{C'}(x,L^N\/K),\n\\]\nwhere $C'$ is the image of $C$ in $G\/N=\\Gal(L^N\/K)$.  \n\\end{romanenum}\n\\end{lemma}\n\\begin{proof}\nPart (\\ref{L:invariance C ii}) is an immediate consequence of Lemma~\\ref{L:invariance under induction}(\\ref{L:invariance under induction ii}); note that $\\delta_{C}$ is equal to the projection $G\\to G\/N$ composed with $\\delta_{C'}$.\n\nWe now prove (\\ref{L:invariance C i}).  By assumption on $C$, there is a set $S\\subseteq H$ for which we have a disjoint union $C=\\cup_{s\\in S} C_G(s)$, where $C_G(s)$ is the conjugacy class of $s$ in $G$.   For each $s\\in S$, let $C_H(s)$ be the conjugacy class of $s$ in $H$.  We have $(\\Ind_H^G \\delta_{C_H(s)})(g)=0$ for $g\\in G -C_G(s)$, so \n\\[\n\\Ind_H^G \\delta_{C_H(s)}=\\lambda_s \\cdot \\delta_{C_G(s)}\n\\] \nfor some $\\lambda_s$.  Therefore, $\\widetilde\\pi_{C_H(s)}(x,L\/L^H) = \\lambda_s \\widetilde\\pi_{C_G(s)}(x,L\/K)$ by Lemma~\\ref{L:invariance under induction}(\\ref{L:invariance under induction i}).   We have \n\\[\n\\widetilde\\pi_{C}(x,L\/K)=\\sum_{s\\in S} \\widetilde\\pi_{C_G(s)}(x,L\/K) =\\sum_{s\\in S} \\lambda_s^{-1} \\widetilde\\pi_{C_H(s)}(x,L\/L^H).\n\\] \nUsing Frobenius reciprocity, cf.~\\cite{MR0450380}*{Theorem~13}, we have\n\\begin{align*} \n\\lambda_s  \\cdot {|C_G(s)|}\/{|G|} = \\big<\\lambda_s \\cdot  \\delta_{C_G(s)}, 1_G  \\big>_G \n= \\big< \\Ind_H^G \\delta_{C_H(s)}, 1_G  \\big>_G = \\big< \\delta_{C_H(s)}, 1_H \\big>_H = {|C_H(s)|}\/{|H|}.\n\\end{align*}\nWe have $|C_G(s)|=|G|\/|\\operatorname{Cent}_G(s)|$ and $|C_H(s)|=|H|\/|\\operatorname{Cent}_H(s)|$, where $\\operatorname{Cent}_H(s)$ and $\\operatorname{Cent}_G(s)$ are the centralizers of $s$ in $G$ and $H$, respectively.  Therefore, $\\lambda_s^{-1} = [\\operatorname{Cent}_G(s): \\operatorname{Cent}_H(s)]^{-1} \\leq 1$ and hence\n\\[\n\\widetilde\\pi_{C}(x,L\/K)\\leq \\sum_{s\\in S} \\widetilde\\pi_{C_H(s)}(x,L\/L^H) \\leq \\widetilde{\\pi}_{C\\cap H}(x,L\/L^H).   \\qedhere\n\\] \n\\end{proof}\n\nIn our applications, we will use Lemma~\\ref{L:invariance C} to reduce our computations of $\\pi_C(x,L\/K)$ to the case where $G$ is abelian.  The following says that $\\widetilde\\pi_C(x,L\/K)$ is a good approximation of $\\pi_C(x,L\/K)$.  \n\n\\begin{lemma} \\label{L:good approximation}\nFor any subset $C$ of $G$ stable under conjugation, we have \n\\[\n\\widetilde\\pi_C(x,L\/K) = \\pi_C(x,L\/K) +O\\big( [K:\\QQ]\\big( \\tfrac{x^{1\/2}}{\\log x} + \\log M(L\/K)\\big) \\big).\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $\\pi_K(x)$ be the number of $\\mathfrak{p} \\in \\Sigma_K$ for which $N(\\mathfrak{p})\\leq x$.    Since there are at most $[K:\\QQ]$ primes $\\mathfrak{p}\\in \\Sigma_K$ dividing any $p$, we have $\\pi_K(x) \\ll [K:\\QQ] \\cdot x\/\\log x$.   For each $\\mathfrak{p} \\in \\Sigma_K$, let $\\deg \\mathfrak{p}$ be the integer for which $N(\\mathfrak{p})=p^{\\deg \\mathfrak{p}}$, where $p$ is the prime divisible by $\\mathfrak{p}$.   Define the sums\n\\[\nB_1:= \\sum_{m\\geq 2} \\sum_{\\substack{\\mathfrak{p}\\in \\Sigma_K \\\\ N(\\mathfrak{p})^m \\leq x}} \\frac{1}{m}, \\quad \nB_2:= \\sum_{\\substack{\\mathfrak{p}\\in \\Sigma_K, \\, \\deg \\mathfrak{p} > 1\\\\ N(\\mathfrak{p})\\leq x}} 1 \\quad \\text{ and } \\quad\nB_3:= \\sum_{\\substack{\\mathfrak{p} \\in \\Sigma_K \\text{ ramified in $L$} \\\\ \\deg \\mathfrak{p} =1 ,\\, N(\\mathfrak{p})\\leq x} } 1.\n\\]\nWe have $0\\leq \\widetilde\\pi_C(x,L\/K)-\\pi_C(x,L\/K) \\leq B_1+B_2+B_3$, so it suffices to bound the $B_1$, $B_2$ and $B_3$.\n\nLet $M\\geq 1$ be the largest integer for which $x^{1\/M} \\geq 2$.  We have\n\\[\nB_1 \\leq \\sum_{m=2}^M \\frac{1}{m} \\pi_K(x^{1\/m}) \\leq [K:\\QQ] \\sum_{m=2}^M \\frac{1}{m} \\frac{x^{1\/m}}{\\log(x^{1\/m})} \\leq [K:\\QQ] \\frac{x^{1\/2}}{\\log x} \\sum_{m=2}^M \\frac{1}{m^2} \\ll [K:\\QQ] \\frac{x^{1\/2}}{\\log x}.\n\\]\nWe have $B_2 \\leq \\sum_{p \\leq \\sqrt{x}} [K:\\QQ] \\ll [K:\\QQ] \\cdot x^{1\/2}\/\\log x$.  If $\\mathfrak{p} \\in \\Sigma_K$ is a prime with $\\deg \\mathfrak{p} =1$ that ramifies in $L$, then $N(\\mathfrak{p}) \\in {\\mathcal P}(L\/K)$, where ${\\mathcal P}(L\/K)$ is the set from the definition of $M(L\/K)$.   Since there are at most $[K:\\QQ]$ primes $\\mathfrak{p}\\in \\Sigma_K$ dividing any $p$, we have $B_3 \\leq [K:\\QQ] \\sum_{p\\in {\\mathcal P}(L\/K)} \\log p \\leq  [K:\\QQ] \\log M(L\/K)$.\n\\end{proof}\n\n\n\\section{Galois representations}   \\label{S:Galois representations}\n\n\\subsection{Elliptic curves}\n\nFix a non-CM elliptic curve $E$ defined over $\\QQ$.  For each prime $\\ell$, let $E[\\ell]$ be the $\\ell$-torsion subgroup of $E({\\overline{\\Q}})$; it is a free ${\\mathcal F}_\\ell$-module of rank $2$.    There is a natural action of $\\Gal_\\QQ$ on $E[\\ell]$ that respects the group structure and can be expressed in terms of a Galois representation\n\\[\n\\rho_{E,\\ell}\\colon \\Gal_\\QQ \\to \\Aut_{{\\mathcal F}_\\ell}(E[\\ell]) \\cong \\operatorname{GL}_2({\\mathcal F}_\\ell).\n\\]\nThe representation $\\rho_{E,\\ell}$ is unramified at all primes $p\\nmid N_E \\ell$ and we have\n\\[\n\\det(xI - \\rho_{E,\\ell}(\\Frob_p))\\equiv x^2- a_p(E) x + p \\pmod{\\ell}.\n\\]\nWe have $\\det \\circ \\rho_{E,\\ell} = \\chi_\\ell$, where $\\chi_\\ell\\colon \\Gal_\\QQ \\to {\\mathcal F}_\\ell^\\times$ is the representation for which $\\sigma(\\zeta)=\\zeta^{\\chi_\\ell(\\sigma)}$ for all $\\ell$-th roots of unity $\\zeta\\in {\\overline{\\Q}}$.  The following is an important theorem of Serre, cf.~\\cite{MR0387283}.\n\n\\begin{theorem}[Serre]   \\label{T:Serre}\nThe representation $\\rho_{E,\\ell}$ is surjective for all but finitely many primes $\\ell$.  \n\\end{theorem}\n\n\\subsection{Some class field theory}  Fix an imaginary quadratic field $k$ and let ${\\mathcal H}$ be its Hilbert class field.   Denote the ring of integers of $k$ by ${\\mathcal O}$.    Take any non-zero prime ideal $\\mathfrak{p}$ of ${\\mathcal O}$.   Let $v_\\mathfrak{p}\\colon k^\\times \\twoheadrightarrow {\\mathbb Z}$ be the surjective discrete valuation corresponding to $\\mathfrak{p}$ which we extend by setting $v_\\mathfrak{p}(0)=+\\infty$.\n\nFix an integer $m\\geq 1$.  Let $I_k^{m}$ be the group of fractional ideals of $k$ generated by prime ideals $\\mathfrak{p} \\nmid m$ of ${\\mathcal O}$.     Let \n\\[\n\\iota\\colon k_m:= \\{ a\\in k^\\times: v_\\mathfrak{p}(a)=0 \\text{ for all }\\mathfrak{p}|m\\} \\to I_k^{m}\n\\]\nbe the homomorphism that takes an element of $k_m$ to the fractional ideal of $k$ it generates.    The \\defi{ray class group modulo $m$} is \nthe group\n\\[\n\\Cl_m := I_k^{m}\/\\iota(k_{m,1}),\n\\]\nwhere $k_{m,1} := \\{ a\\in k_m: v_\\mathfrak{p}(a-1) \\geq v_\\mathfrak{p}(m) \\text{ for all }\\mathfrak{p}|m \\}$.    Note that $\\Cl_{1}$ is the usual class group of ${\\mathcal O}$ which we will also denote by $\\Cl_k$.\n\nBy class field theory, there is a continuous homomorphism \n\\[\n\\widetilde\\psi_{k,m} \\colon \\Gal_k \\to \\Cl_m\n\\]\nsuch that for each prime $\\mathfrak{p}\\nmid m$ of ${\\mathcal O}$, $\\widetilde\\psi_{k,m}$ is unramified at $\\mathfrak{p}$ and $\\widetilde\\psi_{k,m}(\\Frob_\\mathfrak{p})$ is the class of $\\Cl_m$ represented by $\\mathfrak{p}$.   The map $\\widetilde\\psi_{k,m}$ is clearly surjective. \n\n  The homomorphism $\\widetilde\\psi_{k,1}$ is unramified at all non-zero prime ideals $\\mathfrak{p}$ of ${\\mathcal O}$ and has image $\\Cl_k$.   By class field theory, we deduce that the fixed field in ${\\bbar{k}}$ of $\\ker(\\widetilde\\psi_{k,1})$ is ${\\mathcal H}$, i.e., the Hilbert class field of $k$.   \n\n\n\\begin{lemma} \\label{L:class group sequence}\nIf $m\\geq 5$, then there is an exact sequence of groups\n\\[\n1 \\to {\\mathcal O}^\\times \\xrightarrow{\\alpha_m}  ({\\mathcal O}\/m{\\mathcal O})^\\times \\xrightarrow{\\beta_m} \\Cl_{m} \\xrightarrow{\\gamma_m}  \\Cl_k \\to 1,\n\\]\nwhere $\\alpha_m$ is reduction modulo $m$, $\\beta_m$ maps a coset $a+m{\\mathcal O}$ to the class of $\\Cl_{\\m}$ containing $a{\\mathcal O}$, and $\\gamma_m$ is induced by the natural map $I_k^{m}\\to \\Cl_k$.\n\\end{lemma}\n\\begin{proof}\nLet $\\bbar{\\iota}\\colon k_m\/k_{m,1} \\to \\Cl_m$ be the group homomorphism induced from $\\iota\\colon k_m \\xrightarrow{\\iota} I_k^m$.  Since $\\ker(\\iota)={\\mathcal O}^\\times$, we have an exact sequence\n\\[\n{\\mathcal O}^\\times \\to k_m\/k_{m,1} \\xrightarrow{\\bbar \\iota} \\Cl_m.\n\\]\nWe have ${\\mathcal O}^\\times \\cap k_{m,1} = 1$ since $m\\geq 5$.   The group $I_k^m\/\\iota(k_m)$ is naturally isomorphic to $\\Cl_k$.   We thus have an exact sequence\n\\begin{equation} \\label{E:last easy ANT}\n1\\to {\\mathcal O}^\\times \\to k_m\/k_{m,1} \\xrightarrow{\\bbar \\iota} \\Cl_m \\to \\Cl_k \\to 1.\n\\end{equation}\n\nFinally, we explain why $({\\mathcal O}\/m{\\mathcal O})^\\times$ is isomorphic to $k_m\/k_{m,1}$.   We have a natural inclusion $k_m\\hookrightarrow {\\mathcal O}_{m}^\\times$, where ${\\mathcal O}_{m}$ is the $m$-adic completion of ${\\mathcal O}$.  Composing with the reduction modulo $m$ map gives a group homomorphism, $f\\colon k_\\m \\to ({\\mathcal O}_{m}\/m{\\mathcal O}_{m})^\\times = ({\\mathcal O}\/m{\\mathcal O})^\\times$.   The kernel of $f$ is $k_{m,1}$ and it is surjective by weak approximation.   We thus an induced isomorphism $\\overline{f} \\colon k_m\/k_{m,1} \\stackrel{\\sim}{\\to} ({\\mathcal O}\/m{\\mathcal O})^\\times$.  Identifying $k_m\/k_{m,1}$ in (\\ref{E:last easy ANT}) by $({\\mathcal O}\/m{\\mathcal O})^\\times$ via the isomorphism $\\overline{f}$, gives an exact sequence that agrees with the one in the statement of the lemma.\n\\end{proof}\n\n\n\nWe now focus on the case where $m$ is a prime $\\ell\\geq 5$.  By Lemma~\\ref{L:class group sequence}, we may view ${\\mathcal O}^\\times$ as a subgroup of $({\\mathcal O}\/\\ell{\\mathcal O})^\\times$.      With $\\gamma_\\ell$ as in Lemma~\\ref{L:class group sequence}, we have $\\widetilde\\psi_{k,\\ell} = \\gamma_\\ell \\circ \\widetilde\\psi_{k,1}$.   So by restricting $\\widetilde\\psi_{k,\\ell}$ to $\\Gal_{\\mathcal H}$ and using Lemma~\\ref{L:class group sequence}, we obtain a surjective homomorphism\n\\[\n\\psi_{k,\\ell} \\colon \\Gal_{\\mathcal H} \\to  ({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times.\n\\]\n\n\\begin{lemma} \\label{L:psim description}\nFix a prime $p\\nmid \\ell$ that splits completely in ${\\mathcal H}$.   Let $\\P$ be a prime ideal of ${\\mathcal O}_{\\mathcal H}$ that divides $p$.    The representation $\\psi_{k,\\ell}$ is unramified at $\\P$ and satisfies \n\\[\n\\psi_{k,\\ell}(\\Frob_\\P) = (\\pi+ \\ell{\\mathcal O})\\cdot {\\mathcal O}^\\times  \\in \\left({\\mathcal O}\/\\ell{\\mathcal O}\\right)^\\times\/{\\mathcal O}^\\times,\n\\]\nwhere $\\pi$ is a generator of the prime ideal $\\P \\cap {\\mathcal O}$ of ${\\mathcal O}$.\n\\end{lemma}\n\\begin{proof}\nSet $\\mathfrak{p}:=\\P \\cap {\\mathcal O}$; it is a prime that splits completely in ${\\mathcal H}$.  Since $\\mathfrak{p}\\nmid \\ell$, $\\widetilde\\psi_{k,\\ell}$ is unramified at $\\mathfrak{p}$ and  $\\widetilde\\psi_{k,\\ell}(\\Frob_\\mathfrak{p})=[\\mathfrak{p}] \\in \\Cl_\\ell$.    That $\\mathfrak{p}$ splits completely in ${\\mathcal H}$ implies that $\\gamma_\\ell([\\mathfrak{p}])=1$ and hence that $\\mathfrak{p}$ is indeed principal, say $\\mathfrak{p}=\\pi {\\mathcal O}$.   We have $\\gamma_\\ell([\\mathfrak{p}])$, so we may identify $\\widetilde\\psi_{k,\\ell}(\\Frob_\\mathfrak{p})=[\\mathfrak{p}]$ with an element of $({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times$ as viewed above as a subgroup of $\\Cl_\\ell$; in particular, we identify $\\widetilde\\psi_{k,\\ell}(\\Frob_\\mathfrak{p})$ with the coset represented by $\\pi$.     Finally, since $\\mathfrak{p}$ splits completely in ${\\mathcal H}$ we find that $\\widetilde\\psi_{k,\\ell}(\\Frob_\\P)=\\widetilde\\psi_{k,\\ell}(\\Frob_\\mathfrak{p})$.   Therefore, $\\psi_{k,\\ell}(\\Frob_\\mathfrak{p})$ is represented by $\\pi$ as desired.\n\\end{proof}\n\n\n\nLet $N_{k\/\\QQ} \\colon\\left({\\mathcal O}\/\\ell{\\mathcal O}\\right)^\\times\/{\\mathcal O}^\\times \\to {\\mathcal F}_\\ell^\\times$ be the homomorphism induces by the usual norm map $N_{k\/\\QQ}\\colon k\\to \\QQ$; it is well defined since the norm map takes value $1$ on ${\\mathcal O}^\\times$.  \n\n\\begin{lemma} \\label{L:cyclotomic-CFT}\nThe homomorphism $N_{k\/\\QQ}\\circ \\psi_{k,\\ell} \\colon \\Gal_{\\mathcal H} \\to {\\mathcal F}_\\ell^\\times$ agrees with $\\chi_\\ell|_{\\Gal_{\\mathcal H}}$.\n\\end{lemma}\n\\begin{proof}\nTake any prime $\\P | p$ as in the statement of Lemma~\\ref{L:psim description}.   It suffices to show that $N_{k\/\\QQ}(\\psi_{k,\\ell}(\\Frob_\\P))= \\chi_\\ell(\\Frob_\\P)$ since such primes $\\P$ of ${\\mathcal O}_{\\mathcal H}$ have density $1$.    Since $p$ splits completely in ${\\mathcal H}$, we have $\\chi_\\ell(\\Frob_\\P)\\equiv N(\\mathfrak{p})=p \\pmod{\\ell}$.\n\nBy Lemma~\\ref{L:psim description}, we have $N_{k\/\\QQ}(\\psi_{k,\\ell}(\\Frob_\\P))\\equiv N_{k\/\\QQ}(\\pi) \\pmod{\\ell}$, where $\\pi\\in {\\mathcal O}$ is a generator of the ideal $\\P\\cap{\\mathcal O}$.  Since $p$ splits completely in ${\\mathcal H}$, and hence also $k$, we have $N_{k\/\\QQ}(\\pi)=p$.   Therefore, $N_{k\/\\QQ}(\\psi_{K,\\ell}(\\Frob_\\P)) \\equiv p \\equiv \\chi_\\ell(\\Frob_\\P) \\pmod{\\ell}$.\n\\end{proof}\n\n\\subsection{Mixed representations} \\label{SS:mixed}\n\nLet $E$ be a non-CM elliptic curve over $\\QQ$.  Let $k$ be an imaginary quadratic field, let ${\\mathcal O}$ be its ring of integers, and let ${\\mathcal H}$ be its Hilbert class field.  \n\nFix a prime $\\ell\\geq 5$.  We have Galois representations $\\rho_{E,\\ell}$ and $\\psi_{k,\\ell}$ from the previous sections.  By Lemma~\\ref{L:cyclotomic-CFT}, we have $\\det\\circ \\rho_{E,\\ell}|_{\\Gal_{\\mathcal H}} = N_{k\/\\QQ}\\circ \\psi_{k,\\ell}$.   We thus have a well-defined Galois representation\n\\[\n\\Psi_\\ell\\colon \\Gal_{{\\mathcal H}} \\to {\\mathcal G},\\quad \\sigma\\mapsto (\\rho_{E,\\ell}(\\sigma),\\psi_{k,\\ell}(\\sigma)),\n\\]\nwhere ${\\mathcal G}:= \\{ (A,u)\\in\\operatorname{GL}_2({\\mathcal F}_\\ell) \\times \\left(({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times \\right) : \\det(A)= N_{k\/\\QQ}(u) \\}$.\n\nThe trace map $\\Tr_{k\/\\QQ}\\colon k\\to \\QQ$ induces a linear map $\\Tr_{k\/\\QQ}\\colon {\\mathcal O}\/\\ell{\\mathcal O} \\to {\\mathcal F}_\\ell$.  For $u\\in ({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times$, $\\Tr_{k\/\\QQ}(u)$ is a subset of ${\\mathcal F}_\\ell$ of cardinality at most $|{\\mathcal O}^\\times|$.\n\n\\begin{lemma} \\label{L:LT2 key}\nLet $p\\nmid N_E$ be a prime for which $E$ has ordinary reduction at $p$ and for which $\\QQ(\\pi_p)$ is isomorphic to $k$.\n\\begin{romanenum}\n\\item \\label{L:LT2 key i}\nThe prime $p$ splits completely in ${\\mathcal H}$.\n\\item \\label{L:LT2 key ii}\nTake any prime $\\P \\in \\Sigma_{\\mathcal H}$ dividing $\\pi_p {\\mathcal O}$.  Then for any prime $\\ell\\geq 5$ not equal to $p$, the representations $\\rho_{E,\\ell}$ and $\\psi_{k,\\ell}$ are unramified at $\\P$ and \n\\[\n\\tr(\\rho_{E,\\ell}(\\Frob_\\P))\\in \\Tr_{k\/\\QQ}(\\psi_{k,\\ell}(\\Frob_\\P)).\n\\]\n\\end{romanenum}\n\\end{lemma}\n\\begin{proof}\nTo ease notation, set $k=\\QQ(\\pi_p)$.   Since $\\pi_p$ is a root of $x^2-a_p(E)x+p$, we have $a_p(E)=\\Tr_{k\/\\QQ}(\\pi_p)$ and $p=N_{k\/\\QQ}(\\pi_p)$.  The equality $p=N_{k\/\\QQ}(\\pi_p)$ implies that $p$ is either split or ramified in $k$, so  $p{\\mathcal O} = \\mathfrak{p}\\cdot\\mathfrak{p}^\\tau$, where $\\mathfrak{p}:=\\pi_p {\\mathcal O}$ and $\\tau$ is the non-trivial automorphism of $k$.  \n\nWe claim that $p$ splits in $k$.  Suppose otherwise that $p$ is ramified in $k$.  We then have\n\\[\na_p(E) =\\Tr_{k\/\\QQ}( \\pi_p )= \\pi_p+ \\pi_p^\\tau  \\in \\mathfrak{p} + \\mathfrak{p}^\\tau =\\mathfrak{p}\n\\]\nand hence $a_p(E)\\equiv 0 \\pmod{p}$ which contradicts our assumption that $E$ has ordinary reduction at $p$.\n\n\nSince $\\mathfrak{p}$ and $\\mathfrak{p}^\\tau$ are principal ideals in ${\\mathcal O}$, the prime $p$ splits completely in ${\\mathcal H}$.   This completes the proof of (\\ref{L:LT2 key i}).\n\nNow take any prime $\\ell\\nmid 6p$ and any $\\P\\in \\Sigma_{\\mathcal H}$ that divides $\\mathfrak{p} = \\pi_p {\\mathcal O}$.  Since $p$ splits completely in ${\\mathcal H}$, we have ${\\mathcal O}_{\\mathcal H}\/\\P = {\\mathcal F}_p$.   Therefore, $\\rho_{E,\\ell}$ is unramified at $\\P$ and we have \n\\[\n\\tr(\\rho_{E,\\ell}(\\Frob_\\P)) = \\tr(\\rho_{E,\\ell}(\\Frob_p)) \\equiv a_p(E) \\pmod{\\ell}\n\\]\nBy Lemma~\\ref{L:psim description}, we have $\\psi_{k,\\ell}(\\Frob_\\P) = (\\pi_p + \\ell{\\mathcal O}) \\cdot {\\mathcal O}^\\times$.  The image of $a_p(E)=\\Tr_{k\/\\QQ}(\\pi_p)$ in ${\\mathcal F}_\\ell$ thus belongs to $\\Tr_{k\/\\QQ}(\\psi_{k,\\ell}(\\Frob_\\P))$.    Therefore, $\\tr(\\rho_{E,\\ell}(\\Frob_\\P))\\in \\Tr_{k\/\\QQ}(\\psi_{k,\\ell}(\\Frob_\\P))$ as claimed.\n\\end{proof}\n\n\nThe representation $\\Psi_\\ell$ is surjective for all sufficiently large $\\ell$.\n\n\\begin{lemma} \\label{L:intersection} \nIf $\\rho_{E,\\ell}$ is surjective, then the representation $\\Psi_\\ell$ is surjective.\n\\end{lemma}\n\\begin{proof}\nLet $p_1\\colon {\\mathcal G} \\to \\operatorname{GL}_2({\\mathcal F}_\\ell) $ and $p_2\\colon {\\mathcal G} \\to ({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times$ be the projection maps.   Let $H$ be a subgroup of ${\\mathcal G}$ with $p_1(H)\\supseteq\\operatorname{SL}_2({\\mathcal F}_\\ell)$ and $p_2(H)=({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times$.  \n\nWe claim that $H={\\mathcal G}$.  For a finite group $G$, let $G'$ be the commutator subgroups of $G$.  Since $p_1(H)$ contains $\\operatorname{SL}_2({\\mathcal F}_\\ell)$, we have  \n\\[\n\\operatorname{SL}_2({\\mathcal F}_\\ell)' \\subseteq p_1(H)' \\subseteq \\operatorname{GL}_2({\\mathcal F}_\\ell)'=\\operatorname{SL}_2({\\mathcal F}_\\ell).\n\\]\nThe group $\\operatorname{SL}_2({\\mathcal F}_\\ell)$ is perfect since $\\ell\\geq 5$, so  $p_1(H')=p_1(H)' = \\operatorname{SL}_2({\\mathcal F}_\\ell)$.  We have $p_1(H')=\\operatorname{SL}_2({\\mathcal F}_\\ell)$ and $p_2(H')=(({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times)'=\\{1\\}$, so $H' = \\operatorname{SL}_2({\\mathcal F}_\\ell) \\times \\{1\\}$.    The group $H'$ is normal in ${\\mathcal G}$ and $p_2$ induces an isomorphism ${\\mathcal G}\/H' \\xrightarrow{\\sim} ({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times$.   Since $p_2|_H$ is surjective, we deduce that the natural map $H\/H'\\hookrightarrow {\\mathcal G}\/H'$ is surjective and hence $H={\\mathcal G}$.\n\nSet $H:=\\Psi_\\ell(\\Gal_{\\mathcal H})$.  We have $p_2(H)=({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times$ since $\\psi_{k,\\ell}$ is surjective.  We have $p_1(H) = \\rho_{E,\\ell}(\\Gal_{\\mathcal H})$.   Since $\\rho_{E,\\ell}$ is surjective, ${\\mathcal H}\/\\QQ$ is solvable, and $\\operatorname{SL}_2({\\mathcal F}_\\ell)$ is perfect, we have $\\rho_{E,\\ell}(\\Gal_{\\mathcal H}) \\supseteq \\operatorname{SL}_2({\\mathcal F}_\\ell)$.    From our claim, we deduce that $\\Psi_\\ell(\\Gal_{\\mathcal H})=H={\\mathcal G}$.\n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{T:main a}}  \\label{S:main a}\nAssume that GRH holds.  Fix a non-CM elliptic curve $E\/\\QQ$ and an integer $a$.   \n\nFor each prime $\\ell$, we have constructed a representation $\\rho_{E,\\ell}\\colon \\Gal_\\QQ \\to \\operatorname{GL}_2({\\mathcal F}_\\ell)$.    Let $I$ be the set of primes in the interval $[y, 2y]$, where $y$ is a fixed real number that satisfies $c\\leq y \\leq x$ for some constant $c$ depending on $E$.  We will make a more specific choice of $y$ later on.  After increasing $c$, we may assume that $I$ is non-empty and that $\\rho_{E,\\ell}$ is surjective for all primes $\\ell\\in I$.   \n\n\nFor each prime $\\ell$, define \n\\[\nP_{E,a}(x,\\ell):= \\#\\{p\\leq x : p\\nmid N_E,\\, a_p(E)=a \\text{ and $\\ell$ splits in $\\QQ(\\pi_p)$}\\}.\n\\] \nUsing our GRH assumption, Lemma~4.4 of \\cite{MR935007} shows that\n\\begin{equation} \\label{E:PEa break}\nP_{E,a}(x) \\ll_E \\max_{\\ell \\in I}\\, P_{E,a}(x,\\ell).\\\\\n\\end{equation}\n\nNow take any prime $\\ell \\in I$.  Set $L:=\\QQ(E[\\ell])$; it is the fixed field in ${\\overline{\\Q}}$ of $\\ker \\rho_{E,\\ell}$.   Using $\\rho_{E,\\ell}$, we may identify the Galois group $\\Gal(L\/\\QQ)$ with $G:=\\operatorname{GL}_2({\\mathcal F}_\\ell)$.   Define\n\\[\nC:= \\{A \\in G : \\tr(A)\\equiv a \\bmod{\\ell} \\text{ and } \\tr(A)^2- 4\\det(A) \\in {\\mathcal F}_\\ell \\text{ is a square}\\};\n\\]\nit is a subset of $G$ that is stable under conjugation.\n\n\\begin{lemma}\nWe have $P_{E,k}(x,\\ell) \\leq \\pi_C(x,L\/\\QQ) +1$.\n\\end{lemma}\n\\begin{proof}\nTake any prime $p\\nmid N_E \\ell$ such that $a_p(E)=a$ and $\\ell$ splits in $\\QQ(\\pi_p)$.  The representation $\\rho_{E,\\ell}$ is unramified at $p$ and we have $\\tr(\\rho_{E,\\ell}(\\Frob_p))\\equiv a_p(E) = a$ and $\\det(\\rho_{E,\\ell}(\\Frob_p))\\equiv p$ modulo $\\ell$. \n\nSince $\\ell$ splits in $\\QQ(\\pi_p)\\cong \\QQ((a_p(E)^2-4p)^{1\/2}),$ we find that the image of $a_p(E)^2-4p$ in ${\\mathcal F}_\\ell$ is a square.   Therefore, $\\tr(\\rho_{E,\\ell}(\\Frob_p))^2-4\\det(\\rho_{E,\\ell}(\\Frob_p)) \\in {\\mathcal F}_\\ell$ is a square.\n\nWe have thus shown that $\\rho_{E,\\ell}(\\Frob_p) \\subseteq C$.  The bound $P_{E,k}(x,\\ell) \\leq \\pi_C(x,L\/\\QQ) +1$ is now clear; we have added 1 to take into account the excluded prime $p=\\ell$.\n\\end{proof}\n\nLet $B$ be the group of upper triangular matrices in $G$. \n\n\\begin{lemma} \\label{L:second a}\nWe have $P_{E,a}(x,\\ell) \\leq\\widetilde\\pi_{C\\cap B}(x,L\/L^B) +1$.\n\\end{lemma}\n\\begin{proof}\nObserve that every conjugacy class of $G$ in $C$ contains an element from $B$.   Lemma~\\ref{L:invariance C}(\\ref{L:invariance C i})  implies that $P_{E,a}(x,\\ell) \\leq \\widetilde\\pi_{C}(x,L\/\\QQ)+1 \\leq \\widetilde\\pi_{C\\cap B}(x,L\/L^B) +1$.\n\\end{proof}\n\n Let $U$ be the subgroup of $B$ consisting of the upper triangular matrices whose diagonal entries are both $1$.  The group $U$ is normal in $B$ and $B\/U$ is abelian.  Let $C'$ be the image of $C\\cap B$ in $B\/U = \\Gal(L^U\/L^B)$.\n \n\\begin{lemma} \\label{L:third a}\nWe have $P_{E,a}(x,\\ell) \\leq\\widetilde\\pi_{C'}(x,L^U\/L^B) +1$.\n\\end{lemma}\n\\begin{proof}\nWe have $U\\cdot (C\\cap B) = C \\cap B$.  Lemma~\\ref{L:invariance C}(\\ref{L:invariance C ii}) implies that $\\widetilde\\pi_{C\\cap B}(x,L\/L^B) = \\widetilde\\pi_{C'}(x,L^U\/L^B)$.  Therefore, $P_{E,a}(x,\\ell) \\leq\\widetilde\\pi_{C'}(x,L^U\/L^B) +1$ by Lemma~\\ref{L:second a}.\n\\end{proof}\n\nBefore applying our Chebotarev bound to $\\widetilde\\pi_{C'}(x,L^U\/L^B)$, we first bound some of the terms that will occur.\n\n\\begin{lemma} \\label{L:a comps}\nWe have $|C'|\\ll \\ell$, $|C'|\/|B\/U| \\ll 1\/\\ell$, $[L^B:\\QQ] \\ll \\ell$ and $\\log M(L^U\/L^B) \\ll_E \\log \\ell$.\n\\end{lemma}\n\\begin{proof}\nWe have $|G|=(\\ell-1)^2(\\ell+1)\\ell \\asymp \\ell^4$, $|B|=(\\ell-1)^2\\ell \\asymp \\ell^3$, $|U|=\\ell$ and $|B\/U|=(\\ell-1)\\ell \\asymp \\ell^2$.  We have $[L^B:\\QQ] = [G:B] \\asymp \\ell$.  The map $({\\mathcal F}_\\ell^\\times)^2 \\to B,$ $(b_1,b_2)\\mapsto \\big(\\begin{smallmatrix} b_1 & 0\\\\ 0 & b_2\\end{smallmatrix} \\big)$ induces an isomorphism $({\\mathcal F}_\\ell^\\times)^2 \\xrightarrow{\\sim} B\/U$.  Therefore, \n\\[\n|C'|=|\\{(b_1,b_2) \\in {\\mathcal F}_\\ell^2: b_1 b_2 \\neq 0,\\, b_1+b_2\\equiv a \\bmod{\\ell}\\}| \\leq \\ell.\n\\]\nWe thus have $|C'|\/|B\/U| \\ll \\ell\/\\ell^2=1\/\\ell$.  \n\nIf a prime $p$ ramifies in $L$, then $p|N_E\\ell$.  By Proposition~4' of \\cite{MR644559}, we have $\\log(d_{L^B}^{1\/[L^B:\\QQ]})\\ll_E \\log(\\ell\\cdot [L^B:\\QQ]) \\ll \\log \\ell$.  Using the above bounds together, we find that $M(L^U\/L^B)\\ll_E \\ell^2\\cdot \\log \\ell \\cdot \\ell$ and hence $\\log M(L^U\/L^B) \\ll_E \\log \\ell$.\n\\end{proof}\n\nThe AHC conjecture holds for the extension $L^U\/L^B$ since its Galois group $B\/U$ is abelian.  By Theorem~\\ref{T:Chebotarev upper bound} and our GRH assumption, we have\n\\begin{align*}\n\\widetilde\\pi_{C'}(x,L^U\/L^B) & \\ll \\frac{|C'|}{|B\/U|} \\frac{x}{\\log x} + |C'|^{1\/2} [L^B:\\QQ] \\frac{x^{1\/2}}{\\log x} \\log M(L^U\/L^B)\\\\\n& \\ll_E \\frac{1}{\\ell} \\frac{x}{\\log x} + \\ell^{1\/2}\\cdot \\ell \\cdot \\frac{x^{1\/2}}{\\log x} \\log \\ell,\n\\end{align*}\nwhere the last line uses Lemma~\\ref{L:a comps}.  Lemma~\\ref{L:third a} and $\\ell \\in [y,2y]$ implies that\n\\[\nP_{E,a}(x,\\ell) \\ll_E  \\frac{1}{y} \\frac{x}{\\log x} + y^{3\/2}  \\frac{x^{1\/2}}{\\log x} \\log y.\n\\]\nSince this holds for all $\\ell\\in I$, the inequality (\\ref{E:PEa break}) gives\n\\[\nP_{E,a}(x) \\ll_E  \\frac{1}{y} \\frac{x}{\\log x} + y^{3\/2}  \\frac{x^{1\/2}}{\\log x} \\log y.\n\\]\nTake $y := c' \\cdot x^{1\/5}\/(\\log x)^{2\/5}$, where $c'$ is a constant chosen large enough to ensure that $y\\geq c$ for all $x\\geq 2$.   With this choice of $y$, we obtain the bound $P_{E,a}(x) \\ll_E x^{4\/5}\/(\\log x)^{3\/5}$.\\\\\n\nFinally consider the case where $a=0$.  Take any $\\ell \\in I$ and keep notation as above.  Let $H$ be the subgroup of $B$ consisting of the matrices whose eigenvalues are both equal; it is a normal subgroup of $B$ and we have $H\\cdot (C \\cap B) = C \\cap B$ (multiplying a trace $0$ matrix by a scalar does not change the trace).   Lemma~\\ref{L:invariance C}(\\ref{L:invariance C ii}) implies that $\\widetilde\\pi_{C\\cap B}(x,L\/L^B) = \\widetilde\\pi_{C''}(x,L^H\/L^B)$, where $C''$ is the image of $C\\cap H$ in $B\/H$.  Therefore, $P_{E,a}(x,\\ell) \\leq\\widetilde\\pi_{C''}(x,L^H\/L^B) +1$ by Lemma~\\ref{L:second a}.    Arguing as above, and using $|B\/H|=\\ell-1$ and $|C''|=1$, we have\n\\[\nP_{E,0}(x)\\ll \\max_{\\ell \\in I}\\, P_{E,0}(x,\\ell) \\ll \\max_{\\ell\\in I}  \\Big( \\frac{1}{\\ell} \\frac{x}{\\log x} + 1^{1\/2}\\cdot \\ell \\cdot \\frac{x^{1\/2}}{\\log x} \\log \\ell \\Big).\n\\]\nChoosing $y \\asymp x^{1\/4}\/(\\log x)^{1\/2}$, we deduce that $P_{E,0}(x) \\ll  x^{3\/4}\/(\\log x)^{1\/2}$.\n\n\\section{Proof of Theorem~\\ref{T:main k}}  \\label{S:main k}\n\nFix a non-CM elliptic curve $E$ over $\\QQ$ and an imaginary quadratic field $k$.  Let ${\\mathcal O}$ be the ring of integers of $k$. Let ${\\mathcal H}$ be the Hilbert class field of $k$ and let $h_k$ be the class number of $k$.    Fix a prime $\\ell \\geq 5$ such that $\\rho_{E,\\ell}$ is surjective and $\\ell$ \\emph{splits} in $k$; we will make a more specific choice later.\\\\\n\nIn \\S\\ref{SS:mixed}, we constructed a Galois representation\n\\[\n\\Psi_\\ell\\colon \\Gal_{{\\mathcal H}} \\to {\\mathcal G},\n\\]\nwhere ${\\mathcal G}:= \\{ (A,u)\\in\\operatorname{GL}_2({\\mathcal F}_\\ell) \\times \\left(({\\mathcal O}\/\\ell{\\mathcal O})^\\times\/{\\mathcal O}^\\times \\right) : \\det(A)= N_{k\/\\QQ}(u) \\}$.  The representation $\\Psi_\\ell$ is surjective by Lemma~\\ref{L:intersection}.     Let $L$ be the fixed field in $\\bbar{{\\mathcal H}}$ of $\\ker \\Psi_\\ell$.   Using $\\Psi_\\ell$, we will identify the Galois group $\\Gal(L\/{\\mathcal H})$ with ${\\mathcal G}$.  \n\nRecall that the trace map $\\Tr_{k\/\\QQ}\\colon k\\to \\QQ$ induces a linear map $\\Tr_{k\/\\QQ}\\colon {\\mathcal O}\/\\ell{\\mathcal O} \\to {\\mathcal F}_\\ell$.   Define the set\n\\[\n{\\mathcal C} := \\{ (A,u) \\in {\\mathcal G} :  \\tr(A) \\in \\Tr_{k\/\\QQ}(u),\\,  \\tr(A)^2-4\\det(A)\\in {\\mathcal F}_\\ell \\text{ is a square}\\};\n\\]\nit is a subset of ${\\mathcal G}$ stable under conjugacy.   We now give a useful bound for $P_{E,k}(x)$.\n\n\\begin{lemma} \\label{L:first bound k}\nWe have $P_{E,k}(x) \\leq \\frac{1}{h_k} \\,\\pi_{\\mathcal C}(x,L\/{\\mathcal H}) + 4$.\n\\end{lemma}\n\\begin{proof}\nTake any prime $p\\nmid  N_E\\ell$ for which $E$ has ordinary reduction at $p$ and for which $k\\cong \\QQ(\\pi_p)$.  By Lemma~\\ref{L:LT2 key}, the prime $p$ splits completely in ${\\mathcal H}$.   Let $\\P \\in \\Sigma_{\\mathcal H}$ be any of the $h_k$ primes that divide $\\pi_\\mathfrak{p} {\\mathcal O}$.  By Lemma~\\ref{L:LT2 key}, we have $\\tr(\\rho_{E,\\ell}(\\Frob_\\P))\\in \\Tr_{k\/\\QQ}(\\psi_{k,\\ell}(\\Frob_\\P))$.\n\nSince $k\\cong \\QQ(\\pi_p)$ and $\\ell$ splits in $k$, the polynomial $x^2-a_p(E)x +p$ will factor modulo $\\ell$.  Therefore, the image of $a_p(E)^2- 4p$ in ${\\mathcal F}_\\ell$ is a square.  Since $p$ splits completely in ${\\mathcal H}$, we have $\\tr(\\rho_{E,\\ell}(\\Frob_\\P))=\\tr(\\rho_{E,\\ell}(\\Frob_p))$ and $\\det(\\rho_{E,\\ell}(\\Frob_\\P))=\\det(\\rho_{E,\\ell}(\\Frob_p))$.  Therefore, $\\tr(\\rho_{E,\\ell}(\\Frob_\\P))^2- 4\\det(\\rho_{E,\\ell}(\\Frob_\\P)) \\equiv a_p(E)^2-4p \\pmod{\\ell}$ is a square.\n\nWe have verified that $\\Psi_\\ell(\\Frob_\\P) \\subseteq {\\mathcal C}$ for each of the $h_k$ primes $\\P$ dividing $\\pi_p{\\mathcal O}$.   So the set\n\\[\n\\{p\\leq x: p\\nmid N_E,\\, \\QQ(\\pi_p)\\cong k \\} - (S\\cup\\{\\ell\\})\n\\]\nhas cardinality at most $\\frac{1}{h_k} \\pi_C(x,L\/{\\mathcal H})$, where $S$ is the set of primes $p\\nmid N_E$ for which $E$ has supersingular reduction at $p$ and $\\QQ(\\pi_p)\\cong k$.  It thus suffices to show that $|S|\\leq 3$.   \n\nTake any prime $p\\in S$ with $p\\geq 5$.   Since $E$ has supersingular reduction at $p\\geq 5$, we have $a_p(E)=0$.   Therefore, $k$ is isomorphic to $\\QQ(\\pi_p)\\cong \\QQ(\\sqrt{-p})$.    So if $p\\in S$, then $p$ is $2$, $3$ or the unique prime (if it exists) such that $k\\cong \\QQ(\\sqrt{-p})$.  Therefore, $|S|\\leq 3$.\n\\end{proof}\n\nLet $B$ be the group of upper triangular matrices in $\\operatorname{GL}_2({\\mathcal F}_\\ell)$.   Define\n\\[\n\\mathcal B := \\{ (A,u) \\in {\\mathcal G} : A \\in B \\};\n\\]\nit is a subgroup of ${\\mathcal G}$.  We can identify $\\mathcal B$ with the Galois group $\\Gal(L\/L^\\mathcal B)$.\n\n\\begin{lemma} \\label{L:second bound k}\nWe have $P_{E,k}(x) \\leq \\frac{1}{h_k} \\,\\widetilde\\pi_{{\\mathcal C}\\cap \\mathcal B}(x,L\/L^\\mathcal B) + 4$.\n\\end{lemma}\n\\begin{proof}\n Any matrix $A\\in \\operatorname{GL}_2({\\mathcal F}_\\ell)$ with $\\tr(A)^2-4\\det(A)\\in {\\mathcal F}_\\ell$ a square is conjugate to a matrix in $B$.  Therefore, every element of ${\\mathcal C}$ is conjugate in ${\\mathcal G}$ to some element of $\\mathcal B$.  By Lemma~\\ref{L:invariance C}(\\ref{L:invariance C i}), we have $\\widetilde{\\pi}_{\\mathcal C}(x,L\/{\\mathcal H}) \\leq \\widetilde{\\pi}_{C\\cap \\mathcal B}(x,L\/L^{\\mathcal B})$.  The lemma now follows from Lemma~\\ref{L:first bound k} and the easy bound $\\pi_{\\mathcal C}(x,L\/{\\mathcal H})\\leq \\widetilde\\pi_{\\mathcal C}(x,L\/{\\mathcal H})$.\n\\end{proof}\n\nLet ${\\mathcal U}$ be the image of the group \n\\[\n\\{(A,a)\\in \\operatorname{GL}_2({\\mathcal F}_\\ell) \\times {\\mathcal F}_\\ell^\\times: \\text{the eigenvalues of $A$ are both $a$}\\}\n\\]\nin ${\\mathcal G}$ (we can identify ${\\mathcal F}_\\ell^\\times$ with a subgroup of $({\\mathcal O}\/\\ell{\\mathcal O})^\\times$ since ${\\mathcal F}_\\ell$ is a subalgebra of ${\\mathcal O}\/\\ell{\\mathcal O}$).  The group ${\\mathcal U}$ is normal in $\\mathcal B$ and $\\mathcal B\/{\\mathcal U}$ is an abelian group.   We can identify $\\mathcal B\/{\\mathcal U}$ with the Galois group $\\Gal(L^{\\mathcal U}\/L^\\mathcal B)$.  Let ${\\mathcal C}'$ be the image of ${\\mathcal C}\\cap \\mathcal B$ under the homomorphism $\\mathcal B\\to \\mathcal B\/{\\mathcal U}$; it is stable under conjugacy in $\\mathcal B\/{\\mathcal U}$.\n\n\\begin{lemma} \\label{L:third bound k}\nWe have $P_{E,k}(x) \\leq \\frac{1}{h_k} \\,\\widetilde\\pi_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^\\mathcal B) + 4$.\n\\end{lemma}\n\\begin{proof}\nObserve that ${\\mathcal U}\\cdot ({\\mathcal C}\\cap \\mathcal B) = {\\mathcal C} \\cap \\mathcal B$; whether an element $(A,u) \\in \\mathcal B$ belongs to ${\\mathcal C}$ depends only on $u$ and the eigenvalues of $A$, and that $\\tr(A)\\in \\Tr_{k\/\\QQ}(u)$ remains true if $A$ and $u$ are multiplied by a common scalar in ${\\mathcal F}_\\ell^\\times$.  Lemma~\\ref{L:invariance C}(\\ref{L:invariance C ii}) implies that $\\widetilde{\\pi}_{{\\mathcal C}\\cap \\mathcal B}(x,L\/L^\\mathcal B) = \\widetilde{\\pi}_{C'}(x,L^{\\mathcal U}\/L^\\mathcal B)$.   The lemma then follows from Lemma~\\ref{L:second bound k}.   \n\\end{proof}\n\nSince $L^{\\mathcal U}\/L^{\\mathcal B}$ is an abelian extension, we can now apply our Chebotarev bounds to obtain bounds for $P_{E,k}(x)$.   We first bound some terms that will show up.\n\n\\begin{lemma} \\label{L:specific discriminant bounds}\n\\begin{romanenum}\n\\item\nWe have $|{\\mathcal G}| \\asymp \\ell^5$, $|\\mathcal B| \\asymp \\ell^4$ and $|{\\mathcal U}|\\asymp \\ell^2$.\n\\item \nWe have $|{\\mathcal C}'|\\ll \\ell$ and $|{\\mathcal C}'|\/|\\mathcal B\/{\\mathcal U}| \\ll  1\/\\ell$.  \n\\item\nWe have $[L^\\mathcal B: \\QQ] \\ll h_k \\ell$.\n\\item\nWe have $\\log M(L^{\\mathcal U}\/L^\\mathcal B) \\ll_E \\log (d_k\\ell)$.\n\\end{romanenum}\n\\end{lemma}\n\\begin{proof}\nSince $\\ell$ splits in $k$, we have $({\\mathcal O}\/\\ell{\\mathcal O})^\\times \\cong {\\mathcal F}_\\ell^\\times \\times {\\mathcal F}_\\ell^\\times$.  Therefore,\n\\[\n|{\\mathcal C}\\cap \\mathcal B| \\leq |\\{ (A,b,c) \\in B \\times {\\mathcal F}_\\ell^\\times \\times {\\mathcal F}_\\ell^\\times : \\det(A)=bc \\text{ and } \\tr(A)=b+c \\}| \\leq 2 |B|,\n\\]\nwhere the last inequality uses that $x^2-\\tr(A)x +\\det(A)$ has at most two roots $b,c\\in {\\mathcal F}_\\ell$.   We thus have $|{\\mathcal C} \\cap \\mathcal B| \\leq 2 \\ell^3$.  Since ${\\mathcal U}\\cdot ({\\mathcal C}\\cap \\mathcal B) = {\\mathcal C} \\cap \\mathcal B$, we have $|{\\mathcal C}'| = |{\\mathcal C} \\cap \\mathcal B|\/|{\\mathcal U}| \\leq 2\\ell^3\/|{\\mathcal U}| \\ll \\ell$.   We have $|{\\mathcal C}'|\/|\\mathcal B\/{\\mathcal U}| \\ll  1\/\\ell$ since $|\\mathcal B\/{\\mathcal U}| \\asymp \\ell^2$.  We have $[L^\\mathcal B:\\QQ]=[{\\mathcal H}:\\QQ][L^\\mathcal B:{\\mathcal H}]= 2h_k\\cdot [{\\mathcal G}:\\mathcal B]$, so $[L^\\mathcal B:\\QQ] \\ll h_k \\ell$.\n\nLet ${\\mathcal P}$ be the set of rational primes $p$ divisible by some $\\P\\in \\Sigma_{\\mathcal H}$ that ramifies in $L$.    Each prime in ${\\mathcal P}$ divides $N_E \\ell$.    We have $[\\mathcal B:{\\mathcal U}]\\ll \\ell^2$, so\n\\[\n\\log M(L^{\\mathcal U}\/L^\\mathcal B) \\leq \\log\\big([\\mathcal B:{\\mathcal U}] d_{L^\\mathcal B}^{1\/[L^\\mathcal B:\\QQ]}\\cdot N_E \\ell) \\ll_E [L^\\mathcal B:\\QQ]^{-1} \\log(d_{L^\\mathcal B}) + \\log \\ell.\n\\]\nIt thus suffices to prove that $[L^\\mathcal B:\\QQ]^{-1} \\log(d_{L^\\mathcal B}) \\ll_E \\log(d_k\\ell)$.  By Proposition~4 of \\cite{MR644559}, we have\n\\[\n[L^\\mathcal B:\\QQ]^{-1} \\log(d_{L^\\mathcal B}) \\leq [{\\mathcal H}:\\QQ]^{-1} \\log(d_{\\mathcal H}) +  {\\sum}_{p\\in {\\mathcal P}} \\log p + |{\\mathcal P}| \\log([L^\\mathcal B:{\\mathcal H}]).\n\\]\nWe have $[L^\\mathcal B:{\\mathcal H}]\\ll \\ell$, so $[L^\\mathcal B:\\QQ]^{-1} \\log(d_{L^\\mathcal B}) \\ll_E [{\\mathcal H}:\\QQ]^{-1} \\log(d_{\\mathcal H}) + \\log \\ell$.  Since ${\\mathcal H}\/k$ is unramified,  Proposition~4 of \\cite{MR644559} implies that $[{\\mathcal H}:\\QQ]^{-1} \\log(d_{\\mathcal H}) = 2^{-1} \\log d_k$ and hence $[L^\\mathcal B:\\QQ]^{-1} \\log(d_{L^\\mathcal B}) \\ll_E \\log(d_k\\ell)$.\n\\end{proof}\n\n\n\\begin{lemma} \\label{L:easy 0}\nIf $d_k > 4x$, then $P_{E,k}(x)=0$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $d_k > 4x$ and $P_{E,k}(x)>0$.  There is thus a prime $p\\nmid N_E$ satisfying $p\\leq x$ and $k\\cong \\QQ(\\pi_p)$.  We have $\\QQ(\\sqrt{a_p(E)^2-4p})\\cong \\QQ(\\sqrt{-d_k})$, and hence $d_k$ divides $a_p(E)^2-4p$ (the divisibility with respect to the prime $2$ uses that $a_p(E)^2-4p$ is congruent to $0$ or $1$ modulo $4$).  Therefore, $d_k \\leq 4p-a_p(E)^2 \\leq 4x$ which contradicts our assumption.\n\\end{proof}\n\nBy Lemma~\\ref{L:easy 0}, we may assume that $d_k\\leq 4x$; the desired bounds are trivial otherwise.\n\n\\subsection{Conditional bounds}\n\nAssume that GRH holds.  \n\nBy Lemma~\\ref{L:third bound k}, Theorem~\\ref{T:Chebotarev upper bound} and Lemma~\\ref{L:good approximation}, we have\n\\begin{align*}\nP_{E,k}(x) &\\leq \\frac{1}{h_k} \\widetilde{\\pi}_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^\\mathcal B) + 4\\\\\n& \\ll \\frac{1}{h_k} \\Big( \\frac{|{\\mathcal C}'|}{|\\mathcal B\/{\\mathcal U}|} \\frac{x}{\\log x} + |{\\mathcal C}'|^{1\/2}\\, [L^\\mathcal B:\\QQ] \\frac{x^{1\/2}}{\\log x} \\, \\log M(L^{\\mathcal U}\/L^\\mathcal B) \\Big)+4\\\\\n& \\ll_E   \\frac{1}{h_k} \\frac{1}{\\ell} \\frac{x}{\\log x} + \\ell^{3\/2}\\, \\frac{x^{1\/2}}{\\log x} \\log(d_k \\ell).\n\\end{align*}\nUsing Lemma~\\ref{L:specific discriminant bounds} and $d_k\\leq 4x$, we find that\n\\[\nP_{E,k}(x) \\ll_E \\frac{1}{h_k} \\frac{1}{\\ell} \\frac{x}{\\log x} + \\ell^{3\/2}\\, \\frac{x^{1\/2}}{\\log x} \\log(x\\ell).\n\\]\nWe still need to choose our prime $\\ell$.\n\n\\begin{lemma} \\label{L:GRH for split}\nAssuming GRH, there is an absolute constant $\\gamma>0$ such that if $y \\geq \\gamma (\\log d_k)^2$, then there exists a prime $\\ell$ in the interval $[y,2y]$ that splits completely in $k$.\n\\end{lemma}\n\\begin{proof} \nThis follows from Corollary~\\ref{C:cheb existence} with the extension $k\/\\QQ$.\n\\end{proof}\n\nDefine \n\\[\ny:= \\begin{cases}\nC \\cdot h_k^{-2\/5} \\cdot {x^{1\/5}}\/{(\\log x)^{2\/5}} & \\text{if $h_k \\leq x^{1\/2}\/(\\log x)^6$},\\\\\nC \\cdot (\\log x)^2 & \\text{otherwise},\n\\end{cases}\n\\]\nwhere $C>0$ is some constant depending only on $E$.  In both cases, we have $y \\geq C (\\log x)^2$.  \n\nSince $d_k \\leq 4x$, we have, after possibly increasing $C$, that $y\\geq \\gamma(\\log d_k)^2$ with $\\gamma$ as in Lemma~\\ref{L:GRH for split}.  By Lemma~\\ref{L:GRH for split}, there is a prime $\\ell \\in [y,2y]$ that splits completely in $k$.  After possibly increasing the constant $C$ first, we may assume by Theorem~\\ref{T:Serre} that $\\rho_{E,\\ell}$ is surjective and that $\\ell \\geq 5$.   With this prime $\\ell$, we obtain the bound\n\\[\nP_{E,k}(x) \\ll_E \\frac{1}{h_k} \\frac{1}{\\ell} \\frac{x}{\\log x} + \\ell^{3\/2}\\, \\frac{x^{1\/2}}{\\log x} \\log(x\\ell) \\ll \\frac{1}{h_k} \\frac{1}{y} \\frac{x}{\\log x} + y^{3\/2}\\, \\frac{x^{1\/2}}{\\log x} \\log(xy).\n\\]\nSince $y\\ll_E x$, we have \n\\begin{align} \\label{E:balance}\nP_{E,k}(x) \\ll_E \\frac{1}{h_k} \\frac{1}{y} \\frac{x}{\\log x} + y^{3\/2}\\, x^{1\/2}.\n\\end{align}\nIf $h_k \\leq x^{1\/2}\/(\\log x)^6$, and hence $y=Ch_k^{-2\/5}  {x^{1\/5}}\/{(\\log x)^{2\/5}}$,  substituting for $y$ gives the bound \n\\[\nP_{E,k}(x) \\ll_E h_k^{-3\/5} x^{4\/5}\/(\\log x)^{3\/5};\n\\]\nnote that our $y$ was chosen so that both terms in (\\ref{E:balance}) have the same magnitude.   In the case $h_k > x^{1\/2}\/(\\log x)^6$, we obtain\n\\[\nP_{E,k}(x) \\ll_E \\frac{1}{h_k}\\frac{x}{(\\log x)^3} + x^{1\/2}(\\log x)^{3} \\leq 2 x^{1\/2} (\\log x)^3.\n\\]\nThe bound of Theorem~\\ref{T:main k}(\\ref{T:main k i}) follows by adding our two possible bounds for $P_{E,k}(x)$.\n\n\\subsection{Unconditional bounds}\n\nDefine \n\\[\n\\displaystyle y:= C \\frac{1}{h_k} \\frac{\\log x}{(\\log \\log x)^2},\n\\] \nwhere $C>0$ is a constant depending only on $E$ to be chosen later.  Suppose that there is a prime $\\ell$ in the interval $[y,2y]$ such that $\\ell$ splits in $k$, $\\ell\\geq 5$, and $\\rho_{E,\\ell}$ is surjective.    \n\nThe group ${\\mathcal B}\/{\\mathcal U} = \\Gal(L^{\\mathcal U}\/L^{\\mathcal B})$ is abelian.   By Theorem~\\ref{T:effective}(\\ref{T:effective ii}), there are absolute constants $b, c>0$ such that if $\\log x \\geq b [L^{\\mathcal B}:\\QQ]  \\log^2 M(L^{\\mathcal U}\/L^{\\mathcal B})$, then\n\\[\n\\pi_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^{\\mathcal B}) \\ll  \\frac{|{\\mathcal C}'|}{|{\\mathcal B}\/{\\mathcal U}|}\\,  \\frac{x}{\\log x} + |{\\mathcal C}'|^{1\/2}\\, [L^{\\mathcal B}:\\QQ]x \\exp\\bigg( - \n\\frac{c(\\log x)^{1\/2}}{[L^{\\mathcal B}:\\QQ]^{1\/2}} \\bigg) \\log^2 (M(L^{\\mathcal U}\/L^{\\mathcal B})  x),\n\\]\nwhere we have used that $\\beta_{L^{\\mathcal U}}\\leq 1$ if it exists.\n\n\nBy Lemma~\\ref{L:specific discriminant bounds}, we have \n\\[\n[L^{\\mathcal B}:\\QQ]  \\log^2 M(L^{\\mathcal U}\/L^{\\mathcal B}) \\ll h_k \\ell \\log^2(d_k \\ell) \\ll h_k \\ell \\log^2(h_k \\ell),\n\\]\nwhere the last inequality use the Brauer-Seigel theorem.   Using Lemma~\\ref{L:specific discriminant bounds}, we deduce that there are positive absolute constants $b'$ and $c'$ such that if $\\log x \\geq b' \\cdot h_k \\ell\\cdot  \\log^2(h_k \\ell)$, then\n\\[\n\\pi_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^\\mathcal B) \\ll_E  \\frac{1}{\\ell}\\,  \\frac{x}{\\log x} + h_k \\ell^{3\/2} x \\exp\\bigg( - c' \\sqrt{\\frac{\\log x}{h_k\\ell}}\\bigg)\\log^2 (h_k\\ell \\cdot x).\n\\]\nUsing that $\\ell \\in [y,2y]$, we have\n\\[\n\\pi_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^\\mathcal B) \\ll_E  h_k  \\frac{x (\\log \\log x)^2}{(\\log x)^2} + \\frac{1}{\\sqrt{h_k}}\\cdot  \\frac{(\\log x)^{3\/2}}{(\\log\\log x)^3}\\cdot x \\exp\\bigg( - \\frac{c'}{2\\sqrt{C}} \\log \\log x \\bigg)(\\log x)^2.\n\\]\nBy taking our constant $C>0$ sufficiently small, we find that $\\pi_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^\\mathcal B) \\ll_E h_k \\cdot \\frac{x (\\log \\log x)^2}{(\\log x)^2}$.  By Lemmas~\\ref{L:good approximation} and \\ref{L:specific discriminant bounds}, we find that  $\\tilde\\pi_{{\\mathcal C}'}(x,L^{\\mathcal U}\/L^\\mathcal B) \\ll_E h_k \\cdot {x (\\log \\log x)^2}\/{(\\log x)^2}$.    Therefore,\n\\[\nP_{E,k}(x) \\ll_E \\frac{x (\\log \\log x)^2}{(\\log x)^2}\n\\]\nby Lemma~\\ref{L:third bound k}.\n\nFinally, we now need to know that such a prime $\\ell$ exists; at least if $x$ is sufficiently large.  By the Chebotarev density theorem and Theorem~\\ref{T:Serre}, there is a constant $\\gamma \\geq 1$, depending on $E$ and $k$, such that if $y\\geq\\gamma$, then there is a prime $\\ell \\in [y,2y]$ for which $\\ell$ splits in $k$ and $\\rho_{E,\\ell}$ is surjective.  So for $x$ sufficiently large, we will have $y\\geq \\gamma$ and the desired prime $\\ell$ exists.  (One could make this explicit by using an effective version of the Chebotarev density theorem.)  This completes the proof of Theorem~\\ref{T:main k}(\\ref{T:main k ii}). \n\n\n\\section{Proof of Corollary~\\ref{C:D theorem}} \\label{S:D}\n\nLet ${\\mathcal D}_E(x)$ be the set of imaginary quadratic extensions $k$ of $\\QQ$ for which there exists a prime $p\\leq x$ with $\\QQ(\\pi_p) \\cong k$; note that $|{\\mathcal D}_E(x)|=D_E(x)$.  We start with the identity\n\\[\n\\pi(x) = |\\{p \\leq x: p| N_E\\}| + {\\sum}_{k\\in {\\mathcal D}_E(x)} P_{E,k}(x),\n\\]\nTheorem \\ref{T:main k}(\\ref{T:main k i}) then implies that\n\\begin{align*} \nx\/\\log x \\ll_E \\sum_{k\\in {\\mathcal D}_E(x)} P_{E,k}(x)  \\ll_E &  \\sum_{k\\in {\\mathcal D}_E(x)}\\Bigl( \\frac{1}{h_k^{3\/5}}\\frac{x^{4\/5}}{(\\log x)^{3\/5}} + x^{1\/2}(\\log x)^3 \\Bigr)\\\\\n =& \\sum_{k\\in {\\mathcal D}_E(x)} \\frac{1}{h_k^{3\/5}} \\,\\cdot \\, \\frac{x^{4\/5}}{(\\log x)^{3\/5}} + D_E(x) x^{1\/2}(\\log x)^3.\n\\end{align*}\nUsing GRH, one can show that, $h_k \\gg d_k^{1\/2}\/\\log{d_k}.$  By Lemma~\\ref{L:easy 0}, we have $d_k\\leq 4x$ for all $k\\in {\\mathcal D}_E(x)$.  Using these bounds, we have:\n\\begin{align*} \n\\sum_{k\\in {\\mathcal D}_E(x)} \\frac{1}{h_k^{3\/5}} & \\ll  \\sum_{k\\in{\\mathcal D}_E(x)} \\frac{(\\log d_k)^{3\/5}}{d_k^{3\/10}} \\ll  \\sum_{k\\in {\\mathcal D}_E(x)} \\frac{1}{d_k^{3\/10}} (\\log x)^{3\/5}\n\\leq  \\sum_{d=1}^{ D_E(x) } \\frac{1}{d^{3\/10}} (\\log x)^{3\/5}.\n\\end{align*}\nTherefore, $\\sum_{k\\in {\\mathcal D}_E(x)} {h_k^{-3\/5}}  \\ll  D_E(x)^{7\/10} (\\log x)^{3\/5}$.\nCombining with our previous inequality, we have\n\\begin{align*} \nx\/\\log x  &\\ll_E   D_E(x)^{7\/10} x^{4\/5} \\, + \\, D_E(x)x^{1\/2}(\\log x)^3.\n\\end{align*} \nTherefore, we have $x\/\\log x \\ll_E D_E(x)^{7\/10} x^{4\/5}$ or $x\/\\log x \\ll_E D_E(x)x^{1\/2}(\\log x)^3$.   Equivalently, we have $D_E(x) \\gg_E x^{2\/7}\/(\\log x)^{10\/7}$ or $D_E(x) \\gg_E x^{1\/2}\/(\\log x)^4$.  We conclude that $D_E(x)\\gg_E {x^{2\/7}}\/{(\\log x)^{10\/7}}$ since this is the weaker of the two possible bounds.\n\n\\begin{remark}\nIf we had instead used the bound $P_{E,k}(x) \\ll_E {x^{4\/5}}\/{(\\log x)^{3\/5}}$, then we would have deduced that $D_E(x)\\gg_E {x^{1\/5}}\/{(\\log x)^{2\/5}}$.  Thus the factor $h_k^{-3\/5}$ occuring in our bound of $P_{E,k}(x)$ gives a significant improvement.\n\\end{remark}\n\n\\section{Proof of Theorem~\\ref{T:smoothed}} \\label{S:smoothed proof}\n\nFix a real number $x\\geq 2$. For each class function $\\varphi\\colon G\\to \\mathbb C$, we define\n\\[\n\\Theta_\\varphi(x) := \\sum_{\\mathfrak{p} \\in \\Sigma_K, m\\geq 1}  \\varphi(\\Frob_\\mathfrak{p}^m) \\log N(\\mathfrak{p})  \\cdot f(N(\\mathfrak{p})^m\/x).\n\\]\nWe first estimate $\\Theta_\\chi(x)$ for irreducible characters $\\chi\\colon G \\to \\mathbb C$.\n\n\\begin{lemma} \\label{L:key bounds}\nFor any irreducible character $\\chi$ of $G$, we have\n\\[\n\\Theta_\\chi(x) = \\delta_\\chi \\cdot x \\int_0^\\infty f(t)\\, dt +  O_f\\Big(\\chi(1)[K:\\QQ] x^{1\/2} \\log M(L\/K)\\Big),\n\\]\nwhere $\\delta_\\chi=1$ if $\\chi=1$ and $\\delta_\\chi=0$ otherwise.  \n\\end{lemma}\n\\begin{proof}\nLet $L(s,\\chi)$ be the Artin $L$-function arising from $\\chi$; for background on Artin $L$-functions see \\cite{MR0447187}.    By our AHC assumption, the series $L(s,\\chi)$ extends to a function analytic everywhere except at $s=1$ when $\\chi=1$.  We have $\\ord_{s=1} L(s,\\chi)=-\\delta_\\chi$.  \n\nDefine $A_\\chi:=d_K^{\\chi(1)}\\cdot N({\\mathcal F}_\\chi)$, where ${\\mathcal F}_\\chi \\subseteq {\\mathcal O}_K$ is the Artin conductor corresponding to $\\chi$.  We define the completed $L$-function $\\Lambda(s,\\chi) := A_\\chi^{s\/2} \\gamma_\\chi(s) L(s,\\chi)$, where $\\gamma_\\chi(s)$ is a certain product of $\\Gamma$-factors.   See \\cite{MR0447187}*{p.12} for the precise definition of $\\gamma_\\chi$; we simply note that there are explicit positive integers $a$ and $b$ with $a+b = \\chi(1)[K:\\QQ]$ such that\n\\[\n\\gamma_\\chi(s)=(\\pi^{-s\/2} \\Gamma(\\tfrac{s}{2}))^a \\cdot (\\pi^{-(s+1)\/2} \\Gamma(\\tfrac{s+1}{2}))^b.\n\\] \nThe \\defi{functional equation} for $\\Lambda(s,\\chi)$ says that\n\\[\n\\Lambda(s,\\chi) = W_\\chi \\cdot \\Lambda(1-s,\\bbar{\\chi})\n\\]\nfor some $W_\\chi \\in \\mathbb C^\\times$ with absolute value $1$.  The logarithmic derivative of the Artin $L$-series of $L(s,\\chi)$ is\n\\[\n\\frac{L'}{L}(s,\\chi) = -\\sum_{\\mathfrak{p} \\in \\Sigma_K} \\log N(\\mathfrak{p}) \\sum_{m\\geq 1} \\chi(\\Frob_\\mathfrak{p}^m) N(\\mathfrak{p})^{-ms}. \n\\]\nIn particular, $-\\frac{L'}{L}(s,\\chi) = \\sum_{n\\geq 1} \\Lambda_\\chi(n) n^{-s}$, where \n\\[\n\\Lambda_\\chi(n) := \\log n  \\sum_{\\substack{\\mathfrak{p}\\in\\Sigma_K, m\\geq1\\\\ N(\\mathfrak{p})^m = n}}    \\frac{1}{m} \\chi(\\Frob_\\mathfrak{p}^m). \n\\]\n\n\nLet $\\varphi\\colon (0,+\\infty) \\to \\mathbb C$ be a smooth function with compact support. The Mellin transform of $\\varphi$ is\n\\[\n\\widehat{\\varphi}(s) := \\int_0^\\infty \\varphi(t) t^s\\, \\frac{dt}{t}.\n\\]\nDefine the function $\\psi\\colon (0,+\\infty) \\to \\mathbb C$ by $\\psi(t):=t^{-1}\\varphi(t^{-1})$.   The \\defi{explicit formula}, as given by Iwaniec and Kowalski in Theorem~5.11 of \\cite{MR2061214}, says that\n\\begin{align}  \\label{E:explicit formula}\n\\sum_{n\\geq 1} \\Big( \\Lambda_\\chi(n) \\varphi(n) + \\bbar{\\Lambda_\\chi(n)} \\psi(n) \\Big) =&\\,\\, \\varphi(1) \\log A_\\chi + \\delta_\\chi \\int^\\infty_0 \\varphi(t) \\, dt  \\\\\n&+  \\frac{1}{2\\pi i} \\int_{(1\/2)} \\Big(\\frac{\\gamma_\\chi'}{\\gamma_\\chi}(s) + \\frac{\\gamma_{\\bbar{\\chi}}'}{\\gamma_{\\bbar{\\chi}}}(1-s) \\Big) \\widehat{\\varphi}(s)\\, ds - \\sum_\\rho \\widehat{\\varphi}(\\rho),  \\notag\n\\end{align}\nwhere the sum is over the zeros $\\rho$ of $L(s,\\chi)$, with multiplicity, for which $0\\leq \\operatorname{Re}(\\rho) \\leq 1$.   The explicit formula in \\cite{MR2061214} is given for a general $L$-function that satisfies certain properties; they are all known to hold for Artin $L$-function except for the analytic continuation which holds by our ongoing AHC assumption.  \\\\\n\nWe now take $\\varphi\\colon (0,+\\infty)\\to \\mathbb C$ to be the function $\\varphi(t)=f(t\/x)$.  Observe that \n\\[\n\\Theta_\\chi(x) = \\sum_{n\\geq 1} \\Lambda_\\chi(n) \\varphi(n)\n\\] \nand that $\\delta_\\chi \\int^\\infty_0 \\varphi(t) \\, dt = \\delta_\\chi x \\int^\\infty_0 f(t) \\, dt$; it thus remains to bound the other terms occurring in (\\ref{E:explicit formula}).\n\nWe first bound $\\sum_{n\\geq 1} \\bbar{\\Lambda_\\chi(n)} \\psi(n)$.   If $\\Lambda_\\chi(n)$ is non-zero, then $n$ is a prime power.   Since there are at most $[K:\\QQ]$ primes $\\mathfrak{p}\\in \\Sigma_K$ dividing a fixed rational prime, we have $|\\Lambda_\\chi(n)| \\leq \\log n \\cdot \\chi(1) [K:\\QQ]$.  There is a number $c>0$, depending only on $f$, such that $f(t)=0$ for $t\\leq c$.  If $n \\geq c^{-1}$, then $1\/(xn) \\leq c$ and hence $\\psi(n)=n^{-1}f(1\/(xn))=0$.  Therefore,\n\\[\n\\sum_{n\\geq 1} \\bbar{\\Lambda_\\chi(n)} \\psi(n) \\ll\\sum_{n \\leq c^{-1}} |\\Lambda_\\chi(n)| \\, |\\psi(n)|\n\\leq   \\chi(1) [K:\\QQ] \\sum_{n\\leq c^{-1}} \\log n \\cdot \\sup_{t\\in {\\mathbb R}}|f(t)| \\ll_f \\chi(1)[K:\\QQ].\n\\]\n\nWe have $\\log A_\\chi \\ll \\chi(1) [K:\\QQ] \\log M(L\/K)$ by Proposition~2.5 of \\cite{MR935007}.  Therefore, $\\varphi(1) \\log A_\\chi  \\ll_f \\chi(1) [K:\\QQ] \\log M(L\/K)$.\n\n\n\nFor $y\\in {\\mathbb R}$, we have\n\\[\n\\widehat{\\varphi}(1\/2+i y ) = \\int_0^\\infty f(t\/x) t^{1\/2+i y}\\, \\frac{dt}{t} = x^{1\/2} \\cdot x^{i y} \\int^{\\infty}_{-\\infty} f(e^u) e^{u\/2} e^{i\\cdot u y} \\, du,\n\\]\nwhere we have made the substitution $t=xe^u$.   We have $\\int^{\\infty}_{-\\infty} f(e^u) e^{u\/2} e^{i\\cdot u y} \\, du \\ll_f 1\/(|y|+1)^2$ since $f(e^u)e^{u\/2}$, and hence also its Fourier transform, is a Schwartz function.   Therefore, \n\\[\n\\widehat{\\varphi}(1\/2+i y ) \\ll_f  x^{1\/2}\/(|y|+1)^2.\n\\]\nUsing that $\\frac{\\Gamma}{\\Gamma}'(1\/2+i y) \\ll \\log(|y|+2)$ for all real $y$ (cf.~Lemma~6.1 of \\cite{MR0447191}), we have \n\\[\n\\tfrac{\\gamma_\\chi'}{\\gamma_\\chi}(1\/2+iy) \\ll \\chi(1)[K:\\QQ] x^{1\/2}\/(|y|+1)^2.\n\\]\nTherefore,\n\\[\n\\int_{(1\/2)} \\Big(\\tfrac{\\gamma_\\chi'}{\\gamma_\\chi}(s) + \\tfrac{\\gamma_{\\bbar{\\chi}}'}{\\gamma_{\\bbar{\\chi}}}(1-s) \\Big) \\widehat{\\varphi}(s)\\, ds \\ll_f   \\chi(1)[K:\\QQ] x^{1\/2} \\int^\\infty_{-\\infty} \\tfrac{\\log(|y|+2)}{(|y|+1)^2} \\, dy \\ll  \\chi (1) [K:\\QQ] x^{1\/2}.\n\\]\n\nWe now bound the sum $\\sum_\\rho \\widehat{\\varphi}(\\rho)$.  We have $\\zeta_L(s) = \\prod_{\\chi} L(s,\\chi)^{\\chi(1)}$, where the product is over irreducible characters $\\chi$ of $G$ and $\\zeta_L$ is the Dedekind zeta function of $L$.    By assumption, AHC holds for $L\/K$ and GRH holds for $L$, so we deduce that any zero $\\rho$ of $L(s,\\chi)$ with $0\\leq \\operatorname{Re}(\\rho)\\leq 1$ satisfies $\\operatorname{Re}(\\rho)=1\/2$.   Therefore,\n\\[\n\\sum_\\rho \\widehat{\\varphi}(\\rho) \\ll_f x^{1\/2} \\sum_{\\rho=1\/2+i y} 1\/(|y|+1)^2.\n\\]\nFor each real number $t$, let $N(t,\\chi)$ be the number of zeros $\\rho=1\/2+i y$ of $L(s,\\chi)$, counted with multiplicity, such that $|t-y| \\leq 1$.   From equation (3.5.5) and Proposition~2.5 of \\cite{MR935007}, we have \n\\[\nN(t,\\chi)  \\ll \\chi(1) [K:\\QQ] \\log M(L\/K) + \\chi(1)[K:\\QQ] \\log(|t|+2).\n\\]  \nTherefore,\n\\begin{align*}\n\\sum_\\rho \\widehat{\\varphi}(\\rho) \\ll_f x^{1\/2} \\sum_{n\\in {\\mathbb Z}} \\frac{N(n,\\chi)}{(|n|+1)^2} \n&\\ll x^{1\/2} \\chi(1)[K:\\QQ] \\log M(L\/K)  \\sum_{n\\in {\\mathbb Z}} \\frac{\\log(|n|+2)}{(|n|+1)^2}\\\\\n&\\ll x^{1\/2} \\chi(1)[K:\\QQ] \\log M(L\/K).\n\\end{align*}\nUsing the above bounds with (\\ref{E:explicit formula}), we obtain the desired estimate for $\\Theta_\\chi(x) = \\sum_{n\\geq 1} \\Lambda_\\chi(n) \\varphi(n)$.\n\\end{proof}\n\n\n\\begin{lemma} \\label{L:alway Cheb}\nLet $D$ be any subset of $G$ that is stable under conjugation.  Then\n\\[\n\\Big|\\Theta_{\\delta_D}(x) - \\frac{|D|}{|G|} \\Theta_1(x)\\Big| \\leq |D|^{1\/2}\\cdot \\Big(\\frac{1}{|G|} \\sum_{\\chi\\neq 1} |\\Theta_{\\chi}(x)|^2\\Big)^{1\/2},\n\\]\nwhere the sum is over the non-trivial irreducible characters $\\chi$ of $G$.\n\\end{lemma}\n\\begin{proof}\nWe have $\\Theta_{\\delta_D}(x) - \\frac{|D|}{|G|}  \\Theta_1(x) = \\sum_{C\\subseteq D} (\\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_1(x))$, where the sum is over the conjugacy classes $C$ of $G$.  Using the triangle inequality and the Cauchy-Schwartz inequality, we find that $|\\Theta_{\\delta_D}(x) - \\frac{|D|}{|G|}  \\Theta_1(x)|$ is less than or equal to\n\\[\n\\sum_{C\\subseteq D} \\Big|\\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_1(x) \\Big| \\leq \\Big(\\sum_{C\\subseteq D} |C|\\Big)^{1\/2} \\Big( \\sum_C \\frac{1}{|C|} \\Big| \\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_{1}(x) \\Big|^2 \\Big)^{1\/2}.\n\\]\nSince $\\sum_{C\\subseteq D} |C| = |D|$, it suffices to prove that\n\\begin{equation} \\label{E:conj sum 1}\n\\sum_C \\frac{1}{|C|} \\Big| \\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_{1}(x) \\Big|^2\n= \\frac{1}{|G|} \\sum_{\\chi\\neq 1} |\\Theta_{\\chi}(x)|^2,\n\\end{equation}\nwhere the first sum is over the conjugacy classes $C$ of $G$ and the second sum is over the non-trivial irreducible characters of $G$.\n\nFor $C$ and $\\chi$ as above, let $\\chi(C)$ be the common value of $\\chi(g)$ with $g\\in C$.    We have $\\delta_C = \\tfrac{|C|}{|G|} \\sum_{\\chi} \\bbar{\\chi(C)}\\cdot \\chi$, so by linearity $\\Theta_{\\delta_C}(x) = \\frac{|C|}{|G|} \\sum_\\chi \\bbar{\\chi(C)} \\,\\Theta_{\\chi}(x)$.  Therefore,\n\\begin{align*}\n\\sum_{C} \\frac{1}{|C|} \\Big| \\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_{1}(x) \\Big|^2 \n& = \\sum_C \\frac{1}{|C|} \\Big| \\frac{|C|}{|G|} \\sum_{\\chi\\neq 1} \\bbar{\\chi(C)} \\Theta_{\\chi}(x) \\Big|^2 \\\\\n& = \\frac{1}{|G|} \\sum_C  \\frac{|C|}{|G|}   \\sum_{\\chi\\neq 1, \\chi'\\neq 1} \\chi(C) \\bbar{\\chi'(C)} \\Theta_{\\chi}(x) \\bbar{\\Theta_{\\chi'}(x)}\n\\end{align*}\nSince $ \\sum_C \\frac{|C|}{|G|} \\chi(C) \\bbar{\\chi'(C)} = \\frac{1}{|G|} \\sum_{g\\in G} \\chi(g) \\chi'(g)$ is equal to $1$ if $\\chi=\\chi'$ and $0$ otherwise, we have\n\\[\n\\sum_{C} \\frac{1}{|C|} \\Big| \\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_{1}(x) \\Big|^2 \n= \\frac{1}{|G|} \\sum_{\\chi\\neq 1} \\Theta_{\\chi}(x) \\bbar{\\Theta_{\\chi}(x)}\n= \\frac{1}{|G|} \\sum_{\\chi\\neq 1} |\\Theta_{\\chi}(x)|^2. \n\\]\nThis proves (\\ref{E:conj sum 1}).\n\\end{proof}\n\n\n\\begin{lemma} \\label{L:almost there}\nLet $C$ be any subset of $G$ stable under conjugation.   Then\n\\[\n\\Theta_{\\delta_C}(x) = \\frac{|C|}{|G|}\\,x \\int^\\infty_0 f(t) \\, dt + O_f\\Big( |C|^{1\/2} [K:\\QQ] x^{1\/2} \\log M(L\/K) \\Big).\n\\]\n\\end{lemma}\n\\begin{proof}\nThe lemma is trivial if $C=\\emptyset$, so assume that $C\\neq \\emptyset$.  By Lemma~\\ref{L:alway Cheb} and Lemma~\\ref{L:key bounds}, we have\n\\begin{align*}\n\\Big|\\Theta_{\\delta_C}(x) - \\frac{|C|}{|G|} \\Theta_1(x)\\Big| \n&\\leq |C|^{1\/2}\\cdot \\Big(\\frac{1}{|G|} \\sum_{\\chi\\neq 1} \\big(\\chi(1)[K:\\QQ] x^{1\/2} \\log M(L\/K)\\big)^2\\Big)^{1\/2}\\\\\n& \\ll_f |C|^{1\/2} [K:\\QQ] x^{1\/2} \\log M(L\/K) \\cdot \\big(\\frac{1}{|G|}{\\sum}_{\\chi\\neq 1} \\chi(1)^2 \\big)^{1\/2}.\n\\end{align*}\nSince $\\sum_\\chi \\chi(1)^2 = |G|$, we have \n\\[\n\\Theta_{\\delta_C}(x) = \\frac{|C|}{|G|} \\Theta_1(x) +O_f\\Big( |C|^{1\/2} [K:\\QQ] x^{1\/2} \\log M(L\/K) \\Big).\n\\]\nThe lemma follows by using Lemma~\\ref{L:key bounds} with $\\chi=1$ to estimate $\\Theta_1(x)$.\n\\end{proof}\n\nThere is a constant $c>0$, depending only on $f$, such that $f(t)=0$ for all $t\\geq c$.    In particular, we have $f(N(\\mathfrak{p})^m\/x)= 0$ if $N(\\mathfrak{p})^m \\geq  c x$.     Let $S(x)$ be the sum in the statement of Theorem~\\ref{T:smoothed}.    One can readily check that\n\\[\n0 \\leq \\Theta_{\\delta_C}(x) - S(x) \\leq (\\tilde\\pi_C(cx,L\/K)-\\pi_C(cx,L\/K))\\cdot  \\log(cx) \\cdot \\max_{t\\in {\\mathbb R}} |f(t)|.\n\\]\nBy Lemma~\\ref{L:good approximation}, we have $S(x)=\\Theta_{\\delta_C}(x) + O_f( [K:\\QQ] x^{1\/2} \\log M(L\/K))$.  Theorem~\\ref{T:smoothed} now follows directly from Lemma~\\ref{L:almost there}.\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{MR2464027}{article}{\n      author={Cojocaru, Alina~Carmen},\n      author={David, Chantal},\n       title={Frobenius fields for elliptic curves},\n        date={2008},\n        ISSN={0002-9327},\n     journal={Amer. J. Math.},\n      volume={130},\n      number={6},\n       pages={1535\\ndash 1560},\n      review={\\MR{MR2464027 (2009k:11092)}},\n}\n\n\\bib{MR2178556}{article}{\n      author={Cojocaru, Alina~Carmen},\n      author={Fouvry, Etienne},\n      author={Murty, M.~Ram},\n       title={The square sieve and the {L}ang-{T}rotter conjecture},\n        date={2005},\n        ISSN={0008-414X},\n     journal={Canad. J. Math.},\n      volume={57},\n      number={6},\n       pages={1155\\ndash 1177},\n  url={http:\/\/journals.cms.math.ca\/ams\/ams-redirect.php?Journal=CJM&Volume=57&FirstPage=1155},\n      review={\\MR{MR2178556 (2006e:11074)}},\n}\n\n\\bib{MR1144318}{article}{\n      author={Elkies, Noam~D.},\n       title={Distribution of supersingular primes},\n        date={1991},\n        ISSN={0303-1179},\n     journal={Ast\\'erisque},\n      number={198-200},\n       pages={127\\ndash 132 (1992)},\n        note={Journ{\\'e}es Arithm{\\'e}tiques, 1989 (Luminy, 1989)},\n      review={\\MR{1144318 (93b:11070)}},\n}\n\n\\bib{MR2061214}{book}{\n      author={Iwaniec, Henryk},\n      author={Kowalski, Emmanuel},\n       title={Analytic number theory},\n      series={American Mathematical Society Colloquium Publications},\n   publisher={American Mathematical Society},\n     address={Providence, RI},\n        date={2004},\n      volume={53},\n        ISBN={0-8218-3633-1},\n      review={\\MR{MR2061214 (2005h:11005)}},\n}\n\n\\bib{MR0447191}{incollection}{\n      author={Lagarias, J.~C.},\n      author={Odlyzko, A.~M.},\n       title={Effective versions of the {C}hebotarev density theorem},\n        date={1977},\n   booktitle={Algebraic number fields: {$L$}-functions and {G}alois properties\n  ({P}roc. {S}ympos., {U}niv. {D}urham, {D}urham, 1975)},\n   publisher={Academic Press, London},\n       pages={409\\ndash 464},\n      review={\\MR{0447191 (56 \\#5506)}},\n}\n\n\\bib{MR0568299}{book}{\n      author={Lang, Serge},\n      author={Trotter, Hale},\n       title={Frobenius distributions in {${\\rm GL}_{2}$}-extensions},\n      series={Lecture Notes in Mathematics, Vol. 504},\n   publisher={Springer-Verlag},\n     address={Berlin},\n        date={1976},\n        note={Distribution of Frobenius automorphisms in\n  ${{\\rm{G}}L}_{2}$-extensions of the rational numbers},\n      review={\\MR{MR0568299 (58 \\#27900)}},\n}\n\n\\bib{MR0447187}{incollection}{\n      author={Martinet, J.},\n       title={Character theory and {A}rtin {$L$}-functions},\n        date={1977},\n   booktitle={Algebraic number fields: {$L$}-functions and {G}alois properties\n  ({P}roc. {S}ympos., {U}niv. {D}urham, {D}urham, 1975)},\n   publisher={Academic Press, London},\n       pages={1\\ndash 87},\n      review={\\MR{0447187 (56 \\#5502)}},\n}\n\n\\bib{MR935007}{article}{\n      author={Murty, M.~Ram},\n      author={Murty, V.~Kumar},\n      author={Saradha, N.},\n       title={Modular forms and the {C}hebotarev density theorem},\n        date={1988},\n        ISSN={0002-9327},\n     journal={Amer. J. Math.},\n      volume={110},\n      number={2},\n       pages={253\\ndash 281},\n      review={\\MR{935007 (89d:11036)}},\n}\n\n\\bib{MR1694997}{incollection}{\n      author={Murty, V.~Kumar},\n       title={Modular forms and the {C}hebotarev density theorem. {II}},\n        date={1997},\n   booktitle={Analytic number theory ({K}yoto, 1996)},\n      series={London Math. Soc. Lecture Note Ser.},\n      volume={247},\n   publisher={Cambridge Univ. Press, Cambridge},\n       pages={287\\ndash 308},\n      review={\\MR{1694997 (2000g:11094)}},\n}\n\n\\bib{MR819838}{article}{\n      author={Ribet, Kenneth~A.},\n       title={On {$l$}-adic representations attached to modular forms. {II}},\n        date={1985},\n        ISSN={0017-0895},\n     journal={Glasgow Math. J.},\n      volume={27},\n       pages={185\\ndash 194},\n      review={\\MR{819838 (88a:11041)}},\n}\n\n\n\n\\bib{Rouse-Thorner}{article}{\n       author={Rouse, Jeremy},\n       author={Thorner, Jesse},\n       title={The explicit Sato-Tate conjecture and densities pertaining to Lehmer-type questions},\n  \t     journal={arXiv:1305.5283 [math.NT]},\n        date={2013},\n}\n\n\n\\bib{MR0387283}{article}{\n      author={Serre, Jean-Pierre},\n       title={Propri\\'et\\'es galoisiennes des points d'ordre fini des courbes\n  elliptiques},\n        date={1972},\n        ISSN={0020-9910},\n     journal={Invent. Math.},\n      volume={15},\n      number={4},\n       pages={259\\ndash 331},\n      review={\\MR{MR0387283 (52 \\#8126)}},\n}\n\n\\bib{MR0450380}{book}{\n      author={Serre, Jean-Pierre},\n       title={Linear representations of finite groups},\n   publisher={Springer-Verlag},\n     address={New York},\n        date={1977},\n        ISBN={0-387-90190-6},\n        note={Translated from the second French edition by Leonard L. Scott,\n  Graduate Texts in Mathematics, Vol. 42},\n      review={\\MR{MR0450380 (56 \\#8675)}},\n}\n\n\\bib{MR644559}{article}{\n      author={Serre, Jean-Pierre},\n       title={Quelques applications du th\\'eor\\`eme de densit\\'e de\n  {C}hebotarev},\n        date={1981},\n        ISSN={0073-8301},\n     journal={Inst. Hautes \\'Etudes Sci. Publ. Math.},\n      number={54},\n       pages={323\\ndash 401},\n      review={\\MR{MR644559 (83k:12011)}},\n}\n\n\\bib{MR1062334}{article}{\n      author={Wan, Da~Qing},\n       title={On the {L}ang-{T}rotter conjecture},\n        date={1990},\n        ISSN={0022-314X},\n     journal={J. Number Theory},\n      volume={35},\n      number={3},\n       pages={247\\ndash 268},\n      review={\\MR{1062334 (91f:11079)}},\n}\n\n\\end{biblist}\n\\end{bibdiv}\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nSleep Apnea (SA) is a common and strongly under-diagnosed sleep-related respiratory disorder with severe health implications. It affects the natural breathing cycle during sleep and the disruption of normal airflow causes a decrease of oxygen in the blood. If the oxygen level is too low, the brain will force an awakening in order to resume normal breathing. The patient will most likely not remember the continuous awakenings. With such disrupted sleep, the person will in some cases never go into deep sleep, resulting in daytime sleepiness and fatigue. When left untreated, SA can lead to serious medical conditions including cardiovascular diseases such as stroke, metabolic syndromes such as diabetes, and psychological disorders such as depression and anxiety \\cite{young2004risk, punjabi2008epidemiology, huang2008clinical}. SA patients are also more prone to accidents with for instance motor vehicles \\cite{teran1999association}.\n\nOne factor that contributes to SA being severely under-diagnosed is that feeling tired during the day is normal for many people and the patients typically have no recollection of the nightly awakenings. Therefore, patients can typically only describe vague symptoms to the physician that do not necessarily indicate the need for a clinical diagnosis with polysomnography (PSG) in a sleep laboratory (the gold standard for SA diagnosis) or polygraphy (PG) via unattended sleep monitoring at home. A low-cost solution enabling everyone to perform sleep monitoring and analysis independently at home could motivate patients to visit a physician before serious health implications are manifested and provide a better data foundation to physicians for decision making.\n\nIn our earlier research we found that machine learning (ML) can be used to detect SA events in sleep monitoring data. The performance is rather high for PSG and PG data, even if only a single physiological signal is used. For example, using data from a single respiration belt on the abdomen we could achieve an accuracy, sensitivity, and specificity of 0.9330, 0.9570, and 0.9160 with PSG data \\cite{kristiansen2018data} and 0.8349, 0.8244, and 0.8452 with PG data \\cite{kristiansen2020comparing}. Furthermore, we could demonstrate through controlled laboratory experiments that some low-cost respiration belts (based on strain gauge technology) achieve a data quality that comes close to the data quality of the respiration belts that are used for clinical diagnosis. The latter type of respiration belts is much more sophisticated and typically based on respiratory inductance plethysmography (RIP). For example, we achieved with the low-cost sensor from Sweetzpot, called Flow, a positive predictive value for breath detection accuracy of 99.16\\%, using the RIP belt from Nox T3 as reference in the lab experiments \\cite{loberg2018quantifying}.\n\nIn fall 2018, we had the unique opportunity to test the Flow sensor with real patients that underwent each two nights of PG with the Nox T3. This was performed as part of a large clinical study, called A3 study, at the Oslo University Hospital and St. Olavs Hospital, Trondheim \\cite{traaen2020prevalence}. A total of 579 patients were involved in the A3 study and the last 29 patients used the Flow sensor and the Nox T3 for sleep monitoring at the same time. We collected recordings from at least two nights from 15 of the patients, resulting in recordings from a total of 49 nights. A sleep expert scored the Nox T3 data by labelling time periods that contain apneic events. We use these labels for the Flow data after synchronizing the Nox T3 and the Flow recordings. The Nox T3 and Flow data sets are used in this paper to investigate two basic research questions (RQ): (1) How accurately can ML models estimate SA severity with data from the Flow sensor?, and (2) Is the resources required for data analysis with an ML model sufficiently low to be performed on a smartphone?\n\nThis is to the best of our knowledge the first time a strain gauge breathing sensor is used in a clinical study with data from unattended home monitoring, and where an in-depth study of SA detection performance of ML with such data is presented. Several important results and insights are gained in this study. First, despite several issues with the Flow sensor, we achieve an accuracy, sensitivity, and specificity of 0.7609, 0.7833, and 0.7217 of classifying minutes as normal or apneic, with less than an hour of training. In one setting, when the ML-classifier predicts that an individual has moderate SA (more than 15 disrupted breathing events per hour), the probability of this actually being the case doubles relative to the prior probability. A positive test outcome is therefore useful for screening purposes. Second, we find that the resource consumption of data analysis is no issue for today's smartphones. Third, mixed training, i.e., using Flow data together with high-quality Nox-T3 data to train a model for Flow data classification, positively impacts the achievable classification performance; and using only Nox-T3 data as training data leads to comparatively good performance. This implies that to train a good classifier for another strain gauge sensor might not require collecting a large amount of sleep data with that sensor.\n\nThe remainder of this paper is structured as follows: In Section II we present the background and in Section III the data acquisition and pre-processing. In Sections IV and VII we describe the experiments and results. Section VIII concludes this paper with a discussion of the results and impact.\n\n\\section{Background}\n\\label{sec:background}\nSA is a common sleep-related respiration disorder that affects the natural breathing cycle during sleep with periods of reduced respiration (called hypopneas) or no airflow at all (called apneas). The most common type is Obstructive SA (OSA), where the airway is physically obstructed, resulting in a reduction or a complete stop of air passing through. The less common Central SA (CSA) is caused by the lack of proper breathing signals from the brain. Disruptions of normal airflow causes a decrease of oxygen in the blood, commonly referred to as the oxygen de-saturation. When untreated, SA can lead to serious health implications. Hypertension, stroke and other cardiovascular diseases, metabolic syndromes such as diabetes, depression and anxiety have all been linked to SA \\cite{young2004risk, punjabi2008epidemiology, huang2008clinical}. With the daily drowsiness, SA patients are more prone to accidents, especially with motor vehicles \\cite{teran1999association} and other activities that require a high level of concentration.\n\nAn estimated 936 million individuals aged 30-69 years (men and women) are estimated to suffer from OSA and 426 millions of these cases are categorized as moderate or severe \\cite{benjafield2019estimation}. In Norway, 16\\% of the 30-65 aged population suffer from SA and 8\\% are estimated to have moderate or severe SA\\cite{hrubos2011norwegian}. Unfortunately, SA is often diagnosed very late or not at all. It is estimated that 70-80\\% of those affected remain undiagnosed \\cite{punjabi2008epidemiology}. One factor that contributes to SA being severely under-diagnosed is that feeling tired during the day is normal for many people and the patients have typically no recollection of the nightly awakenings. Patients can typically only describe vague symptoms to the physician that do not necessarily indicate the need to perform PSG.\n\n\\subsection{SA Diagnosis}\nThe gold standard for diagnosing SA is via PSG in a sleep laboratory \\cite{punjabi2008epidemiology}. It requires the patient to stay overnight and record various physiological signals during sleep, such as the electrocardiogram, electroencephalogram, electromyogram, electrooculogram, oxygen saturation, heart rate, blood pressure and respiration from the abdomen, chest, and nose. A sleep expert is required to carefully analyze the data according to strict guidelines like those provided by the American Academy of Sleep Medicine (AASM) \\cite{berry2012aasm}. This process involves identifying and denoting the beginning, end, and type of all periods with disrupted breathing. The two types of disrupted breathing are called apneas and hypopneas. Apneas are defined as a time period with a drop of airflow of at least 90\\% lasting for at least 10 seconds. Hypopneas are defined as a time period of at least 10 seconds with a drop of airflow of at least 30\\% breathing followed by a drop of oxygen saturation of at least 3\\%. A diagnosis is given based on the average number of such periods per hour, called the Apnea-Hypopnea Index (AHI). The AHI is used to classify the patient into one of four classes: (1) normal, no SA: AHI $<$ 5, (2) mild SA: 5 $\\leq$ AHI $<$ 15, (3) moderate SA: 15 $\\leq$ AHI $<$ 30, and (4) severe SA: AHI $\\geq$ 30.\n\nPSG is very expensive, time consuming, uncomfortable, and few sleep labs exist. A sleep expert is required to analyze and score the data. Type III and IV portable monitoring systems are intended to alleviate these problems and allow unattended sleep monitoring at home. There exists a wide range of PG devices that are certified for clinical SA diagnosis \\cite{mendoncca2018devices,collop2011obstructive}. With PG many more people can be analyzed. PG is ideal for preliminary screening purposes to determine whether a PSG should be carried out.\n\nRespiration belts are used in PSG and PG to measure respiratory effort from the abdomen and chest. Respiratory Inductance Plethysmography (RIP) is the gold standard \\cite{berry2012aasm}, where coils are sewn into the flexible belts in a sinusoidal pattern and connected to an oscillator. Changes in the lengths of the belts during respiration alters the self-inductance of the coils which results in changes in the oscillations \\cite{cohn1982respiratory}. When properly calibrated, these changes accurately reflect the tidal volume of breathing \\cite{chadha1982validation}. The Flow belt used in our study is based on the much less expensive strain gauge technology \\cite{sweetzpot2020}. Variations in the lengths of the belts during breathing are measured as changes in the tension of the belt at a single point, i.e., as changes in the electrical resistance in the strain gauge at the front of the chest or abdomen. While RIP belts measure actual changes in the belt length, strain gauges only estimate such changes at a single-point and are as such susceptible to errors due to local variations in tension. Unlike the RIP technology, strain gauge has not been subject to rigorous clinical validation for sleep monitoring. \n\n\\subsection{Automatic Analysis with ML}\n\\label{section:autoanalysis}\nThe achievable performance of automatic analysis of sleep data from clinically validated devices is well-known. There is a large body of research on automatic SA detection \\cite{alvarez2015computer, faust2016review, pombo2017classification, uddin2018classification, mendonca2018review, mostafa2019systematic}. Many devices are intended for home monitoring \\cite{collop2011obstructive}, and most have support for (semi-)automatic analysis \\cite{mendoncca2018devices}. They are however much more expensive than the consumer electronics used in this work, and normally use proprietary, custom-tailored analysis solutions that cannot be leveraged for the large and growing variety of sensors in the consumer market. Among the papers that use ML for data analysis, almost all use data from PSG, commonly obtained from one of 23 online databases \\cite{mostafa2019systematic}. We are only aware of three works that use data from unattended home monitoring \\cite{alvarez2016automated, gutierrez2018evaluation, alvarez2020machine}, but these use expensive, clinically validated devices and other signals than in our work.\n\nAccuracy, sensitivity, and specificity are commonly used to quantify the accuracy of classifying subjects into different SA severity groups (called \\emph{subject-based}) or of classifying fixed-size periods of the data as normal or apneic (called \\emph{period-based}). Period-based classification can be used to estimate SA severity by counting the number of apneic periods per hour. A commonly used period size in related works is 60 seconds.\n\nMost of the above mentioned studies use several sensors at once. Using one or two sensors is also relatively well studied for the signals oxygen saturation \\cite{al2014classifying, kaimakamis2009screening, garde2015pulse,hafezi2019sleep} and ECG \\cite{kaimakamis2009screening}. Recently, accelerometers \\cite{al2014classifying,camci2017sleep,ferrer2019automatic,hafezi2019sleep}, microphones \\cite{al2014classifying, nandakumar2015contactless,camci2017sleep,castillo2019entropy,hafezi2019sleep}, radars \\cite{javaid2015towards}, and pressure sensors \\cite{zhang2013real} have been extensively studied. The use of respiratory belts alone has not been equally well studied, especially with only one belt. We are only aware of six such works. In two of these, disrupted breathing is simulated while awake \\cite{nepal2002apnea,dehkordi2012monitoring}, which yields only approximate results. The remaining four use data from sleeping patients in a controlled PSG setting \\cite{elmoaqet2020deep,van2020portable,lin2016sleep,tsouti2020development}, which is not ideal to study the performance for unattended home monitoring. Two of these use RIP belts \\cite{elmoaqet2020deep, tsouti2020development}, and two are based on less expensive piezo and bioimpedance technologies and are thereby more similar to our work \\cite{van2020portable, lin2016sleep}. A period-based accuracy of 72.8\\% and 73.8\\% was achieved with bioimpedance \\cite{van2020portable} (25 subjects) and piezo sensors \\cite{lin2016sleep} (34 subjects) at the abdomen, respectively. A period-based accuracy of 84.4\\% was obtained with a single RIP belt on the abdomen (17 subjects) \\cite{elmoaqet2020deep}, and a subject-based accuracy 74\\% was achieved (12 subjects) \\cite{tsouti2020development}.\n\nOur work contributes to existing knowledge by using data from a clinical study in an unattended home monitoring setting to evaluate the ML performance with data from low-cost respiration belts. In particular, it provides insight into the performance of using strain-gauge technology for this purpose. By conducting a systematic, comparative analysis comprising a wide variety of modern deep ML techniques, we identify the best performing and least resource demanding ML-technique. We also evaluate the feasibility of running this model on a smartphone. We contribute with insights into the performance improvements obtainable by mixing of high- and low-quality data during the training phase, and on the trade-offs between using balanced versus imbalanced training sets for subject-based classification.\n\n\\section{Data Acquisition, Pre-Processing, and Method}\nThis section describes the data acquisition and the pre-processing to (1) handle data quality issues and (2) prepare the data for the ML studies, and the evaluation method.\n\n\\subsection{Data Acquisition}\n\\label{section:dataaquisition}\nThe A3 study is a large clinical study performed at the Oslo University Hospital and St. Olav Hospital, Trondheim which investigates the prevalence of SA among patients with paroxysmal atrial fibrillation. In this study, 579 subjects with paroxysmal atrial fibrillation were recruited to perform unattended sleep monitoring at home (most of them for two nights) with the Type III monitor Nox T3 \\cite{website:nox-t3}. Each individual was given brief training on sensor placement and use of the Nox-T3 device before bringing the device home to undergo unattended sleep monitoring. We got the unique opportunity to additionally collect with the Flow sensor the respiration data from the 29 patients. These patients were instructed on how to attach the Flow sensor side-by-side to the abdomen RIP belt of the Nox-T3 and how to use the SleepRecorder app on an Android phone for data acquisition. The patient then performed the sleep monitoring at home with both monitoring solutions simultaneously. The data from Nox-T3 was scored by a sleep expert and used for the A3 study. Both the Nox-T3 and Flow data was exported as CSV files and a peak detection algorithm was used to synchronize their timelines \\cite{loberg2018quantifying}. This way, the scoring from the sleep expert could be used to label periods of Flow data as either apneic or normal. Among the 29 individuals from which we collected Flow data, 21 are male and 8 are female. The mean age and BMI was 61.1 (SD 10.9) and 28.2 (SD 4.8), respectively. The number in the SA severity classes mild, moderate, and severe is 13, 10, and one, respectively. Five individuals had an AHI below five, and thus belong to the class of normal individuals (no SA).\n\n\\subsection{Pre-Processing}\n\\label{section:preprocessing}\n\nMovement during the night can result in short periods with motion artifacts in the sleep data which make the corresponding data useless for SA detection. This may subsequently result in a permanent change in the tension in the belt causing a sudden baseline shift in the recorded signal. The sensing technology in RIP belts is superior to strain gauge belts for such situations. The Flow sensor in addition suffers from gradual baseline wandering and drift. Nox Medical has spent a substantial effort to eliminate all baseline issues in the Nox-T3 data. We design a simple Baseline Adjustment (BLA) procedure to reduce these issues for the Flow data where we standardize the Flow data using a one-minute sliding window over the entire recording \\cite{hamborg2020}.\n\n\\begin{table}\n  \\centering\n \n  \\begin{tabular}{ c | c | c || c | c | c}\n    \\textbf{Exp.}&\\textbf{Training set}&\\textbf{Test set}&\\textbf{Exp.}&\\textbf{Training set}&\\textbf{Test set}\\\\\n    \\hline\n    1.1&F&F&2.3&NF, FB&FB\\\\\n    1.2&FB&FB&2.4&N, FB&FB\\\\\n    1.3&NF&NF&3&FB&FB\\\\\n    2.1&N&FB&4&FB&FB\\\\\n    2.2&N, FB&FB&&&\\\\\n  \\end{tabular\n  \\caption{Data sets used in the experiments.}\n  \\label{table:datasets}\n\\end{table}\n\nWe refer to data from the Flow sensor as \\emph{Flow data}, and Flow data with and without BLA treatment as $FB$ and $F$, respectively. Each of these contains 235 hours of data. We also use two sub-sets of the data from the Nox-T3 (\\emph{Nox data}). The first ($N$) contains the entire A3 data set from 579 individuals. Since some recordings contain many artifacts, we remove all recordings with more than 20\\% artifacts, leaving a total of 7408.85 hours of data. The second sub-set, called $NF$, contains only the portion of the Nox data that was obtained at the same time as the Flow data (and was synchronized with it). This data set is of the same size as $FB$. Table \\ref{table:datasets} shows the data sets used for training and testing in our experiments. How they are used is described in the experiment descriptions below. The sampling rate of Flow and Nox-T3 sensors is 10Hz\\footnote{In practice, this sampling rate may vary slightly over time \\cite{hamborg2020}.} and 100Hz, respectively. Since breathing movements span several seconds, we can down-sample the data without loss of important respiratory changes. We down-sample the data to 1Hz using the mean value of each second, and standardize the resulting sample values.\n\n\n\n\n\\subsection{Evaluation Method}\n\\label{section:experiments}\nThe experiments in this work are designed to address the two overall research questions (RQ) mentioned in the introduction. Experiments 1, 2, and 3 address RQ1 (ML performance), and Experiment 4 addresses RQ2 (feasibility of using a smartphone for inference). Experiments 1 and 2 study performance in accordance with standard ML methodology for comparability with related works. Experiments 3 and 4 focuses on the performance in our application domain. We describe the aspects common to all experiments in this section, and experiment-specific details in Sections \\ref{section:exp1} to \\ref{section:exp4}. The experiments are conducted on a computer with a 56-core, 2.2Ghz, Intel Xeon Gold 5120 CPU, 128 GB RAM, and four Nvidia GTX 2080 Ti GPUs. In Experiment 4, we also use a Samsung Galaxy Tab S3.\n\nAll experiments except Experiment 3 (see Section \\ref{section:exp3}) are conducted using 10-fold cross-validation (CV). It is common to randomly shuffle the data set before CV (called \\emph{pre-shuffling}). This is not done in our experiments, since it may result in unrealistically high classification performance in our envisioned scenario. We assume that a pre-trained classifier is used to estimate the SA severity of an individual from which data was not available during training. With pre-shuffling, training data will contain a significant amount of samples from the same individuals as those in the test set.\n\nWe estimate SA severity via binary classification of periods. For comparability with related works, we classify each 60-second period as either \\emph{apneic} or \\emph{normal}\\footnote{We do not distinguish between obstructive, central, or mixed apneas and hypopneas in this work, as it does not affect the AHI, which is the main metric to score SA severity.}. Each data set with Flow data ($F$, $FB$, and $NF$) has 2988 apneic periods and 10117 normal periods, while $N$ has 114009 apneic and 330522 normal periods. Due to the imbalanced class distributions, we balance the data set using majority sub-sampling before dividing it into 10 folds, if not explicitly stated otherwise. We discard the final periods to ensure equal-sized folds.\n\nTraining is performed via mini-batch learning with a batch size of 1000 periods. The learning rate fixed at 0.001, as no improvement was experienced with other learning rates during prior testing. 30\\% of the training set is used as a validation set. We train all classifiers for 500 epochs as this was sufficient for all architectures to maximize performance during prior testing. For all Neural Networks (NN), we use the cross-entropy loss function and the Adaptive Moment Estimation (ADAM) optimizer.\n\nWe quantify classification performance using Cohens $\\kappa$ metric \\cite{cohen1960}. It is ideal for comparing classifiers as it accurately summarizes most aspects of prediction performance and accounts for random chance \\cite{cohen1960}. In addition, we present accuracy, sensitivity, specificity since these have more intuitive semantics that can help understand the clinical relevance of the results. The presented results are those from the epoch with the highest $\\kappa$ on the validation set. \n\n\\section{The Impact of ML Technique and BLA}\n\\label{section:exp1}\nIn Experiment 1, we study the classification performance with the Flow sensor and state-of-the-art ML, and how this performance differs with ML-technique and BLA. We show that we can obtain an accuracy of 0.7609 with BLA. The performance is limited by the relatively simple and inexpensive construction of the Flow sensor and the simplicity of BLA. A key finding is that classifiers based on Convolutional NN (CNN) outperform the others in several respects. First, they rank among the best-performing classifiers in terms of classification performance - this is the architecture achieving the above mentioned accuracy. Second, they appear to be the most resilient to baseline shift and wandering. Third, they require an order of magnitude less computational resources than recurrent NN. We also find that BLA leads to substantial performance improvements for nearly all classifiers.\n\nThese findings are obtained by two sub-experiments: one without BLA using $F$ (Experiment 1.1) and one with BLA using $FB$ (Experiment 1.2). As a benchmark, we perform one experiment with $NF$, i.e., with only high-quality Nox data (Experiment 1.3). In each experiment, the different folds from the same data set is used for training and test via 10-fold CV. We include a wide range of ML classifiers that differ in architecture, approach, size, and sophistication. Most are based on modern deep NN. For comparison, we also include two traditional techniques that are commonly used in related works, i.e., Random Forest (RF) and Multi-Layer Perceptrons (MLP). The MLP are feed-forward NN with one hidden layer, and all layers densely connected. The deep NN architectures include CNN, Long Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU). Since we use temporal data, for which LSTM are designed, we include several improvements and modifications of LSTM. These include BI-directional LSTM (BILSTM), Stacked BILSTM (SBILSTM), and SBILSTM With Attention (BIWALSTM) \\cite{wang2016attention}. All classifiers are instantiated in the three sizes small, medium, and large. Classifiers are referenced with the acronyms of their architecture with a post-fix -S, -M, and -L for small, medium, and large sized classifiers, respectively. The parameter values for each classifier are presented in Table \\ref{table:classifiers}. Some parameters are fixed because other values did not lead to performance improvements during prior testing and\/or because the chosen value is commonly used in related works. For the same reasons, we use dropout only in dense layers in MLP and CNN, and no batch normalization.\n\n\\begin{table}\n \n    \\begin{tabular}{ l | l | l | l | l}\n    \\textbf{Classifier}&\\textbf{Hyperparameter}&\\textbf{Small}&\\textbf{Medium}&\\textbf{Large}\\\\\n    \\hline\n    MLP&Hidden nodes&50&100&200\\\\\n    \\emph{Hidden layers: 1}&&&&\\\\\n    \\hline\n    CNN&Conv. layers&3&4&5\\\\\n    \\emph{Kernel size: 5}&Filters\/conv. layer&32-128&32-256&32-512\\\\\n    &Dense layers x nodes&1x128&1x256&2x256\\\\ \n    \\hline\n    LSTM&Hidden state size&50&100&200\\\\\n    \\hline\n    GRU&Hidden state size&50&100&200\\\\\n    \\hline\n    BILSTM&Hidden state size&50&100&200\\\\\n    \\hline\n    BIWALSTM&Hidden state size&50&100&200\\\\\n    \\hline\n    SBILSTM&Layers&1&3&5\\\\\n    \\emph{Hidden state: 100}&&&&\\\\\n    \\hline\n    RF&Trees&50&200&500\\\\\n    \\emph{Max. nodes\/tree: 100}&&&&\\\\\n    \\hline\n   \n   \n   \n   \n    \\end{tabular}\n\\caption{Classifier parametrization.}\n\\label{table:classifiers}\n\\end{table}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.75\\textwidth]{.\/figures\/PerMinCP-Kappa.pdf}\n  \\caption{$\\kappa$ for Experiment 1.}\n  \\label{figure:exp1}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\begin{tabular}{ccc}\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/PerMinCP-Accuracy.pdf}\\\\(a)}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/PerMinCP-Sensitivity.pdf}\\\\(b)}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/PerMinCP-Specificity.pdf}\\\\(c)}\n     \\end{tabular}\n     \\caption{Accuracy, sensitivity, and specificity for Experiment 1.}\n    \\label{figure:exp1accsensspec}\n\\end{figure*}\n\nThe $\\kappa$ results for Experiment 1 are presented in Figure \\ref{figure:exp1}, and the accuracy, sensitivity, and specificity in Figure \\ref{figure:exp1accsensspec}.\n\nOn average, BLA leads to an increase in $\\kappa$ of 0.2889$\\pm$0.1091 points for any given classifier. CNN classifiers benefit far less from BLA than the others, with a mean increase in $\\kappa$ of only 0.1263$\\pm$0.1054. This indicates that CNN is resilient to baseline shifts and wandering. Overall, sensitivity is slightly higher than specificity, indicating that the classifiers are better at correctly classifying apneic than non-apneic periods. Deep NN-based classifiers substantially outperform the older RF and MLP classifiers. The highest $\\kappa$ of 0.5062$\\pm$0.0575 is obtained with BIWALSTMS and BLA, closely followed by CNNM with a $\\kappa$ of 0.5052$\\pm$0.0485. Surprisingly, RF yields better results with Flow data than with Nox data. Since RF nevertheless yields the overall lowest performance for all metrics and data sets, this result is likely a result of RF failing to properly learn the important features of the Nox data. BLA clearly increases accuracy substantially, and has an equalizing effect on sensitivity and specificity. The latter is particularly prominent for MLP and GRU classifiers. The highest accuracy is obtained with CNNM and BLA, i.e., 0.7609$\\pm$0.0284, with a sensitivity of 0.7833 and a specificity of 0.7217. \n\nWe see that the SD are much larger than the differences in mean values. We can therefore not conclude from the mean $\\kappa$ alone that BIWALSTM is the Superior classifier. We must also consider the amount of resources required for training and inference. We measure the average time spent training and testing per epoch. Training involves inference and back-propagation of four batches, and testing the inference of one batch\\footnote{This batch is smaller than 1000 periods, since there are only 598 periods in the test set. Similarly, the final batch in the training set consists of only 766 periods.}. The measurement from the first epoch is excluded since it includes time spent on initialization which is not relevant in this investigation. We find a substantial difference among the classifiers. Recurrent networks spend a factor of 2.40 to 77.46 more time per epoch for training than feed-forward networks. The corresponding numbers for testing time is 3.57 to 41.59, which involves the inference of one batch. BIWALSTM is by far the most resource intensive, requiring just above 224 milliseconds per epoch for training and about 18 milliseconds per epoch for testing. CNNM, which obtain the same classification performance, only requires an average of 33 milliseconds and 2 milliseconds per epoch for training and testing, respectively. These differences can translate into differences in waiting times of hours vs. days with large data sets or models.\n\n\\section{Mixing High- and Low-Quality Data}\nThe data from the Nox-T3 sensor is of much higher quality than that from the Flow sensor. It is well-known that both data quality and quantity affects classification performance. We hypothesize that mixed training with Flow and Nox data can improve the accuracy of classifying Flow data. Similar approaches have previously shown to yield promising results in other domains, e.g., in the field of transfer learning \\cite{pan2009survey}. In Experiment 2, we investigate this hypothesis. Interestingly, the results from Experiment 2 suggest that classifiers that are pre-trained on high-quality data can be used \\emph{as-is} to classify data data from different sensors of lower quality, and still obtain a performance comparable to that of training on data from the target, low quality sensor. This has significant implications, since the collection of large amounts of data is expensive and may not be possible for many producers of inexpensive home electronics due to privacy and resource restrictions. The results furthermore suggest that the lack of large-scale, high-quality data sets may in some cases be compensated for by collecting a much smaller amount of data from high- and low-quality sensors at the same time. We show that this results in substantial savings of training time.\n\nWe perform four sub-experiments 2.1-2.4. Each sub-experiment is repeated 10 times. Due to the large number of repetitions, the total number of experiments is very large. For this reason, and because we focus on the potential improvements beyond what was achieved in Experiment 1.2, we use the CNNM classifier that achieves the overall best performance in Experiment 1.2. In Experiment 2.1, we train the classifier on $N$ and test it on $FB$. This reflects a scenario where a classifier trained on high-quality data is made available and used \\emph{as-is} to classify low-quality data. Experiments 2.2 to 2.4 are trained with a mix of Flow and Nox data. In Experiment 2.2, the training is performed \\emph{sequentially}, i.e., first with data from the Nox-T3 then with data from the Flow sensor. A complete 10-fold CV training procedure is first completed with $N$. The resulting pre-trained models (one per fold) are subsequently used as the initial models for a complete 10-fold CV run using $FB$ as both training and test data. The results are relevant for scenarios where a classifier trained on high-quality, closed data from, e.g., clinical studies is made available for further training on readily available low-quality data. Due to catastrophic forgetting \\cite{french1999catastrophic}, the final classifier in Experiment 2.1 may no longer be able to accurately classify the high-quality Nox-data, only the low-quality Flow data it is intended for. In Experiments 2.3 and 2.4, training is performed \\emph{simultaneously} with Nox-T3 and Flow data. In Experiment 2.3, each of the training sets produced during 10-fold CV is concatenated with $NF$, and only $FB$ is used for testing. This configuration is found in scenarios where both high- and low-quality training data is available from the same set of individuals. Experiment 2.4 is identical to 2.3, except $NF$ is replaced with $N$. This gives an estimate of performance obtainable in scenarios where high-quality data is available from many more individuals than for low-quality data. We compare with results from Experiments 1.2 and 1.3 repeated 10 times where we use only high- and low-quality data from Experiment 1.2 and 1.3 repeated 10 times.\n\nIn Experiments 2.2, 2.3., and 2.4, Flow and Nox data can be weighted differently during training. This is achieved by multiplying the loss of samples in the Nox-data by $w \\in [0,1.0]$ before they are used for back-propagation. We execute these sub-experiments multiple times with varying $w$ to estimate the optimal weight and to study the relative impact of low- and high-quality data during training.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.75\\textwidth]{.\/figures\/mixedtrainingresults.pdf}\n  \\caption{Results from Experiment 2. $w$ is the weight by which the loss from samples in Nox data is multiplied before back-propagation.}\n  \\label{fig:mixed}\n\\end{figure}\n\nThe results from Experiment 2 are presented in Figure \\ref{fig:mixed}. The x-axis shows the different sub-experiments, and the y-axis the mean $\\kappa$ for CNNM. This is not the mean across folds, as in Experiment 1. It is the mean across the results from 10 different repetitions of the complete 10-fold CV procedure with different random seeds. The result from each such repetition is the mean across the 10 folds. This makes visible the variance resulting from different initializations of the weights in the NN. When the training set consists of both Nox and Flow data, the weight on the Nox data is denoted as $w$ in the labels on the x-axis.\n\nThe results show an overall improvement in classification performance when using a mix of Nox and Flow data for training. The $\\kappa$ from Experiments 2.2-2.4 are between 0.0107 and 0.0177 higher than in Experiment 1.2. While the mean accuracy in Experiment 1.2 is 0.7580, the mean accuracies in Experiments 2.2-2.4 are 0.7650, 0.7647, and 0.7647, respectively. There is no clear difference in $\\kappa$ among Experiments 2.2-2.4, since differences in the mean values are much smaller than the corresponding SD. This suggests that it does not matter much for the performance: (1) If we first train on Nox data then on Flow data (Experiment 2.1), or train on both in parallel (Experiments 2.2-2.4), (2) If we use the complete A3 data set (Experiment 2.4) or only the sub-set obtained simultaneously with the Flow data (Experiment 2.3), (3) How the Nox and Flow data is weighted during training. In Experiment 2.3, the training benefits from the pre-liminary synchronization of Nox- and Flow-data. In Experiment 2.4, training benefits from the large amount of training data available. The similarity of the results from these two experiments suggest that these two factors have a similar positive impact. This means that using the complete, large-scale Nox-T3 data from the A3 study yields a similar performance as using the much smaller sub-set that was collected at the same time as the Flow data. Due to the high SD and 95\\% CI we cannot rule out the existence of small differences between these two approaches that would be uncovered with a larger data set.\n\nWe obtain similar $\\kappa$ in Experiment 2.1 as in 1.2, i.e., in the absence of Flow data in the training set. This result suggests that classifiers that are pre-trained on high-quality data can be used \\emph{as-is} to classify data from different sensors of lower quality, and still obtain a performance comparable to that of training on data from the target, low quality sensor.\n\nA proper comparison among the approaches requires to consider the training times. The training data used in Experiments 2.3 is only twice as big as that used in Experiment 1.2. The mean training time per epoch is only about twice as big, i.e., 71 milliseconds. The data set used in Experiments 2.1, 2.2, and 2.4 use the complete A3 data set for training, which comprises 232 batches compared to only four in Experiment 1.2. The mean training times are significantly larger, i.e., 1708 milliseconds per epoch. While Experiments 1.2 and 2.3 complete in minutes, 2.1, 2.2, and 2.3 require several hours to complete. This large difference in training time implies that it is preferable to use $NF$ instead of $N$ to augment the training data from $F$.\n\n\\section{Per-Subject SA Severity Estimation}\n\\label{section:exp3}\nExperiment 1 and 2 are designed with comparability with related works in mind. By balancing the data set, these experiments can provide a good indication of the performance of the classifiers in the general case, i.e., without any assumption of SA prevalence. The goal of Experiment 3 is to study how accurately we can estimate the SA severity of a subject. Our results show a clear correlation between actual (from manual annotation of Nox-T3 data) and predicted (from ML on Flow data) AHI, albeit naturally not as strong as between, e.g., AHI from PG and PSG. Such AHI estimation precludes balancing the test set, causing the class distribution of the test set to be determined by the prevalence of apneic periods. A key insight from Experiment 3 is that, as a result, the decision of whether or not to balance the \\emph{training set} has a significant impact on results, the degree of which depends on the prevalence of SA. Careful attention should therefore be given to properly align this decision with the requirements of the particular screening context. We find that balanced training data leads to fewer false negatives which is generally preferable from a purely medical perspective. But it leads to a consistent over-estimation of SA severity, and may not be preferable from a large-scale, societal perspective, due to the potential of added cost of incorrectly prescribed expensive medical examinations. Conversely, imbalanced training data results in an overall higher prediction accuracy at the cost of a higher number of false negatives. We show that a compromise can be made by averaging the predictions made with balanced and imbalanced data. Averaged predictions, and those from imbalanced data, are better than those from balanced data for discriminating between subjects with AHI above or below 15. This can have significant implications in practice, as an AHI-threshold of 15 is commonly used as a guideline for prescribing a PSG examination.\n\nWe obtain these findings by changing the CV procedure compared to Experiments 1 and 2. It is only possible to estimate the SA severity of an individual by analyzing the entire data set from this individual. For each iteration in the CV procedure, the test set consists of all the recordings of a given subject, while the training set consists of the remaining recordings. To study the effect of balancing the training set, we compare results with classifiers trained on balanced ($C_b$) and imbalanced data ($C_i$). In addition, we create the combined classifier $C_c$ by averaging the predictions from $C_b$ and $C_i$. These classifiers use the best-performing CNNM architecture in Experiment 1. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.75\\textwidth]{.\/figures\/combined.pdf}\n  \\caption{Results from Experiment 3.}\n  \\label{fig:combined}\n\\end{figure}\n\nA proper understanding of classification performance with imbalanced data requires a joint assessment of several metrics \\cite{he2009learning}. Since Experiment 3 focuses on SA-severity estimation, most of the results are based on comparisons between \\emph{predicted} and \\emph{actual AHI}. We compute actual AHI by counting the actual number of disrupted breathing events in the Nox-data\\footnote{Note that this AHI differs from the actual AHI of the subjects presented in Section \\ref{section:dataaquisition}, since a significant portion of the Nox-data is removed during pre-processing (see Section \\ref{section:preprocessing}).}. Predicted AHI is computed based on the total number of predicted apneic periods. Since the number of apneas\/hypopneas may differ from the number of apneic periods, the results in Experiment 3 reflect the negative performance impact of dividing the data into periods. This is not the case in most related works since online data sets often only include per-period labels \\cite{mostafa2019systematic}. As a result, Experiment 3 yields a lower, but more realistic performance.\n\n\\begin{figure*}\n  \\centering\n  \\begin{tabular}{ccc}\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/left\/scatter-withBands-raw.pdf}\\\\(a) $C_i$.}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/combined\/scatter-withBands-raw.pdf}\\\\(b) $C_c$.}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/right\/scatter-withBands-raw.pdf}\\\\(c) $C_b$.}\\\\\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/left\/ahi-BA-raw.pdf}\\\\(d) $C_i$.}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/combined\/ahi-BA-raw.pdf}\\\\(e) $C_c$.}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/right\/ahi-BA-raw.pdf}\\\\(f) $C_b$.}\\\\\n  \\end{tabular}\n  \\caption{Scatter (a-c) and Bland-Altman plots (d-f) for Experiment 3. The results are with $C_i$, $C_c$, and $C_b$ from left to right, respectively.}\n  \\label{figure:exp3scatter}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\begin{tabular}{ccc}\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/left\/cm-col-raw.pdf}\\\\(a) $C_i$.}&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/combined\/cm-col-raw.pdf}\\\\(b) $C_c$. }&\n     \\makecell{\n       \\includegraphics[width=0.30\\textwidth]{.\/figures\/persubj-new\/results-realpersubj-combined\/right\/cm-col-raw.pdf}\\\\(c) $C_b$.}\n  \\end{tabular}\n  \\caption{Confusion matrices for Experiment 3 with the fraction of individuals in predicted SA severity groups that were actually in that severity group.}\n    \\label{figure:exp3confusion}\n\\end{figure*}\n\n\\begin{table*}\n  \\resizebox{\\textwidth}{!}{\n    \\centering\n    \\begin{tabular}{ l l | l | l | l | l || l | l | l | l | l || l | l}\n      &&\\multicolumn{2}{c|}{\\textbf{AHI $<$ TH}}&\\multicolumn{2}{c||}{\\textbf{AHI $\\ge$ TH}}&\\multicolumn{5}{c}{}\\\\\n      &\\textbf{Threshold}&\\textbf{Act.}&\\textbf{Pred.}&\\textbf{Act.}&\\textbf{Pred.}&\\textbf{Acc.}&\\textbf{Sens.}&\\textbf{Spec.}&\\textbf{$\\kappa$}&\\textbf{PPV}&\\textbf{P(A)}&\\textbf{P(A$\\vert$P)}\\\\\n      \\hline\n      &5&4&7&25&22&0.8276&0.8400&0.7500&0.4487&0.9545&0.8621&0.9545\\\\\n      \\textbf{$C_i$:}&15&18&19&11&10&0.7586&0.6364&0.8334&0.4781&0.7000&0.3793&0.7000\\\\\n      &30&29&27&0&2&0.9310&NaN&0.9310&0.0000&0.0000&0.0000&NaN\\\\\n      &&&&&&&&&&&\\\\\n      &5&4&0&25&29&0.8621&1.0&0.0&0.0&0.8621&0.8621&0.9259\\\\\n      \\textbf{$C_b$:}&15&18&9&11&20&0.5517&0.8182&0.3889&0.1786&0.4500&0.3793&0.6000\\\\\n      &30&29&22&0&7&0.7586&NaN&0.7586&0.0000&0.0000&0.0000&NaN\\\\\n      &&&&&&&&&&&\\\\\n      &5&4&2&25&27&0.8621&0.9600&0.2500&0.2658&0.8889&0.8621&0.8276\\\\\n      \\textbf{$C_c$:}&15&18&14&11&15&0.7241&0.8182&0.6667&0.4528&0.6000&0.3793&0.4500\\\\\n      &30&29&27&0&2&0.9310&NaN&0.9310&0.0000&0.0000&0.0000&NaN\\\\\n  \\end{tabular}}\n  \\caption{Results for Experiment 3. Performance of binary classification with thresholds corresponding to the different SA severity classes. P: Predicted positive. A: Actual positive. TH: AHI Threshold. NaN: Not a Number (incomputable).}\n  \\label{table:binarySC}\n\\end{table*}\n\nAs a basis to understand the accuracy of estimating the AHI of individuals, we first investigate how accurately the 60-second periods are classified for each individual. The mean, SD, and CI across all individuals are shown in Figure \\ref{fig:combined}. All remaining results are based on comparisons between the actual AHI of an individual and the AHI estimated using the classified periods for that individual. Figure \\ref{figure:exp3scatter} shows scatter plots (a-c) and Bland-Altman plots (d-f) that compare actual and predicted AHI per individual. The scatter plots include the 95\\% confidence and prediction bands. These outline the 95\\% confidence intervals for the mean and individual AHI for the ground truth given predicted AHI, respectively. Figure \\ref{figure:exp3confusion} shows the confusion matrices for SA severity groups. They denote the probability of a subject belonging to a certain actual severity group given a predicted group. According to the balanced predictions, no subjects belong to the severity group \\emph{mild}, i.e., these values cannot be computed. The actual number of subjects in severity groups normal, mild, moderate, and severe is four, 14, 11, and zero, respectively. Table \\ref{table:binarySC} shows the classification performance where subjects are divided into only two groups, i.e., with AHI above or below the thresholds that separate severity groups. Since no subjects belong to the actual severity group \\emph{severe}, the corresponding sensitivity, $\\kappa$, and PPV values are either not computable or not meaningful. The $P(A)$ column presents the prior probability of a positive based on the SA prevalence among the 29 subjects, and the $P(A|P)$ column presents the posterior probability of an actual positive given a positive test as given by Bayes' formula $P(A \\vert P)=\\frac{P(P \\vert A)P(A)}{P(P)}$. The difference between the two right-most provides an estimate of how much a positive test increases the probability of an actual positive that accounts for SA prevalence (as opposed to PPV).\n\nFigure \\ref{fig:combined} shows that $C_i$ outperforms $C_b$ for all metrics except sensitivity. $C_c$ performs somewhere in between $C_i$ and $C_b$. These findings are a result of the imbalanced test set, which has an over-representation of normal periods. This problem of \\emph{intrinsic imbalance} is common in the biomedical domain \\cite{he2009learning}, and the smaller the data set is, the larger its impact \\cite{weiss2001effect}. This configuration leads to different biases in the training set which is learned by the classifier: a balanced training set has a very different class distribution than the test set, while an imbalanced training set contains many more normal than apneic periods. $C_i$ benefits in terms of accuracy and specificity from the much higher number of non-apneic (10117) than apneic (2988) periods. $C_b$ has seen an equal number of apneic and non-apneic periods during training, while $C_i$ has seen many more non-apneic ones. Therefore, $C_i$ more accurately classifies non-apneic periods, which are much more numerous in the test set than apneic ones. This explanation is confirmed by the fact that $C_b$ outperforms $C_i$ on the balanced validation set in terms of all metrics except accuracy and specificity. These results should be seen in light of the relatively large CI, indicating that a larger data set is desirable to increase the confidence in the mean values.\n\nFigures \\ref{figure:exp3scatter} (a-c) show that we obtain r-values between 0.51 and 0.64 with different prediction approaches. Interestingly, $C_c$ outperforms $C_b$ and $C_i$, in terms of the r-value and the magnitude of the prediction and confidence intervals. There is a clear correlation between predicted and actual AHI. As expected, the r-values are lower than what can be obtained with manual scoring of data from clinically certified PG equipment. For instance, in \\cite{chang2019validation} an r-value of 0.791 is obtained when comparing manual scoring of Nox-T3 and PSG data. This is explained by the results in Figure \\ref{fig:combined}. Values for sensitivity show that only 53.3\\% to 73.3\\% of the apneic periods for each subject are correctly identified as such. We see that the slope of the regression line is lower than 1, which means that predicted AHI are more dispersed than the actual AHI. The fact that the performance in our results is not as good as previous works with clinically certified equipment is not surprising. A major reason is the much lower quality of data from consumer electronics like the Flow sensor. While recent deep NN are known to excel in classification even of noisy data, the presence of irreducible errors in both data and labels imposes an upper limit on the achievable classification performance. This limit is studied in literature both theoretically and empirically \\cite{tumer1996estimating, hestness2017deep}. Furthermore, a large amount of data is required to achieve this upper performance limit \\cite{kristiansen2020machine}. Intuitively, the amount of data required increases with decreased data quality. The Flow data is both of lower quality than the Nox data and much smaller, i.e., it consist of only 49 recordings from 29 subjects with an average length of 4.46 hours of data. With this few subjects, personal differences have a non-neglectable impact on the ability of the classifiers to generalize.\n\nThe BA plots show that $C_b$ is biased towards higher SA severity groups, which is not the case for $C_i$. The effect is reflected in the confusion matrices in Figure \\ref{figure:exp3confusion}. While $C_i$ most often correctly predicts the SA severity group, $C_b$ tends to over-estimate SA severity. $C_b$ has a higher sensitivity and a lower PPV than $C_i$ (see Figure \\ref{fig:combined}), causing $C_b$ to classify more periods as apneic on average.\n\nTable \\ref{table:binarySC} reflects the results above. Since $C_b$ over-estimates SA severity, only half of the subjects are correctly classified at $TH=15$, and only 38.89\\% of negative subjects (AHI $<$ 15) are classified as such. $C_c$ yields good results for $TH=15$, i.e., (1) a relatively high accuracy for $TH=15$, (2) the same sensitivity as with $C_b$, and (3) a relatively modest decrease in specificity compared to $C_i$. We see that $C_i$ outperforms the others for all thresholds (TH) in terms of accuracy, specificity, $\\kappa$, and PPV, with the exception of $TH=5$ for which $C_c$ achieves a slightly higher accuracy. $C_b$ and $C_c$ outperform $C_i$ in terms of sensitivity. Notice that $TH=15$ yields the lowest accuracy but the highest $\\kappa$ among all TH. This is explained by the relatively balanced class distribution at this threshold, i.e., 62.07\\% of the subjects have an actual AHI above this threshold. With $TH=5$ and $TH=30$, 86.2\\% and 0\\% are above the threshold, respectively. Such imbalances result in artificially high accuracies. The usefulness of a test depends on how much a positive test result increases the probability an actual positive (relative to the prior probability). The columns $P(A)$ and $P(A|P)$ show that $C_i$ leads to the largest increase for all TH. Importantly, with $TH=15$, a positive test result almost doubles the probability of SA from 37.93\\% to 70.00\\%.\n\n\\section{Inference on a Smartphone}\n\\label{section:exp4}\nIn Experiment 4, we investigate whether the classifiers can in practice be used in our envisioned target scenario, i.e., on a patient's smartphone. We assume that a classifier is trained on a powerful computer and thereafter made available for use at home. Smartphones might be ideal for this purpose because nearly everyone owns one, but are severely resource limited compared to desktop or cloud computers. We use Tensorflow Lite to convert the classifiers into \\emph{float models} that are suitable for mobile devices. Our results show that it can analyze the data from a complete night mostly within one second on a smartphone from 2012, i.e., the Samsung Galaxy Tab S3. Surprisingly, this is achievable with almost no reduction in classification performance. The classifier is well-suited for use on a smartphone.\n\nThe float models are constructed to be smaller than the original models without a significant loss in prediction performance, and can be executed using the low-overhead \\emph{tflite} operations in Tensorflow Lite. The model can be \\emph{quantized} to reduce inference times further, but at the cost of numerical precision and classification performance. We measure the inference times of both Float and quantized versions of the CNN models from Experiment 1. We measure the time spent classifying data from one complete night, as this reflects the waiting time required for analysis the morning after a night of data collection. To study the impact of model size, we compare results from CNNS, CNNM, and CNNL. After converting the trained models, they are used for inference on a Samsung Galaxy Tab S3 with a 1.6GHz quad-core Qualcomm Snapdragon 820 processor and 4GB of RAM, running the Android 7.0 operating system and using Tensorflow version 2.2.0. Since the main focus of Experiment 4 is inference efficiency and the impact on model reduction on performance, 10 training epochs was sufficient to obtain representative results.\n\\begin{table}\n \n  \\centering\n  \\begin{tabular}{ l | c | c | c | c | c }\n    \\emph{Classifier}&\\emph{Parameters}&\\emph{Size}&\\emph{Inf. time}&\\emph{$\\kappa$}&\\emph{Acc.}\\\\\n    \\hline\n    \\emph{CNNS:}&&&&&\\\\\n    Original&166.658&2-2.2MB&&0.4903&0.7512\\\\\n    Float&166.658&654KB&0.55ms&0.4989&0.7525\\\\\n    Quantized&166.658&168KB&0.34ms&0.4928&0.7492\\\\\n    &&&&&\\\\\n    \\emph{CNNM:}&&&&&\\\\\n    Original&413.058&5-5.2MB&&0.5052&0.7609\\\\\n    Float&413.058&1617KB&1.09ms&0.4820&0.7458\\\\\n    Quantized&413.058&411KB&0.55ms&0.4882&0.7492\\\\\n    &&&&&\\\\\n    \\emph{CNNL:}&&&&&\\\\\n    Original&1.069.186&12.8-13.1MB&&0.5020&0.7554\\\\\n    Float&1.069.186&4181KB&1.93ms&0.4674&0.7375\\\\\n    Quantized&1.069.186&1055KB&0.93ms&0.4531&0.7308\\\\\n  \\end{tabular\n  \\caption{Results from Experiment 4. Mean inference time is the mean time spent for inference per period, i.e., the total amount of time spent for 480 periods divided by 480.}\n  \\label{table:experiment4}\n\\end{table}\n\nTable \\ref{table:experiment4} presents the results for Experiment 4. \\emph{Original} refers to the classifier variant used in Experiments 1-3. Column \\emph{Inf. time} presents the mean time spent for inference per 60-second period of data in milliseconds (ms). Columns $\\kappa$ and \\emph{Acc.} present classification performance in terms of the mean $\\kappa$ and accuracy across the 10 folds, respectively. Classification performance for the original models is obtained from Experiment 1.2, and therefore shows the mean across the 10 models from 10-fold CV. The float and quantized models are based on the first model from 10-fold CV. Since we only execute models intended for mobile devices on the portable device (float and quantized variants), Table \\ref{table:experiment4} does not include inference times for the original models. Instead, we present the size of the original model for comparison. This size depends on the method by which it is stored. We present the two sizes resulting from (1) just saving the weights as a checkpoint, in which case we need to specify the architecture of the model whenever we load it, and (2) saving also the architecture of the model.\n\nAnalysis of data from one complete night is in most cases completed in less than a second. This is true even for the float models. We observe almost no reduction in prediction performance of float models compared to original models. For CNNS, the mean performance of the float model is slightly higher than that of the original model. This difference is most likely attributed to the high SD in our results, as seen in Figure \\ref{fig:mixed}. The low inference times demonstrate that the classification performance obtained in Experiments 1-3 can be achieved on smartphones. We observe almost no performance reduction for quantized models, even if they reduce inference time by 38.18-51.81\\% compared to the float models. This can significantly reduce inference time with larger models or with sensors with higher sampling rates.\n\n\\section{Discussion}\nThe accuracy of classifying Flow data is nearly as high when we train on Nox data as when we train on Flow data. This implies that the need to collect additional data for new, low-quality sensors may not be necessary when high-quality data is available. As a result, the cost of collecting large amounts of sensitive data can be substantially reduced, facilitating low-cost screening based on consumer electronics. Performance improves slightly by training with both low- (Flow) and high-quality (Nox) data, and less high-quality data is required if it is collected simultaneously with the low-quality data.\n\nThe simple BLA for Flow data leads to substantial improvements in performance for nearly all ML methods. BLA merely involves standardizing the data in one-minute intervals, showing that significant performance gains can be achieved with only minimal pre-processing. A promising direction for future work is to investigate if additional pre-processing leads to further performance improvements. CNN is the best-performing ML technique, with an accuracy of 0.7609$\\pm$0.0284 and nearly the lowest training and testing times. The fact that it performs almost as well with and without BLA distinguishes it from the other ML techniques, and shows that it is by far the most robust ML technique against baseline shifts and wandering. This is not an obvious outcome, since CNN is not designed particularly for time series data like RNN. To the best of our knowledge, our results are the first to demonstrate that this applies also to clinical data from consumer electronics. These results emphasize the versatility of CNN and its usefulness even for physiological time series data. We will use CNN for future work in this area. We obtain similar period-based performance as existing works with low-cost respiratory belts with piezo or bioimpedance technology. For instance, a maximum accuracy of 0.728 and 0.7380 is obtained in \\cite{van2020portable, lin2016sleep}. As expected, these results are not at the same level of performance as with PSG and PG with manual scoring, given the much cheaper hardware and the use of generic ML in stead of custom-tailored analysis algorithms. We nevertheless find that automated analysis of Flow data yields predictions that correlate well with actual SA severity, with r-values of up to 0.64, and as such can be used to indicate SA severity. We show that our classifiers can be used on small, resource constrained devices, with almost no loss in classification performance.\n\nThe quality of subject-based classification depends on whether we train the classifier with balanced or imbalanced data. Classifiers trained on balanced data most often correctly identify subjects with SA, but tend to over-estimate SA. This is acceptable when false positives can be tolerated to reduce the number of false negatives. Imbalanced training data may nevertheless be preferrable in a home-screening scenario, even if fewer apneic subjects are identified, since it leads to an overall higher prediction accuracy. This reduces the number of unnecessary visits to the doctor and the use of limited medical resources. Imbalanced data also yields the highest increase in the probability of SA upon a positive test result, which is a key indicator of the usefulness of the classifers. This is noteworthy, since applied ML studies commonly use balanced training data. In general, combining balanced and imbalanced training data constitutes a compromise between using balanced and imbalanced data. This approach outperforms the latter two in several ways, e.g., in terms of the r-value (Figure \\ref{figure:exp3scatter}), the width of the prediction and confidence bands (Figure \\ref{figure:exp3scatter}), and predicting whether a subject has moderate SA (Table \\ref{table:binarySC}).\n\nThere is possible future work in three areas: (1) We wish to investigate whether our results apply to other types of physiological signals and other data types like images. (2) We wish to investigate alternatives for improving classification performance. Our analysis outlines several promising directions. Improvements in hardware are expected to improve performance, since we use sensors of low quality which impacts performance. Our current data set is relatively small, especially for the low-quality data. We expect the performance to improve with more training data. (3) We wish to investigate personalized training where we leverage data from subjects that are made available at an early stage.\n\n\\section{Acknowledgements}\nThis work was funded by the Norwegian Research Council (as part of the CESAR project, no. 250239\/O70), Oslo University Hospital, University of Oslo, and the Norwegian Health Association.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec:introduction}}\nRandom walks, originally introduced in 1921 by G. \nP\\'olya \\cite{Polya1921}, have since been a vital field of research. \nThey are ubiquitous models for physical, biological and social\nprocesses \\cite{vanKampen1992,bergHC1993,hughes1996}. \nExample applications from biology include self-propelled motion of \nbacteria and the diffusion of nutrients \\cite{bergHC1993}, as well as \nanimal motion in general \\cite{Bovet1988,Bartumeus2005} or during \nthe marking of territories or description of home ranges \n\\cite{mohr1947,worton1995,Giuggioli2011}. \nFor the latter case a strong increase of the amount of experimentally available \ndata ocurred after the introduction of\nautomated radio\/GPS tagging of animals \\cite{kenward1987,landguth2010}.\nThe usage of minimum convex polygons, called \\emph{convex hulls}, bordering the\ntrace of an animal \\cite{mohr1947,worton1987} \n is a simple yet versatile \\cite{boyle2009} way to describe\nthe home range and can be used for any type of (random-walk) data. \n In two dimensions, the convex hull of a point \nset is the minimum subset whose elements form a convex polygon in \nsuch a way that (a) all points of the set and (b) the connecting \nlines between all possible pairs lie inside the polygon. The convex hull \nis a suitable measure for time-discretized random walks because it \ncan easily be attributed with geometrical quantities, i.e., hull area \n$A$ and perimeter $L$. Also, for numerical calculations, simple and \nalready quite fast algorithms with running times \n$\\mathcal{O}(N \\cdot \\log N)$ exist \n\\cite{Graham1972, Bykat1978, Andrew1979}.\n\nOn the analytical side, much progress have been made for asymptotic \nresults at long times, when \na random walk (with a finite variance for step sizes) converges to \nthe continuous-time Brownian motion (for a recent review see\nRef. \\cite{Majumdar2010}). For a single \nBrownian motion of length $T$ in two dimensions, the mean \nperimeter~\\cite{Takacs1980,Letac1993} and\nthe mean area~\\cite{ElBachir1983} were computed long back. \n\nRecently, adapting Cauchy's formula~\\cite{Cauchy1832} for convex curves in \ntwo dimensions to random curves, it was \nshown~\\cite{Randon-Furling2009,Majumdar2010} that the problem of computing \nthe mean perimeter and the mean area of the convex hull of an arbitrary \ntwo dimensional stochastic process can be mapped to computing the extremal \nstatistics of the one dimensional component of the process. This procedure \nwas successfully applied recently to compute the mean perimeter and the \nmean area of several two dimensional stochastic processes such as $N$ \nindependent Brownian motions in \n$2$-d~\\cite{Randon-Furling2009,Majumdar2010}, random acceleration process \nin $2$-d~\\cite{Reymbaut2011}, $2$-d branching Brownian motions with absorption \nwith applications to edpidemic outbreak~\\cite{Dumonteil2013} and $2$-d \nanomalous diffusion processes~\\cite{Lukovic2013}. Very recently,\nthis method was also successfully used to compute the exact mean perimeter\nof the convex hull of a planar Brownian motion confined to a half-space~\\cite{Chupeau2014}.\nFinally, using different methods, the mean perimeter and the mean area of the convex hull of a single \nBrownian motion, but in arbitrary dimensions, have been \ncomputed recently in the mathematics literature~\\cite{Eldan2011,Kabluchko2014}.\n\nThe question naturally arises regarding the higher moments or even\nthe full distribution of the perimeter and the area of the\nconvex hulls of two dimensional random walks.\nComputing analytically even the second moment (and hence the variance),  \nfor the perimeter and the area of the convex hull of \na single two dimensional Brownian motion, turns out\nto be highly difficult~\\cite{Snyder1993,Goldman1996}.\nSince so far no analytical results are available concerning the full \ndistributions, performing numerical studies is a natural approach.\nWithin\na straightforward implementation (``simple sampling''), \none generates many times independently a random walk of $T$ \nsteps and constructss the hull polygon and, subsequently computes\nits area $A$ \nand perimeter $L$. Then one records histograms of these\nquantities. Nevertheless, this approach\n only gives insight into a small portion of the \nactual distributions $P(A)$ and $P(L)$, i.e., configurations with \n$P \\propto K^{-1}$, the inverse of the number $K$ of samples. This \ndemands for specific calculations with respect to the \nlarge-deviation properties of this model. So, we use a particular \nlarge-deviation approach~\\cite{align2002,Hartmann2011} which randomly \nalters steps of the random \nwalk, compares the effected change in terms of either of the quantities \nand accepts the alteration according a Metropolis criterion \ninvolving an artifical Monte Carlo ``temperature'' \n$\\Theta$. This is performed for various values of $\\Theta$, giving access\nto different ranges of the distribution under scrutiny.\nAfterwards, the resulting distributions $P_{\\Theta}(A)$ or \n$P_{\\Theta}(L)$ can be merged according to a \ngiven scheme. In the end, we are able to obtain distributions $P(A)$ and \n$P(L)$ which cover a range in principle \nonly limited by the numerical precision of our computers.\n\nThe rest of the  paper is organized as follows: \nSec.\\ \\ref{sec:model} will introduce \nthe used random walk model, the associated convex hull and then \nwill shortly \nallude \nto the algorithms and some heuristics. Following this, in Sec.\\  \n\\ref{sec:large_deviation_scheme} we will introduce the large-deviation \nmethod. Our results are presented in Sec. \\ref{sec:results}, and we \nwill finally conclude in Sec.\\ \\ref{sec:conclusions}, giving an \noutlook to possible future work.\n\n\n\\section{Random Walks, Convex Hulls and Algorithms \\label{sec:model}}\nA time-discretized random walk consists of $T$ step vectors \n$\\vec{\\delta}_i$, and the position $\\vec{x}(\\tau)$ at timestep \n$\\tau < T$ is the sum of all steps up to $\\tau$, i.e.:\n\\begin{equation}\n \\vec{x}(\\tau) = \\vec{x}_0 + \\sum^{\\tau}_{i=1}\\vec{\\delta}_i\n\\end{equation}\nThe walk configuration itself is then the set \n$\\mathcal{W} = \\{\\vec{\\delta}_1, \\vec{\\delta}_1, ..., \\vec{\\delta}_T\\}$ \nof steps \\cite{Feller1950}. The step $\\vec{\\delta}_i = \n(\\delta_{x,i}, \\delta_{y,i})$ itself denotes a displacement of the \nparticle by $\\delta_{x,i}$ in $x$-direction and $\\delta_{y,i}$\nin $y$-direction. We consider two \ndifferent types of steps:\n\n\\begin{enumerate}\n \\item A time-discrete approximation to a Brownian walk, i.e., both \n$\\delta_{x,i}$ and $\\delta_{y,i}$ are, for each $i$, drawn randomly \nfrom a Gaussian distribution with zero mean and variance one.\n\n \\item A walk with discrete steps corresponding to motion on a \nsquare lattice of spacing $J$. This corresponds to the four possible \nsteps $\\vec{x} \\in \\{(0,J),(0,-J),(-J,0),(J,0)\\}$, with each of \nthese possible up\/down\/left\/right steps having probability \n$\\frac{1}{4}$.\n\\end{enumerate}\n\nNote that in the limit of long walks $T \\rightarrow \\infty$ Gaussian \nwalks and discrete lattice walks should statisically agree when setting \n$J = \\sqrt{2}$.  We also consider closed random walks which return to \nthe origin, i.e., $\\vec{x}(T) = \\vec{x}(0)$:\n\n\n\\begin{enumerate}\n \\item For the lattice case, we generate only the first half of the \nwalk randomly. The second half is filled up with inverse steps \n$\\vec{\\delta}_{i+\\frac{T}{2}} = -\\vec{\\delta}_i$. Finally the order of \nthe steps is rando\\-mized via swapping randomly selected pairs of \nsteps.\n \\item For the Gaussian case, we consider Brownian bridges \n\\cite{Mansuy2008}, i.e., from the walk $\\vec{x}(\\tau)$ we \nconstruct\n\\begin{equation}\n \\vec{x}_b(\\tau) = \\vec{x}(\\tau) - \\frac{\\tau}{T} \\vec{x}(T)\n\\end{equation}\n which fulfill $\\vec{x}_b(T) = \\vec{x}(0)$ and have all \nnecessary  statistical properties of random walks.\n\\end{enumerate}\n\n\nThe convex hull $\\mathcal{C} = \\text{conv}(\\mathcal{\\tilde P})$ of a \ntwo-dimensional point set $\\mathcal{\\tilde P} = \\{\\tilde P_i\\}, \n\\tilde P_i \\in \\mathbb{R}^2$ \nis described through a convex set over $\\mathcal{\\tilde P}$. The points $P$ \nwithin $\\mathcal{C}$ are given by all possible combinations \n$P = \\sum \\alpha_i \\tilde P_i$ with $\\tilde P_i \\in \\mathcal{\\tilde \nP}$ and \n$\\sum_i \\alpha_i = 1$ and $\\alpha_i \\in \\mathbb{R}^+_0$ (definition \ngiven according to \\cite{Preparata1985}). This means:\n\\begin{enumerate}\n \\item All points $P_i \\in \\mathcal{P}$ lie within $\\mathcal{C}$.\n \\item All lines $\\overline{P_i P_j}; P_i, P_j \\in \\mathcal{P}$ also \nlie within $\\mathcal{C}$.\n\\end{enumerate}\n\n\nThe boundary \nof the convex set \nis  a \npolygon which connects a subset $\\mathcal{P}\\subset \\mathcal{\\tilde P}$\nof $H$ points from the point set, i.e.,  \n$\\mathcal{P}=\\{P_0,P_1,\\ldots,P_{H-1}\\}$, with $P_i=(x_i,y_i)$ \n($i=0,\\ldots,H-1$). The hull is attributed with area \n$A$ and perimeter $L$ according to (identifying $i=H$ with $i=0$):\n\n\\begin{align}\n A(\\mathcal{C}) = \\frac{1}{2} \\sum_{i=0}^{H-1} (y_i + y_{i+1})\\cdot (x_i-x_{i+1}) \\\\\n L(\\mathcal{C}) = \\sum_{i=0}^{H-1} \\sqrt{(x_i-x_{i+1})^2 + (y_i-y_{i+1})^2}\n\\end{align}\n\nFor our work, we determined the polygons bordering convex hulls\n(for which one uses shortly the  term ``convex hull'') numerically.\nThe most common convex hull algorithms operate in \n$\\mathcal{O}(N \\log N)$. We used Andrew's variant \\cite{Andrew1979} \nof the ``Graham Scan'' algorithm \\cite{Graham1972}, which constructs \nthe convex hull by first drawing a dividing line through the point \nset and then by sorting out those points which don't form monotone chain \nof clockwise\/counter-clockwise turns on each side of the line. In \nusual cases, the application of convex hull algorithms can be \naccelerated by usage of pre-selection heuristics, such as the \none introduced by Akl and Toussaint \\cite{Akl1978}. This heuristic looks \nup extreme points of the set (i.e., those of maximum and minimum $x$- \nand $y$-coordinates) and discards all points which lie inside \nthe quadrilateral formed by these points. We use a custom refinement \nof this heuristic, which is based on iterating the heuristic under \nrotation of the coordinate origin, which eliminates another fraction \nof inert points per each iteration.\n\n\\section{Large-Devation Scheme \\label{sec:large_deviation_scheme}}\n\nFor simple-sampling results, walk configurations $\\mathcal{W}$ \nare generated randomly, and the according convex hulls $\\mathcal{C}$ \nare calculated through the algorithm, resulting in a multitude values of \n$A$ and $L$. As mentioned above, obtaining histograms of these\nvalues only gives access to the high probability regime, where\nthe convex- hull-properties of typical random walks are measured.\nHowever, in order to obtain values of these quantities \nwith especially low probabilities, allowing us the measure the distributions\n$P(A)$ and $P(L)$ over a large range of the supoort, \na certain Markov-Chain Monte \nCarlo (MCMC) scheme can be used \\cite{align2002,Hartmann2011}.\n\n\\begin{figure}[ht!p]\n\\subfigure[ ]{\\includegraphics[width=0.49\\linewidth]{p1a_walks_t1_op}}\n\\subfigure[ ]{\\includegraphics[width=0.49\\linewidth]{p1a_walks_t1_cl}}\n\\subfigure[ ]{\\includegraphics[width=0.49\\linewidth]{p1a_walks_t0_op}}\n\\subfigure[ ]{\\includegraphics[width=0.49\\linewidth]{p1a_walks_t0_cl}}\n\\caption{(color online) \nDifferent renditions of closed walks of length $T=30$ after \n$T_{\\text{MC}} = 3 \\cdot 10^5$ Monte Carlo steps, i.e., $T_s = 10^4$ \nsweeps, at the given temperatures. The top row features lattice \nsteps, while the bottom row shows Gaussian steps, plus the according \nconvex hulls (indicated by the dashed lines). The walks to the left \nare open and those to the right closed. Higher temperatures inhibit\n the acceptance of step alterations which lower the hull area $A$, thus \nthe walks and their hulls experience drastic growth.}\n\\label{fig:walks_t0_cl}\n\\end{figure}\n\nThe MCMC consist of an evolution of random walks $\\mathcal{W}(t)$,\n$t$ being another discrete ``time'' parameter, not to be confused\nwith the time parameter $\\tau$ of the random walks.\nFor the walks, we measure the property $S(t)$, i.e., the area ($S=A$) or \nperimeter ($S=L$), depending on which distribution we are aiming at.\nThe initial configuration $\\mathcal{W}(0)$ is any walk configuration,\ne.g., a randomly chosen one.\nAt each Monte Carlo step $t$, the walk $\\mathcal{W}(t)$ is altered to \n$\\mathcal{W}^*$ by replacing a randomly selected step $\\vec{\\delta}_i$\n($i\\in\\{1,2,\\ldots,T\\}$) \nwith a newly generated step $\\vec{\\delta}'_i$. The new step is generated\naccording the same distribution as all random walks steps\nof the corresponding type, e.g., Gaussian.\nThe convex hull of \n$\\mathcal{W}^*$ is calculated, leading to quantitiy $S^*$. \nThe alteration $\\mathcal{W}^*$ is \\emph{accepted} ($\\mathcal{W}(t+1) = \n\\mathcal{W}^*$) according to the Metropolis probability:\n\n\\begin{equation}\n p_{\\text{Met}} = \\min\\left[1, e^{-(S^*-S(t))\/\\Theta} \\right]\\,.\n\\end{equation}\n\nHere, $\\Theta$ is \nthe (artificial) Monte Carlo ``temperature'', which is just\na parameter used to set the range of the sampled values. \nIf the alteration is not accepted, it is \\emph{rejected}, i.e., \n$\\mathcal{W}(t+1) = \\mathcal{W}(t)$.\nNow, over the course of the \nMC steps, $A$ or $L$ develops according to the effected changes and \n$\\Theta$. Examples are shown in Fig. \\ref{fig:walks_t0_cl}. \nAs usual, the MCMC time is measured in terms of \\emph{sweeps},\ni.e., number of steps per system size, which is here the walk length $T$.\n\nLike in any MCMC simulation one must equilibrate the simulation, i.e.,\ndiscart the initial part of the measured quantities until ``typical''\nvalues are found.\nEquilibration was found to be difficult but possible \nfor simulations regarding hull area $A$. This was in particular difficult for \nGaussian-distributed $\\vec{\\delta}_i$ components, probably due\nto the fact that there exists no \nupper limit for $A$ and $L$, in contrast to the discrete case.\n As demonstrated in Fig.\\ \n\\ref{fig:equilibration_t0} $\\Theta=-16.25$ it still \ntook several $t_s \\propto 10^5$ MC sweeps until equilibration. Opposed \nto this, simulations regarding perimeter $L$ equilibrated rather \nquickly, typically within less than $t_s \\approx 10^3$ sweeps. \nThis differing behavior seems to result from the fact that the \nreplacement of one single step $\\vec{\\delta}_i$ may affect much \nlarger changes in $A$ than in $L$, thereby making the whole model much \nmore sensitive, especially for small value of $T$.\n\n\n\n\n\\begin{figure}[ht!p]\n\\centerline{\\includegraphics[width=1.0\\linewidth]{p1a_equilibration_t0_50_1625}}\n\\caption{Equilibration of the hull area $A$ of open Gaussian\nwalks over the number of Monte \nCarlo steps, normalized to $t_s$ by walk length $T=50$. For \n$\\Theta = -16.25$, the model is fully equilibrated after ca. \n$t_s = 5\\cdot 10^4$ sweeps, and heedless of the differing initial \nconfigurations (see inset), i.e., $A$ behaves similarly within the\nrange of fluctuations.} \n\\label{fig:equilibration_t0}\n\\end{figure}\n\nFor a given walk type (discrete\/Gaussian, open\/closed), a given\nwalk length $T$ and a given quanity ($A$ or $L$) we performed\nsimulations for different values of $\\Theta=\\{\\Theta_1,\\ldots,\\Theta_K\\}$. \nThe numer $K$ of temperatures and the actual values, \nposive and negatives ones, depend heavily on the model, the walk length\nand the measured quantity, see below how the temperatures are chosen. \nThus, we got\ndifferent distributions $P_{\\Theta}(S)$ which are related with the \nactual distribution $P(S)$ according to the following relation\n\\cite{align2002}\n\n\\begin{equation}\n P(S) = e^{S\/\\Theta} \\cdot Z(\\Theta) \\cdot P_{\\Theta}(S)\n\\end{equation}\n\nNote that ``distribution'' here either means ``probability''\nor ``density'', depending on whether the possible values\nof $S$ are discrete or continuous.\nFor different values of $\\Theta$, different ranges of the\nmeasured value $S$ were obtained. This allows for a picewise\nreconstruction of $P(S)$.\nIt only requires knowledge of the normalization constants \n$Z(\\Theta)$.  They can be calculated through inversion of this \nformula whenever for two values $\\Theta_1$ and $\\Theta_2$ the\nranges of the sampled values of $S$ overlap.  Thus, the temperatures\nwere chosen such that for neighbouring temperatures the measured\nhistograms sufficiently overlap.\nIn principle,\nfor each value of $S$ where the measured values $P_{\\Theta_1}(S)>0$\nand $P_{\\Theta_2}(S)>0$ one estimate of $Z(\\Theta_1)\/Z(\\Theta_2)$\nis obtained.\nNote that in case of \nequlibration, this ratio is more or less constant for all values\nof $S$ where the two histograms overlap. One the other hand, \nfor non-equilibrated cases a systematic dependence on $S$ is seen. In this\nway we have another convenient criterion to verify equilibration.\nVia these pairwise comparisons, all ratios $Z(\\Theta_k)\/Z(\\Theta_{k+1})$\ncan be determined. Finally, the overall determination of the\nnormalization constants is obtained from the global normalization\nconstraint $\\sum_S P(S)=1$ (or $\\int P(S)\\,dS=1$ for the continuous case).\n\n\n\n\\section{Results \\label{sec:results}}\n\n\\subsection{Independent Points}\n\nTo verify our simulations, we first simulated convex hulls of \n$n$ independently distributed points in the unit square $[0, 1]^2$.\n For this case some mathematical results are known \n\\cite{Cabo1994,Buchta2005,Groeneboom2012}. In particular the remaining \narea $\\tilde{A}_n = 1 - A(n)$ outside the convex hull is considered. \nThe distribution of the rescaled remaining area \n\\begin{equation}\n{(\\tilde{A}_n - 4b_n})\/2c_n\\,,\n\\label{eq:remainin:area}\n\\end{equation}\n where $b_n = \\frac{2}{3}\\frac{\\log n}{n}$ \nand $c_n = \\sqrt{\\frac{28}{27}\\frac{\\log n}{n^2}}$, should converge for \n$n \\rightarrow \\infty$ in distribution to the standard normal \ndistribution $\\mathcal{N}(0, 1)$.\n\n\\begin{figure}[ht!p]\n\\centerline{\\includegraphics[width=1.0\\linewidth]{squares}}\n\\caption{Average $\\mu$ and variance $\\sigma^2$ (inset) of the \nremaining area $\\tilde{A}_n$ of a unit square in presence of a convex \nhull of $n$ randomly distributed points. The symbols show the \nnumerical results while the lines \nshow $4b_n$ and $2c_n$, respectively.\n}\n\\label{fig:squares}\n\\end{figure}\n\nThus, for the unscaled data $A(n)$, the mean $\\mu$ and the \nstandard deviation $\\sigma$ should follow $4b_n$ and $2c_n$, \nrespectively. This \nis the case for our simulations, as visible in \nFig.\\ \\ref{fig:squares}, where we indeed find a $4b_n$ behavior for the\nmean and for larger sizes a $2c_n$ behavior for the variance.\n\nIn a similar way, we attempted to visualize \nthe predicted convergence of the distributions towards \n$\\mathcal{N}(0, 1)$ by plotting $P(\\tilde{A}_n)\\cdot\\sigma$ over \n$(\\tilde{A}_n-\\mu)\/\\sigma$, cf. Fig. \\ref{fig:squares_rescaled}.\nAs visible, the measured density is very close\nto $N(0,1)$ but no clear convergence is visible. Probably\nthis would be visible only for much larger number $n$ of points,\nthus the convergence is very slow.\nNote that these deviations become more pronounced in the small-probability\ntails, which we also obtained wihthin the MCMC approach for\nselected numbers $n$ (not shown here).\nAnyway, for the purpose of verifying our simulations, the agreement \nwith the predicitions is sufficient.\n\n\n\\begin{figure}[ht!p]\n\\centerline{\\includegraphics[width=1.0\\linewidth]{squares_rescaled}}\n\\caption{As predicted by Cabo \\& Groeneboom \\cite{Cabo1994}, \nthe distributions $P(A_n)$ of the remaining area \\emph{not} covered by \nthe convex hull over $n$ random points inside the unit square \nshould converge towards $\\mathcal{N}(0, 1)$ when being rescaled according\nto Eq.\\ \\ref{eq:remainin:area}.}\n\\label{fig:squares_rescaled}\n\\end{figure}\n\n\\subsection{Random Walks}\n\n\n\\begin{figure*}[htbp]\n\\centerline{\n\\includegraphics[width=0.324\\linewidth]{p1a_t0_area_op} \n\\includegraphics[width=0.325\\linewidth]{p1a_scaled_t0_area_op} \n\\includegraphics[width=0.325\\linewidth]{p1a_scaled_t0_peri_op} \n}\n\\caption{Distributions $P(A)$ and $P(L)$ of area $A$ and perimeter $L$ \nof the convex hull over Gaussian walks of different lengths $T$. \n (Left) ``original'' distribution $P(A)$ for selected small values\nof the walk length $T$. Main: plot log scale, inset: peak region\nin linear scale. (Middle) rescaling of $P(A)$\naccording to Eq.\\ \\eqref{eq:scaling_area}. Main plot: log scale; upper\nright inset: peak region in linear scale; lower left inset:\nfit of Gumbel distribution to $T=1500$ data  (Right)  rescaling of $P(A)$\naccording to Eq.\\ \\eqref{eq:scaling_perimeter}. Main plot: log scale;\nlower left inset: peak region in linear scale; upper right inset: fit of\n$T=1400$ data to ``modified'' Gumbel distribution.\n\\label{fig:distr:P:Gaussian}}\n\\end{figure*}\n\nFor each type of walk and either of the two quantities $A$ and $L$,\nwe considered walks lengths \n$T \\in [30,2000]$ with ca. $K = 10^6$ samples per temperature $\\Theta$. \nThrough sampling multiple values of $A$ or $L$ per MC run (i.e., each \n$t_s = 10$ sweeps), it was possible to obtain the results for each case within \nseveral days of CPU time on a average-size multi-core cluster (using up to few \ndozens of cores per case, corresponding to the number of temperatures\n$\\Theta$).\n\nAn example of the resulting distributions, $P(A)$ for the case of Gaussian\nopen walks of various lengths, is shown in Fig.\\ \\ref{fig:distr:P:Gaussian}.\nEvidently, we were able to obtain the distributions over a large range of the\nsupport, with probability densities as small as $10^{-300}$.\n\nAs the systematic behavior of the distributions $P(A)$ for different \nwalk lengths $T$ hints a common origin at a distribution \n$\\tilde{P}(A)$ (cf. Fig. \\ref{fig:distr:P:Gaussian}), the first property \nwe put under scrutiny  \nis the scaling behavior. The scaling behavior \\cite{Majumdar2010} of the mean \n$\\langle A \\rangle \\sim T$\nsuggests that distributions $P(A)$, which in fact depends on the walk length $T$,\n scales according to the  following relation:\n\n\\begin{equation}\n P(A) = \\frac{1}{T} \\cdot \\widetilde{P}\\left(\\frac{A}{T}\\right)\\,, \n \\label{eq:scaling_area}\n\\end{equation}\nwhere $\\widetilde{P}$ is a $T$-independent distribution.\n The application of the scaling assumption leads \nto a reliable collapse of the distributions $P(A)$ towards \n$\\widetilde{P}\\left(\\frac{A}{T}\\right)$ with the exception of \nfinite-size effects within the tail, as shown in \nthe middle of Fig.\\ \\ref{fig:distr:P:Gaussian}.\n\nFor perimeter distributions, we use a similar scaling assumption, which \nis based upon $\\langle L \\rangle \\propto \\sqrt{T}$ \\cite{Majumdar2010}:\n\n\\begin{equation}\n P(L) = \\frac{1}{\\sqrt{T}} \\cdot \\widetilde{P}\\left(\\frac{L}{\\sqrt{T}}\\right) \n\\label{eq:scaling_perimeter}\n\\end{equation}\n\nThe application of this relation to open walks, shown in Fig.\\ \n \\ref{fig:distr:P:Gaussian}, reveals an almost perfect collapse \neven for large values of $T$ and the latter parts of the rare-event tail. \nWe found similar scaling (not shown) for the discrete cases. \nOpposed to this, closed walks seem to be more strongly affected \nby finite-size effects, i.e., to tend towards \n$\\widetilde{P}\\left(\\frac{L}{\\sqrt{T}}\\right)$ rather slowly. Nevertheless,\nin summary for all 8 cases the corresponding scaling is supported by our data.\n\n\nNext, to extrapolate the leading behavior of the\ndistribution in the large-deviation tail, \nwe calculate the empirical rate function \n$\\Phi(s)$ \\cite{touchette2009} and guard its behavior over walk length \n$T$. Under the assumtition that away from the maximum\nthe the leading behavior of the probability is exponentially small in $T$, i.e., \nthe rate function is defined as:\n\\begin{equation}\n \\Phi(s) \\equiv -\\frac{1}{T} \\log P(s)\\,.\n\\label{eq:rate:function}\n\\end{equation}\nDistributions where the rate function is well defined, i.e., follow\nEq. \\eqref{eq:rate:function} are said to\nobey a ``large-deviation principle''.\nTo allow  a comparison and extrapolation of the rate function, \n$s$ is usually a quantity normalized with the maximum\npossible value, such that $s\\in [0,1]$. Thus,  \none would define $s_A = \\frac{A}{A_{\\max}}$ and \n$s_L = \\frac{L}{L_{\\max}}$, accordingly. For the lattice walk model \nwith $J = \\sqrt{2}$, these maximum possible values were found to be \n$A_{\\max} = \\frac{T^2}{4}$  and $L_{\\max} = 2\\cdot\\sqrt{2}\\cdot T$ for \nopen walks as well as  $A_{\\max} = \\frac{T^2}{8}$  and \n$L_{\\max} = \\sqrt{2}\\cdot T$ for closed walks. But for Gaussian walks \nno actual ``maximum'' values of $A$ and $L$ exist, so in remembrance of \nthe lattice cases to obtain the same scaling, we choose $s_A = \\frac{A}{T^2}$ or \n$s_L = \\frac{L}{T}$, respectively. \n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=0.8\\linewidth]{p1a_rf_t0_peri_op}}\n\\caption{The rate functions $\\Phi(s_L)$ \nof open Gaussian walks for different walk lengths (lines). The symbols show\nthe extrapolated values ($\\Phi_{\\infty}(s)$ see text and \nFig.\\ \\ref{fig:p1a_rf_t0_op_peri_asymptotic}). \nThe inset shows the same on normal scale.\n\\label{fig:p1a_rf_t0_peri_op}}\n\\end{figure}\n\nAn example of the resulting rate functions, for the case of the perimeter\nof the hull of open Gaussian walks, is shown in \nFig.\\ \\ref{fig:p1a_rf_t0_peri_op}.\nApparently, for large values of $s_L$, the rate functions for different\nwalks lengths agree very well, while for small values of $s_L$\nstrong finite length effects are visible. Anyway, from the visual\nimpression, the rate function seems to converge for $T\\to\\infty$ pointwise \nto a limiting function, which starts at the origin. Thus, its seems that\nthe distribution obey the large-deviation principle.\nTo quantify this, we performed fits of the form\n\\begin{equation}\n\\Phi(s,T) = \\Phi_{\\infty}(s) - \\xi \\cdot T^{-\\gamma}\n\\label{eq:power:law:fit}\n\\end{equation}\nfor many\nvalues of $s$, as it is demonstrated \nin Fig. \\ref{fig:p1a_rf_t0_op_peri_asymptotic}. \nWe obtained similar results for all other walk types and measured\nquantities. We found that\n$\\Phi_{\\infty}(s)$ appears to grow according to power laws $s^{\\kappa}$.\n(The parameters \n$\\xi$ and $\\gamma$ are not of interest and show no systematic behavior with $s$).\n The results are listed in \nTab.\\ \\ref{Tab:rf_kappa}. The values for the area are very close to $\\kappa=1$\nand for the perimeter very close to $\\kappa=2$, in particular\nfor the Gaussian walks.\n\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[width=0.86\\linewidth]{p1a_rf_t0_op_peri_asymptotic} }\n\\caption{Dependence of rate function $\\Phi(s_L)$ (for open Gaussian walks)\non walk length $T$ at selected\nvalues $s = 1.2$ and $s = 0.5$ (inset). A power law was used\nto determined the $T\\to\\infty$ convergence towards an asymptotic value \n$\\Phi_{\\infty}$, indicated by horizontal lines. \n\\label{fig:p1a_rf_t0_op_peri_asymptotic}\n}\n\\end{figure}\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{|c||c|c|}\n \\hline\n Model            & $\\kappa_A$, $P(A)$ & $\\kappa_L$, $P(L)$ \\\\\\hline\\hline\n Gaussian, open   & 0.999(1)         & 2.03(2)          \\\\\\hline\n Gaussian, closed & 1.06(2)          & 2.06(1)          \\\\\\hline\n lattice, open    & 1.18(1) &          2.13(1) \\\\\\hline\n lattice, closed  & 1.12(2)          & 2.12(1) \\\\\\hline\n\\end{tabular}\n\\caption{Resulting exponents $\\kappa$ which deterine the power-law growth of the asymptotic value $\\Phi_{\\infty}(s)$ of the rate function $\\Phi(s)$ with $s_A$ or $s_L$, respectively.}\n\\label{Tab:rf_kappa}\n\\end{table}\n\nThis can be already seen when one combines the rate function for the scaled\nvariables with the scaling forms Eqs. \\eqref{eq:scaling_area}\nand \\eqref{eq:scaling_perimeter}. By equating the two expressions\nfor the distribution of the area we obtain\n\n\\begin{equation*}\ne^{-\\Phi(A\/T^2)\\cdot T} \\sim \\frac 1 T \\widetilde P(A\/T)\\,.\n\\end{equation*}\nTo make the argument of the exponential a function of $A\/T$ (the factor $1\/T$\nis of lower order and can be ignored) $\\Phi(s)\\sim s$ \nmust hold, i.e., $\\kappa_A=1$.\nCorrespondingly for the perimeter we get\n\\begin{equation*}\ne^{-\\Phi(L\/T)\\cdot T} \\sim \\frac {1}{\\sqrt{T}} \\widetilde P(L\/\\sqrt{T})\\,,\n\\end{equation*}\nwhich results in $\\phi(s)\\sim x^2$, i.e., indeed $\\widetilde P(L\/\\sqrt{T})\\sim$ \n$e^{-(L^2\/T^2)T} =e^{-(L\/\\sqrt{T})^2}$, and $\\kappa_L=2$.\n\nThe fact that for Gaussian walks, the exponents $\\kappa$ agree very well with \n$\\kappa_A=1$ and $\\kappa_L=2$ but not quite for the lattice walks is probably due\nto the discrete structure of the lattice and due to the limited\naccessible area. These have a strong influence for the given limited\nlength $T$ of the walks and influence the rate function in particular for\nsmall and large values of $s$. In fact the rate functions\nfor the lattice case are continuously\nbending up in a log-log plot (not shown), \nsuch that one easily can find a short interval where the expected exponent\n$\\kappa$ appears. Thus, to conclude this part, the data is well described\nby the rate function, i.e., the distribution obeys the above large-deviation\nprinciple. Hence, in connection with the fulfilled scaling \nforms Eqs. \\eqref{eq:scaling_area}\nand \\eqref{eq:scaling_perimeter}, we see that to leading order and \nasymtotically the distributions are given by $P(A)\\sim e^{-A\/T}$\nand $P(L)\\sim e^{-L^2\/T}$.\n\nThe Gaussian and the exponential\ntails, respectively for the distributions of the perimeter and the area, \nmay be guessed using very simple heuristic arguments. Indeed, \nthe perimeter $L$ of the convex hull of a two dimensional\nstochastic process morally scales as \n$s$, where $s$ represents\nthe span of the one dimensional component process. For the\n$2$-d Brownian motion, $s$ is simply the span of a one dimensional\nBrownian motion of duration $T$. The probability distribution\n$P(s,T)$ of the $1$-d Brownian motion of duration $T$ and\ndiffusion constant $D$ is well \nknown\\cite{hughes1996,Kundu2013} and has a scaling form\n\\begin{equation}\nP(s,T) = \\frac{1}{\\sqrt{4DT}}\\, f\\left(\\frac{s}{\\sqrt{4DT}}\\right)\n\\label{scaling_span.1}\n\\end{equation}\nwhere the scaling function $f(x)$ is given exactly by\n\\begin{equation}\nf(x)= \\frac{8}{\\sqrt{\\pi}}\\, \\sum_{m=1}^{\\infty}\n(-1)^{m+1}\\,'\nm^2\\, e^{-m^2\\, x^2} \\, .\n\\label{scaling_span.2}\n\\end{equation}\nThe scaling function has the following asymptotic behavior\n\\begin{eqnarray}\nf(x)\\to \\left\\{\\begin{array}{ll}\n2\\, \\pi^2\\, x^{-5}\\, e^{-\\pi^2\/{4\\,x^2}}\\;\\;\n&{x\\to 0}\\label{small_x}\\\\\n& \\\\\n\\frac{8}{\\sqrt{\\pi}}\\, e^{-x^2} \\label{large_x}\\;\\;\n&{x\\to \\infty}\n\\end{array}\n\\right.\n\\end{eqnarray}\nand is plotted in Fig.\\ref{fig:span}.\n\\begin{figure}[ht!p]\n\\centerline{\\includegraphics[width=1.0\\linewidth]{span}}\n\\caption{The scaling function $f(x)$ in Eq. (\\ref{scaling_span.2})\nfor the span distribution for a $1$-d Brownian motion, plotted\nas a function of $x$.}  \n\\label{fig:span}\n\\end{figure}\n\n\n\n\n  \nThus, for large $s$, $P(s,T)\\sim \\exp\\left[-s^2\/{4DT}\\right]$ has a\nGaussian tail. Consequently since $L\\sim s$, one would expect\na Gaussian tail for the distribution $P(L)$. Note that for small\n$S$ (and hence for small $L$), one would expect an essential\nsingular behavior $P(L)\\sim \\exp[-b\/L^2]$ from Eq. (\\ref{small_x}),\nwhere $a$ is some constant. The scaling function $f(x)$ in Eq. \n(\\ref{scaling_span.2}), and hence $P(L)$ is almost flat near $L=0$, and \nfalls off\nas a Gaussian for large $L$ with a peak in between, around which \nis has an approximate Gaussian shape (see Fig. \\ref{fig:span})\n\\begin{equation}\nP(L) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\cdot e^{-\\frac{(L-\\mu)^2}{2 \n\\sigma^2}}\n \\label{eq:Gaussian}\n\\end{equation}\n\nSubsequently, for the area we can approximately assume \n$A \\propto L^2$, and thus the distribution of $\\sqrt{A}$ should also \nbe Gaussian, including a factor $1\/\\sqrt{A}$ originating from\n$dL\/dA \\sim 1\/\\sqrt{A}$ : \n\n\n\\begin{equation}\nP(A) = \\frac{1}{\\sqrt{2 \\pi \\sigma_A^2 \\cdot A}} \\cdot \ne^{-\\frac{(\\sqrt{A}-\\mu_A)^2}{2 \\sigma_A^2}}\n \\label{eq:Gaussian_area}\n\\end{equation}\n\nHad the assumption of a Gaussian distribution for the\nperimeter been entirely correct, it would be sufficient to take\nthe average and the variance and plot the resulting Gaussian \ntogether with the data. First, we concentrate on the case of the hull\nperimeter for closed planar Brownian motions, where mean and variance are\navailable analytically~\\cite{Goldman1996}. The mean scaled perimeter is\ngiven by~\\cite{Goldman1996}\n\\begin{equation}\n\\mu \\equiv E\\left(\\frac{L}{\\sqrt{T}}\\right) = \\sqrt{\\frac{\\pi^3 T}{2}}\\cdot \\frac{1}{\\sqrt{T}} = \n\\sqrt{\\frac{\\pi^3}{2}} \\approx 3.937\n\\end{equation}\nwhere $E$ stands for the expectation value.\nFor the variance, we use Goldman's result for the second moment~\\cite{Goldman1996}, \n\\begin{equation}\n E((L\/\\sqrt{T})^2) = \\frac{\\pi^2}{3} \\left(\\pi \\int^{\\pi}_0 \\frac{\\sin u}{u} \n\\text{d}u - 1\\right)\n\\end{equation}\nThe value of the integral of $\\int_0^{\\pi}du \\sin(u)\/u$ is approximately 1.852,  \nknown as the Wilbraham-Gibbs constant. This then leads to the variance:\n\\begin{eqnarray*}\n \\sigma^2 & = & E((L\/\\sqrt{T})^2)-E(L\/\\sqrt{T})^2 \\\\\n& = & \\frac{\\pi^3}{3}(\\pi\\cdot1.852 - 1)-\\frac{\\pi^3}{2} \\approx 0.348\n\\end{eqnarray*}\nBefore comparing the actual distribution, we compare average and variance\nobtained from\nour numerical results. By fitting power laws plus a constant, similar\nto \\eqref{eq:power:law:fit}, we found $\\mu$ to converge asymptotically towards \n$\\mu_{\\infty} = 3.937(1)$, while $\\sigma^2$ fluctuated slightly around \nan average of $\\sigma^2_{\\infty} = 0.347(1)$, see \nFig.\\ \\ref{fig:p1a_average_t0_cl_peri}. Both values agree with \nthe aforementioned predictions. The results of the extrapolations\nfor all considered cases of \nthe different walk models are listed in Tab. \\ref{Tab:Gauss_results}, \nand it can be seen that the asymptotic values of the parameters of \nthe distributions agree rather well \nbetween lattice and Gaussian walks, as expected.\n  Note that for the fitting the results the\nstated error bars are only of statistical nature, thus do not include \nsystematic\ncontributions due to the unkown scaling behavior. Hence, the real\nerror bars should be considerably larger.\nThus the results for the averages $\\mu_{\\infty}$ can be considered  \nto match well the corresponding calculations in \nRefs.\\cite{Randon-Furling2009,Majumdar2010} which read:\n\n\\begin{align}\n \\langle A_{\\text{op}} \\rangle = \\frac{\\pi T}{2} \\Leftrightarrow \\mu_{\\infty} = \\frac{\\pi}{2} \\approx 1.571\\\\\n \\langle A_{\\text{cl}} \\rangle = \\frac{\\pi T}{3} \\Leftrightarrow \\mu_{\\infty} = \\frac{\\pi}{3} \\approx 1.047 \\\\\n \\langle L_{\\text{op}} \\rangle = \\sqrt{8 \\pi T} \\Leftrightarrow \\mu_{\\infty} = \\sqrt{8 \\pi} \\approx 5.013\n\\end{align}\n\n\\begin{table}[htbp]\n\\centering\n\\centerline{\n\\begin{tabular}{|c||c|c||c|c|}\n \\hline\n Model                & Gaussian & Gaussian & lattice  & lattice \\\\\n                      & open     & closed   & open     & closed  \\\\\\hline\\hline\n \\multicolumn{5}{|c|}{Area distributions $P(A)$}  \\\\\\hline\\hline\n $\\mu_{\\infty}$       & 1.585(1) & 1.049(1) & 1.577(1) & 1.049(1) \\\\\\hline\n $\\sigma^2_{\\infty}$  & 0.312(1) & 0.088(1) & 0.309(1) & 0.089(1) \\\\\\hline\\hline\n \\multicolumn{5}{|c|}{Perimeter distributions $P(L)$}  \\\\\\hline\\hline\n $\\mu_{\\infty}$       & 5.023(2) & 3.937(1) & 5.015(1) & 3.938(1) \\\\\\hline\n $\\sigma^2_{\\infty}$  & 1.075(1) & 0.347(1) & 1.075(1) & 0.348(1) \\\\\\hline\n\\end{tabular}\n}\n\\caption{Resulting asymptotic values of average $\\mu$ and variance \n$\\sigma^2$ of rescaled hull area $A$ or perimeter $L$ of the mentioned walk \nmodels. These values have been obtained by fitting to the results of \n$\\mu(T)$ and $\\sigma^2(T)$ with growing walk length, hereby \nextrapolating the $T\\rightarrow\\infty$ case. For equal walk \ntopologies (open\/closed walks) and quantities (area\/perimeter), \nthe asymptotic values agree heedless of the actual step type, i.e., the \nwalk geometry.}\n\\label{Tab:Gauss_results}\n\\end{table}\n\nHowever, as demonstrated in Fig.\\ \\ref{fig:distr:P:Gaussian}, when actually\nplotting the numerical data together with a Gaussian parametrized\nby the analytic values, one sees that distribution only \nmatches the numerics in  the main region of the measured distributions $P(L)$, \nwhile the  tail exhibits a different behaviour. On the other\nhand it is indeed possible to fit \nEq.\\ \\eqref{eq:Gaussian} to the tail of $P(L)$ with good precision.\nThis means the shape of the distribution is Gaussian there. Nevertheless,\nthe resulting value of \n$\\sigma^2\\approx 1.35$ is considerably larger than the aforementioned, and\n the corresponding value of $\\mu\\approx 0.1$ for these fits seems useless. \nIn the same way, for the other cases of perimeter distributions (not shown),\none can fit Gaussians well either to the main region of the data or to the\ntails. But it is not possible to fit the\ndata using one single Gaussian over the full support.\nThus, the approximations which lead to the assumption of a\n Gaussian distribution were\na bit too strong and the true distributions seem to be very Gaussian-like, but\nslightly different.\n\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=0.8\\linewidth]{p1a_average_t0_cl_peri} }\n\\caption{Average (rescaled) perimeter $\\mu$ and variance $\\sigma^2$\nfor closed Gaussian as a function of the walk length $T$. \nA convergence to the analytical expectations $\\mu \\approx 3.937$ and \n$\\sigma^2 \\approx 0.348$ is visible.\n\\label{fig:p1a_average_t0_cl_peri}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[width=0.8\\linewidth]{p1a_scaled_t0_peri_cl} }\n\\caption{Scaled distributions of the \nperimeter $L$. While the scaling assumption collapses the distributions \n$\\tilde{P}(L)$ onto similar behavior within the tail as well as in the \nmain region (large inset), the Gaussian distribution according to \nEq.\\ \\eqref{eq:Gaussian} (small inset) with the values of \n$\\mu$ and $\\sigma^2$ taken from\nthe analytics suits only in the main region. \n\\label{fig:p1a_scaled_t0_peri_cl}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[width=0.8\\linewidth]{p1a_gaussian_t0_area_op} }\n\\caption{ Fit of a Gaussian according to \nEq. \\eqref{eq:Gaussian_area} to $P(A)$ with a focus on either \nthe tail (large plot) or the region around the average (inset).\nIt was not possible to fit one single Gaussian over the full support.\n\\label{fig:p1a_gaussian_t0_area_op}}\n\\end{figure}\n\n\n\nRegarding the area distribution $P(A)$, the mentioned propierties \nare demonstrated in Fig.\\ \\ref{fig:p1a_gaussian_t0_area_op}. Given the\nprevious results for the perimeter, we here just performed fits in\na similar way. A Gaussian \nof the form of Eq. \\eqref{eq:Gaussian_area}  can indeed be employed \nto fit the area distribution either in the high-probability region\nor everywhere away from the central region of the distribution, but again\nnot over the full support. Thus, the same concusion as for the perimeter\ndistribution holds.\n\n\n\n\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[width=0.8\\linewidth]{leftflank_peri_paper} }\n\\caption{(color online) \nMain plot: rescaled distribution (symbols) of the perimeter for the \nopen Gaussian walks for small values of the perimeter, length $T=800$\ntogether with a fit to the function $ae^{-b\/(L^2\/T)}$.\nInset: Scaling behavior of the parameter $b$ as function of walk length,\nfor open and closed Gaussian walks, respectively. The lines represent\nfits to functions of the form $b_{\\infty}+c b^{\\eta}$, resulting\nin $b_{\\infty}=129(9)$ (open) and $b_{\\infty}=250(50)$ (closed),\nwith a very slow convergence ($\\eta=-0.21(3)$ and $\\eta=-0.09(2)$).\n\\label{fig:leftflank_peri}}\n\\end{figure}\n\n\nAlthough the focus of our simulations was on the right tails\nof the distributions, i.e., the large range of support for larger than\ntypical values, we have also performed some simulations to \nstudy very small values of the perimeter, to verify scaling\nfunction (\\ref{small_x}) leading to an expected $\\sim e^{-b\/(L^2\/T)}$\nbehavior.\n\n The main plot of Fig.\\ \\ref{fig:leftflank_peri}\nshows, as example, \nthe resulting rescaled distribution for open Gaussian walks of length\n$T=800$ together with a  fit to this functional form.  \nApparently, the data follows the predicted form very well in the\nrange of small perimeters.\n We have furthermore studied  the asymptotic behavior of the factor $b$\nin the exponent and found (see inset of Fig.\\ \\ref{fig:leftflank_peri})\nthat the behavior is compatible with a convergence to a limiting value.\n\n\n\\section{Conclusions \\label{sec:conclusions}}\nWe succeeded with the application of the above explained large-deviation \nscheme to the study of properties of convex hulls of random walk models.\nWe could obtain the corresponding distributions \nover broad ranges of convex hull area $A$ and perimeter $L$,\ndown to densities and probabilitiues as small as $10^{-300}$.\n\n The \nscaling behavior of these distributions turned out to be linked to \nwalk length $T$ in the same way as the averages $\\langle A \\rangle \\sim T$ and \n$\\langle L \\rangle \\sim \\sqrt{T}$. Also, the examination of the empirical \nrate functions resulted in simple power laws, which are compatible\nwith the scaling behavior. This leads to the natural assumtion\nthat the perimeter $L$ and the square root $\\sqrt{A}$ of the area\nare Gaussian distributed. Indeed either peak region of the data or\nthe tails region fit this form very well, but it is not possible\nto fit over the full range of the support one single function.\nNevertheless, for almost all our results we found asymptotic agreement between\nthe Gaussian and the lattice walk case, as expected.\n\n\nSince the application of large-deviation simulation approaches for this\nproblem turned out to be very useful, it should be applied\nfor similar problems. For future work, \nbesides obvious extensions like considering higher\ndimensions, we want to consider other random walk \nmodels, particularly multiple interacting \nwalkers~\\cite{Acedo2002,Kundu2014}, \nin simple self-avoiding walks as well as in loop-erased \nrandom walks\\cite{Lawler1987,Majumdar1992} and, most \nimportantly, for applications to biology like the formation of animal \nterritories\\cite{Giuggioli2011}.\n\n\n\\begin{acknowledgments}\nWe acknowledge personal support by O. Melchert and \nM. Manssen. Our simulations were carried out on the HERO cluster of \nCarl-von-Ossietzky Universit\\\"at Oldenburg, which has been funded by \nthe DFG through its Major Research Instrumentation Programme \n(INST 184\/108-1 FUGG) and the Ministry of Science and Culture (MWK) of \nthe Lower Saxony State.\nSNM acknowledges support\nby ANR grant 2011-BS04-013-01 WALKMAT and in part by\nthe Indo-French Centre for the Promotion of Advanced Re-\nsearch under Project 4604-3.\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{intro}{Introduction}\nThe K\\\"ahler and complex structure parameters that describe the\npossible\ndeformations of a Calabi-Yau manifold determine the dynamics of the\ncorresponding string compactification. The kinetic energy\nterms and Yukawa couplings in the low-energy effective theory, for\nexample,\ndepend on these moduli. Quantities that depend on the K\\\"ahler\nparameters\nare subject to non-perturbative instanton corrections that, in virtue\nof\nmirror symmetry, may be found by a straightforward calculation\nrelating to the\ncomplex structure sector of the mirror space. In fact, mirror\nsymmetry leads\nto a complete geometrical description of the full moduli space of\nCalabi-Yau manifolds.\nIn this article we continue the study, initiated in a companion\npaper~\n\\REFS{\\rUno}{P.~Candelas, X.~de~la~Ossa, A.~Font, S.~Katz and\nD.~R.~Morrison, ``Mirror Symmetry for Two-Parameter Models--I\",\n\\npb{416} (1994) 481.}\n\\refsend,\nof the mirror map for the moduli spaces of two-parameter Calabi--Yau manifolds. Our\nwork is a\nfurther step towards establishing general results beyond the\none-parameter\ncase considered in Refs.~\n\\REFS\\rCdGP{P.~Candelas, X.~ de la Ossa, P.~Green and L.~Parkes,\n       \\npb{359} (1991) 21.}\n\\REFSCON\\rMorrison{D.~R.~Morrison, ``Picard-Fuchs Equations and\nMirror\nMaps for Hypersurfaces'', in {\\it Essays on Mirror Symmetry}, ed.\\\n       S.-T.~Yau (Intl. Press, Hong Kong, 1992).}\n\\REFSCON\\rFont{A.~Font, \\npb{391} (1993) 358.}\n\\REFSCON\\rKT{A.~Klemm and S.~Theisen, \\npb{389} (1993) 153;\n``Mirror Maps and Instanton Sums for Complete Intersections in\nWeighted\nProjective Space\", preprint LMU-TPW 93-08, hep-th\/9304034.}\n\\REFSCON\\rLT{A.~Libgober and J.~Teitelbaum,  Int. Math. Res.\nNotices\n(1993) 15.}\n\\REFSCON\\rBvS{V.~Batyrev and D.~van Straten, ``Generalized\nHypergeometric\nFunctions and Rational Curves on Calabi-Yau Complete Intersections in\nToric Varieties\", Essen preprint, alg-geom\/9307010.}\n\\refsend.\n\nOur aim here is to apply the methods developed in \\cite{\\rUno}\nto describe the K\\\"ahler moduli of the Calabi--Yau\\ manifold \\Mthree.\nWe first study the complex structure space\nof its mirror to determine its singularities and characterize\nthe point corresponding to the large complex structure limit. In\norder to find\nthe explicit mirror map between the flat K\\\"ahler coordinates and the\ncomplex\nstructure parameters, we obtain and study a basis of\nperiods of the mirror manifold. This basis is constructed\nfrom the fundamental period found\nby direct integration of the holomorphic 3-form.\nThe periods are multivalued functions over the moduli space and this\nimplies the existence of a modular or duality group whose generators\ncorrespond to the monodromy of the period basis about the various\nsingularities in moduli space. Using methods similar to~\n\\REFS\\rCDR{ A. Ceresole, R. D'Auria and  T. Regge,\n``Duality group for Calabi--Yau 2-moduli space'',\npreprint DFTT-34-93, hep-th\/9307151.}\\refsend,\nwe calculate the full duality group for this example.\nThe periods also satisfy certain\ndifferential equations that we use to rederive the monodromy. The\nmonodromy properties permit us to identify the large complex structure limit\\ which is\nthe mirror of the large radius limit of the original manifold.\nWith this information we are then able to choose a basis of flat\ncoordinates in which the duality generators belong to\n$\\hbox{Sp}(6,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$.\nThe relation between the flat coordinates and the periods defines\nthe mirror map that we use to\nderive instanton expansions of the Yukawa couplings and\nthe $N=2$ index of Bershadsky \\hbox{\\it et al.}~\n\\REFS\\rBCOV{M.~Bershadsky, S.~Cecotti, H.~Ooguri and C.~Vafa,\nwith an appendix by S.~Katz,\n\\npb{405} (1993) 279.}\n\\refsend.\nIt is interesting that some of the expansion coefficients, which\ncorrespond\nto numbers of rational and elliptic curves, are negative and we\nexplain\nthis fact as a consequence of the existence of continuous families\nof instantons. A number of two and three parameter examples,\nincluding the\npresent one, have been discussed recently from a somewhat different\nperspective\nin Ref.~\n\\REFS{\\rHKTY}{S.~Hosono, A.~Klemm,\nS.~Theisen and S.-T.~Yau, ``Mirror Symmetry, Mirror Map and\nApplications to\nCalabi-Yau Hypersurfaces\", preprint HUTMP-93\/0801, hep-th\/9308122.}\n\\refsend.\n\n\nThis article closely parallels Ref.~\\cite{\\rUno}. The layout\nis the following. We study in \\SS2 the geometry of Calabi--Yau\\ hypersurfaces\nin\n$\\relax{\\rm I\\kern-.18em P}^{(1,1,1,6,9)}$. The embedding space has an orbifold singularity\nalong a\nsurface which intersects the Calabi--Yau\\ hypersurface along a certain curve\n$C$. The resolution of the singular curve introduces a surface; every\npoint of\n$C$ having\\vadjust{\\vfill\\eject}\nbeen blown up into an instanton. We obtain in this way a\ncontinuous\nfamily of instantons.\nWe find a homology basis and compute certain\ninvariants that are needed later in relation to the instanton\nexpansions. In\n\\SS3 we describe the moduli space of the mirror manifold\nand find the loci where the Calabi--Yau manifold\\ is singular. These loci constitute\nthe\nboundary of the moduli space. We also use the methods of toric\ngeometry to\ndiscuss the compactification of moduli space and locate the point\ncorresponding to the large complex structure limit. In \\SS4 we obtain\na basis of\nperiods and derive the analytic properties that are needed\nin \\SS5 to determine the monodromies under transport around the\nsingularities. In this chapter we also compute monodromy from the\nPicard-Fuchs equations.\nIn \\SS6 we find the flat coordinates and the explicit mirror map\nthat is used in \\SS7 to compute the\ninstanton expansions for the Yukawa couplings and the $N=2$ index.\nWe also investigate an $\\hbox{SL}(2,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$ symmetry of a boundary\ncomponent of the moduli space. Finally, in \\SS8 we verify and explain\nthe significance of some of the instanton numbers that we have found.\n\\vfill\\eject\n %\n\\section{model}{Geometry of CY hypersurfaces in $\\hbox{$\\IP^{(1,1,1,6,9)}$}$}\n\\vskip-20pt\n\\subsection{Linear systems}\nWe consider Calabi-Yau threefolds $\\ca{M}$ which\nare obtained by resolving\nsingularities of degree 18 hypersurfaces $\\relax\\ifmmode\\preMhat\\else $\\preMhat$\\fi\\subset \\hbox{$\\IP^{(1,1,1,6,9)}$}$.\nA typical defining polynomial for such a hypersurface is\n $$\np=x_1^{18}+x_2^{18}+x_3^{18}+x_4^{3}+x_5^{2}~. $$\nThe singularities occur along $x_1=x_2=x_3=0$, which is a curve in\n$\\hbox{$\\IP^{(1,1,1,6,9)}$}$ that meets the $p=0$ hypersurface in the single point\n$[0,0,0,-1,1]$.\nNotice that $\\hbox{$\\IP^{(1,1,1,6,9)}$}$ has quotient singularities by a group\nof order $3$ along this curve.  For example, the chart with $x_4=1$\nis described as points $[x_1,x_2,x_3,1,x_5]$ modulo the equivalences\n $$[x_1,x_2,x_3,1,x_5]\\sim[\\lambda x_1,\n\\lambda x_2,\\lambda x_3,1,\\lambda^9 x_5]~, $$\nwhere $\\lambda^6=1$.  The subgroup of the group of equivalences which\nfixes the curve $x_1=x_2=x_3=0$ is the subgroup generated by\n$\\lambda^2$,\nand this gives rise to quotient singularities by a group of order\n$3$.\nNotice that the polynomial $x_1x_2x_3$ is invariant under this\nsubgroup,\nand describes a divisor which vanishes {\\it simply} along the curve\nonce the quotient has been taken.\nIf this curve is blown up on $\\hbox{$\\IP^{(1,1,1,6,9)}$}$ (which will blow up\nthe corresponding point on the threefold), the singularities of\nthe threefold are resolved.  A single\nexceptional divisor $E$ is created during this process.\n\nWe now study linear systems on $\\ca{M}$. We first consider\nthe linear system $|L|$ generated by the\npolynomials $x_1$, $x_2$, and $x_3$ of degree $1$.  This linear\nsystem maps $\\ca{M}$ to $\\relax{\\rm I\\kern-.18em P}^2$, with the inverse image of a point\nin $\\relax{\\rm I\\kern-.18em P}^2$ being an elliptic curve\n $$x_4^3+x_5^2=\\hbox{constant}~. $$\nConsequently, $L^3=0$.\n\nThe second linear system we study, denoted by\n$|H|$, is generated by\nall polynomials of degree $3$.  One of those polynomials is\n$x_1x_2x_3$,\nand as we have already noticed, that polynomial vanishes simply at\nthe singular point of $\\relax\\ifmmode\\preMhat\\else $\\preMhat$\\fi$.\nIt follows that on $\\ca{M}$, the divisor\n$3L+E$ belongs to our linear system $|H|$.  (The coefficient ``$1$''\nof\n$E$ in that expression corresponds to the ``simple'' vanishing of\nthe polynomial.)  Notice that polynomials of degree 6, 9, \\dots will\ngive divisors in the linear systems $|2H|$, $|3H|$, \\dots.\n %\n\\iffigs\n\\midinsert\n\\def\\hbox{\\def\\epsfsize##1##2{0.9##1}\\epsfbox{fig3.ps}}{\\hbox{\\def\\epsfsize##1##2{0.9##1}\\epsfbox{fig3.ps}}}\n\\figbox{{\\vbox{\\kern0.3truein{\\hbox{\\kern.1truein\\hbox{\\def\\epsfsize##1##2{0.9##1}\\epsfbox{fig3.ps}}}}}}\\vskip10pt}\n{\\figlabel{resolution}}{The space \\Mthree\\ contains a curve $C$ which\nis\nfixed by a $\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_2$ action. In the resolved space the singular curve\nis\nreplaced by a surface $E=H-3L$ that is ruled by instantons.}\n %\n\\place{2.3}{3.5}{$\\twelvemath C$}\n\\place{5.4}{3.95}{$\\twelvemath E$}\n\\place{5.7}{3.9}{$\\twelvemath L$}\n\\endinsert\n\\fi\nIntersection products are now computed as follows.  We work in\nthe affine chart $x_1=1$ (which is a smooth chart on $\\hbox{$\\IP^{(1,1,1,6,9)}$}$).\nIf we set $x_2$, $x_3$, and $x_4$ to constant values, we describe an\nintersection of three divisors in the linear systems\n$|L|$, $|L|$, and $|2H|$,\nrespectively.  The number of points in the intersection is the number\nof solutions to $x_5^2=\\hbox{constant}$, that is, $2$.  We conclude\nthat\n $$L\\cdot L\\cdot (2H)=2~. $$\nA similar argument with $x_2$, $x_4$ and $x_5$ produces\n $$L\\cdot (2H)\\cdot (3H)=18~. $$\nFinally, if we set $x_4$ and $x_5$ to constant values, and also\nimpose\na general cubic equation on $x_2$ and $x_3$, we will calculate the\nintersection of $H$, $2H$, and $3H$.  The intersection number is the\nnumber of common solutions to $x_2^{18}+x_3^{18}=\\hbox{constant}$\nand a general (inhomogeneous)\ncubic in $x_2$, $x_3$; by Bezout's theorem, there are\n$3\\cdot18$ solutions.  We conclude\n $$H\\cdot(2H)\\cdot(3H)=54~. $$\n\nTo summarize, the intersection numbers are:\n $$\nH^3 = 9~,~~~~H^2\\cdot L = 3~,~~~~H\\cdot L^2 = 1~,~~~~L^3 =  0~. $$\n\nLet $l$ be the intersection\nof $L$ and $E$, and\nlet $h$ be the intersection of 2 divisors from $|L|$ (i.e., one\nof the elliptic curves mentioned above).\nUsing the equivalence $l=L\\cdot E=L\\cdot H - 3L^2$,\nwe easily calculate the following intersection numbers,\n $$\\eqalign{\nL\\cdot l&=1\\cr\nH\\cdot l&=0\\cr}\\hskip30pt\n\\eqalign{\nL\\cdot h&=0\\cr\nH\\cdot h&=1~.\\cr} $$\nThis leads to an identification of the K\\\"ahler cone as being the\ncone\ngenerated by $L$ and $H$.\n\n\n\\subsection{Chern classes}\nWe presently compute some intersections that will\nbe of use later in relation to the instantons of genus one.\nFor a smooth divisor $D\\subset\\ca{M}$, we use the notation $c_2(D)$\nto note the second Chern class of the surface $D$.  The notation\n$c_2$ is reserved for the second Chern class of $\\ca{M}$.\nFor the surface $L$ we have $c_2(L)=36$ whereas for the exceptional\ndivisor $E$, we have $c_2(E)=3$.\n\nThe desired results can be obtained by repeated application\nof the formula\n $$\nc_2\\cdot D=c_2(D)-D^3~. $$\nIn particular,\n $$\nc_2\\cdot L=c_2(L)-L^3=36~. $$\nFurthermore, we have\n $$\nc_2\\cdot E=c_2(E)-E^3=3-(H-3L)^3=-6~, $$\nand hence\n $$\nc_2\\cdot H=c_2\\cdot(3L+E)=3(36)-6=102~. $$\n\n\\vfill\\eject\n\\section{moduli1}{The Moduli Space of the Mirror}\n\\vskip-20pt\n\\subsection{Basic facts}\nThe most concrete approach to the moduli space of the mirror\nof $\\Mthree$ begins with the orbifolding construction of~\n\\REFS{\\rGP}{B. R. Greene and M. R. Plesser, Nucl. Phys. {\\bf B338}\n(1990) 15.}\n\\refsend,\naccording to which\nthe mirror of $\\Mthree$ may be identified with the family of\nCalabi-Yau\nthreefolds of the form $\\{p=0\\}\/G$, where\n$G\\cong\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_6 \\times \\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_{18}$\nis the group with generators\n $$\\eqalign{\n&(\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_{6\\hphantom{8}};~0,\\- 1,\\- 3,\\- 2,\\- 0)~,\\cr\n&(\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_{18};~1,-1,\\- 0,\\- 0,\\- 0)~,\\cr} $$\nand $p$ is a $G$-invariant quasi-homogeneous polynomial\nof weighted degree $18$.  The most general possible form of $p$ is\n $$\\eqalign{\np=&a_0 x_1 x_2 x_3 x_4 x_5  +a_1x_1^2x_2^2x_3^2x_4^2\n+a_2x_1^3x_2^3x_3^3x_5 + a_3x_1^4x_2^4x_3^4x_4 \\cr\n&+ a_4 x_1^6 x_2^6 x_3^6\n+a_5x_1^{18} + a_6x_2^{18} + a_7x_3^{18} + a_8x_4^3 + a_9x_5^2.}\n\\eqlabel{generalp}$$\nMultiplying $p$ by a nonzero scalar does not affect the hypersurface\n$\\{p=0\\}$, so we should regard the parameter space of such hypersurfaces\nas forming a $\\cp9$ with homogeneous coordinates $[a_0,\\dots,a_9]$.\n\nThe action of the automorphism group of $\\cp4^{(1,1,1,6,9)}\/G$\nestablishes isomorphisms\namong hypersurfaces defined by different equations.  This automorphism group\nincludes the scaling symmetries $x_j\\mapsto\\lambda_j x_j$,\nwhich induce an action on the coefficients of $p$ of the form\n$a_k\\mapsto \\lambda^{m(k)} a_k$ for  appropriate multi-indices $m(k)$.\nBy using these scaling symmetries, five of the coefficients of $p$ may\nbe set to $1$.\n\nIn the examples studied in  \\cite{\\rUno}, these scaling symmetries\naccounted for the full automorphism group of the ambient space.  This\nis not the case in the present example, however.  There are additional\nautomorphisms of $\\cp4^{(1,1,1,6,9)}\/G$ which arise from the possibility of\nmodifying\nthe homogeneous coordinates of high weight by addition of nonlinear\nterms\ninvolving the homogeneous coordinate of low weight,\nin a $G$-invariant manner. The most general\nautomorphism of this type\ntakes the form\n $$\\eqalign{\nx_1&\\longrightarrow x_1\\cr\nx_2&\\longrightarrow x_2\\cr\nx_3&\\longrightarrow x_3 \\cr\nx_4&\\longrightarrow x_4 + a\\,(x_1x_2x_3)^2\\cr\nx_5&\\longrightarrow x_5 + b\\,(x_1x_2x_3)\\,x_4 +\nc\\,(x_1x_2x_3)^3~.\\cr} \\eqlabel{nilp}$$\nSuch automorphisms should be used to set certain coefficients of $p$\nto zero.  The problem arises of how to select the monomials in $p$\nwhich are to be retained with nonzero coefficients.  A very general\nprocedure for doing this was proposed in~\n\\REFS{\\rMDMM}{P.~S. Aspinwall, B.~R.~Greene and D.~R.~Morrison,\n``The Monomial-Divisor Mirror Map'',\nInt. Math. Res. Notices (1993) 319.}\n\\refsend;\nin the example at hand,\nthat procedure instructs us to attempt to make a transformation\nof the form \\eqref{nilp} which sets $a_1$, $a_2$, and $a_3$ to zero.\nWhen $a_8a_9\\ne0$, this can be done using \\eqref{nilp} with\ncoefficients\n$$\\eqalign{\na =& {-\\xi^2- 4 a_1 a_9   + a_0^2\\over12 a_8 a_9}\\cr\nb =& {\\xi-a_0\\over2a_9}\\cr\nc =& {a_0\\xi^2- 12 a_2 a_8 a_9 + 4 a_0 a_1 a_9  - a_0^3\\over24 a_8 a_9^2},\n}\\eqlabel{nilpcoef}$$\nwhere\n$$ \\xi= \\root4\\of{- 48 a_3 a_8 a_9^2  + 16 a_1^2  a_9^2   + a_0^4\n  + 24 a_0 a_2 a_8 a_9 - 8 a_0^2  a_1 a_9}.$$\nThe fact that these transformations can be found verifies,\nfor this particular example,\nthe ``dominance property'' discussed in \\cite{\\rMDMM}.\n\nApplying \\eqref{nilp} with coefficients \\eqref{nilpcoef}, and then\nusing scaling symmetries, we may reduce \\eqref{generalp} to the form\n $$p=x_1^{18} + x_2^{18} + x_3^{18} + x_4^3 + x_5^2 -\n18\\psi\\, x_1 x_2 x_3 x_4 x_5  - 3\\phi\\, x_1^6 x_2^6 x_3^6\n\\eqlabel{simplep}$$\n(provided that $a_5a_6a_7a_8a_9\\ne0$ after applying \\eqref{nilp}).\nWe have introduced factors of\n $-18$ and $-3$ into \\eqref{simplep}\nfor later convenience; in addition, to\nsimplify some later\nformulas, we sometimes replace $\\psi$ by\n $$\n\\rho\\buildrel\\rm def\\over = (3^42)^{1\/3}\\,\\psi~.$$\n\nThe natural parameter space for polynomials of the form\n\\eqref{simplep} would appear at first sight to be the $\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}^2$ with\ncoordinates $(\\rho,\\psi)$; however, the transformations used to bring\n\\eqref{generalp} to the form \\eqref{simplep} were not unique,\nand this non-uniqueness must now be accounted for.  First, there\nare scaling symmetries which preserve the form of \\eqref{simplep}.\nThese will define an enlargement $\\widehat{G}$ of the group $G$\nconsisting of\nelements $g=(\\alpha^{a_1},\\alpha^{a_2},\\alpha^{a_3},\\alpha^{6a_4},\\alpha^{9a_5})$\nacting as:\n $$(x_1,x_2,x_3,x_4,x_5;\\psi,\\phi)\\mapsto\n(\\alpha^{a_1}x_1,\\alpha^{a_2}x_2,\\alpha^{a_3}x_3,\\alpha^{6a_4}x_4,\\alpha^{9a_5}x_5;\n\\alpha^{-a}\\psi,\\alpha^{-6a}\\phi), $$\nwhere $a=a_1+a_2+a_3+6a_4+9a_5$,\nwhere $\\alpha^{a_i}$, $i=1,2,3$, are 18th roots of unity,\n$\\alpha^{6a_4}$ is a 3rd root of unity ,\nand $\\alpha^{9a_5}$ is a 2nd root of unity\\Footnote{%\nWe do not require that the product of these roots of unity be 1,\nsince we have `corrected'\nthe equation by an appropriate action on the coefficients.}.\nThis group acts on the\n family of weighted projective hypersurfaces $\\{p=0\\}$,\nand induces an action on the parameter\nspace\n$\\{(\\rho,\\phi)\\}$ by a $\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_{18}$ whose generator ${\\cal A}$ acts by\n $$ {\\cal A}:\n(\\rho,\\phi)\\mapsto(\\alpha\\,\\rho,\\alpha^6\\,\\phi) $$\nwhere $\\alpha=e^{2\\pi i\/18}$.\n\nSecond, there are transformations \\eqref{nilp} which preserve the\nform of \\eqref{simplep}.  These transformations are generated by\na transformation $\\ca{I}$\nwhose coefficients are\n $a=54\\psi}\\def\\Ps{\\Psi^2$, $b=9(1-i)\\psi}\\def\\Ps{\\Psi$ and $c=486\\psi}\\def\\Ps{\\Psi$.  The induced action\non the parameter space is\n $$\n\\ca{I}~:~(\\rho}\\def\\vr{\\varrho,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\longrightarrow (i\\rho}\\def\\vr{\\varrho, \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi+\\rho}\\def\\vr{\\varrho^6) . \\eqlabel{Idef}$$\nIt is often convenient to use $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi+{1\\over2}\\rho}\\def\\vr{\\varrho^6$ as a coordinate\nin place of $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$.  For the action of $\\ca{I}$ in such coordinates\nis simply\n$$\n\\ca{I}~:~(\\rho}\\def\\vr{\\varrho,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi+{1\\over2}\\rho}\\def\\vr{\\varrho^6) \\longrightarrow (i\\rho}\\def\\vr{\\varrho, \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi+{1\\over2}\\rho}\\def\\vr{\\varrho^6) .\n$$\n\n It is important to observe\n that the automorphism ${\\cal A}^9\\ca{I}^2$ acts trivially on the\nparameter space.  The corresponding transformation of the\n$x_j$'s is\n$$(x_1,x_2,x_3,x_4,x_5)\\longrightarrow\n(-x_1,x_2,x_3,x_4, x_5 -18\\psi\\,x_1x_2x_3x_4).$$\nThis is an $R$-symmetry, which acts as $-1$ on the holomorphic\n$3$-form of each Calabi--Yau\\ in the family.\nNow it is easy to see that the subgroup of the automorphism group\ngenerated by ${\\cal A}^2$ and $\\ca{I}$ contains all of the actual symmetries\nof the parameter space, and does not contain the $R$-symmetry\n${\\cal A}^9\\ca{I}^2$.  So to construct the moduli space, it would suffice\nto consider the quotient by only the automorphisms from that subgroup.\n\nThe moduli space can  be described explicitly in terms of invariant functions\nfor the actions of ${\\cal A}$ and $\\ca{I}$ on the original parameter space\n$\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}^2=\\{(\\rho,\\phi)\\}$.  First consider the action of\n $${\\cal A}^3:(\\rho,\\phi)\\mapsto(\\alpha^3\\,\\rho,\\phi). $$\nThe invariant functions under this transformation\nare generated by $\\rho^6$ and $\\phi$, and the quotient of the original\n$\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}^2$ by the $\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_6$ generated by\n${\\cal A}^3$ is again a smooth $\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}^2$.\n\nThe action of\nthe transformation ${\\cal A}$ on the invariant functions $\\rho^6$ and $\\phi$\nis via\n $${\\cal A}:(\\rho^6,\\phi)\\mapsto(\\alpha^6\\,\\rho^6,\\alpha^6\\,\\phi), $$\nand this generates a $\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_3$.  The quotient space, which we call the\n``simplified moduli space'' using the terminology of \\cite{\\rMDMM},\n has a singularity at\nthe origin.  To obtain the actual moduli space, we would need to\nquotient this ``simplified'' space by the automorphism $\\ca{I}$.\n\nWe wish to compactify the moduli space in order to study the monodromy\naround boundary divisors.  We will work primarily with compactifications\nof the simplified moduli space, since these can be analyzed using the\nmethods of toric geometry.  Compactifications of the actual moduli space\ncan be obtained by taking the quotient by $\\ca{I}$ of our ``simplified''\ncompactifications.\n\nAn initial compactification of the simplified moduli space can be made\nonce we recognize that the $\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi_3$-quotient singularity in this space\nis precisely the same as the one in the weighted projective plane\n$\\cp2^{(3,1,1)}$.  A compactification can then be made by\nassociating to $(\\rho^6,\\phi)$ the point in $\\cp2^{(3,1,1)}$ with\nhomogeneous coordinates $[1,\\rho^6,\\phi]$.  Other representatives of\nthe same point (when $\\rho$ or $\\phi$ is not zero) are\n$[\\rho^{-18},1,\\rho^{-6}\\phi]$ and $[\\phi^{-3},\\rho^6\\phi^{-1},1]$.\nWe let $[x,y,z]$ denote the general point, in homogeneous\ncoordinates.\n\n\nThe curves in $\\cp2^{(3,1,1)}$ which represent singular Calabi-Yau\nspaces (and so constitute the boundary of the simplified moduli space) are as\nfollows:\n\\vskip10pt\n\\item{1.~~}\n$C_{\\rm con}$---a locus on which the Calabi-Yau acquires a conifold\npoint---described in affine coordinates as\n $$C_{\\rm con}=\\{(\\rho,\\phi)\\ |\\ (\\rho^6+\\phi)^3=1\\} $$\nor in projective\ncoordinates by its homogeneous equation $(y+z)^3=x$;\n\\bigskip\n\\item{2.~~}\n$B_{\\rm con}$---another locus on which the Calabi-Yau acquires a\nconifold\npoint---described in affine coordinates as\n $$B_{\\rm con}=\\{(\\rho,\\phi)\\ |\\ \\phi^3=1\\} $$\nor in projective coordinates by its homogeneous equation $z^3=x$. The\nloci\n$B_{\\rm con}$ and $C_{\\rm con}$ are interchanged under the action of\n$\\ca{I}$.\n\\bigskip\n\\item{3.~~}\n$D_{\\infty}$---the boundary, where $(\\rho}\\def\\vr{\\varrho,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\to\\infty$,\nof the original $(\\rho,\\phi)$\nspace---defined\nby the homogeneous equation $x=0$; and\n\\bigskip\n\\item{4.~~}\n$D_0$ (the fixed point set of $\\ca{A}^3$)---the Calabi-Yau spaces\ncorresponding\nto which will acquire additional singularities during the quotient by\nthe enlarged group $\\widehat{G}$---defined by the affine equation\n$\\rho=0$ or the homogeneous equation $y=0$.\n\n\\noindent These meet in the following points:\n\\vskip10pt\n\\item{$\\bullet$~~}\n$[1,0,1]$, the common point of intersection of $D_0$, $B_{\\rm con}$,\nand\n$C_{\\rm con}$,\n\n\\item{$\\bullet$~~}\n$P_+=[1,\\alpha^6{-}1,1]$ and $P_-=[1,\\alpha^{-6}{-}1,1]$, the two points of\nintersection\n of $C_{\\rm con}$ and $B_{\\rm con}$\nthrough which $D_0$ does not pass,\n\n\\item{$\\bullet$~~}\n$[0,-1,1]$, the point of triple tangency between $C_{\\rm con}$ and\n$D_{\\infty}$,\n\n\\item{$\\bullet$~~}\n$[0,1,0]$, the point of triple tangency between $B_{\\rm con}$ and\n$D_{\\infty}$,\nand\n\n\\item{$\\bullet$~~}\n$[0,0,1]$, the point of intersection of $D_{\\infty}$ and $D_0$.\n\n\n\\vskip10pt\n\\noindent\nAlso of interest is the point $P_0=[1,0,0]$, which is the singular\npoint of $\\cp2^{(3,1,1)}$.\n\nWe sketch the curves showing their intersections in\nFig.~\\figref{modulispace}.  Note that the points of triple tangency\nare depicted as simple tangencies in the diagram.\n %\n\\iffigs\n\\midinsert\n\\def\\hbox{\\epsfbox{fig1.ps}}{\\hbox{\\epsfbox{fig1.ps}}}\n\\figbox{{\\vbox{\\kern0.1truein{\\hbox{\\kern.05truein\\hbox{\\epsfbox{fig1.ps}}}}}}}\n{\\figlabel{modulispace}}{A first sketch of the simplified\nmoduli space showing\nthe\ncomponents of the discriminant locus}\n\\place{5.8}{2}{$\\twelvemath D_\\infty$}\n\\place{3.4}{1.1}{$\\twelvemath D_0$}\n\\place{4.5}{3.2}{$\\twelvemath C_{con}$}\n\\place{1.8}{3.2}{$\\twelvemath B_{con}$}\n\\place{3.2}{3.45}{$\\twelvemath P_-$}\n\\place{3.2}{4.5}{$\\twelvemath P_+$}\n\\place{3.235}{1.45}{$\\twelvemath \\bullet$}\n\\place{2.9}{1.45}{$\\twelvemath P_0$}\n\\endinsert\n\\fi\n %\n\\subsection{Considerations of toric geometry}\nTo describe $\\cp2^{(3,1,1)}$ as a toric variety,\nwe consider first the smooth affine chart whose points are represented by\nhomogeneous coordinates $[\\phi^{-3},\\rho^6\\phi^{-1}]$.  The functions\n $\\phi^{-3}$ and $\\rho^6\\phi^{-1}$ furnish coordinates in this chart,\nand among all rational monomials  $(\\phi^{-3})^a(\\rho^6\\phi^{-1})^b$,\nthe ones which are holomorphic in this first\nchart satisfy\n$a{\\ge}0$, $b{\\ge}0$.  In the other two charts, the corresponding\nconditions\nare\n$a{\\ge}0$, $-3a-b{\\ge}0$ (for the other smooth chart, with\ncoordinates\n$\\rho^{-18}$ and $\\rho^{-6}\\phi$),\nand $b{\\ge}0$, $-3a-b{\\ge}0$ (for the singular chart\ndescribed\nin\nterms of $\\rho^6$ and $\\phi$). The resulting toric diagram is shown\nin\nFig.~\\figref{figA}.\nNote that the vectors $(1,0)$ and $(0,1)$ in the toric diagram\ncorrespond to the divisors $D_\\infty$ and $D_0$, respectively.\n(The divisor corresponding to the vector $(-3,-1)$ does not lie\non the boundary of the moduli space.)\n %\n\\iffigs\n\\midinsert\n\\def\\hbox{\\def\\epsfsize##1##2{0.8##1}\\epsfbox{figA.ps}}{\\hbox{\\def\\epsfsize##1##2{0.8##1}\\epsfbox{figA.ps}}}\n\\figbox{{\\vbox{\\kern0.2truein{\\hbox{\\kern0truein\\hbox{\\def\\epsfsize##1##2{0.8##1}\\epsfbox{figA.ps}}}}}}}\n{\\figlabel{figA}}{Toric diagram for $\\cp2^{(3,1,1)}$.}\n\\endinsert\n\\fi\n %\nAn alternative compactification---the\n compactification of the simplified\nmoduli space described by the ``secondary\nfan''~\n\\REFS{\\rOP}{T. Oda and H.~S. Park, T\\^ohoku Math. J. {\\bf 43} (1991)\n375.}\n\\REFSCON{\\rBFS}{L.J. Billera, P. Filliman, and B. Sturmfels, Adv.\nMath.\n{\\bf 83} (1990) 155.}\\refsend---%\nis well-adapted for rapidly locating the large complex structure\nlimit point(s) (using the ``monomial-divisor mirror map''---cf.~\n\\REF{\\rAGM}{P.~S.~Aspinwall, B.~R.~Greene and D.~R.~Morrison,\n \\plb{303} (1993) 249; \\npb{416} (1994) 414.}\n\\cite{\\rAGM,\\rMDMM}.)\nFirst, the large radius limit points of the mirror moduli space are\nlocated, and then mirror symmetry is used to identify the\ncorresponding  large complex structure limit points.\nThe computation proceeds as follows.\n\nWe give a toric description of the desingularization of \\hbox{$\\IP^{(1,1,1,6,9)}$}\\\n(using the embedding\n $$\n(\\tau_1,\\tau_2,\\tau_3,\\tau_4)\\mapsto [1,\\tau_1,\\tau_2,\\tau_3,\\tau_4]~ $$\nof the  torus $T=(\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}^*)^4$ into \\hbox{$\\IP^{(1,1,1,6,9)}$}) by means of a fan\nin $N_\\relax{\\rm I\\kern-.18em R}={\\rm Hom}(\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$} ^*,T)\\otimes\\relax{\\rm I\\kern-.18em R}$.  The one-dimensional cones in\nthis fan are spanned by the vectors\n $$\\matrix{v_1 =&(  -1,&  -1,&  -6,&  -9)\\cropen{3pt}\n          v_2 =&(\\- 1,&\\- 0,&\\- 0,&\\- 0)\\cropen{3pt}\n          v_3 =&(\\- 0,&\\- 1,&\\- 0,&\\- 0)\\cropen{3pt}\n          v_4 =&(\\- 0,&\\- 0,&\\- 1,&\\- 0)\\cropen{3pt}\n          v_5 =&(\\- 0,&\\- 0,&\\- 0,&\\- 1)\\cropen{3pt}\n          v_6 =&(\\- 0,&\\- 0,&  -2,&  -3)\\cr} $$\nwhich are ordered so that,\nunder the identification of edges in the fan with the ``toric''\ndivisors\nin the toric variety,\n the first five vectors $v_i$, $i=1,\\dots,5$\ncorrespond to the proper transforms\nof $x_i=0$ and the last vector $v_6$\ncorresponds to the exceptional divisor.  (Note that $v_6$ is the\naverage of $v_1$, $v_2$, $v_3$; this corresponds to the\nfact that $x_1=x_2=x_3=0$ has been blown up.)\nThe `big', {\\it i.e.\\ \\\/} top dimensional, cones which describe the blown-up \\hbox{$\\IP^{(1,1,1,6,9)}$}\\\nare:\n $$\n\\eqalign{\n\\mathop{\\rm span}\\nolimits\\{v_1,\\dots,\\widehat{v_i},\\dots,v_5\\}\n\\qquad &\\hbox{for}\\ i=1,2,3\\cr\n\\mathop{\\rm span}\\nolimits\\{v_1,\\dots,\\widehat{v_i},\\dots,\n\\widehat{v_j},\\dots,v_5\\}\n\\qquad &\\hbox{for}\\ i=1,2,3\\ \\hbox{and}\\ j=4,5~.\\cr} $$\n\nTo compute the secondary fan, we need to find a basis for the set of\nall relations $a_1v_1+\\cdots+a_6v_6=0$.  A convenient basis is\nfurnished\nby the rows of\n $$\n\\pmatrix{\n1 &\\- 1 &\\- 1 &\\- 0 &\\- 0 &  -3 \\cr\n0 &\\- 0 &\\- 0 &  -2 &  -3 &  -1 \\cr}\n\\eqlabel{secondaryfan} $$\nThe one-dimensional cones in the {\\sl secondary} fan are then spanned\nby the columns of the matrix \\eqref{secondaryfan}, together with an\nadditional column $\\left({0\\atop 6}\\right)$ whose entries take the\nform\n$-\\sum a_j$ for each relation $\\sum a_jv_j$.  (We may regard this\nadditional column as corresponding to the zero-vector\n$v_0=(0,0,0,0)$.)\nWe may as well take the secondary fan to be spanned by the\nvectors\n $$\n\\pmatrix{1\\cr 0}~,~~~~\\pmatrix{\\- 0\\cr -1}~,~~~~\\pmatrix{-3\\cr\n-1}~,~~~~\n\\pmatrix{0 \\cr 1}~; $$\nthis is illustrated in Fig.~\\figref{figB}.\n %\n\\iffigs\n\\midinsert\n\\def\\hbox{\\def\\epsfsize##1##2{0.8##1}\\epsfbox{figB.ps}}{\\hbox{\\def\\epsfsize##1##2{0.8##1}\\epsfbox{figB.ps}}}\n\\figbox{{\\vbox{\\kern0.2truein{\\hbox{\\kern0truein\\hbox{\\def\\epsfsize##1##2{0.8##1}\\epsfbox{figB.ps}}}}}}}\n{\\figlabel{figB}}{The secondary fan.}\n\\endinsert\n\\fi\n %\nWe see that the divisor $L$ corresponds to $\\left({1\\atop 0}\\right)$,\nthe divisor $E$ to $\\left({-3\\atop -1}\\right)$, and so the divisor\n$H$ to\n$\\left({\\- 0\\atop -1}\\right)$.  It follows that the K\\\"ahler cone\ncorresponds\nto the fourth quadrant.\n\nThis same secondary fan now determines a compactification of the\n(simplified) complex structure moduli space of the mirror.\nUnder the monomial-divisor mirror map \\cite{{\\rAGM,\\rMDMM}},  the\ndivisor determined by the vector $v_j$ corresponds to the monomial\nwith coefficient $c_j$ in the general polynomial\n $$\nc_1\\,x_1^{18} + c_2\\,x_2^{18} + c_3\\,x_3^{18} + c_4\\,x_4^3 +\nc_5\\,x_5^2\n+c_6\\, x_1^6 x_2^6 x_3^6\n+c_0\\, x_1 x_2 x_3 x_4 x_5 .\n $$\nOn the one hand, using the torus action\nto get this polynomial into the form \\eqref{simplep}:\n $$\\eqalign{\nx_1^{18} + x_2^{18} + x_3^{18} + x_4^3 + x_5^2\n&+  c_0c_1^{-1\/18}c_2^{-1\/18}c_3^{-1\/18}c_4^{-1\/3}c_5^{-1\/2}\\,\nx_1 x_2 x_3 x_4 x_5\\cr\n&+ c_1^{-1\/3}c_2^{-1\/3}c_3^{-1\/3}c_6\\, x_1^6 x_2^6 x_3^6\\cr}\n $$\nwe see that\n $$\\eqalign{\n-18\\psi\n&=c_0c_1^{-1\/18}c_2^{-1\/18}c_3^{-1\/18}c_4^{-1\/3}c_5^{-1\/2}~,\\cr\n -3\\phi &=c_1^{-1\/3}c_2^{-1\/3}c_3^{-1\/3}c_6~.\\cr} $$\nOn the other hand, the vectors $\\left({1\\atop 0}\\right)$ and\n$\\left({0\\atop -1}\\right)$ are the edges of the fourth quadrant\nin the diagram, which is the mirror of the K\\\"ahler cone and\nhence should correspond to a large complex structure limit point.\n(We will verify later that this point satisfies the appropriate\nmonodromy conditions.)\nTo find the coordinates near that point, we note that\n$\\left({1\\atop 0}\\right)$ and\n$\\left({0\\atop -2}\\right)$\ncorrespond to the monomials $x_1^{18}$ and $x_4^3$, respectively,\nand we use the torus action to put the polynomial into the form:\n$$\nc_1c_2c_3c_6^{-3}x_1^{18} +\nx_2^{18}+x_3^{18} +\nc_0^{-3}c_4c_5^{3\/2}c_6^{1\/2}x_4^3 +\nx_5^2+x_1^6x_2^6x_3^6+x_1x_2x_3x_4x_5~.\n$$\nThe coordinates near the large complex structure limit point are\nthen given by\n$$ \\eqalign{\n(c_1c_2c_3c_6^{-3},[c_0^{-3}c_4c_5^{3\/2}c_6^{1\/2}]^2)\n&=((-3\\phi)^{-3},(-18\\psi)^{-6}(-3\\phi)) \\cr\n&=(-3^{-3}\\phi^{-3},-2^{-4}3^{-3}\\rho^{-6}\\phi)~.\n} $$\n\nTo relate these coordinates to the toric description of $\\cp2^{(3,1,1)}$\ngiven above,\nwe use the torus action one final time to put the polynomial\ninto the form\n $$\nc_0^{-18}c_1c_2c_3c_4^{6}c_5^{9}\n\\,x_1^{18} + x_2^{18} + x_3^{18} + x_4^3 + x_5^2\n+ c_0^{-6}c_4^{2}c_5^{3}c_6\\, x_1^6 x_2^6 x_3^6\n+ x_1 x_2 x_3 x_4 x_5~.\n $$\nThis time, the remaining monomials $x_1^{18}$ and $x_1^6 x_2^6\nx_3^6$\ncorrespond to the vectors $\\left({1\\atop 0}\\right)$ and\n$\\left({-3\\atop -1}\\right)$ in the secondary fan---precisely the same\nvectors as in the toric diagram for $\\cp2^{(3,1,1)}$.\nMoreover, the corresponding coordinates\n$$(c_0^{-18}c_1c_2c_3c_4^{6}c_5^{9},c_0^{-6}c_4^{2}c_5^{3}c_6)\n=(2^{-12}3^{-12}\\rho^{-18},-2^{-4}3^{-3}\\rho^{-6}\\phi)$$\nagree with those of $\\cp2^{(3,1,1)}$ up to some irrelevant constants.\n\n\\iffigs\n\\midinsert\n %\n\\def\\hbox{\\epsfbox{fig2.ps}}{\\hbox{\\epsfbox{fig2.ps}}}\n\\figbox{{\\vbox{\\kern0.1truein{\\hbox{\\kern.05truein\\hbox{\\epsfbox{fig2.ps}}}}}}}\n{\\figlabel{Modulispace}}{The simplified\nmoduli space resolved so that all the\ncomponents\nof the discriminant locus have normal crossings.}\n\\place{5.8}{1.8}{$\\twelvemath D_\\infty$}\n\\place{3.3}{1.3}{$\\twelvemath D_0$}\n\\place{4.5}{3.2}{$\\twelvemath C_{con}$}\n\\place{1.8}{3.2}{$\\twelvemath B_{con}$}\n\\place{3.2}{3.45}{$\\twelvemath P_-$}\n\\place{3.2}{4.5}{$\\twelvemath P_+$}\n\\place{3.9}{2.15}{$\\twelvemath E_3$}\n\\place{1.45}{1.45}{$\\twelvemath F_1$}\n\\place{4.9}{1.45}{$\\twelvemath F_2$}\n\\place{1.9}{1.2}{$\\twelvemath E_1$}\n\\place{4.5}{1.2}{$\\twelvemath E_2$}\n\\place{2.2}{1.2}{$\\twelvemath G_1$}\n\\place{4.2}{1.2}{$\\twelvemath G_2$}\n\\place{3.6}{1.05}{$\\twelvemath E_0$}\n\\endinsert %\n\\fi\n\n\\subsection{The resolved moduli space}\nIn order to search for possible additional large complex structure\nlimit points, we need\na compactification of the simplified moduli space in which the\nboundary is a divisor with normal crossings.  This can be constructed\nby blowing up the original $\\cp2^{(3,1,1)}$ compactification.\nWe first do toric blowups.  The singular point $P_0$ of $\\cp2^{(3,1,1)}$\ncan be resolved by simply blowing it up; this introduces the vector\n$(-1,0)$ into the toric diagram, Fig.~\\figref{figA}.  We denote\nthe corresponding exceptional divisor by $E_0$.\n\nThe point of intersection of $B_{\\rm con}$ and $D_\\infty$ can then be\nmade into a normal crossings point by additional\ntoric blowups.  We need three such\nblowups to reach normal crossings:  the first has exceptional divisor\n$E_1$ corresponding to the toric vector $(-2,-1)$, the second\nhas exceptional divisor $F_1$ corresponding to the toric vector\n$(-1,-1)$, and the third has exceptional divisor $G_1$ corresponding\nto the toric vector $(0,-1)$.  Since this last vector also\noccurs in the secondary fan, we see that the large complex structure\nlimit point found in the previous subsection lies at the intersection\nof $G_1$ and $D_\\infty$.\n\nThe remaining blowups are non-toric.  There are three blowups to be made\nat the intersection of $C_{\\rm con}$ and $D_\\infty$, leading to\nexceptional divisors $E_2$, $F_2$, and $G_2$.  The final blowup needed\nis of the point $[1,0,1]$ lying at the intersection of $D_0$,\n$B_{\\rm con}$ and $C_{\\rm con}$; the exceptional divisor for this\nblowup is denoted by $E_3$.\n\nThe resolved simplified\nmoduli space is sketched in Fig.~\\figref{Modulispace}.\nThis space is invariant under the action of $\\ca{I}$,\nwhich exchanges $B_{\\rm con}$, $E_1$, $F_1$, and $G_1$ with\n$C_{\\rm con}$, $E_2$, $F_2$, and $G_2$, respectively.\nThe intersection of $G_2$\nand $D_\\infty$ must therefore also be a large complex structure limit point.\n\n\\vfill\\eject\n %\n\\section{vps}{The Periods}\n\\vskip-20pt\n\\subsection{The fundamental period}\nWe take for the holomorphic three--form the quantity\n$$\n\\O = \\psi\\,  {- 2^33^5 \\over (2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i)^3}\\,\n{x_5 dx_1dx_2dx_3 \\over (3x_4^2-18\\psi x_1x_2x_3x_5)}~.$$\n(The expression $3x_4^2-18\\psi x_1x_2x_3x_5$ in the denominator\narises from $\\pd{p}{x_4}$,\nwith $p$ as in \\eqref{simplep}.)\nThe numerical factor has been introduced to simplify later expressions;\nthe factor of $\\psi}\\def\\Ps{\\Psi$ ensures that $\\O$ is\ninvariant under\nthe extended group $\\widehat{G}$.\nThis holomorphic three--form $\\O$ is not invariant under $\\ca{I}$,\nand indeed it is impossible to find a holomorphic three--form which\nis invariant under both $\\ca{I}$ and $\\widehat{G}$, since the\ntransformation ${\\cal A}^9\\ca{I}^2$ is an $R$-symmetry.  However,\nwe {\\it can}\\\/ find a holomorphic three--form which is invariant under\nboth  ${\\cal A}^2$ and $\\ca{I}$; we take it to be\n$$\\widehat{\\O}\\buildrel\\rm def\\over = \\rho^3(\\phi+{1\\over2}\\rho^6)\\,\\O~.$$\nThis one will determine a holomorphic three--form on the actual moduli space\n(which is the quotient of the $(\\rho,\\phi)$ parameter space by\n${\\cal A}^2$ and $\\ca{I}$).\n\n\nA fundamental period\n$\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$\nof the holomorphic 3-form $\\O$\ncan be found by direct integration as explained in~\n\\REFS{\\rperiods}{P.~Berglund, P.~Candelas, X.~de~la~Ossa, A.~Font,\nT.~H\\\"ubsch, D.~Jan\\v{c}i\\'c and F.~Quevedo, ``Periods for Calabi--Yau\\ and\nLandau-Ginzburg Vacua'', preprint\nCERN TH. 6865\/93, HUPAPP-93\/3, NEIP 93-004, NSF-ITP-93-96,\nUTTG-13-93,\nhepth-9308005.}\n\\refsend.\n(Properties of the corresponding fundamental period\n$$\\widehat{\\vp}_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)=\\rho^3(\\phi+{1\\over2}\\rho^6)\\,\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$$\nof the $3$-form\n $\\widehat{\\O}$ can be deduced from a study of $\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$.)\nIn our case we find the expansion\n $$\n\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = \\sum_{n,m = 0}^\\infty\n{(18n+6m)!\\, (-3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^m \\over  (9n+3m)! \\,\n(6n+2m)! \\, (n!)^3\\, m!\\,\n(18\\psi}\\def\\Ps{\\Psi)^{18n+6m}} ~,\\eqlabel{pi0} $$\nwhich converges for sufficiently large $\\psi}\\def\\Ps{\\Psi$.\nA useful form may be obtained by setting $k=3n+m$\nin the sum and summing over $k$ and $n$. The fundamental period can\nthen be\nrewritten in the form\n $$\\eqalign{\n\\vp_0(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi,\\psi}\\def\\Ps{\\Psi)\n&=\\sum_{k=0}^\\infty \\ {{(6k)!}\\over {k! (2k)! (3k)!}}\n\\left(-{3\\over {18^6 \\psi}\\def\\Ps{\\Psi^6}} \\right)^k U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\cr\\noalign{\\medskip}\n&= {1\\over {2\\pi}} \\sum_{k=0}^\\infty \\ { {(-1)^k \\G(k + {1\\over 6})\n\\G(k + {5\\over 6})}\\over {(k!)^2} } \\rho}\\def\\vr{\\varrho^{-6k} U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\cr}\n\\eqlabel{npi} $$\nwhere\n $$\nU_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^k \\,\n\\sum_{n=0}^{\\left[ {k\\over 3}\\right]}\n{ {(-1)^n k!}\\over {(n!)^3 \\G(k-3n+1) (3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{3n}} }\n\\eqlabel{Ukdef} $$\nThe function $U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ is a polynomial of degree\n$k$ but we shall need to extend its definition to\ncomplex values $\\nu$ of $k$. To this end note that the sum\nin \\eqref{Ukdef} can be taken run to $\\infty$ since the factor\n$1\/\\G(k-3n+1)$ vanishes automatically for\n$n>\\left[ {k\\over 3} \\right]$ so we take\n $$\n\\eqalign{\nU_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)&= \\G(\\nu+1)\\,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^\\nu\n\\sum_{m=0}^\\infty  {(-1)^m \\over\\G(\\nu-3m+1)\\, (m!)^3\\,(3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{3m}}\n{}~~,~~|\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3|>1 {}~~,~~0<\\arg\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi<2\\pi\/3 \\cr\\noalign{\\medskip} &=\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^\\nu \\,\n\\FT{-{\\nu\\over 3}}{{{1-\\nu}\\over 3}} {{{2-\\nu}\\over 3}}{1}{1}{\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^{-3}}\n{}~.\\cr}\\eqlabel{uhl} $$\nThe second equality follows by use of the\nmultiplication formula for the  $\\G$-function.\n\nA set of linearly independent periods can be chosen from among\nfrom the functions\n $$\n\\vp_j(\\psi}\\def\\Ps{\\Psi, \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = \\vp_0(\\alpha^j \\psi}\\def\\Ps{\\Psi, \\alpha^{6j}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\quad ; \\quad\nj=0, \\cdots, 17   \\eqlabel{vpj} $$\nTo find explicit expressions for these periods\nwe must first extend $\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$\nto small $\\psi}\\def\\Ps{\\Psi$. This can be done by writing either expansion\nin \\eqref{npi} as a contour integral and then deforming\nthe contour appropriately. The result is\n $$\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = -{1\\over 6} \\sum_{n = 1}^\\infty \\\n{ { \\G({n\\over 6})\n(- 3^{{11\\over 6}} \\, 2 \\psi}\\def\\Ps{\\Psi)^n } \\over\n{\\G(n) \\G(1 - {n\\over 3}) \\G(1 - {n\\over 2}) } } \\\nU_{-{n\\over 6}}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\eqlabel{api}~. $$\nDue to the factors $\\G(1 - {n\\over 3})$ and $\\G(1 - {n\\over 2})$\nin the denominator\nthe summation index actually runs over $n=6k + r$, $r=1,5$.\nHence,\n $$\\vp_0(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = {1\\over {3\\pi}}\n\\sum_{r=1,5} \\sin {{\\pi r}\\over 3} \\ \\sum_{k = 0}^\\infty \\\n{ {(-1)^k \\G^2(k+{r\\over 6}) }\\over\n{\\G(k+1) \\G(k+{r\\over 3})} }\\, \\rho}\\def\\vr{\\varrho^{6k+r}\nU_{-(k+{r\\over 6})}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\eqlabel{pir} $$\n\nThe basis periods are then given by\n $$\\vp_{3a+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = {1\\over {3\\pi}}\n\\sum_{r=1,5} \\alpha^{3ar} \\sin {{\\pi r}\\over 3} \\\n\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma (\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)  \\quad ; \\quad a=0,1~~\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0,1~, \\eqlabel{vpjr} $$\nwhere\n $${\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma} =\\sum_{k = 0}^\\infty \\\n{ {(-1)^k \\G^2(k+{r\\over 6}) }\\over\n{\\G(k+1) \\G(k+{r\\over 3})} }\\,\\rho}\\def\\vr{\\varrho^{6k+r}\\, U^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma_{-(k+{r\\over\n6})}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~~~,\n{}~~~\\left|{\\rho}\\def\\vr{\\varrho^6\\over \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-\\omega}\\def\\O{\\Omega^{-\\tau}}\\right|<1~,\n\\eqlabel{zrs} $$\nand we have also defined\n $$\nU_\\n^\\s(\\ph) = \\omega}\\def\\O{\\Omega^{-\\nu\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}\\, U_\\nu(\\omega}\\def\\O{\\Omega^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~,~~\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0,1,2 ~, \\eqlabel{unus}$$\nwith $\\omega}\\def\\O{\\Omega=e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\/3}$.\nThese latter functions are initially only defined in an angular sector\nof the $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$-plane, but we extend them by analytic continuation\nthroughout a cut\n$\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$-plane  with branch\ncuts chosen to run out radially from the cube roots\nof unity.\n\nThe above expressions imply relations among the $\\vp_j$.\nIt is straightforward to see that\n $$\\eqalign{\n\\vp_j-\\vp_{j+3}+\\vp_{j+6}&=0   \\cr\n        \\vp_j + \\vp_{j+9}&=0 ~.\\cr }\n\\eqlabel{vrels} $$\nAs expected, only six of the $\\vp_j$ are linearly independent.\nthese may be chosen to be the first five and we introduce the vector\n $$\n\\vp = \\vpv $$\n\nIt is worth recording also the fact that the six functions\n$\\vp_j(\\psi}\\def\\Ps{\\Psi,0)~,~j=0,\\ldots,5,$ remain linearly independent even when\n$\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=0$.\nNear $\\psi}\\def\\Ps{\\Psi = \\infty$ these are linear combinations of the six\nfunctions\n $$\n1~,~\\log\\psi}\\def\\Ps{\\Psi~,~\\log^2\\psi}\\def\\Ps{\\Psi~,~\\log^3\\psi}\\def\\Ps{\\Psi~,~\\psi}\\def\\Ps{\\Psi^{-6}~,~\\psi}\\def\\Ps{\\Psi^{-12}~. $$\nNear $\\psi}\\def\\Ps{\\Psi=\\psi}\\def\\Ps{\\Psi_0=(3^4 2)^{-1\/3}$, there are five analytic\ncombinations plus one with leading singular behavior\n$$g(\\psi}\\def\\Ps{\\Psi) \\log (\\psi}\\def\\Ps{\\Psi - \\psi}\\def\\Ps{\\Psi_0) \\quad ,$$\nwhere $g(\\psi}\\def\\Ps{\\Psi) = (\\psi}\\def\\Ps{\\Psi - \\psi}\\def\\Ps{\\Psi_0) + \\cdots$, is itself a period.\n\\subsection{The function $U_\\nu$}\nThe differential equation satisfied by $U_{\\nu}$ follows from\nthe general form of the hypergeometric equation of third order.\nWe find\n $$\n(1-\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3) {{d^3{U_\\nu}}\\over {d\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3}} +\n3(\\nu-1) \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2{{d^2{U_\\nu}}\\over {d\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2}}\n- (3\\nu^2 - 3\\nu + 1) \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi {{d{U_\\nu}}\\over {d\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}} + \\nu^3 {U_\\nu} = 0~.\n\\eqlabel{uneq} $$\nIn order to find a basis of solutions we first write a series for\n$U_\\n(\\ph)$ that\nconverges near $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=0$. This is accomplished by writing the sum in\n\\eqref{uhl}\nas a Barnes' integral and continuing to small $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$:  $$\nU_{\\nu}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) =\n {3^{-1-\\nu} \\omega}\\def\\O{\\Omega^{\\nu\\over 2} \\over \\G(-\\nu) }\n\\sum_{m = 0}^\\infty { \\G({m-\\nu \\over 3})\\, (3\\omega}\\def\\O{\\Omega\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^m  \\over\n\\G^2(1 - {m-\\nu \\over 3}) \\,m! }~~,~~|\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3|<1~. \\eqlabel{smallphi} $$\nWe note in passing some useful properties of $U_{\\nu}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ that\nfollow\nimmediately from this series\n\n\\noindent{$\\bullet$\\ Values at $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=0$}\n $$U_{\\nu}(0) = {{2\\pi}\\over {\\sqrt3}} e^{i\\pi \\nu\/3} \\,\n{1\\over { \\G^2(1 + {\\nu\\over 3}) \\G({{1-\\nu}\\over 3})\n\\G({{2-\\nu}\\over 3})} } \\eqlabel{Un0} $$\nHence, for $n=0,1,2, \\cdots$,\n $$ U_{3n}(0) = { {(-1)^n (3n)!}\\over {3^{3n} (n!)^3}\n}~~~\\hbox{and}~~~\nU_{3n+1}(0) = U_{3n+2}(0) = U_{-(3n+3)}(0) = 0~.\n\\eqlabel{U3n0} $$\n\n\\noindent {$\\bullet$\\ Recurrence relation}\n $${{dU_{\\nu}}\\over {d\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}} = \\nu U_{\\nu -1} \\eqlabel{uder} $$\n\nA set of three solutions to the differential equation is given\nby the functions $U_\\n^\\s(\\ph)$ defined in \\eqref{unus}.\nThe Wronskian of these solutions is\n $$\n\\hbox{Wr}[U_\\nu^0,U_\\nu^1,U_\\nu^2] = -{27i\\over 2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^3}\ne^{-i\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\nu}\\sin^2(\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\nu)\\,\n(1-\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3)^{\\nu-1} $$\nfrom which we see that these solutions are linearly independent\nexcept at the\nintegers where the Wronskian has a double zero. We shall be concerned\nwith\nfinding a basis of solutions that remains linearly independent even\nat the\nintegers. First however it is useful to note that near $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=1$ there\nis a\nmultivalued solution of the form\n $$\ny_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)=  -{\\sqrt{3} \\over 2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\,(\\nu+1)} (\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-1)^{\\nu+1}\\,\n\\{ 1+ \\ca{O}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-1)\\} + \\hbox{analytic} \\eqlabel{ydef}$$\nthe prefactor having been chosen so as to simplify later expressions.\nThere are two other solutions that are single valued and we may\ncomplete the specification of $y_\\nu$ by requiring that it be single\nvalued in\nneighborhoods of $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=\\omega}\\def\\O{\\Omega$ and $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=\\omega}\\def\\O{\\Omega^2$. We set also\n $$\ny_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)=\\omega}\\def\\O{\\Omega^{-\\nu\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}\\, y_\\nu(\\omega}\\def\\O{\\Omega^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\eqlabel{ynus}$$\nwhich are multivalued about the points $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=\\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}$ but single\nvalued at the\nother cube roots of unity. Near $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=\\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}$ we have the asymptotic\nbehavior\n $$\ny_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\sim  -{\\sqrt{3} \\over 2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\,(\\nu+1)} \\,\n\\omega}\\def\\O{\\Omega^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma (\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi - \\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma})^{\\nu+1} + \\hbox{analytic}~. $$\n\nThe solution $U_\\n(\\ph)$ can be expressed in terms of the $y_\\n^\\s(\\ph)$:\n $$\nU_\\n(\\ph) = \\sum_{\\tau=0}^2 \\gamma}\\def\\G{\\Gamma_\\nu^\\tau\\,y_\\nu^\\tau(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) $$\nin terms of certain coefficients $\\gamma}\\def\\G{\\Gamma_\\nu^\\tau$ which may be computed\nfrom the\nlarge $m$ behavior\nof the series \\eqref{smallphi}. In this series we observe that\n $$\\eqalign{\n{\\G({m-\\nu \\over 3}) \\over \\G^2(1 - {m-\\nu \\over 3})} &=\n{1 \\over \\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^2} \\G^3\\left({\\textstyle {m-\\nu \\over 3}}\\right)\\,\n\\sin^2\\left({\\textstyle {m-\\nu \\over 3}}\\right) \\cr\n&\\sim -{3^{{3\\over 2} - m +\\nu} \\over 2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi}\\,\\G(m-\\nu-1)\n\\Big(\\omega}\\def\\O{\\Omega^{m-\\nu} - 2 + \\omega}\\def\\O{\\Omega^{-m+\\nu} \\Big) \\Big[1 + \\ca{O}(m^{-1})\n\\Big]~.\\cr} $$\nSubstituting the leading term into the series \\eqref{smallphi} we\nobtain\nlinear combinations of the binomial series for the quantities\n$(\\omega}\\def\\O{\\Omega^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma - \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{\\nu+1}~,~\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0,1,2$. Remembering that we have defined the\nfunctions in\nthe $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$-plane with cuts that run out radially from the cube roots\nof unity we\nhave\n $$\n(1 - \\omega}\\def\\O{\\Omega^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{\\nu + 1} =\n-e^{-i\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\nu}\\omega}\\def\\O{\\Omega^{(\\nu + 1)\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}\\, (\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi - \\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma})^{\\nu + 1}~~,~~\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0,1,2~.\n$$\nIn this way we see that near the points $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3=1$ we have the\nasymptotic\nbehavior\n $$\nU_\\n(\\ph) \\sim -{\\sqrt{3} \\over 2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\,(\\nu+1)}\n\\Big\\{ (\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-1)^{\\nu+1} - 2\\omega}\\def\\O{\\Omega(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-\\omega}\\def\\O{\\Omega^{-1})^{\\nu+1} +\n\\omega}\\def\\O{\\Omega^2(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-\\omega}\\def\\O{\\Omega^{-2})^{\\nu+1}\n\\Big\\}~. $$\nThus we find\n $$\nU_\\n(\\ph) = y_\\nu^0 - 2y_\\nu^1 + y_\\nu^2 $$\nfrom which we may read off the coefficients $\\gamma}\\def\\G{\\Gamma_\\nu^\\tau$.\nMore generally the functions $U_\\n^\\s(\\ph)$ may be expanded in terms of the\n$y_\\nu^\\tau(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$\n $$\nU_\\n^\\s(\\ph) = \\sum_{\\tau=0}^2 \\gamma}\\def\\G{\\Gamma_\\nu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma,\\tau}\\, y_\\nu^\\tau(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) =\n\\sum_{\\tau=0}^2 \\gamma}\\def\\G{\\Gamma_\\nu^\\tau\\, y_\\nu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma+\\tau}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~.\\eqlabel{gamma} $$\nWe see from their definitions that the $U_\\n^\\s(\\ph)$ and the $y_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$\nare not\nperiodic in $\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma$ in fact\n $$\nU_\\nu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma+3}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i \\nu}\\,U_\\n^\\s(\\ph)~~~,\n{}~~~y_\\nu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma+3}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i \\nu}\\,y_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~. $$\nWith this in mind we are able to read of the coefficients\n$\\gamma}\\def\\G{\\Gamma_\\nu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma,\\tau}$\nfrom \\eqref{gamma}\n $$\n\\gamma}\\def\\G{\\Gamma_\\nu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma,\\tau} =\\pmatrix{\n            \\-          1&           -2&\\quad\\- 1\\cr\n\\hphantom{-2}e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu}&\\-          1&\\quad  -2\\cr\n           -2e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu}& e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu}&\\quad\\- 1\\cr}~.\n\\eqlabel{gammamat} $$\n\nWe are now able to return to the question of finding linearly\nindependent\nsolutions. Note first that the function\n $$\n\\widehat{V}_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = \\sum_{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0}^2 U_\\n^\\s(\\ph) =\n(1 - e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu})\\Big( y_\\nu^0(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) -\ny_\\nu^1(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\Big)\\eqlabel{Vhat}$$\nvanishes at the integers and that\n $$\nV_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = {\\widehat{V}_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\over 1 - e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu}}\n\\eqlabel{Vdef}$$\nhas a nonvanishing limit. Taking say $U_\\nu^0$, $U_\\nu^1$ and $V_\\nu$\nas a basis\nimproves the situation insofar as the Wronskian\n$\\hbox{Wr}[U_\\nu^0,U_\\nu^1,V_\\nu]$ now vanishes only to first order at\nthe\nintegers.\nSince it still vanishes there  must be a linear relation between\nthese\nsolutions at the integers. We define\n $$\n\\widehat{W}_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = 3V_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) - 2U_\\n(\\ph) - U_\\nu^1(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n=  (1 - e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu})\\, y_\\nu^0(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\eqlabel{What}$$\nand we see that $\\widehat{W}_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ also vanishes at the\nintegers but that\n $$\nW_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = {\\widehat{W}_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\over 1 - e^{-2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\nu}}\n\\eqlabel{Wdef}$$\nhas a nonvanishing limit. We now check that\n $$\n\\hbox{Wr}[U_\\nu, V_\\nu, W_\\nu] =\n{27i\\over (2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi)^3}\\, e^{i\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\nu}\\,(1-\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3)^{\\nu-1} $$\nso the solutions $U_\\n(\\ph)$, $V_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ and $W_\\nu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ are always\nlinearly\nindependent.  These functions have nonvanishing limits at\nthe negative integers as we have seen. At the positive integers they\nhave poles although this fact will not concern us in the following.\n\nWe wish now to discuss the asymptotic behavior of the solutions as\n$\\nu\\to\\infty$ for fixed $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$. It is easy to see that the functions\n$(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-\\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma})^{\\nu+1}$ solve the differential equation \\eqref{uneq}\nfor\nlarge $\\nu$. A general solution is then a linear combination of these\nexpressions with coefficients that depend on $\\nu$. The coefficients\nmay be\nfixed from a knowledge of the monodromy of the solution about the\npoints for\nwhich $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3=1$. Thus, for example, the relation $$ y_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n\\sim \\omega}\\def\\O{\\Omega^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma\n(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi - \\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma})^{\\nu+1} $$ which we have already met as a relation\nthat is\nvalid as $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi\\to \\omega}\\def\\O{\\Omega^{-\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}$ {\\it is valid also for all $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$ in the\nasymptotic\nlimit $\\nu\\to\\infty$\\\/}.  The asymptotic behavior of the functions\n$U_\\n^\\s(\\ph)$\nfollows in virtue of \\eqref{gamma}. These considerations will shortly\npermit\nus to write integral representations that may be used to continue the\nperiods\n$\\vp_j(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ to large $\\psi}\\def\\Ps{\\Psi$.\nNotice that\n $$\n\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma = \\int_{\\G_-}{d\\mu \\over 2i \\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi(\\mu+{r\\over 6})}\n{\\G^2(-\\mu) \\over \\G(-\\mu+{1\\over 6})\n\\G(-\\mu+{5\\over 6})}\\rho}\\def\\vr{\\varrho^{-6\\mu}\\,U_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\eqlabel{G-rep} $$\nwith the contour $\\G_-$ enclosing the poles on the negative\n$\\mu$-axis.\n\nIn order to obtain an integral representation valid for large $\\rho}\\def\\vr{\\varrho$ we\nwish to\nrotate the contour so as to run parallel to the imaginary axis. This\nrequires a\nconsideration of the convergence of the integrals and the\ncontribution of the\narcs at infinity. For $\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0$ the arcs at infinity give a vanishing\ncontribution\nso for this case we have\n $$\n\\xi}\\def\\X{\\Xi_r^0 = \\int_\\G {d\\mu \\over 2i\\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi(\\mu+{r\\over 6})}\n{\\G^2(-\\mu) \\over \\G(-\\mu+{1\\over 6})\n\\G(-\\mu+{5\\over 6})}\\rho}\\def\\vr{\\varrho^{-6\\mu}\\,U_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) $$\nFor $\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma\\neq 0$ it is not possible to rotate the contour through the\nsecond quadrant without modifying the integrand. Note however that\nthe value\nof the integral is unchanged if we replace $U_\\m^\\s(\\ph)$ in \\eqref{G-rep}\\\nby\n $$\n\\widetilde{U}_{\\mu,r}^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = U_\\m^\\s(\\ph) -\ne^{i\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi r\/6} {\\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi(\\mu + {r\\over 6}) \\over \\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\mu}\\,f_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) $$\nwith $f_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ a function that has no poles in the left half\n$\\mu$-plane\nand has zeros at the integers. The factor\n $$\ne^{i\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi r\/6} {\\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi(\\mu + {r\\over 6}) \\over \\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\mu} $$\nensures that the new term does not contribute to the poles and tends\nto unity\nas $\\mu\\to\\infty$ along a ray in the second quadrant. By suitable\nchoice of\n$f_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma$ we are able to rotate the contour. The convergence of the\nintegral is\nmost easily studied by writing the $U_\\m^\\s(\\ph)$ in terms of the\n$y_\\m^\\s(\\ph)$-basis. The\nquantities that have to be cancelled arise from the exponential\nentries in the\nmatrix $\\gamma}\\def\\G{\\Gamma_\\mu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma,\\tau}$ \\eqref{gammamat}. It is now easily seen that\nwe should\nchoose\n $$\\eqalign{\nf_\\mu^0 &=\\-  0\\cr\nf_\\mu^1 &=  -\\widehat{W}_\\mu\\cr\nf_\\mu^2 &=\\- \\widehat{V}_\\mu + \\widehat{W}_\\mu~.\\cr} $$\nIn fact in terms of the $y_\\m^\\s(\\ph)$-basis we have\n $$\n\\widetilde{U}_{\\mu,r}^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = \\sum_\\tau\n\\tilde{\\gamma}\\def\\G{\\Gamma}_{\\mu,r}^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma,\\tau}\\,y_\\mu^\\tau(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n{}~~,~~~ \\tilde{\\gamma}\\def\\G{\\Gamma}_{\\mu,r}^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma,\\tau} = \\pmatrix{\n            \\-          1&           -2&\\quad\\- 1\\cr\n\\hphantom{-2}e^{\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi ir\/3}&\\-          1&\\quad  -2\\cr\n           -2e^{\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi ir\/3}& e^{\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi ir\/3}&\\quad\\- 1\\cr}~.\n\\eqlabel{gamtilde} $$\nGathering these results together and performing the sum over $r$ we\nfind the\nfollowing integral representations for the periods:\n $$\\eqalign{\n\\vp_{0\\hphantom{{}+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}}&=\\int_\\G {d\\mu \\over 4\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^2 i}\n{\\G(-\\mu) \\G(\\mu {+} {1\\over 6}) \\G(\\mu {+} {5\\over 6}) \\over\\G(1 {+}\n\\mu)}\n\\rho}\\def\\vr{\\varrho^{-6\\mu}U_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma\\cropen{15pt}\n %\n\\vp_{1\\hphantom{{}+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}}&=\\int_\\G {d\\mu \\over 8\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^3}\n\\G^2(-\\mu) \\G(\\mu {+} {\\textstyle {1\\over 6}}) \\G(\\mu {+} {\\textstyle {5\\over6}})\n\\rho}\\def\\vr{\\varrho^{-6\\mu}\n\\left[ 2i\\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\mu\\,(U_\\mu^1 {+} W_\\mu) + e^{-3\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\mu}W_\\mu\n\\right]\\cropen{15pt}\n %\n\\vp_{2\\hphantom{{}+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}}&=\\int_\\G {d\\mu \\over 8\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^3}\n\\G^2(-\\mu) \\G(\\mu {+} {\\textstyle {1\\over 6}}) \\G(\\mu {+} {\\textstyle {5\\over6}})\n\\rho}\\def\\vr{\\varrho^{-6\\mu}\n\\left[ 2i\\sin\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\mu\\,(U_\\mu^2 {-} V_\\mu {-} W_\\mu)\n- e^{-3\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\mu}(V_\\mu {+} W_\\mu) \\right]\\cropen{15pt} %\n\\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}&=\\int_\\G {d\\mu \\over 8\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^3}\n\\G^2(-\\mu) \\G(\\mu {+} {\\textstyle {1\\over 6}}) \\G(\\mu {+} {\\textstyle {5\\over6}})\n\\rho}\\def\\vr{\\varrho^{-6\\mu}e^{\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\ni\\mu}U_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma~~,\\hskip20pt\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=0,1,2~.\\cr}\\hss\\eqlabel{intreps}$$\nThese representations are valid when\n $$\n-\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi < \\arg\\left({\\rho}\\def\\vr{\\varrho^6\\over \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi - \\omega}\\def\\O{\\Omega^{-\\tau}}\\right) < \\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi~. $$\n %\n\\subsection{Expansions for large $\\psi}\\def\\Ps{\\Psi$}\nWe will see later that the mirror map can be written in terms of the\nquantities\n$\\vp_0$, $\\vp_3$ and $\\sum_\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}$ and we will need the\nexplicit form of\ntheir expansions for large $\\psi}\\def\\Ps{\\Psi$. If\n\\hbox{$\\left|{\\rho}\\def\\vr{\\varrho^6\\over \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi - \\omega}\\def\\O{\\Omega^{-\\tau}}\\right|>1$} the contours may\nbe\ndeformed so as to enclose the poles on the positive $\\mu$-axis and the\nlarge\n$\\psi}\\def\\Ps{\\Psi$ series are obtained by summing over the\nresidues at $\\mu =0,1,\\ldots$.\n\nFor the quantity $\\sum_\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma}$ there is a simple integral\nrepresentation\n $$\n\\sum_\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma} =\n\\int_\\G {d\\mu \\over 4\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi^2 i}\n{\\G(-\\mu) \\G(\\mu {+} {1\\over 6}) \\G(\\mu {+} {5\\over 6}) \\over\\G(1 {+}\n\\mu)}\n\\rho}\\def\\vr{\\varrho^{-6\\mu}V_\\mu~.$$\nOn evaluating the residues we find the following series (for the case\nof\n$\\vp_0$ this merely reproduces the original definition \\eqref{npi})\n $$\\eqalign{\n&\\vp_0 = \\sum_{k=0}^\\infty\n{ (6k)! \\, (-3)^k \\over k! (2k)! (3k)! (18\\psi)^{6k} } U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n\\cropen{10pt}\n %\n&\\sum_\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma} = \\vp_0 - {3\\over 2\\pi i}\\vp_0 \\log\\,(18\\psi}\\def\\Ps{\\Psi)^6\n+ {3\\over 2\\pi i}\n\\sum_{k=0}^\\infty { (6k)! \\, (-3)^k \\over k! (2k)! (3k)! (18\\psi}\\def\\Ps{\\Psi)^{6k}\n}\n\\, \\big [ A_k U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + Y_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)  \\big ] \\cropen{10pt}\n %\n&\\vp_3 - {1\\over 3}\\sum_\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma} =\n{1 \\over 6} \\vp_0 + {1\\over 2\\pi i} \\vp_0 \\log(3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + {1\\over 2\\pi\ni}\n\\sum_{k=0}^\\infty  { (6k)! \\, (-3)^k \\over k! (2k)! (3k)!\n(18\\psi)^{6k} }\n\\, N_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~,  \\cr } \\eqlabel{mlarge} $$\nwhere in these series\n $$\\eqalign{\nA_k &= 6\\Ps(6k+1)-3\\Ps(3k+1)-2\\Ps(2k+1)-\\Ps(k+1) \\cr\\noalign{\\medskip}\nY_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) &= \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^k  k! \\,\\sum_{n=0}^{\\left[ {k\\over 3}\\right]}\n{ (-1)^n\\over {(n!)^3 (k-3n)!  (3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{3n}} }\n[ \\Ps(1+k) - \\Ps(1+n) ]\\cr\\noalign{\\medskip}\nN_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) &= \\pd{U_k}{\\mu} - Y_k - U_k\\log\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi \\cr\\noalign{\\medskip}\n&=\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^k k! \\left\\{\n\\sum_{n=0}^{\\left[ {k\\over 3} \\right]} {(-1)^n [\\Ps(n+1) -\n\\Ps(k-3n+1)]\\over\n(n!)^3\\, (k-3n)!\\, (3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{3n}} +\n\\sum_{\\left[ {k\\over 3} \\right]+1}^\\infty {(-1)^{k+1}\\,(3n-k-1)!\\over\n(n!)^3\\, (3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^{3n}} \\right\\} \\cr}\\hss \\eqlabel{AYdef} $$\nNotice that $A_0 = Y_0 = 0$.\n\nThe following table records the various functions that are associated\nto the\nfundamental period that we have introduced in this section\n\\bigskip\n$$\\vbox{\\offinterlineskip\\halign{\n\\strut # height 15pt depth 8pt&\\hfil\\quad$#$\\quad\\hfil\\vrule\n&\\hfil\\quad$#$\\quad\\hfil\\vrule \\cr\n\\noalign{\\hrule}\n\\vrule& \\hbox{Functions} & \\hbox{Where Defined}\\cr\n\\noalign{\\hrule\\vskip3pt\\hrule}\n\\vrule & U_\\nu~,~U_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma & \\eqref{uhl},\\,\\eqref{unus}\\cr\n\\vrule & y_\\nu~,~y_\\nu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma & \\eqref{ydef},\\,\\eqref{ynus}\\cr\n\\vrule & \\widehat{V}_\\nu~,~V_\\nu & \\eqref{Vhat},\\,\\eqref{Vdef}\\cr\n\\vrule & \\widehat{W}_\\nu~,~W_\\nu & \\eqref{What},\\,\\eqref{Wdef}\\cr\n\\vrule & Y_k~,~N_k& \\eqref{AYdef}\\cr\n\\vrule & \\vp_0~,~\\vp_j~,~\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma &\n\\eqref{pi0},\\,\\eqref{vpj},\\,\\eqref{vpjr},\\,\n\\eqref{zrs}\\cr \\noalign{\\hrule}\n}}\n$$\n\\nobreak\\tablecaption{definitions}{Functions associated to the\nperiods and\ntheir\ndefinitions.}\n\\vfill\\eject\n %\n\\section{monodromy}{Monodromy Calculations}\n\\vskip-20pt\n\\subsection{Generalities}\nWe wish now to study the effect of the various monodromy operations\non the\nperiod vectors $\\vp$ and $\\widehat{\\vp}$ associated to the 3-forms\n$\\O$ and $\\widehat{\\O}$.\nTo do this consistently we must choose a basepoint.\nThat\nis, we make a common choice of basepoint for the curves that encircle the\nvarious\ncomponents of the discriminant locus. A change of basepoint induces a\nconjugation of the matrices. Thus if, with respect to some basepoint,\nthe\nmonodromy matrices are denoted by $M_a~,~a=1,2,\\ldots$ then under a\nchange of\nbasepoint the matrices become $g^{-1} M_a g$ for some $g\\in\n\\hbox{Sp}(6,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$.\nWe choose our basepoint to be a point for which $\\psi}\\def\\Ps{\\Psi$ is large, $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$\nis small\nand $0<\\arg\\,\\psi}\\def\\Ps{\\Psi<{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\over 18}$. We shall refer to such curves as\nhaving a\nbasepoint at $\\infty$. A useful technique for computing some of\nthe matrices, which we illustrate in the following, involves use of\nthe\nintegral\nrepresentations \\eqref{intreps}.\n %\n\\subsection{Monodromy about $\\psi}\\def\\Ps{\\Psi=0$ and $\\psi}\\def\\Ps{\\Psi=\\infty$}\nFix $(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ with $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$ small and consider the curve\n$(e^{i\\theta}\\def\\Th{\\Theta}\\def\\vth{\\vartheta}\\psi}\\def\\Ps{\\Psi,e^{6i\\theta}\\def\\Th{\\Theta}\\def\\vth{\\vartheta}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ as $\\theta}\\def\\Th{\\Theta}\\def\\vth{\\vartheta$ varies in the range\n$\\left [0,{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\over 18}\\right]$. This is a closed curve\non the simplified moduli space in virtue of\nthe\nidentifications on the parameter space. In virtue of the above\ndiscussion we\nare\nmost interested in the case that $\\psi}\\def\\Ps{\\Psi$ is large since this curve has\na\nbasepoint\nat $\\infty$. However before examining this case let us take $\\psi}\\def\\Ps{\\Psi$\nsmall, we can\nsay that this curve has a basepoint at the origin. From \\eqref{vpj}\nand\n\\eqref{api} we see that the monodromy of $\\vp$ along this\ncurve corresponds to the operation\n\\hbox{${\\cal A}:\\vp_j \\to \\vp_{j+1}$}.\nUnder ${\\cal A}$ the period vector $\\vp$ transforms as\n $$\n{\\cal A} : \\vp\\to \\hbox{\\ss A} \\vp \\quad ; \\quad \\hbox{\\ss A} = \\matrixao~.\\eqlabel{am} $$\nWhen we consider the period vector $\\widehat{\\vp}$, however,\nthe monodromy along this same curve is given by\n${\\cal A}'\\buildrel\\rm def\\over ={\\cal A}^{10}\\ca{I}^2$\n(which has the same effect on the parameter space as does ${\\cal A}$,\nbut preserves the 3-form $\\widehat{\\O}$).\nNotice that the action of ${\\cal A}'$ on $\\widehat{\\vp}$ has precisely the same\nmatrix $\\hbox{\\ss A}$ as the action of ${\\cal A}$ on $\\vp$.\n\nConsider now a similar curve with basepoint at $\\infty$. This corresponds\nto an\noperation which we denote by $\\ca{T}_\\infty^{-1}$ (the inverse accounts\nfor the\nfact that a curve in the $\\psi}\\def\\Ps{\\Psi$-plane that winds about $\\psi}\\def\\Ps{\\Psi=0$ in the\npositive\nsense winds about $\\psi}\\def\\Ps{\\Psi=\\infty$ in the negative sense). The matrix\ncorresponding\nto this operation may be computed by computing the effect on the\nintegral\nrepresentations \\eqref{intreps}. We see that after traversing the\ncurve\n $$\n\\eqalign{\n\\rho}\\def\\vr{\\varrho^{-6\\mu}U_\\m^\\s(\\ph) &\\longmapsto \\rho}\\def\\vr{\\varrho^{-6\\mu}\\, U_\\mu^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma+1}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\cr\n\\rho}\\def\\vr{\\varrho^{-6\\mu}\\, V_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)&\\longmapsto \\rho}\\def\\vr{\\varrho^{-6\\mu}\\big( V_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) -\nU_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\big)\\cr\n\\rho}\\def\\vr{\\varrho^{-6\\mu}W_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)&\\longmapsto \\rho}\\def\\vr{\\varrho^{-6\\mu}\\big( W_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) - V_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n\\big)~.\\cr}\n$$\nIn these relations the second and third follow from the first. If we\nnow make\nthese replacements in the integral representations \\eqref{intreps},\ncompute the\nnew residues (it suffices to compute the residues at $\\mu=0,1,2$) and\ncompare\nwith the original residues we find the matrix corresponding to\n$\\ca{T}_\\infty$\n $${\\cal T}_{\\infty} : \\vp\\to \\hbox{\\ss T}_{\\infty} \\vp \\quad ;\n\\quad \\hbox{\\ss T}_{\\infty} = \\matrixti ~. \\eqlabel{timat} $$\nAn identical analysis applies to the periods $\\widehat{\\vp}$,\nand the matrix for the action of $\\ca{T}_\\infty$ on $\\widehat{\\vp}$\nis again $\\hbox{\\ss T}_{\\infty}$.\n\\subsection{The operation $\\ca{I}$}\nTo see the effect of $\\ca{I}:~(\\rho}\\def\\vr{\\varrho,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\mapsto (i\\rho}\\def\\vr{\\varrho,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi+\\rho}\\def\\vr{\\varrho^6)$ recall\nfrom\nSection~\\chapref{vps} that the periods are linear combinations of the\nquantities\n $$\n{\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma} =\\sum_{k = 0}^\\infty \\\n{ {(-1)^k \\G^2(k+{r\\over 6}) }\\over\n{\\G(k+1) \\G(k+{r\\over 3})} }\\,\\rho}\\def\\vr{\\varrho^{6k+r}\\, U^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma_{-(k+{r\\over 6})}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n$$\nfor $r=1$ and $r=5$. The simplest way to proceed is to take $\\rho}\\def\\vr{\\varrho$\nsmall and to\nconsider only the $k=0$ terms in the sums. In this way we see that\n$\\ca{I}:~\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\mapsto i\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma$ and hence that\n $$\n\\ca{I}:~\\vp_j \\longmapsto i\\vp_j~.$$\nThus the effect of $\\ca{I}$ on $\\vp$\nis not to effect a symplectic transformation\nbut to\nmultiply the periods by a gauge factor.\nHowever, when we apply $\\ca{I}$ to the other period vector\n$\\widehat{\\vp}$, we find that $\\ca{I}$ leaves it invariant.\n\nIt is instructive to see the effect of $\\ca{I}$ in detail when $\\rho}\\def\\vr{\\varrho$ is\nnot small.\nTo this end we write the $\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma$ as integrals\n $$\n\\xi}\\def\\X{\\Xi_r^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma = \\rho}\\def\\vr{\\varrho^r\\int_0^1 d\\lambda}\\def\\L{\\Lambda\\, \\lambda}\\def\\L{\\Lambda^{{r\\over 6} - 1} (1 - \\lambda}\\def\\L{\\Lambda)^{{r\\over 6}\n- 1}\nU^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma_{-{r\\over 6}}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + \\lambda}\\def\\L{\\Lambda\\rho}\\def\\vr{\\varrho^6) \\eqlabel{xint}$$\nas is easily verified by expanding the integrand in powers of $\\lambda}\\def\\L{\\Lambda$,\nintegrating\nterm by term by means of the $B$-function formula\n $$\n\\int_0^1 d\\lambda}\\def\\L{\\Lambda\\, \\lambda}\\def\\L{\\Lambda^{a - 1} (1 - \\lambda}\\def\\L{\\Lambda)^{b - 1} = {\\G(a)\\G(b) \\over\n\\G(a+b)} $$\nand using the recurrence relation \\eqref{uder} which is equivalent to\n $$\n(-1)^k\\,\n{\\G\\left( {\\textstyle{r\\over 6}} \\right) \\over \\G\\left( k + {\\textstyle{r\\over\n6}} \\right)}\n\\left( {d\\over d\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi} \\right)^k U_{-{r\\over 6}}^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) =\nU_{-k-{r\\over6}}^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~.  $$\nThe effect of $\\ca{I}$ on the RHS of \\eqref{xint} is to multiply  the\nfactor of\n$\\rho}\\def\\vr{\\varrho^r$ by $i$ and to effect the change \\hbox{$\\lambda}\\def\\L{\\Lambda\\to 1-\\lambda}\\def\\L{\\Lambda$} in the\nintegrand\nbut this leaves the value of the integral unchanged.\n %\n\\subsection{Monodromy about $C_{con}$}\nTo compute the monodromy about the conifold, we notice that\nnear $C_{con}$ the periods have the structure\n $$\\vp_j(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = c_j g(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\log (\\rho}\\def\\vr{\\varrho^6 - \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi -1)\n+ \\hbox{analytic}~, $$\nwhere $g(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)$ is itself a period analytic in the\nneighborhood of $C_{con}$. To evaluate the coefficients\n$c_j$, we set $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=0$, so that $g(\\psi}\\def\\Ps{\\Psi,0)=(\\psi}\\def\\Ps{\\Psi-\\psi}\\def\\Ps{\\Psi_0) + \\cdots$\nas follows from the behavior of the periods around $\\psi}\\def\\Ps{\\Psi_0$.\nThe logarithmic piece in $\\partial \\vp_j\/\\partial \\psi}\\def\\Ps{\\Psi$ is then\nextracted by using Stirling's formula in the series expansion.\nUp to a constant that can be absorbed in $g$, we find\n $$\nc_j = (1,1,-2,1,0,0) \\quad ; \\quad j=0, \\cdots, 5  $$\nThe next step is to express $g$ in terms of the $\\vp_j$.\nAn argument parallel to that made in \\cite{\\rCdGP} shows\nthat $\\vp_0 \\mapsto \\vp_1$ and\n $$\ng = {i\\over 2\\pi c_1}( \\vp_1 - \\vp_0) ~.  $$\nThus, under transport about the conifold\n$\\vp$ transforms as\n $$\n{\\cal T} : \\vp\\to \\hbox{\\ss T}\\vp \\quad ; \\quad \\hbox{\\ss T} = \\matrixto~.\\eqlabel{tmat}\n$$\nThe other period vector $\\widehat{\\vp}$ transforms in the same way,\nagain with matrix $\\hbox{\\ss T}$, since\nthe prefactor $\\rho^3(\\phi+{1\\over2}\\rho^6)$ is single-valued near\nthe conifold locus.\n\nIn this case we get the same result whether we take the basepoint to\nbe at\n$\\infty$ or at the origin.\nBearing in mind that the abstract operations compose in the reverse\norder to\nthe matrices we may write a relation between the operations $\\ca{A}$,\n$\\ca{T}$ and\n$\\ca{T}_\\infty$\n $$\n\\ca{T}_{\\infty} = (\\ca{T}\\ca{A})^{-1}~~~~,~~~~\\hbox{\\ss T}_{\\infty} = (\\hbox{\\ss A}\\hbox{\\ss T})^{-1}%\n{}~.$$\n(A similar relation holds among ${\\cal A}'$, $\\ca{T}$, and $\\ca{T}_{\\infty}$\nacting on $\\widehat{\\vp}$.)\nThis relation has a simple interpretation. It is easy to see, by\nsetting\n$\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=0$, that $\\ca{A}$ and $\\ca{T}_\\infty^{-1}$\ndiffer by a curve that winds about $C_{con}$. Note also that $\\ca{A}$\ncan be\nrealised also as a combination of monodromy operations that have\nbasepoints at\n$\\infty$.\n %\n\\subsection{Monodromy about $B_{con}$}\nWe shall compute the monodromy about $B_{con}$ from the integral\nrepresentations for the periods. To this end we make some\nobservations\nregarding the monodromy of the functions~$U_\\mu^\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma$.\n\nLet us denote by $\\D_\\mu$ the solution to \\eqref{uneq} corresponding\nto the\nfirst term on the RHS of \\eqref{ydef} so that\n $$\ny_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) = \\D_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + \\hbox{analytic}~~~,~~~\n\\D_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)=  -{\\sqrt{3} \\over 2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi\\,(\\mu+1)} (\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-1)^{\\mu+1}\\,\n\\Big\\{ 1+ \\ca{O}(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi-1) \\Big\\}~.$$\nContinuing $y_\\mu$ about $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=1$ gives the result\n $$\ny_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\longmapsto y_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + (e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\mu} - 1)\\D_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) $$\nfrom which it follows that\n $$\nU_\\m^\\s(\\ph) \\longmapsto U_\\m^\\s(\\ph) + \\gamma}\\def\\G{\\Gamma^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma 0}(e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\mu} - 1)\\D_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)~.\n\\eqlabel{Ucont}$$\nOn the other hand we can work through an argument that is by\nnow familiar. We take $x$ real and $x>1$ then\n $$\nU^1_\\mu(x+i\\epsilon}\\def\\ve{\\varepsilon) \\longmapsto U^1_\\mu(x-i\\epsilon}\\def\\ve{\\varepsilon)\n= \\omega}\\def\\O{\\Omega^{-\\mu} U^0_\\mu(\\omega}\\def\\O{\\Omega(x-i\\epsilon}\\def\\ve{\\varepsilon))\n= U^0_\\mu(x+i\\epsilon}\\def\\ve{\\varepsilon) $$\nwith $\\epsilon}\\def\\ve{\\varepsilon$ an infinitesimal and the last equality following from\nthe series\n\\eqref{smallphi}. By comparing this relation with \\eqref{Ucont} we\nfind an\nexpression for $\\D_\\mu$ and hence that\n $$\nU_\\m^\\s(\\ph) \\longmapsto U_\\m^\\s(\\ph) +\n\\gamma}\\def\\G{\\Gamma^{\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma 0} e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i \\mu}\\Big( U^0_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) - U^1_\\mu(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\Big) $$\nand by making these replacements in \\eqref{intreps} we obtain the\nmonodromy\nmatrix\n $$\n{\\cal B} :  \\vp\\to \\hbox{\\ss B} \\vp \\quad ; \\quad \\hbox{\\ss B} = \\matrixbo\n\\eqlabel{bm} $$\nAgain, the same matrix describes the action on $\\widehat{\\vp}$.\n\nNotice that $\\vp_0 \\mapsto \\vp_0$, reflecting the fact that it is\nsingle valued near $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi=1$. We do not expect $\\ca{B}$ to be an\nindependent\ngenerator and we easily see this to be the case. Comparing $\\ca{B}$ with\n$\\ca{T}$ we\nsee that\n $$\\eqalign{\n\\ca{B}:~\\vp_j&\\to \\vp_j + b_j (\\vp_3 - \\vp_4)\\cr\n\\ca{T}:~\\vp_j&\\to \\vp_j + c_j (\\vp_0 - \\vp_1)\\cr} $$\nand from this we see that\n $$\n\\hbox{\\ss B} = \\hbox{\\ss A}^{-3} \\hbox{\\ss T} \\hbox{\\ss A}^3~.$$\n %\n\\subsection{Global considerations}\n\nWe give now a more global interpretation of our monodromy calculations.\nCalculating monodromies around all loops in the moduli space $\\relax{\\rm I\\kern-.18em M}$\n(with a fixed basepoint $P$) determines\nthe {\\it monodromy representation}\\\/ of our theory, which is a homomorphism\nfrom the fundamental group $\\pi_1(\\relax{\\rm I\\kern-.18em M},P)$ to the group of linear\ntransformations $\\hbox{Gl}(6,\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$})$.  (The image will lie in $\\hbox{Sp}(6,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$\nif the basis of periods has been chosen appropriately.)\nThe image is the full group of {\\it duality transformations}\\\/\nof the theory.\n\nWe use the technique of the Zariski--van Kampen theorem (see for example\n Ref.~\n\\REFS{\\rDimca}{ A. Dimca, {\\sl Singularities and Topology of Hypersurfaces},\n Springer-Verlag, 1992.}\n\\refsend)\nto calculate the fundamental group of the moduli space.\n(A similar method was employed in Ref.~\\cite{\\rCDR}.)\nFor this purpose we return to the $\\cp2^{(3,1,1)}$ model of the\ncompactification of the simplified moduli\nspace.  The fundamental group $\\pi_1(\\relax{\\rm I\\kern-.18em M},P)$ is an index~2 subgroup\nof the fundamental group of the simplified moduli space, the extra generator\ncorresponding to $\\ca{I}$.  Since the monodromy transformation corresponding\nto $\\ca{I}$ is trivial on the period vector $\\widehat{\\vp}$, we need only\nconsider the simplified moduli space in calculating the duality\ntransformations.\n\nWe continue to use homogeneous coordinates $[x,y,z]$ on $\\cp2^{(3,1,1)}$,\nand consider the affine\ncoordinate chart with $y=1$ which has coordinates $(x,z)$.  The projection\nto the $z$-axis has as fibers the complex curves $F_{z_0}:=(z=z_0)$ (with\n$z_0$ a constant), and $x$\nserves as a coordinate on any of these curves.  For general values of\n$z_0$, the discriminant locus meets the fiber $F_{z_0}$ in precisely\nthree points:  $F_{z_0}\\cap D_\\infty$ at $x=0$, $F_{z_0}\\cap B_{\\rm con}$\nat $x=z_0^3$, and $F_{z_0}\\cap C_{\\rm con}$ at $x=(z_0+1)^3$.  The\nvalues of $z_0$ at which some of these points come together---namely\n$z_0=0,-1,(-3\\pm\\sqrt{-3})\/6$---will play an important r\\^ole.\nThe latter two correspond to the points $P_-$ and $P_+$, respectively.\n\nLet us choose a base point $P$ for which the corresponding value of\n$z_0$ is not one of the special values.\nThe proof of the Zariski--van Kampen theorem guarantees that the fundamental\ngroup of the complement of the discriminant locus\nis generated by%\n\\Footnote{{\\it A priori}, since our fibration is not generic at $z_0=\\infty$,\nwe should also include a loop in the base of the fibration, around\n$z_0=\\infty$.  But such a loop can be written as a product of loops\naround the finite special $z_0$-values, and such loops in the base\nare equivalent to loops in the fiber.}\nloops in the fiber $F_{z_0}$, based at $P$, around the\n4 points $x=0$, $x=z_0^3$,  $x=(z_0+1)^3$, and $x=\\infty$.\nWhen we choose $P$ to be the basepoint ``at infinity'' (with\n$\\psi$ large, $\\phi$ small, and $0<\\arg\\psi<{2\\pi\\over18}$),\nand choose the loops appropriately, the corresponding monodromy\ntransformations are $\\ca{D}_\\infty$, $\\ca{B}$, $\\ca{T}$, and $\\ca{A}$,\nrepresented by matrices $\\hbox{\\ss D}_\\infty$, $\\hbox{\\ss B}$, $\\hbox{\\ss T}$, and $\\hbox{\\ss A}$.\n($\\ca{D}_\\infty$ has\nnot appeared previously; we shall calculate it shortly.)\nOur choice of loops is indicated in Fig.~\\figref{contA}, valid when\n$z_0=\\varepsilon\\,e^{i\\theta}$ with $\\varepsilon$ and $\\theta$\nsmall and positive.  Also shown in the figure are the branch cuts\nused in our analysis of the periods.\n\n\\iffigs\n\\midinsert\n\\def\\contAone{\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{\ncontA1.ps}}}\n\\figbox{{\\vbox{\\kern0.1truein{\\hbox{\\kern.05truein\\contAone}}}}}\n{\\figlabel{contA}}{The branch cuts, and the choice of loops for\n$z_0=\\varepsilon\\,e^{i\\theta}$, with $\\varepsilon$ and $\\theta$\nsmall and positive}\n\\place{2.95}{2.6}{$P$}\n\\place{1.35}{2.1}{$_{z_0^3}$}\n\\place{0.9}{1.05}{$0$}\n\\place{3.3}{1.05}{$_{(z_0{+}1)^3}$}\n\\place{5.6}{1.1}{$\\infty$}\n\\place{0.9}{2.2}{$\\ca{B}$}\n\\place{0.4}{1.5}{$\\ca{D}_\\infty$}\n\\place{3.85}{1.35}{$\\ca{T}$}\n\\place{5.15}{1.35}{$\\ca{A}$}\n\\endinsert\n\\fi\n\nTo calculate the monodromy transformation $\\ca{D}_\\infty$, recall that\nour convention is that the composite $\\ca{L}_1\\ca{L}_2$ of the monodromy\ntransformations along loops $\\ca{L}_1$ and $\\ca{L}_2$ describes the\nmonodromy along a loop which first traverses $\\ca{L}_1$ and then\ntraverses $\\ca{L}_2$.  If the transformation $\\ca{L}_j$ is represented\nby the matrix $\\hbox{\\ss L}_j$ (i.e., if it maps\nthe period vector $\\Pi$ to $\\hbox{\\ss L}_j\\,\\Pi$), then $\\ca{L}_1\\ca{L}_2$ is\nrepresented by the matrix $\\hbox{\\ss L}_2\\hbox{\\ss L}_1$---the matrices compose in the\nopposite order.  Applying this to our situation, we see that\n$\\ca{D}_\\infty\\ca{B}=\\ca{T}_\\infty$, and so $\\ca{D}_\\infty$ is\nrepresented by the matrix\n$$ \\hbox{\\ss D}_\\infty = \\hbox{\\ss B}^{-1}\\hbox{\\ss T}_\\infty = \\hbox{\\ss B}^{-1}\\hbox{\\ss T}^{-1}\\hbox{\\ss A}^{-1} . $$\nWe have previously observed the relation\n$$ \\hbox{\\ss B}=\\hbox{\\ss A}^{-3}\\hbox{\\ss T}\\hbox{\\ss A}^3 ; $$\nif we combine this with our expression for $\\hbox{\\ss D}_\\infty$ we find\n$$ \\hbox{\\ss D}_\\infty = \\hbox{\\ss A}^{-3}\\hbox{\\ss T}^{-1}\\hbox{\\ss A}^{3}\\hbox{\\ss T}^{-1}\\hbox{\\ss A}^{-1} .$$\nThis demonstrates very explicitly the interesting result\nthat {\\it the duality group for our\nfamily is generated by the matrices $\\hbox{\\ss A}$ and $\\hbox{\\ss T}$.}\n\nIt is instructive to verify the relations in the fundamental group as\ngiven by the Zariski--van Kampen theorem.  First, as we let $z_0$\nwind once about $0$, $z_0^3$ will wind three times about $0$.  The\ncorresponding relation is\n$(\\ca{D}_\\infty\\ca{B})^3\\ca{B}(\\ca{D}_\\infty\\ca{B})^{-3}=\\ca{B}$, or\nin matrix form (using the identity $\\ca{D}_\\infty\\ca{B}=\\ca{T}_\\infty$)\n$$ \\hbox{\\ss T}_\\infty^{-3}\\hbox{\\ss B}\\hbox{\\ss T}_\\infty^3=\\hbox{\\ss B} .$$\nThis is easily verified.\n\n\\iffigs\n\\midinsert\n\\def\\contAtwo{\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{\ncontA2.ps}}}\n\\figbox{{\\vbox{\\kern0.1truein{\\hbox{\\kern.05truein\\contAtwo}}}}}\n{\\figlabel{contAbis}}{Another set of generators at\n$z_0=\\varepsilon\\,e^{i\\theta}$}\n\\place{2.95}{2.5}{$P$}\n\\place{1.35}{1.95}{$_{z_0^3}$}\n\\place{0.9}{0.9}{$0$}\n\\place{3.3}{0.9}{$_{(z_0{+}1)^3}$}\n\\place{5.6}{0.95}{$\\infty$}\n\\place{0.9}{2.15}{$\\ca{B}$}\n\\place{0.4}{1.45}{$\\widehat{\\ca{D}}_\\infty$}\n\\place{3.85}{1.2}{$\\ca{T}$}\n\\place{5.15}{1.2}{$\\ca{A}$}\n\\endinsert\n\\fi\n\nNext, we move $z_0$ to the value $z_0=\\varepsilon\\,e^{i(\\theta+2\\pi\/3)}$,\nthrough values in the upper half plane.\nThe contours will deform as we do so; to describe the deformation it is\nconvenient to change generators and replace $\\ca{D}_\\infty$ by\n$\\widehat{\\ca{D}}_\\infty=\\ca{B}^{-1}\\ca{D}_\\infty\\ca{B}\n=\\ca{B}^{-1}\\ca{T}_\\infty$, as shown in Fig.~\\figref{contAbis}.\nThe new contours deform as shown\nin the top half of Fig.~\\figref{contB}.  To describe the braiding relations,\nit is easiest to\nchange the generators as shown in the bottom half of the figure.  Thus,\nfor this $z_0$-value the generator $\\ca{B}$ is replaced by\n$$\\widetilde{\\ca{B}}=\\widehat{\\ca{D}}_\\infty^{-1}\\ca{B}\\widehat{\\ca{D}}_\\infty\n=\\ca{T}_\\infty^{-1}\\ca{B}\\ca{T}_\\infty .$$\nWe can move $z_0$ from $\\varepsilon\\,e^{i(\\theta+2\\pi\/3)}$ to\n$(-3+\\sqrt{-3})\/6$ without introducing any further braiding; if we then\nlet $z_0$ wind once around $(-3+\\sqrt{-3})\/6$, we find that the intersections\nwith $B_{\\rm con}$ and $C_{\\rm con}$ wind once around each other,\nleading to the relation $\\widetilde{\\ca{B}}\\ca{T}=\\ca{T}\\widetilde{\\ca{B}}$.\nThe matrix form of this relation\n$$\n\\hbox{\\ss T}(\\hbox{\\ss T}_\\infty\\hbox{\\ss B}\\hbox{\\ss T}_\\infty^{-1})=(\\hbox{\\ss T}_\\infty\\hbox{\\ss B}\\hbox{\\ss T}_\\infty^{-1})\\hbox{\\ss T}\n$$\nis easily verified.  The relation at $z_0=(-3-\\sqrt{-3})\/6$ is similar.\n\n\\iffigs\n\\midinsert\n\\def\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contB1.ps}}{\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contB1.ps}}}\n\\def\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contB2.ps}}{\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contB2.ps}}}\n\\figbox{\n{\\kern0.1truein{\\hbox{\\kern.05truein\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contB1.ps}}}}}\n\\vskip24truept\n\\centerline{\\kern0.1truein{\\hbox{\\kern.05truein\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contB2.ps}}}}}\n}\n{\\figlabel{contB}}{The loops\ndeformed to\n$z_0=\\varepsilon\\,e^{i(\\theta+2\\pi\/3)}$ (top half),\nand another set of generators\nat the same $z_0$ value (bottom half)}\n\\place{2.95}{2.65}{$P$}\n\\place{1.35}{2.1}{$_{z_0^3}$}\n\\place{0.9}{1.05}{$0$}\n\\place{3.35}{1.07}{$_{(z_0{+}1)^3}$}\n\\place{5.6}{1.1}{$\\infty$}\n\\place{0.9}{2.25}{$\\widetilde{\\ca{B}}$}\n\\place{0.4}{1.5}{$\\widehat{\\ca{D}}_\\infty$}\n\\place{3.85}{1.4}{$\\ca{T}$}\n\\place{5.15}{1.4}{$\\ca{A}$}\n\\place{3.15}{5.1}{$P$}\n\\place{1.55}{4.575}{$_{z_0^3}$}\n\\place{1.1}{3.525}{$0$}\n\\place{3.5}{3.55}{$_{(z_0{+}1)^3}$}\n\\place{5.8}{3.575}{$\\infty$}\n\\place{1.1}{4.65}{$\\ca{B}$}\n\\place{0.6}{3.95}{$\\widehat{\\ca{D}}_\\infty$}\n\\place{4.05}{3.8}{$\\ca{T}$}\n\\place{5.35}{3.8}{$\\ca{A}$}\n\\endinsert\n\\fi\n\nFinally, we move $z_0$ to the value $z_0=\\varepsilon\\,e^{i\\pi}$, still\nthrough values in the upper half plane.  This time, the contours from\nFig.~\\figref{contAbis} deform\nas shown in the top half of Fig.~\\figref{contC}, and we\nchange generators as shown in the bottom half.  The old\ngenerators $\\ca{B}$ and $\\widehat{\\ca{D}}_\\infty$ are replaced by new\ngenerators $\\widetilde{\\ca{B}}$ and $\\widetilde{\\ca{D}}_\\infty$, with\n$\\widetilde{\\ca{B}}$ as above and\n$$\\widetilde{\\ca{D}}_\\infty\n=\\ca{T}_\\infty^{-1}\\widehat{\\ca{D}}_\\infty\\ca{T}_\\infty\n=\\ca{T}_\\infty^{-1}\\ca{B}^{-1}\\ca{T}_\\infty^2 .$$\nIf we allow $z_0$ to wind once around $-1$, we find that the intersection\nwith $C_{\\rm con}$ winds three times around $0$, leading to the relation\n$(\\widetilde{\\ca{D}}_\\infty\\ca{T})^3\\ca{T}\n(\\widetilde{\\ca{D}}_\\infty\\ca{T})^{-3}=\\ca{T}$, or in matrix form,\n$$(\\hbox{\\ss T} \\hbox{\\ss T}_\\infty^2 \\hbox{\\ss B}^{-1} \\hbox{\\ss T}_\\infty^{-1})^{-3}\\hbox{\\ss T}\n(\\hbox{\\ss T} \\hbox{\\ss T}_\\infty^2 \\hbox{\\ss B}^{-1} \\hbox{\\ss T}_\\infty^{-1})^3=\\hbox{\\ss T} .$$\nThis can also be verified by multiplying the corresponding matrices.\n\n\\iffigs\n\\midinsert\n\\def\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contC1.ps}}{\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contC1.ps}}}\n\\def\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contC2.ps}}{\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contC2.ps}}}\n\\figbox{\n{\\kern0.1truein{\\hbox{\\kern.05truein\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contC1.ps}}}}}\n\\vskip24truept\n\\centerline{\\kern0.1truein{\\hbox{\\kern.05truein\\hbox{\\def\\epsfsize##1##2{0.88##1}\\epsfbox{contC2.ps}}}}}\n}\n{\\figlabel{contC}}{The loops\ndeformed to\n$z_0=\\varepsilon\\,e^{i\\pi}$ (top half),\nand another set of generators\nat the same $z_0$ value (bottom half)}\n\\place{2.95}{2.6}{$P$}\n\\place{0.9}{1.05}{$_{z_0^3}$}\n\\place{2.1}{1.05}{$0$}\n\\place{3.35}{1.06}{$_{(z_0{+}1)^3}$}\n\\place{5.6}{1.1}{$\\infty$}\n\\place{0.4}{1.5}{$\\widetilde{\\ca{B}}$}\n\\place{2.3}{1.45}{$\\widetilde{\\ca{D}}_\\infty$}\n\\place{3.85}{1.4}{$\\ca{T}$}\n\\place{5.15}{1.4}{$\\ca{A}$}\n\\place{3.15}{5.2}{$P$}\n\\place{1.1}{3.65}{$_{z_0^3}$}\n\\place{2.3}{3.65}{$0$}\n\\place{3.52}{3.65}{$_{(z_0{+}1)^3}$}\n\\place{5.8}{3.7}{$\\infty$}\n\\place{1.25}{4.1}{$\\ca{B}$}\n\\place{2.5}{3.9}{$\\widehat{\\ca{D}}_\\infty$}\n\\place{4.05}{3.9}{$\\ca{T}$}\n\\place{5.35}{3.9}{$\\ca{A}$}\n\\endinsert\n\\fi\n\\iffigs\n\\vfill\\eject\n\\fi\n\\subsection{The large complex structure limit}\nFurther to the discussion of~\n\\REF{\\rcompact}{D.~R.~Morrison, ``Compactifications of Moduli Spaces\nInspired by Mirror Symmetry'', preprint DUK-M-93-06, alg-geom\/9304007.}\n\\cite{\\rcompact,\\rUno} we identify a large complex structure limit, for\nthe general\ncase of $n$ parameters, as a point in the parameter space where $n$\ncodimension\n1 hypersurfaces in the (compactification of the) moduli space meet\ntransversely\nin a point and the monodromies $S_i$, of a basis of independent\nperiods, about\nthese boundary divisors satisfy certain characteristic properties.\nThe matrices\n $ R_i = S_i - {\\bf 1} $\nsatisfy the properties\n\\vskip5pt\n $$\n\\llap{\\hbox{$\\eqalign{i.&\\cr ii.&\\cr iii.&\\cr}$\\hskip50pt}}\n\\eqalign{\\left[ R_i,R_j \\right] &= 0\\cr\nR_iR_jR_k &= \\smash{\\hbox{$y\\kern-4pt\\raise1pt\\hbox{${}^\\circ$}$}}_{ijk}Y\\cr\nR_iR_jR_kR_l &= 0\\cr}\\eqlabel{lcsl} $$\n\\vskip5pt\n\\noindent where $Y$ is a nonzero matrix independent of $i$ and the\n$\\smash{\\hbox{$y\\kern-4pt\\raise1pt\\hbox{${}^\\circ$}$}}_{ijk}$\nare the ``topological'' limiting values of the Yukawa couplings,\npredicted to coincide with the intersection numbers in the cohomology\nof the mirror.\nThese relations give then a characterization of the large complex structure limit\\\nand the mirror map that is basis-independent,\nprovided that the cubic form defined by the\ncoefficients $\\smash{\\hbox{$y\\kern-4pt\\raise1pt\\hbox{${}^\\circ$}$}}_{ijk}$ is sufficiently\nnondegenerate to be a candidate for an intersection form on a\nCalabi--Yau threefold.  Among these nondegeneracy conditions is\nthe following:\n\n\\bigskip\n\n\\centerline{$iv.$\\qquad For each $i$, there exist $j$ and $k$\nsuch that $\\smash{\\hbox{$y\\kern-4pt\\raise1pt\\hbox{${}^\\circ$}$}}_{ijk}\\ne0$.}\n\n\\bigskip\n\nWe apply this criterion to locate the large complex structure limit\npoints for our family, using the compactification depicted in\nFig.~\\figref{Modulispace}.  The first boundary\npoint to consider is the point $P_-$,\nat which the local monodromy matrices are\n$\\widetilde{\\hbox{\\ss B}}=\\hbox{\\ss T}_\\infty\\hbox{\\ss B}\\hbox{\\ss T}_\\infty^{-1}$, and $\\hbox{\\ss T}$.\nSince\n$$\\left(\\alpha (\\widetilde{\\hbox{\\ss B}}-{\\bf 1}) + \\beta (\\hbox{\\ss T}-{\\bf 1})\\right)^2=0$$\nfor all $\\alpha$, $\\beta$, these matrices violate condition $iv$, so\n$P_-$ cannot be a large complex structure limit\\ point.  It follows that $P_+=\\ca{I}(P_-)$ is not\na large complex structure limit\\ point either.\n\nWe next consider the boundary points which map to $[0,1,0]$ in $\\cp2^{(3,1,1)}$\n({\\it i.e.\\ \\\/} the origin in the $(x,z)$-plane).  Monodromy calculations for\nthese points are displayed in Table~\\tabref{moncalc}.  For each point,\nloops are described which lie in curves transverse to the divisors\nmeeting at the point in question.  Most of these transverse curves\nmap to curves through the origin in the $(x,z)$-plane, and the\ncorresponding monodromy takes the form $\\hbox{\\ss T}_\\infty^k$ for some $k$\n(depending on how many times the loop winds about the origin).\nThe other transverse curves are of the form $z=\\ve$, and loops on\nthese are identified as in the previous subsection.\n\n\\midinsert\n\\vbox{\n$$\n\\def\\hskip2.5pt{\\hskip10pt}\n\\vbox{\\offinterlineskip\\halign{\\strut # height 15pt depth 8pt\n&\\quad #\\quad\\hfil\\vrule\n&\\quad #\\quad\\hfil\\vrule\n&\\quad #\\quad\\hfil\\vrule\n&\\quad #\\quad\\hfil\\vrule\n&\\quad #\\quad\\hfil\\vrule\\cr\n\\noalign{\\hrule}\n\\vrule &\\hfil Point &\\hfil Divisor &\\hfil Transverse Curve &\\hfil\nLoop\n&\\hfil Monodromy \\cr\n\\noalign{\\hrule\\vskip3pt\\hrule}\n\\vrule &\\hskip2.5pt $E_1\\cap F_1$ &\\quad $E_1$              &\\quad\n$x=\\ve\\,z$\n       &\\quad $|z|=\\ve^2$   &\\qquad $\\hbox{\\ss T}_\\infty$ \\cropen{-3pt}\n\\vrule &                    &\\quad $F_1$              &\\quad\n$x=\\ve^{-1}z^2$\n       &\\quad $|z|=\\ve^2$   &\\qquad $\\hbox{\\ss T}_\\infty^2$\\cr\n\\noalign{\\hrule}\n\\vrule &\\hskip2.5pt $F_1\\cap G_1$ &\\quad $F_1$              &\\quad\n$x=\\ve\\,z^2$\n       &\\quad $|z|=\\ve^2$   &\\qquad $\\hbox{\\ss T}_\\infty^2$ \\cropen{-3pt}\n\\vrule &                    &\\quad $G_1$              &\\quad\n$x=\\ve^{-1}z^3$\n       &\\quad $|z|=\\ve^2$   &\\qquad $\\hbox{\\ss T}_\\infty^3$ \\cr\n\\noalign{\\hrule}\n\\vrule &\\hskip2.5pt $G_1\\cap D_\\infty$ &\\quad $G_1$         &\\quad\n$x=\\ve\\,z^3$\n       &\\quad $|z|=\\ve$     &\\qquad $\\hbox{\\ss T}_\\infty^3$ \\cropen{-3pt}\n\\vrule &                    &\\quad $D_\\infty$ &\\quad $z=\\ve$\n       &\\quad $|x|=\\ve^4$   &\\qquad $\\hbox{\\ss D}_\\infty$ \\cr\n\\noalign{\\hrule}\n\\vrule &\\hskip2.5pt $G_1\\cap B_{\\rm con}$ &\\quad $G_1$      &\\quad\n$x=(1+\\ve)\\,z^3$\n       &\\quad $|z|=\\ve$             &\\qquad $\\hbox{\\ss T}_\\infty^3$\n\\cropen{-3pt}\n\\vrule &                    &\\quad $B_{\\rm con}$      &\\quad $z=\\ve$\n       & $|x-\\ve^3|=\\ve^4$  &\\qquad $\\hbox{\\ss B}$ \\cr\n\\noalign{\\hrule}\n}}\n$$\n\\tablecaption{moncalc}{Monodromy calculations for points mapping to\n$[0,1,0]$.}}\n\\endinsert\n\nCondition $iii$ implies that each monodromy transformation near a\nlarge complex structure limit\\ point must be unipotent.  Now $T_\\infty^2$ includes $e^{2\\pi i\/3}$\namong its eigenvalues, so it cannot be unipotent; therefore,\nneither $E_1\\cap F_1$ nor $F_1\\cap G_1$ can be a large complex structure limit\\ point.\nEliminating $G_1\\cap B_{\\rm con}$ is more tricky, but if we calculate\nthe expressions\n$$(\\hbox{\\ss B}-{\\bf 1})^2=0 \\qquad \\hbox{and} \\qquad\n(\\hbox{\\ss B}-{\\bf 1})(\\hbox{\\ss T}_\\infty^3-{\\bf 1})=0 ,$$\nwe see a violation of condition $iv$.  Moreover, by applying $\\ca{I}$\nwe deduce that none of $E_2\\cap F_2$, $F_2\\cap G_2$ or $G_2\\cap C_{\\rm con}$\nis a large complex structure limit\\ point.\n\nHowever, $G_1\\cap D_\\infty$ is a large complex structure limit\\ point, which we will study further\nin the next section.  Applying $\\ca{I}$, we find another such point\n$G_2\\cap D_\\infty$ with an identical structure.  (In fact, although these\ntwo points appear distinct in our ``simplified'' moduli space, they are\nsimply two different representatives of the same point in the true moduli\nspace.)\n\nTo quickly\nverify that none of the other boundary points in Fig.~\\figref{Modulispace}\nis a large complex structure limit\\ point,\nwe turn to an alternate method.\n\\bigskip\n\\subsection{The Picard--Fuchs equations}\nIn this subsection,\nwe use the differential equations satisfied by the cohomology\nclasses of $\\ca{M}$ to calculate monodromy\naround the  divisors of the compactification of the\nmoduli space described in Section~\\chapref{moduli1},\nand finish the verification that there are no additional\nlarge complex structure limit\\ points.\n\nThese differential equations can be obtained as explained in\nRefs.\n\\REF{\\rCF}{A.~C.~Cadavid and S.~Ferrara, \\plb{267} (1991) 193.}\n\\REF{\\rBV}{B.~Blok and A.~Varchenko, Int. J. Mod. Phys. A{\\bf 7}\n      (1992) 1467.}\n\\REF{\\rLSW}{W.~Lerche, D.~J.~Smit and N.~P.~Warner, \\npb{372} (1992)\n       87.}\n\\REF{\\rCDFLL}{A.~Ceresole, R.~D'Auria, S.~Ferrara, W.~Lerche and\n       J.~Louis, Int. J. Mod. Phys. {\\bf A8} (1993) 79.}\n\\cite{{\\rMorrison ,\\rFont ,\\rCF--\\rCDFLL}}.\nIn the notation of \\cite{\\rFont},\nwe choose the basis for $H^3(\\ca{M})$ corresponding\nto the choice of monomials\n $$\\displaylines{\nx_0x_1x_2x_3x_4x_5\\cr\\noalign{\\medskip}\nx_0^2x_1^7x_2^7x_3^7x_4x_5 ~~~~~~ x_0^2x_1^5x_2^5x_3^5x_4^2x_5 \\cr\\noalign{\\medskip}\nx_0^3x_1^{13}x_2^{13}x_3^{13}x_4x_5 ~~~~~~\nx_0^3x_1^{11}x_2^{11}x_3^{11}x_4^2x_5 \\cr\\noalign{\\medskip}\nx_0^4x_1^{17}x_2^{17}x_3^{17}x_4^2x_5\\cr\\noalign{\\smallskip}} $$\n\nThe differential equations take the matrix form\n $$\n\\pd{R}\\psi = RM_{\\psi}~,~~~~\\pd{R}\\phi = RM_{\\phi}\n\\eqlabel{mms}~. $$\nThe matrices $M_{\\psi}, M_{\\phi}$ can be determined as described\nin Refs. \\cite{{\\rFont,\\rCF}}. We find\n$$\nM_{\\psi} = \\pmatrix{\n\\- 0 &\\-  0 &\\-  0 &\\-  0 &\\-  0 &\n\\-  \\displaystyle{ {{\\psi}\\over {108\\D}} } \\cr\\noalign{\\bigskip}\n\\- 0 &\\-  0 &\\- \\displaystyle{ 54\\psi}\\def\\Ps{\\Psi} &\\-  0 &\\-  0 &\n  \\displaystyle{- {{\\psi(122\\rho}\\def\\vr{\\varrho^6 + 57\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)}\\over {6\\D}} } \\cr\\noalign{\\bigskip}\n\\displaystyle{-2^2 3^7\\psi}\\def\\Ps{\\Psi^3} &\\-  0 &\\-  0 &\\- 0 &\\-  0 &\n\\displaystyle{- {{2^2 3^5 13\\psi^5}\\over {\\D}} } \\cr\\noalign{\\bigskip}\n\\- 0 &\\- 0 &\\-  0 &\\-  0 &\\- \\displaystyle{54\\psi}\\def\\Ps{\\Psi} &\n\\-  \\displaystyle{ {{3\\psi(31\\rho}\\def\\vr{\\varrho^{12} + 57\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi +21\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2)}\\over {\\D}} } \\cr\\noalign{\\bigskip}\n\\- 0 &\\displaystyle{-2^2 3^7 \\psi}\\def\\Ps{\\Psi^3} &\\displaystyle{-2^3 3^{10} \\psi}\\def\\Ps{\\Psi^5} &\\- 0 &\\-  0 &\n\\- \\displaystyle{ {{2~ 3^6\\psi^5(242\\rho}\\def\\vr{\\varrho^6 + 237\\phi)}\\over {\\D}} } \\cr\\noalign{\\bigskip}\n\\- 0 & \\- 0 &\\-  0 & \\displaystyle{- 2^2 3^7\\psi^3 } & \\displaystyle{-2^3 3^{10} \\psi}\\def\\Ps{\\Psi^5} &\n\\displaystyle{- {{2^4 3^{10}\\psi^5(\\rho}\\def\\vr{\\varrho^6 +\\phi)^2}\\over {\\D}} }\n}\\hss$$\n\n\\bigskip\n$$\nM_{\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi} = \\pmatrix{\n\\- 0 &\\-  0 &\\- 0 &\\- \\displaystyle{1\\over 1944Z} &\\-  0\n&\\-  \\displaystyle{{\\psi}\\def\\Ps{\\Psi^2(\\rho}\\def\\vr{\\varrho^{12} + 3\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi +3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2)\\over 648 Z\\D} } \\cr\\noalign{\\bigskip}\n-3 &\\-  0 &\\- 0 &- \\displaystyle{(13\\rho}\\def\\vr{\\varrho^6 + 57\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\over 108Z} &\\-  0\n& \\displaystyle{-{\\alpha\\psi}\\def\\Ps{\\Psi^2\\over 36 Z\\D} } \\cr\\noalign{\\bigskip}\n\\- 0 &\\-  0 &\\- 0 & - \\displaystyle{279\\psi}\\def\\Ps{\\Psi^4\\over 2Z} &\\-  0\n& \\displaystyle{-{\\beta\\over 1944 Z\\D} } \\cr\\noalign{\\bigskip}\n\\- 0 & -3 &\\- 0 &\\- \\displaystyle{(\\rho}\\def\\vr{\\varrho^{12} + 3\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + 21\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2)\\over 6Z} &\\-  0\n& \\- \\displaystyle{{\\gamma}\\def\\G{\\Gamma\\psi}\\def\\Ps{\\Psi^2\\over 2 Z\\D} } \\cr\\noalign{\\bigskip}\n\\- 0 &\\- 0 & -3 &\\- \\displaystyle{2~ 3^6\\psi}\\def\\Ps{\\Psi^4(\\rho}\\def\\vr{\\varrho^6 + 2\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\over Z} &\\-  0\n&\\-  \\displaystyle{{\\delta}\\def\\D{\\Delta\\over 108 Z\\D} } \\cr\\noalign{\\bigskip}\n\\- 0 &\\- 0 &\\- 0 & -\\displaystyle{2~ 3^6\\psi}\\def\\Ps{\\Psi^4(\\rho}\\def\\vr{\\varrho^{12}+3\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi+3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2)\\over Z}\n& -3 & \\displaystyle{-{\\epsilon}\\def\\ve{\\varepsilon\\over 6 Z\\D} }\n\\cr\\noalign{\\smallskip}}$$\n\\vfill\\eject\nwhere we have defined\n$$\\eqalign{\n\\rho}\\def\\vr{\\varrho &= (162)^{1\/3}\\psi}\\def\\Ps{\\Psi \\cr\\noalign{\\medskip}\n\\D &= (\\rho}\\def\\vr{\\varrho^6+\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^3 -1 \\cr\\noalign{\\medskip}\nZ &= 1 - \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 \\cr\\noalign{\\medskip}\n\\alpha &= 13\\rho}\\def\\vr{\\varrho^{18} + 96\\rho}\\def\\vr{\\varrho^{12}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + 210\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2 + 62\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 +109 \\cr\\noalign{\\medskip}\n\\beta &= 31\\rho}\\def\\vr{\\varrho^{18} + 93\\rho}\\def\\vr{\\varrho^{12}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + 93\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2 -125\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 +125 \\cr\\noalign{\\medskip}\n\\gamma}\\def\\G{\\Gamma &=\\rho}\\def\\vr{\\varrho^{24} + 6\\rho}\\def\\vr{\\varrho^{18}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + 33\\rho}\\def\\vr{\\varrho^{12}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2 + 42\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3\n+ 30\\rho}\\def\\vr{\\varrho^6 + 9\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^4 + 54\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi  \\cr\\noalign{\\medskip}\n\\delta}\\def\\D{\\Delta &=18\\rho}\\def\\vr{\\varrho^{24} + 90\\rho}\\def\\vr{\\varrho^{18}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + 162\\rho}\\def\\vr{\\varrho^{12}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2 -116\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3\n+ 224\\rho}\\def\\vr{\\varrho^6 - 201\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^4 + 201\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi  \\cr\\noalign{\\medskip}\n\\epsilon}\\def\\ve{\\varepsilon &= \\rho}\\def\\vr{\\varrho^{30} + 6\\rho}\\def\\vr{\\varrho^{24}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + 15\\rho}\\def\\vr{\\varrho^{18}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2 - 17\\rho}\\def\\vr{\\varrho^{12}\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3\n+35 \\rho}\\def\\vr{\\varrho^{12} -60\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^4 + 69\\rho}\\def\\vr{\\varrho^6\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi -33\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^5 + 33\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^2\\cr\\noalign{\\smallskip}}$$\nThe asymptotic behavior of solutions can be calculated along each of\nthe boundary divisors in the moduli space.  When this is done, we learn\nthat the monodromy around $E_0$ includes $e^{2\\pi i\/18}$ among its\neigenvalues, while the monodromies around $E_3$ and $D_0$ include\n$e^{2\\pi i\/6}$ among their eigenvalues.  It follows that none of\nthese monodromies can be unipotent, so no large complex structure limit\\ points can occur\nalong these divisors.  This takes care of all remaining normal\ncrossing boundary points.\n\\vfill\\eject\n %\n\\section{map}{The Mirror Map}\n\\vskip-20pt\n\\subsection{Flat coordinates and symplectic basis}\nWe wish to find the explicit map between the (extended)\nK\\\"ahler-cone of \\relax\\ifmmode{\\calm}\\else $\\calm$\\fi\\ and the space of complex structures of its mirror.\nTo this end we introduce the period vector\n $$\n\\P=\\pmatrix{\\ca{G}_0\\cr \\ca{G}_1\\cr \\ca{G}_2\\cropen{3pt}\nz^0\\cr z^1\\cr z^2\\cr}~,\\qquad \\ca{G}_a\n=\\pd{\\ca{G}}{z^a}\\eqlabel{Pidef} $$\nsuch that the new periods correspond to a basis that is integral and\nsymplectic. In other words we need to find a homology basis\n$(A^a,B_b),~a,b=0,1,2$ with\n $$\nA^a\\cap A^b=0~~,~~B_a\\cap B_b=0~~,~~A^a\\cap B_b=\\delta}\\def\\D{\\Delta^a_b~. $$\nThe components of $\\P$ are then given by\n $$\nz^a=\\int_{A^a}\\O~~~~,~~~~\\ca{G}_a=\\int_{B_a}\\O~. $$\n(We are working near the toric large complex structure limit point, where\nit is appropriate to use $\\O$ to represent the 3-form, and\n$\\vp$ to describe its\nperiods.)\nWe may choose $A^0$ to be the torus corresponding to our\nfundamental period $\\vp_0$ and $B_0$ to be the three--sphere that\nshrinks to zero at the conifold. Thus\n $$\nz^0=\\vp_0~~\\hbox{and}~~\\ca{G}_0=\\vp_1 - \\vp_0~. $$\nAs explained in \\cite{\\rCdGP},\n$A^0$ and $B_0$ meet in a single point.\nFor a given choice of symplectic basis $(A^a,B_b)$ there will be a\nconstant\nreal matrix $m$ such that\n $$\\P=m\\vp~. $$\n\nOn the K\\\"ahler\\ side the analogue of the decomposition into $A$ and $B$\ncycles is the decomposition of a vector\n $$\n\\amalg = \\pmatrix{\\ca{F}_0\\cr \\ca{F}_1\\cr \\ca{F}_2\\cropen{3pt}\nw^0\\cr w^1\\cr w^2\\cr}~,\\qquad  \\ca{F}_a =\n\\pd{\\ca{F}}{w^a}\\eqlabel{ipdef} $$\nwith respect to $H^0{\\oplus} H^2{\\oplus} H^4{\\oplus} H^6$.\nThe generators of $H^0$ and $H^6$ are special and we identify them\nwith $A^0$ and $B_0$. The flat structure here is identified\nwith the natural flat structure on the K\\\"ahler-cone\n $$\nB+iJ~=~t^j e_j\\eqlabel{flatcds} $$\nwhere the $e_j$ are a basis for $H^2(\\relax\\ifmmode{\\calm}\\else $\\calm$\\fi,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$ and $t^j=w^j\/w^0$.\nFor the case in hand we may take $e_j=(H,L)$.\n\nMirror symmetry implies that the vectors $\\P$ and $\\amalg$ are\nequal up to an $Sp(6,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$ transformation. This observation\nallows the mirror map to be determined. From our discussion\nin Section~\\chapref{monodromy} we know that there must exist\nmonodromies $S_i$ corresponding to $t^j \\to t^j + \\delta}\\def\\D{\\Delta_i^j$.\nHence, there must exist periods such that their ratios\ntranslate by an integer under certain monodromies satisfying\n\\eqref{lcsl}. Since these relations are basis independent,\nwe can work directly with the vector $\\vp$ and use our\nresults in Section (6.4).\n\nWe have argued previously that the relevant monodromies\nare those associated with transport about the curves that\nmeet transversely at the large complex structure point.\nThe monodromies of $\\vp$ about these curves are\n $$\n\\hbox{\\ss G}_1 = \\hbox{\\ss T}_\\infty^3 ~~~~,~~~~\n\\hbox{\\ss D}_{\\infty} = (\\hbox{\\ss A}\\hbox{\\ss T}\\hbox{\\ss B})^{-1} = \\hbox{\\ss B}^{-1}\\hbox{\\ss T}_\\infty\n{}~. $$\nwhich we identify as $\\hbox{\\ss S}_1=\\hbox{\\ss G}_1$\nand $\\hbox{\\ss S}_2=\\hbox{\\ss D}_{\\infty}$. We then set\n $$\n\\hbox{\\ss R}_1 = \\hbox{\\ss T}_\\infty^3 - {\\bf 1}~~~~,~~~~\n\\hbox{\\ss R}_2 = \\hbox{\\ss B}^{-1}\\hbox{\\ss T}_\\infty  - {\\bf 1}\n $$\nand we check that\n $$\\eqalign{\n\\left[ \\hbox{\\ss R}_1 , \\hbox{\\ss R}_2 \\right]& = 0\\cr\\noalign{\\medskip}\n\\hbox{\\ss R}_1^3 = 9\\hbox{\\ss Y}~~,~~\\hbox{\\ss R}_1^2\\hbox{\\ss R}_2 =\n3\\hbox{\\ss Y}~~&,~~\\hbox{\\ss R}_1\\hbox{\\ss R}_2^2=\\hbox{\\ss Y}~~,~~\\hbox{\\ss R}_2^3=0 \\cr\\noalign{\\medskip}\n\\hbox{\\ss R}_1 \\hbox{\\ss Y} = 0~~~&,~~~ \\hbox{\\ss R}_2 \\hbox{\\ss Y} = 0\\cr} $$\nwith $\\hbox{\\ss Y}$ a certain matrix.\nWe see that $\\hbox{\\ss R}_1$ and $\\hbox{\\ss R}_2$ have the same algebra as $H$ and $L$\nand\nconclude that $G_1 \\cap D_{\\infty}$ indeed corresponds to the\nlarge complex structure limit.\n\nTo determine the flat coordinates we look for\nratios of periods, $t^1$ and $t^2$, that under\n$\\hbox{\\ss S}_1$ and $\\hbox{\\ss S}_2$ transform as $t^j \\to t^j + \\delta}\\def\\D{\\Delta_i^j$.\nIn this way we obtain\n $$\nt^1 =  {\\vp_3 - \\vp_0 \\over \\vp_0}~~~~,~~~~\nt^2 = {\\vp_4 + \\vp_5 - 2\\vp_3 + 2\\vp_0 \\over \\vp_0} \\eqlabel{mmap} $$\nwhere we have chosen additive constants so as to simplify\nlater constructions. These relations constitute the mirror map, they\nexpress\nthe\nflat coordinates in terms of the  $\\psi}\\def\\Ps{\\Psi$ and $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$.\n\nOur next task is to find the symplectic basis. Following a procedure\nthat is\npresented in detail in \\cite{\\rUno}, we first identify $\\amalg$ with\n$\\P$.\nThis leaves twelve undetermined parameters in the matrix $m$,\ncorresponding to the unknown periods ${\\cal F}_1$ and ${\\cal F}_2$.\nTo fix these parameters we make use of the prepotential\n $$\\eqalign{\n\\ca{F}&= -{1\\over 6w^0}\\bigg( 9\\, (w^1)^3\n+ 9\\, (w^1)^2 w^2  + 3\\, w^1 (w^2)^2 \\bigg)\n+{1\\over 2}\\bigg(\\alpha\\, (w^1)^2 + 2\\beta\\, w^1 w^2 + \\gamma}\\def\\G{\\Gamma\\, (w^2)^2\\bigg)\\cr\n&\\hskip50pt +\\bigg((\\delta}\\def\\D{\\Delta-{\\textstyle{3\\over 4}})\\, w^0w^1 + \\epsilon}\\def\\ve{\\varepsilon\\, w^0w^2\\bigg)\n+\n\\xi}\\def\\X{\\Xi (w^0)^2 + \\cdots~~,\\cr}\\eqlabel{prepotF} $$\nto construct the vector $\\amalg$ and derive the monodromy matrices\nassociated to $t^j \\to t^j + \\delta}\\def\\D{\\Delta_i^j$.\nThese $S_i$ are related to\n$\\hbox{\\ss S}_i$ in the $\\vp$ basis by $S_i = m\\hbox{\\ss S}_i m^{-1}$.\nImplementing these conditions gives all of the undetermined\nparameters in $m$\nas linear combinations of the parameters $(\\alpha,\\beta,\\ldots,\\epsilon}\\def\\ve{\\varepsilon)$.\n\nThe next step is to require that\nthe monodromy matrix $T=m\\hbox{\\ss T} m^{-1}$ be symplectic.\nThis determines $\\delta}\\def\\D{\\Delta$ and $\\epsilon}\\def\\ve{\\varepsilon$\n $$\n\\delta}\\def\\D{\\Delta=5~~~~,~~~~\\epsilon}\\def\\ve{\\varepsilon={3\\over 2}~. $$\nIt remains to find $\\alpha,\\beta,\\gamma}\\def\\G{\\Gamma$. The matrix $A=m\\hbox{\\ss A} m^{-1}$ must also\nbe integral\nand symplectic. It is symplectic for all values of the parameters but\nis integral only if $\\gamma}\\def\\G{\\Gamma$ is an integer while\n$\\alpha$ and $\\beta$ are half--integers. We take\n $$\n\\alpha={9\\over 2}~~~~,~~~~\\beta={3\\over 2}~~~~,~~~~\\gamma}\\def\\G{\\Gamma=0~. $$\nAny other choice is related to this by an $\\hbox{Sp}(6,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$\ntransformation. The result is that the matrix $m$ is given by\n $$\nm = \\matrixmf ~. $$\n\nFor completeness we also record the monodromies in the\nsymplectic basis,\n $$\nA = \\matrixAf~~,~~~T = \\matrixTf ~.   $$\nThese are the generators of the the duality group\n${\\cal D} \\subset Sp(6,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$.\n\\subsection{Inversion of the mirror map}\nOur aim is to obtain $\\psi}\\def\\Ps{\\Psi(t^1,t^2)$ and $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi(t^1,t^2)$.\nTo begin we express the relations \\eqref{mmap} as expansions valid\nfor large\n$\\rho}\\def\\vr{\\varrho$. To this end, notice that the flat coordinates\n $$\nt^1 = { \\vp_3 - \\vp_0 \\over \\vp_0 } ~~~,~~~\nt^2 = { \\sum_\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma \\vp_{3+\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma} - 3\\vp_3 + 2\\vp_0 \\over \\vp_0 } $$\ncan be expressed entirely in terms of the quantities calculated in\n\\eqref{mlarge} and \\eqref{AYdef}. This leads to the expansions\n $$\\eqalign{\n2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\, t^1 &=- \\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i - \\log \\left( {(18\\psi}\\def\\Ps{\\Psi)^6 \\over 3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}\\right)\n+ {1\\over \\vp_0} \\sum_{k=0}^\\infty \\\n{{(6k)! \\, (-3)^k }\\over {k! (2k)! (3k)! (18\\psi}\\def\\Ps{\\Psi)^{6k}}}\n\\, \\big [A_k U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + Y_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + N_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\big ] \\cr\\noalign{\\bigskip}\n2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\, t^2 &=\\- 3\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i - \\log(3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^3 - {3\\over {\\vp_0}}\n\\sum_{k=0}^\\infty \\\n{{(6k)! \\, (-3)^k }\\over {k! (2k)! (3k)! (18\\psi}\\def\\Ps{\\Psi)^{6k}}}\\, N_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\n\\cr} $$\n\nTo proceed, we introduce `large complex structure' coordinates\n $$X_1= {{(18\\psi}\\def\\Ps{\\Psi)^6}\\over {3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}} \\quad ; \\quad X_2 =(3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)^3 $$\nwhich, up to signs, are the\ninverses of the coordinates found by toric methods in\nSection~\\chapref{moduli1}. We also define functions\n $$\nu_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^{-k}\\, U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)   ~~~;~~~\n\\tilde{u}_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over =\n\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^{-k}\\, \\Big( A_k U_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + Y_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) + N_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\Big)   ~~~,~~~\n\\nu_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi) \\buildrel\\rm def\\over = \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^{-k}\\, N_k(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)  $$\nand set\n $$\nq_1 = e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i t_1}~~~~,~~~~q_2 = e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i t_2}~.$$\nIn this way we are able to rewrite the mirror map in a form that is\namenable to iterative inversion\n $$\\eqalign{\nq_1~&=~-{1 \\over X_1}\\, \\exp\\left\\{\\- {1 \\over\n\\vp_0}\\sum_{k=0}^\\infty\n{(-1)^k\\, (6k)! \\over k!\\, (2k)!\\, (3k)!\\, X_1^k}\\,\n\\tilde{u}_k(X_2)\\right\\}\n\\cropen{10pt}\nq_2~&=~-{1 \\over X_2}\\, \\exp\\left\\{ -{3 \\over \\vp_0}\\sum_{k=0}^\\infty\n{(-1)^k\\, (6k)! \\over k!\\, (2k)!\\, (3k)!\\, X_1^k}\\,\n\\nu_k(X_2)\\right\\}\\cr}\n\\eqlabel{bmap}$$\nand in these expressions $\\vp_0$ is to be expanded in the form\n $$\n\\vp_0~=~\\sum_{k=0}^\\infty\n{(-1)^k\\, (6k)! \\over k!\\, (2k)!\\, (3k)!\\, X_1^k}\\, u_k(X_2)~.$$\nThe coordinates $X_1$ and $X_2$ may be regarded as automorphic\nfunctions of the duality group. The mirror map \\eqref{bmap} is\nnaturally inverted to give these functions as expansions in the\nuniformizing variables $q_1$ and $q_2$. Thus to third order:\n $$\\eqalign{\nX_1 &= -{1\\over q_1}(1 + 312q_1 + 2q_2 + 10260q_1^2 - 540 q_1q_2\n- q_2^2 \\cr\\noalign{\\medskip}\n&\\hskip50pt - 901120q_1^3 + 120420q_1^2q_2 + 20q_2^3 + \\cdots)  \\cr\\noalign{\\bigskip}\nX_2 &= -{1\\over q_2}(1 + 180q_1 - 6q_2 + 11610q_1^2 + 180 q_1q_2\n+ 27q_2^2 \\cr\\noalign{\\medskip}\n&\\hskip50pt + 514680q_1^3 - 150120q_1^2q_2 - 5040q_1q_2^2 -\n164 q_2^3 + \\cdots)\n\\cr\\noalign{\\smallskip}} \\eqlabel{imap} $$\nNotice that the large radius limit $\\Im m\\, t^j \\to \\infty$\nmanifestly corresponds to the large complex structure limit\n$X_j \\to \\infty$.\n\\vfill\\eject\n\\section{inst}{Instanton Expansions}\n\\vskip-20pt\n\\subsection{Yukawa couplings}\nThe four Yukawa couplings $y_{\\alpha\\beta\\gamma}\\def\\G{\\Gamma} $, where\n$\\alpha,\\beta,\\gamma}\\def\\G{\\Gamma$ run over $\\psi}\\def\\Ps{\\Psi$ and $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi$, can be computed\nby means of a calculation in the ring of the\ndefining polynomial ~\n\\REFS\\rPC{P.~Candelas, \\npb{298} (1988) 458.}\n\\refsend.\nMultiplying three deformations of $p$ and\nreducing the result modulo the Jacobian ideal of $p$\nidentifies the couplings through the relation\n $$\n{\\partial_\\alpha p}\\,{\\partial_\\beta p}\\,{\\partial_\\gamma}\\def\\G{\\Gamma p} \\simeq y_{\\alpha\\beta\\gamma}\\def\\G{\\Gamma}\n{{\\bf h}\\over \\langle {\\bf h} \\rangle}\\eqlabel{ring} $$\nwhere ${\\bf h}$ denotes the\ndeterminant of the matrix of second derivatives of $p$.\n\\hbox{$\\langle {\\bf h} \\rangle$} is a normalization factor,\nindependent of the parameters, that can be fixed from our\nknowledge of the periods since we also have the relation\n $$\ny_{\\alpha\\beta\\gamma}\\def\\G{\\Gamma} = - \\Pi^T\\S\\,\\partial_{\\alpha\\beta\\gamma}\\def\\G{\\Gamma}\\Pi = -\n\\vp^T\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma\\,\\partial_{\\alpha\\beta\\gamma}\\def\\G{\\Gamma}\\vp\n\\eqlabel{yper} $$\nwith\n $$\n\\S=\\pmatrix{\\- 0&\\- {\\bf 1}\\cr\n        -{\\bf 1}&\\- 0      \\cr}\n{}~~~~\\hbox{and}~~~~\\sigma}\\def\\S{\\Sigma}\\def\\vs{\\varsigma=m^T\\S m~. $$\nUsing the expansions \\eqref{vpjr} for small $\\psi}\\def\\Ps{\\Psi$ to fix the\nnormalization\nand the ring result \\eqref{ring} we have\n $$\\eqalign{\ny_{\\psi}\\def\\Ps{\\Psi\\ps\\psi}\\def\\Ps{\\Psi}&= -{i\\over 1536\\pi^3}\\,{(18\\psi}\\def\\Ps{\\Psi)^{15}\\over {\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3\n- 1 }\\cr\\noalign{\\bigskip}\ny_{\\psi}\\def\\Ps{\\Psi\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi\\ph}&= -{3i\\over 32\\pi^3}\\,{(18\\psi}\\def\\Ps{\\Psi)^5\\over {\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3 - 1\n}  \\cr\\noalign{\\smallskip}}\n\\hskip20pt\\eqalign{\ny_{\\psi}\\def\\Ps{\\Psi\\ps\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}&=-{i\\over 128\\pi^3}\\,{(18\\psi}\\def\\Ps{\\Psi)^{10}\\over {\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3 -\n1 }  \\cr\\noalign{\\bigskip}\ny_{\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi\\ph\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}&=\n-{9i\\over 8\\pi^3}\\,\\left( {1\\over\\strut {\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3 - 1} -\n{1\\over\\strut \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 - 1 } \\right)\n\\cr\\noalign{\\smallskip}}\n\\rlap{\\hskip20pt\\boxed{$w^0=\\vp_0$}} $$\nwith\n $$\n\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi \\buildrel\\rm def\\over = \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi + \\rho}\\def\\vr{\\varrho^6~.$$\nIf we denote the parameters by $\\vph^\\alpha$ and form the tensor\n${\\bf y} = y_{\\alpha\\beta\\gamma}\\def\\G{\\Gamma}\\,d\\vph^\\alpha d\\vph^\\beta d\\vph^\\gamma}\\def\\G{\\Gamma$ then we find the\nsurprisingly\nsimple expression\n $$\n{\\bf y} = {9i\\over 8\\pi^3}\\,\\left\\{\n{d\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 \\over \\strut \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 - 1} - {d{\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3 \\over{\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3\n- 1}\n\\right\\}~.$$\nWe compute now the couplings in the flat basis, that is in\nthe coordinate basis $(t^1, t^2)$ in which \\hbox{$w^0=1$}.\nThis introduces a factor of \\hbox{$1\/\\vp_0^2$} in addition\nto the usual tensor transformation rules.\n{}From the inverse mirror map we\nfind expansions in the variables $q_j$:\n $$\\eqalign{\ny_{111} &= 9 + 540q_1 + 4860q_1^2 + 15120q_1^3 -\n    1080q_1q_2 + 1146960q_1^2q_2 + 2700q_1q_2^2+ \\cdots \\cr\\noalign{\\medskip}\ny_{112} &= 3 - 1080q_1q_2 + 573480q_1^2q_2 + 5400q_1q_2^2 + \\cdots\n\\cr\\noalign{\\medskip}\ny_{122} &= 1 - 1080q_1q_2 + 286740q_1^2q_2 + 10800q_1q_2^2 + \\cdots\n\\cr\\noalign{\\medskip}\ny_{222} &= 0 + 3q_2(1 - 360q_1 + 47790q_1^2 - 15q_2 + 7200q_1q_2\n+ 244q_2^2 + \\cdots)\n\\cr} $$\nThese expansions are compatible with the general form,\nestablished in Ref.~\n\\REFS{\\rAM}{P.~S.~Aspinwall and D.~R.~Morrison, Comm.\\ Math.\\ Phys.\\\n{\\bf 151} (1993) 245.}\n\\refsend,\nwhich is\n $$\ny_{abc} = \\smash{\\hbox{$y\\kern-4pt\\raise1pt\\hbox{${}^\\circ$}$}}_{abc} +\n\\sum_{j,k=0}^\\infty{c_{abc}(j,k) \\, n_{jk}\\, q_1^j q_2^k\n\\over 1- q_1^j q_2^k}\\eqlabel{instantsum}  $$\nwith the quantities $c_{abc}$ given by\n $$\n\\pmatrix{c_{111}\\cr\n         c_{112}\\cr\n         c_{122}\\cr\n         c_{222}\\cr} =\n\\pmatrix{j^3\\cr\n         j^2k\\cr\n         jk^2\\cr\n         k^3\\cr}~ $$\nValues for the instanton numbers $n_{jk}$ are displayed in\nTable~\\tabref{njk}. These numbers have been found independently in\n\\cite{\\rHKTY}.\n\\bigskip\n $$\n\\font\\sevenrm=cmr7 at 7pt\n\\def\\hskip2.5pt{\\hskip2.5pt}\n\\vbox{\\offinterlineskip\\halign{\n\\strut # height 12pt depth 6pt\n&\\hskip2.5pt\\skip\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\skip\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule \\cr\n\\noalign{\\hrule}\n\\vrule&\\hfil $j$&\\hskip-2.5pt\\hfil $k{=}0$&\\hfil $k=1$&\\hfil\n $k=2$&\\hfil $k=3$&\\hfil $k=4$&\\hfil $k=5$&\\hfil $k=6$&\\hfil $k=7$\\cr\n\\noalign{\\hrule\\vskip3pt\\hrule}\n\\vrule & 0 & *   &{\\-}3 & -6&{\\-}27 & -192 &{\\-}1695 & -17064\n&{\\-}188454\\hskip4.5pt\\cr\n\\vrule & 1 & 540 & -1080 &{\\-}2700 & -17280 &{\\-}154440\n& -1640520  &{\\-}19369800\\hskip1.5pt&\\cr\n\\vrule & 2 & 540 &{\\-}143370 & -574560 &{\\-}5051970 & -57879900\n&{\\-}751684050&&\\cr\n\\vrule & 3 & 540 &{\\-}204071184 &{\\-}74810520 & -913383000\n&{\\-}13593850920&&&\\cr \\vrule & 4 & 540 &{\\-}21772947555 &\n-49933059660\n&{\\-}224108858700&&&&\\cr \\vrule & 5 & 540 &{\\-}1076518252152\n&{\\-}7772494870800&&&&&\\cr \\vrule & 6 & 540\n&{\\-}33381348217290&&&&&&\\cr\n\\vrule & 7 & 540 &&&&&&&\\cr\n\\noalign{\\hrule}\n}}\n$$\n\\nobreak\\tablecaption{njk}{Numbers of instantons of type $(j,k)$ for\n$1\\leq j+k\\leq 7$. The numbers $n_{j0}$ are equal to 540 for all\n$j$.}\n %\n\\bigskip\\bigskip\n %\n\\subsection{An $\\hbox{SL}(2,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$ action on $D_{\\infty}$}\nA curious fact is apparent from Table~\\tabref{njk}; we see that\n$n_{j0}=540$\nfor\nall $j$. Related to this is the fact that when $q_2 = 0$ the\ncouplings take the\nvalues:\n $$\\eqalign{\ny_{111} &= \\smash{9 + 540 \\sum_{k=0}^\\infty {k^3 q^k \\over 1 - q^k}\n             = {27\\over 4} + {9\\over 4}E_2}\\cropen{5pt}\ny_{112} &= 3\\cropen{5pt}\ny_{122} &= 1\\cropen{5pt}\ny_{222} &= 0\\cr}\\hskip30pt\\boxed{$q_2=0$}$$\nWhere\n $$\n\\eqalign{\nE_2(t)\n&\\buildrel\\rm def\\over = 60\\,\\sum_{(m,n)\\neq (0,0)}\\,{1\\over\n(m+nt)^4}\\cropen{5pt}\n&= 1 + 240\\,\\sum_{k=0}^\\infty {k^3 q^k \\over 1 -q^k}~, \\hskip25pt\nq=e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi it}\\cr}$$\nis the Eisenstein function of weight\ntwo \\Ref\\Erdelyi{A.~Erd\\'elyi, F.~Oberhettinger, W.~Magnus and\nF.~G.~Tricomi,\n{\\sl Higher Transcendental Functions} vol.~III, McGraw--Hill 1953.}.\nRecall\nthat a function, $f$, is automorphic of weight $m$ if under an\n$\\hbox{SL}(2,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$ transformation\n $$\nt\\longmapsto \\tilde{t}={at+b\\over ct+d}$$\nit transforms according to the rule\n $$\nf(t)\\longmapsto f(\\tilde t) = (ct+d)^{2m} f(t)~.$$\nIn order to see the $\\hbox{SL}(2,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$ action we set\n $$\\eqalign{\nt~&=~\\hphantom{3\\over 2}t^1\\cr\ns~&=~{3\\over 2} t^1 +t^2\\cr}$$\nand observe that\n $$\\eqalign{\n\\ca{A}^3\\,t~&=~{\\vp_6 - \\vp_3\\over\\vp_3}~=~-{\\vp_0\\over\\vp_3}~\n=~-{1\\over t+1}\\cr\n\\ca{T}_\\infty^3\\, t~&=~t+1~.\\cr}$$\nSo the operations $\\ca{A}^3$ and $\\ca{T}_\\infty^3$ generate an\n$\\hbox{SL}(2,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$\nwhen acting on $t$. We set\n $$\nr~=~e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i s}~=~q_1^{3\\over 2}q_2~.$$\nThe limit $q_2\\to 0$ with $q_1$ finite is the limit $r\\to 0$ with\n$q_1$ finite.\n{}From \\eqref{bmap} we see that when $r$ is small we have the\nasymptotic\nrelation\n $$\nr~\\sim~{\\hbox{const.}\\over \\rho}\\def\\vr{\\varrho^9\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^{3\\over 2}}~.\\eqlabel{smallr}$$\nSo the locus $r=0$ is $D_{\\infty}$ and we see from \\eqref{smallr}\nthat\n $$\n\\ca{A}^3\\,r~=~r~~~~\\hbox{and}~~~~\\ca{T}_\\infty^3\\,r~=~-r$$\nand hence that the locus $r=0$ is preserved by $\\ca{A}^3$ and\n$\\ca{T}_\\infty^3$.\n\nWith respect to the coordinates $t$ and $s$ the couplings become\n $$\\eqalign{\ny_{ttt} &={9\\over 4} E_2(t)\\cropen{5pt}\ny_{tts} &= 0\\cropen{5pt}\ny_{tss} &= 1\\cropen{5pt}\ny_{sss} &= 0~.\\cr}\\hskip30pt\\boxed{$r=0~,~w^0=1$}\\eqlabel{yonDinf}$$\nNow the holomorphic three-form $\\O$ is invariant under the\n$\\hbox{SL}(2,\\relax\\ifmmode\\hbox{\\ss Z\\kern-.4em Z}\\else{\\ss Z\\kern-.4em Z}\\fi)$\ngenerators in the gauge $w^0 = \\vp_0$. Achieving the gauge $w^0=1$\nrequires\ndividing by $\\vp_0$ which is not invariant under $\\ca{A}^3$. In fact\n $$\n\\ca{A}^3\\,\\vp_0~=~\\vp_3~=(t+1)\\vp_0$$\nso in the new gauge $\\O$ has weight $-{1\\over 2}$. Furthermore\n $$\n\\pd{}{t}\\longmapsto \\pd{}{\\tilde t}~=~(ct+d)^2\\pd{}{t}$$\nso each $t$-derivative counts as weight one. Thus we see that\n$y_{ttt}$ has\nweight two. The relations \\eqref{yonDinf} show that the Yukawa\ncoupling, in the\ngauge $w^0=1$, is nonsingular on $D_\\infty$. This was perhaps to be\nexpected\nsince the loci $B_{con}$ and $C_{con}$ where the coupling is\nsingular do not intersect $D_\\infty$ in the resolved moduli space.\nTurning the\nargument around; if we assume that the coupling is nonsingular then\nsince the only automorphic function of weight two that is regular on\nthe upper\nhalf $t$-plane and bounded as $q\\buildrel\\rm def\\over = e^{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i t}\\to 0$ is $E_2$ it\nmust be\nthe\ncase that $y_{ttt}$ is proportional to $E_2$. The coupling $y_{tts}$\nhas weight\none and must vanish since there is no automorphic function of weight\none. The\ncoupling $y_{tss}$ has weight zero and must be a constant since the\nonly\nautomorphic function of weight zero that is bounded as $q\\to 0$ is a\nconstant.\nFinally $y_{sss}$ has weight $-1$ and must vanish since there is no\nautomorphic\nfunction of this weight.\n\nWe may also write the large complex structure variable\n $$\nX_1~=~{(18\\psi}\\def\\Ps{\\Psi)^6 \\over 3\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}~=~432{\\rho}\\def\\vr{\\varrho^6 \\over \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}$$\nin terms of automorphic functions. Clearly $X_1$ is invariant under\nthe\ngenerators $\\ca{A}^3$ and $\\ca{T}_\\infty^3$, when $q_2=0$, so we expect\n$X_1$ to be\nrelated to the $J$-invariant\n $$\n12^3\\,J(t)~=~{1\\over q_1} + 744 + 19688\\,q_1 +\\cdots~. $$\nNow $t_1$, being a ratio of periods, is invariant under $\\ca{I}$ so\n$J(t)$ is\nalso invariant. We see however that $X_1$ is not invariant. In fact\nwe have\n $$\n\\ca{I}X_1~=~-432 {X_1 \\over X_1 + 432}~~~~,~~~~~\\ca{I}^2\nX_1~=~X_1~.$$\nThe basic invariant is\n $$\nX_1 + \\ca{I}X_1~=~{X_1^2 \\over X_1 + 432}$$\nany other invariant combination being a function of this one. We\ntherefore\nexpect a relation of the form\n $$\n{X_1^2 \\over X_1 + 432}~=~f(J) $$\nwith $f$ a rational function of $J$. From \\eqref{mmap} we see that\n$X_1\\sim -q_1^{-1} - 312$ as $q_1\\to 0$ and this information is\nsufficient\nto determine that $f(J)=-12^3J$. Thus the relation is\n $$\n{X_1^2 \\over X_1 + 432}~=~-12^3\\,J(t) $$\nor equivalently\n $$\n{\\rho}\\def\\vr{\\varrho^6 \\over \\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}~=~\n-2J(t)\\left\\{ 1 + \\sqrt{1 - {1\\over J(t)}}\\right\\}~.$$\n %\n\\subsection{Instantons of genus one}\nFollowing Bershadsky {\\it et al.\\\/}\n\\cite{\\rBCOV}\\\nwe consider the index $F_1$\ndefined by a certain path integral and whose topological limit\nfor two-parameter Calabi--Yau\\ threefolds is given by\n $$ F_1^{top} =\n\\log \\left[\\left({\\psi}\\def\\Ps{\\Psi\\over \\vp_0}\\right)^{5-{\\chi\\over 12}}\n\\,{\\partial(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\over \\partial(t^1,t^2)}\\, f \\right] +\n\\hbox{const.} ~, \\eqlabel{topF} $$\nThe holomorphic function $f$ is determined by requiring\nregularity of $F_1^{top}$ at smooth points of moduli space\nand by imposing the large radius limit condition\n $$\\eqalign{\nF_1^{top}&\\sim -{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\over 12}c_2\\cdot(B+iJ)\\cropen{10pt}\n         &= -{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\over 12}c_2\\cdot(t^1H +t^2L)\n\\cr} \\eqlabel{limF}\n $$\nThe regularity conditions are most easily implemented by\nusing the mirror moduli. Thus, $F_1^{top}$ can only diverge\nat the singular loci ${\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3=1$\nand $\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3=1$. Furthermore $F_1^{top}$\nmust be regular at $\\psi}\\def\\Ps{\\Psi=0$ since the corresponding manifold\nis nonsingular.\n\nSince $\\vp_0\\sim\\psi}\\def\\Ps{\\Psi$ for $\\psi}\\def\\Ps{\\Psi$ small,\nwe conclude that $f$ must have the form\n $$\nf=({\\tilde\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi}^3 -1)^a\\,(\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi^3 - 1)^b\\, \\psi}\\def\\Ps{\\Psi^c \\eqlabel{fhol} $$\nwhere the exponent $c$ is fixed by the behavior of the Jacobian\n${\\partial(\\psi}\\def\\Ps{\\Psi,\\phi}\\def\\Ph{\\Phi}\\def\\vph{\\varphi)\\over \\partial(t^1,t^2)}$ at $\\psi}\\def\\Ps{\\Psi=0$.\n{}From the mirror map \\eqref{mmap} and the period expansions\n\\eqref{vpjr} we find that the leading term of this\nJacobian is $\\psi^{-3}$ hence $c=3$.\nThe remaining exponents are then\ndetermined by the large radius limit condition.\n{}From \\eqref{topF}, \\eqref{fhol} and the inverse\nmirror map \\eqref{imap}, we see that\n $$\nF_1^{top} \\sim -{2\\pi i\\over 12}\\left\\{\n\\left[108 + 36a\\right]\\,  t^1\n+ \\left[40 + 12a + 12b \\right]\\,\n t^2\\right\\} ~. $$\nFor the model \\Mthree we have\n $$\\chi=-540~~,~~~~c_2\\cdot H=102~~,~~\\hbox{and}~~c_2\\cdot L=36 $$\nhence we find\n $$a=-1\/6~~\\hbox{and}~~b=-1\/6~. $$\n\nIn virtue of mirror symmetry $F_1^{top}$ enjoys an expansion\n $$\nF_1^{top} = -{2\\pi}\\def\\P{\\Pi}\\def\\vp{\\varpi i\\over 12}c_2\\cdot(B+iJ) + \\hbox{const.} -\n\\,\\sum_{jk}\\left[ 2 d_{jk}\\log\\eta(q_1^j q_2^k) +\n{1\\over 6}n_{jk}\\log(1 - q_1^j q_2^k) \\right]~, \\eqlabel{Finst} $$\nwhere $\\eta$ denotes the Dedekind $\\eta$-function\\Footnote{Note that,\nas observed in \\cite{\\rBCOV}, shifting ${\\partial\\over\\partial t}\\log\\eta$\nby a constant does not affect the final outcome.  We did this for\nsimplicity in \\cite{\\rUno}.} and $d_{jk}$ and $n_{jk}$\nare the numbers of instantons of genus one and genus zero.\n\nComparing this expansion with the expansion that results\nfrom substituting the explicit form for $f$ that we have\nfound in \\eqref{topF} we find values for the $d_{jk}$ displayed in\nTable~\\tabref{djk}.\n\\vfill\\eject\n\\vbox{\n $$\n\\font\\sevenrm=cmr7 at 7pt\n\\def\\hskip2.5pt{\\hskip2.5pt}\n\\vbox{\\offinterlineskip\\halign{\n\\strut # height 12pt depth 6pt\n&\\hskip2.5pt\\skip\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\skip\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule\n&\\hskip2.5pt\\sevenrm #\\hskip2.5pt\\hfil\\vrule \\cr\n\\noalign{\\hrule}\n\\vrule&\\hfil $j$&\\hskip-2.5pt\\hfil $k{=}0$&\\hfil $k=1$&\\hfil\n $k=2$&\\hfil $k=3$&\\hfil $k=4$&\\hfil $k=5$&\\hfil $k=6$&\\hfil $k=7$\\cr\n\\noalign{\\hrule\\vskip3pt\\hrule}\n\\vrule & 0 & * &{\\-}0 &{\\-}0 & -10 &{\\-}231 & -4452&{\\-}80958\n& -1438086\\hskip2.5pt\\skip\\cr\n\\vrule & 1 & 3 & -6 &{\\-}15&{\\-}4764 & -154662 &{\\-}3762246 &\n-82308270\\hskip2.5pt&\\cr\n\\vrule & 2 & 0 &{\\-}2142 & -8568 & -1079298 &{\\-}48907800 &\n-1510850250&&\\cr\n\\vrule & 3 & 0 & -280284 &{\\-}2126358 &{\\-}152278992 &\n-9759419622&&&\\cr\n\\vrule & 4 & 0 & -408993990 &{\\-}521854854 & -16704086880&&&&\\cr\n\\vrule & 5 & 0 & -44771454090 &{\\-}1122213103092&&&&&\\cr\n\\vrule & 6 & 0 & -2285308753398&&&&&&\\cr\n\\vrule & 7 & 0&&&&&&&\\cr\n\\noalign{\\hrule}\n}}\n$$\n\\nobreak\\tablecaption{djk}{Values $d_{jk}$ of genus one instantons\nfor\n$1\\leq j+k\\leq 7$}} %\n\\vfill\\eject\n\\section{true}{Verification of Some Instanton Numbers}\nIn this section, we verify selected instanton numbers which occur in\nTable~\\tabref{njk} and Table~\\tabref{djk}.  In the process, we\nobserve\nsome new phenomena which did not arise in \\cite{\\rUno}: how to\ncalculate some\ninstanton contributions from the topology of singular instanton\nmoduli\nspaces, and the occurence of negative instanton contributions for\nrational\ncurves.\n\nThe first fact we will use is the following.  Suppose that a complete\nfamily of Gorenstein curves\\Footnote{The statement probably remains true even\nif the restriction to Gorenstein curves is removed.} is parameterized by a\n{\\it nonsingular} manifold $B$ of dimension $b$.  Then the instanton\ncontribution\nis the Chern number $c_b(\\Omega^1_B)$, where $\\Omega^1_B$ is the\nholomorphic\ncotangent bundle of $B$.  Equivalently, this is $(-1)^be(B)$, where\n$e(B)$\nis the Euler characteristic of $B$.\n\nThere are a few ingredients used in establishing this fact.  First we\nshow\nthat the deformation-theoretic obstruction bundle in this situation\nis just\n$\\Omega^1_B$.  Next, we show that a deformation of almost complex\nstructure\ngives rise to a $C^\\infty$ section of the obstruction bundle; the\nzero locus\nof such a section gives the parameter values for which the\ncorresponding\ncurve deforms to a pseudo-holomorphic curve on the infinitesimally\nnearby\nalmost complex manifold.  It can be shown by McDuff's transversality\ntheorem\n\\ \\REFS{\\McD}{D.~McDuff, Inv.\\ Math.\\ {\\bf 89} (1987) 13.}\\refsend\\\nthat for rational curves, the generic deformation yields only\nfinitely many\ncurves that deform, and furthermore that there are no higher-order\nobstructions, i.e.\\ the curves that infinitesimally deform actually\ndeform\nin a sufficiently small but finite deformation.\n\nFor the first assertion, we assume for simplicity of exposition that\nwe have\na smooth curve $C\\subset X$.\nConsider the normal bundle $N$ of $C$ in $X$.  For the assertion\nabout the\nform of the obstruction bundle, we first assert\nthat the obstruction bundle is the natural bundle with fiber equal to\n$H^1(N)$  \\Footnote{This was asserted without proof in\n\\ \\REFS{\\rat}\n{S.~Katz,  ``Rational\ncurves on Calabi-Yau threefolds'', in {\\it Essays on Mirror\nManifolds}, ed.\\\nS.-T.~Yau (Intl.\\ Press, Hong Kong, 1992).}\n\\refsend; we\nwill sketch the proof of this presently.}.\nThe Calabi-Yau condition leads\nto $\\wedge^2N\\simeq\\Omega^1_C$.  Also, deformation theory describes\n$H^0(N)$ as the space of first order deformations of $C$ inside $X$,\nand\n$H^1(N)$ as the space of obstructions.  Serre duality gives an\nisomorphism\n$$H^0(N)\\otimes H^1(N)\\to H^1(\\wedge^2N) \\simeq H^1(\\Omega^1_C)\n\\simeq \\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}.$$\nThe last isomorphism is canonically fixed by sending the fundamental\nclass of\na point to 1.\nSince $H^0(N)$ is canonically\nisomorphic to the tangent space to $B$ at the point of\n$B$ corresponding to $C$, it follows that $H^1(N)$ is canonically\nisomorphic to the\ncotangent space of $B$. A generic deformation of almost complex\nstructure of\n$X$ induces an obstruction class in $H^1(N)$ whose vanishing is\nnecessary\nand sufficient for the first-order deformation of $C$ to a\npseudo-holomorphic\ncurve on the nearby almost complex manifold.  As the curve varies\nover the\nparameter space $B$, we get a smooth section of $\\Omega^1_B$ (whose\nvalue\nat $C$ is the corresponding element of $H^1(N)$), with the indicated\nidentifications.  We conclude by observing that $c_b(\\Omega^1_B)$ is\nthe\nnumber of zeros (counted with multiplicity) of a general section of\n$\\Omega_B^1$.  We will see below that negative multiplicities are\npossible.\n\nWe next turn to the calculation of the obstruction bundle.\nTo do this, it will be helpful to adapt the deformation theory of\nalmost\ncomplex structures\n\\ \\REFS{\\kur}{M.~Kuranishi, Ann.\\ Math.\\ {\\bf 75} (1962).}\\refsend\n\\ to our situation.\n\nWe allow the almost complex structure of $X$ to vary by varying the\nholomorphic cotangent space $T^{*1,0}$.  We accomplish this by\nvarying the\nprojection map $T_{\\scriptstyle\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}}^*\\to T^{*1,0}$ to a family of\nprojection maps\n$$\\pi_t : T_{\\scriptstyle\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}}^*\\to T^{*1,0} \\eqlabel{proj}$$\nand for each $t$ defining the deformed holomorphic cotangent space\n$T^{*1,0}_t=\\ker(\\pi_t)$.\n\nThe adjoint map of tangent spaces\n$$\\pi_t^* : T_{0,1} \\to T_{\\scriptstyle\\relax\\hbox{\\kern.25em$\\inbar\\kern-.3em{\\rm C}$}} \\eqlabel{incl}$$\nhas its image annihilated by $T_t^{*1,0}$; hence it is the deformed\nantiholomorphic\ntangent space $T_{01}^t$.  After taking complex conjugates to get\n$T^t_{1,0}$,\nit is easy to write down a projection with kernel $T^t_{1,0}$, giving\na\nconvenient description of $T^t_{1,0}$.\n\nTo describe $T_t$, it suffices to give $\\pi_t|_{T^{*1,0}} :\nT^{*1,0}\\to\nT^{*0,1}$.  In local holomorphic coordinates $z_i$ on $X$ we describe\nthis\ndata by a tensor\n$$A=A^i_{\\overline\\jmath}\\frac{\\partial}{\\partial z^i} \\otimes\ndz^{\\overline\\jmath},\n\\eqlabel{ks}$$\nand almost complex structures near $X$ are parametrized by tensors\n$A$ as\nin \\eqref{ks} which are near 0.\n\nCarrying out the computation outlined above, we find that $T^t_{1,0}$\nis the\nkernel of the operator $P_t$ with\n$$P_t(\\frac{\\partial}{\\partial z^i}) = -{\\overline A}^{\\overline\\jmath}_i\n\\frac{\\partial}{\\partial z^{\\overline\\jmath}} \\qquad\nP_t(\\frac{\\partial}{\\partial z^{\\overline\\jmath}}) =\n\\frac{\\partial}{\\partial z^{\\overline\\jmath}}. \\eqlabel{Pt}$$\n\nFor ease of exposition we only illustrate the deformation theory of\nrational\ncurves, contenting ourselves with a few comments about what changes\nin the\ncase of elliptic curves.\nSo we consider a holomorphic map\n$$f:\\relax{\\rm I\\kern-.18em P}^1\\to X \\eqlabel{map}.$$\nUsing  a local coordinate $w$ on $\\relax{\\rm I\\kern-.18em P}^1$, we express $f$ as\n$z^i=f^i(w)$ locally.  Now, vary the almost complex structure as a\nfunction of a parameter $t$.  We have $A_{\\overline\\jmath}^i =\nA_{\\overline\\jmath}^i(t)$\nand $z^i=f^i(w,t)$.  The pseudo-holomorphicity condition is\n$P_tf_*(\\frac{\\partial}{\\partial w}) = 0$.  This becomes\n$$\\frac{\\partial f^{\\overline\\imath}}{\\partial w}\\frac\\partial{\\partial\nz^{\\overline\\imath}}\n- \\frac{\\partial f^j}{\\partial w} {\\overline A}^{\\overline\\imath}_j\n\\frac\\partial\n{\\partial z^{\\overline\\imath}} = 0. \\eqlabel{pholo}$$\nTaking complex conjugates, multiplying by $d{\\overline w}$, and\ndifferentiating\nat $t=0$, we get\n$${\\overline\\partial}(f^{\\prime i}\\frac\\partial{\\partial z^i}) = f^*(A').\n\\eqlabel{Dolbeault}$$\nHere $f^{\\prime i}$ means $\\frac{\\partial f^i}{\\partial t}(w,0)$ and\n$$A' = \\frac{\\partial A^i_{\\overline\\jmath}}{\\partial t}({\\overline z},0)\n\\frac\\partial{\\partial z^i} \\otimes dz^{\\overline\\jmath}.\n\\eqlabel{ksclass}$$\nThus $A'$ is a $(0,1)$ form on $\\relax{\\rm I\\kern-.18em P}^1$ with values in $f^*(T_{1,0})$.\nEquation~\\eqref{Dolbeault} says that $A'$ represents the zero class\nin\n$$H_{{\\overline \\partial}}^{0,1}(f^*(T_{1,0})) \\simeq\nH^1(f^*T_{1,0}).$$\nHowever, from the exact sequence\n$$0\\to T_{\\relax{\\rm I\\kern-.18em P}^1}\\to f^*(T_{1,0})\\to N \\to 0 \\eqlabel{nbs}$$\nand vanishing of $H^1(T_{\\relax{\\rm I\\kern-.18em P}^1})$, we conclude that\n$H^1(f^*T_{1,0})\\simeq\nH^1(N)$, and so\nthe obstruction section\n$A'$ may be thought of as lying in the claimed obstruction bundle.\n\nIn this way, we get a ($C^\\infty$) obstruction section of\n$\\Omega^1_B$.\nBy McDuff's transversality theorem, this section vanishes at finitely\nmany points if the deformation is generically chosen.  The Euler\nclass\n(or top Chern class) of $\\Omega^1_B$ computes the number of such\nzeros,\nwhere each zero is counted with multiplicity $\\pm 1$ since the\nsection\nis not holomorphic in general.  Note that the space of\npseudoholomorphic maps for the generic almost complex structure\ncarries a preferred orientation\n\\ \\REFS{\\ruan}{Y.~Ruan, ``Topological Sigma model and Donaldson type\ninvariants\nin Gromov Theory'', Preprint.}\\refsend.  There is then an induced\norientation\non the\nlimiting set of maps.  If this orientation differs from the\norientation\ndetermined by the complex structure, then the associated multiplicity\nis\n$-1$.  It can be seen that this multiplicity agrees with the\nmultiplicity\narising from the description of the limiting curves as the zero locus\nof\nthe obstruction section.\n\nFor elliptic curves, there is a complication: elliptic curves have\nmoduli,\nso the elliptic curve $E$ used as the source of a map analogous to\n\\eqref{map}\nmust be allowed to vary with the parameter $t$.\nThis can be accomplished\nby allowing the local parameter $w$ for $E$ depend on $t$.  In doing\nso,\nextra data is introduced\ncorresponding to deformations of the complex structure\nof $E$; these are parametrized by $H^1(T_E)$, where $T_E$ is the\nholomorphic\ntangent bundle of $E$.  Now even if the obstruction element of\n$H^1(f^*(T_{1,0}))$ does not vanish, we may be able to deform $E$ and\nso\nmodify the obstruction section by an element of $H^1(T_E)$.  Thus the\ntrue obstruction data lives in the quotient of $H^1(f^*(T_{1,0}))$ by\nthe\nimage of $H^1(T_E)$ given by the cohomology of the exact sequence\n\\eqref{nbs};\nthis sequence also tells us that the obstruction space is just\n$H^1(N)$.\n\nWe now immediately can verify some of the numbers in\nTable~\\tabref{njk}.\nFor $n_{01}$, we must enumerate curves $C$ with $C\\cdot H=0$ and\n$C\\cdot L=1$.  The first equality implies that $C$ is contained in\nthe\nexceptional divisor $E\\simeq\\relax{\\rm I\\kern-.18em P}^2$; and the second equality shows\nthat\n$C$ is a line in this $\\relax{\\rm I\\kern-.18em P}^2$.  Since the lines in $\\relax{\\rm I\\kern-.18em P}^2$ are\nparametrized\nby (the dual) $\\relax{\\rm I\\kern-.18em P}^2$, we verify\n$$n_{01}=c_2(\\Omega^1_{\\relax{\\rm I\\kern-.18em P}^2})=3.$$\nSimilarly, $n_{02}$ counts the contribution of conics in\n$E\\simeq\\relax{\\rm I\\kern-.18em P}^2$.\nSince conics are parametrized by $\\relax{\\rm I\\kern-.18em P}^5$, we have\n$$n_{02}=c_5(\\Omega^1_{\\relax{\\rm I\\kern-.18em P}^5})=-6.$$\nWe interpret this as follows: given a general deformation of almost\ncomplex\nstructure, at least 6 of the conics will deform to pseudo-holomorphic\ncurves\non the nearby almost complex manifold.  Each deformed curve has a\nmultiplicity\nof $\\pm 1$ determined by its intrinsic orientation, and the algebraic\nsum\nof these multiplicities is $-6$.\nUnfortunately, the moduli space of rational curves of degree $k$ in\n$\\relax{\\rm I\\kern-.18em P}^2$ is more complicated for $k>2$, so this is as much as we can\nsay here\nwithout more work.\n\nWe can also verify the ratios $n_{1k}\/n_{10}$ for $k=1$ and 2.  If\n$C$ satisfies $C\\cdot H=1$ and $C\\cdot L=k$, then $C\\cdot E = 1 - 3k\n< 0$,\nrecalling that $E = H - 3L$.  Thus $C$ has a component which is\ncontained\nin $E$, and it follows immediately that $C$ is a union of a rational\ncurve\n$C'$ with $C'\\cdot H = 1$ and $C'\\cdot L = 0$ and a degree $k$ curve\nin\n$E\\simeq\\relax{\\rm I\\kern-.18em P}^2$.  Note that $C'\\cdot E=1$, so that $C'$ meets $E$ in a\nunique\npoint $p$.  So for each curve $C'$ of type $(1,0)$, we can take a\ndegree $k$\ncurve $D$ in $E$ passing through $p$ to get a connected curve\n$C=C'\\cup D$\n(degenerate instantons must be connected, cf Appendix to\n\\cite{\\rBCOV}).  The\ndegree $k$ curves are parametrized by $\\relax{\\rm I\\kern-.18em P}^1$ for $k=1$ and by\n$\\relax{\\rm I\\kern-.18em P}^4$\nfor $k=2$.  Since we get the same parameter space for any curve of\ntype\n$(1,0)$, we can verify\n$$\\frac{n_{11}}{n_{10}}=c_1(\\relax{\\rm I\\kern-.18em P}^1)=2$$\nand\n$$\\frac{n_{12}}{n_{10}}=c_4(\\relax{\\rm I\\kern-.18em P}^4)=-5.$$\n\nFinally, we check that $n_{10}=540$, and give some supporting\ngeometric evidence for\n$n_{j0}=540$ for all $j\\ge 1$.\nA curve $C$ of type $(1,0)$ satisfies $C\\cdot L=0$, hence is\nan elliptic curve, a fiber of the fibration discussed in\nsection~\\chapref{model}.\nWe want to see when the elliptic curve can acquire a singularity, to\nallow\nit to be the image of a holomorphic map from $\\relax{\\rm I\\kern-.18em P}^1$.  Recalling that\n$C$\nis obtained by fixing the values of $x_1,\\ x_2$, and $x_3$, we get an\nequation for $C$ of the form\n$$ax_5^2+bx_4^3+cx_4x_5+dx_4^2+ex_5+fx_4+g=0,$$\nthe constants being determined by the equation for\n$\\relax\\ifmmode\\preMhat\\else $\\preMhat$\\fi \\subset\\hbox{$\\IP^{(1,1,1,6,9)}$}$ and\nby the $x_1,x_2,x_3$ coordinates.  It is easy to change coordinates\nto arrive\nat the form\n$$x_5^2+x_4^3+fx_4+g=0.$$\nLetting $(x_1,x_2,x_3)$ vary in its parameter space $\\relax{\\rm I\\kern-.18em P}^2$, we\nrealize\nthat $f=f(x_1,x_2,x_3)$ has degree~12, and $g=g(x_1,x_2,x_3)$ has\ndegree~18.\nThe discriminant of our family of curves is\n$$4f^3-27g^2=0 \\eqlabel{discriminant},$$\na plane curve $B$ of degree~36.  $B$ has cuspidal singularities at\nthe\n216 points where $f=g=0$.  Since $B$ therefore does not have a\ncotangent\nbundle, it is easier to perform the obstruction analysis on\n$\\tilde B$, the normalization of $B$.  By the genus formula, $\\tilde\nB$ has\ngenus $(35)(34)\/2-216=379$.  A careful analysis shows that the\nobstruction\nsection of $\\Omega^1_{\\tilde B}$ has extraneous zeros at the points\nof\n$\\tilde B$ which lie over the singularities of $B$.  So the number of\ncurves\nwhich deform in a generic deformation is $c_1(\\Omega^1_{\\tilde\nB})-216=540$.\nAn analogous calculation has been done recently for a different Calabi-Yau\nmanifold in \\ \\REFS{\\des}{D.E.~Sommervoll, Rational curves of low degree on a\ncomplete intersection Calabi-Yau threefold in $\\relax{\\rm I\\kern-.18em P}^3\\times\\relax{\\rm I\\kern-.18em P}^3$, Oslo preprint\nNo.11 June 1993}\\refsend.\n\nFor $j>1$ there are no rational curves of type $(j,0)$, so our instanton\ncalculation is detecting degenerate instantons of type $(j,0)$.\nOne possible description is as follows.  We have a family of rational curves\nparametrized by $B$.  All of these curves have arithmetic genus~1, but are\ninstantons (i.e.\\ rational) because of the singularity that each curve\ncontains.  If we\nassign to each curve the multiplicity $j$ (in other words take the local\nequation of the curve inside the total space of the family, then raise this\nto the power $j$), this gives a family of curves with multiplicity $j$\nparametrized by the same moduli space $B$.  The adjunction formula gives\nthat each of these curves again has arithmetic genus~1.  It is natural\nto speculate that the singularity on each curve gives rise to an interpretation\nof the multiple curves as degenerate instantons.  If this were true, then\nthe geometric calculation of $n_{j0}$ would be identical to the calculation\nof $n_{10}$ above.\nA complete verification of this approach must wait for future work.\n\nLet us now turn to Table~\\tabref{djk}.  For type $(0,k)$, we have\nalready\nseen that they are parametrized by curves of degree $k$ in\n$E\\simeq\\relax{\\rm I\\kern-.18em P}^2$.\nSuch curves are rational if $k<3$, but are elliptic for $k=3$.  The\ncubic curves are parametrized by $\\relax{\\rm I\\kern-.18em P}^9$.  Hence\n$$d_{01}=d_{02}=0,\\ d_{03}=c_9(\\Omega_{\\relax{\\rm I\\kern-.18em P}^9}^1)=-10.\\eqlabel{d0k}$$\nWe have already seen that curves of type $(1,0)$ are elliptic and are\nparametrized by $\\relax{\\rm I\\kern-.18em P}^2$.  So $d_{10}=c_2(\\Omega^1_{\\relax{\\rm I\\kern-.18em P}^2})=3$.  The\nsame argument that we used for rational curves shows that\n$d_{11}=d_{10}\\cdot c_1(\\Omega^1_{\\relax{\\rm I\\kern-.18em P}^1})=-6$ and $d_{12}=d_{10}\n\\cdot c_4(\\Omega^1_{\\relax{\\rm I\\kern-.18em P}^4})\n=15$.\n\nWe also have a consistency check between $d_{13}$ and $n_{13}$.\nCurves of\ntype $(1,3)$ are unions of curves of type $(1,0)$ and of type\n$(0,3)$.\nSince a curve of type $(1,0)$ meets $E$ once, we see that the moduli\nspace\nof curves of type $(1,3)$ with a fixed $(1,0)$ component is\nisomorphic to the\nspace of rational cubic curves containing a fixed point.  This moduli\nspace\ntherefore contributes $n_{13}\/n_{10} = -32$.  But elliptic curves\ncome in\ntwo types: the $(1,0)$ could be rational and the $(0,3)$ elliptic, or\nvice versa.  The elliptic curves through a point are parametrized by\na\n$\\relax{\\rm I\\kern-.18em P}^8$ hence\n$$d_{13} = d_{10} \\cdot (-32) + n_{10} \\cdot c_8(\\Omega^1_{\\relax{\\rm I\\kern-.18em P}^8}) =\n-96 +4860 = 4764.$$\n\\vskip1in\n\\acknowledgements\n\nWe would like to acknowledge fruitful conversations with Xenia de la Ossa,\nand a helpful conversation with Yongbin Ruan.\nA.F. thanks the Centro Cient\\'{\\i}fico IBM-Venezuela for the use\nof its facilities, Fundaci\\'on Polar for providing funds to acquire\nMathematica and the ICTP-Trieste as well as the SLAC Theory Group for\ntheir hospitality while part of this work was done.\n\n\\vfill\\eject\n\\listreferences\n\\end\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\\subsection{Orthogonal polynomials}\n\nOrthogonal polynomials appear in many areas of modern mathematics and mathematical physics \\cite{Chihara, Ismail} (e.g., approximation theory, stochastic processes, random matrix theory and others). In this paper we are interested in discrete $q$-orthogonal polynomials on an exponential lattice. The orthogonality condition for discrete $q$-orthonormal polynomials is given by\n$$\n\\int_a^b p_k(x) p_n(x) w(x) d_q x= \\delta_{k,n},\n$$\nwhere the $q$-integral  \\cite{GR} is defined by\n$$\n\\int_a^b f(x) d_q x=b(1-q)\\sum_{n=0}^{\\infty} q^n f(b q^n)-a(1-q)\\sum_{n=0}^{\\infty}q^n f(a q^n).\n$$\nHere the weight function $w$ is  supported on the exponential lattice $$\\{a q^n,\\;b q^n\\,|\\,n\\in\\mathbb{N}_0\\}$$ and $\\delta_{k,n}$ is the Kronecker delta.\n\n\nThe classical examples include little $q$-Laguerre polynomials, which are orthogonal on the exponential lattice $\\{q^k\\,|\\,k\\in\\mathbb{N}_0\\}$ with respect to the weight function $$w(x)=x^{\\alpha}(qx;q)_{\\infty},\\;\\;\\alpha>-1,\\;\\;q\\in(0,1),$$ where $$(a;q)_{\\infty}=\\prod_{k=0}^{\\infty}(1-a q^k).$$ They can be written in terms of the basic hypergeometric function ${}_2\\phi_1$.\n\nOne of the main features of orthogonal polynomials is the three-term recurrence relation $$x p_n(x)=a_{n+1}p_{n+1}(x)+b_np_n(x)+a_n p_{n-1}(x).$$ Here $\\{p_n(x)\\}$ are orthonormal polynomials of degree $n$ and the coefficients $a_n$ and $b_n$ are usually referred to as the recurrence coefficients. They possess a number of important properties. For instance, they can be expressed in terms of determinants containing the moments of the orthogonality measure \\cite{Chihara}. Moreover, for classical orthogonal polynomials they are known explicitly. For other, non-classical, polynomials the recurrence coefficients are not known explicitly and sometimes they can be expressed in terms of the solutions of discrete (including $q$-discrete)  or continuous Painlev\\'e equations. The Painlev\\'e equations in their turn have many remarkable applications in modern  mathematics and mathematical physics (see for instance \\cite{Clarkson} and the references therein). There are a few examples of relations of $q$-orthogonal polynomials on an exponential lattice to the $q$-discrete Painlev\\'e equations: the weight $$w(x)=(q^4x^4;q^4)_{\\infty}$$ on $\\{\\pm q^k\\,|\\,k\\in\\mathbb{N}_0\\}$ and ${q\\textrm{P}_\\textrm{\\small{I}}}$ \\cite{VanAssche}; the weight $$w(x)=|x|^{\\alpha}(q^2x^2;q^2)_{\\infty}(cq^2x^2;q^2)_{\\infty}$$ on $\\{\\pm q^k \\,|\\,k\\in\\mathbb{N}_0\\}$ and ${\\alpha q\\textrm{P}_\\textrm{\\small{V}}}$ \\cite{Lies, VanAssche}; the weight $$w(x)=x^{\\alpha}(q^2x^2;q^2)_{\\infty}$$ on $\\{ q^k \\,|\\,k\\in\\mathbb{N}_0\\}$ and ${q\\textrm{P}_\\textrm{\\small{V}}}$ \\cite{Lies}. For other examples of relations of the recurrence coefficients for the orthogonal polynomials, not necessarily supported on an exponential lattice, see for instance \\cite{BFV, BV, GF+Walter2, Freud, Magnus, Magnus2} and the references therein. One of the methods to derive the nonlinear discrete equations for the recurrence coefficients is by using the ladder operators.\n\n\n\\subsection{Ladder operators}\nIn the case of discrete $q$-orthogonal polynomials on the exponential lattice, the ladder operators were first\nconsidered in \\cite{Ismailq}.\n We repeat the main statements which we use later on to be self-contained following \\cite{Ismailq} and    \\cite[Section~1.3]{Lies}.\n\nThe $q$-difference operator is given by\n$$\n(D_q f)(x)=\\begin{cases}\n\\frac{f(x)-f(qx)}{x(1-q)},& x\\neq 0,\\\\\nf'(0)&x=0.\n\\end{cases}\n$$\n\nConsider a weight function $w$ on the exponential lattice $\\{a q^n,\\;b q^n\\,|\\,n\\in\\mathbb{N}_0\\},$ such that $w(a\/q)=w(b\/q)=0$ and the sequence of orthonormal polynomials $\\{p_n\\}$ of degree $n$ with respect to this weight. Ismail \\cite{Ismailq} shows that the polynomials satisfy the following relation:\n$$D_q p_n(x)=A_n(x)p_{n-1}(x)-B_n(x)p_n(x)$$\nwith\n\\begin{equation}\\label{An}\nA_n(x)=a_n\\int_a^{b}\\frac{u(qx)-u(y)}{qx-y}p_n(y)p_n(y\/q)w(y)d_q y,\n\\end{equation}\n\\begin{equation}\\label{Bn}\nB_n(x)=a_n\\int_a^{b}\\frac{u(qx)-u(y)}{qx-y}p_n(y)p_{n-1}\n(y\/q)w(y)d_q y.\n\\end{equation}\nHere the function $u$, called the potential, is defined by the following formula:\n\\begin{equation}\\label{potential}\n-u(qx)w(qx)=D_{q}w(x).\n\\end{equation}\nFurthermore, the following relations (compatibility conditions) hold:\n\\begin{equation}\\label{comp1}\nB_{n}+B_{n+1}=(x-b_n)\\frac{A_n}{a_n}+(q-1)x\\,\\sum_{j=0}^n \\frac{A_j}{a_j}-u(qx),\n\\end{equation}\n\\begin{equation}\\label{comp2}\na_{n+1}A_{n+1}-a_n^2\\frac{A_{n-1}}{a_{n-1}}=(x-b_n)B_{n+1}-(qx-b_n)B_n+1.\n\\end{equation}\nRelations (\\ref{comp1}), (\\ref{comp2}) are important in deriving nonlinear discrete equations for the recurrence coefficients, which in some cases can be further reduced to ($q$-)discrete Painlev\\'e equations.\n\n\n\\section{Main results}\n\nIn this paper we study the recurrence coefficients for the weight functions supported on the exponential lattice $\\{ q^k \\,|\\,k\\in\\mathbb{N}_0\\}$ and satisfying the $q$-difference equation (\\ref{potential}) with\n\\begin{equation}\\label{pot gen}\nu(x)=\\frac{k_1 q}{1-q}\\frac{1}{x}+\\frac{k_2 x+k_3}{1-q},\\;\\;k_1\\neq 0,\\;\\;k_2\\neq 0,\n\\end{equation}\nand conditions $w(0)=w(1\/q)=0$. At the end of this section we discuss the existence of such weight functions and consider a few instructive examples. In the following we assume that\nthe sequence of polynomials $\\{p_n\\}$\nis orthonormal with respect to the weight function with potential (\\ref{pot gen}) and hence the orthogonality relation takes the form\n$$\\int_0^1 p_m(x)p_n(x)w(x)d_q x=\\delta_{m,n}.$$\n\n\nIt is straightforward to calculate that\n$$\\frac{u(qx)-u(y)}{qx-y}=\\frac{k_1}{(q-1)xy}+\\frac{k_2}{1-q}.$$\nHence,  the expressions (\\ref{An}) and (\\ref{Bn}) can be computed as follows:\n$$A_n(x)=\\frac{a_n R_n}{x(1-q)}+\\frac{a_n k_2 q^{-n}}{1-q}, $$\n$$B_n(x)=\\frac{r_n}{(1-q)x},$$\nwhere $$R_n=-k_1\\int_0^1  p_n(y)p_{n}(y\/q)\\frac{w(y)}{y}d_q y,\\;\\;r_n=-a_n k_1\\int_0^1 p_n(y)p_{n-1}(y\/q)\\frac{w(y)}{y}d_q y$$\nand we have used orthogonality in computing $$\\int_0^1  p_n(y)p_{n}(y\/q)w(y)d_q y =q^{-n},\\;\\;\\int_0^1  p_n(y)p_{n-1}(y\/q)w(y)d_q y =0.$$\n\nThe compatibility conditions (\\ref{comp1}), (\\ref{comp2}) give rise to the following system (after comparing the coefficients at the powers of $x$):\n\\begin{equation}\\label{eq733g}\nr_{n+1}+r_n=-b_n R_n-k_1,\n\\end{equation}\n\\begin{equation}\\label{eq734g}\nR_n-k_2 q^{-n}b_n-k_3-(1-q)\\sum_{j=0}^nR_j=0,\n\\end{equation}\n\\begin{equation}\\label{eq735g}\na_{n+1}^2R_{n+1}-a_n^2 R_{n-1}=-b_n(r_{n+1}-r_n),\n\\end{equation}\n\\begin{equation}\\label{eq736g}\nk_2 a_{n+1}^2-k_2 q^2 a_n^2=q^{1+n}-q^{2+n}-q^{2+n}r_n+q^{1+n}r_{n+1}.\n\\end{equation}\n\nWe will use these equations to find expressions for the recurrence\ncoefficients $a_n$, $b_n$ of the sequence of orthonormal polynomials\n$\\{p_n\\}$  with respect to $w$. Multiplying (\\ref{eq736g}) by\n$q^{-2n-2}$ and taking a telescopic sum with $a_0=r_0=0$, we get\n\\begin{equation}\\label{eq737g}\na_n^2= \\frac{q^n(1-q^n+r_n)}{k_2}.\n\\end{equation}\n\nMultiplying (\\ref{eq735g}) by $R_n$, substituting $-b_n R_n$ from (\\ref{eq733g}) and taking a telescopic sum, we obtain\n\\begin{equation}\\label{eq738g}\na_n^2 R_n R_{n-1}=r_n(k_1 +r_n).\n\\end{equation}\n\nOn the other hand, if we substitute the expression for $b_n$ from\n(\\ref{eq734g}) and the expression for $a_n^2$ from (\\ref{eq737g})\ninto (\\ref{eq735g}) and we collect the terms in $r_n$ and $r_{n+1}$,\nwe obtain\n\\begin{multline*}r_{n+1}\\left(qR_{n+1}+R_n-k_3-(1-q)\\sum_{j=0}^nR_j\\right)-r_n\\left(qR_n+R_{n-1}-k_3-(1-q)\\sum_{j=0}^{n-1}R_j\\right)\\\\\n=(1-q^n)R_{n-1}-q\\left(1-q^{n+1}\\right)R_{n+1}.\\end{multline*} The\nleft hand side of this expression can easily be summed\ntelescopically and in the resulting equation we recognize the\nexpression for $b_n$ from (\\ref{eq734g}), so we get\n$$r_n\\left(R_{n-1}+k_2b_nq^{-n}\\right)=q^{n+1}R_n+q^nR_{n-1}-R_{n-1}-k_2b_nq^{-n}-k_3.$$\nIf we multiply this by $R_n$, we can use expressions (\\ref{eq733g})\nto substitute $b_nR_n$ and (\\ref{eq737g}), (\\ref{eq738g}) to\nsubstitute $R_nR_{n-1}$.  Eventually we get a quadratic equation for\n$R_n$:\n\\begin{equation}\\label{eq7310g}\nR_n^2-k_3q^{-n-1}R_n=-k_2q^{-2n-1}\\left((1+r_n)(1+r_{n+1})-(1-k_1)\\right).\n\\end{equation}\n\nFrom (\\ref{eq737g})  and (\\ref{eq738g}) we get\n\\begin{equation}\\label{eq738gnew}\nq^{n}(1-q^n+r_n) R_n R_{n-1}=k_2r_n(r_n+k_1).\n\\end{equation}\n\nWe can use equations (\\ref{eq7310g}) and (\\ref{eq738gnew}) to get a second order second degree difference equation for $r_n$.\n  In order to derive such an equation (which we omit here explicitly as it is long and cumbersome) we can first eliminate $R_n$ between equations (\\ref{eq7310g})\n and (\\ref{eq738gnew}) and then eliminate $R_{n-1}$ between the obtained equation and (\\ref{eq7310g}) with $n$ replaced by $n-1$.\n\nAlternatively, we can first   replace equation\n(\\ref{eq734g}) by subtracting from it (\\ref{eq734g}) with $n-1$ to\neliminate the sum of $R_n$. We have the following equation instead\nof (\\ref{eq734g}):\n\\begin{equation}\\label{eq734gnew}\nk_2 q b_{n-1}-k_2 b_n+q^n(q R_n-R_{n-1})=0.\n\\end{equation}\n From (\\ref{eq733g}) we can find $b_n$ in terms of $r_n$ and $R_n$ and substitute this expression into equations (\\ref{eq734g})--(\\ref{eq736g}).\n Further, we use (\\ref{eq737g}).\nSubstituting (\\ref{eq737g}) into (\\ref{eq738g}), we can find $R_{n-1}$ and, by taking (\\ref{eq737g}) and (\\ref{eq738g}) with $n$ replaced by $n+1$,\nwe can find $R_{n+1}$ in terms of $R_n$ and $r_n$ ($r_{n+1}$ respectively). We have \\begin{equation}\\label{R{n-1}g}\nR_{n-1}=-\\frac{k_2q^{-n}r_n(r_n+k_1)}{(q^n-1-r_n)R_n},\n\\end{equation}\n\\begin{equation}\\label{R{n+1}g}\nR_{n+1}=-\\frac{k_2 q^{-1-n}r_{n+1}(r_{n+1}+k_1)}{(q^{n+1}-1-r_{n+1})R_n}.\n\\end{equation}\nSubstituting these expressions into (\\ref{eq734gnew}), we can find\n$R_n^2$:\n\\begin{equation}\\label{Rn2g}\nR_n^2=-\\frac{k_2q^{-1-n}r_n(r_n+k_1)(r_n(q^n-1-r_{n+1})+(q^n-1)(r_{n+1}+k_1))}\n{(q^n-1-r_n)(r_{n-1}(q^n-1-r_n)+(q^n-1)(r_n+k_1))}.\n\\end{equation}\n\nFrom (\\ref{eq737g}) we have an expression of $a_n^2$ in terms of\n$r_n.$ Using (\\ref{Rn2g}) and (\\ref{eq7310g}) we can get an\nexpression of $R_n$ in terms of $r_{n-1},\\;r_n$ and $r_{n+1}$.\nHence, using (\\ref{eq737g}) we can get an expression of $b_n$ in\nterms of $r_{n-1},\\;r_n$ and $r_{n+1}$. Thus, we have proved the\nfollowing theorem.\n\n\n\\begin{theorem}\nThe recurrence coefficients $a_n$, $b_n$ appearing in the three-term recurrence relation for the weight supported on the exponential lattice $\\{ q^k \\,|\\,k\\in\\mathbb{N}_0\\}$ with the potential   satisfying (\\ref{pot gen}) and  conditions $w(0)=w(1\/q)=0$ can be expressed in terms\nof $r_n$ (by using (\\ref{eq737g}), (\\ref{eq733g}) with (\\ref{Rn2g})), which is a solution of  a second order second degree discrete equation (with respect to $n$) given by\n\\begin{equation}\\label{eq:thm1}f^2-2f g+g^2-k_3^2 q^{-2n-2}f=0,\\end{equation} where $f$ denotes the right hand side of (\\ref{Rn2g}) and $g$ is the right hand side of  (\\ref{eq7310g}).\n\\end{theorem}\n\nNote that the discrete equation in the theorem  factorizes if and only if $k_3=0$ and this allows us to express $r_{n+1}$ in terms of $r_{n-1}$ and $r_n$ as a rational function. The connection is given by a $q$-Painlev\\'{e} equation:\n\n\\begin{theorem}\\label{thm:qpV}\nIf $k_3=0$ in (\\ref{pot gen}) then the variable $x_n=(1+r_n)(1-k_1)^{-1\/2}$ satisfies $qP_V$ \\cite{Ramani} given by\n\\begin{equation}\\label{qP5}\n(x_n x_{n-1}-1)(x_n x_{n+1}-1)=\n\\frac{\\gamma\\delta\nq^{2n}(x_n-\\alpha)(x_n-1\/\\alpha)(x_n-\\beta)(x_n-1\/\\beta)}{(x_n-\\gamma\nq^n)(x_n-\\delta q^n)},\n \\end{equation}\nwith \\[\\alpha=\\beta=\\gamma=\\delta=\\frac{1}{p}\\]  where $p=\\sqrt{1-k_1}$.\nThe initial conditions are given by \\[x_0=\\frac{1}{p}\\qquad{\\rm and}\\qquad x_1=p-\\frac{k_2}{qp}\\left(\\frac{\\mu_1}{\\mu_0}\\right)^2.\\]\n$x_n$ is related to the recurrence coefficients $a_n$ and $b_n$ of the orthogonal polynomials by \\begin{equation}\\label{aninThm2}a_n^2=\\frac{q^n}{k_2}\\left(px_n-q^n\\right)\\end{equation} and \\begin{equation}\\label{bninThm2}b_n^2=-\\frac{q^{2n+1}(px_n+px_{n+1}-1-p^2)^2}{k_2p^2(x_nx_{n+1}-1)}.\\end{equation}\\end{theorem}\n\n\\begin{proof}\nIf $k_3=0$ then (\\ref{eq:thm1}) simplifies to $f-g=0$.  Putting $y_n=1+r_n$ and isolating the term containing $y_ny_{n+1}$, we find\n\\begin{eqnarray*}0&=&y_ny_{n+1}(y_n-q^n)^2(y_ny_{n-1}+k_1-1)=(k_1-1)(y_n-q^n)^2y_ny_{n-1}\\\\\n&&+(q^n-y_n)(k_1-1)y_n\\left[(y_n-q^n)+(q^n-1)(y_n-1+k_1)\\right]\\\\\n&&-q^n(y_n-1)(y_n-1+k_1)y_n\\left[q^n(y_n-1)+(q^n-1)(k_1-1)\\right].\\end{eqnarray*}\nWith the substitution $y_n=\\sqrt{1-k_1}x_n=px_n$ the first two terms can be completed to contain $(x_nx_{n-1}-1)(x_nx_{n+1}-1)$ and the equation simplifies to (\\ref{qP5}).\n\nInitial conditions for $a_n$ and $b_n$ (and, hence, $x_n$) can be computed by using the fact that the recurrence coefficients can be expressed in terms of the Hankel determinants containing the moments of the orthogonality measure \\cite[Th. 4.2, p. 19; Ex. 3.1, p. 17]{Chihara}.  In particular, we will need that $b_0=\\mu_1\/\\mu_0$ where $\\mu_k$ is the $k$'th moment of the weight $w$.  Since $a_0=r_0=0$, we immediately find that $y_0=1$ and hence $x_0=1\/p$.\nAs for $x_1$, we know from (\\ref{eq733g}) that $r_1=-b_0R_0-k_1$.  Here, $R_0$ can be obtained from (\\ref{eq734g}) with $n=0$ and $k_3=0$: we find that $R_0=q^{-1}k_2b_0$.  This leads to \\[r_1=-k_1-\\left(\\frac{\\mu_1}{\\mu_0}\\right)^2\\frac{k_2}{q}\\] and hence \\[x_1=p-\\frac{k_2}{pq}\\left(\\frac{\\mu_1}{\\mu_0}\\right)^2.\\]\n\nThe connection (\\ref{aninThm2}) between $x_n$ and $a_n$ follows immediately from (\\ref{eq737g}).  To obtain (\\ref{bninThm2}), we use the squared (\\ref{eq733g}) to write \\[b_n^2=\\frac{(r_{n+1}+r_n+k_1)^2}{R_n^2},\\] where $R_n^2$ can be substituted using (\\ref{Rn2g}).  Finally, substitute $r$ by $x$ using $x_n=(1+r_n)\/p$ and eliminate $x_{n-1}$ using the $q$-Painlev\\'{e} equation (\\ref{qP5}).\n\n\\end{proof}\n\n\n\nClearly (in the case $k_3\\neq 0$), we can also  derive a third order  difference equation for $r_{n}$ from\nthe system (\\ref{eq733g})--(\\ref{eq736g}) as follows. We can  get $R_{n+1}^2$ by either using relation (\\ref{Rn2g}) with $n+1$ or  by squaring  (\\ref{R{n+1}g}) and\nusing (\\ref{Rn2g}). Hence, we can set them equal and in the result, we get a (cumbersome) expression involving only $r_{n-1},\\,r_n,\\,r_{n+1},\\,r_{n+2}$.\n\nTo find out which weights can give rise to a potential of the form\n(\\ref{pot gen}), we notice that it is sufficient if\n$$\\frac{w(x\/q)}{w(x)}=Ax^2+Bx+C$$ for certain constants $A$, $B$,\n$C$, since an easy calculation shows that in that case the potential\nis given by (\\ref{pot gen}) with $k_1=1-C$, $k_2=-Aq$ and $k_3=-Bq$.\n\nIf we define \\begin{eqnarray*}v_1^\\alpha(x)=x^\\alpha,\\quad v_2^c(x)&=&(cx;q)_\\infty,\\quad\nv_3^c(x)=(cx^2;q^2)_\\infty,\\\\\nv_4^c(x)&=&(c\/x;q)_\\infty,\\quad\nv_5^c(x)=(c\/x^2;q^2)_\\infty,\\end{eqnarray*} then\n\\begin{eqnarray*}\n\\frac{v_1^\\alpha(x\/q)}{v_1^\\alpha(x)}=q^{-\\alpha},\\quad\n\\frac{v_2^c(x\/q)}{v_2^c(x)}&=&\\frac{q-cx}{q},\\quad\n\\frac{v_3^c(x\/q)}{v_3^c(x)}=\\frac{q^2-cx^2}{q^2},\\\\\n\\frac{v_4^c(x\/q)}{v_4^c(x)}&=&\\frac{x}{x-c},\\quad\n\\frac{v_5^c(x\/q)}{v_5^c(x)}=\\frac{x^2}{x^2-c}.\\end{eqnarray*}\nHence it is clear which products of $v_i$ lead to a weight for which\nthe potential satisfies (\\ref{pot gen}).  These include, among\nothers, the little $q$-Laguerre weight, the weight in\n\\cite[Sect.~7.3]{Lies}, products of rational functions and the weights above, the  weights in the following examples and others.\n\n\n\n\n\n\n\n\n\n\n\n\n{\\bf Example 1.}  In this example we consider the semi-classical little  $q$-Laguerre weight\n\\begin{equation}\\label{weight}\nw(x)=x^{\\alpha} (qx;q)_{\\infty} (c q x;q)_{\\infty},\\;\\;\\alpha>0,\n\\end{equation}\non the positive exponential lattice $\\{ q^n\\,|\\, n\\in\\mathbb{N}_0\n\\}.$  The case\n$c=-1$ was considered in \\cite[Sect.~7.3]{Lies}.  The case $c=0$\ngives the little $q$-Laguerre weight (and, hence, the recurrence coefficients are known explicitly).  We observe that\n$w(0)=w(1\/q)=0.$\n\nThe potential (\\ref{potential}) is given by\n$$u(x)=\\frac{1}{1-q}\\left(\\frac{q}{x}-\\frac{q^{1-\\alpha}}{x}+q^{1-\\alpha}(1+c)-c q^{1-\\alpha}x\\right)$$\nand, hence, $k_1=1-q^{-\\alpha},\\;\\;k_2=-cq^{1-\\alpha},\\;\\;k_3=(1+c)q^{1-\\alpha}$ in (\\ref{pot gen}). We assume that $c\\neq 0.$\n\nSince $k_3=0$ if and only if $c=-1$, we get that in this case the variable $x_n=q^{\\alpha\/2}(r_n+1)$ satisfies\n\\begin{equation}\\label{qP5part}\n(x_n x_{n-1}-1)(x_n x_{n+1}-1)=\\frac{q^{2n+\\alpha}(x_n-q^{\\alpha\/2})^2(x_n-q^{-\\alpha\/2})^2}{(x_n-q^{n+\\alpha\/2})^2},\n\\end{equation}\n which is a particular case of $qP_V$ (\\ref{qP5})\n(with $\\alpha=\\beta=\\gamma=\\delta=q^{\\alpha\/2}$). This coincides with the result in \\cite[Sect.~7.3]{Lies}. Note that equation (\\ref{qP5part}) for the variable $r_n$ is given by\n\\begin{equation}\\label{eq:rn}\nr_n^2(r_n+1-q^{-\\alpha})^2=q^{-2n}(r_n+1-q^n)^2((r_n+1) (r_{n+1}+1)-q^{-\\alpha})((r_n+1) (r_{n-1}+1)-q^{-\\alpha}).\n\\end{equation}\n\n\nNext we study initial conditions for the recurrence coefficients of the weight (\\ref{weight}) for general $c$.\n Using  (\\ref{eq737g}), we have $$\na_n^2=\\frac{1}{c} \\,\nq^{n-1+\\alpha\/2}\\left(q^{n+\\alpha\/2}-x_n\\right).\n$$\nTo find an expression of $b_n$ in terms of $x_n$ we square the\nexpression of $b_n$ from (\\ref{eq733g}) and substitute (\\ref{Rn2g}).\nWe can also use (\\ref{eq:rn}) to get rid of $x_{n-1}$. We get for\n$n\\geq 1$ $$ c b_n^2(x_n x_{n+1}\n-1)=q^{2n}\\left(1+q^{\\alpha}-q^{\\alpha\/2}(x_n+x_{n+1})\\right)^2.\n$$ We  also get  that $$\nb_0^2=\\frac{1}{c}\\left(q^{\\alpha\/2}x_1-1\\right).\n$$ Recalling that $b_0=\\mu_1\/\\mu_0$, we immediately get that the initial values are given by\n$x_0=q^{\\alpha\/2}$ (since $r_0=0$) and\n\\begin{equation}\\label{x1}\nx_1=q^{-\\alpha\/2}\\left(1+c\\frac{\\mu_1^2}{\\mu_0^2}\\right),\n\\end{equation}\n where $\\mu_k$ is the $k$-th moment of the weight (\\ref{weight}). In fact, we can also calculate $\\mu_k$ by definition\nand get $$\\mu_k=(1-q)(q;q)_{\\infty}(cq;q)_{\\infty} \\,\n{}_2\\phi_1(0,0;cq;q;q^{\\alpha+k+1}),$$ where the basic\nhypergeometric function ${}_2\\phi_1 $ is given by \\cite[Sect. 0.2,\n0.4]{KS}\n$${}_2\\phi_1(a_1,a_2;b_1;q;z)=\\sum_{\\ell=0}^{\\infty}\\frac{(a_1;q)_\\ell (a_2;q)_\\ell}{(b_1;q)_\\ell}\\frac{z^\\ell}{(q;q)_\\ell}.$$\nNote that for $c=q^{\\nu}$ the last expression (up to a factor) can\nbe written in terms of the modified $q$-Bessel function \\cite{Rogov\n$$I^{(1)}_{\\nu}(z,q)=\\frac{(q^{\\nu+1};q)_{\\infty}}{(q;q)_{\\infty}}(z\/2)^{\\nu}{}_2\\phi_1(0,0;q^{\\nu+1};q;z^2\/4)$$ with $z=2q^{(\\alpha+k+1)\/2}.$\nAlso note that a limiting case of Heine's transformation formula \\cite[formula (0.6.9)]{KS} allows us to write this $q$-hypergeometric function as a ${}_0\\phi_1$-function.\n\n\n{\\bf Example 2.}  In this example we consider another semi-classical generalization of the little $q$-Laguerre weight:\n\\begin{equation}\\label{weight2}\nw(x)=x^{\\alpha} \\frac{(qx;q)_{\\infty} \\left(\\frac{c_1}{x};q\\right)_\\infty\\left(\\frac{qx}{c_1};q\\right)_\\infty}{\\left(\\frac{c_2}{x};q\\right)_\\infty},\\;\\;\\alpha>0,\\:c_1<0,\\:c_2<0\n\\end{equation}\non the positive exponential lattice $\\{ q^n\\,|\\, n\\in\\mathbb{N}_0\n\\}.$  The case where $c_1=c_2=1\/c$ gives the weight from the previous example.  Again, it is clear that $w(0)=w(1\/q)=0$.  It is easy to calculate that for this weight we get \\[k_1=1-\\frac{c_2}{c_1}q^{-\\alpha},\\qquad k_2=-\\frac{q^{1-\\alpha}}{c_1}\\qquad{\\rm and}\\qquad k_3=\\frac{c_2+1}{c_1}q^{1-\\alpha}.\\]\n\nAs mentioned earlier, to obtain a Painlev\\'e equation, we need that $k_3=0$, hence $c_2=-1$.  So, following the outline given before, we see that \\[x_n=\\sqrt{-c_1}q^{\\alpha\/2}(r_n+1)\\qquad{\\rm and}\\qquad a_n^2=-c_1q^{n+\\alpha-1}(1-q^n+r_n)\\] where $x_n$ satisfies (\\ref{qP5}) with \\[\\alpha=\\beta=\\gamma=\\delta=\\sqrt{-c_1}q^{\\frac{\\alpha}{2}}.\\]\n\nAs for the initial conditions, we find that \\[x_0=\\sqrt{-c_1}q^{\\frac{\\alpha}{2}}\\] and \\[x_1=\\sqrt{-c_1}q^{\\frac{\\alpha}{2}}\\left(-1+\\frac{c_2}{c_1}q^{-\\alpha}+\\frac{b_0^2}{c_1q^\\alpha}\\right)+1.\\]  Moreover, $b_0=\\mu_1\/\\mu_0$, and it is easily seen that the $k$'th moment of this weight is given by \\[\\mu_k=(1-q)(q;q)_\\infty\\left(\\frac{q}{c_1};q\\right)_\\infty\\frac{(c_1;q)_\\infty}{(c_2;q)_\\infty}{}_2\\phi_1\\left(0,0;\\frac{q}{c_2};q;\\frac{c_1}{c_2}q^{\\alpha+1+k}\\right).\\]  Hence $b_0$ can be written, up to a factor, as a fraction of two modified $q$-Bessel functions with $q^{-\\nu}=c_2$.  Again, the $q$-hypergeometric function can alternatively be written as a ${}_0\\phi_1$-function, using \\cite[formula (0.6.9)]{KS}.\n\n\\section{Discussion}\n\nIn this paper we have shown that it is possible to study simultaneously recurrence coefficients in the three-term recurrence relation for  a large class of weights by using the technique of  ladder operators. The crucial point is to consider the potential (\\ref{pot gen}) with parameters. This allows us to obtain a second degree second order discrete equation, which in some particular cases, can be further reduced to the  discrete Painlev\\'e equation. It is an interesting open problem to try to classify the weights which lead to the appearance of the discrete Painlev\\'e equations for the recurrence coefficients.\n\n\n\n\n\\section*{Acknowledgements}\nGF is supported by MNiSzW Iuventus Plus Grant Nr 0124\/IP3\/2011\/71. Part of this research was done when CS was visiting the University\nof Warsaw, and he is grateful for their hospitality.\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe story of the star-forming history of a molecular cloud is told through its dust, gas, and member stars.  The youngest members, the protostars, { are still forming} and are typically clustered in the densest regions of the cloud.  Stars formed in previous bursts are often still located in small subclusters, while some have been dispersed throughout the cloud or were born in small groups.  In many clouds the older stars are not co-located with the youngest stars, indicating that the site for star formation changes with time within a cloud complex \\citep[e.g.,][]{Getman2014a,Getman2014b, Vazquez-Semadeni2018,Kounkel2018,kristensen18,Liu2021}. \n\nThe Serpens Molecular Cloud is one of the nearest clouds with vigorous ongoing star formation, second to only Orion among active star-forming sites within 500 pc.  Historically the Serpens Main cluster \\citep[see review by][]{eiroa08} and W40 (\\citealt{Kuhn2010}; see also review by \\citealt{rodney08}) were recognized as primary sites of star formation in the complex; mid- and far-IR and X-ray surveys revealed vigorous ongoing star formation in the deeply embedded Serpens South cluster \\citep{Gutermuth2008,povich13,mallick13,konyves15,Dunham2015}.  Beyond these sub-clusters, recently-formed stars are sparse and hard to find \\citep[see review by][]{prato08}.  Accurate astrometry and optical photometry from Gaia DR2 revealed the optical members, including some in new subclusters and some distributed throughout the cloud \\citep{Herczeg2019}, while most candidate members that were outside the main star-forming sites have been identified as contaminants \\citep[see also][]{lee21}.  The optical members that are distributed across the cloud likely formed in past epochs of star formation.\n\n\\begin{deluxetable*}{cccc}[!th]\n\\label{table:photometry_criteria}\n\\tablecaption{Photometry selection criteria}\n\\tablehead{\\nocolhead{Survey} & \\colhead{Pass-band} & \\colhead{Criteria} & \\colhead{Note} }\n\\startdata\n\\textit{Gaia}        & $BP,RP$     &  \\multicolumn{1}{c}{\\begin{tabular}[c]{@{}c@{}}\\texttt{parallax\\_over\\_error>10 } \\\\                                                           \\texttt{astrometric\\_excess\\_noise<2} \\\\ \\texttt{visibility\\_periods\\_used>8} \\\\                                           \\texttt{phot\\_g\\_mean\\_flux\\_over\\_error>50}\\end{tabular}}  & \\\\\\hline\n2MASS       & $J,H,K$     & \\texttt{ph\\_qual=\"AAA\"}   &                                              \\\\ \\hline\nWISE       & $W1,W2$     & \\multicolumn{1}{c}{\\begin{tabular}[c]{@{}c@{}}\\texttt{SNR>10}\\\\ \\texttt{$\\chi^2$<3}\\end{tabular}}  & \\begin{tabular}{p{0.8\\columnwidth}} W1 and W2 are selected and treated independently. \\end{tabular}    \\\\ \\hline\nPan-STARRS  & $g,r,i,z,y$ & \\multicolumn{1}{c}{\\begin{tabular}[c]{@{}c@{}}\\texttt{nDetections>5}\\\\ \\texttt{gQfPerfect>0.85}\\\\ \\texttt{rQfPerfect>0.85}\\\\ \\texttt{iQfPerfect>0.85}\\\\ \\texttt{zQfPerfect>0.85}\\\\ \\texttt{imeanpsfmag - imeankronmag < 0.05} \\\\ \\texttt{OR} \\\\  \\texttt{rmeanpsfmag - rmeankronmag < 0.05} \\end{tabular}} & \\begin{tabular}{p{0.8\\columnwidth}} Pan-STARRS data of a star will be used only if the data satisfy all the criteria. \\end{tabular}    \\\\ \\hline\nAPASS       & $B,V,g',r',i'$ & - & \\begin{tabular}{p{0.8\\columnwidth}} {$r\/i_{\\rm APASS}$ and $r\/i_{\\rm PS1}$} are treated as the same in SED fitting \\tablenotemark{$1$}. If a star has both $r_{\\rm PS1}$ (or $i_{\\rm PS1}$) and $r_{\\rm APASS}$ (or $i_{\\rm APASS}$) data, then Pan-STARRS $r$ (or $i$) will be used. \\end{tabular} \\\\\n\\enddata\n\\tablenotetext{1}{{Discussed in Section \\ref{sec:data}}}\n\\tablecomments{A star will be included only if it passes both \\textit{Gaia} and 2MASS criteria, because the SED fitting method needs at least 4 pass-bands (Section \\ref{subsec:fit_photospheric_SED}) and \\textit{Gaia} and 2MASS pass-bands cover the full temperature range of Pleiades stars (Section \\ref{subsec:selection_Pleiades}, Figure \\ref{fig:color_color_used_in_fit} and Table \\ref{table:color_color_table}). If a star passes both \\textit{Gaia} and 2MASS criteria and fails in WISE\/Pan-STARRS\/APASS criteria, it will be included and its WISE\/Pan-STARRS\/APASS photometry will not be used.}\n\\end{deluxetable*}\n\nA comprehensive description of the star formation history of the Serpens Molecular Cloud requires estimates of when these member stars formed.  However, accurate ages depend on estimates for the stellar properties, which are complicated by the variation of extinction across the cluster.  Cluster ages can be estimated from HR diagrams \\citep[e.g.][]{Erickson2015}, the lithium depletion boundary \\citep[e.g.][]{Basri1996,binks14}, and the X-ray luminosity function \\citep[e.g.][]{Getman2014a}, with varying degrees of accuracy and systematic precision \\citep[see review by][]{soderblom14}. However, { for the most common method, use of HR diagrams,} few young stellar objects in the Serpens Molecular Cloud have measured temperatures and luminosities that could be used to estimate their age \\citep[e.g.][]{Oliveira2013,Erickson2015}.  In the absence of spectroscopic measurements, photospheric properties are usually estimated by fitting SEDs with synthetic spectra \\citep[e.g.,][]{Bayo2008, Robitaille2017, Davies2020}.\nThis approach may introduce biases at cool temperaturesm where synthetic SEDs differ from observed SEDs (e.g. \\citealt{Bell2012} and \\citealt{Lancon2020}; see also analysis of main-sequence M-dwarfs in \\citealt{mann15}).\nBecause nearby star-forming regions are mainly composed of low-mass stars \\citep[e.g.,][]{Luhman2020}, low-mass stars dominate {isochronal} age measurements for most nearby star-forming regions {\\citep[e.g.,][]{Mayne2007,Bell2012,Zari2019}}; any systematic errors in their stellar properties propagate into (often unassessed) uncertainties in ages of star-forming regions.\n\n\n\nIn this paper, we develop a new SED fitting routine designed for young stars and then apply the fitting routine to optical members of Serpens to estimate stellar ages by comparing best-fit temperature and luminosities to model isochrones. In Section \\ref{sec:data}, we present the data, including data for the fitting method and for Serpens. In Section \\ref{sec:fitting_method}, details of the SED fitting method are described. In Section \\ref{sec:Serpens}, we use the method on Serpens stars and derive their ages. In Section \\ref{sec:discussion}, we discuss the results, and the conclusions are summarized in Section \\ref{sec:conclusion}. \n\n\\section{Photometric data and cluster member selection}\n\\label{sec:data}\n\nThe results from this paper are obtained by fitting the SEDs of YSOs in Serpens using a grid of empirically measured SEDs of young stars. The grid is built with known members of the Pleiades and nearby young moving groups, following similar approaches by \\citet{Bell2012} and \\citet{Pecaut2013}.  The Pleiades members are young \\citep[$\\sim$110 Myr,][]{Gaia2018b}, nearby \\citep[$\\sim$136 pc,][]{Gaia2018b}, have solar metallicity ($[Fe\/H] \\sim 0.03$, \\citealp{Soderblom2009}), and for the cooler stars are magnetically active (see the effect of surface gravity in Section \\ref{subsec:method_problems}). The line-of-sight extinction to the Pleiades is negligible  \\citep[$A_V=0.12$,][]{Stauffer1998}, so the photometry does not require any significant correction. This grid and our fitting methods are then tested using known members of Orion.  The photometry used in this paper is obtained from 2MASS \\citep{Cutri2003,2MASS_DOI}, ALLWISE \\citep{Cutri2014,ALLWISE_DOI}, Pan-STARRS1 \\citep[PS1,][]{Chambers2016}, APASS \\citep[the Apache Point Observatory Galactic Evolution Experiment,][]{Henden2015}, and {\\textit{Gaia} EDR3 \\citep{Gaia2021}}, with quality criteria described in Table~\\ref{table:photometry_criteria}. \n\n\n\nThe Pan-STARRS1 and APASS (SDSS-based) filters $r$, $i$ have slightly different transmission curves. \n\\citet{Tonry2012} presents $g_{\\rm SDSS}-g_{\\rm PS1}$, $r_{\\rm SDSS}-r_{\\rm PS1}$, and $i_{\\rm SDSS}-i_{\\rm PS1}$ at $>\\sim4000$ K (their Figure 6, 7), showing that the difference in $g$ band is significant, while $r$ and $i$ bands have negligible difference. At $<\\sim4000$ K, stars in \\citet{kado2016} \n $r_{\\rm APASS}$ is fainter than $r_{\\rm PS1}$ by $\\sim0.07$ mag and $i_{\\rm APASS}$ is fainter than $i_{\\rm PS1}$ by $\\sim0.05$ mag.\nIn addition, the APASS faint completeness limit (V=16) corresponds to $\\sim3700$ K for Pleiades, so this difference is not relevant for most stars. For example, among the selected Pleiades stars under 4000 K (178 stars), only 16 stars have APASS data, and all the 16 stars have Pan-STARRS data. Among the Orion stars used to test the SED fitting method, only 1 star under 4000 K has APASS data, indicating that the magnitude difference has a minor effect on the fitting method.\nIn SED fitting, we use $g_{\\rm APASS}$ and $g_{\\rm PS1}$ separately, and treat $r$ and $i$ as the same. If a star has both $r_{\\rm PS1}$ (or $i_{\\rm PS1}$) and $r_{\\rm APASS}$ (or $i_{\\rm APASS}$) data, then Pan-STARRS $r$ (or $i$) will be used.\n\n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures_revised_2nd\/CMD_and_colorT.png}\n    \\caption{Color-magnitude diagram (top) and color-temperature relation (bottom) of the Pleiades stars in our sample, for cool stars with temperatures derived from \\citet{Pecaut2013} SED fits (deep blue dots), stars with APOGEE temperatures (orange dots), hotter stars with temperatures from other catalogs (light blue square), and (in the bottom panel) stars with APOGEE temperatures that also have temperatures assessed by \\citet{Pecaut2013} (gray triangle).}\n    \\label{fig:CMD_and_colorT}\n\\end{figure}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures_revised_2nd\/color_color_relation_selected.png}\n    \\caption{Color-color relations used in SED fitting (except $i$-band) for the set of {Pleiades stars (orange dots), and young stars from \\citet{Kounkel2019} (light blue dots) and BANYAN \\citep{Gagne2018, Gagne2018b} (deep blue square)}.  Each relationship is shown in the color-color space (top panel) and the residual from a best-fit polynomial. The top panel includes a comparison with colors from BT-Settl spectra with $\\log g=4.5$ (gray dots).}\n    \\label{fig:color_color_used_in_fit}\n\\end{figure*}\n\n\\begin{deluxetable}{lcc}[!bh]\n\\label{table:Orion_disk_info}\n\\tablecaption{Selected catalogs of Orion Stars}\n\\tablehead{\\nocolhead{Catalog} & \\colhead{Region} & \\colhead{Disk ID \\tablenotemark{$1$}} }\n\\startdata\n\\citet{Fang2013} & L1640 & Yes \\\\\n\\citet{Fang2017} & NGC 1980 & Yes \\\\\n\\citet{Getman2017} & Orion A& Partial  \\\\\n & Orion B & Partial\\\\\n & ONC & Partial\\\\\n & Flame Nebula & Partial\\\\\n\\citet{Kuhn2013} & Flame Nebula & No \\\\\n\\citet{Megeath2012} & Orion A, B & No \\\\\n\\citet{Prisinzano2008} & ONC & Yes \\\\\n\\citet{Kounkel2019} \\tablenotemark{$2$} & Orion A, B & Yes \\\\\n& Orion A, B & Yes \\\\\n & $\\sigma$ Ori & Yes \\\\\n & $\\lambda$ Ori & Yes \\\\\n & 25 Ori & Yes \\\\\n\\enddata\n\\tablenotetext{1}{Disk identification criteria differ among catalogs. When no disk information is provided, spectral indices $\\alpha$ are calculated from {\\it Spitzer} photometry.  Stars with $\\alpha<-1.6$ are treated as diskless stars, following Equation 1 and Section 3.2 in \\citet{Dunham2015}.}\n\\tablenotetext{2}{Targets obtained from \\citet{Megeath2012} for Orion A and B, \\citet{Hernandez2007} for $\\sigma$ Ori, \\citet{Suarez2017} for 25 Ori, \\citet{Hernandez2010} for $\\lambda$ Ori.}\n\\end{deluxetable}\n\n\\subsection{Building a Grid of SEDs of young stars}\n\\label{subsec:selection_Pleiades}\n\nThe photometric grid of young stars is built from Pleiades photometry. The samples that are described here cover from $BP-RP=3.2$ to $0$, corresponding to 3000 K to 9500 K. The cool end is sufficient for fitting bright brown dwarfs in Serpens.\n\nThe Pleiades members used for building grids are obtained from several catalogs. First, some Pleiades stars are collected from \\citet{Cottaar2014}, which focused on targets with temperatures of 3200--6000 K, as measured from H-band APOGEE spectra. For stars hotter than 6000 K, membership and temperatures are obtained from previous SED fits \\citep{Bouy2015,Barrado2016,Somers2017,Bouvier2018}; any hot star in our grid appears in at least two of those catalogs with an adopted temperature averaged from those catalogs.\nStars cooler than 3200 K are identified from \\citet{Gaia2018b} and assessed a temperature from the SED fitting method in \\citet{Pecaut2013}, who used synthetic spectra to fit the SEDs of stars with negligible extinction. From these catalogs, we then ensure high confidence in membership by requiring that all stars are contained in a 5 pc sphere around the center of the full Pleiades population, in addition to proper motion constraints applied by \\citet{Gaia2018b}.  The few Pleiades candidates with $BP-RP>3.2$ are faint and have large uncertainties in $BP$ photometry and are therefore excluded. \n\nTo obtain tight color-color relations, a polynomial is fit to the $BP-RP$ and $G-RP$ relationship. Stars with colors that are more than $2\\sigma$ from the fit are removed. We iterate this process until the RMS of the $BP-RP$ and $G-RP$ relation is {$<0.02$ mag}.  A similar procedure is applied to the relationship between $BP-RP$ and $M_G$. These stars form our initial grid for Pleiades (see color-magnitude and color-color diagrams in Figures \\ref{fig:CMD_and_colorT}-- \\ref{fig:color_color_used_in_fit}).\n\nDue to saturation limits, only Pleiades stars with $T_{\\rm{eff}}<4000$ K have Pan-STARRS photometry, while only those with 4000 K $<T_{\\rm{eff}}<$ 6000 K have APASS photometry. To supplement our grid, we add young ($<300$ Myr old) stars within 1000 pc, collected from {\\citet{KounkelCovey2019}, \\citet{Gagne2018}, and \\citet{Gagne2018b}.}  These samples are all expected to have roughly solar metallicities \\citep{spina17}.  To ensure negligible extinction and the lack of a circumstellar disk, {these additional members are fitted only from Pleiades via the same method in Section \\ref{sec:fitting_method} and are required to have:  $| (BP-RP)_{\\rm fit} - (BP-RP)_{\\rm observed} | < 0.06$, $\\chi^2(\\rm fit)<0.5$, and $A_V(\\rm fit)<0.03$. The fitted result is presented in Appendix \\ref{appendix:fit_additional_YSOs}}. This total sample provides Pan-STARRS photometry for stars cooler than 7000 K, sufficiently covering the temperature range of most young stars.\n\n\n\n\\subsection{Stars in Orion to test and apply the SED fits}\n\\label{subsubsec:selection_Orion}\n\nTo test the accuracy of our fitting methods, we apply our SED fits to stars in the Orion Star-forming Complex, leveraging the photometric and spectroscopic analyses by various catalogs (Table \\ref{table:Orion_disk_info}) that include an evaluation of disks from {\\it Spitzer} mid-IR photometry.\\footnote{Catalogs disagree on disk presence for $\\sim 10$ stars because of differences in disk identification methods.  A star is treated as diskless only if all catalogs in which the star is listed identify it as diskless.} Temperatures are obtained from APOGEE\\footnote{Some methodological differences still exist between APOGEE measurements because of differences in synthetic models and abundance assumptions used in different papers, see for example temperature measurements of the same set of stars in \\citet{DaRio2016} and \\citet{Kounkel2018}. {The APOGEE $T_{\\rm eff}$ in \\citet{Cottaar2014} and \\citet{Kounkel2018} is measured from BT-Settl and PHOENIX grid separately.}} spectra \\citep{Kounkel2018}. Samples from catalogs in Table \\ref{table:Orion_disk_info} are cross-matched with \\citet{Kounkel2018} to ensure astrometric membership in Orion. Samples from \\citet{Fang2013, Fang2017} are used to identify stars with and without disks (Section \\ref{subsec:identify_disk}).\n\n\\subsection{The sample of Serpens stars}\n\\label{subsubsec:selection_Serpens}\nThe SED fits are designed to derive the stellar properties of the $\\sim700$ high-confidence members\\footnote{The statistical membership is $\\sim 1167$ optical members, which includes thousands of possible members with low confidence in membership.  The stars selected for this study have high confidence of membership based on astrometry.} of the Serpens star-forming complex with \\textit{Gaia} DR2 astrometry \\citep{Herczeg2019}. \nAbout half of the optical members are located in three distinct clusters, named Serpens Main ($\\sim 440$ pc), Serpens Northeast (Serpens NE, $\\sim 480$ pc) and Serpens far-South ($\\sim380$ pc). Of these clusters, Serpens Main is rich in protostars \\citep{Dunham2015}, while the others have few protostars and disks.  In addition, the Serpens South cluster is rich in protostars but has few optical counterparts.\nA few optical members of the Serpens cloud are located in smaller groups, and the rest are distributed throughout the complex. Among the $\\sim700$ high confidence members in \\citet{Herczeg2019}, $\\sim20$ stars have no \\textit{Gaia} $BP$ or $RP$ photometry, $\\sim20$ stars have no 2MASS data with a crossmatch radius of 2 arcsec, and $\\sim80$ stars do not pass the 2MASS data selection criteria, in most cases because the objects appear elongated by a companion or visual binary.  Thus, over 100 stars are removed from the sample, leaving 569 stars for analysis. \n\n\\subsection{Synthetic Spectra and filter profiles}\n\nSynthetic spectra are used for comparisons to observed color-color relations (e.g. Section \\ref{subsec:selection_Pleiades}) and for estimating the effect of surface gravity and circumstellar disk in SED fitting (Section \\ref{subsec:method_problems}). Synthetic spectra are obtained from BT-Settl models \\citet{Allard2012} with solar abundances from \\citet{Asplund2009}. {Filter profiles and zero points are collected from the SVO Filter Profile Service \\citep{SVO2012,SVO2020} for the following filter sets: GAIA\/GAIA3 (GAIA eDR3 release), 2MASS\/2MASS, WISE\/WISE, PAN-STARRS\/PS1, and Misc\/APASS.} The synthetic spectra from model atmospheres are used to assess the effects of surface gravity (see {Section \\ref{subsubsec:Ple_Ori_color_difference}}).\nMore broadly, differences between the Pleiades colors and synthetic colors from the model spectra, as seen in Figure \\ref{fig:color_color_used_in_fit}, demonstrate why the SED fitting requires empirical colors.  \n\n\n\\subsection{Pre-main-sequence Evolutionary Models}\nTo estimate ages, we adopt the \\citet{Feiden2016} non-magnetic evolutionary models. For comparison, the magnetic isochrones from \\citet{Feiden2016}  and the PARSEC isochrones \\citep{Bressan2012, ChenYang2014, ChenYang2015, TangJing2014} are also used to evaluate uncertainties related to different models of pre-main sequence evolution.\n\n\n\\section{Developing an SED fitting method}\n\\label{sec:fitting_method}\n\n\\subsection{Overview}\n\\label{subsec:overview}\nIn our fitting method, we first create SEDs from the combination of temperature $T_{\\rm{eff}}$, luminosity $L$, and extinction $A_V$ (Section \\ref{subsec:generate_SED}) and then compare these SEDs with the observed stellar SED (Section \\ref{subsec:fit_photospheric_SED}). Since YSOs may contain a dusty accretion disk that affects the observed SED, we separately treat disk and diskless stars (Section~\\ref{subsec:identify_disk}). The fitting method is tested in Section \\ref{subsec:method_test} and compared with other fitting methods in Section \\ref{subsec:compare_other_fitting_method}. Caveats of this method are discussed in Section \\ref{subsec:method_problems}.\n\n\\subsection{Generating SEDs}\n\\label{subsec:generate_SED}\nPleiades stars are used to generate empirical SEDs with photometry in the following filters:  $BP, RP, J, H, K, W1, W2, g, r, i, z, y, g', B, V$. Figure \\ref{fig:CMD_and_colorT} and \\ref{fig:color_color_used_in_fit} show relationships between colors, magnitudes, and temperatures for the Pleiades stars and young stars from other catalogs (see also additional color relationships in Appendix \\ref{appendix:sec:other_color_relations}). In general, young stars from these catalogs are consistent with Pleiades stars in color-color relations. Colors from BT-Settl spectra differ from Pleiades relations, especially at temperatures cooler than $\\sim4000$ K, as expected from previous assessments \\citep[e.g.][]{Bell2012, Herczeg2015,Lancon2020}. \n\nFirst, colors in Figure \\ref{fig:color_color_used_in_fit} are reddened by {visual extinction $A_V$ and relative extinction $A_X\/A_V$}, as obtained from the analysis of red clump stars with total-to-selective extinction coefficient $R_V=3.16$ \\citep{Wang2019} {(except $BP$ and $RP$)}, {and from \\citet{Gange2020} for Gaia $BP$ and $RP$}. \nExtinction corrections are applied only for the \\textit{Gaia} $BP$ and $RP$ photometry, based on the temperature dependence determined by \\citet{Gange2020}\\footnote{{\\citet{Gange2020} uses \\citet{Fitzpatrick1999} extinction law with $R_V=3.1$ to calculate \\textit{Gaia} extinction. The \\citet{Wang2019} extinction law is consistent with \\citet{Fitzpatrick1999} in the optical and is steeper in the infrared. The extinction of both $BP$ and $RP$ in \\citet{Wang2019} corresponds to \\citet{Gange2020} extinction at $\\sim5300$ K.}}. Then, the reddened colors are added by $m_{RP}$, leading to the generated SEDs for use in our fits.\n \nThe extinction-corrected color $BP-RP$ from the best fit is converted to $T_{\\rm eff}$ from the color-temperature relationship established from Pleiades stars (see the bottom panel of Figure \\ref{fig:CMD_and_colorT}). The APOGEE $T_{\\rm eff}$ is preferentially adopted when available, despite systematic differences between the APOGEE scale and the color-temperature relationship of \\citet{Pecaut2013}, as shown in Section \\ref{subsec:selection_Pleiades} and the bottom panel of Figure \\ref{fig:CMD_and_colorT} where the APOGEE $T_{\\rm eff}$ of Pleiades stars only covers 3200-6000 K and differs from \\citet{Pecaut2013} $T_{\\rm eff}$. The SED fit also solves for an apparent magnitude $m_{RP}$, which is then converted to luminosity using bolometric corrections (Section \\ref{subsec:fit_photospheric_SED}), the extinction, and the distance.  The temperatures in our fits range from 3000--9500 K and extinctions range from $0<A_V<6$.\n\nThe broadband SEDs generated from the empirical color-color relations are then fit to the photometry (in mag) of each star.  \n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures_revised_2nd\/BCJ_comparison.png}\n    \\caption{{The relationship between the $J$-band bolometric correction and temperature directly from \\citet{Pecaut2013} in deep blue, the Pleiades APOGEE temperatures with bolometric corrections from \\citet{Pecaut2013} in orange, the Orion APOGEE temperatures with the \\citet{Pecaut2013} bolometric correction in light blue, and \n    Orion APOGEE temperature with $BC_J$ converted from \\citet{Kounkel2018} bolometric fluxes in gray.)}}\n    \\label{fig:BCJ_comparison}\n\\end{figure}\n\n\n\\subsection{Fitting photospheric SEDs}\n\\label{subsec:fit_photospheric_SED}\nIn our fitting procedure, we find the minimum reduced $\\chi_{\\rm red}^2=\\chi^2 \/ (N-3)$, where $N$ is the number of data points, using the function \\texttt{scipy.optimize.minimize} in python \\citep{Virtanen2019}. The free parameters in the fit are an extinction-corrected $BP-RP$ (temperature), $A_V$ (extinction), and $m_{RP}$ (visual magnitude, converted to luminosity after extinction and bolometric corrections). Since these parameters have some degeneracies, the ``minimize\" function starts at 15 equally spaced locations of $BP-RP$ and 10 equally spaced locations of $A_V$. \n\n{ For each photometric band, the uncertainty is estimated from a global evaluation of the average standard between the best-fit model and empirical measurement.  The photometric uncertainty is {first} assumed to be $0.1$ mag to account for variability, unless the listed empirical error is larger than 0.1 mag.  After fitting to all targets, the magnitude difference in all pass-bands is calculated.  The standard deviation for each filter is then adopted as the magnitude uncertainty in the subsequent fitting process.  We iterate one additional sequence to adopt final empirical uncertainties and measurements.}  Errors of the fitted parameters are calculated based on {fits} with $\\chi^2 - \\chi^2_{\\rm min} < 3$ \\citep{Robitaille2007}.  Best-fit results at the edge of the fitting range are included in our tests (Section \\ref{subsec:method_test}) but are excluded from any final results of Serpens members (Section \\ref{subsec:Serpens_ages}).\n\nThe fits use as many of the photometric points as possible. However, as shown in Figure \\ref{fig:color_color_used_in_fit} (or Table \\ref{table:color_color_table}) and described in Section \\ref{subsec:selection_Pleiades}, some photometry is only available over a limited range of $BP-RP$. For example, Pan-STARRS $g$ band photometry only exists at $0.8<BP-RP<3.2$, while APASS $V$ band photometry is available at $0.03<BP-RP<3$. Thus, when calculating $\\chi^2$, $N$ will change with the input $BP-RP$, and a pass-band $X$ will not be used if the input $BP-RP$ exceeds the available $BP-RP$ range of pass-band $X$. Under these circumstances, a new minimum $\\chi^2$ will be calculated for the relevant pass-bands in the previously fitted $BP-RP$. \\footnote{As an example, consider a star with observed photometry of $BP, RP, J, H, K, g$.  Because our grid includes Pan-STARRS $g$ only for $0.8<BP-RP<3.2$, the minimum $\\chi^2$ will include $g$ at $BP-RP>0.8$ and exclude $g$ at $BP-RP<0.8$, changing the number of data points $N$ that lead to the $\\chi^2$.  \nIf the resulting minimum $\\chi^2$ is located at $BP-RP>0.8$, it will be the final minimum $\\chi^2$. Otherwise, a new minimum $\\chi^2$ will be calculated by excluding $g$ in full $BP-RP$ range.}\n\n\nThe extinction-corrected $BP-RP$ of the best-fit SED is converted to temperature using the relationship determined from the Pleiades. \n\n\\subsection{Bolometric corrections}\n\\label{subsec:bolometric_correction}\n\nThe luminosity\\footnote{$M_{bol\\odot}=4.74$ as defined by the IAU Resolution 2015 at https:\/\/www.iau.org\/static\/resolutions\/IAU2015$\\_$English.pdf.} is calculated from $m_{RP}$, corrected for extinction and a $J$ band bolometric correction ($BC_J$) obtained for the relevant temperature from \\citet{Pecaut2013} for stars cooler than 7300 K. For hotter stars, the bolometric correction from $J$ band is obtained from PARSEC isochrones.  \n\n\nSince different $T_{\\rm eff}$ calculation methods affect $BC_J$-$T_{\\rm eff}$ relation, we compare $BC_J$-$T_{\\rm eff}$ relations calculated from \\citet{Pecaut2013} $T_{\\rm eff}$ and APOGEE $T_{\\rm eff}$. \nFor these comparisons, we  with APOGEE $T_{\\rm eff}$ include Pleiades stars (described in Section \\ref{subsec:selection_Pleiades}) and Orion stars (described in Section \\ref{subsubsec:selection_Orion}). The Orion stars are diskless stars with {$A_V<=0.5$}\\footnote{{To avoid extinction issue, stars with negligible extinction are best. Because no stars have $A_V=0$ and only 21 stars have $A_V<=0.2$, we loosen the constraint to $A_V<=0.5$.}} and luminosities calculated by \\citet{Kounkel2018}). Bolometric correction of these Orion stars via \\citet{Kounkel2018} method is also included, where they derived bolometric flux from SED fitting and APOGEE $T_{\\rm eff}$. The result ($BC_J$-$T_{\\rm eff}$ relation) is shown in Figure \\ref{fig:BCJ_comparison}, with the following results.\nWhen using APOGEE $T_{\\rm eff}$ and \\citet{Pecaut2013} bolometric correction method,  Pleiades and Orion relations overlap, with deviations at {$\\sim4000$ K}.\nSome of this difference maybe be caused by the discrepancy between APOGEE $T_{\\rm eff}$ and \\citet{Pecaut2013} temperatures, as seen in the bottom panel of Figure \\ref{fig:CMD_and_colorT}. \n\n\n\\subsection{Identifying stars with disks}\n\\label{subsec:identify_disk}\n\n\\begin{figure}[bht]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures_revised_2nd\/separation_modify_together.png}\n    \\caption{Color-color relationships to separate disk-hosting and diskless stars.  In Orion, stars with disks (deep blue) and without disks (light blue) based on {\\it Spitzer} photometry, separated by the solid gray line.  Most optical members of Serpens (orange) are diskless stars.}\n            \\label{fig:disk_selection_KW2_HW1}\n\\end{figure}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures_revised_2nd\/compare_with_VOSA_R17.png}\n    \\caption{Comparison between temperatures from SED fits and literature (APOGEE) temperatures for diskless (left) and disk (right) stars in Orion, for our SED fitting method (top), VOSA \\citep[middle,][]{Bayo2008}, and \\citet{Robitaille2017} (bottom). \n    The gray solid line represents equal temperature line, while gray dashed lines represent deviation of 300 K and 1000 K. The light blue square symbols are stars with only \\textit{Gaia} and 2MASS data in the fitting. The deep blue asterisk symbols are stars with a nearby source resolved by \\textit{Gaia} and within 2 mag in $G$ band, but unresolved by 2MASS.}\n    \\label{fig:fitting_result_compare}\n\\end{figure*}\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures_revised_2nd\/chi2_contour_001.png}\n    \\caption{{Contour of a diskless Orion star in the fitting calculated from grids of $T_{\\rm eff}$ and $A_V$. Color bar represents $\\log_{10}(\\chi^2)$. The black contours are $\\chi^2-\\chi^2_{min}=5$ and $\\chi^2-\\chi^2_{min}=10$. The orange lines are the best-fit $T_{\\rm eff}$ and $A_V$.}}\n    \\label{fig:chi2_contour}\n\\end{figure}\n\nThe presence of a disk introduces complications to interpreting SEDs. Warm dust in accretion disks produce excess IR emission, especially in the all-sky WISE survey \\citep[see review by][]{Williams2011}.  Accretion produces excess H continuum emission that makes optical colors bluer, in some cases dominating over any photospheric emission \\citep[see review by][]{hartmann16}. For some viewing angles, the disk intercepts our line-of-sight to the star, so the observed emission from the star is extinguished and contaminated by scattered light. These processes are all variable \\citep[e.g.][]{grankin07,guo18gitau,venuti21}, which introduces additional uncertainties when comparing non-simultaneous data.\n\nFigure \\ref{fig:disk_selection_KW2_HW1} shows the 2MASS and WISE color-color relations used to separate disk and diskless stars, based on the identification of disks in the Orion sample from excess emission in {\\it Spitzer} mid-IR imaging \\citep{Fang2013,Fang2017}. To quantify the separation (Figure~\\ref{fig:disk_selection_KW2_HW1}), we follow the ``Class II box\" method in \\citet{Dunham2015} with disk and diskless stars separated by a line that  maximizes the sum of the two kinds of probability: (a) the probability that a diskless star is located under the line (defined as the number of diskless stars under the line divided by the total number of diskless stars), and (b) the probability that a star under the line is classified as diskless (defined as the number of diskless stars under the line divided by the total number of stars under the line). Disks are identified from excess emission at $W2$, as seen in the $K-W2$ (or $H-W1$, if $W2$ photometry is unreliable) versus $J-K$. For stars with disks, $W1$ and $W2$ photometry are excluded from SED fits.\n\n\n\\subsection{Testing the SED fitting results}\n\\label{subsec:method_test}\n\nTo test the fitting method, we fit the SEDs of YSOs in Orion and then compare the resulting temperatures to those measured from APOGEE spectra \\citep{Cottaar2014,Kounkel2018}. Fits to stars with and without disks are analyzed separately.  We exclude stars that only have \\textit{Gaia} and 2MASS photometry and visual binaries that are resolved by Gaia and unresolved by 2MASS.\n\n\nFigure~\\ref{fig:fitting_result_compare} and Table~\\ref{table:fitting_deviation} show the difference and scatter of fitted temperatures \\footnote{Represented by mean and standard deviation of $(T_{\\rm eff, fit}-T_{\\rm eff, APOGEE})$}. In general, the difference and scatter of $T_{\\rm eff}$ are 90 K and 433 K for stars without disks and 533 K and 1246 K for stars with disks. Our method is more reliable for both diskless and disk-hosting stars with $T_{\\rm fit}< \\sim$5000 K, where the difference and scatter are only -13 K and 208 K for diskless stars, and 53 K and 212 K for disk stars. For disk stars with higher fitted temperatures, the fit may be unreliable, affected by variability and by contamination of photospheric emission by emission from the accretion disk. Stars that only have \\textit{Gaia} and 2MASS photometry and also visual binaries that are resolved by \\textit{Gaia} and unresolved by 2MASS are included in Figure~\\ref{fig:fitting_result_compare} to demonstrate that our method is not reliable for these stars.  These stars are excluded in calculating the average temperature differences listed above.\n\n{Figure~\\ref{fig:fitting_result_compare} also shows that there are more stars with $T_{\\rm fit}-T_{\\rm APOGEE}>300~K$ at $T_{\\rm APOGEE}>\\sim4500~K\\ (\\sim3.65$~dex) than at cooler $T_{\\rm APOGEE}$, which is likely due to the larger degeneracy between $T_{\\rm eff}$ and $A_V$ at higher temperature. Figure \\ref{fig:chi2_contour} shows the $\\chi^2$ contours around the best-fit parameters for a diskless star, as calculated from grids of $T_{\\rm eff}$ and $A_V$. The contours extend to high temperatures because of a large degeneracy between temperature and extinction. Such degeneracy is also shown in the bottom panel of Figure \\ref{fig:quantify_RP_J}, where the color-color relation is closer to parallel with the extinction vector at higher temperature. }\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures_revised_2nd\/quantify_RP_J_dots.png}\n    \\caption{Color color diagrams (bottom) and residuals (top) for stars in groups with different ages.  The Pleiades stars (light blue triangles) form the foundation for our fits, and are compared with diskless stars in Orion with $A_V<1$ (gray asterisk), the beta Pic Moving Group (deep blue square),  $\\epsilon$ Chamaeleontis, $\\eta$ Chamaeleontis and TW Hya Association (orange). Deep orange line segments in the bottom panels are extinction vectors ($A_V=1$) at $BP-RP=0.5$ (left) and $BP-RP=2.1$ (right).}\n    \\label{fig:quantify_RP_J}\n\\end{figure}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{figures_revised_2nd\/Orion_T_Av_fitted_literature.png}\n    \\caption{$T_{\\rm eff}$ versus $A_V$ for diskless Orion stars, for (left) our results and (right) fits from \\citet{Kounkel2018}.  The colors and symbols are the same as those in Figure \\ref{fig:fitting_result_compare}.}\n    \\label{fig:Orion_T_Av_fitted_literature}\n\\end{figure*}\n\n\n\n\\begin{deluxetable*}{cccccccc}\n\\label{table:fitting_deviation}\n\\caption{Temperature difference and scatter \\tablenotemark{1} (in unit of K) between fitted and literature result for stars with the highest data quality \\tablenotemark{2}.}\n\\tablehead{\\colhead{$T_{\\rm eff, fit}$ range (K)} & & & \\colhead{Diskless} & & &  \\colhead{Disk}}\n\\startdata\n & &  this work & VOSA & R17 & this work & VOSA & R17 \\\\ \\hline\n\\multirow{2}{*}{\\textless{}3500} & Difference & -40 & -323 & -690 & 47 & -194 & -475 \\\\\n & Scatter & 143 & 225 & 245 & 159 & 206 & 210 \\\\ \\hline\n\\multirow{2}{*}{3500-4000} & Difference & -129 & -369 & -431 & -69 & -344 & -306 \\\\\n & Scatter & 98 & 105 & 104 & 124 & 117 & 210 \\\\ \\hline\n\\multirow{2}{*}{4000-5000} & Difference & 117 & 210 & -21 & 186 & 392 & 605 \\\\\n & Scatter & 240 & 436 & 389 & 234 & 246 & 199 \\\\ \\hline\n\\multirow{2}{*}{\\textgreater{}5000} & Difference & 857 & 2050 & 1565 & 3092 & 4114 & 5524 \\\\\n & Scatter & 779 & 1474 & 2795 & 1350 & 1932 & 6366 \\\\ \\hline\n\\multirow{2}{*}{All} & Difference & 90 & 380 & -109 & 533 & 774 & 571 \\\\\n & Scatter & 433 & 1324 & 1345 & 1246 & 1961 & 3331 \\\\\n\\enddata\n\\tablenotetext{1}{Difference and scatter are the mean and standard deviation of $(T_{\\rm fit}-T_{\\rm APOGEE})$ separately.}\n\\tablenotetext{2}{{Stars with only \\textit{Gaia} and 2MASS data are excluded; stars with a nearby source unresolved by 2MASS are excluded. The rest are marked as ``highest data quality''.}}\n\\tablecomments{The columns marked with ``VOSA'' and ``R17'' are comparison of VOSA and \\citet{Robitaille2017} SED fitting tools with details described in Section \\ref{subsec:method_test} and Section \\ref{subsec:compare_other_fitting_method} separately.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{ccccccccccccccccc}\n\\label{table:banduncertainties}\n\\tablecaption{{Average difference between model and observed magnitude}}\n\\tablehead{\\colhead{Region} & \\colhead{disk} & \\colhead{$T_{\\rm eff, APOGEE}$} & \\colhead{$\\Delta g$} & \\colhead{$\\Delta B$} & \\colhead{$\\Delta BP$} & \\colhead{$\\Delta V$} & \\colhead{$\\Delta r$} & \\colhead{$\\Delta i$} & \\colhead{$\\Delta RP$} & \\colhead{$\\Delta z$} & \\colhead{$\\Delta y$} & \\colhead{$\\Delta J$} & \\colhead{$\\Delta H$} & \\colhead{$\\Delta K$} & \\colhead{$\\Delta W1$} & \\colhead{$\\Delta W2$}}\n\\startdata\n  Orion & diskless & $<=4000$ K & 0.13 & 0.3 & 0.02 & 0.10 & 0.08 & 0.03 & 0.03 & 0.01 & 0.02 & 0.03 & 0.03 & 0.02 & 0.03 & 0.04 \\\\\n   &   & $4000-5000$ K & 0.21 & 0.21 & 0.02 & 0.10 & 0.06 & 0.04 & 0.06 & 0.02 & 0.03 & 0.03 & 0.03 & 0.03 & 0.03 & 0.05 \\\\\n    &   & $>5000$ K & 0.16 & 0.20 & 0.01 & 0.08 & 0.05 & 0.05 & 0.03 & 0.01 & 0.03 & 0.04 & 0.02 & 0.02 & 0.03 & 0.05 \\\\\n    &   & all       & 0.16 & 0.23 & 0.02 & 0.10 & 0.07 & 0.04 & 0.04 & 0.01 & 0.02 & 0.03 & 0.03 & 0.02 & 0.03 & 0.05 \\\\\n  Orion & disk & $<=4000$ K & 0.24 & 0.35 & 0.07 & 0.11 & 0.06 & 0.04 & 0.14 & 0.03 & 0.03 & 0.04 & 0.01 & 0.09 & 0.15 & 0.35  \\\\ \n   &  &  $4000-5000$ K  & 0.47 & 0.21 & 0.07 & 0.07 & 0.18 & 0.08 & 0.17 & 0.05 & 0.05 & 0.08 & 0.04 & 0.15 & 0.15 & 0.34  \\\\ \n    &  &  $>5000$ K  & - & - & - & - & - & - & - & - & - & - & - & - & - & -  \\\\\n    &  &  all  & 0.39 & 0.22 & 0.07 & 0.07 & 0.10 & 0.05 & 0.15 & 0.04 & 0.03 & 0.05 & 0.02 & 0.10 & 0.15 & 0.34 \\\\\n\\enddata\n\\tablecomments{Although the fitting on disk stars excludes $W1$ and $W2$, these pass-bands are still included in the table, representing the difference between model stellar photosphere SEDs and observed SEDs with infrared excess.}\n\\tablecomments{{Only 1 disk star has $T_{\\rm eff,APOGEE}>$5000 K, thus we do not report its magnitude difference.}}\n\\end{deluxetable*}\n\nFigure \\ref{fig:Orion_T_Av_fitted_literature} shows the relationship between extinction and $T_{\\rm eff}$ for diskless Orion stars from our results and from \\citet{Kounkel2018}. The distribution of extinctions in \\citet{Kounkel2018} is $\\sim 1$ mag higher for stars with $\\log (T\/K)\\sim3.6$ than for hotter and cooler stars, possibly due to differences between observed SEDs and synthetic spectra. \\citet{Kounkel2018} first determined temperature from APOGEE spectra and then compared the observed SED with PHOENIX synthetic spectra \\citep{Husser2013} with the same $T_{\\rm eff}$ to calculate $A_V$. If the stellar photosphere is redder than the synthetic spectrum, the calculated $A_V$ will be higher, leading to a gap in $T_{\\rm eff}$ - $A_V$ relation (perhaps caused by the discrepancy seen in color-temperature scale in Figure~\\ref{fig:CMD_and_colorT}). Such a gap is not seen in our fitting method (Figure \\ref{fig:Orion_T_Av_fitted_literature}, left panel), { which is consistent with} Orion stars having colors that are more similar to the Pleiades SEDs than to synthetic spectra.\n\nTable \\ref{table:banduncertainties} lists the average difference between fitted and observed magnitude of these Orion stars after two iterations of SED fits. In general, for diskless stars, the average magnitude difference {is less than $\\sim0.03$ mag at pass-bands redder than $RP$}. The larger uncertainties in blue wavelengths (except $BP$) are likely introduced by differences in spot coverage and by variability.\nFor disk stars, the smallest average magnitude difference is at $H$ band. The $W2$ band, which is excluded in these fits, shows the largest difference, as expected for stars with excess infrared emission from disks.  Optical pass-bands $BP$ and $RP$ have larger differences than the near-infrared passbands, likely because accretion variability and disk obscuration affect the optical more than the near-IR.\n\nAdditional descriptions {and comments} of these results are provided in {Section \\ref{subsec:method_problems}}.\n\n\n\n\n\\subsection{Comparing our SED fits to other fitting methods}\n\\label{subsec:compare_other_fitting_method}\n\nTo compare with other SED fitting methods, photometry of stars in Orion are fit with the commonly used SED codes VOSA \\citep{Bayo2008} and \\citet{Robitaille2017} (hereafter Robitaille).  Our fits with VOSA use the BT-Settl models with solar metallicity and surface gravity $\\log (g)=4.5$, an extinction law from \\citet{Fitzpatrick1999} and improved by \\citet{Indebetouw2005} in the infrared.  The Robitaille fits  use their ``s---s-i\" models (pure stellar photosphere models constructed from \\citet{Castelli2003} above 4000 K and PHOENIX atmospheres \\citep{Brott2005} below 4000 K) with an extinction law consistent with $R_V=5.5$ \\citep[following the description in][]{Forbrich2010}.\n\nFigure \\ref{fig:fitting_result_compare} and Table \\ref{table:fitting_deviation} show that the difference and scatter between the fitted and literature (spectroscopic) temperatures are 380 K and 1324 K for VOSA, -109 K and 1345 K for \\citet{Robitaille2017}, and 90 K and 433 K for our method. All three methods are less reliable for stars that have only \\textit{Gaia} and 2MASS photometry and visual binaries that are resolved by \\textit{Gaia} and unresolved by 2MASS. In addition, at $\\log T\/K<3.6$, both the VOSA and Robitaille fits underestimate temperatures by {$\\sim$300 K to $\\sim$1000~K} relative to literature values. Both models also assess a wide range of temperatures for stars with spectroscopic temperatures of $\\log T\/K\\sim 3.65$. In comparison, temperatures from our fits follow the literature estimates within $\\sim 300$ K until $\\log T\/K=3.65$, where our models suffer from a similar but less severe dispersion.  Possible reasons for this feature are discussed in {Section \\ref{subsec:method_problems}}.\n\nIn these comparisons, the stellar magnitude and fitting ranges are the same as Section \\ref{subsec:method_test}. The \\citet{Robitaille2017} fits do not include additional temperature constraints, thus the fitted temperature can exceed the ranges of our method. Magnitude uncertainties are 0.1 mag in both VOSA and \\citet{Robitaille2017} fits, with no iteration executed in their methods.\n\n\n\n\n\\subsection{Shortcomings and challenges in our SED fitting}\n\\label{subsec:method_problems}\n\nThis SED fitting method uses Pleiades stars to create templates to fit YSOs in the Serpens Molecular Clouds.  However, the Pleiades stars are not perfect templates. Shortcomings caused by Pleiades include differences in surface gravity and spots (see Section \\ref{subsubsec:Ple_Ori_color_difference}). In addition, we comment on our fits to disk stars in Section \\ref{subsubsec:fits_to_disk_stars}, and show how positive extinction affects the fitting result on low extinction clusters (Section \\ref{subsubsec:constrained_extinction}). Finally, we comment the outliers of the fitting result in Section \\ref{subsubsec:outliers_Teff_comparison}.\n\n\n\\subsubsection{Temporal evolution of color-color relations}\n\\label{subsubsec:Ple_Ori_color_difference}\n\nDifferences in color-color relations between Orion and Pleiades, perhaps caused by differences in gravity or spots, may lead to biases in the fitting result. In this subsection, we first give an example on how the difference in color-color relation influences the fitting result.  We then discuss the effect of surface gravity and spots separately.\n\n\nFigure \\ref{fig:quantify_RP_J} shows $(BP-RP)$-$(RP-J)$ for Pleiades, Orion, and members of the beta Pic Moving Group, $\\epsilon$ Cha association, $\\eta$ Cham association, and TW Hya Association from \\citet{Gagne2018,Gagne2018b}. To quantify $(RP-J)-(RP-J)_{\\rm Pleiades}$ at a given $BP-RP$, $(RP-J)-(RP-J)_{\\rm Pleiades}$ is given by setting several bins in $BP-RP$ and calculating the average $RP-J$ in each bin that includes a star. \n\nThe $RP-J$ versus $BP-RP$ colors of the groups differ at $BP-RP\\sim2$ ($\\sim4000$ K or 3.6 dex), which may explain the temperature deviation in Figure \\ref{fig:fitting_result_compare}.  As a simple example, only $BP$, $RP$, and $J$ pass-bands are used in the following demonstration. First, a star in Orion with $A_V \\sim 0$ mag, $BP-RP=2.15$ ($\\sim4000$ K or 3.6 dex) and $RP-J=1.62$ is selected. According to the Pleiades color-color relation and extinction vector in Figure \\ref{fig:quantify_RP_J}, the fitted $BP-RP$ of this star is 1.32 ($\\sim4800$ K or 3.68 dex). While the fitted temperature is higher, the fitted extinction is also higher, leading to a gap in $T_{\\rm eff}$-$A_V$ relation at $\\sim4000$ K. Although this is a simple demonstration using only three pass-bands, redder Orion stars may be one of the reasons for the temperature bump in Figure \\ref{fig:fitting_result_compare}.  \n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures_revised_2nd\/comparison_different_logg.png}\n    \\caption{Comparison between temperatures for spectra with $\\log g$ {=3.5} and temperatures from fits to those spectra using $\\log g$ {=4.5} spectra as templates.}\n    \\label{fig:comparison_different_logg}\n\\end{figure}\n\nThese colors may be caused by the increase in surface gravity as a star contracts, with empirical measurements from APOGEE of $\\log(g)\\sim4.0$  for 2-6 Myr IC 348 stars  to $\\sim4.8$ for $\\sim$110 Myr Pleiades stars \\citep{Cottaar2014}. To simulate the effect of surface gravity, we use BT-Settl models to describe that the lower gravity of younger stars leads to a higher fitted temperature at $\\sim4000$ K.\n\nBT-Settl spectra with $\\log g$ {=4.5} are used to build basic SEDs as templates, while BT-Settl spectra with $\\log g$ {=3.5} are then used as the targets to be fitted. {Offsets in Figure \\ref{fig:comparison_different_logg} (lower $T_{\\rm fit}$ at $\\sim$3.55 dex and higher $T_{\\rm fit}$ at $\\sim$3.61 dex is consistent with the similar offset in Figure \\ref{fig:fitting_result_compare}}. The offset at $3.6<\\log {T_{\\rm{eff}}}<3.7$ in Figure \\ref{fig:fitting_result_compare} is at least partially attributed to the difference in surface gravity.\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures_revised_2nd\/comparison_estimate_disk_effect_veiling_20.png}\n    \\caption{Comparison between the temperature of a BT-Settl spectra versus the fitted temperature of the spectrum plus a warm dust dusk.}\n        \\label{fig:comparison_disk_effect}\n\\end{figure}\n\n\nIn addition to surface gravity, spot properties also vary among YSOs \\citep[e.g.,][]{Grankin2008} and influence SED fitting result. Spots on magnetically active cool stars redden the broadband SED \\citep[e.g.][]{somers20}, leading to errors when fitting with single temperatures \\citep{gully17}. Although the Pleiades stars are heavily spotted \\citep[e.g.][]{stauffer03,Fang2016,Guo2018}, our SED fits may be insufficient in incorporating spots, if they are more important for young stellar objects \\citep{Grankin2008}.\n\n\\subsubsection{{Fits to disk stars}}\n\\label{subsubsec:fits_to_disk_stars}\n\nStars with disks are not well described through photometric fits. Removing a few photometric points may avoid the influence of infrared excess and\/or bluer optical colors caused by a disk, but enough pass-bands are still needed to constrain the photospheric properties.  Most fitting routines, including ours, do not incorporate accretion, which alters colors and in some cases dominates the { optical} emission \\citep[e.g.][]{gahm08}. In such cases, any attempt to measure stellar properties from SEDs or spectra is dubious. More typical accreting stars, where the accretion flow affects but does not dominate the optical emission, have SEDs that are roughly simulated by models but are expected to show excess blue emission.\n\nWe evaluate uncertainties caused by emission from dust by creating SEDs from a combination of BT-Settl spectrum of the photosphere and a 1400 K blackbody for the dust, chosen as the dust sublimation temperature. The blackbody emission flux is scaled so that the $K$-band flux from the blackbody is {two times} of the photosphere flux \\citep[consistent with the ratios in][]{Fischer2011}. The spectrum is then fit with the BT-Settl spectra alone, excluding $W1$ and $W2$. The SED templates are based on interpolating BT-Settl color-color relations. The temperature comparison (Figure \\ref{fig:comparison_disk_effect}) shows that an offset of {$\\sim1000 K$} is present for stars at $\\log T_{\\rm eff}\\sim3.55$, while for the cooler stars the offset is smaller. \n\nAlthough this simple approach indicates that the fits are robust to the presence of steady dust disks, the effect of disks are expected to be more significant than described here. In some cases the disk will obscure the star. Accretion from the disk onto the star contributes and in some cases dominates optical colors. Both disk obscuration and accretion are variable.  These phenomenon are significant confounding factors that are not explored in this paper.\n\n\n\\subsubsection{{Fits to the Pleiades and constrained extinctions}}\n\\label{subsubsec:constrained_extinction}\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures_revised_2nd\/Pleiades_Av_distribution_positive_negative.png}\n   \\caption{The distribution of best-fit $A_V$ (orange histogram; mean indicted with the blue line) of Pleiades stars, for fits that are unconstrained by reality and allowed to be negative (top panel) and fits that are constrained to be $\\geq 0$ (bottom panel).}\n    \\label{fig:Pleiades_Av_distribution}\n\\end{figure}\n\nThe extinction range ($0 \\leq A_V \\leq 6$) affects the extinction distribution of a cluster with low extinction and statistical comparisons of low-extinction clusters to high-extinction clusters.  \nFigure \\ref{fig:Pleiades_Av_distribution} shows the best-fit $A_V$ to stars in the Pleiades.  The average $A_V$ is 0.04 mag when $A_V$ is allowed to be negative and 0.1 mag when $A_V$ cannot be negative.   The spread in estimated $A_V$ and therefore the size of this effect is likely larger for younger stars.\n\n\\subsubsection{Outliers in the temperature comparison}\n\\label{subsubsec:outliers_Teff_comparison}\n\n\nIn this subsection, we comment on stars with the best-fit SED temperature deviating by 1000 K from the previously measured literature temperature, with a total number of 13 stars after excluding those with photometric issues in Section \\ref{subsec:method_test}:\n\n\nFor the four stars in this small sample with Pan-STARRS photometry, three are saturated in the $z$ band. Since $z$ band has the largest weight among the magnitudes when fitting SEDs (Section \\ref{subsec:fit_photospheric_SED}, Table \\ref{table:banduncertainties}), the saturated z-band photometry has a significant influence on the fitting result. The fourth star has a $g$ band image where the star appears to fall at the edge of the detector.\n\nSix stars are variable stars reported by ASAS-SN \\citep[the All-Sky Automated Survey for Supernovae,][]{ASASSN2014,ASASSN2017}, with amplitudes from 0.1 mag to 0.3 mag. A seventh star has not been reported as a variable star but shows a wavy light curve with an amplitude of $\\sim0.1$ mag.\n\nThe previous two sets of stars have some overlap. Of the remaining stars, three with the highest $T_{\\rm fit}$ have $A_V(\\rm fit)<0.5$, indicating a higher degeneracy between $T_{\\rm eff}$ and $A_V$, which is due to the extinction vectors becoming more parallel to color-color relations at higher temperature, as shown in the bottom panel of Figure \\ref{fig:quantify_RP_J}.  Two stars are not explained by the above descriptions.\n\n\n\n\n\\begin{deluxetable*}{cccccccccccccccccc}\n\\label{table:Serpensfits}\n\\tablecaption{Fits of Serpens members}\n\\tablehead{\\colhead{\\textit{Gaia} {EDR3} ID} & \\colhead{RA\/deg} & \\colhead{DEC\/deg} & \\colhead{Region} & \\colhead{d\/pc} & \\colhead{T$_{\\rm eff}$\/K} & \\colhead{$A_V$} & \\colhead{$L$(log$L\/L_{\\odot}$)} & \\colhead{Age\/Myr} & \\colhead{Mass\/$M_{\\odot}$} & \\colhead{Disk} & \\colhead{{Flag}}}\n\\startdata\n4309522600884785536 & 289.69099 & 10.80327 & LDN 673 & 376 & 7014 & 0.84 & 0.69 & 21.8 & 1.50 & - & 1 \\\\\n4309648735482263936 & 290.35284 & 11.11158 & LDN 673 & 414 & 4401 & 2.58 & 0.15 & 1.9 & 1.02 & 0 & 0 \\\\\n4309847540917708416 & 290.04547 & 11.3505 & LDN 673 & 409 & 3013 & 2.03 & -0.51 & 1.0 & 0.18 & 0 & 0 \\\\\n4309847884515108480 & 290.01491 & 11.37616 & LDN 673 & 406 & 3743 & 2.03 & -0.67 & 6.0 & 0.56 & - & 1 \\\\\n4271424759186100352 & 275.28395 & -1.30382 & Distrib & 430 & 4282 & 1.90 & -0.10 & 2.6 & 0.86 & 0 & 1 \\\\\n4271223720358239744 & 275.73095 & -1.40617 & Distrib & 369 & 4322 & 0.28 & -0.91 & 47.8 & 0.68 & 0 & 1 \\\\\n4271528250719881728 & 275.50849 & -1.08444 & Distrib & 443 & 3005 & 1.29 & -1.35 & 3.8 & 0.09 & 0 & 0 \\\\\n4269964371522804224 & 275.3711 & -2.62225 & Distrib & 449 & 4526 & 0.00 & -0.50 & 21.8 & 0.88 & 0 & 1 \\\\\n4269965196156468480 & 275.51853 & -2.60617 & Distrib & 456 & 4215 & 0.12 & -0.66 & 22.9 & 0.82 & 0 & 1 \\\\\n4270725160552584960 & 275.46313 & -2.40182 & Distrib & 446 & 4157 & 0.80 & -0.98 & 47.8 & 0.66 & 0 & 1 \\\\\n\\hline\n\\multicolumn{8}{l}{All photometry in a table for vizier}\\\\\n\\enddata\n\\tablecomments{\"Flag\": \"0\" represents that this star is not included in calculating cluster ages (Section \\ref{subsec:Serpens_ages}).  Individual distances are adopted from \\textit{Gaia} EDR3 parallaxes.}\n\\end{deluxetable*}\n\n\n\n\n{\\catcode`\\&=11\n\\gdef\\2016AandA...593A..99F{\\citet{Feiden2016}}}\n\n\n\\section{SED Fits to Members of the Serpens Star-forming regions}\n\\label{sec:Serpens}\nThe primary motivation for this paper is to characterize the optically bright stars in Serpens. Of the previously identified optical members in Serpens,  569 stars pass the photometric selection criteria and have SEDs fit here. To analyze their SEDs, the first step is identifying if a star has a disk (Section \\ref{subsec:Serpens_disk_star}). The SEDs are fit to the relevant bandpasses, with results and analysis presented in Section \\ref{subsec:Serpens_fit_result_analysis}. We then analyze the cluster ages in Section \\ref{subsec:Serpens_ages}.\n\n\n\\subsection{Disk Presence}\n\\label{subsec:Serpens_disk_star}\nThe presence of disks around young stars is typically assessed from mid-IR photometry.  Since much of the Serpens cloud was imaged by \\textit{Spitzer}, we first cross-match the Serpens sample with \\textit{Spitzer} catalogs \\citep{Evans2003,Gutermuth2008,Dunham2015} and find 123 stars with \\textit{Spitzer} data. A disk is determined to be present in 52 stars based on an IR spectral index $\\alpha>-1.6$, following \\citet{Dunham2015}.  \nFor the Serpens stars without \\textit{Spitzer} data, disk presence is identified by WISE and 2MASS color cuts (Figure \\ref{fig:disk_selection_KW2_HW1}). In total, there are 359 diskless stars (71 from \\textit{Spitzer}, 288 from 2MASS and WISE colors as seen in Figure \\ref{fig:disk_selection_KW2_HW1}), 86 disk stars (52 from \\textit{Spitzer}, 34 from WISE), and 231 stars without disk information due to the lack of both \\textit{Spitzer} data and reliable WISE data. \n\nTo confirm consistency in assessing disk presence or absence, we evaluate 82 members with both {\\it Spitzer} and WISE photometry. Applying these color cuts to regions imaged by {\\it Spitzer} show that the methods are mostly consistent.  The 10\\% of stars that change classification between disk and diskless, all located near the borders of the classification system.  To avoid the appearance of disk stars in fitting diskless stars, stars that have both WISE and {\\it Spitzer} photometry are treated as diskless only if both sets of measurements indicate  the lack of disk.\n\nDisk properties from Figure \\ref{fig:disk_selection_KW2_HW1} are evaluated from the results of SED fits to Serpens members. The fitted $W1$ and $W2$ magnitudes are compared with observed magnitudes to evaluate the disk selection in Figure \\ref{fig:disk_selection_KW2_HW1}. {We would then reclassify the presence of a disk if the $W1$ or $W2$ magnitude is at least $0.78$ mag or $0.98$ mag (the average magnitude difference of disk stars) above the photospheric level, while disk stars would be re-classified as diskless if the $W1$ and $W2$ magnitude are less than $0.05$ mag and $0.09$ mag (the average magnitude difference of diskless stars) of the photospheric level. In this reassessment, only one star is reclassified into the diskless category, and all stars previously identified as diskless remain diskless. We update results following this possible reclassification.}   \n\n\n\\subsection{Isochrone fits to stellar properties}\n\\label{subsec:Serpens_fit_result_analysis}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures_revised_2nd\/temperature_extinction.png}\n    \\caption{Temperature versus extinction (top panels) and the distribution of extinctions (bottom panels) for four major groups in Serpens. The black lines in bottom panels are the mean and standard deviation of $A_V$.}\n    \\label{fig:T_Av_Serpens}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures_revised_2nd\/HRD_Serpens.png}\n    \\caption{The HR diagram of Serpens members in different groups, compared with isochrones from the non-magnetic evolutionary tracks of   \\citet{Feiden2016}, for stars with disks (deep blue triangles), without disks (orange dots), and without disk information (light blue squares). Unfilled symbols are stars which are not used in calculating ages. The gray curves are isochrones with 1 Myr, 2 Myr, 5 Myr, 10 Myr, 15 Myr, 30 Myr, 50 Myr from top right to bottom left.}\n    \\label{fig:HRD_Serpens}\n\\end{figure*}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures_revised_2nd\/age_number_distribution.png}\n    \\caption{The distribution of ages for three groups in Serpens: Main (top panel), NE (medium panel) and far-South (bottom panel), in order of average age for the group.  The age spread in each group may not be significant \\citep[see discussion in][]{soderblom14}.}\n    \\label{fig:age_num_dist}\n\\end{figure}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures_revised_2nd\/age_map.png}\n    \\caption{Spatial location of Serpens members with colors mapped by ages. Orange circles represent the location of three subgroups: Serpens Main, NE and farSouth, with location and radius adopted from Table 1 of \\citet{Herczeg2019}.}\n    \\label{fig:age_map}\n\\end{figure}\n\nTable \\ref{table:Serpensfits} presents the best fit temperatures, luminosities, and extinctions for stars associated with the Serpens Molecular Cloud.   Figure \\ref{fig:T_Av_Serpens} shows color excesses due to extinction versus temperature for stars in four different regions in Serpens (Main, NE, far-South, Distributed). The optical members of Serpens Main have the highest median extinction, while Serpens far-South is the least extincted, consistent with Serpens Main hosting more ongoing star formation than Serpens far-South.\n\n\nThe masses and ages of individual stars are determined by comparing the temperature and luminosity to pre-main sequence evolutionary models for non-magnetic stars calculated by \\citet{Feiden2016}, using the isochrone fitting method in \\citet{Zari2019}. Specifically, the likelihood for a single star to come from an isochrone with an age of $\\tau$ is\n\\begin{equation}\n    \\label{eq:Likelihood_single_star}\n    L(\\tau) = \\int L(\\tau, m)dm,\n\\end{equation}\nwhere $m$ is the mass of a point in the isochrone, and $L(\\tau, m)$ is calculated from\n\\begin{equation}\n    \\label{eq:Likelihood_single_star_mass}\n    L(\\tau, m) = \\frac{1}{(2\\pi)^{1\/2}\\sigma_T} \\frac{1}{(2\\pi)^{1\/2}\\sigma_L} \\exp{(-\\chi^2\/2)},\n\\end{equation}\nwith:\n\\begin{equation}\n    \\label{eq:chi_square_single_star}\n    \\chi^2 = \\left(\\frac{T-T_{iso}}{\\sigma_T}\\right)^2 + \\left(\\frac{Lum-Lum_{iso}}{\\sigma_{Lum}}\\right)^2,\n\\end{equation}\nwhere $T$, $Lum$, $\\sigma_T$, $\\sigma_L$ represent temperature, luminosity and their errors (in log scale) of a star, while $T_{iso}$ and $Lum_{iso}$ are temperature and luminosity of a point in an isochrone which are determined by mass and age. The maximum $L(\\tau)$ corresponds to the best-fit age. The isochrone fitting procedure is based on linear space of the isochrones.\n\n\n\\subsection{Age estimates for Serpens subclusters}\n\\label{subsec:Serpens_ages}\n\nIn this section, ages are estimated via different isochrones and different methods. {Stars with photometric problems or with best-fit parameters at the edge of the parameter space of our grid (Section \\ref{subsec:method_test}) are excluded in calculating ages, with a total number of 455 stars used in age estimation.} The ages adopted in this study is derived in section \\ref{subsubsec:adopted_ages}, while the comparison of ages are presented in Section \\ref{subsubsec:age_other_iso}\n\n\\subsubsection{Ages adopted in this study}\n\\label{subsubsec:adopted_ages}\n\n\\begin{deluxetable*}{lccccccc}\n\\label{table:Serpens_age_iso_fit_std}\n\\tablecaption{Ages and errors (Myr) of Serpens groups with non-magnetic isochrones}\n\\tablehead{\\nocolhead{region} & \\colhead{disk} & \\colhead{error} & \\colhead{diskless} & \\colhead{error} &  \\colhead{total} & \\colhead{error} & \\colhead{disk fraction}}\n\\startdata\nMain      & 3.0  & 0.5  & 4.0 & 0.2 & 4.0 & 0.2 & 0.22  \\\\\nNE        & 7.6 & 2.9 & 8.3  & 0.3 & 8.3 & 0.3 & 0.03 \\\\\nfarSouth  & 9.5 & 9.3 & 11.5 & 0.4 & 13.8 & 0.9 & 0.04 \\\\\nfarS\\_ext & - & - & 21.9 & 1.6 &  21.9 & 1.4 & 0.00  \\\\\nFGAql     & - & - & - & - & - & - & -  \\\\\nLDN673    & 2.9 & 0.5 & - & - &  6.3 & 1.4 & -  \\\\\nDistrib   & 5.5 & 0.6 & 10.0 & 0.3  & 10.0 & 0.4 & 0.12 \\\\\n\\enddata\n\\tablecomments{Estimated with isochrone fitting.  Total includes disk and diskless, and sources with no disk information.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{ccccccccccccc}\n\\label{table:Serpens_age_compare_iso}\n\\tablecaption{Ages and errors (Myr) of Serpens groups with different isochrones and different estimation methods}\n\\tablehead{\\colhead{ } & \\multicolumn{2}{c}{{Feiden, standard}} & \\multicolumn{2}{c}{Feiden, magnetic} & \\multicolumn{2}{c}{Feiden, magnetic} & \\multicolumn{2}{c}{PARSEC} &  \\multicolumn{2}{c}{PARSEC} & \\multicolumn{2}{c}{PARSEC}\\\\\n\\colhead{ } & \\multicolumn{2}{c}{$T_{\\rm eff}-L$}  & \\multicolumn{2}{c}{$T_{\\rm eff}-L$}  & \\multicolumn{2}{c}{$T_{\\rm eff}-L$}  &\\multicolumn{2}{c}{$T_{\\rm eff}-L$} &  \\multicolumn{2}{c}{$T_{\\rm eff}-L$} & \\multicolumn{2}{c}{CMD}\\\\\n\\colhead{ } & \\multicolumn{2}{c}{Cool stars} & \\multicolumn{2}{c}{All stars}   & \\multicolumn{2}{c}{Cool stars} & \\multicolumn{2}{c}{All stars} &  \\multicolumn{2}{c}{Cool stars} & \\multicolumn{2}{c}{Cool stars}\\\\\n\\colhead{Region} & \\colhead{age} & \\colhead{error} & \\colhead{age} & \\colhead{error} & \\colhead{age} & \\colhead{error} & \\colhead{age} & \\colhead{error} & \\colhead{age} & \\colhead{error} & \\colhead{age} & \\colhead{error}}\n\\startdata\nMain      & 4.0 & 0.3 & 7.9 & 0.2 & 9.1 & 1.0 & 4.0 & 0.2 & 4.8 & 0.5 & 7.2 & 0.3 \\\\\nNE        & 6.9 & 0.5 & 17.4 & 0.5 & 18.2 & 1.6 & 7.9 & 0.3 & 7.6 & 0.5 & 9.1 & 0.3  \\\\\nfarSouth  & 13.2 & 1.5 & 31.6 & 0.9 & 26.3 & 3.0 & 13.8 & 0.8 & 20.0 & 2.0 & 19.0 & 0.6 \\\\\nfarS\\_ext & 9.5 & 3.3 & 34.7 & 2.0 & 20.0 & 6.2 & 22.9 & 2.3 & 15.1 & 4.5 & 25.1 & 1.9 \\\\\nFGAql     & - & - & - & - & - & - & - & - & - & - & - & -   \\\\\nLDN673    & 3.8 & 2.1 & 15.1 & 0.7 & 11.5 & 5.3 & 6.3 & 1.3 & 6.6 & 2.0 & 4.8 & 0.3 \\\\\nDistrib   & 6.6 & 0.5 & 22.9 & 0.5 & 15.8 & 1.0 & 10.0 & 0.3 & 10.0 & 0.6 & 10.5 & 0.3  \\\\ \n\\enddata\n\\tablecomments{Estimated with isochrone fitting.}\n\\tablecomments{$T_{\\rm eff}-L$: Ages are estimated in temperature-luminosity space. CMD: Ages are estimated in color-magnitude diagram (Pan-STARRS $r-i$ versus $r$).}\n\\tablecomments{Cool stars: Stars with $r-i>1.2$ ($\\sim3500< T_{\\rm eff} <\\sim4500K$).}\n\\end{deluxetable*}\n\nThe ages adopted in this paper are determined by comparing \\citet{Feiden2016} non-magnetic isochrones to the temperature and luminosity of each source, adopting individual stellar parallax distances for each star (see Figure~\\ref{fig:HRD_Serpens}.\nTable~\\ref{table:Serpens_age_iso_fit_std} lists the ages and errors of different groups within the cloud, as calculated from isochrone fitting. For the major groups in the cloud, Serpens Main is assessed an age of {4 Myr}, Serpens NE is {8.3 Myr}, Serpens far South at {13.8 Myr}, and the extended group  of far-South (labeled as farS\\_ext) is 21.9 Myr. The small LDN 673 Group age of 6.3 Myr is calculated from 5 stars and is therefore not well constrained. \nSerpens South and W40 are deeply embedded and lack sufficient optical members for age estimates, but they are presumably younger than Serpens Main.\n\nFigure \\ref{fig:age_num_dist} shows the age distribution of Serpens Main, Northeast and far-South. To check if age distributions within a single subcluster are significant (see Figure~\\ref{fig:age_num_dist}), age errors of individual stars are calculated via the method in \\citet{Zari2019} {by adopting the 16th and the 84th percentiles of the likelihood (Equation~\\ref{eq:Likelihood_single_star}) distribution. The average age error is $\\sim 0.2$ dex. Since the error relies on temperature and luminosity errors from SED fitting, which are limited by fitting ranges (Section \\ref{subsec:method_test}), the $\\sim 0.2$ dex age error does not reflect the real age probability distribution}. Although some age spread within clusters is likely, it is unclear from our current methods and analysis whether we could subdivide any group according to age.\nAdopting single distances for each cluster does not significantly change the age but would change individual stellar ages.\n\nFigure \\ref{fig:age_map} shows a map of members of the Serpens star-forming region, with stars colored by age.  Age difference among subgroups (Figure \\ref{fig:age_num_dist} and Table \\ref{table:Serpens_age_iso_fit_std}) is also indicated by Figure \\ref{fig:age_map}. The youngest subgroup, Serpens Main, is the smallest and densest group, as shown in Figure \\ref{fig:age_map}, while the oldest stars are scattered in the distributed region.\n\n\\subsubsection{Ages from other estimation methods}\n\\label{subsubsec:age_other_iso}\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\vspace{5mm}\n    \\includegraphics[width=0.95\\columnwidth]{figures_revised_2nd\/CMD_PS1_Main.png}\n    \\caption{Color-magnitude diagram for Pan-STARRS $r$ versus $r-i$, with PARSEC isochrones (gray curves), with stars in Serpens Main (blue and orange dots represent the same set of stars with observed and extinction corrected result separately), and the extinction vector (light blue straight line) that runs parallel to the isochrones for cool stars.}\n    \\label{fig:CMD_PS1_Main}\n\\end{figure}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures_revised_2nd\/E15_and_O13_compare.png}\n    \\caption{Comparison between temperatures derived in this paper and in \\citet{Erickson2015} (left) and \\citet{Oliveira2013} (right). Orange symbols are diskless stars, and blue symbols are disk stars.}\n    \\label{fig:E15_T_compare}\n\\end{figure*}\n\nThe age estimates are sensitive to evolutionary tracks and to the method for isochronal fitting.  In this section, we explore systematic uncertainties in the ages due to the methodological choices. \n\nTable \\ref{table:Serpens_age_compare_iso} compares \n ages from the non-magnetic and magnetic \\citet{Feiden2016} and from PARSEC tracks \\citep{Bressan2012, ChenYang2014, ChenYang2015, TangJing2014}.   In general, the assessed ages {of the magnetic \\citet{Feiden2016} evolutionary models} are older {than ages from non-magnetic models}, due in part to the larger initial stellar radius in the magnetic models. The PARSEC ages are mostly consistent with the \\citet{Feiden2016} non-magnetic ages.\n\nStudies are often limited to age estimates from color-magnitude diagrams and may lack extinction values that are needed for the placement of stars on HR diagrams.  Some color-magnitude diagrams (including $r-i$ versus $r$ (Figure \\ref{fig:CMD_PS1_Main}) have $r$ versus $r-i$ isochrones that are parallel to the extinction vector over some color (temperature) range, allowing for age estimates with limited information \\citep[e.g.][]{sicilia04}. The isochronal fits to these color-magnitude diagrams lead to ages that are older than HR diagram fits (Table \\ref{table:Serpens_age_compare_iso}, the last four columns), likely because the $r$ versus $r-i$ isochrones are not exactly parallel to the extinction vector. For example, a star with $r-i=1.3$ and $M_r=8.8$ is located along the 10 Myr isochrone, but if shifted for $A_V=2$ mag would then have $r-i=0.86$ and $M_r=7.1$, which corresponds to $\\sim7$ Myr. Thus, ages from observed $r-i$ versus $r$ are slightly older than extinction corrected ages.\n\n\n\n\n\\section{A Revised Star Formation History of the Serpens Molecular Cloud}\n\\label{sec:discussion}\n\n\n\nIn Section \\ref{subsec:Serpens_ages}, we estimate the following ages from \\citet{Feiden2016} non-magnetic isochrones for the following sub-clusters: Serpens Main at {4 Myr}, Serpens NE at {8.3 Myr}, Serpens far-South at {13.8 Myr}, Serpens farS\\_ext at {21.9 Myr}, the distributed population at {10 Myr}, and LDN 673 Group at {6.3 Myr}.  The youngest regions, W40 and Serpens South, have few optical members identified for this study, so their ages are not assessed in this work.  Serpens Main also has a large protostar population that is younger than the optical population evaluated here.  In Appendix~\\ref{appendix:coll359}, we apply similar techniques to the nearby open cluster Collinder 359 and assess an age of 21 Myr.  \n\nIn this discussion, we compare our ages with literature ages, place the optical sources in the context of more complete catalogs, and discuss these results in the context of star formation within the Serpens cloud complex.\n\n\\begin{deluxetable*}{ccccccccc}[t]\n\\label{table:Serpens_space_motion}\n\\tablecaption{Proper motion and radial velocity of Serpens groups and Collinder 359.}\n\\tablehead{\\nocolhead{region} & \\colhead{PM(R.A.)} & \\colhead{$\\sigma$ (PM R.A.)} & \\colhead{PM(Dec.)} & \\colhead{$\\sigma$ (PM Dec.)} &  \\colhead{RV} & \\colhead{$\\sigma$ (RV)} & \\colhead{$N_{RV}$\\tablenotemark{$1$} }\\\\\n\\nocolhead{region} & \\colhead{mas\/yr} & \\colhead{mas\/yr} & \\colhead{mas\/yr} & \\colhead{mas\/yr} &  \\colhead{km\/s} & \\colhead{km\/s} & \\colhead{}}\n\\startdata\nMain      & 3.09 & 0.40 & -8.48 & 0.46 & -9.6 & 2.4 & 2  \\\\\nNE        & 2.58 & 0.33 & -8.13 & 0.36 & -12.1 & 5.5 & 4  \\\\\nfarSouth  & 1.94 & 0.36 & -8.90 & 0.31 & -2.4 & 2.1 & 5  \\\\\nfarS\\_ext & 1.76 & 0.50 & -8.66 & 0.70 & - & - & -  \\\\\nFGAql     & 1.67 & 0.19 & -9.74 & 0.28 & - & -  & - \\\\\nLDN673    & 2.56 & 0.55 & -10.23 & 0.23 & - & - & - \\\\\nCollinder 359   & 0.64 & 0.16 & -8.72 & 0.06 & 10.8 & 4.9 & 11 \\\\\n\\enddata\n\\tablenotetext{1}{Number of stars which are used to calculate the RV of a group.}\n\\end{deluxetable*}\n\n\\subsection{Comparisons of stellar parameters to results from other surveys}\n\\label{subsec:compare_other_Serpens}\n\nSeveral Serpens stars in our study have photospheric properties assessed previously with spectroscopic data. This section briefly compares our results to those literature results.\n\n\\citet{Erickson2015} used optical imaging of Serpens Main to select candidate YSOs from locations in the color-magnitude diagram and then evaluated membership of 700 candidates with moderate resolution optical spectra. Of their {63} optical members, {26} have {\\it Gaia} DR2 proper motions or parallaxes that are inconsistent with Serpens, nine were identified as non-members from {\\it Gaia} DR2 astrometry by \\citet{Herczeg2019}, two lack sufficient photometry for our fits (see Table~\\ref{table:photometry_criteria}), {and one has no spectral type in \\citet{Erickson2015}.}  Figure~\\ref{fig:E15_T_compare} compares the temperatures from here and \\citet{Erickson2015} for the { remaining} {25} stars in both studies.  {Three} stars are at the upper limit of $A_V$ ($\\sim6$ mag) of the fitting, which is consistent with their literature $A_V$ ({5.5-6.5} mag in \\citealt{Erickson2015}, 5-9 mag in \\citealt{Getman2017}). For cool stars ($<5000$ K), {the fitted $T_{\\rm eff}$ here generally agrees with \\citet{Erickson2015} measurements, with an average difference of $\\sim$250 K.} {Three} hot stars have temperatures that are {$>$2000 K} cooler here than in \\citet{Erickson2015}.\n With \\citet{Erickson2015} $T_{\\rm eff}$ and luminosity,and \\citet{Feiden2016} non-magnetic isochrones, the members in \\citet{Erickson2015} have an age of 2.8 Myr. \nFor these overlapping stars, the derived ages are 3.7 Myr (\\citet{Erickson2015} $T_{\\rm eff}$ and luminosity) and 3.9 Myr (fitted $T_{\\rm eff}$ and luminosity), indicating consistency in results. The younger overall cluster age obtained by \\citet{Erickson2015} is likely attributed to differences in selection.\n\n\n\n \n\\citet{Oliveira2013} measured spectral types and temperature of candidate Serpens YSOs with optical spectroscopy and used \\textit{Spitzer} infrared photometry to study disk properties. Figure \\ref{fig:E15_T_compare} compares the temperatures of the stars that overlap between our sample and \\citet{Oliveira2013}. Cool stars ($<\\sim5000$ K) have fitted $T_{\\rm eff}$ {$\\sim$500 K} hotter than in \\citet{Oliveira2013} while hot stars have similar $T_{\\rm eff}$. Using the \\citet{Feiden2016} isochrones, the ages of stars in \\citet{Oliveira2013} are 1.8 Myr for all members. For stars that overlap between our samples, the average age is 1.7 Myr with $T_{\\rm eff}$ and $L$ from \\citet{Oliveira2013} and 3.8 Myr for properties measured here, a significant discrepancy { that indicates methodological differences.}\n\n\n\\subsection{Comparison to the distribution of CO gas}\n\\citet{SuYang2020} mapped CO towards the Aquila Rift region, which { covers} the Serpens star-forming region in our study (Serpens NE, farSouth and LDN 673). \nStrong CO emission found by \\citet{SuYang2020} is consistent with the relatively young ages of optical members in Serpens NE and LDN 673. In Serpens far-South (also labeled as Sh 2-62 in \\citealt{SuYang2020}, and also known as MWC 297 region, \\citealt{rumble15}), the strong CO emission region is coincident with the members that are bright in the infrared and faint or undetected at optical wavelengths (see also Figure 4 of \\citealt{Herczeg2019}).   Although these regions are adjacent to each other, they are not necessarily related; the optical members that we associate with Serpens far-South are less extincted and spatially offset from the young stellar objects identified by \\citet{Dunham2015}.\nThe older ({$\\sim13.8$} Myr) optical members are not in the ongoing star-forming region of Serpens far-South and are not associated with molecular gas. The LSR velocities of $^{12}$CO in Serpens NE and far-South is between 8 km\/s and 12 km\/s (Figure 5 of \\citealp{SuYang2020}), corresponding to $\\sim-8$ km\/s to $\\sim-4$ km\/s in heliocentric coordinate. {\\textit{Gaia} radial velocities of the two groups (Table \\ref{table:Serpens_space_motion}) are $<1\\sigma$ away from the boundary of $^{12}$CO velocities.}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures_revised_2nd\/3D_plot_together.png}\n    \\caption{Current position (squares) and inferred position 15 Myr ago (star symbols) (Galactic coordinates, the Sun is at (X,Y,Z) = 0,0,0) of three groups in Serpens and nearby open cluster Collinder 359. Serpens Main is younger than 15 Myr but is still presented. Black arrows represent the space motion of the groups. Colored dashed lines are the projection of space motions into (X,Y), (X,Z) and (Y,Z) planes.}\n    \\label{fig:3D_plot}\n\\end{figure*}\n\n\\subsection{Co-location of disks and protostars}\n\\label{subsec:disk_fraction}\n\nThe disk fraction of a cluster provides an indication of cluster age, relative to the disk fraction in other clusters \\citep[e.g.][]{Haisch2001,hernandez08,Bell2013}.  In Serpens, some optically bright members are in clusters that have protostars, especially in the active star-forming regions Serpens Main and Serpens NE.  Other Serpens members are far from any known protostars but are in groups that still have disks (see Figure 4 of \\citealt{Herczeg2019}).\n\nThe overall disk fraction for the optical members of the Serpens star-forming region is 49 of 406 stars (12\\%), from the sample of high confidence members with reliable photometry (samples without WISE photometry are excluded).  This disk fraction suggests ages of 5--10 Myr, roughly consistent with our age estimates.\n\nThis disk fraction applies only to the optical members and is not necessarily representative of the entire cluster.  An analysis based on the full membership requires significant assumptions and extrapolations.\nFor Serpens Main, the high-confidence members include 17 stars with disks and 62 diskless stars (22\\% disk fraction, as listed in Table \\ref{table:Serpens_age_iso_fit_std}).  The total statistical population consists of 265 members, as measured from the excess population associated with the cluster \\citep{Herczeg2019}.  If we assume that the populations are similar, then the optical population would include 72.7 disk stars and 192.3 diskless stars.  From infrared surveys, \\citet{Dunham2015} identified 222 likely members (excluding AGB stars).  Of the stars with {\\it Gaia} astrometry, 50\\% of the evolved sources and 86\\% of the 73 disks are consistent with Serpens membership.  In follow-up with the envelope tracer HCO$^+$, \\citet{heiderman15} confirmed that 88\\% of candidate protostars in Serpens have an envelope, confirming membership and evolutionary stage.  If we assume that all matches are in the statistical population to avoid double-counting, then we find that Serpens Main consists of 45.5 protostars, 122.6 stars with disks, and 197.8 diskless stars.  This estimate includes significant uncertainties, including complications from binarity and extrapolations from the well-defined optical sample to the infrared sample, which may be younger.  The disk fractions may be underestimated if many diskless sources have high extinctions, either because they have high extinctions or because they are located within or behind the cloud, and were not selected in  the \\citet{Dunham2015} catalog, \n\nBased on the above arguments, we adopt for Serpens Main a disk fraction of 38\\% and a disk+protostar fraction of 46\\%.  Similar estimates lead to disk+protostar fractions of 12\\% for Serpens NE and 16\\% for Serpens far-South.  These disk fractions support our relative ages: Serpens Main is younger than Serpens NE and Serpens far-South.  While these disk fractions indicate average ages for the clusters, the presence of protostars, disks, and diskless stars in the same cluster are likely the consequence of an age spread within the cluster \\citep[see, e.g.][]{kristensen18}.\n\n\\subsection{Dynamics of the subclusters in and around the Serpens Molecular Cloud}\n\\label{subsec:subgroup_dynamic}\n\n\nThe identification of optically bright members of the Serpens molecular cloud complements previous selections from the infrared. The infrared selections identified several star clusters that are deeply embedded in molecular clouds \\citep[e.g.,][]{Winston2009,Kuhn2013,Dunham2015}. The optically bright members are lightly extincted, some still in clusters and some distributed across the region. The infrared members trace the ongoing star formation, while the optical members trace star formation in the recent past. The ages of these different samples help to quantify when and where the bursts of star formation occurred. \n\nTo infer the formation history of Serpens, ages are combined with space velocities of these Serpens groups (Table \\ref{table:Serpens_space_motion}), with proper motions obtained from \\citet{Herczeg2019} and the weighted mean radial velocities and associated uncertainty \\footnote{The radial velocity uncertainty is the larger value between the weighted mean uncertainty and standard deviation.} obtained from members with velocities in \\textit{Gaia} DR2. The radial velocities of Serpens Main, NE, farSouth are $-9.6 \\pm 2.4$ km\/s, $-12.1 \\pm 5.5$ km\/s, and $-2.4 \\pm 2.1$ km\/s separately. However, only few stars have radial velocity data (listed in Table \\ref{table:Serpens_space_motion}), and the radial velocity uncertainties are significantly larger than proper motion uncertainties ($\\sim 0.2$ mas, corresponding to $0.4$ km\/s). Thus, the radial velocities are far less reliable than proper motions, and the radial velocity discrepancy between stars and gas is likely due to the uncertainties of stellar velocity.  The radial velocity of Collinder 359 is calculated from \\citet{Cantat-Gaudin2018} samples with membership probability $P_{memb}\\geq0.9$ and Gaia radial velocity, leading to $10.8 \\pm 4.9$ km\/s.\\footnote{This radial velocity is consistent within uncertainties to the radial velocity of $8.30 \\pm 1.79$ km\/s calculated from the GALAH survey \\citep{Carrera2019} and $5.28 \\pm 3.25$ km\/s from Gaia DR2 using sources in \\citet{Cantat-Gaudin2018} with $P_{memb}\\geq0.4$ \\citep{Soubiran2018}. The \\citet{Soubiran2018} radial velocity rejected samples deviating more than 10 km\/s from the mean.}\n\nWith these space motions, the relative locations of the sub-clusters can be traced back in time (Figure \\ref{fig:3D_plot}). The two youngest groups, Serpens Main and Serpens NE ({4} Myr and {8.3} Myr), have been co-moving from their birth to present, separated by a distance of {$\\sim60$} pc (4 Myr ago) to $\\sim40$ pc now, indicating that they may have similar formation history. The highly embedded regions W40 and Serpens South lack optical members in this study and are  expected to be in the same cloud complex as Serpens Main, as suggested by \\citet{Ortiz-Leon2017}. The older group, Serpens far-South (17 Myr), has larger velocity difference from Serpens Main (e.g., 4.6 km\/s in Galactic X coordinate, 3.2 km\/s in Y, and 3.7 km\/s in Z). Thus, Serpens far-South may have a different origin from Serpens Main ($\\sim 160$ pc away from Serpens Main at the birth of far-South) or had star formation triggered by some interaction that had a different velocity at that location.\nThe open cluster Collinder 359 reached a minimum distance of $\\sim60$ pc to Serpens Main at $\\sim5$ Myr ago, when Serpens Main was born; whether such condition is coincidental or it reflects possible connections between Collinder 359 and Serpens is unclear.\n\n\nThe initial analysis of Gaia DR2 astrometry of the Serpens region by \\citet{Herczeg2019} also includes a discussion of ages and kinematic properties of the sub-clusters. In Serpens NE, \\citet{Herczeg2019} find that some stars are located outside of the molecular cloud, either because stars were born in a centralized space and were flung away (cluster dynamics) or because feedback from the older stars eroded nearby cloud and changed the birth place of younger stars. If these stars were born at the same spatial location, the velocity dispersion would require that they are $\\sim 20$ Myr old.  With the age of Serpens NE of 9 Myr, that scenario is more plausible { than if the stars were much younger.}\n\nIn addition to Serpens, some other nearby young star-forming regions also contain several subgroups with different kinematic properties. For example, \\citet{Krause2018} studied stars and gas of the Scorpius\u2013Centaurus association and described the following sequential scenario to explain the formation history of the subgroups in this association: the first star-forming activity generated superbubbles, which triggered more star-forming events, with subgroups that move in different directions due to gas interaction. \\citet{Krause2018} also suggested that the scenario might apply to many young star-forming regions in the Milky Way. If Serpens has a similar formation scenario, the first star-forming event would start at $\\sim26$ Myr ago (Serpens farS\\_ext and farSouth), which triggered the second star-forming activity at $\\sim9$ Myr ago (Serpens NE and Main), and the current star-forming events are in W40 (the most embedded region), Serpens Main and NE.\n\n Multiple star-forming events in a single cluster appears to be the common mode of star formation.  In a \\textit{Gaia} DR2 study of the Orion OB association , \\citet{Zari2019} found in that multiple star-forming events occurred and leaded to the subgroups. In a search for stellar groups in the Taurus field, \\citet{Liu2021} identified 8 groups younger than $\\sim4$ Myr and 14 older groups (8-49 Myr).  In a smaller region, \\citet{esplin22} found older stars in Corona Australis and inferred a past burst of star formation, likely related to the ongoing star formation in the Coronet Cluster.  These processes seem common to nearby low-mass star formation.\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nWe develop an SED fitting method for diskless young stellar objects based on empirical color-color relationships from the Pleiades and temperatures measured from APOGEE near-IR spectra by \\citet{Cottaar2014}. \nWhen using stars in Orion to test the method, the fitted temperature is consistent with temperatures from APOGEE spectra \\citep{Kounkel2018} to within {$\\sim$240 K} for diskless stars and {$\\sim$630 K ($\\sim$170 K) for disk stars (with $T_{fit}<$5000 K), all substantial improvements on SED fitting from other commonly used methods.}  \nShortcomings in the method include extinction coefficient variation, dust emission from stellar disk, different surface gravity, and different color-color relations among YSOs.\n\n\nWe collect photometry for high-confidence optical members of the Serpens Molecular Cloud and then fit SEDs with our empirical color-color relationships to estimate temperature and luminosity for each star.  The stars belong to several distinct groups in and around the cloud.  Subcluster ages range from {$\\sim 4$ Myr}  for Serpens Main to {$\\sim 22$ Myr} for dispersed stars located to the south of the cloud for standard non-magnetic evolutionary tracks. The optically bright population is much older than the populations that are revealed from infrared selection criteria, such as the Serpens South region.  We then use the ages to discuss possible connections between the distinct subclusters, including a possible explanation similar to the sequential star formation in Sco-Cen OB Association, as inferred by \\citet{Krause2018}.  A deeper understanding of the star formation history of the Serpens Molecular Cloud and any causal connections between different regions will require spectroscopy, especially of infrared-selected members that are not optically visible, and analysis of star and gas motions.\n\n\n\\acknowledgements\n\nWe thank the anonymous referee for the helpful comments and\nsuggestions that helped to significantly improve the analysis and the clarity of the paper. We thank Marina Kounkel for discussion of APOGEE temperature and ages of Serpens and Kevin Covey for discussion of temperature scales.\n\nWe thank Jennifer Hatchell, Carlo Manara, Doug Johnstone, Michael Dunham, Anupam Bhardwaj, Jessy Jose, and Zhen Yuan for their effort in our initial paper that developed the sample used here. \n\nXYZ and GJH are supported by general grants 12173003 and 11773002 awarded by the Natural Science Foundation of China. YL acknowledges financial supports by the Natural Science Foundation of China (Grant No. 11973090) and by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20181513).\n\nThis work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia}\nData Processing and Analysis Consortium (DPAC, \\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC\nhas been provided by national institutions, in particular the institutions\nparticipating in the {\\it Gaia} Multilateral Agreement.\nThis publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center\/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.\nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, funded by the National Aeronautics and Space Administration.\nWe acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research.  This research has made use of NASA's Astrophysics Data System.\nThis research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.\nThis research has made use of the Spanish Virtual Observatory (\\url{http:\/\/svo.cab.inta-csic.es}) supported from the Spanish MICINN\/FEDER through grant AyA2017-84089.  This research made use of Astropy, a community-developed core Python package for Astronomy \\citep{Astropy2013}. \n\n\n\\pagebreak\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:sect1}\nThe discovery of a new type of compact stellar system in the Fornax cluster with luminosities comparable to dwarf galaxies, but of much smaller size, was reported by \\citet{hilker:1999} and \\citet{drinkwater:2000}. The detected objects, named `ultracompact dwarf galaxies' (UCDs) by \\citet{phillipps:2001}, started to fill the mass--size parameter space in between globular clusters (GCs) and compact elliptical galaxies (cEs) in the stellar mass range $M_*$~$=$~$10^6-10^8$~M$_{\\sun}$ and with half-light radii between $r_\\mathrm{h}$~$=$~$3$ and $100$~pc. With the growing number of UCDs, compact galaxies and star clusters are no longer separated, but form instead one sequence of compact stellar systems. One of the main questions driving the investigation of UCDs is thus whether they rather constitute the high-mass, large-size end of the star cluster distribution, or the low-mass and small-size end of the galaxy population.\n\nPrevious studies showed that most UCDs have properties similar to GCs with regard to their ages, metallicities and $\\alpha$-element abundances \\citep[e.g.][]{frank:2011, francis:2012}. Furthermore, on the basis of number counts, \\citet{mieske:2012} concluded that UCDs in the Fornax cluster would statistically be fully consistent with being the brightest members of the central cluster galaxy's GC population. There is an ongoing discussion on the formation scenarios of GCs themselves \\citep[e.g.][]{kruijssen:2014, kruijssen:2015}. One model suggests that the formation of GCs may be triggered during major mergers of gas-rich galaxies \\citep{ashman:1992}. Support for this scenario can be found in observations of the interacting Antennae galaxies \\citep{whitmore:1995}, which show large star cluster complexes residing in giant \\ion{H}{ii} regions, consisting of up to hundreds of individual young massive star clusters. Simulations of the evolution of such young massive star clusters in star cluster complexes showed that the merging of clusters would form a compact object with parameters in the range of typical UCDs \\citep{fellhauer:2002, bruens:2011}. However, hierarchical merging in star cluster complexes is not the only possible formation scenario for UCD-like objects during a galaxy merger. \\citet{renaud:2015} concluded from a hydrodynamical simulation of an Antennae-like galaxy merger that UCD-like objects can also originate from merging of gas clumps during their formation. Based on observations of UCDs in the Perseus cluster, \\citet{penny:2012} suggested a star cluster origin for two UCDs with very blue colours, residing in star-forming filaments of the central cluster galaxy.\n\nSince their discovery, it has also been discussed that UCDs could be related to the population of galaxies. For cEs, which are adjacent to UCDs in magnitude and size, it was already suggested by \\citet{faber:1973} that these could be the remnant bulges of stripped spiral or early-type galaxies. Observational support for this scenario was found for example by \\citet{huxor:2011}. For the lower-mass UCDs it was proposed that they could be the remnant nuclei of nucleated dwarf ellipticals that were tidally stripped while orbiting in the gravitational field of a galaxy cluster \\citep{bekki:2001, drinkwater:2003}. Various simulations of this tidal stripping scenario demonstrated that the stripped galaxy remnants would closely resemble observed UCDs from their structural parameters \\citep{bekki:2003, goerdt:2008, pfeffer:2013}. Observational signatures for a stripped galaxy origin were found by several authors. For example, \\citet{norris:2015} detected an extended star formation history for UCD NGC~4546-UCD1 ($M_*$~$=$~$3.3$~$\\times$~$10^7$~M$_{\\sun}$, $r_\\mathrm{h}$~$=$~$25.5$~pc). Also, \\citet{strader:2013} found strong indications for a galaxy origin of UCD M60-UCD1 ($M_{\\mathrm{dyn}}$~$=$~$2.0$~$\\times$~$10^8$~M$_{\\sun}$, $r_\\mathrm{h}$~$=$~$24.2$~pc), which harbours a central X-ray source and shows strong indications for the presence of a massive black hole that makes up 15~per~cent of the UCD's mass \\citep{seth:2014}. The presence of a massive black hole could also be the cause of the elevated dynamical-to-stellar mass ratio that was inferred for UCD M87-S999 ($M_{\\mathrm{dyn}}\/M_*$~$=$~$8.2$, $M_*$~$=$~$3.9$~$\\times$~$10^6$~M$_{\\sun}$, $r_\\mathrm{h}$~$=$~$20.9$~pc) \\citep{janz:2015}. Another example is UCD NGC~1275-UCD13 ($M_*$~$=$~$4.4$~$\\times$~$10^7$~M$_{\\sun}$, $r_\\mathrm{h}$~$=$~$85$~pc), for which \\citet{penny:2014} concluded that the UCD's colour, size, metallicity, internal velocity dispersion, dynamical mass, and close proximity to the central cluster galaxy likely point to a stripped galaxy origin. \\citet{janz:2016} observed that high-mass UCDs with $M_*$~$\\gtrsim$~$10^7$~M$_{\\sun}$ seem to be generally metal-rich in comparison to early-type galaxies in the same stellar mass range. This may indicate that the objects once were more massive galaxies that were stripped off their stellar material while retaining their central metallicity.\n\nIn general it is believed that both formation channels coexist for the population of UCDs \\citep[e.g.][]{hilker:2009, norris:2014}. For a few UCDs, some of which are mentioned above, either a star cluster or a galaxy origin was found to be more likely. In most cases, however, it was not possible so far to robustly distinguish between the proposed formation scenarios from the observed properties. This is also due to limited resolution and image depth, restricting a detailed analysis of the internal properties mainly to the brighter objects. Furthermore, due to the existence of a luminosity-size relation for the brighter UCDs \\citep[e.g.][]{evstigneeva:2008, caso:2013}, with the fainter objects having smaller half-light radii, most of them appear unresolved or only partly resolved in seeing-limited observations.\n\nAlready in the early phase of UCD research, \\citet{drinkwater:2003} found that the brightest UCD in Fornax (UCD~3) is significantly extended compared to the few other UCDs known at that time. Structural analysis in high resolution {\\it HST} imaging of Virgo and Fornax cluster UCDs showed that some of the brightest UCDs are characterized by a two-component surface-brightness profile with a compact core and an extended low-surface-brightness envelope \\citep{depropris:2005, evstigneeva:2007, evstigneeva:2008}. In ground-based imaging, a large number of UCDs with faint envelopes were detected in the Virgo cluster \\citep{liu:2015}. In the Fornax cluster, \\citet{richtler:2005} and \\citet{voggel:2016} reported that a few extended objects appear peculiar since they show asymmetric structures.\n\nIn deep imaging data of the Fornax cluster core, we search the known population of compact stellar systems for objects that appear significantly extended and\/or exhibit peculiar structures. We visually investigate a large sample of 355 spectroscopically confirmed compact systems with $V-$band equivalent magnitudes between $-14$ and $-10$~mag for the presence of faint structures extending significantly beyond the point spread function (PSF). We quantify structure by introducing parameters that measure the core concentration, as well as the outer asymmetry and ellipticity. Then we relate these parameters to the spatial and phase-space distribution within the Fornax cluster. We limit our analysis to objects brighter than $-10$~mag, which corresponds to the magnitude range of UCDs or bright GCs \\citep{misgeld:2011, norris:2014}. Since there is no unambiguous distinction of objects that have previously been classified as bright GCs and objects that were named UCDs, we will speak of `compact stellar systems' throughout this paper.\n\nWe describe the observations and the sample definition in Section~\\ref{sect:sect2}. Details of the analysis are given in Section~\\ref{sect:sect3}. We present our results in Section~\\ref{sect:sect4} and discuss them in Section~\\ref{sect:sect5}. The conclusions follow in Section~\\ref{sect:sect6}.\n\n\\section{Data}\n\\label{sect:sect2}\n\n\\subsection{Observations}\n\\label{sect:sect2.1}\nThe data are fully characterized by \\citet{lisker:2016}. They were acquired in 2008 and 2010 with the WFI at the ESO\/MPG 2.2m telescope (programmes 082.A-9016 and 084.A-9014, PI A.\\ Pasquali, Guaranteed Time of the Max Planck Institute for Astronomy). We used a transparent filter that nearly equals the no-filter throughput ($>10$~per~cent in the range 350--900~nm) and thus provides a high signal-to-noise ratio. Fig.~\\ref{fig:fig1} shows our deep image of the Fornax cluster core, covering a region of 100~arcmin in east--west and 76~arcmin in north--south direction. This corresponds to 576 and 438~kpc, respectively, at a distance of 20.0 Mpc for Fornax (using 5.76~kpc arcmin$^{-1}$; \\citealt{blakeslee:2009}), which we adopt throughout this work.\n\n\\citet{lisker:2016} determined an approximate `$V-$equivalent' magnitude calibration, based on spectroscopically confirmed foreground stars from \\citet{FCOS, FCOS2}. In the following we denote $V-$equivalent magnitudes and surface brightnesses with the subscript `${Ve}$'. Our data reach a $V-$equivalent depth of 26.58 and 26.76~mag~arcsec$^{-2}$ for the median and 75th percentile, respectively, at S\/N $= 1$ per pixel (1 pixel $\\cor 0.238$~arcsec, or $22.848$~pc). The seeing PSF full width at half-maximum (FWHM) varies over the image (see also Fig.~\\ref{fig:fig2}), but is typically about 1~arcsec. Details of the observations, data reduction, and data characterization are provided in \\citet{lisker:2016}.\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{fig1.eps}\n    \\caption{Deep wide-field imaging of the Fornax cluster core. Image dimensions: $100$~$\\times$~$76$~arcmin$^2$, corresponding to $576$~$\\times$~$438$~kpc$^2$ when assuming a distance of 20.0~Mpc for Fornax. North is up and east is to the left. Red dots indicate our sample of spectroscopically confirmed compact stellar systems with $m_{\\mathrm{Ve}}$~$<$~$21.5$~mag. The central galaxy is NGC~1399.}\n    \\label{fig:fig1}\n\\end{figure}\n\n\\subsection{Catalogue compilation}\n\\label{sect:sect2.2}\n\nBefore defining our working sample, we first compiled a list of published compact stellar systems with spectroscopically confirmed Fornax cluster membership. We used the SIMBAD data base \\citep{SIMBAD} to identify relevant source catalogues, which yielded the following references:\n\\citet{SRH2010}, \\citet{DRG2004}, \\citet{BAL2007}, \\citet{FDE2007}, \\citet{GDE2009}, \\citet[][part of the Fornax Compact Object Survey]{FCOS2}, \\citet[][part of the Fornax Compact Object Survey]{FCOS}, \\citet{FDK2008}, \\citet[][part of the Fornax Cluster Spectroscopic Survey]{DJG2000}, \\citet{KGM1999}, and \\citet{KBS1998}. \nTabulated cluster members from these references, in the given order, were appended to our object list if they were not within 1 arcsec of objects already in the list. Duplications within the same catalogue, which are present in \\citet{SRH2010} and \\citet{DRG2004}, were excluded. Based on this sequence, 695 objects were taken from \\citet{SRH2010}, 113 from \\citet{DRG2004}, 151 from \\citet{BAL2007}, and 123 from the remaining references (with no contributed objects from \\citealt{DJG2000} and \\citealt{KBS1998}), resulting in 1082 confirmed cluster members, of which 1058 were listed as compact systems and 24 as dwarf galaxies.\n\nAfter excluding eight objects that lie outside of our mosaic, we visually inspected the remaining objects on the mosaic at their literature positions, using {\\it SAOImage DS9}. We identified another 75 unambiguous duplications, as well as 20 published object coordinates for which we do not see an optical counterpart in our image within 3 arcsec\\footnote{These are the objects 78:103, GS04-M03:9, GS04-M03:30, and GS04-M03:127 from \\citet{SRH2010}, objects 75:56, 76:112, 78:13, and 78:110 from \\citet{DRG2004}, and the following objects from \\citet{KGM1999}: ntt 201, ntt 203, ntt 407, ntt 410, ntt 414,\nntt 109, ntt 119, ntt 122, ntt 123, ntt 124, ntt 126, and ntt 127. We used the NED Coordinate and Extinction Calculator (http:\/\/ned.ipac.caltech.edu\/forms\/calculator.html) as a check to make sure that our coordinate conversion from the B1950.0 values of\n\\citeauthor{KGM1999} to J2000.0 was done correctly.}. For 50 further published object coordinates, the match to the visible sources is ambiguous --- mostly due to the presence of multiple sources, but in a few cases also due to the fact that the only nearby visible source lies almost 3 arcsec away from the literature position. All of these published objects were excluded from our list, as well as one further object that potentially has wrong published coordinates\\footnote{Object 37 in table 2 of \\citet{GDE2009} is marked there as being the same source as object gc212.2 from \\citet{BAL2007}, but that object actually has different coordinates, offset by about 0.5 arcmin.\nWe therefore excluded object 37 from our list.}. This results in a final catalogue of 904 compact stellar systems that are spectroscopically confirmed cluster members, are covered by our mosaic, and are visually unambiguously identified with an optical source. The catalogue is available online. An excerpt is shown in Table~\\ref{tab:tab1}.\n\nDespite the visually unique match, the literature position may of course still be slightly offset from the image position\\footnote{The image position is based on the astrometric calibration of our mosaic (using the 2MASS catalogue), and obtained by fitting the PSF of the objects using the task {\\it allstar} of the \\texttt{IRAF} package \\texttt{DAOPHOT}.}, and other neighbouring sources may be present. In order to characterize this situation, we determined for each catalogue position whether a single source or multiple image sources are present within a 1~arcsec and 3~arcsec radius. For 792 catalogue positions (88\\%) there is only a single image source within 1~arcsec and no further image source within 3~arcsec. For 95 catalogue positions (11\\%) there is a single image source within 1~arcsec, and at least one further source within 3~arcsec. For 17 catalogue positions (2\\%) there is no image source within 1~arcsec, but either a single image source within 3~arcsec (15 objects) or multiple sources (2 objects\\footnote{The two objects are not part of our working sample, since they have $m_{\\mathrm{Ve}}$~$\\geq$~$21.5$~mag.}). We remind that a \\emph{visually unambiguous} match is present for \\emph{all} 904 catalogue objects, including the 17 just mentioned.\n\n\\begin{table*}\n \\caption{Catalogue of compact stellar systems compiled from the literature. The first five objects are given to illustrate the format of the table. The complete catalogue is provided in the electronic version of the paper. For each object we list our ID, the position based on our astrometry, as well as the SIMBAD identifier. The object IDs are sorted by increasing right ascension. The given velocity corresponds to the velocity with the smallest error from all compiled velocities. The respective literature source is listed in the last column, where 1 = \\citet{SRH2010}, 2 = \\citet{DRG2004}, 3 = \\citet{BAL2007}, 4 = \\citet{FDE2007}, 5 = \\citet{GDE2009}, 6 = \\citet{FCOS2}, 7 = \\citet{FCOS}, 8 = \\citet{FDK2008}, 9 = \\citet{KGM1999}, 10 = \\citet{DJG2000}, 11 = \\citet{KBS1998}.}\n  \\label{tab:tab1}\n \\begin{tabular}{ccccccc}\n  \\hline\n  ID & R.A. (J2000) & Dec. (J2000) & SIMBAD ID & v (km s$^{-1}$) & Lit.\\\\\n   \\hline\n  1 & 03 35 38.87 & -35 21 53.3 & [BAL2007] gc144.6 & 1388 $\\pm$ 32 & 3\\\\\n  2 & 03 35 42.52 & -35 13 51.8 & [BAL2007] gc290.6 & 1901 $\\pm$ 16 & 3\\\\\n  3 & 03 35 50.49 & -35 15 24.2 & [BAL2007] gc302.6 & 1166 $\\pm$ \\phantom{0}6 & 3\\\\\n  4 & 03 35 59.56 & -35 26 56.7 & [BAL2007] gc21.70 & 1272 $\\pm$ 12 & 3\\\\\n  5 & 03 36 01.09 & -35 25 43.0 & [BAL2007] gc69.70 & 1389 $\\pm$ \\phantom{0}8 & 3\\\\\n  \\hline\n \\end{tabular}\n\\end{table*}\n\n\\subsection{Velocity compilation}\n\\label{sect:sect2.3}\nIn the literature compilation described in Section~\\ref{sect:sect2.2} that serves as the basis for our sample, each object only appears once, even if it is listed in several of the references. In order to compile \\emph{all} published heliocentric velocity measurements of a given object, we employed the following steps. First, we went through the list of compiled \\emph{literature positions} of our sample and extracted the nearest matching object within 1 arcsec from each of the literature references given in Section~\\ref{sect:sect2.2}. Secondly, we repeated this for the list of \\emph{image positions} of our sample. Thirdly, we visually inspected the location of all compiled positions, and only kept the unambiguous ones.\n\nThis resulted in 440 objects of our sample having at least two velocity measurements, 80 having at least three, and 37 with four or more. \nFor each object we adopted the velocity with the smallest error from all compiled velocities, which we include in our catalogue (see Table~\\ref{tab:tab1}).\n\n\\subsection{Working sample}\n\\label{sect:sect2.4}\nThe compiled catalogue contains objects in both the UCD and GC magnitude range. In the study presented here we define our working sample to include all objects brighter than $m_{\\mathrm{Ve}} = 21.5$~mag, corresponding to $M_{\\mathrm{Ve}} < -10.0$~mag at the distance of Fornax, i.e. the range of UCDs and bright GCs in the literature \\citep{misgeld:2011, norris:2014}. Due to the depth and seeing-limited resolution of our data, fainter objects (with their smaller intrinsic sizes, see e.g. \\citealt{caso:2013}) could not be analysed robustly. In total our working sample contains 355 compact systems, which are indicated in Fig.~\\ref{fig:fig1}. Their $V$-equivalent magnitudes (`mag best') and the corresponding uncertainties (based on the weight image) were obtained with \\texttt{SExtractor}\\footnote{For five objects from our working sample we adopted the magnitude and corresponding uncertainty obtained from PSF fitting, due to imperfect deblending in \\texttt{SExtractor}.} \\citep{SExtractor} and are provided in Table~\\ref{tab:tab3}.\n\nA rough estimate of the completeness of known compact stellar systems with $M_{V} < -10.3$~mag in Fornax is given by \\citet{mieske:2012}. Between distances of 50--100~kpc from the cluster centre (corresponding to $0.15\\degr$ and $0.29\\degr$, respectively) the authors considered their sample to be complete to 60--70 per cent. At larger cluster-centric distances the completeness is expected to drop below 50 per cent. As discussed in Section~\\ref{sect:sect5.3} this does not affect the conclusions of our study.\n\n\\section{Analysis}\n\\label{sect:sect3}\nThe analysis of our working sample is based on both a visual investigation as well as on a parametrization of selected sample properties. We introduce parameters as a measure for the central core concentration as well as for the shape of the outer light distribution. The analysis is carried out relative to the PSF, which is determined from point sources in our data. Prior to the analysis we have removed large galaxies and stellar haloes by fitting and subtracting their light profile with the \\texttt{IRAF} task \\textit{ellipse} \\citep[see][]{lisker:2016}.\n\n\\subsection{PSF analysis and background correction}\n\\label{sect:sect3.1}\n\nDue to changing seeing conditions over the course of our observing runs, aside from possible intrinsic variations due to the instrument, the PSF varies across the image. We therefore divided the mosaic into regions of similar FWHM, as indicated in Fig.~\\ref{fig:fig2}, and according to the distribution of our sample, i.e., only if an area contains published compact stellar systems, we define it as a PSF region. In each region we determined the PSF from suitable point sources (hereafter `PSF stars') in the magnitude range $17.6$~$<$~$m_{\\mathrm{Ve}}$~$<$~$22.3$~mag, using routines from the \\texttt{IRAF} package \\texttt{DAOPHOT}. The PSF was generated by an iterative approach. In a wide circle with a radius of 30~pixels (7.14~arcsec) around each PSF star, all neighbouring sources were subtracted using a first-estimate PSF. A new PSF model was then calculated based on the neighbour-subtracted PSF stars. We performed three iterations, which significantly reduced the contamination of flux from close neighbours to the PSF. The final PSF was subsequently fitted to our sample and subtracted, where we chose the fitting radius to be slightly larger than the typical FWHM of the respective region. Details on the PSF analysis and properties of the resulting PSFs are summarized in Table~\\ref{tab:tab2}.\n\nFor the following analysis we corrected large-scale background variations using a \\texttt{SExtractor} background map determined from the full image. Remaining local background offsets were corrected for each object individually, by subtracting the median intensity within an annulus 5~pixels wide and with an inner radius of 20~pixels (1 pixel $\\cor 0.238$~arcsec).\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{fig2.eps}\n    \\caption{Central panel: variation of the PSF FWHM across the mosaic. The positions of objects brighter than $m_{\\mathrm{Ve}} = 22$~mag and with an FWHM below 7.5 pixels are shown, which were detected with \\texttt{SExtractor}. The symbols are colour-coded according to the local minimum FWHM, determined within a circle of 500 pixels radius. We measured the FWHM of all \\texttt{SExtractor}-detected objects with the \\texttt{IRAF} task {\\it psfmeasure}, using a Gaussian profile for the PSF. The numbered boxes indicate regions for which we determined an individual PSF. North is up and east is to the left. Side panels: variation of the minimum FWHM along $x$-$\\mathrm{\/}y$-direction. The average of the 10 smallest FWHM values that occur in bins of 500 pixels is plotted.}\n    \\label{fig:fig2}\n\\end{figure}\n\n\\begin{table*}\n \\caption{PSF analysis for regions of similar FWHM on the mosaic (see Fig.~\\ref{fig:fig2}). For each region we specify the size of the region in pixels$^2$, the number of compact stellar systems contained in the region, and the number of PSF stars used to build the PSF. The regions are listed with decreasing size. Working sample: compact objects brighter than $m_{\\mathrm{Ve}} = 21.5$~mag.  Fainter CSS: compact objects fainter than $m_{\\mathrm{Ve}} = 21.5$~mag, which belong to the basis catalogue described in Section~\\ref{sect:sect2.2} and are given here for completeness only. For the resulting PSF we give the FWHM in pixels (1 pixel $\\cor 0.238$~arcsec) and the ellipticity. Both quantities were obtained with the \\texttt{IRAF} task {\\it psfmeasure}, using a Gaussian profile. The last column gives the mean core concentration ($\\overline{\\mathrm{cc}}$) and the corresponding standard deviation of all PSF stars within one region (see Section~\\ref{sect:sect3.2}).}\n  \\label{tab:tab2}\n \\begin{tabular}{lccccccc}\n  \\hline\n  Region & Size & Working sample & Fainter CSS & PSF stars & PSF FWHM & PSF ellipticity & $\\overline{\\mathrm{cc}}$\\\\\n  \\hline\n  \\phantom{0}1 & 8100 $\\times$ 6050 & \\phantom{0}16 & \\phantom{00}4 & 78 & 4.1 & 0.09 & 1.005 $\\pm$ 0.025\\\\\n  \\phantom{0}2 & 5980 $\\times$ 6950 & 132 & 223 & 55 & 4.3 & 0.05 & 1.000 $\\pm$ 0.019\\\\\n  \\phantom{0}3 & 4500 $\\times$ 6840 & \\phantom{0}13 & \\phantom{0}10 & 42 & 5.6 & 0.10 & 1.006 $\\pm$ 0.027\\\\\n  \\phantom{0}4 & 2970 $\\times$ 4980 & \\phantom{0}37 & \\phantom{0}40 & 28 & 4.6 & 0.04 & 1.012 $\\pm$ 0.031\\\\\n  \\phantom{0}5 & 2040 $\\times$ 6950 & 107 & 246 & 35 & 4.5 & 0.05 & 1.000 $\\pm$ 0.037 \\\\\n  \\phantom{0}6 & 3030 $\\times$ 4600 & \\phantom{0}21 & \\phantom{00}2 & 37 & 4.4 & 0.05 & 1.006 $\\pm$ 0.029\\\\\n  \\phantom{0}7 & 2970 $\\times$ 3020 & \\phantom{0}15 & \\phantom{0}24 & 22 & 4.4 & 0.02 & 1.013 $\\pm$ 0.042\\\\\n  \\phantom{0}8 & 2030 $\\times$ 2900 & \\phantom{00}1 & \\phantom{00}0 & 28 & 4.5 & 0.07 & 1.008 $\\pm$ 0.026\\\\\n  \\phantom{0}9 & 2200 $\\times$ 2100 & \\phantom{00}2 & \\phantom{00}0 & \\phantom{0}8 & 4.4 & 0.05 & 1.007 $\\pm$ 0.018\\\\\n  10 & 2000 $\\times$ 1800 & \\phantom{00}1 & \\phantom{00}0 & \\phantom{0}9 & 4.4 & 0.12 & 1.006 $\\pm$ 0.022\\\\\n  11 & 1500 $\\times$ 1700 & \\phantom{00}4 & \\phantom{00}0 & \\phantom{0}7 & 4.9 & 0.12 & 1.012 $\\pm$ 0.040\\\\\n  12 & 1000 $\\times$ 2500 & \\phantom{00}3 & \\phantom{00}0 & 12 & 4.3 & 0.13 & 0.999 $\\pm$ 0.020\\\\\n  13 & 1500 $\\times$ 1500 & \\phantom{00}1 & \\phantom{00}0 & \\phantom{0}8 & 4.7 & 0.10 & 1.011 $\\pm$ 0.024\\\\\n  14 & 1500 $\\times$ 1500 & \\phantom{00}1 & \\phantom{00}0 & 10 & 5.8 & 0.12 & 1.001 $\\pm$ 0.031\\\\\n  15 & 1500 $\\times$ 1500 & \\phantom{00}1 & \\phantom{00}0 & \\phantom{0}8 & 6.2 & 0.11 & 1.027 $\\pm$ 0.049\\\\\n  \\hline\n \\end{tabular}\n\\end{table*}\n\n\\subsection{Core concentration}\n\\label{sect:sect3.2}\n\nThe core concentration is defined as the mean central flux ratio of the object to the fitted PSF and can be expressed in terms of the ratio of the PSF-subtracted residual image to the object image:\n\\begin{eqnarray}\n\\label{eq:eq1}\n\\text{Core concentration} &=& \\frac{1}{n_{\\mathrm{pix}}} \\sum_{r\\leq2\\mathrm{pix}} \\frac{I_{\\mathrm{obj}}}{I_{\\mathrm{PSF}}}\\\\\n                          &=& \\frac{1}{n_{\\mathrm{pix}}} \\sum_{r\\leq2\\mathrm{pix}} \\left[ 1- \\frac{I_{\\mathrm{res}}}{I_{\\mathrm{obj}}} \\right] ^{-1}\\nonumber,\n\\end{eqnarray}\nwhere $I_{\\mathrm{obj}}$ corresponds to the object intensity, $I_{\\mathrm{PSF}}$ to the intensity of the fitted PSF, $I_{\\mathrm{res}}$ to the residual intensity and $n_{\\mathrm{pix}}$ is the number of pixels within the aperture over which is summed. The aperture width is on the order of one FWHM of the typical PSF. The ratio of the residual to the object image was obtained by dividing both images pixel by pixel. We provide the core concentration for our working sample in Table~\\ref{tab:tab3}.\n\nFig.~\\ref{fig:fig3} (upper left panel) shows the distribution of the measured core concentration for our working sample and for the PSF stars used in the analysis. According to equation~\\ref{eq:eq1}, a core concentration close to one indicates a PSF-like core, whereas lower values correspond to a less concentrated core. This is reflected in the distribution of the PSF stars, which is strongly peaked around a value of one. The vast majority of compact stellar systems has lower core concentrations than the PSF stars and follows a broader distribution that is skewed towards lower values. We find that the brighter objects from our working sample reach on average lower core concentrations than the fainter objects. This at least partly reflects the luminosity-size relation that exists for the brighter compact stellar systems, such that fainter systems appear PSF-like since they are unresolved in seeing-limited observations.\n\nDue to the high S\/N of the central pixels, the formal errors in core concentration are very small when considering uncertainties from photon statistics only\\footnote{The relative errors are on the order of 0.04 per cent for objects brighter than $m_{\\mathrm{Ve}} = 20.0$~mag, and 0.13 per cent for fainter objects from our working sample.}. An estimate for the (more relevant) uncertainty that stems from PSF fitting is given by the standard deviation of the core concentration of all PSF stars within one region (provided in column~8 of Table~\\ref{tab:tab2}). This reflects the local FWHM variations as compared to the PSF model of the respective region.\n\n\\begin{table*}\n \\caption{Parameter catalogue for our working sample of spectroscopically confirmed Fornax cluster members with $m_{\\mathrm{Ve}} < 21.5$~mag. The first five objects are printed below to illustrate the format of the table. We provide the complete table in the electronic version of the paper. ID: our object ID from Table~\\ref{tab:tab1}. $m_{\\mathrm{Ve}}$: $V$-equivalent magnitude and corresponding uncertainty. Note that the magnitude uncertainties are purely based on S\/N and do not include the calibration uncertainties mentioned in Section~\\ref{sect:sect2.1}. Flag: 1 = magnitude and uncertainty obtained from \\texttt{SExtractor}; 2 = magnitude and uncertainty obtained from PSF-fitting. cc: core concentration (uncorrected values, not used in the analysis). cc$_{\\mathrm{corr}}$: PSF-corrected core concentration (see Section~\\ref{sect:sect3.4}). An estimate for the uncertainty in core concentration, which is given for each region, can be inferred from column 8 of Table~\\ref{tab:tab2}. ra: residual asymmetry and corresponding uncertainty. el: ellipticity and corresponding uncertainty. A value of --99 for ellipticity or residual asymmetry denotes that it was not possible to determine the respective quantity. Sub.: subsample as defined in Table~\\ref{tab:tab5};  cc+ra=1, cc+RA=2, CC+RA=3, cc+EL=4; Sub.=0 indicates that the object is not part of any subsample. Alt. sub.: alternative subsample as defined in Table~\\ref{tab:tab8}. Region: PSF-region (see Table~\\ref{tab:tab2}).}\n  \\label{tab:tab3}\n \\begin{tabular}{cccccccccc}\n  \\hline\n  ID & $m_{\\mathrm{Ve}}$ (mag) & Flag & cc & cc$_{\\mathrm{corr}}$ & ra & el & Sub. & Alt. sub. & Region\\\\\n   \\hline\n  1 & 20.394 $\\pm$ 0.003 & 1 & 0.946 & 0.946 & 0.504 $\\pm$ 0.094 & 0.161 $\\pm$ 0.067 & 0 & 0 & 12\\\\\n  2 & 20.576 $\\pm$ 0.004 & 1 & 0.929 & 0.929 & 0.429 $\\pm$ 0.114 & 0.129 $\\pm$ 0.097 & 0 & 0 & 12\\\\\n  3 & 21.379 $\\pm$ 0.008 & 1 & 0.908 & 0.908 & 0.403 $\\pm$ 0.125 & 0.066 $\\pm$ 0.078 & 0 & 0 & 12\\\\\n  4 & 20.737 $\\pm$ 0.003 & 1 & 0.942 & 0.942 & 0.520 $\\pm$ 0.101 & 0.325 $\\pm$ 0.084 & 0 & 0 & \\phantom{0}6\\\\\n  5 & 20.645 $\\pm$ 0.004 & 1 & 0.905 & 0.905 & 0.759 $\\pm$ 0.044 & 0.308 $\\pm$ 0.028 & 0 & 0 & \\phantom{0}6\\\\\n  \\hline\n \\end{tabular}\n\\end{table*}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{fig3.eps}\n    \\caption{Distribution of parameter values for compact stellar systems and PSF stars. Our working sample corresponds to the red and blue histograms in the upper panels. For comparison we show the distributions of the artificial compact objects in the lower panels, which are represented by the red and blue histograms with lighter shading. Darker shading is used where histograms overlap. In both core concentration histograms we plot the PSF-corrected values (see Section~\\ref{sect:sect3.4}). Our sample with $m_{\\mathrm{Ve}} < 20.0$~mag contains 35 objects, the sample with $20.0 \\leq m_{\\mathrm{Ve}} < 21.5$~mag includes 320 objects. For three objects from the brighter and fourteen objects from the fainter sample no useful asymmetry value could be determined due to too little flux remaining in the residual image. The total number of artificial objects corresponds to 1170 for objects with $m_{\\mathrm{Ve}} < 20.0$~mag, and to 1215 for objects with $20.0 \\leq m_{\\mathrm{Ve}} < 21.5$~mag. We note that although both real and artificial compact objects span the same magnitude range, the magnitude distribution of the artificial objects is discrete, since the models were created on the basis of magnitudes and sizes from only 18 real UCDs. Therefore the parameter distributions of the artificial objects should be used for guidance only.}\n    \\label{fig:fig3}\n\\end{figure*}\n\n\\subsection{Residual asymmetry and ellipticity}\n\\label{sect:sect3.3}\nFor measuring residual asymmetry and ellipticity, we chose to use the PSF-subtracted residual image with all negative pixels set to zero, to avoid effects from the central oversubtraction of the PSF core (see Fig.~\\ref{fig:fig4}, left-hand panels), as well as noise effects. For simplicity, we refer to this image as the \\emph{residual image} in this subsection.\\\\\nThe residual asymmetry is defined in analogy to the Asymmetry parameter in \\citet{conselice:2000}:\n\\begin{equation}\n\\text{Residual asymmetry} = \\frac{\\sum_{r\\leq12\\mathrm{pix}}\\left| I_{\\mathrm{res,corr}} - I_{\\mathrm{res,corr,180}} \\right|}{\\sum_{r\\leq12\\mathrm{pix}} I_{\\mathrm{res}}}\n\\label{eq:eq2}\n\\end{equation}\nIn order to avoid substantial noise effects due to the small number of pixels (as compared to galaxies), we consider only the flux that exceeds $+1\\sigma$ of the noise level\\footnote{The noise level of the deeper regions of the mosaic corresponds to $\\umu_{\\mathrm{Ve}}=26.76$\\,mag\\,arcsec$^{-2}$ or 0.023 counts, see \\citet{lisker:2016}.}: $I_{\\mathrm{res,corr}}$ is the intensity on the residual image minus the noise level of 0.023 counts, and is set to zero if this is negative. $I_{\\mathrm{res,corr,180}}$ is the corresponding intensity on the image rotated by 180\\degr. The sum of the absolute pixel-by-pixel differences of the original and rotated image is then normalized by the flux of the residual image. We use all pixels within a circular aperture with a radius of 12 pixels ($> 2$ PSF FWHM) around the object centre. Any residual light from extended structures is essentially unaffected by the PSF at a radius larger than 2 FWHM; thus a fixed radius ensures comparability of such structures across the mosaic. All close neighbouring sources (and their counterparts rotated by 180\\degr) are masked, i.e., their pixels are not taken into account in the asymmetry calculation. \n\nThe ellipticity is defined as $1 - b\/a$, where $b\/a$ denotes the axis ratio. The latter is computed on the residual image from the second-order moments of the intensity distribution (as done in \\texttt{SExtractor}), using the above aperture and neighbour masks.\n\nWe estimated the uncertainty in residual asymmetry and ellipticity through error propagation, using the noise level at a given object's position (provided by the weight image) as uncertainty of a pixel's flux. The derived parameter values and corresponding uncertainties are given in Table~\\ref{tab:tab3}. We display the distribution of both residual asymmetry and ellipticity in Fig.~\\ref{fig:fig3} (upper centre and upper right panels).  The relations between core concentration, residual asymmetry and ellipticity are shown in Appendix~\\ref{sect:sectA}.\n\n\\subsection{Comparison to artificial compact objects}\n\\label{sect:sect3.4}\nWe estimated which parameter values can arise due to the shape of the PSF or the profile type of the objects by comparing the parameters of our sample to the parameter range of artificially created compact objects in the same magnitude and size range. We realized the artificial objects using structural parameters of real Fornax and Virgo cluster UCDs from \\citet{evstigneeva:2008}, where we selected 18 one-component UCDs, which have $V-$band magnitudes between $20.90$ and $18.33$~mag and effective radii between $4.0$ and $29.5$~pc. In addition to the observed structural parameters we created further models with different S\\'{e}rsic indices ranging from $n = 1$ to $8$. We generated each artificial model with a 10 times smaller pixel scale (i.e., 10 times better sampling) than our actual data, and then convolved it with each of our PSFs. This resulted in a set of roughly 2400 artificial objects for which we determined the core concentration, residual asymmetry, and ellipticity in the same way as for the real objects.\n\nFor the artificial objects we see a dependence of the core concentration (cc) on the PSF for some of our PSF models. Basically, the more extended (less concentrated) an object's core is intrinsically, the more the cc value gets lowered by the PSF convolution. This effect is enhanced when the PSF is broad, as compared to a reference PSF that is particularly symmetric and peaked. We chose the PSF from {\\it Region~2} (see Table~\\ref{tab:tab2} and  Fig.~\\ref{fig:fig2}) as reference PSF. For those PSFs showing the strongest deviations, the ratio of the cc value obtained with the reference PSF to the cc value obtained with the actual PSF depends roughly linearly on the latter. Therefore, we can determine correction factors, which need to be applied to the real objects to make their core concentration nearly independent of the PSF, and thus comparable to each other. Corrections were only applied for cc values lower than 0.95 and for five different PSFs ({\\it Regions 7, 9, 10, 11} and {\\it 13}, see Table~\\ref{tab:tab2}), according to:\n\n\\begin{equation}\n\\begin{split}\n\\mathrm{cc}_{\\mathrm{corr}} = \\mathrm{cc} \\Big[1 + k(\\mathrm{PSF_i})~\\left(\\mathrm{cc}-t\\right)\\Big],\\\\t=\\begin{cases} 1.0 & \\text{{\\it Regions 10, 13}}\\\\0.95&\\text{{\\it Regions 7, 9, 11,}}\\end{cases}\n\\end{split}\n\\label{eq:eq3}\n\\end{equation}\nwhere $k(\\mathrm{PSF_i})$ is the PSF-dependent correction factor. In total we corrected cc values of 22 objects from our working sample. The difference between the corrected and uncorrected values never exceeds 10 per cent. We include the corrected core concentration in Table~\\ref{tab:tab3}.\n\nWe display the distribution of parameter values for the artificial compact objects in the lower panels of Fig.~\\ref{fig:fig3}. These show magnitude-dependent differences in their parameter range: brighter objects can reach lower core concentration, whereas fainter objects have in general somewhat higher residual asymmetry and ellipticity. As a consequence, we apply slightly different parameter cuts to our working sample for objects with $m_{\\mathrm{Ve}} < 20.0$~mag and for objects with  $20.0$~$\\leq$~$m_{\\mathrm{Ve}}$~$<$~$21.5$~mag when defining subsamples in the following analysis.\n\n\\section{Results}\n\\label{sect:sect4}\n\\subsection{Peculiar compact stellar systems}\n\\label{sect:sect4.1}\n\nWe visually investigated compact stellar systems with high residual asymmetry and ellipticity as compared to the artificial compact objects. From our sample with $m_{\\mathrm{Ve}}~<~20.0$~mag, we examined those objects with residual asymmetry higher than 0.5 or ellipticity higher than 0.2, corresponding to 34 or 25 per cent of all objects with a measured residual asymmetry parameter in that magnitude range. The overlap with the artificial objects that have residual asymmetries in this parameter range is 6 per cent. The overlap with regard to ellipticity is 5 per cent. From our sample with $20.0 \\leq m_{\\mathrm{Ve}} < 21.5$~mag, we looked through all objects with a residual asymmetry higher than 0.55, corresponding to 38 per cent, or ellipticity higher than 0.25, corresponding to 16 per cent of compact stellar systems within that magnitude range. The overlap fractions with the artificial objects are 3 and 15 per cent, respectively.\n\nOur visual investigation revealed peculiar compact stellar systems, which are displayed in Fig~\\ref{fig:fig4}. We compare each peculiar object to an artificial compact object of similar brightness that was convolved with the PSF of the corresponding region (see Fig.~\\ref{fig:fig4}, right-hand panels). The comparison illustrates that the peculiar appearance of the displayed objects is not due to PSF effects.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{fig4.eps}\n\\caption{Peculiar compact stellar systems from our sample (centre sub-panels). The PSF-subtracted residual images are displayed in the left-hand sub-panels. We compare each peculiar object to an artificial compact object of similar brightness that has been convolved with the PSF of the corresponding region (shown in the right-hand sub-panels). The parameters of the artificial objects are based on magnitudes and sizes measured by \\citet{evstigneeva:2008} for 18 real one-component UCDs. In addition to the measured S\\'{e}rsic index, we created further models with a S\\'{e}rsic index in the range of $n = 1-8$. The artificial objects displayed here have an intermediate S\\'{e}rsic index of $n = 4$. For each peculiar object, we selected the artificial object that is closest in magnitude. The displayed objects are sorted by decreasing magnitude ($V-$band equivalent) from panels (a) to (o). The width of a single sub-panel is 13~arcsec (1.2~kpc). In the following we list the ID of each peculiar object, as given in Tables~\\ref{tab:tab1} and \\ref{tab:tab3}:  (a) ID=17, (b) 15, (c) 448, (d) 838, (e) 868, (f) 711, (g) 775, (h) 867, (i) 897, (j) 869, (k) 697, (l) 5, (m) 777, (n) 735, (o) 570. The artificial objects shown in each panel are based on $r_h$ and $m_V$ of the following objects from \\citet{evstigneeva:2008}: (a) UCD~1, (b) UCD~1, (c) UCD~41, (d) UCD~41, (e) UCD~41, (f) UCD~48, (g) UCD~48, (h) UCD~55, (i) UCD~55, (j) UCD~55, (k) UCD~33, (l) UCD~54, (m) UCD~21, (n) UCD~21, (o) UCD~21.}\n    \\label{fig:fig4}\n\\end{figure*}\n\n\\subsection{Parameters and spatial distribution}\n\\label{sect:sect4.2}\n\nIn this section we investigate whether there is a correlation between the parameters defined in Sections~\\ref{sect:sect3.2} and \\ref{sect:sect3.3} and the spatial distribution of the brighter objects in our sample. For each parameter we defined a cut to separate objects with low-value from objects with high-value parameters. We then divided our sample into subsamples by combining different parameter cuts. The definition of the subsamples is specified in Table~\\ref{tab:tab4}. For simplicity, we denote low core concentration with `cc', high core concentration with `CC', and analogous for residual asymmetry (ra\/RA) and ellipticity (el\/EL). The parameter cuts for the defined subsamples are provided in Table~\\ref{tab:tab5}. We adopted the parameter cuts as well as a magnitude limit of $m_{\\mathrm{Ve}} = 20.6$~mag to yield the statistically most significant differences in the distribution of cluster-centric distance between the different subsamples. We note that the cc+ra subsample contains a higher fraction of bright objects than the other subsamples, yet also has the largest spread in luminosity. There are five objects with $m_{\\mathrm{Ve}} < 20.6$~mag that have no measured residual asymmetry and were therefore excluded from the following analysis.\n\nFig.~\\ref{fig:fig5} shows the spatial distribution of our sample in the cluster, where we highlight the subsamples cc+ra, cc+RA, CC+RA and cc+EL. For comparison we also include low-mass cluster galaxies in the magnitude range $-19$~$<$~$M_r$~$<$~$-16$~mag. We probed the differences in the cluster-centric distance distributions (measured from NGC~1399) statistically with a Kolmogorov--Smirnov (KS) test, as summarized in Table~\\ref{tab:tab6}, and provide a comparison of the cumulative distributions of cluster-centric distance for the defined subsamples in Fig. \\ref{fig:fig6}.\n\nWe first compared the subsamples cc+ra, cc+RA, CC+RA and cc+EL to the respective other objects in the same magnitude range with $m_{\\mathrm{Ve}} < 20.6$~mag. We find that both the cc+RA and the cc+EL subsamples are predominantly located at larger cluster-centric distances. This difference is statistically significant according to a KS test, with a probability of 0.0 and 0.5~per~cent, respectively, for both subsamples having the same cluster-centric distance distribution as the respective other objects in the same magnitude range. In the cc+ra and CC+RA subsamples most objects have smaller cluster-centric distances. The distribution of the cc+ra subsample is very similar to the overall distribution of respective other objects with $m_{\\mathrm{Ve}} < 20.6$~mag. The distribution of the CC+RA subsample even appears to be more concentrated\\footnote{This also shows that the centrally concentrated distribution of the cc+ra subsample is not related to the high fraction of bright objects in it, since the CC+RA subsample consists of fainter objects, but is even more concentrated.} than the distribution of objects in the same magnitude range, although this difference has only a low statistical significance. Compared to the low-mass cluster galaxies, we find that the cc+RA and cc+EL subsamples appear similarly distributed, whereas the distribution of the cc+ra and CC+RA subsamples seems to be more centrally concentrated.\n\nWhen comparing the different subsamples to each other, we find the most significant differences between the cluster-centric distance distributions of the cc+RA subsample as compared to the cc+ra and CC+RA subsamples, with a probability of 0.0~per~cent, respectively, for having the same cluster-centric distance distribution. This is also seen for the cc+EL subsample, although with a lower statistical significance. We do not find significant differences between the distributions of the cc+ra and CC+RA subsamples. Also the distributions of the cc+RA and cc+EL subsamples are similar. But for the latter this is mainly due to the fact that the subsamples have some overlap in parameter range, since objects with high residual asymmetry can as well have high ellipticity, or vice versa.\n\nIn addition to the subsamples defined in Table~\\ref{tab:tab5} we provide an alternative subsample definition in Appendix~\\ref{sect:sectB}, where we set the parameter cut for the core concentration to a higher value in order to increase the number of objects in the subsamples with low core concentration. We do not find that our main results change significantly, although they have mainly lower statistical significances. For the CC+RA subsample, which includes fewer objects according to this parameter cut, we find that the more concentrated distribution, compared to the respective other objects of similar magnitude, becomes statistically more significant than previously, with a probability of 0.8~per~cent for the same distribution.\n\n\\begin{table}\n \\caption{Definition of subsamples and abbreviations.}\n \\label{tab:tab4}\n \\begin{tabular}{ll}\n  \\hline\n  Subsample & \\\\\n  \\hline\n  cc+ra & Low core concentration and low res. asymmetry\\\\\n  cc+RA & Low core concentration and high res. asymmetry\\\\\n  CC+RA & High core concentration and high res. asymmetry\\\\\n  cc+EL & Low core concentration and high ellipticity\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{table*}\n \\caption{Parameter ranges for the subsamples cc+ra, cc+RA, CC+RA and cc+EL, defined such that they yield the statistically most significant differences in the distribution of cluster-centric distance. For each subsample we give the fraction of objects in the respective magnitude range and the overlap fractions with the artificial objects.}\n \\label{tab:tab5}\n \\begin{tabular}{lllllll}\n  \\hline\n  Subsample & Cuts for $m_{\\mathrm{Ve}} < 20.0$~mag & Objects & Art. objects & Cuts for $20.0 \\leq m_{\\mathrm{Ve}} < 20.6$~mag & Objects & Art. objects\\\\\n  && (per cent) & (per cent) & & (per cent) & (per cent)\\\\\n  \\hline\n  cc+ra & cc < 0.77 and ra < 0.5 & 46.9 & 21.2 & cc < 0.83 and ra < 0.55 & 17.5 & \\phantom{0}0.0\\\\\n  cc+RA & cc < 0.77 and ra > 0.5 & \\phantom{0}9.4 & \\phantom{0}0.2 & cc < 0.83 and ra > 0.55 & 19.3 & \\phantom{0}0.0\\\\\n  CC+RA & cc > 0.77 and ra > 0.5 & 25.0 & \\phantom{0}5.6 & cc > 0.83 and ra > 0.55 & 36.8 & \\phantom{0}3.7\\\\\n  cc+EL & cc < 0.77 and el\\, > 0.2 & 12.5 & \\phantom{0}0.2 & cc < 0.83 and el\\, > 0.22 & 14.0 & \\phantom{0}0.0\\\\\n  \\hline\n \\end{tabular}\n\\end{table*}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{fig5.eps}\n    \\caption{Spatial distribution of compact stellar systems in the Fornax cluster. For comparison we also show the distribution of the cluster galaxies. Faint CSS: compact objects with $20.6 \\leq m_{\\mathrm{Ve}} < 21.5$~mag. Bright CSS: compact objects with $m_{\\mathrm{Ve}} < 20.6$~mag. Subsamples: cc+ra, cc+RA, CC+RA, cc+EL, as defined in Table \\ref{tab:tab5}. Low-mass galaxies: galaxies with $-19 < M_r < -16$~mag from the Fornax cluster catalogue (FCC, \\citealt{ferguson:1989}; based on the magnitude conversions of \\citealt{weinmann:2011}). Massive galaxies: galaxies with $M_r \\leq -19$~mag from the FCC. Each massive galaxy is represented by a circle with three times its isophotal diameter at $\\mu_B = 25$~mag~arcsec$^{-2}$, $3\\,\\rm{d_{25}}$ (we used the extinction-corrected values for d$_{25}$, obtained from HyperLEDA; \\citealt{makarov:2014}). The two brightest galaxies are NGC~1399 in the centre and NGC~1404 to the south-east from it.}\n    \\label{fig:fig5}\n\\end{figure*}\n\n\\begin{table}\n \\caption{KS test probabilities (percentage) for the null hypothesis that two subsamples have the same cluster-centric distance distribution. In the last row the distributions of the individual subsamples are compared to the respective other compact objects with different parameters in the same magnitude range with $m_{\\mathrm{Ve}} < 20.6$~mag.}\n  \\label{tab:tab6}\n \\begin{tabular}{lllll}\n  \\hline\n  Subsample & cc+ra & cc+RA & CC+RA & cc+EL\\\\\n  \\hline\n  cc+ra & 100.0 & \\phantom{00}0.0 & \\phantom{0}55.3 & \\phantom{00}0.4\\\\\n  cc+RA & & 100.0 & \\phantom{00}0.0 \\phantom{0} & \\phantom{0}98.7\\\\\n  CC+RA & & & 100.0 & \\phantom{00}1.0\\\\\n  cc+EL & & & & 100.0\\\\\n  respective & \\phantom{0}41.7 & \\phantom{00}0.0 & \\phantom{00}6.7 \\phantom{0} & \\phantom{00}0.5\\\\\n  other CSS &&&&\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{fig6.eps}\n    \\caption{Cumulative distribution of projected cluster-centric distance (measured from NGC~1399) for the subsamples cc+ra, cc+RA, CC+RA and cc+EL. The distribution of all objects with $m_{\\mathrm{Ve}} < 20.6$~mag (`Bright CSS') is shown for comparison.}\n    \\label{fig:fig6}\n\\end{figure} \n\n\\subsection{Distribution in phase-space}\n\\label{sect:sect4.3}\n\nIn order to examine the phase-space distribution of compact stellar systems in the Fornax cluster, we define $\\Delta v$ as an object's relative velocity with respect to the cluster mean velocity (1460~km~s$^{-1}$). The latter is the average velocity of Fornax cluster members within a cluster-centric distance of 1\\degr, including all compact objects from our catalogue and galaxies from the Fornax cluster catalogue (FCC, \\citealt{ferguson:1989}). We denote the corresponding standard deviation (324~km~s$^{-1}$) as the cluster velocity dispersion $\\sigma$, $R$ as the cluster-centric distance, and $R_{\\mathrm{vir}}$ as its virial radius (assumed to be 0.85~Mpc or 2.5\\degr, which is the average of the 0.7~Mpc quoted by \\citealt{drinkwater:2001} and 1.0~Mpc quoted by \\citealt{murakami:2011}). The phase-space distribution is shown in Fig.~\\ref{fig:fig7} as $\\Delta v \/ \\sigma$ versus $R\/R_{\\mathrm{vir}}$. We highlight the subsamples cc+ra, cc+RA, CC+RA and cc+EL, and further discriminate between very faint compact stellar systems in the magnitude range of GCs ($m_{\\mathrm{Ve}} \\ge 21.5$~mag, which are not part of our working sample), faint objects ($20.6 \\leq m_{\\mathrm{Ve}} < 21.5$~mag), and bright objects ($m_{\\mathrm{Ve}} < 20.6$~mag), and include low-mass galaxies in the magnitude range $-19 < M_r < -16$~mag for comparison. Table~\\ref{tab:tab7} summarizes the velocity dispersion for the various subsamples and cluster populations. We show the phase-space distribution of the alternative subsamples in Appendix~\\ref{sect:sectB} (Fig.~\\ref{fig:fig10} and Table~\\ref{tab:tab10}).\n\nPhase-space diagrams allow one to study the accretion history of a cluster population. For example, \\citet{noble:2013} used caustic profiles, which are lines of constant $(\\Delta v \/ \\sigma)~\\times~(R\/R_{\\mathrm{vir}})$, to distinguish between infalling and virialized cluster members, where higher values trace systems that were more recently accreted. In Fig.~\\ref{fig:fig7} we plotted caustic lines of constant $(\\Delta v \/ \\sigma)~\\times~(R\/R_{\\mathrm{vir}})$ at $\\pm$0.1 and $\\pm$0.4, respectively. Cosmological simulations show that early accreted systems would be predominantly located within the inner caustic lines of $\\pm$0.1, systems accreted later between the caustics of $\\pm$0.1 and $\\pm$0.4, and recently accreted systems along or outside the caustic lines of $\\pm$0.4 \\citep[cf.][]{haines:2012, noble:2013}.\n\nWe find that essentially all compact stellar systems are located within the inner caustics of $\\pm$0.1. Also some low-mass galaxies are found in this region, but many of them occupy the region in between the caustic lines of $\\pm$0.1 and $\\pm$0.4. This is also reflected in the velocity dispersion, where we find the lowest dispersion in particular for the compact stellar systems with $m_{\\mathrm{Ve}} < 21.5$~mag (the samples comprising the bright and faint CSS) and the highest for the low-mass galaxies.\n\nAmong the bright compact stellar systems with $m_{\\mathrm{Ve}}~<~20.6$~mag, we do not find statistically significant differences in the velocity distribution of the four different subsamples. The main difference seems to be in the spatial distribution. However, with respect to the velocity dispersion, we find that the cc+ra subsample has on average the lowest and the cc+EL subsample the highest velocity dispersion. The two subsamples cc+RA and CC+RA with large residual asymmetry have velocity dispersions in between. In the phase-space diagram we note that especially the cc+EL subsample is predominantly distributed along the inner caustic lines.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{fig7.eps}\n    \\caption{Phase-space distribution of compact stellar systems and low-mass galaxies in the Fornax cluster. Very faint CSS: compact objects with $m_{\\mathrm{Ve}} \\geq 21.5$~mag (not part of our working sample). Faint CSS: compact objects with $20.6 \\leq m_{\\mathrm{Ve}} < 21.5$~mag. Bright CSS: compact objects with $m_{\\mathrm{Ve}} < 20.6$~mag. Subsamples: cc+ra, cc+RA, CC+RA, cc+EL, as defined in Table \\ref{tab:tab5}. Low-mass galaxies: galaxies with $-19 < M_r < -16$~mag from the FCC (\\citealt{ferguson:1989}; based on the magnitude conversions of \\citealt{weinmann:2011}). $\\Delta v$ is the relative velocity of an object with respect to the cluster mean velocity (1460~km~s$^{-1}$). We denote $\\sigma$ as the cluster velocity dispersion (324~km~s$^{-1}$), $R$ as the cluster-centric distance, and $R_{\\mathrm{vir}}$ as its virial radius ($2.5\\degr$, see Section~\\ref{sect:sect4.3}). The mean velocity and dispersion were calculated from all compact stellar systems and FCC galaxies within a cluster-centric distance of 1.0\\degr. The solid lines correspond to caustic lines of constant $(\\Delta v \/ \\sigma)~\\times~(R\/R_{\\mathrm{vir}})$ at $\\pm$0.1 and $\\pm$0.4, respectively.}\n    \\label{fig:fig7}\n\\end{figure*}\n\n\\begin{table}\n \\caption{Velocity dispersion for very faint ($m_{\\mathrm{Ve}} \\geq 21.5$~mag, not part of our working sample), faint ($20.6 \\leq m_{\\mathrm{Ve}} < 21.5$~mag) and bright ($m_{\\mathrm{Ve}} < 20.6$~mag) compact stellar systems, low-mass galaxies (FCC), and the subsamples cc+ra, cc+RA, CC+RA, cc+EL, as defined in Table~\\ref{tab:tab5}. The velocity dispersion of each subsample is calculated as standard deviation within a cluster-centric distance of $R \\leq 1.0\\degr$  ($\\sigma_{\\mathrm{tot}}$), $R \\leq 0.4\\degr$ ($\\sigma_{\\mathrm{in}}$) or $0.4 < R \\leq 1.0\\degr$ ($\\sigma_{\\mathrm{out}}$), based on the velocities given in Table~\\ref{tab:tab1}. $N_{\\mathrm{obj}}$ corresponds to the number of objects from the respective subsample in the inner ($N_{\\mathrm{obj,in}}$) or outer ($N_{\\mathrm{obj,out}}$) cluster region. The velocity dispersion is given in km~s$^{-1}$. $1.0\\degr$ corresponds to $0.346$ and $0.4\\degr$ to $0.138$~Mpc at the distance of Fornax.}\n \\label{tab:tab7}\n \\begin{tabular}{llllll}\n  \\hline\n  Subsample & $\\sigma_{\\mathrm{tot}}$ & $\\sigma_{\\mathrm{in}}$ & $N_{\\mathrm{obj,in}}$& $\\sigma_{\\mathrm{out}}$ & $N_{\\mathrm{obj,out}}$\\\\\n  \\hline\n  Very faint CSS & 337 & 336 & 535 & 341 & 14\\\\\n  Faint CSS & 297 & 302 & 232 & 252 & 29\\\\\n  Bright CSS & 307 & 316 & \\phantom{0}73 & 271 & 21\\\\\n  Low-mass & 400 & 505 & \\phantom{00}4 & 317  & \\phantom{0}7\\\\\n  galaxies&&&&&\\\\\n  cc+ra & 303 & 282 & \\phantom{0}23 & 401 & \\phantom{0}2\\\\\n  cc+RA & 326 & 485 & \\phantom{00}4 & 330 & 10\\\\\n  CC+RA & 328 & 338 & \\phantom{0}25 & 246 & \\phantom{0}4\\\\\n  cc+EL & 414 & 556 & \\phantom{00}5 & 269 & \\phantom{0}7\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\\section{Discussion}\n\\label{sect:sect5}\n\\subsection{Analysis methods and limitations}\n\\label{sect:sect5.1}\n\nThe seeing FWHM varies over the mosaic, since the latter is based on images acquired in multiple nights and observing runs. In the PSF analysis we tried to account for this by determining the PSF in regions of similar FWHM, as indicated in Fig.~\\ref{fig:fig2}. In {\\it Region 3} of our mosaic we observe systematically lower core concentration for our sample with $m_{\\mathrm{Ve}} < 21.5$~mag. We investigated whether this effect could be due to a non-matching PSF. We therefore fitted PSFs from other regions on the mosaic to all objects from our sample that are located in {\\it Region 3}. We find that the core concentration decreases when a PSF of smaller FWHM is used. Thus, if there are variations of the PSF FWHM on scales smaller than the considered region within which the PSF is determined, an artificially low core concentration can occur if an object is located in an area with locally larger FWHM than the PSF FWHM of that region.\n\nIn order to estimate local variations of the FWHM we computed the minimum FWHM around each object. To find the minimum FWHM we first measured the FWHM of all objects in {\\it Region 3} with \\texttt{SExtractor}. From all \\texttt{SExtractor}-detected objects brighter than $m_{\\mathrm{Ve}} = 22$~mag we then determined the minimum FWHM within a circular area of 450~pixels (1.8~arcmin) in radius around each object. Finally we compared the local FWHM to the FWHM of the PSF stars from this region. In {\\it Region 3} we neither find a relation of core concentration with local FWHM, nor are all objects with low core concentration systematically located in areas with large local FWHM.\n\nWe extended this test to the full sample of compact stellar systems brighter than $m_{\\mathrm{Ve}} = 20.6$~mag and investigated how much a locally varying FWHM would affect our results from Section~\\ref{sect:sect4.2}. We first computed the relative local FWHM for each object as difference between the measured local FWHM and the average FWHM of all PSF stars of the respective region. Since we found a slight dependence of the core concentration on the relative local FWHM for objects brighter than $m_{\\mathrm{Ve}} = 20.6$~mag, we defined a linearly varying parameter cut that follows this relation. We then defined new subsamples, using the parameter cuts for the residual asymmetry and ellipticity as specified in Table~\\ref{tab:tab5}, but with a linearly varying parameter cut for the core concentration. According to this definition, our main result that the subsamples with low core concentration and high residual asymmetry (or high ellipticity) are mainly distributed at larger cluster-centric distances remains unchanged\\footnote{With probabilities of 0.0 and 0.8~per~cent, respectively, for the two subsamples having the same cluster-centric distance distribution as objects in the same magnitude range according to a KS test.}.\\\\\n\nThe angular resolution of our data limits our ability to identify possible blends of close neighbouring sources. \\citet{richtler:2005} resolved one of the peculiar compact objects (Fig.~\\ref{fig:fig4}, panel g) into two sources. We therefore attempt to estimate how likely it is that all objects with significant asymmetry or ellipticity are blends. For this purpose we assume that objects that appear asymmetric or elongated could be blends if a close, up to three magnitudes fainter, neighbour source was located within a distance of $4 - 7$ pixels (corresponding to 91 and 160~pc, respectively). When considering all objects from our bright sample ($m_{\\mathrm{Ve}} < 20.6$~mag) with high residual asymmetry or high ellipticity (according to the definition in Table~\\ref{tab:tab5}), which have cluster-centric distances between 20 and 160~kpc, we find an expected number density of faint sources of 0.09 arcsec$^{-2}$ if all those objects were blends. For comparison, the observed number density of faint sources with magnitudes $20.6 < m_{\\mathrm{Ve}} < 23.6$~mag located within the same area is 0.003 objects arcsec$^{-2}$. Thus, only a single one of the 32 bright asymmetric or elongated objects we observe in this area would statistically be expected to be a blend. We note, however, that an overdensity of GCs within 1~kpc of brighter compact stellar systems, including a fraction within 300~pc, was recently reported by \\citet{voggel:2016}. We may thus expect a few blends among the said objects.\n\n\\subsection{Peculiar compact stellar systems in the Fornax cluster}\n\\label{sect:sect5.2}\n\nWe report the discovery of peculiar compact stellar systems in the Fornax cluster, which appear asymmetric or elongated in our images (illustrated in Fig.~\\ref{fig:fig4}). The presence of a few peculiar objects in Fornax was already noted by \\citet{richtler:2005} and \\citet{voggel:2016}\\footnote{Two of our displayed objects were shown in \\citet{richtler:2005} (Fig.~\\ref{fig:fig4}, panels g and n). The object in panel (n) was also pointed out by \\citet{voggel:2016} to be peculiar.}.\n\nSome compact Fornax cluster members have previously been observed with \\textit{HST} \\citep{evstigneeva:2008}. Two of them are also shown in Fig.~\\ref{fig:fig4} (panels d and h), but we note that a direct comparison of the inner structure is not possible due to our much broader PSF. At the same time the \\textit{HST} images are too shallow for a comparison of the outer, low-surface-brightness structure that we are able to measure in our data. The \\emph{azimuthally averaged} UCD surface-brightness profiles of \\citet{evstigneeva:2008} reach 26~mag~arcsec$^{-2}$ at best (see their fig.~1), whereas we reach an image depth of 26.8~mag~arcsec$^{-2}$ at $S\/N=1$ \\emph{per pixel} (see Section~\\ref{sect:sect2.1}).\n\nIn the Virgo cluster the structure of a large sample of UCDs has been investigated by \\citet{liu:2015}, revealing faint envelopes around many of the objects. However, objects with similarly asymmetric appearance as observed in our data were not reported. In the cluster Abell S0740, \\citet{blakeslee:2009} detected faint envelopes around candidate UCDs, which show signs of being elongated in \\textit{HST} images.\n\nThere are two main formation scenarios discussed in the literature, relating UCDs to either the population of galaxies or to the population of star clusters. In the former case, it is suggested that UCDs may be the remnant nuclei of tidally stripped nucleated dwarf galaxies \\citep[e.g.][]{bekki:2003, pfeffer:2013}. In a star cluster origin, UCDs may grow to sizes and masses larger than typical GCs, if they form in star cluster complexes by merging of young massive star clusters \\citep[e.g.][]{fellhauer:2005}.\n\nAccording to simulations by \\citet{bekki:2003}, observable signatures of a tidal stripping origin would be tidal tails as well as relics from the envelope of the progenitor galaxy that has been stripped. \\citet{lisker:2016} did not find any diffuse streams that would be signs of tidal debris around any of the 904 spectroscopically confirmed objects in our compiled catalogue. As noted by \\citet{lisker:2016}, such streams should be visible in most parts of the mosaic if their surface brightness level is in the range $27.5 \\lesssim \\mu_{\\mathrm{Ve}} \\lesssim 28.0$~mag~arcsec$^{-2}$. The absence of visible tidal debris does not necessarily need to be in contradiction to a tidal stripping origin of these objects. \\citet{pfeffer:2014} showed, based on cosmological simulations, that most low-mass cluster galaxies were disrupted already many gigayears ago. Thus, if most of the compact stellar systems resulted from early stripping events, we simply might not be able to observe relics of tidal debris any more today, since tidal tails disperse on time-scales of a few Gyr \\citep{pfeffer:2013}. However, \\citet{bruens:2012} predicted that an object formed via merging of massive star clusters in a star cluster complex would also be surrounded by a faint stellar envelope, thus exhibiting a similar two-component surface-brightness profile as reported in the tidal stripping simulations. Therefore, without the detection of the predicted long tidal streams, we cannot discriminate between a stripping or a merging origin.\n\nIn the star cluster merging scenario described by \\citet{bruens:2011}, the forming object can look quite asymmetric as long as the merger is not yet complete. However, the authors showed that star clusters in a cluster complex typically merge on time-scales of only a few hundred Myr. As a consequence, if the structures we observe for the peculiar systems in Fig.~\\ref{fig:fig4} are signatures of an ongoing star cluster merger, then these should be comparatively young systems. This seems to be in contradiction with the observation that UCDs have in general old stellar populations \\citep{evstigneeva:2007, paudel:2010, francis:2012}.\n\nOn the other hand, \\citet{fellhauer:2005} reported that a merger remnant would be stable over a time-scale of 10~Gyr. Thus, compact objects may have formed from star cluster merging already in very early phases during the formation of the Fornax cluster core, when the merging of gas-rich galaxies possibly offered conditions for strong starbursts in which large star cluster complexes are thought to form. At the same time \\citet{fellhauer:2005} demonstrated that such a merger remnant constantly loses some of its mass with every pericentric passage due to the tidal field of the cluster. Therefore the peculiar structures of some compact systems may not stem from an ongoing merger of star clusters, but may be associated with the deformation or disruption of an extended star cluster that formed via star cluster merging when the Fornax cluster core assembled. Disruption signatures can thus not necessarily discriminate between the proposed formation scenarios of compact stellar systems.\n\nTo estimate whether tidal stripping would be efficient for disturbing the outer structure of an UCD-like compact object orbiting in the Fornax cluster, we calculate the tidal radius according to \\citet{king:1962}:\n\\begin{equation}\nR_{\\mathrm{tidal}} = R_{\\mathrm{peri}} \\left( \\frac{M_{\\mathrm{obj}}}{M_{\\mathrm{cl}}(R_{\\mathrm{peri}})\\,(3 + e)}\\right)^{1\/3},\n\t\\label{eq:eq4}\n\\end{equation}\nwhere $R_{\\mathrm{peri}}$ is the pericentric distance, $M_{\\mathrm{obj}}$ the total mass of the object, $M_{\\mathrm{cl}}(R_{\\mathrm{peri}})$ the enclosed cluster mass within $R_{\\mathrm{peri}}$ and $e$ the eccentricity of the orbit for which we adopt a value of 0.5\\footnote{Bound orbits have an eccentricity of $e < 1.0$. An eccentricity of $e = 0.0$ would correspond to a circular orbit, $e > 0.7$ to a highly eccentric orbit.}. For a typical UCD-like object with a mass of $M_{\\mathrm{obj}} = 10^7$~M$_{\\sun}$, when assuming it reaches an orbital pericentre of $R_{\\mathrm{peri}} = 20$~kpc, the tidal radius would be in the range of $200 - 300$~pc, depending on the adopted mass profile for the Fornax cluster \\citep{drinkwater:2001, richtler:2008, SRH2010}. While this is already on the order of 10 effective radii for a UCD of mass $M_{\\mathrm{obj}} = 10^7$~M$_{\\sun}$ \\citep[cf.][]{norris:2014}, we would be able to observe such an object out to its tidal radius in our deep imaging data, according to the surface-brightness profiles of typical UCDs \\citep[see fig.~1 of][]{depropris:2005}. This estimate shows that distortions of the outer structure due to tidal stripping could be expected for objects with close cluster-centric passages. However, since the tidal radius of an object solely depends on its total mass, and not on how the object mass is distributed, it is not possible to infer the nature of the disturbed object, whether it is the remains of a stripped galaxy or an extended star cluster in process of disruption.\n\nFor the above estimate of the tidal radius we assumed that the objects would be only influenced by the cluster's tidal field. None the less, some objects may be more strongly affected by very close bright galaxies, especially further out from the cluster centre. \\citet{schuberth:2008} investigated whether some compact stellar systems with $M_V < -9.5$~mag were consistent with being associated with any of the ten brightest galaxies in the Fornax cluster core, according to their spatial and velocity distribution. The authors showed that among the compact objects, which are located within a projected distance of $1.5\\,\\rm{d_{25}}$\\footnote{d$_{25}$ is the isophotal diameter of a galaxy at $\\mu_B$~$=$~$25$~mag~arcsec$^{-2}$.} from a bright galaxy, the metal-rich (red) objects have velocities not deviating by more than 100~km~s$^{-1}$ from the velocity of the closest massive galaxy, whereas the metal-poor (blue) objects seem to be characterized by a larger spread in velocities. The remaining objects at distances larger than $1.5\\,\\rm{d_{25}}$ from any bright galaxy seem to be consistent with being kinematically associated with the extended GC system of NGC~1399, out to a cluster-centric distance of 30~arcmin.\n\nIn Fig.~\\ref{fig:fig5}, showing the spatial distribution of known Fornax cluster members, we represented each massive galaxy with $M_r \\leq -19$~mag by three times its isophotal diameter ($3\\,\\rm{d_{25}}$). We find that only few objects from our subsamples lie close in projection to a bright galaxy other than NGC~1399, and of these only a minor fraction also has a similar velocity\\footnote{We find that the following number of objects from our subsamples are located within a distance of  $\\rm{r} = 1.5\\,\\rm{d_{25}}$ from a massive galaxy with $M_r \\leq -19$~mag other than NGC~1399: 4 out of 25 in cc+ra, 3 out of 14 in cc+RA, 7 out of 29 in CC+RA, and 4 out of 12 in cc+EL. When we require additionally a relative velocity smaller than $\\Delta v = 200$~km~s$^{-1}$, the number of objects is: 1 in cc+ra, 1 in cc+RA, 3 in CC+RA, and 2 in cc+EL.}. Therefore we see no indication that a significant fraction of objects from our subsamples is bound to massive galaxies with $M_r \\leq -19$~mag other than NGC~1399.\n\n\\subsection{Spatial and phase-space distribution of compact stellar systems}\n\\label{sect:sect5.3}\n\nIn general it is observed that GCs and galaxies have different spatial distributions in galaxy clusters. For example \\citet{zhang:2015} observed that in the central core region of Virgo, the dwarf ellipticals (dEs) have a much flatter number density profile than the GCs. In Fornax it was also found that the GC population of NGC~1399 is much more centrally concentrated than the surrounding dEs \\citep{GDE2009, hilker:2011}. \\citet{GDE2009} reported that the UCDs in Fornax have lower velocity dispersions than both the GCs and the low-mass galaxies in the core region, which we also find in this work. \\citet{SRH2010} further distinguished between red and blue GCs, and their analysis showed that UCDs have a velocity dispersion lower than that of the blue GCs, but higher than that of the red GCs.\n\nIn Sections~\\ref{sect:sect4.2} and \\ref{sect:sect4.3} we investigated the spatial and phase-space distribution of compact stellar systems in the Fornax cluster, where we focused on the distribution of objects with $m_{\\mathrm{Ve}} < 20.6$~mag ($M_{\\mathrm{Ve}} < -11.1$~mag), corresponding to the brighter UCD magnitude range. We compared the differences in the cluster-centric distance distributions of smaller subsamples, where we categorized the objects according their core concentration, residual asymmetry, and ellipticity. In the following we discuss the results on the spatial and phase-space distribution of our subsamples in terms of the origin and nature of compact stellar systems.\n\nWe find that the objects in the subsample with low core concentration and high residual asymmetry, as well as in the subsample with low core concentration and high ellipticity, are predominantly distributed at larger cluster-centric distances, compared to objects in the same magnitude range but with different parameters\\footnote{We note that the subsamples with larger cluster-centric distances may even be slightly under-represented in our study, as compared to the other subsamples, if the completeness varies with cluster-centric distance (see Section~\\ref{sect:sect2.4}).}. Their extended galaxy-like spatial distribution might favour a formation scenario in which they are remnants of stripped low-mass galaxies\\footnote{This does not exclude that some of them formed as massive star clusters in the tidal tails of a major galaxy merger \\citep[e.g.,][]{gallagher:2001}.}. However, since their location at larger cluster-centric distances implies a currently large tidal radius, their structure could only be explained by tidal stripping if these objects are on rather eccentric orbits with small pericentre distances, such that they possibly approached the cluster centre very closely at earlier times. We observe that especially the subsample with low core concentration and high ellipticity has high velocities relative to the Fornax cluster, which would be expected for radial orbits. This subsample is mainly distributed \\emph{along} the inner caustic lines of constant $(\\Delta v \/ \\sigma)~\\times~(R\/R_{\\mathrm{vir}}) = \\pm0.1$ and seems to integrate smoothly into the phase-space distribution of low-mass cluster galaxies. This could indicate that these objects, or their progenitors, may have been accreted more recently compared to the overall population of compact stellar systems in the UCD and GC magnitude range, which are predominantly confined \\emph{within} those caustics.\n\nIn the Virgo cluster, observations of a large sample of UCDs around M87 point in a similar direction: UCDs at cluster-centric distances larger than 40~kpc have a radially biased orbital structure consistent with that of stripped galaxies on radial orbits \\citep{zhang:2015}. Although no compact stellar systems with obvious distorted outer structures have been reported in Virgo so far, \\citet{liu:2015} observed that compact objects with signs of a faint stellar envelope are found only at distances larger than $0.1\\degr$   (corresponding to 28,8~kpc at a distance of 16.5~Mpc for Virgo; \\citealt{blakeslee:2009}) from the two brightest cluster galaxies M87 and M49, respectively. Their distribution is, however, still more concentrated than that of the cluster's dwarf galaxy population. The authors interpret this sequence of decreasing envelope fraction with decreasing cluster-centric distance as an indication that tidal stripping of nucleated dwarf galaxies plays an important role for the population of compact stellar systems in Virgo.\n\nIn Fornax, when defining a subsample of objects with no signs of distortions\\footnote{By choosing objects with low residual asymmetry and low ellipticity according to Table~\\ref{tab:tab5}, with parameter cuts at ra < 0.5 and el < 0.2 for objects with $m_V < 20.0$~mag, and ra < 0.55 and el < 0.22 for objects with $20.0 \\leq m_V < 20.6$~mag.}, we find the majority of these objects distributed at small cluster-centric distances. This would at least not imply \\emph{recent} stripping, if these objects were stripped galaxies at all. Instead, these objects might already be stripped down to their tidal radius. This could be expected if the objects had been accreted at very early times, and had been exposed for longer to the cluster tidal field than, e.g., the objects with low core concentration and high residual asymmetry (or high ellipticity). This would be supported by their lower velocity dispersion, compared to other subsamples. \\citet{smith:2015}, who studied the effects of harassment on early-type dwarf galaxies in galaxy clusters with numerical simulations, concluded that strongly harassed galaxies would be characterized by low orbital velocities.\n\nHowever, for this subset of objects with no signs of distortions, we would not exclude a star cluster origin either. Based on the spatial and phase-space distribution alone, it may be difficult to disentangle a star cluster from a galaxy origin --- possible progenitor galaxies may have been accreted very early on, or even formed at the same epoch as massive star clusters. For example, \\citet{bournaud:2008} predicted that during the formation of giant ellipticals through mergers of gas-rich galaxies, tidal dwarf galaxies ($M_* = 10^8 - 10^9$~M$_{\\sun}$) could form along with massive star clusters ($M_* = 10^5 - 10^7$~M$_{\\sun}$).\n\nNotwithstanding these considerations, the best candidates for a star cluster origin among the considered subsamples seem to be the objects with high core concentration and high residual asymmetry. Their distribution is very centrally concentrated and looks quite different from the distribution one would expect for galaxies. Moreover, this subsample is characterized by a higher core concentration, i.e. having a more star-like central component. Possibly we observe here a population of bright deformed star clusters instead of remnants of stripped galaxies. Since most of the objects are in close proximity to NGC~1399 the tidal radius would be quite small so that current stripping or deformation could indeed be expected. Some objects of this subsample are located in between NGC~1399 and the close elliptical galaxy NGC~1404 to the south-east from it (see Fig.~\\ref{fig:fig5}). \\citet{kim:2013} and \\citet{dabrusco:2016} took an overabundance of GCs between NGC~1399 and NGC~1404 (as well as other surrounding galaxies) as an indication for interactions in the recent past, which was also simulated by \\citet{bekki:2003b}. Thus, some of our objects with high core concentration and high residual asymmetry may have been distorted and also freed from their parent galaxies during possible interactions of massive cluster galaxies in the Fornax cluster core.\n\n\\section{Conclusions}\n\\label{sect:sect6}\nOur deep imaging data of the Fornax cluster core reveal peculiar compact stellar systems, which appear asymmetric or elongated. From their structure alone we cannot infer their origin,  whether these objects are luminous star clusters in the process of disruption or possible remnants of stripped galaxies.  However, the spatial distribution of objects with low core concentration and high residual asymmetry (or high ellipticity) at mainly larger cluster-centric distances may be explained with a galaxy origin of at least some of these objects. This is also supported by the high relative velocities we observe in particular for the objects with low core concentration and high ellipticity.\n\n\\section*{Acknowledgements}\nThis paper is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (programmes 082.A-9016 and 084.A-9014).\nCW\\ is a member of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD).\nCW\\ acknowledges support by ESO for a research visit while this work was in progress.\nTL\\ acknowledges support by ESO through the Scientific Visitor Programme while this work was in progress.\nTL\\ was supported within the framework of the Excellence Initiative by the German Research Foundation (DFG) through the Heidelberg Graduate School of Fundamental Physics (grant number GSC 129\/1) and through the Innovation Fund FRONTIER of Heidelberg University.\nTL thanks Michael Drinkwater for having shared a Fornax cluster object catalogue while this project was at a preliminary stage.\nWe thank the anonymous referee for useful comments that helped improving the presentation of our results.\nThis research has made use of the SIMBAD data-base, operated at CDS, Strasbourg, France.\nThis research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A\\&AS 143, 23.\nThis research has made use of the HyperLeda data-base (http:\/\/leda.univ-lyon1.fr).\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplemental Material}\n\n\\newlength{\\figwidthhalf}\n\\setlength{\\figwidthhalf}{246.0pt}\n\n\\newcommand{\\secref}[1]{Sec.~\\ref{#1}}\n\\newcommand{\\Secref}[1]{Section~\\ref{#1}}\n\\newcommand{\\figrefs}[1]{Figs.~\\ref{#1}}\n\\newcommand{\\Figrefs}[1]{Figures~\\ref{#1}}\n\n\n\\renewcommand\\thefigure{A.\\arabic{figure}}\n\n\n\n\\subsection{Kinks in domain walls versus kinks in dislocations}\n\nWe show that under certain conditions, DW-kinks may display solitonic behavior.\nIt is interesting to compare the dynamics of DW-kinks to kinks in dislocations in the crystal lattice.\n\\Explcite{Gornostyrev1999} found that solitonic behavior can be observed for dislocation kinks in a two-dimensional rigid-substrate model of the slip plane.\nHowever, Kosevich{} argues that the sine-Gordon equation cannot be a satisfactory model for the dynamics of a free dislocation because dislocation motion couples in an essential way to lattice vibrations \\cite{Kosevich2005}.\nMoreover, it seems that ballistic effects (let alone solitonic behavior) play no role of significance in practical simulations of kink or dislocation dynamics \\cite{Swinburne2013A}.\nIt is important to note that a real dislocation is a line defect in three-dimensional space; its spatial profile depends in an essential way on the perpendicular coordinate. A domain wall, by contrast, is a planar defect. This justifies our approximation of a magnetic thin film containing a domain wall as effectively two-dimensional.\n\n\n\\subsection{Atomistic simulation of a breather (movie)}\n\nA movie file with an atomistic simulation of a breather is provided with this Supplemental Material (see \\figref{fig:snapshot}). It shows a single period of a breather with $\\omega = 0.1$ for $l_1 = l_2 = 1.58a$, $w \\approx 79a$, and no Gilbert damping ($\\alpha = 0$). The simulation box contains $40 \\times 960$ magnetic moments.\nThe translucent yellow strip indicates the evolution of the domain-wall profile according to the analytical breather solution of the sine-Gordon equation.\n\n\\begin{figure}[h]\n  \\includegraphics[width=0.5\\textwidth]{smatfig01mov1snapshot}\n  \\caption{\\label{fig:snapshot}(color online). Atomistic simulation (movie snapshot).}\n\\end{figure}\n\n\n\\subsection{\\label{sec:hoexch}Higher-order exchange}\n\n\\subsubsection{Continuum model}\n\nFor magnetization profiles smooth on the scale of the lattice constant $a$, the atomistic Hamiltonian (see main text) is equivalent to the sum of the continuum energy functionals\n\\begin{subequations}\\label{eq:contenfunc}\n\\begin{align}\nE_\\text{ex}[\\mathbf{m}(x,y)]& = \\int \\! A \\biggl[ {\\biggl\\lVert \\frac{\\partial \\mathbf{m}}{\\partial x} \\biggr\\rVert}^2 + {\\biggl\\lVert \\frac{\\partial\\mathbf{m}}{\\partial y} \\biggr\\rVert}^2 \\biggr] \\, dx \\, dy\\text{,}\\\\\nE_\\text{ani}[\\mathbf{m}(x,y)]& = \\int  \\! \\left[ -K_1 {(\\mathbf{m} \\cdot \\hat{\\mathbf{z}})}^2 + K_2 {(\\mathbf{m} \\cdot \\hat{\\mathbf{x}} )}^2 \\right] \\, dx \\, dy\\text{,}\\\\\nE_\\text{Zee}[\\mathbf{m}(x,y)]& = \\int \\! - M_\\text{S} \\mathbf{H}_\\text{ext} \\cdot \\mathbf{m}\\, dx \\, dy\\text{,}\n\\end{align}\n\\end{subequations}\nwhere $\\mathbf{m}$ represents as before the \\emph{reduced} magnetization ($\\lVert\\mathbf{m}(x,y)\\rVert = 1$).\nIf the characteristic length scale of the magnetization profile, set by the exchange length $l_1$, becomes comparable to $a$, the continuum energy functionals~\\eqref{eq:contenfunc} are no longer a good representation of the atomistic Hamiltonian and correction terms are needed.\nFor our purposes, the most important correction is the Peierls{} potential, which breaks translational symmetry. The Peierls{} potential vanishes exponentially fast in $l_1 \/ a$.\nHowever, there are also corrections to $E_\\text{ex}$ that are \\emph{algebraically} small. These can be expressed in terms of the higher-order spatial derivatives of $\\mathbf{m}(x,y)$.\nThe dominant correction is\n\\begin{equation}\\label{eq:fourthexch}\nE_\\text{ex4}[\\mathbf{m}(x,y)] = \\int \\! B \\biggl[ {\\biggl\\lVert \\frac{\\partial^2\\mathbf{m}}{{\\partial x}^2} \\biggr\\rVert}^2 + {\\biggl\\lVert \\frac{\\partial^2\\mathbf{m}}{{\\partial y}^2} \\biggr\\rVert}^2 \\biggr] \\, dx \\, dy\\text{.}\n\\end{equation}\nWhile the usual exchange parameter $A$ is proportional to the second moment of the atomistic exchange kernel, the parameter $B$ is proportional to the fourth moment. For nearest-neighbor exchange in a square lattice, we derive\n\\begin{equation}\\label{eq:secondexch}\nB = - a^2 A \/ 12\n\\text{.}\n\\end{equation}\nEnergy terms such as \\equaref{eq:fourthexch} do not affect the qualitative features of domain-wall kinks, but they do give certain corrections to kink parameters. Since $l_1 \\sim a$, we may not assume that such corrections can be neglected.\nNotice that \\equaref{eq:secondexch} is specific to systems with only nearest-neighbor exchange; crystals with wider exchange kernels are likely to have a larger parameter $B\/A$, so that the corrections derived here may be more significant.\n\n\n\\subsubsection{Derivation of the domain-wall Hamiltonian}\\label{sec:dwham}\n\nWe now write the Hamiltonian in terms of the collective coordinates $x_0(y),\\vartheta(y)$ of the domain wall. We aim to derive the effective Hamiltonian of the domain wall from the atomistic Hamiltonian in a systematic way, and we apply the approximation that $\\vartheta$ is small only as a final step. In this section, we write $x_0$ for the position of the center of the domain wall to distinguish it from the spatial coordinate $x$.\n\nLet us express the reduced magnetization $\\mathbf{m}(x,y)$ in spherical coordinates,\n\\begin{equation}\\label{eq:coordsys}\n\\mathbf{m} = \\sin(\\theta)\\cos(\\phi)\\hat{\\mathbf{x}} + \\sin(\\theta)\\sin(\\phi)\\hat{\\mathbf{y}} + \\cos(\\theta)\\hat{\\mathbf{z}}\\text{.}\n\\end{equation}\nNotice that we use $\\vartheta$ to denote the collective coordinate of the domain wall, while we use $\\theta$ for the polar angle of the magnetization $\\mathbf{m}$ at a given point in space.\nWe assume that $\\phi(x,y) = \\vartheta(y) - \\pi \/ 2$; the azimuthal angle $\\phi$ is constant in $x$ and depends only on $\\vartheta(y)$. As a result, a domain wall with $\\vartheta \\neq 0$ is equivalent to a Bloch{} domain wall ($\\vartheta = 0$) but with a modified anisotropy.\nThe effective anisotropy $K$ that the domain wall experiences is given by\n\\begin{equation}\nK = K_1+K_2\\sin^2 \\vartheta \\text{,}\n\\end{equation}\nand we can define an effective `exchange length'\n\\begin{equation}\nl(\\vartheta) = \\sqrt{\\frac{A}{K_1+K_2\\sin^2 \\vartheta}} = \\frac{l_1}{\\sqrt{1+(K_2\/K_1)\\sin^2 \\vartheta}}\\text{.}\n\\end{equation}\nWe may now write the Hamiltonian of the domain wall, very generally, as\n\\begin{equation}\\label{eq:effhamfull}\n\\mathcal{H}[x_0(y),\\vartheta(y)] =  \\int \\! \\frac{4A}{l(\\vartheta)} \\biggl[ s_0(x_0,l(\\vartheta)) +  \\frac{1}{2} s_x(x_0,l(\\vartheta)) {\\left( \\frac{\\partial x_0}{\\partial y} \\right)}^2 + \\frac{l(\\vartheta)^2}{2} s_\\vartheta(x_0,l(\\vartheta)) {\\left( \\frac{\\partial \\vartheta}{\\partial y} \\right)}^2 + \\frac{\\pi^2}{24} s_l(x_0,l(\\vartheta)) {\\left( \\frac{\\partial l(\\vartheta)}{\\partial y} \\right)}^2 \\biggr] \\, dy \\text{.}\n\\end{equation}\nHere $s_0$, $s_x$, $s_\\vartheta$, and $s_l$ are dimensionless functions of $x_0\/a$ and $l\/a$ that remain to be determined.\nThe characteristic length scale on which the domain-wall variables $x_0(y),\\vartheta(y)$ vary in $y$ is $w$. Since we do not take into account higher-order derivatives in $y$, the Hamiltonian~\\eqref{eq:effhamfull} is valid up to corrections of $\\mathcal{O}((w\/a)^{-2})$. This is acceptable because $w \\gg a$ while the dominant energy scale $\\lambda$ is of $\\mathcal{O}((w\/a)^{-1})$.\nFor all four functions $s_0$, $s_x$, $s_\\vartheta$, and $s_l$, the dependence on $x_0$ vanishes as $\\sim e^{-\\pi^2 l\/a} \\sim \\mathcal{O}{(w\/a)^{-2}}$. As the factors $(\\partial \\cdot \/ \\partial y)^2$ are already of order $\\mathcal{O}{(w\/a)^{-2}}$, we may neglect the $x_0$-dependence of $s_x$, $s_\\vartheta$, and $s_l$.\nFor $s_0$, we write\n\\begin{equation}\ns_0(x_0,l) = s_0(l) - v(l)\\cos\\frac{2 \\pi x_0}{a}\\text{,}\n\\end{equation}\nwhich defines the Peierls{} potential. Higher Fourier{} components may be neglected (see main text).\nIn conclusion, we get\n\\begin{multline}\\label{eq:effhamfullpeierls}\n\\mathcal{H}[x_0(y),\\vartheta(y)] =  \\int \\! \\frac{4A}{l(\\vartheta)} \\biggl[\n    s_0(l(\\vartheta))\n  + \\frac{1}{2} s_x(l(\\vartheta)) {\\left( \\frac{\\partial x_0}{\\partial y} \\right)}^2\n  - v(l(\\vartheta))\\cos\\frac{2 \\pi x_0}{a}\\\\\n  + \\frac{l(\\vartheta)^2}{2} s_\\vartheta(l(\\vartheta)) {\\left( \\frac{\\partial \\vartheta}{\\partial y} \\right)}^2\n  + \\frac{\\pi^2}{24} s_l(l(\\vartheta)) {\\left( \\frac{\\partial l(\\vartheta)}{\\partial y} \\right)}^2 \\biggr] \\, dy \\text{.}\n\\end{multline}\nIf we expand to second order in $\\vartheta$ and neglect any corrections of $\\mathcal{O}({(w\/a)}^{-2})$, \\equaref{eq:effhamfullpeierls} becomes\n\\begin{equation}\\label{eq:hamfullapprox}\n\\mathcal{H} \\approx \\frac{4A}{l_1} \\int \\! \\biggl[ s_0(l_1) + \\frac{1}{2}\\frac{K_2}{K_1}(s_0(l_1) - l_1 s'_0(l_1)) \\vartheta^2 + \\frac{1}{2}s_x(l_1) {\\biggl( \\frac{\\partial x_0}{\\partial y} \\biggr)}^2\n - v(l_1) \\cos\\frac{2\\pi x_0}{a} + \\frac{l_1^2}{2}s_\\vartheta(l_1) {\\biggl( \\frac{\\partial \\vartheta}{\\partial y} \\biggr)}^2 \\biggr] \\, dy\n\\text{.}\n\\end{equation}\n\\Equaref{eq:hamfullapprox} is equivalent to the domain-wall Hamiltonian given in the main text except for the correction factors $(s_0(l_1) - l_1 s'_0(l_1))$, $s_x(l_1)$, and $s_\\vartheta(l_1)$.\nBy definition, $V_0 = \\epsilon_1 v(l_1)$.\n\n\\subsubsection{Perturbative calculation of the functions $s_0(l)$, $s_x(l)$, $s_\\vartheta(l)$, and $s_l(l)$}\n\nLet us consider a planar domain wall with collective coordinates $x_0,\\vartheta$ and effective exchange length $l$. As is well known, in spherical coordinates~\\eqref{eq:coordsys}, its equilibrium magnetization profile $\\mathbf{m}(x)$, taking into account only the continuum energy functionals~\\eqref{eq:contenfunc}, is given by\n\\begin{subequations}\n\\begin{align}\n\\theta(x)& = 2\\arctan\\Bigl[\\exp\\Bigl( \\frac{x-x_0}{l} \\Bigr)\\Bigr] \\text{,}\\\\\n\\phi(x)& = \\vartheta - \\pi \/ 2 \\text{.}\n\\end{align}\n\\end{subequations}\nWe derive the effect of the fourth-order exchange term~\\eqref{eq:fourthexch} on the equilibrium magnetization profile in first-order perturbation theory.\nIn spherical coordinates~\\eqref{eq:coordsys} and under the assumption that $\\phi$ is constant, we have the identities ${\\lVert \\mathbf{m}' \\rVert}^2 = {(\\theta')}^2$ and ${\\lVert \\mathbf{m}'' \\rVert}^2 = {(\\theta')}^4 + {(\\theta'')}^2$,\nwhere the prime denotes the derivative in $x$. (For constant $\\theta$, we have ${\\lVert \\mathbf{m}' \\rVert}^2 = {(\\phi')}^2 \\sin^2 \\theta$.)\nFor a planar profile, the continuum energy terms~\\eqref{eq:contenfunc} become\n\\begin{subequations}\n\\begin{align}\nE_\\text{ex}& = \\int \\! A {(\\theta')}^2 \\, dx \\text{,}\\\\\nE_\\text{ani}& = \\int  \\! K \\sin^2 \\theta \\, dx\\text{,}\n\\end{align}\n\\end{subequations}\nwhile the fourth-order exchange term~\\eqref{eq:fourthexch} becomes\n\\begin{equation}\nE_\\text{ex4} = \\int \\! B [ {(\\theta')}^4 + {(\\theta'')}^2 ] \\, dx\n\\text{.}\n\\end{equation}\nWe find that, to first order in $B$, the equilibrium configuration is given by\n\\begin{equation}\n\\theta(x) = 2\\arctan\\Bigl[\\exp\\Bigl( \\frac{x-x_0}{l} \\Bigr)\\Bigr] + \\frac{B}{Al^2}\\,\\theta_1\\Bigl(\\frac{x-x_0}{l}\\Bigr)\\text{,}\n\\end{equation}\nwhere\n\\begin{equation}\n\\theta_1(\\zeta) = -3\\tanh\\zeta\\sech\\zeta + \\tfrac{1}{2}\\zeta\\sech\\zeta\\text{.}\n\\end{equation}\n\nWe calculate the coefficients in \\equaref{eq:hamfullapprox} from the (perturbed) magnetization profile of the planar domain wall.\nNotice that the functions $s_0(l)$, $s_x(l)$, $s_\\vartheta(l)$, and $s_l(l)$ are normalized in such a way that they approach $1$ in the continuum limit $B\/(Al^2)\\rightarrow0$.\nWe have\n\\begin{subequations}\n\\begin{align}\ns_0& = \\frac{l}{4A} \\int \\! (A {(\\theta')}^2 + K\\sin^2 \\theta + B[{(\\theta')}^4 + {(\\theta'')}^2] ) \\, dx \\text{,}\\\\\ns_x& = \\frac{l}{2} \\int \\! {\\left\\lVert\\frac{\\partial\\mathbf{m}}{\\partial x_0}\\right\\rVert}^2 \\, dx = \\frac{l}{2} \\int \\! {(\\theta')}^2 \\, dx\n\\text{,}\\\\\ns_\\vartheta& = \\frac{1}{2l} \\int \\! {\\left\\lVert\\frac{\\partial\\mathbf{m}}{\\partial \\vartheta}\\right\\rVert}^2 \\, dx = \\frac{1}{2l} \\int \\! \\sin^2\\theta \\, dx\n\\text{,}\\\\\ns_l& = \\frac{6l}{\\pi^2} \\int \\! {\\left\\lVert\\frac{\\partial\\mathbf{m}}{\\partial l}\\right\\rVert}^2 \\, dx = \\frac{6}{\\pi^2 l} \\int \\! {(x-x_0)}^2 {(\\theta')}^2 \\, dx\n\\text{,}\n\\end{align}\n\\end{subequations}\nwhere we treat $l$ as a constant when taking the derivative in $\\vartheta$.\nWe get\n\\begin{subequations}\n\\begin{align}\ns_0(l)& = 1 + \\tfrac{1}{2} (B\/A)l^{-2} + \\mathcal{O}{({(l\/a)}^{-4})} \\text{,} \\\\\ns_x(l)& = 1 - \\tfrac{3}{2} (B\/A)l^{-2} + \\mathcal{O}{({(l\/a)}^{-4})} \\text{,} \\\\\ns_\\vartheta(l)& = 1 + \\tfrac{3}{2} (B\/A)l^{-2} + \\mathcal{O}{({(l\/a)}^{-4})} \\text{,} \\\\\ns_l(l)& = 1 - (24\/\\pi^2 + \\tfrac{1}{2}) (B\/A)l^{-2} + \\mathcal{O}{({(l\/a)}^{-4})} \\text{.}\n\\end{align}\n\\end{subequations}\n\n\n\\subsubsection{Corrections to kink parameters}\n\nFor convenience, let us define\n\\begin{subequations}\n\\begin{align}\nS_0& = s_0(l_1) - l_1 s'_0(l_1) = 1 + \\tfrac{3}{2} (B\/A) l_1^{-2} + \\mathcal{O}(l_1\/a)^{-4} \\text{,}\\\\\nS_x& = s_x(l_1) = 1 - \\tfrac{3}{2} (B\/A) l_1^{-2} + \\mathcal{O}(l_1\/a)^{-4} \\text{,}\\\\\nS_\\vartheta& = s_\\vartheta(l_1) = 1 + \\tfrac{3}{2} (B\/A) l_1^{-2} + \\mathcal{O}(l_1\/a)^{-4} \\text{,}\n\\end{align}\n\\end{subequations}\nwhich are the correction factors that appear in \\eqref{eq:hamfullapprox}. \\Equaref{eq:secondexch} gives $S_0 \\approx 1 - \\tfrac{1}{8} (l_1\/a)^{-2}$, $S_x \\approx 1 + \\tfrac{1}{8} (l_1\/a)^{-2}$, and $S_\\vartheta \\approx 1 - \\tfrac{1}{8} (l_1\/a)^{-2}$ for a square lattice with nearest-neighbor exchange.\n\nAs for the statical kink parameters, the Hamiltonian \\eqref{eq:hamfullapprox} gives us a sine-Gordon kink solution (see main text) with\n\\begin{equation}\nw = \\frac{a}{2}\\sqrt{\\frac{S_x}{v(l_1)}} = S_x^{1\/2} \\, w_\\text{cont} \\text{,}\n\\end{equation}\nwhere the subscript ${}_\\text{cont}$ denotes the value in the continuum model with only second-order exchange, as used in the main text.\nThe kink energy is given by\n\\begin{equation}\n\\lambda = \\frac{4a}{\\pi} \\epsilon_1 \\sqrt{S_x v(l_1)} = S_x^{1\/2} \\, \\lambda_\\text{cont} \\text{.}\n\\end{equation}\n\n\\begin{figure}\n  \\includegraphics[width=\\figwidthhalf]{smatfig02effmassC}\n  \\caption{\\label{fig:effmassfull}The analytical expression~\\eqref{eq:meffcorr} for the effective mass $m_\\text{eff}$ corrected for the effect of fourth-order exchange (solid black lines) is in almost perfect agreement with the numerical results for the atomistic model (symbols). The dotted lines show the uncorrected analytical expression given in the main text, which differs by $\\sim 5\\%$.\nThe dashed line shows the effective mass in the solitonic limit.\n  We find that the (corrected) analytical expression deviates from the numerically calculated values only in the regime $w \\lesssim 10a$ (not shown). This is not surprising because all our calculations assume $w \\gg a$, as is always verified unless the Peierls{} potential is extremely strong.}\n\\end{figure}\nAs for the dynamical kink parameters, we find\n\\begin{equation}\\label{eq:meffcorr}\nw m_\\text{eff} = \\frac{2 a^2 M_\\text{S}^2 f(w\/l_2)}{\\pi \\gamma^2 K_2 l_1 S_0} = \\frac{f(w\/l_2)}{f({(w\/l_2)}_\\text{cont})} S_0^{-1} \\, {(w m_\\text{eff})}_\\text{cont}\\text{,}\n\\end{equation}\nwhere\n\\begin{equation}\nl_2 = l_1 \\sqrt{\\frac{K_1}{K_2}} \\sqrt{\\frac{S_\\vartheta}{S_0}} = S_0^{-1\/2} S_\\vartheta^{1\/2} \\, l_{2,\\text{cont}} \\text{.}\n\\end{equation}\n\\Figref{fig:effmassfull} shows that this correction to the effective mass $m_\\text{eff}$ is small but significant. With the corrections, the agreement of our analytical expression to our numerical results is almost perfect.\nThe characteristic time scale of the approximate sine-Gordon description is given by\n\\begin{equation}\nT = \\frac{a M_\\text{S}}{\\pi |\\gamma| \\epsilon_1} \\sqrt{\\frac{K_1}{v(l_1) K_2 S_0}} = S_0^{-1\/2} \\, T_\\text{cont} \\text{.}\n\\end{equation}\nThe final velocity (in the linear regime, where $\\vartheta_0 \\ll 1$ and $v_\\text{final} \\ll Y \/ T$) is given by\n\\begin{equation}\n\\frac{v_\\text{final}}{w} = S_x^{-1} \\, {\\Bigl(\\frac{v_\\text{final}}{w}\\Bigr)}_\\text{cont} \\text{.}\n\\end{equation}\n\n\n\n\\subsection{Strength of the Peierls{} potential}\n\nThe kink width $w$ depends on the strength $V_0$ of the Peierls{} potential, which in turn depends on the ratio between the domain-wall width and the lattice periodicity $a$.\nFor the biaxial type of anisotropy defined in the main text, the equilibrium domain-wall width for a fixed value of $\\vartheta$ is given by $\\pi \\sqrt{A \/ (K_1+K_2\\sin^2 \\vartheta)}$. For a Bloch{} domain wall ($\\vartheta=0$), this becomes $\\pi l_1 = \\pi \\sqrt{A \/ K_1}$ \\cite{Brown1963}.\nSince we derive all analytical results in the limit of small $\\vartheta$, $V_0$ can be taken as constant (see also \\secref{sec:dwham}). It must be some dimensionless function of $l_1\/a$ times the Bloch{} domain-wall energy $\\epsilon_1 = 4\\sqrt{A K_1}$.\n\nFor a line of fixed $y = (j+\\tfrac{1}{2})a$, we define the center $x$ of the domain wall as\n\\begin{equation}\nx = \\frac{a}{2} \\sum_i (\\mathbf{m}_{ij} \\cdot \\hat{\\mathbf{z}})\\text{,}\n\\end{equation}\nwhich defines a position relative to the middle of the sample.\nThe magnetic Peierls{} potential $V(x)$ can be obtained by minimizing the atomistic Hamiltonian of some configuration with a domain wall under the constraint of a fixed value of $x$.\nWe find, in agreement with previous considerations \\cite{vandenBroek1971, Hilzinger1972},\n\\begin{equation}\\label{eq:peierlspot}\nV(x) = V_0  \\left[ 1 - \\cos \\frac{2\\pi x}{a} \\right]\\text{,}\n\\end{equation}\nwhere $V$ is an energy per unit area.\nWe have $V_0 \\sim \\epsilon_1 e^{-\\pi^2 l_1 \/ a}$ \\cite{Hilzinger1972}.\nFor a noticeable Peierls{} relief, we must have $l_1 \\lesssim 2.5a$.\nHigher Fourier{} components of $V(x)$ decrease even faster in $l_1\/a$ \\cite{Kosevich2005}, so that \\equaref{eq:peierlspot} is expected to describe the Peierls{} potential almost perfectly unless $l_1 \\lesssim 1.2a$. \nIt is easy to see that an external field $\\mathbf{H}_\\text{app} = H_z \\hat{\\mathbf{z}}$ results in an additional potential $V_{H}(x) = -2 M_\\text{S} H_z x$. Combining these two expressions gives a coercive field $H_\\text{c} = \\pi V_0 \/ (M_\\text{S} a)$ \\cite{Egami1973A}.\nThis provides an alternative way to determine numerically the strength $V_0$ of the Peierls{} potential for a given atomistic model.\n\nFor a domain wall in a $\\lbrace 1 0 0 \\rbrace$ plane of the simple cubic lattice with nearest-neighbor exchange, we find that $V_0$ depends on the domain-wall width $l_1$ as\n\\begin{equation}\\label{eq:peierlsstrength}\nV_0 = (P\\frac{l_1}{a} + Q) \\epsilon_1 \\, e^{-\\pi^2 l_1 \/ a}\\text{.}\n\\end{equation}\nA fit of the coercive fields gives $P \\approx 181$ and $Q\\approx-36$.\nOur expression \\eqref{eq:peierlsstrength} is equivalent to previous results \\cite{Hilzinger1972,Novoselov2003} if we set $Q=0$ and $P=C\/(4\\pi)$.\nWe find that the refinement $Q\\neq0$ is significant and makes the fit to the numerical results almost perfect.\nIf we consider a system with nearest-neighbor and next-nearest-neighbor exchange, \\equaref{eq:peierlsstrength} remains valid but $P$ and $Q$ take different values. The Peierls{} relief might thus be useful as a probe for the strength and type of the exchange interaction on the atomic scale.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\figwidthhalf]{smatfig03kinkwidths}\n  \\caption{\\label{fig:kinkwidth}Kink width $w$ versus exchange length $l_1$. Symbols:~Obtained by numerical minimization of the atomistic Hamiltonian (specified in the main text) starting from a spin configuration containing a domain wall with a single kink. The kink width is extracted from the domain-wall profile $x(y)$ in the final, relaxed configuration. Curves:~Calculated from \\equaref{eq:peierlsstrength} using \\equaref{eq:kinkwidth}, where the parameters $P$ and $Q$ are independently obtained from a fit to the coercive field.\n  Notice that here the corrections to \\equaref{eq:kinkwidth} that result from higher-order terms of the exchange energy (see \\secref{sec:hoexch}) are tiny.\n  }\n\\end{figure}\nGiven $V_0$, we can calculate the kink width $w$ using the relation \\cite{Egami1973A}\n\\begin{equation}\\label{eq:kinkwidth}\nw = \\tfrac{1}{2} a \\sqrt{\\epsilon_1 \/ V_0}\\text{.}\n\\end{equation}\n\\Figref{fig:kinkwidth} shows that the kink widths calculated from \\equaref{eq:peierlsstrength} agree with the results found by numerical relaxation of a domain wall with a kink.\n\n\n\\subsection{Adjoint forms of the kink collective coordinates}\n\nIn the main text, it is derived that the collective coordinates $y_0,p$ of the kink are given by\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial x(y) }{ \\partial y_0}& = -\\frac{a}{w} \\sech \\frac{\\pi y}{ w } \\text{,}\\\\\n\\frac{\\partial \\vartheta(y) }{ \\partial y_0}& = 0, \\\\\n\\frac{\\partial x(y) }{ \\partial p}& = 0\\text{,}\\\\\n\\frac{\\partial \\vartheta(y)}{\\partial p}& = \n- \\frac{a M_\\text{S}}{2 |\\gamma| K_2 l_1 m_\\text{eff} w} {\\Bigl(  1 - l_2^2 \\frac{d^2}{{dy}^2} \\Bigr)}^{-1} \\sech \\frac{\\pi y}{w} \\text{.}\n\\end{align}\n\\end{subequations}\nFor completeness, we mention that, owing to the symplectic structure \\cite{Buijnsters2014} on the domain-wall coordinates $x(y),\\vartheta(y)$, these expressions immediately imply a `direct' definition of $y_0$ and $p$, namely\n\\begin{subequations}\\label{eq:direct}\n\\begin{align}\ny_0& = - \\frac{\\pi}{2a f(w\/l_2)}  \\int \\! \\Delta x(y) {\\Bigl(  1 - l_2^2 \\frac{d^2}{{dy}^2} \\Bigr)}^{-1} \\sech \\frac{\\pi y}{w} \\, dy \\text{,}\\\\\n\\label{eq:pdirect}\np& = - \\frac{2 a M_\\text{S}}{|\\gamma| w} \\int \\! \\vartheta(y)  \\sech \\frac{\\pi y}{w} \\, dy\n \\text{.}\n\\end{align}\n\\end{subequations}\nThis definition is valid up to first order in the deviations $\\Delta x(y), \\vartheta(y)$ from the equilibrium kink configuration (for which $\\vartheta(y)=0$ identically).\nWe have $\\Delta x(y) = x(y) - (2a\/\\pi) \\arctan [ \\exp  (\\pi y \/ w)  ]$.\nA direct definition of collective coordinates is useful when deriving their equations of motion under some non-Hamiltonian perturbation such as Gilbert damping.\nNotice that \\equaref{eq:pdirect} gives\n\\begin{equation}\n\\vartheta_0 = \\frac{\\pi |\\gamma|}{4 a M_\\text{S}} \\, p = - \\frac{\\pi}{2w} \\int \\! \\vartheta(y)  \\sech \\frac{\\pi y}{w} \\, dy \\text{.}\n\\end{equation}\n\n\n\\subsection{Special functions}\n\nThe main text introduces the special functions $r(\\omega)$, $f(\\eta)$, $g(\\eta)$, and $h(\\eta)$. For completeness, we summarize their definitions and limiting behavior here. The functions are plotted in \\figrefs{fig:romega} and~\\ref{fig:fgheta}.\nWe have\n\\begin{equation}\\label{eq:defr}\nr(\\omega) = \\frac{1}{16\\sqrt{1-\\omega^2}} \\int_{0}^{\\pi\/\\omega} \\! \\int_{-\\infty}^\\infty \\! {\\left( \\frac{\\partial \\varphi}{\\partial t} \\right)}^2 \\, dy \\, dt \\text{,}\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:breathersol2}\n\\varphi(y,t) = 4 \\arctan \\left[ \\frac{\\sqrt{1 - \\omega^2}}{\\omega} \\sech{\\left( \\sqrt{1 - \\omega^2}y \\right)} \\cos{\\left( \\omega t \\right)} \\right]\n\\end{equation}\nis a breather solution of the sine-Gordon equation, with parameter $\\omega \\in (0,1)$. We have made all variables dimensionless. The solution~\\eqref{eq:breathersol2} is periodic in $t$ with a period of $2\\pi \/ \\omega$.\n\\Equaref{eq:defr} can be evaluated as\n\\begin{equation}\\label{eq:evalr}\nr(\\omega) = \\frac{2\\pi}{\\sqrt{1-\\omega^2}} \\arctan{\\sqrt{\\frac{1-\\omega}{1+\\omega}}} \\text{.}\n\\end{equation}\nIt is easy to see that $r(\\omega)$ is a bounded function with\n\\begin{subequations}\n\\begin{align}\nr(\\omega \\downarrow 0)& = \\pi^2 \/ 2 \\approx 4.935 \\text{,}\\\\\nr(\\omega \\uparrow 1)& = \\pi \\approx 3.142 \\text{.}\n\\end{align}\n\\end{subequations}\n\\begin{figure}\n  \\centering\n  \\subfloat{\n    \\includegraphics[height=0.5\\figwidthhalf]{smatfig04romega}\n  }\n\n  \\caption{\\label{fig:romega}Special function $r(\\omega)$, with limiting values.}\n\\end{figure}%\n\\begin{figure}\n  \\subfloat{\n    \\includegraphics[height=0.5\\figwidthhalf]{smatfig05afeta}\n  }\n  \\subfloat{\n    \\includegraphics[height=0.5\\figwidthhalf]{smatfig05bfgheta}\n  }\n\n  \\caption{\\label{fig:fgheta}Special functions $f(\\eta)$, $g(\\eta)$, and $h(\\eta)$, with limiting behavior.}\n\\end{figure}%\nFor $\\eta > 0$ we define\n\\begin{equation}\nf(\\eta) = \\frac{1}{2} \\int_{-\\infty}^\\infty \\! \\frac{\\sech^2{x}}{1 + 4 x^2 \/ \\eta^2} \\, dx \\text{,}\n\\end{equation}\nwhich has the following limiting behavior:\n\\begin{subequations}\n\\begin{align}\nf(\\eta)& = \\tfrac{1}{4} \\pi |\\eta| + \\mathcal{O}(\\eta^{2}) \\enspace\\text{for $\\eta \\ll 1$}\\text{,}\\\\\nf(\\eta)& = 1 - \\tfrac{1}{3}\\pi^2\/\\eta^{2} + \\mathcal{O}(\\eta^{-4}) \\enspace\\text{for $\\eta \\gg 1$}\\text{.}\n\\end{align}\n\\end{subequations}\nWe also define two related functions\n\\begin{align}\ng(\\eta)& = 2 f(\\eta) {\\left[\\int_{-\\infty}^\\infty \\! \\frac{\\sech^2{x}}{{(1 + 4 x^2 \/ \\eta^2)}^{2}}   \\, dx\\right]}^{-1}\\text{,} \\\\\nh(\\eta)& = \\frac{1}{\\pi f(\\eta)} \\int_{-\\infty}^\\infty \\! \\frac{\\sech{x}}{1+4x^2\/\\eta^2} \\, dx \\text{,}\n\\end{align}\nfor which we have\n\\begin{subequations}\n\\begin{align}\ng(\\eta \\downarrow 0)& = 2 \\text{,}\\\\\ng(\\eta)& = 1 + \\tfrac{1}{3}\\pi^2\/\\eta^{2} + \\mathcal{O}(\\eta^{-4}) \\enspace\\text{for $\\eta \\gg \n1$}\\text{,}\n\\end{align}\n\\end{subequations}\n\\begin{subequations}\n\\begin{align}\nh(\\eta \\downarrow 0)& = 2\/\\pi \\approx 0.6366 \\text{,}\\\\\nh(\\eta)& = 1 - \\tfrac{2}{3}\\pi^2\/\\eta^{2} + \\mathcal{O}(\\eta^{-4}) \\enspace\\text{for $\\eta \\gg 1$}\n\\text{.}\n\\end{align}\n\\end{subequations}\nNotice that $f(\\eta)$, $g(\\eta)$, and $h(\\eta)$ are defined in such a way that they approach~$1$ for large $\\eta$ (solitonic limit).\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nThe Crab~Nebula is a standard source of\nradiation in the northern sky across a wide energy band, from radio to near 100~TeV\n$\\gamma$-rays. It is well known that the multi-wavelength\nnon-thermal energy spectrum is dominated by synchrotron radiation\nat energies lower than 1~GeV and by the inverse-Compton\nscattering above 1~GeV \\cite{Jag96,Ato96}.\n\nTeV $\\gamma$-rays from the Crab~Nebula were first clearly detected\nby the Whipple collaboration using an imaging air Cherenkov\ntelescope (IACT) in 1989 \\cite{Wee89}.  Since then,\nIACT has become the standard telescope for high-energy\n$\\gamma$-ray observations by virtue of its excellent angular resolution\nand efficiency. Many IACTs have been constructed and are operating around\nthe world, detecting about 70 $\\gamma$-ray sources in total up to\nthe present. On the other hand, air shower arrays have been constructed to\nsearch for $\\gamma$-rays from point sources at high altitude. The\nmerit of this technique is that it can be operated for 24 hours\nevery day, regardless of weather, with a wide field of view of about\n2~sr. The energy threshold of the $\\gamma$-rays detected is higher\nthan that of IACTs, say about 3~TeV at high altitude.  Among these\ninstruments, the Tibet AS$\\gamma$ Collaboration achieved the first successful\nobservation of the Crab~Nebula at a multi-TeV region in\n1999, using a so-called HD (high density) array in which 109 scintillation detectors\nwere deployed at 7.5~m spacing lattice intervals in an area of\n5,175~m$^2$ \\cite{Ame99a}.  The Milagro group also reported the\ndetection of TeV $\\gamma$-ray signals from the Crab~Nebula using a\nwater Cherenkov pool \\cite{Atk03}.\n\nRecently, IACTs have obtained updated information on the Crab~Nebula.  The\nHEGRA experiment has extended the nebula's energy spectrum\nup to 80~TeV with an approximate power-law shape, after patient observation\nfor almost 400 hours in total \\cite{Aha04}.  In contrast, the MAGIC\nexperiment, equipped with the world's largest tessellated reflector (17~m\nin diameter), has successfully observed the lower energy part of the spectrum down to\n77~GeV \\cite{Alb08}.  The H.E.S.S. group examined the energy spectra\nof the Crab~Nebula obtained from various IACTs (Whipple, HEGRA, CAT\nand H.E.S.S.) to evaluate the systematic errors of these instruments\n\\cite{Aha06}.  Their fluxes are well in agreement with one another\nwithin the statistical and systematic errors at the moderate energy\nregion, although the cutoff energy and spectral index seem to\ndiffer somewhat.  Thus, the Crab~Nebula has been well studied by\nvarious techniques and has been used as a standard calibration source\namong the ground-based $\\gamma$-ray experiments in the TeV region.\n\nThe energy of a primary cosmic-ray particle is estimated by observing\nthe number of secondary particles in an air shower experiment as well as the\nnumber of Cherenkov photons for IACTs, and then by comparing these values against the results of detailed Monte\nCarlo simulations, including the detector structure and response.  The\nmost conventional method for estimating the absolute energy scale of\nprimary particles may be to compare the flux values between the direct\n(satellite\/balloon-borne) and indirect observations. The cosmic-ray\nflux, however, depends inevitably on the detection technique,\nanalysis method, detector simulation, and so on.  Therefore, it is\nvery important to develop a new method for directly estimating\nthe absolute energy scale in an air shower experiment at TeV energies.\n\nClark \\cite{Cla57} anticipated in 1957 that the sun and the moon, each with\na finite size of $0\\fdg5$ in diameter, cast shadows in the high-energy \ncosmic-ray flux, respectively.  Actually, the shadowing effect of the sun and\nthe moon (hereafter, we call these the sun's shadow and the moon's\nshadow, respectively) was observed by air shower experiments in\nthe 1990s \\cite{Ale91, Ame93}, and the sharpness of the observed\nshadows was used to estimate their angular resolutions experimentally.  In\nparticular, the Tibet air shower array, with its high event trigger rate\nand good angular resolution, enables us to use the geomagnetic field as\na magnetic spectrometer for primary cosmic rays at multi-TeV energies.\nAs almost all primary cosmic rays are positively charged, they are\nbent eastward by the geomagnetic field at Yangbajing in Tibet.  The\nmoon's shadow should then be observed in the\nwest of the moon's apparent position, although the position of the shadow\nin relation to that of the moon depends on the\ncosmic-ray energy \\cite{Ame00a}. Hence, the position and the\nshape of the moon's shadow allow us to estimate the possible\nsystematic error in the absolute energy scale of observed showers.\nUntil now, the pointing accuracy and angular resolution of the\nTibet-III array have been checked by monitoring the moon's shadow\ncontinuously month by month \\cite{Ame03}.  It is also worthwhile to\nnote that the moon's and sun's shadows provide information about\nthe cosmic-ray $\\bar{p}\/p$ flux ratio \\cite{Ach05,Ame07a} and the\nglobal structure of the interplanetary magnetic field between the sun\nand the earth \\cite{Ame93,Ame94,Ame99b,Ame00a}.\n\nIn this paper, we first discuss the systematic uncertainties of the\nTibet-III array and a new method for calibrating the absolute energy\nof primary particles in the multi-TeV energy region using the moon's\nshadow data.  Based on the results, we report on the $\\gamma$-ray\nobservation of the Crab~Nebula and search for pulsed $\\gamma$-ray\nemissions from the Crab pulsar using the dataset obtained by the\nTibet-III array.\n\n\\section{EXPERIMENT}\n\n\\subsection{Tibet-III Air Shower Array}\n\n\nThe Tibet-III array shown in Figure~\\ref{fig1} was completed on the\nbasis of the success of the Tibet-I, II and II\/HD experiments\n\\citep{Ame92, Ame99a,Ame00b} in the late fall of 1999.  This array\nconsists of 533 scintillation detectors of 0.5~m$^{2}$, and the detectors\non the inner side of the array are placed on a lattice with 7.5 m\nspacing, covering 22,050~m$^2$.  A lead plate 0.5 cm thick is placed on\ntop of each detector to improve the angular resolution.  Using this\narray, we succeeded in observing $\\gamma$-ray flares from Mrk~421 and\nfound a correlation between TeV $\\gamma$-ray and X-ray intensities\n\\cite{Ame03}.  In 2002 and 2003, the inside area of the\nTibet-III array was further enlarged to 36,900~m$^2$ by adding 256\ndetectors. This full Tibet-III array has been successfully operating\nsince 2003.  In this paper, to keep the form of the data the same throughout the\nobservation period from 1999 to 2005, we reconstructed air shower data\nobtained from the detector configuration shown in Figure~\\ref{fig1}\neven for the full Tibet-III array.\n\n\\begin{figure}\n\\epsscale{1.10}\n\\plotone{f1.eps}\n\\caption{ Schematic view of the Tibet-III array operating at\n  Yangbajing.  Open squares: FT detectors equipped with a fast-timing (FT)\n  photomultiplier tube (PMT); filled squares: FT detectors with a wide\n  dynamic-range PMT; filled circles: density detectors with\n  a wide-dynamic range PMT. We have selected air shower events whose cores\n  are located within the detector matrix enclosed by the dotted line.}\n\\label{fig1}\n\\end{figure}\n\n\n\\subsection{Event Reconstruction}\n\nThe raw data obtained from the Tibet-III array system mainly consist\nof the following: a trigger time stamp for each event from a global positioning system (GPS)\nclock supplemented by a computer clock; timing and charge information from\neach hit PMT, digitized by a time-to-digital converter\n(TDC) and a charge-sensitive analog-to-digital converter (ADC); and\ncalibration data taken every 20 minutes.  The ADC and TDC counts are then\nconverted to the number of particles and the relative timing for each\ndetector, respectively, using the calibration data. An air\nshower event is reconstructed as follows.\n\nThe core position of each air shower is estimated using the lateral\ndistribution of the number of shower particles observed in the\narray. The density-weighted position of the air shower core on the surface\nof the Tibet-III array is calculated as $(X_{\\rm core}, Y_{\\rm\n  core})~=~\\left(\\frac{\\sum_{i}~\\rho_{i}^{2}~x_{i}}{\\sum_{i}~\\rho_{i}^{2}},\n\\frac{\\sum_{i}~\\rho_{i}^{2}~y_{i}}{\\sum_{i}~\\rho_{i}^{2}}\\right)$,\nwhere $x_{i}$ and $y_{i}$ are the coordinates of the {\\it i}-th detector and\n$\\rho_{i}$ is the number density (m$^{-2}$) of detected particles.  In\nthis analysis, we regard the shower size $\\sum\\rho_{\\rm FT}$ as the\nprimary energy reference, where size $\\sum\\rho_{\\rm FT}$ is\ndefined as the sum of the number of particles per m$^2$ for each FT\ndetector.  For $\\gamma$-ray-induced air showers, overall core position resolutions\nare then estimated as 12~m and 4~m at median for $\\sum\\rho_{\\rm FT} < 100$ and\n$\\sum\\rho_{\\rm FT} > 100$, respectively.\n\n\nThe arrival direction of each shower is estimated assuming that the front of\nthe air shower is conical shape.  The apex of a cone is then taken to be\nthe estimated core position $(X_{\\rm core}, Y_{\\rm core})$.  The\naverage delay time $T$ (ns) of shower particles is expressed as a\nfunction of the distance $R$ (m) from the core position as $T = 0.075\nR$, which is optimized by simulations.  This gives a\ncone slope of $1\\fdg3$ with respect to the plane perpendicular to the\nair shower arrival direction.\n\n\n\\subsection{Event Selection} \\label{s-2.3}\n\nAn event trigger signal is issued when an any-fourfold coincidence\nappears in the FT detectors that have each recorded more than 0.6 particles within a\ncoincidence gate width of 600~ns, resulting in a trigger rate of\nabout 680~Hz.  We collected 2.0$\\times$10$^{10}$ events during 1318.9\nlive days from 1999 November 18 through 2005 November 15 after some\nquality cuts and event selection based on three simple\ncriteria: (1) each shower event should fire four or more FT detectors\nthat have each recorded 1.25 or more particles; (2) among the 9 hottest FT detectors\nin each event, 8 should be contained in the fiducial area enclosed by\nthe dotted line in Figure~\\ref{fig1}.  If fewer than 9 detectors have been hit,\nthey should all be contained in the fiducial area;\nand (3) the zenith angle of the event arrival direction should be less\nthan 40$\\degr$.  After these criteria have been met, the\noverall angular resolution and the modal energy of air shower events,\nthus obtained, are better than 1~degree and about 3~TeV, respectively\n\\cite{Ame03}, thereby covering the upper portion of the energies measured by\nIACTs.\n\n\n\\section{MOON'S SHADOW AND PERFORMANCE OF THE TIBET-III AIR SHOWER ARRAY}\n\n\\subsection{Analysis} \\label{s-3.1}\n\n\nUsing the dataset described in $\\S$\\ref{s-2.3}, we further select the\nevents within a circle of the radius 5$\\degr$ centered at the moon;\nthis circle is defined as the on-source field. An equatorial coordinate\nsystem is defined, fixing the origin of coordinates at the moon's\ncenter.  To estimate the background against deficits in the moon's\nshadow, we adopt the equi-zenith angle method \\cite{Ame03,Ame05}, which\nis also used for the Crab~Nebula observation described in\n$\\S$\\ref{s-4.1}.  Eight off-source fields are symmetrically aligned on\nboth sides of the on-source field, at the same zenith angle. In order\nto avoid deficit events that are affected by background event contamination \nof the on-source field, the nearest two off-source fields\nare each set at an angular distance $9\\fdg6$ from the on-source field.  Other\noff-source fields are located every $3\\fdg2$ from the nearest\noff-source fields.  The position of each observed event in\nan on-\/off-source field is then specified by the angular distance\n$\\theta$ and the position angle $\\phi$, where $\\theta$ and $\\phi$ are\nmeasured from the center and from the north direction, respectively.\nUsing $\\theta$ and $\\phi$, the on-\/off-source fields are meshed by\n$0\\fdg05$$\\times$$0\\fdg05$ cells, and we count the number of events in\neach cell.  To maximize the $S\/N$ ratio, we group the cells into new\non-\/off-source bins according to the angular resolution, which depends\non the shower size $\\sum\\rho_{\\rm FT}$.  The angular resolution\nbecomes worst at a threshold energy of around 1~TeV. This value is,\nhowever, smaller than half of the angular distance between two\nadjoining off-source bins, i.e., $\\sim1\\fdg6$, so that off-source\nbins never overlap mutually in this background analysis.\n\nWe calculate the statistical significance of deficits or signals using\nthe formula \\cite{Li83} $(N_{\\rm ON} - \\epsilon N_{\\rm\n  OFF}) \/ \\sqrt{\\epsilon (N_{\\rm ON} + N_{\\rm OFF})}$, where $N_{\\rm\n  ON}$ and $N_{\\rm OFF}$ are the number of events in the\non-source bin and the number of background events summed over 8\noff-source bins, respectively, and  $\\epsilon$ is the ratio of the on-source solid angle area to\nthe off-source solid angle area ($\\epsilon$ = 1\/8 in this work).\n\nIn order to investigate the energy dependence, the shower size\n$\\sum\\rho_{\\rm FT}$ is divided by 1\/4 decades in 10$<$$\\sum\\rho_{\\rm\n  FT}$ $\\leq$100, and by 1\/3 decades in 100$<$$\\sum\\rho_{\\rm\n  FT}$$\\leq$1000, where the lowest air shower size bin\nis omitted from the analysis, because it is close to the energy\nthreshold of the Tibet-III array and the trigger efficiency is\nestimated to be very low ($<$1\\%).  Hereafter, this partition is\ncommonly used in observations of both the moon's shadow and the\nCrab~Nebula.\n\nFigure~\\ref{fig2} shows the experimental significance map of the deficit event\ndensities observed with the Tibet-III array for 1318.9 live days.\nThis map is smoothed using the events with $\\sum\\rho_{\\rm\n  FT}$$>$10$^{1.25}$ ($>\\sim$2~TeV) within a circle of radius\n$0\\fdg9$, corresponding to the overall angular resolution for these\nevents.  The maximum deficit reaches the significance level of\n45~$\\sigma$ at the center.  It is seen that the center of the observed\nmoon's shadow is shifted to the west by about $0\\fdg2$ due to the\neffect of the geomagnetic field.\n\n\n\n\n\n\\subsection{Monte Carlo Simulation of the Moon's Shadow} \\label{s-3.2}\n\n\n\\begin{figure}[t]\n\\epsscale{1.20}\n\\plotone{f2.eps}\n\\caption{ Significance map of the deficit event densities observed by\n  the Tibet-III array for 1318.9 live days, made using the events with\n  $\\sum\\rho_{\\rm FT} >10^{1.25}$ ($>\\sim$2~TeV), in the square area of\n  6$\\degr$$\\times$6$\\degr$ whose origin is at the apparent center of the moon.\n  The scale at right shows the level of significance\n  of the deficit event density in terms of the standard deviation\n  $\\sigma$.}\n\\label{fig2}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\epsscale{1.20}\n\\plotone{f3.eps}\n\\caption{ Deficit event density map obtained by the MC simulation.\n  The events with $\\sum\\rho_{\\rm FT} >10^{1.25}$ ($>\\sim$2~TeV) are\n  plotted in the square area of 6$\\degr$$\\times$6$\\degr$.  The map is\n  made in the same way as in Figure~\\ref{fig2}, and the scale at right\n  represents the deficit event density (degree$^{-2}$).}\n\\label{fig3}\n\\end{figure}\n\n\nWe have performed a detailed Monte Carlo (MC) simulation of the moon's\nshadow.  For the geomagnetic field, we adopt the International\nGeomagnetic Reference Field (IGRF) 9th generation model \\cite{Mac03}\nat an altitude $<$~600~km, and connect it to the dipole moment model at\nan altitude $>$ 600~km.  For the primary particles, we use the chemical\ncomposition obtained mainly from the data of direct observations\n\\cite{Asa98,San00,Apa01,Ame08} in the energy range of 0.3~TeV to\n1000~TeV.  The minimum energy of primary particles is set to 0.3~TeV,\nwhich is low enough to cover the threshold energy of our trigger\ncondition.  Air shower events are generated at the top of the\natmosphere along the moon's orbit around the earth, using the CORSIKA\ncode \\cite{Cor98} with QGSJET or SIBYLL interaction models.  The air\nshower core of each simulated event is uniformly distributed over a\ncircular region with a 300~m radius centered at the array; this circle\nsufficiently covers the area where cosmic-ray events are actually\ntriggered in our array.  Air shower particles generated by primary\nparticles in the atmosphere are traced until their energies reach\n1~MeV.  In order to treat the MC events in the same way as the events in the\nexperimental data, these simulated events are distributed among the detectors\nin the same detector configuration as in the Tibet-III array by the\nEpics code \\cite{Ame08,Kas}, and are converted to the same format as\nthe experimental dataset, such as the ADC and TDC values at each\ndetector.  After air shower reconstruction analysis and data\nselection, we assign the opposite charge to the remaining primary\nparticles.  These anti-particles are randomly shot back toward\ndirections within 20$\\degr$$\\times$20$\\degr$ centered at the moon from\nthe first interaction point of the air shower. Hereafter, we call\nthis the initial shooting direction. The particle track\ninfluenced by the geomagnetic field between the earth and the moon is\ncalculated by the Runge-Kutta method of order 4 based on the Lorentz\nforce.  If the primary particle hits the moon, its initial shooting\ndirection should be equivalent to the observed particle direction\nshielded by the moon.  Otherwise, it is shot back in another\ndirection and the particle track is calculated again. This routine\ncontinues until almost all particles have hit the moon.  Finally, these\ninitial shooting directions are smeared by the angular resolution\nevent by event.  In this way, the expected moon's shadow\nis equivalent to the observed moon's shadow.  Figure~\\ref{fig3}\nshows the event map of the moon's shadow calculated by the MC\nsimulation for events with $\\sum\\rho_{\\rm FT}$$>$10$^{1.25}$\n($>\\sim$3~TeV).  This map is smoothed using the events within a circle\nof radius $0\\fdg9$, corresponding to the overall angular resolution\nfor events with $\\sum\\rho_{\\rm FT}$$>$10$^{1.25}$.  The MC simulation\nwell reproduces the observed moon's shadow, as shown in\nFigure~\\ref{fig2}.\n\n\n\n\\subsection{Shape of the Moon's Shadow and Performance of the Tibet-III Array} \\label{s-3.3}\n\n\n\nThe filled circles in Figure~\\ref{fig4}~(a)--(f) show the observed\ndeficit counts around the moon projected onto the east-west axis for\neach $\\sum\\rho_{\\rm FT}$ bin, where $\\sum\\rho_{\\rm FT}$ is the shower\nsize defined as the sum of the number of particles per m$^2$ for each\nFT detector. The representative cosmic-ray energy in each\n$\\sum\\rho_{\\rm FT}$ bin is defined as be the logarithmic mean of the\nenergy $E$ divided by the atomic number $Z$, assuming the cosmic-ray\ncomposition spectrum described in $\\S$\\ref{s-3.2}.  In these figures,\nit is seen that the peak position of the deficit counts gradually shifts\nto the west as primary energy decreases due to the\ninfluence of the geomagnetic field.  Also, the deficit counts\nbecome narrower as primary energy increases, since the angular\nresolution increases roughly in inverse proportion to\n$\\sqrt{\\sum\\rho_{\\rm FT}}$.  The MC results (histograms) are compared\nwith the experimental data at various size intervals in\nFigure~\\ref{fig4}, and are in good agreement with the experimental data.\n\n\nIn order to estimate the peak position of the observed moon's shadow,\nwe use the shadow shape obtained by the MC simulation.\nWe first express the MC shadows shown in Figure~\\ref{fig4} with\nthe superposition of two Gaussian functions using the least $\\chi^{2}$\nmethod as,\n\\begin{eqnarray}\n\\nonumber\nf_{\\rm MC}(\\theta) = G_{1}(\\theta; a_1, m_1, \\sigma_1) \\\\\n+ G_{2}(\\theta; a_2, m_2, \\sigma_2),\n\\label{equ1}\n\\end{eqnarray}\nwhere $G_{i}(\\theta; a_i, m_i, \\sigma_i) = a_{i}e^{\n  -(\\theta-m_{i})^2\/\\sigma_{i}^2 }$ and $\\theta$ is the angular\ndistance from the moon.  Here, $a_1$, $a_2$, $m_1$, $m_2$, $\\sigma_1$ and\n$\\sigma_2$ are the fitting parameters denoting the amplitudes, means\nand one-standard deviations of the double Gaussian, respectively.  It\nis found that the shadows (e) and (f) can be expressed by a single\nGaussian, while the others are expressed by a double Gaussian.\n\nUsing these fitting parameters, we then estimate the peak position of\nthe observed shadow as follows. Keeping the form of function\n$f_{\\rm MC}(\\theta)$, we express the observed shadow by the\nequation,\n\\begin{eqnarray}\n\\nonumber\nf_{\\rm Data}(\\theta)  =  G_{1}(\\theta;A_1, M_1, \\sigma_1) \\\\\n+ G_{2}(\\theta; A_1\\times(a_2\/a_1), M_1+(m_2-m_1), \\sigma_2), &\n\\label{equ2}\n\\end{eqnarray}\nwhere $A_1$ and $M_1$ denote the amplitude and mean of the Gaussian,\nrespectively, and are free parameters, while the others are the\ncoefficients calculated by fitting equation~(\\ref{equ1}).  Using\nequation~(\\ref{equ2}), we can estimate the peak position of the moon's\nshadow.\n\nWe also confirm that the east-west component of the geomagnetic field strength\nis negligible in the part of the sky where the moon is visible by\nthe Tibet-III array. This means that the north-south displacement of the moon's shadow\nobserved by the Tibet-III array does not depend on the geomagnetic field.\n The displacement of the peak of the moon's shadow in the north-south \ndirection then enables us to estimate\nthe magnitude of the array's systematic pointing error.  Figure~\\ref{fig5}\nshows the energy dependence of the displacement of the moon's shadow\nin the north-south direction.  The filled circles denote the\nexperimental data, and the open squares are the MC results. \nThe MC simulation well reproduces the experimental data.  A\n$\\chi^{2}$ fitting to the data gives $0\\fdg008\\pm0\\fdg011$ assuming a\nconstant function independent of energy.  From this, the systematic\npointing error is estimated to be smaller than $0\\fdg011$.\n\n\\begin{figure*}[!t]\n\\epsscale{1.10}\n\\plotone{f4.eps}\n\\caption{Filled circles show experimental data for deficit counts\n  around the moon projected to the east-west axis for each\n  $\\sum\\rho_{\\rm FT}$ bin.  We use the events contained in the angular\n  band, centered at and parallel to the east-west axis, comparable to\n  the $\\sum\\rho_{\\rm FT}$-dependent angular resolution:\n(a): $\\pm1\\fdg4$ for 2.94~TeV\/$Z$ in $10^{1.25} < \\sum\\rho_{\\rm FT} \\le 10^{1.50}$;\n(b): $\\pm1\\fdg0$ for 4.20~TeV\/$Z$ in $10^{1.50} < \\sum\\rho_{\\rm FT} \\le 10^{1.75}$;\n(c): $\\pm0\\fdg7$ for 6.46~TeV\/$Z$ in $10^{1.75} < \\sum\\rho_{\\rm FT} \\le 10^{2.00}$;\n(d): $\\pm0\\fdg5$ for 11.4~TeV\/$Z$ in $10^{2.00} < \\sum\\rho_{\\rm FT} \\le 10^{2.33}$;\n(e): $\\pm0\\fdg3$ for 21.6~TeV\/$Z$ in $10^{2.33} < \\sum\\rho_{\\rm FT} \\le 10^{2.67}$;\n(f): $\\pm0\\fdg2$ for 45.4~TeV\/$Z$ in $10^{2.67} < \\sum\\rho_{\\rm FT} \\le 10^{3.00}$.\nThe solid histograms denote the moon's shadow simulation\nassuming the primary cosmic-ray composition based on direct\nobservations \\cite{Asa98,San00,Apa01,Ame08}.}\n\\label{fig4}\n\\end{figure*}\n\n\n\n\\subsection{Calibration of Primary Particle Energy and Systematic Errors} \\label{s-3.4}\n\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f5.eps}\n\\caption{Dependence of shower size on the displacement of the moon's\n  shadow in the north-south direction.  The filled circles and open\n  squares represent experimental data and the MC simulation,\n  respectively.  The solid line denotes the fitting to the\n  experimental data assuming a constant function, resulting in\n  $0\\fdg008\\pm0\\fdg011$. The upper scale indicates the logarithmic mean\n  of $E\/Z$ (TeV\/$Z$) in each $\\sum\\rho_{\\rm FT}$ bin. }\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f6.eps}\n\\caption{Dependence of shower size on the displacement of the moon's\n  shadow in the east-west direction. The filled circles show the\n  experimental data, and open squares represent the MC\n  simulation. The solid curve is fitted to the MC events, and dashed\n  curves show a $\\pm$10\\% deviation from the solid curve,\n  respectively.  The upper scale indicates the logarithmic mean of $E\/Z$\n  (TeV \/ $Z$) in each $\\sum\\rho_{\\rm FT}$ bin. }\n\\label{fig6}\n\\end{figure}\n\n\n\nFigure~\\ref{fig6} shows the shower size dependence of the displacement\nof the moon's shadow in the east-west direction, obtained by fitting Figure~\\ref{fig4}. In this figure, the\nopen squares show the MC results using the QGSJET model, and are quite\nconsistent with the experimental data. The upper scale indicates the\nlogarithmic mean of the energy $E$ divided by the atomic number $Z$\n(TeV$\/$$Z$), i.e., $<\\log(E\/Z)>$, in each $\\sum\\rho_{\\rm FT}$ bin.\nOne can see that the position of the moon's shadow gradually shifts to\nthe west as the primary energy decreases due to the influence\nof the geomagnetic field.  Hence, the absolute energy scale of cosmic\nrays observed by the Tibet-III array can be directly checked by using\nthe geomagnetic field as a magnetic spectrometer, as we now discuss.\n\nFirst, the MC simulation points are fitted by the function\n$\\kappa(\\sum\\rho_{\\rm FT}\/100)^{\\lambda}$ to define a standard\ncurvature function, resulting in $\\kappa = -0.183$ and $\\lambda = -\n0.720$, as shown by a solid curve in Figure~\\ref{fig6}, where the MC\nstatistical errors are negligible compared with the experimental data.\n\nSecond, the experimental data (filled circles) are fitted by this\nstandard curvature function with the $\\sum\\rho_{\\rm FT}$ shift term\n\\begin{equation}\n-0.183~[(1 - \\Delta R_{\\rm S})(\\sum\\rho_{\\rm FT}\/100)]^{-0.720},\n\\label{equ3}\n\\end{equation}\nto estimate the possible shift in the $\\sum\\rho_{\\rm FT}$ between the\nexperimental data and the MC simulation, as shown by the solid curve in\nFigure~\\ref{fig6}, where $\\Delta$$R_{\\rm S}$ is the $\\sum\\rho_{\\rm\n  FT}$ shift ratio, resulting in $\\Delta$$R_{\\rm S} = (-4.9 \\pm\n9.5)\\%$.  We should then convert $\\Delta$$R_{\\rm S}$ to the energy\nshift ratio $\\Delta$$R_{\\rm E}$ as a final result.  To determine the\nrelationship between $\\Delta$$R_{\\rm S}$ and $\\Delta$$R_{\\rm E}$, and to\nconfirm that this method is sensitive to energies, we prepare six\nkinds of MC event samples in which the energy of the primary particles is\nsystematically shifted event by event in the moon's shadow\nsimulation. These six $\\Delta$$R_{\\rm E}$s are $\\pm$20\\%, $\\pm$15\\%\nand $\\pm$8\\%, respectively.  In each MC event sample, the $\\sum\\rho_{\\rm\n  FT}$ dependence of the displacement of the moon's shadow is\ncalculated in the same way, and the $\\sum\\rho_{\\rm FT}$ shift ratio\n$\\Delta$$R_{\\rm S}$ is estimated by fitting the data to\nequation~(\\ref{equ3}).  Finally, we get the relation $\\Delta$$R_{\\rm E}$\n= $(-0.91\\pm0.05)$ $\\Delta$$R_{\\rm S}$ assuming a linear function.\nHence, the systematic error in the absolute energy scale\n$\\Delta$$R_{\\rm E}$ with statistical error $\\sigma_{\\rm stat}$ is\nestimated to be $(+4.5 \\pm 8.6_{\\rm stat})\\%$.\n\nFurthermore, we investigate two kinds of systematic uncertainties with\nthe proposed method.  One is that the position of the moon's shadow by the MC\nsimulation depends on the assumed primary cosmic-ray composition.  In\nthis simulation, the chemical composition ratio of primary cosmic rays\nis estimated based mainly on the data obtained by direct observations.\nThese datasets should also have statistical and systematic errors.\nThe position of the moon's shadow is dominated by the light component,\nso that the proton ratio is artificially varied by $\\pm$10\\% from a\nstandard chemical composition without changing their spectral index,\nwhile the other components heavier than helium are varied by $\\mp$10\\%\nin total.  Figure~\\ref{fig7} shows the results for the composition\ndependence of primary cosmic rays.  The downward triangles are the\nresults obtained by the proton-rich model (75\\% protons after triggering by the\nTibet-III array), while the upward triangles are the ones for the\nheavy-rich model (P:55\\%).  These models are fitted by\nequation~(\\ref{equ3}). We then obtain $\\sigma_{\\rm syst1}$ = $\\pm$6\\%\nfor the systematic error due to the difference in chemical composition,\nas shown by the dashed curves in Figure~\\ref{fig7}.  Another\nsystematic uncertainty is caused by the difference between hadronic\ninteraction models.  Figure~\\ref{fig8} compares the results for the\nhadronic interaction model dependence by QGSJET with those obtained by SIBYLL.  It\nis found that the results by the SIBYLL model can be well fitted by\nequation~(\\ref{equ3}) obtained using the QGSJET model.  We then\nobtain $\\sigma_{\\rm syst2}$ = 6\\% difference between two models.\nFinally, the difference in the energy dependence of the moon's shadow\nbetween the experimental data and the MC events is estimated to be\n$+4.5\\%~(\\pm 8.6_{\\rm stat} \\pm 6_{\\rm syst1} \\pm 6\/2_{\\rm syst2})\\%$.\nThis value is within the statistical and systematic errors.  Hence,\nthe absolute energy scale error in the Tibet-III array is estimated to\nbe smaller than 12\\% = $\\sqrt{\\Delta R_{\\rm E}^2 + \\sigma_{\\rm stat}^2\n  + \\sigma_{\\rm syst1}^2 + (\\sigma_{\\rm syst2}\/2)^2}$ in total\naveraged from 3 to 45 (TeV$\/$$Z$).\n\n\n\\begin{figure}[t]\n\\epsscale{1.20}\n\\plotone{f7.eps}\n\\caption{Dependence of shower size on the displacement of the moon's\n  shadow in the east-west direction by the MC simulation for the\n  different primary composition models. The filled circles and open\n  squares show the experimental data and MC simulation, respectively.  The\n  solid curve denotes the best-fit curve for the same standard\n  composition ratio as in Fig.~\\ref{fig6} (65\\% P after triggering by\n  the Tibet-III array, where P means protons). The dashed curves show\n  a 6\\% shift, corresponding to $\\sigma_{\\rm syst1}$, from the solid\n  curve (see text).  The downward and upward triangles are the\n  simulated results for the proton-rich model (P:75\\%), and for the\n  heavy-rich model (P:55\\%), respectively. }\n\\label{fig7}\n\\end{figure}\n\n\\begin{figure}[t]\n\\epsscale{1.20}\n\\plotone{f8.eps}\n\\caption{Dependence of shower size on the displacement of the moon's\n  shadow in the east-west direction by the MC simulation for different\n  hadronic interaction models.  The open squares and cross marks are\n  the results obtained by the QGSJET and SIBYLL models, respectively.  The solid\n  and dashed curves are the best-fit results assuming the QGSJET and\n  SIBYLL models, respectively.}\n\\label{fig8}\n\\end{figure}\n\n\n\n\\subsection{On the Energy Estimation of $\\gamma$-Ray Showers} \\label{s-3.5}\n\nWe established a new calibration method of the absolute energy scale\nfor cosmic rays based on the moon's shadow analysis as described\nabove.  The air shower induced by the primary cosmic ray consists of\nhigh-energy hadronic and electromagnetic cascades. Although several\nplausible hadronic interaction models are prepared in the MC\nsimulation, there still remain dependence between these\nmodels. Therefore, the energy reconstruction from the air-shower size\ndepends on hadronic interaction models and also the primary chemical\ncomposition models. These systematic errors were adequately taken into\naccount in the absolute energy scale error in this paper.  On the\nother hand, the air shower induced by the primary $\\gamma$-ray is\npredominated by the theoretically well-known electromagnetic cascades,\nbecause the photon cross section for hadronic interactions is\napproximately two orders of magnitude smaller than that for the pair\ncreation process. Hence, we naturally expect that the absolute energy\nscale error for $\\gamma$-rays is smaller than 12\\% which is deduced\nfrom the moon's shadow observed in the cosmic rays described in\n$\\S$\\ref{s-3.4}. In the next section, we will provide reliable results\non the multi-TeV $\\gamma$-ray observation from the Crab~Nebula with\nthe Tibet-III air shower array finely tuned by the cosmic-ray moon's\nshadow.\n\n\n\n\\section{MULTI-TeV $\\gamma$-RAY OBSERVATION FROM THE CRAB}\n\n\\subsection{Analysis} \\label{s-4.1}\n\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f9.eps}\n\\caption{Number of observed air shower events with $\\sum\\rho_{\\rm FT}\n  >10^{1.25}$ ($>$ $\\sim$1~TeV) after event reduction for the\n  observation time of 1318.9 detector live days as a function of\n  angular distance from the Crab~Nebula in the azimuthal direction. }\n\\label{fig9}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f10.eps}\n\\caption{Contour map of significance distribution around the Crab~Nebula \n  ($\\alpha =83\\fdg63$, $\\delta = 22\\fdg02$) for events with\n  $\\sum\\rho_{\\rm FT} >10^{1.25}$ ($>$ $\\sim$1~TeV). A clear peak\n  excess is seen at the center position\n  $\\Delta\\alpha$cos($\\Delta\\delta$) = $\\Delta\\delta$ = 0\\degr, where\n  $\\Delta\\alpha$ and $\\Delta\\delta$ are the relative right accension and\n  declination, respectively, from the Crab~Nebula.\n  The cross mark indicates the pointing error by a point spread function fitting.}\n\\label{fig10}\n\\end{figure}\n\n\nIn order to extract an excess of TeV $\\gamma$-ray events coming from\nthe direction of the Crab~Nebula ($\\alpha = 83\\fdg63$, $\\delta =\n22\\fdg02$), we adopt the same method as used in the Mrk~421 analysis in\nour previous work \\cite{Ame03}.  We call it the equi-zenith angle\nmethod, which is used also for the moon's shadow analysis described in\n$\\S$\\ref{s-3.1}.  The background is estimated by the number of events\naveraged over eight off-source bins with the same angular radius as\nthe on-source bin, at the same zenith angle, recorded at the same time\nintervals as the on-source bin. The nearest two off-source bins are\nset at an angular distance $6\\fdg4$ from the on-source bin to avoid a\npossible signal tail leaking into these off-source bins.  Other\noff-source bins are located every $3\\fdg2$ step from the nearest\noff-source bins.  The search window radius is expressed by $6.9\/\\sqrt{\n  \\sum\\rho_{\\rm FT} }$ degrees as a function of $\\sum\\rho_{\\rm FT}$,\nwhich is shown to maximize the $S\/N$ ratio by MC study of\n$\\gamma$-ray observation \\cite{Ame03}.\n\nThe number of events after the event reduction is plotted in\nFigure~\\ref{fig9} as a function of angular distance from the\nCrab~Nebula in the azimuthal direction. A clear peak of $\\gamma$-ray\nsignals from the Crab~Nebula is seen at 6.3~$\\sigma$ statistical\nsignificance above the flat cosmic-ray background.  Figure~\\ref{fig10}\nis a significance map around the Crab~Nebula. The peak excess is seen\nat the Crab~Nebula position.  The pointing error as shown by a cross\nmark in Figure~\\ref{fig10} is estimated to be\n$\\Delta\\alpha=+0.13\\pm0.08$ and $\\Delta\\delta=+0.01\\pm0.09$ by the\npoint spread function fitting.  This is consistent with the Crab's\nposition within statistical error.  As difference between $\\gamma$-ray\ninduced air showers and cosmic-ray induced ones does not affect the\npointing accuracy essentially, we estimate our pointing accuracy to\nto be $0\\fdg011$ deduced from the moon's shadow analysis described in \n$\\S$\\ref{s-3.3}.\n\n\n\n\\subsection{Monte Carlo Simulation of $\\gamma$-Ray Observation from the Crab~Nebula}\n\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f11.eps}\n\\caption{Scatter plot of the shower size $\\sum\\rho_{\\rm FT}$\n  and the energy of primary $\\gamma$-rays, where a differential\n  power-law spectrum of the form $E^{-2.6}$ starting at 0.3~TeV is\n  assumed for primary $\\gamma$-rays. For details, see text.}\n\\label{fig11}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1.18}\n\\plotone{f12.eps}\n\\caption{Distribution of excesses as a function of the opening angle\n  relative to the Crab~Nebula direction $\\theta$. The filled circles\n  and shaded histograms stand for the experimental data and the MC\n  events with $\\sum\\rho_{\\rm FT} >10^{1.25}$ ($>$ $\\sim$1~TeV),\n  respectively.}\n\\label{fig12}\n\\end{figure}\n\n\n\n\nThe performance of the Tibet-III array has been studied by a full MC\nsimulation using the CORSIKA code \\cite{Cor98} for event generation in\nthe atmosphere and the Epics code \\cite{Kas} for the response of\nthe scintillation detector \\cite{Ame03}.  These procedures\nare essentially the same as in the case for the moon's shadow described\nin $\\S$\\ref{s-3.2}.  In the simulation for $\\gamma$-ray observation from the Crab~Nebula, primary\n$\\gamma$-rays, assuming the energy spectrum of a power-law type in the\nenergy region of 0.3~TeV to 1000~TeV, are thrown along the diurnal\nmotion of the Crab~Nebula in the sky.  The air shower events generated are\nuniformly distributed over circle with a 300~m radius centered at the\nTibet-III array.  Shown in Figure~\\ref{fig11} is the scatter plot of\nshower size $\\sum\\rho_{\\rm FT}$ and the energy of $\\gamma$-rays coming\nfrom the Crab direction.  The filled circles and error bars stand for\nthe logarithmic mean of the energy and one-standard deviation of the \nlogarithmic Gaussian, respectively.\nThe one-event energy resolution is estimated to be approximately\n$(-40\/+70)\\%$ at 10~TeV, and approximately $\\pm$100\\% in the region of\na few TeV.\n\nThe Crab~Nebula can be treated as a point-like source at the TeV energy\nregion.  To investigate the point spread function of the Tibet-III array, we\ncompared the $\\theta$ distribution of the Crab~Nebula between the\nexperimental data and the MC events, where $\\theta$ is the opening angle\nrelative to the Crab~Nebula direction.  Figure~\\ref{fig12} shows the\ndistribution of the excess events as a function of $\\theta$ for events\nwith $\\sum\\rho_{\\rm FT} >10^{1.25}$. The experimental data agree well\nwith the MC simulation assuming the point-like source.\n\n\n\n\n\\subsection{Energy Spectrum of $\\gamma$-Rays from the Crab Nebula}\n\n\\begin{figure}[t]\n\\epsscale{1.21}\n\\plotone{f13.eps}\n\\caption{Differential energy spectrum of $\\gamma$-rays from the\n  Crab~Nebula obtained using the data collected from 1999 November to\n  2005 November with the Tibet-III array in comparison with the\n  results from IACTs: Whipple \\cite{Hil98}, HEGRA \\cite{Aha04},\n  CANGAROO~III \\cite{Eno06}, H.E.S.S. \\cite{Aha06} and MAGIC \\cite{Alb08}.\n  The Tibet-III upper limit is given at the 90\\%\n  confidence level, according to a statistical prescription\n  \\cite{Hel83}.  }\n\\label{fig13}\n\\end{figure}\n\n\\input{tab1}\n\n\nThe $\\gamma$-ray flux from the Crab~Nebula is estimated by assuming a\npower-law spectrum $f(E) = \\alpha E^{\\beta}$.   The best-fit values \n$\\alpha_{0}$ and $\\beta_{0}$  are given by minimizing a $\\chi^{2}$ function,\nchanging $\\alpha$ and $\\beta$:\n\\begin{equation}\n\\chi^{2} = \\sum_{i=1}^{6} \\left(\\frac{N^{\\rm obs}_i - N^{\\rm sim}_{i}(\\alpha, \\beta)}{\\sigma^{\\rm obs}_{i}} \\right)^2,\n\\label{equ4}\n\\end{equation}\nwhere $N^{\\rm obs}_{i}$, $\\sigma^{\\rm obs}_{i}$ and \n$N^{\\rm sim}_{i}(\\alpha, \\beta)$ are the observed number of excess counts,\nits error and the number of remaining MC events after the analysis\nassuming the spectrum $f(E) = \\alpha E^{\\beta}$, respectively, in the\n{\\it i}-th $\\sum\\rho_{\\rm FT}$ bin among the six $\\sum\\rho_{\\rm FT}$\nbins between $10^{1.25}$ and 10$^{3.00}$ defined in $\\S$\\ref{s-3.1}.\nIn order to estimate $N^{\\rm sim}_{i}(\\alpha, \\beta)$ in the same way\nas experimental data, simulated secondary particles are inputted to\nthe detector response simulation.  Then, we obtain the expected\n$N^{\\rm sim}_{i}(\\alpha, \\beta)$ for the {\\it i}-th $\\sum\\rho_{\\rm FT}$ \nbin after the event reconstruction and event selections in the\nsame way as experimental data.  Here, the expected \n$N^{\\rm sim}_{i}(\\alpha, \\beta)$ includes the energy resolution \neffect by the detector response simulation.\n\nSubsequently, the differential $\\gamma$-ray flux for the {\\it i}-th \n$\\sum\\rho_{\\rm  FT}$ bin is calculated by the following equation:\n\\begin{equation}\nf_{i}(E_{i}) = \\frac{N^{\\rm obs}_{i}}{N^{\\rm sim}_{i}(\\alpha_{0}, \\beta_{0})}~\\frac{N_{\\rm all}^{\\rm sim}(\\alpha_{0}, \\beta_{0})}{\\displaystyle \n\\int_{E_{\\rm min}^{\\rm sim}}^{\\infty} E^{\\beta_{0}}~dE}~\\frac{E_{i}^{\\beta_{0}}}{~S_{\\rm sim}~T_{\\rm obs}},\n\\end{equation}\nwhere $N_{\\rm all}^{\\rm sim}(\\alpha_{0}, \\beta_{0})$ denotes the total\nnumber of MC events generated at the top of the atmosphere along one\ndiurnal motion assuming the spectrum $f(E) = \\alpha_{0} E^{\\beta_{0}}$, \n$\\int_{E_{\\rm min}^{\\rm sim}}^{\\infty} E^{\\beta_{0}}~dE$ \nis the normalization factor of \n$N_{\\rm all}^{\\rm sim}(\\alpha_{0}, \\beta_{0})$, $E_{\\rm min}^{\\rm sim}$ \ndenotes the minimum energy of simulated air shower events (0.3~TeV), \n$S_{\\rm sim}$ denotes the area of core location distribution by the simulation\n(300~m $\\times$ 300~m $\\times$ $\\pi$), $T_{\\rm obs}$ denotes the\nobservation live time, and $E_{i}$ denotes the representative energy\ndefined as the logarithmic mean of the energy calculated by the MC\nsimulation for the {\\it i}-th $\\sum\\rho_{\\rm FT}$ bin.\n\nFigure~\\ref{fig13} shows the differential energy spectrum of the\nCrab~Nebula observed by the Tibet-III array together with the spectra\nobtained by IACTs, including Whipple \\cite{Hil98}, HEGRA \\cite{Aha04},\nCANGAROO~III \\cite{Eno06},\\\\ H.E.S.S. \\cite{Aha06} and MAGIC\n\\cite{Alb08}.  The differential flux for each $\\sum \\rho_{\\rm FT}$ bin\nis presented in Table~\\ref{tbl-1}.  Finally, this energy spectrum is\nfitted by the least $\\chi^{2}$ method assuming $f(E) = \\alpha\n(E\/3~{\\rm TeV})^{\\beta}$, and then we obtain the differential\npower-law spectra as $(dJ\/dE) = (2.09\\pm0.32)\\times10^{-12} (E\/{\\rm\n  3~TeV})^{-2.96\\pm0.14} {\\rm cm}^{-2} {\\rm s}^{-1} {\\rm TeV}^{-1}$ in\nthe energy range of 1.7~TeV to 40~TeV.  \nNote that the absolute\nenergy scale error in the Tibet-III array is experimentally estimated\nto be smaller than $\\pm$12\\% by the moon's shadow observation\ndescribed in $\\S$\\ref{s-3.4}.  The energy scale uncertainty\ncorresponds to $(-28\/+46)\\%$ in the absolute $\\gamma$-ray flux,\nassuming the spectral index $-$2.96, which is our best-fit value.  \nOur energy spectrum in this work\nis consistent with other observations made by IACTs, such as HEGRA and\nH.E.S.S., in the same energy range between 1.7~TeV and 40~TeV.\n\nThe previous flux measurement \\cite{Ame99a}, with the Tibet-HD array\nof 5,175~m$^2$ and an effective running time of 502.1 live days, is\napproximately double this measurement.  In order to properly estimate\nthe difference between the previous work and the present one, we give\na re-fit to both data points from 2.8~TeV to 20 TeV in the overlapping\nenergy region assuming a power-law spectrum.  The previous (Tibet-HD)\nand present (Tibet-III) energy spectra are expressed as\n$(dJ\/dE) = (5.04\\pm0.94)\\times10^{-12} (E\/{\\rm 3~TeV})^{-2.85\\pm0.20} {\\rm cm}^{-2} {\\rm s}^{-1} {\\rm TeV}^{-1}$ and\n$(dJ\/dE) = (2.35\\pm0.49)\\times10^{-12} (E\/{\\rm 3~TeV})^{-3.00\\pm0.25} {\\rm cm}^{-2} {\\rm s}^{-1} {\\rm TeV}^{-1}$,\nrespectively. The flux and spectral index differences between them are estimated to\nbe (2.69$\\pm$1.06) $\\times$10$^{-12} {\\rm cm}^{-2} {\\rm s}^{-1} {\\rm\nTeV}^{-1}$ and 0.15$\\pm$0.32, respectively. As a result, the\ncombined statistical deviation between them is calculated to be\n$\\sqrt{(2.69\/1.06)^2+(0.15\/0.32)^2} \\sigma = 2.6 \\sigma$. Although we have\nupdated the MC simulation in this analysis, we cannot find any\nsystematics to explain this difference.  Hence, we conclude that the\nhigher flux observed in our previous measurement may have been caused\nby a statistical signal fluctuation.  \n\n\n\\subsection{Time Variability}\n\n\n\\begin{figure}[b]\n\\epsscale{1.20}\n\\plotone{f14.eps}\n\\caption{Time variability of the Crab~Nebula and the moon's shadow\n  observed by the Tibet-III array with $\\sum\\rho_{\\rm FT} >10^{1.25}$\n  between 1999 November and 2005 November. (a): Differential flux of\n  the Crab~Nebula at 3~TeV.  (b): Daily deficit event rate averaged\n  over one phase of the moon's shadow. (c): North-south displacement of\n  the moon's shadow.}\n\\label{fig14}\n\\end{figure}\n\n\\input{tab2}\n\nWe divided our dataset from 1999 November to 2005 November into six\nphases, as summarized in Table~\\ref{tbl-2}, to examine the time\nvariability of the flux intensity.  Each phase corresponds to\napproximately one calendar year.  We used slightly different\ncalibration parameters for each phase, because we usually calibrate\nthe scintillation detectors of the Tibet air shower array late in the fall of \nevery year. Unfortunately, some of the blank periods seen in Table~\\ref{tbl-2}\nmostly coincide with the detector calibration periods, periods \nin which the air shower array was upgraded or when the data acquisition system \nwas experiencing problems. The upper panel (a) in Figure~\\ref{fig14} shows the\ntime variability of the $\\gamma$-ray fluxes from the Crab~Nebula at\n3~TeV. We found no evidence for the time variability of flux intensity\nfrom the Crab~Nebula, as we can give a good $\\chi^{2}$ fit to these\nfluxes by a constant function ($\\chi^{2} \/ d.o.f. = 6.55 \/ 5$), where\n$d.o.f.$ means degrees of freedom.  In order to check the possible\nsystematics, the time variability of the deficit event rates of the\nmoon's shadow is also demonstrated as shown by the middle panel (b) in\nFigure~\\ref{fig14}.  The deficit event rates of the moon's shadow from\n1999 November to 2005 November are very stable within a statistical\nerror $\\pm$6\\% year by year.  A fitting to the daily deficit event\nrate averaged over a phase assuming a constant function is consistent\nwith a flat hypothesis ($\\chi^{2} \/ d.o.f.  = 4.82 \/ 5$).  The lower\npanel (c) in Figure~\\ref{fig14} shows the time variability of the\nnorth-south displacement of the moon's shadow, which is a reference to\nthe absolute pointing error described in $\\S$\\ref{s-3.2}.  It is also\nvery stable within $\\pm0\\fdg04$ during our observation period, and is\nconsistent with a flat hypothesis ($\\chi^{2} \/ d.o.f. = 2.09 \/ 5$).\nThese systematics, estimated from observations of the moon's shadow, are\nobviously negligible in the Crab~Nebula observation.\n\n\n\\subsection{Search for $\\gamma$-Rays from the Crab Pulsar}\n\n\nThe rotation period of the Crab pulsar is 33~ms, as inferred from\nradio, optical and X-ray observations.  A pulsed emission with that\nrotation period at the GeV energy region has been detected by EGRET \non board the CGRO satellite \\cite{Fie98}, whereas several observations\nhave reported no evidence for pulsed emissions greater than 10~GeV\n\\cite{Les00,Nau02,Aha04,Aha07,Alb08}.  The emission models of\nhigh-energy pulsed $\\gamma$-rays are mostly based on the outer gap\n\\cite{Che86} and the polar cap \\cite{Dau82} models.  The upscattered\npulsed $\\gamma$-ray flux is also calculated by the inverse-Compton\nprocess and the photon-photon absorption process assuming infrared\nphoton field models.  A model predicts an excessive flux around 1 $\\sim$ \n10~TeV, depending on the infrared photon field models \\cite{Hir01}.\nHere, we present a search for pulsed $\\gamma$-rays from the Crab\npulsar at energies from a few TeV to 100~TeV using the Tibet-III\narray.\n\nThe arrival time of each event is recorded using a quartz clock\nsynchronized with GPS, which has a precision of 1~$\\mu$s.  For the\ntiming analysis, all arrival times are converted to the solar system\nbarycenter frame using the JPL DE200 ephemeris \\cite{Sta82}.  The Crab\npulsar ephemeris is calculated using the Jodrell Bank Crab Pulsar\nMonthly Ephemeris \\cite{JBC0,JBC1}.  The corrected arrival time of\neach event is calculated to the rotation phase of the Crab pulsar, which\ntakes into account of the period derivative $\\dot{P}$ of the period $P$ month by\nmonth.\n\nFigure~\\ref{fig15} shows the distribution of events for each phase in\ntwo rotation periods of the Crab pulsar.  The distribution is\nconsistent with a flat distribution ($\\chi^2\/d.o.f. = 18.1\/19$).\nNo significantly pulsed signal is found in observations for\nevents with $\\sum \\rho_{\\rm FT} > 10^{1.25}$ ($>$ $\\sim$1~TeV).  The\nphase analysis is performed for each $\\sum \\rho_{\\rm FT}$ bin to\nexamine the energy dependence.  Table~\\ref{tbl-3} shows the\nstatistical results by the $Z^2_2$ test \\cite{Buc83}, $H$ test\n\\cite{Jag94} and least $\\chi^2$ test.  Almost all the statistical\ntests show that the phase distributions are uniform within a 3~$\\sigma$\nsignificance level.  We estimate the 3~$\\sigma$ flux upper limit\non the pulsed emission from the Crab pulsar using the $H$ test\n\\cite{Jag94} as\n\\begin{eqnarray}\n\\nonumber\nx_{3\\sigma} & = & (1.5+10.7\\delta)(0.174H)^{0.17+0.14\\delta}\\\\\n\\nonumber\n & & \\times  \\exp \\{(0.08+0.15\\delta) \\\\\n & & \\times (\\log_{10}(0.174H))^2\\},\n\\end{eqnarray}\nwhere $\\delta$ is the duty cycle of the pulsed component, assuming\n$\\delta$ is 21\\% for the Crab pulsar.  Exposure from the Crab pulsar\nto the Tibet-III array is estimated using MC simulation, assuming\nthe differential energy spectrum for $\\gamma$-ray emission has a\nspectral index of $-$2.6.  Upper limits are compared to previous\nresults inferred from other experiments, as shown in\nFigure~\\ref{fig16}.\n\n\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f15.eps}\n\\caption{ Distribution of the event phase of the Crab pulsar.  Phase 0\n  is defined using the timing solution derived from the main pulse of\n  the radio observations.  Upper plot shows our result for events with\n  $\\sum \\rho_{\\rm FT} > 10^{1.25}$ ($>$ $\\sim$1~TeV).  Lower plot\n  shows the $\\gamma$-ray phase histogram above 100~MeV, as measured by\n  EGRET \\cite{Fie98}.}\n\\label{fig15}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1.20}\n\\plotone{f16.eps}\n\\caption{Upper limits on the pulsed $\\gamma$-ray flux from the Crab\n  pulsar observed by the Tibet-III array (arrows with thick solid\n  line), together with results from Whipple \\cite{Les00}, CELESTE\n  \\cite{Nau02}, HEGRA \\cite{Aha04}, H.E.S.S. \\cite{Aha07} and MAGIC\n  \\cite{Alb08}. The long-dashed curve and dashed curve represent the\n  fluxes expected from the outer gap and polar cap models,\n  respectively.}\n\\label{fig16}\n\\end{figure}\n\n\n\\input{tab3}\n\n\n\\section{SUMMARY AND PROSPECTS}\n\nWe have been successfully operating the Tibet-III air shower array at\nYangbajing in Tibet, China since 1999. Using the dataset collected by this array from 1999\nNovember through 2005 November, we obtained the\ndifferential energy spectrum of $\\gamma$-rays from the Crab~Nebula as\n$(dJ\/dE) = (2.09\\pm0.32)\\times10^{-12} (E\/{\\rm 3~TeV})^{-2.96\\pm0.14}\n{\\rm cm}^{-2} {\\rm s}^{-1} {\\rm TeV}^{-1}$ in the energy range of\n1.7~TeV to 40~TeV. This result is consistent with data\nobtained by IACTs, and is statistically consistent with our\nprevious result within 2.6~$\\sigma$.  No evidence is found for time\nvariability of flux intensity from the Crab~Nebula at multi-TeV\nenergies in comparison with the long-term stability of the moon's\nshadow.  We also searched, unsuccessfully, for pulsed $\\gamma$-rays from the Crab\npulsar at multi-TeV energies.\n\nIn this paper, we have carefully analyzed the moon's shadow observed with\nthe Tibet-III array to calibrate the energy of primary cosmic rays\ndirectly.  In general, this energy is indirectly estimated by\nmeasuring shower size in air shower experiments. \nThe cosmic-ray beams coming\nfrom the moon's direction are bent by the geomagnetic field, so\nthat the moon's shadow should shift to the west depending on the\nprimary energy.  We tried to directly estimate the primary energy by measuring\nthe displacement of the moon's shadow.  \nThis energy scale estimation is the first attempt and obtained that the\nsystematic error in the absolute energy scale observed by the\nTibet-III array is estimated to be less than $\\pm$12\\% at energies\naround 10~TeV.  The array's systematic pointing error is also\nestimated to be smaller than $0\\fdg011$. The long-term stability of\nthe deficit rate of the moon's shadow was within\na statistical error $\\pm$6\\% year by year, thus confirming the\nstability of the array operation. This calibration method is very\nunique and will be important to ground-based TeV $\\gamma$-ray\nobservations.\n\nIn the near future, we will set up a 10,000~m$^{2}$\nwater-Cherenkov-type muon detector (MD) array in the ground beneath the\nTibet air shower (AS) array \\cite{Ame07b,Ame07c,Ame07d}.  \nThis Tibet MD array will significantly improve\n$\\gamma$-ray sensitivity of the Tibet air shower array above 10~TeV by\nmeans of $\\gamma$\/hadron separation based on counting the number of\nmuons accompanying each air shower.  The energy spectrum of the\nCrab~Nebula can be surely measured up to several hundred TeV, if extended,\nwith a low background level, using the Tibet AS+MD array. This new array\nwill enable us to survey not only the known sources but also new\nsources in the northern sky above 10~TeV, and may be superior to\nIACTs for observing diffuse $\\gamma$-ray sources \\cite{Ame06,Abd07} and\ndiffuse $\\gamma$-rays from the galactic plane \\cite{Ame02,Atk05},\nowing to its wide field of view and high rejection power for\nhadronic showers.\n\n\n\n\\acknowledgments\n\nThe collaborative experiment of the Tibet Air Shower Arrays has been\nperformed under the auspices of the Ministry of Science and Technology\nof China and the Ministry of Foreign Affairs of Japan. This work was\nsupported in part by a Grant-in-Aid for Scientific Research on Priority\nAreas from the Ministry of Education, Culture, Sports, Science and\nTechnology, by Grants-in-Aid for Science Research from the Japan Society \nfor the Promotion of Science in Japan, and by the Grants \nfrom the National Natural Science Foundation of China and the\nChinese Academy of Sciences.\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\\label{sec:introduction}\n\\IEEEPARstart{M}{ulti-object} tracking (MOT) has become a popular research topic attracting an ever-increasing interest from the computer vision community due to its wide application potential. In general, MOT algorithms can be categorized into the online algorithms and the offline ones based on whether the incorporation of the information from future frames is carried out or not. Compared with their offline counterparts, online algorithms are more suited for real time applications including autonomous driving and mobile robotics. Currently, the mainstream MOT paradigm is tracking-by-detection, which presents a two-step solution to the MOT problem: 1) discover objects of interest through a detector; and 2) form trajectories by using the data association method. It should be mentioned that the tracking-by-detection paradigm is heavily dependent on the detection quality, and an additional feature extractor is usually required to compute the appearance affinity. Another trend in MOT follows the paradigm of tracking-by-SOT, wherein SOT stands for the single object trackers. The correlation filter based tracking methods fall into the tracking-by-SOT paradigm.\n\n\nAs one of the most promising online trackers in MOT, the Tracktor++ proposed in \\cite{Bergmann2019} follows the tracking-by-detection paradigm. In the Tracktor++, a faster region-based convolutional network (Faster R-CNN) is employed as the object detector \\cite{Ren2017a}. The existing tracklets are extended by using the regression head of the detector. Although the Tracktor++ has achieved top performance in view of CLEAR MOT metrics \\cite{Bernardin2008}, there are still some drawbacks in this algorithm. As is shown in Fig.~\\ref{fig1}, one drawback of the Tracktor++ comes from the unreliable judgment of the moment for inactivating a tracklet. Tracklets are inactivated when the tracking hypotheses are missing or unreliable. For online MOT problems, inactivating a tracklet means that a tracklet is temporarily removed from the current tracklet list while keeping its potential qualification to be reidentified. Most current online MOT algorithms would predefine a life span for each tracklet to determine the maximal number of continuous inactivation frames before it is thoroughly deleted.\n\nIn the Tracktor++, a tracklet is inactivated when the corresponding classification score is below a certain threshold, which can be problematic in some cases (especially when a tracking target is leaving the image boundary). We argue that such tracking errors can be alleviated by considering not only the information of individual tracking hypotheses, but also the intra-frame relationship between the tracking hypotheses. In this paper, a discrete conditional random field (CRF) is employed to solve the improper tracklet inactivation problem. Since the tracklet inactivation mechanism is totally decoupled from the main tracking pipeline of Tracktor++, our method can be transferred to trackers with a similar architecture, namely trackers remoulded from detectors, which indicates that our method has wider application potential than Tracktor++.\n\n\nTwo concerns come up when using a CRF in the MOT context: \\romannumeral1) the number of CRF nodes is varying according to the detected objects in a certain frame, whereas the traditional formulation of the CRF is built upon a fixed set of nodes; and \\romannumeral2) the overfitting problem may occur during the training process under the condition that the scale of the training dataset is relatively small with respect to the number of CRF parameters. In this context, two strategies are adopted to address the aforementioned concerns. On the one hand, the hypothesis filtering and dummy nodes are applied to fix the number of CRF nodes. On the other hand, the graph factors are divided into the unary and binary groups. Then, the parameter sharing strategy is utilized within each of the two groups, which effectively reduces the risk of overfitting.\n\nIn the proposed method, the tracking objects in the scene are treated equally (i.e. no object is considered to be more special than others), and thus, the CRF is constructed in a fully connected way. The loopy belief propagation algorithm proposed in \\cite{Kyburg1991} is applied for the inference phase. For the training phase, the maximum likelihood estimation method is adopted and the stochastic gradient descent (SGD) algorithm is utilized to update the parameters.\n\nThe main contributions of this paper can be summarized as follows:\n\\begin{itemize}\n\t\\item A discrete CRF-based framework is developed to handle the unreliable judgment for the tracklet inactivation problem of the Tracktor++.\n\t\\item The feature functions for unary and binary terms are designed to cope with multiple concerns encountered in practical situations.\n\t\\item An effective mechanism is put forward for tracklet inactivation, which outperforms the traditional threshold method.\n\t\\item Experiments are conducted on two benchmarks MOT16 and MOT17. Experimental results demonstrate the superiority of the developed CRF-based method over the baseline method.\n\t\\item The extensibility of the proposed framework is validated by conducting an extensive experiment.\n\\end{itemize}\n\nThe rest of this paper is organized as follows. The background of the MOT problem is introduced in Section \\ref{sec:related_work}. In Section \\ref{sec:method}, the details of the proposed framework are presented. The experimental settings are discussed in Section \\ref{sec:implementation_details}. The details of performance indicators, experimental results and an extensive experiment are provided in Section \\ref{sec:experiments}. Conclusions are drawn in Section \\ref{sec:conclusion}.\n\n\n\\section{Related work}\n\\label{sec:related_work}\nDuring the past few decades, MOT has become a research focus in computer vision. In this section, we briefly review the MOT methods that leverage the power of machine learning and those with the aid of CRF.\n\n\\subsection{MOT with Machine Learning}\n\\label{subsec:mot_with_deep_learning}\nMachine learning approaches have proven to be effective for dealing with the MOT problems through years of practice. The most popular paradigm in this direction is tracking-by-detection \\cite{Ciaparrone2019,Mekonnen2019}. The interested objects are firstly detected by using a deep convolutional neural network. The affinity scores based on appearance and motion information of the objects are then computed. Finally, the objects are correlated through data association. One of the first algorithms that follows this paradigm is the Simple Online and Realtime Tracking (SORT) algorithm \\cite{Bewley2016a} which uses the Kuhn\u2013Munkres algorithm \\cite{Kuhn1955a} for data association and a Kalman filter \\cite{Kalman1960b} for observation correction. DeepSORT has been proposed in \\cite{Wojke2018b} to further improve SORT by learning a deep association metric in visual appearance space. Zhou et al. \\cite{zhou2018deep} have enhanced the tracking accuracy by using a deep alignment network. In addition, a local-to-global strategy for robust data association has been introduced in \\cite{dai2018instance}. In recent years, some dedicated tracking algorithms have been designed either to increase the recall and accuracy of the detector, or to enhance the performance of data association \\cite{bao2020pose,Yu2016,Karunasekera2019,fu2019multi,Lu2017,Sheng2019a}.\n\nAnother line of work follows the paradigm of tracking-by-SOT, which assigns an individual single object tracker for each tracking target. The approaches following this paradigm are effective because single object trackers often pay more attention to the similarity between nearby image patches in consecutive frames than to the integrity of semantic information. Hence, some objects that the detector fails to discover can be recognized by using the single object trackers. This point of view holds particularly well in the trackers based on correlation filters. For example, the kernelized correlation filters (KCF) proposed in \\cite{Henriques2015b} has been employed to solve the MOT problems \\cite{Chu2019a}. In \\cite{Zhu2018b}, the efficient convolution operators (ECOs) proposed in \\cite{Danelljan2017b} have been utilized for dealing with the MOT problems with remarkable performance.\n\n\nThe state-of-the-art tracker, Tracktor++ \\cite{Bergmann2019}, generally falls into the paradigm of tracking-by-detection. It leverages a Faster R-CNN to discover new tracking targets. Notice that the positions of confirmed tracking objects are regressed in later frames by using the regression head of the Faster R-CNN, which differs from the traditional tracking-by-detection approaches. Using such a tracking pipeline, a challenge is the necessity to reasonably inactivate a tracklet in case it drifts to uninterested regions. In Fig.~\\ref{fig1}, we can see that the simple strategy adopted in \\cite{Bergmann2019} is unreliable, which needs to be improved.\n\n\\subsection{MOT with CRF}\n\\label{subsec:mot_with_CRF}\nCRF models are frequently used in Natural Language Processing (NLP) and visual segmentation tasks. The advantage of CRF is its strong ability to model the complex interactions between individuals. Therefore, CRF has been utilized to solve MOT problems.\n\nMost research works on CRF-based methods are geared towards offline MOT problems, i.e. information from future frames is available to be utilized. Yang et al. \\cite{Yang2012a} have built a fully connected CRF model that associated short tracklets to form long ones. The CRF edges have been designed to focus on discriminating spatially close targets with similar appearances. Heili et al. \\cite{heili2014} have exploited long-term connectivity between different detections. Multiple cues have been extracted to measure the similarity and dissimilarity when formulating the energy potentials. A recent work \\cite{Xiang2019a} has followed the basic ideas of \\cite{Yang2012a}, whereas it learns the pairwise potentials by using a bidirectional long short-term memory (LSTM) network, and approximated the inference phase of the CRF by using a recurrent neural network (RNN). The entire tracking pipeline is thus differentiable and can be trained end-to-end.\n\nBy contrast, the application of CRF in online MOT problems has not yet been fully studied. Zhou et al. \\cite{Zhou2019} have proposed a method to solve the MOT problem in an online way. In \\cite{Zhou2019}, the displacements between consecutive frames have been first estimated by using a dedicated neural network. Then, a deep continuous CRF with asymmetric pairwise terms has been utilized to refine the displacements. Note that our method is different from \\cite{Zhou2019} in that we use a discrete CRF to judge the inactivation of tracklets instead of directly interacting with the positions of tracking hypotheses. Besides, since our framework is decoupled from the main tracking pipeline, it is featured with flexibility and extensibility in the sense of being applied to more advanced algorithms in the future.\n\n\\section{Method}\n\\label{sec:method}\nIn this section, we first make a brief description for the baseline tracker, Tracktor++. After that, the formulation of the proposed CRF framework is presented. Then, we show how the feature functions are designed by taking the interaction between different targets into account. Finally, the method for CRF inference and parameter estimation is introduced, including the two techniques employed to keep the number of CRF nodes fixed.\n\\subsection{Baseline Tracker}\nThe tracking pipeline of the baseline tracker, also known as Tracktor++, is depicted in Fig.~\\ref{fig2}. As a tracking-by-detection method, the tracking objects of interest are detected by a Faster R-CNN detector. With the arrival of each new frame, the latest bounding box of each tracklet is treated as the region proposal and sent to the regression and classification heads. The new positions and scales of the tracklets are then provided by the output of the regression head. Once the classification score of a tracklet is lower than a predefined threshold, the tracklet is inactivated. The new detections that are covered by tracklets are filtered out by non-maximum suppression. An individual siamese network is also employed to reidentify the previously inactivated tracklets. Finally, the remaining detections are treated as the starting points of new tracklets.\n\n\\subsection{Problem Formulation}\n\\label{subsec:problem_formulation}\nThe general architecture of our developed method is illustrated in Fig.~\\ref{fig2}, where the state-of-the-art method, Tracktor++, is used as the baseline. Given a set of $ K $ regressing results $ x^t=\\{x_1^t,x_2^t,\\ldots,x_K^t\\} $, our goal is to make a reasonable judgment $ y^t=\\{y_1^t,y_2^t,\\ldots,y_K^t\\} $ labeling whether the tracklets should be inactivated. Here, we use $x_i^t=\\{S_i^t,m_i^t,l_i^t\\}$ as the ensemble tracking information of the tracklet $ i $ at frame $ t $, including the classification score $ S_i^t $, the motion information $ m_i^t $ and the size information $ l_i^t $. We will omit the frame index $ t $ for simplicity in the following formalism.\n\n\\begin{figure}[!t]\n\t\\centerline{\\includegraphics[width=0.5\\textwidth]{fig2.pdf}}\n\t\\caption{The architecture of our method. We pose the judgement for tracklet inactivation as a binary labeling problem, where label $ 0 $ indicates the tracklet should be inactivated and label $ 1 $ indicates the contrary. The labeling output is jointly determined by the unary and binary terms of the CRF model. In this example, the tracking hypothesis represented by the pentagon is inactiavted and removed from the final tracking result. }\n\t\\label{fig2}\n\\end{figure}\n\nWe model the inactivation labeling problem with a CRF which is factorized as a factor graph \\cite{Kschischang2001} $ G=(V,F,\\mathcal{E}) $, where $ V=\\{1,2,\\ldots,K\\} $ is the set of nodes corresponding to the tracklet indices. $ F $ and $ \\mathcal{E} $ are the sets of factors and edges in the graph respectively. The joint conditional distribution can thus be expressed as\n\\begin{equation}\n\\label{eq1}\np(y|x;\\theta)=\\frac{1}{Z(x;\\theta)}\\prod_{f \\in F}\\text{e}^{-E_f(x,y;\\theta)},\n\\end{equation}\nwith\n\\begin{equation}\n\\label{eq2}\nZ(x;\\theta)=\\sum_{y\\in Y} \\prod_{f \\in F}\\text{e}^{-E_f(x,y;\\theta)},\n\\end{equation}\nwhere $ Y $ denotes the domain of $ y $. $ E_f $ is the energy function of the factor $ f $.\n\nNote that the dataset exploited for training the CRF is established by using the improper inactivation instances produced by Tracktor++ as negative samples. Such instances simply produce a small scale dataset. Thereby, parameter sharing is applied to our method to prevent the overfitting problem. This is achieved by grouping the factors in \\eqref{eq1} by a set of two cliques $ C=\\{C_u,C_b\\} $, i.e., a unary one and a binary one. Then, we have\n\n\\begin{equation}\n\\label{eq3}\np(y|x;\\theta)=\\frac{1}{Z(x;\\theta)}\\prod_{C_F \\in C}\\prod_{f \\in C_F}\\text{e}^{-E_f(x_f,y_f;\\theta_f)}.\n\\end{equation}\nDefining $ E_f(x,y_f;\\theta_f)=\\theta_f\\Phi_f(x,y_f)$, we have a more detailed form of \\eqref{eq3} as follows\n\\begin{equation}\n\\label{eq4}\np(y|x;\\theta)=\\frac{1}{Z(x;\\theta)}\\text{e}^{\\theta_u\\sum\\limits_{f\\in C_u}\\Phi_u(x_f,y_f)+\\theta_b\\sum\\limits_{f\\in C_b}\\Phi_b(x_f,y_f)}.\n\\end{equation}\nWe can see that by calculating the energy functions of factors in two cliques, only two variables $ \\theta_u $ and $ \\theta_b $ are required to be optimized.\n\n\\begin{figure*}[!t]\n\t\\centerline{\\includegraphics[width=1.0\\textwidth]{fig3.pdf}}\n\t\\caption{Examples to explain for excluding the width term. The height of a pedestrian is stable regardless of the camera attitude, while the width of a pedestrian may change dramatically even in normal case because of the change in human postures.}\n\t\\label{fig3}\n\\end{figure*}\n\n\\subsection{Feature Function Designing}\n\\label{subsec:feature_function_designing}\nFollowing the above derivation, we now explicitly design the feature functions in the form of a unary term and a binary term. The unary term focuses on the reliability of individual tracklets, whereas the binary term considers the relationship between different pairs of tracklets.\n\n\\subsubsection{Unary Term}\n\\label{subsubsec:unary_term}\nTwo kinds of information are taken into account when designing the feature functions for the unary term: the classification score $ S_f $ and the changing rate of the aspect ratio $ \\Delta R_f $:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq5}\n\\Phi_u(x_f,y_f)=\\textbf{1}_{\\{y_f=0\\}}(\\mid 0-S_f\\mid +\\textbf{1}_{\\{S_f>0.95\\}}\\alpha_1)+\\\\\n\\textbf{1}_{\\{y_f=1\\}}(\\mid 1-S_f\\mid+\\alpha_2 \\mid 1-\\Delta R_f\\mid),\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq6}\n\\textbf{1}_{\\{A=B\\}}= \\left \\{\n\\begin{array}{lr}\n\\displaystyle 1 \\;   \\ \\  , \\;  if \\; A=B \\\\\n0 \\;   \\ \\  , \\; if \\; A \\ne B\n\\end{array} \\ \\ .\n\\right.\n\\end{equation}\nHere, $ \\alpha_1 $ and $ \\alpha_2 $ are hyperparameters that are initialized heuristically and are adjusted by using a validation dataset. $ S_f $ comes directly from the output of the Faster R-CNN. $ \\textbf{1}_{\\{S_f>0.95\\}} $ lays extra punishment for the inactivation with classification scores higher than $ 0.95 $. To define $ \\Delta R_f $ specifically, we replace it by $ \\Delta R_i^t $, where $ i $ denotes the index of the only node in the factor. Then, we have\n\n\\begin{equation}\n\\label{eq7}\n\\Delta R_i^t=\\frac{w_i^t\/h_i^t}{w_i^{t-1}\/h_i^{t-1}}.\n\\end{equation}\n\n\\subsubsection{Binary Term}\n\\label{subsubsec:binary_term}\nIn our method, the binary term formulates the punishment for joint values of the two nodes contained in each factor. In \\cite{Zhou2019}, an assumption is given that the change of velocity should be basically the same between two tracking targets in a short time interval, i.e.\n\n\\begin{equation}\n\\label{eq8}\ndisp_i^t-disp_j^t=disp_i^{t-1}-disp_j^{t-1},\n\\end{equation}\nwhere $ disp_i^t$ is the center displacement of object $ i $ between frame $ t $ and frame $ t-1 $. This assumption have two implications:\n\n(\\romannumeral1) The absolute velocity of pedestrians can be approximately regarded as a constant value during short time intervals. Hence, $ disp_i^t-disp_i^{t-1} $ leads to the magnitude of camera motion.\n\n(\\romannumeral2) Sharp rotation and swing should not be involved in the camera motion.\n\nWe follow a similar assumption and make further improvement in the following four aspects:\n\\begin{enumerate}\n  \\item The units of measure is unified when calculating the velocity by multiplying the frame rate $ \\omega $ of the sequence, i.e., $ V_i^t=\\omega(P_i^t-P_i^{t-1}) $ where $ P $ denotes the center position of the object. Likely, we calculate the changing rate of velocity as $ \\Delta V_i^t=\\omega(V_i^t-V_i^{t-1}) $.\n  \\item A coefficient $ \\tau=\\frac{1}{h_i+h_j} $ is introduced to balance the weights, where $ h $ is the height of a tracked object. This design is proposed due to the observation that the position of a larger object tends to be estimated with worse performance.\n  \\item The above assumption is extended to the changing rate of the object height $ \\Delta L_i^t= \\omega\\frac{\\Delta h_i^t-\\Delta h_i^{t-1}}{\\Delta h_i^{t-1}}$, where $ \\Delta h_i^t=\\omega\\frac{h_i^t-h_i^{t-1}}{h_i^{t-1}}$. Note that the width of object is cast aside when calculating $ \\Delta L $, the reason for which is depicted in Fig.~\\ref{fig3}.\n  \\item When part of a tacking target moves out of sight, $ \\Delta L $ becomes unreliable due to the restriction of the image boundary. Therefore, an additional variable $ \\kappa $ is introduced to address this concern, which takes $ 0 $ if the object is partially beyond the image boundary and takes $ 1 $ otherwise.\n\\end{enumerate}\n\n\nMotivated by the above discussions, the feature functions of the binary term are designed as follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq9}\n\\Phi_b(x_f,y_f)=\\textbf{1}_{\\{y_f=[1,1]\\}} \\bigl(\\sum_{n\\in\\{x,y\\}}\\tau(\\Delta V_{n,i}-\\Delta V_{n,j})^2\\\\\n+\\beta \\kappa\\mid \\Delta L_i-\\Delta L_j \\mid \\; \\bigr),\n\\end{aligned}\n\\end{equation}\nwhere $\\beta$ is a hyperparameter.\n\n\\subsection{Inference and Training}\n\\label{subsec:inference_and_training}\nTo facilitate the inference and training of a CRF, we leverage two techniques to keep the number of CRF nodes fixed. The first technique, termed as hypothesis filtering, filters out extra objects with the highest classification scores when the detected objects outnumber a predefined value $ M $. On the contrary, frames with nodes fewer than $ M $ are complemented by adding dummy nodes. The dummy nodes are artificially created to make the formulation hold, which are designed to exert no influence on the real nodes in a way that the formulae in Section \\ref{sec:method} are regarded as multiplied by a boolean variable (which takes $ 1 $ when a factor contains no dummy nodes and takes $ 0 $ otherwise).\n\n\\subsubsection{CRF Inference}\n\\label{subsubsec:crf_inference}\nThe loopy belief propagation algorithm is adopted for dealing with the CRF inference task \\cite{Kyburg1991}. The main idea of the loopy belief propagation algorithm is demonstrated as follows, and we refer interested readers to \\cite{Kyburg1991,Nowozin2010,Sutton2011} for further mathematical details.\n\nThe message flowing from a factor node to a variable node is defined as\n\\begin{equation}\n\\label{eq10}\nM_{f,v}(x,y_v;\\theta)=\\sum_{\\substack{y_f \\in Y_f(y_v)}}\\bigl(\\text{e}^{-E_f(x,y_f;\\theta)}\\prod\\limits_{\\substack{v^{\\prime}\\sim f\\\\ v^{\\prime} \\ne v}}M_{v^{\\prime},f}(x,y_{v^{\\prime}};\\theta)\\bigr),\n\\end{equation}\nwhere $ v $ and $ f $ denote a variable node and a factor node, respectively. $ Y_f(y_u) $ is a subset of $ y_f $'s domain that requires the node $ u $ to take the value $ y_u $. $ v\\sim f $ signifies that an edge exists between $ v $ and $ f $.\n\nConversely, the message flowing from a variable node to a factor node is defined as\n\\begin{equation}\n\\label{eq11}\nM_{v,f}(x,y_v;\\theta)=\\dfrac{\\prod\\limits_{\\substack{f^{\\prime}\\sim v \\\\ f^{\\prime}\\ne f }}M_{f^{\\prime},v}(x,y_v;\\theta)}{\\sum\\limits_{y_v \\in Y_v}\\prod\\limits_{\\substack{g\\sim v \\\\ f^{\\prime}\\ne f }}M_{f^{\\prime},v}(x,y_v;\\theta)}.\n\\end{equation}\n\nThe marginal distribution is then calculated by\n\n\\begin{equation}\n\\label{eq12}\np(y_v|x;\\theta)=\\dfrac{\\prod\\limits_{f\\sim v}M_{f,v}(x,y_v;\\theta)}{\\sum\\limits_{y_v \\in Y_v}\\prod\\limits_{f\\sim u}M_{f,v}(x,y_v;\\theta)}.\n\\end{equation}\n\nIn order to get the optimal output $ y^{\\star} $ with respect to the maximum posterior probability (MAP), \\eqref{eq10} and \\eqref{eq11} are modified as\n\\begin{equation}\n\\label{eq13}\nM_{f,v}(x,y_v;\\theta)=\\max_{\\substack{y_f \\in Y_f(y_v)}}\\text{e}^{-E_f(x,y_f;\\theta)}\\prod\\limits_{\\substack{v^{\\prime}\\sim f\\\\ v^{\\prime} \\ne v}}M_{v^{\\prime},f}(x,y_{v^{\\prime}};\\theta),\n\\end{equation}\n\\begin{equation}\n\\label{eq14}\nM_{v,f}(x,y_v;\\theta)=\\frac{1}{2}\\dfrac{\\prod\\limits_{\\substack{f^{\\prime}\\sim v \\\\ f^{\\prime}\\ne f }}M_{f^{\\prime},v}(x,y_v;\\theta)}{\\prod\\limits_{y_v \\in Y_v}\\prod\\limits_{\\substack{g\\sim v \\\\ f^{\\prime}\\ne f }}M_{f^{\\prime},v}(x,y_v;\\theta)}.\n\\end{equation}\nFinally, the output labels are obtained as follows:\n\\begin{equation}\n\\label{eq15}\ny^{\\star}=[\\;\\max_{y_1} p(y_1|x;\\theta),\\dots,\\max_{y_M} p(y_M|x;\\theta)\\;]^T.\n\\end{equation}\n\n\\subsubsection{CRF Training}\n\\label{subsubsec:crf_training}\nThe purpose of CRF training is to get an estimation of the parameters using training samples $ \\{(x^1,y^1),\\dots,(x^N,y^N)\\} $. This problem can be naturally solved by Maximum Likelihood Estimation (MLE). Taking the first order derivative of the log likelihood of \\eqref{eq4} and considering all $ N $ training samples, we have\n\\begin{equation}\n\\begin{aligned}\n\\label{eq16}\n\\frac{\\partial l(\\theta)}{\\partial \\theta_{u\/b}}=\\sum_{n=1}^{N}\\bigl(\\sum\\limits_{f\\in C_{u\/b}}\\Phi_{u\/b}(x_f^n,y_f^n)-\\\\\n\\sum\\limits_{f\\in C_{u\/b}}\\sum\\limits_{y_f^{\\prime}}\\Phi_{u\/b}(x_f^n,y_f^{\\prime})p(y_f^{\\prime}|x^n;\\theta)\\;\\bigr)\n\\end{aligned}\n\\end{equation}\nwhere $ p(y_f|x;\\theta) $ is calculated in a similar way to \\eqref{eq12}:\n\\begin{equation}\n\\label{eq17}\np(y_f|x;\\theta)=\\dfrac{e^{-E_f(y_f)}\\prod\\limits_{v\\sim f}M_{v,f}(x,y_v;\\theta)}{\\sum\\limits_{y_f \\in Y_f}\\bigl[e^{-E_f(y_f)}\\prod\\limits_{v\\sim f}M_{v,f}(x,y_v;\\theta)\\bigr]}.\n\\end{equation}\n\nApplying the SGD algorithm with above gradients, the weights are updated with a fixed learning rate $ \\gamma $:\n\\begin{equation}\n\\label{eq18}\n\\theta^{\\prime}=\\theta+\\bigl[\\;\\gamma_u \\frac{\\partial l(\\theta)}{\\partial \\theta_{u}}\\;,\\;\\gamma_b \\frac{\\partial l(\\theta)}{\\partial \\theta_{b}}\\;\\bigr]^T.\n\\end{equation}\n\n\\begin{algorithm}\n\t\\caption{Workflow for tracking}\n\\label{alg1}\n\t\\SetAlgoLined\n\t\\KwIn{Images from frames to be tracked $ I^1,\\dots,I^T $,\n\t\tPublic detection results $ D^1,\\dots,D^t $.}\n\t\\KwOut{Tracklets of objects $ tr_1,...,tr_K $.}\n\tInitialize tracklets $ tr_1,...,tr_K $ by detections $ D^1 $;\\\\\n\t\\For{$t=2,\\dots,T$}\n\t{\n\t\tObtain regressing results $ x^t $ and classification scores $ S^t $ by using Tracktor++;\\\\\n\t\t\\For{tracklet index $ i=1,\\dots,K^t $}\n\t\t{\n\t\t\t\\eIf{$ S_i^t< 0.4$}\n\t\t\t{\n\t\t\t\tInactivate tracklet $ i $;\\\\\n\t\t\t}{\n\t\t\t\t\\If{tracklet length $ C_i \\ge 2$}\n\t\t\t\t{\n\t\t\t\t\tCalculate all the tracking information required in \\ref{subsec:feature_function_designing};\\\\\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\t\t\\If{number of tracklets $ K^t > 10 $ }\n\t\t{Cut down $ K^t $ to $ 10 $ by hypothesis filtering (\\ref{subsec:inference_and_training});\n\t\t}\n\t\tCalculate the unary and binary terms in the CRF (\\ref{subsec:feature_function_designing});\\\\\n\t\tObtain the inactivation labels by using the loopy belief propagation algorithm (\\ref{subsubsec:crf_inference});\\\\\n\t\tInactivate tracklets by using the labels;\\\\\n\t\tDo non-maximum suppression and reidentification.\\\\\n\t\tUpdate $ tr_1,...,tr_k$;\n\t}\n\t\\Return $ tr_1,...,tr_k$.\n\\end{algorithm}\n\n\\section{Implementation details}\n\\label{sec:implementation_details}\nIn this section, we first introduce the datasets used in practical tracking. The details for parameter estimation are then presented. Finally, the workflow and other details of practical tracking are introduced.\n\n\\subsection{Datasets}\n\\label{subsec:datasets}\nTwo popular datasets, MOT16 and MOT17 \\cite{Milan2016}, are employed in this paper. Both these datasets provide official detection results that are called public detections. Researchers can also use private detections produced by their own detectors.\n\\subsubsection{MOT16}\nThe MOT16 dataset contains 14 video sequences of pedestrians with different resolutions, frame rates, lighting conditions, crowd density, and filming angles. They are evenly divided into a training dataset and a testing dataset. Each dataset has 4 sequences with moving cameras and 3 sequences with static cameras. The public detections are produced by using a DPM detector \\cite{Felzenszwalb2010a}.\n\\subsubsection{MOT17}\nThe MOT17 dataset has the same video sequences as the MOT16. Notice that MOT17 provides three groups of public detections, which are produced by Faster R-CNN \\cite{Ren2017a}, DPM \\cite{Felzenszwalb2010a}, and SDP \\cite{Yang2016a} respectively, resulting in 42 sequences altogether. Trackers' performance on all 21 test sequences are averaged to get a generalized evaluation result. When tracking is performed with private detections, there is no difference in using the MOT16 dataset or the MOT17 dataset.\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Parameters in CRF}\n\t\\label{tab1}\n\t\\resizebox{0.35\\textwidth}{!}{%\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\toprule\n\t\t\tParameter\n\t\t\t&$ \\theta_u $&$ \\theta_b $&$ \\alpha_1 $&$ \\alpha_2 $&$ \\beta $\\\\\n\t\t\t\\midrule\n\t\t\tValue\n\t\t\t&0.98&0.12&1.05&1.20&10.80\\\\\n\t\t\t\n\t\t\t\\bottomrule\t\n\t\t\\end{tabular}%\n\t}\n\\end{table}\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Comparison of running speed (Hz)}\n\t\\label{tab2}\n\t\\resizebox{0.44\\textwidth}{!}{%\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\toprule\n\t\t\tHypothesis number\n\t\t\t& 3 & 6 & 10 & 15&Overall\\\\\n\t\t\t\\midrule\n\t\t\tOurs\n\t\t\t&6.80&5.51&2.82&1.67&2.22\\\\\n\n\t\t\tTracktor++\n\t\t\t&7.09&6.35&3.20&1.98&2.58\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}%\n\t}\n\\end{table}\n\n\\begin{table*}[h]\n\t\\centering\n\t\\caption{MOT16 results (public detections)}\n\t\\label{tab3}\n\t\\resizebox{0.75\\textwidth}{!}{%\n\t\t\\begin{tabular}{cccccccccc}\n\t\t\t\\toprule\n\t\t\t&MOTA$\\uparrow$&IDF1$\\uparrow$&MOTP$\\uparrow$&MT$\\uparrow$&ML$\\downarrow$\n\t\t\t&FP$\\downarrow$&FN$\\downarrow$&IDS$\\downarrow$&Frag$\\downarrow$ \\\\\n\t\t\t\\midrule\n\t\t\tOurs\n\t\t\t&\\textbf{57.0}&\\textbf{58.3}&79.1&\\textbf{158}&\\textbf{267}&2610&\\textbf{75337}&538&1212 \\\\\n\t\t\t\n\t\t\tTracktor++v2 \\cite{Bergmann2019}\n\t\t\t&56.2&54.9&\\textbf{79.2}&157&272&\\textbf{2394}&76844&617&1068 \\\\\n\t\t\t\n\t\t\tKCF16\\cite{Chu2019a}\n\t\t\t&48.8&47.2&75.7&120&289&5875&86567&906&1116 \\\\\n\t\t\t\n\t\t\tMOTDT\\cite{Chen2018a}\n\t\t\t&47.6&50.9&74.8&115&291&9253&85431&792&1858 \\\\\n\t\t\t\n\t\t\tJCSTD\\cite{Tian2020}\n\t\t\t&47.4&41.1&74.4&109&276&8076&86638&1266&2697 \\\\\n\t\t\t\n\t\t\tAMIR\\cite{Sadeghian2017a}\n\t\t\t&47.2&46.3&75.8&106&316&2681&92856&\\textbf{370}&\\textbf{598} \\\\\n\t\t\t\n\t\t\tYOONKJ\\cite{Yoon2020}\n\t\t\t&47.0&50.1&75.8&125&317&7901&88179&627&945 \\\\\n\t\t\t\n\t\t\tDD\\_TAMA\\cite{Yoon2019}\n\t\t\t&46.2&49.4&75.4&107&334&5126&92367&598&1127 \\\\\n\t\t\t\n\t\t\tDMAN\\cite{Zhu2018b}\n\t\t\t&46.1&54.8&73.8&132&324&7909&89874&532&1616 \\\\\n\t\t\t\n\t\t\tSTAM\\cite{Chu2017b}\n\t\t\t&46.0&50.0&74.9&111&331&6895&91117&473&1422 \\\\\n\t\t\t\n\t\t\tRAR16pub\\cite{Fang2018}\n\t\t\t&45.9&48.8&74.8&100&318&6871&91173&648&1992 \\\\\n\t\t\t\\bottomrule\t\n\t\t\\end{tabular}%\n\t}\n\\end{table*}\n\n\\begin{table*}[h]\n\t\\centering\n\t\\caption{MOT17 results (public detections)}\n\t\\label{tab4}\n\t\\resizebox{0.75\\textwidth}{!}{%\n\t\t\\begin{tabular}{cccccccccc}\n\t\t\t\\toprule\n\t\t\t&MOTA$\\uparrow$&IDF1$\\uparrow$&MOTP$\\uparrow$&MT$\\uparrow$&ML$\\downarrow$\n\t\t\t&FP$\\downarrow$&FN$\\downarrow$&IDS$\\downarrow$&Frag$\\downarrow$ \\\\\n\t\t\t\\midrule\n\t\t\tOurs\n\t\t\t&\\textbf{56.3}&57.5&\\textbf{78.9}\n\t\t\t&490&839&\\textbf{8672}&236296&\\textbf{1668}&3820 \\\\\n\t\t\t\n\t\t\tTracktor++v2 \\cite{Bergmann2019}\n\t\t\t&\\textbf{56.3}&55.1&78.8&498&831&8866&\\textbf{235449}&1987&3763 \\\\\n\t\t\t\n\t\t\tLSST\\cite{Feng2019}\n\t\t\t&52.7&\\textbf{57.9}&76.2&421&801&15884&246939&3711&8757 \\\\\n\t\t\t\n\t\t\tFAMNet\\cite{Chu2019b}\n\t\t\t&52.0&48.7&76.5&450&787&14138&253616&3072&5318 \\\\\n\t\t\t\n\t\t\tYOONKJ\\cite{Yoon2020}\n\t\t\t&51.4&54.0&77.0&500&878&29051&243202&2118&\\textbf{3072} \\\\\n\t\t\t\n\t\t\tSTRN\\cite{Xu2019}\n\t\t\t&50.9&56.0&75.6&446&797&25295&249365&2397&9363 \\\\\n\t\t\t\n\t\t\tMOTDT\\cite{Chen2018a}\n\t\t\t&50.9&52.7&76.6&413&841&24069&250768&2474&5317 \\\\\n\t\t\t\n\t\t\tDEEP\\_TAMA\\cite{Yoon2019}\n\t\t\t&50.3&53.5&76.7&453&883&25479&252996&2192&3978 \\\\\n\t\t\t\n\t\t\tEDMT\\cite{Chen2017a}\n\t\t\t&50.0&51.3&77.3&\\textbf{509}&855&32279&247297&2264&3260 \\\\\n\t\t\t\n\t\t\tGMPHDOGM\\cite{Song2019}\n\t\t\t&49.9&47.1&77.0&464&895&24024&255277&3125&3540 \\\\\n\t\t\t\n\t\t\tMTDF\\cite{Fu2019}\n\t\t\t&49.6&45.2&75.5&444&\\textbf{779}&37124&241768&5567&9260 \\\\\n\t\t\t\\bottomrule\t\n\t\t\\end{tabular}%\n\t}\n\\end{table*}\n\n\\subsection{Paramater Estimation}\n\\label{subsec:paramater_estimation}\nIt should be mentioned that there is no suitable existing dataset for training our model. In this case, a specific dataset is created on our own to facilitate the training of our CRF module. The training datasets of MOT16 and MOT17 are divided into two parts: one part with $ 40\\% $ video frames is used to generate the training dataset, and the other one serves as the validation dataset. The original Tracktor++ is applied to the first part and pick out the tracking errors produced by improper inactivation for tracklets. Frames with such errors are treated as negative samples. Note that $ 3 $ times more positive samples are randomly selected. The weights of the unary and binary terms are then trained by using the SGD algorithm with a learning rate of $ 1\\times 10^{-2} $ for $ 30 $ epochs.\n\nExperimental results have shown that directly using the trained weights gave rise to unpromising performance on the validation dataset. This phenomenon stems from two sources:\n\n(\\romannumeral1) Tracktor++ itself is a powerful tracker so that the generated training dataset has limited capacity.\n\n(\\romannumeral2) The samples from the dataset do not strictly satisfy the premise of independent and identical distribution (i.i.d.).\n\nThus, we use the weights learned in the training phase as a starting point wherefrom heuristic adjustment is made. Here we first apply the initial values to the hyperparameters and manually select some obvious failing cases from the tracking results on the training dataset. We observe the value of each item in \\eqref{eq5} and \\eqref{eq6}. Then $ \\alpha_1 $ is adjusted to balance the values between normal classification scores and high ($>0.95$) classification scores. $ \\alpha_2 $ is adjusted to balance the values between classifications scores and the changing of aspect ratios. $ \\beta $ is adjusted to balance the values between the changing rate of velocity and object height. Eventually, $ \\theta_u $ and $ \\theta_u $ are adjusted so that neither the unary term nor the binary term should be too much larger then each other. In fact, $ \\alpha_1 $, $ \\alpha_2 $ and $ \\beta $ can also be trained by using the SGD algorithm instead of being treated as hyperparameters. The final values for all the parameters in our method are listed in Table~\\ref{tab2}.\n\n\n\\subsection{Practical Tracking}\n\\label{subsec:practical_tracking}\nThe overall workflow for practical tracking with public detections is shown in Algorithm \\ref{alg1}. Before performing the hypothesis filtering, all the tracklets with classification scores lower than $ 0.4 $ are inactivated. As for the predefined value $ M $ in \\ref{subsec:inference_and_training}, we need to seek a decent balance between the handling capability of the CRF and the tracking speed. Notice that it is scarcely possible for a tracker to simultaneously inactivate more than $ 5 $ objects in a single frame. In this case, we double the number and set $ M=10 $, meaning that $ 10 $ tracking hypotheses with the lowest classification scores are modelled as CRF nodes. Frames with objects less than $10$ will be processed with dummy nodes as mentioned. For tracklets whose lengths are shorter than 3, we follow the convention in Tracktor++, i.e, a threshold (0.5) of classification score is selected to determine the inactivation.\n\nTo present a fair comparison with the baseline tracker, we wrote the experimental code based on the newly released code\\footnote{https:\/\/github.com\/phil-bergmann\/tracking\\_wo\\_bnw} provided by the authors of Tracktor++. Because of a retrained CNN network, the performance of the released code surpasses that of the original paper. When comparing the experimental tracking performance in the following section, we refer to this version of tracker as Tracktor++v2, which is an alias used by its authors on the MOT16 and MOT17 results.\n\n\\begin{table*}[h]\n\t\\centering\n\t\\caption{MOT17 results (private detections)}\n\t\\label{tab5}\n\t\\resizebox{0.75\\textwidth}{!}{%\n\t\t\\begin{tabular}{cccccccccc}\n\t\t\t\\toprule\n\t\t\t&MOTA$\\uparrow$&IDF1$\\uparrow$&MOTP$\\uparrow$&MT$\\uparrow$&ML$\\downarrow$\n\t\t\t&FP$\\downarrow$&FN$\\downarrow$&IDS$\\downarrow$&Frag$\\downarrow$ \\\\\n\t\t\t\\midrule\n\t\t\tOurs\n\t\t\t&58.9&60.4&78.1\n\t\t\t&609&669&20565&208680&2544&6459 \\\\\n\t\t\t\n\t\t\tTracktor++v2\n\t\t\t&58.9&56.9&78.1&603&666&20640&208572&2853&6471 \\\\\n\t\t\t\n\t\t\t\\bottomrule\t\n\t\t\\end{tabular}%\n\t}\n\\end{table*}\n\nThe computational complexity of the CRF module is $\\mathcal{O}(n^2)$ in view of the number of CRF nodes. We test our tracker and the released code of Tracktor++ on a personal computer with an Intel i7-9700 CPU and a Nvidia GTX 2080ti GPU. The average running speed of both trackers with respect to sequence frames containing different number of tracking hypotheses is reported in Table~\\ref{tab1}.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nIn this section, we first introduce the metrics used to evaluate the MOT performance. Then, quantitative results on MOT16 and MOT17 benchmarks are presented. After that, the tracking results are shown qualitatively to provide an intuitive understanding of our method. Finally, an extensive experiment is conducted to exhibit the extensibility of our method.\n\n\\subsection{Evaluation Metrics}\n\\label{subsec:evaluation_metrics}\nEvaluating the performance of an MOT tracker reasonably is a non-trivial task. Three evaluation systems, the CLEAR MOT metrics \\cite{Bernardin2008}, the ID metrics \\cite{Ristani2016} and some other classical metrics are used to verify the validity of our method. These metrics are regrouped into two categories as follows:\n\n(\\romannumeral1) Singular metrics.\n\\begin{itemize}\n\t\\item Multiple Object Tracking Precision (MOTP). MOTP describes the precision of bounding box regression, which mainly relies on the performance of the object detector.\n\t\\item Mostly Tracked (MT). The number of trajectories that are successfully tracked for more than 80\\% of its ground truth boxes.\n\t\\item Mostly Lost (ML). The number of trajectories that fail to be tracked for more than 20\\% of its ground truth boxes.\n\t\\item False Positives (FP).\n\t\\item False Negatives (FN).\n\t\\item Identification Switches (IDS).\n\t\\item Fragments (Frag). The number of fragments in all trajectories.\n\\end{itemize}\n\n(\\romannumeral2) Compound metrics.\n\\begin{itemize}\n\t\\item\nMultiple Object Tracking Accuracy (MOTA). Considering the number of False Positives (FP), False Negatives (FN), and ID Switches (IDS), MOTA is defined as\n\\begin{equation}\n\\label{eq19}\n\\text{MOTA}=1-\\frac{\\text{FP}+\\text{FN}+\\text{IDS}}{\\text{GT}},\n\\end{equation}\nwhere GT denotes the number of ground truth boxes.\n\n\\item\nIdentification $ \\text{F}_1 $ (IDF1). IDF1 stands for the $ \\text{F}_1 $ score of ID Precision (IDP) and ID Recall (IDR):\n\\begin{equation}\n\\label{eq20}\n\\text{IDF}_1=1-\\frac{2}{\\frac{1}{IDP}+\\frac{1}{IDR}}=\\frac{2\\text{IDTP}}{2\\text{IDTP}+\\text{IDFP}+\\text{IDFN}},\n\\end{equation}\nwhere IDTP, IDFP, and IDFN stands for ID True Positives, ID False Positives, and ID False Negatives respectively. Detailed definitions can be found in \\cite{Ristani2016}.\n\\end{itemize}\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{fig4.pdf}\n\t\\caption{The deficiency to use MOTA alone. Tracklet A and tracklet B have the same MOTA score, yet the latter one is intuitively better. This is because the CLEAR MOT metrics pay more attention to the coverage rate of the ground truth boxes, while ID metrics are able to depict more details about the ID information.}\n\t\\label{fig4}\n\\end{figure}\n\nThe compound metrics are chosen as the main indicators to evaluate trackers' performance. Although MOTA is a comprehensive metric, it is not enough to judge a trackers' performance only depending on MOTA. We illustrate the reason why MOTA should be combined with IDF1 in Fig.~\\ref{fig4}. The assumption is that all the objects are perfectly recognized whereas the identifications are improperly judged. Different assumed identifications are represented by different shapes and colors.\n\n\\subsection{Quantitative Analysis}\n\\label{subsec:quantitative_analysis}\nWe compare the proposed tracking method with TOP-10 online MOT trackers\\footnote{We only consider the results on the MOT official website with published works. The TOP-10 list is up to the date when this paper was written.} with public detections on the MOT16 and MOT17 benchmarks. The experimental results are listed in Table~\\ref{tab3} and Table~\\ref{tab4}. The up-arrow denotes the higher the better and the down-arrow means the opposite. The highest scores are marked in bold.\n\nTaking MOTA and IDF1 scores as the first and second important metrics, our method achieves the best performance on both MOT16 and MOT17 benchmarks. Compared with the baseline method Tracktor++ on MOT16, the MOTA and IDF1 scores are increased by $ 0.8\\% $ and $ 3.4\\% $, respectively. On MOT17, we increase the IDF1 score by $ 2.4\\% $. In Table~\\ref{tab4}, the IDF1 score of our method performs better than the others' in evaluating a tracker's ability to maintain the identification of tracklets. This result is coherent with our improved approach for tracklet inactivation, which mitigates the problem of hypothesis drifting.\n\nWe further compare our tracker with Tracktor++ using private detections on the MOT17 benchmark, i.e., objects are detected by employing the retrained Faster R-CNN of both our tracker and Tracktor++. Such comparison is not done on MOT16 since the sequences in MOT16 are identical to those in MOT17. As is shown in Table~\\ref{tab5}, we increase the IDF1 score by $ 3.5\\% $. It is worth mentioning that the ID Switches (IDS) of our approach is less than that of the Tracktor++ in all three experimental setups, which further confirms that our method is important in stabilizing the ID information.\n\nFor a deeper insight of the experimental results, we count the number of IDS in each one of MOT17 test sequences, as is shown in Fig.~\\ref{fig5}. We see that our method alleviates the problem of ID Switch in both sequences with a static camera (01, 03, 08) and sequences with camera motion (06, 07, 12). The only test result with no IDS reduction is MOT17-14, the scenes in which rotate a lot and have a sharp left and right swing. In such a case, our assumption in subsection \\ref{subsec:feature_function_designing} does not hold well as in other video sequences.\n\nWe also make a comparison between the baseline tracker and our improved tracker in view of a significant MOT metric, the IDF1 score, in Fig.~\\ref{fig6}. Compared with IDS, the IDF1 score can better reflect the continuity and uniqueness of ID assignment and thus is an investigation focus of our method. It can be observed that our tracker achieves a higher IDF1 score than the baseline tracker in most test sequences on both MOT16 and MOT17 benchmarks. Note that our tracker is not superior on all the video sequences because of the domain diversity of MOT benchmark. The video sequences in MOT are captured under various environmental settings, such as different illumination conditions, different camera angles, with or without camera motion, etc. This means that we have to make a compromise on the parameters to achieve a generally decent performance for multiple domains. However, in real applications (e.g. intelligent surveillance), we scarcely need a tracker to cope with data from as many domains as in the MOT challenge. In other words, our method has more potential to be exploited than it seems on the benchmark performance.\n\n\\begin{figure}[h]\n\t\\centerline{\\includegraphics[width=0.5\\textwidth]{fig5.pdf}}\n\t\\caption{Statistics of IDS in MOT17 test sequences (with private detections). }\n\t\\label{fig5}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\subfloat[MOT16]{\\includegraphics[width=0.5\\textwidth]{fig6_a.pdf}}\\quad\n\t\\subfloat[MOT17]{\\includegraphics[width=0.5\\textwidth]{fig6_b.pdf}}\\quad \n\t\\caption{Comparision of IDF1 scores on MOT benchmarks.}\n\t\\label{fig6}\n\\end{figure}\n\n\\begin{figure*}[h]\n\t\\centerline{\\includegraphics[width=1.0\\textwidth]{fig7.pdf}}\n\t\\caption{Some visualized tracking results. We pick video clips from MOT17-05, MOT16-06, and MOT16-07, all of which are video sequences with camera motion. The 1st, 3rd, and 5th rows are the tracking results by Tracktor++, while the 2nd, 4th, and 6th rows are our results. By observing the target indicated by the red arrow, we see that drifting could happen when occlusion occurs. Using a discrete CRF, we resolve such problems by reasonable judgement about when to inactivate a tracklet.}\n\t\\label{fig7}\n\\end{figure*}\n\n\\subsection{Qualitative Analysis}\n\\label{subsec:qualitative_analysis}\nWe visualize some of the representative tracking results (with public detections) in Fig.~\\ref{fig7} to provide an intuitive understanding of our method. The tracking hypotheses to be noticed are marked with red arrows. For narrative convenience the hypotheses pointed by red arrows are dubbed as target people. In the 1st row, when the target person is leaving the image boundary at frame 358, its bounding box begins to drift to the man in black shirt. The tracker mistakes the fake target person as the original one and continues to track him without changing the tracking ID in later frames. By contrast, our improved tracker correctly inactivates the target person at frame 358. The newly emerging person in black shirt is then treated as a new tracking object so that a new tracking ID is assigned to him. Likewise, the target person in the 3rd row drifts to the man in suit at frame 680 and the target person in the 5th row drifts to the nearby woman at frame 779. It can be seen from the 4th and the 6th rows that our method kills such drifting trend by inactivating the tracklets in due course.\n\nWhen an occluded object gradually enters the visual field, it is supposed to be either identified as a new track by the detector or reidentified as a previous track by a CNN network. However, neither of these circumstances would be accessible when the object is compulsively assigned a wrong ID due to nearby hypothesis drifting. Even worse, the tracker is unable to be aware of such an ID error since new position is regressed merely using the position in the previous frame. Our CRF model handles this problem by telling the tracker to inactivate the tracklet promptly before drifting happens, making possible new detection and reidentification that would be suppressed otherwise.\n\nIt may be argued about the necessity to consider the inter-object relationship as we do, because hypothesis drifting can be recognized by discovering the sudden change of the position or size of a hypothesis. This argument does not hold when there is camera motion in the video sequences. On the contrary, our algorithm still works out in the existence of camera motion.\n\n\\subsection{Extensive Experiment}\n\\label{subsec:Extensive Experiment}\nDue to the fact that the proposed method is decoupled from the main tracking pipeline of Tracktor++, we can adapt it to other trackers that share a similar architecture with our baseline. To give an example, we conduct an extensive experiment on another state-of-the-art method CenterTrack \\cite{zhou2020tracking} to exhibit the extensibility of our method.\n\nCenterTrack takes both the current frame and the previous frame as input and simultaneously predicts the object locations and the inter-frame offsets. A greedy matching algorithm is utilized to associate the detections and tracklets. To apply our method, we consider the inter-object relationship and remove the associated pairs that are denied by the CRF module. Then we re-associate the remaining unassigned detections and tracklets by using the greedy matching algorithm.\n\nThe unary term of the feature function is adapted into:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq19}\n\\Phi_u(x_f,y_f)=\\textbf{1}_{\\{y_f=0\\}}\\frac{1}{D}+\\textbf{1}_{\\{y_f=1\\}}\\alpha \\mid 1-\\Delta R_f\\mid),\n\\end{aligned}\n\\end{equation}\nwhere $D$ is the calculated distance between the center of the tracked position and the biased center (i.e. taking into account the predicted offset) of the detection. The binary term of the feature function is kept unchanged.\n\nAs is done in \\cite{zhou2020tracking}, the training set of MOT17 is split into a training set and a validation set. We tune the parameters only on the training set and evaluate the performance of our improved method on the validation set. By experiment, it is found that our tracker does not make obvious improvement over the baseline in all video sequences. This is attributed to two reasons: \\romannumeral1) Our method mainly takes effect when the baseline tracker makes ID mistakes. For some of the videos, CenterTrack makes very few such mistakes. \\romannumeral2) Since this extensive experiment is designed to exhibit the potential and extensibility of our tracker, we do not deep dive into it. Therefore, we list the experimental results of three video sequences (05, 10, 11) which are relatively representative in Table~\\ref{tab6}. Compared with the baseline, our method is able to achieve a higher IDF1 score and make fewer ID switches.\n \nFor an intuitive understanding, some of the visualized results are shown in Fig.~\\ref{fig8}. With the consideration of inter-object relationship, the CRF module contributes to killing the wrong ID assignment by detecting the abnormal changes of the tracking hypotheses in a tracklet. This can be seen by observing the behavioral differences between our tracker and CenterTrack at frames 433 $\\rightarrow$ 435, frames 526 $\\rightarrow$ 528, and frames 391 $\\rightarrow$ 393 respectively in the three video clips of Fig.~\\ref{fig8}.\n\n\\begin{figure*}[h]\n\t\\centerline{\\includegraphics[width=1.0\\textwidth]{fig8.pdf}}\n\t\\caption{Some visualized tracking results of the extensive experiment. The first two video clips come from MOT17-05 the third one comes from MOT17-10. The 1st, 3rd, and 5th rows are the tracking results by CenterTrack, while the 2nd, 4th, and 6th rows are our results. The tracking targets to be noticed are indicated by red arrows.}\n\t\\label{fig8}\n\\end{figure*}\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{MOT17 results of the extensive experiment (public detections)}\n\t\\label{tab6}\n\t\\resizebox{0.48\\textwidth}{!}{%\n\t\t\\begin{tabular}{ccccccc}\n\t\t\t\\toprule\n\t\t\t\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{Seq. 05} & \\multicolumn{2}{c}{Seq. 10} & \\multicolumn{2}{c}{Seq. 11} \\\\ \\cline{2-7}\n\t\t\t&IDF1$\\uparrow$&IDS$\\downarrow$&IDF1$\\uparrow$&IDS$\\downarrow$&IDF1$\\uparrow$&IDS$\\downarrow$\\\\ \\midrule\n\t\t\tOurs&61.4&49&56.8&76&59.6&27\\\\\n\t\t\tCenterTrack&60.2&52&55.8&78&58.8&28\\\\ \\bottomrule\n\t\t\\end{tabular}%\n\t}\n\\end{table}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, the improper tracklet inactivation problem in Tracktor++ has been studied by using a discrete conditional random field to model the tracklets. The problem of varying CRF nodes has been handled, and dedicated feature functions have been designed. Our proposed algorithm is meaningful in that it can be extracted as a separate module and conveniently applied to other trackers with a similar architecture to Tracktor++, namely trackers remoulded from detectors. Experiments conducted on MOT16 and MOT17 benchmarks with both public and private detections have corroborated the effectiveness of the proposed method. The extensibility of our method is further validated by an extensive experiment on CenterTrack.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{introduction}\nMore than 30 years after the discovery of cuprate superconductors, the pairing mechanism in these materials still remains unknown. The observation of renormalization effects in the low energy electronic excitations in ARPES has re-ignited the hope that a bosonic mode playing a role in pairing in cuprates could finally be identified, in analogy with how tunneling experiments provided the smoking gun evidence for phononic mechanism in conventional superconductors \\cite{McMillan1965}. However, after two decades of intense research, the debate about the coupling mechanism is still open \\cite{Valla1999,Johnson2001a,Gromko2003,Carbotte2011,Park2013,Li2018,He2018}. One problem was that early studies were focused on the nodal \"kink\" that did not show any significant correlations with superconductivity when the latter was altered by doping or when different cuprate families were compared. Another problem is that cuprates are fundamentally different from simple metals in which superconducting transition occurs from a conventional Fermi liquid metallic state into a state well described by the BCS theory \\cite{Bardeen1957a,Bardeen1957}. Parent compounds of cuprate superconductors are antiferromagnetically ordered Mott insulators wherein conduction and superconductivity are induced by doping additional holes or electrons away from the half filled case \\cite{Lee2006}. The effects of strong correlations extend far away from half filling, deep into the regime that overlaps with superconductivity, where their presence and intertwining with superconductivity complicates the identification of the superconducting mechanism. Therefore, it would be desirable to study superconducting properties in the highly overdoped regime where such effects are absent or strongly reduced.\n\nBi2212 has been a perfect subject of ARPES studies due to its ease of cleaving, a high transition temperature ($T_\\mathrm{c}$), and a large superconducting gap. However, Bi2212 could only be doped within a relatively limited range on the overdoped side, where $T_c$ could not be reduced below $\\sim50$ K, leaving a crucially important region of the phase diagram, where $T_\\mathrm{c}\\rightarrow{0}$, out of reach. Only very recently, has it become possible to extend the overdoped range beyond the point at which superconductivity vanishes by annealing the \\textit{in-situ} cleaved samples in ozone \\cite{Drozdov2018}. For the first time, this has made it possible to monitor the development of electronic excitations as superconductivity weakens and finally completely disappears, allowing a closer look at its origins. \n\n\\section*{Results}\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=9cm]{Fig1.png}\n\\caption{Strongly overdoped regime of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$. (A) Phase diagram near the edge of the superconducting dome, as determined from ref. \\cite{Drozdov2018}. $T_\\mathrm{c}$ and $\\Delta_0$ for the doping levels from this study are indicated by the black and red solid squares, respectively. (B) Fermi surface ($E=0$ contour) of the overdoped, non-superconducting sample, corresponding to $p=0.29$ and (C) of the $T_\\mathrm{c}=72$ K sample, corresponding to $p=0.23$. Maps in (B) and (C) were recorded at $T=12$ K. \n}\n\\label{Fig1}\n\\end{center}\n\\end{figure}\n\nFigure 1(A) shows the overdoped region of the Bi2212 phase diagram from ref. \\cite{Drozdov2018}, along with the four doping levels from the present study. \nIn this region, the pseudogap is no more present, according to the previously published studies \\cite{Renner1998,Ozyuzer2000,Yusof2002,Gomes2007,Benhabib2015} and the remaining  superconductivity becomes more conventional with the gap saturating at the BCS value $2\\Delta_0=4.28k_BT_c$ for $p>0.25$ \\cite{Drozdov2018,He2018}. \nThe as grown OD91 $(p=0.2)$ sample was cleaved in vacuum and annealed in ozone, resulting in increased doping, $p=0.29$, and a complete loss of superconductivity. The Fermi surface of the resulting sample is shown in Fig. 1(B). That same sample is then annealed in vacuum at different temperatures, ranging from 110 to 175$^{\\circ}$ C in order to gradually reduce the doping and increase $T_\\mathrm{c}$ to 38, 50 and 72 K. The intensity at the Fermi level of the same surface after the final annealing is shown in Fig. 1(C). Due to the large superconducting gap ($\\Delta_0=17$ meV), the photoemission intensity is concentrated near the nodes. The doping level in each case is determined independently from the Luttinger count of the area enclosed  by the Fermi contour, $p_\\mathrm{L}=2A_\\mathrm{FS}$. The doping $p$ that serves as the abscissa in phase diagrams of the cuprates, (the doping away from the half-filling) is expressed as $p=p_\\mathrm{L}-1=2A_\\mathrm{FS}-1$ with both the bonding and the antibonding states counted, $A_\\mathrm{FS}=(A_\\mathrm{B}+A_\\mathrm{A})\/2$. The area of the Brillouin zone (BZ) is set to one. \nAlso shown are the Fermi surface contours of the tight-binding (TB) in-plane band structure that best describe the measured ones, as described in the Methods section. \n\nThe antinodal gap magnitude $\\Delta_0$ is determined at the base temperature $(T\\approx 12$ K) from the quasiparticle peak position at $(0, \\pm k_F)$, while the transition temperature $T_\\mathrm{c}$ is determined as the temperature at which the gap closes. The points from the present study shown in Fig. 1(A) follow the trends from our previous study \\cite{Drozdov2018}. Indeed, for the initial ozone annealed surface, that shows no superconductivity within our detection limits, the Van Hove singularity of the antibonding state sits exactly at the Fermi level. \n\n\\begin{figure*}[htpb]\n\\begin{center}\n\\includegraphics[width=12cm]{Fig2.png}\n\\caption{Coupling strength in the overdoped Bi2212 as a function of doping. (A) Electronic structure of Bi2212 near the antinode along the momentum line indicated in Fig.\\ref{Fig1}(B) at low temperature ($T\\sim10$ K) for overdoped, non-superconducting sample. The spectra corresponding to the three overdoped superconducting samples with $T_\\mathrm{c}=38$ K, $T_\\mathrm{c}=50$ K and $T_\\mathrm{c}=72$ K taken in the superconducting state (C, E, G) and normal state (D, F, H). The MDC-fitted dispersions of the bonding state are indicated by the black, blue, red, green and gray curves. The TB dispersions are indicated by the solid white curves. The dashed white curve in (G) represents the TB dispersion gapped by $\\Delta_0=17$ meV. (B) The same measured dispersions, referenced to the corresponding gap value. The momentum scale is referenced to $k_\\mathrm{F}$. The dispersions corresponding to superconducting states are offset in $k$ by 0.01 \\AA, consecutively. Spectra in (A), (C), (E) and (G) were recorded at $T=12$ K and those in (D), (F) and (H) at 45, 60 and 90 K, respectively.\n}\n\\label{Fig2}\n\\end{center}\n\\end{figure*}\n\nThis is also illustrated in Fig. 2(A) that shows the photoemission intensity along the momentum line $k_\\mathrm{y}=\\pi\/a$ indicated by the yellow line in Fig. 1(B). The state at $(0,\\pi\/a)$ is the bottom of the antibonding band that undergoes a Lifshitz transition at that doping level $(p=0.29)$. The remaining state, that crosses the Fermi level at $k_\\mathrm{F}=\\pm 0.144$ \\AA\\  is the bonding state. Its dispersion (black curve), extracted by fitting the momentum distribution curves (MDC), does not show any features that would indicate a structure in the self energy and a renormalization in the form of a \"kink\".  Still, the dispersion is slightly renormalized compared to the TB approximation that was used for the Fermi surface contour (Fig. 1(B)). The state is gapless and does not show any particle-hole mixing expected for Bogoliubov's quasiparticles in the superconducting state. With vacuum annealing and a reduction in hole doping, superconductivity develops and the spectra display the spectral gap at low-temperatures (panels C, E and G). Simultaneously, the photoemission shows a back-folding of the spectral intensity near the $k_\\mathrm{F}$, typical for Bogoliubov's quasiparticles. However, the most important discovery here is an anomaly, or an abrupt change of slope (\"kink\") in the state's dispersion that occurs slightly below the state's maximum at $k_\\mathrm{F}$. This can be seen in the MDC-derived dispersions, represented by blue, red and green curves for the samples with $T_\\mathrm{c}$ of 38, 50 and 72 K, respectively. When plotted on the same scale and referenced to the corresponding gap magnitude, panel (B), these dispersions indicate clear trends in their low-energy behavior: as superconductivity strengthens and $T_\\mathrm{c}$ and $\\Delta_0$ increase, the \"kink\" becomes progressively more pronounced and shifts to higher energies. Notably, the \"kink\" is present only in the superconducting state with no traces of the structure left above $T_\\mathrm{c}$, as can be seen in the corresponding normal state spectra taken approximately 10 K above $T_\\mathrm{c}$ (panels D, F and H). \nThis is highly unusual and, as already noted in previous studies \\cite{Valla2000,Gromko2003,Li2018}, cannot be reconciled with the conventional effects stemming from the electron-phonon coupling. If the \"kink\" was due to the conventional electron-phonon coupling that is at play in 2H-NbSe$_2$ and intercalated graphite, for example, it would have to be present not only in the superconducting state, but also should exist in the normal state \\cite{Valla2004,Valla2009}, as illustrated in Fig. 3(E).  \n \nTo quantify the trends observed in Fig. 2, we plot the kink's characteristic energy, $\\Omega_0$, corresponding to the maximum in Re$\\Sigma$ and its strength, approximated by $\\lambda=-\\frac{\\partial\\mathrm{Re}\\Sigma(\\omega)} {\\partial\\omega} \\vline_{(\\Omega_0<\\omega<\\Delta_0)}$, in Figure 3. In addition, we re-plot the corresponding maximal gap, $\\Delta_0$, and show the energy of the resonance mode, $E_\\mathrm{r}$, and a spin gap, $\\Delta_\\mathrm{Spin}$, from the inelastic neutron scattering studies \\cite{Fong1999,He2001,Capogna2007,Fauque2007,Xu2009,Li2018a}. The energy of the B$_\\mathrm{1g}$ phonon is also indicated, noting that it does not show a significant doping dependence \\cite{Opel1999}. \nWe note that a weak featureless renormalization remains at $p=0.29$ and in the normal state of superconducting samples. That component does not display any doping dependence in the studied range. We call the corresponding slope of Re$\\Sigma$ the critical coupling, $\\lambda_\\mathrm{c}$ as the $p=0.29$ sample sits exactly at the superconducting boundary. \n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=15cm]{Fig3.png}\n\\caption{Doping dependence of the antinodal renormalization effects. (A) Re$\\Sigma$ for four samples shown in Fig.\\ref{Fig2} obtained by subtracting the bare TB dispersion, gapped by the corresponding $\\Delta_0$, from each measured dispersion. The curves are referenced to the Fermi level and those obtained in superconducting state are offset in $y$ by 30 meV for clarity. (B) coupling strength $\\lambda$, approximated as $\\lambda=-\\frac{\\partial\\mathrm{Re}\\Sigma(\\omega)} {\\partial\\omega}\\vline_{(\\Omega_0<\\omega<\\Delta_0)}$ (red diamonds), plotted vs. doping. The normal state  value, $\\lambda_\\mathrm{c}\\approx1.3$, is indicated by the red line. Corresponding $T_\\mathrm{c}$ is also shown (black squares). (C) Kink's energy, $\\Omega_0$, as measured from the corresponding gap value (energy of the maximum in the state's dispersion) (red diamonds). Corresponding gap magnitude, $\\Delta_0$ (red circles) of the studied samples and antiferromagnetic resonance energy, $E_\\mathrm{r}$ (red triangles), and spin gap, $\\Delta_\\mathrm{Spin}$ (red crosses), from references \\cite{Fong1999,He2001,Capogna2007,Fauque2007,Xu2009,Li2018a} are also shown. (D) Dependence of $T_\\mathrm{c}$ on the antinodal coupling strength, $\\lambda$, measured in the superconducting state. The solid curve represents the fit to the power-law behavior, $T_\\mathrm{c}\\propto(\\lambda-\\lambda_\\mathrm{c})^p$ for the four overdoped samples. The dashed curve is the extrapolation from the fitted region. The as grown sample ($T_\\mathrm{c}=91$ K) was not used in fitting. (E) Schematic view of temperature development of the electronic dispersion upon transition from the normal state (NS) to superconducting state (SCS) in the conventional coupling scenario (top, shaded) and the actual one, observed in cuprate superconductors (bottom).\n}\n\\label{Fig3}\n\\end{center}\n\\end{figure*}\n\n\\section*{Discussion}\nIt is obvious that both the strength of the anomaly and its energy are strongly doping dependent, both following $T_\\mathrm{c}$ and and vanishing exactly when superconductivity disappears. This represents very strong evidence that the antinodal kink is very closely related to superconductivity. The fact that $\\Delta_0$ and the observed coupling follow $T_\\mathrm{c}$ and essentially vanish together at the overdoped side is a clear indication that the superconductivity itself turns conventional in that region of the Bi2212 phase diagram and that it is governed by the weakening coupling, rather than by the superfluid density, as suggested by ref. \\cite{Bozovic2016}. \n\nThe antinodal dispersion anomaly also occurs in the $k$-space region where the superconducting gap and pairing are the strongest \\cite{Gromko2003,Li2018}. The fact that it only exists in the superconducting state also provides additional clues for understanding its origin. In that, the antinodal kink is strikingly different from the nodal kink, which does not vary significantly with doping or amongst different cuprate families \\cite{Lanzara2001,Johnson2001a,Ino2013,Park2013}. The apparent lack of correlation of the nodal kink with $T_\\mathrm{c}$ suggests its relative unimportance in superconductivity. The nodal kink is also different in that it exists in both the normal and superconducting states, with only a relatively small change upon the transition,  allowing the possibility that it might be phonon related. In contrast, the strong doping dependence and the simultaneous disappearance of the antinodal kink with superconductivity would require that strong changes in the coupling and in the phonon spectrum itself occur with doping and temperature, if the kink had phononic origin. This has not been observed \\cite{Opel1999}.\n\nThe recent study on the same material reports that the coupling strength has a similar trend with doping \\cite{He2018}. However, that study assigns the observed effects, i.e. the development of the \"peak-dip-hump\" structure in the spectra at ($\\pi,0$), to the coupling to B$_\\mathrm{1g}$ phonon whose energy does not vary with doping ($\\omega_0=37$ meV). Also, the study does not address a lack of the coupling above $T_\\mathrm{c}$. We note that our results, showing strong doping dependence of $\\Omega_0$ and a striking change between the superconducting and normal state spectra rule out the possibility that the involved mode is a phonon. As illustrated in Fig. 3(E), if caused by phonons, kink should be present in both the normal and superconducting states.\n\nThe second bosonic candidate that is often considered as the origin of the observed quasiparticle kink is the so called spin resonance \\cite{Fong1999,Abanov1999,Eschrig2000,He2001,Sandvik2004,Eremin2005,Capogna2007,Dahm2009}. The energy of that mode, $E_\\mathrm{r}$, shows the doping dependence with the same trend as the energy of the kink studied here. Also, its temperature dependence is similar, with both phenomena existing only in the superconducting state. However, as Fig. 3(C) shows, there is a significant mismatch between the energies of the two features. The overlapping point between the neutron scattering and ARPES data, corresponding to the $T_\\mathrm{c}\\approx70$ K sample, would suggest that the $\\Delta_0+\\Omega_0$ scale from ARPES is a better match to $E_\\mathrm{r}$. However, that clearly would not work near the optimal doping. We also note that the momentum and energy conservation rules would have to place the antinodal kink near the energy of the involved mode (as measured from top of electronic dispersion at $\\Delta_0$), particularly if the mode scatters from the antinode to the antinode (small $Q$, or $Q\\approx(\\pi,\\pi)$). This is why a much better candidate for the relevant excitation seems to be the onset of spin fluctuation spectrum, i.e. the spin gap ($\\Delta_\\mathrm{Spin}$), rather than the resonance mode at $E_\\mathrm{r}$ \\cite{Li2018a}. The excitations at the spin gap could explain not only the kink's doping, temperature and momentum dependence, but also the differences between the different families of cuprates - most notably those between Bi2212 and La$_{2-x}$Sr$_x$CuO$_4$. These two materials have very similar scales for $E_\\mathrm{r}$, but a much smaller $\\Delta_\\mathrm{Spin}\\approx8$ meV in La$_{2-x}$Sr$_x$CuO$_4$ near optimal doping would definitely make the observation of a coherent quasiparticle peak and a kink in its dispersion very difficult, in agreement with ARPES measurements \\cite{Yoshida2016}.\n\nAt the end, the remarkable correlation between $T_\\mathrm{c}$ and coupling strength from Fig. 3(B) could offer an interesting insight into the question if the transition temperature in cuprates might reach a limit when coupling gets very strong. When plotted as a function of $\\lambda$, transition temperature displays approximately a square-root behaviour on $(\\lambda-\\lambda_\\mathrm{c})$ in the overdoped regime (Fig. 3(D)). This is a good news and an indication that $T_\\mathrm{c}$ in cuprates does not have a natural limit in the coupling strength itself. However, on the underdoped side, there are many phenomena that limit $T_\\mathrm{c}$, even when coupling is finite, some of these probably being caused by the strong coupling observed here. The point corresponding to the $T_\\mathrm{c}=91$ K sample, laying below the extrapolated curve, indicates that this region might already be affected. \n\n\\section*{Methods}\n\n\\subsection{Sample Preparation}\nThe experiments within this study were done in a new experimental facility that integrates oxide-MBE with ARPES and scanning tunneling spectroscopy (STM) capabilities within the common vacuum system \\cite{Kim2018a}. The starting sample was a slightly overdoped ($T_c=91$ K) single-crystal of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$, synthesized by the traveling floating zone method. It was clamped to the sample holder and cleaved with Kapton tape in the ARPES preparation chamber (base pressure of $3\\times10^{-8}$ Pa). The silver-epoxy glue, commonly used for mounting samples, as well as the need for its processing at elevated temperatures, have been completely eliminated, resulting in perfectly flat cleaved surfaces and unaltered doping level. The cleaved sample was then transfered to the MBE chamber (base pressure of $8\\times10^{-8}$ Pa) where it was annealed in $3\\times10^{-3}$ Pa of cryogenically distilled O$_3$ at 350-480$^{\\circ}$C for $\\approx$ 1 hour. After the annealing, sample was cooled to room temperature in the ozone atmosphere and transfered to the ARPES chamber (base pressure of $8\\times10^{-9}$ Pa). No spectral gap was detected down to the base temperature (12 K) and the doping level determined from the area of the Fermi surface was $p=0.29$. Reduction in doping was achieved by subsequent annealing of the same surface in vacuum to temperatures ranging from 110 to 175$^{\\circ}$C, resulting in development of superconductivity with increasing $T_\\mathrm{c}$. \n\n\\subsection{ARPES}\n\nThe ARPES experiments were carried out on a Scienta SES-R4000 electron spectrometer with the monochromatized HeI (21.22 eV) radiation (VUV-5k). The total instrumental energy resolution was $\\sim$ 4 meV. Angular resolution was better than $\\sim 0.15^{\\circ}$ and $0.3^{\\circ}$ along and perpendicular to the slit of the analyzer, respectively. \n\nThe annealing of cleaved surfaces in ozone results in increased doping only in the near-surface region, while the subsequent annealing in vacuum reduces it. Therefore, aside from the as-grown sample, the only measure of $T_c$ in the near-surface region was spectroscopic: the temperature induced changes in the quasiparticle peak intensity, as well as the leading edge position indicate $T_c$ \\cite{Damascelli2003,Kondo2015}. The leading edge gap and intensities of the QP peak and at the Fermi level all show a prominent change around $T_c$ and the later could be identified as being near the inflection point of these temperature dependencies \\cite{Kondo2015}. The ARPES estimate of $T_\\mathrm{c}$ was within $\\pm4$ K, except for the sample falling outside of the superconducting dome, for which the estimate was limited by the base temperature that could be reached with our cryostat ($T_\\mathrm{c}<12$  K).\n\n\\subsection{As-grown sample}\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig4.png}\n\\caption{As grown Bi2212 sample ($T_\\mathrm{c}=91$ K). (A) Electronic structure near the antinode along the momentum line indicated in Fig. 1(B) at low temperature ($T\\sim12$ K) for the as-grown Bi2212 sample. The MDC-fitted dispersions of the bonding state is indicated by the blue curve. The TB dispersion is indicated by the solid red curve. The dashed red curve represents the TB dispersion gapped by $\\Delta_0=35$ meV. (B) The energy distribution curves corresponding to the $k_\\mathrm{F}$ (black) and the momentum indicated by the red vertical arrow in (A). The horizontal black arrow indicates the ``dip'' in the intensity.\n}\n\\label{Fig4}\n\\end{center}\n\\end{figure}\n\n\nThe spectra for the as-grown, slightly overdoped ($T_c=91$ K) sample (Fig. 4) cannot be reliably analyzed in the same manner as the spectra for highly overdoped samples. The MDC analysis returns a well defined result for the state's dispersion in the low-energy range and in the high-energy range, but not in the vicinity of the kink. This is partially due to the fact that on the particle-like side ($|k|<k_\\mathrm{F}$) of the renormalized Bogoliubov's dispersion, the two sides corresponding to negative and positive momenta, merge and form a continuous renormalized dispersion, with the bottom at $k_x=0$ that could be shallower than the energy of the re-normalizing mode. Also, the intensity from the antibonding state and super-modulation replicas partially overlaps with the fitted state and the MDC fitting is unstable and often shows a sharp discontinuity near the kink energy. Obviously, the energy of the kink cannot be  precisely established by using the MDC analysis, whereas the low-energy slope, that serves for determination of the coupling strength $\\lambda$, can still be correctly determined. Therefore, for the lower limit of the mode's energy we use the energy at which the MDC derived dispersion (blue curve in Fig. 4(A)) shows a discontinuity. As its upper limit, we use the energy at which the energy distribution curves show a ``dip'' (Fig. 4(B)).  This energy coincides with the energy within which the hole-like portion($|k|>k_\\mathrm{F}$) of the Bogoliubov's dispersion shows the ``heavy'', renormalized character.  That part  of the renormalized Bogoliubov's dispersion could be traced all the way to the kink's energy at which the state quickly disappears due to the coherence factors and the onset of strong scattering on the involved mode. We therefore estimate $\\Omega_0=29\\pm4$ meV for the as-grown sample, displayed in Fig. 3(C).\n\n\\subsection{Tight Binding Parameters}\n\nThe bare in-plane band structure of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$ is approximated by the tight-binding formula:\n\n$E_{A,B}(k) =\\mu - 2t (\\cos k_x + \\cos k_y) + 4t' \\cos k_x \\cos k_y - 2t'' (\\cos 2k_x + \\cos 2k_y) \\pm t_{\\perp} (\\cos k_x - \\cos k_y)^2 \/4$, \n\nwhere the index A (B) is for anti-bonding (bonding) state and $\\mu$ is chemical potential. The hopping parameters that best describe the Fermi surfaces of the measured samples are kept fixed at $t=0.36$, $t'=0.108$, $t''=0.036$ and $t_{\\perp}=0.108$ eV, with only the chemical potential being varied from 0.467 eV, for the non-superconducting sample to 0.425 eV, for the $T_\\mathrm{c}=72$ K sample.\nThe TB contours that agree with the experimental contours the best were chosen by eye. By changing them to the point where discrepancies would become clearly visible, we can estimate that the uncertainty in doping, $\\Delta p_\\mathrm{A}$, of this method is very close to that estimated from the experimental momentum width of the Fermi surface, $\\Delta p_\\mathrm{A}\/p_\\mathrm{A}\\sim2\\Delta k_\\mathrm{F}\/k_\\mathrm{F}$ .\n\n\\subsection{Other candidates for the observed renormalization}\n\nIn the following, we discuss some other possibilities for the renormalization effects observed in the antinodal region of Bi2212. One candidate with the proper trend that mimics the kink's energy is the position of van Hove singularity (vHS) of the antibonding band. A significant amount of interband scattering (elastic or inelastic) would affect the lifetime of the probed bonding state as the vHS of the antibonding state moves with doping.  However, the interband scattering would have an opposite effect of what has been seen: the interband channel (if important) would make the state broad(er) where it is open and the state would be narrower where the channel is closed (below the vHS of the antibonding band) Also, as can be seen in Fig. 2G, the kink is significantly deeper than the renormalized bottom of the antibonding band.  In addition, just as with phonons, the effect should not disappear in the normal state.\n\nAnother candidate that could possibly have similar effects on the measured quasiparticle dispersion and its lifetime is the superconducting gap itself. The observed $\\Omega_0$ is very close to $\\Delta_0$ and the reduction of a phase space for scattering related to the opening of the gap, would make the states sharp within a certain energy range, with details depending on the gap symmetry. In the $s$-wave gap, the kink should appear at $\\sim3\\Delta_0$ (or $\\sim2\\Delta_0$, measured from the top of quasiparticle dispersion $\\Delta_0$), if it was caused by the pair-breaking. This might not be strictly valid for the $d$-wave gap, where the scattering could involve the node-antinode mixing.  However, the strength of the antinodal kink weakens rapidly as one moves from the antinode, implying that the mode scatters antinode to the antinode. Therefore, the mode's momentum has to be either $Q\\approx0$, or $Q\\approx(\\pi,\\pi)$, effectively excluding the node to antinode mixing and the pair-breaking as its origin.\n\n\\section*{Data availability} \nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGraphs appear in a variety of situations, ranging from computer science and the physical sciences to economics and social sciences  \\cite{Barabasi1999,clrs2001algorithms,Otte2002}. For example, graphs model the flow of computation in an algorithm, the structure of a molecule or the molecular graph of a photosynthetic light-harvesting organism \\cite{Rebentrost2009}, the supply chain of an online retailer, or the network of people on social media. Graphs consist of vertices representing people, atoms, or warehouses, and edges representing connections between the vertices. Solving problems on graphs can help identifying highly connected influencers in a social network, improving the efficiency of solar cells \\cite{Wodo2012}, or reducing the time till a customer receives an online order.\n\nCommon computational problems on graphs involve finding subgraphs with defined properties such as high connectivity (cliques), routing problems such as the traveling salesman problem, network flow problems, or the coloring of graphs \\cite{west2001introduction}.\nMany graph problems are easy to compute, that is they run in time that is at most a polynomial in the number of vertices and edges. For example, the shortest path in a graph or the maximum matching in a graph can be solved in polynomial time.\nOther problems are famously hard to solve \\cite{karp1972combinatorial}, running in a time that is exponential in either the number of vertices or the number of edges. Examples include the NP-complete clique  and traveling salesman problems.\n\n\nQuantum algorithms promise to solve certain problems much faster than classical algorithms \\cite{Montanaro2016}. Recently, it was discovered that the output probability distribution of photons from a multiport whose input is a Gaussian squeezed state~\\cite{hamilton2016gaussian} depends on the~\\emph{hafnian} of a  sampling matrix. The method, called Gaussian Boson Sampling, is a natural generalization of the original photonic boson sampling~\\cite{aaronson2011computational}, and its modifications~\\cite{lund2014boson,huh2017vibronic,huh2017vibronic,spring2012boson,olson2015sampling}, where single photons or vacuum are input states of a linear interferometer. The hafnian was introduced in the context of interacting quantum field theory~\\cite{caianiello1953quantum} and belongs to the class of matrix functions together with the permanent, Pfaffian or determinant~\\cite{minc1984permanents}. Unlike the latter two functions, the permanent and the hafnian are believed to be hard to calculate since they belong  to the $\\#$P-complete complexity class and both functions (the permanent in particular) have been studied in detail~\\cite{glynn2010permanent,valiant1979complexity,troyansky1996quantum,jerrum2004polynomial,rudelson2016hafnians,bjorklund2012counting,barvinok2017combinatorics,scheel2004permanents}. The class $\\#$P involves counting problems such as how many solutions exist to a\ngiven problem in the class~NP.\n\n\n\nIn this work, we connect Gaussian Boson Sampling to graph theory in order to tackle graph problems that involve the number of perfect matchings~\\cite{lawler1976combinatorial,wolsey1999combinatorial}. This connection is based on the fact that  the number of perfect matchings of an undirected graph\nequals to the hafnian of the graph's adjacency matrix. We propose an efficient way to embed an arbitrary adjacency matrix into a valid Gaussian state.\nThe desired Gaussian state can be generated by injecting thermal states or vacuum states into a circuit consisting of single-mode squeezers and a multiport. The information of the\nrequired input states, single-mode squeezers and the multiport is completely determined by the adjacency matrix. Thus, the whole procedure is programable\nand is straightforward to implement, and therefore can allow the estimation of the number of perfect matchings in arbitrary graphs. \nPromisingly, by appropriate rescaling of the problem and  linearly increasing the size of the circuit we can substantially improve the sampling success probability. \nWe demonstrate our methods for the special cases of complete graphs and one-edge-removed complete graphs.\n\nThis paper is organized as follows. Sec. \\ref{sec:results} summarizes the main results. \nWe first discuss embedding an adjacency matrix into a mixed or pure Gaussian state, and connect \nthe hafnian of the adjacency matrix to the measurement probability. We then discuss the hardness of estimating the hafnian using Gaussian Boson Sampling and propose \nstrategies to amplify the measurement probability. We finally discuss a complete graph example to illustrate the validity of the method. We conclude in Sec. \\ref{sec:conclusion}. \n\n\\section{Results}\\label{sec:results}\n\\subsection{Gaussian Boson Sampling and hafnian}\nAs shown in \\cite{hamilton2016gaussian} the output probability distribution of photons from the multiport with a Gaussian input state is related to the hafnian. In particular, the authors showed that the probability of the photon number distribution, with no more than one photon in each output mode, is related to the hafnian of the submatrices of a matrix $A$, derived from the Gaussian covariance matrices $\\sigma_A$. For an arbitrary $M$-mode Gaussian state,\n$A$ is related to $\\sigma_A$ via \\cite{hamilton2016gaussian}\n\\begin{equation}\\label{eq:sigmaA-1}\nA=X_{2M} \\big[ \\mathbb{I}_{2M}- (\\sigma_A+\\mathbb{I}_{2M}\/2)^{-1} \\big],\n\\end{equation}\nwhere \n$\\mathbb{I}_{2M}$ is the identity matrix and $X_{2M}$ is defined as\n$\nX_{2M}=\\begin{bmatrix}\n           0 & \\mathbb{I}_M \\\\\n           \\mathbb{I}_M & 0 \\\\\n         \\end{bmatrix}.\n$\nIn more detail, assume that $n = \\bigotimes_j^M n_j | n_j\\rangle \\langle n_j|$ represents a certain measurement pattern of photons, where $n_j$ is the photon number in the $j$-th mode. In the case $n_j = \\{ 0, 1\\}$, the measurement probability is given by~\\cite{hamilton2016gaussian}\n\\begin{equation}\\label{eq:ProHafOrig}\n  \\mathrm{Pr} (\\bar n) = {1\\over \\bar n! \\sqrt{\\det{\\sigma_Q}}} \\mathop{{\\mathrm{haf}}}\\nolimits{A_S},\n\\end{equation}\nwhere $\\bar n = n_1! n_2! \\dots n_M!$, $\\sigma_Q \\equiv \\sigma_A + \\mathbb{I}_{2M}\/2$ and $A_S$ is the submatrix of $A$ determined by the position of the output modes registering single photons.\nIn particular, finding the probability of detecting a single photon in every output mode corresponds to estimating the hafnian of the full matrix $A$.\n\n\\begin{figure}[]\n\\includegraphics[width=8.6cm]{fig_PM4.pdf}\n\\caption{ a) Complete graph of four vertices. b) Three perfect matchings of the four vertex complete graph. }\n\\label{fig:PM}\n\\end{figure}\n\nA perfect matching of a graph is when every vertex of the graph is matched up with a single unique other vertex. Often there are multiple ways of perfectly matching vertices, as demonstrated for a small example in Fig.~\\ref{fig:PM}.  Estimating the number of such perfect matchings is known to be difficult for a classical computer as the size of the graph increases \\cite{bjorklund2012counting, Cygan2015}.\nIt is also known that the number of perfect matchings is equal to the hafnian of a graph's adjacency matrix. Motivated from Eq. \\eqref{eq:ProHafOrig},\nwe ask whether it is possible to estimate the number of perfect matchings via measuring the probability of the photon number distribution.\n\nThe key result of this work is to map general graphs to the optical setup of Gaussian Boson Sampling. We start first with inverting Eq.~(\\ref{eq:sigmaA-1}) as\n\\begin{equation}\\label{eq:sigmaA}\n\\sigma_A=(\\mathbb{I}_{2M}-X_{2M}A)^{-1}-{\\mathbb{I}_{2M}\\over2}.\n\\end{equation}\nGiven this relation, a naive idea is to consider $A$ as the adjacency matrix of a graph and\ninsert it into Eq. (\\ref{eq:sigmaA}) to find the corresponding covariance matrix. Unfortunately, this fails because a general adjacency matrix\ndoes not map to a valid covariance matrix through Eq. (\\ref{eq:sigmaA}). An adjacency matrix is a real symmetric matrix and has entries either zero or one. The hafnian\nof a large size adjacency matrix is usually a large number, e.g., a complete graph with $2M$ vertices has $(2M-1)!!$ perfect matchings. \nIntuitively, Eq. (\\ref{eq:ProHafOrig}) does not hold if $A$ is simply regarded as an adjacency matrix\nbecause the probability is a number smaller than one. To make things reasonable one has to rescale the adjacency matrix by a small number.\n\nWe develop a systematic method to map an arbitrary adjacency matrix of an undirected graph to a valid covariance matrix of a Gaussian state (see Appendix \\ref{sec:GraphToCM} for details). We associate the number of perfect\nmatchings with the probability of a photon number distribution. A covariance matrix is by its nature a Hermitian and positive-definite matrix, and satisfies the uncertainty principle \\cite{simon1994quantum}. Following these three constraints, we vary an arbitrary adjacency matrix such that it maps to a valid covariance matrix.\n\n\n\\subsection{Sampling mixed Gaussian covariance matrices}\\label{sec:SampleMixState}\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=15cm]{fig_3DMixedCircuit.pdf}\n\\caption{\\small $(M+N)$-mode circuit that generates $M$-mode mixed Gaussian covariance matrices $\\sigma_{cA}$. $N$ input thermal states are generated by\ntracing out the other $N$ modes of $N$ two-mode squeezed vacuum states with squeezing parameters $\\xi_k$.\nThen $N$ single-mode squeezers squeeze the thermal states and the other\n$(M-N)$ single-mode squeezers with squeezing parameters $r_k$ squeeze the vacua. Photon number detectors measure the state after the linear optics network $K$. }\n\\label{fig:MixCircuit}\n\\end{figure*}\n\nHere, we discuss the conditions under which we can map a graph adjacency matrix $A$ to a mixed Gaussian covariance matrix.\nA $2M\\times 2M$ adjacency matrix $A$ can be written in a block form, with four $M \\times M$ submatrices $A_{11}$, $A_{12}$, $A_{21}$ and $A_{22}$. If these submatrices satisfy\nthe following conditions: (1) $A_{11} = A_{22}$ and $A_{12} = A_{21}$, (2) $A_{11}$ commutes with $A_{12}$, (3) $A_{12}$ is positive definite; then $c A$ maps to a valid\ncovariance matrix $\\sigma_{cA}$ via Eq. (\\ref{eq:sigmaA}), where $c$ is a rescaling parameter satisfying $0 < c < 1\/\\lambda_1$ and $\\lambda_1$ is the maximal eigenvalue of $A$.\nGenerally, $\\sigma_{cA}$ is an $M$-mode mixed Gaussian covariance matrix. \nThis can be seen from its symplectic eigenvalues \\cite{weedbrook2012gaussian} (see also Appendix \\ref{app:PPE}), given by\n\\begin{equation}\\label{SymplecticEV}\n\\nu_k = \\frac{1}{2} \\sqrt{\\frac{(1+c h_k)^2 - c^2 f_k^2}{(1-c h_k)^2 - c^2 f_k^2}} \\ge \\frac{1}{2},\\quad k = 1, 2, \\hdots, M,\n\\end{equation}\nwhere $f_k$ and $h_k$ are eigenvalues of $A_{11}$ and $A_{12}$, respectively. Note that $f_k \\pm h_k$ are eigenvalues of the adjacency matrix $A$. \nA Gaussian state is pure if $\\nu_k = 1\/2$ for all $k$ \\cite{weedbrook2012gaussian}. From Eq. (\\ref{SymplecticEV}), this happens \nwhen all eigenvalues of $A_{12}$ are zero, namely, $A_{12}=0$.\n\nPreparing a general mixed Gaussian state is not usually an easy task. According to Williamson's theorem~\\cite{williamson1936algebraic},\na covariance matrix can be put into a diagonal form by a symplectic transformation.\nThe symplectic transformation can be further decomposed into a product of an orthogonal matrix $K$, a direct sum of $M$ single-mode squeezing matrices\nand another orthogonal matrix $L$. This implies a mixed Gaussian state can be generated by injecting single-mode thermal states (or vacuum) into an $M$-mode\nlinear optics network $L$, passing through $M$ single-mode squeezers and then another linear optics network $K$.\n\nWe observe that to produce the covariance matrix $\\sigma_{cA}$, the orthogonal matrix $L$ can be set to an identity. Therefore the state before entering the linear\noptics network $K$ is a product of squeezed thermal states or squeezed vacuum states. The thermal state is completely characterized\nby the symplectic eigenvalues $\\nu_k$, Eq. \\eqref{SymplecticEV}, which is dependent on the property of the adjacency matrix $A$ and the rescaling number $c$.\nThe squeezing parameters $r_k$ of $M$ single-mode squeezers are also determined by the adjacency matrix and $c$ \nfrom $e^{4r_k} = \\frac{(1+c |f_k|)^2-c^2 h_k^2}{(1-c |f_k|)^2-c^2 h_k^2}$.\nA single-mode thermal state can be purified with another thermal state, forming a pure two-mode squeezed state. The two-mode squeezing parameter $\\xi_k$ is related to the symplectic eigenvalue $\\nu_k$ via $2\\nu_k = \\cosh{[2\\xi_k]}$. Suppose that $A_{12}$ has $N$ nonzero eigenvalues,\nthen $N$ of the $M$ input modes should be injected by thermal states and others by vacuum states. To purify the $N$ thermal states we introduce $N$ two-mode squeezers which squeeze the vacuum. The $(M+N)$-mode circuit that can produce a $M$-mode mixed Gaussian state is shown in Fig. \\ref{fig:MixCircuit}.\n\nIf we already prepared the state $\\sigma_{cA}$ via the circuit in Fig. \\ref{fig:MixCircuit}, we then detect all output modes and evaluate the probability of detecting a single\nphoton in every output mode in order to sample the full adjacency matrix. The probability is related to the hafnian of the adjacency matrix $A$ by\n\\begin{align}\\label{eq:probA}\n&\\mathrm{Pr}_{cA}(\\underbrace{1,\\dots,1}_{M})  \\nonumber\\\\\n&= \\prod_{k=1}^{M} \\sqrt{[1-c(f_k + h_k)] [ 1+c(f_k - h_k)]} ~c^M \\mathop{{\\mathrm{haf}}}\\nolimits{A}. \\nonumber\\\\\n\\end{align}\nTherefore we can obtain the hafnian of $A$ once we know the probability.\n\n\n\n\\subsection{Sampling pure Gaussian covariance matrices}\\label{sec:SamplePureState}\n\n\\begin{figure*}[ht!]\n  \\resizebox{12cm}{!}{\\includegraphics{fig_3DPureCircuit.png}}\n  \\caption{ $2M$-mode canonical circuit that generates pure Gaussian covariance matrices $\\sigma_{c A^{\\oplus 2}}$. $2M$ single-mode squeezed states are injected   into a linear optics network $K$ to produce the target covariance matrix. The squeezing parameters $r_k$ ($r_k$ could be zero and corresponds to a vacuum input)   are completely determined by the eigenvalues of the adjacency matrix $A$ and  the rescaling parameter~$c$. In the output, $2M$ photon number detectors count the photon   number and evaluate the probability of measuring a single photon in every output mode.   }\n  \\label{fig:PureCircuit}\n\\end{figure*}\n\nGenerally, an adjacency matrix does not satisfy the three conditions discussed in Sec. \\ref{sec:SampleMixState} and thus does not map to a mixed Gaussian covariance matrix. The strategy to overcome this difficulty is to embed $A$ into a bigger adjacency matrix which can be mapped to a valid covariance matrix. While there are many ways to do that, one straightforward way is to take two copies of the desired adjacency matrix $A$ and consider a new adjacency matrix $A^{\\oplus 2} = A \\oplus A$ instead. In this case, the new adjacency matrix satisfies the three conditions in Sec. \\ref{sec:SampleMixState} and more importantly it maps to a pure Gaussian covariance matrix $\\sigma_{c A^{\\oplus 2}}$ (see Appendix \\ref{sec:GraphToCM}). \nA pure Gaussian covariance matrix is easier to prepare than a mixed one. The price we have to pay is to double the number of modes, which means the circuit would be twice as large.\n\nAs a special case of the decomposition shown in Fig. \\ref{fig:MixCircuit}, a pure Gaussian covariance matrix can be produced by injecting single-mode squeezed states (or vacuum) into a linear optics network. The squeezing parameter $r_k$ of the $k$-th input single-mode squeezed state is determined by the eigenvalues of the adjacency matrix $A$ and the rescaling parameter $c$ as $e^{2r_k} = (1+c|\\lambda_k|)\/(1-c|\\lambda_k|)$, where $\\lambda_k$ are the eigenvalues of the adjacency matrix $A$. Therefore given an adjacency matrix we can immediately determine the input squeezed states and the linear optics network according to the Euler decomposition, as shown in Fig. \\ref{fig:PureCircuit}. We then detect the output state and evaluate the probability of measuring a single photon in each output mode. The hafnian of the adjacency matrix $A$ is related to the probability via\n\\begin{equation}\\label{eq:probAoplA}\n\\mathrm{Pr}_{cA^{\\oplus2}}(\\underbrace{1,\\dots, 1}_{2M}) = \\prod_{k=1}^{2M} \\sqrt{1-c^2 \\lambda_k^2} ~c^{2M} \\mathop{{\\mathrm{haf}}}\\nolimits^{2}{A}\\,.\n\\end{equation}\n\n\\subsection{Scalability}\n\nThe magnitude of the sampling probability determines the hardness of performing the experiment, e.g., a lower probability will take a longer time to collect the data.\nWe find that the sampling probability depends on the eigenvalues of the adjacency matrix, as well as the rescaling parameter $c$ (see Appendix \\ref{app:SGCM}). In general, $c$ is smaller than one so the probability exponentially decreases as the size of the adjacency matrix \nincreases. To have an optimal probability one has to increase the value of $c$. However, $c$ is restricted by $0 < c < 1\/\\lambda_1$, so the best one can do is to make $c$ as close as possible to $1\/\\lambda_1$. The value of $c$ is critical in terms of how tractable the problem becomes due to the exponential scaling. Such an effect is not unexpected. We probably cannot hope for an efficient quantum algorithm to count the number of perfect matchings for all graphs as a function of the graph size and connectedness. However, the way it manifests itself is intriguing.\n\nThe largest graph eigenvalue appears to be determining the hardness of sampling the corresponding covariance matrix (see the conjecture in Appendix \\ref{app:scalability}). Note that spectral graph theory~\\cite{brouwer2011spectra} is a rich field and the largest eigenvalue of the graph's adjacency matrix can play an important role~\\cite{cvetkovic1990largest}.\nBased on these insights, we can further `hack' the adjacency matrix by i) decreasing the maximal eigenvalue such that the matrix's hafnian remains the same, or ii) back-calculating the hafnian from multiple equivalent sampling probabilities. A more exhaustive list of possible improvements will be subject of future work.\n\nThe first strategy is to add\/subtract a diagonal matrix to the adjacency matrix $A$, e.g., $A \\rightarrow A + d \\, \\mathbb{I}$, which does not change the hafnian since self-loops are irrelevant\nfor the number of perfect matchings. This modification may result in a negative matrix and, correspondingly, we need to modify the constraint of c to be $0 < c < 1\/|\\lambda_1|$\n(see Appendix \\ref{app:scalability} for details).\n\nThe second strategy is to extend the adjacency matrix $A$ of size $2M$ and create a bigger adjacency matrix $\\mathcal{A}$ of size $2nM$, where $n$ is an integer (see Appendix \\ref{app:SubgraphSampling} for details).\n$\\mathcal{A}$ is chosen such that it has many submatrices that are equal to the adjacency matrix $A$. Since each of these submatrices is related to the hafnian of the adjacency\nmatrix $A$, we can sum up all the probabilities corresponding to sampling one of these submatrices. The amplification of the total probability is achieved. The price we have to pay is an increase in the number of modes $n$ times.\n\n\\subsection{Example: complete graphs}\n\nTo illustrate the method we have developed, consider a concrete example: estimating the number of perfect matchings of a complete graph by sampling the probability\nof photon number distribution. Note that the number of perfect matchings in complete graphs is given by an analytical expression.\nHowever, in this section we consider the case of complete graphs to illustrate our methods for scalabilty. Furthermore, in Appendix \\ref{app:SubgraphSampling-2}, we give an explicit example of a noncomplete graph.\nFor a complete graph with $2M$ vertices, denoted as $K_{2M}$, all vertices are connected to each other and all entries of its adjacency matrix\n$A_{K_{2M}}$ are one except the diagonal elements.\nThe number of perfect matchings is known to be $(2M-1)!!$. We estimate\nthe magnitude of the probability and the hardness of sampling.\n\nAccording to the conditions discussed in Sec. \\ref{sec:SampleMixState}, the adjacency matrix of a complete graph maps to a mixed covariance matrix. It is then\nstraightforward to calculate the probability by using Eq. (\\ref{eq:probA}) and the number of perfect matchings. The rescaling parameter $c$ can be chosen to maximize the\nprobability. We can also modify the adjacency matrix by subtracting a diagonal matrix to reduce the absolute value of the maximal eigenvalue. The results for a 20 vertex\ncomplete graph are shown in Fig. \\ref{fig:K20}. Modifying the adjacency matrix significantly increases the probability.\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.6cm]{fig_PrAK20.pdf}\n\\caption{ The probability of photon measurement for the complete graph on twenty vertices as a function of the rescaling parameter $c$, defined by $A_{K_{20}}  \\to c A_{K_{20}} $, where $A_{K_{20}} $ is the graph's adjacency matrix. The black curve corresponds to original adjacency matrix and the dashed orange curve corresponds to modifying  the adjacency matrix as $A_{K_{20}} \\rightarrow A_{K_{20}}- 6\\, \\mathbb{I}_{20}$. }\n\\label{fig:K20}\n\\end{figure}\n\nWe can also take two copies of the adjacency matrix $A_{K_{2M}}$ and sample a $2M$-mode pure Gaussian covariance matrix. However, we found that this is not optimal\nfor complete graphs. In fact, we can extend $K_{2M}$ to a bigger complete graph $K_{2n M}$. We take two copies of $A_{K_{2 n M}}$,\nsubtract a diagonal matrix to $A_{K_{2 n M}} \\oplus A_{K_{2 n M}}$ and then sample a $2 n M $-mode pure covariance matrix. To relate to the hafnian of\n$K_{2M}$, we have to consider the probability of measuring single photons in $2M$ modes and zero photons in the other modes. The total probability increases because we have $\\binom{2nM}{2M}$ ways of selecting $2M$ modes from $2nM$ modes. The price we have to pay is to increase the size of the circuit. \n\nWe find that the maximal squeezing that is required to achieve maximal measurement probability as a function of the size of a complete graph is given by\n$10 \\log_{10}\\bigg[\\frac{\\sqrt{(2m+1)(n+1)}+\\sqrt{2m(n+1)-1}}{\\sqrt{(2m+1)(n+1)}-\\sqrt{2m(n+1)-1}} \\bigg]$ dB. \nFig. \\ref{fig:SqueezingMaxPro} shows an example for $n=16$. It can be seen that even for a relatively small-size graph, the amount of squeezing is demanding. From an experimental perspective, it is interesting to know how much probability one can achieve for a limited amount of squeezing. Fig. \\ref{fig:ProSqueezing} shows the measurement probability as a function of the squeezing for a given complete graph. It can be seen that for a moderate amount of squeezing, e.g., 10 dB, the probability can go up to about $10^{-5}$ for a complete graph with $40$ vertices, and therefore\ncan be sampled in a very short time. Currently, one has access to gigahertz detectors, which means sampling at a rate of $10^4$ per second, where the longer the sampling\ntime the higher the accuracy one can achieve.\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.6cm]{fig_Squeezing-vertices.pdf}\n\\caption{ The maximum squeezing that is required to achieve the maximal measurement probability as a function of graph size $2M$. We assume that a complete graph $K_{2M}$ is embedded in a bigger\ncomplete graph $K_{2nM}$ with $n=16$. Optimal rescaling factors are chosen to obtain the maximum probability. }\n\\label{fig:SqueezingMaxPro}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.6cm]{fig_Pro-Squeezing.pdf}\n\\caption{ The maximum measurement probability as a function of the amount of squeezing for different graph sizes $2M$. Here, we assume that a complete graph $K_{2M}$ is embedded in a bigger\ncomplete graph $K_{2nM}$ with $n=16$. Optimal rescaling factors are chosen to obtain the maximum probability. }\n\\label{fig:ProSqueezing}\n\\end{figure}\n\nSurprisingly, we find that the discussed strategies can significantly increase the sampling probability for an extremely large complete graph (for a complete graph with one edge removed as well).\nFig.~\\ref{fig:K5000} shows an example for a complete graph on 5000 vertices.\nThe maximal probability can go up to about $4\\times 10^{-5}$. The number of modes required to achieve this high probability is $320000$  ($n=64$). \nSuch a large circuit is challenging for current technologies -- the maximal squeezing that is required to achieve the maximal\nprobability is about 42.94 dB. Without this procedure the sampling of a well connected graph with more than one hundred vertices would be hopelessly unlikely.\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.6cm]{fig_K5000.pdf}\n\\caption{ The probability of photon measurement for the complete graph of 5000 vertices. Here, $c$ is the rescaling parameter for the adjacency matrix and $d$ is the scaling factor for subtracting the identity matrix from the adjacency matrix (see also Appendix \\ref{app:SubgraphSampling}). These rescalings correspond to the same graph in terms of perfect matchings but make the probability of detection\nmore favorable. The vertical axis shows the probability. }\n\\label{fig:K5000}\n\\end{figure}\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nIn this work, we have shown how to use a quantum device of squeezed states, optical interferometer, and photon counting measurements to estimate an important, and classically hard to compute, property of arbitrary graphs.\nBased on the relation between Gaussian Boson Sampling and graph theory, we developed  a systematic and programmable method to map an arbitrary adjacency matrix of an undirected graph\nto a continuous-variable Gaussian state. The number of perfect matchings of the graph can be directly estimated by sampling the photon number distribution of the\nGaussian state. The resources (input states and circuit) required to produce the desired Gaussian state are completely determined by the adjacency matrix and its\nvariants. \nAlthough the sampling probability decreases exponentially as the size of the graph increases, we have shown strategies to boost it to reasonably high values\nby a suitable rescaling of the adjacency matrix and by linearly increasing the size of the circuit.\n\nEstimating the number of perfect matchings of a large general graph is difficult for a classical computer. As a consequence of the problem of counting the perfect matchings being in the complexity class \\#P, we expect the time to reach an exact solution to be exponential in the size of the graph on any physical device. \nWe note that the computational hardness of Boson Sampling was conjectured via an embedding of Gaussian random matrices into larger random unitary ensembles \\cite{aaronson2011computational}.\nHowever, as hafnians are harder to compute than permanents, we conjecture that there exist graphs that correspond to important applications such as logistics, social networks, or chemistry and that show a quantum speedup. \nWhile exponential speedups could be possible for certain graphs, we expect that a special purpose quantum device offers a direct route to estimate the number of perfect matchings with better practical performance than a general purpose classical computer. \nThus, the method we have developed may be used to achieve such a task faster and with lower energy consumption using a relatively small photonic device instead of a massive supercomputer.\nHence, our method has the potential to achieve practical quantum supremacy \\cite{harrow2017quantum} for a problem with a wide range of applications.\n\n\\section*{Acknowledgements}\n\nWe thank Juan Miguel Arrazola and Nicol\\'as Quesada for fruitful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNumerical linear algebra with large matrices \nis a common foundation of high-performance scientific computations\nand the present paper focuses on\nquantum material simulations (electronic structure calculations).\nLarge scale electronic structure calculations,  with thousand atoms or more, \nare of great importance both in science and technology.\nSo far, we have developed\nlinear-algebraic algorithms and \nthe code called \\lq ELSES' \\cite{ELSES-URL} \nfor large-scale calculations in nano material studies. \n\\cite{TAKAYAMAI-2004-JPSJ, HOSHI-2005-PRB, TAKAYAMA-2006-PRB, IGUCHI-2007-PRL, SHINAOKA-2008-NEGF, YAMAGEN-2008-SCOCG, SOGABE-2008-ETNA, HOSHI-2009-JPCM-AUNW, HOSHI-AUNW-HANDBOOK, YAMASHITA-2011, TENG-2011, SOGABE-2012-GSQMR, HOSHI-mArnoldi, NISHINO-2012-SSI, HOSHI-2013-KEI-BENCH, LEE-2013-INTERIOR-EIGEN, LEE-2013-EASIAM, HOSHI-2013-KEI-BENCH-APPC}\n\nThe present paper presents a review of methodologies with latest results.\nThe present paper is organized as follows; Sec.~2 is devoted to an overview\nof our study for methodologies and application. \nSec.~3 presents novel linear algebraic solvers and \nthe benchmark of their application on the K computer.\nSeveral methodologies used with ELSES are reviewed \nfor two mode, \\lq novice' and  \\lq expert'  modes, for general users in Sec.~4\nand the use of Extensible Markup Language (XML) for input\/output files in Sec.~5.\nAs our related studies, \na sparse matrix collection called \\lq ELSES matrix library' \\cite{ELSES-MATRIX-LIBRARY}\nand the concept of numerical \\lq engine' \\cite{EIGEN-TEST} are explained in Sec.~6. \nIn Sec.~7, internal eigen-value problem is discussed,\nas a general need from large-scale electronic structure calculations.\nIn Sec.~8, \nphysical (not mathematical) aspects of methodology is focused on,\nin particular, modeled (tight-binding-form) electronic structure theory \nbased on  {\\it ab initio} calculations.\nThe summary and outlook appear in Sec.~9.  \n\n\\section{Overview of methodology and application}\n\nA mathematical foundation of electronic state calculations is \na generalized eigen-value equation\n\\begin{eqnarray}\nA  \\bm{y}_k = \\lambda_k B \\bm{y}_k.\n \\label{EQ-GEV-EQ}\n\\end{eqnarray}\nIn the present paper, as in many cases, \nthe matrices $A$ and $B$ are $M \\times M$ sparse real-symmetric ones and  $B$ is positive definite.\nThe matrix size $M$ is proportional to the number of atoms \nin the calculated material $N$ ($M \\propto N$).\nA standard eigen-value equation also appears among cases \nin which the matrix $B$ is reduced to the unit matrix  ($B = I$).\nAn eigen value $\\lambda_k$ is the energy of one electron and usually called \\lq eigen level',\nwhile an eigen vector $\\bm{y}_k$ represents the shape of wavefunction (electronic 'wave')  for one electron. \nThe use of dense-matrix solvers\nrequires the computational cost is O$(N^3$)\nor proportional to $N^3$, \nwhich will be a bottle neck in large-scale calculations.\nAnother issue is algorithms suitable to massive parallelism\nin modern supercomputers, such as the K computer. \n\n\n\nOur methodologies are based on order-$N$ (O($N$)) theories,\nin which the computational cost is \nproportional to $N$,\nas shown in Fig.~\\ref{fig-BENCH}(a).  \nA fundamental theory of the order-$N$ electronic structure calculation is \nthat the electronic structure calculation can be carried out, \nunlike conventional ones, without eigen-value problems. \n\\cite{KOHN-1996}\nOther references of the order-$N$ calculations are found,\nfor example, in the literature of  Ref.~\\cite{TENG-2011}.\nThe present calculation was carried out with modeled \n(tight-binding-form) electronic structure theory based on {\\it ab initio} calculations. \n\nThe present algorithms are suitable to massive parallelism and \nFig.~\\ref{fig-BENCH}(b) shows \nrecent calculations with one-hundred-million atoms on the K computer. \n\\cite{HOSHI-mArnoldi, HOSHI-2013-KEI-BENCH}\nA one-hundred-million-atom calculation are called  \n\\lq 100-nm-scale calculation',\nbecause one hundred million atoms are \nthose in silicon single crystal with the volume of (126nm)$^3$. \n\n\n\n\\begin{figure}[htbp] \n\\begin{center}\n  \\includegraphics[width=14cm]{graph-bench-100M-aPF-NCCS-mini.eps}\n\n\\end{center}\n\\caption{\\label{fig-BENCH} \n(a) Benchmark for the order-$N$ scaling. \\cite{HOSHI-mArnoldi}\n(b) Parallel efficiency on the K computer with one-hundred-million atoms.\n(\\cite{HOSHI-2013-KEI-BENCH-APPC}, present work). \nThe calculations were carried out \nwith the number of used processor cores $P$ $(T=T(P))$,\nfrom  $P = P_{\\rm min} \\equiv 32,768$ to $P_{\\rm all} \\equiv 663,552$ (all the processor cores).\nSee the text for details.\n}\n\\end{figure}\n\n\\begin{figure}[htbp] \n\\begin{center}\n  \\includegraphics[width=14cm]{fig-ELSES-appl-ol-lt.eps}\n\n\\end{center}\n\\caption{\\label{fig-ELSES-appl} \nExamples of nano material study with ELSES;\n(a) brittleness of solid silicon, \\cite{HOSHI-2005-PRB} \n(b) helical multishell gold nanowire \n\\cite{IGUCHI-2007-PRL, HOSHI-2009-JPCM-AUNW, HOSHI-AUNW-HANDBOOK}, \n(c) sp$^2$-sp$^3$ nano-composite carbon solid \\cite{HOSHI-2013-KEI-BENCH}, \n(d) amorphous-like conjugated polymer, \n\\cite{HOSHI-mArnoldi, HOSHI-2013-KEI-BENCH-APPC}\nas a foundations of opto-electronics, \n(e) ionic liquid of \nN-Methyl-N-propylpiperidinium bis trifluoromethanesulfonyl imide\n\\cite{NISHINO-2012-SSI}, \nas a battery-related problem,\n(f) quantum transport of d-band metal nanowire \n\\cite{SHINAOKA-2008-NEGF}.\n}\n\\end{figure}\n\nFigure \\ref{fig-ELSES-appl} \nshows examples of nano-material studies with ELSES and \nrelated original softwares;\n(a) brittleness of solid silicon \\cite{HOSHI-2005-PRB}, \n(b) helical multishell gold nanowire \n\\cite{IGUCHI-2007-PRL, HOSHI-2009-JPCM-AUNW, HOSHI-AUNW-HANDBOOK}\n(Experimental work is Ref.~\\cite{KONDO-SCIENCE-2000}),\n(c) sp$^2$-sp$^3$ nano-composite carbon solid \\cite{HOSHI-2013-KEI-BENCH}\nin a research of nano-polycrystalline diamond \\cite{IRIFUNE-NPD}, \n(d) amorphous-like conjugated polymer, \n\\cite{HOSHI-mArnoldi, HOSHI-2013-KEI-BENCH-APPC}\n(e) ionic liquid of \nN-Methyl-N-propylpiperidinium bis trifluoromethanesulfonyl imide\n\\cite{NISHINO-2012-SSI}, \n(f) quantum transport of d-band metal nanowire \n\\cite{SHINAOKA-2008-NEGF}.\nThe studies are motivated by the view points of \n(i)  industrial application, \n(ii) new material, especially new material from Japan, \nand (iii) standard material. \n\nA Python-based visualization software \\lq VisBAR' \n\\cite{URL-VISBAR, HOSHI-2013-KEI-BENCH, HOSHI-2013-KEI-BENCH-APPC}\nwas also developed,\nsince a post-simulation analysis of large-scale simulations requires \nseamless procedures of large-data-size numerical analysis and visualization.\nFor example, Fig. \\ref{fig-ELSES-appl}(c) was drawn by VisBAR,\nwhen the sp$^2$-sp$^3$ nano-composite carbon solid \\cite{HOSHI-2013-KEI-BENCH}\nwas analyzed \nfor the distinction between the sp$^2$ (graphite-like) and sp$^3$ (diamond-like) domains.\nThe analysis method is a quantum mechanical analysis called $\\pi$-type \nCrystalline Orbital Hamiltonian Population ($\\pi$COHP).\nThe method is based on the Green's function for electronic wavefunction\nand a theoretical extension of COHP \\cite{COHP}.\n\n \n \n\\section{Novel linear algebraic theory and benchmark on the K computer}\n\n\nThe mathematical foundation of our theories is based on \nthe \\lq generalized shifted linear equation', \nor the set of  linear equations \n\\begin{eqnarray}\n ( z B - A ) \\bm{x} = \\bm{b},\n \\label{EQ-SHIFT-EQ}\n\\end{eqnarray}\ninstead of the original generalized eigen-value equation of Eq.~(\\ref{EQ-GEV-EQ}).\nHere $z$ is a given complex number and has a physical meaning of (complex) energy. \nThe vector $\\bm{b}$ is the input and the vector $\\bm{x}$ is the solution vector. \n\nRecently, we have developed a set of Krylov subspace (iterative) solvers for Eq.~(\\ref{EQ-SHIFT-EQ}); \n(i) generalized shifted conjugate orthogonal conjugate gradient (gsCOCG) method, \\cite{TENG-2011} \n(ii) generalized shifted quasi-minimal residual (gsQMR) method, \\cite{SOGABE-2012-GSQMR}\n(iii) generalized Lanczos (gLanczos) method, \\cite{TENG-2011}\n(iv) generalized Arnoldi (gArnoldi) method, \\cite{TENG-2011}\n(v) Arnoldi ($M,W,G$) method, \\cite{YAMASHITA-2011}\n(vi) multiple Arnoldi method. \\cite{HOSHI-mArnoldi}\nIn the case of $B=I$,\nthe above theories will be reduced to the previous ones.\n\\cite{TAKAYAMAI-2004-JPSJ, TAKAYAMA-2006-PRB}\nThe method is purely mathematical and \nis applicable to other scientific problems. \nFor example, one of the above algorithms\nwas applied to strongly-correlated electrons of La$_{2-x}$Sr$_x$NiO$_4$($x=1\/3, 1\/2$)\n\\cite{YAMAGEN-PRG-2007-LASRNIO,YAMAGEN-2008-SCOCG}.\n\nThe multiple Arnoldi method \\cite{HOSHI-mArnoldi} is mainly used in our simulations\nand Fig.~\\ref{fig-BENCH} shows the result of benchmark. \nThe calculated materials are \namorphous-like conjugated polymer\nof  poly-(9,9 dioctyl-fluorene) (aPF) \\cite{HOSHI-mArnoldi, HOSHI-2013-KEI-BENCH-APPC} and\nsp$^2$-sp$^3$ nano-composite carbon solid (NCCS).\n\\cite{HOSHI-2013-KEI-BENCH}.\nFigure \\ref{fig-BENCH}(a) shows that the calculation \nhas the order-$N$ scaling property \\cite{HOSHI-mArnoldi}.\nFigure \\ref{fig-BENCH}(b) shows \nthe parallel efficiency on the K computer\nwith one hundred million atoms. \nThe MPI\/OpenMP hybrid parallelism is used.\nThe results are shown for \na NCCS case with $N=103,219,200$  \\cite{HOSHI-2013-KEI-BENCH-APPC} \nand aPF case with $N=102,238,848$ (the present work). \nThe elapse time  $T$ of the electronic structure calculation for a given atomic structure \nis measured as the function of \nthe number of used processor cores $P$ $(T=T(P))$,\nfrom  $P = P_{\\rm min} \\equiv 32,768$ to $P_{\\rm all} \\equiv 663,552$ \n(the total number of processor cores on the K computer). \nThe parallel efficiency is defined as $\\alpha(P) \\equiv T(P)\/T(P_{\\rm min})$ and\nsuch a benchmark is called \\lq strong scaling' in the high-performance computation society. \nThe measured parallel efficiency for the NCCS case is\n$\\alpha(P=98,304)=0.98$, \n$\\alpha(P=294,921)=0.90$ and \n$\\alpha(P=P_{\\rm all})=0.73$. \\cite{HOSHI-2013-KEI-BENCH-APPC}  \nThe measured parallel efficiency for the aPF case is\n$\\alpha(P=196,608)=0.98$ and \n$\\alpha(P=P_{\\rm all})=0.56$.\n\n\n\n\n\\begin{figure}[htbp] \n\\begin{center}\n  \\includegraphics[width=8cm]{fig-bench-one-cell-micro-cell-lt.eps}\n\n\\end{center}\n\\caption{\\label{fig-BENCH-FX10} \nBenchmark of the \\lq novice'  mode (square) \nand the \\lq expert' mode (circle) with $N$=132,864 atoms.\nFujitsu FX10 was used\nwith $P$ = 16, 32, 128, 512, 2048 processor cores.  \nThe calculation was carried out with an aPF case. \n}\n\\end{figure}\n\n\\section{\\lq Novice'  and \\lq expert' modes for general users}\n\nA \\lq novice' or easy-to-use mode was developed, as well as an \\lq expert' mode, \nso as to give comfortable user experiences for everyone.\nNowadays, quantum material simulations are popular among various researchers\nand many of them are not familiar to numerical algorithms and detailed procedures of the code.\nSuch \\lq novice' users would like to \nuse the code easily in a satisfactory computational performance\nwith, typically, $10^3$ processor cores or less.\n\\lq Expert' users, on the other hand,\nwould like to achieve the best computational performance \nfor largest problems with a top-class supercomputer. \nTherefore we developed modes for \\lq novice' and \\lq expert' users. \n\nThe \\lq novice' and \\lq expert' modes are different in the computational costs,\nthough they are equivalent mathematically and give the same numerical results.\nThe \\lq expert' mode is a strict order-$N$ procedure and\nrequires users to determine the detailed settings for the best performance,\nwhile the \\lq novice' mode contains\nseveral $O(N^2)$ procedures\nand does not require the detailed settings.\nConsequently, \nthe \\lq novice' mode is easier to use than the \\lq expert' mode\nbut may be worse in the computational performance.\n\nFigure \\ref{fig-BENCH-FX10}  demonstrates that\nthe performance difference\nbetween the two modes is small or moderate \nwith a small number of $N$.\nFigure \\ref{fig-BENCH-FX10} shows a benchmark of $10^5$-atom systems \nwith up to $10^3$ processor cores. \nThe elapse time of the \\lq novice' mode is approximately 50 \\% larger than\nthat of the \\lq expert' mode. \n\n \n\\section{Use of Extensible Markup Language (XML)}\n\nIn our simulations, \nthe main input and output files are written in the format of Extensible Markup Language (XML),\nsince the XML format is simple, flexible and widely used on the internet.\nFor example, \nthe minimum information for an atom is written as follows;\n\\begin{verbatim}\n<atom element=\"C\">\n  <position unit=\"angstrom\">  1.0d0  0.0d0   0.0d0  <\/position>\n<\/atom>\n\\end{verbatim}\nThe above description means that a carbon atom is located \nat the position of $(x,y,z)=(1, 0, 0)$,\nwhere Angstrom unit (1 Angstrom = 10$^{-10}$m) is used.\n\nThe method for reading XML files should be chosen properly,\naccording to the purpose. \nIn our simulations, \nthe  XML files are read by two methods, \nDocument Object Model (DOM) method and Simple API for XML (SAX) method.\nIn general, the DOM method is easier in programming and results in huge\nmemory and time consumption for large-size data,\nwhile the SAX method \nis more difficult in programing and results in tiny memory and time consumption. \n\nTwo input files,  configuration and structure XML files should be prepared for each simulation\nand they are quite different in their file size. \n(i) The configuration XML file describes calculation conditions,\nsuch as temperature of the system.\nA typical file consists of several tens of lines and \nthe file size does not depend on the system size $N$. \nThe configuration file is read by the DOM method, since its file size is always tiny.\n(ii) The structure XML file describes the atomic structure data, as shown in the above example.\nThe structure XML file is read by the DOM method, \nsince the file size is proportional to the system size $N$ and can be huge. \nSince the atomic structure data\ncontains three ($x,y,z$) components in the double precision (8B) value for each atom,\nthe required data size with $10^7$ (=10M) atoms is estimated to be\n$3 \\times 8 {\\rm B} \\times 10$M = 240 MB. \nA typical size of the structure XML file with $10^7$ atoms is one G byte (B).\nIn addition, \nthe parallel file reading is used for large-scale calculations \nwith split XML files for the structure file and \ngives a significant acceleration.\n\\cite{HOSHI-2013-KEI-BENCH}\nThe K computer and FX10 support the parallel file IO,\ncalled \\lq rank directory' function, at the hardware level \nand are suitable to the parallel file reading with split XML files. \n\n\\section{Sparse matrix collection  and numerical \\lq engine'}\n\nThis section explains that\nwe opened a matrix library and are developing a general numerical \\lq engine',\nfor further collaboration between real application and numerical linear algebra.\n\nRecently, \na sparse matrix collection called \\lq ELSES matrix library' \n\\cite{ELSES-MATRIX-LIBRARY} was opened. \nThe stored matrices are sparse and  generated by ELSES \nas the matrices $A$ and $B$ in Eq.~(\\ref{EQ-GEV-EQ}).\nThe maximum matrix size $M$ is one million. \nMatrix data files are written in \nthe Matrix Market format. \\cite{URL-MATRIX-MARKET}\nEach matrix data has its own name and\nappears with a document so as to clarify the physical origin of the matrix. \nFor example, the matrix data \\lq VCNT900' presents\na matrix with the size of $M=900$  \nand its physical origin is thermally vibrating carbon nanotube (VCNT).\n\\footnote{\nThe thermal vibration causes small random deviations on the atomic positions\nand, therefore,  the calculated eigen values are not degenerated. \n}\nThe calculated eigen values are also included in several matrix data. \n\n\nMoreover a general numerical 'engine' for matrix (eigen-value) equations is  under development \nwith a common interface between real applications and numerical solvers. \nA mini application for evaluating the engine, called \\lq Eigen Test', is also under development, \nin which the matrix data in ELSES matrix library \\cite{ELSES-MATRIX-LIBRARY}, or \nsome files on Matrix Market \\cite{URL-MATRIX-MARKET}, can be used as inputs. \nA code in an early version for shared memory systems is available for test users. \\cite{EIGEN-TEST}\nThe code for distributed memory systems is under development, in which two direct solvers, \nScaLAPACK \\cite{URL-SCALAPACK} and EigenExa \\cite{URL-EIGEN-EXA}, are implemented \nfor the standard and generalized eigen-value equations in the Eq.~(\\ref{EQ-GEV-EQ}) with \nreal-symmetric matrices.\n\\footnote{\nAt the present time, \nEigenExa does not support generalized eigen-value problems.\nIn the engine, a generalized eigen-value problem is solved with EigenExa, \nsince the problem is transformed into a standard eigen-value problem\nby the Cholesky decomposition routine in ScaLAPACK.\n}\nThe implementation of Krylov subspace solvers is planned. \nIn near future,\nour application code \\lq ELSES' will be connected to the engine\nand general users can use the solvers without detailed knowledge of the solver algorithms.\nThe engine is general and can be connected to any other real applications.\n\n\n\\begin{figure}[htbp] \n\\begin{center}\n  \\includegraphics[width=12cm]{fig-eigen-engine.eps}\n\n\\end{center}\n\\caption{\\label{fig-eigen-engine} \nIllustration of the role of the general \\lq numerical engine' for matrix equations. \n}\n\\end{figure}\n\n\n\n\n\\section{Internal eigen-value problem}\n\n\\begin{figure}[htbp] \n\\begin{center}\n  \\includegraphics[width=12cm]{0310-Intermidiate-EigenPair-problem-e-lt.eps}\n\n\\end{center}\n\\caption{\\label{FIG-INTERNAL-EIGEN} \nFour types of internal eigen-value problems.\n(I) Point given type,\n(II) Interval given type, \n(III) Order given type,\n(IV) Boundary order given type.\n}\n\\end{figure}\n\nThis section explains internal eigen-value problem\nfrom a general need in large-scale electronic state calculations.\nThe present discussion is limited to real eigen-value problems.\nIn a large-matrix problem, \none should give up the calculation of all the eigen pairs, \nbecause of huge computational cost.\nThen one would like to obtain specific eigen pair(s)  \n($\\lambda_j, \\bm{y}_j$) of Eq.~(\\ref{EQ-GEV-EQ}) \nat an internal eigen-value region ($1 < j < M$),\nbecause \nseveral internal eigen pairs are responsible \nfor electronic and optical properties.\n\nInternal eigen-value problems can be classified \ninto the following four types, \nas illustrated in Fig.~\\ref{FIG-INTERNAL-EIGEN};\n(I) Point given type:\nFor a given real value $a$,\none should find (an) eigen value(s) $\\lambda$ near the given point\n($\\lambda \\approx a$) and its (their) eigen vector(s). \n(II) Interval given type:\nFor a given interval of $[a,b]$  ($a < b$),\none should find eigen values $\\lambda$ in the interval  \n($a < \\lambda < b$) and their eigen vectors. \n(III) Order given type:\nFor a given integer of $j$ ($ 1 < j < M$),\none should find the $j$-th lowest eigen value \n$\\lambda_j$ ($\\cdot \\cdot \\cdot \\lambda_{j-1} \\le \\lambda_j \\le \\lambda_{j+1} \\cdot \\cdot \\cdot$) and its eigen vector. \n(IV) Boundary order given type:\nFor a given integer  $j$ ($ 1 < j$) and a real value $a$,\none should find the $j$-th lowest eigen value $\\lambda_j^{(a)}$\nthat is larger than $a$:\n\\begin{eqnarray}\n\\cdot \\cdot \\cdot \\le \\lambda_{-1}^{(a)} \\le a \\le \\lambda_1^{(a)} \\le \\lambda_2^{(a)} \\le \\cdot \\cdot \\cdot \\lambda_j^{(a)} \\le \\lambda_{j+1}^{(a)} \\cdot \\cdot \\cdot\n\\end{eqnarray}\nOne should find all the (degenerated) eigen vectors\nfor the target eigen value $\\lambda_j^{(a)}$.\nFrom the definitions, \nthe boundary-order-given-type problem with $a < \\lambda_1$\nis reduced to the order-given-type problem.\nIn all the types of problem, \nif the target eigen value is degenerated, one should find all the degenerated eigen vectors.\n\nThe present paper focuses on the order-given-type problem,\nsince the problem appears in many electronic structure calculations.\nIn our problems, \nthe target eigen value(s) is (are) given from the physical viewpoint.\nHere we call eigen value \\lq level', as usual in electronic structure calculations.\nTypically, the two levels, called highest-occupied (HO) and lowest-unoccupied (LU) levels,\nare of fundamental interest.  \nThe HO level, denoted as $j$ hereafter, is defined as\nthe half of the total number of electrons in material $N_{\\rm e}$\n($j \\equiv [ N_{\\rm e} \/ 2 ]$)\n\\footnote{The present case is that of a para-spin material.}\nand the LU level is defined as $j$+1. \nFor example,\nthe difference between the two levels\n$\\Delta \\equiv \\lambda_{j+1} -  \\lambda_{j}$\nis called energy gap\nand is zero in metals and non-zero in semiconductors or insulators.\nTwo notes are added here;\n(a) The levels near the HO or LU level are also of importance \nand are required in many electronic state calculations. \n(b) When $N$ is odd,\nthe HO and LU levels are not defined in a strict physical sense.\nAmong these cases, however,\nthe ($j \\equiv [ N_{\\rm e} \/ 2 ]$])-th  and ($j$+1)-th  levels are still crucial\nfor electronic properties and\nwe call them the HO and LU levels in the present paper.\n\nWe have proposed two approaches for the order-given-type problem:\nOne appears in Refs.~\\cite{LEE-2013-INTERIOR-EIGEN, LEE-2013-EASIAM}\nand the other appears in Ref.~\\cite{HOSHI-2013-KEI-BENCH-APPC}.  \nSee the papers for details.\n\nFurther investigations on efficient algorithms are desired\nfor internal eigen-value problems.\nIt is speculated that a difficulty in numerical treatment \nwill appear among (almost) degenerated eigen pairs.\nAn example is found \nin the matrix of \\lq APF4686'\nof ELSES matrix library\n\\cite{ELSES-MATRIX-LIBRARY},\nin which the 2345-th and 2346-th eigen values \nare almost degenerated \n($\\lambda_{2345}$=-0.356883, $\\lambda_{2346}$= -0.356806).\n\\footnote{The atomic unit is used in the present eigen values}\n\n\n\\begin{figure}[htbp] \n\\begin{center}\n  \\includegraphics[width=14cm]{fig-PF-monomer-dimer-lt.eps}\n\n\\end{center}\n\\caption{\\label{fig-PF-monomer-dimer} \n(a) Structure of poly-(9,9 dioctyl-fluorene).  \nHere $R \\equiv$ C$_8$H$_{17}$ and $n$ is the number of monomer units.\nThe letter \\lq 9' indicates the atom site called \\lq 9 position'. \n(b) DOS of the monomer (red line) and benzene (blue line).\nThe value of the latter graph is magnified twice. \n(c)(d)  Schematic figures of\nthe HO or LU wavefunction  of the dimer, respectively.  \n(e)(f) The calculated dimer structure and the HO or LU wavefunctions, respectively.\nThe large and small balls indicate carbon and hydrogen atoms. \nThe colors indicates the sign of the wavefunction in (c)-(h). \n}\n\\end{figure}\n\n\\section{Physical aspect of the fundamental methodologies}\n\nThe last topic is some physical (not mathematical) aspects of the fundamental methodology,\nin particular, {\\it ab initio} based modelings in electronic structure theory.\nThe present calculation was carried out with modeled \n(tight-binding-form) electronic structure theory based on {\\it ab initio} calculations and \nsometimes a charge-self-consistent theory \\cite{ELSTNER-1998} is also used \n\\cite{NISHINO-2012-SSI, NISHINO-2013-JMODELING}.   \nThe matrices $A$ and $B$ \nin Eqs.(\\ref{EQ-GEV-EQ}) and (\\ref{EQ-SHIFT-EQ})\nare sparse and called \nthe Hamiltonian and overlap matrices \nwritten the linear-combination-of-atomic-orbital (LCAO) representation, respectively.\nThe diagonal elements of $B$ are the unity $(B_{ii}=1)$ and \nthe absolute value of an off-diagonal element is less than the unity\n$(| B_{ij} |  < 1, i \\ne j)$.\n \n\nHere  poly-(9,9 dioctyl-fluorene), \na conjugated polymer depicted in Fig.~\\ref{fig-PF-monomer-dimer}(a),\nis picked out.\nThe material is a basic one for opto-electronics devices.\nA modeled theory in Ref~\\cite{ASED-CALZAFFERI} is used\nand it is called the atomic-superposition and electron-delocalization \ntight-binding (ASED-TB) theory.\nThe monomer unit in Fig.~\\ref{fig-PF-monomer-dimer}(a)\nconsists of two benzene rings \nand alkyl side chains of $R \\equiv$ C$_8$H$_{17}$.  \nThe letter \\lq 9' indicates the atom site called \\lq 9-position',\nlocated at the \\lq root' part of the side chain or\nthe top site of the five-membered ring.\nThe side chains are extended transversely,\nsince the \\lq 9-position' in Fig.~\\ref{fig-PF-monomer-dimer}(a)\nhas an $sp^3$-like electronic configuration and\nthe neighboring four atoms form tetrahedral bonds.\n\nThe monomer and dimer were calculated with the exact solution of Eq.~(\\ref{EQ-GEV-EQ}).\nThe density of states (DOS) function, a eigen-value histogram, \nis shown in Fig.~\\ref{fig-PF-monomer-dimer}(b) for the  monomer and a benzene molecule (C$_6$H$_6$). \nThe HO and LU electronic wavefunctions of the dimer are drawn schematically\nin Fig.~\\ref{fig-PF-monomer-dimer}(c)(d), respectively\nand the calculated HO and LU wavefunctions with the optimized structure\nare shown in Fig.~\\ref{fig-PF-monomer-dimer}(e)(f), respectively.\nThe two monomer units of the dimer\nare connected with twisting, as shown in Fig.~\\ref{fig-PF-monomer-dimer}(e) or (f),  \nmainly because of the repulsion \nbetween the hydrogen atoms on the benzene rings of the neighboring monomers.\nThe half of Fig.~\\ref{fig-PF-monomer-dimer}(c) or (d) \ncorresponds to the schematic figure of \nthe HO or LU state of the monomer, \\cite{QM-TEXT} respectively. \nThe wavefunctions are contributed only by  the $\\pi$-type wavefunctions on the benzene rings.\nThe HO wavefunction of the dimer and the monomer has  two nodes on a benzene ring, \nand the LU wavefunction of the dimer and the monomer has four nodes on a benzene ring.\nThese node structures are the same with those of a benzene,\n\\cite{QM-TEXT}\nbecause the HO and LU levels of the monomer stem from those of benzene,\nwhich is seen in the DOS functions of Fig.~\\ref{fig-PF-monomer-dimer}(b).\nThe monomer and dimer are calculated also \nby the {\\it ab initio} calculation (Gaussian 09$^{\\rm (TM)}$)\nwith the B3LYP functional and  the 6-31G(d,p) basis set\nand the above features of wavefunctions are satisfied \nin the calculated HO and LU wavefunctions shown in Figs.~\\ref{fig-PF-monomer-dimer}(g)(h).\nThe atomic structures and the DOS functions were also calculated\nand satisfy the above features. \nMoreover, detailed data by the present method are added here\nwith those by the {\\it ab initio} calculation in the parentheses; \\cite{HOSHI-mArnoldi} \nThe balance band width $W$ and the band gap $\\Delta$ are\n$W=18.5$ eV (18.3 eV) and $\\Delta =4.25$ eV (4.91 eV) in the monomer\nand $W=19.0$ eV (18.8 eV)  and $\\Delta = 3.58$ eV (4.10 eV) in the dimer.\nThe twisting angle of the dimer $\\theta$ is \n$\\theta=37.3 ^\\circ$ (40.6 $ ^\\circ$).\n\nSeveral notes are posted on the transferability (general applicability) of model theories;\n(i) The present theory is formulated without any data of fluorene cases but\nis formulated with related small molecules such as benzene.\nThe above agreement with the {\\it ab initio} calculations in the fluorene cases shows\nthat the theory is applicable to, at least, materials that have similar binding mechanisms.\n(ii) Quite recently, a modeled van der Waals (vdW) interaction \n\\cite{ORTMANN-2006} is implemented in the code \nfor wider applicability to  materials. \nDetails will be discussed elsewhere.\n(iii) Automated determination methods for obtaining an optimal model among various materials is desired and\nis now developing with ELSES \\cite{NISHINO-2013-JMODELING, OHTANI-2013-MRS}.\n\n\n\n\n\n\\section{Summary and future outlook}\n\nThe present paper reports our methods and results of \nlarge-scale electronic structure calculations\nbased on our novel linear algebraic algorithms. \nA high parallel efficiency was shown \nin one-hundred-million-atom systems \nwith up to all the built-in cores of the K computer. \nThe related methodologies on physics, mathematics and information technology are also discussed. \n\nThe present study is a interdisciplinary one between \nphysics, numerical linear algebra and high-performance computation techniques,\nand such a interdisciplinary study is sometimes called 'Application-Algorithm-Architecture co-design'.\nThe co-design will play a crucial role in exa-scale scientific computations.\n\n\n\\acknowledgments\nThis research is partially supported by Grant-in-Aid \nfor Scientific Research\n(KAKENHI Nos. 23540370 and 25104718)\nfrom the Ministry of Education, Culture, Sports, Science and Technology \n(MEXT) of Japan. \nThe K computer was used \nin the research proposals of hp120170, hp120280 and hp130052.\nSeveral calculations were carried out \nby the supercomputer\nat the Information Technology Center, University of Tokyo,\nin the research proposal of jh130011-NA07.\nSupercomputers were also used \nat the Institute for Solid State Physics, University of Tokyo, \nand at the Research Center for \nComputational Science, Okazaki.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEven though the charmoiumlike resonance $X(3872)$ has been established for several\nyears~\\cite{x3872-1,Aub:2005, Aco:2004, Aba:2004} with $M_X=3871.68\\pm 0.17\\,{\\rm MeV}$ and\n$\\Gamma_X<1.2\\,{\\rm MeV}$~\\cite{Choi:2011, PDG:2012}, the very nature of it has not been fully\nunderstood till now. Its even $C$ parity has been firmly established from its decay to $J\/\\psi\n\\rho$~\\cite{Abu:2006} and to $J\/\\psi \\gamma$~\\cite{Aub:2006}. Further analysis of its decay angular\ndistribution also constrains its total quantum number $J^{PC}$ to be either $1^{++}$ or $2^{-+}$.\nThe discovery of $X(3872)$ has triggered quite a number of theoretical interpretations by assuming\na quantum number $1^{++}$, such as the radial excitation of $\\chi_{c1}$, the $D\\bar{D}^*$\nmolecule~\\cite{Bignamini2009:molecule1, Voloshin2004:molecule2, Lee2009:molecule3}, a tetraquark\nstate~\\cite{Maiani2005:tetra1, Dubnicka2010:tetra2, Dubnicka2011:tetra3}, etc.; however, none of\nthem can accommodate all the observed features of $X(3872)$. The situation became more complicated\nwhen the {\\it BABAR} Collaboration reported in 2009 that $2^{-+}$ is more favored by the study of\nthe decay angular distribution of the process $X(3872)\\rightarrow J\/\\psi\n\\pi^+\\pi^-\\pi^0$~\\cite{delAmoSanchez:2010jr}. In contrast, the same analysis by the Belle\nCollaboration claims that both the $1^{++}$ and $2^{-+}$ assignments are consistent with their\ndata~\\cite{Choi:2011, Lange2011:belle}. Another controversial result comes from the measurements of\nthe radiative decays of $X(3872)$. For the decay mode $X(3872)\\to \\gamma J\/\\psi$, the {\\it BABAR}\nand Belle collaborations reported consistent measurements~\\cite{BaBar:2009, Belle:2011}:\n \\begin{eqnarray}\n {\\rm Br}(B^{\\pm}&\\rightarrow& X(3872)K^{\\pm}){\\rm Br}(X(3872)\\rightarrow J\/\\psi \\gamma)\\nonumber\\\\\n &=&(2.8\\pm 0.8 \\pm 0.1)\\times 10^{-6}~~~(BABAR),\\nonumber\\\\\n {\\rm Br}(B^{\\pm}&\\rightarrow& X(3872)K^{\\pm}){\\rm Br}(X(3872)\\rightarrow J\/\\psi \\gamma)\\nonumber\\\\\n &=&(1.78^{+0.48}_{-0.44}\\pm 0.12)\\times 10^{-6}~~({\\rm Belle}).\n \\end{eqnarray}\nWith the world average value ${\\rm Br}(B^{+}\\rightarrow X(3872)K^{+})<3.2\\times 10^{-4}$, one can\nestimate the branch ratio ${\\rm Br}(X(3872)\\rightarrow J\/\\psi \\gamma)> 0.9\\%~(BaBar)$ or\n$0.6\\%~({\\rm Belle})$. However, for the decay mode $X(3872)\\to \\gamma \\psi'$, {\\it BABAR} measured\na $3.4\\pm 1.4$ times larger branch ratio~\\cite{BaBar:2009}, but Belle found no\nevidence~\\cite{Belle:2011}. This large discrepancy should be reconciled by further experimental\nmeasurements.\n\nTheoretically, if we are constrained to its charmonium assignments, $X(3872)$ can be either the\nradial excitation of $\\chi_{c1}$ (if $1^{++}$), say, $\\chi_{c1}'$, or the ${}^1D_2$ charmonium\n$\\eta_{c2}$ (if $2^{-+}$). The potential quark model predicts the mass of $\\chi_{c1}'$ to be\n$3925\\,{\\rm MeV}$~\\cite{BGS2005:2Pmass}, which deviates from the mass of $X(3872)$ by about\n$50\\,{\\rm MeV}$. There are also many lattice studies predicting a $\\chi_{c1}'$ mass ranging from\n$3850$ to $4060$ MeV~\\cite{Okamoto:2002, Chen:2001, Chen:2007, Liu:2011}, but with various\nuncertainties of their own, where the key difficulty is the challenging task of extracting the\nexcited states. As for the $\\eta_{c2}$, the quark models usually predict the mass to be in the\nrange 3770 to 3830 MeV~\\cite{BGS2005:2Pmass, BG2004:1Dmass, Badalian2000:1Dmass}, which is even\nfurther away from the mass of $X(3872)$. This is also reinforced by recent lattice studies (and\nthis work). At any rate, the mass parameter should not be the unique criterion for the\ninterpretation of $X(3872)$; more information is definitely desired\\--for example, the radiative\ntransition properties of $\\chi_{c1}'$ and $\\eta_{c2}$, which are theoretically accessible and\nhopefully can shed some light on the nature of $X(3872)$.\n\nIn this work, we will focus on the study of the properties of $\\eta_{c2}$, such as its mass and\nradiative transition width to $J\/\\psi$. There are actually several phenomenological studies on this\ntopic~\\cite{Jia:2010jn, Kalash2010, Ke:2011jf}, but they are rather model dependent. In contrast,\nthe lattice QCD approach, as a method from first principles, can provide information that is more\nmodel independent. An additional technical advantage in the study of $\\eta_{c2}$ on the lattice is\nthat it is the ground state in the $2^{-+}$ channel and is free from the uncertainty of the\nextraction of excited states. In view of the notorious bad signal-to-noise ratio for $P$ and $D$\nwave states, we adopt the quenched approximation so as to obtain large enough statistics for\nprecise physical quantities to be derived. As for the quenched approximation, even though long-term\nexperiences show that it is safe for charm quark systems, and the resultant uncertainties can be\nsmall, we still take several steps to check this and be assured of our results. We first calculate\nthe spectrum of the ground state charmonia, such as $1S$, $1P$ states, and make sure that the\nexperimental spectrum patterns are reproduced. As for the radiative transitions, we choose the\ntransition mode of the tensor charmonium $\\chi_{c2}$ to $J\/\\psi$ as a calibration of the systematic\nuncertainties of our formalism by comparing our result to the experimental value. After that, we\ncontinue to the study of the radiative transition of $\\eta_{c2}$ to $J\/\\psi$. All the lattice\ncalculations are carried out on anisotropic lattices which are suitable to the study of heavy\nparticles. The numerical techniques are standard: the mass spectra are extracted from two-point\nfunctions, and the multipole amplitudes contributing to the transition widths are derived from the\ncalculation of relevant three-point functions with a local electromagnetic current insertion. We\napply two anisotropic lattices with different lattice spacings to estimate the lattice artifacts\nowing to the finite lattice spacing.\n\nThis work is organized as follows: The formalism for the calculation of radiative transition widths\non the lattice is briefly introduced in Sec.~\\ref{section2}. In Sec.~\\ref{section3} are the\nnumerical details, where the lattice setup, the extraction of mass spectrum and transition form\nfactors are explained, and the numerical results are presented. Section~\\ref{section4} is the\nconclusion and discussion. The theoretical derivation of the multipole form factors is described in\nthe Appendixes.\n\n\\section{Formalism}\\label{section2}\nAs mentioned above, in this work we aim at the lattice calculation of the radiative transition rate\nof $\\eta_{c2}$ to $J\/\\psi$. The general radiative transition width of an initial particle $i$ to a\nfinal particle $f$ is\n\\begin{eqnarray}\\label{width}\n\\Gamma(i\\rightarrow \\gamma f)&=& \\int d\\Omega_q\n\\frac{1}{32\\pi^2}\\frac{|\\vec{q}|}{M_{i}^2}\\frac{1}{2J_i+1}\\nonumber\\\\\n&\\times&  \\sum\\limits_{r_i,r_j,r_{\\gamma}}\\left|M_{r_i,r_j,r_{\\gamma}}\\right|^2,\n\\end{eqnarray}\nwhere $\\vec{q}=\\vec{p}_i-\\vec{p}_f$ is the decay momentum with the mass-on-shell value\n$|q|=(M_i^2-M_f^2)\/(2M_i)$, $M_{i}$ and $M_f$ are the masses of the particles $i$ and $f$, and\n$M_{r_i,r_f,r_{\\gamma}}$ is the transition amplitude with $r_i, r_f, r_\\gamma$ being the\npolarizations of $i$, $f$, and the photon, respectively. To the lowest order of QED, the amplitude\n$M$ is expressed explicitly as\n\\begin{equation}\nM_{r_i,r_f,r_\\gamma}=\\epsilon_{\\mu}^*(\\vec{q},r_\\gamma)\\langle f(\\vec{p}_f,r_f)|j_{\\rm\nem}^{\\mu}(0)|i(\\vec{p}_i,r_i)\\rangle,\n\\end{equation}\nwhere $\\epsilon_{\\mu}^*(\\vec{q},r_\\gamma)$ is the polarization vector of the photon, and $\\langle\nf(\\vec{p'},r_f)|j_{\\rm em}^{\\mu}(0)|i(\\vec{p},r_i)\\rangle$ gives the on-shell matrix elements of\nthe electromagnetic current $j_{\\rm em}^\\mu(x)=\\bar{\\psi}Q\\gamma^\\mu\\psi(x)$ between the $i$ and\n$f$ states. [Here $\\psi$ refers to an array of all the contributing quark flavors, such as\n$u,d,s,c,\\ldots$, and $Q$ is a diagonal matrix of quark electric charges, say, ${\\rm\ndiag}(Q)=Q_u,Q_d,Q_s,Q_c, \\ldots$.] The hadronic matrix element can be derived directly from the\nlattice QCD calculation of the related three-point functions:\n \\begin{eqnarray}\\label{eq_three}\n\\Gamma^{(3)\\mu}_{mn}(\\vec{p}_f,\\vec{q},t,t')&&=\\sum_{\\vec{x},\\vec{y}}\ne^{-i\\vec{p}_f\\cdot\\vec{x}}e^{-i\\vec{q}\\cdot\\vec{y}}\\times\n\\nonumber\\\\\n&&\\langle O_m^f(\\vec{x},t)j_{\\rm em}^\\mu(\\vec{y},t') O_n^{i\\dagger}(\\vec{0},0)\\rangle,\n\\end{eqnarray}\nwhere $O_{m,n}^{i,f}$ are the interpolating fields for the particles $i$ and $f$, with the indices\n$m,n$ referring to different spatial components for spin nonzero states. The explicit derivation\ncan be expressed as\n\\begin{eqnarray}\\label{three}\n\\Gamma^{(3),\\mu}_{mn}(\\vec{p}_f,\\vec{q},t,t')&=&\\sum_{r_i,r_f}e^{-E_f t}e^{-(E_i-E_f)t'}\\nonumber\\\\\n&\\times&\\frac{\\hat{Z}_{m}^f(\\vec{p}_f,r_f)\\hat{Z}_{n}^{i*}(\\vec{p}_i,r_i)}{2E_i\\ 2E_f}\\nonumber\\\\\n&\\times& \\langle f(\\vec{p}_f,r_f)|j_{\\rm em}^\\mu(0)|i(\\vec{p}_i,r_i)\\rangle\\nonumber\\\\\n&&(t',t-t'\\rightarrow \\infty),\n\\end{eqnarray}\nwhere $\\hat{Z}_{m}^{i,f}$ are the matrix elements like $\\hat{Z}_{m}^X(\\vec{p}_X,r_X)=\\langle\n0|O_m^{X}|X(\\vec{p}_X,r_X)\\rangle$, which can be derived from the relevant two-point functions,\n\\begin{eqnarray}\\label{two}\n\\Gamma_{X,mn}^{(2)}(\\vec{p}_X,t)&=&\\sum\\limits_{-\\vec{x}}e^{i\\vec{p}_X\\cdot\\vec{x}}\\langle\nO_{m}^X(\\vec{x},t)O_{n}^{X\\dagger}(\\vec{0},0)\\rangle\\nonumber\\\\\n&\\rightarrow&\\frac{1}{2E_X}e^{-E_X t}\\sum\\limits_{r_X}\\langle\n0|O_{m}^X|X(\\vec{p}_X,r_X)\\rangle\\nonumber\\\\\n &\\times&\\langle X(\\vec{p}_X,r_X)|O_{n}^{X\\dagger}|0\\rangle~~~(t\\rightarrow \\infty).\n\\end{eqnarray}\nOn the other hand, in the Minkowski space-time, the matrix elements  $\\langle\nf(\\vec{p}_f,r_f)|j_{\\rm em}^{\\mu}(0)|i(\\vec{p}_i,r_i)\\rangle$ can be generally expressed by several\nLorentz-invariant form factors $F_k(Q^2)$ and Lorentz-covariant kinematic factors $\\alpha_k(p_i,\np_f)$ through the multipole decomposition,\n\\begin{equation}\\label{eq_ampth}\n \\langle f(\\vec{p}_f,r_f)|j_{\\rm em}^{\\mu}(0)|i(\\vec{p}_i,r_i)\\rangle=\\sum_k \\alpha_k^{\\mu}(p_i,p_f)F_k(Q^2),\n\\end{equation}\nwhere $p_{i,f}$ are now the four-momenta of particles $i$ and $f$, and $Q^2$ is the squared\ntransfer momentum $Q^2=-(p_i-p_f)^2$. Obviously, the concrete form factors $F_k(Q^2)$ and the\nexplicit expressions of the kinematic factors $\\alpha_k$ depend on the properties of the particles\n$i$ and $f$, and therefore should be worked out case by case. Finally, the decay width with an\non-shell photon ($Q^2=0$) can be expressed as\n\\begin{equation}\n\\Gamma(i\\rightarrow \\gamma f) \\propto \\sum\\limits_k F_k^2(0).\n\\end{equation}\nSo the key problem in this work is to reliably extract these form factors through the lattice\ncalculation of the relevant hadronic two-point functions and three-point functions described above.\n\n\\section{Numerical Details}\\label{section3}\n\nWe use the quenched approximation in this study. The gauge configurations are generated by the\ntadpole improved gauge action~\\cite{morningstar} on anisotropic lattices with the temporal lattice\nmuch finer than the spatial lattice, say, $\\xi=a_s\/a_t\\gg 1$, where $a_s$ and $a_t$ are the spatial\nand temporal lattice spacings, respectively. The much finer lattice in the temporal direction\nyields a higher resolution to hadron correlation functions, such that the masses of heavy particles\ncan be tackled on relatively coarse lattices. We have two anisotropic lattices ($L^3\\times T=\n8^3\\times 96$ and $12^3\\times 144$) with $\\xi=5$. The relevant input parameters are listed in\nTable~\\ref{tab:lattice}, where the lattice spacings, say, $a_s=0.222(2)\\,{\\rm fm}$ for the coarser\nlattice and $a_s=0.138(1)\\,{\\rm fm}$ for the finer lattice, are determined from\n$r_0^{-1}=410(20)\\,{\\rm MeV}$ by calculating the static potential. For each lattice, we generate\n1000 configurations, each of which is separated by 500 heat-bath updating sweeps to avoid the\nautocorrelation. For fermions, we use the tadpole improved clover action for anisotropic\nlattices~\\cite{chuan1}. The parameters in the action are tuned carefully by requiring that the\nphysical dispersion relations of vector and pseudoscalar mesons are correctly reproduced at each\nbare quark mass~\\cite{chuan2}. The bare charm quark masses for the two lattices are set by the\nphysical mass of $J\/\\psi$ $m_{J\/\\psi}=3.097\\,{\\rm GeV}$.\n\nIn this work, we only consider the connected diagrams in the calculation of two-point and three-\npoint functions. The contribution of the disconnected diagrams is assumed to be small for\ncharmonium states due to the OZI suppression.\n\n\\begin{table}\n\\caption{\\label{tab:lattice} Relevant input parameters for this work. The spatial lattice spacing\n$a_s$ is determined from $r_0^{-1}=410(20)\\,{\\rm MeV}$ by calculating the static potential. }\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n$\\beta$ &  $\\xi$  & $a_s$(fm) & $La_s$(fm)&\n $L^3\\times T$ & $N_{conf}$ \\\\\\hline\n   2.4  & 5 & 0.222 & 1.78 &$8^3\\times 96$ & 1000 \\\\\n   2.8  & 5 & 0.138 & 1.66 &$12^3\\times 144$ & 1000  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\n\\end{table}\n\\par\n\\subsection{Ground-state charmonium spectrum}\nAs the first step, we carry out a careful study on the ground-state charmonium spectrum, which can\nillustrate to some extent the systematic uncertainties due to the quenched approximation. For the\nstates $\\eta_c(0^{-+})$, $J\/\\psi(1^{--})$, $h_c(1^{+-})$, $\\chi_{c0}(0^{++})$, and\n$\\chi_{c1}(1^{++})$, we adopt the conventional quark bilinear operators like $\\bar{c}\\Gamma c$,\nwith $\\Gamma=\\gamma_5, \\gamma_i,\\sigma_{ij}, 1$, and $\\gamma_5\\gamma_i$, respectively. For the\ntensor mesons $\\chi_{c2}(2^{++})$ and $\\eta_{c2}(2^{-+})$, since there are not quark bilinear\noperators, we build the corresponding operators by combining the quark bilinear operator with\neither the spatial gauge-covariant derivatives $D_i$ or the color magnetic field strength operator\n$B_i$, which is built from Wilson loops. It is known that the spin $J=2$ states in the continuum\ncorrespond to both the $T_2$ and $E$ irreducible representations (irreps) of the cubic point group\n $O$ on finite lattices, so the interpolating field operators of the two irreps are constructed for\n the tensor charmonia. For example, the $T_2$ operator for the $\\chi_{c2}(2^{++})$ state is taken as\n$|\\epsilon_{ijk}|\\bar{c}\\gamma_j \\overleftrightarrow{D}_k c$ where\n$\\overleftrightarrow{D}=\\overleftarrow{D}-\\overrightarrow{D}$, and the $E$ operator is also built\nto check the restoration of the continuum rotation symmetry.\n\nWe will emphasize the choice of the operators for the $2^{-+}$ state, which is the major object of\nthis work.  The situation for the $\\eta_{c2}$ meson is a little bit more complicated. We try first\nthree types of operators, such as\n\\begin{eqnarray}\\label{eq_opera}\n&&|\\epsilon_{ijk}|\\bar{c}(x)\\Sigma_j\\overleftrightarrow{D}_k c(x)~~(D{\\rm -type}),\\nonumber\\\\\n&&|\\epsilon_{ijk}|\\bar{c}(x)\\gamma_5 \\overleftrightarrow{D}_j \\overleftrightarrow{D}_k c(x)~~(DD{\\rm -type}),\\nonumber\\\\\n&&|\\epsilon_{ijk}|\\bar{c}(x)\\gamma_j B_k c(x)~~(F{\\rm -type}),\\nonumber\n\\end{eqnarray}\nwhere only the $T_2$ operators are presented ($E$ operators can be built similarly, and the details\ncan be found in Ref.~\\cite{Liao:2002rj}).\n\nIt is known that the signal-to-noise ratios of the correlation functions are always bad for\n$P$-wave and $D$-wave states. To circumvent this difficulty, we adopt the Coulomb-gauge fixed wall\nsource techniques in the calculation of the spectrum. The configurations are fixed to the Coulomb\ngauge first, then the charm quark propagators are calculated with uniform wall source vectors. For\nthe spin $J=0,1$ states, the point-sink wall source correlation functions can be constructed\nstraightforwardly with these propagators. For the tensors, we use the $F$-type operators as the\nwall source, which means that additional inversions should be carried out with wall sources\nmultiplied by the local color field strength operators $B_i(x)$. On the other hand, since the gauge\nis fixed, the gauge-covariant derivative operator $\\overleftrightarrow{D}$ is replaced by the\ndirect derivative operator\n$\\overleftrightarrow{\\nabla}=\\overleftarrow{\\nabla}-\\overrightarrow{\\nabla}$ in the practical\ncalculation.\n\nThe masses of $1S$ and $1P$ charmonium states can be neatly derived with the standard data\nanalysis, however, the situation for the $2^{-+}$ channel is very strange.\nFigure~\\ref{fig_2mp_mass} shows the effective masses of various correlation functions of this\nchannel at $\\beta=2.8$. It is seen that the effective mass of the $F$-type point sink and $F$-type\nwall source correlator ($F-F$) saturate at a plateau with the best-fit mass $4.43(8)\\,{\\rm GeV}$,\nwhile that of the $DD$-type point sink and $F$-type wall source correlator ($DD-F$) goes lower and\ndoes not show a perfect plateau. Intuitively, a mass of $4.4\\,{\\rm GeV}$ is too large for the\n$2^{-+}$ ground state charnonium. Thus what one can infer from these behaviors is that the $F$-type\noperator couples predominantly to a higher state but little to the conventional charmonium; in the\nmean time, there must be a lower state which can be accessed by the $DD$-type operator but whose\nspectral weight is relatively small due to the $F$-type wall source. To check this and to dig out\nthe desired $2^{-+}$ charmonium state, we try instead another wall source operator ($T_2$ irreps\nfor example),\n$$|\\epsilon_{ijk}|\\sum\\limits_{\\vec{x},\\vec{y},\\vec{z}}\\bar{c}^a(\\vec{x},0)\\gamma_j\nc'^a(\\vec{x},0)\\bar{c'}^b(\\vec{y},0)\\Sigma_k c^b(\\vec{z},0)~~(Q-{\\rm type}),$$ where\n$\\Sigma_k=\\epsilon_{ijk}\\sigma_{ij}$ and $c'$ stands for a quark field with the same mass as that\nof the charm quark but a different flavor. For simplicity, we call this operator $Q$-type in the\ncontext. With this type of wall source operator, the effective masses of the $F$-type point sink\ncorrelator ($F-Q$) and the $DD$ type point sink correlator ($DD-Q$) are also plotted in\nFig.~\\ref{fig_2mp_mass}, where one can find that the mass plateau of the $F-Q$ correlator coincides\nwith that of the $F-F$ correlator within errors, while the effective mass of the $DD-Q$ correlator\nshows a very nice plateau with the best-fit mass $3.79(3)\\,{\\rm GeV}$. Since the lower state has a\nmass close to the potential model prediction of $2^{-+}$ charmonium and the higher state is much\nheavier, we assign the lower state to the conventional ${}^1D_2$ charmonium state $\\eta_{c2}$. This\nassignment can be reinforced by comparison with the established $1{}^3D_1$ charmonium state\n$\\psi(3770)$: They are both $D$-wave charmonia and are therefore close in mass; the small mass\nsplitting can be attributed to the different spin-spin and spin-orbital interactions.\n\nThe whole spectrum of the lowest-lying charmonium states we extracted in this work is illustrated\nin Fig.~\\ref{spectrum} and listed in Table~\\ref{mass_table}, where the experimental values are also\ngiven for comparison. Since we have only two lattice spacings, we would not carry out a serious\nextrapolation to the continuum limit, but we show all the results, from which one can see that the\neffects of the finite lattice artifacts and the quenched approximation are not that important.\n\\begin{figure}\n\\includegraphics[scale=0.6]{2mp_DD_F.eps}\n\\caption{\\label{fig_2mp_mass} The $2^{-+}$ effective masses of $F-F$, $DD-F$, $F-Q$ and $DD-Q$\ncorrelators at $\\beta=2.8$ are plotted for illustration. $F-F$ and $F-Q$ effective masses lie on\neach other and saturate to the same plateau with the best-fit mass $M=4.43(8)\\,{\\rm GeV}$\n(indicated by the red line with the jackknife error band). The $DD-Q$ effective mass has a plateau\nat $M=3.79(3)$ ( blue line with the jackknife error band ). In contrast, $DD-F$ effective mass does\nnot show perfect plateau, but it evolves gradually from the upper plateau to the lower. }\n\\end{figure}\n\\begin{table}\n\\caption{\\label{mass_table} Listed here are the masses of the lowest-lying charmoinum states\nextracted from the two lattices ($\\beta=2.4$ and $\\beta=2.8$) in this work. The experimental\nresults~\\cite{PDG:2012} and the nonrelativistic quark model predictions~\\cite{BGS2005:2Pmass} are\nalso given for comparison.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\nmeson              & $J^{PC}$ & $M (2.4)$  & $M(2.8)$ &  Expt. & QM\n\n\\\\\\hline\n$\\eta_c({}^1S_0)$    & $0^{-+}$ &  2.989(2)        &  3.007(3)       & 2.981  & 2.982\\\\\n$J\/\\psi({}^3S_1)$    & $1^{--}$ &  3.094(3)        &  3.094(3)       & 3.097  & 3.090 \\\\\n&&&&&\\\\\n$h_c({}^1P_0)      $ & $1^{+-}$ &  3.530(35)       &  3.513(14)      & 3.526  & 3.516 \\\\\n$\\chi_{c0}({}^3P_0)$ & $0^{++}$ &  3.472(34)       &  3.431(30)      & 3.415  & 3.424 \\\\\n$\\chi_{c1}({}^3P_1)$ & $1^{++}$ &  3.508(50)       &  3.499(25)      & 3.511  & 3.505 \\\\\n$\\chi_{c2}({}^3P_2)$ & $2^{++}$ &  3.552(17)       &  3.520(15)      & 3.556  & 3.556 \\\\\n&&&&&\\\\\n$\\psi''({}^3D_1)   $ & $1^{--}$ &     \\--          &   \\--           & 3.770  & 3.785 \\\\\n$\\eta_{c2}({}^1D_2)$ & $2^{-+}$ &  3.777(30)       &  3.789(28)      &   \\--  & 3.799 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nThe goal of the spectroscopy study in this work is twofold. First, the physical spectrum of\nexperimentally established charmonium states can be well reproduced in our formalism. This gives us\nconfidence in our prediciton of the $\\eta_{c2}$ mass. Second, the practical study finds that the\n$DD$-type operator is preferable for producing the $2^{-+}$ charmonium. Therefore, in the study of\nits radiative transition, we choose the $DD$-type operator for $\\eta_{c2}$ in the calculation of\nthe related three-point functions.\n\n\\par\n\\begin{figure}\n\\includegraphics[scale=0.6]{diff_new.eps}\n\\caption{\\label{spectrum} $1S$, $1P$, and $1D$ charmonium spectrum. The red boxes illustrate the\nresults for $\\beta=2.4$, and the blue ones for $\\beta=2.8$. The experimental value are also plotted\nwith points for comparison.}\n\\end{figure}\n\n\\subsection{Renormalization of the vector current}\n\nIn the quenched approximation, since there are no sea quarks, the electromagnetic current\ncontributing to the radiative transitions of charmonia involves only the charm quark, say, $j_{\\rm\nem}(x)=Q_cj^\\mu(x)$ with $j^\\mu(x)=\\bar{c}\\gamma_\\mu c (x)$, which is the one we adopt in this\nstudy. It is a conserved vector current and need not be renormalized in the continuum. However, on\na finite lattice, it is not conserved anymore due to the lattice artifact and receives a\nmultiplicative renormalization factor $Z_V(a_s)$. Following the scheme proposed by\nRef.~\\cite{Dudek:2006ej}, $Z_V(a_s)$ is extracted using the ratio of the $\\eta_c$ two-point\nfunction and the related three-point function evaluated at $Q^2=0$,\n\\begin{equation}\n  Z^{(\\mu)}_V(t) = \\frac{p^\\mu}{E(\\vec{p})} \\frac{\\frac{1}{2} \\Gamma^{(2)}_{\\eta_c}(\\vec{p};\n  t_f=\\frac{n_t}{2}) }\n  {\\Gamma^{(3),\\mu}(\n    \\vec{p},\\vec{p},\\frac{n_t}{2}, t)}, \\nonumber\n\\end{equation}\nwhere the factor $1\/2$ accounts for the effect of the temporal periodic boundary condition, and the\nsuperscript $\\mu$ of $Z_V(a_s)$ is used to differentiate the temporal component from the spatial\nones, since they are not necessarily the same due to the anisotropic lattices we use.\nFigure~\\ref{fig_renorm} plots $Z^{(\\mu)}_V(t)$ with respect to $t$ for the two lattices.  $Z_V$'s\nare extracted from the plateaus and the values are listed in Table~\\ref{renorm}. Obviously, the\nspatial components $Z_V^{(s)}(a)$ deviate from the temporal ones by a few percent. This deviation\ncan be attributed to the imperfect tuning of the bare velocity in the fermion action. In this work,\nonly $Z_V^{(s)}$'s enter the calculation since only the spatial components of the vector current\nare involved in the extraction of the form factors.\n\\begin{table}\n\\caption{The renormalization constants $Z_V^{(s)}$ and $Z_V^{(t)}$ of the spatial and temporal\ncomponents of the vector current for $\\beta=2.4$ and $\\beta=2.8$ lattices. Two momentum modes,\n(0,0,0) and $(1,0,0)$, are used for the derivation.}\\label{renorm}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n$\\beta$ & $Z_V^{(t)}(0,0,0)$ & $Z_V^{(t)}(1,0,0)$ & $Z_V^{(s)}(1,0,0)$  \\\\\\hline\n $2.4$    & $1.288(5)$  & $1.299(11)$   & $1.388(15)$ \\\\\n $2.8$    & $1.155(3)$  & $1.159(3) $   & $1.110(7)$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{figure}[tbh]\\label{fig_renorm}\n\\includegraphics[scale=0.6]{jet_8.eps}\n\\includegraphics[scale=0.6]{jet_12.eps}\n\\caption{\\small The renormalization constant $Z_V$ of the vector current. The upper panel is for\n$\\beta=2.4$ and the lower one for $\\beta=2.8$. The dots are the simulation data with jackknife\nerrors, and the lines show the fit results.}\n\\end{figure}\n\n\\subsection{Three-point functions and form factors}\nWith the prescriptions discussed above, we now give a brief description of the calculation of the\nthree-point functions. In practice, we use local sink and source operators for the initial and the\nfinal states, and insert the vector current $j^\\mu(x)=\\bar{c}\\gamma^\\mu c (x)$ only on the quark\nline. (The current insertion on the antiquark line is numerically equivalent and is taken into\nconsideration by multiplying by a factor of 2 in the final result.) The three-point functions\ncontributed by the connected diagrams (disconnected diagrams are neglected) are calculated by using\nthe standard sequential source technique. (One can refer to Refs.\\cite{Dudek:2006ej,bonnet05} for\nthe details.) In order to increase the statistics, we repeat the same calculations $T$ times (where\n$T$ is the temporal lattice size) by setting a point source on a different time slice each time.\nWith the related two-point functions calculated accordingly, a straightforward way to extract the\ninterested matrix elements $\\langle f(\\vec{p'},r_f)|j^{\\mu}(0)|i(\\vec{p},r_i)\\rangle$ is to fit the\nthree-point function and two-point function simultaneously according to Eq.~(\\ref{three}) and\nEq.~(\\ref{two}). However, it is known that the excited states contribute much to two-point and\nthree-point functions when the time ranges $t$ and $t_f-t$ are not large enough. This situation is\nmore serious for local operators, so it is not trivial to isolate the contribution of ground\nstates. A way around this is to employ the ratios of correlation functions, which can suppress the\ncontribution of excited states substantially. For this purpose, we introduce the functions\n$R^\\mu(t)$,\n\\begin{eqnarray}\\label{eq_amp}\n &&R^\\mu(t)=\\Gamma^{(3)}(\\vec{p_f},\\vec{q},t_f,t)\\times \\nonumber\\\\\n &&\\sqrt{\\frac{2E_i\\Gamma^{(2)}_i(\\vec{p_i},t_f-t)}\n    {\\Gamma^{(2)}_i(\\vec{p_i},t)\\Gamma^{(2)}_i(\\vec{p_i},t_f)}}\n    \\sqrt{\\frac{2E_f\\Gamma^{(2)}_f(\\vec{p_f},t)}\n    {\\Gamma^{(2)}_f(\\vec{p_f},t_f-t)\\Gamma^{(2)}_f(\\vec{p_f},t_f)}},\\nonumber\\\\\n\\end{eqnarray}\nwhich should be insensitive to the variation of $t$ in a time window, so that the desired matrix\nelement $ \\langle f(\\vec{p}_f,r_f)|j^{\\mu}(0)|\\psi|i(\\vec{p}_i,r_i)\\rangle$ can be extracted from\nthe plateau.\n\nIn the data analysis, we divide the 1000 configurations into 100 bins and use each bin average as\nan independent measurement. For the resultant 100 bins, we use the one-eliminating jackknife\nmethod. Since the energies $E_{i,f}$ can be determined very precisely from the two-point functions,\nthey are treated as known parameters in the above equation. Practically, $R^\\mu(t)$ is fitted by\nthe function\n\\begin{equation}\nR^\\mu (t) = \\langle f(\\vec{p}_f,r_f)|j^{\\mu}(0)|\\psi|i(\\vec{p}_i,r_i)\\rangle+\\delta f(t)\n\\end{equation}\nwhere the additional term $\\delta f(t)= a e^{-\\delta m t}$ accounts for the residual contribution\nof excited states.\nThus we can obtain a jackknife ensemble of the matrix elements. The second step of data analysis is\nto extract the form factors that enter the calculation of decay widths. Since these matrix elements\ncan be expressed in terms of form factors through the multipole decomposition,\n\\begin{equation}\n \\langle f(\\vec{p}_f,r_f)|j^{\\mu}(0)|i(\\vec{p}_i,r_i)\\rangle=\\sum_k \\alpha_k^\\mu(p_i,p_f)\\hat{F}_k(Q^2),\n\\end{equation}\nand $\\alpha^\\mu_k (p_i,p_f)$ are theoretically known kinematic functions, the form factors\n$\\hat{F_k}(Q^2)$ can then be derived straightforwardly. Taking into consideration the contribution\nof the current insertion on the antiquark line, the electric charge of the charm quark $Q_c=2\/3$,\nand the renormalization constant of the spatial components of the current operator $Z_V^{(s)}$,\n$\\hat{F_k}$ is related to $F_k$ of Eq.~(\\ref{eq_ampth}) as\n\\begin{equation}\nF_k(Q^2) = 2\\times \\frac{2}{3}e \\times Z_V^{(s)} \\hat{F}_k(Q^2).\n\\end{equation}\nWith this in mind, in the following context, we omit the hat of $\\hat{F}$ and insert $Z_V^{(s)}$\nimplicitly in possible expressions.\n\nIn order to take good care of the correlation between the form factors, we carry out correlated\nminimal $\\chi^2$ fits with the jackknife covariance matrix built from the jackknife ensemble of the\nmatrix elements. On the other hand, for a specific $Q^2$, there may be several symmetric copies of\nthe matrix elements with the same value of $\\alpha_k^\\mu$. These copies are averaged over to\nincrease statistics.\n\nIn the following subsections, we present first the calculation of the process $\\chi_{c2}\\rightarrow\n\\gamma J\/\\psi$ to see how precisely the form factors\\-- and thereby the transition width\\-- can be\nderived, and then the results of $\\eta_{c2}\\rightarrow \\gamma\\psi$.\n\n\n\\subsection{$\\chi_{c2}\\rightarrow \\gamma J\/\\psi$ transition}\nThe Minkowski space-time matrix elements for this transition can be expressed in terms of form\nfactors as follows:\n\\begin{eqnarray}\n&&\\langle V(\\vec{p}_V, \\lambda_V) | j^{\\mu}(0) | T(\\vec{p}_T, \\lambda_T)\\rangle = \\alpha_1^\\mu\nE_1(Q^2) \\nonumber\\\\\n&&+ \\alpha_2^{\\mu}M_2(Q^2) + \\alpha_3^\\mu E_3(Q^2) + \\alpha_4^\\mu C_1(Q^2)+ \\alpha_5^\\mu\nC_2(Q^2)\\nonumber\\\\\n\\end{eqnarray}\nwhere $V$ stands for the $1^{--}$ vector meson $J\/\\psi$, $T$ stands for the $2^{++}$ tensor\n$\\chi_{c2}$, and $\\alpha_i^\\mu$ are Lorentz covariant kinematic functions of $p_V$ and $p_T$ (and\nspecific polarizations of $V$ and $T$), whose explicit expressions are tedious and omitted here.\nAlthough a $J=2$ representation of the rotational symmetry in the continuum breaks into the $E$ and\n$T_2$ irreducible representations (irreps) of the lattice spatial symmetry group $O$, we find that\nthis breaking effect is small in our work, as is manifested by the near degeneracy of the masses\nand spectral weights of the ground states in these two irreps when we study the relevant two-point\nfunctions. Thus, we assume that the rotation symmetry breaking is also negligible for the related\nmatrix elements, and we carry out the multipole decomposition on the basis of $E\\bigoplus T_2$,\nwhich is equivalent to the $J=2$ basis up to an orthogonal transformation. One can find the\ndetailed decomposition procedure in Appendix~\\ref{sec_a_pole2pp} and may also refer to\nRefs.~\\cite{Dudek:2006ej, Dudek2009}. In the practical study, we set $T$ to be at rest and let $V$\nmove with different spatial momenta $\\vec{p}=2\\pi \\vec{n}\/L$. The 27 momentum modes of\n$\\vec{n}=(n_1,n_2,n_3)$ ranging from $(0,0,0)$ to $(2,2,2)$ are calculated for $V$.\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{2pp_8.eps}\n\\includegraphics[scale=0.6]{2pp_12.eps}\n\\caption{\\label{fig_2pp} The extracted form factors $E_1(Q^2)$, $M_2(Q^2)$, and $E_3(Q^2)$ are\nplotted versus $Q^2$ for the two lattices of $\\beta=2.4$ (the upper panel) and $\\beta=2.8$ (the\nlower panel), respectively, where the points are the simulation data, the line the fit function,\nand the error bands the jackknife ones. The PDG values of $E_1(0)$ and $M_2(0)$ are also plotted\nfor comparison.}\n\\end{figure}\n\nThe transition width of $\\chi_{c2}\\rightarrow \\gamma J\/\\psi$ for an on-shell photon ($Q^2=0$)\ninvolves only the form factors $E_1(0)$, $M_2(0)$ and $E_3(0)$, or explicitly,\n\\begin{equation}\n\\Gamma(\\chi_{c2}\\rightarrow \\gamma\nJ\/\\psi)=\\frac{16\\alpha|\\vec{k}|}{45M^2_{\\chi_{c2}}}(|E_1(0)|^2+|M_2(0)|^2+|E_3(0)|^2)\\label{eq_form}\n\\end{equation}\nwhere $|\\vec{q}|=(M_{\\chi_{c2}}^2-M_{J\/\\psi}^2)\/2M_{\\chi_{c2}}$ is the decaying energy of the\nphoton, and $\\alpha=e^2\/(4\\pi)$ is the fine structure constant. Since our simulation data are\nobtained at $Q^2\\neq 0$, these on-shell form factors should be interpolated to $Q^2=0$. In doing\nthis, we adopt the fitting functional form inspired by the nonrelativistic quark\nmodel~\\cite{Dudek:2006ej},\n\\begin{equation}\nF_k(Q^2)=F_k(0)(1+\\lambda_k Q^2)e^{-\\frac{Q^2}{16 \\beta_k^2}}\n\\end{equation}\nwhich has been applied successfully in previous works. Here $F_k(0)$, $\\lambda_k$, and $\\beta_k$\nare the parameters to be fitted though a correlated $\\chi^2$ fitting procedure where the covariance\nmatrix is constructed using the one-eliminating jackknife method. Plotted in Fig.~\\ref{fig_2pp} are\nthe extracted form factors $E_1(Q^2)$, $M_2(Q^2)$, and $E_3(Q^2)$ versus $Q^2$ for the two lattices\nof $\\beta=2.4$ (the upper panel) and $\\beta=2.8$ (the lower panel). The data points are the\nsimulation results, and the lines are the fit function with the jackknife error bands. One can find\nthat the data are very precise owing to the high statistics, and the fit errors are also very\nsmall. We also carry out a simple polynomial fit with respect to $Q^2$, $F_k(Q^2)=F_k(0)+c_1 Q^2 +\nc_2 Q^4$, and get consistent results within errors. Table~\\ref{tab_2pp} lists the results of the\ninterpolation, where the continuum limit extrapolation is also given. It is seen that the electric\ndipole ($E_1$) contribution is dominant in the transition $\\chi_{c2}\\rightarrow \\gamma J\/\\psi$,\nwhile the contribution of the magnetic quadrupole ($M_2$) is drastically suppressed, as depicted by\nthe ratio\n\\begin{equation}\na_2=\\frac{M_2(0)}{\\sqrt{E_1(0)^2+M_2(0)^2+E_3(0)^2}},\n\\end{equation}\nfor which we get a result $a_2=-0.107(3)$ for $\\beta=2.4$ and $a_2=-0.082(7)$ for $\\beta=2.8$.\nAfter a linear extrapolation in $a_s^2$, we get the value in the continuum limit $a_2=-0.067(7)$,\nwhich is consistent with the PDG data, where $a_2=-0.100\\pm 0.015$~\\cite{PDG:2012}. The\ncontribution of the electric octupole $E_3$ is far smaller. For the ratio\n\\begin{equation}\na_3=\\frac{E_3(0)}{\\sqrt{E_1(0)^2+M_2(0)^2+E_3(0)^2}},\n\\end{equation}\nwe obtain $a_3=0.007(2)$ for $\\beta=2.4$, $a_3=0.003(4)$ for $\\beta=2.8$, and the continuum limit\n$a_3=-0.003(6)$, which are also compatible with the PDG data $a_3=0.016\\pm 0.013$~\\cite{PDG:2012}.\nIf we focus on $E_1$, we get the fitting parameters $\\beta_1$ and $\\lambda_1$,\n\\begin{eqnarray}\n\\beta_1&=&0.431(5)\\,{\\rm GeV}\\nonumber\\\\\n\\lambda_1&=& -0.285(2)\\,{\\rm GeV}^{-2}\n\\end{eqnarray}\nfor $\\beta=2.4$ and\n\\begin{eqnarray}\n\\beta_1&=&0.395(4)\\,{\\rm GeV}\\nonumber\\\\\n\\lambda_1&=& -0.336(3)\\,{\\rm GeV}^{-2}\n\\end{eqnarray}\nfor $\\beta=2.8$.\n\n\\begin{table}\n\\caption{Listed here are the results of the interpolated form factors $E_1(0)$, $M_2(0)$, and\n$E_3(0)$, as well as the transition widths.  The continuum limits are also given. All the results\nare in physical units. The widths can be compared with the PDG data $\\Gamma=380(30)\\,{\\rm\nkeV}$~\\cite{PDG:2012}.}\\label{tab_2pp}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n$\\beta$ & $E_1(\\textrm{GeV})$ & $M_2(\\textrm{GeV})$ & $E_3(\\textrm{GeV})$  & $\\Gamma(\\textrm{keV})$\\\\\n    \\hline\n $2.4$    & $2.04(2)$  & $-0.218(4)$   & $0.014(3)$    & $347\\pm 20$   \\\\\n $2.8$    & $2.08(2)$  & $-0.171(10)$  & $0.005(8)$    & $352\\pm 11$   \\\\\n Cont.    & $2.11(2)$  & $-0.141(15)$  & $-0.007(12)$   & $361\\pm 9$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\nUsing the interpolated form factors $F_k(0)$ and taking the fine structure constant $\\alpha=1\/137$,\nthe transition width can be calculated directly. As shown in Table~\\ref{tab_2pp}, the partial decay\nwidth of $\\Gamma(\\chi_{c2} \\to J\/\\psi\\ \\gamma)$ is predicted to be $347\\pm20$ keV or $352\\pm11$ keV\nfor the two lattices, respectively. The continuum extrapolation gives $\\Gamma=361\\pm 9\\,{\\rm keV}$.\nAll these results can be compared with the PDG average of $380(30)\\,{\\rm keV}$. The agreement with\nexperimental data of the $\\chi_{c2} \\to J\/\\psi\\ \\gamma$ transition indicates that our method for\nthe $\\chi_{c2}\\rightarrow J\/\\psi$ transition is reliable. Then we can turn to transition the\n$\\eta_{c2}\\rightarrow \\gamma J\/\\psi$.\n\n\\subsection{$\\eta_{c2} \\to J\/\\psi\\gamma$ transition}\nThe general Lorentz decomposition of the Minkowski matrix elements responsible for the transition\n$\\eta_{c2} \\to \\gamma J\/\\psi$ can be expressed as\n\\begin{eqnarray}\\label{2-+deco1}\n &&\\langle V(\\vec{p}_V, \\lambda_V) | j^{\\mu}(Q^2) | T(\\vec{p}_T, \\lambda_T) \\rangle =\n {a(Q^2)}A^{\\mu}\\nonumber\\\\\n &&+{b(Q^2)}B^{\\mu}+{c(Q^2)}C^{\\mu}+{d(Q^2)}D^{\\mu}+{e(Q^2)}E^{\\mu}\n\\end{eqnarray}\nwhere $T$ stands now for the tensor meson $\\eta_{c2}$;  $a(Q^2)$, $b(Q^2)$, $c(Q^2)$, $d(Q^2)$, and\n$e(Q^2)$ are Lorentz-invariant scalar functions of $Q^2$, and $A^\\mu$, $B^\\mu$, $C^\\mu$, $D^\\mu$\nare kinematic functions whose explicit expressions can be found in Appendix~\\ref{sec_a_pole2mp}.\nWith the multipole decomposition, the matrix elements can be also expressed in terms of form\nfactors $M_1$, $E_2$, $M_3$, and $C_2$:\n\\begin{eqnarray}\\label{eq_mTV}\n&&\\langle V(\\vec{p}_V, \\lambda_V) | j^{\\mu}(0) |T(\\vec{p}_T, \\lambda_T) \\rangle = i\\alpha_1^\\mu {\\\nM_1(Q^2)}\\nonumber\\\\ &&+ i\\alpha_2^\\mu {\\ E_2(Q^2)} + i\\alpha_3^\\mu {\\ M_3(Q^2)} -i \\alpha_4^\\mu {\\\nC_2(Q^2)}\n\\end{eqnarray}\nwhere $\\alpha_i^\\mu$ are also kinematic functions which can be expressed in terms of the kinematic\nfunctions in Eq.~(\\ref{2-+deco1}). (See Appendix~\\ref{sec_a_pole2mp}.) With real photons in the\ntransition $\\eta_{c2} \\to \\gamma J\/\\psi$, only three multipoles are contributing: the magnetic\ndipole ($M_1$), the electric quadrupole ($E_2$), and $M_3$. The transition width is written as\n\\begin{equation}\\label{width_2mp}\n\\Gamma(\\eta_{c2}\\rightarrow \\gamma\nJ\/\\psi)=\\frac{16\\alpha|\\vec{q}|}{45M^2_{\\eta_{c2}}}(|M_1(0)|^2+|E_2(0)|^2+|M_3(0)|^2).\n\\end{equation}\nSince they are calculated at $Q^2\\neq 0$, the multipole amplitudes should be interpolated to\n$Q^2=0$. The form factor $C_2(Q^2)$ corresponds to the emission of longitudinal photons and does\nnot contribute at $Q^2=0$. In extracting the amplitudes, we take the standard procedure as\ndescribed in Sec. II. The three-point functions are calculated by setting the tensor at rest and\nmaking the vector moving. In analogy with the $2^{++}$ case, the effect of rotational symmetry\nbreaking between $T_2$ irreps and $E$ irreps is found to be small in this case and is neglected in\nthe data analysis. The form factors $M_1(Q^2)$, $E_2(Q^2)$, and $M_3(Q^2)$ with various $Q^2$ are\nextracted jointly by a correlated fitting with a one-eliminating jackknife covariance matrix, and\nthe results are illustrated in Fig.~\\ref{fig_2mp} as data points with jackknife errors. In the\nfollowing discussion, we will focus on the interpolation procedure. What is interesting is the\nrelation between the two sets of form factors. We should first mention that the form factors, given\nthe fact that they are functions of $Q^2$, can also be written in terms of another Lorentz\ninvariant variable, $\\Omega$:\n\\begin{eqnarray}\n\\Omega &\\equiv& (p_V\\cdot p_T)^2-m_V^2 m_T^2\\nonumber\\\\\n&=&\\frac{1}{4}[(m_V+m_T)^2+Q^2][(m_V-m_T)^2+Q^2].\n\\end{eqnarray}\nThus the two sets of the form factors are related to each other as follows,\n\\begin{widetext}\n\\bea \\label{relation}M_1(\\Omega)&=&i\\frac{\\sqrt{\\Omega}}{m_T} \\left[\\sqrt{\\frac{5}{12}}\\left(a\\\nm_V+a\\ m_T-2 c\\ m_T\\right)+\\frac{2 a\\ m_V-3 c\\  m_T-4 d\\  m_V^2 m_T+6 e\\  m_V m_T^2 }{4\n\\sqrt{15}}\\left(\\frac{\\Omega}{ m_V^2 m_T^2}\\right)\\right.\\nonumber\\\\\n&& +\\left.\\frac{-2 a\\ m_V+3 c\\ m_T }{16 \\sqrt{15}}\\left(\\frac{\\Omega}{m_T^2 m_V^2}\\right)^2\n+O\\left(\\left(\\frac{\\Omega}{m_T^2 m_V^2}\\right)^3\\right)\\right],\n\\nonumber\\\\\nE_2(\\Omega)&=&i\\frac{\\sqrt{\\Omega}}{m_T}\\left[-\\sqrt{\\frac{3}{4}} \\left(a\\ m_T-a\\ m_V\n\\right)+\\frac{2 a\\ m_V-c\\ m_T- 4 d\\ m_V^2 m_T+2 e\\  m_V m_T^2}{4 \\sqrt{3}}\\left(\\frac{\\Omega}{\nm_V^2 m_T^2}\\right)\\right.\\nonumber\\\\&&\n+\\left.\\frac{-2 a\\ m_V+c\\  m_T }{16 \\sqrt{3}}\\left(\\frac{\\Omega}{m_T^2 m_V^2}\\right)^2+\nO\\left(\\left(\\frac{\\Omega}{m_T^2 m_V^2}\\right)^3\\right)\\right],\\nonumber\\\\\nM_3(\\Omega)&=&i\\frac{\\sqrt{\\Omega}}{m_T}\\left[-\\frac{-a\\ m_V-c\\  m_T+2 d\\  m_V^2 m_T+ 2 e\\  m_V\nm_T^2}{\\sqrt{15}}\\left(\\frac{\\Omega}{ m_V^2 m_T^2}\\right)-\\frac{a\\ m_V+ c\\  m_T}{4\n\\sqrt{15}}\\left(\\frac{\\Omega}{m_T^2 m_V^2}\\right)^2 \\right.\\nonumber\\\\\n&&+\\left.O\\left(\\left(\\frac{\\Omega}{m_T^2m_V^2}\\right)^3\\right)\\right]. \\eea\n\\end{widetext}\nIt is seen that each multipole form factor can be expressed as a series of $\\Omega\/(m_V^2m_T^2)$\nwith a prefactor $\\sqrt{\\Omega}\/m_T$. In the rest frame of $T$ (as is the case in our calculation),\nthe expression of $\\Omega$ is simplified as $\\Omega=(m_T |\\vec{p}_V|)^2$, such that\n$\\Omega\/(m_V^2m_T^2)=v^2$, with $v=|\\vec{p}_V|\/m_V$ being the spatial velocity of $V$. The\nconvergence of the series in $v$ is guaranteed if the form factors $a$, $b$, $c$, $d$, $e$ are not\nsingular in $Q^2$, since $v<1$ (For our calculation, the largest value of $v$ is approximately\n$0.5$). So, for the decaying $T$ at rest, we have the simplified expression of the form factors\n$M_1$, $E_2$, and $M_3$:\n\\begin{eqnarray}\\label{relation2}\nM_1&=&|\\vec{p}_V|(A_1(Q^2)+B_1(Q^2)v^2+C_1(Q^2)v^4+O(v^6))\\nonumber\\\\\nE_2&=&|\\vec{p}_V|(A_2(Q^2)+B_2(Q^2)v^2+C_2(Q^2)v^4+O(v^6))\\nonumber\\\\\nM_3&=&|\\vec{p}_V|(B_3(Q^2)v^2+C_3(Q^2)v^4+O(v^6)).\n\\end{eqnarray}\nWith these expressions, the following information can be inferred: (i) The desired $|\\vec{p}_V|$\nprefactor accounting for the $P$-wave decay of $\\eta_{c2}\\to \\gamma J\/\\psi$ is explicitly derived.\n(ii) The leading contribution to $M_1$ and $E_2$ is of order $O(1)$, while that of $M_3$ is of\norder $O(v^2)$. For the case of this study, since $v_{\\rm max}\\sim 0.5$, it is reasonable that the\nnonsingular $A_i,B_i$ and $C_i$ can be expanded with respect to $v$, such that we can take the\nfollowing functions to do the interpolation:\n\\begin{eqnarray}\\label{fit_2mp}\nF_i(v)&=& Av+Bv^3+Cv^5+O(v^6)(F_i\\to M_1,~E_2)\\nonumber\\\\\nF_i(v)&=& Bv^3+Cv^5+Dv^7 +O(v^9)(F_i\\to M_3),\n\\end{eqnarray}\nand the on-shell amplitudes $M_1(Q^2=0)$, $E_2(Q^2=0)$, and $M_3(Q^2=0)$ can be reached by\n$F_i(v_0)$ with $v_0=(m_T^2-m_V^2)\/(2m_Tm_V)$. The extracted form factor and the interpolation are\n shown in Fig.~\\ref{fig_2mp}, where the data points are the simulated results with jackknife errors.\n One can see that at $v=0$ [corresponding to $Q^2=-(m_T-m_V)^2\\sim 0.5\\,{\\rm GeV}^2$] the form\n factors $M_1$, $E_2$, and $M_3$ are surely consistent with zero. The fits using\n Eq.~(\\ref{fit_2mp})\n are also shown as curves with jackknife error bands. The interpolated values of these form factors\n at $Q^2=0$ for both $\\beta=2.4$ and $\\beta=2.8$ are listed in Table~\\ref{tab_2mp}, where the\n resultant transition widths and the corresponding continuum limits are also given. It is surprising\n that, for both lattices, the obtained $|M_3|$ is unexpectedly large and comparable to $M_1$. This\n may be qualitatively attributed to recoiling effects of the charm quark or charm antiquark by emitting the hard\n photon with an energy $E_\\gamma\\sim 0.6\\,{\\rm GeV}$ in this transition, which may result in large\n form factors $d(Q^2)$ and $e(Q^2)$ (see the discussion below). In contrast to the mild dependence of $M_1$ and\n $M_3$ on the lattice spacing, the form factor $E_2$ is very sensitive to the lattice spacing.\n The reason for this is unclear and under investigation. Anyway, after a naive continuum\n extrapolation using the data from the two lattices in this work, we get the continuum results of\n the form factors as follows:\n\\begin{eqnarray}\\label{eq_formus}\nM_1&=&0.104(10)\\textrm{GeV},\\nonumber\\\\\nE_2&=&-0.071(20)\\textrm{GeV},\\nonumber\\\\\nM_3&=&-0.132(10)\\textrm{GeV}.\n\\end{eqnarray}\nApplying these results to Eq.~(\\ref{width_2mp}), the transition width of $\\eta_{c2}\\to \\gamma\nJ\/\\psi$ is predicted to be\n\\begin{equation\n\\Gamma(\\eta_{c2}\\rightarrow \\gamma J\/\\psi)=3.8\\pm 0.9\\,{\\rm keV}.\n\\end{equation}\n\\begin{figure}[tbh!]\n\\includegraphics[scale=0.6]{2mp_8.eps}\n\\includegraphics[scale=0.6]{2mp_12.eps}\n\\caption{\\label{fig_2mp} The $\\eta_{c2}-J\/\\psi$ transition form factors $M_1(Q^2)$, $E_2(Q^2)$, and\n$M_3(Q^2)$ are plotted versus $Q^2$ for the two lattices of $\\beta=2.4$ (the upper panel) and\n$\\beta=2.8$ (the lower panel), respectively. The points are the simulation data, and the lines\nillustrate the fit function with jackknife error bands.}\n\\end{figure}\n\\begin{table}[t]\n\\caption{\\label{tab_2mp} Listed here are the interpolated values of the form factors $M_1$, $E_2$,\nand $M_3$ at $Q^2=0$ for both $\\beta=2.4$ and $\\beta=2.8$. The\n resultant transition widths and the corresponding continuum limits are also given.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n$\\beta$ & $M_1(\\textrm{GeV})$ & $E_2(\\textrm{GeV})$ & $M_3(\\textrm{GeV})$  & $\\Gamma(\\textrm{keV})$\\\\\n    \\hline\n $2.4$    & $0.133(13)$  & $0.111(17)$   & $-0.093(9)$    & $4.4\\pm0.9$   \\\\\n $2.8$    & $0.115(11)$  & $-0.0007(14)$ & $-0.117(9)$    & $3.1\\pm0.6$   \\\\\n Cont.    & $0.104(10)$  & $-0.071(20)$  & $-0.132(10)$   & $3.8\\pm0.9$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nThere have also been several phenomenological studies on this transition, one of which is in the\nframework of the light-front quark model~\\cite{Ke:2011jf}, where the on-shell transition amplitude\nis decomposited as\n\\bea\n&&\\langle V(\\vec{p}_V, \\lambda_V) | j^{\\mu}(0) | \\eta_{2}(\\vec{p}_T, \\lambda_T) \\rangle = \\nonumber \\\\\n&&\\Big[2f_1\\epsl^{\\mu\\nu\\rho\\sigma}p^T_{\\nu} p^V_{\\rho}\\epsilon^{\\beta}_{\\sigma}(\\vec{p}_T,\n\\lambda_T) \\epsilon^*_{\\beta}(\\vec{p}_V, \\lambda_V)\n\\nonumber\\\\\n&&+(f_2+f_3)\\epsl^{\\mu\\nu\\rho\\sigma}p^T_{\\nu} p^V_{\\rho}\\epsilon^*_{\\sigma}(\\vec{p}_V, \\lambda_V)\n\\epsilon^{\\alpha \\beta}(\\vec{p}_T, \\lambda_T)p^V_{\\alpha}p^V_{\\beta}\\ \\nonumber \\\\&&\n+2f_4\\epsl^{\\mu\\nu\\rho\\sigma}p^T_{\\nu} p^V_{\\rho}\\epsilon^{\\beta}_{\\sigma}(\\vec{p}_T,\n\\lambda_T)p^V_{\\beta} \\epsilon^{*\\alpha}(\\vec{p}_V, \\lambda_V)p^T_{\\alpha}\\Big],\\eea\nand the effective couplings are determined to be\n\\begin{eqnarray}\\label{eq_lifn}\nf_1&=&-0.0140(2)\\textrm{GeV}^{-1},\\nonumber\\\\\nf_2&=& 0.146(3)\\textrm{GeV}^{-3},\\nonumber\\\\\nf_3&=&-0.092(1)\\textrm{GeV}^{-3},\\nonumber\\\\\nf_4&=&0.0180(1)\\textrm{GeV}^{-3}.\n\\end{eqnarray}\nSince this decomposition is equivalent to Eq.~(\\ref{2-+deco1}) by the relation  \\bea c(0)=2f_1,\\\nd(0)=-(f_2+f_3),\\ e(0)=-2f_4 \\eea  [It should be notified that $a(Q^2)$ and $b(Q^2)$ are equal to\nzero when $Q^2=0$ because they are proportion to $C_2(Q^2)$.], the corresponding multipole\namplitudes can be calculated from Eq.~(\\ref{relation}) as\n\\begin{eqnarray}\n\\label{eq_licasea}\nM_1&=&0.079(2)\\textrm{GeV},\\nonumber\\\\\nE_2&=&-0.086(2)\\textrm{GeV},\\nonumber\\\\\nM_3&=&-0.125(3)\\textrm{GeV},\n\\end{eqnarray}\nwhich gives a width of $\\Gamma$=3.54(12) keV. Taking into consideration the uncertainty of the\nchoice of parameters such as the charm quark mass $m_c$ and the wave function parameter, etc., one\ncan find that the lattice results and the LFQM results are surprisingly in excellent agreement. On\nthe other hand, with the values in Eq.~(\\ref{eq_lifn}), we find that the coefficients $B_i(Q^2)$ of\nthe $v^2$ term in Eq.~(\\ref{relation2}) are surely much larger than $A_i(Q^2)$ at $Q^2=0$ so as to\ncompensate for the suppression of $v^2$. This explains to some extent the fact that the $M_3$ in\nthis transition competes $M_1$ and $E_2$.\n\nThe other phenomenological study~\\cite{Jia:2010jn} applying the nonrelativistic QCD (NRQCD) gives\nthe transition width as\n\\begin{equation} \\Gamma(\\eta_{c2}\\to \\gamma\nJ\/\\psi)=\\frac{8\\alpha|\\vec{k}|^3}{675m^2_c}(a_1^2+a_2^2+a_3^2)\\label{eq_mTV_ph}\n\\end{equation}\nwhere $a_1$, $a_2$, and $a_3$ are equivalent to the standard formfactors $M_1$, $E_2$, and $M_3$ up\nto a constant factor and are calculated explicitly in NRQCD. By comparing this equation with\nEq.~(\\ref{eq_form}), the factor is approximately $0.33\\,{\\rm GeV}$, say, $F_i\\simeq (0.33\\,{\\rm\nGeV}) a_i$. Thus, their work gives the predictions\n\\begin{equation}\nM_1\\sim 0.026-0.045\\,{\\rm GeV} ,E_2\\simeq M_3 \\simeq -0.13\\,{\\rm GeV},\n\\end{equation}\nwhich are also in reasonable agreement with our results.\n\n\\section{Conclusion}\\label{section4}\nWe calculate in the quenched approximation the mass of $J^{PC}=2^{-+}$ charmonium $\\eta_{c2}$, as\nwell as its radiative transition width to $J\/\\psi$. The computations are carried out on two\nanisotropic lattices with different lattice spacings, such that the lattice artifacts can be\ncontrolled to some extent. As a calibration, we calculate first the spectrum of the lowest-lying\ncharmonia, such as $1S$ and $1P$ states, and reproduce the physical pattern of the spectrum. In\naddition, we calculate the transition width of $\\chi_{c2}\\rightarrow \\gamma J\/\\psi$ and get the\nresult $361\\pm 9\\,{\\rm keV}$, which is in good agreement with the experimental value of $380\\pm\n30\\,{\\rm keV}$. Both of these facts manifest the small systematic uncertainties due to the quenched\napproximation and the finite lattice spacings.\n\nThere are two states observed in the $2^{-+}$ channel, with masses $3.80(3)\\,{\\rm GeV}$ and\n$4.43(8)\\,{\\rm GeV}$. The lower state has a mass similar to that of the well-established\n$\\psi(3770)$, which is always assigned to be mainly the $1{}^3D_1$ charmonium, and therefore can be\nnaturally identified as the conventional $1{}^1D_2$ charmonium $\\eta_{c2}$. Obviously, it is about\n70 MeV lower in mass than $X(3872)$, and this difference cannot be attributed to the systematic\nuncertainties of our work.\n\nAs for the transition rate of $\\eta_{c2} \\to J\/\\psi\\gamma$, we get a small partial width of roughly\n3.8(9) keV, which is in agreement with the phenomenological studies. Taking the branch ratio ${\\rm\nBr}(X(3872)\\rightarrow J\/\\psi \\gamma)> 0.9\\%~(BABAR)$ or $0.6\\%~({\\rm Belle})$, the full width of\n$X(3872)$ is estimated to be $<420-630$ keV, which is smaller than, but not in contradiction with\nthe experimental upper limit $\\Gamma_X<1.2\\,{\\rm MeV}$. Obviously, a reliable calculation of the\npartial width $\\eta_{c2}\\to \\psi' \\gamma $ is also crucial for the ${}^1D_2$ charmonium assignment\nof $X(3872)$, but unfortunately there are difficulties in the unambiguous extraction of excited\nstates on the lattice. However, we can still infer some useful information from the calculation of\n$\\eta_{c2} \\to J\/\\psi\\ \\gamma$. In the potential quark model, it is known that ${}^1D_2\\to \\gamma\nV$ is a hindered transition with $M_1(0)=0$; therefore, the observed nonzero $M_1(0)$ and the\nappearance of the higher multipoles $E_2$ and $M_3$ can be understood as the relativistic\ncorrection and the recoil effects of the emission of a hard photon ($E_\\gamma\\sim 0.65\\,{\\rm GeV}$\nfor the final $J\/\\psi$ and $E_\\gamma\\sim 0.11\\,{\\rm GeV}$ for $\\psi'$). Intuitively, this kind of\neffect for the final $\\psi'$ can be similar to that for the final $J\/\\psi$, or even milder; thus,\nthe width of $\\eta_{c2}\\to \\gamma \\psi'$ will be suppressed by a kinematic factor of\n$(0.65\/0.11)^3\\sim 200$ when compared with the transition $\\eta_{c2}\\to \\gamma J\/\\psi$. With this\nfact in mind, the $\\eta_{c2}$ assignment of $X(3872)$ can be ruled out if {\\it BABAR}'s observation\nof ${\\rm Br}(X(3872)\\to \\gamma \\psi')\/{\\rm Br}(X(3872)\\to \\gamma J\/\\psi)=3.4\\pm 1.4$ is confirmed.\n\n\n\\section*{ACKNOWLEDGEMENTS}\n\n\nThis work was supported in part by the National Science Foundation of China (NSFC) under Grants No.\n10835002, No. 11075167, No. 11021092, No. 10975076, and No. 11105153. Y. Chen and C. Liu also thank\nthe support of NSFC and DFG (CRC110).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\n\nMonad bundles~\\cite{Witten:1993yc, Distler:1993mk, Blumenhagen:1996vu,\n  Anderson:2007nc, Anderson:2008uw, Anderson:2008ex, Guffin:2008pi}\nare the largest known class of $(0,2)$-compactifications. However, so\nfar only monad bundles on simply-connected Calabi-Yau manifolds were\nexplicitly constructed. However, just as in the heterotic standard\nembedding, free quotients are, amongst many other aspects, important\nin reducing the particle spectrum. For example, the net number of\ngenerations was found~\\cite{Anderson:2008uw} to peak somewhere around\n$60$. By dividing out the free action of a discrete group $G$, the\nnumber of generations would be divided by $|G|$.\n\nIn this paper, I will construct a slope-stable rank $5$ vector bundle\non the $\\ensuremath{{\\Z_5\\times\\Z_5}}$-quotient of the quintic via the monad construction. The\nbest way to work with non-simply connected manifolds is, as usual, to\nconstruct everything on the universal cover (the quintic). However,\ncare has to be taken to make everything symmetric under the group\naction, and as we will see this imposes purely topological\nrestrictions on the Chern classes of the constituents of the\nmonad. Via the so-called ``non-standard embedding'', this bundle of\nvanishing first Chern class defines a $(0,2)$-compactification of the\n$E_8$ heterotic string, giving rise to a low-energy $SU(2)$ gauge\ngroup. The resulting matter spectrum will be three generations of\n$\\Rep{10}+\\barRep{5}$ together with an (optional) vector-like pair of\n$\\Rep{5}$--$\\barRep{5}$. A hopefully useful \\textsc{Singular}\\xspace worksheet\ndemonstrating the spectrum computation is in\n\\autoref{sec:singular}. Although $\\mathbb{Z}_5$ Wilson lines cannot be used to\nbreak $SU(5)$ to the standard model gauge group, a mechanism like\n$U(1)_Y$-flux could conceivably be employed. In any case, the\ntechnology for constructing equivariant monads will certainly be\nuseful for more thorough searches for heterotic standard models.\n\n\n\n\\section{The Quintic Quotient}\n\\label{sec:quintic}\n\nThe quintic $Q$ is the simplest Calabi-Yau manifold. As the name\nsuggests, it is given by the zero set of a sufficiently generic\ndegree-$5$ hypersurface $Q(z_0,z_1,z_2,z_3,z_4)=0$ in projective space\n$\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$. Using this description of a smooth hypersurface, one can show\nthat there are precisely two\\footnote{The $\\mathbb{Z}_5$ is a subgroup of\n  $\\ensuremath{{\\Z_5\\times\\Z_5}}$, so there is one \\emph{maximal} free group action.}  possible\nfree group actions on the quintic: $\\mathbb{Z}_5$ and $\\ensuremath{{\\Z_5\\times\\Z_5}}$. However, the\ncomplex structure of $Q$, that is, the quintic polynomial $Q(z)$, has\nto be chosen suitably to admit this symmetry. Without symmetry, the\nquintic has $h^{2,1}(Q)=101$ complex structure moduli. Imposing the\nsymmetries restricts the complex structure moduli to a $25$ and\n$5$-dimensional stratum, respectively.\n\nFor the purposes of this paper, I will focus solely on the free $\\ensuremath{{\\Z_5\\times\\Z_5}}$\ngroup action~\\cite{Candelas:1985en}. It acts projectively linear on the homogeneous\ncoordinates of the ambient projective space; The two group generators\nare ($\\zeta = e^{\\frac{2\\pi i}{5}}$)\n\\begin{equation}\n  \\label{eq:gzact}\n  g_1(z_i) = z_{i+1}\n  ,\\qquad\n  g_2(z_i) = \\zeta^i z_i  \n  .\n\\end{equation}\nThey satisfy\n\\begin{equation}\n  \\label{eq:g1g2commutator}\n  g_1^5=1=g_2^5\n  ,\\quad\n  g_1 g_2 = \\zeta^{-1} g_2 g_1\n\\end{equation}\nand, therefore, define a $\\ensuremath{{\\Z_5\\times\\Z_5}}$ group action on $\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$. If the\nquintic is invariant, then this action defines a group action on the\nhypersurface, too. The invariant polynomials are best described by a\nHironaka decomposition\n\\begin{equation}\n  \\label{eq:HironakaZ5Z5}\n  \\ensuremath{{\\mathbb{C}}}[z_0,z_1,z_2,z_3,z_4]^\\ensuremath{{\\Z_5\\times\\Z_5}} =\n  \\bigoplus_{i=1}^{100}\n  \\eta_i \n  \\, \n  \\ensuremath{{\\mathbb{C}}}[\\theta_1,\\theta_2,\\theta_3,\\theta_4,\\theta_5 ]\n  .\n\\end{equation}\nHere, the primary invariants are~\\cite{Braun:2007sn}\n\\begin{equation}\n  \\label{eq:thetadef}\n  \\begin{array}{rcl}\n    \\theta_1 =&\\; \n    z_0^5+z_1^5+z_2^5+z_3^5+z_4^5 \n    \\;&= z_0^5 + \\ensuremath{(\\text{cyc})}\n    \\\\\n    \\theta_2 =&\\; \n    z_0 z_1 z_2 z_3 z_4 \n    \\\\\n    \\theta_3 =&\\; \n    z_0^3 z_1 z_4+z_0 z_1^3 z_2+z_0 z_3 z_4^3\n    +z_1 z_2^3 z_3+z_2 z_3^3 z_4 \n    \\;&= z_0^3 z_1 z_4 + \\ensuremath{(\\text{cyc})}\n    \\\\\n    \\theta_4 =&\\; \n    z_0^{10}+z_1^{10}+z_2^{10}+z_3^{10}+z_4^{10} \n    \\;&= z_0^{10} + \\ensuremath{(\\text{cyc})}\n    \\\\\n    \\theta_5 =&\\; \n    z_0^8 z_2 z_3+z_0 z_1 z_3^8+z_0 z_2^8 z_4+z_1^8 z_3 z_4+z_1 z_2\n    z_4^8\n    \\;&= z_0^8 z_2 z_3 + \\ensuremath{(\\text{cyc})}\n  \\end{array}\n\\end{equation}\nand the secondary invariants are, in degrees $< 10$, \n\\begin{equation}\n\\begin{gathered}\n  \\begin{aligned}\n    \\eta_1 =&\\; 1    ,\\\\ \n  \\end{aligned}\n  \\\\[1ex]\n  \\begin{aligned}\n    \\eta_2 =&\\; z_0^2 z_1 z_2^2 + \\ensuremath{(\\text{cyc})}    ,&\n    \\eta_3 =&\\; z_0^2 z_1^2 z_3 + \\ensuremath{(\\text{cyc})}    ,&\n    \\eta_4 =&\\; z_0^3 z_2 z_3 + \\ensuremath{(\\text{cyc})}      \n    .\n  \\end{aligned}\n\\end{gathered}\n\\end{equation}\nHence, an invariant quintic is of the form\n\\begin{equation}\n  Q(z) = c_0 \\theta_1 + c_1 \\theta_2 + c_2 \\theta_3 + \n  c_3 \\eta_2 + c_4 \\eta_3 + c_5 \\eta_4\n  .\n\\end{equation}\nThe complex structure moduli space of the invariant quintics is\nan open subset of $\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^5$ parametrized by $[c_0:\\cdots:c_5]$. Finally,\none can easily check that a generic member of this family of quintics\nis smooth and fixed-point free. Therefore, the quotient\n\\begin{equation}\n  X = Q \\Big\/ \\big(\\ensuremath{{\\Z_5\\times\\Z_5}}\\big)\n\\end{equation}\nis a smooth Calabi-Yau threefold with fundamental group $\\pi_1(X) =\n\\ensuremath{{\\Z_5\\times\\Z_5}}$.\n\n\n\\section{Monadology}\n\\label{sec:monad}\n\n\\subsection{Overview}\n\\label{sec:monadintro}\n\nA monad~\\cite{Monadology, Witten:1993yc, Distler:1993mk, Blumenhagen:1996vu} is a\nthree-step filtration of a vector bundle $V$. That is, a complex\n\\begin{equation}\n  \\xymatrix{\n    0 \\ar[r] & \n    A \\ar[r]^-a & \n    B \\ar[r]^-b & \n    C \\ar[r] & \n    0\n  }\n\\end{equation}\nwhose cohomology is a vector bundle in the middle, at the $B$\nentry. In other words, $a$ is injective, $b$ is surjective, $b \\circ\na = 0$, and $V = \\ker(b)\/\\img(a)$ is the vector bundle. For example, a\nshort exact sequence corresponds to the zero vector bundle.\n\nFor the purposes of this paper, I will only consider \\emph{positive\n  monads} where $A=0$ and $B$, $C$ are very ample bundles. In this\ncase, $V$ \\emph{is} defined by a short exact sequence\n\\begin{equation}\n  \\label{eq:posmonad1}\n  \\xymatrix{\n    0 \\ar[r] & \n    V \\ar[r] & \n    B \\ar[r]^-f & \n    C \\ar[r] & \n    0\n  }  \n\\end{equation}\nIn particular, I will always take the base space to be $Q\\subset\n\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$ and \n\\begin{equation}\n  \\label{eq:posmonad2}\n  B = \\bigoplus_{i=1}^n \\ensuremath{\\mathscr{O}}(b_i)\n  ,\\quad\n  C = \\bigoplus_{j=1}^m \\ensuremath{\\mathscr{O}}(c_j)\n\\end{equation}\nwith $b_i$, $c_j>0$. The positive monads for rank $3$, $4$, and $5$\nbundles satisfying heterotic anomaly cancellation without anti-branes\nwere classified in~\\cite{Anderson:2007nc, Anderson:2008uw,\n  Anderson:2008ex}. There are $43$ such monads, none of which gives\nrise to $3$ generations\\footnote{The net number of generations, that\n  is the difference between generations and anti-generations, equals\n  the index $\\Ind(V)$ of the vector bundle.}. Adding the additional\nconstraint that the number of generations is at least a multiple of\n$3$, the authors\n\\begin{table}[htb]\n  \\begin{center}\n    \\begin{tabular}{|c|c|c|c|c|} \\hline\n      $\\rank V$ & $\\{b_i\\}$ & $\\{c_i\\}$ & $\\Ind V$ \n      & $G_\\text{3-gen}$ \\\\ \\hline\\hline\n      3 & \n      \\textcolor{red}{$\\{$2, 2, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$4, 3$\\}$} & \n      \\textcolor{red}{$-60$} & -- \\\\\n      3 &\n      \\textcolor{red}{$\\{$2, 2, 2, 1, 1$\\}$} &\n      \\textcolor{red}{$\\{$5, 3$\\}$} & \n      \\textcolor{red}{$-105$} & -- \\\\ \n      3 & \n      \\textcolor{red}{$\\{$3, 2, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$4, 4$\\}$} & \n      $-75$ & -- \\\\ \n      3 & \n     \n      \\textcolor{red}{$\\{$1, 1, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$2, 2, 2$\\}$} & \n      \\textcolor{red}{$-15$} & $\\mathbb{Z}_5$ \\\\ \n      3 & \n      \\textcolor{red}{$\\{$2, 2, 2, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$3, 3, 3$\\}$} & \n      \\textcolor{red}{$-45$} & -- \\\\ \n      3 & \n      \\textcolor{red}{$\\{$3, 3, 3, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$4, 4, 4$\\}$} & \n      \\textcolor{red}{$-90$} & -- \\\\ \n      3 & \n      \\textcolor{red}{$\\{$2, 2, 2, 2, 2, 2, 2, 2$\\}$} & \n      \\textcolor{red}{$\\{$4, 3, 3, 3, 3$\\}$} &\n      \\textcolor{red}{$-90$} & -- \\\\ \n      3 & \n      \\textcolor{red}{$\\{$2, 2, 2, 2, 2, 2, 2, 2, 2$\\}$} & \n      \\textcolor{red}{$\\{$3, 3, 3, 3, 3, 3$\\}$} & \n      $-75$ & -- \\\\ \n      4 & \n      \\textcolor{red}{$\\{$2, 2, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$4, 4$\\}$} & \n      \\textcolor{red}{$-90$} & -- \\\\ \n      4 & \n      \\textcolor{red}{$\\{$1, 1, 1, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$3, 2, 2$\\}$} & \n      \\textcolor{red}{$-30$} & -- \\\\ \n      4 & \n      \\textcolor{red}{$\\{$2, 2, 2, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$4, 3, 3$\\}$} & \n      $-75$ & -- \\\\ \n      4 & \n      \\textcolor{red}{$\\{$2, 2, 2, 2, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$3, 3, 3, 3$\\}$} & \n      \\textcolor{red}{$-60$} & -- \\\\ \n      5 & \n      \\textcolor{red}{$\\{$1, 1, 1, 1, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$3, 3, 2$\\}$} & \n      \\textcolor{red}{$-45$} & -- \\\\ \n      5 & \n      \\textcolor{red}{$\\{$1, 1, 1, 1, 1, 1, 1, 1$\\}$} & \n      \\textcolor{red}{$\\{$4, 2, 2$\\}$} & \n      \\textcolor{red}{$-60$} & -- \\\\ \n      5 & $\\{$2, 2, 2, 2, 2, 1, 1, 1, 1, 1$\\}$ & $\\{$3, 3, 3, 3, 3$\\}$ &\n      $-75$ & $\\mathbb{Z}_5\\times\\mathbb{Z}_5$\\\\ \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Positive monad bundles on the quintic. The entries marked\n    in \\textcolor{red}{red} are obstructions to a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-action, see\n    \\autoref{sec:topology}. The last column $G_\\text{3-gen}$ are\n    free group actions such that the quotient has three generations.}\n  \\label{tab:monads}\n\\end{table}\nfound $15$ monads on the quintic, which are reproduced in\n\\autoref{tab:monads} for convenience. As I will discuss shortly, most\nof these monad bundles do not have a suitable symmetry for the\nquotient to yield a three generation model. However, two of them\ndo. The first one is a rank $3$ bundle which we can divide by\n$\\mathbb{Z}_5$. This yields a low-energy $E_6$ gauge group with three\ngenerations of $\\Rep{27}$ and no anti-generations. The $\\mathbb{Z}_5$\nfundamental group can be used to break the $E_6$ to the standard model\ngauge group plus two extra $U(1)$~\\cite{Goodsell:2009xc}, though it\nwould be difficult to remove the exotic matter coming from the\ndecomposition of the $\\Rep{27}$. The other model, and subject of this\npaper, is in the last row of \\autoref{tab:monads}. As we will see in\nthe remainder of this section, being able to divide out the non-cyclic\ngroup $\\ensuremath{{\\Z_5\\times\\Z_5}}$ will pose many more restrictions on the monad bundle than\njust $\\mathbb{Z}_5$. At the same time, the larger group makes it much easier\nto project out unwanted matter states.\n\n\n\\subsection{Symmetry Considerations}\n\\label{sec:symmetry}\n\nFor the reasons just mentioned, in the following I will only consider\nthe monad bundle\\footnote{I will always write $r \\ensuremath{\\mathscr{O}}(n)$ for the\n  rank-$r$ bundle $\\oplus_{i=1}^r \\ensuremath{\\mathscr{O}}(n) = \\ensuremath{\\mathscr{O}}(n)^{\\oplus r} =\n  \\Rep{r} \\otimes \\ensuremath{\\mathscr{O}}(n)$.}\n\\begin{equation}\n  \\label{eq:monad}\n  \\xymatrix{\n    0 \\ar[r] & \n    V \\ar[r] & \n    5 \\ensuremath{\\mathscr{O}}_Q(1) \\oplus 5 \\ensuremath{\\mathscr{O}}_Q(2) \\ar[r]^-f & \n    5 \\ensuremath{\\mathscr{O}}_Q(3) \\ar[r] & \n    0\n    .\n  }  \n\\end{equation}\nHowever, eq.~\\eqref{eq:monad} only defines a vector bundle on\n$Q\\subset \\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$. In order to divide out the group action and obtain a\nbundle on $X$, we need to first \\emph{specify} a group action on $V$.\n\nThe choice of a group action on the vector bundle is called an\nequivariant structure. This is a choice bundle map $\\gamma(g):V\\to V$\ncovering the group action $g:Q\\to Q$ on the base of the bundle for all\ngroup elements $g\\in G$. In other words, we need to pick a linear map\n$\\gamma_p(g):V_p\\to V_{gp}$ holomorphically varying over each point\n$p\\in Q$. Similarly, an equivariant map $f:(V,\\gamma^V)\\to\n(W,\\gamma^W)$ between equivariant bundles is a map of the vector\nbundles that intertwines the equivariant structure in the obvious way,\n\\begin{equation}\n  \\label{eq:intertwine}\n  \\vcenter{\\xymatrix{\n      V_p \\ar[r]^-{f_p(g)} \\ar[d]_-{\\gamma^V_p(g)} & \n      W_{p} \\ar[d]^-{\\gamma^W_p(g)} \\\\\n      V_{gp} \\ar[r]^-{f_{gp}(g)} & \n      W_{gp}\n    }}\n  \\quad\\Leftrightarrow\\quad\n  \\gamma^W_p(g) \\circ f_p(g) = f_{gp}(g) \\circ \\gamma^V_p(g)\n  .\n\\end{equation}\nDirect sums and tensor products of equivariant bundles are equivariant\nin the obvious way, turning the category $\\Vect_G$ of $G$-equivariant\nvector bundles into a ring. Finally, a monad is equivariant if the\nobjects in the complex and the maps are equivariant; The cohomology of\nan equivariant monad is an equivariant vector bundle.\n\nFor example, consider the trivial line bundle $\\ensuremath{\\mathscr{O}}$ on a compact\ncomplex manifold. Up to an overall factor, there is a unique section\n$s$ which is nowhere vanishing. Hence, the choice of a $G$-equivariant\nstructure on $\\ensuremath{\\mathscr{O}}$ is equivalent to the choice of a\n$G$-representation on $\\Gamma(\\ensuremath{\\mathscr{O}})=\\ensuremath{{\\mathbb{C}}} s$, that is, a\nmultiplicative character of $G$. By abuse of\nnotation~\\cite{Braun:2005zv}, we denote by $\\chi \\in\n\\Hom(G,\\ensuremath{{\\mathbb{C}}}^\\times)$ also the corresponding line bundle. Moreover, we\nwrite $\\chi V$ for the tensor product of the line bundle $\\chi$ and\nthe vector bundle $V$.\n\nSince we are particularly interested in slope-stable bundles, let us\nnote that the equivariant structure on such a bundle is unique up to\nmultiplication with a multiplicative character. The proof is as\nfollows, assume that you have two different equivariant structures\n$V_1=(V,\\gamma_{(1)})$ and $V_2=(V,\\gamma_{(2)})$. Then\n$\\gamma_{(2)}^{-1}\\circ \\gamma_{(1)}$ is a nontrivial automorphism of\n$V$. But the only automorphisms of stable bundles are multiplication\nby a constant~\\cite{MR1600388}. The constant can depend on $g\\in G$,\nbut must be a $1$-dimensional representation $\\chi$. Therefore, $V_1 =\n\\chi V_2$.\n\n\n\n\n\n\\subsection{Schur Covers}\n\\label{sec:Schur}\n\nNot every vector bundle can carry an equivariant structure. The\nsimplest example would be the line bundle $\\ensuremath{\\mathscr{O}}_Q(1)$. Let us look\nat the problem in some detail. First, let us identify the sections\nwith the homogeneous coordinates on $\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$ in the usual way,\n\\begin{equation}\n  \\Gamma \\ensuremath{\\mathscr{O}}(1) = \n  \\langle z_0, z_1, z_2, z_3, z_4 \\rangle\n  .\n\\end{equation}\nThe naive guess for an equivariant structure would be the tautological\naction\n\\begin{equation}\n  \\label{eq:tautological}\n  \\gamma(g_1)(z) = g_1(z)\n  ,\\quad\n  \\gamma(g_2)(z) = g_2(z)  \n  .\n\\end{equation}\nThis fails to be a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure because the two\nactions do not commute,\n\\begin{equation}\n  \\gamma(g_2) \\gamma(g_1) \\gamma(g_2)^{-1} \\gamma(g_1)^{-1} \n  = \\zeta \\not= 1\n  ,\n\\end{equation}\nsee eq.~\\eqref{eq:g1g2commutator}. We can define our way out of this\nproblem by introducing another generator $g_3$ such that\n$\\gamma(g_3)(z)=\\zeta z$ and $g_2 g_1 g_2^{-1} g_1^{-1} = g_3$. This\nenlarged group is the Heisenberg group\n\\begin{equation}\n  H_5 = \\langle g_1, g_2, g_3 \\rangle = \\big( \\ensuremath{{\\Z_5\\times\\Z_5}} \\big) \\rtimes \\mathbb{Z}_5\n  ,\n\\end{equation}\nand we just defined a $H_5$-equivariant structure on\n$\\ensuremath{\\mathscr{O}}_Q(1)$. We note that the Heisenberg group is a Schur cover of\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$,\n\\begin{equation}\n  \\label{eq:SchurCover}\n  \\xymatrix{\n    0 \\ar[r] &\n    \\mathbb{Z}_5 \\ar[r] &\n    H_5 \\ar[r] &\n    \\ensuremath{{\\Z_5\\times\\Z_5}} \\ar[r] &\n    0\n  }\n  ,\n\\end{equation}\nand that this construction can be generalized: Given a projective\naction on the homogeneous coordinates of a projective space,\n$\\ensuremath{\\mathscr{O}}(1)$ is equivariant with respect to a Schur cover.\n\nBut we wanted a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure, and not a\n$H_5$-equivariant structure! Note that a $H_5$-equivariant structure\n$\\gamma$ is, in fact, a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure if and only if\nthe kernel $\\langle g_3 \\rangle = \\mathbb{Z}_5$ of the cover is represented\ntrivially. Therefore, we arrive at the following characterization of\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant bundles: \n\\begin{itemize}\n\\item Every vector bundle $(V,\\gamma)$ is $H_5$-equivariant.\n\\item The vector bundle $(V,\\gamma)$ is $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant if and only\n  if\\footnote{Note that $g_3$ acts trivially on the base space, so its\n    action on the bundle is just a linear map of the fiber to itself.}\n  $\\gamma(g_3)=\\Id$.\n\\end{itemize}\nThis shows that eq.~\\eqref{eq:tautological} does not define an\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure on $\\ensuremath{\\mathscr{O}}_Q(1)$. Could there be another\n$H_5$-equivariant structure that \\emph{does} descend to a\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure? To rule out this possibility first note\nthat, like all line bundles, $\\ensuremath{\\mathscr{O}}_Q(1)$ is a slope-stable vector\nbundle. By the argument in \\autoref{sec:symmetry}, any other\nequivariant structure must differ from~\\eqref{eq:tautological} by a\nmultiplicative character of $H_5$. As we will see in more detail in\n\\autoref{sec:rep}, the multiplicative characters of $H_5$ are just the\n$1$-d representations of $\\ensuremath{{\\Z_5\\times\\Z_5}}$. Therefore, any character $\\chi$ of\n$H_5$ satisfies $\\chi(g_3)=1$ and we clearly cannot use this freedom\nto turn eq.~\\eqref{eq:tautological} into a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant\nstructure. Hence, there cannot be any $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure on\n$\\ensuremath{\\mathscr{O}}_Q(1)$.\n\n\n\n\\subsection{Topological Restrictions}\n\\label{sec:topology}\n\nAlthough we have used the holomorphic structure in the above argument,\none can exclude the existence of a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure on\n$\\ensuremath{\\mathscr{O}}_Q(1)$ on grounds of topology alone. Recall that the\ntopological isomorphism class of a line bundle is classified by its\nfirst Chern class. In the case at hand, it is\n\\begin{equation}\n  c_1\\big( \\ensuremath{\\mathscr{O}}_Q(1) \\big) = 1 \\in \\mathbb{Z} = H^2\\big(Q,\\mathbb{Z}\\big)\n  .\n\\end{equation}\nThe cohomology of the cover $Q$ and the quotient $X=Q\/(\\ensuremath{{\\Z_5\\times\\Z_5}})$ is\nrelated via the Leray-Serre spectral sequence,\n\\begin{equation}\n  E_2^{p,q} = H^q\\Big(Q, H^p(\\ensuremath{{\\Z_5\\times\\Z_5}},Z) \\Big)\n  \\quad \\Rightarrow \\quad\n  H^{p+q}_{\\ensuremath{{\\Z_5\\times\\Z_5}}}(Q,\\mathbb{Z}) = H^{p+q}(X,\\mathbb{Z})\n  .\n\\end{equation}\nFor our purposes, the important part is the\nnonvanishing~\\cite{Aspinwall:1994uj, Braun:2007tp, Braun:2007xh,\n  Braun:2007vy} differential $d_3$ in the tableau\n\\begin{equation}\n  \\label{eq:GLSssE2}\n  E_2^{p,q} =\n  \\vcenter{\n    \\xymatrix@=2mm{\n      {\\scriptstyle \\vdots}\\hspace{1.6mm} &\n      \\vdots & \\vdots & \\vdots &\\vdots & \\vdots & \\vdots & \\mathinner{\\mkern2mu\\raise\\p@\\hbox{. \\\\\n      {\\scriptstyle q=2}\\hspace{1.6mm} &\n      \\mathbb{Z} \\ar[ddrrr]|(0.51){d_3} & 0 & \\mathbb{Z}_5^2 & \\mathbb{Z}_5 & \n      \\mathbb{Z}_5^3 & \\mathbb{Z}_5^2\n      & \\cdots \\\\\n      {\\scriptstyle q=1}\\hspace{1.6mm} &\n      0 & 0 & 0 & 0 & 0 & 0\n      & \\cdots \\\\\n      {\\scriptstyle q=0}\\hspace{1.6mm} &\n      \\mathbb{Z} & 0 & \\mathbb{Z}_5^2 & \\mathbb{Z}_5 & \n      \\mathbb{Z}_5^3 & \\mathbb{Z}_5^2\n      & \\cdots \\\\\n      \\ar[]+\/r 3.4mm\/+\/u 1.7mm\/;[rrrrrrr]+\/r 3mm\/+\/r 3.4mm\/+\/u 1.7mm\/\n      \\ar[]+\/r 3.4mm\/+\/u 1.7mm\/;[uuuu]+\/u  2mm\/+\/r 3.4mm\/+\/u 1.7mm\/\n      & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle p=0} & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle p=1} & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle p=2} & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle p=3} & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle p=4} & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle p=5} & \n      {\\vbox{\\vspace{3.5mm}}\\scriptstyle \\cdots }&\n    }}\n  .\n\\end{equation}\nTherefore, only the multiples of $5 \\in H^2(Q,\\mathbb{Z})$ survive to the\n$E_3$ tableau\\footnote{And, since the Leray-Serre spectral sequence is\n  a first quadrant spectral sequence, $E_3^{0,2}=E_\\infty^{0,2}$.}. In\nparticular, we obtain\n\\begin{equation}\n  H^2\\big(X,\\mathbb{Z})\n  = \n  \\ker(d_3) \\oplus \\mathbb{Z}_5 \\oplus \\mathbb{Z}_5\n  ~\\simeq~\n  \\mathbb{Z} \\oplus \\mathbb{Z}_5 \\oplus \\mathbb{Z}_5\n\\end{equation}\non the quotient\\footnote{That is, the line bundles on the quotient are\n  classified by the de Rham part of the first Chern class\n  $\\tfrac{1}{2\\pi}[F]$ together with two $\\mathbb{Z}_5$ phases for the\n  discrete Wilson lines.}. Another way of expressing this factor of\n$5$ is the following. Consider the quotient map $q:Q\\to X$, and pull\nback the cohomology groups. Then\n\\begin{equation}\n  \\label{eq:q2}\n  q^\\ast: \n  H^2\\big(X,\\mathbb{Z})\n  \\longrightarrow\n  H^2\\big(Q,\\mathbb{Z})\n  ,\\quad\n  (f,t_1,t_2)\\mapsto 5 f\n\\end{equation}\nis multiplication by $5$.\n\nNow we apply the usual tautology \n\\begin{displaymath}\n  \\xymatrix@C=3cm{\n    *++[F-:<3pt>]+{\\text{Equivariant bundles on $Q$}}\n    \\ar@\/^8mm\/[r]_-{\\text{quotient}}\n    &\n    *++[F-:<3pt>]+{\\text{Bundles on $X$}}\n    \\ar@\/^8mm\/[l]_-{\\text{pullback $q^*$}}\n  }\n\\end{displaymath}\nthat equivariant bundles on $Q$ are the same as bundles on the\nquotient $X$. The Chern classes are natural, that is, any bundle $W$\non $X$ satisfies $c_i \\big( q^*W \\big) = q^* \\big( c_i(W)\n\\big)$. Therefore, a necessary condition for a bundle $V$ on $Q$ to\nhave a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure is that its Chern classes are in the\nimage of $q$, that is,\n\\begin{equation}\n  V\\in \\Vect_\\ensuremath{{\\Z_5\\times\\Z_5}}(Q) \n  \\quad\\Rightarrow\\quad\n  c_i(V) \\in \\img(q^*)\n  .\n\\end{equation}\nIn particular, $c_1(V) \\in 5 \\mathbb{Z} \\subset H^2(Q,\\mathbb{Z})$ is a necessary\ncondition for a bundle on $Q$ to be $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant. This\nrestriction on the first Chern class already forbids equivariant\nstructures on most of the $B$, $C$ bundles in \\autoref{tab:monads}.\n\nIn fact, next to the monad bundle that I am investigating in this\npaper, only the monad with $B=3 \\ensuremath{\\mathscr{O}}(2) \\oplus 4 \\ensuremath{\\mathscr{O}}(1)$ and\n$C=\\ensuremath{\\mathscr{O}}(4)\\oplus 2\\ensuremath{\\mathscr{O}}(3)$ seems to be allowed. However, the\nlatter is excluded by its second Chern classes. For completeness, the\nexpressions for the integrally normalized Chern classes and the\npullback $q^*: H^\\text{ev}(X)\\to H^\\text{ev}(Q)$ are in\n\\autoref{sec:Chern}.\n\n\n\n\\subsection{Heterotic Anomaly Cancellation}\n\\label{sec:anomaly}\n\nOf course, the heterotic anomaly cancellation condition\n\\begin{equation}\n  c_2(X) - c_2(W) - c_2(W_\\text{hidden}) = PD(C) \n  \\quad \\in H^4\\big(X,\\mathbb{Z}\\big)\n\\end{equation}\nhas to be satisfied on the quotient manifold $X=Q\/(\\mathbb{Z}_5\\times\\mathbb{Z}_5)$\nand bundle $W=V\/\\gamma$, where $PD(C)$ is the Poincar\\'e-dual of the\ncurve wrapped by five-branes. \n\nA necessary but not sufficient criterion for the anomaly cancellation\nis that the image of both sides under the pull-back is the same,\n\\begin{equation}\n  c_2(Q) - c_2(V) - c_2(V_\\text{hidden}) = q^* PD(C)   \n  \\quad \\in H^4\\big(Q,\\mathbb{Z}\\big).\n\\end{equation}\nHowever, on the quintic we are in the favorable circumstance that \n\\begin{equation}\n  q^*: \n  \\underbrace{H^4(X,\\mathbb{Z})}_{\\simeq \\mathbb{Z}} \n  \\to\n  \\underbrace{H^4(Q,\\mathbb{Z})}_{\\simeq \\mathbb{Z}}\n  ,\\quad\n  n\\mapsto 25 n\n\\end{equation}\nis injective\\footnote{Modulo torsion (the finite part in $H^4$), the\n  pull-back is always injective. However, it is important to cancel it\n  in integral cohomology on the quotient manifold as there is a danger\n  of a discrete anomaly. Moreover, the existence of holomorphic curves\n  can depend on the torsion part of their homology\n  class~\\cite{Braun:2007tp, Braun:2007xh, Braun:2007vy}.}. Hence, the\nanomaly cancellation condition on the covering space and on the\nquotient are equivalent. In particular, the monad bundle\neq.~\\eqref{eq:monad} satisfies $c_2(V)=c_2(Q)$, and therefore cancels\nthe heterotic anomaly on the cover $Q$ as well as on the quotient $X$\nwithout hidden bundle or branes.\n\n\n\n\n\n\\subsection{Representation Theory}\n\\label{sec:rep}\n\nIn order to better understand the Schur cover\n\\begin{equation}\n  H_5 = \\big< g_1, g_2, g_3\n  \\big|\n  g_1^5=g_2^5=g_3^5=1\n  ,~\n  g_2g_1g_2^{-1}g_1^{-1}=g_3\n  ,~\n  g_1g_3=g_3g_1\n  ,~\n  g_2g_3=g_3g_2\n  \\big>\n\\end{equation}\nlet us quickly discuss its representation theory~\\cite{GAP4}. First of\nall, its Abelianisation is $\\ensuremath{{\\Z_5\\times\\Z_5}}$. Therefore, each $1$-dimensional\nrepresentation has to factor through $\\ensuremath{{\\Z_5\\times\\Z_5}}$, see\neq.~\\eqref{eq:SchurCover}, and $H_5$ has $25$ one-dimensional\nrepresentations $r_1^i r_2^j$, $0\\geq i,j<5$. In addition, there are\nfour\\footnote{In the following, we will use the notation where these\n  four representations are indexed by $\\mathbb{Z}\\text{ mod }5$, with $0$\n  being disallowed. In other words, $R_{i+5}= R_i$.} useful to\ntake irreducible representations $R_1$,~$\\dots$,~$R_4$ of dimension\n$5$, distinguished by the weight\n\\begin{equation}\n  R_i(g_3) = \\zeta^i \\diag(1,1,1,1,1)\n  .\n\\end{equation}\nFor example, we defined the representation carried by the $5$\nhomogeneous variables of $\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$ to be $R_1$, see\neq.~\\eqref{eq:gzact}. Together, these are all irreducible\nrepresentations:\n\\begin{equation}\n  \\sum_{i,j=0}^4 \\dim\\big(r_1^i r_2^j\\big)^2 +\n  \\sum_{i=1}^4 \\dim\\big(R_i\\big)^2 \n  = 125 = |H_5|\n  .\n\\end{equation}\nThe tensor products of the irreducible representations can be\nsummarized in the representation ring\n\\begin{multline}\n  R(H_5) = \\mathbb{Z}\\big[r_1,r_2,R_1,R_2,R_3,R_4\\big]\n  \\Big\/\n  \\Big< \n  r_i^5=1\n  ,\\quad\n  r_i R_j=R_j\n  ,~\n  \\\\\n  R_a R_{5-a} = \\sum r_1^ir_2^j\n  ,\\quad\n  R_a R_b = 5 R_{a+b}\\text{ if $a+b\\not=0$ mod $5$}\n  \\Big>\n  .\n\\end{multline}\nOne observes that\n\\begin{itemize}\n\\item Take one of the genuinely $H_5$ representations $R_i$.\n\\item Tensor with $R_{5-i}$.\n\\item The result is a $\\ensuremath{{\\Z_5\\times\\Z_5}}$ representation, that is, $R_i\n  R_{5-i}(g_3)=\\Id$.\n\\end{itemize}\nThis will be important in the following to construct\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant rank-$5$ vector bundles. \n\nFinally, since we will be interested in polynomials, we will need the\nsymmetric powers of $R_1$. They are\n\\begin{equation}\n  \\label{eq:SymR1}\n  \\Sym^k(R_1) = \n  \\begin{cases}\n    \\frac{1}{5}\\binom{4+k}{k} R_k\n    &\n    k\\not=0\\text{ mod }5\n    ,\n    \\\\\n    1 + \n    \\frac{1}{25}\\Big[\\binom{4+k}{k}-1\\Big]\n    \\sum_{i,j} r_1^i r_2^j\n    &\n    k=0\\text{ mod }5\n    .\n  \\end{cases}\n\\end{equation}\n\n\n\n\\subsection{Equivariance}\n\\label{sec:equivariant}\n\nEvery representation $\\rho \\in R(H_5)$ of dimension $r=\\dim(\\rho)$\ndefines an $H_5$-equivariant vector bundle on $Q$ by taking the\ntrivial vector bundle $r \\ensuremath{\\mathscr{O}}_Q$ with its global sections\n$\\vec{s}=(s_\\alpha)$, $\\alpha\\in\\{0,\\dots,r-1\\}$, and defining an\nequivariant structure\n\\begin{equation}\n  \\gamma(g)(\\vec{s}) = \\rho(g) \\vec{s}\n  .\n\\end{equation}\nBy abuse of notation, I will denote the corresponding vector bundle by\n$\\rho$ as well. In particular, taking $\\rho=R_{5-n}$ and tensoring\nwith a line bundle defines the rank $5$ vector bundles\n\\begin{equation}\n  \\label{eq:Phi}\n  \\Phi(n) = R_{5-n} \\otimes \\ensuremath{\\mathscr{O}}_Q(n)\n  ,\\qquad\n  n\\not= 0\\text{ mod }5\n  .\n\\end{equation}\nPreviously, in \\autoref{sec:Schur}, I defined a $H_5$-equivariant\nstructure on $\\ensuremath{\\mathscr{O}}_Q(1)$ and, hence, on\n$\\ensuremath{\\mathscr{O}}_Q(n)=\\ensuremath{\\mathscr{O}}_Q(1)^{\\otimes n}$. Therefore, $\\Phi(n)$ is\n$H_5$-equivariant as the tensor product of equivariant\nbundles. Moreover, $g_3$ acts trivially, and $\\Phi(n)$ is actually a\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant vector bundle. Topologically $\\Phi(n)=5\n\\ensuremath{\\mathscr{O}}_Q(n)$ is decomposable, but as an equivariant bundle it is not.\n\nLet us take a closer look at this definition. A basis for\n$H^0\\big(Q,\\Phi(n)\\big)$ is the same as for $5\\ensuremath{\\mathscr{O}}_Q(n)$, namely\nthe $5\\cdot \\binom{4+n}{n}$ sections\n\\begin{equation}\n  s_{\\alpha,(i_1,\\dots,i_n)}\n  =\n  \\begin{pmatrix}\n    0 \\\\ \\vdots \\\\ 0 \\\\ \n    z_{i_1} z_{i_2} \\cdots z_{i_n} \n    \\mathpalette\\mathrlapinternal{\\textcolor{blue}{\n        \\qquad\\longleftarrow\\text{row $\\alpha$}\n      }}\n    \\\\ 0 \\\\ \\vdots \\\\ \n    0\\mathpalette\\mathrlapinternal{\\hspace{13mm}, \\qquad  0\\leq \\alpha, i_1,\\dots, i_n<5.}\n  \\end{pmatrix}\n  \\phantom{, \\qquad  0\\leq \\alpha, i_1,\\dots, i_n<5.}\n\\end{equation}\nThe generators $g_1$, $g_2$ act in an obvious way on the homogeneous\ncoordinates $z_0$, $\\dots$, $z_4$, this is the $H_5$ action on\n$\\ensuremath{\\mathscr{O}}_Q(1)$. The action on the sections of $\\Phi(n)$ is slightly\ndifferent, and a convenient basis choice is\n\\begin{equation} \n  \\label{eq:g1g2s}\n  \\begin{split}\n    g_1\\big( s_{\\alpha,(i_1,~\\dots,~i_n)} \\big)\n    =&\\;\n    s_{\\alpha-n, (i_1+1,~\\dots,~ i_n+1)}\n    ,\n    \\\\\n    g_2\\big( s_{\\alpha,(i_1,~\\dots,~i_n)} \\big)\n    =&\\;\n    \\zeta^{\\alpha+i_1+\\cdots+i_n}\n    s_{\\alpha, (i_1,~\\dots,~ i_n)}    \n    .\n  \\end{split}\n\\end{equation}\nOne can easily check that\n\\begin{equation}\n  g_1\\circ g_2 \n  \\big( s_{\\alpha,(i_1,~\\dots,~i_n)} \\big) \n  = \n  \\zeta^{\\alpha+i_1+\\cdots+i_n}\n  s_{\\alpha-n, (i_1+1,~\\dots,~ i_n+1)}\n  =\n  g_2\\circ g_1\n  \\big( s_{\\alpha,(i_1,~\\dots,~i_n)} \\big)   \n  .\n\\end{equation}\nTherefore, eq.~\\eqref{eq:g1g2s} defines a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant structure\non $\\Phi(n)$. \n\nTo summarize, we have now defined equivariant structures on the\nentries of the monad under consideration. They are\n\\begin{equation}\n  B = \\Phi(1) \\oplus \\Phi(2)\n  ,\\quad\n  C = \\Phi(3)\n  .\n\\end{equation}\nIf we can find a equivariant morphism from $B$ to $C$, then we have\ndefined an equivariant monad.\n\n\n\\subsection{Morphisms}\n\\label{sec:map}\n\nA morphism $B\\to C$ is simply given by a $\\rank(C)\\times \\rank(B)$\nmatrix of polynomials such that the $(j,i)$ entry is of degree\n$c_j-b_i$. Keeping track of the equivariant structure on the bundles\n$\\Phi(n)$ and assuming\\footnote{I will allways assume this in the\n  following. For the purposes of this paper, only the case where\n  $1\\geq n,m\\geq 3$ is relevant.} $m-n\\text{ mod }5\\not=0$, one finds\n\\begin{multline}\n  \\Hom\\big(\\Phi(n),\\Phi(m) \\big) =\n  \\Hom\\Big(R_{5-n}\\ensuremath{\\mathscr{O}}_Q(n),R_{5-m}\\ensuremath{\\mathscr{O}}_Q(m) \\Big) \n  \\\\\n  =\n  R_{5-n}^\\ensuremath{\\vee} R_{5-m} \\Sym^{m-n}(R_1) \n  = \n  \\binom{4+m-n}{m-n} \\sum_{i,j=0}^4 r_1^i r_2^j\n  .\n\\end{multline}\n\nLet me describe the invariant homomorphisms in more detail. First,\nnote the obvious basis\n\\begin{multline}\n  \\Hom\\big( \\Phi(n),\\Phi(n+k) \\big) =\n  \\big<\n  f^\\alpha_{\\beta,(i_1,\\dots,i_k)}\n  \\big>\n  ,\n  \\\\[3ex]\n  f^\\alpha_{\\beta,(i_1,\\dots,i_k)} =\n  (\\delta^\\alpha_\\beta) \\;\n  z_{i_1}\\cdots z_{i_k}\n  =\n  \\begin{pmatrix}\n    0      & \\cdots & \n    \\mathpalette\\mathclapinternal{\\textcolor{blue}{\\smash{\n          \\begin{array}[b]{ccc}\n            \\text{column $\\alpha$}\\\\\n            \\downarrow\\\\\n            \\\\\n          \\end{array}\n        }}}\n    \\mathpalette\\mathclapinternal{0}                  & \\cdots & 0 \\\\\n    \\vdots &   &                    &        & \\vdots \\\\\n    0      &        & z_{i_1}\\cdots z_{i_k} &        & 0\n    \\mathpalette\\mathrlapinternal{\n      \\textcolor{blue}{\n        \\quad \\longleftarrow \\text{ row $\\beta$}\n      }\n      ,\n    } \\\\\n    \\vdots &        &                    &  & \\vdots \\\\\n    0      & \\cdots & 0                  & \\cdots & 0 \n  \\end{pmatrix}\n  \\phantom{\\quad \\longleftarrow \\text{ row $\\alpha$}}\n\\end{multline}\nwhere $(\\delta^\\alpha_\\beta)$ is the $5\\times 5$-matrix with a single\nnon-zero entry $=1$ and the $z_i$ are again the homogeneous\ncoordinates of $\\ensuremath{\\mathop{\\null {\\mathbb{P}}}}\\nolimits^4$. The action on a section of $\\Phi(n)$ is simply\nmatrix-vector multiplication, that is,\n\\begin{equation}\n  f^\\alpha_{\\beta,(i_{n+1},\\dots,i_{n+k})} \n  \\big( s_{\\epsilon,(i_1,\\dots,i_n)} \\big)\n  = \n  \\delta^\\alpha_\\epsilon \n  s_{\\beta,(i_1,\\dots,i_{n+k})}\n  .\n\\end{equation}\nFinally, the $g_1$, $g_2$ action on the space of maps is the usual\naction on the homogeneous coordinates combined with a matrix action to\ncorrectly intertwine between $\\Phi(n)$ and $\\Phi(n+k)$, see\neq.~\\eqref{eq:intertwine}. Explicitly, the action is\n\\begin{equation}\n  \\begin{split}\n    g_1\\big( f^\\alpha_{\\beta,(i_1,\\dots,i_k)}  \\big)\n    =&\\; \n    f^{\\alpha-n}_{\\beta-n-k,(i_1+1,\\dots,i_k+1)}  \n    ,\\\\\n    g_2\\big( f^\\alpha_{\\beta,(i_1,\\dots,i_k)}  \\big)\n    =&\\; \n    \\zeta^{\\beta-\\alpha+i_1+\\cdots+i_k}\n    f^\\alpha_{\\beta,(i_1,\\dots,i_k)}  \n    .\n  \\end{split}\n\\end{equation}\nThe $g_1$ and $g_2$ actions commute,\n\\begin{equation}\n  g_2 g_1\\big( f^\\alpha_{\\beta,(i_1,\\dots,i_k)}  \\big)\n  =\n  \\zeta^{\\beta-\\alpha+i_1+\\cdots+i_k}\n  f^{\\alpha-n}_{\\beta-n-k,(i_1+1,\\dots,i_k+1)}  \n  =\n  g_1 g_2 \\big( f^\\alpha_{\\beta,(i_1,\\dots,i_k)}  \\big)\n  ,\n\\end{equation}\nand therefore decompose into $\\ensuremath{{\\Z_5\\times\\Z_5}}$-representations as we argued\nabove. For example, the $5$-dimensional space of invariant\nhomomorphisms from $\\Phi(2)$ to $\\Phi(3)$ is spanned by\n\\begin{equation}\n  \\left(\\begin{smallmatrix}\n      z_0& 0& 0& 0& 0 \\\\ \n      0& 0& 0& 0& z_3 \\\\ \n      0& 0& 0& z_1& 0 \\\\ \n      0& 0& z_4& 0& 0 \\\\ \n      0& z_2& 0& 0& 0\n    \\end{smallmatrix}\\right)\n  ,\n  \\left(\\begin{smallmatrix}\n      0& z_1& 0& 0& 0 \\\\ \n      z_4& 0& 0& 0& 0 \\\\ \n      0& 0& 0& 0& z_2 \\\\ \n      0& 0& 0& z_0& 0 \\\\ \n      0& 0& z_3& 0& 0\n    \\end{smallmatrix}\\right)\n  ,\n  \\left(\\begin{smallmatrix}\n      0& 0& z_2& 0& 0 \\\\ \n      0& z_0& 0& 0& 0 \\\\ \n      z_3& 0& 0& 0& 0 \\\\ \n      0& 0& 0& 0& z_1 \\\\ \n      0& 0& 0& z_4& 0\n    \\end{smallmatrix}\\right)\n  ,\n  \\left(\\begin{smallmatrix}\n      0& 0& 0& z_3& 0 \\\\ \n      0& 0& z_1& 0& 0 \\\\ \n      0& z_4& 0& 0& 0 \\\\\n      z_2& 0& 0& 0& 0 \\\\ \n      0& 0& 0& 0& z_0\n    \\end{smallmatrix}\\right)\n  ,\n  \\left(\\begin{smallmatrix}\n      0& 0& 0& 0& z_4 \\\\\n      0& 0& 0& z_2& 0 \\\\\n      0& 0& z_0& 0& 0 \\\\ \n      0& z_3& 0& 0& 0 \\\\ \n      z_1& 0& 0& 0& 0  \n    \\end{smallmatrix}\\right)\n  .\n\\end{equation}\nWe now have all ingredients to define a $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant monad\nbundle on the quintic. For explicitness, I will from now on take the\nquintic to be the Fermat quintic\n\\begin{equation}\n  Q = z_0^5+z_1^5+z_2^5+z_3^5+z_4^5\n  .\n\\end{equation}\nLet the $\\ensuremath{{\\Z_5\\times\\Z_5}}$-equivariant rank-$5$ vector bundle be the kernel of the\npositive monad\n\\begin{equation}\n  \\label{eq:ZZZmonad}\n  \\xymatrix{\n    0 \\ar[r] & \n    V \\ar[r] & \n    \\Phi(1) \\oplus \\Phi(2) \\ar[r]^-f & \n    \\Phi(3) \\ar[r] & \n    0\n  }  \n\\end{equation}\nwith an invariant map given by the polynomial matrix\n\\begin{equation}\n  \\label{eq:fdef}\n  f =\n  \\begin{pmatrix}\n    0 &  z_3^2 & 0 &  0 &  z_2^2 & 0 & z_1 & 0 & z_3 & z_4 \\\\\n    0 &  z_0^2 & 0 &  z_1^2 & 0 &  z_4 & 0 & z_1 & z_2 & 0 \\\\\n    z_4^2 & 0 &  0 &  z_3^2 & 0 &  0 & z_4 & z_0 & 0 & z_2 \\\\\n    z_1^2 & 0 &  z_2^2 & 0 &  0 &  z_2 & z_3 & 0 & z_0 & 0 \\\\\n    0 &  0 &  z_4^2 & 0 &  z_0^2 & z_1 & 0 & z_3 & 0 & z_0 \n   \n   \n   \n   \n   \n  \\end{pmatrix}\n  .\n\\end{equation}\nAs we will see soon, the map $f$ has been chosen generic enough so\nthat the monad is, indeed, a vector bundle. Yet it is special enough\nso that one $\\Rep{5}$--$\\barRep{5}$ pair survives\\footnote{The\n  observation of~\\cite{Anderson:2007nc, Anderson:2008uw,\n    Anderson:2008ex} that a completely generic map leads to no\n  vector-like pairs whatsoever remains true.}.\n\n\n\n\n\n\\section{Particle Spectrum}\n\\label{sec:spectrum}\n\n\\subsection{Machinery}\n\\label{sec:SS}\n\nMassless fields in $10$ dimensions give rise to massless fields in the\n$4$-dimensional effective action if their dependence on the Calabi-Yau\ncoordinates is a zero mode of the Dirac operator. This boils down to a\nquestion about harmonic forms on the vector bundle $V$ and its\nexterior powers $\\wedge^i V$, $1\\geq i\\geq 4$. By changing our model\nfor the cohomology to sheaf cohomology, this becomes a tractable\ncomputation. In particular, we can use the defining monad to convert\n$\\wedge^i V$ into something that depends only on $B$ and $C$ and the\nmap in-between. But since $B$ and $C$ are only sums of line bundles,\nall cohomology groups are now given explicitly as vector spaces\nspanned by polynomials.\n\nExplicitly, we first replace $\\wedge^i V$ by equivalent objects in the\nderived category,\\footnote{The underlined entry marks the zero\n  position of the complex.}\n\\begin{equation}\n  \\label{eq:DerV}\n  \\begin{split}\n    V =&\\;\n    \\left[\\strut\n      0 \\longrightarrow\n      \\underline{B} \\stackrel{f}{\\longrightarrow} C\n      \\longrightarrow\n      0\n    \\right]\n    ,\n    \\\\\n    \\wedge^2 V =&\\;\n    \\left[\\strut\n      0 \\longrightarrow\n      \\underline{\\wedge^2B} \n      \\xrightarrow{b_1\\wedge b_2 \\mapsto [b_1 \\otimes f(b_2)]}\n      B\\otimes C  \\\n      \\xrightarrow{(b,c)\\mapsto \\{f(b)\\otimes c\\} }\n      \\Sym^2 C\n      \\longrightarrow\n      0\n    \\right]\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    .\n  \\end{split}\n\\end{equation}\nThe remaining exterior powers are just duals, namely\n\\begin{equation}\n \n \n  \\wedge^3 V = \\wedge^2V^\\ensuremath{\\vee}\n  ,\\quad\n  \\wedge^4 V = V^\\ensuremath{\\vee}\n  ,\\quad\n  \\wedge^5 V = \\det V = \\ensuremath{\\mathscr{O}}_Q\n  ,\n\\end{equation}\nand their cohomology can be determined via Serre duality. As a\nnecessary evil we have to deal with cohomology for complexes, the\nso-called hypercohomology. Note that there are two ways to ``take\ncohomology'' here: There is the cohomology of a complex\n$\\ensuremath{\\mathscr{K}}^\\bullet$, and the cohomology of the objects in the\ncomplex. Here, we only need the case where the cohomology of the\ncomplex is a vector bundle (or sheaf) $V$ located at a single\nposition,\n\\begin{equation}\n  V = \n  \\ker(\\ensuremath{\\mathscr{K}}^0\\to \\ensuremath{\\mathscr{K}}^1) \\Big\/\n  \\img(\\ensuremath{\\mathscr{K}}^{-1})\n  ,\\qquad\n  \\ker(\\ensuremath{\\mathscr{K}}^p\\to \\ensuremath{\\mathscr{K}}^{p+1}) = \\img(\\ensuremath{\\mathscr{K}}^{p-1})\n  \\text{ if }p\\not=0\n  .\n\\end{equation}\nIn this case, the hypercohomology $H(\\ensuremath{\\mathscr{K}}^\\bullet)$ is simply the\ncohomology of $V$. The other way of taking cohomologies gives rise to\nthe hypercohomology spectral sequence\n\\begin{equation}\n  E_1^{p,q} = H^q \\big( Q, \\ensuremath{\\mathscr{K}}^p \\big)\n  \\quad \\Rightarrow \\quad \n  H^{p+q}(V)\n\\end{equation}\nThanks to Kodaira vanishing, only the $q=0$ row will be\nnonzero. Moreover, the first and only non-vanishing differential\n$d_1:E_1^{p,0}\\to E_1^{p+1,0}$ is just multiplication by $H(f)$\ninduced from $f:B\\to C$ with suitable (anti-)symmetrization.\n\nTo summarize, using the defining map $f$, see eq.~\\eqref{eq:fdef},\ndefines maps between spaces of polynomials\n\\begin{equation}\n  \\begin{gathered}\n    H^0\\big( Q, B \\big)\n    \\xrightarrow{H(f)}\n    H^0\\big( Q, C \\big)\n    ,\n    \\\\[1ex]\n    H^0\\big( Q, \\wedge^2B \\big)\n    \\stackrel{F_B}{\\longrightarrow}\n    H^0\\big( Q, B\\otimes C \\big)\n    \\stackrel{F_C}{\\longrightarrow}\n    H^0\\big( Q, \\Sym^2 C \\big)\n    .\n  \\end{gathered}\n\\end{equation}\nHere, $H(f)$ is tautologically the same matrix as in\neq.~\\eqref{eq:fdef}. The polynomial matrices\\footnote{Of course they\n  satisfy $F_C F_B = 0$, as required for the $d_1$ differentials in\n  the Hypercohomology spectral sequence.} $F_B$ and $F_C$ are of\ndimension $50\\times 45$ and $15\\times 50$, respectively, and are\nexplicitly constructed by the script in \\autoref{sec:singular}. Taking\nthe cohomology, one finds\n\\begin{equation}\n  H^i\\big(Q,V\\big) = \n  \\begin{cases}\n    0   \\\\\n    0   \\\\\n    \\coker H(f) \\\\\n    \\ker H(f)  \n  \\end{cases}\n  , \\quad\n  H^i\\big(Q,\\wedge^2 V\\big)  =\n  \\begin{cases}\n    0  & i=3 \\\\\n    \\coker F_C  & i=2 \\\\ \n    \\ker F_C \\big\/ \\img F_B & i=1 \\\\\n    \\ker F_B & i=0 \\\\ \n  \\end{cases}\n  .\n\\end{equation}\nFinally, using Serre duality,\n\\begin{equation}\n  H^i\\big(Q,\\wedge^k V\\big) = H^{3-i}\\big(Q,\\wedge^{5-k} V\\big)\n  ,\n\\end{equation}\nwe determined all cohomology groups of exterior powers of $V$. The\nrelevant computations with polynomials can be easily done with\n\\textsc{Singular}\\xspace~\\cite{GPS05} and are recorded in\n\\autoref{sec:singular}.\n\n\n\n\n\\subsection{Slope Stability}\n\\label{sec:stab}\n\nSince the quintic has only a one-dimensional $H^2(Q,\\mathbb{Z})$ we are in the\nlucky case where we can apply Hoppe's\ncriterion~\\cite{MR757476}. Specifically, we have\n\\begin{itemize}\n\\item The Fermat quintic $Q$ is a smooth manifold with\n  $\\dim\\,H^2(Q,\\mathbb{Z})=1$.\n\\item The monad defines a \\emph{vector bundle} $V$; This requires a\n  short computation that $Q$ and the $5\\times 5$ minors of $f$ do not\n  vanish simultaneously.\n\\item The first Chern class of $V$ vanishes by construction.\n\\item Finally, $H^0(Q,\\wedge^iV)$ must vanish for $1\\geq i\\geq 4$. The\n  potential contributions are $\\ker H(f)$ and $\\ker F_B$, and a short\n  computation shows that they indeed vanish, see again\n  \\autoref{sec:singular}. This can also be argued more generally using\n  the Koszul resolution~\\cite{Anderson:2008uw, Anderson:2008ex}.\n\\end{itemize}\nHoppe's criterion then guarantees that the bundle $V$ is slope-stable\nand, therefore, admits a Hermitian Yang-Mills connection.\n\n\n\\subsection{Light Matter}\n\\label{sec:matter}\n\nUsing the well-known embedding of $SU(5)\\times SU(5) \\subset E_8$, the\nmatter spectrum of the $E_8\\times E_8$ heterotic string compactified\non a slope-stable vector bundle over a Calabi-Yau manifold is\ndetermined by the multiplicities\\footnote{The last equation follows\n  from $\\Ind(V)=\\Ind(\\wedge^2V)$ for a rank-$5$ bundle or anomaly\n  cancellation in the low-energy action.}\n\\begin{equation}\n  \\begin{gathered}\n    n_\\Rep{10} = h^1 (V)\n    ,\\quad\n    n_\\barRep{10} = h^1 (V^\\ensuremath{\\vee}) = h^2(V)\n    ,\\quad\n    \\\\[1ex]\n    n_\\Rep{5} = h^1 (\\wedge^2 V^\\ensuremath{\\vee}) = h^2 (\\wedge^2 V)\n    ,\\quad\n    n_\\barRep{5} = h^1 (\\wedge^2 V ) = n_\\Rep{10} - n_\\barRep{10} + n_\\Rep{5}\n    .\n  \\end{gathered}\n\\end{equation}\nWe already noted that $h^0(V)$ had to vanish for\nslope-stability. Therefore, index theory determines the only\nnon-vanishing cohomology group $h^1(V)=75$. Moreover, the\n$\\ensuremath{{\\Z_5\\times\\Z_5}}$-action is uniquely determined by the corresponding\ncharacter-valued index, and one obtains\n\\begin{equation}\n  H^1\\big( Q, V \\big) = 3 \\sum_{i,j=0}^4 r_1^i r_2^j\n  .\n\\end{equation}\nNext, consider $H^2(Q,\\wedge^2V) = \\coker F_C$. We can think of it as\nsymmetric tensors in $H^0(Q,C\\otimes C)$ with the basis\n\\begin{equation}\n  \\Big\\{\n  \\ensuremath{{\\mathbb{C}}}\\big[z_0,z_1,z_2,z_3,z_4\\big]_6 \\Big\/ \n  \\langle Q=0 \\rangle\n  \\Big\\}\n  \\otimes \n  \\big\\{ \\vec{e}_{(\\alpha,\\beta)} \\big|~ 0\\geq \\alpha\\geq \\beta \\geq 4 \\big\\}\n\\end{equation}\nThe $\\ensuremath{{\\Z_5\\times\\Z_5}}$-group action can easily be identified as the one coming\nfrom\n\\begin{equation}\n  \\Gamma \\Phi(3) \\times \\Gamma \\Phi(3)\n  \\longrightarrow\n  H^0\\big( Q, \\Sym^2 C \\big)\n  ,\\quad\n  \\big( s_{\\alpha,(i_1,i_2,i_3)}, s_{\\beta,(i_4,i_5,i_6)} \\big)\n  \\mapsto\n  e_{(\\alpha,\\beta)} \\prod_{k=1}^6 z_{i_k}\n  .\n\\end{equation}\nWhen asked nicely, \\textsc{Singular}\\xspace can compute a basis for the\ncokernel, see \\autoref{sec:singular}. One obtains\n\\begin{equation}\n  \\coker F_C = \n  \\big<\n  z_4^6       \\vec{e}_{(1,4)},~\n  z_0^3 z_2 z_4^2 \\vec{e}_{(0,0)},~\n  z_3^6       \\vec{e}_{(1,1)},~\n  z_1 z_3^5     \\vec{e}_{(2,2)},~\n  z_4^6       \\vec{e}_{(3,3)},~\n  z_2 z_4^5     \\vec{e}_{(4,4)}\n  \\big>.\n\\end{equation}\nIn general, the $\\ensuremath{{\\Z_5\\times\\Z_5}}$-action will map the representatives to\ndifferent representatives of the same quotient space. These must be\nprojected back onto the chosen representatives to read off the group\naction. Decomposing this $6$-dimensional $\\ensuremath{{\\Z_5\\times\\Z_5}}$-representation into\nirreducible representations yields\n\\begin{equation}\n  H^1\\big( Q, \\wedge^2 V \\big) = \n  H^2\\big( Q, \\wedge^2 V \\big) = \n  r_2^4 + 1+r_1+r_1^2+r_1^3+r_1^4\n  .\n\\end{equation}\n\nTo summarize, we defined an equivariant bundle $(V,\\gamma)$ on $Q$ and\ncomputed the $\\ensuremath{{\\Z_5\\times\\Z_5}}$-action on its cohomology. Now, finally, we extract\nthe invariants to obtain the cohomology of the quotient bundle\n$W = V\/\\gamma$ on the quotient space $X=Q\/(\\ensuremath{{\\Z_5\\times\\Z_5}})$,\n\\begin{equation}\n  H^i\\big(X,W\\big) = \n  \\begin{cases}\n    0   \\\\\n    0   \\\\\n    3   \\\\\n    0\n  \\end{cases}\n  , \\quad\n  H^i\\big(X,\\wedge^2 W\\big)  =\n  \\begin{cases}\n    0  & i=3 \\\\\n    1  & i=2 \\\\ \n    1+3  & i=1 \\\\\n    0  & i=0 \\\\ \n  \\end{cases}\n  .\n\\end{equation}\nTherefore, the ensuing $SU(5)$ GUT has a matter spectrum of three\n$\\Rep{10}+\\barRep{5}$ and a single vector-like pair\n$\\Rep{5}$--$\\barRep{5}$. Perturbing the bundle moduli $\\phi$ by\nchanging the map eq.~\\eqref{eq:fdef} removes this vector-like pair, so\nthere must be a $\\mu$-term $\\sim \\phi \\psi_\\Rep{5}\n\\psi_\\barRep{5}$ in the superpotential.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn  this paper we develop  the Kleinian construction of\nhyperelliptic Abelian functions,  which is a natural\ngeneralization of the Weierstrass approach in the elliptic\nfunctions theory to the case of a hyperelliptic curve of genus\n$g>1$.  Kleinian $ \\zeta$ and $ \\wp$--functions are defined as\n \\[\n  \\zeta_{i} ( \\boldsymbol{ u}) = \\frac{ \\partial}{ \\partial\nu_i} \\mathrm{ln} \\; \\sigma ( \\boldsymbol{ u}) ,   \\quad\n  \\wp_{ij} ( \\boldsymbol{ u}) =- \\frac{ \\partial^2}{ \\partial\nu_i \\partial u_j}  \\mathrm{ln} \\; \\sigma ( \\boldsymbol{ u}) ,   \\qquad\ni, j=1,  \\ldots, g,   \\]\nwhere the vector $ \\boldsymbol{ u}$ belongs to Jacobian\n$ \\mathrm{Jac} (V) $ of the\nhyperelliptic curve\n$V= \\{ (y, x)  \\in  \\mathbb{ C}^2:y^2- \\sum_{i=0}^{2g+2}\n\\lambda_{i}x^i=0 \\}$\nand the $ \\sigma ( \\boldsymbol{ u}) $ is the Kleinian $ \\sigma$--function.\n\nThe systematical study of the $ \\sigma$--functions,\nwhich may be related\nto the paper of Klein  \\cite{kl88},  was an alternative to\nthe developments of\nWeierstrass  \\cite{w54a, w54}  (the hyperelliptic generalization of\nthe Jacobi elliptic functions\n$ \\mathrm{sn},  \\mathrm{cn},  \\mathrm{dn}$)  and the purely\n$ \\theta$--functional theory G{\\\"o}ppel \\cite{go47} and Rosenhain\n \\cite{ro51} for genus $2$,  generalized further by Riemann.  The\n$ \\sigma$ approach was contributed by Burkhardt  \\cite{bur88},\nWiltheiss  \\cite{wi88},  Bolza  \\cite{bo95},  Baker  \\cite{ba98} and\nothers; the detailed bibliography may be found in  \\cite{kw15}.  We\nwould like to cite separately H.F. Baker's monographs\n \\cite{ba97, ba07},  worth special attention.\n\nThe paper is organized as follows. We recall the basic\nfacts about hyperelliptic curves in the\nSection~ \\ref{prelims}. In the Section~ \\ref{sigma-defs} we\nconstruct  the explicit expression for the fundamental\n$2$--differential of the second kind and derive the solution of\nthe Jacobi inversion problem in terms of the hyperelliptic\n$ \\wp$--functions.  We\ngive in the Section~3 the proof and the analysis of basic\nrelations for $ \\wp$--functions.  It is given  an explicit\ndescription of the $ \\mathrm{Jac} (V) $ in $ \\mathbb{\nC}^{g+ \\frac{g (g+1) }{2}}$ as the intersection of cubics. We\nintroduce coordinates $h_{ij}$  (see below  \\eqref{variables}) ,  in terms\nof which these cubics are the determinants of\n$3 \\times3$--matrices,  inheriting in such a way the structure of\nWeierstrass  elliptic  cubic. The Kummer variety\n$ \\mathrm{Kum} (V) = \\mathrm{Jac} (V) \/ \\pm$ appears to be the\nintersection of  quartics in\n$ \\mathbb{ C}^{ \\frac{g (g+1) }{2}}$ and is described in a whole by\nthe condition  $  \\mathrm{rank} \\left ( \\{h_{ij}\n \\}_{i, j=1,  \\ldots, g+2} \\right) <4$.\nThe Section~4 describes some natural applications of the Kleinian\nfunctions theory.\n\nThe paper is based on the recent results  partially announced\nin  \\cite{le95, bel96, el95, be96b}.  The given\nresults are already used to describe a $2$--dimensional\nSchr{\\\"o}dinger equation  \\cite{be95}.\n\n\n\\section{Preliminaries} \\label{prelims}\n\nWe recall some basic definition from the theory of the\nhyperelliptic curves and $ \\theta$--functions; see e.g.\n \\cite{ba97, ba07, fa73, mu75, gh78, fk80} for the detailed exposition.\n \\subsection{Hyperelliptic curves}\nThe set of points $V (y, x) $ satisfying the\n \\begin{eqnarray}\ny^2= \\sum_{i=0}^{2g+2} \\lambda_{i}x^{i}\n= \\lambda_{2g+2} \\prod_{k=1}^{2g+2} (x-e_{k}) =f (x)\n \\label{curve}\n \\end{eqnarray}\nis a model of a plane { \\em hyperelliptic curve} of genus $g$,\nrealized as a $2$--sheeted\ncovering over Riemann sphere with the { \\em branching points}\n$e_1,  \\ldots, e_{2g+2}$. Any pair  $ (y, x) $ in $V (y, x) $\nis called an { \\em analytic point}; an analytic point,  which is not\na  branching point is called a { \\em regular point}.\n The { \\em hyperelliptic involution} $ \\phi ( \\;) $  (the swap of the\nsheets of covering)  acts as $ (y, x)   \\mapsto  (-y, x) $,  leaving the\nbranching points fixed.\n\nTo make  $y$ the singlevalued function of $x$ it suffices to\ndraw $g+1$ cuts,  connecting pairs of branching points\n$e_i$---$e_{i'}$ for some partition of $ \\{1,  \\ldots,  2g+2 \\}$ into\nthe set of $g+1$ disjoint pairs ${i,  i'}$.  Those of $e_j$,  at\nwhich the cuts start we will denote ${a_i}$,  ending points of the\ncuts we will denote $b_i$,  respectively; except for one of the\ncuts which is denoted by starting point $a$ and ending point $b$.\nIn the case $ \\lambda_{2g+2}  \\mapsto 0$ this point $a  \\mapsto  \\infty$. The\nequation of the curve,  in case $ \\lambda_{2g+2}=0$ and\n$ \\lambda_{2g+1}=4$  can be rewritten as  \\begin{eqnarray} &y^2=4\nP (x) Q (x) ,  \\label{alt-curve}  \\\\ &P (x) = \\prod_{i=1}^{g} (x-a_i) ,\n \\quad Q (x) = (x-b)  \\prod_{i=1}^{g} (x-b_i) . \\nonumber\n \\end{eqnarray}\n\nThe local parametrisation of the point $ (y, x) $ in the  vicinity of\na point $ (w, z) $:   \\[ x=z+ \\left \\{ \\begin{array}{ll}  \\xi,  \\quad\n& \\text{near regular point} \\,  ( \\pm w, z) ;  \\\\  \\xi^2, & \\text{near\nbranching point} \\,  (0, e_i) ;  \\\\  \\frac{1}{ \\xi}, & \\text{near regular\npoint} \\,   ( \\pm \\infty,  \\infty) ; \\\\  \\frac{1}{ \\xi^2}, & \\text{near\nbranching point} \\,   ( \\infty,  \\infty)    \\end{array} \\right.   \\]\nprovides the structure of the { \\em hyperelliptic Riemann surface}\n--- a one-dimensional compact complex manifold. We will employ the\nsame notation for the plane curve and the Riemann surface ---\n$V (y, x) $ or $V$. All curves and Riemann surfaces through the paper\nare assumed to be hyperelliptic,  if the converse not stated.\n\nA { \\em marking} on $V (y, x) $ is given by the base\npoint $x_0$ and the canonical basis of cycles\n$ (A_1,  \\ldots, A_g;B_1,  \\ldots, B_g) $ --- the basis in the group of\none-dimensional homologies $H_1 (V (y, x) ,  \\mathbb{ Z}) $ on the\nsurface $V (y, x) $  with the symplectic intersection matrix\n$I= \\left ( \\begin{array}{cc}0&- \\mathbf{ 1}_g \\\\ \\mathbf{\n1}_g&0 \\end{array} \\right) $,  where $ \\mathbf{ 1}_g$ is the unit\n$g \\times g$--matrix.\n\n \\subsection{Differentials}\nTraditionally three kinds of differential $1$--forms are\ndistingui \\-s \\-h \\-ed on a Riemann surface.\n\n \\subsubsection{Holomorphic differentials}\nor the differentials of the first kind,\nare the differential $1$--forms\n$ \\mathrm{d}u$,  which can be locally given as\n$ \\mathrm{d}u= ( \\sum_{i=0}^ \\infty \\alpha_i \\xi^i) d \\xi$ in the vicinity\nof any point $ (y, x) $ with some constants $ \\alpha_i \\in \\mathbb{ C}$.\nIt can be checked directly,  that forms satisfying such a condition\nare all of the form $ \\sum_{i=0}^{g-1}  \\beta_i\nx^i \\frac{ \\mathrm{d}x}{y}$.  Forms $ \\{ \\mathrm{d}u_i \\}_{i=1}^g$,\n \\[ \\mathrm{d}u_i= \\frac{x^{i-1}  \\mathrm{d}x}{ y},  \\quad\ni \\in 1,  \\ldots, g  \\] are the set of { \\em canonical\nholomorphic differentials} in $H^1 (V,  \\mathbb{ C}) $.  The\n$g \\times g$--matrices of their $A$ and $B$--periods,\n \\[2 \\omega= \\left ( \\oint_{A_k} \\mathrm{d}u_l \\right) ,  \\quad\n2 \\omega'= \\left ( \\oint_{B_k} \\mathrm{d}u_l \\right)   \\]\nare nondegenerate. Under the action of the transformation\n$ (2 \\omega) ^{-1}$  the vector\n$ \\mathrm{d} \\mathbf{ u}= ( \\mathrm{d}u_1,  \\ldots,  \\mathrm{d}u_g) ^T$ maps\nto the vector of normalized holomorphic differentials\n$ \\mathrm{d} \\mathbf{ v}= ( \\mathrm{d}v_1,  \\ldots,  \\mathrm{d}v_g) ^T$\n --- the   vector in\n$ H^1 (V,  \\mathbb{ C}) $  to satisfy the conditions\n$ \\oint_{A_k} \\mathrm{d}v_k= \\delta_{kl}, k, l=1 \\ldots, g$. It is known,\nthat $g \\times g$ matrix,\n \\[ \\tau= \\left ( \\oint_{B_k} \\mathrm{d}v_l \\right) = \\omega^{-1} \\omega' \\]\nbelongs to the { \\em upper Siegel halfspace} $ \\mathcal{ S}_g$ of degree\n$g$,  i.e. it is symmetric  and has a positively defined imaginary\npart.\n\n\n\nLet us denote by $ \\mathrm{Jac} (V) $ the { \\em Jacobian} of the curve\n$V$,  i.e.  the factor $ \\mathbb{ C}^g\/ \\Gamma$,  where\n$ \\Gamma=2 \\omega \\oplus 2 \\omega'$ is the lattice generated by the\nperiods of canonical holomorphic differentials.\n\n{ \\em Divisor} $ \\mathcal{D}$ is a formal sum of subvarieties of\ncodimension $1$ with coefficients from $ \\mathbb{ Z}$. Divisors on\nRiemann surfaces are given by formal sums of analytic points\n$ \\mathcal{ D}= \\sum_i^n m_i (y_i, x_i) $,  and $ \\mathrm{deg} \\mathcal{\nD}= \\sum_i^n m_i$. The { \\em effective divisor} is such that $m_i>0\n \\forall i$.\n\nLet $ \\mathcal{D}$ be a divisor of degree $0$,  $ \\mathcal{\nD}= \\mathcal{ X}- \\mathcal{ Z}$,  with $ \\mathcal{ X}$ and\n$ \\mathcal{ Z}$ --- the\neffective divisors $ \\mathrm{ deg } \\; \\mathcal{ X}= \\mathrm{ deg } \\;\n \\mathcal{Z}=n$ presented by $ \\mathcal{ X}= \\{ (y_1,  x_1) ,\n\\ldots,   (y_n,  x_n)  \\}$ and $ \\mathcal{ Z}= \\{ (w_1,  z_1) ,\n\\ldots,  (w_n,  z_n)  \\}  \\in  (V) ^{n}$,  where $ (V) ^n$ is the\n$n$--th symmetric power of $V$.\n\nThe {  \\em Abel map} $  \\mathfrak{ A}: (V) ^n   \\rightarrow   \\mathrm{Jac}\n(V) $ puts into correspondence the divisor $  \\mathcal{ D}$,  with\nfixed $  \\mathcal{ Z}$,  and the point $  \\boldsymbol{ u}= (u_1  \\ldots,\nu_g) ^T  \\in   \\mathrm{Jac} (V) $ according to the   \\[  \\boldsymbol{\nu}=  \\int_{  \\mathcal{ Z}}^  \\mathcal{ X}   \\mathrm{d}  \\mathbf{ u},\n  \\quad  \\text{or}  \\quad u_i=  \\sum_{k=1}^n  \\int_{z_k}^{x_k}  \\mathrm{d}u_i,\n  \\quad i=1,   \\ldots, g.  \\]\n\nThe {  \\em Abel's theorem} says that the points of the\ndivisors $  \\mathcal{ Z}$ and $  \\mathcal{ X}$ are respectively the poles and\nzeros of a meromorphic function on $V (y, x) $ iff\n$  \\int^  \\mathcal{ X}_  \\mathcal{ Z}  \\mathrm{d}  \\mathbf{ u}=0  \\mod\n \\Gamma$.\nThe {  \\em Jacobi inversion problem} is\nformulated as the problem of inversion of the map $  \\mathfrak{ A}$,\nwhen $n=g$ the $  \\mathfrak{ A}$ is $1  \\to1$,  except for so called\n{  \\em special divisors}. In our case special divisors  of degree\n$g$ are such that  at least for one pair $j$ and $k   \\in 1  \\ldots\ng$ the point $ (y_j, x_j) $ is the image of the hyperelliptic\ninvolution of the  point $ (y_k, x_k) $.\n\n  \\subsubsection{Meromorphic differentials} or the differentials of the\nsecond kind,  are the differential $1$-forms $  \\mathrm{d}r$ which can\nbe locally given as\n$  \\mathrm{d}r= (  \\sum_{i=-k}^  \\infty  \\alpha_i  \\xi^i) d  \\xi$ in the\nvicinity\nof any point $ (y, x) $ with some constants $  \\alpha_i$,  and\n$  \\alpha_{ (-1) }=0$.  It can be also checked directly,  that forms\nsatisfying such a condition are all of the form $\n  \\sum_{i=0}^{g-1}   \\beta_i x^{i+g}  \\frac{  \\mathrm{d}x}{y}$  ($  \\mod$\nholomorphic differential) .\nLet us introduce the following {  \\em canonical Abelian\ndifferentials of the second kind}   \\begin{equation}\n\\mathrm{d}r_j=  \\sum_{k=j}^{2g+1-j} (k+1-j)   \\lambda_{k+1+j}\\frac{x^k\n  \\mathrm{d}x}{\n4y},   \\quad j=1,   \\ldots, g.   \\end{equation} We denote their\n matrices of $A$ and $B$--periods,\n  \\[2  \\eta=  \\left (-  \\oint_{A_k}  \\mathrm{d}r_l  \\right) ,   \\quad 2  \\eta'\n=  \\left (-  \\oint_{B_k}  \\mathrm{d}r_l  \\right) .  \\]\n From {  \\em Riemann bilinear\nidentity},  for the period matrices of the differentials of the\nfirst and second kind follows:    \\begin{lemma} $2g  \\times\n2g$--matrix $  \\mathcal{ G}=  \\left (  \\begin{array} {cc}  \\omega&\n\\omega'  \\\\\n  \\eta&  \\eta'  \\end{array}  \\right) $ belongs to $PSp_{2g}$:\n  \\[  \\mathcal{ G}  \\left (  \\begin{array}{cc}0&-  \\mathbf{\n1}_g  \\\\  \\mathbf{ 1}_g&0   \\end{array}  \\right)    \\mathcal{ G}^T=-\n \\frac{  \\pi\ni}{2}  \\left (  \\begin{array}{cc} 0&-  \\mathbf{ 1}_g  \\\\  \\mathbf{\n1}_g&0  \\end{array}  \\right) .   \\]    \\label{LPSR}\n  \\end{lemma}\n\n\n\n  \\subsubsection{Differentials of the third kind}\nare the differential 1-forms $  \\mathrm{d}  \\Omega$ to have only\npoles of order $1$ and $0$ total residue,  and so are locally\ngiven in the vicinity of any of the poles as\n$  \\mathrm{d}  \\Omega= (  \\sum_{i=-1}^  \\infty  \\alpha_i  \\xi^i) d  \\xi$\nwith some constants $  \\alpha_i$,  $  \\alpha_{-1}$ being nonzero. Such\nforms  ($  \\mod$  holomorphic differential)  may be presented as:\n  \\[\n  \\sum_{i=0}^{n}   \\beta_i\n  \\left (  \\frac{y+y^+_i}{x-x^+_i}-  \\frac{y+y^-_i}{x-x^-_i}  \\right)\n  \\frac{  \\mathrm{d}x}{y},\n  \\]\nwhere $ (y^  \\pm_i, x^  \\pm_i) $ are the analytic points of the poles of\npositive  (respectively,  negative)  residue.\n\nLet us introduce the canonical differential of the third kind\n  \\begin{equation}\n  \\mathrm{d}   \\Omega (x_1, x_2) =\n  \\left (  \\frac{y+y_1}{x-x_1}-  \\frac{y+y_2}{x-x_2}  \\right)\n  \\frac{  \\mathrm{d} x}{2y},   \\label{third}\n  \\end{equation}\nfor this differential we have $  \\int_{x_3}^{x_4}  \\mathrm{d}\n  \\Omega (x_1, x_2) =  \\int_{x_1}^{x_2}  \\mathrm{d}   \\Omega (x_3, x_4) $.\n\n\n  \\subsubsection{Fundamental $2$--differential of the second kind}\nFor\n$  \\{ (y_1, x_1) ,  (y_2, x_2)   \\}  \\in  (V) ^2$\nwe introduce function  $F (x_1, x_2) $ defined by the conditions\n  \\begin{eqnarray}\n (  \\text{i}) .  &&F (x_1, x_2) =F (x_2, x_1) ,    \\nonumber   \\\\\n (  \\text{ii}) .  &&F (x_1, x_1) =2f (x_1) ,   \\nonumber   \\\\\n (  \\text{iii}) .  &&  \\frac{  \\partial\nF (x_1, x_2) }{   \\partial x_2}   \\big|_{x_2=x_1}\n=  \\frac{  \\mathrm{d}f (x_1) }{   \\mathrm{d}x_1}.  \\label{propf}\n  \\end{eqnarray}\n\nSuch $F (x_1, x_2) $ can be presented in the following\nequivalent forms\n  \\begin{eqnarray}\nF (x_1, x_2) &=&2y_2^2+2 (x_1-x_2) y_2  \\frac{  \\mathrm{d} y_2}{\n  \\mathrm{d}x_2}  \\nonumber  \\\\\n&+& (x_1-x_2) ^2  \\sum_{j=1}^gx_1^{j-1}  \\sum_{k=j}^{2g+1-j} (k-j+1)\n   \\lambda_{k+j+1}x_2^k,\n  \\label{Formf-1}  \\\\\nF (x_1, x_2) &=&2  \\lambda_{2g+2}x_1^{g+1}x_2^{g+1}+\n  \\sum_{i=0}^{g}x_1^ix_2^i (2  \\lambda_{2i}+  \\lambda_{2i+1} (x_1+x_2) ) .\n  \\label{Formf-2}   \\end{eqnarray}\n\n\nProperties   \\eqref{propf} of  $F (x_1, x_2) $ permit to construct\nthe {  \\em global Abelian $2$--differential of the second\nkind} with the unique  pole of order $2$  along $x_1=x_2$ :\n  \\begin{equation}\n  \\omega (x_1, x_2) =  \\frac{2y_1 y_2+F (x_1, x_2) }{\n4 (x_1-x_2) ^2}  \\frac{  \\mathrm{d}x_1}{ y_1}  \\frac{  \\mathrm{d}x_2}{y_2},\n  \\label{omega-1}   \\end{equation} which expands in the vicinity of\nthe pole as   \\[\n  \\omega (x_1, x_2) =  \\left (  \\frac{1}{2 (  \\xi-  \\zeta) ^2}+O (1)\n  \\right)   \\mathrm{d}  \\xi\n  \\mathrm{d}  \\zeta,\n  \\]\nwhere $  \\xi$ and $  \\zeta$ are the local\ncoordinates at the points $x_1$ and $x_2$ correspondingly.\n\nUsing the   \\eqref{Formf-1},  rewrite the   \\eqref{omega-1} in the form\n  \\begin{equation}\n  \\omega (x_1, x_2) =  \\frac{  \\partial}{   \\partial x_2}\n\\left (  \\frac{y_1+y_2}{\n2y_1 (x_1-x_2) }   \\right)   \\mathrm{d}x_1  \\mathrm{d}x_2\n+  \\mathrm{d}  \\mathbf{\nu}^T (x_1)   \\mathrm{d}  \\mathbf{ r} (x_2) ,   \\label{omega-2}  \\end{equation}\nwhere the differentials $  \\mathrm{d}  \\mathbf{ u},   \\mathrm{d}  \\mathbf{ r}$\nare as above. So,  the  periods of this $2$-form  (the double\nintegrals $  \\oint  \\oint  \\omega (x_1, x_2) $)  are expressible in terms of\n$ (2  \\omega, 2  \\omega') $ and $ (-2  \\eta, -2  \\eta') $,  e.g.,  we\nhave for $A$-periods:\n  \\[\n  \\left  \\{  \\oint_{A_i}  \\oint_{A_k}  \\omega (x_1, x_2)   \\right  \\}_{i, k=1,   \\ldots, g}=\n-4  \\omega^T  \\eta.\n  \\]\n\n  \\subsection{Riemann $  \\theta$-function}\n\nThe standard $  \\theta$--function $  \\theta (  \\boldsymbol{ v}|  \\tau) $ on $  \\mathbb{\nC}^g  \\times   \\mathcal{ S}_g$ is defined by its Fourier series,\n\n  \\[\n  \\theta (  \\boldsymbol{ v}|  \\tau)\n=  \\sum_{  \\boldsymbol{ m}  \\in   \\mathbb{ Z}^g}  \\mathrm{\nexp}  \\;  \\pi i  \\left  \\{   \\boldsymbol{ m}^T  \\tau   \\boldsymbol{ m}\n+2  \\boldsymbol{ v}^T\n  \\boldsymbol{ m}  \\right  \\}  \\]\nThe $  \\theta$--function\npossesses the periodicity properties $  \\forall k   \\in 1,   \\ldots, g$\n  \\begin{eqnarray*} &&  \\theta (v_1,   \\ldots, v_k+1,   \\ldots, v_g|  \\tau)\n=  \\theta (  \\boldsymbol{ v}|  \\tau) ,   \\\\\n&&  \\theta (v_1+  \\tau_{1k},   \\ldots, v_k+  \\tau_{kk},   \\ldots, v_g\n+  \\tau_{gk}|  \\tau)\n=  \\mathrm{ e}^{i  \\pi   \\tau_{kk}-2  \\pi i v_k}  \\theta (  \\boldsymbol{ v}|  \\tau) .\n  \\end{eqnarray*}\n\n$  \\theta$--functions with characteristics $[  \\varepsilon]=\n  \\left[   \\begin{array}{c}  \\varepsilon'  \\\\\n\\varepsilon   \\end{array}  \\right]\n=  \\left[   \\begin{array}{ccc}  \\varepsilon_1'&\n\\ldots&  \\varepsilon_g'\n  \\\\  \\varepsilon_1&  \\ldots&  \\varepsilon_g\n\\end{array}  \\right]   \\in   \\mathbb{\nC}^{2g}$\n  \\[\n  \\theta[  \\varepsilon] (  \\boldsymbol{ v}|  \\tau) =  \\sum_{  \\boldsymbol{ m}  \\in   \\mathbb{\nZ}^g}  \\mathrm{ exp}  \\;  \\pi i  \\left  \\{  (  \\boldsymbol{ m}+  \\varepsilon') ^T  \\tau\n (  \\boldsymbol{ m}+  \\varepsilon') +2 (  \\boldsymbol{ v}+  \\varepsilon) ^T\n (  \\boldsymbol{ m}+  \\varepsilon')   \\right  \\},\n  \\] for which\nthe periodicity properties are\n  \\begin{eqnarray*}\n&&  \\theta[  \\varepsilon] (v_1,   \\ldots, v_k+1,   \\ldots, v_g|  \\tau)\n=  \\mathrm{ e}^{2  \\pi i  \\varepsilon_k'}  \\theta (  \\boldsymbol{\nv}|  \\tau) ,   \\\\\n&&  \\theta[  \\varepsilon] (v_1+  \\tau_{1k},   \\ldots, v_k+  \\tau_{kk},\\ldots, v_g\n+  \\tau_{gk}|  \\tau)\n=  \\mathrm{ e}^{i  \\pi   \\tau_{kk}-2  \\pi i v_k-2  \\pi i\n  \\varepsilon_k}  \\theta (  \\boldsymbol{ v}|  \\tau) .\n  \\end{eqnarray*}\n\nFurther,  consider half-integer characteristics $[  \\varepsilon]$;\nthe $  \\theta$--function $  \\theta[  \\varepsilon] (  \\boldsymbol{ v}|\\tau) $\nis even or odd whenever $4{  \\varepsilon'}^T  \\varepsilon=0$ or 1\nmodulo 2.  There are $  \\frac{1}{2} (4^g+2^g) $ even characteristics\nand $  \\frac{1}{2} (4^g-2^g) $ odd.\n\n\nLet $  \\boldsymbol{ w}^T= (w_1,   \\ldots, w_g)   \\in   \\mathrm{Jac} (V) $ be\nsome fixed vector,  the function,\n  \\[   \\mathcal{\n R} (x) =  \\theta  \\left (   \\int_{x_0}^x   \\mathrm{d}  \\mathbf{ v} -  \\boldsymbol{\nw}|  \\tau  \\right) ,   \\quad x  \\in V\n  \\] is called {  \\em Riemann $  \\theta$--function}.\n\nThe Riemann $  \\theta$--function $  \\mathcal{ R} (x) $ is either identically\n$0$,  or it has exactly $g$ zeros $x_1,   \\ldots, x_g  \\in V$,  for which\nthe {  \\em Riemann vanishing theorem} says that\n  \\[\n  \\sum_{k=1}^g\n  \\int_{x_0}^{x_i}   \\mathrm{d}  \\mathbf{ v}=  \\boldsymbol{ w}+  \\mathbf{\nK}_{x_0},   \\] where $  \\mathbf{ K}^T_{x_0}= (K_1,   \\ldots, K_g) $ is the\nvector of Riemann constants with  respect to the base point\n$x_0$ and is defined by the formula\n  \\begin{equation}\nK_j=  \\frac{1+  \\tau_{jj}}{ 2}-\n  \\sum_{l  \\neq j}\n  \\oint_{A_l} \\mathrm{d}v_l (x)   \\int_{x_0}^x \\mathrm{d}v_j,\n  \\quad j=1,   \\ldots,  g.\n  \\label{Rconst}   \\end{equation}\n\n\n\n  \\section{Kleinian functions}  \\label{sigma-defs}\nLet $  \\boldsymbol{ m},   \\boldsymbol{ m}'  \\in   \\mathbb{ Z}^g$ be two\narbitrary vectors; denote  periods $  \\mathbf{ E} (  \\boldsymbol{ m},\n  \\boldsymbol{\nm}') =2  \\eta   \\boldsymbol{ m}+2  \\eta'  \\boldsymbol{ m}'$,  $  \\boldsymbol{   \\Omega} (\n  \\boldsymbol{\nm},   \\boldsymbol{ m}') =2  \\omega   \\boldsymbol{ m}+2  \\omega'  \\boldsymbol{ m}'$.\n  \\subsection{$  \\sigma$--function}\n\nIn    \\cite{kl88, ba98}  it  was shown,  that the properties\n  \\eqref{period} and   \\eqref{0-expan}\n define the function,  which plays the central role in the\ntheory of Kleinian functions.\n\n  \\begin{definition} An integral function\n$  \\sigma (  \\boldsymbol{ u}) $ is the Kleinian {  \\em fundamental\n$  \\sigma$--function} iff\n  \\begin{enumerate}   \\item for any vector\n$  \\boldsymbol{ u}  \\in   \\mathrm{Jac} (V) $   \\begin{equation}   \\sigma (\n  \\boldsymbol{\nu}+  \\boldsymbol{   \\Omega} (  \\boldsymbol{  m},   \\boldsymbol{  m}') )\n=  \\mathrm{\nexp}  \\left  \\{   \\mathbf{ E}^T (  \\boldsymbol{  m},   \\boldsymbol{  m}')  (  \\boldsymbol{\nu}+  \\tfrac{1}{2}  \\boldsymbol{   \\Omega} (  \\boldsymbol{\nm},   \\boldsymbol{ m}') )  +  \\pi i  \\boldsymbol{  m}^T   \\boldsymbol{\nm}'  \\right  \\}  \\sigma (  \\boldsymbol{  u}) .    \\label{period}\n  \\end{equation}\n  \\item $  \\sigma (  \\boldsymbol{  u}) $ has $0$ of\n$  \\left[  \\frac{g+1}{2}  \\right]$ order at $  \\boldsymbol{  u}=0$ and\n  \\begin{equation}   \\lim_{  \\boldsymbol{  u}  \\to 0}  \\frac{  \\sigma (  \\boldsymbol{\nu}) }{  \\delta (  \\boldsymbol{  u}) }=1,   \\label{0-expan}   \\end{equation} where\n$  \\delta (  \\boldsymbol{  u}) =\n  \\det  \\left (-  \\{u_{i+j-1}  \\}_{i, j=1,   \\ldots,   \\left[  \\frac{g+1}{2}  \\right]}  \\right) $.\n  \\end{enumerate}\n  \\end{definition}\n\nFor small genera we have,\n$  \\sigma=u_1+  \\ldots$ for $g=1$ and $2$;\n$  \\sigma=u_1u_3-u_2^2+  \\ldots$ for $g=3$ and $4$;\n$  \\sigma=-u_3^3+2u_2u_3u_4-u_1u_4^2-u_2^2u_5+u_1u_3u_5+  \\ldots$ for $g=5$\nand $6$  etc.\n\nWe introduce the $  \\sigma$-{  \\em functions with characteristic},\n$  \\sigma_{  \\boldsymbol{  r},   \\boldsymbol{  r}'}$ for vectors\n $   \\boldsymbol{  r}^T,    \\boldsymbol{  r}'  \\in   \\frac12   \\mathbb{ Z}^g\/  \\mathbb{ Z}^g$ defined\nby the formula\n  \\[\n  \\sigma_{  \\boldsymbol{  r},   \\boldsymbol{  r}'} (  \\boldsymbol{  u}) =  \\mathrm{\ne}^{-  \\mathbf{E}^T (  \\boldsymbol{  r},   \\boldsymbol{  r}')   \\boldsymbol{\nu}}  \\frac{  \\sigma (  \\boldsymbol{ u}+  \\boldsymbol{   \\Omega} (  \\boldsymbol{\nr},   \\boldsymbol{  r}') ) } {  \\sigma (  \\boldsymbol{   \\Omega} (  \\boldsymbol{\nr},   \\boldsymbol{ r}') ) }.   \\] These functions are completely\nanalogous to the Weierstrass' $  \\sigma_  \\alpha$  appearing in the\nelliptic theory   \\cite{ba55}.\n\n  \\subsubsection{ $  \\sigma$--function as\n $  \\theta$--function}\nFundamental hyperelliptic Kleinian $  \\sigma$--function belongs to\nthe class of generalized $  \\theta$--functions.  We give the\nexplicit expression of the $  \\sigma $ in terms of standard\n$  \\theta$--function as follows:\n\\begin{equation}   \\sigma (  \\boldsymbol{\nu}) =C  \\mathrm{ e}^{  \\boldsymbol{  u}^T   \\varkappa    \\boldsymbol{\nu}}\n\\theta ( (2  \\omega) ^{-1}\n\\boldsymbol{u}-\n\\mathbf{K}_{a}|\\tau) ,\n\\label{sigma}\n\\end{equation}\nwhere $   \\varkappa = (2  \\omega) ^{-1}  \\eta$,\n$  \\mathbf{ K}_{a}$ is the vector of Riemann constants with the base\npoint $a$ and the constant    \\[\nC=  \\frac{  \\epsilon_4}{  \\theta (0|  \\tau) }\n  \\prod_{r=1}^g  \\frac{  \\sqrt{{P}' (a_r) }}{  \\sqrt[4]{f' (a_r) }}\n  \\frac{1}{  \\prod_{k<l}  \\sqrt{e_k-e_l}},   \\]\nwhere $ (  \\epsilon_4) ^4=1$.\n\nDirect calculation shows,  that the function defined by\n  \\eqref{sigma} satisfies   \\eqref{period} and\n  \\eqref{0-expan}, we only note that,\nin  our case the vector of Riemann constants\n  \\eqref{Rconst} is, as follows from Riemann vanishing theorem,\n  \\begin{equation}\n  \\mathbf{ K}_{a}=  \\sum_{k=1}^g  \\int_{a}^{a_i}\n  \\mathrm{d}  \\mathbf{ v}.   \\label{rvector}\n\\end{equation}\n\n\nPutting $g=1$ and fixing the elliptic curve $y^2=f (x) =4x^3-g_2\nx-g_3$ in   \\eqref{sigma},  we see that the function\n  \\[  \\sigma (u) =  \\frac{1}{   \\vartheta_3 (0|  \\tau)\n  \\sqrt[4]{ (e_1-e_2)  (e_2-e_3) }}   \\mathrm{ e}^  \\frac{  \\eta u^2}{\n2  \\omega}  \\vartheta_1   \\left (  \\frac{u}{ 2  \\omega}  \\big|  \\tau  \\right)   \\]\n is the standard Weierstrass $  \\sigma$--function,  were we have used\nthe standard notation for Jacobi $  \\vartheta$--functions  (see e.g.\n  \\cite{ba55} ) .\n\n\n  \\subsection{Functions $  \\zeta$ and $  \\wp$}\nKleinian $  \\zeta$ and $  \\wp$-functions  are defined as logarithmic\nderivatives of the fundamental $  \\sigma$\n  \\begin{eqnarray*}&&  \\zeta_i (  \\boldsymbol{  u}) =  \\frac{  \\partial\n\\mathrm{\nln}  \\;  \\sigma (  \\boldsymbol{  u})  }{   \\partial u_i},   \\quad\ni  \\in 1,   \\ldots, g;  \\\\ &&  \\wp_{ij} (  \\boldsymbol{  u}) =-  \\frac{  \\partial^2\n  \\mathrm{ ln}  \\;   \\sigma (  \\boldsymbol{ u}) }{   \\partial u_i  \\partial\nu_j},   \\;   \\wp_{ijk} (  \\boldsymbol{ u}) =-  \\frac{  \\partial^3   \\mathrm{\nln}  \\;  \\sigma (  \\boldsymbol{  u}) }{   \\partial u_i  \\partial\n u_i  \\partial  u_k}  \\ldots,  i, j, k,   \\ldots  \\in 1,   \\ldots, g.\n  \\end{eqnarray*}\n\nThe functions $  \\zeta_i (  \\boldsymbol{  u}) $ and $  \\wp_{ij} (  \\boldsymbol{  u}) $ have the\nfollowing periodicity properties\n  \\begin{eqnarray*}   \\zeta_i (  \\boldsymbol{\nu}+  \\boldsymbol{   \\Omega} (  \\boldsymbol{  m},   \\boldsymbol{  m}') ) &=\n&  \\zeta_i (  \\boldsymbol{  u})\n+E_i (  \\boldsymbol{  m},   \\boldsymbol{  m}') ,    \\quad i  \\in 1,   \\ldots, g,\n  \\\\   \\wp_{ij} (  \\boldsymbol{  u}+\n  \\boldsymbol{   \\Omega} (  \\boldsymbol{  m},   \\boldsymbol{  m}') )\n&=&  \\wp_{ij} (  \\boldsymbol{  u}) ,    \\quad\ni, j   \\in 1,   \\ldots, g.   \\end{eqnarray*}\n\n\\subsubsection{Realization of the fundamental $2$--differential of\nthe second kind by Kleinian functions}\nThe construction is based on the following\n  \\begin{theorem}\n  \\label{the-S} Let $ (y (a_0),  a_0) $,  $ (y, x) $ and $ (  \\nu,\n\\mu) $  be arbitrary distinct  points on $V$ and let $  \\{ (y_1,\nx_1),   \\ldots,  (y_g, x_g)   \\}$  and $  \\{ (  \\nu_1, \\mu_1) ,\n \\ldots, (  \\nu_g,   \\mu_g)   \\}$ be arbitrary sets of distinct\n points  $  \\in (V) ^g$.  Then the following relation is valid\n \\begin{eqnarray} \\lefteqn{   \\int_{  \\mu}^x \\sum_{i=1}^g   \\int_{\n  \\mu_i}^{x_i} \\frac{2yy_i+F (x, x_i) }{4 (x-x_i) ^2}   \\frac{\n  \\mathrm{d}x}{y} \\frac{ \\mathrm{d}x_i}{y_i}}  \\nonumber  \\\\ &&=\n  \\mathrm{ln }   \\; \\left  \\{ \\frac{  \\sigma   \\left (\n\\int_{a_0}^x  \\mathrm{d} \\mathbf{ u} -  \\sum_{i=1}^g\n  \\int_{a_i}^{x_i}  \\mathrm{d} \\mathbf{ u}  \\right)  }{  \\sigma\n\\left (  \\int_{a_0}^x  \\mathrm{d} \\mathbf{ u}-  \\sum_{i=1}^g\n  \\int_{a_i}^{  \\mu_i}  \\mathrm{d} \\mathbf{ u}  \\right) } \\right\n\\}-  \\mathrm{ ln }   \\;   \\left  \\{ \\frac{  \\sigma \\left (\n  \\int_{a_0}^{  \\mu}  \\mathrm{d}  \\mathbf{ u}-  \\sum_{i=1}^g\n\\int_{a_i}^{x_{i}}  \\mathrm{d}  \\mathbf{ u}  \\right) }{  \\sigma\n  \\left (  \\int_{a_0}^{  \\mu}  \\mathrm{d}  \\mathbf{\nu}-  \\sum_{i=1}^g  \\int_{a_i}^{  \\mu_i}  \\mathrm{d}  \\mathbf{\nu}  \\right) }  \\right  \\} ,   \\label{r11}  \\end{eqnarray} where the function\n$F (x, z) $ is given by   \\eqref{Formf-2}.    \\end{theorem}\n\n  \\begin{proof} Let us consider the sum\n  \\begin{equation}\n  \\sum_{i=1}^g  \\int_{  \\mu}^x  \\int_{  \\mu_i}^{x_i}   \\left[\n  \\omega (x, x_i)  +\n  \\mathrm{d}  \\mathbf{ u}^T (x)\n  \\varkappa\n  \\mathrm{d}  \\mathbf{ u} (x_i)   \\right],\n  \\label{r10}\n  \\end{equation}\nwith $  \\omega (  \\cdot,   \\cdot) $ given by   \\eqref{omega-2}.\nIt is the normalized Abelian\nintegral of the third kind with the logarithmic residues in the\npoints $x_i$ and $  \\mu_i$. By Riemann vanishing theorem  we can\nexpress   \\eqref{r10} in terms of Riemann $  \\theta$--functions as\n  \\begin{equation}\n  \\mathrm{ln}\n  \\left  \\{\n  \\tfrac{{  \\displaystyle{  \\theta}}   \\left (  \\int_{a_0}^x   \\mathrm{d}  \\mathbf{\nv}- (  \\sum_{i=1}^g  \\int_{a_0}^{x_i}  \\mathrm{d}  \\mathbf{ v}-  \\mathbf{\nK}_{a_0})   \\right) }\n{{  \\displaystyle{  \\theta}}   \\left (  \\int_{a_0}^{x}  \\mathrm{d}  \\mathbf{\nv}- (  \\sum_{i=1}^g  \\int_{a_0}^{  \\mu_i}  \\mathrm{d}  \\mathbf{ v}-  \\mathbf{\nK}_{a_0})   \\right) }\n  \\right  \\}\n-  \\mathrm{ln}\n  \\left  \\{\n  \\tfrac{{  \\displaystyle{  \\theta}}\n  \\left (  \\int_{a_0}^{  \\mu}  \\mathrm{d}  \\mathbf{\n v}- (  \\sum_{i=1}^g  \\int_{a_0}^{x_i}  \\mathrm{d}  \\mathbf{ v}-  \\mathbf{\nK}_{a_0})   \\right) }{{  \\displaystyle{  \\theta}}\n  \\left (  \\int_{a_0}^{  \\mu}  \\mathrm{d}  \\mathbf{\nv}- (  \\sum_{i=1}^g  \\int_{a_0}^{  \\mu_i}  \\mathrm{d}  \\mathbf{ v}-  \\mathbf{\nK}_{a_0})   \\right) }  \\right  \\},   \\label{in-depends}\n  \\end{equation}\nand to obtain right hand side of   \\eqref{r11} we have to combine the\n  \\eqref{sigma},   expression of the vector $  \\mathbf{ K}_{a_0}$\n  \\eqref{rvector},  matrix $   \\varkappa = (2  \\omega) ^{-1}  \\eta$\nand Lemma   \\ref{LPSR}. Left hand side  of\n  \\eqref{r11} is obtained using   \\eqref{omega-1}.   \\end{proof}\n\n The fact,  that right hand side of the\n  \\eqref{r11} is independent on the arbitrary point $a_0$,  to be\n employed further,  has its origin in the properties of the vector\n of Riemann constants. Consider the difference $  \\mathbf{ K}_{a_0}-\n   \\mathbf{ K}_{a_0'}$ of vectors of Riemann constants with arbitrary\n base points $a_0$ and $a_0'$ by   \\eqref{Rconst} we find\n   \\[\n   \\mathbf{ K}_{a_0}-\n   \\mathbf{ K}_{a_0'} = (g-1)   \\int_{a_0'}^{a_0}  \\mathrm{d}  \\mathbf{ v},\n   \\]\nthis property provides that\n   \\[\n  \\int_{a_0}^{x_0}  \\mathrm{d}  \\mathbf{\n v}- (  \\sum_{i=1}^g  \\int_{a_0}^{x_i}  \\mathrm{d}  \\mathbf{ v}-  \\mathbf{\nK}_{a_0}) =\n   \\int_{a_0'}^{x_0}  \\mathrm{d}  \\mathbf{\n v}- (  \\sum_{i=1}^g  \\int_{a_0'}^{x_i}  \\mathrm{d}  \\mathbf{ v}-  \\mathbf{\nK}_{a_0'})\n   \\]\nfor arbitrary $x_i, $ with $i  \\in 0,   \\ldots, g$  on $V$,  so the\narguments of $  \\sigma$'s in   \\eqref{r11} which are linear\ntransformations by $2  \\omega$ of the arguments of $  \\theta$'s in\n\\eqref{in-depends},  do not depend on $a_0$.\n\n  \\begin{cor}  \\label{cor-P}  From Theorem   \\ref{the-S} for arbitrary\ndistinct $ (y (a_0) , a_0) $ and $ (y, x) $ on $V$ and  arbitrary set of\ndistinct points $  \\{ (y_1, x_1)   \\ldots,  (y_g, x_g)   \\}  \\in  (V) ^g$ follows:\n  \\begin{equation}   \\sum_{i, j=1}^g  \\wp_{ij}   \\left (  \\int_{a_0}^x\n  \\mathrm{d}  \\mathbf{ u}+   \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}  \\mathbf{\nu}  \\right)  x^{i-1}x_r^{j-1}=  \\frac{F (x, x_r) -2yy_r}{ 4 (x-x_r) ^2},   \\;\nr=1,   \\ldots, g.  \\label{principal}\n  \\end{equation}\n  \\end{cor}\n  \\begin{proof}\nTaking the partial derivative $  \\partial^2\/  \\partial x_r  \\partial x$\nfrom the both sides of   \\eqref{r11} and using the hyperelliptic\ninvolution $  \\phi (y, x) = (-y, x) $ and\n$  \\phi (y (a_0) , a_0) = (-y (a_0) , a_0) ) $ we obtain   \\eqref{principal}.  \\end{proof}\n\nIn the case $g=1$ the formula   \\eqref{principal} is\nactually the addition theorem for the Weierstrass elliptic\nfunctions,\n  \\[\n  \\wp (u+v) =-  \\wp (u) -  \\wp (v) +  \\frac{1}{4}  \\left[  \\frac{  \\wp' (u) -  \\wp' (v) }{\n  \\wp (u) -  \\wp (v)  }  \\right]^2\n  \\]\non the  elliptic curve  $y^2=f (x) =4x^3-g_2\nx-g_3$.\n\n\nNow we can give the expression for $  \\omega (x, x_r) $ in terms of\nKleinian functions.  We send the base point $a_0$ to the branch\n place $a$,  and  for $r  \\in 1,   \\ldots, g $ the fundamental\n$2$--differential of the second kind is given by   \\[   \\omega (x, x_r) =\n  \\sum_{i, j=1}^g  \\wp_{ij}   \\left (  \\int_a^x   \\mathrm{d}  \\mathbf{ u}-\n  \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}  \\mathbf{\nu}  \\right)   \\frac{x^{i-1}  \\mathrm{d}x}{y}  \\frac{x_r^{j-1}\n\\mathrm{d}x_r}{y_r}. \\]\n  \\begin{cor}\n  $  \\forall r   \\neq\ns   \\in 1,   \\ldots, g$\n  \\begin{eqnarray}\n  \\sum_{i, j=1}^g  \\wp_{ij}\n  \\left (  \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}  \\mathbf{ u}  \\right)\nx_s^{i-1}x_r^{j-1} =  \\frac{F (x_s, x_r) -2y_sy_r}{ 4 (x_s-x_r) ^2}\n.  \\label{principal11}\n  \\end{eqnarray}\n  \\end{cor}\n  \\begin{proof}\nIn   \\eqref{principal} we have for $s  \\neq r$\n  \\begin{eqnarray*}\n&  \\int  \\limits_{  \\phi (a_0) }^x\n  \\mathrm{d}  \\mathbf{ u}+   \\sum  \\limits_{k=1}^g\n  \\int  \\limits_{a_k}^{x_k}  \\mathrm{d}  \\mathbf{\nu}\n =-2  \\omega (  \\int  \\limits_{a_0}^{  \\phi (x) }  \\mathrm{d}  \\mathbf{\n v}- (  \\sum  \\limits_{i=1}^g  \\int  \\limits_{a_0}^{x_i}  \\mathrm{d}\n\\mathbf{ v}-  \\mathbf{\nK}_{a_0}) ) =  \\\\\n&-2  \\omega (  \\int  \\limits_{x_s}^{  \\phi (x) }  \\mathrm{d}  \\mathbf{\n v}- (  \\sum  \\limits_{  \\begin{subarray}{l}i=1  \\\\\ni  \\neq s  \\end{subarray}}^g  \\int  \\limits_{x_s}^{x_i}  \\mathrm{d}  \\mathbf{\nv}-  \\mathbf{ K}_{x_s}) ) =  \\int  \\limits_{a_s}^{x}  \\mathrm{d}  \\mathbf{\n u}+  \\sum  \\limits_{  \\begin{subarray}{l}i=1  \\\\\ni  \\neq s  \\end{subarray}}^g  \\int  \\limits_{a_i}^{x_i}  \\mathrm{d}  \\mathbf{\nu}\n  \\end{eqnarray*}\nand the change of notation $x   \\to x_s$ gives\n  \\eqref{principal11}.    \\end{proof}\n\n  \\subsubsection{Solution of the Jacobi inversion problem}\nThe equations of Abel map in conditions of Jacobi\ninversion problem\n  \\begin{equation}\nu_i=  \\sum_{k=1}^g  \\int_{a_k}^{x_k}   \\frac{x^{i-1}  \\mathrm{d}x}{y},\n  \\label{Abel-map}\n  \\end{equation}\nare invertible if the points $ (y_k,  x_k) $ are distinct and\n$  \\forall j, k   \\in 1,   \\ldots, g  \\;   \\phi (y_k, x_k)   \\neq (y_j, x_j)  $.  Using\n  \\eqref{principal} we find the solution of Jacobi inversion problem\non the curves with $a=  \\infty$ in a very effective form.\n\n  \\begin{theorem} The Abel preimage of the point $  \\boldsymbol{\nu}  \\in  \\mathrm{Jac} (V) $ is given  by the set\n$  \\{ (y_1, x_1) ,   \\ldots,  (y_g, x_g)   \\}  \\in  (V) ^g$,  where\n$  \\{x_1,   \\ldots, x_g  \\}$ are the zeros of the polynomial\n  \\begin{equation}   \\mathcal{ P} (x;  \\boldsymbol{  u}) =0  \\label{x},\n \\end{equation}\nwhere   \\begin{equation}   \\mathcal{ P} (x;  \\boldsymbol{  u}) =x^g-x^{g-1}\n \\wp_{g, g}\n (  \\boldsymbol{ u}) -x^{g-2}  \\wp_{g, g-1} (  \\boldsymbol{ u}) -  \\ldots-\n\\wp_{g, 1} (  \\mathbf{\nu}) ,   \\label{p}\n  \\end{equation}\nand  $  \\{y_1,   \\ldots, y_g  \\}$ are given by\n  \\begin{equation} y_k=-  \\frac{  \\partial   \\mathcal{\nP} (x;  \\boldsymbol{  u}) }{   \\partial u_g}  \\Bigl\\lvert_{x=x_k},\n  \\; \\label{y} \\end{equation} \\end{theorem} \\begin{proof} We tend\nin \\eqref{principal} $a_0  \\to a=  \\infty$. Then we take\n  \\begin{equation}\n  \\lim_{x   \\to  \\infty}\n  \\frac{F (x, x_r) }{4x^{g-1} (x-x_r) ^2}=\n  \\sum_{i=1}^g  \\wp_{gi} (  \\boldsymbol{ u}) x_r^{i-1}.  \\label{limit}\n  \\end{equation} The limit in the left hand side of  \\eqref{limit} is\nequal to $x_r^g$,  and we obtain   \\eqref{x}.\n\nWe find from   \\eqref{Abel-map},\n  \\[  \\sum_{i=1}^g  \\frac{x^{k-1}_i}{ y_i}\n  \\frac{  \\partial x_i}{   \\partial u_j}=  \\delta_{jk},   \\qquad\n\\frac{  \\partial x_k}{   \\partial u_g}=  \\frac{y_k\n}{  \\prod_{i  \\neq\nk} (x_k-x_i) }.  \\]\nOn the other hand we have\n  \\[\n  \\frac{  \\partial   \\mathcal{ P}}{   \\partial u_g}\n\\Bigl\\lvert_{x=x_k}= -  \\frac{  \\partial x_k}{   \\partial u_g}\n\\prod_{i  \\neq k} (x_i-x_k) ,   \\] and we obtain   \\eqref{y}.\n\\end{proof}\n\nLet us denote by $  \\boldsymbol{  \\wp}$,  $  \\boldsymbol{  \\wp}'$ the\n$g$--dimensional vectors,\n  \\[\n  \\boldsymbol{   \\wp}=  \\left (  \\wp_{g1},   \\ldots,   \\wp_{gg}\n  \\right) ^T,\n  \\quad\n  \\boldsymbol{   \\wp}'=  \\frac{  \\partial   \\boldsymbol{   \\wp}}{\n\\partial u_g}\n  \\] and the companion matrix   \\cite{hj86} of the polynomial\n$  \\mathcal{ P} (z;  \\boldsymbol{ u}) $,  given by   \\eqref{p}\n  \\[  \\mathcal{ C}=  \\mathcal{ B}+   \\boldsymbol{   \\wp}   \\mathbf{ e}_g^T,\n  \\quad  \\text{where}  \\quad  \\mathcal{ B}=   \\sum_{k=1}^g   \\mathbf{ e}_k\n \\mathbf{\ne}^T_{k-1}.  \\]   The companion matrix $  \\mathcal{\nC}$ has the property   \\begin{equation} x_k^n=  \\mathbf{ X}^T_k  \\mathcal{\nC}^{n-g+1}  \\mathbf{ e}_g=  \\mathbf{ X}^T_k  \\mathcal{ C}^{n-g}  \\boldsymbol{\n  \\wp},    \\quad   \\forall n  \\in   \\mathbb{ Z},    \\label{prop-c}\n\\end{equation} with the vector $  \\mathbf{ X}_k^T= (1, x_k,\n\\ldots, x_k^{g-1}) $,  where $x_k$ is one of the roots of\n\\eqref{x}.  From   \\eqref{principal11} we find $-2y_ry_s=4\n(x_r-x_s) ^2  \\sum_{i=1}^{g}  \\wp_{ij} (  \\boldsymbol{ u})\nx_r^{i-1}x_s^{j-1}-F (x_r, x_s) $.  Introducing matrices $  \\Pi= (\n\\wp_{ij}) $, $  \\Lambda_0=  \\mathrm{diag} (  \\lambda_{2g-2},\n\\ldots,   \\lambda_0) $ and $  \\Lambda_1=  \\mathrm{diag}  \\; (\n\\lambda_{2g-1},   \\ldots,   \\lambda_1) $, we have,  taking into\n account   \\eqref{prop-c}, \\begin{eqnarray*} \\lefteqn{2  \\mathbf{\n  X}^T_r  \\boldsymbol{ \\wp}'{  \\boldsymbol{   \\wp}'}^T  \\mathbf{\n  X}_s=-4  \\mathbf{ X}^T_r ({  \\mathcal{ C}}^2  \\Pi-2   \\mathcal{\nC}  \\Pi   \\mathcal{ C}^T+  \\Pi {  \\mathcal{ C}^T}^2 )    \\mathbf{\nX}_s }   \\nonumber  \\\\&&+ 4  \\mathbf{ X}^T_r (  \\mathcal{ C}\n\\boldsymbol{ \\wp}  \\boldsymbol{   \\wp}^T+  \\boldsymbol{   \\wp}\n  \\boldsymbol{   \\wp}^T  \\mathcal{ C}^T)   \\mathbf{ X}_s +2\n\\mathbf{ X}^T_r  \\Lambda_0  \\mathbf{ X}_s+  \\mathbf{ X}^T_r (\n\\mathcal{ C}  \\Lambda_1+  \\Lambda_1  \\mathcal{ C}^T)   \\mathbf{\nX}_s.  \\end{eqnarray*} Whence,   (see   \\cite{be96b}) :\n  \\begin{cor}\nThe relation\n  \\begin{equation}\n2  \\boldsymbol{\n  \\wp}'{  \\boldsymbol{   \\wp}'}^T=-4 ({  \\mathcal{\nC}}^2  \\Pi-2   \\mathcal{ C}  \\Pi   \\mathcal{ C}^T+  \\Pi {  \\mathcal{ C}^T}^2 )\n+ 4 (  \\mathcal{ C}  \\boldsymbol{\n  \\wp}  \\boldsymbol{   \\wp}^T+  \\boldsymbol{   \\wp}  \\boldsymbol{   \\wp}^T\n  \\mathcal{\nC}^T) +  \\mathcal{ C}  \\Lambda_1+  \\Lambda_1  \\mathcal{\nC}^T+2  \\Lambda_0.  \\label{principium}\n  \\end{equation}\nconnects odd functions $  \\wp_{ggi}$ with poles order $3$ and  even\nfunctions $  \\wp_{jk}$ with poles of order $2$ in the field of\nmeromorphic functions on $  \\mathrm{Jac} (V) $.\n  \\end{cor}\n\n  \\begin{definition}  \\label{umbral_D} The {  \\em umbral derivative}\n \\cite{ro84}\n  $D_s (p (z) ) $ of a polynomial   \\newline $p (z) =  \\sum_{k=0}^n p_k z^k$\nis given by   \\[ D_s p (z) =\n  \\left (  \\frac{p (z) }{z^s}  \\right) _+=  \\sum_{k=s}^{n}p_kx^{k-s},\n  \\]\nwhere $ (  \\cdot) _+$ means taking the purely polynomial part.\n  \\end{definition}\n\nConsidering polynomials $p=  \\prod_{k=1}^{n} (z-z_k) $\nand ${  \\tilde p}= (z-z_0) p$,  the elementary properties of $D_s$ are\nimmediately deduced:\n  \\begin{eqnarray}\n&D_s (p) =z D_{s+1} (p) +p_s=z D_{s+1} (p)  +\nS_{n-s} (z_1,   \\ldots, z_n) ,   \\nonumber  \\\\\n&D_s ({  \\tilde\np}) = (z-z_0) D_s (p) +p_{s-1}= (z-z_0) D_s (p) +S_{n+1-s} (z_1,\n  \\ldots, z_n) ,   \\label{umbral_2}\n  \\end{eqnarray}\nwhere $S_l (  \\cdots) $ is the $l$--th order elementary symmetric\nfunction of its variables  times $ (-1) ^l$  (we assume\n$S_0 (  \\cdots) =1$) .\n\nFrom   \\eqref{umbral_2} we see that $S_{n-s} (z_0,   \\ldots,   \\hat\nz_l,   \\ldots, z_n) =   \\left (D_{s+1} ({  \\tilde\np}) |_{z=z_l}  \\right) $. This is particularly useful to  write\ndown the inversion of   \\eqref{y}\n  \\begin{equation}\n  \\wp_{ggk} (  \\boldsymbol{ u}) =  \\sum_{l=1}^g y_l\n  \\left (  \\frac{D_k (P (z) ) }\n{  \\frac{  \\partial}{  \\partial z}P (z) }  \\Bigg|_{z=x_l}  \\right) ,\n  \\label{inversiony}\n  \\end{equation}\nwhere $P (z) =  \\prod_{k=1}^{g} (z-x_k) $.\n\n\nIt is of importance to describe the set of common zeros of the\nfunctions $  \\wp_{ggk} (  \\boldsymbol{ u}) $.\n\n  \\begin{cor}\nThe vector function $  \\boldsymbol{   \\wp}' (  \\boldsymbol{ u}) $\nvanishes iff\n$  \\boldsymbol{ u}$ is a halfperiod.   \\end{cor}\n\n  \\begin{proof} The equations $  \\wp_{ggk} (  \\boldsymbol{ u}) =0,\n  \\;k  \\in 1,   \\ldots,  g$ yield due to   \\eqref{inversiony} the equalities\n$y_i=0,   \\forall i  \\in 1,   \\ldots, g$. The latter is possible if and\nonly if the points $x_1,   \\ldots,  x_g$ coincide with any $g$ points\n$e_{i_1},   \\ldots, e_{i_g}$ from the set  branching points\n$e_1,   \\ldots, e_{2g+2}$. So the point\n  \\[\n  \\boldsymbol{ u}=  \\sum_{l=1}^{g}  \\int_{a_l}^{e_{i_l}}  \\mathrm{d}  \\mathbf{\nu}  \\in   \\mathrm{Jac} (V)    \\] is of the second order in Jacobian and\nhence is a halfperiod.  \\end{proof}\n\n\n\n\n  \\section{Basic relations}\n\nIn this section  we are going to derive the explicit algebraic\nrelations between the generating functions in the field of\nmeromorphic functions on $  \\mathrm{Jac} (V) $. After some\npreparations just below,  we will in the section~  \\ref{hyper-Jac}\nfind the explicit cubic relations between $  \\wp_{ggi}$\nand $  \\wp_{ij}$.  These,  in turn,  lead\nto very special corollaries:  the variety\n$  \\mathrm{Kum} (V) =  \\mathrm{Jac} (V) \/  \\pm$ is mapped into the space of\nsymmetric matrices of rank not greater than $3$.\n\n\nWe start with,  the conditions\n$  \\lambda_{2g+2}=0,   \\,   \\lambda_{2g+1}=4$ being imposed,\nthe following Theorem   \\ref{the-Z},  which is the starting point for\nderivation of the basic relations.\n\n\n\n  \\begin{theorem}  \\label{the-Z}\n Let $(y_0, x_0)\\in V$ be an arbitrary point and\n$  \\{ (y_1, x_1) ,   \\ldots,  (y_g, x_g)   \\}  \\in  (V)^g$ be the\nAbel preimage of the point $  \\boldsymbol{ u}  \\in\n\\mathrm{Jac} (V) $. Then \\begin{equation} -  \\zeta_j \\left (\n\\int_{a}^{x_0} \\mathrm{d}  \\mathbf{ u} +  \\boldsymbol{ u}  \\right)\n  = \\int_{a}^{x_0}  \\mathrm{d}r_j + \\sum_{k=1}^g  \\int_{a_k}^{x_k}\n  \\mathrm{d}r_j -  \\frac{1}{2}  \\sum_{k=0}^g y_k  \\left (\n\\frac{D_j (R' (z) ) }{R' (z) }  \\Bigg|_{z=x_k}  \\right) ,\n  \\label{principalz}\n  \\end{equation}\nwhere $ R (z) =  \\prod_0^g (z-x_j) $ and $\nR' (z) =  \\frac{  \\partial}{  \\partial z} R (z) $.\n\nAnd\n  \\begin{eqnarray}\n-  \\zeta_j (  \\boldsymbol{ u}) =   \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}\nr_j-  \\frac{1}{2}  \\wp_{gg, j+1} (  \\boldsymbol{ u}).\n  \\label{principalz1} \\end{eqnarray} \\end{theorem}\n\n  \\begin{proof} Putting in   \\eqref{r11} $  \\mu_i=a_i$ we have\n\n  \\begin{equation}   \\mathrm{ln}\n  \\left  \\{  \\frac{  \\sigma (  \\int_{a_0}^{x}  \\mathrm{d}   \\mathbf{ u}-  \\boldsymbol{ u}) }\n{  \\sigma (  \\int_{a_0}^{x}  \\mathrm{d}  \\mathbf{ u}) }   \\right  \\}\n-  \\left  \\{  \\frac{  \\sigma (  \\int_{a_0}^{  \\mu}  \\mathrm{d}   \\mathbf{ u}-  \\boldsymbol{ u}) }\n{  \\sigma (  \\int_{a_0}^{  \\mu}  \\mathrm{d}  \\mathbf{ u}) }   \\right  \\} =\n  \\int_{  \\mu}^{x}  \\mathrm{d}  \\mathbf{ r}^T  \\,   \\boldsymbol{\nu}+  \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}  \\Omega (x,   \\mu) ,\n  \\label{r20}  \\end{equation}\nwhere $  \\mathrm{d}  \\Omega$ is as in   \\eqref{third}. Taking derivative over\n$u_j$ from the both sides of the equality   \\eqref{r20},  after that\nletting $a_0  \\to  \\mu$ and  applying $  \\phi (y, x) = (-y, x) $ and\n$  \\phi (  \\nu,   \\mu) = (-  \\nu,   \\mu) $,  we have\n  \\[\n  \\zeta_j  \\left (\n  \\int_{  \\mu}^x  \\mathrm{d}  \\mathbf{ u}+  \\boldsymbol{ u}\n  \\right)  +  \\int_{  \\mu}^{x}   \\mathrm{d}  r_j   -\n  \\frac12  \\sum_{k=1}^g  \\frac{1}{y_k}  \\frac{  \\partial x_k}{  \\partial\nu_j}   \\frac{y_k-y}{x_k-x}\n=\n  \\zeta_j  \\left (\n  \\boldsymbol{ u}\n  \\right) -  \\frac12  \\sum_{k=1}^g  \\frac{1}{y_k}  \\frac{  \\partial x_k}{\n \\partial\nu_j}   \\frac{y_k-  \\nu}{x_k-  \\mu}.\n  \\]\nPut $x=x_0$. Denoting $P (z) =  \\prod_1^g (z-x_j) $ we find\n  \\begin{eqnarray*}\n&&  \\sum_{k=1}^g  \\frac{1}{y_k}  \\frac{  \\partial x_k}{  \\partial\nu_j}   \\frac{y_k-y}{x_k-x}=\n  \\sum_{k=1}^g  \\left (  \\frac{D_{j} (P (z) ) }{P' (z) }  \\Bigg|_{z=x_k}\n\\right)\n  \\frac{y_k-y}{x_k-x}  \\\\=\n&&  \\sum_{k=0}^g\ny_k  \\left (  \\frac{D_j (R' (z) ) }{R' (z) }  \\Bigg|_{z=x_k}  \\right)\n-  \\sum_{k=1}^g\ny_k  \\left (  \\frac{D_{j+1} (P (z) ) }{P' (z) }  \\Bigg|_{z=x_k}  \\right) .\n  \\end{eqnarray*}\nHence,  using   \\eqref{inversiony} and adding to both sides\n$  \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}r_j$,  we deduce\n  \\begin{eqnarray}\n&&  \\zeta_j  \\left (   \\int_{  \\mu}^{x_0}  \\mathrm{d}  \\mathbf{\nu}+  \\boldsymbol{ u}   \\right)  +  \\int_{  \\mu}^{x_0}   \\mathrm{d}  r_j\n+  \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}r_j  -\n  \\frac12  \\sum_{k=0}^g\ny_k  \\left (  \\frac{D_j (R' (z) ) }{R' (z) }  \\Bigg|_{z=x_k}  \\right)\n  \\nonumber  \\\\\n&&=\n  \\zeta_j  \\left (\n  \\boldsymbol{ u}\n  \\right) +  \\sum_{k=1}^g  \\int_{a_k}^{x_k}  \\mathrm{d}r_j\n-  \\frac12  \\sum_{k=1}^g  \\frac{1}{y_k}  \\frac{  \\partial x_k}{  \\partial\nu_j}   \\frac{y_k-  \\nu}{x_k-  \\mu}-  \\frac12  \\wp_{gg, j+1}.  \\label{deduce}\n  \\end{eqnarray}\nNow see,  that the left hand side of the   \\eqref{deduce}  is\nsymmetrical in $x_0,  x_1,    \\ldots,  x_g$,  while the right hand side\ndoes not depend on $x_0$. So,  it does not depend on any of $x_i$.\nWe conclude,  that it is a constant  depending only on $  \\mu$.\nTending $  \\mu  \\to a$ and applying the hyperelliptic involution  to\nthe whole aggregate,  we find this constant to be $0$.   \\end{proof}\n\n\n  \\begin{cor} For $ (y, x)   \\in V$ and $  \\boldsymbol{\n  \\alpha}=  \\int_a^{x}  \\mathrm{d}  \\mathbf{ u}$ :\n  \\begin{equation}\n  \\zeta_{j} (  \\boldsymbol{ u}+  \\boldsymbol{   \\alpha})\n-  \\zeta_{j} (  \\boldsymbol{ u}) -  \\zeta_{j} (  \\boldsymbol{\n  \\alpha}) =  \\frac{ (-y D_j+  \\partial_j)    \\mathcal{ P} (x;  \\boldsymbol{\nu}) } {2  \\mathcal{ P} (x;  \\boldsymbol{ u}) }\n,   \\label{stickelberger}   \\end{equation}\nwhere $  \\partial_j=  \\partial\/  \\partial u_j$.\n  \\end{cor}\n\n  \\begin{proof} To find $  \\zeta_{j} (  \\boldsymbol{\n  \\alpha}) $ take the limit\n$  \\{x_1,   \\ldots, x_g  \\}  \\to  \\{a_1,   \\ldots, a_g  \\}$ in\n  \\eqref{principalz}. The right hand side of\n  \\eqref{stickelberger} is obtained by rearranging $\n  \\frac12  \\sum_{k=1}^g  \\frac{1}{y_k}  \\frac{  \\partial x_k}{  \\partial\nu_j}   \\frac{y_k-y}{x_k-x}$.  \\end{proof}\n\n\n\n  \\begin{cor}  The functions\n$  \\wp_{gggk}$,  for $k=1,   \\ldots, g$ are given by   \\begin{eqnarray}\n  \\wp_{gggi}= (6  \\wp_{gg}+  \\lambda_{2g})   \\wp_{gi}+6  \\wp_{g, i-1}-2  \\wp_{g-1, i}\n+  \\frac{1}{2}  \\delta_{gi}  \\lambda_{2g-1}.  \\label{wpgggi}\n  \\end{eqnarray}\n  \\end{cor}\n\n  \\begin{proof} Consider the relation\n  \\eqref{principalz1}. The differentials $  \\mathrm{d}  \\zeta_i$,\n$i=1,   \\ldots, g$ can be presented in the following forms\n  \\[\n-  \\mathrm{d}  \\zeta_i=  \\sum_{k=1}^g  \\wp_{ik}  \\mathrm{d}u_k=\n  \\sum_{k=1}^g  \\mathrm{d}r_i (x_k) -   \\frac12\n  \\sum_{k=1}^g  \\wp_{gg, i+1, k}  \\mathrm{d}u_k.\n  \\]\nPut $i=g-1$.\nWe obtain for each of the $x_k$,  $k=1,   \\ldots, g$\n  \\[\n  \\left (12x_k^{g+1}+2  \\lambda_{2g}x_k^g+\n  \\lambda_{2g-1}x_k^{g-1}\n-4  \\sum_{j=1}^g  \\wp_{g-1, j}x_k^{j-1}  \\right)   \\frac{  \\mathrm{d}x_k}{ y_k}\n=2  \\sum_{j=1}^g  \\wp_{gggj}x_k^{j-1}  \\frac{  \\mathrm{d}x_k}{\ny_k}.  \\]\nApplying the formula   \\eqref{p} to eliminate the\npowers of $x_k$ greater than $g-1$,  and taking into account,  that\nthe differentials $  \\mathrm{d}x_k$ are independent,  we come to\n  \\[  \\sum_{i=1}^g  \\left[ (6  \\wp_{gg}+  \\lambda_{2g})   \\wp_{gi}+6  \\wp_{g, i-1}\n-2  \\wp_{g-1, i}\n+  \\frac{1}{2}  \\delta_{gi}  \\lambda_{2g-1}  \\right]x_k^{i-1}=\n  \\sum_{j=1}^g  \\wp_{gggj}x_k^{j-1}.  \\]  \\end{proof}\n\nLet us\ncalculate the difference\n$  \\frac{  \\partial  \\wp_{gggk}}{  \\partial\nu_i}-  \\frac{  \\partial  \\wp_{gggi}}{  \\partial u_k}$ according to the\n  \\eqref{wpgggi}. We obtain\n  \\begin{cor}\n  \\begin{equation}\n  \\wp_{ggk}  \\wp_{gi}-  \\wp_{ggi}  \\wp_{gk}+  \\wp_{g, i-1, k}-  \\wp_{gi, k-1}=0.\n  \\label{wp3}\n  \\end{equation}\n  \\end{cor}\nThis means that the $1$--form\n$  \\sum_{i=1}^g (  \\wp_{gg}  \\wp_{gi}+  \\wp_{g, i-1})   \\mathrm{d} u_i$ is\nclosed. We can rewrite this as $  \\mathrm{d}  \\boldsymbol{ u}^T  \\mathcal{\nC}  \\boldsymbol{  \\wp}$.\n\nDifferentiation of   \\eqref{wp3} by\n$u_g$ yields\n  \\begin{equation}\n  \\wp_{gggk}  \\wp_{gi}-  \\wp_{gggi}  \\wp_{gk}+  \\wp_{gg, i-1, k}-  \\wp_{ggi, k-1}=0.\n  \\label{i-1, k-1}\n  \\end{equation}\nAnd the corresponding closed $1$--form is\n$  \\mathrm{d}   \\boldsymbol{ u}^T  \\mathcal{ C}  \\boldsymbol{   \\wp}'$.\n\n\n\n  \\subsection{Fundamental cubic and quartic\nrelations}  \\label{hyper-Jac} We are going to find the relations\nconnecting the odd functions $  \\wp_{ggi}$ and even functions\n$  \\wp_{ij}$. These relations take in hyperelliptic theory the place\nof the Weierstrass cubic relation   \\[ {  \\wp'}^2=4  \\wp^3-g_2  \\wp-g_3,\n  \\]\nfor elliptic functions,  which establishes the meromorphic map\nbetween the elliptic Jacobian $  \\mathbb{ C}\/ (2  \\omega, 2  \\omega') $ and\nthe plane cubic.\n\nThe theorem below is based\non the property of an Abelian function to be constant if  any\ngradient  of it is identically $0$,  or,  if for Abelian\nfunctions $G (  \\boldsymbol{ u}) $ and $F (  \\boldsymbol{ u}) $ there exist such a\nnonzero vector $  \\boldsymbol{\n  \\alpha}  \\in   \\mathbb{ C}^g$,  that\n$  \\sum_{i=1}^g  \\alpha_i  \\frac{  \\partial}{  \\partial u_i} (G (  \\mathbf{\nu}) -F (  \\boldsymbol{ u}) ) $  vanishes,  then\n$G (  \\boldsymbol{ u}) -F (  \\boldsymbol{ u}) $ is a constant.\n\n\n  \\begin{theorem} The  functions\n$  \\wp_{ggi}$ and $  \\wp_{ik}$ are related by\n  \\begin{eqnarray}\n  \\lefteqn{  \\wp_{ggi}  \\wp_{ggk}=4  \\wp_{gg}  \\wp_{gi}  \\wp_{gk}-\n2 (  \\wp_{gi}  \\wp_{g-1, k}+  \\wp_{g, k}  \\wp_{g-1, i}) }  \\nonumber  \\\\\n&&+\n4 (  \\wp_{gk}  \\wp_{g, i-1}+  \\wp_{gi}  \\wp_{g, k-1}) +\n4  \\wp_{k-1, i-1}-2 (  \\wp_{k, i-2}+  \\wp_{i, k-2})    \\nonumber  \\\\\n&&+\n  \\lambda_{2g}  \\wp_{gk}  \\wp_{gi}+  \\frac{  \\lambda_{2g-1}}2\n (  \\delta_{ig}  \\wp_{kg}+  \\delta_{kg}  \\wp_{ig}) +c_{ (i, k) },\n  \\label{product3}\n  \\end{eqnarray}\nwhere\n  \\begin{equation}c_{ (i, k) }=  \\lambda_{2i-2}  \\delta_{ik}\n+  \\frac12 (  \\lambda_{2i-1}  \\delta_{k, i+1}\n+  \\lambda_{2k-1}  \\delta_{i, k+1}) .\n  \\label{cij}\n  \\end{equation}\n  \\end{theorem}\n\n  \\begin{proof}\nWe are looking for such a function $G (  \\boldsymbol{ u}) $ that\n$  \\frac{  \\partial}{  \\partial u_g} (  \\wp_{ggi}  \\wp_{ggk}-G) $ is $0$.\nDirect check using   \\eqref{wp3} shows that\n  \\begin{multline*}\n  \\frac{  \\partial}{  \\partial u_g} (  \\wp_{ggi}  \\wp_{ggk}- (4  \\wp_{gg}  \\wp_{gi}  \\wp_{gk}-\n2 (  \\wp_{gi}  \\wp_{g-1, k}+  \\wp_{g, k}  \\wp_{g-1, i})   \\\\\n+\n4 (  \\wp_{gk}  \\wp_{g, i-1}+  \\wp_{gi}  \\wp_{g, k-1}) +\n4  \\wp_{k-1, i-1}-2 (  \\wp_{k, i-2}+  \\wp_{i, k-2})    \\\\\n+\n  \\lambda_{2g}  \\wp_{gk}  \\wp_{gi}+  \\frac{  \\lambda_{2g-1}}2\n (  \\delta_{ig}  \\wp_{kg}+  \\delta_{kg}  \\wp_{ig}) ) ) =0.\n  \\end{multline*}\nIt remains to determine $c_{ij}$.  From   \\eqref{principium},  we\nconclude,  that $c_{ (i, k) }$  for $k=i$ is equal to\n$  \\lambda_{2i-2}$,  for $k=i+1$ to $  \\frac12{  \\lambda_{2i-1}}$,\notherwise $0$. So $c_{ij}$ is given by   \\eqref{cij}.  \\end{proof}\n\n\\noindent Consider $  \\mathbb{ C}^{g+ \\frac{g (g+1) }{2}}$ with\ncoordinates $ (\\boldsymbol{ z}, p=\\{ p_{i, j}  \\}_{i, j=1\n\\ldots g}) $ with $  \\boldsymbol{ z}^T= (z_1, \\ldots, z_g)$ and\n $p_{ij}=p_{ji}$,  then we\nhave\n  \\begin{cor} The map   \\[ \\varphi:\n  \\mathrm{Jac} (V)   \\backslash (  \\sigma)   \\to   \\mathbb{\nC}^{g+  \\frac{g (g+1) }{2}},   \\quad   \\varphi (  \\boldsymbol{\nu}) = (  \\boldsymbol{   \\wp}' (  \\boldsymbol{ u}) ,    \\Pi (  \\boldsymbol{ u}) ) ,\n  \\]\nwhere $  \\Pi=  \\{  \\wp_{ij}  \\}_{i, j=1,   \\ldots, g}$,  is  meromorphic\nembedding.\n\nThe image\n$  \\varphi (  \\mathrm{Jac} (V)   \\backslash (  \\sigma) )   \\subset\n   \\mathbb{ C}^{g+  \\frac{g (g+1) }{2}}$ is the intersection of\n$  \\frac{g (g+1) }{2}$ cubics,  induced by   \\eqref{product3}.\n  \\end{cor}\n$ (  \\sigma) $ denotes the divisor of $0$'s of $  \\sigma$.\n\nConsider projection   \\[\n  \\pi:  \\mathbb{ C}^{  \\frac{g+g (g+1) }{2}}  \\to   \\mathbb{\nC}^{  \\frac{g (g+1) }{2}},   \\quad   \\pi (  \\boldsymbol{ z}, p) =p.\n  \\]\n  \\begin{cor}\nThe restriction  $  \\pi  \\circ  \\varphi$ is the meromorphic embedding of\nthe Kummer variety\n$  \\mathrm{Kum} (V) =\n(  \\mathrm{Jac} (V)   \\backslash (  \\sigma) ) \/  \\pm$  into\n$  \\mathbb{ C}^{  \\frac{g (g+1) }{2}}$. The image\n$  \\pi (  \\varphi (  \\mathrm{Jac} (V)   \\backslash (  \\sigma) ) )\n\\linebreak[3]  \\subset   \\mathbb{ C}^{  \\frac{g (g+1) }{2}}$ is\nthe intersection of quartics,  induced by \\begin{equation} (\n \\wp_{ggi}  \\wp_{ggj})  (  \\wp_{ggk}  \\wp_{ggl}) - (  \\wp_{ggi}\n\\wp_{ggk})  (  \\wp_{ggj}  \\wp_{ggl}) =0, \\label{Kijkl}\n  \\end{equation} where the parentheses mean,  that substitutions\nby   \\eqref{product3} are made before expanding.  \\end{cor} The\nquartics   \\eqref{Kijkl}  have no analogue in the elliptic theory.\nThe first example is given by genus $2$, where the celebrated\nKummer surface   \\cite{hu94} appears.\n\n  \\subsection{Analysis of fundamental relations}  \\label{hyper-Kum}\nLet us take a second look at the fundamental cubics\n  \\eqref{product3} and quartics   \\eqref{Kijkl}.\n\n  \\subsubsection{Sylvester's identity}\nFor any matrix $K$ of entries $k_{ij}$ with $i, j=1,   \\ldots, N$ we\nintroduce the symbol $K[{}^{i_1}_{j_1}  \\cdots{}^{i_m}_{j_n}]$ to\ndenote the $m  \\times n$ submatrix:\n  \\[\nK[{}^{i_1}_{j_1}  \\cdots{}^{i_m}_{j_n}]\n=  \\{k_{i_k, j_l}  \\}_{k=1,   \\ldots, m;  \\, l=1,   \\ldots, n}\n  \\]\nfor subsets of rows $i_k$ and columns $j_l$.\n\n\nWe will need here the {  \\em Sylvester's\nidentity}  (see,  for instance~  \\cite{hj86}) . Let us fix a subset of\nindices $  \\boldsymbol{  \\alpha}=  \\{i_1,   \\ldots, i_k  \\}$,  and make up the\n$N-k  \\times N-k$ matrix $S (K,   \\boldsymbol{  \\alpha}) $ assuming that\n  \\[\nS (K,   \\boldsymbol{  \\alpha}) _{  \\mu,   \\nu}=  \\det\nK[{}^{  \\mu,   \\boldsymbol{  \\alpha}}_{  \\nu,   \\boldsymbol{  \\alpha}}]\n  \\] and\n$  \\mu,   \\nu$ are not in $  \\boldsymbol{  \\alpha}$,  then\n  \\begin{equation}\n  \\det S (K,   \\boldsymbol{  \\alpha}) =  \\det\nK[{}^{  \\boldsymbol{  \\alpha}}_{  \\boldsymbol{  \\alpha}}]^{ (N-k-1) }\n \\det K .\n  \\label{Sylvester}   \\end{equation}\n\n  \\subsubsection{Determinantal form}\n\nWe introduce  (cf.   \\cite{le95})  new functions  $h_{ik}$ defined by\nthe formula   \\begin{eqnarray} h_{ik}&=&4  \\wp_{i-1, k-1} -2\n  \\wp_{k, i-2} -2  \\wp_{i, k-2}   \\nonumber  \\\\ &+&\n  \\frac{1}{2}  \\left (  \\delta_{ik} (  \\lambda_{2i-2}+  \\lambda_{2k-2})\n+  \\delta_{k, i+1}  \\lambda_{2i-1}\n+  \\delta_{i, k+1}  \\lambda_{2k-1}  \\right) ,\n  \\label{variables}\n  \\end{eqnarray}\nwhere the indices\n$i, k  \\in 1,   \\ldots, g+2$.\nWe assume that $  \\wp_{nm}=0$ if $n$ or $m$ is $<1$ and\n$  \\wp_{nm}=0$ if $n$ or $m$ is $>g$. It is evident that\n$h_{ij}=h_{ji}$. We shall denote the matrix of $h_{ik}$ by $H$.\n\nThe map   \\eqref{variables} from $  \\wp$'s and $  \\lambda$'s to $h$'s\nrespects the grading\n  \\[  \\mathrm{deg}  \\;h_{ij}=i+j,   \\qquad\n  \\mathrm{deg}  \\;  \\wp_{ij}=i+j+2,   \\qquad\n  \\mathrm{deg}  \\;  \\lambda_{i}=i+2,   \\] and on a fixed level $L$\n  \\eqref{variables} is\nlinear and invertible. From the definition follows\n  \\[\n  \\sum_{i=1}^{L-1}h_{i, L-i}=  \\lambda_{L-2}   \\Rightarrow   \\boldsymbol{\nX}^T H  \\boldsymbol{ X}=  \\sum_{i=0}^{2g+2}  \\lambda_{i}x^{i}\n  \\]\nfor $  \\boldsymbol{ X}^T= (1, x,   \\ldots, x^{g+1}) $ with\narbitrary $x  \\in  \\mathbb{ C}$. Moreover, for any roots $x_r$ and\n$x_s$ of the equation $  \\sum_{j=1}^{g+2} h_{g+2, j}x^{j-1}=0$ we\nhave  (cf.   \\eqref{principal11})  $ y_ry_s =  \\boldsymbol{ X}_r^T\nH  \\boldsymbol{ X}_s$.\n\nFrom   \\eqref{variables} we have\n\\begin{gather}\n-2  \\wp_{ggi}=  \\tfrac{  \\partial}{  \\partial\nu_g}h_{g+2, i}=  \\tfrac{  \\partial}{  \\partial\nu_i}h_{g+2, g}=-  \\tfrac12  \\tfrac{  \\partial}{  \\partial\nu_i}h_{g+1, g+1},   \\notag  \\\\\n2 ( \\wp_{gi, k-1}- \\wp_{g, i-1, k}) =  \\tfrac{  \\partial}{\n  \\partial u_k}h_{g+2, i-1}- \\tfrac{  \\partial}{  \\partial\nu_i}h_{g+2, k-1}= \\tfrac12 \\tfrac{  \\partial}{  \\partial\nu_k}h_{g+1, k}\n-\\tfrac12  \\tfrac{  \\partial}{  \\partial\nu_i}h_{g+2, i}, \\notag  \\\\\n  \\intertext{$  \\ldots$ etc.,    \\,   and  (see   \\eqref{wpgggi}) :}\n-2  \\wp_{gggi}= \\tfrac{ \\partial^2}{\\partial\nu_g^2}h_{g+2, i}=\n-  \\det H[{}^{i, }_{g+1 }{}^{g+1 }_{g+2}] -\n  \\det H[{}^{i-1, }_{g+1,}{}^{g+2 }_{g+2}] -\n  \\det H[{}^{i, }_{g,}{}^{g+2}_{g+2}].\\label{dg_H}\n\\end{gather}\nUsing \\eqref{variables},  we write  \\eqref{product3} in\nmore effective form:\n\\begin{eqnarray}\n4  \\wp_{ggi}  \\wp_{ggk}=  \\tfrac{  \\partial}{  \\partial\nu_g}h_{g+2, i}  \\tfrac{  \\partial}{  \\partial\nu_g}h_{g+2, k}=- \\det\nH[{}^{i, }_{k, }{}^{g+1, }_{g+1, }{}^{g+2}_{g+2}]\n  \\label{wpggiwpggk}\n  \\end{eqnarray}\n\n\nConsider,  as an example,  the case of genus $1$. We define on the\nJacobian of a curve   \\[ y^2=  \\lambda_4 x^4+  \\lambda_3 x^3 +  \\lambda_2\nx^2+  \\lambda_1 x+  \\lambda_0    \\] the Kleinian functions:\n$  \\sigma_{K} (u_1) $ with expansion $u_1+  \\ldots$,  its second  and\nthird logarithmic derivatives $-  \\wp_{11}$ and $-  \\wp_{111}$. By\n  \\eqref{wpggiwpggk} and following the definition   \\eqref{variables}\n  \\[\n  -4\\wp_{111}^2=\\det H\n  \\left[{}_{1, 2, 3}^{1, 2, 3}  \\right]= \\det  \\left (\n  \\begin{array}{ccc}   \\lambda_0&   \\tfrac{1}{2}  \\lambda_1&-2\n  \\wp_{11}  \\\\ \\tfrac12 \\lambda_1&4  \\wp_{11}+  \\lambda_2&\n  \\tfrac12  \\lambda_3  \\\\ -2  \\wp_{11}&  \\tfrac12\n\\lambda_3&  \\lambda_4 \\end{array} \\right) ; \\] the determinant\n expands as:  \\[ \\wp_{111}^2=4  \\wp_{11}^3+  \\lambda_2\n  \\wp_{11}^2+ \\wp_{11}  \\frac{  \\lambda_1   \\lambda_3-4  \\lambda_4\n  \\lambda_0}{4}+ \\frac{  \\lambda_0   \\lambda_3^2+  \\lambda_4 (\n \\lambda_1^2-4  \\lambda_2  \\lambda_0) }{16}, \\] and the\n  \\eqref{dg_H},  in complete accordance, gives \\[ \\wp_{1111}=6\n  \\wp_{11}^2 +  \\lambda_2   \\wp_{11}+ \\frac{  \\lambda_1\n  \\lambda_3-4  \\lambda_4  \\lambda_0}{8} \\] These equations show\nthat $  \\sigma_K$ differs by $  \\mathrm{exp} (-  \\frac1{12}\n\\lambda_2 u_1^2) $ from standard Weierstrass $  \\sigma_W$ built by\nthe invariants $g_2=  \\lambda_4  \\lambda_0+  \\frac1{12}\n\\lambda_2^2-  \\frac14  \\lambda_3  \\lambda_1$ and $g_3=  \\det\n\\left (  \\begin{smallmatrix} \\lambda_0&  \\frac14  \\lambda_1&\n  \\frac16  \\lambda_2  \\\\ \\frac14  \\lambda_1&  \\frac16  \\lambda_2&\n  \\frac14  \\lambda_3  \\\\ \\frac16  \\lambda_2&  \\frac14  \\lambda_3&\n  \\lambda_4 \\end{smallmatrix}  \\right) $  (see,  e.g.\n  \\cite{ba55, ww73}) .\n\nFurther,\nwe find,  that $  \\mathrm{rank}  \\; H=3$ in generic point of Jacobian,\n$  \\mathrm{rank}  \\;  H=2$ in halfperiods.  At $u_1=0$,  where\n$  \\sigma_K$ has is $0$ of order $1$,  we have $  \\mathrm{rank}  \\;\n  \\sigma_K^2 H=3$.\n\n\nConcerning the general case,  on the ground of   \\eqref{wpggiwpggk},\nwe prove the following:\n  \\begin{theorem}\n$  \\mathrm{rank}  \\;H=3$ in generic point $  \\in  \\mathrm{Jac} (V) $  and\n$  \\mathrm{rank}  \\;H=2$ in the halfperiods.\n$  \\mathrm{rank}  \\;  \\sigma (  \\boldsymbol{ u}) ^2 H=3$ in generic\npoint $  \\in  (  \\sigma) $ and\n$  \\mathrm{rank}  \\;  \\sigma (  \\boldsymbol{ u}) ^2 H$ in the  points\nof $ (  \\sigma) _{  \\mathrm{sing}}$.\n  \\end{theorem}\nHere $ (  \\sigma)   \\subset   \\mathrm{Jac} (V) $ denotes the divisor of\n$0$'s of $  \\sigma (  \\boldsymbol{ u}) $. The\n$ (  \\sigma) _{  \\mathrm{sing}}  \\subset  (  \\sigma) $ is the so-called\nsingular set of $ (  \\sigma) $. $ (  \\sigma) _{  \\mathrm{sing}}$ is the set\nof points where $  \\sigma$ vanishes and all its first partial\nderivatives vanish.  $ (  \\sigma) _{  \\mathrm{sing}}$ is known  (see\n  \\cite{fa73} and references therein)  to be a subset of dimension\n$g-3$ in hyperelliptic Jacobians of $g>3$,  for  genus $2$ it is\nempty and consists of single point for $g=3$. Generally,  the\npoints of $ (  \\sigma) _{  \\mathrm{sing}}$ are presented by\n$  \\{ (y_1, x_1) ,   \\ldots,  (y_{g-3}  \\, , x_{g-3}  \\, )   \\}  \\in (V) ^{g-3}$ such that\nfor all $i  \\neq j  \\in 1,   \\ldots,  g-3, $ $  \\phi (y_i, x_i)   \\neq (y_j, x_j) $.\n\n  \\begin{proof} Consider the Sylvester's\nmatrix $S=S   \\left (H\n [{}^{i, j, g+1, g+2}_{k, l, g+1, g+2}],   \\{g+1, g+2  \\}\n  \\right) $.\nBy   \\eqref{wpggiwpggk} we have\n$S=-4  \\left (   \\begin{array}{ll}\n  \\wp_{ggi}  \\wp_{ggk}&  \\wp_{ggi}  \\wp_{ggl}  \\\\\n  \\wp_{ggj}  \\wp_{ggk}&  \\wp_{ggj}  \\wp_{ggl}\n  \\end{array}  \\right) $\nand $  \\det S=0$,  so by   \\eqref{Sylvester} we see,  that\n$  \\det H  [{}^{i, j, g+1, g+2}_{k, l, g+1, g+2} ]\n  \\det H  [{}^{g+1, g+2}_{g+1, g+2} ]$ vanishes\nidentically. As  $  \\det H  [{}^{g+1, g+2}_{g+1, g+2} ]=\n  \\lambda_{2g+2} (4  \\wp_{gg}+  \\lambda_{2g}) -  \\frac14  \\lambda_{2g+1}^2$\nis not an identical $0$,  we infer that\n  \\begin{equation}\n  \\det H  \\left[{}^{i, j, g+1, g+2}_{k, l, g+1, g+2}  \\right]=0.\n  \\label{subHdet}\n  \\end{equation}\nRemark,  that this equation is actually the   \\eqref{Kijkl} rewritten\nin terms of $h$'s. Now from the   \\eqref{subHdet},  putting $j=l=g$,\nwe obtain for any $i, k$,  except for such $  \\boldsymbol{ u}$,  that\n$H  \\left[{}^{g, g+1, g+2}_{g, g+1, g+2}  \\right]$ becomes degenerate,  and\nthose  where the entries become singular i.e.\n$  \\boldsymbol{u}  \\in (  \\sigma) $,    \\begin{equation}\nh_{ik}= (h_{i, g}, h_{i, g+1}, h_{i, g+2})\n  \\left (H  \\left[{}^{g, g+1, g+2}_{g, g+1, g+2}  \\right]  \\right) ^{-1}\n  \\left (  \\begin{array}{l}\nh_{k, g}  \\\\\nh_{k, g+1}  \\\\\nh_{k, g+2}\n  \\end{array}\n  \\right) .  \\label{hik}\n  \\end{equation}\nThis leads to the skeleton decomposition of the matrix $H$\n  \\begin{equation}\nH=H  \\left[{}^{1,   \\, {  \\displaystyle   \\ldots}  \\, , g+2}_{g, g+1, g+2}\n \\right]\n  \\left (H  \\left[{}^{g, g+1, g+2}_{g, g+1, g+2}  \\right]  \\right) ^{-1}\nH  \\left[{}^{g, g+1, g+2}_{1,   \\, {  \\displaystyle   \\ldots}  \\, , g+2}\n \\right],\n  \\label{skeleton}\n  \\end{equation}\nwhich shows,  that in generic point of $  \\mathrm{Jac} (V) $ rank\nof $H$ equals $3$.\n\nConsider the case $  \\det\nH  [{}^{g, g+1, g+2}_{g, g+1, g+2}]=0$. As by\n  \\eqref{wpggiwpggk} we have $   \\det\nH [{}^{g, g+1, g+2}_{g, g+1, g+2}]= 4 \\wp_{ggg}^2$,  this may\nhappen only iff $  \\boldsymbol{u}$ is a halfperiod.  And therefore\n  we have instead of   \\eqref{subHdet} the equalities $H\n[{}^{i, g+1, g+2}_{k, g+1, g+2}]=0$ and consequently\nin halfperiods matrix $H$ is decomposed as\n  \\[\nH=H  \\left[{}^{1,   \\, {  \\displaystyle   \\ldots}  \\, , g+2}_{g+1, g+2}\n \\right]\n  \\left (H  \\left[{}^{g+1, g+2}_{g+1, g+2}  \\right]  \\right) ^{-1}\nH  \\left[{}^{g+1, g+2}_{1,   \\, {  \\displaystyle   \\ldots}  \\, , g+2}\n \\right],\\]\nhaving the rank $2$.\n\nNext,  consider $  \\sigma (  \\boldsymbol{ u}) ^2 H$ at the\n$  \\boldsymbol{ u}  \\in (  \\sigma) $. We have\n$  \\sigma (  \\boldsymbol{ u}) ^2\nh_{i, k}=4  \\sigma_{i-1}  \\sigma_{k-1}-2  \\sigma_{i}  \\sigma_{k-2}\n-2  \\sigma_{i-2}  \\sigma_{k}, $  where\n$  \\sigma_{i}=  \\frac{  \\partial}{  \\partial u_i}  \\sigma (  \\boldsymbol{\nu}) $,  and,  consequently,  the\ndecomposition\n  \\[\n   \\sigma (  \\boldsymbol{ u}) ^2 H|_{  \\boldsymbol{\nu}  \\in (  \\sigma) }=2 (  \\boldsymbol{ s}_1,    \\boldsymbol{ s}_2,\n  \\boldsymbol{ s}_3)    \\left (   \\begin{array}{rrr} 0&0&-1  \\\\ 0&2&0  \\\\\n-1&0&0   \\end{array}   \\right)    \\left (   \\begin{array}{r}   \\boldsymbol{\ns}_1^T  \\\\   \\boldsymbol{ s}_2^T  \\\\   \\boldsymbol{ s}_3^T   \\end{array}\n  \\right) ,    \\] where $  \\boldsymbol{\ns}_1= (  \\sigma_1,   \\ldots,   \\sigma_g, 0, 0) ^T$,  $  \\boldsymbol{\ns}_2= (0,   \\sigma_1,   \\ldots,   \\sigma_g, 0) ^T$  and $  \\boldsymbol{\ns}_3= (0, 0,   \\sigma_1,   \\ldots,   \\sigma_g) ^T$. We infer,  that\n$  \\mathrm{rank} (  \\sigma (  \\boldsymbol{ u}) ^2 H) $ is $3$ in generic\npoint of $ (  \\sigma) $,  and\nbecomes $0$ only when $  \\sigma_1=  \\ldots=  \\sigma_g=0$,   is in the\npoints $  \\in  (  \\sigma) _{  \\mathrm{sing}}$,  while no other values are\npossible.\n  \\end{proof}\n\n{  \\em Conclusion}. The map   \\begin{align*}\nh:  \\boldsymbol{ u}  \\mapsto&\n  \\{4  \\sigma_{i-1}  \\,   \\sigma_{k-1}-2  \\sigma_{i}  \\,   \\sigma_{k-2}\n-2  \\sigma_{i-2}  \\,   \\sigma_{k}  \\\\&-  \\sigma\n (4  \\sigma_{i-1, k-1}  \\, -2  \\sigma_{i, k-2}  \\,\n-2  \\sigma_{i-2, k}) +\n  \\tfrac{1}{2}  \\sigma^2 (  \\delta_{ik} (  \\lambda_{2i-2}+  \\lambda_{2k-2})   \\\\\n&+  \\delta_{k, i+1}  \\lambda_{2i-1}\n+  \\delta_{i, k+1}  \\lambda_{2k-1})   \\}_{i, k  \\in 1,   \\ldots,  g+2},\n   \\end{align*}  induced by\n  \\eqref{variables} establish the meromorphic map of the\n$  \\big (  \\mathrm{Jac} (V)   \\backslash (  \\sigma) _{  \\mathrm{sing}}  \\big) \/  \\pm\n$ into the space $Q_3$\nof complex symmetric $ (g+2)   \\times (g+2) $ matrices of\n$  \\mathrm{rank}$ not greater than $3$.\n\nWe give the example of genus   $2$ with $  \\lambda_6=0$ and\n$  \\lambda_5=4$:\n\n  \\begin{equation}\nH=  \\left (  \\begin{array}{cccc}  \\lambda_0&  \\frac{1}{2}  \\lambda_1&-2\n \\wp_{11}&-2  \\wp_{12}  \\\\\n  \\frac{1}{2}  \\lambda_1&  \\lambda_2+4  \\wp_{11}&  \\frac{1}{2}  \\lambda_3-\n2  \\wp_{12}&-2  \\wp_{22}  \\\\-2  \\wp_{11}&  \\frac{1}{2}  \\lambda_3-2\n \\wp_{12}&  \\lambda_4+4  \\wp_{22}&2  \\\\\n-2  \\wp_{12}&-2  \\wp_{22}&2&0  \\end{array}  \\right) .\n  \\label{kum}  \\end{equation}\nIn this case $ (  \\sigma) _{  \\mathrm{sing}}=  \\{  \\varnothing  \\}$,\nso the Kummer surface in $  \\mathbb{ C}  \\mathbb{ P}^3$ with coordinates\n \\newline\n$ (X_0,  X_1,  X_2,  X _3)  =  (  \\sigma^2,   \\sigma^2  \\wp_{11},   \\sigma^2  \\wp_{12},   \\sigma^2  \\wp_{22}) $ is\ndefined by the equation $  \\det  \\sigma^2 H=0$.\n\n  \\subsubsection{Extended cubic relation}\nThe extension   \\cite{le95} of   \\eqref{wpggiwpggk} is given by\n  \\begin{theorem}\n  \\begin{eqnarray}\n  \\mathbf{ R}^T  \\boldsymbol{   \\pi}_{jl}  \\boldsymbol{   \\pi}_{ik}^T\n \\mathbf{\nS}=  \\frac{1}{4}  \\det   \\left (  \\begin{array}{cc} H\n  \\left[{}^i_j{}^k_l{}^{g+1}_{g+1}{}^{g+2}_{g+2}  \\right]\n&  \\mathbf{ S}  \\\\  \\mathbf{ R}^T&0\n  \\end{array}  \\right) ,   \\label{bakergen}\n  \\end{eqnarray}\nwhere $  \\mathbf{ R},   \\,    \\mathbf{ S}  \\in   \\mathbb{  C}^4$ are\narbitrary vectors and\n  \\[  \\boldsymbol{   \\pi}_{ik}=  \\left (\n  \\begin{array}{c} -  \\wp_{ggk}  \\\\\n  \\wp_{ggi}  \\\\\n  \\wp_{g, i, k-1}-  \\wp_{g, i-1, k}  \\\\\n  \\wp_{g-1, i, k-1}-\n  \\wp_{g-1, k, i-1}+\n  \\wp_{g, k, i-2}-\n  \\wp_{g, i, k-2}\n  \\end{array}\n  \\right)   \\]\n  \\end{theorem}\n\n  \\begin{proof}\nVectors $  \\boldsymbol{ {  \\tilde  \\pi}}=  \\boldsymbol{   \\pi}_{ik}$ and\n$  \\boldsymbol{   \\pi}=  \\boldsymbol{   \\pi}_{jl}$ solve the equations    \\[\nH  \\left[{}^i_j{}^k_l{}^{g+1}_{g+1}{}^{g+2}_{g+2}  \\right]  \\boldsymbol{\n  \\pi}=0;   \\quad   \\boldsymbol{\n{  \\tilde  \\pi}}^T{H}  \\left[{}^i_j{}^k_l{}^{g+1}_{g+1}{}^{g+2}_{g+2}\n  \\right]=0.\n  \\]\nThe theorem follows.\n  \\end{proof}\n\nThe case of genus $2$,  when $  \\boldsymbol{\n  \\pi}_{21}= (-  \\wp_{222},   \\wp_{221}, -  \\wp_{211},   \\wp_{111}) ^T$\nexhausts all the possible $  \\wp_{ijk}$--functions,  the relation\n  \\eqref{bakergen} was thoroughly studied by Baker   \\cite{ba07}.\n\n\n\n\n  \\section{Applications}\n\n  \\subsection{Matrix realization of hyperelliptic Kummer varieties}\nHere we present the explicit matrix realization  (see   \\cite{bel96})\nof hyperelliptic Jacobians $  \\mathrm{Jac} (V) $  and Kummer varieties\n$  \\mathrm{Kum} (V) $ of the curves $V$ with the fixed branching point\n$e_{2g+2}=a=  \\infty$.  Our approach is based on the results of\nSection   \\ref{hyper-Kum}.\n\nLet us consider the space $  \\mathcal{ H}$ of complex symmetric\n$ (g+2)   \\times (g+2) $--matrices  $  \\mathrm{ H}=  \\{  \\mathrm{\nh}_{k, s}  \\}$,   with $  \\mathrm{ h}_{g+2, g+2}=0$ and $  \\mathrm{\nh}_{g+1, g+2}=2$.  Let us put in correspondence  to $  \\mathrm{\nH}  \\in   \\mathcal{ H}$  a symmetric $g  \\times g$--matrix $  \\mathrm{\nA} (  \\mathrm{ H}) $,  with entries $a_{k, s}=  \\det   \\mathrm{ H}\n  \\left[{}_{s, g+1, g+2}^{k, g+1, g+2}  \\right]$.\n\nFrom the Sylvester's identity   \\eqref{Sylvester} follows that\n rank of the matrix $  \\mathrm{ H}  \\in   \\mathcal{ H}$\ndoes not exceed $3$ if and only if rank of the  matrix\n$  \\mathrm{ A} (  \\mathrm{ H}) $ does not exceed $1$.\n\n\nLet us put  $K  \\mathcal{ H}=   \\left  \\{  \\mathrm{ H}  \\in   \\mathcal{ H}:\n  \\mathrm{rank}  \\mathrm{ H}  \\leq 3  \\right  \\} $. For each complex\nsymmetric $g  \\times g$--matrix $  \\mathrm{ A}=  \\{a_{k, s}  \\}$ of rank not\ngreater $1$,  there exists,  defined up to sign,\na $g$--dimensional column vector $\n  \\mathbf{  z}=  \\mathbf{  z} (  \\mathrm{ A}) $,  such that\n$  \\mathrm{ A}=-4  \\mathbf{  z}  \\cdot   \\mathbf{  z}^T$.\n\nLet us introduce vectors\n$  \\mathbf{  h}_k=  \\{  \\mathrm{ h}_{k, s};  \\;s=1,   \\ldots, g   \\}  \\,\n \\in   \\mathbb{\nC}^g$.\n\n  \\begin{lemma}  \\label{geom-2}\n  Map\n  \\begin{eqnarray*}\n&  \\gamma: K  \\mathcal{ H}   \\to  (  \\mathbb{ C}^g\/  \\pm)   \\times\n \\mathbb{ C}^g  \\times  \\mathbb{\nC}^g  \\times  \\mathbb{ C}^1  \\\\\n&  \\gamma (  \\mathrm{ H}) =\n-  \\left (  \\mathbf{  z}  \\left (  \\mathrm{ A} (  \\mathrm{ H})   \\right) ,\n   \\mathbf{\nh}_{g+1},    \\mathbf{  h}_{g+2},   \\mathrm{h}_{g+1, g+1}  \\right)\n  \\end{eqnarray*}\n is a homeomorphism.\n  \\end{lemma}\n\n  \\begin{proof} follows from the relation:\n  \\[\n4   \\hat{  \\mathrm{ H}}=4   \\mathbf{  z}  \\cdot   \\mathbf{  z}^T+2\n  \\left (  \\mathbf{  h}_{g+2}  \\mathbf{  h}_{g+1}^T+  \\mathbf{  h}_{g+1}\n  \\mathbf{  h}_{g+2}^T  \\right) -  \\mathrm{h}_{g+1, g+1}  \\mathbf{  h}_{g+2}  \\mathbf{  h}_{g+2}^T\n  \\]\nwhere $  \\hat{  \\mathrm{ H}}$ is the matrix composed of the column\nvectors $  \\mathbf{  h}_k,    \\,  k=1,   \\ldots, g$,  and $  \\mathbf{  z}= (  \\mathbf{\nz}  \\left (  \\mathrm{ A} (  \\mathrm{ H})   \\right) $.\n  \\end{proof}\n\n\nLet us introduce  the $2$--sheeted ramified\n covering\n$  \\pi:J  \\mathcal{ H}  \\to K  \\mathcal{ H}$,   which the\ncovering $  \\mathbb{ C}^g  \\to (  \\mathbb{ C}^g\/  \\pm) $ induces by the map\n $  \\gamma$.\n\n  \\begin{cor}\n $  \\hat   \\gamma: J  \\mathcal{ H}   \\cong   \\mathbb{ C}^{3g+1}$.\n  \\end{cor}\n\nNow let us consider the  universal space\n$W_g$ of $g$--th\nsymmetric powers of hyperelliptic curves\n  \\[V=\n  \\left  \\{  (y, x)    \\in   \\mathbb{ C}^2:   \\, y^2=4 x^{2g+1} +\n  \\sum_{k=0}^{2g}  \\lambda_{2g-k}x^{2g-k}  \\right  \\}   \\]  as an algebraic\nsubvariety in $ (  \\mathbb{ C}^2) ^g  \\times\n  \\mathbb{ C}^{2g+1}$ with coordinates\n  \\[  \\left  \\{\n  \\left ( (y_1, x_1) ,   \\ldots,  (y_g, x_g)   \\right) ,   \\;  \\lambda_{2g},   \\ldots,\n  \\lambda_0   \\right  \\},   \\] where $ (  \\mathbb{ C}^2) ^g$ is  $g$--th\nsymmetric power of the space $  \\mathbb{ C}^2$.\n\nLet us define the map   \\[  \\lambda: J  \\mathcal{ H}  \\cong\n \\mathbb{ C}^{3g+1}  \\to\n (  \\mathbb{ C}^2) ^g  \\times   \\mathbb{ C}^{2g+1}  \\] in the following way:\n  \\begin{itemize}\n  \\item for\n$  \\boldsymbol{ G}= (  \\mathbf{  z},   \\mathbf{  h}_{g+1},   \\mathbf{\nh}_{g+2},   \\mathrm{h}_{g+1, g+1})    \\in   \\mathbb{ C}^{3g+1}$ construct\nby Lemma\n  \\ref{geom-2} the matrix $  \\pi (  \\boldsymbol{ G}) =  \\mathrm{\nH}=  \\{  \\mathrm{ h}_{k, s}  \\}  \\in K  \\mathcal{ H}$\n  \\item  put\n  \\[  \\lambda (  \\boldsymbol{ G}) =  \\{ (y_k, x_k) ,\n  \\lambda_r;  \\;k=1,   \\ldots, g,   \\,  r=0,   \\ldots,  2g,\n  \\}  \\]\nwhere $  \\{x_1,    \\ldots,  x_g  \\}$ is the set of roots of the equation\n$2 x^g+   \\mathbf{  h}_{g+2}^T   \\mathbf{ X}=0$,  and\n$y_k=  \\mathbf{  z}^T   \\mathbf{ X}_k$,  and\n$  \\lambda_r=  \\sum_{i+j=r+2}   \\mathrm{h}_{i, j}$.\n  \\end{itemize}\nHere\n$  \\mathbf{ X}_k= (1, x_k,   \\ldots, x_k^{g-1}) ^T$.\n\n  \\begin{theorem}. Map  $  \\lambda$ induces map\n$J  \\mathcal{ H}  \\cong   \\mathbb{ C}^{3g+1}  \\to W_g$ .  \\end{theorem}\n\n  \\begin{proof} Direct check shows,  that  the identity is valid\n  \\[\n  \\mathbf{ X}_k^T   \\mathrm{ A}\n  \\mathbf{ X}_s+4  \\sum_{i, j=1}^{g+2}\n  \\mathrm{h}_{i, j}x_{k}^{i-1}x_s^{j-1}=0,    \\]\nwhere $  \\mathrm{\nA}=  \\mathrm{ A} (  \\mathrm{ H}) $  and $  \\mathrm{ H}=  \\pi (  \\boldsymbol{\n G}) $.  Putting $k=s$ and using $  \\mathrm{ A}=4   \\mathbf{  z}  \\cdot\n  \\mathbf{  z}^T$,  we have $y_k^2=4\nx_k^{2g+1}+  \\sum_{s=0}^{2g}  \\lambda_{2g-s}x_k^{2g-s}$.  \\end{proof}\n\nNow it is all ready to give the description of our realization\nof varieties $T^g=  \\mathrm{Jac} (V) $ and\n$K^g=  \\mathrm{Kum} (V) $ of the hyperelliptic curves.\n\n\nFor each nonsingular curve\n$V=   \\left  \\{\n (y, x) ,  y^2=4\nx^{2g+1} +   \\sum_{s=0}^{2g}  \\lambda_{2g-s}x^{2g-s}  \\right  \\} $\ndefine the map\n  \\[  \\gamma:  \\;T^g  \\backslash (  \\sigma)   \\to   \\mathcal{\nH}:  \\gamma (u) =  \\mathrm{ H}=  \\{  \\mathrm{h}_{k, s}  \\},   \\] where\n$  \\mathrm{h}_{k, s}=4  \\wp_{k-1, s-1}-2 (  \\wp_{s, k-2}+  \\wp_{s-2, k})\n+  \\frac12\n[  \\delta_{ks} (  \\lambda_{2s-2}+  \\lambda_{2k-2}) +  \\delta_{k+1, s}\n  \\lambda_{2k-1}+  \\delta_{k, s+1}  \\lambda_{2s-1}]$.\n\n\n  \\begin{theorem} The map  $  \\gamma$  induces map\n$T^g  \\backslash (  \\sigma)   \\to K  \\mathcal{ H}$,  such that\n$  \\wp_{ggk}  \\wp_{ggs}=  \\dfrac14 a_{ks} (  \\gamma (u) ) $,   i.e\n$  \\gamma$  is lifted to\n  \\[\n{  \\tilde   \\gamma}:T^g  \\backslash (  \\sigma)   \\to J  \\mathcal{ H}  \\cong   \\mathbb{\nC}^{3g+1}   \\;   \\text{with }  \\;   \\mathbf{  z}= (  \\wp_{gg1},   \\ldots,\n \\wp_{ggg} ) ^T.    \\]\nComposition of maps $  \\lambda{  \\tilde\n  \\gamma}:  \\;T^g  \\backslash (  \\sigma)   \\to W_g$  defines\nthe inversion of the Abel map $  \\mathfrak{ A}:  \\;  (V) ^g  \\to T^g$ and,\ntherefore,  the map $  \\tilde  \\gamma$ is an embedding.\n  \\label{geom-3}\n  \\end{theorem}\n\nSo we have obtained the explicit realization of the Kummer variety\n$T^g  \\backslash (  \\sigma) \/  \\pm$  of the hyperelliptic\ncurve  $V$ of genus $g$ as a subvariety in the\nvariety of matrices $K  \\mathcal{ H}$.  As a consequence of the  Theorem\n  \\ref{geom-3},  particularly,  follows the new proof of the theorem\nby B.A. Dubrovin and S.P. Novikov  about rationality of the\nuniversal space of the Jacobians of hyperelliptic curves\n$V$ of genus  $g$ with the fixed branching point $e_{2g+2}=  \\infty$\n  \\cite{dn74}.\n\n\n\n  \\subsection{Hyperelliptic $  \\Phi$--function}\n In this section we construct the linear\ndifferential operators,  for which the hyperelliptic curve\n$V (y, x) $ is the spectral variety.\n\n  \\begin{definition}\n$  \\Phi$--function of the curve $V (y, x) $ with fixed\npoint $a$\n  \\begin{eqnarray*}\n&  \\Phi:  \\mathbb{ C}  \\times  \\mathrm{Jac} (V)   \\times V\n \\to   \\mathbb{ C}  \\\\\n&  \\Phi (u_0,   \\boldsymbol{\nu}; (y, x) ) =  \\dfrac{  \\sigma (  \\boldsymbol{\n  \\alpha}-  \\boldsymbol{ u}) }{  \\sigma (  \\boldsymbol{\n  \\alpha})   \\sigma (  \\boldsymbol{ u}) }  \\exp (-  \\frac{1}{2}y u_0+\n \\boldsymbol{\n  \\zeta}^T (  \\boldsymbol{\n  \\alpha})   \\boldsymbol{ u}) ,\n  \\end{eqnarray*}\nwhere $  \\boldsymbol{\n  \\zeta}^T (  \\boldsymbol{\n  \\alpha}) = (\n  \\zeta_1 (  \\boldsymbol{\n  \\alpha}) ,   \\ldots,\n  \\zeta_g (  \\boldsymbol{\n  \\alpha}) ) $ and $ (y, x)   \\in V$,\n$  \\boldsymbol{ u}$ and $  \\boldsymbol{\n  \\alpha}=  \\int_a^x  \\mathrm{d}  \\mathbf{ u}$ $  \\in   \\mathrm{Jac} (V) $.\n  \\end{definition}\n\n\nParticularly,  $  \\Phi (0,   \\boldsymbol{\nu}; (y, x) ) $ is the Baker function   (see   \\cite[page 421]{ba97} and\n  \\cite{kr77}) .\n\n  \\begin{theorem} The function $  \\Phi=  \\Phi (u_0,   \\boldsymbol{\nu}; (y, x) ) $ solves the Hill's equation\n  \\begin{equation}\n (  \\partial_g^2-2  \\wp_{gg})   \\Phi= (x+  \\frac{  \\lambda_{2g}}{4})   \\Phi,\n  \\label{Hill}  \\end{equation}\nwith respect to $u_g$,  for all $ (y, x)   \\in V$.\n  \\end{theorem}\n\n  \\begin{proof} From   \\eqref{stickelberger}\n  \\[\n  \\partial_g  \\Phi=  \\frac{y+  \\partial_g{  \\mathcal{ P}} (x;\n\\boldsymbol{ u}) } {2\n  \\mathcal{ P} (x;  \\boldsymbol{ u}) }   \\Phi,\n  \\]\nwhere\n$  \\mathcal{ P} (x;  \\boldsymbol{  u}) $ is given by   \\eqref{p},\nhence:\n  \\[\n  \\frac{  \\partial_g^2  \\Phi}{  \\Phi}=  \\frac{y^2- (  \\partial_g  \\mathcal{\nP} (x;  \\boldsymbol{  u}) ) ^2 +2  \\mathcal{ P} (x;  \\boldsymbol{\nu})   \\partial_g^2  \\mathcal{ P} (x;  \\boldsymbol{ u}) }{4  \\mathcal{\nP}^2 (x;  \\boldsymbol{  u}) }   \\]   and by   \\eqref{product3} and\n  \\eqref{wpgggi} we obtain the theorem.  \\end{proof}\n\nLet us introduce the vector $  \\boldsymbol{   \\Psi}= (  \\Phi,   \\Phi_g) $,\nwhere $  \\Phi_g$ stands for $  \\partial_g    \\Phi$. Then equation\n  \\eqref{Hill} may be written as\n  \\begin{equation}\n  \\partial_g  \\boldsymbol{   \\Psi}= L_g   \\boldsymbol{   \\Psi},\n  \\,   \\text{where}   \\quad L_g=   \\begin{pmatrix}\n0&1  \\\\\nx+2  \\wp_{gg}+  \\frac{  \\lambda_{2g}}{4}&0\n  \\end{pmatrix}.   \\label{matrix-Hill}   \\end{equation}\n\n\nIn regard of   \\eqref{matrix-Hill}  and   \\eqref{stickelberger},  it is\nnatural to introduce the family of $g+1$ operators,  presented by\n$2   \\times 2$ matrices,    \\[  \\{ L_0, L_1,   \\ldots, L_g  \\},    \\quad L_k=\n  \\left (   \\begin{array}{rr} V_k&U_k  \\\\W_k&-V_k   \\end{array}   \\right)\n   \\]\nand defined by the equalities\n  \\[\nL_k  \\boldsymbol{   \\Psi}={  \\partial_k}   \\boldsymbol{\n  \\Psi},   \\quad k  \\in 0,   \\ldots, g.\n  \\]\nThe theory developed in previous sections leads to the following\ndescription of this family of operators.\n  \\begin{prop} Entries of\nthe matrices $L_k$ are polynomials in $x$ and $2g$--periodic in\n$  \\boldsymbol{ u}$:    \\begin{gather} L_k=D_k L_0-  \\frac12\n  \\begin{pmatrix} 0&0  \\\\h_{g+2, k}&0   \\end{pmatrix},   \\notag  \\\\\n  \\intertext{with}\nU_0=  \\frac12  \\sum_{i=1}^{g+2}x^{i-1}h_{g+2, i},   \\quad\nV_0=-  \\frac14  \\sum_{i=1}^{g+2}x^{i-1}  \\partial_g h_{g+2, i},    \\\\\n  \\quad  \\text{and}  \\quad W_0=  \\frac14  \\sum_{i=1}^{g+2}x^{i-1}  \\det\n  \\begin{pmatrix} h_{g+1, i}&h_{g+2, g}  \\\\h_{g+2, i}&h_{g+2,\n  g+1}  \\notag \\end{pmatrix}.  \\label{entries} \\end{gather} And\nthe compatibility conditions \\[ [L_k, L_i]=  \\partial_k L_i -\n        \\partial_i L_k   \\] are satisfied.  \\end{prop} Here $D_k$ is\numbral derivative  (see page~  \\pageref{umbral_D}) .  Proof is\nstraightforward due to   \\eqref{stickelberger}, \\eqref{variables},\n  \\eqref{dg_H} and   \\eqref{wpggiwpggk}.\n\n  \\begin{theorem}  \\label{the-F} The function\n$  \\Phi=  \\Phi (u_0,   \\boldsymbol{ u}; (y, x) ) $ solves the system of\nequations   \\begin{gather}   \\left (  \\partial_k   \\partial_l\n-  \\gamma_{kl} (x,   \\boldsymbol{ u})   \\partial_g +\n  \\beta_{kl} (x,   \\boldsymbol{ u})\n  \\right)   \\Phi=  \\tfrac{1}{4}D_{k+l}  \\big (f (x)   \\big)   \\Phi\n  \\notag  \\\\ \\intertext{with polynomials in $x$} \\begin{aligned}\n  \\gamma_{kl} (x,   \\boldsymbol{ u}) &=\n  \\tfrac{1}{4}   \\left[\n  \\partial_k D_l +\n  \\partial_l D_k\n  \\right]  \\sum_{i=1}^{g+2}x^{i-1}h_{g+2, i}  \\quad  \\text{and}  \\\\\n  \\beta_{kl} (x,   \\boldsymbol{ u}) &=  \\tfrac{1}{8}   \\left[\n (  \\partial_g  \\partial_k +h_{g+2, k} )  D_l +\n (  \\partial_g  \\partial_l +h_{g+2, l} )  D_k\n  \\right]  \\sum_{i=1}^{g+2}x^{i-1}h_{g+2, i}  \\\\\n&-  \\frac14   \\sum  \\limits_{j=k+l+2}^{2g+2}x^{j- (k+l+2) }\n  \\left[ \\left (  \\sum  \\limits_{  \\nu=1}^{k+1} h_{  \\nu, j-  \\nu}\n  \\right) + \\left (  \\sum  \\limits_{  \\mu=1}^{l+1} h_{j-  \\mu,\n  \\mu}  \\right)    \\right] \\end{aligned}  \\notag \\end{gather} for\nall $k, l   \\in 0,   \\ldots,  g$ and arbitrary $ (y, x)    \\in V$.\n  \\end{theorem}\nHere $f (x) $ is as given in   \\eqref{curve} with\n$  \\lambda_{2g+2}=0$ and $  \\lambda_{2g+1}=4$.\n  \\begin{proof}\nConstruction of operators $L_k$ yields\n   \\[\n  \\Phi_{lk}=  \\tfrac12   \\left (\n  \\partial_l U_k+\n  \\partial_k U_l  \\right) {  \\Phi_g}+  \\left (V_lV_k +  \\tfrac12\n (   \\partial_l V_k+  \\partial_k\nV_l+U_k W_l+W_k U_l)   \\right)   \\Phi.\n  \\]\nTo prove the theorem we use   \\eqref{entries},  and it only remains\nto notice,  that   (cf. Lemma   \\ref{geom-2}) :\n  \\begin{eqnarray*}\n&&D_k (V_0) D_l (V_0) +  \\tfrac12 D_k (U_0) D_l (W_0)\n+  \\tfrac12 D_l (U_0) D_k (W_0) =  \\\\&&\n-  \\tfrac{1}{16}  \\left (  \\det\nH   \\left[{}_{g+1}^{g+1}{}_{g+2}^{g+2}  \\right]\n  \\right)\n (1, x,   \\ldots, x^{g+1-k})\nH   \\left[{}_{k}^{l}{}_{  \\ldots}^{  \\ldots}{}_{g+2}^{g+2}  \\right]\n (1, x,   \\ldots, x^{g+1-l}) ^T,\n  \\end{eqnarray*}\nhaving in mind that $h_{g+2, g+2}=0$ and $h_{g+2, g+1}=2$,  we obtain\nthe theorem due to properties of matrix $H$.\n  \\end{proof}\n\nConsider as an example the case of genus $2$.\n  \\begin{eqnarray*}\n&& (  \\partial_2^2-2  \\wp_{22})\n  \\Phi=  \\tfrac{1}{4} (4 x+  \\lambda_4)   \\Phi,   \\\\\n&& (  \\partial_2  \\partial_1+  \\tfrac{1}{2}  \\wp_{222}  \\partial_2\n-  \\wp_{22} (x+  \\wp_{22}+\n  \\tfrac{1}{4}  \\lambda_4)  +2  \\wp_{12})\n  \\Phi=  \\tfrac{1}{4} (4 x^2+  \\lambda_4 x +  \\lambda_3)   \\Phi,    \\\\\n&& (  \\partial_1^2+  \\wp_{122}  \\partial_2\n-2  \\wp_{12} (x+\n  \\wp_{22}+  \\tfrac{1}{4}  \\lambda_4) )\n  \\Phi=\n  \\tfrac{1}{4} (4 x^3+  \\lambda_4 x^2 +  \\lambda_3 x+  \\lambda_2)   \\Phi.\n   \\end{eqnarray*}\nAnd the $  \\Phi=  \\Phi (u_0, u_1, u_2; (y, x) ) $ of the curve $y^2=4\nx^5+  \\lambda_4 x^4 +  \\lambda_3 x^3+  \\lambda_2x^2+  \\lambda_1\nx+  \\lambda_0$ solves these equations for all $x$.\n\nThe most remarkable of the equations of Theorem   \\ref{the-F} is the\nbalance of powers of the polynomials $  \\gamma_{kl}$,  $  \\beta_{kl}$\nand of the ``spectral part'' --- the umbral derivative\n$D_{k+l} (f (x) ) $:\n  \\begin{align*}\n&  \\mathrm{deg}_x  \\gamma_{kl} (x,   \\boldsymbol{ u})   \\leqslant\ng-1-  \\mathrm{min} (k, l) ,    \\\\&  \\mathrm{deg}_x  \\beta_{kl} (x,   \\boldsymbol{\nu})   \\leqslant 2g- (k+l) ,   \\\\\n&  \\mathrm{deg}_x D_{k+l} (f (x) ) =2g+1- (k+l) .\n  \\end{align*}\n\n\n  \\subsection{Solution of KdV equations\n by Kleinian functions}\nThe KdV system is the infinite hierarchy of differential equations\n  \\[\nu_{t_{k}} =  \\mathcal{ X}_{k}[u],\n  \\]\nthe first two are\n  \\[u_{t_{1}} =u_x ,   \\quad  \\text{and}  \\quad u_{t_{2}} =\n-  \\tfrac12 (u_{xxx}-6u u_{x}) ,   \\]\nand the higher ones are defined by the relation\n  \\[\n  \\mathcal{ X}_{k+1}[u]=  \\mathcal{ R}  \\mathcal{ X}_{k}[u],\n  \\]\nwhere $  \\mathcal{ R}=-  \\frac12  \\partial_x^2+2u+u_x  \\partial_x^{-1}$ is the Lenard's\nrecursion operator.\n\nIdentifying time variables $ (t_1=x,  t_2,   \\ldots,  t_g)   \\to (u_{g},\nu_{g-1},   \\ldots, u_1) $ we have\n\n  \\begin{prop}\nThe function $u=2  \\wp_{gg} (  \\boldsymbol{ u}) $  is a $g$--gap\nsolution of the KdV system.\n  \\end{prop}\n\n  \\begin{proof}\nReally,  we have $u_x=  \\partial_g 2  \\wp_{gg}$  and\nby   \\eqref{wpgggi}\n  \\[u_{t_{2}}=  \\partial_{g-1} 2  \\wp_{gg}=\n-  \\wp_{ggggg}+12  \\wp_{gg}  \\wp_{ggg}.  \\]\nThe action of $  \\mathcal{ R}$\n  \\[   \\partial_{g-i-1}2  \\wp_{gg}=   \\left[-\n  \\partial_{g}^2+8  \\wp_{gg}   \\right]   \\wp_{gg, g-i}\n+4  \\wp_{g, g-i}  \\wp_{ggg}  \\]\nis verified by   \\eqref{wpgggi} and    \\eqref{wp3}.\n\nOn the $g$--th step of recursion the ``times''  $u_i$ are\nexhausted and the stationary equation \\[ \\mathcal{ X}_{g+1}[u]=0\n\\] appears.  A periodic solution of $g+1$ higher stationary\nequation is a $g$--gap potential  (see   \\cite{dmn76}) .\n\\end{proof}\n\n\\section*{Concluding remarks}\n\nThe Kleinian theory of hyperelliptic\nAbelian functions as, the authors hope, this paper shows is an\nimportant approach alternative to the generally adopted formalism\nbased directly on the multidimensional $\\theta$--functions in\nvarious branches of mathematical physics.  Still, a number of\nremarkable properties of the Kleinian functions were left beyond\nthe scope of our paper. We give some instructive examples for the\ncase of genus two ${\\boldsymbol u}=\\{u_1, u_2\\}$).\n\n\\begin{itemize} \\item the\naddition theorem \\[ \\frac{\\sigma({\\boldsymbol u}+{\\boldsymbol\nv})\\sigma({\\boldsymbol u}-{\\boldsymbol v})}{\\sigma^2({\\boldsymbol\nu})\\sigma^2({\\boldsymbol v})}= \\wp_{22}({\\boldsymbol\nu})\\wp_{12}({\\boldsymbol v})-\\wp_{12}({\\boldsymbol u})\\wp_{22}({\\boldsymbol\nv})+\\wp_{11}({\\boldsymbol v})-\\wp_{11}({\\boldsymbol u}) ,\n\\]\n\n\\item the equation, capable of being interpreted as the\nHirota bilinear relation:\n\\[\n\\left\\{\\frac13\\Delta\\Delta^T+\\Delta^T\\left(\\begin{array}{ccc}\n0&0&1\\\\0&-\\frac12&0\\\\1&0&0\\end{array}\\right)\\epsilon_{\\eta,\\eta}\n\\epsilon_{\\eta,\\eta}\\cdot\\epsilon_{\\eta,eta}^T-(\\xi-\\eta)^4\n\\epsilon_{\\eta,\\xi}\\epsilon_{\\eta,\\xi}^T\\right\\}\\sigma({\\boldsymbol u})\n\\sigma({\\boldsymbol u}')\\Big|_{{\\boldsymbol u}'={\\boldsymbol u}},\n\\]\nis identically $0$, where\n$\\Delta^T=(\\Delta_1^2,2\\Delta_1\\Delta_2,\\Delta_2^2)$ with\n$\\Delta_i=\\frac{\\partial}{\\partial u_i}-\\frac{\\partial}{\\partial\nu_i'}$ and also $\\epsilon^T_{\\xi,\\eta}=(1,\\eta+\\xi,\\eta\\xi)$.\nAfter evaluation the powers of parameters $\\eta$ and $\\xi$ are\nreplaced according to rules $\\eta^k,\\xi^k \\to\n\\lambda_k\\frac{k!(6-k)!}{6!}$ by the constants defining the curve.\n\n\\item  for the Kleinian $\\sigma$--functions the operation is\ndefined\n\\[ \\sigma(u_1,u_2)=\\mathrm{exp}\n\\left\\{\\frac{u_2}{u_1} \\sum_{k=1}^6  k\n\\lambda_k\\frac{\\partial}{\\partial \\lambda_{k-1}}\\right\\}\n\\sigma(u_1,0), \\]\nwhich resembles the function executed by vertex operators.\n\\end{itemize}\nWe give these formulas with reference to \\cite{ba07}.\n\n\nAnother interesting problem is the reduction of\nhyperelliptic $\\wp$--functions to lower genera. In the case of\ngenus two it, happens according to the Weierstrass theorem when the\nperiod matrix $\\tau$ can be transformed to the form (see e.g.\n \\cite{ba97,hu94})\n\\[\n\\tau=\\left(\\begin{array}{cc}\\tau_{11}&\\frac{1}{N}\\\\\n\\frac{1}{N}&\\tau_{22}\\end{array}\\right),\n\\]\nwhere so called {\\it Picard number} $N>1$ is a positive integer.\nThe associated Kummer surface turns in this case to {\\it\nPl\\\"ucker surface}. The reductions of the like were studied in\n\\cite{bbeim94} in order to single out elliptic potentials among\nthe finite gap ones. The problems of this kind were treated in\n\\cite{gw95, gw95a,gw95b}  by means of the spectral theory. We\nremark that the formalism of Kleinian functions extremely\nfacilitates the related calculations and makes the solution more\ndescriptive.\n\n\nThese and other problems of hyperelliptic abelian functions will\nbe discussed in our forthcoming publications.\n\nConcluding we emphasize, that the Kleinian construction of\nthe hyperelliptic Abelian functions  does not\nexclude the theta functional realization but complements it, and\nto the authors' experience the combination of the both approaches\nmakes the whole picture more complete and descriptive.\n\n\n\\section*{Acknowledgments}\nThe authors are grateful to S.P. Novikov for the attention and\nstimulating discussions; we are also grateful to I.M. Krichever,\nS.M. Natanson and A.P. Veselov for the valuable discussions.\nSpecial thanks to G. Thieme for the help in the collecting\nthe classical German mathematical literature.\n\nThe research described in this publication was supported in\npart by grants no. M3Z000  (VMB)   and no. U44000  (VZE)  from the\nInternational Science Foundation and also the INTAS grant\nno.  93-1324  (VZE and DVL),\nand  grant no.  94-01-01444 from Russian Foundation of Fundamental\nResearches.\n\n\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe main topic of this expository article is the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces. \n\nMore precisely, we consider a connected closed manifold $M$ and a smooth {\\em Tonelli Lagrangian} $L$ on the tangent bundle $TM$ of $M$: The Tonelli assumption means that $L$ is fiberwise uniformly convex and superlinear. It is a very natural assumption: For instance, it guarantees that the Legendre transform is well defined and produces a diffeomorphism between the tangent and the cotangent bundle of $M$, it allows to prove that every pair of points in $M$ is connected by a curve $\\gamma$ which minimizes the Lagrangian action \n\\[\n\\int_{t_0}^{t_1} L(\\gamma(t),\\gamma'(t)) \\, dt\n\\]\nand is a solution of the Euler-Lagrange equation associated to $L$, which is unique whenever the two points are sufficiently close to each other (see e.g. \\cite{bgh98} or \\cite{maz11}). Typical examples of Tonelli Lagrangians are {\\em electromagnetic Lagrangians}, that is functions of the form\n\\[\nL(x,v) = \\frac{1}{2} |v|_x^2 + \\theta(x)[v] - V(x), \\qquad \\forall (x,v)\\in TM,\n\\]\nwhere $|\\cdot|_x$ denotes the norm associated to a Riemannian metric on $M$ (the kinetic energy), $\\theta$ is a smooth one-form (the magnetic potential) and $V$ is a smooth function (the scalar potential) on $M$. \n\nThe Euler-Lagrange equations associated to $L$ induce a smooth flow on $TM$ which preserves the energy function $E:TM \\rightarrow {\\mathbb{R}}$,\n\\[\nE(x,v) := d_v L(x,v)[v] - L(x,v), \\qquad \\forall (x,v)\\in TM.\n\\]\nGiven a number $\\kappa\\in [\\min E,+\\infty)$, the problem under considerations is the existence of a periodic orbit on the energy level $E^{-1}(\\kappa)$.\n\nSuch a problem has been studied by several authors, for several classes of Tonelli Lagrangians and energy ranges, and by several techniques. For instance, in the case of Lagrangians of the form \n\\[\nL(x,v) = \\frac{1}{2} |v|_x^2 - V(x),\n\\]\none can use the Maupertuis-Jacobi metric as in \\cite{ben84} and reduce the problem to the existence of closed Riemannian geodesics either on $M$, if $\\kappa$ is larger than the maximum of $V$, or on the domain $\\{V\\leq \\kappa\\}\\subset M$, which is endowed with a metric which degenerates on the boundary $V^{-1}(\\kappa)$, if $\\kappa$ is smaller than the maximum of $V$. For a general Tonelli Lagrangian, the role of the maximum of $V$ is played by the {\\em Ma\\~{n}\\'e critical values}. More precisely, important values of the energy are the numbers\n\\[\n\\min E \\leq e_0(L) \\leq c_u(L) \\leq c_0(L),\n\\]\nwhere $e_0(L)$ is the maximal critical value of $E$, $c_u(L)$ is minus the infimum of the mean Lagrangian action\n\\[\n\\frac{1}{T} \\int_0^T L(\\gamma(t)\\, \\gamma'(t))\\, dt\n\\]\nover all contractible closed curves $\\gamma$, and  $c_0(L)$ is minus the infimum of the mean Lagrangian action over all null-homologous closed curves. In the case of electromagnetic Lagrangians, $e_0(L)=c_u(L)=c_0(L)$ when the magnetic potential $\\theta$ vanishes, but the first and the second values are in general distinct when $\\theta$ does not vanish ($c_u(L)$ and $c_0(L)$ can be distinct only when the fundamental group of $M$ is sufficiently non-abelian). The importance of $e_0(L)$ is clear, since it marks a change in the topology of $E^{-1}(\\kappa)$: If $\\kappa>e_0(L)$, then $E^{-1}(\\kappa)$ is diffeomorphic to the unit tangent bundle of $M$, if $\\kappa<e_0(L)$ then the projection of $E^{-1}(\\kappa)$ to $M$ is not surjective anymore. The {\\em lowest} Ma\\~{n}\\'e critical value $c_u(L)$ affects directly the behavior of the   \n{\\em free period Lagrangian action functional}\n\\[\n{\\mathbb{S}}_{\\kappa}(\\gamma) := \\int_0^T \\Bigl( L\\bigl(\\gamma(t),\\gamma'(t)\\bigr) + \\kappa \\Bigr) \\, dt, \\qquad \\gamma: {\\mathbb{R}}\/T{\\mathbb{Z}} \\rightarrow M.\n\\]\nThe critical points of this functional, whose domain is a suitable space of closed curves $\\gamma$ in $M$ of arbitrary period $T$, are exactly the closed orbits of energy $\\kappa$. The functional ${\\mathbb{S}}_{\\kappa}$ is bounded from below on every connected component of the free loop space whenever $\\kappa\\geq c_u(L)$, and it is unbounded from below on every such connected component when $\\kappa< c_u(L)$.\nThe {\\em strict} Ma\\~{n}\\'e critical value $c_0(L)$ is not directly related to the topology of ${\\mathbb{S}}_{\\kappa}$, but it has dynamical and geometric significance: For $\\kappa>c_0(L)$ the energy surface $E^{-1}(\\kappa)$ is of {\\em restricted contact type}, and the Euler-Lagrangian flow on it is conjugated, up to a time reparametrization, to a {\\em Finsler geodesic flow} on $M$, whereas both facts are in general false for $\\kappa\\leq c_0(L)$. Furthermore, the Ma\\~{n}\\'e critical values are related to compactness properties of the functional ${\\mathbb{S}}_{\\kappa}$, such as the {\\em Palais-Smale condition}. \n\nBy exploiting these facts, the free period action functional ${\\mathbb{S}}_{\\kappa}$ can be effectively used as a variational principle for our problem and allows to prove various results, which we summarize into the following theorem.\n\n\\begin{Thm*}\nLet $L$ be a Tonelli Lagrangian on the tangent bundle of the closed manifold $M$.\n\\begin{enumerate}\n\\item If $\\kappa>c_u(L)$ and $M$ is not simply connected, then the energy level $E^{-1}(\\kappa)$ has a ${\\mathbb{S}}_{\\kappa}$-minimizing periodic orbit in each non-trivial homotopy class of the free loop space of $M$.\n\\item If $\\kappa>c_u(L)$ and $M$ is simply connected, then the energy level $E^{-1}(\\kappa)$ has a periodic orbit with positive  action ${\\mathbb{S}}_{\\kappa}$.\n\\item For almost every $\\kappa\\in (\\min E,c_u(L))$  the energy level $E^{-1}(\\kappa)$ has a periodic orbit with positive  action ${\\mathbb{S}}_{\\kappa}$.\n\\item If the energy level $E^{-1}(\\kappa)$ is stable then $E^{-1}(\\kappa)$ has a periodic orbit.\n\\end{enumerate}\n\\end{Thm*}\n\nNotice that in (iii) only existence for {\\em almost every} energy level in the interval $(\\min E,c_u(L))$ (in the sense of Lebesgue measure) is stated: existence for {\\em all} energy levels in this range is still unknown, although no counterexamples have been found so far. This issue is related to the fact that the Palais-Smale condition does not hold anymore below $c_u(L)$. The {\\em stability condition} which is assumed in (iv) is a weaker form of the contact type condition.\n\nThe above theorem was first proved in this form by G.~Contreras \\cite{con06} (assuming contact type instead of stable in (iv)), building on previous geometric ideas of I.~A.~Taimanov \\cite{tai83,tai92b}. Contreras' long paper \\cite{con06} contains many other beautiful results, such as the study of the invariant probability measures which one obtains as limits of Palais-Smale sequences which do not converge in the free loop space. This article is meant to be a gentle introduction, including some technical simplifications, to the part of \\cite{con06} which concerns periodic orbits.\n\nUnlike in a typical survey article, we are more concerned with detailed proofs, which we try to make accessible to a large audience including students, than with a systematic overview of the literature, for which we refer to the beautiful survey of Taimanov \\cite{tai92b}, to \\cite{cmp04} and to the already cited \\cite{con06}. In particular, we start by proving well known abstract results, such as the mountain pass theorem, the general minimax principle, and the construction of the structure of infinite dimensional Hilbert manifold on the space of closed loops on $M$ of Sobolev class $W^{1,2}$ (Sections \\ref{mpsec} and \\ref{hmlsec}). In Section \\ref{sec3} we introduce the already mentioned free period action functional ${\\mathbb{S}}_{\\kappa}$, which plays a fundamental role in this article and gives it its title: Unlike in \\cite{con06}, we use it also to get existence of closed orbits for energies below $e_0(L)$. The Ma\\~{n}\\'e critical values which are relavant for this article are introduced in Section \\ref{mcv}, together with some of their characterizations and with the discussion of two geometric properties of an energy level, namely the contact type and the stability condition. The analysis of Palais-Smale sequences is carried out in Section \\ref{pss}, and in Section \\ref{pohe} we prove statements (i) and (ii) of the above theorem. The topology of the free period action functional for $\\kappa<c_u(L)$ is studied in  Section \\ref{tfpafle}, and in Section \\ref{pole} we finally prove statements (iii) and (iv), using {\\em Struwe's monotonicity argument}, together with a weaker version of (iii), using an alternative argument.\n\n\\medskip\n\n\\paragraph{\\bf Acknowledgments} This expository article is the outcome of two series of lectures that the author gave at two summer schools at the Korea Institute for Advanced Study of Seul in 2010 and at the Universit\\'e de Neuch\\^atel in 2011, respectively. I am grateful to Urs Frauenfelder and Felix Schlenk for organizing these two events, and to Gabriel Paternain, who was also a speaker at the first school, for many fruitful discussions. I would like to thank Jungsoo Kang, who participated to the first school, organized the material and typed a first version of the notes which eventually became this article. I would like to thank also Luca Asselle, who participated to the second school and suggested Lemma \\ref{PST0} below, which allows to avoid extra technicalities and the detailed analysis of Palais-Smale sequences with infinitesimal periods which was present in the previous notes. \n\n\\numberwithin{equation}{section}\n\n\\section{The minimax principle}\n\\label{mpsec}\n\n\\paragraph{\\bf The mountain pass theorem} Let $H$ be a real Hilbert space and let $f$ be a continuously differentiable real function on $H$. The symbol ${\\rm Crit} f$ denotes the set of critical points of $f$.\nWe assume that a certain open sublevel $\\{f<a\\}$ is\nnot connected, say $\\{f<a\\}=A\\cup B$, with $A$ and $B$ disjoint non-empty open sets. We may think of $A$ and $B$ as two valleys, and consider the set of paths going from one valley to the other one, that is the set\n$$\n\\Gamma:=\\{\\textrm{curves in $H$ with one end in $A$ and the other in $B$}\\}.\n$$\nWe can define the minimax value of $f$ on $\\Gamma$ as\n$$\nc:=\\inf_{\\gamma\\in\\Gamma}\\max_{x\\in\\gamma} f(x),\n$$\nand we notice that $a\\leq c < +\\infty$, because $\\Gamma$ is non empty and each of its elements intersects the set $H\\setminus (A\\cup B) = \\{f\\geq a\\}$.\nOne would expect this mountain pass level $c$ to be a critical value of $f$.\nThe next simple example shows that this is not always the case.\n\n\\begin{Ex}\nConsider the smooth function $f$ on ${\\mathbb{R}}^2$ defined by\n\\[\nf(x,y)=e^x-y^2.\n\\]\nThen $\\{f<0\\}$ has two connected components, $c=0$, but $f$ has no critical points. The problem here is that the critical point is pushed to infinity: Indeed, $f(-n,0)=e^{-n}$ converges to the mountain pass level $c=0$ and $df(-n,0)=e^{-n} dx$ tends to zero.\n\\end{Ex}\n\nThis example suggests the following definition.\n\n\\begin{Def}\nA sequence $(x_n)_{n\\in{\\mathbb{N}}}\\subset H$ is called a {\\em Palais-Smale sequence} at level $c$ ($(\\mathrm{PS})_c$ for short) if\n$$\n\\lim_{n\\to\\infty}f(x_n)=c\\quad\\mbox{and}\\quad \\lim_{n\\to\\infty}df(x_n)=0.\n$$\nThe function $f$ is said to satisfy $(\\mathrm{PS})_c$ if all $(\\mathrm{PS})_c$ sequences are compact. It is said to satisfy $(\\mathrm{PS})$ if it satisfies $(\\mathrm{PS})_c$ for every $c\\in {\\mathbb{R}}$.\n\\end{Def}\n\nNotice that limiting points of $(\\mathrm{PS})_c$ sequences are critical points at level $c$. We can now state the celebrated mountain pass theorem of Ambrosetti and Rabinowitz \\cite{ar73} in the following form:\n\n\\begin{Thm}[Mountain Pass Theorem]\n\\label{mp}\nLet $f\\in C^{1,1}(H)$ be such that  $\\{f<a\\}$ is not connected and let $c$ be defined as above. Then $f$ admits a $(\\mathrm{PS})_c$ sequence. In particular, if $f$ satisfies $(\\mathrm{PS})_c$, then $c$ is a critical value.\n\\end{Thm}\n\nHere $C^{1,1}$ denotes the set of functions whose differential is locally Lipschitz-continuous.\n\n\\begin{proof}\nBy contradiction, suppose that there exists $\\epsilon>0$ such that $||df|| \\geq\\epsilon$ on the set $\\{|f-c|\\leq\\epsilon\\}$. We denote by $\\nabla f$ the gradient of $f$ and we assume for sake of simplicity that the locally Lipschitz vector field $-\\nabla f$ is positively complete, meaning that its flow $\\phi$, that is the solution of\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{\\partial}{\\partial t}\\phi_t (u)=-\\nabla f\\bigl(\\phi_t(u)\\bigr),\\\\[1ex]\n&\\phi_0 (u)=u,\n\\end{aligned}\n\\;\\;\\right.\n\\]\nis defined for every $t\\geq 0$ and every $u\\in H$. This holds, for instance, if $\\nabla f$ is globally Lipschitz (in this case the flow of $-\\nabla f$ is defined on the whole ${\\mathbb{R}}\\times H$). See Remark \\ref{noncomp} below for a hint on how to remove this extra assumption. Notice that\n\\begin{equation}\n\\label{decr}\n\\frac{d}{dt} f\\bigl(\\phi_t(u)\\bigr) = df\\bigl( \\phi_t(u) \\bigr) \\bigl[-\\nabla f\\bigl(\\phi_t(u)\\bigr) \\bigr] = - \\bigl\\|df\\bigl(\\phi_t(u)\\bigr)\\bigr\\|^2,\n\\end{equation}\nso the function $t\\mapsto f(\\phi_t(u))$ is decreasing. If $|f(\\phi_t(u))-c|\\leq\\epsilon$ for all $t\\in[0,T]$, we have\n\\[\n2\\epsilon \\geq f(u)-f(\\phi_T(u)) =-\\int_0^T\\frac{d}{dt}f(\\phi_t(u))dt\n=\\int_0^T \\bigl\\|d f\\bigl(\\phi_t(u)\\bigr)\\bigr\\|^2dt\n\\geq\\epsilon^2T,\n\\]\nfrom which we conclude that $T\\leq 2\/\\epsilon$. Choose $\\gamma\\in\\Gamma$ such that $\\max_\\gamma f\\leq c+\\epsilon$ and set\n$$\n\\tilde\\gamma=\\phi_T(\\gamma), \\qquad\\textrm{for some } T>\\frac{2}{\\epsilon}.\n$$\nThe fact that $f$ decreases along the orbits of $\\phi$ implies that $\\tilde\\gamma$ belongs to $\\Gamma$. Since $f\\leq c+\\epsilon$ on $\\gamma$, any $x\\in\\gamma$ satisfies either (i) $|f(x)-c|\\leq\\epsilon$ or (ii) $f(x)<c-\\epsilon$. Let $x\\in \\gamma$.\nIf (i) holds, then $f(\\phi_T(x))<c-\\epsilon$ because $T>2\/\\epsilon$. If (ii) holds, then $f(\\phi_T(x))<c-\\epsilon$ because $f$ decreases along the orbits of $\\phi$. Therefore we conclude that $\\tilde\\gamma\\subset\\{f< c-\\epsilon\\}$, which contradicts the definition of $c$.\n\\end{proof}\n\n\\begin{Rmk}\n\\label{noncomp}\nIf the vector field $-\\nabla f$ is not positively complete, we can replace it by the complete one\n $-\\nabla f\/\\sqrt{||\\nabla f||^2+1}$. The above proof goes through with minor adjustments.\n\\end{Rmk}\n\n\\begin{Rmk}\nThe mountain pass theorem holds also for $f\\in C^{1,1}(\\mathcal{M})$ where $(\\mathcal{M},g)$ is a Hilbert manifold equipped with a complete Riemannian metric $g$.\nIn this case, $(x_n)_{n\\in{\\mathbb{N}}}\\subset\\mathcal{M}$ is said to be a $(\\mathrm{PS})_c$ sequence if $\\lim_{n\\to\\infty}f(x_n)=c$ and $\\lim_{n\\to\\infty}||df(x_n)||=0$, where $\\|\\cdot\\|$ denotes the dual norm induced by $g$. Notice that the (PS) condition and the completeness of $g$ are somehow antagonist requirements: One may always achieve the completeness of an arbitrary Riemannian metric $g$ by multiplying it by a positive function which diverges at infinity (such an operation reduces the set of the Cauchy sequences), while the (PS) condition could be achieved by multiplying $g$ by a positive function which is infinitesimal at infinity (since the dual norm is multiplied by the inverse of this function, this operation reduces the set of the (PS) sequences).\n\\end{Rmk}\n\n\\begin{Rmk}\nThe mountain pass theorem holds also if $f$ is just continuously differentiable. In this case, its negative gradient vector field is just continuous and may not induce a continuous flow. In order to prove the above theorem, one needs to construct a locally Lipschitz pseudo-gradient vector field for $f$, see for instance \\cite[Lemma 3.2]{str00}. The same construction allows to prove the mountain pass theorem for continuously differentiable functions on Banach spaces, or more generally on Banach manifolds.\n\\end{Rmk}\n\n\\begin{Rmk}\nWhen dealing with functions on manifolds, it is sometimes useful to have a formulation of the mountain pass theorem which does not involve the choice of a metric. Here is such a formulation. Assume that $f$ is a continuously differentiable function on a Hilbert manifold $\\mathcal{M}$ and that $V$ is a positively complete locally Lipschitz vector field such that $df[V]<0$ on $\\mathcal{M} \\setminus{\\rm Crit} f$. Then the mountain pass theorem holds, provided that we define $(x_n)_{n\\in{\\mathbb{N}}}\\subset \\mathcal{M}$ to be a $(\\mathrm{PS})_c$ sequence if $f(x_n)$ tends to c and $df(x_n)[V(x_n)]$ is infinitesimal. Now the antagonism is between this form of the (PS) condition and the positive completeness of $V$.\n\\end{Rmk}\n\n\n\\paragraph{\\bf The general minimax principle} In the proof of Theorem \\ref{mp} we have not used the fact that $\\Gamma$ is a set of curves, but rather that $\\Gamma$ is  positively invariant with respect to the negative gradient flow $\\phi$ of $f$, meaning that  $\\phi_t(\\gamma)\\in\\Gamma$ for all $\\gamma\\in\\Gamma$ and $t\\geq 0$. Here $\\phi$ is either the flow of $-\\nabla f$, when this vector field is positively complete, or the flow of some conformally equivalent positively complete vector field, such as\n$-\\nabla f\/\\sqrt{\\|\\nabla f\\|^2+1}$, in the general case.\nThis simple observation leads to the following powerful generalization of the mountain pass theorem.\n\n\\begin{Thm}[\\bf General Minimax Principle]\n\\label{thm:finite c induces PS}\nLet $f$ be a $C^{1,1}$ function on the complete Riemannian Hilbert manifold $(\\mathcal{M},g)$ and let $\\Gamma$ be a set of subsets of $\\mathcal{M}$ which is positively invariant with respect to the negative gradient flow of $f$. If the number\n\\[\nc= \\inf_{\\gamma\\in\\Gamma}\\sup_\\gamma f\n\\]\nis finite, then $f$ admits a $(\\mathrm{PS})_c$ sequence. In particular, if $f$ satisfies $(\\mathrm{PS})_c$, then $c$ is a critical value.\n\\end{Thm}\n\nThe proof is a straightforward modification of the proof of Theorem \\ref{mp}.\n\n\\begin{Ex}\nLet $f\\in C^{1,1}(H)$, where $H$ is a Hilbert space. If $\\pi_k(\\{f<a\\})\\ne 0$ for some $k\\geq 0$ and $f$ satisfies $(\\mathrm{PS})$,  then $f$ has a critical point. Indeed, we can consider the set\n\\[\n\\begin{split}\n\\Gamma := \\bigl\\{ z(\\overline{B}^{k+1}) \\; \\Big| \\; & z: (\\overline{B}^{k+1},\\partial B^{k+1}) \\rightarrow (H,\\{f<a\\}) \\mbox{ continuous map such that }\\\\  & [z|_{\\partial B^{k+1}}] \\neq 0 \\mbox{ in } \\pi_k(\\{f<a\\})\\Bigr\\},\n\\end{split}\n\\]\nwhere $B^{k+1}$ denotes the unit open ball of dimension $k+1$. By applying Theorem \\ref{thm:finite c induces PS} with such a $\\Gamma$ we get the existence of a critical point at level $c\\geq a$. The case $k=0$ is precisely the Mountain Pass Theorem \\ref{mp}.\n\\end{Ex}\n\n\\begin{Rmk}\n\\label{mini}\nIf $\\Gamma$ is the class of all one-point sets in $\\mathcal{M}$, then $c$ is the infimum of $f$. Therefore, the general minimax principle has as a particular case the following existing result for minimizers: Assume that $f\\in C^{1,1}(\\mathcal{M})$ is bounded from below, has complete sublevels and satisfies $(\\mathrm{PS})_c$ at the level $c=\\inf f$; then $f$ has a minimizer.\n\\end{Rmk}\n\n\\begin{Rmk}\n\\label{trunc}\nIt is sometimes useful to replace the negative gradient flow by a flow which fixes a certain sublevel of $f$. Let $\\rho:{\\mathbb{R}}\\longrightarrow{\\mathbb{R}}^+$ be a smooth bounded function such that $\\rho=0$ on $(-\\infty,b]$ and $\\rho>0$ on $(b,+\\infty)$. Then we consider the vector field $V=-\\rho(f)\\cdot\\nabla f$ (or $V= - \\rho(f) \\nabla f\/\\sqrt{\\|\\nabla f\\|^2+1}$ in the non-positively complete case) and denote its flow by $\\phi$. It is a negative gradient flow truncated below level $b$: The function $t\\mapsto f(\\phi_t(u))$ is constant if $u\\in {\\rm Crit} f \\cup \\{f\\leq b\\}$ and it is strictly decreasing otherwise.\nIf $\\Gamma$ is positively invariant with respect to this negative gradient flow truncated below level $b$ and the minimax value $c$ is strictly larger than $b$, then $f$ has a $(\\mathrm{PS})_c$ sequence.\n\\end{Rmk}\n\n\\section{A Hilbert manifold of loops}\n\\label{hmlsec}\n\nLet $(M,g)$ be a closed Riemannian manifold of dimension $n$ and consider the Sobolev space of loops\n$$\nW^{1,2}({\\mathbb{T}},M):=\\Big\\{x:{\\mathbb{T}}\\longrightarrow M\\,\\Big|\\,x\\textrm{ is absolutely continuous and } \\int_{\\mathbb{T}}|x'(s)|^2_{x(s)}ds<\\infty\\Big\\},\n$$\nwhere ${\\mathbb{T}}:={\\mathbb{R}}\/{\\mathbb{Z}}$ and $|\\cdot|_{\\cdot}$ denotes the norm induced by $g$. This set of loops is clearly independent from the choice of the Riemannian metric $g$.\n\n\\medskip\n\n\\paragraph{\\bf The smooth structure of $\\mathbf{W^{1,2}(T,M)}$} Let us recall the construction of the smooth Hilbert manifold structure on $W^{1,2}({\\mathbb{T}},M)$.\nFix $x_0\\in C^\\infty({\\mathbb{T}},M)$. Assume for simplicity that $x_0$ preserves the orientation, so that $x_0^*(TM)$ has a trivialization\n$$\n\\Phi: {\\mathbb{T}}\\times{\\mathbb{R}}^n\\longrightarrow x_0^*(TM).\n$$\nLet $B_r$ be the open ball of radius $r$ about $0$ in ${\\mathbb{R}}^n$. Consider a smooth map\n\\[\n\\varphi:{\\mathbb{T}}\\times B_r \\longrightarrow M,\n\\]\nsuch that $\\varphi(t,0)=x_0(t)$ and $\\varphi(t,\\cdot)$ is a diffeomorphism onto an open subset in $M$, for every $t\\in {\\mathbb{T}}$. For instance, the map\n\\[\n\\varphi(t,\\xi)= \\exp_{x_0(t)}\\big(\\Phi(t,\\xi)\\big),\n\\]\nsatisfies the above requirements if $r$ is small enough.\n\nThe map $\\varphi$ induces the following parameterization:\n\\begin{equation}\n\\label{eq:chart of the loop space}\n\\varphi_*: W^{1,2}({\\mathbb{T}},B_r)\\longrightarrow W^{1,2}({\\mathbb{T}},M), \\quad\n\\zeta\\mapsto\\varphi\\big(\\cdot,\\zeta(\\cdot)\\big),\n\\end{equation}\nwhere $W^{1,2}({\\mathbb{T}},B_r)$ denotes the open subset of the Hilbert space $W^{1,2}({\\mathbb{T}},{\\mathbb{R}}^n)$ which consists of loops taking values into $B_r$. The collection of all these parameterizations, for every $x_0\\in C^{\\infty}({\\mathbb{T}},M)$ and every $\\varphi$ as above, defines a smooth atlas for $W^{1,2}({\\mathbb{T}},M)$, which is then a smooth manifold modeled on the Hilbert space $W^{1,2}({\\mathbb{T}},{\\mathbb{R}}^n)$. Indeed, the smoothness of the transition maps is an immediate consequence of the chain rule.\nIt is worth noticing that the image of the parameterization $\\varphi_*$ is $C^0$-open.\n\n\\begin{Rmk}\nIf $x_0$ is not orientation preserving, the natural model for the connected component of $W^{1,2}({\\mathbb{T}},M)$ which contains $x_0$ is the space of $W^{1,2}$ sections of the vector bundle $x_0^*(TM)$. Alternatively, one can define a manifold structure on $W^{1,2}([0,1],M)$ without encountering topological problems, and then see $W^{1,2}({\\mathbb{T}},M)$ as the inverse image of the diagonal of $M \\times M$ by the smooth submersion \n\\[\nW^{1,2}([0,1],M) \\rightarrow M \\times M, \\qquad x \\mapsto (x(0),x(1)).\n\\]\n\\end{Rmk}\n\nThe tangent space of $W^{1,2}({\\mathbb{T}},M)$ at $x$ is naturally identified with the space of $W^{1,2}$ sections of $x^*(TM)$. Therefore, we can define a Riemannian metric on $W^{1,2}({\\mathbb{T}},M)$ by setting\n\\begin{equation}\n\\label{metr}\n\\langle \\xi,\\eta\\rangle_x:=\\int_{{\\mathbb{T}}} \\Bigl( g(\\xi,\\eta)+g(\\nabla_t\\xi,\\nabla_t\\eta) \\Bigr) \\,dt, \\quad \\forall \\xi,\\eta\\in T_x W^{1,2}({\\mathbb{T}},M),\n\\end{equation}\nwhere $\\nabla_t$ denotes the Levi-Civita covariant derivative along $x$. The distance induced by this Riemannian metric is compatible with the topology of $W^{1,2}({\\mathbb{T}},M)$.\nThe fact that $M$ is compact implies that this metric on $W^{1,2}({\\mathbb{T}},M)$ is complete (more generally, this metric is complete whenever $g$ is complete).\n\nThe gradient of functionals on $W^{1,2}({\\mathbb{T}},M)$ is the one which is associated to such a Riemannian metric.\n\n\\begin{Rmk}\nIf $\\varphi$ is the restriction of a smooth map $B_{r'}\\times {\\mathbb{T}}\\rightarrow M$ with the same properties, for some $r'>r$, then the parameterization $\\varphi_*$ is bi-Lipschitz.\n\\end{Rmk}\n\nSee e.g. \\cite{kli82} for more details on the Hilbert manifold structure of $W^{1,2}({\\mathbb{T}},M)$.\n\n\\medskip\n\n\\paragraph{\\bf The homotopy type of $\\mathbf{W^{1,2}(T,M)}$} The inclusions\n$$\nC^\\infty({\\mathbb{T}},M)\\hookrightarrow W^{1,2}({\\mathbb{T}},M)\\hookrightarrow C({\\mathbb{T}},M)\n$$\nare dense homotopy equivalences. These facts can be proved by embedding $M$ into a Euclidean space ${\\mathbb{R}}^N$, by regularizing the loops $x:{\\mathbb{T}}\\rightarrow M\\subset {\\mathbb{R}}^N$ by convolution, and by projecting the regularized loop back to $M$ using the tubular neighborhood theorem. In particular, the connected components of $W^{1,2}({\\mathbb{T}},M)$ are in one-to-one correspondence with the conjugacy classes of\n$\\pi_1(M)$. See, e.g., \\cite[Chapter 10]{lee03} for more details.\n\n\\section{The free period action functional}\n\\label{sec3}\n\n\\paragraph{\\bf Tonelli Lagrangians} Let $M$ be a connected closed manifold.\nA function $L\\in C^\\infty(TM)$ is called a {\\em Tonelli Lagrangian} if:\n\\begin{enumerate}\n\\item $L$ is fiberwise uniformly convex, i.e.\\ $d_{vv}L(x,v)>0$ for every $(x,v)\\in TM$, where $d_{vv}L$ denotes the fiberwise second differential of $L$;\n\\item $L$ has superlinear growth on each fiber, i.e.\n\\[\n\\lim_{|v|\\to +\\infty}\\frac{L(x,v)}{|v|_x}=+\\infty.\n\\]\n\\end{enumerate}\nThe main example of Tonelli Lagrangians is given by the {\\em electromagnetic Lagrangians}, that is functions of the form\n\\begin{equation}\n\\label{elmag}\nL(x,v)=\\frac{1}{2}|v|^2_x+\\theta(x)[v]-V(x),\n\\end{equation}\nwhere $|\\cdot|_x$ denotes the norm associated to a Riemannian metric (the kinetic energy), $\\theta$ is a smooth one-form (the magnetic potential) and $V$ is a smooth function (the scalar potential) on $M$. We shall omit the subscript $x$ in $|\\cdot|_x$ when the point $x$ is clear from the context.\nThe Tonelli assumptions imply that the Euler-Lagrange equation, which in local coordinates can be written as\n\\begin{equation}\n\\label{EL}\n\\frac{d}{dt}\\bigl( \\partial_v L (\\gamma(t),\\gamma'(t)) \\bigr)= \\partial_x L (\\gamma(t),\\gamma'(t)),\n\\end{equation}\nis well-posed and defines a smooth flow on $TM$. This flow preserves the energy\n\\[\nE:TM\\rightarrow {\\mathbb{R}}, \\quad\nE(x,v):=d_v L(x,v)[v]-L(x,v),\n\\]\nwhere $d_v$ denotes the fiberwise differential.\nWhen $L$ has the form (\\ref{elmag}), then\n\\begin{equation}\n\\label{ene}\nE(x,v) = \\frac{1}{2} |v|^2 + V(x).\n\\end{equation}\nIn general, the energy function of a Tonelli Lagrangian satisfies the following properties:\n\\begin{itemize}\n\\item[(i)] $E$ is fiberwise uniformly convex and superlinear.\n\\item[(ii)] For any $x\\in M$, the restriction of $E$ to $T_x M$ achieves its minimum at $v=0$.\n\\item[(iii)] The point $(\\bar x,0)$ is singular for the Euler-Lagrange flow if and only if $(\\bar x,0)$ is a critical point of $E$.\n\\end{itemize}\n\nWe are interested in proving the existence of periodic orbits on a given energy level $E^{-1}(\\kappa)$. Since such an energy level is compact, up to the modification of $L$ far away from it, we may assume that the Tonelli Lagrangian $L(x,v)$ is electromagnetic for $|v|$ large. In particular, we have the inequalities\n\\begin{eqnarray}\n\\label{boundsonL}\nL(x,v) \\geq L_0 |v|^2 - L_1, \\qquad & \\forall (x,v)\\in TM,\\\\\n\\label{bd2L}\nd^2_{vv} L(x,v)[u,u] \\geq 2 L_0 |u|^2, \\qquad &\\forall (x,v)\\in TM, \\; u\\in T_x M,\n\\end{eqnarray}\nfor some numbers $L_0>0$ and $L_1\\in {\\mathbb{R}}$. Moreover, $E$ has the form (\\ref{ene}) for $|v|$ large.\n\n\\medskip\n\n\\paragraph{\\bf The free period action functional} We would like to study the Lagrangian action on the space of closed curves of arbitrary period. The latter space can be given a manifold structure by reparametrizing each curve on ${\\mathbb{T}}$ and by keeping track of its period as a second variable:\nLet $\\gamma :{\\mathbb{R}}\/{T{\\mathbb{Z}}}\\longrightarrow M$ be an absolutely continuous $T$-periodic curve and define $x:{\\mathbb{T}} \\rightarrow M$ as $x(s):=\\gamma(sT)$. The closed curve $\\gamma$ is identified with the pair $(x,T)$.\nThe action of $\\gamma$ on the time interval $[0,T]$ is the number\n\\[\n\\int_0^TL\\bigl(\\gamma(t),\\gamma'(t)\\bigr)\\, dt= T\\int_{{\\mathbb{T}}} L\\bigl(x(s),x'(s)\/{T}\\bigr)\\, ds.\n\\]\nFix a real number $\\kappa$, the value of the energy for which we would like to find periodic solutions. Consider the {\\em free period action functional} corresponding to the energy $\\kappa$\n\\[\n{\\mathbb{S}}_{\\kappa}(\\gamma) =\n{\\mathbb{S}}_{\\kappa} (x,T) := T\\int_{{\\mathbb{T}}} \\Bigl( L\\bigl(x(s),x'(s)\/T\\bigr) + \\kappa \\Bigr)\\,ds = \\int_0^T \\Bigl( L\\bigl( \\gamma(t),\\gamma'(t)\\bigr)+\\kappa\\Bigr)\\, dt.\n\\]\nThe fact that $L$ is electromagnetic outside a compact subset of $TM$ implies that ${\\mathbb{S}}_{\\kappa}(x,T)$ is well-defined when $x\\in W^{1,2}({\\mathbb{T}},M)$. Therefore, we obtain a functional\n\\[\n{\\mathbb{S}}_{\\kappa} : W^{1,2}({\\mathbb{T}},M) \\times (0,+\\infty) \\rightarrow {\\mathbb{R}}.\n\\]\nThe Hilbert manifold $W^{1,2}({\\mathbb{T}},M) \\times (0,+\\infty)$ is denoted by $\\mathcal{M}$.\n\n\\begin{Lemma}(Regularity properties of ${\\mathbb{S}}_{\\kappa}$)\n\\begin{itemize}\n\\item[(i)] ${\\mathbb{S}}_{\\kappa}$ is in $C^{1,1}(\\mathcal{M})$ and is twice Gateaux differentiable at every point.\n\\item[(ii)] ${\\mathbb{S}}_{\\kappa}$ is twice Fr\\'ech\\'{e}t differentiable at every point if and only if $L$ is electromagnetic on the whole $TM$. In this case, ${\\mathbb{S}}_{\\kappa}$ is actually smooth on $\\mathcal{M}$.\n\\end{itemize}\n\\end{Lemma}\n\nSee e.g.\\ \\cite{as09b} for a detailed proof.\n\nIf $d_x$ denotes the horizontal differential with respect to some horizontal-vertical splitting of $TTM$, the differential of ${\\mathbb{S}}_{\\kappa}$  with respect to the first variable at some $(x,T)\\in \\mathcal{M}$ has the form\n\\begin{equation}\n\\label{diff1}\n\\begin{split}\nd{\\mathbb{S}}_{\\kappa} (x,T) \\bigl[(\\xi,0)\\bigr] & = T \\int_0^1 \\Bigl( d_x L\\bigl( x, x'\/T \\bigr) [ \\xi ] + d_v L\\bigl( x, x'\/T \\bigr) \\bigl[ \\nabla_s \\xi\/T \\bigr] \\Bigr)\\, ds \\\\ & = \\int_0^T \\Bigl( d_x L \\bigl(\\gamma,\\gamma'\\bigr) [\\zeta] +   d_v L \\bigl(\\gamma,\\gamma'\\bigr) [\\nabla_t \\zeta] \\Bigr)\\, dt,\n\\end{split}\\end{equation}\nwhere $\\xi\\in T_x W^{1,2}({\\mathbb{T}},M)$, $\\gamma(t)= x(t\/T)$ and $\\zeta(t):=\\xi(t\/T)$. \nLet $(x,T)$ be a critical point of ${\\mathbb{S}}_{\\kappa}$. The above formula and an integration by parts imply that $\\gamma$ is a $T$-periodic solution of (\\ref{EL}). Moreover\n\\begin{equation}\n\\label{eq:diff A w.r.t. T}\n\\begin{split}\n\\frac{\\partial {\\mathbb{S}}_{\\kappa}}{\\partial T}  (x,T) &= \\int_{{\\mathbb{T}}} \\Bigl( L\\bigl( x(s),x'(s)\/T\\bigr) + \\kappa + T \\, d_v L\\bigl( x(s),x'(s)\/T\\bigr) \\bigl[-x'(s)\/T^2\\bigr] \\Bigr)\\, ds \\\\ &= \\int_{{\\mathbb{T}}} \\Bigl( \\kappa - E\\bigl( x(s),x'(s)\/T\\bigr) \\Bigr)\\, ds = \\frac{1}{T}\\int_0^T \\Bigl( \\kappa - E\\bigl( \\gamma(t),\\gamma'(t)\\bigr) \\Bigr) \\, dt.\n\\end{split}\n\\end{equation}\nTogether with the fact that $E$ is constant along the orbits of the Euler-Lagrange flow, the above identity shows that the $T$-periodic orbit $\\gamma$ belongs to the energy levek $E^{-1}(\\kappa)$. We conclude that $(x,T)$ is a critical point of ${\\mathbb{S}}_{\\kappa}$ on $\\mathcal{M}$ if and only if $\\gamma(t):= x(t\/T)$ is a $T$-periodic orbit of energy $\\kappa$ ($T$ is not necessarily the minimal period).\n\n\\medskip\n\n\\paragraph{\\bf Behavior of ${\\mathbb{S}}_{\\kappa}$ for $\\mathbf{T\\to 0}$} The Hilbert manifold $\\mathcal{M}=W^{1,2}({\\mathbb{T}},M)\\times (0,+\\infty)$ is endowed with the product Riemannian structure of (\\ref{metr}) and the Euclidean metric of $(0,+\\infty) \\subset {\\mathbb{R}}$. As such, it is not complete, the non-converging Cauchy sequences being the sequences $(x_h,T_h)$ with $x_h \\rightarrow x\\in W^{1,2}({\\mathbb{T}},M)$ and $T_h \\rightarrow 0$. Therefore, we need to understand the behavior of ${\\mathbb{S}}_{\\kappa}$ on such sequences.\n\nWe decompose $\\mathcal{M}$ as $\\mathcal{M} = \\mathcal{M}^{\\mathrm{contr}} \\sqcup \\mathcal{M}^{\\mathrm{noncontr}}$, where $\\mathcal{M}^{\\mathrm{contr}}$ denotes the connected component consisting of contractible loops.\n\n\\begin{Lemma}\n\\label{Lem:3}\n\\begin{enumerate}\n\\item On $\\mathcal{M}^\\mathrm{noncontr}$ the sublevels $\\{{\\mathbb{S}}_{\\kappa}\\leq c\\}$ are complete. More precisely, if $(x_h,T_h)\\in \\mathcal{M}^\\mathrm{noncontr}$ and $T_h \\rightarrow 0$, then ${\\mathbb{S}}_{\\kappa}(x_h,T_h)\\rightarrow +\\infty$.\n\\item If $(x_h,T_h)\\in\\mathcal{M}^\\mathrm{contr}$ and $T_h\\to 0$, then $\\liminf_h{\\mathbb{S}}_{\\kappa}(x_h,T_h)\\geq0$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nBy (\\ref{boundsonL}), we have the chain of inequalities\n\\begin{equation}\n\\label{lowbdonA}\n\\begin{split}\n{\\mathbb{S}}_{\\kappa} (x,T) &= T \\int_{{\\mathbb{T}}} \\Bigl( L\\bigl(x,x'\/T\\bigr) + \\kappa \\Big)\\, ds \\geq T \\int_{{\\mathbb{T}}} \\Bigl( L_0 \\frac{|x'|^2}{T^2} -L_1 + \\kappa \\Bigr)\\, ds\n\\\\ &= \\frac{L_0}{T} \\int_{{\\mathbb{T}}} |x'|^2\\, ds - (L_1-\\kappa) T\n\\geq \\frac{L_0}{T} \\ell(x)^2 - (L_1 - \\kappa)T,\n\\end{split} \\end{equation}\nwhere $\\ell(x)$ denotes the length of the loop $x$. The length of the non-contractible loops in $M$ is bounded away from zero. Therefore, the estimate (\\ref{lowbdonA}) implies statement (i). Statement (ii) is also an immediate consequence of (\\ref{lowbdonA}).\n\\end{proof}\n\nSince $\\mathcal{M}$ is not complete, we cannot expect the vector field $-\\nabla {\\mathbb{S}}_{\\kappa}$ to be positively complete. However, the only sources of non-completeness is the second component approaching zero. The next result says that this may happen only at level zero:\n\n\\begin{Lemma}\\label{Lem:4}\nLet $(x,T):[0,\\sigma^*)\\longrightarrow\\mathcal{M}^\\mathrm{contr}$, $0<\\sigma^*<\\infty$, be a flow line of $-\\nabla {\\mathbb{S}}_{\\kappa}$ such that\n\\[\n\\liminf_{\\sigma\\to\\sigma^*}T(\\sigma)=0.\n\\]\nThen\n\\[\n\\lim_{\\sigma\\to\\sigma^*}{\\mathbb{S}}_{\\kappa} \\big(x(\\sigma),T(\\sigma)\\big)=0.\n\\]\n\\end{Lemma}\n\n\\begin{proof}\nSince both $E$ and $L$ are quadratic in $v$ for $|v|$ large, we have the estimate\n\\[\nE(x,v) \\geq C_0 \\, L(x,v) - C_1,\n\\]\nfor some $C_0>0$ and $C_1\\in {\\mathbb{R}}$. From (\\ref{eq:diff A w.r.t. T}) we obtain the inequality\n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial {\\mathbb{S}}_{\\kappa}}{\\partial T} ( x,T) &= \\frac{1}{T} \\int_0^T \\bigl( \\kappa - E(\\gamma,\\gamma') \\bigr)\\, dt\n\\leq \\frac{1}{T} \\int_0^T \\bigl( \\kappa - C_0 \\,L(\\gamma,\\gamma') + C_1 \\bigr)\\, dt \\\\\n&= \\kappa + C_1 - \\frac{C_0}{T} \\int_0^T \\bigl( L(\\gamma,\\gamma')+\\kappa \\bigr)\\, dt + C_0 \\kappa = (C_0+1) \\kappa + C_1 - \\frac{C_0}{T} {\\mathbb{S}}_{\\kappa} (x,T),\n\\end{split} \\end{equation*}\nwhich can be rewritten as\n\\begin{equation}\n\\label{boundsondAdt}\n{\\mathbb{S}}_{\\kappa} (x,T) \\leq \\frac{T}{C_0} \\Bigl( C - \\frac{\\partial {\\mathbb{S}}_{\\kappa}}{\\partial T}  (x,T) \\Bigr),\n\\end{equation}\nfor a suitable constant $C$.\nBy the assumption, there exists an increasing sequence $(\\sigma_h)$ which converges to $\\sigma^*$ and satisfies $T'(\\sigma_h)\\leq 0$ and\n$T(\\sigma_h)\\rightarrow 0$. Since $\\sigma\\mapsto (x(\\sigma),T(\\sigma))$ is a flow line of $-\\nabla {\\mathbb{S}}_{\\kappa}$,\n\\[\n0 \\geq T'(\\sigma_h) = - \\frac{\\partial {\\mathbb{S}}_{\\kappa}}{\\partial T} \\bigl(x(\\sigma_h),T(\\sigma_h) \\bigr),\n\\]\nand by (\\ref{boundsondAdt}) we have\n\\[\n{\\mathbb{S}}_{\\kappa} \\bigl(x(\\sigma_h),T(\\sigma_h) \\bigr) \\leq \\frac{T(\\sigma_h)}{C_0} \\Bigl( C - \\frac{\\partial {\\mathbb{S}}_{\\kappa}}{\\partial T}  \\bigl(x(\\sigma_h),T(\\sigma_h) \\bigr) \\Bigr) \\leq \\frac{C}{C_0} T(\\sigma_h).\n\\]\nSince $T(\\sigma_h)$ is infinitesimal, we obtain\n\\[\n\\limsup_{h\\rightarrow \\infty} {\\mathbb{S}}_{k} \\bigl(x(\\sigma_h),T(\\sigma_h) \\bigr) \\leq 0.\n\\]\nTogether with statement (ii) of Lemma \\ref{Lem:3} and the monotonicity of the function $\\sigma \\longmapsto {\\mathbb{S}}_{\\kappa}(x(\\sigma),T(\\sigma))$,\nthis concludes the proof.\n\\end{proof}\n\n\\section{Ma\\~n{\\'e} critical values, contact type and stability conditions}\n\\label{mcv}\n\n\\paragraph{\\bf The critical values} The following numbers should be interpreted as energy levels and mark important dynamical and geometric changes for the Euler-Lagrange flow induced by the Tonelli Lagrangian $L$:\n\\begin{equation*}\n\\begin{split}\nc_0(L)&:=\\inf\\{\\kappa\\in{\\mathbb{R}} \\, | \\,{\\mathbb{S}}_{\\kappa}(x,T)\\geq 0 \\; \\forall (x,T)\\in \\mathcal{M} \\mbox{ with }x\\mbox{ homologous to zero} \\} \\\\\n&= - \\inf \\Bigl\\{ \\frac{1}{T} \\int_0^T L(\\gamma(t),\\gamma'(t)) \\, dt \\, \\Big| \\, \\gamma\\in C^{\\infty}({\\mathbb{R}}\/T{\\mathbb{Z}},M) \\mbox{ homologous to zero, } T>0\\Bigr\\}, \\\\\nc_u(L)&:=\\inf\\{\\kappa\\in{\\mathbb{R}} \\, | \\,{\\mathbb{S}}_{\\kappa}(x,T)\\geq 0 \\; \\forall (x,T)\\in \\mathcal{M} \\mbox{ with }x\\mbox{ contractible} \\}\\\\\n&= - \\inf \\Bigl\\{ \\frac{1}{T} \\int_0^T L(\\gamma(t),\\gamma'(t)) \\, dt \\, \\Big| \\, \\gamma\\in C^{\\infty}({\\mathbb{R}}\/T{\\mathbb{Z}},M) \\mbox{ contractible, } T>0 \\Bigr\\}, \\\\\ne_0(L)&:=\\max_{x\\in M} E(x,0) = \\max \\set{E(x,v)}{(x,v) \\in {\\rm Crit} E}.\n\\end{split} \\end{equation*}\nThe number $c_0(L)$ is known as the {\\em strict Ma\\~n\\'e critical value}, while $c_u(L)$ is the {\\em lowest Ma\\~{n}\\'e critical value} (see \\cite{man97}). When the fundamental group of $M$ is rich, there are other Ma\\~n\\'e critical values, which are associated to the different coverings to $M$, but the above ones are those which are more relevant for the question of existence of periodic orbits.\nIt is easy to see that\n\\[\n\\min E\\leq e_0(L)\\leq c_u(L)\\leq c_0(L).\n\\]\nWhen $L$ has the form \\eqref{elmag}, $\\min E$ is the minimum of the scalar potential $V$, while $e_0(L)$ is its maximum. When the magnetic potential $\\theta$ vanishes, the identities $e_0(L)=c_u(L)=c_0(L)$ hold, but in general $e_0(L)$ is strictly lower than the other two numbers. The values $c_u(L)$ and $c_0(L)$ coincide when the fundamental group of $M$ is Abelian and, more generally, when it is ameanable (see \\cite{fm07}).\n\nThe lowest Ma\\~{n}\\'e critical value $c_u(L)$ plays a decisive role in the geometry of the action functional ${\\mathbb{S}}_{\\kappa}$, as the next result shows:\n\n\\begin{Lemma}\\label{Lem:5}\nIf $\\kappa\\geq c_u(L)$, then ${\\mathbb{S}}_{\\kappa}$ is bounded from below on every connected component of $\\mathcal{M}$. If $\\kappa<c_u(L)$, then the functional ${\\mathbb{S}}_{\\kappa}$ is unbounded from below on each connected component of $\\mathcal{M}$.\n\\end{Lemma}\n\n\\begin{proof}\nChoose $\\gamma:{\\mathbb{R}}\/T{\\mathbb{Z}}\\rightarrow M$ in some connected component of the free loop space and let $\\tilde\\gamma:[0,T]\\longrightarrow\\widetilde M$ be the its lift to the universal covering $\\pi:\\widetilde M \\rightarrow M$. We lift the metric of $M$ to $\\widetilde{M}$ and notice that the fact of having fixed the connected component of the free loop space implies that  the quantity $\\mathrm{dist}\\big(\\tilde\\gamma(T),\\tilde\\gamma(0)\\big)$ is uniformly bounded. Therefore, there exists a path $\\tilde{\\alpha}:[0,1]\\rightarrow \\widetilde M$ which joins $\\tilde{\\gamma}(T)$ to $\\tilde{\\gamma}(0)$ and has uniformly bounded action\n\\[\n\\widetilde{{{\\mathbb{S}}}}_{\\kappa} (\\tilde{\\alpha}) := \\int_0^1 \\Bigl( \\widetilde L\\bigl(\\tilde{\\alpha}(t), \\tilde{\\alpha}'(t) \\bigr) + \\kappa \\Bigr) \\, dt \\leq C,\n\\]\nwhere $\\widetilde{L}$ denotes the Lagrangian on $T\\widetilde M$ which is obtained by lifting $L$. If $\\alpha := \\pi\\circ \\tilde{\\alpha}$, the juxtaposition $\\gamma\\# \\alpha$ is a contractible loop in $M$. Since $\\kappa\\geq c_u(L)$, we have\n\\[\n0 \\leq {\\mathbb{S}}_{\\kappa} (\\gamma \\# \\alpha) = {\\mathbb{S}}_{\\kappa} (\\gamma) + {\\mathbb{S}}_{\\kappa} (\\alpha) = {\\mathbb{S}}_{\\kappa} (\\gamma) + \\widetilde{{{\\mathbb{S}}}}_{\\kappa} (\\tilde{\\alpha})\n\\leq {\\mathbb{S}}_{\\kappa} (\\gamma)+C,\n\\]\nfrom which ${\\mathbb{S}}_{\\kappa}(\\gamma)\\geq -C$.\n\nWhen $\\kappa<c_u(L)$, the functional ${\\mathbb{S}}_{\\kappa}$ is unbounded from below on each connected component of $\\mathcal{M}$. In fact, if $\\alpha$ is a contractible closed curve with ${\\mathbb{S}}_{\\kappa}(\\alpha)<0$, we can modify any closed curve $\\gamma$ within its free homotopy class and make it have arbitrarily low action ${\\mathbb{S}}_{\\kappa}$: Join $\\gamma(0)$ to $\\alpha(0)$ by some path, wind around $\\alpha$ several times, come back to $\\gamma(0)$ by the inverse path, and finally go once around $\\gamma$.\n\\end{proof}\n\nThe strict Ma\\~n\\'e critical value $c_0(L)$ is not directly related to the geometry of ${\\mathbb{S}}_{\\kappa}$, but has the following important characterization (see \\cite{fat97} and \\cite{cipp98}):\n\\begin{equation}\n\\label{cara}\nc_0(L) = \\inf \\left\\{ \\max_{x\\in M} H(x,\\alpha(x)) \\, \\Big| \\, \\alpha \\mbox{ smooth closed one-form on } M\\right\\},\n\\end{equation}\nwhere $H:T^*M \\rightarrow {\\mathbb{R}}$ is the Hamiltonian associated to the Lagrangian $L$ via Legendre duality:\n\\[\nH(x,p) := \\max_{v\\in T_x M} \\bigl( p[v] - L(x,v) \\bigr).\n\\]\nThen $H$ is a Tonelli Hamiltonian, meaning that it is fiberwise superlinear and uniformly convex (see the beginning of Section \\ref{sec3}). \nLet $X_H$ be the induced Hamiltonian vector field on $T^*M$, which is defined by the identity\n\\[\n\\omega(X_H(z),\\zeta) = - dH(z)[\\zeta], \\quad \\forall z\\in T^*M, \\; \\zeta\\in T_z T^*M,\n\\]\nwhere $\\omega=dp\\wedge dx$ is the standard symplectic form on $T^*M$. The flow of $X_H$ preserves each level $H^{-1}(\\kappa)$, where it is conjugated to the Euler-Lagrange flow of $L$ on $E^{-1}(\\kappa)$ by the Legendre transform\n\\[\nTM \\rightarrow T^*M, \\quad (x,v) \\mapsto \\bigl( x , d_v L(x,v) \\bigr).\n\\]\nAssume that $\\kappa$ is a regular value of $H$, so that $\\Sigma:= H^{-1}(\\kappa)$ is a hypersurface. Up to a time reparametrization, the dynamics on $\\Sigma$ is determined only by the geometry of $\\Sigma$ and not by the Hamiltonian of which $\\Sigma$ is an energy level: In fact the nowhere vanishing  vector field $X_H|_{\\Sigma}$ belongs to the one-dimensional distribution \n\\[\n\\mathcal{L}_{\\Sigma}:=\\ker \\omega|_{\\Sigma},\n\\]\nwhose integral curves are hence the orbits of $X_H|_{\\Sigma}$.\n\nThe characterization (\\ref{cara}) has a dynamical and a geometric consequence. We begin with the dynamical one: \n\n\\begin{thm}\n\\label{finsler}\nIf $\\kappa>c_0(L)$, then the Euler-Lagrange flow on $E^{-1}(\\kappa)$ is conjugated up to a time-reparametrization to the geodesic flow which is induced by a Finsler metric on $M$.\n\\end{thm}\n\n\\begin{proof}\nSince $\\kappa>c_0(L)$, there is a smooth closed one-form $\\alpha$ whose image is contained in the sublevel $\\{H<\\kappa\\}$. Since $\\alpha$ is closed, the diffeomorphism of $T^*M$ defined by $(x,p) \\mapsto (x,p+\\alpha(x))$ is symplectic and conjugates the Hamiltonian flow of $H$ to that of $K(x,p):= H(x,p+\\alpha(x))$. The energy level $K^{-1}(\\kappa)$ is now the boundary\nof a fiberwise uniformly convex bounded open set which contains the zero section of $T^*M$. Therefore, there exists a fiberwise convex and 2-homogeneous function $F:T^*M \\rightarrow [0,+\\infty)$ such that $F^{-1}(1)=K^{-1}(\\kappa)$. Thus, the Hamiltonian flow of $F$ on $F^{-1}(1)=K^{-1}(\\kappa)$ is related to that of $K$ - hence to that of $H$ on $H^{-1}(\\kappa)$ - by a time reparametrization. But the Legendre dual of the fiberwise convex and 2-homogeneous Hamiltonian $F$ is the square of a Finsler structure on $M$. We conclude that the orbits of the Euler-Lagrange flow of $L$ of energy $\\kappa$ are reparametrized Finsler geodesics.\n\\end{proof}\n\nIn order to derive the geometric consequence of (\\ref{cara}), we need to recall some notions from symplectic topology.\n\n\\medskip\n\n\\paragraph{\\bf Contact type and stable energy hypersurfaces} \nThe energy level $\\Sigma$ is said to be of {\\em contact type} if there is a one-form $\\eta$ on $\\Sigma$ which is a primitive of $\\omega|_{\\Sigma}$ and is such that $\\eta$ does not vanish on $\\mathcal{L}_{\\Sigma}$. Equivalently, there is a smooth vector field $Y$ in a neighborhood of $\\Sigma$ which is transverse to $\\Sigma$ and such that $L_Y \\omega = \\omega$ (the vector field $Y$ and the one-form $\\eta$ are related by the identity $\\imath_Y \\omega|_{\\Sigma} = \\eta$). The energy level $\\Sigma$ is said to be of {\\em restricted contact type} if the one-form $\\eta$ extends to a primitive of $\\omega$ on the whole $T^*M$. \n\nIf the surface $\\Sigma\\subset T^*M$ bounds an open fiberwise convex set which contains the zero-section (or more generally an open set which is starshaped with respect to the zero section), then it is of restricted contact type: As $\\eta$ one can take the Liouville form $p\\, dq$. Therefore, arguing as in the first part of the proof of Theorem \\ref{finsler}, we obtain the following geometric consequence of the characterization (\\ref{cara}):\n\n\\begin{thm}\n\\label{contact}\nIf $\\kappa>c_0(L)$, then $H^{-1}(\\kappa)$ is of restricted contact type.\n\\end{thm}\n \n \n If $c_u(L) \\leq \\kappa \\leq c_0(L)$ and $M$ is not the 2-torus, $H^{-1}(\\kappa)$ is not of contact type (see \\cite[Proposition B.1]{con06}), and it is conjectured that the same is true for $e_0(L)<\\kappa<c_u(L)$. If $\\min E < \\kappa < e_0(L)$, $H^{-1}(\\kappa)$ might or might not be of contact type: For instance, if the one-form $\\theta(x)[v]:=d_vL(x,0)[v]$ is closed, then every regular energy level is of contact type (see \\cite[Proposition C.2]{con06}, in this case $e_0(L)=c_u(L)=c_0(L)$).\n\nThe contact condition has the following dynamical consequence: If $\\Sigma$ is a contact type compact hypersurface in a symplectic manifold $(W,\\omega)$ (in our case, $W=T^*M$), then there is a diffeomorphism\n\\[\n(-\\epsilon, \\epsilon) \\times \\Sigma \\rightarrow W, \\qquad (r,x) \\mapsto \\psi_r(x),\n\\]\nonto an open neighborhood of $\\Sigma$ such that $\\psi_0$ is the identity on $\\Sigma$ and \n\\[\n\\psi_r : \\Sigma \\rightarrow \\Sigma_r := \\psi_r(\\Sigma)\n\\]\ninduces an isomorphism between the line bundles $\\mathcal{L}_{\\Sigma}$ and $\\mathcal{L}_{\\Sigma_r}$ (the hypersurface $\\Sigma_r$ is the image of $\\Sigma$ by the flow at time $r$ of the vector field $Y$ given by the contact condition, see \\cite[page 122]{hz94}). Therefore, if the hypersurfaces $\\Sigma_r$ are level sets of a Hamiltonian $K$, the dynamics of $X_K$ on $\\Sigma_r$ is conjugate to the one on $\\Sigma_0=\\Sigma$ up to a time reparametrization.\n\nHypersurfaces with the above propery are called {\\em stable} (see \\cite[page 122]{hz94}). The stability condition is weaker than the contact condition, as the following characterization, which is due to K.\\ Cieliebak and K.\\ Mohnke \\cite[Lemma 2.3]{cm05}, shows:\n\n\\begin{Prop}   \nLet $\\Sigma$ be a compact hypersurface in the symplectic manifold $(W,\\omega)$. Then the following facts are equivalent:\n\\begin{enumerate}\n\\item $\\Sigma$ is stable;\n\\item there is a vector field $Y$ on a neighborhood of $\\Sigma$ which is transverse to $\\Sigma$ and satisfies $\\mathcal{L}_{\\Sigma} \\subset \\ker ( L_Y \\omega|_{\\Sigma})$;\n\\item there is a one-form $\\eta$ on $\\Sigma$ such that $\\mathcal{L}_{\\Sigma} \\subset \\ker d\\eta$ and $\\eta$ does not vanish on $\\mathcal{L}_{\\Sigma}$.\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n(i) $\\Rightarrow$ (ii). By stability, a neighborhood of $\\Sigma$ can be identified with $(-\\epsilon,\\epsilon)\\times \\Sigma$ in such a way that $\\mathcal{L}_{\\{r\\} \n\\times \\Sigma}$ does not depend on $r$. Set $Y:= \\partial\/\\partial r$ and denote by $\\phi_t(r,x)=(r+t,x)$ its flow. Then $\\ker ( \\phi_t^* \\omega|_{\\{0\\} \\times \\Sigma} )$ does not depend on $t$ and differentiating in $t$ at $t=0$ we get\n\\[\n\\mathcal{L}_{\\Sigma} = \\ker \\omega|_{\\Sigma} \\subset \\ker ( L_Y \\omega|_{\\Sigma}).\n\\]\n(ii) $\\Rightarrow$ (iii). If we set $\\eta:= \\imath_Y \\omega|_{\\Sigma}$, by Cartan's identity we have\n\\[\nd\\eta = d \\imath_Y \\omega|_{\\Sigma} = (L_Y \\omega - \\imath_Y d \\omega)|_{\\Sigma} =  L_Y \\omega|_{\\Sigma},\n\\]\nso $\\mathcal{L}_{\\Sigma} \\subset \\ker(L_Y \\omega|_{\\Sigma}) = \\ker d\\eta$. If $\\xi\\neq 0$ is a vector in $\\mathcal{L}_{\\Sigma}$, then\n\\[\n\\eta(\\xi) = \\omega(Y,\\xi) \\neq 0,\n\\]\nbecause $Y\\notin T\\Sigma$.\n\n\\noindent (iii) $\\Rightarrow$ (i). Consider the closed two-form on $(-\\epsilon,\\epsilon) \\times \\Sigma$\n\\[\n\\tilde{\\omega} = \\omega|_{\\Sigma} + d(r\\eta) = \\omega|_{\\Sigma} + rd\\eta + dr\\wedge \\eta.\n\\]\nIf $\\epsilon$ is small enough, the form $\\omega|_{\\Sigma} + rd\\eta$ is non-degenerate on $\\ker \\eta$ for every $r\\in (-\\epsilon,\\epsilon)$, from which we deduce that $\\tilde{\\omega}$ is a symplectic form. Since $\\tilde{\\omega}|_{\\{0\\} \\times \\Sigma}$ coincides with $\\omega|_{\\Sigma}$, by the coisotropic neighborhood theorem (see \\cite{got82}, or \\cite[Exercise 3.36]{ms98} for the particular case of a hypersurface), a neighborhood of $\\Sigma$ in $W$ is symplectomorphic to $((-\\epsilon,\\epsilon) \\times \\Sigma,\\tilde{\\omega})$, up to the choice of a smaller $\\epsilon$. Since for $\\xi\\in \\mathcal{L}_{\\Sigma}(x)$ and $\\zeta \\in T_{(r,x)} (\\{r\\} \\times \\Sigma) = (0) \\times T_x \\Sigma$ there holds\n\\[\n\\tilde{\\omega}(\\xi,\\zeta) = \\omega(\\xi,\\zeta) + r d\\eta (\\xi,\\zeta) = 0,\n\\]\nwe deduce that $\\ker (\\tilde{\\omega}|_{\\{r\\} \\times \\Sigma}) = \\mathcal{L}_{\\Sigma}$ does not depend on $r$. Therefore, $\\{0\\} \\times \\Sigma$ is stable in $((-\\epsilon,\\epsilon) \\times \\Sigma,\\tilde{\\omega})$ and hence $\\Sigma$ is stable in $(W,\\omega)$.\n\\end{proof}\n\n\\begin{Rmk}\nL.~Macarini and G.~P.~Paternain have constructed examples of Tonelli Lagrangians on the tangent bundle of $\\mathbb{T}^n$ such that $H^{-1}(\\kappa)$ is stable for $\\kappa=c_u(L)=c_0(L)$, see \\cite{mp10}.\n\\end{Rmk}\n\n\n\\section{Palais-Smale sequences}\n\\label{pss}\n\nPalais-Smale sequences $(x_h,T_h)$ with $T_h\\rightarrow 0$ are a possible source of non-compactness, but the next result shows that they occur only at level zero.\n\n\\begin{Lemma}\n\\label{PST0}\nLet $(x_h,T_h)$ be a $(\\mathrm{PS})_c$ sequence for ${\\mathbb{S}}_{\\kappa}$ with $T_h\\to0$. Then $c=0$.\n\\end{Lemma}\n\n\\begin{proof}\nSet $\\gamma_h(t):= x_h(t\/T_h)$.\nSince $(x_h,T_h)$ is a $(\\mathrm{PS})$ sequence, the identity (\\ref{eq:diff A w.r.t. T}) shows that the sequence\n\\[\n\\alpha_h := \\frac{1}{T_h} \\int_0^{T_h} \\bigl( E(\\gamma_h,\\gamma_h') - \\kappa \\bigr) \\, dt = - \\frac{\\partial {\\mathbb{S}}_{\\kappa}}{\\partial T} (x_h,T_h)\n\\]\nis infinitesimal and, in particular, bounded. Since $L(x,v)$ is electromagnetic for $|v|$ large, $E(x,v)$ has the form (\\ref{ene}) for $|v|$ large and hence satisfies the estimate\n\\[\nE(x,v) \\geq E_0 |v|^2 - E_1, \\qquad \\forall (x,v)\\in TM,\n\\]\nfor suitable numbers $E_0>0$ and $E_1\\in {\\mathbb{R}}$. It follows that\n\\[\n\\alpha_h \\geq \\frac{1}{T_h} \\int_0^{T_h} \\bigl( E_0 |\\gamma_h'|^2 - E_1 - \\kappa \\bigr)\\, dt = \\frac{E_0}{T_h} \\int_0^{T_h} |\\gamma_h'|^2\\,dt - (E_1 + \\kappa),\n\\]\nand hence\n\\[\n\\int_0^{T_h}  |\\gamma_h'|^2\\,dt \\leq \\frac{T_h}{E_0} ( \\alpha_h + E_1 + \\kappa) = O(T_h)\n\\]\nfor $h\\rightarrow \\infty$. From the lower bound (\\ref{boundsonL}) and from the analogous upper bound\n\\[\nL(x,v) \\leq L_2 |v|^2 + L_3, \\qquad \\forall (x,v)\\in TM,\n\\]\nfor suitable positive number $L_2,L_3$, we find\n\\[\n{\\mathbb{S}}_{\\kappa}(x_h,T_h) = \\int_0^{T_h} \\bigl( L(\\gamma_h,\\gamma_h') + \\kappa \\bigr)\\, dt \\geq L_0 \\int_0^{T_h} |\\gamma_h'|^2\\, dt + T_h (\\kappa - L_1) = O(T_h)\n\\]\nand\n\\[\n{\\mathbb{S}}_{\\kappa}(x_h,T_h) = \\int_0^{T_h} \\bigl( L(\\gamma_h,\\gamma_h') + \\kappa \\bigr)\\, dt \\leq L_2 \\int_0^{T_h} |\\gamma_h'|^2\\, dt + T_h (L_3 + \\kappa) = O(T_h)\n\\]\nfor $h\\rightarrow \\infty$. Since $T_h\\rightarrow 0$, we conclude that ${\\mathbb{S}}_{\\kappa}(x_h,T_h)$ is infinitesimal and hence $c=0$.\n\\end{proof}\n\n\\begin{Rmk}\nBy choosing a suitable metric on the Hilbert manifold $\\mathcal{M}$, one can also obtain that for every Palais-Smale sequences $(x_h,T_h)$ with $T_h\\rightarrow 0$ the sequence $(x_h)$ converges to an equilibrium solution of the Euler-Lagrange equations which has energy $\\kappa$ (see \\cite[Proposition 3.8]{con06}). In particular, such Palais-Smale sequences can exist only when $\\kappa$ is a critical value of $E$.\n\\end{Rmk}\n\nThe next result says that Palais-Smale sequences $(x_h,T_h)$ with $(T_h)$ bounded and bounded away from zero are always compact.\n\n\n\\begin{Lemma}\\label{Lem;2}\nLet $(x_h,T_h)$ be a $(\\mathrm{PS})_c$ sequence for ${\\mathbb{S}}_{\\kappa}$ with $0<T_*\\leq T_h\\leq T^*<\\infty$. Then $(x_h,T_h)$ is compact in $\\mathcal{M}$.\n\\end{Lemma}\n\n\\begin{proof}\nUp to a subsequence, we may assume that $(T_h)$ converges to some $T\\in [T_*,T^*]$.\nBy (\\ref{boundsonL}) we have\n\\begin{equation}\n\\label{isbd}\n\\begin{split}\nc+o(1) = {\\mathbb{S}}_{\\kappa}(x_h,T_h) = T_h \\int_0^1 \\Bigr( L\\bigl(x_h,x_h'\/T_h \\bigr)+\\kappa \\Bigr)\\, ds \\\\ \\geq T_h \\int_0^1\\Bigr( L_0 \\frac{|x_h'|^2}{T_h^2}-(L_1-\\kappa)\\Bigr)\\, ds \\geq \\frac{L_0}{T^*}\\|x_h'\\|_2^2-T^* |L_1-\\kappa|,\n\\end{split} \\end{equation}\nwhere $\\|\\cdot\\|_2$ denotes the $L^2$ norm with respect to the fixed Riemannian metric on $M$. Therefore, $\\|x_h'\\|_2$ is uniformly bounded and $(x_h)$ is 1\/2-equi-H\\\"older-continuous:\n\\[\n\\mathrm{dist}\\big(x_h(s'),x_h(s)\\big)\\leq\\int_s^{s'}|x_h'(r)|\\, dr \\leq |s'-s|^{1\/2} \\|x_h'\\|_2.\n\\]\nBy the Ascoli-Arzel\\`a theorem, up to a subsequence $(x_h)$ converges uniformly to some $x\\in C({\\mathbb{T}},M)$.  In particular, $(x_h)$ eventually belongs to the image of the parameterization $\\varphi_*$ induced by a smooth map\n\\[\n\\varphi: {\\mathbb{T}}\\times B_r \\rightarrow M.\n\\]\nSee \\eqref{eq:chart of the loop space} and recall that the image of this parameterization is $C^0$-open.\nThen $x_h=\\varphi_*(\\xi_h)$, where $\\xi_h$ belongs to $W^{1,2}({\\mathbb{T}},B_r)$ and is a $(\\mathrm{PS})$ sequence for the functional\n$$\n\\widetilde{{\\mathbb{S}}}(\\xi,T)=T\\int_{{\\mathbb{T}}}\\widetilde L\\bigr(s,\\xi,\\xi'\/T \\bigr)\\,ds,\n$$\nwith respect to the standard Hilbert product on $W^{1,2}({\\mathbb{T}},{\\mathbb{R}}^n)$,\nwhere the Lagrangian $\\widetilde{L}\\in C^{\\infty}({\\mathbb{T}}\\times B_r \\times {\\mathbb{R}}^n)$ is obtained by pulling back $L+\\kappa$ by $\\varphi$. Moreover, $(\\xi_h)$ converges uniformly and, since $\\|\\xi_h'\\|_2$ is bounded, weakly in $W^{1,2}$ to some $\\xi$ in\n$W^{1,2}({\\mathbb{T}},B_r)$. We must prove that this convergence is actually strong in $W^{1,2}$.\n\nSince $\\widetilde{L}(s,x,v)$ is electromagnetic for $|v|$ large, we have the bounds\n\\begin{equation}\n\\label{bddL}\n\\bigl| d_x \\widetilde L (s,x,v) \\bigr| \\leq C(1+|v|^2),\\quad\\bigl| d_v \\widetilde L (s,x,v)\\bigr| \\leq C(1+|v|),\n\\end{equation}\nfor a suitable constant $C$. Since $(\\xi_h,T_h)$ is a $(\\mathrm{PS})$ sequence with $(\\xi_h)$  bounded in $W^{1,2}$, we have by (\\ref{diff1})\n\\begin{equation*}\n\\begin{split}\no(1) &= d\\widetilde{{\\mathbb{S}}}(\\xi_h,T_h)[(\\xi_h-\\xi,0)] \\\\ &= T_h \\int_{{\\mathbb{T}}} d_x \\widetilde L \\bigl(s,\\xi_h,\\xi_h'\/T_h\\bigr) [\\xi_h-\\xi]\\, ds + T_h \\int_{{\\mathbb{T}}} d_v \\widetilde{L} \\bigl(s,\\xi_h,\\xi_h'\/T_h\\bigr) \\bigl[ (\\xi_h'-\\xi')\/T_h \\bigr] \\, ds.\n\\end{split} \\end{equation*}\nBy the first bound in (\\ref{bddL}) and the uniform convergence $\\xi_h\\rightarrow \\xi$, the first integral is infinitesimal. Together with the fact that $(T_h)$ is bounded away from zero, we obtain\n\\begin{equation}\n\\label{uno}\n\\int_{{\\mathbb{T}}} d_v \\widetilde{L} \\bigl(s,\\xi_h,\\xi_h'\/T_h\\bigr) \\bigl[ (\\xi_h'-\\xi')\/T_h \\bigr] \\, ds=o(1).\n\\end{equation}\nFrom the fiberwise uniform convexity of $\\widetilde{L}$, we have the bound\n\\[\nd_{vv} \\widetilde{L}  (s,x,v) [u, u] \\geq \\delta |u|^2, \\quad \\forall (s,x,v)\\in {\\mathbb{T}}\\times B_r \\times {\\mathbb{R}}^n, \\; u\\in {\\mathbb{R}}^n,\n\\]\nfor a suitable positive number $\\delta$. It follows that\n\\begin{equation*}\n\\begin{split}\nd_v \\widetilde L \\left(s,\\xi_h,\\frac{\\xi_h'}{T_h}\\right) & \\left[ \\frac{\\xi_h'-\\xi'}{T_h} \\right] - d_v\\widetilde L  \\left(s,\\xi_h,\\frac{\\xi'}{T_h} \\right)\\left[\\frac{\\xi_h'-\\xi'}{T_h} \\right] \\\\\n&= \\int_0^1 d_{vv} \\widetilde L \\left(s,\\xi_h,\\frac{\\xi'}{T_h}+\\sigma \\frac{\\xi_h'-\\xi'}{T_h} \\right) \\left[ \\frac{\\xi_h'-\\xi'}{T_h}, \\frac{\\xi_h'-\\xi'}{T_h} \\right] \\, d\\sigma \\geq \\frac{\\delta}{T_h^2} |\\xi_h'- \\xi'|^2.\n\\end{split} \\end{equation*}\nBy integrating this inequality over $s\\in {\\mathbb{T}}$ and by (\\ref{uno}), we obtain\n\\[\no(1)- \\int_{{\\mathbb{T}}} d_v \\widetilde L \\bigl(s,\\xi_h,\\xi'\/T_h)\\bigl[ (\\xi_h'-\\xi')\/T_h \\bigr] \\, ds \\geq \\frac{\\delta}{T_h^2} \\|\\xi_h' - \\xi'\\|_2^2.\n\\]\nBy the second bound in (\\ref{bddL}), the sequence\n\\[\nd_v \\widetilde L \\bigl(s,\\xi_h,\\xi'\/T_h\\bigr)\n\\]\nconverges strongly in $L^2$. By the weak $L^2$ convergence to $0$ of $(\\xi_h'-\\xi)$, we deduce that the integral on the left-hand side of the above inequality is infinitesimal. We conclude that $(\\xi_h)$ converges to $\\xi$ strongly in $W^{1,2}$.\n\\end{proof}\n\nIn general, ${\\mathbb{S}}_{\\kappa}$ might have Palais-Smale sequences $(x_h,T_h)$ with $(T_h)$ unbounded. However, this does not occur when $\\kappa$ is larger than the lowest Ma\\~{n}\\'e critical value $c_u(L)$:\n\n\\begin{Lemma}\\label{Lem:6}\nIf $\\kappa>c_u(L)$, then any $(\\mathrm{PS})$ sequence $(x_h,T_h)$ in a given connected component of $\\mathcal{M}$ with $T_h\\geq T_*>0$ is compact.\n\\end{Lemma}\n\n\\begin{proof}\nBy Lemma \\ref{Lem;2}, it is enough to show that $(T_h)$ is bounded from above. Since\n\\[\n{\\mathbb{S}}_{\\kappa} (x,T)={\\mathbb{S}}_{c_u(L)} (x,T) + \\big(\\kappa -c_u(L)\\big)T,\n\\]\nthe period\n\\[\nT_h = \\frac{1}{\\kappa -c_u(L)} \\bigr({\\mathbb{S}}_{\\kappa} (x_h,T_h) - {\\mathbb{S}}_{c_u(L)} (x_h,T_h)\\bigr)\n\\]\nis bounded from above, because ${\\mathbb{S}}_{\\kappa}$ is bounded on the $(\\mathrm{PS})$ sequence $(x_h,T_h)$ and ${\\mathbb{S}}_{c_u(L)}(x_h,T_h)$ is bounded from below by Lemma \\ref{Lem:5}.\n\\end{proof}\n\n\\begin{Rmk}\nBy choosing a suitable metric on $\\mathcal{M}$, it is possible to characterize $c_u(L)$ as the infimum of all $\\kappa_0$'s such that ${\\mathbb{S}}_{\\kappa}$ satisfies the Palais-Smale condition for every $\\kappa\\in [\\kappa_0,+\\infty)$. See \\cite{cipp00}.\n\\end{Rmk} \n\n\\section{Periodic orbits with high energy}\n\\label{pohe}\n\nThe following result shows that the energy levels above $c_u(L)$ have always periodic orbits and proves statements (i) and (ii) of the theorem in the Introduction.\n\n\\begin{Thm}\n\\label{highthm}\nAssume that $\\kappa>c_u(L)$. Then the following facts hold.\n\\begin{enumerate}\n\\item If $M$ is not simply connected, then the energy level $E^{-1}(\\kappa)$ has a $\\kappa$-action-minimizing periodic orbit in each non-trivial homotopy class of the free loop space.\n\\item If $M$ is simply connected, then the energy level $E^{-1}(\\kappa)$ has a periodic orbit with positive  action ${\\mathbb{S}}_{\\kappa}$.\n\\end{enumerate}\n\\end{Thm}\n\n\\begin{proof}\n(i) Assume that $M$ is not simply connected. Let $\\alpha \\in[{\\mathbb{T}},M]$ be a non-trivial homotopy class and let\n$\\mathcal{M}_{\\alpha}$ be the connected component of $\\mathcal{M}^\\mathrm{noncontr}$ corresponding to $\\alpha$. By Lemma \\ref{Lem:5}, the functional ${\\mathbb{S}}_{\\kappa}$ is bounded from below on $\\mathcal{M}_{\\alpha}$. By Lemma \\ref{Lem:3} (i), the sublevels\n\\[\n\\{ (x,T)\\in \\mathcal{M}_{\\alpha} \\, | \\, {\\mathbb{S}}_{\\kappa}(x,T)\\leq c \\}\n\\]\nare complete. Let $(x_h,T_h)\\subset \\mathcal{M}_{\\alpha}$ be a (PS) sequence for ${\\mathbb{S}}_{\\kappa}$. By Lemma \\ref{Lem:3} (i), $(T_h)$ is bounded away from zero, so Lemma \\ref{Lem:6} implies that ${\\mathbb{S}}_{\\kappa}$ satisfies the (PS) condition on $\\mathcal{M}_{\\alpha}$. By Remark \\ref{mini}, we conclude that ${\\mathbb{S}}_{\\kappa}$ has a minimizer on $\\mathcal{M}_{\\alpha}$, as we wished to prove.\n\n\\medskip\n\n(ii) Assume that $M$ is simply connected, so that $\\mathcal{M}=\\mathcal{M}^{\\mathrm{contr}}$. In this case, ${\\mathbb{S}}_{\\kappa}$ is strictly positive everywhere, because $\\kappa>c_u(L)$, but the infimum of ${\\mathbb{S}}_{\\kappa}$ is zero, as one readily checks by looking at sequences of the form $(x_0,T_h)$, with $x_0$ a constant loop and $T_h\\rightarrow 0$. So the infimum is not achieved. We will find the periodic orbit by considering the same minimax class which Lusternik and Fet \\cite{lf51} considered in their proof of the existence of a closed geodesic on a simply connected compact manifold.\n\nSince the closed manifold $M$ is simply connected, there exists $l\\geq 2$ such that $\\pi_l(M)\\ne 0$ (a manifold all of whose homotopy groups vanish is contractible, but closed manifolds are never contractible, for instance because their $n$-dimensional homology group with ${\\mathbb{Z}}_2$ coefficients does not vanish). We fix a non-zero homotopy class $\\mathcal{G} \\in\\pi_l(M)$. Thanks to the isomorphism\n$\\pi_{l-1}(C({\\mathbb{T}},M))\\cong\\pi_l(M)$, we have an induced non-zero homotopy class\n\\[\n\\mathcal{H} \\in [S^{l-1},C({\\mathbb{T}},M)] \\cong [S^{l-1},\\mathcal{M}],\n\\]\nand we consider the minimax value\n\\[\nc=\\inf_{\\substack{h:S^{l-1}\\to \\mathcal{M}\\\\ h\\in \\mathcal{H}}} \\max_{\\xi\\in S^{l-1}} {\\mathbb{S}}_{\\kappa} (h(\\xi)).\n\\]\nLet us show that $c>0$. Since $\\H$ is non-trivial, there exists a positive number $a$ such that for every map $h=(x,T): S^{l-1} \\rightarrow \\mathcal{M}$ belonging to the class $\\H$ there holds\n\\[\n\\max_{\\xi \\in S^{l-1}} \\ell(x(\\xi)) \\geq a,\n\\]\nwhere $\\ell(x(\\xi))$ denotes the length of the loop $x(\\xi)$ (see \\cite[Theorem 2.1.8]{kli78}). If $(x,T)$ is an element of $\\mathcal{M}$ with $\\ell(x)\\geq a$, then (\\ref{boundsonL}) implies\n\\begin{equation*}\n\\begin{split}\n{\\mathbb{S}}_{\\kappa}(x,T) &= T\\int_{{\\mathbb{T}}} \\Bigl(L\\bigl(x,x'\/T\\bigr)+\\kappa \\Bigr) \\,ds\n\\geq T \\int_{{\\mathbb{T}}} \\Bigl( L_0\\frac{|x'|^2}{T^2} -L_1+\\kappa \\Bigr)\\, ds \\\\\n& \\geq \\frac{L_0}{T}\\ell(x)^2-T(L_1-\\kappa)\n\\geq \\frac{L_0}{T}a^2-T(L_1-\\kappa).\n\\end{split} \\end{equation*}\nSince $a>0$, the above chain of inequalities implies that there exists $T_0>0$ such that for every $(x,T)\\in \\mathcal{M}$ with $\\ell(x)\\geq a$ and ${\\mathbb{S}}_{\\kappa}(x,T)\\leq c+1$, the period $T$ is at least $T_0$. Now let $h\\in \\H$ be such that the maximum of ${\\mathbb{S}}_{\\kappa}$ on $h(S^{l-1})$ is less than $c+1$. By the above considerations, there exists $(x,T)$ in $h(S^{l-1})$ with $T\\geq T_0$, whence\n\\[\n{\\mathbb{S}}_{\\kappa} (x,T)={\\mathbb{S}}_{c_u(L)} (x,T)+ \\bigl(\\kappa-c_u(L)\\bigr)T \\geq \\bigl(\\kappa-c_u(L)\\bigr) T_0.\n\\]\nThis shows that the minimax value $c$ is strictly positive.\n\nTheorem \\ref{thm:finite c induces PS}, together with Remark \\ref{trunc} and Lemma \\ref{Lem:4}, implies the existence of a $(\\mathrm{PS})_c$ sequence $(x_h,T_h)$. Lemma \\ref{PST0} guarantees that $(T_h)$ is bounded away from zero, so by Lemma \\ref{Lem:6} the sequence $(x_h,T_h)$ has a limiting point in $\\mathcal{M}$, which gives us the required periodic orbit.\n\\end{proof}\n\n\\begin{Rmk}\nIf $M$ is not simply connected and $\\kappa>c_u(L)$, the energy level $E^{-1}(\\kappa)$ might have no contractible periodic orbits. Consider for instance the Lagrangian $L(x,v)=|v|^2\/2$ on the torus ${\\mathbb{T}}^n$, equipped with the flat metric. The corresponding Euler-Lagrange flow is the geodesic flow on $T{\\mathbb{T}}^n$, $c_u(L)=0$, and all the non-constant closed geodesics on the flat torus are non-contractible.\n\\end{Rmk}\n\n\\begin{Rmk}\n\\label{veryhigh}\nIf $\\kappa>c_0(L)$, the existence of a periodic orbit on $E^{-1}(\\kappa)$ also follows from Theorem \\ref{finsler}, because every Finsler metric on a closed manifold has a closed geodesic.  \n\\end{Rmk}\n\n\\begin{Rmk}\nTheorem \\ref{finsler} implies that the multiplicity results for closed Finsler geodesics hold also for Hamiltonian orbits at energy levels $\\kappa>c_0(L)$. Actually, most of these results remain true for $\\kappa>c_u(L)$.\nIn fact, as the proof of Theorem \\ref{highthm} suggests, the (PS) condition and the topology of the sublevels of the functional ${\\mathbb{S}}_{\\kappa}$ are analogous to the corresponding properties of the Finsler geodesic energy functional (with the notable exception of the zero level). \n\\end{Rmk}\n\n\\section{Topology of the free period action functional for low energies}\n\\label{tfpafle}\n\nAs we have seen, when $\\kappa<c_u(L)$ the functional ${\\mathbb{S}}_{\\kappa}$ is unbounded from below on each connected component of $\\mathcal{M}$. The aim of this section is to show that, nevertheless, some sublevels of ${\\mathbb{S}}_{\\kappa}$ have a sufficiently rich topology.\n\nWe start by proving a simple lemma about the integral of a one form. The integral of a given one-form on a curve $x$ is clearly bounded by a constant times the length of $x$. When the support of the curve is contained in a ball of $M$, one may also take the square of the length in this bound, which is a better estimate for short curves. More precisely, we have the following:\n\n\\begin{Lemma}\\label{Lemma:isoper}\nLet $\\theta$ be a smooth one-form on $M$ and let $U\\subset M$ be an open set whose closure is diffeomorphic to a closed ball in ${\\mathbb{R}}^n$. Then there exists a number $\\Theta>0$ such that\n\\[\n\\bigg| \\int_{\\mathbb{T}} x^*(\\theta) \\bigg| \\leq \\Theta \\cdot\\ell(x)^2,\n\\]\nfor every closed curve $x:{\\mathbb{T}}\\rightarrow U$.\n\\end{Lemma}\n\n\\begin{proof}\nUp to the change of the constant $\\Theta$,\nwe may assume that $U=B_r$ is the ball of radius $r$ around the origin in ${\\mathbb{R}}^n$, equipped with the Euclidean metric. Given the closed curve $x:{\\mathbb{T}}\\rightarrow B_r$, we consider the map\n\\[\nX:[0,1]\\times {\\mathbb{T}} \\rightarrow B_r, \\quad X(s,t) = x(0) + s \\bigl( x(t) - x(0) \\bigr).\n\\]\nThen $X(1,t)=x(t)$ and $X(0,t)=x(0)$, hence by Stokes theorem\n\\begin{equation*}\n\\begin{split}\n\\left| \\int_{{\\mathbb{T}}} x^*(\\theta) \\right| &= \\bigg| \\int_{[0,1] \\times {\\mathbb{T}}} X^* (d\\theta) \\bigg| = \\left| \\int_0^1 ds \\int_{{\\mathbb{T}}} d\\theta\\bigl( X(s,t) \\bigr) \\Bigl[ \\frac{\\partial X}{\\partial s} , \\frac{\\partial X}{\\partial t} \\Bigr] \\, dt \\right| \\\\ &\\leq \\|d\\theta\\|_{\\infty} \\int_0^1 ds \\int_{{\\mathbb{T}}} \\left| \\frac{\\partial X}{\\partial s} \\right| \\, \\left| \\frac{\\partial X}{\\partial t} \\right| \\, dt\n= \\|d\\theta\\|_{\\infty} \\int_0^1 ds \\int_{{\\mathbb{T}}} \\bigl| x(t) - x(0) \\bigr| s \\bigl| x'(t) \\bigr|\\, dt \\\\ &\\leq \\frac{1}{2} \\|d\\theta\\|_{\\infty}\\, \\ell(x) \\int_0^1 ds \\int_{{\\mathbb{T}}} s \\bigl| x'(t)\\bigr|\\, dt = \\frac{1}{4} \\|d\\theta\\|_{\\infty}\\, \\ell(x)^2,\n\\end{split} \\end{equation*}\nas claimed.\n\\end{proof}\n\n\\paragraph{\\bf The energy range $\\mathbf{(e_0(L),c_u(L))}$} If $\\kappa<c_u(L)$, there are contractible closed curves with negative action ${\\mathbb{S}}_{\\kappa}$. Since the space of contractible loops is connected, we can consider the following class of continuous paths in $\\mathcal{M}$:\n\\begin{equation}\n\\label{curve}\n\\mathcal{Z}_0 := \\bigl\\{ (x,T):[0,1] \\rightarrow \\mathcal{M}\\, \\big| \\, x(0) \\mbox{ is a constant loop and } {\\mathbb{S}}_{\\kappa}(x(1),T(1)) <0 \\bigr\\}.\n\\end{equation}\nNotice that if $x_0$ is a constant loop and $T>0$, then\n\\begin{equation}\n\\label{constants}\n{\\mathbb{S}}_{\\kappa}(x_0,T) = T \\bigl( L(x_0,0) + \\kappa \\bigr) = T \\bigl( \\kappa - E(x_0,0) \\bigr).\n\\end{equation}\nWhen $\\kappa>e_0(L)=\\max_{x\\in M} E(x,0)$, the above quantity is strictly positive (and tends to zero for $T\\rightarrow 0$). The next lemma shows that when $e_0(L) < \\kappa < c_u(L)$, ${\\mathbb{S}}_{\\kappa}$ has a sort of mountain pass geometry:\n\n\\begin{Lemma}\n\\label{Lem:mountain pass}\nAssume that $e_0(L) <\\kappa < c_u(L)$. Then there exists $a>0$ such that for every $z\\in \\mathcal{Z}_0$ there holds\n\\[\n\\max_{\\sigma\\in[0,1]} {\\mathbb{S}}_{\\kappa} \\bigl(z(\\sigma)\\bigr)\\geq a.\n\\]\n\\end{Lemma}\n\n\\begin{proof}\nConsider the smooth one-form on $M$,\n\\[\n\\theta(x) [v] := d_v L(x,0)[v].\n\\]\nBy taking a Taylor expansion and by using the bound (\\ref{bd2L}), we get the estimate\n\\begin{equation}\n\\label{bbb}\nL(x,v) = L(x,0)+d_vL(x,0)[v]+\\frac{1}{2}d_{vv}L(x,s v)[v,v]\n\\geq -E(x,0)+\\theta(x)[v]+ L_0 |v|^2,\n\\end{equation}\nwhere $s\\in [0,1]$. Let $\\{U_1,\\dots,U_N\\}$ be a finite covering of $M$ consisting of open sets whose closures are diffeomorphic to closed Euclidean balls, and let $\\Theta >0$ be such that the conclusion of Lemma\n\\ref{Lemma:isoper} holds for the one-form $\\theta$, for each of the open sets $U_j$'s. Let $r_0$ be a Lebesgue number for this covering, meaning that every ball of radius $r_0$ is contained in one of the $U_j$'s.\n\nWe claim that if ${\\mathbb{S}}_{\\kappa}(x,T)<0$ then\n\\begin{equation}\n\\label{claim}\n\\ell(x)>\\min\\bigg\\{r_0,\\frac{\\sqrt{L_0(\\kappa-e_0(L))}}{\\Theta}\\bigg\\}=:r_1.\n\\end{equation}\nIn fact, assuming that $\\ell(x)\\leq r_0$, we have that $x({\\mathbb{T}})$ is contained in some $U_j$, for $1\\leq j \\leq N$. Set as usual $\\gamma(t)=x(t\/T)$.\nBy Lemma \\ref{Lemma:isoper} and by (\\ref{bbb}), we obtain the chain of inequalities\n\\begin{equation}\n\\label{stma}\n\\begin{split}\n0 &>{\\mathbb{S}}_{\\kappa}(x,T) ={\\mathbb{S}}_{\\kappa}(\\gamma) = \\int_0^T \\bigr(L(\\gamma,\\gamma')+\\kappa \\bigr)\\,dt\n\\\\ &\\geq \\int_0^T\\bigr( -E(\\gamma,0) + \\theta(\\gamma)[\\gamma'] + L_0 |\\gamma'|^2 + \\kappa \\bigr)\\, dt\n\\\\ &= \\int_0^T \\bigl( \\kappa - E(\\gamma,0)\\bigr)\\, dt + \\int_{{\\mathbb{R}}\/T{\\mathbb{Z}}} \\gamma^*(\\theta) + L_0 \\int_0^T |\\gamma'|^2\\, dt \\\\\n&\\geq \\bigl(\\kappa-e_0(L)\\bigr)T-\\Theta\\cdot \\ell(\\gamma)^2+\\frac{L_0}{T}\\ell(\\gamma)^2.\n\\end{split} \\end{equation}\nSince we are assuming $\\kappa>e_0(L)$, the above estimate implies that\n$T> L_0\/\\Theta$ and that\n\\[\n\\ell(\\gamma)^2>\\frac{\\bigl(\\kappa-e_0(L)\\bigr)T}{\\Theta-L_0\/T}>\\frac{\\bigl(\\kappa-e_0(L)\\bigr)T}{\\Theta}>\\frac{\\bigl(\\kappa-e_0(L)\\bigr)L_0}{\\Theta^2},\n\\]\nwhich proves (\\ref{claim}).\n\nFix some number $r$ in the open interval $(0,r_1)$.\nSince $z=(x,T)\\in \\mathcal{Z}_0$, ${\\mathbb{S}}_{\\kappa}(x(1),T(1))$ is negative, so by (\\ref{claim}) the length of $x(1)$ is larger than $r_1$. By continuity, using the fact that $x(0)$ is a constant loop, we get the existence of $\\sigma\\in(0,1)$ for which $\\ell(x(\\sigma))=r$. Then (\\ref{stma}) implies\n\\[\n{\\mathbb{S}}_{\\kappa}(x(\\sigma),T(\\sigma)) \\geq \\bigl(\\kappa-e_0(L)\\bigr) T +\\Bigr(\\frac{L_0}{T}-\\Theta\\Bigr)r^2.\n\\]\nMinimization in $T$ yields\n\\[\n{\\mathbb{S}}_{\\kappa}(x(\\sigma),T(\\sigma)) \\geq\nr \\bigr(\\sqrt{L_0(\\kappa-e_0(L))}-\\Theta r \\bigr)=: a.\n\\]\nThe number $a$ is positive because $r<r_1$. This concludes the proof.\n\\end{proof}\n\n\\paragraph{\\bf The energy range $\\mathbf{(\\min E,e_0(L))}$} When $\\kappa<e_0(L)$, the identity (\\ref{constants}) shows that ${\\mathbb{S}}_{\\kappa}$ takes negative values on some constant loops, and the conclusion of Lemma \\ref{Lem:mountain pass} cannot hold. Instead than considering the class of paths which go from some constant loop to a loop of negative action, one has to consider the class of deformations of the space of constant loops - which is diffeomorphic to $M$ - into the space of loops with negative action. More precisely, we consider the set of continuous maps\n\\[\n\\mathcal{Z}_M = \\bigl\\{ (x,T): [0,1]\\times M \\rightarrow \\mathcal{M} \\, \\big| \\, x(0,x_0) = x_0 \\mbox{ and } {\\mathbb{S}}_{\\kappa}\\bigl(x(1,x_0),T(1,x_0)\\bigr)<0, \\; \\forall x_0\\in M\\bigr\\}.\n\\]\n\n\\begin{Lemma}\n\\label{notempty}\nIf $\\kappa<c_u(L)$, then the set $\\mathcal{Z}_M$ is not empty.\n\\end{Lemma}\n\nWe just sketch the proof, referring to \\cite{tai83} for more details (see also \\cite{tai10}). The argument follows closely  Bangert's technique of ``pulling one loop at a time'' (see \\cite{ban80} and \\cite{bk83}).\n\nLet $M_0\\subset M_1 \\subset \\dots \\subset M_n = M$ be a CW-complex decomposition of $M$. Since $\\kappa<c_u(L)$ and since the 0-skeleton $M_0$ is a finite set, it is easy to construct a continuous map\n\\[\nz_0:[0,1]\\times M \\rightarrow \\mathcal{M}, \\quad z_0(\\sigma,x_0) = \\bigl(y_0(\\sigma,x_0),T_0(\\sigma,x_0)\\bigr),\n\\]\nsuch that\n\\begin{enumerate}\n\\item $y_0(0,x_0) = x_0$ for every $x_0\\in M$;\n\\item ${\\mathbb{S}}_{\\kappa}\\circ z_0(1,x_0)<0$ for every $x_0\\in M_0$.\n\\end{enumerate}\nGiven a positive integer $h$, we may iterate each loop $h$ times and obtain the map\n\\[\nz_0^h: [0,1]\\times M\\rightarrow \\mathcal{M}, \\quad z_0^h(\\sigma,x_0) = \\bigl(y_0^h(\\sigma,x_0),hT_0(\\sigma,x_0)\\bigr),\n\\]\nwhere\n\\[\ny_0^h(\\sigma,x_0)(s) := y_0(\\sigma,x_0)(hs), \\quad \\forall (\\sigma,x_0)\\in [0,1]\\times M, \\; \\forall s\\in {\\mathbb{T}}.\n\\]\nConsider an edge $S$ in $M_1$ with end-points $x_0,x_1\\in M_0$. The map $z_0^h(1,\\cdot)$ maps the the end-points of $S$ into the $h$-th iterates $\\alpha^h$ and $\\beta^h$ of two loops $\\alpha$ and $\\beta$ with negative action ${\\mathbb{S}}_{\\kappa}$. By pulling one of the $h$ loops at a time from $\\alpha^h$ to $\\beta^h$, one can construct a new map from $S$ into $\\mathcal{M}$ with end-points $\\alpha^h$ and $\\beta^h$ and such that ${\\mathbb{S}}_{\\kappa}$ is negative on its image, provided that $h$ is large enough. By relying on the map $z_0^h$, this construction can be done globally, and one ends up with a continuous map\n\\[\nz_1:[0,1]\\times M\\rightarrow \\mathcal{M}, \\quad z_1(\\sigma,x_0) = \\bigl(y_1(\\sigma,x_0),T_1(\\sigma,x_0)\\bigr),\n\\]\nsuch that\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\renewcommand{\\labelenumi}{(\\theenumi')}\n\\begin{enumerate}\n\\item $y_1(0,x_0) = x_0$ for every $x_0\\in M$;\n\\item ${\\mathbb{S}}_{\\kappa}\\circ z_1(1,x_0)<0$ for every $x_0\\in M_1$.\n\\end{enumerate}\nIterating this process, one can construct continuous maps\n\\[\nz_k:[0,1]\\times M\\rightarrow \\mathcal{M}, \\quad z_k(\\sigma,x_0) = \\bigl(y_k(\\sigma,x_0),T_k(\\sigma,x_0)\\bigr),\n\\]\nsuch that\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\renewcommand{\\labelenumi}{(\\theenumi'')}\n\\begin{enumerate}\n\\item $y_k(0,x_0) = x_0$ for every $x_0\\in M$;\n\\item ${\\mathbb{S}}_{\\kappa}\\circ z_k(1,x_0)<0$ for every $x_0\\in M_k$.\n\\end{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\renewcommand{\\labelenumi}{(\\theenumi)}\nThe map $z_n$ is an element of $\\mathcal{Z}_M$. This concludes our sketch of the proof of Lemma \\ref{notempty}.\nThe proof of the following result is analogous to the proof of Lemma \\ref{Lem:mountain pass}.\n\n\\begin{Lemma}\n\\label{mtpass2}\nAssume that $\\min E <\\kappa < c_u(L)$. Then there exists $a>0$ such that for every $z\\in \\mathcal{Z}_M$ there holds\n\\[\n\\max_{(\\sigma,x_0) \\in[0,1]\\times M} {\\mathbb{S}}_{\\kappa} \\bigl(z(\\sigma,x_0)\\bigr)\\geq a.\n\\]\n\\end{Lemma}\n\n\\section{Periodic orbits with low energy}\n\\label{pole}\n\n\\paragraph{\\bf The Struwe monotonicity argument} When $\\kappa\\leq c_u(L)$, the periods in a (PS) sequence need not be bounded anymore. Because of this fact, the question of the existence of periodic orbits for every energy $\\kappa$ in the interval $[\\min E,c_u(L)]$ is open, although no counterexamples are known. \nThe known system which is closer to being a counterexample is the horocycle flow on a closed surface $M$ with constant negative curvature (see e.g. \\cite{man91, cmp04}): Such a flow has no periodic orbits (actually, every orbit is dense) and it is the restriction of a Hamiltonian flow to an energy surface at a Ma\\~{n}\\'e critical value, but the corresponding Lagrangian is well defined only on the (non compact) universal cover of $M$ (such a system belongs to the family of {\\em non-exact magnetic flows}, whereas only {\\em exact} magnetic flows can be described by a Tonelli Lagrangian on $TM$). \n\n\nThe following argument is a version of an argument of Struwe, which says that when dealing with a minimax value associated to a family of functionals depending on a real parameter in a suitable monotone way, there exist compact (PS) sequences for almost every value of the parameter. This argument has applications both to Hamiltonian periodic orbits and to semilinear elliptic equations\n(see \\cite{str90}, \\cite[section II.9]{str00} and references therein).\n\nLet us assume that $\\min E<c_u(L)$, otherwise the interval of low energies collapses to a single level and there is nothing to prove.\nGiven $\\kappa\\in (\\min E,c_u(L))$, let $\\Gamma$ be the set of the images of the maps either in $\\mathcal{Z}_0$ or in $\\mathcal{Z}_M$, which were introduced in the previous section: If $e_0(L)< \\kappa < c_u(L)$ we may take $\\mathcal{Z}_0$, while in general we should take $\\mathcal{Z}_M$. Let $I$ be either the interval $(e_0(L),c_u(L))$ - if we are dealing with $\\mathcal{Z}_0$ - or the interval $(\\min E,c_u(L))$ - if we are dealing with $\\mathcal{Z}_M$. For every $\\kappa\\in I$, consider the minimax value\n\\begin{equation}\n\\label{minimax}\nc(\\kappa) := \\inf_{K \\in \\Gamma} \\max_{(x,T)\\in K} {\\mathbb{S}}_{\\kappa} (x,T).\n\\end{equation}\nBy Lemmas \\ref{Lem:mountain pass}, \\ref{notempty}, and \\ref{mtpass2}, $c(\\kappa)$ is finite and positive, and since ${\\mathbb{S}}_{\\kappa}$ depends monotonically on $\\kappa$, the function\n\\[\nc:I \\rightarrow (0,+\\infty)\n\\]\nis weakly increasing. By Lebesgue Theorem, the set of points of $I$ at which $c$ has a linear modulus of continuity, that is\n\\[\nJ := \\bigl\\{ \\bar\\kappa\\in I\\, \\big| \\, \\exists \\delta>0, \\; \\exists M>0 \\mbox{ s.t. } |c(\\kappa) - c(\\bar{\\kappa})| \\leq M |\\kappa - \\bar{\\kappa}| \\mbox{ for every } \\kappa \\in I \\mbox{ with }|\\kappa - \\bar\\kappa| < \\delta\\bigr\\},\n\\]\nhas full Lebesgue measure in $I$.\n\n\\begin{Lemma}\n\\label{strarg}\nIf $\\bar \\kappa\\in J$, then ${\\mathbb{S}}_{\\bar\\kappa}$ admits a bounded (PS) sequence at level $c(\\bar\\kappa)$, which consists of contractible loops.\n\\end{Lemma}\n\n\\begin{proof}\nFirst recall that $\\Gamma$ is a class of subsets of $\\mathcal{M}^{\\mathrm{contr}}$.\nLet $(\\kappa_h)\\subset I$ be a strictly decreasing sequence which converges to $\\bar\\kappa$, and set $\\epsilon_h:=\\kappa_h-\\bar \\kappa\\downarrow0$. We pick $K_h \\in \\Gamma$ such that\n\\[\n\\max_{K_h}{\\mathbb{S}}_{\\kappa_h}\\leq c(\\kappa_h)+\\epsilon_h.\n\\]\nLet $z=(x,T)\\in K_h$ be such that ${\\mathbb{S}}_{\\bar \\kappa}(z)>c(\\bar \\kappa)-\\epsilon_h$. Since $\\bar\\kappa$ belongs to $J$, we have\n\\[\nT = \\frac{{\\mathbb{S}}_{\\kappa_h}(z)-{\\mathbb{S}}_{\\bar \\kappa}(z)}{\\kappa_h-\\bar\\kappa} \\leq \\frac{c(\\kappa_h)+\\epsilon_h-c(\\bar\\kappa)+\\epsilon_h}{\\epsilon_h}\\leq M+2.\n\\]\nMoreover,\n\\[\n{\\mathbb{S}}_{\\bar \\kappa}(z) \\leq {\\mathbb{S}}_{\\kappa_h}(z) \\leq c(\\kappa_h)+\\epsilon_h \\leq c(\\bar \\kappa)+(M+1)\\epsilon_h.\n\\]\nBy the above considerations,\n\\[\nK_h \\subset A_h \\cup\\big\\{{\\mathbb{S}}_{\\bar \\kappa}\\leq c(\\bar \\kappa)-\\epsilon_h\\big\\},\n\\]\nwhere\n\\[\nA_h := \\big\\{(x,T)\\,\\big|\\, T\\leq M+2\\mbox{ and }{\\mathbb{S}}_{\\bar \\kappa} (x,T) \\leq c(\\bar \\kappa)+(M+1)\\epsilon_h\\big\\}.\n\\]\nIf $(x,T)$ belongs to $A_h$, we have the estimate\n\\[\n{\\mathbb{S}}_{\\bar\\kappa}(x,T)  \\geq \\frac{L_0}{M+2}\\|x'\\|_2^2-(M+2)(L_1-\\bar\\kappa),\n\\]\n(see (\\ref{isbd})), which shows that $A_h$ is bounded in $\\mathcal{M}$, uniformly in $h$. Let $\\phi$ be the flow of the vector field obtained by multiplying $-\\nabla{\\mathbb{S}}_{\\bar \\kappa}$ by a suitable non-negative function, whose role is to make the vector field bounded on $\\mathcal{M}$ and vanishing on the sublevel $\\{{\\mathbb{S}}_{\\bar\\kappa} \\leq c(\\bar\\kappa)\/4\\}$, while keeping the uniform decrease condition\n\\begin{equation}\n\\label{ud}\n\\frac{d}{d\\sigma} {\\mathbb{S}}_{\\bar\\kappa} \\bigl(\\phi_{\\sigma}(z)\\bigr) \\leq - \\frac{1}{2} \\min \\bigl\\{\n\\|d{\\mathbb{S}}_{\\bar\\kappa} (\\phi_{\\sigma}(z)) \\|^2,1 \\bigr\\}, \\quad \\mbox{if } {\\mathbb{S}}_{\\bar\\kappa} (\\phi_{\\sigma}(z)) \\geq c(\\bar\\kappa)\/2.\n\\end{equation}\nSee (\\ref{decr}) and Remarks \\ref{noncomp}, \\ref{trunc}. Then Lemma \\ref{Lem:4} implies that $\\phi$ is well-defined on $[0,+\\infty[\\times \\mathcal{M}$, and the class of sets $\\Gamma$ is positively invariant with respect to $\\phi$. Since $\\phi$ maps bounded sets into bounded sets, we have\n\\begin{equation}\n\\label{dove}\n\\phi([0,1]\\times K_h) \\subset B_h \\cup \\bigl\\{{\\mathbb{S}}_{\\bar \\kappa}\\leq c(\\bar \\kappa)-\\epsilon_h \\bigr\\},\n\\end{equation}\nfor some uniformly bounded set\n\\begin{equation}\n\\label{bdd}\nB_h\\subset \\bigl\\{{\\mathbb{S}}_{\\bar \\kappa}\\leq c(\\bar \\kappa)+(M+1)\\epsilon_h \\bigr\\}.\n\\end{equation}\nWe claim that there exists a sequence $(z_h)\\subset \\mathcal{M}^{\\mathrm{contr}}$ with\n\\[\nz_h \\in B_h\\cap \\bigl\\{{\\mathbb{S}}_{\\bar \\kappa}\\geq c(\\bar \\kappa)-\\epsilon_h \\bigr\\},\n\\]\nand $\\| d{\\mathbb{S}}_{\\bar \\kappa}(z_h)\\|$ infinitesimal. Such a sequence is clearly a bounded (PS) sequence at level $c(\\bar\\kappa)$.\nAssume, by contradiction, the above claim to be false. Then there exists $0<\\delta<1$ which satisfies\n\\[\n\\|d{\\mathbb{S}}_{\\bar \\kappa}\\| \\geq \\delta \\quad \\mbox{on } B_h \\cap \\bigl\\{{\\mathbb{S}}_{\\bar \\kappa}\\geq c(\\bar \\kappa)-\\epsilon_h\\bigr\\},\n\\]\nfor every $h$ large enough. Together with (\\ref{ud}), (\\ref{dove}) and (\\ref{bdd}), this implies that, for $h$ large enough, for any $z\\in K_h$ such that\n\\[\n\\phi([0,1]\\times \\{z\\}) \\subset \\bigl\\{ {\\mathbb{S}}_{\\kappa} \\geq c(\\bar \\kappa)-\\epsilon_h\\bigr\\},\n\\]\nthere holds\n\\[\n{\\mathbb{S}}_{\\bar \\kappa} \\big(\\phi_1(z)\\big) \\leq {\\mathbb{S}}_{\\bar \\kappa}(z)- \\frac{1}{2} \\delta^2 \\leq c(\\bar \\kappa)+ (M+1) \\epsilon_h- \\frac{1}{2} \\delta^2.\n\\]\nIt follows that\n\\[\n\\max_{\\phi_1(K_h)} {\\mathbb{S}}_{\\bar \\kappa} \\leq c(\\bar \\kappa)-\\epsilon_h,\n\\]\nfor $h$ large enough. Since $\\phi_1(K_h)$ belongs to $\\Gamma$, this contradicts the definition of $c(\\bar\\kappa)$ and concludes the proof.\n\\end{proof}\n\n\\paragraph{\\bf Existence of periodic orbits of low energy} We are finally ready to prove the following result, which is statement (iii) in the theorem of the Introduction:\n\n\\begin{Thm}\n\\label{ae}\nFor almost every $\\kappa \\in (\\min E,c_u(L))$, there is a contractible periodic orbit $\\gamma$ of energy $E(\\gamma,\\gamma')=\\kappa$ and positive action ${\\mathbb{S}}_{\\kappa}(\\gamma)=c(\\kappa)$.\n\\end{Thm}\n\n\\begin{proof}\nLet $\\kappa$ be an element of the full measure set $J\\subset I$. \nBy Lemma \\ref{strarg}, ${\\mathbb{S}}_{\\kappa}$ admits a (PS)$_{c(\\kappa)}$ sequence $(x_h,T_h)\\subset \\mathcal{M}^{\\mathrm{contr}}$ with $(T_h)$ bounded. By Lemma \\ref{PST0}, $(T_h)$ is bounded away from zero, because $c(\\kappa)>0$.\nBy Lemma \\ref{Lem;2}, the sequence $(x_h,T_h)$ has a limiting point in $\\mathcal{M}^{\\mathrm{contr}}$, which gives us the required contractible periodic orbit.\n\\end{proof}\n\n\\begin{Rmk}\nThe existence of a periodic orbit for almost energy level in $(\\min E,e_0(L))$ can be proved also by an argument from symplectic topology. In fact, let $H:T^*M \\rightarrow {\\mathbb{R}}$ be the Hamiltonian which is Legendre dual to $L$.\nThe fact that $\\kappa< e_0(L)$ implies that the restriction of the projection $T^*M\\rightarrow M$ to $H^{-1}(\\kappa)$ is not surjective. Therefore, one can build a Hamiltonian diffeomorphism of $T^*M$ which displaces $H^{-1}(\\kappa)$ from itself (see \\cite[Proposition 8.2]{con06}). Sets which are displaceable by a Hamiltonian diffeomorphism have finite $\\pi_1$-sensitive Hofer-Zehnder capacity (see \\cite{sch06} and \\cite{fs07}) and this fact implies the almost everywhere existence result for periodic orbits (see \\cite{hz94}). See \\cite[Corollary 8.3]{con06} for more details on such a proof.\n\\end{Rmk}\n\nThe next result shows that stable energy levels of Tonelli Hamiltonians posses periodic orbits, proving statement (iv) of the theorem in the Introduction. In particular, the same is true for contact type energy levels.\n\n\\begin{Cor}\nAssume that $\\kappa$ is a regular value of the Tonelli Hamiltonian $H\\in C^{\\infty}(T^*M)$ and that the hypersurface\n$\\Sigma=H^{-1}(\\kappa)$ is stable. Then $\\Sigma$ carries a periodic orbit.\n\\end{Cor}\n\n\\begin{proof}\nBy stability, we can find a diffeomorphism \n\\[\n(-\\epsilon, \\epsilon) \\times \\Sigma \\rightarrow T^*M, \\qquad (r,x) \\mapsto \\psi_r(x),\n\\]\nonto an open neighborhood of $\\Sigma$ such that $\\psi_0$ is the identity on $\\Sigma$ and \n\\[\n\\psi_r : \\Sigma \\rightarrow \\Sigma_r := \\psi_r(\\Sigma)\n\\]\ninduces an isomorphism between the line bundles $\\mathcal{L}_{\\Sigma}$ and $\\mathcal{L}_{\\Sigma_r}$. Up to the choice of a smaller $\\epsilon$, we may assume that all the hypersurfaces $\\Sigma_r$ are levels of a uniformly convex function. Therefore, they are the level sets of a Tonelli Hamiltonian $K\\in C^{\\infty}(T^* M )$ (see \\cite{mp10} for a detailed construction of $K$). Since the Legendre transform of $K$ is a Tonelli Lagrangian, Theorems \\ref{highthm} and \\ref{ae} imply that $K^{-1}(\\kappa)$ has periodic orbits for almost every $\\kappa$. In particular, $\\Sigma_r$ has periodic orbits for almost every $r$, but since the dynamics on $\\Sigma_r$ and on $\\Sigma$ are conjugated, the same is true for $\\Sigma$. \n\\end{proof}\n\n\\begin{Rmk}\nThe above proof shows the usefulness of having a theory which works with Tonelli Lagrangians, rather than just electromagnetic ones. In fact even if the stable hypersurface $\\Sigma$ is the level set of an electromagnetic Hamiltonian (that is, it is fiberwise an ellipsoid), the hypersurfaces $\\Sigma_r$ given by the stability assumption may be more general fiberwise uniformly convex hypersurfaces. \n\\end{Rmk}\n\n\\begin{Rmk}\nIf $E^{-1}(\\kappa)$ is of contact type and $\\pi_* : H_1(E^{-1}(\\kappa),{\\mathbb{R}}) \\rightarrow H_1(M,{\\mathbb{R}})$ is injective, then ${\\mathbb{S}}_{\\kappa}$ satisfies the Palais-Smale condition (with a suitable choice of the metric of $\\mathcal{M}$). See \\cite[Proposition F]{con06}. Therefore, in this case the existence of a periodic orbit can be obtained also without using stability.\n\\end{Rmk}\n\n\\begin{Rmk}\nIt can be proved that when $M$ is a closed surface and $L$ is of the form (\\ref{elmag}) with $V=0$ (that is, in the case of exact magnetic flows), there are periodic orbits on {\\em every} energy level below $c_0(L)$ (see  \\cite{tai92b}, \\cite{tai92c}, \\cite{tai92} and \\cite{cmp04}). In fact, the advantage of dealing with a surface is that when $\\kappa<c_0(L)$ one can minimize ${\\mathbb{S}}_{\\kappa}$ on a suitable space of {\\em embedded} closed curves. In the same setting, one can prove that for almost every energy level below $c_u(L)$ there are infinitely many periodic orbits, at least if all periodic orbits are assumed to be non-degenerate (see \\cite{amp13}).\n\\end{Rmk}\n\n\\paragraph{\\textbf{The two Lyapunov functions argument}} We conclude these notes by discussing an alternative argument to deal with the lack of (PS) which is exhibited by ${\\mathbb{S}}_{\\kappa}$ when $\\kappa<c_u(L)$. It allows to prove that the set of energy levels $\\kappa$ such that the Euler-Lagrange flow has a periodic orbit of energy $\\kappa$ is {\\em dense} in $(\\min E,c_u(L))$, a weaker statement than Theorem \\ref{ae}. However, it has some advantages, which are discussed in Remark \\ref{adv} below. This argument is used, in a different context, in \\cite{ama08}. Here we shall use it in order to prove the following weaker version of Theorem \\ref{ae}:\n\n\\begin{Thm}\n\\label{dense}\nLet $\\min E<\\bar \\kappa<c_u(L)$ and assume that there are no contractible periodic orbits of energy $\\bar\\kappa$ and non-negative action ${\\mathbb{S}}_{\\bar\\kappa}$.\nThen there exists a strictly decreasing sequence $(\\kappa_h)$ which converges to $\\bar\\kappa$ and is such that the Euler-Lagrange flow has a contractible periodic orbit $\\gamma_h$ with energy $\\kappa_h$ and period $T_h$, which satisfies ${\\mathbb{S}}_{\\kappa_h}(\\gamma_h)\/T_h \\downarrow 0$.\n\\end{Thm}\n\n\\begin{proof}\nWe argue by contradiction and  assume that there exists $\\tilde\\kappa>\\bar\\kappa$ and $\\delta>0$ such that for any $\\kappa \\in[\\bar \\kappa,\\tilde \\kappa]$ all the periodic orbits $\\gamma$ of\nenergy $\\kappa$ and period $T$ satisfy either ${\\mathbb{S}}_{\\kappa}(\\gamma)\/T\\geq\\delta$ or ${\\mathbb{S}}_{\\kappa} (\\gamma)\\leq 0$. Fix real numbers $a>c(\\bar \\kappa)$ and  $\\kappa^*\\in(\\bar \\kappa,\\tilde \\kappa]$. Assume that we can find $\\lambda\\in[0,1]$ and $(x,T)\\in\\mathcal{M}$ such that\n\\[\n\\lambda \\, d{\\mathbb{S}}_{\\bar \\kappa}(x,T)+(1-\\lambda)\\, d{\\mathbb{S}}_{\\kappa^*}(x,T)=0, \\quad 0<{\\mathbb{S}}_{\\bar \\kappa}(x,T)\\leq a.\n\\]\nThen $(x,T)$ is a critical point of ${\\mathbb{S}}_{\\lambda\\bar \\kappa+(1-\\lambda)\\kappa^*}$, hence it is a $T$-periodic orbit with energy $\\lambda\\bar \\kappa+(1-\\lambda)\\kappa^*$. By what we have assumed at the beginning, we have\n\\[\n\\delta \\leq \\frac{1}{T}{\\mathbb{S}}_{\\lambda\\bar \\kappa+(1-\\lambda)\\kappa^*}(x,T)\n= \\frac{1}{T} {\\mathbb{S}}_{\\bar \\kappa}(x,T)+(1-\\lambda)(\\kappa^*-\\bar \\kappa)\n\\leq \\frac{a}{T} + \\kappa^*-\\bar\\kappa.\n\\]\nUp to the choice of a smaller $\\kappa^*>\\bar\\kappa$, we may assume that $\\kappa^*-\\bar \\kappa\\leq \\delta\/2$. Then the above estimate implies that\n\\[\nT\\leq \\frac{2a}{\\delta} =: T^*.\n\\]\nWith these choices of $\\kappa^*$ and $T^*$, we can restate what we have proved so far as:\n\n\\begin{Lemma}\n\\label{oppo}\nIf $T>T^*$ and $0<{\\mathbb{S}}_{\\bar \\kappa}(x,T)\\leq a$, then the segment\n\\[\n\\mathrm{conv} \\bigl\\{ d{\\mathbb{S}}_{\\bar \\kappa}(x,T), d{\\mathbb{S}}_{\\kappa^*}(x,T) \\bigr\\} \\subset T_{(x,T)}^* \\mathcal{M}\n\\]\ndoes not contain $0$.\n\\end{Lemma}\n\nThe above lemma allows us to construct a negative pseudo-gradient vector field for ${\\mathbb{S}}_{\\bar\\kappa}$ which has all the good properties of $-\\nabla {\\mathbb{S}}_{\\bar\\kappa}$ and moreover has ${\\mathbb{S}}_{\\kappa^*}$ as a Lyapunov function on the open set\n\\[\nA:= \\bigl\\{T>T^*\\} \\cap \\{0< {\\mathbb{S}}_{\\bar\\kappa} < a \\bigr\\}.\n\\]\nIn fact, the only obstruction to finding a vector field $W$ whose flow make both ${\\mathbb{S}}_{\\bar\\kappa}$ and ${\\mathbb{S}}_{\\kappa^*}$ decrease in $A$, is that the differentials of ${\\mathbb{S}}_{\\bar\\kappa}$ and ${\\mathbb{S}}_{\\kappa^*}$ point in opposite directions in some point of $A$, and this is precisely what is excluded by Lemma \\ref{oppo}.\nMore precisely, one can prove the following:\n\n\\begin{Lemma}\n\\label{pg}\nThere exists a locally Lipschitz vector field $W$ on $\\mathcal{M}$ such that:\n\\begin{enumerate}\n\\item $d{\\mathbb{S}}_{\\bar\\kappa}[W]<0$ on $\\{{\\mathbb{S}}_{\\bar\\kappa}>0\\}$;\n\\item $W$ is forward complete and vanishes on $\\{{\\mathbb{S}}_{\\bar\\kappa}\\leq 0\\}$;\n\\item let $z_h=(x_h,T_h)$ be a sequence in $\\mathcal{M}^{\\mathrm{contr}}$ such that\n\\[\n0< \\inf {\\mathbb{S}}_{\\bar{\\kappa}}(z_h) \\leq \\sup {\\mathbb{S}}_{\\bar{\\kappa}}(z_h) < +\\infty, \\quad \\lim_{h\\rightarrow \\infty} d{\\mathbb{S}}_{\\bar\\kappa}(z_h)[W(z_h)] = 0,\n\\]\nand $(T_h)$ is bounded from above; then $(z_h)$ has a subsequence which converges in $\\mathcal{M}^{\\mathrm{contr}}$;\n\\item $d{\\mathbb{S}}_{\\kappa^*}[W]<0$ on $A$.\n\\end{enumerate}\n\\end{Lemma}\n\nIn fact, one can choose $W$ to be given by the vector field\n\\[\n\\nabla{\\mathbb{S}}_{\\bar \\kappa}+\\chi \\frac{\\|\\nabla{\\mathbb{S}}_{\\bar \\kappa}\\|}{\\|\\nabla{\\mathbb{S}}_{\\kappa^*}\\|}\\nabla{\\mathbb{S}}_{\\kappa^*}\n\\]\nmultiplied by a suitable non-positive function. Here $\\chi$ is a suitable cut-off function. See \\cite[Lemmas 5.1 and 5.4]{ama08} for a similar construction.\n\nWe can now prove Theorem \\ref{dense}.\nBy the definition of $c(\\bar\\kappa)$, there is a set $K$ in $\\Gamma$ such that\n\\[\n\\max_{K} {\\mathbb{S}}_{\\bar \\kappa} <a.\n\\]\nBy Lemma \\ref{pg} (i) and (ii), for every $\\sigma_0>0$ we have\n\\[\n\\inf_{\\sigma\\in \\sigma_0} \\Bigl| d{\\mathbb{S}}_{\\bar\\kappa}\\bigl(\\phi_{\\sigma}(z)\\bigr)\\bigl[W(\\phi_{\\sigma}(z))\\bigr] \\Bigr| \\leq \\frac{1}{\\sigma_0} \\int_0^{\\sigma_0} \\Bigl| d{\\mathbb{S}}_{\\bar\\kappa}\\bigl(\\phi_{\\sigma}(z)\\bigr)\\bigl[W(\\phi_{\\sigma}(z))\\bigr]\\Bigr| \\, d\\sigma = \\frac{\n{\\mathbb{S}}_{\\bar\\kappa}(z) - {\\mathbb{S}}_{\\bar\\kappa}(\\phi_{\\sigma_0}(z))}{\\sigma_0},\n\\]\nand, by the definition of $c(\\bar\\kappa)$,\n\\[\n\\max_{z\\in K} {\\mathbb{S}}_{\\bar\\kappa} \\bigl( \\phi_{\\sigma_0}(z) \\bigr) \\geq c(\\bar\\kappa).\n\\]\nBy taking a limit for $\\sigma_0\\rightarrow +\\infty$, thanks to Lemma \\ref{pg} (ii), the above facts imply that $\\phi_{{\\mathbb{R}}^+}(K)\\cap \\{{\\mathbb{S}}_{\\bar\\kappa}>0\\}$ contains a sequence $z_h=(x_h,T_h)$ such that\n\\[\n0< c(\\bar\\kappa) \\leq \\inf {\\mathbb{S}}_{\\bar{\\kappa}}(z_h) \\leq \\sup {\\mathbb{S}}_{\\bar{\\kappa}}(z_h) < a \\quad \\mbox{and} \\quad \\lim_{h\\rightarrow \\infty} d{\\mathbb{S}}_{\\bar\\kappa}(z_h)[W(z_h)] = 0.\n\\]\nIt is enough to show that $(T_h)$ is bounded from above: Indeed, in this case Lemma \\ref{pg} (iii) implies that $(z_h)$ has a limiting point, which is a critical point of ${\\mathbb{S}}_{\\bar\\kappa}$ with positive action, contradicting the hypothesis of Theorem \\ref{dense}.\n\nThe upper bound on $(T_h)$ is a consequence of the following claim: the period $T$ is bounded on the set $\\phi_{{\\mathbb{R}}^+}(K)\\cap \\{{\\mathbb{S}}_{\\bar\\kappa}>0\\}$.\nIn order to prove this claim, we first notice that\n\\begin{equation}\n\\label{hhh}\n{\\mathbb{S}}_{\\bar \\kappa}(x,T)\\leq a, \\;\\; T\\leq T^* \\quad \\Rightarrow \\quad\n{\\mathbb{S}}_{\\kappa^*}(x,T)\\leq a+(\\kappa^*-\\bar \\kappa)T^*=:b.\n\\end{equation}\nSince $K$ is compact, we can find $d>b$ such that $K\\subset \\{{\\mathbb{S}}_{\\kappa^*} < d\\}$. Let $\\phi$ be the flow of the vector field $W$. We claim that\n\\begin{equation}\n\\label{cc}\n\\phi_{{\\mathbb{R}}^+}(K) \\cap \\bigl\\{{\\mathbb{S}}_{\\bar\\kappa} > 0\\bigr\\} \\subset \\bigl\\{{\\mathbb{S}}_{\\kappa^*} < d\\bigr\\}.\n\\end{equation}\nIn fact, let $z\\in K$ and let $\\sigma_0>0$ be the first instant such that ${\\mathbb{S}}_{\\kappa^*}(\\phi_{\\sigma_0}(z))=d$, while ${\\mathbb{S}}_{\\bar{\\kappa}}(\\phi_{\\sigma_0}(z))> 0$. By Lemma \\ref{pg} (i), ${\\mathbb{S}}_{\\bar\\kappa}(\\phi_{\\sigma_0}(z)) \\leq {\\mathbb{S}}_{\\bar\\kappa}(z) < a$. By Lemma \\ref{pg} (iv), the point $\\phi_{\\sigma_0}(z)$ cannot belong to $A$, so $\\phi_{\\sigma_0}(z)=(x,T)$ with $T\\leq T^*$ and (\\ref{hhh}) implies that ${\\mathbb{S}}_{\\kappa^*}(\\phi_{\\sigma_0}(z)) \\leq b < d$. This contradiction proves (\\ref{cc}).\n\nIf ${\\mathbb{S}}_{\\bar\\kappa}(x,T)>0$ and ${\\mathbb{S}}_{\\kappa^*}(x,T)<d$, then\n\\[\nd >  {\\mathbb{S}}_{\\kappa^*}(x,T) = {\\mathbb{S}}_{\\bar\\kappa}(x,T) + (\\kappa^*-\\bar\\kappa) T > (\\kappa^*-\\bar\\kappa) T.\n\\]\nThis shows that the period $T$ is bounded on the set\n\\[\n\\bigl\\{{\\mathbb{S}}_{\\bar\\kappa} > 0\\bigr\\} \\cap \\bigl\\{{\\mathbb{S}}_{\\kappa^*} < d\\bigr\\},\n\\]\nand by (\\ref{cc}) it is bounded also on\n\\[\n\\phi_{{\\mathbb{R}}^+}(K) \\cap \\bigl\\{{\\mathbb{S}}_{\\bar\\kappa}>0\\},\n\\]\nas claimed.\n\\end{proof}\n\n\\begin{Rmk}\n\\label{adv}\nIn the Struwe monotonicity argument, one gets the existence of bounded (PS) sequences at level $c(\\bar{\\kappa})$, but has no control on the (PS) sequences at other levels. Therefore, it is not clear whether the space of negative gradient flow lines for ${\\mathbb{S}}_{\\bar\\kappa}$ which connect two given critical points - say with positive action - is bounded.\nAn advantage of the two Lyapunov functions argument, is that the latter fact is true for the flow lines of the vector field $W$ constructed in Lemma \\ref{pg}: the second Lyapunov function ${\\mathbb{S}}_{\\kappa^*}$ allows to exclude the existence of flow lines which go arbitrarily far and come back. This fact would allow to develop some global critical point theory for ${\\mathbb{S}}_{\\bar\\kappa}$, such as Morse theory or Lusternik-Schnirelmann theory. This is not useful here, because the a priori estimates which lead to the existence of the pseudo-gradient vector field $W$ come from a contradiction argument. However, it might be useful in situations where these a priori bounds have a different origin, such as for example in the case of tame energy levels (see \\cite{cfp10} for the definition of tameness and for motivating examples).\n\\end{Rmk}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:01}\nOn earth in liquid $^4$He the gravity is an external force which causes a space dependent \npressure $p=p(z)$ depending on the altitude coordinate $z$. Since the critical temperature \nof the superfluid transition $T_\\lambda=T_\\lambda(p)$ depends on the pressure $p$, in the \nhelium the critical temperature $T_\\lambda(z)=T_\\lambda(p(z))$ varies with the altitude $z$. \nIn leading approximation it is a linear function of the altitude\n\\begin{equation} \nT_\\lambda(z) = T_\\lambda(z_0) + (\\partial T_\\lambda \/ \\partial z) \\, (z-z_0)\n\\label{Eq:A01} \n\\end{equation} \nwhere the gradient is determined experimentally as \\cite{Al68}\n$\\partial T_\\lambda \/ \\partial z=+1.273\\,\\mathrm{\\mu K\/cm}$. The sign is positive \nwhich means that the critical temperature increases with the altitude $z$.\n\nIn thermal equilibrium, the local temperature of the helium $T(\\mathbf{r},t)=T$ is \nconstant with respect to any space and time variable. If in an experiment we choose \nthe temperature $T=T_\\lambda(z_0)$ we find an interface at $z=z_0$, which separates \nsuperfluid $^4$He in the upper region $z>z_0$ where $T<T_\\lambda(z)$ from normal-fluid \n$^4$He in the lower region $z<z_0$ where $T>T_\\lambda(z)$. This interface is the main \nconcern of the present paper.\n\nCorrelation effects imply an interface which is not sharply defined but smeared out \nover a certain length scale $\\xi_g$. Ginzburg and Sobyanin \\cite{GS76} have calculated \nthe order parameter profile $\\psi(z)$ for liquid $^4$He in gravity within their $\\psi$ \ntheory which is a mean-field theory modified by scaling functions in order to \nincorporate the effects of critical fluctuations and the critical exponents to some \nextent. They find the characteristic length scale $\\xi_g= 67\\,\\mathrm{\\mu m}$ \n(see Fig.\\ 4 and Eq.\\ (3.49) in Ref.\\ \\onlinecite{GS76}).\n\nA heat current $Q$ flowing from bottom to top in the direction of $z$ enhances the formation \nof the superfluid\/normal-fluid interface. Heat transport phenomena imply a space-dependent \ntemperature $T(z)$ with a negative gradient $\\partial T \/ \\partial z < 0$ which acts \nopposed to the positive gradient of the critical temperature $T_\\lambda(z)$. Onuki \n\\cite{On83,On87} has investigated the interface under a heat flow $Q$ within a dynamic \nmean-field theory modified by scaling functions. He finds that the thickness of the \ninterface decreases according to $\\xi_Q\\sim Q^{-1\/2}$ with increasing heat current $Q$.\n\nWhile on earth the gravity acceleration $g=9.81\\,\\mathrm{m\/s^2}$ is constant, the \nheat current $Q$ can be varied in the experiment. For large heat currents $Q\\gtrsim Q_0$, \nthe heat-current effects dominate, where on the other hand for small heat currents \n$Q\\lesssim Q_0$ gravity effects dominate. The heat current which separates both regimes, \nis about $Q_0=70\\,\\mathrm{nW\/cm^2}$. In this paper we focus on small heat currents where \ngravity is the main effect.\n\nThe critical dynamics of liquid $^4$He near the superfluid transition $T_\\lambda$ is \ndescribed by a hydrodynamic model with Gaussian fluctuating forces which is called \nmodel $F$ in the classification of Hohenberg and Halperin \\cite{HH77}. This model has \noriginally been derived by Halperin, Hohenberg, and Siggia \\cite{HHS74} in order to \ndescribe the critical dynamics of a planar ferromagnet, which is in the same universality \nclass as liquid $^4$He. The field-theoretic renormalization-group theory of model $F$ has \nbeen elaborated by Dohm \\cite{Do85,Do91}. The specific heat and the thermal conductivity \nhave been calculated up to two-loop order \\cite{Do85} and compared with very accurate \nexperimental data \\cite{Al84,Al85}. In this way, the renormalized coupling parameters \nand some other parameters have been adjusted \\cite{Do91} so that all parameters of \nmodel $F$ are known. Thus model $F$ is ready for application without any further adjustable \nparameters.\n\nIn this paper we present a renormalization-group (RG) calculation of the \nsuperfluid\/normal-fluid interface based on model $F$. The calculation is technically very \ndifficult and challenging for two reasons. First the Green functions and Feynman \ndiagrams must be evaluated in a spatially inhomogeneous system. Secondly, the \nrenormalization factors depend on space and time coordinates via the RG flow parameter \nso that the partial derivatives with respect to space and time must be replaced by \nappropriate \\emph{covariant} derivatives.\n\nThe first challenge was overcome step by step in several previous papers. On the \nnormal-fluid side of the interface the Green function was calculated \\cite{HD91} \nfor a zero order parameter $\\langle\\psi(z)\\rangle=0$ and a linear temperature \nparameter $r_0(z) = a_0 + b_0 z$. The local thermal conductivity $\\lambda_T(T,Q)$ \nand the related temperature profile $T(z)$ was calculated. On the superfluid side \nof the interface the Green function and related thermodynamic quantities were \ncalculated \\cite{HD92} for a plane-wave order parameter \n$\\langle\\psi(z)\\rangle=\\eta\\,e^{ikz}$ and a constant temperature parameter $r_0$. \nHere a critical superfluid current was found which implies a depression of the \nsuperfluid transition temperature $T_\\lambda(Q)<T_\\lambda$ by a nonzero heat \ncurrent $Q$.\n\nLater the normal-fluid approach \\cite{HD91} was extended beyond the interface into \nthe superfluid region \\cite{Ha99,Ha99a}. The calculation was made self consistent by \na lowest-order $1\/n$ expansion which is equivalent to the Hartree approximation of \nquantum many-particle physics. In this way the superfluid region could be reached \nwhere in the whole system the average order parameter $\\langle\\psi(z)\\rangle=0$ is \nzero due to phase fluctuations related to the motion of vortices where however the \ncondensate density $n_\\mathrm{s}=\\langle |\\psi(z)|^2 \\rangle$ and the superfluid \ncurrent $\\mathbf{J}_\\mathrm{s}=\\langle \\mathrm{Im}[\\psi^*\\nabla\\psi] \\rangle$ are \nmacroscopically large. The RG theory was applied locally using a local flow parameter \nwhich depends on the altitude coordinate $z$. The specific heat $C_Q$ and the thermal \nconductivity $\\lambda_T$ were calculated for the whole superfluid\/normal-fluid \ninterface where the effects of the gravity acceleration $g$ and the heat current $Q$ \nwere included. The temperature profile $T(z)$ was obtained by integrating the \nheat-transport equation $\\mathbf{Q}=-\\lambda_T \\nabla T$.\n\nIn the superfluid region a nonzero temperature gradient $\\nabla T$ was found which \nis due to a nonzero thermal resistivity induced by the motion of vortices and \nquantum turbulence. The theory was especially successful to describe the so called \nself-organized critical state, which was predicted by Onuki \\cite{On87} and which \nwas discovered in the experiment by Moeur \\emph{et al.}\\ \\cite{Du97}. In this \nstate the temperature gradient $\\nabla T$ is equal to the gravity induced gradient \n$\\nabla T_\\lambda$ of \\eqref{Eq:A01}, i.e.\\ $\\nabla T=\\nabla T_\\lambda$, so that \nthe system is homogeneous over a large area in space.\n\nHowever, for the superfluid\/normal-fluid interface the self-consistent approach \n\\cite{Ha99,Ha99a} works only for large heat currents \n$Q\\gtrsim Q_0=70\\,\\mathrm{nW\/cm^2}$ where the heat current $Q$ dominates over the \neffects of gravity $g$. For smaller heat currents this approach does not yield a result. \nThe existence and motion of vortices is essential for phase fluctuations in order to \nhave a zero average order parameter $\\langle\\psi(z)\\rangle=0$. \n\nFor small heat currents $Q\\lesssim Q_0=70\\,\\mathrm{nW\/cm^2}$ vortices are not present\nso that the average order parameter $\\langle\\psi(z)\\rangle$ is nonzero. In this case, \na local calculation is not possible. Instead, the full model-$F$ equations must be solved \nas partial differential equations. Here the second challenge arises if the RG theory is \ninvolved. The RG flow parameter is determined locally by a constraint condition so that \nit will depend on space and time. This fact requires the definition of \\emph{covariant} \ndifferential operators. A first step for this kind of theory was made by the author \nand Nikodem \\cite{HN08}. The interface was investigated in thermal equilibrium where \nonly the gravity acceleration $g$ is present but no heat current. The covariant \nderivatives were defined for the renormalized order parameter and for the renormalized \ntemperature parameter. The renormalized Ginzburg-Landau equation was solved numerically \nas a boundary value problem by the multiple-shooting algorithm. Results for the \norder-parameter profile $\\langle\\psi(z)\\rangle$, the correlation length $\\xi$, and \nthe heat capacity $C$ were obtained. However, the calculations \\cite{HN08} were not \nfinished and not published.\n\nThe present paper is devoted to continue, extend, and publish our recent calculations \n\\cite{HN08}. We develop a local and time-dependent RG theory for small heat currents \n$Q\\lesssim Q_0=70\\,\\mathrm{nW\/cm^2}$ in order to fill the gap which our previous \ntheory \\cite{Ha99,Ha99a} has left. We solve the partial differential equations of \nmodel $F$ together with a local constraint condition for the RG flow parameter. We \ncalculate the average order parameter profile $\\langle\\psi(z)\\rangle$ and the \ntemperature profile $T(z)$ for the superfluid\/normal-fluid interface. Furthermore, \nwe calculate the related thermodynamic and transport quantities, i.e.\\ the specific \nheat $C_Q$ and the thermal conductivity $\\lambda_T$ or thermal resistivity \n$\\rho_T=1\/\\lambda_T$. The calculations are not restricted to a stationary state of a \nconstant heat current $Q$. More generally, we solve the model-$F$ equations as \ntime-dependent partial differential equations, so that time-dependent and transient \neffects can be handled like the propagation of second sound.\n\nThe paper is organized as follows. In Sec.\\ \\ref{Sec:02} we briefly describe model $F$, \nthe reduced hydrodynamic model for the critical dynamics of liquid $^4$He near the \nsuperfluid transition. Furthermore, we explain the approximations that we use. In \nSec.\\ \\ref{Sec:03} we develop our method in order to solve the model-$F$ equations together \nwith local constraint conditions for the local RG theory. In Sec.\\ \\ref{Sec:04} we present \nour numerical results for the superfluid\/normal-fluid interface in gravity where small \nheat currents are flowing upward or downward. We compare our results with the experiment \nof Chatto \\emph{et al.}\\ \\cite{Du07} and find good agreement for the local thermal \nresistivity. In Sec.\\ \\ref{Sec:05} we compare our small-heat-current results with the \nlarge-heat-current results of our previous approach \\cite{Ha99,Ha99a}. We discuss the \nstability of the solutions of our present and our previous approach. Finally, in \nSec.\\ \\ref{Sec:06} we compare our present and our previous approach with other theories \nand recent experiments. We discuss and conclude to which extent our theory can describe \nmutual friction effects for larger heat currents due to the motion of vortices and quantum \nturbulence.\n\n\n\\section{Model and approximation}\n\\label{Sec:02}\nThe local thermodynamic properties of liquid $^4$He are described by the three standard \nhydrodynamic variables, the mass density $\\rho(\\mathbf{r},t)$, the mass-current density \n$\\mathbf{j}(\\mathbf{r},t)$, and the entropy density $\\sigma(\\mathbf{r},t)$. Since $^4$He \nbecomes superfluid below the critical temperature $T_\\lambda\\approx 2\\,K$, there exists \nan additional fourth hydrodynamic variable, the macroscopic wave function \n$\\psi(\\mathbf{r},t)$, which is the order parameter of the superfluid phase transition. \nThe full hydrodynamic equations for superfluid $^4$He described by all these four \nvariables have been derived long ago by Pitaevski \\cite{Pi59}.\n\nFor the critical dynamics near $T_\\lambda$ the mass density $\\rho$ and the mass-current \ndensity $\\mathbf{j}$ are irrelevant variables, because the related hydrodynamic modes,\nfirst sound and viscosity effects, are fast. $\\rho$ and $\\mathbf{j}$ can be eliminated or \nintegrated out, so that the remaining relevant variables for the critical slow modes near \nthe transition (second sound and order parameter relaxation) are the order parameter \n$\\psi$ and the entropy density $\\sigma$. For these two relevant variables, the \nhydrodynamic equations are given by model $F$ \\cite{HH77} and read\n\\begin{eqnarray}\n\\frac{\\partial\\psi}{\\partial t} &=& - 2 \\Gamma_0 \\frac{\\delta H}{\\delta\\psi^*}\n+ i g_0\\, \\psi \\frac{\\delta H}{\\delta m} + \\theta_\\psi \\ ,\n\\label{Eq:B01} \\\\\n\\frac{\\partial m}{\\partial t} &=& \\lambda_0 \\nabla^2 \\frac{\\delta H}{\\delta m}\n- 2 g_0\\, \\mathrm{Im} \\Bigl( \\psi^* \\frac{\\delta H}{\\delta\\psi^*} \\Bigr) \n+ \\theta_m \\ .\n\\label{Eq:B02} \n\\end{eqnarray}\nFor convenience and historical reasons, the dimensionless entropy density is denoted \nby $m=\\sigma\/k_\\mathrm{B}$. In the equations\n\\begin{eqnarray}\nH &=&\\  \\int d^dr\\, \\bigl[ \\textstyle{\\frac{1}{2}} \\tau_0 |\\psi|^2\n+ \\textstyle{\\frac{1}{2}} |\\nabla\\psi|^2 + \\tilde{u}_0 |\\psi|^4 \\nonumber\\\\\n&&\\hspace{12mm} + \\textstyle{\\frac{1}{2}} \\chi_0^{-1} m^2 + \\gamma_0 m |\\psi|^2 \n- h_0 m \\bigr]\n\\label{Eq:B03} \n\\end{eqnarray}\nis the free energy functional divided by $k_B T$. The Gaussian stochastic forces \n$\\theta_\\psi$ and $\\theta_m$ incorporate the fluctuations. They are defined by the \naverages $\\langle \\theta_\\psi\\rangle=0$, $\\langle \\theta_m\\rangle=0$, and by the \ncorrelations\n\\begin{eqnarray}\n\\langle \\theta_\\psi(\\mathbf{r},t)\\theta^*_\\psi(\\mathbf{r}^\\prime,t^\\prime)\\rangle &=&\n4\\,\\Gamma_0\\, \\delta(\\mathbf{r}-\\mathbf{r}^\\prime)\\,\\delta(t-t^\\prime) \\ ,\n\\label{Eq:B04} \\\\\n\\langle \\theta_m(\\mathbf{r},t)\\theta_m(\\mathbf{r}^\\prime,t^\\prime)\\rangle &=&\n-2\\,\\lambda_0\\,\\nabla^2\\, \\delta(\\mathbf{r}-\\mathbf{r}^\\prime)\\,\\delta(t-t^\\prime) \\ . \\qquad\n\\label{Eq:B05} \n\\end{eqnarray}\nThe dimension of the space $d$ is assumed to be arbitrary and continuous in the \ngeneral calculations. However, eventually we set $d=3$ when evaluating explicit \nresults for liquid $^4$He in a three-dimensional cell. For the calculations in the \ncritical regime the model-$F$ equations \\eqref{Eq:B01}-\\eqref{Eq:B03} are treated by \nfield-theoretic means, i.e.\\ perturbation series expansion with respect to Feynman \ndiagrams, renormalization, and the renormalization group. For example, the heat \ncapacity and the thermal conductivity were evaluated up to two-loop order \\cite{Do85}.\n\nIn this paper, we use an approximation following our previous work \\cite{Ha99,Ha99a}. In \nmany-particle physics this approximation is known as the \\emph{Hartree approximation}\n(see e.g.\\ Ref.\\ \\onlinecite{FW71}). It is a self-consistent approximation including \nonly a single one-loop diagram which is the tadpole diagram. Alternatively, the approximation \nis obtained by the $1\/n$ expansion in leading order, where $n$ is the number of complex \nfields in a generalized model with a generalized order parameter \n$\\Psi=(\\psi_1,\\ldots,\\psi_n)$.\n\nFor the model-$F$ equations \\eqref{Eq:B01} and \\eqref{Eq:B02} the approximation is obtained \nby taking the nonequilibrium average $\\langle\\cdots\\rangle$ for all terms and by \nperforming appropriate factorizations of the averages of products of the fluctuating \nhydrodynamic variables $\\psi$, $\\psi^*$, and $m$. The factorizations are justified by \ninspection of the Feynman diagrams of the Hartree approximation which are shown in Fig.\\ 2 \nof Ref.\\ \\onlinecite{Ha99a}. We factorize the nonlinear terms according to\n\\begin{eqnarray}\n2 \\Bigl\\langle \\frac{\\delta H}{\\delta \\psi^*} \\Big\\rangle &\\approx& \n\\bigl[\\tau_0 - \\nabla^2 + 4 \\tilde{u}_0 \\langle |\\psi|^2 \\rangle \n+ 2 \\gamma_0 \\langle m \\rangle \\bigr] \\langle\\psi\\rangle \\ , \\qquad\n\\label{Eq:B06} \\\\\n\\Bigl\\langle \\psi \\frac{\\delta H}{\\delta m} \\Big\\rangle &\\approx&\n\\langle \\psi \\rangle\\, \\Bigl\\langle\\frac{\\delta H}{\\delta m} \\Big\\rangle \\ ,\n\\label{Eq:B07} \n\\end{eqnarray}\nwhere\n\\begin{equation} \n\\Bigl\\langle\\frac{\\delta H}{\\delta m} \\Big\\rangle = \\chi_0^{-1} \\langle m \\rangle \n+ \\gamma_0 \\langle |\\psi|^2 \\rangle - h_0 \\ .\n\\label{Eq:B08} \n\\end{equation}\nWithout an approximation we obtain \n\\begin{equation} \n-2\\,\\Bigl\\langle \\mathrm{Im} \\Bigl( \\psi^* \\frac{\\delta H}{\\delta \\psi^*} \\Bigr) \\Bigr\\rangle \n= \\nabla \\langle \\mathrm{Im}[ \\psi^* \\nabla \\psi ] \\rangle \\ .\n\\label{Eq:B09} \n\\end{equation}\nConsequently, from \\eqref{Eq:B01} and \\eqref{Eq:B02} we obtain the approximate equations\n\\begin{eqnarray}\n\\frac{\\partial \\langle\\psi\\rangle}{\\partial t} &=& - \\Gamma_0 \\bigl[ \\tau_0 - \\nabla^2 \n+ 4 \\tilde{u}_0 n_\\mathrm{s} + 2 \\gamma_0 \\langle m \\rangle \\bigr] \\langle\\psi\\rangle \\nonumber\\\\\n&&+ i g_0 \\bigl[ \\chi_0^{-1} \\langle m \\rangle + \\gamma_0 n_\\mathrm{s} - h_0 \\bigr] \n\\langle\\psi\\rangle \\ ,\n\\label{Eq:B10} \\\\\n\\frac{\\partial \\langle m\\rangle}{\\partial t} &=& \\lambda_0 \\nabla^2 \n\\bigl[ \\chi_0^{-1} \\langle m \\rangle + \\gamma_0 n_\\mathrm{s} - h_0 \\bigr] \n+ g_0\\, \\nabla \\mathbf{J}_\\mathrm{s} \\ , \\qquad\n\\label{Eq:B11} \n\\end{eqnarray}\nwhere we define the condensate density $n_\\mathrm{s}$ and the superfluid current density \n$\\mathbf{J}_\\mathrm{s}$ by\n\\begin{eqnarray}\nn_\\mathrm{s} &=& \\langle |\\psi|^2 \\rangle \\ ,\n\\label{Eq:B12} \\\\\n\\mathbf{J}_\\mathrm{s} &=& \\langle \\mathrm{Im}[ \\psi^* \\nabla \\psi ] \\rangle \\ ,\n\\label{Eq:B13} \n\\end{eqnarray}\nrespectively. \n\nNext, for convenience and simplification of the equations we define the temperature \nparameters\n\\begin{eqnarray}\n\\Delta r_0 &=& 2 \\chi_0 \\gamma_0\\, \\Bigl\\langle\\frac{\\delta H}{\\delta m} \\Big\\rangle \n\\nonumber\\\\\n&=& 2 \\chi_0 \\gamma_0\\, \\bigl[ \\chi_0^{-1} \\langle m \\rangle + \\gamma_0 n_\\mathrm{s} - h_0 \\bigr] \\ ,\n\\label{Eq:B14} \\\\\nr_0 &=& \\tau_0 + 2 \\chi_0 \\gamma_0 h_0 + \\Delta r_0 \\nonumber\\\\\n&=& \\tau_0 + 2 \\chi_0 \\gamma_0\\, \\bigl[ \\chi_0^{-1} \\langle m \\rangle + \\gamma_0 n_\\mathrm{s}\\bigr] \\ ,\n\\label{Eq:B15}\n\\end{eqnarray}\nand the modified temperature parameter \n\\begin{equation}\nr_1 = r_0 + 4 u_0 n_\\mathrm{s} \\ ,\n\\label{Eq:B16}\n\\end{equation}\nwhere $u_0=\\tilde{u}_0 - \\frac{1}{2} \\chi_0 \\gamma_0^2$ is a combined coupling constant\nfollowing Ref.\\ \\onlinecite{Do85}. Thus, the model-$F$ equations can be written in the \nsimple form\n\\begin{eqnarray}\n\\frac{\\partial \\langle\\psi\\rangle}{\\partial t} &=& - \\Gamma_0 \\bigl[ r_1 - \\nabla^2 \\bigr] \n\\langle\\psi\\rangle + i \\frac{g_0}{2 \\chi_0 \\gamma_0} \\Delta r_0 \\langle\\psi\\rangle\\ , \\qquad\n\\label{Eq:B17} \\\\\n\\frac{\\partial \\langle m\\rangle}{\\partial t} &=& - \\nabla \\mathbf{q} \\ .\n\\label{Eq:B18}\n\\end{eqnarray}\nThe last equation is the heat transport equation where \n$\\langle m \\rangle = \\langle\\sigma\\rangle \/ k_\\mathrm{B}$ is the dimensionless \nentropy density and\n\\begin{equation} \n\\mathbf{q} = - \\frac{\\lambda_0}{2 \\chi_0 \\gamma_0} \\nabla \\Delta r_0 - g_0 \\mathbf{J}_\\mathrm{s}\n\\label{Eq:B19} \n\\end{equation}\nis the dimensionless entropy current density. The latter is related to the heat \ncurrent $\\mathbf{Q}$ in standard physical units by \n$\\mathbf{q}=\\mathbf{Q}\/k_\\mathrm{B} T\\approx \\mathbf{Q}\/k_\\mathrm{B} T_\\lambda$. \n\nThe order-parameter equation \\eqref{Eq:B17} can be written in the form \n$L \\langle\\psi\\rangle=0$ where the operator $L$ is defined in (3.13) of our previous \npaper \\cite{Ha99a} and related to the off-diagonal matrix elements of the inverse \nGreen function. This observation shows that the factorizations of the present \napproach are equivalent to the self-consistent approximation in our previous paper.\nWe note that the factorization is applied only in the order parameter equation \n\\eqref{Eq:B17}. The heat transport equation \\eqref{Eq:B18} is derived without \nany factorization or approximation.\n\nThe parameters $\\Delta r_0$ and $r_0$ are related to the local space and time \ndependent temperature $T=T(\\mathbf{r},t)$ and to the critical temperature \n$T_\\lambda=T_\\lambda(z)$ of \\eqref{Eq:A01} according to \n\\begin{eqnarray}\n\\Delta r_0 &=& 2 \\chi_0 \\gamma_0\\, \\Bigl\\langle\\frac{\\delta H}{\\delta m} \\Big\\rangle \n= 2 \\chi_0 \\gamma_0\\, \\frac{T-T_0}{T_\\lambda} \\ , \\qquad\n\\label{Eq:B20} \\\\\nr_0 - r_{0\\mathrm{c}} &=& 2 \\chi_0 \\gamma_0\\, \\frac{T-T_\\lambda}{T_\\lambda} \\ ,\n\\label{Eq:B21}\n\\end{eqnarray}\nwhere $T_0$ is a constant reference temperature. These equations have been derived in our \nprevious paper \\cite{Ha99a}. The critical value of $r_0$ is $r_{0\\mathrm{c}}=0$ in one-loop \napproximation \\cite{Do85} and hence also in our self-consistent approximation. The factor \n$T_\\lambda$ in the denominators is easily explained. Since $H$ is the free energy divided \nby $k_\\mathrm{B} T_\\lambda$ and since $m$ is the entropy density divided by $k_\\mathrm{B}$, \nwe find that the functional derivative $\\delta H \/ \\delta m$ is a temperature divided by \n$T_\\lambda$. We note that the critical temperature $T_\\lambda=T_\\lambda(z)$ defined in \n\\eqref{Eq:A01} depends on the altitude $z$. Since the gradient is very small, the $z$ \ndependence is very weak. Thus, in the denominator we may approximately use a constant \naverage value which may be the critical temperature at the interface $z=z_0$, \ni.e.\\ $T_\\lambda=T_\\lambda(z)\\approx T_\\lambda(z_0)$.\n\nUntil now, the condensate density $n_\\mathrm{s}$ and the superfluid current density \n$\\mathbf{J}_\\mathrm{s}$ defined in \\eqref{Eq:B12} and \\eqref{Eq:B13} are unknown. Since \nthey are defined by an average of two fields $\\psi$ and $\\psi^*$ they are related to \nthe equal-time Green function \n\\begin{eqnarray} \nG(\\mathbf{r},t;\\mathbf{r}^\\prime,t) &=& \\langle \\psi(\\mathbf{r},t) \n\\psi^*(\\mathbf{r}^\\prime,t)\\rangle \\nonumber\\\\\n&=& \\langle\\psi(\\mathbf{r},t)\\rangle\\, \\langle\\psi^*(\\mathbf{r}^\\prime,t)\\rangle \\nonumber\\\\\n&&+ \\langle \\delta\\psi(\\mathbf{r},t) \\delta \\psi^*(\\mathbf{r}^\\prime,t)\\rangle \\ .\n\\label{Eq:B22} \n\\end{eqnarray}\nThis Green function was evaluated in the Appendix of Ref.\\ \\onlinecite{Ha99a}. However,\nwhile in our previous paper the average order parameter $\\langle\\psi\\rangle$ was zero,\nin the present paper it is nonzero. Hence, we must split the Green function into two \ncontributions, a mean-field term and a fluctuating term where \n$\\delta\\psi= \\psi- \\langle\\psi\\rangle$ is the fluctuating field. While the mean-field term \nis expressed in terms of the average order parameter $\\langle\\psi\\rangle$, the fluctuating \nterm is given by the result of our previous paper. Consequently, the condensate density \n$n_\\mathrm{s}$ and the superfluid current density $\\mathbf{J}_\\mathrm{s}$ are split into \ntwo contributions, too. From Eq.\\ (3.24) and (3.25) of Ref.\\ \\onlinecite{Ha99a} we obtain\n\\begin{eqnarray} \nn_\\mathrm{s} &=& | \\langle\\psi\\rangle |^2\n- \\frac{2}{\\varepsilon} A_d\\, \\Phi_{-1+\\varepsilon\/2}(X)\\, r_1^{1-\\varepsilon\/2} \\ ,\n\\label{Eq:B23} \\\\\n\\mathbf{J}_\\mathrm{s} &=& \\mathrm{Im}[ \\langle\\psi^*\\rangle \\nabla \\langle\\psi\\rangle ]\n\\nonumber\\\\\n&&+ \\frac{g_0}{2 \\Gamma_0^\\prime} \\frac{\\nabla \\Delta r_0}{2 \\chi_0 \\gamma_0} \\,\n\\frac{1}{\\varepsilon} A_d\\, \\Bigl( 1 - \\frac{\\varepsilon}{2} \\Bigr) \n\\, \\Phi_{\\varepsilon\/2}(X)\\, r_1^{-\\varepsilon\/2} \\ . \\qquad\n\\label{Eq:B24} \n\\end{eqnarray}\nHere it is $\\varepsilon=4-d$ where $d$ is the dimension of the space. Furthermore, \n$A_d=S_d\\,\\Gamma(1-\\varepsilon\/2)\\Gamma(1+\\varepsilon\/2)$ is a geometrical factor \nwhich is related via $S_d=\\Omega_d\/(2\\pi)^d$ to the surface of the $d$ dimensional \nunit sphere $\\Omega_d=2\\,\\pi^{d\/2}\/\\Gamma(d\/2)$. The function $\\Phi_\\alpha(X)$ is \ndefined by the divergent series\n\\begin{equation} \n\\Phi_\\alpha(X) = \\sum_{N=0}^\\infty \\frac{\\Gamma(\\alpha+3N)}{\\Gamma(\\alpha)}\n\\,\\frac{X^N}{N!}\n\\label{Eq:B25} \n\\end{equation}\nwhere the argument $X$ is related to the square of the gradients of the parameters\n$r_1$ and $\\Delta r_0$ according to\n\\begin{eqnarray} \nX &=& \\frac{1}{12\\, r_1^3} \\Bigl[ (\\nabla r_1)^2 \n+ 2 \\frac{\\Gamma_0^{\\prime\\prime}}{\\Gamma_0^\\prime} \n\\Bigl( \\frac{g_0}{4 \\chi_0 \\gamma_0 \\Gamma_0^\\prime} \\nabla \\Delta r_0 \\Bigr)\n \\cdot \\nabla r_1 \\nonumber\\\\\n&&- \\Bigl( \\frac{g_0}{4 \\chi_0 \\gamma_0 \\Gamma_0^\\prime} \\nabla \\Delta r_0 \\Bigr)^2\n\\Bigr] \\ .\n\\label{Eq:B26} \n\\end{eqnarray}\nThe Green function \\eqref{Eq:B22} was evaluated locally for a spatial inhomogeneous \nsystem where the temperature parameters $r_1$ and $\\Delta r_0$ depend on the space \ncoordinate $\\mathbf{r}$. Gradient terms $\\nabla r_1$ and $\\nabla \\Delta r_0$ are \nincluded but curvature terms and higher derivatives are omitted. This fact is \nclearly seen in the function \\eqref{Eq:B25} and its argument \\eqref{Eq:B26}.\n\nNow, all quantities are determined. The approximate model-$F$ equations \n\\eqref{Eq:B17}-\\eqref{Eq:B18} together with the entropy current density \\eqref{Eq:B19},\nthe temperature parameters \\eqref{Eq:B20}, \\eqref{Eq:B21}, \\eqref{Eq:B16}, and the \nquantities \\eqref{Eq:B23}-\\eqref{Eq:B26} are closed equations, which in principle \ncan be solved numerically. We insert the condensate density \\eqref{Eq:B23} into the \nequation for the modified temperature parameter \\eqref{Eq:B16}. After reordering \nthe terms we obtain\n\\begin{equation} \nr_1 \\Bigl\\{ 1 + 8 u_0 \\frac{1}{\\varepsilon} A_d\\, \n\\Phi_{-1+\\varepsilon\/2}(X)\\, r_1^{-\\varepsilon\/2} \\Bigr\\}\n= r_0 + 4 u_0 | \\langle\\psi\\rangle |^2 \\ .\n\\label{Eq:B27} \n\\end{equation}\nThe left-hand side shows clearly that this is an implicit equation for the parameter \n$r_1$. Furthermore, we insert the superfluid current \\eqref{Eq:B24} into the formula \nfor the entropy current \\eqref{Eq:B19}. After  reordering the terms we obtain\n\\begin{eqnarray} \n\\mathbf{q} &=& - \\frac{\\lambda_0}{2 \\chi_0 \\gamma_0} \\Bigl\\{ 1 + \n\\frac{g_0^2}{2 \\lambda_0 \\Gamma_0^\\prime} \n\\frac{1}{\\varepsilon} A_d\\, \\Bigl( 1 - \\frac{\\varepsilon}{2} \\Bigr) \n\\, \\Phi_{\\varepsilon\/2}(X)\\, r_1^{-\\varepsilon\/2} \\Bigr\\} \\nonumber\\\\\n&&\\times \\nabla \\Delta r_0 \n- g_0 \\mathrm{Im}[ \\langle\\psi^*\\rangle \\nabla \\langle\\psi\\rangle ] \\ .\n\\label{Eq:B28} \n\\end{eqnarray}\nEqs.\\ \\eqref{Eq:B27} and \\eqref{Eq:B28} of the present paper should be compared \nwith Eqs.\\ (3.32) and (3.35) of our previous paper \\cite{Ha99a}, respectively. \nNew contributions are those terms on the right hand sides which involve the \naverage order parameter $\\langle\\psi\\rangle$. The last term in \\eqref{Eq:B28}\nmay be interpreted as the mean-field contribution of the superfluid current.\nThe fluctuating term of the superfluid current \\eqref{Eq:B24} is proportional to \nthe temperature-parameter gradient $\\nabla\\Delta r_0$. For this reason, the \nfluctuating term is integrated into the first term of \\eqref{Eq:B28} and hence\ncontributes to the normal-fluid term. Similarly, in Eq.\\ \\eqref{Eq:B27} the \nmean-field contribution of the condensate density is put on the right-hand side \nwhile the fluctuating contribution is put on the left-hand side of the equation.\n\n\n\\section{Local renormalization-group theory for partial differential equations}\n\\label{Sec:03}\nThe liquid $^4$He is considered in the critical regime for temperatures $T$ close to \nthe superfluid transition at $T_\\lambda$. In order to treat the critical fluctuations \ncorrectly, we must renormalize the equations of the previous section and apply the \nrenormalization-group (RG) theory. Since we consider local physical quantities which \nare functions of space and time, the RG flow parameter will be local and depend on \nspace and time. The derivatives with respect to space and time in the model-$F$ \nequations are in conflict with a local RG flow parameter because they do not commute \nwith this parameter. For this reason, the development of the local RG theory for the \nmodel-$F$ equations which are partial differential equations is a very challenging task.\n\n\n\\subsection{Renormalization}\n\\label{Sec:03A}\nWe start with the renormalization of the average order parameter $\\langle\\psi\\rangle$, \nthe temperature parameters $\\Delta r_0$, $r_0$, and the coupling constant $u_0$. \nFollowing Ref.\\ \\onlinecite{Do85} we have\n\\begin{eqnarray}\n\\langle\\psi\\rangle &=& Z_\\phi^{1\/2} \\langle\\psi_\\mathrm{ren}\\rangle \\ ,\n\\label{Eq:C01} \\\\\n\\Delta r_0 &=& Z_r\\, \\Delta r \\ ,\n\\label{Eq:C02} \\\\\nr_0 - r_{0\\mathrm{c}} &=& Z_r\\, r \\ ,\n\\label{Eq:C03} \\\\\nu_0 &=& Z_u Z_\\phi^{-2} (\\mu^\\varepsilon\/A_d) \\, u \\ .\n\\label{Eq:C04}\n\\end{eqnarray}\nIn these and the following renormalization equations we use the convention that the \nbare quantities are always on the left-hand side while renormalized quantities are \nalways on the right hand side. The $Z$ factors are the renormalization factors. \nIn the Hartree approximation, which we use in the present paper and in our previous \npaper \\cite{Ha99a}, these $Z$ factors are\n\\begin{equation}\nZ_\\phi = 1 \\ , \\qquad Z_r = Z_u = 1 \/ [ 1 - 8u \/ \\varepsilon] \\ ,\n\\label{Eq:C05}\n\\end{equation}\nwhere it is $r_{0\\mathrm{c}}=0$. The modified temperature parameter $r_1$ is not \nrenormalized. We apply the renormalizations to Eq.\\ \\eqref{Eq:B27}, multiply both \nsides with the inverse factor $Z_r^{-1}$, and reorder the terms. Without any further \napproximation we obtain\n\\begin{eqnarray} \n&&r_1 \\Bigl\\{ 1 + \\frac{8 u}{\\varepsilon} \\Bigl[ \\Phi_{-1+\\varepsilon\/2}(X)\\, \n\\Bigl( \\frac{r_1}{\\mu^2} \\Bigr)^{-\\varepsilon\/2} - 1 \\Bigr] \\Bigr\\} \\nonumber\\\\\n&&= r + 4 u \\frac{\\mu^\\varepsilon}{A_d} | \\langle\\psi\\rangle |^2 \\ .\n\\label{Eq:C06} \n\\end{eqnarray}\n\nThe average entropy density $\\langle m \\rangle$, the entropy current density \n$\\mathbf{q}$, and the remaining model-$F$ parameters are renormalized by \\cite{Do85}\n\\begin{eqnarray}\n\\langle m \\rangle &=& (\\chi_0 Z_m)^{1\/2}\\, \\langle m_\\mathrm{ren} \\rangle \\ ,\n\\label{Eq:C07} \\\\\n\\mathbf{q} &=& (\\chi_0 Z_m)^{1\/2}\\, \\mathbf{q}_\\mathrm{ren} \\ ,\n\\label{Eq:C08} \\\\\n\\chi_0 \\gamma_0 &=& (\\chi_0 Z_m)^{1\/2} Z_r (\\mu^\\varepsilon\/A_d)^{1\/2}\\, \\gamma \\ ,\n\\label{Eq:C09} \\\\\ng_0 &=& (\\chi_0 Z_m)^{1\/2} (\\mu^\\varepsilon\/A_d)^{1\/2}\\, g \\ ,\n\\label{Eq:C10} \\\\\n\\lambda_0 \/ \\chi_0 &=& Z_\\lambda^{-1}\\, \\lambda \\ ,\n\\label{Eq:C11} \\\\\n\\Gamma_0 &=& Z_\\Gamma^{-1}\\, \\Gamma \\ .\n\\label{Eq:C12}\n\\end{eqnarray}\nThe dimensionless renormalized parameters are defined by the ratios\n\\begin{eqnarray}\nw &=& \\Gamma \/ \\lambda \\ ,\n\\label{Eq:C13} \\\\\nF &=& g \/ \\lambda \\ ,\n\\label{Eq:C14} \\\\\nf &=& F^2\/w^\\prime = g^2 \/ \\lambda \\Gamma^\\prime \\ .\n\\label{Eq:C15}\n\\end{eqnarray}\nWe note that $\\Gamma=\\Gamma^\\prime + i \\Gamma^{\\prime\\prime}$ and \n$w=w^\\prime + i w^{\\prime\\prime}$ are complex parameters. The $Z$ factors, which \nwe need explicitly in our calculation, are given in Hartree approximation \n\\cite{Ha99a} by\n\\begin{equation}\nZ_m Z_\\lambda = 1 \/ [ 1 - f \/ 2 \\varepsilon] \\ , \\qquad Z_\\Gamma = 1 \\ .\n\\label{Eq:C16}\n\\end{equation}\nThe factor $\\chi_0 Z_m$ will cancel out in all our equations. Hence this \nlatter factor is not needed explicitly. Applying the renormalizations to \nEq.\\ \\eqref{Eq:B28} we obtain the renormalized heat current \n\\begin{eqnarray} \n\\mathbf{q}_\\mathrm{ren} &=& - \\frac{\\lambda}{2 \\gamma}\n\\Bigl( \\frac{A_d}{\\mu^\\varepsilon} \\Bigr)^{1\/2} \\Bigl\\{ 1 + \\frac{f}{2 \\varepsilon} \n\\Bigl[ \\Bigl( 1 - \\frac{\\varepsilon}{2} \\Bigr) \\nonumber\\\\\n&&\\times \\Phi_{\\varepsilon\/2}(X) \\Bigl( \\frac{r_1}{\\mu^2} \\Bigr)^{-\\varepsilon\/2}\n- 1 \\Bigr] \\Bigr\\} \\, \\nabla \\Delta r \\nonumber\\\\\n&&- g\\, \\Bigl( \\frac{\\mu^\\varepsilon}{A_d} \\Bigr)^{1\/2} \n\\mathrm{Im}[ \\langle\\psi_\\mathrm{ren}^*\\rangle \\nabla \\langle\\psi_\\mathrm{ren}\\rangle ] \\ .\n\\label{Eq:C17} \n\\end{eqnarray}\nAgain no further approximation is made when reordering the terms. In order to \nevaluate the function $\\Phi_\\alpha(X)$ we need the argument $X$ expressed in terms \nof the dimensionless renormalized parameters. From \\eqref{Eq:B26} we obtain\n\\begin{eqnarray} \nX &=& \\frac{1}{12\\, r_1^3} \\Bigl[ (\\nabla r_1)^2 \n+ 2 \\frac{w^{\\prime\\prime}}{w^\\prime} \n\\Bigl( \\frac{F}{4 \\gamma w^\\prime} \\nabla \\Delta r \\Bigr) \\cdot \\nabla r_1 \\nonumber\\\\\n&&- \\Bigl( \\frac{F}{4 \\gamma w^\\prime} \\nabla \\Delta r \\Bigr)^2\n\\Bigr] \\ .\n\\label{Eq:C18} \n\\end{eqnarray}\nThe renormalization of the model-$F$ equations is straight forward. From \\eqref{Eq:B17} \nand \\eqref{Eq:B18} we obtain\n\\begin{eqnarray}\n\\frac{\\partial \\langle\\psi_\\mathrm{ren}\\rangle}{\\partial t} &=& \n- \\Gamma \\bigl[ r_1 - \\nabla^2 \\bigr] \\langle\\psi_\\mathrm{ren}\\rangle \n+ i \\frac{g}{2 \\gamma} \\Delta r \\langle\\psi_\\mathrm{ren}\\rangle\\ , \\qquad\n\\label{Eq:C19} \\\\\n\\frac{\\partial \\langle m_\\mathrm{ren} \\rangle}{\\partial t} &=& - \\nabla \n\\mathbf{q}_\\mathrm{ren} \\ .\n\\label{Eq:C20}\n\\end{eqnarray}\nWe furthermore need a relation between the entropy density $\\langle m_\\mathrm{ren} \\rangle$ \nand the temperature parameters $r$ or $\\Delta r$ in renormalized form. We solve \nEq.\\ \\eqref{Eq:B15} with respect to the entropy density $\\langle m\\rangle$, eliminate the \ncondensate density $n_\\mathrm{s}$ by \\eqref{Eq:B16}, and then perform the renormalization.\nAs a result we obtain\n\\begin{equation}\n\\langle m_\\mathrm{ren} \\rangle = m_\\mathrm{c,ren}\n+ \\Bigl( \\frac{A_d}{\\mu^\\varepsilon} \\Bigr)^{1\/2} \\frac{r}{2 \\gamma} \n\\Bigl\\{ 1 + \\frac{\\gamma^2}{2u} \\Bigl[ 1 - \\frac{r_1}{r} \\Bigr] \\Bigr\\} \\ . \n\\label{Eq:C21}\n\\end{equation}\nWe have separated the constant value $m_\\mathrm{c,ren}$ which is the entropy at \nthe critical point with temperature $T=T_\\lambda$, zero heat current \n$\\mathbf{Q}=\\mathbf{0}$ and zero gravity $g=0$. We need not know this constant value \nexplicitly. Another useful quantity is the derivative of $\\langle m_\\mathrm{ren} \\rangle$ \nwith respect to the temperature parameter $r$. It is related to the renormalized specific \nheat \\cite{Do85} according to\n\\begin{equation}\nC_\\mathrm{ren} = 2 \\gamma \\Bigl( \\frac{\\mu^\\varepsilon}{A_d} \\Bigr)^{1\/2} \n\\, \\frac{\\partial \\langle m_\\mathrm{ren} \\rangle}{\\partial r} = \n1 + \\frac{\\gamma^2}{2u} \\Bigl[ 1 - \\frac{\\partial r_1}{\\partial r} \\Bigr] \\ . \n\\label{Eq:C22}\n\\end{equation}\nIn this way, the time derivative of the renormalized entropy density can be expressed \nin terms of a time derivative of a temperature parameter. We find\n\\begin{equation}\n\\frac{\\partial \\langle m_\\mathrm{ren} \\rangle}{\\partial t} \n= \\frac{C_\\mathrm{ren}}{2\\gamma} \\Bigl( \\frac{A_d}{\\mu^\\varepsilon} \\Bigr)^{1\/2}\n\\, \\frac{\\partial r}{\\partial t}\n= \\frac{C_\\mathrm{ren}}{2\\gamma} \\Bigl( \\frac{A_d}{\\mu^\\varepsilon} \\Bigr)^{1\/2}\n\\, \\frac{\\partial \\Delta r}{\\partial t} \\ . \n\\label{Eq:C23}\n\\end{equation}\nSince the critical temperature $T_\\lambda(z)$ does not depend on the time, the two \ntemperature parameters $r$ and $\\Delta r$ differ by a time-independent value. For \nthis reason, the time derivatives of $r$ and $\\Delta r$ are equal. In the present \npaper we prefer the latter time derivative. In this way, we reformulate the second \nmodel-$F$ equation \\eqref{Eq:C20} as\n\\begin{equation}\n\\frac{C_\\mathrm{ren}}{2\\gamma} \\Bigl( \\frac{A_d}{\\mu^\\varepsilon} \\Bigr)^{1\/2}\n\\, \\frac{\\partial \\Delta r}{\\partial t} \n= - \\nabla \\mathbf{q}_\\mathrm{ren} \\ .\n\\label{Eq:C24}\n\\end{equation}\nIn the renormalized specific heat \\eqref{Eq:C22} the remaining derivative \n$\\partial r_1 \/ \\partial r$ may be obtained as the proportionality factor of the \ngradients $\\nabla r_1$ and $\\nabla r$ according to \n\\begin{equation}\n\\nabla r_1 = \\frac{\\partial r_1}{\\partial r} \\, \\nabla r \\ .\n\\label{Eq:C25}\n\\end{equation}\nIn order to find a relation between the two gradients we apply the nabla operator \n$\\nabla$ to Eq.\\ \\eqref{Eq:C06}. Thus, we find\n\\begin{eqnarray} \n&&(\\nabla r_1) \\Bigl\\{ 1 + \\frac{8 u}{\\varepsilon} \\Bigl[ \n\\Bigl( 1 - \\frac{\\varepsilon}{2} \\Bigr) \\Phi_{\\varepsilon\/2}(X)\n\\, \\Bigl( \\frac{r_1}{\\mu^2} \\Bigr)^{-\\varepsilon\/2} - 1 \\Bigr] \\Bigr\\} \\nonumber\\\\\n&&= \\nabla r + 4 u \\frac{\\mu^\\varepsilon}{A_d} \n\\, \\nabla | \\langle\\psi_\\mathrm{ren}\\rangle |^2 \\ .\n\\label{Eq:C26} \n\\end{eqnarray}\nIn this result the derivative has increased the index $\\alpha$ of the function \n$\\Phi_\\alpha(X)$ by one. Furthermore, the function is multiplied by a factor \n$(1 - \\varepsilon\/2)$. These facts are well known from the calculations in our \nprevious paper \\cite{Ha99a}. By comparing Eqs.\\ \\eqref{Eq:C25} and \\eqref{Eq:C26} \nwe extract $\\partial r_1 \/ \\partial r$. Since we consider the space dependence only \nin one dimension $z$ which is the altitude, we obtain a unique result. We conclude \nthat in this subsection we have derived all equations in renormalized form which \nare needed for a numerical calculation to solve the model-$F$ equations as partial \ndifferential equations with respect to space and time.\n\n\n\\subsection{Dimensionless renormalized quantities}\n\\label{Sec:03B}\nIn the renormalization equations \\eqref{Eq:C01}-\\eqref{Eq:C04} and \n\\eqref{Eq:C07}-\\eqref{Eq:C12} the new arbitrary parameter $\\mu$ occurs which \nhas the unit of an inverse length scale. Consequently, this parameter may be used \nto fix the length scale. On the other hand in the renormalized model-$F$ equations \n\\eqref{Eq:C19} and \\eqref{Eq:C24} together with \\eqref{Eq:C17} the dynamic parameters \n$\\Gamma=\\Gamma^\\prime+i\\Gamma^{\\prime\\prime}$, $\\lambda$, and $g$ all have the \nunit of a diffusion constant, i.e.\\ length square divided by time. Hence, these \nparameters multiplied by $\\mu^2$ may be used to fix the time scale. The \ndimensionless ratios \\eqref{Eq:C13}-\\eqref{Eq:C15} imply that only one of these \nparameters is needed. Thus, we will use $g \\mu^2$ to fix the time scale.\n\nWe rewrite the renormalized model-$F$ equations and the related renormalized variables \nand parameters in a dimensionless form using $\\mu$ and $g \\mu^2$ for the scales. \nFollowing our previous paper \\cite{Ha99a} we define the dimensionless temperature \nparameters\n\\begin{eqnarray}\n\\Delta\\rho &=& \\Delta r \/ \\mu^2 = \\tau^{-1} \\, (T-T_0)\/T_\\lambda \\ ,\n\\label{Eq:C27} \\\\\n\\rho &=& r \/ \\mu^2 = \\tau^{-1} \\, (T-T_\\lambda)\/T_\\lambda \\ ,\n\\label{Eq:C28} \\\\\n\\rho_1 &=& r_1 \/ \\mu^2 \\ .\n\\label{Eq:C29}\n\\end{eqnarray}\nThe last equality sign in \\eqref{Eq:C27} and \\eqref{Eq:C28} is obtained by \nrenormalizing the bare equations \\eqref{Eq:B20} and \\eqref{Eq:B21}. The renormalization \nfactors are combined into the dimensionless parameter \n\\begin{equation}\n\\tau = \\Bigl( \\frac{A_d \\mu^d}{\\chi_0 Z_m} \\Bigr)^{1\/2} \\frac{1}{2\\gamma}\n\\label{Eq:C30}\n\\end{equation}\nwhich may be viewed as a renormalization-group (RG) flow parameter. \nA change of the length scale by replacing $\\mu\\to \\mu\\ell$ causes a change of $\\tau$. \nWhile $\\ell$ is the conventional RG flow parameter related to the length scale,\n$\\tau=\\tau(\\ell)$ is a RG flow parameter related to the temperature scale. In the \nliterature on the dynamic RG theory for liquid $^4$He both flow parameters have been \nused \\cite{Do85,Do91,Ha99,Ha99a,Ha99b}. For the dimensionless coupling parameters of \nmodel $F$ the notations $u(\\ell)$, $\\gamma(\\ell)$, \\emph{etc.}\\ and $u[\\tau]$, $\\gamma[\\tau]$, \n\\emph{etc.}\\ have been used. In the present paper we will use $\\tau$ as the RG flow\nparameter.\n\nWe define the dimensionless renormalized order parameter $Y$ and the dimensionless \nrenormalized heat current $\\tilde{\\mathbf{q}}$ by\n\\begin{eqnarray}\nY &=& \\langle \\psi_\\mathrm{ren} \\rangle \/ \\mu^{(d-2)\/2} \\ ,\n\\label{Eq:C31} \\\\\n\\tilde{\\mathbf{q}} &=& \\Bigl( \\frac{A_d}{\\mu^\\varepsilon} \\Bigr )^{1\/2} \n\\frac{\\mathbf{q}_\\mathrm{ren}}{g} \\, \\frac{1}{\\mu^{d-1}}\n= \\frac{\\mathbf{q}}{g_0} \\, \\frac{1}{\\mu^{d-1}} \\ ,\n\\label{Eq:C32}\n\\end{eqnarray}\nrespectively. For convenience of the notation, following Ref.\\ \\onlinecite{Ha99a} \nwe define the dimensionless amplitudes\n\\begin{eqnarray}\nA &=& \\varepsilon^{-1}[ \\Phi_{-1+\\varepsilon\/2}(X)\\, \\rho_1^{-\\varepsilon\/2} - 1 ] \\ ,\n\\label{Eq:C33} \\\\\nA_1 &=& \\varepsilon^{-1}[ (1 - \\varepsilon\/2) \\Phi_{\\varepsilon\/2}(X)\\, \n\\rho_1^{-\\varepsilon\/2} - 1 ] \\ .\n\\label{Eq:C34}\n\\end{eqnarray}\nConsequently, the renormalized heat current \\eqref{Eq:C17} can be rewritten in the \ndimensionless simple form\n\\begin{equation}\n\\tilde{\\mathbf{q}} = \\frac{1}{\\mu} \\Bigl[ - \\frac{A_d}{2\\gamma F}\\, \n\\Bigl\\{ 1 + \\frac{f}{2} A_1 \\Bigr\\} \\, \\nabla \\Delta \\rho \n- \\mathrm{Im}[ Y^* \\nabla Y] \\Bigr] \\ .\n\\label{Eq:C35}\n\\end{equation}\nThe overall factor $1\/\\mu$ is needed to keep the nabla operators dimensionless. We \nnote that $A_d$ is a geometrical factor related to surface of the $d$ dimensional \nunit sphere \\cite{Do85}. It should not be confused with the amplitudes $A$ and $A_1$. \nIn an analogous way Eqs.\\ \\eqref{Eq:C06} and \\eqref{Eq:C26} for the modified \ntemperature parameter $\\rho_1$ and its derivative $\\nabla \\rho_1$ can be written \nin a dimensionless form. We obtain\n\\begin{eqnarray}\n\\rho_1 \\{ 1 + 8u A \\} &=& \\rho + (4u\/A_d) \\, |Y|^2 \\ ,\n\\label{Eq:C36} \\\\\n(\\nabla\\rho_1) \\{ 1 + 8u A_1 \\} &=& \\nabla\\rho + (4u\/A_d) \\, \\nabla |Y|^2 \\ , \\qquad\n\\label{Eq:C37}\n\\end{eqnarray}\nwhere the second equation should be multiplied by an overall factor $1\/\\mu$ \nto make the nabla operators dimensionless. Finally, the renormalized specific heat \n$C_\\mathrm{ren}$ defined in \\eqref{Eq:C22} and the parameter $X$ defined in \n\\eqref{Eq:C18} are already dimensionless, so that we can keep them unchanged. \nWe must only insert the dimensionless temperature parameters \n\\eqref{Eq:C27}-\\eqref{Eq:C29} and use the dimensionless nabla operator \n$\\mu^{-1}\\nabla$.\n\nNow, all variables and parameters are expressed in a dimensionless form. Thus,\nwe are ready to rewrite the renormalized model-$F$ equations in dimensionless forms. \nFrom Eqs.\\ \\eqref{Eq:C19} and \\eqref{Eq:C24} we obtain\n\\begin{eqnarray}\n\\frac{1}{g\\mu^2} \\, \\frac{\\partial Y}{\\partial t} &=& \n- \\frac{w}{F} \\bigl[ \\rho_1 - \\mu^{-2}\\nabla^2 \\bigr] Y \\nonumber\\\\ \n&&+ \\frac{i}{2 \\gamma} \\Delta \\rho \\, Y \\ , \\qquad\n\\label{Eq:C38} \\\\\n\\frac{C_\\mathrm{ren}}{2\\gamma} \\frac{A_d}{g\\mu^2}\n\\, \\frac{\\partial \\Delta \\rho}{\\partial t} \n&=& - \\mu^{-1}\\nabla \\tilde{\\mathbf{q}} \\ ,\n\\label{Eq:C39}\n\\end{eqnarray}\nwhere $w=w^\\prime + i w^{\\prime\\prime}$ is a complex parameter.\nIn these equations we clearly see that $\\mu$ defines the length scale \nand $g \\mu^2$ defines the time scale. We may interpret $\\mu^{-1}\\nabla$ \nas a dimensionless nabla operator and $(g \\mu^2)^{-1} \\partial \/ \\partial t$ \nas a dimensionless time derivative.\n\n\n\\subsection{Evaluation of the perturbative amplitudes}\n\\label{Sec:03C}\nThe amplitudes $A$ and $A_1$, defined in \\eqref{Eq:C33} and \\eqref{Eq:C34}, \nrespectively, represent the contributions of the perturbation series expansion \nwhich in our case is the Hartree term. In order to solve the model-$F$ equations \nas partial differential equations we must have explicit expression to evaluate \nthese amplitudes. The non-trivial contribution in the amplitudes is the function \n$\\Phi_\\alpha(X)$ together with its variable $X$ which are defined in \nEqs.\\ \\eqref{Eq:B25} and \\eqref{Eq:C18}. This function was first derived in \nRef.\\ \\onlinecite{HD91}. Unfortunately, the function is a divergent infinite \nseries so that it is not well defined in this form. However, in thermal equilibrium \nat zero heat current $Q=0$ and zero gravity $g=0$ this function can be omitted \nbecause it is just unity. In this case all temperature gradients are zero, so \nthat the variable $X$ is zero which implies $\\phi_\\alpha(X=0)=1$. Hence, the \namplitudes reduce to \n\\begin{eqnarray}\nA &=& \\varepsilon^{-1}[ \\rho_1^{-\\varepsilon\/2} - 1 ] \\ ,\n\\label{Eq:C40} \\\\\nA_1 &=& \\varepsilon^{-1}[ (1 - \\varepsilon\/2) \\rho_1^{-\\varepsilon\/2} - 1 ] \\ .\n\\label{Eq:C41}\n\\end{eqnarray}\nSince $\\rho_1\\sim \\rho\\sim (T-T_\\lambda)$ must be positive, these amplitudes are \nvalid for the \\emph{normal-fluid} equilibrium state only and agree with former \nresults \\cite{Do85}.\n\nIn the nonequilibrium state the variable $X$ is nonzero. In this case the \ninfinite series \\eqref{Eq:B25} must be resummed to obtain a well-defined \nexpression which can be evaluated numerically. Following our previous papers \n\\cite{HD91,Ha99a} we write\n\\begin{equation}\n\\Phi_\\alpha(X) = [\\Gamma(\\alpha)]^{-1} \\, \\zeta^\\alpha \\, \n\\mathcal{F}_\\alpha(\\zeta)\n\\label{Eq:C42}\n\\end{equation}\nwhere $\\zeta=(-X)^{-1\/3}$ and \n\\begin{equation}\n\\mathcal{F}_\\alpha(\\zeta) = \\int_0^\\infty dv \\, v^{\\alpha-1} \\, \n\\exp(-v^3 - v\\zeta) \\  .\n\\label{Eq:C43}\n\\end{equation}\nThe integral is well defined for $\\alpha>0$ and obtained by analytical continuation \nfor $\\alpha<0$. The new variable $\\zeta$ is defined by a third root. Consequently, \n$\\zeta$ is not unique a priori and may be complex. We must specify the root which \nshould be taken. For this purpose we define the dimensionless parameter\n\\begin{eqnarray} \n\\sigma &=& - \\frac{1}{12 \\, \\mu^2} \\Bigl[ (\\nabla \\rho_1)^2 \n+ 2 \\frac{w^{\\prime\\prime}}{w^\\prime} \n\\Bigl( \\frac{F}{4 \\gamma w^\\prime} \\nabla \\Delta \\rho \\Bigr) \n\\cdot \\nabla \\rho_1 \\nonumber\\\\\n&&- \\Bigl( \\frac{F}{4 \\gamma w^\\prime} \\nabla \\Delta \\rho \\Bigr)^2\n\\Bigr] \\ .\n\\label{Eq:C44} \n\\end{eqnarray}\nso that $X=-\\sigma\/\\rho_1^3$. Hence, the new variable can be written in the form \n$\\zeta = \\rho_1\/\\sigma^{1\/3}$.\n\nThe transition from normal-fluid to superfluid $^4$He is related to a change of \nsign of $\\rho_1\\sim\\rho\\sim(T-T_\\lambda)$. Consequently, also the new variable \n$\\zeta$ changes sign. The nontrivial third root to be evaluated is $\\sigma^{1\/3}$. \nFor this reason, we must distinguish two cases which are related to the two possible \nsigns of $\\sigma$. This distinction has important physical consequences. There will \nbe two kinds of nonequilibrium superfluid phases of liquid $^4$He which are related \nto the two regimes where either the heat current $Q$ or the gravity $g$ is the \ndominating external influence. We discuss these two cases in the following.\n\n\\subsubsection{Heat current dominated regime: $\\sigma>0$}\n\\label{Sec:03C1}\nThe self-organized critical state observed in the experiment by Moeur \n\\emph{et al.}\\ \\cite{Du97} implies linear temperature profiles $T(z)$ and \n$T_\\lambda(z)$ as function of the altitude $z$. The temperature difference \n$T(z)-T_\\lambda(z)=\\Delta T$ is constant over a large range of the altitude. \nConsequently, the related gradient parameters \n$\\nabla \\rho_1\\sim \\nabla \\rho\\sim \\nabla (T-T_\\lambda)$ are zero. On the other \nhand the heat current $\\mathbf{Q}$ causes a nonzero constant gradient \n$\\nabla \\Delta\\rho\\sim \\nabla T$. Thus, in the formula \\eqref{Eq:C44} only the \nlast term is nonzero which yields a positive result for $\\sigma$. Hence, for \nthe self-organized critical state the dimensionless parameter $\\sigma$ is \nalways constant and positive.\n\nFor the inhomogeneous nonequilibrium state we may conclude that $\\sigma$ is also \npositive whenever the heat current $\\mathbf{Q}$ and hence the related gradient \n$\\nabla \\Delta\\rho$ is large compared to the effects of gravity. In our previous \npaper \\cite{Ha99a} we confirm $\\sigma>0$ for heat transport in liquid $^4$He on \nearth for heat currents $Q\\gtrsim 70\\,\\mathrm{nW\/cm^2}$. Moreover, for an \nexperiment in zero gravity conditions in space, $\\sigma$ is positive for all \nheat currents.\n\nWhenever $\\sigma$ is positive, the root $\\sigma^{1\/3}$ is straight forward. We \njust take the real positive root. Consequently, the variable $\\zeta$ is real \nand changes sign at the superfluid transition. We find $\\zeta>0$ in the normal \nfluid regime and $\\zeta<0$ is the superfluid regime. The function \\eqref{Eq:C42} \nand the integral \\eqref{Eq:C43} can be evaluated directly. As a result we obtain \nthe amplitudes\n\\begin{eqnarray}\nA &=& \\frac{1}{\\varepsilon} \\Bigl[ \n\\frac{\\sigma^{-\\varepsilon\/6}}{\\Gamma(-1+\\varepsilon\/2)} \n\\, \\zeta^{-1} \\mathcal{F}_{-1+\\varepsilon\/2}(\\zeta) - 1 \\Bigr] \\ , \\qquad\n\\label{Eq:C45} \\\\\nA_1 &=& \\frac{1}{\\varepsilon} \\Bigl[ \n- \\frac{\\sigma^{-\\varepsilon\/6}}{\\Gamma(-1+\\varepsilon\/2)} \n\\, \\mathcal{F}_{\\varepsilon\/2}(\\zeta) - 1 \\Bigr] \\ ,\n\\label{Eq:C46}\n\\end{eqnarray}\nwhich we have derived and used in our previous paper \\cite{Ha99a}.\n\nWe investigate the asymptotic behaviors of the function \n$\\mathcal{F}_\\alpha(\\zeta)$ and find\n\\begin{equation}\n\\mathcal{F}_\\alpha(\\zeta) \\approx \\Gamma(\\alpha) \\, \\zeta^{-\\alpha}\n\\label{Eq:C47}\n\\end{equation}\nfor $\\zeta\\gg+1$ in the normal-fluid regime and\n\\begin{equation}\n\\mathcal{F}_\\alpha(\\zeta) \\approx (\\pi\/3)^{1\/2} \\, (-\\zeta\/3)^{\\alpha\/2-3\/4}\n\\, \\exp\\{ 2 (-\\zeta\/3)^{3\/2} \\}\n\\label{Eq:C48}\n\\end{equation}\nfor $\\zeta\\ll-1$ in the superfluid regime, respectively \\cite{Ha99a}. In the first \nasymptotic case \\eqref{Eq:C47} we recover the amplitudes \\eqref{Eq:C40} and \n\\eqref{Eq:C41} of the normal-fluid equilibrium state. In the second asymptotic case \n\\eqref{Eq:C48} we obtain exponentially large amplitudes $A$ and $A_1$ for the \nnonequilibrium superfluid state.\n\nThe latter asymptotic case has an important physical consequence. We consider \nEq.\\ \\eqref{Eq:C36} which is a constraint to define $\\rho_1$. In the original \nform related to \\eqref{Eq:B16} this equation is rewritten as\n\\begin{equation}\n\\rho_1 = \\rho - 8u A \\rho_1 + (4u\/A_d) \\, |Y|^2 \\ .\n\\label{Eq:C49}\n\\end{equation}\nThe last term is the contribution of the renormalized complex order parameter $Y$ \nwhich is nonzero only in the superfluid state. However, in the superfluid regime \nthe second term may be a competing term because the amplitude $A$ may be \nexponentially large. Thus, in the nonequilibrium system there may be two \ncompeting superfluid phases which have different physical properties. In Eq.\\ \n\\eqref{Eq:B22} we have split the order-parameter Green function into two terms,\na mean-field term and a fluctuating term. The third and the second term in \n\\eqref{Eq:C49} refer to these two terms of the Green function, respectively.\n\nThe complex order parameter $Y$ may be decomposed into a modulus $\\eta$ and \na phase $\\varphi$ according to $Y=\\eta\\, e^{i\\varphi}$. In our previous papers \n\\cite{Ha99,Ha99a} we argue that in the superfluid regime the modulus $\\eta$ \nand hence the average order parameter $Y$ is zero because of large fluctuations of \nthe phase $\\varphi$. These large phase fluctuations are related to vortices \nand quantum turbulence. In the present paper we consider a nonzero average order \nparameter $Y$ in the superfluid regime and solve the renormalized model-$F$ equations \nnumerically as partial differential equations. We find a competition between the \n\\emph{mean-field superfluid phase}, described by the average order parameter $Y$, \nand the \\emph{fluctuating superfluid phase}, described by the exponentially large \namplitude $A$.\n\n\n\\subsubsection{Gravity dominated regime: $\\sigma<0$}\n\\label{Sec:03C2}\nIn thermal equilibrium for zero heat currents $Q=0$ the temperature $T$ is\nconstant. Consequently, the gradient $\\nabla \\Delta \\rho\\sim \\nabla T$ is zero.\nOn the other hand, gravity on earth implies a nonzero gradient of the \ncritical temperature $\\nabla T_\\lambda$. Hence the other gradients \n$\\nabla \\rho_1\\sim \\nabla \\rho \\sim \\nabla (T-T_\\lambda)=-\\nabla T_\\lambda$ \nare nonzero. In Eq.\\ \\eqref{Eq:C44} only the first term is nonzero which implies \na negative dimensionless parameter $\\sigma$. A small heat current $Q$ will not \nchange the situation. In our numerical calculations we find $\\sigma<0$ for \n$Q\\lesssim 20\\,\\mathrm{nW\/cm^2}$. \n\nAn exception is the self-organized critical state which always implies $\\sigma > 0$\nand which exists for arbitrary small heat currents $Q$ where the temperature difference \n$T(z)-T_\\lambda(z)=\\Delta T$ is constant. Nevertheless, for small heat currents \n$\\Delta T$ is positive so that the system is normal fluid and the sign of $\\sigma$ \nis irrelevant.\n\nFor negative $\\sigma$ the third root is always complex. We find \n$\\sigma^{1\/3}=e^{\\mp i\\pi\/3} (-\\sigma)^{1\/3}$, so that the variable of the function \n\\eqref{Eq:C43} is complex, i.e.\\ $\\zeta = e^{\\pm i\\pi\/3} \\rho_1 \/ (-\\sigma)^{1\/3}$.\nFor convenience we introduce the new real parameter \n$\\bar\\zeta = \\rho_1 \/ (-\\sigma)^{1\/3}$ which is related to the old parameter via \n$\\zeta = e^{\\pm i\\pi\/3} \\bar\\zeta$. We furthermore define the new complex function\n\\begin{equation}\n\\mathcal{G}_\\alpha(\\bar\\zeta) = e^{\\pm i\\alpha\\pi\/3} \\, \\mathcal{F}_\\alpha(\\zeta)\n= e^{\\pm i\\alpha\\pi\/3} \\, \\mathcal{F}_\\alpha(e^{\\pm i\\pi\/3} \\bar\\zeta) \\ .\n\\label{Eq:C50}\n\\end{equation}\nwhich can be decomposed into real and imaginary parts according to \n$\\mathcal{G}_\\alpha(\\bar\\zeta) = \\mathcal{G}^\\prime_\\alpha(\\bar\\zeta) \n\\pm i \\, \\mathcal{G}^{\\prime\\prime}_\\alpha(\\bar\\zeta)$. The new complex function \nis not uniquely defined because there are two complex roots which can be chosen. \nThis fact causes two possible signs for the imaginary part. However, we choose \nthe so called \\emph{principal} part, which is obtained as the average of the two \ncases so that the imaginary part cancels. Thus, we simply omit the imaginary part \n$\\mathcal{G}^{\\prime\\prime}_\\alpha(\\bar\\zeta)$. In the normal-fluid region \n$\\bar\\zeta>0$ this assumption is plausible because the imaginary part converges\nto zero exponentially for increasing $\\bar\\zeta$. As a result, we rewrite the \namplitudes \\eqref{Eq:C45} and \\eqref{Eq:C46} in terms of the new function \n\\eqref{Eq:C50} as\n\\begin{eqnarray}\nA &=& \\frac{1}{\\varepsilon} \\Bigl[ \n\\frac{(-\\sigma)^{-\\varepsilon\/6}}{\\Gamma(-1+\\varepsilon\/2)} \n\\, \\bar\\zeta^{-1} \\mathcal{G}^\\prime_{-1+\\varepsilon\/2}(\\bar\\zeta) - 1 \\Bigr] \\ , \\qquad\n\\label{Eq:C51} \\\\\nA_1 &=& \\frac{1}{\\varepsilon} \\Bigl[ \n- \\frac{(-\\sigma)^{-\\varepsilon\/6}}{\\Gamma(-1+\\varepsilon\/2)} \n\\, \\mathcal{G}^\\prime_{\\varepsilon\/2}(\\bar\\zeta) - 1 \\Bigr] \\ .\n\\label{Eq:C52}\n\\end{eqnarray}\nOnce again, we consider the asymptotic behaviors of the function \n$\\mathcal{G}^\\prime_\\alpha(\\bar\\zeta)$. We find\n\\begin{equation}\n\\mathcal{G}^\\prime_\\alpha(\\bar\\zeta) \\approx \\Gamma(\\alpha) \n\\, \\bar\\zeta^{-\\alpha}\n\\label{Eq:C53}\n\\end{equation}\nfor $\\bar\\zeta\\gg+1$ in the normal-fluid regime and\n\\begin{eqnarray}\n\\mathcal{G}^\\prime_\\alpha(\\bar\\zeta) &\\approx& (\\pi\/3)^{1\/2} \n\\, (-\\bar\\zeta\/3)^{\\alpha\/2-3\/4} \\nonumber\\\\\n&&\\times \\cos\\{ 2 (-\\bar\\zeta\/3)^{3\/2} + (\\pi\/4)(2\\alpha-1) \\} \\qquad\n\\label{Eq:C54}\n\\end{eqnarray}\nfor $\\bar\\zeta\\ll-1$ in the superfluid regime. In the first asymptotic case \n\\eqref{Eq:C53} we recover the amplitudes \\eqref{Eq:C40} and \\eqref{Eq:C41} of \nthe normal-fluid equilibrium state. Since here the amplitudes do not depend \non the dimensionless parameter $\\sigma$ at all, in the normal-fluid regime \nthe sign of $\\sigma$ is irrelevant. In the second asymptotic case \\eqref{Eq:C54}\nthe function $\\mathcal{G}_\\alpha(\\bar\\zeta)$ and hence the amplitudes $A$ and \n$A_1$ oscillate but remain of order unity.\n\nAgain, the latter asymptotic case has an important physical consequence. \nIn Eq.\\ \\eqref{Eq:C49} the second term is always small because the amplitude \n$A$ never becomes large. Hence the superfluid phase is unique. It is the \n\\emph{mean-field superfluid phase} where the average order parameter $Y$ is \nnonzero. Vortices due to fluctuations effects and a fluctuating superfluid \nphase do not exist for $\\sigma<0$. \n\n\n\\subsection{Renormalization-group theory \\newline and flow parameter condition}\n\\label{Sec:03D}\nIn the renormalization procedure the parameter $\\mu$ is introduced which fixes \nthe length scale. This parameter generates a transformation group which is known \nas the renormalization group. Following Ref.\\ \\onlinecite{Do85} it can be changed \nby the substitution $\\mu\\to\\mu\\ell$, where the dimensionless parameter $\\ell$ is \ncalled the renormalization-group (RG) flow parameter \\cite{Do85}. However, for \nsimplicity and consistency of the following calculations, in this paper we do not \nuse the above substitution. We avoid the use of the flow parameter $\\ell$ and \nthus change the length scale parameter $\\mu$ directly. We use the alternative \ndimensionless RG flow parameter $\\tau$ which is defined in \\eqref{Eq:C30}.\nAll quantities of the renormalized theory can be expressed in terms of this RG \nflow parameter. The dimensionless coupling parameters are $u[\\tau]$, \n$\\gamma[\\tau]$, $w[\\tau]=w^\\prime[\\tau]+iw^{\\prime\\prime}[\\tau]$, $F[\\tau]$, \nand $f[\\tau]$. This is a notation which was defined in Refs.\\ \\onlinecite{Do85} \nand \\onlinecite{Do91}. \n\nA differential relation between the flow parameter $\\tau$ and the length-scale \nparameter $\\mu$ can be obtained by a logarithmic differentiation of \nEq.\\ \\eqref{Eq:C30}, which reads\n\\begin{equation}\nd\\ln\\tau = \\Bigl[ \\frac{1}{2} \\Bigl( d + \\frac{\\partial\\ln Z_m^{-1}}{\\partial\\ln\\mu} \\Bigr) \n- \\frac{\\partial\\ln\\gamma}{\\partial\\ln\\mu} \\Bigr]\\, d\\ln\\mu \\ .\n\\label{Eq:C55}\n\\end{equation}\nUsing the definitions of the RG zeta functions \\cite{Do85}\n\\begin{eqnarray}\n\\zeta_\\phi &=& \\partial\\ln Z_\\phi^{-1} \/ \\partial \\ln\\mu \\ , \n\\label{Eq:C56} \\\\\n\\zeta_r &=& \\partial\\ln Z_r^{-1} \/ \\partial \\ln\\mu \\ , \n\\label{Eq:C57} \\\\\n\\zeta_m &=& \\partial\\ln Z_m^{-1} \/ \\partial \\ln\\mu \\ , \n\\label{Eq:C58}\n\\end{eqnarray}\nusing the RG equation for the parameter $\\gamma$ \\cite{Do85}\n\\begin{equation}\n\\partial\\ln\\gamma \/ \\partial\\ln\\mu = [ - \\varepsilon + 2\\zeta_r + \\zeta_m ] \/ 2 \\ ,\n\\label{Eq:C59}\n\\end{equation}\nand using $\\varepsilon=4-d$, Eq.\\ \\eqref{Eq:C55} can be simplified into\n\\begin{equation}\nd\\ln\\tau = [ 2 - \\zeta_r ] \\, d\\ln\\mu \\ .\n\\label{Eq:C60}\n\\end{equation}\nThe zeta function $\\zeta_r=\\zeta_r(u)$ is explicitly available as a function of \n$u=u[\\tau]$ \\cite{SD89}. Thus Eq.\\ \\eqref{Eq:C60} enables an explicit numerical \ncalculation of $\\tau$ as a function of $\\mu$ and vice versa.\n\nSince the renormalization procedure implies a reordering of the perturbation \nseries, the RG flow parameter $\\tau$ should be chosen in an optimum way so that \nthe convergence behavior of the series is optimized. To do this we choose the \nconstraint condition \n\\begin{equation}\n3\\,\\rho_1 - 2\\,\\rho + 3 (4u\/A_d) f_Y ( \\nabla Y \/ \\mu )^2 \n+ f_{\\Delta\\rho} ( \\nabla \\Delta\\rho \/ \\mu)^2 = 1 \\ . \n\\label{Eq:C61}\n\\end{equation}\nThe modified temperature parameter $\\rho_1$ is defined in Eq.\\ \\eqref{Eq:C36} \nwhich may be viewed as a second constraint equation. The first two terms on the \nleft-hand side of \\eqref{Eq:C61} guarantee the standard flow parameter conditions \nof normal-fluid and superfluid $^4$He in thermal equilibrium and zero gravity \nwhich have been formulated in Ref.\\ \\onlinecite{Do85}. The latter two terms \nare gradient terms which stabilize the intermediate region of the \nsuperfluid\/normal-fluid interface. The two parameters $f_Y$ and $f_{\\Delta\\rho}$ \nare dimensionless and control the influence of the gradient terms. In our \ncalculations we have used $f_Y=5$ and $f_{\\Delta\\rho}=1$ as an optimum choice.\n\nIn thermal equilibrium and zero gravity all quantities and parameters are \nconstant in space and time. An exception is the renormalized order parameter \n$Y=Y(t)=\\eta\\, e^{i\\varphi(t)}$ together with the constant modulus $\\eta$ and \nthe time-dependent phase $\\varphi(t) = - \\omega t + \\varphi_0$. Since all gradient \nterms are zero, the model-$F$ equations \\eqref{Eq:C38}, \\eqref{Eq:C39} and the \nflow-parameter equation \\eqref{Eq:C61} reduce to \n\\begin{eqnarray}\n\\omega &=& - g\\mu^2 (2\\gamma)^{-1} \\, \\Delta\\rho \\ ,\n\\label{Eq:C62} \\\\\n\\rho_1\\, Y &=& 0 \\ ,\n\\label{Eq:C63} \\\\\n3\\,\\rho_1 - 2\\,\\rho &=& 1 \\ .\n\\label{Eq:C64}\n\\end{eqnarray}\nThe first equation is always satisfied because it defines the order-parameter \nfrequency $\\omega$ in terms of the dimensionless renormalized temperature \ndifference $\\Delta\\rho$ where the time scale is ruled by the parameter \ncombination $g\\mu^2$.\n\nIn the normal-fluid state, the second equation \\eqref{Eq:C63} implies the \nzero order parameter $Y=0$, where $\\rho_1$ may be nonzero. The flow-parameter \nequation \\eqref{Eq:C64} together with the constraint \\eqref{Eq:C36} and \nthe amplitudes \\eqref{Eq:C40} and \\eqref{Eq:C41} imply $\\rho=\\rho_1=1$, \n$A=0$, and $A_1=-1\/2$. These results are compatible with the equilibrium \ntheory of Ref.\\ \\onlinecite{Do85}. The resulting flow parameter condition is\n$\\rho=1$ which in the notation of Ref.\\ \\onlinecite{Do85} reads \n$r(l)\/(\\mu\\ell)^2=1$. Consequently, from Eq.\\ \\eqref{Eq:C28} we obtain \nthe flow parameter $\\tau=(T-T_\\lambda)\/T_\\lambda$ which just is the reduced \ntemperature as known from earlier work \\cite{Do85}.\n\nIn the superfluid state, Eq.\\ \\eqref{Eq:C63} implies $\\rho_1=0$ where the \norder parameter $Y$ is nonzero. Consequently, Eq.\\ \\eqref{Eq:C64} yields \nthe flow-parameter condition $\\rho=-1\/2$ which is well known from \nRef.\\ \\onlinecite{Do85} in the notation $r(\\ell)\/(\\mu\\ell)^2=-1\/2$. Again, \nEq.\\ \\eqref{Eq:C28} relates the flow parameter to the reduced temperature according \nto $\\tau=2(T_\\lambda-T)\/T_\\lambda$. From the constraint \\eqref{Eq:C36} we obtain \nthe modulus of the order parameter $\\eta=|Y|$. Since the left-hand side is zero, \nwe obtain $\\eta=(A_d\/8u)^{1\/2}$.\n\nThe above investigation of the normal-fluid and superfluid equilibrium states in \nzero gravity shows, that in our numerical calculations for the superfluid\/normal-fluid \ninterface the dimensionless renormalized temperature variables $\\rho$, $\\rho_1$, \nand the modulus of the dimensionless renormalized order parameter $\\eta=|Y|$ must \napproach constant asymptotic values on both sides far away from the interface. In \nthe intermediate region near the interface, the variables will interpolate the \nasymptotic values. The RG flow-parameter condition \\eqref{Eq:C61} guarantees the \nasymptotic values and yields an appropriate interpolation in the intermediate \ninterface region. The gradient terms in this condition will stabilize the interpolation.\n\nThe RG flow-parameter condition \\eqref{Eq:C61} is designed for the \nsuperfluid\/normal-fluid interface at small heat currents where gravity is \nthe dominating external influence and where in the superfluid phase the order\nparameter $Y$ is nonzero. In our classification of Sec.\\ \\ref{Sec:03C} this\nsuperfluid phase is the mean-field superfluid phase. The other case is the\nfluctuating superfluid phase where the order parameter $Y$ is zero and vortices \nare present. This latter case has been investigated in our previous publications \n\\cite{Ha99,Ha99a} where the RG flow-parameter condition is given by Eqs.\\ (11) \nand (4.39) of Refs.\\ \\onlinecite{Ha99} and \\onlinecite{Ha99a}, respectively. \nThis latter flow parameter condition can be compared with our present condition \n\\eqref{Eq:C61} if $\\rho$ is eliminated by using the second constraint \\eqref{Eq:C36} \nand if we use $\\rho_1=\\sigma^{1\/2}\\zeta$. Then, the first and second term \nof our present condition \\eqref{Eq:C61} are identified with the second and third \nterm in Eqs.\\ (11) and (4.39) of Ref.\\ \\onlinecite{Ha99} and \\onlinecite{Ha99a}. \nThe gradient terms of Eq.\\ \\eqref{Eq:C61} are replaced by the first term in \nEqs.\\ (11) and (4.39) of Refs.\\ \\onlinecite{Ha99} and \\onlinecite{Ha99a}, which \nis also a gradient term because $\\sigma$ is depends on the gradients following \n\\eqref{Eq:C44}. We note that the RG flow-parameter condition of \nRef.\\ \\onlinecite{Ha99} and \\onlinecite{Ha99a} is designed for the fluctuating \nsuperfluid phase where the order parameter $Y$ is zero and vortices are present.\n\n\n\\subsection{Covariant derivatives}\n\\label{Sec:03E}\nThe flow parameter equation \\eqref{Eq:C61} and the constraint condition \\eqref{Eq:C36} \nare local equations. Consequently, the flow parameter $\\tau$, the renormalization $Z$ \nfactors, and the dimensionless coupling parameters are local and depend on space \nand time. This fact will affect the space and time derivatives in the renormalized \nequations. We must replace the partial differential operators by covariant \nderivatives. To do this, we write the renormalization equations in a form like \nEqs.\\ \\eqref{Eq:C01}-\\eqref{Eq:C03}, so that the bare quantities are on the left-hand \nside and all renormalized quantities are on the right hand side. Then we apply the \ndifferential operator. We start with the renormalization of the temperature \n\\eqref{Eq:C27} which is equivalent to \\eqref{Eq:C02}. We apply the nabla operator \nand obtain\n\\begin{eqnarray}\n\\nabla [(T-T_0)\/T_\\lambda] &=& \\nabla [\\tau\\, \\Delta\\rho] \n= \\tau \\, [ \\nabla + (\\nabla\\ln\\tau) ] \\Delta\\rho \\nonumber\\\\\n&=& \\tau \\, \\mathbf{D} \\Delta\\rho \\ .\n\\label{Eq:C65}\n\\end{eqnarray}\nThe last equality sign defines the covariant derivative. We continue with the \ntemperature difference \\eqref{Eq:C28} which is equivalent to \\eqref{Eq:C03} \nand proceed in the same way. As a result we obtain the covariant derivatives \n\\begin{eqnarray}\n\\mathbf{D} \\Delta\\rho &=& [ \\nabla + (\\nabla\\ln\\tau) ] \\Delta\\rho \\ ,\n\\label{Eq:C66} \\\\\n\\mathbf{D} \\rho &=& [ \\nabla + (\\nabla\\ln\\tau) ] \\rho \\ .\n\\label{Eq:C67}\n\\end{eqnarray}\nThe covariant derivative of $\\rho_1$ is more complicated. Generalizing \nEq.\\ \\eqref{Eq:C37} we obtain\n\\begin{equation}\n(\\mathbf{D}\\rho_1) \\{ 1 + 8u A_1 \\} = \\mathbf{D}\\rho \n+ (4u\/A_d) \\, \\mathbf{D} |Y|^2\n\\label{Eq:C68}\n\\end{equation}\nwhich can be resolved with respect to $\\mathbf{D}\\rho_1$.\n\nNext we consider the renormalization of the order parameter \\eqref{Eq:C01}.\nWe write this equation in terms of the dimensionless renormalized order parameter \n$Y$ by using \\eqref{Eq:C31}, apply the nabla operator, use \\eqref{Eq:C56}, and obtain\n\\begin{eqnarray}\n\\nabla \\langle\\psi\\rangle &=& \\nabla [ Z_\\phi^{1\/2} \\mu^{(d-2)\/2}\\, Y ]\n\\nonumber\\\\\n&=& Z_\\phi^{1\/2} \\mu^{(d-2)\/2}\\, \\Bigl[ \\nabla + \\frac{1}{2}\n( d - 2 - \\zeta_\\phi ) (\\nabla\\ln\\mu) \\Bigr] \\, Y \\nonumber\\\\\n&=& Z_\\phi^{1\/2} \\mu^{(d-2)\/2}\\, \\Bigl[ \\nabla + \\frac{1}{2}\n\\frac{ d - 2 - \\zeta_\\phi }{ 2 - \\zeta_r } \n(\\nabla\\ln\\tau) \\Bigr] \\, Y \\nonumber\\\\\n&=& Z_\\phi^{1\/2} \\mu^{(d-2)\/2}\\, \\mathbf{D} Y \\ .\n\\label{Eq:C69}\n\\end{eqnarray}\nAgain, the last equality sign defines the covariant derivative. We define \nthe running critical exponents \\cite{SD89}\n\\begin{eqnarray}\n\\nu &=& 1 \/ ( 2 - \\zeta_r ) \\ ,\n\\label{Eq:C70} \\\\\n\\eta &=& - \\zeta_\\phi \\ ,\n\\label{Eq:C71} \\\\\n\\beta &=& \\nu ( d - 2 + \\eta ) \/ 2 \\ .\n\\label{Eq:C72}\n\\end{eqnarray}\nThese exponents are called \\emph{running} exponents because they depend on the \nRG flow parameter $\\tau$ via the zeta functions and thus carry all the Wegner \ncorrections. In the asymptotic limit $\\tau\\to 0$ they converge to the universal \ncritical exponents. Then from Eq.\\ \\eqref{Eq:C69} we obtain the covariant derivative \nof the dimensionless renormalized order parameter\n\\begin{equation}\n\\mathbf{D} Y = [ \\nabla + \\beta (\\nabla\\ln\\tau) ] Y \\ .\n\\label{Eq:C73}\n\\end{equation}\nWe note that $\\beta$ is the running critical exponent of the order parameter.\nThis result makes clear, how the general structure of a covariant derivative of \na dimensionless renormalized quantity looks like: It is the partial derivative \nof the quantity plus the critical exponent times the partial derivative of $\\ln\\tau$ \ntimes the quantity. In the model-$F$ equations we also need the second covariant \nderivative of the order parameter. It is obtained by applying the operator twice, \ni.e.\\ \n\\begin{equation}\n\\mathbf{D}^2 Y = [ \\nabla + \\beta (\\nabla\\ln\\tau) ]^2 Y \\ .\n\\label{Eq:C74}\n\\end{equation}\n\nFurthermore we consider the renormalization of the heat current \\eqref{Eq:C32}. \nApplying the nabla operator we obtain\n\\begin{eqnarray}\n\\nabla [ \\mathbf{Q}\/g_0 k_B T_\\lambda ] &=& \\nabla[ \\mu^{d-1} \\, \\tilde{\\mathbf{q}} ] \n\\nonumber\\\\\n&=& \\mu^{d-1} \\, [ \\nabla + (d-1) (\\nabla\\ln\\mu) ] \\, \\tilde{\\mathbf{q}}\n\\nonumber\\\\\n&=& \\mu^{d-1} \\, [ \\nabla + (d-1) \\nu (\\nabla\\ln\\tau) ] \\, \\tilde{\\mathbf{q}}\n\\nonumber\\\\\n&=& \\mu^{d-1} \\, \\mathbf{D} \\tilde{\\mathbf{q}} \\ .\n\\label{Eq:C75}\n\\end{eqnarray}\nThus, we find the covariant derivative of the dimensionless renormalized heat current\n\\begin{equation}\n\\mathbf{D} \\tilde{\\mathbf{q}} = [ \\nabla + (d-1)\\nu (\\nabla\\ln\\tau) ] \n\\tilde{\\mathbf{q}} \\ .\n\\label{Eq:C76}\n\\end{equation}\nWe identify $(d-1)\\nu$ as the running critical exponent of the heat current. The \ninverse exponent $x=1\/[(d-1)\\nu]$ is known from the depression of the critical \ntemperature $T_\\lambda$ by a nonzero heat current $Q$ \\cite{On83,HD92}.\nWe note that the covariant derivatives \\eqref{Eq:C66}-\\eqref{Eq:C68} and \n\\eqref{Eq:C73}-\\eqref{Eq:C74} have been derived already in our previous unpublished \napproach \\cite{HN08} for the interface in thermal equilibrium at zero heat current.\n\nAbove, we have defined the covariant derivatives with respect to the space coordinates \n$\\mathbf{D}$. We also need the covariant derivatives with respect to time $D_t$. \nTo obtain them we replace the nabla operator by the partial time derivative \n$\\partial_t=\\partial\/\\partial t$. Thus, as results we obtain e.g.\\ \n\\begin{eqnarray}\nD_t Y &=& [ \\partial_t + \\beta (\\partial_t \\ln\\tau) ] Y \\ ,\n\\label{Eq:C77} \\\\\nD_t \\Delta\\rho &=& [ \\partial_t + (\\partial_t \\ln\\tau) ] \\Delta\\rho \\ .\n\\label{Eq:C78}\n\\end{eqnarray}\n\nNow, we are ready to rewrite the model-$F$ equations in terms of covariant \nderivatives. From Eqs.\\ \\eqref{Eq:C38} and \\eqref{Eq:C39} we obtain\n\\begin{eqnarray}\n\\frac{1}{2\\gamma\\, \\tau} \\, \\frac{1}{g_0} \\, D_t Y \n&=& - \\frac{w}{F} \\bigl[ \\rho_1 - \\xi^2 \\mathbf{D}^2 \\bigr] Y \\nonumber\\\\\n&&+ \\frac{i}{2 \\gamma} \\Delta \\rho \\, Y \\ ,\n\\label{Eq:C79} \\\\\n\\frac{C_\\mathrm{ren}}{(2\\gamma)^2\\tau} \\frac{A_d}{g_0}\n\\, D_t \\Delta \\rho &=& - \\xi\\, \\mathbf{D} \\tilde{\\mathbf{q}} \\ .\n\\label{Eq:C80}\n\\end{eqnarray}\nIn these equations we have performed some further substitutions which \nare known from our previous paper \\cite{Ha99a}, i.e.\\ \n\\begin{eqnarray}\n\\mu = \\xi^{-1} \\ , \\qquad  g \\mu^2 = g_0 \\, 2 \\gamma \\, \\tau \\ .\n\\label{Eq:C81}\n\\end{eqnarray}\nHere $\\xi=\\xi[\\tau]$ is the correlation length. Close to criticality it has the \nasymptotic form $\\xi=\\xi_0\\, \\tau^{-\\nu}$. The identification $\\mu=\\xi^{-1}$ is \ncorrect in our Hartree approximation which is a self-consistent one-loop \napproximation. Corrections appear in higher orders \\cite{SD89}. The renormalized \ntime-scale parameter $g \\mu^2$ is expressed in terms of the bare parameter $g_0$ \nby using the renormalization equation \\eqref{Eq:C10} where $\\chi_0 Z_m$ has been \neliminated in favor of $\\tau$ by \\eqref{Eq:C30}. As a result the renormalized \nmodel-$F$ equations \\eqref{Eq:C79} and \\eqref{Eq:C80} are dimensionless equations \nfor the dimensionless quantities. There are two parameters, which control the \nscales of space and time. They are $\\xi_0$ and $g_0$, respectively.\n\nThe model-$F$ equations \\eqref{Eq:C79} and \\eqref{Eq:C80} are supplemented by some \nfurther equations including the dimensionless renormalized entropy current \n\\eqref{Eq:C35} and the two constraint conditions \\eqref{Eq:C36} and \\eqref{Eq:C61} \nwhere all nabla operators $\\nabla$ are replaced by respective covariant derivatives \n$\\mathbf{D}$. Thus, we obtain the dimensionless renormalized entropy current\n\\begin{equation}\n\\tilde{\\mathbf{q}} = - \\frac{A_d}{2\\gamma F}\\, \\Bigl\\{ 1 + \\frac{f}{2} A_1 \\Bigr\\} \n\\, \\xi\\, \\mathbf{D} \\Delta \\rho - \\mathrm{Im}[ Y^* \\xi\\, \\mathbf{D} Y] \\ ,\n\\label{Eq:C82}\n\\end{equation}\nthe constraints\n\\begin{eqnarray}\nK_1 &=& 3\\,\\rho_1 - 2\\,\\rho + 3 (4u\/A_d) f_Y ( \\xi\\,\\mathbf{D} Y )^2 \\nonumber\\\\\n&&+ f_{\\Delta\\rho} ( \\xi\\,\\mathbf{D} \\Delta\\rho )^2 - 1 = 0 \\ ,\n\\label{Eq:C83} \\\\\nK_2 &=& \\rho_1 \\{ 1 + 8u A \\} - [ \\rho + (4u\/A_d) \\, |Y|^2 ] = 0 \\ , \\qquad\n\\label{Eq:C84}\n\\end{eqnarray}\nand furthermore the dimensionless variables\n\\begin{eqnarray} \n\\sigma &=& - \\frac{1}{12} \\Bigl[ (\\xi\\,\\mathbf{D} \\rho_1)^2 \n+ 2 \\frac{w^{\\prime\\prime}}{w^\\prime} \n\\Bigl( \\frac{F}{4 \\gamma w^\\prime} \\xi\\,\\mathbf{D} \\Delta \\rho \\Bigr) \n\\cdot ( \\xi\\,\\mathbf{D} \\rho_1 ) \\nonumber\\\\\n&&- \\Bigl( \\frac{F}{4 \\gamma w^\\prime} \\xi\\,\\mathbf{D} \\Delta \\rho \\Bigr)^2\n\\Bigr] \\ ,\n\\label{Eq:C85} \n\\end{eqnarray}\nand $\\zeta = \\rho_1\/\\sigma^{1\/3}$ or $\\bar\\zeta=\\rho_1 \/ (-\\sigma)^{1\/3}$ which \nare needed to calculate the dimensionless amplitudes \\eqref{Eq:C45}-\\eqref{Eq:C46} \nor \\eqref{Eq:C51}-\\eqref{Eq:C52}.\n\n\n\\subsection{Numerical algorithm}\n\\label{Sec:03F}\nThe numerical algorithm for solving the renormalized model-$F$ equations \n\\eqref{Eq:C79} and \\eqref{Eq:C80} together with the constraints \\eqref{Eq:C83} \nand \\eqref{Eq:C84} is implemented by two iterations. First on the left-hand sides \nof the model-$F$ equations the partial time derivatives within the covariant \nderivatives are replaced by discrete forward differences\n\\begin{eqnarray}\n\\partial_t Y &\\to& [ Y(\\mathbf{r},t+\\Delta t) - Y(\\mathbf{r},t) ] \/ \\Delta t \\ ,\n\\label{Eq:C86} \\\\\n\\partial_t \\Delta\\rho &\\to& [ \\Delta\\rho(\\mathbf{r},t+\\Delta t) \n- \\Delta\\rho(\\mathbf{r},t) ] \/ \\Delta t \\ .\n\\label{Eq:C87}\n\\end{eqnarray}\nSecondly, the constraints are solved by a Newton method. The two iterations \nare performed in parallel, i.e.\\ alternatively one time step and one Newton \nstep. In this way starting with appropriate initial functions at an initial \ntime the dimensionless renormalized quantities $Y(\\mathbf{r},t)$, \n$\\Delta\\rho(\\mathbf{r},t)$ and $\\rho_1(\\mathbf{r},t)$, $\\ln\\tau(\\mathbf{r},t)$ \nare obtained as functions of space and time. All the other dimensionless renormalized \nquantities which are needed on the right and sides of the iteration equations \ncan be calculated from the four quantities by formulas we have derived above. \nThe covariant derivatives $\\mathbf{D}Y$, $\\mathbf{D}^2Y$, $\\mathbf{D}\\Delta\\rho$,\n$\\mathbf{D}\\rho$, and $\\mathbf{D}\\tilde{q}$ are calculated with discrete \nnabla and Laplace operators on an equidistant grid of the space-coordinates \n$\\mathbf{r}$. The covariant derivatives of further quantities can by related to \nthose five by equations like \\eqref{Eq:C68}.\n\nFor the Newton-iteration step we need the derivatives of the constraint \nfunctions $K_1$ and $K_2$ with respect to $\\rho_1$ and $\\ln\\tau$. We use \nthe derivatives \n\\begin{eqnarray}\n\\partial K_1 \/ \\partial \\rho_1 &=& 3 \\ ,\n\\label{Eq:C88} \\\\\n\\partial K_2 \/ \\partial \\rho_1 &=& 1 + 8u\\,A_1 \\ ,\n\\label{Eq:C89} \\\\\n\\partial K_1 \/ \\partial \\ln\\tau &\\approx& - \\bigl\\{ -2\\rho \n+ 3 (4u\/A_d) f_Y ( \\xi\\, \\mathbf{D} Y ) ^2 2 ( \\beta + \\nu ) \\nonumber\\\\\n&&+ f_{\\Delta\\rho} ( \\xi\\, \\mathbf{D} \\Delta\\rho )^2 2 ( 1 + \\nu ) \\bigr\\} \\ ,\n\\label{Eq:C90} \\\\\n\\partial K_2 \/ \\partial \\ln\\tau &\\approx& - \\bigl\\{ 8u\\, E_1 \\, 2 ( 1 + \\nu )\n\\nonumber\\\\\n&&- [ \\rho + (4u\/A_d) Y^2 2\\beta ] \\bigr\\}\n\\label{Eq:C91}\n\\end{eqnarray}\ntogether with the amplitude\n\\begin{equation}\nE_1 = (\\rho_1 \/ 6) \\bigl[ (2-\\varepsilon) A - 2 A_1 - 1 \\bigr] \\ .\n\\label{Eq:C92}\n\\end{equation}\nThe latter two derivatives are approximations, because we omitted the weak \ndependence of the dimensionless coupling parameters $u[\\tau]$, $\\gamma[\\tau]$, \netc.\\ on the logarithmic RG flow parameter $\\ln\\tau$. Nevertheless, our numerical \ncalculation works. There is no significant influence of this approximation.\n\nOur numerical calculations are performed very close to criticality where $\\tau< 10^{-5}$. \nConsequently, for the running exponents we can use the universal critical \nexponents as a good approximation. We use the experimental value $\\nu=0.671$ of \nLipa \\emph{et al.}\\ \\cite{Li96,Li03} and the theoretical value $\\eta=0.038$ of Schloms \nand Dohm \\cite{SD89}. The exponent $\\beta=0.348$ is calculated from the scaling \nrelation \\eqref{Eq:C72}, where the dimension of space is $d=3$. Finally, we can \nuse the asymptotic formula for the correlation length $\\xi=\\xi_0\\, \\tau^{-\\nu}$ \nas a good approximation.\n\nOur numerical calculations show that the iterations are stable for small heat \ncurrents $Q=|\\mathbf{Q}|\\lesssim 20\\,\\mathrm{nW\/cm^2}$ where the gravity is the \ndominating external force and the dimensionless parameter $\\sigma$ defined in \n\\eqref{Eq:C85} is always negative. For lager heat currents the parameter $\\sigma$ \nwill have a sign change locally in space, which causes numerical troubles. We can \nstabilize the calculations up to a maximum heat current \n$Q_\\mathrm{max}=160\\,\\mathrm{nW\/cm^2}$ by adding a small imaginary constant \nto the right-hand side of Eq.\\ \\eqref{Eq:C85}. However, for larger heat currents \nwhere the heat flow is the major and the gravity is the minor external influence \nthe iteration is unstable so that no results can be obtained.\n\n\n\\section{Numerical results}\n\\label{Sec:04}\nMost experiments with liquid $^4$He close to the superfluid transition are performed \nat saturated vapor pressure. The temperature $T$ is varied in the region near \n$T_\\lambda$ where the pressure is kept at the value of the liquid-gas transition. \nIn this case the critical temperature is $T_\\lambda=2.172\\,\\mathrm{K}$. The parameters \nwhich specify the scales of length and time are $\\xi_0=1.44\\times10^{-8}\\,\\mathrm{cm}$ \nand $g_0=2.164\\times 10^{11}\\,\\mathrm{s^{-1}}$, respectively \\cite{Al85,Li96,Li03}. The \ndimensionless renormalized coupling parameters $u[\\tau]$, $\\gamma[\\tau]$, \n$w[\\tau]=w^\\prime[\\tau]+i w^{\\prime\\prime}[\\tau]$, $F[\\tau]$, and $f[\\tau]$ as functions \nof the RG flow parameter $\\tau$ are taken from Ref.\\ \\onlinecite{Do91}.\n\nWe perform the numerical calculations for liquid $^4$He in $d=3$ dimensions. The \nsystem is assumed to be homogeneous in the two horizontal directions $x$ and $y$. \nThus, all quantities and functions depend only on the altitude coordinate $z$ and \nthe time $t$. The model-$F$ equations reduce to partial differential equations with \nthe two variables $z$ and $t$. The size of the experimental cells which contain the \nliquid $^4$He is usually some millimeters in $z$ direction. We use a cell length \n$L=2.0\\,\\mathrm{mm}$ and discretize the $z$ coordinate into $500$ points. Consequently, \nthe discretization is $\\Delta z=4.0\\,\\mathrm{\\mu m}$ in the altitude coordinate.\n\nThe discretization of the time $\\Delta t$ in the partial derivatives \\eqref{Eq:C86} \nand \\eqref{Eq:C87} must be sufficiently small so that the iteration converges. On the \nother hand $\\Delta t$ should be sufficiently large, so that the calculation time on \nthe computer is not too long. We find $\\Delta t = 4.0\\times10^{-6}\\,\\mathrm{s}$ as \nan optimum choice. Starting the calculations in any nonequilibrium state, we first \nobserve space and time dependent oscillations which are related to second sound. \nThese oscillations relax on a time scale of about one second. After a time interval\n$\\delta t=2.0\\,\\mathrm{s}$ the system reaches a stationary state with a constant \nhomogeneous heat current $\\mathbf{Q}$ where all oscillations are disappeared. This \nmeans we need $5\\times 10^5$ iteration steps on the computer until the system \nconverges to the steady state.\n\nFor a heat flow in $z$ direction there must be a heat source and a heat sink \nat the boundaries of the cell $z_1=-L\/2=-1.0\\,\\mathrm{mm}$ and \n$z_2=+L\/2=+1.0\\,\\mathrm{mm}$, respectively. Thus, a source and sink term must \nbe added to the heat transport equation of model $F$ \\eqref{Eq:B02} which is\ngiven by\n\\begin{equation}\nW(\\mathbf{r},t) = 2\\, [ Q_1\\,\\delta(z-z_1) - Q_2\\,\\delta(z-z_2) ] \\ .\n\\label{Eq:D01}\n\\end{equation}\nIn dimensionless renormalized form the source and sink term is\n\\begin{equation}\n\\tilde{w}(\\mathbf{r},t) = 2\\, [ \\tilde{q}_1\\,\\delta([z-z_1]\/\\xi) \n- \\tilde{q}_2\\,\\delta([z-z_2]\/\\xi) ] \\ .\n\\label{Eq:D02}\n\\end{equation}\nThis latter term must be added to the dimensionless renormalized model-$F$ \nequation \\eqref{Eq:C80} on the right-hand side. The relation between the \ndimensionless renormalized heat currents $\\tilde{q}$ and the physical heat \ncurrents $Q$ is obtained from the renormalization equations \\eqref{Eq:C08},\n\\eqref{Eq:C10}, and \\eqref{Eq:C32}. We obtain\n\\begin{equation}\n\\tilde{q} = \\frac{q}{g_0}\\, \\xi^{d-1}\n= \\frac{Q\\, \\xi^{d-1}}{g_0 k_\\mathrm{B} T_\\lambda}\n\\label{Eq:D03}\n\\end{equation}\nwhich should be applied to both heat currents in \\eqref{Eq:D01} and \n\\eqref{Eq:D02}. It is important to note, that the RG flow parameter \n$\\tau=\\tau(\\mathbf{r},t)$ and hence the correlation length $\\xi=\\xi[\\tau]$ \ndepend on space and time. This fact is important for Eqs.\\ \\eqref{Eq:D01} \nand \\eqref{Eq:D02}.\n\nWe perform the calculations in the following way. First the system is \nstabilized in the thermal equilibrium. Then at time $t=t_0$ the external \nheat source and sink \\eqref{Eq:D01} or \\eqref{Eq:D02} is switched on \nwhere we chose equal values $Q_1=Q_2=Q$. Then after a time interval \n$\\delta t=2.0\\,\\mathrm{s}$ all oscillations are relaxed and the system \nreaches a steady state. The local heat current \n$\\mathbf{Q}(\\mathbf{r},t)=Q\\,\\mathbf{e}_z$ will be homogenous in space, \nconstant in time, and directed vertically along the $z$ axis.\n\nThe boundary conditions at $z_1$ and $z_2$ are important for the stability of \nthe iterations. There should be no boundaries at all. This means we need periodic \nboundary conditions. The system can be made periodic in the following way.\nWe mirror the cell at one of the boundaries. Then we obtain a periodic structure \nof length $2L$. Furthermore, for the discretization the delta functions in \nEqs.\\ \\eqref{Eq:D01} and \\eqref{Eq:D02} must be replaced by smooth peaks of a \nsmall width $\\delta z$. We choose $\\delta z=3\\,\\Delta z$ which is a few \ndiscretization lengths. From the heat source at $z=z_1$ the heat current $Q_1$ \nwill flow away in \\emph{both} directions, where on the other hand a heat current \n$Q_2$ will flow from \\emph{both} directions to the heat sink at $z_2$. This fact \nexplains the factor $2$ in Eqs.\\ \\eqref{Eq:D01} and \\eqref{Eq:D02}.\n\n\n\\subsection{Dimensionless renormalized quantities}\n\\label{Sec:04A}\nThe direct results of the numerical calculation are the dimensionless renormalized \ntemperature parameters $\\Delta\\rho$, $\\rho$, $\\rho_1$, and the dimensionless \nrenormalized order parameter $Y$ as functions of the altitude coordinate $z$ and \nthe time $t$. In Fig.\\ \\ref{Fig:01} the results are shown for the \nsuperfluid\/normal-fluid interface of liquid $^4$He at zero heat current $Q=0$ in \nthermal equilibrium. The interface is induced by the gravitational acceleration \n$g=9.81\\,\\mathrm{m\/s^2}$ on earth. Since in thermal equilibrium the temperature \nis constant we may choose it equal to the reference temperature so that $T=T_0$. \nHence Eq.\\ \\eqref{Eq:C27} implies $\\Delta\\rho=0$. This is a trivial result which \nis shown by the black dotted line. The parameter $\\rho$ is related to the temperature \ndifference $T-T_\\lambda$ by \\eqref{Eq:C28} and shown as green solid line. The \nmodified temperature parameter $\\rho_1$ is defined in \\eqref{Eq:C36} and shown as \nblue dashed line. Finally, the modulus of the dimensionless renormalized order \nparameter $\\eta=|Y|$ is shown as red dash-dotted line.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig01.eps}\n\\caption{(Color online) The dimensionless renormalized temperature parameters \n$\\rho$ (green solid line), $\\rho_1$ (blue dashed line), and the modulus of the \norder parameter $\\eta=|Y|$ (red dash-dotted line) for the superfluid\/normal-fluid\ninterface.}\n\\label{Fig:01}\n\\end{figure}\n\nIn Fig.\\ \\ref{Fig:01} we observe three different regions. For low altitudes \n$z\\lesssim -100\\,\\mathrm{\\mu m}$ we find the asymptotic values $\\rho\\to 1$, \n$\\rho_1\\to 1$, and $\\eta=|Y|\\to 0$. Hence, in this region the $^4$He is normal fluid. \nWe recover the related flow parameter condition $\\rho=1$ of Ref.\\ \\onlinecite{Do85} \nin the asymptotic limit $z\\to-\\infty$. For high altitudes \n$z\\gtrsim +100\\,\\mathrm{\\mu m}$ we find the asymptotic values $\\rho\\to -1\/2$, \n$\\rho_1\\to 0$, and $\\eta=|Y|\\to (A_d\/8u)^{1\/2}$ where $A_d=1\/4\\pi$ for $d=3$.\nHence, in this latter region the order parameter is nonzero and the $^4$He is \nsuperfluid. Again, we recover the related flow parameter condition $\\rho=-1\/2$ of \nRef.\\ \\onlinecite{Do85} in the asymptotic limit $z\\to+\\infty$. The third region \nis the interface region $-100\\,\\mathrm{\\mu m}\\lesssim z \\lesssim +100\\,\\mathrm{\\mu m}$. \nHere the curves interpolate between the asymptotic values. We clearly see that the \ninterface induced by gravity has a thickness of about \n$\\Delta z_{I,g}\\approx 200\\,\\mathrm{\\mu m}$.\n\nSince the system is constant with respect to the horizontal coordinates $x$, $y$, \nand with respect to the time $t$, the covariant derivatives of the dimensionless \nrenormalized quantities are nonzero only for the altitude coordinate $z$. In most \ncases these covariant derivatives are calculated by numerical differentiation \nusing the formulas derived in section \\ref{Sec:03E}. An exception is \n$\\xi D_z \\rho_1$ which is expressed in terms of other covariant derivatives by \nformula \\eqref{Eq:C68}. The result is shown in Fig.\\ \\ref{Fig:02} by the blue \ndashed line. Alternatively, we apply Eq.\\ \\eqref{Eq:C67} to the modified \ntemperature parameter $\\rho_1$ and calculate the covariant derivative directly by \nnumerical differentiation. This latter procedure is not correct in the interface \nregion where the renormalization factors depend on the altitude coordinate because \n$\\rho_1$ is not renormalized as $\\rho$. Nevertheless, in Fig.\\ \\ref{Fig:02} the \nresult is shown by the blue solid line. Surprisingly, the two blue lines, the solid \none and the dashed one, are close to each other. Hence, Eq.\\ \\eqref{Eq:C67} is not \nthat bad for calculating the covariant derivative $\\xi D_z \\rho_1$.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig02.eps}\n\\caption{(Color online) The blue lines show the dimensionless covariant \nderivative $\\xi D_z\\rho_1$ calculated in two ways: by numerical differentiation \n(blue solid line) and by formula \\eqref{Eq:C68} (blue dashed line). Furthermore \nthe parameter $\\sigma$ defined in \\eqref{Eq:C85} and multiplied by a factor $12$ \nis shown as orange dash-dotted line. Finally, the argument of the function \n\\eqref{Eq:C50} $\\bar\\zeta$ is shown as magenta double-dash-dotted line.}\n\\label{Fig:02}\n\\end{figure}\n\nThe blue lines in Fig.\\ \\ref{Fig:02} represent the covariant derivative of the \nblue dashed line in Fig.\\ \\ref{Fig:01}. However, the latter line represents \n$\\rho_1=\\rho_1(z)$ and shows a negative minimum value $\\rho_{1,\\mathrm{min}}<0$ \nat the position $z_\\mathrm{min}\\approx 60\\,\\mathrm{\\mu m}$. Consequently for \n$\\xi D_z \\rho_1$ we expect a zero at this position related to a sign change. \nIn Fig.\\ \\ref{Fig:02} the solid blue line does show this zero and sign change \nbut the dashed blue line does not. In this way, the apparently incorrect formula \n\\eqref{Eq:C67} for $\\rho_1$ appears to be more realistic than the generic \nformula \\eqref{Eq:C68}.\n\nThe existence of the sign change is supported by the following argument. In a small \n$z$ interval close to the interface we may modify the renormalization-group theory \nby choosing a constant flow parameter $\\tau$. In this case the covariant derivatives \nreduce to the partial derivatives so that Eqs.\\ \\eqref{Eq:C67} and \\eqref{Eq:C68} \nwould yield identical results for $\\xi D_z \\rho_1=\\xi \\partial_z \\rho_1$ and the \ntwo blue lines in Fig.\\ \\ref{Fig:02} would collapse to a single line. As a result \nthe sign change would be found at $z_\\mathrm{min}$ if we evaluate the partial \nderivative explicitly by differentiation of the blue dashed line in Fig.\\ \\ref{Fig:01}. \n\nHowever, the sign change of the solid blue line in Fig.\\ \\ref{Fig:02} would have \na dramatic consequence for the numerical procedure when calculating $\\sigma$ \nand the amplitudes $A$ and $A_1$. In thermal equilibrium we have $\\Delta\\rho=0$ \nso that Eq.\\ \\eqref{Eq:C85} reduces to $\\sigma= -(\\xi D_z \\rho_1)^2\/12$. \nConsequently, $\\sigma$ will be negative everywhere except at a point close to \n$z_\\mathrm{min}$. At this point we have $\\sigma=0$ so that the formulas for the \namplitudes $A$ and $A_1$ reduce to \\eqref{Eq:C40} and \\eqref{Eq:C41}, respectively. \nHowever, close to the minimum position $z_\\mathrm{min}$ the modified temperature \nparameter $\\rho_1$ is negative which implies an imaginary result for \n$\\rho_1^{-\\varepsilon\/2}$ in Eqs.\\ \\eqref{Eq:C40} and \\eqref{Eq:C41} where \n$\\varepsilon=4-d=1$ for $d=3$. Hence, the amplitudes $A$ and $A_1$ are not well \ndefined if the solid line in Fig.\\ \\ref{Fig:02} and the formula \\eqref{Eq:C67} is used.\n\nThe problem arises due to the fact that we evaluate the Green function \\eqref{Eq:B22},\nthe condensate density $n_\\mathrm{s}$, the superfluid current $\\mathbf{J}_\\mathrm{s}$, \nand hence the amplitudes $A$ and $A_1$ in an approximation where only the covariant \ngradients of the temperature parameters $\\rho_1$ and $\\Delta \\rho$ are taken into \naccount. If we could do the calculation for the full space dependence all these \nquantities would be well defined. In the unpublished work \\cite{HN08} we extended \nthe calculation by including also the curvatures of the temperature parameters. \nWhile the problem at $z_\\mathrm{min}$ was abolished, the calculation was much more \ncomplicated and restricted to the thermal equilibrium at zero heat current. Moreover, \nother mathematical difficulties appeared. Hence, this more sophisticated calculation \ncould not be realized in practice for our purpose.\n\nHowever, our fortune is a small inaccuracy of the approximation in our numerical \ncalculation in practice which implies that the blue dashed line in Fig.\\ \\ref{Fig:02} \nis completely negative and does not show a zero and a sign change for the covariant \nderivative $\\xi D_z \\rho_1$. The related parameter $\\sigma= -(\\xi D_z \\rho_1)^2\/12$ \nis shown in Fig.\\ \\ref{Fig:02} as orange dash-dotted line where it has been enhanced \nby a factor $12$. Clearly, this curve is negative and never zero for all altitudes \n$z$ in the interface region. For this reason we can apply our formulas \\eqref{Eq:C51} \nand \\eqref{Eq:C52} for the amplitudes $A$ and $A_1$ without a problem if we use the \ngeneric formula \\eqref{Eq:C68} for the dimensionless gradient $\\xi D_z \\rho_1$. We \nobtain smooth and stable results which are within the accuracy of our approximation.\n\nIn order to evaluate the amplitudes $A$ and $A_1$ we need the function \n\\eqref{Eq:C50} and its argument $\\bar\\zeta=\\rho_1 \/ (-\\sigma)^{1\/3}$. Consequently, \nfrom the dashed blue line in Fig.\\ \\ref{Fig:01} and the orange dash-dotted line in \nFig.\\ \\ref{Fig:02} we obtain the dimensionless variable $\\bar\\zeta$ as a function \nof the altitude coordinate $z$ which is shown in Fig.\\ \\ref{Fig:02} by the magenta \ndouble-dash-dotted line. In the normal-fluid region for $z<0$ the variable \n$\\bar\\zeta$ increases quickly for decreasing altitude $z$. Consequently, in this \ncase the asymptotic formula \\eqref{Eq:C53} can be used so that the amplitudes \n$A$ and $A_1$ reduce to the simple formulas \\eqref{Eq:C40} and \\eqref{Eq:C41} of \nthe normal-fluid equilibrium state. In the superfluid region near the interface \nthe variable $\\bar\\zeta$ is negative. However, it is bounded from below by the \nvalue $-1$. Consequently, the asymptotic formula \\eqref{Eq:C54} is not needed. \nThis means that the variable $\\bar\\zeta$ never comes in the large negative region \nwhere the function \\eqref{Eq:C50} oscillates and possesses a significant imaginary \npart. This observation is very important for the consistency of our theory because \nthe oscillations would be unphysical and the imaginary part would be related to \nan instability.\n\n\n\\subsection{Temperature profiles}\n\\label{Sec:04B}\nUntil now, the calculations are restricted to the thermal equilibrium at zero \nheat current $Q=0$. Here the phase of the order parameter $Y=\\eta\\,e^{i\\varphi}$ \nis constant, so that we can chose $\\varphi=0$. We have extended our numerical \ncalculations to small nonzero heat currents $Q$ in the interval \n$-70\\,\\mathrm{nW\/cm^2}\\leq Q \\leq +160\\,\\mathrm{nW\/cm^2}$. In this latter case \nthe phase of the order parameter $\\varphi=\\varphi(z,t)$ will be a nontrivial \nfunction of the altitude coordinate $z$ and the time $t$. A positive heat current \n$Q>0$ means a heat flow $\\mathbf{Q}=Q\\,\\mathbf{e}_z$ in the $z$ direction which means \nthat the heat current flows upward from bottom to top. The original experiment \nby Duncan \\emph{et al.}\\ \\cite{DAS88} and succeeding experiments investigating \nthe superfluid\/normal-fluid interface induced by a heat current $Q$ were \nperformed in this configuration. On the other hand a negative heat current $Q<0$ \nmeans a downward heat flow from top to bottom. This latter configuration was \ninvestigated much later in the experiment by Moeur \\emph{et al.}\\ \\cite{Du97}. \n\nIn the nonequilibrium system with a nonzero heat flow the dimensionless \nrenormalized temperature parameter $\\Delta\\rho$ will be nonzero. Once the local \nspace and time dependent RG flow parameter $\\tau=\\tau(z,t)$ is known, the space \nand time dependent temperature profile $T=T(z,t)$ is calculated from $\\Delta\\rho$ \nby Eq.\\ \\eqref{Eq:C27}. Furthermore, the local space and time dependent heat \ncurrent $Q=Q(z,t)$ is calculated from the dimensionless renormalized heat \ncurrent $\\tilde q$ by Eq.\\ \\eqref{Eq:D03}. After a time difference of about \n$\\delta t=2\\,\\mathrm{s}$ the system will relax in a stationary state where all \nquantities are constant in time. If in Eq.\\ \\eqref{Eq:D01} the source and sink \nparameters $Q_1=Q_2=Q$ are chosen, a vertical heat current $Q$ will be found in \nthe whole system which is constant in the space variable $z$. Consequently, the \ndifferent temperature profiles we obtain in our numerical calculations can be \nlabeled by this constant heat current.\n\nOur numerical results are shown in Fig.\\ \\ref{Fig:03}. The temperature profile \n$T(z)$ is shown by the colored solid lines for several values of the heat current \n$Q$ which are specified in the caption of the figure. On the other hand, the \nsuperfluid transition temperature $T_\\lambda(z)$ as a function of the altitude \ncoordinate $z$ is shown by the straight black dashed line. The slope of this \nlatter line is the effect of the gravity on earth.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig03.eps}\n\\caption{(Color online) The temperature profiles $T(z)$ of the \nsuperfluid\/normal-fluid interface of liquid $^4$He in gravity are shown for several \nheat currents $Q$ as colored solid lines. The solid lines on the left-hand (normal-fluid)\nside are ordered from top to bottom with respect to decreasing heat currents $Q=160$, \n$130$, $100$, $70$, $40$, $20$, $0$, $-20$, $-40$, $-60$, $-70\\,\\mathrm{nW\/cm^2}$.\nOn the right-hand (superfluid) side they are ordered from bottom to top.\nThe horizontal black solid line represents the temperature in thermal equilibrium \nfor $Q=0$. The straight black dashed line represents the superfluid transition \ntemperature $T_\\lambda(z)$.}\n\\label{Fig:03}\n\\end{figure}\n\nThe altitude $z_0$ at which the temperature profiles $T(z)$ and $T_\\lambda(z)$ \nintersect each other so that $T(z_0)= T_\\lambda(z_0)$ may be viewed as a reference \naltitude to specify the position of the superfluid\/normal-fluid interface. We \nrealize that the system is translation invariant in the sense that we can move \nthe curves parallel along the straight dashed line. Thus, for convenience and \nsimplicity we select a coordinate system so that all curves intersect at the same \naltitude $z=z_0=0$. This choice is no physical restriction and has been applied \nin Fig.\\ \\ref{Fig:03}.\n\nOn the left hand side for low altitudes $z\\lesssim -100\\,\\mathrm{\\mu m}$ the \nsystem is normal fluid. Here the heat transport equation $Q=-\\lambda_T \\partial_z T$ \nimplies that the temperature gradient $\\partial_z T$ is negative for positive \nheat currents $Q$ and positive for negative heat currents. The values of the \ngradients are considerably large. On the right hand side for high altitudes \n$z\\gtrsim +100\\,\\mathrm{\\mu m}$ the system is superfluid. Here the heat is \ntransported convectively following the two-fluid model so that the temperatures \n$T(z)$ are nearly constant and the gradients are nearly zero. The intermediate \nregion $-100\\,\\mathrm{\\mu m} \\lesssim z\\lesssim +100\\,\\mathrm{\\mu m}$ is the \nsuperfluid\/normal-fluid interface. Here the temperature profiles interpolate \nthe two outer regions.\n\nFor positive heat currents $Q>0$ (heat flow upward) the slope of the temperature \ncurve $T(z)$ increases without a limit on the normal-fluid side for $z\\to -\\infty$. \nHowever, for negative heat currents $Q<0$ (heat flow downward) the slope increases \nup to a limiting value which is the slope of $T_\\lambda(z)$ so that in the limit \n$z\\to -\\infty$ the temperature profile $T(z)$ approaches a straight line parallel \nto the straight dashed line $T_\\lambda(z)$. This latter fact is clearly observed \nin the lower left part of Fig.\\ \\ref{Fig:03}. It represents the \n\\emph{self-organized critical state} predicted by Onuki \\cite{On87} \nand discovered in the experiment by Moeur \\emph{et al.}\\ \\cite{Du97}.\n\nWhile Figs.\\ \\ref{Fig:01} and \\ref{Fig:02} are calculated for the thermal \nequilibrium at zero heat current $Q=0$, we have calculated the related curves also \nfor the nonequilibrium state at the nonzero heat currents of Fig.\\ \\ref{Fig:03}. \nMost curves do not change very much, the characteristic forms remain qualitatively. \nAn exception is the parameter $\\sigma$ defined in Eq.\\ \\eqref{Eq:C85} and shown as \norange dash-dotted line in Fig.\\ \\ref{Fig:02}. This parameter is negative in the \nwhole system only for small heat currents in the interval \n$-10\\,\\mathrm{nW\/cm^2}\\lesssim Q \\lesssim +20\\,\\mathrm{nW\/cm^2}$. For larger heat \ncurrents outside this interval the parameter $\\sigma$ will change the sign from \nnegative to positive at specific altitudes $z$. For even larger negative heat \ncurrents $Q\\lesssim -20\\,\\mathrm{nW\/cm^2}$ and even larger positive heat currents \n$Q\\gtrsim +40\\,\\mathrm{nW\/cm^2}$ the parameter $\\sigma$ is positive in the whole \nsystem.\n\n\n\\subsection{Order parameter}\n\\label{Sec:04C}\nThe order parameter in physical units $\\langle\\psi\\rangle$ is calculated from the \ndimensionless renormalized order parameter $Y$ via the renormalization formulas \n\\eqref{Eq:C01} and \\eqref{Eq:C31}. Putting these equations together and replacing \n$\\mu\\to\\xi^{-1}$ we obtain\n\\begin{equation}\n\\langle\\psi\\rangle = Z_\\phi^{1\/2}\\, Y \\, \\xi^{-(d-2)\/2} \\ .\n\\label{Eq:D04}\n\\end{equation}\nIntegrating the defining equation \\eqref{Eq:C56} for the zeta function $\\zeta_\\phi$, \nwe obtain an integral representation for the renormalization factor\n\\begin{equation}\nZ_\\phi = \\exp\\Bigl\\{ \\int_\\mu^\\infty \\zeta_\\phi \\, \\frac{d\\mu^\\prime}{\\mu^\\prime} \\Bigr\\} \n= \\exp\\Bigl\\{ - \\int_\\tau^\\infty \\nu\\eta \\, \\frac{d\\tau^\\prime}{\\tau^\\prime} \\Bigr\\} \\ .\n\\label{Eq:D05}\n\\end{equation}\nThe second equality sign is implied by the flow-parameter transformation \\eqref{Eq:C60} \ntogether with the running exponents $\\nu$ and $\\eta$, defined in \\eqref{Eq:C70} and \n\\eqref{Eq:C71}. The upper infinite integration boundaries guarantee $Z_\\phi=1$ in the \nlimits $\\mu\\to\\infty$ and $\\tau\\to\\infty$ which represent the mean-field or Gaussian \nfix point of the RG flow. If we use the correlation length $\\xi=\\xi_0\\, \\tau^{-\\nu}$ \nwe obtain the asymptotic formula for the order parameter \n$\\langle\\psi\\rangle\\sim \\tau^{\\nu(d-2+\\eta)\/2}=\\tau^\\beta$ with the correct critical \nexponent $\\beta$ defined in \\eqref{Eq:C72}.\n\nEqs.\\ \\eqref{Eq:D04} and \\eqref{Eq:D05} are suited for a numerical calculation once \nthe dimensionless renormalized order parameter $Y$, the RG flow parameter $\\tau$, \nand the running exponents \\eqref{Eq:C70}-\\eqref{Eq:C72} are known. We have calculated \nthe order parameter $\\langle\\psi\\rangle= M\\, e^{i\\varphi}$ in the stationary state \nfor all those heat currents $Q$ for which we have calculated the temperature profiles \nin the previous subsection. We obtain the modulus $M$ and the phase $\\varphi$ of \nthe order parameter. Our results for the modulus $M$ are shown in Fig.\\ \\ref{Fig:04} \nfor positive heat currents $Q\\geq 0$ (heat flow upward). The colors of the solid lines \ncorrespond to those in Fig.\\ \\ref{Fig:03}. Here and in the following figures we omit \nthe lines for negative heat currents $Q<0$ (heat flow downward) because they make the \nfigures complicated and involved but do now show new physics.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig04.eps}\n\\caption{(Color online) The modulus of the order parameter $M=|\\langle\\psi\\rangle|$ \nas a function of the temperature difference $T-T_\\lambda$ for the superfluid\/normal-fluid \ninterface in gravity. The colored solid lines from left to right represent the heat \ncurrents $Q=160$, $130$, $100$, $70$, $40$, $20$, $0\\,\\mathrm{nW\/cm^2}$.}\n\\label{Fig:04}\n\\end{figure}\n\nClose to criticality $T=T_\\lambda$ the curves are smooth. This is an effect of \ngravity and related to the superfluid\/normal-fluid interface. The width of the \nsmooth region is $\\Delta T_{I,g}=25\\,\\mathrm{nK}$ which corresponds to the \nthickness of the interface $\\Delta z_{I,g}=200\\,\\mathrm{\\mu m}$. The ratio is \napproximately the gradient of the superfluid transition temperature, i.e.\\ \n$\\Delta T_{I,g}\/\\Delta z_{I,g} \\approx \\partial T_\\lambda\/\\partial z \n= 1.273\\,\\mathrm{\\mu K\/cm}$. For increasing heat currents $Q$ the smooth curves \nare shifted to the left to lower temperatures. This fact is related to the \ndepression of the superfluid transition to lower temperatures by a heat current \nwhich has been observed and investigated in the experiment by Duncan, Ahlers, \nand Steinberg \\cite{DAS88}.\n\nAway from criticality for lower temperatures $T-T_\\lambda\\lesssim -30\\,\\mathrm{nK}$ \nthe curves approach asymptotically a single line which corresponds to the \nsingular order parameter $M=|\\langle\\psi\\rangle|\\sim (T_\\lambda - T)^\\beta$ for \n$T<T_\\lambda$ in thermal equilibrium and zero gravity. In Fig.\\ \\ref{Fig:04} the \nasymptotic curves do not fall perfectly on a single line. This observation is \na numerical error in our calculation. In order to stabilize the numerical iterations \nwe must add an imaginary part to the parameter $\\sigma$ defined in \\eqref{Eq:C85}. \nThis imaginary part increases with increasing heat current $Q$ and influences \nslightly the curves on the superfluid side.\n\nThe physical units $\\mathrm{cm}^{-1\/2}$ of the order parameter arising from \nthe formula \\eqref{Eq:D04} for $d=3$ dimensions appear to be artificial and \nunphysical. However, since the order parameter can not be observed in physical \nexperiments, this artifact is not important and no matter of concern.\n\nThe phase of the order parameter $\\varphi$ is dimensionless. Its gradient is \nrelated to the superfluid velocity $\\mathbf{v}_\\mathrm{s} = (\\hbar\/m_4)\\nabla\\varphi$. \nFor nonzero heat currents $Q$ we find nontrivial results for the superfluid velocity \n$v_\\mathrm{s}$. If we approach the interface from the superfluid side, $v_\\mathrm{s}$ \nincreases monotonically. However, on the normal-fluid side, the phase $\\varphi$ \nand the superfluid velocity $v_\\mathrm{s}$ are irrelevant because the modulus $M$ \napproaches zero.\n\n\n\\subsection{Correlation length}\n\\label{Sec:04D}\nThe correlation length $\\xi$ has been calculated by Schloms and Dohm \\cite{SD89} \nin thermal equilibrium and zero gravity. In the renormalized perturbation \ntheory up to two-loop order they obtain $\\xi^{-2}=\\mu^2 A_\\xi$ with an amplitude \nfunction $A_\\xi=1+\\mathcal{O}(u^2)$. However, since our Hartree approximation is \nfirst order in $u$ we may approximate $A_\\xi\\approx 1$, so that the correlation \nlength is just $\\xi=\\mu^{-1}$. This quantity is provided by our numerical calculation. \nOur results are shown in Fig.\\ \\ref{Fig:05} by the colored solid lines for the same \npositive heat currents as in the previous figures. In the interface region close \nto criticality $T=T_\\lambda$ the colored solid curves are smooth. The correlation \nlength has a maximum value $\\xi_g\\approx 50\\,\\mathrm{\\mu m}$ which is implied by \nthe gravity acceleration $g=9.81\\,\\mathrm{m\/s^2}$ on earth. The effect of a small \nnonzero heat current $Q$ is weak. For increasing heat currents $Q$ the position \nof the maximum of the correlation length is shifted slightly to lower temperatures. \nWe note that our maximum correlation length $\\xi_g$ is of the same order of magnitude \nas the characteristic length $l_g=67\\,\\mathrm{\\mu m}$ which was used by Ginzburg \nand Sobyanin \\cite{GS76} within their $\\psi$ theory.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig05.eps}\n\\caption{(Color online) The correlation length $\\xi$ as a function of the \ntemperature difference $T-T_\\lambda$ for the superfluid\/normal-fluid interface \nin gravity. The colored solid lines from left to right represent the heat currents \n$Q=160$, $130$, $100$, $70$, $40$, $20$, $0\\,\\mathrm{nW\/cm^2}$. As a reference \nthe black dashed line shows the singular correlation length in thermal equilibrium \nand zero gravity.}\n\\label{Fig:05}\n\\end{figure}\n\nFrom Fig.\\ \\ref{Fig:01} we have inferred the interface thickness \n$\\Delta z_{I,g}=200\\,\\mathrm{\\mu m}$. Thus we calculate the ratio \n$\\Delta z_{I,g} \/ \\xi_g \\approx 4$ which means that the interface thickness \nis four times the maximum of the correlation length. While in a nonequilibrium \nand\/or gravity environment the correlation length $\\xi$ is finite and a smooth \nfunction, in equilibrium and zero gravity it shows the well known singular \nbehavior $\\xi\\sim |T-T_\\lambda|^{-\\nu}$ near criticality for $T\\to T_\\lambda$\nwith an exponent $\\nu=0.671$. This latter singular correlation length is \nshown by the black dashed line which diverges at $T=T_\\lambda$. Far away from \ncriticality which means far away from the interface all solid lines converge \nto a single line which is identical with the black dashed line. Thus, far away \nfrom the interface the gravity $g$ and the heat current $Q$ do not have an \ninfluence on the correlation length $\\xi$. Finally, here we do not see an influence \nof the imaginary part of the parameter $\\sigma$ we introduce in our calculation \nin order to stabilize the numerical iterations.\n\n\n\\subsection{Specific heat}\n\\label{Sec:04E}\nThere are two possibilities to calculate the specific heat. First, we may calculate \nthe entropy $S$ within our renormalization-group theory and then calculate the \nderivative $C_X=T ( \\partial S \/ \\partial T )_X$ numerically where any quantity $X$ \nmay be kept constant. This has been done in our previous paper \\cite{Ha99a} where \n$X=Q$ or $X=\\nabla T$. The entropy $S$ is given by Eqs.\\ (8.10) or (8.12) of Ref.\\ \n\\onlinecite{Ha99a}. Secondly, we calculate the specific heat directly by \n$C=k_\\mathrm{B}\\chi_0 Z_m C_\\mathrm{ren}$ where the renormalized specific heat \n$C_\\mathrm{ren}$ is defined in \\eqref{Eq:C22} and the renormalization factor \n$\\chi_0 Z_m$ is defined implicitly in \\eqref{Eq:C30}. Thus, we obtain\n\\begin{equation}\nC_X = k_\\mathrm{B} \\frac{A_d}{4\\tau^2 \\xi^d} \\, \\Bigl\\{ \\frac{1}{\\gamma^2}\n+ \\frac{1}{2u} \\Bigl[ 1 - \\Bigl( \\frac{\\partial\\rho_1}{\\partial\\rho} \\Bigr)_X \n\\Bigr] \\Bigr\\} \\ ,\n\\label{Eq:D06}\n\\end{equation}\na formula which should be compared with the entropy (8.10) in Ref.\\ \\onlinecite{Ha99a}.\nThe formula can be simplified if we use the asymptotic formulas for the correlation \nlength $\\xi=\\xi_0\\, \\tau^{-\\nu}$ and for the coupling parameter \n$\\gamma^{-2}=(4\\nu\/\\alpha)(1-b \\tau^\\alpha)$ where $\\nu=0.671$ and \n$\\alpha=2-d\\nu=-0.013$ are critical exponents and $b$ is a known constant. \nAs a result we obtain the specific heat\n\\begin{equation}\nC_X = B + \\tilde A \\{ (4\\nu\/\\alpha) + F_X[u] \\} \\tau^{-\\alpha}\n\\label{Eq:D07}\n\\end{equation}\ntogether with the amplitude\n\\begin{equation}\nF_X[u] = \\frac{1}{2u} \\Bigl[ 1 - \\Bigl( \\frac{\\partial\\rho_1}{\\partial\\rho} \n\\Bigr)_X \\Bigr] \\ .\n\\label{Eq:D08}\n\\end{equation}\nThis formula should be compared with the entropy (8.12) in our previous paper \n\\cite{Ha99a} together with Eqs.\\ (8.13)-(8.16). Here $\\tilde A$ and $B$ are \nnonuniversal constants which can be expressed in the forms \n$\\tilde A= k_\\mathrm{B} A_d \/ 4 \\xi_0^d$ and $B= \\tilde A (-4\\nu\/\\alpha) b$.\nAlternatively, these constants can be obtained by fitting the formula to the \nexperimental data for liquid $^4$He in a micro gravity environment in space \n\\cite{Li96,Li03}. In this way we obtain $\\tilde A= 2.22\\,\\mathrm{J\/mol\\, K}$ and \n$B=456\\,\\mathrm{J\/mol\\, K}$ where the constants are multiplied additionally \nby the molar volume of liquid $^4$He at saturated vapor pressure \\cite{Al85} \n$V_\\lambda=27.38\\,\\mathrm{cm^3\/mol}$.\n\nThe amplitude $F_X[u]$ can be compared directly with the amplitudes $F_\\pm[u]$ \nof Dohm \\cite{Do85}, if we consider the asymptotic limits far away from the \ninterface. The temperature parameters $\\rho$ and $\\rho_1$ are related to each \nother by \\eqref{Eq:C36}. In the normal-fluid region far away from the interface \nthe renormalized order parameter is $Y=0$ and the amplitudes $A$ and $A_1$ are \ngiven by \\eqref{Eq:C40} and \\eqref{Eq:C41}. The partial derivative can be performed \neasily so that we obtain $\\partial\\rho\/\\partial\\rho_1 = 1 + 8u\\, A_1= 1 - 4u$ \nwhich does not depend on the variable $X$ that is kept constant. Thus we obtain \nthe amplitude\n\\begin{equation}\nF_+[u] = (2u)^{-1} [ 1 - 1 \/ ( 1 - 4u ) ] = - 2 + \\mathcal{O}(u)  \\ .\n\\label{Eq:D09}\n\\end{equation}\nIn the superfluid region $\\rho_1$ approaches $0$ more rapidly than $\\rho$ \napproaches $-1\/2$. Consequently, in the superfluid region far away from the \ninterface the partial derivative is $\\partial\\rho_1\/\\partial\\rho=0$ which again \ndoes not depend on the variable $X$ that is kept constant. Thus we obtain \nthe amplitude $F_-[u] = (2u)^{-1}$. If we compare our results for $F_\\pm[u]$ \nwith those of Dohm \\cite{Do85} we find agreement for the leading terms in \npowers of $u$ in both cases $+$ and $-$, respectively.\n\nWe have calculated the specific heat numerically with both methods described \nabove using the entropy formula (8.12) of our previous paper \\cite{Ha99a} and \nthe specific-heat formula \\eqref{Eq:D07} of the present paper. The results \nagree with each other within the accuracy of our Hartree approximation which \nis a self-consistent one-loop approximation combined with the \nrenormalization-group theory. This agreement is a test for the validity and \nthe accuracy of our method presented in this paper. While in the previous \npaper we have used the first method, in this paper we prefer the second method,\ni.e.\\ formula \\eqref{Eq:D07} together with \\eqref{Eq:D08}. The reason is that \nin the present calculation the second method provides curves looking smoother \nand more nice. \n\nOur results are shown in Fig.\\ \\ref{Fig:06} by the colored solid lines for the \nsame heat currents as in the previous figures. We have calculated the specific \nheat $C_Q$ where the heat current $Q$ and the gravity acceleration \n$g=9.81\\,\\mathrm{cm\/s^2}$ are kept constant. Clearly, in the interface region \nnear criticality we find smooth curves. The specific heat has a maximum slightly \nbelow the critical temperature. For increasing heat currents $Q$ this maximum is \nshifted to lower temperatures which is related to the depression of the superfluid \ntransition temperature observed in the experiment by Duncan, Ahlers, and Steinberg \n\\cite{DAS88}. Furthermore, the maximum of the specific heat is strongly enhanced \nfor increasing heat currents $Q$. This enhancement is an effect of the constant \nheat current $Q$ when calculating the specific heat $C_Q$. It has been observed \nalready in our previous paper \\cite{Ha99a}, where $C_Q$ has be calculated for \nthe much higher heat current $Q=42.9\\,\\mathrm{\\mu W\/cm^2}$ where gravity effects \nare negligible. The strong enhancement of the maximum is also compatible with \nexperimental measurements of $C_Q$ by Harter \\emph{et al.}\\ \\cite{HL00}.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig06.eps}\n\\caption{(Color online) The specific heat $C_Q$ as a function of the \ntemperature difference $T-T_\\lambda$ for the superfluid\/normal-fluid interface \nin gravity. The colored solid lines from left to right represent the heat currents \n$Q=160$, $130$, $100$, $70$, $40$, $20$, $0\\,\\mathrm{nW\/cm^2}$. The black dashed \nline represents the singular specific heat in microgravity fitted to the data of \nthe experiment by Lipa \\emph{et al.}\\ \\cite{Li96,Li03}. The black crosses show the \nexperimental data for zero heat current in gravity on earth by Lipa \\cite{Li11} \nwhere the black dotted line is the related theoretical curve for the average \nspecific heat $\\bar C_Q$.}\n\\label{Fig:06}\n\\end{figure}\n\nFar away from criticality and the interface on both sides the colored solid curves \nconverge to a single line, respectively. These single lines represent the asymptotic \nspecific heat $C=B+(A_\\pm\/\\alpha)|t|^{-\\alpha}$ where $t=(T-T_\\lambda)\/T_\\lambda$ \nis the reduced temperature and $\\alpha$ is the critical exponent. On the \nnormal-fluid side the single line is perfect. However, on the superfluid side it \nis slightly influenced by the imaginary part of the parameter $\\sigma$ which we \nmust add in our numerical calculation in order to have stable iterations. This \nfact is related to the similar observation in our results for the order parameter \nshown in Fig.\\ \\ref{Fig:04}.\n\nThe smooth colored solid lines in Fig.\\ \\ref{Fig:06} show that the critical \nsingularity is rounded by the gravity $g$ and the heat current $Q$. The \ntemperature scale for this rounding is $\\Delta T_{g,I}=25\\,\\mathrm{nK}$ if \ngravity is the dominating effect. We have obtained this value from the thickness \nof the interface $\\Delta z_{I,g}=200\\,\\mathrm{\\mu m}$. Consequently, the asymptotic \ncritical behavior of the specific heat and all other singular quantities can be \nobserved only for temperatures $|T-T_\\lambda|\\gtrsim \\Delta T_{g,I}=25\\,\\mathrm{nK}$\naway from criticality. Hence, the gravity implies that on earth the critical point \ncan never be reached. For this reason, experiments to measure the asymptotic behavior \ncloser to the critical point must be performed in a micro-gravity environment in \nspace.\n\nLipa \\emph{et al.}\\ \\cite{Li96,Li03} have performed a space experiment which was \ncalled \\emph{Lambda Point Experiment} (LPE) and which flew aboard the space shuttle \nColumbia (STS-52) in 1992. They obtained data for the specific heat up to \n$|T-T_\\lambda|=1\\,\\mathrm{nK}$. They fitted an asymptotic formula to the data and \ndetermined the exponent $\\alpha=-0.013$, the amplitudes $A_\\pm$ and $B$ and some \nfurther parameters. The resulting fit curve is shown in Fig.\\ \\ref{Fig:06} by the \nblack dashed line. This curve shows the typical lambda of the specific heat with \na singularity at $T=T_\\lambda$. Away from criticality on both sides for \n$|T-T_\\lambda|\\gtrsim 50\\,\\mathrm{nK}$ the solid lines and the dashed line come \nclose to each other which demonstrates the agreement between theory and experiment. \nHowever, the agreement is not perfect. There remains a small discrepancy which is \ndue to the amplitude ratio $A_+\/A_-$ because our theory provides an approximate \nvalue for this amplitude ratio which can never be identical to the experimental \nvalue.\n\nWe note that in our calculation the specific heat is defined locally. It depends \non the altitude $z$ so that $C_Q=C_Q(z)$. The rounding of the critical singularity \nis caused by the gradient of $T_\\lambda(z)$ and by the heat current only. However, \nin experiments the $^4$He is in a cell of a finite extension. The cell is usually \nconfined by two horizontal plates at altitudes $z_1$ and $z_2$, where the vertical \nheight $L=z_2-z_1$ is small and the horizontal extensions are large. For this reason \nthe experiment measures an average specific heat which is defined by the integral\n$\\bar{C}_Q=(z_2-z_1)^{-1} \\int_{z_1}^{z_2} dz\\, C_Q(z)$. This average process \nsmooths the curve additionally. The maximum of the average specific heat \n$\\bar{C}_Q$ will be broader than the maximum of the related local specific heat \n$C_Q$.\n\nIn order to minimize the averaging effects the cell height $L$ should be chosen \nas small as possible. However, it must be considerably larger than the maximum \ncorrelation length $\\xi_g=50\\,\\mu m$ because for small $L$ finite size effects \noccur which again smooth and round the critical singularity. Consequently, for \nthe cell height $L$ there will be an optimum range to obtain best measurements \non earth. \n\nIn Fig.\\ \\ref{Fig:06} the black crosses represent the experimental data of a \nmeasurement on earth at zero heat current performed by Lipa \\cite{Li11}. In this \ncase the cell height is $L=0.38\\, mm$ which causes considerable average effects. \nThe maximum of the experimental data is much broader than the maximum of the black \nsolid line which represents the local specific heat $C_Q$ for $Q=0$. We have \ncalculated the related average specific heat $\\bar C_Q$ for $Q=0$ which is shown \nby the black dotted line. This latter curve shows a much broader maximum at \ncriticality which agrees with the experimental data. Since the ratio $L\/\\xi_g=7.6$ \nis large, finite size effects are small. However, the Dirichlet boundary \nconditions of the order parameter at the cell walls imply that nevertheless the \nfinite size effects cause a depression of the data which is clearly observed in \nFig.\\ \\ref{Fig:06} because the experimental data (black crosses) are below the \ntheoretical curve (black dotted line). Thus we conclude that the experimental \ndata obtained in a measurement in gravity on earth agree with our theory.\n\n\n\\subsection{Thermal conductivity and resistivity}\n\\label{Sec:04F}\nThe thermal conductivity $\\lambda_\\mathrm{T}$ is defined locally by the heat \ntransport equation $\\mathbf{Q}=-\\lambda_\\mathrm{T} \\nabla T$. Resolving this \nequation we obtain the thermal conductivity explicitly as \n$\\lambda_\\mathrm{T} = |\\mathbf{Q}|\/|\\nabla T|$. Next we insert the renormalization \nequations for the heat current \\eqref{Eq:D03} and for the temperature gradient \n\\eqref{Eq:C65}. Thus, we obtain\n\\begin{equation}\n\\lambda_\\mathrm{T} = \\frac{g_0 k_\\mathrm{B}}{\\tau\\,\\xi^{d-2}} \n\\,\\frac{|\\tilde{\\mathbf{q}}|}{|\\xi\\mathbf{D}\\Delta\\rho|} \\ .\n\\label{Eq:D10}\n\\end{equation}\nThe dimensionless heat current $\\tilde{\\mathbf{q}}$ and the dimensionless \nrenormalized gradient $\\xi\\mathbf{D}\\Delta\\rho$ are variables in our numerical \ncalculation. Hence Eq.\\ \\eqref{Eq:D10} is well suited for an explicit calculation \nof the thermal conductivity.\n\nThe dimensionless renormalized heat current is defined in \\eqref{Eq:C82}.\nFar away from the interface in the normal-fluid region the order parameter $Y$ \nand hence the last term of \\eqref{Eq:C82} is zero. On the other hand, the amplitude \n$A_1$ in the first term of \\eqref{Eq:C82} reduces to $A_1=-1\/2$ following \n\\eqref{Eq:C41}. Thus Eq.\\ \\eqref{Eq:D10} provides the thermal conductivity \n\\begin{equation}\n\\lambda_\\mathrm{T} = \\frac{g_0 k_\\mathrm{B}}{\\tau\\,\\xi^{d-2}} \n\\,\\frac{A_d}{2\\gamma F}\\, \\Bigl\\{ 1 - \\frac{f}{4} \\Bigr\\} \\ .\n\\label{Eq:D11}\n\\end{equation}\nin the normal-fluid region. This result is well known and agrees with the \nlinear-response calculation of Dohm \\cite{Do85}. Since our result is derived in \nHartree approximation, the agreement is up to one-loop order.\n\nFar away from the interface in the superfluid region the heat current $\\mathbf{Q}$ \nis nonzero where the temperature gradient $\\nabla T$ is zero. Hence the thermal \nconductivity $\\lambda_\\mathrm{T}$ is infinite. This fact is also seen in the formulas \n\\eqref{Eq:C82} and \\eqref{Eq:D10}. A zero dimensionless renormalized temperature \ngradient $\\xi\\mathbf{D}\\Delta\\rho=\\mathbf{0}$ implies a zero first term in \n\\eqref{Eq:C82} and a zero denominator in \\eqref{Eq:D10}. On the other hand, the \nsecond term in \\eqref{Eq:C82} is minus the dimensionless renormalized superfluid \ncurrent which is nonzero. Consequently, Eq.\\ \\eqref{Eq:D10} provides an infinite \nthermal conductivity once again in the superfluid region.\n\nA related quantity is the thermal resistivity $\\rho_\\mathrm{T}=1\/\\lambda_\\mathrm{T}$ \nwhich is the inverse of the thermal conductivity. In the normal-fluid region the \nthermal resistivity $\\rho_\\mathrm{T}$ is finite. On the other hand, in the superfluid \nregion it is zero. For this reason the thermal resistivity $\\rho_\\mathrm{T}$ is well \nsuited for a graphical representation. In Fig.\\ \\ref{Fig:07} the results of our \nnumerical calculation are shown as colored solid lines for several positive values \nof the heat current $Q$. These heat currents are the same as those in the previous \nfigures. Clearly, in the interface region near criticality the colored solid curves \nare smooth lines. The critical singularity at $T=T_\\lambda$ is smoothed by the nonzero \nvalues of gravity $g$ and the heat current $Q$. For increasing heat currents $Q$ the \ncolored solid lines are shifted to lower temperatures. This fact is again related \nto the depression of the superfluid transition temperature by nonzero heat currents \nfollowing Duncan, Ahlers, and Steinberg \\cite{DAS88}.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth,clip]{fig07.eps}\n\\caption{(Color online) The thermal resistivity as a function of the \ntemperature difference $T-T_\\lambda$ for the superfluid\/normal-fluid interface \nin gravity. The colored solid lines from left to right represent the heat currents \n$Q=160$, $130$, $100$, $70$, $40$, $20$, $0\\,\\mathrm{nW\/cm^2}$. The data points \nrepresent the experimental data by Day \\emph{et al.}\\ \\cite{Du98} taken for the \nsame heat currents except $Q=0$. Related to the theoretical curves the data are \nordered from left to right.}\n\\label{Fig:07}\n\\end{figure}\n\nOn both sides far away from criticality $T=T_\\lambda$ and from the interface the \ncolored curves asymptotically approach single lines, respectively. In the normal-fluid \nregion the asymptotic thermal resistivity is given by \\eqref{Eq:D11}, where in the \nsuperfluid region it is just zero.\n\nThe local thermal resistivity $\\rho_\\mathrm{T}$ has been measured for the \nsuperfluid\/normal-fluid interface in the experiment by Day \\emph{et al.}\\ \\cite{Du98}. \nWhile gravity on earth is $g=9.81\\,\\mathrm{m\/s^2}$, the heat current is flowing \nupward from bottom to top so that $Q$ is positive. The values of heat current $Q$ \nare the same as in our calculation. For this reason, our colored solid lines can \nbe compared directly with the experimental data. In Fig.\\ \\ref{Fig:07} the experimental \ndata for several heat currents $Q$ are represented by points of several symbol types. \nThe data points can be related to the solid curves by their color or alternatively \nby there order from left to right. An exception is the black solid line for zero \nheat current $Q=0$. In this latter case no experimental data are available. However, \nthe black solid line is very close to the red solid line for $Q=20\\,\\mathrm{nW\/cm^2}$. \nConsequently for $Q\\lesssim 20\\,\\mathrm{nW\/cm^2}$ the effect of the heat current is \nvery small.\n\nQualitatively, the experimental data agree with the theoretical solid lines. However,\nthere are quantitative discrepancies. First of all, in the normal-fluid region well \nabove criticality for temperature differences $T-T_\\lambda\\gtrsim 50\\,\\mathrm{nK}$ \nthe thermal resistivity converges to a single line represented by the formula \n\\eqref{Eq:D11}. We have multiplied the theoretical results with a correction factor \nwhich slightly differs from unity. In this way we achieve that far away from \ncriticality the experimental data are lying on top of the theoretical curves. This \ncorrection factor is justified, because in Ref.\\ \\onlinecite{Do85} the model-$F$ \nparameters were adjusted for the theoretical specific heat and thermal resistivity \nin \\emph{two-loop} order where in the present paper and also in our previous papers \n\\cite{Ha99,Ha99a} the quantities were calculated in the Hartree approximation which \nis a self-consistent \\emph{one-loop} approximation.\n\nFor larger heat currents $Q$ and temperatures $T$ slightly below $T_\\lambda$ our \nsolid lines show some bumps which are probably artifacts of our approximation. \nThe magnitude of the artifacts is within the accuracy of our approach.\nIn the superfluid region well below criticality for temperature differences \n$T-T_\\lambda\\lesssim -50\\,\\mathrm{nK}$ the thermal resistivity $\\rho_\\mathrm{T}$ \nis very close to zero both in theory and experiment. This fact represents the \nfrictionless heat transport by the superfluid\/normal-fluid counterflow in \nsuperfluid $^4$He. Vortices are not present in our calculation presented here.\n\n\n\\subsection{Second sound and time dependent phenomena}\n\\label{Sec:04G}\nUntil now we have considered only stationary nonequilibrium states where a \nconstant heat current $\\mathbf{Q}=Q\\,\\mathbf{e}_z$ is flowing vertically in the \n$^4$He and where all time-dependent phenomena are relaxed. However, our numerical \ncalculation solves the time-dependent model-$F$ equations so that time-dependent \nphenomena can be treated explicitly. In this case the order parameter \n$\\langle\\psi(z,t)\\rangle$, the temperature $T(z,t)$, and the heat current density \n$\\mathbf{Q}(z,t)=Q(z,t)\\,\\mathbf{e}_z$ are functions of altitude $z$ and time $t$. \nIn superfluid $^4$He the most important dynamic phenomenon is second sound. We can \ngenerate a second sound pulse on the upper cell boundary $z_2$ if in the external \nheat source \\eqref{Eq:D01} we choose a time-dependent upper source function \n$Q_2(t)$. Then the phase of the order parameter $\\varphi(z,t)$, the temperature \n$T(z,t)$, and the local heat current $Q(z,t)$ show a pulse which is traveling \ndownward toward the superfluid\/normal-fluid interface. Approaching gradually \nthe interface the width of the pulse increases. Once the interface is reached \nthe pulse disappears by broadening where nearly nothing is reflected. In the end \nthe second-sound pulse is absorbed nearly completely by the interface.\n\nA heat pulse can be generated also on the lower cell boundary $z_1$ if the lower \nsource function $Q_1(t)$ is chosen time dependent. However, in this case the \nresponse of the system is less spectacular because on the normal-fluid side \nthe heat is transported diffusively. Nevertheless, our time-dependent investigations \nyield an important result. All time-dependent perturbations of the stationary \nstates with a constant heat flow relax and disappear. Thus we conclude that the \nsuperfluid\/normal-fluid interface in gravity and in the presence of a vertical heat \nflow is a stable physical configuration.\n\n\n\\section{Comparison with our previous approach for large heat currents}\n\\label{Sec:05}\n\n\\subsection{Solutions}\n\\label{Sec:05A}\nWe have found two different solutions of the model-$F$ equations for superfluid \n$^4$He in the nonequilibrium state where a heat current is flowing. In the first \ncase the order parameter $\\langle\\psi\\rangle=M\\, e^{i\\varphi}$ is nonzero \nand no vortices are present. The heat is transported convectively without any \nfriction by the superfluid\/normal-fluid counterflow so that the temperature \ngradient is zero. This solution is investigated in the present paper. In the \nsecond case the order parameter is zero due to fluctuations of the phase $\\varphi$ \nby moving vortices and quantum turbulence. Here the moving vortices imply a small \nthermal resistivity which causes a small temperature gradient. This latter solution \nwas investigated in our previous papers \\cite{Ha99,Ha99a}. The two different \nsolutions exist only in the superfluid and interface region where a nonzero order \nparameter is possible. In the normal-fluid region the solution is unique because \nhere the order parameter is always zero.\n\nThe two solutions are controlled by the parameter $\\sigma$ defined in \nEq.\\ \\eqref{Eq:C85}. The first solution exists for negative and positive values \nof $\\sigma$ where the second solution exists only for positive $\\sigma$. For the \nsuperfluid\/normal-fluid interface in gravity on earth this means that the first \nsolution exist for small and large heat currents. On the other hand the second \nsolution exists only for large heat currents $|Q| \\gtrsim 70\\,\\mathrm{nW\/cm^2}$ \nwhere the heat current is the major and gravity is the minor influence. However, \nthe numerical calculations of our present paper show that in practice the first \nsolution is stable only for small heat currents in the interval \n$-70\\,\\mathrm{nW\/cm^2} \\lesssim Q \\lesssim +160\\,\\mathrm{nW\/cm^2}$. For larger \nheat currents outside the interval we do not obtain a stable solution. We do not \nknow if this instability is a property of our iteration procedure only which we \ndescribe in Subsec.\\ \\ref{Sec:03F}. However we guess that there is a physical \ninstability beyond a certain critical heat current. This means we expect a \ndiscontinuous first-order like transition in the nonequilibrium state near the\nheat current $Q_0\\approx 70\\,\\mathrm{nW\/cm^2}$ which separates the gravity dominated \nregime (first solution) from the heat current dominated regime (second solution). \nNevertheless, there will be an overlap region where both solutions exist.\n\n\n\\subsection{Stability}\n\\label{Sec:05B}\nThe time-dependent nature of our numerical calculation provides a test for the \nphysical stability of our stationary solutions. We may add a small perturbation \nto the solution and then start the calculation. After a time difference \n$\\delta t\\approx 2\\,\\mathrm{s}$ the perturbations relax and disappear so that the \nsystem returns to the stationary state. We find this behavior for all stationary \nsolutions with a constant heat flow which we have presented in this paper. Thus, \nwe conclude that the \\emph{first-type solutions are stable} which describe \nphysical states with a nonzero order parameter and no vortices. This result \nis expected because the order-parameter variation and second sound are damped. \nWe note that we have a numerical instability for larger heat currents. However, \nthis latter instability is unphysical and has a completely different nature \nbecause it appears on the short length scales $\\Delta z$ of the discretization of \nthe altitude coordinate.\n\nIn our previous papers \\cite{Ha99,Ha99a} we did not prove the stability of the \nsecond-type solutions which describe superfluid states with vortices and a zero \naverage order parameter. The reason is that in our previous papers we did not solve \nthe model-$F$ equations as partial differential equations. Consequently, we do this \nnow and provide the proof in the following. However, we note that in this case \nthe stability is nontrivial because the system is in a superfluid state where the \naverage order parameter is $\\langle\\psi\\rangle=0$ and the temperature is \n$T<T_\\lambda$. We solve the time-dependent renormalized model-$F$ equations for a \nsecond-type solution with a small perturbation. For simplicity we consider a \nself-organized critical state with a constant heat $Q$ current and constant \ntemperature gradients $\\nabla T=\\nabla T_\\lambda$ because this state is spatially \nhomogeneous before the perturbation is applied. Unfortunately, the state is not \nperiodic because we cannot require periodic boundary conditions for all quantities.  \nExceptions are the temperatures $T(z.t)$, $T_\\lambda(z)$ and the phase of the order \nparameter $\\varphi(z,t)$. Nevertheless, we can generalize the boundary conditions. For \nthe latter three quantities we require pseudoperiodic boundary conditions in the sense \nof the \\emph{impossible objects} of the famous Dutch graphic artist M.C. Escher \n\\cite{Ho79}, i.e.\n\\begin{eqnarray}\nT(z+L,t) &=& T(z,t) + \\Delta T_L \\ ,\n\\label{Eq:E01} \\\\\nT_\\lambda(z+L) &=& T_\\lambda(z) + \\Delta T_L \\ ,\n\\label{Eq:E02} \\\\\n\\varphi(z+L,t) &=& \\varphi(z,t) + \\Delta\\varphi_L(t) \\ .\n\\label{Eq:E03}\n\\end{eqnarray}\nSince the critical temperature $T_\\lambda(z)$ is linear in $z$, from \nEq.\\ \\eqref{Eq:E02} we obtain $\\Delta T_L=(\\partial_z T_\\lambda) L$. On the other \nhand from Eq.\\ \\eqref{Eq:E01} we obtain $\\Delta T_L=-(Q\/\\lambda_\\mathrm{T})L$ where \n$\\lambda_\\mathrm{T}$ is the thermal conductivity. We note that for the self-organized \ncritical state both results for the period constant $\\Delta T_L$ must be equal. \nThis fact implies $Q=-\\lambda_\\mathrm{T}(\\partial_z T_\\lambda)$.\nThe period constant of the order-parameter phase $\\Delta\\varphi_L(t)$ can be \ndetermined by investigating the renormalized model-$F$ equation \\eqref{Eq:C79}. On the \nright-hand side we replace $\\Delta\\rho(z,t)$ in favor of the temperature $T(z,t)$ by \ninserting \\eqref{Eq:C27}. We insert the complex order parameter $Y=\\eta\\,e^{i\\varphi}$ \nand derive an equation for the order parameter phase $\\varphi(z,t)$. We consider \nthis equation for the altitudes $z=z_0$ and $z=z_0+L$ and then subtract the resulting \nequations. Thus, as a result we obtain \n$\\partial_t \\Delta\\varphi_L(t) = g_0 (\\Delta T_L\/T_\\lambda)$. We integrate this \nequation and obtain the period constant\n\\begin{equation}\n\\Delta\\varphi_L(t) = \\Delta\\varphi_L(t_0) + g_0 (\\Delta T_L\/T_\\lambda) (t-t_0) \\ .\n\\label{Eq:E04}\n\\end{equation}\nClearly, the period of the order-parameter phase depends linearly on the time $t$. We \nconclude and find that for the self-organized critical state with small perturbations \nthe renormalized model-$F$ equations can be solved numerically using Escher pseudoperiodic \nboundary conditions which are defined by Eqs.\\ \\eqref{Eq:E01}-\\eqref{Eq:E03}. Our \nnumerical test provides the following result. Any small perturbation of the \nself-organized critical state relaxes and disappears after a time difference of \n$\\delta t\\approx 2\\,\\mathrm{s}$. Thus we conclude that the \\emph{second-type solutions \nare stable}. This means that the nonequilibrium states considered in the previous \npaper are stable.\n\nA matter of special interest is the relaxation of the order parameter. We start \nat time $t_0$ with a small constant perturbation $Y(z,t_0)=\\eta_0\\,e^{i \\varphi_0}$.\nThis means $\\eta_0$ is small but nonzero, and $\\varphi_0$ is constant. We consider \nthe renormalized model-$F$ equation \\eqref{Eq:C79} which describes the time evolution \nof the order parameter $Y(z,t)=\\eta(z,t)\\,e^{i\\varphi(z,t)}$. Decomposing the \nequation with respect to the modulus $\\eta(z,t)$ and the phase $\\varphi(z,t)$ \nfrom the first term on the right hand side we infer the damping for the modulus\n\\begin{equation}\nD= g_0 (2\\gamma\\,\\tau) (w^\\prime\/F) [ \\rho_1 + (\\xi\\nabla\\varphi)^2 ] \\ .\n\\label{Eq:E05}\n\\end{equation}\nThis damping is an inverse relaxation time. The solution of the model-$F$ equation \nis stable whenever this damping is positive and unstable otherwise. The prefactors \nare always positive so that the crucial quantity is the expression in the \nsquare brackets. For times $t$ shortly after the beginning of the calculation $t_0$ \nthe phase is expected to be $\\varphi(z,t)\\approx \\varphi_0$ so that \n$\\nabla\\varphi\\approx \\mathbf{0}$. Consequently, the main contribution is the \ndimensionless modified temperature parameter $\\rho_1$. The stability of the \nsolution depends on its sign. In Fig.\\ \\ref{Fig:01} $\\rho_1$ is plotted as the \nblue dashed line for the superfluid\/normal-fluid interface. A similar curve is \nobtained for the self-organized critical state if $\\rho_1$ is plotted as a \nfunction of the heat current $Q$. While in the normal-fluid region $\\rho_1$ \nis positive, in the superfluid region it is negative. For the heat current \n$Q=170\\,\\mathrm{nW\/cm^2}$ we find the minimum value $\\rho_{1,\\mathrm{min}}=-0.39$. \nThus, we conclude that the solution is stable in the normal-fluid region but \nunstable in the superfluid region.\n\nHowever, the instability is true only for short times where $t-t_0$ is small. \nFor longer times we must investigate the space and time dependence of the \norder-parameter phase $\\varphi(z,t)$. From Eq.\\ \\eqref{Eq:E04} and from the \nrenormalized model-$F$ equation \\eqref{Eq:C79} we infer\n\\begin{eqnarray}\n\\varphi(z,t) &=& \\varphi_0 + \\Delta\\varphi_L(t)\\,(z-z_0)\/L \\nonumber\\\\\n&=& \\varphi_0 + g_0\\,(\\Delta T_L\/T_\\lambda)\\,(t-t_0)\\,(z-z_0)\/L \\ . \\qquad\n\\label{Eq:E06}\n\\end{eqnarray}\nHere $z_0$ is the altitude where the temperature equals the reference temperature, \ni.e. $T(z_0,t_0)=T_0$. For the last equality sign we have used \\eqref{Eq:E04} \nrealizing that in our case at the beginning $t=t_0$ the Escher period constant \nis $\\Delta\\varphi_L(t_0)=0$. Differentiating with respect to the altitude \ncoordinate $z$ and multiplying by $\\xi$ we obtain the dimensionless gradient of \nthe order-parameter phase\n\\begin{equation}\n\\xi\\nabla\\varphi = g_0\\,(\\Delta T_L\/T_\\lambda)\\,(t-t_0)\\,(\\xi\/L)\\,\\mathbf{e}_z \\ .\n\\label{Eq:E07}\n\\end{equation}\nConsequently, we find $(\\xi\\nabla\\varphi)^2 \\sim (t-t_0)^2$. This means that in \nEq.\\ \\eqref{Eq:E05} the second term in the square bracket increases with time. \nEven though at the beginning the square bracket may be negative because of the \nfirst term, after a short time the second term makes the square bracket positive. \nThus we conclude: Even though there may be an instability for short times, the \nincrease of the gradient of the phase \\eqref{Eq:E07} makes the damping \n\\eqref{Eq:E05} finally positive so that the time evolution of the order parameter \nis finally stable.\n\nThe stability of the second model-$F$ equation \\eqref{Eq:C80} is easily proven. \nWe insert the dimensionless renormalized heat current \\eqref{Eq:C82} and \nneglect its last term because it is squared in the small nonzero order parameter. \nAs a result we obtain a diffusion equation for the dimensionless renormalized \ntemperature parameter $\\Delta\\rho(z,t)$. Since the related diffusion constant \nis positive, this equation is always stable. The covariant derivatives in this \nequation do not affect the stability.\n\nWe summarize that we have presented an explicit proof for the stability of the \nself-organized critical state which is a spatially homogeneous second-type solution \nof the model-$F$ equations. We expect that also the more general second-type \nsolutions for spatially inhomogeneous systems are stable, which describe the \nsuperfluid\/normal-fluid interface in our previous papers \\cite{Ha99,Ha99a}. Our \nnumerical calculations of the present paper support this expectation. However, \nwe note that the stability is nontrivial because the system is in a superfluid \nstate where the average order parameter is $\\langle\\psi\\rangle=0$ and the \ntemperature is $T<T_\\lambda$.\n\n\n\\section{Discussion and conclusion}\n\\label{Sec:06}\nOnuki \\cite{On83,On87} and and later Weichman and Miller \\cite{WM00} have \nalso investigated the superfluid\/normal-fluid interface within model $F$. They \nobtain temperature profiles which agree qualitatively with our results shown \nin Fig.\\ \\ref{Fig:03}. However, they did not use the renormalization-group \ntheory and the related coupling parameters which have been determined by Dohm \n\\cite{Do91}. For this reason, it is not possible to compare the results \nquantitatively. Weichman and Miller \\cite{WM00} furthermore considered the \nself-organized critical state for a heat current flowing downward. While \nin the superfluid region usually the temperature profile is flat, they obtain \nphase slips in the order parameter which produce a stair-case like temperature \nprofile. In this way they obtain a temperature gradient $\\nabla T$ which on \naverage equals the gradient $\\nabla T_\\lambda$ as required for the self-organized \ncritical state. More recently Yabunaka and Onuki \\cite{On10} performed a \nthree-dimensional numerical simulation based on model $F$ in order to investigate \nthe self-organized critical state and the superfluid\/normal-fluid interface. \nThey observed the formation and motion of vortices and phase slips which produce \na nonzero temperature gradient $\\nabla T$ on average in the superfluid region \nwhich compensates $\\nabla T_\\lambda$.\n\nA sophisticated theory for mutual friction, quantum turbulence, and the dynamics \nof vortices in superfluid $^4$He was developed long time ago by Vinen \\cite{Vi57}. \nA measure for the quantum turbulence is the density of the vortices which is \ndefined as the total length of the vortex lines per volume. For this vortex density \na rate equation is derived. On the right hand side of this equation there is a term \nfor the generation and a term for the decay of vortices and quantum turbulence. \nVortices are usually generated by a nucleation process. This means that an energy \nbarrier must be overcome which strongly reduces the generation rate.\n\nWe believe that model-$F$ includes the effects of vortices and quantum turbulence \ncorrectly so that the Vinen theory can be derived if the model-$F$ equations can \nbe solved exactly without any approximation. However, our two solutions which are \nderived within the Hartree approximation are idealized solutions of model $F$. \nThey describe the two phases of a first-order nonequilibrium transition but do \nnot include metastability and the nucleation process. \n\nFor the check between theory and experiment an important quantity is the thermal \nresistivity $\\rho_\\mathrm{T}$ in superfluid $^4$He for $T<T_\\lambda$ and larger \nheat currents $Q\\gtrsim 100\\,\\mathrm{nW\/cm^2}$ induced by the effect of vortices. \nIn Fig.\\ 3 of our previous paper \\cite{Ha99a} we have compared the result of our \nsecond-type solution with experimental data by Baddar \\emph{et al.}\\ \\cite{Al00}. \nThe experimental thermal resistivity is lower by a factor of $20$ than our theoretical \nresult. A plausible explanation for this discrepancy is the following. In the \nexperiment the heat current is flowing upward from bottom to top. The related \nsuperfluid current is flowing in the opposite direction, i.e.\\ downward from top \nto bottom. Consequently, the vortices are transported together with the superfluid \ncurrent downward. Since the order parameter $\\langle\\psi\\rangle$ increases and \nthe vortex density decreases with the altitude $z$, the downward transport of the \nvortices together with the metastability of the nucleation process can reduce \nthe vortex density considerably. Since the hopping over energy barriers in the \nnucleation process causes exponential factors, a reduction of the thermal \nresistivity by a factor $20$ is plausible. \n\nThe situation is different for the self-organized critical state. Here the system \nis spatially homogeneous because the heat current $Q$ and the temperature difference \n$T(z)-T_\\lambda(z)=\\Delta T$ are constant and do not depend the altitude $z$. The \nnucleation process reaches an equilibrium state, so that our theory will predict \nthe density of the vortices and the related thermal resistivity correctly. A first \nagreement with the theory was found in the experiment by Moeur \\emph{et al.}\\ \n\\cite{Du97}. Here the temperature difference $\\Delta T$ was measured as a function \nof the heat current $Q$. The comparison with our theory is shown in Fig.\\ 3 of our \nprevious paper \\cite{Ha99}. The agreement is encouraging. However, for very large \nheat currents $Q\\gtrsim 2\\,\\mathrm{\\mu W\/cm^2}$ deep in the superfluid region a \ndeviation was found. This deviation may be a problem of the temperature measurement \nbecause the temperature $T$ is never measured in the bulk of the system but always \non the surface. In order to avoid the Kapitza resistance which implies a temperature \njump on the surface, the temperature is usually measured by thermometers on the side \nwalls. However, in cases where vortices are present, a superfluid flow parallel \nalong a side wall may cause a transverse Kapitza resistance so that there is a \ntemperature jump also on a side wall. \n\nIn a recent experiment Chatto \\emph{et al.}\\ \\cite{Du07} performed an experiment \nto measure the thermal conductivity\/resistivity indirectly where the explicit \nmeasurement of the temperature is avoided. Instead they measured the velocity $v$ \nof a propagating thermal mode as a function of the heat current $Q$ in the interval \n$30\\,\\mathrm{nW\/cm^2}\\lesssim Q\\lesssim 15\\,\\mathrm{\\mu W\/cm^2}$. On the other \nhand, they derived a theoretical curve for the velocity $v$ from our theoretical \nresults for the thermal conductivity \\cite{Ha99,Ha99a}. They find very good agreement \nbetween theory and experiment in the whole range of heat currents $Q$, even for \nthe largest values $Q\\approx 15\\,\\mathrm{\\mu W\/cm^2}$ deep in the superfluid \nregion. We conclude that this experiment is an important verification of our \ntheory. This means our theory \\cite{Ha99,Ha99a} describes the effects of vortices, \nmutual friction, and the thermal conductivity\/resistivity correctly on a quantitative \nlevel for the self-organized critical state.\n\n\n\\acknowledgments\n\\noindent\nWe would like to thank Prof.\\ Dr.\\ V.\\ Dohm for helpful comments on the manuscript \nand Prof.\\ Dr.\\ J.A. Lipa for experimental data of the specific heat.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Approximating spherical projections}\n\nFor convex bodies $K$,\nthe set $S(K)$ is a manifold\\footnote{\n    Technically,\n    $S(K)$ will be a manifold-with-corners\n    (see~\\cite[p.~137]{SpivakManifolds}).\n    However,\n    their interiors relative to the $(d-1)$-dimensional sphere $S^{d-1}$\n    are manifolds,\n    and since we're dealing with areas\n    there will be no harm in ignoring these boundaries.\n}.\nIf $K$ does not contain the origin in its interior,\nthen $S(K)$ may be parameterized with a single coordinate system;\nthat is,\nthere is a set $U \\subseteq \\mathbb R^{d-1}$\nand a continuously differentiable function $\\varphi: U \\to S(K)$\nwhich is a bijection between $U$ and $S(K)$.\nSince we want to move $P$ towards infinity,\nthis shall always be the case\nif the translation vector is long enough.\nIn this case,\nwe define its area to be~\\cite[p.~126]{SpivakManifolds}\n\\begin{align*}\n    \\area S(K) &= \\int_U \\norm{D_1 \\varphi \\times \\dots \\times D_{d-1} \\varphi} \\\\\n        &= \\int_U \\norm{\n            \\frac{\\partial \\varphi}{\\partial x_1}\n            \\times \\dots \\times\n            \\frac{\\partial \\varphi}{\\partial x_{d-1}} dx_1 \\dots dx_{d-1}\n        }.\n\\end{align*}\n\nThe following theroem states that\nthe spherical projection approximates,\nin a sense,\nthe orthogonal projection,\nfor large enough translation vectors.\n\n\\begin{theorem}\n    \\label{thm:spherical-approaches-orthogonal}\n    Let $v$ be a unit vector\n    and $K \\subseteq \\mathbb R^d$ a convex body.\n    Then\n    \\begin{equation*}\n        \\lim_{\\mu \\to \\infty} \\mu^{d-1} \\area S(P + \\mu v) = V_K(v).\n    \\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\n    By rotating all objects involved if needed,\n    we may assume that $v = (0, \\dots, 0, 1)$.\n    Let $N$ be large enough that $K \\subseteq B_N(0)$;\n    we'll assume that $\\mu > N$,\n    so that $K + \\mu v$ lies strictly above the hyperplane $x_d = 0$.\n\n    Let $K'$ be the orthogonal projection of $K$ into $\\{v\\}^\\perp$.\n    We'll think of $K'$ as being a subset of $\\mathbb R^{d-1}$.\n    Denote by $K_\\mu$ the projection of the set $\\mu S(K + \\mu v)$ on $\\mathbb R^{d-1}$\n    (Figure~\\ref{fig:spherical-orthogonal-projection});\n    that is,\n    first project $K + \\mu v$ to the sphere with radius $\\mu$,\n    then discard the last coordinate.\n    Nnote this is similar to project it to the hyperplane $x_d = 0$.\n    We'll show that,\n    as $\\mu$ goes to infinity,\n    both the Hausdorff distance between $K_\\mu$ and $K'$\n    and the difference between the volume of $K_\\mu$ and the area of $\\mu S(K + \\mu v)$\n    tend to zero.\n\n    \\begin{figure}[t]\n        \\centering\n        \\begin{tikzpicture}\n            \\pgfmathsetmacro{\\Mu}{4}\n            \\pgfmathsetmacro{\\N}{2}\n            \\draw[gray, thin] (-\\Mu - 1, 0) -- (\\Mu + 1, 0);\n\n           \n            \\draw[dashed] (\\Mu, 0) arc [radius = \\Mu, start angle = 0, end angle = 180];\n            \\draw[|<->|] (-\\Mu, 0) -- node [below] {$\\mu$} (0, 0);\n\n           \n            \\draw[dashed] (0, \\Mu) circle [radius = \\N];\n            \\draw[|<->|] (-\\N, \\Mu) -- node [above] {$N$} (0, \\Mu);\n\n           \n            \\coordinate (A) at (0.6, \\Mu + 1.0);\n            \\coordinate (B) at (1.6, \\Mu + 0.8);\n            \\coordinate (C) at (1.8, \\Mu + 0.4);\n            \\coordinate (D) at (0.8, \\Mu + 0.4);\n            \\draw [thick, blue, fill = blue!40] (A) -- (B) -- (C) -- (D) -- cycle;\n            \\node [blue] at (\\N + 0.5, \\Mu + 0.5) {$K + \\mu v$};\n\n           \n            \\draw [blue, line width = 2pt, yshift = 1pt]\n                (A |- 0, 0) -- node [above] {$K'$} (C |- 0, 0);\n\n           \n            \\pgfmathanglebetweenpoints{\\pgfpointorigin}{\\pgfpointanchor{C}{center}}\n            \\edef\\startangle{\\pgfmathresult}\n            \\pgfmathanglebetweenpoints{\\pgfpointorigin}{\\pgfpointanchor{A}{center}}\n            \\edef\\endangle{\\pgfmathresult}\n            \\draw [very thick, red] (\\startangle:\\Mu)\n                coordinate (S-right)\n                arc [radius = \\Mu, start angle = \\startangle, end angle = \\endangle]\n                coordinate (S-left);\n            \\node [red, right = 0.5] at (S-right) {$\\mu S(K + \\mu v)$};\n\n           \n            \\draw [green!30!black, line width = 2pt, yshift = -1pt]\n                (S-left |- 0, 0) -- node [below] {$K_\\mu$} (S-right |- 0, 0);\n\n            \\begin{scope}[very thin]\n               \n                \\draw [red] (0, 0) -- (A);\n                \\draw [red] (0, 0) -- (C);\n\n               \n                \\draw [blue] (A) -- (A |- 0, 0);\n                \\draw [blue] (C) -- (C |- 0, 0);\n                \\draw [green!30!black] (S-left) -- (S-left |- 0, 0);\n                \\draw [green!30!black] (S-right) -- (S-right |- 0, 0);\n            \\end{scope}\n        \\end{tikzpicture}\n        \\caption[\n            Spherical and orthogonal projections.\n        ]{\n            The spherical projection $\\mu S(K + \\mu v)$,\n            when projected orthogonally to the plane $x_d = 0$\n            (the set $K_\\mu$),\n            approaches the volume of the projection $K'$.\n        }\n        \\label{fig:spherical-orthogonal-projection}\n    \\end{figure}\n\n    First, let's bound the Hausdorff distance between $K'$ and $K_\\mu$.\n    If $x \\in K + \\mu v$,\n    then $x$ gets projected to a point $x_0 \\in K'$\n    by just discarding the last coordinate;\n    however, to be projected to a point $x_1 \\in K_\\mu$,\n    first we replace $x$ by $x' = \\frac{\\mu}{\\norm x} x$\n    to get a point $x' \\in \\mu S(K + \\mu v)$,\n    and then the last coordinate of $x'$ is discarded.\n    Note that $x_1 = \\frac{\\mu}{\\norm x} x_0$;\n    therefore,\n    the distance between these two points is\n    \\begin{equation*}\n        \\norm{x_0 - x_1}\n            = \\left| 1 - \\frac{\\mu}{\\norm x} \\right| \\norm{x_0}\n            = \\frac{|\\norm x - \\mu|}{\\norm x} \\norm{x_0}\n    \\end{equation*}\n\n    We have $x \\in K + \\mu v \\subseteq B_N(\\mu v)$,\n    so $\\mu - N \\leq \\norm x \\leq \\mu + N$.\n    As $v = (0, \\dots, 0, 1)$ and $x_0$ is $x$ without its last coordinate,\n    we have $\\norm{x_0} \\leq N$\n    (because, in $\\mathbb R^{d-1}$,\n    we have $x_0 \\in B_N(0)$).\n    So,\n    the distance between $x_0$ and $x_1$ is at most $\\frac{N^2}{\\mu + N}$.\n\n    Every point in $K'$ and in $K_\\mu$ is obtained through these projections.\n    This means that,\n    given any point $x_0$ in one of the sets,\n    we may find another point $x_1$ in the other set\n    which is at a distance of at most $\\frac{N^2}{\\mu + N}$ from the former,\n    because we can just pick a point $x$ whose projection is $x_0$;\n    then its other projection $x_1$ will be close to $x_0$.\n    Thus\n    \\begin{equation*}\n        \\rho(K', K_\\mu) \\leq \\frac{N^2}{\\mu + N},\n    \\end{equation*}\n    so by Theorem~\\ref{thm:hausdorff-distance}\n    the volumes of $K_\\mu$ converges to $\\vol K'$.\n\n    Now,\n    let's relate $\\vol K_\\mu$ with $\\mu^{d-1} \\area S(K + \\mu v)$.\n    If $y = (y_1, \\dots, y_d)$ is a point in $\\mu S(K + \\mu v)$,\n    we know that $\\norm y = \\mu$\n    and that $y_d > 0$\n    (because we're assuming $\\mu > N$).\n    Therefore,\n    if we define $\\varphi: K_\\mu \\to \\mu S(K + \\mu v)$ by\n    \\begin{equation*}\n        \\varphi(y_1, \\dots, y_{d-1}) = \\left(\n            y_1, \\dots, y_{d-1}, \\sqrt{\\mu^2 - y_1^2 - \\dots - y_{d-1}^2}\n        \\right),\n    \\end{equation*}\n    then $\\varphi$ will be a differentiable bijection\n    between $K_\\mu$ and $\\mu S(K + \\mu v)$,\n    so that $\\varphi$ is a parametrization for $\\mu S(K + \\mu v)$.\n\n    For the partial derivatives,\n    we have $\\frac{\\partial \\varphi_i}{\\partial y_j} = [i = j]$ if $i < d$;\n    that is,\n    the partial derivatives behave like the identity.\n    For $i = d$,\n    we have\n    \\begin{equation*}\n        \\frac{\\partial \\varphi_d}{\\partial y_j} =\n            \\frac{y_j}{\\sqrt{\\mu^2 - y_1^2 - \\dots - y_{d-1}^2}}.\n    \\end{equation*}\n\n    Now, by definition of $N$, we have\n    \\begin{equation*}\n        \\left| \\frac{\\partial \\varphi_d}{\\partial y_j} \\right|\n            \\leq \\frac{N}{\\sqrt{\\mu^2 - N^2}},\n    \\end{equation*}\n    so the vectors $D_i \\varphi$ converge uniformly to $e_i$ for large $\\mu$.\n    Since the generalized cross product is linear in each entry,\n    the vector $D_1 \\varphi \\times \\dots \\times D_{d-1} \\varphi$\n    converges uniformly to $e_d$,\n    and thus the number\n    \\begin{align*}\n        \\left| \\vol K_\\mu - \\area \\mu S(K + \\mu v) \\right|\n        &=\n            \\left| \\int_{K_\\mu} 1 -\n                \\int_{K_\\mu} \\norm{D_1 \\varphi \\times \\dots \\times D_{d-1} \\varphi}\n            \\right| \\\\\n        &\\leq\n            \\int_{K_\\mu} \\Big|\n                1 - \\norm{D_1 \\varphi \\times \\dots \\times D_{d-1} \\varphi}\n            \\Big|\n    \\end{align*}\n    converges to zero.\n\n    Now, since $\\area (\\mu S(K + \\mu v)) = \\mu^{d-1} \\area S(K + \\mu v)$,\n    combining these two convergence results gives the theorem.\n\\end{proof}\n\n\\section{Final remarks}\n\nTheorems \\ref{thm:rational-reconstruction} and~\\ref{thm:convex-body-translation-variant}\nboth assume that $L_{K + w}(s) = L_{H + w}(s)$\nfor all integer $w$ and real $s > 0$,\nand both conclude that $K = H$.\nThe first theorem assume that the objects being considered are rational polytopes,\nbut no symmetry condition is imposed;\nthe second theorem assumes that the objects are symmetric,\nbut otherwise permits arbitrary convex bodies.\n\nThis suggest the following common generalization of these theorems:\n\n\\begin{conjecture}\n    Let $K$ and $H$ be any convex bodies,\n    and assume that $L_{K + w}(s) = L_{H + w}(s)$\n    for all real $s > 0$ and all integer $w$.\n    Then $K = H$.\n\\end{conjecture}\n\n\\section{Introduction}\n\nGiven a polytope $P \\subseteq \\mathbb R^d$,\nthe classical Ehrhart lattice point enumerator $L_P(t)$ is defined as\n\\begin{equation*}\n    L_P(t) = \\#(tP \\cap \\mathbb Z^d), \\qquad \\text{integer $t \\geq 0$.}\n\\end{equation*}\nHere,\n$\\#(A)$ is the number of elements in $A$\nand $tP = \\{tx \\mid x \\in P\\}$ is the dilation of $P$ by $t$.\nThe above definition may be extended\nto allow for $P$ to be an arbitrary convex body,\nand for $t$ to be an arbitrary real number.\n\nTo minimize confusion,\nwe will denote real dilation parameters with the letter $s$,\nso that $L_P(t)$ denotes the classical Ehrhart function\nand $L_P(s)$ denotes the extension considered in this paper.\nSo,\nfor example,\n$L_P(t)$ is just the restriction of $L_P(s)$ to integer values.\n\nIt is clear from the definition that\nthe classical Ehrhart function is invariant under integer translations;\nthat is,\nfor every polytope $P$ and every integer vector $w$,\nwe have\n\\begin{equation*}\n    L_{P + w}(t) = L_P(t)\n\\end{equation*}\nfor all $t$.\nThis is not true for the real Ehrhart function $L_P(s)$.\nIn fact,\nan earlier paper~\\cite{Eu}\nshows that the list of functions $L_{P + w}(s)$,\nfor integer $w$,\nis a complete set of invariants for full-dimensional rational polytopes.\nMore precisely:\n\n\\begin{theorem}\n    \\label{thm:rational-reconstruction}\n    Let $P$ and $Q$ be full-dimensional rational polytopes,\n    and suppose that $L_{P + w}(s) = L_{Q + w}(s)$\n    for all $s > 0$ and all integer $w$.\n    Then $P = Q$.\n\\end{theorem}\n\nIt is also conjectured that\nthe rationality hypothesys may be dropped.\nIn this paper,\nwe will show a special case of this conjecture,\nwhere we assume that the polytopes are symmetric\n(that is, $x \\in P$ if and only if $-x \\in P$).\nIn fact,\nwe never use the fact that $P$ and $Q$ are polytopes;\nwe just need convexity.\nSo we have the following.\n\n\\begin{theorem}\n    \\label{thm:convex-body-translation-variant}\n    Let $K$ and $H$ be symmetric convex bodies,\n    and assume that $L_{K + w}(s) = L_{H + w}(s)$\n    for all real $s > 0$ and all integer $w$.\n    Then $K = H$.\n\\end{theorem}\n\n\\subsection{Limit behavior of pseudopyramids}\n\nNow we'll show how to use the pseudopyramids to compute these projections.\n\nLet $K$ be a pseudopyramid.\nDefine the \\emph{outer radius} $R(K)$ of $K$\nto be the smallest number such that\nthe ball of radius $R(K)$ around the origin contains $K$.\nThat is,\n\\begin{equation*}\n    R(K) = \\inf \\{R \\geq 0 \\mid K \\subseteq B_R(0) \\}.\n\\end{equation*}\n\nDefine the \\emph{front shell} of $K$\nto be the set of points in the boundary of $K$\nwhich are not contained in any facet passing through the origin;\nthat is,\nthe set of points $x$ in the boundary of $K$ such that\n$\\lambda x$ is contained in the interior of $K$ for all $0 < \\lambda < 1$.\nDefine, then,\nthe \\emph{inner radius} $r(K)$ of $K$\nto be the largest number such that\nthe ball of radius $r(K)$ around the origin\ncontains no points of the front shell of $K$\n(Figure~\\ref{fig:inner-outer-radius}).\nNote that this is equivalent to $r(K) S(K)$ to be contained in $K$;\nthat is,\n\\begin{equation*}\n    r(K) = \\sup \\{r \\geq 0 \\mid r S(K) \\subseteq K\\}.\n\\end{equation*}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n        \\draw (-1, 0) -- (3, 0);\n        \\draw (0, -1) -- (0, 3);\n        \\draw [thick, blue, fill = blue!40]\n            (0, 0) -- (2, 1) -- (2, 2) -- (1, 2) -- cycle;\n\n        \\pgfmathsetmacro{\\innerradius}{veclen(2, 1)}\n        \\pgfmathsetmacro{\\outerradius}{veclen(2, 2)}\n        \\draw [dashed] (\\innerradius, 0) arc\n            [radius = \\innerradius, start angle = 0, end angle = 90];\n        \\draw [dashed] (\\outerradius, 0) arc\n            [radius = \\outerradius, start angle = 0, end angle = 90];\n\n        \\draw [xshift = -2pt, |<->|]\n            (0, 0) -- node[left] {$r(K)$} (0, \\innerradius);\n        \\draw [yshift = -2pt, |<->|]\n            (0, 0) -- node[below] {$R(K)$} (\\outerradius, 0);\n    \\end{tikzpicture}\n    \\qquad\n    \\begin{tikzpicture}\n        \\draw (-1, 0) -- (3, 0);\n        \\draw (0, -1) -- (0, 3);\n        \\draw [thick, blue, fill = blue!40]\n            (0, 0) -- (2.5, 0.5) -- (0.5, 2.5) -- cycle;\n\n        \\pgfmathsetmacro{\\innerradius}{veclen(1.5, 1.5)}\n        \\pgfmathsetmacro{\\outerradius}{veclen(2.5, 0.5)}\n        \\draw [dashed] (\\innerradius, 0) arc\n            [radius = \\innerradius, start angle = 0, end angle = 90];\n        \\draw [dashed] (\\outerradius, 0) arc\n            [radius = \\outerradius, start angle = 0, end angle = 90];\n\n        \\draw [xshift = -2pt, |<->|]\n            (0, 0) -- node[left] {$r(K)$} (0, \\innerradius);\n        \\draw [yshift = -2pt, |<->|]\n            (0, 0) -- node[below] {$R(K)$} (\\outerradius, 0);\n    \\end{tikzpicture}\n    \\caption{\n        Inner and outer radius for two pseudopyramids.\n    }\n    \\label{fig:inner-outer-radius}\n\\end{figure}\n\nWe leave the following lemma to the reader.\nIt relates the inner and outer radii\nwith the area of the spherical projection.\n\n\\begin{proposition}\n    \\label{thm:radii-vs-spherical-projection}\n    Let $K \\subseteq \\mathbb R^d$ be a convex body\n    which does not contain the origin.\n    Then\n    \\begin{equation*}\n        \\frac{\\vol \\ppyr K}{R(\\ppyr K)^d}\n        \\leq\n        \\frac{\\area S(K)}{d}\n        \\leq\n        \\frac{\\vol \\ppyr K}{r(\\ppyr K)^d}.\n    \\end{equation*}\n\\end{proposition}\n\nThis lemma,\ncombined with Lemma~\\ref{thm:spherical-approaches-orthogonal},\nshows how to calculate the volume of the orthogonal projection\nknowing only the volumes of the pseudopyramids.\n\n\\begin{lemma}\n    \\label{thm:volume-projection-from-volume-ppyr}\n    Let $K \\subseteq \\mathbb R^d$ be a convex body,\n    and $v$ any unit vector.\n    Then\n    \\begin{equation*}\n        \\lim_{\\mu \\to \\infty}\n            \\frac{\\vol \\ppyr(K + \\mu v)}{\\mu}\n            = \\frac{V_K(v)}{d}.\n    \\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n    Let $N$ be large enough so that $K \\subseteq B_N(0)$.\n    For any $\\mu$,\n    as $v$ is a unit vector,\n    we have $K + \\mu v \\subseteq B_{N + \\mu}(0)$,\n    so\n    \\begin{equation*}\n        R(\\ppyr(K + \\mu v)) \\leq \\mu + N.\n    \\end{equation*}\n\n    Since all the points in the front shell of $\\ppyr(K + \\mu v)$\n    are points of $K$,\n    all of them must have norm greater or equal to $\\mu - N$.\n    Therefore,\n    no origin-centered ball with radius smaller than that can contain these points.\n    Thus,\n    \\begin{equation*}\n        r(\\ppyr(K + \\mu v)) \\geq \\mu - N.\n    \\end{equation*}\n\n    Using these two inequalities\n    and Proposition~\\ref{thm:radii-vs-spherical-projection}\n    gives\n    \\begin{equation*}\n        \\frac{\\vol \\ppyr(K + \\mu v)}{(\\mu + N)^d}\n        \\leq\n        \\frac{\\area S(K + \\mu v)}{d}\n        \\leq\n        \\frac{\\vol \\ppyr(K + \\mu v)}{(\\mu - N)^d},\n    \\end{equation*}\n    which may be rewritten as\n    \\begin{equation*}\n        \\begin{adjustbox}{center}\n           \n            $\\displaystyle\n                \\frac{(\\mu - N)^d}{\\mu^d} \\frac{\\mu^{d-1} \\area S(K + \\mu v)}{d}\n                \\leq\n                \\frac{\\vol \\ppyr(K + \\mu v)}{\\mu}\n                \\leq\n                \\frac{(\\mu + N)^d}{\\mu^d} \\frac{\\mu^{d-1} \\area S(K + \\mu v)}{d}.\n            $\n        \\end{adjustbox}\n       \n    \\end{equation*}\n\n    Now Theorem~\\ref{thm:spherical-approaches-orthogonal}\n    and the squeeze theorem finish the proof.\n\\end{proof}\n\nFor example,\nfor $K = [0, 1]^2$ and $v = (1, 0)$,\nwe have $\\vol \\ppyr(K + \\mu v) = 1 + \\frac{\\mu}{2}$,\nso $\\lim_{\\mu \\to \\infty} \\frac{\\vol \\ppyr(K + \\mu v)}{\\mu} = \\frac12$,\nwhich is precisely one-half of the area of $\\{0\\} \\times [0, 1]$,\nthe projection $K'$ of $K$ on the $y$-axis.\nThis highlights that,\nfor large $\\mu$,\nthe pseudopyramid $\\ppyr(K + \\mu v)$ ``behaves like'' an actual pyramid,\nwith height $\\mu$ and base $K'$.\n\n\n\\section{Notation and basic results}\n\nThe punchline is Alexsandrov's projection theorem.\nLet $K \\subseteq \\mathbb R^d$ be a convex body\n(that is,\na covex, compact set with nonempty interior).\nFor any unit vector $v$,\nwe will denote by $V_K(v)$\nthe $(d-1)$-dimensional area of the orthogonal projection of $K$ in $\\{v\\}^\\perp$.\n\nFor example,\nlet $K = [0, 1] \\times [0, 1] \\subseteq \\mathbb R^2$,\n$v = (0, 1)$ and $v' = (\\frac{\\sqrt 2}{2}, \\frac{\\sqrt 2}{2})$.\nThen $V_K(v) = 1$ and $V_K(v') = \\sqrt 2$.\n\nA convex body $K$ is said to be \\emph{symmetric}\nif $x \\in K$ if and only if $-x \\in K$.\nAn importan reconstruction theorem for symmetric convex bodies\nis Aleksandrov's projecion theorem\n(see e.g.~\\cite[p.~115]{GeometricTomography}).\n\n\\begin{theorem}[Aleksandrov's projection theorem]\n    Let $K$ and $H$ be two symmetric convex bodies in $\\mathbb R^d$\n    such that $V_K(v) = V_H(v)$ for all unit vectors $v$.\n    Then $K = H$.\n\\end{theorem}\n\nSo,\nthe goal is to compute the function $V_K$ using the Ehrhart functions $L_{K + w}$.\nThe two main tools are the Hausdorff distance and pseudopyramids.\n\nFor $\\lambda \\geq 0$ and $x \\in \\mathbb R^d$,\ndenote by $B_\\lambda(x)$ the ball with radius $\\lambda$ centered at $x$.\nIf $K$ is a convex body and $\\lambda \\geq 0$,\ndefine $K_\\lambda$ by\n\\begin{equation*}\n    K_\\lambda = \\bigcup_{x \\in K} B_\\lambda(x).\n\\end{equation*}\n\nThe Hausdorff distance $\\rho(K, H)$ between two convex bodies $K$ and $H$\nis defined to be (Figure~\\ref{fig:hausdorff-distance})\n\\begin{equation*}\n    \\rho(K, H) = \\inf \\{ \\lambda \\geq 0 \\mid\n        K \\subseteq H_\\lambda \\text{ and } H \\subseteq K_\\lambda\n    \\}.\n\\end{equation*}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n        \\draw[line width = 2pt, blue] (0, 0) -- (0, 2) -- (2, 0) -- cycle;\n        \\draw[line width = 2pt, green!30!black]\n            (0.5, 0.5) -- (2, 0.5) -- (2, 2) -- (0.5, 2) -- cycle;\n\n        \\pgfmathsetmacro{\\rho}{sqrt(2)}\n        \\draw[thin, blue]\n            (-\\rho, 0) arc [radius = \\rho, start angle = 180, end angle = 270]\n            -- ++(2, 0) arc [radius = \\rho, start angle = -90, end angle = 45]\n            -- ++(-2, 2) arc [radius = \\rho, start angle = 45, end angle = 180]\n            -- cycle;\n\n        \\draw[thin, green!30!black] (0.5, 0.5)\n            ++(-\\rho, 0) arc [radius = \\rho, start angle = 180, end angle = 270]\n            -- ++(1.5, 0) arc [radius = \\rho, start angle = -90, end angle = 0]\n            -- ++(0, 1.5) arc [radius = \\rho, start angle = 0, end angle = 90]\n            -- ++(-1.5, 0) arc [radius = \\rho, start angle = 90, end angle = 180]\n            -- cycle;\n    \\end{tikzpicture}\n    \\caption[\n        Hausdorff distance between two convex sets.\n    ]{\n        Hausdorff distance between two convex sets.\n        The thick lines are the boundaries of the sets $K$ and $H$;\n        the thin lines are the boundaries of the sets\n        $K_\\lambda$ and $H_\\lambda$.\n    }\n    \\label{fig:hausdorff-distance}\n\\end{figure}\n\nIt can be shown that\nthe set of convex sets in $\\mathbb R^d$\nis a metric space under the Hausdorff distance\nand that the Euclidean volume is continuous in this space\n(see e.g.~\\cite[p.~9]{GeometricTomography}),\nbut we just need the following special case of this theory.\n\n\\begin{lemma}\n    \\label{thm:hausdorff-distance}\n    Let $K$ and $A_1, A_2, \\dots$ be convex bodies.\n    If $\\lim_{i \\to \\infty} \\rho(K, A_i) = 0$,\n    then $\\lim_{i \\to \\infty} \\vol A_i = \\vol K$.\n\\end{lemma}\n\nThe concept of pseudopyramid was introduced in~\\cite{Eu}.\nIf $K$ is a convex body,\nthe pseudopyramid $\\ppyr K$ is defined to be\n(Figure~\\ref{fig:pseudopyramid})\n\\begin{equation*}\n    \\ppyr K = \\bigcup_{0 \\leq \\lambda \\leq 1} \\lambda K.\n\\end{equation*}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n        \\draw (-1, 0) -- (3, 0);\n        \\draw (0, -1) -- (0, 3);\n        \\filldraw [thick, blue, fill = blue!40] (1, 1) rectangle (2.5, 2.5);\n        \\node at (1.75, 1.75) {$P$};\n\n        \\begin{scope}[xshift = 5cm]\n            \\draw (-1, 0) -- (3, 0);\n            \\draw (0, -1) -- (0, 3);\n            \\filldraw [thick, blue, fill = blue!40]\n                (0, 0) -- (1, 2.5) -- (2.5, 2.5) -- (2.5, 1) -- cycle;\n            \\node at (1.75, 1.75) {$\\ppyr(P)$};\n        \\end{scope}\n    \\end{tikzpicture}\n    \\caption{\n        Pseudopyramid of a polytope.\n    }\n    \\label{fig:pseudopyramid}\n\\end{figure}\n\nWe will use the following lemma,\nwhich is a consequence of Lemma 1 of~\\cite{Eu}.\n\n\\begin{lemma}\n    \\label{thm:volume-of-pseudopyramid}\n    Let $K$ and $H$ be convex bodies,\n    and suppose that $L_K(s) = L_H(s)$ for all $s > 0$.\n    Then $\\vol \\ppyr H = \\vol \\ppyr K$.\n\\end{lemma}\n\nIn other words,\nwhenever we know $L_K$,\nwe may assume we also know $\\vol \\ppyr K$.\n\n\\begin{proof}\n    Lemma 1 of~\\cite{Eu} states that,\\footnote{\n        Tecnically,\n        Lemma 1 of~\\cite{Eu} only states this for polytopes,\n        but the proof holds verbatim for convex bodies.\n    }\n    if $L_K(s) = L_H(s)$,\n    then $L_{\\ppyr K}(s) = L_{\\ppyr H}(s)$.\n    Since\n    \\begin{equation*}\n        \\lim_{s \\to \\infty} \\frac{L_{\\ppyr K}(s)}{s^d} = \\vol \\ppyr K\n    \\end{equation*}\n    (and similarly for $H$),\n    we have $\\vol \\ppyr K = \\vol \\ppyr H$.\n\\end{proof}\n\n\\subsection{Piecing everything together}\n\n\\begin{restatetheorem}{thm:convex-body-translation-variant}\n    Let $K$ and $H$ be symmetric convex bodies,\n    and assume that $L_{K + w}(s) = L_{H + w}(s)$\n    for all real $s > 0$ and all integer $w$.\n    Then $K = H$.\n\\end{restatetheorem}\n\n\\begin{proof}\n    By Lemma~\\ref{thm:volume-of-pseudopyramid},\n    we have $\\vol \\ppyr (K + w) = \\vol \\ppyr (H + w)$\n    for all integer $w$.\n\n    If $w$ is a nonzero integer vector,\n    let $v = \\frac{w}{\\norm w}$;\n    then,\n    by Lemma~\\ref{thm:volume-projection-from-volume-ppyr},\n    we have\n    \\begin{align*}\n        \\frac{V_K(v)}{d}\n            &= \\lim_{\\mu \\to \\infty} \\frac{\\vol \\ppyr(K + \\mu v)}{\\mu} \\\\\n            &= \\lim_{\\mu \\to \\infty} \\frac{\\vol \\ppyr(H + \\mu v)}{\\mu} \\\\\n            &= \\frac{V_H(v)}{d}\n    \\end{align*}\n\n    This shows that,\n    whenever $v$ is a multiple of a rational vector,\n    we have $V_K(v) = V_H(v)$.\n    Since the functiosn $V_K$ and $V_H$ are continuous,\n    we have $V_K = V_H$,\n    and thus by Aleksandrov's projection theorem\n    we conclude that $K = H$.\n\\end{proof}\n\n\\section{Pseudopyramid volumes and areas of projections}\n\\label{sec:projection-areas}\n\nIn this section,\nwe will show how to compute the function $V_K$\nin terms of the numbers $\\vol \\ppyr(P + w)$ for integer $w$.\n\nGiven a convex body $K$,\ndefine its spherical projection $S(K)$ by\n(Figure~\\ref{fig:spherical-projection})\n\\begin{equation*}\n    S(K) = \\left\\{ \\frac{x}{\\norm x}\n        \\,\\middle|\\, x \\in K \\text{ and } x \\neq 0 \\right\n    \\}.\n\\end{equation*}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n        \\draw[->] (-1, 0) -- (6, 0);\n        \\draw[->] (0, -1) -- (0, 4);\n\n        \\draw (1, 0) arc [start angle = 0, end angle = 90, radius = 1];\n\n        \\filldraw [thick, blue, fill = blue!40]\n            (5, 3) -- (6, 3) -- (6, 4) -- (5, 4) -- cycle;\n        \\node[blue] at (6.5, 3.5) {$K + v$};\n\n        \\draw [gray, thin] (5, 4) -- (0, 0) -- (6, 3);\n\n        \\pgfmathsetmacro{\\startangle}{atan(3\/6)}\n        \\pgfmathsetmacro{\\endangle}{atan(4\/5)}\n        \\pgfmathsetmacro{\\radius}{veclen(5, 3)}\n\n        \\draw [red, very thick] (\\startangle:1)\n            arc [start angle = \\startangle, end angle = \\endangle, radius = 1];\n        \\draw [red] (\\startangle:1)\n            +(1, 0) node {$S(K + v)$};\n\n        \\draw [dotted] (\\startangle - 10:\\radius)\n            arc [start angle = \\startangle - 10,\n                end angle = \\endangle + 10, radius = \\radius];\n        \\draw [red, thick] (\\startangle : \\radius)\n            arc [start angle = \\startangle, end angle = \\endangle, radius = \\radius]\n            ++(-1.5, 0) node {$\\norm v S(K + v)$};\n    \\end{tikzpicture}\n    \\caption{\n        Spherical projection of $K + v$.\n    }\n    \\label{fig:spherical-projection}\n\\end{figure}\n\nThe connection between pseudopyramid volumes\nand areas of projections\ncan be seen in Figure~\\ref{fig:spherical-projection}.\nThe set $\\norm v S(K + v)$ is a dilation of the projection $S(K + v)$ of $K$.\nNote that the shape of $\\norm v S(K + v)$\n``looks like'' the orthogonal projection of $K$ in $\\{v\\}^\\perp$;\nthat is,\nthe area of $\\norm v S(K + v)$\napproximates $V_K(v)$.\n\nIf the pseudopyramid were an actual pyramid\n(with base $\\norm v S(K + v)$),\nthen using the formula $v = \\frac{Ah}{d}$ for the volume of a pyramid\nwould allow us to discover what is the area of the projection,\nwhich would give an approximation to $V_K(v)$.\nWe will show that this formula is true ``in the infinity'';\nthat is,\nusing limits,\nwe can recover the area of the projection\nusing taller and taller pseudopyramids.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nHamilton introduced the Ricci flow to study the global structures and classification of Riemannian manifolds in his seminal work \\cite{H1}. It was used to solve Thurston's geometrization conjecture for $3$-manifolds \\cite{P1, P2, P3, KL, MT}. The Ricci flow preserves the K\\\"ahlerian metrics: If $(X,g_0)$ is compact K\\\"ahler manifold of complex dimension $n$, then any solution $g(t)$ of the Ricci flow with initial condition $g_0$ must be K\\\"ahler. This leads to the K\\\"ahler-Ricci flow which has been very useful in K\\\"ahler geometry: \n \\begin{equation}\\label{unkrflow}\n\\left\\{\n\\begin{array}{l}\n{ \\displaystyle \\ddt{g} = -Ric(g) ,}\\\\\n\\\\\ng|_{t=0} =g_0 .\n\\end{array} \\right.\n\\end{equation}\nIn \\cite{Cao1}, adapting certain arguments in \\cite{Y1}, Cao first studied the K\\\"ahler-Ricci flow and used this to give an alternative proof of  the Calabi conjecture.\n \nThe Ricci flow always has a solution for $t$ small. It was proved in \\cite{TZha} that \\eqref{unkrflow} admits a maximal solution on $[0, T)$, where \n\\begin{equation} \\label{T}\nT \\,=\\, \\sup \\{ t \\in \\mathbb{R} \\ | \\ [\\omega_0] + t  [K_X] >0 \\}.\n\\end{equation}\nwhere $\\omega_0$ denotes the K\\\"ahler form of $g_0$. This gives a sharp local existence theorem. Therefore, the K\\\"ahler-Ricci flow admits a long time solution if and only if the canonical class is nef.  It was also shown in \\cite{Cao1} that the K\\\"ahler-Ricci flow after normalization always converges exponentially fast to a K\\\"ahler-Einstein metric if the first Chern class is negative or zero. If the first Chern class is positive, $T$ is finite and one can study finer behavior of $g(t)$ as $t$ tends to $T$. This has been extensively studied (see \\cite{P4, SeT, TZhu1,  PSSW, CS, TiZ1, B, CW} etc.). One would hope that the K\\\"ahler-Ricci flow should deform any initial K\\\"ahler metric to a K\\\"ahler-Einstein metric, however, most K\\\"ahler manifolds do not admit definite or vanishing first Chern class and so the flow will in general develop singularities. An Analytic Minimal Model Program (AMMP) through Ricci flow was initiated by Song-Tian more than a decade ago to study birational classification of compact K\\\"ahler manifolds, including algebraic manifolds. We refer the readers to \\cite{SoT3} for a description of the AMMP.\nOne crucial problem in this program is to study formation of singularities along the K\\\"ahler-Ricci flow. It is conjectured by Song-Tian in \\cite{SoT3} that the K\\\"ahler-Ricci flow will deform a projective variety $X$ of nonnegative Kodaira dimension, to its minimal model via finitely many divisorial metric contractions and metric flips in Gromov-Hausdorff topology, then eventually converge to a unique canonical metric of Einstein type on its unique canonical model. The existence and uniqueness is proved in \\cite{SoT3} for the analytic solutions of the K\\\"ahler-Ricci flow on algebraic varieties with log terminal singularities and the K\\\"ahler-Ricci flow can be analytically and uniquely extended through divisorial contractions and flips \\cite{SoT3}. Finite time geometric surgeries in terms of Gromov-Hausdorff topology are introduced and established in the case of K\\\"ahler surfaces and more generally, flips induced by Mumford quotients in \\cite{SW1, SW2, SW3, SY, S1}. An alternative approach to understand the K\\\"ahler-Ricci flow through singularities was proposed in \\cite{LT} in the frame work of K\\\"ahler quotients by transforming the parabolic complex Monge-Amp\\`ere equation into an elliptic  $V$-soliton equation.\n\n\n\nIn this paper, we are interested in geometric behavior of long time solutions of the K\\\"ahler-Ricci flow. We consider the normalized K\\\"ahler-Ricci flow on an $n$-dimensional projective manifold $X$ defined by\n \\begin{equation}\\label{krflow}\n\\left\\{\n\\begin{array}{l}\n{ \\displaystyle \\ddt{g} = -Ric(g) - g,}\\\\\n\\\\\ng|_{t=0} =g_0 .\n\\end{array} \\right.\n\\end{equation}\nwith the initial K\\\"ahler metric $g_0$. The long time existence of the flow (\\ref{krflow}) is equivalent to the canonical class $K_X$ being nef. A projective manifold of nef canonical model is called a minimal model. The well-known abundance conjecture in birational geometry predicts that  $K_X$ being nef is equivalent to $K_X$ being semi-ample and the conjecture holds up to complex dimension $3$. In this paper we will always assume the canonical bundle $K_X$ to be semi-ample, and a uniform scalar curvature bounded is established for the global solutions of the flow (\\ref{krflow}).  \n\nThe Kodaira dimension is the algebraic dimension measuring the size of the pluricanonical system of the underlying K\\\"ahler manifold $X$. When the canonical bundle $K_X$ is semi-ample or nef, the Kodaira dimension of $X$ must be a nonnegative integer no greater than $\\dim_\\mathbb{C} X=n$.\nWe will discuss some of the known results by Kodaira dimension ${\\rm Kod}(X)$.\n\n When the ${\\rm Kod}(X) =n$,  the canonical class of $X$ is big and nef and it was proved in \\cite{Ts, TZha} that  the flow (\\ref{krflow})  converges weakly to the unique K\\\"ahler-Einstein current on the canonical model $X_{can}$ of $X$ which is smooth on $X_{can}^\\circ$, the regular set of $X_{can}$. Recently, it was proved in \\cite{S2} that the metric completion of such a smooth K\\\"ahler-Einstein metric on $X_{can}$ is homeomorphic to the projective variety $X_{can}$ itself. This result was used in \\cite{W} to obtain a uniform diameter bound for the flow (\\ref{krflow}). In the special case when $K_X$ is ample,  it was shown in \\cite{Cao1} that the flow will always converge smoothly to the unique K\\\"ahler-Einstein metric on $X$.\n\n\nWhen ${\\rm Kod}(X)=0$, $X$ must a Calabi-Yau manifold, as we mentioned above, the unnormalized flow (\\ref{unkrflow}) converges smoothly \\cite{Cao1} to the unique Ricci-flat K\\\"ahler metric in the initial K\\\"ahler class.\n\nWhen  ${\\rm Kod} (X)=\\kappa$ and $1\\leq \\kappa \\leq n-1$,  for sufficiently large $m$, the canonical ring is finitely generated and the pluricanonical system $|mK_X|$ induces a unique projective morphism\n$$\\Phi: X \\rightarrow X_{can}$$\nwhere $X_{can}$ is the unique canonical model of $X$ and $\\dim X_{can} = \\kappa$. $X_{can}$ has canonical singularities and the map $\\Phi$ is a holomorphic fibration of $(n-\\kappa)$-dimensional manifolds of vanishing first Chern class over $X_{can}^\\circ$, the set of smooth points on $X_{can}$ over which $\\Phi$ is submersion.  $X_{can}^\\circ$ is a Zariski open set of $X_{can}$. In \\cite{ST1,ST2}, the twisted K\\\"ahler-Einstein (possibly singular) metric $g_{can}$ on $X_{can}$ is defined to be\n\\begin{equation}\\label{twke}\nRic(g_{can}) = - g_{can} + g_{WP},\n\\end{equation}\nwhere $g_{WP}$ is the Weil-Petersson current induced from the variation of Calabi-Yau fibres of $X$ over $X_{can}$. In fact, $g_{can}$ has bounded local potentials and it is smooth on $X_{can}^\\circ$.  It was shown in \\cite{ST1, ST2} that the solution of (\\ref{krflow}) converges in current to the canonical twisted K\\\"ahler-Einstein current $g_{can}$ on $X_{can}$ and the local potential converges in $C^{1,\\alpha}$ on any compact subset of $X^\\circ= \\Phi^{-1}(X_{can}^\\circ)$. The local convergence is improved to the $C^0$-topology for $g(t)$ to $g_{can}$ in \\cite{TWY}. \n\nWe now state our first result of the paper.\n\n\n\\begin{theorem} \\label{main1} Let $X$ be a projective manifold with semi-ample canonical bundle. Let $g(t)$ be the solution of the normalized K\\\"ahler-Ricci flow (\\ref{krflow}) on $X$ with any initial K\\\"ahler metric $g_0$. If there exists an open domain $U$ of $X$ containing a fibre of $X$ over $X_{can}$ and $\\Lambda>0$  such that \n\\begin{equation}\\label{curvature bound}\n\\sup_{U\\times [0, \\infty)} |Ric(g(t))|_{g(t)} <\\Lambda .\n\\end{equation} Then there exists $D>0$ such that for all $t\\in [0, \\infty)$ we have\n\\begin{equation}\\label{diambd}\ndiam(X, g(t)) < D ,\n\\end{equation}\nwhere $diam(X, g(t))$ is the diameter of $(X, g(t))$.\n\\end{theorem}\n\nThe assumption on the Ricci curvature in Theorem \\ref{main1} is used to apply the relative volume comparison for the Ricci flow proved in \\cite{TiZ3}. It is expected that such a Ricci bound holds on any open domain $U$ compactly supported  in $X^\\circ= \\Phi^{-1}(X_{can}^\\circ)$,  but it is still open at the moment. The recent work of \\cite{HT} might give some technical hints on how to bound the Ricci curvature locally in $X^\\circ$. In fact, when the general fibre of $\\Phi: X\\rightarrow X_{can}$ is a complex torus, then the full curvature tensors are uniformly bounded on any compact subset of $X_{can}^\\circ$ \\cite{TWY} and the following corollary immediately follows from Theorem \\ref{main1}. Combined with the uniform bounds for the scalar curvature\n$$\\sup_{X\\times [0, \\infty)} |R(g(t))| < \\infty$$\nfor all time (\\cite{SoT4}), the estimate in Theorem \\ref{main1} for long time collapsing solutions can be compared to Perelman's diameter and scalar curvature bounds for non-collapsed solutions of the K\\\"ahler-Ricci flow on Fano manifolds. The diameter estimate can also be viewed as a boundedness result in view of algebraic geometry and topology. Theorem \\ref{main1} can be largely extended to the K\\\"ahler case since the semi-ampleness of the canonical bundle already implies that the canonical model is projective. \n\n\n\n\\begin{corollary}\\label{mainc1} Let $g(t)$ be the solution of the normalized K\\\"ahler-Ricci flow (\\ref{krflow}) on $X$ with any initial K\\\"ahler metric $g_0$. If ${\\rm Kod}(X)\\geq \\dim X - 1$,  or more generally if the general fibre of $\\Phi: X \\rightarrow X_{can}$ is a complex torus, then there exists $D>0$ such that for all $t\\in [0, \\infty)$ we have\n$$diam(X, g(t)) < D .$$\n\n\\end{corollary}\n\n\n\nBy Theorem \\ref{main1}, one can always extract a convergent sequence  along time for the solution of the normalized K\\\"ahler-Ricci flow in Gromov-Hausdorff topology, however, more delicate analysis is required to show the uniqueness of such limits. We naturally propose the following conjecture about the convergence of the flow (\\ref{krflow}) on smooth minimal models, i.e. projective manifolds with nef canonical bundle.\n\n\\begin{conjecture} \\label{conj} Let $X$ be a smooth K\\\"ahler manifold with nef canonical bundle. Then for any initial K\\\"ahler metric $g_0$ on $X$, the solution $g(t)$ of the unnormalized K\\\"ahler-Ricci flow (\\ref{krflow}) has uniformly bounded diameter and scalar curvature for all time. Furthermore, $(X, g(t))$ converges in the Gromov-Hausdorff topology to a unique compact metric space $(\\mathcal{Z}, d_\\mathcal{Z}) $ homeomorphic to the canonical $X_{can}$.\n\n\n\\end{conjecture}\n We confirm the conjecture in the following theorem  when $K_X$ is semi-ample, the Kodaira dimension is $1$and the general fibre of $X$ over $X_{can}$ is a complex torus. In particular, it confirms the conjecture for the K\\\"ahler-Ricci flow on minimal elliptic surfaces of Kodaira dimension $1$ in \\cite{ST1}. \n\n\n\\begin{theorem}\\label{KRF: minimal model}\nUnder the same assumptions including (\\ref{curvature bound}) in Theorem \\ref{main1}, if the projective manifold $X$ has Kodaira dimension $1$, then any solution of the normalized K\\\"ahler-Ricci flow (\\ref{krflow}) converges in the Gromov-Hausdorff topology to the metric completion of  $(X_{can}^\\circ, g_{can})$, which is homeomorphic to $X_{can}$. In particular, the conclusion holds for projective manifolds of Kodaira dimension $1$ and semi-ample canonical bundle whose general fibre over its canonical model is a complex torus. \n\\end{theorem}\n\nWe remark that the result in Theorem \\ref{KRF: minimal model} still holds in the case of higher Kodaira dimension if $X_{can}\\setminus X_{can}^\\circ$  is a set of finitely many isolated points.\n\nThe proof of Theorem \\ref{main1} relies on the diameter estimate for certain family of twisted K\\\"ahler-Einstein metrics established in \\cite{FGS} and the relative volume comparison established in \\cite{TiZ3}. The main technical contribution of the paper is to prove that the evolving metrics $g(t)$ of the K\\\"ahler- Ricci flow has suitable convexity on $\\Phi^{-1}(X_{can}^\\circ)$. Such a convexity result is built on the almost  convexity of the twisted K\\\"ahler-Einstein metric $g_{can}$ on $X_{can}^\\circ$ in the following theorem for the continuity method improving the results in  \\cite{FGS}.\n\n\\begin{theorem} \\label{main2}Let $X$ be a projective K\\\"ahler manifold of $\\dim_{\\mathbb{C}} X =n$ with semi-ample canonical line bundle $K_X$ with $X_{can}$ being the canonical model of $X$. Let $A$ be an ample line bundle and $g_t\\in [tA + K_X]$ be the unique K\\\"ahler metrics for $t\\in (0, \\infty)$ satisfying\n$$Ric(g_t) = -g_t + t g_A, $$\nfor any fixed K\\\"ahler metric $\\omega_A\\in [A]$ on $X$. Then the following hold.\n\n\\begin{enumerate}\n\n\\item $(X, g_t)$ converges in the Gromov-Hausdorff topology to a compact metric space $(\\mathcal{Z}, d_\\mathcal{Z})$ as $t\\rightarrow 0^+$.\n\n\\item  $g_t$ converges in the $C^0$-topology on $\\Phi^{-1}(X_{can}^\\circ)$ to the pullback of $g_{can}$ on $X_{can}^\\circ$ as $t\\rightarrow 0^+$.\n\n\\item The metric completion of $(X_{can}^\\circ, g_{can})$ is isomorphic to $(\\mathcal{Z}, d_\\mathcal{Z})$.\n\n\\end{enumerate}\nIn particular, if $\\dim_{\\mathbb{C}} X_{can} \\leq 2$, or more generally if $X_{can}$ has only orbifold singularities, $(\\mathcal{Z}, d_\\mathcal{Z})$ is homeomorphic to $X_{can}$.\n\n\n\n\n\\end{theorem}\n\n\n\nThe advantage of using the continuity method over the Ricci flow is that the Ricci curvature is naturally bounded below uniformly and one can apply many existing techniques in comparison geometry.   A more natural adaption of Theorem \\ref{main2} is for the collapsing behavior of Ricci-flat K\\\"ahler metrics on a Calabi-Yau manifold as a holomorphic fibration of Calabi-Yau manifolds. This topic has been extensively studied in \\cite{Tos, GTZ, ToZh, HT}.\n\n\n\\begin{theorem} \\label{main3}Let $X$ be a projective K\\\"ahler manifold of $\\dim_{\\mathbb{C}} X =n$ with $c_1(X)=0$. Suppose $L$ is a semi-ample line bundle over $X$. The linear system $H^0(X, mL)$ induces a holomorphic map $\\Phi: X \\rightarrow Y$ for sufficiently large $m$ from $X$ to  a projective variety $Y$. Let $A$ be an ample line bundle and $g_t\\in [tA + L]$ be the unique Calabi-Yau metrics for $t\\in (0, \\infty)$. Then the followings hold.\n\n\\begin{enumerate}\n\n\\item $(X, g_t)$ converges in the Gromov-Hausdorff topology to a compact metric space $(\\mathcal{Z}, d_\\mathcal{Z})$ as $t\\rightarrow 0^+$,\n\n\\item  $g_t$ converges in the $C^0$-topology on $\\Phi^{-1}(Y^\\circ)$ to the pullback of a unique smooth K\\\"ahler metric $g_Y$ on $Y^\\circ$ as $t\\rightarrow 0^+$.  Here $Y^\\circ$ is the set of smooth points of $Y$ over which $\\Phi$ is submersion. \n\n\\item $g_Y$ extends to a K\\\"ahler current on $Y$ with bounded local potentials and on $Y^\\circ$, we have\n$$Ric(g_Y) = g_{WP},$$\nwhere $g_{WP}$ is the Weil-Petersson metric of the variation of the smooth Calabi-Yau fibres of $X$ over $Y^\\circ$.\n\n\\item The metric completion of $(Y^\\circ, g_Y)$ is isomorphic to $(\\mathcal{Z}, d_\\mathcal{Z})$.\n\n\\end{enumerate}\nIn particular, if $\\dim_{\\mathbb{C}} Y \\leq 2$ or more generally if $Y$ has only orbifold singularities, $(\\mathcal{Z}, d_\\mathcal{Z})$ is homeomorphic to $Y$.\n\n\n\\end{theorem}\n\n\nThe twisted Ricci-flat K\\\"ahler metric $g_Y$ in Theorem \\ref{main3} was proposed in \\cite{ST1, ST2} as a special case of the twisted K\\\"ahler-Einstein metrics and it had already been implicitly studied in \\cite{GW, Fi} in the case of complex surfaces. The statement (4) in Theorem \\ref{main3} confirms the conjecture proposed in \\cite{ToZh} (Conjecture 1.1 (a) and (b)) related to an analogous conjecture by Gross \\cite{Gr}, Kontsevich-Soilbelman \\cite{KS} and Todorov \\cite{Ma} for collapsing limits of Ricci-flat K\\\"ahler metrics near complex structure limits. The statement (3) in Theorem \\ref{main3} is shown in \\cite{ST1, ST2}. The  sequential convergence in statements (1) is proved in \\cite{Tos, GTZ} and the statement (2) is proved in \\cite{TWY}. The special case for the statement (4)  is proved in \\cite{GTZ, ToZh} when $\\dim_{\\mathbb{C}} Y=1$ or $X$ is Hyperkahler. The main contribution of our work in this paper is the statement (4) for identifying the intrinsic and extrinsic geometric limits of collapsing Calabi-Yau metrics. \n\nWe would also like to point out that the projective assumption for the K\\\"ahler manifold $X$ in this paper is for conveniences and is not essential in the proof. In fact, the semi-ampleness condition already implies $X_{can}$ is projective. \n\nWe give a brief outline of the paper. In section 2, we prove Theorem \\ref{main2} and in particular, we show that $(X_{can}^\\circ, g_{can})$ is almost geodesically convex in $(\\mathcal{Z}, d_\\mathcal{Z})$. In section 3, Theorem \\ref{main3} is proved by slight modification of the proof of Theorem \\ref{main2}. In section 4, we prove our main result Theorem \\ref{main1} and its corollaries by using the result and proof of section 2. Finally, we prove Theorem \\ref{KRF: minimal model} in section 5. \n\n\\section{Proof of Theorem \\ref{main2}}\n\nIn this section, we will study deformation of a family of collapsing twisted K\\\"ahler-Einstein metrics and prove   Theorem \\ref{main2}.\n\n\nLet $X$ be a projective manifold of complex dimension $n$. Suppose the canonical line bundle $K_X$ is semi-ample and the Kodaira dimension of $X$ is $\\kappa$, i.e., $\\dim_{\\mathbb{C}}X_{can} = \\kappa$. In this section we will always assume $0<\\kappa < n$. Then the pluricanonical system $|mK_X|$ induces a holomorphic morphism\n$$\\Phi: X \\rightarrow X_{can}\\hookrightarrow \\mathbb{CP}^{N_m}, $$\nfor sufficiently large $m$ and $X_{can}$ is the unique canonical model of $X$.\n\n\n\nLet $\\omega_A$ be a K\\\"ahler metric in a K\\\"ahler class $A$ on $X$. We will now consider a continuous family of K\\\"ahler metrics $ \\omega (t)$ defined by\n\\begin{equation}\\label{contin1}\nRic(  \\omega(t)) = -   \\omega(t) + t \\omega_A, ~ t\\in (0, 1].\n\\end{equation}\nWe let $\\Omega$ be a smooth volume form on $X$ such that\n\\begin{equation}\n\\chi = \\sqrt{-1}\\partial\\dbar\\log \\Omega =\\frac{1}{m} \\Phi^*\\omega_{FS}\\in [K_X],\n\\end{equation}\nwhere $\\omega_{FS}$ is the Fubini-Study metric of \n$\\mathbb{CP}^{N_m}$. \nTherefore \n$$[  \\omega(t)] = [\\chi] + t [\\omega_A].$$ Throughout the paper, we abuse the notation by identifying  $\\chi$ with $\\frac{1}{m}\\omega_{FS}|_{X_{can}}$ on $X_{can}$ as well.\nIf we write $$  \\omega(t) = \\chi +t \\omega_A +\\sqrt{-1}\\partial\\dbar \\psi (t)$$\nfor some $\\psi = \\psi(t) \\in C^\\infty(X)$,\nthen by straightforward deductions, the equation (\\ref{contin1}) becomes\n\\begin{equation}\\label{contin2}\nt^{-(n-\\kappa)} (\\chi + t \\omega_A+ \\sqrt{-1}\\partial\\dbar \\psi)^n = e^{\\psi} \\Omega.\n\\end{equation}\n\nEquation (\\ref{contin2}) has a smooth solution for all $t >  0$ by \\cite{Y1, A} since $[\\chi+t\\omega_A ]$ is a K\\\"ahler class and we are interested in the the limiting behavior of $  \\omega(t)$ as $t \\rightarrow 0$.   We first state some basic estimates for $\\psi$.\n\n\n\\begin{theorem} There exists $C>0$ such that for all $t\\in (0,1]$, we have\n$$|\\psi| \\leq C. $$\n\n\\end{theorem}\n\n\\begin{proof} By the maximum principle, there exist $C_1, C_2>0$ such that for any $t\\in (0,1]$, we have\n$$\\sup_X \\psi \\leq \\sup_X \\left(\\log \\frac{  t^{-(n-\\kappa)}(\\chi + t \\omega_A )^n}{\\Omega} \\right)\\leq C_1  \\sup_X \\left(\\log \\frac{ \\chi^\\kappa \\wedge \\omega_A^{n-\\kappa}}{\\Omega} \\right)\\leq C_2. $$\nThen the right hand side of the equation (\\ref{contin2}) is uniformly bounded above and the lower bound of $\\psi$ follows directly from $L^\\infty$-estimate for degenerate complex Monge-Amp\\`ere equations by Demailly-Pali \\cite{DP} (see also \\cite{Kol1, EGZ}).\n\n\n\\end{proof}\n\n\nWe notice that for any $t\\in (0,1]$, the Ricci curvature of $  \\omega(t)$ is uniformly bounded below by $-1$. We can then apply the following diameter estimate proved in \\cite{FGS}.\n\n\n\\begin{lemma} Let $  g(t)$ be the K\\\"ahler metric associated to $  \\omega(t)$.\nThere exists $L>0$ such that for all $t\\in (0,1]$,\n$$Diam(X,   g(t)) \\leq L.$$\n\n\\end{lemma}\n\n\nBy the volume comparison, we immediately have the following volume estimate.\n\n\\begin{corollary} \\label{volr} There exists $C>0$ such that for any point $p\\in X$ and $t\\in (0,1]$ and $0< r < L$,\n$$Vol(B_{  g(t)} (p, r),   g(t) ) \\geq Cr^{2n}  t^{n-\\kappa}.$$\n\n\n\n\\end{corollary}\n\n\n\nThe following lemma is due to \\cite{G} (also see \\cite{CC, Da}) as a consequence of the volume comparison and it is very useful to prove geometric convexity for certain family of metric spaces.\n\n\\begin{lemma} \\label{gromov} Let $(M, g)$ be a Riemannian manifold of dimension $n$ satisfying\n$$Ric(g) \\geq - g, ~ Diam(M, g) \\leq L.$$\nLet  $E\\subset M$ be any compact set with a smooth boundary. If there are two points $p_1, p_2\\in M$ with $$B_g(p_i, r) \\cap E = \\emptyset, ~ i =1, 2$$\nand every minimal geodesic from $p_1$ to points in $B(p_2, r)$ intersects $E$, then there exists $c=c(n, r, L)>0$ such that\n$$Vol(\\partial E, g) \\geq c Vol(B_g(p_2, r), g). $$\n\n\n\\end{lemma}\n\n\nWe will construct the set $E$ for the family of metrics $  \\omega(t)$.\nFirst, by semi-ampleness of $K_X$ we can assume $K_X$ is the pullback of an ample line bundle $\\mathcal{L}$ on $X_{can}$. We can pick an effective $\\mathbb{Q}$-divisor $\\sigma$ on $X_{can}$ such that\n\n\\begin{enumerate}\n\n\\item $\\sigma$ lies in the class of $[\\mathcal{L}]$,\n\n\\item  $X_{can}\\setminus X_{can}^\\circ$ is contained in the support of $\\sigma$.\n\n\n\\end{enumerate}\nWe let $\\sigma'=\\Phi^*\\sigma$.\n\nSecond, we consider a log resolution of $X_{can}$ defined by\n$$\\Psi: W \\rightarrow X_{can}$$\nsuch that\n\n\\begin{enumerate}\n\n\\item $W$ is smooth and the exceptional locus of $\\Psi$ is a union of smooth divisors of simple normal crossings.\n\n\\item $\\tilde \\sigma$, the pullback of $\\sigma$, is a union of smooth divisors of simple normal crossings.\n\n\\end{enumerate}  The Fubini-Study metric $\\chi$ on $X_{can}$ also lies in $[\\sigma]$.\nLet $\\tilde X$ be the blow-up of $X$ induced by $\\Psi: W \\rightarrow X_{can}$ and we let $  \\Psi':   \\tilde X \\rightarrow X$. We also pick the hermitian metric $h$ on $\\mathcal{L}$ such that $Ric(h) = \\chi$. Let $\\tilde \\sigma' = \\Psi^* \\sigma'$. Away from $\\tilde \\sigma'$, $\\tilde X$ can be identified as $X$ by assuming the blow-ups take place at the support of $ \\sigma'$. We also let $ h' = \\Phi^*h$ and $\\tilde h' = \\Psi^* h'$.\n\n\nThe following is an analogue of the Schwarz lemma.\n\\begin{lemma} \\label{schwa} There exists $c>0$ such that for all $t\\in (0,1]$ we have on $X$,\n$$  \\omega(t) \\geq c \\chi. $$\n\n\\end{lemma}\n\n\\begin{proof} We can directly apply the maximum principle to the following quantity\n$$\\log tr_{ \\omega}(\\chi) - K \\psi$$\nfor sufficiently large $K$ and the estimate of the lemma will immediately follows as $\\psi$ is uniformly bounded.\n\n\n\\end{proof}\n\n\nFor simplicity, we assume that   \n$$|\\tilde \\sigma'|^2_{\\tilde h'} \\leq 1$$\neverywhere.\nLet $F$ be the standard decreasing smooth cut-off function defined on $[0, \\infty)$ satisfying\n\n\\begin{enumerate}\n\\item $F(x)=3$,  if $x\\in [0, 1\/2]$,\n\n\\item  $F(x)=0$,  if $x\\in [3, \\infty)$,\n\\item $F(x) = 3-x$, if $x\\in [1,2]$.\n\\end{enumerate}\nLet $$\\eta_\\epsilon = \\max \\left( \\log |\\tilde\\sigma'|^2_{\\tilde h'}, \\log \\epsilon \\right)$$\nfor some sufficiently small $\\epsilon>0$ to be determined later.\nBy the construction of $\\tilde\\sigma'$ and $h$, we have\n$$ \\sqrt{-1}\\partial\\dbar \\log |\\tilde \\sigma'|^2_{\\tilde h'} +  \\chi \\geq 0,$$\ntherefore  $$\\eta_\\epsilon \\in PSH(X, \\chi) \\cap C^0(X). $$\nIn particular, for sufficiently small $\\epsilon>0$, we have $$ \\log \\epsilon \\leq \\eta_\\epsilon \\leq 0. $$\nWe define $\\rho_\\epsilon$ by\n$$\\rho_\\epsilon = F\\left( \\frac{100\\eta_\\epsilon}{\\log \\epsilon} \\right). $$\n\n\nThe following estimate is based on the calculations in \\cite{S2} (see ).\n\\begin{lemma} \\label{cutoff1}  There exists $C>0$ such that for any $t\\in (0,1]$ and  any $0<\\epsilon<1$, we have\n$$\\int_{  X} |\\nabla \\rho_\\epsilon|^2 \\wedge  \\omega(t)^n \\leq C(-\\log \\epsilon)^{-1}  t^{n-\\kappa}. $$\n\n\\end{lemma}\n\n\\begin{proof} There exist $C_1, C_2 >0$ such that\n\\begin{eqnarray*}\n&& \\sqrt{-1} \\int_X \\partial \\rho_\\epsilon \\wedge \\overline{\\partial} \\rho_\\epsilon \\wedge  \\omega^{n-1}\\\\\n&=& 10000(\\log \\epsilon)^{-2}\\sqrt{-1} \\int_X (F')^2 \\partial \\eta_\\epsilon \\wedge \\overline{\\partial} \\eta_\\epsilon \\wedge  \\omega^{n-1}\\\\\n&\\leq & C_1  (\\log \\epsilon)^{-2} \\int_X  (- \\eta_\\epsilon) \\sqrt{-1}\\partial\\dbar \\eta_\\epsilon \\wedge  \\omega^{n-1}\\\\\n&=& C_1  (\\log \\epsilon)^{-2} \\left( \\int_X  (- \\eta_\\epsilon)(\\chi + \\sqrt{-1}\\partial\\dbar \\eta_\\epsilon) \\wedge  \\omega^{n-1} +   \\int_X \\eta_\\epsilon \\chi \\wedge   \\omega^{n-1} \\right) \\\\\n&\\leq & C_1  (-\\log \\epsilon)^{-1}   \\int_X (\\chi + \\sqrt{-1}\\partial\\dbar \\eta_\\epsilon) \\wedge  \\omega^{n-1} \\\\\n&= & C_1  (-\\log \\epsilon)^{-1}   \\int_X \\chi  \\wedge  \\omega^{n-1} \\\\\n&\\leq &C_2 (-\\log \\epsilon)^{-1} t^{n-\\kappa }.\n\\end{eqnarray*}\n\n\n\\end{proof}\n\nWe will pick one of the level set of $|\\tilde\\sigma'|^2_h$ to be the hypersurface $E$ in Lemma \\ref{gromov}.\n\n\\begin{lemma} \\label{bdyarea}  There exists $C>0$ such that for any $0<\\epsilon_0<1$ and any $t\\in (0,1]$, there exists $\\epsilon_0^2 \\leq   \\epsilon \\leq \\epsilon_0$, such that \n$$ Vol_{ \\omega(t)}(\\{|\\tilde\\sigma'|^{200}_{\\tilde h'} =   \\epsilon \\}) \\leq C(-\\log \\epsilon)^{-1\/2}   t^{n-\\kappa}  . $$\n\n\n\\end{lemma}\n\n\n\\begin{proof} We apply the co-area formula\n$$\\int_X H dg = \\int_{-\\infty}^\\infty \\int_{\\{G=u\\}} \\frac{ H}{|\\nabla G|} dg|_{G=u} d u $$\nby letting $H= |\\nabla G|$ and $G= \\rho_\\epsilon$.   Applying  the previous lemma, there exists $C>0$ such that for all sufficiently small $\\epsilon>0$ and $t\\in (0, 1]$,\n$$\\int_X |\\nabla \\rho_\\epsilon|  \\omega^n \\leq \\left(\\int_X |\\nabla \\rho_\\epsilon|^2  \\omega^n \\right)^{1\/2} \\left(\\int_X \\omega^n \\right)^{1\/2} \\leq C(-\\log\\epsilon)^{-1\/2} t^{n-\\kappa}. $$\nWe consider the region $$B_{\\epsilon_0} = \\{ \\epsilon_0^2\\leq |\\sigma|^{200}_h \\leq \\epsilon_0  \\}.$$ In $B_{\\epsilon_0}$,\n$$1\\leq \\rho_{\\epsilon_0}=F(100\\eta_{\\epsilon_0}\/\\log \\epsilon_0) \\leq 2$$\nand\n$$\\int_1^2\nVol(\\{\\rho_{\\epsilon_0} = u\\})  d u \\leq C(-\\log\\epsilon_0)^{-1\/2} t^{n-\\kappa} .$$\nBy mean value theorem, there is $a\\in[1,2]$ such that\n$$Vol(\\{\\rho_{\\epsilon_0} = a\\})   \\leq C(-\\log\\epsilon_0)^{-1\/2} t^{n-\\kappa} .$$\nIn other words,\n$$Vol(\\{\\Phi^*|\\sigma|_h^{200}=\\epsilon_0^{3-a}\\})   \\leq C(-\\log\\epsilon_0)^{-1\/2} t^{n-\\kappa} .$$\n\\end{proof}\n\nLet \n\\begin{equation}\\label{deset}\nD_\\epsilon = X \\setminus \\{ |\\tilde\\sigma'|^{200}_{\\tilde h'} <\\epsilon\\}. \n\\end{equation}\nFor sufficiently large $N>0$\n\\begin{equation}\\label{gradchi}\n\\sup_X \\left |\\partial |\\tilde \\sigma'|_{\\tilde h'} ^{2N}\\right|_{\\chi}  <\\infty\n\\end{equation}\n because there exists $C=C(N)>0$ such that\n$$ \\chi\\geq |\\tilde \\sigma' |^{2N}_{\\tilde h'} \\geq C g_A$$\nfor some fixed K\\\"ahelr metric $g_A$ on $X$. Without loss of generality, we can assume $N=100$ for simplicity.\nThe previous lemma indicates that for almost every sufficiently small $\\epsilon>0$, $\\partial D_\\epsilon$ has very small volume. The following lemma also shows that $\\{ |\\tilde \\sigma'|^{200}_h < \\epsilon\\}$ has very small volume.\n\n\\begin{lemma} \\label{smvol} For any $\\delta>0$, there exists $\\epsilon>0$ such that for all $t\\in (0,1]$,\n$$\\int_{|\\tilde\\sigma'|^{200}_{\\tilde h'} \\leq \\epsilon}  \\omega(t)^n \\leq \\delta t^{n-\\kappa}.$$\n\n\n\\end{lemma}\n\n\\begin{proof} First we notice that $\\rho_\\epsilon \\geq 1$ when $|\\sigma|^{200}_h \\leq \\epsilon$ and so\n$$\\int_{|\\tilde\\sigma'|^{200}_{\\tilde h'}  \\leq \\epsilon}  \\omega(t)^n \\leq \\int_X \\rho_\\epsilon  \\omega(t)^n. $$\nAlso $$\\lim_{\\epsilon\\rightarrow 0} \\int_{|\\tilde\\sigma'|^{200}_{\\tilde h'}  \\leq \\epsilon} \\Omega = 0 $$\nand if we let $\\theta(t) = \\chi+ t\\omega_A$,\n$$ \\int_X \\rho_\\epsilon \\theta^n \\leq t^{n-\\kappa} \\int_X \\rho_\\epsilon\\Omega$$\nand\n$$\\int_X \\rho_\\epsilon ( \\omega(t)^n - \\theta^n) = \\sum_{l=0}^{n-1} \\int_X \\rho_\\epsilon \\sqrt{-1}\\partial\\dbar \\psi \\wedge  \\omega(t)^l \\wedge \\theta^{n-1-l}.$$\nNow by similar calculations as in the proof of Lemma \\ref{cutoff1}, there exist $C_1, C_2, C_3, C_4>0$ such that\n\n\\begin{eqnarray*}\n&& \\int_X \\rho_\\epsilon \\sqrt{-1}\\partial\\dbar \\psi\\wedge    \\omega \\wedge \\theta^{n-1-l}\\\\\n&=& \\int_X \\psi \\sqrt{-1}\\partial\\dbar \\rho_\\epsilon \\wedge  \\omega \\wedge \\theta^{n-1-l} \\\\\n&=&  \\int_X \\psi  \\left( 10^2(\\log \\epsilon)^{-1}F' \\sqrt{-1}\\partial\\dbar \\eta_\\epsilon + 10^4( \\log \\epsilon)^{-2}F'' \\partial \\eta_\\epsilon \\wedge \\overline{\\partial} \\eta_\\epsilon \\right)   \\wedge  \\omega \\wedge \\theta^{n-1-l} \\\\\n&\\leq& C_1(-\\log \\epsilon)^{-1} \\int_X (\\sqrt{-1}\\partial\\dbar \\eta_\\epsilon + \\chi)    \\wedge  \\omega \\wedge \\theta^{n-1-l} + C_1(-\\log \\epsilon)^{-1} \\int_X    \\chi \\wedge  \\omega \\wedge \\theta^{n-1-l} \\\\\n&&+C_1  (\\log\\epsilon)^{-2}\\int_X   \\partial \\eta_\\epsilon \\wedge \\overline{\\partial} \\eta_\\epsilon   \\wedge  \\omega \\wedge \\theta^{n-1-l} \\\\\n&\\leq& C_2(-\\log\\epsilon)^{-1} [\\chi]\\wedge[ \\omega]^{n-1}-  C_1  (\\log\\epsilon)^{-2}\\int_X  \\eta_\\epsilon \\sqrt{-1}\\partial\\dbar \\eta_\\epsilon   \\wedge  \\omega \\wedge \\theta^{n-1-l}\\\\\n&\\leq& C_3 (-\\log\\epsilon)^{-1} [\\chi]\\cdot [ \\omega(t)]^{n-1}\\\\\n&\\leq& C_4(-\\log\\epsilon)^{-1}  t^{n-\\kappa}.\n\\end{eqnarray*}\n\nThe lemma easily follows by combining the above estimates.\n\n\\end{proof}\n\nRecall that there exists  $L>0$ such that for all $t\\in (0,1]$,\n$$ diam(X,  \\omega(t)) \\leq L. $$\n\n\\begin{lemma} \\label{distance1} For any $\\delta>0$, there exists $0<\\epsilon <\\delta$ such that for any  $t\\in (0,1]$ and any two points\n$p_1, p_2 \\in D_\\delta$, \n there exists  a smooth path $\\gamma_t \\subset D_ \\epsilon $ joining $p_1$ and $p_2$ satisfying\n %\n$$\\mathcal{L}_{g(t)}(\\gamma_t)   \\leq d_{g(t)}(p_1, p_2) + \\delta. $$\nwhere $\\mathcal{L}_{g(t)}(\\gamma_t)  $ is the arc length of $\\gamma_t$ with respect to the metric $g(t)$.\n\\end{lemma}\n\n\\begin{proof} \nLet\n$$K_{\\epsilon_1}(t)=\\{ x\\in X ~|~ d_{g(t)}(x, \\partial D_\\delta)< \\epsilon_1\\} . $$\nFor any $x\\in K_{\\epsilon_1}(t)$, there exists $x'\\in \\partial \\tilde D_\\delta$ such that\n\\begin{eqnarray*}\n|\\tilde \\sigma|_{\\tilde h}^{200}(x) &\\geq& |\\tilde \\sigma|_{\\tilde h}^{200}(x') - \\left(\\sup_X \\left|\\nabla |\\sigma|_{\\tilde h'}^{200}\\right|_{g(t)}\\right) d_{g(t)}(x, x') \\\\\n&\\geq& |\\tilde \\sigma|_{\\tilde h}^{200}(x') - C_1 \\left(\\sup_X \\left|\\nabla |\\sigma|_{\\tilde h'}^{200}\\right|_{\\chi}\\right) d_{g(t)}(x, x')\\\\\n&\\geq& \\delta- C_2 \\epsilon_1\n\\end{eqnarray*}\nby Lemma \\ref{schwa} and (\\ref{gradchi}), where $C_1, C_2$ do not depend on $\\delta$, $\\epsilon_1$ or $t$. By choosing $\\epsilon_1<<\\delta$, we have for all $t\\in (0,1)$\n$$K_{\\epsilon_1} (t)\\subset D_{\\epsilon_1}.$$\nWe  choose $\\epsilon< \\epsilon_1$ with \n$$K_{\\epsilon_1} (t) \\subset D_{\\epsilon_1} \\subset D_\\epsilon$$ and by Lemma \\ref{bdyarea}, \nand there exists $C_3>0$ such that \n$$Vol_{g(t)}(\\partial  D_\\epsilon ) \\leq C_3 (-\\log \\epsilon)^{-\\frac{1}{2}} t^{n-\\kappa} $$\nfor all $t\\in (0,1)$.\n\nBy Corollary \\ref{volr}, there exists $c>0$ such that for all $t\\in (0,1)$ and $r<1$\n$$Vol_{  g (t)} (B_{  g (t)}(p_i, r) \\geq  c_1 r^{2n}  t^{n-\\kappa} $$\nand so \n$$Vol_{  g (t)} (B_{  g (t)}(p_i, \\epsilon_1)  \\geq c_1 (\\epsilon_1)^{2n} t^{n-\\kappa}, ~ i=1, 2.$$\n\nSince $B_{  g (t)}(p_i, \\epsilon_1) \\subset\\subset   D_\\epsilon$, $i=1, 2$, we can apply Lemma \\ref{gromov} by choosing sufficiently small $\\epsilon>0$ with \n$$ (-\\log \\epsilon)^{\\frac{1}{2}}  (\\epsilon_1)^{2n} >>1$$\nand letting $E = D_\\epsilon$ and $r=\\epsilon_1$. \nHence  for all $t\\in (0, 1)$, there exist $q\\in B_{g(t)}(p_2, \\epsilon)$ and a minimal geodesic $\\hat\\gamma_t \\subset D_\\epsilon$ (with respect to $g(t)$) joining $p_1$ and  $q$ and\n$$ \\mathcal{L}_{g(t)}(\\hat\\gamma_t) =d_{g(t)}(p_1, q) \\leq d_{g(t)} (p_1, p_2) + \\epsilon.$$\nNow we can complete the proof of the lemma by letting $\\gamma_t$ be the curve combining $\\hat\\gamma_t$ and a minimal geodesic joining $p_2$ and $q$ because\n$$\\mathcal{L}_{g(t)}(\\gamma_t) \\leq d_{g(t)}(p_1, p_2) + 2\\epsilon  \\leq d_{g(t)}(p_1, p_2) + 2\\delta. $$\n\n\\end{proof}\n\n\nWe will also need the $C^0$ regularity of metrics. Let $\\omega_{can} = \\chi+\\sqrt{-1}\\partial\\dbar \\psi_{can}$ be the twisted K\\\"ahler-Einstein metric on $X_{can}$\n$$Ric(\\omega_{can})= -\\omega_{can} + \\omega_{WP}. $$\nWe now define a semi-flat closed $(1,1)$-current on $X$ introduced in \\cite{ST1, ST2} by the following\n$$\\omega_{SF} = \\omega_A + \\sqrt{-1}\\partial\\dbar \\phi_{SF}$$ such that for any $z \\in X_{can}^{\\circ}$,\n$$Ric(\\omega_{SF} |_{X_z}) =0, ~~ \\int_{X_z} \\phi_{SF} \\omega_0^{n-\\kappa} |_{X_z}=0. $$\n\nThe following lemma is due to \\cite{FGS, TWY}.\n\n\n\\begin{lemma} \\label{equivalence} For any $\\epsilon>0$, there exists $h_\\epsilon (t)\\geq 0$ with $\\lim_{t\\rightarrow 0} h_\\epsilon(t) =0$ such that for all $t\\in (0,1] $, we have on $D_\\epsilon$,\n$$ (1-h_\\epsilon(t)) (\\Phi^*\\omega_{can} + t \\omega_{SF}) \\leq        \\omega(t) \\leq (1+h_\\epsilon(t)) (\\Phi^*\\omega_{can} + t \\omega_{SF}) .$$\nIn particular, $\\omega_t$ converges in $C^0$-topology on $\\Phi^{-1}(X_{can}^\\circ)$ to $\\Phi^*\\omega_{can}$ as $t\\rightarrow 0^+$.\n\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNow we are ready to prove the main result of this section.\n\n\n\\begin{proposition} \\label{convex1} For any $\\delta>0$, there exist $\\epsilon_1 > \\epsilon_2 >0$ such that for all $t\\in (0,1]$,\n\n\\begin{enumerate}\n\n\\item $Vol( X\\setminus D_{\\epsilon_1},  \\omega(t)) < \\delta$.\n\n\\item for any two points $p, q \\in D_{ \\epsilon _1}$, there exists a continuous path $\\gamma_t \\subset D_{\\epsilon _2}$ joining $p$ and $q$ such that\n$$\\mathcal{L}_{g(t)}(\\gamma_t)   \\leq d_{ \\omega(t)}(p, q) + \\delta \\leq L+\\delta.$$\n\n\\end{enumerate}\n\n\\end{proposition}\n\n\n\\begin{proof}   For any $\\delta>0$, we can always choose $\\epsilon_1$ sufficiently small so that the first estimate in the proposition holds by Lemma \\ref{smvol}.  The second statement follows directly by Lemma \\ref{distance1} after choosing both $\\epsilon_1$ and $\\epsilon_2$ sufficiently small.\n\\end{proof}\n\n\n\nProposition \\ref{convex1} has many geometric consequences. We will use Proposition \\ref{convex1} to prove  Theorem \\ref{main2}. The following proposition proves the first three statements in Theorem \\ref{main2}.\n\n\\begin{proposition} \\label{contpro}The following hold.\n\\begin{enumerate}\n\n\\item $(X, g(t))$ converges in Gromov-Hausdorff topology to a compact metric space $(\\mathcal{Z}, d_\\mathcal{Z})$ as $t\\rightarrow 0^+$,\n\n\n\\item  $\\omega_t$ converges in $C^0$-topology on $\\Phi^{-1}(X_{can}^\\circ)$ to the pullback of a smooth K\\\"ahler metric $g_{can}$ on $X_{can}^\\circ$ as $t\\rightarrow 0^+$,\n\n\\item the metric completion of $(X_{can}^\\circ, g_{can})$ is isomorphic to $(\\mathcal{Z}, d_\\mathcal{Z})$.\n\n\\end{enumerate}\n\n\n\n\\end{proposition}\n\n\n\\begin{proof} It is proved in \\cite{FGS} that for any sequence $t_j\\rightarrow 0$, $(X, g(t_j))$ converges in Gromov-Hausdorff topology to a compact metric space $(\\mathcal{Z}, d_{\\mathcal{Z}})$  after possibly passing to a subsequence and so (1) follows.\n\n(2) follows directly from Lemma \\ref{equivalence}.\n\nIn order to prove (3), we again apply the result of  \\cite{FGS}   that $(X_{can}^\\circ, g_{can})$ can be locally isometrically embedded into $(\\mathcal{Z}, d_{\\mathcal{Z}})$ as a dense open set in $\\mathcal{Z}$. For any two points $p, q \\in X_{can}^\\circ$, we choose arbitary $p'$ and $q'$  in the fibre $X_p=\\Phi^{-1}(p) $ and $X_q=\\Phi^{-1}(q)$. By Proposition \\ref{convex1}, for any $\\delta>0$, there exists $\\epsilon >0$ such that for all $t_j $,\n %\n %\n$$p', q' \\in  D_{\\epsilon}  $$\nand there exists a continuous path $\\gamma_{t_j} \\subset D_{\\epsilon}$ joining $p'$ and $q'$ such that\n$$\\mathcal{L}_{g(t_j)}(\\gamma_{t_j})  \\leq d_{ g(t_j)}(p', q') + \\delta. $$\nSince the fibre diameter uniformly tends to $0$ away from singular fibres and $g_{t_j}$ converges uniformly in $C^0$ to $g_{can}$ on $D_\\epsilon$, $p$ and $g$ must converge in Gromov-Hausdorff distance to $p$ and $q$ on $(\\mathcal{Z}, d_\\mathcal{Z})$. Then there exists $J>0$ such that for all $j\\geq T$, we have \n$$\\mathcal{L}_{g(t_j)}(\\gamma_{t_j})    \\leq d_\\mathcal{Z}(p, q) + 2\\delta.$$\nWe can also assume for all $j\\geq J$, we have \n$$g_{can} \\leq (1+\\delta)g_{t_j} $$\non $D_\\epsilon$ by the $C^0$-estimate and convergence of $g(t_j)$ to $g_{can}$.\nTherefore for $j\\geq J$, we have\n\\begin{eqnarray*}\nd_{g_{can}|_{X_{can}^\\circ}}(p, q) &\\leq&  |\\Phi(\\gamma_{t_j})|_{g_{can}}  \\leq  (1+\\delta) \\mathcal{L}_{g(t_j)}(\\gamma_{t_j})   \\\\\n&\\leq & (1+\\delta) (d_\\mathcal{Z}(p, q) + 2\\delta)\\\\\n&\\leq& d_\\mathcal{Z}(p, q)  +  \\delta(diam(\\mathcal{Z}, d_\\mathcal{Z}) + 2+2\\delta)\n\\end{eqnarray*}\nwhere $d_{g_{can}|_{X_{can}^\\circ}}$ is the distance function on $X_{can}^\\circ$ induced by $g_{can}|_{X_{can}^\\circ}$.  By letting $\\delta \\rightarrow 0$, we have on $X_{can}^\\circ$,\n\\begin{equation}\\label{d1side}\nd_{g_{can}|_{X_{can}^\\circ}} \\leq d_\\mathcal{Z}.\n\\end{equation}\nOn the other hand, for any $\\delta>0$, by definition there exists a continuous path $\\gamma_\\delta$  in $X_{can}^\\circ$ such that\n$$ \\left| \\mathcal{L}_{g_{can}} (\\gamma_\\delta) -  d_{g_{can}|_{X_{can}^\\circ}} (p, q) \\right|< \\delta, $$\nwhile for sufficiently large $j>0$,\n$$\\mathcal{L}_{g_{can}} (\\gamma_\\delta)   \\geq  \\mathcal{L}_{g(t_j)} (\\gamma'_\\delta)- \\delta \\geq d_{ g(t_j)}(p', q') - \\delta \\geq d_\\mathcal{Z}(p, q) - 2\\delta,$$\nwhere $\\gamma'_\\delta$ is a lift of $\\gamma_\\delta$ with $p'$ and $q'$ as the end points.\nTherefore we have on $X_{can}^\\circ$,\n$$d_{g_{can}|_{X_{can}^\\circ}} \\geq d_\\mathcal{Z}$$\nand so combined with (\\ref{d1side}), we have $$d_{g_{can}|_{X_{can}^\\circ}}  = d_\\mathcal{Z}|_{X_{can}}. $$\nIt immediately implies that the identity map on $X_{can}^\\circ$ induces a Lipschitz map\n$$\\mathcal{F}: (\\mathcal{Z}, d_\\mathcal{Z}) \\rightarrow (Z_{can}, d_{Z_{can}}), $$\nwhere $(Z_{can}, d_{Z_{can}})$ is the metric completion of $(X_{can}^\\circ, g_{can})$.\n\nSince $X_{can}^\\circ$ is a dense open set in $\\mathcal{Z}$, for any point $z\\in \\mathcal{Z}$, there exist a sequence $z_i \\in X_{can}^\\circ$ converging to $z$ with respect to $d_\\mathcal{Z}$. Since $d_{\\omega_{can}|_{X_{can}^\\circ}}  = d_\\mathcal{Z} $ on $X_{can}^\\circ$, $z_j$ must also converge in $(Z, d_Z)$ and so $\\mathcal{F}$ must be injective. Same argument implies $\\mathcal{F}$ is also surjective. This completes the proof of the proposition.\n\n\\end{proof}\n\n\nWe have completed the proof for  the statements (1), (2), (3) of Theorem \\ref{main2} by Proposition \\ref{contpro}. \n\n\n\n\nNow we will prove the last part of Theorem \\ref{main2}. First, we state the following H\\\"older estimate in \\cite{Kol2} for complex Monge-Amp\\`ere equations on K\\\"ahler orbifolds.\n\n\\begin{lemma}\\label{holder1}Let $X$ be an $n$-dimensional K\\\"ahler orbifold. Let $\\omega_{orb}$ be an orbifold K\\\"ahler metric on $X$ and $\\Omega$ be an orbifold volume form on $X$.  We consider the following complex Monge-Amp\\`ere equation\n$$(\\omega_{orb} + \\sqrt{-1}\\partial\\dbar\\varphi)^n = F \\Omega, $$\nwhere $F$ is a non-negative function with $\\int_X F \\Omega = \\int_X \\omega_{orb}^n$.\nIf $||F||_{L^p(X, \\Omega)} <\\infty$ for some $p>1$, then there exist $\\alpha=\\alpha(X, p) \\in (0,1)$ and $C=C(X, p, \\omega_{orb},  ||F||_{L^p(X, \\Omega)})>0$ such that\n$$||\\varphi -\\sup_X \\varphi||_{C^\\alpha(X, \\omega_{orb})} \\leq C. $$\n\n\n\\end{lemma}\n\n\nThe following lemma is a slight orbifold generalization of a distance estimate obtained in \\cite{Li}.\n\n\n\\begin{lemma} \\label{holder2} Let $X$ be an $n$-dimensional K\\\"ahler orbifold. Let $\\omega_{orb}$ be an orbifold K\\\"ahler metric on $X$. Suppose $\\omega \\in [\\omega_{orb}]$ is an orbifold K\\\"ahler metric on $X$ satisfying\n$$\\left\\|\\frac{\\omega^n }{\\omega_{orb}^n} \\right\\|_{L^p(X, \\omega_{orb})} <\\infty. $$\nThen there exist $\\alpha=\\alpha(X, p)\\in (0,1)$ and $C=C\\left(X, p, \\omega_{orb},  \\left\\|\\omega^n \/\\omega_{orb}^n \\right\\|_{L^p(X, \\omega_{orb})} \\right)$ such that\n$$ d_{\\omega}(p, q) \\leq  Cd_{\\omega_{orb}}(p, q)^\\alpha $$\nfor any two points $p, q \\in X$.\n\\end{lemma}\n\n\n\\begin{proof} The proof follows the argument of \\cite{Li}. We include the argument for the sake of completeness since it is fairly short and effective.\n\n\nFor any $p\\in X$ we consider the distance function $f(z)=d_{\\omega}(p,z)$. Let $\\pi:\\tilde B\\rightarrow B$ be a local orbifold uniformization at $p$ on a metric ball $B=B_{\\omega_{orb}}(p,4r_0)$ for some uniform radius $r_0$. Let $\\tilde{\\omega}_{orb}=\\pi^*\\omega_{orb}$.  Denote $\\tilde B=\\tilde B_{\\tilde{orb}}(\\tilde{p},4r_0)$ the lifting metric ball and $\\tilde f(\\tilde z)=d_{\\tilde \\omega_{orb}}(\\tilde p,\\tilde z)$ the distance function accordingly.\n\n\n\n\nFor any $\\tilde{q}\\in \\tilde B_{\\tilde{\\omega}_{orb}}(\\tilde{p},r_0)$ and radius $r\\le r_0$, we define a cut-off function $\\tilde\\rho_r$ via\n$$\\tilde\\rho_r = F(d_{\\tilde\\omega_{orb}}(\\tilde q, \\cdot)\/r)$$\nwhere $d_{\\tilde\\omega_{orb}}$ is the distance function with respect to $\\tilde\\omega_{orb}$ and $F$ is one smooth nonnegative cut-off function with $F(x)=1$ for $x\\in [0, 1]$ and $F(x)=0$ for $x\\geq 2.$ Then $-C_1 r^{-2} \\tilde \\omega_{orb}\\leq \\sqrt{-1}\\partial\\dbar \\tilde \\rho_r \\leq C_1 r^{-2} \\tilde \\omega_{orb}$ in $\\tilde B_{\\tilde{\\omega}_{orb}}(\\tilde{q},2r)\\subset \\tilde B$ for some fixed $C_1>0$, and for $r<r_0$ there exist $C_2, C_3, C_4 >0 $ such that\n\\begin{eqnarray*}\n\\int_{\\tilde B_{\\tilde\\omega_{orb}}(\\tilde q, r)} tr_{\\tilde \\omega_{orb}}(\\tilde \\omega) \\tilde\\omega_{orb}^n &\\leq & \\int_{\\tilde B_{\\tilde{\\omega}_{orb}}(\\tilde{q},2r)} \\tilde \\rho_r tr_{\\tilde \\omega_{orb}}(\\tilde \\omega)\\tilde \\omega_{orb}^n\\\\\n&=&\\int_{\\tilde B_{\\tilde{\\omega}_{orb}}(\\tilde{q},2r)} \\tilde \\rho_r \\tilde \\omega_{orb}^n + n\\int_{\\tilde B_{\\tilde{\\omega}_{orb}}(\\tilde{q},2r)} \\tilde \\rho_r \\sqrt{-1}\\partial\\dbar \\tilde \\varphi \\wedge \\tilde \\omega_{orb}^{n-1}\\\\\n&\\leq& C_2 r^{2n} + n\\int_{\\tilde B_{\\tilde{\\omega}_{orb}}(\\tilde{q},2r)}(\\tilde\\varphi -\\tilde\\varphi(\\tilde p)) \\sqrt{-1}\\partial\\dbar \\tilde \\rho_r  \\wedge \\tilde \\omega_{orb}^{n-1}\\\\\n&\\leq& C_2 r^{2n} + C_3r^{-2+\\alpha}\\int_{\\tilde B_{\\tilde\\omega_{orb}}(\\tilde p, 2r)} \\omega_{orb}^n\\\\\n&\\leq& C_4 r^{2n-2+\\alpha}.\n\\end{eqnarray*}\nObviously, $|\\nabla\\tilde f|^2_{\\tilde \\omega} =1$. Therefore $|\\nabla\\tilde  f |_{\\tilde \\omega_{orb}} \\leq tr_{\\tilde\\omega_{orb}}(\\tilde\\omega).$ It follows that for all $r\\leq r_0$, we have\n$$\\int_{\\tilde B_{\\tilde \\omega_{orb}}( \\tilde q, r)} |\\nabla \\tilde f|^2_{\\tilde\\omega_{orb}} \\tilde \\omega_{orb}^n  \\leq C_4 r^{2n-2+\\alpha}. $$\nThen by Morrey's embedding theorem and the fact that $\\tilde f(\\tilde p)=0$, we have\n$$|\\tilde f|_{C^{\\alpha\/2}(B_{\\tilde\\omega_{orb}}(\\tilde p, \\beta r_0))} \\leq C_5$$\nfor some fixed $C_5>0$. In particular, we have\n$$d_{\\tilde{\\omega}}(\\tilde p,\\tilde q)\\le C_5 d_{\\tilde \\omega_{orb}}(\\tilde p,\\tilde q)^{\\alpha\/2}$$\nfor all $\\tilde q\\in \\tilde B_{\\tilde \\omega_{orb}}(\\tilde p,r_0). $\nThe corresponding distance function on $X$ satisfies\n$$d_\\omega(p,q)\\le d_{\\tilde{\\omega}}(\\tilde p,\\tilde q)\\le C_5 d_{\\tilde \\omega_{orb}}(\\tilde p,\\tilde q)^{\\alpha\/2} =C_5 d_{\\omega_{orb}}(p,q)^{\\alpha\/2}$$\nfor all $q\\in B_{\\omega_{orb}}(p,r_0).$ \nThis completes the proof for the lemma.\n\\end{proof}\n\n\n\\begin{proposition} \\label{contpro2} If $X_{can}$ has only orbifold singulairties, then $(\\mathcal{Z}, d_\\mathcal{Z})$ is homeomorphic to $X_{can}$.\n\n\\end{proposition}\n\n\\begin{proof} We will break the proof into the following steps.\n\n\\noindent {\\bf Step 1.}  Let $\\omega_{orb}$ be a smooth orbifold K\\\"ahler metric on $X_{can}$ in the same class of $\\omega_{can}$.  Let $F=-\\log \\left( \\frac{\\omega_{can}^\\kappa}{(\\omega_{orb})^\\kappa} \\right)$. We claim that $F$ is bounded above. For simplicity, we assume $X_{can}$ is smooth. We let $\\omega(t)$, $\\chi$, $\\omega_A$ and $\\psi(t)$ be defined as in (\\ref{contin1}). We then consider the following quantity\n$$H= \\log tr_{\\omega(t)}(\\chi) - B \\psi(t) $$\nfor $t\\in (0, 1]$ and some sufficiently large $B>0$ to be determined. Straightforward calculations show that there exist $C_1, C_2>0$ such that for all $t\\in (0, 1]$\n$$\\Delta_t \\log H \\geq (B-C_1)tr_{\\omega(t)}(\\chi) - C_2,$$\nusing the fact that $\\psi(t)$ is uniformly bounded in $L^\\infty(X)$ for $t\\in (0,1]$ and the curvature of $\\chi$ is uniformly bounded, where $\\Delta_t$ is the Laplace operator associated to $\\omega(t)$. After applying the maximum principle, we conclude that $H$ is uniformly bounded above and so there exists $C_3>0$ such that\n$$tr_{\\omega(t)}(\\chi) \\leq C_3$$\nand we immediately obtain the upper bound for $F$ by letting $t\\rightarrow 0$ and using the mean value inequality since $\\omega(t)$ converges to $\\omega_{can}$ on $X^\\circ$ and $\\chi$ is equivalent to $\\omega_{orb}$. \n\n\nLet $\\Omega$ be the smooth volume form on $X$ such that $$\\sqrt{-1}\\partial\\dbar\\log \\Omega = \\chi \\in [K_X]$$\nas before.\nIf we let $\\omega_{can} = \\chi + \\sqrt{-1}\\partial\\dbar \\varphi_{can}$,\n$$\\omega_{can}^\\kappa= (\\chi + \\varphi_{can})^\\kappa = e^{\\varphi_{can}}\\Phi_*\\Omega= e^{-F} (\\omega_{orb})^\\kappa$$\nfor some $\\varphi_{can}\\in PSH(X_{can}, \\chi)\\cap C^0(X_{can})$, where $\\Phi_*\\Omega$ is the pushforward or fibre-integration of $\\Omega$. In particular, $\\varphi_{can}$ is smooth on $X_{can}^\\circ$.\nBy \\cite{ST2}, there exists $p>1$ such that $$||e^{-F}||_{L^p(X_{can}, \\omega_{orb})}<\\infty .$$\nAlso we have$$\\sqrt{-1}\\partial\\dbar F = - \\omega_{can} +\\omega_{WP} - Ric(\\omega_{orb}) .$$\nwhere the Weil-Petersson current $\\omega_{WP}$ is a  $(1,1)$-current from the variation of the complex structure of smooth Calabi-Yau fibres. Since $\\omega_{WP}$ is semi-positive on $X_{can}^\\circ$, we have\n$$Ric(\\omega_{orb})+\\omega_{can}+\\sqrt{-1}\\partial\\dbar F \\geq 0$$\n on $X_{can}^\\circ$ and $\\omega_{can}$ has bounded local potentials. This implies that $F$ is quasi-plurisubharmonic on $X_{can}^\\circ$ with respect to $Ric(\\omega_{orb})+\\omega_{can}$, by extension theorem of quasi-plurisubharmonic function, $F$ must be also quasi-plurisubharmonic on $X_{can}$ since $F$ is bounded above and $X_{can}\\setminus X_{can}^\\circ$ is an analytic subvariety of $X_{can}$, i.e., \n$$F\\in PSH(X_{can}, Ric(\\omega_{orb})+\\omega_{can}).$$ \nIn particular, it implies that $\\omega_{WP}$ is a global semi-positive current on $X_{can}$. In fact, the semi-positivity of $\\omega_{WP}$ holds in general for any canonical models (\\cite{GS}). By the standard approximation theory for plurisubharmonic functions, there exist a sequence orbifold smooth functions. $F_j \\in PSH(X_{can}, B\\omega_{orb}+\\omega_{can})$ for some fixed constant $B>0$ such that $F_j$ converges to $F$ decreasingly. Then immediately, we have\n$$||e^{-F_j}||_{L^p(X_{can}, \\omega_{orb})}\\leq ||e^{-F}||_{L^p(X_{can}, \\omega_{orb})}<\\infty. $$\nWe then consider the following complex Monge-Amp\\`ere equations\n$$(\\omega_{orb}+\\sqrt{-1}\\partial\\dbar \\varphi_j)^\\kappa = c_j e^{-F_j} (\\omega_{orb})^\\kappa, $$\nwhere $c_j$ is the norming constant with\n$$[\\omega_{orb}]^\\kappa = c_j \\int_{X_{can}} e^{-F_j} (\\omega_{orb})^\\kappa. $$\n$c_j$ is uniformly bounded below because $F_j$ decreases to $F_j$ and it is also uniformly bounded above because $F_j$ are uniformly bounded from above.  Therefore \n$$||c_j e^{-F_j}||_{L^p(X_{can}, \\omega_{orb})}$$ is uniformly bounded above for all $j$.\n\n\n\\noindent {\\bf Step 2.} By Lemma \\ref{holder1}, there exist $\\alpha$ and  $C_1>0$ such that for all $j$,\n$$||\\varphi_j||_{C^\\alpha (X_{can}, \\omega_{orb})} \\leq C_1. $$\nand by Lemma \\ref{holder2} that there exists $C_2$ such that on $X$,\n$$d_{g_j}(\\cdot, \\cdot)  \\leq C_2 d_{g_{orb}}(\\cdot, \\cdot)^\\alpha.$$\nWe then apply the maximum principle to the following quantity for sufficiently large $K>0$\n$$H = \\log tr_{\\omega_j}(\\omega_{orb}) - K \\varphi_j,$$\nwhere $\\omega_j = \\omega_{orb}+ \\varphi_j$. \nThen similarly to the Schwarz lemma, there exists $C_3>0$ such that for all $j$,\n$$tr_{\\omega_j}(\\omega_{orb})\\leq C_3, $$\nor equivalently,\n$$\\omega_j \\geq (nC_3)^{-1} \\omega_{orb}$$\nsince the curvature of $\\omega_{orb}$ is bounded and $\\varphi_j$ is uniformly bounded in $L^\\infty$ for all $j$.\n\nIn particular, this implies that there exists $C_4, C_5>0$ such that\n$$Ric(\\omega_j) = \\sqrt{-1}\\partial\\dbar F_j +Ric(\\omega_{orb}) \\geq - C_4 \\omega_{orb} \\geq -  C_5\\omega_j.$$\nTherefore the Ricci curvature of $g_j$ are uniformly bounded below. Let $g_j$ and $g_{orb}$ be the orbifold K\\\"ahler metrics associated to $\\omega_j$ and $\\omega_{orb}$. We can apply the result of \\cite{FGS} and there exists $D>0$ such that\n$$diam_{g_j}(X_{can}) \\leq D. $$\n\n\n\n\\noindent {\\bf Step 3.} Let $\\sigma$ be an effective $\\mathbb{Q}$-divisor in the class $[\\omega_{can}]$ such that $X_{can}\\setminus X_{can}^\\circ$ is contained in the support of $\\sigma$ and $h$ is the smooth hermitian metric on $[\\sigma]$ with $Ric(h) = \\omega_{orb}$. We define $D_\\epsilon = \\{ |\\sigma|^2_h < \\epsilon \\}.$  The same argument in   Proposition \\ref{convex1} implies that for for any $\\delta>0$, there exist $\\epsilon_1 > \\epsilon_2 >0$ such that for all $j\\geq 0$,\n\n\\begin{enumerate}\n\n\\item $Vol( X\\setminus D_{\\epsilon_1}, g_j) < \\delta$.\n\n\\item for any two points $p, q \\in D_{ \\epsilon _1}$, there exists a continuous path $\\gamma_j \\subset D_{\\epsilon _2}$ joining $p$ and $q$ such that\n$$\\mathcal{L}_{g_j}(\\gamma_j)\\leq d_{g_j}(p, q) + \\delta \\leq D+\\delta. $$\n\n\\end{enumerate}\n\n\n\n\n\n\n\nWe apply  Lemma \\ref{holder2} again. There exist $\\alpha\\in (0,1)$ and $C_\\alpha>0$ such that for any $j$ and any two points $p, q\\in X_{can}$\n$$d_{g_j}(p, q) \\leq C_\\alpha d_{g_{orb}}(p, q)^\\alpha. $$\nAfter letting $j\\rightarrow \\infty$, $g_j$ converges to $g_{can}$ smoothly on $X_{can}^\\circ$\nand for any $p, q\\in X_{can}^\\circ$, we have\n$$d_{g_{can}|_{X_{can}^\\circ}}(p, q) \\leq C_\\alpha d_{g_{orb}}(p, q)^\\alpha, $$\nwhere $d_{g_{can}|_{X_{can}^\\circ}}$ is the distance function on $X_{can}^\\circ$ induced by $g_{can}$.\n\n\n\n\nSince $g_{can}\\geq cg_{orb}$ for some $c>0$ on $X_{can}^\\circ$ and $(\\mathcal{Z}, d_\\mathcal{Z})$ is the metric completion of $(X_{can}^\\circ, \\omega_{can})$,   the local isometry map from $X_{can}^\\circ$ into $\\mathcal{Z}$ extends to a surjective Lipschitz map\n$$\\mathcal{F}: (\\mathcal{Z}, d_\\mathcal{Z}) \\rightarrow (X_{can}, g_{orb}).$$\nWe claim that $\\mathcal{F}$ must be injective. For any point $z\\in X_{can}$, there exist a sequence $\\{z_i\\} \\in X_{can}^\\circ$ converging to $z$ with respect to $g_{orb}$. The inequality\n$$d_{g_{can}|_{X_{can}^\\circ}}(\\cdot, \\cdot) \\leq C \\left(d_{g_{orb}}(\\cdot, \\cdot)|_{X_{can}^\\circ }\\right)^\\alpha$$ implies that\n$\\{z_i\\}$ is also a Cauchy sequence in $(X_{can}^\\circ, g_{can})$. Since $(\\mathcal{Z}, d_\\mathcal{Z})$ is the metric completion of $(X_{can}^\\circ, g_{can})$, $\\{z_i\\}$ must also be a Cauchy sequence of $(\\mathcal{Z}, d_\\mathcal{Z})$ and so $\\mathcal{F}$ must also be injective. This completes the proof of the proposition.\n\n\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{main3} }\nIn this section, we will prove Theorem \\ref{main3} by the same argument in the proof of Theorem \\ref{main2}.\nLet $X$ be an $n$-dimensional projective K\\\"ahler manifold of $c_1(X)=0$.  Suppose $L$ is a semi-ample line bundle and the linear system $|mL|$ induces a holomorphic map\n$$\\Phi: X \\rightarrow Y$$\nfor sufficiently large $m$. We assume $0< \\dim_{\\mathbb{C}}Y=k <n$. The general fibre of $X$ over $Y$ is a smooth Calabi-Yau manifold of dimension $n-\\kappa$. Let $Y^\\circ$ be the set of all smooth points of $Y$ over which $\\Phi$ is submersion.  Let $A$ be an ample line bundle over $X$ then by Yau's theorem, there exists a unique Ricci-flat K\\\"ahler metric $\\omega(t) \\in [tA+L]$.\nIn other words, let $\\Omega$ be a smooth volume form on $X$ with\n$$\\sqrt{-1}\\partial\\dbar\\log \\Omega=0,~ \\int_X \\Omega =1,$$ then\n$$\\omega(t)^n = c_t \\Omega, $$\nwhere $c(t)$ is the normalization constant defined by $c_t = [tA+L]^n$. In particular, $c_t$ is uniformly bounded above and below away from $0$ for all $t\\in (0, 1]$.\n\nLet $\\omega_A$ be a fixed K\\\"ahler form in $[A]$ and $\\chi$ be pullback of the Fubini-Study metric on $Y$ from the projective embedding of the linear system $|mL|$. Then we have\n$$t^{-(n-\\kappa)} (t \\omega_A + \\chi + \\sqrt{-1}\\partial\\dbar\\varphi(t))^n =c_t\\Omega.$$\n\nIt is shown in \\cite{ST1, ST2} that $\\omega(t)= t \\omega_A + \\chi + \\sqrt{-1}\\partial\\dbar\\varphi(t)$ converges weakly as $t\\rightarrow 0$ to the twisted K\\\"ahler-Einstein metric $\\omega_Y$ on $Y$ defined in \\cite{ST1, ST2} is given by\n$$Ric(\\omega_Y) = \\omega_{WP},$$\nwhere $\\omega_{WP}$ is the Weil-Petersson current from the variation of the complex structures of the Calabiy-Yau fibres over $Y$. More precisely, $\\omega_Y = \\chi + \\sqrt{-1}\\partial\\dbar\\varphi_Y$ for some $\\varphi_Y\\in PSH(Y, \\chi)\\cap C^0(Y)$ and\n$$\\omega_Y^\\kappa= (\\chi+ \\sqrt{-1}\\partial\\dbar \\varphi_Y)^\\kappa = c_0\\Phi_* \\Omega,$$\nwhere $\\Phi_*\\Omega$ is the pushforward of $\\Omega$, or the integration over the fibres.  In particular, $\\omega_Y$ is smooth on $Y^\\circ$. The following proposition is proved by exactly the same argument in the proof of Proposition \\ref{contpro}.\n\n\\begin{proposition}\nLet $\\omega_t= t\\omega_A + \\chi + \\sqrt{-1}\\partial\\dbar\\varphi(t)$ and $g(t)$ be the associated K\\\"ahler metric on $X$. Then the following hold.\n\\begin{enumerate}\n\n\\item $(X, g_t)$ converges in Gromov-Hausdorff topology to a compact metric space $(\\mathcal{Z}, d_\\mathcal{Z})$ as $t\\rightarrow 0^+$,\n\n\\item  $\\omega_t$ converges in $C^0$-topology on $\\Phi^{-1}(Y^\\circ)$ to the pullback of a smooth K\\\"ahler metric $\\omega_Y$ on $Y^\\circ$ as $t\\rightarrow 0^+$,\n\n\\item the metric completion of $(Y^\\circ, \\omega_{orb})$ is isomorphic to $(\\mathcal{Z}, d_\\mathcal{Z})$.\n\n\\end{enumerate}\n\n\n\n\n\\end{proposition}\n\n\n\nThe same argument in Proposition \\ref{contpro2} also gives the following proposition.\n\\begin{proposition} If $Y$ has only orbifold singulairties, then $(\\mathcal{Z}, d_\\mathcal{Z})$ is homeomorphic to $X_{can}$.\n\n\\end{proposition}\n\nWe have now proved Theorem \\ref{main3} by combining the above two propositions.\n\n\n\\section{Proof of Theorem \\ref{main1} for long time collapsing solutions of the K\\\"ahler-Ricci flow }\n\nIn this section, we will prove Theorem \\ref{main1} using the results in Section 2 to derive a uniform diameter estimate for the normalized K\\\"ahler-Ricci flow on projective manifolds with semi-ample canonical bundle. As before, $X$ is an $n$-dimensional projective manifold with semi-ample $K_X$ and its canonical model $X_{can}$ has dimension $0<\\kappa <n$. We will keep the same notations as in section 2. Let $\\omega_0$ be any initial K\\\"ahler form associated to $g_0$.\nThen the normalized K\\\"ahler-Ricci flow (\\ref{krflow}) on $X$ %\nstarting with $g_0$ can be expressed as\n \\begin{equation}\\label{maflow}\n\\left\\{\n\\begin{array}{l}\n{ \\displaystyle \\ddt{\\varphi} = \\log \\frac{ e^{(n-\\kappa)t} (\\chi + e^{-t} (\\omega_0 - \\chi) +\\sqrt{-1}\\partial\\dbar \\varphi)^n}{\\Omega} - \\varphi,}\\\\\n\\\\\n\\varphi(0) = 0.\n\\end{array} \\right.\n\\end{equation}\nIn particular, the K\\\"ahler class of $[\\omega(t)]$ associated to $g(t)$ satisfies\n\\begin{equation}\\label{classcon}\n[\\omega(t)] = [\\chi] + e^{-t}[\\omega_0 - \\chi]\n\\end{equation}\nfor all $t\\in [0, \\infty)$. \n\nThe following lemma is obtained in \\cite{ST1, ST2, SoT4}.\n\\begin{lemma} \\label{0est} There exists $C>0$ such that for all $t\\geq 0$, we have\n$$||\\varphi ||_{L^\\infty(X)} + \\left|\\left| \\ddt{\\varphi} \\right|\\right|_{L^\\infty(X)}\\leq C. $$\nFurthermore,  for any compact set $K \\subset X^\\circ$, $\\varphi$ converges in $C^{1, \\alpha}$ to $\\varphi_{can}$ for any $\\alpha \\in [0, 1)$.\n\n\\end{lemma}\n\n\n\nThe following lemma  is proved in \\cite{TWY} for the local $C^0$-convergence of the evolving collapsing metrics. \n\\begin{lemma}   Let $\\omega(t)$ be the solutions of the K\\\"ahler-Ricci flow (\\ref{krflow}). Then the following hold.\n\n\\begin{enumerate}\n\n\\item For any compact subset $K \\subset X^\\circ$,\n\n$$\\lim_{t\\rightarrow \\infty} ||\\omega(t) - \\omega_{can}||_{C^0(K, \\omega_0)} =0. $$\n\n\\item For any compact subset $K' \\subset X_{can}^\\circ$, the fibre metric over any point in $K'$ converges after rescaling to a Ricci-flat K\\\"ahler metric uniformly. More precisely, let $\\omega_{CY, y}$ be the unique Ricci-flat K\\\"ahler metric in the K\\\"ahler class $[\\omega_0|_{X_y}$. Then\n$$\\lim_{t\\rightarrow \\infty} \\sup_{y\\in K'} || e^t \\omega(t)|_{X_y} - \\omega_{CY, y}||_{C^0(X_y, \\omega_0|_{X_y}} = 0. $$\n\n\n\n\\end{enumerate}\n\n\n\n\\end{lemma}\n\nThe uniform bound for the scalar curvature of global solutions of the normalized K\\\"ahler-Ricci flow is established in \\cite{SoT4} as in the following lemma. \n\\begin{lemma} The scalar curvature $R$ of $g(t)$ is uniformly bounded, i.e., there exists $C>0$ such that for all $t\\geq 0$,\n\n$$\\sup_{X} |R(\\cdot, t)| \\leq C. $$\n\n\n\\end{lemma}\n\nWe now state the relative volume comparison established in \\cite{TiZ3}.\n\n\\begin{lemma} \\label{rvolcom}For any $B\\geq 1$, there exists $k=k(m, B)>0$ such that the following hold. Let $g(t)$ be a solution to the Ricci flow on a compact $n$-dimensional Riemannian manifold $M$ over time $0\\leq t\\leq r_0^2$. If\n$$|Ric| \\leq r_0^{-2}, ~in~ B_{g(0)}(x_0, r_0)\\times [0, r_0^2], $$\n then for any $B_{g(r_0^2)}(x, r) \\subset B_{g(r_0^2)}(x_0, B r_0)$ satisfying\n$$R|_{t=r_0^2}\\leq r^{-2}, ~in~ B_{g(r_0^2)}(x, r), $$\nwe have\n$$\\frac{Vol(B_{g(r_0^2)}(x, r), g(r_0^2))}{r^m} \\geq k \\frac{Vol(B_{g(0)}(x_0, r_0), g(r_0^2))}{r_0^m} .$$\n\n\n\n\\end{lemma}\n\n\n\nWe let $D_\\epsilon$ be the set defined as (\\ref{deset}).\n\\begin{lemma} \\label{convex32} Let $g(t)$ be the global solution of the normalized K\\\"ahler-Ricci flow on $X$. There exists $L>0$ such that for any $\\delta>0$ there exist $\\epsilon_1, \\epsilon_2 >0$ and $T>0$   so that the following hold.\n\\begin{enumerate}\n\n\\item $Vol( X\\setminus D_{\\epsilon_1}, g(t)) < \\delta e^{-(n-\\kappa)t}$ for all $t\\geq 0$.\n\n\\item For any two points $p, q \\in D_{\\epsilon_1 }$ and $t>T$, there exists a continuous path $\\gamma_t \\subset D_{ \\epsilon_2 }$ joining $p$ and $q$ such that\n$$\\mathcal{L}_{g(t)}(\\gamma_t) \\leq  L. $$\n\\end{enumerate}\n\n\\end{lemma}\n\n\n\\begin{proof}  Using the fact that $\\varphi(t)$ is uniformly bounded, the same argument in the proof of Proposition \\ref{convex1} can show that for any $\\delta>0$ there exists $\\epsilon_1>0$ such that\n$$Vol( X\\setminus D_{\\epsilon_1}, g(t)) < \\delta e^{-(n-\\kappa)t} $$ for all $t\\geq 0$. Let $p' = \\Phi(p)$ and $q'=\\Phi(q)$ for any two points $p, q \\in X\\setminus D_{\\epsilon_1}$. Since $ (X_{can}^\\circ, g_{can})$ is almost geodesically convex in $(X_{can}, g_{can})$, there exists a continuous path $\\gamma'$ in $\\Phi(X\\setminus D_{\\epsilon_2})$ joining $p'$ and $q'$ such that\n$$\\mathcal{L}_{g_{can}}(\\gamma') \\leq D+\\delta$$\nby choosing sufficiently small $\\epsilon_2>0$, where $D$ is the diameter of $(X_{can}, \\omega_{can})$.\nWe lift $\\gamma'$ to a continuous path $\\gamma$ in $ X\\setminus D_{\\epsilon_1}$ joining $p$ and $q$. Since $\\omega(t)$ converges to $\\omega_{can}$ on $\\Phi^{-1}(X\\setminus D_{\\epsilon_1})$ to $\\omega_{can}$ in $C^0$-topology uniformly for as $t\\rightarrow \\infty$,  there exists $T>0$ such that\n$$\\mathcal{L}_{g(t)}(\\gamma)  \\leq\\mathcal{L}_{g_{can}}(\\gamma')  + \\delta  \\leq D+2\\delta. $$\nThis completes the proof of the lemma.\n\n\\end{proof}\n\n\n\nNow we can prove Theorem \\ref{main1} as a consequence of the following proposition.\n\\begin{proposition} \\label{mainprop} If there exists an open domain $U$ in $X$ and $\\Lambda>0$ such that such that $\\sup_{U\\times [0, \\infty)} |Ric(g(t))|_{g(t)} <\\Lambda $. Then there exists $D>0$ such that for all $t\\in [0, \\infty)$ we have\n$$diam(X, g(t)) < D .$$\n\n\\end{proposition}\n\n\\begin{proof} By the result in \\cite{SoT4}, the scalar curvature is uniformly bounded along the flow and so we can apply Lemma \\ref{rvolcom} for  relative volume comparison. We then apply Lemma \\ref{convex32} with the same notations. We fixed a base point $P\\in X^\\circ$. Suppose there exist a point $Q\\in X$ and $t>0$ such that\n$$2L <d_{\\omega(t)}(P, Q) < 4 L. $$\nBy Lemma \\ref{rvolcom} for the relative volume comparison, there exists $k=k(X, g_0)>0$ such that\n$$Vol_{\\omega(t)}(B_{\\omega(t)}(Q, L) \\geq k L^{2\\kappa} e^{-(n-\\kappa)t}. $$\nWe will apply Lemma \\ref{convex32} by choosing\n$$\\delta = 100^{-1}k L^{2\\kappa} $$ and $T>0$ accordingly.\nSuppose %\n$$\\limsup_{t\\rightarrow \\infty} diam(X, g(t)) = \\infty. $$\nThen there exist $t'>T$ and $Q'\\in X$ such that\n$$2L <d_{\\omega(t')}(P, Q') < 4 L. $$\nObviously,\n$$B_{\\omega(t')}(Q', L) \\subset X\\setminus D_{\\epsilon_1}.$$\nThen applying Lemma \\ref{convex32}  again, we have\n$$ k L^{2\\kappa} e^{-(n-\\kappa)t}\\leq Vol_{\\omega(t')}(B_{\\omega(t')}(Q', L) )\\leq \\delta e^{-(n-\\kappa)t}.$$\nContradiction. \\end{proof}\n\n\nCorollary \\ref{mainc1} immediately follows from Proposition \\ref{mainprop} and the result proved in \\cite{TWY} that the curvature tensor $g(t)$ is uniformly bounded on any compact subset of $X^\\circ$ when the general fibre of $\\Phi: X \\rightarrow X_{can}$ is a complex torus. \n\n\n\n\\section{Proof of Theorem \\ref{KRF: minimal model}}\n\nWe will prove Theorem \\ref{KRF: minimal model} in this section. Let $X$ be a projective manifold with semi-ample canonical bundle of Kodaira dimension $1$. The canonical model $X_{can}$ of $X$ must be a nonsingular Riemann surface and we let $\\Phi: X \\rightarrow X_{can}$ be the morphism induced by the pluricanonical system. We assume that the general fibre is a complex torus as in the assumptions of Theorem \\ref{KRF: minimal model}.  The set of  critical values of $\\Phi$  coincide with $\\mathcal{S} = X_{can}\\setminus X_{can}^\\circ$ and we let\n$$\\mathcal{S} = \\{ p_1, ..., p_N\\}. $$\nLet $g_{can}$ be the twisted K\\\"ahler-Einstein metric on $X_{can}^\\circ$ induced by the twisted K\\\"ahler-Einstein current $\\omega_{can}$ satisfying \n$$Ric(\\omega_{can}) = - \\omega_{can} + \\omega_{WP}$$\non $X_{can}$. \nFor each $p_i\\in \\mathcal{S}$, we define for $\\delta>0$, \n$$B_{i, \\delta}= \\{ p\\in X_{can}~|~ d_\\chi(p, p_i) < \\delta\\}, $$\nwhere $d_\\chi$ is the distance function with respect to the smooth K\\\"ahler metric $\\chi$. \nLet $g(t)$ be the solution of the normalized K\\\"ahler-Ricci flow on $X$ and $$f_{i,\\delta} =  \\int_{ \\Phi^{-1}(B_{i, \\delta})}\\Omega.$$ Then the following holds as $\\Phi^{-1}(B_{i, \\delta})$ converges to a single fibre as $\\delta \\rightarrow 0$.\n\\begin{lemma} \\label{smallvol}\n\n$$\n\\lim_{\\delta\\rightarrow 0} f_{i,\\delta} = 0. \n$$\n\n\\end{lemma} \nWe now estimate the volume of $\\Phi^{-1}(B_{i, \\delta})$.\n\n\\begin{lemma} \\label{smallvo}There exists $C>0$ such that \n\n$$Vol_{g(t)} \\left( \\Phi^{-1}(B_{i, \\delta}) \\right) \\leq C  e^{-(n-1)t} f_{i, \\delta}. $$\n\n\\end{lemma}\n\n\\begin{proof} By the uniform $L^\\infty$-estimate of $\\varphi$ and $\\ddt{\\varphi}$ in Lemma \\ref{0est}, there exists $C>0$ such that for all $t\\geq 0$, we have \n$$\\omega(t)^n \\leq C e^{-(n-1)t} \\Omega,$$\nwhere $\\Omega$ is a smooth volume form on $X$.  Then the lemma follows by the following calculations \n$$Vol_{g(t)} \\left( \\Phi^{-1}(B_{i, \\delta}) \\right) =  \\int_{ \\Phi^{-1}(B_{i, \\delta})} \\omega(t)^n\n\\leq  C e^{-(n-1)t}\\int_{ \\Phi^{-1}(B_{i, \\delta})}\\Omega. $$\n\n\n\n\n\\end{proof}\n\n\nThe following lemma follows immediately from Proposition \\ref {contpro2} (also see \\cite{Zh17})  $X_{can}$ is nonsingular.\n\\begin{lemma} The metric completion of $(X_{can}^\\circ, g_{can})$ is homeomorphic to $X_{can}$. \n\n\\end{lemma}\n\nWe define $$\\gamma_\\delta (t)= \\sup_{p, q \\in \\partial B_{i, \\delta}} d_{g(t)}(p, q)$$ as the diameter of $\\partial B_{i, \\delta}$ with respect to $g(t)$. \n\n\\begin{lemma}\\label{bdl} For any $\\epsilon>0$, there exist $\\delta_0>0$ and $T(\\delta)>0$ for $0< \\delta <\\delta_0$ such that for all $0<\\delta<\\delta_0$ and $t>T(\\delta)$, we have \n$$ \\gamma_\\delta(t) < \\frac{\\epsilon}{8} .$$\n\\end{lemma}\n\\begin{proof} The lemma follows from the geodesic convexity of $(X_{can}^\\circ, g_{can})$ in its metric completion and the fact that $g(t)$ converges uniformly to $g_{can}$ uniformly away from singular fibres. \n\n\n\\end{proof}\n\nThe following lemma is the key estimate in this section.\n\n\n\\begin{lemma} \\label{sec5key} For any $\\epsilon>0$, there exist $\\delta>0$ and $T>0$ such that for all $t\\geq T$, we have \n$$ diam( \\Phi^{-1}(B_{i, \\delta}), g(t)) < \\epsilon, $$\nwhere the distance is calculated in $(X, g(t))$ and $i=1, 2, ..., N$. \n\n\\end{lemma}\n\n\\begin{proof}  %\nLet\n %\n $$d_{i,\\delta} (t) = \\frac{1}{2} \\sup_{q \\in \\Phi^{-1} (B_{i, \\delta})} d_{g(t)}\\left(q, \\partial \\left( \\Phi^{-1} (B_{i, \\delta}) \\right)\\right),$$\nwhere $d_{g(t)}$ is the distance function with respect to $g(t)$.\nThen  one can find a geodesic ball $B(t)$ of radius $d_{i,\\delta} (t)$ with respect to the metric $g(t)$ lying entirely in $\\Phi^{-1} (B_{i, \\delta})$.  By Lemma \\ref{smallvo} there exists $C>0$ independent of $t$ and $\\delta$ such that  \n$$Vol_{g(t)} (B(t)) \\leq Vol_{g(t)} (\\Phi^{-1}(B_{i,\\delta})) \\leq C e^{-(n-1)t} f_{i, \\delta}.$$\n\nSince the diameter of $g(t)$ is uniformly bounded by Theorem \\ref{main1} and the Ricci curvature of $g(t)$ is uniformly bounded away from $\\mathcal{S}$ for all $t$, we can apply the relative volume comparison in Lemma \\ref{rvolcom}: there exists $c>0$ independent of $t$ and $\\delta$ such that \n$$ce^{-(n-1)t} d_{i, \\delta}(t)^{2n}  \\leq Vol_{g(t)} (B(t)) < C e^{-(n-1)t} f_{i, \\delta}.$$\nThis implies that \n$$ d_{i, \\delta}(t) \\leq (c^{-1} C f_{i, \\delta})^{\\frac{1}{2n}} \\rightarrow 0 $$\n as $\\delta \\rightarrow 0$, in other words, for any $\\epsilon>0$, there exists $\\delta>0$ such that for all $t\\geq 0$, we have \n\\begin{equation}\\label{smball}\nd_{i, \\delta} (t) < \\frac{\\epsilon}{8}.\n\\end{equation}\n\nLet $X_y = \\Phi^{-1}(y)$ be the fibre of $\\Phi$ over $y\\in X_{can}$. For fixed $\\delta$ chosen above, we can choose $T>0$ such that for all $t\\geq T$, \n\\begin{equation}\\label{fibresm}\n \\sup_{y\\notin \\cup_{i=1}^N B_{i, \\delta}} diam( X_y, g(t)) < \\frac{\\epsilon}{8}\n %\n \\end{equation}\nbecause the fibre diameter uniformly goes to $0$ away from singular fibres.  Combining (\\ref{smball}), (\\ref{fibresm}) and Lemma \\ref{bdl}, we have for all $t\\geq T$, \n$$diam(\\Phi^{-1}(B_{i, \\delta}), g(t)) < \\epsilon, $$\nwhere the diameter is computed in the ambient space $(X, g(t))$. \n\n\n\\end{proof}\n\nThe following lemma holds since $g(t)$ converges in $C^0$ to $g_{can}$ away from singular fibres and the fibre diameter  with respect to $g(t)$ uniformly tends to zero away from $\\Phi^{-1} (B_{i, \\delta} )$ as $t\\rightarrow \\infty$.\n\n\\begin{lemma} \\label{appraw} \n\n\\begin{equation}\n\\lim_{\\delta \\rightarrow 0} \\limsup_{t\\rightarrow \\infty} d_{GH}\\big((X\\setminus \\left( \\cup_{i=1}^N \\Phi^{-1} (B_{i, \\delta} )\\right) ,g(t)),~(X_{can}\\setminus  \\left( \\cup_{i=1}^N  B_{i, \\delta} \\right),g_{can})\\big) =0.\n\\end{equation}\n\n\n\n\\end{lemma}\n\nTheorem \\ref{KRF: minimal model} immediately follows from Lemma \\ref{sec5key} and Lemma \\ref{appraw} since the metric completion $(X_{can}^\\circ, g_{can})$ is homeomorphic to $X_{can}$. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Where is the particle?}\n\nGiven two wave functions, the one shown in Fig.~\\ref{fig1}(a) and the one shown in Fig.~\\ref{fig1}(b), which is the correct eigenstate for a double well potential?\n\nThe actual correct answer is that we do not have enough information. We do not have enough information because we are only given a picture of the double well potential, and not an explicit definition. An explicit definition of the potential used in Fig.~\\ref{fig1} will be provided below, and this definition will reveal an asymmetry not apparent in the figure, that the potential well on the right hand side is slightly lower than that on the left hand side. The potential is actually drawn with this asymmetry, but it is so small as to be invisible to the eye on this scale (by many orders of magnitude); \nthe net result, however, is that the actual ground state is that pictured in \nFig.~\\ref{fig1}(b), with the wave function located entirely in the right side well. Hence, a tiny perturbation (the `flea')\nresults in a state very different from the familiar symmetric superposition of `left' and `right' well occupancy (shown in Fig.~\\ref{fig1}(a)).\n\\par\n\\bigskip\n\n\\section{The Asymmetric Double Well Potential}\n\n\\subsection{Introduction}\n\nThe double well potential is often used in quantum mechanics to illustrate \\hypertarget{hrazavy03}{situations} \\hypertarget{hjelic12}{in} which more than one state is accessible\nin a system, with a coupling from one to the other through tunnelling.\\cite{razavy03,jelic12} For example, in the \\hypertarget{hfeynman65}{Feynman Lectures} the\nammonia molecule is used to illustrate a physical system that has a double well potential for the Nitrogen atom.\\cite{feynman65}\nHe used an effective two-state model to illustrate these ideas, and in most undergraduate textbooks a two state system is utilized\nfor a similar purpose. Here instead we will first focus on a full solution to a double well potential; the features inherent in a two state\nsystem will emerge from our calculations. Indeed, we will also present a refined two-state model to\ncapture the essence of the asymmetry in a microscopic double well potential, as we are using in Fig.~\\ref{fig1}.\n\n\\begin{figure}[here]\n\\includegraphics[width =8.0cm,angle=0]{Dauphinee_Fig01a.eps}\n\\includegraphics[width =8.0cm,angle=0]{Dauphinee_Fig01b.eps}\n\\caption{(a) State A, ``Left'' \\textbf{and} ``Right'', or (b) State B: ``Just Left'' \\textbf{or} ``Just Right''. In the first state the wave function\nhas equal amplitude to be in either well, a linear superposition of states so common in quantum mechanics, while in the second state the wave function\nis entirely in the right side well. Which is the ground state for the double well potential shown?}\n\\label{fig1}\n\\end{figure}\n\nThe notion that a slight asymmetry can result in a  \\hypertarget{hjona-lasinio81a}{drastic} \\hypertarget{hjona-lasinio81b}{change} \\hypertarget{hsimon85}{in} \nthe wave function is not new --- it was\nfirst discussed in Refs. [\\onlinecite{jona-lasinio81a,jona-lasinio81b,simon85}], but in a manner and context inaccessible to\n\\hypertarget{hreuvers12}{undergraduate} \\hypertarget{hlandsman12}{students}. More recently the topic has been revisited \\cite{reuvers12,landsman12} to illustrate an emerging\nphenomenon in the semiclassical limit ($\\hbar \\rightarrow 0$). These authors refer to the `Flea' in reference to\nthe very minor\nperturbation in the potential (as in ours) and to the `Elephant' \\cite{simon85} (the very deep double well potential). \nSchr\\\"odinger's Cat has crept into the discussion \\cite{reuvers12,landsman12} because the `flea' disrupts the entangled\ncharacter (Fig.~\\ref{fig1}(a)) of the usual Schr\\\"odinger cat-like double well wave function (Fig.~\\ref{fig1}(a)).\n\nThe purpose of this paper is to utilize a very simple model of an asymmetric double well potential, \\hypertarget{hmarsiglio09}{solvable} either analytically \nor through an application of matrix mechanics, \\cite{marsiglio09,jelic12} to demonstrate the rather potent effect of a rather tiny\nimperfection in the otherwise symmetric double well potential. Contrary to the impression one might get from the references on this subject, there\nis nothing `semiclassical' about the asymmetry of the wave function illustrated in Fig.~\\ref{fig1}(b). We will show,\nusing a slight modification of Feynman's ammonia example, \\cite{feynman65,landsman12} that the important parameter to\nwhich the asymmetry should be compared is the tunnelling probability; this latter parameter can be arbitrarily small. This correspondence applies for excited states as well, along with other asymmetric double well shapes.\n\n\\subsection{Square Double Well with Asymmetry}\n\nA variety of symmetric double well potentials was used in Ref. \\onlinecite{jelic12} to illustrate the universality of the\nenergy splitting described there. Here instead we will use perhaps the simplest model to exhibit the impact of asymmetry --- we have checked with other versions and the same physics applies universally. This model uses two square wells, with left and\nright wells having base levels $V_L$ and $V_R$, respectively, separated by a barrier of width $b$ and height $V_0$\nand enclosed within an infinite square well extending from $x=0$ to $x=a$. For our purposes we assign the two wells\nequal width, $w = (a-b)\/2$. Mathematically it is described by\n\\begin{equation}\nV(x) = \\begin{cases} \\infty & \\text{if $x < 0$ or $x > a$} \\\\ \nV_0 & \\text{if $(a-b)\/2 <  x < (a+b)\/2$} \\\\ \nV_L & \\text{if $0 < x < (a-b)\/2$} \\\\ \nV_R & \\text{if $(a+b)\/2 < x < a$},\n\\end{cases}\n\\label{asy_doublewell_potential} \n\\end{equation}\nand is shown in Fig.~\\ref{fig2}. We can readily recover the symmetric double well by using $V_L = V_R$. Units of \nenergy are those of the ground state for the infinite square well of width $a$, $E_1^0 = \\pi^2 \\hbar^2\/(2m_0a^2)$, \nwhere $m_0$ is mass of the particle, and units of length are expressed in terms of the width of the infinite square well, $a$.\nWe will typically use a barrier width $b = a\/5$ so that the individual wells have widths $w = 2a\/5$. The height of the barrier\nthen controls the degree to which tunnelling from one well to the other occurs, and $\\delta \\equiv (V_L - V_R)\/2$ controls\nthe asymmetry.\n\nIn the following subsection we will work through detailed solutions to this problem, first analytically, and then numerically.\nIt is important to realize that these solutions retain the full Hilbert space in the problem. A `toy' model is introduced in a later\nsubsection, and reduces this complex problem to a two-state problem. We then proceed to illustrate how the two-state problem\nreproduces remarkable features of the complex problem, as a function of the asymmetry in the two wells.\n\n\\begin{figure}[here]\n\\includegraphics[width=0.55\\textwidth]{Dauphinee_Fig02.eps}\n\\caption{A schematic of the generic asymmetric square double well potential. The well widths are the same but the\nleft- and right-side levels can be independently adjusted.}\n\\label{fig2}\n\\end{figure}\n\n\\subsection{Preliminary Analysis}\n\nFig.~\\ref{fig1} was produced with $v_0 \\equiv V_0\/E_1^0 = 500$, $b\/a = 0.2$, $v_L \\equiv V_L\/E_1^0 = 0$, and\n$v_R \\equiv V_R\/E_1^0 = 0$ in part (a) and $v_R \\equiv V_R\/E_1^0 = -0.00001$ in part (b). The change in potential strength\ncompared with the barrier between these two cases is $1$ part in $50,000,000$. This results in ground state energies of\n$e_1 \\equiv E_1\/E_1^0 = 5.827 034$ and $e_1 \\equiv E_1\/E_1^0 = 5.827 025$ for the symmetric and asymmetric case,\nrespectively. Needless to say, either the difference in potentials or the difference in ground state energies represent minute\nchanges compared to the tremendous qualitative change in the wave function evident in parts (a) vs (b) of Fig.~\\ref{fig1}.\n\nThe potential used is simple enough that an `analytical' solution is also possible. The word `analytical' is in quotations here\nbecause, in reality, the solution to the equation for the energy for each state must be obtained graphically, i.e. numerically. \nWhile this poses no significant difficulty,\nit is sufficient \\hypertarget{hremark1}{work} that essentially all textbooks stop here, and do not examine the wave function.\\cite{remark1} \n\nAssuming that $E<V_0$, the analytical solution is\n\\begin{align}\n\\psi_{I}(x) & = A\\sin{kx} &  k  = \\left( 2m(E - V_L)\/\\hbar^{2}\\right)^{\\frac{1}{2}} \\notag \\\\\n\\psi_{II}(x) & = Be^{\\kappa x} + Ce^{-\\kappa x} & \\kappa  = \\left( 2m(V_{0}-E)\/\\hbar^{2} \\right)^{\\frac{1}{2}} \\notag \\\\\n\\psi_{III}(x) & = D\\sin{q(a-x)} & q = \\left( 2m(E - V_R)\/\\hbar^{2}\\right)^{\\frac{1}{2}}.\n\\label{psi_regions}\n\\end{align}\nwhere the regions I, II, and III are depicted in Fig.~\\ref{fig2}.\nApplying the matching conditions at $x = (a \\pm b)\/2$ leads to an equation for the allowed energies: \n\\begin{align}\n  \\left( \\sin(kw) +  \\frac{k}{\\kappa}\\cos(kw) \\right) & \\left( \\sin(q w) + \\frac{q}{\\kappa}\\cos(q w)\\right) = \\notag \\\\\n  e^{-2 \\kappa b} \\left( \\sin(kw) - \\frac{k}{\\kappa}\\cos(kw)  \\right) & \\left( \\sin(q w) - \\frac{q}{\\kappa}\\cos(q w) \\right),\n\\label{allowed}\n\\end{align}\nwhere $w \\equiv (a-b)\/2$ is the width of the individual wells and $b$ is the width of the barrier. The matching conditions also\nprovide expressions for the relative amplitude $D\/A$:\n\\begin{equation}\n\\left| \\frac{D}{A} \\right| =  e^{-\\kappa b}  \\left| \\frac{\\kappa \\sin(kw) - k\\cos(kw) }{\\kappa \\sin(qw) + q\\cos(qw)} \\right|,\n\\label{relamp}\n\\end{equation}\nor, alternatively,\n\\begin{equation}\n\\left| \\frac{D}{A} \\right| =  e^{+\\kappa b} \\left| \\frac{\\kappa \\sin(kw) + k\\cos(kw) }{\\kappa \\sin(qw) - q\\cos(qw)} \\right|.\n\\label{relampinv}\n\\end{equation}\nTo proceed further, Eq.~(\\ref{allowed}) is solved numerically for the allowed energy values. \nAs might be expected the low energy solutions come in pairs. \nOnce $E$ is known,\nthen so are $k$, $q$ and $\\kappa$, and the $D\/A$ ratio can be determined through either of Eqs.~(\\ref{relamp}) or (\\ref{relampinv}).\nIn addition, through similar relations and normalization, all other coefficients in Eq.~(\\ref{psi_regions}) can be determined.\nNotice that at first glance Eq.~(\\ref{relamp}) suggests that $|D| << |A|$, while  Eq.~(\\ref{relampinv}) suggests the opposite.\nIn reality both equations provide the correct answer, though with limited numerical precision one is generally more accurate than the other. \nWhich is more accurate depends on whether\n$V_R < V_L$ or vice-versa. For $V_R {{ \\atop <} \\atop {>\\atop }} V_L$, we expect $|D| {{ \\atop >} \\atop {<\\atop }} |A|$.\n\nAlternatively, we solve the original Schr\\\"odinger Equation numerically right from the start\nby expanding in the infinite square well basis [$\\phi_n(x)\n= \\sqrt{2 \\over a} \\sin{\\left({n \\pi x \\over a}\\right)}$ for $n = 1,2,3....$], and `embed' the double well part in this basis.\nMore specifically, we write\n\\begin{equation}\n|\\psi\\rangle = \\sum_n c_n |\\phi_n\\rangle,\n\\label{expansion}\n\\end{equation}\nand insert this into the Schr\\\"odinger Equation to obtain the matrix equation,\n\\begin{equation}\n\\sum_m H_{nm} c_m = E_n c_n,\n\\label{matrix}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n&&H_{nm} = \\delta_{nm} \\left( n^2E_1^{0} +(V_L+V_R)w\/a + V_0 b\/a \\right) \\nonumber \\\\\n&& \\phantom{aaaaa}+\\delta_{nm}  [2V_0-V_L-V_R] \\ {\\rm sinc}(2n) \\nonumber \\\\\n&&+\\left(1 - \\delta_{nm} \\right) D_{nm} \\left(V_L - V_0 + [V_R - V_0] (-1)^{n+m}]\\right) \\nonumber \\\\\n\\label{ham}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nD_{nm} \\equiv  {\\rm sinc}(n-m)  - {\\rm sinc}(n+m),\n\\label{defn}\n\\end{equation}\nand\n\\begin{equation}\n{\\rm sinc}(n) \\equiv {{\\rm sin}(\\pi n w) \\over \\pi n}.\n\\label{sinc}\n\\end{equation}\nAs before,   $w \\equiv (a-b)\/2$ and $b$ are the widths of the wells and barrier, respectively.\nThe general procedure is provided in Refs. [\\onlinecite{marsiglio09}] and [\\onlinecite{jelic12}], and the reader is referred \nto these papers for more details. Using either the analytical expressions or the numerical diagonalization, the results are identical. The advantage of the latter method is that the study is not confined to simple well geometries consisting of boxes, and\nstudents can easily explore a variety of double well potential shapes.\n\n\\subsection{Results}\nIn Fig.~3 the resulting wave function is shown for a variety of asymmetries.\n\\begin{figure}[here]\n\\includegraphics[width =0.5\\textwidth]{Dauphinee_Fig03.eps}\n\\caption{Progression of the wave function as the right well level is lowered from $0$ (same as the left well level) to\n$-10^{-5}$ (in units of $E_1^0 = \\hbar^2 \\pi^2\/(2ma^2)$). When the double well is symmetric the probability \ndensity ($\\left|\\psi(x)\\right|^2$) is symmetric (shown in red); the degree of asymmetry in the probability density increases\nmonotonically as $V_R\/E_1^0$ decreases from $0$ to $-10^{-5}$. The actual values of $V_R$ used in this plot, in units of \n$E_1^0$, are indicated in the legend. Any of these values is absolutely\nindistinguishable on the energy scale of the barrier ($V_0\/E_1^0 = 500$) or the ground state energies ($E_{\\rm GS}\/E_1^0\n\\approx 5.827$).\n}\n\\label{fig3}\n\\end{figure}\nThe result is remarkable. As $V_R\/E_1^0$ decreases from $0$ to a value of $-0.00001$, the probability density changes from a\nsymmetric profile (equal probability in left and right wells) to an entirely asymmetric profile (entire probability localized\nto the right well). The other `obvious' energy scales in the problem are the barrier height ($V_0\/E_1^0 = 500$)  and the\nground state energies ($E_{\\rm GS}\/E_1^0 \\approx 5.827$), so these changes in the potential are minute in comparison. \nEven more remarkable \nis that as far as the energies are concerned, these minute changes give rise to equally minute changes in the ground state\nenergy: $E_{\\rm GS}\/E_1^0 \\approx 5.827034$) ($ \\approx 5.827025$) for $V_R\/E_1^0 = 0.0$ ($V_R\/E_1^0 = -0.000 01$),\nrespectively, while the changes in the wave functions are qualitatively spectacular. \n\n\\subsection{Discussion}\n\nThat such enormous qualitative changes can result from such minute asymmetries in the double well potential is of\ncourse important for experiments in this area, where it would be very difficult to control deviations from perfect symmetry in a typical double well potential\nat the $10^{-5}$ level. Why is this phenomenon not widely disseminated in textbooks? And what precisely controls the\nenergy scale for the `flea-like' perturbation that eventually results in a completely asymmetric wave function situated in only one\nof the two wells [Fig.~(1b)]? The answer to the first question is undoubtedly connected to the lack of a straightforward analytical\ndemonstration of the strong asymmetry in the wave function. Certainly Eqs. (\\ref{relamp},\\ref{relampinv}) exist, but it is difficult to coax out of \neither of these equations an explicit demonstration of the resulting asymmetry apparent in Fig.~(3) as a \nfunction of lowering (or raising) the level of the potential well on the right. To shed more light on this phenomenon, \nand to provide an answer to the second question, we resort to a `toy model' slightly modified from the one used by \nFeynman \\cite{feynman65} to explain tunnelling in a symmetric double well system, and introduced more \nrecently by Landsman and Reuvers \\cite{landsman12,reuvers12} in a perturbative way. Such a model is also \\hypertarget{hcohentannoudji77}{used} \nin standard  \\hypertarget{htownsend00}{textbooks}\nto discuss `fictitious' spins interacting with a magnetic field\\cite{cohentannoudji77} and two level systems subject to an electric field.\\cite{townsend00}\n\n\\section{A Toy model for the Asymmetric Double Well}\n\nAn aid towards understanding the results of our `microscopic' calculations is provided by an `effective' model. The tact is to\nstrip the system of its complexity and focus on the essential ingredients. In this instance, the key features amount to whether the\nparticle is in the right well, or left well, or a combination thereof.\nFollowing Feynman, we begin with two isolated wells, each with a particular energy level, and with each coupled to the other\nthrough some matrix element, $t$:\n\\begin{eqnarray}\nH\\psi_L &=& E_L \\psi_L - t\\psi_R \\nonumber \\\\\nH\\psi_R &=& E_R \\psi_R - t\\psi_L,\n\\label{feynman}\n\\end{eqnarray}\nwhere, in the absence of coupling, the left (right) well would have a ground state energy $E_L$ ($E_R$), and $\\psi_L$\n($\\psi_R$) represents a wave function localized in the left-side (right-side) well. A straightforward solution of this two state\nsystem results in an energy splitting, as in the symmetric case:\n\\begin{equation}\nE_{\\pm} = {E_L + E_R \\over 2} \\pm \\sqrt{\\left({E_L - E_R \\over 2}\\right)^2 + t^2}.\n\\label{energy_split}\n\\end{equation}\nFor typical barriers ($t <<(E_L + E_R)\/2$) and small asymmetries ($E_L - E_R << (E_L + E_R)\/2$), very little difference occurs in the energies, in agreement with the results from our more microscopic calculations above.\nIf we define $\\delta \\equiv (E_L - E_R)\/2$ [$ \\equiv (V_L - V_R)\/2$ for the square double well potential], then the \nground state wave function becomes\n\\begin{equation}\n\\psi = {1 \\over \\sqrt{2}} \\sqrt{1 - {\\delta \\over \\sqrt{\\delta^2 + t^2}}}\\psi_L + {1 \\over \\sqrt{2}} \n\\sqrt{1 + {\\delta \\over \\sqrt{\\delta^2 + t^2}}}\\psi_R.\n\\label{wave_function_asym}\n\\end{equation}\nIn the symmetric case we recover the (symmetric) linear superposition of the state with the particle in the left well, along\nwith the state with the particle in the right well (see the remark in Ref. [\\onlinecite{remark1}]). With increasing \nasymmetry, however, say with $V_R < V_L$, i.e. $\\delta > 0$,\nthe amplitude for the particle being in the right well rises to unity, while that for the particle in the left well decreases to zero.\nOur toy model illustrates that the energy scale for this cross-over is the tunnelling matrix element, $t$. This energy scale must be\nclearly present in the microscopic model defined in Eq.~(\\ref{asy_doublewell_potential}), but it is not there explicitly.\n\n\\subsection{Comparison of the toy model to the microscopic model}\n\nTo see how well the toy model defined by the two-state system in Eq.~(\\ref{feynman}) reproduces properties of the\nmicroscopic calculations, we make an attempt to\ncompare the results from the two calculations. This is most readily accomplished by the following procedure. First,\nthe solid curves displayed in Fig.~4 are readily obtained by plotting the two amplitudes in Eq.~(\\ref{wave_function_asym}), \n\\begin{eqnarray}\n|c_L|^2 &\\equiv &{1 \\over 2} \\left( 1 - {\\delta \\over \\sqrt{\\delta^2 + t^2}} \\right) \\nonumber \\\\\n|c_R|^2 &\\equiv &{1 \\over 2} \\left( 1 + {\\delta \\over \\sqrt{\\delta^2 + t^2}} \\right)\n\\label{cs}\n\\end{eqnarray}\nas a function of $\\delta\/t$. Then, for one of the results shown in Fig.~(\\ref{fig3}),\n\\begin{figure}[here]\n\\includegraphics[width =0.56\\textwidth]{Dauphinee_Fig04.eps}\n\\caption{Plot of the amplitude in each well vs the asymmetry parameter, $\\delta\/t$. The circles indicate the area integrated from Fig.~3 on each side of the barrier, as indicated. The large circle shows the point used to establish a value of $t \\approx 6.84\\cdot 10^{-7} E_1^0$, so that the expression in Eq.~(\\ref{cs}) matches precisely the value determined by the more microscopic calculation of the previous subsection. With this value of $t$ the curves corresponding to the expressions in Eq.~(\\ref{cs}) are also plotted, and the agreement is excellent over the entire range of $\\delta\/t$. This indicates that the `toy model' phenomenology \nis very accurate. }\n\\label{fig4}\n\\end{figure}\nwe compute the area under the curve on the left; it will correspond to an amplitude $|c_L|^2$ in Fig.~4. By placing\nthis value on the appropriate curve in Fig.~4 we are able to extract a value of $\\delta\/t$ and hence an effective value\nof $t$ (since $\\delta \\equiv (V_L- V_R)\/2$ is known). This is marked by a large circle in Fig.~4.\nWe have thus identified a value of $t$, strictly only defined for the\ntoy model, with a specific barrier height and width in the more microscopic calculations connected with Eqs. (\\ref{psi_regions}-\n\\ref{relampinv}) or their numerical counterparts. We can then vary the value of $\\delta$ (as was done to generate the \ncurves shown in Fig.~3) and plot the values of the total probability density in the left and right wells as a function of $\\delta\/t$. \nThe smaller circles in Fig.~4 are the results of these calculations, and they almost perfectly lie on the curves generated\nfrom the toy model, thus showing that the asymmetric double well system indeed behaves like a two-state system\ndescribed phenomenologically by Eq.~(\\ref{feynman}). We have done this for other barrier heights and widths and similar\nvery accurate agreement between the two approaches is achieved. \nWe have also carried out such comparisons for excited states, and\nalso for other kinds of double wells (e.g. so-called Gaussian wells), with similar agreement.\n\n\\begin{figure}[here]\n\\includegraphics[width =0.5\\textwidth]{Dauphinee_Fig05.eps}\n\\caption{As in Fig.~3, progression of the wave function as the right well level is lowered from $0$ (same as the left well level) to\n$-3.0\\cdot 10^{-8}$ (in units of $E_1^0 = \\hbar^2 \\pi^2\/(2ma^2)$), but now with a barrier height of $V_0 = 1000E_1^0$. When the double well is symmetric the probability \ndensity ($\\left|\\psi(x)\\right|^2$) is symmetric (shown in red); the degree of asymmetry in the probability density increases\nmonotonically as $V_R\/E_1^0$ decreases from $0$ to $-3.0\\cdot 10^{-8}$. The actual values of $V_R$ used in this plot, in units of \n$E_1^0$, are indicated in the legend. Even more so than before, any of these values is absolutely\nindistinguishable on the energy scale of the barrier ($V_0\/E_1^0 = 1000$) or the ground state energies ($E_{\\rm GS}\/E_1^0\n\\approx 5.947$).}\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}[here]\n\\includegraphics[width =0.56\\textwidth]{Dauphinee_Fig06.eps}\n\\caption{As in Fig.~4, plot of the amplitude in each well vs the asymmetry parameter, $\\delta\/t$, but now for a much higher barrier potential, $V_0\/E_1^0 = 1000$. The circles indicate the area integrated from Fig.~5 on each side of the barrier, as indicated. The large circle shows the point used to establish a value of $t \\approx 1.45\\cdot 10^{-9} E_1^0$, so that the expression in Eq.~(\\ref{cs}) matches precisely the value determined by the more microscopic calculation of the previous subsection. With this value of $t$ the curves corresponding to the expressions in Eq.~(\\ref{cs}) are also plotted, and the agreement is excellent over the entire range \nof $\\delta\/t$. As long as the barrier is sufficiently high to delineate two very distinct states (`left' and `right,' or `dead' and `alive',\nthe two state model works very well.}\n\\label{fig6}\n\\end{figure}\n\nAs an example, in Fig.~5 and Fig.~6 we show results analogous to those of Fig.~3 and Fig.~4, but for a double well with a barrier\nwith the same width, but with a significantly increased height. The sequence of probability densities in Fig.~5 is similar\nto those shown in Fig.~3 except that the changes in the potential asymmetry are orders of magnitude smaller. Fig.~6 then\nconfirms that the significantly enhanced sensitivity is due to the significantly reduced effective `hopping' amplitude $t$\nbetween the two wells, such that the asymmetry in probability densities as a function of potential asymmetry in Fig.~6\nis as it is in Fig.~4 as a function of $\\delta\/t$, with both of these parameters greatly reduced.\n\n\\subsection{Origin of the coupling $t$}\n\n\nThe origin of the coupling parameter $t$ in the two-state  toy model is clearly the possibility of tunnelling that exists from one well into\nthe other. It is rather involved to `derive' this parameter $t$ from parameters of the original double well potential \n\\hypertarget{hmerzbacher98}{specified} in Eq.~(\\ref{asy_doublewell_potential}). \nIn fact it suffices to provide an estimate based only on the symmetric case ($V_L = V_R = 0$), \nand we provide a brief exposition here, following Merzbacher.\\cite{merzbacher98}\n\n\nOne starts with a variational wave function, of the form \n\\begin{equation}\n\\psi_{\\pm}(x) = {N_{\\pm} \\over \\sqrt{2}}\\bigl( \\psi_L(x) \\pm \\psi_R(x) \\bigr),\n\\label{variational}\n\\end{equation}\nwhere the $\\pm$ refers to the ground state ($+$) or first excited state ($-$), respectively, and the subscript $L$ ($R$)\nrefers to the left (right) well, respectively. First taking the two wells in isolation, we obtain $\\psi_L(x)$, for example, with a solution\nsimilar to that in Eq.~(\\ref{psi_regions}),\n\\begin{align}\n \\psi_L(x) & = A\\sin{(kx)} & 0<x < w \\notag \\\\\n & = A\\sin{(kw)} e^{-\\kappa(x-w)}  & x > w,\n\\label{psi_L_region}\n\\end{align}\nand similarly for the well on the right; with all coordinates displaced a distance $b$ to the right, it forms a mirror image of the one on the left. \nThis distance can be considered\nto be very large at first. In these equations we have included an unimportant normalization constant, $N_{\\pm}$ in Eq.~(\\ref{variational})\nand $A$ in Eq.~(\\ref{psi_L_region}). The energy splitting between the two states is determined by the `overlap' between the two\nwells. A straightforward calculation gives\n\\begin{equation}\nE_{\\rm split} = \\int \\ dx \\ \\psi_L(x) H \\psi_R(x) \\propto e^{-\\kappa b} \\approx e^{-b\\sqrt{V_0}},\n\\label{split}\n\\end{equation}\nso as $V_0$, the height of the barrier, increases for fixed width $b$, the splitting becomes less and less. Note, however, that the\nparameter $t$, first introduced in Eq.~(\\ref{feynman}) is proportional to this same quantity:\n\\begin{equation}\nt \\propto e^{-b\\sqrt{V_0}}.\n\\label{t}\n\\end{equation}\nThis factor is equal to $8.6 \\times 10^{-7}$ and $2.5 \\times 10^{-9}$ for $V_0\/E_1^0 = 500$ and $1000$, respectively. The actual values\nof $t$ obtained phenomenologically through the fitting procedure described above are $6.8 \\times 10^{-7}$ and $1.5 \\times 10^{-9}$,\nrespectively, which tracks very closely these exponentially decaying factors.\n\n\n\n\n\n\\section{Summary}\n\nWe have examined the simplest asymmetric double well potential and explored the behavior of the wave function\nas a function of asymmetry. In the symmetric case the ground state is a linear superposition of the particle in the left\nwell and the particle in the right well.\nAs the floor level of the potential on the right side ($V_R$) decreases, the probability\nfor the particle to be on the right side `slowly' increases. The remarkable result of our calculations is that the energy\nscale over which the transition from symmetric ground state to completely asymmetric ground state can be made\narbitrarily small. As our `toy model' calculation demonstrated, this energy scale is controlled by the tunnelling probability\nbetween the two wells, which is an energy scale that is not obviously present in the microscopic parameters (height and\nwidth of the barrier). In fact, the better well-defined the two well system is, the smaller this energy scale. Fig.~3 (or Fig.~5)\ndemonstrates this quite dramatically, where imperceptibly small asymmetries in the potential give rise to a completely asymmetric\nwave function.\nThere is very little indication of this failure from the ground state energy; instead, it\nrequires a calculation of the wave function to demonstrate this. \n\nThese calculations serve to demonstrate a number of important principles for the novice. First, the numerical calculation is\n`simpler' than the analytical calculation, and less likely to lead to error. By this we mean that solving for the wave function\nthrough Eqs.~(\\ref{allowed},\\ref{relamp},\\ref{relampinv}) is a little subtle and, for example, the wrong choice of using either\nEq.~(\\ref{relamp}) or Eq.~(\\ref{relampinv}) can lead to inaccuracies. In contrast the numerical solution is straightforward.\nThen, the solution obtained here can be tied to perturbation theory. An accurate calculation of the energy can be achieved with\nperturbation theory, but not so with the wave function! This ties into the variational principle, and teaches the important lesson that\na (very) accurate estimate of the energy certainly does not imply an even qualitatively correct wave function.\n\n\\begin{acknowledgments}\n\nThis work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), by the Alberta\niCiNano program, and by a University of Alberta Teaching and Learning Enhancement Fund (TLEF) grant. \n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe study of quantum walks, tracing back to \\cite{Gudder,Meyer},\nhas been accelerated from various aspects during the last decade,\nsee e.g., \\cite{Ambainis,KFCSWAMW,KonnoBook,Venegas} and references cited therein.\nFrom a mathematical viewpoint sharp contrast\nbetween quantum walks and random walks is of particular importance.\nFor example, the ballistic spreading is observed in a wide class of\nquantum walks \\cite{ABNVW,CHKS,IKS,Konno2002,Konno2005,KLS,Katori},\ni.e., the speed of a quantum walker's spreading is proportional to the time $n$\nwhile the typical scale for a random walk is $\\sqrt{n}$.\nMoreover, the limit distributions of quantum walks are obtained \\cite{CHKS,IKS,Konno2002,Konno2005,KLS,TS,Katori}\nwith a significant contrast with the normal Gaussian law in the case of random walks.\nIn this paper we focus on the phenomenon called \\textit{localization},\nwhich is also considered as a typical property of quantum walks, \nsee \\cite{CGMV2,IKS,KS} among others.\nWe introduce the Grover walk on a particular infinite graph called a \\textit{spidernet},\nconsider an isotropic initial state,\nand determine the class of spidernets which exhibits localization.\nA spidernet is not only a new example for the localization\nbut also is expected to be a clue to understand localization from graph structure.\nOur method is based on quantum probabilistic spectral analysis of graphs \\cite{HO}.\n\nA spidernet is obtained by adding large cycles to an almost regular tree,\nsee Subsection \\ref{subsec:Spidernets} for definition and\nsee Fig.~\\ref{fig:illestration of spidernets} for illustration.\nIt is expected to have intermediate properties between\ntrees and lattices,\nand its spectral properties have been studied to some extent, \nsee e.g., \\cite{IO} for the spectral distribution of the adjacency matrix\nand \\cite{Urakawa2003} for estimates of the Cheeger constant and Green kernel \nin terms of spectra.\nThen the standard application of the Karlin-McGregor formula\n(see e.g., \\cite{Obata2012}) \nyields an explicit formula for the $n$-step transition probability of the isotropic random walk on a spidernet,\nwhere the free Meixner law appears as the spectral distribution.\nThis argument is along a natural extension of the result on random walks \non a homogeneous tree due to Kesten \\cite{Kesten}.\nOur attempt in this paper is to establish the quantum counterpart.\n\nIn Section~\\ref{sec:Grover Walks on Graphs} we introduce the Grover walk on a general graph\nafter the standard literatures, see e.g., \\cite{Ambainis,Watrous}.\nThen we formulate two concepts of localization,\nthat is, \\textit{initial point localization} and \n\\textit{exponential localization}.\nSeveral quantum walks are known to exhibit the localization,\nsee e.g., \\cite{CGMV2,CHKS,IKS,KLS,KS,Katori}.\nFor relevant discussion see also \\cite{S}.\n\nIn Section~\\ref{sec:Main Results} we introduce the spidernet $S(a,b,c)$ and\nmention the main results.\nWe first obtain the integral representation of the $n$-step transition amplitude\nfor the Grover walk on a spidernet:\n\\begin{equation}\\label{1eqn:main expression of probability amplitude}\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n=\\int_{-1}^{1} \\cos n\\theta \\,\\mu(d\\lambda),\n\\quad\nn=0,\\pm1,\\pm2,\\dots,\n\\end{equation}\nwhere $\\lambda=\\cos\\theta$ and $\\mu$ is the free Meixner law of which the parameters\nare determined by $a,b,c$ of the spidernet under consideration,\nsee Theorem \\ref{mainthm:probability amplitude} for the precise statement.\nThe free Meixner law is a probability distribution on $[-1,1]$\nwhich is the sum of absolutely continuous part and at most two point masses.\nIt is then rather easy to derive from \\eqref{1eqn:main expression of probability amplitude}\nthe asymptotic behavior of the transition amplitude as $n\\rightarrow\\infty$.\nIn fact, only the effect of the point masses remains in the limit\nand the asymptotic results follow.\nIn particular, we prove that the initial point localization occurs\nif and only if $b>c+\\sqrt{c}$, see Theorem \\ref{3thm:localization for spidernet} for details.\n\nIt is instructive to consider the family of spidernets $S(\\kappa,\\kappa+2,\\kappa-1)$,\n$\\kappa\\ge2$.\nThese are obtained by suitably adding a large cycle to the homogeneous tree with degree $\\kappa$.\nWe see from Theorem \\ref{3thm:localization for spidernet} that\nthe initial point localization occurs for $2\\leq\\kappa< 10$\nand no initial point localization occurs for $\\kappa\\ge10$\n(Corollaries \\ref{3cor:k<10} and \\ref{3thm:estimate kappa ge10}).\nWhile, Corollary \\ref{tree} asserts that no initial point localization occurs\non a homogeneous tree either.\nIn the recent work \\cite{Katori} we know that the Grover walk on two-dimensional lattice exhibits\nthe initial point localization.\nThese results suggest the effect of cycles for the localization of the Grover walk.\n\nIn Section \\ref{sec:One-dimensional reduction} we introduce the one-dimensional\nreduction of our Grover walk, called a $(p,q)$-quantum walk on $\\mathbb{Z}_+$.\nWe determine the eigenvalues of the $(p,q)$-quantum walk with space cutoff\nby extending the quantum probabilistic method together with theory of Jacobi matrices.\n\nIn Section \\ref{sec:Proofs of main results} we obtain the integral expression of\nthe $n$-step transition amplitude of the $(p,q)$-quantum walk on $\\mathbb{Z}_+$\n(Theorem \\ref{5thm:integral representation of four amplitudes})\nand the asymptotic behavior of the transition amplitude\n(Theorem \\ref{5thm:localization criteria}).\nWith these preparations we prove the main results.\n\nIn Appendix we recall the definition of the free Meixner law\nand derive the associated orthogonal polynomials.\nThe explicit form of the orthogonal polynomials is used to derive \nthe somehow amazing result (Lemma \\ref{5lem:special value}) \nwhich plays a key role in deriving the exponential localization.\n\n\nFinally, we mention some relevant works.\nThe so-called CGMV method \\cite{CGMV,CGMV2,GVWW,KS} is also based on the spectral analysis on the unit circle\nand seems to have close connection with our approach. \nThe technique to get the eigensystem of some class of quantum walks on a finite system including the Grover walk \nis established in \\cite{Sze}. \nOur result is an extension to an infinite system, where the orthogonal polynomials \nwith respect to the free Meixner play a key role.\nConservation of probability is an interesting question for a quantum walk,\nsee e.g., \\cite{CHKS,IKS,KLS,SK,Katori}. \nThe quantum walks studied in \\cite{CHKS,IKS,KLS} are non-conservative\nand the ``missing probability\" is found through the weak convergence theorem\nin such a way that the limit distribution is a convex combination of \na point mass at the origin corresponding to localization and the Konno density function \n\\cite{Konno2002,Konno2005} coming from ballistic spreading, see \\cite{KLS} for details. \nIt is not yet checked whether our Grover walks are conservative or not.\nThere is a large number of literatures under the name of quantum graphs, \nsee e.g., \\cite{GS2006} and references cited therein,\nwhich are expected to have a profound relation to quantum walks but not yet very clear.\n\n\n\n\\section{Grover Walks on Graphs}\n\\label{sec:Grover Walks on Graphs}\n\nLet $G$ be a graph with vertex set $V=V(G)$ and edge set $E=E(G)$,\ni.e., $V$ is a non-empty (finite or infinite) set \nand $E$ is a subset of $\\{\\{u,v\\}\\,;\\, u,v\\in V, u\\neq v\\}$.\nWe often write $u\\sim v$ for $\\{u,v\\}\\in E$. \nThroughout the paper a graph is always assumed to be \\textit{locally finite}, \ni.e., $\\deg(u)=|\\{v\\in V\\,;\\, v\\sim u\\}| <\\infty$ for all $u\\in V$,\nand \\textit{connected}, i.e., every pair of vertices are connected by a walk.\nAn ordered pair $(u,v)\\in V\\times V$ is called \na \\textit{half-edge} extending from $u$ to $v$ if $u\\sim v$.\nLet $A(G)$ denote the set of half edges of $G$.\n\nThe state space of our Grover walk will be given by the Hilbert space\n$\\mathcal{H}=\\mathcal{H}(G)=\\ell^2(A(G))$ of square-summable functions on $A(G)$.\nThe inner product is defined by\n\\[\n\\langle \\bs{\\phi},\\bs{\\psi}\\rangle\n=\\sum_{(u,v)\\in A(G)} \\overline{\\bs{\\phi}(u,v)}\\,\\bs{\\psi}(u,v),\n\\qquad\n\\bs{\\phi},\\bs{\\psi}\\in \\mathcal{H}.\n\\]\nIn general, a unit vector in $\\mathcal{H}$ is called a \\textit{state}.\nThe canonical orthonormal basis is denoted by \n$\\{\\bs{\\delta}_{(u,v)}\\,;\\, (u,v)\\in A(G)\\}$.\nFor $u\\in V$ let $\\mathcal{H}_u$ be the closed subspace spanned by\n$\\{\\bs{\\delta}_{(u,v)}\\,;\\, v\\sim u\\}$.\nObviously, we have $\\dim \\mathcal{H}_u=\\deg(u)$ and\nthe orthogonal decomposition:\n\\[\n\\mathcal{H}=\\sum_{u\\in V} \\oplus \\mathcal{H}_u\\,.\n\\]\n\nWe next introduce unitary operators on $\\mathcal{H}$.\nWith each $u\\in V$ we associate a \\textit{Grover operator} $H^{(u)}$ on $\\mathcal{H}_u$\ndefined by means of the actions on the orthonormal basis\n$\\{\\bs{\\delta}_{(u,v)}\\,;\\, v\\sim u\\}$:\n\\begin{equation}\n(H^{(u)})_{vw}\n\\equiv \\langle \\bs{\\delta}_{(u,v)}, H^{(u)}\\bs{\\delta}_{(u,w)}\\rangle\n=\\frac{2}{\\deg(u)}-\\delta_{vw}\\,.\n\\end{equation} \nAs is easily verified,\nthe Grover operator $H^{(u)}$ is a real symmetric, unitary operator on $\\mathcal{H}_u$.\nThen the \\textit{coin flip operator} $C$ on $\\mathcal{H}$ is defined by\n\\begin{equation}\nC\\bs{\\delta}_{(u,v)}\n=\\sum_{w\\sim u}(H^{(u)})_{vw}\\bs{\\delta}_{(u,w)}.\n\\end{equation}\nThe \\textit{shift operator} $S$ is defined by\n\\[\nS\\bs{\\delta}_{(u,v)}=\\bs{\\delta}_{(v,u)}\\,.\n\\]\nNote that $C^2=S^2=I$ (the identity operator).\nSince both $C$ and $S$ are unitary operators on $\\mathcal{H}$,\nso is  \n\\[\nU=SC,\n\\]\nwhich is called the \\textit{Grover walk} on the graph $G$.\n\nThe time evolution of the Grover walk with an initial state $\\bs{\\Phi}_0\\in\\mathcal{H}=\\ell^2(A(G))$\nis given by the sequence of unit vectors:\n\\[\n\\bs{\\Phi}_n=U^n\\bs{\\Phi}_0\\,,\n\\qquad n=0,1,2,\\dots.\n\\]\nSince $U^n$ is unitary, we have\n\\[\n1=\\|\\bs{\\Phi}_n\\|^2=\\sum_{u\\in V}\\sum_{v\\sim u}|\\bs{\\Phi}_n(u,v)|^2,\n\\qquad n=0,1,2,\\dots.\n\\]\nTherefore, the function\n\\[\nu\\mapsto \\sum_{v\\sim u}|\\bs{\\Phi}_n(u,v)|^2,\n\\qquad u\\in V,\n\\]\ndefines a probability distribution on $V$,\nwhich is interpreted as the probability\nof finding a Grover walker at $u\\in V$ at time $n$.\nFollowing convention we write\n\\begin{equation}\\label{2eqn:distribution of GW}\nP(X_n=u)=\\sum_{v\\sim u}|\\bs{\\Phi}_n(u,v)|^2,\n\\qquad u\\in V.\n\\end{equation}\nIt is noted, however, that\n$X_n$ is merely defined as a random variable for each $n$.\nIt is an interesting question to construct a discrete-time stochastic process\n$\\{X_n\\,;\\, n=0,1,2,\\dots\\}$ with state space $V$ reasonably reflecting \nprobabilistic properties of the Grover walk.\nThe quantity $\\bs{\\Phi}_n(u,v)=\\langle\\bs{\\delta}_{(u,v)},U^n\\bs{\\Phi}_0\\rangle$\nappearing in \\eqref{2eqn:distribution of GW},\nor more generally $\\langle\\bs{\\Phi},U^n\\bs{\\Phi}_0\\rangle$ for \ntwo states $\\bs{\\Phi},\\bs{\\Phi}_0$ is\ncalled a \\textit{transition amplitude}.\nThis is a quantum counterpart of transition probability of a Markov chain.\n\n\nSince the sequence $\\{P(X_n=u)\\,;\\, n=0,1,2,\\dots\\}$ defined in \n\\eqref{2eqn:distribution of GW} is oscillating in general,\nit is essential to study the time average:\n\\[ \n\\overline{q}^{(\\infty)}(u)\n=\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}P(X_n=u),\n\\qquad u\\in V,\n\\]\nwhen the limit exists.\nFor a state $\\bs{\\Phi}\\in\\mathcal{H}=\\ell^2(A(G))$ we denote by\n$\\mathrm{supp\\,}\\bs{\\Phi}$ the set of vertices $u\\in V$\nsuch that $\\bs{\\Phi}(u,v)=\\langle \\bs{\\delta}_{(u,v)}, \\bs{\\Phi}\\rangle\\neq0$\nfor some $v\\sim u$.\n\n\\begin{definition}[Initial point localization]\n\\label{def:initial point localization}\n\\normalfont\\rm\nLet $o\\in V$ be a distinguished vertex\nand $\\bs{\\Phi}_0\\in\\mathcal{H}=\\ell^2(A(G))$ a state with $\\mathrm{supp\\,}\\bs{\\Phi}=\\{o\\}$.\nWe say that the Grover walk on $G$ with an initial state $\\bs{\\Phi}_0$ exhibits\n\\textit{initial point localization} if $\\overline{q}^{(\\infty)}(o)>0$.\n\\end{definition}\n\n\\begin{definition}[Exponential localization]\n\\label{def:exponential localization}\n\\normalfont\\rm\nLet $o\\in V$ and $\\bs{\\Phi}_0$ be the same as in Definition \\ref{def:initial point localization}.\nWe say that the Grover walk with an initial state $\\bs{\\Phi}_0$ exhibits\n\\textit{exponential localization} if there exist constant numbers $C>0$ and $0<r<1$ such that\n\\begin{equation}\\label{2eqn:exponential localization}\n\\overline{q}^{(\\infty)}(u)\\ge Cr^{\\partial(o,u)},\n\\qquad u\\in V,\n\\end{equation}\nwhere $\\partial(o,u)$ stands for the graph distance between $o$ and $u$,\ni.e., the length of the shortest path connecting them.\n\\end{definition}\n\n\\begin{remark}\nIn some literatures, e.g., \\cite{Ahlbrecht,Joye}, \n``exponential localization\" is defined when the opposite inequality\n$\\overline{q}^{(\\infty)}(u)\\le Cr^{\\partial(o,u)}$ is satisfied\ninstead of \\eqref{2eqn:exponential localization}.\nThis concept is more likely referred to as ``exponentially bounded\" or ``exponential decay\"\nand is used for a different purpose.\n\\end{remark}\n\nNote that the concepts of localization in Definitions \n\\ref{def:initial point localization} and \n\\ref{def:exponential localization} depend on the choice of an initial state.\n\n\n\\section{Main Results}\n\\label{sec:Main Results}\n\n\\subsection{Spidernets}\n\\label{subsec:Spidernets}\n\nLet $G=(V,E)$ be a (locally finite and connected) graph with a distinguished vertex $o\\in V$.\nWe introduce a stratification of $G$ by\n\\[\nV=\\bigcup_{j=0}^\\infty V_j\\,,\n\\qquad V_j=\\{u\\in V\\,;\\, \\partial(u,o)=j \\}.\n\\]\nThen for $\\epsilon\\in\\{+,-,\\circ\\}$ we define a function $\\omega_\\epsilon$ on $V$ by\n\\[ \n\\omega_\\epsilon(u)\n=|\\{v\\in V_{j+\\epsilon}\\,;\\, v \\sim u\\}|,\n\\qquad u\\in V_j\\,,\n\\]\nwhere we understand $j+\\epsilon=j+1,j-1,j$ for $\\epsilon=+,-,\\circ$, respectively.\nNote that $\\deg(u)=\\omega_+(u)+\\omega_\\circ(u)+\\omega_-(u)$ for $u\\in V$.\n\nA graph is called a \\textit{spidernet}\nif there exist a distinguished vertex $o\\in V$ and integers $a,b,c$ with\n\\begin{equation}\\label{3eqn:constraint abc}\na\\ge1, \\qquad b\\ge2, \\qquad 1\\le c\\le b-1\n\\end{equation}\nsuch that\n\\begin{gather*}\n\\omega_+(u)=\n\\begin{cases}\n\ta, & \\text{for  $u=o$}, \\\\ \n    c, & \\text{otherwise},\n\\end{cases}\n\\quad\n\\omega_-(u)=\n\\begin{cases}\n\t0, & \\text{for $u=o$}, \\\\\n    1, & \\text{otherwise},\n\\end{cases}\n\\\\\n\\omega_\\circ(u)=\n\\begin{cases}\n\t0, & \\text{for $u=o$}, \\\\ \n    b-c-1, & \\text{otherwise}.\n\\end{cases}\n\\end{gather*}\nSuch a spidernet is denoted by $S(a,b,c)$.\nIt is noted that $S(a,b,c)$ is not necessarily determined uniquely by the parameters $a,b,c$.\nBy definition we have\n\\begin{equation}\n\\deg(u)=\n\\begin{cases}\n\ta, & \\text{for $u=o$}, \\\\ \n    b, & \\text{otherwise},\n\\end{cases}\n\\end{equation}\nand \n\\[\n|V_0|=1, \n\\qquad\n|V_j|=ac^{j-1}\\,,\\quad j=1,2,\\dots.\n\\]\nHence a spidernet is an infinite graph.\n\nA spidernet $S(a,b,b-1)$ is a tree.\nIn particular, $S(\\kappa,\\kappa,\\kappa-1)$ with $\\kappa\\ge2$ is the homogeneous tree of degree $\\kappa$.\nWhile, a spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ is obtained by adding a large cycle to each stratum\nof the homogeneous tree of degree $\\kappa$.\nA typical example is shown in Fig.~\\ref{fig:illestration of spidernets}; however,\nnote that a spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ is not uniquely determined by\nthe parameter $\\kappa\\ge2$.\n\\begin{figure}[htbp]\n \\begin{center}\n  \\includegraphics[width=300pt]{spider543and463.eps}\n \\end{center}\n\\caption{$S(5,4,3)$ and $S(4,6,3)$}\n\\label{fig:illestration of spidernets}\n\\end{figure}\n\\subsection{Grover walks on spidernets}\nWe focus on the Grover walk $U$ on a spidernet $G=S(a,b,c)$.\nRecall that the state space is given by $\\mathcal{H}=\\ell^2(A(G))$, \nof which the canonical orthonormal basis is denoted by $\\{\\bs{\\delta}_{(u,v)}\\,;\\, (u,v)\\in A(G)\\}$.\nDefine a state $\\bs{\\psi}^+_0\\in\\mathcal{H}$ by\n\\begin{equation}\\label{3eqn:initial state}\n\\bs{\\psi}^+_0\n= \\frac{1}{\\sqrt{a}} \\sum_{v\\sim o} \\bs{\\delta}_{(o,v)}\\,,\n\\end{equation}\nwhich is taken to be the initial state of our Grover walk.\nNote that $\\bs{\\psi}^+_0$ is characterized by\n$\\mathrm{supp\\,}\\bs{\\psi}^+_0=\\{o\\}$ and being isotropic.\n\nWe now list the main results of this paper.\n\n\\begin{theorem}[Integral representation of transition amplitude]\n\\label{mainthm:probability amplitude}\nLet $U$ be the Grover walk on a spidernet $S(a,b,c)$ with an initial state $\\bs{\\psi}^+_0$\ndefined by \\eqref{3eqn:initial state}.\nLet $\\mu$ be the free Meixner law with parameters $q,pq,r$,\nwhere  \n\\begin{equation}\\label{3eqn:p and q from spidernet}\np=\\frac{c}{b}\\,,\n\\qquad\nq=\\frac{1}{b}\\,,\n\\qquad\nr=\\frac{b-c-1}{b}\\,.\n\\end{equation}\nThen for all $n=0,\\pm1,\\pm2,\\dots$ it holds that\n\\begin{equation}\\label{3eqn:main expression of probability amplitude}\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n=\\int_{-1}^{1} \\cos n\\theta \\,\\mu(d\\lambda),\n\\quad\n\\lambda=\\cos\\theta.\n\\end{equation}\n\\end{theorem}\n\nFor the definition of the free Meixner law, see Appendix.\nIt is also noted that \\eqref{3eqn:main expression of probability amplitude} admits\nan alternative expression:\n\\[\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n=\\int_{-1}^{1} T_{|n|}(\\lambda) \\,\\mu(d\\lambda),\n\\quad\nn=0,\\pm1,\\pm2,\\dots,\n\\]\nwhere $T_n$ is the \\textit{Chebyshev polynomials of the first kind} defined by\n\\[\n\\cos n\\theta=T_n(\\cos\\theta),\n\\qquad n=0,1,2,\\dots,\n\\]\nsee e.g., \\cite{Chihara,HO}.\n\nFrom Theorem \\ref{mainthm:probability amplitude} we will derive some results \non initial point localization of our Grover walks.\nIn fact, we determine the class of spidernets $S(a,b,c)$ which exhibit\ninitial point localization as follows.\n\n\\begin{theorem}\\label{3thm:localization for spidernet}\nLet $U$ be the Grover walk on a spidernet $S(a,b,c)$ with an initial state $\\bs{\\psi}^+_0$\ndefined by \\eqref{3eqn:initial state}.\nIt then holds that\n\\[\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n\\sim\nw \\cos n \\tilde\\theta,\n\\qquad\n\\text{as $n\\rightarrow\\infty$},\n\\]\nwhere\n\\[\nw=\\max\\left\\{\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\,,0\\right\\},\n\\quad\n\\cos\\tilde\\theta=-\\frac{1}{b-c}\\,,\n\\quad\n0<\\tilde\\theta<\\pi.\n\\]\nIn particular, if $b>c+\\sqrt{c}$, then the initial point localization occurs:\n\\[\n\\overline{q}^{(\\infty)}(o)\n=\\lim_{N\\rightarrow\\infty}\\frac1N\\sum_{n=0}^{N-1} P(X_n=o)\n=\\frac{w^2}{2}>0.\n\\]\nIf $b\\le c+\\sqrt{c}$, then no localization occurs:\n\\[\n\\lim_{n\\rightarrow\\infty}P(X_n=o)=0\n\\quad{and}\\quad\n\\overline{q}^{(\\infty)}(o)=0.\n\\]\n\\end{theorem}\n\nIt is instructive to consider the family of spidernets $S(\\kappa,\\kappa+2,\\kappa-1)$, $\\kappa\\ge2$.\nNote that $S(\\kappa,\\kappa+2,\\kappa-1)$ is obtained by adding a large cycle to each stratum\nof $S(\\kappa,\\kappa,\\kappa-1)$, which is the homogeneous tree of degree $\\kappa$.\nBelow we list some results obtained immediately from Theorem \\ref{3thm:localization for spidernet}.\n\n\\begin{corollary}\\label{3cor:k<10}\nLet $2\\leq\\kappa< 10$.\nFor the Grover walk on a spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ with \ninitial state $\\bs{\\psi}_0^+$ it holds that\n\\[\nP(X_n=o)\n=|\\langle\\bs{\\psi}_0^+,U^n\\bs{\\psi}_0^+\\rangle|^2\n\\sim \n\\left(\\frac{10-\\kappa}{12}\\right)^2\\cos^2 n\\tilde\\theta,\n\\quad n\\rightarrow\\infty,\n\\]\nwhere $\\cos\\tilde\\theta=-1\/3$, $0\\le \\tilde\\theta\\le \\pi$.\nMoreover, \n\\begin{equation}\n\\overline{q}^{(\\infty)}(o)\n=\\frac12 \\left(\\frac{10-\\kappa}{12}\\right)^2,\n\\end{equation}\nwhich means that the Grover walk under consideration exhibits initial point localization.\n(An example for $\\kappa=4$ is shown in Fig.~\\ref{fig:one}.)\n\\end{corollary}\n\n\n\\begin{corollary}\\label{3thm:estimate kappa ge10}\nLet $\\kappa\\geq 10$.\nFor the Grover walk on a spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ with \nan initial state $\\bs{\\psi}_0^+$ it holds that\n\\[\n\\lim_{n\\rightarrow\\infty}P(X_n=o)=0,\n\\quad{hence}\\quad\n\\overline{q}^{(\\infty)}(o)=0.\n\\]\n\\end{corollary}\n\n\\begin{corollary}\\label{tree}\nFor the Grover walk $U$ on a spidernet $S(a,b,b-1)$ with an initial state $\\bs{\\psi}^+_0$\nwe have\n\\[\n\\lim_{n\\rightarrow\\infty}P(X_n=o)=0,\n\\quad{hence}\\quad\n\\overline{q}^{(\\infty)}(o)=0.\n\\]\n\\end{corollary}\n\nFrom Corollaries \\ref{3cor:k<10}--\\ref{tree} we see \nthat the localization occurs when the ``density\" of large cycles is high.\nFurther study in this direction is now in progress.\n\nCorollary \\ref{tree} follows directly from Theorem \\ref{mainthm:probability amplitude}\nas a homogeneous tree is a special case of spidernets.\nWhile, quantum walks on a tree have been studied from various aspects and\nthe result in Corollary \\ref{tree} is already known \\cite{CHKS}.\nNote also that localization may occur for the Grover walk on a tree with \na non-isotropic initial state.\n\nIn relation to Theorem \\ref{3thm:localization for spidernet} we have the following\n\n\\begin{theorem}\\label{exponential localization}\nConsider a spidernet $S(a,b,c)$ with $b>c+\\sqrt{c}$.\nThen for the Grover walk $U$ with an initial state $\\bs{\\psi}_0^+$ it holds that\n\\[\n\\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}P(X_n\\in V_l)\n\\ge \\frac{b}{2c}\\left\\{\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\right\\}^2\n    \\left\\{\\frac{c}{(b-c)^2}\\right\\}^l,\n\\quad l\\ge1.\n\\]\nIf the spidernet $S(a,b,c)$ is rotationally symmetric around $o$, we have\n\\[\n\\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}P(X_n=u)\n\\ge \\frac{b}{2a}\\left\\{\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\right\\}^2\n    \\left\\{\\frac{1}{(b-c)^2}\\right\\}^{\\partial(u,o)}\\,,\n\\]\nfor all $u\\in V$, $u\\neq o$.\nNamely, the Grover walk under consideration exhibits exponential localization.\n\\end{theorem}\n\nSpecializing the parameters in Theorem \\ref{exponential localization}, we\nobtain the following result with no difficulty.\n\n\\begin{corollary}\\label{local}\nFor $2\\leq\\kappa< 10$ the Grover walk on a spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ with \nan initial state $\\bs{\\psi}_0^+$ it holds that\n\\[\n\\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}P(X_n\\in V_l)\n\\ge \\frac{\\kappa+2}{2(\\kappa-1)}\\left(\\frac{10-\\kappa}{12}\\right)^2 \n \\left(\\frac{\\kappa-1}{9}\\right)^l,\n\\quad l\\ge1.\n\\]\nIf the spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ is rotationally symmetric\naround $o$, we have\n\\[\n\\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}P(X_n=u)\n\\ge \\frac{\\kappa+2}{2\\kappa}\\left(\\frac{10-\\kappa}{12}\\right)^2 \n \\left(\\frac{1}{9}\\right)^{\\partial(0,u)},\n\\quad u\\in V, \\,\\, u\\neq o.\n\\]\nNamely, the Grover walk under consideration exhibits exponential localization.\n\\end{corollary}\n\n\\begin{remark}\nBy changing variable as $\\lambda=\\cos\\theta$,\nthe right-hand side of \\eqref{3eqn:main expression of probability amplitude} becomes an \nintegral over $[0,\\pi]$. Then using symmetric extension we can write\n\\[\n\\int_{-1}^{1} \\cos n\\theta \\,\\mu(d\\lambda)\n=\\int_{-\\pi}^{\\pi} \\cos n\\theta \\,\\nu(d\\theta)\n\\]\nwith a suitable probability distribution $\\nu$ on $[-\\pi,\\pi]$\nsuch that $\\nu(-d\\theta)=\\nu(d\\theta)$,\nwhere no point mass at $\\pm\\pi$.\nThus, we have an alternative expression for the transition amplitude:\n\\[\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n=\\int_{-\\pi}^{\\pi} e^{in\\theta} \\,\\nu(d\\theta),\n\\qquad n=0,\\pm1,\\pm2,\\dots,\n\\]\nwhich is directly related to the spectral decomposition of the unitary\noperator $U$.\n\\end{remark}\n\n\\begin{remark}\nLet $\\{X_n\\,;n=0,1,2,\\dots\\}$ be the isotropic random walk on $S(a,b,c)$ with\ntransition matrix $T$. It then follows from the well-established general theory that\n\\begin{equation}\\label{3eqn:RW}\nP(X_n=o|X_0=o)\n=\\langle\\bs{\\delta}_o,T^n\\bs{\\delta}_o\\rangle\n=\\int_{-1}^{1} \\lambda^n \\,\\mu(d\\lambda),\n\\quad n=0,1,2,\\dots,\n\\end{equation}\nwhere $\\mu$ is the same probability distribution as in Theorem \\ref{mainthm:probability amplitude},\nsee also \\cite{Obata2012} for relevant discussion along quantum probability.\nWe see that \\eqref{3eqn:RW} makes a good contrast to the transition amplitude \n\\eqref{3eqn:main expression of probability amplitude}.\n\\end{remark}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=225pt]{origin.eps}\n\\caption{The Grover walk on $S(4,6,3)$ from time $n=620$ to $n=650$\n(see Corollary \\ref{3cor:k<10}): \nThe dots stand for $P(X_n=o)$ calculated by numerical simulation. \nThe curve is the graph of $(1\/4)\\cos^2(t\\tilde\\theta)$ \nand the horizontal line depicts the time averaged limit probability $\\overline{q}^{(\\infty)}(o)$.}\n\\label{fig:one}      \n\\end{center}\n\\end{figure}\n\n\n\\section{One-dimensional reduction}\n\\label{sec:One-dimensional reduction}\n\n\n\\subsection{$(p,q)$-Quantum walk on $\\mathbb{Z}_+$}\n\\label{subsec:4.1}\n\nLet $U=SC$ be the Grover walk on the spidernet $G=S(a,b,c)$.\nDefine orthonormal vectors in $\\mathcal{H}=\\ell^2(A(G))$ by\n\\begin{alignat}{2}\n\\bs{\\psi}^+_n \n&= \\frac{1}{\\sqrt{ac^n}}\n\\sum_{u\\in V_n}\\sum_{\\substack{v\\in V_{n+1} \\\\ v\\sim u}} \\bs{\\delta}_{(u,v)}\\,,\n&\\quad &n\\ge0, \n\\label{3eqn:def of psi+} \\\\\n\\bs{\\psi}^\\circ_n \n&= \\frac{1}{\\sqrt{a(b-c-1)c^{n-1}}}\n\\sum_{u\\in V_n}\\sum_{\\substack{v\\in V_n \\\\ v\\sim u}} \\bs{\\delta}_{(u,v)}\\,,\n&\\quad &n\\ge1, \n\\label{3eqn:def of psio} \\\\\n\\bs{\\psi}^-_n \n&= \\frac{1}{\\sqrt{ac^{n-1}}}\n\\sum_{u\\in V_n}\\sum_{\\substack{v\\in V_{n-1} \\\\ v\\sim u}} \\bs{\\delta}_{(u,v)}\\,,\n&\\quad &n\\ge1.\n\\label{3eqn:def of psi-}\n\\end{alignat}\nWe keep the same notations as in \\eqref{3eqn:p and q from spidernet}:\n\\begin{equation}\\label{4eqn:pqr}\np=\\frac{c}{b}\\,,\n\\qquad\nq=\\frac{1}{b}\\,,\n\\qquad\nr=\\frac{b-c-1}{b}\\,,\n\\end{equation}\nverifying that\n\\[\np>0,\n\\quad q>0, \n\\quad r=1-p-q\\ge0.\n\\]\n\n\\begin{lemma}\\label{lem:actions of C on psi}\nIt holds that\n\\begin{align}\nC\\bs{\\psi}^+_n\n&=\\begin{cases}\n\\bs{\\psi}^+_0\\,, & n=0, \\\\\n(2p-1)\\bs{\\psi}^+_n+2\\sqrt{pr}\\,\\bs{\\psi}^\\circ_n+2\\sqrt{pq}\\,\\bs{\\psi}^-_n\\,, & n\\ge1,\n\\end{cases}\n\\label{eqn:def of psi +n} \\\\\nC\\bs{\\psi}^\\circ_n\n&=2\\sqrt{pr}\\,\\bs{\\psi}^+_n+(2r-1)\\bs{\\psi}^\\circ_n+2\\sqrt{qr}\\,\\bs{\\psi}^-_n\\,, \n\\,\\,\\,\\,\\quad n\\ge1,\n\\label{eqn:def of psi on} \\\\\nC\\bs{\\psi}^-_n\n&=2\\sqrt{pq}\\,\\bs{\\psi}^+_n +2\\sqrt{qr}\\, \\bs{\\psi}^\\circ_n +(2q-1)\\bs{\\psi}^-_n\\,,\n\\quad n\\ge1.\n\\label{eqn:def of psi -n}\n\\end{align}\n\\end{lemma}\n\n\\noindent{\\it Proof.} \nBy definition we have\n\\begin{align*}\nC\\bs{\\delta}_{(x,y)}\n&=\\sum_{w\\sim x}(H^{(x)})_{yw}\\bs{\\delta}_{(x,w)}\\,,\n\\qquad (x,y)\\in A(G), \\\\\n(H^{(x)})_{yw}\n&=\n\\begin{cases}\n\\dfrac{2}{a}-\\delta_{yw}\\,,& x=o, \\\\[8pt]\n\\dfrac{2}{b}-\\delta_{yw}\\,,& \\textit{otherwise}.\n\\end{cases}\n\\end{align*}\nWe first show \\eqref{eqn:def of psi +n} for $n=0$.\nSuppose $(o,y)\\in A(G)$.\nThen, \n\\begin{align*}\nC\\bs{\\delta}_{(o,y)}\n&=\\sum_{w\\sim o}(H^{(o)})_{yw}\\bs{\\delta}_{(o,w)} \\\\\n&=\\sum_{w\\sim o}\\left(\\dfrac{2}{a}-\\delta_{yw}\\right)\\bs{\\delta}_{(o,w)} \\\\\n&=\\frac{2}{a}\\sum_{w\\sim o}\\bs{\\delta}_{(o,w)}-\\bs{\\delta}_{(o,y)}\\,.\n\\end{align*}\nTaking the summation over $y\\sim o$, we obtain\n\\[\n\\sum_{y\\sim o}C\\bs{\\delta}_{(o,y)}\n=2\\sum_{w\\sim o}\\bs{\\delta}_{(o,w)}-\\sum_{y\\sim o}\\bs{\\delta}_{(o,y)}\n=\\sum_{w\\sim o}\\bs{\\delta}_{(o,w)}\\,,\n\\]\nfrom which the desired relation follows by\ndividing both sides by $\\sqrt{a}$.\n\nWe next prove \\eqref{eqn:def of psi +n} for $n\\ge1$.\nSuppose $x\\in V_n$ with $n\\ge1$.\nThen by definition,\n\\begin{align*}\n\\sum_{\\substack{y\\in V_{n+1} \\\\ y\\sim x}} C\\bs{\\delta}_{(x,y)}\n&=\\sum_{\\substack{y\\in V_{n+1} \\\\ y\\sim x}} \\sum_{w\\sim x}\\left(\\dfrac{2}{b}-\\delta_{yw}\\right)\\bs{\\delta}_{(x,w)} \\\\\n&=\\frac{2c}{b} \\sum_{w\\sim x}\\bs{\\delta}_{(x,w)} \n - \\sum_{\\substack{y\\in V_{n+1} \\\\ y\\sim x}} \\bs{\\delta}_{(x,y)}\\,,\n\\end{align*}\nwhere $|\\{y\\in V_{n+1}\\,;\\, y\\sim x\\}|=c$ is taken into account.\nTaking the summation over $x\\in V_n$, we obtain\n\\begin{align*}\n\\sum_{x\\in V_n}\\sum_{\\substack{y\\in V_{n+1} \\\\ y\\sim x}} C\\bs{\\delta}_{(x,y)}\n&=\\frac{2c}{b} \\sum_{x\\in V_n}\\sum_{w\\sim x}\\bs{\\delta}_{(x,w)} \n - \\sum_{x\\in V_n}\\sum_{\\substack{y\\in V_{n+1} \\\\ y\\sim x}} \\bs{\\delta}_{(x,y)} \\\\\n&=\\frac{2c}{b}\\Big(\\sqrt{ac^n}\\, \\bs{\\psi}^+_n\n + \\sqrt{a(b-c-1)c^{n-1}}\\,\\bs{\\psi}^\\circ_n \\\\\n&\\qquad\\qquad  +\\sqrt{ac^{n-1}}\\,\\bs{\\psi}^-_n\\Big) \n - \\sqrt{ac^n}\\,\\bs{\\psi}^+_n(x)\n\\end{align*}\nand then, dividing both sides by $\\sqrt{ac^n}$, we come to\n\\[\nC\\bs{\\psi}^+_n\n=\\left(\\frac{2c}{b}-1\\right)\\bs{\\psi}^+_n\n +\\frac{2\\sqrt{c(b-c-1)}}{b}\\,\\bs{\\psi}^\\circ_n\n +\\frac{2\\sqrt{c}}{b}\\,\\bs{\\psi}^-_n\\,,\n\\]\nwhich shows \\eqref{eqn:def of psi +n}.\nThe rest of the relations is proved in a similar manner.\n\\begin{flushright} $\\square$ \\end{flushright}\n\n\\begin{lemma}\\label{docomo}\nIt holds that\n\\begin{alignat}{2}\nS\\bs{\\psi}^+_n &=\\bs{\\psi}^-_{n+1}\\,, &\\quad &n\\ge0,\n\\label{eqn:def of S psi +n} \\\\\nS\\bs{\\psi}^\\circ_n &=\\bs{\\psi}^\\circ_n\\,, &\\quad &n\\ge1, \n\\label{eqn:def of S psi on} \\\\\nS\\bs{\\psi}^-_n &= \\bs{\\psi}^+_{n-1}\\,, &\\quad &n\\ge1.\n\\label{eqn:def of S psi -n}\n\\end{alignat}\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nBy Straightforward calculation similar to the proof of Lemma \\ref{lem:actions of C on psi}.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\noindent It is convenient to study the actions of $C$ and $S$\ndescribed in Lemmas \\ref{lem:actions of C on psi} and \\ref{docomo}\nin a slightly more general context.\nWe consider the Hilbert space $\\mathcal{H}(\\mathbb{Z}_+)$ of the form:\n\\[\n\\mathcal{H}(\\mathbb{Z}_+)\n=\\mathbb{C}\\bs{\\psi}^+_0\\oplus \\sum_{n=1}^\\infty\\oplus \n (\\mathbb{C}\\bs{\\psi}^+_n\\oplus \\mathbb{C}\\bs{\\psi}^\\circ_n\\oplus \\mathbb{C}\\bs{\\psi}^-_n),\n\\]\nwhere $\\bs{\\psi}^+_0, \\bs{\\psi}^+_1,\\bs{\\psi}^\\circ_1, \\bs{\\psi}^-_1,\\dots$ form an \northonormal basis of $\\mathcal{H}(\\mathbb{Z}_+)$.\nLet $p,q,r$ be constant numbers satisfying\n\\[\np>0, \\quad q>0, \\quad r=1-p-q\\ge0.\n\\]\nWe then define the coin operator $C$ and the shift operator $S$ on $\\mathcal{H}(\\mathbb{Z}_+)$\nby \\eqref{eqn:def of psi +n}--\\eqref{eqn:def of psi -n} and by \\eqref{eqn:def of S psi +n}--\\eqref{eqn:def of S psi -n},\nrespectively. \nIt is easily seen that both $C$ and $S$ are unitary operators.\nHence $U=SC$ is also a unitary operator on $\\mathcal{H}(\\mathbb{Z}_+)$,\nwhich is called the \\textit{$(p,q)$-quantum walk} on $\\mathbb{Z}_+$.\n\nThus the Grover walk on a spidernet $G=S(a,b,c)$ restricted to\nthe closed subspace spanned by\n$\\{\\bs{\\psi}^+_n,\\,n\\ge0\\}\\cup\n \\{\\bs{\\psi}^\\circ_n\\,;\\, n\\ge1\\}\\cup\n \\{\\bs{\\psi}^-_n\\,;\\,n\\ge1\\}$ is a $(p,q)$-quantum walk on $\\mathbb{Z}_+$,\nwhere $p,q$ are given by \\eqref{4eqn:pqr}.\n\nWe define orthonormal vectors in $\\mathcal{H}(\\mathbb{Z}_+)$ by\n\\begin{align*}\n\\bs{\\Psi}_0 &=\\bs{\\psi}^+_0, \\\\\n\\bs{\\Psi}_n &=\\sqrt{p}\\,\\bs{\\psi}^+_n+\\sqrt{r}\\,\\bs{\\psi}^\\circ_n+\\sqrt{q}\\,\\bs{\\psi}^-_n\\,, \\quad n\\ge1,\n\\end{align*}\nand set\n\\[\n\\Gamma(\\mathbb{Z}_+)=\\sum_{n=0}^\\infty \\oplus \\mathbb{C} \\bs{\\Psi}_n\\,.\n\\]\nThen $\\Gamma(\\mathbb{Z}_+)\\subset \\mathcal{H}(\\mathbb{Z}_+)$ is a closed subspace.\nLet $\\Pi:\\mathcal{H}(\\mathbb{Z}_+)\\rightarrow\\Gamma(\\mathbb{Z}_+)$ denote the orthogonal projection.\n\n\\begin{lemma}\\label{3lem:C=2pi-I on Z}\nIt holds that\n\\[\nC=C^*=2\\Pi-I.\n\\]\nIn particular, $C$ is the reflection with respect to $\\Gamma(\\mathbb{Z}_+)$ and\nacts on $\\Gamma(\\mathbb{Z}_+)$ as the identity.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nStraightforward by definition.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\n\\subsection{$(p,q)$-Quantum walk on a path of finite length}\n\nLet $U$ be a $(p,q)$-quantum walk on $\\mathbb{Z}_+$ as in the previous section.\nWe will introduce a $(p,q)$-quantum walk on the path of length $N\\ge2$,\nobtained from the $(p,q)$-quantum walk on $\\mathbb{Z}_+$ by cutoff.\n\nFor $N\\ge2$ we define a Hilbert space:\n\\[\n\\mathcal{H}(N)\n=\\mathbb{C}\\bs{\\psi}^+_0\\oplus \\sum_{n=1}^{N-1}\\oplus \n (\\mathbb{C}\\bs{\\psi}^+_n\\oplus \\mathbb{C}\\bs{\\psi}^\\circ_n\\oplus \\mathbb{C}\\bs{\\psi}^-_n)\n \\oplus \\mathbb{C}\\bs{\\psi}^-_N\n\\]\nand unitary operators $C=C_N$ and $S=S_N$ respectively \nas in \\eqref{eqn:def of psi +n}--\\eqref{eqn:def of psi -n}\nand in \\eqref{eqn:def of S psi +n}--\\eqref{eqn:def of S psi -n}, except\n\\begin{equation}\\label{3eqn:exception C}\nC\\bs{\\psi}^-_N=\\bs{\\psi}^-_N.\n\\end{equation}\nThen we obtain a unitary operator $U=U_N=S_NC_N$ on $\\mathcal{H}(N)$,\nwhich is called the \\textit{$(p,q)$-quantum walk} on the path of length $N$.\nBoth endpoints play as reflection barriers in analogy of random walks.\nFrom now on we omit the suffix $N$ whenever there is no danger of confusion.\n\nIn view of \\eqref{eqn:def of psi +n}--\\eqref{eqn:def of psi -n}, \\eqref{3eqn:exception C}\nand \\eqref{eqn:def of S psi +n}--\\eqref{eqn:def of S psi -n}\nthe explicit actions of $U$ on $\\bs{\\psi}^\\epsilon_j$ are easily written down as follows:\n\\begin{align}\nU\\bs{\\psi}^+_j\n&=\\begin{cases}\n\\bs{\\psi}^-_1\\,, & j=0, \\\\\n(2p-1)\\bs{\\psi}^-_{j+1}+2\\sqrt{pr}\\,\\bs{\\psi}^\\circ_j+2\\sqrt{pq}\\,\\bs{\\psi}^+_{j-1}\\,, & 1\\le j\\le N-1,\n\\end{cases}\n\\label{eqn:action of U psi +n} \\\\\nU\\bs{\\psi}^\\circ_j\n&=2\\sqrt{pr}\\,\\bs{\\psi}^-_{j+1}+(2r-1)\\bs{\\psi}^\\circ_j+2\\sqrt{qr}\\,\\bs{\\psi}^+_{j-1}\\,, \n\\,\\,\\,\\,\\quad 1\\le j\\le N-1,\n\\label{eqn:action of U psi on} \\\\\nU\\bs{\\psi}^-_j\n&=\\begin{cases}\n2\\sqrt{pq}\\,\\bs{\\psi}^-_{j+1} +2\\sqrt{qr}\\, \\bs{\\psi}^\\circ_j +(2q-1)\\bs{\\psi}^+_{j-1}\\,, & 1\\le j\\le N-1, \\\\\n\\bs{\\psi}^+_{N-1}, & j=N,\n\\end{cases}\n\\label{eqn:action of U psi -n}\n\\end{align}\n\nThe goal of this subsection is to determine the spectra (eigenvalues) of $U$.\nWe start with the following result.\n\n\\begin{lemma}\\label{3lem:Trace of U}\n$\\mathrm{Tr\\,}U=(2r-1)(N-1)$.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nWe see from \\eqref{eqn:action of U psi +n}--\\eqref{eqn:action of U psi -n} that\n\\[\n\\mathrm{Tr\\,}U=\\sum_{\\epsilon,n}\\langle \\bs{\\psi}^\\epsilon_n, U \\bs{\\psi}^\\epsilon_n\\rangle\n=\\sum_{j=1}^{N-1} \\langle \\bs{\\psi}^\\circ_j, U \\bs{\\psi}^\\circ_j\\rangle\n=(2r-1)(N-1)\n\\]\nas desired.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\noindent Define orthonormal vectors in $\\mathcal{H}(N)$ by\n\\begin{align*}\n&\\bs{\\Psi}_0 =\\bs{\\psi}^+_0, \\\\\n&\\bs{\\Psi}_j =\\sqrt{p}\\,\\bs{\\psi}^+_j+\\sqrt{r}\\,\\bs{\\psi}^\\circ_j+\\sqrt{q}\\,\\bs{\\psi}^-_j\\,, \\quad 1\\le j\\le N-1, \\\\\n&\\bs{\\Psi}_N =\\bs{\\psi}^-_N\n\\end{align*}\nand set\n\\[\n\\Gamma(N)=\\sum_{j=0}^N \\oplus\\mathbb{C} \\bs{\\Psi}_j\\,.\n\\]\nThen $\\Gamma(N)$ is a closed subspace of $\\mathcal{H}(N)$\nand let $\\Pi=\\Pi_N:\\mathcal{H}(N)\\rightarrow\\Gamma(N)$ denote the orthogonal projection.\nThe assertion of Lemma \\ref{3lem:C=2pi-I on Z} remains true, i.e., \nit holds that \n\\[\nC=C^*=2\\Pi-I.\n\\]\n\nFor the $(p,q)$-quantum walk $U=U_N$ we consider $\\Pi U \\Pi$ as an operator on $\\Gamma(N)$, \nwhich is denoted by $T=T_N$.\nThus,\n\\begin{equation}\\label{4eqn:T and Pi}\nT=\\Pi U \\Pi \\!\\!\\restriction_{\\Gamma(N)}\n=\\Pi SC \\Pi \\!\\!\\restriction_{\\Gamma(N)}\n=\\Pi S \\!\\!\\restriction_{\\Gamma(N)}\\,.\n\\end{equation}\nMoreover, by direct calculation we obtain its matrix expression \nwith respect to the orthonormal basis $\\{\\bs{\\Psi}_j\\,;\\,0\\le j\\le N\\}$ as follows:\n\\[\nT=T_N=\n\\begin{bmatrix}\n0        & \\sqrt{q} &                                \\\\\n\\sqrt{q} & r        & \\sqrt{pq} &                    \\\\  \n         & \\sqrt{pq}& r         & \\sqrt{pq} &        \\\\  \n         &          & \\ddots    & \\ddots    & \\ddots \\\\  \n& & &  \\sqrt{pq} & r        &\\sqrt{pq} \\\\  \n& & & & \\sqrt{pq} & r        &\\sqrt{p} \\\\  \n& & & &           & \\sqrt{p} & 0 \\\\  \n\\end{bmatrix}.\n\\]\nFor example,\n\\begin{align*}\n&T \\bs{\\Psi}_0 =\\sqrt{q}\\, \\bs{\\Psi}_1\\,, \\\\\n&T \\bs{\\Psi}_1 =\\sqrt{pq}\\, \\bs{\\Psi}_2+ r \\bs{\\Psi}_1 + \\sqrt{q}\\,\\bs{\\Psi}_0\\,,\n\\quad \\text{etc.}\n\\end{align*}\n\n\\begin{lemma}\nEvery eigenvalue of $T$ is simple.\nMoreover, $\\mathrm{Spec}(T)\\subset [-1,1]$ and $1\\in \\mathrm{Spec}(T)$.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nThat every eigenvalue of $T$ is simple follows from general theory of Jacobi matrices\n(see e.g., \\cite{HO,Deift}).\nSince $T=\\Pi S\\!\\!\\restriction_{\\Gamma(N)}$ by \\eqref{4eqn:T and Pi}, the operator norm of $T$ is bounded by one.\nHence every eigenvalue of $T$ lies in $[-1,1]$.\nFinally, it is easily verified by expansion that $\\det(T-I)=0$.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\begin{lemma}\\label{3lem:3.6}\n{\\upshape (1)} If $r>0$, there exists no non-zero $v\\in\\Gamma(N)$ such that $Sv=-v$.\n\n{\\upshape (2)} If $r=0$, there exists non-zero $v\\in\\Gamma(N)$ such that $Sv=-v$.\nMoreover, such a non-zero vector $v$ is determined uniquely up to a constant factor.\n\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nEvery $v\\in \\Gamma(N)$ is in the form:\n\\begin{align*}\nv\n&=\\sum_{j=0}^N \\gamma_j \\bs{\\Psi}_j \\\\\n&=\\gamma_0 \\bs{\\psi}_0^+\n  +\\sum_{j=1}^{N-1} \\gamma_j (\\sqrt{p}\\,\\bs{\\psi}_j^+ +\\sqrt{r}\\, \\bs{\\psi}_j^\\circ + \\sqrt{q}\\, \\bs{\\psi}_j^-)\n  +\\gamma_N \\bs{\\psi}_N^-,\n\\end{align*}\nwhere $\\gamma_0,\\dots,\\gamma_N$ are constant numbers.\nThen the equation $Sv=-v$ is equivalent to the one for these constant numbers,\nwhich is obtained by direct calculation:\n\\begin{align}\n&\\gamma_1=-\\frac{1}{\\sqrt{q}}\\,\\gamma_0\\,,\n\\quad\n\\gamma_N=-\\sqrt{p}\\,\\gamma_{N-1}\\,, \n\\label{4eqn:in proof (000)} \\\\\n&\\gamma_j=-\\sqrt{\\frac{p}{q}}\\,\\gamma_{j-1}\\,,\n\\quad 2\\le j\\le N-1,\n\\label{4eqn:in proof (001)} \\\\\n&\\gamma_j\\sqrt{r}=-\\gamma_j \\sqrt{r}\\,,\n\\quad 1\\le j\\le N-1.\n\\label{4eqn:in proof (002)} \n\\end{align}\nIf $r>0$, it follows from \\eqref{4eqn:in proof (002)} that $\\gamma_1=\\dots=\\gamma_{N-1}=0$.\nThen in view of \\eqref{4eqn:in proof (000)} we also have $\\gamma_0=\\gamma_N=0$,\nwhich implies $v=0$.\nIf $r=0$, the recurrence relations \\eqref{4eqn:in proof (000)} and \\eqref{4eqn:in proof (001)} determine\nthe sequence $\\gamma_0,\\gamma_1,\\dots,\\gamma_N$ uniquely by the initial value $\\gamma_0$.\nHence $\\dim\\{v\\in\\Gamma(N)\\,;\\, Sv=-v\\}=1$ as desired.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\begin{lemma}\\label{3lem:3.7}\nIf $Tv=\\pm v$ for $v\\in\\Gamma(N)$, then $Sv=Uv=\\pm v$.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nIt is sufficient to consider the case of $v\\neq0$.\nSuppose that $Tv=\\pm v$ for $v\\in\\Gamma(N)$.\nFrom $T=\\Pi S\\!\\!\\restriction_{\\Gamma(N)}$ and $\\Pi v=v$ we obtain $\\Pi(Sv\\mp v)=0$.\nHence $\\langle Sv \\mp v,v\\rangle=0$, which implies that\n\\[\n\\langle Sv,v \\rangle\n=\\pm \\langle v,v \\rangle.\n\\]\nSince $S$ is unitary, the above relation implies the Schwartz equality\nand $Sv=\\alpha v$ with some constant $\\alpha\\in\\mathbb{C}$.\nIt follows by applying $\\Pi$ that $\\alpha=\\pm1$ and $Sv=\\pm v$.\nFinally, since $U=SC$ by definition and $C$ acts on $\\Gamma(N)$ as the identity,\nwe have $Uv=SCv=Sv$.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\begin{lemma}\nWe have $-1 \\not\\in\\mathrm{Spec}(T)$ for $r>0$,\nand $-1 \\in \\mathrm{Spec}(T)$ for $r=0$.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nSuppose that $r>0$ and $Tv=-v$ for $v\\in\\Gamma(N)$.\nWe see from Lemma \\ref{3lem:3.7} $Sv=-v$,\nthen applying Lemma \\ref{3lem:3.6} we come to $v=0$.\nThis means that $-1$ is not an eigenvalue of $T$.\n\nWe next suppose that $r=0$. \nBy Lemma \\ref{3lem:3.6} there exists a non-zero vector $v\\in\\Gamma(N)$ such that $Sv=-v$.\nThen $Tv=\\Pi Sv=-v$, which means that $-1$ is an eigenvalue of $T$.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\noindent Thus, the eigenvalues of $T$ are arranged in such a way that\n\\begin{gather}\n\\lambda_0=1=\\cos\\theta_0,\n\\quad\n\\lambda_1=\\cos\\theta_1\\,,\n\\quad\n\\lambda_2=\\cos\\theta_2\\,,\n\\quad \\dots,\n\\quad\n\\lambda_N=\\cos\\theta_N\\,, \n\\label{3eqn:arrangement of lambda} \\\\\n0=\\theta_0<\\theta_1<\\theta_2<\\dots<\\theta_N\\le \\pi,\n\\nonumber\n\\end{gather}\nwhere $\\theta_N<\\pi$ for $r>0$ and $\\theta_N=\\pi$ for $r=0$.\nFor each eigenvalue $\\lambda_j$ we take a normalized eigenvector $\\bs{\\Omega}_j\\in\\Gamma(N)$,\ni.e.,\n\\[\nT\\bs{\\Omega}_j=\\lambda_j \\bs{\\Omega}_j\\,,\n\\quad\n\\|\\bs{\\Omega}_j\\|=1.\n\\]\nThen we have the orthogonal decomposition of $\\Gamma(N)$ in two ways:\n\\[\n\\Gamma(N)\n=\\sum_{j=0}^N \\oplus \\mathbb{C}\\bs{\\Psi}_j\n=\\sum_{j=0}^N \\oplus \\mathbb{C}\\bs{\\Omega}_j.\n\\]\nWe next study the subspace\n\\begin{equation}\\label{4eqn:def of L(N)}\n\\mathcal{L}(N)=\\Gamma(N)+S\\Gamma(N),\n\\end{equation}\nwhich is invariant under the actions of $S$ and $U$.\nIn fact, for $\\bs{\\phi}, \\bs{\\psi}\\in \\Gamma(N)$ we have\n\\begin{align*}\nU(\\bs{\\phi}+S\\bs{\\psi})\n&=SC\\bs{\\phi}+SCS\\bs{\\psi}\n=S\\bs{\\phi}+S(2\\Pi-I)S\\bs{\\psi} \\\\\n&=S\\bs{\\phi}+2S\\Pi S\\bs{\\psi}-S^2\\bs{\\psi}\n=S(\\bs{\\phi}+2\\Pi S\\bs{\\psi})-\\bs{\\psi},\n\\end{align*}\nwhich shows that $\\mathcal{L}(N)=\\Gamma(N)+S\\Gamma(N)$ is invariant under $U$.\n\n\\begin{lemma}\\label{3lem:orthogonality}\n{\\upshape (1)} If $r>0$, then the vectors $\\bs{\\Omega}_0,\\bs{\\Omega}_1,\\dots, \\bs{\\Omega}_N,\nS\\bs{\\Omega}_1,\\dots, S\\bs{\\Omega}_N$ are linearly independent.\nMoreover,\n\\begin{equation}\\label{3eqn:inner product 3.9}\n\\langle \\bs{\\Omega}_j, S\\bs{\\Omega}_k\\rangle=\\lambda_k\\delta_{jk}\\,,\n\\quad 0\\le j\\le N, \n\\quad 1\\le k\\le N.\n\\end{equation}\n\n{\\upshape (2)} If $r=0$, then $S\\bs{\\Omega}_N=-\\bs{\\Omega}_N$ and \n$\\bs{\\Omega}_0,\\bs{\\Omega}_1,\\dots, \\bs{\\Omega}_N$,\n$S\\bs{\\Omega}_1,\\dots, S\\bs{\\Omega}_{N-1}$ are linearly independent.\nMoreover, \\eqref{3eqn:inner product 3.9} remains valid where\n$0\\le j\\le N$ and $1\\le k\\le N-1$.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\n(1) Suppose that\n\\begin{equation}\\label{3eqn:in proof 3.9 (1)}\n\\alpha_0 \\bs{\\Omega}_0 \n+\\sum_{j=1}^N \\alpha_j \\bs{\\Omega}_j\n+\\sum_{j=1}^N \\beta_j S\\bs{\\Omega}_j=0.\n\\end{equation}\nTaking $\\Pi S=T$ in mind, we apply $\\Pi$ to both sides to obtain\n\\begin{equation}\\label{3eqn:in proof 3.9 (2)}\n\\alpha_0 \\bs{\\Omega}_0 +\\sum_{j=1}^N (\\alpha_j+\\beta_j\\lambda_j) \\bs{\\Omega}_j=0.\n\\end{equation}\nSimilarly, applying $\\Pi S$ to both sides of \\eqref{3eqn:in proof 3.9 (1)}, we obtain\n\\begin{equation}\\label{3eqn:in proof 3.9 (3)}\n\\alpha_0 \\bs{\\Omega}_0 +\\sum_{j=1}^N (\\alpha_j\\lambda_j+\\beta_j) \\bs{\\Omega}_j=0,\n\\end{equation}\nwhere $S^2=I$ and $S\\bs{\\Omega}_0=\\bs{\\Omega}_0$ from Lemma \\ref{3lem:3.7} are taken into account.\nIt then follows from \\eqref{3eqn:in proof 3.9 (2)} and \\eqref{3eqn:in proof 3.9 (3)} that\n\\[\n\\alpha_0=0, \\quad \\alpha_j+\\beta_j\\lambda_j=\\alpha_j\\lambda_j+\\beta_j=0,\n\\quad 1\\le j\\le N.\n\\]\nSince $\\lambda_j\\neq \\pm1$ for $1\\le j\\le N$, we see that $\\alpha_j=\\beta_j=0$ for all $j$.\nThe inner product \\eqref{3eqn:inner product 3.9} is computed as follows:\n\\[\n\\langle \\bs{\\Omega}_j, S\\bs{\\Omega}_k\\rangle\n=\\langle \\bs{\\Omega}_j, \\Pi S\\bs{\\Omega}_k\\rangle\n =\\langle \\bs{\\Omega}_j, T\\bs{\\Omega}_k\\rangle\n=\\lambda_k \\langle \\bs{\\Omega}_j, \\bs{\\Omega}_k\\rangle\n=\\lambda_k\\delta_{kj}\\,.\n\\]\n\n(2) is proved similarly by\nusing Lemma \\ref{3lem:3.7} and $\\lambda_N=-1$. \n\\begin{flushright}$\\square$\\end{flushright}\n\n\\begin{lemma}\\label{4lem:U on omegas}\n{\\upshape (1)} If $r>0$ we have\n\\begin{align*}\n&U \\bs{\\Omega}_0=\\bs{\\Omega}_0, \\\\\n&U \\bs{\\Omega}_j=S\\bs{\\Omega}_j, \\quad\nU S\\bs{\\Omega}_j=-\\bs{\\Omega}_j+2\\lambda_j S\\bs{\\Omega}_j, \n\\quad 1\\le j\\le N.\n\\end{align*}\n\n{\\upshape (2)} If $r=0$, the above relations hold except $U \\bs{\\Omega}_N=-\\bs{\\Omega}_N$.\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nSince the proofs are similar we prove only (1).\nWe first observe that\n\\[\nU\\bs{\\Omega}_j\n=SC\\bs{\\Omega}_j\n=S(2\\Pi-I)\\bs{\\Omega}_j\n=2S\\Pi\\bs{\\Omega}_j -S\\bs{\\Omega}_j\n=S\\bs{\\Omega}_j\\,,\n\\quad 0\\le j\\le N.\n\\]\nFor $j=0$ we have $T\\bs{\\Omega}_0=\\bs{\\Omega}_0$ so \n$S\\bs{\\Omega}_0=\\bs{\\Omega}_0$ by Lemma \\ref{3lem:3.7}.\nHence\n\\[\nU\\bs{\\Omega}_0=SC\\bs{\\Omega}_0=\\bs{\\Omega}_0\\,.\n\\]\nWe next calculate $US\\bs{\\Omega}_j$ for $1\\le j\\le N$. \nUsing $C=2\\Pi-I$ and $S^2=I$ we obtain\n\\begin{align*}\nUS\\bs{\\Omega}_j\n&=SCS\\bs{\\Omega}_j\n=S(2\\Pi-I)S\\bs{\\Omega}_j \\\\\n&=2ST\\bs{\\Omega}_j-\\bs{\\Omega}_j\n=2\\lambda_j S\\bs{\\Omega}_j-\\bs{\\Omega}_j\\,.\n\\end{align*}\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\noindent Thus, we obtain the orthogonal decomposition of $\\mathcal{L}(N)$ defined\nin \\eqref{4eqn:def of L(N)}:\n\\begin{alignat}{2}\n\\mathcal{L}(N)\n&=\\mathbb{C}\\bs{\\Omega}_0 \\oplus\n \\sum_{j=1}^N \\oplus (\\mathbb{C}\\bs{\\Omega}_j+\\mathbb{C}S\\bs{\\Omega}_j),\n&\\qquad &r>0, \n\\label{4eqn:decomposition of L(N)}\\\\\n\\mathcal{L}(N)\n&=\\mathbb{C}\\bs{\\Omega}_0 \\oplus\n \\sum_{j=1}^{N-1} \\oplus (\\mathbb{C}\\bs{\\Omega}_j+\\mathbb{C}S\\bs{\\Omega}_j)\n \\oplus \\mathbb{C}\\bs{\\Omega}_N\\,,\n&\\quad &r=0,\n\\label{4eqn:decomposition of L(N)(r=0)}\n\\end{alignat}\nwhere each factor is invariant under the action of $U$.\n\n\\begin{theorem}\\label{3thmspectra of U}\n{\\upshape (1)} If $r>0$, the eigenvalues of $U$ are\n\\[\n1, \n\\quad e^{\\pm i\\theta_j} \\quad (1\\le j\\le N),\n\\quad -1,\n\\]\nwhere $0<\\theta_1<\\theta_2<\\dots<\\theta_N<\\pi$\nare obtained in \\eqref{3eqn:arrangement of lambda}.\nAll the eigenvalues except $-1$ are multiplicity free\nand the multiplicity of the eigenvalue $-1$ is $N-2$.\n\n{\\upshape (2)} If $r=0$, the eigenvalues of $U$ are\n\\[\n1, \n\\quad e^{\\pm i\\theta_j} \\quad (1\\le j\\le N-1),\n\\quad -1,\n\\]\nwhere $0<\\theta_1<\\theta_2<\\dots<\\theta_{N-1}<\\pi$.\nAll the eigenvalues except $-1$ are multiplicity free\nand the multiplicity of the eigenvalue $-1$ is $N$.\n\\end{theorem}\n\n\\noindent{\\it Proof.}\nSince the proofs are similar we prove only (1).\nThe orthogonal decomposition \\eqref{4eqn:decomposition of L(N)} gives rise to \na blockwise diagonalization of $U$.\nIt is obvious from Lemma \\ref{4lem:U on omegas} \nthat $U$ restricted to $\\mathbb{C}\\bs{\\Omega}_0$ is the identity operator.\nNext suppose that $1\\le j\\le N$.\nWe see from Lemma \\ref{4lem:U on omegas} that $U$\nrestricted to $\\mathbb{C}\\bs{\\Omega}_j+\\mathbb{C}S\\bs{\\Omega}_j$ \nadmits a matrix representation:\n\\[\n\\begin{bmatrix}\n0 & -1 \\\\\n1 & 2\\lambda_j\n\\end{bmatrix},\n\\]\nof which the eigenvalues are \n\\[\n\\lambda_j\\pm i\\sqrt{1-\\lambda_j^2}=e^{\\pm i\\theta_j}.\n\\]\nDenoting by $\\mathcal{M}$ the orthogonal complement of $\\mathcal{L}(N)$ in $\\mathcal{H}(N)$,\nwe have\n\\begin{align*}\n\\mathrm{Tr\\,}(U)\n&=1+\\sum_{j=1}^N2\\lambda_j + \\mathrm{Tr\\,}(U\\!\\!\\restriction_{\\mathcal{M}})\n=1+2(\\mathrm{Tr\\,}(T)-1)+\\mathrm{Tr\\,}(U\\!\\!\\restriction_{\\mathcal{M}}) \\\\\n&=2\\mathrm{Tr\\,}(T)-1+\\mathrm{Tr\\,}(U\\!\\!\\restriction_{\\mathcal{M}})\n=2r(N-1)-1+\\mathrm{Tr\\,}(U\\!\\!\\restriction_{\\mathcal{M}}).\n\\end{align*}\nOn the other hand, $\\mathrm{Tr\\,}(U)=(2r-1)(N-1)$ by Lemma \\ref{3lem:Trace of U}.\nHence\n\\[\n\\mathrm{Tr\\,}(U\\!\\!\\restriction_{\\mathcal{M}})\n=(2r-1)(N-1)-(2r(N-1)-1)\n=-(N-2).\n\\]\nSince $\\dim \\mathcal{M}=(3N-1)-(2N+1)=N-2$,\nwe see that $U\\!\\!\\restriction_{\\mathcal{M}}=-I$.\nTherefore $\\mathcal{M}$ is the eigenspace of $U$ with eigenvalue $-1$\nso that the multiplicity of the eigenvalue $-1$ coincides with $\\dim \\mathcal{M}=N-2$.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\begin{theorem}\n{\\upshape (1)} Let $r>0$ and set \n\\[\n\\bs{\\Omega}_j^{\\pm}=\\frac{1}{\\sqrt2\\, \\sin\\theta_j}\\,(\\bs{\\Omega}_j-e^{\\pm i\\theta_j}S\\bs{\\Omega}_j),\n\\quad 1\\le j\\le N.\n\\]\nThen $\\bs{\\Omega}_j^{\\pm}\\in (\\mathbb{C}\\bs{\\Omega}_j+\\mathbb{C}S\\bs{\\Omega}_j)$, \n$\\|\\bs{\\Omega}_j^{\\pm}\\|=1$ and\n\\[\nU\\bs{\\Omega}_j^{\\pm}=e^{\\pm i\\theta_j} \\bs{\\Omega}_j^{\\pm}.\n\\]\nIn other words,\n$\\bs{\\Omega}_j^{\\pm}$ is a normalized eigenvector of $U$ with eigenvalue $e^{\\pm i\\theta_j}$.\n\n{\\upshape (2)} If $r=0$, the above assertion remains valid for $1\\le j\\le N-1$.\n\\end{theorem}\n\n\\noindent{\\it Proof.}\nThat $\\|\\bs{\\Omega}_j^{\\pm}\\|=1$ is verified by using Lemma \\ref{3lem:orthogonality}.\nThat $U\\bs{\\Omega}_j^{\\pm}=e^{\\pm i\\theta_j} \\bs{\\Omega}_j^{\\pm}$ follows \nfrom Lemma \\ref{4lem:U on omegas}.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\n\\section{Proofs of main results}\n\\label{sec:Proofs of main results}\n\n\\subsection{Transition amplitudes of the $(p,q)$-quantum walk on $\\mathbb{Z}_+$}\n\nFor Theorem \\ref{mainthm:probability amplitude} we need to calculate\nthe transition amplitude:\n\\begin{equation}\\label{5eqn:goal amplitude}\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n=\\langle \\bs{\\Psi}_0, U^n \\bs{\\Psi}_0\\rangle\n\\end{equation}\nfor the $(p,q)$-quantum walk $U$ on $\\mathbb{Z}_+$.\nA key observation here is that \\eqref{5eqn:goal amplitude} is \ncalculated after cutoff.\nMore generally, if $\\bs{\\phi}, \\bs{\\psi}\\in\\mathcal{H}(\\mathbb{Z}_+)$ have\nfinite supports, \n\\begin{equation}\\label{5eqn:goal amplitude finite}\n\\langle \\bs{\\phi}, U^n \\bs{\\psi}\\rangle\n=\\langle \\bs{\\phi}, U_N^n \\bs{\\psi}\\rangle_{\\mathcal{L}(N)}\n\\end{equation}\nholds for all sufficiently large $N$.\nIn fact, if $\\mathrm{supp\\,}\\bs{\\phi}\\subset\\{0,1,\\dots,l\\}$ and\n$\\mathrm{supp\\,} \\bs{\\psi}\\subset\\{0,1,\\dots,m\\}$, then\n\\eqref{5eqn:goal amplitude finite} holds for $N>\\min\\{n+l,n+m\\}$.\nThe purpose of this subsection is to derive an integral formula for \\eqref{5eqn:goal amplitude finite}.\n\nNow let $N\\ge2$ be fixed and start with \nthe $(p,q)$-quantum walk $U$ on the path of length $N\\ge2$.\n\n\\begin{lemma}\\label{4lem:4.33}\nFor $r>0$ it holds that\n\\begin{align}\n\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^{\\pm} \\rangle \n&= \\frac{\\mp ie^{\\pm i\\theta_j}}{\\sqrt{2}}\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle, \n\\qquad 1\\le j\\le N,\n\\label{4eqn:in 4.33 (1)}\\\\\n\\langle S\\bs{\\Psi}_l,\\bs{\\Omega}_0 \\rangle\n&=\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0 \\rangle, \n\\label{4eqn:in 4.33 (3)}\\\\\n\\langle S\\bs{\\Psi}_l,\\bs{\\Omega}_j^{\\pm} \\rangle \n&= \\frac{\\mp i}{\\sqrt{2}}\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle,\n\\qquad 1\\le j\\le N,\n\\label{4eqn:in 4.33 (2)}\n\\end{align}\nFor $r=0$ the above relations remain valid for $1\\le j\\le N-1$ and\n\\begin{equation}\\label{4eqn:in 4.33 (6)}\n\\langle S\\bs{\\Psi}_l,\\bs{\\Omega}_N \\rangle\n=-\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_N \\rangle. \n\\end{equation}\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nBy definition we have\n\\begin{equation}\\label{4eqn:in proof 4.2 (1)}\n\\langle\\bs{\\Psi}_l,\\bs{\\Omega}_j^{\\pm}\\rangle\n=\\frac{1}{\\sqrt2\\,\\sin\\theta_j}\\,\\left(\n \\langle\\bs{\\Psi}_l,\\bs{\\Omega}_j\\rangle\n -e^{\\pm i\\theta_j}\\langle\\bs{\\Psi}_l,S\\bs{\\Omega}_j\\rangle\n \\right).\n\\end{equation}\nSince $\\Pi\\bs{\\Psi}_l=\\bs{\\Psi}_l$ we have\n\\[\n\\langle\\bs{\\Psi}_l,S\\bs{\\Omega}_j\\rangle\n= \\langle\\bs{\\Psi}_l,\\Pi S\\bs{\\Omega}_j\\rangle\n= \\langle\\bs{\\Psi}_l,T\\bs{\\Omega}_j\\rangle\n= \\lambda_j \\langle\\bs{\\Psi}_l,\\bs{\\Omega}_j\\rangle.\n\\]\nThen \\eqref{4eqn:in proof 4.2 (1)} becomes\n\\begin{equation}\\label{4eqn:in proof 4.2 (2)}\n\\langle\\bs{\\Psi}_l,\\bs{\\Omega}_j^{\\pm}\\rangle\n=\\frac{1}{\\sqrt2\\,\\sin\\theta_j}\\,(1-e^{\\pm i\\theta_j}\\lambda_j) \\langle\\bs{\\Psi}_l,\\bs{\\Omega}_j\\rangle.\n\\end{equation}\nWe see easily from  $\\cos\\theta_j=\\lambda_j$ that\n\\[\n1-e^{\\pm i\\theta_j}\\lambda_j=\\mp i (\\sin \\theta_j) e^{\\pm i\\theta_j}.\n\\]\nInserting the above relation into \\eqref{4eqn:in proof 4.2 (2)}, we obtain \\eqref{4eqn:in 4.33 (1)}. \n\nWe next show \\eqref{4eqn:in 4.33 (2)}.\nIn view of the definition of $\\bs{\\Omega}_j^{\\pm}$ and using $S^2=I$ we have\n\\begin{align*}\n\\langle S\\bs{\\Psi}_l,\\bs{\\Omega}_j^{\\pm}\\rangle\n&=\\frac{1}{\\sqrt2\\,\\sin\\theta_j}\\,\\left(\n \\langle S\\bs{\\Psi}_l,\\bs{\\Omega}_j\\rangle\n -e^{\\pm i\\theta_j}\\langle S\\bs{\\Psi}_l,S\\bs{\\Omega}_j\\rangle\n \\right) \\\\\n&=\\frac{1}{\\sqrt2\\,\\sin\\theta_j}\\,\\left(\n \\langle \\bs{\\Psi}_l, S\\bs{\\Omega}_j\\rangle\n -e^{\\pm i\\theta_j}\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j\\rangle\n \\right). \n\\end{align*}\nThen, applying a similar consideration as above, we obtain\n\\eqref{4eqn:in 4.33 (2)} with no difficulty.\n\nFinally, since $U\\bs{\\Omega}_0=S\\bs{\\Omega}_0=\\bs{\\Omega}_0$, we have \n\\[\n\\langle S\\bs{\\Psi}_l,\\bs{\\Omega}_0 \\rangle\n=\\langle \\bs{\\Psi}_l, S\\bs{\\Omega}_0 \\rangle\n=\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0 \\rangle, \n\\]\nwhich shows \\eqref{4eqn:in 4.33 (3)}.\nFor \\eqref{4eqn:in 4.33 (6)} we need only to note that $S\\bs{\\Omega}_N=-\\bs{\\Omega}_N$.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\n\\begin{lemma}\\label{5lem:four inner products}\nFor $0\\le l,m\\le N$ and $n=0,\\pm1,\\pm2,\\dots$ it holds that\n\\begin{align}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\sum_{j=0}^N (\\cos n\\theta_j)\n \\langle \\bs{\\Psi}_l, \\bs{\\Omega}_j\\rangle\n \\langle \\bs{\\Omega}_j, \\bs{\\Psi}_m\\rangle,  \n\\label{4eqn:5-2-01} \\\\\n\\langle S\\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\langle \\bs{\\Psi}_l,U^{n-1}\\bs{\\Psi}_m \\rangle, \n\\label{4eqn:5-2-02} \\\\\n\\langle \\bs{\\Psi}_l,U^n S\\bs{\\Psi}_m \\rangle\n&=\\langle \\bs{\\Psi}_l,U^{n+1}\\bs{\\Psi}_m \\rangle, \n\\label{4eqn:5-2-03} \\\\\n\\langle S\\bs{\\Psi}_l, U^n S\\bs{\\Psi}_m \\rangle\n&=\\langle \\bs{\\Psi}_l,U^{n}\\bs{\\Psi}_m \\rangle.\n\\label{4eqn:5-2-04}\n\\end{align}\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nBecause the proofs are similar, we prove the assertions under $r>0$. \nSince $\\bs{\\Psi}_l$ and $U^n\\bs{\\Psi}_m$ are vectors in $\\mathcal{L}(N)$, \nthe left-hand side of \\eqref{4eqn:5-2-01}\nis expanded in terms of the orthonormal basis\n$\\bs{\\Omega}_0, \\bs{\\Omega}_1^\\pm,\\dots, \\bs{\\Omega}_N^\\pm$ as follows:\n\\begin{align}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0\\rangle\\langle \\bs{\\Omega}_0, U^n\\bs{\\Psi}_m \\rangle\n +\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^+\\rangle\\langle \\bs{\\Omega}_j^+, U^n\\bs{\\Psi}_m \\rangle \n\\nonumber \\\\\n&\\qquad  +\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^-\\rangle\\langle \\bs{\\Omega}_j^-, U^n\\bs{\\Psi}_m \\rangle\n\\label{5eqn: in proof 5.2}\n\\end{align}\nThe first term becomes \n\\begin{equation}\\label{5eqn:in proof 5.2 101}\n\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0\\rangle\\langle \\bs{\\Omega}_0, U^n\\bs{\\Psi}_m \\rangle\n=\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0\\rangle\\langle U^{-n}\\bs{\\Omega}_0, \\bs{\\Psi}_m \\rangle\n=\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0\\rangle\\langle \\bs{\\Omega}_0, \\bs{\\Psi}_m \\rangle.\n\\end{equation}\nFor the second term of \\eqref{5eqn: in proof 5.2} we see that\n\\begin{align*}\n\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^+\\rangle\\langle \\bs{\\Omega}_j^+, U^n\\bs{\\Psi}_m \\rangle\n&=\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^+\\rangle\n \\langle U^{-n}\\bs{\\Omega}_j^+, \\bs{\\Psi}_m \\rangle \\\\\n&=\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^+\\rangle\n \\langle e^{-in\\theta_j}\\bs{\\Omega}_j^+, \\bs{\\Psi}_m \\rangle \\\\\n&=\\sum_{j=1}^N e^{in\\theta_j}\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^+\\rangle\n \\langle \\bs{\\Omega}_j^+, \\bs{\\Psi}_m \\rangle.\n\\end{align*}\nThen we apply Lemma \\ref{4lem:4.33} to have\n\\begin{align}\n\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^+\\rangle\\langle \\bs{\\Omega}_j^+, U^n\\bs{\\Psi}_m \\rangle\n&=\\sum_{j=1}^N e^{in\\theta_j}\n  \\frac{- ie^{i\\theta_j}}{\\sqrt{2}}\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle\n  \\frac{ie^{-i\\theta_j}}{\\sqrt{2}}\\langle\\bs{\\Omega}_j, \\bs{\\Psi}_m \\rangle \n\\nonumber \\\\\n&=\\sum_{j=1}^N \\frac{e^{in\\theta_j}}{2}\\,\n  \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle\n  \\langle\\bs{\\Omega}_j, \\bs{\\Psi}_m \\rangle.\n\\label{5eqn:in proof 5.2 102}\n\\end{align}\nApplying a similar argument to the third term of \\eqref{5eqn: in proof 5.2}, we obtain \n\\begin{equation}\\label{5eqn:in proof 5.2 103}\n\\sum_{j=1}^N \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j^-\\rangle\\langle \\bs{\\Omega}_j^-, U^n\\bs{\\Psi}_m \\rangle\n=\\sum_{j=1}^N \\frac{e^{-in\\theta_j}}{2}\\,\n  \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle\n  \\langle\\bs{\\Omega}_j, \\bs{\\Psi}_m \\rangle.\n\\end{equation}\nSumming up \\eqref{5eqn:in proof 5.2 101}--\\eqref{5eqn:in proof 5.2 103}, we see that\n\\eqref{5eqn: in proof 5.2} becomes\n\\begin{align*}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\langle \\bs{\\Psi}_l,\\bs{\\Omega}_0\\rangle\\langle \\bs{\\Omega}_0, \\bs{\\Psi}_m \\rangle\n+\\sum_{j=1}^N \\frac{e^{in\\theta_j}+e^{-in\\theta_j}}{2}\\,\n  \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle\n  \\langle\\bs{\\Omega}_j, \\bs{\\Psi}_m \\rangle \\\\\n&=\\sum_{j=0}^N (\\cos n\\theta_j)\n  \\langle \\bs{\\Psi}_l,\\bs{\\Omega}_j \\rangle\n  \\langle\\bs{\\Omega}_j, \\bs{\\Psi}_m \\rangle,\n\\end{align*}\nwhere $\\theta_0=0$ is taken into account.\nThus, \\eqref{4eqn:5-2-01} is proved.\n\nNoting that $U=SC$ and $C$ acts on $\\Gamma(N)$ as the identity, we see that\n\\begin{align*}\n\\langle S\\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\langle S\\bs{\\Psi}_l, SCU^{n-1}\\bs{\\Psi}_m \\rangle \\\\\n&=\\langle \\bs{\\Psi}_l, CU^{n-1}\\bs{\\Psi}_m \\rangle \\\\\n&=\\langle C\\bs{\\Psi}_l, U^{n-1}\\bs{\\Psi}_m \\rangle \\\\\n&=\\langle \\bs{\\Psi}_l, U^{n-1}\\bs{\\Psi}_m \\rangle,\n\\end{align*}\nwhich proves \\eqref{4eqn:5-2-02}.\nWe next observe that\n\\[\n\\langle \\bs{\\Psi}_l,U^n S\\bs{\\Psi}_m \\rangle\n=\\langle \\bs{\\Psi}_l,U^n SC\\bs{\\Psi}_m \\rangle \n=\\langle \\bs{\\Psi}_l,U^{n+1}\\bs{\\Psi}_m \\rangle,\n\\]\nwhich proves \\eqref{4eqn:5-2-03}.\nFinally, we see that\n\\[\n\\langle S\\bs{\\Psi}_l, U^n S\\bs{\\Psi}_m \\rangle\n=\\langle SC\\bs{\\Psi}_l, U^n SC\\bs{\\Psi}_m \\rangle\n=\\langle U\\bs{\\Psi}_l, U^{n+1}\\bs{\\Psi}_m \\rangle\n=\\langle \\bs{\\Psi}_l,U^{n}\\bs{\\Psi}_m \\rangle,\n\\]\nwhich shows \\eqref{4eqn:5-2-04}.\n\\begin{flushright}$\\square$\\end{flushright}\n\nLet $\\{P_n\\,;\\,n=0,1,2,\\dots\\}$ be the orthogonal polynomials with respect to the free Meixner law with\nparameters $q,pq,r$, i.e.,\nthe polynomials defined by the Jacobi parameters\n\\[\n\\omega_1=q,\n\\,\\,\\,\n\\omega_2=\\omega_3=\\dots=pq;\n\\quad\n\\alpha_1=0,\n\\,\\,\\,\n\\alpha_2=\\alpha_3=\\dots=r,\n\\]\nsee also Appendix.\nWe set\n\\begin{align}\np_0(x)\n&=P_0(x)=1, \n\\nonumber \\\\\np_j(x)\n&=\\frac{P_j(x)}{\\sqrt{\\mathstrut\\omega_1\\dots \\omega_j}}\n =\\frac{P_j(x)}{\\sqrt{q(pq)^{j-1}}}\\,,\n\\quad j=1,2,\\dots,N.\n\\label{5eqn:def of p_j}\n\\end{align}\nIt is shown that $\\{p_j\\,;\\, 0\\le j\\le N\\}$ satisfies the recurrence relations\ndetermined by $T_N$.\nWe define \n\\[\n\\mu_N=\\sum_{j=0}^N \\rho(j)\\delta_{\\lambda_j}\\,,\n\\qquad\n\\rho(j)=\\rho_N(j)=\\bigg(\\sum_{n=0}^N p_n(\\lambda_j)^2\\bigg)^{-1}.\n\\]\nThe following results are known by general theory of Jacobi matrices and orthogonal polynomials. \\cite{Deift,HO}\n\n\\begin{lemma}\n$\\mu_N$ is a probability distribution uniquely determined by the Jacobi matrix $T_N$.\nMoreover, $\\{p_j\\,;\\,j=0,1,2,\\dots,N\\}$ is the orthogonal polynomials with respect to $\\mu_N$, normalized \nso as to have norm one, i.e.,\n\\[\n\\int_{-1}^1 p_j(x)p_k(x)\\mu_N(dx)=\\delta_{jk}\\,.\n\\]\n\\end{lemma}\n\n\\begin{lemma}\\label{5lem:explicit form of Omega_j}\nFor $j=0,1,\\dots,N$ \nlet $\\bs{\\Omega}_j$ be the normalized eigenvector of $T_N$ with eigenvalue $\\lambda_j$ such that\n$\\langle \\bs{\\Omega}_j, \\bs{\\Psi}_0\\rangle>0$.\nThen, \n\\[\n\\bs{\\Omega}_j=\\sqrt{\\rho_N(j)}\\sum_{n=0}^N p_n(\\lambda_j)\\bs{\\Psi}_n\\,,\n\\]\nor equivalently,\n\\[\n\\langle \\bs{\\Omega}_j, \\bs{\\Psi}_n\\rangle\n=\\sqrt{\\rho_N(j)}\\, p_n(\\lambda_j).\n\\]\n\\end{lemma}\n\nThe next result is a key for removing the cutoff.\n\n\\begin{lemma}\\label{4lem:limit free Meixner law}\nThe sequence of probability distributions $\\mu_N$ converges weakly to the\nfree Meixner law with parameters $q,pq,r$.\nIn particular, for any continuous function $f$ on $[-1,1]$ we have\n\\[\n\\lim_{N\\rightarrow\\infty} \\int_{-1}^1 f(x)\\mu_N(dx)= \\int_{-1}^1 f(x)\\mu(dx).\n\\]\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nWe first note that\n\\begin{equation}\\label{eqn:convergence in moments}\n\\lim_{N\\rightarrow\\infty} \\int_{-\\infty}^{+\\infty} x^m\\mu_N(dx)= \\int_{-1}^1 x^m\\mu(dx),\n\\qquad m=0,1,2,\\dots.\n\\end{equation}\nIn fact, the $m$-th moment of $\\mu_N$ is a polynomial in the\nfirst $m$ terms of the Jacobi coefficients of $\\mu_N$,\nwhich are identical with the first $m$ terms of the Jacobi coefficients of the free\nMeixner law $\\mu$ if $m<N$.\nSince the free Meixner law has a compact support,\nit is uniquely determined by the moment sequence.\nTherefore, it follows by general theory  \nthat \\eqref{eqn:convergence in moments} implies the weak convergence\nof $\\mu_N$ to $\\mu$.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\n\\begin{theorem}[Integral representation of transition amplitude]\n\\label{5thm:integral representation of four amplitudes}\nLet $U$ be the $(p,q)$-quantum walk on $\\mathbb{Z}_+$\nand $\\mu$ the free Meixner law with parameters $q,pq,r$.\nFor any $l,m\\in\\mathbb{Z}_+$ and $n=0,\\pm1,\\pm2,\\dots$ it holds that\n\\begin{equation}\\label{4eqn:5-2-11}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n=\\int_{-1}^1 (\\cos n\\theta)\\,p_l(\\lambda)p_m(\\lambda)\\,\\mu(d\\lambda),\n\\quad \\cos\\theta=\\lambda.\n\\end{equation}\nMoreover,\n\\begin{align*}\n\\langle S\\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\int_{-1}^1 (\\cos (n-1)\\theta)\\,p_l(\\lambda)p_m(\\lambda)\\,\\mu(d\\lambda),\n\\\\\n\\langle \\bs{\\Psi}_l,U^n S\\bs{\\Psi}_m \\rangle\n&=\\int_{-1}^1 (\\cos (n+1)\\theta)\\,p_l(\\lambda)p_m(\\lambda)\\,\\mu(d\\lambda),\n\\\\\n\\langle S\\bs{\\Psi}_l, U^n S\\bs{\\Psi}_m \\rangle\n&=\\int_{-1}^1 (\\cos n\\theta)\\,p_l(\\lambda)p_m(\\lambda)\\,\\mu(d\\lambda).\n\\end{align*}\n\\end{theorem}\n\n\\noindent{\\it Proof.}\nSince $\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle$ coincides with the\nsimilar expression for $(p,q)$-quantum walk on the path of length $N> \\min\\{l+n, m+n\\}$.\nWe take such a sufficiently large $N$.\nBy Lemmas \\ref{5lem:four inner products} and \\ref{5lem:explicit form of Omega_j} we have\n\\begin{align}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\sum_{j=0}^N (\\cos n\\theta_j)\n \\langle \\bs{\\Psi}_l, \\bs{\\Omega}_j\\rangle\n \\langle \\bs{\\Omega}_j, \\bs{\\Psi}_m\\rangle \n\\nonumber \\\\\n&=\\sum_{j=0}^N (\\cos n\\theta_j)\\, p_l(\\lambda_j) p_m(\\lambda_j) \\rho(j) \n\\nonumber \\\\\n&=\\int_{-1}^1 (\\cos n\\theta)\\, p_l(\\lambda)p_m(\\lambda) \\mu_N(d\\lambda),\n\\end{align}\nwhich holds for all sufficiently large $N$.\nThen, taking Lemma \\ref{4lem:limit free Meixner law} into account, we come to\n\\begin{align*}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_m \\rangle\n&=\\lim_{N\\rightarrow\\infty}\\int_{-1}^1  (\\cos n\\theta)\\,p_l(\\lambda)p_m(\\lambda) \\mu_N(d\\lambda) \\\\\n&=\\int_{-1}^1 (\\cos n\\theta)\\, p_l(\\lambda)p_m(\\lambda) \\mu(d\\lambda).\n\\end{align*}\nThis completes the proof of \\eqref{4eqn:5-2-11}.\nThe rest is proved by combination of\nLemma \\ref{5lem:four inner products} and \\eqref{4eqn:5-2-11}.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\\begin{theorem}\\label{5thm:localization criteria}\nLet $U$ be the $(p,q)$-quantum walk on $\\mathbb{Z}_+$ with parameters satisfying\n\\begin{equation}\\label{5eqn:condition for pqr}\np+q+r=1,\n\\qquad p\\ge q>0,\n\\qquad r\\ge0.\n\\end{equation}\nThen it holds that\n\\[\n\\langle \\bs{\\Psi}_l, U^n \\bs{\\Psi}_0\\rangle\n\\sim\nw p_l(\\xi)\\cos n \\tilde\\theta,\n\\qquad\n\\text{as $n\\rightarrow\\infty$},\n\\]\nwhere\n\\[\nw=\\max\\left\\{\\frac{(1-p)^2-pq}{(1-p)(1-p+q)}\\,,0\\right\\},\n\\quad\n\\xi=-\\frac{q}{1-p}=\\cos\\tilde\\theta,\n\\quad\n0<\\tilde\\theta <\\pi.\n\\]\nTherefore,\n\\[\n\\lim_{N\\rightarrow\\infty}\\frac1N\\sum_{n=0}^{N-1}\n|\\langle \\bs{\\Psi}_0, U^n \\bs{\\Psi}_0\\rangle|^2\n=\\frac{w^2}{2}\\,.\n\\]\nIn particular, $U$ exhibits the initial point localization \nif and only if $w>0$, i.e., $(1-p)^2-pq>0$.\n\\end{theorem}\n\n\\noindent{\\it Proof.}\nBy \\eqref{4eqn:5-2-11} we have\n\\begin{equation}\\label{5eqn:in proof Theorem 5.7} \n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_0 \\rangle\n=\\int_{-1}^1 (\\cos n\\theta)\\,p_l(\\lambda)\\,\\mu(d\\lambda),\n\\quad \\cos\\theta=\\lambda.\n\\end{equation}\nUnder the assumption \\eqref{5eqn:condition for pqr}\nthe free Meixner law with parameters $q,pq,r$ is of the form:\n\\[\n\\mu(dx)=\\rho(x)dx+w\\delta_\\xi \\,,\n\\]\nwhere $\\rho$ is a continuous function on $[r-2\\sqrt{pq}\\,, r+2\\sqrt{pq}\\,]\\subset [-1,1]$,\nan explicit form is deferred in Appendix, and\n\\[\n\\xi=-\\frac{q}{1-p}\\,,\n\\quad\nw=\\max\\left\\{\\frac{(1-p)^2-pq}{(1-p)(1-p+q)}\\,,0\\right\\}.\n\\]\nSince $\\rho$ is an integrable function, the Riemann--Lebesgue lemma implies that\n\\[\n\\lim_{n\\rightarrow\\infty} \\int_{-1}^{1} (\\cos n\\theta)\\,p_l(\\lambda) \\rho(\\lambda) \\, d\\lambda=0.\n\\]\nHence in \\eqref{5eqn:in proof Theorem 5.7} only contribution by the point mass\nremains in the limit, i.e.,\n\\[\n\\langle \\bs{\\Psi}_l, U^n \\bs{\\Psi}_0\\rangle\n\\sim\nw p_l(\\xi) \\cos n \\tilde\\theta,\n\\qquad\n\\text{as $n\\rightarrow\\infty$},\n\\]\nas desired.\nThe rest is straightforward.\n\\begin{flushright}$\\square$\\end{flushright}\n\n\n\\subsection{Proof of Theorem \\ref{mainthm:probability amplitude}}\n\nLet $U$ be the Grover walk on a spidernet $G=S(a,b,c)$ and consider\nthe initial state $\\bs{\\psi}^+_0$ defined by \\eqref{3eqn:initial state}.\nLet\n\\[\n\\Gamma(\\mathbb{Z}_+)\\subset\\mathcal{H}(\\mathbb{Z}_+)\\subset \\mathcal{H}(G)\n\\]\nbe the subspaces defined in Subsection \\ref{subsec:4.1}.\nThen ${\\mathcal{H}(\\mathbb{Z}_+)}$ is invariant under $U$ and\n$U\\!\\!\\restriction_{\\mathcal{H}(\\mathbb{Z}_+)}$ is the $(p,q)$-quantum walk on $\\mathbb{Z}_+$,\nwhere\n\\begin{equation}\\label{5eqn:pqr by abc}\np=\\frac{c}{b}\\,,\n\\qquad\nq=\\frac{1}{b}\\,,\n\\qquad\nr=\\frac{b-c-1}{b}\\,.\n\\end{equation}\nSince the initial state $\\bs{\\psi}^+_0=\\bs{\\Psi}_0$ belongs to $\\mathcal{H}(\\mathbb{Z}_+)$,\n$\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle$ is obtained from the\n$(p,q)$-quantum walk on $\\mathbb{Z}_+$.\nIn fact, by Theorem \\ref{5thm:integral representation of four amplitudes}\nwe have\n\\begin{equation}\\label{5eqn:main representation}\n\\langle \\bs{\\psi}_0^+, U^n \\bs{\\psi}_0^+\\rangle\n=\\int_{-1}^{1} (\\cos n\\theta) \\,\\mu(d\\lambda),\n\\quad\n\\lambda=\\cos\\theta,\n\\end{equation}\nwhere $\\mu$ be the free Meixner law with parameters $q,pq,r$.\nThis completes the proof of Theorem \\ref{mainthm:probability amplitude}.\n\n\n\n\\subsection{Proof of Theorem \\ref{3thm:localization for spidernet}}\n\nFor a spidernet $G=S(a,b,c)$ the parameters $p,q,r$ defined by\n\\eqref{5eqn:pqr by abc} satisfies the condition in\nTheorem \\ref{5thm:localization criteria}.\nSo it holds that\n\\[\n\\langle \\bs{\\Psi}_0, U^n \\bs{\\Psi}_0\\rangle\n\\sim\nw \\cos n \\tilde\\theta,\n\\qquad\n\\text{as $n\\rightarrow\\infty$},\n\\]\nwhere\n\\begin{align}\nw&=\\max\\left\\{\\frac{(1-p)^2-pq}{(1-p)(1-p+q)}\\,,0\\right\\},\n\\label{5eqn: theta and w} \\\\\n\\xi&=-\\frac{q}{1-p}=\\cos\\tilde\\theta,\n\\quad 0<\\tilde\\theta <\\pi.\n\\label{5eqn: c and tilde theta}\n\\end{align}\nFor the first half of Theorem \\ref{3thm:localization for spidernet} \nit is sufficient to apply the following obvious relations:\n\\[\n\\frac{(1-p)^2-pq}{(1-p)(1-p+q)}=\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\,,\n\\qquad\n-\\frac{q}{1-p}=-\\frac{1}{b-c}\\,.\n\\]\nFor the second half we need only to note that\n$(b-c)^2-c>0$ is equivalent to $b>c+\\sqrt{c}$ under the assumption\n\\eqref{3eqn:constraint abc} posed at the beginning.\n\n\\subsection{Proofs of Corollaries \\ref{3cor:k<10}--\\ref{tree}}\n\nThese follow immediately from Theorem \\ref{3thm:localization for spidernet}.\nWe need only to check the parameters.\nFor a spidernet $S(\\kappa,\\kappa+2,\\kappa-1)$ we have\n\\begin{align*}\n\\xi&=\\cos\\tilde\\theta=-\\frac{1}{b-c}=-\\frac{1}{3}\\,, \\\\\nw&=\\max\\left\\{\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\,,0\\right\\}\n=\\max\\left\\{\\frac{10-\\kappa}{12}\\,,0\\right\\}.\n\\end{align*}\nWhile, for a spidernet $S(a,b,b-1)$ we have\n\\[\n(b-c)^2-c=-(b-2)\\le 0,\n\\qquad \\kappa\\ge2,\n\\]\nwhich implies $w=0$.\n\n\n\\subsection{Proof of Theorem \\ref{local}}\n\nIn a similar manner as in the proof of Theorem \\ref{3thm:localization for spidernet}\nwe see that\n\\begin{equation}\\label{4eqn:5-2-110}\n\\langle \\bs{\\Psi}_l,U^n\\bs{\\Psi}_0 \\rangle\n\\sim w p_l(\\xi) \\cos n\\tilde\\theta,\n\\quad\\text{as $n\\rightarrow\\infty$},\n\\end{equation}\nwhere $w$, $\\xi$, $\\tilde\\theta$ are given by \\eqref{5eqn: theta and w} and \\eqref{5eqn: c and tilde theta}.\nThe value $p_l(\\xi)$ is known explicitly from Lemma \\eqref{5lem:special value} below:\n\\[\np_l(\\xi)=\\frac{1}{\\sqrt{p}}\n\\bigg(-\\frac{\\sqrt{\\mathstrut pq}}{1-p}\\bigg)^l\n=\\sqrt{\\frac{b}{c}}\\left(-\\frac{\\sqrt{c}}{b-c}\\right)^l,\n\\qquad l=1,2,\\dots.\n\\]\nThen the time averaged limit probability is given by\n\\begin{align}\n&\\lim_{N\\rightarrow\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}\n |\\langle \\bs{\\Psi}_l, U^n\\bs{\\Psi}_0 \\rangle |^2\n=\\frac{w^2}{2}p_l(\\xi)^2\n\\nonumber \\\\\n&\\qquad\n=\\frac12\\left\\{\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\right\\}^2\n \\times \\frac{b}{c} \\left\\{\\frac{c}{(b-c)^2}\\right\\}^l.\n\\label{exponential(0)} \n\\end{align}\nWe here use the following rather obvious result.\n\n\\begin{lemma}\\label{5lem:5.7}\nLet $U$ be the Grover walk on a spidernet $S(a,b,c)$ with an initial state $\\bs{\\psi}_0^+$.\nThen we have\n\\begin{equation}\\label{5eqn:in lem 5.7(0)}\nP(X_n\\in V_l)\n\\ge|\\langle \\bs{\\Psi}_l, U^n\\bs{\\Psi}_0\\rangle|^2,\n\\qquad l=1,2,\\dots.\n\\end{equation}\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nWe first note the obvious inequality:\n\\begin{align}\nP(X_n\\in V_l)\n&=\\sum_{u\\in V_l} \\sum_{v\\sim u} \n |\\langle \\bs{\\delta}_{(u,v)}, U^n\\bs{\\psi}_0^+\\rangle|^2 \n\\nonumber \\\\\n&\\ge \\left|\\left\\langle  \\frac{1}{\\sqrt{b |V_l|}}\\sum_{u\\in V_l} \\sum_{v\\sim u} \n  \\bs{\\delta}_{(u,v)}, U^n\\bs{\\Psi}_0\\right\\rangle\\right|^2.\n\\label{5eqn:in proof 5.7}\n\\end{align}\nOn the other hand, from the definitions \\eqref{3eqn:def of psi+}--\\eqref{3eqn:def of psi-} we see that\n\\[\n\\sum_{u\\in V_l} \\sum_{v\\sim u} \\bs{\\delta}_{(u,v)}\n=\\sqrt{\\mathstrut ac^{l}}\\,\\bs{\\psi}_l^+ \n +\\sqrt{\\mathstrut a(b-c-1)c^{l-1}}\\,\\bs{\\psi}_l^\\circ\n  +\\sqrt{\\mathstrut ac^{l-1}}\\, \\bs{\\psi}_l^-.\n\\]\nThen, noting that $|V_l|=ac^{l-1}$ for $l\\ge1$, we obtain\n\\[\n\\frac{1}{\\sqrt{b |V_l|}}\\sum_{u\\in V_l} \\sum_{v\\sim u} \\bs{\\delta}_{(u,v)}\n=\\sqrt{p}\\,\\bs{\\psi}_l^+ +\\sqrt{r}\\,\\bs{\\psi}_l^\\circ+\\sqrt{q}\\, \\bs{\\psi}_l^-\n=\\bs{\\Psi}_l\\,.\n\\]\nInserting the above relation into \\eqref{5eqn:in proof 5.7},\nwe obtain \\eqref{5eqn:in lem 5.7(0)}.\n\\begin{flushright}$\\square$\\end{flushright}\n\nApplying Lemma \\ref{5lem:5.7} to \\eqref{exponential(0)}, we obtain\n\\begin{equation}\\label{5eqn:1 st of theorem}\n\\liminf_{N\\rightarrow\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1}P(X_n\\in V_l)\n\\ge \\frac{b}{2c}\\left\\{\\frac{(b-c)^2-c}{(b-c)(b-c+1)}\\right\\}^2\n \\left\\{\\frac{c}{(b-c)^2}\\right\\}^l,\n\\end{equation}\nwhich proves the first half of Theorem \\ref{exponential localization}.\nIf the spidernet $S(a,b,c)$ is rotationally symmetric around $o$, we have\n\\[\nP(X_n=u)\n=\\frac{1}{|V_l|}\\, P(X_n\\in V_l),\n\\qquad \\partial(o,u)=l.\n\\]\nThen the second half of Theorem \\ref{exponential localization} follows by\ndividing \\eqref{5eqn:1 st of theorem} by $|V_l|=ab^{l-1}$.\n\nFinally, we calculate the value of $p_l(x)$ at $x=\\xi$.\nThe result is somehow amazing and plays a key role in showing the exponential localization.\n\n\\begin{lemma}\\label{5lem:special value}\nLet $p,q,r$ be constant numbers satisfying\n\\[\np>0,\\quad\nq>0, \\quad\nr=1-p-q\\ge0, \\quad\n(1-p)^2-pq>0.\n\\]\nLet $\\{p_n\\}$ be the orthogonal polynomials \nassociated with the free Meixner law with parameters $q,pq,r$,\nnormalized to have norm one as before, see \\eqref{5eqn:def of p_j}.\nThen we have\n\\[\np_n\\left(-\\frac{q}{1-p}\\right)=\\frac{1}{\\sqrt{p}}\n\\left(-\\frac{\\sqrt{\\mathstrut pq}}{1-p}\\right)^n,\n\\quad n=1,2,\\dots.\n\\]\n\\end{lemma}\n\n\\noindent{\\it Proof.}\nWe see from Theorem \\ref{Athm: OP for free Meixner} that\nthe orthogonal polynomials $\\{P_n\\}$ associated with the free Meixner law with parameters $q,pq,r$\nverify\n\\begin{equation}\\label{5eqn:in 5.9 (1)}\nP_n(x)=\\frac{(xR_+(x)-2q)R_+(x)^{n-1}-(xR_-(x)-2q)R_-(x)^{n-1}}{2^{n-1}(R_+(x)-R_-(x))}\\,,\n\\quad n\\ge1,\n\\end{equation}\nwhere\n\\[\nR_{\\pm}(x)=x-r \\pm\\sqrt{\\mathstrut (x-r)^2-4pq}\\,,\n\\quad\n(x-r)^2-4pq>0.\n\\]\nWe need to compute the value of $P_n(x)$ at $\\xi=-q\/(1-p)$. \nNoting first that\n\\[\n(\\xi-r)^2-4pq\n=\\left(-\\frac{q}{1-p}-r\\right)^2-4pq\n=\\left\\{\\frac{(1-p)^2-pq}{1-p}\\right\\}^2,\n\\]\nwe obtain\n\\[\nR_+(\\xi)=-\\frac{2pq}{1-p}\\,,\n\\qquad\nR_-(\\xi)=-2(1-p),\n\\]\nand hence \n\\begin{gather*}\n\\xi R_+(\\xi)-2q=\\frac{2q(pq-(1-p)^2)}{(1-p)^2}\\,,\n\\qquad\n\\xi R_-(\\xi)-2q=0, \\\\\n\\xi(R_+(\\xi)-R_-(\\xi))=\\xi R_+(\\xi)-2q.\n\\end{gather*}\nThen putting $x=\\xi$ in \\eqref{5eqn:in 5.9 (1)} we have\n\\begin{align*}\nP_n(\\xi)\n&=\\frac{(\\xi R_+(\\xi)-2q)R_+(\\xi)^{n-1}}{2^{n-1}(R_+(\\xi)-R_-(\\xi))} \\\\\n&=\\frac{\\xi}{2^{n-1}}\\,R_+(\\xi)^{n-1} \\\\\n&=\\frac{1}{p}\\left(-\\frac{pq}{1-p}\\right)^n,\n\\qquad n=1,2,\\dots.\n\\end{align*}\nFinally, in view of \\eqref{5eqn:def of p_j} we have\n\\[\np_n(x)\n=\\frac{P_n(x)}{\\sqrt{\\mathstrut q(pq)^{n-1}}}\n=\\frac{1}{\\sqrt{\\mathstrut p}}\\left(-\\frac{\\sqrt{\\mathstrut pq}}{1-p}\\right)^n,\n\\quad n=1,2,\\dots.\n\\]\nThis completes the proof.\n\\begin{flushright}$\\square$\\end{flushright}\n\\par\n\\par\\noindent\n\\noindent\n{\\bf Acknowledgments.}\nNK was partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the \nPromotion of Science (Grant No. 21540118). \nNO was partially supported by the CREST project ``A Mathematical Challenge to a New Phase of Material Sciences\" (2008--2014)\nof Japan Science and Technology Agency.\n\\par\n\\\n\\par\n\n\\begin{small}\n\\bibliographystyle{jplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nModelling insurance claim sizes is not only essential in various actuarial applications including pricing and risk management, but is also very challenging due to several peculiar characteristics of claim severity distributions. Claim size distributions often exhibit multimodality for small and moderate claims, when there exists unobserved heterogeneities possibly reflected by different claim types and accident causes, or when the observed samples come from a contaminated distribution. Also, the distribution is usually heavy-tail in nature, where very large claims occur with a small but non-negligible probability.\nDue to the highly complex distributional characteristics, we have to admit the impossibility to perfectly capture all the distributional features using a parametric model without excessively large number of parameters (which results in over-fitting). When model misspecification is unavoidable, correct specification of the tail part is more important than finely capturing the distributional nodes of smaller claims which are rather immaterial to the insurance portfolio, because the large claims are the losses which can severely damage the portfolio. As a result, we need to specify an appropriate distributional model with a justifiable statistical inference approach which not only preserves sufficient flexibility to appropriately capture the whole severity distribution, but also puts a particular emphasis on robust estimation of the tail.\n\n\nExisting actuarial loss modelling literature focus a lot on the model specifications, by introducing various distributional classes to capture the peculiar characteristics of claim severity distributions. Notable directions include extreme value distributions (EVD, see e.g. \\cite{embrechts1999extreme}) to capture the heavy-tailedness, composite loss modelling (see e.g. \\cite{cooray2005modeling}, \\cite{scollnik2007composite}, \\cite{bakar2015modeling} and \\cite{grun2019extending}) to cater for mismatch between body and tail behavior of claim severity distributions, and finite mixture model (FMM, see e.g. \\cite{LEE2010modeling} and \\cite{MILJKOVIC2016387}) to capture distributional multimodality.\nIn particular, FMM is becoming an increasingly useful smooth density estimation tool in insurance claim severity modelling perspective, due to its high versatility theoretically justified by denseness theorems (\\cite{LEE2010modeling}).\nThe mismatch between its body and tail behavior can also be easily modelled by FMM by selecting varying distributional classes among mixture component functions (see e.g. \\cite{BLOSTEIN201935} and \\cite{fung2021mixture}), including both light-tailed and heavy-tailed distributions. In both actuarial research and practice, statistical inferences of FMM are predominantly based on maximum likelihood estimation (MLE) with the use of Expectation-Maximization (EM) algorithm.\n\nNonetheless, MLE would often cause tail-robustness issues where the tail part of the fitted model is very sensitive to model misspecifications -- when the observations are generated from a perturbed and\/or contaminated distribution.\nAs evidenced by several empirical studies including \\cite{fung2021mixture} and \\cite{wuthrich2021statistical}, the estimated tail part of the FMM obtained by MLE can be unreliable and highly unstable in most practical cases. This is mainly due to the overlapping density regions between mixture components modelling small to moderate claims (body) and those modelling large claims (tail). Hence, the estimated tail distribution will be heavily influenced by some smaller claims if FMM is not able to fully explain those small claims, which is always the case in practice due to the distributional complexity of real dataset impossible to be perfectly captured even by flexible density approximation tools including FMM. Under MLE approach, FMM may fail to extrapolate well the large claims, and this would lead to serious implications to insurance pricing and risk management perspectives. It is therefore natural to question whether or not MLE is still a plausible approach in modelling actuarial claim severity data, and whether or not there exists an alternative statistical inference tool which better addresses our modelling challenges and outperforms the MLE.\n\nRobust statistical inference methods for heavy tail distributions are relatively scarce in actuarial science literature. \nNotable contributions include \\cite{brazauskas2000robust}, \\cite{serfling2002efficient}, \\cite{brazauskas2003favorable} and \\cite{dornheim2007robust} who adopt various kinds of statistical inference tools, such as quantile, trimmed mean, trimmed-M and generalized median estimators, to robustly estimate Gamma, Pareto and Log-normal distributions.\nRecent actuarial works study several variations of the method of moments (MoM) for robust estimations of Pareto and Log-normal loss models. Notable contributions in this direction include \\cite{brazauskas2009robust}, \\cite{poudyal2021robust} (trimmed moments), \\cite{zhao2018robust} (winsorized moments) and \\cite{poudyal2021truncated} (truncated moments).\nNote that these research works address robustness issue against the upper outliers by reducing the influence of few extreme observations to the estimated model parameters. This outlier-robustness issue is however very different from the tail-robustness issue mentioned above as the key motivation of this paper, where the contaminations on the body part affects the tail extrapolations. Very few research works look into this ``non-standard\" tail-robustness issue. Notable contributions are \\cite{beran2012robust} and \\cite{gong2018robust} who propose a huberization of the MLE, which protects against perturbations and misspecifications in the body part of distribution, to robustly estimate the tail index of Pareto and Weibull distributions. \nAll of the above existing approaches focus solely on one or two-parameter distributions. A general robust tail estimation strategy under multi-parameter flexible models like FMM is lacking.\n\nMotivated by the aforementioned tail-robustness issue in insurance context, we propose a new maximum weighted likelihood estimation (MWLE) approach for robust heavy-tail modelling of FMM. \nUnder the MWLE, an observation-dependent weight function is introduced to the log-likelihood, de-emphasizing the contributions of smaller claims and hence reducing their influence to the estimated tail part of FMM. Down-weighting small claims is also natural in insurance loss modelling perspective, as mentioned in the beginning of this section, accurate modelling of the large claims is more important than the smaller claims. \nTo offset the bias caused by the weighting scheme, we also include an adjustment term in the weighted log-likelihood, which can be interpreted as the effects of randomly truncating the observations. With the bias adjustment term, we prove that estimated parameters under the proposed MWLE is consistent and asymptotically normal with any pre-specified choices of weight functions.\nAlso, under some specific choices of weight functions, we will show that the MWLE tail index, which determines the tail heaviness of a distribution, is consistent, even under model misspecifications where the true model is not FMM. Therefore, MWLE can be regarded as a generalized alternative framework of Hill estimator (\\cite{hill1975simple}). Furthermore, with a probabilistic interpretation of the proposed MWLE approach, it is still possible to derive a Generalized EM (GEM) algorithm to efficiently estimate parameters which maximize the weighted log-likelihood function.\n\nNote that the proposed MWLE is different from the existing statistics papers which adopt weighting schemes for likelihood-based inference. The existing literature are mainly motivated by one of the following two aspects very different from the focus of this paper: (i) Robustness against upper and lower outliers, where related research works include e.g. \\cite{fieldsmith1994}, \\cite{markatou1997weighted}, \\cite{markatou2000mixture}, \\cite{dupuis2002robust}, \\cite{ahmed2005robust}, \\cite{wong2014robust} and \\cite{aeberhard2021robust}; (ii) Bringing in more relevant observations for statistical inference to increase precision while trading off some biases, studied by e.g. \\cite{wang2001maximum}, \\cite{hu2002weighted}, \\cite{wang2004asymptotic} and \\cite{wang2005selecting}. Note also that our proposed MWLE stems differently from the existing statistics literature in terms of mathematical technicality, since none of the above papers incorporate the truncation-based bias adjustment as included in the proposed WMLE.\n\nThe rest of this paper is structured as follows. In Section \\ref{sec:fmm}, we briefly revisit the class of FMM for heavy tail modelling. Section \\ref{sec:MWLE} introduces the proposed MWLE for robust heavy-tail modelling of FMM and explains its motivations in terms of insurance claims modelling. Section \\ref{sec:theory} explores several theoretical properties to understand and justify the proposed MWLE. After that, we present in Section \\ref{sec:em} two types of GEM algorithms for efficient parameter estimations under the MWLE approach on FMM. In Section \\ref{sec:ex}, we analyze the performance of the proposed MWLE through three empirical examples: a toy example, a simulation study and a real insurance claim severity dataset. After showing the superior performance of MWLE compared to MLE, we finally summarize our findings in Section \\ref{sec:discussion} with a brief discussion how the proposed MWLE can be extended to a regression framework.\n\n\n\n\\section{Finite mixture model} \\label{sec:fmm}\nThis section provides a very brief review on finite mixture model (FMM) which serves as a flexible density estimation tool. Suppose that there are $n$ i.i.d. claim severities $\\bm{Y}=(Y_1,\\ldots, Y_n)$ with realizations $\\bm{y}=(y_1,\\ldots, y_n)$. $Y_i$ is generated by a probability distribution of $G(\\cdot)$ with density function $g(\\cdot)$ which is unknown. In insurance context, claim severity distribution often exhibits multimodality, which results from the unobserved heterogeneity stemming from the amalgamation of different types of claims unobserved in advance. Also, claim sizes are often heavy-tail in nature, which can be attributed to a few large losses from a portfolio of policies which usually represent the greatest part of the indemnities paid by the insurance company. The mismatch between body and tail behavior often poses difficulties to fit the data well using only a standard parametric distribution.\n\nMotivated by the challenges of modelling insurance claim severities, we aim to model the dataset using finite mixture model (FMM). Define a class of finite mixture distributions $\\mathcal{H}=\\{H(\\cdot;\\bm{\\Phi}):\\bm{\\Phi}\\in\\Omega\\}$, where $\\bm{\\Phi}=(\\psi_1,\\ldots,\\psi_P)^T$ is a column vector with length $P$ representing the model parameters and $\\Omega$ is the parameter space. Its density function $h(y;\\bm{\\Phi})$ is given by the following form:\n\n\\begin{equation} \\label{eq:model}\nh(y_i;\\bm{\\Phi})=\\sum_{j=1}^{J}\\pi_j f_b(y_i;\\bm{\\varphi}_j)+\\pi_{J+1}f_t(y_i;\\bm{\\eta}),\\qquad y_i>0,\n\\end{equation}\nwhere the parameters $\\bm{\\Phi}$ can alternatively be written as $\\bm{\\Phi}=(\\bm{\\pi},\\bm{\\varphi},\\bm{\\eta})$. Here, $\\bm{\\pi}=(\\pi_1,\\ldots,\\pi_{J+1})$ are the mixture probabilities for each of the $J+1$ components with $\\sum_{j=1}^{J+1}\\pi_j=1$. $\\bm{\\varphi}=(\\bm{\\varphi}_1,\\ldots,\\bm{\\varphi}_J)$ and $\\bm{\\eta}$ are the parameters for the mixture densities $f_b$ and $f_t$ respectively.\n\nThe $J$ mixture components with densities $f_b$ mainly serve as modelling the multimodality for the body part of the distribution. $f_b$ is naturally chosen as a light-tailed distribution like Gamma, Weibull and Inverse-Gaussian.\nThe remaining mixture component $f_t$ is designed to capture the large observations and hence the tail distribution can be properly extrapolated. The possible choices of heavy-tail distribution for $f_t$ include Log-normal, Pareto and Inverse-Gamma.\n\n\n\\section{Maximum weighted log-likelihood estimator} \\label{sec:MWLE}\nWith the maximum likelihood estimation (MLE) approach, parameter estimations require maximizing the log-likelihood function\n\n\\begin{equation} \\label{eq:loglik}\n\\mathcal{L}_n(\\bm{\\Phi};\\bm{y})=\\sum_{i=1}^{n}\\log h(y_i;\\bm{\\Phi})\n\\end{equation}\nwith respect to the parameters $\\bm{\\Phi}$. Under this approach, each claim has the same relative influence to the estimated parameters, but in insurance loss modelling and ratemaking perspective, correct specification and projection of larger claims are more important than those of smaller claims. More importantly, as explained by \\cite{fung2021mixture} and \\cite{wuthrich2021statistical}, MLE of FMM in Equation (\\ref{eq:model}) would fail in most practical cases due to incorrectly estimations of tail heaviness under model misspecification. Precisely, because of the overlapping region between the body parts $f_b$ and tail part $f_t$ of the distribution, the small claims may distort the estimated tail distribution $f_t$ if they are not fully captured by the $J$ mixture densities in the body. However, due to the highly complex multimodality characteristics of the body distribution which often appears in real insurance claim severity data, it is impossible to capture all the body distributional patterns without prohibitively large $J$ which causes over-fitting and loss of model interpretability. Therefore, it is often the case in practice that MLE of FMM would result in unstable estimates of tail distribution, causing unreliable tail extrapolation.\n\nOne way to mitigate the aforementioned model misspecification effect is to impose observation-dependent weights to the log-likelihood function, where a larger claim $y$ is assigned to a larger weight. This will reduce the influence of smaller observed values to the estimated tail parameter. For parameter estimations, we propose maximizing the weighted log-likelihood as follows instead\n\n\\begin{equation} \\label{eq:loglik_weight}\n\\mathcal{L}^*_n(\\bm{\\Phi};\\bm{y})=\\sum_{i=1}^{n}W(y_i)\\log \\frac{h(y_i;\\bm{\\Phi})W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du},\n\\end{equation}\nwhere $0\\leq W(\\cdot)\\leq 1$ is the weight of the log-likelihood function. We call the resulting parameters maximum weighted likelihood estimators (MWLE). To allow for greater relative influence of larger claims, we construct $W(u)$ as a monotonically non-decreasing function of $u$. In this case, we may interpret the weighted log-likelihood function as follows: First, we pretend that each claims $y_1,\\ldots,y_n$ are only observed respectively by $W(y_1),\\ldots,W(y_n)$ times. However, this alone will introduce bias to a heavier estimated tail because this implies more large claims are effectively included due to the weighting effect. To remove such a bias, we pretend to model $y_i$ by a random truncation model $\\tilde{h}(y_i;\\bm{\\Phi}):=h(y_i;\\bm{\\Phi})W(y_i)\/\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du$ instead of the original modelling distribution $h(y_i;\\bm{\\Phi})$\n\n\\begin{remark}\nThe proposed MWLE can be viewed as a form of M-estimator, where the optimal parameters are determined through maximizing a function. We here discuss two special cases of MWLE. (i) MLE: When $W(\\cdot)=1$, MWLE is reduced to a standard MLE; (ii) Truncated MLE: When $W(y)=1\\{y\\geq\\tau\\}$ for some threshold $\\tau>0$, then MWLE is reduced to truncated MLE introduced by \\cite{marazzi2004adaptively}, where a hard rejection is applied to all samples smaller than $\\tau$.\n\\end{remark}\n\n\n\n\n\\section{Theoretical Properties} \\label{sec:theory}\nThis section presents several theoretical properties of the proposed MWLE to theoretically justify the use of the proposed MWLE. Unless specified otherwise, throughout this section the estimated model parameters $\\hat{\\bm{\\Phi}}$ are obtained by maximizing the proposed weighted log-likelihood function given by Equation (\\ref{eq:loglik_weight}).\n\\subsection{Asymptotic behavior with fixed weight function}\n\\subsubsection{Consistency and asymptotic normality}\nWe first want to show that the proposed weighted log-likelihood approach leads to correct convergence to true model parameters as $n\\rightarrow\\infty$. The proof is presented in Section 2 of the supplementary materials.\n\\begin{theorem} \\label{thm:asym_tru}\nSuppose that $G(\\cdot)=H(\\cdot;\\bm{\\Phi}_0)\\in\\mathcal{H}$. Assume that the density function $h(y;\\bm{\\Phi})$ satisfies a set of regularity conditions outlined in Section 1 of supplementary materials\\footnote{Note that the set of regularity conditions are equivalent to those required for consistent and asymptotic normal estimations of MLE.}. Then, there exists a local maximizer $\\hat{\\bm{\\Phi}}_n$ of the weighted log-likelihood $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})$ such that\n\\begin{equation}\n\\sqrt{n}(\\hat{\\bm{\\Phi}}_n-\\bm{\\Phi}_0)\\overset{d}{\\rightarrow}\\mathcal{N}(\\bm{0},\\bm{\\Sigma}),\n\\end{equation}\nwhere $\\bm{\\Sigma}=\\bm{\\Gamma}^{-1}\\bm{\\Lambda}\\bm{\\Gamma}^{-1}$, with $\\bm{\\Lambda}$ and $\\bm{\\Gamma}$ being $P\\times P$ matrices given by\n\\begin{align} \\label{eq:asym:lambda}\n\\bm{\\Lambda} \n&=E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad-\\frac{1}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}\\Bigg\\{E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\nonumber\\\\\n&\\hspace{8em}+E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\Bigg\\}\\nonumber\\\\\n&\\quad+\\frac{E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]^2}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\n\\end{align}\nand\n\\begin{align} \\label{eq:asym:gamma}\n\\bm{\\Gamma}\n&=-E_{\\bm{\\Phi}_0}\\left[W(Y)\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad+\\frac{1}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T,\n\\end{align}\nwhere $E_{\\bm{\\Phi}_0}[Q(Y)]=\\int_{0}^{\\infty}Q(u)h(u;\\bm{\\Phi}_0)du$ represents the expectation under density $h(\\cdot;\\bm{\\Phi}_0)$ for any functions $Q$, and the derivative ${\\partial}\/{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})$ is assumed to be a column vector with length $P$.\n\\end{theorem}\n\n\\begin{remark}\nWhen $W(\\cdot)=1$, all except the first term in the right hand side of Equations (\\ref{eq:asym:lambda}) and (\\ref{eq:asym:gamma}) vanish. As a result, the asymptotic variance $\\bm{\\Sigma}=\\bm{\\Gamma}^{-1}\\bm{\\Lambda}\\bm{\\Gamma}^{-1}$ is reduced to the inverse of Fisher information matrix under standard MLE approach.\n\\end{remark}\n\n\\begin{remark}\nTheorem \\ref{eq:asym:gamma} only asserts the existence of local maximizer instead of global maximizer, because in FMM it is common that the likelihood function has multiple critical points and\/or is unbounded (\\cite{Mclachlan2004Finite}).\n\\end{remark}\n\nThe above theorem suggest that for large sample size, the estimated parameters are approximately unbiased and we may approximate the variance of estimated parameters as\n\\begin{equation} \\label{eq:asym_var}\n\\widehat{\\text{Var}}(\\hat{\\bm{\\Phi}}_{n})\\approx \\frac{1}{n}\\hat{\\bm{\\Gamma}}_n^{-1}\\hat{\\bm{\\Lambda}}_n\\hat{\\bm{\\Gamma}}_n^{-1}\n\\end{equation}\nwhere $\\hat{\\bm{\\Lambda}}_n$ and $\\hat{\\bm{\\Gamma}}_n$ are given by $\\bm{\\Lambda}$ and $\\bm{\\Gamma}$ in Equations (\\ref{eq:asym:lambda}) and (\\ref{eq:asym:gamma}) except that the expectations are changed to empirical means and $\\bm{\\Phi}_0$ is changed to $\\hat{\\bm{\\Phi}}_n$. Then, it is easy to construct a two-sided Wald-type confidence interval (CI) for $\\psi_p$ ($p=1,\\ldots,P$) as\n\\begin{equation} \\label{eq:asym_CI}\n\\left[\\hat{\\psi}_{n,p}-\\frac{z_{1-\\kappa\/2}}{\\sqrt{n}}\\sqrt{\\left[\\hat{\\bm{\\Gamma}}_n^{-1}\\hat{\\bm{\\Lambda}}_n\\hat{\\bm{\\Gamma}}_n^{-1}\\right]_{p,p}},\\hat{\\psi}_{n,p}+\\frac{z_{\\kappa\/2}}{\\sqrt{n}}\\sqrt{\\left[\\hat{\\bm{\\Gamma}}_n^{-1}\\hat{\\bm{\\Lambda}}_n\\hat{\\bm{\\Gamma}}_n^{-1}\\right]_{p,p}}\\right],\n\\end{equation}\nwhere $\\hat{\\psi}_{n,p}$ is the estimated $\\psi_p$, $z_{\\kappa}$ is the $\\kappa$-quantile of the standard normal distribution and $\\left[\\bm{M}\\right]_{p,p}$ is the $(p,p)$-th element of $\\bm{M}$ for some matrices $\\bm{M}$. For other quantities of interest (e.g.~mean, VaR and CTE of claim amounts), one may apply a delta method or simulate parameters from ${\\cal N}(\\hat{\\bm{\\Phi}}_{n},\\widehat{\\text{Var}}(\\hat{\\bm{\\Phi}}_{n}))$ to analytically or empirically approximate their CIs.\n\n\\medskip\n\nNext, we examine the asymptotic property of MWLE dropping the assumption that $G(\\cdot)\\in\\mathcal{H}$ (i.e. we may misspecify the model class).\n\n\\begin{theorem} \\label{thm:asym_mis}\nAssume that the density function $h(y;\\bm{\\Phi})$ satisfies the same set of regularity conditions as in the previous theorem. Further assume that there is a local maximizer $\\bm{\\Phi}_0^{*}$ of\n\\begin{equation}\n\\tilde{E}\\left[\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]:=\n\\tilde{E}\\left[W(Y)\\log \\frac{h(Y;\\bm{\\Phi})W(Y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}\\right],\n\\end{equation}\nwhere $\\tilde{E}\\left[Q(Y)\\right]=\\int_0^{\\infty}Q(u)dG(u)$ represents the expectation under distribution $G(y)$ for any functions $Q$. Then, there exists a local maximizer $\\hat{\\bm{\\Phi}}_n$ of the weighted log-likelihood $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})$ such that\n\\begin{equation}\n\\sqrt{n}(\\hat{\\bm{\\Phi}}_n-\\bm{\\Phi}_0^{*})\\overset{d}{\\rightarrow}\\mathcal{N}(\\bm{0},\\tilde{\\bm{\\Sigma}}),\n\\end{equation}\nwhere $\\tilde{\\bm{\\Sigma}}=\\tilde{\\bm{\\Gamma}}^{-1}\\tilde{\\bm{\\Lambda}}\\tilde{\\bm{\\Gamma}}^{-1}$, with $\\tilde{\\bm{\\Lambda}}$ and $\\tilde{\\bm{\\Gamma}}$ given by\n\\begin{align} \\label{eq:asym:lambda_mis}\n\\tilde{\\bm{\\Lambda}} \n&=\\tilde{E}\\left[W(Y)^2\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad-\\frac{1}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}\\Bigg\\{\\tilde{E}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\nonumber\\\\\n&\\hspace{8em}+E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\tilde{E}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\Bigg\\}\\nonumber\\\\\n&\\quad+\\frac{\\tilde{E}\\left[W(Y)^2\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]^2}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\n\\end{align}\nand\n\\begin{align} \\label{eq:asym:gamma_mis}\n\\tilde{\\bm{\\Gamma}}\n&=\\tilde{E}\\left[W(Y)\\frac{\\partial^2}{\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T}\\log h(Y;\\bm{\\Phi})\\right]\n-\\frac{\\tilde{E}\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial^2}{\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T}\\log h(Y;\\bm{\\Phi})\\right]\\nonumber\\\\\n&\\quad-\\frac{\\tilde{E}\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad+\\frac{\\tilde{E}\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]^2}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T.\n\\end{align}\n\\end{theorem}\n\nAs shown by the above theorem, the MWLE is still asymptotically convergent and normally distributed. As a result, it is still justifiable to evaluate the parameter uncertainties and CI of parameters in the forms of Equations (\\ref{eq:asym_var}) and (\\ref{eq:asym_CI}). However, in the context of modelling heavy-tail distributions as an example, there could be an asymptotic bias on the estimated tail index. As a result, it is important to theoretically examine how the choice of weight functions influence the impacts of model misspecifications. These will be leveraged to the next subsections on the robustness studies and asymptotics under varying weight functions.\n\n\\subsubsection{Robustness}\nIt is well known that MLE is the most efficient estimator under all asymptotically unbiased estimators. Therefore, with an attempt to reduce the bias of estimated tail distribution under misspecified models through MWLE approach with $W(\\cdot)\\neq 1$, there will be a trade-off between bias regularizations and loss in efficiencies. This subsection will analyze such a trade-off, which may provide guidance to choose an appropriate weight function $W(\\cdot)$. In light of Theorem \\ref{thm:asym_tru}, it is easy to show the following proposition by applying delta method.\n\n\\begin{proposition}\nSuppose that $G(\\cdot)=H(\\cdot;\\bm{\\Phi}_0)\\in\\mathcal{H}$ with the same set of regularity conditions as previous theorems. Define $\\hat{\\bm{\\Phi}}_n$ and $\\hat{\\bm{\\Phi}}_n^{(0)}$ as the MWLE and MLE respectively. Then, for some differentiable functions $U(\\cdot)$, the relative asymptotic efficiency (AEFF) of $U(\\hat{\\bm{\\Phi}}_n)$ is given by\n\\begin{equation}\n\\text{AEFF}(W;\\bm{\\Phi}_0):=\\lim_{n\\rightarrow\\infty}\\frac{\\text{Var}(\\hat{\\bm{\\Phi}}_n^{(0)})}{\\text{Var}(\\hat{\\bm{\\Phi}}_n)}=\\frac{U'(\\bm{\\Phi}_0)^T\\bm{\\Sigma}^{(0)}U'(\\bm{\\Phi}_0)}{U'(\\bm{\\Phi}_0)^T\\bm{\\Sigma}U'(\\bm{\\Phi}_0)},\n\\end{equation}\nwhere $U'(\\bm{\\Phi})$ is the gradient of $U(\\bm{\\Phi})$ w.r.t. $\\bm{\\Phi}$, and $\\bm{\\Sigma}^{(0)}$ is the inverse of Fisher information matrix under standard MLE approach.\n\\end{proposition}\n\nNext, we need to quantify robustness by some statistical measures. In a theoretical setting, we follow e.g. \\cite{huber1981robust}, \\cite{beran2012robust} and \\cite{gong2018robust} to consider the case that $Y_i$ is generated by a contamination model, given by\n\n\\begin{equation} \\label{eq:asym_contam}\nG(y):=G(y;\\epsilon,M,\\bm{\\Phi}_0)=(1-\\epsilon)H(y;\\bm{\\Phi}_0)+\\epsilon M(y), \\quad y>0,\n\\end{equation}\nfor some contamination distribution function $M$. Then, the asymptotic bias can be analyzed through evaluating the influence function (IF), a column vector with length $P$ given by\n\n\\begin{equation}\n\\text{IF}(\\bm{\\Phi}_0; H, M)=\\lim_{\\epsilon\\rightarrow 0}\\frac{\\tilde{\\bm{\\Phi}}^{\\epsilon,M}-\\bm{\\Phi}_0}{\\epsilon},\n\\end{equation}\nwhere $\\tilde{\\bm{\\Phi}}^{\\epsilon,M}$ is the asymptotic estimated parameters if $H^{\\epsilon,M}$ given by Equation (\\ref{eq:asym_contam}) is distribution generating $Y_i$, contrasting to $\\bm{\\Phi}_0$ which are the true model parameters. IF can be interpreted as the infinitesimal asymptotic bias of estimated parameters by perturbing the model generating $Y_i$. Smaller $|\\text{IF}_p(\\bm{\\Phi}_0; H, M)|$ (with $\\text{IF}_p$ being the $p^{\\text{th}}$ element of $\\text{IF}$) means a more robust estimation of $\\phi_p$ under model misspecification. Our goal is to demonstrate the potential of the proposed MWLE to reduce such a bias and hence improve the robustness. We have the following proposition which derives the IF under the MWLE approach:\n\n\\begin{proposition}\nThe IF is given by\n\\begin{align}\n\\text{IF}(\\bm{\\Phi}_0; H, M)\n&=-\\bm{\\Gamma}^{-1}\\Bigg\\{E_M\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\hspace{5em}-\\frac{E_M\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\Bigg\\},\n\\end{align}\nwhere $E_M[Q(Y)]=\\int_0^{\\infty}Q(u)dM(u)$ for some functions $Q$, and $\\bm{\\Gamma}$ is given by Equation (\\ref{eq:asym:gamma}).\n\\end{proposition}\n\nWe will show empirically how the choice of weight functions $W(\\cdot)$ affects the AEFF and IF in Section \\ref{sec:ex:toy}, which will help us understand the bias-variance tradeoff of our proposed MWLE approach.\n\n\\subsection{Asymptotic behavior of tail index with varying weight functions} \\label{sec:theory:tail_idx}\n\nTail index measures the tail-heaviness of a probability distribution. Correctly specifying the tail index is a critical task of modelling insurance data with heavy-tail nature, as insurance companies often care more about large claims which are more material than small ones. In this section, we show that under some sequences of weight functions $W_n(\\cdot)$ which depend on the number of observations $n$, the estimated tail index will be consistent under the proposed MWLE even if the model class is misspecified. \nThis result theoretically justifies how the proposed MWLE addresses the tail-robustness issue caused by model misspecification and distributional contamination, by showing that reduced influence of smaller claims through downweighting can be useful for producing a plausible tail index estimate. \nAlso, the result may provide some theoretical guidance on selecting an appropriate weight function.\n\nDenote $\\mathcal{R}_{-\\gamma}$ be a class of regularly varying distributions with tail index $\\gamma>0$, such that $\\bar{H}\\in\\mathcal{R}_{-\\gamma}$ if and only if $\\bar{H}(y)\\sim y^{-\\gamma}L_0(y)$ as $y\\rightarrow\\infty$ for some slowly varying functions $L_0(y)\\in\\mathcal{R}_0$ satisfying $L_0(ty)\/L_0(y)\\rightarrow 1$ as $y\\rightarrow\\infty$ for any $t>0$. Smaller $\\gamma$ implies heavier tail. Note that regularly varying distributions include many distributions that capture heavy-tail behaviors of loss random variables and we refer the readers to \\citet{cooke2014fat} for more explanation on these distributions. Also define the following transformed density functions\n\\begin{equation}\n\\tilde{g}_{n}(y)=\\frac{g(y)W_n(y)}{\\int_{0}^{\\infty}g(u)W_n(u)du},\\qquad \\tilde{h}_{n}(y;\\bm{\\Phi})=\\frac{h(y;\\bm{\\Phi})W_n(y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W_n(u)du},\n\\end{equation}\nand $\\tilde{G}_{n}(\\cdot)$ and $\\tilde{H}_{n}(\\cdot)$ are the corresponding distribution functions. We further put a bar to any function $Q$ to denote its survival function (i.e. $\\bar{Q}:=1-Q$). We then make the following assumptions:\n\n\\begin{enumerate}[font={\\bfseries},label={A\\arabic*.}]\n\\item $\\bar{G}\\in\\mathcal{R}_{-\\gamma_0}$ with tail index $\\gamma_0>0$.\n\\item $\\bar{H}(y;\\bm{\\Phi})=y^{-\\gamma}L(y;\\bm{\\Phi})$ for some slowly varying functions $L$, so that $\\bar{H}\\in\\mathcal{R}_{-\\gamma}$. Here, $\\gamma$ is the only model parameter within $\\bm{\\Phi}$ that governs the tail index. Also, both $L(yt;\\bm{\\Phi})\/L(y;\\bm{\\Phi})$ and its derivative w.r.t. $\\bm{\\Phi}$ converges uniformly as $y\\rightarrow\\infty$ for any fixed $t>1$.\n\\item There exists some sequences of thresholds $\\{\\tau_n\\}_{n=1,2,\\ldots}$ with $\\tau_n\\rightarrow\\infty$ as $n\\rightarrow\\infty$ such that $\\tau_nW_n(\\tau_n)\\rightarrow 0$ as $n\\rightarrow\\infty$.\n\\item $\\tilde{E}_n[(\\log \\tilde{h}_{n}(Y;\\bm{\\Phi}))^2]\/(n\\tilde{E}[W_n(Y)])\\rightarrow 0$ as $n\\rightarrow\\infty$, where $\\tilde{E}_n[Q(Y)]=\\int_{0}^{\\infty}Q(u)d\\tilde{G}_{n}$ and $\\tilde{E}[Q(Y)]=\\int_{0}^{\\infty}Q(u)dG$ for some functions $Q$.\n\\item The density functions $h(y;\\bm{\\Phi})$ and $g(y)$ are ultimately monotone (i.e. monotone on $y\\in (z,\\infty)$ for some $z>0$), uniformly on $\\bm{\\Phi}$.\n\\end{enumerate}\n\nAssumptions \\textbf{A1} and \\textbf{A2} ensure that both the model generating the observations and the fitted model class are heavy tail in nature, with tail heaviness quantified by tail indices $\\gamma_0$ and $\\gamma$ respectively. In finite mixture context, see Section \\ref{sec:fmm}, \\textbf{A2} can be easily satisfied choosing $h$ in Equation (\\ref{eq:model}) as any standard regularly varying distributions such as Pareto and Inverse-Gamma with compact parameter space. Assumption \\textbf{A3} asserts that all observations other than the extreme ones are greatly down-weighted. This assumption provides a theoretical guidance of choosing the weight function such that small to moderate claims should only be allocated by small weights, while substantial weights should be assigned only to large claims. \\textbf{A4} requires that the effective number of MWLE observations $n\\tilde{E}[W_n(Y)]\\rightarrow\\infty$ such that large sample theories hold. The numerator $\\tilde{E}_n[(\\log \\tilde{h}_{n}(Y;\\bm{\\Phi}))^2]$ grows much slower than the denominator as a logarithm is involved. Assumption \\textbf{A5} is of no practical concern. Now, we have the following theorem which asserts the consistency of estimated tail index. The proof is leveraged to Section 3 of the supplementary material.\n\n\\begin{theorem} \\label{thm:asym:tail_idx}\nAssume \\textbf{A1} to \\textbf{A5} hold for the settings under the MWLE, and the regularity conditions outlined in Section 1 of Supplementary materials are satisfied. Then, there exists a local maximizer $\\hat{\\bm{\\Phi}}_n$ of the weighted log-likelihood function $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ with the estimated tail index $\\hat{\\gamma}_n$ such that $\\hat{\\gamma}_n\\rightarrow\\gamma_0$ as $n\\rightarrow\\infty$. Further, the local maximizer $\\hat{\\gamma}_n$ is unique in probability as $n\\rightarrow\\infty$.\n\\end{theorem}\n\n\n\\begin{remark}\nConsider a special case where: (i) the weight functions $W_n(y)=1\\{y>\\tau_n\\}$ are step functions for some sequences of $\\tau_n\\rightarrow\\infty$; and (ii) the fitted model class $H(y;\\bm{\\Phi})$ is chosen as a Generalized Pareto distribution (GPD) or equivalently Lomax distribution which will be described in Section \\ref{sec:ex} (i.e. $H(y;\\bm{\\Phi})$ is an FMM in Equation (\\ref{eq:model}) with $J=0$ and $f_t$ is a GPD). Theorem \\ref{thm:asym:tail_idx} is then asserting the consistency of tail index obtained by excess over threshold method on GPD (\\cite{smith1987estimating}), which has a very close connection with the consistency property of the Hill estimator (\\cite{hill1975simple}) (see Section 4 of \\cite{smith1987estimating}). Therefore, we can regard the proposed MWLE approach as a generalized framework of the Hill-type estimator by \\cite{hill1975simple}.\n\\end{remark}\n\n\\section{Parameter estimation} \\label{sec:em}\n\\subsection{GEM algorithm} \\label{sec:em:gem}\nSince there is a probabilistic interpretation of the weighted log-likelihood given by Equation (\\ref{eq:loglik_weight}), it is feasible to construct a generalized Expectation-Maximization (GEM) algorithm for efficient parameter estimations. In this paper, we will present two distinctive approaches of complete data constructions which result to two different kinds of GEM algorithms.\n\n\\subsubsection{Method 1: Hypothetical data approach}\n\\paragraph{Construction of complete data}\nTo address the challenges of optimizing directly the ``observed data\" weighted log-likelihood in Equation (\\ref{eq:loglik_weight}), we extend the ``hypothetical complete data\" method proposed by \\cite{FUNG2020MoECensTrun}, by defining the complete data\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i,k_i,\\{\\bm{z}'_{is},y'_{is}\\}_{s=1,\\ldots,k_i})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $k_i$ is the number of missing sample points ``generated\" by observation $i$, due to the probabilistic interpretation that each sample $i$ is removed with a probability of $1-W(y_i)$. As an auxiliary tool for efficient computations we assume that $k_i$ follows geometric distribution with probability mass function\n\\begin{equation} \\label{eq:em:k}\np(k_i;\\bm{\\Phi})=\\left[1-\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du\\right]^{k_i}\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du, \\qquad k_i=0,1,\\ldots,\n\\end{equation}\nand $\\{y'_{is}\\}_{s=1,\\ldots,k_i}$ are i.i.d. variables representing the missing samples. We assume that $Y'_{is}$ (with realization $y'_{is}$) is independent of $y_i$ and $k_i$, and follows a distribution with the following density function\n\\begin{equation} \\label{eq:em:y}\n\\tilde{h}^{*}(y'_{is};\\bm{\\Phi})=\\frac{h(y'_{is};\\bm{\\Phi})(1-W(y'_{is}))}{\\int_0^{\\infty}h(u;\\bm{\\Phi})(1-W(u))du},\\qquad y'_{is}>0.\n\\end{equation}\n\nFurther, $\\bm{z}_i=(z_{i1},\\ldots,z_{i(J+1)})$ are the latent mixture components assignment labels, where $z_{ij}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ latent class and $z_{ij}=0$ otherwise. Similarly, $\\bm{z}'_i=(z'_{is1},\\ldots,z'_{is(J+1)})$ are the labels for missing data, where $z'_{isj}=1$ if the $s^{\\text{th}}$ missing sample generated by observation $i$ belongs to the $j^{\\text{th}}$ latent class, and $z'_{isj}=0$ otherwise. \n\nThe complete data weighted log-likelihood function is then given by\n\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^{n}W(y_i)\\log\\Bigg\\{\\frac{\\left\\{\\prod_{j=1}^{J}[\\pi_jf_b(y_i;\\bm{\\varphi}_j)]^{z_{ij}}\\right\\}\\left(\\pi_{J+1}f_t(y_i;\\bm{\\eta})\\right)^{z_{i(J+1)}}W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du} \\nonumber\\\\\n&\\hspace{8em} \\times \\left[1-\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du\\right]^{k_i}\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du \\nonumber\\\\\n&\\hspace{8em} \\times \\prod_{s=1}^{k_i}\\frac{\\left\\{\\prod_{j=1}^{J}[\\pi_jf_b(y'_{is};\\bm{\\varphi}_j)]^{z'_{isj}}\\right\\}\\left(\\pi_{J+1}f_t(y'_{is};\\bm{\\eta})\\right)^{z'_{is(J+1)}}W(y_i)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})(1-W(u))du}\\Bigg\\} \\nonumber\\\\\n&=\\sum_{i=1}^nW(y_i)\\left\\{\\left[\\sum_{j=1}^{J}z_{ij}\\log \\pi_jf_b(y_i;\\bm{\\varphi}_j)\\right]+z_{i(J+1)}\\log\\pi_{J+1} f_t(y_i;\\bm{\\eta})\\right\\}\\nonumber\\\\\n&\\quad +\\sum_{i=1}^{n}\\sum_{s=1}^{k_i}W(y_i)\\left\\{\\left[\\sum_{j=1}^J z'_{ijs}\\log\\pi_j f_b(y'_{is};\\bm{\\varphi}_j)\\right]+z'_{i(J+1)s}\\log\\pi_{J+1} f_t(y'_{is};\\bm{\\eta})\\right\\}+\\text{const.},\n\\end{align}\n\\endgroup\nwhich is more computationally tractable. In the following we will omit the constant term which is irrelevant for calculations.\n\n\n\n\n\\paragraph{Iterative procedures}\nIn the $l^{\\text{th}}$ iteration of the E-step, we compute the expectation of the complete data weighted log-likelihood as follows:\n\\begin{align} \\label{eq:em:q}\nQ^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\n&=\\sum_{i=1}^nW(y_i)\\left\\{\\left[\\sum_{j=1}^{J}z_{ij}^{(l)}\\log \\pi_jf_b(y_i;\\bm{\\varphi}_j)\\right]+z_{i(J+1)}^{(l)}\\log\\pi_{J+1} f_t(y_i;\\bm{\\eta})\\right\\}\\nonumber\\\\\n&\\quad +\\sum_{i=1}^{n}k_i^{(l)}W(y_i)\\Big\\{\\left[\\sum_{j=1}^J {z'}^{(l)}_{ij}\\left(\\log\\pi_j +E\\left[\\log f_b(Y';\\bm{\\varphi}_j)|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}\\right]\\right)\\right] \\nonumber\\\\\n& \\hspace{8em} +{z'}^{(l)}_{i(J+1)}\\left(\\log\\pi_{J+1}+ E\\left[\\log f_t(Y';\\bm{\\eta})|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}\\right]\\right)\\Big\\},\n\\end{align}\nwhere $z_{ij}^{(l)}=E[z_{ij}|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$, ${z'}^{(l)}_{ij}=E[z'_{ijs}|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$ and $k_i^{(l)}=E[K_i|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$. Also, $K$ follows $p(\\cdot;\\bm{\\Phi}^{(l-1)})$ in Equation (\\ref{eq:em:k}) and $Y'$ follows $\\tilde{h}^{*}(\\cdot;\\bm{\\Phi}^{(l-1)})$ in Equation (\\ref{eq:em:y}). The precise expressions of the above expectations are presented in Section 4.2 of supplementary materials, under a particular specification of Gamma distribution for $f_b$ and Lomax distribution for $f_t$. This specification will also be studied in the illustrating examples (Section \\ref{sec:ex}).\n\nIn the M-step, we attempt to find the updated parameters $\\bm{\\Phi}^{(l)}$ such that $Q^{*}(\\bm{\\Phi}^{(l)}|\\bm{\\Phi}^{(l-1)})\\geq Q^{*}(\\bm{\\Phi}^{(l-1)}|\\bm{\\Phi}^{(l-1)})$. Note in Equation (\\ref{eq:em:q}) that $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ is linearly separable w.r.t. parameters $(\\bm{\\pi}, \\bm{\\varphi}_1,\\ldots,\\bm{\\varphi}_J,\\bm{\\eta})$. Therefore, the optimization can be done separately w.r.t. each subset of parameters. Details are leveraged to Section 4.3 of supplementary materials. \n\n\\subsubsection{Method 2: Parameter transformation approach}\n\\paragraph{Construction of complete data}\nMotivated by the mixture probability transformation approach adopted by e.g. \\cite{lee2012algorithms} and \\cite{VERBELEN2015Censor} for truncated data, we here rewrite the random truncation distribution $\\tilde{h}(y_i;\\bm{\\Phi})$ in Equation (\\ref{eq:loglik_weight}) as\n\\begin{equation}\n\\tilde{h}(y_i;\\bm{\\Phi})\n=\\frac{h(y_i;\\bm{\\Phi})W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}\n=\\sum_{j=1}^J\\pi_j^{*}\\frac{f_b(y_i;\\bm{\\varphi}_j)W(y_i)}{\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}+\\pi_{J+1}^{*}\\frac{f_t(y_i;\\bm{\\eta})W(y_i)}{\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du},\n\\end{equation}\nwhere $\\bm{\\pi}^{*}:=(\\pi_1^{*},\\ldots,\\pi_{J+1}^{*})$ are the transformed mixing weight parameters given by\n\\begin{equation} \\label{eq:em:pi_trans}\n\\pi_j^{*}=\\frac{\\pi_j\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du},~j=1,\\ldots,J;\\qquad\n\\pi_{J+1}^{*}=\\frac{\\pi_{J+1}\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}.\n\\end{equation}\n\nAs a result, the problem is reduced to maximizing the weighted log-likelihood of finite mixture of random truncated distributions. In this case, define the complete data\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i^{*})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $\\bm{z}_i^{*}=(z_{i1}^{*},\\ldots,z_{i(J+1)}^{*})$ are the labels where $z_{ij}^{*}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ (transformed) latent mixture component and $z_{ij}^{*}=0$ otherwise. The complete data weighted log-likelihood function is reduced to\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\left[\\sum_{j=1}^{J}z_{ij}^{*}\\left(\\log\\pi_j^{*}+\\log\\frac{f_b(y_i;\\bm{\\varphi}_j)W(y_i)}{\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}\\right)\\right]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*}\\left(\\log\\pi_{J+1}^{*}+\\log\\frac{f_t(y_i;\\bm{\\eta})W(y_i)}{\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du}\\right)\\Bigg\\}.\n\\end{align}\n\n\\paragraph{Iterative procedures}\nIn the $l^{\\text{th}}$ iteration of the E-step, the expectation of the complete data weighted log-likelihood is:\n\\begin{align} \\label{eq:em:q2}\nQ^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\left[\\sum_{j=1}^{J}z_{ij}^{*(l)}\\left(\\log\\pi_j^{*}+\\log\\frac{f_b(y_i;\\bm{\\varphi}_j)W(y_i)}{\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}\\right)\\right]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*(l)}\\left(\\log\\pi_{J+1}^{*}+\\log\\frac{f_t(y_i;\\bm{\\eta})W(y_i)}{\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du}\\right)\\Bigg\\},\n\\end{align}\nwhere $z_{ij}^{*(l)}=E[z_{ij}^{*}|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$ is provided in Section 5.2 of supplementary materials.\n\nIn the M-step, similar to Method 1 that $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ is linearly separable w.r.t. parameters $(\\bm{\\pi}^{*}, \\bm{\\varphi}_1,\\ldots,\\bm{\\varphi}_J,\\bm{\\eta})$, we can maximize $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ sequentially w.r.t. each subset of parameters. Details are presented in Section 5.3 of supplementary materials. Note that the M-step of this approach is slightly more computationally more intensive than Method 1 as the target function $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ here involves numerical integrals.\n\nAfter completing the iterative procedures, we will obtain an estimate of the transformed mixing weights $\\bm{\\pi}^{*}$ instead of $\\bm{\\pi}$. One can revert Equation (\\ref{eq:em:pi_trans}) to get back the estimated original mixing weights as follows:\n\\begin{equation}\n\\pi_j=\\frac{\\pi_j^{*}[\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du]^{-1}}{\\pi_j^{*}\\sum_{j'=1}^J[\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_{j'})W(u)du]^{-1}+\\pi_{J+1}^{*}[\\int_{0}^{\\infty}f_t(u;\\bm{\\eta})W(u)du]^{-1}},~j=1,\\ldots,J;\n\\end{equation}\n\\begin{equation}\n\\pi_{J+1}=\\frac{\\pi_{J+1}^{*}[\\int_{0}^{\\infty}f_t(u;\\bm{\\eta})W(u)du]^{-1}}{\\pi_j^{*}\\sum_{j'=1}^J[\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_{j'})W(u)du]^{-1}+\\pi_{J+1}^{*}[\\int_{0}^{\\infty}f_t(u;\\bm{\\eta})W(u)du]^{-1}}.\n\\end{equation}\n\n\n\\subsection{Ascending property of the GEM algorithm}\nIt is well known from \\cite{DEMPSTER1977EM} that an increase of complete data log-likelihood implies an increase of observed data log-likelihood (Equation (\\ref{eq:loglik})). This can be analogously extended to the proposed weighted log-likelihood framework where we have the following proposition. The proof is leveraged to Section 6 of the supplementary material.\n\n\\begin{proposition} \\label{prop:ascend}\nIf the expected complete data weighted log-likelihood is increased during the $l^{\\text{th}}$ iteration (i.e. $Q^{*}(\\bm{\\Phi}^{(l)}|\\bm{\\Phi}^{(l-1)})\\geq Q^{*}(\\bm{\\Phi}^{(l-1)}|\\bm{\\Phi}^{(l-1)})$), then the observed data weighted log-likelihood is also increased (i.e. $\\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l)};\\bm{y})\\geq \\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l-1)};\\bm{y})$).\n\\end{proposition}\n\n\\subsection{Parameter Initialization, convergence acceleration and stopping criterion} \\label{sec:em:init}\nInitialization of parameters is an important issue, in the sense that poor initializations may lead to slow convergence, numerical instability and even convergence to spurious local maximum. We suggest to determine the initial parameters $\\bm{\\Phi}^{(0)}$ using a modified version of clusterized method of moments (CMM) approach by \\cite{gui2018fit}. Under this approach, we first determine a threshold $\\tau$ which classifies observations $y_i$ into either body ($y_i\\leq\\tau$) or tail ($y_i>\\tau$) part of the distribution. We then apply a $K$-means clustering method to assign ``body\" observations $y_i$ with $y_i\\leq\\tau$ to one of the $J$ mixture components for the body, with moment matching method for each mixture components to determine the initial parameters $(\\bm{\\pi}^{(0)},\\bm{\\varphi}^{(0)})$. Moment matching technique is also applied to ``tail\" observations $y_i$ with $y_i>\\tau$ to initialize $\\bm{\\eta}^{(0)}$. For details, we direct readers to Section 7 of the supplementary material.\n\nAs EM algorithm often converges slowly with small step sizes, we further apply a step lengthening procedure for every two GEM iterations to accelerate the algorithm. This is described by \\cite{jamshidian1997acceleration} and its references therein as a ``pure accelerator\" for the EM algorithm.\n\n\\sloppy The GEM algorithm is iterated until the relative change of iterated parameters $\\Delta^{\\text{rel}}\\bm{\\Phi}^{(l)}:=|\\log(\\bm{\\Phi}^{(l)}\/\\bm{\\Phi}^{(l-1)})|\/P$ is smaller than a threshold of $10^{-5}$ or the maximum number of iterations of 1000 is reached.\n\n\\subsection{Specification of weight function}\nOur proposed MWLE is rather flexible by allowing us to pre-specify any weight functions $W(\\cdot)$ prior to fitting the GEM algorithm. The appropriate choice of $W(\\cdot)$ depends on some decision rules beyond what statistical inference can do. In insurance loss modelling perspective, such decision rule includes the relative importance of insurance company to correctly specify the tail distribution (to evaluate some tail measures such as Value-at-risk (VaR)) compared to that of more accurately modelling the smaller attritional claims. If accurate extrapolation of huge claims are way more important than modelling the smaller claims, then one may consider $W(y)$ to be close to zero unless $y$ is large, aligning with Assumption \\textbf{A3} in Section \\ref{sec:theory:tail_idx} to ensure near-consistent tail index estimations (Theorem \\ref{thm:asym:tail_idx}). Otherwise, one may consider a flatter $W(y)$ across $y$.\n\nThroughout the entire paper, we analyze the following general form of weight function\n\\begin{equation} \\label{eq:em:wgt_func}\nW(y):=W(y;\\xi,\\tilde{\\mu},\\tilde{\\phi})=\\xi+(1-\\xi)\\int_0^y\\frac{(\\tilde{\\phi}\\tilde{\\mu})^{-1\/\\tilde{\\phi}}}{\\Gamma(1\/\\tilde{\\phi})}u^{1\/\\tilde{\\phi}-1}e^{-u\/(\\tilde{\\phi}\\tilde{\\mu})}du,\\quad y>0,\n\\end{equation}\nwhich is the distribution function of a zero-inflated Gamma distribution. The above weight function has the following characteristics:\n\\begin{itemize}\n\\item $W(y)$ is a non-decreasing function of $y$, meaning that smaller observations are down-weighted.\n\\item $\\xi\\in[0,1]$ is the minimum weight assigned to each observation.\n\\item $\\tilde{\\mu}$ and $\\tilde{\\phi}$ are the location and dispersion hyperparameters of Gamma distribution respectively. Larger $\\tilde{\\mu}$ means more (small to moderate) claims are under-weighted by a larger extent, while $\\tilde{\\phi}$ controls the shape of weight function, or how the observations are under-weighted.\n\\item If $\\xi=1$ or $\\tilde{\\mu}=0$, then the weight function is reduced to $W(\\cdot)=1$, leading to standard MLE approach.\n\\item If $\\xi=0$ and $\\tilde{\\phi}\\rightarrow 0$, then $W(y)=1\\{y\\geq\\tilde{\\mu}\\}$, meaning that only observations greater than $\\tilde{\\mu}$ are informative in determining the estimated parameters.\n\\end{itemize}\n\nOverall, smaller $\\xi$, larger $\\tilde{\\mu}$ and smaller $\\tilde{\\phi}$ represent greater under-weightings to more small claims, where we will expect more robust tail estimation by sacrificing more efficiencies on body estimations. In this paper, instead of quantifying decision rules to select the hyperparameters, in the subsequent sections we empirically test various (wide range) combinations of $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$ to study how these hyperparameters affect the trade-off between tail-robustness and estimation efficiency. These provide practical guidance and assessments to determine the suitable hyperparameters.\n\n\\begin{remark}\nThere are many possible ways to quantify the decision rule to select the ``optimal\" weight function hyperparameters. We here briefly discuss two possible ways: (1) Consider a goodness-of-fit test statistic for heavy-tailed distributions, such as the modified AD test (\\cite{ahmad1988assessment}). Then select weight function hyperparameters which optimizes the test statistic; (2) Define an acceptable range of estimated parameter uncertainty of tail index, e.g. two times as the uncertainty obtained by MLE. Then select the hyperparameters with the greatest distortion metric (e.g. the average downweighting factor $\\sum_{i=1}^{n}(1-W(y_i))\/n$) where the tail index uncertainty is still within the acceptable range.\n\\end{remark}\n\n\n\\subsection{Choice of model complexity} \\label{sec:em:complex}\nThe above GEM algorithm assumes a fixed number of mixture component $J$. However, it is important to control the model complexity by choosing an appropriate $J$ which allows enough flexibility to capture the distributional characteristics without over-fitting. \n\nThe first criterion is motivated by maximizing the expected weighted log-likelihood\n\\begin{equation} \\label{eq:em:e_wgt_ll}\nn\\times\\tilde{E}[\\mathcal{L}^{*}(\\bm{\\Phi};\\bm{Y})]=n\\times\\tilde{E}\\left[W(Y)\\log\\frac{h(Y;\\bm{\\Phi})W(Y)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du}\\right],\n\\end{equation}\nwhere the expectation is taken on $Y$ under the true model generating the observations. Without knowing the true model (in real data applications), Equation (\\ref{eq:em:e_wgt_ll}) is approximated by $\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})$ in Equation (\\ref{eq:loglik_weight}) with fitted model parameters $\\hat{\\bm{\\Phi}}$. Note that it is positively biased with correction term $\\text{tr}(-\\bm{\\Gamma}^{-1}\\bm{\\Lambda})$ shown by \\cite{konishi1996generalised}. This leads to a robustified AIC\n\\begin{equation}\n\\text{RAIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+2\\times\\text{tr}(-\\hat{\\bm{\\Gamma}}^{-1}\\hat{\\bm{\\Lambda}}).\n\\end{equation}\n\nAnalogous and naturally, since AIC-type criteria often choose excessively complex models, we also consider the robustified BIC given by\n\\begin{equation}\n\\text{RBIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+(\\log n)\\times\\text{tr}(-\\hat{\\bm{\\Gamma}}^{-1}\\hat{\\bm{\\Lambda}}).\n\\end{equation}\n\nWe choose $J$ that minimizes either the RAIC or RBIC, and the $(p,p)$-th element of $-\\hat{\\bm{\\Gamma}}^{-1}\\hat{\\bm{\\Lambda}}$ can be interpreted as the effective number of parameter attributed by the $p^{\\text{th}}$ parameter.\n\nInsurance loss dataset is often characterized by very complicated and multimodal distribution on very small claims, yet it is not meaningful to capture all these small nodes by choosing an overly complex mixture distribution with large $J$. However, the above RAIC and RBIC cannot effectively reduce those mixture components as the effective number of parameters for those capturing the smaller claims could be very small if $W(\\cdot)$ is chosen very small over the region of small claims. To effectively remove components which excessively capture the small claims, we propose treating all parameters as ``full parameters\", which results to the following truncated AIC and BIC:\n\\begin{equation}\n\\text{TAIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+2\\times P,\n\\end{equation}\n\\begin{equation}\n\\text{TBIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+\\left(\\log \\sum_{i=1}^{n}W(y_i)\\right)\\times P.\n\\end{equation}\n\n\\begin{remark} \\label{rmk:tic}\nThe above TAIC and TBIC are motivated by the bias of approximating $n\\times\\tilde{E}[\\mathcal{L}^{*}(\\bm{\\Phi};\\bm{Y})]$ by the empirical truncated log-likelihood $\\mathcal{L}^{**}_n(\\hat{\\bm{\\Phi}};\\bm{y}):=\\sum_{i=1}^{n}V_i(y_i)\\log \\frac{h(y_i;\\bm{\\Phi})W(y_i)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du}$ instead of $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$, where $V_i(y)\\sim\\text{Bernoulli}(W(y))$ is an indicator randomly discarding some observations. It can be easily shown (details in Section 8 of supplementary material) that the asymptotic bias is simply $P$ with effective number of observations $\\sum_{i=1}^{n}W(y_i)$. Note also that the weighted log-likelihood $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ is asymptotically equivalent to the truncated log-likelihood $\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})$, except that the former produces more accurate estimated parameters than the latter. This motivates why in TAIC and TBIC we choose to evaluate $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ instead of $\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})$.\n\\end{remark}\n\n\\section{Illustrating examples} \\label{sec:ex}\nIn this section, we analyze the performance of our proposed MWLE approach (Equation (\\ref{eq:loglik_weight})) on FMM given by Equation (\\ref{eq:model}). In the following examples, we select Gamma density for the body components $f_b$, a light-tailed distribution to capture the distributional multimodality of small to moderate claims, and Lomax density for the tail component $f_t$ to extrapolate well the tail-heaviness of larger observations. Then, Equation (\\ref{eq:model}) becomes\n\\begin{equation} \\label{eq:em:density_mixture}\nh(y;\\bm{\\Phi})\n=\\sum_{j=1}^J\\pi_jf_b(y;\\mu_j,\\phi_j)+\\pi_{J+1}f_t(y;\\theta,\\gamma),\n\\end{equation}\nwhere the parameter set is re-expressed as $\\bm{\\Phi}=(\\bm{\\pi},\\bm{\\mu},\\bm{\\phi},\\gamma)$ while $\\bm{\\varphi}_j=(\\mu_j,\\phi_j)$ and $\\bm{\\eta}=(\\theta,\\gamma)$, and the Gamma and Lomax densities $f_b$ and $f_t$ are respectively given by\n\\begin{equation} \\label{eq:em:comp}\nf_b(y;\\mu,\\phi)=\\frac{(\\phi\\mu)^{-1\/\\phi}}{\\Gamma(1\/\\phi)}y^{1\/\\phi-1}e^{-y\/(\\phi\\mu)}\\quad \\text{and} \\quad f_t(y;\\theta,\\gamma)=\\frac{\\gamma\\theta^{\\gamma}}{\\left(y+\\theta\\right)^{\\gamma+1}},\n\\end{equation}\nwhere $\\mu$ and $\\phi$ are the mean and dispersion parameters of Gamma distribution, while $\\gamma$ is the tail index parameter for the Lomax distribution. $\\theta$ is scale of the Lomax distribution. Note that the above model is a regular varying distribution with the tail behavior predominately explained by the tail index $\\gamma$. As a result, tail-robustness is highly determined by how stable and accurate the estimated tail index $\\gamma$ is.\n\nThe specifications of body and tail component functions are mainly motivated by the key characteristics of insurance claim severity distributions (multimodal distribution of small claim, existence of extremely large claims, mismatch between body and tail behavior etc.) which will be illustrated in the real insurance data application section. While we do not preclude the existence of other specifications, such as Weibull for the body and Inverse-Gamma for the tail, plausible for insurance applications, in this section we simply focus on studying Gamma-Lomax combination to focus on the scope of this paper -- demonstrating the usefulness of the proposed MWLE, instead of performing distributional comparisons under FMM.\n\n\\subsection{Toy example} \\label{sec:ex:toy}\n\nWe demonstrate how the proposed MWLE framework works through a simple toy example of one-parameter Lomax distribution $H(y;\\gamma)=1-(y+1)^{-\\gamma}$ ($y>0$), which is a special case of Equation (\\ref{eq:em:density_mixture}) with $J=0$ and $\\theta=1$.\n\nConsider the first case where the true model $G(\\cdot)$ is a Lomax with $\\gamma=\\gamma_0=1$. For the weight function for the MWLE, we consider the form of Equation (\\ref{eq:em:wgt_func}) with $\\xi=0$ for simplicity. We will test across a wide range of $\\tilde{\\mu}$ and across $\\tilde{\\phi}\\in\\{0.1, 0.2, 0.5, 1\\}$. Figure \\ref{fig:thm_aeff} presents how the choices of these hyperparameters affect the AEFF. Starting from $\\text{AEFF}=1$ when $\\tilde{\\mu}=0$ which is equivalent to standard MLE, the AEFF decrease monotonically as $\\tilde{\\mu}$ increases. This is intuitive because under-weighting smaller observations with MWLE means effectively discarding some observed information, leading to larger parameter uncertainties compared to MLE. Since the MLE estimated tail index is unbiased under the true model, there is obviously no benefit of using the proposed MWLE to fit the true model.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_aeff_1.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_aeff_2.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{AEFF as a function of the weight location hyperparameter $\\tilde{\\mu}$ (left panel) or Pareto (Lomax) quantile of $\\tilde{\\mu}$ (right panel) under Lomax true model.}\n\\label{fig:thm_aeff}\n\\end{figure}\n\nNow, consider the second case where the true model is perturbed by the contamination function $M$, as presented in Equation (\\ref{eq:asym_contam}). In this demonstration example, we consider two following two choices for the contamination function $M$:\n\\begin{itemize}\n\\item Degenerate perturbation: One-point distribution on $y=1\/4$\n\\item Pareto perturbation: Lomax distribution with tail index $\\gamma=\\gamma^{*}=4>\\gamma_0$\n\\end{itemize}\nNote that the contamination function is relatively lighter tailed and hence it would not affect the tail behavior of the perturbed distribution. In Figure \\ref{fig:thm_if}, we present the IF as a function of the AEFF (determined as a function of chosen $\\tilde{\\mu}$) under the two choices of $M$. We find that as the AEFF reduces (by choosing a larger $\\tilde{\\mu}$), the IF would shrink towards zero. This reflects that a more robust estimation of tail index can be achieved using the proposed MWLE approach by trading off some efficiencies.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_if_1.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_if_2.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{IF as a function of AEFF under degenerate (left panel) and Pareto (right panel) contaminations.}\n\\label{fig:thm_if}\n\\end{figure}\n\n\n\n\\subsection{Simulation studies} \\label{sec:ex:sim}\n\\subsubsection{Simulation settings}\nWe here simulate $n=10,000$ claims (the sample size is motivated by the size of a typical insurance portfolio) from the aforementioned $J$-Gamma Lomax distribution for each of the following two parameter settings with $\\theta=1000$:\n\\begin{itemize}\n    \\item Model 1: $J=2$, $\\bm{\\pi}=(0.4,0.4,0.2)$, $\\bm{\\mu}=(100,300)$, $\\bm{\\phi}=(0.25,0.25)$ and $\\gamma=2$.\n    \\item Model 2: $J=3$, $\\bm{\\pi}=(0.4,0.3,0.1,0.2)$, $\\bm{\\mu}=(50,200,600)$, $\\bm{\\phi}=(0.2,0.2,0.2)$ and $\\gamma=2$.\n\\end{itemize}\n\nWe also consider the zero-inflated Gamma distribution given by Equation (\\ref{eq:em:wgt_func}) as the weight function, with $\\tilde{\\mu}\\in\\{q_{0},q_{0.9},q_{0.95},q_{0.99},q_{0.995}\\}$, $\\tilde{\\phi}\\in\\{0.025,0.1,0.25,1\\}$ and $\\xi\\in\\{0.001,0.01,0.05,0.25\\}$, where $q_{\\alpha}$ is the empirical quantile of the data with $0\\leq \\alpha\\leq 1$. Recall that the choice of $\\tilde{\\mu}=q_0=0$ implies that $W(y;\\xi,\\tilde{\\mu},\\tilde{\\phi})= 1$ and hence the MWLE is equivalent to standard MLE. For each combinations of models and weight function hyperparameters, the simulations of sample points are repeated by 100 times to enable thorough analysis of the results using the proposed weighted log-likelihood approach under various settings. Each simulated sample is then fitted to the $J$-Gamma Lomax mixture in Equation (\\ref{eq:em:density_mixture}) with $J=2$. Note that for simplicity, in the simulation studies we do not examine the choice of $J$ as outlined by Section \\ref{sec:em:complex}. As a result, we have the following research goals in the simulation studies:\n\\begin{itemize}\n\\item Under Model 1, the data is fitted to the true class of models. Hence, we empirically verify the consistencies of estimating model parameters (Theorem \\ref{thm:asym_tru}) using the MWLE. We also study how the selection of weight function hyperparameters affect the estimated parameter uncertainties. Further, we compare the computational efficiency of the two kinds of proposed GEM algorithms.\n\\item Under Model 2, the data is fitted to a misspecified class of models. Hence, we demonstrate how this would distort the estimation of the tail under the MLE, and study how the proposed MWLE produces a more robust tail estimation.\n\\end{itemize}\n\n\\subsubsection{Results of fitting Model 1 (true model)}\n\nConsidering the case where we fit the true class of model to the data generated by Model 1, we first compare the computational efficiencies between the two construction methods of the GEM algorithm as presented by Section \\ref{sec:em:gem}. In general, around 100 iterations are needed under parameter transformation approach (Method 2), as compared to at least 300 iterations under hypothetical data approach (Method 1), revealing relatively faster convergences under Method 2. Figure \\ref{fig:sim_tru_rel} plots the relative change of iterated parameters $\\Delta^{\\text{rel}}\\bm{\\Phi}^{(l)}:=|\\log(\\bm{\\Phi}^{(l)}\/\\bm{\\Phi}^{(l-1)})|\/P$ versus the GEM iteration $l$ under two example choices of weight function hyperparameters, where the division operator is applied element-wise to the vector of parameters. It is apparent that the curve drops much faster under Method 2 than Method 1, confirming faster convergence under Method 2. The main reason is that the construction of hypothetical missing observations under Method 1 will generally effectively reduce the learning rates of the optimization algorithms. As both methods produce very similar estimated parameters while Method 2 is more computationally efficient, from now on we only present the results produced by the GEM algorithm under Method 2.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_rel1.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_rel2.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{The relative change of iterated parameters in the first 100 GEM iterations under two example choices of weight function hyperparameters: Left panel -- $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.01,q_{0.95},0.1)$; Right panel -- $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.05,q_{0.99},0.25)$.}\n\\label{fig:sim_tru_rel}\n\\end{figure}\n\nFigure \\ref{fig:sim_tru_gamma} demonstrates the how the biasedness and uncertainty of the estimated tail index $\\hat{\\gamma}$ differ among various choices of weight functions and their corresponding hyperparameters $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$. From the left panel, the median estimated parameters are very close to the true model parameters (differ by less than 1-2\\%) under most settings of the weight functions, except for few extreme cases where both $\\xi$ and $\\tilde{\\phi}$ are chosen to be very small. This empirically justifies the asymptotic unbiasedness of the MWLE. As expected from the right panel, the uncertainties of MLE parameters are the smallest, verifying that MLE is the asymptotically most efficient estimator among all unbiased estimators if we are fitting the correct model class. The parameter uncertainties generally slightly increase as we choose larger $\\tilde{\\mu}$ to de-emphasize the impacts of smaller observations. In some extreme cases where $\\xi$ and $\\tilde{\\phi}$ very small and $\\tilde{\\mu}$ is very large, the standard error can grow dramatically, reflecting that a lot of information are effectively discarded. \n\nSimilarly in Figure \\ref{fig:sim_tru_mu1} where the biasedness and uncertainty of an estimated mean parameter $\\hat{\\mu_1}$ from the body distribution are displayed, we observe that the proposed MWLE approach behaves properly for fitting the body distribution unless when $\\xi$ and $\\tilde{\\phi}$ are both chosen to be extremely small (in those cases, the estimated body parameters would become unstable with inflated uncertainties). Hence, these extreme choices of hyperparameters are deemed to be inappropriate.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_gamma_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_gamma_sd.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and standard deviation of the estimated tail index $\\hat{\\gamma}$ versus various weight function hyperparameters under the true model.}\n\\label{fig:sim_tru_gamma}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_mu1_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_mu1_sd.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and standard deviation of the estimated mean parameter of the first mixture component $\\hat{\\mu_1}$ versus various weight function hyperparameters under the true model.}\n\\label{fig:sim_tru_mu1}\n\\end{figure}\n\n\\subsubsection{Results of fitting Model 2 (misspecified model)}\n\nWe now turn to the case where we fit a misspecified model (with $J=2$) to the simulated data generated from Model 2 (with $J=3$). The left panels of Figures \\ref{fig:sim_mis_gamma} and \\ref{fig:sim_mis_tailp} examine how the robustness of the estimated tail index $\\hat{\\gamma}$ and tail probability $\\hat{\\pi}_{J+1}$ differs among different choices of hyperparameters $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$. From the left panel, the MLE of the tail index is around $\\hat{\\gamma}=2.48$ which largely over-estimates the true tail index $\\gamma=2$, indicating that the heavy-tailedness of the true distribution is under-extrapolated. On the other hand, with the incorporation of weight functions to under-weight the smaller claims, the biases of the MWLE of $\\gamma$ are greatly reduced compared to that of the MLE under most choices of weight function hyperparameters. In particular, the bias reduction for tail index is more effective using smaller $\\xi$ (i.e. $\\xi\\leq 0.05$). This is intuitive as smaller $\\xi$ means smaller claims are under-weighted by a larger extent, reducing the impacts of smaller claims on the tail index estimations. Similarly from the right panel, the proposed MWLE approach effectively reduces the bias of the estimated tail probability $\\hat{\\pi}_{J+1}$. \n\nThe analysis of bias-variance trade-off is also conducted through computing the mean-squared errors (MSE) of both estimated tail index $\\hat{\\gamma}$ and tail probability $\\hat{\\pi}_{J+1}$. From the right panels of Figures \\ref{fig:sim_mis_gamma} and \\ref{fig:sim_mis_tailp}, as evidenced by smaller MSEs under most choices of weight function hyperparameters, MWLE is much more preferable than MLE approach even after accounting for the increased parameter uncertainties through down-weighting the importance of smaller claims.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_gamma_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_gamma_mse.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and MSE of the estimated tail index $\\hat{\\gamma}$ versus various weight function hyperparameters under the misspecified model.}\n\\label{fig:sim_mis_gamma}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_tailp_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_tailp_mse.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and MSE of the estimated tail probability $\\hat{\\pi}_{J+1}$ versus various weight function hyperparameters under the misspecified model.}\n\\label{fig:sim_mis_tailp}\n\\end{figure}\n\n\\subsubsection{Summary remark on the choice of weight function hyperparameters}\nFrom the above two simulation studies, we find that under a wide range of choices of weight function hyperparameters, the proposed MWLE not only produces plausible model estimations under true model (Model 1), but is also effective in mitigating the bias of tail estimation inherited from model misspecifications (Model 2).\n\nAmong the three hyperparameters $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$, the choice of minimum weight hyperparameter $\\xi$ plays a particularly vital role on the bias-variance trade-off of the estimated parameters. Under misspecified model (Model 2), smaller $\\xi$ (i.e. $\\xi\\leq 0.05$) is more effective in reducing the biases of both estimated tail index $\\hat{\\gamma}$ and tail probability $\\hat{\\pi}_{J+1}$. However, as evidenced by the results produced under the true model (Model 1), the estimated parameters of the body distributions (i.e. $\\hat{\\bm{\\mu}}$ and $\\hat{\\bm{\\phi}}$) may become prohibitively unstable if $\\xi$ is chosen to be extremely small (i.e. $\\xi\\leq 0.001$) such that smaller observations are effectively almost fully discarded. It is therefore important to compare parameter uncertainties of MWLE to that of MLE, and select\/ consider only the weight function hyperparameters where the corresponding MWLE parameter uncertainties are within an acceptable range (i.e. not too off from the MLE parameter uncertainties). Overall, the choices of $\\xi$ between 0.01 and 0.05 are deemed to be suitable.\n\n\\subsection{Real data analysis} \\label{sec:ex:real}\n\\subsubsection{Data description and background} \\label{sec:ex:real:dat}\nIn this section, we study an insurance claim severity dataset kindly provided by a major insurance company operating in Greece. It consists of 64,923 motor third-party liability (MTPL) insurance policies with non-zero property claims for underwriting years 2013 to 2017. This dataset is also analyzed by \\cite{fung2021mixture} using a mixture composite model, with an emphasis on selecting various policyholder characteristics (explanatory variables) which significantly influence the claim severities. The empirical claim severity distribution exhibits several peculiar characteristics including multimodality and tail-heaviness. The excessive number of distributional nodes for small claims reflects the possibility of distributional contamination, which cannot be and should not be perfectly captured and over-fitted by parametric models like FMM. Preliminary analyses also suggest that the estimated tail index is around 1.3 to 1.4, but note that these are only rough and subjective estimates. The details of preliminary data analysis are provided in Section 9 of the supplementary materials. The key goals of this real data analysis are as follows:\n\\begin{enumerate}\n\\item Illustrate that MLE of FMM would produce highly unstable and unrobust estimates to the tail part of the claim severity distribution. This confirms that tail-robustness is an important research problem in real insurance claim severity modelling which needs to be properly addressed.\n\\item Demonstrate how the proposed MWLE approach leads to superior fittings to the tail and more reliable estimates of tail index as compared to MLE, without much sacrificing its ability to adequately capture the body.\n\\end{enumerate}\n\nTo avoid diverging the focus of this paper, in this analysis we solely examine the distributional fitting of the claim sizes without considering the explanatory variables. Note however that the proposed MWLE can be extended to a regression framework, with the discussions being leveraged to Section \\ref{sec:discussion}.\n\n\n\\subsubsection{Fitting results}\nThe claim severity dataset is fitted to the mixture Gamma-Lomax distribution with density given by Equation \\ref{eq:em:density_mixture} under the proposed MWLE approach. The fitting performances will be examined thoroughly across different number of Gamma (body) mixture components $J\\in\\{1,2,\\ldots,10\\}$ and various choices of weight function hyperparameters ($\\tilde{\\mu}\\in\\{q_{0},q_{0.9},q_{0.95},q_{0.99},q_{0.995}\\}$, $\\tilde{\\phi}\\in\\{0.025,0.1,0.25,1\\}$ and $\\xi\\in\\{0.001,0.01,0.05,0.25\\}$). The MWLE fitted parameters are also compared to the standard MLE across various $J$.\n\nWe first present in Figure \\ref{fig:greek:gamma} the fitted tail index $\\hat{\\gamma}$ versus the number of body components $J$ under all combinations of selected weight function hyperparameters. Each of the four sub-figures corresponds to a particular choice of $\\xi\\in\\{0.001,0.01,0.05,0.25\\}$. The black thick trends for each sub-figure are the MLE estimated tail indexes for comparison purpose. The MLE tail indexes are rather unstable as evidenced by great fluctuations across different number of body components $J$, showing that MLE may not be reliable in extrapolating the heavy-tailedness of complex claim distributions.\nFor instance, with a slight change of model specification from $J=5$ to $J=6$, the estimated tail index largely drops from about 1.8 to 1.5. This is rather unnatural because the change from $J=5$ to $J=6$ should only reflect a slight change in specifying the body.\nThe large drop of the estimated tail index reflects that the Lomax tail part of FMM is not specialized in extrapolating the tail-heaviness of the distribution, but instead is very sensitive to the small claims and the model specifications of the body part.\nTherefore, we conclude that the mixture Gamma-Lomax FMM is not achieving its modelling purpose under the MLE.\n\nOn the other hand, looking individually at each path under MWLE, we find that the estimated $\\hat{\\gamma}$ is much more stable across different $J$ under most choices of weight function hyperparameters, especially when $J\\geq 5$. Also, the estimated MWLE $\\hat{\\gamma}$ is in general smaller than the $\\hat{\\gamma}$ obtained by MLE, moving closer to the values roughly determined by the preliminary data analysis in Section \\ref{sec:ex:real:dat}. Note in the figure that there are a few black solid dots, which appear when the estimated $\\hat{\\gamma}$ under MWLE is outside the range of the plots. These unstable estimates of $\\hat{\\gamma}$ are rare and only occur under one the following two situations: (i) $J$ is chosen be very small (i.e. $J\\leq 2$) in the sense that the models would severely under-fit the distributional complexity of the dataset; (ii) extreme choices of weight function hyperparameters (very small $\\xi$ and $\\tilde{\\phi}$) aligned to the results of the simulation studies.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma1.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma2.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma3.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma4.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Estimated tail index versus the number of Gamma mixture components under MLE and MWLE with various choices of weight function hyperparameters.}\n\\label{fig:greek:gamma}\n\\end{figure}\n\nThe optimal choice of $J$ is tricky as evidenced by the excessive number of small distributional nodes for very small claim sizes described in Section \\ref{sec:ex:real:dat}, which should not be over-emphasized or excessively modelled as these very small claims are almost irrelevant for pricing and risk management. However, both AIC and BIC decrease slowly and steadily for MLE models as $J$ increases. The optimal $J$ in this case goes way beyond $J=10$. Under the proposed MWLE approach with various choices of weight function hyperparameters, the same model selection problem exists using RAIC and RBIC, with the reasons already explained in Section \\ref{sec:em:complex}. On the other hand, using TAIC and TBIC (especially for TBIC), a majority selections of weight function hyperparameters lead to an optimal $J=5$, aligning with the heuristic arguments by \\cite{fung2021mixture} that $J=5$ is enough for capturing all the distributional nodes except for the very small claims which are smoothly approximated by a single mixture component.\n\nTo better understand how the use of proposed MWLE affects the estimations of all parameters (not just the tail index but also parameters affecting the body distributions such as $\\bm{\\mu}$), we showcase in Table \\ref{tab:greek:est_prm} all the estimated parameters and their standard errors (based on Equation (\\ref{eq:asym_var})) using MWLE under two distinguishable example choices of hyperparameters (MWLE 1: $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.01,q_{0.99},0.1)$; MWLE 2: $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.05,q_{0.995},0.25)$) as compared to MLE parameters with $J=5$, the optimal number of body components under TBIC for both MWLE 1 and MWLE 2. Note that the two selected examples are for demonstration purpose -- generally the following findings and conclusions are also valid for other choices of weight function hyperparameters under the proposed MWLE.\n\nWe first observe that the estimated parameters influencing the body (i.e. $\\bm{\\pi}$, $\\bm{\\mu}$ and $\\bm{\\phi}$) under MWLE are very close to those under MLE, even if the smaller claims are greatly down-weighted. MWLE generally results to larger parameter uncertainties as compared with MLE -- reflecting a bias-variance trade-off, but these standard errors are of the same order of magnitude and are still relatively immaterial compared to the estimates as the sample size $n=64,923$ is large. \n\nComparing between the above two MWLE examples, we further notice that the parameter uncertainties under MWLE 1 are greater than those under MWLE 2. This is expected because the influences of smaller claims are down-weighted more under MWLE 1 than those under MWLE 2 (as reflected by smaller minimum weight hyperparameter $\\xi$ chosen under MWLE 1). On the other hand, the estimated tail index $\\hat{\\gamma}$ under MWLE 1 is slightly closer to the heuristic values (i.e. 1.3 to 1.4) than MWLE 2. These may also reflect the bias-variance trade-off among various choices of weight function hyperparameters.\n\n\\begin{table}[!h]\n\\centering\n\\begin{tabular}{lrrrrrr}\n\\toprule\n & \\multicolumn{2}{c}{MWLE 1} & \\multicolumn{2}{c}{MWLE 2} & \\multicolumn{2}{c}{MLE} \\\\\n\\cmidrule(l{3pt}r{3pt}){2-3} \\cmidrule(l{3pt}r{3pt}){4-5} \\cmidrule(l{3pt}r{3pt}){6-7}\n & \\multicolumn{1}{c}{Estimates} & \\multicolumn{1}{c}{Std. Error} & \\multicolumn{1}{c}{Estimates} & \\multicolumn{1}{c}{Std. Error} & \\multicolumn{1}{c}{Estimates} & \\multicolumn{1}{c}{Std. Error} \\\\\n\\midrule\n$\\pi_1$ & 0.3787 & 0.0053 & 0.3829 & 0.0031 & 0.3878 & 0.0022 \\\\\n$\\pi_2$ & 0.0380 & 0.0036 & 0.0404 & 0.0021 & 0.0444 & 0.0014 \\\\\n$\\pi_3$ & 0.1117 & 0.0024 & 0.1134 & 0.0020 & 0.1161 & 0.0017 \\\\\n$\\pi_4$ & 0.0221 & 0.0059 & 0.0192 & 0.0021 & 0.0153 & 0.0008 \\\\\n$\\pi_5$ & 0.2173 & 0.0036 & 0.2163 & 0.0022 & 0.2130 & 0.0019 \\\\\n\\hline\n$\\mu_1$ & 1,303.21 & 50.30 & 1,322.10 & 16.70 & 1,348.68 & 11.11 \\\\\n$\\mu_2$ & 9,171.42 & 145.83 & 9,165.92 & 64.64 & 9,165.36 & 49.38 \\\\\n$\\mu_3$ & 27,590.46 & 125.68 & 27,571.75 & 64.36 & 27,538.88 & 52.41 \\\\\n$\\mu_4$ & 317,274.90 & 2,410.93 & 323,827.70 & 2,159.37 & 322,872.40 & 2,372.68 \\\\\n$\\mu_5$ & 89,007.07 & 170.41 & 88,979.12 & 112.01 & 88,895.92 & 99.20 \\\\\n\\hline\n$\\phi_1$ & 0.9945 & 0.0175 & 0.9996 & 0.0121 & 1.0062 & 0.0113 \\\\\n$\\phi_2$ & 0.0264 & 0.0089 & 0.0284 & 0.0030 & 0.0324 & 0.0020 \\\\\n$\\phi_3$ & 0.0154 & 0.0015 & 0.0158 & 0.0007 & 0.0164 & 0.0005 \\\\\n$\\phi_4$ & 0.0472 & 0.0033 & 0.0333 & 0.0025 & 0.0186 & 0.0020 \\\\\n$\\phi_5$ & 0.0127 & 0.0007 & 0.0126 & 0.0003 & 0.0122 & 0.0002 \\\\\n\\hline\n$\\gamma$ & 1.5353 & 0.0707 & 1.6153 & 0.0586 & 1.7963 & 0.0471 \\\\\n$\\theta$ & 62,637.42 & 7,829.79 & 73,604.51 & 6,630.65 & 101,107.20 & 5,088.29\\\\\n\\hhline{=======}\n\\end{tabular}\n\\caption{\\label{tab:greek:est_prm}Estimated parameters and standard errors under MLE and MWLE approaches with $J=5$.}\n\\end{table}\n\nThe Q-Q plot in Figure \\ref{fig:greek:qq} suggests that the fitting results are satisfactory under both MWLE and MLE except for the very immaterial claims (i.e. $y<100$). Note however that due to the log-scale nature of the Q-Q plot, it is hard to examine from the plot how well the fitted models extrapolate the tail-heaviness of the claim severity data. To examine the tail behavior of the fitted models, we present the log-log plot in the left panel of Figure \\ref{fig:greek:loglog}, with the axis shifted to include large claims only. We observe that for extreme claims (i.e. claim amounts greater than about 0.5 millions, or $\\log y>13$), the logged survival probability produced by MLE fitted model diverges quite significantly from that of empirical observations. Such a divergence can effectively be mitigated by using MWLE with either of the hyperparameter settings. \n\nWe further compute the value-at-risk (VaR) and conditional tail expectation (CTE) at $100q^{\\text{th}}$ security level (denoted as $\\text{VaR}_q(Y;\\hat{\\bm{\\Phi}})$ and $\\text{CTE}_q(Y;\\hat{\\bm{\\Phi}})$ respectively) from the fitted models, and compare them to the empirical values from the severity data (denoted as $\\widehat{\\text{VaR}}_q(Y)$ and $\\widehat{\\text{CTE}}_q(Y)$ respectively). The results are summarized in Table \\ref{tab:greek:risk}. Both MLE and MWLE produce plausible estimates of VaR and CTE up to security levels of 95\\% and 75\\% respectively, reflecting the ability of both approaches in capturing the body part of severity distribution. Nonetheless, the MLE fitted model shows significant divergences of VaR and CTE from the empirical data at higher security levels. In particular, the 99\\%-CTE and 99.5\\%-CTE are largely underestimated by the MLE approach. Such a divergence is effectively reduced by the proposed MWLE approach where superior fittings to the tail are obtained. Further, MWLE 1 seems to perform slightly better than MWLE 2 in terms of tail fitting, as reflected by smaller underestimations of CTEs at high security levels. This provides a plausible trade-off to the increased parameter uncertainties under MWLE 1 as previously mentioned.\n\nTo visualize the results, we further plot the relative misfit of VaR, given by $\\log(\\text{VaR}_q(Y;\\hat{\\bm{\\Phi}})\/\\widehat{\\text{VaR}}_q(Y))$, versus the log survival probability $\\log (1-q)\\in(-4.7,-7.5)$, equivalent to the range of security level from 99\\% to 99.95\\%, in the right panel of Figure \\ref{fig:greek:loglog}. We observe that the MLE fitted model over-estimates the VaR of large claims (security level between 99\\% to 99.8\\%) but then largely under-extrapolates the extreme claims (security level beyond 99.8\\%). This issue is well mitigated by the MWLE where the misfits of VaR are smaller in both regions. Therefore, we conclude that the proposed MWLE effectively improves the goodness-of-fit on the tail part of distribution (as compared to the MLE) without much sacrificing its flexibly to adequately capture the body part. \n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_qq.jpg}\n\\end{subfigure}\n\\end{center}\n\\caption{Q-Q plot under MLE and MWLE with two selected combinations of weight function hyperparameters.}\n\\label{fig:greek:qq}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_loglog.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_misfit.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Left panel: log-log plot of fitted models compared to empirical data; Right panel: misfit of logged claim amounts versus logged survival probabilities under three fitted models.}\n\\label{fig:greek:loglog}\n\\end{figure}\n\n\\begin{table}[!h]\n\\centering\n\\begin{tabular}{rrrrrlrrrrr}\n\\toprule\n \\multicolumn{5}{c}{VaR ('000)} &  & \\multicolumn{5}{c}{CTE ('000)} \\\\\n\\cmidrule(l{3pt}r{3pt}){1-5} \\cmidrule(l{3pt}r{3pt}){7-11}\n \\multicolumn{1}{c}{Level} & \\multicolumn{1}{c}{MLE} & \\multicolumn{1}{c}{MWLE 1} & \\multicolumn{1}{c}{MWLE 2} & \\multicolumn{1}{c}{Empirical} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{Level} & \\multicolumn{1}{c}{MLE} & \\multicolumn{1}{c}{MWLE 1} & \\multicolumn{1}{c}{MWLE 2} & \\multicolumn{1}{c}{Empirical} \\\\\n\\midrule\n50\\% & 21 & 21 & 21 & 21 &  & 0\\% & 109 & 112 & 111 & 116 \\\\\n75\\% & 83 & 82 & 82 & 82 &  & 50\\% & 174 & 180 & 177 & 187 \\\\\n95\\% & 190 & 191 & 187 & 182 &  & 75\\% & 468 & 505 & 489 & 536 \\\\\n99\\% & 461 & 450 & 445 & 452 &  & 90\\% & 1,140 & 1,326 & 1,242 & 1,498 \\\\\n99.5\\% & 719 & 693 & 674 & 676 &  & 95\\% & 1,711 & 2,115 & 1,948 & 2,455 \\\\\n99.75\\% & 1,149 & 1,031 & 1,046 & 1,075 &  & 99\\% & 2,533 & 3,379 & 3,063 & 4,057 \\\\\n99.95\\% & 2,787 & 3,163 & 3,220 & 3,348 &  & 99.5\\% & 5,956 & 10,243 & 8,572 & 13,329\\\\\n\\hhline{===========}\n\\end{tabular}\n\\caption{VaR and CTE (in thousands) estimated by MLE, MWLE and empirical approaches.}\n\\label{tab:greek:risk}\n\\end{table}\n\n\n\n\\section{Discussions} \\label{sec:discussion}\nIn this paper, we introduce a maximum weighted log-likelihood estimation (MWLE) approach to robustly estimate the tail part of finite mixture models (FMM) while preserving the capability of FMM to flexibly capture the complex distributional phenomena from the body part. Asymptotic theories justify the unbiasedness and robustness of the proposed estimator. In computational aspect, the applicability of EM-based algorithm for efficient estimation of parameters makes the proposed MWLE distinctive compared to the existing literature on weighted likelihood approach. Through several simulation studies and real data analyses, we empirically confirm that the proposed MWLE approach is more appropriate in specifying the tail part of the distribution compared to MLE, and at the same time it still preserves the flexibility of FMM in fitting the smaller observations.\n\nAnother advantage of the MWLE not yet mentioned throughout this paper is its extensibility. First, it is obvious that the proposed MWLE is not restricted to FMM but it is also applicable to any continuous or discrete distributions. Second, MWLE can be easily extended to regression settings, which is crucial for insurance pricing perspective as insurance companies often determine different premiums across policyholders based on individual attributes (e.g. age, geographical location and past claim history). In regression settings, we define $\\bm{x}=(\\bm{x}_1,\\ldots,\\bm{x}_n)$ as the covariates vectors for each of the $n$ observations. Then, the weighted log-likelihood function in Equation (\\ref{eq:loglik_weight}) is then re-expressed as\n\\begin{equation} \\label{eq:loglik_weight_reg}\n\\mathcal{L}^*_n(\\bm{\\Phi};\\bm{y},\\bm{x})=\\sum_{i=1}^{n}W(y_i)\\log \\frac{h(y_i;\\bm{\\Phi},\\bm{x}_i)W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi},\\bm{x}_i)W(u)du}\n\\end{equation}\nfor some regression models with density function $h(y_i;\\bm{\\Phi},\\bm{x}_i)$. Obviously, the asymptotic properties still hold subject further to some regularity conditions on covariates $\\bm{x}_i$. For parameter estimations using the GEM algorithm, only the hypothetical data approach (Method 1, which converge slower than Method 2 in Section \\ref{sec:ex}) works, because the transformed mixing probabilities in Equation (\\ref{eq:em:pi_trans}) under Method 2 are assumed to be homogeneous across all observations. We leave all theoretical details with more empirical studies and applications to the future research direction.\n\n\n\n\n\\bibliographystyle{abbrvnat}\n\n\\section{Regularity conditions for asymptotic theory} \\label{apx:asym_reg}\nLet $h(y;\\bm{\\Phi})$ be the density function of $Y$ with parameter space of $\\bm{\\Phi}\\in\\bm{\\Omega}$. For a more concise presentation on the regularity conditions, we here write $\\bm{\\Phi}=(\\psi_1,\\ldots,\\psi_P)$ where $P$ is the total number of parameters in the model. The regularity conditions are:\n\n\\begin{enumerate}[font={\\bfseries},label={R\\arabic*.}]\n\\item $h(y;\\bm{\\Phi})$ has common support in $y$ for all $\\bm{\\Phi}\\in\\bm{\\Omega}$, $h(y;\\bm{\\Phi})$ is identifiable in $\\bm{\\Phi}$ up to a permutation of mixture components.\n\\item $h(y;\\bm{\\Phi})$ admits third partial derivatives with respect to $\\bm{\\Phi}$ for each $\\bm{\\Phi}\\in\\bm{\\Omega}$ and for almost all $y$.\n\\item For all $j_1,j_2=1,\\ldots,P$, the first two derivatives of $h(y;\\bm{\\Phi})$ satisfy\n\\begin{equation}\nE\\left[\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\right]=0;\n\\end{equation}\n\\begin{equation}\nE\\left[\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right]=E\\left[-\\frac{\\partial^2}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right].\n\\end{equation}\n\\item The Fisher information matrix is finite and positive definite at $\\bm{\\Phi}=\\bm{\\Phi}_0$:\n\\begin{equation}\n\\mathcal{I}(\\bm{\\Phi})=E\\left[\\left(\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(y;\\bm{\\Phi})\\right)\\left(\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(y;\\bm{\\Phi})\\right)^T\\right].\n\\end{equation}\n\\item There exists an integrable function $\\mathcal{M}(y)$ such that\n\\begin{equation}\n\\hspace{-1cm}\n\\left|\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\\quad\n\\left|\\frac{\\partial^2}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\\quad\n\\left|\\frac{\\partial^3}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}\\partial\\psi_{j_3}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\n\\end{equation}\n\\begin{equation}\n\\hspace{-1cm}\n\\left|\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\\quad\n\\left|\\frac{\\partial^2}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_3}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\n\\end{equation}\n\\begin{equation}\n\\hspace{-1cm}\n\\left|\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_3}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y).\n\\end{equation}\n\\end{enumerate}\n\n\\section{Proof of Theorems 1 and 2} \\label{apx:asym_proof1}\nWe first focus on Theorem 1. Denote the weighted log-likelihood of a single observation\n\\begin{equation}\n\\mathcal{L}^{*}(\\bm{\\Phi};y)=W(y)\\log \\frac{h(y;\\bm{\\Phi})W(y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}.\n\\end{equation}\n\nThe consistency and asymptotic normality can be proved by applying Theorems 5.41 and 5.42 of \\cite{van2000asymptotic}. The theorems require the regularity conditions that $E\\left[\\|\\partial\/\\partial\\bm{\\Phi}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\|^2\\right]<\\infty$, the matrix $E\\left[\\partial^2\/\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]$ exists and that $|\\partial^3\/\\partial\\psi_{j_1}\\partial\\psi_{j_2}\\partial\\psi_{j_3}\\mathcal{L}^{*}(\\bm{\\Phi};y)|$ is dominated by a fixed integrable function of $y$, $j_1,j_2,j_3=1,\\ldots,P$ and $\\psi_j$ is the $j^{\\text{th}}$ element of $\\bm{\\Phi}$. Through a direct computation of differentiations, the aforementioned equations can all be expressed as functions of $\\kappa(u;\\bm{\\Phi})$ and $\\int_{0}^{\\infty}\\kappa(u;\\bm{\\Phi})h(u;\\bm{\\Phi})W(u)du$ only, where $\\kappa(\\bm{\\Phi})$ can be the six terms presented in regularity condition \\textbf{R5} (the left hand side of the six equations underneath \\textbf{R5} without the absolute sign). Given \\textbf{R5} that $\\kappa(u;\\bm{\\Phi})$ is bounded by an integrable function and since $|\\int_{0}^{\\infty}\\kappa(u;\\bm{\\Phi})h(u;\\bm{\\Phi})W(u)du|\\leq \\int_{0}^{\\infty}|\\kappa(u;\\bm{\\Phi})|h(u;\\bm{\\Phi})du$, the aforementioned regularity conditions required by \\cite{van2000asymptotic} hold.\n\n\\medskip\n\nFor consistency, it suffices from Theorem 5.42 of \\cite{van2000asymptotic} to show that $\\bm{\\Phi}_0$ is the maximizer of\n\\begin{align}\nE_{\\bm{\\Phi}_0}\\left[\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]\n&=\\int_{0}^{\\infty}W(y)\\log \\frac{h(y;\\bm{\\Phi})W(y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}h(y;\\bm{\\Phi}_0)dy\\nonumber\\\\\n&=c_1\\int_{0}^{\\infty}\\tilde{h}(y;\\bm{\\Phi}_0)\\log\\frac{\\tilde{h}(y;\\bm{\\Phi})}{\\tilde{h}(y;\\bm{\\Phi}_0)}dy+c_2\\nonumber\\\\\n&=-c_1D_{\\text{KL}}\\left(\\tilde{h}(y;\\bm{\\Phi})\\|\\tilde{h}(y;\\bm{\\Phi}_0)\\right)+c_2,\n\\end{align}\nwhere $c_1= \\int_0^\\infty h(y; \\bm{\\Phi}_0)W(y) dy >0$ and $c_2=c_1 \\int_0^\\infty \\tilde{h}(y;\\bm{\\Phi}_0) \\log \\tilde{h}(y;\\bm{\\Phi}_0) dy$ are constants and $D_{\\text{KL}}(Q_1\\|Q_2)\\geq 0$ is the KL divergence between $Q_1$ and $Q_2$. Since $D_{\\text{KL}}\\left(\\tilde{h}(y;\\bm{\\Phi})\\|\\tilde{h}(y;\\bm{\\Phi}_0)\\right)=0$ as $\\bm{\\Phi}=\\bm{\\Phi}_0$, the result follows.\n\n\\medskip\n\nFor asymptotic normality, from Theorem 5.41 of \\cite{van2000asymptotic}, we have $\\sqrt{n}(\\hat{\\bm{\\Phi}}_n-\\bm{\\Phi}_0)\\overset{d}{\\rightarrow}\\mathcal{N}(\\bm{0},\\bm{\\Sigma})$ with \n$\\bm{\\Sigma}=\\bm{\\Gamma}^{-1}\\bm{\\Lambda}\\bm{\\Gamma}^{-1}$, where \n\\begin{equation} \\label{eq:asym:lambda_proof}\n\\bm{\\Lambda}=E_{\\bm{\\Phi}_0}\\left[\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\n\\end{equation}\nand\n\\begin{equation} \\label{eq:asym:gamma_proof}\n\\bm{\\Gamma}=-E_{\\bm{\\Phi}_0}\\left[\\frac{\\partial^2}{\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right].\n\\end{equation}\n\nPerforming the derivatives and algebra manipulations from Equations (2.3) and (2.4) would result to Equations (4.2) and (4.3) respectively, which prove the asymptotic normality result.\n\n\\medskip\n\nProof idea of Theorem 2 is completely identical as the above, except that the expectations in Equations (2.3) and (2.4) are taken as $\\tilde{E}[\\cdot]$ instead of $E_{\\bm{\\Phi}_0}[\\cdot]$.\n\n\n\n\\section{Proof of Theorem 3} \\label{apx:asym_proof2}\nWe begin with the following lemmas:\n\\begin{lemma} \\label{apx:lem:asym1}\nTo prove Theorem 3, it suffices to show that\n\\begin{equation}\nT_n(\\bm{\\Phi}):=\\frac{\\partial}{\\partial\\gamma}\\tilde{E}_n[\\log \\tilde{h}_{n}(Y;\\bm{\\Phi})]\n\\end{equation}\nis asymptotically a strictly decreasing function of $\\gamma$ as $n\\rightarrow\\infty$, with $T_n(\\bm{\\Phi})|_{\\gamma=\\gamma_0}\\rightarrow 0$ as $n\\rightarrow\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nIf we keep the weight function $W(\\cdot)$ fixed (independent of $n$), applying Theorem 5.7 of \\cite{van2000asymptotic} we have that maximizing the weighted log-likelihood function $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ is asymptotically equivalent to maximizing $\\tilde{E}_n[\\log \\tilde{h}_{n}(Y;\\bm{\\Phi})]$ (which is indeed independent of $n$).\n\nNow that the weight function $W_n(\\cdot)$ depends on $n$, and as $n$ increases, the increasing distortion (more down-weighting) of the relative importance of observations would cause reduction of the effective number of observations. Heuristically, we need the number of observations $n$ to increase faster than the distortion impacts of $W_n(\\cdot)$, so that effective number of observations grows to infinity and large sample theory still applies. Quantitatively, we require that the variance of (scaled) empirical weighted log-likelihood\n\\begin{equation}\nV_n(\\bm{\\Phi}):=\\text{Var}\\left(\\frac{1}{n\\int_{0}^{\\infty}W_n(u)g(u)du}\\sum_{i=1}^{n}W_n(Y)\\log\\tilde{h}_{n}(Y;\\bm{\\Phi})\\right)\\rightarrow 0\n\\end{equation}\nas $n\\rightarrow\\infty$, such that the (scaled) empirical weighted log-likelihood function converges to its expectation which is $\\tilde{E}_n[\\log \\tilde{h}_{n}(Y;\\bm{\\Phi})]$. Now, $V_n(\\bm{\\Phi})$ is evaluated as follows:\n\\begin{align}\nV_n(\\bm{\\Phi})\n&=\\frac{1}{n(\\int_{0}^{\\infty}W_n(u)g(u)du)^2}\\text{Var}\\left(W_n(Y)\\log\\tilde{h}_{n}(Y;\\bm{\\Phi})\\right)\\nonumber\\\\\n&\\leq\\frac{1}{n(\\int_{0}^{\\infty}W_n(u)g(u)du)^2}\\tilde{E}\\left[W_n(Y)(\\log\\tilde{h}_{n}(Y;\\bm{\\Phi}))^2\\right]\\nonumber\\\\\n&=\\frac{1}{n\\int_{0}^{\\infty}W_n(u)g(u)du}\\int_{0}^{\\infty}\\frac{W_n(y)g(y)}{\\int_{0}^{\\infty}W_n(u)g(u)du}(\\log\\tilde{h}_{n}(y;\\bm{\\Phi}))^2dy\\nonumber\\\\\n&=\\frac{\\tilde{E}_n[(\\log \\tilde{h}_{n}(Y;\\bm{\\Phi}))^2]}{n\\tilde{E}[W_n(Y)]}\\rightarrow 0,\n\\end{align}\nwhere the convergence is based on Assumption \\textbf{A4}.\n\\end{proof}\n\n\\begin{lemma} \\label{apx:lem:asym2}\n(Monotone density theorem -- Theorem 1.7.2 of \\cite{bingham1989regular}) Denote $H$ as a probability distribution function with $h$ being the corresponding probability density function. Assume $h$ is ultimately monotone (i.e. $h$ is monotone on $(z,\\infty)$ for some $z>0$). If\n\\begin{equation}\n\\bar{H}(y)\\sim y^{-\\gamma}L(y)\n\\end{equation}\nas $y\\rightarrow\\infty$ for some $\\gamma>0$ and slowly varying functions $L$, then\n\\begin{equation}\nh(y)\\sim \\gamma y^{-\\gamma-1}L(y)\n\\end{equation}\nas $y\\rightarrow\\infty$.\n\\end{lemma}\n\nWe proceed to the proof of Theorem 3 as follows. Using the result from Lemma \\ref{apx:lem:asym1}, it suffices to evaluate\n\\begin{align} \\label{apx:eq:proof2:Tn}\nT_n(\\bm{\\Phi})\n&=\\frac{\\partial}{\\partial\\gamma}\\int_{0}^{\\infty}\\tilde{g}_n(y)\\log \\tilde{h}_{n}(y;\\bm{\\Phi})dy\\nonumber\\\\\n&=\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\tilde{g}^{*}_n(y)\\log\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy + o(1)\\nonumber\\\\\n&=\\frac{\\partial}{\\partial\\gamma}\\log\\tilde{h}^{*}_{n}(\\tau_n;\\bm{\\Phi})\n+\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\left[\\frac{\\partial}{\\partial y}\\log\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})\\right]\\times\\bar{\\tilde{G}}^{*}_{n}(y)dy + o(1)\\nonumber\\\\\n&:=M_1(\\tau_n;\\bm{\\Phi})+M_2(\\tau_n;\\bm{\\Phi}) + o(1),\n\\end{align}\nwhere\n\\begin{equation}\n\\tilde{g}^{*}_{n}(y)=\\frac{g(y)W_n(y)}{\\int_{\\tau_n}^{\\infty}g(u)W_n(u)du}1\\{y\\geq\\tau_n\\},\\qquad \\tilde{h}^{*}_{n}(y;\\bm{\\Phi})=\\frac{h(y;\\bm{\\Phi})W_n(y)}{\\int_{\\tau_n}^{\\infty}h(u;\\bm{\\Phi})W_n(u)du}1\\{y\\geq\\tau_n\\},\n\\end{equation}\nare the proper transformed density functions, and $\\tilde{G}^{*}_{n}$ and $\\tilde{H}_{n}^{*}$ are the corresponding distribution functions. The second equality of Equation (\\ref{apx:eq:proof2:Tn}) is resulted from Assumption \\textbf{A3}, while the third equality is followed by integration by parts. Now, we evaluate $M_1(\\tau_n;\\bm{\\Phi})$ and $M_2(\\tau_n;\\bm{\\Phi})$ as follows:\n\n\\begin{align}\nM_1(\\tau_n;\\bm{\\Phi})\n&=\\frac{\\partial}{\\partial\\gamma}\\log\\tilde{h}^{*}_{n}(\\tau_n;\\bm{\\Phi})\\nonumber\\\\\n&=\\frac{\\partial}{\\partial\\gamma}\\left[\\log\\gamma-(\\gamma+1)\\log\\tau_n+\\log L(\\tau_n;\\bm{\\Phi})\\right]\\nonumber\\\\\n&\\hspace{3em}-\\int_{\\tau_n}^{\\infty}\\frac{\\partial}{\\partial\\gamma}\\left[\\log\\gamma-(\\gamma+1)\\log y+\\log L(y;\\bm{\\Phi})\\right]\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy+o(1)\\nonumber\\\\\n&=\\frac{1}{\\gamma}-\\log\\tau_n-\\frac{1}{\\gamma}+\\int_{\\tau_n}^{\\infty}(\\log y)\\times\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy\n-\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\log\\frac{L(y;\\bm{\\Phi})}{L(\\tau_n;\\bm{\\Phi})}\\times\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy+o(1)\\nonumber\\\\\n&=-\\log\\tau_n+\\log\\tau_n\n+\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{H}}^{*}_{n}(y;\\bm{\\Phi})dy\n-\\frac{\\partial}{\\partial\\gamma}\\int_{1}^{\\infty}\\log\\frac{L(\\tau_nt;\\bm{\\Phi})}{L(\\tau_n;\\bm{\\Phi})}\\times\\tau_n\\tilde{h}^{*}_{n}(\\tau_nt;\\bm{\\Phi})dt+o(1)\\nonumber\\\\\n&=\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{H}}^{*}_{n}(y;\\bm{\\Phi})dy + o(1),\n\\end{align}\nwhere dominated convergence theorem and integration by parts are repeatedly used. The second equality involves monotone density theorem (Lemma \\ref{apx:lem:asym2}) with Assumption \\textbf{A5} being satisfied. The last term of the second last equality converges to zero uniformly on $\\bm{\\Phi}$ due to dominated convergence theorem and the uniform convergence conditions in Assumption \\textbf{A2}. Using similar techniques as the above, $M_2(\\tau_n;\\bm{\\Phi})$ can be evaluated as\n\\begin{align}\nM_2(\\tau_n;\\bm{\\Phi})\n&=-\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{G}}^{*}_{n}(y)dy\n+\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\frac{\\partial}{\\partial y}(\\log L(y;\\bm{\\Phi}))\\times\\bar{\\tilde{G}}^{*}_{n}(y)dy\\nonumber\\\\\n&=-\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{G}}^{*}_{n}(y)dy-\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\log\\frac{L(y;\\bm{\\Phi})}{L(\\tau_n;\\bm{\\Phi})}\\times\\tilde{g}^{*}_{n}(y)dy\\nonumber\\\\\n&=-\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{G}}^{*}_{n}(y)dy + o(1).\n\\end{align}\n\nTo sum up, we have\n\\begin{equation}\nT_n(\\bm{\\Phi})\n=\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\left[\\bar{\\tilde{H}}^{*}_{n}(y;\\bm{\\Phi})-\\bar{\\tilde{G}}^{*}_{n}(y)\\right]dy\n=\\int_{1}^{\\infty}\\frac{1}{t}\\left[\\bar{\\tilde{H}}^{*}_{n}(\\tau_nt;\\bm{\\Phi})-\\bar{\\tilde{G}}^{*}_{n}(\\tau_nt)\\right]dt.\n\\end{equation}\n\nInvestigating each term inside the integrand, we have\n\\begin{align}\n\\bar{\\tilde{H}}^{*}_{n}(\\tau_nt;\\bm{\\Phi})\n&=\\frac{\\int_t^{\\infty}h(\\tau_nv;\\bm{\\Phi})W_n(\\tau_nv)dv}{\\int_1^{\\infty}h(\\tau_nv;\\bm{\\Phi})W_n(\\tau_nv)dv}\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)[L(\\tau_nv;\\bm{\\Phi})\/L(\\tau_n;\\bm{\\Phi})]dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)[L(\\tau_nv;\\bm{\\Phi})\/L(\\tau_n;\\bm{\\Phi})]dv} + o(1)\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv} + o(1),\n\\end{align}\nand\n\\begin{align}\n\\bar{\\tilde{G}}^{*}_{n}(\\tau_nt)\n&=\\frac{\\int_t^{\\infty}g(\\tau_nv)W_n(\\tau_nv)dv}{\\int_1^{\\infty}g(\\tau_nv)W_n(\\tau_nv)dv}\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)[L_0(\\tau_nv)\/L_0(\\tau_n)]dv}{\\int_1^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)[L_0(\\tau_nv)\/L_0(\\tau_n)]dv} + o(1)\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv} + o(1),\n\\end{align}\nwhere $\\tilde{W}_n(v)=W_n(\\tau_nv)$. Therefore, it is clear that\n\\begin{equation}\nT_n(\\bm{\\Phi})=\\int_{1}^{\\infty}\\frac{1}{t}\\left[\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}-\\frac{\\int_t^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv}\\right]dt+o(1)\n\\end{equation}\nconverges to zero for $\\gamma=\\gamma_0$ as $n\\rightarrow\\infty$. To show that $T_n(\\bm{\\Phi})$ is a strictly decreasing function of $\\gamma$ as $n\\rightarrow\\infty$, it suffices to evaluate\n\\begin{align}\n\\frac{\\partial}{\\partial\\gamma}\\log\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}\n&=-\\frac{\\int_t^{\\infty}(\\log v)v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}+\\frac{\\int_1^{\\infty}(\\log v)v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv},\n\\end{align}\nwhich is negative if and only if\n\\begin{equation}\n\\int_{1}^{t}(\\log v)k_{n,1,t}(v;\\gamma)dv<\\int_{t}^{\\infty}(\\log v)k_{n,t,\\infty}(v;\\gamma)dv,\n\\end{equation}\nwhere\n\\begin{equation}\nk_{n,t_1,t_2}(v;\\gamma)=\\frac{v^{-\\gamma-1}\\tilde{W}_n(v)}{\\int_{t_1}^{t_2}v^{-\\gamma-1}\\tilde{W}_n(v)dv}1\\{t_1<v\\leq t_2\\}\n\\end{equation}\nis a proper probability density function with $1\\leq t_1<t_2\\leq\\infty$. Since $k_{n,1,t}(v;\\gamma)$ and $k_{n,t,\\infty}(v;\\gamma)$ are both proper densities, it is clear that $\\int_{1}^{t}(\\log v)k_{n,1,t}(v;\\gamma)dv=\\log v_1$ for some $v_1\\in (1,t)$ and $\\int_{t}^{\\infty}(\\log v)k_{n,t,\\infty}(v;\\gamma)dv=\\log v_2$ for some $v_2\\in (t,\\infty]$. The result then follows.\n\n\n\\section{GEM algorithm for MWLE under J-Gamma Lomax mixture model: Hypothetical data approach} \n\n\\subsection{Construction of complete data}\nThe complete data is given by\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i,k_i,\\{\\bm{z}'_{is},y'_{is}\\}_{s=1,\\ldots,k_i})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $\\bm{z}_i=(z_{i1},\\ldots,z_{i(J+1)})$ with $z_{ij}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ latent class and $z_{ij}=0$ otherwise. Similarly, $\\bm{z}'_i=(z'_{is1},\\ldots,z'_{is(J+1)})$ and $z'_{isj}=1$ if the $s^{\\text{th}}$ missing sample generated by observation $i$ belongs to the $j^{\\text{th}}$ latent class, and $z'_{isj}=0$ otherwise. The complete data weighted log-likelihood function is given by\n\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^nW(y_i)\\left\\{\\left[\\sum_{j=1}^{J}z_{ij}\\log \\pi_jf_b(y_i;\\mu_j,\\phi_j)\\right]+z_{i(J+1)}\\log\\pi_{J+1} f_t(y_i;\\theta,\\gamma)\\right\\}\\nonumber\\\\\n&\\quad +\\sum_{i=1}^{n}\\sum_{s=1}^{k_i}W(y_i)\\left\\{\\left[\\sum_{j=1}^J z'_{ijs}\\log\\pi_j f_b(y'_{is};\\mu_j,\\phi_j)\\right]+z'_{i(J+1)s}\\log\\pi_{J+1} f_t(y'_{is};\\theta,\\gamma)\\right\\}.\n\\end{align}\n\n\\subsection{E-step} \\label{supp:sec:em_e}\nThe expectation of the complete data weighted log-likelihood is given by the following for the $l^{\\text{th}}$ iteration:\n\\begin{align}\n&Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\\nonumber\\\\\n&=\\sum_{i=1}^nW(y_i)\\Bigg\\{\\sum_{j=1}^{J}z^{(l)}_{ij}\\left\\{\\log\\pi_j-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\log y_i-\\frac{y_i}{\\phi_j\\mu_j}\\right\\}\\nonumber\\\\\n&\\hspace{8em}+ z^{(l)}_{i(J+1)}\\left\\{\\log\\pi_{J+1}+\\log\\gamma+\\gamma\\log\\theta-(\\gamma+1)\\log(y_i+\\theta)\\right\\}\\Bigg\\}\\nonumber\\\\\n&\\quad +k^{(l)}\\left(\\sum_{i=1}^{n}W(y_i)\\right)\\Bigg\\{\\sum_{j=1}^J z^{'(l)}_{j}\\left\\{\\log\\pi_j-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\widehat{\\log y'}^{(l)}_j-\\frac{\\widehat{y'}^{(l)}_j}{\\phi_j\\mu_j}\\right\\}\\nonumber\\\\\n&\\hspace{12em}+z^{'(l)}_{(J+1)}\\left\\{\\log\\pi_{J+1}+\\log\\gamma+\\gamma\\log\\theta-(\\gamma+1)\\widehat{\\log(y'+\\theta)}^{(l)}_{J+1}\\right\\}\\Bigg\\},\n\\end{align}\nwhere\n\\begin{equation}\nz^{(l)}_{ij}=P(z_{ij}=1|\\bm{y},\\bm{\\Phi}^{(l-1)})=\n\\begin{cases}\n\\dfrac{\\pi_j^{(l-1)}f_b(y_i;\\mu_j^{(l-1)},\\phi_j^{(l-1)})}{h(u;\\bm{\\Phi}^{(l-1)})},\\quad j=1,\\ldots,J\\\\\n\\dfrac{\\pi^{(l-1)}_{J+1}f_t(y_i;\\theta,\\gamma^{(l-1)})}{h(u;\\bm{\\Phi}^{(l-1)})},\\quad j=J+1,\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nk^{(l)}=E(k_i|\\bm{y},\\bm{\\Phi}^{(l-1)})=\\frac{\\int_0^{\\infty}f_t(u;\\bm{\\Phi}^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}h(u;\\bm{\\Phi}^{(l-1)})W(u)du},\n\\end{equation}\n\\begin{equation}\nz_{j}^{'(l)}=P(z'_{ijs}=1|\\bm{y},\\bm{\\Phi}^{(l-1)})\n\\begin{cases}\n\\dfrac{\\pi_j^{(l-1)}\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}h(u;\\bm{\\Phi}^{(l-1)})(1-W(u))du},\\quad j=1,\\ldots,J\\\\\n\\dfrac{\\pi_j^{(l-1)}\\int_0^{\\infty}f_t(u;\\theta,\\gamma^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}h(u;\\bm{\\Phi}^{(l-1)})(1-W(u))du}, \\quad j=J+1,\n\\end{cases}\n\\end{equation}\n\\begin{equation}\n\\widehat{y'}^{(l)}_j=E(y_{is}'|\\bm{y},\\bm{\\Phi}^{(l-1)},z'_{ijs}=1)\n=\\frac{\\int_0^{\\infty}uf_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}, \\quad j=1,\\ldots,J,\n\\end{equation}\n\\begin{equation}\n\\widehat{\\log y'}^{(l)}_j=E(\\log y_{is}'|\\bm{y},\\bm{\\Phi}^{(l-1)},z'_{ijs}=1)\n=\\frac{\\int_0^{\\infty}\\log uf_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}, \\quad j=1,\\ldots,J,\n\\end{equation}\n\\begin{equation}\n\\widehat{\\log(y'+\\theta)}^{(l)}_{J+1}=E(\\log (y_{is}'+\\theta)|\\bm{y},\\bm{\\Phi}^{(l-1)},z'_{i(J+1)s}=1)\n=\\frac{\\int_0^{\\infty}\\log (u+\\theta) f_t(u;\\theta,\\gamma^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}f_t(u;\\theta,\\gamma^{(l-1)})(1-W(u))du}.\n\\end{equation}\n\nNote that the above integrals all have analytical form of solutions under the example choice of the following weight functions (for the generalized weight function form as presented in the main paper, we do a numerical integration instead):\n\\begin{itemize}\n    \\item Case 1: Exponential distribution with $W(y;\\tilde{\\mu})=1-\\exp\\{-y\/\\tilde{\\mu}\\}$;\n    \\item Case 2: Two-point discrete distribution with $W(y;\\tilde{\\mu},\\tilde{\\phi})=(1-\\tilde{\\phi})1\\{y>\\tilde{\\mu}\\}+\\tilde{\\phi}$.\n\\end{itemize}\n\nRe-parameterize the gamma distribution with $\\alpha=1\/\\phi_j$ and $\\beta=1\/\\phi_j\\mu_j$, we are to compute\n\\begin{equation}\n\\int_{0}^{\\infty}q(u)\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du\n\\end{equation}\nfor $q(u)=1$, $q(u)=u$ and $q(u)=\\log u$; and\n\\begin{equation}\n\\int_{0}^{\\infty}r(u)\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du\n\\end{equation}\nfor $r(u)=1$ and $r(u)=\\log(u+\\theta)$.\n\n\\textbf{Case 1}. We have the following analytical results:\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\left(\\frac{\\beta}{\\beta+1\/\\tilde{\\mu}}\\right)^{\\alpha},\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\alpha\\beta^{\\alpha}}{(\\beta+1\/\\tilde{\\mu})^{\\alpha+1}},\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}\\frac{\\partial}{\\partial\\alpha}\\frac{\\Gamma(\\alpha)}{(\\beta+1\/\\tilde{\\mu})^{\\alpha}},\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=\\gamma\\left(\\frac{\\theta}{\\tilde{\\mu}}\\right)^{\\gamma}\\exp\\left\\{\\frac{\\theta}{\\tilde{\\mu}}\\right\\}\\Gamma(-\\gamma;\\frac{\\theta}{\\tilde{\\mu}},\\infty),\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log(u+\\theta)\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=-\\gamma\\theta^{\\gamma}\\exp\\left\\{\\frac{\\theta}{\\tilde{\\mu}}\\right\\}\\frac{\\partial}{\\partial\\gamma}\\Gamma(-\\gamma;\\frac{\\theta}{\\tilde{\\mu}},\\infty),\n\\end{equation}\nwhere $\\Gamma(m;c_1,c_2)=\\int_{c_1}^{c_2}u^{m-1}\\exp\\{-u\\}du$ is an incomplete gamma function.\n\n\\textbf{Case 2}. We have the following analytical results:\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\tilde{\\phi}\\frac{\\Gamma(\\alpha;\\beta\\tilde{\\mu},\\infty)}{\\Gamma(\\alpha)}+(1-\\tilde{\\phi}),\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\alpha}{\\beta}\\left[\\tilde{\\phi}\\Gamma(\\alpha+1;\\beta\\tilde{\\mu},\\infty)+(1-\\tilde{\\phi})\\right],\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}\\left[\\tilde{\\phi}\\frac{\\partial}{\\partial\\alpha}\\frac{\\Gamma(\\alpha;\\beta\\tilde{\\mu},\\infty)}{\\beta^{\\alpha}}+(1-\\tilde{\\phi})\\frac{\\partial}{\\partial\\alpha}\\frac{\\Gamma(\\alpha)}{\\beta^{\\alpha}}\\right],\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=\\tilde{\\phi}\\left(\\frac{\\theta}{\\tilde{\\mu}+\\theta}\\right)^{\\gamma}+(1-\\tilde{\\phi}),\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log(u+\\theta)\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=-\\gamma\\theta^{\\gamma}\\frac{\\partial}{\\partial\\gamma}\\left[\\tilde{\\phi}\\frac{1}{\\gamma(\\tilde{\\mu}+\\theta)^{\\gamma}}+(1-\\tilde{\\phi})\\frac{1}{\\gamma\\theta^{\\gamma}}\\right].\n\\end{equation}\n\n\\subsection{M-step} \\label{supp:sec:em_m}\nMaximizing $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ with respect to $\\bm{\\Phi}$ yields the following parameter updates:\n\\begin{equation}\n\\pi_j^{(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{ij}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_j^{'(l)}}{\\sum_{j'=1}^{J+1}\\left\\{\\sum_{i=1}^{n}W(y_i)z_{ij'}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_{j'}^{'(l)}\\right\\}},\\quad j=1,\\ldots,J+1,\n\\end{equation}\n\\begin{equation}\n\\mu_j^{(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{ij}^{(l)}y_i+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_j^{'(l)}\\widehat{y'}^{(l)}_j}{\\sum_{i=1}^{n}W(y_i)z_{ij}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_j^{'(l)}},\\quad j=1,\\ldots,J,\n\\end{equation}\n\\begin{align}\n\\phi_j^{(l)}\n&=\\underset{\\phi_j>0}{\\text{argmax}}\\Bigg\\{\\sum_{i=1}^nW(y_i)z^{(l)}_{ij}\\left\\{-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j^{(l)}-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\log y_i-\\frac{y_i}{\\phi_j\\mu_j}\\right\\}\\nonumber\\\\\n&\\hspace{5em} +k^{(l)}\\left(\\sum_{i=1}^{n}W(y_i)\\right)z^{'(l)}_{j}\\left\\{-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j^{(l)}-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\widehat{\\log y'}^{(l)}_j-\\frac{\\widehat{y'}^{(l)}_j}{\\phi_j\\mu_j^{(l)}}\\right\\}\\Bigg\\},\\nonumber\\\\\n\\end{align}\n\\begin{equation}\n\\gamma^{(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{i(J+1)}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_{J+1}^{'(l)}}{\\sum_{i=1}^{n}W(y_i)z_{i(J+1)}^{(l)}\\left[\\log(y_i+\\theta)-\\log\\theta\\right]+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_{J+1}^{'(l)}\\left[\\widehat{\\log(y'+\\theta)}^{(l)}_{J+1}-\\log\\theta\\right]}.\n\\end{equation}\n\nNote here that $\\theta$ is treated as a fixed hyperparameter not involved in estimation procedure. To estimate $\\theta$ as a parameter, we may need to take a further step to numerically maximize the observed data weighted log-likelihood $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})$ w.r.t. $\\theta$.\n\n\\section{GEM algorithm for MWLE under J-Gamma Lomax mixture model: Parameter transformation approach}\n\\subsection{Construction of complete data}\nThe complete data is given by\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i^{*})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $\\bm{z}_i^{*}=(z_{i1}^{*},\\ldots,z_{i(J+1)}^{*})$ are the labels where $z_{ij}^{*}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ (transformed) latent mixture component and $z_{ij}^{*}=0$ otherwise. The complete data weighted log-likelihood function is given by\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\left[\\sum_{j=1}^{J}z_{ij}^{*}\\left(\\log\\pi_j^{*}+\\log f_b(y_i;\\mu_j,\\phi_j) -\\log\\int_0^{\\infty}f_b(u;\\mu_j,\\phi_j)W(u)du\\right)\\right]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*}\\left(\\log\\pi_{J+1}^{*}+\\log f_t(y_i;\\theta,\\gamma)W(y_i)-\\log\\int_0^{\\infty}f_t(u;\\theta,\\gamma)W(u)du\\right)\\Bigg\\}.\n\\end{align}\n\n\n\\subsection{E-step} \\label{supp:sec:em_e2}\nThe expectation of the complete data weighted log-likelihood is given by the following for the $l^{\\text{th}}$ iteration:\n\\begin{align}\nQ^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\Bigg[\\sum_{j=1}^{J}z_{ij}^{*(l)}\\Bigg(\\log\\pi_j^{*}-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\log y_i-\\frac{y_i}{\\phi_j\\mu_j}\\nonumber\\\\\n&\\hspace{12em}-\\log\\int_0^{\\infty}f_b(u;\\mu_j,\\phi_j)W(u)du\\Bigg)\\Bigg]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*(l)}\\Bigg(\\log\\pi_{J+1}^{*}+\\log\\gamma+\\gamma\\log\\theta-(\\gamma+1)\\log(y_i+\\theta)\\nonumber\\\\\n&\\hspace{13em}-\\log\\int_0^{\\infty}f_t(u;\\theta,\\gamma)W(u)du\\Bigg)\\Bigg\\},\n\\end{align}\nwhere\n\\begin{equation}\nz^{*(l)}_{ij}=P(z^{*}_{ij}=1|\\bm{y},\\bm{\\Phi}^{(l-1)})=\n\\begin{cases}\n\\dfrac{\\pi_j^{*(l-1)}f_b(y_i;\\mu_j^{(l-1)},\\phi_j^{(l-1)})W(y_i)}{\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})W(u)du\\times h(y_i;\\bm{\\Phi}^{(l-1)})},\\quad j=1,\\ldots,J\\\\\n\\dfrac{\\pi^{*(l-1)}_{J+1}f_t(y_i;\\theta,\\gamma^{(l-1)})W(y_i)}{\\int_0^{\\infty}f_t(u;\\theta,\\gamma^{(l-1)})W(u)du\\times h(y_i;\\bm{\\Phi}^{(l-1)})},\\quad j=J+1.\n\\end{cases}\n\\end{equation}\n\n\\subsection{M-step} \\label{supp:sec:em_m2}\nMaximizing $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ with respect to $\\bm{\\Phi}$ yields the following parameter updates:\n\\begin{equation}\n\\pi_j^{*(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{ij}^{*(l)}}{\\sum_{j'=1}^{J+1}\\sum_{i=1}^{n}W(y_i)z_{ij'}^{*(l)}},\\quad j=1,\\ldots,J+1,\n\\end{equation}\nand the other parameters $(\\bm{\\mu},\\bm{\\phi},\\theta,\\gamma)$ are sequentially updated by numerically maximizing $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ w.r.t. each of the parameters.\n\n\\section{Proof of Proposition 3} \\label{supp:sec:ascend}\nWrite $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})=\\sum_{i=1}^{n}W(y_i)\\log p(y_i|\\bm{\\Phi})$ and $\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi}^{(l)};\\mathcal{D}^{\\text{com}})=\\sum_{i=1}^{n}W(y_i)\\left[\\log p(y_i|\\bm{\\Phi}) +\\log p(\\mathcal{D}^{\\text{mis}}_i|\\bm{\\Phi},y_i)\\right]$ for some probability density $p$ and missing data from sample $i$ given by $\\mathcal{D}^{\\text{mis}}_i$. Then, we have\n\\begin{align}\n\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})\n&=\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi}^{(l)};\\mathcal{D}^{\\text{com}})-\\sum_{i=1}^{n}W(y_i)\\log p(\\mathcal{D}^{\\text{mis}}_i|\\bm{\\Phi},y_i)\\nonumber\\\\\n&=Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})-\\sum_{i=1}^{n}W(y_i)\\int p(\\bm{v}_i|\\bm{\\Phi}^{(l-1)},y_i)\\log p(\\bm{v}_i|\\bm{\\Phi},y_i)d\\bm{v}_i,\n\\end{align}\nwhere the second equality results from expectation of both sides on the missing data under parameters $\\bm{\\Phi}^{(l-1)}$. Then, we have\n\\begin{align}\n\\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l)};\\bm{y})-\\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l-1)};\\bm{y})\n&=Q^{*}(\\bm{\\Phi}^{(l)}|\\bm{\\Phi}^{(l-1)})-Q^{*}(\\bm{\\Phi}^{(l-1)}|\\bm{\\Phi}^{(l-1)})\\nonumber\\\\\n&\\quad+\\sum_{i=1}^{n}W(y_i)\\int p(\\bm{v}_i|\\bm{\\Phi}^{(l-1)},y_i)\\log\\frac{p(\\bm{v}_i|\\bm{\\Phi}^{(l-1)},y_i)}{p(\\bm{v}_i|\\bm{\\Phi}^{(l)},y_i)}d\\bm{v}_i\\geq 0.\n\\end{align}\n\n\\section{Initialization of parameters} \\label{apx:em:init}\nAs briefly described in Section 5.3 of the paper, parameter initialization $\\bm{\\Phi}^{(0)}$ is done using the CMM approach by \\cite{gui2018fit}. This comes with the following steps:\n\\begin{enumerate}\n\\item Determine a threshold $\\tau$ which classifies observations $y_i$ into either body (when $y_i\\leq\\tau$) or tail (when $y_i>\\tau$) part of distribution. This can be done by plotting the log of empirical data survival function against $\\log y_i$, which is called the log-log plot. For regular varying distributions, the log-log plot is asymptotically linear. $\\tau$ is approximated by the point where the curve turns linear onwards.\n\\item Perform K-means clustering on $\\{y_i\\}_{i:y_i\\leq\\tau}$ with $J$ clusters, and obtain the clustering mean $\\{\\mu^{\\text{cluster}}_j\\}_{j=1,\\ldots,J}$, variance $\\{(\\sigma^{\\text{cluster}}_j)^2\\}_{j=1,\\ldots,J}$ and weights $\\{\\tilde{\\pi}_j^{\\text{cluster}}\\}_{j=1,\\ldots,J}$.\n\\item Set $\\mu_j^{(0)}=\\mu^{\\text{cluster}}_j$, $\\phi_j^{(0)}=({\\sigma^{\\text{cluster}}_j})^2\/{\\mu^{\\text{cluster}}_j}^2$.\n\\item Obtain $\\theta^{(0)}$ and $\\gamma^{(0)}$ by matching the first two moments of observations belonging to the tail component (i.e. $\\{y_i\\}_{i:y_i>\\tau}$).\n\\item Set $\\pi_{J+1}^{(0)}$ as the proportion of observations satisfying $y_i>\\tau$.\n\\item Set the remaining weight parameters as $\\pi_{j}^{(0)}=\\tilde{\\pi}_j^{\\text{cluster}}(1-\\pi_{J+1}^{(0)})$.\n\\end{enumerate}\n\n\\section{Truncated log-likelihood function} \\label{sec:supp:tll}\nThis section includes more details for Remark 6 in the paper. Denote $g(y)$ as the true distribution generating the observations and $\\tilde{h}(y;\\bm{\\Phi})=\\frac{h(y;\\bm{\\Phi})W(y)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du}$ as the truncated distribution. The expected weighted log-likelihood can be alternatively written as\n\\begin{align}\nn\\times\\tilde{E}[\\mathcal{L}^{*}(\\bm{\\Phi};\\bm{Y})]\n&=n\\int_{0}^{\\infty}W(u)\\log \\tilde{h}(u;\\bm{\\Phi})\\times g(u)du\\nonumber\\\\\n&=n\\int_{0}^{\\infty}g(u)W(u)du\\times\\int_{0}^{\\infty}\\log \\tilde{h}(u;\\bm{\\Phi})\\times\\frac{g(u)W(u)}{\\int_{0}^{\\infty}g(t)W(t)dt}du\\nonumber\\\\\n&=n\\int_{0}^{\\infty}g(u)W(u)du\\times\\tilde{E}^*[\\log \\tilde{h}(u;\\bm{\\Phi})],\n\\end{align}\nwhere the expectation $\\tilde{E}^*$ is taken on $Y$ under the random truncated distribution $\\frac{g(u)W(u)}{\\int_{0}^{\\infty}g(t)W(t)dt}$. Next, denote a random set $S_n=\\{i:V_i(y_i)=1\\}$, such that $\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})$ can be written as\n\\begin{equation}\n\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})=\\sum_{i\\in S_n}\\log \\tilde{h}(u;\\bm{\\Phi}),\n\\end{equation}\nwith effective number of terms $\\|S_n\\|\\approx n\\int_{0}^{\\infty}g(u)W(u)du\\approx \\sum_{i=1}^{n}W(y_i)$ in probability as $n\\rightarrow\\infty$. Comparing the above two equations, they simply correspond to standard MLE with bias term of $P$.\n\n\\section{Preliminary analysis of the motivating Greek dataset} \\label{apx:prelim_data}\nModelling the property damage claim size distribution is very challenging. Observing from Figures \\ref{fig:density} and \\ref{fig:loglogplot} which are also presented by \\cite{fung2021mixture}, the claim size distribution is not only heavy-tailed but also multi-modal. The key complexity of the empirical distribution is that there are many small distributional nodes for smaller claims, as evidenced by the right panel of Figure \\ref{fig:density}. On the other hand, it is undesirable to model all these nodes using excessive number of mixture components as (i) precise predictions of small claims are of less relevance of insurance pricing and risk management; (ii) this impedes the model interpretability. Further, the heavy-tailedness of empirical distribution is evidenced by asymptotic linearity of both log-log plot and mean excess plot in Figure \\ref{fig:loglogplot}. The asymptotic slope of log-log plot suggests that the estimated tail index is $\\gamma\\approx 1.3$ while the Lomax tail index obtained by \\cite{fung2021mixture} is about $\\gamma=1.38$, under a subjective choice of splicing threshold. Note however that these only provide a very rough guidance on the true tail index.\n\nNote that distributional multimodality and contamination are indeed prevalent not only to the aforementioned Greek dataset, but also to many publicly available insurance data sets. Notable examples include the French business interruption losses (\\textbf{frebiloss}), French motor third party liability claims (\\textbf{fremotor2sev9907} and \\textbf{freMPL8}) and Norwegian car claims (\\textbf{norauto}) which can all be retrived from the \\textbf{R} package \\textbf{CASdatasets}. This suggests that the modelling challenges emphasized in this paper is not only valid for the Greek data set we are analyzing, but is also applicable in many insurance claim severity data sets.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/density_original.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/density_log.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Empirical density of claim amounts (left panel) and log claim amounts (right panel); the orange vertical lines represent amounts of 10,000, 20,000, 50,000 and 100,000 respectively.}\n\\label{fig:density}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/loglogplot.jpg}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/me_plot.jpg}\n\\end{subfigure}\n\\end{center}\n\\caption{Left panel: log-log plot of the claim amounts; right panel: mean excess plot.}\n\\label{fig:loglogplot}\n\\end{figure}\n\n\n\\bibliographystyle{abbrvnat}\n\n\\section{Introduction}\nModelling insurance claim sizes is not only essential in various actuarial applications including pricing and risk management, but is also very challenging due to several peculiar characteristics of claim severity distributions. Claim size distributions often exhibit multimodality for small and moderate claims, when there exists unobserved heterogeneities possibly reflected by different claim types and accident causes, or when the observed samples come from a contaminated distribution. Also, the distribution is usually heavy-tail in nature, where very large claims occur with a small but non-negligible probability.\nDue to the highly complex distributional characteristics, we have to admit the impossibility to perfectly capture all the distributional features using a parametric model without excessively large number of parameters (which results in over-fitting). When model misspecification is unavoidable, correct specification of the tail part is more important than finely capturing the distributional nodes of smaller claims which are rather immaterial to the insurance portfolio, because the large claims are the losses which can severely damage the portfolio. As a result, we need to specify an appropriate distributional model with a justifiable statistical inference approach which not only preserves sufficient flexibility to appropriately capture the whole severity distribution, but also puts a particular emphasis on robust estimation of the tail.\n\n\nExisting actuarial loss modelling literature focus a lot on the model specifications, by introducing various distributional classes to capture the peculiar characteristics of claim severity distributions. Notable directions include extreme value distributions (EVD, see e.g. \\cite{embrechts1999extreme}) to capture the heavy-tailedness, composite loss modelling (see e.g. \\cite{cooray2005modeling}, \\cite{scollnik2007composite}, \\cite{bakar2015modeling} and \\cite{grun2019extending}) to cater for mismatch between body and tail behavior of claim severity distributions, and finite mixture model (FMM, see e.g. \\cite{LEE2010modeling} and \\cite{MILJKOVIC2016387}) to capture distributional multimodality.\nIn particular, FMM is becoming an increasingly useful smooth density estimation tool in insurance claim severity modelling perspective, due to its high versatility theoretically justified by denseness theorems (\\cite{LEE2010modeling}).\nThe mismatch between its body and tail behavior can also be easily modelled by FMM by selecting varying distributional classes among mixture component functions (see e.g. \\cite{BLOSTEIN201935} and \\cite{fung2021mixture}), including both light-tailed and heavy-tailed distributions. In both actuarial research and practice, statistical inferences of FMM are predominantly based on maximum likelihood estimation (MLE) with the use of Expectation-Maximization (EM) algorithm.\n\nNonetheless, MLE would often cause tail-robustness issues where the tail part of the fitted model is very sensitive to model misspecifications -- when the observations are generated from a perturbed and\/or contaminated distribution.\nAs evidenced by several empirical studies including \\cite{fung2021mixture} and \\cite{wuthrich2021statistical}, the estimated tail part of the FMM obtained by MLE can be unreliable and highly unstable in most practical cases. This is mainly due to the overlapping density regions between mixture components modelling small to moderate claims (body) and those modelling large claims (tail). Hence, the estimated tail distribution will be heavily influenced by some smaller claims if FMM is not able to fully explain those small claims, which is always the case in practice due to the distributional complexity of real dataset impossible to be perfectly captured even by flexible density approximation tools including FMM. Under MLE approach, FMM may fail to extrapolate well the large claims, and this would lead to serious implications to insurance pricing and risk management perspectives. It is therefore natural to question whether or not MLE is still a plausible approach in modelling actuarial claim severity data, and whether or not there exists an alternative statistical inference tool which better addresses our modelling challenges and outperforms the MLE.\n\nRobust statistical inference methods for heavy tail distributions are relatively scarce in actuarial science literature. \nNotable contributions include \\cite{brazauskas2000robust}, \\cite{serfling2002efficient}, \\cite{brazauskas2003favorable} and \\cite{dornheim2007robust} who adopt various kinds of statistical inference tools, such as quantile, trimmed mean, trimmed-M and generalized median estimators, to robustly estimate Gamma, Pareto and Log-normal distributions.\nRecent actuarial works study several variations of the method of moments (MoM) for robust estimations of Pareto and Log-normal loss models. Notable contributions in this direction include \\cite{brazauskas2009robust}, \\cite{poudyal2021robust} (trimmed moments), \\cite{zhao2018robust} (winsorized moments) and \\cite{poudyal2021truncated} (truncated moments).\nNote that these research works address robustness issue against the upper outliers by reducing the influence of few extreme observations to the estimated model parameters. This outlier-robustness issue is however very different from the tail-robustness issue mentioned above as the key motivation of this paper, where the contaminations on the body part affects the tail extrapolations. Very few research works look into this ``non-standard\" tail-robustness issue. Notable contributions are \\cite{beran2012robust} and \\cite{gong2018robust} who propose a huberization of the MLE, which protects against perturbations and misspecifications in the body part of distribution, to robustly estimate the tail index of Pareto and Weibull distributions. \nAll of the above existing approaches focus solely on one or two-parameter distributions. A general robust tail estimation strategy under multi-parameter flexible models like FMM is lacking.\n\nMotivated by the aforementioned tail-robustness issue in insurance context, we propose a new maximum weighted likelihood estimation (MWLE) approach for robust heavy-tail modelling of FMM. \nUnder the MWLE, an observation-dependent weight function is introduced to the log-likelihood, de-emphasizing the contributions of smaller claims and hence reducing their influence to the estimated tail part of FMM. Down-weighting small claims is also natural in insurance loss modelling perspective, as mentioned in the beginning of this section, accurate modelling of the large claims is more important than the smaller claims. \nTo offset the bias caused by the weighting scheme, we also include an adjustment term in the weighted log-likelihood, which can be interpreted as the effects of randomly truncating the observations. With the bias adjustment term, we prove that estimated parameters under the proposed MWLE is consistent and asymptotically normal with any pre-specified choices of weight functions.\nAlso, under some specific choices of weight functions, we will show that the MWLE tail index, which determines the tail heaviness of a distribution, is consistent, even under model misspecifications where the true model is not FMM. Therefore, MWLE can be regarded as a generalized alternative framework of Hill estimator (\\cite{hill1975simple}). Furthermore, with a probabilistic interpretation of the proposed MWLE approach, it is still possible to derive a Generalized EM (GEM) algorithm to efficiently estimate parameters which maximize the weighted log-likelihood function.\n\nNote that the proposed MWLE is different from the existing statistics papers which adopt weighting schemes for likelihood-based inference. The existing literature are mainly motivated by one of the following two aspects very different from the focus of this paper: (i) Robustness against upper and lower outliers, where related research works include e.g. \\cite{fieldsmith1994}, \\cite{markatou1997weighted}, \\cite{markatou2000mixture}, \\cite{dupuis2002robust}, \\cite{ahmed2005robust}, \\cite{wong2014robust} and \\cite{aeberhard2021robust}; (ii) Bringing in more relevant observations for statistical inference to increase precision while trading off some biases, studied by e.g. \\cite{wang2001maximum}, \\cite{hu2002weighted}, \\cite{wang2004asymptotic} and \\cite{wang2005selecting}. Note also that our proposed MWLE stems differently from the existing statistics literature in terms of mathematical technicality, since none of the above papers incorporate the truncation-based bias adjustment as included in the proposed WMLE.\n\nThe rest of this paper is structured as follows. In Section \\ref{sec:fmm}, we briefly revisit the class of FMM for heavy tail modelling. Section \\ref{sec:MWLE} introduces the proposed MWLE for robust heavy-tail modelling of FMM and explains its motivations in terms of insurance claims modelling. Section \\ref{sec:theory} explores several theoretical properties to understand and justify the proposed MWLE. After that, we present in Section \\ref{sec:em} two types of GEM algorithms for efficient parameter estimations under the MWLE approach on FMM. In Section \\ref{sec:ex}, we analyze the performance of the proposed MWLE through three empirical examples: a toy example, a simulation study and a real insurance claim severity dataset. After showing the superior performance of MWLE compared to MLE, we finally summarize our findings in Section \\ref{sec:discussion} with a brief discussion how the proposed MWLE can be extended to a regression framework.\n\n\n\n\\section{Finite mixture model} \\label{sec:fmm}\nThis section provides a very brief review on finite mixture model (FMM) which serves as a flexible density estimation tool. Suppose that there are $n$ i.i.d. claim severities $\\bm{Y}=(Y_1,\\ldots, Y_n)$ with realizations $\\bm{y}=(y_1,\\ldots, y_n)$. $Y_i$ is generated by a probability distribution of $G(\\cdot)$ with density function $g(\\cdot)$ which is unknown. In insurance context, claim severity distribution often exhibits multimodality, which results from the unobserved heterogeneity stemming from the amalgamation of different types of claims unobserved in advance. Also, claim sizes are often heavy-tail in nature, which can be attributed to a few large losses from a portfolio of policies which usually represent the greatest part of the indemnities paid by the insurance company. The mismatch between body and tail behavior often poses difficulties to fit the data well using only a standard parametric distribution.\n\nMotivated by the challenges of modelling insurance claim severities, we aim to model the dataset using finite mixture model (FMM). Define a class of finite mixture distributions $\\mathcal{H}=\\{H(\\cdot;\\bm{\\Phi}):\\bm{\\Phi}\\in\\Omega\\}$, where $\\bm{\\Phi}=(\\psi_1,\\ldots,\\psi_P)^T$ is a column vector with length $P$ representing the model parameters and $\\Omega$ is the parameter space. Its density function $h(y;\\bm{\\Phi})$ is given by the following form:\n\n\\begin{equation} \\label{eq:model}\nh(y_i;\\bm{\\Phi})=\\sum_{j=1}^{J}\\pi_j f_b(y_i;\\bm{\\varphi}_j)+\\pi_{J+1}f_t(y_i;\\bm{\\eta}),\\qquad y_i>0,\n\\end{equation}\nwhere the parameters $\\bm{\\Phi}$ can alternatively be written as $\\bm{\\Phi}=(\\bm{\\pi},\\bm{\\varphi},\\bm{\\eta})$. Here, $\\bm{\\pi}=(\\pi_1,\\ldots,\\pi_{J+1})$ are the mixture probabilities for each of the $J+1$ components with $\\sum_{j=1}^{J+1}\\pi_j=1$. $\\bm{\\varphi}=(\\bm{\\varphi}_1,\\ldots,\\bm{\\varphi}_J)$ and $\\bm{\\eta}$ are the parameters for the mixture densities $f_b$ and $f_t$ respectively.\n\nThe $J$ mixture components with densities $f_b$ mainly serve as modelling the multimodality for the body part of the distribution. $f_b$ is naturally chosen as a light-tailed distribution like Gamma, Weibull and Inverse-Gaussian.\nThe remaining mixture component $f_t$ is designed to capture the large observations and hence the tail distribution can be properly extrapolated. The possible choices of heavy-tail distribution for $f_t$ include Log-normal, Pareto and Inverse-Gamma.\n\n\n\\section{Maximum weighted log-likelihood estimator} \\label{sec:MWLE}\nWith the maximum likelihood estimation (MLE) approach, parameter estimations require maximizing the log-likelihood function\n\n\\begin{equation} \\label{eq:loglik}\n\\mathcal{L}_n(\\bm{\\Phi};\\bm{y})=\\sum_{i=1}^{n}\\log h(y_i;\\bm{\\Phi})\n\\end{equation}\nwith respect to the parameters $\\bm{\\Phi}$. Under this approach, each claim has the same relative influence to the estimated parameters, but in insurance loss modelling and ratemaking perspective, correct specification and projection of larger claims are more important than those of smaller claims. More importantly, as explained by \\cite{fung2021mixture} and \\cite{wuthrich2021statistical}, MLE of FMM in Equation (\\ref{eq:model}) would fail in most practical cases due to incorrectly estimations of tail heaviness under model misspecification. Precisely, because of the overlapping region between the body parts $f_b$ and tail part $f_t$ of the distribution, the small claims may distort the estimated tail distribution $f_t$ if they are not fully captured by the $J$ mixture densities in the body. However, due to the highly complex multimodality characteristics of the body distribution which often appears in real insurance claim severity data, it is impossible to capture all the body distributional patterns without prohibitively large $J$ which causes over-fitting and loss of model interpretability. Therefore, it is often the case in practice that MLE of FMM would result in unstable estimates of tail distribution, causing unreliable tail extrapolation.\n\nOne way to mitigate the aforementioned model misspecification effect is to impose observation-dependent weights to the log-likelihood function, where a larger claim $y$ is assigned to a larger weight. This will reduce the influence of smaller observed values to the estimated tail parameter. For parameter estimations, we propose maximizing the weighted log-likelihood as follows instead\n\n\\begin{equation} \\label{eq:loglik_weight}\n\\mathcal{L}^*_n(\\bm{\\Phi};\\bm{y})=\\sum_{i=1}^{n}W(y_i)\\log \\frac{h(y_i;\\bm{\\Phi})W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du},\n\\end{equation}\nwhere $0\\leq W(\\cdot)\\leq 1$ is the weight of the log-likelihood function. We call the resulting parameters maximum weighted likelihood estimators (MWLE). To allow for greater relative influence of larger claims, we construct $W(u)$ as a monotonically non-decreasing function of $u$. In this case, we may interpret the weighted log-likelihood function as follows: First, we pretend that each claims $y_1,\\ldots,y_n$ are only observed respectively by $W(y_1),\\ldots,W(y_n)$ times. However, this alone will introduce bias to a heavier estimated tail because this implies more large claims are effectively included due to the weighting effect. To remove such a bias, we pretend to model $y_i$ by a random truncation model $\\tilde{h}(y_i;\\bm{\\Phi}):=h(y_i;\\bm{\\Phi})W(y_i)\/\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du$ instead of the original modelling distribution $h(y_i;\\bm{\\Phi})$\n\n\\begin{remark}\nThe proposed MWLE can be viewed as a form of M-estimator, where the optimal parameters are determined through maximizing a function. We here discuss two special cases of MWLE. (i) MLE: When $W(\\cdot)=1$, MWLE is reduced to a standard MLE; (ii) Truncated MLE: When $W(y)=1\\{y\\geq\\tau\\}$ for some threshold $\\tau>0$, then MWLE is reduced to truncated MLE introduced by \\cite{marazzi2004adaptively}, where a hard rejection is applied to all samples smaller than $\\tau$.\n\\end{remark}\n\n\n\n\n\\section{Theoretical Properties} \\label{sec:theory}\nThis section presents several theoretical properties of the proposed MWLE to theoretically justify the use of the proposed MWLE. Unless specified otherwise, throughout this section the estimated model parameters $\\hat{\\bm{\\Phi}}$ are obtained by maximizing the proposed weighted log-likelihood function given by Equation (\\ref{eq:loglik_weight}).\n\\subsection{Asymptotic behavior with fixed weight function}\n\\subsubsection{Consistency and asymptotic normality}\nWe first want to show that the proposed weighted log-likelihood approach leads to correct convergence to true model parameters as $n\\rightarrow\\infty$. The proof is presented in Section 2 of the supplementary materials.\n\\begin{theorem} \\label{thm:asym_tru}\nSuppose that $G(\\cdot)=H(\\cdot;\\bm{\\Phi}_0)\\in\\mathcal{H}$. Assume that the density function $h(y;\\bm{\\Phi})$ satisfies a set of regularity conditions outlined in Section 1 of supplementary materials\\footnote{Note that the set of regularity conditions are equivalent to those required for consistent and asymptotic normal estimations of MLE.}. Then, there exists a local maximizer $\\hat{\\bm{\\Phi}}_n$ of the weighted log-likelihood $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})$ such that\n\\begin{equation}\n\\sqrt{n}(\\hat{\\bm{\\Phi}}_n-\\bm{\\Phi}_0)\\overset{d}{\\rightarrow}\\mathcal{N}(\\bm{0},\\bm{\\Sigma}),\n\\end{equation}\nwhere $\\bm{\\Sigma}=\\bm{\\Gamma}^{-1}\\bm{\\Lambda}\\bm{\\Gamma}^{-1}$, with $\\bm{\\Lambda}$ and $\\bm{\\Gamma}$ being $P\\times P$ matrices given by\n\\begin{align} \\label{eq:asym:lambda}\n\\bm{\\Lambda} \n&=E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad-\\frac{1}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}\\Bigg\\{E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\nonumber\\\\\n&\\hspace{8em}+E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\Bigg\\}\\nonumber\\\\\n&\\quad+\\frac{E_{\\bm{\\Phi}_0}\\left[W(Y)^2\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]^2}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\n\\end{align}\nand\n\\begin{align} \\label{eq:asym:gamma}\n\\bm{\\Gamma}\n&=-E_{\\bm{\\Phi}_0}\\left[W(Y)\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad+\\frac{1}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T,\n\\end{align}\nwhere $E_{\\bm{\\Phi}_0}[Q(Y)]=\\int_{0}^{\\infty}Q(u)h(u;\\bm{\\Phi}_0)du$ represents the expectation under density $h(\\cdot;\\bm{\\Phi}_0)$ for any functions $Q$, and the derivative ${\\partial}\/{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})$ is assumed to be a column vector with length $P$.\n\\end{theorem}\n\n\\begin{remark}\nWhen $W(\\cdot)=1$, all except the first term in the right hand side of Equations (\\ref{eq:asym:lambda}) and (\\ref{eq:asym:gamma}) vanish. As a result, the asymptotic variance $\\bm{\\Sigma}=\\bm{\\Gamma}^{-1}\\bm{\\Lambda}\\bm{\\Gamma}^{-1}$ is reduced to the inverse of Fisher information matrix under standard MLE approach.\n\\end{remark}\n\n\\begin{remark}\nTheorem \\ref{eq:asym:gamma} only asserts the existence of local maximizer instead of global maximizer, because in FMM it is common that the likelihood function has multiple critical points and\/or is unbounded (\\cite{Mclachlan2004Finite}).\n\\end{remark}\n\nThe above theorem suggest that for large sample size, the estimated parameters are approximately unbiased and we may approximate the variance of estimated parameters as\n\\begin{equation} \\label{eq:asym_var}\n\\widehat{\\text{Var}}(\\hat{\\bm{\\Phi}}_{n})\\approx \\frac{1}{n}\\hat{\\bm{\\Gamma}}_n^{-1}\\hat{\\bm{\\Lambda}}_n\\hat{\\bm{\\Gamma}}_n^{-1}\n\\end{equation}\nwhere $\\hat{\\bm{\\Lambda}}_n$ and $\\hat{\\bm{\\Gamma}}_n$ are given by $\\bm{\\Lambda}$ and $\\bm{\\Gamma}$ in Equations (\\ref{eq:asym:lambda}) and (\\ref{eq:asym:gamma}) except that the expectations are changed to empirical means and $\\bm{\\Phi}_0$ is changed to $\\hat{\\bm{\\Phi}}_n$. Then, it is easy to construct a two-sided Wald-type confidence interval (CI) for $\\psi_p$ ($p=1,\\ldots,P$) as\n\\begin{equation} \\label{eq:asym_CI}\n\\left[\\hat{\\psi}_{n,p}-\\frac{z_{1-\\kappa\/2}}{\\sqrt{n}}\\sqrt{\\left[\\hat{\\bm{\\Gamma}}_n^{-1}\\hat{\\bm{\\Lambda}}_n\\hat{\\bm{\\Gamma}}_n^{-1}\\right]_{p,p}},\\hat{\\psi}_{n,p}+\\frac{z_{\\kappa\/2}}{\\sqrt{n}}\\sqrt{\\left[\\hat{\\bm{\\Gamma}}_n^{-1}\\hat{\\bm{\\Lambda}}_n\\hat{\\bm{\\Gamma}}_n^{-1}\\right]_{p,p}}\\right],\n\\end{equation}\nwhere $\\hat{\\psi}_{n,p}$ is the estimated $\\psi_p$, $z_{\\kappa}$ is the $\\kappa$-quantile of the standard normal distribution and $\\left[\\bm{M}\\right]_{p,p}$ is the $(p,p)$-th element of $\\bm{M}$ for some matrices $\\bm{M}$. For other quantities of interest (e.g.~mean, VaR and CTE of claim amounts), one may apply a delta method or simulate parameters from ${\\cal N}(\\hat{\\bm{\\Phi}}_{n},\\widehat{\\text{Var}}(\\hat{\\bm{\\Phi}}_{n}))$ to analytically or empirically approximate their CIs.\n\n\\medskip\n\nNext, we examine the asymptotic property of MWLE dropping the assumption that $G(\\cdot)\\in\\mathcal{H}$ (i.e. we may misspecify the model class).\n\n\\begin{theorem} \\label{thm:asym_mis}\nAssume that the density function $h(y;\\bm{\\Phi})$ satisfies the same set of regularity conditions as in the previous theorem. Further assume that there is a local maximizer $\\bm{\\Phi}_0^{*}$ of\n\\begin{equation}\n\\tilde{E}\\left[\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]:=\n\\tilde{E}\\left[W(Y)\\log \\frac{h(Y;\\bm{\\Phi})W(Y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}\\right],\n\\end{equation}\nwhere $\\tilde{E}\\left[Q(Y)\\right]=\\int_0^{\\infty}Q(u)dG(u)$ represents the expectation under distribution $G(y)$ for any functions $Q$. Then, there exists a local maximizer $\\hat{\\bm{\\Phi}}_n$ of the weighted log-likelihood $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})$ such that\n\\begin{equation}\n\\sqrt{n}(\\hat{\\bm{\\Phi}}_n-\\bm{\\Phi}_0^{*})\\overset{d}{\\rightarrow}\\mathcal{N}(\\bm{0},\\tilde{\\bm{\\Sigma}}),\n\\end{equation}\nwhere $\\tilde{\\bm{\\Sigma}}=\\tilde{\\bm{\\Gamma}}^{-1}\\tilde{\\bm{\\Lambda}}\\tilde{\\bm{\\Gamma}}^{-1}$, with $\\tilde{\\bm{\\Lambda}}$ and $\\tilde{\\bm{\\Gamma}}$ given by\n\\begin{align} \\label{eq:asym:lambda_mis}\n\\tilde{\\bm{\\Lambda}} \n&=\\tilde{E}\\left[W(Y)^2\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad-\\frac{1}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}\\Bigg\\{\\tilde{E}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\nonumber\\\\\n&\\hspace{8em}+E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\tilde{E}\\left[W(Y)^2\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\\Bigg\\}\\nonumber\\\\\n&\\quad+\\frac{\\tilde{E}\\left[W(Y)^2\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]^2}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T\n\\end{align}\nand\n\\begin{align} \\label{eq:asym:gamma_mis}\n\\tilde{\\bm{\\Gamma}}\n&=\\tilde{E}\\left[W(Y)\\frac{\\partial^2}{\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T}\\log h(Y;\\bm{\\Phi})\\right]\n-\\frac{\\tilde{E}\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial^2}{\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T}\\log h(Y;\\bm{\\Phi})\\right]\\nonumber\\\\\n&\\quad-\\frac{\\tilde{E}\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\quad+\\frac{\\tilde{E}\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]^2}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]^T.\n\\end{align}\n\\end{theorem}\n\nAs shown by the above theorem, the MWLE is still asymptotically convergent and normally distributed. As a result, it is still justifiable to evaluate the parameter uncertainties and CI of parameters in the forms of Equations (\\ref{eq:asym_var}) and (\\ref{eq:asym_CI}). However, in the context of modelling heavy-tail distributions as an example, there could be an asymptotic bias on the estimated tail index. As a result, it is important to theoretically examine how the choice of weight functions influence the impacts of model misspecifications. These will be leveraged to the next subsections on the robustness studies and asymptotics under varying weight functions.\n\n\\subsubsection{Robustness}\nIt is well known that MLE is the most efficient estimator under all asymptotically unbiased estimators. Therefore, with an attempt to reduce the bias of estimated tail distribution under misspecified models through MWLE approach with $W(\\cdot)\\neq 1$, there will be a trade-off between bias regularizations and loss in efficiencies. This subsection will analyze such a trade-off, which may provide guidance to choose an appropriate weight function $W(\\cdot)$. In light of Theorem \\ref{thm:asym_tru}, it is easy to show the following proposition by applying delta method.\n\n\\begin{proposition}\nSuppose that $G(\\cdot)=H(\\cdot;\\bm{\\Phi}_0)\\in\\mathcal{H}$ with the same set of regularity conditions as previous theorems. Define $\\hat{\\bm{\\Phi}}_n$ and $\\hat{\\bm{\\Phi}}_n^{(0)}$ as the MWLE and MLE respectively. Then, for some differentiable functions $U(\\cdot)$, the relative asymptotic efficiency (AEFF) of $U(\\hat{\\bm{\\Phi}}_n)$ is given by\n\\begin{equation}\n\\text{AEFF}(W;\\bm{\\Phi}_0):=\\lim_{n\\rightarrow\\infty}\\frac{\\text{Var}(\\hat{\\bm{\\Phi}}_n^{(0)})}{\\text{Var}(\\hat{\\bm{\\Phi}}_n)}=\\frac{U'(\\bm{\\Phi}_0)^T\\bm{\\Sigma}^{(0)}U'(\\bm{\\Phi}_0)}{U'(\\bm{\\Phi}_0)^T\\bm{\\Sigma}U'(\\bm{\\Phi}_0)},\n\\end{equation}\nwhere $U'(\\bm{\\Phi})$ is the gradient of $U(\\bm{\\Phi})$ w.r.t. $\\bm{\\Phi}$, and $\\bm{\\Sigma}^{(0)}$ is the inverse of Fisher information matrix under standard MLE approach.\n\\end{proposition}\n\nNext, we need to quantify robustness by some statistical measures. In a theoretical setting, we follow e.g. \\cite{huber1981robust}, \\cite{beran2012robust} and \\cite{gong2018robust} to consider the case that $Y_i$ is generated by a contamination model, given by\n\n\\begin{equation} \\label{eq:asym_contam}\nG(y):=G(y;\\epsilon,M,\\bm{\\Phi}_0)=(1-\\epsilon)H(y;\\bm{\\Phi}_0)+\\epsilon M(y), \\quad y>0,\n\\end{equation}\nfor some contamination distribution function $M$. Then, the asymptotic bias can be analyzed through evaluating the influence function (IF), a column vector with length $P$ given by\n\n\\begin{equation}\n\\text{IF}(\\bm{\\Phi}_0; H, M)=\\lim_{\\epsilon\\rightarrow 0}\\frac{\\tilde{\\bm{\\Phi}}^{\\epsilon,M}-\\bm{\\Phi}_0}{\\epsilon},\n\\end{equation}\nwhere $\\tilde{\\bm{\\Phi}}^{\\epsilon,M}$ is the asymptotic estimated parameters if $H^{\\epsilon,M}$ given by Equation (\\ref{eq:asym_contam}) is distribution generating $Y_i$, contrasting to $\\bm{\\Phi}_0$ which are the true model parameters. IF can be interpreted as the infinitesimal asymptotic bias of estimated parameters by perturbing the model generating $Y_i$. Smaller $|\\text{IF}_p(\\bm{\\Phi}_0; H, M)|$ (with $\\text{IF}_p$ being the $p^{\\text{th}}$ element of $\\text{IF}$) means a more robust estimation of $\\phi_p$ under model misspecification. Our goal is to demonstrate the potential of the proposed MWLE to reduce such a bias and hence improve the robustness. We have the following proposition which derives the IF under the MWLE approach:\n\n\\begin{proposition}\nThe IF is given by\n\\begin{align}\n\\text{IF}(\\bm{\\Phi}_0; H, M)\n&=-\\bm{\\Gamma}^{-1}\\Bigg\\{E_M\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\nonumber\\\\\n&\\hspace{5em}-\\frac{E_M\\left[W(Y)\\right]}{E_{\\bm{\\Phi}_0}\\left[W(Y)\\right]}E_{\\bm{\\Phi}_0}\\left[W(Y)\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(Y;\\bm{\\Phi})\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\\Bigg\\},\n\\end{align}\nwhere $E_M[Q(Y)]=\\int_0^{\\infty}Q(u)dM(u)$ for some functions $Q$, and $\\bm{\\Gamma}$ is given by Equation (\\ref{eq:asym:gamma}).\n\\end{proposition}\n\nWe will show empirically how the choice of weight functions $W(\\cdot)$ affects the AEFF and IF in Section \\ref{sec:ex:toy}, which will help us understand the bias-variance tradeoff of our proposed MWLE approach.\n\n\\subsection{Asymptotic behavior of tail index with varying weight functions} \\label{sec:theory:tail_idx}\n\nTail index measures the tail-heaviness of a probability distribution. Correctly specifying the tail index is a critical task of modelling insurance data with heavy-tail nature, as insurance companies often care more about large claims which are more material than small ones. In this section, we show that under some sequences of weight functions $W_n(\\cdot)$ which depend on the number of observations $n$, the estimated tail index will be consistent under the proposed MWLE even if the model class is misspecified. \nThis result theoretically justifies how the proposed MWLE addresses the tail-robustness issue caused by model misspecification and distributional contamination, by showing that reduced influence of smaller claims through downweighting can be useful for producing a plausible tail index estimate. \nAlso, the result may provide some theoretical guidance on selecting an appropriate weight function.\n\nDenote $\\mathcal{R}_{-\\gamma}$ be a class of regularly varying distributions with tail index $\\gamma>0$, such that $\\bar{H}\\in\\mathcal{R}_{-\\gamma}$ if and only if $\\bar{H}(y)\\sim y^{-\\gamma}L_0(y)$ as $y\\rightarrow\\infty$ for some slowly varying functions $L_0(y)\\in\\mathcal{R}_0$ satisfying $L_0(ty)\/L_0(y)\\rightarrow 1$ as $y\\rightarrow\\infty$ for any $t>0$. Smaller $\\gamma$ implies heavier tail. Note that regularly varying distributions include many distributions that capture heavy-tail behaviors of loss random variables and we refer the readers to \\citet{cooke2014fat} for more explanation on these distributions. Also define the following transformed density functions\n\\begin{equation}\n\\tilde{g}_{n}(y)=\\frac{g(y)W_n(y)}{\\int_{0}^{\\infty}g(u)W_n(u)du},\\qquad \\tilde{h}_{n}(y;\\bm{\\Phi})=\\frac{h(y;\\bm{\\Phi})W_n(y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W_n(u)du},\n\\end{equation}\nand $\\tilde{G}_{n}(\\cdot)$ and $\\tilde{H}_{n}(\\cdot)$ are the corresponding distribution functions. We further put a bar to any function $Q$ to denote its survival function (i.e. $\\bar{Q}:=1-Q$). We then make the following assumptions:\n\n\\begin{enumerate}[font={\\bfseries},label={A\\arabic*.}]\n\\item $\\bar{G}\\in\\mathcal{R}_{-\\gamma_0}$ with tail index $\\gamma_0>0$.\n\\item $\\bar{H}(y;\\bm{\\Phi})=y^{-\\gamma}L(y;\\bm{\\Phi})$ for some slowly varying functions $L$, so that $\\bar{H}\\in\\mathcal{R}_{-\\gamma}$. Here, $\\gamma$ is the only model parameter within $\\bm{\\Phi}$ that governs the tail index. Also, both $L(yt;\\bm{\\Phi})\/L(y;\\bm{\\Phi})$ and its derivative w.r.t. $\\bm{\\Phi}$ converges uniformly as $y\\rightarrow\\infty$ for any fixed $t>1$.\n\\item There exists some sequences of thresholds $\\{\\tau_n\\}_{n=1,2,\\ldots}$ with $\\tau_n\\rightarrow\\infty$ as $n\\rightarrow\\infty$ such that $\\tau_nW_n(\\tau_n)\\rightarrow 0$ as $n\\rightarrow\\infty$.\n\\item $\\tilde{E}_n[(\\log \\tilde{h}_{n}(Y;\\bm{\\Phi}))^2]\/(n\\tilde{E}[W_n(Y)])\\rightarrow 0$ as $n\\rightarrow\\infty$, where $\\tilde{E}_n[Q(Y)]=\\int_{0}^{\\infty}Q(u)d\\tilde{G}_{n}$ and $\\tilde{E}[Q(Y)]=\\int_{0}^{\\infty}Q(u)dG$ for some functions $Q$.\n\\item The density functions $h(y;\\bm{\\Phi})$ and $g(y)$ are ultimately monotone (i.e. monotone on $y\\in (z,\\infty)$ for some $z>0$), uniformly on $\\bm{\\Phi}$.\n\\end{enumerate}\n\nAssumptions \\textbf{A1} and \\textbf{A2} ensure that both the model generating the observations and the fitted model class are heavy tail in nature, with tail heaviness quantified by tail indices $\\gamma_0$ and $\\gamma$ respectively. In finite mixture context, see Section \\ref{sec:fmm}, \\textbf{A2} can be easily satisfied choosing $h$ in Equation (\\ref{eq:model}) as any standard regularly varying distributions such as Pareto and Inverse-Gamma with compact parameter space. Assumption \\textbf{A3} asserts that all observations other than the extreme ones are greatly down-weighted. This assumption provides a theoretical guidance of choosing the weight function such that small to moderate claims should only be allocated by small weights, while substantial weights should be assigned only to large claims. \\textbf{A4} requires that the effective number of MWLE observations $n\\tilde{E}[W_n(Y)]\\rightarrow\\infty$ such that large sample theories hold. The numerator $\\tilde{E}_n[(\\log \\tilde{h}_{n}(Y;\\bm{\\Phi}))^2]$ grows much slower than the denominator as a logarithm is involved. Assumption \\textbf{A5} is of no practical concern. Now, we have the following theorem which asserts the consistency of estimated tail index. The proof is leveraged to Section 3 of the supplementary material.\n\n\\begin{theorem} \\label{thm:asym:tail_idx}\nAssume \\textbf{A1} to \\textbf{A5} hold for the settings under the MWLE, and the regularity conditions outlined in Section 1 of Supplementary materials are satisfied. Then, there exists a local maximizer $\\hat{\\bm{\\Phi}}_n$ of the weighted log-likelihood function $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ with the estimated tail index $\\hat{\\gamma}_n$ such that $\\hat{\\gamma}_n\\rightarrow\\gamma_0$ as $n\\rightarrow\\infty$. Further, the local maximizer $\\hat{\\gamma}_n$ is unique in probability as $n\\rightarrow\\infty$.\n\\end{theorem}\n\n\n\\begin{remark}\nConsider a special case where: (i) the weight functions $W_n(y)=1\\{y>\\tau_n\\}$ are step functions for some sequences of $\\tau_n\\rightarrow\\infty$; and (ii) the fitted model class $H(y;\\bm{\\Phi})$ is chosen as a Generalized Pareto distribution (GPD) or equivalently Lomax distribution which will be described in Section \\ref{sec:ex} (i.e. $H(y;\\bm{\\Phi})$ is an FMM in Equation (\\ref{eq:model}) with $J=0$ and $f_t$ is a GPD). Theorem \\ref{thm:asym:tail_idx} is then asserting the consistency of tail index obtained by excess over threshold method on GPD (\\cite{smith1987estimating}), which has a very close connection with the consistency property of the Hill estimator (\\cite{hill1975simple}) (see Section 4 of \\cite{smith1987estimating}). Therefore, we can regard the proposed MWLE approach as a generalized framework of the Hill-type estimator by \\cite{hill1975simple}.\n\\end{remark}\n\n\\section{Parameter estimation} \\label{sec:em}\n\\subsection{GEM algorithm} \\label{sec:em:gem}\nSince there is a probabilistic interpretation of the weighted log-likelihood given by Equation (\\ref{eq:loglik_weight}), it is feasible to construct a generalized Expectation-Maximization (GEM) algorithm for efficient parameter estimations. In this paper, we will present two distinctive approaches of complete data constructions which result to two different kinds of GEM algorithms.\n\n\\subsubsection{Method 1: Hypothetical data approach}\n\\paragraph{Construction of complete data}\nTo address the challenges of optimizing directly the ``observed data\" weighted log-likelihood in Equation (\\ref{eq:loglik_weight}), we extend the ``hypothetical complete data\" method proposed by \\cite{FUNG2020MoECensTrun}, by defining the complete data\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i,k_i,\\{\\bm{z}'_{is},y'_{is}\\}_{s=1,\\ldots,k_i})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $k_i$ is the number of missing sample points ``generated\" by observation $i$, due to the probabilistic interpretation that each sample $i$ is removed with a probability of $1-W(y_i)$. As an auxiliary tool for efficient computations we assume that $k_i$ follows geometric distribution with probability mass function\n\\begin{equation} \\label{eq:em:k}\np(k_i;\\bm{\\Phi})=\\left[1-\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du\\right]^{k_i}\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du, \\qquad k_i=0,1,\\ldots,\n\\end{equation}\nand $\\{y'_{is}\\}_{s=1,\\ldots,k_i}$ are i.i.d. variables representing the missing samples. We assume that $Y'_{is}$ (with realization $y'_{is}$) is independent of $y_i$ and $k_i$, and follows a distribution with the following density function\n\\begin{equation} \\label{eq:em:y}\n\\tilde{h}^{*}(y'_{is};\\bm{\\Phi})=\\frac{h(y'_{is};\\bm{\\Phi})(1-W(y'_{is}))}{\\int_0^{\\infty}h(u;\\bm{\\Phi})(1-W(u))du},\\qquad y'_{is}>0.\n\\end{equation}\n\nFurther, $\\bm{z}_i=(z_{i1},\\ldots,z_{i(J+1)})$ are the latent mixture components assignment labels, where $z_{ij}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ latent class and $z_{ij}=0$ otherwise. Similarly, $\\bm{z}'_i=(z'_{is1},\\ldots,z'_{is(J+1)})$ are the labels for missing data, where $z'_{isj}=1$ if the $s^{\\text{th}}$ missing sample generated by observation $i$ belongs to the $j^{\\text{th}}$ latent class, and $z'_{isj}=0$ otherwise. \n\nThe complete data weighted log-likelihood function is then given by\n\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^{n}W(y_i)\\log\\Bigg\\{\\frac{\\left\\{\\prod_{j=1}^{J}[\\pi_jf_b(y_i;\\bm{\\varphi}_j)]^{z_{ij}}\\right\\}\\left(\\pi_{J+1}f_t(y_i;\\bm{\\eta})\\right)^{z_{i(J+1)}}W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du} \\nonumber\\\\\n&\\hspace{8em} \\times \\left[1-\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du\\right]^{k_i}\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du \\nonumber\\\\\n&\\hspace{8em} \\times \\prod_{s=1}^{k_i}\\frac{\\left\\{\\prod_{j=1}^{J}[\\pi_jf_b(y'_{is};\\bm{\\varphi}_j)]^{z'_{isj}}\\right\\}\\left(\\pi_{J+1}f_t(y'_{is};\\bm{\\eta})\\right)^{z'_{is(J+1)}}W(y_i)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})(1-W(u))du}\\Bigg\\} \\nonumber\\\\\n&=\\sum_{i=1}^nW(y_i)\\left\\{\\left[\\sum_{j=1}^{J}z_{ij}\\log \\pi_jf_b(y_i;\\bm{\\varphi}_j)\\right]+z_{i(J+1)}\\log\\pi_{J+1} f_t(y_i;\\bm{\\eta})\\right\\}\\nonumber\\\\\n&\\quad +\\sum_{i=1}^{n}\\sum_{s=1}^{k_i}W(y_i)\\left\\{\\left[\\sum_{j=1}^J z'_{ijs}\\log\\pi_j f_b(y'_{is};\\bm{\\varphi}_j)\\right]+z'_{i(J+1)s}\\log\\pi_{J+1} f_t(y'_{is};\\bm{\\eta})\\right\\}+\\text{const.},\n\\end{align}\n\\endgroup\nwhich is more computationally tractable. In the following we will omit the constant term which is irrelevant for calculations.\n\n\n\n\n\\paragraph{Iterative procedures}\nIn the $l^{\\text{th}}$ iteration of the E-step, we compute the expectation of the complete data weighted log-likelihood as follows:\n\\begin{align} \\label{eq:em:q}\nQ^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\n&=\\sum_{i=1}^nW(y_i)\\left\\{\\left[\\sum_{j=1}^{J}z_{ij}^{(l)}\\log \\pi_jf_b(y_i;\\bm{\\varphi}_j)\\right]+z_{i(J+1)}^{(l)}\\log\\pi_{J+1} f_t(y_i;\\bm{\\eta})\\right\\}\\nonumber\\\\\n&\\quad +\\sum_{i=1}^{n}k_i^{(l)}W(y_i)\\Big\\{\\left[\\sum_{j=1}^J {z'}^{(l)}_{ij}\\left(\\log\\pi_j +E\\left[\\log f_b(Y';\\bm{\\varphi}_j)|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}\\right]\\right)\\right] \\nonumber\\\\\n& \\hspace{8em} +{z'}^{(l)}_{i(J+1)}\\left(\\log\\pi_{J+1}+ E\\left[\\log f_t(Y';\\bm{\\eta})|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}\\right]\\right)\\Big\\},\n\\end{align}\nwhere $z_{ij}^{(l)}=E[z_{ij}|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$, ${z'}^{(l)}_{ij}=E[z'_{ijs}|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$ and $k_i^{(l)}=E[K_i|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$. Also, $K$ follows $p(\\cdot;\\bm{\\Phi}^{(l-1)})$ in Equation (\\ref{eq:em:k}) and $Y'$ follows $\\tilde{h}^{*}(\\cdot;\\bm{\\Phi}^{(l-1)})$ in Equation (\\ref{eq:em:y}). The precise expressions of the above expectations are presented in Section 4.2 of supplementary materials, under a particular specification of Gamma distribution for $f_b$ and Lomax distribution for $f_t$. This specification will also be studied in the illustrating examples (Section \\ref{sec:ex}).\n\nIn the M-step, we attempt to find the updated parameters $\\bm{\\Phi}^{(l)}$ such that $Q^{*}(\\bm{\\Phi}^{(l)}|\\bm{\\Phi}^{(l-1)})\\geq Q^{*}(\\bm{\\Phi}^{(l-1)}|\\bm{\\Phi}^{(l-1)})$. Note in Equation (\\ref{eq:em:q}) that $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ is linearly separable w.r.t. parameters $(\\bm{\\pi}, \\bm{\\varphi}_1,\\ldots,\\bm{\\varphi}_J,\\bm{\\eta})$. Therefore, the optimization can be done separately w.r.t. each subset of parameters. Details are leveraged to Section 4.3 of supplementary materials. \n\n\\subsubsection{Method 2: Parameter transformation approach}\n\\paragraph{Construction of complete data}\nMotivated by the mixture probability transformation approach adopted by e.g. \\cite{lee2012algorithms} and \\cite{VERBELEN2015Censor} for truncated data, we here rewrite the random truncation distribution $\\tilde{h}(y_i;\\bm{\\Phi})$ in Equation (\\ref{eq:loglik_weight}) as\n\\begin{equation}\n\\tilde{h}(y_i;\\bm{\\Phi})\n=\\frac{h(y_i;\\bm{\\Phi})W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}\n=\\sum_{j=1}^J\\pi_j^{*}\\frac{f_b(y_i;\\bm{\\varphi}_j)W(y_i)}{\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}+\\pi_{J+1}^{*}\\frac{f_t(y_i;\\bm{\\eta})W(y_i)}{\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du},\n\\end{equation}\nwhere $\\bm{\\pi}^{*}:=(\\pi_1^{*},\\ldots,\\pi_{J+1}^{*})$ are the transformed mixing weight parameters given by\n\\begin{equation} \\label{eq:em:pi_trans}\n\\pi_j^{*}=\\frac{\\pi_j\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du},~j=1,\\ldots,J;\\qquad\n\\pi_{J+1}^{*}=\\frac{\\pi_{J+1}\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}.\n\\end{equation}\n\nAs a result, the problem is reduced to maximizing the weighted log-likelihood of finite mixture of random truncated distributions. In this case, define the complete data\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i^{*})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $\\bm{z}_i^{*}=(z_{i1}^{*},\\ldots,z_{i(J+1)}^{*})$ are the labels where $z_{ij}^{*}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ (transformed) latent mixture component and $z_{ij}^{*}=0$ otherwise. The complete data weighted log-likelihood function is reduced to\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\left[\\sum_{j=1}^{J}z_{ij}^{*}\\left(\\log\\pi_j^{*}+\\log\\frac{f_b(y_i;\\bm{\\varphi}_j)W(y_i)}{\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}\\right)\\right]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*}\\left(\\log\\pi_{J+1}^{*}+\\log\\frac{f_t(y_i;\\bm{\\eta})W(y_i)}{\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du}\\right)\\Bigg\\}.\n\\end{align}\n\n\\paragraph{Iterative procedures}\nIn the $l^{\\text{th}}$ iteration of the E-step, the expectation of the complete data weighted log-likelihood is:\n\\begin{align} \\label{eq:em:q2}\nQ^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\left[\\sum_{j=1}^{J}z_{ij}^{*(l)}\\left(\\log\\pi_j^{*}+\\log\\frac{f_b(y_i;\\bm{\\varphi}_j)W(y_i)}{\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du}\\right)\\right]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*(l)}\\left(\\log\\pi_{J+1}^{*}+\\log\\frac{f_t(y_i;\\bm{\\eta})W(y_i)}{\\int_0^{\\infty}f_t(u;\\bm{\\eta})W(u)du}\\right)\\Bigg\\},\n\\end{align}\nwhere $z_{ij}^{*(l)}=E[z_{ij}^{*}|\\mathcal{D}^{\\text{com}},\\bm{\\Phi}^{(l-1)}]$ is provided in Section 5.2 of supplementary materials.\n\nIn the M-step, similar to Method 1 that $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ is linearly separable w.r.t. parameters $(\\bm{\\pi}^{*}, \\bm{\\varphi}_1,\\ldots,\\bm{\\varphi}_J,\\bm{\\eta})$, we can maximize $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ sequentially w.r.t. each subset of parameters. Details are presented in Section 5.3 of supplementary materials. Note that the M-step of this approach is slightly more computationally more intensive than Method 1 as the target function $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ here involves numerical integrals.\n\nAfter completing the iterative procedures, we will obtain an estimate of the transformed mixing weights $\\bm{\\pi}^{*}$ instead of $\\bm{\\pi}$. One can revert Equation (\\ref{eq:em:pi_trans}) to get back the estimated original mixing weights as follows:\n\\begin{equation}\n\\pi_j=\\frac{\\pi_j^{*}[\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_j)W(u)du]^{-1}}{\\pi_j^{*}\\sum_{j'=1}^J[\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_{j'})W(u)du]^{-1}+\\pi_{J+1}^{*}[\\int_{0}^{\\infty}f_t(u;\\bm{\\eta})W(u)du]^{-1}},~j=1,\\ldots,J;\n\\end{equation}\n\\begin{equation}\n\\pi_{J+1}=\\frac{\\pi_{J+1}^{*}[\\int_{0}^{\\infty}f_t(u;\\bm{\\eta})W(u)du]^{-1}}{\\pi_j^{*}\\sum_{j'=1}^J[\\int_0^{\\infty}f_b(u;\\bm{\\varphi}_{j'})W(u)du]^{-1}+\\pi_{J+1}^{*}[\\int_{0}^{\\infty}f_t(u;\\bm{\\eta})W(u)du]^{-1}}.\n\\end{equation}\n\n\n\\subsection{Ascending property of the GEM algorithm}\nIt is well known from \\cite{DEMPSTER1977EM} that an increase of complete data log-likelihood implies an increase of observed data log-likelihood (Equation (\\ref{eq:loglik})). This can be analogously extended to the proposed weighted log-likelihood framework where we have the following proposition. The proof is leveraged to Section 6 of the supplementary material.\n\n\\begin{proposition} \\label{prop:ascend}\nIf the expected complete data weighted log-likelihood is increased during the $l^{\\text{th}}$ iteration (i.e. $Q^{*}(\\bm{\\Phi}^{(l)}|\\bm{\\Phi}^{(l-1)})\\geq Q^{*}(\\bm{\\Phi}^{(l-1)}|\\bm{\\Phi}^{(l-1)})$), then the observed data weighted log-likelihood is also increased (i.e. $\\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l)};\\bm{y})\\geq \\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l-1)};\\bm{y})$).\n\\end{proposition}\n\n\\subsection{Parameter Initialization, convergence acceleration and stopping criterion} \\label{sec:em:init}\nInitialization of parameters is an important issue, in the sense that poor initializations may lead to slow convergence, numerical instability and even convergence to spurious local maximum. We suggest to determine the initial parameters $\\bm{\\Phi}^{(0)}$ using a modified version of clusterized method of moments (CMM) approach by \\cite{gui2018fit}. Under this approach, we first determine a threshold $\\tau$ which classifies observations $y_i$ into either body ($y_i\\leq\\tau$) or tail ($y_i>\\tau$) part of the distribution. We then apply a $K$-means clustering method to assign ``body\" observations $y_i$ with $y_i\\leq\\tau$ to one of the $J$ mixture components for the body, with moment matching method for each mixture components to determine the initial parameters $(\\bm{\\pi}^{(0)},\\bm{\\varphi}^{(0)})$. Moment matching technique is also applied to ``tail\" observations $y_i$ with $y_i>\\tau$ to initialize $\\bm{\\eta}^{(0)}$. For details, we direct readers to Section 7 of the supplementary material.\n\nAs EM algorithm often converges slowly with small step sizes, we further apply a step lengthening procedure for every two GEM iterations to accelerate the algorithm. This is described by \\cite{jamshidian1997acceleration} and its references therein as a ``pure accelerator\" for the EM algorithm.\n\n\\sloppy The GEM algorithm is iterated until the relative change of iterated parameters $\\Delta^{\\text{rel}}\\bm{\\Phi}^{(l)}:=|\\log(\\bm{\\Phi}^{(l)}\/\\bm{\\Phi}^{(l-1)})|\/P$ is smaller than a threshold of $10^{-5}$ or the maximum number of iterations of 1000 is reached.\n\n\\subsection{Specification of weight function}\nOur proposed MWLE is rather flexible by allowing us to pre-specify any weight functions $W(\\cdot)$ prior to fitting the GEM algorithm. The appropriate choice of $W(\\cdot)$ depends on some decision rules beyond what statistical inference can do. In insurance loss modelling perspective, such decision rule includes the relative importance of insurance company to correctly specify the tail distribution (to evaluate some tail measures such as Value-at-risk (VaR)) compared to that of more accurately modelling the smaller attritional claims. If accurate extrapolation of huge claims are way more important than modelling the smaller claims, then one may consider $W(y)$ to be close to zero unless $y$ is large, aligning with Assumption \\textbf{A3} in Section \\ref{sec:theory:tail_idx} to ensure near-consistent tail index estimations (Theorem \\ref{thm:asym:tail_idx}). Otherwise, one may consider a flatter $W(y)$ across $y$.\n\nThroughout the entire paper, we analyze the following general form of weight function\n\\begin{equation} \\label{eq:em:wgt_func}\nW(y):=W(y;\\xi,\\tilde{\\mu},\\tilde{\\phi})=\\xi+(1-\\xi)\\int_0^y\\frac{(\\tilde{\\phi}\\tilde{\\mu})^{-1\/\\tilde{\\phi}}}{\\Gamma(1\/\\tilde{\\phi})}u^{1\/\\tilde{\\phi}-1}e^{-u\/(\\tilde{\\phi}\\tilde{\\mu})}du,\\quad y>0,\n\\end{equation}\nwhich is the distribution function of a zero-inflated Gamma distribution. The above weight function has the following characteristics:\n\\begin{itemize}\n\\item $W(y)$ is a non-decreasing function of $y$, meaning that smaller observations are down-weighted.\n\\item $\\xi\\in[0,1]$ is the minimum weight assigned to each observation.\n\\item $\\tilde{\\mu}$ and $\\tilde{\\phi}$ are the location and dispersion hyperparameters of Gamma distribution respectively. Larger $\\tilde{\\mu}$ means more (small to moderate) claims are under-weighted by a larger extent, while $\\tilde{\\phi}$ controls the shape of weight function, or how the observations are under-weighted.\n\\item If $\\xi=1$ or $\\tilde{\\mu}=0$, then the weight function is reduced to $W(\\cdot)=1$, leading to standard MLE approach.\n\\item If $\\xi=0$ and $\\tilde{\\phi}\\rightarrow 0$, then $W(y)=1\\{y\\geq\\tilde{\\mu}\\}$, meaning that only observations greater than $\\tilde{\\mu}$ are informative in determining the estimated parameters.\n\\end{itemize}\n\nOverall, smaller $\\xi$, larger $\\tilde{\\mu}$ and smaller $\\tilde{\\phi}$ represent greater under-weightings to more small claims, where we will expect more robust tail estimation by sacrificing more efficiencies on body estimations. In this paper, instead of quantifying decision rules to select the hyperparameters, in the subsequent sections we empirically test various (wide range) combinations of $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$ to study how these hyperparameters affect the trade-off between tail-robustness and estimation efficiency. These provide practical guidance and assessments to determine the suitable hyperparameters.\n\n\\begin{remark}\nThere are many possible ways to quantify the decision rule to select the ``optimal\" weight function hyperparameters. We here briefly discuss two possible ways: (1) Consider a goodness-of-fit test statistic for heavy-tailed distributions, such as the modified AD test (\\cite{ahmad1988assessment}). Then select weight function hyperparameters which optimizes the test statistic; (2) Define an acceptable range of estimated parameter uncertainty of tail index, e.g. two times as the uncertainty obtained by MLE. Then select the hyperparameters with the greatest distortion metric (e.g. the average downweighting factor $\\sum_{i=1}^{n}(1-W(y_i))\/n$) where the tail index uncertainty is still within the acceptable range.\n\\end{remark}\n\n\n\\subsection{Choice of model complexity} \\label{sec:em:complex}\nThe above GEM algorithm assumes a fixed number of mixture component $J$. However, it is important to control the model complexity by choosing an appropriate $J$ which allows enough flexibility to capture the distributional characteristics without over-fitting. \n\nThe first criterion is motivated by maximizing the expected weighted log-likelihood\n\\begin{equation} \\label{eq:em:e_wgt_ll}\nn\\times\\tilde{E}[\\mathcal{L}^{*}(\\bm{\\Phi};\\bm{Y})]=n\\times\\tilde{E}\\left[W(Y)\\log\\frac{h(Y;\\bm{\\Phi})W(Y)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du}\\right],\n\\end{equation}\nwhere the expectation is taken on $Y$ under the true model generating the observations. Without knowing the true model (in real data applications), Equation (\\ref{eq:em:e_wgt_ll}) is approximated by $\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})$ in Equation (\\ref{eq:loglik_weight}) with fitted model parameters $\\hat{\\bm{\\Phi}}$. Note that it is positively biased with correction term $\\text{tr}(-\\bm{\\Gamma}^{-1}\\bm{\\Lambda})$ shown by \\cite{konishi1996generalised}. This leads to a robustified AIC\n\\begin{equation}\n\\text{RAIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+2\\times\\text{tr}(-\\hat{\\bm{\\Gamma}}^{-1}\\hat{\\bm{\\Lambda}}).\n\\end{equation}\n\nAnalogous and naturally, since AIC-type criteria often choose excessively complex models, we also consider the robustified BIC given by\n\\begin{equation}\n\\text{RBIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+(\\log n)\\times\\text{tr}(-\\hat{\\bm{\\Gamma}}^{-1}\\hat{\\bm{\\Lambda}}).\n\\end{equation}\n\nWe choose $J$ that minimizes either the RAIC or RBIC, and the $(p,p)$-th element of $-\\hat{\\bm{\\Gamma}}^{-1}\\hat{\\bm{\\Lambda}}$ can be interpreted as the effective number of parameter attributed by the $p^{\\text{th}}$ parameter.\n\nInsurance loss dataset is often characterized by very complicated and multimodal distribution on very small claims, yet it is not meaningful to capture all these small nodes by choosing an overly complex mixture distribution with large $J$. However, the above RAIC and RBIC cannot effectively reduce those mixture components as the effective number of parameters for those capturing the smaller claims could be very small if $W(\\cdot)$ is chosen very small over the region of small claims. To effectively remove components which excessively capture the small claims, we propose treating all parameters as ``full parameters\", which results to the following truncated AIC and BIC:\n\\begin{equation}\n\\text{TAIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+2\\times P,\n\\end{equation}\n\\begin{equation}\n\\text{TBIC}=-2\\times\\mathcal{L}_n^{*}(\\hat{\\bm{\\Phi}};\\bm{y})+\\left(\\log \\sum_{i=1}^{n}W(y_i)\\right)\\times P.\n\\end{equation}\n\n\\begin{remark} \\label{rmk:tic}\nThe above TAIC and TBIC are motivated by the bias of approximating $n\\times\\tilde{E}[\\mathcal{L}^{*}(\\bm{\\Phi};\\bm{Y})]$ by the empirical truncated log-likelihood $\\mathcal{L}^{**}_n(\\hat{\\bm{\\Phi}};\\bm{y}):=\\sum_{i=1}^{n}V_i(y_i)\\log \\frac{h(y_i;\\bm{\\Phi})W(y_i)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du}$ instead of $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$, where $V_i(y)\\sim\\text{Bernoulli}(W(y))$ is an indicator randomly discarding some observations. It can be easily shown (details in Section 8 of supplementary material) that the asymptotic bias is simply $P$ with effective number of observations $\\sum_{i=1}^{n}W(y_i)$. Note also that the weighted log-likelihood $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ is asymptotically equivalent to the truncated log-likelihood $\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})$, except that the former produces more accurate estimated parameters than the latter. This motivates why in TAIC and TBIC we choose to evaluate $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ instead of $\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})$.\n\\end{remark}\n\n\\section{Illustrating examples} \\label{sec:ex}\nIn this section, we analyze the performance of our proposed MWLE approach (Equation (\\ref{eq:loglik_weight})) on FMM given by Equation (\\ref{eq:model}). In the following examples, we select Gamma density for the body components $f_b$, a light-tailed distribution to capture the distributional multimodality of small to moderate claims, and Lomax density for the tail component $f_t$ to extrapolate well the tail-heaviness of larger observations. Then, Equation (\\ref{eq:model}) becomes\n\\begin{equation} \\label{eq:em:density_mixture}\nh(y;\\bm{\\Phi})\n=\\sum_{j=1}^J\\pi_jf_b(y;\\mu_j,\\phi_j)+\\pi_{J+1}f_t(y;\\theta,\\gamma),\n\\end{equation}\nwhere the parameter set is re-expressed as $\\bm{\\Phi}=(\\bm{\\pi},\\bm{\\mu},\\bm{\\phi},\\gamma)$ while $\\bm{\\varphi}_j=(\\mu_j,\\phi_j)$ and $\\bm{\\eta}=(\\theta,\\gamma)$, and the Gamma and Lomax densities $f_b$ and $f_t$ are respectively given by\n\\begin{equation} \\label{eq:em:comp}\nf_b(y;\\mu,\\phi)=\\frac{(\\phi\\mu)^{-1\/\\phi}}{\\Gamma(1\/\\phi)}y^{1\/\\phi-1}e^{-y\/(\\phi\\mu)}\\quad \\text{and} \\quad f_t(y;\\theta,\\gamma)=\\frac{\\gamma\\theta^{\\gamma}}{\\left(y+\\theta\\right)^{\\gamma+1}},\n\\end{equation}\nwhere $\\mu$ and $\\phi$ are the mean and dispersion parameters of Gamma distribution, while $\\gamma$ is the tail index parameter for the Lomax distribution. $\\theta$ is scale of the Lomax distribution. Note that the above model is a regular varying distribution with the tail behavior predominately explained by the tail index $\\gamma$. As a result, tail-robustness is highly determined by how stable and accurate the estimated tail index $\\gamma$ is.\n\nThe specifications of body and tail component functions are mainly motivated by the key characteristics of insurance claim severity distributions (multimodal distribution of small claim, existence of extremely large claims, mismatch between body and tail behavior etc.) which will be illustrated in the real insurance data application section. While we do not preclude the existence of other specifications, such as Weibull for the body and Inverse-Gamma for the tail, plausible for insurance applications, in this section we simply focus on studying Gamma-Lomax combination to focus on the scope of this paper -- demonstrating the usefulness of the proposed MWLE, instead of performing distributional comparisons under FMM.\n\n\\subsection{Toy example} \\label{sec:ex:toy}\n\nWe demonstrate how the proposed MWLE framework works through a simple toy example of one-parameter Lomax distribution $H(y;\\gamma)=1-(y+1)^{-\\gamma}$ ($y>0$), which is a special case of Equation (\\ref{eq:em:density_mixture}) with $J=0$ and $\\theta=1$.\n\nConsider the first case where the true model $G(\\cdot)$ is a Lomax with $\\gamma=\\gamma_0=1$. For the weight function for the MWLE, we consider the form of Equation (\\ref{eq:em:wgt_func}) with $\\xi=0$ for simplicity. We will test across a wide range of $\\tilde{\\mu}$ and across $\\tilde{\\phi}\\in\\{0.1, 0.2, 0.5, 1\\}$. Figure \\ref{fig:thm_aeff} presents how the choices of these hyperparameters affect the AEFF. Starting from $\\text{AEFF}=1$ when $\\tilde{\\mu}=0$ which is equivalent to standard MLE, the AEFF decrease monotonically as $\\tilde{\\mu}$ increases. This is intuitive because under-weighting smaller observations with MWLE means effectively discarding some observed information, leading to larger parameter uncertainties compared to MLE. Since the MLE estimated tail index is unbiased under the true model, there is obviously no benefit of using the proposed MWLE to fit the true model.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_aeff_1.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_aeff_2.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{AEFF as a function of the weight location hyperparameter $\\tilde{\\mu}$ (left panel) or Pareto (Lomax) quantile of $\\tilde{\\mu}$ (right panel) under Lomax true model.}\n\\label{fig:thm_aeff}\n\\end{figure}\n\nNow, consider the second case where the true model is perturbed by the contamination function $M$, as presented in Equation (\\ref{eq:asym_contam}). In this demonstration example, we consider two following two choices for the contamination function $M$:\n\\begin{itemize}\n\\item Degenerate perturbation: One-point distribution on $y=1\/4$\n\\item Pareto perturbation: Lomax distribution with tail index $\\gamma=\\gamma^{*}=4>\\gamma_0$\n\\end{itemize}\nNote that the contamination function is relatively lighter tailed and hence it would not affect the tail behavior of the perturbed distribution. In Figure \\ref{fig:thm_if}, we present the IF as a function of the AEFF (determined as a function of chosen $\\tilde{\\mu}$) under the two choices of $M$. We find that as the AEFF reduces (by choosing a larger $\\tilde{\\mu}$), the IF would shrink towards zero. This reflects that a more robust estimation of tail index can be achieved using the proposed MWLE approach by trading off some efficiencies.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_if_1.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/thm_if_2.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{IF as a function of AEFF under degenerate (left panel) and Pareto (right panel) contaminations.}\n\\label{fig:thm_if}\n\\end{figure}\n\n\n\n\\subsection{Simulation studies} \\label{sec:ex:sim}\n\\subsubsection{Simulation settings}\nWe here simulate $n=10,000$ claims (the sample size is motivated by the size of a typical insurance portfolio) from the aforementioned $J$-Gamma Lomax distribution for each of the following two parameter settings with $\\theta=1000$:\n\\begin{itemize}\n    \\item Model 1: $J=2$, $\\bm{\\pi}=(0.4,0.4,0.2)$, $\\bm{\\mu}=(100,300)$, $\\bm{\\phi}=(0.25,0.25)$ and $\\gamma=2$.\n    \\item Model 2: $J=3$, $\\bm{\\pi}=(0.4,0.3,0.1,0.2)$, $\\bm{\\mu}=(50,200,600)$, $\\bm{\\phi}=(0.2,0.2,0.2)$ and $\\gamma=2$.\n\\end{itemize}\n\nWe also consider the zero-inflated Gamma distribution given by Equation (\\ref{eq:em:wgt_func}) as the weight function, with $\\tilde{\\mu}\\in\\{q_{0},q_{0.9},q_{0.95},q_{0.99},q_{0.995}\\}$, $\\tilde{\\phi}\\in\\{0.025,0.1,0.25,1\\}$ and $\\xi\\in\\{0.001,0.01,0.05,0.25\\}$, where $q_{\\alpha}$ is the empirical quantile of the data with $0\\leq \\alpha\\leq 1$. Recall that the choice of $\\tilde{\\mu}=q_0=0$ implies that $W(y;\\xi,\\tilde{\\mu},\\tilde{\\phi})= 1$ and hence the MWLE is equivalent to standard MLE. For each combinations of models and weight function hyperparameters, the simulations of sample points are repeated by 100 times to enable thorough analysis of the results using the proposed weighted log-likelihood approach under various settings. Each simulated sample is then fitted to the $J$-Gamma Lomax mixture in Equation (\\ref{eq:em:density_mixture}) with $J=2$. Note that for simplicity, in the simulation studies we do not examine the choice of $J$ as outlined by Section \\ref{sec:em:complex}. As a result, we have the following research goals in the simulation studies:\n\\begin{itemize}\n\\item Under Model 1, the data is fitted to the true class of models. Hence, we empirically verify the consistencies of estimating model parameters (Theorem \\ref{thm:asym_tru}) using the MWLE. We also study how the selection of weight function hyperparameters affect the estimated parameter uncertainties. Further, we compare the computational efficiency of the two kinds of proposed GEM algorithms.\n\\item Under Model 2, the data is fitted to a misspecified class of models. Hence, we demonstrate how this would distort the estimation of the tail under the MLE, and study how the proposed MWLE produces a more robust tail estimation.\n\\end{itemize}\n\n\\subsubsection{Results of fitting Model 1 (true model)}\n\nConsidering the case where we fit the true class of model to the data generated by Model 1, we first compare the computational efficiencies between the two construction methods of the GEM algorithm as presented by Section \\ref{sec:em:gem}. In general, around 100 iterations are needed under parameter transformation approach (Method 2), as compared to at least 300 iterations under hypothetical data approach (Method 1), revealing relatively faster convergences under Method 2. Figure \\ref{fig:sim_tru_rel} plots the relative change of iterated parameters $\\Delta^{\\text{rel}}\\bm{\\Phi}^{(l)}:=|\\log(\\bm{\\Phi}^{(l)}\/\\bm{\\Phi}^{(l-1)})|\/P$ versus the GEM iteration $l$ under two example choices of weight function hyperparameters, where the division operator is applied element-wise to the vector of parameters. It is apparent that the curve drops much faster under Method 2 than Method 1, confirming faster convergence under Method 2. The main reason is that the construction of hypothetical missing observations under Method 1 will generally effectively reduce the learning rates of the optimization algorithms. As both methods produce very similar estimated parameters while Method 2 is more computationally efficient, from now on we only present the results produced by the GEM algorithm under Method 2.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_rel1.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_rel2.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{The relative change of iterated parameters in the first 100 GEM iterations under two example choices of weight function hyperparameters: Left panel -- $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.01,q_{0.95},0.1)$; Right panel -- $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.05,q_{0.99},0.25)$.}\n\\label{fig:sim_tru_rel}\n\\end{figure}\n\nFigure \\ref{fig:sim_tru_gamma} demonstrates the how the biasedness and uncertainty of the estimated tail index $\\hat{\\gamma}$ differ among various choices of weight functions and their corresponding hyperparameters $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$. From the left panel, the median estimated parameters are very close to the true model parameters (differ by less than 1-2\\%) under most settings of the weight functions, except for few extreme cases where both $\\xi$ and $\\tilde{\\phi}$ are chosen to be very small. This empirically justifies the asymptotic unbiasedness of the MWLE. As expected from the right panel, the uncertainties of MLE parameters are the smallest, verifying that MLE is the asymptotically most efficient estimator among all unbiased estimators if we are fitting the correct model class. The parameter uncertainties generally slightly increase as we choose larger $\\tilde{\\mu}$ to de-emphasize the impacts of smaller observations. In some extreme cases where $\\xi$ and $\\tilde{\\phi}$ very small and $\\tilde{\\mu}$ is very large, the standard error can grow dramatically, reflecting that a lot of information are effectively discarded. \n\nSimilarly in Figure \\ref{fig:sim_tru_mu1} where the biasedness and uncertainty of an estimated mean parameter $\\hat{\\mu_1}$ from the body distribution are displayed, we observe that the proposed MWLE approach behaves properly for fitting the body distribution unless when $\\xi$ and $\\tilde{\\phi}$ are both chosen to be extremely small (in those cases, the estimated body parameters would become unstable with inflated uncertainties). Hence, these extreme choices of hyperparameters are deemed to be inappropriate.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_gamma_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_gamma_sd.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and standard deviation of the estimated tail index $\\hat{\\gamma}$ versus various weight function hyperparameters under the true model.}\n\\label{fig:sim_tru_gamma}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_mu1_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_tru_mu1_sd.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and standard deviation of the estimated mean parameter of the first mixture component $\\hat{\\mu_1}$ versus various weight function hyperparameters under the true model.}\n\\label{fig:sim_tru_mu1}\n\\end{figure}\n\n\\subsubsection{Results of fitting Model 2 (misspecified model)}\n\nWe now turn to the case where we fit a misspecified model (with $J=2$) to the simulated data generated from Model 2 (with $J=3$). The left panels of Figures \\ref{fig:sim_mis_gamma} and \\ref{fig:sim_mis_tailp} examine how the robustness of the estimated tail index $\\hat{\\gamma}$ and tail probability $\\hat{\\pi}_{J+1}$ differs among different choices of hyperparameters $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$. From the left panel, the MLE of the tail index is around $\\hat{\\gamma}=2.48$ which largely over-estimates the true tail index $\\gamma=2$, indicating that the heavy-tailedness of the true distribution is under-extrapolated. On the other hand, with the incorporation of weight functions to under-weight the smaller claims, the biases of the MWLE of $\\gamma$ are greatly reduced compared to that of the MLE under most choices of weight function hyperparameters. In particular, the bias reduction for tail index is more effective using smaller $\\xi$ (i.e. $\\xi\\leq 0.05$). This is intuitive as smaller $\\xi$ means smaller claims are under-weighted by a larger extent, reducing the impacts of smaller claims on the tail index estimations. Similarly from the right panel, the proposed MWLE approach effectively reduces the bias of the estimated tail probability $\\hat{\\pi}_{J+1}$. \n\nThe analysis of bias-variance trade-off is also conducted through computing the mean-squared errors (MSE) of both estimated tail index $\\hat{\\gamma}$ and tail probability $\\hat{\\pi}_{J+1}$. From the right panels of Figures \\ref{fig:sim_mis_gamma} and \\ref{fig:sim_mis_tailp}, as evidenced by smaller MSEs under most choices of weight function hyperparameters, MWLE is much more preferable than MLE approach even after accounting for the increased parameter uncertainties through down-weighting the importance of smaller claims.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_gamma_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_gamma_mse.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and MSE of the estimated tail index $\\hat{\\gamma}$ versus various weight function hyperparameters under the misspecified model.}\n\\label{fig:sim_mis_gamma}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_tailp_m.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/sim_mis_tailp_mse.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Median and MSE of the estimated tail probability $\\hat{\\pi}_{J+1}$ versus various weight function hyperparameters under the misspecified model.}\n\\label{fig:sim_mis_tailp}\n\\end{figure}\n\n\\subsubsection{Summary remark on the choice of weight function hyperparameters}\nFrom the above two simulation studies, we find that under a wide range of choices of weight function hyperparameters, the proposed MWLE not only produces plausible model estimations under true model (Model 1), but is also effective in mitigating the bias of tail estimation inherited from model misspecifications (Model 2).\n\nAmong the three hyperparameters $(\\xi,\\tilde{\\mu},\\tilde{\\phi})$, the choice of minimum weight hyperparameter $\\xi$ plays a particularly vital role on the bias-variance trade-off of the estimated parameters. Under misspecified model (Model 2), smaller $\\xi$ (i.e. $\\xi\\leq 0.05$) is more effective in reducing the biases of both estimated tail index $\\hat{\\gamma}$ and tail probability $\\hat{\\pi}_{J+1}$. However, as evidenced by the results produced under the true model (Model 1), the estimated parameters of the body distributions (i.e. $\\hat{\\bm{\\mu}}$ and $\\hat{\\bm{\\phi}}$) may become prohibitively unstable if $\\xi$ is chosen to be extremely small (i.e. $\\xi\\leq 0.001$) such that smaller observations are effectively almost fully discarded. It is therefore important to compare parameter uncertainties of MWLE to that of MLE, and select\/ consider only the weight function hyperparameters where the corresponding MWLE parameter uncertainties are within an acceptable range (i.e. not too off from the MLE parameter uncertainties). Overall, the choices of $\\xi$ between 0.01 and 0.05 are deemed to be suitable.\n\n\\subsection{Real data analysis} \\label{sec:ex:real}\n\\subsubsection{Data description and background} \\label{sec:ex:real:dat}\nIn this section, we study an insurance claim severity dataset kindly provided by a major insurance company operating in Greece. It consists of 64,923 motor third-party liability (MTPL) insurance policies with non-zero property claims for underwriting years 2013 to 2017. This dataset is also analyzed by \\cite{fung2021mixture} using a mixture composite model, with an emphasis on selecting various policyholder characteristics (explanatory variables) which significantly influence the claim severities. The empirical claim severity distribution exhibits several peculiar characteristics including multimodality and tail-heaviness. The excessive number of distributional nodes for small claims reflects the possibility of distributional contamination, which cannot be and should not be perfectly captured and over-fitted by parametric models like FMM. Preliminary analyses also suggest that the estimated tail index is around 1.3 to 1.4, but note that these are only rough and subjective estimates. The details of preliminary data analysis are provided in Section 9 of the supplementary materials. The key goals of this real data analysis are as follows:\n\\begin{enumerate}\n\\item Illustrate that MLE of FMM would produce highly unstable and unrobust estimates to the tail part of the claim severity distribution. This confirms that tail-robustness is an important research problem in real insurance claim severity modelling which needs to be properly addressed.\n\\item Demonstrate how the proposed MWLE approach leads to superior fittings to the tail and more reliable estimates of tail index as compared to MLE, without much sacrificing its ability to adequately capture the body.\n\\end{enumerate}\n\nTo avoid diverging the focus of this paper, in this analysis we solely examine the distributional fitting of the claim sizes without considering the explanatory variables. Note however that the proposed MWLE can be extended to a regression framework, with the discussions being leveraged to Section \\ref{sec:discussion}.\n\n\n\\subsubsection{Fitting results}\nThe claim severity dataset is fitted to the mixture Gamma-Lomax distribution with density given by Equation \\ref{eq:em:density_mixture} under the proposed MWLE approach. The fitting performances will be examined thoroughly across different number of Gamma (body) mixture components $J\\in\\{1,2,\\ldots,10\\}$ and various choices of weight function hyperparameters ($\\tilde{\\mu}\\in\\{q_{0},q_{0.9},q_{0.95},q_{0.99},q_{0.995}\\}$, $\\tilde{\\phi}\\in\\{0.025,0.1,0.25,1\\}$ and $\\xi\\in\\{0.001,0.01,0.05,0.25\\}$). The MWLE fitted parameters are also compared to the standard MLE across various $J$.\n\nWe first present in Figure \\ref{fig:greek:gamma} the fitted tail index $\\hat{\\gamma}$ versus the number of body components $J$ under all combinations of selected weight function hyperparameters. Each of the four sub-figures corresponds to a particular choice of $\\xi\\in\\{0.001,0.01,0.05,0.25\\}$. The black thick trends for each sub-figure are the MLE estimated tail indexes for comparison purpose. The MLE tail indexes are rather unstable as evidenced by great fluctuations across different number of body components $J$, showing that MLE may not be reliable in extrapolating the heavy-tailedness of complex claim distributions.\nFor instance, with a slight change of model specification from $J=5$ to $J=6$, the estimated tail index largely drops from about 1.8 to 1.5. This is rather unnatural because the change from $J=5$ to $J=6$ should only reflect a slight change in specifying the body.\nThe large drop of the estimated tail index reflects that the Lomax tail part of FMM is not specialized in extrapolating the tail-heaviness of the distribution, but instead is very sensitive to the small claims and the model specifications of the body part.\nTherefore, we conclude that the mixture Gamma-Lomax FMM is not achieving its modelling purpose under the MLE.\n\nOn the other hand, looking individually at each path under MWLE, we find that the estimated $\\hat{\\gamma}$ is much more stable across different $J$ under most choices of weight function hyperparameters, especially when $J\\geq 5$. Also, the estimated MWLE $\\hat{\\gamma}$ is in general smaller than the $\\hat{\\gamma}$ obtained by MLE, moving closer to the values roughly determined by the preliminary data analysis in Section \\ref{sec:ex:real:dat}. Note in the figure that there are a few black solid dots, which appear when the estimated $\\hat{\\gamma}$ under MWLE is outside the range of the plots. These unstable estimates of $\\hat{\\gamma}$ are rare and only occur under one the following two situations: (i) $J$ is chosen be very small (i.e. $J\\leq 2$) in the sense that the models would severely under-fit the distributional complexity of the dataset; (ii) extreme choices of weight function hyperparameters (very small $\\xi$ and $\\tilde{\\phi}$) aligned to the results of the simulation studies.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma1.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma2.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma3.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_gamma4.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Estimated tail index versus the number of Gamma mixture components under MLE and MWLE with various choices of weight function hyperparameters.}\n\\label{fig:greek:gamma}\n\\end{figure}\n\nThe optimal choice of $J$ is tricky as evidenced by the excessive number of small distributional nodes for very small claim sizes described in Section \\ref{sec:ex:real:dat}, which should not be over-emphasized or excessively modelled as these very small claims are almost irrelevant for pricing and risk management. However, both AIC and BIC decrease slowly and steadily for MLE models as $J$ increases. The optimal $J$ in this case goes way beyond $J=10$. Under the proposed MWLE approach with various choices of weight function hyperparameters, the same model selection problem exists using RAIC and RBIC, with the reasons already explained in Section \\ref{sec:em:complex}. On the other hand, using TAIC and TBIC (especially for TBIC), a majority selections of weight function hyperparameters lead to an optimal $J=5$, aligning with the heuristic arguments by \\cite{fung2021mixture} that $J=5$ is enough for capturing all the distributional nodes except for the very small claims which are smoothly approximated by a single mixture component.\n\nTo better understand how the use of proposed MWLE affects the estimations of all parameters (not just the tail index but also parameters affecting the body distributions such as $\\bm{\\mu}$), we showcase in Table \\ref{tab:greek:est_prm} all the estimated parameters and their standard errors (based on Equation (\\ref{eq:asym_var})) using MWLE under two distinguishable example choices of hyperparameters (MWLE 1: $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.01,q_{0.99},0.1)$; MWLE 2: $(\\xi,\\tilde{\\mu},\\tilde{\\phi})=(0.05,q_{0.995},0.25)$) as compared to MLE parameters with $J=5$, the optimal number of body components under TBIC for both MWLE 1 and MWLE 2. Note that the two selected examples are for demonstration purpose -- generally the following findings and conclusions are also valid for other choices of weight function hyperparameters under the proposed MWLE.\n\nWe first observe that the estimated parameters influencing the body (i.e. $\\bm{\\pi}$, $\\bm{\\mu}$ and $\\bm{\\phi}$) under MWLE are very close to those under MLE, even if the smaller claims are greatly down-weighted. MWLE generally results to larger parameter uncertainties as compared with MLE -- reflecting a bias-variance trade-off, but these standard errors are of the same order of magnitude and are still relatively immaterial compared to the estimates as the sample size $n=64,923$ is large. \n\nComparing between the above two MWLE examples, we further notice that the parameter uncertainties under MWLE 1 are greater than those under MWLE 2. This is expected because the influences of smaller claims are down-weighted more under MWLE 1 than those under MWLE 2 (as reflected by smaller minimum weight hyperparameter $\\xi$ chosen under MWLE 1). On the other hand, the estimated tail index $\\hat{\\gamma}$ under MWLE 1 is slightly closer to the heuristic values (i.e. 1.3 to 1.4) than MWLE 2. These may also reflect the bias-variance trade-off among various choices of weight function hyperparameters.\n\n\\begin{table}[!h]\n\\centering\n\\begin{tabular}{lrrrrrr}\n\\toprule\n & \\multicolumn{2}{c}{MWLE 1} & \\multicolumn{2}{c}{MWLE 2} & \\multicolumn{2}{c}{MLE} \\\\\n\\cmidrule(l{3pt}r{3pt}){2-3} \\cmidrule(l{3pt}r{3pt}){4-5} \\cmidrule(l{3pt}r{3pt}){6-7}\n & \\multicolumn{1}{c}{Estimates} & \\multicolumn{1}{c}{Std. Error} & \\multicolumn{1}{c}{Estimates} & \\multicolumn{1}{c}{Std. Error} & \\multicolumn{1}{c}{Estimates} & \\multicolumn{1}{c}{Std. Error} \\\\\n\\midrule\n$\\pi_1$ & 0.3787 & 0.0053 & 0.3829 & 0.0031 & 0.3878 & 0.0022 \\\\\n$\\pi_2$ & 0.0380 & 0.0036 & 0.0404 & 0.0021 & 0.0444 & 0.0014 \\\\\n$\\pi_3$ & 0.1117 & 0.0024 & 0.1134 & 0.0020 & 0.1161 & 0.0017 \\\\\n$\\pi_4$ & 0.0221 & 0.0059 & 0.0192 & 0.0021 & 0.0153 & 0.0008 \\\\\n$\\pi_5$ & 0.2173 & 0.0036 & 0.2163 & 0.0022 & 0.2130 & 0.0019 \\\\\n\\hline\n$\\mu_1$ & 1,303.21 & 50.30 & 1,322.10 & 16.70 & 1,348.68 & 11.11 \\\\\n$\\mu_2$ & 9,171.42 & 145.83 & 9,165.92 & 64.64 & 9,165.36 & 49.38 \\\\\n$\\mu_3$ & 27,590.46 & 125.68 & 27,571.75 & 64.36 & 27,538.88 & 52.41 \\\\\n$\\mu_4$ & 317,274.90 & 2,410.93 & 323,827.70 & 2,159.37 & 322,872.40 & 2,372.68 \\\\\n$\\mu_5$ & 89,007.07 & 170.41 & 88,979.12 & 112.01 & 88,895.92 & 99.20 \\\\\n\\hline\n$\\phi_1$ & 0.9945 & 0.0175 & 0.9996 & 0.0121 & 1.0062 & 0.0113 \\\\\n$\\phi_2$ & 0.0264 & 0.0089 & 0.0284 & 0.0030 & 0.0324 & 0.0020 \\\\\n$\\phi_3$ & 0.0154 & 0.0015 & 0.0158 & 0.0007 & 0.0164 & 0.0005 \\\\\n$\\phi_4$ & 0.0472 & 0.0033 & 0.0333 & 0.0025 & 0.0186 & 0.0020 \\\\\n$\\phi_5$ & 0.0127 & 0.0007 & 0.0126 & 0.0003 & 0.0122 & 0.0002 \\\\\n\\hline\n$\\gamma$ & 1.5353 & 0.0707 & 1.6153 & 0.0586 & 1.7963 & 0.0471 \\\\\n$\\theta$ & 62,637.42 & 7,829.79 & 73,604.51 & 6,630.65 & 101,107.20 & 5,088.29\\\\\n\\hhline{=======}\n\\end{tabular}\n\\caption{\\label{tab:greek:est_prm}Estimated parameters and standard errors under MLE and MWLE approaches with $J=5$.}\n\\end{table}\n\nThe Q-Q plot in Figure \\ref{fig:greek:qq} suggests that the fitting results are satisfactory under both MWLE and MLE except for the very immaterial claims (i.e. $y<100$). Note however that due to the log-scale nature of the Q-Q plot, it is hard to examine from the plot how well the fitted models extrapolate the tail-heaviness of the claim severity data. To examine the tail behavior of the fitted models, we present the log-log plot in the left panel of Figure \\ref{fig:greek:loglog}, with the axis shifted to include large claims only. We observe that for extreme claims (i.e. claim amounts greater than about 0.5 millions, or $\\log y>13$), the logged survival probability produced by MLE fitted model diverges quite significantly from that of empirical observations. Such a divergence can effectively be mitigated by using MWLE with either of the hyperparameter settings. \n\nWe further compute the value-at-risk (VaR) and conditional tail expectation (CTE) at $100q^{\\text{th}}$ security level (denoted as $\\text{VaR}_q(Y;\\hat{\\bm{\\Phi}})$ and $\\text{CTE}_q(Y;\\hat{\\bm{\\Phi}})$ respectively) from the fitted models, and compare them to the empirical values from the severity data (denoted as $\\widehat{\\text{VaR}}_q(Y)$ and $\\widehat{\\text{CTE}}_q(Y)$ respectively). The results are summarized in Table \\ref{tab:greek:risk}. Both MLE and MWLE produce plausible estimates of VaR and CTE up to security levels of 95\\% and 75\\% respectively, reflecting the ability of both approaches in capturing the body part of severity distribution. Nonetheless, the MLE fitted model shows significant divergences of VaR and CTE from the empirical data at higher security levels. In particular, the 99\\%-CTE and 99.5\\%-CTE are largely underestimated by the MLE approach. Such a divergence is effectively reduced by the proposed MWLE approach where superior fittings to the tail are obtained. Further, MWLE 1 seems to perform slightly better than MWLE 2 in terms of tail fitting, as reflected by smaller underestimations of CTEs at high security levels. This provides a plausible trade-off to the increased parameter uncertainties under MWLE 1 as previously mentioned.\n\nTo visualize the results, we further plot the relative misfit of VaR, given by $\\log(\\text{VaR}_q(Y;\\hat{\\bm{\\Phi}})\/\\widehat{\\text{VaR}}_q(Y))$, versus the log survival probability $\\log (1-q)\\in(-4.7,-7.5)$, equivalent to the range of security level from 99\\% to 99.95\\%, in the right panel of Figure \\ref{fig:greek:loglog}. We observe that the MLE fitted model over-estimates the VaR of large claims (security level between 99\\% to 99.8\\%) but then largely under-extrapolates the extreme claims (security level beyond 99.8\\%). This issue is well mitigated by the MWLE where the misfits of VaR are smaller in both regions. Therefore, we conclude that the proposed MWLE effectively improves the goodness-of-fit on the tail part of distribution (as compared to the MLE) without much sacrificing its flexibly to adequately capture the body part. \n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_qq.jpg}\n\\end{subfigure}\n\\end{center}\n\\caption{Q-Q plot under MLE and MWLE with two selected combinations of weight function hyperparameters.}\n\\label{fig:greek:qq}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_loglog.pdf}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/greek_misfit.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Left panel: log-log plot of fitted models compared to empirical data; Right panel: misfit of logged claim amounts versus logged survival probabilities under three fitted models.}\n\\label{fig:greek:loglog}\n\\end{figure}\n\n\\begin{table}[!h]\n\\centering\n\\begin{tabular}{rrrrrlrrrrr}\n\\toprule\n \\multicolumn{5}{c}{VaR ('000)} &  & \\multicolumn{5}{c}{CTE ('000)} \\\\\n\\cmidrule(l{3pt}r{3pt}){1-5} \\cmidrule(l{3pt}r{3pt}){7-11}\n \\multicolumn{1}{c}{Level} & \\multicolumn{1}{c}{MLE} & \\multicolumn{1}{c}{MWLE 1} & \\multicolumn{1}{c}{MWLE 2} & \\multicolumn{1}{c}{Empirical} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{Level} & \\multicolumn{1}{c}{MLE} & \\multicolumn{1}{c}{MWLE 1} & \\multicolumn{1}{c}{MWLE 2} & \\multicolumn{1}{c}{Empirical} \\\\\n\\midrule\n50\\% & 21 & 21 & 21 & 21 &  & 0\\% & 109 & 112 & 111 & 116 \\\\\n75\\% & 83 & 82 & 82 & 82 &  & 50\\% & 174 & 180 & 177 & 187 \\\\\n95\\% & 190 & 191 & 187 & 182 &  & 75\\% & 468 & 505 & 489 & 536 \\\\\n99\\% & 461 & 450 & 445 & 452 &  & 90\\% & 1,140 & 1,326 & 1,242 & 1,498 \\\\\n99.5\\% & 719 & 693 & 674 & 676 &  & 95\\% & 1,711 & 2,115 & 1,948 & 2,455 \\\\\n99.75\\% & 1,149 & 1,031 & 1,046 & 1,075 &  & 99\\% & 2,533 & 3,379 & 3,063 & 4,057 \\\\\n99.95\\% & 2,787 & 3,163 & 3,220 & 3,348 &  & 99.5\\% & 5,956 & 10,243 & 8,572 & 13,329\\\\\n\\hhline{===========}\n\\end{tabular}\n\\caption{VaR and CTE (in thousands) estimated by MLE, MWLE and empirical approaches.}\n\\label{tab:greek:risk}\n\\end{table}\n\n\n\n\\section{Discussions} \\label{sec:discussion}\nIn this paper, we introduce a maximum weighted log-likelihood estimation (MWLE) approach to robustly estimate the tail part of finite mixture models (FMM) while preserving the capability of FMM to flexibly capture the complex distributional phenomena from the body part. Asymptotic theories justify the unbiasedness and robustness of the proposed estimator. In computational aspect, the applicability of EM-based algorithm for efficient estimation of parameters makes the proposed MWLE distinctive compared to the existing literature on weighted likelihood approach. Through several simulation studies and real data analyses, we empirically confirm that the proposed MWLE approach is more appropriate in specifying the tail part of the distribution compared to MLE, and at the same time it still preserves the flexibility of FMM in fitting the smaller observations.\n\nAnother advantage of the MWLE not yet mentioned throughout this paper is its extensibility. First, it is obvious that the proposed MWLE is not restricted to FMM but it is also applicable to any continuous or discrete distributions. Second, MWLE can be easily extended to regression settings, which is crucial for insurance pricing perspective as insurance companies often determine different premiums across policyholders based on individual attributes (e.g. age, geographical location and past claim history). In regression settings, we define $\\bm{x}=(\\bm{x}_1,\\ldots,\\bm{x}_n)$ as the covariates vectors for each of the $n$ observations. Then, the weighted log-likelihood function in Equation (\\ref{eq:loglik_weight}) is then re-expressed as\n\\begin{equation} \\label{eq:loglik_weight_reg}\n\\mathcal{L}^*_n(\\bm{\\Phi};\\bm{y},\\bm{x})=\\sum_{i=1}^{n}W(y_i)\\log \\frac{h(y_i;\\bm{\\Phi},\\bm{x}_i)W(y_i)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi},\\bm{x}_i)W(u)du}\n\\end{equation}\nfor some regression models with density function $h(y_i;\\bm{\\Phi},\\bm{x}_i)$. Obviously, the asymptotic properties still hold subject further to some regularity conditions on covariates $\\bm{x}_i$. For parameter estimations using the GEM algorithm, only the hypothetical data approach (Method 1, which converge slower than Method 2 in Section \\ref{sec:ex}) works, because the transformed mixing probabilities in Equation (\\ref{eq:em:pi_trans}) under Method 2 are assumed to be homogeneous across all observations. We leave all theoretical details with more empirical studies and applications to the future research direction.\n\n\n\n\n\\bibliographystyle{abbrvnat}\n\n\\section{Regularity conditions for asymptotic theory} \\label{apx:asym_reg}\nLet $h(y;\\bm{\\Phi})$ be the density function of $Y$ with parameter space of $\\bm{\\Phi}\\in\\bm{\\Omega}$. For a more concise presentation on the regularity conditions, we here write $\\bm{\\Phi}=(\\psi_1,\\ldots,\\psi_P)$ where $P$ is the total number of parameters in the model. The regularity conditions are:\n\n\\begin{enumerate}[font={\\bfseries},label={R\\arabic*.}]\n\\item $h(y;\\bm{\\Phi})$ has common support in $y$ for all $\\bm{\\Phi}\\in\\bm{\\Omega}$, $h(y;\\bm{\\Phi})$ is identifiable in $\\bm{\\Phi}$ up to a permutation of mixture components.\n\\item $h(y;\\bm{\\Phi})$ admits third partial derivatives with respect to $\\bm{\\Phi}$ for each $\\bm{\\Phi}\\in\\bm{\\Omega}$ and for almost all $y$.\n\\item For all $j_1,j_2=1,\\ldots,P$, the first two derivatives of $h(y;\\bm{\\Phi})$ satisfy\n\\begin{equation}\nE\\left[\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\right]=0;\n\\end{equation}\n\\begin{equation}\nE\\left[\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right]=E\\left[-\\frac{\\partial^2}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right].\n\\end{equation}\n\\item The Fisher information matrix is finite and positive definite at $\\bm{\\Phi}=\\bm{\\Phi}_0$:\n\\begin{equation}\n\\mathcal{I}(\\bm{\\Phi})=E\\left[\\left(\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(y;\\bm{\\Phi})\\right)\\left(\\frac{\\partial}{\\partial\\bm{\\Phi}}\\log h(y;\\bm{\\Phi})\\right)^T\\right].\n\\end{equation}\n\\item There exists an integrable function $\\mathcal{M}(y)$ such that\n\\begin{equation}\n\\hspace{-1cm}\n\\left|\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\\quad\n\\left|\\frac{\\partial^2}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\\quad\n\\left|\\frac{\\partial^3}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}\\partial\\psi_{j_3}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\n\\end{equation}\n\\begin{equation}\n\\hspace{-1cm}\n\\left|\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\\quad\n\\left|\\frac{\\partial^2}{\\partial\\psi_{j_1}\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_3}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y),\n\\end{equation}\n\\begin{equation}\n\\hspace{-1cm}\n\\left|\\frac{\\partial}{\\partial\\psi_{j_1}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_2}}\\log h(y;\\bm{\\Phi})\\frac{\\partial}{\\partial\\psi_{j_3}}\\log h(y;\\bm{\\Phi})\\right|\\leq \\mathcal{M}(y).\n\\end{equation}\n\\end{enumerate}\n\n\\section{Proof of Theorems 1 and 2} \\label{apx:asym_proof1}\nWe first focus on Theorem 1. Denote the weighted log-likelihood of a single observation\n\\begin{equation}\n\\mathcal{L}^{*}(\\bm{\\Phi};y)=W(y)\\log \\frac{h(y;\\bm{\\Phi})W(y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}.\n\\end{equation}\n\nThe consistency and asymptotic normality can be proved by applying Theorems 5.41 and 5.42 of \\cite{van2000asymptotic}. The theorems require the regularity conditions that $E\\left[\\|\\partial\/\\partial\\bm{\\Phi}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\|^2\\right]<\\infty$, the matrix $E\\left[\\partial^2\/\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]$ exists and that $|\\partial^3\/\\partial\\psi_{j_1}\\partial\\psi_{j_2}\\partial\\psi_{j_3}\\mathcal{L}^{*}(\\bm{\\Phi};y)|$ is dominated by a fixed integrable function of $y$, $j_1,j_2,j_3=1,\\ldots,P$ and $\\psi_j$ is the $j^{\\text{th}}$ element of $\\bm{\\Phi}$. Through a direct computation of differentiations, the aforementioned equations can all be expressed as functions of $\\kappa(u;\\bm{\\Phi})$ and $\\int_{0}^{\\infty}\\kappa(u;\\bm{\\Phi})h(u;\\bm{\\Phi})W(u)du$ only, where $\\kappa(\\bm{\\Phi})$ can be the six terms presented in regularity condition \\textbf{R5} (the left hand side of the six equations underneath \\textbf{R5} without the absolute sign). Given \\textbf{R5} that $\\kappa(u;\\bm{\\Phi})$ is bounded by an integrable function and since $|\\int_{0}^{\\infty}\\kappa(u;\\bm{\\Phi})h(u;\\bm{\\Phi})W(u)du|\\leq \\int_{0}^{\\infty}|\\kappa(u;\\bm{\\Phi})|h(u;\\bm{\\Phi})du$, the aforementioned regularity conditions required by \\cite{van2000asymptotic} hold.\n\n\\medskip\n\nFor consistency, it suffices from Theorem 5.42 of \\cite{van2000asymptotic} to show that $\\bm{\\Phi}_0$ is the maximizer of\n\\begin{align}\nE_{\\bm{\\Phi}_0}\\left[\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]\n&=\\int_{0}^{\\infty}W(y)\\log \\frac{h(y;\\bm{\\Phi})W(y)}{\\int_{0}^{\\infty}h(u;\\bm{\\Phi})W(u)du}h(y;\\bm{\\Phi}_0)dy\\nonumber\\\\\n&=c_1\\int_{0}^{\\infty}\\tilde{h}(y;\\bm{\\Phi}_0)\\log\\frac{\\tilde{h}(y;\\bm{\\Phi})}{\\tilde{h}(y;\\bm{\\Phi}_0)}dy+c_2\\nonumber\\\\\n&=-c_1D_{\\text{KL}}\\left(\\tilde{h}(y;\\bm{\\Phi})\\|\\tilde{h}(y;\\bm{\\Phi}_0)\\right)+c_2,\n\\end{align}\nwhere $c_1= \\int_0^\\infty h(y; \\bm{\\Phi}_0)W(y) dy >0$ and $c_2=c_1 \\int_0^\\infty \\tilde{h}(y;\\bm{\\Phi}_0) \\log \\tilde{h}(y;\\bm{\\Phi}_0) dy$ are constants and $D_{\\text{KL}}(Q_1\\|Q_2)\\geq 0$ is the KL divergence between $Q_1$ and $Q_2$. Since $D_{\\text{KL}}\\left(\\tilde{h}(y;\\bm{\\Phi})\\|\\tilde{h}(y;\\bm{\\Phi}_0)\\right)=0$ as $\\bm{\\Phi}=\\bm{\\Phi}_0$, the result follows.\n\n\\medskip\n\nFor asymptotic normality, from Theorem 5.41 of \\cite{van2000asymptotic}, we have $\\sqrt{n}(\\hat{\\bm{\\Phi}}_n-\\bm{\\Phi}_0)\\overset{d}{\\rightarrow}\\mathcal{N}(\\bm{0},\\bm{\\Sigma})$ with \n$\\bm{\\Sigma}=\\bm{\\Gamma}^{-1}\\bm{\\Lambda}\\bm{\\Gamma}^{-1}$, where \n\\begin{equation} \\label{eq:asym:lambda_proof}\n\\bm{\\Lambda}=E_{\\bm{\\Phi}_0}\\left[\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]\\left[\\frac{\\partial}{\\partial\\bm{\\Phi}}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\right]^T\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right]\n\\end{equation}\nand\n\\begin{equation} \\label{eq:asym:gamma_proof}\n\\bm{\\Gamma}=-E_{\\bm{\\Phi}_0}\\left[\\frac{\\partial^2}{\\partial\\bm{\\Phi}\\partial\\bm{\\Phi}^T}\\mathcal{L}^{*}(\\bm{\\Phi};Y)\\Bigg|_{\\bm{\\Phi}=\\bm{\\Phi}_0}\\right].\n\\end{equation}\n\nPerforming the derivatives and algebra manipulations from Equations (2.3) and (2.4) would result to Equations (4.2) and (4.3) respectively, which prove the asymptotic normality result.\n\n\\medskip\n\nProof idea of Theorem 2 is completely identical as the above, except that the expectations in Equations (2.3) and (2.4) are taken as $\\tilde{E}[\\cdot]$ instead of $E_{\\bm{\\Phi}_0}[\\cdot]$.\n\n\n\n\\section{Proof of Theorem 3} \\label{apx:asym_proof2}\nWe begin with the following lemmas:\n\\begin{lemma} \\label{apx:lem:asym1}\nTo prove Theorem 3, it suffices to show that\n\\begin{equation}\nT_n(\\bm{\\Phi}):=\\frac{\\partial}{\\partial\\gamma}\\tilde{E}_n[\\log \\tilde{h}_{n}(Y;\\bm{\\Phi})]\n\\end{equation}\nis asymptotically a strictly decreasing function of $\\gamma$ as $n\\rightarrow\\infty$, with $T_n(\\bm{\\Phi})|_{\\gamma=\\gamma_0}\\rightarrow 0$ as $n\\rightarrow\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nIf we keep the weight function $W(\\cdot)$ fixed (independent of $n$), applying Theorem 5.7 of \\cite{van2000asymptotic} we have that maximizing the weighted log-likelihood function $\\mathcal{L}_n^{*}(\\bm{\\Phi};\\bm{y})$ is asymptotically equivalent to maximizing $\\tilde{E}_n[\\log \\tilde{h}_{n}(Y;\\bm{\\Phi})]$ (which is indeed independent of $n$).\n\nNow that the weight function $W_n(\\cdot)$ depends on $n$, and as $n$ increases, the increasing distortion (more down-weighting) of the relative importance of observations would cause reduction of the effective number of observations. Heuristically, we need the number of observations $n$ to increase faster than the distortion impacts of $W_n(\\cdot)$, so that effective number of observations grows to infinity and large sample theory still applies. Quantitatively, we require that the variance of (scaled) empirical weighted log-likelihood\n\\begin{equation}\nV_n(\\bm{\\Phi}):=\\text{Var}\\left(\\frac{1}{n\\int_{0}^{\\infty}W_n(u)g(u)du}\\sum_{i=1}^{n}W_n(Y)\\log\\tilde{h}_{n}(Y;\\bm{\\Phi})\\right)\\rightarrow 0\n\\end{equation}\nas $n\\rightarrow\\infty$, such that the (scaled) empirical weighted log-likelihood function converges to its expectation which is $\\tilde{E}_n[\\log \\tilde{h}_{n}(Y;\\bm{\\Phi})]$. Now, $V_n(\\bm{\\Phi})$ is evaluated as follows:\n\\begin{align}\nV_n(\\bm{\\Phi})\n&=\\frac{1}{n(\\int_{0}^{\\infty}W_n(u)g(u)du)^2}\\text{Var}\\left(W_n(Y)\\log\\tilde{h}_{n}(Y;\\bm{\\Phi})\\right)\\nonumber\\\\\n&\\leq\\frac{1}{n(\\int_{0}^{\\infty}W_n(u)g(u)du)^2}\\tilde{E}\\left[W_n(Y)(\\log\\tilde{h}_{n}(Y;\\bm{\\Phi}))^2\\right]\\nonumber\\\\\n&=\\frac{1}{n\\int_{0}^{\\infty}W_n(u)g(u)du}\\int_{0}^{\\infty}\\frac{W_n(y)g(y)}{\\int_{0}^{\\infty}W_n(u)g(u)du}(\\log\\tilde{h}_{n}(y;\\bm{\\Phi}))^2dy\\nonumber\\\\\n&=\\frac{\\tilde{E}_n[(\\log \\tilde{h}_{n}(Y;\\bm{\\Phi}))^2]}{n\\tilde{E}[W_n(Y)]}\\rightarrow 0,\n\\end{align}\nwhere the convergence is based on Assumption \\textbf{A4}.\n\\end{proof}\n\n\\begin{lemma} \\label{apx:lem:asym2}\n(Monotone density theorem -- Theorem 1.7.2 of \\cite{bingham1989regular}) Denote $H$ as a probability distribution function with $h$ being the corresponding probability density function. Assume $h$ is ultimately monotone (i.e. $h$ is monotone on $(z,\\infty)$ for some $z>0$). If\n\\begin{equation}\n\\bar{H}(y)\\sim y^{-\\gamma}L(y)\n\\end{equation}\nas $y\\rightarrow\\infty$ for some $\\gamma>0$ and slowly varying functions $L$, then\n\\begin{equation}\nh(y)\\sim \\gamma y^{-\\gamma-1}L(y)\n\\end{equation}\nas $y\\rightarrow\\infty$.\n\\end{lemma}\n\nWe proceed to the proof of Theorem 3 as follows. Using the result from Lemma \\ref{apx:lem:asym1}, it suffices to evaluate\n\\begin{align} \\label{apx:eq:proof2:Tn}\nT_n(\\bm{\\Phi})\n&=\\frac{\\partial}{\\partial\\gamma}\\int_{0}^{\\infty}\\tilde{g}_n(y)\\log \\tilde{h}_{n}(y;\\bm{\\Phi})dy\\nonumber\\\\\n&=\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\tilde{g}^{*}_n(y)\\log\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy + o(1)\\nonumber\\\\\n&=\\frac{\\partial}{\\partial\\gamma}\\log\\tilde{h}^{*}_{n}(\\tau_n;\\bm{\\Phi})\n+\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\left[\\frac{\\partial}{\\partial y}\\log\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})\\right]\\times\\bar{\\tilde{G}}^{*}_{n}(y)dy + o(1)\\nonumber\\\\\n&:=M_1(\\tau_n;\\bm{\\Phi})+M_2(\\tau_n;\\bm{\\Phi}) + o(1),\n\\end{align}\nwhere\n\\begin{equation}\n\\tilde{g}^{*}_{n}(y)=\\frac{g(y)W_n(y)}{\\int_{\\tau_n}^{\\infty}g(u)W_n(u)du}1\\{y\\geq\\tau_n\\},\\qquad \\tilde{h}^{*}_{n}(y;\\bm{\\Phi})=\\frac{h(y;\\bm{\\Phi})W_n(y)}{\\int_{\\tau_n}^{\\infty}h(u;\\bm{\\Phi})W_n(u)du}1\\{y\\geq\\tau_n\\},\n\\end{equation}\nare the proper transformed density functions, and $\\tilde{G}^{*}_{n}$ and $\\tilde{H}_{n}^{*}$ are the corresponding distribution functions. The second equality of Equation (\\ref{apx:eq:proof2:Tn}) is resulted from Assumption \\textbf{A3}, while the third equality is followed by integration by parts. Now, we evaluate $M_1(\\tau_n;\\bm{\\Phi})$ and $M_2(\\tau_n;\\bm{\\Phi})$ as follows:\n\n\\begin{align}\nM_1(\\tau_n;\\bm{\\Phi})\n&=\\frac{\\partial}{\\partial\\gamma}\\log\\tilde{h}^{*}_{n}(\\tau_n;\\bm{\\Phi})\\nonumber\\\\\n&=\\frac{\\partial}{\\partial\\gamma}\\left[\\log\\gamma-(\\gamma+1)\\log\\tau_n+\\log L(\\tau_n;\\bm{\\Phi})\\right]\\nonumber\\\\\n&\\hspace{3em}-\\int_{\\tau_n}^{\\infty}\\frac{\\partial}{\\partial\\gamma}\\left[\\log\\gamma-(\\gamma+1)\\log y+\\log L(y;\\bm{\\Phi})\\right]\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy+o(1)\\nonumber\\\\\n&=\\frac{1}{\\gamma}-\\log\\tau_n-\\frac{1}{\\gamma}+\\int_{\\tau_n}^{\\infty}(\\log y)\\times\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy\n-\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\log\\frac{L(y;\\bm{\\Phi})}{L(\\tau_n;\\bm{\\Phi})}\\times\\tilde{h}^{*}_{n}(y;\\bm{\\Phi})dy+o(1)\\nonumber\\\\\n&=-\\log\\tau_n+\\log\\tau_n\n+\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{H}}^{*}_{n}(y;\\bm{\\Phi})dy\n-\\frac{\\partial}{\\partial\\gamma}\\int_{1}^{\\infty}\\log\\frac{L(\\tau_nt;\\bm{\\Phi})}{L(\\tau_n;\\bm{\\Phi})}\\times\\tau_n\\tilde{h}^{*}_{n}(\\tau_nt;\\bm{\\Phi})dt+o(1)\\nonumber\\\\\n&=\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{H}}^{*}_{n}(y;\\bm{\\Phi})dy + o(1),\n\\end{align}\nwhere dominated convergence theorem and integration by parts are repeatedly used. The second equality involves monotone density theorem (Lemma \\ref{apx:lem:asym2}) with Assumption \\textbf{A5} being satisfied. The last term of the second last equality converges to zero uniformly on $\\bm{\\Phi}$ due to dominated convergence theorem and the uniform convergence conditions in Assumption \\textbf{A2}. Using similar techniques as the above, $M_2(\\tau_n;\\bm{\\Phi})$ can be evaluated as\n\\begin{align}\nM_2(\\tau_n;\\bm{\\Phi})\n&=-\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{G}}^{*}_{n}(y)dy\n+\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\frac{\\partial}{\\partial y}(\\log L(y;\\bm{\\Phi}))\\times\\bar{\\tilde{G}}^{*}_{n}(y)dy\\nonumber\\\\\n&=-\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{G}}^{*}_{n}(y)dy-\\frac{\\partial}{\\partial\\gamma}\\int_{\\tau_n}^{\\infty}\\log\\frac{L(y;\\bm{\\Phi})}{L(\\tau_n;\\bm{\\Phi})}\\times\\tilde{g}^{*}_{n}(y)dy\\nonumber\\\\\n&=-\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\bar{\\tilde{G}}^{*}_{n}(y)dy + o(1).\n\\end{align}\n\nTo sum up, we have\n\\begin{equation}\nT_n(\\bm{\\Phi})\n=\\int_{\\tau_n}^{\\infty}\\frac{1}{y}\\left[\\bar{\\tilde{H}}^{*}_{n}(y;\\bm{\\Phi})-\\bar{\\tilde{G}}^{*}_{n}(y)\\right]dy\n=\\int_{1}^{\\infty}\\frac{1}{t}\\left[\\bar{\\tilde{H}}^{*}_{n}(\\tau_nt;\\bm{\\Phi})-\\bar{\\tilde{G}}^{*}_{n}(\\tau_nt)\\right]dt.\n\\end{equation}\n\nInvestigating each term inside the integrand, we have\n\\begin{align}\n\\bar{\\tilde{H}}^{*}_{n}(\\tau_nt;\\bm{\\Phi})\n&=\\frac{\\int_t^{\\infty}h(\\tau_nv;\\bm{\\Phi})W_n(\\tau_nv)dv}{\\int_1^{\\infty}h(\\tau_nv;\\bm{\\Phi})W_n(\\tau_nv)dv}\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)[L(\\tau_nv;\\bm{\\Phi})\/L(\\tau_n;\\bm{\\Phi})]dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)[L(\\tau_nv;\\bm{\\Phi})\/L(\\tau_n;\\bm{\\Phi})]dv} + o(1)\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv} + o(1),\n\\end{align}\nand\n\\begin{align}\n\\bar{\\tilde{G}}^{*}_{n}(\\tau_nt)\n&=\\frac{\\int_t^{\\infty}g(\\tau_nv)W_n(\\tau_nv)dv}{\\int_1^{\\infty}g(\\tau_nv)W_n(\\tau_nv)dv}\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)[L_0(\\tau_nv)\/L_0(\\tau_n)]dv}{\\int_1^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)[L_0(\\tau_nv)\/L_0(\\tau_n)]dv} + o(1)\\nonumber\\\\\n&=\\frac{\\int_t^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv} + o(1),\n\\end{align}\nwhere $\\tilde{W}_n(v)=W_n(\\tau_nv)$. Therefore, it is clear that\n\\begin{equation}\nT_n(\\bm{\\Phi})=\\int_{1}^{\\infty}\\frac{1}{t}\\left[\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}-\\frac{\\int_t^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma_0-1}\\tilde{W}_n(v)dv}\\right]dt+o(1)\n\\end{equation}\nconverges to zero for $\\gamma=\\gamma_0$ as $n\\rightarrow\\infty$. To show that $T_n(\\bm{\\Phi})$ is a strictly decreasing function of $\\gamma$ as $n\\rightarrow\\infty$, it suffices to evaluate\n\\begin{align}\n\\frac{\\partial}{\\partial\\gamma}\\log\\frac{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}\n&=-\\frac{\\int_t^{\\infty}(\\log v)v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_t^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv}+\\frac{\\int_1^{\\infty}(\\log v)v^{-\\gamma-1}\\tilde{W}_n(v)dv}{\\int_1^{\\infty}v^{-\\gamma-1}\\tilde{W}_n(v)dv},\n\\end{align}\nwhich is negative if and only if\n\\begin{equation}\n\\int_{1}^{t}(\\log v)k_{n,1,t}(v;\\gamma)dv<\\int_{t}^{\\infty}(\\log v)k_{n,t,\\infty}(v;\\gamma)dv,\n\\end{equation}\nwhere\n\\begin{equation}\nk_{n,t_1,t_2}(v;\\gamma)=\\frac{v^{-\\gamma-1}\\tilde{W}_n(v)}{\\int_{t_1}^{t_2}v^{-\\gamma-1}\\tilde{W}_n(v)dv}1\\{t_1<v\\leq t_2\\}\n\\end{equation}\nis a proper probability density function with $1\\leq t_1<t_2\\leq\\infty$. Since $k_{n,1,t}(v;\\gamma)$ and $k_{n,t,\\infty}(v;\\gamma)$ are both proper densities, it is clear that $\\int_{1}^{t}(\\log v)k_{n,1,t}(v;\\gamma)dv=\\log v_1$ for some $v_1\\in (1,t)$ and $\\int_{t}^{\\infty}(\\log v)k_{n,t,\\infty}(v;\\gamma)dv=\\log v_2$ for some $v_2\\in (t,\\infty]$. The result then follows.\n\n\n\\section{GEM algorithm for MWLE under J-Gamma Lomax mixture model: Hypothetical data approach} \n\n\\subsection{Construction of complete data}\nThe complete data is given by\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i,k_i,\\{\\bm{z}'_{is},y'_{is}\\}_{s=1,\\ldots,k_i})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $\\bm{z}_i=(z_{i1},\\ldots,z_{i(J+1)})$ with $z_{ij}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ latent class and $z_{ij}=0$ otherwise. Similarly, $\\bm{z}'_i=(z'_{is1},\\ldots,z'_{is(J+1)})$ and $z'_{isj}=1$ if the $s^{\\text{th}}$ missing sample generated by observation $i$ belongs to the $j^{\\text{th}}$ latent class, and $z'_{isj}=0$ otherwise. The complete data weighted log-likelihood function is given by\n\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^nW(y_i)\\left\\{\\left[\\sum_{j=1}^{J}z_{ij}\\log \\pi_jf_b(y_i;\\mu_j,\\phi_j)\\right]+z_{i(J+1)}\\log\\pi_{J+1} f_t(y_i;\\theta,\\gamma)\\right\\}\\nonumber\\\\\n&\\quad +\\sum_{i=1}^{n}\\sum_{s=1}^{k_i}W(y_i)\\left\\{\\left[\\sum_{j=1}^J z'_{ijs}\\log\\pi_j f_b(y'_{is};\\mu_j,\\phi_j)\\right]+z'_{i(J+1)s}\\log\\pi_{J+1} f_t(y'_{is};\\theta,\\gamma)\\right\\}.\n\\end{align}\n\n\\subsection{E-step} \\label{supp:sec:em_e}\nThe expectation of the complete data weighted log-likelihood is given by the following for the $l^{\\text{th}}$ iteration:\n\\begin{align}\n&Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\\nonumber\\\\\n&=\\sum_{i=1}^nW(y_i)\\Bigg\\{\\sum_{j=1}^{J}z^{(l)}_{ij}\\left\\{\\log\\pi_j-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\log y_i-\\frac{y_i}{\\phi_j\\mu_j}\\right\\}\\nonumber\\\\\n&\\hspace{8em}+ z^{(l)}_{i(J+1)}\\left\\{\\log\\pi_{J+1}+\\log\\gamma+\\gamma\\log\\theta-(\\gamma+1)\\log(y_i+\\theta)\\right\\}\\Bigg\\}\\nonumber\\\\\n&\\quad +k^{(l)}\\left(\\sum_{i=1}^{n}W(y_i)\\right)\\Bigg\\{\\sum_{j=1}^J z^{'(l)}_{j}\\left\\{\\log\\pi_j-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\widehat{\\log y'}^{(l)}_j-\\frac{\\widehat{y'}^{(l)}_j}{\\phi_j\\mu_j}\\right\\}\\nonumber\\\\\n&\\hspace{12em}+z^{'(l)}_{(J+1)}\\left\\{\\log\\pi_{J+1}+\\log\\gamma+\\gamma\\log\\theta-(\\gamma+1)\\widehat{\\log(y'+\\theta)}^{(l)}_{J+1}\\right\\}\\Bigg\\},\n\\end{align}\nwhere\n\\begin{equation}\nz^{(l)}_{ij}=P(z_{ij}=1|\\bm{y},\\bm{\\Phi}^{(l-1)})=\n\\begin{cases}\n\\dfrac{\\pi_j^{(l-1)}f_b(y_i;\\mu_j^{(l-1)},\\phi_j^{(l-1)})}{h(u;\\bm{\\Phi}^{(l-1)})},\\quad j=1,\\ldots,J\\\\\n\\dfrac{\\pi^{(l-1)}_{J+1}f_t(y_i;\\theta,\\gamma^{(l-1)})}{h(u;\\bm{\\Phi}^{(l-1)})},\\quad j=J+1,\n\\end{cases}\n\\end{equation}\n\\begin{equation}\nk^{(l)}=E(k_i|\\bm{y},\\bm{\\Phi}^{(l-1)})=\\frac{\\int_0^{\\infty}f_t(u;\\bm{\\Phi}^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}h(u;\\bm{\\Phi}^{(l-1)})W(u)du},\n\\end{equation}\n\\begin{equation}\nz_{j}^{'(l)}=P(z'_{ijs}=1|\\bm{y},\\bm{\\Phi}^{(l-1)})\n\\begin{cases}\n\\dfrac{\\pi_j^{(l-1)}\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}h(u;\\bm{\\Phi}^{(l-1)})(1-W(u))du},\\quad j=1,\\ldots,J\\\\\n\\dfrac{\\pi_j^{(l-1)}\\int_0^{\\infty}f_t(u;\\theta,\\gamma^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}h(u;\\bm{\\Phi}^{(l-1)})(1-W(u))du}, \\quad j=J+1,\n\\end{cases}\n\\end{equation}\n\\begin{equation}\n\\widehat{y'}^{(l)}_j=E(y_{is}'|\\bm{y},\\bm{\\Phi}^{(l-1)},z'_{ijs}=1)\n=\\frac{\\int_0^{\\infty}uf_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}, \\quad j=1,\\ldots,J,\n\\end{equation}\n\\begin{equation}\n\\widehat{\\log y'}^{(l)}_j=E(\\log y_{is}'|\\bm{y},\\bm{\\Phi}^{(l-1)},z'_{ijs}=1)\n=\\frac{\\int_0^{\\infty}\\log uf_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})(1-W(u))du}, \\quad j=1,\\ldots,J,\n\\end{equation}\n\\begin{equation}\n\\widehat{\\log(y'+\\theta)}^{(l)}_{J+1}=E(\\log (y_{is}'+\\theta)|\\bm{y},\\bm{\\Phi}^{(l-1)},z'_{i(J+1)s}=1)\n=\\frac{\\int_0^{\\infty}\\log (u+\\theta) f_t(u;\\theta,\\gamma^{(l-1)})(1-W(u))du}{\\int_0^{\\infty}f_t(u;\\theta,\\gamma^{(l-1)})(1-W(u))du}.\n\\end{equation}\n\nNote that the above integrals all have analytical form of solutions under the example choice of the following weight functions (for the generalized weight function form as presented in the main paper, we do a numerical integration instead):\n\\begin{itemize}\n    \\item Case 1: Exponential distribution with $W(y;\\tilde{\\mu})=1-\\exp\\{-y\/\\tilde{\\mu}\\}$;\n    \\item Case 2: Two-point discrete distribution with $W(y;\\tilde{\\mu},\\tilde{\\phi})=(1-\\tilde{\\phi})1\\{y>\\tilde{\\mu}\\}+\\tilde{\\phi}$.\n\\end{itemize}\n\nRe-parameterize the gamma distribution with $\\alpha=1\/\\phi_j$ and $\\beta=1\/\\phi_j\\mu_j$, we are to compute\n\\begin{equation}\n\\int_{0}^{\\infty}q(u)\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du\n\\end{equation}\nfor $q(u)=1$, $q(u)=u$ and $q(u)=\\log u$; and\n\\begin{equation}\n\\int_{0}^{\\infty}r(u)\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du\n\\end{equation}\nfor $r(u)=1$ and $r(u)=\\log(u+\\theta)$.\n\n\\textbf{Case 1}. We have the following analytical results:\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\left(\\frac{\\beta}{\\beta+1\/\\tilde{\\mu}}\\right)^{\\alpha},\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\alpha\\beta^{\\alpha}}{(\\beta+1\/\\tilde{\\mu})^{\\alpha+1}},\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}\\frac{\\partial}{\\partial\\alpha}\\frac{\\Gamma(\\alpha)}{(\\beta+1\/\\tilde{\\mu})^{\\alpha}},\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=\\gamma\\left(\\frac{\\theta}{\\tilde{\\mu}}\\right)^{\\gamma}\\exp\\left\\{\\frac{\\theta}{\\tilde{\\mu}}\\right\\}\\Gamma(-\\gamma;\\frac{\\theta}{\\tilde{\\mu}},\\infty),\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log(u+\\theta)\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=-\\gamma\\theta^{\\gamma}\\exp\\left\\{\\frac{\\theta}{\\tilde{\\mu}}\\right\\}\\frac{\\partial}{\\partial\\gamma}\\Gamma(-\\gamma;\\frac{\\theta}{\\tilde{\\mu}},\\infty),\n\\end{equation}\nwhere $\\Gamma(m;c_1,c_2)=\\int_{c_1}^{c_2}u^{m-1}\\exp\\{-u\\}du$ is an incomplete gamma function.\n\n\\textbf{Case 2}. We have the following analytical results:\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\tilde{\\phi}\\frac{\\Gamma(\\alpha;\\beta\\tilde{\\mu},\\infty)}{\\Gamma(\\alpha)}+(1-\\tilde{\\phi}),\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\alpha}{\\beta}\\left[\\tilde{\\phi}\\Gamma(\\alpha+1;\\beta\\tilde{\\mu},\\infty)+(1-\\tilde{\\phi})\\right],\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log u\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}u^{\\alpha-1}\\exp\\{-\\beta u\\}(1-W(u))du=\\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}\\left[\\tilde{\\phi}\\frac{\\partial}{\\partial\\alpha}\\frac{\\Gamma(\\alpha;\\beta\\tilde{\\mu},\\infty)}{\\beta^{\\alpha}}+(1-\\tilde{\\phi})\\frac{\\partial}{\\partial\\alpha}\\frac{\\Gamma(\\alpha)}{\\beta^{\\alpha}}\\right],\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=\\tilde{\\phi}\\left(\\frac{\\theta}{\\tilde{\\mu}+\\theta}\\right)^{\\gamma}+(1-\\tilde{\\phi}),\n\\end{equation}\n\\begin{equation}\n\\int_{0}^{\\infty}\\log(u+\\theta)\\frac{\\gamma\\theta^{\\gamma}}{(u+\\theta)^{\\gamma+1}}(1-W(u))du=-\\gamma\\theta^{\\gamma}\\frac{\\partial}{\\partial\\gamma}\\left[\\tilde{\\phi}\\frac{1}{\\gamma(\\tilde{\\mu}+\\theta)^{\\gamma}}+(1-\\tilde{\\phi})\\frac{1}{\\gamma\\theta^{\\gamma}}\\right].\n\\end{equation}\n\n\\subsection{M-step} \\label{supp:sec:em_m}\nMaximizing $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ with respect to $\\bm{\\Phi}$ yields the following parameter updates:\n\\begin{equation}\n\\pi_j^{(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{ij}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_j^{'(l)}}{\\sum_{j'=1}^{J+1}\\left\\{\\sum_{i=1}^{n}W(y_i)z_{ij'}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_{j'}^{'(l)}\\right\\}},\\quad j=1,\\ldots,J+1,\n\\end{equation}\n\\begin{equation}\n\\mu_j^{(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{ij}^{(l)}y_i+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_j^{'(l)}\\widehat{y'}^{(l)}_j}{\\sum_{i=1}^{n}W(y_i)z_{ij}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_j^{'(l)}},\\quad j=1,\\ldots,J,\n\\end{equation}\n\\begin{align}\n\\phi_j^{(l)}\n&=\\underset{\\phi_j>0}{\\text{argmax}}\\Bigg\\{\\sum_{i=1}^nW(y_i)z^{(l)}_{ij}\\left\\{-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j^{(l)}-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\log y_i-\\frac{y_i}{\\phi_j\\mu_j}\\right\\}\\nonumber\\\\\n&\\hspace{5em} +k^{(l)}\\left(\\sum_{i=1}^{n}W(y_i)\\right)z^{'(l)}_{j}\\left\\{-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j^{(l)}-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\widehat{\\log y'}^{(l)}_j-\\frac{\\widehat{y'}^{(l)}_j}{\\phi_j\\mu_j^{(l)}}\\right\\}\\Bigg\\},\\nonumber\\\\\n\\end{align}\n\\begin{equation}\n\\gamma^{(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{i(J+1)}^{(l)}+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_{J+1}^{'(l)}}{\\sum_{i=1}^{n}W(y_i)z_{i(J+1)}^{(l)}\\left[\\log(y_i+\\theta)-\\log\\theta\\right]+\\left(\\sum_{i=1}^{n}W(y_i)\\right)k^{(l)}z_{J+1}^{'(l)}\\left[\\widehat{\\log(y'+\\theta)}^{(l)}_{J+1}-\\log\\theta\\right]}.\n\\end{equation}\n\nNote here that $\\theta$ is treated as a fixed hyperparameter not involved in estimation procedure. To estimate $\\theta$ as a parameter, we may need to take a further step to numerically maximize the observed data weighted log-likelihood $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})$ w.r.t. $\\theta$.\n\n\\section{GEM algorithm for MWLE under J-Gamma Lomax mixture model: Parameter transformation approach}\n\\subsection{Construction of complete data}\nThe complete data is given by\n\\begin{equation}\n\\mathcal{D}^{\\text{com}}=\\{(y_i,\\bm{z}_i^{*})\\}_{i=1,\\ldots,n},\n\\end{equation}\nwhere $\\bm{z}_i^{*}=(z_{i1}^{*},\\ldots,z_{i(J+1)}^{*})$ are the labels where $z_{ij}^{*}=1$ if observation $i$ belongs to the $j^{\\text{th}}$ (transformed) latent mixture component and $z_{ij}^{*}=0$ otherwise. The complete data weighted log-likelihood function is given by\n\\begin{align}\n\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi};\\mathcal{D}^{\\text{com}})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\left[\\sum_{j=1}^{J}z_{ij}^{*}\\left(\\log\\pi_j^{*}+\\log f_b(y_i;\\mu_j,\\phi_j) -\\log\\int_0^{\\infty}f_b(u;\\mu_j,\\phi_j)W(u)du\\right)\\right]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*}\\left(\\log\\pi_{J+1}^{*}+\\log f_t(y_i;\\theta,\\gamma)W(y_i)-\\log\\int_0^{\\infty}f_t(u;\\theta,\\gamma)W(u)du\\right)\\Bigg\\}.\n\\end{align}\n\n\n\\subsection{E-step} \\label{supp:sec:em_e2}\nThe expectation of the complete data weighted log-likelihood is given by the following for the $l^{\\text{th}}$ iteration:\n\\begin{align}\nQ^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})\n&=\\sum_{i=1}^{n}W(y_i)\\Bigg\\{\\Bigg[\\sum_{j=1}^{J}z_{ij}^{*(l)}\\Bigg(\\log\\pi_j^{*}-\\frac{1}{\\phi_j}\\log\\phi_j-\\frac{1}{\\phi_j}\\log\\mu_j-\\log\\Gamma(\\frac{1}{\\phi_j})+(\\frac{1}{\\phi_j}-1)\\log y_i-\\frac{y_i}{\\phi_j\\mu_j}\\nonumber\\\\\n&\\hspace{12em}-\\log\\int_0^{\\infty}f_b(u;\\mu_j,\\phi_j)W(u)du\\Bigg)\\Bigg]\\nonumber\\\\\n&\\hspace{8em}+z_{i(J+1)}^{*(l)}\\Bigg(\\log\\pi_{J+1}^{*}+\\log\\gamma+\\gamma\\log\\theta-(\\gamma+1)\\log(y_i+\\theta)\\nonumber\\\\\n&\\hspace{13em}-\\log\\int_0^{\\infty}f_t(u;\\theta,\\gamma)W(u)du\\Bigg)\\Bigg\\},\n\\end{align}\nwhere\n\\begin{equation}\nz^{*(l)}_{ij}=P(z^{*}_{ij}=1|\\bm{y},\\bm{\\Phi}^{(l-1)})=\n\\begin{cases}\n\\dfrac{\\pi_j^{*(l-1)}f_b(y_i;\\mu_j^{(l-1)},\\phi_j^{(l-1)})W(y_i)}{\\int_0^{\\infty}f_b(u;\\mu_j^{(l-1)},\\phi_j^{(l-1)})W(u)du\\times h(y_i;\\bm{\\Phi}^{(l-1)})},\\quad j=1,\\ldots,J\\\\\n\\dfrac{\\pi^{*(l-1)}_{J+1}f_t(y_i;\\theta,\\gamma^{(l-1)})W(y_i)}{\\int_0^{\\infty}f_t(u;\\theta,\\gamma^{(l-1)})W(u)du\\times h(y_i;\\bm{\\Phi}^{(l-1)})},\\quad j=J+1.\n\\end{cases}\n\\end{equation}\n\n\\subsection{M-step} \\label{supp:sec:em_m2}\nMaximizing $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ with respect to $\\bm{\\Phi}$ yields the following parameter updates:\n\\begin{equation}\n\\pi_j^{*(l)}=\\frac{\\sum_{i=1}^{n}W(y_i)z_{ij}^{*(l)}}{\\sum_{j'=1}^{J+1}\\sum_{i=1}^{n}W(y_i)z_{ij'}^{*(l)}},\\quad j=1,\\ldots,J+1,\n\\end{equation}\nand the other parameters $(\\bm{\\mu},\\bm{\\phi},\\theta,\\gamma)$ are sequentially updated by numerically maximizing $Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})$ w.r.t. each of the parameters.\n\n\\section{Proof of Proposition 3} \\label{supp:sec:ascend}\nWrite $\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})=\\sum_{i=1}^{n}W(y_i)\\log p(y_i|\\bm{\\Phi})$ and $\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi}^{(l)};\\mathcal{D}^{\\text{com}})=\\sum_{i=1}^{n}W(y_i)\\left[\\log p(y_i|\\bm{\\Phi}) +\\log p(\\mathcal{D}^{\\text{mis}}_i|\\bm{\\Phi},y_i)\\right]$ for some probability density $p$ and missing data from sample $i$ given by $\\mathcal{D}^{\\text{mis}}_i$. Then, we have\n\\begin{align}\n\\mathcal{L}^{*}_n(\\bm{\\Phi};\\bm{y})\n&=\\tilde{\\mathcal{L}}^{*}_n(\\bm{\\Phi}^{(l)};\\mathcal{D}^{\\text{com}})-\\sum_{i=1}^{n}W(y_i)\\log p(\\mathcal{D}^{\\text{mis}}_i|\\bm{\\Phi},y_i)\\nonumber\\\\\n&=Q^{*}(\\bm{\\Phi}|\\bm{\\Phi}^{(l-1)})-\\sum_{i=1}^{n}W(y_i)\\int p(\\bm{v}_i|\\bm{\\Phi}^{(l-1)},y_i)\\log p(\\bm{v}_i|\\bm{\\Phi},y_i)d\\bm{v}_i,\n\\end{align}\nwhere the second equality results from expectation of both sides on the missing data under parameters $\\bm{\\Phi}^{(l-1)}$. Then, we have\n\\begin{align}\n\\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l)};\\bm{y})-\\mathcal{L}^{*}_n(\\bm{\\Phi}^{(l-1)};\\bm{y})\n&=Q^{*}(\\bm{\\Phi}^{(l)}|\\bm{\\Phi}^{(l-1)})-Q^{*}(\\bm{\\Phi}^{(l-1)}|\\bm{\\Phi}^{(l-1)})\\nonumber\\\\\n&\\quad+\\sum_{i=1}^{n}W(y_i)\\int p(\\bm{v}_i|\\bm{\\Phi}^{(l-1)},y_i)\\log\\frac{p(\\bm{v}_i|\\bm{\\Phi}^{(l-1)},y_i)}{p(\\bm{v}_i|\\bm{\\Phi}^{(l)},y_i)}d\\bm{v}_i\\geq 0.\n\\end{align}\n\n\\section{Initialization of parameters} \\label{apx:em:init}\nAs briefly described in Section 5.3 of the paper, parameter initialization $\\bm{\\Phi}^{(0)}$ is done using the CMM approach by \\cite{gui2018fit}. This comes with the following steps:\n\\begin{enumerate}\n\\item Determine a threshold $\\tau$ which classifies observations $y_i$ into either body (when $y_i\\leq\\tau$) or tail (when $y_i>\\tau$) part of distribution. This can be done by plotting the log of empirical data survival function against $\\log y_i$, which is called the log-log plot. For regular varying distributions, the log-log plot is asymptotically linear. $\\tau$ is approximated by the point where the curve turns linear onwards.\n\\item Perform K-means clustering on $\\{y_i\\}_{i:y_i\\leq\\tau}$ with $J$ clusters, and obtain the clustering mean $\\{\\mu^{\\text{cluster}}_j\\}_{j=1,\\ldots,J}$, variance $\\{(\\sigma^{\\text{cluster}}_j)^2\\}_{j=1,\\ldots,J}$ and weights $\\{\\tilde{\\pi}_j^{\\text{cluster}}\\}_{j=1,\\ldots,J}$.\n\\item Set $\\mu_j^{(0)}=\\mu^{\\text{cluster}}_j$, $\\phi_j^{(0)}=({\\sigma^{\\text{cluster}}_j})^2\/{\\mu^{\\text{cluster}}_j}^2$.\n\\item Obtain $\\theta^{(0)}$ and $\\gamma^{(0)}$ by matching the first two moments of observations belonging to the tail component (i.e. $\\{y_i\\}_{i:y_i>\\tau}$).\n\\item Set $\\pi_{J+1}^{(0)}$ as the proportion of observations satisfying $y_i>\\tau$.\n\\item Set the remaining weight parameters as $\\pi_{j}^{(0)}=\\tilde{\\pi}_j^{\\text{cluster}}(1-\\pi_{J+1}^{(0)})$.\n\\end{enumerate}\n\n\\section{Truncated log-likelihood function} \\label{sec:supp:tll}\nThis section includes more details for Remark 6 in the paper. Denote $g(y)$ as the true distribution generating the observations and $\\tilde{h}(y;\\bm{\\Phi})=\\frac{h(y;\\bm{\\Phi})W(y)}{\\int_0^{\\infty}h(u;\\bm{\\Phi})W(u)du}$ as the truncated distribution. The expected weighted log-likelihood can be alternatively written as\n\\begin{align}\nn\\times\\tilde{E}[\\mathcal{L}^{*}(\\bm{\\Phi};\\bm{Y})]\n&=n\\int_{0}^{\\infty}W(u)\\log \\tilde{h}(u;\\bm{\\Phi})\\times g(u)du\\nonumber\\\\\n&=n\\int_{0}^{\\infty}g(u)W(u)du\\times\\int_{0}^{\\infty}\\log \\tilde{h}(u;\\bm{\\Phi})\\times\\frac{g(u)W(u)}{\\int_{0}^{\\infty}g(t)W(t)dt}du\\nonumber\\\\\n&=n\\int_{0}^{\\infty}g(u)W(u)du\\times\\tilde{E}^*[\\log \\tilde{h}(u;\\bm{\\Phi})],\n\\end{align}\nwhere the expectation $\\tilde{E}^*$ is taken on $Y$ under the random truncated distribution $\\frac{g(u)W(u)}{\\int_{0}^{\\infty}g(t)W(t)dt}$. Next, denote a random set $S_n=\\{i:V_i(y_i)=1\\}$, such that $\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})$ can be written as\n\\begin{equation}\n\\mathcal{L}^{**}_n(\\bm{\\Phi};\\bm{y})=\\sum_{i\\in S_n}\\log \\tilde{h}(u;\\bm{\\Phi}),\n\\end{equation}\nwith effective number of terms $\\|S_n\\|\\approx n\\int_{0}^{\\infty}g(u)W(u)du\\approx \\sum_{i=1}^{n}W(y_i)$ in probability as $n\\rightarrow\\infty$. Comparing the above two equations, they simply correspond to standard MLE with bias term of $P$.\n\n\\section{Preliminary analysis of the motivating Greek dataset} \\label{apx:prelim_data}\nModelling the property damage claim size distribution is very challenging. Observing from Figures \\ref{fig:density} and \\ref{fig:loglogplot} which are also presented by \\cite{fung2021mixture}, the claim size distribution is not only heavy-tailed but also multi-modal. The key complexity of the empirical distribution is that there are many small distributional nodes for smaller claims, as evidenced by the right panel of Figure \\ref{fig:density}. On the other hand, it is undesirable to model all these nodes using excessive number of mixture components as (i) precise predictions of small claims are of less relevance of insurance pricing and risk management; (ii) this impedes the model interpretability. Further, the heavy-tailedness of empirical distribution is evidenced by asymptotic linearity of both log-log plot and mean excess plot in Figure \\ref{fig:loglogplot}. The asymptotic slope of log-log plot suggests that the estimated tail index is $\\gamma\\approx 1.3$ while the Lomax tail index obtained by \\cite{fung2021mixture} is about $\\gamma=1.38$, under a subjective choice of splicing threshold. Note however that these only provide a very rough guidance on the true tail index.\n\nNote that distributional multimodality and contamination are indeed prevalent not only to the aforementioned Greek dataset, but also to many publicly available insurance data sets. Notable examples include the French business interruption losses (\\textbf{frebiloss}), French motor third party liability claims (\\textbf{fremotor2sev9907} and \\textbf{freMPL8}) and Norwegian car claims (\\textbf{norauto}) which can all be retrived from the \\textbf{R} package \\textbf{CASdatasets}. This suggests that the modelling challenges emphasized in this paper is not only valid for the Greek data set we are analyzing, but is also applicable in many insurance claim severity data sets.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/density_original.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/density_log.pdf}\n\\end{subfigure}\n\\end{center}\n\\caption{Empirical density of claim amounts (left panel) and log claim amounts (right panel); the orange vertical lines represent amounts of 10,000, 20,000, 50,000 and 100,000 respectively.}\n\\label{fig:density}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/loglogplot.jpg}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{figure\/me_plot.jpg}\n\\end{subfigure}\n\\end{center}\n\\caption{Left panel: log-log plot of the claim amounts; right panel: mean excess plot.}\n\\label{fig:loglogplot}\n\\end{figure}\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFinding elements of high multiplicative order in a finite field is an interesting problem in computational number theory and has applications in cryptography (for instance: Discrete Logarithm Problem). A general method to find high order elements was given in \\cite{gao1999elements}, later improved in \\cite{conflitti2001} and \\cite{popovych2014gao}. Another general result in this area is an algorithmic technique for finding primitive elements which is devised in \\cite{huang2015primitive}.\nSuch technique is efficient in finite fields of small characteristic. \nOther strategies which allow to construct elements of high order usually address specific sequences of finite fields. In this regard, methods involving Gauss periods were first proposed in the results summarized in \\cite{vonzurgathen_shparlinski2001gauss}. After that, an extensive literature followed with works such as  \\cite{ahmadi_etal2010gauss}, \\cite{popovych2012}, \\cite{popovych2013}, \\cite{chang2013gauss} and \\cite{popovych2014sharpening}. Recently, Artin-Schreier extensions were also effectively used in \\cite{popovych2015artin} and \\cite{brochero2016artin}. \nAnother interesting approach is to look for high order elements which arise as coordinates of points on an algebraic curve defined over a finite field (see for example \\cite{voloch2007curves}, \\cite{voloch2010elliptic} and \\cite{chang2013curves}). One way which has been explored for generating elements of this type is through the iterative use of polynomial equations of type $f(x_{n - 1},x_n) = 0$, defining suitable towers of fields, which we address as \\textit{recursive towers} in this work. Examples of this can be found in \\cite{burkhart2009finite}, \\cite{voloch2010elliptic}, \\cite{popovych2015wiedmann} and \\cite{popovych2018conway}. \n\nIn \\cite{burkhart2009finite}, a recursive tower defined by $f(x_{n - 1},x_n)$ is used to produce elements $\\delta_n$ with high multiplicative order in $\\mathrm{GF}(q,2^n)$, for $q$ odd, and in $\\mathrm{GF}(q,3^n)$, for $q \\neq 3$. \nThe choice of the polynomial $f$ for the recursive process to generate high order elements in finite field extensions, was limited to the equations of the modular curve towers in \\cite{elkies2001explicit}. \n\n\nIn this work, we attempt to generalize the choice of the polynomials.\nWe illustrate in detail several interesting towers of fields defined by $x_n^2 + x_n = v(x_{n - 1})$, where $v(x) \\in \\mathrm{GF}(q,1)[x]$, for $q$ odd, or $x_n^3 + x_n = v(x_{n - 1})$, for $v(x) \\in \\mathrm{GF}(2,1)[x]$. These towers\ngenerate elements of high orders in $\\mathrm{GF}(q,2^n)$ and in $\\mathrm{GF}(2,2 \\cdot 3^n)$, for $n \\geq 1$. \nWe also give a recipe for finding other towers of the same form which have similar properties. The simple algebraic conditions given in Sections \\ref{sec:odd} and \\ref{sec:even}, \nwhich differ \npartially from the conditions required in \\cite{burkhart2009finite} (Remark \\ref{BurkhartConditionsComparison} below), seem to play an important role for this purpose. In fact, in many \nof the cases we studied, these conditions are \nuseful to prove the existence of high order elements\n$x_n$, in the field extension. \n\nThroughout this paper, $\\delta_n$ in $\\mathrm{GF}(q,2^n)$ is the discriminant of the polynomial $f(x_{n},y)$ in $\\mathrm{GF}(q,2^n)[y]$. In Corollary \\ref{bound-odd}, we prove that the multiplicative orders of $x_n$ and $\\delta_n$ grow very fast if $x_{n - j}^2$ and $\\delta_{n-j}^2$ do not belong to $\\mathrm{GF}(q,2^{n - j - 1})$, for all $j < n-1$. Similar results hold also in even characteristic, see Corollary \\ref{bound-even}. Notably, despite the bounds obtained are similar, the even characteristic case turns out to be completely new\nwith respect to \\cite{burkhart2009finite}. \nIn particular no additional conditions on the discriminant are required, and the details of the proof are worked out in a different manner. Furthermore the numerical performance of some of our examples are better than \\cite{burkhart2009finite}, in the iterations we were able to compute. As already mentioned above, the polynomials used in \\cite{burkhart2009finite} are the models of certain modular curves given in \\cite{elkies2001explicit}. Despite this fact, a possible relation of the construction of high order elements with the arithmetic properties of such curves does not seem to play a role in the proof of the lower bounds. \nInstead, in one case, we do make use of some arithmetic properties of the algebraic curve considered\nby us (Lemma \\ref{lem:nonsqnear5}).\n\nA comparative study with other relevant literature has also been carried-out.\nFor example, a specific\nconstruction of high order elements in the same type of fields of odd characteristic $q$ can be found in \\cite{cohen1992explicit}, and some variations on it are in \\cite{meyn1995explicit} and \\cite{chapman1997completely}. Comparing the numerical performance of their construction with our variety of examples, we observe that the results are similar for $q \\equiv 1 \\pmod  4$, while for $q \\equiv 3 \\pmod  4$ our construction performs better (see Section \\ref{sec:numres} for examples with $q=3,11$).\n\nIn Section \\ref{sec:notation}, we introduce the notation that we use in the paper. In Sections \\ref{sec:odd} and \\ref{sec:even}, we give the main results\nwhich allow us to obtain the lower bounds on the order of $x_n$ and $\\delta_n$. \nSection \\ref{sec:odd} deals with odd characteristic and Section \\ref{sec:even} deals with even characteristic.\nThe lists of towers satisfying the properties given in Sections \\ref{sec:odd} and \\ref{sec:even} are provided in Sections \\ref{sec:list-odd} and \\ref{sec:list-even}, respectively. Finally, in Section \\ref{sec:numres}, we list numerical results obtained using MAGMA \\cite{MAGMA}, about the seven towers listed in Sections \\ref{sec:list-odd} and \\ref{sec:list-even}.\n\n\n\\section{Background and notation}\\label{sec:notation}\n\nLet $q$ be an odd prime. By \\emph{tower of fields}, or simply a \\emph{tower}, we mean a sequence of field extensions\n\t$$K_1\\subset K_2\\subset \\ldots \\subset K_n\\subset \\ldots.$$\nWe are interested in infinite towers, namely towers such that the degree $[K_n : K_1]$ grows to infinity. All the towers considered in this paper are actually finite, normal and separable, i.e., each extension $K_n\/K_{n - 1}$ is finite, normal and separable, for every $n > 1$. For each positive integer $n$, let $K_n = \\mathrm{GF}(q,1)(x_n)$, where the element $x_n \\in \\mathrm{GF}(q,2^n)$ is given by a recursive formula $f(x_{n - 1},x_n) = 0$, for a polynomial $f(x,y) \\in \\mathrm{GF}(q,1)[x,y]$. In this case, we say that the tower $K_1 \\subset K_2 \\subset \\ldots \\subset K_n \\subset \\ldots$ is defined by $f(x_{n - 1},x_n)$ and we address this kind of towers as \\emph{recursive towers}. We focus on towers defined by $f(x_{n - 1},x_n) = x_n^2 + x_n - v(x_{n - 1})$, for $n \\ge 2$, with $x_1 \\in \\mathrm{GF}(q,2)$, and where $v(x)$ is a polynomial in $\\mathrm{GF}(q,1)[x]$. We denote by $\\delta_n$ the discriminant $\\delta_n = 1 + 4v(x_n)$, for $n \\ge 1$. We point out that both elements $x_n$ and $\\delta_n$ belong to $\\mathrm{GF}(q,2^n)$, but they could also lie in a smaller extension $\\mathrm{GF}(q,2^k)$ for some $k < n$. Given the tower defined by $f(x_{n-1},x_n)$, we denote by $g(x,y) \\in \\mathrm{GF}(q,1)[x,y]$ a polynomial giving the relation between two consecutive discriminants $\\delta_{n-1}$ and $\\delta_{n}$, namely $g(\\delta_{n-1},\\delta_{n})=0$. In the case of even characteristic (Sections \\ref{sec:even} and \\ref{sec:list-even}), we deal with towers defined by $f(x_{n-1},x_n) = x_n^3 + x_n + v(x_{n - 1})$, with $x_n \\in \\mathrm{GF}(2,2 \\cdot 3^n)$, for $n \\ge 1$, and $v(x)$ being a polynomial in $\\mathrm{GF}(2,1)[x]$.\n\nGiven two positive integers $j$ and $n$, such that $j < n$, we denote the norm of the field extension $\\mathrm{GF}(q,2^n)\/ \\mathrm{GF}(q,2^{n - j})$ by, $\\mathrm{N}_{n,j} \\colon \\mathrm{GF}(q,2^n) \\to \\mathrm{GF}(q,2^{n - j})$. The norm in the odd case is $\\mathrm{N}_{n,j}(x) = x^{\\prod_{i = 1}^j (q^{2^{n - i}}+1)}$. In order to apply the same techniques to even characteristic, we also denote by $\\mathrm{N}_{n,j}\\colon\\mathrm{GF}(2,2\\cdot 3^n)\\to \\mathrm{GF}(2,2 \\cdot 3^{n-j})$ the norm of the extension $\\mathrm{GF}(2,2 \\cdot 3^n)\/ \\mathrm{GF}(2,2 \\cdot 3^{n - j})$, namely $\\mathrm{N}_{n,j}(x) = x^{\\prod_{i = 1}^j (4^{2 \\cdot 3^{n-i}} + 4^{3^{n - i}} + 1)}$. For every characteristic, we use the conventions $\\mathrm{N}(x):=\\mathrm{N}_{n,1}(x)$ and $\\mathrm{N}_{n,0}(x)=x$.\n\n\nWe use the following lemma for estimating the order of the elements in finite fields. \n\\begin{lem}\\label{lem:estimate}\nLet $\\ell$ be a prime and let\n$a$, $b$ and $c$ be positive integers such that $b<c$. Assume $a\\equiv 1 \\bmod \\ell$.\nLet $p$ be a prime dividing $\\frac{1}{\\ell}\\sum_{j=1}^{\\ell}a^{\\ell^b(\\ell-j)}$.\nThen $p>\\ell^{b+1}$ and\n$\\gcd\\left(\\sum_{j=1}^{\\ell}a^{\\ell^b(\\ell-j)},\\sum_{j=1}^{\\ell}a^{\\ell^c(\\ell-j)}\\right)=\\ell$. In particular $\\frac{1}{\\ell}\\sum_{j=1}^{\\ell}a^{\\ell^b(\\ell-j)}$ and\n$\\frac{1}{\\ell}\\sum_{j=1}^{\\ell}a^{\\ell^c(\\ell-j)}$ are coprime.\n\\end{lem}\n\\begin{proof}\nSee \\cite[Lemmas 1 and 2]{burkhart2009finite}.\n\\end{proof}\nIn order to compute the Galois group of a cubic polynomial in characteristic 2, the following result is useful.\n\\begin{lem}\\label{lem:quadratic_resolvent}\nLet $f(x)=x^3+ax^2+bx+c$ be a separable\nirreducible polynomial\nover a field $K$.\nThe Galois group of the extension given by the roots of $f$ is the alternating group $A_3$ if its quadratic resolvent\n$R(x)=x^2+(ab-3c)x+a^3c+b^3+9c^2-6abc$\nis reducible over $K$ and it is the symmetric group $S_3$ otherwise.\n\\end{lem}\n\\begin{proof}\nSee \\cite[Section 1, pag.53]{kap1972fields}.\n\\end{proof}\n\n\nIn order to prove that a cubic polynomial is irreducible, we also need the following results.\n\\begin{lem}\\label{lem:cubicroots1}\nIf $u\\in \\mathrm{GF}(2,2\\cdot 3^{n})$\nand $c:=u+u^{-1}\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$, then $u\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$.\n\\end{lem}\n\\begin{proof}\nIf $u\\notin \\mathrm{GF}(2,2\\cdot 3^{n-1})$, then $x^2+cx+1$ is the minimum polynomial of\n$u$ over $\\mathrm{GF}(2,2\\cdot 3^{n-1})$. So $u\\in \\mathrm{GF}(2,2\\cdot 3^{n})\\cap\\mathrm{GF}(2,4\\cdot 3^{n-1})=\\mathrm{GF}(2,2\\cdot 3^{n-1})$ and we get a contradiction.\n\\end{proof}\n\\begin{lem}\\label{lem:cubicroots2}\nLet $u^3\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$ be a root of the quadratic polynomial $x^2+tx+1$, with $t\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$. Then $y:=u+u^{-1}\\in \\mathrm{GF}(2,2\\cdot 3^{n})$ is a root of the cubic polynomial $x^3+x+t$, and furthermore $y\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$ if and only if $u\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$.\n\\end{lem}\n\\begin{proof}\nThis is Cardano's formula for solving cubic equations in even characteristic. The second statement follows by Lemma \\ref{lem:cubicroots1} taking $y=c=u+u^{-1}$.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Towers in odd characteristic}\\label{sec:odd}\n\nIn order to find good towers we restrict our search to polynomials $f(x,y)=y^2+y-v(x)$, with $v(x)\\in\\mathrm{GF}(q,1)[x]$ being a non-zero polynomial, which satisfy Condition \\textbf{(1)} below and at least one of the last two conditions:\n\\begin{description}\n\t\\item[(1)] $\\frac{f(x_{n - 1},0)}{x_{n - 1}}$ is a square in $ \\mathrm{GF}(q,2^{n-1})$ for $n \\geq 2$;\n\t\\item[(2)] $\\frac{g(\\delta_{n - 1},0)}{x_{n - 1}}$ is a square in $\\mathrm{GF}(q,2^{n-1})$ for $n \\geq 2$;\n\t\\item[(2')] $\\frac{g(\\delta_{n - 1},0)}{\\delta_{n - 1}}$ is a square in $\\mathrm{GF}(q,2^{n-1})$ for $n \\geq 2$.\n\\end{description}\n\\begin{rem}\\label{BurkhartConditionsComparison}\nCondition \\textbf{(2')} above is satisfied by other towers of fields in the literature, see for example \\cite[Section 4, formula (5)]{burkhart2009finite}. We don't know whether the corresponding tower (see \\cite[Section 2, equation (2)]{burkhart2009finite}), which does not satisfies Condition \\textbf{(1)} above, satisfies a\nsuitable analog of this condition which ensure\nthat Proposition \\ref{general-odd} below holds.\n\\end{rem}\n\\begin{rem} These conditions are not sufficient for obtaining high order elements from each tower, but, for our particular choices of $f$, they are sufficient to construct a recursive tower defined by $f(x_{n - 1},x_{n})$ as Proposition\n\\ref{general-odd} below shows.\n\\end{rem}\nThe following key proposition ensures that all the polynomials $f(x_{n - 1}, x_{n})$ listed in Section \\ref{sec:list-odd} define infinite towers of fields. In particular it shows that $[K_n : K_{n - 1}] = 2$, for all $n > 1$. The argument of the proof is the corresponding analogue of\n\\cite[Proposition~1]{burkhart2009finite}\nbut it could be applied to many different towers.\n\\begin{prop}\\label{general-odd}\nLet $v(x) \\in \\mathrm{GF}(q,1)[x]$ be a polynomial and assume that $f(x_{n - 1},x_n) = x_n^2 + x_n - v(x_{n - 1})$ satisfies Conditions \\textbf{(1)} \\textit{and} \\textbf{(2)}, or Conditions \\textbf{(1)} \\textit{and} \\textbf{(2')}. If $x_{n - 1}$ and $\\delta_{n - 1}$ are not squares in the multiplicative group $\\mathrm{GF}(q,2^{n - 1})^*$ for a suitable $n \\ge 2$, then $x_j$ and $\\delta_j$ are not squares in the multiplicative group $\\mathrm{GF}(q,2^j)^*$, for $j \\ge n$.\n\\end{prop}\n\\begin{proof}\nThe element $x_{n}$ is not in $\\mathrm{GF}(q,2^{n - 1})$ because $\\delta_{n - 1}$ is not a square in $\\mathrm{GF}(q,2^{n - 1})$, so $f(x_{n - 1},y)$ is the minimal polynomial of $x_n$. We need to ensure that $x_{n}^{(q^{2^n} - 1)\/2} = -1$. As in \\cite[Proposition 1]{burkhart2009finite}, we obtain:  \n\\begin{align*}\nx_{n}^{(q^{2^n} - 1)\/2} &= (x_{n}^{q^{2^{n - 1}} + 1})^{(q^{2^{n - 1}} - 1)\/2} = \\mathrm{N}(x_n)^{(q^{2^{n - 1}} - 1)\/2} = \\\\ &= f(x_{n - 1},0)^{(q^{2^{n - 1}} - 1)\/2}= -1,\n\\end{align*}\nwhere $\\mathrm{N}(x_n) = x_n^{q^{2^{n - 1}} + 1} = f(x_{n - 1},0)$ is the norm of $x_n$ over $\\mathrm{GF}(q,2^{n - 1})$ and we use Condition \\textbf{(1)} in last equality\nto show that $f(x_{n - 1},0)$ is not a square in $\\mathrm{GF}(q,2^{n - 1})$ for $n > 1$.\n\nConsider the discriminant $\\delta_n$. Again $g(\\delta_{n - 1},y)$ is the minimal polynomial of $\\delta_n = 1 + 4v(x_n)$. Since, in $\\mathrm{GF}(q,2^{n})$, we know that $\\frac{f(x_n,0)}{x_n}$ is a square by Condition \\textbf{(1)}, $-1$ is a square and $x_n$ is not a square as above, then $v(x_n) = -f(x_n,0)$ is not a square\nin $\\mathrm{GF}(q,2^{n })$. \nHence, $\\delta_n \\notin \\mathrm{GF}(q,2^{n - 1})$. The same computation as above yields:\n\\begin{align*}\n\\delta_{n}^{(q^{2^n} - 1)\/2} &= (\\delta_{n}^{q^{2^{n - 1}} + 1})^{(q^{2^{n-1}}-1)\/2} = \\mathrm{N}(\\delta_n)^{(q^{2^{n - 1}} - 1)\/2}= \\\\ \n&= g(\\delta_{n - 1},0)^{(q^{2^{n - 1}} -1)\/2} = -1,\n\\end{align*}\nwhere we use Condition \\textbf{(2)}, respectively \\textbf{(2')}, in last equality to show that $g(\\delta_n,0)$ is not a square in $\\mathrm{GF}(q,2^{n - 1})$, because $x_{n - 1}$, (respectively $\\delta_{n - 1}$), is a non-square by hypothesis. It follows that $x_{n}$ and $\\delta_{n}$ are non-squares in $\\mathrm{GF}(q,2^n)$. Repeating the same argument, we find that $x_{j}$ and $\\delta_{j}$ are not squares in $\\mathrm{GF}(q,2^{j})$, for all $j > n$, which completes the proof.\n\\end{proof}\n\nThe importance of this proposition is evident if we consider Corollary~\\ref{bound-odd} below, which is an analogue of \\cite[Proposition 2]{burkhart2009finite}. We first state the following property of the norm  that is used in the proof of the corollary.\n\\begin{lem}\\label{lem:factor_odd} Let $n\\geq 2$ and $j<n$ be positive integers, then\n$$\\frac{\\mathrm{N}_{n,j}(x_n)}{x_{n - j}} =\\prod_{k=1}^{j}\\mathrm{N}_{n-k,j-k}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right).$$\nMoreover $\\frac{\\mathrm{N}_{n,j}(x_n)}{x_{n - j}}$ is a square in $\\mathrm{GF}(q,2^{n-j})$.\n\\end{lem}\n\\begin{proof} The case $j=1$ is trivial. By induction on $j$, let $j\\geq 2$ and assume the result holds for $j-1$, then \n\\allowdisplaybreaks\n\\begin{align*}\n&\\frac{\\mathrm{N}_{n,j}(x_n)}{x_{n-j}}= \\frac{x_n^{(q^{2^{n-1}}+1)\\prod_{i=2}^{j}(q^{2^{n-i}}+1)}}{x_{n-j}}= \\\\\n&=\\left(\\frac{x_n^{q^{2^{n-1}}+1}}{x_{n-1}}\\right)^{\\prod_{i=2}^{j}(q^{2^{n-i}}+1)}\\frac{x_{n-1}^{\\prod_{i=2}^{j} (q^{2^{n-i}}+1)}}{x_{n-j}}= \\\\\n&=\\left(\\frac{\\mathrm{N}_{n,1}(x_n)}{x_{n-1}}\\right)^{\\prod_{i=2}^{j}(q^{2^{n-i}}+1)}\\frac{\\mathrm{N}_{n-1,j-1}(x_{n-1})}{x_{n-j}}= \\\\\n&=\\left(\\frac{\\mathrm{N}_{n,1}(x_n)}{x_{n-1}}\\right)^{\\prod_{i=1}^{j-1}(q^{2^{n-1-i}}+1)}\n\\prod_{k=1}^{j-1}\\mathrm{N}_{n-k-1,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k,1}(x_{n-k})}{x_{n-k-1}}\\right)= \\\\\n&=\\mathrm{N}_{n-1,j-1}\\left(\\frac{\\mathrm{N}_{n,1}(x_{n})}{x_{n-1}}\\right)\n\\prod_{k=2}^{j}\\mathrm{N}_{n-k,j-k}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n-k}}\\right).\n\\end{align*}\n\\allowdisplaybreaks[0]\nThe remaining part of the proof follows by Condition \\textbf{(1)}.\n\\end{proof}\n\\begin{cor}\\label{bound-odd}\nLet $v(x)$ be a polynomial in $\\mathrm{GF}(q,1)[x]$ and assume that \\\\  $f(x_{n - 1}, x_n) = x_n^2 + x_n - v(x_{n - 1})$ satisfies Conditions \\textbf{(1)} and \\textbf{(2)}, or Conditions \\textbf{(1)} and \\textbf{(2')}, and that $x_1$ and $\\delta_1$ are not squares in $\\mathrm{GF}(q,2)$. Then $x_n^2 \\notin \\mathrm{GF}(q,2^{n-1})$ and the order of $x_n$ is greater than\n\t$$2^{\\frac{1}{2}(n^2 + 3n) + \\mathrm{ord}_2(q - 1) - 2},$$\nfor all $n> 1$. The same lower bound also holds for the order of $\\delta_n$ if $\\delta_n^2 \\notin \\mathrm{GF}(q,2^{n - 1})$  for all $n > 1$.\n\\end{cor}\n\\begin{proof}\nWe know that $x_n\\not\\in \\mathrm{GF}(q,2^{n-1})$ by Proposition \\ref{general-odd}, therefore $x_n^2=-x_n+v(x_{n-1})\\not\\in \\mathrm{GF}(q,2^{n-1})$ for all $n>1$. We show that the order of $x_n$ has a common factor with the odd number $\\frac{q^{2^{n - j}} + 1}{2}$ proving that $x_n^{\\frac{2(q^{2^n} - 1)}{q^{2^{n - j}} + 1}} \\ne 1$, for $j = 1,2,\\ldots,n-1$. \nFor $j=1$, we have\n\\[\nx_n^{\\frac{2(q^{2^n} - 1)}{q^{2^{n - 1}} + 1}}=x_n^{2(q^{2^{n-1}}-1)} \\ne 1,\n\\]\nsince $x_n^2\\not\\in \\mathrm{GF}(q,2^{n - 1})$, as we have just seen. For $j\\geq 2$, we get\n\\begin{align*}\nx_n^{\\frac{2(q^{2^n} - 1)}{q^{2^{n - j}} + 1}} &= \\left(x_n^{\\prod_{k = 1}^{j - 1}( q^{2^{n - k}}+1)}\\right)^{2(q^{2^{n - j}} - 1)}= \\mathrm{N}_{n,j - 1}(x_n)^{2(q^{2^{n - j}} - 1)}\n\\end{align*}\nand the last member above is 1 only if $\\mathrm{N}_{n,j - 1}(x_n)^2\\in\\mathrm{GF}(q,2^{n-j})$. We show that this is not possible. Consider $\\mathrm{N}_{n,j }(x_n)=\\mathrm{N}_{n-j+1,1 }(\\mathrm{N}_{n,j-1 }(x_n))$. \nIf $\\mathrm{N}_{n,j - 1}(x_n)^2\\in\\mathrm{GF}(q,2^{n-j})$, then either $\\mathrm{N}_{n,j }(x_n)=\\mathrm{N}_{n,j - 1}(x_n)^2$ or $\\mathrm{N}_{n,j }(x_n)=\\mathrm{N}_{n,j - 1}(x_n)$. The latter equality is not possible since $\\mathrm{N}_{n,j-1 }(x_n)$ is not a square in \n$\\mathrm{GF}(q,2^{n-j+1})$ by Lemma \\ref{lem:factor_odd} but \n$\\mathrm{N}_{n,j }(x_n)\\in \\mathrm{GF}(q,2^{n-j})$ is a square in \n$\\mathrm{GF}(q,2^{n-j+1})$. The former equality, by Lemma \\ref{lem:factor_odd}, gives:\n\\allowdisplaybreaks\n\\begin{align*}\n1&=\\frac{x_{n - j} \\prod_{k=1}^{j}\\mathrm{N}_{n-k,j-k}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)}{x_{n - j+1}^2 \\prod_{k=1}^{j-1}\\left(\\mathrm{N}_{n-k,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\right)^2}=\\\\\n&=\\frac{x_{n - j}\\frac{\\mathrm{N}_{n-j+1,1}(x_{n-j+1})}{x_{n - j}} \\prod_{k=1}^{j-1}\\mathrm{N}_{n-k,j-k}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)}{x_{n - j+1}^2 \\prod_{k=1}^{j-1}\\left(\\mathrm{N}_{n-k,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\right)^2}=\\\\\n&=\\!\\frac{\\mathrm{N}_{n-j+1,1}(x_{n-j+1}) \\!\\prod_{k=1}^{j-1}\\!\\mathrm{N}_{n-j+1,1}\\!\\left(\\!\\mathrm{N}_{n-k,j-k-1}\\!\\left(\\!\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\!\\right)}{x_{n - j+1}^2 \\prod_{k=1}^{j-1}\\left(\\mathrm{N}_{n-k,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\right)^2}\\!=\\\\\n&=\\frac{(x_{n-j+1})^{q^{2^{n-j}}+1} \\prod_{k=1}^{j-1}\\left(\\mathrm{N}_{n-k,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\right)^{q^{2^{n-j}}+1}}{x_{n - j+1}^2 \\prod_{k=1}^{j-1}\\left(\\mathrm{N}_{n-k,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\right)^2}=\\\\\n& =x_{n - j+1}^{q^{2^{n-j}}-1} \\prod_{k=1}^{j-1}\\left(\\mathrm{N}_{n-k,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)\\right)^{q^{2^{n-j}}-1}.\n\\end{align*}\n\\allowdisplaybreaks[0]\nSince the last term is 1, then\n$$x_{n-j+1} \\prod_{k=1}^{j-1}\\mathrm{N}_{n-k,j-k-1} \\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right) \\in\\mathrm{GF}(q,2^{n-j}),$$\nbut this is impossible because $x_{n-j+1}$ is a non-square in $\\mathrm{GF}(q,2^{n-j+1})$, by Proposition \\ref{general-odd}, but \n$$\\mathrm{N}_{n-k,j-k-1}\\left(  \\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n - k}}\\right)=\\mathrm{N}_{n-k,j-k-1}\\left(  \\frac{f(x_{n-k},0)}{x_{n - k}}\\right)$$ \nis a square in $\\mathrm{GF}(q,2^{n-j+1})$, for each $k<j$, by Condition \\textbf{(1)} and by multiplicativity of the norm.\n\nThis odd common factor ensures, by Lemma \\ref{lem:estimate} with $a=q$, $b=n-j$ and $\\ell=2$, the existence of a lower bound on the order of $x_n$, namely $p_j>2^{n - j + 1}$, for every $j = 1,2,\\ldots,n-1$. Hence, the order is bounded below by $$2^{\\frac{n(n + 1)}{2} - 1}=\\prod_{j = 1}^{n - 1} 2^{n - j + 1} <\\prod_{j = 1}^{n - 1}p_j .$$\n\nThe remaining term $2^{n + \\mathrm{ord}_2(q - 1) - 1}$ follows as in \\cite[Proposition 2]{burkhart2009finite}. \nBy the repetition of the difference of squares formula, we get:\n$$\\mathrm{ord}_2 \\left(\\frac{q^{2^n} - 1}{2}\\right) = \\sum_{j = 0}^{n - 1} \\mathrm{ord}_2(q^{2^j} + 1) + \\mathrm{ord}_2(q - 1) - 1 = n + \\mathrm{ord}_2(q - 1) - 1,$$\nfor all $n\\geq 1$. It follows that $2^{n + \\mathrm{ord}_2(q - 1) - 1}$ divides the order of $x_n$ because $x_n^{\\frac{q^{2^n} - 1}{2}} = -1$ by Proposition \\ref{general-odd}. The proof for $\\delta_n$ is similar. \n\\end{proof}\n\n\n\n\n\n\\section{Towers in even characteristic}\\label{sec:even}\n\n\nThe even analogue of Conditions \\textbf{(1)} and \\textbf{(2)} in the odd case for polynomials $f(x,y)=y^3+y+v(x)$, with $v(x)\\in \\mathrm{GF}(2,1)[x]$, is:\n\\begin{description}\n\\item[(3)] There exists an integer $e\\geq 0$ such that $f(x_{n-1},0)=x_{n-1}^{2^e}$ for all $n\\geq 2$.\n\\end{description}\nThis means that we can restrict our study to polynomials in the form $f(x,y)=y^3+y+x^{2^e}$, with $e\\geq 0$, and\ndeduce similar results as in the previous section. In Section \\ref{sec:list-even}, we find some cases where the towers defined by  polynomials $f(x_{n - 1},x_n)$ are infinite and Galois. \nThis is achieved by finding a suitable initial element $x_1\\in \\mathrm{GF}(2,6)$.\nUnder these hypotheses we have an analogue of Proposition \\ref{general-odd}.\n\\begin{prop}\\label{general-even}\nConsider an infinite normal tower defined by $f(x_{n - 1},x_n) = x_n^3 + x_n + x_{n - 1}^{2^e}$ for a certain $e\\geq 0$, for all $n>1$.\nLet $p$ be a prime divisor of $|\\mathrm{GF}(2,2\\cdot 3^{n - 1})^*|$, for a suitable $n>1$, and assume that $x_{n - 1}$ is not a $p$-th power in the multiplicative group $\\mathrm{GF}(2,2\\cdot 3^{n - 1})^*$. Then $x_j$ is not a $p$-th power in the multiplicative group $\\mathrm{GF}(2,2\\cdot 3^{j})^*$, for $j \\geq n$. \n\\end{prop}\n\\begin{proof}\nBy assumption $f(x_{n-1},y)$ is irreducible, so $x_{n}\\not\\in\\mathrm{GF}(2,2 \\cdot 3^{n - 1})$ and $f(x_{n-1},y)$ is the minimum polynomial of $x_{n}$. We need to check that $x_{n}^{(4^{3^n}-1)\/p}\\not =1$. As in the proof of Proposition~\\ref{general-odd}, we obtain:  \n\\begin{align*}\nx_{n}^{(4^{3^n} - 1)\/p} &= (x_{n}^{4^{2 \\cdot 3^{n - 1}} + 4^{3^{n - 1}} + 1})^{(4^{3^{n - 1}} - 1)\/p}= \\\\\n&= \\mathrm{N}(x_n)^{(4^{3^{n - 1}} - 1)\/p} = f(x_{n - 1},0)^{(4^{3^{n - 1}} - 1)\/p},\n\\end{align*}\nwhere $\\mathrm{N}(x_n) = x_{n}^{4^{2 \\cdot 3^{n - 1}} + 4^{3^{n - 1}} + 1} = f(x_{n - 1},0)$ is the norm of $x_n$ over $\\mathrm{GF}(2,2\\cdot 3^{n-1})$. The last term is not equal to $1$ because $x_{n - 1}$ is not a $p$-th power in $\\mathrm{GF}(2,2 \\cdot 3^{n - 1})$, hence, by Condition \\textbf{(3)}, \n$f(x_{n - 1},0)$ is not a $p$-th power as well.\n\\end{proof}\nThe analogue of Lemma \\ref{lem:factor_odd} in even characteristic is the following: \n\\begin{lem}\\label{lem:factor_even} Let $e\\geq 0$, $n\\geq 2$ and $j<n$ be positive integers, then\n$$\\frac{\\mathrm{N}_{n,j}(x_n)}{x_{n - j}^{2^{ej}}} = \\prod_{k = 1}^{j} \\mathrm{N}_{n - k,j-k} \\left(\\frac{\\mathrm{N}_{n - k+1,1}(x_{n - k+1})}{x^{2^e}_{n - k }}\\right)^{2^{e(k-1)}}.$$\nIn particular, if the tower defined by $f(x_{n-1},x_n)$ satisfies Condition \\textbf{(3)} for a certain $e\\geq 0$, then \n$\\mathrm{N}_{n,j}(x_n)=x_{n - j}^{2^{ej}}.$\n\\end{lem}\n\\begin{proof} By induction on $j$. For $j=1$ the result is trivial. Let $j\\geq 2$ and assume the result holds for $j-1$, then:\n\\begin{align*}\n&\\frac{\\mathrm{N}_{n,j}(x_n)}{x^{2^{ej}}_{n-j}}\\!=\\! \\frac{x_n^{(4^{2\\cdot 3^{n-1}}+4^{3^{n-1}}+1)\\prod_{i=2}^{j}(4^{2\\cdot 3^{n-i}}+4^{3^{n-i}}+1)}}{x^{2^{ej}}_{n-j}}\\!= \\\\\n&=\\!\\left(\\!\\frac{x_n^{4^{2\\cdot 3^{n-1}}+4^{3^{n-1}}+1}}{x_{n-1}^{2^e}}\\!\\right)^{\\prod_{i=2}^{j}(4^{2\\cdot 3^{n-i}}+4^{3^{n-i}}+1)}\\!\\!\\!\\left(\\frac{x_{n-1}^{\\prod_{i=2}^{j} (4^{2\\cdot 3^{n-i}}+4^{3^{n-i}}+1)}}{x^{2^{e(j-1)}}_{n-j}}\\right)^{2^e}\\!\\!\\!= \\\\\n&=\\!\\left(\\frac{\\mathrm{N}_{n,1}(x_n)}{x_{n-1}^{2^e}}\\right)^{\\prod_{i=2}^{j}(4^{2\\cdot 3^{n-i}}+4^{3^{n-i}}+1)}\\left(\\frac{\\mathrm{N}_{n-1,j-1}(x_{n-1})}{x_{n-j}^{2^{e(j-1)}}}\\right)^{2^e}= \\\\\n&=\\!\\mathrm{N}_{n-1,j-1}\\left(\\frac{\\mathrm{N}_{n,1}(x_{n})}{x_{n-1}^{2^e}}\\right)\n\\prod_{k=1}^{j-1}\\mathrm{N}_{n-k-1,j-k-1}\\left(\\frac{\\mathrm{N}_{n-k,1}(x_{n-k})}{x_{n-k-1}^{2^e}}\\right)^{2^{ek}}= \\\\\n&=\\!\\mathrm{N}_{n-1,j-1}\\left(\\frac{\\mathrm{N}_{n,1}(x_{n})}{x_{n-1}^{2^e}}\\right)\\prod_{k=2}^{j}\\mathrm{N}_{n-k,j-k}\\left(\\frac{\\mathrm{N}_{n-k+1,1}(x_{n-k+1})}{x_{n-k}^{2^e}}\\right)^{2^{e(k-1)}}.\n\\end{align*}\nThe remaining part of the proof follows by Condition \\textbf{(3)}. \\end{proof}\n\\begin{cor}\\label{bound-even}\nConsider an infinite normal tower defined by\n$f(x_{n-1},x_n)=x^3_n+x_n+x_{n-1}^{2^e}$, \nfor a certain $e\\geq 0$, for all $n>1$. If $x_1$ is not a cube in $\\mathrm{GF}(2,6)$, then $x_n^3 \\notin \\mathrm{GF}(2,2 \\cdot 3^{n - 1})$ for all $n\\geq 2$ and the order of $x_n$ in the tower defined by $f(x_{n - 1},x_n)$ is greater than $$3^{\\frac{1}{2}(n^2 + 3n)-1}.$$\n\\end{cor}\n\\begin{proof}\nThe proof is similar to the proofs of Corollary \\ref{bound-odd} and \\cite[Proposition 4]{burkhart2009finite}. \nWe know that $x_n\\not\\in \\mathrm{GF}(2,2\\cdot 3^{n - 1})$ by\nProposition~\\ref{general-even}, therefore $x^3_n=x_n+v(x_{n-1})\\not \\in \\mathrm{GF}(2,2\\cdot 3^{n - 1})$.\nIt follows that $\\left(x^3_n\\right)^{2^e}$ does not belong to $\\mathrm{GF}(2,2\\cdot 3^{n - 1})$.\nIn order to show that the order of $x_n$ has a common factor with $\\textstyle \\frac{1}{3}(4^{2 \\cdot 3^{n - j}} + 4^{3^{n - j}} + 1)$, we show that $x_n^\\frac{3(4^{3^n} - 1)}{4^{2 \\cdot 3^{n - j}} + 4^{3^{n - j}} + 1} \\ne 1$, for $j = 1,2,\\ldots,n-1$. \nWe have:\n\\begin{align*}\nx_n^\\frac{3(4^{3^n} - 1)}{4^{2 \\cdot 3^{n - j}} + 4^{3^{n - j}} + 1} &= x_n^{\\frac{4^{3^n} - 1}{4^{3^{n - j}} - 1} \\cdot \\frac{3(4^{3^{n - j}} - 1)}{4^{2 \\cdot 3^{n - j}} + 4^{3^{n - j}} + 1}}= x_n^{\\frac{3(4^{3^n} - 1)(4^{3^{n - j}} - 1)}{4^{3^{n - j + 1}} - 1}} = \\\\\n&=x_n^{3(4^{3^{n - j}} - 1)\\prod_{i = 1}^{j - 1} (4^{2 \\cdot 3^{n - i}} + 4^{3^{n - i}} + 1)}=\n\\mathrm{N}_{n,j - 1}(x_n)^{3(4^{3^{n - j} - 1})}.\n\\end{align*}\nBy Lemma \\ref{lem:factor_even} \nwe have that $\\mathrm{N}_{n,j}(x_n)=x_{n-j}^{2^{ej}}$, for $j = 1,2,\\ldots,n-1$. But $\\left(x_{n - j + 1}^{2^{e(j-1)}}\\right)^3$ does not belong to $\\mathrm{GF}(2,2 \\cdot 3^{n - j})$ for all $j \\geq 1$. \nIt follows that $\\mathrm{N}_{n,j - 1}(x_n)^{3(4^{3^{n - j} - 1})}$ cannot be equal to $1$. This ensures, by Lemma \\ref{lem:estimate} with $a=4$, $b=n-j$ and $\\ell=3$, the existence of a lower bound on the order of $x_n$, namely $p_j>3^{n - j + 1}$, for every $j = 1,2,\\ldots,n-1$. Hence, we get a lower bound for the order of $x_n$, which is $3^{\\frac{n(n + 1)}{2}-1}=\\prod_{j = 1}^{n - 1} 3^{n - j + 1}<\\prod_{j = 1}^{n - 1}p_j$.\n\t\nThe remaining term $3^{n}$ follows by the computation of the power of 3 dividing the order of $x_n$. By the repetition of the difference of cubes formula, we have:\n$$\\mathrm{ord}_3 \\left(\\frac{4^{3^n} - 1}{3}\\right) = \\sum_{j = 0}^{n - 1} \\mathrm{ord}_3(4^{2 \\cdot 3^j} + 4^{3^j} + 1) + \\mathrm{ord}_3(4 - 1) - 1 = n,$$\nfor all $n \\geq 1$. This term divides the order of $x_n$, since $x_n^{\\frac{4^{3^n} - 1}{3}} \\neq 1$, by Proposition~\\ref{general-even}.\n\\end{proof}\n\n\n\n\n\n\\section{Examples of good towers in odd characteristic}\\label{sec:list-odd}\n\nIn this section we find high order elements in $\\mathrm{GF}(q,2^n)$, for odd $q$, using five good towers. In this section, we denote by $\\varepsilon$ the element $4^{-1}$ inside $\\mathrm{GF}(q,1)$. We consider the polynomials $f_i(x_{n - 1}, x_n) := x_n^2 + x_n - v_i(x_{n - 1})$, for $i \\in\\{1, 2,\\ldots, 5\\}$,\nwhere $v_i(x)$ is a polynomial chosen as follows:\n\\begin{enumerate}\n    \\item $v_1(x) := \\varepsilon x;$\n    \\item $v_2(x) := 4x{(x + 3\\varepsilon)}^2;$\n    \\item $v_3(x) := 2\\varepsilon x;$\n    \\item $v_4(x) := 8x{(2x + 3\\varepsilon)}^2;$\n    \\item $v_5(x) := 8x{(x + 3\\varepsilon)}^2.$\n\\end{enumerate}\n\\begin{rem}\\label{rem:conditions}\nCondition \\textbf{(1)} holds for all the previous polynomials and the relation between two consecutive discriminants is given respectively by:\n\\begin{align*}\ng_1(\\delta_{n - 1}, \\delta_n) &= \\delta_n^2 - \\delta_n- \\varepsilon \\delta_{n - 1} + \\varepsilon; \\\\\ng_2(\\delta_{n - 1}, \\delta_n) &= \\delta_n^2 - \\delta_n -4 \\delta_{n - 1}^3 + 6\\delta_{n - 1}^2 - 9\\varepsilon\\delta_{n - 1} + \\varepsilon; \\\\\ng_3(\\delta_{n - 1}, \\delta_n) &= \\delta_n^2 - \\delta_{n - 1};\n\\\\g_4(\\delta_{n - 1}, \\delta_n) &= \\delta_n^2 + 48\\delta_{n - 1}\\delta_n - 256\\delta_{n - 1}^3 + 288\\delta_{n-1}^2 - 81\\delta_{n - 1}; \\\\\ng_5(\\delta_{n-1}, \\delta_n) &= \\delta_n^2 - 16\\delta_{n - 1}^3 + 24\\delta_{n - 1}^2 - 9\\delta_{n - 1}.\n\\end{align*}\nThe first two towers satisfy Condition \\textbf{(2)}. \nIn fact\n\\begin{align*}\ng_1(\\delta_{n-1},0)&=-\\varepsilon (1+4x_{n-1})+\\varepsilon=-x_{n-1};\\\\\ng_2(\\delta_{n-1},0)&=x_{n-1}(x_{n-1}+3\\varepsilon)^2(x_{n-1}^3+6x_{n-1}^2+9\\varepsilon^2 x_{n-1}+3\\varepsilon^3)^2.\n\\end{align*}\nSimilarly the last three towers satisfy Condition \\textbf{(2')}. In fact,\n\\begin{align*}\ng_3(\\delta_n, 0) &= -\\delta_{n};\\\\\ng_4(\\delta_n, 0) &= -256\\delta_n(\\delta_n - 9 \\varepsilon^2)^2; \\\\\ng_5(\\delta_n, 0) &= -16\\delta_n(\\delta_n - 3\\varepsilon)^2.\n\\end{align*}\nHence, Proposition \\ref{general-odd} applies to $f_i(x_{n - 1}, x_n)$, for $i \\in\\{ 1, 2, \\ldots, 5\\}$ once we have some starting points. \n\\end{rem}\nThe next two lemmas ensures the existence of a non-square $x_1$ such that $\\delta_1$ is a non-square in $\\mathrm{GF}(q,2)$ as well. This would be the corresponding analogue of \\cite[Lemma 3]{burkhart2009finite}, but \nhere we also need that \\textit{both} $x_1$ and $\\delta_1$ must be non-squares. This requires more effort, especially for the last tower $f_5(x_{n-1},x_n)$ below, but, \nas a balance, this gives \na lower bound for the order of $x_n$, also. \n\nThe present proof relies mainly on elementary combinatorial arguments. \n\\begin{lem}\\label{lem:nonsqnear}\nLet $c\\in \\mathrm{GF}(q,1)$ be a non-zero element. There is at least a non-square $x_1 \\in \\mathrm{GF}(q,2)$ such that $x_1 + c$ is a non-square as well.\n\\end{lem}\n\\begin{proof}\nConsider the action $\\rho$ of $\\mathrm{GF}(q,1)$ on $\\mathrm{GF}(q,2)$ as an additive group, namely $\\rho_g(x) = x + g$, for $g \\in \\mathrm{GF}(q,1)$ and $x \\in \\mathrm{GF}(q,2)$. Then, $\\mathrm{GF}(q,2)$ is partitioned into $q$ orbits. There are exactly $\\textstyle\\frac{1}{2}(q^2 + 1)$ squares in $\\mathrm{GF}(q,2)$. Among these, there are all the elements of the orbit $\\mathrm{GF}(q,1)$. It follows that there are exactly $\\textstyle\\frac{1}{2}(q^2 - 2q + 1)$ square elements in $q - 1$ orbits. Hence, there is at least one orbit with at most $\\textstyle\\frac{1}{2}(q - 1)$ square elements and at least $\\textstyle\\frac{1}{2}(q + 1)$ non-square elements. We denote this orbit by $S$. It follows that there are at least two consecutive non-squares in $S$ under the repeated action of $\\rho_c$, namely $a$ and $\\rho_c(a) = a + c$. The lemma follows by choosing $x_1 = a$.\n\\end{proof}\n\\begin{exa}\\label{exa1}\nConsider, $q = 3$ and $c=1$. Denote by $z$ a generator of $\\mathrm{GF}(3,2)^*$ satisfying $z^2 = z + 1$. There are exactly $5$ squares in $\\mathrm{GF}(3,2)^*$, but $3$ of them are in the same orbit $\\mathrm{GF}(3,1)$. The remaining ones are $z^2 = z + 1$ and $z^6 = 2z + 2$. One can check that they belong to the orbits $S_1 = (z; \\quad z + 1 = z^2; \\quad z + 2 = z^7)$ and $S_2 = (2z = z^5; \\quad 2z + 1 = z^3; \\quad 2z + 2 = z^6)$. As $x_1$ we can choose the element $2z$ or $z + 2$. They are both roots of the polynomial $x^2 = 2x + 1$, so we use this polynomial for $q = 3$ in Table \\ref{tab:res1} in Section~\\ref{sec:numres}.\n\\end{exa}\n\nIn order to show the existence of a suitable initial element $x_1$ for the tower defined by $f_5(x_{n-1},x_n)$ we prove the following lemma. \n\\begin{lem}\\label{lem:nonsqnear5}Let $q$ be an odd prime and let $p(x)$ be a cubic polynomial in $\\mathrm{GF}(q,1)[x]$ without multiple roots, such that $p(0)\\not = 0$. Then:\n\\begin{enumerate}[i)]\\item\\label{nonsqnear5_C1} The curve $C_1: y^2 = p(x)$ has at most $q^2 + 2q$ affine $\\mathrm{GF}(q,2)$-rational points and the curve $C_2: y^2 = p(x^2)$ has at least $q^2 - 4q - 1$ affine $\\mathrm{GF}(q,2)$-rational points.\\item\\label{nonsqnear5_exists} If $q\\geq 11$, then there is at least a non-square $x_1\\in\\mathrm{GF}(q,2)$ such that $p(x_1)$ is a non-square in $\\mathrm{GF}(q,2)$ as well.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof} \\ref{nonsqnear5_C1}) We observe that $p(x^2)$ is square-free since $p(x)$ is square-free and $p(0)\\ne 0$ by hypothesis. The first statement follows by Weil bound $|N-(q^2 + 1)| \\leq 2gq$, for every smooth projective curve of genus $g$ with $N$ points over $\\mathrm{GF}(q,2)$, since $C_1$ is an elliptic curve and $C_2$ has genus at most 2, see \\cite[Propositions 6.1.3 (a) and 6.2.3 (b)]{stichtenoth2009functionfields}. It is well known that the number of points at infinity is $1$ in an elliptic curve and it is at most $2$ in a genus $2$ curve. Hence, \\ref{nonsqnear5_C1}) is proved.\n\n\\ref{nonsqnear5_exists}) By contradiction we assume that $p(\\alpha)$ is a square for all non-square $\\alpha \\in \\mathrm{GF}(q,2)$. Let $\\beta\\in \\mathrm{GF}(q,2)$ be a square root of $p(\\alpha)$. Since there are exactly $\\textstyle\\frac{1}{2}(q^2 - 1)$ non-squares in $\\mathrm{GF}(q,2)$ and $\\beta \\neq 0$, except at most for $3$ choices of $\\alpha$, then the pairs $(\\alpha,\\beta)$ and $(\\alpha,-\\beta)$ produce at least $q^2 - 4$ distinct points of $C_1$. We show that such points are too many. We estimate the number of squares $\\alpha$ such that $p(\\alpha)$ is also a square in $\\mathrm{GF}(q,2)$. Each point $(t,y)$ in $C_2$ corresponds to the point $(x,y)$ in $C_1$ with $x = t^2$. This correspondence is not $1-1$ because, when $t\\ne 0$, the point $(-t,y)$ determines the same point in $C_1$. Let $N$ be the number of affine $\\mathrm{GF}(q,2)$-rational points of $C_2$, then $C_1$ must have more than $\\textstyle\\frac{N}{2}$ affine $\\mathrm{GF}(q,2)$-rational points $(x,y)$ with $x$ being a square in $\\mathrm{GF}(q,2)$. By Part \\ref{nonsqnear5_C1}), we have $N\\ge q^2 - 4q -1$. Counting the points of $C_1$ we get, again by Part \\ref{nonsqnear5_C1}), $\\textstyle q^2 - 4 +\\frac{1}{2}(q^2 - 4q -1) \\le q^2 + 2q$ which yields, after a straightforward computation, $q^2 - 8q - 9 \\le 0$. It follows that $q \\le 9$, which is contrary to our assumption on $q$. Hence, there is at least one non-square $x_1 \\in \\mathrm{GF}(q,2)$ such that $p(x_1)$ is a non-square too.\n\\end{proof}\n\\begin{rem}\\label{rem5}For suitable polynomials $p(x)$, Part \\ref{nonsqnear5_exists}) of the previous lemma also holds for odd primes $q < 11$. In the sequel, we are interested in \\begin{align*}p(x)\\! &= \\!1 + 4v_5(x) \\!=\\! 1 + 32x(x + 3\\varepsilon)^2\\!=\\! 32\\!\\left(x + \\frac{1}{2}\\right)\\!\\!\\left(x^2 + x + \\frac{1}{16}\\right)\\!= \\\\ &= 32\\left(x +\\frac{1}{2}\\right)\\left(x + \\frac{1}{2} - a\\right)\\left(x + \\frac{1}{2} + a\\right),\n\\end{align*} \nwhere $a$ is a square root of $\\textstyle\\frac{3}{16}$ in $\\mathrm{GF}(q,2)$. We are interested in  this polynomial because $\\delta_1=p(x_1)$, for $f_5$, so we need that both $x_1$ and $\\delta_1$ are non-squares. It follows that $p(x)$ is square-free for $q = 5$ and $q = 7$. For $q = 5$, if we choose $x_1$ being a root of the polynomial $x^2 + 4x + 2$, then $p(x_1) = x_1^5$. Hence, both $x_1$ and $p(x_1)$ are non-square in $\\mathrm{GF}(5,2)$. \nSimilarly, for $q = 7$, if we choose $x_1$ as a root of $x^2 + 5x + 5$, then both $x_1$ and $p(x_1)$ are non-squares in $\\mathrm{GF}(7,2)$. \nFinally, for $q = 3$, we have that $p(x)$ has multiple roots, but Part \\ref{nonsqnear5_exists}) of Lemma \\ref{lem:nonsqnear5} still holds. In fact, if we choose $x_1$ as a root of the polynomial $x^2 + 2x + 2$, as in Example \\ref{exa1}, then $p(x_1) = x_1$, hence $p(x_1)$ is a non-square as well. Kindly refer to Remark \\ref{rem_f4=f5} for further explanations. We use the aforementioned examples in Table \\ref{tab:res5} in Section~\\ref{sec:numres}.\n\\end{rem}\nThe following corollary ensures the existence of towers defined by $f_i(x_{n - 1},x_n)$ generating high order elements for $i \\in\\{ 1,2,\\ldots,5\\}$.\n\\begin{cor}\\label{bound1-4}\nThe polynomials $f_i(x_{n - 1},x_n)$, for $i \\in\\{1,2,\\ldots,5\\}$, define infinite towers of fields. Moreover, for a suitable choice of $x_1$, the order of $x_n$ in $\\mathrm{GF}(q,2^n)$ is greater than $2^{\\frac{1}{2}(n^2 + 3n) + \\mathrm{ord}_2(q - 1) - 2}$. The same bound holds for $\\delta_n$ in the towers defined by $f_1(x_{n - 1},x_n)$ and $f_2(x_{n - 1},x_n)$ and, when $q > 3$, for $\\delta_n$ in the tower defined by $f_4(x_{n - 1},x_n)$.\n\\end{cor}\n\\begin{proof}\nFirst, for each tower considered, we show the existence of a non-square starting point $x_1$ such that the discriminant $\\delta_1$ is a non-square as well. A straightforward computation shows that $\\delta_1 = x_1 + 1$ for $f_1$ and that $\\delta_1 = 16x_1^3 + 24x_1^2 + 9x_1 + 1 = (x_1 + 1)(4x_1 + 1)^2$ for $f_2$. Hence, for the first two polynomials, it is enough to choose $x_1$ as in Lemma \\ref{lem:nonsqnear} with $c=1$. A straightforward computation also shows that $\\textstyle \\delta_1 = 2\\left(x_1 + \\frac{1}{2}\\right)$ for $f_3$ and that $\\textstyle \\delta_1 = 128\\left(x_1 + \\frac{1}{2}\\right)(x_1 + 2\\varepsilon^2)^2$ for $f_4$. Hence, for the third and the fourth polynomial, it is enough to choose $x_1$ as in Lemma \\ref{lem:nonsqnear} with $\\textstyle c=\\frac{1}{2}$. For the last tower, by Remark~\\ref{rem5}, we can take $x_1$ as in Remark \\ref{rem5} for $q \\leq 7$ and we can take $x_1$ as in Lemma~\\ref{lem:nonsqnear5} for $q \\geq 11$.\n\nNow, we know, by Remark \\ref{rem:conditions}, that all the considered towers satisfy Conditions \\textbf{(1)} and \\textbf{(2)}, or Conditions \\textbf{(1)} and \\textbf{(2')}. Therefore, the result for $x_n$ follows by Corollary \\ref{bound-odd}. For $\\delta_n$ we have to check that $\\delta_n^2 \\notin \\mathrm{GF}(q,2^{n-1})$ for $n > 1$, in the tower defined by $f_1(x_{n-1},x_n)$ and $f_2(x_{n-1},x_n)$, for $q\\geq 3$, and by $f_4(x_{n-1},x_n)$ for $q > 3$. But this follows by the expression of $g_1(\\delta_{n - 1},\\delta_{n })$,\n $g_2(\\delta_{n - 1},\\delta_{n })$ and $g_4(\\delta_{n - 1},\\delta_{n })$ in Remark~\\ref{rem:conditions}.\n\\end{proof}\nAs in \\cite{burkhart2009finite}, the bound of the previous corollary does not seem to be sharp, in fact in many cases we were able to construct generators of the multiplicative group $\\mathrm{GF}(q,2^n)^*$, whose order is $q^{2^n} - 1$, which is much higher than $2^{\\frac{n^2}{2}}$. The interested reader can compare the tables in Section \\ref{sec:numres} with the experimental results of \\cite{burkhart2009finite}.\n\\begin{rem}\\label{rem_f4=f5} \nThe bound in the Corollary \\ref{bound1-4} above, does not hold for $\\delta_n$ in the tower defined by $f_3(x_{n - 1}, x_n)$ and $f_5(x_{n - 1}, x_n)$. \nIn fact, $\\delta_n^2 \\in \\mathrm{GF}(q,2^{n - 1})$, for all $n > 1$, which can be verified easily. The interested reader can see the numerical results of Tables \\ref{tab:res3} and \\ref{tab:res5} in Section \\ref{sec:numres}.\nA careful comparison between these two tables reveals an interesting difference when $q > 3$. In fact, the order of the discriminant $\\delta_n$ turns out to grow very slowly in Table \\ref{tab:res3} in comparison to Table \\ref{tab:res5}. The reason is that in the former tower the discriminants satisfy the relation\n$g_3(\\delta_{n - 1}, \\delta_n) = \\delta_n^2 - \\delta_{n - 1} = 0,$\nwhich yields $\\delta_n^{2^{n - 1}} = \\delta_{n - 1}^{2^{n - 2}} = \\ldots = \\delta_{1} \\in \\mathrm{GF}(q,2).$  This implies that we can estimate the order of $\\delta_n$, which turns out to be lower than $2^{n - 1 + \\mathrm{ord}_2(q^2 - 1)}$. In the tower defined by $f_5(x_{n - 1},x_n)$, we have that $\\delta_n^{2^j} \\in \\mathrm{GF}(q,2^{n - j})$ holds for $j = 1$, but not for all $j < n$. This explains why the order grows comparatively faster when $q > 3$. In the case $q = 3$ the polynomial equation $g_5(\\delta_{n - 1},\\delta_n) = \\delta_n^2 - \\delta_{n - 1}^3 = 0$ gives $\\delta_n^{2^{n - 1}} = \\delta_{1}^{3^{n - 1}} \\in \\mathrm{GF}(3,2)$. This explains why the numerical results for the order of $\\delta_n$ are similar to the tower defined by $f_3(x_{n - 1},x_n)$.\n\\end{rem}\n\\begin{rem}\nFrom the relation $g_4(\\delta_{n - 1},\\delta_n)=0$ between $\\delta_n$ and $\\delta_{n - 1}$ in the fourth tower, for $q = 3$, we get $g_4(\\delta_{n - 1}, \\delta_n) = \\delta_n^2 - \\delta_{n - 1}^3 = 0$. Hence, we observe that the proof of last corollary does not work when $q = 3$. We also point out that $f_4(x_{n - 1}, x_n) = f_5(x_{n - 1}, x_n)$ when $q = 3$. This fact explains why the numerical results in Tables \\ref{tab:res4} and \\ref{tab:res5} have the same values in the first two columns.\n\\end{rem}\nOf course could exist other towers satisfying analogues of Conditions \\textbf{(1)} and \\textbf{(2)} or Conditions \\textbf{(1)} and \\textbf{(2')} above. \nAn extensive computer search could show the non-existence of similar examples of the form $f(x_{n - 1},x_n) = x_n^2 + x_n + v(x_{n - 1})$, with $\\deg(v(x)) \\leq 3$, at least for small prime fields.\n\n\n\n\n\n\n\n\\section{Examples of good towers in even characteristic}\\label{sec:list-even}\n\n\nIn this section we \nlist polynomials generating high order elements, as in\nSection \\ref{sec:list-odd}.\nWe have to adapt some proofs in even characteristic, since we have to prove that our cubic polynomials $f(x_{n-1},y)$ are Galois and irreducible in $\\mathrm{GF}(2,2\\cdot 3^{n-1})[y]$. Let $e$ be a non-negative integer. In the following results, we prove that $f(x_{n - 1},x_n):= x_n^3+x_n+x_{n-1}^{2^e}$ actually defines an infinite normal separable tower.\n\n\n\n\\begin{lem}\\label{lem:cubicroots4}\nLet $e$ and $n$ be integers such that $e\\geq 0$ and $n\\ge 2$, and let $x_{n-1}\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$. Assume that $u_n^3\\in \\mathrm{GF}(2,2\\cdot 3^{n-1})$ is a root of the quadratic polynomial $y^2+x_{n-1}^{2^e}y+1$ and that $x_n:=u_n+u^{-1}_n\\notin \\mathrm{GF}(2,2\\cdot 3^{n-1})$ is a root of the cubic polynomial $y^3+y+x_{n-1}^{2^e}$. \nLet $u_{n+1}\\in \\mathrm{GF}(2,2\\cdot 3^{n+1})$ be a third root of $u_n^{2^e}$. Then:\n\\begin{enumerate}[i)] \n\\item\\label{item_cond1}\n $u_{n+1}\\notin \\mathrm{GF}(2,2\\cdot 3^{n})$;\n\n\\item\\label{item_cond2}\n$u_{n+1}^3$ and $u_{n+1}^{-3}$ are the roots of $y^2+x_n^{2^e}y+1$;\n\n\\item\\label{item_cond3}\n $x_{n+1}:=u_{n+1}+u_{n+1}^{-1}$ is a root of $y^3+y+x_n^{2^e}$ and $x_{n+1}\\notin \\mathrm{GF}(2,2\\cdot 3^{n})$.\n\\end{enumerate} \n\\end{lem}\n\\begin{proof}\nPart \\ref{item_cond1}) follows since $u_{n+1}^9=(u_{n+1}^3)^3=(u_n^{2^e})^3=(u_n^3)^{2^e}$ belongs to $\\mathrm{GF}(2,2\\cdot 3^{n-1})$ and since $\\mathrm{GF}(2,2\\cdot 3^{n})$ does not contain any 9-th root of non-cubic elements in $\\mathrm{GF}(2,2\\cdot 3^{n-1})$ because 9 does not divide\n$$\\frac{|\\mathrm{GF}(2,2\\cdot 3^{n})^*|}{|\\mathrm{GF}(2,2\\cdot 3^{n-1})^*|}=1+4^{3^{n-1}}+4^{2\\cdot 3^{n-1}},$$ for all $n\\ge 1$.\n\nPart \\ref{item_cond2}) follows by straightforward verification.\n\nThe last part follows by Lemma \\ref{lem:cubicroots2} and by Parts \\ref{item_cond1}) and \\ref{item_cond2}).\n\\end{proof}\n\nPart iii) in the previous Lemma shows \nby induction that if $f(x_1,y)=y^3+y+x_{1}^{2^e}$\nis irreducible over  \n$\\mathrm{GF}(2,6)$, then \n$f(x_n,y)=y^3+y+x_{n}^{2^e}$ is also irreducible over  \n$\\mathrm{GF}(2,2\\cdot 3^{n})$ for all $n>1$.\nIt follows that the Galois groups of the splitting field of $f(x_n,y)$ is either the cyclic group\n$\\mathbb{Z}\/3 \\mathbb{Z}$ or the symmetric group $\\mathrm{S}_3$, at each iteration. \nIn the same spirit of the results above, we show that if the first polynomial $f(x_1,y)$ is normal (i.e., the Galois group of the splitting field is\n$\\mathbb{Z}\/3 \\mathbb{Z}$) then \n$f(x_n,y)$ is normal at each iteration.\n\n\\begin{lem}\\label{lem:galois}\nLet $e\\geq 0$ and $x_{n - 1} \\in \\mathrm{GF}(2,2 \\cdot 3^{n - 1})$. Assume that $f(x_{n - 1},y)=y^3+y+x_{n-1}^{2^e}$ splits into linear factors in $\\mathrm{GF}(2,2 \\cdot 3^{n})[y]$. Then $f(x_{n},y)$ splits into linear factors in $\\mathrm{GF}(2,2 \\cdot 3^{n + 1})[y]$.\n\\end{lem}\n\\begin{proof}\n Let $r_1, r_2, r_3 \\in \\mathrm{GF}(2,2 \\cdot 3^{n})$ be the roots of $f(x_{n - 1},y)$ and choose $x_n = r_1$. Let $R(x_n,y) = y^2 + x_n^{2^e} y + x_n^{2^{e+1}} + 1$ be the quadratic resolvent of $f(x_{n},y)$. \nApplying Frobenius automorphism, we know that the roots satisfy $r_2^{2^e} + r_3^{2^e} = r_1^{2^e}$ and $r_1^{2^e}r_2^{2^e}r_3^{2^e}=(x_{n-1}^{2^e})^{2^e}$. Hence, we get\n\\begin{align*}\nR(x_n,y)&=y^2 + r_1^{2^e}y + r_1^{2^{e+1}} + 1=y^2 + r_1^{2^e}y + \\frac{(r_1^3 + r_1)^{2^e}}{r_1^{2^e}}= \\\\\n&= y^2 + r_1^{2^e}y + \\frac{x_{n-1}^{2^{2e}}}{r_1^{2^e}}=(y + r_2^{2^e})(y + r_3^{2^e}).\n\\end{align*}\nIt follows that $R(x_n,y)$ splits in $\\mathrm{GF}(2,2 \\cdot 3^{n})[y]$. Therefore, $f(x_{n},x_{n + 1})$ splits into linear factors in $\\mathrm{GF}(2,2 \\cdot 3^{n + 1})[y]$ by Lemma \\ref{lem:quadratic_resolvent}.\n\\end{proof}\n\nWe summarize the results above in the following Corollary,\nwhich provides a good initial choice for $x_1$, resulting in\n$f(x_{n-1},x_n)$ to be a normal separable recursive tower.\n\n\\begin{cor}\\label{cor:eventowerorder}\nLet $e\\geq 0$ be an integer. Then $f(x_{n-1},x_n):= x_n^3+x_n+x_{n-1}^{2^e}$ defines an infinite tower of fields and, for a suitable choice of $x_1$, the order of $x_n\\in \\mathrm{GF}(2,2\\cdot 3^n)$, for $n \\ge 2 $, is greater than $3^{\\frac{1}{2}(n^2 + 3n)-1}.$\n\\end{cor}\n\\begin{proof}\nLet $x_1$ be one of the roots of $h(x) := x^6 + x^5 + x^3 + x^2 + 1.$ The reader can verify that each root of this polynomial is not a cube in $\\mathrm{GF}(2,18)$. Moreover, the quadratic resolvent $R(x_1,y)=y^2+x_1y+x_1+1$ of $f(x_{1},y)= y^3+y+x_{1}$ is reducible in $\\mathrm{GF}(2,6)[y]$ and the roots of $y^2+x_1y+1$ are not cubes in $\\mathrm{GF}(2,6)$. Applying the Frobenius automorphism, this implies that, for all $e\\ge 0$, the roots of $y^2+x_1^{2^e}y+1$ are not cubes and the quadratic resolvent $R(x_1^{2^e},y)=y^2+x_1^{2^e}y+x_1^{2^{e+1}}+1$ of $f(x_{1},y)= y^3+y+x_{1}^{2^e}$ is reducible in $\\mathrm{GF}(2,6)[y]$.\n\nBy Lemma~\\ref{lem:cubicroots4}, Part \\ref{item_cond3}), the fact that the roots of $y^2+x_1^{2^e}y+1$ are not cubes implies that $f(x_{n},y)= y^3+y+x_{n}^{2^e}$ is irreducible for each $n\\ge 1$. Hence $f(x_{n-1},x_n)$ defines an infinite tower of fields. By Lemma~\\ref{lem:galois}, the condition on the resolvent implies that this tower is Galois.\n\nSince $f$ clearly satisfies Condition \\textbf{(3)} of Section \\ref{sec:even}, so the proof follows by Corollary~\\ref{bound-even}.\n\\end{proof}\nIn Table \\ref{tab:res6} of Section~\\ref{sec:numres} we collated the numerical results for $f_6(x_{n - 1},x_n) := x_n^3 + x_n + x_{n - 1}$ and $f_7(x_{n - 1},x_n) := x_n^3 + x_n + x_{n - 1}^2$ corresponding to $e=0$ and $e=1$, respectively.\nThe initial element $x_1$ is one of the roots of $h(x) := x^6 + x^5 + x^3 + x^2 + 1$ as explained in the proof of Corollary \\ref{cor:eventowerorder}.\n\n\n\n\n\\section{Numerical results}\\label{sec:numres}\n\nIn this section, we have collated the multiplicative orders $o(x_n)$ (and $o(\\delta_n )$ for $q$ odd) for small $n$ in the towers defined by $f_i (x_{n - 1}, x_n)$, for $i = 1, 2,\\ldots,7$. In most of the cases we obtained generators of the multiplicative groups $\\mathrm{GF}(q, 2^n)^*$ and $\\mathrm{GF}(2, 2 \\cdot 3^n)^*$. We tabulated base 2 logarithm of the orders as they grow   \nexponentially. The interested reader can also find the lower and upper bounds for $o(x_n)$ and $o(\\delta_n)$ listed in Tables \\ref{tab:bounds} and \\ref{tab:res6}, for odd and even characteristic respectively. Finally, in Table \\ref{tab:ComparativeAnalysis}, we compare one of our example in characteristic $3$, with the constructions of \\cite{burkhart2009finite} and \\cite{cohen1992explicit}.\n\nMAGMA \\cite{MAGMA} computational algebra system was used for the experiments and a sample MAGMA code and output, for $q = 11$ can be found in \\cite{CODE}. The performance of the code depends on the efficiency of the \\textit{root} finding algorithm that one uses. We have used the standard function of MAGMA \\cite{MAGMA} for finding roots.\n\n\n\\begingroup\n\\tiny{\n\\begin{table}[ht]\n\\caption{Results for $f_1(x_{n - 1},x_n)$ for odd $q\\leq 11$.}\n\\begin{center}\n\\begin{adjustwidth}{-0.35in}{-0.35in}\n\\begin{tabular}{|c||c|c||c|c||c|c||c|c|}\n\\hline              \n\n$q$ & 3 & 3 & 5 & 5 & 7 & 7 & 11 & 11 \\\\\n\\hline\n\n$x_1^2=$ & $2x_1 + 1$ & $2x_1 + 1$  & $3x_1 + 2$ & $3x_1 + 2$ & $x_1 + 4$ & $x_1 + 4$ & $4x_1 + 9$ & $4x_1 + 9$ \\\\\n\\hline\n\\hline\n$n$ & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) \\\\ \n\\hline  \n\n$1$ & 3.0 & 3.0 & 4.6 & 3.0 & 5.6 & 5.6 & 6.9 & 5.3 \\\\\n\\hline\n\n$2$ & 6.3 & 6.3 & 9.3 & 9.3 & 11.2 & 11.2 & 13.8 & 13.8 \\\\\n\\hline\n\n$3$ & 12.7 & 12.7 & 18.6 & 18.6 & 22.5 & 22.5 & 27.7 & 27.7 \\\\\n\\hline\n\n$4$ & 25.4 & 25.4  & 37.2  & 37.2 & 44.9 & 44.9 & 55.4 & 55.4 \\\\\n\\hline\n\n$5$ & 50.7 & 50.7 & 74.3 & 74.3 & 89.8 & 89.8 & 110.7 & 110.7 \\\\\n\\hline\n\n$6$ & 101.4 & 101.4 & 148.6 & 148.6 & 179.7 & 179.7 & 221.4 & 221.4 \\\\\n\\hline\n\n$7$ & 202.9 &  202.9 & 297.2 & 297.2 & 359.3 & 359.3 & 442.8 & 442.8 \\\\\n\n\\hline\n$8$ & 405.8 & 405.8 & 594.4 & 594.4 & 718.7 & 718.7 & 883.3 & 885.6 \\\\ \n\\hline\n\n$9$ &  811.5 &  811.5 & 1188.8 & 1188.8 & 1437.4 & 1437.4 & 1771.2 & 1771.2 \n\\\\\n\n\\hline\n\\end{tabular}\\label{tab:res1}\n\\end{adjustwidth}\n\\end{center}\n\\end{table}\n}\n\\endgroup\n\\begingroup\\tiny{\n\\begin{table}[ht]\n\\caption{Results for $f_2(x_{n - 1},x_n)$ for odd $q\\leq 11$.}\n\\begin{center}\n\\begin{adjustwidth}{-0.35in}{-0.35in}\n\\begin{tabular}{|c||c|c||c|c||c|c||c|c|}\\hline              \n$q$ & 3 & 3 & 5 & 5 & 7 & 7 & 11 & 11 \\\\\n\\hline\n$x_1^2=$ & $2x_1+1$ & $2x_1+1$  & $3x_1+2$ & $3x_1+2$ & $x_1+4$ & $x_1+4$ & $4x_1+9$& $4x_1+9$ \\\\\n\\hline \n\\hline\n$n$ & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$))& $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$))\\\\ \n\\hline  \n\n$1$ & 3.0 & 3.0 & 4.6 & 4.6 & 5.6 & 5.6 & 6.9 & 4.6 \\\\\\hline\n\n$2$ & 6.3 & 6.3 & 9.3 & 9.3 & 11.2 & 11.2 & 13.8 & 12.3 \\\\\\hline\n\n$3$ & 12.7 & 12.7 & 18.6 & 18.6 & 20.9 & 22.5 & 26.1 & 27.7 \\\\\\hline\n\n$4$ & 25.4 & 25.4  & 37.2  & 37.2 & 44.9 & 44.9 & 55.4 & 48.9 \\\\\\hline\n\n$5$ & 50.7 & 50.7 & 74.3& 74.3 & 88.3 & 89.8 & 106.6 & 110.7 \\\\\\hline\n\n$6$ & 101.4 & 101.4 & 148.6 & 148.6 & 179.7 & 179.7 & 221.4 & 219.8 \\\\\\hline\n\n$7$ &  202.9 &  202.9 & 297.2 & 297.2 & 357.8 & 359.3 & 441.2 & 442.8 \\\\\\hline\n\n$8$ & 405.8 & 405.8 & 594.4 & 594.4 & 718.7 & 718.7 & 885.6 & 879.2 \\\\ \\hline\n\n$9$ &  811.5 &  811.5 & 1188.8 & 1188.8 & 1435.8 & 1437.4 & 1767.1 & 1771.2 \\\\\\hline\n\n\\end{tabular}\\label{tab:res2}\n\\end{adjustwidth}\n\\end{center}\n\\end{table}}\n\\endgroup\n\\begingroup\n\\tiny{\n\\begin{table}[ht]\n\\caption{Results for $f_3(x_{n - 1},x_n)$ for odd $q \\leq 11$.}\n\\begin{center}\n\\begin{adjustwidth}{-0.35in}{-0.35in} \n\\begin{tabular}{|c||c|c||c|c||c|c||c|c|}\n\\hline             \n\n$q$ & 3 & 3 & 5 & 5 & 7 & 7 & 11 & 11 \\\\\n\\hline\n\n$x_1^2=$ & $x_1+1$& $x_1+1$  & $2x_1+2$ & $2x_1+2$ & $3x_1+2$& $3x_1+2$ & $4x_1+9$& $4x_1+9$ \\\\\n\\hline \n\\hline\n$n$ & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$))& $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) \\\\ \n\\hline  \n\n$1$ & 3.0 & 3.0 & 4.6 & 4.6 & 5.6 & 4.0 & 6.9 & 5.3 \\\\\n\\hline\n\n$2$ & 6.3 & 4.0 & 9.3 & 5.6 & 11.2 & 5.0 & 13.8 & 6.3 \\\\\n\\hline\n\n$3$ & 12.7 & 5.0 & 18.6 & 6.6 & 22.5 & 6.0 & 25.4 & 7.3 \\\\\n\\hline\n\n$4$ & 25.4& 6.0  & 37.2  & 7.6 & 44.9 & 7.0 & 51.3 & 8.3 \\\\\n\\hline\n\n$5$ & 50.7 & 7.0 & 74.3 & 8.6 & 89.8 & 8.0 & 106.6 & 9.3 \\\\\n\\hline\n\n$6$ & 101.4& 8.0 & 148.6 & 9.6 & 179.7 & 9.0 & 217.3 & 10.3 \\\\\n\\hline\n\n$7$ &  202.9 &  9.0 & 297.2 & 10.6 & 359.3 & 10.0 & 436.4 & 11.3 \\\\\n\\hline\n\n$8$ & 405.8 & 10.0 & 594.4& 11.6 & 718.7 & 11.0 & 881.5 & 12.3 \\\\ \n\\hline\n\n$9$ &  811.5 &  11.0 & 1188.8 & 12.6 & 1437.4 & 12.0 & 1767.1 & 13.3 \\\\\n\\hline\n\n\\end{tabular}\\label{tab:res3}\n\\end{adjustwidth}\n\\end{center}\n\\end{table}\n}\n\\endgroup\n\\begingroup \\tiny{\n\\begin{table}[ht]\n\\caption{Results for $f_4(x_{n - 1},x_n)$ for odd $q \\leq 11$.}\n\\begin{center}\n\\begin{adjustwidth}{-0.35in}{-0.35in} \\begin{tabular}{|c||c|c||c|c||c|c||c|c|}\\hline             \n$q$ & 3 &3 & 5 &5 & 7 &7 &11 & 11\\\\\n\\hline\n$x_1^2=$ & $x_1+1$& $x_1+1$  & $4x_1+3$ & $4x_1+3$ & $2x_1+4$& $2x_1+4$ & $7x_1+4$& $7x_1+4$ \\\\\n\\hline \n \\hline\n$n$ & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$))& $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) \\\\ \\hline  $1$ & 3.0 & 3.0 & 4.6 & 4.6 & 5.6 & 5.6 & 6.9 & 4.6 \\\\\n\\hline\n$2$ & 6.3 & 4.0 & 9.3 & 7.7 & 11.2 & 11.2 & 13.8 & 13.8 \\\\\n\\hline\n$3$ & 12.7 & 5.0 & 17.0 & 17.0 & 22.5 & 20.9 & 27.7 & 27.7 \\\\\n\\hline\n$4$ & 25.4 & 6.0  & 35.6  & 37.2 & 41.0 & 43.3 & 55.4 & 55.4 \\\\\n\\hline$5$ & 50.7 & 7.0 & 72.7 & 72.7 & 89.8 & 89.8 & 110.7 & 109.1 \\\\\n\\hline$6$ & 101.4& 8.0 & 147.0 & 148.6 & 177.3 & 179.7 & 221.4 & 219.8 \\\\\n\\hline$7$ &  202.9 &  9.0 & 295.6 & 295.6 & 359.3 & 357.8 & 442.8 & 441.2 \\\\\n\\hline$8$ & 405.8 & 10.0 & 592.8  & 594.4 & 717.1 & 717.1 & 885.6 & 884.0 \\\\ \\hline$9$ &  811.5 &  11.0 & 1187.2 & 1188.8 & 1435.8 & 1435.8 & 1769.6 & 1771.2 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:res4}\n\\end{adjustwidth}\n\\end{center}\n\\end{table}}\n\\endgroup\n\\begingroup\\tiny{\n\\begin{table}[ht]\n\\caption{Results for $f_5(x_{n - 1},x_n)$ for odd $q\\leq 11$.}\n\\begin{center}\\begin{adjustwidth}{-0.35in}{-0.35in} \\begin{tabular}{|c||c|c||c|c||c|c||c|c|}\n\\hline              $q$ & 3 &3 & 5 &5 & 7 &7 &11 & 11 \\\\\n\\hline$x_1^2=$ & $x_1+1$& $x_1+1$  & $x_1+3$ & $x_1+3$ & $2x_1+2$& $2x_1+2$ & $4x_1+4$& $4x_1+4$ \\\\\n\\hline \n\\hline$n$ & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$))& $\\log_2$(o($\\delta_n$)) & $\\log_2$(o($x_n$)) & $\\log_2$(o($\\delta_n$)) \\\\ \\hline  $1$ & 3.0 & 3.0 & 4.6 & 4.6 & 5.6 & 5.6 & 6.9 & 3.0 \\\\\n\\hline$2$ & 6.3& 4.0 & 9.3& 5.6 & 8.9 & 5.0 & 13.8& 5.6 \\\\\n\\hline$3$ & 12.7& 5.0 & 18.6 & 8.7 & 22.5& 12.2 &27.7& 14.8 \\\\\n\\hline$4$ & 25.4& 6.0  & 35.6  & 18.0 & 42.6 & 23.5 & 55.4 & 28.7 \\\\\n\\hline$5$ & 50.7 & 7.0 & 72.7 & 38.2 & 89.8 & 45.9 & 110.7 & 56.4 \\\\\n\\hline$6$ & 101.4& 8.0 & 147.0 & 73.7 & 179.7 & 89.3 & 221.4 & 110.1 \\\\\n\\hline$7$ &  202.9 &  9.0 & 295.6 & 149.6 & 359.3 & 179.1 & 442.8 & 220.8 \\\\\n\\hline$8$ & 405.8 & 10.0 & 592.8& 296.6 & 718.7 & 358.8 & 883.3 & 442.8 \\\\ \n\\hline$9$ &  811.5 &  11.0 & 1187.2 & 595.4 & 1437.4 & 718.1 & 1771.2 & 885.0 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:res5}\n\\end{adjustwidth}\\end{center}\n\\end{table}}\n\\endgroup\n\n\\begingroup\n\\tiny{\n\\begin{table}[ht]\n\\caption{Upper bounds  for odd $q\\leq 11$ and lower bound.}\n\\begin{center}\n\\begin{tabular}{|c||c||c|c|c|c|}\n\\hline \n              \n$q$  & 3 & 5 &7  & 11 & Lower bound\\\\\n\\hline              \n\n$n$  & $\\log_2(q^{2^n}-1)$ & $\\log_2(q^{2^n}-1)$ & $\\log_2(q^{2^n}-1)$ & $\\log_2(q^{2^n}-1)$& $\\log_2(2^{(n^2+3n)\/2})$ \\\\\n\\hline  \n\n$1$  & 3.0 & 4.6 & 5.6 & 6.9 & 2.0\\\\\n\\hline\n\n$2$ & 6.3 & 9.3 & 11.2 & 13.8& 5.0  \\\\\n\\hline\n\n$3$  & 12.7 &  18.6 & 22.5 & 27.7& 9.0 \\\\\n\\hline\n\n$4$ & 25.4  & 37.2 & 44.9  & 55.4 & 14.0 \\\\\n\\hline\n\n$5$  & 50.7 & 74.3 & 89.8  & 110.7 & 20.0\\\\\n\\hline\n\n$6$  & 101.4 & 148.6 & 179.7 & 221.4 & 27.0\\\\\n\\hline\n\n$7$  &  202.9 & 297.2 & 359.3 & 442.8 &  35.0\\\\\n\\hline\n\n$8$  & 405.8 & 594.4 & 718.7 & 885.6 & 44.0\\\\ \n\\hline\n\n$9$  &  811.5 & 1188.8 & 1437.4 & 1771.2 &  54.0\\\\\n\\hline\n\n\\end{tabular}\\label{tab:bounds}\n\\end{center}\n\\end{table}\n}\n\\endgroup\n\n\\clearpage\n\n\\begingroup\n\\tiny{\n\\begin{table}[ht]\n\\caption{Results for $f_6(x_{n - 1},x_n)$ and $f_7(x_{n - 1},x_n)$ for $q=2$ and related lower and upper bounds. }\n\\begin{adjustwidth}{-0.50in}{-0.50in} \n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline \n\n$f(x_{n-1},x_n)=$ & $f_6(x_{n-1},x_n)$ & $f_7(x_{n-1},x_n)$& Lower bound & Upper bound \\\\\n\\hline \\hline              \n\n$n$ & $\\log_2$(o($x_n$)) & $\\log_2$(o($x_n$)) & $\\log_2(3^{n(n+3)\/2})$ & $\\log_2(4^{3^n}-1) $\\\\ \n\\hline  \n\n$1$ & 6.0  & 6.0 &  3.2 & 6.0 \\\\\n\\hline  \n\n$2$ & 18.0& 18.0 & 7.9 & 18.0 \\\\\n\\hline\n\n$3$ & 54.0  & 54.0 & 14.3 & 54.0\\\\\n\\hline\n\n$4$ & 162.0  & 162.0 & 22.2 & 162.0 \\\\\n\\hline\n\n$5$ & 486.0   & 486.0 &  31.7 & 486.0  \\\\\n\\hline\n\n$6$ & 1458.0   & 1458.0   & 42.8  & 1458.0 \\\\\n\\hline\n\n\\end{tabular}\\label{tab:res6}\n\\end{center}\n\\end{adjustwidth}\n\\end{table}\n}\n\\endgroup\n\n\n\\begingroup\n\\tiny{\n\\begin{table}[ht]\n\\caption{Comparative Analysis}\n\\begin{center}\n\\begin{tabular}{|c||c||c|c|c|c|}\n\\hline \n              \n$n$  & $\\log_2 (|{{\\mathbb{F}}^{*}}_{3^{2^n}}|))$ & Our Model & Burkhart's Model \\cite{burkhart2009finite}  & Cohen's Model \\cite{cohen1992explicit}  & McNay's Model \\cite{mcnay1995topics} \\\\\n              \n\\hline              \n\n$1$  & 3.0 & 3.0 & 3.0 & 3.0 & 3.0 \\\\\n\\hline\n\n$2$ & 6.3 & 6.3 & 5.3 & 4.0 & 4.3 \\\\\n\\hline\n\n$3$  & 12.7 &  12.7 & 10.7 & 5.0 & 7.4  \\\\\n\\hline\n\n$4$ & 25.4  & 25.4 & 22.4  & 6.0 & 13.7 \\\\\n\\hline\n\n$5$  & 50.7 & 50.7 & 46.8  & 7.0 & 26.4 \\\\\n\\hline\n\n$6$  & 101.4 & 101.4 & 96.5 & 8.0 & 51.7 \\\\\n\\hline\n\n$7$  &  202.9 & 202.9 & 197.0 & 9.0 & 102.4 \\\\\n\\hline\n\n$8$  & 405.8 & 405.8 & 399.0 & 10.0 & 203.9 \\\\ \n\\hline\n\n$9$  &  811.5 & 811.5 & 804.0 & 11.0 & 406.8 \\\\\n\\hline\n\n\\end{tabular}\\label{tab:ComparativeAnalysis}\n\\end{center}\n\\end{table}\n}\n\\endgroup\n\n\n\n\n\\section{Conclusion and future work}\n\n\nIn~\\cite{burkhart2009finite}, the choice of polynomials for the recursive process to generate high order elements in finite field extensions, was limited to the equations of the modular curve towers in \\cite{elkies2001explicit}. In this work, we attempted to generalize the choice of the polynomials. This provides us with more examples with similar properties. A central theme of this research work is to find a recipe to choose polynomials to use the recursive process. There might be other equations which could help to attain similar bounds. It would be interesting to understand in general which equations are good and which ones are not. We also point out that there could be other explicit towers satisfying similar properties. We were in fact attracted previously by other interesting examples with $v(x)$ being a polynomial of higher degree over $\\mathrm{GF}(q, 1)$, which turned out to give high order elements, although the proof seems to be much harder. A possible relation linking together these equations could allow to obtain other families of towers with good parameters. We also expect to improve our results by\nextending the construction of Section \\ref{sec:odd} to higher degree polynomials and\nextending the construction of Section \\ref{sec:even} to odd characteristic $q > 3$.\n\nAnother question that would be interesting to explore is the possible relation with some geometric construction.\nIn fact, since the tower in \\cite{burkhart2009finite} is obtained from the equation of a modular curve, it is a natural question to ask whether our results have a geometric interpretation or not. We hope that a finer understanding of the subject might also possibly provide a recipe for finding high order elements from towers obtained from different forms.\n\n\\section{Acknowledgements}\n\nThe second author was partially supported by the research grant \"Ing. Giorgio Schirillo\" of the Istituto Nazionale di Alta Matematica \"F. Severi\", Rome. The third author would like to thank the High Performance Computing facility of University of L'Aquila (UAQ), which enabled us to implement the algorithm on MAGMA \\cite{MAGMA} computer algebra system,  and run the experiments to validate our results. The third author is grateful to the fruitful and illuminating discussions with Professor Norberto Gavioli, UAQ. The third author is thankful to Professor Kalyan Chakraborty for arranging his research visit to Harish-Chandra Institute (HRI), Prayagraj, India, where the possible modularity aspects of this research work were explored. The third author thanks Dr. Kalyan Banerjee (Post-Doctoral Fellow, HRI) for his interest in exploring the geometric meaning of such towers and the modularity connections.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n{\\let\\thefootnote\\relax\\footnotetext{Author contributions: Hao Zhu designed the research; Weize Chen prepared the data, and organized data annotation; Hao Zhu and Xu Han designed the experiments; Weize Chen performed the experiments; Hao Zhu, Weize Chen and Xu Han wrote the paper; Zhiyuan Liu and Maosong Sun proofread the paper. Zhiyuan Liu is the corresponding author.}}\n\nRelations\\footnote{Sometimes relations are also named properties.}, representing various types of connections between entities or arguments, are the core of expressing relational facts in most general knowledge bases (KBs)~\\cite{suchanek2007yago,bollacker2008freebase}. Hence, identifying relations is a crucial problem for several information extraction tasks. Although considerable effort has been devoted to these tasks, some nuances between similar relations are still overlooked, (\\cref{tab:similarity_example} shows an example); on the other hand, some distinct surface forms carrying the same relational semantics are mistaken as different relations. These severe problems motivate us to quantify the similarity between relations in a more effective and robust method. \n\n\\begin{table}\n\\centering\n\\resizebox{0.9\\linewidth}{!}{\n\\begin{tabular}{p{0.35\\columnwidth} p{0.69\\columnwidth}}\n\\toprule\nSentence & The crisis didn't influence his two daughters OBJ and SUBJ. \\\\ \\midrule\nCorrect & per:siblings \\\\ \\midrule\nPredicted & per:parents \\\\ \\midrule\nSimilarity Rank & 2 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{An illustration of the errors made by relation extraction models. The sentence contains obvious patterns indicating the two persons are siblings, but the model predicts it as parents. We introduce an approach to measure the similarity between relations. Our result shows ``siblings'' is the second most similar one to ``parents''. By applying this approach, we could analyze the errors made by models, and help reduce errors.}\n\\label{tab:similarity_example}\n\\end{table}\n\nIn this paper, we introduce an adaptive and general framework for measuring similarity of the pairs of relations. Suppose for each relation $r$, we have obtained a conditional distribution, $P(h, t \\mid r)$ ($h,t\\in \\mathcal{E}$ are head and tail entities, and $r\\in \\mathcal{R}$ is a relation), over all head-tail entity pairs given $r$. We could quantify similarity between a pair of relations by the divergence between the conditional probability distributions given these relations. In this paper, this conditional probability is given by a simple feed-forward neural network, which can capture the dependencies between entities conditioned on specific relations. Despite its simplicity, the proposed network is expected to cover various facts, even if the facts are not used for training, owing to the good generalizability of neural networks. For example, our network will assign a fact a higher probability if it is ``logical'': e.g., the network might prefer an athlete has the same nationality as same as his\/her national team rather than other nations. \n\nIntuitively, two similar relations should have similar conditional distributions over head-tail entity pairs $P(\\,h, t \\mid r\\,)$, e.g., the entity pairs associated with \\textit{be trade to} and \\textit{play for} are most likely to be athletes and their clubs, whereas those associated with \\textit{live in} are often people and locations. In this paper, we evaluate the similarity between relations based on their conditional distributions over entity pairs. Specifically, we adopt Kullback--Leibler (KL) divergence of both directions as the metric. However, computing exact KL requires iterating over the whole entity pair space $\\mathcal{E} \\times \\mathcal{E}$, which is quite intractable. Therefore, we further provide a sampling-based method to approximate the similarity score over the entity pair space for computational efficiency. \n\nBesides developing a framework for assessing the similarity between relations, our second contribution is that we have done a survey of applications. We present experiments and analysis aimed at answering five questions:\n\n(1) How well does the computed similarity score correlate with human judgment about the similarity between relations? How does our approach compare to other possible approaches based on other kinds of relation embeddings to define a similarity? (\\cref{sec:relationship} and \\cref{sec:human-judgment})\n\n(2) Open IE models inevitably extract many redundant relations. How can our approach help reduce such redundancy? (\\cref{sec:openie})\n\n(3) To which extent, quantitatively, does best relational classification models make errors among similar relations? (\\cref{sec:error-analysis})\n\n(4) Could similarity be used in a heuristic method to enhance negative sampling for relation prediction? (\\cref{sec:training-guidance-relation-prediction})\n\n(5) Could similarity be used as an adaptive margin in softmax-margin training method for relation extraction? (\\cref{sec:training-guidance-relation-extraction})\n\n\nFinally, we conclude with a discussion of valid extensions to our method and other possible applications.\n\n\\cutforspace{\nWe conduct experiments on various tasks related to relations. The experimental results show that our relation similarity computed with fact distribution significantly outperforms other baseline approaches. Moreover, an ideal quantification should be easy to understand, and strongly correlates to human perception of ``similar''. Therefore, we also conduct human evaluation with guidance of given grading criterion. The results clearly show that our computed scores are interpretable and meet human perception. According to the experiment results, we conclude some typical scenarios can be handled well by our similarity:\n\n\\paragraph{Redundant relation removal (\\cref{sec:openie})} The relation patterns extracted by OpenIE~\\cite{angeli2015leveraging,saha2017bootstrapping} are always redundant, as distinct semantic patterns can express the same meaning. For example, for (\\emph{Cricket}, \\texttt{is also played in}, \\emph{County Donegal}) and (\\emph{Football}, \\texttt{is a popular sport in}, \\emph{Macedonia}), both \\texttt{is also played in} and \\texttt{is a popular sport in} means a sport played in a place. Therefore, these two relations should be combined into one. To eliminate the redundant relations would help us enrich the knowledge base, and thus benefit many downstream applications like question answering. Based on our proposed similarity quantification, we can filter out those redundant relation pairs beyond a threshold.\n\n\n\n\\paragraph{Error analysis (\\cref{sec:error-analysis})} As we proved the correlation between similarity and human perception (\\cref{sec:human-judgment}), we further use the computed score between each pair to analyze errors made in predicting relations. By comparing the similarity between the wrongly predicted relation and ground-truth relations, we find that even the best models on these tasks still make errors among the most similar relations, which highlights the needs to guide the model to focus on the nuances between these relations. \n\n\\paragraph{Training guidance (\\cref{sec:training-guidance})} Since similar relations are harder to discriminate, it is intuitive to learn more on these pairs. In this paper, we propose two methods to incorporate the similarity into negative sampling and softmax-margin methods, on relation prediction and relation extraction tasks respectively. For negative sampling, we use similarity-based probability to sampling contrastive samples to help learn the boundary between similar relations; for softmax-margin methods, we use similarity as cost function to force methods learn a relative large margin between regions of similar relations. In both of the application, we gradually reduce the temperature of exponential function, during which the probability move from flat to peaky accordingly. This can warm up our models at first, and help tell the fine-grained difference at last.\n}\n\n\\section{Learning Head-Tail Distribution}\n\\label{sec:fact-distribution}\nJust as introduced in \\cref{sec:introduction}, we quantify the similarity between relations by their corresponding head-tail entity pair distributions. Consider the typical case that we have got numbers of facts, but they are still sparse among all facts in the real world. How could we obtain a well-generalized distribution over the whole space of possible triples beyond the training facts? This section proposes a method to parameterize such a distribution.\n\n\\subsection{Formal Definition of Fact Distribution}\n\nA \\textit{fact} is a triple $(h, r, t)\\in \\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}$, where $h$ and $t$ are called \\textit{head} and \\textit{tail} entities, $r$ is the relation connecting them, $\\mathcal{E}$ and $\\mathcal{R}$ are the sets of entities and relations respectively. We consider a score function $F_{\\theta}:\\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}\\rightarrow \\mathbb{R}$ maps all triples to a scalar value. As a special case, the function can be factorized into the sum of two parts: $F_{\\theta}(\\,h,t; r\\,)\\mathrel{\\stackrel{\\mbox{\\tiny def}}{=}} u_{\\theta_1}(h; r) + u_{\\theta_2}(t; h, r)$. We use $F_{\\theta}$ to define the unnormalized probability.\n\\begin{equation}\n\\small\n\\tilde{P}_{\\theta}(\\,h, t\\mid r\\,) \\mathrel{\\stackrel{\\mbox{\\tiny def}}{=}} \\exp F_{\\theta}(\\,h, r; t\\,)\n\\end{equation}\nfor every triple $(\\, h, r, t\\,)$. The real parameter $\\theta$ can be adjusted to obtain difference distributions over facts.\n\nIn this paper, we only consider \\textit{locally normalized} version of $F_{\\theta}$:\n\\begin{equation}\n\\label{eq:local-normalization}\n\\small\n\\begin{aligned}\nu_{\\theta_1}(h; r) &= \\log \\frac{\\exp \\tilde{u}_{\\theta_1}(h; r)}{\\sum_{h'} \\exp \\tilde{u}_{\\theta_1}(h'; r)},\\\\\nu_{\\theta_2}(t; h, r) &= \\log \\frac{\\exp \\tilde{u}_{\\theta_2}(t;h, r)}{\\sum_{t'} \\exp \\tilde{u}_{\\theta_2}(t';h, r)},\\\\\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{u}_{\\theta_1}$ and $\\tilde{u}_{\\theta_2}$ are directly parameterized by feed-forward neural networks. Through local normalization, $\\tilde{P}_{\\theta}(\\,h,t \\mid r\\,)$ is naturally a valid probability distribution, as the partition function $\\sum_{h,t} \\exp F_{\\theta}(\\,h,t;r\\,) = 1$. Therefore, $P_{\\theta}(\\,h,t\\mid r\\,)=\\tilde{P}_{\\theta}(\\,h,t\\mid r\\,)$.\n\n\\subsection{Neural architecture design}\nHere we introduce our special design of neural networks. \nFor the first part and the second part, we implement the scoring functions introduced in \\cref{eq:local-normalization} as\n\\begin{equation}\n\\small\n\\begin{aligned}\n\\tilde{u}_{\\theta_1}(h; r) &= \\mathtt{MLP}_{\\theta_1}(\\bm{r})^{\\top}\\bm{h},\\\\\n\\tilde{u}_{\\theta_2}(t; h, r) &= \\mathtt{MLP}_{\\theta_2}([\\bm{h}; \\bm{r}])^{\\top}\\bm{t},\\\\\n\\end{aligned}\n\\end{equation}\nwhere each $\\mathtt{MLP}_{\\theta}$ represents a multi-layer perceptron composed of layers like $\\bm{y} = \\mathtt{relu}(\\bm{W}\\bm{x} + \\bm{b})$, $\\bm{h}, \\bm{r}$, $\\bm{t}$ are embeddings of $h,r$, $t$, and $\\theta$ includes weights and biases in all layers.\n\n\\subsection{Training}\nNow we discuss the method to perform training. In this paper, we consider joint training. By minimizing the loss function, we compute the model parameters $\\theta^*$:\n\n\\begin{equation}\n\\small\n\\begin{aligned}\n\\theta^*  &= \\argmin_{\\theta} \\mathcal{L}(G) \\\\\n&=\\argmin_{\\theta} \\sum_{(\\,h,r,t\\,)\\in G} - \\log P_{\\theta}(\\,h, t\\mid r\\,),\n\\end{aligned}\n\\end{equation}\nwhere $G \\subset \\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}$ is a set of triples.\\footnote{In our applications, the set of triples could be a knowledge base or a set of triples in the training set etc.} The whole set of parameters, $\\theta = \\{\\theta_1, \\theta_2, \\{\\bm{e}, \\forall e\\in \\mathcal{E}\\}, \\{\\bm{r}, \\forall r\\in \\mathcal{R}\\}\\}$. We train these parameters by Adam optimizer \\cite{kingma2014adam}. Training details are shown in \\cref{sec:training_detail}.\n\n\\begin{table*}[!t]\n\\small\n\\center\n\\begin{tabular}{c c c}\nRelation Representation & Method & Similarity Quantification \\\\ \\toprule\nVectors & TransE \\cite{bordes2013translating} & $S(r_1, r_2) = \\exp \\left(\\bm{r}_1^{\\top} \\bm{r}_2\/\\lVert \\bm{r}_1\\rVert_2\\lVert\\bm{r}_2\\rVert_2\\right)$\\\\\nVectors & DistMult \\cite{yang2014embedding} & $S(r_1, r_2) = \\exp \\left(\\bm{r}_1^{\\top} \\bm{r}_2\/\\lVert \\bm{r}_1\\rVert_2\\lVert\\bm{r}_2\\rVert_2\\right)$\\\\\nMatrices & RESCAL \\cite{nickel2011three} & $S(r_1, r_2) = \\exp (\\lVert M_{r_1} - M_{r_2}\\rVert_F)$\\\\\nAngles & RotatE \\cite{sun2018rotate} & $S(r_1, r_2) = \\exp (- \\sum_{i=1}^n \\lvert \\bm{r}_{1,i} - \\bm{r}_{2,i} \\rvert_1)$\\\\ \\midrule\nProbability Distribution & Ours & \\cref{eq:similarity-defination} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Methods to define a similarity function with different types of relation representations}\n\\label{tab:other-similarity}\n\\end{table*}\n\n\n\\section{Quantifying Similarity}\n\\label{sec:quantify}\nSo far, we have talked about how to use neural networks to approximate the natural distribution of facts. The center topic of our paper, quantifying similarity, will be discussed in detail in this section.\n\n\\subsection{Relations as Distributions}\nIn this paper, we provide a probability view of relations by representing relation $r$ as a probability distribution $P_{\\theta^*}(\\,h, t \\mid r\\,)$. After training the neural network on a given set of triples, the model is expected to generalize well on the whole $\\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}$ space. \n\nNote that it is very easy to calculate $P_{\\theta^*}(\\,h, t \\mid r\\,)$ in our model thanks to local normalization (\\cref{eq:local-normalization}). Therefore, we can compute it by\n\\begin{equation}\n\\small\nP_{\\theta^*}(\\,h, t \\mid r\\,) = \\exp (u_{\\theta_1}(h; r) + u_{\\theta_2}(t; h, r)).\n\\end{equation}\n\n\\subsection{Defining Similarity}\n\\label{sec:defining-similarity}\nAs the basis of our definition, we hypothesize that the similarity between $P_{\\theta^*}(\\,h, t\\mid r\\,)$ reflects the similarity between relations.\\footnote{\\cref{sec:human-judgment} provides empirical results to corroborate this hypothesis.} For example, if the conditional distributions of two relations put mass on similar entity pairs, the two relations should be quite similar. If they emphasize different ones, the two should have some differences in meaning. \n\nFormally, we define the similarity between two relations as a function of the divergence between the distributions of corresponding head-tail entity pairs:\n\\begin{equation}\n\\small\n\\label{eq:similarity-defination}\n\\begin{aligned}\n    S(r_1, r_2) = g\\Big(&\\KLD{P_{\\theta^*}(\\,h,t\\mid r_1\\,)}{P_{\\theta^*}(\\,h,t\\mid r_2\\,)},\\\\\n    &\\KLD{P_{\\theta^*}(\\,h,t\\mid r_2\\,)}{P_{\\theta^*}(\\,h,t\\mid r_1\\,)}\\Big),\n\\end{aligned}\n\\end{equation}\nwhere $\\KLD{\\cdot}{\\cdot}$ denotes Kullback--Leibler divergence, \n\\begin{equation}\n\\begin{aligned}\n\\KLD{P_{\\theta^*}(\\,h,t\\mid r_1\\,)}{P_{\\theta^*}(\\,h,t\\mid r_2\\,)}\\\\\n=\\mathbb{E}_{h, t\\sim P_{\\theta^*}(\\,h,t\\mid r_1\\,)} \\log \\frac{P_{\\theta^*}(\\,h,t\\mid r_1\\,)}{P_{\\theta^*}(\\,h,t\\mid r_2\\,)}\n\\end{aligned}\n\\end{equation}\nvice versa, and function $g(\\cdot, \\cdot)$ is a symmetrical function. To keep the coherence between semantic meaning of ``similarity'' and our definition, $g$ should be a monotonically decreasing function. Through this paper, we choose to use an exponential family\\footnote{We view KL divergences as energy functions.} composed with max function, i.e., $g(x, y) = e^{-\\max(x, y)}$. Note that by taking both sides of KL divergence into account, our definition incorporates both the entity pairs with high probability in $r_1$ and $r_2$. Intuitively, if $P_{\\theta^*}(\\,h, t\\mid r_1\\,)$ mainly distributes on a proportion of entities pairs that $P_{\\theta^*}(\\,h, t\\mid r_2\\,)$ emphasizes, $r_1$ is only hyponymy of $r_2$. Considering both sides of KL divergence could help model yield more comprehensive consideration. We will talk about the advantage of this method in detail in \\cref{sec:relationship}.\n\n\\subsection{Calculating Similarity}\nJust as introduced in \\cref{sec:introduction}, it is intractable to compute similarity exactly, as involving $\\mathcal{O}(|\\mathcal{E}|^2)$ computation. Hence, we consider the monte-carlo approximation:\n\\begin{equation}\n\\small\n\\begin{aligned}\n&\\KLD{P_{\\theta^*}(\\,h, t \\mid r_1\\,)}{P_{\\theta^*}(\\,h, t\\mid r_2\\,)}\\\\\n=\\, & \\mathbb{E}_{h, t\\sim P_{\\theta^*}(\\,h, t \\mid r_1\\,)} \\log\\frac{P_{\\theta^*}(\\,h, t \\mid r_1\\,)}{P_{\\theta^*}(\\,h, t\\mid r_2\\,)}\\\\\n\\approx\\, & \\frac{1}{|\\mathcal{S}|}\\sum_{h, t\\in \\mathcal{S}} \\log\\frac{P_{\\theta^*}(\\,h, t \\mid r_1\\,)}{P_{\\theta^*}(\\,h, t\\mid r_2\\,)},\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{S}$ is a list of entity pairs sampled from $P_{\\theta^*}(\\,h, t \\mid r_1\\,)$. We use sequential sampling\\footnote{Sampling $h$ and $t$ at the same time requires $\\mathcal{O}(|\\mathcal{E}|^2)$ computation, while sequential sampling requires only $\\mathcal{O}(|\\mathcal{E}|)$ computation.} to gain $\\mathcal{S}$, which means we first sample $h$ given $r$ from $u_{\\theta_1}(h;r)$, and then sample $t$ given $h$ and $r$ from $u_{\\theta_2}(t;h, r)$.\\footnote{It seems to be a non-symmetrical method, and sampling from the mixture of both forward and backward should yield a better result. Surprisingly, in practice, sampling from single direction works just as well as from both directions.}\n\n\\subsection{Relationship with other metrics}\n\\label{sec:relationship}\nPrevious work proposed various methods for representing relations as vectors \\cite{bordes2013translating, yang2014embedding}, as matrices \\cite{nickel2011three}, even as angles \\cite{sun2018rotate}, etc. Based on each of these representations, one could easily define various similarity quantification methods.\\footnote{Taking the widely used vector representations as an example, we can define the similarity between relations based on cosine distance, dot product distance, L1\/L2 distance, etc.} We show in \\cref{tab:other-similarity} the best one of them in each category of relation presentation.\n\n\n\nHere we provide two intuitive reasons for using our proposed probability-based similarity: (1) the capacity of a single fixed-size representation is limited --- some details about the fact distribution is lost during embedding; (2) directly comparing distributions yields a better interpretability --- you can not know about how two relations are different given two relation embeddings, but our model helps you study the detailed differences between probabilities on every entity pair. \\cref{fig:head-tail-distribution} provides an example. Although the two relations talk about the same topic, they have different meanings. TransE embeds them as vectors the closest to each other, while our model can capture the distinction between the distributions corresponds to the two relations, which could be directly noticed from the figure. \n\n\n\\begin{figure}[!]\n\\centering\n\\resizebox{0.8\\linewidth}{!}{\n\\includegraphics{scatter.pdf}}\n\\caption[Caption for LOF]{Head-tail entity pairs of relation ``be an unincorporated community in'' (in blue) and ``be a small city in'' (in red) sampled from our fact distribution model. The coordinates of the points are computed by t-sne \\cite{maaten2008visualizing} on the concatenation of head and tail embeddings\\protect\\footnotemark. The two \\textbf{larger} blue and red points indicate the embeddings of these two relations.}\n\\label{fig:head-tail-distribution}\n\\end{figure}\n\n\\footnotetext{Embeddings used in this graph are from a trained TransE model.}\n\n\\begin{figure}[!t]\n\\centering\n\\resizebox{0.86\\linewidth}{!}{\n\\includegraphics[trim={0 0 0 0},clip]{correlation.pdf}}\n\\caption{Spearman correlations between human judgment and models' outputs. The inter-subject correlation is also shown as a horizontal line with its standard deviation as an error band. Our model shows the strongest positive correlation with human judgment, and, in other words, the smallest margin with human inter-subject agreement. Significance: ***\/**\/* := p < .001\/.01\/.05.}\n\\label{fig:correlation}\n\\end{figure}\n\n\\section{Dataset Construction}\nWe show the statistics of the dataset we use in \\cref{tab:statistics}, and the construction procedures will be introduced in this section.\n\\subsection{Wikidata}\n\\label{sec:construction_wikidata}\n\nIn Wikidata \\cite{vrandevcic2014wikidata}, facts can be described as (Head item\/property, \\textit{Property}, Tail item\/property). To construct a dataset suitable for our task, we only consider the facts whose head entity and tail entity are both items. We first choose the most common 202 relations and 120000 entities from Wikidata as our initial data. Considering that the facts containing the two most frequently appearing relations (\\textit{P2860: cites}, and \\textit{P31: instance of}) occupy half of the initial data, we drop the two relations to downsize the dataset and make the dataset more balanced. Finally, we keep the triples whose head and tail both come from the selected 120000 entities as well as its relation comes from the remaining 200 relations. \n\n\\subsection{ReVerb Extractions}\n\\label{sec:construction_reverb}\nReVerb \\cite{fader2011identifying} is a program that automatically identifies and extracts binary relationships from English sentences. We use the extractions from running ReVerb on Wikipedia\\footnote{http:\/\/reverb.cs.washington.edu\/}. We only keep the relations appear more than 10 times and their corresponding triples to construct our dataset.\n\n\\subsection{FB15K and TACRED}\n\\label{sec:construction_fb15k_tacred}\nFB15K \\cite{bordes2013translating} is a subset of freebase. TACRED \\cite{zhang2017position} is a large supervised relation extraction dataset obtained via crowdsourcing. We directly use these two dataset, no extra processing steps were applied. \n\n\\section{Human Judgments}\n\\label{sec:human-judgment}\nFollowing \\citet{miller1991contextual,resnik1999semantic} and the vast amount of previous work on semantic similarity, we ask nine undergraduate subjects to assess the similarity of 360 pairs of relations from a subset of Wikidata \\cite{vrandevcic2014wikidata}\\footnote{Wikidata provides detailed descriptions to properties (relations), which could help subjects understand the relations better.} that are chosen to cover from high to low levels of similarity. In our experiment, subjects were asked to rate an integer similarity score from 0 (no similarity) to 4 (perfectly the same)\\footnote{The detailed instruction is attached in the \\cref{sec:wikidata_annotation}.} for each pair. The inter-subject correlation, estimated by leaving-one-out method \\cite{weiss1991computer}, is r = $0.763$, standard deviation = $0.060$. This important reference value (marked in \\cref{fig:correlation}) could be seen as the highest expected performance for machines \\cite{resnik1999semantic}. \n\nTo get baselines for comparison, we consider other possible methods to define similarity functions, as shown in \\cref{tab:other-similarity}. We compute the correlation between these methods and human judgment scores. As the models we have chosen are the ones work best in knowledge base completion, we do expect the similarity quantification approaches based on them could measure some degree of similarity. As shown in \\cref{fig:correlation}, the three baseline models could achieve moderate ($0.1\\text{--}0.5$) positive correlation. On the other hand, our model shows a stronger correlation ($0.63$) with human judgment, indicating that considering the probability over whole entity pair space helps to gain a similarity closer to human judgments. These results provide evidence for our claim raised in \\cref{sec:defining-similarity}. \n\n\\section{Redundant Relation Removal}\n\\label{sec:openie}\n\nOpen IE extracts concise token patterns from plain text to represent various relations between entities, e.g.,, (Mark Twain, \\textit{was born in}, Florida). As Open IE is significant for constructing KBs, many effective extractors have been proposed to extract triples, such as Text-Runner~\\cite{yates2007textrunner}, ReVerb~\\cite{fader2011identifying}, and Standford Open IE~\\cite{angeli2015leveraging}. However, these extractors only yield relation patterns between entities, without aggregating and clustering their results. Accordingly, there are a fair amount of redundant relation patterns after extracting those relation patterns. Furthermore, the redundant patterns lead to some redundant relations in KBs.\n\nRecently, some efforts are devoted to Open Relation Extraction (Open RE)~\\cite{Lin2001DIRT,Yao2011Structured,marcheggiani2016discrete,Elsahar2017Unsupervised}, aiming to cluster relation patterns into several relation types instead of redundant relation patterns. Whenas, these Open RE methods adopt distantly supervised labels as golden relation types, suffering from both false positive and false negative problems on the one hand. On the other hand, these methods still rely on the conventional similarity metrics mentioned above.\n\nIn this section, we will show that our defined similarity quantification could help Open IE by identifying redundant relations. To be specific, we set a toy experiment to remove redundant relations in KBs for a preliminary comparison (\\cref{sec:toy-experiment}). Then, we evaluate our model and baselines on the real-world dataset extracted by Open IE methods (\\cref{sec:real-experiment}). Considering the existing evaluation metric for Open IE and Open RE rely on either labor-intensive annotations or distantly supervised annotations, we propose a metric approximating recall and precision evaluation based on operable human annotations for balancing both efficiency and accuracy.\n\n\\begin{table}[!t]\n\\small\n\\resizebox{1.0\\linewidth}{!}{\n\\begin{tabular}{l r r r l}\nTriple Set & $|\\mathcal{R}|$& $|\\mathcal{E}|$ & \\#Fact & Section \\\\\\toprule\nWikidata & 188 & 112,946 & 426,067 & \\cref{sec:human-judgment} and \\cref{sec:toy-experiment}\\\\\nReVerb Extractions & 3,736 & 194,556 & 266,645 & \\cref{sec:real-experiment}\\\\\nFB15K & 1,345 & 14,951 & 483,142 & \\cref{sec:error-analysis-relation-prediction} and \\cref{sec:training-guidance-relation-prediction}\\\\\nTACRED & 42 & 29,943 & 68,124 & \\cref{sec:error-analysis-relation-extraction} and \\cref{sec:training-guidance-relation-extraction}\\\\ \\bottomrule\n\\end{tabular}}\n\\caption{Statistics of the triple sets used in this paper.}\n\\label{tab:statistics}\n\\end{table}\n\n\\subsection{Toy Experiment}\n\\label{sec:toy-experiment}\n\nIn this subsection, we propose a toy environment to verify our similarity-based method. Specifically, we construct a dataset from Wikidata\\footnote{The construction procedure is shown in \\cref{sec:construction_wikidata}.} and implement Chinese restaurant process\\footnote{Chinese restaurant process is shown in \\cref{sec:Chinese_restaurant}.} to split every relation in the dataset into several sub-relations. Then, we filter out those sub-relations appearing less than 50 times to eventually get 1165 relations. All these split relations are regarded as different ones during training, and then different relation similarity metrics are adopted to merge those sub-relations into one relation. As Figure~\\ref{fig:correlation} shown that the matrices-based approach is less effective than other approaches, we leave this approach out of this experiment. The results are shown in Table~\\ref{tab:toy-environment}. \n\n\\begin{table}[!t]\n\\small\n\\center\n\\begin{tabular}{c c c c}\n Method & P & R & $F_1$ \\\\ \n \\toprule\n Vectors (TransE) &0.28 & 0.14& 0.18\\\\\n Vectors (DistMult) &0.44 & 0.41 & 0.42\\\\\n Angles & 0.48 & 0.43 & 0.45 \\\\ \n Ours & \\textbf{0.65}& \\textbf{0.50} & \\textbf{0.57}\\\\\n \\bottomrule\n\\end{tabular}\n\\caption{The experiment results on the toy dataset show that our metric based on probability distribution significantly outperforms other relation similarity metrics.}\n\\label{tab:toy-environment}\n\\end{table}\n\n\\subsection{Real World Experiment}\n\\label{sec:real-experiment}\n\nIn this subsection, we evaluate various relation similarity metrics on the real-world Open IE patterns. The dataset are constructed by ReVerb. Different patterns will be regarded as different relations during training, and we also adopt various relation similarity metrics to merge similar relation patterns. Because it is nearly impossible to annotate all pattern pairs for their merging or not, meanwhile it is also inappropriate to take distantly supervised annotations as golden results. Hence, we propose a novel metric approximating recall and precision evaluation based on minimal human annotations for evaluation in this experiment.\n\n\\subsubsection*{Approximating Recall and Precision}\n\n\\paragraph{Recall} \n\nRecall is defined as the yielding fraction of true positive instances over the total amount of real positive\\footnote{Often called relevant in information retrieval field.} instances. However, we do not have annotations about which pairs of relations are synonymous. Crowdsourcing is a method to obtain a large number of high-quality annotations. Nevertheless, applying crowdsourcing is not trivial in our settings, because it is intractable to enumerate all synonymous pairs in the large space of relation (pattern) pairs $\\mathcal{O}(|\\mathcal{R}|^2)$ in Open IE. A promising method is to use rejection sampling by uniform sampling from the whole space, and only keep the synonymous ones judged by crowdworkers. However, this is not practical either, as the synonymous pairs are sparse in the whole space, resulting in low efficiency. Fortunately, we could use normalized importance sampling as an alternative to get an unbiased estimation of recall. \n\n\\begin{theorem}\\footnote{See proof in \\cref{sec:proof_recall}}\nSuppose every sample $x \\in X$ has a label $f(x)\\in\\{0, 1\\}$, and the model to be evaluated also gives its prediction $\\hat{f}(x)\\in\\{0, 1\\}$. The recall can be written as\n\\begin{equation}\n\\small\n\\label{eq:recall-expectation}\nRecall =  \\mathbb{E}_{x\\sim U} \\mathbb{I}[\\hat{f}(x)=1],\n\\end{equation}\nwhere $U$ is the uniform distribution over all samples with $f(x)=1$. If we have a proposal distribution $q(x)$ satisfying $\\forall x,  f(x)=1\\wedge \\hat{f}(x)=1\\Rightarrow q(x) \\neq 0$, we get an unbiased estimation of recall:\n\\begin{equation}\n\\small\n\\label{eq:recall}\nRecall \\approx\\sum_{i=1}^{n} \\mathbb{I}[\\hat{f}(x_i)=1] \\hat{w}_i,\n\\end{equation}\nwhere $\\hat{w}_i$ is a normalized version of $w_i = \\frac{\\mathbb{I}[f(x_i)=1]}{\\tilde{q}(x_i)}$, where $\\tilde{q}$ is the unnormalized version of q, and $\\{x_i\\}_{i=1}^{n}$ are i.i.d. drawn from $q(x)$.\n\\end{theorem}\n\n\\paragraph{Precision} Similar to \\cref{eq:recall-expectation}, we can write the expectation form of precision:\n\\begin{equation}\n\\small\nPrecision = \\mathbb{E}_{x\\sim U'}\\mathbb{I}[f(x) = 1],\n\\end{equation}\nwhere $U'$ is the uniform distribution over all samples with $\\hat{f}(x)=1$. As these samples could be found out by performing models on it. We can simply approximate precision by Monte Carlo Sampling:\n\\begin{equation}\n\\small\n\\label{eq:precision}\nPrecision \\approx \\frac{1}{n} \\sum_{i=1}^{n} \\mathbb{I}[f(x_i)=1],\n\\end{equation}\nwhere $\\{x_i\\}_{i=1}^{n} \\stackrel{i.i.d.}{\\sim} U'$.\n\nIn our setting, $x = (r_1, r_2)\\in \\mathcal{R}\\times\\mathcal{R}$, $f(x) = 1$ means $r_1$ and $r_2$ are the same relations, $\\hat{f}(x) = 1$ means $S(r_1, r_2)$ is larger than a threshold $\\lambda$.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{PR_OpenIE.pdf}\n\\caption{Precision-recall curve on Open IE task comparing our similarity function with vector-based and angle-based similarity. Error bar represents $95\\%$ confidential interval. Bootstraping is used to calculate the confidential interval.}\n\\label{fig:precision-recall-openie}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\begin{minipage}{0.6\\textwidth}\n\\begin{subfigure}{0.49\\textwidth}\n  \\centering\n  \\resizebox{\\linewidth}{!}{\n  \\includegraphics{aveRank_rp.pdf}}\n  \\caption{FB15K}\n  \\label{fig:ave_rank_rp}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\textwidth}\n  \\centering\n\\resizebox{\\linewidth}{!}{\n\\includegraphics[trim={0 0 0 0},clip]{aveRank_re.pdf}}\n\\caption{TACRED}\n\\label{fig:ave_rank_re}\n\\end{subfigure}\n\\caption{Similarity rank distributions of distracting relations on different tasks and datasets. Most of the distracting relations have top similarity rank. Distracting relations are, as defined previously, the relations have a higher rank in the relation classification result than the ground truth.}\n\\label{fig:ave_rank}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.38\\textwidth}\n\\resizebox{\\linewidth}{!}{\n\\includegraphics[trim={0 0 1.2cm 0},clip]{relation_prediction.pdf}}\n\\caption{Improvement of using similarity in a heuristic method for negative sampling. MRR denotes the mean reciprocal\nrank.}\n\\label{fig:relation_prediction}\n\\vspace*{45pt}\n\\end{minipage}\n\\end{figure*}\n\n\n\\subsubsection*{Results}\n\nThe results on the ReVerb Extractions dataset that we constructed are described in \\cref{fig:precision-recall-openie}. To approximate recall, we use the similarity scores as the proposal distribution $\\tilde{q}$. 500 relation pairs are then drawn from $\\tilde{q}$. To approximate precision, we set thresholds at equal intervals. At each threshold, we uniformly sample 50 to 100 relation pairs whose similarity score given by the model is larger than the threshold. We ask 15 undergraduates to judge whether two relations in a relation pair have the same meaning. A relation pair is viewed valid only if 8 of the annotators annotate it as valid. We use the annotations to approximate recall and precision with \\cref{eq:recall} and \\cref{eq:precision}. Apart from the confidential interval of precision shown in the figure, the largest $95\\%$ confidential interval among thresholds for recall is $0.04$\\footnote{The figure is shown in \\cref{fig:recall_std}}. From the result, we could see that our model performs much better than other models' similarity by a very large margin.\n\n\n\n\\section{Error Analysis for Relational Classification}\n\\label{sec:error-analysis}\n\nIn this section, we consider two kinds of relational classification tasks: (1) relation prediction and (2) relation extraction. Relation prediction aims at predicting the relationship between entities with a given set of triples as training data; while relation extraction aims at extracting the relationship between two entities in a sentence. \n\n\\subsection{Relation Prediction}\n\\label{sec:error-analysis-relation-prediction}\nWe hope to design a simple and clear experiment setup to conduct error analysis for relational prediction. Therefore, we consider a typical method TransE \\cite{bordes2013translating} as the subject as well as FB15K~\\cite{bordes2013translating} as the dataset. TransE embeds entities and relations as vectors, and train these embeddings by minimizing\n\\begin{equation}\n\\small\n\\mathcal{L} = \\sum_{(h, r, t) \\in \\mathcal{D}} [d(\\bm{h}+\\bm{r},\\bm{t}) - d(\\bm{h}'+\\bm{r}',\\bm{t}') + \\gamma]_{+},\n\\end{equation}\nwhere $\\mathcal{D}$ is the set of training triples, $d(\\cdot, \\cdot)$ is the distance function, $(h', r', t')$\\footnote{Note that only head and tail entities are changed in the original TransE when doing link prediction. But changing $r'$ results in better performance when doing relation prediction.} is a negative sample with one element different from $(h,r,t)$ uniformly sampled from $\\mathcal{E}\\times\\mathcal{R}\\times\\mathcal{E}$, and $\\gamma$ is the margin.\n\nDuring testing, for each entity pair $(h, t)$, TransE rank relations according to $d(\\bm{h}+\\bm{r}, \\bm{t})$. For each $(h,r,t)$ in the test set, we call the relations with higher rank scores than $r$ \\textbf{\\textit{distracting relations}}. We then compare the similarity between the golden relation and \\textbf{\\textit{distracting relations}}. Note that some entity pairs could correspond to more than one relations, in which case we just do not see them as distracting relations.\n\n\\subsection{Relation Extraction}\n\\label{sec:error-analysis-relation-extraction}\nFor relation extraction, we consider the supervised relation extraction setting and TACRED dataset \\cite{zhang2017position}. As for the subject model, we use the best model on TACRED dataset --- position-aware neural sequence model. This method first passes the sentence into an LSTM and then calculate an attention sum of the hidden states in the LSTM by taking positional features into account. This simple and effective method achieves the best in TACRED dataset. \n\n\\subsection{Results}\n\\label{sec:error_analysis_result}\n\\cref{fig:ave_rank} shows the distribution of similarity ranks of distracting relations of the above mentioned models' outputs on both relation prediction and relation extraction tasks. From \\cref{fig:ave_rank_rp,fig:ave_rank_re}, we could observe the most distracting relations are the most similar ones, which corroborate our hypothesis that even the best models on these tasks still make mistakes among the most similar relations. This result also highlights the importance of a heuristic method for guiding models to pay more attention to the boundary between similar relations. We also try to do the negative sampling with relation type constraints, but we see no improvement compared with uniform sampling. The details of negative sampling with relation type constraints are presented in \\cref{sec:relation-type-constraints}.\n\n\n\\begin{table}[!t]\n\\center\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l l c c c}\n&Model & P & R & $F_1$\\\\\n\\toprule\nTraditional &Patterns & \\textbf{86.9} & 23.2 &  36.6\\\\\n&LR &73.5 &49.9 &59.4 \\\\\\midrule\nNeural&CNN & 75.6 & 47.5 & 58.3 \\\\\n&CNN-PE & 70.3 & 54.2 & 61.2\\\\\n&SDP-LSTM \\cite{xu2015classifying} & 66.3 & 52.7 & 58.7\\\\\n&LSTM & 65.7 & 59.9 & 62.7\\\\\n&PA-LSTM \\cite{zhang2017position} & 65.7 & 64.5 & 65.1\\\\ \n\\midrule\nNeural+Ours&PA-LSTM (Softmax-Margin Loss) & 68.5 & \\textbf{64.7} & \\textbf{66.6}\\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Improvement of using similarity in softmax-margin loss.}\n\\label{tab:relation_extraction}\n\\end{table}\n\n\n\\section{Similarity and Negative Sampling}\n\\label{sec:training-guidance-relation-prediction}\nBased on the observation presented in \\cref{sec:error_analysis_result}, we find out that similar relations are often confusing for relation prediction models. Therefore, corrupted triples with similar relations can be used as high-quality negative samples.\n\nFor a given valid triple $(h, r, t)$, we corrupt the triple by substituting $r$ with $r'$ with the probability,\n\\begin{equation}\np=\\frac{S(r,r')^{1\/\\alpha}}{\\sum_{r''\\in {\\mathcal{R}\\backslash \\{r\\}}} S(r,r'')^{1\/\\alpha}}, \n\\end{equation}\nwhere $\\alpha$ is the temperature of the exponential function, the bigger the $\\alpha$ is, the flatter the probability distribution is. When the temperature approaches infinite, the sampling process reduces to uniform sampling. \n\nIn training, we set the initial temperature to a high level and gradually reduce the temperature. Intuitively, it enables the model to distinguish among those obviously different relations in the early stage and gives more and more confusing negative triples as the training processes to help the model distinguish the similar relations. This can be also viewed as a process of curriculum learning\\cite{bengio2009curriculum}, the data fed to the model gradually changes from simple negative triples to hard ones. \n\nWe perform relation prediction task on FB15K with TransE. Following \\citet{bordes2013translating}, we use the \"Filtered\" setting protocol, i.e., filtering out the corrupted triples that appear in the dataset. Our sampling method is shown to improve the model's performance, especially on Hit@1 (\\cref{fig:relation_prediction}). Training details are described in \\cref{sec:training_detail}.\n\n\\section{Similarity and Softmax-Margin Loss}\n\\label{sec:training-guidance-relation-extraction}\nSimilar to \\cref{sec:training-guidance-relation-prediction}, we find out that relation extraction models often make wrong preditions on similar relations. In this section, we use similarity as an adaptive margin in softmax-margin loss to improve the performance of relation extraction models.\n\nAs shown in \\cite{gimpel2010softmax}, Softmax-Margin Loss can be expressed as\n \\begin{equation}\n \\small\n \\begin{aligned}\n     \\mathcal{L}=&\\sum_{i=1}^n -\\theta^T f(x^{(i)},r^{(i)})+\\\\\n     &\\log\\sum_{r\\in \\mathcal{R}(x^{(i)})}\\exp\\{\\theta^Tf(x^{(i)},r)+\\text{cost}(r^{(i)},r)\\},\n \\end{aligned}\n \\end{equation}\nwhere $\\mathcal{R}(x)$ denotes a structured output space for $x$, and $\\langle x^{(i)}, r^{(i)}\\rangle$ is $i^{th}$ example in training data.\n\nWe can easily incorporate similarity into cost function $\\text{cost}(r^{(i)},r)$. In this task, we define the cost function as $\\alpha S(r^{(i)},r)$, where $\\alpha$ is a hyperparameter. \n\nIntuitively, we give a larger margin between similar relations, forcing the model to distinguish among them, and thus making the model perform better. We apply our method to Position-aware Attention LSTM (PA-LSTM)\\cite{zhang2017position}, and \\cref{tab:relation_extraction} shows our method improves the performance of PA-LSTM. Training details are described in \\cref{sec:training_detail}.\n\n\\section{Related Works}\n\nAs many early works devoted to psychology and linguistics, especially those works exploring semantic similarity~\\cite{miller1991contextual,resnik1999semantic}, researchers have empirically found there are various different categorizations of semantic relations among words and contexts. For promoting research on these different semantic relations, \\newcite{bejar1991cognitive} explicitly defining these relations and \\newcite{miller1995wordnet} further systematically organize rich semantic relations between words via a database. For identifying correlation and distinction between different semantic relations so as to support learning semantic similarity, various methods have attempted to measure relational similarity~\\cite{turney2005measuring,turney2006similarity,zhila2013combining,pedersen2012duluth,rink2012utd,mikolov2013distributed,mikolov2013efficient}.\n\nWith the ongoing development of information extraction and effective construction of KBs~\\cite{suchanek2007yago,bollacker2008freebase,bizer2009dbpedia}, relations are further defined as various types of latent connections between objects more than semantic relations. These general relations play a core role in expressing relational facts in the real world. Hence, there are accordingly various methods proposed for discovering more relations and their facts, including open information extraction~\\cite{brin1998extracting,agichtein2000snowball,ravichandran2002learning,banko2007open,zhu2009statsnowball,etzioni2011open,saha2017bootstrapping} and relation extraction~\\cite{riedel2013relation,liu2013convolution,zeng2014relation,santos2015classifying,zeng2015distant,lin2016neural}, and relation prediction~\\cite{bordes2013translating,wang2014knowledge,lin2015learning,lin2015modeling,xie2016representation}.\n\nFor both semantic relations and general relations, identifying them is a crucial problem, requiring systems to provide a fine-grained relation similarity metric. However, the existing methods suffer from sparse data, which makes it difficult to achieve an effective and stable similarity metric. Motivated by this, we propose to measure relation similarity by leveraging their fact distribution so that we can identify nuances between similar relations, and merge those distant surface forms of the same relations, benefitting the tasks mentioned above.\n\n\\section{Conclusion and Future Work}\n\nIn this paper, we introduce an effective method to quantify the relation similarity and provide analysis and a survey of applications. We note that there are a wide range of future directions: (1) human prior knowledge could be incorporated into the similarity quantification; (2) similarity between relations could also be considered in multi-modal settings, e.g., extracting relations from images, videos, or even from audios; (3) by analyzing the distributions corresponding to different relations, one can also find some ``meta-relations'' between relations, such as hypernymy and hyponymy. \n\n\\section*{Acknowledgements}\nThis work is supported by the National Natural Science Foundation of China (NSFC No. 61572273, 61532010), the National Key Research and Development Program of China (No. 2018YFB1004503). Chen and Zhu is supported by Tsinghua University Initiative Scientific Research Program, and Chen is also supported by DCST Student Academic Training Program. Han is also supported by 2018 Tencent Rhino-Bird Elite Training Program.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Introduction}\nMixed-valence or fluctuating valence behaviour are usually found in lanthanide-based compounds due to the intermixing of the \\textit{s}$-$\\textit{d} band with the localised \\textit{f} band near the Fermi level. Therefore, they exhibit unique magnetic, thermal and electrical properties \\cite{Varma1976a}. Eu cations in Eu-based compounds mostly occur in the 2\\textsuperscript{+} valence. However, in trieuropium tetroxide (Eu$_3$O$_4$) Eu ions exhibit a mixed-valence of Eu$^{2+}$ and Eu$^{3+}$.\n\nEu$_3$O$_4$ crystallises into an orthorhombic structure (space group \\textit{Pnma}) similar to CaFe\\textsubscript{2}O\\textsubscript{4}  with the lattice parameters \\textit{a}$=10.085$ \\AA, \\textit{b}$=3.502$ {\\AA} and \\textit{c}$=12.054$ {\\AA} \\cite{Rau1966a,Ahn2009a}. Figure~\\ref{Eu3O4} shows the Eu$_3$O$_4$ structure, where the Eu$^{2+}$ and Eu$^{3+}$ ions are occupying the Ca$^{2+}$ and Fe$^{3+}$ sites, respectively. The oxygen ions form a six- and eight-fold coordination around the Eu$^{3+}$ and Eu$^{2+}$ ions, respectively. The coordination is then completed with two oxygen ions lying at the corner of the crystal \\cite{Rau1966a,Holmes1966a}. Therefore, Eu$_3$O$_4$ has two Eu$^{3+}$ and one Eu$^{2+}$ ions per unit formula \\cite{Holmes1966a}.\n \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{Eu3O4_colour.jpg}\n\\caption{The orthorhombic structure of Eu$_3$O$_4$, in which the purple, magenta and green spheres represent the Eu$^{3+}$, Eu$^{2+}$ and O$^{2-}$ ions, respectively. The black box represents one unit cell of the Eu$_3$O$_4$ crystal.\n\\label{Eu3O4}\n\\end{figure}\n\nThe compound Eu$_3$O$_4$ has an antiferromagnetic arrangement below 5 K. Its bulk and powder forms show a metamagnetic behaviour below the N\u00e9el temperature (T\\textsubscript{N}), at a critical field of 2.4 kOe. Therefore, Eu$_3$O$_4$ is considered a potential material for magnetic refrigeration applications \\cite{Ahn2009a,Holmes1966,Holmes1966a}. Although Eu$_3$O$_4$ is a mixed-valence compound, its magnetic ordering is mainly determined by the Eu$^{2+}$ ions at low-temperature due to the high magnetic moment of Eu$^{2+}$ (total angular momentum, \\textit{J}$=7\/2$) in comparison to the Eu$^{3+}$ ions (\\textit{J}$=0$) \\cite{Holmes1966,Holmes1966a,Ahn2009a}. It has been proposed that the nearest neighbouring Eu$^{2+}$ ions are strongly coupled by ferromagnetic interactions at low temperature, whereas the distant ions are coupled by weaker antiferromagnetic coupling, resulting in the overall antiferromagnetic state of Eu$_3$O$_4$ \\cite{Holmes1966}.\n\nGraphene is a promising material for spintronics applications due to its desirable properties such as its long spin-diffusion length and high electron mobility. So far, no study on the graphene\/Eu$_3$O$_4$ system has been reported. This may well be due to the difficulty of growing Eu$_3$O$_4$, which is the unstable high-temperature phase of Eu-oxides. Therefore, this study presents one of the first fundamental steps towards understanding the exchange coupling between a Eu$_3$O$_4$ and graphene\n\nIn this study, 20 nm Eu$_3$O$_4$ thick films were grown on a Si\/SiO$_2$ substrate and graphene sheet supported on Si\/SiO$_2$ by molecular beam epitaxy (MBE) and capped with 5 nm of Au. Eu was deposited at high temperatures (300 - 600 $^{\\circ}$C) in an oxygen flux. The growth parameters such as the oxygen partial pressure, temperature and deposition rate were optimised to achieve a crystalline Eu$_3$O$_4$(001) phase. The\nstructural characterisation of the films was studied by X-ray diffraction (XRD) and reflection (XRR), where a superconducting quantum interference device magnetometer (SQUID) was used to study their magnetic properties. The results show a successful growth of crystalline, highly-textured Eu$_3$O$_4$ (001) films with a Curie temperature (\\textit{T}\\textsubscript{C}) of $\\sim\\,5$ K, which is in agreement with the value reported in Ref. [3]. Depth-profile X-ray photon electron spectroscopy (XPS) scans were performed to prove the mixed-valence of Eu cations in Eu$_3$O$_4$. Furthermore, Raman spectroscopy measurements on the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$ sample showed that although the growth of Eu$_3$O$_4$ film induced defects in the graphene sheet, the graphene retains its hexagonal lattice structure.\n\n\\section{Sample Preparation} \\label{SamplePrep}\n20 nm Eu$_3$O$_4$(001) films were deposited on cleaned Si\/SiO$_2$ and commercially purchased Si\/SiO$_2$\/graphene substrates by MBE with a base pressure of 4 $\\times$ 10$^{-10}$ mbar. The substrates were heated to 400$^{\\circ}$C, while the Eu was evaporated at a rate of 1.2 nm\/min. Oxygen was then introduced into the growth chamber, resulting in a partial pressure of 1.1 $\\times$ 10$^{-8}$ mbar to deposit Eu$_3$O$_4$(001) at a rate of 1.11 nm\/min. A 5 nm film of Au was grown subsequently on the Eu$_3$O$_4$(001) films to prevent them from oxidising to the most stable oxide phase of Eu (Eu$_2$O$_3$). The Au films were deposited at 45$^{\\circ}$C, with a rate of 0.057 nm\/min at a pressure of 1.9 $\\times$ 10$^{-10}$ mbar.\n\nA quartz crystal microbalance was used during the deposition to monitor the growth rate and thus the thicknesses of the layers. Room temperature (RT) XRD scans were used to study the crystallinity of the grown films. The XRR measurements were then performed to confirm the results of the microbalance readings and deduce the density and roughness between the layers. These acquisitions were carried out using a Bruker D8 Discover HRXRD with a Cu K$\\alpha$ monochromatic beam with a voltage of 40 kV and a current of 40 mA. The magnetic properties of the Eu$_3$O$_4$ films were studied using a Quantum Design SQUID.\n\nDepth-profile XPS scans using Al K$\\alpha$ X-ray source (1486.68 eV, beam width of 500 $\\mu$m) were performed on the Si\/SiO$_2$\/Eu$_3$O$_4$\/Au sample. This was done to study the homogeneity of the Eu$_3$O$_4$ film, determine the atomic ratio of Eu$^{2+}$ and Eu$^{3+}$ and confirm the mixed-valence character of the grown film. Furthermore, the effect of depositing Eu$_3$O$_4$ film on the graphene sheet was investigated by Raman spectroscopy measurements using a Renishaw InVia spectrometer (100$\\times$ objective, 10\\% laser power, spot size of $\\sim\\,1\\,\\mu$m, 0.5 s exposure time, a wavelength of 532 nm). However, the Au capping layer was selectively etched in KI\/I$_2$ solution before the measurements to eliminate the Au interference with the Raman measurements. The Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample was cleaved into $\\sim 2\\times 2$ mm square and placed in the etchant solution for 5 minutes at RT, rinsed with DI water twice, then with IPA and dried with dry N$_2$. Raman scans were then taken at every $50\\,\\mu$m in a grid pattern over an area of $1000\\,\\mu$m $\\times1000\\,\\mu$m for the region coated with Eu$_3$O$_4$, and every $10\\,\\mu$m over an area of $140\\,\\mu$m $\\times140\\,\\mu$m for the bare graphene surface.\n\n\\section{Results and Discussion} \n\\subsection{X-ray Diffraction (XRD)}\n\\begin{figure}[t!]\n\\includegraphics[width=1.0\\textwidth]{XrayEu3O4monochromatic.jpg}\n\\caption{(a) The RT XRD scan of  Si\/SiO$_2$\/Eu$_3$O$_4$(001)\/Au (black line) and Si\/SiO$_2$\/C\/Eu$_3$O$_4$(001)\/Au (red line) samples carried out between 20$^{\\circ}$ - 100$^{\\circ}$ using a monochromator and a 1D detector. The scan for the Si\/SiO$_2$\/C\/Eu$_3$O$_4$(001)\/Au sample is down-shifted by a factor of five for ease of comparison. (b) The XRR measurement of the Si\/SiO$_2$\/Eu$_3$O$_4$(001)\/Au sample (black line) and the corresponding fit (red line). The table lists the thickness, roughness and density of the deposited films as deduced from the fit.}\n\\label{Xray}\n\\end{figure}\nThe RT XRD scans (from 20$^{\\circ}$ - 100$^{\\circ}$) of the Eu$_3$O$_4$(001) films grown on the Si\/SiO$_2$ substrate and on graphene are shown in Figure~\\ref{Xray} (a). The XRD scans show highly textured Eu$_3$O$_4$(002) films with no sign of other oxide-phase of Eu or unreacted Eu within the detection limit of the set-up. Additional Eu$_3$O$_4$(004) and (008) peaks are observed in the scan of the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample, indicating that the underlying graphene layer improves the crystallinity of the Eu$_3$O$_4$ film. This is also proven by the smaller full-width at half-maximum (FWHM) of the Eu$_3$O$_4$ peaks of the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample compared to the Si\/SiO$_2$\/Eu$_3$O$_4$\/Au. Figure~\\ref{Xray} (b) shows the XRR scan and the corresponding fit for the Si\/SiO$_2$\/Eu$_3$O$_4$\/Au sample, whereas the deduced values for the thickness, density and roughness of the layers are listed in the inset table.\n\n\\subsection{SQUID}\nFigure~\\ref{SQUID} shows the field-cooled (FC) and zero field-cooled (ZFC) measurements of the Si\/SiO$_2$\/Eu$_3$O$_4$(001)\/Au and Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$(001)\/Au samples, respectively. Both show a \\textit{T}\\textsubscript{C} of $\\sim 5.5 \\pm 0.1$ K as can be deduced from the d\\textit{M}\/d\\textit{T} vs \\textit{T} (insets), which agrees with values reported in the literature \\cite{Holmes1966,Holmes1966a,Ahn2009a}. Therefore, care has to be given to check for impurities of Eu$_3$O$_4$ phase in EuO$_{1-x}$ thin films which sometimes show a pronounce bump at \\textit{T}$<20$ K \\cite{Liu2012e,Liu2013}.\n\\begin{figure}[t!]\n\\includegraphics[width=1.0\\textwidth]{Eu3O4_SQUID_both_samples.jpg}\n\\caption{(a) The 20 Oe FC \\textit{M} vs \\textit{T} measurement for Si\/SiO$_2$\/Eu$_3$O$_4$\/Au. The inset presents the d\\textit{M}\/d\\textit{T} vs \\textit{T} plot used to determine the \\textit{T}\\textsubscript{C}. (b) The ZFC \\textit{M} vs \\textit{T} measurement for the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample. The inset shows the d\\textit{M}\/d\\textit{T} vs \\textit{T} graph used to deduce the \\textit{T}\\textsubscript{C} of the Eu$_3$O$_4$.}\n\\label{SQUID}\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{ZFC_MvsH.jpg}\n\\caption{ZFC Isothermal magnetisation hysteresis loops of the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample measured as a function of the temperature at 2 K, 5 K and 10 K. The inset highlights the virgin magnetisation curves at these temperatures.}\n\\label{SQUID2}\n\\end{figure}\n\nThe ZFC isothermal magnetisation measurements as a function of the applied magnetic field for the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample at 2 K, 5K and 10 K are shown in Figure ~\\ref{SQUID2}. The hysteresis curves show that the grown Eu$_3$O$_4$ films exhibit ferromagnetic behaviour with a coercive field of 22 Oe. The inset highlights the virgin magnetisation curves at these temperatures. Although the XRD scans (Figure ~\\ref{Xray} (a)) and the \\textit{M} vs \\textit{T} measurements (Figure ~\\ref{SQUID}) prove the growth of Eu$_3$O$_4$(001) thin films, surprisingly, the virgin \\textit{M}$-$\\textit{H} curves show no metamagnetic transition even with an applied in-plane magnetic field of 3 kOe as reported for crystal and powder Eu$_3$O$_4$ \\cite{Holmes1966a,Ahn2009a}. This could be attributed to the strain from the substrate which could be resolved by growing a thicker film.\n\n\\subsection{XPS}\n\\begin{figure}[b!]\n\\includegraphics[width=1.0\\textwidth]{AtomicProfileSurvey.jpg}\n\\caption{(a) XPS etch profile of the Si\/SiO$_2$\/Eu$_3$O$_4$\/Au sample by Ar$^+$ plasma etching. The yellow shaded region highlights the area considered for the analysis of Eu cations, whereas the dashed vertical lines indicate $t=$210 s, 360 s, 450 s and 510 s. (b) The XPS survey spectra collected at \\textit{t}$=60$ s, \\textit{t}$=360$ s and \\textit{t}$=510$ s of the Si\/SiO$_2$\/Eu$_3$O$_4$\/Au highlighting the Au, C, Eu, Si and O peaks.}\n\\label{atomicProfile}\n\\end{figure}\n\\begin{figure}[b!]\n\\includegraphics[width=1.0\\textwidth]{XPS_Eu3d_merged.jpg}\n\\caption{The deconvoluted 3\\textit{d} XPS spectra of the Eu$_3$O$_4$ film on Si\/SiO$_2$ substrate using Al K$\\alpha$ source (1486.68 eV) after the Shirley background subtraction, measured at etching time (a) \\textit{t}=210 s, (b) \\textit{t}=360 s, (c) \\textit{t}=450 s and (d) \\textit{t}=510 s. The raw data (black line), fitting curve (red line), Eu$^{2+}$ 3\\textit{d} (blue shaded peaks), Eu$^{2+}$ multiplet satellites (yellow-shaded peaks), Eu$^{3+}$3\\textit{d} (green shaded peaks), Eu$^{3+}$ multiplet satellites (magenta shaded peaks) and plasmon excitations (grey shaded peaks).}\n\\label{XPS}\n\\end{figure}\nThe existence of mixed-valence Eu cations was investigated by performing depth-profile XPS scans while measuring the Eu 3\\textit{d} and 4\\textit{d} spectra simultaneously after Ar$^+$ plasma etching. Figure~\\ref{atomicProfile} (a) shows the XPS etch profile of the sample, whereas the XPS survey collected at \\textit{t}$=$210 s, 360 s, 450 s and 510 s, highlighting the different detected elements are shown in Figure ~\\ref{atomicProfile} (b). The 4\\textit{d} XPS spectra have a complicated structure (not shown) due to the strong unfilled 4\\textit{f}$-4$\\textit{d} hole interaction, whereas the 3\\textit{d} states have a weaker multiplet splitting and broader photoexcitation cross-section. Therefore, the latter is usually used to analyse the Eu XPS spectra and obtain a better estimation of the Eu initial valence \\cite{Caspers2011c,EnJinCho1996,Mariscal2018,EnJinCho1995}.     \n\nFigure~\\ref{XPS} (a) - (d) shows the Eu 3\\textit{d} XPS spectra after subtracting an optimised Shirley background, measured at \\textit{t}$=$210 s, 360 s, 450 s and 510 s. The peaks were deconvoluted using Gaussian-Lorentzian fitting, while the $\\chi^2$ value indicates the quality of the fit. Although the Eu$_3$O$_4$ layer was etched fully, only these scans were considered for the analysis of the Eu cation valency (the yellow shaded area of Figure~\\ref{atomicProfile} (a)) to minimise the effect of interdiffusion at the SiO$_2$\/Eu$_3$O$_4$ and Eu$_3$O$_4$\/Au interfaces and\nincrease the intensity of the Eu 3\\textit{d} and 4\\textit{d} peaks.\n\n\\begin{table}\n\\centering\n\\caption{The positions of the Eu$^{2+}$ and Eu$^{3+}$ 3\\textit{d} peaks, their FWHM and the position of their corresponding multiplet satellites. The atomic ratios of the Eu$^{2+}$ to Eu$^{3+}$ were deduced from the areas of the peaks. The table also lists the $\\chi^2$ value of the fittings.}\n\\begin{tabular}{ P{2cm} P{1.3cm} P{1.2cm} P{2.8cm} P{1.3cm} P{1.2cm} P{2.8cm} }\n\\hline\n {Spectrum} & \\multicolumn{6}{c}{Eu$^{2+}$ (eV)} \\\\\n \\cline{2-7}\n & \\centering{3\\textit{d}$_{5\/2}$} & \\centering{FWHM} & \\centering{3\\textit{d}$_{5\/2}$ Satellites} & \\centering{3\\textit{d}$_{3\/2}$} & \\centering{FWHM} & 3\\textit{d}$_{3\/2}$ Satellites \\\\\n\\hline\n\\textit{t}$=210$ s & 1125.54 & \\centering{5.39} & \\centering{1131.26} & \\centering{1155.14} & \\centering{5.29} & 1160.16 \\\\\n\\textit{t}$=360$ s & 1124.88 & \\centering{5.17} & \\centering{1130.64} & 1154.75 & \\centering{5.76} & 1160.04 \\\\\n\\textit{t}$=450$ s & 1124.78 & \\centering{5.01} & \\centering{1131.19} & 1154.65 & \\centering{5.64} & 1159.96 \\\\\n\\textit{t}$=410$ s & 1124.56 & \\centering{4.58} & \\centering{1131.07} & 1154.21 & \\centering{5.36} & 1159.58 \\\\\n\n \\multirow{2}{2cm}{} & \\multicolumn{6}{c}{Eu$^{3+}$(eV)} \\\\\n \\cline{2-7}\n & \\centering{3\\textit{d}$_{5\/2}$} & FWHM & \\centering{3\\textit{d}$_{5\/2}$ Satellites} & \\centering{3\\textit{d}$_{3\/2}$} & FWHM & 3\\textit{d}$_{5\/2}$ Satellites \\\\\n\\hline\n\\textit{t}$=210$ s & 1134.07 & \\centering{4.56} & \\centering{1142.39} & 1163.46 & \\centering{5.03} & 1166.92 \\\\\n\\textit{t}$=360$ s & 1133.49 & \\centering{4.69} & \\centering{1141.90} & 1162.96 & \\centering{4.85} & 1166.00 \\\\\n\\textit{t}$=450$ s & 1133.93 & \\centering{4.46} & \\centering{1142.29} & 1163.15 & \\centering{4.80} & 1166.00 \\\\\n\\textit{t}$=410$ s & 1133.83 & \\centering{4.56} & \\centering{1142.23} & 1163.15 & \\centering{5.38} & 1167.48 \\\\\n\\\\\n & \\multicolumn{2}{c}{\\centering{Atomic ratio 3\\textit{d}$_{5\/2}$}} & & \\multicolumn{2}{c}{\\centering{Atomic ratio 3\\textit{d}$_{3\/2}$}} & $\\chi^2$ \\\\\n \\cline{2-3} \\cline{5-6}\n & \\centering{Eu$^{2+}$} & \\centering{Eu$^{3+}$} & & \\centering{Eu$^{2+}$} & \\centering{Eu$^{3+}$} & \\\\\n \\hline\n\\textit{t}$=210$ s & \\centering{49.81} & \\centering{50.19} & & \\centering{21.39} & \\centering{78.61} & 9.30 \\\\\n\\textit{t}$=360$ s & \\centering{14.63} & \\centering{85.37} & & \\centering{26.84} & \\centering{73.16} & 11.30 \\\\\n\\textit{t}$=450$ s & \\centering{21.58} & \\centering{78.42} & & \\centering{31.61} & \\centering{68.39} & 18.20 \\\\\n\\textit{t}$=410$ s & \\centering{26.80} & \\centering{73.20} & & \\centering{31.96} & \\centering{68.04} & 20.70 \\\\\n\\cline {1-6}\nAverage & \\centering{28.20} & \\centering{71.80} & & \\centering{27.95} & 72.05 \\\\\n \\hline\n\\end{tabular}\n\\label{Table}\n\\end{table}\n\nAll spectra in Figure~\\ref{XPS} show the spin-orbit coupling (SOC) components, 3\\textit{d}$_{5\/2}$ and 3\\textit{d}$_{3\/2}$, for Eu$^{2+}$ and Eu$^{3+}$ separated by $\\Delta\\sim 29.5$ eV, which agrees with previously reported values \\cite{Mariscal2018,Caspers2011c,Orlowski2005}. They also show additional peaks at slightly higher binding energy (BE) to the SOC peaks for the Eu$^{2+}$ and Eu$^{3+}$. These shake-up satellite peaks arise as a result of the multiplet structures of the 4\\textit{f}$^7-3$\\textit{d} hole in the final state \\cite{EnJinCho1995}. Furthermore, the fast 3\\textit{d} photoelectrons create plasmon excitation structures observed as broad peaks at BE $\\sim1146$ eV and $\\sim1170$ eV \\cite{Caspers2011c}. The XPS spectra shown in Figure~\\ref{XPS} prove the mixed-valency of Eu cations as they agree well with previous work reported for Eu$^{2+}$ and Eu$^{3+}$ \\cite{Caspers2011c,Mariscal2018,Ohno2002,Kim2019,Cho1999}. Moreover, the average atomic ratio of the Eu$^{2+}$ to Eu$^{3+}$ in the 3\\textit{d}$_{5\/2}$ and 3\\textit{d}$_{3\/2}$ $\\sim\\,28:72$ is consistent with the values reported in Eu-doped ZnO \\cite{Lupan2013} and Eu-doped GaN nanowires \\cite{Faye2019a}. Table~\\ref{Table} summarises the positions of the Eu$^{2+}$ and Eu$^{3+}$ 3\\textit{d} peaks, their FWHM, their corresponding multiplet satellites, the ratio of Eu$^{2+}$\/Eu$^{3+}$ and the fits $\\chi^2$ values of the four spectra. \n\n\\subsection{Raman Spectroscopy}\nRaman spectroscopy is a versatile and non-destructive technique widely used to study the structural and electronic properties of graphene \\cite{Wang2008,Childres2013}. A good-quality monolayer of graphene has two main characteristic Raman peaks; the \\textit{G} and 2\\textit{D} peaks at $\\sim\\,1582$ cm$^{-1}$ and $\\sim\\,2700$ cm$^{-1}$, respectively. It can also possesses other disorder-induced peaks such as the \\textit{D} peak at $\\sim 1350$ cm$^{-1}$ \\cite{Allard2010,Malard2009,Shlimak2015a}. Therefore, the presence or absence of these peaks and the ratio of the intensity of the \\textit{D} peak to the intensity of the \\textit{G} peak (\\textit{I}$_D$\/\\textit{I}$_G$), which represents the defect density in the graphene structure, were mostly used to assess the quality of our graphene underlayer \\cite{Aboljadayel2021}.\n\nFigure~\\ref{RamanOptical} (a) and (b) show the microscopic optical images of the sample highlighting three different regions of the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample taken before etching the Au capping layer. The zoom-in image collected with a $\\times 100$ objective lens (Figure~\\ref{RamanOptical} (b)) shows that the graphene layer consists of a mixture of mono- and multilayer graphene domains rather than a continuous homogeneous monolayer, which could be either a result of the growth of the Eu$_3$O$_4$ film or the pristine quality of the commercial graphene. Therefore, one would expect the presence of defect-induced peaks in the Raman scans \\cite{Aboljadayel2021}.    \n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.75\\textwidth]{OpticalImages.jpg}\n\\caption{The microscopic optical images of the Si\/SiO$_2$\/graphene\/Eu$_3$O$_4$\/Au sample highlighting the different regions of the sample. (a) the bare and coated area of the Si\/SiO$_2$ substrate before the etching process, (b) a zoom-in view of the surface using a 100X objective lens showing the mixed domain structures of the graphene underlayer, (c) the graphene under the Eu$_3$O$_4$ layer before and (d) after etching the Au capping layer.}\n\\label{RamanOptical}\n\\end{figure}\n\nFigure~\\ref{RamanOptical} (c) and (d) show the microscopic optical images of the graphene edge under the Eu$_3$O$_4$ film before and after removing the Au layer, respectively. No significant change in the contrast is observed between the two images suggesting that the etching process did not remove or affect the graphene underlayer. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{RamanScans.jpg}\n\\caption{RT Raman spectroscopy measurements for (a) the bare graphene sheet on the Si\/SiO$_2$ substrate and (b) graphene under the Eu$_3$O$_4$ film after etching the Au capping layer.} \n\\label{RamanScans}\n\\end{figure}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{ID_IG_new.jpg}\n\\caption{The frequency distribution of the intensity of the \\textit{D} peak to the intensity of the \\textit{G} peak of (a) the bare graphene and (b) the graphene under the Eu$_3$O$_4$ film. Raman spectra of the regions with no graphene signal were removed from the statistics.} \n\\label{ID_IG}\n\\end{figure}\nFour random Raman scans taking on two different areas of the sample's surface; bare graphene and the graphene\/Eu$_3$O$_4$ region after removing the Au layer are shown in Figure~\\ref{RamanScans} (a) and (b), respectively. The emergence of the additional defect-induced peaks in the spectra of the graphene\/Eu$_3$O$_4$ area and the increase in their intensities (Figure~\\ref{RamanScans} (b)) compared to the scans of the bare graphene (Figure~\\ref{RamanScans} (a)) indicates that the growth of Eu$_3$O$_4$ film increased the defect density in the graphene structure. This is also seen by the shift in the \\textit{I}$_D$\/\\textit{I}$_G$ ratio of the graphene under the Eu$_3$O$_4$ film towards the higher values, compared to the bare graphene sheet (Figure~\\ref{ID_IG}). This is because \\textit{I}$_D$\/\\textit{I}$_G$ is known to be small for low-defect-density graphene \\cite{Shlimak2015a,Cabrero-Vilatela2016a}. However, since the underlayer graphene is visible with the optical microscope (Figure ~\\ref{RamanOptical}) and that the Raman characteristic features of graphene are maintained after the growth of the Eu$_3$O$_4$ film (Figure ~\\ref{RamanScans}), one would expect the graphene underlayer to retain its properties.\n\n\\section{Conclusion}    \\label{conclusion}\nIn summary, we discussed the experimental work carried out to study the growth of Eu$_3$O$_4$ thin film by MBE on Si\/SiO$_2$ and on a graphene sheet. The structural and magnetic characterisations show successful deposition of crystalline, highly-textured Eu$_3$O$_4$(001) films with a \\textit{T}\\textsubscript{C} of $\\sim 5.5\\pm0.1$ K. However, the films show no metamagnetic behaviour which could be attributed to the strain from the substrate. Furthermore, a qualitative analysis of the XPS scans confirms the mixed-valency of the Eu cation.\n\nRaman measurements show that the graphene layer retained its hexagonal lattice structure under the Eu$_3$O$_4$ film. Therefore, this study represents the first successful step towards integrating a Eu$_3$O$_4$ thin film in two widely used electronic substrates for future spintronics applications.  \n\n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nAuthors would like to acknowledge David Love and Pedro M S Monteiro for their support.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplementary Material: Learning Embeddings from Knowledge Graphs With Numeric Edge Attributes}\n\n\\section{Preliminaries}\\label{sec:background} \n\n\\paragraph{Knowledge Graph.}\nA knowledge graph $\\mathcal{G}=\\{ (s,p,o)\\} \\subseteq \\mathcal{E} \\times \\mathcal{R} \\times  \\mathcal{E}$ is a set of triples $t=(s,p,o)$ each including a subject $s \\in \\mathcal{E}$, a predicate $p \\in \\mathcal{R}$, and an object $o \\in \\mathcal{E}$. $\\mathcal{E}$ and $\\mathcal{R}$ are the sets of all entities and relation types of $\\mathcal{G}$.\n\n\n\\paragraph{Knowledge Graph with numeric-enriched triples.}\nIn a numeric-enriched knowledge graph $\\mathcal{G}$, each triple is assigned a numeric attribute $w \\in \\mathbb{R}$, leading to $\\mathcal{G}=\\{ t=(s,p,o,w)\\}$. It is worth noting that we are not tied to a specific numeric attribute semantics, as these numbers may encode importance, uncertainty, strength, etc. \nFor example, Figure~\\ref{fig:kg} assumes that numeric values indicate the importance of a link. The triple \\texttt{(MLEngineer, requiresSkill, Python, 0.98)} is therefore more important than \\texttt{(MLEngineer, requiresSkill, JavaScript, 0.13)}. \n\nNote $w$ can be either defined at a predicate level or at triple level. In this paper, we assume that $w$ is triple-specific. %\n\n\n\n\\paragraph{Knowledge Graph Embedding Models.}\nKnowledge graph embedding models (KGE) are neural architectures designed to predict missing links between entities. KGE encode both entities $\\mathcal{E}$ and relations $\\mathcal{R}$ into low-dimensional, continuous vectors $\\in \\mathbb{R}^k$ (i.e, the embeddings). \nKnowledge graph embeddings are learned by training a neural architecture over a training knowledge graph: an input layer feeds training triples to an embedding lookup layer that retrieves embeddings of entities and relations. \n\nA scoring layer $f(t)$ assigns plausibility scores to each triple. The scoring layer is designed to assign high scores to positive triples and low scores to negative corruptions. \nMost of literature differs in the design rationale of $f(t)$.\nFor example, the scoring function of TransE~\\cite{bordes2013translating} computes a similarity between the embedding of the subject $\\mathbf{e}_{s}$ translated by the embedding of the predicate $\\mathbf{e}_{p}$ and the embedding of the object $\\mathbf{e}_{o}$. %\n\n\\textit{Corruptions} are synthetic negative triples generated by a corruption generation layer that follows the protocol proposed in~\\cite{bordes2013translating}:\nwe define a corruption of $t$ as $t^-=(s,p,o')$ or $t^-=(s',p,o)$ where $s', o'$ are respectively subject or object corruptions (i.e. other entities randomly selected from $\\mathcal{E}$). We generate synthetic negatives by corrupting one side of the triple at a time to comply with the local closed world assumption~\\cite{nickel2016review}.\n\nFinally, a loss layer optimizes the embeddings by maximizing the margin between positive triples $t$ and corruptions $t^-$. In other words, the goal of the optimization procedure is learning optimal embeddings, such that at inference time the scoring function $f(t)$ assigns high scores to triples likely to be correct and low scores to triples unlikely to be true.\n\n\n\n\\textbf{Link Prediction.} \nThe task of predicting unseen triples in knowledge graphs is formalized in literature as a learning to rank problem, where the objective is learning a scoring function $f(t=(s, p, o)): \\mathcal{E} \\times \\mathcal{R} \\times \\mathcal{E} \\rightarrow \\ensuremath{\\mathbb{R}}$ that given an input triple $t=(s,p,o)$ assigns a score $f(t) \\in \\ensuremath{\\mathbb{R}}$ proportional to the likelihood that the fact $t$ is true. \nSuch predictions are ranked against predictions from synthetic corruptions, to gauge how well the model tells positives from negatives.\n\n\\paragraph{Link Prediction with numeric-enriched triples.}\nIn this paper we predict probability estimates for unseen numeric-enhanced triples $t=(s, p, o, w)$. \nThe task is formalized as the same learning to rank problem described above for conventional link prediction.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe show that by plugging an additional layer we can make a conventional KGE architecture aware of numeric values associated to triples. This leads to models that better discriminate high-valued and low-valued triples, regardless of the semantics of the numeric attributes, and without requiring additional out-of-band rules (unlike UKGE). \n\n\nFuture work will investigate the capability of predicting numeric values associated to unseen triples. We will also extend our approach to support multiple numeric attributes associated to the same triple.\n\n\n\\section{Experiments}\\label{sec:eval}\nWe assess the predictive power of FocusE on the link prediction task with numeric-enriched triples. Experiments show that FocusE outperforms conventional KGE models and its closest direct competitor UKGE~\\cite{chen2019embedding} in discriminating low-valued triples from high-valued ones.\n\n\\textbf{Datasets}\nWe experiment with three publicly available benchmark datasets originally proposed by~\\cite{chen2019embedding}. Triples are associated to numeric values $w$ interpreted as uncertainty values. We also introduce a fourth dataset, where numeric values $w$ encode the \\textit{importance} of each link. \nAll datasets include triple-specific $w$ values.\nTable~\\ref{table:kgstats} shows the statistics of all the datasets used for the experiments.\n\n\n\\begin{itemize}\n  \\item \\textbf{CN15K}~\\cite{chen2019embedding}. A subset of ConceptNet~\\cite{CN}, a common sense knowledge graph built to represent general human knowledge. Numeric values on triples represent uncertainty.\n  \\item \\textbf{NL27K}~\\cite{chen2019embedding}. A subset of the Never-Ending Language Learning (NELL)~\\cite{mitchell2018never} dataset, which collects data from web pages. Numeric values on triples represent link uncertainty.\n  \\item \\textbf{PPI5K}~\\cite{chen2019embedding}. Knowledge graph of protein-protein interactions~\\cite{PPI}. Numeric values represent the confidence of the link based on existing scientific literature evidence.\n\n  \\item \\textbf{O*NET20K}\\footnote{\\texttt{\\url{https:\/\/docs.ampligraph.org\/en\/latest\/ampligraph.datasets.html}}}. \nWe introduce a subset of O*NET~\\footnote{\\texttt{\\url{https:\/\/www.onetonline.org\/}}}, a dataset that includes job descriptions, skills and labeled, binary relations between such concepts. Each triple is labeled with a numeric value that indicates the \\textit{importance} of that link. \nUnlike the other datasets, we built a test set that includes a single predicate type that connects jobs to skills, under the assumption that we are interested in predicting that type of links only. This is done to mock up single-target link prediction tasks adopted in applied knowledge discovery scenarios.\n\n\n\\end{itemize} \n\n\\begin{table}[t]\n  \\centering\n  \\footnotesize\n  \\setlength{\\tabcolsep}{5pt}\n  \\begin{tabular}{l cccc}\n      \\toprule           & O*NET20K & CN15K & NL27K & PP15K\\\\ \n      \\midrule\n      Training              & 461,932 & 204,984 & 149,100 & 230,929            \\\\ \n      Validation$^\\ddagger$  & 138 & 3532 & 8161 & 1940            \\\\ \n      \\multirow{2}{*}{\\shortstack[l]{Test \\\\(top 10\\%)$^*$}}    & \\multirow{2}{*}{200} & \\multirow{2}{*}{1,929} & \\multirow{2}{*}{1,402} & \\multirow{2}{*}{2,172}       \\\\ \n      &&&& \\\\\n      \\multirow{2}{*}{\\shortstack[l]{Test \\\\(bottom 10\\%)$^*$}} & \\multirow{2}{*}{200} & \\multirow{2}{*}{1,929} & \\multirow{2}{*}{1,402} & \\multirow{2}{*}{2,172}       \\\\ \n      &&&& \\\\\n\n      Entities              & 20,643 & 15,000 & 27,221 & 5,000 \\\\ \n      Relations             & 19 & 36 & 404 & 7 \\\\ \n      \\bottomrule\n  \\end{tabular}\n  \\caption{Datasets used in experiments. ($^*$) test sets include either top-10\\% valued triples or bottom 10\\%. All $w$ are normalized, so that $w \\in [0-1]$. ($^\\ddagger$) validation sets only include high-valued triples where $w \\geq 0.8$.}\n  \\label{table:kgstats}\n\\end{table}\n\n\\textbf{Implementation Details and Baselines.}\nFocusE and all baselines are implemented with the AmpliGraph library~\\cite{ampligraph} version 1.4.0, using TensorFlow 1.15.2 and Python 3.7. Code and experiments are available at \\texttt{\\url{https:\/\/github.com\/Accenture\/AmpliGraph}}.\nWe experiment with three popular KGE baselines: TransE, DistMult, ComplEx. For each baseline and for FocusE, we carried out extensive grid search, over the following ranges of hyperparameter values: embedding dimensionality $k=[200-600]$, with a step of 100; baseline losses=\\{negative log-likelihood, multiclass-NLL, self-adversarial\\}; synthetic negatives ratio $\\eta=\\{5, 10, 20, 30\\}$; learning rate$=\\{\\num{1e-3}, \\num{5e-3}, \\num{1e-4}\\}$; epochs$=[100-800]$, step of 100; L3 regularizer, with weight $\\gamma=\\{\\num{1e-1}, \\num{1e-2}, \\num{1e-3}\\}$.\nFor FocusE we also tuned the decay $\\lambda=[100-800]$, with increments of 100. \nBest combinations for all experiments are reported in Appendix~\\ref{app:hyper}.\n\nFor experiments with UKGE we modified\\footnote{The original UKGE codebase does not perform a fair rank computation: the authors assign rank=1 when $t$ and $t^-$ have the same score, whereas we assign a lower rank, as per agreed upon practice in the community. For example, if we have $\\eta=1,000$ corruptions $t^-$ which are all assigned the same score as $t$, we assign $t$ a rank=1,000 whereas UKGE ranks it first. To guarantee a fair comparison, we aligned UKGE to our procedure.} the original codebase\\footnote{\\texttt{\\url{https:\/\/github.com\/stasl0217\/UKGE}}} provided by the authors to generate learning to rank metrics required by our experimental protocol. We used the best hyperparameter configuration proposed by the authors.\nNote we do not evaluate UKGE on O*NET20K, since the model requires additional logical rules which are not available for this dataset.\n\nAll experiments were run under Ubuntu 16.04 on an Intel Xeon Gold 6142, 64 GB, equipped with a Tesla V100 16GB.\n\n\n\\textbf{Evaluation protocol.}\nWe adopt the standard evaluation protocol described by \\cite{bordes2013translating} to all datasets described above. We predict whether each triple $t=(s,p,o) \\in \\mathcal{T}$ is a positive fact, where $\\mathcal{T}$ is a disjoint held-out test set that includes only positives triples. \nWe cast the problem as a learning-to-rank task: for each $t=(s,p,o) \\in \\mathcal{T}$, we generate synthetic negatives $t^- \\in \\mathcal{N}_t$ by corrupting one side of the triple at a time (i.e. either the subject or the object). \nWe predict a score for each $t$ and all its negatives $t^- \\in \\mathcal{N}_t$. We then rank the only positive $t$ against all the negatives $\\mathcal{N}_t$ \\footnote{As for standard protocol, we used distinct entities twice, leading to CN15K=$\\sim$30k, NL27K=$\\sim$54k, PP15K=10k, O*NET20K=$\\sim$40k synthetic negatives (Table~\\ref{table:kgstats}).}. \nWe report learning to rank metrics such as mean rank (MR), mean reciprocal rank (MRR), and Hits at $n$ (where $n=1, 10$) by filtering out spurious ground truth positives from the list of generated corruptions (i.e. ``filtered'' metrics).\n\n\n\\subsection{Predicting High-Valued Links}\\label{sec.top10}\nFirst, we assess how well FocusE predicts triples associated to \\textit{high} numeric values. This is an important task in many applicative scenarios: high numeric values usually imply low uncertainty, high strength or importance, thus leading to more valuable newly-discovered knowledge.\nTable~\\ref{table:high-p} reports, for each dataset, the best mean rank, MRR, and Hits@\\{1,10\\} computed over test sets that include only triples associated to the top-10\\% numeric attributes. Our goal is learning to assign low ranks to test triples (i.e. rank=1 being the best outcome).\n\nResults show that FocusE brings better or very similar MRR to traditional, numeric-unaware baselines:\non O*NET20K, FocusE increases MRR for all models, and it outperforms the best baseline by 14 base points (MRR marginally differs on CN15K, NL27K, and PPI5K).\n\nExperiments show that FocusE outperforms UKGE: on CN15K, MRR is 15 base points higher, 19 points higher on NL27K, and up to 30 points better for PPI15K (UKGE$_{rect}$ fails to provide actionable results\\footnote{Due to a problem in the UKGE codebase, UKGE predictive power is MRR=0, hence \"not actionable\".} on PPI15K). \nIt is worth mentioning that UKGE requires additional external rules, hence the absence of results for O*NET20K. FocusE achieves better predictive power, without requiring additional out-of-band rules.\n\n\\begin{table}\n  \\centering\n  \\footnotesize\n  \\setlength\\tabcolsep{2pt}\n  \\begin{tabular}{ll cccc}\n\n      \\toprule           \n      & & \\multicolumn{4}{c}{\\textbf{MRR} (bottom 10\\%)} \\\\\n      & & O*NET20K & CN15K & NL27K & PP15K\\\\ \n      \\midrule\n      TransE &         & .13 & .05 & .36 & .26            \\\\ \n      DistMult &       & .15 & .05 & .83 & .97             \\\\ \n      ComplEx&         & \\textbf{.12} & .08 & .81 & .95       \\\\ \n      UKGE$_{rect}$ &  &  - & \\textbf{.02} & .28 & .00$^*$ \\\\ \n      UKGE$_{logi}$ &  &  - & \\textbf{.02} & .36 & .56 \\\\\n      \\multirow{3}{*}{\\textbf{FocusE}}               \n        & TransE       & .14  & .04 & \\textbf{.26} & \\textbf{.08}       \\\\ \n        & DistMult     & .15 & .05 & .70 & .72       \\\\ \n        & ComplEx      & .13 & .08 & .53 & .49       \\\\  \n      \\bottomrule\n  \\end{tabular}\n  \\caption{Predictive power of FocusE on the 10\\% low-valued triples. Lower = better. Best results in bold. ($^*$) UKGE$_{rect}$ fails to produce actionable results on PPI5K.}\n  \\label{table:low-p}\n\\end{table}\n\n\n\\begin{table*}\n  \\centering\n    \\small\n\n  \\begin{tabular}{ll @{\\extracolsep{8pt}} cccc  cccc}\n      & & \\multicolumn{3}{c}{\\textbf{O*NET20K} (top 10\\%)}  &  & & \\multicolumn{3}{c}{\\textbf{CN15K} (top 10\\%)}       \\\\\n      \\cline{3-6} \\cline{7-10}\n\n      & & & & \\multicolumn{2}{c}{Hits@}  &    & & \\multicolumn{2}{c}{Hits@}    \\\\\n      \\cline{5-6} \\cline{9-10}\n\n      & & MR & MRR & 1  & 10 & MR & MRR & 1  & 10  \\\\\n\n       \\midrule\n\n        \\multicolumn{2}{l}{TransE} \n        & \\underline{8}      & .37          & .00       & .88\n        & \\underline{623}     & .19          & .03       & \\textbf{.47}  \\\\\n\n        \\multicolumn{2}{l}{DistMult}  \n        & 13      & \\underline{.49}         & \\underline{.33}       & .82\n        & 942      & .22      & .15       & .37     \\\\\n\n        \\multicolumn{2}{l}{ComplEx} \n        & 26      & .32          & .19      & .59\n        & 863      & \\underline{.28}      & \\textbf{.22}       & .40     \\\\\n\n        \\multicolumn{2}{l}{UKGE$_{rect}$} \n        & -      & -          & -       & -\n        & 984      & .14         & .05       & .32  \\\\\n\n        \\multicolumn{2}{l}{UKGE$_{logi}$} \n        & -      & -          & -       & -        \n        & 2214      & .13         & .08       & .22  \\\\\n        \n        \\midrule\n\n        \\multirow{3}{*}{\\textbf{FocusE}}\n        & TransE\n        & \\textbf{4}      & .41      & .0       & \\textbf{.94}        \n        & \\textbf{438}      & .18      & .02     & \\textbf{.47}     \\\\\n        \n        & DistMult\n        & 102      & \\textbf{.63}      & \\textbf{.52}      & \\underline{.85}         \n        & 1552      & .24      & .17      & .37     \\\\\n        \n        & ComplEx\n        & 9      & .46      & .23       & .83     \n        & 840      & \\textbf{.29}      & \\underline{.21}       & \\underline{.45}     \\\\\n\n  \\end{tabular}\n  \\caption*{}\n\n\n  \\begin{tabular}{ll @{\\extracolsep{8pt}} cccc  cccc}\n      & & \\multicolumn{3}{c}{\\textbf{NL27K} (top 10\\%)}  &  & & \\multicolumn{3}{c}{\\textbf{PP15k} (top 10\\%)}       \\\\\n      \\cline{3-6} \\cline{7-10}\n\n      & & & & \\multicolumn{2}{c}{Hits@}    &  & & \\multicolumn{2}{c}{Hits@}    \\\\\n      \\cline{5-6} \\cline{9-10}\n\n      & & MR & MRR & 1  & 10 & MR & MRR & 1  & 10  \\\\\n\n       \\midrule\n\n        \\multicolumn{2}{l}{TransE} \n        & \\underline{91}      & .56          & .38       & .86         \n        & 6      & .42          & .00       & .94     \\\\\n\n        \\multicolumn{2}{l}{DistMult}  \n        & 130      & .83      & .77       & \\underline{.93}        \n        & 6      & \\textbf{.97}      & \\textbf{.96}       & \\textbf{.99}     \\\\\n\n        \\multicolumn{2}{l}{ComplEx} \n        & \\textbf{90}      & \\textbf{.87}      & \\textbf{.82}       & \\textbf{.94}        \n        & 10      & \\textbf{.97}      & \\underline{.95}      & .\\textbf{.99}     \\\\\n        \n        \\multicolumn{2}{l}{UKGE$_{rect}$} \n\n        & 169      & .61      & .50      & .80     \n        & 4943    & .00        & .00         & .00 \\\\\n\n        \\multicolumn{2}{l}{UKGE$_{logi}$} \n        & 299      & .65         & .56       & .80\n        & \\textbf{2}      & .65        & .37      & \\textbf{.99}  \\\\\n\n\n        \\midrule\n\n        \\multirow{3}{*}{\\textbf{FocusE}}\n        & TransE\n        & \\textbf{90}     & .57      & .37       & .88         \n        & \\underline{4}      & .43      & $<.01$       & \\underline{.96}     \\\\\n        \n        & DistMult\n        & 140      & .82      & .77       & .92         \n        & 16      & \\underline{.96}      & .93       & \\textbf{.99}     \\\\\n        \n        & ComplEx\n        & 224     & \\underline{.84}      & \\underline{.79}     & .92         \n        & 20      & .95      & .92       & \\textbf{.99}     \\\\\n\n  \\end{tabular}\n\n\n\\caption{Predicting high-valued links: ranking metrics computed on test sets that consist in the top-valued 10\\% triples. Filtered metrics. Best results in bold, second best underlined.}\n\\label{table:high-p}\n\\end{table*}\n\n\n\n\n\n\n\\begin{table*}\n  \\centering\n    \\small\n\n  \\begin{tabular}{ll @{\\extracolsep{8pt}} ccc  ccc}\n      & & \\multicolumn{3}{c}{\\textbf{O*NET20K}} & \\multicolumn{3}{c}{\\textbf{CN15K}}       \\\\\n      \\cline{3-5} \\cline{6-8}\n\n      & & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(top 10\\%)}} & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(bottom 10\\%)}}   & \\multirow{2}{*}{$\\Delta$MRR} \n      & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(top 10\\%)}} & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(bottom 10\\%)}}   & \\multirow{2}{*}{$\\Delta$MRR}\\\\ \n      &&&& \\\\\n\\midrule\n        \\multicolumn{2}{l}{TransE} \n        & .37      & .13          & .24  \n        & .19      & .05          & \\textbf{.14}   \\\\\n\n        \\multicolumn{2}{l}{\\textbf{FocusE} TransE} \n        & .41      & .14          & \\textbf{.27 (+13\\%)} \n        & .18      & .04          & \\textbf{.14}   \\\\\n        \n        \\midrule\n\n        \\multicolumn{2}{l}{DistMult}  \n        & .49      & .15          & .34  \n        & .22      & .05          & .17   \\\\\n        \n        \\multicolumn{2}{l}{\\textbf{FocusE} DistMult} \n        & .63      & .15          & \\textbf{.48 (+41\\%)} \n        & .24      & .05          & \\textbf{.19 (+12\\%)}   \\\\\n        \n        \\midrule\n\n        \\multicolumn{2}{l}{ComplEx} \n        & .32      & .12          & .20  \n        & .28      & .08         & .20   \\\\\n        \n         \\multicolumn{2}{l}{\\textbf{FocusE} ComplEx} \n        & .46      & .13         & \\textbf{.33 (+65\\%)} \n        & .29      & .08          & \\textbf{.21 (+5\\%)}   \\\\\n        \n\n        \\midrule\n\n        \\multicolumn{2}{l}{UKGE$_{rect}$} \n        & -      & -         & - \n        & .14      & .02         & .12   \\\\\n\n        \\multicolumn{2}{l}{UKGE$_{logi}$} \n        & -      & -         & - \n        & .13      & .02         & .11   \\\\\n        \n         \\multicolumn{2}{l}{\\textbf{FocusE} ComplEx} \n        & .46      & .13         & .33 \n        & .29      & .08          & \\textbf{.21 (+75\\%)}   \\\\\n    \n\n\n\n\n  \\end{tabular}\n    \\caption*{}\n  \n  \\begin{tabular}{ll @{\\extracolsep{8pt}} ccc  ccc}\n      & & \\multicolumn{3}{c}{\\textbf{NL27K}} & \\multicolumn{3}{c}{\\textbf{PPI5K}}       \\\\\n      \\cline{3-5} \\cline{6-8}\n\n      & & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(top 10\\%)}} & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(bottom 10\\%)}}   & \\multirow{2}{*}{$\\Delta$MRR} \n      & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(top 10\\%)}} & \\multirow{2}{*}{\\shortstack[c]{MRR\\\\(bottom 10\\%)}}   & \\multirow{2}{*}{$\\Delta$MRR}\\\\ \n      &&&& \\\\\n\\midrule\n        \\multicolumn{2}{l}{TransE} \n        & .56      & .36          & .20  \n        & .42      & .26         & .16   \\\\\n        \n        \\multicolumn{2}{l}{\\textbf{FocusE} TransE} \n        & .57      & .26          & \\textbf{.31 (+55\\%)}\n        & .43      & .08          & \\textbf{.35 (+119\\%)}   \\\\\n        \n        \\midrule\n        \n        \\multicolumn{2}{l}{DistMult}  \n        & .83      & .83          & $<.01$  \n        & .97      & .97         & $<.01$   \\\\\n       \n        \\multicolumn{2}{l}{\\textbf{FocusE} DistMult}  \n        & .82      & .70          & \\textbf{.12 (+1,100\\%)}  \n        & .96      & .72          & \\textbf{.24 (+2,300\\%)}   \\\\\n        \n        \\midrule\n        \n        \\multicolumn{2}{l}{ComplEx} \n        & .87      & .81          & .06  \n        & .97      & .95          & .02   \\\\\n        \n        \\multicolumn{2}{l}{\\textbf{FocusE} ComplEx} \n        & .84      & .53          & \\textbf{.31 (+417\\%)}  \n        & .95      & .49          & \\textbf{.46 (+2,200\\%)}   \\\\\n \n\n         \\midrule\n\n        \\multicolumn{2}{l}{UKGE$_{rect}$} \n        & .61      & .28         & \\textbf{.33} \n        & .00      & .00         & .00   \\\\\n\n        \\multicolumn{2}{l}{UKGE$_{logi}$} \n        & .65      & .36         & .29 \n        & .65      & .56         & .09   \\\\\n        \n         \\multicolumn{2}{l}{\\textbf{FocusE} ComplEx} \n        & .84      & .53          & .31 (-6\\%)  \n        & .95      & .49          & \\textbf{.46 (+411\\%)}   \\\\\n          \n  \\end{tabular}\n\n\n\\caption{High-valued vs low-valued triples discriminative power: FocusE brings larger differences in MRR across the board, showing better capabilities at correctly ranking the top-10\\% high-valued triples vs the bottom 10\\%. Best results in bold.}\n\n\\label{table:deltaMRR}\n\\end{table*}\n\n\n\\subsection{Discriminating High and Low-Valued Links}\nThe primary goal of FocusE is achieving a clear separation between scores assigned to high and low-valued triples. %\nTo assess how well FocusE can tell high-valued links from low-valued, we look at two complementary tasks: how the model performs on test triples with high values (top 10\\%) and on triples with low values (bottom 10\\%). We collect the respective MRR results and we compute the difference $\\Delta$MRR. The higher the difference, the better. \nNote this task differs from what traditionally done by KGE models, which are designed to tell positives from synthetic negatives only, thus not being able to discriminate between low and high-valued triples.\n\n\nSuch prior art shortcoming is evident from results in Table~\\ref{table:deltaMRR}. Models trained using FocusE have a much wider $\\Delta$MRR compared to traditional baselines. \nUsing link numeric values of training triples, FocusE better discriminates triples with higher values in the test set from the ones with lower values (see also Table~\\ref{table:low-p}  for details on the bottom 10\\% links). \nIn particular, on O*NET20K, NL27K and PPI5K, FocusE models perform exceptionally better, with $\\Delta$MRR being consistently higher than traditional KGE baselines. \nFocusE also outperforms UKGE by a large margin (75\\%) on CN15K. It provides a large margin of 40 base points on PPI5K as well, i.e. a 411\\% increase (UKGE$_{rect}$ fails to provide actionable results on PPI15K).\n\nAppendix \\ref{app:violin} shows additional evidence of FocusE outperforming baselines, by comparing distributions of scores (logits).\n\n \n\n\n\\subsection{Decay Impact}\\label{sec:decay}\n\nThe structural influence $\\beta$ affects the impact of numeric values $w$. If $\\beta=1$, FocusE falls back to the underlying KGE model. If $\\beta=0$, the original numeric values $w$ have maximum impact throughout the entire training. \nDuring training we decay $\\beta$ linearly, until it reaches 0. \n\nIn this experiment we vary the decay $\\lambda$, while freezing the other hyperparameters to ($k = 300$, $\\eta=30$, NLL of normalized softmax scores loss (Equation~\\ref{eq:loss}), Adam optimizer with learning rate \\num{1e-3}, L3 regularizer, with weight $\\gamma=\\num{1e-2}$. \nWe choose different decay epoch values $\\lambda=\\{0, 100, 200, 300, 400, 500, 600, 700, 800\\}$.\nWe report changes in MRR on NL27K's top 10\\% test set by experimenting on TransE, DistMult, and ComplEx. \nFigure~\\ref{fig:decay} shows that performance improves if $\\lambda$ increases. In most cases model performance saturates when $\\lambda > 400$ epochs. %\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.7\\columnwidth]{img\/decay_impact.png}\n\\caption{Impact of $\\lambda$, the decay of structural influence $\\beta$ on model performance over n-epochs for NL27k dataset.}\n\\label{fig:decay}\n\\end{figure}\n\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\nKnowledge graphs are graph-based knowledge bases whose facts are modeled as labeled, directed edges between entities. Whether it is a social network, a bioinformatics dataset, or retail purchase data, modelling knowledge as a graph lets organizations capture patterns that would otherwise be overlooked. Research led to broad-scope graphs such as DBpedia~\\cite{auer2007dbpedia}, WordNet, and YAGO~\\cite{suchanek2007yago}. Countless domain-specific knowledge graphs have also been published on the web, giving birth to the so-called Web of Data~\\cite{bizer2011linked}.\n\nKnowledge graph embeddings (KGE) are a family of graph representation learning methods that learn vector representations of nodes and edges of a knowledge graph. They are applied to graph completion, knowledge discovery, entity resolution, and link-based clustering, just to cite a few~\\cite{nickel2016review}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.7\\columnwidth]{img\/kg_eg.png}\n\\caption{A Knowledge graph with numeric attributes associated to triples.}\n\\label{fig:kg}\n\\end{figure}\n\n\n\n\n\nIn \\textit{multimodal} knowledge graphs, entity nodes are associated to attributes from different data modalities, such as numbers, text, images. \nIn this paper we deal with a specific flavor of multimodal knowledge graphs, that is graphs with numeric-enriched edges, either at predicate-level (i.e. a number assigned to each predicate \\textit{type}), or specific to each triple (this latter case is shown in Figure~\\ref{fig:kg}).\nNumeric-enriched triples have a prominent role in a growing number of applicative scenarios, from protein networks to workforce knowledge bases. Such numeric values may represent edge uncertainty (i.e. uncertain knowledge graphs) as in ConceptNet~\\cite{speer2017conceptnet}; trust in the automated or semi-automated data extraction workflow used to build the graph~\\cite{mitchell2018never}; out-of-band knowledge from wet lab experiments~\\cite{PPI}; edge importance or link strength.\n\nA number of works in knowledge graph representation learning literature support multimodal information and leverage numeric values associated to node entities to generate better embeddings for enhanced link prediction~\\cite{garcia2017kblrn,kristiadi2019incorporating,wu2018knowledge}.\nNevertheless, such models are not designed to learn from numeric values associated to \\textit{edges} of a knowledge graph. With the notable exception of \\cite{chen2019embedding}, that however is designed for uncertain graphs only, supporting numeric is to date still an under-researched direction.\n\nIn this work, we focus on the task of predicting probability estimates of missing links in knowledge graphs with numeric-enhanced triples. We claim that a model must take into account such numeric literals. Regardless of their semantics, we operate under the assumption that such numeric values intensify or mitigate the probability of existence of a link. That implies treating triples with low-numeric values as ``pseudo-negatives''. Traditional embedding models put at the same level low-valued and high-valued triples. However, in many real-world scenarios it is important to make such distinction (e.g. predicting interactions in protein networks~\\cite{PPI}).\n\n\n\nWe propose FocusE, an add-on layer for knowledge graph embeddings to enhance link prediction with edge-related numeric literals. FocusE works with any existing KGE model that adopts the standard negatives generation protocol~\\cite{bordes2013translating}. \nWe use edge numeric literals to modulate the margin between the scores of true triples and their corresponding negative corruptions.\nInspired by Focal Loss~\\cite{Lin2017} that aims at sparser hard examples by modulating the loss function, we leverage numeric literals to ``focus'' traditional KGE models on triples with higher numeric values.\n\nExperiments show that models trained using FocusE outperform numeric-unaware baselines, in particular in discriminating triples with high-numeric attributes from those associated to low values.\n\n\n\n\n\\section{FocusE}\\label{sec:method}\n\nWe present FocusE, an add-on layer for knowledge graph embedding architectures designed for link prediction with numeric-enriched triples.\nFocusE takes into account numeric literals associated to each link. Regardless of their semantics, we operate under the assumption that numeric values intensify or mitigate the probability of existence of a link.\nFor example, given numeric values $w$ in the $[0-1]$ range, we assume that high values identify triples with higher chances of being true, low scores single out weak or unlikely relations, and $w=0$ triples are considered negative samples. \n\nFocusE consists in a plug-in layer that fits between the scoring and loss layers of a conventional KGE method and it is designed to be used during training (Figure~\\ref{fig:arch}).\nUnlike traditional architectures, before feeding the scoring layer to a loss function, we modulate its output based on numeric values associated to triples, to obtain ``focused'' scores. \nWe leverage numeric values associated to triples so that during training the model focuses on triples with higher numeric values. \nWe want our model to learn from training triples with high numeric values, and at the same time use edge numeric values to maximise the margin between scores assigned to true triples and those assigned to their corruptions. This increases the loss of the model and helps it focus on triples with higher values.\nOur contribution is described below in detail.\n\n\nLet $t=(s,p,o)$ be a positive triple and $w>0$ its numeric-value. We define a corruption of $t$ as $t^-=(s,p,o')$ or $t^-=(s',p,o)$. where $s', o'$ are respectively subject or object corruptions.\n\nLet $f(t)$ be the scoring function of a KGE model. In the case of TransE~\\cite{bordes2013translating} this is:\n\n\\begin{equation}\nf(t) = -||\\mathbf{e}_{s} + \\mathbf{r}_{p} - \\mathbf{e}_{o}||_n\n\\end{equation}\n\nwhere $\\mathbf{e}_{s}$, $\\mathbf{r}_{p}$ and $\\mathbf{e}_{o}$ are the embeddings of the subject $s$, predicate $p$, and object $o$.\n\nWe use a softplus non-linearity $\\sigma$ to make sure the scores returned by $f(t)$ are greater or equal to zero, without introducing excessive distortion: %\n\n\\begin{equation}\ng(t) = \\sigma(f(t))= \\ln{(1+e^{f(t)})} \\geq 0\n\\end{equation}\n\n\nTo take into account the effect of numeric values associated to triples, we define a modulating factor $\\alpha \\in \\mathbb{R}$ which is responsible to strike a balance between the influence of the structure of the graph and the impact of the numeric values associated to each triple:\n\n\\begin{equation}\n\\alpha =\n\\begin{cases}\n \\beta + (1-w)(1-\\beta) & \\mbox{if \\,} t \\\\\n \\beta + w(1-\\beta) & \\mbox{if \\,} t^-\n\\end{cases}\n\\label{eq:alpha}\n\\end{equation}\n\nwhere $\\beta \\in [0,1]$ is the \\textit{structural influence}, an hyperparameter that modulates the influence of graph topology, and $w \\in \\mathbb{R}$ is the numeric value associated to the \\textit{positive} triple $t$. \n$\\beta$ is used to re-weigh the triple value $w$. If $\\beta=0$ the original numeric values $w$ are used. If $\\beta=1$, numeric values $w$ are ignored and the model is equivalent to a conventional KGE architecture.\nNote that positive and negative triples are assigned different $\\alpha$ equations.\nThis is done to lower the margin between the scores for triples and their respective corruptions when the triple numeric value is high. \n\nFinally, the FocusE layer $h(t)$ is defined as:\n\n\\begin{equation}\nh(t) = \\alpha \\, g(t)\n\\label{eq:ht}\n\\end{equation}\n\nPutting all together, the FocusE layer $h(t)$ is then used in the loss function\\footnote{Note that the FocusE layer described in Eq.\\ref{eq:ht} is compatible with other losses used in KGE literature (e.g. pairwise, negative log-likelihood, etc.)} $L$. \nThis is a modified, more numerically stable version of the negative log-likelihood of normalized softmax scores proposed in \\cite{kadlec2017knowledge}:\n\n\\begin{equation}\nL = - \\sum_{t^+,t^-} log \\frac{e^{h(t^+)}}{e^{h(t^+)} + e^{h(t^-)}} \\label{eq:loss}\n\\end{equation}\n\nAs seen in \\eqref{eq:loss}, the modulation between structural influence and numeric values increases the margin between high and low-valued triples. Hence the model learns to focus on triples with higher numeric attributes. \n\nDuring training, we decay the structural influence $\\beta$ from 1 to 0. This is controlled by the hyperparameter $\\lambda$ (decay). Initially the model gives equal importance to all training triples. When $\\beta$ decays to zero (linearly, over $\\lambda$ epochs), the model exclusively relies on numeric attributes ($w$).\nSection~\\ref{sec:decay} shows empirical evidence of the effectiveness of $\\lambda$.\n\n\n\n\n\\section{Related Work}\\label{sec:relatedwork}\n\n\\textbf{Knowledge Graph Embeddings.} Although a comprehensive survey is out of the scope of this work (recent surveys provide a good coverage of the landscape~\\cite{bianchi2020knowledge}), it is worth listing the most popular knowledge graph embedding models proposed to date.\nTransE~\\cite{bordes2013translating} is the forerunner of distance-based models, and inspired a number of models commonly referred to as TransX.\nThe symmetric bilinear-diagonal model DistMult~\\cite{yang2014embedding} paved the way for its asymmetric evolutions in the complex space, ComplEx~\\cite{trouillon2016complex} and RotatE~\\cite{sun2019rotate}.\nSome models such as RESCAL~\\cite{nickel2011three}, TuckER~\\cite{balavzevic2019tucker}, and SimplE~\\cite{kazemi_simplE} rely on different tensor decomposition techniques. \nModels such as ConvE~\\cite{DBLP:conf\/aaai\/DettmersMS018} or ConvKB~\\cite{nguyen2018novel} leverage convolutional layers. \nAttention is used by~\\cite{nathani2019learning}. \n\n\\textbf{Numeric-aware models (node attributes).}\nNone of the models listed above leverage numeric attributes of any kind. \nHowever, a number of recent works support multimodal knowledge graphs and learn from numeric values associated to node entities.\nLiteralE enriches node embeddings with numeric information before scoring the triples~\\cite{kristiadi2019incorporating}. KBLRN combines latent, relational and numeric features using product of experts model~\\cite{garcia2017kblrn}. TransEA learns a vanilla structural model using TransE scoring, and an attribute model for attributed triples, using regression over the attribute values, which is jointly trained~\\cite{wu2018knowledge}. \nNevertheless, such models are not designed to learn from numeric values associated to \\textit{edges}. \n\n\\textbf{Numeric-aware models (edge attributes).}\nTo the best of our knowledge\\footnote{We limit to Knowledge Graph Embeddings literature and knowledge graphs, i.e. directed, labeled, multi graphs. Other numeric-aware models such as graph neural networks that operate on different data modalities are out of the scope of this paper (they are either designed for tasks other than link prediction, most of them do not support multi-relational graphs, and cannot currently be applied at the scale KGE models operate). \n}, the only work designed to work with numeric-aware edges is UKGE~\\cite{chen2019embedding}. \nUKGE generates confidence scores for known triples by squashing numeric values in the $[0-1]$ interval. \nIt then uses probabilistic soft logic~\\cite{kimmig2012short} to predict probability estimates for unseen triples, by jointly training a model to regress over the confidence values. A limitation of this approach is that out-of-band logical rules are required as additional input. %\nIt is also worth noting that UKGE design rationale aims at supporting uncertain knowledge graphs, i.e. graphs whose edge numeric values represent uncertainty. In this paper we aim at supporting generic numeric values, regardless of their semantics (uncertainty, strength, importance, etc).\nMoreover, we claim that triples with high numeric values should have higher impact, and this should be explicitly modeled in the scoring or loss computation.\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{img\/FocusEArchitecture.png}\n\\caption{A Knowledge Graph Embedding model architecture enhanced with FocusE. The add-on acts as an intermediate layer between the traditional scoring layer and the loss.}\n\\label{fig:arch}\n\\end{figure*}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Recollection of the canonical odd Laplacian}\n\nIn this section we review the construction of the odd Laplacian on\nhalf-densities due to~\\cite{hov:max}. See also~\\cite{hov:semi,\nhov:proclms} and \\cite{tv:laplace1}.\n\nLet $M$ be a supermanifold endowed with an odd symplectic structure,\ngiven by an odd $2$-form $\\omega$. We shall refer to such supermanifolds\nas  \\textit{odd symplectic manifolds}. (We always skip the prefix\n`super-' unless required to avoid confusion.) Later we shall discuss\nthe more general case of an odd Poisson manifold. A brief definition\nof the odd Laplacian acting on half-densities on $M$  follows.\n\nConsider a cover of $M$ by Darboux charts, in which the symplectic\nform takes the canonical expression $\\omega=dx^id{\\xi}_i$. Here $x^i$,\n${\\xi}_i$ are canonically conjugate variables of opposite parity. We\nassume that the $x^i$ are even; hence the ${\\xi}_i$, odd. Let $Dy$, for\nany kind of variables $y$, stand for the Berezin volume element.\nThen half-densities on $M$ locally look like\n$\\sigma=s(x,{\\xi})(D(x,{\\xi}))^{1\/2}$. (Notice that we skip  questions related\nwith orientation.) We set\n\\begin{equation}\\label{eq:delta}\n    \\Delta \\sigma:=\\dder{s}{x^i}{{\\xi}_i}\\,\\bigl(D(x,{\\xi})\\bigr)^{1\/2},\n\\end{equation}\nin Darboux coordinates, and call $\\Delta$, the \\textit{canonical odd\nLaplacian on half-densities}.\n\nThe simplicity of formula~\\eqref{eq:delta} is very deceptive. The\nexpression $\\der{}{x^i}\\der{f}{{\\xi}_i}$ was originally suggested by\nBatalin and Vilkovisky, and is the famous `BV operator'. However,\nthe trouble is that it is not well-defined on functions (actually,\non any objects) unless we fix a volume form, which should therefore\nenter the definition.  The geometrically invariant construction for\nfunctions, using a volume form, was first given\nin~\\cite{hov:deltabest}. \\textit{There is no canonical volume form\non an odd symplectic manifold} (unlike even symplectic manifolds,\nenjoying the Liouville form). In particular, the coordinate volume\nform $D(x,{\\xi})$ for Darboux coordinates is \\textbf{not} preserved by\nthe (canonical) coordinate transformations (see later). Hence the\ninvariance of the operator $\\Delta$ given by~\\eqref{eq:delta} is a deep\ngeometric fact.\n\nAs we showed in~\\cite{tv:laplace1}, on any odd Poisson, in\nparticular, odd symplectic,  manifold there is a natural\n\\textit{master groupoid} of `changes of volume forms' $\\r\\mapsto\ne^S\\r$ satisfying the master equation $\\Delta_{\\r}e^{S\/2}=0$ (note $1\/2$\nin the exponent; without it there would be no groupoid). Here\n$\\Delta_{\\r}$ is the odd Laplacian on functions with respect to the\ngiven volume form $\\r$. It is defined by $\\Delta_{\\r}f:=\\div_{\\r}X_f$,\nwhere $X_f$ is the Hamiltonian vector field corresponding to $f$.\n(See~\\cite{hov:deltabest}; note also~\\cite{yvette:divergence} for\nanother approach.) In a similar way one can define the odd Laplacian\non any densities\n--- again, depending on a chosen volume form.  Now, half-densities are\ndistinguished from densities of other weights precisely by the fact\nthat for them the corresponding odd Laplacian would depend only on\nthe orbit of a volume form with respect to the action of the above\ngroupoid~\\cite{tv:laplace1}. It turns out that on an odd symplectic\nmanifold, all Darboux coordinate volume forms belong to the same\norbit of the master groupoid. We can regard it as a `preferred\norbit'; hence, in the absence of an invariant volume form, the odd\nLaplacian on half-densities defined by an arbitrary  Darboux\ncoordinate volume form  is invariant. It is just~\\eqref{eq:delta}.\n\n\\section{Homological interpretation of the odd Laplacian}\n\nNow we are going to approach $\\Delta$ on half-densities from a very\ndifferent angle.\n\nLet $\\Omega(M)$ be the space of all pseudodifferential forms on $M$,\ni.e., functions on $\\P TM$. (As usual, $\\P$ stands for the parity\nreversion functor on vector spaces, vector bundles, etc.) In\ncoordinates such functions have the form  $s=s(x,{\\xi},dx,d{\\xi})$, where\nthe differentials of coordinates are commuting variables of parity\nopposite to that of the respective coordinate. In our case $dx^i$\nare odd and $d{\\xi}_i$ are even. We do not assume that functions\n$s(x,{\\xi},dx,d{\\xi})$ are polynomial in $d{\\xi}_i$.  Of course they are\n(Grassmann) polynomial in $dx^i$, because these variables are odd.\n\n\n\nConsider the odd symplectic form $\\omega$. Since $\\omega^2=0$,\nmultiplication by $\\omega$ can be considered as a differential. Define\nthe operator $D=d+\\omega$, where $d$ is the de Rham differential. Since\n$d\\omega=0$, it follows that $D^2=0$ and we have a `double complex'\n$\\bigl(\\Omega(M),D=\\omega+d\\bigr)$. \\textbf{Warning:} here a\n\\textit{complex} means just a ${\\mathbb Z_{2}}$-graded object.\n\nThe reader should bear in mind that since $\\omega=d\\Theta$ for some even\n$1$-form $\\Theta$, which is true globally, we have\n$D=e^{-\\Theta}\\circ d \\circ e^{\\Theta}$ and the multiplication by\nthe inhomogeneous differential form $e^{\\Theta}$ sets an isomorphism\nbetween the complexes $\\bigl(\\Omega(M), D\\bigr)$ and $\\bigl(\\Omega(M),\nd\\bigr)$. It follows that $H(\\Omega(M), D)$ is isomorphic to $H(\\Omega(M),\nd)$, which is just the de Rham cohomology of the underlying manifold\n$M_0$. (Note that the isomorphism $e^{\\Theta}$ preserves only\nparity, but not ${\\mathbb Z}$-grading, even if we restrict it to\n{differential} forms on $M$, i.e., polynomials in $dx,d{\\xi}$.)\n\n\nThe operator $D=\\omega+d$ was introduced in~\\cite{severa:originbv}. The\nidea was to consider the spectral sequence for $\\bigl(\\Omega(M),D\\bigr)$\nregarded as a double complex. We shall follow it in a form best\nsuiting our purposes and which is slightly different\nfrom~\\cite{severa:originbv}. (In particular, we do not assume\ngrading in the space of forms.)\n\nAlthough there is no ${\\mathbb Z}$-grading present, single or double, one\ncan still develop the machinery of spectral sequences as follows.\n\nWe define linear relations (see~\\cite{maclane:homology}) on $\\Omega(M)$:\n\\begin{equation*}\n    \\partial_0:=\\bigl\\{(\\a,\\b) \\in \\Omega(M)\\times \\Omega(M)\\ |\\  \\omega\\a=\\b\\bigr\\},\n\\end{equation*}\nand\n\\begin{multline*}\n    \\partial_r:=\\bigl\\{(\\a,\\b) \\in \\Omega(M)\\times \\Omega(M)\\ |\\  \\exists \\a_1,\\ldots, \\a_r\\in \\Omega(M) \\ :\n    \\ \\omega\\a=0, \\\\\n    d\\a+ \\omega\\a_1=0, \\ \\ldots,\n    \\ d\\a_{r-2}+ \\omega\\a_{r-1}=0, \\ d\\a_{r-1}+ \\omega\\a_{r}=\\b\\bigr\\}\n\\end{multline*}\nfor all $r=1,2, 3, \\ldots \\ {}$  We also set $\\partial_{-1}:=\\{(\\a,0)\\}$.\nWe have subspaces $\\Ker\\partial_r$, $\\Def\\partial_r$ (the \\textit{domain of\ndefinition}), $\\Ind\\partial_r$ (the \\textit{indeterminancy}), and\n$\\Im\\partial_r$ in $\\Omega(M)$, and by a direct check\n\\begin{align*}\n    \\Im \\partial_r&\\subset \\Ker \\partial_r\\,,\\\\\n\\Def \\partial_r&=\\Ker\\partial_{r-1}\\,,\\\\\n\\Ind \\partial_r&=\\Im\\partial_{r-1}\\,.\n\\end{align*}\nThat is, we have a sequence of differential relations on $\\Omega(M)$,\ndefining a spectral sequence $(E_r,d_r)$  where\n\\begin{equation*}\n    E_{r}:=\\frac{\\Ker \\partial_{r-1}}{\\Im\\partial_{r-1}}=\\frac{\\Def \\partial_{r}}{\\Ind\\partial_{r}}\n\\end{equation*}\nand the homomorphism $d_r\\colon\\thinspace E_r\\to E_r$ is induced by $\\partial_r$ in the\nobvious way. (In fact, differential relations like this is the\nshortest way of defining spectral sequences,\nsee~\\cite[p.~340]{maclane:homology}.)\n\nClearly $E_0=\\Omega(M)$. The relation $\\partial_0$ is simply the graph of the\nlinear map $d_0\\colon\\thinspace \\Omega(M)\\to \\Omega(M)$, $d_0\\a=\\omega\\a$. What is $E_1$?\n\n\\begin{thm} The space $E_1$ can be naturally identified with the space\nof half-densities on $M$.\n\\end{thm}\n\nA proof consists of two independent steps. First, we find the\ncohomology of $d_0$ using algebra. Second, we identify the result\nwith a geometrical object. The first part goes as follows.\n\nThe operator $d_0=\\omega$ is a Koszul type differential, since in an\narbitrary Darboux chart $\\omega=dx^id{\\xi}_i$. Introduce a ${\\mathbb Z}$-grading by\nthe degree in the odd variables $dx^i$. The operator $d_0$ increases\nthe degree by one. (This grading is \\textbf{not} preserved by\nchanges of coordinates.) From general theory it follows that the\ncohomology should be concentrated in the ``maximal degree''. Indeed,\nsuppose that $\\dim M=n|n$ and consider the linear operator $H$ on\npseudodifferential forms defined as follows. For\n$\\sigma=\\sigma(x,{\\xi},dx,d{\\xi})$,\n\\begin{equation*}\n    H\\sigma(x,{\\xi},dx,d{\\xi}):=\\int_0^1 \\!dt\\,t^{n-1}\\, \\dder{\\sigma}{dx^i}{d{{\\xi}}_i}\n    (x,{\\xi},t^{-1}dx,t\\,d{\\xi})\\,,\n\\end{equation*}\n--- notice the similarity with the $\\Delta$-operator. The operator $H$ is\nwell defined on all forms of degree  less than $n$ in $dx^i$ and on\nforms of `top' degree if they vanish at $d{\\xi}_i=0$. (In both cases\nthere will be no problem with division by $t$.) For forms on which\n$H$ makes sense one can check that\n\\begin{equation*}\n    (Hd_0+d_0H)\\sigma=\\sigma\\,.\n\\end{equation*}\nIn particular, if a form $\\sigma$ is $d_0$-closed and of degree less\nthan $n$ in $dx^i$, then $\\sigma=d_0 H\\sigma$. The same applies for a top\ndegree form taking a non-zero value at $d{\\xi}_i=0$. Hence the\n$d_0$-cohomology ``sits on'' pseudodifferential forms of degree $n$\nin $dx^i$ that do not depend on $d{\\xi}_i$:\n\\begin{equation*}\n    \\sigma=s(x,{\\xi})\\,dx^1\\ldots dx^n.\n\\end{equation*}\nNo non-zero form of this appearance can be cohomologous to zero:\nindeed, any $d_0$-exact form, $d_0\\tau=\\omega\\tau$, vanishes at\n$d{\\xi}_i=0$.\n\n\\textit{Hence, each $d_0$-cohomology class has a unique\nrepresentative in a given Darboux coordinate system $x^i,{\\xi}_i$. It\nis obtained by taking an arbitrary form from the class, extracting\nits component of degree $n$ in $dx^i$ and evaluating  at $d{\\xi}_i=0$.}\nBy applying this to the class of $dx^1\\ldots dx^n$, we immediately\narrive at\n\\begin{lm} \\label{lem1} Elements of the cohomology space $E_1=H(\\Omega(M),\\omega)$\nare represented in Darboux coordinates as classes\n\\begin{equation*}\n    \\sigma=s(x,{\\xi})\\,[dx^1\\ldots dx^n],\n\\end{equation*}\nwhere under a change of Darboux coordinates\n\\begin{align*}\n    x^i&=x^i(x',{\\xi}'),\\\\\n{\\xi}_i&={\\xi}_i(x',{\\xi}')\n\\end{align*}\nthe class $[dx^1\\ldots dx^n]$ transforms as follows:\n\\begin{equation*}\n    [dx^1\\ldots dx^n]=\\det J_{00}\\cdot[dx^{1'}\\ldots\n    dx^{n'}]\\,.\n\\end{equation*}\nHere $J_{00}=\\der{x}{x'}$ is the even-even block of the Jacobi\nmatrix $J=\\der{(x,{\\xi})}{(x',{\\xi}')}$.\n\\end{lm}\n\n\nTo  better appreciate the statement, notice that\n\\begin{equation*}\n    dx^i=dx^{i'}\\der{x^i}{x^{i'}}+d{\\xi}_{i'}\\der{x^i}{{\\xi}_{i'}}.\n\\end{equation*}\nHence\n\\begin{equation*}\n    dx^1\\ldots dx^n=dx^{1'}\\ldots\n    dx^{n'}\\cdot \\det\\left(\\der{x^i}{x^{i'}}\\right)+ \\text{terms containing\n    $d{\\xi}_{i'}$}\\,.\n\\end{equation*}\nPassing to cohomology is equivalent to discarding these lower order\nterms.\n\nWhat kind of geometrical object is this?\n\n\n\\begin{lm} \\label{lem2} Objects of the form\n    $\\sigma=s(x,{\\xi})\\,[dx^1\\ldots dx^n]$,\nin Darboux coordinates, with the transformation law given in\nLemma~\\ref{lem1} can be identified with half-densities on $M$.\n\\end{lm}\n\nThis is the crucial claim. There is a simple but fundamental fact\nfrom linear algebra behind Lemma~\\ref{lem2}, which will be proved in\nthe next section.\n\nThe transformation law for $[dx^1\\ldots dx^n]$ can be obtained from\nthe formal ``law'' $[dx^i]=[dx^{i'}]\\der{x^i}{x^{i'}}$.\nUnfortunately, it does not define a geometric object, because it\ndoes not obey the cocycle condition. In a way, it is only a\n`virtual' transformation law, which will make sense only if an extra\nstructure is imposed on $M$.\n\n\nNow as we have the space $E_1$, let us check the differential $d_1$\non it. It is induced  by the differential relation $\\partial_1$ on\n$\\Omega(M)$. Take an element $\\sigma=s(x,{\\xi})\\,[dx^1\\ldots dx^n]\\in E_1$,\ntake its representative $\\a=s(x,{\\xi})\\, dx^1\\ldots dx^n$ and consider\n$\\b\\in\\Omega(M)$  such that $d\\a+\\omega\\a_1=\\b$, for $\\a_1\\in \\Omega(M)$. We\nwill have $[\\b]=d_1\\sigma$ for the class $[\\b]$ in $E_1$. Notice that\n$d\\a=d{\\xi}_i\\der{s}{{\\xi}_i}\\, dx^1\\ldots dx^n$ and it will vanish at\n$d{\\xi}_i=0$, therefore it is an $\\omega$-exact form, according to our\nprevious analysis. \\textit{Thus $d_1=0$ identically and $E_2=E_1$.}\n\nConsider $d_2$ on $E_1=E_2=H(\\Omega(M),\\omega)$. By definition, $d_2$ maps\nthe class $\\sigma=s(x,{\\xi})\\,[dx^1\\ldots dx^n]$, with a local\nrepresentative $\\a=s(x,{\\xi})\\, dx^1\\ldots dx^n$, to the class of\n$\\b\\in \\Omega(M)$ such that $d\\a+\\omega\\a_1=0$, $d\\a_1+\\omega\\a_2=\\b$, for some\n$\\a_1$ and $\\a_2$.  We may set $\\a_1:=-Hd\\a$, where $H$ is the\nhomotopy operator defined above, and  $\\b:=d\\a_1=-dHd\\a$. Directly:\n\\begin{multline*}\n    Hd\\a=H\\Bigl(d{\\xi}_i\\der{s}{{\\xi}_i}\\, dx^1\\ldots dx^n\\Bigr)=\\sum (-1)^{i +\\tilde\ns} \\der{s}{{\\xi}_i}\\, dx^1\\ldots \\widehat{dx^i} \\ldots dx^n\n\\end{multline*}\nand\n\\begin{multline*}\n    \\b=-dHd\\a=-d \\sum (-1)^{i-1+\\tilde\ns} \\der{s}{{\\xi}_i}\\, dx^1\\ldots \\widehat{dx^i} \\ldots dx^n=\\\\\n-dx^j\\der{}{x^j}\\sum (-1)^{i +\\tilde s} \\der{s}{{\\xi}_i}\\, dx^1\\ldots\n\\widehat{dx^i} \\ldots dx^n + \\text{\\ lower order terms in $dx$}=\\\\\n-\\dder{s}{x^i}{{\\xi}_i}\\,dx^1\\ldots dx^n + \\text{\\ lower order terms in\n$dx$}\\,.\n\\end{multline*}\nHence in $E_1$ we get:\n\\begin{equation*}\n    d_2\\sigma=d_2 \\bigl(s(x,{\\xi})\\,[dx^1\\ldots dx^n]\\bigr)=-\\dder{s}{x^i}{{\\xi}_i}\\,[dx^1\\ldots\n    dx^n]=-\\Delta\\sigma\\,,\n\\end{equation*}\nwhich is quite remarkable. What about the space $E_3$ and the\ndifferential $d_3$, and so on?\n\nIt is not hard to notice that the cohomology of the $\\Delta$-operator on\nhalf-densities on $M$ is isomorphic to the de Rham cohomology of the\nunderlying ordinary manifold $M_0$ (we shall say more about this\nlater). Locally the cohomology vanishes except for constants:\n$\\sigma=\\const\\cdot [dx^1\\ldots dx^n]$. Thus $d_3=0$, and $E_4=E_3$; the\nsame continues for $d_4=0$,  $E_5=E_4=E_3$, and so on. We arrive at\nthe following statement (which was the main result\nof~\\cite{severa:originbv}):\n\n\n\\begin{thm} \\label{thm2} With the identification of the space $E_1=H(\\Omega(M),\\omega)$\nwith half-densities on $M$, the differential $d_1$ vanishes and the\nnext differential $d_2$ coincides up to a sign with the canonical\nodd Laplacian. The spectral sequence $(E_r,d_r)$ degenerates at the\nterm $E_3$, which is the cohomology of the operator $\\Delta$.\n\\end{thm}\n\n\n\nThe importance of Theorem~\\ref{thm2}  is in the fact that it gives\nan alternative proof of the invariance of the odd Laplacian on\nhalf-densities $\\Delta$, by identifying  it with an operator in a\nspectral sequence invariantly associated with the odd symplectic\nstructure.\n\n\n\n\\section{Berezinian of a canonical transformation}\n\nConsider a vector space $V=V_0\\oplus V_1$ with an odd symplectic\nstructure, i.e., an odd non-degenerate antisymmetric bilinear form.\n(A choice of `antisymmetric' or `symmetric' does not make any\ndifference.) Necessarily $\\dim V=n|n$. We call matrices preserving\nthis form, \\textit{symplectic}. This should not cause problems; when\ncomparing them with ordinary symplectic matrices  corresponding to\nan even symplectic structure, we shall make the reference to the\nparity of the bilinear form explicit.\n\n\\begin{thm} \\label{thm:block} Suppose that $J$ is a symplectic matrix for an odd symplectic space.\nLet\n\\begin{equation*}\n    J=\\begin{pmatrix}\n        J_{00} & J_{01} \\\\\n        J_{10} & J_{11} \\\\\n      \\end{pmatrix}\n\\end{equation*}\nbe its standard block decomposition. Then\n\\begin{equation}\n    \\Ber J =(\\det J_{00})^2\\,.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof} We can write the matrix of our symplectic form as\n\\begin{equation*}\n    B=\\begin{pmatrix}\n        0 & 1 \\\\\n        1 & 0 \\\\\n      \\end{pmatrix}\\,.\n\\end{equation*}\nThe relation for $J$ is $JBJ\\,^T=B$, where the operation of matrix\ntranspose takes into account the parities of the blocks:\n\\begin{multline*}\n    \\begin{pmatrix}\n        J_{00} & J_{01} \\\\\n        J_{10} & J_{11} \\\\\n      \\end{pmatrix}\n      \\begin{pmatrix}\n        0 & 1 \\\\\n        1 & 0 \\\\\n      \\end{pmatrix}\n      \\begin{pmatrix}\n        J_{00} & J_{01} \\\\\n        J_{10} & J_{11} \\\\\n      \\end{pmatrix}^T=\n\\begin{pmatrix}\n        J_{00} & J_{01} \\\\\n        J_{10} & J_{11} \\\\\n      \\end{pmatrix}\n      \\begin{pmatrix}\n        0 & 1 \\\\\n        1 & 0 \\\\\n      \\end{pmatrix}\n      \\begin{pmatrix}\n        J_{00}^{\\,T} & J_{10}^{\\,T} \\\\\n        -J_{01}^{\\,T} & J_{11}^{\\,T} \\\\\n      \\end{pmatrix}.\n     \n\n\n\n\\end{multline*}\nHence we obtain\n\\begin{align\n    J_{00}J_{01}^{\\,T}=\\left(J_{00}J_{01}^{\\,T}\\right)^T, \\label{eq:symp1}\\\\\n    J_{11}J_{10}^{\\,T}=-\\left(J_{11}J_{10}^{\\,T}\\right)^T, \\label{eq:symp2}\\\\\n    J_{00}J_{11}^{\\,T}+J_{01}J_{10}^{\\,T}=1 \\label{eq:symp3}.\n\\end{align}\nFrom~\\eqref{eq:symp3} we may express\n\\begin{multline*}\n    J_{11}={J_{00}^{\\,T}}^{-1}+J_{10}{J_{01}^{\\,T}}{J_{00}^{\\,T}}^{-1} \\\\\n    ={J_{00}^{\\,T}}^{-1}+J_{10}J_{00}^{-1}J_{01}, \\text{\\quad   taking into\n    account~\\eqref{eq:symp1}}.\n\\end{multline*}\nWe arrive at the identity\n\\begin{equation}\n    J_{11}-J_{10}J_{00}^{-1}J_{01}={J_{00}^{\\,T}}^{-1}.\\label{eq:id}\n\\end{equation}\nTherefore\n\\begin{equation*}\n    \\Ber J=\\frac{\\det\n    J_{00}}{\\det\\bigl(J_{11}-J_{10}J_{00}^{-1}J_{01}\\bigr)}=\n    \\frac{\\det\n    J_{00}}{\\det{J_{00}^{\\,T}}^{-1}}=\\left(\\det\n    J_{00}\\right)^2.\n\\end{equation*}\n\\end{proof}\n\n\n\nNotice that the steps we followed in the proof are the same as may\nbe used for proving the classical Liouville theorem, i.e., that\n$\\det J=1$ for an ordinary symplectic matrix $J\\in \\Sp(2n)$. The\ndecomposition of $V$ in that case will be the decomposition into the\nsum of Lagrangian subspaces and an identity similar to~\\eqref{eq:id}\nwill be valid. The difference will arise only when calculating the\ndeterminant: instead of the ratio  of the determinants of the\nblocks, there will the product, which will give $1$ instead of $\\det\nJ_{00}^2$.\n\nIt is easy to generalize. Let $V$ be a vector (super)space with a\nsymplectic structure, even or odd. Consider its decomposition into\nthe sum of two Lagrangian subspaces. In the even case they will have\nthe same dimensions; in the odd case, the opposite, i.e., $p|{n-p}$\nand ${n-p}|p$. Denote the chosen decomposition by $V=V_0\\oplus V_1$.\nHere the indices have nothing to do with parity. By picking\n`canonically conjugate' bases in $V_0$ and $V_1$ we arrive at a\npicture formally the same as above. The Berezinian can be calculated\nusing the corresponding block decompositions. It will be either\n\\begin{equation*}\n    \\Ber J=\\Ber J_{00}\\cdot \\Ber \\bigl(J_{11}-J_{10}J_{00}^{-1}J_{01}\\bigr)\n\\end{equation*}\nor\n\\begin{equation*}\n    \\Ber J=\\frac{\\Ber J_{00}}{\\Ber\n    \\bigl(J_{11}-J_{10}J_{00}^{-1}J_{01}\\bigr)}\n\\end{equation*}\ndepending on the parity of the symplectic form (in the odd case the\n`formats' of the matrix blocks will be the opposite, hence\ndivision). Then the analog of the identity~\\eqref{eq:id} should be\napplied. For even symplectic structure we thus obtain the analog of\nLiouville's theorem, and for odd, we arrive at\n\n\n\\begin{thm} \\label{thm:lagrange} Let $J$ be a symplectic transformation of an odd symplectic space $V$.\nThen for an arbitrary decomposition into the sum of Lagrangian\nsubspaces, $V=V_0\\oplus V_1$ \\emph{(indices not indicating parity)},\nthe identity\n\\begin{equation}\\label{eq:berj}\n    \\Ber J=\\left(\\Ber J_{00}\\right)^2\n\\end{equation}\nholds.\n\\end{thm}\n\n\n\\begin{rem} Theorem~\\ref{thm:block}  gives, in particular, that the\nBerezinian of a symplectic matrix is a polynomial in the matrix\nentries and, moreover, a complete square. This is somewhat masked in\nthe more general Theorem~\\ref{thm:lagrange}.\n\\end{rem}\n\n\nThere is an `abstract' argument parallel to the calculation above.\nConsider a decomposition into Lagrangian subspaces $V=V_0\\oplus\nV_1$, for an even or odd symplectic form. (This works in the same\nway for symmetric forms.) Consider the dual space $V_1^*$. We have\n$V_1^*\\cong \\Ann V_0\\subset V^*$ and, using the form, can replace\nthe annihilator by the orthogonal complement:\n\\begin{equation*}\n    V_1^*\\cong V_0^{\\perp}\\subset V\n\\end{equation*}\nfor the even form or\n\\begin{equation*}\n    V_1^*\\cong \\Pi V_0^{\\perp}\\subset \\Pi V\n\\end{equation*}\nfor the odd form. Recalling that $V_0$ and $V_1$ are Lagrangian\nsubspaces, we get $V_1^*\\cong V_0$ or $V_1^*\\cong \\Pi V_0$. Hence\n\\begin{align*}\n    \\Ber V=\\Ber (V_0\\oplus V_1)&=\n    \\Ber V_0\\otimes \\Ber V_1\\\\\n    &=\\Ber\n    V_0\\otimes \\Ber V_0^*=1\n    \\intertext{(even symplectic form) or}\n&=\\Ber V_0\\otimes \\Ber \\Pi V_0^*=(\\Ber V_0)^{\\otimes 2}\n\\end{align*}\n(odd symplectic form). Equalities here mean  natural isomorphisms.\n\nA different abstract argument based on the well-known interpretation\nof the space $\\Ber V$ as the cohomology of a Koszul complex and\njustifying the equality $\\Ber V=(\\Ber V_0)^{\\otimes 2}$ for an odd\nsymplectic space was given in~\\cite{severa:originbv}.\n\nA weak point of abstract arguments is that they do not really give\ninformation about matrices, which is necessary in applications such\nas a proof of Lemma~\\ref{lem2}.\n\n\n\\section{``Supersymmetries'' of differential forms}\n\nIn this section we change   viewpoint. We would like to phrase the\nprevious constructions entirely in the language of `classical'\ndifferential-geometric objects. In this way we shall see how the\ncanonical odd Laplacian on half-densities on an odd symplectic\nmanifold can be seen as a `classical' object equipped with extra\nsymmetries.\n\nLet $M$ now stand for an arbitrary manifold or supermanifold.\nPreviously we worked with   odd symplectic manifolds. It is known\nthat any such odd symplectic manifold can be  non-canonically\nidentified with $\\Pi T^*M$ considered with the natural odd bracket,\nfor some $M$. A change of   identifying symplectomorphism is\nequivalent to a symplectomorphism (or canonical transformation) of\nthe space $\\Pi T^*M$. Therefore, we can restrict ourselves to\nobjects on $\\Pi T^*M$, but should analyze them from   the viewpoint\nof the larger supergroup of all canonical transformations of $\\Pi\nT^*M$, not just that of diffeomorphisms of $M$.\n\nWe shall consider multivector fields (and multivector densities)\nand differential forms on $M$. When $M$ is a supermanifold, we\nactually speak of pseudodifferential forms.\n\nMultivector fields on $M$ are identified with functions on $\\Pi\nT^*M$. In local coordinates, we have $X=X(x,x^*)$, where $x^a$ are\ncoordinates on $M$ and $x^*_a$ are the corresponding coordinates on\nthe fibers, of the opposite parity, transforming as\n$x^*_a=\\der{x^{a'}}{x^a}\\,x^*_{a'}$. There is no problem with\ncanonical transformations of $\\Pi T^*M$ acting on multivector fields\non $M$ --- it is just the pull-back of functions.\n\nMultivector densities on $M$ have the form $\\sigma=s(x,x^*)\\,Dx$ and at\nfirst glance it is not obvious how a transformation mixing $x$ and\n$x^*$ can be applied to them. However, in view of\nTheorems~\\ref{thm:block} and \\ref{thm:lagrange}, for  a canonical\ntransformation $F\\colon\\thinspace \\Pi T^*M\\to \\Pi T^*M$ one can set\n\\begin{equation*}\n    F^*\\sigma:=s\\bigl(y(x,x^*),y^*(x,x^*)\\bigr)\\,\\Ber\\left(\\der{y}{x}\\right)\\,Dx\n\\end{equation*}\nif  in coordinates $F\\colon\\thinspace (x,x^*)\\mapsto \\bigl(y=y(x,x^*),\ny^*=y^*(x,x^*)\\bigr)$. This is a well-defined action. In other\nwords, we identify multivector densities on $M$ with half-densities\non $\\Pi T^*M$ and apply the natural  action, taking into account\nidentity~\\eqref{eq:berj}. In integration theory, multivector\ndensities are known as integral forms (more precisely,\npseudointegral, if we insist on   differentiating between arbitrary\nsmooth functions and polynomials). Therefore we can make a  remark:\nintegral forms on an arbitrary supermanifold $M$ are the same  as\nhalf-densities on the odd symplectic manifold $\\Pi T^*M$. In this\nlanguage we see that integral forms have more symmetries than those\nobvious ones  given by diffeomorphisms of $M$.\n\n\nConsider  now pseudodifferential forms on $M$, i.e., functions on\n$\\Pi TM$. They are related with (pseudo)integral forms, i.e.,\nmultivector densities on $\\Pi T^*M$ by the Fourier transform:\n\\begin{equation}\\label{eq:fourier}\n    \\omega(x,dx)=\\int\\limits_{\\Pi T^*_xM}\\! Dx^*\\, e^{\\i dx^ax^*_a}\\,s(x,x^*)\n\\end{equation}\nand conversely\n\\begin{equation}\\label{eq:invfourier}\n    s(x,x^*)=\\const\\int\\limits_{\\Pi T_xM}\\! D(dx)\\,\n    e^{-\\i dx^ax^*_a}\\,\\omega(x,dx)\\,.\n\\end{equation}\nFrom here we obtain the action of the canonical transformations of\nthe odd symplectic manifold $\\Pi T^*M$ on forms on $M$ as follows:\n\\begin{multline*}\n    (F^*\\omega)(x,dx)=\\const\\iint_{\\Pi T^*_xM\\times \\Pi T_yM} Dx^* D(dy)\n    \\,e^{\\i \\left(dx^ax^*_a-dy^ay^*_a(x,x^*)\\right)}\\cdot \\\\\n    \\omega\\bigl(y(x,x^*),dy\\bigr)\\,\\Ber\\der{y}{x}(x,x^*)\\,,\n\\end{multline*}\nwhere, as above, $F\\colon\\thinspace (x,x^*)\\mapsto \\bigl(y=y(x,x^*),\ny^*=y^*(x,x^*)\\bigr)$. In general, this action is non-local.\n\nWe shall consider the representation of the infinitesimal canonical\ntransformations of $\\Pi T^*M$ on forms and multivector densities on\n$M$. As it  turns out, the description in both cases will be very\nsimple.\n\nFor the odd symplectic manifold $\\Pi T^*M$, the canonical odd\nLaplacian on half-densities $\\Delta$ on $\\Pi T^*M$ is just the familiar\ndivergence of multivector densities $\\delta$ on $M$. Indeed,\n\\begin{equation*}\n    \\delta\\sigma= \\dder{s}{x^a}{x^*_a} \\,(x,x^*)\\,Dx\\,,\n\\end{equation*}\nif $\\sigma=s(x,x^*)\\,Dx$. Consider the infinitesimal canonical\ntransformation of $\\Pi T^*M$ generated by a function\n(``Hamiltonian'') $H=H(x,x^*)$. Denote  by $L_H$ the corresponding\nLie derivative. Notice that from the viewpoint of $M$, the function\n$H$ is a multivector field.\n\\begin{thm} On multivector densities (= integral forms) on $M$, the\nLie derivative w.r.t. the infinitesimal canonical transformation of\n$\\Pi T^*M$ generated by  $H$ is given by the formula:\n\\begin{equation*}\n    L_H=[\\delta,H],\n\\end{equation*}\nwhere at the r.h.s. stands the commutator of the divergence operator\n$\\delta$ and  multiplication by the multivector field $H$.\n\\end{thm}\n\nA proof can be given by a direct computation. It fact, the statement\nmimics a similar and more general statement concerning odd Laplace\noperators acting on densities of various weights,\nsee~\\cite{tv:laplace1}.\n\n\\begin{cor} \\label{cor:invar} The operator $\\delta$ on multivector densities on $M$ is\ninvariant under all canonical transformations of the odd symplectic\nmanifold $\\Pi T^*M$. \\emph{(At least those given by a Hamiltonian.)}\n\\end{cor}\n\\begin{proof} We need to show that $\\delta$ commutes with all Lie derivatives\n$L_H$. Indeed, $[\\delta,L_H]=[\\delta,[\\delta,H]]=[\\delta^2,H]=0$, since $\\delta^2=0$.\n\\end{proof}\n\nWe can adopt the following viewpoint. Suppose we do not know\nanything about the operator $\\Delta$ on half-densities in odd symplectic\ngeometry. Instead we concentrate on a familiar object, the operator\n$\\delta$ on multivector densities on a manifold $M$. The operator $\\delta$,\nas shown, is invariant under much larger group of transformations\nthan just diffeomorphisms of $M$. It is invariant under\nsymplectomorphisms of $\\Pi T^*M$. We can then take $\\delta$ as the\ndefinition of $\\Delta$ for $\\Pi T^*M$. Since any odd symplectic manifold\n$N$ is symplectomorphic to some $\\Pi T^*M$, we can use this to\n\\textbf{define} $\\Delta$ on $N$. The invariance of $\\Delta=\\delta$ for $\\Pi\nT^*M$ under symplectomorphisms of $\\Pi T^*M$ shows that $\\Delta$ on $N$\nis well-defined, i.e., its action on half-densities on $N$ does not\ndepend on an arbitrary choice of the identifying symplectomorphism\n$N\\cong \\Pi T^*M$.\n\nIt may seem that there is a gap in such an argument as the\ninvariance was proved only infinitesimally or, equivalently,  for\ntransformations that can be included into a Hamiltonian flow. In\nfact, there is no gap. Consider the supergroup $\\Can \\Pi T^*M$ of\nall canonical transformations. If $M$ is an ordinary manifold, the\nstructure of this supergroup was described in~\\cite{hov:semi}. It is\nthe product of the three subgroups:\n\n\\begin{enumerate}\n  \\item Transformations induced by diffeomorphisms of $M$;\n  \\item Shifts in the fibers of $\\Pi T^*M$  of the form\n\\begin{equation*}\n    F^*x^a=x^a, \\quad\n    F^*x^*_a=x^*_a+\\der{\\Phi}{x^a}\\,,\n\\end{equation*}\nwhere $\\Phi=\\Phi(x)$ is an odd function on $M$;\n  \\item Transformations identical on the submanifold $M\\subset \\Pi\n  T^*M$.\n\\end{enumerate}\n\nSince $\\delta$ is invariant under diffeomorphisms of $M$, all that\nremains is to study transformations of types 2 and 3. Canonical\ntransformations of types 2 and 3  can be included into Hamiltonian\nflows. Indeed, for type 2  one can take the flow with the\nHamiltonian $\\Phi$. For type 3 there is also a Hamiltonian flow,\nwith the Hamiltonian of the form $\\Psi=\\Psi^{ab}(x,x^*)x^*_a x^*_b$,\nas shown in~\\cite{hov:semi}. Therefore,  transformations of types 2\nand 3  are covered by the argument in the proof of\nCorollary~\\ref{cor:invar}, and this completes the proof.\n\nNow let us turn our attention to (pseudo)differential forms on $M$.\n\nUnder the Fourier\ntransform~\\eqref{eq:fourier},\\eqref{eq:invfourier}, the divergence\noperator $\\delta$ becomes the exterior differential $d$, up to a\nmultiple of $i$. The multiplication by a multivector field\n$H=H(x,x^*)$ becomes the `convolution' (or `cap product'):\n\\begin{multline*}\n    (H*\\omega)(x,dx)=\\const\\iin\n    Dx^* D(\\overline{dx})\\,e^{\\i(dx^a-\\overline{dx}^a)x^*_a}\\,\n    H(x,x^*)\\,\\omega(x,\\overline{dx})\\\\=\n    \\int D(\\overline{dx})\\,\\check\n    H(x,dx-\\overline{dx})\\,\\omega(x,\\overline{dx})\\,.\n\\end{multline*}\nwhere $\\check H=\\check H(x,dx)$ is the inverse Fourier transform of\n$H$. In other words, if we denote\n\\begin{equation*}\n    i_H\\omega:=H*\\omega\\,,\n\\end{equation*}\nwe have\n\\begin{equation}\\label{eq:inter}\n    i_H=H\\Bigl(x,-\\i\\der{}{dx}\\Bigr)\\,\n\\end{equation}\nthe differential operator, w.r.t. the variables $dx^a$, with the\nsymbol $H$. It is clear that up to $\\i$'s, it is just the classical\ninternal product of a form by a multivector field, if we deal with\nordinary differential forms and multivectors on an ordinary\nmanifold.\n\nWe immediately get\n\\begin{thm} On (pseudo)differential forms on $M$, the\nLie derivative w.r.t. the infinitesimal canonical transformation of\n$\\Pi T^*M$ generated by  $H$ is given by the `Cartan like formula':\n\\begin{equation}\\label{eq:lie}\n    \\i L_H=[d,i_H],\n\\end{equation}\nwhere at the r.h.s. stands the commutator of the de Rham\ndifferential  and the interior product by the multivector field $H$\nas defined by~\\eqref{eq:inter}.\n\\end{thm}\n\nThis is very remarkable. Suppose $M$ is an ordinary manifold. The\noperation $i_H$, up to the imaginary unit,  is the familiar interior\nproduct with a multivector field, generalizing the interior product\nwith a vector field. For Lie derivatives along vector fields one\nproves the Cartan formula $L_X=[d,i_X]$. For multivector fields, as\nopposed to vector fields, this equation  is taken as the definition\nof a `Lie derivative of a differential form along a multivector\nfield'. In the classical picture it is not seen how these\nderivatives corresponds to actual transformations. Now we see that\nthey are generators of odd canonical transformations acting on\ndifferential forms.\n\nNotice that in general $L_H$ is not a derivation of the algebra\n$\\Omega(M)$. Of course,\n\\begin{equation*}\n    L_{{[\\![} H,G {]\\!]}}=[L_H,L_G]\n\\end{equation*}\nwhere at the l.h.s. stands the Schouten bracket of multivector\nfields.  Equation~\\eqref{eq:lie} implies that the de Rham\ndifferential on $M$ is invariant under the canonical transformations\nof $\\Pi T^*M$. Again, one can see the $\\Delta$ operator as the de Rham\ndifferential considered together with these extra symmetries.\n\nSome of the arguments of this section were implicit in our earlier\nworks~\\cite{hov:deltabest, hov:max, hov:semi, hov:proclms}.\n\n\n\n\\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCarbon is one of the most important elements in materials science, chemistry and biology. However, its versatile chemical bonding presents a formidable challenge for the development of accurate and transferable atomistic simulation models. The development of interatomic potentials for carbon started in the 1980's with empirical formulations of the bond order~\\cite{abell1985empirical, tersoff_C_1988_PhysRevLett.61.2879} and the effort culminated with the rigorous derivation of bond-order potentials (BOP) from the electronic structure \\cite{oleinik_pettifor_prl}.  When it became possible to carry out large numbers of electronic structure calculations, mainly using density functional theory (DFT) \\cite{hohenberg_kohn_DFT_orig1,kohn_sham_DFT_orig2}, machine learning (ML) potentials superseded the classical potentials since they were able to reproduce the DFT data with minimal errors. Yet the inherent dependence on the reference data and poor extrapolative capabilities limit the transferability of ML potentials and often lead to nonphysical predictions for atomic configurations not included in the training dataset.  Emerging alternatives to ML potentials are polynomial\/tensorial expansions, in particular the moment tensor potentials (MTP) \\cite{Shapeev16} and the atomic cluster expansion (ACE)~\\cite{ACE_Drautz_PhysRevB.99.014104}. By employing a mathematically complete basis of the atomic environment \\cite{Dusson22}, it was demonstrated that traditional ML frameworks like neural networks or kernel-based Gaussian process regression are not necessary for obtaining accurate interatomic potentials. In fact, ACE was shown to be not only superior in terms of accuracy but also computational efficiency~\\cite{PACE_Lysogorskiy2021}.\n\nThe difficulty of modeling carbon is reflected by the fact that the pioneering developments of Tersoff~\\cite{tersoff_C_1988_PhysRevLett.61.2879} and Brenner~\\cite{REBO1_PhysRevB.42.9458} followed the Finnis-Sinclair (FS) \\cite{Finnis84} and embedded atom method (EAM) \\cite{Daw84} potentials for metals. Different from the metallic potentials, which focus on atomic energies, Tersoff introduced an empirical expression for angularly dependent bond order \\cite{Tersoff86} to model the formation of covalent directional bonds in C and Si.\nThe Tersoff potential as well as the family of reactive empirical bond order (REBO) potentials~\\cite{REBO1_PhysRevB.42.9458, REBO2_Brenner_2002} were applied widely to study properties of crystalline, amorphous and molecular carbon structures~\\cite{tersoff_application1, tersoff_application2, tersoff_application3}. Nevertheless, these early potentials had a number of limitations, such as short interaction ranges, neglect of $\\pi$ bonding or missing van der Waals (vdW) interactions ~\\cite{Pastewka2012_BOP_for_fracture, marks1_transferability, marks2_graphitization}. Some of these deficiencies were improved in subsequent modifications, for example, by introducing screening functions for a better description of bond breaking ~\\cite{Pastewka_SiC_PhysRevB.87.205410,Pastewka_SiC_PhysRevB.87.205410,rebo_S_pastewka_PhysRevB.78.161402}, or explicit terms to capture long-range dispersion forces~\\cite{airebo_doi:10.1063\/1.481208, AIREBO_M_doi:10.1063\/1.4905549,LCBOP_PhysRevB.68.024107}.  Other successful carbon potentials, such as the environment dependent interaction potential (EDIP) \\cite{edip_marks_PhysRevB.63.035401,Justo98} or ReaxFF~\\cite{reax_ff_vanDuin2001, reaxff_Srinivasan2015}, followed similar strategies of employing suitable functional forms based on physical and chemical intuition, yet still in a mostly empirical way. Rigorous derivations of the $\\sigma$ and $\\pi$ bond orders were eventually carried out by Pettifor and co-workers~\\cite{Alinaghian94,oleinik_pettifor_prb2} based on a quantum-mechanical tight-binding model~\\cite{Oleinik_pettifor_aBOP_carbon,oleinik_pettifor_prb2,oleinik_pettifor_prl,MROVEC2007230}. \n\nDespite their relatively short history, several ML potentials have already been developed for carbon. They can be differentiated by how the local atomic environment is sensed via descriptor functions~\\cite{musil2021physics}. To fulfill fundamental physical symmetries, these functions should be invariant under translation, rotation, inversion, and permutation of atoms of the same chemical species. For every atom, hundreds or thousands of different descriptor functions typically need to be evaluated and their numerical values form the input for an ML algorithm that predicts the atomic property. The most prominent ML approaches include neural network potentials (NNP) \\cite{NN_Behler_parinello_sym_PhysRevLett.98.146401} and kernel based methods, in particular the Gaussian approximation potentials (GAP) \\cite{kernel_regression_GAP_PhysRevLett.104.136403}, with descriptors based on the atom centered symmetry functions (ACSF) \\cite{ml_behler_C1CP21668F} or the smooth overlap of atomic positions (SOAP)~\\cite{SOAP_PhysRevB.87.184115}. \n\nFirst  NNP models were tailored for specific applications, such as transformations between graphite and diamond or behavior of multilayered graphene~\\cite{nnp_carbon_gradia_PhysRevB.81.100103,nnp_carbon_gradia_Khaliullin2011, NNP_Gr_PhysRevB.100.195419}. Recent NNPs aim at improved transferability by employing comprehensive reference datasets. These models include PANNA~\\cite{Shaidu2021_PaNNA2021}, which was built using an iterative self-consistent workflow, and DeepMD~\\cite{deep_MD_carbonpot_WANG20221}, which was trained on a large dataset of  bulk and low dimensional phases as well as snapshots from ab initio molecular dynamics (AIMD).  Parallel to the NNPs, several GAP parametrizations for carbon were also developed. The first GAPs focused on simulations of liquid and amorphous carbon~\\cite{gap17_PhysRevB.95.094203} and pristine graphene~\\cite{GAP_graphene_PhysRevB.97.054303}. In 2020, a large number of diverse carbon structures were employed to obtain a widely transferable GAP (GAP20)~\\cite{gap20_doi:10.1063\/5.0005084}. A closely related TurboGAP parameterization used a more efficient implementation of the SOAP descriptor~\\cite{TurboGAP, soap_turbo_PhysRevB.100.024112}. These GAPs were applied to study complex phenomena, such as vapour deposition of amorphous carbon films~\\cite{GAP_application_thinfilm_PhysRevB.102.174201} or the effect of defects on the corrugation of graphene~\\cite{Thiemann_2021_corrugation_graphene}. Overall, GAPs were found to be the most accurate models among fourteen carbon potentials in predicting realistic amorphous structures~\\cite{marks1_transferability}.\n\nDifferent from ML potentials, which often employ empirical descriptor functions, the basis of ACE is mathematically complete. This means that ACE parameterizations can be improved and converged systematically. Furthermore, the hierarchical basis not only enables ACE to represent many ML potentials~\\cite{ACE_Drautz_PhysRevB.99.014104} but also to relate ACE to physically and chemically intuitive classical models. The physically motivated representation, which can be linear or mildly non-linear (see below), in combination with a consistent reference dataset helps to ensure that ACE asserts genuine transferability and is not plagued by the reproducibility-crisis of ML-based science \\cite{Kapoor22}.\n\nHere we present a first ACE parametrization for carbon which is not only more accurate and transferable than any of the previous potentials but also significantly more computationally efficient. We compare it in detail to the best available ML potentials for carbon and provide performance indicators for several other potentials. The excellent transferability and predictive power of ACE is highlighted on three distinct applications - a brittle crack propagation in diamond,  a formation of amorphous carbon structures at different quench rates, and a nucleation and growth of fullerene clusters from gas phase at high pressures and temperatures.\n\nThe paper is organized as follows. In the following two sections, we provide a brief theoretical overview of the ACE formalism and elucidate how common classical potentials can be understood as simplified representations of a general ACE descriptor. In section \\ref{sec:training}, we describe the training dataset and the fitting protocol we employed for construction of our carbon model. A detail assessment of the parametrization with respect to a test dataset to determine the general quality of the parametrization is also provided.  In section \\ref{sec:validation}, we subject the potential to multiple validation tests that show the model's ability to predict structural, elastic, vibrational and thermodynamic properties of perfect bulk phases as well as defects. The ACE predictions are compared with those of the reference electronic structure calculations and the best available ML carbon potentials, namely GAP20, TurboGAP and PANNA. In section~\\ref{sec:applications}, we apply the model to perform three large scale simulations to investigate the brittle crack propagation in diamond, the formation of amorphous carbon structures, and the growth of fullerene molecules from the gas phase.\n\n\n\\section{Basics of the atomic cluster expansion}\\label{sec:ace_basics}\n\nThe atomic cluster expansion provides a complete set of basis functions that span the space of local atomic environments. We summarize only the essentials of ACE here and direct interested readers to Refs.~\\cite{ACE_Drautz_PhysRevB.99.014104, dusson2019atomic, drautz2020atomic, PACE_Lysogorskiy2021, pace_bochkarev_PhysRevMaterials.6.013804}.\n\nAn atomic property $p$ that is a function of the local atomic environment of atom $i$ is expanded as\n\\begin{equation}\n\\varphi_i^{(p)} = \\sum_{{\\bm v}} \\cBB_{{\\bm v}}^{(p)} {\\bm B}_{i {\\bm v}} \\,, \\label{eq:ACE}\n\\end{equation}\nwith expansion coefficients $\\cBB_{{\\bm v}}^{(p)}$, and basis functions ${\\bm B}_{i {\\bm v}}$ with multi-indices ${{\\bm v}}$. The energy of atom $i$ can then be evaluated as\n\\begin{equation}\n    E_i = \\varphi_i^{(1)} \\,,\n\\end{equation}\nfor only one atomic property ($p=1$).  If more properties are used,\n\\begin{equation}\nE_i = \\mathcal{F}(\\varphi_i^{(1)},  \\varphi_i^{(2)},  \\dots, \\varphi_i^{(P)}) \\,,\n\\end{equation}\nwhere $\\mathcal{F}$ in general is a non-linear function.\nIn the present ACE model, the energy is expressed using two contributions, a linear term and a square root term,\n\\begin{equation}\nE_i =  \\varphi_i^{(1)} + \\sqrt{\\varphi_i^{(2)}}  \\,. \\label{eq:EFS}\n\\end{equation}\n\nThe basis functions ${\\bm B}_{i {\\bm v}}$ depend on atomic positions and are ordered hierarchically, which enables a systematic convergence of ACE by increasing the number of basis functions. The basis functions fulfill the fundamental translation, rotation, inversion and permutation (TRIP) invariances for the representation of scalar variables, or equivariances for the expansion of vectorial or tensorial quantities. This is achieved by taking linear combinations of basis functions, which do not necessarily fulfill any particular symmetries, as\n\\begin{equation}\n{\\bm B}_{i {\\bm v}} = \\sum_{{\\bm v}'} \\pmb{C}_{{\\bm v} {\\bm v}'} {\\bm A}_{i {\\bm v}'} \\,. \n\\end{equation}\nwhere the generalized Clebsch-Gordan coefficients $\\pmb{C}$ act as a filter and remove basis functions that are not invariant under rotation or permutation.\n\nFor numerical efficiency the basis functions ${\\bm A}$ are constructed recursively \\cite{dusson2019atomic, PACE_Lysogorskiy2021}. The order of the product $\\nu$ determines the body order of a basis function\n\\begin{equation}\n    {{\\bm A}}_{i{\\bm v}} = \n    \\prod_{t = 1}^{\\nu} A_{i {\\bm v}_t} \\,. \\label{eq:AA}\n\\end{equation}\nThe atomic base $A$ is obtained by projecting local basis functions on the atomic density\n\\begin{equation}\n    A_{i{\\bm v}} = \\braket{\\phi_{{\\bm v}}}{\\rho_i} \\,,\n\\end{equation}\nwith the atomic density centered on atom $i$\n  \\begin{equation}\n\\rho_i  = \\sum_j^{j \\neq i} \\delta( \\pmb{r} - \\pmb{r}_{ji}) \\,.\n\\end{equation} \nThe local basis functions are expressed as\n\\begin{equation}\n\\phi_{{\\bm v}} = R_{nl}(r_{ji}) Y_{lm}(\\hat{{\\bm r}}_{ji}) \\,, \\label{eq:LCAO}\n\\end{equation}\nwith $r_{ji}$ being the distance from atom $i$ to $j$ that enters in the radial functions $R_{nl}$, while the spherical harmonics $Y_{lm}$ depend on the direction $\\hat{\\pmb{r}}$. The index ${\\bm v} = (nlm)$ is cumulative, where $n$ differentiates between orbitals with the same angular quantum number $l$ and $m$. \n\nThe basis functions can represent local descriptor functions that are used in ML potentials as well as density and angular functions from classical potentials \\cite{ACE_Drautz_PhysRevB.99.014104, drautz2020atomic}. If decomposed into explicit many-atom functions, the two-body basis functions are given by radial functions\n\\begin{equation}\nB_{i{\\bm v}} = R_{n0} (r_{ji}) \\,, \\label{eq:B1exp}\n\\end{equation}\nwhile the three-body terms have the form\n\\begin{equation}\nB_{i{\\bm v}} = \\frac{1 }{2l +1} \\sum_{jk} R_{n_1 l} (r_{ji}) R_{n_2 l} (r_{ki}) P_l( \\cos \\theta_{jik}) \\,,  \\label{eq:B2exp}\n\\end{equation}\nwith Legendre polynomials $P_l(x)$. Expressions for higher body orders can also be obtained but are more complex~\\cite{ACE_Drautz_PhysRevB.99.014104}.\n\n\n\\section{ACE as a generalization of classical potentials}\\label{sec:ace_classical_pots}\n\n\nTo elucidate that ACE is not only a formally complete expansion but that it also can be considered as a systematic generalization of classical interatomic potentials, we sketch the link between ACE and the second-moment approximation (SMA) of electronic structure~\\cite{pettifor1995bonding, finnis2003interatomic,Friedel69,Ducastelle70}. \n\nStarting from DFT, the energy functional can be decomposed into the band energy and  a double-counting contribution,\n\\begin{equation}\nE^{(\\text{DFT})} = E_{band} + E_{dc} \\,.\n\\end{equation}\nThe band energy is given by $E_{band} = \\sum_n f_n \\epsilon_n$, with occupation numbers $f_n$ and eigenvalues $\\epsilon_n$.  The eigenstates can then be expanded using local orbitals $\\phi_{i \\alpha}$ [cf. Eq.~(\\ref{eq:LCAO})] as \n\\begin{equation}\n\\psi_n = \\sum_{i \\alpha} \\braket{i \\alpha}{n} \\phi_{i \\alpha} \\,.\n\\end{equation}\nWe assume the basis to be complete and for ease of notation consider the local orbitals to be orthonormal. \nThe band energy is then represented as\n\\begin{equation}\nE_{band} = \\sum_{i \\alpha} \\int^{\\epsilon_F} \\epsilon\\, n_{i\\alpha}(\\epsilon) \\,d\\epsilon = \\sum_{i\\alpha \\, j \\beta} \\Theta_{i \\alpha j \\beta} H_{j \\beta i \\alpha } \\,.  \\label{eq:onin}\n\\end{equation}\nThe first identity is the the so-called onsite representation of the band energy, where the local density of states $n_{i\\alpha}(\\epsilon)$ of orbital $\\alpha$ on atom $i$ is filled with electrons up to the Fermi level $\\epsilon_F$.\\footnote{For a discrete spectrum the local density of states is given as $n_{i\\alpha}(\\epsilon) = \\sum_n |\\braket{i \\alpha}{n}|^2 \\delta(\\epsilon - \\epsilon_n)$.}\nThe second identity, the so-called intersite representation, involves the density matrix\/bond order\n\\begin{equation}\n\\Theta_{i \\alpha j \\beta} = \\sum_n f_n \\braket{i \\alpha}{n} \\braket{n}{j \\beta} \\,,\n\\end{equation}\nand Hamiltonian matrix elements $H_{i \\alpha j \\beta} = \\braHket{i \\alpha}{\\hat{H}}{j \\beta}$. \nA formal expansion of the DFT functional with respect to charge density \\cite{harris1985simplified, sutton1988tight, Foulkes89, frauenheim2000self, finnis2003interatomic, Drautz_PhysRevB.84.214114_2011, Drautz15} presents the basis of modern tight-binding (TB) models and results in a partitioning of the energy as\n\\begin{equation}\nE^{(\\text{TB)}} = E_{bond} + E_{prom} + E_{rep} \\,,\n\\end{equation}\nwhere for simplicity we neglected charge transfer. If there is no promotion of electrons and the energy scale is set such that $H_{i \\alpha i \\alpha} = 0$, the band and bond energies are identical, $E_{band} = E_{bond}$. The repulsive energy $E_{rep}$, comprising the double-counting contribution and Coulomb interactions between the atomic cores, is often approximated by a pair potential\n\\begin{equation}\nE_{rep} = \\frac{1}{2} \\sum_{ij} V(r_{ij})\\,.\n\\end{equation}\nFor the derivation of the SMA for metals, we utilize a local expansion of the band energy for an atom \n\\begin{equation}\nE_{band,i} = \\int^{\\epsilon_F} \\epsilon\\, n_{i}(\\epsilon) \\,d\\epsilon \\,\\,\\,\\,\\, \\text{with} \\,\\,\\,\\,\\, n_{i}(\\epsilon)  = \\sum_{\\alpha} n_{i \\alpha}(\\epsilon) \\,, \n\\end{equation}\nWe next construct the local density of states from the information about the local atomic environment. This is achieved by the recursion method~\\cite{Ducastelle70,Friedel69,Haydock80}. If the recursion is continued with constant coefficients after the first recursion level, only information up to the second moment of the density of states enters the expansion. The second moment, given by\n\\begin{align}\n\\mu_i^{(2)} &= \\int \\epsilon^2 n_{i}(\\epsilon) \\,d\\epsilon = \\sum_{\\alpha\\, j \\beta} H_{i \\alpha j \\beta} H_{ j \\beta i \\alpha} \\, , \n\\end{align}\nis determined by Hamiltonian matrix elements that rapidly decay with increasing distance between atoms $j$ and $i$. As the zeroth moment $\\mu_i^{(0)}=1$ and the first moment was set to $\\mu_i^{(1)} =  \\sum_{\\alpha} H_{i \\alpha i \\alpha} = 0$, it can be viewed merely as a consequence of appropriate scaling that in SMA~\\cite{Gupta81, pettifor1995bonding, Ackland_1988}\n\\begin{equation}\nE_{band,i}^{(\\text{SMA})} = C \\sqrt{\\mu_i^{(2)} }  \\propto \\sqrt{\\mathcal{Z}_{i}} \\,, \\label{eq:2Mband}\n\\end{equation}\nwhere the pre-factor $C$ is a function of band filling and $\\mathcal{Z}_{i}$ is the coordination, the number of nearest neighbors, of atom $i$. The total atomic energy can then be written as \n\\begin{equation}\nE_{i}^{(\\text{SMA})} = C \\sqrt{\\mu_i^{(2)} } + \\sum_j V(r_{ij})\\,.\n\\end{equation}\nAs $\\mu_i^{(2)}$ is strictly positive, it may also be understood as an atomic density $\\rho_i$ computed from pairwise functions to neighboring atoms. In this way, one obtains immediately the FS potential~\\cite{Finnis84}.  If the square root function is replaced by a general, concave embedding function one arrives at the EAM formulation~\\cite{Daw84}. Therefore, the cohesion in metals in SMA does not increase linearly with increasing coordination and correctly reflects the unsaturated nature of the metallic bond~\\cite{pettifor1995bonding, heine1991many}.\n\n\nThe derivation of SMA expressions for covalent elements is somewhat more involved and requires explicit consideration of the angular character of atomic orbitals. By equivalence of the onsite and intersite representations of the band energy [cf. Eq.~(\\ref{eq:onin})], SMA for the band energy also implies a corresponding expression for the bond order. We assume the Slater-Koster two-center approximation \\cite{Slater54} with the $z$-axis of the coordinate system aligned along the bond $i -j$.  For a d-valent atom, the Hamiltonian matrix is diagonal with the matrix elements equal to two-center distance-dependent bond integrals dd$\\sigma(r_{ij})$, dd$\\pi(r_{ij})$ and dd$\\delta(r_{ij})$.\nThe second moment is then by construction invariant under rotation and given by\n\\begin{equation}\n \\mu_i^{(2)} = \\sum_{j} \\left[ \\mathrm{dd}\\sigma(r_{ij})^2 + 2\\, \\mathrm{dd}\\pi(r_{ij})^2 + 2\\, \\mathrm{dd}\\delta(r_{ij})^2 \\right] \\,, \\label{eq:dSMA}\n\\end{equation}\nwhere the summation is over the $\\mathcal{Z}_{i}$ neighbors of atom $i$. \nThe $\\sigma$ bond order for the $i-j$ bond (with analogous expressions for the $\\pi$ and $\\delta$ bond orders) is expressed as~\\cite{pettifor1995bonding, finnis2003interatomic}\n\\begin{equation}\n| \\Theta_{ij}^{(\\sigma)} | = \n\\frac{C }{ \\sqrt{ \\mu_i^{(2)}  } } \\propto \\frac{1}{\\sqrt{\\mathcal{Z}_{i} }}\\,, \\label{eq:BO}\n\\end{equation}\nand is thus inversely proportional to the square root of the number of neighbors of atoms $i$. Note that the bond order is not symmetric with respect to exchange of atoms $i$ and $j$ and therefore the denominator needs to be replaced by $\\sqrt{( \\mu_i^{(2)} + \\mu_j^{(2)})\/2}$.\n\n\nNext, since the bond (or band) energy of atom $i$ \n\\begin{equation}\nE_{bond,i}^{(\\sigma)} =  \\sum_j \\Theta_{ij}^{(\\sigma)} dd\\sigma(r_{ij}) \\propto \\sqrt{\\mathcal{Z}_{i}}  \\,. \\label{eq:ebond_dds}\n\\end{equation}\nis obtained as the sum over all $\\mathcal{Z}_{i}$ neighbors, it attains the square root dependence with the number of neighbors as in Eq.~(\\ref{eq:2Mband}).\n\nThe derivation for sp-valent elements follows along the same lines but needs to take into account the directionality of the hybrid orbitals and the energy splitting $\\Delta E_{sp}$ of the s and p orbitals. The hybrid $\\sigma$ orbitals oriented along the $z$-axis are formed as\n\\begin{equation}\n\\ket{i\\sigma} = \\frac{1}{\\sqrt{1+\\lambda^2}} \\left ( \\ket{is} + \\lambda \\, \\ket{i p_z} \\right ) \\,,\n\\end{equation}\nwhere $\\lambda=1$, $\\sqrt{2}$ and $\\sqrt{3}$ correspond to sp, sp$^2$ and sp$^3$ hybrids, respectively. The bond integral of the $\\sigma$ hybrid is given as\n\\begin{equation}\nh_{\\sigma}(r_{ij}) = \\frac{ \\mathrm{ss}\\sigma(r_{ij}) - 2 \\lambda\\, \\mathrm{sp}\\sigma(r_{ij}) - \\lambda^2 \\, \\mathrm{pp}\\sigma(r_{ij}) }{1+\\lambda^2}  \\, ,\n\\end{equation}\nwhere ss$\\sigma$, pp$\\sigma$ and sp$\\sigma$ are Slater-Koster two-center bond integrals.\nThe second moment takes the form \\cite{pettifor1995bonding}\n\\begin{equation}\n\\begin{split}\n \\mu^{(2)}_{i\\to j} &=  c \\, \\Delta E_{sp}^2 + h_\\sigma(r_{ij})^2 + \\\\\n & \\sum_{k \\ne i,j} \\frac{1}{2} \\left(  h_\\sigma(r_{ik})^2 \\, g(\\theta_{jik}) + h_\\sigma(r_{jk})^2 \\, g(\\theta_{ijk}) \\right)  \\,.\n\\end{split}\n\\end{equation}\nwhich differs from Eq.~(\\ref{eq:ebond_dds}) due to the angular functions $g_{jik}$ and $g_{ijk}$ which depend on the angle $\\theta$ between bonds~\\cite{pettifor1995bonding, Alinaghian94}.  Different from the d-valent case, the second moment is not rotationally invariant and the orientation of the $z$-axis along the bond $i \\to j$ has to be given explicitly.\nThe bond order retains a form analogous to Eq.~(\\ref{eq:BO}),\n\\begin{equation}\n |\\Theta_{i \\to j}^{(\\sigma)} | = \\frac{C}{ \\sqrt{ \\mu_{i\\to j}^{(2)} } } \\,.\n\\end{equation}\nThis expression for the bond order is not dissimilar from the empirical bond order introduced by Tersoff \\cite{Tersoff86}. Derivations of the $\\pi$ bond order and the promotion energy, which are important for the bond formation in carbon, can be found in Refs.~\\cite{Alinaghian94,Oleinik_pettifor_aBOP_carbon, pettifor1995bonding, finnis2003interatomic}. \n\nThe aim of this analysis was to show that the most important classical potentials for metals and covalent semiconductors can be understood from the second-moment approximation. The crucial point is that for both these materials, SMA predicts the bond energy to scale as a square root of the local atomic density. This suggests that a physically based model of the atomic energy should comprise an attractive part with a square-root dependence on the number of neighbors, and a pair-wise repulsive part which scales linearly with the number of neighbors. This expression for the energy,\n\\begin{equation}\nE_i = - A \\sqrt{\\mathcal{Z}_i} + B \\mathcal{Z}_i \\,\n\\end{equation}\nmimics the ACE formulation in Eq.~(\\ref{eq:EFS}). If we limit ACE to two-body basis functions, the representation of the energy is closely related to FS models. If 2-body and 3-body contributions are included in the ACE basis, we expect ACE to reproduce not only the original Tersoff formulations but also Stillinger-Weber \\cite{Stillinger85}, EDIP \\cite{Justo98} and other empirical bond order potentials. Incorporation of basis functions with higher body orders then represents a systematic generalization. The higher body orders are similar to including higher moments of the density of states as required for more accurate structural differentiation~\\cite{pettifor1995bonding}. An ACE parametrization with contributions up to the body order of six was developed recently for metallic Cu \\cite{PACE_Lysogorskiy2021}. Here we show that this approach also works extremely well for covalent carbon, where directional bonding is much more important and delicate.\n\n\n\\section{Training of the ACE for carbon}\n\\label{sec:training}\n\n\n\\begin{figure*}\n\\centering\n    \\includegraphics[width=0.99\\textwidth]{dataset_together.png} \n   \n    \\caption{\\label{fig:dataset} Training dataset visualized as a scatter plot of energy per atom with respect to the nearest interatomic distance within each structure and (b) the distribution of cohesive energies.}\n\\end{figure*}\n\nAn accurate and consistent reference dataset that covers a large part of the phase space of atomic configurations is critical for the construction of an ACE model. Such a dataset consists of a series of atomic structures and their corresponding energies, forces and stresses, typically evaluated using electronic structure methods such as DFT. In this section, we describe the details of our DFT calculations including peculiarities related to carbon, present our strategy to generate an exhaustive and balanced set of reference structures, and show how the ACE parametrization is carried out.\n\n\n\\subsection{DFT reference and dispersion interactions}\n\\label{sec:dft_settings}\n\nThe DFT reference calculations were performed using the Vienna Ab-initio Simulation Package (VASP)~\\cite{dft1,dft2,dft3}, version 5.4.4. The exchange-correlation energy was computed using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)~\\cite{gga} and the core electrons were modeled by the Projector-Augmented Wave (PAW)~\\cite{paw1,paw2} method (C: s2p2).  We carried out highly converged calculations with tight settings of the principal parameters in order to obtain accurate results for the energy and forces. Specifically, the energy cutoff for the plane-wave basis was set to 500 eV and the convergence threshold for the energy to 10$^{-6}$ eV.  Gaussian smearing with a width of 0.1 eV was applied.\nFor periodic structures, the Brillouin zone was sampled using a dense $\\Gamma$-centered k-point mesh with the spacing between the k-point of 0.125  \nper \\AA{}$^{-1}$ while for non-periodic clusters only the $\\Gamma$ point was used. An additional support grid was employed for the calculation of forces to ensure that forces and energies are consistent.\n\nWhile the PBE functional describes accurately covalent bonds, it does not capture long-range dispersion interactions. As the vdW interaction plays a crucial role in the stabilization of many important carbon structures, such as graphite and its derivatives, PBE data alone is not suitable for the parametrization of a fully transferable carbon model. There exist various approaches to account for dispersion interactions within DFT~\\cite{vdw_d2_coorection, vdw_d3_coorection,vdw_d4_coorection,vdw_TS_correction, vdw_TS_MBD, vdw_TS_MBD2}. We decided to employ additive corrections, which allow us to parameterize ACE based on standard PBE data, which is significantly shorter ranged than the vdW interactions, and then to amend the ACE model with a correction term in analogy to most dispersion-corrected DFT approaches~\\cite{grimme2011density}. This approach not only results in an efficient model with the correct description at long interatomic distances, but it also gives the flexibility to employ correction terms of different complexity or even switch off the long-range interactions if they are not needed.\n\nFinally, ACE is parametrized such that the interaction between C atoms approaches zero at infinite separation to ensure that the fitted energies correspond to cohesive energies. This is done by taking the energy of an isolated spin-unpolarized atom as the reference zero energy. \n\n\n\\begin{table*}\n\\caption{A description of datasets used in this work.}\\label{tab:dataset_description}\n\\centering\n\\begin{tabular}{lccccc}\n\\toprule\n            Category  &  Description &  \\makecell{Number\\\\of structures} &  \\makecell{Number\\\\ of atoms} & \\makecell{[$E_{min}$, $E_{max}$]\\\\ (eV\/atom)} &  \\makecell{NNB range\\\\ (\\AA)} \\\\\n\\midrule\n      sp$^2$ structures &  \\makecell{graphene, graphite, fullerenes, nanotubes, incl. defects} &     3532 &  88358 & [-9.07, 78.50] & [0.7, 4.4] \\\\[5pt] \\hline\n      sp$^3$ structures &  \\makecell{cubic and hexagonal diamond, \\\\ high-pressure phases (bc8, st12, m32, etc.), incl. defects} &     3407 &  84290 & [-8.93, 36.99] & [0.9, 4.9] \\\\[5pt] \\hline\n    amorphous\/liquid &  \\makecell{selected from available datasets;\\\\ amorphous and liquid phases \\cite{gap20_doi:10.1063\/5.0005084} \\\\ MD trajectories of multilayered graphene \\cite{NNP_Gr_PhysRevB.100.195419}} &     2642 &  146188 & [-9.06, -3.18] & [1.0, 1.7] \\\\[5pt] \\hline\ngeneral bulk & \\makecell{basic crystals; fcc, hcp, bcc, sc, A15, etc.\\\\ over broad range of volume and \\\\random displacements of atoms\/cell deformations }   &     5342 &   39 126 & [-8.06, 82.17] & [0.9, 4.4] \\\\[5pt] \\hline\ngeneral clusters & \\makecell{non-periodic clusters with 2-6 atoms} &     2370 &    8801 & [-6.19, 83.28] & [0.6, 5.0] \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\n\n\\subsection{Reference dataset}\n\\label{sec:dataset}\n\nWe constructed an extensive reference dataset consisting of 17,293 structures with a total number of 366,763 atoms. The reference structures were chosen to sample a broad range of atomic configurations for carbon to ensure both accuracy and transferability of the ACE parametrization.\n\n\nWe divided the reference structures into five categories. The first category contains carbon structures with prototypical sp$^2$ bonding, including a variety of bulk graphite structures, 2D graphene sheets, molecular fullerenes and nanotubes. The second category contains sp$^3$ four-fold coordinated crystalline diamond structures and their high pressure variants. In both categories, we sampled the structures over a broad range of interatomic distances, shape distortions, random displacements of atoms, and incorporated point and planar defects such as vacancies, Stone-Wales defects, surfaces, interfaces, etc. These two categories were complemented by structures from MD simulations of amorphous and liquid carbon~\\cite{gap20_doi:10.1063\/5.0005084} and  multilayered graphene~\\cite{NNP_Gr_PhysRevB.100.195419}. As the MD structures are highly correlated, we included only a relatively small number of atomic configurations from these two collections.  \nThe last two categories, referred to as `general bulk' and `general clusters', consist of general crystal structures, e.g., fcc, bcc, sc, hcp, A15, and isolated random clusters containing up to six carbon atoms. These two categories serve to sample a broader region of configurational space and allow the potential to model close-packed atomic environments. A summary of the five categories is provided in Table~\\ref{tab:dataset_description}.\n\nFigure~\\ref{fig:dataset} shows the cohesive energy for the reference structures as a function of the shortest bond length within each structure (left panel) and the distribution of structures within the corresponding energy range (right panel). The standard PBE functional without any vdW correction predicts the graphene phase to have the lowest cohesive energy of $-9.12$~eV\/atom and an equilibrium bond length of 1.45~\\AA{}. The energies of most structures from the first three categories lie within 3 eV\/atom above the ground state energy, while the remaining two categories are characterized by cohesive energies mostly greater than $-6$~eV\/atom.  The six-dimensional configuration space of  clusters with four atoms was sampled uniformly~\\cite{ACE_Drautz_PhysRevB.99.014104} with interatomic distances ranging from $0.7$ to $3.0\\, r{_s}$, where $r{_s}$ = 1.45~\\AA{} corresponds to the NNB distance in graphene. \n\n\n\\subsection{ACE implementation and efficiency}\n\\label{sec:software}\n\nFor the parameterization of ACE we employed the software package {\\verb+pacemaker+} \\cite{pace_bochkarev_PhysRevMaterials.6.013804}. The simulations for validation and applications were carried out using {\\texttt{\\detokenize{LAMMPS}} } \\cite{LAMMPS} with the {\\texttt{\\detokenize{PACE}} } package \\cite{PACE_Lysogorskiy2021}. Both {\\verb+pacemaker+} and {\\texttt{\\detokenize{LAMMPS}} }+ {\\texttt{\\detokenize{PACE}} } can be executed on CPU and GPU architectures. The computational performance of ACE in comparison with other ML methods is shown in Fig.~\\ref{fig:walltime}. The graph displays the CPU\/GPU times for a single MD time step per atom from representative $NVT$ MD simulations of liquid carbon at 4000 K with density of 2 g\/cm$^{3}$ (using a periodic supercell containing 1000 atoms for 1000 time steps). For TurboGAP we used the value reported in Ref.~\\cite{TurboGAP}, since it is not available in {\\texttt{\\detokenize{LAMMPS}} }.  On CPU, ACE is more than 1-2 orders of magnitude faster than the other models in accordance with our previous benchmarks~\\cite{PACE_Lysogorskiy2021}. On GPU, ACE reaches efficiency comparable to that of classical interatomic potentials.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{walltime_withGPU.png}\n  \\caption{\n    A comparison of typical CPU (AMD Ryzen 5 3600X) and GPU (Tesla V100S-PCIE 32 GB) times for ACE, TurboGAP, PANNA and GAP20 (for TurboGAP the value was taken from Ref.~\\cite{TurboGAP}). CPU times are given per core and GPU times are per divice.}\n  \\label{fig:walltime}\n\\end{figure}\n\n\n\n\\subsection{Training}\n\\label{sec:fitting_process}\n\nTraining a carbon potential for an optimal balance between accuracy and transferability is more challenging than for many other materials. The potential must simultaneously reproduce minute energy differences between the most stable phases and allotropes while being able to capture large energy changes associated with breaking and re-arrangement of the strong directional covalent bonds that occur during phase transformations or in the vicinity of structural defects.\n\nDuring the ACE training, we employed a hierarchical optimization strategy implemented in the {\\verb+pacemaker+} code that sequentially adds more basis functions. The radial basis functions represented by exponentially-scaled Chebshev polynomials were also included in the optimization. Structures with low cohesive energies (3 eV above the ground state) were assigned higher weights in the loss function. The employed training dataset together with the input for the {\\verb+pacemaker+} code is provided in the supplementary material~\\cite{suppl}.\n\nThe presented ACE parametrization comprises of 488 basis functions with terms up to the fifth body order, which is sufficient to attain an outstanding overall accuracy while remaining computationally very efficient. As shown on the feature curve in Fig.~\\ref{fig:timing_n_funcs}(a), the accuracy can be further improved by the increasing number of basis functions~\\cite{pace_bochkarev_PhysRevMaterials.6.013804}, while the scaling of the computational time, shown in Fig.~\\ref{fig:timing_n_funcs}(b), remains linear~\\cite{PACE_Lysogorskiy2021}.\n\n\n\\begin{figure}\n\\centering\n    \\subfloat[]{\\includegraphics[width=0.25\\textwidth]{learning_curves.png}}\n    \\subfloat[]{\\includegraphics[width=0.25\\textwidth]{time_funcs.png}} \n    \\caption{(a) The feature curves showing the RMSE of energy and force as a function of the number of basis functions; (b) the scaling of ACE computational time on GPU with the number of basis functions; vertical dashed lines mark the presented ACE parametrization with 488 basis functions.}\n    \\label{fig:timing_n_funcs} \n\\end{figure}\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrrr}\n\\toprule\n    Category (train\/test) & $E_\\text{train}$ & $E_\\text{test}$ & $F_\\text{train}$ &  $F_\\text{test}$ \\\\\n                       \n\\midrule\n   sp$^2$ structures (4549\/528) &        29 &   28 &   465 &  499 \\\\\n   sp$^3$ structures (6130\/623) &        56 &   54 &   587 &  567 \\\\\n   amorphous\/liquid (2615\/315) &        59 &   74 &   307 &  376 \\\\\n   general bulk (5926\/652)      &        97 &  114 &  1332 &  1420 \\\\\n   general clusters (2360\/276)  &       154 &  186 &  1120 &  1195 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[fit stats]{Energy and force RMSE for each category of the reference dataset. Numbers in brackets correspond to the number of structures for training and testing, respectively. Energies are in meV\/atom and forces in meV\/\\AA{}. \\label{tab:rmse_dataset}}\n\\end{table}\n\n\nThe root mean square errors (RMSE) for the different subsets are given in Table~\\ref{tab:rmse_dataset}. The fact that RMSE for train and test sets are comparable demonstrates that the model is not overfitted. The optimized potential has an energy RMSE of 21 meV\/atom for the structures within 3 eV\/atom from the ground state and 166 meV\/atom for the complete dataset.  The corresponding force RMSE amount to 218 and 689 meV\/\\AA{}, respectively. Most large force errors arise from high energy structures from the `general bulk' and `general clusters' categories (see supplementary material~\\cite{suppl}) and structures with short interatomic distances, which show forces up to 100 eV\/\\AA{}. As reported in the supplementary material~\\cite{suppl}, previous carbon models exhibit even larger errors for both forces and energies.\n\n\n\\begin{figure}\n\\centering\n    \n    \\begin{tabular}{@{}c@{}}\n    \\includegraphics[width=1\\columnwidth]{crossval_en_test.png}\n   \n  \\end{tabular}\n  \n  \\vspace{\\floatsep}\n\n  \\begin{tabular}{@{}c@{}}\n    \\includegraphics[width=1\\columnwidth]{crossval_f_test.png}\n   \n  \\end{tabular}\n    \n    \n    \\caption{\\label{fig:crossref} Energy and force errors for a split test dataset consisting of 1,912 structures; the dashed vertical line marks the boundary of differently weighted data.}\n\\end{figure}\n\n\nAn overall assessment of the accuracy of presented ACE parametrization is given in Fig.~\\ref{fig:crossref}. Figure~\\ref{fig:crossref}(a) shows the predicted ACE energies with respect to the reference PBE data for a 10\\% split test set with 1,912 structures. The effect of higher weighting of the low energy structures is clearly visible. Most structures with energies below $-6.2$ eV\/atom, indicated by the vertical line, match the reference very clesely. For structures with higher energies and lower weights, deviations are larger. Figure~\\ref{fig:crossref}(b) details the distribution of energy errors for the different categories. As these are not normalized distributions, the area under the peaks corresponds to the total number of structures within each category. The standard deviations associated with each category are considerably different, with the sp$^2$ and sp$^3$ structures having the most narrow distributions. The force cross-correlation and error distribution are plotted for all categories in Figs.~\\ref{fig:crossref}(c) and (d), respectively. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{graphite_eb.png}\n\\caption{Energy as a function of interlayer separation in graphene. Result for other models are shown in the supplementary material~\\cite{suppl}.}\n\\label{fig:graphite_eb}\n\\end{figure}\n\n\nFigure~\\ref{fig:graphite_eb} shows the binding energy between the layers of graphite as a function of the interlayer separation. The standard PBE functional gives a negligible binding energy of 1 meV\/atom at around 4.5 \\AA{}, i.e., essentially only a short range repulsive interactions between the graphene sheets. This PBE reference is reproduced accurately by the base ACE (referred to as ACE$_\\text{PBE}$), that was trained on the uncorrected PBE data.  By adding the D2 correction with a long range cutoff of 9 \\AA{}, ACE achieves an excellent description of cohesion in graphite, very close to that of TurboGAP. While qualitatively similar, GAP20 shows rather oscillatory behavior whereas for PANNA the range of vdW interactions is significantly underestimated. Additional results for the hNN\\_Gr$_x$ and DeepMD potentials are given in the supplementary material~\\cite{suppl}. \n\n\n\n\\section{Validation}\n\\label{sec:validation}\n\nWe carried out multiple validation tests to assess the performance of the carbon ACE. In the following we present key tests and compare the predictions of ACE to those of the best available potentials. Further validations are provided in the supplementary material~\\cite{suppl}.\n\n\n\\subsection{Stability of bulk phases} \\label{sec:basic_tests}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{ev_groundstate_4pots_acegaps.png}\n\\caption{Relative stability of graphite, graphene and diamond as predicted by PBE+D2, ACE and the best available models. $\\Delta E$ with respect to graphite as given by the respective potential. Result for other NNPs are presented~\\cite{suppl}.}\n\\label{fig:ev_groundstate}\n\\end{figure}\n\n\nOne of the peculiarities of bonding in carbon is that the energy of three-fold coordinated graphite and graphene is nearly the same as that of four-fold coordinated diamond, while the energy of the carbon dimer is much higher. This illustrates the importance of strong angular bond contributions as well as weak dispersion interactions. According to our PBE+D2 calculations, the graphite ground state is separated from those of diamond and graphene by only 29 and 50~meV\/atom, respectively. This energy ordering is in agreement with experimental~\\cite{wagman1945heats} as well as recent theoretical predictions using a high-level coupled cluster theory, according to which diamond lies less than 30 meV\/atom above graphite~\\cite{Popov_2019_C8CP07592A, Gruber_PhysRevX.8.021043}. These subtle energy differences are, however, not captured correctly by all DFT functionals.  While our PBE+D2 results agree well with those of the PBE+MBD \\cite{mbd1_PhysRevLett.108.236402,mbd2_doi:10.1063\/1.4865104} functional, which was used to generate the TurboGAP reference data, the hybrid  optB88-vdW \\cite{opt1_PhysRevB.82.081101,opt2_PhysRevB.83.195131,opt3_PhysRevLett.92.246401,opt4_Klime__2009} and rVV10 \\cite{rVV10_PhysRevB.87.041108} functionals, which were used for the GAP20 and PANNA datasets, respectively, predict diamond to have a significantly larger energy than both graphite and graphene. The consequence of different training data is illustrated in Fig.~\\ref{fig:ev_groundstate}, which shows the relative stability of the three structures as a function of nearest-neighbor (NNB) distance obtained by ACE, TurboGAP, GAP20 and PANNA. While ACE and TurboGAP match closely the PBE+D2 reference data, GAP20 and PANNA predict that diamond is substantially less stable than both sp$^2$ allotropes. \n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.5\\textwidth]{gradia_transformation_2.png}\n    \\caption{Energy barrier associated with the graphite to diamond transformation; $\\Delta E$ with respect to graphite. \\label{fig:gradia}\n    }\n\\end{figure}\n\nFigure~\\ref{fig:gradia} shows the energy barrier associated with the transformation between AB-stacked rhombohedral graphite and diamond, which proceeds by simultaneous buckling and lateral compression of graphene sheets~\\cite{gradia_transform_PhysRevB.34.1191}. According to PBE+D2, the barrier is 350 meV\/atom with respect to graphite. ACE reproduces the barrier within a few meV. TurboGAP, GAP20 and PANNA overestimate it by 164, 97 and 95 meV\/atom, respectively.\n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.44\\textwidth]{ev_all_virtical.png}\n    \n    \\caption{\\label{fig:ev_curves} Binding energy vs NNB distance for various carbon structures predicted by ACE, TurboGAP, GAP20 and PANNA. Symbols in ACE panel represent the PBE+D2 reference. Binding energies for other carbon potentials cen be found in the supplementary material~\\cite{suppl}.}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:ev_curves} we compare binding energy curves for various carbon structures obtained by ACE, TurboGAP, GAP20 and PANNA.  Results for other carbon potentials are provided in the supplementary material~\\cite{suppl}. ACE describes the binding energy curves in excellent agreement with the DFT+D2 reference over the whole range of considered NNB distances. This may be attributed to the extensive reference dataset and the extrapolation capabilities of ACE basis functions~\\cite{PACE_Lysogorskiy2021}. In contrast, TurboGAP, GAP20 and PANNA reveal the well known inability of most ML potentials to extrapolate outside of the reference dataset. The three models were fitted mostly to configurations with densities close to those of equilibrium graphite and diamond, and they clearly fail to describe the low-density structures. The problem is more severe for GAP20, as it predicts large unphysical oscillations for nearly all structures beyond the NNB distance of about 1.7 \\AA{}. For all structures there exist multiple local minima. These can lead to occurrence of spurious phases, for instance, in the vicinity of defects or when the system is subject to external loads. The local minima and maxima can further affect forces and undermine the description of bond making and breaking~\\cite{rebo_S_pastewka_PhysRevB.78.161402}.  Finally, even though carbon does not readily form structures like fcc, bcc or sc, it is advisable to reproduce properties of these structures as well, as they may occur in MD simulations under non-equilibrium conditions or at high pressure. For example, atomistic models of amorphous carbon are often generated by melting a sc lattice of C atoms \\cite{marks1_transferability,marks2_graphitization,Jana_2019}, and the sc phase is found to be stable at extreme pressures\\cite{Oleynik_2022_c_extrem,sun_2009}. \n\nIt is worth mentioning that the energy of an isolated atom, which represents the reference for the cohesive energy, is different for the four models. For ACE, the energy of a non-magnetic free atom is subtracted from the reference data, as discussed in Sec.~\\ref{sec:dft_settings}, and therefore the energy tends to zero as the atoms are pulled apart beyond the chosen cutoff. For TurboGAP, GAP20 and PANNA, this limit is not zero (cf. Fig.~\\ref{fig:ev_curves}) but $-0.51$, $+0.94$ and $-249.96$ eV\/atom, respectively. \n\n\n\n\\subsection{Elastic and vibrational properties} \\label{sec:elc_phon}\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.5\\textwidth]{phonons_3_phases.png}\n    \\caption{Phonon band structures and densities of states for (a) graphene, (b) graphite, and (c) diamond predicted by ACE (solid lines) and PBE+D2 (dashed lines). \\label{fig:phonons} }\n\\end{figure}\n\n\n\\begin{table*}[]\n\\caption{Elastic moduli of graphite (Gr-ite), graphene (Gr-ene) and diamond (Dia); all valueas are in GPa. }\\label{tab:elastic_prop}\n\\begin{tabular}{l|ccc|ccc|ccc|ccc|ccc|}\\hline\n    & \\multicolumn{3}{c|}{PBE+D2}   & \\multicolumn{3}{c|}{ACE}      & \\multicolumn{3}{c|}{TurboGAP} & \\multicolumn{3}{c|}{GAP20} & \\multicolumn{3}{c|}{PANNA}     \\\\ \n    & Gr-ite & Dia & Gr-ene & Grite & Dia & Gr-ene & Gr-ite & Dia & Gr-ene & Gr-ite & Dia & Gr-ene & Gr-ite & Dia & Gr-ene \\\\ \\hline\n$C_{11}$ &   1095 $\\pm$ 25 &  1042 $\\pm$ 8 &    238 $\\pm$ 12 &   1045 &  1004 &    275 &    945 &  1024 &    261 &   1022 &   924 &    273 &   1045 &   1054 &    235 \\\\\n$C_{12}$ &    179 $\\pm$ 51 &   131 $\\pm$ 6 &     39 $\\pm$ 25 &    182 &   141 &     44 &    142 &   104 &     39 &    210 &    24 &     47 &    213 &    131 &     58 \\\\\n$C_{13}$ &     -5 $\\pm$ 25 &               &                 &     13 &       &        &     12 &       &        &     24 &       &        &     -8 &       &  \\\\\n$C_{33}$ &     58 $\\pm$ 10 &               &                 &     27 &       &        &    186 &       &        &    110 &       &        &     33 &       &  \\\\\n$C_{44}$ &      1 $\\pm$ 5  &  556 $\\pm$ 23 &                 &     9  &   537 &        &  10 &    524 &         &     35 &    474 &        &     -9 &    461 &       \\\\\n$C_{66}$ &    457 $\\pm$ 36 &               &     96 $\\pm$ 18 &   431  &       &    115 &    402 &       &    111 &    406 &        &    113 &   403 &        &    89 \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\nThe computed elastic moduli for graphene, diamond and graphite are listed in Table~\\ref{tab:elastic_prop} and the phonon spectra plotted along high symmetry directions of the Brillouin zone and the corresponding phonon densities of states are shown in Fig.~\\ref{fig:phonons}. The ACE predictions agree closely with the PBE+D2 reference for all three structures. As pointed out by the PANNA developers~\\cite{Shaidu2021_PaNNA2021} ab: authors maybe?, long wave length undulations of graphene or graphite sheets are sensitive to numerical details and can induce slightly negative phonon branches close to the $\\Gamma$ point as well as slightly negative elastic moduli. Elastic and phonon properties for other bulk structures are provided in the supplementary material~\\cite{suppl}. Most of these phases show a number of elastic and phonon instabilities that are accurately captured by ACE but not by TurboGAP, GAP20 or PANNA.\n\n\n\\subsection{Point defects} \\label{sec:point_defects}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{gra_monovac_compiled.png}\n\\caption{Three-fold $D_{3h}$ (left) and reconstructed $C_{2v}$ (right) monovacancy predicted by ACE. Bottom panel: side view of graphene sheet, out-of-plane displacement of the marked atom is about 0.4 \\AA. \n}\n\\label{fig:gra_monovac_restructuring}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{defects.png}\n\\caption{Point defects in graphene (left panel) and surfaces in diamond (right panel). See text for details on DFT calculations ; the empty bars for the (110) surface correspond to values reported in the original GAP20 reference~\\cite{gap20_doi:10.1063\/5.0005084}.}\n\\label{fig:defects}\n\\end{figure*}\n\nThe directional covalent bonds in carbon imply large point defect energies and significant local reconstructions. Vacancies and their clusters influence a broad range of electronic, physical and mechanical properties in graphene~\\cite{han2014graphene, lopez2015increasing, skowron2015energetics, zambudio2021, lui2019}. An unreconstructed monovacancy in graphene, formed by removing a single carbon atom, has a three-fold $D_{3h}$ symmetry with three dangling bonds. It undergoes a Jahn-Teller distortion~\\cite{skowron2015energetics} and reconstructs to a configuration with lower $C_{2v}$ symmetry consisting of 5- and 9-membered rings. In Fig.~\\ref{fig:gra_monovac_restructuring} we show that ACE correctly reproduces this reconstruction, including an out of plane displacement of the central atom. The energy difference between unreconstructed and reconstructed configuration presents an upper bound of the monovacancy migration barrier, predicted to be 0.37 eV by ACE, in close agreement with a range of DFT values~\\cite{skowron2015energetics}.\nMost classical as well as the considered ML potentials are unable to account for the reconstruction and predict the three-fold symmetric structure as the only stable monovacancy configuration~\\cite{qian2021comprehensive}. \n\nThe removal of two neighbouring atoms in graphene allows for a better saturation of dangling bonds. There exist three stable divacancy configurations. The simplest one consists of one 8-membered ring with two adjacent 5-membered rings (5-8-5 configuration). A rotation of a pair of bonds in the 8-member ring results in two further configurations with lower energies, the 555-777 and 5555-6-7777 configurations, with increased numbers of the adjacent rings. The formation energies of these configurations are reported in the supplementary material~\\cite{suppl}. ACE predicts correctly that the 555-777 divacancy is more stable than the 5-8-5 and 5555-6-7777 configurations by about 0.8 and 0.5 eV, respectively.  Since, the formation energies of divacancies are comparable to that of the monovacancy, it is favorable for two monovacancies to coalesce.\n\nFigure~\\ref{fig:defects} (left panel) shows a comparison of computed energies for unreconstructed vacancy defects in graphene. For a fair comparison, the DFT references are those used for the construction of the potentials, namely, PBE+D2 for ACE, PBE+MBD for TurboGAP, and optB88-vdw for GAP20~\\cite{gap20_doi:10.1063\/5.0005084}. The DFT data for PANNA were not provided, but we assume them to be similar to those for GAP20. It should be noted that the differences between different DFT methods are comparable to the differences between the fitted and reference values. Energies and structural reconstructions of additional point defects are provided in the supplementary material~\\cite{suppl}.\n\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{stone_Wales_rotation.png}\n\\caption{Bond rotation for formation of Stone-Wales (SW) defect.}\n\\label{fig:stone_Wales_rotatation}\n\\end{figure}\n\nThe point defect in graphene with the lowest energy is the Stone-Wales (SW) defect. It is formally generated by a 90$^{\\circ}$ rotation of one C-C bond, transforming four 6-member rings into two 5-member rings and two 7-member rings. A consistent description of the SW defect and its formation mechanism implies that a potential models local changes in hybridization correctly and is critical for simulating more complex structures, such as the recently reported monolayer amorphous graphene~\\cite{monolayer_a_grapheneToh2020_nature}. We computed the energy barrier associated with C-C bond rotation to be $\\sim 8$ eV and the energy of the SW defect as $4.91$ eV, both in excellent agreement with PBE+D2 reference as shown in Fig.~\\ref{fig:stone_Wales_rotatation}. \n\nOverall, ACE predicts structures and energies of basic point defects in graphene in close agreement with the reference DFT values and a comparable level of accuracy is obtained also for other defects~\\cite{suppl}. \n\n\n\\subsection{Diamond surfaces} \\label{sec:surfaces}\n\nThe right panel of Fig.~\\ref{fig:defects} shows the energy of relaxed but unreconstructed low-index diamond surfaces. As for the point defects in the left panel, the DFT reference energies correspond to the values obtained with the exchange correlation functional that was used for the training of each potential.  The empty bars for the (110) surface correspond to the values reported in the original GAP20 publication~\\cite{gap20_doi:10.1063\/5.0005084}, which we were not able to reproduce. ACE predicts the energies of all surfaces within 3\\% error, while the errors of TurboGAP and GAP20 are larger.\n\nReconstructions of diamond surfaces driven by relaxation of dangling bonds induce subtle atomic displacement patterns that are a challenging test for interatomic potentials. For example, due to dangling bonds the unreconstructed $(111)$ and $(100)$ surfaces have significantly higher energies than the $(110)$ surface. Two distinct surface terminations, with three (3db) and with one (1db) dangling bonds, exist for the $(111)$ surface. We examined surface reconstructions by relaxing atomic positions to minimize the energy of larger supercells. The simulations were initialized to break the symmetry of the unreconstructed surfaces, either manually or by short MD simulations. \n\nFigure~\\ref{fig:surface_reconstruction} shows side and top views of the ideal and reconstructed $(100)$, 3db-$(111)$ and 1db-$(111)$ surfaces. Surface atoms with low coordinations tend to rearrange to achieve a more favourable sp$^2$ (green) hybridization. The 3db-$(111)$ surface undergoes the so-called Pandey-chain reconstruction~\\cite{Pandey_reconstruction_PhysRevB.25.4338}, which rearranges surface atoms into $\\pi$-bonded chain structures. It is a delicate reconstruction and the surface layers show a strong tendency to graphitize~\\cite{chadi_dia_111_surfs}. The 1db-$(111)$ surface instead creates $\\pi$-bonded chains between the surface atoms. All reconstructions have a dramatic effect on the surface energies, which are reduced by 2.4, 7.3 and 0.3 J\/m$^2$ for the $(100)$, 3db-$(111)$ and 1db-$(111)$ surfaces, respectively. The surface energies are summarized in Table~\\ref{tab:surface_energies}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{surface_reconstruction_all_view_just3.png}\n\\caption{Top views of the (a) $(100)$  and  (b,c) $(111)$  surfaces with reconstructions predicted by ACE. White, red, green and blue corresponds to 1-fold, 2-fold, 3-fold and 4-fold coordinated atoms.\n}\n\\label{fig:surface_reconstruction}\n\\end{figure}\n\n\n\\begin{table}[]\n\\begin{tabular}{lccccc}\n\\hline\nSurface  & DFT    & ACE                  & TurboGAP             & GAP20           & PANNA          \\\\ \\hline\n\\bf{(100)}     &        & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n ~~Ideal       & 9.33 (9.33) & 9.10      & 11.85              & 10.09       & 8.97 \t     \\\\\n ~~Relaxed     & 9.04 (9.08) & 8.78      & 10.25              & 9.13        & 5.92 \t      \\\\\n ~~Reconstr.   & 4.97 (4.83) & 6.08      & 5.45               & 4.96        & 5.92 \t      \\\\ \\hline\n\\bf{3db-(111)} &        & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n ~~Ideal       & 13.01 (13.02) & 12.72              & 11.37              & 13.45   &   12.81         \\\\\n ~~Relaxed     & 12.93 (12.98) & 12.71              & 11.37              & 12.17   &   12.65          \\\\\n ~~Reconstr.   &               & 7.36 \t            & 6.56               & 6.24    &    6.72      \\\\ \\hline\n\\bf{1db-(111)} &        & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n ~~Ideal       & 7.25 (8.10) & 6.72                & 6.56               & 5.12      &  4.80         \\\\\n ~~Relaxed     & 5.72 (6.46) & 5.28                & 4.48               & 3.68      &  3.84       \\\\\n ~~Reconstr.   & 3.51 (3.76) & 4.96 \t            & 4.32               & 3.68     &  4.00         \\\\ \\hline\n\\bf{(110)}     &        & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} & \\multicolumn{1}{l}{} \\\\\n ~~Ideal       & 6.77 (7.46) & 7.13                & 6.88               & 5.76      & 5.12         \\\\\n ~~Relaxed    & 5.76  (6.50) & 5.73                & 4.80               & 4.16      & 3.52        \\\\ \\hline\n\\end{tabular}\n\\caption{Surface energies in J\/m$^2$ for ideal, relaxed and reconstructed low-index diamond surfaces. DFT: PBE+D2, in brackets: B3LYP from~\\cite{delapierre_2013_doi:10.1080\/00268976.2013.829250}}\\label{tab:surface_energies}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{aitts.png}\n\\caption{Decohesion of unrelaxed, unreconstructed diamond crystals along the $(111)$, $(110)$ and $(100)$ crystallographic orientations. \n}\n\\label{fig:aitts}\n\\end{figure}\n\nIn addition to static surface calculations, we carried out cleavage simulations to assess smoothness of the energy landscape during bond breaking. A bulk diamond cubic crystal was rigidly separated, excluding relaxations and reconstructions, in the direction normal to the surface using periodic supercells with 32 (for the $(100)$ surface) or 48 (for the $(110)$ and $(111)$ surfaces) atoms. The corresponding variations of the energy as a function of the  separation are shown for all orientations in Fig.~\\ref{fig:aitts}. The energy increase is steepest for the $(111)$ surface orientation, for which one set of the C-C bonds is oriented parallel to the loading direction.  DFT (PBE+D2) predicts that the energy plateaus at a separation of  $\\approx 2$ \\AA{} with a shallow maximum between 1.0 and 1.5 \\AA{}. For the $(100)$ and $(110)$ surface orientations, with bonds inclined with respect to the loading direction, the energy increases less steeply and without any barrier.  For the three orientations, ACE captures the PBE+D2 reference closely. TurboGAP and especially GAP20 exhibit oscillations, while PANNA predicts relatively smooth decohesion curves.  \n\n\n\n\\section{Applications}\n\\label{sec:applications}\n\n\\subsection{Brittle fracture of diamond} \\label{sec:decohesion_crack}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{crack_propagation.png}\n\\caption{Relative distance between atoms at crack tip as function of applied stress intensity factor $K_\\text{I}$. }\n\\label{fig:crack_propagation_plot}\n\\end{figure}\n\n\nFracture simulations of brittle materials are very challenging as they require transferable models which are able to describe bond breaking processes under large and inhomogeneous stresses. At the same time, the models need to remain numerically efficient to be able to simulate large supercells with complex crack geometries at finite temperatures and over realistic time scales~\\cite{andric2018atomistic, bitzek2015atomistic}. Here we present MD simulations of brittle cleavage of diamond performed using ACE, GAP20 and PANNA potentials, mostly intended to compare the predictions of these three models. More extended study of diamond fracture will be presented elsewhere.\n\nWe simulated semi-infinite cracks with periodic boundary conditions applied along the crack front. The atomic configurations were generated by applying a given stress intensity factor, $K_\\text{I}$, to model the asymptotic crack tip region in linear elastic fracture mechanics without applying traction on the outer cell boundaries~\\cite{andric2018atomistic} using the code {\\texttt{\\detokenize{atomsk}} }~\\cite{atomsk_HIREL2015212}.  \nDepending on the magnitude of the applied $K_\\text{I}$, the crack either tends to heal or to propagate during the simulation.  By varying K$_\\text{I}$, one can estimate the critical stress intensity factor, $K_\\text{IC}$.\n\nWe simulated nine cracks with different magnitudes of the stress intensity factor using LAMMPS \\cite{LAMMPS}. The simulation cells contained 5280 atoms with the crack plane normal oriented along the $\\langle 011 \\rangle$ direction and the $ \\langle 111 \\rangle$ propagation direction~\\cite{suppl}. The initial crack tip was always located in between two $ \\langle 011 \\rangle$ planes in the center of the simulation cell. We followed the crack evolution for 2 ps using $NVT$ MD simulations at $T=300$ K. The change of the interatomic distance $r$ between two atoms at the initial crack tip  (see the supplementary material~\\cite{suppl}) was used to quantify the healing or propagation of the crack. The variation $r\/r_0$, where $r_0$ is the initial bond length, as a function of the applied $K_\\text{I}$ is plotted for ACE, GAP20 and PANNA in Fig.~\\ref{fig:crack_propagation_plot}. We were not able to run equivalent simulations using TurboGAP since it is not implemented in LAMMPS. \n\nOur simulations show that below the critical loading all models predict a closing of the crack along the crack plane. However, ACE is the only model that sustains brittle cleavage, when the loading exceeds the critical value of about 4.2 MPa m$^{1\/2}$.  Both GAP20 and PANNA show instead local structural transformations into graphitic structures which lead to blunting of the crack tip, as displayed in Fig.~\\ref{fig:compiled_crack_600}. The formation of the graphitic structures explains the sharp drop of $r\/r_0$ for both potentials visible in Fig.~\\ref{fig:crack_propagation_plot}. To our best knowledge this behavior has not been observed in any theoretical or experimental studies. As discussed in Sec.~\\ref{sec:basic_tests}, the phase transformation may be related to the overestimated energy of the diamond phase with respect to those of graphene and graphite due to the DFT reference data employed in the construction of the GAP20 and PANNA potentials.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.98\\textwidth]{fracture_final_snapshots.png}\n\\caption{Snapshots of crack configurations at $K_\\text{I} = 6$ MPa m$^{1\/2}$ after 1~ps as predicted by different models; Green color highlights the atoms with sp$^2$ bonding. GAP20 and PANNA show local structural transformations into graphitic structures at the crack tip, while ACE demostrates brittle cleavage.}\n\\label{fig:compiled_crack_600}\n\\end{figure*}\n\n\n\n\\subsection{Amorphous carbon}\n\\label{sec:amorphous_C}\n\n\nThe great variability of amorphous carbon (a-C) networks, governed by competing sp, sp$^2$ and sp$^3$ hybridizations, poses another difficult challenge for atomistic simulations. Two extensive comparative studies of fourteen interatomic potentials~\\cite{marks1_transferability, marks2_graphitization} showed that there exist marked differences in the predictions of structural and physical properties of a-C systems. Among the investigated models, GAP17 \\cite{gap17_PhysRevB.95.094203} was found to provide the most reliable description (except of some unphysical predictions of 5-fold coordinated atoms at high densities). However, it had by far the highest computational cost of all potentials, limiting its use in large-scale simulations. Even though GAP17 was successfully applied to study the deposition of thin a-C films~\\cite{Caro2018_PhysRevLett.120.166101, GAP_application_thinfilm_PhysRevB.102.174201}, larger system sizes and extended simulation times are crucial to achieve realistic amorphous networks. For instance,  Jana et al.~\\cite{Jana_2019} investigated in detail the effect of quench rates on the formation and properties of a-C structures using GAP17, the screened Tersoff (Tersoff-S) potential~\\cite{Pastewka_SiC_PhysRevB.87.205410} and DFT. The simulations revealed a crucial role of the quench rate on the resulting a-C morphology. Slower cooling rates allowed the atoms to achieve energetically more favourable configurations, thus giving rise to structures with lower cohesive energies, while fast cooling rates resulted in more distorted and less stable forms of a-C.\n\nWe studied properties of bulk a-C samples prepared with the liquid-quench MD protocol from Ref.~\\onlinecite{Jana_2019}. Simple cubic supercells containing 8000 atoms were melted during 4.0 ps using $NVT$ MD at 12000 K, employing the Nos\\'e-Hoover thermostat and time step of 1 fs. The liquid phase was then equilibrated at 8000 K for 10 ps before quenching it to 300 K by linearly decreasing the temperature. We employed three different quench rates: fast at 1000 K\/ps, medium at 100 K\/ps, and slow at 10 K\/ps.  The final structures were optimized by relaxing either the atomic positions only or both the atomic positions and the cell vectors to minimize the stresses in the cells. For the latter protocol, we observed negligible changes of densities in most samples. For each quench rate, we generated ten a-C samples with densities ranging from 1.8 to 3.5 g\/cm$^3$. This range encompasses low-density nanoporous structures, crystalline graphite ($\\rho_\\text{gra} = 2.24$ g\/cm$^3$), and diamond ($\\rho_\\text{dia} = 3.54$ g\/cm$^3$).\n\n \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{v3_aC_strucs.png}\n\\caption{Representative a-C structures at three densities using 1000 K\/ps (left) and 10 K\/ps (right) quench rate. Shown are slices of 1 nm thickness that were cut out of the simulation cell. White, red, green and blue colors correspond to 1-fold, 2-fold, 3-fold and 4-fold coordinated atoms, respectively. Coordination from 1.85 \\AA{} cutoff.}\n\\label{fig:a_C_snapshots}\n\\end{figure}\n\n\nSnapshots of representative a-C structures with three different densities generated using the fast and slow quench rates are depicted in Fig.~\\ref{fig:a_C_snapshots}. Samples generated with the fast and medium quench rates exhibit uniformly disordered structures that differ in the fraction of the sp$^2$ and sp$^3$ bonded atoms, in agreement with Ref.~\\cite{Jana_2019}. The structures of lowest density ($\\rho$ = 1.8 g\/cm$^3$) are composed of highly distorted and defective graphene sheets with a considerable number of nanovoids and sp-bonded carbon chains connecting the sheets. The structures of intermediate densities ($\\rho$ = 2.2 to 2.9 g\/cm$^3$) correspond to commonly synthesized a-C structures and the fast and medium quench rates result in disordered glassy networks with a homogeneous mixture of sp$^2$ and sp$^3$ bonded atoms. The structures with the highest density ($\\rho$ = 3.4 g\/cc) contain mostly sp$^3$ bonded, diamond-like atoms.\n\nIn the thermodynamic limit of infinitesimally slow cooling we expect graphite, diamond or coexisting graphite and diamond at relative phase fractions that are determined by density. At the slow cooling rate we observe the onset of phase separation and the occurrence of ordered structures with separated sp$^2$ and sp$^3$ regions. For the lowest density, slow quenching leads to more extended sheets with fewer defects. Such graphitized nanostructures have been observed in previous simulations with GAP17~\\cite{marks2_graphitization,Jana_2019}, TurboGAP~\\cite{TurboGAP} and experimentally motivated simulations by Bhattarai et al.~\\cite{exp_aG_BHATTARAI2018168,exp_aG_C8CP02545B}. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{aC_sp3_atrel.png}\n\\caption{Variation of sp$^3$ bond fraction in a-C with density for different quench rates and methods (see text for details).\n}\n\\label{fig:sp3_fraction}\n\\end{figure}\n\n\n\n\nFigure~\\ref{fig:sp3_fraction} displays how the fraction of sp$^3$ bonded atoms changes as a function of density for different quench rates and potentials. Results for GAP17 (4087 atoms) and DFT (216 atoms) were taken from Ref.~\\cite{Jana_2019}, PANNA (216 atoms) was taken from Ref.~\\cite{Shaidu2021_PaNNA2021}.  As expected the sp$^3$ fraction increases with density for all methods and quench rates. For the first time we observe a clear influence of quench rate on the a-C morphology with a non-classical potential. For the fast quench at 1000 K\/ps, which is also feasible with DFT, the sp$^3$ fraction increases almost linearly with density and leads to a homogeneous amorphous network, see Fig.~\\ref{fig:a_C_snapshots}. In contrast, at the slow quench rate of 10 K\/ps the sp$^3$ fraction is almost negligible up to $\\rho$ = 2.55 g\/cm$^3$ and then increases sharply to reach the high values of diamond-like a-C structures. In Fig.~\\ref{fig:sp3_fraction}, we mark the equilibrium densities of graphite and diamond by dashed vertical lines. The solid vertical line indicates the density at which the free energies of homogeneously compressed graphite and dilated diamond coincide at $T=0$ K. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{aC_vorvol_adf.png}\n  \\caption{\n    Distribution of (a) Voronoi atomic volumes and (b) bond angles in a-C structures at densities 2.0, 2.9 and 3.4 g\/cc, quenched at 10 K\/ps with ACE.}\n  \\label{fig:voronoi_adf}\n\\end{figure}\n\nThe distributions of Voronoi atomic volumes for the slowly quenched samples with densities of 2.0, 2.9 and 3.4 g\/cm$^3$ are plotted in Fig.~\\ref{fig:voronoi_adf} (a). For the highest density sample, there is a single peak (dark blue) centered about the atomic volume of diamond. In contrast, the sample with the intermediate density is characterized by two peaks (dark red). The smaller peak coincides again with the volume of diamond while the higher peak is located in between the diamond and graphite volumes. This results is in accordance with the phase separation into sp$^2$ and sp$^3$ dominated regions observed in Fig.~\\ref{fig:a_C_snapshots}. The position of the higher peak indicates that the sp$^2$ phase is a compressed form of amorphous graphite as it is located below the equilibrium volume of crystalline graphite. The distribution of the low density sample (green) is broadest and skewed towards larger volumes due to the existence of nanovoids.\n\nA qualitatively similar outcome can be seen on the distributions of bond angles displayed Fig.~\\ref{fig:voronoi_adf} (b). The angular distribution function (ADF) enables to characterize lattice distortions by comparing the positions and widths of the ADF peaks with bond angles in ideal graphite and diamond. The ADFs are centered around $\\phi$ = 109.5$^{\\circ}$ for the high-density sample and $\\phi$ = 120$^{\\circ}$ for the low-density sample. For the intermediate-density sample, the distribution is significantly broader and clearly composed of two overlapping peaks centered at the angles mentioned above. The ADFs for structures generated using the medium and high quench rates are shown in the supplementary material~\\cite{suppl}.\n\n\\subsection{Fullerene formation}\n\n\\begin{figure*}\n\\centering\n    \\includegraphics[width=0.99\\textwidth]{fuller_growth.png}\n    \\caption{Formation of fullerenes from gas phase during combustion simulation. Small dots are Ar atoms.\\label{fig:fullerene_growth}}\n\\end{figure*}\n\nCarbon clusters are often formed during combustion of carbon-rich materials at high temperatures and pressures. The nucleation and growth process of carbon was the subject of various experimental and theoretical studies\\cite{Greiner1988,Millicent2017,Korets2010,Pineau2008,Qin2020,Los2009}.  Here, we present long-time MD simulations of the nucleation and growth of molecular fullerenes from gas phase carbon at high pressure and temperature.\n\nAs the starting configuration, in a cubic supercell of side 47.6 \\AA{} 402 C atoms and 2973 Ar atoms were arranged randomly. The Ar-Ar and Ar-C interactions were modeled using a simple Lennard-Jones potential with parameters taken from Refs.~\\cite{Pineau2008,Shashkov_1979}, while the carbon interactions were modelled using ACE. The Ar atoms serve as a proxy to exert pressure in the cell and to induce collisions between the C atoms, but do not participate in chemical reactions. The $NVT$ ensemble was used to run MD at 3000 K for 12 nanoseconds. Snapshots from the simulation are shown in Fig.~\\ref{fig:fullerene_growth}.\n\nSimilar to the findings of Pineau et al.~\\cite{Pineau2008}, early in the simulation the gas phase carbon atoms bond together to form small buckyball molecules. As the system evolves, the buckyballs interact and coalesce into larger fullerenes. Eventually, at around 12 ns all 402 atoms have merged into a large sp2 bonded fullerene cluster.\n\n\n\\section{Summary and conclusions} \n\\label{sec:conclusions}\n\nWe developed a general purpose ACE parametrization for carbon that surpasses the accuracy and transferability of state-of-the-art ML models at a fraction of their computational cost. The outstanding predictive power of ACE stems from its physically and chemically motivated formulation, smooth extrapolative properties of the ACE basis, and carefully chosen and internally consistent training data.\n\nWe validated extensively the potential by a number of challenging tests. ACE predicts accurately structural and thermodynamic properties for a broad range of ideal and defective carbon polytypes and captures the complex bonding of carbon including bond distortions and bond breaking and making. We showed several exemplary cases where the best available ML potentials fail while ACE predictions are correct.\n\nThe efficiency and robustness of ACE was demonstrated on three distinct applications. In simulations of diamond fracture we showed that ACE maintains brittle cleavage when the system is strained beyond the critical load. In contrast, GAP20 and PANNA both predict a graphitic phase transformation at the crack tip. Simulations of non-equilibrium amorphous carbon structures reveal that their  structural morphology depends not only on the density but also strongly on the quenching rate. This result could only be achieved due to the outstanding computational efficiency of ACE that enables much slower quenching rates than was possible with other ML potentials.  Lastly, we examined the capability of ACE to describe the  evolution of large fullerene clusters during combustion at high temperatures and pressures. The nucleation and growth of these clusters requires not only long-time MD simulations but also a reliable description of bond formation under highly non-equilibrium conditions.\n\nIn summary, our carbon ACE opens new possibilities for structural modeling of carbon at the atomic scale. It not only describes the fundamental properties of carbon allotropes with DFT accuracy, but is also able to maintain this accuracy in large-scale simulations. If necessary, the ACE accuracy and transferability can be further improved systematically, either by tailoring of the training dataset for the required application or by extending the ACE basis. The increased complexity of the ACE parametrization increases the computational costs only linearly which is in stark contrast to the much steeper scaling behaviors of most ML models. Finally, the elemental ACE models can be readily extended or combined to address multi-component systems, such as hydrocarbon systems or transition metal carbides.\n\n\n\n\\section*{Acknowledgements}\nThe authors would like to acknowledge valuable discussions with Lars Pastewka,  Romain Perriot and Bernd Meyer. MQ acknowledges funding through a scholarship from the International Max Planck Research School for Interface Controlled Materials for Energy Conversion (IMPRS-SurMat).  This work was in part supported by the German Science Foundation (DFG), projects 405621081 and 405621217.\n\n\n\n\\newpage\n\n\n\\section{D2 correction}\nThe D2 term in ACE is expressed as a simple pair potential\n\n\\begin{equation}\n    E_{disp} = -\\frac{1}{2} \\sum_{i} \\sum_{j} \\frac{C_6}{r_{ij}^6} f_{d}(r_{ij})  ,\n\\end{equation}\nwhere the damping function $f_d$ is given by\n\\begin{equation}\n    f_d = \\frac{s_6}{1+\\exp(-d (r_{ij}\/r_c - 1))} .\n\\end{equation}\nThe dispersion coefficient $C_{6}$, the damping factor $d$, and the vdW radius $r_c$ were parameterized for all elements up to Xe~\\cite{vdw_d2_coorection}. The global scaling parameter $s_{6}$ depends on the specific DFT functional used and is set to 0.75 for PBE.\n\n\n\\section{Validation of neural network models}\n\nThis section includes additional validation of the recent NN models hNN\\_Gr$_x$ and DeepMD.\n\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.45\\columnwidth]{graphite_eb_nnps.png}\n    \\hfill\n    \\includegraphics[width=0.45\\columnwidth]{gradia_transformation_nnps.png}\n     \\caption{Binding of graphite as a function of interlayer separation (a), and the ransformation path from graphite to diamond (b), as predicted by hNN\\_Gr$_x$ and DeepMD potentials.\\label{fig:nnp_graphite_eb} }\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{ev_groundstate_NNpots.png}\n\\caption{Relative stability of the three lowest energy phases of carbon, as predicted by DFT and NN-based models. $\\Delta$E refers to the energy difference with respect to graphite given by the respective potential. Solid, dashed and dotted lines refer to graphite, graphene and diamond phase respectively.}\n\\label{fig:ev_groundstate}\n\\end{figure}\n\n\n\\clearpage\n\\section{Comparison of RMSE values for energies and forces}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[height=0.9\\textheight]{all_Crossvals.png}\n\\caption{Fit metrics for 10\\% split test datasets corresponding to different categories. }\n\\label{fig:all_crossvalidations}\n\\end{figure*}\n\n\\begin{table}[h]\n\\caption{Comparison of RMSE values for energies and forces for the considered models computed from a test dataset. RMSE values are calculated with respect to PBE data. Energies are expressed in terms of relative cohesive energy with respect to ground state of the given potential to account for different references.}\\label{tab:s_rmse_pots}\n\\begin{tabular}{lcc}\n\\hline\n Potenial    & E$_\\text{RMSE}$ (meV\/atom) & F$_\\text{RMSE}$ (meV\/\\AA{}) \\\\ \\hline\nACE      & 102   & 539   \\\\\nTurboGAP & 238   & 883   \\\\\nDeepMD   & 841   & 942  \\\\\nGAP20    & 1083  & 725   \\\\\nhNN\\_Gr$_\\text{x}$ & 1552  & 1982  \\\\\nPANNA    & 1793  & 1327 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n    \n    \\begin{tabular}{@{}c@{}}\n    \n    \\includegraphics[width=0.9\\columnwidth]{pots_testset_E_separated.png}\n   \n  \\end{tabular}\n  \n  \\small (a)\n  \n  \\vspace{\\floatsep}\n\n  \\begin{tabular}{@{}c@{}}\n    \\includegraphics[width=0.9\\columnwidth]{pots_testset_F_separated.png} \n   \n  \\end{tabular}\n  \n  \\small (b)\n    \n    \n    \\caption{\\label{fig:s_pots_crossref} (a) Cohesive energy and (b) force validation plots for considered models for a split test dataset with 1912 structures. RMSE values are given in Table~\\ref{tab:s_rmse_pots}. Points are colored according to the legend given in Fig. 1 in the main paper.}\n\\end{figure}\n\n\\clearpage\n\\section{Validation of additional models and properties}\\label{fig:all_Ev}\n\n\n\\begin{table}[h]\n\\caption{Formation energies for point defects in graphene and diamond lattice }\n\\begin{tabular}{lcccccc}\n\\hline\n                        & DFT* & ACE  & GAP20 & PANNA & hNN\\_Gr$_\\text{x}$ & DeepMD \\\\ \\hline\nGraphene                &     &      &     &     &     & \\\\ \\hline\nStone-Wales             & 4.9 & 4.9  & 4.8 & 4.6 & 5.3 & 4.8  \\\\\nMonovacancy             & 7.7 & 5.8  & 7.0 & 6.5 & 7.9 & 6.2 \\\\\nDivacancy (5-8-5)       & 7.4 & 7.2  & 7.9 & 7.0 & 8.0 & 7.2 \\\\\nDivacancy (555-777)     & 6.6 & 6.4  & 6.9 & 6.0 & 6.9 & 6.5 \\\\\nDivacancy (5555-6-7777) & 6.9 & 6.9  & 7.4 & 6.3 & 7.3 & 8.0 \\\\\nAdatom                  & 6.4 & 6.4  & 5.9 & 10.8 & 8.3 & 6.7 \\\\ \\hline\nDiamond                 &     &      &     &     &    & \\\\ \\hline\nMonovacancy             & 6.6 & 5.8  & 4.3 & 5.0 &    & 5.7 \\\\\nDivacancy               & 9.1 & 12.0 & 6.6 & 11.0 &   & 11.6\\\\ \\hline\n\\end{tabular}\n\n{*DFT reference is taken from~\\cite{gap20_doi:10.1063\/5.0005084} and is given for optB88-vdW DFT functional}\n\\end{table}\n\n\n\n\\begin{figure}\n\n\\centering\n    \\subfloat{\\includegraphics[width=0.33\\textwidth]{ev_pot_deepMD.png}} \n    \\subfloat{\\includegraphics[width=0.33\\textwidth]{ev_pot_tb.png}}\n    \\subfloat{\\includegraphics[width=0.33\\textwidth]{ev_pot_reaxff.png}}\n\n\n    \\subfloat{\\includegraphics[width=0.33\\textwidth]{ev_pot_gr_nn.png}} \n    \\subfloat{\\includegraphics[width=0.33\\textwidth]{ev_pot_airebo.png}}\n    \\subfloat{\\includegraphics[width=0.33\\textwidth]{ev_pot_lcbop.png}}\n    \n\\caption{Energy volume curves for some bulk phases as predicted by (a) DeepMD NNP~\\cite{deep_MD_carbonpot_WANG20221}, (b) Tight Binding model parametrized by Xu et al.~\\cite{Xu_1992}, (c) ReaxFF for hydrocarbon oxidation~\\cite{reaxff_CHO_Chenoweth2008}. (d) NNP for multilayered graphene~\\cite{NNP_Gr_PhysRevB.100.195419}, (e) AIREBO with Morse modification~\\cite{AIREBO_M_doi:10.1063\/1.4905549} and (f) LCBOP potentials~\\cite{LCBOP_PhysRevB.68.024107} \\label{fig:ev_suppl} }    \n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[height=0.9\\textheight]{phonons_otherbulk.png}\n    \\caption{Computed phonon band structures for fcc, bcc and sc bulk phases; all three phases are dynamically unstable according to DFT.\\label{fig:other_bulk_phonons} }\n\\end{figure}\n\n\n\\clearpage\n\\section{Diamond fracture}\n\n\\begin{figure}[h]\n  \\centering\n  \\begin{tabular}{@{}c@{}}\n    \\includegraphics[width=0.5\\linewidth]{crack_480_full.png} \\\\[\\abovecaptionskip]\n    \\small (a)\n  \\end{tabular}\n\n  \\vspace{\\floatsep}\n\n  \\begin{tabular}{@{}c@{}}\n    \\includegraphics[width=0.5\\linewidth]{cracktip_480.png} \\\\[\\abovecaptionskip]\n    \\small (b)\n  \\end{tabular}\n\n  \\caption{An example of simulation cell consisting of 5280 atoms used to study the brittle crack propagation in diamond (a) and a close-up of the crack tip region (b).}\\label{fig:crack_tip}\n\\end{figure}\n\n\\clearpage\n\\section{Amorphous carbon}\n\n\\begin{figure}[h]\n\\centering\n    \\includegraphics[width=0.9\\textwidth]{adf_all.png}\n    \\caption{A comparison of angular distribution functions for a-C structures at different quench rates\\label{fig:all_adfs} }\n\\end{figure}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nClifford analysis is, from the very beginning, considered to be a refinement of harmonic analysis\nfor Clifford algebra (or spinor) valued functions. The perfect description of this statement is\nthe Fischer decomposition of spinor-valued polynomials.\n\nLet us first recall the Fischer decomposition of the space\n$\\cP$ of complex-valued polynomials in the Euclidean space $\\bR^m.$ \nDenote by $\\cH_k$ the space of $k$-homogeneous harmonic polynomials in $\\bR^m.$ \nThen, under a~natural action of the orthogonal group $O(m),$ the space \n$\\cP$ has an irreducible (not multiplicity free) decomposition\n\\begin{equation}\\label{Fischer}\n\\cP=\\bigoplus_{k=0}^{\\infty}\\bigoplus_{p=0}^{\\infty}r^{2p}\\cH_k\n\\end{equation}\nwhere $r^2=x_1^2+\\cdots +x_m^2$ for the vector variable $\\underline{x}=(x_1,\\ldots,x_m)\\in\\bR^m.$\n\nFor spinor-valued polynomials,\nthere is a refinement of this decomposition.\nLet $\\bC_m$ be the complex Clifford algebra generated by vectors of the standard basis $(e_1,\\ldots,e_m)$ of $\\bR^m.$\nRecall that the $L$-action of the Pin group $Pin(m)$ on functions $f:\\bR^m\\to\\bC_m$ is defined by\n\\begin{equation}\\label{Laction}\n[L(s)(f)](\\underline{x}) = s\\,f(s^{-1}\\underline{x}s),\\ s\\in Pin(m)\\text{\\ \\ and\\ \\ }\\underline{x}=(x_1,\\ldots,x_m)\\in\\bR^m.\n\\end{equation}\nDenote by $\\bS$ a~basic spinor representation for $Pin(m).$ \nAs is well-known, the spinor space $\\bS$ can be realized inside the Clifford algebra $\\bC_m.$ \nLet us remark that, under the $L$-action, the space $\\cP(\\bS)=\\cP\\otimes\\bS$ of spinor valued polynomials forms a~$Pin(m)$-module.\nDenote by $\\cM_k(\\bS)$ the space of $k$-homogeneous polynomials $P\\in\\cP(\\bS)$ which are (left) monogenic,  \ni.e., which satisfy the Dirac equation $\\underline{D}P=0$ where $$\\underline{D}=e_1\\pa_{x_1}+\\cdots+e_m\\pa_{x_m}.$$ \nNow we are ready to state the Fischer decomposition (sometimes called also Almansi decomposition) for this case. Namely, under the $L$-action, the space \n$\\cP(\\bS)$ has an irreducible (not multiplicity free) decomposition \n\\begin{equation}\\label{FischerPin}\n\\cP(\\bS)=\\bigoplus_{k=0}^{\\infty}\\bigoplus_{p=0}^{\\infty}\\underline{x}^p\\cM_k(\\bS)\n\\end{equation}\nwith $\\underline{x}=e_1x_1+\\cdots+e_mx_m.$ See \\cite{BSES,MR,rya}.\nAs $\\cH_k\\otimes\\bS=\\cM_k(\\bS)\\oplus\\underline{x}\\cM_{k-1}(\\bS)$ and $\\underline{x}^2=-r^2,$\nit is easy to see that (\\ref{FischerPin}) is a~real refinement of (\\ref{Fischer}).\n\nThe main aim of the underlying paper is to show that\nthere exists a natural further refinement of the monogenic Fischer decomposition \\eqref{FischerPin}. It is \nquite surprising that such a finer Fischer decomposition was not described earlier.\nIt was the study of special solutions of the Dirac equation which led to such a~refinement.\nBy special solutions we mean just solutions having their values\nin a chosen subspace $V$ of the Clifford algebra $\\bC_m$. There are a lot of possibilities for a choice \nof $V,$ but it is clearly preferable to choose the subspace $V$ having some special properties. \n\nTypical examples\nare solutions of the Dirac equation having values in  spinor\nsubspaces of the Clifford algebra. \nThis case is closely related to the \n$L$-action \\eqref{Laction}.\nIndeed, it is well-known that the Clifford algebra $\\bC_m$, considered as a \n$Pin(m)$-module by left multiplication, decomposes into many equivalent spinor submodules. \nMoreover, for every choice of the spinor\nsubmodule, the $Pin(m)$-module of spinor-valued solutions has quite analogous properties.\n \nAnother interesting example of special solutions of the Dirac equation is given by the so-called generalized Moisil-Th\\'eodoresco system (GMT system for short).\nA~lot of interest has recently been paid to GMT systems (see \\cite{DLS} and the references there).\nIn this case, the space $V$ is supposed to be invariant under another (both side) action of the Pin group, namely the so-called $H$-action.\nThe $H$-action on Clifford algebra valued functions $f:\\bR^m\\to\\bC_m$ is given by\n\\begin{equation}\\label{Haction}\n[H(s)(f)](\\underline{x}) = s\\,f(s^{-1}\\underline{x}s)s^{-1},\\ s\\in Pin(m)\\text{\\ \\ and\\ \\ }\\underline{x}\\in\\bR^m.\n\\end{equation}\nIn what follows, we shall use the language of differential forms.  \nIndeed, following \\cite{BDS}, we identify naturally the Clifford algebra $\\bC_m$ with the Grassmann algebra $\\La^*(\\bC^m)$ \nand we study the space $\\cP^*=\\cP\\otimes\\La^*(\\bC^m)$ of polynomial differential forms instead of Clifford algebra valued polynomials.\nThen the $H$-action translates into a~natural action of the orthogonal group $O(m)$ on $\\cP^*$ and the Dirac operator $\\underline{D}$ corresponds to the operator $d+d^*.$ Here $d$ and $d^*$ are, respectively, the standard de Rham differential and its adjoint (see \\eqref{dd^*} below). \nAs an $O(m)$-module, the \nGrassmann algebra $\\La^*(\\bC^m)$ has a multiplicity free irreducible decomposition\n$$\\Lambda^*(\\bC^m)=\\bigoplus_{s=0}^m\\Lambda^s(\\bC^m)$$\nwith $\\Lambda^s(\\bC^m)$ being the space of $s$-vectors over $\\bC^m.$ A~GMT system is then defined as the homogeneous system obtained by restricting the operator $d+d^*$ to functions having values in the space\n$$V=\\bigoplus_{s\\in S}\\Lambda^s(\\bC^m)$$\nfor some (suitable) subset $S\\subset\\{0,1,\\ldots,m\\},$ i.e.\n$$(d+d^*)P=0\\mbox{\\ \\ \\ for $V$-valued $P.$}$$\nIn particular, for $V=\\Lambda^s(\\bC^m),$ the corresponding GMT system coincide with\nthe so-called Hodge-de Rham system\n\\begin{equation}\\label{Hodge}\nd P=0,\\, d^* P=0.\n\\end{equation}\n\n\n\nVarious versions of GMT systems of PDE's were studied for a long time (in particular in low dimensions) and they \nwere used in many different applications. \nApplications in numerical analysis and engineering sciences can be found in \\cite{gs1,gs2}.\nIn a~review paper \\cite{spr}, you can find various generalizations of the well-known Hodge-de Rham decomposition of smooth 1-forms, including decompositions for quaternionic and Clifford algebra valued functions and the Almansi (i.e., monogenic Fischer) decomposition. For applications in theory of electromagnetic fields we can refer to \\cite{S_Maxwell}.\n\nThe Fischer decomposition always played a key role in Clifford analysis. \nIn \\cite{lav_Fischer}, the Fischer decomposition for the $H$-action has been recently applied to inframonogenic functions introduced in \\cite{MPS}. Moreover, in \\cite{lav_Fischer}, the obtained results for the space $\\cP^*$ are translated back into the framework of Clifford analysis. \nFor yet another application, we can refer to \\cite{lavSL3}.\n\nRecently, the Fischer decomposition (together with the Cauchy-Kovalevskaya extension) was used systematically for construction of orthogonal bases in the spaces of homogeneous polynomial solutions.\nIn the classical Clifford analysis, it has a quite long history (see \\cite{BGLS} for historical account,\nvarious results can be found in \\cite{Bock2010c,Bock2010a,Bock2009,BG,CacGueMal,CacGueBock,BockCacGue,cac,CM06,CM07,CM08,\nFCM,FM,Gurlebeck1999,NGue2009,lavSL2,step2,Malonek1987,MS,mor09,som,van}).\nAnalogous results in Hermitean Clifford analysis are described in \\cite{ckH, kerH,GTinH, GT2H}.\nFinally, in \\cite{DLS4}, the Fischer decomposition for the $H$-action plays a~key role in constructing orthogonal bases not only for the spaces $H^s_k$ of solutions to the Hodge-de Rham system but even for the spaces of homogeneous solutions of an arbitrary generalized\nMoisil-Th\\'eodoresco system. \n\n\n\nIn this paper, we will establish (using results from \\cite{hom}) a form of the Fischer decomposition  appropriate for the $H$-action. The theory of the Howe duality developed in \\cite{how} shows us that we may\nexpect a further refinement of the monogenic Fischer decomposition (\\ref{FischerPin}), see \\cite{DLS3} for details.\nIndeed, this is visible from the form of invariants contained in the polynomial spaces considered.\nFor scalar valued functions, invariant polynomials are generated by powers of $r^2,$\nand the basic equation is the Laplace equation.\nFor spinor-valued polynomials with the left action, we have to look for invariants with\nvalues in the space of endomorphisms of the spinor space (which is, basically, the corresponding\nClifford algebra) and there is just a new invariant $\\underline{x},$ acting as a refinement of $r^2.$\nThe corresponding basic equation is the Dirac equation.\n\nIn the case of the both side action, \nwe deal with the space $\\cP^*$ of $\\La^*(\\bC^m)$-valued\npolynomials. The space of invariants with values in the space of endomorphisms of the Grassmann algebra $\\La^*(\\bC^m)$ is now much richer.  It is generated by two elements ${x}$ and ${x}^*$ which  correspond to the differential operators $d^*$ and $d$ by the Fischer duality (see \\eqref{xx^*} below). \nConsequently, the corresponding basic system of equations is the Hodge-de Rham system (\\ref{Hodge})\nand the space of invariants consists of polynomials in $x$ and $x^*.$ Actually,  due to the fact\nthat ${x}^2=({x}^*)^2=0,$ such invariants are generated by the set \n\\begin{equation}\\label{Omega}\n\\Omega=\\{1, x, x^*, xx^*, x^*x, xx^*x, x^*xx^*, \\ldots\\}.\n\\end{equation}\nMoreover, denote by $\\cP^*_k$ the space of $k$-homogeneous polynomial forms $P\\in\\cP^*$\nand by $H^s_k$ the space of $\\La^s(\\bC^m)$-valued polynomial forms $P\\in\\cP^*_k$ which satisfy the Hodge-de Rham system (\\ref{Hodge}).  \nThen, using results from \\cite{hom}, we shall deduce in Section 2 the corresponding Fischer decomposition for the $H$-action.\n\n\n\\begin{thm}\\label{tFischer}\nThe space $\\cP^*=\\cP\\otimes\\La^*(\\bC^m)$ decomposes as follows:\n\\begin{equation}\\label{etFischer}\n\\cP^*=\\cP^*_{(0,0)}\\oplus\\left(\\bigoplus_{s=1}^{m-1}\\bigoplus_{k=0}^{\\infty}\\cP^*_{(s,k)}\\right)\\oplus\\cP^*_{(m,0)}\\text{\\ \\ \\ with\\ \\ \\ }\n\\cP^*_{(s,k)}=\\bigoplus_{w\\in\\Omega}wH^s_k.\n\\end{equation}\nMoreover, in (\\ref{etFischer}), all\n$O(m)$-modules $H^s_k$ are non-trivial, irreducible and mutually inequivalent and all\n$\\cP^*_{(s,k)}$ are corresponding $O(m)$-isotypic components of $\\cP^*.$\n\\end{thm}\n\nNow we show that the Fischer decomposition of the space $\\cP^*$ given in Theorem \\ref{tFischer} is a~refinement of\nthe monogenic Fischer decomposition (\\ref{FischerPin}).\nIndeed, when we identify the Clifford algebra $\\bC_m$ with the Grassmann algebra $\\La^*(\\bC^m)$ on the space $\\cP^*$ we know that \n$$\\underline{D}=d+d^*\\text{\\ \\ \\ and\\ \\ \\ }-\\underline{x}=x+x^*.$$\nConsequently, the space of spherical monogenics of order $k$ is given by $$\\cM_k=\\{P\\in\\cP^*_k;\\ (d+d^*)P=0\\}.$$ \nMoreover, recall that the Laplace operator $\\Delta$ is given by $\\Delta=\\sum_{j=1}^m\\pa^2_{x_j}$ and put \n$$\\Ker_k\\Delta=\\{P\\in\\cP^*_k;\\ \\Delta P=0\\}.$$\n\nAs we mentioned before, under the $L$-action, $\\bC_m\\simeq\\La^*(\\bC^m)$ decomposes into many copies of basic spinor representations $\\bS$ of $Pin(m)$ and so\nthe whole space $\\cM_k$ of spherical monogenics  is not irreducible. Indeed,\nunder the $L$-action,\nthe space $\\cM_k$\ndecomposes into many copies of irreducible modules $\\cM_k(\\bS)$.\nIn particular, we have thus that $\\Ker_k\\Delta=\\cM_k\\oplus(x+x^*)\\cM_{k-1}$\nand, by (\\ref{Fischer}), we get easily the following decomposition of the space $\\cP^*$\n\\begin{equation}\\label{FischerPin*}\n\\cP^*=\\bigoplus_{k=0}^{\\infty}\\bigoplus_{p=0}^{\\infty}r^{2p}(\\cM_k\\oplus(x+x^*)\\cM_{k-1}).\n\\end{equation}\nIn an obvious sense, the decomposition (\\ref{FischerPin*}) is equivalent to (\\ref{FischerPin}). \n\nIn Section 3, we shall prove the following theorem which tells us that, under the $H$-action,\nthe spaces $\\cM_k$ and $(x+x^*)\\cM_{k-1}$ decompose again into many irreducible pieces \nbut, in this case, these pieces are not equivalent and \nthey have a~different representation character.\n\n \n\\begin{thm}\\label{tdecompM}\nFor $k\\geq 1,$ the following statements hold:\n\\begin{itemize}\n\\item[(a)] $\\cM_k=\\left(\\bigoplus_{s=0}^m H^s_k\\right)\\oplus\\left(\\bigoplus_{s=1}^{m-1}M_{s,k}\\right)$\n\\item[] where $M_{s,k}=[(k-1+m-s)x^*-(k-1+s)x]H^s_{k-1}.$ \n\\item[(b)]\n$(x+x^*)\\cM_{k-1}=\\left(\\bigoplus_{s=0}^m(x+x^*)H^s_{k-1}\\right)\\oplus\\left(\\bigoplus_{s=1}^{m-1}W^s_k\\right)$\n\\item[] where $W^s_k=[(k-2+m-s)xx^*-(k-2+s)x^*x]H^s_{k-2}.$\n\\end{itemize}\n\\end{thm}\n\nUsing Theorem \\ref{tdecompM}, we get from the decomposition \\eqref{FischerPin*} directly a~finer decomposition of the space $\\cP^*$ which is irreducible with respect to the $H$-action.\n \nThe results stated in Theorems \\ref{tFischer} and \\ref{tdecompM} remain valid also for real valued polynomial forms, that is,\nin the case when the complex Grassmann algebra $\\La^*(\\bC^m)$ is replaced with the real one $\\La^*(\\bR^m).$ \nIndeed, it is sufficient to realize that irreducible $O(m)$-representations $\\La^s(\\bC^m)$ are all of real type, see \\cite[p. 163]{GM}. \n\n \n\n\n \n\\section{A proof of the Fischer decomposition for the $H$-action}\\label{stFischer}\n\nIn this section, we give a~proof of Theorem \\ref{tFischer} stated in Introduction. \nLet $\\cP_k$ stand for the space of $k$-homogeneous (complex-valued) polynomials of $\\cP$ and let\n$\\cP^s_k=\\cP_k\\otimes\\Lambda^s(\\bC^m).$\nThen\nit is easy to see that\n\\begin{equation}\\label{decompP}\n\\cP^*=\\bigoplus_{s=0}^m\\bigoplus_{k=0}^{\\infty}\\cP^s_k.\n\\end{equation}\nLet us remark that a~polynomial form $P$ belongs to $\\cP^s_k$ if and only if\n\\begin{equation}\n\\label{skform}\nP=\\sum_I P_I\\; dx_I\n\\end{equation}\nwhere the sum is taken over all finite strictly increasing sequences $I=\\{i_j\\}_{j=1}^s$ of numbers of the set $\\{1,\\cdots, m\\},$ $P_I\\in\\cP_k$ and $dx_I=dx_{i_1}\\wedge\\cdots\\wedge dx_{i_s}.$\nThe contraction $dx_j\\krat$ is defined as\n$$dx_j\\krat\\; dx_I=\\sum_{k=1}^s(-1)^{k-1}\\delta_{ji_k}dx_{I\\setminus\\{i_k\\}}\\text{ and }\ndx_j\\krat\\; P=\\sum_I P_I\\; dx_j\\krat\\; dx_I$$ for a~polynomial form $P.$ \nThen we have that\n\\begin{equation}\n\\label{dd^*}\nd=\\sum_{j=1}^m\\; \\pa_{x_j} dx_j\\wedge\\text{\\ \\ \\ and\\ \\ \\ }\nd^*=-\\sum_{j=1}^m\\;\\pa_{x_j} dx_j\\krat, \n\\end{equation}\n\\begin{equation}\n\\label{xx^*}\nx=-\\sum_{j=1}^m x_j\\; dx_j\\wedge\\text{\\ \\ \\ and\\ \\ \\ }\nx^*=\\sum_{j=1}^m x_j\\; dx_j\\krat.\n\\end{equation}\nIt is easy to see that $d,$ $d^*,$ $x$ and $x^*$ are $O(m)$-invariant operators on the space $\\cP^*.$\n\nNow we describe explicitly an irreducible decomposition of $O(m)$-modules\n$$\\Ker_k^s\\;\\Delta=\\{P\\in\\cP_k^s;\\ \\Delta P=0\\}.$$\nThe following key result is obtained in \\cite{hom}.\n\n\\begin{lem}{\\label{thforms}}\nGiven $0\\leq s\\leq m$ and $k\\in\\NN_0,$ we have that\n$$\\Ker_k^s\\Delta= H^s_k\\oplus U^s_k\\oplus V^s_k\\oplus W^s_k$$ where\n$H^s_k,$\n$U^s_k,$ $V^s_k$ and $W^s_k$ are irreducible $O(m)$-modules\nwith the following properties:\n\\begin{itemize}\n\\item[(a1)] $H^s_k=\\{P\\in \\cP^s_k;\\ dP=0,\\ d^*P=0\\}$ and $\\Ker_0^s\\Delta= H^s_0=\\cP_0^s.$\n\\item[(a2)] In addition, $H^s_k=\\{0\\}$ for $s\\in\\{0,m\\}$ and $k\\geq 1.$ Otherwise, all $O(m)$ modules $H^s_k$ are non-trivial, irreducible and mutually inequivalent.\n\\item[(b)]  $U^s_k=xH^{s-1}_{k-1}\\simeq H^{s-1}_{k-1}$ for $1\\leq s\\leq m$ and $k\\geq 1,$ and $U^s_k=\\{0\\}$ otherwise.\n\\item[(c)] $V^s_k=x^*H^{s+1}_{k-1}\\simeq H^{s+1}_{k-1}$ for $0\\leq s\\leq m-1$ and $k\\geq 1,$ and $V^s_k=\\{0\\}$ otherwise.\n\\item[(d)] $W^s_k=[(k-2+m-s)xx^*-(k-2+s)x^*x]H^s_{k-2}\\simeq H^s_{k-2}$\\\\\nfor $1\\leq s\\leq m-1$ and $k\\geq 2,$ and $W^s_k=\\{0\\}$ otherwise.\n\\end{itemize}\n\\end{lem}\n\nNow we are ready to prove Theorem \\ref{tFischer}. \n\n\\begin{proof}[Proof of Theorem \\ref{tFischer}]\nAs $\\Ker^s_k\\Delta=\\cH_k\\otimes\\La^s(\\bC^m)$ the Fischer decomposition (\\ref{Fischer}) yields\n$$\\cP_k^s=\\bigoplus_{p=0}^{[k\/2]}r^{2p}\\Ker^s_{k-2p}\\Delta,$$\nwhere for $y\\in\\bR,$ $[y]$ denotes the greatest integer not greater than $y.$\nConsequently, by Lemma \\ref{thforms}, we get the decomposition\n\\begin{equation}\\label{decompPsk}\n\\cP_k^s= H^s_k\\oplus\n\\bigoplus_{p=0}^{[k\/2]}r^{2p}U^s_{k-2p}\\oplus\n\\bigoplus_{p=0}^{[k\/2]}r^{2p}V^s_{k-2p}\\oplus\n\\bigoplus_{p=0}^{[k\/2]}r^{2p}Z^s_{k-2p}\n\\end{equation}\nwhere $Z^s_k=r^2 H^s_{k-2}\\oplus W^s_k.$\nSince $r^2=-(xx^*+x^*x)$ Lemma \\ref{thforms} implies that,\nfor $0\\leq s\\leq m$ and $k\\geq 2,$   \n$$Z^s_k=(xx^*)H^s_{k-2}\\oplus (x^*x)H^s_{k-2}.$$\nMoreover, it is easy to see that \n$$r^{2p}U^s_k=(xx^*)^pxH^{s-1}_{k-1},\\ \\ \\\nr^{2p}V^s_k=(x^*x)^px^*H^{s+1}_{k-1}\\text{\\ \\ and}$$\n$$r^{2p}Z^s_k=(xx^*)^{p+1}H^s_{k-2}\\oplus (x^*x)^{p+1}H^s_{k-2}.$$\nNow to complete the proof\nit suffices to use the decompositions (\\ref{decompP}) and (\\ref{decompPsk}).\n\\end{proof}\n\nAt the end of this section we collect the well-known relations we need later on. \nPut, for linear operators $T_1$ and $T_2$ on $\\cP^*,$\n$\\{T_1,T_2\\}=T_1T_2+T_2T_1$ and $[T_1,T_2]=T_1T_2-T_2T_1.$  Then we have that (see e.g. \\cite{hom} or \\cite{BDS})\n\n\\begin{lem}\\label{lrels} Let $E$ be the Euler operator and $\\hat{E}$ be the skew Euler operator, i.e. $$E=\\sum_{j=1}^m x_j\\pa_{x_j}\\text{\\ \\ \\ and\\ \\ \\ }\n\\hat{E}=\\sum_{j=1}^m(dx_j\\;\\wedge)(dx_j\\krat).$$\nThen we have that $EP=kP$ and $\\hat{E}P=sP$ for each $P\\in\\cP^s_k.$\n\nFurthermore, the following relations hold:\n\\begin{equation*}\n\\begin{array}{lll}\n\\{x,x\\}=0, &\\{x^*,x^*\\}=0, &\\{x,x^*\\}=-r^2,\\medskip\\\\{}\n\\{d,d\\}=0, &\\{d^*,d^*\\}=0, &\\{d,d^*\\}=-\\Delta,\\medskip\\\\{}\n\\{x^*,d\\}=E+\\hat{E}, &\\{x,d^*\\}=E-\\hat{E}+m, &\\{x^*,d^*\\}=0=\\{x,d\\}.\n\\end{array}\n\\end{equation*}\n\\end{lem}\n\n\nUsing Lemma \\ref{lrels}, we may give, for example, an explicit description of the projections of the space $\\Ker_k^s\\Delta$ onto the pieces $H^s_k,$ $U^s_k,$ $V^s_k$ and $W^s_k.$ \n\n\\begin{prop}\\label{chforms}\nGiven $0\\leq s\\leq m$ and $k\\in\\NN_0,$ put $c_1=k-2+s$ and $c_2=k-2+m-s.$\nFurthermore, let $\\pi_1,$ $\\pi_2,$ $\\pi_3$ and $\\pi_4$ be the projections of the space $\\Ker_k^s\\Delta$ onto the subspaces $H^s_k,$\n$U^s_k,$ $V^s_k$ and $W^s_k,$ respectively. Then we have that\n\\begin{equation*\n\\pi_4=\\left\\{\n\\begin{array}{ll}\n\\frac{c_2xx^*-c_1x^*x}{c_1c_2(c_1+c_2+2)}\\;dd^* &\\text{for\\ \\ }1\\leq s\\leq m-1\\text{\\ and\\ }k\\geq 2,\\medskip\\\\{}\n0, &\\text{otherwise.}\n\\end{array}\n\\right.\n\\end{equation*}\nMoreover, denoting $\\pi=1-\\pi_4,$ we have that\n\\begin{equation*}\n\\begin{array}{ll\n&\\pi_2 =\\left\\{\n\\begin{array}{ll}\n\\frac{1}{c_2+2}\\;xd^*\\pi &\\text{for\\ \\ }1\\leq s\\leq m\\ \\text{\\ and\\ }k\\geq 1,\\medskip\\\\{}\n0, &\\text{otherwise;}\n\\end{array}\n\\right.\\medskip\\\\{}\n\n&\\pi_3 =\\left\\{\n\\begin{array}{ll}\n\\frac{1}{c_1+2}\\;x^*d\\pi &\\text{for\\ \\ }0\\leq s\\leq m-1\\text{\\ and\\ }k\\geq 1,\\medskip\\\\{}\n0, &\\text{otherwise;}\n\\end{array}\n\\right.\\medskip\\\\{}\n\n&\\pi_1 =1-\\pi_2-\\pi_3-\\pi_4.\n\\end{array}\n\\end{equation*} \n\\end{prop}\n\n\\begin{proof}\nLet $P\\in\\Ker^s_k\\Delta$ be given. Then, by Lemma \\ref{thforms}, there are uniquely determined $P_1\\in H^s_k,$ $P_2\\in H^{s-1}_{k-1},$ $P_3\\in H^{s+1}_{k-1}$ and $P_4\\in H^s_{k-2}$ such that $$P=P_1+xP_2+x^*P_3+(c_2xx^*-c_1x^*x)P_4.$$\nBy Lemma \\ref{lrels},\nit is easy to see that\n$$dd^*P=c_1c_2(c_1+c_2+2)P_4=-d^*dP,$$ which easily implies the formula for $\\pi_4.$\n\nMoreover, $\\pi(P)=P_1+xP_2+x^*P_3.$ By Lemma \\ref{lrels}, we have that \n$$d^*\\pi(P)=d^*xP_2=(c_2+2)P_2\\text{\\ \\ \\ and\\ \\ \\ }d\\pi(P)=dx^*P_3=(c_1+2)P_3,$$\nfrom which the formulae for the projections $\\pi_2$ and $\\pi_3$ may be derived.\n\\end{proof}\n\n\n\n\\section{Decomposition of monogenic polynomial forms}\n \nIn this section, we give a~proof of Theorem \\ref{tdecompM} stated in Introduction.\nTo prove Theorem \\ref{tdecompM} we need some lemmas.\n\n\n\\begin{lem}\\label{lMsk}\nFor $1\\leq s\\leq m-1$ and $k\\geq 1,$\nwe have that\n$$\\left(xH^s_{k-1}\\oplus x^*H^s_{k-1}\\right)\\cap\\cM_k=M_{s,k}.$$\nHere $M_{s,k}=[(k-1+m-s)x^*-(k-1+s)x]H^s_{k-1}.$\n\\end{lem}\n\n\\begin{proof}\nLet $P_1,P_2\\in H^s_{k-1}$ and put $P=xP_1+x^*P_2.$ It suffices to show that $(d+d^*)P=0$\nif and only if $$P_1=-\\frac{k-1+s}{k-1+m-s}P_2.$$\nBy virtue of Lemma \\ref{lrels}, it is easy to see that\n$$(d+d^*)P=(E+m-\\hat E)P_1+(E+\\hat E)P_2=(k-1+m-s)P_1+(k-1+s)P_2,$$\nwhich completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{ldUV}\nFor $1\\leq s\\leq m-1$ and $k\\geq 1,$ we have that\n$$xH^s_{k-1}\\oplus x^*H^s_{k-1}=(x+x^*)H^s_{k-1}\\oplus M_{s,k}. $$\n\\end{lem}\n\n\\begin{proof}\nObvious.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{tdecompM}]\nPut$$\\tilde\\cM_k=\\left(\\bigoplus_{s=0}^m H^s_k\\right)\\oplus\\left(\\bigoplus_{s=1}^{m-1}M_{s,k}\\right).$$\nThen, by Lemma \\ref{lMsk}, it is easy to see that\n$\\tilde\\cM_k\\subset\\cM_k.$\nMoreover, by Lemma \\ref{thforms}, $W^s_k=(x+x^*)M_{s,k-1}.$\nFinally, using Lemma \\ref{ldUV} and Lemma \\ref{thforms},\nwe obtain that\n$$\n\\Ker_k\\Delta=\\tilde\\cM_k\\oplus(x+x^*)\\tilde\\cM_{k-1}\\subset\\cM_k\\oplus(x+x^*)\\cM_{k-1}=\\Ker_k\\Delta,\n$$\nwhich completes the proof.\n\\end{proof}\n\n\n\\subsection*{Acknowledgments}\n\nR. L\\'avi\\v cka and V. Sou\\v cek acknowledge the financial support from the grant GA 201\/08\/0397.\nThis work is also a part of the research plan MSM 0021620839, which is financed by the Ministry of Education of the Czech Republic.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Acknowledgements}\nThis work is supported by PPARC's Particle Physics Peer Review Panel. Grateful thanks must also go to Peter Blythe, Claudia Eberlein and Robert Smith for useful discussion. \n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA basic problem in optimal transportation is the question on whether optimal plans are unique and induced by maps. The crucial result in this direction is the celebrated one of Brenier \\cite{Brenier87} granting that for $\\mu,\\nu\\in\\probt{\\mathbb{R}^d}$ with $\\mu$ absolutely continuous w.r.t. the Lebesgue measure and cost=squared-distance, indeed optimal plans are unique and induced by maps. An important generalization has been given by McCann \\cite{McCann01} for the same problem on Riemannian manifolds: he shows that the unique optimal map can be written as  $\\exp(-\\nabla\\varphi)$, where $\\varphi$ is a Kantorovich potential. As a byproduct of McCann's argument, we also know that for $\\mu$-a.e. $x$ the geodesic connecting $x$ to $\\exp(-\\nabla\\varphi(x))$ is unique. We can express both the fact that the unique  optimal plan is induced by a map and the uniqueness of geodesics by looking at the optimal transport problem as a dynamical problem, i.e. by minimizing\n\\[\n\\iint_0^1|\\dot\\gamma_t|^2\\,\\d t\\,{\\mbox{\\boldmath$\\pi$}}(\\gamma), \n\\] \nover all measures ${\\mbox{\\boldmath$\\pi$}}$ on $C([0,1],M)$ such that $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\mu$, $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\nu$, ${\\rm{e}}_t:C([0,1],M)\\to M$ being the evaluation map given by ${\\rm{e}}(\\gamma):=\\gamma_t$. Then McCann's result can be read as the uniqueness of the minimizer ${\\mbox{\\boldmath$\\pi$}}$ and the fact that such ${\\mbox{\\boldmath$\\pi$}}$ is induced by some map $T:M\\to C([0,1],M)$ (which in fact takes its values in the space of constant speed minimizing geodesics) in the sense that ${\\mbox{\\boldmath$\\pi$}}=T_\\sharp\\mu$. We refer to \\cite{Villani09} and \\cite{AmbrosioGigli11} for an overview of the subject.\n\n\\medskip\n\nIn the pioneering works of Lott-Villani \\cite{Lott-Villani09} and Sturm \\cite{Sturm06I}, \\cite{Sturm06II}, an abstract notion of lower Ricci curvature bound on metric measure spaces has been given, and since then a great interest has been given by the community to the understanding of the geometric\/analytic properties of these spaces. In \\cite{AmbrosioGigliSavare11-2}, a strengthening of the original ${\\sf CD}(K,\\infty)$ condition as defined by Lott-Sturm-Villani has been proposed: this new condition, called Riemannian Ricci curvature bound and denoted by ${\\sf RCD}(K,\\infty)$, enforces in some weak sense a Riemannian-like behavior of the space. According to the analysis done in \\cite{Erbar-Kuwada-Sturm13}, \\cite{AMS}, \\cite{Gigli13} a natural finite-dimensional analogue of the ${\\sf RCD}(K,\\infty)$ notion can be given by requiring that the space is both ${\\sf RCD}(K,\\infty)$ and satisfies the (reduced) curvature-dimension condition ${\\sf CD}^*(K,N)$ as defined in \\cite{BacherSturm10}.\n\nAim of this short note is to prove the analogue of  Brenier-McCann's theorem  on ${\\sf RCD}^*(K,N)$ spaces, our result being:\n\\begin{theorem}[Optimal maps]\\label{thm:1}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$ and $(X,{\\sf d},\\mathfrak m)$  an ${\\sf RCD}^*(K,N)$ space. Then for every $\\mu,\\nu\\in\\probt X$ with $\\mu\\ll\\mathfrak m$ there exists a unique plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$. Furthermore, this plan is induced by a map and concentrated on a set of non-branching geodesics, i.e. there is a Borel set $\\Gamma\\subset C([0,1],X)$ such that ${\\mbox{\\boldmath$\\pi$}}(\\Gamma)=1$ and for every $t\\in[0,1)$ the map ${\\rm{e}}_t:\\Gamma\\to X$ is injective.\n\\end{theorem}\nHere by ${\\rm{OptGeo}}(\\mu,\\nu)$ we are denoting the set of minimizers of the dynamical version of the optimal transport as discussed above.\nTo some extent, the `hard work' needed for the proof of this result has been already carried out in \\cite{RajalaSturm12} and \\cite{Gigli12a} where it has been proved the following theorem:\n\\begin{theorem}[Optimal maps on ${\\sf RCD}(K,\\infty)$ spaces]\\label{thm:2a}\nLet $K\\in\\mathbb{R}$ and $(X,{\\sf d},\\mathfrak m)$  an ${\\sf RCD}(K,\\infty)$ space. Then for every $\\mu,\\nu\\in\\probt X$ with $\\mu,\\nu\\ll\\mathfrak m$ there exists a unique plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and  this plan is induced by a map and concentrated on a set of non-branching geodesics.\n\\end{theorem}\nMore precisely,  in \\cite{RajalaSturm12} it has been worked around the delicate issue concerning the non-branching assumption, showing that on ${\\sf RCD}(K,\\infty)$ spaces every optimal geodesic plan between absolutely continuous measures must be concentrated on a set of non-branching geodesics. Then, still in \\cite{RajalaSturm12}, it has been observed how such  result coupled with the technique used in \\cite{Gigli12a} to prove existence and uniqueness of optimal maps in the non-branching case yield Theorem \\ref{thm:2a}.\n\n\nHere we  start from this results and obtain Theorem \\ref{thm:1} using the enhanced compactness granted  by the finite dimensionality together with quite standard ideas in optimal transport theory.\n\nAn  interesting fact about Theorem \\ref{thm:1} is that it can be equivalently reformulated in the following way:\n\\begin{theorem}[Exponentiation]\\label{thm:2b}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$, $(X,{\\sf d},\\mathfrak m)$  an ${\\sf RCD}^*(K,N)$ space and $\\varphi:X\\to\\mathbb{R}$ a $c$-concave function ($c=\\frac{{\\sf d}^2}{2}$). Then  for $\\mathfrak m$-a.e. $x\\in X$ there exists exactly one geodesic $\\gamma$ such that $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$.\n\\end{theorem}\nThis result can be naturally interpreted as a definition of what is the exponential map evaluated at  `minus the gradient of a $c$-concave function $\\varphi$': for every $x\\in X$ such that the geodesic $\\gamma$  with $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$ is unique, we put $\\exp(-t\\nabla\\varphi):=\\gamma_t$, thus somehow `reversing' the proof of Brenier-McCann theorem. The role of Theorem \\ref{thm:2b} is to ensure that this map  is well defined for $\\mathfrak m$-a.e. $x\\in X$.\n\nNotice that to some extent Theorem \\ref{thm:2b} is the best one we can expect about exponentiation on a metric measure space. To see why just consider the case of a smooth complete Riemannian manifold $M$ with boundary. Then given $x\\in M$ and $v\\in T_xM$, the value of $\\exp(v)$ is well defined only if there is $y\\in M$ such that $\\nabla\\frac{{\\sf d}^2(\\cdot,y)}2=-v$ (neglecting smoothness issues), and functions of the kind $\\frac{{\\sf d}^2(\\cdot,y)}2$ are the prototype of $c$-concave functions. \n\nTheorem \\ref{thm:1} has some simple but interesting consequences, the first being:\n\\begin{corollary}\\label{cor:1}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$ and $(X,{\\sf d},\\mathfrak m)$  an ${\\sf RCD}^*(K,N)$ space. Then for every $x\\in \\mathop{\\rm supp}\\nolimits(\\mathfrak m)$ the following holds: for $\\mathfrak m$-a.e. $y$ there is only one geodesic connecting $y$ to $x$.\n\\end{corollary} \nThis can be easily seen choosing $\\nu:=\\delta_x$ in Theorem \\ref{thm:1}.\nIn \\cite{R2012a} the conclusion of Corollary \\ref{cor:1} was proven under the assumption that the ${\\sf CD}(K,N)$ condition holds along every geodesic.\nHowever, ${\\sf RCD}^*(K,N)$ a priori only gives the ${\\sf CD}^*(K,N)$ condition along every geodesic between any two measures with bounded densities, see \\cite{Erbar-Kuwada-Sturm13}.\nThus Corollary \\ref{cor:1} is not a direct consequence of \\cite[Theorem 4]{R2012a}.\nA further consequence of this corollary is the following:\n\\begin{corollary}\\label{cor:2}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$ and $(X,{\\sf d},\\mathfrak m)$  an ${\\sf RCD}^*(K,N)$ space. Then the space satisfies the ${\\sf MCP}(K,N)$ condition.\n\\end{corollary} \nFrom \\cite{R2012b} we know that every ${\\sf CD}(K,N)$ space satisfies the ${\\sf MCP}(K,N)$ condition in the sense of \\cite{O2007},\nmeaning that between any absolutely continuous measure and a dirac mass there exists a geodesic that satisfies the ${\\sf MCP}(K,N)$ condition.\nIn Corollary \\ref{cor:2} we obtain a more strict version of the ${\\sf MCP}(K,N)$ condition, considered in \\cite{Sturm06II}, with\na global selection of distributions of geodesics between points such that using these geodesics\nthe ${\\sf MCP}(K,N)$ condition always holds. Since by Corollary \\ref{cor:1} the geodesics are essentially unique, in fact\nany choice of geodesics in an ${\\sf RCD}^*(K,N)$ space will work for the ${\\sf MCP}(K,N)$ condition.\n\nThe difficult part in proving Corollary \\ref{cor:2} relies in proving a sort of self-improving property for the ${\\sf CD}^*(K,N)$ condition: this has been the scope of \\cite{Cavalletti-Sturm12}, where such result has been proved under the non-branching assumption. Yet, such additional hypothesis was made only to get the result of Corollary \\ref{cor:1} above. Given that in the ${\\sf RCD}^*(K,N)$ it holds without the a priori non-branching assumption, Corollary \\ref{cor:2} follows.\n\nA final remark which is worth to make, in particular in connection with Sobolev calculus as developed in \\cite{AmbrosioGigliSavare11}, is the following:\n\\begin{corollary}\nLet $K\\in\\mathbb{R}$, $N\\in[1,\\infty)$, $(X,{\\sf d},\\mathfrak m)$  an ${\\sf RCD}^*(K,N)$ space,  $\\mu,\\nu\\in\\probt X$ with $\\mu\\ll\\mathfrak m$ and ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ be the  unique optimal geodesic plan given by Theorem \\ref{thm:1}. Then $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\ll\\mathfrak m$ for every $t\\in[0,1)$. \n\nFurthermore, if $\\mu,\\nu$ have bounded support (resp. $K=0$) and $\\mu$ and has density bounded above by some constant $C$, then $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\leq C(t)\\mathfrak m$ for any $t\\in[0,1)$ and some constant $C(t)$ depending only on $C,t,K,N$ and the supports of $\\mu,\\nu$ (resp. on $C,t,N$). If $K<0$ and either $\\mu$ or $\\nu$ have unbounded support, then the optimal geodesic plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ can be written as ${\\mbox{\\boldmath$\\pi$}}=\\sum_{n\\in\\mathbb{N}}{\\mbox{\\boldmath$\\pi$}}_n$ with ${\\mbox{\\boldmath$\\pi$}}_n$ non negative Borel measures on ${\\rm{Geo}}(X)$ such that  $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\leq C_n(t)\\mathfrak m$ for any $t\\in[0,1)$ and some constants $C_n(t)$ depending only on $C,t,K,N,n$.\n\\end{corollary}\nThe simple proof follows by localizing the ${\\sf CD}^*(K,N)$ condition along the optimal geodesic plan.\n\n\\smallskip\n\\noindent {\\bf Acknowledgement.}\nThis paper was partly written during the program ``Interactions Between Analysis and Geometry'' at the Institute for Pure and Applied Mathematics (IPAM)\nat University of California, Los Angeles. The authors thank the institute for the excellent research environment.\nT.R. also acknowledges the support of the Academy of Finland project no. 137528.\n\n\n\\section{Preliminaries}\n\nWe assume the reader to be familiar with optimal transport and the definition of spaces with Ricci curvature bounded from below in the sense of Lott-Sturm-Villani. Here we just recall some basic notation.\n\nGiven a geodesic, complete and separable metric space $(X,{\\sf d})$, the set $\\probt X$ is the set of Borel probability measures on it with finite second moment. By ${\\rm{Geo}}(X)$ we denote the space of constant speed minimizing geodesics on $X$ endowed with the $\\sup$-distance.\n\nGiven such metric space $(X,{\\sf d})$ and $\\mu,\\nu\\in\\probt X$, a Borel probability measure ${\\mbox{\\boldmath$\\pi$}}$ on ${\\rm{Geo}}(X)$ is called optimal geodesic plan from $\\mu$ to $\\nu$ provided $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\mu$, $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\nu$  and it achieves the minimum of\n\\[\n\\int {\\sf d}^2(\\gamma_0,\\gamma_1)\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma),\n\\]\namong all Borel probability measure ${\\mbox{\\boldmath$\\pi$}}'$ on ${\\rm{Geo}}(X)$ such that $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}'=\\mu$, $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}'=\\nu$.  The set of all optimal geodesic plans is denoted by ${\\rm{OptGeo}}(\\mu,\\nu)$. Notice that ${\\rm{OptGeo}}(\\mu,\\nu)$ is never empty under the above assumption.\n\nA function $\\varphi:X\\to\\mathbb{R}\\cup\\{-\\infty\\}$ not identically $-\\infty$ is called $c$-concave provided there is $\\psi:X\\to\\mathbb{R}\\cup\\{-\\infty\\}$ such that\n\\[\n\\varphi(x)=\\inf_{y\\in X}\\frac{{\\sf d}^2(x,y)}{2}-\\psi(y).\n\\]\nGiven a $c$-concave function $\\varphi$, its $c$-transform $\\varphi^c:X\\to\\mathbb{R}\\cup\\{+\\infty\\}$ is defined by\n\\[\n\\varphi^c(y):=\\inf_{x\\in X}\\frac{{\\sf d}^2(x,y)}{2}-\\varphi(x).\n\\] \nIt turns out that for every $c$-concave function $\\varphi$ it holds $\\varphi^{cc}=\\varphi$. The $c$-superdifferential $\\partial^c\\varphi$ of a $c$-concave function $\\varphi$ is the subset of $X^2$ of those couples $(x,y)$ such that\n\\[\n\\varphi(x)+\\varphi^c(y)=\\frac{{\\sf d}^2(x,y)}2,\n\\]\nand for $x\\in X$, the set $\\partial^c\\varphi(x)\\subset X$ is the set of those $y$'s such that $(x,y)\\in\\partial^c\\varphi(y)$.\n\nIt can be proved that a Borel probability measure ${\\mbox{\\boldmath$\\pi$}}$ on ${\\rm{Geo}}(X)$ belongs to ${\\rm{OptGeo}}(({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}},({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}})$ if and only if there is a $c$-concave function $\\varphi$ such that $\\mathop{\\rm supp}\\nolimits({\\rm{e}}_0,{\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\subset\\partial^c\\varphi$. Any such $\\varphi$ is called Kantorovich potential from $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}$ to $({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}$. It is then easy to check that for any Kantorovich potential $\\varphi$ from $\\mu$ to $\\nu$, every  ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and every $t\\in[0,1]$,  the function $t\\varphi$ is a Kantorovich potential from $\\mu$ to $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$.\n\n\n\nNotice that Kantorovich potentials can be chosen to satisfy the following property, slightly stronger than $c$-concavity:\n\\[\n\\varphi(x)=\\inf_{y\\in\\mathop{\\rm supp}\\nolimits(\\nu)}\\frac{{\\sf d}^2(x,y)}{2}-\\varphi^c(y),\n\\]\nwhich shows in particular that if $\\mathop{\\rm supp}\\nolimits(\\nu)$ is bounded, then $\\varphi$ can be chosen to be locally Lipschitz.\n\n\n\nWe turn to the formulation of the ${\\sf CD}^*(K,N)$ condition, coming from  \\cite{BacherSturm10}, to which we also refer for a detailed discussion of its relation with the ${\\sf CD}(K,N)$ condition\n (see also \\cite{Cavalletti-Sturm12} and \\cite{Cavalletti12}).\n\nGiven $K \\in \\mathbb{R}$ and $N \\in [1, \\infty)$, we define the distortion coefficient $[0,1]\\times\\mathbb{R}^+\\ni (t,\\theta)\\mapsto \\sigma^{(t)}_{K,N}(\\theta)$ as\n\\[\n\\sigma^{(t)}_{K,N}(\\theta):=\\left\\{\n\\begin{array}{ll}\n+\\infty,&\\qquad\\textrm{ if }K\\theta^2\\geq N\\pi^2,\\\\\n\\frac{\\sin(t\\theta\\sqrt{K\/N})}{\\sin(\\theta\\sqrt{K\/N})}&\\qquad\\textrm{ if }0<K\\theta^2 <N\\pi^2,\\\\\nt&\\qquad\\textrm{ if }K\\theta^2=0,\\\\\n\\frac{\\sinh(t\\theta\\sqrt{K\/N})}{\\sinh(\\theta\\sqrt{K\/N})}&\\qquad\\textrm{ if }K\\theta^2 <0.\n\\end{array}\n\\right.\n\\]\n\\begin{definition}[Curvature dimension bounds]\nLet $K \\in \\mathbb{R}$ and $ N\\in[1,  \\infty)$. We say that a m.m.s.  $(X,{\\sf d},\\mathfrak m)$\n is a ${\\sf CD}^*(K,N)$-space if for any two measures $\\mu_0, \\mu_1 \\in \\prob X$ with support  bounded and contained in $\\mathop{\\rm supp}\\nolimits(\\mathfrak m)$ there\nexists a measure ${\\mbox{\\boldmath$\\pi$}} \\in {\\rm{OptGeo}}(\\mu_0,\\mu_1)$ such that for every $t \\in [0,1]$\nand $N' \\geq  N$ we have\n\\begin{equation}\\label{eq:CD-def}\n-\\int\\rho_t^{1-\\frac1{N'}}\\,\\d\\mathfrak m\\leq - \\int \\sigma^{(1-t)}_{K,N'}({\\sf d}(\\gamma_0,\\gamma_1))\\rho_0^{-\\frac1{N'}}+\\sigma^{(t)}_{K,N'}({\\sf d}(\\gamma_0,\\gamma_1))\\rho_1^{-\\frac1{N'}}\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma)\n\\end{equation}\nwhere for any $t\\in[0,1]$ we  have written $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\rho_t\\mathfrak m+\\mu_t^s$  with $\\mu_t^s \\perp \\mathfrak m$.\n\\end{definition}\nNotice that if $(X,{\\sf d},\\mathfrak m)$ is a ${\\sf CD}^*(K,N)$-space, then so is $(\\mathop{\\rm supp}\\nolimits(\\mathfrak m),{\\sf d},\\mathfrak m)$, hence it is not restrictive to assume that $\\mathop{\\rm supp}\\nolimits(\\mathfrak m)=X$, a hypothesis that we shall always implicitly do from now on. Also, for any ${\\sf CD}^*(K,N)$ space $(X,{\\sf d},\\mathfrak m)$ we have that $(X,{\\sf d})$ is geodesic and proper.\n\nIn \\cite{AmbrosioGigliSavare11-2} (see also \\cite{AmbrosioGigliMondinoRajala12}) an enforcement of the curvature condition ${\\sf CD}(K,\\infty)$ as defined by Lott-Villani and Sturm in \\cite{Lott-Villani09} and \\cite{Sturm06I} has been proposed. This condition, called Riemannian Ricci curvature bound and denoted by ${\\sf RCD}(K,\\infty)$, enforces in some weak sense a Riemannian-like structure of the space. For our purposes, it is not necessary to recall the quite technical definition, but only the following crucial result, proved in  \\cite{RajalaSturm12} (see also \\cite{Gigli12a}):\n\\begin{theorem}[Optimal maps in ${\\sf RCD}(K,\\infty)$ spaces]\\label{thm:optmap}\nLet $(X,{\\sf d},\\mathfrak m)$ be an  ${\\sf RCD}(K,\\infty)$ space and $\\mu,\\nu\\in\\probt X$ two measures absolutely continuous w.r.t. $\\mathfrak m$. \n\nThen there exists a unique ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and this plan is induced by a map and concentrated on a set of non-branching geodesics, i.e. for any $t\\in[0,1]$ there exists a Borel map $T_t:X\\to{\\rm{Geo}}(X)$ such that ${\\mbox{\\boldmath$\\pi$}}=(T_t)_\\sharp({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$. \n\\end{theorem}\n\nFinally we recall the definition of ${\\sf RCD}^*(K,N)$ spaces as given in \\cite{Erbar-Kuwada-Sturm13} (see also \\cite{AMS}):\n\\begin{definition}[${\\sf RCD}^*(K,N)$ spaces]\nLet $K\\in \\mathbb{R}$ and $N\\in[1,\\infty)$. We say that $(X,{\\sf d},\\mathfrak m)$ is an ${\\sf RCD}^*(K,N)$ space provided it is both ${\\sf CD}^*(K,N)$ and ${\\sf RCD}(K,\\infty)$.\n\\end{definition}\n\n\n\n\\section{Exponentiation and optimal maps}\nWe start with the following simple result which shows how the use of Theorem \\ref{thm:optmap} allows for the localization of the ${\\sf CD}^*(K,N)$ condition along a geodesic connecting two absolutely continuous measures.\n\\begin{proposition}\\label{prop:varphi}\nLet $(X,{\\sf d},\\mathfrak m)$ be an ${\\sf RCD}^*(K,N)$ space and  $\\mu_i=\\rho_i\\mathfrak m\\in\\probt X$, $i=0,1$,  two given measures. Let ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu_0,\\mu_1)$ be the unique optimal geodesic plan from $\\mu_0$ to $\\mu_1$ given by Theorem \\ref{thm:optmap} and put $\\mu_t:=({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$. Then $\\mu_t\\ll\\mathfrak m$ for every $t\\in[0,1]$ and writing $\\mu_t=\\rho_t\\mathfrak m$ for every $0\\leq t\\leq r\\leq s\\le 1$ we have\n\\begin{equation}\n\\label{eq:boundpoint}\n\\rho_r(\\gamma_r)^{-\\frac1N}\\geq \\rho_t(\\gamma_t)^{-\\frac1N}\\sigma^{(\\frac{s-r}{s-t})}_{K,N}({\\sf d}(\\gamma_t,\\gamma_s))+\\rho_s(\\gamma_s)^{-\\frac1N}\\sigma^{(\\frac{r-t}{s-t})}_{K,N}({\\sf d}(\\gamma_t,\\gamma_s)),\\qquad{\\mbox{\\boldmath$\\pi$}}-a.e.\\ \\gamma.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof} We start by proving that $\\mu_t\\ll\\mathfrak m$ for every $t\\in[0,1]$.  Fix $\\bar x\\in X$ and for $M>0$ let $G_M\\subset {\\rm{Geo}}(X)$ be defined by \n\\[\nG_M:=\\Big\\{\\gamma\\in {\\rm{Geo}}(X)\\ :\\ \\rho_0(\\gamma_0),\\rho_1(\\gamma_1),{\\sf d}(\\gamma_0,\\bar x),{\\sf d}(\\gamma_1,\\bar x)\\leq M\\Big\\}.\n\\]\nFor $M$ large enough we have ${\\mbox{\\boldmath$\\pi$}}(G_M)>0$, thus the plan ${\\mbox{\\boldmath$\\pi$}}_M:=c_M{\\mbox{\\boldmath$\\pi$}}\\restr{G_M}$ is well defined,  $c_M:={\\mbox{\\boldmath$\\pi$}}(G_M)^{-1}$ being the normalizing constant. Put $\\mu_0^M:=({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M$, $\\mu_1^M:=({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M$ and notice that $\\mu_0^M,\\mu_1^M\\ll\\mathfrak m$ and that by construction and since  optimality is stable by restriction we get  ${\\mbox{\\boldmath$\\pi$}}_M\\in{\\rm{OptGeo}}( \\mu_0^M, \\mu_1^M)$. Hence the uniqueness part of Theorem \\ref{thm:optmap} yields that  ${\\mbox{\\boldmath$\\pi$}}_M$ is the only optimal plan from $\\mu_0^M$ to $\\mu_1^M$. Being $(X,{\\sf d},\\mathfrak m)$ a ${\\sf CD}(K,N)$ space it is also a  ${\\sf CD}(K,\\infty)$ space and thus fact that ${\\rm Ent}_\\mathfrak m(\\mu_0^M),{\\rm Ent}_\\mathfrak m(\\mu_1^M)<\\infty$ (because both have bounded densities) give ${\\rm Ent}_\\mathfrak m(({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M)<\\infty$ for every $t\\in[0,1]$. In particular, $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M\\ll\\mathfrak m$ for every $t\\in[0,1]$. Since $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_M\\uparrow ({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}=\\mu_t$ as $M\\to\\infty$, we deduce $\\mu_t\\ll\\mathfrak m$ for every $t\\in[0,1]$.\n\nWe turn to \\eqref{eq:boundpoint}. Assume for a moment $t=0$, $s=1$ and that the supports of $\\mu_0,\\mu_1$ are bounded and notice that in this case to prove \\eqref{eq:boundpoint} is equivalent to prove that for any Borel set $G\\subset{\\rm{Geo}}(X)$ it holds\n\\begin{equation}\n\\label{eq:intcd}\n\\begin{split}\n-\\int_G\\rho_{r}^{-\\frac1N}(\\gamma_{r})\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma)\\leq& -\\int_G\\rho_0(\\gamma_0)^{-\\frac1N}\\sigma^{(1-t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))+\\rho_1(\\gamma_1)^{-\\frac1N}\\sigma^{(t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))\\,\\d{\\mbox{\\boldmath$\\pi$}}(\\gamma).\n\\end{split}\n\\end{equation}\nFix such Borel set $G\\subset{\\rm{Geo}}(X)$, assume without loss of generality that ${\\mbox{\\boldmath$\\pi$}}(G)>0$ and define ${\\mbox{\\boldmath$\\pi$}}_G:={\\mbox{\\boldmath$\\pi$}}(G)^{-1}{\\mbox{\\boldmath$\\pi$}}\\restr G$.  Let $T_t:X\\to{\\rm{Geo}}(X)$ be the maps given by Theorem \\ref{thm:optmap} and notice that the identity ${\\mbox{\\boldmath$\\pi$}}=(T_t)_\\sharp({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$ ensures $({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_G={\\mbox{\\boldmath$\\pi$}}(G)^{-1}\\chi_G\\circ T_t({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}$. In other words, letting $\\rho_{G,t}\\mathfrak m=({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}_G$, a direct consequence of the fact that ${\\mbox{\\boldmath$\\pi$}}$ is concentrated on a set of non-branching geodesics is that we have\n\\begin{equation}\n\\label{eq:simpleresc}\n\\rho_{G,t}(\\gamma_t)={\\mbox{\\boldmath$\\pi$}}(G)^{-1}\\rho_t(\\gamma_t),\\qquad{\\mbox{\\boldmath$\\pi$}}-a.e.\\ \\gamma\\in G.\n\\end{equation}\nIt is clear that ${\\mbox{\\boldmath$\\pi$}}_G$ is optimal from $\\rho_{G,0}\\mathfrak m$ to $\\rho_{G,1}\\mathfrak m$ and by the uniqueness part of Theorem \\ref{thm:optmap} we know that it is the only optimal plan, hence the ${\\sf CD}^*(K,N)$ condition and the fact that $\\rho_{G,0}\\mathfrak m,\\rho_{G,1}\\mathfrak m$ have bounded support (because we assumed $\\mu_0,\\mu_1$ to have bounded support), yield\n\\[\n-\\int\\rho_{r}^{-\\frac1N}(\\gamma_{r})\\,\\d{\\mbox{\\boldmath$\\pi$}}_G(\\gamma)\\leq -\\int\\rho_0(\\gamma_0)^{-\\frac1N}\\sigma^{(1-t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))+\\rho_1(\\gamma_1)^{-\\frac1N}\\sigma^{(t)}_{K,N}({\\sf d}(\\gamma_0,\\gamma_1))\\,\\d{\\mbox{\\boldmath$\\pi$}}_G(\\gamma),\n\\]\nwhich, taking into account \\eqref{eq:simpleresc}, is \\eqref{eq:intcd}.\n\nThe assumption that $\\mu_0,\\mu_1$ have bounded support  can be removed with the same truncation argument used at the beginning of the proof. To deal with the case of arbitrary $0\\leq t<s\\leq 1$ use the uniqueness part of Theorem \\ref{thm:optmap} again to deduce that the only optimal plan from $\\mu_t$ to $\\mu_s$ is given by $({\\rm Restr}_t^s)_\\sharp{\\mbox{\\boldmath$\\pi$}}$, where ${\\rm Restr}_t^s:{\\rm{Geo}}(X)\\to{\\rm{Geo}}(X)$ is defined by\n\\[\n{\\rm Restr}_t^s(\\gamma)_r:=\\gamma_{(1-r)t+rs}.\n\\]\nThen repeat the argument used for the case $t=0$, $s=1$.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\\label{le:ac}\nLet $\\mu,\\nu\\in\\probt X$ be with bounded support and such that  $\\mu\\leq C\\mathfrak m$ for some $C>0$. Then there exists a geodesic $(\\mu_t)$ from $\\mu$ to $\\nu$ such that $\\mu_t\\ll\\mathfrak m$ for every $t\\in[0,1)$.\n\\end{lemma}\n\\begin{proof}\nLet $(\\nu^n)\\subset\\probt X$ be a sequence of absolutely continuous measures weakly converging to $\\nu$ and with uniformly bounded supports and ${\\mbox{\\boldmath$\\pi$}}^n\\in{\\rm{OptGeo}}(\\mu,\\nu^n)$ the unique optimal plan given by Theorem \\ref{thm:optmap}. Then the bound \\eqref{eq:boundpoint} gives, after dropping the term involving $\\rho_1$, the inequality\n\\begin{equation}\n\\label{eq:sar}\n\\rho_t(\\gamma_t)\\leq \\rho_0(\\gamma_0)(\\sigma_{K,N}^{(1-t)}({\\sf d}(\\gamma_0,\\gamma_1)))^{-N},\\qquad{\\mbox{\\boldmath$\\pi$}}^n-a.e.\\ \\gamma .\n\\end{equation}\nBy the definition of the distortion coefficients $\\sigma_{K,N}^{(1-t)}(\\theta)$ we see that for some function $f:[0,1)\\to\\mathbb{R}^+$ depending on $K$, $N$ and ${\\rm diam}(\\mathop{\\rm supp}\\nolimits(\\mu)\\cup(\\cup_n\\mathop{\\rm supp}\\nolimits(\\nu^n)))$, it holds $(\\sigma_{K,N}^{(1-t)}({\\sf d}(\\gamma_0,\\gamma_1)))^{-N}\\leq f(t)$ and thus \\eqref{eq:sar} and the bound $\\mu\\leq C\\mathfrak m$ give\n\\[\n\\mu^n_t:=({\\rm{e}}_t)_\\sharp{\\mbox{\\boldmath$\\pi$}}^n\\leq Cf(t)\\mathfrak m,\\qquad\\forall t\\in[0,1).\n\\]\nThis bound is independent on $n\\in\\mathbb{N}$, hence with a simple compactness argument based on the fact that $(X,{\\sf d},\\mathfrak m)$ is proper we get the conclusion by letting $n\\to\\infty$.\n\\end{proof}\n\nWe shall also use the following lemma, whose proof was given in \\cite{FigalliGigli11} (see also \\cite{Gigli11}) for the case of Riemannian manifolds; yet, the argument is only metric and can be repeated without any change. We report it just for completeness.\n\\begin{lemma}\\label{le:lip}\nLet $(X,{\\sf d})$ be a proper geodesic space,   $\\varphi$ a $c$-concave function and $\\Omega\\subset X$ the interior of $\\{\\varphi>-\\infty\\}$. Then $\\varphi$ is locally bounded and locally Lipschitz on $\\Omega$ and for every compact set $K\\subset\\Omega$ the set $\\cup_{x\\in K}\\partial^c\\varphi(x)$ is bounded and not empty.\n\\end{lemma}\n\\begin{proof} \nBeing $c$-concave, $\\varphi$ is the infimum of a family of continuous functions, hence upper-semicontinuous and thus locally bounded from above. We prove that it is locally bounded from below by contradiction. Thus, recall that $(X,{\\sf d})$ is proper, assume that there exists a sequence $(x_n)\\subset \\Omega$ converging to some $x_\\infty\\in\\Omega$ such that $\\varphi(x_n)\\to-\\infty$ as $n\\to\\infty$. For every $n\\in\\mathbb{N}$ let $y_n\\in X$ be such that\n\\begin{equation}\n\\label{eq:yn}\n\\varphi(x_n)\\geq \\frac{{\\sf d}^2(x_n,y_n)}{2}-\\varphi^c(y_n)-1,\\qquad\\forall n\\in\\mathbb{N},\n\\end{equation}\nand notice that this bound and the fact that $\\varphi(x_n)\\to-\\infty$ yield that $\\varphi^c(y_n)\\to+\\infty$ as $n\\to\\infty$. Thus from \n\\[\n\\mathbb{R}\\ni \\varphi(x_\\infty)\\leq \\frac{{\\sf d}^2(x_\\infty,y_n)}{2}-\\varphi^c(y_n),\\qquad\\forall n\\in\\mathbb{N},\n\\]\nwe deduce that $\\frac{{\\sf d}^2(x_\\infty,y_n)}{2}\\to+\\infty$ as well as $n\\to\\infty$ and therefore also that\n\\[\n\\lim_{n\\to\\infty}\\frac{{\\sf d}^2(x_n,y_n)}{2}\\to+\\infty.\n\\]\nIn particular, without loss of generality we can assume ${\\sf d}(x_n,y_n)\\geq 1$ for every $n\\in\\mathbb{N}$. Now let $\\gamma^n:[0,{\\sf d}(x_n,y_n)]\\to X$ be a geodesic from $x_n$ to $y_n$ parametrized by arc-length.  We claim that \n\\begin{equation}\n\\label{eq:claimcn}\n\\sup_{B_1(\\gamma^n_1)}\\varphi\\to-\\infty,\\qquad \\textrm{as } n\\to\\infty.\n\\end{equation}\nIndeed, for $x\\in B_1(\\gamma^n_1) $  we have\n\\[\n\\begin{split}\n\\varphi(x)&\\leq \\frac{{\\sf d}^2(x,y_n)}2-\\varphi^c(y_n)\\leq \\frac{({\\sf d}(x,\\gamma^n_1)+{\\sf d}(\\gamma^n_1,y_n))^2}2-\\varphi^c(y_n)\\\\\n&\\leq \\frac{{\\sf d}^2(x_n,y_n)}2-\\varphi^c(y_n)\\leq\\varphi(x_n)+1,\n\\end{split}\n\\]\nhaving used \\eqref{eq:yn} in the last inequality. Given that the $x_n$'s were chosen so that $\\varphi(x_n)\\to-\\infty$ as $n\\to\\infty$, our claim \\eqref{eq:claimcn} is proved.  \n\nUp to pass to a subsequence, we can assume that $(\\gamma^n_1)$ converges to some $z\\in X$. From \\eqref{eq:claimcn} it easily follows that in the internal part of $B_1(z)$ the function $\\varphi$ is identically $-\\infty$. Given that ${\\sf d}(x,z)=1$, this fact contradicts the assumption that $x\\in\\Omega$. Hence $\\varphi$ is locally bounded.\n\nNow let $\\bar x\\in\\Omega$ and $r>0$ be such that $B_{2r}(x)\\subset\\Omega$. Pick  $x\\in B_r(\\bar x)$ and let $(y_n)$ be such that $\\varphi(x)=\\lim_{n}\\frac{{\\sf d}^2(x,y_n)}{2}-\\varphi^c(y_n)$. We claim that there exists a constant $C$ depending only on $\\bar x, r$ and $\\varphi$ such that $(y_n) \\subset B_C(\\bar x)$. In proving this we may assume that ${\\sf d}(x,y_n)>r$ for all $n$. \nPick unit speed geodesics $\\gamma^n:[0,{\\sf d}(x,y_n)]\\to X$ from $x$ to $y_n$ and notice that\n\\[\n\\begin{split}\n\\varlimsup_{n\\to\\infty}\\varphi(x)-\\varphi(\\gamma^n_r)\\geq \\varlimsup_{n\\to\\infty}\\frac{{\\sf d}^2(x,y_n)}{2}-\\frac{{\\sf d}^2(\\gamma^n_r,y_n)}{2}=\\varlimsup_{n\\to\\infty}r{\\sf d}(x,y_n)-\\frac{r^2}{2}.\n\\end{split}\n\\]\nBy construction we have $x,\\gamma^n_r\\in B_{2r}(\\bar x)\\subset \\Omega$ thus by what we previously proved we know that the leftmost side of the above inequality is bounded by some constant depending only on $\\bar x, r$ and $\\varphi$. Hence the sequence $(y_n)$ is bounded and we directly get that any limit point belongs to $\\partial^c\\varphi(x)$, which therefore is non-empty. The very same argument also shows that $C:=\\cup_{x\\in B_r(\\bar x)}\\partial^c\\varphi(x)$ is bounded. In particular we get \n\\[\n\\varphi(x)=\\min_{y\\in C}\\frac{{\\sf d}^2(x,y)}{2}-\\varphi^c(y),\\qquad\\forall x\\in B_r(\\bar x),\n\\]\nand since for $y\\in C$ the functions $x\\mapsto \\frac{{\\sf d}^2(x,y)}{2}-\\varphi^c(y)$ are uniformly Lipschitz, we deduce the local Lipschitz continuity of $\\varphi$ as well.\n\\end{proof}\n\\begin{theorem}[Exponentiation and optimal maps] Let $K\\in \\mathbb{R}$, $N\\in[1,\\infty)$, $(X,{\\sf d},\\mathfrak m)$ an ${\\sf RCD}^*(K,N)$ space,  $\\varphi$ a $c$-concave function and $\\Omega\\subset X$ the interior of $\\{\\varphi>-\\infty\\}$. Then for $\\mathfrak m$-a.e. $x\\in\\Omega$ there exists a unique geodesic $\\gamma$ with $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$.\n\nIn particular, for every $\\mu,\\nu\\in\\probt X$ with $\\mu\\ll\\mathfrak m$ there exists a unique optimal geodesic plan ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ and this plan is induced by a map and concentrated on a set of non-branching geodesics.\n\\end{theorem}\n\\begin{proof} Existence trivially follows from the fact that $\\partial^c\\varphi(x)$ is non-empty for every $x\\in\\Omega$ and the fact that $(X,{\\sf d})$ is geodesic. For uniqueness we argue by contradiction. For $x\\in\\Omega$ let  $G(x)\\subset{\\rm{Geo}}(X)$ be the set of $\\gamma$'s such that $\\gamma_0=x$ and $\\gamma_1\\in\\partial^c\\varphi(x)$ and assume that there is a compact set $K_1\\subset \\Omega$ such that $\\mathfrak m(K_1)>0$ and $\\#G(x)\\geq 2$ for every $x\\in K_1$. By Lemma \\ref{le:lip} we know that for some $L>0$ we have ${\\sf d}(\\gamma_0,\\gamma_1)\\leq L$ for any $x\\in K_1$ and $\\gamma\\in G(x)$ so that the geodesics in $\\cup_{x\\in K_1}G(x)$ are equi-Lipschitz.\n\n\nFor some $a>0$ the compact set $K_2\\subset K_1$ of $x$'s such that ${\\rm diam}G(x)\\geq a$ is such that $\\mathfrak m(K_2)>0$. Pick such $a$ and $K_2$. For $t\\in[0,1]$ put  $G_t(x):=\\{\\gamma_t:\\gamma\\in G(x)\\}\\subset X$ and consider the set $\\mathcal K\\subset K_2\\times [0,1]$ of $(x,t)$'s such that  ${\\rm diam}G_t(x)\\geq\\frac a2$. It is easy to check that $\\mathcal K$ is closed and the continuity of geodesics grants that for any $x\\in K_2$ the set of $t$'s such that $(x,t)\\in \\mathcal K$  has positive $\\mathcal L^1$-measure. By Fubini's theorem, there is $t_0\\in[0,1]$ such that the compact set $K_3\\subset K_2$ of $x$'s such that ${\\rm diam}G_{t_0}(x)\\geq\\frac a2$ has positive $\\mathfrak m$-measure. Notice that necessarily $t_0>0$. With a Borel selection argument we can find a Borel map $T:K_3\\to X$ such that $T(x)\\in G_{t_0}(x)$ for every $x\\in K_3$. Let $x_0\\in X$ be such that  $T_\\sharp(\\mathfrak m\\restr {K_3})(B_{\\frac a6}(x_0))>0$ and put  $A:=T^{-1}(B_{\\frac a6}(x_0))$, so that $\\mathfrak m(A)>0$. By construction, the map $A\\ni x\\mapsto G_{t_0}(x)\\setminus B_{\\frac{a}3}(x_0)$ is Borel and has non-empty values, thus again with a Borel selection argument we can find Borel map $S:A\\to X$ such that $S(x)\\in G_{t_0}(x)\\setminus B_{\\frac{a}3}(x_0) $ for every $x\\in A$.\n\nLet $\\mu:=\\mathfrak m(A)^{-1}\\mathfrak m\\restr A$, $\\nu_1:=T_\\sharp\\mu$ and $\\nu_2:=S_\\sharp\\mu$. By construction $\\nu_1$ and $\\nu_2$ have disjoint support, and in particular $\\nu_1\\neq \\nu_2$. Furthermore, the function $t_0\\varphi$ is a Kantorovich potential both from $\\mu$ to $\\nu_1$ and from $\\mu$ to $\\nu_2$. Apply Lemma \\ref{le:ac} to both $(\\mu,\\nu_1)$ and $(\\mu,\\nu_2)$ to find geodesics $(\\mu^i_t)$, $i=1,2$, from $\\mu$ to $\\nu_1,\\nu_2$ respectively such that $\\mu^i_t\\ll\\mathfrak m$ for every $t\\in[0,1)$, $i=1,2$. By construction, for $t$ sufficiently close to 1 we have $\\mu^1_t\\neq \\mu^2_t$. Fix such $t$, let ${\\mbox{\\boldmath$\\pi$}}^i\\in{\\rm{OptGeo}}(\\mu,\\mu^i_t)$, $i=1,2$ and notice that ${\\mbox{\\boldmath$\\pi$}}^1\\neq {\\mbox{\\boldmath$\\pi$}}^2$ and that $\\mathop{\\rm supp}\\nolimits(({\\rm{e}}_0,{\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}^i)\\subset\\partial^c(tt_0\\varphi)$, $i=1,2$. \n\nThus for the plan ${\\mbox{\\boldmath$\\pi$}}:=\\frac12({\\mbox{\\boldmath$\\pi$}}^1+{\\mbox{\\boldmath$\\pi$}}^2)$ it also holds $\\mathop{\\rm supp}\\nolimits(({\\rm{e}}_0,{\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}})\\subset \\partial^c(tt_0\\varphi)$ and thus is optimal. Moreover it satisfies $({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}},({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}\\ll\\mathfrak m$ and, by construction, is not induced by a map. This contradicts Theorem \\ref{thm:optmap}, concluding the proof of the first part of the statement.\n\nFor the second part, notice that if the optimal geodesic plan is not unique or not induced by a map, there must be ${\\mbox{\\boldmath$\\pi$}}\\in{\\rm{OptGeo}}(\\mu,\\nu)$ which is not induced by a map. With a restriction argument we can then assume that $\\mu:=({\\rm{e}}_0)_\\sharp{\\mbox{\\boldmath$\\pi$}}$, and $\\nu:=({\\rm{e}}_1)_\\sharp{\\mbox{\\boldmath$\\pi$}}$ have bounded support, with $\\mu\\ll\\mathfrak m$. But in this case there is a locally Lipschitz Kantorovich potential from $\\mu$ to $\\nu$ and the first part of the statement gives the conclusion.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiven a sequence of independent and identically distributed standard complex Gaussian random variables $(\\xi_n)$, consider the Gaussian power series \n   $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ defined in the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C}: |z| < 1\\}$ and the zero set of $f(z)$, that is, \n$$\\mathscr{Z} = \\{z \\in \\mathbb{D}: f(z) = 0\\}.$$\nThe zero set $\\mathscr{Z}$ constitutes a point process on $\\mathbb{D}$. The joint intensity $p$ of the process $\\mathscr{Z}$ is defined as\n   $$p(z_1,z_2, \\ldots, z_n) = \\lim_{\\epsilon\\to 0} \\frac{\\mathbb{P}_{\\epsilon}(z_1, z_2, \\ldots, z_n)}{\\pi^n \\epsilon^{2n}}$$ where \n$\\mathbb{P}_\\epsilon(z_1, z_2, \\ldots, z_n)$ is the probability that simultaneously for all $1\\leq i \\leq n$, the function $f(z)$ has a zero in the disc of centre $z_i$ and radius $\\epsilon>0$. \nRecently, Peres and Vir\\'ag \\cite{Peres_Virag} obtained the striking fact that the point process $\\mathscr{Z}$ is a determinantal process. In fact, they proved that, for all $z_1, z_2, \\ldots, z_n$ in $\\mathbb{D}$, \n         $$p(z_1, z_2, \\ldots, z_n) = \\det\\left(\\frac{1}{\\pi(1 - z_k \\overline{z_j})^2}\\right)_{k,j=1}^n.$$\nThat is, \n            $$p(z_1, z_2, \\ldots, z_n) = \\det\\left(K(z_k, z_j)\\right)_{1\\leq k,j\\leq n}$$  where $K(z, w) = \\pi^{-1}(1 - z \\overline{w})^{-2}$ is the classical Bergman kernel in $\\mathbb{D}$. (A thorough discussion on determinantal point processes can be found in the book by Hough et al. \\cite{Hough}.)\nThe study of the zeros of the general random series \n $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ where the restrictions of independence and identical distribution are relaxed upon the random variables $(\\xi_n)$ is an important open problem. This paper considers this question in the following context: {\\it Determine sequences of dependent Gaussian random variables $(\\xi_n)$ such that the zero set of the random series $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ is a determinantal point process as in the case of i.i.d random variables.} We recall that a matrix $(a_{k,j})$ is called a Toeplitz matrix if $a_{k,j}$ depends only on the difference $k-j$, that is, for all $k,j$ and for any integer $\\ell$ such that $a_{k+\\ell, j+\\ell}$ is defined, $a_{k+\\ell, j+\\ell} = a_{k,j}.$ \nWe consider a complex infinite Toeplitz matrix $G$ that is hermitian and positive definite and we  assume that $G$ admits a classical inverse in the sense that there exists a hermitian positive definite matrix $G^{-1}$ such that $G G^{-1} = G^{-1} G = I$. Since $G$ is positive definite and Toeplitz, then there exits a positive definite function $\\gamma$ on the integers such that $G_{k,j} = \\gamma(k-j) = {\\overline{\\gamma(j-k)}}$ for all $k,j$. (We shall assume without loss of generality that $G_{k,k} = \\gamma(0) = 1$ for all $k$.)\nThen by the classical Bochner theorem, one can associate to $G$ a (unique) probability measure $\\mu$ on the unit circle $\\mathbb{T}$ such that \n\\begin{eqnarray*}\n\\gamma(n) = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} d\\mu(\\theta),\\,\\, \\mbox{ for all } n \\in \\mathbb{Z}.\n\\end{eqnarray*}\nWe shall assume throughout that the probability measure $\\mu$ satisfies the following condition: \n{\\bf Condition (C):} The measure $\\mu$ is absolutely continuous and its density $\\varphi$ is strictly positive almost everywhere on $\\mathbb{T}$ with respect to the Lebesgue measure on $\\mathbb{T}$.  \n\nWe now consider a discrete-time complex Gaussian process $(\\xi_n)_{n\\in \\mathbb{N}}$ with zero mean, covariance matrix  $G^{-1}$ and zero pseudo-covariance matrix, that is, for all $n,m\\in \\mathbb{N}$,\n      $$\\mathbb{E}(\\xi_n) = 0, \\,\\,\\,\\mathbb{E}(\\xi_n \\overline{\\xi_m}) = \\left(G^{-1}\\right)_{n,m} \n\\mbox{    and   } \\mathbb{E}(\\xi_n \\xi_m)  = 0.$$\nThe existence of such process $(\\xi_n)$ is classical (see for example Miller \\cite{Miller} and references therein.) In the case where $G$ (and hence $G^{-1})$ is a real matrix one can simply take a real Gaussian process $(\\zeta_n)$ of covariance matrix $G^{-1}$ and write $\\xi_n = (\\zeta_n + i \\zeta_n')\/\\sqrt{2}$ where $(\\zeta_n')$ is an independent copy of $(\\zeta_n)$.\nWe shall consider the Gaussian analytic function \n         $$f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1},\\,\\, z\\in \\mathbb{D}.$$\nOur main finding is that  the zero set of the Gaussian analytic function $f(z)$\nis a determinantal point process governed by the Bergman kernel just as it is for the case of i.i.d random variables.         \nThe main result of this paper is the following.\n \\begin{theorem}\\label{mainth} \n Let $G$ be an infinite, invertible, hermitian and positive definite Toeplitz matrix such that the associated probability measure $\\mu$ satisfies condition $(C)$ and the inverse $G^{-1}$ is such that\n\\begin{eqnarray} \\label{sdse34rfs}\n\\sup_{n,m} |(G^{-1})_{n,m}| < \\infty.\n\\end{eqnarray}\nIf $(\\xi_n)_{n \\in \\mathbb{Z}}$ is a centred complex Gaussian process with covariance matrix $G^{-1}$ and zero pseudo-covariance matrix, then the  zero set of the Gaussian analytic function\n$$f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1},\\,\\, z\\in \\mathbb{D}$$\n  is a determinantal point process governed by the Bergman kernel. That is the joint intensity $p$ of the zeros of $f(z)$ is given by\n       $$p(z_1, z_2, \\ldots, z_n) =  \\det\\left(\\frac{1}{\\pi(1 - z_k \\overline{z_j})^2}\\right)_{1\\leq k,j\\leq n},\\,\\,\\, z_1, z_2, \\ldots, z_n \\in \\mathbb{D}.$$ \n\\end{theorem}\nIf $(\\zeta_n)$ is a centred  Gaussian process with covariance matrix $G$ (that is a Toeplitz matrix), the zero set of $f(z) =  \\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ does not necessarily have the same distribution as $\\sum_{n=1}^\\infty \\chi_n z^{n-1}$ for i.i.d Gaussian variables $(\\chi_n)$. (This is discussed in \\cite{Mukeru_al}. An example is also given in section 6.1.) However if we take instead of $(\\zeta_n)$  a sequence $(\\xi_n)$ with covariance matrix $G^{-1}$ then the corresponding zero set has the same distribution as for i.i.d. Gaussian variables.   \n This implies that in terms of the corresponding zero sets sequences of Gaussian variables whose covariance matrix is the inverse of a Toeplitz matrix are more closer to the sequence of i.i.d Gaussian variables than sequences of variables whose covariance matrix is a Toeplitz matrix. This looks awkward but one should remember that the distribution of a Gaussian vector depends more directly on the inverse of its covariance matrix rather than the covariance matrix itself. \nLemma \\ref{lemmaone} in section \\ref{sdsefer} gives an interesting property of sequences of Gaussian variables whose covariance is an inverse of a Toeplitz matrix. It looks like such sequences are of some independent interest that requires further investigation. \n\nThe question of dependent random variables can be raised in connection with other determinantal point processes, for instance the point processes obtained by Krishnapur \\cite{Krishnapur}. It is also relevant in the context of \nPfaffian processes studied by  Matsumoto and Shirai \\cite{Matsumoto}. \n The rest of the paper is organised as follows. Section 2 contains some basic well-known facts about the zeros of the $f(z)$ and the Szeg\\\"o kernel. Section 3 contains a connection between the covariance kernel of $f(z)$ and a Hardy space defined by the spectral measure of $(\\xi_n)$. In section 4 we obtain some  properties of the sequence $(\\xi_n)$. In section 5 we provide an important connection between the classical Mobius transformation and the covariance kernel of $f(z)$ which is key to the proof of the main result. The last section contains some examples that illustrate the main result.   \n\n\n\n\n\n\n\\section{Szeg\\\"o kernel and Hardy spaces}   \n                  \nThe starting point in the study of the zeros of any zero-mean Gaussian analytic function $f$ in a planar domain is the following general expression for its joint intensity function (Peres and Vir\\'ag \\cite{Peres_Virag}): \n    \\begin{eqnarray} \\label{eqnads23001}\n     p(z_1, z_2, \\ldots, z_n) = \\frac{\\mathbb{E}\\left(|f'(z_1) f'(z_2) \\ldots f'(z_n)|^2\\,|f(z_1) = f(z_2) = \\ldots=f(z_n) = 0\\right)}{\\pi^n \\det(A)}\n     \\end{eqnarray}\n or equivalently  \n   \\begin{eqnarray} \\label{eqnads231}\n     p(z_1, z_2, \\ldots, z_n) = \\frac{\\mbox{perm}(C - B A^{-1} B^*)}{\\pi^n \\det(A)}\n     \\end{eqnarray}\n  where $A, B$ and $C$ are the $n \\times n$ matrices\n   $$A = (\\mathbb{E}(f(z_k) \\overline{f(z_j)})), B = (\\mathbb{E}(f'(z_k) \\overline{f(z_j)})) \\mbox{ and } C = (\\mathbb{E}(f'(z_k) \\overline{f'(z_j)}))$$ \nand $\\mbox{perm}$ denotes the permanent of a matrix. For the classical Gaussian power series $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ with independent and identically distributed (i.i.d) random variables $(\\xi_n)$,\n    $$ \\mathbb{E}(f(z) \\overline{f(w)})) = \\sum_{n=0}^\\infty (z \\overline{w})^{n-1} = \\frac{1}{1 - z \\overline{w}}, \\,\\, z, w \\in \\mathbb{D}.$$\nThat is, $$ \\mathbb{E}(f(z) \\overline{f(w)})) = \\mathbb{K}(z, w)$$ where $\\mathbb{K}$ is the classical Szeg\\\"o kernel. This means that the covariance kernel of $f(z)$  is the Szeg\\\"o kernel.\nThe classical Hardy space\n $\\mathbb{H}^2(\\mathbb{D})$  is  the class of holomorphic functions $f$ in the unit disc $\\mathbb{D}$ for which \n           $$\\sup_{0 \\leq r < 1} \\int_{\\mathbb{T}} \\left|f(r e^{2\\pi i \\theta})\\right|^2 d\\theta = \\lim_{r \\to 1} \\int_{\\mathbb{T}} \\left|f(r e^{2\\pi i \\theta})\\right|^2 d\\theta  < \\infty$$ where \n      $\\mathbb{T}$ is the unit circle $\\mathbb{R}\/\\mathbb{Z}$. Equivalently $\\mathbb{H}^2(\\mathbb{D})$ is the class of holomorphic functions  $f(z) = \\sum_{n=0}^\\infty a_n z^n$,  $z \\in \\mathbb{D}$, $a_n \\in \\mathbb{C}$ such that $\\sum_{n=0}^\\infty |a_n|^2 < \\infty.$ \nIt is a Hilbert space with the inner product: \n \\begin{eqnarray} \\label{ew23edfr1e1}\n\\langle f, g\\rangle = \\sum_{n=0}^\\infty a_n \\overline{b_n},\\,\\,\\,\\, f = \\sum_{n=0}^\\infty a_n z^n, \\,\\,g = \\sum_{n=0}^\\infty b_n z^n.\n\\end{eqnarray}\nAny function $f(z) = \\sum_{n=0}^\\infty a_n z^n$ in $\\mathbb{H}^2(\\mathbb{D})$ is such that its radial limit\n$$\\tilde{f}(\\theta) = \\lim_{r\\to 1} f(r e^{2\\pi i \\theta}) = f(e^{2\\pi i \\theta})=  \\sum_{n=0}^\\infty a_n e^{2\\pi i n \\theta}$$\nexists almost everywhere in $\\mathbb{T}$ and $\\tilde{f} \\in L^2(\\mathbb{T})$. Moreover\n$$\\langle f, g\\rangle = \\int_{\\mathbb{T}} \\tilde{f}(\\theta) \\overline{\\tilde{g}(\\theta)} d\\theta =  \\int_{\\mathbb{T}} f(e^{2\\pi i \\theta}) \\overline{g(e^{2\\pi i \\theta})} d\\theta$$ and \n $$\\|f\\|^2_{H^2(\\mathbb{D})} = \\|\\tilde{f}\\|_{L^2(\\mathbb{T})} = \\sum_{n=0}^\\infty |a_n|^2.$$\n(See Katznelson \\cite[p 98]{Katznelson}.)    \n\nIt is well-known that $\\mathbb{H}^2(\\mathbb{D})$  is a reproducing kernel Hilbert space whose kernel is the Szeg\\\"o kernel. This means that for each $y \\in \\mathbb{D}$ and for each $f \\in \\mathbb{H}^2(\\mathbb{D})$, the function\n$\\mathbb{K}(.,y): \\mathbb{D} \\to \\mathbb{C}$ defined by $\\mathbb{K}(.,y)(x) = \\mathbb{K}(x, y)$ is such that\n     $$f(y) = \\langle f, \\mathbb{K}(.,y) \\rangle.$$\n(We refer to the book by Paulsen \\cite{Paulsen} for a background on reproducing kernel Hilbert spaces.)\nThe main argument of Peres and Vir\\'ag is to make use of these connections between the Hardy space $\\mathbb{H}^2(\\mathbb{D})$ and the Szeg\\\"o kernel.\n\n\n\n\n\\section{Inverse Toeplitz matrices and weighted Hardy spaces}  \nIn the general case where the covariance matrix of the variables $(\\xi_n)$ is the matrix $G^{-1}$ (where $G$ is an infinite hermitian and positive definite Toeplitz matrix), the covariance kernel of the function   $f(z)$ is given by\n  \\begin{eqnarray} \\label{sdds34rewdss}\n\\mathbb{K}_G(z,w) = \\mathbb{E}\\left(f(z) \\overline{f(w)}\\right) =  \\sum_{k,j=1}^\\infty \\left(G^{-1}\\right)_{k,j} z^{k-1} \\left(\\overline{w}\\right)^{j-1}, \\,\\, z, w \\in \\mathbb{D}.\n\\end{eqnarray}\nFor the convergence of the series in   (\\ref{sdds34rewdss}) it is enough to assume that \n $\\sup_{k,j} |(G^{-1})_{k,j}| < \\infty$\n(that is condition (\\ref{sdse34rfs}) in Theorem \\ref{mainth}.)\nOne can write        $$ \\mathbb{K}_G(z,w) =  Z^{T} G^{-1} \\overline{W}$$\nwhere $Z^T = (1, z, z^2, \\ldots)$ and $W = (1, w, w^2, \\ldots)^T$. \nClearly in the particular case where $G$ is the identity matrix $\\mathbb{K}_G$ is the Szeg\\\"o kernel . An important tool in the proof of the main result is the fact that there is a reproducing kernel Hilbert space whose kernel is the covariance kernel $\\mathbb{K}_G$. Assume that the Toeplitz matrix $G$ is given by $G_{k,j} = \\gamma(k-j) = {\\overline{\\gamma(j-k)}}$ for  a function $\\gamma$ defined on the integers. We shall assume without loss of generality that $G_{k,k} = \\gamma(0) = 1$ for all $k$.\nSince $G$ is a Toeplitz matrix and it is positive definite then by the classical Bochner theorem, there exists a probability measure $\\mu$ on the unit circle $\\mathbb{T}$ such that \n\\begin{eqnarray} \\label{sdwdsdserfds}\n\\gamma(n) = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} d\\mu(\\theta),\\,\\, \\mbox{ for all } n \\in \\mathbb{Z}.\n\\end{eqnarray}\nWe have assumed throughout that $\\mu$ is absolutely continuous. Its density $\\varphi$ is called the spectral density function of the matrix $G$ and is such that \n\\begin{eqnarray} \\label{dsdsedererss}\n\\gamma(n) = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\varphi(\\theta)  d\\theta,\\,\\, \\mbox{ for all } n \\in \\mathbb{Z}\n\\end{eqnarray}\nwhich (under some conditions)  implies in return that \n\\begin{eqnarray} \\label{dsdswedwew334}\n\\varphi(\\theta) = \\sum_{n \\in \\mathbb{Z}} \\gamma(n) e^{-2\\pi  i n \\theta},\\,\\, \\theta \\in \\mathbb{T}. \n\\end{eqnarray}\nConsider the sub-space $H^2_G(\\mathbb{D})$ of the Hardy space $H^2(\\mathbb{D})$ of functions $f(z) = \\sum_{n=0}^\\infty a_n z^n$, $z \\in \\mathbb{D}$, $a_n \\in \\mathbb{C}$ such that \n $$\\|f(e^{2\\pi i \\theta})\\|^2_{L^2(\\mu)} = \\int_{\\mathbb{T}} |f(e^{2\\pi i \\theta})|^2 d\\mu(\\theta)  < \\infty$$ and set\n  $$\\|f\\|_{H^2_G(\\mathbb{D})} = \\|f(e^{2\\pi i \\theta})\\|_{L^2(\\mu)}.$$\nClearly if $$\\|f\\|^2_{H^2_G(\\mathbb{D})} = \\int_{\\mathbb{T}} |f(e^{2\\pi i \\theta})|^2 \\varphi(\\theta) d\\theta = 0,$$ then the fact that $\\varphi > 0$ almost everywhere in $\\mathbb{T}$ yields\n$$\\|f\\|^2_{H^2(\\mathbb{D})} = \\int_{\\mathbb{T}} |f(e^{2\\pi i \\theta})|^2 d\\theta = 0.$$ \nThis implies that $f = 0$ everywhere in $\\mathbb{D}$.  Moreover the norm $\\|.\\|_{H^2_G(\\mathbb{D})}$ is complete. \n Indeed, let $(f_k)$  be a sequence of functions in $H^2_G(\\mathbb{D})$ such that \n$$\\lim_{k,j\\to \\infty} \\|f_k - f_j\\|^2_{H^2_G(\\mathbb{D})} = \\lim_{k,j\\to \\infty} \\|f_k(e^{2\\pi i \\theta}) - f_j(e^{2\\pi i \\theta})|^2 _{L^2(\\mu)}=  0.$$ Since $L^2(\\mu)$ is complete, then the sequence $(f_k(e^{2\\pi i \\theta}))$ has a limit $\\ell: \\mathbb{T} \\to \\mathbb{C}$ in $L^2(\\mu)$. That is,\n \\begin{eqnarray*} \n\\lim_{k\\to \\infty} \\int_{\\mathbb{T}} |f_k(e^{2\\pi i \\theta}) - \\ell(\\theta)|^2 \\varphi(\\theta) d\\theta =  0.\n\\end{eqnarray*}\nAgain since $\\varphi >0$ almost everywhere, then \n\\begin{eqnarray} \\label{kaziba1}\n\\lim_{k\\to \\infty} \\int_{\\mathbb{T}} |f_k(e^{2\\pi i \\theta}) - \\ell(\\theta)|^2 d\\theta =  0.\n\\end{eqnarray}\n This means that $\\ell$ is also the limit of the sequence  $(f_k(e^{2\\pi i \\theta}))$ in \n$L^2(\\mathbb{T})$. It follows that $\\ell \\in L^2(\\mathbb{T})$. We can consider the Fourier series of $\\ell$ and write\n $\\ell(\\theta) = \\sum_{n = -\\infty}^{\\infty} \\hat \\ell(n) e^{2\\pi i n \\theta} $ (this series converges in $L^2(\\mathbb{T})$) and $$\\sum_{n = -\\infty}^{\\infty} |\\hat \\ell(n)|^2  = \\|\\ell\\|^2_{L^2(\\mathbb{T})} < \\infty.$$ \nSet\n    $$f_{n,k}(z) = \\sum_{n=0}^\\infty a_{n,k} z^n.$$ \n  Then (\\ref{kaziba1}) implies\n               $$\\lim_{k \\to \\infty} \\sum_{n \\geq 0} |a_{n,k} - \\hat \\ell(n)|^2 + \\sum_{n < 0} |\\hat \\ell(n)|^2 = 0.$$\nHence $ \\hat \\ell(n) = 0$ for all $n < 0$. Then consider the function \n $g(z) = \\sum_{n = 0}^\\infty \\hat \\ell(n) z^n$. It is now clear that $(f_k)$ converges to $g$ both in $H^2(\\mathbb{D})$ and $H^2_G(\\mathbb{D})$. \n\nWe define the inner product on $H^2_G(\\mathbb{D})$ by:\n \\begin{eqnarray} \\label{ew23edfr1eswe}\n\\langle f, g\\rangle =  \\int_{\\mathbb{T}} f(e^{2\\pi i \\theta}) \\overline{g(e^{2\\pi i \\theta})} d\\mu(\\theta).\n\\end{eqnarray}\nClearly $H^2_G(\\mathbb{D})$ is a Hilbert space. We want to show that $\\mathbb{H}^2_G(\\mathbb{D})$ is in fact a reproducing kernel Hilbert space whose kernel is $\\mathbb{K}_G$.  \n   \\begin{theorem} \\label{sdsd34edsds}\nThe space $\\mathbb{H}^2_G(\\mathbb{D})$ is a reproducing kernel Hilbert space whose kernel is \n  $\\mathbb{K}_G$ given by\n  $$ \\mathbb{K}_G(z,w) =  Z^{T} G^{-1} \\overline{W},\\,\\,z, w \\in \\mathbb{D}$$\n    with $Z = (z^n)_{n \\in \\mathbb{N}}$ and $W = (w^n)_{n \\in \\mathbb{N}}$.\n\\end{theorem}\n\n\\begin{proof}\nWe shall first prove that the monomials $z^n$ are in the reproducing kernel Hilbert space associate to $\\mathbb{K}_G$. That is, \n    $$z^n = \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\,\\,\\overline{\\mathbb{K}_G(e^{2\\pi i \\theta}, z)} d\\mu(\\theta).$$\nSince for all $w, y \\in \\mathbb{D}$,\n  $$ \\mathbb{K}_G(w, y) = \\sum_{k, j = 1}^\\infty (w)^{k-1} (\\overline{y})^{j-1} (G^{-1})_{k,j} = \\overline{\\mathbb{K}_G(y, w)},$$ then \n\n\\begin{eqnarray*}\n\\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\,\\,\\overline{\\mathbb{K}_G(e^{2\\pi i \\theta}, z)} d\\mu(\\theta) & = & \\int_{\\mathbb{T}} e^{2\\pi i n \\theta} \\,\\,\\mathbb{K}_G(z, e^{2\\pi i \\theta}) d\\mu(\\theta) \\\\\n & = &  \\sum_{j=1}^{\\infty} z^{j-1} \\sum_{k=1}^\\infty (G^{-1})_{k,j} \\int_{\\mathbb{T}}  e^{2\\pi i (n - k+1) \\theta} d\\mu(\\theta) \\\\\n& = & \\sum_{j=1}^{\\infty} z^{j-1} \\sum_{k=1}^\\infty (G^{-1})_{k,j} \\,\\gamma(n-k+1) \\\\\n& = & \\sum_{j=1}^{\\infty} z^{j-1} \\sum_{k=1}^\\infty (G^{-1})_{k,j}\\, G_{n+1,k} \\\\\n& = & z^n.\\\\\n\\end{eqnarray*}\nTo complete of the proof it suffices to determine an orthonormal basis $\\{P_k(z): k=1,2,\\ldots\\}$ of $H^2_G(\\mathbb{D})$ and prove that it is the case that\n          $$\\sum_{k=1}^\\infty P_k(z) \\overline{P_k(w)} = \\mathbb{K}_G(z, w),\\,\\,\\mbox{ for all } z, w \\in \\mathbb{D}.$$\nFirst, it is clear that  in the Hilbert space $H^2_G(\\mathbb{D})$, \nfor all $k,j \\in \\mathbb{N}$, \n           \\begin{eqnarray*}\n         \\langle z^k,  z^j \\rangle & = & \\int_{\\mathbb{T}}  e^{2\\pi i(k-j) \\theta}  d\\mu(\\theta)  \\\\\n                  & = & \\gamma(k-j).\n \\end{eqnarray*}\nNext we shall take $(P_k)$ to be the orthonormal basis deduced from \nthe sequence of polynomials $(1, z, z^2, \\ldots, z^k, \\ldots)$ by the classical Gram--Schmidt process.  \nConsider an infinite lower-triangular matrix $A = (a_{k,j})$ (that is $a_{k,j} = 0$ for $j>k$) such that \n for each $k$,\n\\begin{eqnarray} \\label{sdewewe3reew}\nz^{k-1} = a_{k,1} P_1(z) + a_{k,2} P_2(z) + \\ldots+ a_{k,k} P_k(z).\n\\end{eqnarray}\nThen since $\\{P_k: k=1,2,\\ldots\\}$ is orthonormal, then for $j\\leq k$, \n\\begin{eqnarray*}\n\\langle z^{k-1},  z^{j-1} \\rangle  = a_{k,1} \\overline{a_{j,1}} + a_{k,2} \\overline{a_{j,2}} + \\ldots+ a_{k,j} \\overline{a_{j,j}}.\n\\end{eqnarray*}\nMoreover, using \n$ \\langle z^{k-1},  z^{j-1} \\rangle  = \\gamma(k-j)$, it follows that\n  $$a_{k,1} \\overline{a_{j,1}} + a_{k,2} \\overline{a_{j,2}} + \\ldots+ a_{k,j} \\overline{a_{j,j}} = \\gamma(k-j) = G_{k,j}.$$\nThis yields\n       $A A^* = G$ where $A^*$ is the conjugate transpose of $A$.\nIt follows from (\\ref{sdewewe3reew}) that \n   \\begin{eqnarray*}\n  \\begin{pmatrix}\n   P_1(z)\\\\\n   P_2(z)\\\\\n   P_3(z)\\\\\n   \\vdots\\\\\n   \n   P_k(z)\\\\\n   \\vdots\n  \\end{pmatrix}\n    = A^{-1} \n   \\begin{pmatrix}\n   1\\\\\n   z\\\\\n   z^2\\\\\n   \\vdots\\\\\n   z^{k-1}  \\\\\n\\vdots \n   \\end{pmatrix} = A^{-1} Z\n  \\end{eqnarray*}\n for a lower-triangular matrix $A$ such that $A A^* = G$. It is clear that since $A$ is a lower triangular matrix, then $A^{-1}$ is also a lower triangular matrix and moreover $P_k(z)$ is fully determined by the first $k$ rows of $A$.         \nNow clearly,\n     \\begin{eqnarray*}\n      \\sum_{k=1}^\\infty P_k(z) \\overline{P_k(w)} &  = & \\lim_{n \\to \\infty} \\sum_{k=1}^n P_k(z) \\overline{P_k(w)}  \\\\\n& = &\\lim_{n \\to \\infty} \\left(\\left(A_n\\right)^{-1} Z_n\\right)^T \\overline{\\left(A_n\\right)^{-1} W_n} \\\\\n& = & \\lim_{n\\to \\infty} Z_n^T (G_n)^{-1} \\overline{W_n} \\\\\n& = & Z^T G^{-1} \\overline{W}\n     \\end{eqnarray*}\nwhere $A_n$ (resp. $G_n$)  is the block of $A$ (resp. $G$) consisting of the first $n$ rows and columns of $A$ (resp. $G$) and\n$Z_n = (1, z, z^2, \\ldots, z^{n-1})$ and $W_n  = (1, w, w^2,  \\ldots, w^{n-1})$.\nIt follows that \n $$      \\sum_{k=1}^\\infty P_k(z) \\overline{P_k(w)}  = \\mathbb{K}_G(z, w)$$ which concludes the proof. \n\n\\end{proof}\n\n\\begin{corollary}\nThe Gaussian analytic function $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ (where $(\\xi_n)$ has covariance matrix $G^{-1}$) has the same distribution with the function\n                 $g(z) = \\sum_{n=1}^\\infty \\chi_k P_n(z)$ where $(\\chi_n)$ is a sequence of standard i.i.d complex Gaussian random variables and $(P_n(z))$ are the polynomials defined by the matrix $G$ as in the proof of Theorem \\ref{sdsd34edsds}.              \n\\end{corollary}\nIt is so because the two random functions have the same covariance kernel:\n $$\\mathbb{E}(f(z) \\overline{f(w)}) = \\mathbb{E}(g(z) \\overline{g(w)}) = \\sum_{k=1}^\\infty P_n(z) \\overline{P_n(w)} = Z^T G^{-1} \\overline{W}.$$\n\n\n\n\n\n \\section{Some properties of the sequence $(\\xi_n)$ of covariance $G^{-1}$} \\label{sdsefer}\nHere we obtain important properties of the sequence $(\\xi_n)$ of covariance matrix $G^{-1}$ that will be useful for the proof of main result. \n\n\\begin{lemma}\\label{lemmaone}\nAssume that  $(\\xi_n)_{n\\in \\mathbb{N}}$ is a centred complex Gaussian process with zero \\\\ pseudo-covariance and  covariance matrix $G^{-1}$ where $G$ is an infinite hermitian positive definite Toeplitz matrix. Then for each $n \\geq 2$, the conditional joint distribution of the sequence $(\\xi_n, \\xi_{n+1}, \\xi_{n+2}, \\ldots)$ under the condition $\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0$ is equal to the unconditional joint distribution of $(\\xi_{1}, \\xi_{2}, \\xi_{3}, \\ldots)$. \nThat is,\n $$((\\xi_n, \\xi_{n+1}, \\xi_{n+3}, \\ldots)|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0) \\stackrel{d}{=} (\\xi_{1}, \\xi_{2}, \\xi_3, \\ldots)$$\n \\end{lemma}\n\n\\begin{proof}.\nSet $$S_1 = (\\xi_1, \\xi_2, \\ldots), \\,\\, S_n = (\\xi_{n}, \\xi_{n+1}, \\ldots),\\,\\,\\mbox{ for all } n\\geq 1.$$\nThen it is well-known that the covariance matrix of \n$$(S_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$$ is the Schur complement of $\\mbox{Cov}(\\xi_1, \\xi_2, \\ldots, \\xi_{n-1})$ in the overall covariance matrix $\\mbox{Cov}(S_1) = G^{-1}.$ It is obtained by taking the matrix $G^{-1}$, take its inverse, that is, $G$, delete the rows and columns corresponding to the random variables $\\xi_1, \\xi_2, \\ldots, \\xi_{n-1}$ and take the inverse of the resulting matrix. Now deleting the first $n-1$ rows and columns of the infinite Toeplitz matrix $G$ yields the very same matrix $G$. It follows that \nthe covariance matrix of $(S_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$ is just $G^{-1}$. This implies that \n $(S_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$ has the same distribution as $(\\xi_1, \\xi_2, \\ldots)$. \n \\hfill \\qed\n\\end{proof}\nAn immediate consequence of this lemma is: \n\\begin{corollary} \\label{corol1}\nFor any sequence $(\\alpha_k)$ of complex numbers and for any integer $n \\geq 1$,\n $$\\left.\\left(\\sum_{k=1}^\\infty \\alpha_{k} \\xi_{n+k-1}\\right)\\right|\\xi_1 = \\xi_2 = \\ldots=\\xi_{n-1} = 0)$$ has the same distribution with $$\\sum_{k=1}^\\infty \\alpha_k \\xi_{k}$$ provided the involved series converge almost surely. \nIn particular,\n           $$(\\xi_n|\\xi_1 = \\xi_2 = \\ldots=\\xi_{k} = 0) \\stackrel{d}{=} \\xi_{n-k} \\,\\, \\mbox{ for all } 1 \\leq k < n.$$\n           \\end{corollary}\n\n             \n\n\n\\section{Mobius transformation and the kernel $\\mathbb{K}_G$}\n\\subsection{Mobius transformation}\nFor $w \\in \\mathbb{D}$, consider as in Peres and Vir\\'ag \\cite{Peres_Virag}, the Mobius transformation of the unit disc $\\mathbb{D}$:\n           $$T_w(z) = \\frac{z-w}{1-z \\overline{w}},\\,\\, z\\in \\mathbb{D}.$$\nIn the case where $G$ is the identity matrix (or equivalently the random variables $(\\xi_k)$ are independent and identically distributed), Peres and Vir\\'ag \\cite{Peres_Virag} proved the following lemma:\n\\begin{lemma}[Peres and Vir\\'ag] \\label{lemmaPV}\nAssume that $(\\xi_k)$ is the sequence of standard i.i.d complex Gaussian random variables. Then for any $w$ fixed in $\\mathbb{D}$, under the condition $f(w) = 0$, the random function $f(z)$ has the same distribution with \n          $$T_{w}(z) f(z) = \\left(\\frac{z-w}{1- z \\overline{w}}\\right) f(z),\\,\\,z \\in \\mathbb{D},$$ that is, \n$$(f(z)| f(w) = 0) \\stackrel{d}{=} T_{w}(z) f(z)$$ where\n   $\\stackrel{d}{=}$ denotes equality in distribution.  \n In general for $w_1, w_2, \\ldots, w_n$ fixed in $\\mathbb{D}$, \n        \\begin{eqnarray} \n(f(z)| f(w_1)=0, f(w_2)=0, \\ldots, f(w_n) = 0) \\stackrel{d}{=} T_{w_1}(z) T_{w_2}(z) \\ldots T_{w_n}(z) f(z).\n\\end{eqnarray}   \n\\end{lemma}\nA closer look at Peres and Vir\\'ag's proofs reveals that their main result (that is Theorem \\ref{mainth} in the  case where $G$ is the identity matrix) is a consequence of Lemma \\ref{lemmaPV}. This implies that if we prove that Lemma \\ref{lemmaPV} holds true in the  general case of a  Toeplitz matrix $G$, then the same argument as in Peres and Vir\\'ag \\cite{Peres_Virag} will complete the proof of Theorem \\ref{mainth}.\nIn other words, in order to prove our main result, it is sufficient to prove that the following lemma holds.  \n\n\\begin{lemma} \\label{lemmaSM}\nLet $G$ be an invertible infinite hermitian Toeplitz matrix such that its associated measure $\\mu$ is absolutely continuous with density $\\varphi >0$ almost everywhere on $\\mathbb{T}$. Assume that $(\\xi_k)_{k\\in \\mathbb{N}}$ is a centred Gaussian process  with covariance matrix $G^{-1}$ and zero pseudo-covariance. Let\n      $$f(z) = \\sum_{k=1}^\\infty \\xi_k z^{k-1},\\,\\,\\, z\\in \\mathbb{D}.$$\nThen for any $w$ fixed in $\\mathbb{D}$, \n$$(f(z)| f(w) = 0) \\stackrel{d}{=} T_{w}(z) f(z).$$ Moreover, \n for $w_1, w_2, \\ldots, w_n$ fixed in $\\mathbb{D}$, \n        \\begin{eqnarray} \n(f(z)| f(w_1)=0, f(w_2)=0, \\ldots, f(w_n) = 0) \\stackrel{d}{=} T_{w_1}(z) T_{w_2}(z) \\ldots T_{w_n}(z) f(z). \n\\end{eqnarray}   \n\\end{lemma}\nIt is now an easy matter to prove that Lemma \\ref{lemmaSM} yields Theorem \\ref{mainth} based on Peres and Vir\\'ag arguments. \n\\subsection{Proof of Theorem \\ref{mainth}.}\nFor all fixed $z_1, z_2, \\ldots, z_n$, $w_1, w_2, \\ldots, w_n$ in $\\mathbb{D}$,\n   the conditional joint distribution of $$(f(z_1), f(z_2), \\ldots, f(z_n)|f(w_1) = f(w_2) = \\ldots = f(w_n)=0)$$ is equal to the non-conditional joint distribution of $$\\left(T_{w_1}(z_1) f(z_1), T_{w_2}(z_2) f(z_2), \\ldots, T_{w_n}(z_n) f(z_n)\\right).$$ Taking the derivatives, it follows as in \\cite[corollary 13]{Peres_Virag} \nthat the conditional joint distribution of \n   $$(f{'}(z_1), f{'}(z_2), \\ldots, f'(z_n)|f(z_1) = f(z_2) = \\ldots = f(z_n) = 0)$$is the same as the unconditional joint distribution of \n   $$(\\Upsilon'(z_1) f(z_1), \\Upsilon'(z_2) f(z_2), \\ldots, \\Upsilon'(z_n) f(z_n))$$\nwhere \n           $$\\Upsilon(z) = T_{z_1}(z) T_{z_2}(z) \\ldots T_{z_n}(z).$$    \nThis follows from the fact that\n         $$T_{z}'(z) = \\frac{1}{1- |z|^2}\\,\\,\\mbox{ and } \\,\\,T_z(z) = 0,\\,\\,\\,\\,z\\in \\mathbb{D}. $$\nAt this stage, we make use of relation (\\ref{eqnads23001}) to obtain\n \\begin{eqnarray*} \n     p_0(z_1, z_2, \\ldots, z_n) & = & \\frac{\\mathbb{E}\\left(|f'(z_1) f'(z_2) \\ldots f'(z_n)|^2\\,|f(z_1) = f(z_2) = \\ldots=f(z_n) = 0\\right)}{\\pi^n \\det(A)} \\\\\n     & = & \\frac{\\mathbb{E}\\left(|\\Upsilon'(z_1) f(z_1) \\Upsilon'(z_2) f(z_2) \\ldots \\Upsilon'(z_n) f(z_n)|^2\\right)}{\\pi^n \\det(A)} \\\\\n     & = & \\frac{\\mathbb{E}\\left(|f(z_1)  f(z_2) \\ldots  f(z_n)|^2\\right) \\prod_{k=1}^n |\\Upsilon'(z_k)|^2 }{\\pi^n \\det(A)} \\\\\n         \\end{eqnarray*}\nUsing the classical Cauchy determinant formula, Peres and Vir\\'ag \\cite{Peres_Virag}  showed that \n     $$ \\prod_{k=1}^n |\\Upsilon'(z_k)|  = \\det(A_0)$$     \nwhere  \n                $$A_0 = \\left(\\frac{1}{1- z_k \\overline{z_j}}\\right)_{k,j=1}^n.$$\nMoreover since it is well-known that if $X_1, X_2, \\ldots, X_n$ are random variables with joint Gaussian distribution with mean 0 and covariance matrix $\\Sigma$, \n    $$\\mathbb{E}\\left(|X_1 X_2 \\ldots X_n|^2\\right)  = \\mbox{perm}(\\Sigma),$$ it follows that, \n \\begin{eqnarray} \\label{dsdefesdws}\n    p_0(z_1, z_2, \\ldots, z_n)  = \\frac{\\mbox{perm}(A)(\\det(A_0))^2 }{\\pi^n \\det(A)}.\n     \\end{eqnarray}\nNow elementary operations on the matrix $A$ yields\n      \\begin{eqnarray*}\n\\mbox{perm}(A) & = &  \\mbox{perm}(A_0) \\prod_{k=1}^n \\left(\\frac{1}{|1-z_k|^2}\\right)\\\\\n\\det(A) & = &  \\det(A_0) \\prod_{k=1}^n \\left(\\frac{1}{|1-z_k|^2}\\right).\n\\end{eqnarray*}\nHence (\\ref{dsdefesdws}) yields\n            \\begin{eqnarray*} \n    p_0(z_1, z_2, \\ldots, z_n)  = \\frac{\\mbox{perm}(A_0)\\det(A_0) }{\\pi^n}\n     \\end{eqnarray*}\nand it is proven in Peres and Vir\\'ag \\cite[rel. (27)]{Peres_Virag} that \n  $$\\mbox{perm}(A_0)\\det(A_0)  = \\det\\left(\\frac{1}{(1 - z_k \\overline{z_j})^2}\\right)_{k,j=1}^n.$$\nThis concludes the proof. \\hfill \\qed\n\n\\subsection{Proof of Lemma \\ref{lemmaSM}.} Peres and Virag's proof is based on the invariance property of the Szeg\\\"o kernel with respect to Mobius transformations that are conformal mappings. This property does not hold for the general kernel $\\mathbb{K}_G$. Our proof is more general. \n (a) In the case where $w = 0$, it is an immediate consequence of Corollary \\ref{corol1}. Indeed, \n\\begin{eqnarray*}\n(f(z)| f(w) = 0)  & = & (f(z)|f(0) = 0) = (f(z)|\\xi_1=0) = \\left(\\sum_{k=2}^\\infty \\xi_k z^{k-1}|\\xi_1 = 0\\right)\\\\\n& = & z \\left(\\sum_{k=2}^\\infty \\xi_k z^{k-2}|\\xi_1 = 0\\right)  =  z \\left(\\sum_{k=1}^\\infty \\xi_{k+1} z^{k-1}|\\xi_1 = 0\\right)\\\\\n&\\stackrel{d}{=} & z \\sum_{k=1}^\\infty \\xi_{k} z^{k-1} \n\\end{eqnarray*}\nwhere the equality in distribution follows from Corollary \\ref{corol1}.\\\\\n(b)  For a general $w \\in \\mathbb{D}$, set \n            $$F(z) = (f(z)|f(w)  = 0), \\,z \\in \\mathbb{D}.$$ \nClearly, the covariance kernel of the random function $F(z)$ is given by\n   \\begin{eqnarray*}\n\\mathbb{E}(F(z) \\overline{F(y)}) & = & \\mathbb{E}\\left(f(z) \\overline{f(y)}\\right) - \\frac{\\mathbb{E}\\left(f(z) \\overline{f(w)}\\right) \\mathbb{E}\\left(f(w) \\overline{f(y)}\\right)}{\\mathbb{E}\\left(f(w) \\overline{f(w)}\\right)}\\\\\n&  = & \\mathbb{K}_G(z, y)-\\frac{\\mathbb{K}_G(z, w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)}\n\\end{eqnarray*}\nwhere $z, y \\in \\mathbb{D}$. This is clearly a kernel function and we shall denote it by $\\mathscr{K}_1$, that is:\n  $$\\mathscr{K}_1(z, y) = \\mathbb{K}_G(z, y)-\\frac{\\mathbb{K}_G(z, w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)},\\,\\,z, y\\in \\mathbb{D}.$$\nWe  also denote by $\\mathscr{K}_2$ the covariance kernel of the random function $T_w(z) f(z)$, that is,\n\\begin{eqnarray*}\n\\mathscr{K}_2(z, y) &  = & \\mathbb{E}\\left(T_w(z) f(z) \\overline{T_w(y) f(y)}\\right)\\\\\n                    & =  & T_w(z) \\mathbb{K}_G(z, y)  \\overline{T_w(y)}.\n\\end{eqnarray*}\nTo show that the (Gaussian) random functions $F(z)$ and  $T_w(z) f(z)$ have the same distribution, it is sufficient to show that their  covariance kernels $\\mathscr{K}_1$ and $\\mathscr{K}_2$ are identical. So we shall prove the following important identity: For all $w, z, y \\in \\mathbb{D}$,\n     $$\\mathbb{K}_G(z, y)-\\frac{\\mathbb{K}_G(z, w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)}  = T_w(z) \\mathbb{K}_G(z, y)  \\overline{T_w(y)}.$$      \nWe shall prove that  $\\mathscr{K}_1$ and $\\mathscr{K}_2$ are both reproducing kernels of a certain Hilbert space, namely the subspace $\\mathscr{H}$ of $H^2_G(\\mathbb{D})$ of the functions that vanish at the point $w$. That is, \n          $$\\mathscr{H} = \\{g \\in H^2_G(\\mathbb{D}): g(w) = 0\\}.$$  \n(To see that  $\\mathscr{H}$ is closed in  $H^2_G(\\mathbb{D})$, assume that $g_k = \\sum_{n=0}^\\infty a_{n,k} z^n$ converges to $g = \\sum_{n=0}^\\infty b_n z^n$. Then as discussed earlier, this convergence also holds with respect to the $H^2(\\mathbb{D})$-norm and hence $\\lim_{k \\to \\infty} \\sum_{n=0}^\\infty |a_{n,k} - b_n|^2 = 0$. Then \nfor each $z \\in \\mathbb{D}$ (by the Cauchy-Schwarz inequality)\n $$\\lim_{k \\to \\infty} |g_{k}(z) - g(z)|^2 = \\lim_{k \\to \\infty} |\\sum_{n=0}^\\infty (a_{n,k} - b_n) z^k|^2 \\leq \\lim_{k \\to \\infty} \\sum_{n=0}^\\infty |a_{n,k} - b_n|^2 \\sum_{n=0}^\\infty |z|^{2n} = 0.$$ In particular since $g_{k}(w) = 0$ for all $n$, then $g(w) = 0$ and hence $g \\in \\mathscr{H}$.)\n \nTo prove the claim that $\\mathscr{K}_1$ and $\\mathscr{K}_2$ are reproducing kernels $\\mathscr{H}$, we shall prove that for any function $g \\in \\mathscr{H}$ and any $y \\in \\mathbb{D}$,\n\\begin{eqnarray} \\label{sdsde34rfds}\ng(y) = \\langle g,  \\mathscr{K}_1(.,y)\\rangle  = \\langle g,  \\mathscr{K}_2(.,y)\\rangle\n\\end{eqnarray}\n where\n$\\mathscr{K}_1(.,y)$ is  the function defined by\n   $$\\mathscr{K}_1(.,y): \\mathbb{D} \\to \\mathbb{C},\\, z \\mapsto \\mathscr{K}_1(z,y)$$ and similarly for $\\mathscr{K}_2.$\nClearly,\n         \\begin{eqnarray*}\n\\langle g,  \\mathscr{K}_1(.,y)\\rangle  & = & \\left\\langle g,  \\mathbb{K}_G(., y)-\\frac{\\mathbb{K}_G(., w)\\mathbb{K}_G(w, y)}{\\mathbb{K}_G(w, w)}\\right \\rangle  \\\\\n& = &  \\left\\langle g,  \\mathbb{K}_G(., y)\\right\\rangle -  \\left(\\frac{\\overline{\\mathbb{K}_G(w, y)}}{\\overline{\\mathbb{K}_G(w, w)}}\\right)  \\left\\langle g, \\mathbb{K}_G(., w)\\right \\rangle\\\\\n& = & g(y) - \\left(\\frac{\\overline{\\mathbb{K}_G(w, y)}}{\\mathbb{K}_G(w, w)}\\right)\\, g(w)\\\\\n& = & g(y)  \n  \\end{eqnarray*}       \nsince $g(w) = 0$.\\\\\nFor the kernel $\\mathscr{K}_2$, we shall make use of the explicit  inner product in $H^2_G(\\mathbb{D})$ in relation (\\ref{ew23edfr1eswe}), and show that\n\\begin{eqnarray}\\label{sdsdesafa1}\ng(y) = \\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\overline{\\mathscr{K}_2(e^{2 \\pi i \\theta},y)} d\\mu(\\theta).\n\\end{eqnarray}\nClearly, by definition of the kernel $\\mathscr{K}_2$, \n\\begin{eqnarray} \\label{ewdsdaswe45}\n\\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\overline{\\mathscr{K}_2(e^{2 \\pi i \\theta},y)} d\\mu(\\theta)\n= T_w(y) \\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\overline{T_w(e^{2 \\pi i \\theta})} \\, \\overline{\\mathbb{K}_G(e^{2 \\pi i \\theta},y)} d\\mu(\\theta).\n\\end{eqnarray}\nNow note that \n  $$\\overline{T_w(e^{2 \\pi i \\theta})} =  \\frac{e^{-2 \\pi i \\theta} -\\overline{w}}{1 -  w  e^{-2 \\pi i \\theta}} = \\frac{1 -\\overline{w} e^{2 \\pi i \\theta} }{e^{2 \\pi i \\theta} -  w}.$$\n Therefore\n     \\begin{eqnarray*}\n  \\langle g,  \\mathscr{K}_2(.,y)\\rangle & = &  T_w(y)  \\int_{\\mathbb{T}} g(e^{2 \\pi i\\theta}) \\left(\\frac{1 -\\overline{w}\\  e^{2 \\pi i \\theta} }{e^{2 \\pi i \\theta} -  w}\\right) \\, \\overline{\\mathbb{K}_G(e^{2 \\pi i \\theta},y)} d\\mu(\\theta)\\\\\n      & = &  T_w(y) \\left \\langle g\\ p, \\mathbb{K}_G(., y)\\right\\rangle\n     \\end{eqnarray*} \nwhere $p$ is the function defined by \n $$p(z) = \\frac{1 -\\overline{w}  z }{z -  w} = \\frac{1}{T_w(z)},\\,\\,\\,z \\in \\mathbb{D}.$$ \n \nSince $g \\in H^2_G(\\mathbb{D})$, then the product $g p$ is also in $H^2_G(\\mathbb{D})$. Indeed,\n  $$\\int_{\\mathbb{T}} |g(e^{2\\pi i \\theta})|^2 |p(e^{2\\pi i \\theta})|^2 d\\mu(\\theta) \\leq C \\int_{\\mathbb{T}} |g(e^{2\\pi i \\theta})|^2  d\\mu(\\theta) < \\infty$$ where\n  $$C = \\sup_{\\theta \\in \\mathbb{T}} \\left|p(e^{2 \\pi i \\theta})\\right|^2 =  \\sup_{\\theta \\in \\mathbb{T}} \\left|\\frac{1 -\\overline{w}  e^{2\\pi i \\theta }}{e^{2\\pi i \\theta} -  w}\\right|^2 < \\infty$$\n(because $w \\in \\mathbb{D}$ and hence $e^{2\\pi i \\theta} -  w \\ne 0$). \nIt follows that  \n $$\\left \\langle g p, \\mathbb{K}_G(., y)\\right\\rangle = g(y) p(y) = g(y) \\left(T_w(y)\\right)^{-1}.$$\n Hence\n       $$ \\langle g,  \\mathscr{K}_2(.,y)\\rangle =  T_w(y) g(y) \\left(T_w(y)\\right)^{-1} = g(y).$$ \nThis yields (\\ref{sdsde34rfds}) and concludes the proof of \n           $$F(z) = (f(z)| f(w) = 0) \\stackrel{d}{=} T_w(z) f(z).$$ \nNow the general case that\n$$(f(z)| f(w_1)=0, f(w_2)=0, \\ldots, f(w_n) = 0) \\stackrel{d}{=} T_{w_1}(z) T_{w_2}(z) \\ldots T_{w_n}(z) f(z)$$\n             follows immediately by an induction argument.   This concludes the proof of  Lemma \\ref{lemmaSM}. \\hfill \\qed  \n \n\n\n\\section{Illustrating examples}\n\n\n\n\\subsection{Explicit inverse of tridiagonal Toeplitz matrices}\\label{exampl1}\nGiven a real number $q$ such that $|q|< 1\/2$, consider the Toeplitz matrix\n  $$G = \\left(\\gamma(k-j)\\right)_{k,j=1}^\\infty$$ where \n  \\begin{eqnarray*}\n         \\gamma(k) = \\left \\{ \\begin{array}{cc}\n                                               1 &  \\mbox{ if } k=0 \\\\\n                                                q &  \\mbox{ if } |k| = 1\\\\\n                                                0 & \\mbox{ otherwise. }\n                                               \\end{array}\n                     \\right.     \n \\end{eqnarray*}\nThe spectral density function $\\varphi$ of $G$ (i.e. the density of the corresponding measure $\\mu$) is given by\n$$\\varphi(\\theta) = 1 + q e^{2 \\pi i \\theta} + q e^{-2 \\pi i \\theta} = 1 + 2 q \\cos(2\\pi \\theta),\\,\\, \\theta \\in \\mathbb{T}.$$\nLet $G_n$ be the submatrix of $G$ formed by its first $n$ rows and first $n$ columns. Then the inverse $G_n^{-1}$ of $G_n$ is the symmetric matrix given by (see  da Fonseca and Petronilho \\cite{da Fonseca}): \n  $$\\left(G_n^{-1}\\right)_{k, j} = (-1)^{k+j} \\frac{q^{j-k}}{|q|^{j-k+1}} \\frac{U_{k-1}(\\alpha) U_{n-j}(\\alpha)}{U_n(\\alpha)},\\,\\,\\,1\\leq k \\leq j\\leq n$$\nwhere $$\\alpha = \\frac{1}{2 |q|}$$ and $(U_k)$ is the sequence of Chebyshev polynomials of second kind given by:\n     \\begin{eqnarray*}\n      U_0 & = &  1\\\\\n      U_1(x) & = & 2 x\\\\\n      U_{k+1} (x) & = & 2 x U_k(x) - U_{k-1}(x),\\,\\,k = 1,2,\\ldots\n          \\end{eqnarray*}\nExplicitly, for $|x| > 1$, \n       $$U_k(x) = \\frac{\\left(x + \\sqrt{x^2-1}\\right)^{k+1} - \\left(x - \\sqrt{x^2-1}\\right)^{k+1}}{2 \\sqrt{x^2-1}}.$$\nTaking the limit of $G_n^{-1}$ as $n \\to \\infty$, it can be easily checked that the infinite matrix $G$ is indeed invertible and its inverse is the infinite symmetric matrix given for $k \\leq j$ by\n  \\begin{eqnarray*}\n\\left(G^{-1}\\right)_{k,j} & = &  \\lim_{n\\to \\infty} \\left(G_n^{-1}\\right)_{k,j}\\\\\n&  = & \\frac{(-2 q)^{j - k} \\left(1 + \\sqrt{1 - 4 |q|^2}\\right)^{-j} \\left(\\left(1 + \\sqrt{1 - 4 |q|^2}\\right)^k - \\left(1 - \\sqrt{1 - 4 |q|^2}\\right)^k\\right)}{\\sqrt{1 - 4|q|^2}}. \n   \\end{eqnarray*}\nFor example, if $q = -\\frac{1}{3}$, the inverse of the infinite matrix $G$  is given by \n $$\\left(G^{-1}\\right)_{k, j} = \\left(\\frac{3}{\\sqrt{5}}\\right) \\left(\\frac{3 - \\sqrt{5}}{2}\\right)^j \\left(\\left(\\frac{3 + \\sqrt{5}}{2}\\right)^\n   k-\\left(\\frac{3 - \\sqrt{5}}{2}\\right)^{k}\\right),\\,\\,\\mbox{ for } j \\geq k.$$\nNow some involved (but elementary) calculations yield that the kernel function $\\mathbb{K}_G$ defined by the matrix $G$ is given by\n       \\begin{eqnarray}\\label{sdwwe34r}\n\\mathbb{K}_G(z, w) = Z^T G^{-1} \\overline{W} = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}}\n\\end{eqnarray}\n where  $\\psi$ is the function defined in the unit disc $\\mathbb{D}$ \nby\n  $$\\psi(z) =\\left(\\frac{2}{|q|}\\right)^{1\/2} \\left(\\frac{1}{a + b z}\\right)$$ with\n \\begin{eqnarray*}\n a & = & \\sqrt{|q|^{-1} + \\sqrt{q^{-2} - 4}} \\\\\n b & = & (2\/a)\\sign(q).\n \\end{eqnarray*}\nFor example for $q = -1\/3$,       \n$$\\psi(z) = \\frac{5^{1\/4}(3 + \\sqrt{5}) \\left(\\frac{3}{2(5 + 3\\sqrt{5})}\\right)^{1\/2}}{z - \\frac{3+ \\sqrt{5}}{2}}.$$\nNote that relation (\\ref{sdwwe34r}) means that\n        $$\\mathbb{K}_G(z, w) = \\psi(z) \\mathbb{K}(z, w) \\overline{\\psi(w)}$$ where $\\mathbb{K}$ is the classical Szeg\\\"o kernel (which is the covariance kernel associated to $\\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ for i.i.d $(\\zeta_n)$).      \nThen for a sequence $(\\xi_n)$ of Gaussian random variables with covariance matrix $G^{-1}$, this implies that  the random functions $\\sum_{n=1}^\\infty \\xi_n z^{n-1}$  and  $\\psi(z) \\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ have the same distribution, that is,\n $$\\sum_{n=1}^\\infty \\xi_n z^{n-1}  \\stackrel{d}{=}  \\psi(z) \\sum_{n=1}^\\infty \\zeta_k z^{k-1}$$\n (for i.i.d $(\\zeta_n)$). Since clearly $\\psi(z) \\ne 0$ everywhere in $\\mathbb{D}$, it follows that the zeros of $\\sum_{n=1}^\\infty \\xi_n z^{n-1}$ have the same distribution as the zeros of $\\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ and therefore they constitute a determinantal point process as predicted by Theorem 1. \n\nIn particular the intensity of the zeros of $f(z)$ is also given \n$$p(z) = \\frac{1}{\\pi(1 - |z|^2)^2},\\,\\,z \\in \\mathbb{D}$$ as it is the case for i.i.d variables.\nTo emphasize that we need to consider the $G^{-1}$ as the covariance matrix instead of $G$, consider a sequence of Gaussian variables $(\\tau_n)$ with covariance matrix $G$ and the function \n$g(z) = \\sum_{n=1}^\\infty \\tau_n z^n$. Using relation (\\ref{eqnads231}), it is easy to derive that the intensity of the corresponding zero set is\n  $$p(z) = \\frac{1}{\\pi(1 - |z|^2)^2} \\left(1 - \\frac{q^2(1 - |z|^2)^2}{(1 + q z + q \\overline{z})^2}\\right),\\,\\,\\,\\, z \\in \\mathbb{D}.$$\nHence clearly the zeros of  $g(z)$ do not have the same distribution as the zeros of  $f(z) = \\sum_{n=1}^\\infty \\zeta_n z^{n-1}$ for i.i.d $(\\zeta_n)$ since the corresponding intensity is $p(z) = \\pi^{-1}(1 - |z|^2)^{-2}$.  \n\nFinally since the function $\\varphi(\\theta) = 1 + 2 q \\cos(2 \\pi \\theta)$ is bounded on $\\mathbb{T}$, then the set \n$H^2_G(\\mathbb{D})$ is equal to $H^2(\\mathbb{D})$ but with a different norm: \n    $$\\|g\\|^2_{H^2_G(\\mathbb{D})} = \\int_{\\mathbb{T}} |g(\\theta)|^2 \\varphi(\\theta) d\\theta = \\int_{\\mathbb{T}} |g(\\theta)|^2 (1 + 2 q \\cos(2 \\pi \\theta))  d\\theta.$$\nTherefore with this norm, $H^2(\\mathbb{D})$ is the reproducing kernel Hilbert space given by the kernel\n          $$K_G(z, w) = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}} $$ where the function $\\psi(z)$ is given above. \n\n\n\n\\subsection{Inverse of the Kac-Murdock-Szeg\\\"o matrix}\nThe same property is also observed for  the classical Kac-Murdock-Szeg\\\"o matrix.  It is the Toeplitz matrix $G$ defined for a complex number $q$ by \n        $$ (G)_{k,j} =   q^{|k-j|}.$$ \nFor $q$ real with $|q| < 1$, the spectral density function $\\varphi$ of $G$ is given by\n       $$\\varphi(\\theta) = \\sum_{n=-\\infty}^\\infty q^{|n|} e^{2\\pi i n \\theta} = \\frac{1 - q^2}{1 - 2q \\cos(2 \\pi \\theta) + q^2}. $$\n(Here $\\varphi(\\theta)$ can be seen as the classical Poisson kernel in the unit disc.)\n Under these conditions ($q$ real and $|q|< 1$), $G$ is symmetrical and invertible and its inverse is well-known and given by:\n    \\begin{eqnarray*}\n\\left(G^{-1}\\right)_{k,j}  = \\left\\{ \\begin{array} {ccc}\n            (1-q^2)^{-1} & \\mbox{ if } k=j=1\\\\\n (1+q^2)(1-q^2)^{-1} & \\mbox{ if } k=j\\geq 2\\\\\n -q (1-q^2)^{-1} & \\mbox{ if } |k-j| = 1\\\\\n 0 & \\mbox{ otherwise.} \n \\end{array}\n \\right.\n \\end{eqnarray*}\n(More details on the spectral properties of Kac-Murdock-Szeg\\\"o matrix for a complex parameter are given in Fikioris \\cite{Fikioris}.)\n Then clearly the corresponding kernel is \n$$\\mathbb{K}_G(z, w) = \\frac{(1- q z)(1 - q \\overline{w})}{(1-q^2)(1 - z \\overline{w})} = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}} $$ \n  where $$\\psi(z) = \\frac{1- q z}{\\sqrt{1-q^2}}.$$  \nIn general for $q \\in \\mathbb{C}$ with $|q| < 1$, let\n    \\begin{eqnarray*}\nG_{k,j}  = \\left\\{ \\begin{array} {ccc}\n            q^{|k-j|} \\mbox{ if } k \\geq j \\\\\n            (\\overline{q})^{|k-j|} \\mbox{ otherwise. }\n \\end{array}\n \\right.\n \\end{eqnarray*}  \nThen $G$ is hermitian and invertible and its inverse is given by\n      \\begin{eqnarray*}\n\\left(G^{-1}\\right)_{k,j}  = \\left\\{ \\begin{array} {ccc}\n            (1-|q|^2)^{-1} & \\mbox{ if } k=j=1\\\\\n (1+|q|^2)(1-|q|^2)^{-1} & \\mbox{ if } k=j\\geq 2\\\\\n -q (1-|q|^2)^{-1} & \\mbox{ if } k-j = 1\\\\\n -\\overline{q}(1-|q|^2)^{-1} & \\mbox{ if } k - j = -1\\\\\n 0 & \\mbox{ otherwise.} \n \\end{array}\n \\right.\n \\end{eqnarray*}\nThis yields \n $$\\mathbb{K}_G(z, w) = Z G^{-1} \\overline{W} = \\frac{(1- q z)(1 - \\overline{q} \\overline{w})}{(1-|q|^2)(1 - z \\overline{w})} = \\frac{\\psi(z) \\overline{\\psi(w)}}{1 - z \\overline{w}} $$ \n  where $$\\psi(z) = \\frac{1- q z}{\\sqrt{1-|q|^2}}.$$ \n\\subsection{Inverse fractional Gaussian noise} \nGiven $0 < h < 1$, the classical complex fractional Gaussian noise of Hurst index $h$  is a sequence $\\{\\Delta_n\\}_{n=1}^\\infty$ of centred Gaussian random variables such that $\\mathbb{E}(\\Delta_n \\Delta_m) = 0$ and with covariance structure\n                \\begin{eqnarray} \\label{ew32wswaws2121}\n\\gamma(k) := \\mathbb{E}(\\Delta_n \\overline{\\Delta_{n+k}}) = {\\scriptstyle\\frac{1}{2}}|k+1|^{2h} + {\\scriptstyle\\frac{1}{2}}|k-1|^{2h} -|k|^{2h},\\,\\,\\,\\, k, n\\in \\mathbb{N}.\n\\end{eqnarray}\nThe covariance matrix of $\\{\\Delta_n\\}_{n=0}^\\infty$ is the Toeplitz matrix $G$ given by\n     $$G = \\left(\\gamma(k-j)\\right)_{k,j=1}^\\infty.$$ \n(The particular case $h = 1\/2$ corresponds i.i.d random variables.)\nIt is well-known that the matrix $G$ is invertible (see for example \\cite{Dambrogi-ola}.) Unfortunately an explicit inverse of $G$ is not known. Consider its inverse matrix $G^{-1}$. A sequence of Gaussian random variables with covariance matrix $G^{-1}$ shall be called the inverse fractional Gaussian noise of index $h$.  \nSome properties of the zeros of the random polynomial $\\sum_{k=0}^n \\Delta_k x^k$ and the power series $\\sum_{k=0}^{\\infty} \\Delta_n x^n$ where $(\\Delta_n)$ is the fractional Gaussian noise are given in \\cite{Mukeru_2018} and \\cite{Mukeru_al}. Here we are interested in the function $f(z) = \\sum_{n=1}^\\infty \\xi_n z^{n-1}$ where $(\\xi_n)$ is the inverse fractional Gaussian noise. \nIt is well-known (using an argument by Sinai \\cite[Theorem 2.1]{Sinai}) that the matrix $G$ admits a spectral density function $\\varphi_h$ given by\n   \\begin{eqnarray} \\label{dsdfesw23}\n\\varphi_h(\\theta) = C(h)|e^{2 \\pi i \\theta}-1|^2\\left(\\sum_{n=-\\infty}^\\infty \\frac{1}{|\\theta +n|^{2h+1}}\\right),\\, \\theta \\in \\mathbb{T}, \\theta \\ne 0\n\\end{eqnarray}\n       where $C(h)$ is a normalising constant given by\n        $$C(h) = -\\frac{\\zeta(-2h)}{2 \\zeta(1+2h)}$$ where $\\zeta(.)$ is the Riemann zeta function. \nClearly, \n\\begin{eqnarray} \\label{dsdfesw2312s}\n\\varphi_h(\\theta) &=& 4 C(h) \\left(\\sin^2 \\pi \\theta\\right) \\sum_{n=0}^\\infty \\left(\\frac{1}{(n+\\theta)^{2h+1}} + \\frac{1}{(n+ 1-\\theta)^{2h+1}}\\right) \\nonumber\\\\\n           & = &4 C(h) \\left(\\sin^2 \\pi \\theta\\right) (\\zeta(2h+1, \\theta) + \\zeta(2h+1, 1-\\theta))\n\\end{eqnarray}\n where $\\zeta(.,.)$ is the classical Hurwitz zeta function. \n\nIt is not difficult to see that the function $\\varphi_h$ is continuous on $(0, 1)$ and satisfies \n    \\begin{eqnarray*} \\label{sddsdwdwwsas}\n\\varphi_h(t)  = O(t^{1-2h} (1-t)^{1-2h}),\\,\\, \\mbox{ for } t \\mbox{ near } 0 \\mbox{ or } 1.\n\\end{eqnarray*}\nThis implies that both functions $\\varphi(t)$ and $1\/\\varphi(t)$ are integrable on the unit circle. The inverse matrix $G^{-1}$ is therefore such that \n     $$\\left(G^{-1}\\right)_{k,j} =  \\int_{\\mathbb{T}} \\frac{e^{-2\\pi i (k-j) t}}{\\varphi_h(t)} dt,\\,\\,\\mbox{ for } j+ k \\to \\infty.$$\n(See D'Ambrogi-Ola \\cite{Dambrogi-ola}.)\nThis yields that $G^{-1}$ is asymptotically a Toeplitz matrix in the sense that for each $k,j$ fixed,\n         $$\\lim_{n\\to \\infty} \\left(G^{-1}\\right)_{k+n,j+n} = \\int_{\\mathbb{T}} \\frac{e^{-2\\pi i (k-j) t}}{\\varphi_h(t)} dt = \\widehat{\\left(1\/\\varphi_h\\right)} (k-j).$$\nThis implies in particular that $\\left(G^{-1}\\right)_{k,j} \\to 0$ for $k+j \\to \\infty$ and hence \n$\\sup_{k,j}|(G^{-1})_{k,j}| < \\infty$ which guarantees that \nfor each $z, w \\in \\mathbb{D}$ the series $Z^{T} G^{-1} \\overline{W}$ converges (for $Z = (1, z, z^2, \\ldots)$  and $W = (1, w, w^2, \\ldots))$. \nThe corresponding space $H^2_G(\\mathbb{D})$ is the class of functions $ g \\in H^2(\\mathbb{D})$ such that\n  $$\\int_{\\mathbb{T}} \\left|g(e^{2 \\pi i \\theta})\\right|^2 \\varphi_h(\\theta) d\\theta < \\infty.$$ \nFor $h$ varying in $(0,1)$, this yields a family of sub-spaces of the Hardy space $H^2(\\mathbb{D})$. \n The exact entries of the matrix $G^{-1}$ are not known and therefore we do not have an explicit representation  of the kernel $\\mathbb{K}_G(z, w)$ as  in the first two examples. However Theorem \\ref{mainth} yields that if $(\\xi_n)_{n\\in \\mathbb{N}}$ is a zero-mean complex Gaussian sequence of covariance matrix $G^{-1}$ and zero pseudo-covariance, then the zeros of $f(z) = \\sum_{n =1}^\\infty \\xi_n z^{n-1}$ constitute a determinantal point process. \n \nAs in the general case one can compute from the sequence of polynomials $1, z, z^2, \\ldots$ a sequence of  orthonormal polynomials $\\{P_n(z): n=1,2,\\ldots\\}$ and deduce that if $(\\chi_n)$ is a sequence of i.i.d standard Gaussian random variables then the zeros of   $f(z) = \\sum_{n=1}^\\infty \\chi_n P_n(z)$ constitute a determinantal point process. \n      \nIn the limit case where $h = 0$, the fractional Gaussian noise with index $h$ is such that the covariance matrix $G$ is given by \n      $$G_{k,k} = 1, G_{k,k+1} = G_{k+1, k} = -1\/2 \\mbox{ and } G_{k,j} = 0 \\mbox{ for } |k-j| \\geq 2,$$ and  it is not invertible. However it still determines a determinantal point process.  \nThe spectral density function of $G$ is \n           $$\\varphi_0(\\theta) = 1 - \\cos(2\\pi \\theta),\\,\\, 0 \\in \\mathbb{T}.$$\n From the sequence of polynomials  $(1, z, z^2, z^3, \\ldots)$, we derive the orthonormal sequence: \n  \\begin{eqnarray*}\n  P_n(z) = \\left(\\frac{2}{n(n+1)}\\right)^{1\/2}(1 + 2 z + 3 z^2 + \\ldots+ n z^{n-1}),\\,\\,\\,n = 1,2,\\ldots\n \\end{eqnarray*}\nIn this case, the kernel of $\\mathbb{H}^2_G(\\mathbb{D})$ is explicitly given by:\n\\begin{eqnarray*} \\label{Saf_Kernel}\n\\mathbb{K}_0(z, w) = \\sum_{n=1}^\\infty P_n(z) \\overline{P_n(w)} = \\frac{2}{(1-z)(1-\\overline{w})(1- z \\overline{w})},\\,\\,\\,z, w \\in \\mathbb{D}.\n \\end{eqnarray*}\n This is exactly the limit case of the kernel given in Example \\ref{exampl1} when the parameter $q$ approaches $ -1\/2$. \nThen for a sequence $(\\chi_n)$ of i.i.d standard Gaussian variables, the zeros of \n   \\begin{eqnarray*}\n   f(z) = \\sum_{n=1}^\\infty \\chi_n P_n(z),\\,\\,\\,\\, z \\in \\mathbb{D}\n   \\end{eqnarray*}\nconstitute a determinantal point process. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\n\\noindent Spontaneous symmetry breaking is usually treated as a\nphenomenon that arises from properties of an asymmetric quantum mechanical vacuum\nstate. In particular, the non-invariance of the vacuum\nstate with respect to a symmetry is said to lead to spontaneous\nsymmetry breakdown. While this picture is clearly valid and useful, it\nis not generally appreciated that in relativistic theories of quantum mechanics, \nit is strictly a matter of convention which\narises from the (usually implicit) choice of quantization\nsurface~\\cite{Dirac:1949cp}. Indeed, the standard viewpoint ---the\ninstant form--- arises from choosing to view dynamics in Minkowski\nspace as the evolution of families of parallel spaces at various\ninstants of time. An alternate view of dynamics is to consider the\nevolution of families of parallel spaces tangent to the light cone;\ni.e.~null planes~\\cite{Dirac:1949cp,Weinberg:1966jm,Susskind:1967rg,Bardakci:1969dv,Kogut:1969xa,Leutwyler:1977vy}. In this\nviewpoint ---the front form--- the momentum operator has a spectrum\nconfined to the open positive half-line and therefore the vacuum of\nthe interacting theory may be regarded as the structureless Fock-space\nvacuum, which is an invariant with respect to all internal symmetries,\nand spontaneous symmetry breaking must be attributed to properties of\nthe dynamical Poincar\\'e generators. Therefore in the front form,\nspontaneous chiral symmetry breaking is a property of operators rather\nthan of a complicated vacuum state.  Naturally one expects that\nphysics is independent of the choice of quantization surface. However,\nfor theories like QCD where the detailed dynamics are largely\nintractable, one may suppose that the two forms of dynamics lead to\ndistinct insights into the behavior of the theory at strong coupling.\nOur goal in this paper is to argue that this is indeed the case.\n\nThe fundamental point which we wish to emphasize in this paper is\nthat, in contrast to the instant form, where spontaneous symmetry\nbreaking lives entirely in the non-trivial vacuum, in the front form,\nsymmetry breaking is expressed entirely through the fact that the\nHamiltonians, or dynamical Poincar\\'e generators, do not commute with\nthe internal symmetry charges. The resulting commutation relations\namong space-time generators and internal symmetry generators in QCD\nimply powerful constraints on the spectrum and spin of the hadronic\nworld~\\cite{Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn,Weinberg:1994tu}.\nThere have been many studies of spontaneous chiral symmetry breaking\non null\nplanes~\\cite{Jersak:1969zg,Leutwyler:1969av,Weinberg:1969hw,Weinberg:1969db,Feinberg:1973qb,Eichten:1973ip,Casher:1973vh,Casher:1974xd,Carlitz:1974sg,Sazdjian:1974gk,Wilson:1994fk,Wilson:1994gn,Susskind:1994wr,Kim:1994rm,Burkardt:1995eb,Burkardt:1996pa,Burkardt:1998dd,Yamawaki:1998cy,Itakura:2001yt,Burkardt:2002yf,Wu:2003vn,Lenz:2004tw,Dalley:2004re,Brodsky:2008xm,Brodsky:2008xu,Brodsky:2009zd,Ji:2009jc,Strikman:2010pu,Chang:2011mu,Alberg:2012wr,Brodsky:2010xf,Schweitzer:2012hh,Brodsky:2012ku}.\nIn many cases the emphasis has been on learning detailed information\nabout the dynamical mechanism of chiral symmetry breaking in QCD and\nin models. Here our approach is much less ambitious; we assume that\nchiral symmetry is broken spontaneously by complicated and not-well\nunderstood dynamics, and we then determine the constraints that follow\nfrom this assumption. In particular, we are interested primarily in\nformulating the model-independent consequences of chiral symmetry\nbreaking on null-planes. A fundamental assumption we make is that\nphysics must be independent of the choice of quantization\nsurface. Nowhere in this study do we find anything resembling a\ncontradiction of this basic assumption. Indeed, this assumption of\nwhat one might call ``form invariance'' leads to various constraints\nwhich reveal a great deal about the nature and consistency of chiral\nsymmetry breaking on null-planes. On general grounds, the null-plane\nchiral symmetry charges annihilate the vacuum. Therefore, in order\nthat spontaneous chiral symmetry breaking take place, the chiral\nsymmetry axial-vector current on the null-plane cannot be\nconserved~\\cite{Jersak:1969zg,Leutwyler:1969av}.  This property leads\nto a simple proof of Goldstone's theorem on a null-plane, which is\ncompletely decoupled from any assumptions about the formation of\nsymmetry-breaking condensates. A second consistency condition is that\nthe part of the QCD vacuum energy that is dependent on the quark\nmasses should be invariant with respect to the choice of\ncoordinates. This condition recovers the Gell-Mann-Oakes-Renner\nrelation~\\cite{GellMann:1968rz} in the null-plane description, and\nleads to a general prescription for relating all instant-form\nchiral-symmetry breaking condensates to the vacuum expectation values\nof chiral singlet null-plane QCD operators.\n\nIt is difficult to find a general solution of the null-plane operator\nalgebra~\\cite{Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn,Weinberg:1994tu}. However,\nthere is a non-trivial limit in which a solution can be found. One\nexpects that, in general, the chiral symmetry breaking part of the\nnull-plane energy has an energy scale comparable to $\\Lambda_{QCD}$\nand therefore is not parametrically small. However, assuming that this\nis small (which is the case parametrically for baryon operators at\nlarge-$N_c$), while the chiral symmetry breaking part of the spin\nHamiltonians is of natural size, allows a non-trivial solution of the\noperator algebra which closes to the Lie brackets of $SU(2N)$, thus\nrecovering the basic group theoretical structure of the constituent\nquark model. This result, originally found by\nWeinberg~\\cite{Weinberg:1994tu} working with current-algebra sum rules\nin special Lorentz frames, is shown in this context to be a general\nconsequence of the null-plane QCD Lie algebraic constraints which are\nvalid in any Lorentz frame.\n\nThe paper is organized as follows.  In section~\\ref{sec:poincare}, the\nnull-plane coordinates and conventions are introduced, and the\nfront-form Poincar\\'e algebra is obtained. The null-plane Hamiltonians\nand the Lie brackets that they satisfy are identified, and the\nmomentum eigenstates are constructed.  In\nsection~\\ref{sec:chiralityintro} the null-plane internal symmetry\ncharges are introduced, and the commutators that mix Poincar\\'e and\nchiral generators are obtained. Using these commutators, a general\nproof of Goldstone's theorem is given, and a polology analysis is\ngiven which elucidates the structure of the axial-vector current on\nthe null-plane.  The special case of QCD with $N$ flavors of light\nquarks is considered in section~\\ref{sec:qcd}. The QCD Lagrangian is\nexpressed in the null-plane coordinates, and the chiral symmetry\nbreaking Hamiltonians and the constraints that they satisfy are\nderived.  The issue of condensates in the null-plane formulation is\naddressed in detail; the Gell-Mann-Oakes-Renner formula is recovered\nin the front-form and a general method for relating instant-form\ncondensates to front-form condensates is presented.\nSection~\\ref{sec:consequences} explores the consequences of the QCD\nnull-plane operator algebra. In particular, a simple solution of the\noperator algebra is given which contains the spin-flavor symmetries of\nthe constituent quark model.  In section~\\ref{sec:conc} we summarize\nour findings and conclude.  \\vskip0.3in\n\\noindent {\\it Nota bene}: \nWe have made use of the many general\nreviews of null-plane (or light-front)\nquantization~\\cite{Hornbostel:1990ya,Perry:1994kp,Zhang:1994ti,Burkardt:1995ct,Harindranath:1996hq,Brodsky:1997de,Perry:1997uv,Miller:1997cr,Venugopalan:1998zd,Heinzl:2000ht,Miller:2000kv,Diehl:2003ny,Belitsky:2005qn}, as well as reviews that focus\nprimarily on chiral symmetry related\nissues~\\cite{Mustaki:1994mf,Yamawaki:1998cy,Itakura:2001yt}.\nIn order to provide a self-contained description of the subject\nof chiral-symmetry breaking on a null-plane, there is a significant \namount of review material in this paper. \n\n\n\n\\section{Space-time symmetry in the front form}\n\\label{sec:poincare}\n\n\\subsection{A null plane defined}\n\n\\noindent In the front-form of relativistic Hamiltonian dynamics, one\nchooses the initial state of the system to be on a light-like plane,\nor null-plane, which is a hypersurface of points $x$ in Minkowski\nspace such that $x\\cdot n =\\tau$ (see fig.~\\ref{fig:nullplane}). Here\n$n$ is a light-like vector which will be chosen below, and $\\tau$ is a\nconstant which plays the role of time. We will refer to a null-plane\nas $\\Sigma_n^\\tau$.  The subgroup of the Poincar\\'e group that maps\n$\\Sigma_n^\\tau$ to itself is called the stability group of the\nnull-plane and determines the kinematics within the null-plane. The\nremaining three Poincar\\'e generators map $\\Sigma_n^\\tau$ to a new\nsurface, $\\Sigma_n^{\\tau'}$, and therefore describe the evolution of the\nsystem in time. The front-form is special in that it has seven\nkinematical generators, the largest stability group of all of the\nforms of dynamics~\\cite{Dirac:1949cp}. It stands to reason that in\ncomplicated problems in relativistic quantum mechanics one would\nprefer a formulation which has the fewest number of Hamiltonians to\ndetermine.\n\\begin{figure}[!t]\n  \\centering\n     \\includegraphics[scale=0.54]{nullplane2.pdf}\n     \\caption{A null plane is a surface tangent to the light cone. The\n       null-plane Hamiltonians map the initial light-like surface onto\n       some other surface and therefore describe the dynamical\n       evolution of the system. The energy $P^-$ translates the system\n       in the null-plane time coordinate $x^+$, whereas the spin\n       Hamiltonians $F_r$ rotate the initial surface about the surface\n       of the light cone.}\n  \\label{fig:nullplane}\n\\end{figure}\n\\subsection{Choice of coordinates}\n\n\\noindent Consider the light-like vectors $n^\\mu$ and $n^{\\ast\\mu}$\nwhich satisfy $n^2 = n^{\\ast 2} = 0$ and $n \\cdot n^\\ast = 1$. Here we will\nchoose these vectors such that \n\\begin{equation}\nn^\\mu \\equiv \\ft{1}{\\sqrt{2}} (1, 0, 0, -1) \\quad , \\quad\nn^{\\ast \\mu} \\equiv \\ft{1}{\\sqrt{2}} (1, 0, 0, 1)\n\\, .\n\\label{nullvectors}\n\\end{equation}\nWe will take the initial surface to be the null-plane\n$\\Sigma_n^0$. A coordinate system adapted to null-planes is then given by\n\\begin{eqnarray}\nx^+ \\equiv x \\cdot n = \\ft{1}{\\sqrt{2}} (x^0 + x^3)\n\\, , \\qquad\nx^- \\equiv x \\cdot n^\\ast = \\ft{1}{\\sqrt{2}} (x^0 - x^3)\n\\label{eq:LCcoordinates}\n\\end{eqnarray} which we take as the time variable and ``longitudinal'' position, respectively~\\footnote{This\n  is known as the Kogut-Soper convention~\\cite{Kogut:1969xa}. Our\n  metric and other notational conventions can be found in\n  Appendix~\\ref{npconventions} and in Ref.~\\cite{Brodsky:1997de}.}.  The remaining coordinates, ${\\bf\n  x}_\\perp=( {x}^1, {x}^2)$ provide the ``transverse''\nposition.  Denoting the null-plane contravariant coordinate\nfour-vector by ${\\tilde x}^\\mu=(x^+,{x}^1,{x}^2,x^-)=(x^+,{\\bf x}_\\perp,x^-)$, then\none can write\n\\begin{eqnarray}\n{\\tilde x}^\\mu\\ = \\ {\\cal C}^\\mu_\\nu\\ x^\\nu \\ .\n\\label{eq:ETtoLC}\n\\end{eqnarray} The matrix ${\\cal C}^\\mu_\\nu$, given explicitly in\nAppendix~\\ref{npconventions}, allows one to transform all Lorentz\ntensors from instant-form to front-form coordinates. In particular,\nthe null-plane metric tensor is given by\n\\begin{eqnarray}\n{\\tilde g}_{\\mu\\nu}\\ =\\ ({\\cal C}^{-1})^\\alpha_\\mu\\ g_{\\alpha\\beta}\\ ({\\cal C}^{-1})^\\beta_\\nu \\ .\n\\label{eq:LCmetric1}\n\\end{eqnarray}\n\nThe energy, canonical to the null-plane time variable $x^+$ is\n$p^-=p_+$ , and the momentum canonical to the longitudinal position\nvariable $x^-$ is $p^+=p_-$.  Therefore, the on-mass-shell condition\nfor a relativistic particle of mass $m$ yields the null-plane\ndispersion relation:\n\\begin{eqnarray}\np^-\\ =\\ \\frac{{\\bf p}^2_\\perp\\ +\\ m^2}{2p^+} \\ .\n\\label{eq:LCdisp}\n\\end{eqnarray} This dispersion relation reveals several interesting generic\nfeatures of the null-plane formulation. Firstly, the dispersion\nrelation resembles the non-relativistic dispersion relation of a\nparticle of mass $p^+$ in a constant potential. Secondly, we see that\nthe positivity and finiteness of the null-plane energy of a free\nmassive particle requires $p^+>0$. Only massless particles with\nstrictly vanishing momentum can have $p^+=0$. This implies that pair\nproduction is subtle, and the vacuum state is in some sense simple,\nwith the exception of contributions that are strictly from $p^+=0$\nmodes~\\cite{Hornbostel:1990ya,Perry:1994kp,Zhang:1994ti,Burkardt:1995ct,Harindranath:1996hq,Brodsky:1997de,Perry:1997uv,Venugopalan:1998zd,Heinzl:2000ht,\n  Diehl:2003ny,Belitsky:2005qn}.\n\n\n\\subsection{The null-plane Poincar\\'e generators}\n\n\\noindent In this section we will review the\nLie brackets of the Lorentz generators in the front form~\\footnote{Here\n  we follow closely the development of\n  Refs.~\\cite{Bardakci:1969dv,Kogut:1969xa,Leutwyler:1977vy}. See also Ref.~\\cite{Ballesteros:1995mi}.}.  The Poincar\\'e algebra\nin our convention is:\n\\begin{eqnarray}\n& & \\lbrack\\, P^\\mu \\, ,\\, P^\\nu\\, \\rbrack\\ =\\ 0 \\qquad , \\qquad \\lbrack\\, { M}_{\\mu\\nu} \\, ,\\, P_\\rho\\, \\rbrack\\ \n=\\ i\\, (\\,  g_{\\nu\\rho} P_\\mu\\, -\\, g_{\\mu\\rho} P_\\nu \\, ) \n\\nonumber \\\\\n& &\\lbrack\\, { M}_{\\mu\\nu} \\, ,\\, { M}_{\\rho\\sigma}\\, \\rbrack\\  = \n\\ i\\, (\\, g_{\\mu\\sigma} { M}_{\\nu\\rho}\\, + \\, g_{\\nu\\rho} { M}_{\\mu\\sigma}\\, - \\, g_{\\mu\\rho} { M}_{\\nu\\sigma}\\, -\n\\, g_{\\nu\\sigma} { M}_{\\mu\\rho}\\, ) \\ ,\n\\label{eq:ETpoincare}\n\\end{eqnarray} where ${ M}_{ij}=\\epsilon_{ijk}{ J}_k$ and ${\n  M}_{i0}={ K}_i$ with ${ J}_i$ and ${ K}_i$ the generators\nof rotations and boosts, respectively.  Using ${\\cal C}^\\mu_\\nu$ we\ncan transform from the instant-form to the front-form giving ${\\tilde\n  { P}}^\\mu=({ P}^+,{ P}^1,{ P}^2,{ P}^-)$, \n${\\tilde {  M}}_{r+}=-{\\tilde {  M}}_{+r}={F}_r$, \n${\\tilde {  M}}_{r-}=-{\\tilde {  M}}_{-r}={E}_r$, \n${\\tilde { M}}_{rs}=\\epsilon_{rs}{J}_3$, and\n${\\tilde { M}}_{+-}=-{\\tilde { M}}_{-+}={ K}_3$, where we have\ndefined\n\\begin{eqnarray}\n{ P}^+ &=& \\textstyle{1\\over\\sqrt{2}}\\, (\\, { P}^0\\, +\\, { P}^3\\, )\\qquad , \\qquad { P}^-\\ =\\ \\textstyle{1\\over\\sqrt{2}}\\, (\\, { P}^0\\, -\\, { P}^3\\, )\\ ;\n\\nonumber \\\\\n{ E}_r &=& \\textstyle{1\\over\\sqrt{2}}\\, (\\, { K}_r\\, + \\, \\epsilon_{rs}{ J}_s\\, )\\quad , \\quad \n\\quad{ F}_r\\ =\\ \\textstyle{1\\over\\sqrt{2}}\\, (\\, { K}_r\\, - \\, \\epsilon_{rs}{ J}_s\\, )\\ .\n\\label{eq:LCpoingens}\n\\end{eqnarray} Here ${ P}_+={ P}^-$ is the null-plane energy while ${ P}_- = { P}^+$ is the longitudinal momentum.\n(Note that the indices $r,s,t,\\ldots$ are transverse indices that range over $1,2$.\nSee Appendix~\\ref{npconventions}.)\n\nIt is straightforward to show that ${ P}^+$, ${ P}_r$, ${ K}_3$, ${ E}_r$,\nand ${ J}_3$ are kinematical generators that leave the null plane $x^+=0$\nintact. These seven generators form the stability group of the null plane. \nIt is useful to classify the subgroups of the Poincar\\'e algebra by\nconsidering the transformation properties of the generators with respect\nto longitudinal boosts, which serve to rescale the generators. Writing\n\\begin{eqnarray}\n& & \\lbrack\\, { K}_3 \\, ,\\, A\\, \\rbrack\\ =\\ -i \\gamma\\, A\n\\label{eq:goodness}\n\\end{eqnarray}\nwhere $A$ is a generator, one finds ${ E}_r$ and ${ P}^+$ have $\\gamma=1$,\n${ J}_3$, ${ K}_3$ and ${ P}_r$ have $\\gamma=0$, and ${ P}^-$ and ${ F}_r$ have $\\gamma=-1$.\nThe Poincar\\'e generators have subgroups $G_\\gamma$ labeled by $\\gamma$, and\nthere exist two seven-parameter subgroups $S_\\pm$ with a semi-direct product\nstructure $S_\\pm=G_0\\times G_\\pm$. Therefore the stability group coincides\nwith the subgroup $S_+$.\nThe non-vanishing commutation relations among these generators are:\n\\begin{eqnarray}\n& & \\lbrack\\, { K}_3 \\, ,\\, { E}_r\\, \\rbrack\\ =\\ -i { E}_r \\quad , \\quad \\lbrack\\, { K}_3 \\, ,\\, { P}^+\\, \\rbrack\\ =\\ -i { P}^+  \\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { J}_3 \\, ,\\, { E}_r\\, \\rbrack\\ =\\ i\\epsilon_{rs} { E}_s \\quad , \\quad \n\\lbrack\\, { J}_3 \\, ,\\, { P}_r\\, \\rbrack\\ =\\ i\\epsilon_{rs} { P}_s \\ ;\n\\nonumber \\\\\n& & \\qquad \\qquad \\quad\n\\lbrack\\, { E}_r  \\, ,\\,{ P}_s \\, \\rbrack\\ =\\ -i\\delta_{rs} { P}^+ \\ .\n\\label{eq:LCSGcomms}\n\\end{eqnarray}\n\nBy contrast, ${ P}^-$ and ${ F}_r$ are the Hamiltonians which consist\nof the subgroup $G_{-1}$; they are\nthe dynamical generators which move physical states away from the\n$x^+=0$ surface (see fig.~\\ref{fig:nullplane}). The non-vanishing commutators among the stability\ngroup generators and the Hamiltonians are:\n\\begin{eqnarray}\n& & \\lbrack\\, { K}_3 \\, ,\\, { P}^-\\, \\rbrack\\ =\\ i { P}^- \\quad , \\quad \\lbrack\\, { E}_r \\, ,\\, { P}^-\\, \\rbrack\\ =\\ -i { P}_r \\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { K}_3 \\, ,\\, { F}_r\\, \\rbrack\\ =\\ i { F}_r \\quad , \\quad \\lbrack\\, { J}_3 \\, ,\\, { F}_r\\, \\rbrack\\ =\\ i\\epsilon_{rs} { F}_s \\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { P}_r \\, ,\\, { F}_s\\, \\rbrack\\ =\\ i\\delta_{rs}{ P}^- \\quad , \\quad \\lbrack\\, { P}^+ \\, ,\\, { F}_r\\, \\rbrack\\ =\\ i{ P}_r\\ ;\n\\nonumber \\\\\n& & \\lbrack\\, { E}_r \\, ,\\, { F}_s\\, \\rbrack\\ =\\ -i\\left(\\,\\delta_{rs}\\,{ K}_3\\,+\\,\\epsilon_{rs} { J}_3\\,\\right)\\ .\n\\label{eq:LCHSGcomms}\n\\end{eqnarray} \nThis algebraic structure is isomorphic to the Galilean group of\ntwo-dimensional quantum mechanics where one identifies\n$\\lbrace\\,P^-,E_r,P_r,J_3,P^+\\,\\rbrace$ with the Hamiltonian, Galilean\nboosts, momentum, angular momentum, and mass, respectively. This\nisomorphism is responsible for the similarities between the front form\nand nonrelativistic quantum mechanics that we noted in the dispersion\nrelation, and was originally noted in the context of the\ninfinite momentum frame of instant-form\ndynamics~\\cite{Weinberg:1966jm,Susskind:1967rg} which has a similar\ndispersion relation.\n\n\\subsection{Null-plane momentum states and reduced Hamiltonians}\n\n\\noindent As momentum is a kinematical observable, it is convenient to work with momentum eigenstates,\nsuch that\n\\begin{eqnarray}\nP_r\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle  &=& p_r|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ ;\\\\\nP^+\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle  &=& p^+|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:momonmom}\n\\end{eqnarray}\nThe action of the boosts on momentum states follows directly from the commutation relations in eq.~\\ref{eq:LCSGcomms}\nand is given by\n\\begin{eqnarray}\ne^{-iv_r E_r}e^{-i\\omega K_3}|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, \\rangle = |\\, e^\\omega\\,p^+\\, ,\\, {\\bf p}_\\perp+p^+{\\bf v}_\\perp\\,\\rangle \\ .\n\\label{eq:boostsdefined}\n\\end{eqnarray}\nOne can then define the unitary boost operator\n\\begin{eqnarray}\n{\\cal U}(p^+,p_r)=e^{-i\\beta_r E_r}e^{-i\\beta_3 K_3} \\ ,\n\\label{eq:boostop}\n\\end{eqnarray}\nwith $\\beta_r\\equiv p_r\/p^+$ and $\\beta_3\\equiv\\log(\\sqrt{2}p^+\/M)$ \nwhich boosts the state at rest to one with arbitrary momentum:\n\\begin{eqnarray}\n{\\cal U}(p^+,p_r)|\\, M\/\\sqrt{2}\\, ,\\, {\\bf 0}\\, \\rangle = |\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:boostfromrest}\n\\end{eqnarray}\nThe action of the boosts on the momentum states is then easily found to be\n\\begin{eqnarray}\nE_r\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle  &=& ip^+\\frac{d}{dp_r}|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ ;\\\\\nK_3\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle  &=& ip^+\\frac{d}{dp^+}|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:boostonmom}\n\\end{eqnarray}\n\nUnitarity of the boost operators fixes the normalization of the momentum states up to a constant. We assume the covariant\nnormalization:\n\\begin{eqnarray}\n\\langle\\, p^{+{\\prime}}\\, ,\\, {\\bf p}_\\perp^{\\,\\prime}\\, | \\, p^+\\, ,\\, {\\bf p}_\\perp\\, \\rangle\\  =\\ \n(2\\pi)^3\\,2\\,p^+\\,\n\\delta(\\,p^{+{\\prime}}\\,-\\,p^+\\,)\\,\\delta^2(\\,{\\bf p}_\\perp^{\\,\\prime}\\,-\\,{\\bf p}_\\perp \\,) \\ ,\n\\label{eq:HBnormalization}\n\\end{eqnarray}\nand the corresponding completeness relation\n\\begin{eqnarray}\n{\\bf 1}\\, =\\, \n\\int\\,\\frac{dp^+d^2{\\bf p}_\\perp}{(2\\pi)^3 2p^+}\\,\n|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, \\rangle\\, \\langle\\, p^+\\, ,\\, {\\bf p}_\\perp\\, | \\ .\n\\label{eq:HBcomplete}\n\\end{eqnarray}\n\n\n\nWe can now find angular momentum operators, ${\\cal J}_r$ and ${\\cal J}_3$, that are\nvalid in any frame by boosting from an arbitrary momentum state to a state at rest,\nacting with the angular momentum generators $J_r=\\epsilon_{rs}(F_s-E_s)\/\\sqrt{2}$ and $J_3$, and then boosting back\nto the arbitrary momentum state. That is,\n\\begin{eqnarray}\n{\\cal J}_i|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ =\\ {\\cal U}(p^+,p_r)\\;J_i\\;{\\cal U}^{-1}(p^+,p_r)\\,|\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle \\ .\n\\label{eq:generalAM}\n\\end{eqnarray}\nUsing this procedure one finds angular momentum operators that are valid in any frame:\n\\begin{eqnarray}\n{\\cal J}_3 &=& { J}_3\\ +\\ \\epsilon_{rs}\\,{ E}_r{ P}_s\\,\\left(1\/{ P}^+\\right) \\ ; \\label{eq:npJ3generalA} \\\\\n{\\cal J}_r &=& \\epsilon_{rs}\\,\\big\\lbrack\\,{ P}^+{ F}_s\\ -\\ { P}^-{ E}_s\\ +\\ \\epsilon_{st}{ P}_t{\\cal J}_3\\ +\\ { P}_s{ K}_3\\,\\big\\rbrack\\,\\left(1\/M\\right) \\ .\n\\label{eq:npJ3generalB}\n\\end{eqnarray}\nInverting eq.~\\ref{eq:npJ3generalB} one then finds the following expressions for the Hamiltonians:\n\\begin{eqnarray}\n{ P}^- &=& \\left(1\/2{ P}^+\\right)\\,\\big\\lbrack\\, { P}_1^2\\ +\\ { P}_2^2\\ +\\ M^2 \\,\\big\\rbrack \\ ; \\nonumber \\\\\n{ F_1}\\ &=& \\left(1\/{ P}^+\\right)\\,\\big\\lbrack\\,-{ P}_1{ K}_3\\ +\\ { P}^-{ E}_1\\ -\\ { P}_2{\\cal J}_3\\ -\\ M{\\cal J}_2\\,\\big\\rbrack  \n\\ ; \\nonumber \\\\\n{ F}_2\\ &=& \\left(1\/{ P}^+\\right)\\,\\big\\lbrack\\,-{ P}_2{ K}_3\\ +\\ { P}^-{ E}_2\\ +\\ { P}_1{\\cal J}_3\\ +\\ M{\\cal J}_1\\, \\big\\rbrack \\ .\n\\label{eq:npHams}\n\\end{eqnarray} \nA striking feature of the null-plane formulation is that the\nfundamental dynamical objects are the {\\it products} $M^2$ and $M{\\cal\n  J}_{r}$, rather than the generators themselves. Following\nRef.~\\cite{Leutwyler:1977vy}, we will refer to these objects as\nreduced Hamiltonians. The reduced Hamiltonians, together with \n${\\cal J}_3$, commute with all kinematical generators and\nsatisfy the algebra of $U(2)$. This is conveniently demonstrated\nby making use of the Pauli-Lubanski vector\n\\begin{eqnarray}\n{ W}^\\mu = \\ft12 \\varepsilon^{\\mu\\nu\\rho\\sigma} { P}_\\nu { M}_{\\rho\\sigma} \\ ,\n\\end{eqnarray}\nwhich satisfies $W^\\mu P_\\mu=0$ and the non-trivial commutation relations:\n\\begin{eqnarray}\n\\lbrack\\, { M}_{\\mu\\nu} \\, ,\\, W_\\rho\\, \\rbrack & =& i\\, (\\,  g_{\\nu\\rho} W_\\mu\\, -\\, g_{\\mu\\rho} W_\\nu \\, ) \\ ;\\\\\n\\lbrack\\, W^\\mu \\, ,\\,W^\\nu\\, \\rbrack &=& -i\\varepsilon^{\\mu\\nu\\rho\\sigma} W_\\rho P_\\sigma \\ .\n\\label{eq:ETpoincareW}\n\\end{eqnarray}\nOne then finds general, compact expressions for the angular momentum operators:\n\\begin{eqnarray}\n{\\cal J}_3 \\ = \\ { W}^+\/{ P}^+ \\quad , \\quad\nM{\\cal J}_r \\ = \\ { W}_r\\ -\\ { P}_r\\;{ W}^+\/{ P}^+ \\ .\n\\end{eqnarray}\nBy considering the commutation relations among $W_\\mu$, $P^\\mu$ and $M_{\\mu\\nu}$\none confirms that\n\\begin{eqnarray}\n[\\, {\\cal J}_3 \\, ,\\,  M{\\cal J}_r  \\, ] \\ =\\   i\\,\\epsilon_{rs} M{\\cal J}_s\n\\ \\  \\ & , &\\  \\ \\ \n[\\, {\\cal J}_3 \\, ,\\,  M^2  \\, ]\\, = \\, 0 \\ ; \\nonumber \\\\\n{} [\\, M{\\cal J}_r \\, ,\\,  M{\\cal J}_s  \\, ] \\ =\\  i\\,\\epsilon_{rs} M^2 {\\cal J}_3  \n\\ \\ \\ & ,& \\ \\ \\\n[\\, M^2 \\, ,\\,  M{\\cal J}_r  \\, ]\\, = \\, 0 \\ .\n\\label{eq:dynalgarbB}\n\\end{eqnarray} \nHence, the reduced Hamiltonians together with the stability group\ngenerator ${\\cal J}_3$ satisfy the algebra of $U(2)$, and the\nproblem of finding a Lorentz invariant description of a relativistic\nquantum mechanical system is thus equivalent to finding a representation\nof the three reduced Hamiltonians which satisfy this algebra~\\footnote{Since \nthe mass operator, $M=\\sqrt{p_\\mu p^\\mu}$, commutes with the spin operators, this\nalgebra can clearly be expressed in the canonical form: \n$[\\, {\\cal J}_i \\, ,\\,  {\\cal J}_j  \\, ]\\, = \\, i\\epsilon_{ijk}J_k$ and $[\\, M \\, ,\\,  {\\cal J}_i  \\, ]\\, = \\, 0$.}.  \nSince the essence of Lorentz invariance resides in these Lie brackets, and they\nrequire knowledge of the reduced Hamiltonians, in theories with complicated\ndynamics like QCD, the formulation of the theory at weak coupling ---where QCD is\ndefined as a continuum quantum field theory--- will\nlack manifest Lorentz invariance, which is tied up with the detailed\ndynamics of the theory, and is as complicated to achieve as finding\nthe spectrum of the theory.\n\nWe can write a general momentum eigenstate as:\n\\begin{eqnarray}\n|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle = |\\, p^+\\, ,\\, {\\bf p}_\\perp\\,\\rangle\\otimes | \\lambda \\, ,\\, n\\, \\rangle \\ .\n\\label{eq:HBdefined}\n\\end{eqnarray}\nHere $n$ are additional variables that may be needed to specify the\nstate of a system at rest, and $\\lambda$ is helicity, the eigenvalue of ${\\cal J}_3$:\n\\begin{eqnarray}\n{\\cal J}_3\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle  \\ =\\ \\lambda\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ ,\n\\label{eq:J3eigen}\n\\end{eqnarray} \nand therefore, using eq.~\\ref{eq:npJ3generalA}, we have\n\\begin{eqnarray}\nJ_3\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle  \n\\ =\\ \\left(\\,\\lambda+i\\epsilon_{rs}\\,p_r\\frac{d}{dp_s}\\,\\right)\\;|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ ,\n\\label{eq:J3atrest}\n\\end{eqnarray} \nwhich completes the catalog of the action of the stability group generators on the momentum states.\nIt is useful to write\n\\begin{eqnarray}\n|\\, p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\, ,\\,n\\, \\rangle \\ =\\  {\\cal U}(p^+,p_r)|\\, M\/\\sqrt{2}\\, ,\\, {\\bf 0}\\, ; \\lambda\\, ,\\,n\\,\\rangle\n\\ \\equiv \\ a_n^\\dagger\\left(p^+\\, ,\\, {\\bf p}_\\perp\\, ; \\lambda\\right)\\;|\\, 0\\,\\rangle \\ ,\n\\label{eq:vacuumdefined}\n\\end{eqnarray}\nwhere $a_n^\\dagger$ is an operator that creates the momentum state when acting on the null-plane vacuum, $|\\, 0\\,\\rangle$.\nWhat is special about the null-plane description is that the\nkinematical generators (with the exception of ${\\cal J}_3$) act on\nstates in a manner independent of the inner variables $n$. And the\nreduced Hamiltonians act exclusively on the inner variables in a\nmanner independent of the momentum. Therefore, one may view the\nPoincar\\'e algebra by the direct sum of ${\\cal K}$ and ${\\cal D}$, where\n${\\cal K}=\\lbrace\\,E_r,P_r,K_3,P^+\\,\\rbrace$ contains all stability\ngroup generators with the exception of ${\\cal J}_3$ which is grouped\nwith the reduced Hamiltonians, ${\\cal D}=\\lbrace\\,{\\cal J}_3,M{\\cal\n  J}_r,M^2\\rbrace$~\\cite{Leutwyler:1977vz}.  \n\nThe structure of the Poincar\\'e algebra in the front-form is well\nsuited to the study of systems with complicated dynamics like QCD, as\nthe dynamical generators are directly related to the most important\nobservable quantities, namely the energy and the angular momentum of\nthe system, while momenta and boosts are purely kinematical and\ntherefore are easy to implement~\\footnote{By contrast, in the instant form\n  of dynamics, the energy and the boosts are dynamical. As boosts are\n  not among the observables, one refers only to the one Hamiltonian\n  corresponding to energy.}. The reduced Hamiltonians will have a\nfundamental role to play in the description of chiral symmetry\nbreaking on null planes.\n\n\n\n\\section{Chiral symmetry in the front form}\n\\label{sec:chiralityintro}\n\n\\subsection{Null plane charges and the chiral algebra}\n\n\\noindent Consider a Lagrangian field theory that has an\n$SU(N)_R\\otimes SU(N)_L$ chiral symmetry.  Let us assume that this\nsystem has a null-plane Lagrangian formulation which allows one, by\nthe standard Noether procedure, to obtain the currents ${\\tilde\n  J}_\\alpha^\\mu(x)$ and ${\\tilde J}_{5\\alpha}^\\mu(x)$, which are\nrelated to the symmetry currents via ${\\tilde J}^\\mu_{L\\alpha} =\n({\\tilde J}^\\mu_\\alpha - {\\tilde J}^\\mu_{5\\alpha})\/2$ and ${\\tilde\n  J}^\\mu_{R\\alpha} = ({\\tilde J}^\\mu_\\alpha + {\\tilde\n  J}^\\mu_{5\\alpha})\/2$.  We will further assume that the Lagrangian\ncontains an operator that explicitly breaks the chiral symmetry in the\npattern $SU(N)_R\\otimes SU(N)_L\\rightarrow SU(N)_F$ and is governed by\nthe parameter $\\epsilon_\\chi$ such that as $\\epsilon_\\chi\\rightarrow\n0$, the symmetry is restored at the classical level. The general\nrelation between currents and their associated charges is given by\n\\begin{eqnarray}\n{Q}(n\\cdot x)\\, =\\, \\int\\, d^4y\\, \\delta(n\\cdot (x\\, -\\, y\\,)\\, )\\, n\\cdot {J}(y) \\ ,\n\\label{eq:Qgendef}\n\\end{eqnarray} \nwhere the vector $n_\\mu$ selects the initial quantization\nsurface, which we take to be the null plane $\\Sigma_n^\\tau$. Therefore, the null-plane chiral symmetry charges are \n\\begin{eqnarray}\n{\\tilde Q}_\\alpha \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{\\alpha}(x^-, {\\vec x}_\\perp) \\ ; \\label{eq:npchargesgenA}\\\\\n{\\tilde Q}^5_\\alpha(x^+) \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ,\n\\label{eq:npchargesgenB}\n\\end{eqnarray}\nwhere the axial charges have been given explicit null-plane time dependence as they are not conserved\ndue to the explicit breaking operator in the Lagrangian.\nThese charges satisfy the $SU(N)_R\\otimes SU(N)_L$ chiral algebra,\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha\\,  ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\, i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma\n\\ & , &\\  \n[\\, {\\tilde Q}_5^\\alpha(x^+)\\, ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\,  i\\,f^{\\alpha\\beta\\gamma}\\,  {\\tilde Q}_5^\\gamma(x^+) \\ ;\n\\label{eq:LCalga}\n\\end{eqnarray}\n\\begin{eqnarray}\n[\\, {\\tilde Q}_5^\\alpha(x^+) \\, ,\\, {\\tilde Q}_5^\\beta(x^+)\\, ] \\, =\\,  i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma \\ .\n\\label{eq:LCalgb}\n\\end{eqnarray}\nWe further assert that both types of chiral charges annihilate the vacuum. That is,\n\\begin{eqnarray}\n{\\tilde Q}^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, {\\tilde Q}_5^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, 0 \\ .\n\\label{eq:vacuumgen}\n\\end{eqnarray} \nThis is the statement that the front-form vacuum is invariant with\nrespect to the full $SU(N)_R\\otimes SU(N)_L$ symmetry. In particular,\nthis implies that there can be no vacuum condensates that break\n$SU(N)_R\\otimes SU(N)_L$ on a null-plane. This may seem to be an odd\nassumption, since the chiral charge is directly related to the\naxial-vector current through eq.~\\ref{eq:npchargesgenB}, and in\ngeneral one would expect that this current has a Goldstone boson pole\ncontribution, in turn implying that the chiral charges acting on the\nvacuum state excite massless Goldstone bosons.  Below we will confirm the assertion,\neq.~\\ref{eq:vacuumgen}, by using standard current-algebra polology to\nshow that indeed the Goldstone boson pole contribution to the null-plane\naxial-vector current is absent.\n\n\\subsection{Symmetries of the reduced Hamiltonians}\n\n\\noindent Mixed commutators among the Poincar\\'e generators and internal symmetry generators\ncan be expressed generally as~\\cite{Feinberg:1973qb}:\n\\begin{eqnarray}\n\\lbrack\\, {Q}_\\alpha(n\\cdot x) \\, ,\\, P^\\mu\\rbrack \\, &=&\\, -i\\;n^\\mu  \n\\int\\, d^4y\\, \\delta( n\\cdot (x\\, -\\, y\\,)\\, )\\, \\partial_\\nu J^\\nu_\\alpha(y)  \\ ; \\\\\n\\lbrack\\, {Q}_\\alpha(n\\cdot x) \\, ,\\, M^{\\mu\\nu}\\rbrack \\,&=&\\,\ni\\int\\, d^4y\\, \\delta( n\\cdot (x\\, -\\, y\\,)\\, )\\,\\left(n^\\mu y^\\nu \\ -\\ n^\\nu y^\\mu \\right) \\partial_\\kappa J^\\kappa_\\alpha(y)  \\ .\n\\end{eqnarray}\nFrom these expressions one then obtains the mixed commutator between the Pauli-Lubanski vector\nand the internal symmetry charges:\n\\begin{eqnarray}\n\\hskip-1.3em\\lbrack{Q}_\\alpha(n\\cdot x)  , W_\\nu\\rbrack = \\ft i2 \\varepsilon_{\\nu\\delta\\rho\\sigma}\n\\int d^4y \\delta( n\\cdot (x - y))\n\\bigg\\lbrack M^{\\delta\\rho}n^\\sigma - \\left(n^\\delta y^\\rho -n^\\rho y^\\delta \\right)P^\\sigma\n\\bigg\\rbrack \\partial_\\kappa J^\\kappa_\\alpha(y).\n\\end{eqnarray}\nUsing these expressions, one finds the commutation relations between null-plane chiral charges and the\nreduced Hamiltonians:\n\\begin{eqnarray}\n\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack \\, &=&\\, -2i\\,P^+\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ; \\label{eq:mixedchiralc}\\\\\n\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M{\\cal J}_r\\rbrack \\, &=&\\, i\\,\\epsilon_{rs}\\,P^+\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\Gamma}_s\\,\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ,\n\\label{eq:mixedchirald}\n\\end{eqnarray}\nwhere $\\Gamma_s\\equiv E_s - P^+ x_s $. Here and in what follows, we are assuming that $SU(N)_F$ is \nunbroken and therefore $\\partial_\\mu {\\tilde J}^\\mu_{\\alpha}=0$ and\nthe reduced Hamiltonians commute with the $SU(N)_F$ charges:\n\\begin{eqnarray}\n\\lbrack\\, {\\tilde Q}_\\alpha\\, ,\\, M^2\\rbrack &=& \\lbrack\\, {\\tilde Q}_\\alpha\\, ,\\, M{\\cal J}_r\\rbrack  \\ = \\ 0\\ .\n\\label{eq:mixedvector}\n\\end{eqnarray}\n\n\\subsection{Goldstone's theorem on a null plane}\n\n\\noindent In the instant form, a symmetry has three possible fates in\nthe quantum theory: the symmetry remains exact and the current is\nconserved, the symmetry is spontaneously broken and again the current\nis conserved, or the symmetry is anomalous and the current is not\nconserved. The front form realizes a fourth possibility: the symmetry\nis spontaneously broken and the associated current in not conserved.\nThis fourth possibility is necessary in the front form because the\nvacuum is invariant with respect to all internal symmetries.  In general, we\ncan write\n\\begin{eqnarray}\n\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ = \\ \\epsilon_{\\chi}\\ {\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+) \\ ,\n\\label{eq:genpcac}\n\\end{eqnarray}\nwhere $\\epsilon_{\\chi}$ is the parameter that gauges the amount of chiral symmetry breaking that is present in the Lagrangian.\nUsing the short hand,\n\\begin{eqnarray}\n|\\,h\\,\\rangle\\equiv \\,|\\, p^+\\, ,\\, {\\vec p}_\\perp\\,;\\, \\lambda\\,,\\, h\\,\\rangle \\ ,\n\\label{eq:HME2}\n\\end{eqnarray} for the momentum eigenstates, we take the matrix element of\neq.~\\ref{eq:mixedchiralc} between momentum eigenstates, which gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle\\, &=&\\, -2i\\,p^+\\,\n\\epsilon_{\\chi}\\, \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, | {\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+) |\\, h\\, \\rangle \\ ; \n\\label{eq:mixedchiral1}\n\\end{eqnarray} If the right hand side of this equation vanishes for all $h$ and\n$h'$, then there can be no chiral symmetry breaking of any kind. \nTherefore, in order that the chiral symmetry be spontaneously broken, the\nchiral current cannot be conserved and we have the following\nconstraint~\\cite{Kim:1994rm,Yamawaki:1998cy} in the limit $\\epsilon_\\chi\\rightarrow 0$:\n\\begin{eqnarray}\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, |{\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+)|\\, h\\, \\rangle &\\longrightarrow & \\frac{1}{\\epsilon_\\chi}\\ +\\ \\ldots \\ ,\\label{eq:pizeromodea}\n\\end{eqnarray}\nwhere the dots represent other possible terms that are non-singular in the limit $\\epsilon_\\chi\\rightarrow 0$.\nNow we will show that this condition implies the existence of $N^2-1$ Goldstone bosons~\\footnote{Note that if we took eq.~\\ref{eq:pizeromodea}\nas a constraint on the operator ${\\tilde P}_\\alpha$ rather than on its matrix elements, then this constraint would be viewed\nas a constraint on the zero-modes of the operator~\\cite{Kim:1994rm,Yamawaki:1998cy}. Here we work entirely with matrix elements.}. \nWe will assume that ${\\tilde P}_\\alpha$ is an interpolating operator for Lorentz-scalar fields $\\phi_\\alpha^i$, and \ntherefore we can write\n\\begin{eqnarray}\n{\\tilde P}_\\alpha(x) \\ =\\ \\sum_i {\\cal Z}_i\\,\\phi_\\alpha^i(x) \\ \n\\label{eq:zfactors}\n\\end{eqnarray}\nwhere the ${\\cal Z}_i$ are overlap factors. Using the reduction\nformula we relate the matrix elements of field operators between\nphysical states to transition amplitudes. Of course here it is understood\nthat there is no selection rule which would forbid these transitions.\nThe S-matrix element for the transition $h(p)\\rightarrow h'(p')+\\phi^i_\\alpha(q)$ can be defined by\n\\begin{eqnarray}\n\\langle \\, h'\\,;\\, \\phi_\\alpha^i(q)\\, |\nS |\\, h\\, \\rangle  &\\equiv&  i(2\\pi)^4\\,\\delta^4(\\, p\\,-\\, p'-q)\\,\n{\\cal M}^i_\\alpha (\\,p',\\,\\lambda',\\, h'\\,;\\,p,\\,\\lambda,\\, h\\,) \\nonumber \\\\\n&=& i\\int d^4x\\,e^{-iq\\cdot x}\\,\\left(-q^2+M_{\\phi^i}^2\\right)\\,\\langle\\, h'\\, |\\, \\phi_\\alpha^i(x)\\,|\\, h\\, \\rangle\n\\label{eq:HME3}\n\\end{eqnarray}\nwhere ${\\cal M}^i_\\alpha$ is the Feynman amplitude and in the second line we have used\nthe reduction formula. It then follows that\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, \\phi_\\alpha^i(x)\\,|\\, h\\, \\rangle & =& \n-e^{iq\\cdot x}\\,{1\\over{q^2-M_{\\phi^i}^2}}\\, {\\cal M}^i_\\alpha(q) \\ .\n\\label{eq:reductiongen}\n\\end{eqnarray}\nUsing this formula together with eq.~\\ref{eq:zfactors} in eq.~\\ref{eq:mixedchiral1} then gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle &=& 2i\\,p^+\\,\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\sum_i \\frac{\\epsilon_\\chi\\, {\\cal Z}_i}\n{2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2}-M_{\\phi^i}^2}\\,{\\cal M}^i_\\alpha(q) \\nonumber \\\\ \n&=&{} -2i\\,p^+\\, (2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\sum_i \\frac{\\epsilon_\\chi\\, {\\cal Z}_i}\n{M_{\\phi^i}^2}\\,{\\cal M}^i_\\alpha(q^-) \\ ,\n\\label{eq:mixedchiralproof}\n\\end{eqnarray} \nwhere in the second line we have used the momentum delta\nfunctions.  In order that the right hand side not vanish in the\nsymmetry limit, there must be at least one field $\\phi_\\alpha^i$ whose\nmass-squared vanishes proportionally to $\\epsilon_\\chi$ as\n$\\epsilon_\\chi\\rightarrow 0$. We will denote this field as\n$\\pi^\\alpha\\equiv \\phi_\\alpha^1$ with \n\\begin{eqnarray} M_\\pi^2 \\ =\\ c_p\\,\n\\epsilon_\\chi \\ ,\n\\label{eq:pionmass}\n\\end{eqnarray} \nwhere $c_p$ is a constant of proportionality. There are therefore\n$N^2-1$ massless fields $\\pi_\\alpha$ in the symmetry limit, which we\nidentify as the Goldstone bosons.  It is noteworthy that this proof\nrelies entirely on physical matrix elements; i.e. there is no need to\nassume the existence of a vacuum condensate that breaks the chiral\nsymmetry. Of course, in instant-form QCD, we know that the\nproportionality constant in eq.~\\ref{eq:pionmass} contains the quark\ncondensate. This issue will be resolved below in\nsection~\\ref{sec:qcd}.  While we have carried out this proof in the\ncase of $SU(N)_R\\otimes SU(N)_L$ broken to $SU(N)_V$, it is clearly\neasily generalized to other systems.\n\nWe can now write ${\\tilde P}_\\alpha \\ =\\ {\\cal Z}\\,\\pi_\\alpha\\ +\\ \\ldots$\nwhere the dots represent non-Goldstone boson fields, and \n\\begin{eqnarray}\n\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \n\\langle\\, h'\\, |\\,  {\\bar{\\cal Z}}\\,M_\\pi^2\\,\\pi_\\alpha(x)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:pcacgen}\n\\end{eqnarray}\nwhere ${\\bar{\\cal Z}}\\equiv{\\cal Z}\/c_p$. Here, as in the usual current algebra\nmanipulations, we have assumed that only the Goldstone bosons couple to the\naxial-vector current, and it is now a standard exercise to determine the overlap factor. First, define\nthe Goldstone-boson decay constant, $F_\\pi$, via\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, \\pi_\\beta\\, \\rangle & \\equiv& -i\\,p^\\mu\\;F_\\pi\\,\\delta_{\\alpha\\beta}\\;e^{ip\\cdot x} \\ ,\n\\label{eq:piondecaydef}\n\\end{eqnarray}\nwhere $|\\, \\pi_\\beta\\, \\rangle \\equiv |\\, p^+\\, ,\\, {\\vec p}_\\perp\\,;\\, 0,,\\, \\pi_\\beta\\, \\rangle$.\nTaking the divergence of the current and raising eq.~\\ref{eq:pcacgen} to an operator relation yields\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,{\\bar{\\cal Z}}\\,M_\\pi^2\\,\\pi_\\alpha(x) \\,|\\, \\pi_\\beta\\, \\rangle & =& F_\\pi\\,M_\\pi^2\\,\\delta_{\\alpha\\beta}\\;e^{ip\\cdot x} \\ .\n\\label{eq:piondecaydef2}\n\\end{eqnarray}\nThe normalization of the Goldstone-boson field,\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,\\pi_\\alpha(x)\\,|\\, \\pi_\\beta\\, \\rangle &= & \\delta_{\\alpha\\beta}\\;e^{ip\\cdot x} \\ ,\n\\label{eq:pionfieldnorm}\n\\end{eqnarray}\nthen gives $\\bar{\\cal Z}=F_\\pi$ and we recover the standard operator relation\n\\begin{eqnarray}\n\\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x) & =&  F_\\pi\\,M_\\pi^2\\,\\pi_\\alpha(x) \\ .\n\\label{eq:pcacgen2}\n\\end{eqnarray}\nWe can now express the mixed Lie bracket, eq.~\\ref{eq:mixedchiralproof}, as \n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle \n&=& -2i\\,p^+\\, (2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}F_\\pi\\,{\\cal M}_\\alpha(q^-) \\ ,\n\\label{eq:mixedfinal}\n\\end{eqnarray}\nwhere here ${\\cal M}_\\alpha(q^-)$ is the Feynman amplitude for the transition $h(p)\\rightarrow h'(p')+\\pi_\\alpha(q)$.\nWe see that while the chiral current is not conserved, its divergence is proportional to an S-matrix element.\nNoting that \n\\begin{eqnarray}\n\\hskip-1.3em\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\, \\rangle \n&=& 2p^+q^-\\langle\\, h'\\, |{\\tilde Q}^5_\\alpha(x^+) |\\, h\\, \\rangle \\ =\\  -2p^+\\langle\\, h'\\, |i\\,\\frac{d}{dx^+}{\\tilde Q}^5_\\alpha(x^+) |\\, h\\, \\rangle \\ ,\n\\label{eq:mixeddef}\n\\end{eqnarray}\nand from the definition of the chiral charge, eq.~\\ref{eq:npchargesgenB},\n\\begin{eqnarray}\n\\langle\\, h'\\, |{\\tilde Q}^5_\\alpha(x^+) |\\, h\\, \\rangle &=& \n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\langle\\, h'\\, |{\\tilde J}^+_{5\\alpha} (0) |\\, h\\, \\rangle \\ ,\n\\label{eq:chiralchargecurrent}\n\\end{eqnarray}\none finds, using eq.~\\ref{eq:mixedfinal},\n\\begin{eqnarray}\n{\\cal M}_\\alpha(q^-)\n& =& \n\\frac{i\\,q^-}{F_\\pi}\\, \\langle\\, h'\\, |\\,  {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:matcurr}\n\\end{eqnarray}\nor, in Lorentz-invariant form,\n\\begin{eqnarray}\n{\\cal M}_\\alpha(q) & =& \n\\frac{i\\ q_\\mu}{F_\\pi}\\, \\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:HME8a}\n\\end{eqnarray}\nwhich is the standard current-algebra result.  In order to confirm\nsome of these properties in a better-known fashion, and to address the\nassumption we have made that the chiral charges annihilate the vacuum,\nwe will now consider current algebra polology on the\nnull-plane.\n\n\\subsection{Polology and the chiral invariant vacuum}\n\n\\begin{figure}[!t]\n  \\centering\n     \\includegraphics[scale=0.25]{polology.pdf}\n     \\caption{Above shows the standard instant-form polology; the matrix element of the chiral current\nhas a Goldstone-boson pole piece, and a non-pole piece. These two contributions cancel in the symmetry limit\nensuring a conserved chiral current. Below shows the standard front-form polology; the Goldstone-boson pole contribution\nis absent and therefore the current is not conserved but rather has a divergence which is proportional to the matrix element\nfor the emission or absorption of a Goldstone boson.}\n  \\label{fig:polo}\n\\end{figure}\n\n\\noindent  Our starting point is the matrix element between hadronic states $h$\nand $h'$ of the axial-vector current, which can be written in a general\nway as~\\cite{Weinberg:1995mt,Weinberg:1996kr}\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\, =\\, \n\\frac{i F_\\pi\\, q^\\mu}{q^2-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \n\\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N}\n\\label{eq:HME1}\n\\end{eqnarray}\nwhere as before $q=p-p^\\prime$. Using translational invariance, we have\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle \\, =\\, \ne^{iq\\cdot x}\\,\\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ .\n\\label{eq:HME4}\n\\end{eqnarray}\nIt follows that \n\\begin{eqnarray}\n\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \ni\\, q_\\mu\\,\\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle  \\nonumber \\\\\n& =& \ne^{iq\\cdot x}\\, \\Bigl\\lbrack\\,\n\\frac{-F_\\pi\\, q^2}{q^2-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N}\\, \\Bigr\\rbrack \\ ,\n\\label{eq:HME5}\n\\end{eqnarray}\nand using\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \n\\langle\\, h'\\, |\\,  F_\\pi\\,M_\\pi^2\\,\\pi_\\alpha(x)\\,|\\, h\\, \\rangle \\ ,\n\\label{eq:HME6}\n\\end{eqnarray}\nand the reduction formula, eq.~\\ref{eq:reductiongen}, reproduces eq.~\\ref{eq:HME8a}.\nNote that in null-plane coordinates eq.~\\ref{eq:HME1} gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\,  {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\, =\\, \n\\frac{i F_\\pi\\, q^+}{2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2}-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \n\\langle\\, h'\\, |\\,  {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\ .\n\\label{eq:LCC2}\n\\end{eqnarray}\nWe therefore have\n\\begin{eqnarray}\n\\lim_{{q+,{\\vec q}_\\perp}\\rightarrow 0}\\langle\\, h'\\, |\\,  {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ =\\ \n\\langle\\, h'\\, |\\,  {\\tilde J}^+_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\ .\n\\label{eq:LCC3}\n\\end{eqnarray}\nBy comparing with eq.~\\ref{eq:chiralchargecurrent}, it is clear that\nthe null-plane chiral charges, by construction, do not excite the\nGoldstone boson states. The property, eq.~\\ref{eq:vacuumgen}, of\nvacuum annihilation which we assumed above, is therefore a general\nproperty of the null-plane chiral charges.\n\nAgain consider the space-integrated current divergence in the front-form, but now using eq.~\\ref{eq:HME5}. One finds\n\\begin{eqnarray}\n&&\\int\\, d x^-\\, d^2 {\\bf x_\\perp}\\,\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle \\  = \\\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\,\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle  \\nonumber \\\\\n&&\\qquad = \n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\,\n\\Bigl\\lbrack\\,\n\\frac{-F_\\pi\\, \\left( 2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2} \\right)}{2q^+q^-\\,-\\,{\\vec q}_\\perp^{\\;2}-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\,\n\\Bigl\\rbrack \\nonumber \\\\\n&&\\qquad =\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\, i\\ q_\\mu\\, \\langle\\, h'\\, |\\,  {\\tilde J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\nonumber \\\\\n&&\\qquad =\n(2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\n\\, F_\\pi\\,{\\cal M}_\\alpha(q^-) \\ ,\n\\label{eq:HME9a}\n\\end{eqnarray}\nwhere in the third line the momentum delta functions have been used,\nand in the last line we have used eq.~\\ref{eq:HME8a} and eq.~\\ref{eq:LCC3}. Now using\neq.~\\ref{eq:mixedchiralc}, we see that we have recovered\neq.~\\ref{eq:mixedfinal}. In this derivation we see explicitly that the\nGoldstone-boson pole does not contribute to the divergence of the\naxial-current.  It is for this reason that the current cannot be\nconserved.  For purposes of comparison, recall that in the instant\nform, one has\n\\begin{eqnarray}\n\\int\\, d^3 x\\,\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{J}^\\mu_{5\\alpha}(x)\\,|\\, h\\, \\rangle & =& \n(2\\pi)^3\\,\\delta^3(\\,{\\vec q}\\,)\n\\,\\langle\\, h'\\, |\\,  \\partial_\\mu\\,{J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle \\ ; \\nonumber \\\\\n& =& \n(2\\pi)^3\\,\\delta^3(\\,{\\vec q}\\,)\\,\n\\Bigl\\lbrack\\,\n\\frac{-F_\\pi\\, q_0^2}{q_0^2-M_\\pi^2}\\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\,  {J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\,\n\\Bigl\\rbrack \\ ; \\nonumber \\\\\n& \\mapright{M_\\pi\\rightarrow 0}& \n(2\\pi)^3\\,\\delta^3(\\,{\\vec q}\\,)\\,\n\\Bigl\\lbrack\\,\n-F_\\pi \\, {\\cal M}_\\alpha\n\\, +\\, \ni\\ q_\\mu\\, \\langle\\, h'\\, |\\,  {J}^\\mu_{5\\alpha}(0)\\,|\\, h\\, \\rangle_{N} \\,\n\\Bigr\\rbrack \\ ; \\nonumber \\\\\n& =& 0 \\ ,\n\\label{eq:HME9}\n\\end{eqnarray}\nwhere in the last line, eq.~\\ref{eq:HME8a} has once again been used.\nHere there is a cancellation between the pole and non-pole parts of the matrix element\nwhich ensure that the integrated current divergence vanishes in the chiral limit.\nThis analysis, which is expressed pictorially in fig.\\ref{fig:polo},  suggests that the front-form and instant-form  axial-vector currents\nare related, at the operator level, through\n\\begin{eqnarray}\n{\\tilde J}^\\mu_{5\\alpha}\\ =\\ {J}^\\mu_{5\\alpha}\\ -\\ ({J}^\\mu_{5\\alpha})_{\\it GB\\,pole}\n\\label{eq:HME11}\n\\end{eqnarray} \nwhere the second term on the right is the purely Goldstone-boson pole part of the axial-vector current.\nWe will see that this peculiar realization of chiral symmetry does indeed emerge in QCD.\n\nIt is useful to define new objects which give a matrix-element representation of the \ninternal-symmetry charges~\\cite{Weinberg:1969hw,Weinberg:1969db}:\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde Q}^5_\\alpha(x^+) \\,|\\, h\\, \\rangle \\, =\\, \n(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\n\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\,\n\\delta_{\\lambda'\\lambda} \\ ;\n\\label{eq:HME40}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\, {\\tilde Q}_\\alpha \\,|\\, h\\, \\rangle \\, =\\, \n(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\n\\lbrack\\, T_\\alpha \\,\\rbrack_{h}\\, \\delta_{h h'}\\,\n\\delta_{\\lambda'\\lambda} \\ . \n\\label{eq:HME41}\n\\end{eqnarray}\nThese definitions are particularly useful as they allow the preservation of the Lie-algebraic structure of the operator algebra in the\ncase where correlation functions are given purely by single-particle states. The matrix element for Goldstone boson emission and absorption is:\n\\begin{eqnarray}\n{\\cal M}_\\alpha (\\,p',\\,\\lambda',\\, h'\\,;\\,p,\\,\\lambda,\\, h\\,) \\ =\\ \\frac{i}{F_\\pi}\\,\n(\\,M^2_h\\ -\\ M^2_{h'}\\,)\\, \\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\,\n\\delta_{\\lambda'\\lambda} \\ . \n\\label{eq:HME42}\n\\end{eqnarray}\nAs one might expect, in the limit that chiral symmetry is restored through a second-order phase transition,\nthe matrix $\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}$ becomes a true symmetry generator~\\cite{Weinberg:1995fe}. \nIn this limit, one also expects that the states $h'$ and $h$ become degenerate. In order that the matrix element of\neq.~\\ref{eq:HME42} not vanish in this limit, ${F_\\pi}$ must approach zero in the symmetry limit in precisely the same way~\\cite{Weinberg:1995fe}. \nThe role of ${F_\\pi}$ as an order parameter of chiral symmetry breaking is then apparent in eq.~\\ref{eq:mixedfinal},\nas the mixed-Lie bracket vanishes as ${F_\\pi}\\rightarrow 0$. Therefore, ${F_\\pi}$ is an order parameter of chiral \nsymmetry breaking on the null-plane.\n\n\n\\subsection{Broken chiral symmetry and spin}\n\\label{susec:spin}\n\n\\noindent Using the results of the previous two sections one finds\n\\begin{eqnarray}\n&&\\hspace{-.4in}\\langle\\, h'\\,,\\,\\lambda' |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M^2\\rbrack |\\, h\\,,\\,\\lambda \\rangle = \n\\delta_{\\lambda',\\lambda}(2\\pi)^3\\,2\\,p^+\\,\\delta(q^+)\\delta^2({\\vec q}_\\perp)\n\\left(M_h^2-M_{h'}^2\\right)\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\n\\label{eq:mixedchiral1out}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&&\\hspace{-.92in}\\langle\\, h'\\,,\\,\\lambda' |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, M{\\cal J}_\\pm\\rbrack |\\, h\\,,\\,\\lambda \\rangle\\ =\\ \n\\delta_{\\lambda',\\lambda\\pm 1}\\;(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\nonumber\\\\\n&&\\times\\;\\bigg\\lbrack M_h\\,c_\\pm(h,\\lambda)\\,\\lbrack\\, X_\\alpha(\\lambda\\pm 1) \\,\\rbrack_{h' h}\\,-\\,\nM_h'\\,c_\\mp(h',\\lambda')\\,\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\bigg\\rbrack,\n\\label{eq:mixedchiral2out}\n\\end{eqnarray}\nwhere ${\\cal J}_\\pm\\equiv{\\cal J}_1\\pm i{\\cal J}_2$ and $c_\\pm(h,\\lambda)\\equiv\\sqrt{j_h(j_h+1)-\\lambda(\\lambda\\pm1)}$.\nEq.~\\ref{eq:mixedchiral2out} has been obtained by a direct evaluation of the left-hand side using the usual\nangular momentum ladder relations and eq.~\\ref{eq:HME40}.\nWritten in this form, it is clear that in the presence of spontaneous symmetry breaking, the mixed Lie brackets between\nthe reduced Hamiltonians and the chiral charge are directly related to Goldstone-boson transition amplitudes and are\nnon-vanishing in the symmetry limit. The spin reduced Hamiltonians imply constraints on Goldstone-boson transitions that \nchange the helicity by one unit.\n\nAn important consequence of eqs.~\\ref{eq:mixedchiral1out} and ~\\ref{eq:mixedchiral2out} which will \nprove useful below is that chiral symmetry breaking remains relevant even when there are no mass \nsplittings. If we take $M_h\\;=\\;M_{h'}$, then chiral symmetry breaking arises solely through the\ntransverse spin operator, ${\\cal J}_r$, which is dynamical on the null-plane. That is,\n\\begin{eqnarray}\n&&\\hspace{-.92in}\\langle\\, h'\\,,\\,\\lambda' |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, {\\cal J}_\\pm\\rbrack |\\, h\\,,\\,\\lambda \\rangle\\ =\\ \n\\delta_{\\lambda',\\lambda\\pm 1}\\;(2\\pi)^3\\,2\\,p^+\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\nonumber\\\\\n&&\\times\\;\\bigg\\lbrack \\,c_\\pm(h,\\lambda)\\,\\lbrack\\, X_\\alpha(\\lambda\\pm 1) \\,\\rbrack_{h' h}\\,-\\,\n\\,c_\\mp(h',\\lambda')\\,\\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\bigg\\rbrack \\ .\n\\label{eq:mixedchiral2outb}\n\\end{eqnarray}\nIn this case, Goldstone's theorem must be obtained from the relation\n\\begin{eqnarray}\n\\hspace{-.32in}\\langle\\, h'\\, |\\;M\\;\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, {\\cal J}_r\\rbrack |\\, h\\, \\rangle\\, &=&\\, -i\\,\\epsilon_{rs}\\,p^+\\,\n\\epsilon_{\\chi}\\, \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, |{x}_s\\,  {\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+) |\\, h\\, \\rangle  ,\n\\label{eq:mixedchiral2}\n\\end{eqnarray}\nand its corresponding constraint\n\\begin{eqnarray}\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\langle\\, h'\\, |{x}_s\\,{\\tilde P}_{\\alpha}(x^-, {\\vec x}_\\perp, x^+)|\\, h\\, \\rangle &\\longrightarrow & \\frac{1}{\\epsilon_\\chi}\\ +\\ \\ldots \n\\label{eq:pizeromodeb}\n\\end{eqnarray}\nin the symmetry limit, ${\\epsilon_\\chi}\\rightarrow 0$.\nFollowing the same steps as for the mass-squared reduced Hamiltonian, we have\n\\begin{eqnarray}\n\\hskip-1.8em\\langle\\, h'\\, |\\lbrack\\, {\\tilde Q}^5_\\alpha(x^+) \\, ,\\, {\\cal J}_r\\rbrack |\\, h\\, \\rangle\n= \n-\\epsilon_{rs}\\,\\frac{p^+}{M_h}\\, (2\\pi)^3\\,\\delta(\\,q^+\\,)\\,\\delta^2(\\,{\\vec q}_\\perp \\,)\\,e^{ix^+q^-}\\sum_i \\frac{\\epsilon_\\chi\\, {\\cal Z}_i}\n{M_{\\phi^i}^2}\\left(\\frac{\\partial}{\\partial q_s}\\;{\\cal M}^i_\\alpha(q)\\right)\n\\label{eq:mixedchiralproof2}\n\\end{eqnarray} which again leads, via the same logic presented above, to\nGoldstone's theorem. Therefore, even if $M^2$ commutes with the chiral\ncharges, the chiral symmetry breaking contained in the spin\nHamiltonians implies the presence of massless states. Evaluating\neq.~\\ref{eq:mixedchiralproof2} in the rest frame, where\n$p^+\\rightarrow M_h\/\\sqrt{2}$ and ${\\cal J}_r\\rightarrow{J_r}$, and using eq.~\\ref{eq:HME8a}, gives\n\\begin{eqnarray}\n\\langle\\, h'\\, |\\lbrack\\, {\\tilde J}^+_{5\\alpha}\\, ,\\, {J}^r\\rbrack |\\, h\\, \\rangle\n&=& \ni\\frac{1}{\\sqrt{2}}\\epsilon^{rs}\\langle\\, h'\\, |\\; {\\tilde J}^s_{5\\alpha}\\; |\\, h\\, \\rangle \\ ,\n\\label{eq:wetheorem}\n\\end{eqnarray}\nwhich is simply the statement that the axial current transform as a vector operator.\n\n\\subsection{General operator algebra and the chiral basis}\n\n\\noindent A physical system with an $SU(N)_R\\otimes SU(N)_L$ chiral\nsymmetry broken to the vector subgroup $SU(N)_F$ may be expressed as a\ndynamical Hamiltonian system which evolves with null-plane time,\nwhose reduced Hamiltonians satisfy the $U(2)$ algebra of eq.~\\ref{eq:dynalgarbB},\nand in addition have non-vanishing Lie brackets with the non-conserved\nchiral charges. In operator form the reduced Hamiltonians satisfy:\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\beta_5(x^+)\\, ,\\, M^2\\, ] \\ \\neq \\ 0 \\quad ; \\quad\n[\\, {\\tilde Q}^\\beta_5(x^+)\\, ,\\, M {\\cal J}_\\pm\\, ]\\ \\neq \\ 0 \\ , \n\\label{eq:Qcomm2agen}\n\\end{eqnarray}\nwhich express the spontaneous breaking of the chiral symmetry. \nEq.~\\ref{eq:Qcomm2agen} has the general operator solution\n\\begin{eqnarray}\nM^2\\, &=&\\, M_{\\bf 1}^2\\, +\\, \\sum_{\\cal R}\\, M_{\\cal R}^2 \\ \\ ; \\nonumber \\\\\nM {\\cal J}_\\pm\\, &=&(M {\\cal J}_\\pm)\\, _{\\bf 1}\\, +\\, \\sum_{\\cal R}\\, (M {\\cal J}_\\pm)\\, _{\\cal R}\n\\label{eq:MESONScommgen}\n\\end{eqnarray} \nwhere ${\\bf 1}$ denotes the singlet $SU(N)_R\\otimes SU(N)_L$\nrepresentation, $({\\bf 1},{\\bf 1})$, and ${\\cal R}=({\\bf {\\cal\n    R}_{R}},{\\bf {\\cal R}_{L}})$ is a non-trivial representation. Note\nthat all three symmetry-breaking reduced Hamiltonians must transform\nin the same way. This follows directly from eqs.~\\ref{eq:mixedchiralc}\nand~\\ref{eq:mixedchirald}.  \n\nIt is useful to give a heuristic description of the consequences of\nthis algebraic structure.  Consider an interpolating field operator,\n$a_h^\\dagger$ which creates a momentum state $h$ out of the vacuum;\nthat is,\n\\begin{eqnarray}\na_h^\\dagger\\, |\\, 0\\,\\rangle\\, =\\, |\\, h\\,\\rangle\\, \\ .\n\\label{eq:hdef}\n\\end{eqnarray}\nHere and below for simplicity we will suppress the flavor indices.\nBecause the null-plane chiral charges annihilate the vacuum, ${\\tilde Q}_5\\, |\\, 0\\,\\rangle\\, =\\, 0 $,\none has\n\\begin{eqnarray}\n{\\tilde Q}_5 \\, |\\, h\\,\\rangle\\, &=&\\, \n\\lbrack\\, {\\tilde Q}_5 \\, ,\\, a_h^\\dagger\\, \\rbrack \\, |\\, 0\\,\\rangle\\ . \n\\label{eq:htransf}\n\\end{eqnarray}\nNow we will assume that the interpolating field operator $a_h^\\dagger$ has definite chiral transformation properties with respect \nto the chiral charge in the sense that \n\\begin{eqnarray}\n\\lbrack\\, {\\tilde Q}_5 \\, ,\\, a_h^\\dagger\\, \\rbrack \\, =\\, C'\\, a_{h'}^\\dagger\\, +\\, C''\\, a_{h''}^\\dagger\\, +\\, \\ldots \\ ,\n\\label{eq:htransfprops}\n\\end{eqnarray}\nwhere $C',C'',\\ldots$ are group-theoretic factors.\nThis is simply the statement that the field operators $\\lbrace a_h, a_{h'},a_{h''},\\ldots\\rbrace$ are in a non-trivial $SU(N)_R\\otimes SU(N)_L$ representation. It\nthen follows from eq.~\\ref{eq:htransf} that\n\\begin{eqnarray}\n{\\tilde Q}_5 \\, |\\, h\\,\\rangle\\, &=&\\, C'\\,|\\, h'\\,\\rangle\\, + C''\\,|\\, h''\\,\\rangle\\, +\\,\\ldots \\ ,\n\\label{eq:htransf2}\n\\end{eqnarray} \nand therefore the states $\\lbrace h, {h'},{h''},\\ldots\\rbrace$ are also in an $SU(N)_R\\otimes SU(N)_L$ representation~\\footnote{\nNote that the instant-form interpolating operators also fill out \n$SU(N)_R\\otimes SU(N)_L$ representations. However, the instant-form charges do not annihilate the vacuum,\n  i.e. $Q_5|\\,0\\, \\rangle \\equiv |\\, {\\bf\\omega}\\, \\rangle$, it follows\n  that ${Q}_5\\, |\\, h\\,\\rangle\\, =\\, \\lbrack\\, {Q}_5\\, ,\\,\n  a_h^\\dagger\\, \\rbrack \\, |\\, 0\\,\\rangle\\, +\\, |\\, h\\,;\\, {\\bf\\omega}\n  \\,\\rangle$.  Therefore ${Q}^5 \\, |\\, h\\,\\rangle\\, =\\, C'\\,|\\,\n  h'\\,\\rangle\\, + C''\\,|\\, h''\\,\\rangle\\, +\\,\\ldots\\,+ \\, |\\, h\\,;\\,\n  {\\bf\\omega} \\,\\rangle$ and the utility of chiral symmetry as a\n  classification symmetry is lost.}.\n\nOne then has, for instance, \n\\begin{eqnarray}\n\\langle\\,h'\\,|\\,{\\tilde Q}_5\\, |\\, h\\,\\rangle & =& C'  \\ ; \\\\\n\\langle\\,h''\\,|\\,{\\tilde Q}_5\\, |\\, h\\,\\rangle &=& C'' \\ ,\n\\label{eq:piontrans}\n\\end{eqnarray} \nwhich are, via eq.~\\ref{eq:HME40}, Goldstone-boson transition matrix elements.\nIf $\\lbrace h, h',h'',\\ldots\\rbrace$ are in an irreducible\nrepresentation, then the $C$'s are completely determined by the\nsymmetry (i.e. are Clebsch-Gordon coefficients), while if the\nrepresentation is reducible, then the $C$'s will depend on the mixing\nangles which mix the irreducible representations.  Therefore through\nthe study of Goldstone-boson transitions one learns about the chiral\nrepresentations filled out by the physical states~\\footnote{Here it should be\nstressed that the chiral multiplet structure of the states is useful\nonly when the null-plane chiral charges mediate transitions\nbetween single-particle states~\\cite{Weinberg:1969hw,Weinberg:1969db}. Multi-particle\nstates obscure the algebraic consequences of null-plane chiral symmetry.}.\nTo learn more about the chiral representations, one considers the\nmixed Lie brackets, eqs.~\\ref{eq:mixedchiral1out} and \\ref{eq:mixedchiral2out}.\nKnowledge of the transformation properties of\nthe chiral-symmetry breaking reduced Hamiltonians gives\ninformation about how the hadron masses and spins are related, and therefore in\nhow the irreducible representations mix with each other when the\nsymmetry is broken. \n\nA natural null-plane basis can be written as\n\\begin{eqnarray}\n|\\, k^+\\, ,\\, {\\vec k}_\\perp\\, ;\\, \\lambda\\,, h\\, ,\\, ({\\bf {\\cal R}_{R}},{\\bf {\\cal R}_{L}})\\, \\rangle \\ .\n\\label{eq:HBpluschiraldefined}\n\\end{eqnarray} \nWhile the mass eigenstates are eigenstates of helicity, they clearly\nare not eigenstates of $SU(N)_R\\otimes SU(N)_L$ when the symmetry is\nspontaneously broken. Nevertheless, the chiral basis is useful when\nthe state $h$ can only appear in a finite number of chiral\nrepresentations, even though $h$ may be in an infinite-dimensional\nreducible chiral representation, as is the case generally in QCD at large-${N_c}$~\\cite{'tHooft:1973jz,Weinberg:1990xn}. In the\nchiral basis, the reduced Hamiltonian matrix $M^2$ is then of finite\nrank, even though there can be submatrices of infinite rank (and\ntherefore the Fock expansion in the number of constituents is\ninfinite). Ultimately, the utility of the chiral basis is\ndetermined by comparison with experiment~\\cite{Adler:1965ka,Weisberger:1965hp,Gilman:1967qs,Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn,Weinberg:1994tu,Beane:1999hp,Beane:2002ud,Beane:2002td}.\n\n\\section{QCD in the front form}\n\\label{sec:qcd}\n\n\\subsection{Basic instant-form conventions}\n\n\\noindent In this section, we will review the relevant symmetry\nproperties of the instant-form QCD Lagrangian for purposes of\nestablishing conventions which will clarify the null-plane\ndescription. Consider the QCD Lagrangian with $N$ flavors of light\nquarks and $N_c$ colors:\n\\begin{equation}\n{\\cal L}_{\\scriptstyle\\rm QCD} (x)\n=\n \\bar\\psi (x)\\Big\\lbrack\\ft{i}2\\left(\n\\stackrel{{}_\\rightarrow}{D}_\\mu - \\stackrel{{}_\\leftarrow}{D}_\\mu\n\\right)\n\\gamma^\\mu  - \\mathbb{M} \\Big\\rbrack\\psi (x)\n-\n\\ft14 F_{\\mu\\nu}^a (x) F^{\\mu\\nu}_a (x)\n\\,\n\\label{QCDET}\n\\end{equation}\nwhere $\\mathbb{M}$ is the quark mass matrix, for now taken as a diagonal $N\\times N$ matrix, and\nthe covariant derivatives are \n\\begin{equation}\n\\stackrel{{}_\\rightarrow}{D}_\\mu\n\\, = \\,\n\\stackrel{{}_\\rightarrow}{\\partial}_\\mu\n- \\, i g \\, t^a A_\\mu^a (x)\n\\, , \\qquad\n\\stackrel{{}_\\leftarrow}{D}_\\mu\n\\, = \\,\n\\stackrel{{}_\\leftarrow}{\\partial}_\\mu\n+ \\, i g \\, t^a A_\\mu^a (x)\n\\, ,\n\\end{equation}\nwhere $g$ is the strong coupling constant, and indices $a,b,\\ldots$ are taken as adjoint indices of the $SU(3)$-color gauge group.\nThe Lagrangian is invariant with respect to baryon number and \nsinglet axial transformations\n\\begin{equation}\n\\psi\\rightarrow e^{-i\\theta} \\psi \\ \\ \\ ,\\ \\ \\  \\psi\\rightarrow e^{-i\\theta \\gamma_5} \\psi \\ ,\n\\end{equation}\nwith associated currents\n\\begin{equation}\n{J}^\\mu\\ =\\ \\bar\\psi \\gamma^\\mu\\psi \\ \\ \\ , \\ \\ \\ {J}^\\mu_{5}\\ =\\ \\bar\\psi \\gamma^\\mu\\gamma_5 \\psi \\ ,\n\\end{equation}\nand with divergences\n\\begin{equation}\n\\partial_\\mu {J}^\\mu\\ =\\ 0 \\ \\ \\ ,\\ \\ \\ \n\\partial_\\mu {J}^\\mu_{5}\\ =\\ 2i \\bar\\psi\\, \\mathbb{M}\\, \\gamma_5 \\psi \\ -\\  N \\frac{g^2}{16 \\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\;\ntr\\left( F_{\\mu\\nu}F_{\\rho\\sigma}\\right)  \\ ,\n\\end{equation}\nwhere the singlet axial symmetry is of course anomalous. In addition, the Lagrangian is invariant with respect to \nthe symmetry transformations\n\\begin{equation}\n\\psi\\rightarrow e^{-i\\theta_\\alpha T_\\alpha} \\psi \\ \\ \\ , \\ \\ \\ \\psi\\rightarrow e^{-i\\theta_\\alpha T_\\alpha \\gamma_5} \\psi \\ ,\n\\end{equation}\nwhere the $T_\\alpha$  are $SU(N)$ generators (see appendix).\nBy the standard Noether procedure one defines the associated currents,\n\\begin{equation}\n{J}^\\mu_\\alpha\\ =\\ \\bar\\psi \\gamma^\\mu T_\\alpha \\psi \\ \\ \\ , \\ \\ \\ \n{J}^\\mu_{5\\alpha}\\ =\\ \\bar\\psi \\gamma^\\mu\\gamma_5 T_\\alpha \\psi \\ ,\n\\end{equation}\nrespectively, with divergences\n\\begin{equation}\n\\partial_\\mu {J}^\\mu_\\alpha\\ =\\ -i \\bar\\psi\\, [\\, \\mathbb{M}\\, ,\\, T_\\alpha \\, ]\\, \\psi \\ \\ \\ , \\ \\ \\ \n\\partial_\\mu {J}^\\mu_{5\\alpha}\\ =\\ i \\bar\\psi\\, \\{ \\, \\mathbb{M}\\, ,\\, T_\\alpha \\, \\}\\, \\gamma_5 \\psi \\ .\n\\end{equation}\nTherefore, with $N$ degenerate flavors the QCD Lagrangian is $SU(N)_F$ invariant and\nin the chiral limit where $\\mathbb{M}$ vanishes, there is an\n $SU(N)_R\\otimes SU(N)_L$ chiral symmetry generated by the currents\n${J}^\\mu_{L\\alpha} = ({J}^\\mu_\\alpha - {J}^\\mu_{5\\alpha})\/2$ and ${J}^\\mu_{R\\alpha} = ({J}^\\mu_\\alpha + {J}^\\mu_{5\\alpha})\/2$.\n\nThe energy-momentum tensor may be written as\n\\begin{equation}\n{T}^{\\mu\\nu}\\ =\\ - g^{\\mu\\nu} {\\cal L}_{\\rm\\scriptstyle QCD} \\ -\\ \nF^{\\mu\\rho}_a \\, F^{\\nu}_{a \\, \\rho}\n\\ +\\ \n\\frac{i}{2}\n\\bar\\psi\n\\stackrel{{}_\\leftrightarrow}{D}{\\!}^\\mu \\gamma^\\nu\n\\psi\n\\, .\n\\label{EMtensorQCD}\n\\end{equation}\nFrom the energy-momentum tensor we can form the Hamiltonian,\n\\begin{eqnarray}\nP^0 \\ =\\ \\int\\, d^3 \\bit{x}\\, {T}^{00}\\ .\n\\end{eqnarray}\nHere we will assume that chiral symmetry is spontaneously broken through the formation of the condensate\n\\begin{eqnarray}\n\\mathbb{M}\\,\\langle\\, \\Omega \\, |\\,\\bar\\psi\\psi  \\,|\\, \\Omega\\, \\rangle\\ =\\ \n\\mathbb{M}\\,\\langle\\, \\Omega \\, |\\,\\frac{\\partial {\\mit T}^{00}}{\\partial \\mathbb{M}}\\,|\\, \\Omega\\, \\rangle\\ =\\\n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0}{\\partial \\mathbb{M}}\\ \\neq\\  0 \\ ,\n\\end{eqnarray}\nwhere we have used the Feynman-Hellmann theorem, $|\\, \\Omega\\, \\rangle$ represents the (complicated) instant-form QCD vacuum state,\nand ${\\cal E}_0$ is the QCD vacuum energy. It is straightforward to show that the condensate transforms as \nthe $(\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ representation of $SU(N)_R\\otimes SU(N)_L$. \nWe can compute the vacuum energy in the low-energy effective field theory; i.e. chiral perturbation theory ($\\chi$PT)~\\cite{Weinberg:1978kz,Gasser:1983yg},\nas well. And therefore, \n\\begin{eqnarray} \n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0}{\\partial \\mathbb{M}}\\ =\\\n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0^{\\chi{\\rm PT}}}{\\partial\n  \\mathbb{M}} \\ , \n\\end{eqnarray} \nwhere ${\\cal E}_0^{\\chi{\\rm PT}}$ is the $\\chi$PT vacuum energy.  In\nthe non-linear realization of the chiral group\nthe Goldstone boson field may be written as\n$U(x)=\\exp{\\left(i\\pi_\\alpha(x) T_\\alpha\/F_\\pi\\right)}$, and the\nleading quark mass contribution to the $\\chi$PT Lagrangian is\n\\begin{eqnarray}\n{\\cal L}^{\\chi{\\rm PT}}_{\\rm\\scriptstyle QCD} \\ =\\ v\\;tr\\left( U \\mathbb{M}^\\dagger + U^\\dagger  \\mathbb{M}\\right) \\ +\\ \\ldots \\ ,\n\\end{eqnarray}\nwith $v=M_\\pi^2\\,F_\\pi^2\/\\mathbb{M}$ and with $M_\\pi$ the Goldstone boson mass. One then obtains the Gell-Mann-Oakes-Renner formula~\\cite{GellMann:1968rz}.\n\\begin{eqnarray}\n-{\\mathbb{M}\\,\\langle\\, \\Omega \\, |\\,\\bar\\psi\\psi  \\,|\\, \\Omega\\, \\rangle}\\ =\\ \\textstyle{1\\over 2} \\,N\\,M_\\pi^2\\,F_\\pi^2 \\ \\ +\\ \\dots \\ .\n\\label{eq:ifgmor}\n\\end{eqnarray}\nIt will be a principle goal in what follows to determine what takes the place of this relation in null-plane QCD.\n\n\\subsection{Null plane representation}\n\n\\noindent The QCD Lagrangian in the null-plane coordinates is obtained by generalizing the results\ngiven in Appendices~\\ref{FFdecompose} and \\ref{FGdecompose} to the interacting case~\\footnote{We follow the notation\nand conventions of Ref.~\\cite{Belitsky:2005qn}.}. (Note that we work in light-cone gauge, $A^+=0$, throughout.) The \nQCD equations of constraint for the non-dynamical degrees of freedom are\n\\begin{equation}\n\\psi_- \\ =\\ \\frac{1}{2 i\\! \\stackrel{{}_\\rightarrow}{\\partial^+}}\\left( -i \\bit{\\gamma}^r\\!\\stackrel{{}_\\rightarrow}{D^r}\\! \\ +\\ \\mathbb{M} \\right) \\gamma^+\\psi_+\n\\quad , \\quad\n\\psi^\\dagger_- \\ =\\  \\psi^\\dagger_+ \\gamma^- \\left( i \\bit{\\gamma}^r\\!\\stackrel{{}_\\leftarrow}{D^r}\\! \\ -\\ \\mathbb{M} \\right) \\frac{1}{2 i\\! \\stackrel{{}_\\leftarrow}{\\partial^+}}\n\\end{equation}\nfor the redundant quark degrees of freedom, and\n\\begin{eqnarray}\n\\partial^+ A^-_a \\ =\\ \\frac{1}{\\partial^+}\nD^r_{ab} \\partial^+ \\bit{A}^r_{b} \\ -\\ g \\frac{1}{\\partial^+} \\bar\\psi_+ \\gamma^+ t^a \\psi_+ \\ ,\n\\label{eq:freegaugeEOMsol1withI}\n\\end{eqnarray}\nfor the redundant gauge degrees of freedom.\nThe null-plane QCD Lagrangian can then be expressed in terms of the dynamical degrees of freedom as\n\\begin{eqnarray}\n{\\tilde {\\cal L}}_{\\scriptstyle\\rm QCD} & = &\ni \\bar\\psi_+ \\gamma^+ \\partial^- \\psi_+\n-\n\\ft{i}{2}\n\\bar\\psi_+ \\bit{\\gamma}^r \\gamma^+ \\bit{\\gamma}^s\nD^r \\frac{1}{\\partial^+} D^s \\psi_+\n\\nonumber\\\\\n&+&\\!\\!\\!\n\\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\mathbb{M}^2 \\frac{1}{\\partial^+} \\psi_+\n+\n\\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\mathbb{M}\\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)\\frac{1}{\\partial^+}\\psi_+\n-\n\\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\mathbb{M}\\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\psi_+\\right)\n\\nonumber\\\\\n&-&\\!\\!\\!\n\\ft{1}{4} \\bit{F}_{a}^{rs} \\bit{F}_{a}^{rs}\n\\ +\\ \n\\left( \\partial^+ \\bit{A}^r_{a} \\right)\n\\left( \\partial^- \\bit{A}^r_{a} \\right)\n-\n\\ft12\n\\left(\n\\frac{1}{\\partial^+}\nD^r_{ab} \\partial^+ \\bit{A}^r_{b}\n-\ng \\frac{1}{\\partial^+} \\bar\\psi_+ \\gamma^+ t^a \\psi_+\n\\right)^2\n\\, .\n\\end{eqnarray}\nThe price to pay for working with the physical degrees of freedom in\nthe null-plane coordinates is a loss of manifest Lorentz covariance, as well as\nthe appearance of operators which that appear to be non-local in the longitudinal\ncoordinate.  As in the instant-form, one should view this Lagrangian\nas providing a perturbative definition of QCD at large momentum\ntransfers, where the longitudinal zero modes play no role.\nNotice that in null-plane QCD there are two kinds of operators that\ndepend on the quark-mass matrix~\\footnote{To minimize clutter, it will prove convenient to \ndefine the operator\n\\begin{eqnarray}\n\\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ \\equiv \\\n\\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)\\frac{1}{\\partial^+}\\psi_+\n-\n\\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\psi_+\\right) \\ .\n\\end{eqnarray}\n}. One is a kinetic term, quadratic in the quark masses, and the other\nis a spin-flip quark-gluon interaction that is linear in the quark masses.\n\nNaturally we expect that null-plane QCD has the same\nsymmetries as instant-form QCD.  Consider the $U(1)_R\\otimes U(1)_L$ transformations,\n\\begin{equation}\n\\psi_+\\rightarrow e^{-i\\theta} \\psi_+ \\ \\ \\ , \\ \\ \\ \\psi_+\\rightarrow e^{-i\\theta \\gamma_5} \\psi_+ \\ .\n\\label{chiraltransu1}\n\\end{equation}\nWhile baryon number is unaltered in moving to the null-plane\ncoordinates, this is clearly not the same chiral transformation that\nwe had in the instant form, as that transformation acts on the\nnon-dynamical degrees of freedom, $\\psi_-$, in a distinct manner and\nis therefore complicated on the null-plane. That the chiral symmetry transformations\nare different in the two forms of dynamics is essential for what follows. We will return below to\nthe relation between the chiral symmetries in the instant-form and the\nfront form, as this will be important in understanding the role of\ncondensates on the null-plane. The $U(1)_A$ current and its divergence\nare~\\cite{Wu:2003vn}\n\\begin{equation}\n{\\tilde J}^\\mu_{5}\\ =\\ {J}^\\mu_{5}  \\ -\\  i \\bar\\psi \\gamma^\\mu \\gamma^+\\gamma_5\\, \\mathbb{M}\\,  \\frac{1}{\\partial^+} \\psi_+\\ ;\n\\label{eq:HME11qcdu1}\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_{5}\\ =\\ \n\\bar\\psi_+ \\gamma^+\\gamma_5 \\, \\mathbb{M}\\,  \\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \n\\ -\\  N \\frac{g^2}{16 \\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\;\ntr\\left( F_{\\mu\\nu}F_{\\rho\\sigma}\\right)  \\ .\n\\end{equation}\nConsider the $SU(N)_R\\otimes SU(N)_L$ transformations,\n\\begin{equation}\n\\psi_+\\rightarrow e^{-i\\theta_\\alpha T_\\alpha} \\psi_+ \\ \\ \\ , \\ \\ \\ \\psi_+\\rightarrow e^{-i\\theta_\\alpha T_\\alpha \\gamma_5} \\psi_+ \\ .\n\\label{chiraltrans}\n\\end{equation}\nThe currents associated with eq.~\\ref{chiraltrans} are\n\\begin{equation}\n{\\tilde J}^\\mu_\\alpha\\ =\\  {J}^\\mu_\\alpha  \\ -\\  \\ft{i}{2} \\bar\\psi \\gamma^\\mu \\gamma^+ \\, [\\, \\mathbb{M}\\, ,\\, T_\\alpha \\, ]\\, \\frac{1}{\\partial^+} \\psi_+\\ ;\n\\end{equation}\n\\begin{equation}\n{\\tilde J}^\\mu_{5\\alpha}\\ =\\ {J}^\\mu_{5\\alpha}  \\ -\\  \\ft{i}{2} \\bar\\psi \\gamma^\\mu \\gamma^+\\gamma_5 \\, \\{\\, \\mathbb{M}\\, ,\\, T_\\alpha \\, \\}\\, \\frac{1}{\\partial^+} \\psi_+ \\ ,\n\\end{equation}\nwith divergences\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_\\alpha\\ =\\ \\ft{1}{2} \\bar\\psi \\gamma^+ \\, [\\, \\mathbb{M}^2\\, ,\\, T_\\alpha \\, ]\\, \\frac{1}{\\partial^+} \\psi_+ \\ ;\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}\\ =\\ \\ft{1}{2} \\bar\\psi \\gamma^+ \\gamma_5 \\, [\\, \\mathbb{M}^2\\, ,\\, T_\\alpha \\, ]\\, \\frac{1}{\\partial^+} \\psi_+\n+\\ft{1}{2}\n\\bar\\psi_+ \\gamma^+\\gamma_5 \\{ \\, \\mathbb{M}\\, ,\\, T_\\alpha \\, \\}\\, \\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ .\n\\end{equation}\nFor $N$ degenerate flavors, the quark mass matrix is proportional to the identity, the vector current is conserved, and the axial current\nand the divergence of the axial current are \n\\begin{equation}\n{\\tilde J}^\\mu_{5\\alpha}\\ =\\ {J}^\\mu_{5\\alpha}  \\ -\\  i \\bar\\psi \\gamma^\\mu \\gamma^+\\gamma_5\\, T_\\alpha \\, \\mathbb{M}\\,  \\frac{1}{\\partial^+} \\psi_+\\ ;\n\\label{eq:HME11qcd}\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}\\ =\\ \n\\bar\\psi_+ \\gamma^+\\gamma_5 \\, T_\\alpha \\, \\mathbb{M}\\,  \\frac{1}{\\partial^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ .\n\\label{eq:chidivqcd}\n\\end{equation}\nHere note in particular that the null-plane axial-vector current in null-plane QCD evidently takes the form, eq.~\\ref{eq:HME11}, expected\non general grounds. \n\n\\subsection{Null-plane charges}\n\n\\noindent The null-plane singlet axial charge is defined as\n\\begin{eqnarray}\n{\\tilde Q}^5 \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{5} \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\bar\\psi_+\\gamma^+ \\gamma_5 \\, \\psi_+  \\ ,\n\\label{eq:QCDu1a}\n\\end{eqnarray}\nwhere we have used eq.~\\ref{eq:HME11qcdu1}. Using the momentum-space representation of $\\psi_+$, given in eq.~\\ref{eq:freediracmomspace}, one finds\n\\begin{eqnarray}\n{\\tilde Q}^5  &=& \\sum_{\\lambda = \\uparrow\\downarrow}2\\lambda\n\\int \\frac{d k^+ d^2 \\bit{k}_\\perp}{2 k^+ (2 \\pi)^3}\n \\left\\{  b^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) b_\\lambda (k^+ , \\bit{k}_\\perp)   +\nd^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) d_\\lambda (k^+ , \\bit{k}_\\perp) \\right\\}.\n\\end{eqnarray}\nComparison with eq.~\\ref{eq:freehelquark} one sees that the singlet axial charge coincides (up to a factor of two) with the free-fermion helicity operator.\nThis of course explains why the quark mass term in the free-fermion theory is a chiral invariant; on the null-plane, chiral symmetry breaking\nin the free-fermion theory implies breaking of rotational invariance in the transverse plane.\n\nSimilarly, the null-plane non-singlet vector and chiral charges are, respectively, \n\\begin{eqnarray}\n{\\tilde Q}_\\alpha \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{\\alpha} \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\bar\\psi_+\\gamma^+ \\, T_\\alpha \\psi_+  \\ ; \\\\\n{\\tilde Q}^5_\\alpha \\ &=& \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde J}^+_{5\\alpha} \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\bar\\psi_+\\gamma^+ \\gamma_5 \\, T_\\alpha \\psi_+  \\ ,\n\\label{eq:QCDcc}\n\\end{eqnarray}\nand using the momentum-space representation of $\\psi_+$, given in eq.~\\ref{eq:freediracmomspace}, one finds\n\\begin{eqnarray}\n\\hskip-1.9em{\\tilde Q}_\\alpha  &=& \\sum_{\\lambda = \\uparrow\\downarrow}\n\\int \\frac{d k^+ d^2 \\bit{k}_\\perp}{2 k^+ (2 \\pi)^3}\n \\left\\{  b^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T_\\alpha b_\\lambda (k^+ , \\bit{k}_\\perp)   -\nd^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T^T_\\alpha d_\\lambda (k^+ , \\bit{k}_\\perp) \\right\\} \\ ; \\\\\n\\hskip-1.9em{\\tilde Q}^5_\\alpha  &=& \\sum_{\\lambda = \\uparrow\\downarrow}2\\lambda\n\\int \\frac{d k^+ d^2 \\bit{k}_\\perp}{2 k^+ (2 \\pi)^3}\n \\left\\{  b^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T_\\alpha b_\\lambda (k^+ , \\bit{k}_\\perp)   +\nd^\\dagger_\\lambda (k^+ , \\bit{k}_\\perp) T^T_\\alpha d_\\lambda (k^+ , \\bit{k}_\\perp) \\right\\}.\n\\end{eqnarray}\nOne readily checks that the null-plane chiral algebra, eqs.~\\ref{eq:LCalga} and \\ref{eq:LCalgb}, is satisfied by these charges. \nAs these charge are written as sums of number operators that count the number\nof quarks and anti-quarks, both chiral charges annihilate the vacuum, and we have\n\\begin{eqnarray}\n{\\tilde Q}^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, {\\tilde Q}_5^\\alpha\\, |\\, 0\\,\\rangle\\, =\\, 0 \\ ,\n\\label{eq:vacuumann2}\n\\end{eqnarray}\nas expected on the general grounds presented above. One then has\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha\\, ,\\, \\psi_+ \\, ]\\, =\\,  -T^\\alpha\\,\\psi_+ \\ \\ ; \\ \\ \n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, \\psi_+ \\, ]\\, =\\,  -\\gamma_5\\,T^\\alpha\\,\\psi_+ \\ .\n\\end{eqnarray}\nBreaking down the fields into left- and right-handed components, \n\\begin{eqnarray}\n\\psi_{+R} \\ =\\ \\ft{1}{2} (1+\\gamma_5) \\psi_+ \\qquad , \\qquad \n\\psi_{+L} \\ =\\ \\ft{1}{2} (1-\\gamma_5) \\psi_+ \n\\end{eqnarray}\nand, using the results of Appendix~\\ref{FFdecompose}, one verifies the fermion transformation properties with respect to $SU(N)_R\\otimes SU(N)_L$:\n\\begin{eqnarray}\n\\psi_{+R}\\ =\\ \\psi_{+\\uparrow}\\ &\\in& \\ ({\\bf 1},{\\bf N}) \\qquad , \\qquad \\psi_{+R}^\\dagger \\ =\\ \\psi_{+\\downarrow}^\\dagger\\ \\in \\ ({\\bf 1},\\bar{\\bf N})\\ ; \\\\\n\\psi_{+L}\\ =\\ \\psi_{+\\downarrow}\\ &\\in& \\ ({\\bf N},{\\bf 1}) \\qquad , \\qquad \\psi_{+L}^\\dagger \\ =\\ \\psi_{+\\uparrow}^\\dagger\\ \\ \\in \\ (\\bar{\\bf N},{\\bf 1})\\ ,\n\\label{eq:chiralassign}\n\\end{eqnarray}\nand the helicity eigen-equations of the quarks\n\\begin{eqnarray}\n\\Sigma_{12}\\;\\psi_{+\\uparrow}& =& {\\textstyle\\frac{1}{2}}\\psi_{+\\uparrow} \\ ;\\\\\n\\Sigma_{12}\\;\\psi_{+\\downarrow}& =& -{\\textstyle\\frac{1}{2}}\\psi_{+\\uparrow} \\ ,\n\\end{eqnarray}\nwhere the helicity operator, $\\Sigma_{12}$, is defined in Appendix~\\ref{FFdecompose}.\n\n\\subsection{Chiral symmetry breaking Hamiltonians}\n\n\\noindent Using the results of the previous section, it is\nstraightforward to find the transformation properties of the\nsymmetry-breaking parts of the reduced Hamiltonians.  Define the\noperators~\\footnote{From here forward we will use the definition:\n\\begin{equation}\n\\frac{1}{\\partial_{\\mathbb{M}}^+} \\equiv \\mathbb{M}\\,\\frac{1}{\\partial^+} \\ .\\nonumber\n\\end{equation}\n}:\n\\begin{equation}\n{\\tilde D}_{5\\alpha} \\ \\equiv \\ \\partial_\\mu {\\tilde J}^\\mu_{5\\alpha}\\ =\\ \n\\bar\\psi_+ \\gamma^+\\gamma_5 \\, T_\\alpha \\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ;\n\\label{eq:axcurr2}\n\\end{equation}\n\\begin{equation}\n{\\tilde D}_{5} \\ \\equiv \\ \\bar\\psi_+ \\gamma^+\\gamma_5 \\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ;\n\\end{equation}\n\\begin{equation}\n{\\tilde D}_{\\alpha} \\ \\equiv \\ \\bar\\psi_+ \\gamma^+\\, T_\\alpha \\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ ;\n\\end{equation}\n\\begin{equation}\n{\\tilde D} \\ \\equiv \\ \\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \\ .\n\\end{equation}\nIt is then a textbook exercise to find\n\\begin{eqnarray}\n&&\\hfill[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}^\\beta_5 \\, ] \\, =\\,  \\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\;{\\tilde D} \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}^\\gamma \\ ; \n\\label{eq:algsola}\\\\\n&&\\hfill[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}\\, ] \\, =\\,  2\\;{\\tilde D}_5^\\alpha \\ ; \n\\label{eq:algsolb}\\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}^\\beta \\, ]\\, =\\,  \\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\;{\\tilde D}_5 \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}_5^\\gamma \\ ; \n\\label{eq:algsolc}\\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde D}_5\\, ]\\, =\\,  2\\;{\\tilde D}^\\alpha \\ .\n\\label{eq:algsold}\n\\end{eqnarray}\nIt follows that the $2N^2$ operators $({\\tilde D}_{5\\alpha},{\\tilde D}_{5},{\\tilde D}_{\\alpha},{\\tilde D})$ fill out the\n$(\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ representation of $SU(N)_R\\otimes SU(N)_L$. \n\nThe null-plane Hamiltonian $P^-$ is:\n\\begin{eqnarray}\nP^- \\ = \\ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {T}^{-+}\\  ,\n\\end{eqnarray}\nand therefore the chiral-symmetry breaking part of this Hamiltonian is given by:\n\\begin{eqnarray}\nP_{({\\bf N},{\\bf N})}^- \\ \\equiv \\ -\\ft{i}{2}\\,\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde D} \\ .\n\\end{eqnarray}\nOne readily checks that this is consistent with eqs.~\\ref{eq:mixedchiralc} and \\ref{eq:axcurr2}.\n\nOne then finds the symmetry breaking parts of the reduced QCD Hamiltonians:\n\\begin{eqnarray}\nM^2_{({\\bf N},{\\bf N})} & = & -i P^+ \\,\n\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, \\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+  \\ ; \n\\label{eq:symmbreakhamsa}\\\\\n\\left(M{\\cal J}_r\\right)_{({\\bf N},{\\bf N})} & = &   i{\\textstyle{\\frac{1}{2}}}\\epsilon_{rs}\\,P^+\n{\\int}\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\Gamma}_s\\, \\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+  \\ ,\n\\label{eq:symmbreakhamsb}\n\\end{eqnarray}\nwhere, in addition, we have used eqs.~\\ref{eq:mixedchirald} and \\ref{eq:chidivqcd} to obtain the reduced Hamiltonian for spin.\nAll chiral symmetry breaking in null-plane QCD is contained in these two operators. \n\nUsing eqs.~\\ref{eq:mixedchiralc},  \\ref{eq:mixedchirald} and \\ref{eq:algsola} one finds\n\\begin{eqnarray}\n{}[{\\tilde Q}^\\beta_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M^2\\, ]] &=& -2iP^+\\;\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\;\\left(\\;\n\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,{\\tilde D} \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}^\\gamma\\; \\right) \\ ; \\label{eq:fundcona}\\\\\n{}[{\\tilde Q}^\\beta_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M{\\cal J}_r\\, ]] &=& i\\epsilon_{rs}P^+\\;\\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\,\\Gamma_s\\;\\left(\\;\n\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,{\\tilde D} \\ +\\ d^{\\alpha\\beta\\gamma}\\,{\\tilde D}^\\gamma\\; \\right) \\ .\n\\label{eq:fundconb}\n\\end{eqnarray}\nActing on these equations with $\\delta^{\\alpha\\beta}$ and $d^{\\alpha\\beta\\gamma}$, and using the identities in Appendix~\\ref{suN} gives\n\\begin{eqnarray}\n&&-2iP^+ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde D} \\ =\\ \\frac{N}{N^2-1}\\, [{\\tilde Q}^\\alpha_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M^2\\, ]]\\ ; \\\\\n&&-2iP^+ \\int\\, d x^-\\, d^2 \\bit{x}_\\perp\\, {\\tilde D}_\\gamma \\ =\\ d_{\\alpha\\beta\\gamma}\\frac{N}{N^2-4}\\, [{\\tilde Q}^\\beta_5\\, ,\\, [\\, {\\tilde Q}^\\alpha_5\\, ,\\, M^2\\, ]]\\ .\n\\end{eqnarray}\nTherefore, eq.~\\ref{eq:fundcona} can be written as\n\\begin{eqnarray}\n\\hskip-1.4em[{\\tilde Q}^\\beta_5 , [ {\\tilde Q}^\\alpha_5 , M^2]]=\\frac{1}{N^2-1}\\delta^{\\alpha\\beta}[{\\tilde Q}^\\gamma_5 , [ {\\tilde Q}^\\gamma_5 , M^2 ]]\n+ \\frac{N}{N^2-4}d^{\\alpha\\beta\\gamma}d^{\\mu\\nu\\gamma}[ {\\tilde Q}^\\mu_5 , [ {\\tilde Q}^\\nu_5 , M^2 ]]\\ ,\n\\end{eqnarray} \nand eq.~\\ref{eq:fundconb} takes the same form but with $M^2$ replaced by $M {\\cal J}_\\pm$.  Defining the projection operator\n\\begin{eqnarray} {\\cal P}^{\\alpha\\beta ; \\mu\\nu}\\ \\equiv\\\n\\delta^{\\alpha\\nu}\\delta^{\\beta\\mu}\\ -\\\n\\frac{1}{N^2-1}\\;\\delta^{\\alpha\\beta}\\delta^{\\mu\\nu} \\ -\\\n\\frac{N}{N^2-4}\\;d^{\\alpha\\beta\\gamma}d^{\\mu\\nu\\gamma} \\ , \n\\end{eqnarray} \nwe can express the constraints on the reduced Hamiltonians in compact notation as:\n\\begin{eqnarray}\n{\\cal P}^{\\alpha\\beta ;\\mu\\nu}\\;[{\\tilde Q}^\\mu_5\\, ,\\, [\\, {\\tilde Q}^\\nu_5\\, ,\\, M^2]]\\ =\\ {\\cal P}^{\\alpha\\beta ;\\mu\\nu}\\;[{\\tilde Q}^\\mu_5\\, ,\\, [\\, {\\tilde Q}^\\nu_5\\, ,\\, M {\\cal J}_\\pm]]\\ =\\ 0 \\ .\n\\label{eq:fundmix}\n\\end{eqnarray} \nThese are quite possibly the most important equations in null-plane\nQCD, as they are the mathematical expression of the specific way in\nwhich the internal symmetries and Poincar\\'e symmetries intersect.\nThese equations were obtained originally in\nRefs.~\\cite{Weinberg:1969hw,Weinberg:1969db,Weinberg:1990xn} by\nconsidering the most general form of Goldstone-boson-hadron scattering\namplitudes in specially-designed Lorentz frames, and using input from Regge-pole\ntheory expectations of their high-energy behavior.  Note that the\nprojection operator, ${\\cal P}^{\\alpha\\beta ; \\mu\\nu}$, has four\nadjoint indices and is, as shown in Ref.~\\cite{Weinberg:1969hw}\nrelated to the interactions of Goldstone bosons (in the t-channel of\nGoldstone-boson-hadron scattering), which are in the adjoint of\n$SU(N)_F$ and whose scattering amplitudes therefore transform as the\nproduct of two adjoints. In the case of two flavors, where ${\\bf\n  3}\\otimes{\\bf 3}={\\bf 1}\\oplus{\\bf 3}\\oplus{\\bf 5}$, it projects out\nthe ${\\bf 5}$-dimensional representation ($I=2$) and in the case of\nthree flavors, where ${\\bf 8}\\otimes{\\bf 8}={\\bf 1}\\oplus{\\bf\n  8}\\oplus{\\bf 8}\\oplus{\\bf 10}\\oplus\\bar{\\bf 10}\\oplus{\\bf 27}$, it\nprojects out the ${\\bf 10}$, $\\bar{\\bf 10}$, and ${\\bf\n  27}$-dimensional representations.  As shown above, these are the\nrepresentations that cannot be formed from a single quark bilinear;\ni.e. they are not contained in $(\\bar{\\bf N},{\\bf N})\\oplus({\\bf\n  N},\\bar{\\bf N})$, as is clear from direct inspection of\neqs.~\\ref{eq:fundcona} and \\ref{eq:fundconb}.\n\n\n\\subsection{Gell-Mann-Oakes-Renner relation recovered}\n\n\\noindent We are now in a position to address the fate of instant-form QCD chiral-symmetry breaking condensates\nin null-plane QCD. Again using the Feynman-Hellmann theorem we find\n\\begin{eqnarray} \n\\mathbb{M}\\,\\langle\\, 0 \\, |\\,\\frac{\\partial T^{-+}}{\\partial \\mathbb{M}}\\,|\\, 0\\, \\rangle &=& \\mathbb{M}\\,\\frac{\\partial {\\tilde {\\cal E}}_0}{\\partial\n  \\mathbb{M}}\\ =\\ \n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0}{\\partial \\mathbb{M}}\\ =\\\n\\mathbb{M}\\,\\frac{\\partial {\\cal E}_0^{\\chi{\\rm PT}}}{\\partial\n  \\mathbb{M}} \\ , \n\\end{eqnarray} \nwhere $|\\, 0\\, \\rangle$ represents the null-plane QCD vacuum\nstate, and ${\\tilde {\\cal E}}_0$ is the null-plane QCD vacuum energy. In this equation we have also expressed\nthat physics is independent of the choice of coordinates. Therefore calculation\nof the leading quark-mass contribution to the vacuum energy must be independent of the quantization surface,\nand should be the same whether one works with the fundamental degrees of freedom, or with the Goldstone bosons\nin the infrared. One then has\n\\begin{eqnarray} \n\\mathbb{M}\\,\\frac{\\partial {\\tilde {\\cal E}}_0}{\\partial \\mathbb{M}}\\ =\\ -\\mathbb{M}\\,\\langle\\, 0\\, |\\,i\\;\\bar\\psi_+ \\gamma^+\n\\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\,|\\, 0\\, \\rangle  +\n\\langle\\, 0\\, |\\,  \\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \n\\,|\\, 0\\, \\rangle \\ .\n\\end{eqnarray} \nThe second term must vanish as the chiral charges annihilate the vacuum and therefore there can be no\nchiral-symmetry breaking condensates. Operationally one sees this directly by \ntaking the vacuum expectation value of eq.~\\ref{eq:algsola} which gives\n\\begin{eqnarray} \n\\langle\\, 0\\, |\\,  \\ft{i}{2}\n\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+ \n\\,|\\, 0\\, \\rangle \\ =\\ 0.\n\\end{eqnarray} \nWe are then left with the null-plane expression of the Gell-Mann-Oakes-Renner relation:\n\\begin{eqnarray} {\\mathbb{M}\\,\\langle\\, 0\\, |\\,i\\;\\bar\\psi_+ \\gamma^+\n  \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\,|\\, 0\\, \\rangle}\\ =\\ \\textstyle{1\\over 2} \\,\nN\\,M_\\pi^2\\,F_\\pi^2 \\ \\ +\\ \\dots \\ .  \n\\label{eq:npgoar}\n\\end{eqnarray} Hence, a chiral-symmetry breaking condensate in the instant-form\nformulation of QCD has been replaced by a chiral-symmetry conserving\ncondensate in the null-plane formulation. Note that while the operator\nnaively vanishes in the chiral limit, the matrix element is infrared\nsingular and therefore it need not, and indeed cannot, vanish in the\nchiral limit~\\footnote{This expression of the Gell-Mann-Oakes-Renner\n  formula was found previously in Ref.~\\cite{Wu:2003vn} using the\n  methods that will be described below.}.  It would be very\ninteresting to define the relevant operator non-perturbatively and\ncalculate this condensate directly, perhaps using transverse lattice\ngauge theory\nmethods~\\cite{Bardeen:1979xx,Burkardt:1998ws,Dalley:1998bj,Dalley:2001gj,Burkardt:2001mf,Burkardt:2001dy,Burkardt:2001jg,Dalley:2002nj,Bratt:2004wq}. Note\nthat {\\it a priori} knowledge of the singlet condensate in\neq.~\\ref{eq:npgoar} is not very different to {\\it a priori} knowledge of the symmetry-breaking quark condensate in\neq.~\\ref{eq:ifgmor}. In both cases, it is necessary to keep the quark masses finite and only\nat the very end take the chiral limit~\\cite{Banks:1979yr}.\n\n\n\\subsection{Condensates on a null-plane}\n\n\\noindent We will now derive the Gell-Mann-Oakes-Renner relation in a different way\nwhich will suggest a general prescription for expressing all\ninstant-form condensates with null-plane condensates.\nWhile the left- and right-handed components of $\\psi_+$ transform irreducibly with respect\nto the null-plane chiral charges, the transformation properties of $\\psi$ are complicated\nby the presence of the non-dynamical component $\\psi_-$. Indeed one finds\n\\begin{eqnarray} [\\, {\\tilde Q}^\\alpha_5\\, ,\\, \\psi \\, ]\\, =\\,\n-\\gamma_5\\,T^\\alpha\\,\\psi \\ -\\\ni\\;\\gamma_5\\;\\gamma^+\\;T^\\alpha\\;\\frac{1}{\\partial^+_\\mathbb{M}} \\psi \\ ,  \n\\label{eq:psitrans}\n\\end{eqnarray} \nfrom which it follows that \n\\begin{eqnarray}\n\\psi_{R}\\, ,\\, \\psi_{L}\\ \\in \\ ({\\bf 1},{\\bf N})\\oplus ({\\bf N},{\\bf 1}) \\quad , \\quad\n\\psi_{R}^\\dagger\\, ,\\,\\psi_{L}^\\dagger \\ \\in \\ ({\\bf 1},\\bar{\\bf\n  N})\\oplus(\\bar{\\bf N},{\\bf 1}) \\ .  \\end{eqnarray} \nSince the left- and right-handed components of the quark field\ntransform reducibly with respect to the chiral group, generally\nproducts of bilinear operators of the form $\\bar\\psi\\Gamma\\psi$ will have\ncomplicated reducible chiral transformation properties. However, QCD\noperators built out of these bilinears will always have a component that\ntransforms as a chiral singlet. We will now see, for the simplest\nexample, that this is essential to the consistency of the null-plane\nformulation. Consider the transformation properties of the following\nset of bilinears:\n\\begin{eqnarray}\n{D}^\\alpha_{5} \\ &\\equiv& \\ \\bar\\psi\\,\\gamma_5\\,T^\\alpha\\,\\psi\\ \\ , \\ \\\n{D}_{5} \\ \\equiv \\ \\bar\\psi\\,\\gamma_5\\,\\psi\\ ; \\\\\n{D}^{\\alpha} \\ &\\equiv& \\ \\bar\\psi\\,T^\\alpha\\,\\psi\\ \\ \\ , \\ \\ \\ \\\n{D} \\ \\equiv \\ \\bar\\psi\\,\\psi\\ .\n\\end{eqnarray}\nIs is again simple to check that these operators fill out the\n$(\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})$ representation of $SU(N)_R\\otimes SU(N)_L$\nwith respect to the instant-form chiral charges $Q_5^\\alpha$.\nNow consider the transformation properties of these operators \nwith respect to the null-plane chiral charges. One finds\n\\begin{eqnarray} \n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D}^\\beta_{5} \\, ]\\,\n=\\, -\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,\\left(\\,D + \\\n  i\\;\n  \\bar\\psi_+ \\gamma^+  \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\right) \\label{eq:GORf} \\nonumber \\\\\n&&\\quad\\qquad\\qquad\\qquad-\\,d^{\\alpha\\beta\\gamma}\\left(\\,D^\\gamma \\ +\\\n  i\\;\\bar\\psi_+ \\gamma^+ T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}}\n  \\psi_+\\right) \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D} \\, ]\\,\n=\\, -2\\,D^\\alpha_5 \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D}^\\beta \\, ]\\,\n=\\, -\\frac{1}{N}\\,\\delta^{\\alpha\\beta}\\,D_5 \\ -\\ d^{\\alpha\\beta\\gamma}\\,D_5^\\gamma\\nonumber\\\\\n&&\\quad\\qquad\\qquad\\qquad\n  +\\,f^{\\alpha\\beta\\gamma}\\,\\bar\\psi_+ \\gamma^+ \\gamma_5 T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}}\n  \\psi_+\\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,{D}_5 \\, ]\\,\n=\\, -2\\,D^\\alpha\\ -\\ 2i  \\bar\\psi_+ \\gamma^+ T^\\alpha \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ .\n\\end{eqnarray} \nTo close the algebra we must add, in addition, the commutation relations:\n\\begin{eqnarray} \n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,\\bar\\psi_+ \\gamma^+  \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\, ]\\ =\\ 0 \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,\\bar\\psi_+ \\gamma^+ T^\\beta \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\, ]\\ =\\ \ni\\,f^{\\alpha\\beta\\gamma} \\, \\bar\\psi_+ \\gamma^+ \\gamma_5 T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ ; \\\\\n&&[\\, {\\tilde Q}^\\alpha_5\\, ,\\,\\bar\\psi_+ \\gamma^+ \\gamma_5 T^\\beta \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\, ]\\ =\\ \ni\\,f^{\\alpha\\beta\\gamma} \\, \\bar\\psi_+ \\gamma^+ T^\\gamma \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ .\n\\end{eqnarray}\nHence the full set of operators transform as the reducible $4N^2$-dimensional\n$({\\bf 1},{\\bf 1})\\oplus({\\bf 1},{\\bf {\\cal A}})\\oplus({\\bf {\\cal A}},{\\bf 1})\\oplus (\\bar{\\bf N},{\\bf  N})\\oplus({\\bf N},\\bar{\\bf N})$ \nrepresentation of $SU(N)_R\\otimes SU(N)_L$, where here ${\\bf {\\cal A}}$ denotes the $SU(N)$ adjoint\nrepresentation. In particular one see that\n\\begin{eqnarray}\n\\bar\\psi\\psi \\in \\ (\\bar{\\bf N},{\\bf N})\\oplus({\\bf N},\\bar{\\bf N})\\oplus({\\bf 1},{\\bf 1})\\oplus\\ldots \\ ,\n\\end{eqnarray} \nand therefore transforms reducibly. This is verified by direct calculation which gives~\\footnote{Note that the second term, which is breaks chiral symmetry\nand is independent of the interaction does not appear in the free fermion Lagrangian as it is cancelled by a piece coming from the other kinetic term, as must be the\ncase in order that the Lagrangian commute with the helicity operator. }\n\\begin{eqnarray}\n\\hskip-1.3em\\mathbb{M}\\,\\bar\\psi\\psi \\ =\\ -i\\,\\mathbb{M}\\,\\bar\\psi_+ \\gamma^+ \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+ \\ +\\  \n\\ft{i}{2}\\bar\\psi_+ \\gamma^+\\, \\frac{1}{\\partial_\\mathbb{M}^+} \\left( \\bit{\\gamma}^r g\\, t^a\\bit{A}^{r}_a\\right)'\\psi_+\\ .\n\\label{eq:psibarpsiexpl}\n\\end{eqnarray} \nTaking the vacuum expectation value of eq.~\\ref{eq:psibarpsiexpl} (or eq.~\\ref{eq:GORf}) gives the general\nsolution~\\cite{Wu:2003vn}\n\\begin{eqnarray}\n\\langle\\, 0\\, |\\,\\bar\\psi\\psi\\,|\\, 0\\, \\rangle\\ = \\ -\\langle\\, 0\\, |\\,i\\;\\bar\\psi_+ \\gamma^+  \\frac{1}{\\partial^+_\\mathbb{M}} \\psi_+\\,|\\, 0\\, \\rangle \\ .\n\\label{eq:WuZhang}\n\\end{eqnarray} \nTherefore only the singlet part of $\\bar\\psi\\psi$ can acquire a vacuum\nexpectation value on the null plane, as must be the case since\n$SU(N)_R\\otimes SU(N)_L$ is a symmetry of the null-plane vacuum\nstate. This argument readily generalizes to any chiral symmetry\nbreaking Lorentz scalar operator, ${\\mathcal O}$, that one can build\nout of products of quark bilinears in instant-form QCD. One can write\n\\begin{eqnarray}\n{\\mathcal O} \\ =\\  \\sum_{\\bf{\\mathcal R}}\\,{\\mathcal O}_{\\bf{\\mathcal R}} \\ =\\  \\sum_{\\tilde{\\bf{\\mathcal R}}}\\,{\\cal O}_{\\tilde{\\bf{\\mathcal R}}}\\ + \\ {\\mathcal O}_{\\tilde{\\bf 1}} \n\\label{eq:gencondensate}\n\\end{eqnarray} \nwhere $\\bf{\\mathcal R}$ is a non-trivial chiral representation with respect to the instant-form chiral charges, $Q_{5\\alpha}$,\nand $\\tilde{\\bf{\\mathcal R}}$ (${\\tilde{\\bf 1}}$) is a non-trivial (the singlet) representation with respect to the \nfront-form chiral charges, ${\\tilde Q}_{5\\alpha}$. Unless protected by another symmetry, ${\\mathcal O}$ has a non-vanishing \nvacuum expectation value, which can be expressed as\n\\begin{eqnarray}\n\\langle\\, \\Omega \\, |\\,{\\mathcal O} \\,|\\, \\Omega\\, \\rangle\\ =\\ \n\\langle\\, \\Omega \\, |\\,\\sum_{\\bf{\\mathcal R}}\\,{\\mathcal O}_{\\bf{\\mathcal R}}\\,|\\, \\Omega\\, \\rangle\\ =\\ \n\\langle\\, 0 \\, |\\,{\\mathcal O}_{\\tilde{\\bf 1}} \\,|\\, 0\\, \\rangle \\ \\neq \\ 0 \\ .\n\\label{eq:gencond}\n\\end{eqnarray}\nNote that the final equality expresses an equivalence between a matrix element evaluated\nin the instant form and one in the front form. This equality ensures that physics is\nunmodified in moving between the two forms of dynamics.\nTherefore all instant-form chiral symmetry breaking QCD condensates map to chiral\nsymmetry conserving condensates in the front-form. The presence of the singlet\npart of the operator can always be traced to the reducible chiral transformation\nproperty of $\\psi$ given in eq.~\\ref{eq:psitrans}. For the case at hand, with \n${\\mathcal O}=\\bar\\psi\\psi$, we have\n\\begin{eqnarray}\n\\langle\\, \\Omega\\, |\\,\\bar\\psi\\psi\\,|\\, \\Omega\\, \\rangle \\ = \\ \\langle\\, 0\\, |\\,\\bar\\psi\\psi\\,|\\, 0\\, \\rangle \\ ,\n\\label{eq:psibarpsicond}\n\\end{eqnarray} \nwhich together with eq.~\\ref{eq:WuZhang}, provides the desired link between the instant-form and front-form expressions of\nthe Gell-Mann-Oakes-Renner relation.\n\nThe general relation, eq.~\\ref{eq:gencond} is important for the\nconsistency of null-plane QCD, as it demonstrates that, as expected,\nthe QCD vacuum energy is unaltered in moving from the instant-form to\nthe front-form description, and these relations must, of course, exist\nin order that the operator product expansion be independent of the\nchoice of quantization surface.  We see that a symmetry-breaking\ncondensate can form in the instant-form coordinates with an asymmetric\nvacuum which is equal to a corresponding symmetry-preserving\ncondensate in the null-plane description with a symmetric vacuum. The\ncondensate relation eq.~\\ref{eq:psibarpsicond} is one of an infinite\nnumber of relations which translates condensates which break chiral\nsymmetry in the instant form to null-plane condensates which transform\nas chiral singlets.\n\n\\section{Consequences of the operator algebra}\n\\label{sec:consequences}\n\n\\subsection{Summary of the null-plane QCD description}\n\n\\noindent Before considering the consequences of the null-plane QCD\noperator algebra, we will summarize the picture of chiral symmetry\nbreaking that we have so far established. While the null-plane QCD\nvacuum state is chirally invariant, chiral symmetry is spontaneously\nbroken by the three reduced Hamiltonians that have contributions,\n$M^2_{({\\bf N},{\\bf N})}$ and $\\left(M{\\cal J}_r\\right)_{({\\bf N},{\\bf\n    N})}$, which transform as $(\\bar{\\bf N},{\\bf N})\\oplus({\\bf\n  N},\\bar{\\bf N})$ with respect to $SU(N)_R\\otimes SU(N)_L$.  The\nthree reduced Hamiltonians satisfy the constraints,\neq.~\\ref{eq:fundmix}.  In addition to these signatures of chiral\nsymmetry breaking, the three reduced Hamiltonians, together with the\ngenerator of rotations on the transverse plane together generate the\n$U(2)$ dynamical sub-group of the null-plane Poincar\\'e algebra,\neq.~\\ref{eq:dynalgarbB}.  And finally, the null-plane vector and\nchiral charges satisfy the $SU(N)_R\\otimes SU(N)_L$ algebra,\neqs.~\\ref{eq:LCalga} and \\ref{eq:LCalgb}.  The entire set of\nLie-brackets provide all of the constraints that exist among the\ngenerators of the internal and space-time symmetries in null-plane\nQCD.  The consequences of chiral symmetry breaking for the spectrum\nand spin of QCD are contained in the symmetry-breaking parts of the\nreduced Hamiltonians.\n\n\\subsection{Recovery of spin-flavor symmetries}\n\n\\noindent In searching for solutions of the algebraic system that\nmixes the chiral charges and the reduced Hamiltonians, one may worry\nabout the existence of no-go theorems that forbid non-trivial algebras\nthat mix space-time and internal symmetries. In the null-plane\nformulation the no-go theorems are avoided because it is only the\ndynamical part, ${\\cal D}$, of the null-plane Poincar\\'e algebra that\nmixes with the internal symmetry generators~\\cite{Leutwyler:1977vz}.\nUnfortunately, a direct general solution of the null-plane QCD\noperator algebra in the general case appears difficult. However, there\nis a limiting case in which the algebra yields an important\nnon-trivial solution.  Here we will treat the QCD operator algebra as\nan abstract operator algebra and consider the limit in which the\nchiral-symmetry breaking part of the reduced Hamiltonian $M^2$ can be\ntreated as a perturbation. However, one should keep in mind that\nmatrix elements of the operator relations between hadronic states\nmust eventually be taken in order to extract observables.\nWe first define\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, M\\, ] \\, \\equiv \\,  \\epsilon^\\alpha \\ ,\n\\end{eqnarray}\nand neglect terms of ${\\cal O}(\\epsilon)$. This implies that all\nchiral symmetry breaking occurs in the spin Hamiltonians.\nThis limit is non-trivial, as we have shown above in \nsection~\\ref{susec:spin} that the spin Hamiltonians alone imply the presence\nof Goldstone bosons. In this limit, the QCD operator algebra reduces to\n\\begin{eqnarray}\n[\\, {\\cal J}_i \\, ,\\,   {\\cal J}_j  \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\,  {\\cal J}_k\n\\end{eqnarray}\nwhich generates $SU(2)$ spin, and the $SU(N)_R\\otimes SU(N)_L$ algebra,\n\\begin{eqnarray}\n[\\, {\\tilde Q}^\\alpha\\,  ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\, i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma \\ \\ , \\ \\ \n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde Q}^\\beta\\, ]\\, =\\,  i\\,f^{\\alpha\\beta\\gamma}\\,  {\\tilde Q}^\\gamma_5 \\ \\ , \\ \\\n[\\, {\\tilde Q}^\\alpha_5\\, ,\\, {\\tilde Q}^\\beta_5\\, ] \\, = \\,  i\\,f^{\\alpha\\beta\\gamma} \\, {\\tilde Q}^\\gamma  \\ .\n\\label{eq:QCDoa}\n\\end{eqnarray}\nThe remaining non-trivial mixed commutator is for the spin Hamiltonian:\n\\begin{eqnarray} \n{\\cal P}^{\\alpha\\beta ;\\mu\\nu}\\;[{\\tilde Q}^\\mu_5\\, ,\\, [\\, {\\tilde\n  Q}^\\nu_5\\, ,\\, {\\cal J}_\\pm]]\\ =\\ 0 \\ .  \n\\label{eq:OPQcomm2b}\n\\end{eqnarray} \nNow this simplified algebra can be put into a more familiar form. Consider\nan operator $G_{\\alpha i}$ which transforms in the adjoint of  $SU(N)$ and as a rotational\nvector in the sense that\n\\begin{eqnarray}\n&&[\\, {\\cal J}_i \\, ,\\,   {G}_{\\alpha j}  \\, ] \\ = \\ i\\,\\epsilon_{ijk}\\,  {G}_{\\alpha k} \\ ; \\\\\n&&[\\, {\\tilde Q}_\\alpha \\, ,\\,   {G}_{\\beta i}  \\, ]  \\ = \\ i\\,f_{\\alpha\\beta\\gamma}\\,  {G}_{\\gamma i} \\ .\n\\end{eqnarray}\nIn general, the commutator of $G^{\\alpha i}$ with itself may be expressed as\n\\begin{eqnarray}\n[\\, G_{\\alpha i} \\, ,\\, G_{\\beta j} \\, ] \\, =\\, \ni\\,f_{\\alpha\\beta\\gamma}\\, {\\cal A}_{ij,\\gamma}\\, +\\, i\\,\\epsilon_{ijk}\\,  {\\cal B}_{\\alpha\\beta,k} \\ ,\n\\label{eq:su4commgen}\n\\end{eqnarray}\nwhere ${\\cal A}_{ij,\\gamma}={\\cal A}_{ji,\\gamma}$ and ${\\cal\n  B}_{\\alpha\\beta,k}={\\cal B}_{\\beta\\alpha,k}$.  Now we identify\n$G^{\\alpha 3}\\ \\equiv \\ {\\tilde Q}^\\alpha_5$. From\neq.~\\ref{eq:QCDoa} it then follows that ${\\cal\n  A}_{33,\\alpha}={\\tilde Q}_\\alpha$. Rotational invariance then\nimplies ${\\cal A}_{ij,\\alpha}=\\delta_{ij}{\\tilde Q}_\\alpha$.  By\nconsidering Jacobi identities of ${\\cal J}_i$ and ${\\tilde Q}_\\alpha$\nwith the commutator in eq.~\\ref{eq:su4commgen} one finds, respectively,\n\\begin{eqnarray}\n&&[\\, {\\tilde Q}_\\gamma \\, ,\\,   {\\cal B}_{\\alpha\\beta,i}  \\, ] \\, =\\, i\\,f_{\\gamma\\beta\\mu}\\, {\\cal B}_{\\alpha\\mu,i} \\, +\\, i\\,f_{\\gamma\\alpha\\mu}\\, {\\cal B}_{\\beta\\mu,i}\\ ; \\\\\n&&[\\, {\\cal J}_i \\, ,\\,   {\\cal B}_{\\alpha\\beta,j}  \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\, {\\cal B}_{\\alpha\\beta,k}\\ ,\n\\end{eqnarray}\nwhich simply indicate that ${\\cal B}_{\\alpha\\beta,i}$ transforms as a rank-two $SU(N)$ tensor and a\nrotational vector.\n\nTo obtain ${\\cal B}_{\\alpha\\beta,i}$ we use eq.~\\ref{eq:OPQcomm2b} to find:\n\\begin{eqnarray}\n{\\cal P}_{\\alpha\\beta ;\\mu\\nu}\\;[\\, G_{\\alpha 3}\\, ,\\, G_{\\beta 1}\\, \\pm\\, i\\, G_{\\beta 2} \\, ] \\ = \\ 0 \\ , \n\\label{eq:su4commgen3}\n\\end{eqnarray} \nfrom which it follows that  ${\\cal B}_{\\alpha\\beta,2}$ and ${\\cal B}_{\\alpha\\beta,1}$ have a piece proportional\nto $\\delta_{\\alpha\\beta}$ and a piece proportional to $d_{\\alpha\\beta\\gamma}$. \nRotational invariance then determines that ${\\cal B}_{\\alpha\\beta,i}$ is a linear combination of\n$\\delta_{\\alpha\\beta}{\\cal J}_i$ and  $d_{\\alpha\\beta\\gamma}G_{\\gamma i}$. The coefficients of these\nterms are determined by considering the Jacobi identity of $G_{\\alpha i}$ with the commutator \nin eq.~\\ref{eq:su4commgen}, together with the relation among $SU(N)$ structure constants\ngiven in Appendix~\\ref{suN}. Finally, one obtains\n\\begin{eqnarray}\n[\\, G_{\\alpha i} \\, ,\\, G_{\\beta j} \\, ] \\, =\\, \ni\\,\\delta_{ij}\\,f_{\\alpha\\beta\\gamma}\\, {\\tilde Q}_\\gamma\\, +\\, \\frac{2}{N}\\,i\\,\\delta_{\\alpha\\beta}\\,\\epsilon_{ijk}\\,{\\cal J}_k  \\, + \\, i\\epsilon_{ijk}\\,d_{\\alpha\\beta\\gamma}\\, {G}_{\\gamma k} \\ ,\n\\label{eq:su2N}\n\\end{eqnarray}\nwhich together with \n\\begin{eqnarray} &&[\\, {\\tilde Q}_\\alpha\\, ,\\, G_{\\beta i}\\, ]\\, =\\,\ni\\,f_{\\alpha\\beta\\gamma} \\, G_{\\gamma i} \\ \\ \\ , \\ \\ \\\n[\\, {\\cal J}_i \\, ,\\,   G_{\\alpha j}  \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\,  G_{\\alpha k} \\ ; \\\\\n&&[\\, {\\tilde Q}_\\alpha\\, ,\\, {\\tilde Q}_\\beta\\, ]\\, =\\,\ni\\,f_{\\alpha\\beta\\gamma} \\, {\\tilde Q}_\\gamma \\ \\ \\ , \\ \\ \\ [\\, {\\cal\n  J}_i \\, ,\\, {\\cal J}_j \\, ] \\, =\\, i\\,\\epsilon_{ijk}\\, {\\cal J}_k\n\\end{eqnarray} \nclose the algebra of the symmetry group $SU(2N)$. To find the consequences\nof this algebra for observable quantities like the mass-squared matrix and\nthe matrix elements for Goldstone boson emission and absorption, one takes matrix elements of this\nalgebra between hadron states $h'$ and $h$, and neglecting transitions from single-particle to multi-particle\nstates in the completeness sums over intermediate states, one recovers the same algebra but with the replacements\n${\\tilde Q}_\\alpha\\rightarrow \\lbrack\\, T_\\alpha \\,\\rbrack_{h' h}\\,$ and ${\\tilde Q}_{5\\alpha}\\rightarrow \\lbrack\\, X_\\alpha(\\lambda) \\,\\rbrack_{h' h}\\,$,\nand corresponding replacements for $G_{\\beta i}$ and ${\\cal J}_k$.\nThis result, originally found by Weinberg~\\cite{Weinberg:1994tu}, is here shown to\nbe a general consequence of the null-plane QCD operator algebra, valid\nin any Lorentz frame.\n\nIt is important to emphasize that the $SU(2N)$ symmetry found here is\nonly operative in the full interacting field theory.  It is therefore\nunrelated to the $SU(2N)$ invariance of the QCD Lagrangian in the\nlimit of no interaction.  Indeed we have show above in\nsection~\\ref{susec:spin} that eq.~\\ref{eq:OPQcomm2b}, the main\ningredient in the derivation of $SU(2N)$, in itself implies the\nexistence of Goldstone bosons.  In addition, in a special case, this\nsymmetry does emerge in a well-defined limit of QCD.  As\n$\\langle\\,h'\\,|\\epsilon^\\alpha|\\, h\\,\\rangle\\sim M_h-M_{h'}$, and\nbaryons within a given large-$N_c$ multiplet have mass splittings that\nscale as $1\/N_c$~\\cite{Witten:1979kh}, the large-$N_c$ QCD scaling\nrules suggest that for baryons $\\epsilon^\\alpha\\sim 1\/N_c$.  Of\ncourse, as the matrix element of chiral charges between baryon states\nscales as $N_c$, the $SU(2N)$ symmetry reduces to the contracted\n$SU(2N)$~\\cite{Weinberg:1994tu,Beane:1998xp} for baryons in the\nlarge-$N_c$ limit, as one expects on general\ngrounds~\\cite{Gervais:1983wq,Dashen:1993as,Dashen:1993jt}.\n\nIt is instructive to consider a simple example. Consider the case\n$N=3$. Using the chiral transformation properties of the quarks,\neq.~\\ref{eq:chiralassign}, one sees that a $\\lambda=3\/2$ baryonic\noperator $\\psi_{+\\uparrow}\\psi_{+\\uparrow}\\psi_{+\\uparrow}$ transforms\nas $({\\bf 1},{\\bf 1})$, $({\\bf 1},{\\bf 8})$, or $({\\bf 1},{\\bf 10})$\nwith respect to $SU(3)_R\\otimes SU(3)_L$. Therefore, if the baryon is\na decuplet of $SU(3)_F$ with its $\\lambda=3\/2$ part in the $({\\bf 1},{\\bf 10})$, then one easily checks that its $\\lambda=1\/2$\npart must transform as $({\\bf 3},{\\bf 6})$ or $({\\bf 6},{\\bf 3})$.\nHowever, the different helicity states are unrelated by chiral\nsymmetry in itself. It is the mixed Lie-bracket,\neq.~\\ref{eq:OPQcomm2b}, the expression of broken chiral symmetry in\nthe spin Hamiltonian, that relates the helicities. Indeed taking the\n$\\lambda=1\/2$ decuplet to transform as $({\\bf 3},{\\bf 6})$ together\nwith an octet spin-$1\/2$ field and their negative-helicity partners in\n$({\\bf 10},{\\bf 1})\\oplus({\\bf 6},{\\bf 3})$ together fill out the\n${\\bf 56}$-dimensional representation of $SU(6)$ as is familiar from\nthe quark model. The difference here is that this symmetry arises from\nQCD symmetries and their pattern of breaking, and, in particular, has\nnothing to do with the non-relativistic limit.  Hence we see that\nstarting from the formal null-plane QCD operator algebra, the simple\nassumption that the part of the null-plane reduced Hamiltonian, $M^2$,\nthat breaks chiral symmetry is small implies all of the usual\nconsequences of the non-relativistic quark model, without the need of\nany further assumption like the existence of constituent quark degrees\nof freedom~\\cite{Weinberg:1994tu}.\n\n\\section{Conclusion}\n\\label{sec:conc}\n\n\\noindent Usually one views the spontaneous breaking of a symmetry as\nthe non-invariance of the vacuum state with respect to the\nsymmetry. However, in relativistic theories of quantum mechanics, this\npicture is purely a matter of convention. We have seen that the\nfront-form vacuum is a singlet with respect to all symmetries and yet\nspontaneous symmetry breaking can occur via non-conserved currents\nwhose divergences are directly proportional to S-matrix elements for\nthe emission and absorption of Goldstone bosons.  One may view the\nnull-plane description as a change of coordinates which moves\ndynamical information out of the vacuum state and into the interaction\noperators of the theory. The primary advantage of working with the\nnull-plane description is that broken chiral symmetry constraints\nbecome manifest in the sense that there are non-trivial Lie brackets\nbetween the Poincar\\'e generators and the broken symmetry\ngenerators. In the instant-form, the chiral constraints that appear\nnaturally in the front-form are present, but require one to work in\nspecial Lorentz frames and to make assumptions about the asymptotic\nbehavior of Goldstone-boson scattering amplitudes.\n\\vskip0.2in\n\nHere we will restate the main conclusions of this paper:\n\n\\vskip0.2in\n\\noindent$\\bullet$ In the front-form, spontaneous chiral symmetry\nbreaking is contained entirely in the three null-plane reduced\nHamiltonians, which encode the mass spectrum and spin content of a\ngiven theory. This must be the case as the null-plane chiral charges\nannihilate the vacuum state, and therefore chiral symmetry breaking\ncannot be attributed to the formation of chiral-symmetry breaking\ncondensates. In null-plane QCD, all chiral symmetry breaking\narises from the symmetry breaking parts of the reduced Hamiltonians,\ngiven explicitly in eqs.~\\ref{eq:symmbreakhamsa} and \\ref{eq:symmbreakhamsb}.\n\n\\vskip0.2in\n\\noindent$\\bullet$ Goldstone's theorem on the null-plane follows\ndirectly from the Lie-brackets between the null-plane Hamiltonians and\nthe chiral charges. A consistent null-plane description of spontaneous\nsymmetry breaking requires that a small explicit symmetry-breaking\noperator be included and that this explicit symmetry breaking be taken\nto zero only at the level of matrix elements of operators. The\ndivergence of the axial-vector current is proportional to the explicit\nsymmetry breaking. Therefore, as the current cannot be conserved in\nthe symmetry limit, the existence of massless states arises as a\nconsequence of the need to cancel the explicit breaking parameter that\nappears in its divergence.\n\n\\vskip0.2in\n\\noindent$\\bullet$ The Gell-Mann-Oakes-Renner relation is recovered in\nnull-plane QCD and a general prescription exists for translating all\nchiral-symmetry breaking condensates in instant-form QCD to\nchiral-singlet condensates in null-plane QCD.  It is therefore\nsimplistic to say that the vacuum is trivial in the front-form, since\nthere are necessarily symmetry-preserving condensates which arise from\nmodes with strictly zero longitudinal momentum. In particular,\nin contrast with claims in the literature~\\cite{Brodsky:2008xm,Brodsky:2008xu,Brodsky:2009zd,Brodsky:2010xf},\nwe expect that the QCD vacuum energy is unaltered in moving from\nthe instant-form to the front-form descriptions of QCD, as\nis essential for the consistency of null-plane QCD.\n\n\\vskip0.2in\n\\noindent$\\bullet$ A simple solution of the null-plane operator\nalgebra recovers the spin-flavor symmetry of the constituent quark\nmodel.  This result was obtained originally in\nRef.~\\cite{Weinberg:1994tu}, which obtained the algebra of charges and\nHamiltonians by working with sum rules obtained in special Lorentz\nframes, and using input from Regge-pole theory expectations of the\nasymptotic behavior of scattering amplitudes involving Goldstone\nbosons.  The results of the present work may be viewed as an attempt to clarify this\noriginal work by formulating it in a Lorentz frame-independent manner\nwhich follows directly from null-plane QCD.\n\n\\vskip0.2in\n\nIn the null-plane formulation of QCD, the loss of manifest Lorentz\ninvariance and locality are, operationally, a result of integrating\nout non-dynamical degrees of freedom. Physically, it is clear that the\nloss of Lorentz invariance is tied to the fact that the essence of\nLorentz invariance lies in the Poincar\\'e Lie brackets that must be\nsatisfied by the spin generators, and, of course, on the null-plane\nspin is dynamical and therefore requires the solution of the theory to\nproperly implement. By contrast, the non-locality of the theory would\nappear to be related to the fact that the null-plane chiral symmetry\nconstraints on observables are properly formulated as sum rules which\nspan many energy scales, and therefore do not exhibit the separation\nof scales that allows a useful description in terms of local\nLagrangian effective field theory.  Indeed, it appears that, in some\nsense, scattering amplitudes are the fundamental objects in the\nnull-plane formulation. This is particularly clear from the\nLie-brackets that mix the Poincar\\'e and chiral symmetry generators,\nwhich are given by the S-matrix elements for Goldstone boson emission\nand absorption.  From a theoretical standpoint, the most interesting\nconsequences of the results obtained in this paper are apparent only\nin the large-$N_c$ limit, which will be treated separately.\n\n\\vskip0.3in\n\n\\noindent \nI thank Ulf-G.~Mei\\ss ner for valuable comments on the manuscript, and\nT.~Becher, G.~Colangelo, H.~Leutwyler, F.~Niedermayer, and U.~Wenger\nfor useful discussions. I am particularly grateful to the Institute\nfor Theoretical Physics at the University of Bern for providing a\nstimulating work environment during academic year 2010\/2011.  The\nAlbert Einstein Center for Fundamental Physics is supported by the\n\"Innovations- und Kooperationsprojekt C-13\" of the \"Schweizerische\nUniversit\\\"atskonferenz SUK\/CRUS\". I gratefully acknowledge the\nhospitality of HISKP-theorie and the support of the Mercator programme\nof the Deutsche Forschungsgemeinschaft during academic year\n2012\/2013. This work was supported in part by NSF CAREER Grant\nPHY-0645570 and continuing grant PHY1206498.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nThe concept of deformation quantization has been introduced by Bayen,\nFlato, Fronsdal, Lichnerowicz and Sternheimer in their seminal paper\n\\cite{bayen.et.al:1978a} based on the theory of associative\ndeformations of algebras \\cite{gerstenhaber:1964a}.  A formal star\nproduct on a Poisson manifold $M$ is defined as a formal associative\ndeformation of the algebra of smooth functions $\\Cinfty (M)$\non $M$ (the name comes from the notation $\\star$ for the deformed product) and its existence has been proved as a corollary of the\nso-called \\emph{formality theorem} in \\cite{kontsevich:2003a} (for\nmore details in deformation quantization we refer to the textbooks\n\\cite{esposito:2015a, waldmann:2007a}). On the other hand, Drinfeld\nintroduced the notion of quantum groups in the setting of formal\ndeformations, see e.g. the textbooks \\cite{chari.pressley:1994a,\n  etingof.schiffmann:1998a} for a detailed discussion.  Drinfeld also\nintroduced the idea of using symmetries to get formal\ndeformations. More explicitely, given an action by derivations of a\nLie algebra $\\lie{g}$ on an associative algebra $(\\algebra{A},\nm_{\\algebra A})$, the definition of the so-called \\emph{Drinfeld\n  twist} \\cite{drinfeld:1983a, drinfeld:1988a} $J \\in\n(\\algebra{U}({\\lie{g}}) \\tensor \\algebra{U}(\\lie g))[\\![\\hbar]\\!]$ allows us\nto obtain an associative formal deformation of $\\algebra{A}$ by means\nof a \\emph{universal deformation formula}\n\\begin{equation}\n    \\label{eq:TheUDF}\n    a \\star_{J} b\n    =\n    m_{\\algebra A} (J\\acts (a \\tensor b))\n\\end{equation}\nfor $a, b \\in \\algebra{A}[\\![\\hbar]\\!]$. Here $\\acts$ is the action of\n$\\lie{g}$ extended to the universal enveloping algebra\n$\\algebra{U}(\\lie g)$ and then to $\\algebra{U}(\\lie g) \\tensor\n\\algebra{U}(\\lie g)$ acting on $\\algebra{A} \\tensor \\algebra{A}$.  The\ndeformed algebra $(\\algebra A[\\![\\hbar]\\!], \\star_J)$ is then a\nmodule-algebra for the quantum group:\n\\begin{equation}\n   \\algebra U_J(\\lie{g}) \n   := (\\algebra U(\\lie{g})[\\![\\hbar]\\!], \\Delta_J \n   := J \\Delta J^{-1}).\\end{equation}\nIn other words, Drinfeld obtains a quantized action. \nWe mention here that the relevance of deformations induced via symmetries has been deeply investigated in\n\\cite{giaquinto.zhang:1998a} and in a non-formal setting in\n\\cite{bieliavsky.gayral:2015a}. \n\nThe aim of this paper consists in obtaining a more general notion of\ndeformation through symmetry, by using formality theory. We focus on\nthe quantization of Lie algebra actions in the particular case of\ntriangular Lie algebras. Such actions can be regarded as the\ninfinitesimal version of Poisson Lie group actions (see e.g.\n\\cite{Kosmann-Schwarzbach2004,Semenov1985}) and they are very\nimportant in the context of integrable systems. Triangular Lie\nalgebras and their quantizations have been studied by many authors,\nsee e.g. \\cite{Calaque2006,enriquez.etingof:2005,xu:2002a}.\nThe idea of applying formality to actions has also been used \nin \\cite{Arnal2007}, where\nthe authors use the Kontsevich formality on a Poisson manifold to construct\nfor each Poisson vector field a derivation of the star product. We recover this result.\n\nThe formality theorem states the existence of an\n$L_\\infty$-quasi-isomorphism from polyvectorfields to polydifferential\noperators on a manifold $M$. In \\cite{Dolgushev2005, Dolgushev2005a}\nDolgushev proves the theorem for general $M$ using the proof for\n$M=\\mathbb{R}^n$. In order to construct such $L_\\infty$-quasi-isomorphisms,\nDolgushev uses Fedosov's methods \\cite{fedosov:1994a} concerning\nformal geometry, Kontsevich's quasi-isomorphism\n\\cite{kontsevich:2003a} and the twisting procedure inspired by Quillen\n\\cite{Quillen69}. Following the construction provided by Dolgushev,\nCalaque proved a formality theorem for Lie algebroids\n\\cite{Calaque2005}.\nWe consider an infinitesimal action of $\\lie g$ on $M$, i.e. a Lie\nalgebra homomorphism $\\varphi\\colon\n\\mathfrak{g}\\rightarrow\\Secinfty(TM)$. This can immediately be\nextended to a DGLA morphism\n\\begin{equation}\n  \\Tpoly{\\mathfrak{g}}{} \\longrightarrow \\Tpoly{}{}(M),\n\\end{equation}\nwhere $\\Tpoly{\\mathfrak{g}}{} = \\wedge^\\bullet \\lie g$ and\n$\\Tpoly{}{}(M) = \\Secinfty(\\wedge^\\bullet TM)$  with the brackets extended via a Leibniz rule. From the formality\ntheorem we know that we have the following $L_\\infty$-quasi-isomorphisms\n\\begin{equation}\n  \\Tpoly{\\mathfrak{g}}{} \\longrightarrow \\Dpoly{\\mathfrak{g}}{}\n  \\quad \\mbox{and} \\quad\n  \\Tpoly{}{}(M) \\longrightarrow \\Dpoly{}{}(M).\n\\end{equation}\nUsing the quasi-invertibility of $L_\\infty$-quasi-isomorphisms we obtain\nthe existence of an $L_\\infty$-morphism \n\\begin{equation}\n\\label{eq:totwist}\n  \\Dpoly{\\mathfrak{g}}{} \\longrightarrow \\Dpoly{}{}(M).\n\\end{equation}\nIf the Lie algebra $\\lie g$ is endowed with an $r$-matrix, i.e.  an\nelement $r \\in \\lie g \\wedge \\lie g$ satisfying the Maurer--Cartan\nequation $\\Schouten{r, r} = 0$, the action always induces a Poisson\nstructure on $M$ and it is automatically a Poisson action.\n\\begin{lemma}\n  \\leavevmode\n  \\begin{lemmalist}\n  \\item Given the formal Maurer--Cartan element $\\hbar r \\in\n    \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]$, we obtain via formality a\n    Maurer--Cartan element $\\rho_\\hbar$. This yields a quantum group\n    $\\algebra U_{\\rho_\\hbar} (\\lie g)$ with deformed coproduct $\\Delta_{1\\otimes 1+\\rho_\\hbar}$.\n  \\item Given the  Maurer--Cartan element $\\hbar \\pi =\n    \\varphi\\wedge \\varphi(r) \\in \\Tpoly{}{}(M)[\\![\\hbar]\\!]$, we obtain via\n    formality a Maurer--Cartan element $B_\\hbar$. This induces a\n    formal deformation $(\\Cinfty(M)[\\![\\hbar]\\!], \\star_{B_\\hbar})$ of the Poisson\n    algebra $(\\Cinfty(M), \\pi)$.\n  \\end{lemmalist}\n\\end{lemma}\nThe DGLA obtained from twisting the DGLA of $\\lie g$-polydifferential\noperators on the point $\\Dpoly{\\lie g}{}$ as given in \\cite{Calaque2005} turns\nout to be a special case of a DGLA canonically associated to any Hopf algebra\n$H$, which we call $H_{poly}$. Given any Maurer-Cartan element $F\\in H_{poly}$ \nthere is the associated Drinfeld twist $J=1\\otimes 1 + F$ (in the formal\nsense). It turns out that the twisted DGLA $H_{poly}^F$ is canonically\nisomorphic to $(H_J)_{poly}$ where $H_J$ denotes the Hopf algebra twisted by\n$J$. Thus, using the twisting procedure on the $L_\\infty$-morphism\n\\eqref{eq:totwist} we prove the following theorem.\n\n\\begin{theorem}\n  Let $\\lie g$ be a Lie algebra endowed with a classical $r$-matrix\n  and a Lie algebra action $\\varphi\\colon\n  \\mathfrak{g}\\rightarrow\\Secinfty(TM)$ inducing a Poisson structure\n  on $M$ by $\\pi := \\varphi\\wedge\\varphi(r)$.  Then, there exists an\n  $L_\\infty$-morphism $(\\algebra U_F(\\lie g)[\\![\\hbar]\\!])_{poly} \\to\n  C(\\algebra A_\\hbar;\\algebra A_\\hbar)$ between the DGLA associated to the quantum group $\\algebra U_{\\rho_\\hbar}(\\lie g)[\\![\\hbar]\\!]$ and the Hochschild complex of the\n  deformation quantization $\\algebra A_\\hbar$ of $\\Cinfty(M)$.\n\\end{theorem}\nThis theorem motivates a definition, which generalizes \nDrinfeld quantized action. \n \\begin{definition}[Deformation Symmetry]\n   A deformation symmetry of a Hopf algebra $H$ in \n   a unital associative algebra $\\algebra A$ is a map\n   \\begin{equation} \n     \\Phi\n     \\colon \n     H_{poly}\\longrightarrow C(\\algebra A)\n   \\end{equation}\n   of $L_\\infty$-algebras. \n\\end{definition}\nComparing the quantized structures obtained with our approach, it is \neasy to see that we recover Drinfeld's universal deformation formulas. \n\n\\vspace{0.3cm}\n\nThe paper is organized as follows. In Section~\\ref{sec:Preliminaries}\nwe recall the language of $L_\\infty$-algebras and the theorem, due to\nKontsevich, stating the existence of an $L_\\infty$-quasi-isomorphism\nbetween polyvector fields and polydifferential operators on the formal\ncompletion at $0\\in \\mathbb{R}^d$.  In Section~\\ref{sec:formality} we briefly\ndiscuss the proof of formality for Lie algebroids, following \n\\cite{Calaque2006, Dolgushev2005, Dolgushev2005a}. In particular, we \nrecall the twisting procedure in the curved context.\nSection~\\ref{sec:FormalityActions}\ncontains the main\nresults of the paper, i.e. the construction of an $L_\\infty$-morphism\nout of a Poisson action and the discussion on twisted structures and\ndeformation symmetry. Finally we compare our approach with Drinfeld's\ndeformation formulas.\n\n\\section*{Acknoledgments}\nThe authors are grateful to Ryszard Nest and Stefan Waldmann \nfor the inspiring discussions. \n\n\n\n  \n\n\\section{Preliminaries}\n\\label{sec:Preliminaries}\n\nGiven a graded vector space $V^\\bullet$ over $\\mathbb{K}$ \nwe denote the $k$-\\emph{shifted} vector space by $V[k]^\\bullet$, it\nis given by\n\\begin{equation}\n  V^\\bullet[k]^l=V^{l+k}\n\\end{equation}\n\n\n\n\n\\subsection{$L_\\infty$-setting}\n\\label{sec:Linfty}\n\nWe shall recall the definitions of $L_\\infty$-algebra and\n$L_\\infty$-morphisms for the convenience of the reader (and to fix\ncertain conventions). For the rest of this section we consider a\nfield $\\mathbb{K}$ of characteristic $0$. Although many constructions \nwill also allow for replacement of $\\mathbb{K}$ by a PID containing the rationals.\n\\begin{definition}[$L_\\infty$-algebra]\n  A degree $+1$ coderivation $Q$ on the co-unital conilpotent\n  cocommutative coalgebra $S^c(\\mathfrak{L})$ cofreely cogenerated by\n  the graded vector space $\\mathfrak{L}[1]^\\bullet$ over $\\mathbb{K}$ is\n  called an $L_\\infty$-structure on the graded vector space $\\mathfrak{L}$ if\n  $Q^2=0$.\n\\end{definition}\nIn more explicit terms we have\n\\begin{equation}\n  S^c(\\mathfrak{L})=\\bigoplus_{k=0}^\\infty \\Anti^{k}\n  \\left(\\mathfrak{L}[1]\\right)\n\\end{equation}\nequipped with the coproduct $\\Delta$ given by \n\\begin{align}\n  \\Delta(1)\n  &=\n  1\\otimes 1\n  \\qquad \\mbox{and}\n  \\\\\n  \\Delta(\\gamma_1\\wedge\\ldots\\wedge \\gamma_k)\n  &=\n  1\\otimes\\gamma_1\\wedge\\ldots\\wedge\\gamma_k\n  +\n  \\gamma_1\\wedge\\ldots\\wedge\\gamma_k\\otimes 1 \n  +\n  \\overline{\\Delta}(\\gamma_1\\wedge\\ldots\\wedge\\gamma_k)\n\\end{align}\nfor $k\\geq1$ and any $\\gamma_i\\in \\mathfrak{L}[1]$. Here we have\n\\begin{equation}\n  \\overline{\\Delta}(\\gamma_1\\wedge\\ldots\\wedge \\gamma_k)=\n  \\sum_{i=1}^{k-1}\\sum_{\\sigma\\in\\mbox{\\tiny Sh($i$,$k-i$)}}\\epsilon(\\sigma)\n  \\gamma_{\\sigma(1)}\\wedge\\ldots\\wedge \\gamma_{\\sigma(i)}\\bigotimes\n  \\gamma_{\\sigma(i+1)}\\wedge\\ldots\\wedge \\gamma_{\\sigma(k)},\n\\end{equation}\nwhere $\\mathrm{Sh}(i,k-i)$ denotes the $(i,k-i)$ shuffles in the symmetric\ngroup $S_k$ in $k$ letters and the \\emph{Koszul sign}\n$\\epsilon(\\sigma)=\\epsilon(\\sigma, \\gamma_1,\\ldots, \\gamma_k)$ is\ndetermined by the rule\n\\begin{equation}\n  \\gamma_1\\wedge\\ldots\\wedge\\gamma_k=\\epsilon(\\sigma)\n  \\gamma_{\\sigma(1)}\\wedge\\ldots\\wedge\\gamma_{\\sigma(k)}.\n\\end{equation}\nRecall that $S^c(\\mathfrak{L})$ is given by the co-invariants of the\ntensor algebra for the action of the symmetric groups generated by\n\\begin{equation}\n  (i \\ i+1)(\\gamma_1\\otimes\\ldots\\otimes \\gamma_k)\n  =\n  (-1)^{|\\gamma_i||\\gamma_{i+1}|}\\gamma_{1}\\otimes\\ldots\\otimes \n  \\gamma_{i-1}\\otimes \\gamma_{i+1}\\otimes \\gamma_i\\otimes \\gamma_{i+2}\\otimes\\ldots \\gamma_{k},\n\\end{equation}\nwhere we use the vertical bars to denote the shifted degree, i.e. the\ndegree of $\\gamma_i$ in $\\mathfrak{L}[1]$. The co-unit is given by the\nprojection $\\pr_\\mathbb{K}$ onto the ground field $\\mathbb{K}$.\n\\begin{remark}\n  \\label{rem:bialgebra}\n  A direct computation shows that, denoting the flip $a\\otimes\n  b\\mapsto (-1)^{|a||b|}b\\otimes a$ by $\\tau$, we have\n  %\n  \\begin{equation}\n    \\Delta\\circ\n    (\\argument\\wedge\\argument)\n    =\n    (\\argument\\wedge\\argument)\n    \\otimes\n    (\\argument\\wedge\\argument)  \n    \\circ\n    (\\id\\otimes\\tau\\otimes\\id)\\circ\\Delta\\otimes\\Delta.\n  \\end{equation}\n \n  So we obtain the unital and co-unital bialgebra\n  $(S^c(\\mathfrak{L}),\\argument\\wedge\\argument, 1\\in \\mathbb{K},\n  \\Delta,\\pr_\\mathbb{K})$, i.e. $1\\wedge X=X$ for all $X\\in\n  S^c(\\mathfrak{L})$.  We sometimes abuse notation by omitting\n  $\\wedge$ in favor of simple concatenation or superscripts, e.g.  $ab\n  := a\\wedge b$ and $x^3 := x\\wedge x\\wedge x$.\n\\end{remark}\n\\begin{lemma}[Characterization of coderivations]\n  Every degree $+1$ coderivation $Q$ on $S^c(\\mathfrak{L})$ is uniquely\n  determined by the components\n  %\n  \\begin{equation}\n    Q_n\\colon \\Anti^n(\\mathfrak{L}[1])\\longrightarrow \\mathfrak{L}[2]\n  \\end{equation}\n  %\n  by the formula \n  %\n  \\begin{equation}\n    Q(\\gamma_1\\wedge\\ldots\\wedge\\gamma_n)\n    =\n    \\sum_{k=0}^n\\sum_{\\sigma\\in\\mbox{\\tiny Sh($k$,$n-k$)}}\n    \\epsilon(\\sigma)Q_k(\\gamma_{\\sigma(1)}\\wedge\\ldots\\wedge\n    \\gamma_{\\sigma(k)})\\wedge\\gamma_{\\sigma(k+1)}\\wedge\n    \\ldots\\wedge\\gamma_{\\sigma(n)},\n  \\end{equation} \n  %\n  where we use the conventions that $\\mathrm{Sh}(n,0) = \\mathrm{Sh}(0,n) = \\{\\id\\}$\n  and that the empty product equals the unit. \n\\end{lemma}\n\\begin{proof}\n  It follows by simply writing out both sides of the defining equation\n  \\begin{equation}\n    \\Delta\\circ Q\n    =\n    (Q\\otimes \\id+\\id\\otimes Q)\\circ\\Delta.\n  \\end{equation}\n\\end{proof}\nNote that $Q_0(1)$ is of degree $1$ in $\\mathfrak{L}[1]$ (thus of\ndegree $2$ in $\\mathfrak{L}$).  The condition $Q^2=0$ can now be\nexpressed in terms of a quadratic equation in the components $Q_n$.\n\\begin{example}[Curved Lie algebras]\n  \\label{ex:CurvedLie}\n  Our main example of an $L_\\infty$-algebra is given by (curved) Lie\n  algebras, i.e. the tuple $(\\mathfrak{L},R,\\D,[\\argument,\\argument])$ where\n  we set $Q_0(1) = R$, $Q_1 = \\D$, $Q_2 = [\\argument,\\argument]$ and $Q_i=0$\n  for all $i\\geq 3$. The condition $Q^2=0$ amounts to:\n  \\begin{itemize} \n  \\item $\\D R=0$, \n  \\item $\\D^2(\\argument)=[R,\\argument]$,\n  \\item $\\D$ is a derivation of $[\\argument,\\argument]$,\n  \\item The graded Jacobi identity for $[\\argument,\\argument]$.\n  \\end{itemize}\n\\end{example}\n\\begin{remark}\n  We should note that the our definition of $L_\\infty$-algebra is\n  usually called \\emph{curved} $L_\\infty$-algebra (see e.g.~\n  \\cite{Markl100}).  Although this definition is also not set in\n  stone, see for instance \\cite{GradyGwilliam} for yet another notion\n  of curved $L_\\infty$-algebra.  For the purpose of this paper it is,\n  however, more convenient to call the curved version simply\n  $L_\\infty$-algebra.  The only $L_\\infty$-algebras playing a role in this paper are, however, the flat $L_\\infty$-algebras, i.e.\n  those having $Q_0=0$.  The usual definition for an\n  $L_\\infty$-algebra thus coincides with our definition of flat\n  $L_\\infty$-algebra.\n\\end{remark}\n\\begin{remark}\n  \\label{rem:filtration}\n  In the following we have to deal with various infinite sums. In\n  order for this to make sense, we always consider only\n  $L_\\infty$-algebras $\\mathfrak{L}$ that are equipped with a\n  decreasing filtration\n  \\begin{equation}\n    \\mathfrak{L}\n    =\n    \\mathcal{F}^0\\mathfrak{L}\n    \\supset\\mathcal{F}^1\\mathfrak{L}\n    \\supset\\ldots\\supset\\mathcal{F}^k\\mathfrak{L}\\supset\\ldots,\n  \\end{equation}\n  respecting the $L_\\infty$-structure and which is moreover\n  \\emph{complete}, i.e.\n  \\begin{equation}\n    \\bigcap_k\\mathcal{F}^k\\mathfrak{L}\n    =\n    \\{0\\}.\n  \\end{equation}\n  This yields a corresponding complete metric topology and we consider\n  convergence of infinite sums in terms of this topology.\n\\end{remark}\n\\begin{definition}[$L_\\infty$-morphisms]\n  \\label{def:Linftymorph}\n  Let $\\mathfrak{L}$ and $\\widetilde{\\mathfrak{L}}$ be\n  $L_\\infty$-algebras.  A degree $0$ filtration preserving co-unital\n  co-algebra morphism\n  \\begin{equation}\n    F\\colon \n    S^c(\\mathfrak{L})\n    \\longrightarrow \n    S^c(\\widetilde{\\mathfrak{L}})\n  \\end{equation}\n  such that $FQ = \\widetilde{Q}F$ is called\n  an $L_\\infty$-morphism.\n\\end{definition}\n\\begin{lemma}[Characterization of co-algebra morphisms]\n  A co-algebra morphism $F$ from\\\\ $S^c(\\mathfrak{L})$ to\n  $S^c(\\widetilde{\\mathfrak{L}})$ is uniquely determined by its\n  components, also called Taylor coefficients,\n  \\begin{equation}\n    F_n\n    \\colon \n    \\Anti^n(\\mathfrak{L}[1])\n    \\longrightarrow \n    \\widetilde{\\mathfrak{L}}[1],\n  \\end{equation}\n  where $n\\geq 1$. Namely, we set $F(1)=1$ and use the formula \n  \\begin{equation}\n    \\label{eq:coalgebramorphism}\n    \\begin{gathered}\n      F(\\gamma_1\\wedge\\ldots\\wedge\\gamma_n)= \n      \\\\     \n      \\sum_{p\\geq1}\\sum_{\\substack{k_1,\\ldots, k_p\\geq1\\\\k_1+\\ldots+k_p=n}}\n      \\sum_{\\sigma\\in \\mbox{\\tiny Sh($k_1$,..., $k_p$)}}\\frac{\\epsilon(\\sigma)}{p!}\n      F_{k_1}(\\gamma_{\\sigma(1)}\\wedge\\ldots\\gamma_{\\sigma(k_1)})\\wedge\\ldots\\wedge \n      F_{k_p}(\\gamma_{\\sigma(n-k_p+1)}\\wedge\\ldots\\wedge\\gamma_{\\sigma(n)}),\n    \\end{gathered}\n  \\end{equation}\n  where $Sh(k_1,...,k_p)$ denotes the set of $(k_1,\\ldots,\n  k_p)$-shuffles in $S_n$ and $Sh(n) = \\{\\id\\}$.\n\\end{lemma}\n\\begin{proof} \n  It simply follows by writing out the defining equation\n  \\begin{equation}\n    \\Delta\\circ F\n    =\n    F\\otimes F\\circ \\Delta.\n  \\end{equation}\n\\end{proof}\n\\begin{example} \n  Let $(\\mathfrak{L},R,\\D,[\\argument,\\argument])$ and\n  $(\\mathfrak{L}',R',\\D',[\\argument,\\argument]')$ be two curved Lie\n  algebras and consider the morphism $C\\colon\n  \\mathfrak{L}\\longrightarrow \\mathfrak{L}'$ of curved Lie algebras,\n  i.e. $C(R)=R'$, $C\\D = \\D'C$ and $C$ is a morphism of the underlying\n  Lie algebras.  Then the map $F$ given by applying the formula\n  \\eqref{eq:coalgebramorphism} to the components $F_1=C$ and $F_i=0$\n  for $i\\geq 2$ is an $L_\\infty$-morphism. In general, if $F\\colon\n  \\mathfrak{L}\\longrightarrow \\mathfrak{L}'$ is an\n  $L_\\infty$-morphism, then $F_1(R)=R'$, but we only have\n  $\\D'F_1(\\gamma)=F_1\\D(\\gamma)+F_2(R\\wedge\\gamma)$.\n\\end{example}\nNote that, given an $L_\\infty$-morphism of \\emph{flat}\n$L_\\infty$-algebras $\\mathfrak{L}$ and $\\widetilde{\\mathfrak{L}}$, we\nobtain the map of complexes\n\\begin{equation}\n  F_1\n  \\colon \n  (\\mathfrak{L},Q_1)\n  \\longrightarrow \n  (\\widetilde{\\mathfrak{L}},\\widetilde{Q}_1).\n\\end{equation}\n\\begin{definition}[$L_\\infty$-quasi-isomorphism]\n  \\label{def:Linftyquis}\n  An $L_\\infty$-morphism $F$ is called $L_\\infty$-quasi-isomorphism if\n  $F_1$ is a quasi-isomorphism of complexes.\n\\end{definition}\nThe $L_\\infty$-quasi-isomorphisms we deal with in this paper happen to\nbe the ones witnessing \\emph{formality}, let us therefore introduce\nthe notion of formal $L_\\infty$-algebras here.\n\\begin{definition}[Formal $L_\\infty$-algebra]\n  \\label{defformal}\n  An $L_\\infty$-algebra $\\mathfrak{L}$ is called formal if it is flat\n  and admits an $L_\\infty$-quasi-isomorphism\n  \\begin{equation}\n    F\n    \\colon \n    \\mathrm{H}(\\mathfrak{L})\\longrightarrow \\mathfrak{L}\n  \\end{equation}\n  for the $L_\\infty$-structure canonically induced on the cohomology\n  $\\mathrm{H}(\\mathfrak{L})$ of $\\mathfrak{L}$.\n\\end{definition}\nFinally, a crucial concept for this paper is the one of Maurer--Cartan\nelements, that we define below.\n\\begin{definition}[Maurer-Cartan element]\n  \\label{MCdef} \n  Given an $L_\\infty$-algebra $(\\mathfrak{L}, Q)$, an element $\\pi\\in\n  \\mathcal{F}^1\\mathfrak{L}[1]^0$ is called a Maurer-Cartan or MC\n  element if it satisfies the following equation\n  \\begin{equation}\n    \\label{eq:MC}\n    \\sum_{n=0}^\\infty \\frac{Q_n(\\pi^n)}{n!}=0.\n  \\end{equation}\n\\end{definition}\n\n\n\n\n\n\n\n\\subsection{Local Formality}\n\\label{sec:LocalFormality}\n\nLet us denote the formal completion at $0\\in \\mathbb{R}^d$ by $\\Rf$. The\nsmooth functions $\\Cinfty(\\Rf)$ on $\\Rf$ are given by the algebra\n\\begin{equation}\n  \\Cinfty(\\Rf)\n  :=\n  \\varprojlim_{k\\rightarrow \\infty} \\Cinfty(\\mathbb{R}^d)\/\\mathcal{I}_0^k,\n\\end{equation} \nwhere $\\mathcal{I}_0$ denotes the ideal of functions vanishing at\n$0\\in\\mathbb{R}^d$.  Note that $\\Cinfty(\\Rf)$ comes equipped with the complete\ndecreasing filtration\n\\begin{equation}\n  \\Cinfty(\\Rf)\\supset\\mathcal{I}_0\\supset\\mathcal{I}_0^2\\supset\\ldots\n\\end{equation}\nand its corresponding (metric) topology. The Lie algebra of continuous\nderivations of $\\Cinfty(\\Rf)$ is denoted by\n$\\Tpoly{}{0}(\\Rf)$. By setting $\\Tpoly{}{-1} := \\Cinfty(\\Rf)$ we obtain\nthe Lie--Rinehart pair $(\\Tpoly{}{-1},\\Tpoly{}{0})$ and the graded\nvector space\n\\begin{equation}\n  \\Tpoly{}{} (\\Rf)\n  :=\n  \\bigoplus_{k\\geq -1}\\Tpoly{}{k} (\\Rf),\n\\end{equation}\nwhere $\\Tpoly{}{k} (\\Rf) := \\Anti^{k+1}\\Tpoly{}{0} (\\Rf)$ for $k\\geq\n0$.  Here the tensor product is understood to be over\n$\\Tpoly{}{-1}(\\Rf)$ and completed.  Notice that there is no confusion\nabout grading here although it may seem unnatural at first glance. It\nis actually obtained by shifting the natural grading.  The natural\nstructure is that of \\emph{Gerstenhaber algebra}, but we are only\nconsidering the underlying graded Lie algebra.  The Lie bracket\n$\\Schouten{\\argument,\\argument}$ on $\\Tpoly{}{0}(\\Rf)$ extends to a\ngraded Lie algebra structure on $\\Tpoly{}{}(\\Rf)$ by the rules\n\\begin{equation}\n  \\begin{aligned}\n    \\Schouten{f,g} \n    &= \n    0 ,\n    \\\\\n    \\Schouten{X_0,f}  \n    &=  \n    X_0(f), \n    \\\\\n    \\Schouten{X_0\\wedge\\ldots\\wedge X_k,Y} \n    &=      \n    \\sum_{j=0}^k(-1)^{kl+j}\\Schouten{X_j,Y}\\wedge X_0\\wedge\\ldots\\wedge\n    X_{j-1}\\wedge X_{j+1}\\wedge\\ldots\\wedge X_k\n  \\end{aligned}\n\\end{equation} \nfor all $f,g \\in \\Tpoly{}{-1}(\\Rf)$, $X_0,\\ldots, X_k \\in\n\\Tpoly{}{0}(\\Rf)$ and $Y \\in \\Tpoly{}{l}(\\Rf)$.\n\nThe universal enveloping algebra of the Lie-Rinehart pair\n$(\\Tpoly{}{-1}(\\Rf),\\Tpoly{}{0}(\\Rf))$ is denoted by\n$\\Dpoly{}{0}(\\Rf)$. Recall that $\\Dpoly{}{0}(\\Rf)$ is naturally\nequipped with the structures of a bialgebra (see\ne.g. \\cite{Moerdijk2010}). More precisely, $\\Dpoly{}{0}(\\Rf)$ allows\nan $\\mathbb{R}$-algebra structure $\\cdot$ and an $\\mathbb{R}$-coalgebra structure\n$\\Delta$. We extend the algebra structure in the obvious\n(componentwise) way to\n\\begin{equation}\n  \\Dpoly{}{}(\\Rf)\n  :=\n  \\bigoplus_{k\\geq -1}\\Dpoly{}{k}(\\Rf),\n\\end{equation} \nwhere $\\Dpoly{}{-1} (\\Rf):= \\Tpoly{}{-1}(\\Rf)$ and $\\Dpoly{}{k} (\\Rf)\n:= \\left(\\Dpoly{}{0}(\\Rf)\\right)^{\\otimes k+1}$. Again the\ntensor product is understood to be over $\\Dpoly{}{-1}(\\Rf)$ and\ncompleted. This allows us to define two $\\mathbb{R}$-bilinear operations\n$\\bullet$ and $[\\argument,\\argument]_G$ given by\n\\begin{equation}\n\\label{eq:Pre-Lie}\n  P_1\\bullet P_2\n  :=\n  \\sum_{i=0}^{k_1}\n  (-1)^{ik_2}(\\id^{\\otimes i}\\otimes\\Delta^{(k_2)}\\otimes \\id^{\\otimes k_1-i})(P_1)\\cdot (1^{\\otimes i}\\otimes P_2\\otimes 1^{\\otimes k_1-i})\n\\end{equation}\nand \n\\begin{equation}\n \\label{eq:Lie}\n  [P_1,P_2]_G\n  :=\n  P_1\\bullet P_2-(-1)^{k_1k_2}P_2\\bullet P_1\n\\end{equation}\nwhere $P_1\\in \\Dpoly{}{k_1}(\\Rf)$, $P_2\\in\\Dpoly{}{k_2}(\\Rf)$ and\n$\\Delta^{(k)}$ denotes the $k$-th iteration of $\\Delta$ given by \n$(\\Delta\\otimes\\id^{\\otimes k-1})(\\Delta\\otimes \\id^{\\otimes\n  k-2})\\ldots(\\Delta\\otimes \\id)\\Delta$. Note that the\nbracket $[\\argument,\\argument]_G$ defines a graded Lie algebra\nstructure on $\\Dpoly{}{}(\\Rf)$.\n\\begin{theorem}[Kontsevich\\cite{kontsevich:2003a}]\n  \\label{thm:kontsevich}\n  There exists an $L_\\infty$-quasi-isomorphism between DGLA's\n  \\begin{equation}\n    \\label{eq:kontsevich}\n    \\mathscr{K}\n    \\colon \n    \\left(\\Tpoly{}{}(\\Rf),0,\\Schouten{\\argument,\\argument}\\right)\n    \\longrightarrow \n    \\left(\\Dpoly{}{}(\\Rf),\\partial,[\\argument,\\argument]_G\\right)\n  \\end{equation} \n  where $\\partial = [\\mu,\\argument]_G$ for $\\mu=1\\otimes 1\\in\n  \\Dpoly{}{1}(\\Rf)$. Moreover\n  \\begin{enumerate} \n  \\item $\\mathscr{K}$ is $\\group{GL}(d,\\mathbb{R})$ equivariant;\n  \\item\\label{thm:kontsevich2} $\\mathscr{K}_n(X_1,\\ldots, X_n)=0$ for all $X_i\\in\n    \\Tpoly{}{0}(\\Rf)$ and $n>1$;\n  \\item $\\mathscr{K}_n(X,Y_2,\\ldots, Y_n)=0$ for all\n    $Y_i\\in\\Tpoly{}{}(\\Rf)$ and $n\\geq 2$ whenever\n    $X\\in\\Tpoly{}{0}(\\Rf)$ is induced by the action of\n    $\\mathfrak{gl}(d,\\mathbb{R})$.\n  \\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Formality for Lie algebroids}\n\\label{sec:formality}\n\nIn this section we recall the formality theorem for Lie\nalgebroids, which is due to Calaque, see \\cite{Calaque2005}. The proof\nof this theorem follows the lines of Dolgushev's construction\n\\cite{Dolgushev2005a, Dolgushev2005} of the\n$L_\\infty$-quasi-isomorphism from polyvectorfields to polydifferential\noperators. The main ingredients are Fedosov's methods\n\\cite{fedosov:1994a} concerning formal geometry, Kontsevich's\nquasi-isomorphism \\cite{kontsevich:2003a} and the twisting procedure\ninspired by Quillen \\cite{Quillen69} (although we use Dolgushev's\nversion \\cite{Dolgushev2005}).  Since we only need\nthe result and not in fact the details of the construction we are \nrather brief here and refer to \\cite{Calaque2005} for details.\n\n\\subsection{Fedosov resolutions}\n\n\nAs a first step, Calaque constructs\nFedosov resolutions of polyvector fields and polydifferential operators\nof Lie algebroids.\n\nLet us recall that a Lie algebroid is a vector bundle $E \\to M$ over a\nmanifold $M$, equipped with a Lie bracket on sections $\\Secinfty (E)$\nand an anchor map $\\rho \\colon E \\to TM$, preserving the Lie bracket,\nsuch that\n\\begin{equation}\n  [v, f w]_E\n  =\n  f[v, w]_E + (\\rho(v)f)w,\n\\end{equation}\nfor any $v, w \\in \\Secinfty (E)$ and $f\\in \\Cinfty (M)$.\nEquivalently, we can consider the algebra of $E$-differential forms\n$\\Secinfty(\\wedge^\\bullet E^*)$ endowed with the differential $\\D_E$\ngiven by $(\\D_Ef)(v)=\\rho(v)(f)$ for $f\\in\\Cinfty(M)$ and $X\\in\n\\Secinfty(E)$, by\n$(\\D_E\\alpha)(v,w)=\\rho(v)(\\alpha(w))-\\rho(w)(\\alpha(v))-\\alpha([v,w]_E)$\nfor $X,Y\\in \\Secinfty(E)$ and $\\alpha\\in\\Secinfty(E^*)$ and extended\nas a derivation for the wedge product.\n\nThe definitions of the DGLA's $\\Tpoly{}{}(\\Rf)$ and $\\Dpoly{}{}(\\Rf)$\ngiven in Section~\\ref{sec:LocalFormality}\ngo through mutatis mutandis to define the DGLA's $ \\Tpoly{E}{}(M)$ and\n$ \\Dpoly{E}{}(M)$ starting from the Lie-Rinehart pair\n$\\left(\\Cinfty(M), \\Secinfty(E)\\right)$. Notice that the resulting\nspaces $ \\Dpoly{E}{k}(M)$ can be identified with the spaces of\n$E$-polydifferential operators of order $k+1$.\nIn order to extend the result of Theorem~\\ref{thm:kontsevich} to any\nLie algebroid, we need to consider the so-called \\emph{Fedosov\n  resolutions}.  The idea (coming from formal\ngeometry) consists in replacing the DGLA's $\\Tpoly{}{}(\\Rf)$ and\n$\\Dpoly{}{}(\\Rf)$ by quasi-isomorphic DGLA's (using DGLA morphisms in\nthis case). For the rest of this section we consider a Lie algebroid\n$E$ of rank $d$.\nWe denote by $ ^E\\FibT{}$ the bundle of formal fiberwise $E$-polyvector\nfields over $M$, this is the bundle \nassociated to the principal bundle of general linear frames in\n$E$ with fiber $\\Tpoly{}{}(\\Rf)$.\nSimilarly, the bundle $ ^E\\FibD{}$ of formal fiberwise\n$E$-polydifferential operators is the bundle over $M$ associated to the\nprincipal bundle of general linear frames in $E$ with fiber\n$\\Dpoly{}{}(\\Rf)$.\nThe Fedosov resolutions are given on the level of vector spaces by the\n$E$-differential forms with values in \n$^E\\FibT{}$ and $^E\\FibD{}$ respectively. We denote these spaces\nby $\\FormsT{}{M}{}$ and $\\FormsD{}{M}{}$ respectively. Note that these\nspaces carry a natural DGLA structure, namely the one induced by the\nstructure on fibers (which is $\\group{GL}(d,\\mathbb{R})$-equivariant).\n\\begin{lemma}\n  \\label{explicit}\n  There exist $\\group{GL}(d,\\mathbb{R})$-equivariant\n  isomorphisms of algebras\n  \\begin{equation}\n    \\Cinfty(\\Rf)\\simeq \\prod_{k=0}^\\infty \\Sym^kT^*_0\\mathbb{R}^d\\simeq \\mathbb{R}\\Schouten{\\hat{x}_1,\\ldots, \\hat{x}_d}\n  \\end{equation}\n  \\begin{equation}\n    \\Tpoly{}{0}(\\Rf)\\simeq \\Cinfty(\\Rf)\\otimes T_0\\mathbb{R}^d.\n  \\end{equation}\n  Here the Lie algebra structure on $\\Cinfty(\\Rf)\\otimes T_0\\mathbb{R}^d$ is\n  induced from the action of $T_0\\mathbb{R}^d$ as derivations at $0$ on\n  $\\Cinfty(\\mathbb{R}^d)$.\n\\end{lemma}\nThe proof of the above lemma can be found in \\cite[Prop. 2.1.10]{Niek} and \\cite[Theorem\n  1.1.3]{MoerdijkReyes}. It implies that\n\\begin{equation}\n  \\FormsT{}{M}{}\n  \\simeq \n  \\Secinfty\\left(\\prod_{k=0}^\\infty \\Anti^\\bullet E\\otimes \\Sym^k E^*\\otimes \\Anti^\\bullet E^*\\right)\n\\end{equation}\nand similarly \n\\begin{equation}\n  \\FormsD{}{M}{}\n  \\simeq \n  \\Secinfty\\left(\\prod_{k=0}^\\infty\\left(\\bigoplus_{l=0}^\\infty \\Sym^l E\\right)^{\\otimes\\bullet}\\otimes \\Sym^k E^*\\otimes\\Anti^\\bullet E^*\\right),\n\\end{equation} \nwhere $\\Anti$ and $\\Sym$\ndenote the anti-symmetric and symmetric algebra, respectively.\n\nThe next step consists in finding a differential on the Fedosov\nresolutions which is compatible with the graded Lie algebra structure\nand which makes them into DGLA's quasi-isomorphic to $\\Tpoly{E}{}(M)$\nand $\\Dpoly{E}{}(M)$, respectively.\nGiven some trivializing coordinate neighborhood $V\\subset M$ of $E$, a\nlocal frame $e_1, \\dots, e_d$ and its dual frame $(x_1,\\ldots, x_d)$,\nwe can define the operators\n\\begin{equation}\n  \\delta\\colon \\FormsT{}{V}{}\\rightarrow \\FormsT{}{V}{}\n\\end{equation} \nby the formula \n\\begin{equation}\n  \\label{delta}\\delta(Y)\n  =\n  \\sum_{i=1}^d x_i\\wedge\\Schouten{{e_i},Y},\n\\end{equation} \nIn other words $\\delta=\\Schouten{A_{-1},\\argument}$ where\n$A_{-1}\\in\\FormsT{1}{V}{0}$ is the one-form\n$A_{-1}=\\sum_{i=1}^dx_i\\otimes e_i$.  One easily checks that $\\delta$\ndoes not depend on the choice of coordinates, since $A_{-1}$ is\nindependent of coordinates, and therefore extends to all of $M$.\nBy replacing $\\Schouten{\\argument,\\argument}$ by\n$[\\argument,\\argument]_G$ in \\eqref{delta} we obtain the operators\n\\begin{equation}\n  \\delta\\colon \\FormsD{}{M}{}\\longrightarrow\\FormsD{}{M}{}.\n\\end{equation}\nNote that, since $\\Schouten{A_{-1},A_{-1}}=[A_{-1},A_{-1}]_G = 0$, we\nhave $\\delta^2 = 0$. Furthermore, since it is given by an inner\nderivation and $\\delta\\mu = 0$, $\\delta$ is compatible with the\nfiberwise Lie structures and thus yields DGLA structures.\n\n\nThe cohomology of the complexes $\\left( \\FormsT{l}{M}{},\\delta\\right)$\nand $\\left( \\FormsD{l}{M}{},\\delta\\right)$ is given by the following\nproposition (proved e.g in \\cite[Prop.~2.1]{Calaque2005}).\n\\begin{proposition}\n  We have that \n  \\begin{equation}\n    \\Cohom{0}( \\FormsT{}{M}{},\\delta)\n    \\simeq\n    \\Gamma^\\infty(\\Anti^\\bullet E)\n    \\hspace{0.3cm} \\mbox{and}\\hspace{0.3cm}\n    \\Cohom{0}( \\FormsD{}{M}{},\\delta)\n    \\simeq\n    \\Secinfty\\left(\\left(\\bigoplus_{l=0}^\\infty S^lE\\right)^{\\otimes\\bullet}\\right)\n  \\end{equation} \n  while \n  \\begin{equation}\n    \\Cohom{>0}( \\FormsT{}{M}{},\\delta)=0\n    \\hspace{0.3cm}\\mbox{and}\\hspace{0.3cm}\n    \\Cohom{>0}( \\FormsD{}{M}{},\\delta)=0\n  \\end{equation}\n\\end{proposition}\nNotice that\n\\begin{equation}\n  \\Cohom{0}( \\FormsT{}{M}{},\\delta)\n  \\simeq \\hspace*{0.05cm}\n   \\Tpoly{E}{}(M)\n  \\hspace{0.3cm}\\mbox{and}\\hspace{0.3cm}\n   \\Cohom{0}( \\FormsD{}{M}{},\\delta)\n  \\simeq \\hspace*{0.05cm}\n   \\Dpoly{E}{}(M)\n\\end{equation}\nas vector spaces. This does not provide us with the quasi-isomorphisms\nwe are looking for, since $\\Cohom{0}( \\FormsT{}{M}{},\\delta)$ carries\nthe trivial Lie algebra structure. To correct it, the idea is to\nconstruct a perturbation of the differential $\\delta$ that does not\naffect the size of the cohomology, but only the Lie\nalgebra structure on cohomology.\nNotice that the operator $\\delta$ is of degree $-1$ in\nterms of the filtration and so we may start perturbing at order $0$,\ni.e. adding a connection in the bundle $E$.  The fact that the resulting\nperturbation should square to zero forces us to choose a torsion-free\nconnection $\\nabla\\!\\!\\!\\!\\nabla$.  This gives us the operators\n\\begin{equation}\n  \\nabla\\!\\!\\!\\!\\nabla\n  \\colon \n  \\FormsT{}{M}{} \\longrightarrow \\FormsT{}{M}{}\n  \\hspace{0.3cm}\\mbox{and}\\hspace{0.3cm} \n  \\nabla\\!\\!\\!\\!\\nabla\n  \\colon \n  \\FormsD{}{M}{}\\longrightarrow\\FormsD{}{M}{}.\n\\end{equation} \nThus we consider the corresponding operators $-\\delta+\\nabla\\!\\!\\!\\!\\nabla$ and\n$-\\delta+ \\nabla\\!\\!\\!\\!\\nabla +\\partial$. This leads to the problem that there\nis no reason to assume that we can find $\\nabla\\!\\!\\!\\!\\nabla$ such that $\\nabla\\!\\!\\!\\!\\nabla^2=0$\n(since not every Lie algebroid is flat). Following the idea of\nFedosov, we correct $-\\delta + \\nabla\\!\\!\\!\\!\\nabla$ by an inner derivation and make\nthe ansatz\n\\begin{equation}\n  D\n  :=\n  -\\delta + \\nabla\\!\\!\\!\\!\\nabla + [A, \\argument]\n\\end{equation}\nwith $A \\in \\FormsT{1}{M}{0}\\hookrightarrow \\FormsD{1}{M}{0}$ and\nwhere $[A,\\argument]$ means $\\Schouten{A,\\argument}$ or\n$[A,\\argument]_G$ depending on the situation.  The trick is to find\n$A$ such that $D^2 = 0$, as proved in \\cite[Prop.~2.2]{Calaque2005}.\n\\begin{lemma}\n\\label{lem:A}\n  There exists a unique $A$ such that\n  \\begin{lemmalist}\n   \\item $\\delta A = R + \\nabla\\!\\!\\!\\!\\nabla A + \\frac{1}{2} [A, A]$\n   \\item $\\delta ^{-1} A = 0$.\n  \\end{lemmalist}\n\\end{lemma}\nHere $R$ denotes the curvature of $\\nabla\\!\\!\\!\\!\\nabla$ expressed in terms of the\nbundle $^E\\FibT{}(M)$ (or $ ^E\\FibD{}(M)$), i.e. it is given by the\nequation\n\\begin{equation}\n  \\nabla\\!\\!\\!\\!\\nabla^2Y\n  =\n  [R,Y]\n\\end{equation}\nand $\\delta^{-1}$ is a particular $\\delta$-homotopy from the\nprojection onto degree $0$, denoted $\\sigma$, to the identity, i.e.\n\\begin{equation}\n  \\delta^{-1}\\delta + \\delta\\delta^{-1} + \\sigma\n  =\n  \\id.\n\\end{equation} \nThe condition $\\delta^{-1}A = 0$ is simply a normalization condition\nensuring uniqueness of the solution.\n\\begin{proposition}\n  \\label{prop:DcohomM}\n  We have  \n  \\begin{equation}\n    \\Cohom{>0}( \\FormsT{}{M}{},D)\n    =\n    0\n    \\qquad \\mbox{and} \\qquad\n   \n    \\Cohom{>0}( \\FormsD{}{M}{},D)\n    =\n    0.\n  \\end{equation} \n  Furthermore we have \n  \\begin{equation}\n  \\begin{gathered}\n    \\Cohom{0}( \\FormsT{}{M}{},D)\\simeq \\Cohom{0}( \\FormsT{}{M}{},\\delta)\n    \\\\\n    \\Cohom{0}( \\FormsD{}{M}{},D)\\simeq \\Cohom{0}( \\FormsD{}{M}{},\\delta).\n  \\end{gathered}\n  \\end{equation}\n\\end{proposition}\n\\begin{proof}\n  \\cite[Thm.~2.3]{Calaque2005}\n\\end{proof}\nLet us denote the isomorphisms from the above Proposition by $\\tau$.\nThen, using a Poincar\\'e-Birkhoff-Witt-type isomorphism, Calaque constructs an\nisomorphism (see \\cite[Sec.~2.3]{Calaque2005})\n\\begin{equation}\n  \\nu \n  \\colon \n  \\Ker\\delta\\cap\\FormsD{0}{M}{}\\longrightarrow \\Dpoly{E}{}(M)\n\\end{equation}\nof filtered vector spaces. \n\nSimilarly, but in an easier way, we obtain an isomorphism \n\\begin{equation}\n  \\nu\n  \\colon\n  \\Ker\\delta\\cap \\FormsT{0}{M}{}\\longrightarrow \\Tpoly{E}{}(M)\n\\end{equation}\nof graded vector spaces.\nFinally, as proved in \\cite[Prop.~2.4-2.5]{Calaque2005}, we get:\n\\begin{theorem}[Fedosov Resolutions]\n  \\label{thm:FedRes}\n  The maps \n  \\begin{equation}\n    \\lambda_D\n    \\colon \n    \\left(\\Dpoly{E}{}(M),\\partial\\right)\n    \\longrightarrow\n    \\left(\\FormsD{}{M}{},\\partial + D\\right)\n  \\end{equation} \n  and \n  \\begin{equation}\n    \\lambda_T\n    \\colon \n    \\left(\\Tpoly{E}{}(M),0\\right)\n    \\longrightarrow \n    \\left(\\FormsT{}{M}{},D\\right)\n  \\end{equation} \n  both given by $\\tau\\circ \\nu^{-1}$  are DGLA quasi-isomorphisms.\n\\end{theorem}\nLet us sketch the remaining steps necessary to obtain the\n$L_\\infty$-quasi-isomorphisms from $\\Tpoly{E}{}(M)$ to $\\Dpoly{E}{}(M)$.\nAs a second step, one notices that in a trivializing neighborhood\n$U\\subset M$ of\n$E$ the connection $\\nabla\\!\\!\\!\\!\\nabla$ on both $\\FormsT{}{M}{}$ and\n$\\FormsD{}{M}{}$ is given by $\\D_E + [B_U,\\argument]$ for some element $B_U\\in \\FormsT{1}{M}{0}\\hookrightarrow\\FormsD{1}{M}{0}$. \nThus, in this neighborhood, we have\n$D = \\D_E + [\\Gamma,\\argument]$, where $\\Gamma$ is a Maurer-Cartan element.  \nWe now observe that\nthe map\n\\begin{equation}\n  \\mathscr{U}\n  \\colon \n  \\FormsT{}{U}{}\n  \\longrightarrow \n  \\FormsD{}{U}{}\n\\end{equation} \ngiven by applying the map $\\mathscr{K}$ from\nTheorem~\\ref{thm:kontsevich} fiberwise\ncommutes with $\\D_E$. \nThe next step consists in twisting this map by\n$\\Gamma$ to obtain $L_\\infty$-quasi-isomorphisms\n\\begin{equation}\n  \\mathscr{U}^\\Gamma\\circ\\lambda_T\n  \\colon \\Tpoly{E}{}(U)\n  \\longrightarrow\n  \\FormsD{}{U}{}.\n\\end{equation}\nThe twisting procedure is essential in our paper and will be discussed\nin full detail in Section~\\ref{sec:Twisting}.  By using the properties\nof Kontsevich's quasi-isomorphism \\eqref{eq:kontsevich} and the fact\nthat $\\nabla\\!\\!\\!\\!\\nabla$ is a $\\mathfrak{gl}(d,\\mathbb{R})$ connection we find that these\nquasi-isomorphisms coincide on intersections and thus we obtain\n\\begin{equation}\n  \\mathscr{U}^\\Gamma\\circ\\lambda_T\n  \\colon \n  \\Tpoly{E}{}(M)\\longrightarrow\\FormsD{}{M}{}.\n\\end{equation}\n\n\\begin{remark} \n  Although it may seem that we are being sloppy with notation by\n  writing $\\mathscr{U}^\\Gamma$, since it is not a twist a priori, it\n  is still possible to consider it as a twist in the context of curved\n  $L_\\infty$-algebras. This construction will be discussed in the\n  upcoming paper \\cite{sisters:2017b}.\n\\end{remark}\nFinally we would like to define the $L_\\infty$-quasi-isomorphism $\n\\lambda_D^{-1}\\circ\\mathscr{U}^\\Gamma\\circ\\lambda_T\\colon\n\\Tpoly{E}{}\\longrightarrow \\Dpoly{E}{}.  $ One problem remains and it\nis that, although $\\lambda_D$ is obviously injective, we cannot be\nassured that $\\mathscr{U}^\\Gamma\\circ\\lambda_T$ maps $\\Tpoly{E}{}$\ninto the image of $\\lambda_D$. However, Dolgushev\n\\cite[Prop.~5]{Dolgushev2005a} shows that we can always \nmodify $\\mathscr{U}^\\Gamma\\circ\\lambda_T$ using a so-called\n\\emph{partial homotopy} to obtain a new quasi-isomorphism\n$\\overline{\\mathscr{U}}$ which maps into the image of\n$\\lambda_D$. Thus we obtain the $L_\\infty$-quasi-isomorphism\n\\begin{equation}\n  \\label{eq:Q}\n  F_E\n  :=\n  \\lambda_D^{-1}\\circ\\overline{\\mathscr{U}}\\colon \\Tpoly{E}{}\\longrightarrow \\Dpoly{E}{}.\n\\end{equation} \nAs a consequence, we obtain the formality theorem for a generic\nmanifold $M$ by considering the case $E = TM$ and formality for Lie\nalgebras by considering the case $E = \\lie g$ over a point.\n\\begin{remark}\n  \\label{rem:choice}\n  Note that the constructions of $D$, $\\tau$ and so on are not unique,\n  but they depend only on the choice of the torsion-free\n  $E$-connection $\\nabla\\!\\!\\!\\!\\nabla$.\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Twisting procedure}\n\\label{sec:Twisting}\n\nIn the following we recall the notions of twisting DGLA's and\n$L_\\infty$-morphisms by Maurer--Cartan elements.  The idea of such\ntwisting procedures comes from Quillen's seminal work\n\\cite{Quillen69}. Here we follow Dolgushev's approach as\nlaid out in \\cite{Dolgushev2005}. As an example we show how one\nobtains the local $L_\\infty$-quasi-isomorphisms $\\mathscr{U}^\\Gamma$\nmentioned above.  \n\\begin{lemma}\n\\label{lem:exppi}\n  Suppose $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, then the element\n  \\begin{equation}\n    \\exp(\\pi):=\\sum_{n=0}^\\infty\\frac{\\pi^k}{k!}\n  \\end{equation}\n  is well-defined, invertible and group-like. \n\\end{lemma}\n\\begin{proof} \n  $\\exp(\\pi)$ is well-defined, since the\n  partial sums converge by virtue of $\\pi$ being in the first\n  filtration (the filtration is respected by $\\wedge$).\n  Invertibility follows from the usual direct computations showing\n  that $\\exp(-\\pi)\\exp(\\pi)=1=\\exp(\\pi)\\exp(-\\pi)$.  The fact that\n  $\\exp(\\pi)$ is group-like can similarly be deduced from a direct\n  computation using the definition of $\\Delta$ given in Section~\\ref{sec:Linfty}.\n\\end{proof}\nGiven $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$ we define the\n$\\pi$-twist of the $L_\\infty$-algebra $(\\mathfrak{L},Q)$ as the\n$L_\\infty$-algebra $\\mathfrak{L}^\\pi$ given by the pair\n$(\\mathfrak{L}, Q^\\pi)$ with\n\\begin{equation}\n  Q^\\pi(a)\n  :=\n  \\exp(-\\pi)\\wedge Q(\\exp(\\pi)\\wedge a).\n \\end{equation}\n %\n\\begin{corollary}\nSuppose $(\\mathfrak{L},Q)$ is an $L_\\infty$-algebra and $\\pi\\in \\mathcal{F}^1\\mathfrak{L}[1]^0$, then the $\\pi$-twist $(\\mathfrak{L}, Q^\\pi)$ is an $L_\\infty$-algebra.\n\\end{corollary}\n\\begin{example} \n  Given a curved Lie algebra $(\\mathfrak{L}, R,\\D,\n  [\\argument,\\argument])$ we find the twisted curved Lie algebra\n  $(\\mathfrak{L}, R^\\pi,\n  \\D + [\\pi,\\argument],[\\argument,\\argument])$, where\n  \\begin{equation}\n  \\label{eq:Rpi}\n    R^\\pi\n    :=\n    R+\\D\\pi + \\frac{1}{2}[\\pi,\\pi].\n  \\end{equation}\n  Note in particular that the $\\pi$-twist is flat exactly when $\\pi$ satisfies the Maurer-Cartan equation.\n\\end{example}\n\n\n\\begin{proposition}\n  \\label{prop:MCtwist} \n  Suppose $\\mathfrak{L}$ is an $L_\\infty$-algebra and $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, then the $\\pi$-twist $\\mathfrak{L}$ is flat if and only if $\\pi$ is a Maurer-Cartan element. \n\\end{proposition}\n\\begin{proof} \n  We have \n  \\begin{equation} \n    Q^\\pi(1)\n    =\n    \\exp(-\\pi)\\wedge Q(\\exp(\\pi))\n    =\n    \\sum_{n=0}^\\infty \\frac{Q_n(\\pi^n)}{n!}\n    ,\n  \\end{equation}\n  since all terms in $\\bigoplus_{k=2}^\\infty\\Anti^k(\\mathfrak{L}[1])$\n  cancel out by virtue of the fact that\n  $Q^\\pi(1)=Q^\\pi_0(1)\\in\\mathfrak{L}[1]$.\n\\end{proof}\n\\begin{example} \n  For a DGLA $(\\mathfrak{L}, \\D,[\\argument,\\argument])$ Eq.~\\eqref{eq:Rpi}\n  boils down to the usual Maurer--Cartan equation\n  \\begin{equation}\n    \\D\\pi+\\frac{1}{2}[\\pi,\\pi]\n    =\n    0.\n  \\end{equation}\n  If we have similarly a curved Lie algebra with curvature $-R$ it\n  comes down to the non-homogeneous equation\n  \\begin{equation}\n    \\D\\pi+\\frac{1}{2}[\\pi,\\pi]\n    =\n    R.\n  \\end{equation}\n\\end{example}\n\\begin{lemma}\n  \\label{lem:MCcondition}\n  Suppose $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, then $\\pi$ is an MC\n  element if and only if $Q(\\exp(\\pi))=0$.\n\\end{lemma}\n\\begin{proof}\n  The proof follows from the following equation\n  \\begin{align*}\n    Q(\\exp(\\pi))\n    &=\n    \\sum_{n=0}^\\infty\\sum_{k=0}^n\\sum_{\\sigma\\in \\mbox{\\tiny Sh($k$,$n-k$)}}\\epsilon(\\sigma)\\frac{1}{n!}Q_k(\\pi^k)\\wedge\n    \\pi^{n-k}\n    \\\\\n    &=\n    \\sum_{n=0}^\\infty\\sum_{k=0}^n\\frac{1}{k!(n-k)!}Q_k(\\pi^k)\n    \\wedge\\pi^{n-k}\n    \\\\\n    &=\n    \\left(\\sum_{n=0}^\\infty \\frac{Q_n(\\pi^n)}{n!}\\right)\n    \\wedge\\exp(\\pi).\n  \\end{align*}\n\\end{proof}\n\n\\begin{lemma}\n  \\label{lem:Fep=epF}\n  Given an L$_\\infty$-morphism $F$ from $\\mathfrak{L}$ to\n  $\\mathfrak{L}'$ and an element\n  $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, we define the $F$-associated\n  element $\\pi_F\\in\\mathcal{F}^1\\mathfrak{L}'[1]^0$ by the formula\n  \\begin{equation}\n    \\pi_F\n    :=\n    \\sum_{n=1}^\\infty\\frac{F_n(\\pi^n)}{n!}.\n  \\end{equation}\n  We have\n  \\begin{equation}\n    F(\\exp(\\pi))=\\exp(\\pi_F)\n  \\end{equation}\n\\end{lemma}\n\\begin{proof} \n  It follows from explicit computation using the formula\n  \\eqref{eq:coalgebramorphism}.\n\\end{proof}\nLemmas \\ref{lem:Fep=epF} and \\ref{lem:MCcondition} imply the following corollary.\n\\begin{corollary}\n   If $\\pi$ is an MC element, then $\\pi_F$ is also an MC element.\n\\end{corollary}\nLet $F\\colon (\\mathfrak{L},Q)\\longrightarrow (\\widetilde{\\mathfrak{L}},\\widetilde{Q})$\nbe an $L_\\infty$-morphism and $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$. \n\\begin{definition}[$\\pi$-twist morphism]\n  The $\\pi$-twist of $F$ is a map\n  \\begin{equation}\n    F^\\pi\n    \\colon \n    (\\mathfrak{L},Q^\\pi)\\longrightarrow (\\widetilde{\\mathfrak{L}},\\widetilde{Q}^\\pi)\n  \\end{equation}\n  defined by \n  \\begin{equation}\n    F^\\pi(a)\n    :=\n    \\exp(\\pi_F)\\wedge F(\\exp(\\pi)\\wedge a).\n  \\end{equation}\n\\end{definition}\n\\begin{corollary}\n  The $\\pi$-twist of an $L_\\infty$-morphism $F$ is an\n  $L_\\infty$-morphism.\n\\end{corollary}\n\\begin{proof}\n  Note that, by Lemma~\\ref{lem:exppi} and Remark~\\ref{rem:bialgebra},\n  the operators of multiplication by $\\exp(\\pi)$ and $\\exp(\\pi_F)$ are\n  co-algebra morphisms. Thus $F^\\pi$ is a co-algebra morphism. The\n  relation $F^\\pi Q^\\pi=\\widetilde{Q}^\\pi F^\\pi$ follows from\n  the definitions.\n\\end{proof}\n\\begin{remark}\n  \\label{comptwist}\n  Given two $L_\\infty$-morphisms $F$ and $G$ from $\\mathfrak{L}$\n  to $\\mathfrak{L}'$ and $\\mathfrak{L}'$ to $\\mathfrak{L}''$,\n  respectively, and the elements $\\pi, B\\in\n  \\mathcal{F}^1\\mathfrak{L}[1]^0$, we have that\n  \\begin{align} \n    (Q^{\\pi})^B\n    &=\n    Q^{\\pi+B}=(Q^B)^\\pi,\n    \\\\ \n    (F^\\pi)^B\n    &=\n    F^{\\pi+B}=(F^B)^\\pi,\n    \\\\\n    \\pi_F+B_{F^\\pi}\n    &=(\\pi+B)_F\n    =\n    B_F+\\pi_{F^B},\n    \\\\\n    (\\pi_F)_G\n    &=\n    \\pi_{G\\circ F}.\n  \\end{align}\n\\end{remark}\nFor the proof of the following proposition we refer to\n\\cite[Prop. 1]{Dolgushev2005}.\n\\begin{proposition}\n  \\label{prop:twistquis}\n  Let $F\\colon\\mathfrak{L}\\rightarrow\\widetilde{\\mathfrak{L}}$ be\n  an $L_\\infty$-quasi-isomorphism such that the induced morphisms\n  \\begin{equation}\n    F|_{\\mathcal{F}^k\\mathfrak{L}}\n    \\colon \n    \\mathcal{F}^k\\mathfrak{L}\\longrightarrow \\mathcal{F}^k\\widetilde{\\mathfrak{L}}\n  \\end{equation}\n  are also $L_\\infty$-quasi-isomorphisms for all $k$. Suppose further\n  that $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$ is an MC element. Then\n  the $\\pi$-twist\n  \\begin{equation}\n    F^\\pi\\colon \\mathfrak{L}^\\pi\\longrightarrow \\widetilde{\\mathfrak{L}}^{\\pi_F}\n  \\end{equation}\n  of $F$ is also a quasi-isomorphism. \n\\end{proposition}\n\\begin{remark}\n  \\label{rem:thepoint}\n  The proposition above says that the class of\n  $L_\\infty$-quasi-isomorphisms is closed under the operation of\n  twisting by a Maurer--Cartan element. This provides the method of\n showing that an $L_\\infty$-morphism is an $L_\\infty$-quasi-isomorphism by showing\n that it is the twist of a known $L_\\infty$-quasi-isomorphism.\n\\end{remark}\n\\begin{example}[Formality for $\\mathbb{R}^d$]\n  \\label{ex:formalityRd}\n  Here we generalize the result of Theorem \\ref{thm:kontsevich} from\n  $\\Rf$ to $\\mathbb{R}^d$ by providing an example of the claim in\n  Remark~\\ref{rem:thepoint}. From now on we set $E = TM$ and drop the\n  $E$ for notational convenience. Proposition \\ref{prop:twistquis}\n  allows us to obtain an $L_\\infty$-quasi-isomorphism witnessing the\n  formality of $\\Dpoly{}{}(M)$ for any manifold by twisting the formal\n  quasi-isomorphism of Theorem~\\ref{thm:kontsevich}.\n  We set $M=\\mathbb{R}^d$\n \n \n  and recall that we are looking for an $L_\\infty$-quasi-isomorphism\n  \\begin{equation}\n    \\mathscr{U}^\\delta\n    \\colon\n    (\\Omega(\\mathbb{R}^d;\\FibT{}),D)\n    \\longrightarrow \n    (\\Omega(\\mathbb{R}^d,\\FibD{}),\\partial+D),\n  \\end{equation}\n  since this would complete \n  the diagram \n  \\begin{equation}\n  \\label{eq:seqquisRd}\n    (\\Tpoly{}{}(\\mathbb{R}^d),0)\n    \\stackrel{\\lambda_T}{\\longrightarrow} \n    (\\Omega(\\mathbb{R}^d;\\FibT{}),D)\n    \\stackrel{\\mathscr{U}^\\delta}{\\longrightarrow} \n    (\\Omega(\\mathbb{R}^d,\\FibD{}),\\partial+D)\n    \\stackrel{\\lambda_D}{\\longleftarrow} (\\Dpoly{}{}(\\mathbb{R}^d),\\partial)\n  \\end{equation}\n  of $L_\\infty$-quasi-isomorphisms. \n \n \n \n  Also, recall that $D:=-\\delta+\\D$. We obtain this map\n  $\\mathscr{U}^\\delta$ as follows.\n  First we note that, by applying the map $\\mathscr{K}$ from \n  Theorem~\\ref{thm:kontsevich} fiberwise, we obtain the $L_\\infty$-morphism\n  \\begin{equation}\n    \\mathscr{U}\n    \\colon \n    (\\Omega(\\mathbb{R}^d;\\FibT{}),\\D)\n    \\longrightarrow \n    (\\Omega(\\mathbb{R}^d;\\FibD{}),\\partial+\\D).\n  \\end{equation}\n  By considering the filtrations by exterior degree on both these\n  algebras we construct spectral sequences which show that\n  $\\mathscr{U}$ is a quasi-isomorphism. Using this same filtration we\n  may consider the MC element\n  $-A_{-1}\\in\\mathcal{F}^1\\Omega(\\mathbb{R}^d;\\FibT{})$. Now note that\n  $\\Omega(\\mathbb{R}^d;\\FibT{})^{-A_{-1}}$ is exactly\n  $(\\Omega(\\mathbb{R}^d;\\FibT{}),D)$ and\n  $\\Omega(\\mathbb{R}^d;\\FibD{})^{(-A_{-1})_\\mathscr{U}}$ is exactly\n  $(\\Omega(\\mathbb{R}^d;\\FibD{}),D)$, since $(-A_{-1})_{\\mathscr{U}}=-A_{-1}$\n  by point \\refitem{thm:kontsevich2} of Theorem~\\ref{thm:kontsevich}. So we obtain the\n  diagram \\eqref{eq:seqquisRd} by setting $\\mathscr{U}^\\delta:=\n  \\mathscr{U}^{-A_{-1}}$. This concludes the example of the claim in\n  Remark~\\ref{rem:thepoint}.\n  In order to obtain the quasi-isomorphism \n  \\begin{equation}\n    \\Tpoly{}{}(\\mathbb{R}^d)\\longrightarrow \\Dpoly{}{}(\\mathbb{R}^d)\n  \\end{equation} \n  we need to invert the final arrow of diagram \\eqref{eq:seqquisRd}.\n  This arrow is actually an identification (by\n  DGLA-morphism) with the kernel of $D$ in exterior degree $0$.  Thus\n  it can be inverted without problems if we can guarantee that the map\n  $\\mathscr{U}^\\delta\\circ\\lambda_T$ maps $\\Tpoly{}{}(\\mathbb{R}^d)$ into this\n  kernel.  We refer to \\cite[Sect.~4.2]{Dolgushev2005a} for an\n  explanation of a way to correct $\\mathscr{U}^\\delta$ to have this\n  property.\n\\end{example}\n\n\\begin{example}[Formality for Lie algebras]\n  Let us conclude this section by providing the equivalent of the\n  proof of formality for the case where $M=\\{\\mbox{pt}\\}$ is the\n  connected $0$-dimensional manifold and $E$ is a $d$-dimensional Lie\n  algebra $\\mathfrak{g}$. The DGLA of polyvector fields is given by\n  $\\mathrm{CE}_\\bullet(\\mathfrak{g})$, the Chevalley-Eilenberg complex\n  with the trivial differential. The complex of $E$-differential forms\n  with values in the fiberwise polyvector fields is thus given by\n  \\begin{equation}\n    \\mathrm{CE}^\\bullet(\\mathfrak{g};\\mathrm{CE}_\\bullet(\\mathfrak{g};\n    \\widehat{\\mathcal{S}}(\\mathfrak{g}^*))),\n  \\end{equation}\n  where we have denoted $\\widehat{\\mathcal{S}}(\\mathfrak{g}^*)=\n  \\prod_{k\\geq 0}S^k\\mathfrak{g}^*$ and the differential\n  $\\delta-d_E$ coincides with the usual Chevalley-Eilenberg\n  differential. A linear $E$-connection $\\nabla\\!\\!\\!\\!\\nabla$ is simply given by a\n  linear map\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla\\colon \\mathfrak{g}\\otimes\\mathfrak{g}\\longrightarrow \\mathfrak{g}.\n  \\end{equation}\n  The corresponding map\n  $\\nabla\\!\\!\\!\\!\\nabla\\colon\\Anti^\\bullet\\mathfrak{g}^*\\rightarrow \\Anti^{\\bullet\n    +1}\\mathfrak{g}^*$ is given by extending the formula\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla\\alpha(X, Y)=-\\alpha(\\nabla\\!\\!\\!\\!\\nabla(\\frac{1}{2}(X\\otimes Y-Y\\otimes X)))\n  \\end{equation}\n  from one-forms as a $\\wedge$-derivation. Note that the\n  $\\mathfrak{g}$-differential $\\D_\\mathfrak{g}$ is simply given by\n  $X\\otimes Y\\mapsto [X,Y]$. Suppose $\\{e_i\\}_{i=1}^d$ is a basis for\n  $\\mathfrak{g}$ with dual basis $\\{e^i\\}_{i=1}^d$. Then the\n  torsion-freeness of the connection $\\nabla\\!\\!\\!\\!\\nabla$ can be expressed  as\n  $\\tilde{\\Gamma}_{ij}^k=\\tilde{\\Gamma}_{ji}^k$ in terms\n  of the Christoffel symbols $\\tilde{\\Gamma}_{ij}^k\\in \\mathbb{R}$ defined by\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla(e_i\\otimes e_j)=\\tilde{\\Gamma}_{ij}^ke_k,\n  \\end{equation}\n  where we have used the Einstein summation convention. Let us consider\n  also the \\emph{relative} Christoffel symbols\n  $\\Gamma_{ij}^k$ defined by\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla(e_i\\otimes e_j)-[e_i,e_j]=\\Gamma_{ij}^ke_k,\n  \\end{equation}\n  i.e. $\\Gamma_{ij}^k=\\tilde{\\Gamma}_{ij}^k-\\frac{1}{2}c_{ij}^k$ where\n  $c_{ij}^k$ are the structure constants. In terms of these torsion-freeness is\n  equivalent to the equation\n  \\begin{equation}\n    \\Gamma_{ij}^k-\\Gamma_{ji}^k-c_{ij}^k=0.\n  \\end{equation}\n  Note in particular that the connection $\\D_\\mathfrak{g}$ is\n  \\underline{not} torsion-free. The most obvious choice of\n  torsion-free connection is given by\n  $\\Gamma_{ij}^k=\\frac{1}{2}c_{ij}^k$, but we leave the choice of\n  symmetric part open. Given any connection $\\nabla\\!\\!\\!\\!\\nabla$ it is given on\n  $\\mathrm{CE}^\\bullet(\\mathfrak{g};\\mathrm{CE}_\\bullet(\\mathfrak{g};\n  \\widehat{\\mathcal{S}}(\\mathfrak{g}^*)))$ by the formula\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla=\\D_\\mathfrak{g}+[\\Gamma_{ij}^k e^i\\hat{e}^je_k,\\argument]\n  \\end{equation}\n  where we have used the hat to signify that we consider $\\hat{e}^j\\in\n  \\widehat{S}(\\mathfrak{g}^*)$. Similar statements hold for\n  $\\Dpoly{\\mathfrak{g}}{}$.\n  Now the example proceeds identically to the previous one. \n\\end{example}\n\n\n\n\n\n\\section{Formality and Deformation Symmetries}\n\\label{sec:FormalityActions}\n\nIn this section we prove the main result of this paper, which \nleads to a new perspective on Drinfeld's approach to deformation\nquantization. First we construct certain $L_\\infty$-algebras related\nto a Hopf algebra or more generally a unital bialgebra and show how\none obtains deformations from Drinfeld twists and maps into a\nHochschild cochain complex. Then we briefly recall the basic notions\nof Poisson action and triangular Lie algebra.  We consider the\nparticular case of a Poisson action of a triangular Lie algebra $(\\lie\ng, r)$ on a manifold $M$ and we show that we can construct a\ncorresponding $L_\\infty$-morphism between polydifferential operators\n$\\Dpoly{\\mathfrak{g}}{}$ and $\\Dpoly{}{}(M)$.  This morphism induces a\nDGLA morphism between a quantum group associated to our Lie algebra\nand a deformed algebra of smooth functions on $M$.\n\n\\subsection{Deformation Symmetries}\n\nIn the following we define the concept of a deformation\nsymmetry. This notion is inspired by Drinfeld's work on deformation\nthrough quantum actions and Drinfeld twists. Let us start by\nrecalling the definition of Drinfeld twist. In this section we shall\nfix the Hopf algebra $(H,\\Delta, \\epsilon, S)$ over the PID $\\ring R$\ncontaining $\\mathbb{Q}$.\n\n\\begin{definition}[Drinfeld twist, \\cite{drinfeld:1983a, drinfeld:1988a}]\n  \\label{def:TwistUEA}%\n  An element $J\\in H\\otimes H$ is said to be a twist on $H$ if the\n  following three conditions are satisfied.\n  \\begin{definitionlist}\n  \\item\\label{def:twist1} $J$ is invertible;\n  \\item \\label{def:twist2} \n    $(\\Delta\\tensor 1)(J)(J\\tensor 1) = (1\\tensor\\Delta)(J)(1\\tensor J)$\n    and\n  \\item \\label{def:twist3}\n    $(\\epsilon\\tensor 1)J =\n    (1\\tensor\\epsilon)J = 1$.\n  \\end{definitionlist}\n\\end{definition}\nIn the following we consider formal deformations. If we consider\ntwists in $H[\\![\\hbar]\\!]$, the condition of invertibility and\n``co-invertibility'' (condition \\refitem{def:twist3} in the above definition) \nmay be replaced by a stronger condition which is\neasier to check. In fact this condition may be formulated for any Hopf\nalgebra equipped with a complete filtration $H = \\mathcal{F}^0 H\n\\supset \\mathcal{F}^1H\\supset\\ldots $.\n\\begin{definition}[Formal Drinfeld twist]\n  \\label{def:FTwistUEA}%\n  Let $H$ be equipped with the complete filtration\n  \\\\$H = \\mathcal{F}^0 H \\supset \\mathcal{F}^1 H \\supset \\ldots$. Then an\n  element $J \\in H \\otimes H$ is said to be a formal twist on $H$\n  if $J$ satisfies \\refitem{def:twist2} of Definition \\ref{def:TwistUEA} and\n  $J-1\\otimes 1\\in \\mathcal{F}^1(H\\otimes H)$.\n\\end{definition}\n\\begin{corollary}\n  \\label{cor:ft=t}\n  A formal twist on $H$ is a twist on $H$.\n\\end{corollary}\n\\begin{proof} \n  This follows immediately from the compatibility of the Hopf\n  algebra structure with the complete filtration.\n\\end{proof}\nIt turns out that the definition of formal twist coincides exactly\nwith the definition of Maurer-Cartan element on a certain DGLA that we\nshall now define. The main observation is that the formulas\n\\eqref{eq:Pre-Lie} and \\eqref{eq:Lie} for the Gerstenhaber bracket on\n$\\Dpoly{E}{}$ only involve the structure of a unital bialgebra.\nFrom now on we denote \n\\begin{equation}\n  TH = \\bigoplus_{k=0}^\\infty T^k H\n  \\qquad\n  \\mbox{with}\n  \\qquad\n  T^k H := H^{\\otimes k}.\n\\end{equation}\nFor $P_1\\in T^{k_1+1}H$ and $P_2\\in\nT^{k_2+1}H$ set\n\\begin{equation}\n  \\label{H-Pre-Lie}\n  P_1\\bullet P_2\n  :=\n  \\sum_{i=0}^{k_1}\n  (-1)^{ik_2}(\\id^{\\otimes i}\\otimes\\Delta^{(k_2)}\\otimes \n  \\id^{\\otimes k_1-i})(P_1)\\cdot (1^{\\otimes i}\\otimes P_2\\otimes 1^{\\otimes k_1-i})\n\\end{equation}\nand \n\\begin{equation}\n  \\label{H-Lie}\n  [P_1,P_2]_H\n  :=\n  P_1\\bullet P_2-(-1)^{k_1k_2}P_2\\bullet P_1\n\\end{equation}\n\\begin{proposition} \n  The graded vector space $TH[1]$ equipped with the bracket\n  $[\\argument,\\argument]_H$ is a graded Lie algebra.\n\\end{proposition}\n\\begin{proof}\n\n  \\leavevmode\n\n  \\noindent We can immediately extend $[\\argument,\\argument]_H$ to\n  non-homogeneous elements, since $\\bullet$ can be extended by bilinearity.  Thus\n  the bilinearity and anti-symmetry of $[\\argument,\\argument]_H$\n  follow immediately from the bilinearity of $\\bullet$, which follows\n  in turn from the linearity of the coproduct and the bilinearity of\n  the product. Finally denote the associator of $\\bullet$ by $\\alpha$,\n  i.e.\n  \\begin{equation} \n    \\alpha(A,B,C)=A\\bullet(B\\bullet C)- (A\\bullet B)\\bullet C.\n  \\end{equation}\n  Then the average of $\\alpha$ over the symmetric group $S_3$ is $0$, i.e \n  \\begin{equation}\\label{real-pre-lie}\n    \\sum_{\\sigma\\in S_3}\\sigma^*\\alpha =0.\n  \\end{equation}\n  Here $S_3$ acts on $(TH[1])^{\\otimes 3}$ through the usual signed\n  permutation of tensor legs. The last equation is obviously\n  equivalent to the Jacobi identity for $[\\argument,\\argument]_H$.\n\\end{proof}\n\\begin{remark}\n  \\label{rem:braces}\n  The structure $\\bullet$ on $TH[1]$ is\n  actually the pre-Lie structure coming from a brace algebra\n  structure.  As such the identity \\eqref{real-pre-lie} can actually\n  be proved by showing the pre-Lie identity\n  \\begin{equation}\n    \\alpha(A,B,C)\n    =\n    (-1)^{|A||B|}\\alpha(B,A,C).\n  \\end{equation}\n  The braces underlying the brace algebra structure are given by \n  \\begin{align*}\n    &P \\langle Q_1,\\ldots, Q_r\\rangle \n    =\n    \\sum_{0\\leq i_1< i_2<\\ldots<i_r\\leq k}(-1)^{i_1k_1+i_2k_2+\\ldots+i_rk_r}\n    \\left(\n    \\id^{\\otimes i_1}\\otimes\\Delta^{(k_1)}\\otimes \\id^{\\otimes (i_2-i_1)}\\otimes\\ldots \\right.\n    \\\\\n    & \\left. \\ldots \\otimes\\Delta^{(k_r)}\\otimes \\id^{\\otimes (k-i_r)}\n    \\right)(P)\\cdot \n    \\left(1^{\\otimes i_1}\\otimes Q_1\\otimes 1^{\\otimes (i_2-i_1)}\\otimes\\ldots\\otimes 1^{\\otimes (i_{r}-i_{r-1})}\\otimes Q_r\\otimes 1^{\\otimes (k-i_r)}\\right),\n  \\end{align*}\n  where $P\\in T^{k+1}H$ and $Q_j\\in T^{k_j+1}H$ for all $j$. Note that\n  restricting this brace algebra structure to $H=T^1H=(TH[1])^0$ we\n  find the brace algebra given in \\cite[Section~6]{Aguiar}.\n\\end{remark}\n \n \nLet us denote the twist of $TH[1]$ by the ``Maurer-Cartan\" element $1\\otimes 1$ as\n\\begin{equation}\n  \\left(H^\\bullet_{poly},[\\argument,\\argument],\\partial\\right)\n  :=\n  \\left(T^{\\bullet +1}H,[\\argument,\\argument]_H,[1\\otimes1,\\argument]_H\\right).\n\\end{equation}\n\\begin{lemma}\n  An element $J\\in T^2H$ is a formal twist on $H$ if and only if\n  $J-1\\otimes 1$ is a Maurer-Cartan element in $H_{poly}$.\n\\end{lemma}\n\\begin{proof} \n\n  \\leavevmode\n\n  \\noindent Suppose first that $J\\in T^2H$ is a formal twist on\n  $H$. Then, by definition, $F := J-1\\otimes 1$ is an element of\n  $\\mathcal{F}^1H_{poly}^1$ and\n  \\begin{equation}\n    \\partial F+\\frac{1}{2}[F,F]_H\n    =\n    \\frac{1}{2}[J,J]_H\n    =\n    (\\Delta\\otimes \\id)(J)(J\\otimes 1)-(\\id\\otimes \\Delta)(J)(1\\otimes J)=0\n  \\end{equation}\n  by condition \\refitem{def:twist2} of Definition \\ref{def:TwistUEA}.  Conversely,\n  suppose $F$ is a Maurer-Cartan element in $H_{poly}$, then, for $J =\n  1\\otimes 1 + F$,\n  \\begin{equation} \n    (\\Delta\\otimes \\id)(J)(J\\otimes 1)-(\\id\\otimes \\Delta)(J)(1\\otimes J)\n    =\n    \\frac{1}{2}[J,J]_H\n    =\n    \\partial F +\\frac{1}{2}[F,F]_H\n    =\n    0\n  \\end{equation}\n  by the Maurer-Cartan equation. So $J$ satisfies \\refitem{def:twist2} of\n  \\ref{def:TwistUEA}, while clearly $J-1\\otimes\n  1=F\\in\\mathcal{F}^1(H\\otimes H)$.\n \\end{proof}\nDrinfeld discovered \\cite{drinfeld:1988a,drinfeld:1983b} that one can\ntwist the Hopf algebra structure on $H$ by any (formal) twist $J$. \nMore explicitely, one obtains the twisted Hopf algebra $H_J$\nby changing only the coproduct $\\Delta$ to $\\Delta_J$ given by\n\\begin{equation} \n  \\Delta_J(X)\n  =\n  J^{-1}\\Delta(X)J\n\\end{equation} \nLet us fix the (formal) twist $J$ on $H$. It\nis convenient to introduce the following notation\n\\begin{align*}\n  J_k\n  &:=\n  \\prod_{i=1}^{k}(\\Delta^{(k-i)}\\otimes \\id^{\\otimes i})(J\\otimes 1^{\\otimes i-1})\n  \\\\\n  &=\n  (\\Delta^{(k-1)}\\otimes \\id)(J)\\cdot(\\Delta^{(k-2)}\\otimes \\id\\otimes \\id)(J\\otimes 1)\n  \\cdot\\ldots\\cdot\n  (\\Delta\\otimes \\id^{\\otimes k-1})(J\\otimes 1^{\\otimes k-2})\\cdot (J\\otimes 1^{\\otimes k-1})\n\\end{align*}\nand we set $J_0=1$.  Note that $J_k\\in H^{\\otimes k+1}$ is invertible\nwith $J_k^{-1}$ given by reversing\nthe order of terms above and replacing $J$ by $J^{-1}$.\n\\begin{lemma}\n  \\label{lem:DeltaJk}\n  The iterates of $\\Delta_J$ are given by the formula \n  \\begin{equation} \n    \\Delta_J^{(k)}(X)\n    =\n    J_k^{-1}\\Delta^{(k)}(X)J_k.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof} \n  The proof is given by straightforward computation.  \n\\end{proof}\nWe also need the following (slightly technical) lemma.\n\\begin{lemma}\n  \\label{lem:tech}\n  The $J_k$'s satisfy the relation \n  \\begin{equation} \n  \\label{eq:tech}\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})(J_k)\n    \\cdot\n    (1^{\\otimes i}\\otimes J_l\\otimes 1^{\\otimes k-i})\n    =\n    J_{k+l}\n  \\end{equation}\n  for all $0\\leq i\\leq k\\in\\mathbb{Z}$ and $l\\in \\mathbb{Z}_{\\geq0}$. \n\\end{lemma}\n\\begin{proof}\n  For $k=i=0$, the formula \\eqref{eq:tech} reads\n  \\begin{equation} \n    \\Delta^{(l)}(1)\\cdot J_l\n    =\n    J_l\n  \\end{equation}\n  which is obviously satisfied. For $k=1$ and $i=0$ we get\n  \\begin{equation} \n    (\\Delta^{(l)}\\otimes \\id)(J)\\cdot(J_l\\otimes 1)\n    =\n    J_{l+1},\n  \\end{equation}\n  which follows immediately from the definition of $J_{l+1}$. For $k=1$, $i=1$ and $l=0$ we find \n  \\begin{equation}\n  J\\cdot(1\\otimes 1)=J.\n  \\end{equation}\n  We will fully establish the $k=1$ case by induction now. So suppose the formula \\ref{eq:tech} holds for $k=1$, $i=1$, $l-1\\in \\mathbb{Z}_{\\geq0}$. Then \n  \\begin{align*}\n    ((\\id\\otimes \\Delta^{(l)})(J)\\cdot(1\\otimes J_l)\n    &= (\\id\\otimes \\Delta^{(l-1)}\\otimes\\id)\\left((\\id\\otimes \\Delta)(J)\\cdot(1\\otimes J)\\right)\\cdot(1\\otimes J_{l-1}\\otimes 1)\n    \\\\\n    &=\n    (\\id\\otimes\\Delta^{(l-1)}\\otimes \\id)\\left((\\Delta\\otimes \\id)(J)\\cdot\n    (J\\otimes 1)\\right)\\cdot(1\\otimes J_{l-1}\\otimes 1)\n    \\\\\n    &=\n    (\\Delta^{(l)}\\otimes \\id)(J)\\cdot(\\id\\otimes \\Delta^{(l-1)}\\otimes \\id)(J\\otimes 1)\\cdot (1\\otimes J_{l-1}\\otimes 1)\n    \\\\\n    &=\n    (\\Delta^{(l)}\\otimes \\id)(J)\\cdot\\left(((\\id\\otimes \\Delta^{(l-1)})(J)\\cdot(1\\otimes J_{l-1}))\\otimes 1\\right)\n    \\\\\n    &=(\\Delta^{(l)}\\otimes \\id)(J)\\cdot (J_l\\otimes 1)=J_{l+1}.\n  \\end{align*}\n  Thus we have established the $k=0$ and $k=1$ cases completely. \n \n \n To establish the $k\\geq 2$ cases, we proceed by induction. Suppose that the formula \\eqref{eq:tech} is\n  satisfied for all triples $(\\kappa,i,l)\\in(\\mathbb{Z}_{\\geq 0})^3$ where $i\\leq \\kappa$ and $\\kappa<k$. Then we\n  have\n  \\begin{align*} \n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})(J_k)&\\cdot (1^{\\otimes i}\\otimes J_l\\otimes 1^{\\otimes k-i})\n    =\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{k-i})\\left((\\id\\otimes \\Delta^{(k-1)})(J)\\cdot(1\\otimes J_{k-1})\\right)\\cdot\\\\ \n    &\\hspace{0.4cm}\\cdot(1^{\\otimes i}\\otimes J_l\\otimes 1^{\\otimes k-i})\n    \\\\\n    &=\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes\\id^{\\otimes k-i})((\\id\\otimes \\Delta^{(k-1)})(J))\\cdot\n    \\\\\n    &\\hspace{0.4cm}\\cdot(1\\otimes \\left((\\id^{\\otimes (i-1)}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})(J_{k-1})\\right)\\cdot(1^{\\otimes i-1}\\otimes J_l\\otimes 1^{k-i}))\n    \\\\\n    &=\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})((\\id\\otimes \\Delta^{(k-1)})(J))\\cdot(1\\otimes J_{k+l-1})\n    \\\\\n    &=(\\id\\otimes \\Delta^{(k+l-1)})(J)\\cdot(1\\otimes J_{k+l-1})\n    \\\\\n    &=\n    J_{k+l}.\n  \\end{align*}\n  Thus, Eq.~\\eqref{eq:tech} is satisfied for all triples $(k,i,l)$. \n\\end{proof}\n\\begin{proposition}\n  \\label{prop:twistJ}\n  The map $\\mathscr{J}\\colon (H_J)_{poly}\\longrightarrow\n  (H_{poly})^{J-1\\otimes 1}$ given by $\\mathscr{J}(P)=J_k\\cdot P$ for\n  $P\\in (H_J)_{poly}^k$ is a DGLA isomorphism.\n\\end{proposition}\n\\begin{proof} \n  First we note that $\\mathscr{J}^{-1}(P)=J_k^{-1}P$ for $P\\in\n  H_{poly}^k$ is obviously an inverse of $\\mathscr{J}$. Furthermore we\n  observe that $\\mathscr{J}(1\\otimes 1) = J$, which shows that we only need\n  to check that $\\mathscr{J}$ preserves the brackets. This follows by\n  direct computation from Lemmas \\ref{lem:DeltaJk} and \\ref{lem:tech}\n\\end{proof}\n\\begin{definition}[Deformation Symmetry]\n  \\label{Defsym}\n  Let $\\algebra A$ be a unital associative algebra over $\\ring R$\n  equipped with a complete filtration $\\algebra A =\\mathcal{F}^0\n  \\algebra A\\supset\\mathcal{F}^1 \\algebra A\\supset\\ldots$ and consider\n  the Hochschild DGLA structure on the complex $C^\\bullet(\\algebra\n  A) = C^\\bullet(\\algebra A; \\algebra A)[1]$. A deformation symmetry of\n  $H$ in $\\algebra A$ is a map\n  \\begin{equation} \n    \\Phi\n    \\colon \n    H_{poly}\\longrightarrow C(\\algebra A)\n  \\end{equation}\n  of $L_\\infty$-algebras. \n\\end{definition}\nThe previous proposition implies the following claim. \n\\begin{corollary}\n  Any formal twist on $H$ produces\n  deformations of all algebras $\\algebra A$ equipped with a\n  deformation symmetry of $H$.\n\\end{corollary}\nThe notion of deformation symmetry is a generalization of the\nstandard notion of universal deformation via Drinfeld twist, see e.g.\n\\cite{bieliavsky.gayral:2015a,ESW2016,giaquinto.zhang:1998a}. Universal deformation formula \nrelies on the notion of Hopf\nalgebra action that we recall in the following definition.\n\\begin{definition}[Hopf algebra action]\n  Let $\\algebra A$ be as in Definition \\ref{Defsym}. Then an action of\n  the Hopf algebra $H$ on $\\algebra A$ is defined as a map\n  \\begin{equation}\n    \\phi\\colon H\\otimes \\algebra A\\longrightarrow \\algebra A\n  \\end{equation}\n  such that \n  \\begin{align}\n    \\phi\\circ(\\id_H\\otimes \\mu) \n    &= \n    \\mu\\circ (\\phi\\otimes\\phi)\n    \\circ\n    (\\id_H\\otimes \\tau_{H\\otimes \\algebra A}\\otimes\\id_{\\algebra A})\n    \\circ \n    (\\Delta\\otimes\\id_{\\algebra A\\otimes \\algebra A}), \n    \\\\\n    \\phi\\circ(\\mu_H\\otimes \\id_{\\algebra A})\n    &=\n    \\phi\\circ(\\id_H\\otimes \\phi)\n    \\\\\n    \\phi\\circ(\\eta_H\\otimes \\id_{\\algebra A})\n    &=\n    \\mu_{\\algebra A}\\circ(\\eta_{\\algebra A}\\otimes \\id_{\\algebra A})\n    \\\\\n    \\phi\\circ(\\id_H\\otimes \\eta_{\\algebra A}) &= \\eta_{\\algebra A}\\circ\\mu_{\\ring R}\\circ(\\epsilon\\otimes \\id_{\\ring R}).\n  \\end{align}\n  Here $\\mu$ denotes the multiplication of $\\algebra A$, $\\mu_{\\ring R}$\n  denotes the multiplication of $\\ring R$, $\\mu_H$ denotes the\n  multiplication of $H$, $\\eta_{\\algebra A}\\colon \\ring R\\rightarrow\n  A$ denotes the unit of $A$, $\\eta_H\\colon \\ring R\\rightarrow H$\n  denotes the unit of $H$ and $\\tau_{H\\otimes A}$ denotes the flip\n  $h\\otimes a\\mapsto a\\otimes h$.\n\\end{definition}\nNotice that $\\phi$ can be regarded as a map $H\\rightarrow \\End_{\\ring\n  R}(A) = C^1(A;A)$ satisfying certain conditions.\n\\begin{proposition}\n  \\label{actionimpliesdefsym}\n  Given a Hopf algebra action $\\phi$ of $H$ on $A$, the map $\\Phi$\n  defined by\n  \\begin{equation}\n    h_0\\otimes \\ldots \\otimes h_k\\mapsto \\mu_{\\algebra A}^{(k)}\\circ(\\phi(h_0)\\otimes\n    \\phi(h_1)\\otimes \\ldots\\otimes \\phi(h_k))\n  \\end{equation}\n  is a deformation symmetry. Here $\\mu_{\\algebra A}^{(k)}$ denotes the\n  $k$-th iteration $\\mu_{\\algebra A}\\circ(\\id\\otimes \\mu_{\\algebra\n    A})\\circ\\ldots\\circ(\\id^{\\otimes k-1}\\otimes \\mu_{\\algebra A})$ of\n  $\\mu_{\\algebra A}$.\n\\end{proposition}\n\\begin{proof} \n\n  \\leavevmode\n\n  \\noindent We prove this proposition by showing that $\\Phi$ is a\n  map of DGLA's.  Note that for $P=P_0\\otimes \\ldots\\otimes\n  P_k$, $Q=Q_0\\otimes \\ldots\\otimes Q_l$ and $a=a_0\\otimes\n  \\ldots\\otimes a_{k+l}$ we have\n  \\begin{align*}\n    \\Phi(P\\bullet Q)(a)\n    &=\n    \\Phi\\left(\\sum_{i=0}^k(-1)^{il}P_0\\otimes \\ldots\\otimes P_{i-1}\\otimes \\Delta^{(l)}(P_i)Q\\otimes P_{i+1}\\otimes \\ldots\\otimes P_k\\right)(a)\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\n    \\phi(P_0)(a_0)\\cdot\\ldots\\cdot\n    \\phi(P^{(0)}_iQ_0)(a_i)\\cdot\\ldots\\cdot\\phi(P^{(l)}_iQ_l)(a_{i+l})\\cdot\n    \\\\\n    &\\qquad\\cdot\n    \\phi(P_{i+1})(a_{i+1+l})\\cdot\\ldots\\cdot\\phi(P_k)(a_{k+l})\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\\phi(P_0)(a_0)\n    \\cdot\\ldots\\cdot\n    \\phi(P_i)(\\phi(Q_0)(a_i)\\cdot\\ldots\\cdot\\phi(Q_l)(a_{l+i}))\\cdot\n    \\\\\n    &\\qquad\\cdot \\phi(P_{i+1}(a_{l+i+1})\\cdot\\ldots\\cdot\\phi(P_k)(a_{l+k})\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\\phi(P_0)(a_0)\n    \\cdot\\ldots\\cdot\n    \\phi(P_{i-1})(a_{i-1})\\cdot\\phi(P_i)(\\Phi(Q)(a_{i}\\otimes\\ldots\\otimes a_{i+l})\\cdot\n    \\\\\n    &\\qquad\\cdot \\phi(P_{i+1})(a_{l+i+1})\\cdot \\ldots\\cdot\\phi(P_{k})(a_{k+l})\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\\Phi(P)(a_0\\otimes \n    \\ldots\\otimes a_{i-1}\\otimes \\Phi(Q)(a_i\\otimes\\ldots\\otimes a_{i+l})\\otimes a_{i+l+1}\\otimes \\ldots\\otimes a_{k+l})\n    \\\\\n    &=\n    \\Phi(P)\\{\\Phi(Q)\\}(a)\n  \\end{align*}\n  where we have used the \\emph{brace} notation, see e.g.\n  \\cite{GerstVoron1995}. Note that\n  $[A,B]_G=A\\{B\\}-(-1)^{|A||B|}B\\{A\\}$. In short the above computation \n  boils down to\n  \\begin{equation}\n    \\Phi(P\\bullet Q)\n    =\n    \\Phi(P)\\{\\Phi(Q)\\}.\n  \\end{equation} \n  Thus $\\Phi$ respects the brackets and observing that $\\Phi(1\\otimes\n  1)=\\mu_{\\algebra A}$ the proposition is proved.\n\\end{proof}\n\\begin{remark}\n  Recall that $TH[1]$ is endowed with a brace algebra structure, as described in\n  Remark~\\ref{rem:braces}.  The above proposition actually follows\n  from the fact that a Hopf algebra action $\\phi$ of $H$ on\n  $\\mathscr{A}$ actually induces a brace algebra morphism\n  \\begin{equation}\n    \\Phi\\colon TH[1]\\longrightarrow C^\\bullet(\\mathscr{A}),\n  \\end{equation}\n  where the braces on $C^\\bullet(\\mathscr{A})$ are defined as usual by \n  \\begin{align*}\n    P\\{Q_1,\\ldots, Q_r\\}&(a_0\\otimes \\ldots\\otimes a_{k+k_1+\\ldots+k_r})\n    =\n    \\sum_{0\\leq i_1<i_2<\\ldots<i_r\\leq k}(-1)^{i_1k_1+i_2k_2+\\ldots+i_rk_r}\n    P(a_0\\otimes \\ldots\\otimes \n    \\\\\n    &\\otimes\n    Q_1(a_{i_1}\\otimes \n    \\ldots \\otimes a_{i_1+k_1})\\otimes a_{i_1+k_1+1}\\otimes\\ldots \n    \\otimes a_{i_2+k_1-1}\\otimes \n    Q_2(a_{i_2+k_1}\\otimes \\ldots\\otimes a_{i_2+k_1+k_2})\\otimes \n    \\\\\n    &\\ldots\\otimes \n    Q_r(a_{i_r+k_1+\\ldots+k_{r-1}}\\otimes a_{i_r+k_1+\\ldots+k_r})\\otimes \\ldots \\otimes a_{k+k_1+\\ldots+k_r}),\n  \\end{align*}\n  for $P\\in C^{k+1}(\\mathscr{A};\\mathscr{A})$, $Q_j\\in\n  C^{k_j+1}(\\mathscr{A};\\mathscr{A})$ for all $j$ and $a_i\\in\n  \\mathscr{A}$ for all $i$. So we have\n  \\begin{equation} \n    \\Phi(P\\langle Q_1,\\ldots, Q_r\\rangle)\n    =\n    \\Phi(P)\\{\\Phi(Q_1),\\ldots, \\Phi(Q_r)\\}\n  \\end{equation}\n  for all $P, Q_1,\\ldots, Q_r\\in TH[1]$. \n\\end{remark}\n\n\n\n\n\n\n\\subsection{Twisting Poisson actions}\n\n\nA Lie bialgebra is a pair $(\\lie g, \\gamma)$ where $\\lie g$ is a Lie\nalgebra and $\\gamma$ is a $1$-cocycle, $ \\gamma \\colon \\lie g \\to \\lie\ng\\wedge \\lie g$.  In this paper we consider a particular class of Lie\nbialgebras.  Recall that an element $r \\in \\lie g\\wedge \\lie g$ is\ncalled $r$-matrix if it satisfies the Maurer--Cartan equation\n$\\Schouten{r, r}= 0$.  It can be proved that $r$-matrices always\ninduce a Lie bialgebra structure on $\\lie g$, by setting $\\gamma =\n\\Schouten{r, \\argument }$. We refer to the pair $(\\lie g, r)$ as\n\\emph{triangular Lie algebra}. For further details on Lie bialgebras\nwe refer to \\cite{Kosmann-Schwarzbach2004}.\n\\vspace{0.3cm}\n\nLet us consider a Lie algebra action $\\varphi : \\lie{g} \\to\n\\Secinfty(TM)$.\n\\begin{definition}[Poisson action]\n  The action $\\varphi$ is Poisson if it satisfies\n  \\begin{equation}\n    \\D_\\pi \\varphi(X)\n    =\n    \\varphi \\wedge \\varphi \\circ \\gamma (X)\n  \\end{equation}\n  where $\\D_\\pi = \\Schouten{\\pi , \\argument}$. \n\\end{definition}\n\\begin{proposition}\n  \\label{prop:PoissonR}\n  Let $\\lie{g}$ be a finite dimensional triangular Lie algebra and\n  $\\varphi \\colon \\lie{g} \\to \\Secinfty(TM)$ a Lie algebra action.\n  \\begin{propositionlist}\n  \\item The bitensor $\\pi$ defined as the image of $r$ via $\\varphi$\n    is a Poisson tensor.\n  \\item $\\varphi$ is a Poisson action wrt $\\pi = \\varphi \\wedge\n    \\varphi (r)$.\n  \\end{propositionlist}\n\\end{proposition}\n\\begin{proof}\n  Let us consider the $r$-matrix $r = \\frac{1}{2} \\sum_{i,j} r^{ij}\n  e_i \\wedge e_j$ and define\n  \\begin{equation}\n    \\label{eq:PoissonR}\n    \\pi\n    :=\n    \\frac{1}{2} \\sum_{i,j} r^{ij} \\varphi(e_i) \\wedge \\varphi(e_j).\n  \\end{equation}\n  From $\\Schouten{r, r} = 0$, using the fact that $\\varphi$ is a Lie\n  algebra morphism, it follows that $\\Schouten{\\pi, \\pi} = 0$. The second\n  claim is a straightforward computation.\n\\end{proof}\nIt is easy to see that \nthe notion of Poisson action can be extended to a morphism of\nDGLA's\n\\begin{equation}\n  \\label{eq:PoissonActionExt}\n  \\wedge^{\\bullet}\\varphi\n  : \n  (\\wedge^{\\bullet}\\lie g ,\\delta, [\\argument ,\\argument])\n  \\longrightarrow \n  (\\Secinfty (\\wedge^{\\bullet} TM), \\D_\\pi, \\Schouten{\\argument , \\argument}).\n\\end{equation}\n\nWe now show how one may use the formality of $\\Dpoly{\\mathfrak{g}}{}$\nand $\\Dpoly{}{}(M)$ to obtain a deformation symmetry of\n$\\mathscr{U}(\\mathfrak{g})$ in $\\Cinfty(M)$ given an infinitesimal\naction $\\varphi$ of $\\mathfrak{g}$ on $M$. Thus, given an r-matrix of\n$\\mathfrak{g}$, we obtain deformations of all relevant structures and\nwe comment on these. In Section~\\ref{sec:formality} we have obtained\nthe \\emph{formal} DGLA's $\\Dpoly{\\mathfrak{g}}{}$ and $\\Dpoly{}{}(M)$.\nThe classical $r$-matrix $r\\in\\Tpoly{\\mathfrak{g}}{1}$ yields a\nMaurer-Cartan element. Although it satisfies the Maurer-Cartan\nequation $\\Schouten{r,r}=0$ it is in fact not an MC element according\nto our Definition~\\ref{MCdef} as we have neglected to consider any\nfiltration. The filtration is needed since we go through formality,\nwhich means we encounter infinite sums. More precisely, we obtain a\nfiltration by considering the formal power series ring $\\mathbb{R}[\\![\\hbar]\\!]$. \nGiven a DGLA $\\mathfrak{L}$ we denote the DGLA obtained by\nextending scalars to the formal power series ring by\n$\\mathfrak{L}[\\![\\hbar]\\!]$.  We consider these DGLA's as filtered by the\ndegree in $\\hbar$. Note that, given $L_\\infty$-morphisms of DGLA's\n$\\mathfrak{L}\\rightarrow \\mathfrak{L}'$ we obtain also\n$L_\\infty$-morphisms of the extended DGLA's\n$\\mathfrak{L}[\\![\\hbar]\\!]\\rightarrow\\mathfrak{L}'[\\![\\hbar]\\!]$.  In this way\n$L_\\infty$-quasi-isomorphism go to $L_\\infty$-quasi-isomorphisms.\n\n\nFrom Section~\\ref{sec:formality} it follows that we\nhave a \\emph{horse-shoe} diagram in the category of\n$L_\\infty$-algebras:\n\\begin{equation}\n  \\label{horse-shoe}\n  \\begin{tikzcd}[row sep=3em,column sep=4em]\n    & \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n    \\arrow[r, \"\\varphi\"]\\arrow[d, swap, \"F_{\\mathfrak{g}}\"] \n    & \\Tpoly{}{}(M)[\\![\\hbar]\\!] \\arrow[d,  \"F_M\"]\n    \\\\\n    & \\Dpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n    & \\Dpoly{}{}(M)[\\![\\hbar]\\!].\n  \\end{tikzcd}\n\\end{equation}\nHere the vertical maps $F_\\mathfrak{g}$ and $F_M$ are\n$L_\\infty$-quasi-isomorphisms constructed as discussed in\nSection~\\ref{sec:formality} and the horizontal arrow is induced by\n$\\varphi$ as in \\eqref{eq:PoissonActionExt}.\nThus we obtain, given an r-matrix $r\\in\\mathfrak{g}\\wedge\\mathfrak{g}$, the MC elements:\n\\begin{equation}\n  \\begin{array}{l}\n    \\hbar r\\in \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!];\n    \\\\\n    \\hbar\\pi\n    =\n    (\\hbar r)_\\varphi\\in \\Tpoly{}{}(M)[\\![\\hbar]\\!];\n    \\\\\n    \\rho_\\hbar\n    :=\n    (\\hbar r)_{F_\\mathfrak{g}}\\in \\Dpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n    \\\\\n    B_\\hbar\n    :=\n    (\\hbar \\pi)_{F_M}\\in \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!].\n  \\end{array}\n\\end{equation}\n\n\\begin{remark} \n  \\label{rem:WrongCandidate}\n  Notice that we obtain the deformed versions of $\\Cinfty(M)$ and\n  $\\mathscr{U}(\\mathfrak{g})$ by twisting by the Maurer-Cartan\n  elements listed above. We would like then to obtain a deformation\n  symmetry completing the square in the horse-shoe diagram above, such\n  that it commutes.  Commutativity ensures that the two formal\n  deformations of $\\Cinfty(M)$ induced by $\\hbar r$ by transporting it\n  along the bottom or the top to $\\Dpoly{}{}(M)[\\![\\hbar]\\!]$ coincide. An\n  obvious candidate for such a map would be\n  \\begin{equation}\n    X_1\\otimes\\ldots\\otimes X_k\n    \\mapsto \n    \\varphi(X_1)\\otimes\\ldots\\otimes\\varphi(X_k),\n  \\end{equation}\n  i.e. the map induced by the map of Lie-Rinehart pairs\n  $(\\mathfrak{g}, \\mathbb{R})\\rightarrow (\\Secinfty(TM),\\Cinfty(M))$.  In\n  other terms the deformation symmetry induced by the obvious Hopf\n  algebra action of $\\mathscr{U} (\\mathfrak{g})$ on $\\Cinfty(M)$\n  through proposition \\ref{actionimpliesdefsym}. However this map may\n  not make the diagram commute. In fact it is the opinion of the\n  authors that such commutation would involve some condition of\n  compatibility of the connections used in defining these maps, see\n  Remark~\\ref{rem:choice}.\n\\end{remark}\nWe complete the horse-shoe diagram \\eqref{horse-shoe} to a commuting\nsquare by observing that $L_\\infty$-quasi-isomorphisms are\n\\emph{invertible}. The following lemma is essentially contained in\n\\cite[Chapt.~10.4]{AO}. We remind the reader that the field underlying\n$L_\\infty$-algebras is of characteristic $0$.\n\\begin{lemma}\n  \\label{invert}\n  Suppose $\\mathfrak{L}$ and $\\mathfrak{L}'$ are formal\n  $L_\\infty$-algebras and $f\\colon H(\\mathfrak{L})\\rightarrow\n  H(\\mathfrak{L'})$ is an $L_\\infty$-morphism, then there exists a\n  lift $\\tilde{f}\\colon \\mathfrak{L}\\rightarrow \\mathfrak{L}'$ of\n  $f$. In other words there exists a commuting diagram\n  \\begin{equation}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & H(\\mathfrak{L})\n      \\arrow[r, \"f\"]\\arrow[d, swap, \"i_l\"] \n      & H(\\mathfrak{L}') \\arrow[d,  \"i_r\"]\n      \\\\\n      & \\mathfrak{L} \\arrow[r, swap, \"\\tilde{f}\"] \n      & \\mathfrak{L}'.\n    \\end{tikzcd}\n  \\end{equation}\n  in the category of $L_\\infty$-algebras such that the vertical arrows\n  are quasi-isomorphisms.\n\\end{lemma}\n\\begin{proof}\n  Note that we start by hypothesis with the horse-shoe\n  \\begin{equation}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & H(\\mathfrak{L})\n      \\arrow[r, \"f\"]\\arrow[d, swap, \"i_l\"] \n      & H(\\mathfrak{L}') \\arrow[d,  \"i_r\"]\n      \\\\\n      & \\mathfrak{L}\n      & \\mathfrak{L'}.\n    \\end{tikzcd}\n  \\end{equation}\n  in the category of $L_\\infty$-algebras such that the vertical arrows\n  are quasi-isomorphisms. As shown in \\cite[Sect.~10.4.4]{AO} we can\n  always find a quasi-inverse $l_i\\colon\\mathfrak{L}\\rightarrow\n  H(\\mathfrak{L})$ of $i_l$ such that the induced maps in cohomology\n  are inverse to each other. \\ We define $\\tilde{f}:=i_r\\circ f\\circ\n  l_i$. Note that this already proves that $\\tilde{f}\\circ\n  i_l=i_r\\circ f$ in cohomology (and thus ``up to homotopy''). To get\n  the stronger statement in the lemma we will need to consider the\n  construction of the map $l_i$. This construction involves the notion\n  of a so-called $\\infty$-isomorphism. An $\\infty$-isomorphism is a\n  morphism of $L_\\infty$-algebras $F$ such that $F_1$ is an\n  \\emph{isomorphism}. The main observations for the construction of\n  $l_i$ are two-fold. First, the homotopy transfer theorem\n  \\cite[Sect.~10.3]{AO} yields an $L_\\infty$-structure on\n  $H(\\mathfrak{L})$ which is unique up to $\\infty$-isomorphism. It is\n  obtained by picking a retraction of $\\mathfrak{L}$ onto\n  $H(\\mathfrak{L})$, we may pick the retraction given by\n  $i_l$. Secondly in section 10.4.2 of \\cite{AO} it is shown that any\n  $L_\\infty$-algebra $\\mathfrak{L}$ is $\\infty$-isomorphic to the sum\n  $H(\\mathfrak{L})\\oplus K$ where $K$ is an acyclic chain complex\n  (with trivial $Q_0$, $Q_2$, $Q_3$ and so on). Now the construction\n  of $l_i$ follows by the fact that $\\infty$-isomorphisms are\n  invertible (shown in section 10.4.1 of \\cite{AO}). Our lemma follows\n  from the fact that, for a formal $L_\\infty$-algebra the\n  $L_\\infty$-structure induced on $H(\\mathfrak{L})$ by homotopy\n  transfer equals the canonically induced structure up to\n  $\\infty$-isomorphism. Thus we may simply consider the splittings\n  $\\mathfrak{L}\\simeq H(\\mathfrak{L})\\oplus K_{\\mathfrak{L}}$ and\n  $\\mathfrak{L}'\\simeq H(\\mathfrak{L}')\\oplus K_{\\mathfrak{L}'}$,\n  where $\\simeq$ means $\\infty$-isomorphism, given by the retractions\n  induced by $i_l$ and $i_r$. Then the map $\\tilde{f}$ simply maps\n  $H(\\mathfrak{L})$ to $H(\\mathfrak{L}')$ by $f$.\n\\end{proof}\n\\begin{corollary}\n\\label{twistaction}  The diagrams \n  \\begin{equation}\n    \\label{square}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n      \\arrow[r, \"\\varphi\"]\\arrow[d, swap, \"F_{\\mathfrak{g}}\"] \n      & \\Tpoly{}{}(M)[\\![\\hbar]\\!] \\arrow[d,  \"F_M\"]\n      \\\\\n      & \\Dpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!] \\arrow[r, \"\\tilde{\\varphi}\"]\n      & \\Dpoly{}{}(M)[\\![\\hbar]\\!]\n    \\end{tikzcd}\n  \\end{equation}\n  and\n  \\begin{equation}\n    \\label{twistsquare}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & \\Tpoly{\\mathfrak{g}}{\\hbar r}[\\![\\hbar]\\!]\n      \\arrow[r, \"\\varphi^{\\hbar r}\"]\\arrow[d, swap, \"F^{\\hbar r}_{\\mathfrak{g}}\"] \n      & \\Tpoly{}{\\hbar\\pi}(M)[\\![\\hbar]\\!] \\arrow[d,  \"F^{\\hbar\\pi}_M\"]\n      \\\\\n      & \\Dpoly{\\mathfrak{g}}{\\rho_\\hbar}[\\![\\hbar]\\!] \\arrow[r, \"\\tilde{\\varphi}^{\\rho_\\hbar}\"]\n      & \\Dpoly{}{B_\\hbar}(M)[\\![\\hbar]\\!]\n    \\end{tikzcd}\n  \\end{equation}\n  commute.\n\\end{corollary}\n\\begin{proof}\n  Applying the above lemma to our situation we immediately find that\n  the diagram \\eqref{square} commutes. Also, by applying the results\n  of Section~\\ref{sec:Twisting} we obtain diagram \\eqref{twistsquare},\n  which commutes thanks to Remark~\\ref{comptwist}.\n\\end{proof}\n\n\\subsection{Twisted structures}\n\nIn the following we show that the twisted complexes obtained above are\ncoming from a formal deformation quantization of $\\Cinfty(M)$ (in the\ncase of $\\Dpoly{}{B_\\hbar}(M)$) and a deformation of\n$\\mathscr{U}(\\mathfrak{g})$ into a quantum group (in the case of\n$\\Dpoly{\\mathfrak{g}}{\\rho_\\hbar}$).\n\\begin{proposition}\n  \\label{defHoch}\n  There is a formal deformation quantization $\\mathscr{A}_\\hbar$ of\n  $(\\Cinfty(M),\\pi)$ such that\n  \\begin{equation}\n    \\Dpoly{}{B_\\hbar}(M)\n    \\hookrightarrow \n    C^\\bullet(\\algebra{A}_\\hbar,\\algebra{A}_\\hbar),\n  \\end{equation}\n  i.e. $\\Dpoly{}{B_\\hbar}(M)$ is a subcomplex of the Hochschild\n  cochain complex of $\\algebra{A}_\\hbar$.\n\\end{proposition}\n\\begin{proof}\n  Note that the Maurer--Cartan equation \\eqref{eq:MC} matches exactly\n  the associativity condition of $m + B_\\hbar$, where $m$ denotes the\n  pointwise multiplication in $\\Cinfty(M)[\\![\\hbar]\\!]$. Since\n  \\begin{equation}\n    B_\\hbar\n    =\n    \\sum_{n\\geq 1}\\frac{1}{n!}(F_M)_n(\\hbar\\pi,\\ldots,\\hbar\\pi)\n    =\n    \\sum_{n\\geq 1}\\frac{\\hbar^n}{n!}(F_M)_n(\\pi,\\ldots,\\pi),\n  \\end{equation}\n  we see that $m+B_\\hbar$ defines a deformation quantization\n  $\\algebra{A}_\\hbar$. The differential on the Hochschild complex\n  $C^\\bullet(\\algebra A, \\algebra A)$ is given by taking the\n  Gerstenhaber bracket with the multiplication for any associative\n  algebra $\\algebra A$.  Thus the twisted differential on\n  $\\Dpoly{}{B_\\hbar}(M)$ coincides with the differential of\n  $C^\\bullet(\\algebra{A}_\\hbar,\\algebra{A}_\\hbar)$.  Finally we recall\n  that\n  \\begin{equation}\n    B_\\hbar\n    =\n    \\sum_{n\\geq 1}\\frac{\\hbar^n}{n!}(F_M)_n(\\pi,\\pi,\\ldots,\\pi)\n    =\n    \\hbar(F_M)_1(\\pi)  \n    \\hspace{0.4cm}\\mbox{mod}\\hspace{0.2cm}\\hbar^2.\n  \\end{equation}\n  Since $(F_M)_1$ is a quasi-isomorphism we find that the alternating\n  part of $B_\\hbar$ is $\\hbar\\pi$ modulo $\\hbar^2$ and\n  \\begin{equation}\n    \\frac{[f,g]_{\\star}}{\\hbar}\n    =\n    \\pi(f,g)+O(\\hbar)\n  \\end{equation}\n  for all $f,g\\in \\Cinfty(M)$. Here $[\\argument, \\argument]_\\star$\n  denotes the commutator bracket of $\\algebra{A}_\\hbar$.\n\\end{proof}\nThus we obtain in particular the deformation quantization\n$\\algebra{A}_\\hbar$.  On the other hand we also obtain the MC element\n$\\rho_\\hbar$ in $\\Dpoly{\\mathfrak{g}}{}$. This yields the\ncoproduct $\\Delta_{J_\\hbar}$ (where $J_\\hbar=1\\otimes 1 +\\rho_\\hbar)$\nthus establishing a quantum group, obtained by quantization of the Lie\nbialgebra $(\\mathfrak{g},\\Schouten{r,\\argument})$.\nFinally we obtain the following theorem as a corollary of\nProp.~\\ref{prop:twistJ} and Corollary~\\ref{twistaction}. \n\n\\begin{theorem}\n  Suppose $(\\lie g, r)$ is a triangular Lie algebra and $\\varphi\\colon\n  \\lie g\\rightarrow \\Secinfty(TM)$ is an action on the manifold\n  $M$. Then there exist a formal deformation quantization\n  $\\mathscr{A}_\\hbar$ of $(M,\\pi)$ (where\n  $\\varphi\\wedge\\varphi(r)=\\pi$) and a quantization\n  $\\mathscr{U}_\\hbar(\\lie g)$ of $(\\lie g, r)$ which allow a\n  deformation symmetry\n  \\begin{equation} \n    \\tilde{\\varphi}^{\\rho_\\hbar}\\circ\\mathscr{J}\n    \\colon \n    \\mathscr{U}_\\hbar(\\lie g)_{poly}\\longrightarrow C(\\mathscr{A}_\\hbar;\\mathscr{A}_\\hbar)\n  \\end{equation}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Comparison with Drinfeld's construction}\n\n\n\n\n\n \n \n \n\nFirst, let us briefly recall the original construction of Drinfeld\n(see\n\\cite{aschieri.schenkel:2014a,drinfeld:1989a,giaquinto.zhang:1998a}). Consider\na formal twist $J$ on $\\mathscr{U}(\\mathfrak{g})[\\![\\hbar]\\!]$ and a generic\n$\\mathscr{U}(\\lie{g})$-module algebra $\\algebra A$. Drinfeld proved\nthat we can then always define an associative star product on\n$\\algebra A$. In particular, consider $\\algebra A = \\Cinfty(M)$ with\npointwise multiplication $m$.  Given a Lie algebra action $\\varphi:\n\\lie g \\to \\Secinfty(TM)$ we obtain a Hopf algebra action\n\\begin{equation}\n  \\label{eq:Action}\n  \\acts\n  \\colon\n  \\mathscr{U}(\\lie{g}) \\times \\Cinfty(M)\n  \\longrightarrow\n  \\Cinfty(M),\n\\end{equation}\nwhich makes $\\Cinfty(M)$ into a left $\\mathscr{U}(\\lie{g})$-module\nalgebra. More precisely, $X \\acts f = \\Lie_{\\varphi(X)} f$ where\n$\\Lie$ denotes the Lie derivative. The action $\\acts$ extends to\nformal power series \n\\begin{equation}\n  \\acts \\colon\n  \\mathscr{U}(\\lie{g})[\\![\\hbar]\\!] \\times \\Cinfty(M)[\\![\\hbar]\\!]\\longrightarrow\n  \\Cinfty(M)[\\![\\hbar]\\!].\n\\end{equation}\nThus the product defined by\n\\begin{align}\n  \\label{eq:StarTwist}\n  f\\star_J g\n  =\n  m(J\\acts(f\\tensor g))\n  =\n  m (\\Lie_{\\varphi \\tensor \\varphi(J)}(f \\tensor g))\n\\end{align}\nfor $f, g \\in \\Cinfty(M)[\\![\\hbar]\\!]$ is a star product.  The classical\nlimit of \\eqref{eq:StarTwist} is given by\n\\begin{equation}\n  \\label{eq:ThePoissonStructure}\n  \\{f, g\\}\n  =\n  m (r \\acts (f \\tensor g)),\n\\end{equation}\nwhere $r:=\\frac{J-\\tau(J)}{\\hbar}|_{\\hbar=0}$ is the $r$-matrix\nassociated to the twist $J$, here $\\tau$ denotes the flip $X\\otimes\nY\\mapsto Y\\otimes X$.  It is important to underline that the deformed\nalgebra $(\\Cinfty (M)[\\![\\hbar]\\!], \\star_J)$ is then a module-algebra for\nthe quantum group:\n\\begin{equation}\n  \\algebra U_J(\\lie{g}) \n  := (\\algebra U(\\lie{g})[\\![\\hbar]\\!], \\Delta_J).\n\\end{equation}\nIn other words, $\\acts$ is a Hopf agebra action of the twisted Hopf\nalgebra $\\algebra U_J(\\lie{g})$ on $(\\Cinfty (M)[\\![\\hbar]\\!], \\star_J)$.\n\n\n\nThus, the construction takes a formal twist $J\\in\n\\mathscr{U}(\\mathfrak{g})[\\![\\hbar]\\!]$ and an infinitesimal action of $\\lie\ng$ on $M$ as input and produces a deformation quantization $\\algebra\nA_\\hbar$ together with an action of the quantum group\n$\\mathscr{U}_J(\\lie g)$ on it. \nIn our approach one starts with\nan $r$-matrix $r\\in\\lie g\\wedge\\lie g$ and an infinitesimal action of\n$\\lie g$ on $M$ and obtains a formal twist $J$ and a deformation\nsymmetry $\\tilde{\\varphi}$. These then also yield a deformation\nquantization given by\n\\begin{equation}\n  f\\star_r g=\\tilde{\\varphi}(J)(f\\otimes g),\n\\end{equation}\nand another deformation symmetry $\\tilde{\\phi}^{\\rho_\\hbar}\\circ\\mathscr{J}$ \nof the quantum group $\\algebra U_J(\\lie g)$ in the deformed algebra $\\algebra A_\\hbar$.\n %\n \n \n \n \nThe main difference of the two approaches is that the formal twist is taken as\ngiven in Drinfeld's approach while we obtain it through quantization of an $r$-matrix. The trade-off\nis however that we do not obviously obtain an action of a quantum\ngroup anymore, instead we obtain the deformation symmetry. A direct comparison of the two approaches will be\nnontrivial and it implies a study of the compatibility condition\nbetween connections mentioned in Remark \\ref{rem:WrongCandidate}. In\nparticular it is of interest whether the process of quantization so\nobtained can be made functorial for equivariant maps between the\nmanifolds. We will come back to this in a future project.\n\n\n\n\n\n\n\n{\n  \\footnotesize\n  \\renewcommand{\\arraystretch}{0.5}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nWithin the past two years, three neutron star binaries, PSR\nJ1824$-$2452I in M28 \\citep{Pap13}, PSR J1023+0038 \\citep{Pat14}, and\nXSS J12270$-$4859 \\citep{Bassa14}, have been observed to\nswitch between accreting and rotation-powered pulsar states. These\ndiscoveries have confirmed the standard model for the formation of\nrotation-powered millisecond pulsars (MSPs) via spin-up by accretion\nin a low-mass X-ray binary (LMXB).\n\nThe two confirmed transitional MSPs in the field of the Galaxy, PSR\nJ1023+0038 and XSS J12270$-$4859, exhibit strong $\\gamma$-ray emission\nin \\textit{Fermi} Large Area Telescope data even during their\naccreting states. This implies that other similar objects may be\nhiding among the unidentified \\textit{Fermi} LAT sources \\citep[see,\n  e.g.,][for the case of 1FGL J1417.7$-$4407]{Stra15}.  While targeted\nradio pulsation searches of \\textit{Fermi} LAT sources have proven to\nbe particularly fruitful for discovering new rotation-powered MSPs,\nyielding at least 38 published discoveries to date\\footnote{See\n  https:\/\/confluence.slac.stanford.edu\/display\/GLAMCOG\/\n  \\\\ Public+List+of+LAT-Detected+Gamma-Ray+Pulsars for an up-to-date\n  list.}, transitional objects may elude detection at radio\nfrequencies since the pulsar emission mechanism appears to be\nextinguished when an accretion disk is present in the system\n\\citep[e.g.,][]{Stap14}.\n\nIn their low-luminosity ($\\sim$$10^{33}$\\,erg s$^{-1}$ in the soft\nX-ray band) accreting states, PSR J1023+0038, XSS\nJ12270$-$4859, and PSR J1824--2452I exhibit sudden, unpredictable\ndrops in flux and occasional intense flares\n\\citep{deM10,deM13,Lin14,Ten14,Bog14b}. This characteristic X-ray\nbehavior is a unique signature that can be used to identify additional\ntransition systems and distinguish them from other varieties of\naccreting objects such as cataclysmic variables (CVs).\n\nWe have identified a particularly promising candidate for an LMXB\/MSP\ntransition object, namely, 3FGL J1544.6$-$1125, previously known as\n2FGL J1544.5$-$1126. The only X-ray-bright object within the 95\\%\nconfidence error ellipse of this 3FGL source is the \\textit{ROSAT}\nsource 1RXS J154439.4$-$112820 \\citep{Step10}.  Optical identification of\n\\textit{ROSAT}-selected candidates for unassociated \\textit{Fermi} LAT\nsources by \\citet{Mas13} revealed this object to have an optical\nspectrum in 2012 consistent with those of accretion-disk-dominated\nsystems, with prominent emission lines of H and He. On this basis, it\nwas classified as a CV and was dismissed as the counterpart of 3FGL\nJ1544.6$-$1125 because no CVs are known to exhibit persistent\n$\\gamma$-ray emission.  However, given the fact that both PSR\nJ1023+0038 and XSS J12270--4859 were mis-classified as CVs upon their\ndiscovery suggests that similar objects may also be masquerading as\nCVs.  Based on this, we suspected that 1RXS J154439.4$-$112820 is\nindeed the counterpart of the \\textit{Fermi} LAT source. Here, we\npresent an analysis of X-ray and UV data from \\textit{XMM-Newton} and\noptical time-series photometry obtained at MDM Observatory of 1RXS\nJ154439.4$-$112820, which reveal the true nature of this object.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.42\\textwidth]{finding_chart.ps}\n\\end{center}\n\\caption{Finding chart for \\rxs\\ (tic marks) at position \\pos\\ from\n  MDM 1.3m images in the GG420 filter.  The size is\n  $4^{\\prime}\\!.3\\times4^{\\prime}\\!.3$.  North is up and east is to\n  the left.  The star marked ``C'' has magnitude $R=14.25$ in the USNO\n  B1.0 catalog, and is used for appoximate calibration in GG420.\n  Faint fringing is a detector artifact due to the filter transmission\n  at long wavelengths.  }\n\\label{fig:finder}\n\\end{figure} \n\n\n\\begin{deluxetable*}{cccccl}\n\\tabletypesize{\\small}\n\\tablewidth{0pt}\n\\tablecaption{Log of MDM Observatory Time Series of \\rxs\\tablenotemark{a}}\n\\tablehead{\n\\colhead{Telescope} & \\colhead{Date (UT)} & \\colhead{Time (UT)} &\n\\colhead{Exp. time (s)} &  \\colhead{$N_{\\rm exp}$} & Conditions\n}\n\\startdata\n  2.4-m  &  2014 Mar 22  & 09:53--12:42  &  10  &  677  & Clear \\\\\n  2.4-m  &  2014 Mar 23  & 07:15--12:40  &  10  & 1331  & Photometric \\\\\n  1.3-m  &  2015 Feb 13  & 09:44--13:14  &  30  &  379  & Cirrus, near moon \\\\\n  1.3-m  &  2015 Feb 14  & 09:45--13:13  &  30  &  376  & Thin clouds \\\\\n  1.3-m  &  2015 Feb 17  & 09:19--13:13  &  30  &  425  & Partly cloudy \\\\\n  1.3-m  &  2015 Feb 18  & 09:14--11:52  &  30  &  288  & Photometric \n\\enddata\n\\label{tab:log}\n\\tablenotetext{a}{All observations in the GG420 filter.}\n\\end{deluxetable*}\n\n\n\n\n\\section{OBSERVATIONS AND DATA REDUCTION}\n\n\\subsection{XMM-Newton}\nThe field around 1RXS J154439.4$-$112820 was observed with\n\\textit{XMM-Newton} on 2014 February 16 (ObsID 072080101). The EPIC MOS1\ninstrument was configured in ``full frame'' mode The MOS2 and PN\ndetectors were used in ``timing'' mode, which provide time resolution\nof 1.75 ms and 30 $\\mu$s, respectively. In this mode, the fine time\nresolution is enabled by imaging in only one spatial dimension (RAWX),\nwith the other dimension (RAWY\/TIME) used for fast read out.  The\nmedium optical blocking filter was used for all three instruments.\nThe effective exposures for the MOS1, MOS2, and pn instruments were\n40.9 ks, 40.6 ks, and 38.9 ks, respectively. All three data sets were\nprocessed with the SAS\\footnote{The XMM-Newton SAS is developed and\n  maintained by the Science Operations Centre at the European Space\n  Astronomy Centre and the Survey Science Centre at the University of\n  Leicester.} version {\\tt xmmsas\\_20141104\\_1833}.  The MOS1 source\nevents were extracted from a circular region of radius 36$''$. The\nMOS2 and pn data were extracted from regions of width 60 and 7\ndetector pixels, respectively, in the imaging (RAWX) direction and the\nfull detector width in the readout (RAWY\/TIME) direction. The standard\nflag and pattern event screening criteria were applied to all three\ndata sets.\n\nThe \\textit{XMM-Newton} Optical Monitor (OM) was used with the $U$\nfilter (centered on 3440 {\\AA} with a band pass of 840 {\\AA}) in place\nand in \"fast\" mode, which permits rapid readout and photon counting\ncapabilities.  A photometric light curve was obtained using the {\\tt\n  omfchain} processing pipeline with the default parameter settings.\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[height=5.8cm]{j1544_lc_new.ps}\n\\includegraphics[height=5.8cm]{j1544_histo_final.ps}\n\\end{center}\n\\caption{Top left: Exposure-corrected, background-subtracted 0.3--10\n  keV X-ray light curve of 1RXS J154439.4$-$112820 from the\n  \\textit{XMM-Newton} EPIC observation obtained by co-adding the\n  MOS1\/2 and pn light curves binned at a resolution of 20\n  seconds. Note the sudden jumps in count rate between $\\approx$0.2\n  and $\\approx$2 counts s$^{-1}$. Top right: Histogram of\n    number of 20 second time bins ($N$) as a function of X-ray count\n    rate, showing strong bimodality in flux.  Bottom:\n  \\textit{XMM-Newton} OM $U$ filter photometric light curve binned at\n  20 minute resolution. The dotted line shows the best-fit sinusoid\n  with period 5.2 hours.}\n\\end{figure*}\n\n\n\n\\subsection{MDM Observatory}\nWe obtained time-series optical photometry of \\rxs\\ using the MDM\nObservatory's 2.4-m Hiltner telescope and 1.3-m McGraw-Hill telescope\non Kitt Peak during the 2014 and 2015 observing seasons.\nTable~\\ref{tab:log} is a summary of the data obtained on six nights.\nThe detector was the thinned, backside illuminated SITe CCD\n``Templeton''.  It has $1024\\times1024$ pixels, with a scale of\n$0.\\!^{\\prime\\prime}275$ pixel$^{-1}$ on the 2.4-m and\n$0.\\!^{\\prime\\prime}509$ pixel$^{-1}$ on the 1.3-m.  The filter used\nwas a GG420, to maximize throughput.  In order to reduce the readout\ntime, the detector was windowed down and the pixels were binned\n$2\\times2$.  Exposure times were 10~s with a read\/prep time of\n5\\,s, or 30~s with a read\/prep time of 3\\,s.\nFigure~\\ref{fig:finder} is a finding chart from the MDM GG420 filter\nimages, showing the object at position \\pos\\ from the USNO B1.0\ncatalog. \n\n\n\n\\section{X-ray Analysis}\nTo maximize the photon statistics of the data for the variability\nanalysis, we extracted a total exposure-corrected and\nbackground-subtracted X-ray light curve of 1RXS J154439.4$-$112820 by\ncombining the individual MOS1\/2 and pn time series over the period\nwhen the three instruments are collecting data simultaneously. This\nwas done using the {\\tt epiclccorr} task in SAS. For this relatively\nbright source, removal of periods contaminated by background flares\nwas not neccessary. We verified that the observed variability in the\nbackground light curve is not present in the background-subtracted\nsource light curves.\n\nThe total 0.3--10 keV light curve, binned at a resolution of\n20 s to look for rapid variability, is shown in Figure 2.  The X-ray\nlight curve exhibits obvious short-timescale, large-amplitude\nvariability. The X-ray flux alternates irregularly between two\nclearly distict modes that differ in flux by a factor of\n$\\approx$10. Indeed, we find clear bimodality in the X-ray count rate\nwith two peaks at $\\approx$0.2 and $\\approx$2 counts s$^{-1}$.  The\ntransition between the two flux levels is very rapid, with a typical\nduration of order 10 s. There is no evidence for periodicity in the\nobserved variability pattern.\n\nArchival \\textit{Swift} XRT data obtained in 2006, 2007, and 2012 also\nshow X-ray variability comparable to that observed with\n\\textit{XMM-Newton}. However, due to the brief \\textit{Swift}\nexposures it is not possible to determine whether the variability\npattern is the same.\n\n\nWe carried out a spectroscopic analysis of the \\textit{XMM-Newton}\nEPIC data in XSPEC\\footnote{Available at\n  \\url{http:\/\/heasarc.nasa.gov\/docs\/xanadu\/xspec\/index.html}.}  12.7.1\n\\citep{Arnaud96}. Due to known issues with the spectral calibration of\nthe pn instrument in timing mode, we only considered the MOS1\/2\ndata. The time-averaged spectrum of 1RXS J154439.4$-$112820 is\nwell-described by a simple absorbed power-law. The best fit parameters\nare $\\Gamma=1.68\\pm0.04$ for the spectral photon index, $N_{\\rm H}=(1.4\n\\pm 0.1)\\times 10^{21}$ cm$^{-2}$ for the hydrogen column density\nalong the line of sight, and $\\chi^2_{\\nu}=0.95$ for 399 degrees of\nfreedom. The unabsorbed flux in the 0.3--10 keV band is $F_X=(3.51 \\pm\n0.07) \\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$. All uncertainties\nquoted are at a 90\\% confidence level.  The derived $N_{\\rm H}$ is in\nfull agreement with the value measured through the Galaxy of $1.3\n\\times 10^{21}$ cm$^{-2}$ in the direction of this source\n\\citep{Kalb05}.\nFitting the high and low emission separately (by making a cut\n  at 0.9 counts s$^{-1}$ in the light curve from Figure 2) produces\n  $\\Gamma=1.67\\pm0.04$ and $\\Gamma=1.97\\pm0.28$, implying only\n  marginally significant evidence for spectral variations as a\n  function of source brightness. As the distance to 1RXS\nJ154439.4$-$112820 is unconstrained, it is not possible to determine\nthe X-ray luminosity levels of the high and low modes.\n\nAlthough the 30 $\\mu$s time resolution of the pn permits the search for\nX-ray pulsations at millisecond periods, the lack of an orbital period\nand ephemeris, combined with the relatively low source count rate,\nrender a blind periodicity search infeasible at this time.\n\n\n\n\\section{UV and Optical Variability}\nThe bottom panel of Figure 2 shows the \\textit{XMM-Newton} OM $U$\nfilter photometric data at a 20 minute resolution, which shows\nvariability by up to $\\approx$0.5 mag.  A fit with a sinusoid\n  indicates a periodicity of 5.2 hours. However, it should be noted\n  that the low points in the U filter data may not correspond to a\n  binary modulation as they could be related to the intervals\n  dominated by low mode X-ray emission that occur at the same time.\nDue to the faint nature of 1RXS J154439.4$-$112820, it is not possible\nto probe variability on timescales of tens of seconds with the OM\ndata.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{time_series.ps}\n\\end{center}\n\\caption{MDM light curves of \\rxs.\n  Magnitude is based on differential photometry\n  in the GG420 filter with respect to the comparison star indicated\n  in Figure~\\ref{fig:finder}.\n}\n\\label{fig:phot}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{psrj1023.ps}\n\\end{center}\n\\caption{MDM 1.3m $R$-band light curve of \\psr\\ with 13~s cadence.\n  The orbital period of the binary is 0.198~d, which is evident\n  as the long wave in the light curve. The magnitude\n  is based on differential photometry with respect to a\n  calibrated comparison star as described in \\citet{Bog14b}.\n}\n\\label{fig:psrj1023}\n\\end{figure} \n\n\nDifferential photometry on the much more sensitive MDM time series was\nperformed with respect to field stars, and the results are shown in\nFigure~\\ref{fig:phot}.  Although not calibrated in the GG420 filter,\nwe adopt the $R$ magnitudes from stars in the USNO B1.0 catalog as an\napproximate reference, in particular, $R=14.25$ for the bright star\nmarked ``C'' on the finding chart.  On this scale, the magnitude of\n\\rxs\\ varies from about 18.3 to 19.1, similar to $R=18.53$ as listed\nin the USNO B1.0 catalog.\n\nThe nature of the optical variability of \\rxs\\ observed at MDM is\nsubtly different from that of most cataclysmic binaries.  Rather than\na random walk or flickering, the brightness appears to be confined\nbetween minimum and maximum values separated by $\\approx$$0.5$\nmagnitudes. This is especially obvious in the higher-quality 2.4m\ndata. In this regard, it resembles the limit-cycle behavior\noccasionally seen from \\psr, an example of which is shown in\nFigure~\\ref{fig:psrj1023}.  There is no obvious candidate for an\norbital or any other period in these light curves of \\rxs. A\n  periodicity search of the longest light curve on 2014 March 23 shows\n  a peak in the power spectrum at $\\sim$5.4 hours, but this is unreliable\n  because it is comparable to the length of the observation, and it\n  cannot be confirmed by any of the shorter time series. In contrast,\nin all light curves of \\psr, e.g., Figure~\\ref{fig:psrj1023}, its\n4.75~hr orbital period is clearly manifest as a modulation due to\nheating of the side of the companion facing the neutron star.\n\n\nThe Catalina Real-time Transient Survey \\citep{Dra09} contains 324\nphotometric measurements of \\rxs\\ over a span of 7 years.  We searched\nthese for periodicity in the range 1--24 hours, such as may arise from\norbital modulation, but did not find a significant detection.\n\n\\section{DISCUSSION AND CONCLUSIONS} \nWe have presented an X-ray, UV, and optical investigation of 1RXS\nJ154439.4$-$112820, the most likely counterpart of the \\textit{Fermi}\nLAT source 3FGL J1544.6$-$1125.  When combined with previous studies,\nwe can establish the following observational properties of the system:\ni) strong emission lines of H and He, suggestive of an accretion disk\n\\citep[see][]{Mas13}; ii) fast optical variaiblity within limits; iii)\npower-law X-ray emission with $\\Gamma \\approx 1.7$ that exhibits rapid\nmoding between two distinct X-ray flux levels; iv) persistent\n$\\gamma$-ray emission with no evidence for a spectral cut-off\n\\citep[see][]{Fermi15}.  These characteristics are strikingly similar\nto what is observed for the transitional MSPs PSR J1824$-$2452I\n\\citep{Pap13,Pal13,Lin14,Fer14}, PSR J1023+0038\n\\citep{Stap14,Cot14,Tak14,Bog14b} and XSS J12270--4859\n\\citep{deM10,deM13,Pap14,Bogd14a,Roy15,Pap15} in their low-luminosity accreting states.\nTherefore, 1RXS J154439.4$-$11282 is the counterpart of 3FGL\nJ1544.6$-$1125 and is almost certainly a binary that hosts an MSP and\nis presently in a low-luminosity LMXB state.\n\nAs argued by \\citet{Del14} for the case of PSR J1023+0038, the\n$\\gamma$-ray emission seen by \\textit{Fermi} LAT is likely generated\nin a jet\/wind outflow, perhaps driven by propeller-mode accretion.\nThe peculiar X-ray moding appears to be caused by rapid transitions\nbetween channelled accretion onto the neutron star surface (the high\nmode) and no accretion (the low mode).  As with PSR J1023+0038\n\\citep{Arch14} and XSS J12270--4859 \\citep{Pap14}, we predict that\n1RXS J154439.4$-$112820 should exhibit coherent X-ray pulsations at\nthe neutron star spin period during the high flux mode. However, due\nto the lack of a precise orbital ephemeris, at present, a blind\nperiodicity search of the X-ray data is not practical.  In the absence\nof evidence for the orbit of \\rxs\\ in either X-ray or optical light\ncurves, a spectroscopic period should be obtained from its optical\nemission lines.\n\nThis system can serve as an additional laboratory for studying\nthe peculiar phenomenology of transitional MSP systems, especially the\nhighly unusual X-ray flux switching, and the relation, if any, with\nthe optical moding we see in 1RXS J154439.4$-$112820. In turn, this\nmay shed light on the little-understood low-luminosity (``quiescent'')\nregime in neutron star X-ray binaries.  In addition, extending the\npresently small sample of transitional MSPs has important implication\nfor understanding key aspects of the evolution of compact neutron star\nbinaries.\n\nBased on the recent history of the three confirmed LMXB\/RMSP\ntransition objects, we expect 1RXS J154439.4$-$112820 to transform\ninto a disk-free state in which the rotation-powered millisecond\npulsar activates. Therefore, this object warrants intensive\nmulti-wavelength monitoring in order to catch such an occurence.\n\n\n\\acknowledgements\nA portion of the results presented were based on observations obtained\nwith \\textit{XMM-Newton}, an ESA science mission with instruments and\ncontributions directly funded by ESA Member States and NASA.  This\nwork used observations obtained at the MDM Observatory, operated by\nDartmouth College, Columbia University, Ohio State University, Ohio\nUniversity, and the University of Michigan. We have made\nextensive use of the NASA Astrophysics Data System (ADS) and the arXiv.\n\nFacilities: \\textit{XMM}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn Random Matrix Theory (RMT) many statistical quantities can be\ndescribed as determinants. This is especially true in the cases of\nwhat are known as the classical ensembles, for example, the Gaussian\nUnitary Ensemble (GUE) or the Laguerre Ensembles. Since the\ngroundbreaking work of Tracy and Widom \\cite{TW1994}, which\ncharacterized the largest eigenvalue distribution for the GUE, one\ncommon approach is to write some statistical quantity as a\ndeterminant and then express the determinant as something involving\na Painlev\\'e transcendent, a solution to one of the classical\nPainlev\\'e second order non-linear differential equations.\n\nTechniques to find the determinant and then the resulting\ndifferential equation are quite complicated. Tracy and Widom used\nmore of an operator theory approach, others have used an integrable\nsystem approach, whilst others still have used a stochastic equation\napproach.\n\nThis paper highlights a technique known as the ``ladder operator''\napproach and is, in some sense, one long example illustrating the\ntechnique and describing the application of interest. The specific\napplication relates to the performance of multiple-input\nmultiple-output (MIMO) wireless communication systems, in which both\nthe transmitter and receiver devices are equipped with multiple\nantennas. Such systems have been the subject of intense interest\nsince the key papers \\cite{Telatar} and \\cite{Foschini}, and now\nform the cornerstone of most modern day wireless systems (Wi-Fi,\ncellular networks, etc). Here, a key fundamental performance measure\nis considered, the so-called ``outage probability'',  and this is\nshown to involve the probability distribution of a certain ``linear\nstatistic'' in a Laguerre random matrix ensemble. The problem thus\nfalls naturally within the realm of the ladder operator framework.\n\nHere is the idea of the approach. For the example at hand, the\nlinear statistic of interest can be described through its moment\ngenerating function as a Hankel determinant, which using the theory\nof orthogonal polynomials (for a certain nonstandard weight\ngenerally) can be computed via a product of norms of monic\northogonal polynomials. Now it is well-known that orthogonal\npolynomials satisfy three term recurrence equations. In fact, the\ntwo coefficients in the recurrence combined with initial conditions,\ncompletely determine the polynomials. Thus information about the\ncoefficients in the recurrence equations should yield information\nabout the Hankel determinants.\n\nThe path to this information is from the ladder operators, two\nformulas that connect the polynomials one index apart to their\nderivatives. These yield, using basic complex analysis, a set of\nequations in the coefficients along with two additional auxiliary\nquantities that arise.  The story would end here, except for the\nfact that often there is a ``time'' parameter implicit in the\noriginal weight and thus in the polynomials themselves. Using\n``time'' evolution one can then, using only elementary means, find a\npair of coupled Ricatti equations in the two auxiliary quantities.\nThese then lead directly to a Painlev\\'e equation. It should be\npointed out that this method works at least in principle, if the\n``time'' parameter is present and if the derivative of the logarithm\nof the weight is a rational function. Then one can in many cases\nfollow the steps illustrated in this paper.\n\n\nHere is an outline of the paper. The next section contains the\npreliminaries of the theory including the ladder operator equations.\nSection III shows how the application of interest, the outage\nprobability performance measure which arises in the application of\nMIMO wireless communication, can be described using the RMT\nframework. Section IV provides the details of the path to the\ndifferential equation solutions, which are presented Theorems 1 and\n2.\n\n\\section{Preliminaries}\n\n\n\\subsection{Linear Statistics of Hermitian Random Matrices and Hankel Determinants}\n\nFor the MIMO capacity application, it will be seen that the problem\nof interest falls within the general theory of linear statistics of\nHermitian random matrices, with a close connection to the theory of\northogonal polynomials. Here a brief introduction to the general\ntheory is given, and preliminaries are established for later use.\n\nWe will require the distribution of a certain linear statistic\n\\begin{align} \\label{eq:LinStat}\n\\sum_{k=1}^N f(x_k)\n\\end{align}\nin the eigenvalues $\\{ x_k \\}$ of a $N \\times N$ Hermitian random\nmatrix, with joint eigenvalue density of the form\n\\begin{align} \\label{eq:EigDensGeneric}\np(x_1, \\ldots, x_N) \\propto \\prod_{k = 1}^N  w_0(x_k) \\prod_{1 \\leq\ni <  j \\leq N} ( x_j - x_i )^2  ,\n \\quad  x_k \\in (a, b)\n\\end{align}\nfor some weight function $w_0(\\cdot)$. It is convenient to attempt\nto characterize the distribution of the linear statistic\n(\\ref{eq:LinStat}) through its moment generating\nfunction\\footnote{The parameter $\\lambda$ is an indeterminate which\ngenerates the random variable\n$\\sum_{k=1}^N f(x_k).$\\\\\n},\n\\begin{align} \\label{eq:MGFFirst}\n\\mathcal{M}(\\lambda) = E \\left[ \\exp \\left( \\lambda \\sum_{k=1}^N f (x_k)\n\\right) \\right] \\; = \\; E \\left[ \\prod_{k=1}^N e^{\\lambda f(x_k)}\n\\right]\n\\end{align}\nwhich upon substituting for (\\ref{eq:EigDensGeneric}) gives\n\\begin{align} \\label{eq:DnGeneric}\n\\mathcal{M}(\\lambda) &= \\frac{ \\frac{1}{N!} \\int_{(a,b)^N}\\prod_{1\\leq\ni<j\\leq N}(x_j-x_i)^2\\prod_{k=1}^{N} w(x_k) dx_k} {\\frac{1}{N!}\n\\int_{(a,b)^N}\\prod_{1\\leq i<j\\leq\nN}(x_j-x_i)^2\\prod_{k=1}^{N}w_0(x_k) dx_k}\n\\end{align}\nwhere\n$$\nw(x) := w_0(x) e^{\\lambda f(x) }\n$$\ndenotes the \\emph{deformed} version of the reference weight\n$w_0(\\cdot)$. Application of the Andreief-Heine identity \\cite{Sze}\nnow directly leads to\n\\begin{align} \\label{eq:MGFRatio}\n\\mathcal{M}(\\lambda) = \\frac{D_N[ w ]}{D_N[w_0 ]} \\; = \\; \\frac{ \\det\n\\left( \\int_a^b x^{i+j-2} w(x) dx \\right)_{i,j=1}^{N} } {\\det \\left(\n\\int_a^b x^{i+j-2}w_0(x) dx \\right)_{i,j=1}^{N}}\\; \\; ,\n\\end{align}\nwhich is a ratio of Hankel determinants. By virtue of the Selberg\nintegral, for most ``classical'' weight functions of interest, the\nHankel determinant in the denominator of (\\ref{eq:DnGeneric}) admits\nan explicit closed-form (non determinantal) representation. The\nnumerator, on the other hand, is much more difficult to\ncharacterize, since it involves the more complicated deformed weight\n$w$. To proceed, methods based on orthogonal polynomials will be\nintroduced in the sequel.\n\n\\subsection{Orthogonal Polynomials and their Ladder Operators}\n\\label{sec:LadderOperators}\n\nWe start by noting that\n\\begin{align}\n\\prod_{1\\leq i < j \\leq N}(x_j-x_i) = \\det \\left( x_j^{i-1}\n\\right)_{i,j=1}^N = \\det \\left( P_{i-1}(x_j) \\right)_{i,j=1}^N\n\\end{align}\nwhere $P_j (\\cdot)$ represents any monic polynomial of degree $j$,\n {\\small \\begin{eqnarray} \\label{eq:MonicDefn}\nP_j(z)=z^{j}+\\textsf{p}_1(j)\\:z^{j-1}+... \\end{eqnarray}} Applying this in\n(\\ref{eq:DnGeneric}) and once again integrating via the\nAndreief-Heine identity, the numerator evaluates to\n\\begin{align} \\label{eq:DnIntermed}\nD_N[w] = \\det \\left( \\int_a^b P_{i-1}(x) P_{j-1}(x) w(x) dx\n\\right)_{i,j=1}^{N} \\; .\n\\end{align}\nIf we orthogonalize the polynomial sequence $\\{P_{n}(x)\\}$ with\nrespect to $w(x)$ over the interval $[a,b],$ i.e.,\n\\begin{align}\n\\label{eq:OrthRel} \\int_{a}^{b}\\:P_{j}(x)P_k(x)\nw(x)dx=h_{j}\\delta_{j,k},\\;\\; j,k=0,1,2,...\n\\end{align}\nwith $h_j$ denoting the square of the $L^2$ norm of $P_j$ over\n$[a,b],$ then (\\ref{eq:DnIntermed}) reduces to\n\\begin{align} \\label{eq:DetRelation}\nD_N[ w ] = \\prod_{k=0}^{N-1} h_k \\; .\n\\end{align}\n\nThe key challenge is how to characterize the class of polynomials\nwhich obey the orthogonality constraints in (\\ref{eq:OrthRel}) or,\nmore importantly, the norms of such polynomials required to evaluate\n(\\ref{eq:DetRelation}).\n\n\nIf all the moments of the weight $w$ exist, then the theory of\northogonal polynomials states that the $P_n(z)$ for $n=0,1,2,...$\nsatisfy the three term recurrence relations,\n\\begin{align}\n\\label{eq:RecRel}\nzP_n(z)=P_{n+1}(z)+\\alpha_n\\:P_n(z)+\\beta_n\\:P_{n-1}(z).\n\\end{align}\nThe above sequence of polynomials can be generated from the\northogonality conditions (\\ref{eq:OrthRel}), the recurrence\nrelations (\\ref{eq:RecRel}), and the initial conditions,\n\\begin{align}\nP_0(z)=1,\\;\\;\\beta_0P_{-1}(z)=0.\n\\end{align}\nSubstituting (\\ref{eq:MonicDefn}) into the recurrence relations, an\neasy computation shows that \\begin{eqnarray} \\label{eq:p1Defn}\n\\textsf{p}_1(n)-\\textsf{p}_1(n+1)=\\alpha_n, \\end{eqnarray} with\n$\\textsf{p}_1(0):=0.$ A telescopic sum of (\\ref{eq:p1Defn}) gives\n\\begin{align}\n\\textsf{p}_1(n) =-\\sum_{j=0}^{n-1} \\alpha_j .\n\\end{align}\n>From the recurrence relation (\\ref{eq:RecRel}) and the orthogonality\nrelations (\\ref{eq:OrthRel}), we find {\\small \\begin{eqnarray}\\label{eq:betah}\n \\beta_n=\\frac{h_n}{h_{n-1}}. \\end{eqnarray}}\n\nWe shall see that $\\textsf{p}_1(n)$ plays an important role in later\ndevelopments. For more information on orthogonal polynomials, we\nrefer the reader to Szeg\\\"o's treatise \\cite{Sze}.\n\nNext, we present three Lemmas which are concerned with the ``ladder\noperators'' associated with orthogonal polynomials, as well as\ncertain supplementary conditions. Note that these have been known\nfor quite sometime; we reproduce them here for the convenience of\nthe reader using the notation of \\cite{chen+its}, where one can also\nfind a list of references to the literature. We also mention that\nMagnus \\cite{Magnus} was perhaps the first to apply these\nlemmas---albeit in a slightly different form---to random matrix\ntheory and the derivation of Painlev\\'e equations. Tracy and Widom\nalso made use of the compatibility conditions in their systematic\nstudy of finite $n$ matrix models \\cite{twdet}. See also\n\\cite{for1}.\n\\\\\n\\noindent {\\bf Lemma 1} \\emph{Suppose $\\v(x)=-\\log w(x)$ has a\nderivative in some Lipshitz class with positive exponent. The\nlowering and raising operators satisfy the differential-difference\nformulas:} {\\small \\begin{eqnarray}\nP_n'(z)&=&-B_n(z)P_n(z)+\\beta_n\\:A_n(z)P_{n-1}(z)\\\\\nP_{n-1}'(z)&=&[B_n(z)+{\\mathsf{v}^{\\prime}}(z)]P_{n-1}(z)-A_{n-1}(z)P_n(z), \\end{eqnarray}} where\n{\\small \\begin{eqnarray}\nA_n(z)&:=&\\frac{1}{h_n}\\int_{a}^{b}\\frac{{\\mathsf{v}^{\\prime}}(z)-{\\mathsf{v}^{\\prime}}(y)}{z-y}\\:P_n^2(y)w(y)dy \\label{eq:AnNewDefn}\\\\\nB_n(z)&:=&\\frac{1}{h_{n-1}}\\int_{a}^{b}\\frac{{\\mathsf{v}^{\\prime}}(z)-{\\mathsf{v}^{\\prime}}(y)}{z-y}P_n(y)P_{n-1}(y)w(y)dy.\n\\label{eq:BnNewDefn} \\end{eqnarray}} A direct computation produces two\nfundamental supplementary or compatibility conditions valid for all\n$z\\in \\mathbb{C}\\cup\\{\\infty\\}$. These are stated in the next Lemma.\n\\\\\n{\\bf Lemma 2} \\emph{The functions $A_n(z)$ and $B_n(z)$ satisfy the\nsupplementary conditions:} {\\small $$\nB_{n+1}(z)+B_n(z)=(z-\\alpha_n)A_n(z)-{\\mathsf{v}^{\\prime}}(z)\\eqno(S_1)\n$$\n$$\n1+(z-\\alpha_n)[B_{n+1}(z)-B_n(z)]=\\beta_{n+1}A_{n+1}-\\beta_nA_{n-1}(z).\\eqno(S_2)$$}\n\\\\\nIt turns out that there is an equation which gives better insight\ninto the coefficients $\\alpha_n$ and $\\beta_n$, if $(S_1)$ and $(S_2)$\nare suitably combined to produce a ``sum rule'' on $A_n(z).$ We\nstate this in the next lemma. The sum rule, we shall see later,\nprovides important information about the logarithmic derivative of\nthe Hankel determinant.\n\\\\\n{\\bf Lemma 3} \\emph{The functions $A_n(z),$ $B_n(z)$, and the sum}\n{\\small $$ \\sum_{j=0}^{n-1}A_j(z),\n$$}\n\\emph{satisfy the condition:} {\\small $$\nB_n^2(z)+{\\mathsf{v}^{\\prime}}(z)B_n(z)+\\sum_{j=0}^{n-1}A_j(z)=\\beta_n\\:A_n(z)\\:A_{n-1}(z).\\eqno(S_2')\n$$}\n\n\n\\section{Information Theory of MIMO Wireless Systems}\n\nIn this section we introduce the wireless communication problem of\ninterest, and connect it with the general linear statistics\nframework introduced previously.\n\nWe consider a MIMO communication system in which a transmitter\nequipped with $n_t$ antennas communicates with a receiver equipped\nwith $n_r$ antennas. Denoting the transmitted signal vector as ${\\bf x}\n\\in \\mathbb{C}^{n_t}$ and the received signal vector as ${\\bf y} \\in\n\\mathbb{C}^{n_r}$, under a certain assumption on the channel (known\nas ``flat fading''), these signals are related via the linear model\n{\\small \\begin{eqnarray} \\label{eq:LinearModel} {\\bf y} = {\\bf H} {\\bf x} + {\\bf n} \\; , \\end{eqnarray}}\nwhere ${\\bf n}  \\in \\mathbb{C}^{n_r}$ the receiver noise vector, is\ncomplex Gaussian with zero mean and covariance $E( \\mathbf{n}\n\\mathbf{n}^\\dagger ) = \\mathbf{I}_{n_r}$. The matrix \\emph{channel\nmatrix}, ${\\bf H} \\in \\mathbb{C}^{n_r \\times n_t}$,\n represents the wireless fading coefficients\nbetween each transmit and receive antenna. The channel is modeled\nstochastically, with distribution depending on the specific wireless\nenvironment. Under the realistic assumption that there are\nsufficient scatterers surrounding the transmit and receive\nterminals, the channel matrix ${\\bf H}$ is well modeled by a\ncomplex Gaussian distribution with independent and identically\ndistributed (i.i.d.) elements having zero mean and unit variance.\nThis matrix is assumed to be known at the receiver\\footnote{In\npractice, this information can be obtained using standard estimation\ntechniques.}, but the transmitter only has access to its\ndistribution. The transmitted signal ${\\bf x}$ is designed to meet a\npower constraint:\n{\\small \\begin{align}\n \\label{eq:PowerConstraint} E({\\bf x}^{\\dag}{\\bf x})\\leq P \\; .\n\\end{align}}\n\nOur objective is to study the fundamental capacity limits of a MIMO\ncommunication system. Such limits are described by the field of\ninformation theory, founded by Claude Shannon in 1948\n\\cite{Shannon}. Specifically, information-theoretic measures allow\none to precisely determine the highest data rate that can be\ncommunicated with negligible errors by any transmission scheme.\nConsequently, information theory offers a benchmark for the design\nof practical transmission technologies, and has become an\nindispensable tool for modern communication system design.\n\nThe capacity of a communication link is determined by the so-called\n``mutual information'' between the input and output signals. For the\nMIMO model (\\ref{eq:LinearModel}) it is given by: {\\small \\begin{eqnarray}\n\\label{eq:MIDefn}\nI({\\bf x};{\\bf y} | {\\bf H} \n={\\cal H}({\\bf y} | {\\bf H})-{\\cal H}({\\bf n}) \\end{eqnarray} } with ${\\cal H}({\\bf y} | {\\bf H})$\ndenoting the conditional entropy of ${\\bf y}$, {\\small \\begin{eqnarray} {\\cal H}({\\bf y}\n| {\\bf H})= -\\int_{\\mathbb{C}^{n_r} }p({\\bf y} | {\\bf H})\\log p({\\bf y} | {\\bf H})d {\\bf y}\n, \\end{eqnarray}} where $p({\\bf y} | {\\bf H})$ denotes the conditional density of ${\\bf y}$\ngiven ${\\bf H}$.  This formula represents the maximum amount of\ninformation that can be reliably transported between the transmitter\nand receiver (i.e., it represents the rate which is ``supportable''\nby a given realization of the MIMO channel). It was proved in\n\\cite{Telatar} that the conditional mutual information $I({\\bf x};{\\bf y} |\n{\\bf H} )$ is maximized by choosing the input signal vector ${\\bf x}$\naccording to a zero-mean circularly-symmetric complex Gaussian\ndistribution with covariance ${\\bf Q}_{\\rm x} = E \\left( {\\bf x} {\\bf x}^{\\dag}\n\\right)$ satisfying ${\\rm tr} \\left({\\bf Q}_{\\rm x} \\right) \\leq P$.\nIn this case, the mutual information (\\ref{eq:MIDefn}) was shown to\nbe {\\small \\begin{align}\n \\label{eq:MIDefn2} I({\\bf x};{\\bf y} | {\\bf H} ) = \\log \\det \\left( {\\bf I}_{n_r}  + {\\bf H} {\\bf Q}_{\\rm x} {\\bf H}^{\\dag}\n \\right) \\; .\n\\end{align}}\n\nIn this paper, we will consider a scenario in which the channel is\nselected randomly at the beginning of a transmission, and remains\n\\emph{fixed} during the transmission.  In this scenario, it is\nimpossible to guarantee that the communication will be completely\nreliable, since no matter what transmission rate $R$ we choose\n(which is assumed fixed) there is always a non-zero probability that\nthe rate may not be supportable by the channel. In other words,\nthere is always a chance that the mutual information $I({\\bf x};{\\bf y} |\n{\\bf H} )$ falls below $R$, and thus communicating at rate $R$ becomes\nimpossible. This is referred to as an ``outage event'', and the\nprobability of this occurring is called the \\emph{outage\nprobability}, {\\small \\begin{align} \\label{eq:outProb} P_{\\rm out} (\nR ) &:= {\\rm Pr} \\left( I({\\bf x};{\\bf y} | {\\bf H} ) < R  \\right)\n.\n\\end{align}}\n\n\nHere, we will make the common assumption that {\\small \\begin{align}\n\\label{eq:xCov} \\mathbf{Q}_{\\rm x} = \\frac{P}{n_t} {\\bf I}_{n_t} \\; ,\n\\end{align}}\ncorresponding to sending independent complex Gaussian signals from\neach transmit antenna, each with power $P\/n_t$. Hence, with this\ninput signal covariance, the quantity $P \\, (>0)$ will also\nrepresent the signal-to-noise ratio (SNR).  With $\\mathbf{Q}_{\\rm\nx}$ given by (\\ref{eq:xCov}), the mutual information $I({\\bf x};{\\bf y} |\n{\\bf H} )$ becomes\n{\\small \\begin{align} I({\\bf x};{\\bf y} | {\\bf H} ) = \\log\\det \\left( {\\bf I}_{n_r}\n+ \\frac{1}{t} {\\bf H} {\\bf H}^{\\dag} \\right) , \\; \\quad t := \\frac{n_t}{P} .\n\\end{align}}\nTo fix notation, let $M := \\textsf{max}\\{n_{r},n_{t}\\}, N :=\n\\textsf{min}\\{n_{r},n_{t}\\} ,\\alpha := M - N$ and define {\\small \\begin{eqnarray}\n{\\bf W} &:=& \\left\\{\n\\begin{array}{lr}\n {\\bf H}{\\bf H}^{\\dag},  &  n_{r}<n_{t}\\nonumber\\\\\n{\\bf H}^{\\dag}{\\bf H}, &    n_{r} \\geq n_{t} \\nonumber\n\\end{array}\n\\right. \\end{eqnarray}} with positive eigenvalues $\\{x_{k}\\}_{k=1}^{N}$. With\nthese definitions, and with $\\det({\\bf I}+ {\\bf A} {\\bf\nB})=\\det({\\bf I}+{\\bf B}{\\bf A})$, we can further evaluate {\\small\n\\begin{align} \\label{eq:linStat}\nI({\\bf x};{\\bf y} | {\\bf H} ) &= \\sum_{k=1}^N \\log \\left( 1 + \\frac{1}{t} x_k\n\\right)  \\;  \\nonumber \\\\\n&= - N \\log t + \\sum_{k=1}^N \\log \\left( t + x_k \\right) .\n\\end{align}}\n\nComputation of the outage probability (\\ref{eq:outProb}) requires\nthe probability distribution of $I({\\bf x};{\\bf y} | {\\bf H} )$. From\n(\\ref{eq:linStat}), this is clearly a linear statistic in the\neigenvalues of the Hermitian random matrix ${\\bf W}$ (with a constant\nshift of $-N \\log t$). Moreover, ${\\bf W}$ is complex Wishart\ndistributed \\cite{Muirhead}, thus the eigenvalues\n$\\{x_{k}\\}_{k=1}^{N}$ are well-known to admit the joint density\n{\\small\n\\begin{align}\n \\label{eq:EigPDF_Wishart} p(x_1,x_2,...,x_N)\\propto\n\\prod_{l=1}^{N} w_{{\\rm Lag}} (x_l) \\prod_{1\\leq j<k\\leq\nN}(x_k-x_j)^2 , \\quad \\quad  x_l\\in[0,\\infty)\n\\end{align}\n} where {\\small\n\\begin{align} w_{{\\rm Lag}} (x):=x^{\\alpha}\\:{\\rm e}^{-x},\n\\;\\;\\alpha>-1,\\;\\;x\\in[0,\\infty)\n\\end{align}}\nis the classical Laguerre weight.\n\nOur aim will be to compute the moment generating function of the\nlinear statistic, {\\small\n\\begin{align}\n{\\tilde {\\cal M}}(\\lambda) := E_{{\\bf H}} \\left( e^{\\lambda I({\\bf x};{\\bf y} |\n{\\bf H}) } \\right)  \\; = \\; t^{-N \\lambda} {\\cal M}(\\lambda)\n\\end{align}\n} where ${\\cal M}(\\lambda)$ is identified by (\\ref{eq:MGFFirst}) but\nwith the following particularizations:\n\\begin{align}\n\\left( f(x), w_0(x), w(x), a, b \\right) \\Longrightarrow \\left(\n\\log(t+x), w_{\\rm Lag}(x), w_{\\rm dLag}(x, t), 0, \\infty \\right) \\;\n\\end{align}\nwhere $w_{\\rm dLag}(x,t)$ is a deformed Laguerre weight, {\\small\n\\begin{align}\n\\label{eq:DefLagWeight} w_{\\rm dLag}(x,t):= (x+t)^{\\lambda} w_{{\\rm\nLag}} (x) ,\\;\\;\\; t>0 \\;\\;\\;x>0 \\; .\n\\end{align}\n}\nThus, (\\ref{eq:MGFRatio}) immediately gives {\\small\n\\begin{align}\n \\label{eq:MGF_MainResultsSec} {\\cal M}(\\lambda) = \\frac{D_N(t,\\lambda)}{D_N[w_{\\rm Lag}]} \\;\n\\end{align}}\nwhere \\begin{eqnarray} D_N(t,\\lambda)=\\det\\left( \\mu_{i+j-2}(t, \\lambda)\n\\right)_{i,j=1}^{N} \\label{eq:HankelDefin1} \\end{eqnarray} is the Hankel\ndeterminant generated from $w_{\\rm dLag}(x,t)$ with moments {\\small\n\\begin{align} \\label{eq:MomentDefn}\n \\mu_{k}(t, \\lambda) := \\int_{0}^{\\infty}x^{k} w_{\\rm dLag}(x) dx \\;\n, \\hspace*{1cm} k = 0, 1, 2, \\ldots .\n\\end{align}\n}\n\n\nThe quantity $D_N[w_{{\\rm Lag}}]$ in the denominator of\n(\\ref{eq:MGF_MainResultsSec}) is the Hankel determinant generated\nfrom the classical Laguerre weight, $w_{{\\rm Lag}} (x)$, and can be\ncomputed in terms of the Barnes $G$--function as {\\small\n\\begin{align} \\label{eq:DnLaguerre_LambdaZero} D_N[w_{{\\rm\nLag}}]=\\frac{G(N+1)G(N+\\alpha+1)}{G(\\alpha+1)},\\quad G(1)=1.\n\\end{align}}\n\nOur next objective will be to compute a non-determinantal\nrepresentation for the (scaled) moment generating function\n(\\ref{eq:MGF_MainResultsSec}).  This, in turn, will require\nevaluation of the Hankel determinant $D_N(t,\\lambda)$ in\n(\\ref{eq:HankelDefin1}).  We will address this problem in the sequel\nby appealing to the orthogonal polynomial framework introduced in\nSection \\ref{sec:LadderOperators}.  We should like to mention here\nthat, unlike the classical ladder operators, the ``coefficients'' in\nour ladder operators are $``x\"$ dependent, as we shall see later.\n\n\n\\setcounter{equation}{0}\n\n\n\n\n\\section{Painlev\\'e V and the Continuous and Discrete $\\sigma$-Form}\n\n\\subsection{Main Results}\n\nThe following two theorems present the main results of the paper:\n\n{\\bf Theorem 1\\;\\;} \\emph{The Hankel determinant $D_N(t,\\lambda)$\nadmits the following integral representation:}\n\\begin{align}\n\\label{eq:Intrep} \\frac{D_N(t,\\lambda)}{D_N[w_{\\rm Lag}]} =\nt^{N\\lambda} \\exp \\left( \\int^{\\infty}_{t}f(y(s),y'(s),s) ds \\right)\n\\end{align}\n\\emph{where}\n\\begin{align} &f(y(s),y'(s),s):=\n\\frac{\\lambda^2+2(s+\\alpha-\\lambda)y+(4 N\ns+(s+\\alpha)^2-2(s+2\\alpha)\\lambda+\\lambda^2)y^2\n}{4y(y-1)^2} \\\\\n& \\hspace*{4cm} + \\frac{-2(2 N\ns+\\alpha(s+\\alpha-\\lambda))y^3+\\alpha^2y^4-[s\\;y'(s)]^2}{4y(y-1)^2}\n\\; .\n\\end{align}\n\n\n\\noindent {\\bf Theorem 2\\;\\;} \\emph{Equivalently, the Hankel\ndeterminant $D_N(t,\\lambda)$ also admits the following integral\nrepresentation:} \\begin{eqnarray} \\label{eq:HankelDet_ContSigma}\n\\frac{D_N(t,\\lambda)}{D_N[w_{\\rm Lag}]}= t^{N\\lambda} \\exp\n\\left(\\int_{\\infty}^t \\frac{H_N(x)-N\\lambda}{x} dx \\right)\n\\label{eq:tmp} \\end{eqnarray} \\emph{where} $H_N(t)$ \\emph{ satisfies the\nJimbo-Miwa-Okamoto $\\sigma-$form of Painlev\\'e V:} {\\small\n\\begin{align}\n & (t H_N'')^2=\\left[t H_N'- H_N + H_N'(2N+\\alpha+\\lambda) + N \\lambda \\right]^2 \\nonumber \\\\ & \\hspace*{2cm} - 4(t\nH_N'- H_N + {\\delta_N} )\\left[( H_N')^2+\\lambda H_N'\\right] \\label{eq:JimboPV}\n\\end{align}}\n\\emph{with} ${\\delta_N} := N(N+\\alpha+\\lambda)$.\n\nBefore presenting the proof of these results, we would like to point\nout that Painlev\\'e equations first appeared in the early 1900's\nthrough the work of Painlev\\'e and his collaborators \\cite{book}. In\nthe mid 1970's, Painlev\\'e equations first appeared in\ncharacterizing the correlation function of an Ising model through\nthe pioneering work of Barouch, McCoy, Tracy and Wu, see\n\\cite{Tracy}. The 1-particle reduced density matrix was shown in\n1980 to satisfy a particular Painlev\\'e V, see \\cite{Jimbo}.  For a\nrecent review on this and other related problems in matrix\nensembles, see \\cite{TW2011}. Another Painlev\\'e V appeared in the\nHankel determinant associated with the ``time evolved\" Jacobi\npolynomials, see \\cite{Basor}.\n\n\\subsection{Proof of Theorems 1 and 2}\n\n\\subsubsection{Compatibility Conditions, Recurrence Coefficients and\nDiscrete Equations}\n\n\n\n\nFor the purpose of applying the ladder operator framework introduced\nin Lemmas 1--3, an easy computation shows that {\\small \\begin{eqnarray}\n\\v(z,t)&:=&-\\log w_{\\rm dLag}(z,t)=-\\alpha \\log z-\\lambda \\log(z+t)+z , \\nonumber\\\\\n\\v'(z,t)&=&-\\frac{\\alpha}{z}-\\frac{\\lambda}{z+t}+1\\nonumber \\end{eqnarray} and therefore \\begin{eqnarray}\n\\frac{\\v'(z,t)-\\v'(y,t)}{z-y}&=&\\frac{\\alpha}{zy}+\\frac{\\lambda}{(z+t)(y+t)}.\\nonumber\n\\end{eqnarray}}\nSubstituting the above into (\\ref{eq:AnNewDefn}) and\n(\\ref{eq:BnNewDefn}), followed by integration by parts, we obtain\n{\\small \\begin{align}\n\\label{eq:AnDefn}\nA_n(z)&=\\frac{1-R_n(t)}{z}+\\frac{R_n(t)}{z+t}\\\\\nB_n(z)&=-\\frac{n+r_n(t)}{z}+\\frac{r_n(t)}{z+t}\n\\end{align}}\nwhere we have introduced the auxiliary quantities:\n{\\small \\begin{align}\nR_n(t)&:=\\frac{\\lambda}{h_n}\\int_{0}^{\\infty}\\frac{[P_n(x)]^2}{x+t}w_{{\\rm dLag}}(x,t)dx\\\\\nr_n(t)&:=\\frac{\\lambda}{h_{n-1}}\\int_{0}^{\\infty}\\frac{P_n(x)P_{n-1}(x)}{x+t}w_{{\\rm dLag}}(x,t)dx.\n\\label{eq:rnDefn}\n\\end{align}}\nThe next Lemma gives a representation of the recurrence coefficients\n$\\alpha_n,$ $\\beta_n,$ $\\sum_{j}\\:R_j,$ and $\\textsf{p}_1(n)$---the\ncoefficient of $z^{n-1}$ of $P_n(z)$--- in terms of the auxiliary\nvariables $r_n$ and $R_n.$ Note that $\\textsf{p}_1(n)$ also depends\non $t$ but we do not display this if there is no confusion.\n\\\\\n{\\bf Lemma 4} \\emph{The recurrence coefficients $\\alpha_n$ and $\\beta_n$\nare expressed in terms of the auxiliary quantities $r_n$ and $R_n$\nas:}\n{\\small \\begin{eqnarray} \\alpha_n&=&2n+1+\\alpha+\\lambda-tR_n\\\\ \\label{eq:AlphanRelation}\n\\beta_n&=&\\frac{1}{1-R_n}\\left[r_n(2n+\\alpha+\\lambda)+\\frac{r_n^2-\\lambda\nr_n}{R_n}+n(n+\\alpha)\\right]. \\label{eq:BetanRelation}\n\\end{eqnarray}}\n\\emph{Furthermore,}\n{\\small \\begin{eqnarray}\nt\\sum_{j=0}^{n-1}R_j&=&n(n+\\alpha+\\lambda)-\\beta_n-tr_n, \\label{eq:SumRjRelation} \\\\\n\\textsf{p}_1(n)&=&-\\beta_n-tr_n. \\label{eq:p1Relation}\n\\end{eqnarray}}\n{\\bf Proof\\;\\;} We start from  $(S_1)$. Equating the coefficients of $z^{-1}$ and $(z+t)^{-1}$, we obtain\nthe following difference equations relating $\\alpha_n$ to $r_n$ and $R_n$:\n{\\small \\begin{align}\n\\label{eq:S1Diff1}\n-(2n+1+r_{n+1}+r_n)&=\\alpha-\\alpha_n(1-R_n)\\\\\nr_{n+1}+r_n &= \\lambda-R_n(t+\\alpha_n).\n\\label{eq:S1Diff2}\n\\end{align}}\nTo proceed further, we take note of\n(\\ref{eq:AnDefn})--(\\ref{eq:rnDefn}), and derive\nidentities based on the supplementary condition $(S_2')$, which will be of particular interest.\nA straightforward (but long)  computation shows that the r.h.s. of $(S_2')$ becomes\n{\\small \\begin{align}\n& B_n^2(z)+\\v'(z)B_n(z)+\\sum_{j=0}^{n-1}A_j(z) \\nonumber \\\\\n& \\hspace*{2cm} = z^{-2}[(n+r_n)^2+\\alpha(n+r_n)]\\nonumber\\\\\n& \\hspace*{2cm} +z^{-1}\\Big\\{n-\\sum_{j=0}^{n-1}R_j+r_n[\\lambda-\\alpha-t-2(n+r_n)]\/t+(n-\\lambda)\/t\\Big\\}\\nonumber\\\\\n& \\hspace*{2cm} +(z+t)^{-1}\\Big\\{\\sum_{j=0}^{n-1}R_j+r_n[t+\\alpha-\\lambda+2(n+r_n)]\/t-n\\lambda\/t\\Big\\}\\nonumber\\\\\n& \\hspace*{2cm} +(z+t)^{-2}[r_n^2-\\lambda r_n] . \\nonumber\n\\end{align}}\nNow focusing on $(S_2')$ as presented above and equating the coefficients\nof $z^{-2},$ $z^{-1},$ $(z+t)^{-1},$ $(z+t)^{-2}$,\ngive rise to the following difference equations involving $\\beta_n$, $r_n$, $R_n$ and $\\sum_j\\:R_j$ :\n{\\small \\begin{align}\n\\label{eq:S2Diff1} (n+r_n)^2+\\alpha(n+r_n) =\\beta_n(1-R_n)(1-R_{n-1})\n\\end{align}}\n{\\small \\begin{align}\n\\small n-\\sum_{j=0}^{n-1}R_j+\\frac{r_n}{t}[\\lambda-\\alpha-t-2(n+r_n)]+\\frac{n(\\lambda-t)}{t} =\n\\frac{\\beta_n}{t}\\left[(1-R_{n-1})R_n+(1-R_{n-1})R_n\\right]\n\\label{eq:S2Diff2a}\n\\end{align}}\n{\\small \\begin{align}\n\\sum_{j=0}^{n-1}R_{j}+\\frac{r_n}{t}[t+\\alpha-\\lambda+2(n+r_n)]-\\frac{n\\lambda}{t}\n =\n-\\frac{\\beta_n}{t}\\left[(1-R_n)R_{n-1}+(1-R_{n-1})R_n\\right]\n\\label{eq:S2Diff2b}\n\\end{align}}\n{\\small \\begin{align}\nr_n^2-\\lambda r_n =\\beta_nR_nR_{n-1}.\\quad \\label{eq:S2Diff3}\n\\end{align}}\nObserve that (\\ref{eq:S2Diff2a}) and\n(\\ref{eq:S2Diff2b}) are equivalent. We shall see later that (\\ref{eq:S2Diff2a}), when combined with\ncertain identities, performs the sum {\\large $\\sum_{j=0}^{n-1}R_j$}\nautomatically in closed-form. This sum will provide an important\nlink between the logarithmic derivative of the Hankel determinant\nwith respect to $t$, and $\\beta_n$, $r_n$, which is an essential step\nin establishing the Painlev\\'e equation. Whilst the difference relations\n(\\ref{eq:S1Diff1})--(\\ref{eq:S2Diff1}) and (\\ref{eq:S2Diff3}) look\nrather complicated, these can be manipulated to give us insight into\nthe recurrence coefficients $\\alpha_n$ and $\\beta_n.$ To this end,\nsumming (\\ref{eq:S1Diff1}) and (\\ref{eq:S1Diff2}) gives us a simple\nexpression for the recurrence coefficient $\\alpha_n$ in terms of $R_n$:\n{\\small \\begin{eqnarray}\n\\label{eq:alrn} \\alpha_n=2n+1+\\alpha+\\lambda-tR_n.\n\\end{eqnarray}}\nFrom\n(\\ref{eq:S2Diff1}) and (\\ref{eq:S2Diff3}) we find after a minor re-arrangement\n{\\small \\begin{eqnarray} \\label{eq:RnBnCond}\n\\beta_n(R_{n}+R_{n-1})=\\beta_n-n(n+\\alpha)-r_n(\\alpha+\\lambda+2n).\n\\end{eqnarray}}\nNow substituting (\\ref{eq:S2Diff3}) and (\\ref{eq:RnBnCond}) into either\n(\\ref{eq:S2Diff2a}) or (\\ref{eq:S2Diff2b}) to eliminate $R_n$ and\n$R_{n-1}$ leaves us the following very simple form for\n$\\sum_{j=0}^{n-1}R_j,$ which will play a crucial role later,\n{\\small \\begin{eqnarray}\n\\label{eq:RSum} t\\sum_{j=0}^{n-1}R_j=n(n+\\alpha+\\lambda)-\\beta_n-tr_n.\n\\end{eqnarray}}\nBut in view of (\\ref{eq:alrn}), we also have an alternative representation of $\\sum_{j=0}^{n-1}R_j,$ namely,\n{\\small \\begin{eqnarray}\nt\\sum_{j=0}^{n-1}R_j&=&n(n+\\alpha+\\lambda)-\\sum_{j=0}^{n-1}\\alpha_j\n=n(n+\\alpha+\\lambda)+\\textsf{p}_1(n) \\label{eq:RSumb}.\n\\end{eqnarray}}\nIn summary, we have obtained two different ways to express $\\sum_{j}R_j,$ and comparing\n(\\ref{eq:RSum}) with (\\ref{eq:RSumb}) gives us the important relation (\\ref{eq:p1Relation}).\nlinking $\\textsf{p}_1(n)$ to $\\beta_n$  and $r_n.$\nWe are now in the position to find an expression for $\\beta_n$ in terms\nof $r_n$ and $R_n$. This is found by eliminating $R_{n-1}$ from\n(\\ref{eq:RnBnCond}) and (\\ref{eq:S2Diff3}) resulting in (\\ref{eq:BetanRelation}).\n{\\bf End of Proof.}\n\n\\subsubsection{$t$ Evolution and Painlev\\'e V}\nIn the next stage of the development, we vary $t$ and $n.$ The\ndifferential-difference relations generated here when combined with the\ndifference relations obtained previously will give us the desired\nPainlev\\'e equation. A straightforward computation shows that\n{\\small \\begin{eqnarray} \\frac{d}{dt}\\log\nh_n=R_n. \\end{eqnarray}}\n But, from (\\ref{eq:betah}), it follows that\n {\\small \\begin{eqnarray}\n\\frac{d\\beta_n}{dt}&=&\\beta_n(R_n-R_{n-1})\\\\\n&=&\\beta_n R_n-\\frac{r_n^2-\\lambda r_n}{R_n}, \\label{eq:BetaDiff} \\end{eqnarray}}\nwhere the last equality follows from (\\ref{eq:S2Diff3}).\n\nDifferentiating\n{\\small $$\n0=\\int_{0}^{\\infty}x^{\\alpha}(x+t)^{\\lambda}{\\rm e}^{-x}P_{n}(x)P_{n-1}(x)dx\n$$}\nwith respect to $t$ produces\n{\\small \\begin{align}\n0&=\\lambda\\int_{0}^{\\infty}\\:x^{\\alpha}\\:(x+t)^{\\lambda-1}{\\rm e}^{-x}P_n(x)P_{n-1}(x)dx+\n\\int_{0}^{\\infty}x^{\\alpha}(x+t)^{\\lambda}{\\rm e}^{-x}\\left[\\frac{d}{dt}\\textsf{p}_1(n)\\:x^{n-1}+...\\right]P_{n-1}(x)dx\\nonumber\\\\\n&=\\lambda\\int_{0}^{\\infty}\\frac{P_{n-1}(x)P_{n}(x)}{x+t}w_{{\\rm dLag}}(x,t)dx+h_{n-1}\\:\\frac{d}{dt}\\textsf{p}_1(n),\\nonumber\n\\end{align}}\nresulting in\n{\\small \\begin{eqnarray} \\label{eq:p1Diff}\n\\frac{d}{dt}\\textsf{p}_1(n)&=&-r_n .\n\\end{eqnarray}}\nUpon noting\n(\\ref{eq:p1Defn}), this implies\n\\begin{eqnarray} \\label{eq:AlphaDiff}\n\\frac{d\\alpha_n}{dt}&=&r_{n+1}-r_{n} .\n\\end{eqnarray}\nNow differentiating\n(\\ref{eq:p1Relation}) with respect to $t$ and noting\n(\\ref{eq:p1Diff}), we find\n{\\small \\begin{eqnarray}\n\\frac{d}{dt}\\textsf{p}_1(n)&=&-\\frac{d\\beta_n}{dt}-\\frac{d}{dt}(tr_n)\n=-\\frac{d\\beta_n}{dt}-r_n-t\\frac{dr_n}{dt}\n=-r_n.\\nonumber\n\\end{eqnarray}}\nThe above result combined with (\\ref{eq:BetaDiff})\ngives\n{\\small \\begin{eqnarray}\n\\frac{d\\beta_n}{dt}=-t\\frac{dr_n}{dt}=\\beta_nR_n-\\frac{r_n^2-\\lambda\nr_n}{R_n}.\n\\end{eqnarray}}\nWe now come to a key Lemma which gives the first order derivative of $r_n(t)$ and $R_n(t)$ with respect to $t,$ and where $n$\nappears as a parameter.\n\\\\\n\\\\\n{\\bf Lemma 5} \\emph{The auxiliary variables} $r_n$ \\emph{and} $R_n$\n\\emph{satisfy the following coupled Riccatti equations,}\n{\\small \\begin{align}\n\\label{eq:RicEqr} t\\frac{dr_n}{dt}=\\frac{r_n^2-\\lambda\nr_n}{R_n}-\\frac{R_n}{1-R_n}\\left[r_n(2n+\\alpha+\\lambda)+\\frac{r_n^2-\\lambda\nr_n}{R_n}+n(n+\\alpha)\\right],\n\\end{align}}\n\\emph{and}\n{\\small \\begin{align}\n\\label{eq:RicEqR} 2r_n=t\\frac{dR_n}{dt}+\\lambda-R_n\\:(t+2n+\\alpha+\\lambda-t\\:R_n).\n\\end{align}}\n\\emph{Furthermore,}\n{\\small $$\ny(t)=y(t, n):=1-\\frac{1}{1-R_n(t)},\n$$}\n\\emph{satisfies the following second order non-linear ode,}\n\\begin{align}\ny''&=\\frac{3y-1}{2y(y-1)}\\:(y')^2- \\frac{y'}{t}+\\frac{(y-1)^2}{t^2}\n\\: \\left(\\frac{\\alpha^2}{2}y-\\frac{\\lambda^2}{2y}\\right)+\\frac{(2n+1+\\alpha+\\lambda)\\:y}{t}-\\frac{y(y+1)}{2(y-1)},\n\\label{eq:PVyt}\n\\end{align}\n\\emph{which is recognized to be a} {\\small\n$$P_V\\left(\\frac{\\alpha^2}{2},-\\frac{\\lambda^2}{2},2n+1+\\alpha+\\lambda,-1\/2\\right).$$}\n\n{\\bf Proof\\;}Because (\\ref{eq:BetanRelation}) expresses $\\beta_n$ as a\nquadratic in $r_n,$ we see that $r_n$ satisfies the Riccatti\nequation (\\ref{eq:RicEqr}). Eliminating $r_{n+1}$ from\n(\\ref{eq:S1Diff2}) and (\\ref{eq:AlphaDiff}), and upon referring to\n(\\ref{eq:AlphanRelation}), we obtain (\\ref{eq:RicEqR}). Next, we\nsimply substitute  $r_n(t)$ from (\\ref{eq:RicEqR}) into\n(\\ref{eq:RicEqr}), to see that $R_n(t)$ satisfies a second order\nnon-linear ode in $t$, in which $n$, $\\alpha$, and $\\lambda$ appear as\nparameters. A further linear fractional change of variable {\\small\n$$ R_n(t)=1-\\frac{1}{1-y(t)}\\quad{\\rm or}\\quad\ny(t) =1-\\frac{1}{1-R_n(t)},\n$$}\nestablishes that $y(t)$ satisfies the Painlev\\'e V displayed in the Lemma.{\\;\\bf End of Proof.}\n\nWe begin here a series of computations which ultimately give rise\ntwo integral representations of the Hankel determinant of interest,\ni.e., {\\small\n$$ D_N(t,\\lambda)=\n\\det\\left(\\int_{0}^{\\infty}x^{j+k-2}(x+t)^{\\lambda}x^{\\alpha}{\\rm\ne}^{-x}dx\\right)_{1\\leq j,k\\leq N} \\; ,$$} given in Theorems 1 and\n2. To this end, an easy computation shows that {\\small \\begin{eqnarray}\nH_N(t)&:=&t\\frac{d}{dt}\\log D_N(t,\\lambda)=t\\frac{d}{dt}\\sum_{j=0}^{N-1}\\log h_j=t\\sum_{j=0}^{N-1}R_j\\nonumber\\\\\n&=& N(N+\\alpha+\\lambda)-\\beta_N-tr_N \\label{eq:Hna} \\\\\n&=& N(N+\\alpha+\\lambda)+\\textsf{p}_1(N) \\label{eq:Hnb}, \\end{eqnarray}} where the last\ntwo equations follow from (\\ref{eq:SumRjRelation}) and\n(\\ref{eq:p1Relation}) of Lemma 4. Integrating (\\ref{eq:Hna}) with\nrespect to $t$, while noting (\\ref{eq:BetanRelation}),\n(\\ref{eq:RicEqR}) and $R_N(t)=1-1\/(1-y(t)),$ we obtain the result\nstated\nin Theorem 1.\n\n\nTo obtain the second integral representation for $D_n(t,\\lambda)$\nstated in Theorem 2 (i.e., in terms of $H_n(t)$), we note that from\n(\\ref{eq:p1Diff}), (\\ref{eq:Hna}), and (\\ref{eq:Hnb}), we obtain\nexpressions for $\\beta_N$ and $r_N$ in terms of $H_N$ and $H_N'$,\n \\begin{eqnarray}\n\\beta_N&=&N(N+\\alpha+\\lambda)+tH_N'-H_N \\label{eq:BetanHn} \\\\\nr_N &=& -H_N'. \\end{eqnarray} What we need to do is to eliminate $R_N$ to find\na functional equation satisfied by $H_N,$ $H_N'$ and $H_N''.$ For\nthis purpose, we examine two quadratic equations satisfied by $R_N$,\none of which is simply a rearrangement of (\\ref{eq:BetanRelation})\nand reads \\begin{eqnarray} \\label{eq:Frac1} \\frac{r_N^2-\\lambda r_N}{R_N}+\\beta_N\nR_N=\\beta_N-r_N(2N+\\alpha+\\lambda)-N(N+\\alpha). \\end{eqnarray} The other follows from a\nderivative of the first equation of (\\ref{eq:BetanHn}) with respect\nto $t$ and (\\ref{eq:BetaDiff}), \\begin{eqnarray} \\label{eq:Frac2} \\beta_N\nR_N-\\frac{r_N^2-\\lambda r_N}{R_N} &=& tH_N''. \\end{eqnarray} Solving for $R_N$ and\n$1\/R_N$ from the linear system (\\ref{eq:Frac1}) and\n(\\ref{eq:Frac2}), we find\n\\begin{eqnarray}\n2R_N&=&1+\\frac{tH_N''-(2N+\\lambda+\\lambda)r_N-N(N+\\alpha)}{tH_N'-H_N+N(N+\\alpha+\\lambda)} \\label{eq:Rna} \\\\\n\\frac{2}{R_N}&=&\\frac{-tH_N''+(t+2N+\\alpha+\\lambda)H_N'-H_N+N\\lambda}{(H_N')^2+\\lambda\\:H_N'}\n\\label{eq:Rnb}, \\end{eqnarray} where we have replaced $\\beta_N$ and $r_N$ in\nterms of $H_N,\\;H_N' {\\rm and\\;} H_N''$ with (\\ref{eq:BetanHn}). The\nproduct (\\ref{eq:Rna}) and (\\ref{eq:Rnb}) gives us the desired\n$\\sigma-$form (\\ref{eq:JimboPV}).\n\nIt is finally worth noting that with\n$D_N(t,\\lambda)=:t^{{\\delta_N}}\\tilde{D}_N,$\nwe find, after a little computation that $\\tilde{D}_N$ satisfies the Toda\nmolecule equation \\cite{Toda} \\begin{eqnarray}\n\\frac{d^2}{dt^2}\\log\\tilde{D}_N=\\frac{\\tilde{D}_{N+1}\\tilde{D}_{N-1}}{\\tilde{D}_N^2} . \\end{eqnarray}\n\n\n\n\\subsection{The Discrete $\\sigma-$Form}\n\nAs an alternative to the Jimbo-Miwa-Okamoto $\\sigma-$form satisfied\nby $H_N$ in Theorem 2, we state (without proof) a non-linear\ndifference equation satisfied by $H_N$. We believe this to be new.\nSpecifically, we find that the logarithmic derivative of the Hankel\ndeterminant in (\\ref{eq:HankelDefin1}), generated by the deformed\nLaguerre weight in (\\ref{eq:DefLagWeight}), satisfies a second order\nnon-linear difference equation, which we call the discrete\n$\\sigma-$form,\n\\begin{align}\n&\\left[\\frac{N(N+\\alpha)t+(\\Delta^2H_N+t)(H_N-\\delta_N)}{\\Delta^2H_N+2N+\\alpha+\\lambda+t}\\right]^2\n-\\lambda\\:\\frac{N(N+\\alpha)t+(\\Delta^2H_N+t)(H_N-\\delta_N)}{\\Delta^2H_N+2N+\\alpha+\\lambda+t}\\nonumber\\\\\n& \\hspace*{0.5cm}= \\left[\\delta_N-H_N\n+\\frac{N(N+\\alpha)t+(\\Delta^2H_N-t)(H_N-\\delta_N)}{\\Delta^2H_N+2N+\\alpha+\\lambda+t}\\right]\n(H_{N+1}-H_N)(H_N-H_{N-1}) \\quad \\nonumber\n\\end{align}\nwhere $\\Delta^2H_N:=H_{N+1}-H_{N-1}$. The initial conditions are\n$H_1(t) = \\frac{d}{dt}\\log D_1(t,\\lambda),\\: H_2(t) = \\frac{d}{dt}\\log\nD_2(t,\\lambda)$ with $ D_1(t,\\lambda) = \\mu_0(t),\\: D_2(t,\\lambda) =\n\\mu_0(t)\\mu_2(t)-\\mu_1^2(t),$ and the moments are defined in\n$(\\ref{eq:MomentDefn}).$\n\n\nWe do not further elaborate on this result, but present it here\nsince we believe that it may provide a useful tool for efficiently\ncomputing $H_N$ in an iterative manner, and give further insight\ninto the mutual information of a single-user MIMO system. A thorough\nanalysis of the implications of this non-linear difference equation\nin regards to wireless communication applications is the subject of\non-going work.\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\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\n\n\n\n\n\n\n\\section{Concluding Remarks}\n\nThis is a short review is about Hankel determinants that arise in\nthe information-theoretic study of MIMO communication systems.  We\nconsider the capacity of single-user MIMO systems, in which case the\ndeterminant of interest is generated from a certain deformed\nLaguerre weight. We have obtained two exact integral representations\nfor this Hankel determinant, one of which is described in terms of\nthe $\\sigma-$form of a particular Painlev\\'{e} V differential\nequation. We have also stated an alternative representation,\ninvolving a second-order non-linear difference equation.\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","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOn a landline telephone, the headset is usually connected with the phone via a helical cord. Such a helical cord, although initially nice and straight [Fig.~\\ref{fig:cords}(a)], often tangles up after use [Fig.~\\ref{fig:cords}(b)]. A careful examination would reveal that a tangled helical cord is sharply different from tangled hair or ropes. The later come from knotted structures, while a tangled helical cord is usually knot free. Instead, it is due to kinks, at which a helical cord changes its chirality and turns sharply. \nWhere do these kinks come from? Why do they turn a nice and straight cord into a tangled mess?  Why are they so hard to get rid of? \nInterestingly,  answers to these questions, as we show below, have roots in the fascinating physics and mathematics of materials and structures at the verge of mechanical instability, to which these helical cords belong.\n\n\n\n\nMechanical stability is concerned with the response of a material to external loads---whether it holds its shape like a solid or flows like a liquid.  At the verge between these two types of behaviors, interesting physics arises, such as critical phenomena~\\cite{Jacobs1995,Liu2010,Ellenbroek2011,Ellenbroek2014,Zhang2017} and topological states~\\cite{Kane2014,Lubensky2015,Paulose2015a,Stenull2016,mao2018maxwell,Zhou2018}.  One peculiar feature of these materials is that they can exhibit a small number of floppy modes, i.e., normal modes of deformation that cost little elastic energy, but at the same time remain stable under other types of loads.  By controlling these floppy modes, which are the low energy excitations of the system, mechanical response of a material can be precisely programmed, which has broad applications such as switchable, actuatable, deployable materials~\\cite{bertoldi2017flexible,Mullin2007,schenk2013geometry,Florijn2014,Paulose2015,filipov2015origami,Chen2015,dudte2016programming,Rocklin2017,tang2017programmable}.  \n\nAmong these fascinating systems right at the verge of mechanical instability, a subcategory, known as Maxwell networks\/lattices~\\cite{Lubensky2015}, has been extensively studied recently, and provided deep insight into many problems in soft matter~\\cite{Mao2010,Mao2011a,Broedersz2011,Mao2013b,Mao2013c,Dennison2013,Mao2015,Zhang2015a,Zhang2016,Zhang2017,Liarte2019}, metamaterials~\\cite{Chen2014,Paulose2015,Rocklin2017,Zhang2018,Ma2018} and mechanobiology~\\cite{Alvarado2013,Feng2015,Feng2016,Sharma2016,Ronceray201514208,feng2019cell}. \nThanks to the discreteness of the degrees of freedom and constraints in these lattices\/networks, \nwithin linear elasticity, a universal theorem, known as the the Maxwell-Calladine index theorem~\\cite{Calladine1978,PellegrinoCal1986,Kapko2009,Sun2012} can be rigorously proved, which reveals a profound relation between floppy modes and states of self stress.\nStates of self stress, as we define rigorously below,  describes modes of distributing stress on  a network leaving all components in force balance and thus characterize the ability of a network to carry stress.  In contrast to floppy modes which describe kinematics of a structure, states of self stress focus on the statics.\nIn addition to demonstrating a universal connection between these two seemingly unrelated quantities, this theorem also played a crucial role in the discovery of topological mechanics in Maxwell lattices and networks~\\cite{Kane2014,Lubensky2015}. Moreover, there is a rigorous duality relation between the geometry of  floppy modes and states of self stress via Maxwell reciprocal diagrams~\\cite{maxwell1864xlv,crapo1994spaces,mitchell2016mechanisms,Zhou2019}.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.8\\columnwidth]{cords.pdf}\n\t\\caption{Kinks in helical cords. (a) A kink-free helical cord is straight.  (b) Kinks sharply turn the helical cord.  Chirality (L or R) of domains and topological charge of kinks are marked (see discussions in Secs.~\\ref{SEC:MiniSurf} and \\ref{SEC:FracRibb}).\n\t}\n\t\\label{fig:cords}\n\\end{figure}\n\nIn contrast, the verge of mechanical instability in continuous elastic media is much less understood, especially in terms nonlinear elasticity which naturally arises when low dimensional elastic objects are embedded in a higher dimensional space~\\cite{Witten2007,Klein1116,sharon2010mechanics,Efrati2011,EFRATI2009762,Armon1726,Kim1201,chen2012nonlinear,guo2014shape,gladman2016biomimetic}.  It is worth pointing out that here mechanical instability refers to intrinsic instability coming from material properties and geometry, rather than instabilities caused by external stress, such as buckling.  In a nonlinear continuous elastic medium, is there a connection between floppy modes and states of self stress, similar to that in lattices\/networks? If so, does it lead to states with nontrivial topological features?\n\nIn continuum, floppy modes correspond to vanishing elastic moduli for certain types of load,  \nwhereas states of self stress dictate the possibility for an elastic body to carry residual stress. \nStress can be generated in elastic bodies in many different ways, such as frozen in stress from manufacturing, topological defects in crystals, and inhomogeneous growth in biological tissue~\\cite{Alexander1998,Withers2001,goriely2007definition}.  When the elastic body is allowed to relax in absence of external load, these stresses may or may not be fully released, and the stress that remains becomes residual stress.  Physically, the existence of residual stress implies that if we cut the system into two parts, neither of them will retain its original shape. Instead, the two pieces will both deform to reduce elastic energy associated with the residual stress. Residual stress often results in ``weak points'' for mechanical failure, but they are also very useful in toughening brittle materials and can be introduced intentionally.  \nStress-free solid materials, on the contrary, usually requires fine tuning, e.g., through careful annealing. Once achieved, cutting the material into pieces will not affect its shape and\/or spacial configuration, in contrast to stressed materials.\n\nIn this manuscript, we examine floppy modes and states of self stress in continuum elasticity, and demonstrate that Maxwell's counting argument still applies, and surprisingly, even in the nonlinear regime.  Based on Maxwell's counting, we classify elastic bodies at different dimensions and study their ground states and low-energy excitations. In light of the counting argument and insights gained from the study of quantum topological states, we identify two types of  thin elastic objects, where topological floppy modes\/low-energy excitations arise. The first class is called Maxwell plates. Such a plate is stress-free and contains sub-extensive number of holographic floppy modes, in strong analogy to 2D Maxwell\/isostatic lattices. The second class is also stress-free and features only one floppy mode. In addition, low-energy excitations in these system are fractional particles, in strong analogy to a $Z_2$  spin liquid and\/or a dimerized spin chain. The nature of these fractional excitations and their connection with topological degeneracy are also discussed. In addition to common features of fractional excitations, fractional excitations in these elastic systems show holographic feature, i.e., the location and configuration of a bulk fractional excitation are be fully dictated by the edge. Interestingly, these topological and holographic features naturally answer the three questions about kinks in a telephone cord mentioned above.\n\n\n\\section{Maxwell lattices and networks}\\label{SEC:counting}\nIn this section, we provide a brief review about elasticity in discrete systems, e.g., Maxwell lattices and networks. Mechanical properties of discrete networks can be conveniently analyzed via a counting argument due to Maxwell~\\cite{Maxwell1864,Calladine1978,Sun2012}.  \n\nThe idea of Maxwell's counting is based on a simple insight about the interplay between degrees of freedom and constraints. If the number of constraints $N_c$ exceeds the\nnumber of the degrees of freedom $N_{\\textrm{d.o.f.}}$, the system is often ``rigid''. On the contrary,\nif $N_c<N_{\\textrm{d.o.f.}}$, the system is often ``floppy''. In this picture, the verge of rigidity is marked by the Maxwell condition\n\\begin{align}\nN_{\\textrm{d.o.f.}}=N_{c},\n\\label{eq:z2D}\n\\end{align}\nand lattices satisfying this condition are called ``Maxwell lattices''. This Maxwell condition also plays an important role in a \nwide range of elastic systems at the verge of mechanical stability~\\cite{Liu2010,Mao2010,Broedersz2011,During2013,Kane2014}.\n\nBefore applying to elasticity,  here we first demonstrate these ideas by considering a set of linear equations, where $N_{\\textrm{d.o.f.}}$ is just the number of variables and \n$N_{c}$ is the number of equations, because each equation enforces one linear constraint on these variables.\nIt is easy to realize that the number of  ``un-fixed'' degrees of freedom can be obtained as\n\\begin{align}\nN_{\\textrm{un-fixed d.o.f.}} = N_{\\textrm{d.o.f.}}-N_{c}+N_{\\textrm{r.c.}}\n\\label{eq:N0}\n\\end{align}\nHere, because constraints may not all be independent of one another, redundant constraints needs to be removed in this counting \nand  $N_{\\textrm{r.c.}}$ here represent the number of redundant constraints.\nFor linear equations, this counting relation can be rigorously proved. However, it must be emphasized that when nonlinearity are taken into account, such\na counting argument is not generically expected, due to the complicated nature of nonlinear constraints. \n\nFor discrete lattices\/networks, the counting relation Eq.~\\eqref{eq:N0}  can be applied if we focus on linear response around stress-free ground states.\nHere, it is easy to realize that the number of  ``un-fixed'' degrees of freedom $N_{\\textrm{un-fixed d.o.f.}}$ gives the number of zero modes $N_0$, i.e. normal modes that don't\ncost any elastic energy. As for redundant constraints, they have a one-to-one correspondence with states of self stress and thus $N_{\\textrm{r.c.}}$ must \ncoincide with the number of states of self stress $N_s$.\nThis is because  by definition a redundant constraint tries to fix a degree of freedom which is already pinned by other constraints. Thus, in order to keep the network stress-free, this redundant constraint must be ``fine-tuned'' such that it is compatible with other constraints. If not compatible, such a redundant constraint will result in conflict constraints and thus generates residual stress,   giving rise to a state of self-stress. \nAs a result, the counting argument discussed above leads to the Maxwell-Calladine index theorem.\n\\begin{align}\nN_0 -N_{s} = N_{\\textrm{d.o.f.}}-N_{c}\n\\label{eq:N0s}\n\\end{align}\nwhich shows that the number of zero modes $N_0$ and the number of states of self-stress $N_{\\textrm{s}}$ are directly related. \nIn particular, for networks\/lattices at the verge of mechanical instability, i.e., Maxwell lattices\/networks with $N_{\\textrm{d.o.f.}}=N_{c}$, these two numbers must coincide ($N_0 =N_{s}$).\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\columnwidth]{counting.pdf}\n\t\\caption{Counting degrees of freedom and constraints.  (a-c) 2D discrete networks composed of sites (black dots) and bonds (black solid lines), which are \n\t(a) underconstraint, (b) isostatic and (c) over-constraint. In (a), $N_{\\textrm{d.o.f.}}=8, N_{c}=4, N_{s}=0$, so $N_0=4$ and thus the network has one floppy mode (green dashed configuration), i.e., $N_f=1$ according to Eq.~\\eqref{eq:Nm}. (b) Introducing one more constraint makes $N_{c}=5$ and $N_f=0$, and thus the network is isostatic and at the verge of mechanical instability.  (The dashed line is not a constraint.)  (c) The addition of more constraints into the network leads to states of self stress: when the new diagonal bond is not at the prescribed length [black dashed line in (b)] the network has residual stress on bonds (yellow for tension and blue for compression). (d-f) $d$ dimensional elastic bodies embedded in a $D=3$ dimensional space.  (d) A curve of $d=1$, with bending and torsional stiffness ignored, is under constrained, in analogy to (a).  (e) A surface of $d=2$, with bending stiffness ignored, is at the verge of mechanical instability, in analogy to (b).  (f) A body of $d=3$ is always over-constrained, in analogy to (c), and able to host residual stress (blue and yellow colors).}\n\t\\label{fig:counting}\n\\end{figure}\n\nIn Fig.~\\ref{fig:counting}(a-c), we demonstrate the Maxwell-Calladine theorem by considering a network consisting of point-like sites (hinges) connected by central-force bonds (struts) and assume that all bonds are at their rest-lengths in the reference state. Here, each site has $D$ degrees of freedom, where $D$ is the spatial dimension, and each bond provides one constraint. Thus the Maxwell-Calladine theorem takes the form of \n\\begin{align}\nN_0-N_{s}= N D - N_b   = ND - N\\langle z \\rangle\/2  .\n\\label{eq:QC}\n\\end{align}\nHere $N D$ is the total number of degrees of freedom $N_{\\textrm{d.o.f.}}$, with $N$ being the number of sites.\nThe total number of constraint $N_{\\textrm{c}}$ is now given by the number of bonds $N_b$, which can be written as $N \\langle z \\rangle\/2$ where $\\langle z \\rangle$ is the average coordination number (number of bonds connecting to one site) of the network. \nFor a periodic lattice, the Maxwell condition [Eq.~\\eqref{eq:z2D}] here takes the form of $\\langle z \\rangle =2D$.\nIn Fig.~\\ref{fig:counting}(c), we present one example demonstrating the relation between redundant constraints and states of self stress. When the last bond is added to the network, if it is not fine-tuned to the length prescribed by other bonds, residual stress is introduced.  Analogous continuous elastic media (d-f) are discussed in Sec.~\\ref{SEC:ContinuumCounting}.\n\nOne important feature of Maxwell lattices\/networks lies in its mechanical properties under open boundary conditions. An infinitely large Maxwell lattice is perfectly constrained everywhere (unless some constraints are redundant).  When a finite piece of Maxwell lattice is cut off, under open boundary conditions, the lattice will have a subextensive number of zero modes, coming from the cut bonds on the surface, inducing a deficit of constraints.  These zero modes exhibit very interesting properties, and their localization has been shown to be a topologically protected property, dictated by the bulk phonon structure~\\cite{Kane2014,Lubensky2015}.\n\nWe conclude this section by discussing the relation between the Maxwell condition [Eq.~\\eqref{eq:z2D}] and isostaticity. For an elastic system, if is often useful \nto exclude trivial degrees of freedom associated with rigid translations and rotations of the whole system, and thus\nthe number of floppy modes is defined as\n\\begin{align}\nN_f = N_0 - \\frac{D(D+1)}{2} ,\n\\label{eq:Nm}\n\\end{align}\nwhich describes elastic deformations that cost no elastic energy. Networks that satisfy $N_f=N_s=0$ are called ``isostatic'' structures, which are both stable (no floppy modes) and stress-free [See Fig.~\\ref{fig:counting}(b) for an example]. According to the Maxwell-Calladine theorem, isostatic systems should satisfy the relation $N_{\\textrm{d.o.f.}}-D(D+1)\/2=N_{c}$. This relation differs slightly from the \nMaxwell condition [Eq.~\\eqref{eq:z2D}], due to the extra term $D(D+1)\/2$ which comes from excluding trivial degrees of freedom (i.e. rigid body translations\/rotations).\nThis difference often becomes negligible for large systems or in the continuum, where $N_{\\textrm{d.o.f.}}$ and $N_c$ diverge.  \n\n\\section{Continuous elasticity of non-flat media}\nIn this section, we write down the elastic theory for a $d$-dimensional elastic medium embedded in a $D$-dimensional Euclidean space ($d \\le D$). In this convention, the typical 3D elastic theory corresponds to $d=D=3$, while a 2D plate embedded in the 3D space has $d=2$ and $D=3$.\nFor 2D plates, unless stated otherwise, in this manuscript, we focus on plates that are homogenous along the thickness direction \n(e.g. a plate formed by identical 2D layers stacked vertically on top of each other), and thus the two sides of each plate are always fully equivalent, in contrast to elastic shells. \nIt turns out that this symmetry plays an important role in the fractional excitations we discuss below.\n\n\nThe spatial configuration of a $d$-dimensional object in a $D$-dimensional Euclidean space is described by the mapping $\\mathbf{r} \\to \\mathbf{R}$, where $\\mathbf{r}$ and $\\mathbf{R}$ are the $d$- and $D$-dimensional coordinates of the original space and the target space respectively. For $d=D$, the elastic energy depends on the  deformation of the elastic medium, characterized by the metric tensor (defined below). For a homogenous or nearly homogenous system (i.e., elastic moduli remain constants but stress can slowly vary in space), the elastic energy takes the following functional form\n\\begin{align}\nE_s=\\int dr \\sqrt{\\det g_0} \\left\\{ \\frac{B-\\frac{2}{d}G}{2} \\tr(g-g_0)^2+ G \\tr[(g-g_0)^2]\\right\\}\n\\label{eq:Es}\n\\end{align}\nwhere $B$ ($G$) is the bulk (shear) modules  and $g$ is the metric tensor\n$g_{\\alpha,\\beta}=\\partial_\\alpha \\mathbf{R}\\cdot \\partial_\\beta \\mathbf{R}$,\nalso known as the first fundamental form, with $\\partial_\\alpha=\\partial\/\\partial \\mathbf{r}_\\alpha$ ($\\alpha=1$, $2$, $\\ldots$, $d$). Same as $g$, $g_0$ is also a symmetric rank-2 tensor ($d\\times d$), which is usually called the reference metric tensor. As shown in Eq.~\\eqref{eq:Es}, $g_0$ is the target for $g$, if we try to minimize the elastic energy.  In terms of discrete elastic networks defined in Sec.~\\ref{SEC:counting}, $g=g_0$ corresponds to the configuration where all bonds are at their rest length.\n\nWhen $g_0=I$, the conventional $D$-dimensional elastic theory is recovered with the strain tensor $\\epsilon=g-I$. In this manuscript, we consider more generic cases where $g_0$ may deviate from $I$. For $d=D$, such a generic $g_0$ describes prestressed materials, in which $g_0$ can never be reduced to identity in any coordinate systems. \nFrom Eq.~\\eqref{eq:Es}, it is easy to realize that for $g_0 \\ne I$, deforming a system (i.e. making $\\mathbf{R}\\ne \\mathbf{r}$ and thus $g\\ne I$) may actually reduce the elastic energy, which corresponds to fully or partially releasing of build-in stress via deformation.\n\nFor $d<D$ (e.g. $d=2$ and $D=3$), there is one more reason to consider a generic $g_0$ beyond identity: if the ground state of the 2D plate is curved, $g_0$ may not be reduced to identity in any coordinate systems, and thus it is necessary to explore this more generic $g_0 \\ne I$ case. For $d<D$, e.g., a 2D sheet embedded in 3D, the elastic energy can be separated into two parts\n\\begin{align}\nE=E_s+E_b.\n\\label{eq:d=2D=3}\n\\end{align}\nwhere the first term $E_s$ describes in plane stress while the second term $E_b$ describes the energy cost associated with bending.\n(For even lower $d<2$ or higher $D>3$, more terms can arise, such as the torsion term, which can be treated along the same line but will not be considered explicitly here).\nFor $d=2$ and $D=3$, this elastic energy [Eq.~\\eqref{eq:d=2D=3}] can be obtained by considering a thin 3D plate with thickness $h$, whose elastic energy takes the form of Eq.~\\eqref{eq:Es}~\\cite{willmore1993riemannian,Klein1116}. \nWhen the thickness of the plate $h$ is small, the total elastic energy can be written as a power-law expansion of $h$. By keeping the leading order terms, Eq.~\\eqref{eq:d=2D=3} is obtained with $E_s \\propto h$ and $E_b \\propto h^3$, while higher order terms (e.g. $O(h^5)$) will be ignored in this manuscript. For the special case of a flat plate, this construction is demonstrated in Landau's book~\\cite{Landau1986}. Here, we consider more generic 2D plates which are not flat.\n\nThe first term of Eq.~\\eqref{eq:d=2D=3} takes the same form as Eq.~\\eqref{eq:Es} but with two modifications\n\\begin{align}\nE_s=h \\int dr{\\sqrt{\\det g_0}}\\large\\{ \\frac{B_0-G_0}{2} & \\tr(g-g_0)^2\n\\nonumber\\\\\n&+ G_0 \\tr[(g-g_0)^2]\\large\\}\n\\label{eq:Es_2D}\n\\end{align}\nFirst, this $E_s$ has an extra pre-factor $h$, which is the thickness of the plate, and secondly, $g$ and $g_0$ are now $d\\times d$ matrices, instead of $D\\times D$.\nThe elastic moduli here are $B_0=\\frac{9 B G}{4(3B+4G)}$ and $G_0=G\/4$. \n\nFor plates that are homogenous along the thickness direction, deformations that make the plate non-flat will cost bending energy, which is the $E_b$ term here. Mathematically, bending is described by the so-called the second fundamental form, and thus $E_b$ shall be in general a functional of the second fundamental form. \nAs will be shown below, for 2D plates, it is always possible to reach $g=g_0$ at least locally, and thus, for a homogenous or nearly-homogenous 2D plate, \n$E_b$ only depends on two qualities, the mean and Gaussian curvatures ($H$ and $K$)\n\\begin{align}\nE_b=h^3  \\int dr{\\sqrt{\\det g_0}}\\frac{G}{12}\\left[ \\frac{8(3B+G)}{3B+4G} H^2 -2 K \\right]\n\\label{eq:Eb}\n\\end{align}\nwhich scales as $h^3$. For a flat plate, this bending energy recovers the text-book example of the elastic theory of thin plates~\\cite{Landau1986}. \n\n\nIt is worthwhile to emphasize that because although the $E_b$ term prefers a plate to be flat, the $E_s$ term may prefer a non-flat ground state, if $g_0$ gives a non-zero curvature, as we discuss below.\n\nFor thin plates (small $h$), because $E_s\\propto h$ while $E_b\\propto h^3$, $E_b \\ll E_s$. Thus, $E_s$ dominates the elastic energy and $E_b$ can be treated as a small perturbation. \nIn the rest of the manuscript, we will follow this perturbative approach and all the conclusion would be accurate to the first order of $E_b$ (i.e., $~h^3$). Higher order contributions, e.g. $~h^5$ will be ignored.\n\n\n\\section{Generalized Maxwell's counting and Janet-Cartan theorem}\\label{SEC:ContinuumCounting}\n\n\\begin{table*}\\label{tab:class}\n\\centering\n\\begin{tabular}[c]{| c | c | >{\\columncolor[rgb]{0.88,1,1}}c || c | c | c | >{\\columncolor[rgb]{0.88,1,1}}c |}\n\\hline\n  &\\multicolumn{2}{c||}{Maxwell systems}   \t&\\multicolumn{4}{c| }{ Over-constrained systems} \\\\\n  \\hline\n  &Maxwell Lattices \t& Maxwell Plates\t&Typically solids\t& \\multicolumn{3}{ c|}{ Plates with bending stiffness}\\\\\n  & \n  $D=2$ or $3$\t& $d=2$, $D=3$\t& $d=D=2$ or $3$\t& \\multicolumn{3}{ c| }{ $d=2$, $D=3$}\\\\\n\\hline\nStress free & Yes$^1$  & Yes$^2$   & No unless $g_0$ is flat  & No in general & \\multicolumn{2}{ c| }{ Yes if criterion in Sec.~\\ref{SEC:PlatBend} is met}\\\\\n\\hline\nGaussian curvature & - & Any & - & Any & Non-negative  & Negative \\\\\n\\hline\nFloppy modes & Subextensive  & Subextensive & None  & None  & None & 1\\\\\n\\hline\nHolographic & \tYes\t & Yes \t& No    & No  & No & Yes \\\\\n \\hline\nShapes &  Any & Any & Any & Any & Sphere or flat & Minimal surface \\\\\n \\hline\nFractional excitations &  No & No  & No\t& No & No & Yes\\\\\n \\hline\n\\end{tabular}\n\\caption{Summary of results. Two types of continuum media with nontrivial topological features are highlighted in cyan. $^1$ if the lattice doesn't contain redundant constraints.  $^2$ if the plate is under open boundary conditions and does not hit singularity points.}\n\\end{table*}\n\n\nIn this section and the following section, we generalize Maxwell's counting argument to continuum elasticity. Similar to discrete systems, this generalized counting argument provides a simple principle for identifying systems with floppy modes and\/or states of self-stress. In addition, we also show that predictions from this generalized Maxwell's counting is in full alignment with the mathematical theorems on local embedding\n~\\cite{janet1926possibilite,cartan1927possibilite}, global embedding and rigidity.\n\nIn this section, we focus on systems with elastic energy $E=E_s$, while more general elastic energy (e.g. with bending $E_b$) will be studied in Sec.~\\ref{SEC:PlatBend}.  \nSystems with $E=E_s$ include two possible cases. The first one is $d=D$, where $E_s$ is naturally the only term in the elastic energy. The second case is for $d<D$, e.g. a 2D plate in a 3-dimensional space. Here, in general, the elastic energy contains more terms beyond $E_s$, e.g., $E_b$. However, as shown above, when the plate is thin enough, $E_b$ becomes negligible, and thus only $E_s$ needs to considered. \n\n\\subsection{Counting argument and its predictions}\nDue to nonlinearity, the counting argument [Eq.~\\eqref{eq:N0s}] is not generically expected here. In this section, we first perform the counting analysis while ignoring \npossible problems caused by nonlinearity. These predictions will then be checked in the next section. Surprisingly, as shown below, for all generic cases considered here, \nwe found that this counting argument happens to provide correct predictions. Of course, it must be emphasized that because such a coincidence cannot be universally proved,\nthere could exist special cases where such a relation is invalid.\n\n\nIn a $D$-dimensional space, each real-space point has $D$ degrees of freedom. As for constraints, following the same spirit of Maxwell's counting argument, $E_s$ can be treated\nas constraints $g=g_0$ , i.e. the elastic energy vanishes if $g=g_0$, in analogy to the a spring network where all springs are at its rest length. The total number of constraints enforced by this relation is $d(d+1)\/2$ per real-space point, because $g=g_0$ is a $d$-dimensional symmetric rank-2 tensor with $d(d+1)\/2$ independent variables. In mathematical literature, this quantity $d(d+1)\/2$ is often referred to as the Janet dimension.\n\nIf $D>d(d+1)\/2$ (e.g. a 1D chain in 3D space with $E=E_s$), the system is under-constrained, and thus it must be floppy with extensive number of floppy modes. In general, unless some incompatible constraints happen to arise in the system, we expect no state of self-stress, which implies that in general, the ground state is stress-free and can reach $g=g_0$.\n\nIf $D<d(d+1)\/2$ (e.g. $d=D>1$), the system is over-constrained and thus we expect extensive number of states of self-stress. These systems don't have to contain floppy modes, and thus in general we expect the system to be rigid with one unique ground state.\n\nAs for the marginal case where the spatial dimension and the Janet dimension coincides, $D=d(d+1)\/2$, the system is at the Maxwell point, where constraints and degrees of freedom have exactly the same number. One such example is a 2D plate embedded in  the 3D space ($d=2$ and $D=3$). If the bending energy is negligible, such a plate will be called a Maxwell plate. \n\n\n\n\\subsection{Local embedding and the Janet-Cartan theorem}\n\\label{sec:janetcartan}\nThe counting argument discussed above is in full alignment with the Janet-Cartan theorem~\\cite{janet1926possibilite,cartan1927possibilite}.\n\nFor any given metric $g=g_0$ (which minimizes $E_s$ and thus fully releases built-in stress), whether a mapping $\\mathbf{r} \\to \\mathbf{R}$ exist such that $(g_{0})_{ij}=\\partial_i \\mathbf{R}\\cdot \\partial_j \\mathbf{R}$, is known as the isometric embedding problem.  $g=g_0$ may not always be achievable, because actual variables of the system are $\\mathbf{R}(\\mathbf{r})$, and different components of $g$ are related with one another.  \nThe  Janet-Cartan theorem of analytic local embedding states tells us that a \\emph{local embedding} always exists for $D\\ge d(d+1)\/2$, where local embedding means that for any real space point, we can always find a region around this point such that $g=g_0$ for every point in this region for any analytic $g_0$. In other words, as long as our elastic system is small enough and has open boundaries, any under-constrained or Maxwell systems ($D\\ge d(d+1)\/2$) must have a stress-free ground state for any analytic $g_0$, i.e., the system has no states of self stress. In contrast, for over-constrained systems $D>d(d+1)\/2$, an isometric embedding doesn't exist in general. Physically, this means that for a generic $g_0$, it is impossible to reach $g=g_0$ and fully released stress. Thus, the system contains states of self-stress.  \n\nThis is in good agreement with conclusions in the previous section following from Maxwell's counting argument.  It is worth noting that the Janet-Cartan theorem incorporates nonlinearities in the constraint $g=g_0$, and interestingly it supports the linear counting argument.\n\nBelow, we consider two examples: an over-constrained system with $d=D$ and a Maxwell plate with $d=2$ and $D=3$, in order to verify relations between zero modes and states of self stress from Eq.~\\eqref{eq:N0s}].\n\n\n\n\\subsection{Over-constrained systems with $d=D$}\nAccording to the counting argument outlined above, a system with $d=D>1$ is always over-constrained, and thus we expect extensive number of states of self stress. \nAs shown above, this conclusion qualitatively agrees with the Janet-Cartan theorem.\nIn this section, we further quantitatively verify the counting argument by directly counting the number of states of self stress and then comparing \nit with the value predicted by the counting argument [Eq.~\\eqref{eq:N0s}]. Same as in discrete systems, we start from a fine-tuned stress-free ground state \n(with $g=g_0$) and ask how many different ways residual stress can arise in such a system, which is a linear elasticity problem and thus a definite answer can be obtained.\n\nBecause the number of floppy modes per real space point is in general zero without fine tuning in an over-constrained system, \naccording to  Eq.~\\eqref{eq:N0s}, the number of states of self stress should equal to the number of constraints minus the number of degrees of freedom, \nwhich is $D(D-1)\/2$ for $d=D$. This is indeed correct. A $D$-dimensional stress tensor field $\\sigma$  has $D(D+1)\/2$ independent components. \nAt the same time, force equilibrium condition requires the stress field to be divergenceless,\n\\begin{align}\n \\partial_i \\sigma_{ij}=0,\n\\end{align}\nwhich provides $D$ partial differential equations and thus fixes $D$ of the $D(D+1)\/2$ independent components at each real space point. As a result, the number of free components becomes $D(D-1)\/2$, matching exactly the expected number of states of self stress of the counting argument. It is worthwhile to mention that to count the states of self stress, in \nprinciple, force-free boundary conditions also need to be enforced in addition to the divergenceless condition. However, because the boundary is subextensive, \nits impact on an extensive quantity (i.e. the number of states of self stress here) is negligible and thus would not change the counting. \n\nWe conclude this subsection by discussing the sufficient and necessary condition, under which this over-constrained system has a stress free ground state. \nSimilar to over-constrained networks\/lattices, although generically our over-constrained system shall contain residual stress, the ground state could be made stress free if we fine tune the constraints to make them fully compatible with each other. For lattices and networks this fine tuning is about the rest length of each bond, while in continuum, we need to fine tune the reference metric tensor $g_0$. For $d=D$, the sufficient and necessary condition for an elastic body to be stress free is that $g_0$ must be flat, because the $D$ dimensional Euclidean space we try to embed this body into is flat. More precisely, if we treat $g_0$ as a metric tensor and compute its Gaussian curvature (as show in App.~\\ref{app:sec:2Din3D}, for a Riemannian manifold, its Gaussian curvature is uniquely determined by the metric tensor), this Gaussian curvature must vanish at every point, which implies that the system is globally flat. \n\n\\subsection{Maxwell plates and holographic floppy modes}\nIf $E_b$ is ignored, a 2D plate ($d=2$ and $D=3$) satisfies the Maxwell condition [Eq.~\\eqref{eq:z2D}]  with $D=d(d+1)\/2$, and thus is right at the Maxwell point, which is the reason\nwhy they are called Maxwell plates.\n\nIn this section, we consider small plates with open boundary conditions, where the Janet-Cartan theorem applies. More generic cases, e.g., large or infinite plates and closed plates without boundary, will be discussed in Sec.~\\ref{SEC:global}.\n\n\nAs shown above, the number of degrees of freedom and constraints perfectly match in the bulk of a Maxwell plate.  \nThus, edge degrees of freedom become important. Same as Maxwell lattices\/networks, a point on an open edge has the same number of \ndegrees of freedom but fewer constraints, because its constraints only comes from one side of the edge, while the other side is empty and thus enforces \nno constraint. This deficit of constraints on the boundary leads to floppy modes in Maxwell plates when they are under open boundary conditions, and according to Eq.~\\eqref{eq:N0s}, the number of these floppy modes is proportional to the length of the boundary, and is a sub-extensive quantity.\nIn addition, same as Maxwell lattices, the edge origin implies that floppy modes in Maxwell plates are \\emph{holographic}, i.e., for any low-energy deformations, the deformation field at the edge fully dictates the bulk deformation field. The existence of sub-extensive holographic floppy modes is a key property of Maxwell plates, in strong analogy to Maxwell lattices.\n\nThese counting-based conclusions on holographic floppy modes can be rigorously proved using  mathematical tools of isometric embedding. As show in Ref.~\\onlinecite{Han2006isometric}, \nmathematically, the problem of isometric embedding is translated into partial differential equations, known as the Gauss-Codazzi equations (See App.~\\ref{app:sec:2Din3D} for\nmore details). \nFor plates considered here, solutions to these partial differential equations alway exists, as long as suitable boundary conditions are applied and the plate is not too large to hit singularity points. In particular, solutions to these equations have a one-to-one correspondence with the boundary conditions, i.e. boundary conditions fully determine the solution in the bulk, which is the mathematical origin of the holographic floppy modes. This is in strong analogy to static electricity in a cavity, which is also governed by a set of partial differential equations. There, the field configuration is also holographic, i.e. it is fully dictated by boundary conditions, e.g. charge or field configurations on the boundary uniquely determines the charge\/field configuration in the bulk.\n\n\n\n\n\\subsection{Global embedding and systems beyond the Janet-Cartan theorem}\\label{SEC:global}\nIn this section, we explore infinitely large 2D plates and\/or closed 2D plates without open boundaries. Here a global embedding is required to fully release the stress,\nwhich is a much stronger requirement in comparison with a local embedding. A global embedding requires $g=g_0$ even if we extends the system (through analytic continuation) to infinity or until the system forms a closed manifold . A global embedding cannot always be achieved for $D=d(d+1)\/2$. \nIn the counting argument, this means that at $D=d(d+1)\/2$ , although $g=g_0$ introduces no redundant constraints for any small regions, if the elastic system is large enough, or if the system form a closed manifold (e.g. by enforcing periodic boundary conditions), redundant constraints may arise and thus the ground state may not be able to fully release stress unless we fine tune these constraints to make them compatible with one another. \n\nIn this section, we consider two different families of Maxwell plates with (1) positive or (2) negative Gaussian curvature. Remarkably, for both cases, their elastic properties obey the counting predictions.\n\nFirst, we consider a 2D closed plate with positive Gaussian curvature and zero genus, which means that the plate is topologically equivalent to a sphere $S^2$. Because such a plate has no boundary, the number of constraints and degrees of freedom perfectly match with each other at every point, and thus the counting argument requires that the density of floppy modes and states of self stress must coincide.  \nThis is indeed true for such a plate. Mathematically, the embedding of such a plate \n(positive Gaussian curvature and zero genus) is known as the Weyl problem~\\cite{Weyl1916uber}. Based on the Nirenberg-Pogorelov theorem~\\cite{Nirenberg1953,Pogorelov1952}\nsuch an embedding with $g=g_0$ always exists and the embedding is ``rigid'',  i.e., any deformation from the ground state will make $g\\ne g_0$ and thus cost energy.\nThe existence of embedding for any $g_0$ implies that the ground state is always stress free, and thus there is no state of self stress $N_s=0$. At the same time, the rigidity\npart of the Nirenberg-Pogorelov theorem implies the absence of floppy mode ($N_f=0$), and thus indeed $N_f=N_s$ as the counting argument predicts.\nIn addition, because such a plate has $N_f=N_s=0$, this system is not only ``Maxwell'', but also ``isostatic''.\n\nFor plates with negative Gaussian curvature, it cannot form a closed manifold and floppy modes can emerge from edge degrees of freedom as shown in the section above. Here, we consider an infinitely-large plate. If the Gaussian curvature decays to zero fast enough at infinity, a global embedding always exits~\\cite{Hong1993} and thus \n$g=g_0$ can always be reached. Here, conclusions of local embedding remain and we shall have no state of self-stress and sub-extensive number of floppy modes. \nIf the Gaussian curvature doesn't approach zero or approaches zero too slowly at infinity, an isometric embedding may hit singularity and thus make a global embedding\nimpossible (e.g. the Efimov theorem~\\cite{Efimov1966surfaces}). Because $g$ cannot reach $g_0$, the ground state carriers residual stress, i.e., state of self stress emerges.\nAs shown in Ref.~\\cite{marder2006geometry,sharon2010mechanics},  wrinkles will start to develop beyond these singular points. According to Maxwell's counting argument, this emergence of state of self-stress must be accompanied by new floppy modes. For this example, these floppy modes correspond to translations of the wrinkles.\n\n\n\n\n\n\\section{2D plates with bending energy}\\label{SEC:PlatBend}\nIn this section, we consider the impact of bending energy $E_b$ for $d<D$. As mentioned above, we will focus on the thin plate limit (small $h$) and thus $E_b \\ll E_s$.\nIn the counting analysis, $E_b$ can also be treated as constraints. \nSame as $E_s$, generally, the number of constraints enforced by $E_b$ is an extensive quantity. For 2D plates in a 3D space, as shown in the previous section, without $E_b$, the number of degrees of freedom exceeds the number of constraints by a small (sub-extensive) margin, and as a result, after adding extensive number of constraints from $E_b$, the system is over-constrained. Therefore, same as the case of $d=D$, we shall in general expect one unique ground state (no floppy mode) and extensive number of states of self-stress.  Thus, unless we fine tune $g_0$, the system shall in general carry residual stress.\n\nIn this section, we answer two questions:\n\\begin{itemize}\n\\item Is it possible to make a 2D plate with bending stiffness stress free? \n\\item Is it possible to make a 2D plate with bending stiffness flexible, i.e. introduce one or more floppy mode(s) into this over-constrained system? \n\\end{itemize}\nIt turns out that the answer to the second question relies on the first one, and thus we will start by exploring the first question. \n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{sphere.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{Enneper.pdf}}\n\t\\caption{Two types of stress-free 2D plates with (a) positive or (b) negative Gaussian curvature. (a) With a positive Gaussian curvature, a sphere is the only configuration for 2D stress-free plates. (b) With a negative Gaussian curvature, a 2D plate must form a minimal surface to be stress-free. Here, we present one such example, an Enneper surface.}\n\t\\label{fig:}\n\\end{figure}\n\n\nHere we show that a 2D plate becomes stress free, if and only if $g_0$ satisfies either one of the following two criteria:\n\\begin{itemize}\n\\item the sphere criterion: the Gaussian curvature $K_0$ calculated from the metric tensor $g_0$ is a non-negative constant over the entire plate.\n\\item the minimal surface criterion: if $K_0<0$, we treat the product $\\sqrt{|K_0|}g_0$ as a new metric tensor and compute its Gaussian curvature $K_1$. $K_1$ must vanish at every point on the plate.\n\\end{itemize}\nIf the first criterion is met, the ground state of such a 2D plate is (part of) a sphere with radius $\\rho_0=1\/\\sqrt{K_0}$. Here, the system is stress free and it has a unique ground state (no floppy mode). \nAs proved by Ricci in 1894, the second criterion is the sufficient and necessary condition for the existence of a minimal-surface with $g=g_0$~\\cite{Ricci1894}. \nIf $g_0$ satisfies the second criterion, the ground state is a minimal surface. Remarkably, such a 2D plate is not only stress free but also has one floppy mode [see Sec.~\\ref{SEC:MiniSurf} below]. One interesting example of minimal-surface plates \ncan be found in Ref.~\\onlinecite{Efrati2011}, which applies to any narrow ribbon with negative Gaussian curvature.\nIf none of these two criteria holds, the corresponding 2D plate must have residual stress, which cannot be released.\n\nHere, we just provide a brief outline of the proof, while details can be found in App.~\\ref{app:sec:stress:free:2D}. Within the perturbative approach described above (keeping terms up to $O(h^3)$), in order to find the ground state, we first minimize the leading-order term $E_s$, which enforces the constraint $g=g_0$, and then, under the constraint of $g=g_0$, $E_b$ is then minimized. The constraint $g=g_0$ not only fixes $g$ but also the Gaussian curvature $K$ of the 2D plate setting $K=K_0$, because the Gaussian curvature  is uniquely determined by the metric tensor and its spatial derivatives (known as Gauss's  \\emph{theorema egregium}, see App.~\\ref{app:sec:2Din3D} for more details). For $E_s$, this means that the value of $K$ in Eq.~\\eqref{eq:Eb} is already pinned as we set $g=g_0$. As a result, the $K$ term in $E_b$ does not participle in the minimization of $E_b$ and thus can be ignored. After dropping $K$,  it is straightforward that $E_s$ prefers to minimize the square of the mean curvature $H=(k_1+k_2)\/2$, where $k_1$ and $k_2$ are the two principle curvatures. This minimization needs to obey the constraint on the Gaussian curvature $K=k_1 \\times k_2=K_0$. With a fixed $k_1\\times k_2$, the minimization of $H^2$ has two possible scenarios: (1) $k_1=k_2=\\sqrt{\\kappa_0}$ for $K \\ge 0$ or (2) $k_1=-k_2$ for $K<0$. This is the constraint enforced by $E_b$.\n\nAs shown in App.~\\ref{app:sec:stress:free:2D}, $E_s$ enforces constraints on the metric tensor, which is also known as the first fundamental form, while constraints enforced \nby $E_b$ are about curvatures, which are mainly associated with the second fundamental form. Thus, in order to ensure constraints enforced by $E_s$ and $E_b$ are \ncompatible with each other, relations between first and second fundamental forms play a crucial role. It is known that the first and second fundamental forms are related by a set of partial differential equations, known the Gauss-Codazzi equations (in addition to the constraint of Gaussian curvature discussed above). As shown in App.~\\ref{app:sec:stress:free:2D}, the Gauss-Codazzi equations are satisfied if and only if one of the two criteria mentioned above is met.\n\n\n\\section{Minimal surfaces and their floppy mode}\\label{SEC:MiniSurf}\nAs shown in the previous section, when bending energy is taken into account, 2D plates are generically over-constrained and only two special cases can be stress free (a sphere or a minimum surface).  Elastic properties of these two cases are very different. Most importantly, a sphere has a unique ground state, while a minimal-surface 2D plate has infinitely many degenerate ground states, which are connected with each other via a floppy mode. In this section, we focus on the case of minimal surfaces.\n\n\\subsection{Floppy mode}\nMinimal surfaces are 2D surfaces embedded in a 3D space with their local area minimized. The key signature of a minimal surface is the vanishing mean curvature $H=0$. For $E_b$, because the Gaussian curvature $K$ is fixed by $g_0$ as we enforce the constraint $g=g_0$, a minimal surface with $H=0$ offers a natural minimization of $E_b$. This is the key reason why minimal surface can make constraints from $E_b$ and $E_s$ compatible. \n\nAs shown in App.~\\ref{app:sec:minimal:surface}, for any minimal surface, an isometric deformation always exists (i.e. $g$ stays invariant), under which the surface remains minimal (i.e. $H$ remains $0$). One example of such an isometric deformation is shown in Fig.~\\ref{fig:associate:family}.\nFor elasticity, it is easy to notice that this isometric deformation doesn't cost any elastic energy, as $E=E_s+E_b$ stays the same, and thus is a floppy mode. All minimal surfaces that are connected by this deformation is called an associate family, and this floppy mode transfers one minimal surface to other minimal surfaces, which belong to the same associated family.\n\nMinimal surfaces in an associate family can be labeled by a phase angle $\\varphi$ (see App.~\\ref{app:sec:minimal:surface} for details). As a phase angle, only $\\varphi \\mod 2\\pi$ has a real physical meaning. \nIn the  Weierstrass-Enneper parameterization, minimal surfaces are mapped into complex functions, and the phase angle $\\varphi$ is an overall complex phase in the Weierstrass-Enneper parameterization. By varying this phase angle (and keeping the rest of the complex functions invariant), a minimal surface is deformed into other ones in its associate family.\nIn this parameterization, the floppy mode corresponds to adiabatically adjusting the value of $\\varphi$.\n\nIt is worthwhile to emphasize that although for a minimal surface 2D plate $E=E_s+E_b$ has infinite degenerate ground states, in reality, this infinite degeneracy will be lifted by higher order terms in the elastic energy (e.g. if one introduces anisotropy into the elastic medium and one example is shown in App.~\\ref{app:sec:higher:order}). \nAs a result, a real system may prefer one (or two) specific minimal surface(s) as its ground state, while the floppy mode becomes a soft mode, i.e., it costs a small amount of energy to deform the ground state to another minimal surface in its associate family. However, as long as we are focusing on low-energy soft deformation, other configurations beyond the associate family can still be excluded, which cost much higher energies to reach.\n\n\\subsection{Narrow ribbons}\\label{SEC:Ribb}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.9\\columnwidth]{assoicatefamiliy.pdf}\n\t\\caption{Ground state configurations of 2D narrow ribbon with negative Gaussian curvature. Any minimal surface in the helicoid-catenoid associate family is a ground state. These ground states\n\tare labeled by a phase angle $\\varphi$ as marked in the figure.}\n\t\\label{fig:associate:family}\n\\end{figure}\n\nHere, we present one example of stress-free 2D plates, by considering one easy realization of the stress-free criteria. As shown in App.~\\ref{app:sec:ribbons}, for  2D ribbons, as long as the width of a ribbon is narrow enough, the system is guaranteed to satisfy one of the two stress-free criteria regardless of microscopic details, and thus their ground states must be either part of a sphere or a minimal surface. Interesting examples of of minimal surface hyperbolic thin ribbons have been discussed in Ref.~\\onlinecite{Efrati2011}.\n\nIn this section, we focus on 2D narrow ribbons with minimal surface ground states.\nAs discussed above, such a system has infinite number of ground states characterized by a phase angle $\\varphi$. As shown in in Fig.~\\ref{fig:associate:family}, \n$\\varphi=\\pm \\pi\/2$ represent the helicoid ground states with left-hand and right-hand chirality respectively, while $\\varphi=0$ and $\\pi$ represent two catenoid ground states. Other values of $\\varphi$ represent other ground states in this minimal-surface associate family, known as the helicoid-catenoid family.\nIt is worthwhile to mention that as shown in App.~\\ref{app:sec:ribbons}, depending on the linear term of $u$ in $g_{22}$ [Eq.~\\eqref{app:g22_expansion}],\na ribbon's helicoid ground state may or may not contain the rotational axis, i.e. helicoids shown in Fig.~\\ref{fig:associate:family} at $\\varphi=\\pm \\pi\/2$ vs. helicoids in Fig.~\\ref{fig:spin:chain}(e) and (f). These two cases may look different, but they are part of the same helicoid and thus share the same elastic properties.  Specifically, helicoids in Fig.~\\ref{fig:spin:chain}(e) and (f) can be seen as cutting a thin strip from the edge of helicoids in Fig.~\\ref{fig:associate:family}.\n\nAs can be seen from the figure, except for the two catenoid ground states ($\\varphi = 0$ or $\\pi$), all other ground states of the 2D plate simultaneously break the chiral symmetry (of the target space). \nThe two catenoid ground states don't break spatial symmetries of the target space. Instead, they break an internal symmetry of the 2D plate. This can be made more visible if we color the two sides of the plate with two different colors [yellow and blue in Fig.~\\ref{fig:associate:family}]. In our elastic energy, the two sides are fully equivalent and thus the elastic energy is invariant if we swap these two colors. However, a catenoid ground state breaks this symmetry making the yellow and blue inequivalent. Because swapping the two colors is a $Z_2$ symmetry, we should expect two such ground states when this symmetry is broken, which are the two catenoids at $\\varphi= 0$ and $\\pi$. If we swap the two colors, the two catenoids are mapped into each other. This is in analogy to an $Z_2$ internal symmetry breaking, e.g. a ferromagnetic state in the Ising spin model.\n\n\n\n\n\\subsection{Fractional solitons in thin ribbons}\nThe floppy mode in 2D minimal-surface plates has one interesting implication. It means that the system supports fractional topological excitations, in strong analogy to \n fractional topological states such as a $Z_2$ spin liquid.\n\nBecause the system has infinite degenerate ground states $0 \\le \\varphi<2\\pi$, low-energy excitations in such a ribbon shall be characterized as a $\\varphi$ field slowly varying   along the ribbon.\nThe elastic energy for such a low-energy excitation can be described by the following phenomenological theory\n\\begin{align}\nE=\\int d v (\\partial_v \\varphi)^2\n\\end{align}\nwhere $v$ is the coordinate along the ribbon direction.\nBecause any constant $\\varphi$ produces a degenerate ground state, this elastic energy can only depend on derivatives of $\\varphi$, instead of $\\varphi$ itself.  The quadratic term included here is the leading order contribution. Without loss of generality, the elastic modulus for this term is set to unity, which describes the energy cost for an inhomogeneous $\\varphi$.\n\nAs mentioned above, in a real system, the infinite number of minimal-surface states in the associate family cannot be perfectly degenerate, due to higher order corrections in the elastic energy, which can be described by a function of $\\varphi$, $E_{c}(\\varphi)$. Because for a phase angle, $\\varphi$ and $\\varphi+2\\pi$ are fully equivalent as mentioned above, this function must be a periodic function with $E_c(\\varphi)=E_c(\\varphi+2\\pi)$. Thus, in general, the elastic energy shall take the following form\n\\begin{align}\nE=\\int d v \\left[ (\\partial_v \\varphi)^2+ \\sum_n \\gamma_n \\cos  n (\\varphi-\\varphi_{n})\\right]\n\\label{eq:sine_gordon_general}\n\\end{align}\nwith the second term describing higher order terms that lift the infinite degeneracy of the ground state. Here, $n$ are positive integers, while $\\gamma_n$ and $\\varphi_{n}$ are constants for each $n$. This effective theory is a generalized sine-Gordon theory. It has long been known that the sine-Gordon theory supports soliton excitations, i.e., topological excitations \nwith $\\varphi$ changes from one value to another over a finite length scale. A particularly interesting example of a soliton in a mechanical chain described by a sine-Gordon equation is the nonlinear soliton in a 1D topological rotor chain, as discussed in Ref.~\\onlinecite{Chen2014}, where the kink soliton is exactly zero energy due to a boundary term.  \n\nIt must be emphasized that the sine-Gordon theory we use here  is \\emph{compact}, i.e. $\\varphi$ and $\\varphi+2\\pi$ are identical. As will be discussed below, solitons are quantized in a compact sine-Gordon theory, and each soliton has a well-defined quantized charge. \nHere, we first introduce the definition of soliton charge, while its quantization will be addressed below. For a compact field $\\varphi$, the periodicity $2\\pi$ offers a natural reference to measure the varition of $\\varphi$. For a soliton where $\\varphi$ changes by $\\Delta \\varphi$, its charge is defined as $\\Delta \\varphi \/2\\pi$. Solitons with integer charge are called integer solitons, i.e., $\\Delta \\varphi = 2 n \\pi$ with $n$ being an integer, while a fractionally charged  soliton will be called a fractional soliton.\n\nFor a (generalized) sine-Gordon theory shown in Eq.~\\eqref{eq:sine_gordon_general}, generically, solitons  have integer charge (i.e., $\\varphi$ changes by $2 n \\pi$). \nThis charge quantization is because generically these cosine terms will select a unique ground state with a specific value of $\\varphi$ (say $\\varphi=\\varphi_0$). Thus, for a soliton, in order to ensure that the system is at the ground state away from the soliton, $\\varphi$ must change by $2\\pi$ times an integer (e.g. from $\\varphi_0$ to $\\varphi_0+ 2 n \\pi$). This leads to the quantization of the soliton charge. Here, we demonstrate one such example by keeping only the $n=1$ term in the elastic energy, where the standard sine-Gordon theory is recovered $E=\\int d v \\left[ (\\partial_v \\varphi)^2+\\gamma \\cos \\varphi\\right]$. Such a theory is known to support soliton excitations with $\\Delta \\varphi =2\\pi$, i.e. charge-$1$ solitons. \nBy combining multiple charge-1 solitons together, solitons with higher integer charge can be created, but solitons with non-integer (fractional or irrational) charge are prohibited.\n\nFor 2D plates that we considered here, however, the situation differs sharply from the generic case discussed above. \nAs shown in App.~\\ref{app:sec:minimal:surface}, the transformation $\\varphi \\to \\varphi+\\pi$ is a chiral symmetry transformation ($\\mathbf{R} \\to -\\mathbf{R}$). \nBecause the elastic energy of a 2D plate is invariant under a chiral transformation, our elastic energy must be invariant under $\\varphi \\to \\varphi+\\pi$ (although the ground states can spontaneously break this symmetry, as we discuss in Sec.~\\ref{SEC:Ribb}), and thus \n$E_c(\\varphi)=E_c(\\varphi+\\pi)$, in contrast to a generic compact sine-Gordon theory where $E_c(\\varphi)=E_c(\\varphi+2\\pi)$. This means that with the help of the \nchiral symmetry, our system always contains two degenerate ground states (with $\\varphi=\\varphi_0$ and $\\varphi=\\varphi_0+\\pi$), and thus one can introduce\nhalf-charged solitons, which is the domain wall between these two ground states. Because the values of $\\varphi$ for these two ground states differ by $\\pi$, such a soliton \nmust have $\\Delta \\varphi =(2n+1)\\pi$, and thus a fractional charge $n+1\/2$. This fractionalization is in analogy to nematic liquid crystals, where the molecules (and the order parameter)\nare invariant under a $\\pi$ rotation. Such a symmetry also results in fractional topological defects in nematic liquid crystals, i.e., disinclinations or disinclination lines, which can be viewed as\nhalf of a vortex or a vortex line~\\cite{degennes1993, Lubensky2000}.\n\nThe configuration of these fractional solitons can be obtained using the elastic energy. Because the chiral symmetry requires $E_c(\\varphi)=E_c(\\varphi+\\pi)$, \n odd-integer cosine terms in Eq.~\\eqref{eq:sine_gordon_general} are prohibited and thus we have\n\\begin{align}\nE=\\int d v \\left[ (\\partial_v \\varphi)^2+ \\sum_n \\gamma_{2n} \\cos  2 n (\\varphi-\\varphi_{2n})\\right] .\n\\label{eq:sine_gordon_general_chiral}\n\\end{align}\nHere, we demonstrate a fractional soliton by considering the minimal model, which only contains the $\\cos  2 \\varphi$ term\n\\begin{align}\nE=\\int d v \\left[ (\\partial_v \\varphi)^2+\\gamma \\cos 2 \\varphi\\right] .\n\\label{eq:sine_gordon}\n\\end{align}\nThis is similar to  the standard sine-Gordon theory, but with an extra factor of $2$ in the cosine term.  For $\\gamma>0$, the system has two ground states $\\varphi=\\pm \\pi\/2$,  while for $\\gamma<0$, the two ground states are $\\varphi=0$ or $\\pi$. Here, we demonstrate the physics by focusing on the case of $\\gamma>0$. \nFollowing the standard sine-Gordon theory approach,  soliton solutions are expected. Because of the extra factor of $2$ in the cosine term, $\\varphi$ changes by $\\pi$ for such a soliton, instead of $2\\pi$, and thus it contains half charge. Here, the soliton is the domain boundary between the left- and right- handed helicoid ground states. \n\nThis physic picture of fractional excitations can be generalized.  As mention above, for a system whose low-energy \nphysics is described by a compact sine-Gordon theory,  generically, soliton charge is quantized to integer values. However,\nif the Hamiltonian is invariant under the $Z_N$ transformation $\\varphi\\to \\varphi +2\\pi\/N$  ($N$ is an integer),\nthis $Z_N$ symmetry will change the charge quantization from integer to fractional values (integer times $1\/N$). A minimal-surface 2D plate offers such an example with $N=2$, and the $Z_N$ symmetry here is the $Z_2$ \nchiral symmetry $\\varphi\\to \\varphi +\\pi$.\n\n\n\n\\subsection{Numerical verification}\\label{SEC:Nume}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=.2\\columnwidth]{numericdomain1.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.5\\columnwidth]{numericdomain2.pdf}}\n\t\\caption{Soliton configuration from finite-element analysis. By minimizing the elastic energy, here we show the soliton configuration for ribbons with (a)  helicoid or (b) catenoid ground states.}\n\t\\label{fig:numerical}\n\\end{figure}\n\nThese fractional solitons indeed exist in narrow elastic ribbons. \nIn Fig.~\\ref{fig:numerical}, we simulate a narrow ribbon with $E=E_s + E_b$ and a small perturbation is added to lift the infinite degeneracy of the ground states.\nIn this simulation, 10-node triangular elements are utilized. The shape function of such an element preserves the three-fold rotational symmetry, which help minimizing \nanisotropy induced by the shape function. The entire ribbon is composed of $60\\times 4$ nodes. The elastic moduli are set to (in arbitrary units) \n$h B_0 = 6 \\times 10^5$ and $G_0 =B_0\/2$  [Eq.~\\eqref{eq:Es_2D}]. The bending stiffness \n$D_1=\\frac{2 h^3 G (3B+G)}{3(3B+4G)}$ is set to $ 2.4 \\times 10^3$ [Eqs.~\\eqref{eq:Eb} and~\\eqref{eq:Eb:app}].\nWe also added a small perturbation to favor the helicoid (or catenoid) ground states [as shown in Eq.~\\eqref{eq:deltaEb:app}], \nwhose coefficient $\\delta D=0.02 D_1$ in Fig.~\\ref{fig:numerical}.(a) and $-0.01D_1$ in Fig.~\\ref{fig:numerical}(b).\nAs shown in App.~\\ref{app:sec:ribbons}, $g_0$ can always be written in the form of \nEqs.~\\eqref{app:g0_general} and~\\eqref{app:g22_expansion} and here we choose $a_1=0$ and $a_2= 2\\pi\/10$ [Eq.~\\eqref{app:g22_expansion}].\nAll qualitative features that we observed are insensitive to microscopic details and remain stable as we vary the control parameters and the system size.\nThe simulation didn't enforce the excluded-volume condition, and thus the ribbon may intersect with itself. \nEnforcing excluded volume or not doesn't change any qualitative conclusions.\n\nFrom the finite-element analysis, we found that a fractional soliton is indeed a local energy minimum. \nIn particular, for the helicoid ground states, \nby minimizing the elastic energy, we find that such a domain structure\nalways bends the ribbon by nearly $180^\\circ$, i.e. if we move along the direction of the helicoid ribbon, each soliton excitation implies a sharp U-turn. The reason for this sharp turn can be understood by looking at a section of catenoid and trying to connect it with two helicoids with opposite chirality to its two ends. As shown in Fig.~\\ref{fig:associate:family}, if one tries to smoothly connect them (by slowly varying $\\varphi$ along the chain), a $180^\\circ$ degree turn shall naturally arise. \nThis sharp U-turn associated with such a fractional soliton is the key for the formation of kinks in tangled phone-cords.\n\n\\section{Majumdar-Ghosh model and Z$_2$ topological spin liquid}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\columnwidth]{sl.pdf}\n\t\\caption{Fractional solitons in dimerized spin chains. (a-d) and narrow ribbons (e-h). The quantum spin system has two degenerate ground states (a) and (b). A spin-1 excitation\n\tcan be created via local perturbations (c), which splits into two deconfined spin-1\/2 excitations (d). (e) and (f) show two degenerate ground states of a helicoid ribbon. \n\t(g) shows a local excitation, which decays into a pair of fractional solitons in (f).}\n\t\\label{fig:spin:chain}\n\\end{figure}\n\nIt turns out that the elastic system we discuss here and its fractional soliton excitations show strong analogy to fractional excitations in $Z_2$ spin liquids and other similar quantum dimer systems.  Thus we can utilize insights obtained from the study of fractional excitations in quantum systems to help understanding fractional excitations in 2D plates.\nIn this section, we provide a brief review about some key properties of 2D $Z_2$ spin liquids and 1D quantum dimer systems, which will be compared with 2D plates in the\nnext section.\n\nA $Z_2$ spin liquid in 2D is one of the most important and well-studied fractional topological states, which exhibit exotic properties such as $Z_2$ topological order,  long-range entanglement, fractional excitations and topological degeneracy (see for example a recent review Ref.~\\onlinecite{Wen2017} and references therein).\nThe study of $Z_2$ spin liquids originates from Anderson's resonating-valence-bond (RVB) senario~\\cite{Anderson1973, Fazekas1974} in frustrated quantum spin systems and quantum dimer models~\\cite{Kivelson1987, Rokhsar1988, Moessner2001}. \nThis exotic quantum phase of matter is characterized by a topological Ising gauge theory and gives rise to deconfined fractional excitations, e.g. spinons which carry spin-$1\/2$ \nbut no charge~\\cite{Read1991,Wen1991,Mudry1994, Senthil2000, Moessner2001b}. Later, an exactly sovable model with the same topological order was introduced, known as the toric code model of Kitaev~\\cite{Kitaev2003}.\n\n\\begin{figure}[t]\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid1.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid1.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid2.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid2.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid3.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid3.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid4.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid4.pdf}}\n\t\\caption{Fractional excitations and topological degeneracy. The left column demonstrates a quantum $Z_2$ spin liquid (e.g. a RVB spin liquid) defined on an annulus, which has two degenerate ground states (a) and (g) due to topological degeneracy. Via local perturbation, a spin-1 excitation can be introduced to the first ground state, which can split into two spin-1\/2 fractional excitations as show in Fig.~(c). If these two fractional excitations are moved around the annuls (e) and then annihilated with each other (g), the system turns into the other ground state, different from the original one where we start from. A catenoid with the same geometry setup shows the same property as show in the second column. Here, we also have two degenerate ground states, which correspond to swap the two sides of the 2D manifold. One can create two charge-1\/2 solitons (d) and move them around the catenoid (f) before annihilate them. This procedure also flips a ground state into the other one.}\n\t\\label{fig:spin:liquid}\n\\end{figure}\n\n\nHere, instead of providing a comprehensive review about spin liquids in 2D, we examine a highly-simplified 1D version of dimer states, which exhibits a lot of the interesting ingredients of 2D spin liquids. Consider the one-dimensional Majumdar-Ghosh model~\\cite{Majumdar1969},\nwhich studies a 1D chain of spin-$1\/2$ Heisenberg spins with frustrated nearest and next-nearest-neighbor anti-ferromagnetic couplings. This model has two degenerate ground states as shown in Fig.~\\ref{fig:spin:chain}(a) and (b), where neighboring spins form singlet pairs and each spin can only participate in the formation of one spin singlet with one of its two neighbors. Such a singlet pair is called a ``dimer'' and the ground states of the 1D Majumdar-Ghosh model are called dimer states. This 1D chain is in strong analogy to\n$Z_2$ spin liquids in 2D, whose ground states are also composed of dimerized  singlet pairs.\nIn the 1D Majumdar-Ghosh model, the two-fold ground-state degeneracy is due to spontaneous breaking of the lattice translational symmetry. \nFor a $Z_2$ spin liquid defined on a 2D annulus  [Fig.~\\ref{fig:spin:liquid} (a) and (g)], although the ground state breaks no symmetry, a similar \ntwo-fold degeneracy is expected due to topological reasons, which is known as topological degeneracy (see e.g. Ref.~\\onlinecite{Wen2017} and references therein). In contrast to degeneracy from spontaneous symmetry breaking,\ntopological degeneracy implies that the number of degenerate ground states varies  according to the topology of \nthe underlying manifold of the system in real space. A $Z_2$ spin liquid has $2^{n_L}$ ground states, where $n_L$ counts the number of \nindependent non-contractible loops of the underlying manifold, e.g. a sphere or a disk has $n_L=0$. An annulus has $n_L=1$, while a torus or a double torus has $n_L=2$ or $n_L=3$ respectively. In this manuscript, to compare with narrow ribbons,\nwe focus on spin liquid defined on an annulus, with $n_L=1$ and thus $2$ degenerate ground states.\n\nNow, we consider excitations in the Majumdar-Ghosh model. The obvious excitation in a dimerized ground state is to break a dimer, i.e. transfer a singlet into a triplet. Such a triplet excitation carries spin-1, which is a local excitation and can be introduced via a local perturbation. In the Majumdar-Ghosh model, such a local excitations can fractionalize into two spin-1\/2 fractional excitations, as shown in Fig~\\ref{fig:spin:chain} (c) and (d). Similar fractional excitations arise in 2D spin liquids. In both 1D and 2D, such a fractional excitation cannot be directly created in the bulk. Instead, they need to be created in pairs. More importantly, although each spin-1\/2 excitation may look like an individual particle, fractional excitations in each pair are connected by a ``string'', which distinguish them form ordinary local excitations.\n\nIn a conventional material, the energy cost increases rapidly as one tries to split an integer-charged excitation into two parts and separate them away from each other.  \nIn a $Z_2$ spin liquid or a 1D Majumdar-Ghosh chain, however, the energy cost for separating two fractional excitations saturate quickly as their separation gets large and thus the attraction between the two fractional excitations decreases to zero as an exponential function of the separation. Therefore, once far apart, such a fractional excitation behaves just like a point particle, and is called a deconfined fractional excitation.\nIn addition to $Z_2$ spin liquids, this deconfinement mechanism is applicable to a wide variety of fractional excitations as well, such as fractional and non-abelian particles in fractional quantum Hall systems~\\cite{Nayak2008}.\n\nIn topological states such as a Z$_2$ spin liquid, there exists one important connection between topological degeneracy and fractional excitations, known as braiding. \nThis terminology usually refers to move a fractional particle around another one in 2D~\\cite{Nayak2008}. However, similar phenomena often arise as long as \nthe trajectory of a fractional particle forms a non-contractible loop, i.e. a closed loop which cannot smoothly shrink into a point, not necessarily due to the existence of another particle. Thus, in this paper, we will use this terminology loosely to refer to any non-contractible loops.\nImagine that we create a set of fractional excitations via certain local perturbation in a fractional topological state (e.g. a $Z_2$ spin \nliquid or a fractional quantum Hall system), and then adiabatically move their locations in a non-contractible way, i.e., the path of certain fractional excitation form a non-contractible closed loop. Afterwards, these fractional excitations are annihilated with one another and thus the system goes back to the ground state. Although this procedure starts from a ground states and ends also as a ground state, the initial and final states may be two distinct quantum states orthogonal to each other. \nOne such example in a $Z_2$ spin liquid is demonstrated in Fig.~\\ref{fig:spin:liquid}(a) (c) (e) and (g). \nAs mentioned above, on a 2D annulus, this system has two degenerate ground states due to topological degeneracy. \nBy creating a pair of fractional excitations and moving them around the annulus, the system is transformed from one ground state\nto the other. In other words, braiding offers a pathway for topologically degenerate ground states to evolve into each other. This phenomenon plays a crucial role in topological quantum computing. If we consider two degenerate ground states as a quantum two-level system, these two states \nforms a q-bit and braiding serves as logical gates that flips and control this q-bit (For more details, see a review article Ref.~\\onlinecite{Nayak2008} and references therein). \nFor a Z$_2$ spin liquid, this braiding relation makes it possible to use this topological states as a topological quantum memory, where information stored in such a memory is robust against any local perturbations or quantum decoherence. The same principle can also be used for topological quantum computing. However, for such an objective, topological states with more complicated fractional particles and braiding algebra are needed, such as Majorana or Fibonacci anyons, where the later one can even achieve \nuniversal quantum computation~\\cite{Freedman2002}.\n\nThis relation between degeneracy and fractional excitations also arises in 1D dimer states, and in Fig.~\\ref{fig:spin:chain} (c) and (d), we can already see that moving\nfractional particles flip the ground state (from blue to yellow).\n\nFinally, it needs to be highlighted that their exists a deep connection between fractional excitations in a quantum spin chain and those in a minimal-surface narrow ribbon, \nif one realizes that low-energy physics in both systems are described the compact sine-Gordon theory. \nIn the study of spin chains, one well-known and very powerful mathematical tool is the Luttinger liquid approach~\\cite{Emery1979, \nAffleck1989, Francesco2012,Fradkin2013}. As shown by Haldane~\\cite{Haldane1982}, in this approach, \nthe low-energy effective theory of a dimerized spin chain is a compact sine-Gordon field theory, same as these ribbon systems. \nRemarkably, in the same work, Haldane also pointed out that spin-1\/2 and spin-1 excitations there indeed correspond \nto charge-1\/2 and charge-1 solitons respectively.\nThis analogy is the fundamental reason why these elastic ribbons share common properties with exotic quantum states formed by frustrated quantum spins.\n\n\n\\section{Fractional excitations in narrow ribbons}\\label{SEC:FracRibb}\nIn this section, we discuss physical properties of fractional excitations in minimal-surface narrow ribbons.\n\n\\subsection{Integer and fractional solitons}\nWe first consider a minimal-surface narrow ribbon and assume that higher order terms in the elastic energy  favor the helicoid ground state.\nAs discussed above, key properties of such a ribbon can be characterized by the phenomenological theory of Eq.~\\eqref{eq:sine_gordon} with $\\gamma>0$. Most of our\nconclusions can be easily generalized to other cases, e.g. for $\\gamma<0$, where the ground state is a catenoid.\n\nAs shown in Fig.~\\ref{fig:spin:chain} (e) and (f), this system has two degenerate ground states, i.e. left- or right- handed helicoids.\nNow, we introduce soliton excitations to one of the ground states (e.g. right-handed) as shown in Fig.~\\ref{fig:spin:chain} (g) and (h). \nWith local deformations, only integer charged solitons can be introduced [Fig.~\\ref{fig:spin:chain} (g)], which can fractionalize into two 1\/2-charge fractional solitons\n[Fig.~\\ref{fig:spin:chain} (h)]. It is worthwhile to point out that for better visualization, we keep these helicoids straight even after solitons are introduced, whose \ntrue lowest-energy configuration should involve a kink as shown in Fig.~\\ref{fig:numerical}.  \nSame as the dimer chain [Fig.~\\ref{fig:spin:chain} (a-d)], moving a pair of fractional solitons here flips one ground state into the other one.\nOnce created and becomes deconfined, a fractional soliton can no longer be removed by any local adjustment. Instead, it can only be neutralized by globally adjust \nthe entire chain or by annihilating with an anti-soliton.\n\nIn Fig.~\\ref{fig:catenoid:solitons}, we consider a ribbon with catenoid ground states ($\\gamma<0$) and assume that the catenoid form a closed loop as shown in the figure. \nInterestingly, creating a half-integer-charge fractional soliton must sacrifice the orientability, turning the orientable cylinder-like structure into an non-orientable surface.\nwhich offers another hint about the fractional nature of these solitons.\n\n\n\\begin{figure}[t]\n\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-no-sol.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-half-sol.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-one-sol.pdf}}\n\t\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-threehalf-sol.pdf}}\n\t\\caption{Integer and fractional excitations in a catenoid. (a) shows a catenoid ground state without soliton excitations. (b-d) contain solitons. In contrast to an integer charged soliton (c), which preserves the orientability, a half-integer charged soliton makes the manifold non-orientable as shown in (b) and (d).}\n\t\\label{fig:catenoid:solitons}\n\\end{figure}\n\n\n\\subsection{Two unique features}\n\\label{sec:sub:two:features}\nFractional solitons in a narrow helical ribbon have two unique features, which are not generally expected for most other fractional excitations.\nFirs of all, these solitons are \\emph{holographic}, which means that if there is only one charge-1\/2 soliton in a helicoid, we can pin-point and control its location via \ncontrolling the two ends of the helicoid.\nThis is because the soliton here is the domain boundary between left- and right- handed sections. For a helicoid with length $L$, we define the left-handed section length to be $x$, and thus the right-handed section shall have length $L-x$. For simplicity, here we ignore the width of the soliton, which can be included easily without alternating any main conclusions. For a helicoid, we can define the total helicity of the entire ribbon as $(L-2x)\/\\lambda$ where $\\lambda$ is the pitch of the helicoid. This quantity describes how many times the ribbon twists,\nwith right-handed twists defined as positive. It is easy to notice that this quantity directly connects helicity with the location of the fraction soliton $x$. For an elastic ribbon, the helicity can be adjust by twisting the two ends of the ribbon in the opposite direction and each $2\\pi$ twist increases\/decreases the helicity by $1$, which moves the fractional soliton by one \npitch $\\lambda$. This holographic control is not a general property of fractional solitons, but a special feature for solitons in helicoids. \nIn addition to control fractional solitons, the holographic property also provide a natural way to generate these fractional excitations. If we twist the two ends of a helicoid such that \nthe helicity decreases from the ground state value ($L\/\\lambda$), this process will eventually create a fractional soliton (i.e. a non-zero $x$) to reduce energy. As will be discussed \nin the next section, this is the mechanism how solitons in telephone cords is generated.\n\n\nThe second feature of these fractional solitons is that they kink the helical ribbon by turning its direction by almost $180^\\circ$ as discussed in Sec.~\\ref{SEC:Nume}. Again, this is a special feature, not generally expected for fractional excitations.\nFor a telephone cord, this is the reason why solitons  tangle up the cord.\n\n\\subsection{Braiding}\nIn this section we demonstrate how braiding changes the ground state in narrow ribbons by considering a catenoid ground state $g<0$. As mentioned early on, here\nwe are using this terminology loosely, which includes any non-contractable loop trajectory.\n\nHere, we consider one orientable cylinder-like catenoid [Fig.~\\ref{fig:spin:liquid}(b)] and introduce a pair of fractional solitons by locally flipping a section of \nthe the catenoid inside out [Fig.~\\ref{fig:spin:liquid}(d)]. Then, we gradually move the pair of fractional solitons around the catenoid [Fig.~\\ref{fig:spin:liquid}(f)], where they meet\nagain and annihilate with each other. It is easy to realize that such a procedure changes the ground state, in analogy to the same procedure in a $Z_2$ spin liquid. \n\nIt is worthwhile to mention that here, as we annihilate two solitons, large dissipation is assumed, such that a system can quickly release its elastic energy and dissipate from an excited state (with a pair of solitons) to  the ground state (without solitons). In the absence of dissipation, the system will remain in excited states and thus solitons cannot be annihilated, even if a pair of solitons  collide. In such a scenario, the solitons will pass through each other, which is in fact part of the definition of solitons. \n\n\n\n\n\\section{discussion}\nNow we come back to telephone cords. \nA helical telephone cord is not a thin 2D helicoid ribbon, but we can consider it as a stack of 2D helicoid ribbons and thus they are expected to share similar qualitative features\nas our minimal-surface ribbons.  The insight we gain from the discussions above on fractional solitions provide us with answers to the three questions we raised in the Introduction.\n\nFirst, when we use a telephone headset, we often unintentionally twist\/rotate the headset before we put it back. As shown in Sec.~\\ref{sec:sub:two:features}, \ndue to the holographic property, this type of twisting introduces fractional solitons.  This is the origin, from which these solitons emerge in  phone cords.\nSecond, in Sec.~\\ref{sec:sub:two:features}, we showed that such a soliton bend the ribbon, which is why they result in kinks in phone cords and turn them into a tangled mess. \nThird, because these fractional excitations are topological defects, they cannot be removed by any local deformations, and this is the reason why these annoying kinks are \nso hard to get rid off.\nWith the origin of the kinks understood, we can use this knowledge as guidance to remove\/avoid these kinks. By avoiding rotating the headset, these kinks would not be \ncreated. For telephone cords that already have these kinks, we just need to twist the headset, which moves the solitons, making them annihilate with each other or driving them out\nof the cord.\n\nSimilar types of kinks\/solitons between domains of helical structures with opposite handedness have been observed previously in various situations such as perversion of tendrils on climbing plants~\\cite{Goriely1998}, intrinsically curved rods~\\cite{domokos2005multiple}, self-assembled structures of Janus colloidal particles~\\cite{Chen2011supra},\nelastic bi-strips~\\cite{liu2014structural}, helical strings~\\cite{Nisoli2015}, and minimal surface liquid films~\\cite{Machon2016}.  In this paper we show how this type of solitons arises  in narrow elastic ribbons and are fractional excitations flipping the ribbon between ground states that belong to the same minimal surface associate family.  We further demonstrate that they share the same topological description as fractional excitations in $Z_2$ spin liquids.\n\nThe fractional soliton excitations we discussed in the helicoid-catenoid family are protected by the chiral symmetry of the elastic theory.  Interestingly, when this symmetry is weakly broken, e.g., the left- and right- handed helicoids have slightly different elastic energy but both remain local energy minima, the soliton propagation between these two domains will have a preferred direction, such that it flips the higher energy domain into the lower energy domain.  Interestingly, because this soliton is holographic, its propagation requires the rotation of the domains, the sizes of which change as the soliton propagates.  \nThis results in a locked speed for the soliton propagation, as the released elastic energy from the domain-energy difference \n  turns into kinetic energy and fuels the rotation of the domain (the size of which increases).  This mechanism of locked soliton speed was first discussed in the context of solitons in helical strings, in Ref.~\\onlinecite{Nisoli2015}.  In contrast, if the chiral symmetry is preserved and the two domains have exactly the same elastic energy, the speed of the soliton changes at it propagates, because the domain size changes but the total energy is conserved.  This phenomenon of locked soliton speed may have broad applications as a constant actuation\/propulsion mechanism without the need of fine-tuning.\n  \n\nTopological solitons have also been found in discrete 1D rotor chains, which are  Maxwell lattices with exactly one floppy mode under open boundary conditions~\\cite{Chen2014}.  Interestingly, solitons in these 1D chains exhibit a kink-antikink asymmetry~\\cite{Zhou2017}.  \nIf we follow the same soliton charge defined above, this asymmetry implies the breaking of the charge conjugation symmetry for such solitons, i.e., positively and negatively charged solitons are no longer equivalent,  in sharp contrast to conventional solitons. \nWhether similar type of excitations can arise in continuum would be an interesting question for future studies.\n\nIn addition to narrow ribbons, similar physics of associate family arises in any minimal-surface 2D plates and their low-energy properties share the same sine-Gordon\ndescription, from which fractional excitations can also arise. Generalizing this knowledge about fractional excitations to other 2D plates will be an interesting subject for future studies,\nfrom which a universal understanding about fractional excitations in systems at the verge of mechanical stability may eventually emerge. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the most important tasks in high energy physics in the next decades\nwill be to determine if Higgs boson(s) exist, and to measure the\nproperties of any that are found. Only then will a full understanding\nof the Higgs sector and its role in the complete underlying\ntheory be possible.\nHere we outline the very substantial\nand, in some areas, unmatched capabilities of a muon collider \nfor these tasks.~\\cite{bbgh} We restrict our discussions\nto the Standard Model (SM) with a single neutral \nHiggs boson, $h_{SM}$, and to the minimal\nsupersymmetric extension of the SM (MSSM) with a light SM-like\nHiggs boson, $h^0$,\nand two heavier neutral Higgs scalars, $H^0$ and $A^0$.\n\nAlthough muon colliders have only received substantial attention in\nthe last few years, so far it seems~\\cite{muonreports} that there are\n``no barriers'' to either of the following two designs:\n\\begin{itemize}\n\\item $\\sqrt s\\alt 500~{\\rm GeV}$, \n${\\cal L}\\sim {\\rm few} \\times 10^{33}$cm$^{-2}$s$^{-1}$,\n$L_{\\rm tot}=50~{\\rm fb}^{-1}$\/yr;\n\\item $\\sqrt s\\alt 4~{\\rm TeV}$, ${\\cal L}\\sim {\\rm few} \\times 10^{35}$\ncm$^{-2}$s$^{-1}$, $L_{\\rm tot}=200~{\\rm fb}^{-1}$\/yr.\n\\end{itemize}\nA $\\mu^+\\mu^-$ collider has some natural advantages \nas compared to an $e^+e^-$ collider, including some\nthat are crucial for Higgs boson physics:\n\\begin{itemize}\n\\item there is essentially no beamstrahlung; \n\\item there is substantially reduced bremsstrahlung;~\\footnote{Still,\nbremsstrahlung depletes the central Gaussian peak in $\\sqrt s$\nby $\\sim 40\\%$ --- this effect is included in the computations.~\\cite{bbgh}}\n\\item there is no final focus problem (storage rings are used to build up\nthe effective instantaneous luminosity);\n\\item as required for detailed Higgs studies in the $s$-channel,\nexcellent beam energy resolution of $R=0.01\\%$ and the ability \nto set $E_{\\rm beam}$ to 1 part in $10^6$ using spin\nprecession techniques are both possible if the necessary technology is\nbuilt into the machine;\n\\item $\\sqrt s_{\\rm max}> 500~{\\rm GeV}$ can probably be reached more easily; and\n\\item there is hope that the cost might be lower.\n\\end{itemize}\nThe negatives regarding a muon collider include:\n\\begin{itemize}\n\\item the design is immature, and five years of research and development\nprojects are needed before a full-fledged proposal would be possible ---\nin particular, cooling tests are required to see if multistage \ncooling will be sufficiently efficient;\n\\item the exact nature of the detector backgrounds, and how to manage them,\nis still under investigation --- certainly the detector will be more\nexpensive due to higher shielding and segmentation requirements;\n\\item significant polarization probably implies significant loss in ${\\cal L}$\n --- for $s$-channel Higgs production, \n$S\/\\sqrt B\\propto \\sqrt{L_{\\rm tot}}\\sqrt{(1+P^2)^2\/(1-P^2)}$\nimplies that\n$[P=0,L_{\\rm tot}]$ is equivalent to $[P=0.8,L_{\\rm tot}\/10]$;\n\\item it is not possible to have a $\\gamma\\gam$ collider facility.\n\\end{itemize}\n\n\n\\begin{figure}[h]\n\\leavevmode\n\\begin{center}\n\\centerline{\\psfig{file=schannelhiggsdiagram.ps,width=5cm}}\n\\end{center}\n\\fcaption{Feynman diagram for $s$-channel production of a Higgs boson.}\n\\label{schanfig}\n\\end{figure}\n\nA muon collider can explore Higgs physics in two modes. \nAway from the $s$-channel Higgs pole, {\\it e.g.}\\ at\nthe maximum available energy $\\sqrt s_{\\rm max}$, a $\\mu^+\\mu^-$ collider\ncan discovery and study Higgs bosons in exactly \nthe same ways as an $e^+e^-$ collider\nwith the same $\\sqrt s$ and ${\\cal L}$ (barring\nunmanageable detector backgrounds at the muon collider).\nOn the $s$-channel pole at $\\sqrt s\\sim m_{\\h}$, \n$\\mu^+\\mu^-\\to h$ collisions (see Fig.~\\ref{schanfig}) imply\nunique capabilities for:\n\\begin{itemize}\n\\item precision studies of $(b\\overline bh)^2:(WWh)^2:(ZZh)^2$ coupling-squared\n ratios and of $\\Gamma_{\\h}^{\\rm tot}$ for a SM-like Higgs (either $h_{SM}$ or $h^0$)\nwith $m_{\\h}\\alt 2m_W$; and\n\\item discovery and study\nof the heavier $H^0,A^0$ of the MSSM up to $\\sqrt s_{\\rm max}$.\n\\end{itemize}\nFor $s$-channel Higgs physics,\nthe size of the $s$-channel cross section, $\\overline \\sigma_{\\h}$, is crucial.\nWe obtain $\\overline \\sigma_{\\h}$ \nby convoluting the standard Breit-Wigner shape for the Higgs\nwith a Gaussian distribution of width \n$\\sigma_{\\tiny\\sqrt s}^{}$ centered at $\\sqrt s=m_{\\h}$; $\\overline \\sigma_{\\h}$ is given by \n$\\overline \\sigma_{\\h}\\sim 4\\pim_{\\h}^{-2}BF(h\\to\\mu^+\\mu^-)$ if $\\sigma_{\\tiny\\sqrt s}^{}\\ll\\Gamma_{\\h}^{\\rm tot}$\nand by\n$\\overline \\sigma_{\\h}\\sim2\\pi^2m_{\\h}^{-2}\\Gamma(h\\to\\mu^+\\mu^-)\/(\\sqrt{2\\pi}\\sigma_{\\tiny\\sqrt s}^{})$\nif $\\sigma_{\\tiny\\sqrt s}^{}\\gg\\Gamma_{\\h}^{\\rm tot}$. To get near maximal $\\overline \\sigma_{\\h}$ and\nto have sensitivity to $\\Gamma_{\\h}^{\\rm tot}$ via scanning in $\\sqrt s$ it\nis important that $\\sigma_{\\tiny\\sqrt s}^{}$ be no larger than $2-3\\times\\Gamma_{\\h}^{\\rm tot}$.\nVery small $\\Gamma_{\\h}^{\\rm tot}$ is not uncommon.\nFig.~\\ref{hwidths} shows that $\\Gamma_{\\h}^{\\rm tot}< 1-10~{\\rm MeV}$ is typical of the $h_{SM}$\nfor $m_{\\hsm}\\alt 140~{\\rm GeV}$ and of {\\it all} \nthe MSSM Higgs bosons if $\\tan\\beta\\alt 2$ and $m_{\\ha}\\alt 2m_W$.\nUsing the parameterization \n$\\sigma_{\\tiny\\sqrt s}^{}\\simeq 7~{\\rm MeV}\\left({R\\over 0.01\\%}\\right)\\left({\\sqrt s\\over \n100~{\\rm GeV}}\\right)$ for $\\sigma_{\\tiny\\sqrt s}^{}$ in terms of the beam energy resolution, $R$,\nwe see that very excellent resolution $R\\sim0.01\\%$ is required\nfor $\\sigma_{\\tiny\\sqrt s}^{}< 2-3\\times\\Gamma_{\\h}^{\\rm tot}$ in the above cases.\n\n\\begin{figure}[h]\n\\leavevmode\n\\begin{center}\n\\centerline{\\psfig{file=hwidths.ps,width=3.25in}}\n\\end{center}\n\\fcaption{\nTotal width versus mass of the SM and MSSM Higgs bosons\nfor $m_t=175~{\\rm GeV}$.\nIn the case of the MSSM, we have plotted results for\n$\\tan\\beta =2$ and 20, taking $m_{\\stop}=1~{\\rm TeV}$ and\nincluding two-loop\/RGE-improved Higgs mass corrections and\nneglecting squark mixing; SUSY decay channels are assumed to be absent.}\n\\label{hwidths}\n\\end{figure}\n\nA final important note is that high ${\\cal L}$ is\nneeded at all $\\sqrt s$ values where a Higgs boson \nis discovered or might exist.  This will possibly imply that several\nfinal storage rings (which fortunately are relatively cheap),\ndesigned to maintain near-optimal ${\\cal L}$ over a span of $\\sqrt s$ values, \nwill need to be constructed\n\n\n\\section{{\\boldmath $s$}-channel studies of a Standard-Model-like Higgs boson}\n\nMost probably, one would first\ndiscover the $h$ at the LHC (e.g. in the $h\\to\\gamma\\gam$ or $h\\to 4\\ell$\ndiscovery modes), \nor at the NLC  (e.g. in the $Z^\\star\\to Zh$ production mode)\nand then set up the muon collider for running at $\\sqrt s\\simm_{\\h}$. \nAt the LHC or NLC, $m_{\\h}$ will be quite well determined. \nFor $m_{\\h}\\alt 2m_W$ one finds:~\\cite{snowmass96}\n$\\Deltam_{\\h}\\sim 100~{\\rm MeV}\\left({600~{\\rm fb}^{-1}\\over L}\\right)^{1\/2}$ at the LHC, \nusing $h\\to\\gamma\\gam,4\\ell$ decays and reconstructing the\nresonance peak in $m_{\\gamma\\gam},m_{4\\ell}$;\n$\\Deltam_{\\h}\\sim 100~{\\rm MeV}\\left({50~{\\rm fb}^{-1}\\over L}\\right)^{1\/2}$ for NLC $\\sqrt s=500~{\\rm GeV}$\nrunning and reconstructing the $m_{b\\overline b}$ resonance peak\nin the $Zh$ ($h\\to b\\overline b$) mode; and $\\Deltam_{\\h}< 100~{\\rm MeV}$ at the\nNLC, assuming $L=50~{\\rm fb}^{-1}$ devoted to a $Zh$ threshold study\nat $\\sqrt s=m_Z+m_{\\h}+0.5~{\\rm GeV}$. If there is no LHC or NLC, one would accumulate\n$L\\sim 1~{\\rm fb}^{-1}$ at $\\sqrt s=500~{\\rm GeV}$ at the muon collider\nin order to observe $\\mu^+\\mu^-\\to Zh$ and\ndetermine $m_{\\h}$ to within $\\Deltam_{\\h}\\sim 1~{\\rm GeV}$, and then turn to $\\sqrt s\\simm_{\\h}$\nrunning.\n\nFor a SM-like Higgs with $m_{\\h}\\agt 2m_W$, \n$\\Gamma_{\\h}^{\\rm tot}$ is large (as a result\nof $h\\to WW,ZZ$ decays), see Fig.~\\ref{hwidths}, \n$\\overline \\sigma_{\\h}\\proptoBF(h\\to\\mu^+\\mu^-)$ is tiny,\nand $\\mu^+\\mu^-\\toh$ will not be useful.~\\cite{bbgh}\nBut, if $m_{\\h}\\alt 2m_W$ then $\\Gamma_{\\h}^{\\rm tot}$ is very small,\n$\\sigma_{\\tiny\\sqrt s}^{}\\sim 2-3\\times\\Gamma_{\\h}^{\\rm tot}$ is typical\nfor $R=0.01\\%$, $\\overline \\sigma_{\\h}\\propto\\Gamma(h\\to\\mu^+\\mu^-)\/\\sigma_{\\tiny\\sqrt s}^{}$ \nwill  be much larger,  and the $\\mu^+\\mu^-$ collider\nbecomes a Higgs factory. There are several reasons to suppose\nthat the SM-like Higgs will have mass $\\alt 2m_W$:\na) precision electroweak measurements currently\nfavor (but only weakly) a relatively light SM-like Higgs boson;\nb) light Higgs masses are preferred in the SM \nif the SM couplings are required to remain perturbative up to the GUT scale;\nand c) in the minimal supersymmetric model (MSSM), the light SM-like Higgs has \n$m_{\\hl}\\alt 130~{\\rm GeV}$.~\\cite{dpfreport}\n\nIf a SM-like $h$ with $m_{\\h}\\leq 2m_W$ has been discovered, \nthe FMC final ring would\nbe optimized for $\\sqrt s\\sim m_{\\h}$ and a scan of the region\nof size $2\\Deltam_{\\h}$ in the vicinity of the known $m_{\\h}$ \nwould be performed in order to pin down $m_{\\h}$ more\nprecisely (roughly within $\\pm \\sigma_{\\tiny\\sqrt s}^{}\/2$).~\\footnote{Details of\nthe strategy\nfor this scan~\\cite{bbgh} will not be discussed here;\nthe scan is most efficient for small $R$ (as desirable in any case)\nin which case $\\sqrt s$ must be reset rapidly with high precision.}~\nIn the  ``typical case'' of $m_{\\h}\\sim 110~{\\rm GeV}$, $\\sigma_{\\tiny\\sqrt s}^{}\\sim 8~{\\rm MeV}$\n(for $R\\sim 0.01\\%$) and\n$2\\Deltam_{\\h}\\sim 200~{\\rm MeV}$ (see above), we would require $\\agt 25$ scan points\nwith $\\sim 0.01~{\\rm fb}^{-1}$ per scan point (the luminosity for a $5\\sigma$\nsignal if $\\sqrt s=m_{\\h}$), corresponding to a total luminosity\nof $L_{\\rm tot}\\sim 0.25~{\\rm fb}^{-1}$.\n(One must reset $\\sqrt s$ with $\\Delta\\sqrt s<\\sigma_{\\tiny\\sqrt s}^{}$ hourly.)\nCentering on $\\sqrt s\\simeqm_{\\h}$ does not require much luminosity      \nfor such Higgs masses. The worst mass is $m_{\\h}\\simeqm_Z$.\nFor the $R=0.01\\%$ value of $\\sigma_{\\tiny\\sqrt s}^{}\\sim 6.5~{\\rm MeV}$,\n$L_{\\rm tot}\\sim 25~{\\rm fb}^{-1}$ would be needed to center on $\\sqrt s\\simeq m_{\\h}$\nby scanning. A smaller $\\Deltam_{\\h}$, such as could be obtained\nusing ``super'' tracking or other improvements\nat the NLC,~\\cite{snowmass96} would be very helpful for avoiding wasting\nluminosity on the  centering process in this case.\n\nOnce we have determined $m_{\\h}$ to within $\\sim \\pm\\sigma_{\\tiny\\sqrt s}^{}\/2$, we would \nperform a three point scan of the Higgs resonance peak employing\nluminosity distributed as follows:\n$L_1$ at $\\sqrt s\\simeqm_{\\h}$; $2.5L_1$ at\n$\\sqrt s\\simeq m_{\\h}+2\\sigma_{\\tiny\\sqrt s}^{}$; $2.5L_1$ at\n$\\sqrt s\\simeq m_{\\h}-2\\sigma_{\\tiny\\sqrt s}^{}$.\nThe total luminosity for the scan is then $L=6L_1$.  \nThis scan would simultaneously yield measurements of $\\Gamma_{\\h}^{\\rm tot}$\nand of the $\\mu^+\\mu^-\\toh\\to b\\overline b$, $WW^\\star$ and\n$ZZ^\\star$ rates.  Assuming that $L=200~{\\rm fb}^{-1}$ is devoted over a period\nof several years to this three point scan (presumably beginning with\n$\\sqrt s\\simeqm_{\\h}$), one obtains accuracies for\nthe event rates as given in Table~\\ref{fmcsigbrerrors} and\nerrors for $\\Gamma_{\\h}^{\\rm tot}$ and coupling-squared ratios as given in\nTable~\\ref{fmcerrors}.\n\n\\begin{table}[h]\n\\tcaption{Errors for\n$\\sigma(\\mu^+\\mu^-\\toh_{SM})BF(h_{SM}\\to b\\overline b, WW^\\star, ZZ^\\star)$,\nfor $R=0.01\\%$ and $L_{\\rm scan}=200~{\\rm fb}^{-1}$ \n(equivalent to $L_{\\protect\\sqrt s=m_{\\h}}=50~{\\rm fb}^{-1}$).}\n\\footnotesize\n\\begin{center}\n\\small\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n Quantity & \\multicolumn{5}{c|}{Errors} \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 80} & {\\bf 90} & {\\bf 100} & {\\bf 110} & {\\bf 120} \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( b\\overline b) $ & \n$\\pm 0.2\\%$ & $\\pm 1.6\\%$ & $\\pm 0.4\\%$ & $\\pm 0.3\\%$ & $\\pm 0.3\\%$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( WW^\\star) $ &\n$-$ & $-$ & $\\pm 3.5\\%$ & $\\pm 1.5\\%$ & $\\pm 0.9\\%$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( ZZ^\\star) $ &\n$-$ & $-$ & $-$ & $\\pm 34\\%$ & $\\pm 6.2\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 130} & {\\bf 140} & {\\bf 150} & {\\bf 160} & {\\bf 170} \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( b\\overline b) $ & \n$\\pm 0.3\\%$ & $\\pm 0.5\\%$ & $\\pm 1.1\\%$ & $\\pm 59\\%$ & $-$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( WW^\\star) $ &\n$\\pm 0.7\\%$ & $\\pm 0.5\\%$ & $\\pm 0.5\\%$ & $\\pm 1.1\\%$ & $\\pm 9.4\\%$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( ZZ^\\star) $ &\n$\\pm 2.8\\%$ & $\\pm 2.0\\%$ & $\\pm 2.1\\%$ & $\\pm 22\\%$ & $\\pm 34\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 180} & {\\bf 190} & {\\bf 200} & {\\bf 210} & {\\bf 220} \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( WW^\\star) $ &\n $\\pm 18\\%$ & $\\pm 38\\%$ & $\\pm 58\\%$ & $\\pm 79\\%$ & $-$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( ZZ^\\star) $ &\n$\\pm 25\\%$ & $\\pm 27\\%$ & $\\pm 35\\%$ & $\\pm 45\\%$ & $\\pm 56\\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{fmcsigbrerrors}\n\\end{table}\n\n\\begin{table}[h]\n\\tcaption{Errors for \ncoupling-squared ratios and $\\Gamma_{\\hsm}^{\\rm tot}$ for\n$s$-channel Higgs production at the FMC, assuming $L=6L_1=200~{\\rm fb}^{-1}$\ntotal scan luminosity and $R=0.01\\%$.}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Quantity & \\multicolumn{4}{c|}{Errors} \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 80} & {\\bf $m_Z$} & {\\bf 100} & {\\bf 110} \\\\\n\\hline\n$(W\\wstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & $-$ & $-$ & $\\pm 3.5\\%$ & $\\pm 1.6\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & $-$ & $-$ & $-$ & $\\pm 34\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(W\\wstarh_{SM})^2$ & $-$ & $-$ & $-$ & $\\pm 34\\%$ \\\\\n\\hline\n $\\Gamma_{\\hsm}^{\\rm tot}$ & $\\pm 2.6\\%$ & $\\pm 32\\%$ & $\\pm 8.3\\%$ & \n  $\\pm 4.2\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 120} & {\\bf 130} & {\\bf 140} & {\\bf 150} \\\\\n\\hline\n$(W\\wstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & \n $\\pm 1\\%$ & $\\pm 0.7\\%$ & $\\pm 0.7\\%$ & $\\pm 1\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & \n $\\pm 6\\%$ & $\\pm 3\\%$ & $\\pm 2\\%$ & $\\pm 2\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(W\\wstarh_{SM})^2$ & \n $\\pm 6\\%$ & $\\pm 3\\%$ & $\\pm 2\\%$ & $\\pm 2\\%$ \\\\\n\\hline\n $\\Gamma_{\\hsm}^{\\rm tot}$ & $\\pm 3.6\\%$ & $\\pm 3.6\\%$ & $\\pm 4.1\\%$ &\n  $\\pm 6.5\\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{fmcerrors}\n\\end{table}\nVery good errors in the $c\\overline c$ and $\\tau^+\\tau^-$ channels\nmight also possible, depending on the detector.\nFor example, $c\\overline c$ isolation requires\ntopological tagging in which one must\ndistinguish primary, secondary, and \ntertiary vertices for a $b$-jet vs. primary and\nsecondary only for a $c$-jet. The ability to do so\ndepends on how close to the beam the first layer \nof the vertex detector can be placed.\n\nWhat do the errors of Table~\\ref{fmcerrors} \nmean in terms of our ability to discriminate between the SM $h_{SM}$\nand the MSSM $h^0$? Plots~\\cite{dpfreport,snowmass96}\nof the $WW^\\star\/b\\overline b$ coupling-squared\nratio as computed in the MSSM divided by that computed in the SM\nshow: a) there is almost no dependence of the ratio \nupon the squark mixing scenario; \nb) the ratio is essentially independent of $\\tan\\beta$ in the\nallowed portion of the standard $(m_{\\ha},\\tan\\beta)$ parameter space;\nand c) roughly $-50\\%$ ($-20\\%$) deviations from the SM result\nare predicted for $m_{\\ha}\\sim 250~{\\rm GeV}$ ($m_{\\ha}\\sim 400~{\\rm GeV}$).\nBy the time the muon collider is in operation, theoretical systematic\nerrors (primarily from uncertainty in the running mass, $m_b(m_{\\h})$)\nin the ratio should be in the $\\pm 5\\%$ to $\\pm 10\\%$ range.\nTable~\\ref{fmcerrors} shows that statistical errors will be much smaller\nover most of the relevant $m_{\\h}$ range.  Thus, the $WW^\\star\/b\\overline b$\nevent rate ratio will distinguish between the SM and the MSSM\nat the $\\geq 2\\sigma$ level for $m_{\\ha}\\alt 400~{\\rm GeV}$. For $m_{\\ha}< 400~{\\rm GeV}$,\na rough determination of $m_{\\ha}$ will be possible. (This will be important\nfor the $H^0,A^0$ discussion in the next section.) \nSimilar results apply for the $ZZ^\\star\/b\\overline b$ ratio.  \n\nIn contrast to the event rate ratios, deviations in $\\Gamma_{\\h}^{\\rm tot}$ \nin the MSSM vs. the SM as a function of $(m_{\\ha},\\tan\\beta)$ are very dependent\nupon the squark mixing scenario, the amount of SUSY decays present,\nand so forth; large deviations in $\\Gamma_{\\h}^{\\rm tot}$ \nfrom SM expectations are the rule, but do not pin down either $m_{\\ha}$\nor $\\tan\\beta$. However, if one~\\cite{snowmass96}\nuses the very accurate direct determination\nof $\\Gamma_{\\h}^{\\rm tot}$ from the FMC scan in combination with other measurements\nperformed with $L=200~{\\rm fb}^{-1}$ and $\\sqrt s=500~{\\rm GeV}$ at the NLC~\\footnote{The same\nmeasurements are also possible at an FMC, but we use NLC notation\nin what follows.}~\na large variety of very important coupling-squared magnitudes can be extracted.\nThe ratio of the MSSM prediction to the SM prediction for a squared coupling\nis always very squark-mixing independent and gives new opportunities\nfor determining $m_{\\ha}$.\nAs an example, there are four ways to determine\n$\\Gamma(h\\to\\mu^+\\mu^-)$ by combining NLC and FMC data:~\\footnote{Note\nthat since $\\sigma_{\\tiny\\sqrt s}^{}\\sim 2-3\\times\\Gamma_{\\h}^{\\rm tot}$, \nthe measured $\\mu^+\\mu^-\\toh\\to X$ rate is more or less proportional\nto $\\Gamma(h\\to\\mu^+\\mu^-)\/\\sigma_{\\tiny\\sqrt s}^{}$ so that $\\Gamma(h\\to\\mu^+\\mu^-)$\ncan be computed given the known $\\sigma_{\\tiny\\sqrt s}^{}$; small corrections\nfrom the influence of $\\Gamma_{\\h}^{\\rm tot}$ can be made using the measured $\\Gamma_{\\h}^{\\rm tot}$.}~\n$\\Gamma(h\\to\\mu^+\\mu^-)=$\n\n\\begin{tabular}{ll}\n 1)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nb\\overline b)]_{\\rm FMC}\\over BF(h\\to b\\overline b)_{\\rm NLC}}$;&\n 2)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nWW^\\star)]_{\\rm FMC}\\overBF(h\\to WW^\\star)_{\\rm NLC}}$;\\\\\n 3)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nZZ^\\star)\\Gamma_{\\hsm}^{\\rm tot}]_{\\rm FMC}\\over\\Gamma(h\\to ZZ^\\star)_{\\rm NLC}}$;&\n 4)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nWW^\\star)\\Gamma_{\\hsm}^{\\rm tot}]_{\\rm FMC}\\over\\Gamma(h\\to WW^\\star)_{\\rm NLC}}$.\\\\\n\\end{tabular}\n\n\\noindent\nIn the above, $\\Gamma_{\\h}^{\\rm tot}|_{\\rm FMC}$ refers to the scan measurement at the FMC.\nResulting errors for $h=h_{SM}$ are tabulated in Table~\\ref{nlcfmcerrors},\nlabelled $(\\mupmumh_{SM})^2|_{\\rm NLC+FMC}$.  The predicted\n$(\\mupmumh_{SM})^2|_{\\rm MSSM}\/(\\mupmumh_{SM})^2|_{\\rm SM}$ coupling-squared ratio\nis independent of $\\tan\\beta$: one finds\nvalues of 1.5, 1.2, $1.15$ for $m_{\\ha}=300$, 475, $600~{\\rm GeV}$, respectively (for\ntheoretically allowed $\\tan\\beta$ values at the given $m_{\\ha}$).\nSince there is\nno systematic error due to uncertainty in the muon mass\nand since experimental systematics should be well below 10\\%, \nthe above numbers imply that the expected $\\alt \\pm 5\\%$ statistical\nerror probes out to very high $m_{\\ha}$.\n\n\n\\begin{table}[h]\n\\tcaption{Errors for combining NLC ($L=200~{\\rm fb}^{-1}$) data, \nthe NLC + LHC ($L=600~{\\rm fb}^{-1}$,\nATLAS+CMS) $BF(h_{SM}\\to\\gamma\\gam)$ determination, \n$\\gamma\\gam$ collider ($L=50~{\\rm fb}^{-1}$) data and FMC $s$-channel ($L=200~{\\rm fb}^{-1}$) data.}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Quantity & \\multicolumn{4}{c|}{Errors} \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 80} & {\\bf 100} & {\\bf 110} & {\\bf 120} \\\\\n\\hline\n $(b\\overline bh_{SM})^2|_{\\rm NLC+FMC} $ & $\\pm6\\%$ & $\\pm 9\\%$ & $\\pm 7\\%$ &\n  $\\pm6\\%$ \\\\\n\\hline\n $(c\\overline ch_{SM})^2|_{\\rm NLC+FMC} $ & $\\pm9\\%$ & $\\pm 11\\%$ & $\\pm 10\\%$ &\n  $\\pm9\\%$ \\\\\n\\hline\n $(\\mupmumh_{SM})^2|_{\\rm NLC+FMC}$ & \n$\\pm 5\\%$ & $\\pm 5\\%$ & $\\pm 4\\%$ & $\\pm 4\\%$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm FMC}$ & $\\pm 15\\%$ & $\\pm 16\\%$ & $\\pm 14\\%$ &\n $\\pm 13\\%$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm NLC+FMC}$ & $\\pm 9\\%$ & $\\pm 10\\%$ & $\\pm 9\\%$ &\n $\\pm 9\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 130} & {\\bf 140} & {\\bf 150} & {\\bf 170} \\\\\n\\hline\n $(b\\overline bh_{SM})^2|_{\\rm NLC+FMC}$ & $\\pm7\\%$ & $\\pm7\\%$ & $\\pm10\\%$ &\n $\\pm23\\%$ \\\\\n\\hline\n $(c\\overline ch_{SM})^2|_{\\rm NLC+FMC} $ & $\\pm10\\%$ & \\multicolumn{3}{c|}{$?$} \\\\\n\\hline\n $(\\mupmumh_{SM})^2|_{\\rm NLC+FMC}$ & \n$\\pm 3\\%$ & $\\pm 3\\%$ & $\\pm 4\\%$ &  $\\pm 10\\%$ \\\\\n\\hline\n $(W\\wstarh_{SM})^2|_{\\rm FMC}$ & $\\pm 16\\%$ & $\\pm 9\\%$ & $\\pm 9\\%$ &\n $-$ \\\\\n\\hline\n $(W\\wstarh_{SM})^2|_{\\rm NLC+FMC}$ & $\\pm 5\\%$ & $\\pm 4\\%$ & $\\pm 6\\%$ &\n $\\pm 10\\%$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm FMC}$ & $\\pm 14\\%$ & $\\pm 18\\%$ & $\\pm 36\\%$ & $-$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm NLC+FMC}$ & \n    $\\pm 10\\%$ & $\\pm 13\\%$ & $\\pm 23\\%$ & $-$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{nlcfmcerrors}\n\\end{table}\n\nTable~\\ref{nlcfmcerrors} gives a number of other quantities that are\ndetermined with remarkable precision by combining $\\sqrt s=500~{\\rm GeV}$ NLC data\nand $s$-channel FMC data\n[and, in the case of $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2$, including the (NLC+LHC) determination\nof $BF(h_{SM}\\to\\gamma\\gam)$].~\\cite{snowmass96}  The advantages\nof having high $L$ data from all three machines are enormous. \nIn particular, \nif there is a SM-like Higgs bosons with $m_{\\h}\\alt 2m_W$, \nit would be much more beneficial to have \nboth an $e^+e^-$ collider operating at full energy, $\\sqrt s\\sim 500~{\\rm GeV}$,\n{\\it and} a $\\mu^+\\mu^-$  collider operating at $\\sqrt s\\sim m_{\\h}$, as opposed\nto two NLC's.\n\n\\section{The Heavy MSSM Higgs Bosons}\n\nColliders other than the FMC offer various mechanisms\nto directly search for the $A^0,H^0$, but have significant limitations:\n\\begin{itemize}\n\\item There are regions in $(m_{\\ha},\\tan\\beta)$ parameter space at moderate\n$\\tan \\beta$, $m_{\\ha}\\agt 200~{\\rm GeV}$ in which the $H^0,A^0$ cannot be detected\nat the LHC.\n\\item At the NLC one can use the mode $e^+e^-\\to Z^\\star\\to H^0A^0$,\nbut it is limited to $m_{\\hh}\\sim m_{\\ha}\\alt \\sqrt{s}\/2$.\n\\item A $\\gamma \\gamma$ collider could probe heavy Higgs up to masses of\n$m_{\\hh}\\sim m_{\\ha}\\sim 0.8\\sqrt s$, but this would quite likely require\n$L> 100~{\\rm fb}^{-1}$, especially if the Higgs bosons are at the upper\nend of the  $\\gamma \\gamma$ collider energy spectrum.~\\cite{ghgamgam}\n\\end{itemize}\nIn contrast, there is an excellent chance of being able to detect\nthe $H^0,A^0$ at a $\\mu^+\\mu^-$ collider provided only that $m_{\\ha}$ is smaller\nthan the maximal $\\sqrt s$ available. This could prove to be very important\ngiven that GUT MSSM models usually predict $m_{\\ha}\\agt 200~{\\rm GeV}$.\n\nA detailed study of $s$-channel production\nof the $H^0,A^0$ has been made.~\\cite{bbgh} \nThe signals become viable when $\\tan\\beta>1$\n(as favored by GUT models) since the $\\mupmumH^0$ and $\\mupmumA^0$\ncouplings are proportional to $\\tan\\beta$. In particular, \neven though $\\Gamma_{\\hh}^{\\rm tot},\\Gamma_{\\ha}^{\\rm tot}$ are big (see Fig.~\\ref{hwidths}) at high $\\tan\\beta$, \ndue to large $b\\overline b$ decay widths, $BF(H^0,A^0\\to\\mu^+\\mu^-)$\napproaches a constant value that is large enough to imply\nsubstantial $\\overline \\sigma_{\\hh},\\overline \\sigma_{\\ha}$. The optimal strategy \nfor $H^0,A^0$ detection and study depends upon the circumstances.\nFirst, it could be that the $H^0$ and\/or $A^0$ will already have been\ndiscovered at the LHC.  \nWith $L=300~{\\rm fb}^{-1}$ (ATLAS+CMS) of integrated           \nluminosity, this would be the case if $\\tan\\beta\\alt 3$ or $\\tan\\beta$ \nis above an $m_{\\ha}$-dependent lower bound ({\\it e.g.}\\ $\\tan\\beta\\agt 10$ for $m_{\\ha}\\sim\n400~{\\rm GeV}$).~\\footnote{For \n$\\tan\\beta\\alt 3$, one makes use of modes such as $H^0\\toh^0\\hl\\to\nb\\overline b \\gamma\\gam$ and $H^0 \\to ZZ^{(*)}\\to 4\\ell$, when $m_{\\hh}\\alt 2m_t$,\nor $H^0,A^0\\to t\\overline t$, when $m_{\\hh},m_{\\ha}\\agt 2m_t$.  At high $\\tan\\beta$,\nthe enhanced production rates for $b\\overline b H^0,b\\overline bA^0$ with\n$H^0,A^0\\to\\tau^+\\tau^-$ are employed.}~\nEven if the $H^0,A^0$ have not been detected,\nstrong constraints on $m_{\\ha}$ are possible if precision measurements\nof the properties of the $h^0$ (such as the $b\\overline b\/WW^\\star$\nand $c\\overline c\/b\\overline b$ event rate ratios and the $(\\mupmumh^0)^2$\ncoupling-squared, as discussed earlier) are made via $s$-channel production\nat the FMC or in $\\sqrt s=500~{\\rm GeV}$ running at the NLC~\\cite{snowmass96} or\nby combining these two types of data.\nBy limiting the $\\sqrt s$ scan for the $H^0$ and $A^0$\nin the $s$-channel to the $m_{\\ha}\\simm_{\\hh}$ \nmass region preferred by $h^0$ measurements,\nwe would greatly reduce the luminosity needed to\nfind the $A^0$ and $H^0$ via an $s$-channel scan as compared to that required\nif $m_{\\ha}$ is not constrained.\n\nWith such pre-knowledge\nof $m_{\\ha}$, it will be possible to detect and perform detailed\nstudies of the $H^0,A^0$ for all $\\tan\\beta\\geq 1$\nprovided only that $m_{\\ha}\\alt \\sqrt s_{\\rm max}$.~\\footnote{We\nassume that a final ring optimized for maximal luminosity at $\\sqrt s\\sim m_{\\ha}$\nwould be constructed.}~ If $\\tan\\beta\\alt 3$ and $m_{\\hh},m_{\\ha}\\alt 2m_t$, then\nexcellent resolution, $R\\sim 0.01\\%$, will be necessary for detection\nsince the $A^0$ and $H^0$ are relatively narrow (see Fig.~\\ref{hwidths}).\nFor higher $\\tan\\beta$ values, $R\\sim 0.1\\%$ is adequate\nfor $H^0,A^0$ detection, but $R\\sim 0.01\\%$ would be required in order\nto separate the rather overlapping $H^0$ and $A^0$ peaks (as a function\nof $\\sqrt s$) from one another.\n\n\nEven without pre-knowledge of $m_{\\ha}$, \nthere would be an excellent chance for discovery of the $A^0,H^0$\nHiggs bosons in the $s$-channel at a $\\mu^+\\mu^-$ collider if they\nhave not already been observed at the LHC.\nThis is because non-observation at the LHC implies \nthat $\\tan\\beta\\agt 3$ while it is precisely for\n$\\tan\\beta\\agt 2.5-3$ that detection of the $A^0,H^0$ is possible~\\cite{bbgh}\nin the mass range from 200 to 500 GeV via an $s$-channel scan in \n$\\mu^+\\mu^-$ collisions. (The lower $\\tan\\beta$ reach given\nassumes that $L_{\\rm tot}=200~{\\rm fb}^{-1}$ is devoted to the scan.\nA detailed strategy as to how\nmuch luminosity to devote to different $\\sqrt s$ scan settings\nin the $200-500~{\\rm GeV}$ range must be employed.~\\cite{bbgh})\nThat the LHC and the FMC are complementary in this\nrespect is a very crucial point. Together, the LHC and FMC\nguarantee discovery of the $A^0,H^0$ after 3 to 4 years of\nhigh luminosity operation each, provided $m_{\\ha}\\alt 500~{\\rm GeV}$.\nOnce $m_{\\ha},m_{\\hh}$ are known, very precise measurements of some of\nthe crucial properties of the $H^0,A^0$ \n(including a scan determination of their\ntotal widths) become possible.~\\cite{bbgh}\n\n\nIn the event that the NLC has not been constructed, it could be\nthat the first mode of operation of the FMC would be to optimize\nfor and accumulate luminosity at, say, $\\sqrt s=500~{\\rm GeV}$.\nIn this case, there is still a high probability\nfor detecting the $H^0,A^0$ if they have not been\nobserved at the LHC. First, if $m_{\\ha}\\simm_{\\hh}\\alt \\sqrt s\/2\\sim 250~{\\rm GeV}$ then\n$\\mu^+\\mu^-\\to H^0A^0$ (and $H^+H^-$) pair production will be observed. Second,\nalthough reduced in magnitude compared to an electron\ncollider, there is a long low-energy bremsstrahlung tail \nat a muon collider that provides a\nself-scan over the full range of $\\sqrt s$ values below \nthe nominal operating energy.\nObservation of $A^0,H^0$ $s$-channel peaks in the $b\\overline b$ mass \n($m_{b\\overline b}$) distribution\ncreated by this bremsstrahlung tail may be possible.\nThe region of the $(m_{\\ha},\\tan\\beta)$ parameter space plane for which\na peak is observable depends strongly on the $m_{b\\overline b}$\nresolution. For excellent $m_{b\\overline b}$\nresolution of order $\\pm 5~{\\rm GeV}$ and integrated luminosity\nof $L=200~{\\rm fb}^{-1}$ at $\\sqrt s=500~{\\rm GeV}$, \nthe $A^0,H^0$ peak(s) are observable for $\\tan\\beta\\agt 4-5$\nif $500~{\\rm GeV}\\geqm_{\\ha}\\geq 250~{\\rm GeV}$.~\\footnote{Required $\\tan\\beta$ values increase\ndramatically as one moves into the $m_{\\ha}\\sim m_Z$ zone,\nbut this region is covered by $H^0A^0$ pair production.}\n\n\n\nFinally, if neither the LHC nor a FMC scan of the $\\leq 500~{\\rm GeV}$ region\nhas discovered the $H^0,A^0$, but supersymmetric particles and the $h^0$\nhave been observed, we would believe that the $H^0,A^0$ must exist\nbut have $m_{\\ha}\\simm_{\\hh}\\geq\n500~{\\rm GeV}$ . Analyses of the SUSY spectrum in the GUT context\nand precision $h^0$ studies\nmight have yielded some prejudice for the probable $m_{\\ha}$, and\nan extension of the FMC energy up to the appropriate $\\sqrt s\\sim m_{\\ha}$ \nfor $s$-channel discovery of the $H^0,A^0$\ncould be considered. But,\na machine with much higher $\\sqrt s$, such as the earlier-mentioned $\\sqrt s=4~{\\rm TeV}$,\nmight be most worthwhile.\nIt has been shown~\\cite{gk} \nthat such an energy with appropriately matched luminosity\nwould allow discovery of $\\mu^+\\mu^-\\to\\haH^0$ \nand $H^+H^-$ pair production,\nvia the $b\\overline b$ or $t\\overline t$ decay channels of the $H^0,A^0$\nand $t\\overline b,\\overline t b$ decay channels of the $H^+,H^-$,\nup to masses very close to $m_{\\ha}\\sim m_{\\hh}\\sim m_{\\hpm} \\sim\n2~{\\rm TeV}$, even if SUSY decays of the $H^0,A^0,H^{\\pm}$ are substantial. \n(This mass range certainly includes that expected in\nany supersymmetric model that provides a solution to the\nnaturalness and hierarchy problems.) \nDetailed studies of the $H^0,A^0,H^{\\pm}$ would be possible \nonce they were discovered.~\\cite{gk,fengmoroi}\n\n\n\\section{Conclusions and Discussion}\n\nIn this report, we have briefly reviewed \nthe capabilities of a muon collider to explore Higgs physics\nin the Standard Model and its minimal supersymmetric extension\nvia direct $s$-channel Higgs production. If there is a light ($m_{\\h}\\alt 2m_W$)\nSM-like Higgs boson, an $s$-channel scan of the Higgs resonance\npeak provides direct measurements of a number of its important \nproperties (in particular $\\Gamma_{\\h}^{\\rm tot}$)\nthat cannot be duplicated using $e^+e^-$ (or $\\mu^+\\mu^-$) \ncollisions at high $\\sqrt s$. Thus, if a light SM-like Higgs boson\nhas already been observed at the LHC or in early operation at the first NLC,\nit would be much more useful to follow the first NLC with an FMC,\nrather than a second NLC.\nIn the context of the minimal supersymmetric extension of the SM,\nthe FMC takes on added value.  At the FMC, \nan appropriately designed $s$-channel scan will allow $H^0,A^0$ discovery \nin the $200\\alt m_{\\ha}\\alt 500~{\\rm GeV}$ range for all $\\tan\\beta\\agt 3$. Thus, if\n$H^0A^0$ pair production is not seen in $\\sqrt s=500~{\\rm GeV}$\nrunning at the NLC (or FMC), implying $m_{\\ha}\\agt 200-240~{\\rm GeV}$,\nnor at the LHC, implying $\\tan\\beta\\agt 3$ (and below an \n$m_{\\ha}$-dependent upper limit) if $m_{\\ha}\\agt 200~{\\rm GeV}$,\nthe $H^0,A^0$ {\\it will} be observed at the FMC (provided $m_{\\ha}\\alt 500~{\\rm GeV}$).\nOnce discovered (at any collider), the FMC $s$-channel \nproduction mode allows (for any $\\tan\\beta>1$)\na detailed study of some of their key properties\nand a scan determination of their total widths. As reviewed\nelsewhere,~\\cite{bbgh,snowmass96,dpfreport,perspectives97} a muon collider will\nbe at least as valuable if the Higgs sector is still more\nexotic than the constrained two-doublet MSSM sector. Particularly noteworthy\nis the ability of $\\mu^-\\mu^-$ collisions to probe the\n$\\mu^-\\mu^-$ coupling of a doubly-charged\nHiggs boson~\\footnote{Such a Higgs is\nvery likely to be seen at the LHC if it has mass below 1 TeV.~\\cite{glp}}~\n(as present in many Higgs triplet models)\ndown to an extraordinarily small coupling magnitude. Overall,\nif there are elementary Higgs bosons, a muon collider will\nalmost certainly be mandated\npurely on the basis of its ability to explore the Higgs sector\nvia $s$-channel factory-like production of Higgs bosons.\n\n\n\\section{Acknowledgements}\nI would like to acknowledge the many contributions\nof my collaborators, V. Barger, M. Berger, and T. Han,\nto the muon collider Higgs physics results. This work was supported\nin part by the Department of Energy and by the Davis Institute for\nHigh Energy Physics.\n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe inflationary paradigm~\\cite{Guth:1980zm,Starobinsky:1980te} is a part of the standard cosmology. Especially, in slow-roll inflation models~\\cite{Linde:1983gd}, the primordial curvature perturbations are naturally produced by the quantum fluctuation of (at least) one scalar field called inflaton during inflation. The spectrum of scalar perturbation predicted by a slow-roll model is almost flat, which is compatible with the cosmic microwave background (CMB) observation results.\n\nInflation models in supergravity (SUGRA) have been studied through many works. It is interesting that SUGRA is an effective theory of superstring, which is a possible candidate for a unified theory including gravity. \n\nIn this work, we propose a new class of inflation models in SUGRA including higher derivative interactions. Higher derivative terms always arise from e.g. the Dirac-Born-Infeld action describing D-brane dynamics in string theory. Some models with higher derivative terms have been constructed in SUGRA~\\cite{Koehn:2012ar,Farakos:2012je}. Higher derivative terms in SUGRA can be described by using the supersymmetry~(SUSY) covariant derivative operator of SUSY multiplets. We call a SUGRA term including SUSY derivative operators a SUSY derivative term in this letter.  The components of SUSY derivative terms contain the higher derivative interactions. \n\nIn the standard SUGRA action without SUSY derivative terms, dynamical chiral multiplets should be contained in a K\\\"ahler potential. However, as we will show, the kinetic terms of chiral multiplets also arise from SUSY derivative terms.\n\n We will discuss inflation models in which the inflaton multiplet is not included in the K\\\"ahler potential. Surprisingly, the so-called $\\eta$ problem, which spoils the flatness of the inflation potential, does not exist in our models. It is also remarkable that the effects of the higher derivative terms of the inflaton become negligibly small on the slow-roll trajectory. Consequently, the inflation is effectively driven by a single scalar field with a standard kinetic term and a scalar potential.  We will show two inflation models, the chaotic inflation\\cite{Linde:1983gd} and the Starobinsky inflation~\\cite{Starobinsky:1980te}. Especially, the chaotic inflation in our model predicts a characteristic spectrum of the perturbation, which can be tested by forthcoming observations.\n\nThe remaining parts of this letter are consisted as follows. First, in Sec.~\\ref{setup}, we will show the action we will discuss. After that, choosing the two types of superpotential, we will construct the chaotic and the Starobinsky like inflation models, and show the cosmological parameters in those models in Sec.~\\ref{chaotic} and \\ref{Starobinsky}. Finally we will conclude in Sec.~\\ref{con}. In Appendix.~\\ref{A}, the cosmological parameters in our models are discussed.\n\\section{Inflation with higher derivative terms}\\label{model}\n\\subsection{Setup}\\label{setup}\nTo clarify the SUGRA system discussed below, we first show the corresponding action in superspace. It is described by \n\\begin{align}\nS=&\\int d^4\\theta K(\\hat{S},\\bar{\\hat{S}})+\\left(\\int d^2\\theta W(\\hat{\\Phi},\\hat{S})+{\\rm h.c.}\\right)\\nonumber\\\\\n&+\\int d^4\\theta C D_\\alpha\\hat{\\Phi}  D^{\\alpha}\\hat{S}\\bar{D}_{\\dot{\\alpha}}\\bar{\\hat{\\Phi}}\\bar{D}^{\\dot{\\alpha}}\\bar{\\hat{S}},\\label{gSUSY}\n\\end{align}\nwhere $\\hat{S}$ and $\\hat{\\Phi}$ are chiral multiplets, $K$ and $W$ are K\\\"ahler and super-potential respectively, $C$ is a constant, $D_\\alpha$ and $\\bar{D}_{\\dot{\\alpha}}$ denote SUSY covariant derivatives, and $\\alpha,\\dot{\\alpha}$ are spinor indices. Here and the following, we use the unit~$M_{P}=1$ where $M_P=2.4\\times10^{18}$GeV is the reduced Planck mass. We will denote the scalar component of each multiplet by the character without a hat below. \n\nThe last term in Eq.~(\\ref{gSUSY}) is a SUSY derivative term. It leads ghost-free interactions with respect to the bosonic sector discussed in Ref.~\\cite{Koehn:2012ar}. In the well-known SUGRA models without SUSY derivative terms, a dynamical chiral multiplet should be contained in a K\\\"ahler potential $K$ because superpotential terms does not give kinetic terms for chiral multiplets. However, in our case, the SUSY derivative term in Eq.~(\\ref{gSUSY}) gives $S$ and $\\Phi$ not only higher derivative terms, but also standard kinetic terms in non-canonical forms as we will show. \n\nHere, we choose the following $K$ and $W$~\\cite{Kawasaki:2000yn,Kallosh:2010xz}, \n\\begin{eqnarray}\nK&=&|\\hat{S}|^2,\\label{K}\\\\\nW&=&f(\\hat{\\Phi}) \\hat{S},\n\\end{eqnarray}\nwhere $f(\\hat{\\Phi})$ is a holomorphic function of $\\hat{\\Phi}$. We emphasize that the K\\\"ahler potential~(\\ref{K}) does not contain $\\hat{\\Phi}$, which we identify as the inflaton multiplet. $\\hat{S}$ becomes the Goldstino multiplet during inflation. With a minimal K\\\"ahler potential $K=|\\hat{S}|^2$, $S$ often has its light or tachyonic mass during inflation, and therefore the quantum fluctuation of $S$ may become a source of the curvature perturbation as discussed in Ref.~\\cite{Kawasaki:2000yn,Demozzi:2010aj}. Recently, the authors of Ref.~\\cite{Antoniadis:2014oya} showed the absence of such a situation in the case that $\\hat{S}$ is the Volkov and Akulov supermultiplet~\\cite{Volkov:1973ix}, which satisfies $\\hat{S}^2=0$.\\footnote{We can construct a similar model with an unconstrained multiplet instead of $\\hat{S}$ if its scalar component has a sufficiently heavy mass. However, in that case, the equations of motions of auxiliary fields are complicated and the solutions are not determined uniquely~\\cite{Koehn:2012ar}.} Then, $S$ is identically $0$ because it is the bilinear of the fermionic component of $\\hat{S}$~\\cite{Antoniadis:2014oya}. \n\nIn conformal SUGRA~\\cite{Kugo:1982cu,Kugo:1983mv}, the action corresponding to one in Eq.~(\\ref{gSUSY}) can be expressed as follows, \n\\begin{align}\nS_{\\rm SG}=&\\frac{1}{2}\\left[S_0\\bar{S}_0 (-3e^{-\\frac{K}{3}})\\right]_D\\nonumber\\\\\n&+\\left[S_0^3W\\right]_F+[C\\Da \\Phi\\uDa S\\Dad \\bar{\\Phi}\\uDad \\bar{S}]_D,\\label{Sconf}\n\\end{align} \nwhere $[\\cdots]_{D,F}$ denote the superconformal D- and F-density formulae~\\cite{Kugo:1982cu}, $S_0$ is a chiral compensator, and $\\Da$ denotes the superconformal spinor derivative corresponding to $D_\\alpha$ in Eq.~(\\ref{gSUSY})~\\cite{Kugo:1983mv}.\\footnote{We note that the last term in Eq.~(\\ref{Sconf}) is manifestly K\\\"ahler invariant. The K\\\"ahler transformation, $K\\rightarrow K+\\Lambda+\\bar{\\Lambda}$, $W\\rightarrow We^{\\Lambda}$, is the redefinition of the compensator $S_0$ in conformal SUGRA. Obviously, the last term in Eq.~(\\ref{Sconf}) is not coupled to the compensator, and therefore it is inert under the K\\\"ahler transformation. However, the sigma model diffeomorphism, implies that $C$ in Eq.~(\\ref{Sconf}) should be the tensor of the target space.} In the following, we only discuss the bosonic action, and then we can put the usual superconformal gauge fixing conditions~\\cite{Kugo:1982cu,Kugo:1983mv}, e.g. $S_0=\\bar{S}_0=e^{K\/6}$ and obtain the SUGRA action in Einstein frame. \n\nAfter eliminating the compensator $S_0$ and the auxiliary field $A_\\mu$, which is one of the gauge fields of the superconformal symmetry, we obtain the following action, \n\\begin{align}\n\\mathcal{L}=&K_{S\\bar{S}}F^S\\bar{F}^{\\bar{S}}-3F^0\\bar{F}^0\\nonumber\\\\\n&+\\left(W_SF^S+{\\rm h.c.}\\right)\\nonumber\\\\\n&+32CF^\\Phi F^S\\bar{F}^{\\bar{\\Phi}}\\bar{F}^{\\bar{S}}-16C\\partial_\\mu\\Phi \\partial ^\\mu\\bar{\\Phi}F^S\\bar{F}^{\\bar{S}},\n\\end{align}\nwhere $F^{0,S,\\Phi}$ are the auxiliary components of $\\hat{S}_0,\\hat{S},$ and $\\hat{\\Phi}$ respectively and we used the condition $K=K_S=W_\\Phi=W=0$ because $S\\equiv 0$. The solutions for the equations of motion of the auxiliary fields are simply given by \n\\begin{align}\n&F^0=0,\\\\\n&F^{\\Phi}=0,\\\\\n&F^S=-\\frac{\\bar{W}_{\\bar{S}}}{1-16C\\partial_\\mu\\Phi\\partial^\\mu\\bar{\\Phi}},\n\\end{align}\nwhere we have used $K_{S\\bar{S}}=1$.\n\nBy substituting the on-shell expressions of F-terms into the action, we obtain the effective Lagrangian as follows,\n\\begin{align}\n\\mathcal{L}=\\frac{V}{A^2}\\tilde{X}-\\frac{V}{A}\\left(2-\\frac{1}{A}\\right),\\label{Leff}\n\\end{align}\nwhere $V\\equiv|W_S|^2=|f(\\Phi)|^2$, $\\tilde{X}\\equiv-16C \\partial_\\mu\\Phi\\partial^\\mu\\bar{\\Phi}$ and $A\\equiv (1+\\tilde{X})$. We can redefine the complex scalar $\\Phi$ as follows,\n\\begin{align}\n\\varphi\\equiv4\\sqrt{C}\\int f(\\Phi)d\\Phi.\n\\end{align}\nThen the Lagrangian~(\\ref{Leff}) can be rewritten as\n\\begin{align}\n\\mathcal{L}=\\frac{X}{A^2}-\\frac{V}{A}\\left(2-\\frac{1}{A}\\right),\n\\end{align}\nwhere $X\\equiv \\partial_\\mu \\varphi\\partial^\\mu\\bar{\\varphi}=V\\tilde{X}$. \n\nAlthough the action contains the higher order terms of $X$, the cosmological parameters are same with ones in the models without higher derivative action as shown in Appendix.~\\ref{A}. Therefore, on the slow-roll trajectory, we can approximate the Lagrangian as follows, \n\\begin{align}\n\\mathcal{L}\\sim X-V,\n\\end{align}\nwhere we used the approximation $A\\sim 1$. The approximated action is one with the standard kinetic and the potential terms of $\\varphi$, and we will use this action in the following discussion.\n\nIt is worth to remark the absence of the $\\eta$ problem in this model. In the standard SUGRA models, the F-term scalar potential is given by,\n\\begin{eqnarray}\nV_F=e^K\\left(K^{I\\bar{J}}D_IWD_{\\bar{J}}\\bar{W}-3|W|^2\\right),\n\\end{eqnarray} \nwhere $D_IW=W_I +K_IW$, and $K^{I\\bar{J}}$ is the inverse of $K_{I\\bar{J}}$. In the case that K\\\"ahler potential is given by $K=|\\Phi|^2+\\cdots$, the scalar potential becomes steep due to the factor $e^K$. In our case, however, the inflaton $\\Phi$ is not contained in the K\\\"ahler potential, and therefore the $\\eta$ problem does not exist.\\footnote{From the conformal SUGRA viewpoint, we can also understand the reason why the $\\eta$ problem does not exist in our model. In our case, the last term in Eq.(\\ref{Sconf}) does not contains the mixing between Ricci scalar and the inflaton. In this case, the $\\eta$ problem does not occur~\\cite{Abe:2014opa}. } \n\\subsection{Chaotic inflation}\\label{chaotic}\nWe consider the following function $f(\\Phi)$,\n\\begin{align}\nf(\\Phi)=\\lambda_n\\Phi^n,\n\\end{align} \nwhere $n\\geq1$, $\\lambda_n$ is a coupling constant. Then the scalar $\\varphi$ can be written as\n\\begin{align}\n\\varphi=\\frac{4}{n+1}\\sqrt{C}\\lambda_n\\Phi^{n+1}.\n\\end{align}\nWe identify $\\sqrt{2}|\\varphi|$ as the inflaton denoted by $\\phi$, and we can write down the scalar potential $V$ as follows,\n\\begin{align}\nV=\\tilde{\\lambda}_n^2\\phi^{2n\/(n+1)},\\label{Vc}\n\\end{align}\nwhere $\\tilde{\\lambda}_n^2\\equiv \\left(2^{-n}(16C)^{-n}(n+1)^{2n}\\lambda_n^2\\right)^{1\/(n+1)}$.  Note that this model resembles to the running kinetic inflation model~\\cite{Nakayama:2010kt}, however, the scalar potential is highly restricted in our case.\n\nSurprisingly enough, the effective potential is restricted from the linear potential to the quadratic one, even when the power of $f(\\Phi)$ becomes higher. The predicted spectral tilt $n_s$ and tensor-to-scalar ratio $r$ are given by,\n\\begin{align}\nn_s&\\sim1-\\frac{2n+1}{n+1}\\frac{1}{N},\\\\\nr&\\sim\\frac{8n}{n+1}\\frac{1}{N},\n\\end{align}\nwhere $N$ denotes the number of e-foldings and we have omitted smaller corrections. With a sufficiently large $n$, the scalar potential~(\\ref{Vc}) asymptotes to $\\phi^2$, which is favored by the BICEP2 data~\\cite{Ade:2014xna}. This model can be tested by forthcoming experiments.\n\\subsection{Starobinsky inflation }\\label{Starobinsky}\nNext, we choose the following function $f(\\Phi)$~\\cite{Kallosh:2013lkr},\n\\begin{align}\nf(\\Phi)=\\lambda(1-e^{-a\\Phi}),\n\\end{align}\nwhere $a$ and $\\lambda$ are real constant parameters. Then, the canonical normalized complex scalar $\\varphi$ is\n\\begin{align}\n\\varphi=4\\sqrt{C}\\lambda\\left(\\Phi+\\frac{1}{a}e^{-a\\Phi}\\right)\\sim4\\sqrt{C}\\lambda\\Phi.\n\\end{align} \nHere, we identify $\\sqrt{2}{\\rm Re}~\\varphi=\\phi$ as the inflaton. Then, the effective potential is given by\n\\begin{align}\nV\\sim\\lambda^2 (1-e^{-\\frac{\\tilde{a}}{\\sqrt{2}}\\phi})^2,\n\\end{align}\nwhere $\\tilde{a}=(4\\sqrt{C} \\lambda)^{-1}a$ and we only write down the leading terms. The resultant potential is same with one in the Starobinsky model~\\cite{Kallosh:2013lkr}, and the predicted spectral tilt~$n_s$ and tensor-to-scalar ratio~$r$ are as follows,\n\\begin{align}\nn_s&\\sim 1-\\frac{2}{N},\\label{Stilt}\\\\\nr&\\sim \\frac{16}{\\tilde{a}^2N^2} \\label{Sr},\n\\end{align} \nwhere $N$ denotes the number of e-foldings. The predicted $r$ is very small when $\\tilde{a}\\sim\\mathcal{O}(1)$. For $N\\sim 60$, the values of $(n_s,r)$ are compatible with the Planck2013 result~\\cite{Ade:2013uln}, however, this model may be excluded if the result from BICEP2~\\cite{Ade:2014xna} is confirmed by other experiments.\n\nWe also note that the mass of ${\\rm Im}\\varphi$ is much smaller than the Hubble scale during inflation although its minimum is located its origin. Then, the light direction can be a curvaton as in the case of the simplest chaotic inflation in SUGRA~\\cite{Kawasaki:2000yn,Demozzi:2010aj}, if the decay of ${\\rm Im}\\varphi$ occurs after that of the inflaton~${\\rm Re}\\varphi$. Further investigation of such a case would be interesting but it is beyond the scope of this letter.\n\\section{Conclusion}\\label{con}\nWe have proposed a new class of inflation models with a SUSY derivative term in SUGRA.  We have found that the kinetic terms of chiral multiplets are included in SUSY derivative terms, even if the multiplets are absent in K\\\"ahler potential terms. By virtue of the absence of the inflaton in $K$, the $\\eta$ problem does not exist in our models. It is also remarkable that the action contains the higher order terms of $X=-\\partial_\\mu \\phi \\partial^\\mu \\phi\/2$, however, their contributions are negligible and the effective action is given by the scalar system with the standard kinetic term and the scalar potential. \n\nWe have shown two inflation models in our setup. In Sec.~\\ref{chaotic}, we discussed the chaotic inflation model. It is surprising that the effective potential is restricted from the linear to the quadratic potential regardless of the superpotential containing the arbitrary power of $\\Phi$. Therefore this model can be tested by forthcoming experiments.\n\nOn the other hand, we have also constructed the Starobinsky type inflation model in Sec.~\\ref{Starobinsky}. In contrast to the chaotic type model, it predicts the very small value of the tensor-to-scalar ratio $r$ which is compatible with the Planck2013 result. As discussed in Sec.~\\ref{Starobinsky}, this model contains the light scalar ${\\rm Im}\\varphi$ which may produce additional adiabatic perturbations. If it the case, the predicted cosmological parameters shown in Eq.(\\ref{Stilt}) and (\\ref{Sr}) can be changed. That is an interesting possibility but it requires more detailed investigation of other sectors. \n\nIn this work, we have only discussed the dynamics of the scalar sector, however we need to take fermions into account to discuss the universe after inflation. That will be our future work. \n\\section*{Acknowledgements}\nThe authors would like to thank Hiroyuki Abe for useful discussion and comments.\nY.Y. was supported by JSPS Research Fellowships for Young Scientists No. 26-4236 in Japan.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nEllis \\cite{ellis3} gave a description of the free crossed square of\ngroups of a CW-complex using topological methods. In contexts other\nthan groups, he has defined crossed squares, 2-crossed modules etc.\nin \\cite{ellis3}. For commutative algebras, 2-crossed modules have\nbeen defined by Grandjean and Vale \\cite{gv}.\n\nCombining earlier work \\cite{porter1} of Porter with  Arvasi and Porter's joint papers\n\\cite{ap1,ap2,ap3}, one starts to see how a study\nof the links between simplicial commutative algebras and classical\nconstructions of homological algebra can be strengthened by\ninterposing crossed algebraic models for the homotopy types of\nsimplicial algebras. In this note, we continue this process using\nthese methods to give a description in terms of tensor products,\nof the top corner of a free crossed square of commutative algebras\nand the top term of the corresponding free 2-crossed module. The\nmethods are slightly different, but these results are\n2-dimensional analogues of the description of the free crossed\nmodule on a `presentation' of an algebra in terms of Kozsul-like\nterms, given in \\cite{porter1}.\n\nWe end with a section looking at possible links of this with\nAndr\\'e-Quillen homology and a squared complex form of the\ncontangent complex.\n\nThe results and general methods we use are inspired by those given for the\ncorresponding case of groups in \\cite{mp}. Some of the methods of that paper,\nof course, go across almost verbatim to this commutative algebra case,\nbut, as the audience for this paper is probably more or less disjoint from\nthat for \\cite{mp} it seems advisable to repeat arguments from that paper in\nthis algebra case even when they might be safely `left to the reader'. Of\ncourse, there are all times when the translation between the two contexts\nis less easy.\n\n\n\\textbf{Acknowledgements}. The authors wishes to thank Professor\nTimothy Porter\nfor his helpful comments.\n\n\n\\section{Preliminaries}\n\nAll algebras will be commutative and will be over the same fixed but\nunspecified ground ring.\n\n\\subsection{Simplicial algebras}\n\nDenoting the usual category of finite ordinals by $\\Delta $, we obtain for\neach $k\\geqslant 0$ a subcategory $\\Delta _{\\leqslant k}$ determined by the objects $%\n[j]$ of $\\Delta $ with $j\\leqslant k$. A simplicial algebra is a\nfunctor from the opposite category $\\Delta ^{op}$ to \\textbf{Alg};\na $k$-truncated simplicial algebra is a functor from $(\\Delta\n_{\\leqslant k}^{op})$ to \\textbf{Alg}. We denote the category of\nsimplicial algebras by \\textbf{SimpAlg} and the category of\n\\textit{k}-truncated simplicial algebras by\n$\\mathbf{Tr_{k}SimpAlg}$. By a \\textit{k-truncation of a\nsimplicial algebra}, we mean a $k$-truncated simplicial algebra\n\\textbf{tr}$_{k}\\mathbf{E}$ obtained by forgetting dimensions of\norder $>k$ in a simplicial algebra $\\mathbf{E},$ that is\nrestricting \\textbf{E} to $\\Delta _{\\leqslant k}^{op}$. This gives a\ntruncation functor\n\\begin{equation*}\n\\mathbf{tr_{k}:SimpAlg\\longrightarrow Tr_{k}SimpAlg}\n\\end{equation*}\nwhich admits a right adjoint\n\\begin{equation*}\n\\mathbf{cosk_{k}:Tr_{k}SimpAlg\\longrightarrow SimpAlg}\n\\end{equation*}\ncalled the \\textit{k-coskeleton functor}, and a left adjoint\n\\begin{equation*}\n\\mathbf{sk_{k}:Tr_{k}SimpAlg\\longrightarrow SimpAlg},\n\\end{equation*}\ncalled the \\textit{k-skeleton functor}. For explicit construction\nof these see \\cite{dusk}. We will say that a simplicial algebra\n$\\mathbf{E}$ is \\textit{\\ k-skeletal} if the natural morphism\n\\textbf{sk$_{k}\\mathbf{E}$}$  \\rightarrow \\mathbf{E}$ is an\nisomorphism.\n\nRecall that given a simplicial algebra \\textbf{E}, \\emph{the Moore\ncomplex} $(\\mathbf{NE},\\partial )$ \\emph{of} \\textbf{E} is the\nchain complex defined by\n\\begin{equation*}\n(NE)_n=\\bigcap_{i=0}^{n-1}\\mbox{\\rm Ker}d_i^n\n\\end{equation*}\nwith $\\partial _n:NE_n\\rightarrow NE_{n-1}$ induced from $d_n^n$ by\nrestriction.\n\nThe \\emph{n$^{th}$ homotopy module} $\\pi _{n}$(\\textbf{E}) of \\textbf{E} is\nthe n$^{th}$ homology of the Moore complex of \\textbf{E}, i.e.,\n\\begin{equation*}\n\\begin{array}{rcl}\n\\pi _{n}(\\mathbf{E}) & \\cong & H_{n}(\\mathbf{NE},\\partial ) \\\\\n& = & \\bigcap\\limits_{i=0}^{n}\\mbox{\\rm\nKer}d_{i}^{n}\/d_{n+1}^{n+1}(\\bigcap \\limits_{i=0}^{n}\\mbox{\\rm\nKer}d_{i}^{n+1}).\n\\end{array}\n\\end{equation*}\nWe say that the Moore complex \\textbf{NE} of a simplicial algebra is\nof \\emph{length} $k$ if $NE_{n}=0$ for all $n\\geqslant k+1$ so that a\nMoore complex is of length $k$ also of length $r$ for $r\\geqslant k.$ For\nexample, if \\textbf{E} has Moore complex of length 1, then\n$(NE_{1},NE_{0},\\partial _{1})$ is a crossed module and conversely.\nIf \\textbf{NG} is of length 2, the corresponding Moore complex gives\na 2-crossed module (cf. \\cite{ap3}).\n\n\\subsection{Free Simplicial Algebras}\n\nRecall from \\cite{ap1} the definition of free simplicial algebra\ngiven by the `step-by-step' construction of Andr\\'e \\cite{andre}.\n\nLet \\textbf{E} be a simplicial algebra and $k\\geqslant 1$, $k$-skeletal\nbe  fixed. A simplicial algebra \\textbf{F} is called a\n\\textit{free} if\n\ni) $F_{n}=E_{n}$ for $n<k,$\n\nii) $F_{k}=$ a free $E_{k}$-algebra over a set of non-degenerate\nindeterminates, all of whose faces are zero except the $k^{th}$,\n\niii) $F_{n}$ is a free $E_{n}$-algebra over the degenerate\nelements for $n>k$.\n\n\n\\textbf{Remark}: if \\textbf{A} is a simplicial algebra, then there\nexists a free simplicial algebra \\textbf{E} and an epimorphism\n$\\mathbf{E\\rightarrow A}$ which induces isomorphisms on all\nhomotopy modules. The details are omitted as they are\n'well-known'.\n\n\\subsection{Crossed Modules of Algebras}\nThroughout this paper we denote an action of $r\\in R$ on $c\\in C$\nby $r\\cdot c$.\n\nA \\emph{crossed module} is an algebra morphism $\\partial\n:C\\rightarrow R$ with an action of $R$ on $C$ satisfying (i)\n$\\partial (r\\cdot c)=r\\partial c$ and (ii) $\\partial (c)\\cdot\nc^{\\prime }=cc^{\\prime }$ for all $c,c^{\\prime }\\in M,r\\in R$. For\nthe weaker notion in which condition (ii) is not required, the\nmodels are called \\textit{pre-crossed modules}.\n\nExamples of crossed modules are:\n\ni) Any ideal, $I$, in $R$ gives an inclusion map $I\\longrightarrow\nR,$ which is a crossed module. Conversely, given any crossed\n$R$-module $\\mu :C\\longrightarrow R$, the image $I=\\mu (C)$ of $C$\nis an ideal in $R$.\n\nii) Any $R$-module $M$ can be considered as an $R$-algebra with\nzero multiplication and hence $M\\overset{0}{\\longrightarrow }R$ is\na crossed $R$-module. Conversely, if $\\mu :C\\longrightarrow R$ is\na crossed $R$-module, Ker$\\mu $ is an $R\/\\mu (C)$-module.\n\n\\section{Crossed Squares and Simplicial Algebras}\nAlthough we will be mainly concerned with crossed squares in this\npaper, some of the arguments either clearly apply or would seem to\napply in the more general case of crossed $n$-cubes and $n$-cube\ncomplexes.\n\nCrossed $n$-cubes in algebraic settings such as commutative\nalgebras, Jordan algebras, Lie algebras have been defined by\nEllis, \\cite{ellis2}.\n\nA \\emph{crossed n-cube of commutative algebras} is a family of\ncommutative algebras, $M_A$ for $A\\subseteq \\langle\nn\\rangle=\\{1,...,n\\}$ together with homomorphisms $\\mu\n_i:M_A\\rightarrow M_{A-\\{i\\}}$ for $i\\in \\langle n \\rangle$ and for\n$A,B\\subseteq \\langle n \\rangle$, functions\n\\begin{equation*}\nh:M_A\\times M_B\\longrightarrow M_{A\\cup B}\n\\end{equation*}\nsuch that for all $k\\in \\mathbf{k},\\ a,a^{\\prime }\\in M_A,\\\nb,b^{\\prime }\\in M_B,\\ c\\in M_C,$ $i,j\\in \\langle n \\rangle$ and\n$A\\subseteq B$\n\\begin{equation*}\n\\begin{array}{lrcl}\n1) & \\mu _ia & = & a\\ \\quad \\mbox{\\rm {\\rm if}}\\ i\\not \\in A \\\\\n2) & \\mu _i\\mu _ja & = & \\mu _j\\mu _ia \\\\\n3) & \\mu _ih(a,b) & = & h(\\mu _ia,\\mu _ib) \\\\\n4) & h(a,b) & = & h(\\mu _ia,b)=h(a,\\mu _ib)\\ \\quad \\mbox{\\rm {\\rm if}}\\ i\\in\nA\\cap B \\\\\n5) & h(a,a^{\\prime }) & = & aa^{\\prime } \\\\\n6) & h(a,b) & = & h(b,a) \\\\\n7) & h(a+a^{\\prime },b) & = & h(a,b)+h(a^{\\prime },b) \\\\\n8) & h(a,b+b^{\\prime }) & = & h(a,b)+h(a,b^{\\prime }) \\\\\n9) & k\\cdot h(a,b) & = & h(k\\cdot a,b)=h(a,k\\cdot b) \\\\\n10) & h(h(a,b),c) & = & h(a,h(b,c))=h(b,h(b,c)). \\\\\n&  &  &\n\\end{array}%\n\\end{equation*}\n\nA \\emph{morphism of crossed n-cubes} is defined in the obvious way:\nIt is a family of commutative algebra homomorphisms, for $A\\subseteq\n\\langle n \\rangle$, $\\ f_A:M_A\\longrightarrow M_A^{\\prime } $\ncommuting with the $\\mu _i^{\\prime}$s and $h^{\\prime}s$. We thus\nobtain a category of crossed $n$-cubes denoted by $\\mathbf{Crs^n} $.\n\n\\vspace{0.5cm}\n\n{\\noindent \\textbf{Examples.}} ${\\noindent(1)}$ For $n=1,$ a crossed 1-cube\nis the same as a crossed module.\n\n{\\noindent } For $n=2$ one has a crossed square:\n\\begin{equation*}\n\\xymatrix{M_{\\langle 2 \\rangle}\\ar[r]^{\\mu_2}\\ar[d]_{\\mu_1}&\nM_{\\{1\\}}\\ar[d]^{\\mu_1}\\\\\nM_{\\{2\\}}\\ar[r]_{\\mu_2}&M_{\\emptyset}. }\n\\end{equation*}\nEach $\\mu _i$ is a crossed module as is $\\mu _1\\mu _2$. The\n$h$-functions give actions and a pairing\n\\begin{equation*}\nh:M_{\\{1\\}}\\times M_{\\{2\\}}\\longrightarrow M_{\\langle 2\\rangle}.\n\\end{equation*}\nThe maps $\\mu _2$ (or $\\mu _1)$ also define a map of crossed\nmodules from $(M_{\\langle 2 \\rangle},M_{\\{2\\}},\\mu _1)$ to\n$(M_{\\langle 2 \\rangle},M_\\emptyset ,\\mu _1)$. In fact a crossed\nsquare can be thought of as a crossed module in the category of\ncrossed modules.\n\n(2) Let $I_{1}$ and $I_{2}$ be ideals of an algebra $E$. The\ncommutative square diagram of inclusions;\n\\begin{equation*}\n\\xymatrix{I_1 \\cap I_2 \\ar[r]^-{inc.}\\ar[d]_{inc.}&\nI_2\\ar[d]^{inc.}\\\\\nI_1\\ar[r]_{inc.}&E}\n\\end{equation*}\nnaturally comes together with actions of $E$ on $I_{1},I_{2}$ and\n$I_{1}\\cap I_{2}$ given by multiplication, and functions\n\\begin{equation*}\n\\begin{array}{lrll}\nh: & I_{A}\\times I_{B} & \\longrightarrow & I_{A}\\cap I_{B}=I_{A\\cup B} \\\\\n& (a,b) & \\longmapsto & ab.\n\\end{array}%\n\\end{equation*}%\nThat this is a crossed square is easily checked.\n\n(3) Let $\\mathbf{E}$ be a simplicial algebra. Let $M(\\mathbf{E},2)$\ndenote the following diagram\n\\begin{equation*}\n\\xymatrix{NE_2\/\\partial_3(NE_3)\\ar[r]^-{\\partial_2}\\ar[d]_{\\partial'_2}&NE_1\\ar[d]^{\\mu}\\\\\n\\overline{NE_1}\\ar[r]_{\\mu'}&E_1.}\n\\end{equation*}\nThen this is the underlying square of a crossed square. The extra\nstructure is given as follows: $NE_1=$\\text{Ker}$d_0^1$ and\n$\\overline{NE}_1=$\\textrm{Ker}$d_1^1$. Since $E_1$ acts on\n$NE_2\/\\partial _3NE_3,\\ \\overline{NE}_1$ and $NE_1,$ there are\nactions of $\\overline{NE}_1$ on $NE_2\/\\partial _3NE_3$ and $NE_1$\nvia $\\mu ^{\\prime },$ and $NE_1$ acts on $NE_2\/\\partial _3NE_3$ and\n$\\overline{NE}_1$ via $\\mu .$ As $\\mu $ and $\\mu ^{\\prime }$ are\ninclusions, all actions can be given by multiplication. The $h$-map\nis\n\\begin{equation*}\n\\begin{array}{rcl}\nNE_1\\times \\overline{NE}_1 & \\longrightarrow & NE_2\/\\partial _3NE_3 \\\\\n(x,\\overline{y}) & \\longmapsto & h(x,\\overline{y})=s_1x(s_1y-s_0y)+\\partial\n_3NE_3,%\n\\end{array}%\n\\end{equation*}\nwhich is bilinear. Here $x$ and $y$ are in $NE_1$ as there is a\nnatural bijection between $NE_1$ and $\\overline{NE}_1$ (by\n\\cite[Lemma 2.1]{ap2}). The element $\\bar y$ is the image of $y$\nunder this. This example effectively introduces the functor\n$$\n\\mathbf{M}(-,2):\\mathbf{SimpAlg}\\rightarrow \\mathbf{Crs^2}.\n$$\nThis is the case $n=2$ of a general construction of a crossed\n$n$-cube from a simplicial algebra given by the first author in\n\\cite{a1} where the reader may find the verification of the\naxioms. (This notational convention will be revisited at the end\nof section \\ref{sec5})\n\nNote if we consider the above crossed square as a vertical\nmorphism of crossed modules, we can take its kernel and cokernel\nwithin the category of crossed modules. In the above the morphisms\nin the top left hand corner are induced from $d_2$ so\n\\begin{equation*}\n\\text{Ker}\\left( \\partial'_2:\\frac{NE_2}{\\partial _3NE_3}\\longrightarrow \\text{Ker%\n}d_1\\right) =\\frac{NE_2\\cap \\text{Ker}d_2}{\\partial _3NE_3}\\cong \\pi _2(%\n\\mathbf{E})\n\\end{equation*}\nwhilst the other map labeled $\\mu $ is an inclusion so has trivial\nkernel, hence the kernel of this morphism of crossed modules is\n\\begin{equation*}\n\\pi _2(\\mathbf{E})\\longrightarrow 0.\n\\end{equation*}\nThe image of $\\partial_2$ (and $\\mu' $) is an ideal in both the\nalgebras on the bottom line and as Ker$d_0=NE_1$ with the\ncorresponding Im$\\mu$ being $d_2NE_2,$ the cokernel is\n$NE_1\/\\partial _2NE_2,$ whilst $E_1\/$Ker$d_0\\cong E_0,$ i.e, the\ncokernel of $\\mu $ is $M(\\mathbf{E},1)$.\n\nIn fact of course $\\mu$ is not only a morphism of crossed modules,\nit is a crossed module. This means that $\\pi\n_2(\\mathbf{E})\\rightarrow 0$ is in some sense a\n$M(\\mathbf{E},1)$-module, (cf. \\cite{ae}), and that\n$M(\\mathbf{E},2)$ can be thought of as a crossed extension of\n$M(\\mathbf{E},1)$ by $\\pi _2(\\mathbf{E})$.\n\n\\section{ Free Crossed Squares}\n\nEllis, \\cite{ellis3}, in 1993 presented the notion of a free\ncrossed square for the case of groups. In this section, we\nintroduce a commutative algebra version of this definition and\ngive a construction of a free crossed square by using the second\norder Peiffer elements and the 2-skeleton of a `step-by-step'\nconstruction of a free simplicial algebra.\n\nWe firstly adapt Ellis's definition of the free crossed square on a pair of\nfunctions $(f_2,f_3)$ to the algebra context:\n\nLet $\\mathbf{S_{1},S_{2}}$ and $\\mathbf{S_{3}}$ be sets which for\nsimplicity we assume are finite. Suppose given a function\n$f_{2}:\\mathbf{S_{2}\\rightarrow }R$ from a set $\\mathbf{S_{2}}$ to a\nfree algebra $R$ on $\\mathbf{S_{1}}$. Let $\\partial :M\\rightarrow R$\nbe the free pre-crossed module on $f_{2}$. Using the action of $R$\non $M$ we can form the semi-direct product $M\\rtimes R$. The\ninclusion $\\mu :M\\rightarrow M\\rtimes R$ given by $m\\mapsto (m,0)$\nenables us to take $M$ as an ideal of $M\\rtimes R$. (Recall from\nexamples of crossed modules that any ideal inclusion is a crossed\nmodule with action by multiplication.) There is also another ideal\nof $M\\rtimes R$ coming from $M$, namely\n\\begin{equation*}\n\\overline{M}=\\{(m,r)\\in M\\rtimes R:\\partial m=-r\\}\n\\end{equation*}\nwith inclusion denoted $\\bar{\\mu}:\\overline{M}\\rightarrow M\\rtimes R$.\n\nAssume given a function from a set $\\mathbf{S_{3}}$ to $M$,\n$f_{3}:\\mathbf{S_{3}}\\rightarrow M$, which is to satisfy $\\partial\nf_{3}=0$. Then there is a corresponding function\n$\\bar{f}_{3}:\\mathbf{S_{3}}\\rightarrow \\overline{M}$ given by\n$y\\mapsto (f_{3}(y),0)$.\n\nWe say a crossed square\n\\begin{equation*}\n\\xymatrix{L\\ar[r]^{\\partial'_2}\\ar[d]_{\\partial_2}&\\overline{M}\\ar[d]^{\\mu'}\\\\\nM\\ar[r]_-{\\mu}&M\\rtimes R}\n\\end{equation*}\nis \\emph{totaly free} on the pair of functions $(f_2,f_3)$ if\n\n{\\noindent (i) }$(M,R,\\partial )$ is the free pre-crossed module on $f_2;$\n\n{\\noindent (ii)} $\\mathbf{S_3}$ is a subset of $L$ with $f_3$ and $\\bar f_3$\nthe restrictions of $\\partial _2$ and $\\partial _2^{\\prime }$ respectively;\n\n{\\noindent (iii)} for any crossed square\n\\begin{equation*}\n\\xymatrix{L'\\ar[r]^{\\tau'}\\ar[d]_{\\tau}&\\overline{M}\\ar[d]^{\\mu'}\\\\\nM\\ar[r]_{\\mu}&M\\rtimes R}\n\\end{equation*}\nand any function $\\nu :\\mathbf{S_3}\\rightarrow L'$ satisfying $\\tau \\nu =f_3$%\n, there is a unique morphism $\\Phi =(\\phi $,id,id,id) of crossed\nsquares:\n$$\n\\xymatrix@!0{\n  & L \\ar[rrrr] \\ar[dddd]\\ar[ddl]_{\\phi}\n      &  & & &\\overline{M} \\ar[dddd]\\ar@{=}[ddl]        \\\\\n      \\\\\n  L' \\ar[rrrr]\\ar[dddd]\n      &  &  & &\\overline{M} \\ar[dddd] \\\\\n      \\\\\n  & M \\ar[rrrr]\\ar@{=}[ddl]\n      &  & &   &M\\rtimes R \\ar@{=}[ddl]               \\\\\n      \\\\\n  M \\ar[rrrr]\n      &  &  &  & M\\rtimes R    }\n$$\nsuch that $\\phi \\nu ^{\\prime }=\\nu ,$ where $\\nu ^{\\prime\n}:\\mathbf{S_3}\\to L $ is the inclusion.\n\nWe denote such a free crossed square of algebras by\n$(L,M,\\overline{M},M\\rtimes R). $\n\nWe know the free pre-crossed module on\n$f:\\mathbf{S_{2}}\\rightarrow R$ is $\\partial\n:R^{+}[\\mathbf{S_{2}}]\\rightarrow R$, so the function\n$$f_{3}:\\mathbf{S_{3}}\\rightarrow M,\\ \\ (\\text{with}\\ \\ M=R^{+}[\\mathbf{S_{2}}])$$\nis precisely the data $(\\mathbf{S_{3}},f_{3})$ for 2-dimensional\nconstruction data in the simplicial context. We thus need to recall\nthe 2-dimensional construction data in a free simplicial algebra\n(cf. \\cite{ap2}). This 2-dimensional form can be pictured by the\ndiagram\n\\begin{equation*}\n\\xymatrix@C=3pc{\\mathbf{E^{(2)}}:\\cdots (R[s_0 (\\mathbf{S_2}),s_1\n\\mathbf{S_2}])[\\mathbf{S_3}]\\ar@<2ex>[r]^-{d_0,d_1,d_2}\\ar@<1ex>[r]\\ar@<0pt>[r]\n&\nR[\\mathbf{S_2}]\\ar@<1ex>[r]^-{d_0,d_1}\\ar@<0ex>[r]\\ar@<1ex>[l]\\ar@<2ex>[l]^-{s_0,s_1}&R\n\\ar@<1ex>[l]^-{s_0}}\n\\end{equation*}\nwith the simplicial morphisms given as in \\cite{ap3}. Here\n$\\mathbf{S_{2}} =\\{S_{1},\\ldots ,S_{n}\\}$ and\n$\\mathbf{S_{3}}=\\{S_{1}^{\\prime },\\ldots ,S_{m}^{\\prime }\\}$ are\nfinite sets with $\\pi _{0}(\\mathbf{E}^{(2)})\\cong\nB=R\/(t_{1},\\ldots ,t_{n})$ as an $R$-algebra where $t_i=\\partial\nS_i$.\n\n\\subsection{Free crossed squares exist}\n\n\\begin{thm}\nA totally free crossed square $(L,M,\\overline{M},M\\rtimes R)$ exists\non the 2-dimensional construction data and is given by\n$\\mathbf{M(E^{(2)},2)}$ where $\\mathbf{E^{(2)}}$ is the 2-skeletal\nfree simplicial algebra defined by the construction data.\n\\end{thm}\n\n\\begin{pf}\nSuppose given the 2-dimensional construction data for a free\nsimplicial algebra, \\textbf{E}, which we will take as above as the\ndata for a totally free crossed square. We will not assume\ndetailed knowledge of \\cite{ap1} so we start with $R$ and $\\\nf_{2}:\\mathbf{S_{2}}\\rightarrow R$ and form$\\\nM=R^{+}[\\mathbf{S_{2}}]=(S_{1},\\ldots ,S_{n}).$ This gives\n$\\partial _{1}:R^{+}[\\mathbf{S_{2}}]\\rightarrow R$ as free\npre-crossed module on $f_{2}.$ The semidirect product gives us\nback\n\\begin{equation*}\nR[\\mathbf{S_{2}}]\\cong M\\rtimes R\n\\end{equation*}\nand we can identify this with $\\mathbf{E_{1}^{(2)}}$. This identification\nalso makes\n\\begin{equation*}\nM\\cong \\mbox{\\rm Ker}d_{0}^{1}\n\\end{equation*}\nfor the $d_{0}^{1}$ of $\\mathbf{E^{(2)}}$.\n\nNext form $\\overline{M}=\\{(m,r)\\in M\\rtimes R:\\partial m=-r\\}$. As\n$m\\in R^{+}[\\mathbf{S_{2}}]$, writing $m=\\sum r_{\\alpha }S^{\\alpha\n}$ for multi-indices $\\alpha $, we get $\\partial m=\\sum r_{\\alpha\n}t^{\\alpha }$ where $t_{i}=f_{2}(S_{i})$. Thus we can identify\n$\\overline{M}$ with $(S_{1}-t_{1},\\ldots ,S_{n}-t_{n})$ which is\nexactly Ker$d_{1}^{1}$ (for this see \\cite{ap1}).\n\nNow $\\ f_{3}:\\mathbf{S_{3}}\\rightarrow \\mbox{\\rm Ker}\\partial\n_{1}= \\mbox{\\rm Ker}(\\partial :NE_{1}^{(2)}\\rightarrow\nNE_{0}^{(2)})\\subset R^{+}[\\mathbf{ S_{2}}].$ We know that this\nallows us to construct $\\mathbf{E_{2}^{(2)}}$, and hence\n$\\mathbf{E_{n}^{(2)}}$ for $n\\geqslant 3$, and in addition that taking\n\\begin{equation*}\nL=NE_{2}^{(2)}\/\\partial _{3}(NE_{3}^{(2)}),\n\\end{equation*}\ngives a crossed square\n\\begin{equation*}\n\\xymatrix{L\\ar[r]^{\\partial'_2}\\ar[d]_{\\partial_2}&\\overline{M}\\ar[d]^{\\lambda'}\\\\\nM\\ar[r]_{\\lambda}&E^{(2)}_1}\n\\end{equation*}\nwhich is $M(\\mathbf{E}^{(2)},2)$. We claim this is the totally free crossed\nsquare on the construction data.\n\nAt this stage it is worth nothing that there would seem to be no simple\nadjointness statement between $\\mathbf{M(-,2)}$ and some functor that would\ngive a quick proof of freeness. The problem being that $\\mathbf{M(-,2)}$\nseems to be an adjoint only up to some sort of coherent homotopy. To avoid\nthis difficulty we use a more combinatorial approach involving the higher\norder Peiffer elements and a more concrete description of $L$.\n\nIn \\cite{ap1}, the first author and Porter analysed in general the\nstructure of algebras of boundaries such as $\\partial\n_{3}(NE_{3}^{(2)}).$ There they showed that $NE_{1}^{(2)}$ is\ngenerated as an ideal by elements of the following forms:\n\nFor all $x\\in NE_{1}^{(2)},~y\\in NE_{2}^{(2)}$,\n\\begin{align*}\nC_{(1,0)(2)}(x\\otimes y) & =  (s_{1}s_{0}x-s_{2}s_{0}x)s_{2}y, \\\\\nC_{(2,0)(1)}(x\\otimes y) & =  (s_{2}s_{0}x-s_{2}s_{1}x)(s_{1}y-s_{2}y), \\\\\nC_{(2,1)(0)}(x\\otimes y) & = s_{2}s_{1}x(s_{0}y-s_{1}y+s_{2}y);\\\\\n\\intertext{ whilst for all $x,~y\\in NE_{2}$,}\nC_{(1)(0)}(x\\otimes y) & =  s_{1}x(s_{0}y-s_{1}y)+s_{2}(xy), \\\\\nC_{(2)(0)}(x\\otimes y) & =  (s_{2}x)(s_{0}y), \\\\\nC_{(2)(1)}(x\\otimes y) & =  s_{2}x(s_{1}y-s_{2}y).\n\\end{align*}\nWe know that $\\partial _{3}(NE_{3}^{(2)})$ is generated by elements\nof the forms\n\\begin{equation*}\n(s_{1}s_{0}d_{1}S_{i}-s_{0}S_{i})S_{j}^{\\prime },\\ \\\n(s_{0}S_{i}-s_{1}S_{i})(s_{1}d_{2}S_{j}^{\\prime }-S_{j}^{\\prime }),\\ \\\ns_{1}S_{i}(s_{0}d_{2}S_{j}^{\\prime }-s_{1}d_{2}S_{j}^{\\prime }+S_{j}^{\\prime\n}),\n\\end{equation*}\nand for $S_{i}^{\\prime },$ $S_{j}^{\\prime }\\in \\mathbf{S_{2}}$,\n\\begin{equation*}\nS_{i}^{\\prime }(s_{1}d_{2}S_{j}^{\\prime }-S_{j}^{\\prime }),\\ \\ S_{i}^{\\prime\n}(S_{j}^{\\prime }+s_{0}d_{2}S_{j}^{\\prime }-s_{1}d_{2}S_{j}^{\\prime }),\\ \\\n(s_{0}d_{2}S_{i}^{\\prime }-s_{1}d_{2}S_{i}^{\\prime }+S_{i}^{\\prime\n})(s_{1}d_{2}S_{j}^{\\prime }-S_{j}^{\\prime }),\n\\end{equation*}\nwhich are the second order Peiffer elements defined in \\cite{ap1},\nwhere $S_{i}\\in NE_{1}=\\mbox{\\rm Ker}d_{0}=R^{+}[\\mathbf{S_{2}}]$\nand $S_{i}^{\\prime }\\in\nNE_{2}=R[s_{0}(\\mathbf{S_{2}})]^{+}[s_{1}(\\mathbf{S_{2}}),\\mathbf{S_{3}}]\\cap\n(s_{0}(\\mathbf{S_{2}})-s_{1}(\\mathbf{S_{2}})).$\n\nThe above diagram can thus be realised as\n\\begin{equation*}\n\\xymatrix{\\dfrac{R[s_0({\\bf S_2})]^+[s_1({\\bf S_2}),{\\bf S_3}]\\cap\n(s_0({\\bf S_2})-s_1({\\bf\nS_2}))}{P_2}\\ar[r]^-{\\partial'_2}\\ar[d]_-{\\partial_2}&\\overline{R^+\n[{\\bf S_2}]}\n\\ar[d]^{\\lambda'}\\\\\nR^{+}[\\mathbf{S_2}]\\ar[r]_{\\lambda}&R[\\mathbf{S_2}]}\n\\begin{array}{cc}\n&\\\\\n&\\\\\n&(\\ast)\\\\\n\\end{array}\n\\end{equation*}\nwhere $P_{2}$ is the second order Peiffer ideal which is in fact\njust $\\partial _{3}(NE_{3}^{(2)}).$\n\nGiven any crossed square $(T,M,\\overline{M},M\\rtimes R)$ with a\nfunction $\\nu :\\mathbf{S_{3}}\\rightarrow T$, then there exists a\nmorphism\n\\begin{equation*}\n\\phi :(L,M,\\overline{M},M\\rtimes R)\\longrightarrow\n(T,M,\\overline{M},M\\rtimes R)\n\\end{equation*}\ngiven by\n\\begin{equation*}\n\\phi (S_{i}^{\\prime }+P_{2})=\\nu (S_{i}^{\\prime })\n\\end{equation*}\nsuch that $\\phi \\nu ^{\\prime }=\\nu $ where $\\nu\n:\\mathbf{S_{3}}\\rightarrow L$ is a function. The existence of\n$\\phi $ follows by using the freeness property of the algebra\n$NE_{2}^{(2)}$ and then restricting to\n$R[s_{0}(\\mathbf{S_{2}})]^{+}[s_{1}(\\mathbf{S_{2}}),\n\\mathbf{S_{3}}]\\cap\n(s_{0}(\\mathbf{S_{2}})-s_{1}(\\mathbf{S_{2}})).$ The ideal\ngenerating elements of $P_{2}$ are then easily shown to have\ntrivial image in $T$ as that algebra is part of the second crossed\nsquare.\n\nThus diagram $(\\ast )$ is the desired totally free crossed square on\nthe 2-dimensional construction data. The crossed square properties\nof $(L,M,\\overline{M},M\\rtimes R)$ may be easily verified or derived\nfrom the fact that this is exactly $\\mathbf{M(E}^{(\\mathbf{2})},2).$\n\\end{pf}\n\n\n\\vspace{0.5cm}\n\n\\textbf{Remark}:\n\n At this stage, it is important to note that nowhere\nin the argument was use made of the freeness of the 1-skeleton. If\n\\textbf{E} is any 1-skeletal simplicial algebra and we form a new\nsimplicial algebra \\textbf{F} by adding in a set $\\mathbf{S}_{3}$\nof new generators in dimension 2, so that for instance,\n$F_{2}=E_{2}^{+}[\\mathbf{S}_{3}],$ a free algebra on\n$\\mathbf{S}_{3}$, then we can use \\thinspace\n$M=NE_{1}=$Ker$d_{0}^{1}$ as before even though it need not be\nfree. The corresponding $\\overline{M}$ is then isomorphic to\nKer$d_{1}^{1}$ with the bottom right hand corner being $E_{1}$.\nThe `construction data' is now replaced by data for killing some\nelements of $\\pi _{1}(E),$ specified by\n$f_{3}:\\mathbf{S}_{3}\\rightarrow M.$ We introduce the term\n`totally free crossed square' for the type of free crossed square\nconstructed in the above theorem, using free crossed square for\nthe more general situation in which $(M,E,\\partial )$ and $f_{3}$\nare specified and no requirement $(M,E,\\partial )$ to be a free\npre-crossed module is made.\n\n\n\\subsection{ The $n$-type of the $k$-skeleton}\n\nAs in the other paper in this series, we will use the\n`step-by-step' construction of a free simplicial algebra to\nobserve the way in which the models react to the various steps of\nthe construction.\n\nBy a `step-by-step' construction of a free simplicial algebra,\nthere are simplicial inclusions\n\\begin{equation*}\n\\mathbf{E}^{(0)}\\subseteq \\mathbf{E}^{(1)}\\subseteq\n\\mathbf{E}^{(2)}\\subseteq \\cdots\n\\end{equation*}%\nThe functor, $\\mathbf{M(\\ ,\\ }n\\mathbf{)}$, from the category of\nsimplicial algebras to that of crossed $n$-cubes gives the\ncorresponding inclusions\n\\begin{equation*}\n\\mathbf{M}(\\mathbf{E}^{(0)},\\ n)\\hookrightarrow\n\\mathbf{M}(\\mathbf{E}^{(1)},\\ n)\\hookrightarrow\n\\mathbf{M}(\\mathbf{E}^{(2)},\\ n)\\hookrightarrow \\mathbf{\\cdots }\n\\end{equation*}%\nWe investigate $\\mathbf{M(E^{(i)}},n)$, for $n=0,1,2$, and varying\n$i$.\n\nFirstly look at $\\mathbf{M(E^{(0)}},n),$where the 0-skeleton\n$\\mathbf{E}^{(0)}\\,$ is\n\\begin{equation*}\n\\begin{array}{lccc}\n\\mathbf{E}^{(0)}: & \\cdots \\longrightarrow R\\longrightarrow R\\longrightarrow\nR & \\overset{f}{\\longrightarrow } & B%\n\\end{array}%\n\\end{equation*}\nwith the $d_i^n=s_j^n=\\ $identity homomorphisms.\n\nFor $n=0,\\,$there is an equality\n$\\mathbf{M(E^{(0)}},0)=E_0^{(0)}\/d_1(\\mbox{\\rm Ker}d_0)=R, $\n and so $\\mathbf{M(E^{(0)},}0)$ is just an algebra of $0$-simplices of \\textbf{E}.\n\nFor $n=1, \\mathbf{M(E^{(0)}},1)$ is $NE_1^{(0)}\/\\partial\n_2NE_2^{(0)}\\rightarrow E_0. $ It is easy to show that\n$NE_1^{(0)}\/\\partial _2NE_2^{(0)}$ is trivial in the 0-skeleton\n$\\mathbf{E}^{(0)}$ and hence\n\\begin{equation*}\n\\mathbf{M(E^{(0)}},1)\\cong (0\\longrightarrow R).\n\\end{equation*}\n\nFor $n=2$, \\ $\\mathbf{M(E^{(0)}},2)$ is the trivial crossed square\n\\begin{align*}\n\\begin{aligned}\\xymatrix{ NE_2 \/ d^{3}_{3} (NE_3 ) \\ar[d]\\ar[r]&\\text{Ker}d_0\n   ^1\\ar[d]\\\\\n   \\text{Ker}d_1^1\\ar[r]&E_1}\\end{aligned}&= \\begin{aligned}\\xymatrix{0 \\ar[d]\\ar[r]&0\\ar[d]\\\\\n   0\\ar[r]&R.}\\end{aligned}\n\\end{align*}\nNext take $\\mathbf{M(E^{(1)}},n)$ and recall that the 1-skeleton\n\\textbf{E}$^{(1)}$ is\n\\begin{equation*}\n\\xymatrix@C=3pc{\\mathbf{E^{(1)}}:\\cdots R[s_0 (\\mathbf{S_2}),s_1(\n\\mathbf{S_2})]\\ar@<2ex>[r]^-{d_0,d_1,d_2}\\ar@<1ex>[r]\\ar@<0pt>[r] &\nR[\\mathbf{S_2}]\\ar@<1ex>[r]^-{d_0,d_1}\\ar@<0ex>[r]\\ar@<1ex>[l]\\ar@<2ex>[l]^-{s_0,s_1}&R\n\\ar@<1ex>[l]^-{s_0}\\ar[r]^{f}&R\/I.}\n\\end{equation*}\nFor $n=0$, it follows that $\\mathbf{M(E^{(1)},} 0)$ is $\nE_0^{(1)}\/d_1(\\mbox{\\rm Ker}d_0)\\cong R\/I $ which is $\\pi\n_0(\\mathbf{E} ^{(1)})\\cong\\pi _0(\\mathbf{E})$.\n\nLet $n=1.$ We have that%\n\\begin{align*}\n\\mathbf{M(E}^{(1)}\\mathbf{,\\ }1\\mathbf{)} & = NE_{1}^{(1)}\/\\partial\n_{2}NE_{2}^{(1)}\\rightarrow E_{0}^{(1)} \\\\\n& =  R^{+}[\\mathbf{S_{2}}]\/P_{1}\\rightarrow R%\n\\end{align*}\nwhich is the free crossed module. In fact this is the free crossed\nmodule on the (generalised) presentation\n$(\\mathbf{S_{1};S_{2}},f_{2})$. As pointed out in \\cite{ap1}, it is\noften convenient to generalise the notion of a presentation\n$P=(R\\,;x_{1},\\ldots ,x_{n})$ of an $R$-algebra $B$ in this way and\n$\\mathbf{E}^{(1)}$ is the 1-skeleton of the free simplicial algebra\ngenerated by this presentation, then\n\\begin{equation*}\n\\delta :NE_{1}^{(1)}\/\\partial _{2}(NE_{2}^{(1)})\\longrightarrow NE_{0}^{(1)}\n\\end{equation*}%\nis the free crossed module on $\\{S_{1},\\ldots ,S_{n}\\}\\rightarrow\nR$. This has a neat description (cf. \\cite{porter1}) as follows: The\nPeiffer ideal $P_1 =\\partial_2(NE^{(1)}_2)$ contains all such terms\nas $S_iS_j-\\delta(S_i)S_j$, so any polynomial in the $S_i$'s can be\nreduced mod $P_1$ to a linear form, hence each coset has a\nrepresentative of the form $\\sum r_iS_i$. As $S_iS_j=S_jS_i$, these\nrepresentatives are nonunique and so the free crossed $C$ is $R^n$\nfactored by all $\\delta(S_i)S_j-\\delta(S_j)S_i$ we thus have\n\\begin{equation*}\n\\pi _{1}(\\mathbf{M}(\\mathbf{E}^{(1)},1))\\cong \\mathrm{Ker}(C\\longrightarrow\nR)\\cong H_{2}(B,B)\n\\end{equation*}\nthe second Andr\\'{e}-Quillen homology group, where $C\\cong\nR^{n}\/\\mathrm{Im}d, $ for $d: \\Lambda ^{2}R^{n}\\rightarrow R$, the\nsecond Kozsul differential, see \\cite{porter1} for details. Thus\n\\begin{align*}\n\\pi _{0}(\\mathbf{M}(\\mathbf{E}^{(1)},1)) & \\cong  B \\\\\n\\pi _{1}(\\mathbf{M}(\\mathbf{E}^{(1)},1)) & \\cong  H_{2}(B,B)\n\\intertext{whilst} \\pi _{i}(\\mathbf{M}(\\mathbf{E}^{(1)},1))&\\cong\n0.\n\\end{align*}\n\nFor $n=2$,\n$$\n NE_{2}^{(1)}=(R[s_{0}(\\mathbf{S_{2}})]^{+}[s_{1}(\\mathbf{S_{2}})])\\cap\n(s_{0}(\\mathbf{S_{2}})-s_{1}(\\mathbf{S_{2}})),\n$$\n\n$\\mathbf{M(E}^{(1)},2)$ simplifies to give (up to isomorphism)\n\\begin{align*}\n\\begin{aligned}\n   \\xymatrix{ NE_2 \/ d^{3}_{3} (NE_3 ) \\ar[d]\\ar[r]&\\text{Ker}d_0\n   ^1\\ar[d]\\\\\n   \\text{Ker}d_1^1\\ar[r]&E_1}\n   \\end{aligned}&=\n\\begin{aligned}\n   \\xymatrix{\\dfrac{(R[s_0({\\bf S_2})]^+[s_1({\\bf S_2})])\\cap\n(s_0({\\bf S_2})-s_1({\\bf S_2}))}{P_2}\n\\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n   R^+[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}]}\n   \\end{aligned}\n  \\end{align*}\nwhich is a crossed square.\n\nLet us next look at $\\mathbf{M(E}^{(2)},n).$ Recall the 2-skeleton\n$\\mathbf{E}^{(2)}$\n\\begin{equation*}\n\\xymatrix@C=3pc{(R[s_0 (\\mathbf{S_2}),s_1\n\\mathbf{S_2}])[\\mathbf{S_3}]\\ar@<2ex>[r]^-{d_0,d_1,d_2}\\ar@<1ex>[r]\\ar@<0pt>[r]\n&\nR[\\mathbf{S_2}]\\ar@<1ex>[r]^-{d_0,d_1}\\ar@<0ex>[r]\\ar@<1ex>[l]\\ar@<2ex>[l]^-{s_0,s_1}&R\n\\ar@<1ex>[l]^-{s_0}\\ar[r]^f&R\/I.}\n\\end{equation*}\nThe following equalities can be easily obtained by direct\ncalculation: \\thinspace for $n=0,$\n\\begin{equation*}\n\\mathbf{M(E^{(2)}},0)=E_{0}\/d_{1}(\\mbox{\\rm Ker} d_{0})\\cong \\pi\n_{0}(\\mathbf{E}^{(2)})=\\mathbf{M(E}^{(1)},0).\n\\end{equation*}\nFor $n=1,$\n\\begin{equation*}\n\\mathbf{M(E}^{(2)},1)\\cong (R^{+}[\\mathbf{S_{2}}\n]\/P_{1}\\rightarrow R)=\\mathbf{M(E}^{(1)},1).\n\\end{equation*}%\nand there is an isomorphism\n\\begin{equation*}\n\\pi _{2}(\\mathbf{E}^{(2)})\\cong \\mbox{\\rm Ker}\\left(\nNE_{2}^{(2)}\/\\partial _{3}(NE_{3}^{(2)})\\longrightarrow\nE_{1}^{(2)}\\right).\n\\end{equation*}\n Finally, let $n=2.$ Since by an earlier result of this section,\n $\\mathbf{M(E}^{(2)},2)$ corresponds to the free\ncrossed square, we obtain:\n\\begin{align*}\n\\begin{aligned}\n   \\xymatrix{ NE_2^{(2)} \/ d^{3}_{3} (NE_3^{(2)} ) \\ar[d]\\ar[r]&\\text{Ker}d_0\n   ^1\\ar[d]\\\\\n   \\text{Ker}d_1^1\\ar[r]&E_1}\n         \\end{aligned}&=\n         \\begin{aligned}\n    \\xymatrix{\\dfrac{R[s_0({\\bf S_2})]^+[s_1({\\bf S_2}),\\bf{S_3}]\\cap\n(s_0({\\bf S_2})-s_1({\\bf S_2}))}{P_2}\n\\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n   R^+[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}].} \\\\\n  \\end{aligned}\n  \\end{align*}\nThis reduces to the earlier case if $\\mathbf{S_3}$ is empty. Thus we\nhave the following relations\n$$\n\\mathbf{M(E}^{(2)},0)=\\mathbf{M(E}^{(1)},0), \\quad\n\\mathbf{M(E}^{(2)},1)=\\mathbf{M(E}^{(1)},1)\n$$\nbut $\\mathbf{M(E}^{(2)},2)$ and $\\mathbf{M(E}^{(3)},2)$ need not to\nbe the same due to the additional influence of $\\mathbf{S_3}$. Of\ncourse it is clear that, in general:\n$$\n\\mathbf{M(E}^{(i)},n)=\\mathbf{M(E}^{(i+1)},n) \\quad \\text{if}\\ i\\geqslant\nn+1.\n$$\nClearly these top left hand corner terms are unwieldy to handle\nand we will seek in section \\ref{sec5} an alternative description.\n\n\n\n\\section{Squared Complexes}\n\nThe first author and M. Ko\\c{c}ak defined $n$-crossed complexes of\nalgebras in \\cite{ak} as the analogue for commutative algebras of\nthe notion introduced by Ellis, \\cite{ellis3}, in homotopy theory.\nIn this paper we will only need the case $n=2$, which we shall\ncall a \\emph{squared complex}; it consists of a diagram of algebra\nhomomorphisms\n$$\n\\xymatrix{(*)&&\\cdots\\ar[r]&C_5\\ar[r]^{\\partial_5}&C_4\\ar[r]^{\\partial_4}&L\\ar[r]^{\\lambda'}\\ar[d]_{\\lambda}&N\n\\ar[d]^{\\mu'}\\\\\n&&&&&M\\ar[r]_{\\mu}&R}\n$$\ntogether with an action of $R$ on $L,N,M$ and $C_i$ for $i\\geqslant 4,$\nand a function $h:M\\times N\\rightarrow L.$ The following axioms need\nto be satisfied.\n\n$(i)$ The square\n\\begin{equation*}\n\\left(\n\\begin{array}{cc}\n  \\xymatrix{L\\ar[r]\\ar[d]&N\\ar[d]\\\\\n M\\ar[r]&R}\n\\end{array}\n\\right)\n\\end{equation*}\nis a crossed square.\n\n$(ii)$  $C_{n}$ is an $A$-module for $n\\geqslant 4$ with $A=R\/\\left\\{ \\mu\n(M)+\\mu ^{\\prime }(N)\\right\\} .$\n\n$(iii)$ The action of $R$ on $C_{n},n\\geqslant 4,$ is such that $\\mu (M)$\nand $\\mu ^{\\prime }(N)$ operate trivially. Thus each $C_{n}$ is an\n$A$-module.\n\n$(iv)$ each $\\partial _{n}$ is $A$-module homomorphism and for\n$n\\geqslant 4$, $\\partial _{n}\\partial _{n+1}=0$.\n\nA \\emph{morphism} of square complexes\n\\begin{equation*}\n\\phi :\\left(\n\\begin{array}{cc}\nC_i, & \\left(\n\\begin{array}{cc}\nL & N \\\\\nM & R%\n\\end{array}\n\\right)%\n\\end{array}\n\\right) \\longrightarrow \\left(\n\\begin{array}{cc}\nC_i^{\\prime }, & \\left(\n\\begin{array}{cc}\nL^{\\prime } & N^{\\prime } \\\\\nM^{\\prime } & R^{\\prime }\n\\end{array}\n\\right)%\n\\end{array}\n\\right)\n\\end{equation*}\nconsists of a morphism of crossed squares $(\\phi _L,\\phi _M,\\phi\n_N,\\phi _R)$ together with a family of $\\phi _R$-equivariant\nhomomorphisms $\\phi _i$,  $i\\geqslant 4$ satisfying $\\phi _L\\partial\n_4=\\partial _4^{\\prime }\\phi _4$ and $\\phi _{i-1}\\partial\n_i=\\partial _i^{\\prime }\\phi _i$ for $i\\geqslant 5.$ There is clearly a\ncategory \\textbf{SqComp} of squared complexes.\n\nBy a (totally) free squared complex, we will mean one in which the\ncrossed square is (totally) free, and in which each $C_n$ is free\nas a $\\pi _0$-module for $i\\geqslant 3.$\n\n\\begin{prop}\nThere is a functor\n\\begin{equation*}\nC(\\ ,2):\\mathbf{SimpAlg} \\longrightarrow \\mathbf{SqComp}\n\\end{equation*}\nsuch that free simplicial algebras are sent to totally free\nsquared complexes.\n\\end{prop}\n\n\\begin{pf}\nLet $\\mathbf{E}$ be a simplicial algebra. We will define a squared\ncomplex $ C(\\mathbf{E},2)$ by specifying $C(\\mathbf{E},2)_{A}$ for\neach $A\\subseteq \\langle 2\\rangle$ and for $n\\geqslant\n3,C(\\mathbf{E},2)_{n}.$ As usual, (cf. the other paper in this\nseries {\\cite{ap1, ap2, ap3}), we will denote by $D_{n}$ the ideal\nof $NE_{n}$ generated by degenerate elements. }\n\nFor $A\\subseteq \\langle 2\\rangle$, we define in particular\n\\begin{equation*}\nC(\\mathbf{E},2)_{\\langle\n2\\rangle}=\\mathbf{M}(\\mathbf{sk}_{2}\\mathbf{E},2)_{\\langle\n2\\rangle}=\\dfrac{NE_2}{\\partial_3(NE_3 \\cap D_3)}.\n\\end{equation*}%\nWe do not need to define $\\mu _{i}$ and the $h$-maps relative to\nthese algebras as they are already defined in the crossed square\n$\\mathbf{M}(\\mathbf{sk}_{2} \\mathbf{E},2)_{A}.$\n\nFor $n\\geqslant 3,$ we set\n\\begin{equation*}\nC(\\mathbf{E},2)_{n}=\\frac{NE_{n}}{(NE_{n}\\cap D_{n})+d_{n+1}(NE_{n+1}\\cap\nD_{n+1})}.\n\\end{equation*}\nAs this is part of the crossed complex associated to \\textbf{E},\nwe can take the structure maps to be those of that crossed\ncomplex, cf. \\cite{ap1}. The terms are all modules over the\ncorresponding, $\\pi _{0}$ as is easily checked. The final missing\npiece, $\\partial _{3},$ of the structure is induced by the\ndifferential $\\partial _{3}$ of $NE.$\n\nThe axioms for a squared complex can now be verified the known\nresults for crossed squares and for crossed complexes with a direct\nverificatiton of those axioms relating to the interaction of the two\nparts of the structure, much as in \\cite{ap1}.\n\nNow suppose the simplicial algebra is free. The proof of the\nfreeness of $\\mathbf{M}(sk_{2}\\mathbf{E},2)$ together with the\nfreeness of the crossed complex of a free simplicial algebra,\n\\cite{ap1}, now completes the proof.\n\\end{pf}\n\nSuppose that $\\rho $ is a general squared complex. The \\emph{homotopy modules%\n} $\\pi _{n}(\\rho ),n\\geqslant 0$ of $\\rho $ are defined in \\cite{ak} to\nbe the homology modules of the complex\n\\begin{equation*}\n\\xymatrix{\\cdots\\ar[r]^{\\partial_6}&C_5\\ar[r]^{\\partial_5}&C_4\n\\ar[r]^{\\partial_4}& L\\ar[r]^-{\\partial_3}&M\\rtimes\nN\\ar[r]^-{\\partial_2}\\ar[r]&0}\n\\end{equation*}%\nwith $\\partial _{3}(l)=(-\\lambda ^{\\prime }l,\\lambda l)$ and\n$\\partial _{2}(m,n)=\\mu (m)+\\mu ^{\\prime }(n)$. The axioms of a\ncrossed square guarantee (see \\cite{ap3}) that $\\partial _{3}$ and\n$\\partial _{2}$ are homomorphisms with $\\partial _{4}(C_{4})$ an ideal in Ker$(\\partial _{3})$, \\ $%\n\\partial _{3}(L)$ an ideal in Ker$(\\partial _{2})$, and $\\partial _{2}(M\\rtimes\nN)$ an ideal in $R$.\n\n\\begin{prop}\nThe homotopy groups of $C(\\mathbf{E},2)$ are isomorphic to those of $\\mathbf{%\nE}$ itself.\n\\end{prop}\n\n\\begin{pf}\nAgain this is a consequence of well-known results on the two parts of the\nstructure.\n\\end{pf}\n\n\n\n\\section{Alternative Description of Freeness \\label{sec5}}\n\nIn the context of CW-compexes, Ellis \\cite{ellis3} gave a neat\ndescription of the top algebra $L$ in (totally) free crossed\nsquares.  A free simplicial algebra is the algebraic analogue of a\nCW-complex so one would expect a similar result to hold in this\nsetting. For this we need two constructions.\n\n\n\\subsection{Tensor Products}\n\nSuppose that $\\mu :M\\rightarrow R$ and $\\nu :N\\rightarrow R$ are crossed\nmodules of commutative algebras over $R$. The algebras $M$ and $N$ act on\neach other, and themselves, via the action of $R$. The \\emph{tensor product}\n$M\\otimes N$ is the algebra generated by the symbols $m\\otimes n$ for $m\\in\nM,\\ n\\in N$ and $r\\in R$ subsect to the relations%\n\\begin{equation*}\n\\begin{array}{rrll}\n\\text{(i)} & r\\left( m\\otimes n\\right) & = & rm\\otimes n=m\\otimes rn \\\\\n\\text{(ii)} & \\left( m+m^{\\prime }\\right) \\otimes n & = & m\\otimes\nn+m^{\\prime }\\otimes n \\\\\n& m\\otimes (n+n^{\\prime }) & = & m\\otimes n+m\\otimes n^{\\prime } \\\\\n\\text{(iii)} & \\left( m\\otimes n\\right) \\left( m^{\\prime }\\otimes n^{\\prime\n}\\right) & = & \\left( mm^{\\prime }\\otimes nn^{\\prime }\\right)%\n\\end{array}\n\\end{equation*}\nwhere $m^{\\prime }\\in M$ and $n^{\\prime }\\in N.$ There are\nmorphisms $ \\lambda :M{\\otimes }N\\rightarrow M,\\ m{\\otimes\n}n\\mapsto m\\cdot n=m\\nu (n)$ and $\\lambda ^{\\prime }:M{\\otimes\n}N\\rightarrow N,\\ m{\\otimes }n\\mapsto n\\cdot m=\\mu (m)n$. The\nalgebra $R$ acts on $M\\otimes N$ by $r\\cdot (m\\otimes n)=r\\cdot\nm\\otimes n=m\\otimes r\\cdot n,$ and there is a function $ h:M\\times\nN\\rightarrow M\\otimes N,$ \\ $(m,n)\\mapsto m{\\otimes }n$. It is\nverified in \\cite{ak} that this structure is a crossed square\n\\begin{equation*}\n\\xymatrix{M\\otimes N \\ar[r]^-{\\lambda}\\ar[d]_{\\lambda'}&N\\ar[d]^{\\nu}\\\\\nM\\ar[r]_{\\mu}&R}\n\\end{equation*}%\nwith the universal property of extending the corner\n\\begin{equation*}\n\\xymatrix{&N\\ar[d]^{\\nu}\\\\\nM\\ar[r]_{\\mu}&R.}\n\\end{equation*}\n\n\n\\subsection{Coproducts}\n\nThe following construction is due to Shammu \\cite{nizar}.\n\nLet $(M,R,\\partial _{1}),(N,R,\\partial _{2})$ be crossed\n$R$-modules. Then $N $ acts on $M$, and $M$ acts on $N$, via the\ngiven actions of $R$. Let $ M\\rtimes N$ denote the semidirect\nproduct with the multiplication given by\n\\begin{equation*}\n(m,n)(m^{\\prime },n^{\\prime })=(mm^{\\prime },n\\partial _{2}(m^{\\prime\n})+\\partial _{2}(m)n^{\\prime }+nn^{\\prime })\n\\end{equation*}\nand injections\n\\begin{equation*}\n\\begin{array}{ccccc}\ni^{\\prime }\\colon & M\\rightarrow M\\rtimes N & \\mbox{ and } & j^{\\prime\n}\\colon & N\\rightarrow M\\rtimes N \\\\\n& m\\mapsto (m,0) &  &  & n\\mapsto (0,n).\n\\end{array}%\n\\end{equation*}\nWe define the pre-crossed module\n\\begin{equation*}\n\\begin{array}{cccl}\n\\underline{\\delta }\\colon & M\\rtimes N & \\longrightarrow & R \\\\\n& (m,n) & \\longmapsto & \\partial _{1}(m)+\\partial _{2}(n).\n\\end{array}\n\\end{equation*}\nLet $P$ be the ideal of $M\\rtimes N$ generated by elements of the\nform\n\\begin{equation*}\n(m,n)(m^{\\prime }n^{\\prime })-\\underline{\\delta }(m,n)(m^{\\prime },n^{\\prime\n})=(-\\partial _{1}(m)n,m\\partial _{2}(n))\n\\end{equation*}\nfor all $(m,n),(m^{\\prime },n^{\\prime })\\in M\\rtimes N.$ Thus we are able to\nform the quotient algebra $M\\rtimes N\/P$ and obtain an induced morphism\n\\begin{equation*}\n\\partial \\colon M\\rtimes N\/P\\longrightarrow R\n\\end{equation*}%\ngiven by\n\\begin{equation*}\n\\partial (m,n)+P=\\partial _{1}m+\\partial _{2}n.\n\\end{equation*}%\nLet $q\\colon M\\rtimes N\\rightarrow M\\rtimes N\/P$ be projection and\nlet $ i=qi^{\\prime },j=qj^{\\prime }.$ Then $M\\sqcup N=M\\rtimes\nN\/P,$ with, the morphism $i,j,$ is the \\emph{coproduct } of $M,N$\nin the category $\\mathbf{ XMod_{k}}$. The above notation can be\nsummarised in the following diagram:\n\\begin{equation*}\n\\xymatrix{M\\ar[dr]_{i}\\ar[dd]_{\\partial_1}\\ar[rr]^{i'}&&M\\rtimes\nN\\ar[dl]^{q}\\\\\n&M\\rtimes N\/P \\ar[dl]^{\\partial}&\\\\\nR&&N.\\ar[uu]^{j'}\\ar[ll]^{\\partial_2}\\ar[ul]^{j}}\n\\end{equation*}\n\n\\begin{prop}\nLet\n\\begin{equation*}\n\\left(\n\\begin{array}{cc}\n\\xymatrix{L\\ar[r]\\ar[d]&\\overline{M}\\ar[d]\\\\\nM\\ar[r]&M\\rtimes R}\n\\end{array}%\n\\right)\n\\end{equation*}\nbe a free crossed square on the 2-dimensional construction data or\non functions $(f_{2},f_{3})$ as described above. Let $\\partial\n:C\\rightarrow M\\rtimes R$ be the free crossed module on the\nfunction \\textbf{S}$_{3}\\rightarrow M\\rtimes R$ given by $y\\mapsto\n(f_{3}y,0)$. Form the crossed module $\\partial ^{\\prime }:M\\otimes\n\\overline{M}\\rightarrow M\\rtimes R$, then\n\\begin{equation*}\nL\\cong \\{(M\\otimes \\overline{M})\\sqcup C\\}\/\\thicksim\n\\end{equation*}\nwhere $\\thicksim$ corresponds to the relations\n\\begin{equation*}\n\\begin{array}{lllll}\n1) &  & i_{M\\otimes \\overline{M}}(\\partial c\\otimes \\overline{n}) & \\thicksim & j(c)-j(%\n\\overline{n}\\cdot c) \\\\\n2) &  & i_{M\\otimes \\overline{M}}(m\\otimes \\partial c) & \\thicksim & j(m\\cdot c)-j(c)%\n\\end{array}%\n\\end{equation*}%\nfor $c\\in C,m\\in M$ and $\\overline{n}\\in \\overline{M.}$\n\nThe homomorphisms $L\\rightarrow M,L\\rightarrow \\overline{M}$ are\ngiven by the homomorphisms\n\\begin{equation*}\n\\lambda :M\\otimes \\overline{M}\\rightarrow M\\text{ and }\\lambda ^{\\prime\n}:M\\otimes \\overline{M}\\rightarrow \\overline{M}\n\\end{equation*}%\nand $\\partial :C\\rightarrow M\\cap \\overline{M}.$ The $h$-map of the crossed\nsquare is given by\n\\begin{equation*}\nh(m,\\overline{n})=i(m\\otimes \\overline{n})\n\\end{equation*}%\nfor $m,n\\in M.$\n\\end{prop}\n\n\\begin{pf}\nThis comes by direct verification using the universal properties of tensors\nand coproducts.\n\\end{pf}\n\n\\bigskip\n\n\\textbf{Remark}: For future applications it is again important to\nnote that the result is not dependent on the crossed square being\n\\emph{totaly} free. If $M\\rightarrow R$ is any pre-crossed module,\none can form the corner\n\\begin{equation*}\n\\xymatrix{&M\\ar[d]^{\\nu}\\\\\n\\overline{M}\\ar[r]_-{\\mu}&M\\rtimes R}\n\\end{equation*}\ncomplete it to a crossed square via $M\\otimes \\overline{M}$ and then add in $%\n\\mathbf{S_{3}}\\rightarrow M.$ Nowhere does this use freeness of\n$M\\rightarrow R$.\n\n\\begin{cor}\n\\label{c} Let \\textbf{E}$^{(1)}$ be the 1-skeleton of a free\nsimplicial algebra. Given the free crossed square\n$\\mathbf{M}(\\mathbf{E}^{(1)},2)$ described above, then\n\\begin{equation*}\nNE_{2}^{(1)}\/\\partial _{3}NE_{3}^{(1)}\\cong \\mbox{\\rm\nKer}d_{1}^{1}\\otimes _{E_{1}}\\mbox{\\rm Ker}d_{0}^{1}.\n\\end{equation*}\n\\end{cor}\n\n\\begin{pf}\nIn the 1-skeleton of a free simplicial algebra \\textbf{E}$^{(1)},$ the set\n\\textbf{S}$_{3}$ is empty. Thus this is clear from the previous proposition.\n\\end{pf}\n\n\\textbf{Remark}: If we set $M=\\ker d_{0}^{1}=NE_{1}^{(1)}$, then the\nidentification given by the Corollary gives\n\\begin{equation*}\nNE_{2}^{(1)}\/\\partial _{3}(NE_{3}^{(1)})\\cong M\\otimes \\overline{M}.\n\\end{equation*}%\nThis uses the fact that $\\ker d_{0}^{1}$ and $\\ker d_{1}^{1}$ are linked via\nthe map sending $m$ to $\\left( m-s_{0}d_{1}m\\right) $ for $m\\in \\ker\nd_{0}^{1}.$ The $h$-map\n\\begin{equation*}\nh:M\\times \\overline{M}\\longrightarrow NE_{2}^{(1)}\/\\partial _{3}NE_{3}^{(1)}\n\\end{equation*}%\ngiven by\n\\begin{equation*}\nh(x,\\overline{y})=s_{1}x(s_{1}y-s_{0}y)+\\partial _{3}NE_{3}^{(1)}.\n\\end{equation*}%\nBut this is also $h(x,\\overline{y})=x\\otimes \\overline{y}$ . Thus\n\\begin{equation*}\nx\\otimes \\overline{y}=s_{1}x(s_{1}y-s_{0}y)+\\partial\n_{3}NE_{3}^{(1)}\n\\end{equation*}%\nunder the identification via the isomorphism of the above corollary.\n\nThis explains the mysterious formula of \\cite{ap2} in the discussion\nbefore Proposition 2.6 of that paper.\n\n\\section{Applications}\n\n\\subsection{2-crossed complexes}\n\nA notion of $2$-crossed complex of commutative algebras is defined\nby Grandjean and Vale in \\cite{gv}. We have considered freeness\nconditions in \\cite{ap3} and this generalises easily to 2-crossed\ncomplexes.\n\nA \\emph{2-crossed complex} of commutative algebras is a sequence\nof $k$-algebras\n\\begin{equation*}\n\\xymatrix{C:\\cdots \\ar[r]^-{\\partial_{n+1}}&C_n\n\\ar[r]^-{\\partial_n}& C_{n-1}\\ar[r]^-{\\partial_{n-1}}&\\cdots\n\\ar[r]^{\\partial_3}&C_2 \\ar[r]^{\\partial_2 }&C_1\n\\ar[r]^{\\partial_1}&C_0}\n\\end{equation*}\ntogether with a 2-crossed module structure given by the pairing\n\\begin{equation*}\n\\left\\{ \\quad \\otimes \\quad \\right\\} :C_{1}\\otimes\n_{C_{0}}C_{1}\\longrightarrow C_{2}\n\\end{equation*}\nsuch that\n\n(i) $C_{n}$ is an $A$-module for $n\\geqslant 3$ with $A=C_{0}\/\\partial\n_{1}(C_{1});$\n\n(ii) $C_{0}$ acts on $C_{n},$ $n\\geqslant 1,$ the action of $\\partial _{1}(C_{1})$\nbeing trivial on $C_{n}$ for $n\\geqslant 3;$\n\n(iii) each $\\partial _{n}$ is an $A$-module homomorphism and $\\partial\n_{n}\\partial _{n+1}=0$ for all $n\\geqslant 1.$\n\n\\bigskip\n\nNote that $K=$ Ker $\\partial _{2}$ is an $C_{0}\/\\partial _{1}C_{1}$-module\nas $\\partial _{2}$ is a crossed module.\n\nThe notion of a morphism for 2-crossed complexes should be clear.\nSuch a morphism will be a morphism `chain complexes of algebras'\nrestricting to a morphism of 2-crossed modules on the bottom three\nterms and compatible with the action. This gives the category,\n\\textbf{2-CrsComp}, of 2-crossed complexes and morphisms between\nthem.\n\\begin{prop}\nThere is a functor%\n\\begin{equation*}\n\\mathbf{C}^{(2)}:\\mathbf{SimpAlg}\\longrightarrow\n\\mathbf{2\\text{-}CrsComp}.\n\\end{equation*}\n\\rm{(We will usually omit the superfix $(2)$ writing simply\n$\\mathbf{C}$ for this.)}\n\\end{prop}\n\n\\begin{pf}\n\\bigskip Given a simplicial algebra $\\mathbf{E},$ define\n\\begin{equation*}\nC_{n}=\n\\begin{cases}\nNE_n & \\text{for } n=0,1;\\\\\n\\dfrac{NE_{2}}{\\partial _{3}(NE_{3}\\cap D_{3})} &   \\text{for }  n=2; \\\\\n\\dfrac{NE_{n}}{(NE_{n}\\cap D_{n})+\\partial _{n+1}(NE_{n+1}\\cap\nD_{n+1})} & \\text{for }  n\\geqslant 3,%\n\\end{cases}%\n\\end{equation*}%\nwith $\\partial _{n}$ induced by the differential of \\textbf{NE}.\nNote that the bottom three terms (for $n=0,1$ and $2$) form the\n2-crossed module considered in \\cite{ap3} and that for $n\\geqslant 3$ the algebras are all $%\nC_{0}\/\\partial _{1}(C_{1})$-modules, since in these dimensions\n$C_{n}$ is the same as the corresponding crossed complex term (cf.\n\\cite{ap1}). The only thing remaining is to check that $\\partial\n_{2}\\partial _{3}$ is trivial which is straightforward.\n\\end{pf}\n\nSince\n\\begin{equation*}\n\\mathbf{C^{(2)}}(\\mathbf{E})_{2}=NE_{2}\/\\partial _{3}(NE_{3}\\cap\nD_{3}),\n\\end{equation*}\nthe same formula as that for $\\mathbf{C(E},2)_{\\langle 2 \\rangle}$,\nwe obtain the following result.\n\n\\begin{cor}\nIf $\\mathbf{E}^{(1)}$ is the $1$-skeleton of a free simplicial\nalgebra $\\mathbf{E}$ then the $2$-crossed complex of\n$\\mathbf{E}^{(1)}$ satisfies\n\\begin{equation*}\n\\mathbf{C^{(2)}}(\\mathbf{E}^{(1)})_{2}\\cong\n\\text{Ker}d_{1}^{1}\\otimes \\text{Ker}d_{0}^{1}.\n\\end{equation*}%\nMoreover%\n\\begin{equation*}\n\\mathbf{C^{(2)}}(\\mathbf{E}^{(2)})_{2}\\cong \\left( \\left( \\text{Ker}d_{1}^{1}\\otimes \\text{%\nKer}d_{0}^{1}\\right) \\sqcup C\\right) \/\\sim\n\\end{equation*}%\nwhere as in Proposition $5.1,$ this $C$ is a free crossed module on the\n\\textquotedblleft new generators\\textquotedblright\\ in dimension $2.$\n\\end{cor}\n\n\n\n\\begin{lem}\nIf  $\\mathbf{E}$ is a simplicial resolution of $B$  then, for $k\\geqslant\n3,\\mathbf{C^{(2)}}(\\mathbf{E}^{(2)})_{k}$ a free $B$-module on the\ngiven data.\n\\end{lem}\n\nTo sum up we have the following result.\n\n\\begin{thm}\nThe `step-by-step' construction of a simplicial resolution of an\nalgebra $B$ gives a  `step-by-step' construction of a 2-crossed\nresolution of $B$ via the 2-crossed complex construction\n$\\mathbf{C}^{(2)}$.\n\\end{thm}\n\n\\subsection{`Quadratic' analogues of the cotangent complex?}\n\nIn this final section we take the $1$-skeleton of a simplicial\nalgebra and see how it relates to other algebraic construction,\nsuch as Andr\\'{e}-Quillen homology and a squared complex form of\nthe cotangent complex. Here our results are less conclusive than\nwe would like.\n\n\\textbf{Remark}: We will assume that rings and algebras are\nNoetherian for convenience and thus that ideals are finitely\ngenerated. This means $\\mathbf{S_2, S_3}$ etc will all be finite.\n\nFrom Proposition 4.1 we have the free crossed square\n$$\n \\left(\n\\begin{array}{cc}\n\\xymatrix{\\dfrac{(R[s_0({\\bf S_2})]^+[s_1({\\bf S_2})])\\cap (s_0({\\bf\nS_2})-s_1({\\bf S_2}))}{P_2} \\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n   R^+[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}]}\n\\end{array}\n\\right)\n$$\nso using corollary 5.2 there is the following isomorphism:\n$$\n\\dfrac{(R[s_0({\\bf S_2})]^+[s_1({\\bf S_2})])\\cap (s_0({\\bf\nS_2})-s_1({\\bf S_2}))}{P_2} \\cong R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}.\n$$\nThus the free crossed square becomes\n$$\n \\left(\n\\begin{array}{cc}\n\\xymatrix{ R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n   R^{+}[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}].}\n\\end{array}\n\\right)\n$$\nIn section 4 we saw that there is a 2-crossed module\n$$\n\\mathbb{X}:\\xymatrix{&R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[r]^{\\partial_3}&R^{+}[\\mathbf{S_2}]\\rtimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[r]^-{\\partial_2}&R[\\mathbf{S_2}]}\n$$\nwhere\n$$\n\\partial_3(x\\otimes y)=(-\\lambda(x\\otimes y),\\lambda'(x\\otimes\ny))\n$$\nand\n$$\n\\partial_2(x,y)=\\mu(x)+\\mu'(x).\n$$\nThe axioms of a squared complex ensure that $\\partial_3$ and\n$\\partial_2$ are homomorphisms and $\\partial_3$ is a module.\n\nThe 2-crossed complex $\\mathbf{C^{(2)}(E^{(2)})}$ has a smaller\n2-crossed module at its base namely\n$$\n\\mathbb{Y}:\\xymatrix{&R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[r]&R^{+}[\\mathbf{S_2}]\\ar[r]^-{\\partial_2}&R}\n$$\nand it is important to compare the two. In fact there is a split\nepimorphism from $\\mathbb{X}$ to $\\mathbb{Y}$ with kernel\n$$\n\\xymatrix{0\\ar[r]&R^{+}[\\mathbf{S_2}]\\ar[r]^{=}&R^{+}[\\mathbf{S_2}]}\n$$\nwhich has, of course, trivial homotopy. Thus $\\mathbb{X}$ and\n$\\mathbb{Y}$ encode the same information about the presentation of\n$R\/I$, $I=$Im $\\partial_2$.\n\nCrossed complexes form a category \\textbf{Crs} which can be\nconsidered as a full subcategory of both the categories of 2-crossed\ncomplexes and of squared complexes. In the case of 2-crossed\ncomplexes, any crossed complex\n$$\n\\mathbb{C}:\\xymatrix{\\cdots\\ar[r]&C\\ar[r]&M\\ar[r]&R}\n$$\nyields a 2-crossed complex with the same terms at each level and\nwith trivial Peiffer lifting $\\{\\otimes\\}:M\\otimes_{R} M\\rightarrow\nC$ whilst considered as a squared complex we get $\\mathbb{C}$ yields\n$$\n\\xymatrix{\\cdots\\ar[r]&C\\ar[r]\\ar[d]&M\\ar[d]\\\\\n&0\\ar[r]&R.}\n$$\nIn both cases higher dimensional terms are left unchanged. Both\nthese inclusions have left adjoints, i.e., the embeddings give\nreflective subcategories. The proofs are quite easily (and will be\ngiven elsewhere).\n\nThe functors from \\textbf{SimpAlg} to \\textbf{2-CrsComp} and\n\\textbf{SqComp} used above, when composed with the reflections to\n\\textbf{Crs} yield the associated crossed complex functor mentioned\nearlier (see \\cite{ap1}).\n\nFinally the category of chain complexes over $R\/I$ embeds as a\nreflexive subcategory of \\textbf{Crs} and the reflection  sends a\ncrossed resolution to the (intermediate stage of the) cotangent\ncomplex (see \\cite{ap1} and \\cite{porter2}). Thus given a simplicial\nresolution of $R\/I$, constructed as in \\cite{andre} by a\nstep-by-step method, the 2-crossed and squared resolutions it gives\ncan be considered as `quadratic' analogues of the cotangent complex,\nin the same way that the crossed complex is a `linear' homotopy\nanalogue of the `homological' cotangent complex. (Here we are using\n`quadratic' and `linear' in the analogous way to that used by Baues\nin \\cite{b1} for the group based theory.)\n\nGiven this it is of interest to study the complex $\\mathbb{X}$ (or\nequivalently  $\\mathbb{Y}$) and their analogues when $\\mathbf{S_3}$\ninformation is added in. Here we have no definitive results, only\nproblems.\n\nThe idea will be to try to provide algorithms for calculating and\nthus controlling, the kernel of $\\partial_3$  in  $\\mathbb{X}$ (or\nequivalently  $\\mathbb{Y}$). We know these give\n$\\pi_3(\\mathbf{E}^{(1)})$ and it is hoped that if these algorithms\nworked, they would allow an analysis of $\\pi_3(\\mathbf{E}^{(2)})$\nand thus to study the effect of adding in $\\mathbf{S_3}$ information\nto the higher terms of the simplicial resolution. As yet we are not\nsure if a general analysis will be possible or whether it will be\nnecessary to limit ourselves to specific classes of example, using,\nfor instance, methods from Gr\\\"{o}bner base theory.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we initiate the study of the characteristic initial value problem for impulsive gravitational spacetimes in general relativity. These were considered to be solutions of the vacuum Einstein equations\n$$R_{\\mu\\nu}=0$$\nwith a delta singularity in the Riemann curvature tensor supported on a null hypersurface. Their historical origin can be traced back to the cylindrical waves of Einstein-Rosen \\cite{EinsteinRosen}, the plane waves of Brinkmann \\cite{Brinkmann} and the explicit impulsive gravitational spacetimes of Penrose \\cite{Penrose72}.\n\nImpulsive gravitational waves have often been studied within the class of plane fronted gravitational waves. This class of explicit solutions to the vacuum Einstein equations has been first studied by Brinkmann \\cite{Brinkmann} and interest in them has been revived in later decades by the work of Bondi-Pirani-Robinson \\cite{BPR}. They were later classified geometrically by Jordon-Ehlers-Kundt \\cite{JEK}. Among this class are the \\emph{pp}-waves (plane fronted waves with parallel rays) that were discovered by Brinkmann \\cite{Brinkmann}, for which the metric takes the form\n$$g=2d\\ub dr+H(\\ub,X,Y)d\\ub^2+dX^2+dY^2,$$\nand the Einstein vacuum equations imply that \n\\begin{equation}\\label{H}\n\\frac{\\partial^2 H}{\\partial X^2}+\\frac{\\partial^2 H}{\\partial Y^2}=0.\n\\end{equation}\nThese include the special case of sandwich waves, where $H$ is compactly supported in $\\ub$. Since (\\ref{H}) is linear, it follows that \\emph{pp}-waves enjoy a principle of linear superposition.\n\n\\emph{pp}-waves have a plane symmetry and originally impulsive gravitational waves have been thought of as a limiting case of the \\emph{pp}-wave with the function $H$ admitting a delta singularity in the variable $\\ub$. Precisely, explicit impulsive gravitational spacetimes were discovered and studied by Penrose \\cite{Penrose72}, who gave the metric in the following double null coordinate form:\n$$g=-2dud\\ub+(1-\\ub\\Theta(\\ub))dx^2+(1+\\ub\\Theta(\\ub))dy^2,$$\nwhere $\\Theta$ is the Heaviside step function. In the Brinkmann coordinate system, the metric has the \\emph{pp}-wave form and an obvious delta singularity:\n$$g=-2d\\ub dr-\\delta(\\ub)(X^2-Y^2)d\\ub^2+dX^2+dY^2,$$\nwhere $\\delta(\\ub)$ is the Dirac delta. Despite the presence of the delta singularity for the metric in Brinkmann coordinates, the corresponding spacetime is Lipschitz and it is only the Riemann curvature tensor (specifically, the only non-trivial $\\alpha$ component of it) that has a delta function supported on the plane null hypersurface $\\ub=0$. This spacetime turns out to possess remarkable global geometric properties \\cite{Penrose65}. In particular, it exhibits a strong focusing property such that the whole null cone emanating from a point in the past of $\\ub=0$ refocuses to a single line in the future of $\\ub=0$ (See Figure 1). In \\cite{Penrose65}, this property forms the basis of Penrose's argument that global hyperbolicity fails in this spacetime. Penrose's impulsive spacetime is plane symmetric, non-asymptotically flat and the delta curvature singularity is supported on the 3-dimensional infinite plane $\\{(u,\\ub,x,y) : \\ub=0\\}$. It has been long debated whether the strong focussing property and the resulting lack of global hyperbolicity is directly tied to the infinite extent of the impulsive gravitational wave (See Yurtsever \\cite{Yurtsever}).\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{focus.pdf_t}\n \n\\caption{Focusing in Penrose's impulsive gravitational spacetime}\n\\end{center}\n\\end{figure}\n\nWhile the interest in impulsive gravitational spacetimes had been high due to the availability of explicit solutions, their global geometric properties, superposition properties and their limiting relation to general spacetimes (see Penrose \\cite{Penrose72} , Aichelburg-Sexl \\cite{AichSexl}), the first study of general spacetimes satisfying the Einstein equations and admitting possible three dimensional delta singularities was undertaken by Taub \\cite{Taub}, who derived a system of consistency relations linking the metric, curvature tensor and the geometry of the singular hypersurface. Impulsive gravitational spacetimes also arise as high-frequency limiting spacetimes considered by Choquet-Bruhat \\cite{Bruhat}. We refer the readers to \\cite{Gr}, \\cite{GrPo}, \\cite{BaHo}, \\cite{Bicak} and the references therein for a more detailed exposition on the physics literature.\n\nIn this paper, we begin the study of impulsive gravitational spacetimes viewed in the context of an (characteristic) initial value problem. We consider the data, prescribed on an outgoing null hypersurface $u=0$ and assume that the $\\alpha$ component of the curvature has a delta singularity supported on a two dimensional surface $S_{0,\\ub_s}$, which can be thought of as the intersection of the hypersurfaces $u=0$ and $\\ub=\\ub_s$. Observe that in the Penrose's explicit impulsive solution, the $\\alpha$ curvature component has precisely this type of behavior when restricted to $u=0$. Unlike the explicit impulsive spacetimes, which have only been constructed in plane symmetry and thus have infinite spatial extent, we will consider the case of $S_{0,\\ub_s}$ with compact topology, more precisely a sphere. The data on the incoming null hypersurface is prescribed to be smooth but otherwise without any smallness assumptions.\n\nWith this data, we show that a unique spacetime satisfying the vacuum Einstein equations can be constructed locally\\footnote{Notice that while the Riemann curvature tensor admits a delta function singularity, we show that the Ricci curvature tensor is well-defined in $L^2$, allowing us to make sense of the vacuum Einstein equations.}. Moreover, the delta singularity propagates along a null hypersurface emanating from the initial singularity on $S_{0,\\ub_s}$ and the spacetime is smooth away from this null hypersurface (See Figure 2). \n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{propagation.pdf_t}\n\n\\caption{Propagation of Singularity}\n\\end{center}\n\\end{figure}\n\nOur main result on the propagation of an impulsive gravitational wave is described by\n\\begin{theorem}\\label{giwthmv1}\nSuppose the following hold for the initial data set:\n\\begin{itemize}\n\\item The data on $\\Hb_0$ is smooth. \n\\item The data on $H_0$ is smooth except across a two sphere $S_{0,\\ub_s}$, where the traceless part of the second fundamental form $\\chih$ has a jump discontinuity.\n\\end{itemize}\nThen\n\\begin{enumerate}[(a)]\n\\item Given such initial data and $\\epsilon$ sufficiently small, there exists a unique spacetime $(\\mathcal M,g)$ endowed with a double null foliation $u$, $\\ub$ that solves the characteristic initial value problem for the vacuum Einstein equations in the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq\\ub_*$ whenever $u_*, \\ub_* \\leq \\epsilon$.\n\\item Define $\\Hb_{\\ub_s}$ to be the incoming null hypersurface emanating from $S_{0,\\ub_s}$. Then the curvature components $\\alpha_{AB}=R(e_A,e_4,e_B,e_4)$ are measures with a singular atom supported on the hypersurface $\\Hb_{\\ub_s}$. All other components of the curvature tensor can be defined in $L^2$. Moreover, the solution is smooth away from $\\Hb_{\\ub_s}$.\n\\end{enumerate}\n\\end{theorem}\n\nFor an impulsive gravitational wave, standard local existence shows that the spacetime is smooth in $0\\leq \\ub <\\ub_s$. The problem of constructing an impulsive gravitational wave lies in making sense of the solution in the whole region with $\\ub_s\\leq \\ub\\leq \\ub_*$ and showing that the singularity propagates along the characteristic hypersurface $\\Hb_{\\ub_s}$ and that the spacetime is smooth after the impulse.\n\n\\begin{remark}[Larger class of initial data]\nThe proof introduces a new type of energy estimates for the Einstein vacuum equations which allows some components of the Riemann curvature tensor not to be in $L^2$. This discovery allows us to consider the problem of local existence and uniqueness for a larger class of non-regular data. This includes data with a Riemann curvature tensor that can only be understood as a conormal distribution. \n\\end{remark}\n\n\\begin{remark}[Uniqueness]\nPenrose's construction of explicit impulsive gravitational spacetimes is based on the gluing approach, which has later been used to generate other explicit solutions of the Einstein equations. In the general case of the characteristic initial value problem for an impulsive gravitational wave, an appropriate adaptation of the gluing philosophy would allow us to construct {\\bf weak} solutions with undetermined uniqueness. In this paper, however, we construct {\\bf strong} solutions in the sense that {\\bf uniqueness} can also be established. To prove uniqueness, we establish a priori estimates for a larger class of admissible initial data. This allows us to show that the solution is unique among all $C^0$ limits of smooth solutions to the vacuum Einstein equations.\n\\end{remark}\n\n\\begin{remark}[Propagation of singularity]\nOur theorem gives a precise description of the propagation of the initial singularity. Such problems have been extensively studied for semilinear equations (see for example \\cite{Beals}, \\cite{Metivier}). Our result can be formally compared to the works of Majda on the propagation of shocks \\cite{Maj1}, \\cite{Maj2} for systems of conservation laws and the subsequent \\cite{Alinhac}, \\cite{Metivier2}, \\cite{Metivier3}, \\cite{CS}. In these works, a short time existence, uniqueness and regularity result was established for initial data with a jump discontinuity across a surface and a precise description of the propagation of the singularity was also given. We establish an analogous result of propagation of singularity for a nonlinear system of quasilinear hyperbolic equations. However, contrary to \\cite{Maj1}, \\cite{Maj2} the singularity that is considered in the present paper is not a shock, as it propagates along the characteristics. Moreover, unlike in \\cite{Maj1}, \\cite{Maj2}, \\cite{Alinhac},\\cite{Metivier2}, \\cite{Metivier3}, \\cite{CS}, where the problem is reformulated as an initial-boundary value problem and uniqueness is known only within the class of piecewise smooth solutions, our solution is also unique among limits of smooth solutions. In order to achieve this, the special structure of the Einstein equations in the double null foliation gauge has been heavily exploited. \n\\end{remark}\n\nThe construction of spacetimes from non-regular characteristic initial data consistent with that of an impulsive gravitational wave had been known only under symmetry assumptions \\cite{ChrSph1}, \\cite{LeSm}, \\cite{LeSte2}. The work of Christodoulou, who solved the characteristic initial value problem for data with bounded variation for the spherically symmetric Einstein-scalar field system \\cite{ChrSph1}, can be thought of as a first result in that direction. In particular, in this work, the second derivatives of the scalar field, which formally is analogous to the curvature, is allowed to have a delta singularity. The study of data with bounded variation turned out to have important consequences in the global structure of spacetimes and the resolution of the cosmic censorship conjecture for the spherically symmetric Einstein-scalar field system \\cite{ChrSph2}, \\cite{ChrSph3}. The construction of distributional solutions for the vacuum Einstein equations that include \\emph{plane} impulsive gravitational wave was carried out in \\cite{LeSm}, \\cite{LeSte2}. The present paper is the first work that provides a consistent study of the initial value problem of impulsive gravitational spacetimes, including their existence, uniqueness and propagation of singularity\/regularity.\n\nGoing beyond spacetimes which represent a single impulsive gravitational wave, colliding impulsive gravitational waves had been studied by Khan-Penrose \\cite{KhanPenrose} and Szekeres \\cite{Szekeres}. In these explicit solutions, the spacetimes possess two null hypersurfaces with curvature delta singularity with a transverse intersection, representing the nonlinear interaction of two impulsive gravitational waves. The study of the characteristic initial value problem for the colliding impulsive gravitational waves will be carried out by the authors in a subsequent paper.\n\n\\subsection{First Version of the Theorem}\n\nOur general approach is based on energy estimates and transport equations in the double null foliation gauge. This general approach in the double null foliation gauge has been carried out in \\cite{KN}, \\cite{Chr} and \\cite{KlRo}.\n\nThe spacetime in question will be foliated by families of outgoing and incoming null hypersurfaces $H_u$ and $\\Hb_{\\ub}$ respectively. Their intersection is assumed to be a 2-sphere denoted by $S_{u,\\ub}$. Define a null frame $\\{e_1,e_2,e_3,e_4\\}$, where $e_3$ and $e_4$ are null, as indicated in Figure 3, and $e_1$, $e_2$ are tangent to the two spheres $S_{u,\\ub}$. $e_4$ is tangent to $H_u$ and $e_3$ is tangent to $\\Hb_{\\ub}$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{frame.pdf_t}\n \n\\caption{The Basic Setup and the Null Frame}\n\\end{center}\n\\end{figure}\nDecompose the Riemann curvature tensor with respect to this frame:\n\\begin{equation*}\n\\begin{split}\n\\a_{AB}&=R(e_A, e_4, e_B, e_4),\\quad \\, \\,\\,   \\ab_{AB}=R(e_A, e_3, e_B, e_3),\\\\\n\\b_A&= \\frac 1 2 R(e_A,  e_4, e_3, e_4) ,\\quad \\bb_A =\\frac 1 2 R(e_A,  e_3,  e_3, e_4),\\\\\n\\rho&=\\frac 1 4 R(e_4,e_3, e_4,  e_3),\\quad \\sigma=\\frac 1 4  \\,^*R(e_4,e_3, e_4,  e_3)\n\\end{split}\n\\end{equation*}\nIn the context of impulsive gravitational spacetimes, the $\\alpha$ component can only be understood as a measure. \n\n\\noindent Define also the following Ricci coefficients with respect to the null frame:\n\\begin{equation*}\n\\begin{split}\n&\\chi_{AB}=g(D_A e_4,e_B),\\, \\,\\, \\quad \\chib_{AB}=g(D_A e_3,e_B),\\\\\n&\\eta_A=-\\frac 12 g(D_3 e_A,e_4),\\quad \\etab_A=-\\frac 12 g(D_4 e_A,e_3)\\\\\n&\\omega=-\\frac 14 g(D_4 e_3,e_4),\\quad\\,\\,\\, \\omegab=-\\frac 14 g(D_3 e_4,e_3),\\\\\n&\\zeta_A=\\frac 1 2 g(D_A e_4,e_3)\n\\end{split}\n\\end{equation*}\n\nDefine $\\chih$ (resp. $\\chibh$) to be the traceless part of $\\chi$ (resp. $\\chib$). For the problem of the propagation of impulsive gravitational waves, we prescribe initial data on $H_0$ such that $\\chih$ has a jump discontinuity across $S_{0,\\ub_s}$ but smooth otherwise. On $\\Hb_0$, we prescribe the initial data to be smooth but without any smallness assumptions.\n\nAs mentioned before, we will prove local existence and uniqueness for a class of data more general than that for the impulsive gravitational wave. More precisely, we require that $\\chih$ and its angular derivatives are merely bounded. However, we do not require the derivative of $\\chih$ in the $e_4$ direction even to be defined.\n\n\\begin{theorem}\\label{rdthmv1}\nSuppose the characteristic initial data are smooth on $\\Hb_0$. On $H_0$, the characteristic initial data are determined by $\\chi$ and $\\zeta_A$. Let $\\th^A$ be the coordinates on the two sphere foliating $H_0$. Suppose, in every coordinate patch\n$$\\sum_{i\\leq 3}|(\\frac{\\partial}{\\partial\\th})^i\\chih_{AB}|,\\sum_{k\\leq 1}\\sum_{i\\leq 3} |(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i(\\trch,\\zeta_A)|\\leq C.$$\nThen there exists $\\epsilon$ sufficiently small such that the unique solution to the vacuum Einstein equations $(\\mathcal M,g)$ endowed with a double null foliation $u$, $\\ub$ exists in $0\\leq u\\leq \\epsilon$, $0\\leq \\ub\\leq \\epsilon$. Associated to the spacetime a coordinate system $(u,\\ub,\\th^1,\\th^2)$ exists, relative to which the spacetime is in particular Lipschitz and retains higher regularity in the angular directions.\n\\end{theorem}\n\n\\begin{remark}\nWe will also explicitly construct a class of initial data that satisfy the assumptions of Theorem \\ref{giwthmv1} in Section \\ref{initialcondition}. This construction of initial data will indicate an easy modification leading to an even more general class satisfying the assumptions of Theorem \\ref{rdthmv1}.\n\\end{remark}\n\nThis Theorem in particular implies Theorem \\ref{giwthmv1}(a). An additional argument, based on the estimates derived in the proof of Theorem \\ref{rdthmv1}, will be carried out in Section \\ref{limitgiw} to prove part (b) of Theorem \\ref{giwthmv1}. The statement of Theorem \\ref{rdthmv1} will be made precise in Section \\ref{thmstatement}. In particular, the smoothness assumptions of Theorem \\ref{rdthmv1} can be weakened and we will define in what sense the vacuum Einstein equations are satisfied.\n\n\\begin{remark}[Comparison to local existence results]\nWithout symmetry assumptions, all known proofs of existence of spacetimes satisfying the Einstein equations are based on $L^2$-type estimates for the curvature tensor and its derivatives or the metric components and their derivatives. Even with the recent resolution of the $L^2$ curvature conjecture by Klainerman-Rodnianski-Szeftel \\cite{L21}, \\cite{L22}, \\cite{L23}, \\cite{L24}, \\cite{L25}, the Riemann curvature tensor has to be at least in $L^2$. (The classical local existence result of Hughes-Kato-Marsden \\cite{HKM} requires the metric to be in $H^s$, with $s> \\frac 52$.)\n\nIn this paper, we prove local existence and uniqueness under the assumption that the spacetime is merely Lipschitz, which in terms of differentiability is even one derivative weaker than the $L^2$ curvature conjecture. Of course the Lipschitz assumption refers to the worst possible behavior observed in our data and our result heavily relies on the structure of the Einstein equations which allows us to efficiently exploit the better behavior of the other components.\n\\end{remark}\n\n\\subsection{Strategy of the Proof}\n\nFor an impulsive gravitational wave, the curvature tensor can only be defined as a measure and is not in $L^2$. This is one of the main challenges of this work.  Let $\\Psi$ denote the curvature components and $\\psi$ denote the Ricci coefficients. The key new observation of this paper is that the $L^2$-type energy estimates for the components of the Riemann curvature tensor\n$$\\int_{H_u} \\Psi^2+\\int_{\\Hb_{\\ub}} \\Psi^2\\leq \\int_{H_0} \\Psi^2+\\int_{\\Hb_0} \\Psi^2 +\\int_0^{\\ub} \\int_0^u\\int_{S_{u\n',\\ub'}} \\psi\\Psi\\Psi du' d\\ub'$$\ncoupled together with the null transport equations for the Ricci coefficients \n$$\\nab_3\\psi=\\Psi+\\psi\\psi,\\quad\\nab_4\\psi=\\Psi+\\psi\\psi$$\ncan be renormalized and closed avoiding the $L^2$-non-integrable components of curvature.\n\n\\subsubsection{Renormalized Energy Estimates}\\label{eeoutline}\n\nThe difficulty in carrying out the above argument is that in our setting, not all curvature components are defined in $L^2$. Even at the level of initial data, $\\alpha$ is only defined as a measure. Therefore, we need to prove $L^2$ energy estimates without involving $\\alpha$. To this end, we need to introduce and estimate the renormalized curvature components. The idea of renormalizing the curvature components has been introduced in \\cite{KlRo:causal}. Unlike this work, there the renormalization was used to obtain estimates for the Ricci coefficients.\n\nThe classical way to derive energy estimates for the Einstein equations is via the Bel-Robinson tensor. In view of our renormalization, we avoid the use of the Bel-Robinson tensor and instead prove the energy estimates directly from the Bianchi identities (\\ref{eq:null.Bianchi}). We note that the derivation of the energy estimates directly from the Bianchi equations without using the Bel Robinson tensor has also appeared in the work of Holzegel \\cite{Holzegel} in a different setting. The challenge and motivation of this method is the derivation of estimates not involving the singular component of curvature $\\alpha$.\n\nTo illustrate this, we first prove the energy estimates for $\\beta$ on $H_u$ and for $(\\rho,\\sigma)$ on $\\Hb_{\\ub}$ by considering the following set of Bianchi equations:\n$$\\nabla_4\\rho=\\div\\beta - \\frac 12 \\chibh\\cdot\\alpha+...$$\n$$\\nab_4\\sigma=-\\div ^*\\beta+\\frac 12\\chibh\\cdot\\alpha+...$$\n$$\\nab_3\\beta=\\nab\\rho+\\nab^*\\sigma+...$$\nHowever, the curvature component $\\alpha$ still appears in the nonlinear terms in these equations. In order to deal with this problem, we renormalize $\\rho$ and $\\sigma$. Define\n$$\\rhoc=\\rho-\\frac 12 \\chih\\cdot\\chibh,\\quad \\sigmac=\\sigma+\\frac 12 \\chih\\wedge\\chibh.$$\nUsing the equation\n$$\\nabla_4\\chih=-\\alpha+...,$$\nwe notice that the first two equations become\n$$\\nabla_4\\rhoc=\\div\\beta+...,\\quad\\nab_4\\sigmac=-\\div ^*\\beta+...$$\nAt the same time, the equation for $\\beta$ can be re-written in terms of of $\\rhoc$ and $\\sigmac$:\n$$\\nab_3\\beta=\\nab\\rhoc+\\nab^*\\sigmac+\\psi\\nab\\psi+...$$\nNow we have a set of renormalized Bianchi equations that does not contain $\\alpha$. Using these equations, we derive the renormalized energy estimate\n$$\\int_{H_u} \\beta^2+\\int_{\\Hb_{\\ub}} (\\rhoc,\\sigmac)^2\\leq \\int_{H_0} \\beta^2+\\int_{\\Hb_0} (\\rhoc,\\sigmac)^2 +\\int_0^{\\ub} \\int_0^u\\int_{S_{u\n',\\ub'}} \\psi\\Psi\\Psi+\\psi\\psi\\nab\\psi du' d\\ub',$$\nin which $\\alpha$ does not appear in the error term.\n\nThe same philosophy can be applied for the remaining curvature components ($\\rho,\\sigma,\\betab,\\alphab$). As a consequence, we obtain a set of $L^2$ curvature estimates which do not explicitly couple to the singular curvature component $\\alpha$. We say explicitly that there is still a remaining possibility that the Ricci coefficients $\\psi$ appearing in the nonlinear error terms for the energy estimates may depend on $\\alpha$.\n\n\\subsubsection{Estimates for the Ricci Coefficients}\\label{Riccioutline}\n\nIn order to close the estimates, it is necessary to show that all the Ricci coefficients can be estimated without \\emph{any} knowledge of $\\alpha$. The most dangerous component is $\\chih$, which naively would have to be estimated using the transport equation\n$$\\nabla_4\\chih+\\trch\\chih=-2\\omega\\chih-\\alpha.$$\nWe take an alternate route and estimate $\\chih$ from the equation\n$$\\nab_3\\chih+\\frac 1 2 \\trchb \\chih=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta,$$\nor\n$$\\div\\chih=\\frac 12 \\nabla \\trch - \\frac 12 (\\eta-\\etab)\\cdot (\\chih -\\frac 1 2 \\trch) -\\beta.$$\n\nAs we shall see, the loss of information of $\\alpha$ is accompanied by the loss of information of $\\beta$ on $\\Hb_{\\ub}$. This presents yet an additional challenge in estimating the Ricci coefficients.\n\n\\subsubsection{Higher Order Energy Estimates}\\label{higherorderoutline}\n\nAnother difficulty arises considering the fact that in order to close the energy estimates, we need to prove higher derivative estimates for the curvature components. However, the derivatives of some curvature components along $e_4$ are not defined in $L^2$ initially. We will therefore only use angular covariant derivatives $\\nab$ as commutators and will prove estimates only for the $L^2$ norms of the angular covariant derivatives of the renormalized curvature components. We will show firstly that this procedure does not introduce terms that cannot be estimated in $L^2$ (in particular $\\alpha$ will not appear) and secondly that all the energy estimates can be closed only using the estimates of the angular derivatives of the renormalized curvature components.\n\n\\subsubsection{Existence and Uniqueness}\n\nSince we work with initial data with very low regularity, the a priori estimates that we have described above in Sections \\ref{eeoutline}, \\ref{Riccioutline} and \\ref{higherorderoutline} do not immediately imply the existence and uniqueness of the solutions. Instead, we need to approximate the data by a sequence of smooth data and show first that they have a common domain of existence and second that they converge in this domain. The fact that the solutions to the sequence of smooth data have a common domain of existence follows from the a priori estimates that do not involve $\\alpha$ and its derivatives as outlined above. Since the approximating data are smooth, we can conclude that the approximating solutions are also smooth and exist in a common domain.\n\nOnce we have proved the uniform a priori estimates, we proceed to prove that the sequence of solutions converges. To this end, we consider the equations for the difference of the Ricci coefficients and curvature components. We identify the spacetimes in this sequence by the value of their coordinate functions and derive equations for the difference of the Ricci coefficients and curvature components. Our a priori estimates heavily relies on the structure of the Einstein equations which allows us to eliminate any dependence of the $\\alpha$ component. A priori, there is no reason to think that this structure is preserved when considering difference of these solutions, which is necessary to show that our end result is a strong solution to the Einstein equations. It is a remarkable fact that it turns out that in the equations for the difference of the Ricci coefficients and curvature components, $\\alpha$ indeed does not appear. Thus the a priori estimates we have proved are sufficient to control the difference of the Ricci coefficients and curvature components from the difference of the initial value. This proves that the sequence of solutions converge. Even though we face the standard challenge of loss of derivative in the quasilinear equations, our estimates are still sufficient to show that the sequence converges.\n\nThe constructed limiting spacetime is not smooth. In particular, second derivative of the metric in the $e_4$ direction is not even defined. We will, however, show that the Ricci curvature tensor is better behaved than a general second derivative of the metric and that the limiting spacetime satisfies the vacuum Einstein equations $R_{\\mu\\nu}=0$ in the $L^2$ sense.\n\nThe estimates for the difference of the Ricci coefficients and curvature components imply that the constructed spacetime is the unique solution to the Einstein equations among the class of spacetimes admitting a double null foliation and satisfying strong enough a priori bounds. Moreover, we can prove uniqueness of the constructed solution among all limits of smooth spacetimes, i.e, any spacetime that arises as a $C^0$ limit of smooth solutions to the vacuum Einstein equations with initial data converging to the given initial data must coincide with the constructed spacetime.\n\nIt can be observed that the above argument for the a priori estimates, as well as that for showing existence and uniqueness, does not use the fact that initially, $\\alpha$ is a measure whose singular support is on $S_{0,\\ub_s}$. In effect, the argument avoids $\\alpha$ completely, and can be used for data such that $\\alpha$ is much rougher. Since the argument used only estimates on the Ricci coefficients and the curvature components other than $\\alpha$, it can be used to handle initial data satisfying only the assumptions of Theorem \\ref{rdthmv1}.\n\n\\subsubsection{Regularity and Propagation of Singularity}\n\nIn the setting of Theorem \\ref{giwthmv1}, i.e., that of an impulsive gravitational wave, the theorem gives a precise description of the propagation of singularity. The a priori estimates imply that all the curvature components other than $\\alpha$ are bounded. Here, we are interested in proving two additional statements: firstly, $\\alpha$ is a measure that indeed has a delta singularity on $\\Hb_{\\ub_s}$; secondly, the spacetime is smooth away from $\\Hb_{\\ub_s}$.\n\nTo show that $\\alpha$ is a measure, we approximate the data for $\\alpha$ by a sequence of smooth data $\\alpha_n$ such that $\\alpha_n$ is of size $2^n$ in a region $|\\ub-\\ub_s|\\leq 2^{-n}$. The spacetimes constructed for such data are smooth and therefore allow us to use the previously avoided $L^2$ estimate for the $\\alpha$ component of curvature. This estimate imply that in the constructed spacetime, in the region $|\\ub-\\ub_s|\\geq 2^{-n}$, the $L^2_{\\ub}$ norm of $\\alpha_n$ are uniformly bounded; while in the region $|\\ub-\\ub_s|\\leq 2^{-n}$, the $L^2_{\\ub}$ norm of $\\alpha_n$ is bounded by $2^{\\frac n2}$. By Cauchy-Schwarz, we have that the $L^1_{\\ub}$ norms of $\\alpha_n$ are uniformly bounded. This allows us to show that in the limiting spacetime, the curvature component $\\alpha$ is a measure.\n\nTo show that the singular part of $\\alpha$ is a delta function supported on the null hypersurface $\\Hb_{\\ub_s}$, we notice that\n$$\\alpha=-\\nab_4\\chih-\\trch \\chih-2 \\omega \\chih.$$\nTherefore, it suffices to show that $\\chih$ has a jump discontinuity across $\\Hb_{\\ub_s}$ and smooth everywhere else. This can be proved using the equation\n$$\\nab_3\\chih+\\frac 1 2 \\trchb \\chih=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta$$\nand the fact that on the initial hypersurface $H_0$, $\\chih$ has a jump discontinuity across $S_{0,\\ub_s}$ and smooth everywhere else.\n\nIn order to show that the spacetime is smooth away from $\\Hb_{\\ub_s}$, we will estimate the higher regularity of all the curvature components in that region. We first use the Bianchi equations\n$$\\nab_3\\beta+\\trchb\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma).$$\nIntegrating this equation and using Gronwall's inequality, we obtain for any $\\ub\\neq\\ub_s$ that\n$$\\sup_u\\sum_{i\\leq I}||\\nab^i\\beta||_{L^2(S_{u,\\ub})}\\leq (\\sum_{i\\leq I}||\\nab^i\\beta||_{L^2(S_{0,\\ub})}+...)\\exp(\\int \\sum_{i\\leq I}||\\nab^i(\\trch,\\omegab)||_{L^\\infty(S)}).$$\nThe regularity of $\\beta$ is thus inherited from the initial data, which is smooth away from $\\ub=\\ub_s$.\nOnce we have estimates for $\\beta$, we consider the equation\n$$\\nab_3\\alpha+\\frac 12 \\trchb \\alpha=\\nabla\\hot \\beta+ 4\\omegab\\alpha-3(\\chih\\rho+^*\\chih\\sigma)+\n(\\zeta+4\\eta)\\hot\\beta.$$\nIntegrating as before, we see that $\\alpha$ also inherits the regularity from the initial data. Higher derivatives estimates for $\\alpha$ can be derived analogously by differentiating this equation. The other components of curvature can be controlled in a similar fashion. Notice that this procedure results in a loss of derivatives. In particular, in order to control the $N$-th derivative of the curvature, one needs $\\sim 2N$ derivatives initially.\n\n\\subsection{Outline of the Paper}\n\nFinally, we give the outline of the remainder of the paper. In the next Section, we give a careful introduction of the setting, describing the double null foliation, the coordinate system, the equations and relevant notations. We will state a more precise version of Theorem \\ref{rdthmv1}, which we will call Theorem \\ref{rdthmv2}. In Section \\ref{initialcondition}, we provide a construction of the initial data set satisfying the conditions in Theorem \\ref{giwthmv1} and exhibit a sequence of smooth data approximating the data with a curvature delta singularity. In Sections \\ref{estimates} and \\ref{convergence}, we prove Theorem \\ref{rdthmv2}. In Section \\ref{estimates}, we prove that for smooth initial data satisfying the assumptions of Theorem \\ref{rdthmv2}, a unique spacetime exists in a region depending only on the constants in the assumptions. This in particular implies that the approximating sequence of initial data constructed in Section \\ref{initialcondition} gives rise to spacetimes with a common and uniform region of existence (identified by a choice of a double null coordinate system). In Section \\ref{convergence}, we study the equations for the difference of two spacetimes. This allows us to show convergence of solutions with converging initial data and conclude the existence part of Theorem \\ref{rdthmv2}. In Section \\ref{limit}, we examine the regularity of the limiting spacetime and show that it is a solution to the vacuum Einstein equations. In Section \\ref{uniquenesssec}, we conclude the uniqueness part of Theorem \\ref{rdthmv2}. Finally, in Section \\ref{limitgiw}, we return to the proof of part (b) of Theorem \\ref{giwthmv1}, giving a precise description of the propagation of singularity.\\\\\n\n\\noindent{\\bf Acknowledgments:} The authors would like to thank Mihalis Dafermos for valuable discussions. We also thank Dejan Gajic, Joe Keir, Jan Sbierski, Martin Taylor, as well as an anonymous referee, for helpful comments. J. Luk is supported by the NSF Postdoctoral Fellowship DMS-1204493. I. Rodnianski is supported by the NSF grant DMS-1001500 and the FRG grant DMS-1065710.\n\n\\section{Setting and Equations}\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{frame.pdf_t}\n \n\\caption{The Basic Setup and the Null Frame}\n\\end{center}\n\\end{figure}\n\nOur setting is the characteristic initial value problem with data given on the two characteristic hypersurfaces $H_0$ and $\\Hb_0$ intersecting at the sphere $S_{0,0}$ (see Figure 4). The spacetime will be a solution to the Einstein equations constructed in a neighborhood of $S_{0,0}$ bounded by the two hypersurfaces.\n\n\\subsection{Double Null Foliation}\nFor a spacetime in a neighborhood of $S_{0,0}$, we define a double null foliation as follows: Let $u$ and $\\ub$ be solutions to the eikonal equation\n$$g^{\\mu\\nu}\\partial_\\mu u\\partial_\\nu u=0,\\quad g^{\\mu\\nu}\\partial_\\mu\\ub\\partial_\\nu \\ub=0,$$\nsatisfying the initial conditions $u=0$ on $H_0$ and $\\ub=0$ on $\\Hb_0$.\nLet\n$$L'^\\mu=-2g^{\\mu\\nu}\\partial_\\nu u,\\quad \\Lb'^\\mu=-2g^{\\mu\\nu}\\partial_\\nu \\ub.$$ \nThese are null and geodesic vector fields. Define\n$$2\\Omega^{-2}=-g(L',\\Lb').$$\nDefine\n$$e_3=\\Omega\\Lb'\\mbox{, }e_4=\\Omega L'$$\nto be the normalized null pair such that \n$$g(e_3,e_4)=-2$$\nand\n$$\\Lb=\\Omega^2\\Lb'\\mbox{, }L=\\Omega^2 L'$$\nto be the so-called equivariant vector fields.\n\nWe will denote the level sets of $u$ as $H_u$ and the level sets of $\\ub$ and $\\Hb_{\\ub}$. By virtue of the eikonal equations, $H_u$ and $\\Hb_{\\ub}$ are null hypersurface. Notice that the sets defined by fixed values of $(u,\\ub)$ are 2-spheres. We denote such spheres by $S_{u,\\ub}$. They are intersections of the hypersurfaces $H_u$ and $\\Hb_{\\ub}$. The integral flows of $L$ and $\\Lb$ respect the foliation $S_{u,\\ub}$.\n\n\\subsection{The Coordinate System}\\label{coordinates}\nOn a spacetime in a neighborhood of $S_{0,0}$, we define a coordinate system $(u,\\ub,\\th^1,\\th^2)$ as follows:\nOn the sphere $S_{0,0}$, define a coordinate system $(\\th^1,\\th^2)$ for the sphere such that on each coordinate patch the metric $\\gamma$ is smooth, bounded and positive definite. Then we define the coordinates on the initial hypersurfaces by requiring \n$$\\frac{\\partial}{\\partial u}\\th^A=0\\mbox{ on $\\Hb_0$, and }\\frac{\\partial}{\\partial \\ub}\\th^A=0\\mbox{ on $H_0$}.$$\nWe now define the coordinate system in the spacetime in a neighborhood of $S_{0,0}$ by letting $u$ and $\\ub$ to be solutions to the eikonal equations:\n$$g^{\\mu\\nu}\\partial_\\mu u\\partial_\\nu u=0,\\quad g^{\\mu\\nu}\\partial_\\mu\\ub\\partial_\\nu \\ub=0,$$\nand define $\\th^1, \\th^2$ by\n$$\\Ls_L \\th^A=0,$$ \nwhere $\\Ls_L$ denote the restriction of the Lie derivative to $TS_{u,\\ub}$ (See \\cite{Chr}).\nRelative to the coordinate system, the null pair $e_3$ and $e_4$ can be expressed as\n$$e_3=\\Omega^{-1}\\left(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial \\th^A}\\right), e_4=\\Omega^{-1}\\frac{\\partial}{\\partial \\ub},$$\nfor some $b^A$ such that $b^A=0$ on $\\Hb_0$, while the metric $g$ takes the form\n$$g=-2\\Omega^2(du\\otimes d\\ub+d\\ub\\otimes du)+\\gamma_{AB}(d\\th^A-b^Adu)\\otimes (d\\th^B-b^Bdu).$$ \n\n\\subsection{The Equations}\\label{sec.eqns}\nWe will recast the Einstein equations as a system for Ricci coefficients and curvature components associated to a null frame $e_3$, $e_4$ defined above and an orthonormal frame ${e_1,e_2}$ tangent to the 2-spheres $S_{u,\\ub}$. Using the indices $A,B$ to denote $1,2$, we define the Ricci coefficients relative to the null fame:\n \\begin{equation}\n\\begin{split}\n&\\chi_{AB}=g(D_A e_4,e_B),\\, \\,\\, \\quad \\chib_{AB}=g(D_A e_3,e_B),\\\\\n&\\eta_A=-\\frac 12 g(D_3 e_A,e_4),\\quad \\etab_A=-\\frac 12 g(D_4 e_A,e_3)\\\\\n&\\omega=-\\frac 14 g(D_4 e_3,e_4),\\quad\\,\\,\\, \\omegab=-\\frac 14 g(D_3 e_4,e_3),\\\\\n&\\zeta_A=\\frac 1 2 g(D_A e_4,e_3)\n\\end{split}\n\\end{equation}\nwhere $D_A=D_{e_{(A)}}$. We also introduce the  null curvature components,\n \\begin{equation}\n\\begin{split}\n\\a_{AB}&=R(e_A, e_4, e_B, e_4),\\quad \\, \\,\\,   \\ab_{AB}=R(e_A, e_3, e_B, e_3),\\\\\n\\b_A&= \\frac 1 2 R(e_A,  e_4, e_3, e_4) ,\\quad \\bb_A =\\frac 1 2 R(e_A,  e_3,  e_3, e_4),\\\\\n\\rho&=\\frac 1 4 R(e_4,e_3, e_4,  e_3),\\quad \\sigma=\\frac 1 4  \\,^*R(e_4,e_3, e_4,  e_3)\n\\end{split}\n\\end{equation}\nHere $\\, ^*R$ denotes the Hodge dual of $R$.  We denote by $\\nab$ the \ninduced covariant derivative operator on $S_{u,\\ub}$ and by $\\nab_3$, $\\nab_4$\nthe projections to $S_{u,\\ub}$ of the covariant derivatives $D_3$, $D_4$, see\nprecise definitions in \\cite{KN}. \n\nObserve that,\n\\begin{equation}\n\\begin{split}\n&\\omega=-\\frac 12 \\nab_4 (\\log\\Omega),\\qquad \\omegab=-\\frac 12 \\nab_3 (\\log\\Omega),\\\\\n&\\eta_A=\\zeta_A +\\nab_A (\\log\\Omega),\\quad \\etab_A=-\\zeta_A+\\nab_A (\\log\\Omega)\n\\end{split}\n\\end{equation}\n\nLet $\\phi^{(1)}\\cdot\\phi^{(2)}$ denote an arbitrary contraction of the tensor product of $\\phi^{(1)}$ and $\\phi^{(2)}$ with respect to the metric $\\gamma$. We also define\n$$(\\phi^{(1)}\\hot\\phi^{(2)})_{AB}:=\\phi^{(1)}_A\\phi^{(2)}_B+\\phi^{(1)}_B\\phi^{(2)}_A-\\gamma_{AB}(\\phi^{(1)}\\cdot\\phi^{(2)}) \\quad\\mbox{for one forms $\\phi^{(1)}_A$, $\\phi^{(2)}_A$,}$$\n$$\\phi^{(1)}\\wedge\\phi^{(2)}:=\\eps^{AB}(\\gamma^{-1})^{CD}\\phi^{(1)}_{AC}\\phi^{(2)}_{BD}\\quad\\mbox{for symmetric two tensors $\\phi^{(1)}_{AB}$, $\\phi^{(2)}_{AB}$}.$$\nFor totally symmetric tensors, the $\\div$ and $\\curl$ operators are defined by the formulas\n$$(\\div\\phi)_{A_1...A_r}:=\\nabla^B\\phi_{BA_1...A_r},$$\n$$(\\curl\\phi)_{A_1...A_r}:=\\eps^{BC}\\nabla_B\\phi_{CA_1...A_r},$$\nwhere $\\eps$ is the volume form associated to the metric $\\gamma$.\nDefine also the trace to be\n$$(\\mbox{tr}\\phi)_{A_1...A_{r-1}}:=(\\gamma^{-1})^{BC}\\phi_{BCA_1...A_{r-1}}.$$\n\nWe separate the trace and traceless part of $\\chi$ and $\\chib$. Let $\\chih$ and $\\chibh$ be the traceless parts of $\\chi$ and $\\chib$ respectively. Then $\\chi$ and $\\chib$ satisfy the following null structure equations:\n\\begin{equation}\n\\label{null.str1}\n\\begin{split}\n\\nab_4 \\trch+\\frac 12 (\\trch)^2&=-|\\chih|^2-2\\omega \\trch\\\\\n\\nab_4\\chih+\\trch \\chih&=-2 \\omega \\chih-\\alpha\\\\\n\\nab_3 \\trchb+\\frac 12 (\\trchb)^2&=-2\\omegab \\trchb-|\\chibh|^2\\\\\n\\nab_3\\chibh + \\trchb\\,  \\chibh&= -2\\omegab \\chibh -\\alphab\\\\\n\\nab_4 \\trchb+\\frac1 2 \\trch \\trchb &=2\\omega \\trchb +2\\rho- \\chih\\cdot\\chibh +2\\div \\etab +2|\\etab|^2\\\\\n\\nab_4\\chibh +\\frac 1 2 \\trch \\chibh&=\\nab\\widehat{\\otimes} \\etab+2\\omega \\chibh-\\frac 12 \\trchb \\chih +\\etab\\widehat{\\otimes} \\etab\\\\\n\\nab_3 \\trch+\\frac1 2 \\trchb \\trch &=2\\omegab \\trch+2\\rho- \\chih\\cdot\\chibh+2\\div \\eta+2|\\eta|^2\\\\\n\\nab_3\\chih+\\frac 1 2 \\trchb \\chih&=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta\n\\end{split}\n\\end{equation}\nThe other Ricci coefficients satisfy the following null structure equations:\n\\begin{equation}\n\\label{null.str2}\n\\begin{split}\n\\nabla_4\\eta&=-\\chi\\cdot(\\eta-\\etab)-\\b\\\\\n\\nabla_3\\etab &=-\\chib\\cdot (\\etab-\\eta)+\\bb\\\\\n\\nabla_4\\omegab&=2\\omega\\omegab+\\frac 34 |\\eta-\\etab|^2-\\frac 14 (\\eta-\\etab)\\cdot (\\eta+\\etab)-\n\\frac 18 |\\eta+\\etab|^2+\\frac 12 \\rho\\\\\n\\nabla_3\\omega&=2\\omega\\omegab+\\frac 34 |\\eta-\\etab|^2+\\frac 14 (\\eta-\\etab)\\cdot (\\eta+\\etab)- \\frac 18 |\\eta+\\etab|^2+\\frac 12 \\rho\\\\\n\\end{split}\n\\end{equation}\nThe Ricci coefficients also satisfy the following constraint equations\n\\begin{equation}\n\\label{null.str3}\n\\begin{split}\n\\div\\chih&=\\frac 12 \\nabla \\trch - \\frac 12 (\\eta-\\etab)\\cdot (\\chih -\\frac 1 2 \\trch) -\\beta,\\\\\n\\div\\chibh&=\\frac 12 \\nabla \\trchb + \\frac 12 (\\eta-\\etab)\\cdot (\\chibh-\\frac 1 2   \\trchb) +\\betab\\\\\n\\curl\\eta &=-\\curl\\etab=\\sigma +\\frac 1 2\\chibh \\wedge\\chih\\\\\nK&=-\\rho+\\frac 1 2 \\chih\\cdot\\chibh-\\frac 1 4 \\trch \\trchb\n\\end{split}\n\\end{equation}\nwith $K$ the Gauss curvature of the surfaces $S$.\nThe null curvature components satisfy the following null Bianchi equations:\n\\begin{equation}\n\\label{eq:null.Bianchi}\n\\begin{split}\n&\\nab_3\\alpha+\\frac 12 \\trchb \\alpha=\\nabla\\hot \\beta+ 4\\omegab\\alpha-3(\\chih\\rho+^*\\chih\\sigma)+\n(\\zeta+4\\eta)\\hot\\beta,\\\\\n&\\nab_4\\beta+2\\trch\\beta = \\div\\alpha - 2\\omega\\beta +  \\eta \\alpha,\\\\\n&\\nab_3\\beta+\\trchb\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma),\\\\\n&\\nab_4\\sigma+\\frac 32\\trch\\sigma=-\\div^*\\beta+\\frac 12\\chibh\\cdot ^*\\alpha-\\zeta\\cdot^*\\beta-2\\etab\\cdot\n^*\\beta,\\\\\n&\\nab_3\\sigma+\\frac 32\\trchb\\sigma=-\\div ^*\\betab+\\frac 12\\chih\\cdot ^*\\alphab-\\zeta\\cdot ^*\\betab-2\\eta\\cdot \n^*\\betab,\\\\\n&\\nab_4\\rho+\\frac 32\\trch\\rho=\\div\\beta-\\frac 12\\chibh\\cdot\\alpha+\\zeta\\cdot\\beta+2\\etab\\cdot\\beta,\\\\\n&\\nab_3\\rho+\\frac 32\\trchb\\rho=-\\div\\betab- \\frac 12\\chih\\cdot\\alphab+\\zeta\\cdot\\betab-2\\eta\\cdot\\betab,\\\\\n&\\nab_4\\betab+\\trch\\betab=-\\nabla\\rho +^*\\nabla\\sigma+ 2\\omega\\betab +2\\chibh\\cdot\\beta-3(\\etab\\rho-^*\\etab\\sigma),\\\\\n&\\nab_3\\betab+2\\trchb\\betab=-\\div\\alphab-2\\omegab\\betab+\\etab \\cdot\\alphab,\\\\\n&\\nab_4\\alphab+\\frac 12 \\trch\\alphab=-\\nabla\\hot \\betab+ 4\\omega\\alphab-3(\\chibh\\rho-^*\\chibh\\sigma)+\n(\\zeta-4\\etab)\\hot \\betab\n\\end{split}\n\\end{equation}\nwhere $^*$ denotes the Hodge dual on $S_{u,\\ub}$.\n\nIn the sequel, we will use capital Latin letters $A\\in \\{1,2\\}$ for indices on the spheres $S_{u,\\ub}$ and Greek letters $\\mu\\in\\{1,2,3,4\\}$ for indices in the whole spacetime.\n\nIn the following it will be useful to apply a schematic notation. We will let $\\phi$ denote an arbitrary tensorfield, $\\psi$ a Ricci coefficient and $\\Psi$ a null curvature component different from $\\alpha$. We will simply write $\\psi\\psi$ or $\\psi^2$ (or $\\psi\\Psi$, etc.) to denote an arbitrary contraction. Moreover, we will denote by $\\nab^i\\psi^j$ the sum of all terms which are products of $j$ factors, with each factor being $\\nab^{i_k}\\psi$ and that the sum of all $i_k$'s being $i$, i.e., \n$$\\nab^i\\psi^j=\\displaystyle\\sum_{i_1+i_2+...+i_j}\\underbrace{\\nab^{i_1}\\psi\\nab^{i_2}\\psi...\\nab^{i_j}\\psi}_\\text{j factors}.$$\nThe use of the schematic notation is reserved for the cases when the precise nature of the contraction is not important to the argument. In particular, when using this schematic notation, we will neglect all constant factors.\n\n\\subsection{Integration and Norms}\n\nLet $U$ be a coordinate patch on $S_{0,0}$ and $p_U$ be a partition of unity in $D_U$ such that $p_U$ is supported in $D_U$. Given a function $\\phi$, the integration on $S_{u,\\ub}$ is given by the formula:\n$$\\int_{S_{u,\\ub}} \\phi :=\\sum_U \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi p_U\\sqrt{\\det\\gamma}d\\th^1 d\\th^2.$$\nLet $D_{u_*,\\ub_*}$ by the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$. The integration on $D_{u,\\ub}$ is given by the formula\n\\begin{equation*}\n\\begin{split}\n\\int_{D_{u,\\ub}} \\phi :=&\\sum_U \\int_0^u\\int_0^{\\ub}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi p_U\\sqrt{-\\det g}d\\th^1 d\\th^2d\\ub du\\\\\n=&2\\sum_U \\int_0^u\\int_0^{\\ub}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi p_U\\Omega^2\\sqrt{-\\det \\gamma}d\\th^1 d\\th^2d\\ub du.\n\\end{split}\n\\end{equation*}\nSince there are no canonical volume forms on $H_u$ and $\\Hb_{\\ub}$, we define integration by\n$$\\int_{H_{u}} \\phi :=\\sum_U \\int_0^{\\epsilon}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi2 p_U\\Omega\\sqrt{\\det\\gamma}d\\th^1 d\\th^2d\\ub,$$\nand\n$$\\int_{H_{\\ub}} \\phi :=\\sum_U \\int_0^\\epsilon\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi2p_U\\Omega\\sqrt{\\det\\gamma}d\\th^1 d\\th^2du.$$\n\nWith these definitions of integration, we can define the norms that we will use. Let $\\phi$ be a tensorfield. For $1\\leq p<\\infty$, define\n$$||\\phi||_{L^p(S_{u,\\ub})}^p:=\\int_{S_{u,\\ub}} <\\phi,\\phi>_\\gamma^{p\/2},$$\n$$||\\phi||_{L^p(H_u)}^p:=\\int_{H_{u}} <\\phi,\\phi>_\\gamma^{p\/2},$$\n$$||\\phi||_{L^p(\\Hb_{\\ub})}^p:=\\int_{\\Hb_{\\ub}} <\\phi,\\phi>_\\gamma^{p\/2}.$$\nDefine also the $L^\\infty$ norm by\n$$||\\phi||_{L^\\infty(S_{u,\\ub})}:=\\sup_{\\th\\in S_{u,\\ub}} <\\phi,\\phi>_\\gamma^{1\/2}(\\th).$$\n\n\\subsection{Precise Statement of the Main Theorem}\\label{thmstatement}\nWith the notations introduced in this Section, we give a precise version of the statement of Theorem \\ref{rdthmv1}:\n\\begin{theorem}\\label{rdthmv2}\nSuppose the initial data set for the characteristic initial value problem is given on $H_0$ for $0\\leq \\ub\\leq \\ub_*$ and on $\\Hb_0$ for $0\\leq u\\leq u_*$ such that\n$$c\\leq |\\det\\gamma \\restriction_{S_{u,0}} |, |\\det\\gamma \\restriction_{S_{0,\\ub}} |\\leq C,$$\n$$\\sum_{i\\leq 3}\\left(|(\\frac{\\partial}{\\partial\\th})^i\\gamma \\restriction_{S_{u,0}}|+|(\\frac{\\partial}{\\partial\\th})^i\\gamma \\restriction_{S_{0,\\ub}}|\\right)\\leq C,$$\n$$\\mathcal O_0:= \\sum_{i\\leq 3} \\sup_{\\ub}||\\nabla^i\\psi||_{L^2(S_{0,\\ub})}+\\sum_{i\\leq 3} \\sup_{u}||\\nabla^i\\psi||_{L^2(S_{u,0})}\\leq C,$$\n$$\\mathcal R_0:=\\sum_{i\\leq 2}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab^i\\Psi||_{L^2(S_{0,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_u||\\nab^i\\Psi||_{L^2(S_{u,0})}\\right)\\leq C.$$\nThen for $\\epsilon$ sufficiently small depending only on $C$ and $c$, there exists a spacetime $(\\mathcal M,g)$ endowed with a double null foliation $u$, $\\ub$ that solves the characteristic initial value problem to the vacuum Einstein equations in $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$ for $u_*,\\ub_*\\leq \\epsilon$. The metric is continuous and takes the form\n$$g=-2\\Omega^2(du\\otimes d\\ub+d\\ub\\otimes du)+ \\gamma_{AB}(d\\th^A-b^A du)\\otimes(d\\th^B-b^B du).$$\n$(\\mathcal M,g)$ is a $C^0$ limit of smooth solutions to the vacuum Einstein equations and is the unique spacetime solving the characteristic initial value problem among all $C^0$ limits of smooth solutions. Moreover, \n$$\\frac{\\partial}{\\partial \\th}g,\\frac{\\partial}{\\partial u}g\\in C^0_u C^0_{\\ub} L^4(S),$$\n$$\\frac{\\partial^2}{\\partial \\th^2}g,\\frac{\\partial^2}{\\partial u\\partial\\th}g,\\frac{\\partial^2}{\\partial u^2}g\\in C^0_u C^0_{\\ub} L^2(S),$$\n$$\\frac{\\partial}{\\partial \\ub}g, \\frac{\\partial}{\\partial\\ub}((\\gamma^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma)_{AB}) \\in L^\\infty_u L^\\infty_{\\ub} L^\\infty(S),$$\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g,\\frac{\\partial^2}{\\partial u\\partial\\ub}g,\\frac{\\partial^2}{\\partial \\ub^2}b^A\\in L^\\infty_u L^\\infty_{\\ub} L^4(S).$$\nIn the $(u,\\ub,\\th^1,\\th^2)$ coordinates, the Einstein equations are satisfied in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Furthermore, the higher angular differentiability in the data results in higher angular differentiability.\n\\end{theorem}\nWe prove this Theorem in two steps. First, we show that if the initial data are smooth and the bounds for the initial data in the assumption of Theorem \\ref{rdthmv2} hold, then an $\\epsilon$ can be chosen depending only on the bounds in the assumption of Theorem \\ref{rdthmv2} such that a smooth spacetime solving the Einstein equations exists in $0\\leq u\\leq\\epsilon$ and $0\\leq \\ub\\leq \\epsilon$. This is formulated as Theorem \\ref{timeofexistence}, and is proved in Section \\ref{estimates}. Then, we show that a sequence of solutions to the Einstein equations with a converging sequence of smooth initial data, satisfying the assumptions of Theorem \\ref{rdthmv2} with uniform constants $C$ and $c$, converges. The limit spacetime satisfies the Einstein equations and has the properties stated in Theorem \\ref{rdthmv2}. This is formulated as Theorem \\ref{convergencethm2} and is proved in Section \\ref{convergence}. We furthermore show that the solution is unique and that the regularity in the angular directions persists in Section \\ref{convergence}.\n\n\\section{The Initial Data}\\label{initialcondition}\n\nIn this Section, we construct data satisfying the constraint equations such that $\\chih$ has a jump discontinuity across a two sphere. We also construct a sequence of smooth data satisfying the constraint equations approaching the data with discontinuous $\\chih$. We derive precise bounds for the Ricci coefficients and the curvature components for the initial data in this approximating sequence.\n\nWe fix\n$$\\Omega=1$$\nidentically on the initial hypersurfaces $H_0$ and $\\Hb_0$. \nOn $H_0$, $\\gamma$ and $\\chi$ have to satisfy the equations\n\\begin{equation}\\label{con1}\n\\Ls_L \\gamma=2\\chi,\\quad \\Ls_L \\trch= -\\frac 12 (\\trch)^2-|\\chih|_\\gamma^2,\n\\end{equation}\nwhile on $\\Hb_0$, $\\gamma$ and $\\chib$ have to satisfy the equations\n\\begin{equation}\\label{con2}\n\\Ls_{\\Lb}\\gamma=2\\chib,\\quad\\Ls_{\\Lb} \\trchb= -\\frac 12 (\\trchb)^2-|\\chibh|_\\gamma^2\n\\end{equation}\nHere $\\Ls$ denotes the restriction of the Lie derivative to $TS_{u,\\ub}$.\n\nFollowing \\cite{Chr}, we can obtain initial data satisfying the above constraint equations by prescribing $\\gamma_{AB}$, $\\zeta_A$, $\\trch$ and $\\trchb$ on the two sphere $S_{0,0}$ and prescribing the conformal class of the metric $\\hat{\\gamma}_{AB}$ satisfying $\\sqrt{\\det \\hat{\\gamma}_{AB}}=1$ on each of the initial hypersurfaces. Relative to the coordinate system $(\\th^1,\\th^2)$, we require that on $S_{0,0}$,\n$$\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB}, \\zeta_A, tr\\chi, tr\\chib)|\\leq C.$$\nOn the initial incoming hypersurface $\\Hb_0$, in the coordinate system $(u,\\th^1,\\th^2)$ as in Section \\ref{coordinates}, we require the conformal class of the metric $\\hat{\\gamma}_{AB}$ to be smooth, satisfying $\\sqrt{\\det \\hat{\\gamma}_{AB}}=1$ and obeying the estimates\n$$\\sum_{j\\leq J+1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial u})^j(\\frac{\\partial}{\\partial\\th})^i\\hat{\\gamma}_{AB}|\\leq C.$$\n\nOn the initial outgoing hypersurface $H_0$, in the coordinate system $(u,\\th^1,\\th^2)$ as in Section \\ref{coordinates}, we now prescribe the conformal class of the metric $\\hat{\\gamma}_{AB}$ satisfying $\\sqrt{\\det \\hat{\\gamma}_{AB}}=1$ such that its $\\ub$ derivative has a jump discontinuity. To this end we define smooth matrices $((\\hat{\\gamma})_1)_{AB}$ and $((\\hat{\\gamma})_2)_{AB}$, such that $((\\hat{\\gamma})_1)_{AB}$ is positive definite with determinant equals to $1$ and both $((\\hat{\\gamma})_1)_{AB}$ and $((\\hat{\\gamma})_2)_{AB}$ satisfy\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i((\\hat{\\gamma})_j)_{AB}|\\leq C.$$\nFix $\\ub_s\\leq\\frac{\\epsilon}{2}$, where $\\epsilon>0$ is a small parameter depending on the constants $C$ above and will be determine later. \nLet\n$$\\underline{\\hat{\\gamma}}_{AB}=((\\hat{\\gamma})_1)_{AB}+(\\ub-\\ub_s)((\\hat{\\gamma})_2)_{AB}\\mathbbm 1_{\\{\\ub\\geq\\ub_s\\}}.$$\nFor $\\epsilon$ sufficiently small depending on $C$, $\\underline{\\hat{\\gamma}}_{AB}$ is positive definite for $0\\leq\\ub\\leq\\epsilon$. We then define \n$$\\hat{\\gamma}_{AB}=\\frac{1}{\\sqrt{\\det \\underline{\\hat{\\gamma}}}}\\underline{\\hat{\\gamma}}_{AB}.$$\n\nAccording to the procedure in \\cite{Chr}, there exists $\\epsilon$ sufficiently small depending only on $C$ such that there exists initial data for $0\\leq u\\leq \\epsilon$ on $\\Hb_0$ satisfying \\eqref{con2} and initial data for $0\\leq \\ub\\leq \\epsilon$ on $H_0$ obeying \\eqref{con1}. We refer the readers to \\cite{Chr} for details.\n\nWe note that according to \\cite{Chr}, in order to obtain the initial data set on $H_0$, we need to solve for the conformal factor $\\Phi$ defined by \n$$\\gamma_{AB}=\\Phi^2 \\hat{\\gamma}_{AB},$$\nwhich obeys the ODE\n\\begin{equation}\\label{PhiODE}\n\\frac{\\partial^2\\Phi}{\\partial \\ub^2}+\\frac 18 ({\\hat\\gamma}^{-1})^{AC}({\\hat\\gamma}^{-1})^{BD}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{CD}\\Phi=0,\n\\end{equation}\nwith initial data\n$$\\Phi |_{S_{0,0}}=1.$$\nThe time of existence $\\epsilon$ for this ODE depends only on the size of $({\\hat\\gamma}^{-1})^{AC}({\\hat\\gamma}^{-1})^{BD}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{CD}$. Therefore, even though $\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB}$ is discontinuous, we can prescribe the discontinuity at $\\ub=\\ub_s$ such that $\\ub_s=\\frac{\\epsilon}{2}$.\n\nGiven the conformal part of the metric $\\hat{\\gamma}$ and the conformal factor $\\Phi$, we can identify\n\\begin{equation}\\label{chihPhi}\n\\chih_{AB}=\\frac{1}{2}\\Phi^2\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB},\n\\end{equation}\nand\n\\begin{equation}\\label{trchPhi}\n\\trch=\\frac{2}{\\Phi}\\frac{\\partial\\Phi}{\\partial\\ub}.\n\\end{equation}\nWith this identification, $\\chih$ has a jump discontinuity at $\\ub=\\ub_s$ while $\\trch$ is continuous.\n\n\\subsection{Approximation Procedure}\n\nWe now introduce an approximation procedure to construct a sequence of smooth initial data approaching the data described above. Consider a $C^\\infty_0(\\mathbb R)$ function $\\tilde{h}_0$ that is supported in $[-1,1]$ and is identically 1 in $[-\\frac{1}{2},\\frac{1}{2}]$. Let\n$$\\tilde{h}(x)=\\left\\{\\begin{array}{clcr}\\tilde{h}_0(x)-\\tilde{h}_0(2x)&x\\ge 0\\\\0&x< 0\\end{array}\\right.$$\nNote that $\\tilde{h}$ is smooth. Now let\n$$h_n(x)=\\mathbbm 1_{\\{x\\geq 0\\}}+\\sum_{j=-\\infty}^{n}\\tilde{h}(2^j x).$$\nWe note that $h_n$ is supported in $\\{x\\geq 2^{-(n+1)}\\}$ and $h'_n:=h_n-h_{n-1}$ is supported in $\\{2^{-(n+1)}\\leq x\\leq 2^{-n}\\}$. Moreover, $h_{n}\\to\\mathbbm 1_{\\{x\\geq 0\\}}$ in $L^p$ for every $p<\\infty$ as $n\\to\\infty$ and $x h_n\\to x\\mathbbm 1_{\\{x\\geq 0\\}}$ in $L^p$ for every $p\\leq\\infty$ as $n\\to\\infty$. At the $n$-th step, we define \n$$(\\gamma_n,\\zeta_n,\\trch_n)=(\\gamma,\\zeta,\\trch)\\mbox{  on $S_{0,0}$}$$\nand\n$$(\\chibh_n,\\trchb_n)=(\\chibh,\\trchb)\\mbox{  on $\\Hb_{0}$}$$\nas before. Define\n$$(\\underline{\\hat{\\gamma}}_n)_{AB}=((\\hat{\\gamma})_1)_{AB}+(\\ub-\\ub_s)h_n(\\ub-\\ub_s)((\\hat{\\gamma})_2)_{AB}\\mbox{  on $H_{0}$,}$$\nand\n$$(\\hat{\\gamma}_n)_{AB}=\\frac{1}{\\sqrt{\\det (\\underline{\\hat{\\gamma}})_n}}(\\underline{\\hat{\\gamma}}_n)_{AB}\\mbox{  on $H_{0}$}.$$\nNotice that for $\\epsilon$ sufficiently small, we have uniform upper bounds for each component of  $\\hat{\\underline{\\gamma}}$ and their angular derivatives and a uniform lower bound for $\\det\\hat{\\underline{\\gamma}}$. Moreover, it is easy to see that for \n$${\\hat{\\gamma}}'_n:={\\hat{\\gamma}}_n-{\\hat{\\gamma}}_{n-1},$$\nwe have\n$$\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i{\\hat{\\gamma}}'_n|\\leq C2^{(-1+k)n},\\quad\\mbox{ for }\\ub_s\\leq \\ub\\leq\\ub_s+2^{-n}, k\\leq K+1, $$\n$$(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i{\\hat{\\gamma}}'_n=0,\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n},\\mbox{ for $i\\leq I+3$, $k\\leq K+1$}, $$\nThus\n$$(\\int_0^\\epsilon|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i{\\hat{\\gamma}}'_n|^p d\\ub)^{\\frac 1p}\\leq C2^{(-1+k-\\frac 1p) n}\\mbox{ for }k\\leq K+1.$$\nMoreover, we know that ${\\hat{\\gamma}}_n\\to {\\hat{\\gamma}}$ in the sense that\n$$(\\int_0^\\epsilon|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i({\\hat{\\gamma}}_n-{\\hat{\\gamma}})|^p d\\ub)^{\\frac 1p}\\leq C2^{(-1+k-\\frac 1p) n}\\mbox{ for }k\\leq K+1.$$\n\n\nWith the definition of $(\\hat{\\gamma}_n)_{AB}$, we can solve (\\ref{PhiODE}) for $\\Phi_n$ and use (\\ref{chihPhi}) and (\\ref{trchPhi}) to solve for $\\chih_n$ and $(\\trch)_n$. This then allows us to use the null structure equations and the Bianchi identities to solve for all Ricci coefficients and curvature components, as well as their derivatives. In the following we will use the notation that a subscript $n$ denotes that metric component, Ricci coefficient or curvature component associated to $\\hat{\\gamma}_n$. Moreover, we will use the notation\n$$\\Phi'_n=\\Phi_n-\\Phi_{n-1}.$$\nWe now derive the bounds for each of the $\\Phi_n$'s and $\\Phi'_n$'s in the initial data. By uniqueness of the constraint ODEs, all \n$$\\Phi'_n=0 \\mbox{ for }\\ub\\leq\\ub_s.$$\nWe therefore only derive bounds for $\\ub_s\\leq\\ub\\leq \\epsilon$.\n\nFrom (\\ref{PhiODE}), we have\n$$c\\leq \\Phi_n\\leq C,\\quad \\sum_{k\\leq K+2}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\Phi_n|\\leq C$$\nthus\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\th})^i(\\chih,\\trch)_n|\\leq C.$$\nThen using the equation (\\ref{PhiODE}) again, \n$$\\sum_{k\\leq K+2}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\Phi'_n|\\leq C2^{(-2+k)n},\\quad\\mbox{ for }\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n},$$\nand\n$$(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\Phi'_n=0,\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n}\\mbox{ and }2\\leq k\\leq K+2,\\mbox{ for all }i\\leq I+3.$$\nThen by (\\ref{chihPhi}) and (\\ref{trchPhi}), we have for $\\chih'_n$,\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i(\\chih'_n)_{AB}|\\leq C2^{kn},\\quad\\mbox{ for }\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i(\\chih'_n)_{AB}|\\leq C2^{-n},\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n};$$\nand for $\\trch'_n$:\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\trch'_n|\\leq C2^{(-1+k)n},\\quad\\mbox{ for }\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n}$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\trch'_n|\\leq C2^{-n},\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n}.$$\nFrom the equation for $\\zeta$ on $H_0$ together with the Codazzi equation, we have\n$$\\nabla_4\\zeta=\\div\\chih-\\frac 12 \\nab tr\\chi-\\chih\\zeta-\\frac 32 \\trch\\zeta.$$\nNotice that since $\\Omega=1$ on $H_0$, we can write\n$$\\nab_4\\zeta=\\frac{\\partial}{\\partial \\ub}\\zeta+\\chi\\zeta.$$\nThus, we can integrate to get the estimate\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\zeta_n|\\leq C.$$\nFor the difference $\\zeta'_n=\\zeta_n-\\zeta_{n-1}$,\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\zeta'_n|\\leq C2^{-n}.$$\nNotice that we have used the control on the metric to relate $\\div$ to the coordinates derivatives. Since on $H_0$,\n$$\\zeta=\\eta=-\\etab,$$\nwe have\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i(\\eta_n,\\etab_n)|\\leq C.$$\nand\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i(\\eta'_n,\\etab'_n)|\\leq C2^{-n}.$$\nBy the equation of $\\trchb$ on $H_0$\n$$\\nabla_4(\\trchb)+\\frac 12\\trch \\trchb=2\\rho-\\chih\\cdot\\chibh-2\\div\\zeta+|\\zeta|^2_\\gamma,$$\nand the Gauss equation\n$$K=-\\rho+\\frac 12 \\chih\\cdot\\chibh-\\frac 14 \\trch\\trchb,$$\nwe have\n$$\\nab_4(\\trchb)+\\trch \\trchb=-2K-2\\div\\zeta+|\\zeta|^2_\\gamma.$$\nSince the estimates for the Gauss curvature $K$ can be derived from the estimates for $\\gamma$ and its derivatives alone, by the estimates derived above and the initial estimate for $\\trchb$ on $S_{0,0}$,\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\trchb_n|\\leq C.$$\nand\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i \\trchb'_n|\\leq C2^{-n}.$$\nBy the equation of $\\chibh$ on $H_0$\n$$\\nabla_4 \\chibh=-\\frac{1}{2}\\trch\\chibh-\\nabla\\hot\\zeta+\\zeta\\hot\\zeta-\\frac{1}{2}\\trchb\\chih,$$\nwe have\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\chibh_n|\\leq C.$$\nand\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\chibh'_n|\\leq C2^{-n}.$$\nBy the following equation on $H_0$\n$$\\nabla_4\\chih=-\\trch \\chih-\\alpha,$$\nwe have\n$$\\sum_{k\\leq K-1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\alpha_n|\\leq C2^{(1+k)n},\\quad\\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K-1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\alpha_n|\\leq C2^{-n},\\quad\\mbox{ for }\\ub\\geq \\ub_s+2^{-n},$$\nBy the Codazzi equation on $H_0$\n$$\\div\\chi=\\frac{1}{2}\\nabla \\trch-\\zeta\\cdot(\\chih-\\frac{1}{2}\\trch)-\\beta,$$\nwe have\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\beta_n|\\leq C \\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i \\leq I+2}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\beta'_n|\\leq C2^{-n} \\mbox{ for }\\ub\\geq \\ub_s+2^{-n}.$$\nBy the Gauss equation on $H_0$\n$$K=-\\rho+\\frac{1}{2}\\chih\\cdot\\chibh-\\frac{1}{4}tr\\chi tr\\chib,$$\nwe have\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\rho_n|\\leq C \\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\rho'_n|\\leq C2^{-n} \\mbox{ for }\\ub\\geq \\ub_s+2^{-n}.$$\nBy the equation on $H_0$\n$$\\curl \\zeta=\\sigma+\\frac 12\\chibh\\wedge\\chih,$$\nwe have\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\sigma_n|\\leq C \\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\th})^i\\sigma'_n|\\leq C2^{-n} \\mbox{ for }\\ub\\geq \\ub_s+2^{-n}.$$\nOn $S_{0,0}$, using the Codazzi equation\n$$\\div\\chibh=\\frac{1}{2}\\nabla \\trchb+\\zeta\\cdot(\\chih-\\frac{1}{2}\\trch)+\\beta,$$\nwe have\n$$\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial\\th})^i\\betab_n|\\leq C \\mbox{ on $S_{0,0}$}.$$\nBy the following Bianchi equation on $H_0$\n$$\\nabla_4\\betab=-\\nab\\rho+^*\\nab\\sigma-tr\\chi\\betab+2\\chibh\\cdot\\beta-3(^*\\zeta\\sigma-\\zeta\\rho),$$\nwe have\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\betab_n|\\leq C,$$\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\betab'_n|\\leq C2^{-n}.$$\nBy the null structure equation on $H_0$\n$$\\nabla_4\\omegab=3|\\zeta|^2+\\frac 12\\rho,$$\nwe have \n$$\\sum_{k\\leq K+1}\\sum_{i=0}^{I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\omegab_n|\\leq C.$$\n$$\\sum_{k\\leq K+1}\\sum_{i=0}^{I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\omegab'_n|\\leq C2^{-n}.$$\nTogether with $\\omega=0$ which follows from $\\omega=-\\frac{1}{2}\\nabla_4(\\log\\Omega)$, we have the bounds in $L^\\infty$ for all the Ricci coefficients and null curvature components on $H_0$.\n\nA similar procedure allows us to solve for the Ricci coefficients and the null curvature components on $\\Hb_0$. Recall that for an impulsive gravitational wave, we take the data on $\\Hb_0$ to be fixed, i.e., $\\psi_n=\\psi$ and $\\Psi_n=\\Psi$ on $\\Hb_0$.\n\nFinally, we see that on $H_0$, we can define $\\nab_3^j\\psi_n$ and $\\nab_3^j\\Psi_n$ for $j\\leq J$ by taking the values of $\\nab_3^j\\psi_n$ and $\\nab_3^j\\Psi_n$ restricted to $S_{0,0}$ and solve the ODEs as above. Notice that they can be solved as long as $I$ is sufficiently large depending on $J$.\n\nThis allows us to conclude\n\\begin{proposition}\\label{dataprop}\nFor every $I\\geq 2$, $J,K\\geq 0$, there exists characteristic initial data for an impulsive gravitational wave such that on $\\Hb_0$:\n\\begin{equation}\\label{data1}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nabla_3^j\\nab_4^k\\nabla^i\\psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},\n\\end{equation}\n\\begin{equation}\\label{data2}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_u||\\nabla_3^j\\nab_4^k\\nabla^i\\Psi||_{L^2(S_{u,0})}\\leq C_{I,J,K},\\end{equation}\nOn $H_0$, $\\chih$ has a jump discontinuity across $S_{0,\\ub_s}$, and\n\\begin{equation}\\label{data3}\n\\sum_{j\\leq J}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J},\n\\end{equation}\n\\begin{equation}\\label{data4}\n\\sum_{j\\leq J}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\Psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J},\n\\end{equation}\nand for $H_0\\setminus\\{|\\ub-\\ub_s|\\geq\\eta\\}$ for every $\\eta>0$,\n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},$$\n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\Psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},$$\nand for $\\epsilon$ sufficiently small, $\\gamma$ is positive definite on $H_0$ and $\\Hb_0$ with bounded angular derivatives up to the $(I+2)$-nd derivative. \n\nThere exists a sequence of of smooth initial data approaching the characteristic initial data described above such that (\\ref{data1})-(\\ref{data4}) hold for $\\psi_n$ and $\\Psi_n$, \nand for $H_0\\setminus\\{\\ub_s\\leq\\ub\\leq \\ub_s+2^{-n}\\}$,\n\\begin{equation}\\label{data5}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\psi_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},\n\\end{equation}\n\\begin{equation}\\label{data6}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\Psi_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},\n\\end{equation}\nand for $H_0\\cap\\{\\ub_s\\leq\\ub\\leq \\ub_s+2^{-n}\\}$,\n\\begin{equation}\\label{data7}\n\\sum_{j\\leq J}\\sum_{i\\leq I}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K}2^n.\n\\end{equation}\nThis sequence of data converges in the sense that on $\\Hb_0$, the data is fixed; and on $H_0$, for $\\psi'_n=\\psi_n-\\psi_{n-1}$ and $\\Psi'_n=\\Psi_n-\\Psi_{n-1}$,\n\\begin{equation}\\label{data8}\n\\sum_{j\\leq J}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J}2^{-n},\n\\end{equation}\n\\begin{equation}\\label{data9}\n\\sum_{j\\leq J}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\Psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J}2^{-n},\n\\end{equation}\non $H_0\\setminus\\{\\ub_s\\leq\\ub\\leq \\ub_s+2^{-n}\\}$,\n\\begin{equation}\\label{data10}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K}2^{-n},\n\\end{equation}\n\\begin{equation}\\label{data11}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\Psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K}2^{-n}.\n\\end{equation}\n\\end{proposition}\n\\begin{remark}\nIn the context of Theorem \\ref{rdthmv2}, corresponding sequences of $\\psi_n$ and $\\Psi_n$ can be constructed. As opposed to the case of an impulsive gravitational wave, such sequences will satisfy (\\ref{data1})-(\\ref{data4}) and (\\ref{data8})-(\\ref{data9}), but they will not obey the improved estimates of Proposition \\ref{dataprop} which holds on $H_0\\setminus\\{\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n}\\}$. This additional regularity is only relevant for the argument that the impulsive gravitational wave is smooth away from $\\Hb_{\\ub_s}$.\n\\end{remark}\nProposition \\ref{dataprop} in particular constructs the set of initial data satisfying the assumptions of Theorem \\ref{giwthmv1}. In the following, we will focus on the case $I=2$, $J=K=0$. This level of regularity is sufficient to prove all the estimates and to construct a spacetime metric satisfying the Einstein equations.\n\n\\section{A Priori Estimates}\\label{estimates}\n\nIn this section, we begin the proof of Theorem \\ref{rdthmv2}. We show that for any smooth characteristic initial data satisfying the estimates in the assumption of Theorem \\ref{rdthmv2}, there is an $\\epsilon$ depending only on the constants in the estimates in the assumptions such that the unique resulting spacetime admits a double null foliation $(u, \\ub)$ and remains smooth with precise estimates on its Ricci coefficients and curvature components in $0\\leq u\\leq u_*$ and $0\\leq \\ub\\leq\\ub_*$ for $u, \\ub\\leq \\epsilon$. In particular, no assumptions have to be made on the size of $\\alpha$.\n\nIn the context of Theorem \\ref{rdthmv2}, our Theorem in this Section asserts that a region of existence can be found independent of $n$ for any smooth approximating sequence of data satisfying the bounds in the assumptions of Theorem \\ref{rdthmv2}.\n\nWe first define the norms that we will work with. Let\n$$\\mathcal R=\\sum_{i\\leq 2}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_u||\\nabla^i\\Psi||_{L^2(H_u)} +\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_{\\ub}||\\nabla^i\\Psi||_{L^2(\\underline{H}_{\\ub})}\\right),$$\n$$\\mathcal O_{i,p}=\\sup_{u,\\ub}||\\nabla^i\\psi||_{L^p(S_{u,\\ub})},$$\nand\n$$\\tilde{\\mathcal O}_{i,2}=\\sum_{\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}}\\sup_u||\\nabla^i\\psi_H||_{L^2(H_u)}+ \\sum_{\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}}\\sup_{\\ub}||\\nabla^i\\psi_{\\Hb}||_{L^2(\\Hb_{\\ub})}.$$\nWe write\n$$\\mathcal O=\\sum_{i=0}^{2} \\mathcal O_{i,2}+\\sum_{i=0}^{1}\\mathcal O_{i,4}+\\mathcal O_{0,\\infty}.$$\nThe following is the main theorem in this section:\n\\begin{theorem}\\label{timeofexistence}\nSuppose the initial data set for the characteristic initial value problem is smooth and satisfies\n$$c\\leq |\\det\\gamma \\restriction_{S_{u,0}} |\\leq C,\\quad \\sum_{i\\leq 3}|(\\frac{\\partial}{\\partial\\th})^i\\gamma \\restriction_{S_{u,0}}|\\leq C,$$\n$$\\mathcal O_0:= \\sum_{i\\leq 2} \\sup_{\\ub}||\\nabla^i\\psi||_{L^2(S_{0,\\ub})}+\\sum_{i\\leq 2} \\sup_{u}||\\nabla^i\\psi||_{L^2(S_{u,0})}+||\\nabla^3\\psi||_{L^2(H_0)}+||\\nabla^3\\psi||_{L^2(\\Hb_0)}\\leq C,$$\n$$\\mathcal R_0:=\\sum_{i\\leq 2}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab^i\\Psi||_{L^2(H_0)}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab^i\\Psi||_{L^2(\\Hb_0)}\\right)\\leq C.$$\nThen for $\\epsilon$ sufficiently small depending only on $C$ and $c$, there exists a unique spacetime $(\\mathcal M, g)$ endowed with a double null foliation $(u,\\ub)$ that solves the Einstein equations in the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$, whenever $u_*, \\ub_*\\leq \\epsilon$. Moreover, in this region, the following norms are bounded above by a constant $C'$ depending only on $C$ and $c$:\n$$\\mathcal O, \\tilde{\\mathcal O}_{3,2}, \\mathcal R < C'.$$\n\\end{theorem}\nIt follows by \\cite{Rendall} that for every smooth initial data set, a unique smooth spacetime satisfying the Einstein equations exists in a small neighborhood of $S_{0,0}$. In order to show that the size of this region only depends on the constants in the assumption of Theorem \\ref{timeofexistence}, it suffices to establish a priori control in the region $0\\leq u\\leq\\epsilon$, $0\\leq \\ub\\leq \\epsilon$. The Theorem then follows from a standard last slice argument (See \\cite{L}).\n\nIn this section, all estimates will be proved under the following bootstrap assumption:\n\\begin{equation}\\tag{A}\\label{BA1}\n\\mathcal O_{0,\\infty}\\leq \\Delta_0.\n\\end{equation}\n\nWe now outline the steps in proving a priori estimates. In Section \\ref{metric}, we will first estimate the metric components under the bootstrap assumption (\\ref{BA1}). In Section \\ref{transportsec}-\\ref{elliptic}, we will derive some preliminary estimates and formulae. In Section \\ref{transportsec}, we provide Propositions which gives $L^p$ estimates for general quantities satisfying transport equations. They will be used to control the Ricci coefficients and curvature components. In Section \\ref{Embedding}, we establish Sobolev Embedding Theorems in our setting. In Section \\ref{commutation}, we state some formulae for the commutators $[\\nabla_4,\\nabla]$ and $[\\nabla_3,\\nabla]$. They will then be used to obtain higher order estimates for general quantities satisfying transport equations. In Section \\ref{elliptic}, we prove elliptic estimates for general quantities obeying Hodge systems. \n\nAfter these preliminary estimates, we estimate the Ricci coefficients and the curvature components. This proceeds in three steps:\\\\\n\n\\noindent {\\bf STEP 1}: In Section \\ref{Riccisec}, we prove that $\\mathcal R<\\infty$ and $\\tilde{\\mathcal O}_{3,2}<\\infty$ together imply that $\\mathcal O\\leq C(\\mathcal O_0)$, where $C(\\mathcal O_0)$ is a constant depending only on the initial norm $\\mathcal O_0$. These estimates are proved via the null structure equations.\\\\\n\n\\noindent {\\bf STEP 2}: In Section \\ref{Ricciellipticsec}, we use elliptic estimates to prove that $\\mathcal R<\\infty$ implies $\\tilde{\\mathcal O}_{3,2}\\leq C(\\mathcal O_0,\\mathcal R_0,\\mathcal R)$, where $C(\\mathcal O_0,\\mathcal R_0,\\mathcal R)$ is a constant depending both on the initial norm and $\\mathcal R$. \\\\\n\n\\noindent {\\bf STEP 3}: In Section \\ref{energyestimatessec}, we derive the energy estimates and prove the boundedness of $\\mathcal R$ by the initial data.\n\\\\\n\nThese steps provide bounds for the Ricci coefficients up to three angular derivatives and the curvature components up to two angular derivatives depending only on the constants in the assumptions of Theorem \\ref{timeofexistence}. In Section \\ref{Propregsec}, we show that this implies that higher regularity propagates and that the spacetime remains smooth in a region depending only on the constants in the assumptions of Theorem \\ref{timeofexistence}. This allows us to conclude the proof of Theorem \\ref{timeofexistence} in Section \\ref{EndofProof}.\n\nFinally, in Section \\ref{AddEst}, we return to the sequence of smooth spacetimes arising from the sequence of data approximating the data of an impulsive gravitational wave as described in Section \\ref{initialcondition}. We show that these spacetimes obey estimates in addition to those given in Theorem \\ref{timeofexistence}. These bounds will be applied in Section \\ref{limitgiw} to establish the regularity and singularity properties of an impulsive gravitational spacetime.\n\n\\subsection{Estimates for Metric Components}\\label{metric}\nWe first show that we can control $\\Omega$ under the bootstrap assumption (\\ref{BA1}):\n\\begin{proposition}\\label{Omega}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that for every $\\epsilon\\leq\\epsilon_0$,\n$$\\frac 12\\leq \\Omega\\leq 2.$$\n\\end{proposition}\n\\begin{proof}\nConsider the equation\n\\begin{equation}\\label{Omegatransport}\n \\omega=-\\frac{1}{2}\\nabla_4\\log\\Omega=\\frac{1}{2}\\Omega\\nabla_4\\Omega^{-1}=\\frac{1}{2}\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}.\n\\end{equation}\nFix $\\ub$. Notice that both $\\omega$ and $\\Omega$ are scalars and therefore the $L^\\infty$ norm is independent of the metric. We can integrate equation (\\ref{Omegatransport}) using the fact that $\\Omega^{-1}=1$ on $H_0$ to obtain\n$$||\\Omega^{-1}-1||_{L^\\infty(S_{u,\\ub})}\\leq C\\int_0^{u}||\\omega||_{L^\\infty(S_{u',\\ub})}du\\leq C\\Delta_0\\epsilon.$$\nThis implies both the upper and lower bounds for $\\Omega$ for sufficiently small $\\epsilon$.\n\\end{proof}\n\n\nWe then show that we can control $\\gamma$ under the bootstrap assumption (\\ref{BA1}):\n\\begin{proposition}\\label{gamma}\nConsider a coordinate patch $U$ on $S_{0,0}$ and define $U_{u,0}$ to be a coordinate patch on $S_{u,0}$ given by the one-parameter diffeomorphism generated by $\\Lb$. Define $U_{u,\\ub}$ to be the image of $U_{u,0}$ under the one-parameter diffeomorphism generated by $L$. Define also $D_U=\\bigcup_{0\\leq u\\leq \\epsilon ,0\\leq \\ub\\leq \\epsilon} U_{u,\\ub}$. For $\\epsilon$ small enough depending on initial data and $\\Delta_0$, there exists $C$ and $c$ depending only on initial data such that the following pointwise bounds hold for $\\gamma$ in $\\mathcal D_U$:\n$$c\\leq \\det\\gamma\\leq C. $$\nMoreover, in $D_U$,\n$$|\\gamma_{AB}|,|(\\gamma^{-1})^{AB}|\\leq C.$$\n\\end{proposition}\n\\begin{proof}\nThe first variation formula states that\n$$\\Ls_L\\gamma=2\\Omega\\chi.$$\nIn coordinates, this means\n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB}.$$\nFrom this we derive that \n$$\\frac{\\partial}{\\partial \\ub}\\log(\\det\\gamma)=\\Omega\\trch.$$\nDefine $\\gamma_0(u,\\ub,\\th^1,\\th^2)=\\gamma(0,\\ub,\\th^1,\\th^2)$. \n$$|\\det\\gamma-\\det(\\gamma_0)|\\leq C\\int_0^{\\ub}|\\trch|d\\ub'\\leq C\\Delta_0\\epsilon.$$\nThis implies that the $\\det \\gamma$ is bounded above and below. Let $\\Lambda$ be the larger eigenvalue of $\\gamma$. Clearly,\n\\begin{equation}\\label{La}\n\\Lambda\\leq C\\sup_{A,B=1,2}\\gamma,\n\\end{equation}\nand \n$$\\sum_{A,B=1,2}|\\chi_{AB}|^2\\leq C\\Lambda ||\\chi||_{L^\\infty(S_{u,\\ub})}.$$\nThen\n$$|\\gamma_{AB}-(\\gamma_0)_{AB}|\\leq C\\int_0^{\\ub}|\\chi_{AB}|d\\ub'\\leq C\\Lambda\\Delta_0\\epsilon.$$\nUsing the upper bound (\\ref{La}), we thus obtain the upper bound for $|\\gamma_{AB}|$. The upper bound for $|(\\gamma^{-1})^{AB}|$ follows from the upper bound for $|\\gamma_{AB}|$ and the lower bound for $\\det\\gamma$.\n\\end{proof}\n\nA consequence of the previous Proposition is an estimate on the surface area of each two sphere $S_{u,\\ub}$.\n\\begin{proposition}\\label{area}\n$$\\sup_{u,\\ub}|\\mbox{Area}(S_{u,\\ub})-\\mbox{Area}(S_{u,0})|\\leq C\\Delta_0\\epsilon.$$\n\\end{proposition}\n\\begin{proof}\nThis follows from the fact that $\\sqrt{\\det\\gamma}$ is pointwise only slightly perturbed if $\\epsilon$ is chosen to be appropriately small.\n\\end{proof}\nWith the estimate on the volume form, we can now show that the $L^p$ norms defined with respect to the metric and the $L^p$ norms defined with respect to the coordinate system are equivalent.\n\\begin{proposition}\\label{eqnorm}\nGiven a covariant tensor $\\phi_{A_1...A_r}$ on $S_{u,\\ub}$, we have\n$$\\int_{S_{u,\\ub}} <\\phi,\\phi>_{\\gamma}^{p\/2} \\sim \\sum_{i=1}^r\\sum_{A_i=1,2}\\iint |\\phi_{A_1...A_r}|^p \\sqrt{\\det\\gamma} d\\th^1 d\\th^2.$$\n\\end{proposition}\nWe can also control $b$ under the bootstrap assumption, thus controlling the full spacetime metric: \n\\begin{proposition}\\label{b}\nIn the coordinate system $(u,\\ub,\\th^1,\\th^2)$,\n$$|b^A|\\leq C\\Delta_0\\epsilon.$$\n\\end{proposition}\n\\begin{proof}\n$b^A$ satisfies the equation\n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A.$$\nThis can be derived from \n$$[L,\\Lb]=\\frac{\\partial b^A}{\\partial \\ub}\\frac{\\partial}{\\partial \\th^A}.$$\nNow, integrating and using Proposition \\ref{eqnorm} gives the result.\n\\end{proof}\n\n\\subsection{Estimates for Transport Equations}\\label{transportsec}\nThe estimates for the Ricci coefficients and the null curvature components are derived from the null structure equations and the null Bianchi equations respectively. In order to use the equations, we need a way to obtain estimates from the null transport type equations. Such estimates require the boundedness of $\\trch$ and $\\trchb$, which is consistent with our bootstrap assumption (\\ref{BA1}). Below, we state two Propositions which provide $L^p$ estimates for general quantities satisfying transport equations either in the $e_3$ or $e_4$ direction.\n\\begin{proposition}\\label{transport}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that for all $\\epsilon \\leq \\epsilon_0$ and for every $2\\leq p<\\infty$, we have\n\\[\n ||\\phi||_{L^p(S_{u,\\ub})}\\leq C(||\\phi||_{L^p(S_{u,\\ub'})}+\\int_{\\ub'}^{\\ub} ||\\nabla_4\\phi||_{L^p(S_{u,\\ub''})}d{\\ub''}),\n\\]\n\\[\n ||\\phi||_{L^p(S_{u,\\ub})}\\leq C(||\\phi||_{L^p(S_{u',\\ub})}+\\int_{u'}^{u} ||\\nabla_3\\phi||_{L^p(S_{u'',\\ub})}d{u''}).\n\\]\n\\end{proposition}\n\n\\begin{proof}\n\nThe following identity holds for any scalar $f$:\n\\[\n \\frac{d}{d\\ub}\\int_{\\mathcal S_{u,\\ub}} f=\\int_{\\mathcal S_{u,\\ub}} \\left(\\frac{df}{d\\ub}+\\Omega \\trch f\\right)=\\int_{\\mathcal S_{u,\\ub}} \\Omega\\left(e_4(f)+ \\trch f\\right).\n\\]\nSimilarly, we have\n\\[\n \\frac{d}{du}\\int_{\\mathcal S_{u,\\ub}} f=\\int_{\\mathcal S_{u,\\ub}} \\Omega\\left(e_3(f)+ \\trchb f\\right).\n\\]\nHence, taking $f=|\\phi|_{\\gamma}^p$, we have\n\\begin{equation}\\label{Lptransport}\n\\begin{split}\n ||\\phi||^p_{L^p(S_{u,\\ub})}=&||\\phi||^p_{L^p(S_{u,\\ub'})}+\\int_{\\ub'}^{\\ub}\\int_{S_{u,\\ub''}} p|\\phi|^{p-2}\\Omega\\left(<\\phi,\\nabla_4\\phi>_\\gamma+ \\frac{1}{p}\\trch |\\phi|^2_{\\gamma}\\right)d{\\ub''}\\\\\n ||\\phi||^p_{L^p(S_{u,\\ub})}=&||\\phi||^p_{L^p(S_{u',\\ub})}+\\int_{u'}^{u}\\int_{S_{u'',\\ub}} p|\\phi|^{p-2}\\Omega\\left(<\\phi,\\nabla_3\\phi>_\\gamma+ \\frac{1}{p}\\trchb |\\phi|^2_{\\gamma}\\right)d{u''}\n\\end{split}\n\\end{equation}\nThe Proposition is proved using Cauchy-Schwarz on the sphere and the $L^\\infty$ bounds for $\\Omega$ and $\\trch$ ($\\trchb$) which are provided by Proposition \\ref{Omega} and the bootstrap assumption (\\ref{BA1}) respectively.\n\\end{proof}\nThe above estimates also hold for $p=\\infty$:\n\\begin{proposition}\\label{transportinfty}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that for all $\\epsilon \\leq \\epsilon_0$, we have\n\\[\n ||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\left(||\\phi||_{L^\\infty(S_{u,\\ub'})}+\\int_{\\ub'}^{\\ub} ||\\nabla_4\\phi||_{L^\\infty(S_{u,\\ub''})}d{\\ub''}\\right)\n\\]\n\\[\n ||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\left(||\\phi||_{L^\\infty(S_{u',\\ub})}+\\int_{u'}^{u} ||\\nabla_3\\phi||_{L^\\infty(S_{u'',\\ub})}d{u''}\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nThis follows simply from integrating along the integral curves of $L$ and $\\Lb$, and the estimate on $\\Omega$ in Proposition \\ref{Omega}.\n\\end{proof}\n\n\\subsection{Sobolev Embedding}\\label{Embedding}\nUnder the bootstrap assumption (\\ref{BA1}), the Sobolev Embedding hold on a 2-sphere $S_{u,\\ub}$. \n\\begin{proposition}\\label{L4}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that as long as $\\epsilon\\leq \\epsilon_0$, we have\n$$||\\phi||_{L^4(S_{u,\\ub})}\\leq C\\sum_{i=0}^1||\\nabla^i\\phi||_{L^2(S_{u,\\ub})}. $$\n\\end{proposition}\n\\begin{proof}\nWe first prove this for scalars. Given our coordinate system, the desired Sobolev Embedding follows from standard Sobolev Embedding Theorems and the lower and upper bound of the volume form. Thus, the proposition holds for scalars. Now, for $\\phi$ being a tensor, let $f=\\sqrt{|\\phi|_{\\gamma}^2+\\delta^2}$. Then\n$$||f||_{L^4(S_{u,\\ub})}\\leq C\\left(||f||_{L^2(S_{u,\\ub})}+||\\frac{<\\phi,\\nabla\\phi>_{\\gamma}}{\\sqrt{|\\phi|_{\\gamma}^2+\\delta^2}}||_{L^2(S_{u,\\ub})}\\right)\\leq C\\left(||f||_{L^2(S_{u,\\ub})}+||\\nabla\\phi||_{L^2(S_{u,\\ub})}\\right).$$\nThe Proposition can be achieved by sending $\\delta\\to 0$.\n\\end{proof}\nWe can also prove the Sobolev Embedding Theorem for the $L^\\infty$ norm: \n\\begin{proposition}\\label{Linfty}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that as long as $\\epsilon\\leq \\epsilon_0$, we have\n$$||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\left(||\\phi||_{L^2(S_{u,\\ub})}+||\\nabla\\phi||_{L^4(S_{u,\\ub})}\\right). $$\nAs a consequence,\n$$||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\sum_{i=0}^2||\\nabla^i\\phi||_{L^2(S_{u,\\ub})}. $$\n\\end{proposition}\n\\begin{proof}\nThe first statement follows from coordinate considerations as in Proposition \\ref{L4}. The second statement follows from applying the first and Proposition \\ref{L4}.\n\\end{proof}\n\n\\subsection{Commutation Formulae}\\label{commutation}\nWe have the following formula from \\cite{KN}:\n\\begin{proposition}\nThe commutator $[\\nabla_4,\\nabla]$ acting on an $(0,r)$ S-tensor is given by\n\\begin{equation*}\n \\begin{split}\n[\\nabla_4,\\nabla_B]\\phi_{A_1...A_r}=&[D_4,D_B]\\phi_{A_1...A_r}+(\\nabla_B\\log\\Omega)\\nabla_4\\phi_{A_1...A_r}-(\\gamma^{-1})^{CD}\\chi_{BD}\\nabla_C\\phi_{A_1...A_r} \\\\\n&-\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chi_{BD}\\etab_{A_i}\\phi_{A_1...\\hat{A_i}C...A_r}+\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chi_{A_iB}\\etab_{D}\\phi_{A_1...\\hat{A_i}C...A_r}.\n \\end{split}\n\\end{equation*}\nSimilarly, the commutator $[\\nabla_3,\\nabla]$ acting on an $(0,r)$ S-tensor is given by\n\\begin{equation*}\n \\begin{split}\n[\\nabla_3,\\nabla_B]\\phi_{A_1...A_r}=&[D_3,D_B]\\phi_{A_1...A_r}+(\\nabla_B\\log\\Omega)\\nabla_3\\phi_{A_1...A_r}-(\\gamma^{-1})^{CD}\\chib_{BD}\\nabla_C\\phi_{A_1...A_r} \\\\\n&-\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chib_{BD}\\eta_{A_i}\\phi_{A_1...\\hat{A_i}C...A_r}+\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chib_{A_iB}\\eta_{D}\\phi_{A_1...\\hat{A_i}C...A_r}.\n \\end{split}\n\\end{equation*}\n\\end{proposition}\n\nBy induction, we get the following schematic formula for repeated commutations:\n\\begin{proposition}\\label{commuteeqn}\nSuppose $\\nabla_4\\phi=F_0$. Let $\\nabla_4\\nabla^i\\phi=F_i$.\nThen\n\\begin{equation*}\n\\begin{split}\nF_i\\sim &\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3} F_0+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\\\\n&+\\sum_{i_1+i_2+i_3+i_4=i-1} \\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\beta\\nabla^{i_4} \\phi.\n\\end{split}\n\\end{equation*}\nwhere we have applied our schematic notation and by $\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}$ we mean the sum of all terms which is a product of $i_2$ factors, each factor being $\\nabla^j \\eta$ or $\\nabla^j\\underline{\\eta}$ for some $j$ and that the sum of all $j$'s is $i_1$, i.e., $$\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}=\\displaystyle\\sum_{j_1+...+j_{i_2}=i_1}\\sum_{\\psi_1,...,\\psi_{i_2}\\in\\{\\eta,\\etab\\}}\\nabla^{j_1}\\psi_1...\\nabla^{j_{i_2}}\\psi_{i_2}.$$ Similarly, suppose $\\nabla_3\\phi=G_{0}$. Let $\\nabla_3\\nabla^i\\phi=G_{i}$.\nThen\n\\begin{equation*}\n\\begin{split}\nG_{i}\\sim &\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3} G_{0}+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\underline{\\chi}\\nabla^{i_4} \\phi\\\\\n&+\\sum_{i_1+i_2+i_3+i_4=i-1} \\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\underline{\\beta}\\nabla^{i_4} \\phi.\n\\end{split}\n\\end{equation*}\n\n\\end{proposition}\n\\begin{proof}\nThe proof is by induction. We will prove it for the first statement. The proof of second one is analogous. The formula obviously holds for $i=0$. Assume that the statement is true for $i<i_0$.\n\\begin{equation*}\n\\begin{split}\nF_{i_0}=&[\\nabla_4,\\nabla]\\nabla^{i_0-1}\\phi+\\nabla F_{i_0-1}\\\\\n\\sim &\\chi\\nabla^{i_0}\\phi+(\\eta,\\underline{\\eta})\\nabla_4\\nabla^{i_0-1}\\phi+\\beta\\nabla^{i_0-1}\\phi+\\chi(\\eta,\\underline{\\eta})\\nabla^{i_0-1}\\phi\\\\\n&+\\sum_{i_1+i_2+i_3=i_0}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3} F_{0}\\\\\n&+\\sum_{i_1+i_2+i_3+i_4=i_0}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\\\\n&+\\sum_{i_1+i_2+i_3+i_4=i_0-1} \\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\beta\\nabla^{i_4} \\phi.\n\\end{split}\n\\end{equation*}\nFirst notice that the first, third and fourth terms are acceptable. Then plug in the formula for $F_{i_0-1}=\\nabla_4\\nabla^{i_0-1}\\phi$ and get the result.\n\\end{proof}\nThe following further simplified version is useful for our estimates in the next section:\n\\begin{proposition}\nSuppose $\\nabla_4\\phi=F_0$. Let $\\nabla_4\\nabla^i\\phi=F_i$.\nThen\n\\begin{equation*}\n\\begin{split}\nF_i\\sim &\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} F_0+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi.\\\\\n\\end{split}\n\\end{equation*}\nSimilarly, suppose $\\nabla_3\\phi=G_{0}$. Let $\\nabla_3\\nabla^i\\phi=G_{i}$.\nThen\n\\begin{equation*}\n\\begin{split}\nG_{i}\\sim &\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} G_{0}+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\underline{\\chi}\\nabla^{i_4} \\phi.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nWe replace $\\beta$ and $\\betab$ using the Codazzi equations, which schematically looks like\n$$\\beta=\\nabla\\chi+\\psi\\chi,$$\n$$\\betab=\\nabla\\chib+\\psi\\chib.$$\n\\end{proof}\n\n\\subsection{General Elliptic Estimates for Hodge Systems}\\label{elliptic}\nWe recall the definition of the divergence and curl of a symmetric covariant tensor of an arbitrary rank:\n$$(\\div\\phi)_{A_1...A_r}=\\nabla^B\\phi_{BA_1...A_r},$$\n$$(\\curl\\phi)_{A_1...A_r}=\\eps^{BC}\\nabla_B\\phi_{CA_1...A_r},$$\nwhere $\\eps$ is the volume form associated to the metric $\\gamma$.\nRecall also that the trace is defined to be\n$$(tr\\phi)_{A_1...A_{r-1}}=(\\gamma^{-1})^{BC}\\phi_{BCA_1...A_{r-1}}.$$\nThe following elliptic estimate is standard (See for example \\cite{CK} or \\cite{Chr}):\n\\begin{proposition}\\label{ellipticthm}\nLet $\\phi$ be a totally symmetric $r+1$ covariant tensorfield on a 2-sphere $(\\mathbb S^2,\\gamma)$ satisfying\n$$\\div\\phi=f,\\quad \\curl\\phi=g,\\quad \\mbox{tr}\\phi=h.$$\nSuppose also that\n$$\\sum_{i\\leq 1}||\\nabla^i K||_{L^2(S)}< \\infty.$$\nThen for $i\\leq 3$,\n$$||\\nabla^{i}\\phi||_{L^2(S)}\\leq C(\\sum_{k\\leq 1}||\\nabla^k K||_{L^2(S)})(\\sum_{j=0}^{i-1}(||\\nabla^{j}f||_{L^2(S)}+||\\nabla^{j}g||_{L^2(S)}+||\\nabla^{j}h||_{L^4(S)}+||\\phi||_{L^2(S)})).$$\n\\end{proposition}\nFor the special case that $\\phi$ a symmetric traceless 2-tensor, we only need to know its divergence:\n\\begin{proposition}\\label{elliptictraceless}\nSuppose $\\phi$ is a symmetric traceless 2-tensor satisfying\n$$\\div\\phi=f.$$\nSuppose moreover that $$\\sum_{i\\leq 1}||\\nabla^i K||_{L^2(S)}< \\infty.$$\nThen, for $i\\leq 3$,\n$$||\\nabla^{i}\\phi||_{L^2(S)}\\leq C(\\sum_{k\\leq 1}||\\nabla^k K||_{L^2(S)})(\\sum_{j=0}^{i-1}(||\\nabla^{j}f||_{L^2(S)}+||\\phi||_{L^2(S)})).$$\n\\end{proposition}\n\\begin{proof}\nIn view of Proposition \\ref{ellipticthm}, this Proposition follows from\n$$\\curl\\phi=^*f.$$\nThis is a direct computation using that fact that $\\phi$ is both symmetric and traceless.\n\\end{proof}\n\n\n\\subsection{$L^p(S)$ Estimates for the Ricci Coefficients}\\label{Riccisec}\nWe continue to work under the bootstrap assumptions (\\ref{BA1}). In this Subsection, we pursue step 1 of the proof as outlined in the beginning of the Section. We show if that the curvature norm $\\mathcal R$ and the third order Ricci coefficient norm $\\tilde{\\mathcal O}_{3,2}$ are bounded, then so is the Ricci coefficient norm $\\mathcal O$. In particular, our bootstrap assumption (\\ref{BA1}) and all the estimates in the last section are verified as long as $\\mathcal R$ and $\\tilde{\\mathcal O}_{3,2}$ are controlled.\n\\begin{proposition}\\label{Ricci}\nSuppose\n$$\\mathcal R<\\infty,\\quad\\tilde{\\mathcal O}_{3,2}<\\infty.$$\nThen there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R,\\tilde{\\mathcal O}_{3,2})$ such that for $\\epsilon\\leq \\epsilon_0$, \n\\[\n \\mathcal O\\leq C(\\mathcal O_0),\n\\]\nwhere $\\mathcal O_0$ is the initial norm for the Ricci coefficients defined in Theorem \\ref{timeofexistence}.\n\\end{proposition}\n\\begin{proof}\nTo prove the Proposition, we need to estimate $\\displaystyle\\sum_{i\\leq 1}{\\mathcal O}_{i,4}$ and $\\displaystyle\\sum_{i\\leq 2}{\\mathcal O}_{i,2}$. First, we estimate the $\\mathcal O_{i,4}$ norms. Using the null structure equations for all Ricci coefficients except $\\chih$, we have\n\\begin{equation*}\n\\sum_{\\psi\\in\\{\\trchb,\\chibh,\\etab,\\omega\\}}\\nabla_3\\psi=\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\Psi+\\psi\\psi,\n\\end{equation*}\nand\n\\begin{equation*}\n\\sum_{\\psi\\in\\{\\trch,\\eta,\\omegab\\}}\\nabla_4\\psi=\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\Psi+\\psi\\psi,\n\\end{equation*}\nwhere the $\\psi$ on the right hand sides denote any Ricci coefficients. We will also use the null structure equations commuted with angular derivatives:\n\\begin{equation}\\label{Riccieqn}\n\\begin{split}\n\\sum_{\\psi\\in\\{\\trchb,\\chibh,\\etab,\\omega\\}}\\nabla_3\\nabla^i\\psi=&\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\Psi+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi,\\\\\n\\sum_{\\psi\\in\\{\\trch,\\eta,\\omegab\\}}\\nabla_4\\nabla^i\\psi=&\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\Psi+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi,\n\\end{split}\n\\end{equation}\nwhere we have applied the notation defined at the end of Section \\ref{sec.eqns} to represent the sum of products of derivatives of $\\psi$ by $\\nab^{i_1}\\psi^{i_2}$.\nBy Sobolev Embedding in Proposition \\ref{L4} and the bootstrap assumption (\\ref{BA1}), we have\n$$\\sum_{i_1+i_2+i_3\\leq 1}||\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\Psi||_{L^4(S_{u,\\ub})}\\leq \\Delta_0\\sum_{i=0}^2||\\nabla^i\\Psi||_{L^2(S_{u,\\ub})}.$$\n\\begin{equation}\\label{Riccinl}\n\\sum_{i_1+i_2+i_3+i_4\\leq 1}||\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi||_{L^4(S_{u,\\ub})}\\leq (\\Delta_0+\\Delta_0^2)\\sum_{i=0}^1\\mathcal O_{i,4}.\n\\end{equation}\nHence, by Proposition \\ref{transport} applied to (\\ref{Riccinl}), we obtain\n\\begin{equation}\\label{RicciL41}\n\\begin{split}\n&\\sum_{i=0}^{1}\\mathcal O_{i,4}[\\trch,\\eta,\\etab,\\omega,\\omegab,\\trchb,\\chibh]\\\\\n\\leq &\\mathcal O_0+C\\epsilon^{\\frac{1}{2}}\\Delta_0\\mathcal R+C(\\Delta_0)\\epsilon \\sum_{i=0}^{1}\\mathcal O_{i,4}.\n\\end{split}\n\\end{equation}\nIn order to close this estimate, we need to estimate $\\chih$. We like to avoid the use of the equation $\\nabla_4\\chih=-\\alpha+\\psi\\psi$ since we do not have an estimate for singular curvature component $\\alpha$. Instead, we use the equation\n$$\\nabla_3\\chih=\\nabla\\hot\\eta+\\psi\\psi.$$\nCommuting with angular derivatives, we get\n\\begin{equation}\\label{Riccieqn2}\n\\nabla_{3}\\nabla^i\\chih=\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3+1}\\eta+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi.\n\\end{equation}\nApplying Proposition \\ref{transport} to (\\ref{Riccieqn2}) and using the estimates in (\\ref{Riccinl}), we have\n\\begin{equation*}\n\\sum_{i=0}^1||\\nabla^i\\chih||_{L^4(S_{u,\\ub})}\\leq \\mathcal O_0+C\\int_0^u ||\\nabla^2\\eta||_{L^4(S_{u',\\ub})}du'+C(\\Delta_0)\\epsilon \\sum_{i=0}^{1}\\mathcal O_{i,4}.\n\\end{equation*}\nBy Cauchy-Schwarz and the Sobolev Embedding in Proposition \\ref{L4},\n$$\\int_0^u ||\\nabla^2\\eta||_{L^4(S_{u',\\ub})}du' \\leq C\\epsilon \\mathcal O_{2,2}+C\\epsilon^{\\frac 12} \\tilde{\\mathcal O}_{3,2}.$$\nTherefore,\n\\begin{equation}\\label{RicciL42}\n\\sum_{i=0}^1||\\nabla^i\\chih||_{L^4(S_{u,\\ub})}\\leq C\\epsilon \\mathcal O_{2,2}+C\\epsilon^{\\frac 12} \\tilde{\\mathcal O}_{3,2}+C(\\Delta_0)\\epsilon \\sum_{i=0}^{1}\\mathcal O_{i,4}.\n\\end{equation}\nPutting (\\ref{RicciL41}) and (\\ref{RicciL42}) together, we get\n\\begin{equation}\\label{RicciL4}\n\\begin{split}\n\\sum_{i=0}^{1}\\mathcal O_{i,4}\\leq \\mathcal O_0+C\\epsilon^{\\frac{1}{2}}\\Delta_0\\mathcal R+C\\epsilon \\mathcal O_{2,2}+C\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2}+ C(\\Delta_0)\\epsilon \\sum_{i=0}^{1}\\mathcal O_{i,4}. \\\\\n\\end{split}\n\\end{equation}\nBy choosing $\\epsilon$ sufficiently small depending on $\\Delta_0$, $\\mathcal O_0$, $\\mathcal R$ and $\\tilde{\\mathcal O}_{3,2}$, we have\n$$\\sum_{i=0}^{1}\\mathcal O_{i,4}\\leq C(\\mathcal O_0)+C\\epsilon \\mathcal O_{2,2},$$\nwhere $C(\\mathcal O_0)$ does not depend on $\\Delta_0$, once $\\epsilon$ is chosen to be small enough.\n\nWe now move to estimate the $\\mathcal O_{i,2}$ norms. Applying Proposition \\ref{transport} to (\\ref{Riccieqn}), we obtain\n\\begin{equation}\\label{RicciL21}\n\\begin{split}\n&\\sum_{i=0}^{2}\\mathcal O_{i,2}[\\trch,\\eta,\\etab,\\omega,\\omegab,\\trchb,\\chibh]\\\\\n\\leq &\\mathcal O_0+C\\epsilon^{\\frac{1}{2}}\\Delta_0\\mathcal R+\\epsilon\\Delta_0 \\sum_{i=0}^{2}\\mathcal O_{i,2}+\\epsilon(\\sum_{i=0}^{1}\\mathcal O_{i,4})^2.\n\\end{split}\n\\end{equation}\nFor $\\chih$, we apply Proposition \\ref{transport} to (\\ref{Riccieqn2}) to get\n\\begin{equation}\\label{RicciL22}\n\\sum_{i=0}^2||\\nabla^i\\chih||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)+ C\\epsilon^{\\frac{1}{2}}\\tilde{\\mathcal O}_{3,2}+C\\epsilon\\Delta_0 \\sum_{i=0}^{2}\\mathcal O_{i,2}+\\epsilon(\\sum_{i=0}^{1}\\mathcal O_{i,4})^2.\n\\end{equation}\n(\\ref{RicciL21}) and (\\ref{RicciL22}) together give\n$$\\sum_{i=0}^2\\mathcal O_{i,2}\\leq \\mathcal O_0+C\\epsilon^{\\frac{1}{2}}\\Delta_0\\mathcal R+\\epsilon\\Delta_0 \\sum_{i=0}^{2}\\mathcal O_{i,2}+\\epsilon(\\sum_{i=0}^{1}\\mathcal O_{i,4})^2.$$\nChoosing $\\epsilon$ to be appropriately small depending on $\\Delta_0$, $\\mathcal O_0$, $\\mathcal R$ and $\\tilde{\\mathcal O}_{3,2}$, we can absorb $\\displaystyle\\epsilon\\Delta_0 \\sum_{i=0}^{2}\\mathcal O_{i,2}$ to the left hand side to obtain\n\\begin{equation}\\label{RicciL2}\n\\sum_{i=0}^2\\mathcal O_{i,2}\\leq C(\\mathcal O_0)+\\epsilon(\\sum_{i=0}^{1}\\mathcal O_{i,4})^2.\n\\end{equation}\nFor $\\epsilon$ sufficiently small, (\\ref{RicciL4}) and (\\ref{RicciL2}) imply that\n$$\\sum_{i=0}^1\\mathcal O_{i,4}+\\sum_{i=0}^2\\mathcal O_{i,2}\\leq C(\\mathcal O_0).$$\n\nFinally, we use Sobolev embedding in Proposition \\ref{Linfty} to get the necessary bounds for $\\mathcal O_{0,\\infty}$ from the control for $\\displaystyle\\sum_{i=0}^1\\mathcal O_{i,4}$. Moreover, $\\Delta_0$ can be chosen so that $\\Delta_0\\gg C(\\mathcal O_0)$ and the bootstrap assumption (\\ref{BA1}) is verified as long as $\\mathcal R$ and $\\tilde{\\mathcal O}_{3,2}$ are controlled. Furthermore, the since the choice of $\\Delta_0$ depend only on $\\mathcal O_0$, the $\\epsilon$ can be set to depend only on $\\mathcal O_0$, $\\mathcal R$ and $\\tilde{\\mathcal O}_{3,2}$. \n\\end{proof}\n\n\\subsection{$L^2(S)$ Estimates for Curvature}\nIn this Subsection, we prove that $\\beta,\\rho,\\sigma,\\betab$ and their first derivatives can be controlled in $L^2(S)$. The argument relies on Proposition \\ref{transport} and the Bianchi equations (\\ref{eq:null.Bianchi}). The results of this Section will give the bounds for the Gauss curvature and its first derivatives appearing in Proposition \\ref{ellipticthm}.\n\\begin{proposition}\\label{RS}\nSuppose\n$$\\mathcal R<\\infty,\\quad\\tilde{\\mathcal O}_{3,2}<\\infty.$$\nThen there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R,\\tilde{\\mathcal O}_{3,2})$ such that for $\\epsilon\\leq \\epsilon_0$, \n$$\\sum_{i\\leq 1}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nabla^i\\Psi||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal R_0).$$\n\\end{proposition}\n\\begin{proof}\nBy the Bianchi equations, we have\n$$\\nabla_3(\\beta,\\rho,\\sigma,\\betab)=\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\nabla\\Psi+\\psi\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab,\\alphab\\}}\\Psi$$\nCommuting the above with $\\nabla$, we have\n$$\\nabla_3\\nabla(\\beta,\\rho,\\sigma,\\betab)=\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\nabla^2\\Psi+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab,\\alphab\\}}(\\Psi\\nabla\\psi+\\psi\\nabla\\Psi).$$\nEstimating directly, we get\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla_3\\nabla^i(\\beta,\\rho,\\sigma,\\betab)||_{L^2(\\Hb_{\\ub})}\\\\\n\\leq &\\sum_{i\\leq 2}\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nabla^i\\Psi||_{L^2(\\Hb_{\\ub})}+C\\sup_u\\sum_{i_1\\leq 1}||\\nabla^{i_1}\\psi||_{L^4(S_{u,\\ub})}\\sum_{i_2\\leq 2}\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nabla^{i_2}\\Psi||_{L^2(\\Hb_{\\ub})}\\\\\n&+C\\epsilon\\sup_u\\sum_{i_1\\leq 1}||\\nabla^{i_1}\\psi||_{L^4(S_{u,\\ub})}\\sum_{i_2\\leq 2}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nabla^{i_2}\\Psi||_{L^2(S_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\nTherefore, by the definition of $\\mathcal R$ and Proposition \\ref{Ricci}, we have\n$$\\sum_{i=0}^1||\\nabla_3\\nabla^i(\\beta,\\rho,\\sigma,\\betab)||_{L^2(\\Hb_{\\ub})}\\leq C(\\mathcal O_0)\\mathcal R+C(\\mathcal O_0)\\epsilon\\sum_{i\\leq 1}||\\nabla^i(\\beta,\\rho,\\sigma,\\betab)||_{L^2(S_{u,\\ub})}.$$\nTherefore, by Proposition \\ref{transport}, we have\n\\begin{equation*}\n\\sum_{i\\leq 1}||\\nabla^i(\\beta,\\rho,\\sigma,\\betab)||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal R_0)+ \\epsilon^{\\frac 12} C(\\mathcal O_0)\\mathcal R+C(\\mathcal O_0)\\epsilon^{\\frac 32}\\sum_{i\\leq 1}||\\nabla^i(\\beta,\\rho,\\sigma,\\betab)||_{L^2(S_{u,\\ub})}.\n\\end{equation*}\nThe conclusion follows from choosing $\\epsilon$ sufficiently small and absorbing the last term to the left hand side.\n\\end{proof}\nWe now derive $L^2(S)$ estimate for $K$ and its derivative, which will in particular allow us to apply Proposition \\ref{ellipticthm}.\n\\begin{proposition}\\label{Kest}\nSuppose\n$$\\mathcal R<\\infty,\\quad\\tilde{\\mathcal O}_{3,2}<\\infty.$$\nThen there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R,\\tilde{\\mathcal O}_{3,2})$ such that for $\\epsilon\\leq \\epsilon_0$, \n\\[\n\\sum_{i\\leq 1}||\\nabla^i K||_{L^2(S)}\\leq C(\\mathcal O_0, \\mathcal R_0)\n\\]\n\\end{proposition}\n\\begin{proof}\nWe use the equation\n$$K=-\\rho+\\frac{1}{2} \\chih\\cdot\\chibh-\\frac 1 4 tr\\chi\\cdot \\trchb,$$\nand bound term by term using Propositions \\ref{Ricci} and \\ref{RS}.\n\\end{proof}\n\n\\subsection{Elliptic Estimates for Ricci Coefficients}\\label{Ricciellipticsec}\n\nThis Subsection contains step 2 of the proof of the a priori estimates outlined in the beginning of the Section. We bound the third derivatives of the Ricci coefficients. They cannot be estimated by transport equations alone as before because that would require the control of three derivatives of curvature, which we do not have. Instead, we need to use elliptic estimates, following ideas in \\cite{CK}, \\cite{KN}, \\cite{Chr}, \\cite{KlRo}. We hope to find carefully chosen combination of $\\Theta:=\\nabla\\psi+\\Psi$ such that $\\Theta$ satisfies transport equations\n\\begin{equation}\\label{thtransport3}\n\\nabla_3\\Theta=\\psi\\nabla\\psi_{\\Hb}+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb},\n\\end{equation}\nor\n\\begin{equation}\\label{thtransport4}\n\\nabla_4\\Theta=\\psi\\nabla\\psi_H+\\psi\\psi\\psi+\\psi\\Psi_H,\n\\end{equation}\nwhere $\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}$, $\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}$, $\\Psi_H\\in\\{\\beta,\\rho,\\sigma,\\betab\\}$ and $\\Psi_{\\Hb}\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}$. These transport equations will then be coupled to the estimates from Hodge systems of the type\n$$\\mathcal D\\psi=\\Theta+\\Psi+\\psi\\psi$$\nto recover the control for the third derivatives of the Ricci coefficients. In order to apply this strategy, it is desirable for (\\ref{thtransport3}) and (\\ref{thtransport4}) to have two special structures. Firstly, $\\nabla\\Psi$ should not appear on the right hand side of the equation so that $\\Theta$ can be estimated by these transport equations without a loss of derivative. Secondly, we hope that the terms $\\psi\\nab\\psi_H$ and $\\psi\\Psi_H$ do not enter the $\\nabla_3\\Theta$ equation (respectively, $\\psi\\nab\\psi_{\\Hb}$ and $\\psi\\Psi_{\\Hb}$ do not enter the $\\nabla_4\\Theta$ equation). If this is satisfied for equations (\\ref{thtransport3}) and (\\ref{thtransport4}), then the integrated error terms can be controlled by the flux term that are bounded on the corresponding hypersurface. Indeed, the third derivatives of most the Ricci coefficients can be obtained in this way by constructing approximate $\\Theta$ variables which satisfy either (\\ref{thtransport3}) or (\\ref{thtransport4}).\n\nAn extra difficulty arises when we estimate $\\nab^3\\omega$. More precisely, the $\\Theta$ variable that allows us to retrieve estimates for derivatives of $\\omega$ satisfies an equation of the form\n\\begin{equation}\\label{probTh}\n\\nabla_3\\Theta=\\psi\\nabla\\psi_{\\Hb}+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb}+\\psi\\beta,\n\\end{equation}\nwhich has an undesirable term $\\psi\\beta$. Since we prove estimates without any information on the singular curvature component $\\alpha$, we automatically lose control of $\\nab^2\\beta$ in $L^2(\\Hb)$. Therefore, when obtaining the estimates from this transport equation using Proposition \\ref{transport}, the error term arising from $\\psi\\beta$ in (\\ref{probTh}) cannot be bounded. In order to overcome this problem, we prove a weaker bound for $\\nab^3\\omega$. More precisely, we will only control it in $L^2(H)$ but will not obtain any bounds of its $L^2(\\Hb)$ norm. We will moreover show that this is sufficient to close the energy estimates.\n\nWe first bound the $\\Theta$ variables that satisfy equations of the form (\\ref{thtransport3}) and (\\ref{thtransport4}).\n\\begin{proposition}\\label{Theta}\nLet $\\Theta$ satisfy the equation\n$$\\nabla_3\\Theta=\\psi\\nabla\\psi_{\\Hb}+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb},$$\nor\n$$\\nabla_4\\Theta=\\psi\\nabla\\psi_H+\\psi\\psi\\psi+\\psi\\Psi_H,$$\nwhere $\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}$, $\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}$, $\\Psi_H\\in\\{\\beta,\\rho,\\sigma,\\betab\\}$ and $\\Psi_{\\Hb}\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}$. \nSuppose $$\\mathcal R<\\infty,\\quad \\tilde{\\mathcal O}_{3,2}<\\infty.$$ \nThen there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R,\\tilde{\\mathcal O}_{3,2})$ such that for $\\epsilon\\leq \\epsilon_0$.\n$$\\sum_{i=0}^2\\sup_{u,\\ub}||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0).$$\n\\end{proposition}\n\\begin{proof}\nNotice that the assumption allows us to choose $\\epsilon_0$ such that the conclusion of Proposition \\ref{Ricci} holds. For $\\Theta$ satisfying\n$$\\nabla_3\\Theta=\\psi\\nabla\\psi_{\\Hb}+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb},$$\nwe commute with angular derivatives to obtain\n$$\\nabla_{3}\\nabla^{i}\\Theta=\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4+1}\\psi_{\\Hb}+\\nabla^{i_1}\\psi^{i_2+1}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi+\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi_{\\Hb}).$$\nTake $i\\leq 2$. We estimate term by term. For the first two terms, except for the factor with $\\nab^3\\psi_{\\Hb}$, can be estimated using the $\\mathcal O$ norm, which is bounded in view of Proposition \\ref{Ricci}. The term $\\psi\\nab^3\\psi_H$ can be controlled by the $\\tilde{\\mathcal O}_{3,2}$ norm. We have\n$$\\sum_{i_1+i_2+i_3+i_4\\leq 2}||\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4+1}\\psi_{\\Hb}+\\nabla^{i_1}\\psi^{i_2+1}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi||_{L^1_uL^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)(1+\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2}).$$\nThe curvature term can be estimated using Proposition \\ref{Ricci} and the definition of the curvature norm, since the component $\\beta$ does not appear:\n$$\\sum_{i_1+i_2+i_3+i_4\\leq 2}||\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi_{\\Hb}||_{L^1_uL^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)(1+\\epsilon^{\\frac 12}\\mathcal R).$$\nWe then estimate using Proposition \\ref{transport},\n$$\\sum_{i=0}^2\\sup_u||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)+C(\\mathcal O_0)\\epsilon^{\\frac{1}{2}}(\\mathcal R+\\tilde{\\mathcal O}_{3,2}).$$\nSimilarly, for $\\Theta$ satisfying\n$$\\nabla_4\\Theta=\\psi\\nabla\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_H,$$\nwe commute with angular derivatives to obtain\n$$\\nabla_{3}\\nabla^{i}\\Theta=\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4+1}\\psi_{H}+\\nabla^{i_1}\\psi^{i_2+1}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi+\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi_H),$$\nWe then estimate as above and use Proposition \\ref{transport} to get,\n$$\\sum_{i=0}^2\\sup_{\\ub}||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)+C(\\mathcal O_0)\\epsilon^{\\frac{1}{2}}(\\mathcal R+\\tilde{\\mathcal O}_{3,2}).$$\nThe Proposition follows from choosing $\\epsilon$ sufficiently small depending on $\\mathcal R$ and $\\tilde{\\mathcal O}_{3,2}$.\n\\end{proof}\nWe now write down the explicit $\\Theta$ quantities following \\cite{KlRo} and show that the estimates for some of the Ricci coefficients can be recovered from $\\Theta$ via elliptic equations.\n\\begin{proposition}\\label{ellipticTheta}\nAssume \n$$\\mathcal R<\\infty,\\quad \\tilde{\\mathcal O}_{3,2}< \\infty.$$\nLet $\\Theta$ denote any of the following quantities:\n$$\\nab\\trch,\\quad \\nab\\trchb,\\quad \\mu:=-\\div\\eta-\\rhoc,\\quad \\mub:=-\\div\\etab-\\rhoc.$$ \nThen $\\Theta$ satisfies (\\ref{thtransport3}) or (\\ref{thtransport4}). Moreover, there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R,\\tilde{\\mathcal O}_{3,2})$ such that for $\\epsilon\\leq \\epsilon_0$, we have\n$$\\sup_{u,\\ub}||\\nab^3(\\trch,\\trchb)||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0),$$\n$$\\sup_u||\\nabla^3(\\chih,\\eta,\\etab)||_{L^2(H_u)} \\leq C(\\mathcal O_0)(\\epsilon^{\\frac 12}+\\mathcal R),$$\n$$\\sup_{\\ub}||\\nabla^3(\\chibh,\\eta,\\etab)||_{L^2(\\Hb_{\\ub})} \\leq C(\\mathcal O_0)(\\epsilon^{\\frac 12}+\\mathcal R),$$\nwhere, as before, $\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}$, $\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}$.\n\\end{proposition}\n\\begin{proof}\nConsider the following equations:\n$$\\nabla_3 \\trchb=\\psi_{\\Hb}\\psi_{\\Hb},$$\n$$\\nabla_4 \\trch=\\psi_H\\psi_H.$$\nThese equations do not involve a curvature term. After commuting with an angular derivative, we have\n$$\\nabla_{3}\\nabla\\trchb=\\psi\\nabla\\psi_{\\Hb}+\\psi\\psi\\psi,$$\n$$\\nabla_{4}\\nabla\\trch=\\psi\\nabla\\psi_H+\\psi\\psi\\psi.$$\nThus $\\nabla\\trch$ and $\\nabla\\trchb$ satisfy the requirement to be one of the $\\Theta$ quantities.\nThus, by definition,\n$$||\\nabla^3\\trch||_{L^2(S_{u,\\ub})} \\leq C||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}.$$\nMoreover,\n\\begin{equation}\\label{elliptictrch}\n||\\nabla^3\\trch||_{L^2(H_u)} \\leq C\\epsilon^{\\frac 12}\\sup_{\\ub}||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}.\n\\end{equation}\nand\n\\begin{equation}\\label{elliptictrchb}\n||\\nabla^3\\trchb||_{L^2(\\Hb_{\\ub})} \\leq C\\epsilon^{\\frac 12}\\sup_u||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}.\n\\end{equation}\nFrom the estimates for $\\nabla^3\\trch$ and $\\nabla^3\\trchb$, we can obtain the estimates for $\\nabla^3\\chih$ and $\\nabla^3\\chibh$. This is because of the Codazzi equations:\n$$\\div\\chih=\\frac{1}{2}\\nabla\\trch-\\beta+\\psi\\psi,$$\n$$\\div\\chibh=\\frac{1}{2}\\nabla\\trchb+\\betab+\\psi\\psi,$$\nSince $\\chih$ and $\\chibh$ are symmetric traceless 2-tensors and we have the estimates for the Gauss curvature in Proposition \\ref{Kest}, we can apply the elliptic estimates for symmetric traceless 2-tensors in Proposition \\ref{elliptictraceless}\n\\begin{equation*}\n\\begin{split}\n||\\nabla^3\\chih||_{L^2(H_u)} \n\\leq &C(||\\nabla^3\\trch||_{L^2(H_u)}+||\\nabla^2\\beta||_{L^2(H_u)}+\\sum_{i_1+i_2\\leq 2}||\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi||_{L^2(H_u)}).\n\\end{split}\n\\end{equation*}\nSince, by Proposition \\ref{Ricci},\n\\begin{equation}\\label{quadratic}\n\\begin{split}\n&\\sum_{i_1+i_2\\leq 2}||\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi||_{L^2(H_u)}\\\\\n\\leq &C\\epsilon^{\\frac 12}\\sup_{\\ub}(||\\nabla\\psi||_{L^4(S_{u,\\ub})}||\\nabla\\psi||_{L^4(S_{u,\\ub})}+||\\psi||_{L^\\infty(S_{u,\\ub})}\\sum_{i\\leq 2}||\\nabla^i\\psi||_{L^2(S_{u,\\ub})})\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12},\n\\end{split}\n\\end{equation}\nwe have\n\\begin{equation}\\label{ellipticchih}\n||\\nabla^3\\chih||_{L^2(H_u)} \\leq C\\epsilon^{\\frac 12}||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}+C\\mathcal R+C(\\mathcal O_0)\\epsilon^{\\frac 12}.\n\\end{equation}\nSimilarly, using the Codazzi equation for $\\chibh$, we get\n\\begin{equation}\\label{ellipticchibh}\n||\\nabla^3\\chibh||_{L^2(\\Hb_{\\ub})} \\leq C\\epsilon^{\\frac 12}||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}+C\\mathcal R+C(\\mathcal O_0)\\epsilon^{\\frac 12}.\n\\end{equation}\n\nDefine\n$$\\mu=-\\div\\eta-\\rhoc,$$\n$$\\mub=-\\div\\etab-\\rhoc,$$\nwhere as before $\\rhoc=\\rho-\\frac 12 \\chibh\\cdot\\chih$.\nNotice that $\\div\\eta$ and $\\div\\etab$ satisfy the following equations:\n$$\\nabla_4\\div\\eta=-\\div\\beta+\\psi\\nabla\\psi+\\psi\\psi\\psi+\\psi\\Psi_H,$$\n$$\\nabla_3\\div\\etab=\\div\\betab+\\psi\\nabla\\psi+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb},$$\nRecall the Bianchi equation:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\rho+\\frac 32\\trch\\rho=\\div\\beta-\\frac 12\\chibh\\cdot\\alpha+\\zeta\\cdot\\beta+2\\etab\\cdot\\beta,\\\\\n&\\nab_3\\rho+\\frac 32\\trchb\\rho=-\\div\\betab- \\frac 12\\chih\\cdot\\alphab+\\zeta\\cdot\\betab-2\\eta\\cdot\\betab,\\\\\n\\end{split}\n\\end{equation*}\nIn the $\\nabla_4$ equation for $\\rho$, the term $\\alpha$ appears. This is undesirable because we do not have good estimates for $\\alpha$. Instead we renormalize the equations using\n$$\\rhoc:=\\rho-\\frac{1}{2}\\chibh\\cdot\\chih.$$\nNow, we write down the equations for $\\rhoc$ instead of $\\sigma$ and $\\rho$. Since $\\nabla_4\\chih=-\\alpha+\\psi\\psi$, the $\\nab_4$ equations now do not contain $\\alpha$. Schematically, we have\n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\rhoc=\\div\\beta+\\psi\\nabla\\psi+\\psi\\Psi_H+\\psi\\psi\\psi,\\\\\n&\\nabla_3\\rhoc=-\\div\\betab+\\psi\\nabla\\psi+\\psi\\Psi_{\\Hb}+\\psi\\psi\\psi,\n\\end{split}\n\\end{equation*}\nwhere all the curvature components are now acceptable. Thus $\\mu$ and $\\mub$ satisfy the following equations:\n$$\\nabla_4 \\mu=\\psi\\Psi_H+\\psi\\nabla\\psi_H+\\psi\\psi\\psi,$$\n$$\\nabla_3 \\mub=\\psi\\Psi_{\\Hb}+\\psi\\nabla\\psi_{\\Hb}+\\psi\\psi\\psi,$$\nThus $\\mu$ and $\\mub$ also satisfy the schematic equation for $\\Theta$. We now show that we can get the estimates $\\nabla^3\\eta$ and $\\nabla^3\\eta$ from that of $\\nabla^2\\mu$ and $\\nabla^3\\mub$. To do this, we consider the following div-curl systems\n$$\\div\\eta=-\\mu-\\rho+\\psi\\psi,$$\n$$\\curl\\eta=\\sigma+\\psi\\psi,$$\n$$\\div\\etab=-\\mub-\\rho+\\psi\\psi,$$\n$$\\curl\\etab=-\\sigma+\\psi\\psi.$$\nNotice that $\\rho$ and $\\sigma$ can be estimated by $\\mathcal R$ on both $H_u$ and $\\Hb_{\\ub}$. Then, using (\\ref{quadratic}) and Proposition \\ref{ellipticthm}, we have\n\\begin{equation}\\label{ellipticetau}\n||\\nabla^3(\\eta,\\etab)||_{L^2(H_u)} \\leq C\\epsilon^{\\frac 12}||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}+C\\mathcal R+C(\\mathcal O_0)\\epsilon^{\\frac 12},\n\\end{equation}\nand\n\\begin{equation}\\label{ellipticetaub}\n||\\nabla^3(\\eta,\\etab)||_{L^2(\\Hb_{\\ub})} \\leq C\\epsilon^{\\frac 12}||\\nabla^2\\Theta||_{L^2(S_{u,\\ub})}+C\\mathcal R+C(\\mathcal O_0)\\epsilon^{\\frac 12}.\n\\end{equation}\nThe Proposition thus follows from (\\ref{elliptictrch}), (\\ref{elliptictrchb}), (\\ref{ellipticchih}), (\\ref{ellipticchibh}), (\\ref{ellipticetau}), (\\ref{ellipticetaub}) and Proposition \\ref{Theta}.\n\\end{proof}\n\nThe above construction of $\\Theta$ provides the desired estimates for the third derivatives of all the Ricci coefficients with the exception of $\\omega$ and $\\omegab$. Therefore, to estimate the $\\tilde{\\mathcal O}_{3,2}$ norm, it remains to prove estimates for $\\nab^3\\omega$ and $\\nab^3\\omegab$. To this end we follow the strategy in \\cite{KlRo}. Define two auxiliary quantities $\\omega^\\dagger$ and $\\omegab^\\dagger$ to be solutions to \n$$\\nabla_3\\omega^\\dagger=\\frac{1}{2}\\sigma,$$\n$$\\nabla_4\\omegab^\\dagger=\\frac{1}{2}\\sigma,$$\nwith zero boundary conditions along $H_0$ and $\\Hb_0$ respectively. For the remainder of the Section, $\\psi$ will also be used to denote $\\omega^\\dagger$ and $\\omegab^\\dagger$; $\\psi_H$ will also be used to denote $\\omega^\\dagger$; and $\\psi_{\\Hb}$ will also be used to denote $\\omegab^\\dagger$. It is easy to see that\n\\begin{proposition}\\label{daggerest}\nSuppose\n$$\\mathcal R<\\infty.$$\nThen there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R)$ such that for every $\\epsilon\\leq \\epsilon_0$,\n$$\\sum_{i\\leq 2}||\\nab^i(\\omega^\\dagger,\\omegab^\\dagger)||_{L^2(S_{u,\\ub})}\\leq C.$$\n\\end{proposition}\n\\begin{proof}\nSince $\\omega^\\dagger$ and $\\omegab^\\dagger$ obey the equations\n$$\\nabla_3\\omega^\\dagger=\\frac{1}{2}\\sigma,$$\n$$\\nabla_4\\omegab^\\dagger=\\frac{1}{2}\\sigma,$$\nwith zero data, it follows from Proposition \\ref{transport} that \n$$\\sum_{i\\leq 2}||\\nab^i(\\omega^\\dagger,\\omegab^\\dagger)||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12}\\mathcal R.$$\nThe Proposition follows from choosing $\\epsilon$ appropriately.\n\\end{proof}\nWe are now ready to use elliptic estimates to control $\\nab^3\\omega$ and $\\nab^3\\omegab$. As before, our strategy is to use a combination $\\Theta=\\nab\\psi+\\Psi$ that can be estimated using a transport equation. However, $\\Theta$ in this case does not satisfy an equation of the type in Proposition \\ref{Theta}.\n\\begin{proposition}\\label{omegaelliptic}\nSuppose \n$$\\mathcal R<\\infty,\\quad\\tilde{\\mathcal O}_{3,2}<\\infty.$$\nThen there exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R,\\tilde{\\mathcal O}_{3,2})$ such that whenever $\\epsilon\\leq \\epsilon_0$, we have\n$$||\\nab^3(\\omega,\\omega^\\dagger)||_{L^2(H_u)}\\leq C(\\mathcal O_0)(\\epsilon^{\\frac 12}+\\mathcal R),$$\n$$||\\nab^3(\\omegab,\\omegab^\\dagger)||_{L^2(\\Hb_{\\ub})}\\leq C(\\mathcal O_0)(\\epsilon^{\\frac 12}+\\mathcal R),$$\n\\end{proposition}\n\\begin{proof}\nLet\n$$\\kappa:=\\nabla\\omega+^*\\nabla\\omega^{\\dagger}-\\frac 12 \\beta,$$\n$$\\kappab:=-\\nabla\\omegab+^*\\nabla\\omegab^{\\dagger}-\\frac 12 \\betab.$$\nWe have \n$$\\nabla_3\\nabla\\omega=\\frac 12 \\nabla\\rho+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,$$\n$$\\nabla_3{ }^*\\nabla\\omega^\\dagger=\\frac {1}{2} {^*\\nabla}\\sigma+\\psi\\Psi+\\psi\\nabla\\omega^\\dagger,$$\nand the Bianchi equation\n$$\\nabla\\rho+^*\\nabla\\sigma=\\nabla_3\\beta+\\psi\\Psi.$$\nTherefore\n\\begin{equation}\\label{kappaeqn}\n\\nabla_3\\kappa=\\psi\\Psi+\\psi\\nabla\\psi+\\psi\\psi\\psi.\n\\end{equation}\nSimilar, using\n$$\\nabla_4\\nabla\\omegab=\\frac 12 \\nabla\\rho+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,$$\n$$\\nabla_4{ }^*\\nabla\\omegab^\\dagger=\\frac 12 {^*\\nabla}\\sigma+\\psi\\Psi+\\psi\\nabla\\omega^\\dagger,$$\nand the Bianchi equation\n$$-\\nabla\\rho+^*\\nabla\\sigma=\\nabla_4\\betab+\\psi\\Psi,$$\nwe have\n\\begin{equation}\\label{kappabeqn}\n\\nabla_4\\kappab=\\psi\\Psi+\\psi\\nabla\\psi+\\psi\\psi\\psi.\n\\end{equation}\nThe equations for $\\kappa$ and $\\kappab$ do not have the favorable structure of the equations of the other $\\Theta$ variables. In particular, for the $\\nab_3\\kappa$ equation, $\\beta$ and $\\nab\\omega$ appear as source terms. The loss of the $L^2$ energy estimate for $\\alpha$ is automatically accompanied by the loss of estimate for $\\beta$ in $L^2(\\Hb)$. Therefore, $\\nab^2\\beta$ cannot be controlled in $L^2(\\Hb_{\\ub})$. This means that we cannot estimate $\\nab^2\\kappa$ in $L^2(\\Hb_{\\ub})$. Nevertheless, as we now show, we can estimate $\\nab^2\\kappa$ in $L^2(H_u)$. This estimate will be sufficient in the sequel.\n\nCommuting (\\ref{kappaeqn}) with angular derivatives, we get\n$$\\nabla_3\\nab^2\\kappa=\\sum_{i_1+i_2+i_3\\leq 2}\\psi^{i_1}\\nab^{i_2}\\psi\\nab^{i_3}\\Psi+\\sum_{i_1+i_2+i_3\\leq 3}\\psi^{i_1}\\nab^{i_2}\\psi\\nabla^{i_3}\\psi.$$\nNotice that we now allow $\\psi$ to represent the terms $\\omega^\\dagger$ and $\\omegab^\\dagger$. Since by Proposition \\ref{daggerest} $\\nab^i\\omega^\\dagger$ and $\\nab^i\\omegab^\\dagger$ satisfy the same estimates as any other $\\psi$ for $i\\leq 2$, those terms can be estimated in the same way as in Proposition \\ref{Theta}. The new terms that have not been estimated in Proposition \\ref{Theta} are $\\psi\\nab^3\\omegab^\\dagger$, $\\displaystyle\\sum_{i_1+i_2+i_3\\leq 2}\\psi^{i_1}\\nab^{i_2}\\psi\\nab^{i_3}\\beta$ and $\\psi\\nab^3(\\chih,\\omega,\\omega^\\dagger)$. We estimate them by\n$$||\\psi\\nab^3\\omegab^\\dagger||_{L^1_uL^2(S)}\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12}||\\nab^3\\omegab^\\dagger||_{L^2_uL^2(S)},$$\nand\n$$||\\sum_{i_1+i_2+i_3\\leq 2}\\psi^{i_1}\\nab^{i_2}\\psi\\nab^{i_3}\\beta||_{L^1_uL^2(S_{u,\\ub})}\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12}\\sum_{i\\leq 2}||\\nab^i\\beta||_{L^2_uL^2(S_{u,\\ub})},$$\nand\n$$||\\psi\\nab^3(\\chih,\\omega,\\omega^\\dagger)||_{L^1_uL^2(S)}\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12}||\\nab^3(\\chih,\\omega,\\omega^\\dagger)||_{L^2_uL^2(S)}.$$\nThen, using Proposition \\ref{transport} and the estimates in the proof of Proposition \\ref{Theta}, we have\n\\begin{equation*}\n\\begin{split}\n&||\\nab^2\\kappa||_{L^2(S_{u,\\ub})}\\\\\n\\leq& C(\\mathcal O_0)(1+\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2}+\\epsilon^{\\frac 12}\\mathcal R+\\epsilon^{\\frac 12}||\\nab^3\\omegab^\\dagger||_{L^2_uL^2(S_{u,\\ub})}\\\\\n&\\qquad+\\epsilon^{\\frac 12}\\sum_{i\\leq 2}||\\nab^i\\beta||_{L^2_uL^2(S_{u,\\ub})}+\\epsilon^{\\frac 12}||\\nab^3(\\chih,\\omega,\\omega^\\dagger)||_{L^2_uL^2(S_{u,\\ub})}).\n\\end{split}\n\\end{equation*}\nThis holds for any fixed $u,\\ub$. Integrating this in $L^2_{\\ub}$, we get\n\\begin{equation*}\n\\begin{split}\n&||\\nab^2\\kappa||_{L^2(H_u)}\\\\\n\\leq &C(\\mathcal O_0)\\epsilon^{\\frac 12}(1+\\epsilon^{\\frac 12}(\\tilde{\\mathcal O}_{3,2}+\\mathcal R+||\\nab^3\\omegab^\\dagger||_{L^\\infty_{\\ub}L^2_uL^2(S)})\\\\\n&\\qquad+\\sum_{i\\leq 2}||\\nab^i\\beta||_{L^2_{\\ub}L^2_uL^2(S)}+||\\nab^3(\\chih,\\omega,\\omega^\\dagger)||_{L^2_{\\ub}L^2_uL^2(S)}).\n\\end{split}\n\\end{equation*}\nUsing Cauchy-Schwarz in the $u$ variable, we have\n$$\\sum_{i\\leq 2}||\\nab^i\\beta||_{L^2_{\\ub}L^2_uL^2(S_{u,\\ub})}\\leq \\epsilon^{\\frac 12}\\mathcal R,$$\nand\n$$||\\nab^3(\\chih,\\omega,\\omega^\\dagger)||_{L^2_{\\ub}L^2_uL^2(S_{u,\\ub})})\\leq \\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2}+\\epsilon^{\\frac 12}||\\nab^3\\omega^\\dagger||_{L^\\infty_{u}L^2_{\\ub}L^2(S_{u,\\ub})}.$$\nTherefore,\n\\begin{equation}\\label{kappa}\n||\\nab^2\\kappa||_{L^2(H_u)}\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12}(1+\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2}+\\epsilon^{\\frac 12}\\mathcal R+\\epsilon^{\\frac 12}||\\nab^3\\omegab^\\dagger||_{L^\\infty_{\\ub}L^2_uL^2(S_{u,\\ub})}+\\epsilon^{\\frac 12}||\\nab^3\\omega^\\dagger||_{L^\\infty_{u}L^2_{\\ub}L^2(S_{u,\\ub})}).\n\\end{equation}\nSimilarly, we can use (\\ref{kappabeqn}) to conclude that \n\\begin{equation}\\label{kappab}\n||\\nab^2\\kappab||_{L^2(\\Hb_{\\ub})}\\leq C(\\mathcal O_0)\\epsilon^{\\frac 12}(1+\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2}+\\epsilon^{\\frac 12}\\mathcal R+\\epsilon^{\\frac 12}||\\nab^3\\omegab^\\dagger||_{L^\\infty_{\\ub}L^2_uL^2(S_{u,\\ub})}+\\epsilon^{\\frac 12}||\\nab^3\\omega^\\dagger||_{L^\\infty_{u}L^2_{\\ub}L^2(S_{u,\\ub})}).\n\\end{equation}\nWe show that we can recover $\\nabla^3\\omega$, $\\nabla^3\\omegab$, $\\nabla^3\\omega^\\dagger$ and $\\nabla^3\\omegab^\\dagger$ by considering the div-curl system for $\\nabla\\omega$ and $\\nabla\\omega^\\dagger$. We have\n$$\\div\\nabla\\omega=\\div\\kappa+\\frac 12\\div\\beta,$$\n$$\\curl\\nabla\\omega=0,$$\n$$\\div\\nabla\\omega^\\dagger=\\curl\\kappa+\\frac 12\\curl\\beta,$$\n$$\\curl\\nabla\\omega^\\dagger=0,$$\n$$\\div\\nabla\\omegab=-\\div\\kappab-\\frac 12\\div\\betab,$$\n$$\\curl\\nabla\\omegab=0,$$\n$$\\div\\nabla\\omegab^\\dagger=-\\curl\\kappab-\\frac 12\\curl\\betab,$$\n$$\\curl\\nabla\\omegab^\\dagger=0.$$\nThus, by Proposition \\ref{ellipticthm} and \\ref{Kest}, we have\n\\begin{equation*}\\label{ellipticomega}\n||\\nabla^3(\\omega,\\omega^\\dagger)||_{L^2(H_u)}\\leq C(||\\nabla^2\\kappa||_{L^2(H_u)}+\\sum_{i\\leq 2}||\\nabla^i\\beta||_{L^2(H_u)}),\n\\end{equation*}\nand\n\\begin{equation*}\\label{ellipticomegab}\n||\\nabla^3(\\omegab,\\omegab^\\dagger)||_{L^2(\\Hb_{\\ub})}\\leq C(||\\nabla^2\\kappab||_{L^2(\\Hb_{\\ub})}+\\sum_{i\\leq 2}||\\nabla^i\\betab||_{L^2(\\Hb_{\\ub})}).\n\\end{equation*}\nUsing (\\ref{kappa}) and (\\ref{kappab}), we can thus choose $\\epsilon$ sufficiently small to conclude the Proposition.\n\\end{proof}\n\nPutting these all together gives\n\\begin{proposition}\\label{Riccielliptic}\nAssume\n$$\\mathcal R<\\infty.$$\nThen there exists $\\epsilon_0=(\\mathcal O_0,\\mathcal R)$ such that \n\\[\n\\tilde{\\mathcal O}_{3,2}\\leq C(\\mathcal O_0)(\\epsilon^{\\frac 12}+\\mathcal R).\n\\]\n\\end{proposition}\n\\begin{proof}\nImpose the bootstrap assumption\n\\begin{equation}\\label{BA2}\n\\tilde{\\mathcal O}_{3,2}\\leq \\Delta_1.\n\\end{equation}\nUnder this assumption, we can choose $\\epsilon$ depending on $\\mathcal O_0$, $\\mathcal R$ and $\\Delta_1$ such that the conclusions in Propositions \\ref{Theta}, \\ref{ellipticTheta} and \\ref{omegaelliptic} hold. \nThus, \n$$\\tilde{\\mathcal O}_{3,2}\\leq C(\\mathcal O_0)(\\epsilon^{\\frac 12}+\\mathcal R).$$\nThis improves the bootstrap assumption (\\ref{BA2}) for $\\Delta_1$ sufficiently large. Moreover, $\\Delta_1$ can be chosen to depend only on $\\mathcal O_0$ and $\\mathcal R$. Thus, $\\epsilon_0$ depends only on $\\mathcal O_0$ and $\\mathcal R$.\n\\end{proof}\n\n\\subsection{Energy Estimates for Curvature}\\label{energyestimatessec}\n\nIn this Subsection, we carry out step 3 in the proof of the a priori estimates as outlined in the beginning of the Section. We prove energy estimates for the curvature components and their first two derivatives, thus providing bounds for $\\mathcal R$ and closing all the estimates.  These energy estimates will be derived after renormalization directly from the Bianchi equations. This is analogous to \\cite{Holzegel}, where a similar procedure without curvature renormalization has been carried out to derive energy estimates for all components of curvature. We remind the reader of our challenge to avoid any information on the non-$L^2$-integrable curvature component $\\alpha$ in our context. \n\nWe will need the following two Propositions, which can be proved by a direct computation.\n\\begin{proposition}\\label{intbyparts34}\nSuppose $\\phi_1$ and $\\phi_2$ are $r$ tensorfields, then\n$$\\int_{D_{u,\\ub}} \\phi_1 \\nabla_4\\phi_2+\\int_{D_{u,\\ub}}\\phi_2\\nabla_4\\phi_1= \\int_{\\Hb_{\\ub}(0,u)} \\phi_1\\phi_2-\\int_{\\Hb_0(0,u)} \\phi_1\\phi_2+\\int_{D_{u,\\ub}}(2\\omega-\\trch)\\phi_1\\phi_2,$$\n$$\\int_{D_{u,\\ub}} \\phi_1 \\nabla_3\\phi_2+\\int_{D_{u,\\ub}}\\phi_2\\nabla_3\\phi_1= \\int_{H_{u}(0,\\ub)} \\phi_1\\phi_2-\\int_{H_0(0,\\ub)} \\phi_1\\phi_2+\\int_{D_{u,\\ub}}(2\\omegab-\\trchb)\\phi_1\\phi_2.$$\n\\end{proposition}\n\\begin{proposition}\\label{intbypartssph}\nSuppose we have an $r$ tensorfield $^{(1)}\\phi$ and an $r-1$ tensorfield $^{(2)}\\phi$.\n$$\\int_{D_{u,\\ub}}{ }^{(1)}\\phi^{A_1A_2...A_r}\\nabla_{A_r}{ }^{(2)}\\phi_{A_1...A_{r-1}}+\\int_{D_{u,\\ub}}\\nabla^{A_r}{ }^{(1)}\\phi_{A_1A_2...A_r}{ }^{(2)}\\phi^{A_1...A_{r-1}}= -\\int_{D_{u,\\ub}}(\\eta+\\etab){ }^{(1)}\\phi{ }^{(2)}\\phi.$$\n\\end{proposition}\nBy using the Bianchi equations and the integration by parts formula in Proposition \\ref{intbyparts34} and \\ref{intbypartssph}, we can prove the following energy estimates. The main new ingredient is the observation that $\\alpha$ can be renormalized away in the energy estimates. We can therefore control $\\beta,\\rho,\\sigma,\\betab,\\alphab$ without any knowledge of $\\alpha$\n\\begin{proposition}\\label{ee}\nThe following $L^2$ estimates for the curvature components hold for $i\\leq 2$:\n\\begin{equation*}\n\\begin{split}\n&\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} |\\nabla^i\\Psi|^2_\\gamma+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psi|^2_\\gamma  \\\\\n\\leq &\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_{u'}} |\\nabla^i\\Psi|^2_\\gamma+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub'}} |\\nabla^i\\Psi|^2_\\gamma+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0).\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nWe first consider the following sets of Bianchi equations:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta+tr\\chib\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma),\\\\\n&\\nabla_4\\sigma+\\frac 32tr\\chi\\sigma=-\\div^*\\beta+\\frac 12\\chibh\\cdot ^*\\alpha-\\zeta\\cdot^*\\beta-2\\etab\\cdot\n^*\\beta,\\\\\n&\\nabla_4\\rho+\\frac 32tr\\chi\\rho=\\div\\beta-\\frac 12\\chibh\\cdot\\alpha+\\zeta\\cdot\\beta+2\\etab\\cdot\\beta,\\\\\n\\end{split}\n\\end{equation*}\nIn the $\\nabla_4$ equations for $\\sigma$ and $\\rho$, the term $\\alpha$ appears. This is undesirable because we do not have good estimates for $\\alpha$. Instead we renormalize both $\\sigma$ and $\\rho$ as in the proof of Proposition \\ref{ellipticTheta}. Define\n$$\\sigmac:=\\sigma+\\frac{1}{2}\\chibh\\wedge\\chih,$$\n$$\\rhoc:=\\rho-\\frac{1}{2}\\chibh\\cdot\\chih.$$\nNow, we write down the equations for $\\sigmac$ and $\\rhoc$ instead of $\\sigma$ and $\\rho$. Since $\\nabla_4\\chih=-\\alpha+\\psi\\psi$, the equations now do not contain $\\alpha$. Schematically, we have\n\\begin{equation}\\label{rcsc}\n\\begin{split}\n&\\nabla_4\\sigmac=-\\div^*\\beta+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,\\\\\n&\\nabla_4\\rhoc=\\div\\beta+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,\n\\end{split}\n\\end{equation}\nwhere $\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab,\\alphab\\}$ is now one of the acceptable curvature components.\nRewrite also the first equation as\n\\begin{equation}\\label{bc}\n\\nabla_3\\beta=\\nabla\\rhoc+^*\\nabla\\sigmac+\\psi\\Psi+\\psi\\nabla\\psi.\n\\end{equation}\nApplying Proposition \\ref{intbypartssph} yields the following: \n\\begin{equation*}\n\\begin{split}\n\\int_{D_{u,\\ub}} <\\beta,\\nabla_3\\beta>_\\gamma =&\\int <\\beta,\\nabla\\rhoc+^*\\nabla\\sigmac>_\\gamma+<\\beta,\\psi\\Psi>_\\gamma \\\\\n=&\\int_{D_{u,\\ub}} -<\\div\\beta,\\rhoc>_\\gamma+<\\div ^*\\beta,\\sigmac>_\\gamma +<\\beta,\\psi\\Psi+\\psi\\nabla\\psi>_\\gamma\\\\\n=&\\int_{D_{u,\\ub}} -<\\nabla_4\\rhoc,\\rhoc>_\\gamma-<\\nabla_4\\sigmac,\\sigmac>_\\gamma +<\\Psi,\\psi\\Psi+\\psi\\nabla\\psi+\\psi\\psi\\psi>_\\gamma.\n\\end{split}\n\\end{equation*}\nApplying Proposition \\ref{intbyparts34} then yields the energy estimates for the $0$-th derivative of the curvature:\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub}} \\rhoc^2+\\sigmac^2  \\\\\n\\leq &\\int_{H_{u'}} |\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub'}} \\rhoc^2+\\sigmac^2\n+\\int_{D_{u,\\ub}}\\Psi(\\psi\\Psi+\\psi\\nabla\\psi+\\psi\\psi\\psi).\n\\end{split}\n\\end{equation*}\nWe now commute equations (\\ref{rcsc}) and (\\ref{bc}) with $\\nabla^i$ to get\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nabla^i\\beta- \\nabla \\nabla^i\\rhoc-\\nabla\\nabla^i\\sigmac \\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}(\\rhoc,\\sigmac)+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nabla^i\\rhoc- \\div\\nabla^i\\beta\\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\beta+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nabla^i\\sigmac- \\div^*\\nabla^i\\beta\\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\beta+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nwhere \n$$(\\div\\nabla^i\\beta)_{A_1...A_i}=(\\nabla^{i+1}\\beta)^B{ }_{A_1...A_iB}.$$\nWe integrate by parts using Proposition \\ref{intbyparts34} and \\ref{intbypartssph} as in the $0$-th derivative case to get\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i(\\rhoc,\\sigmac)|^2_\\gamma  \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i(\\rhoc,\\sigmac)|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi.\n\\end{split}\n\\end{equation*}\nWe then consider the following set of Bianchi equations. Notice that $\\alpha$ does not appear in any equations in this set.\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\sigma+\\frac 32tr\\chib\\sigma=-\\div ^*\\betab+\\frac 12\\chih\\cdot ^*\\alphab-\\zeta\\cdot ^*\\betab-2\\eta\\cdot \n^*\\betab,\\\\\n&\\nabla_3\\rho+\\frac 32tr\\chib\\rho=-\\div\\betab- \\frac 12\\chih\\cdot\\alphab+\\zeta\\cdot\\betab-2\\eta\\cdot\\betab,\\\\\n&\\nabla_4\\betab+tr\\chi\\betab=-\\nabla\\rho +^*\\nabla\\sigma+ 2\\omega\\betab +2\\chibh\\cdot\\beta-3(\\etab\\rho-^*\\etab\\sigma),\\\\\n\\end{split}\n\\end{equation*}\nFrom these we can derive the following estimates:\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i(\\rho,\\sigma)|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\betab|^2_\\gamma  \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i(\\rho,\\sigma)|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\betab|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi.\n\\end{split}\n\\end{equation*}\nFinally, we look at the following set of Bianchi equations:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\betab+2tr\\chib\\betab=-\\div\\alphab-2\\omegab\\betab+\\etab \\cdot\\alphab,\\\\\n&\\nabla_4\\alphab+\\frac 12 tr\\chi\\alphab=-\\nabla\\hot \\betab+ 4\\omega\\alphab-3(\\chibh\\rho-^*\\chibh\\sigma)+\n(\\zeta-4\\etab)\\hot \\betab\n\\end{split}\n\\end{equation*}\nAgain, we notice the absence of the $\\alpha$ terms. Hence we have the estimate\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\betab|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\alphab|^2_\\gamma  \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\betab|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\alphab|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi.\n\\end{split}\n\\end{equation*}\nWe have thus established that \n$$\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} |\\nabla^i\\Psi|^2_\\gamma+\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psic|^2_\\gamma$$\ncan be bounded by the right hand side in the statement of the Proposition. To conclude, we use the fact that\n$$\\Psic=\\Psi+\\psi\\psi,$$\nand thus\n\\begin{equation*}\n\\begin{split}\n&\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psi|^2_\\gamma\\\\\n\\leq &\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psic|^2_\\gamma+\\sum_{i_1+i_2\\leq 2}||\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi||_{L^2(\\Hb_{\\ub})}\\\\\n\\leq& \\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psic|^2_\\gamma+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0).\n\\end{split}\n\\end{equation*}\n\\end{proof}\nFinally, we prove the boundedness of the norm $\\mathcal R$:\n\\begin{proposition}\\label{curvature}\nThere exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R_0)$ such that for every $\\epsilon\\leq\\epsilon_0$,\n\\[\n \\mathcal R\\leq C(\\mathcal O_0, \\mathcal R_0).\n\\]\n\\end{proposition}\n\\begin{proof}\nAssume as a bootstrap assumption:\n\\begin{equation}\\label{BA3}\n\\mathcal R\\leq \\Delta_2.\n\\end{equation}\nThus we can apply Propositions \\ref{Ricci}, \\ref{Kest} and \\ref{Riccielliptic}.\nBy Proposition \\ref{ee}, we have\n\\begin{equation}\\label{eenonlinear}\n\\begin{split}\n&\\sum_{i=0}^{2}(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} (\\nabla^i\\Psi)^2+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\underline{H}_{\\underline{u}}} (\\nabla^i\\Psi)^2)\\\\\n\\leq& C\\mathcal R_0^2+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0)+\\int_{\\mathcal D_{u,\\underline{u}}}\\sum_{i\\leq 2}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4\\leq 2}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi \\\\\n&+\\int_{\\mathcal D_{u,\\underline{u}}}\\sum_{i\\leq 2}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4\\leq 1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi \\\\\n&+\\int_{\\mathcal D_{u,\\underline{u}}}\\sum_{i\\leq 2}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4\\leq 3}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n\\leq &C\\mathcal R_0^2+C\\sum_{i\\leq 2}\\int_{D_{u,\\ub}} (\\nab^i\\Psi)^2 +\\sum_{i_1+i_2+i_3+i_4\\leq 2}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\right)^2\\\\\n&+\\sum_{i_1+i_2+i_3+i_4\\leq 1}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} K\\nabla^{i_4}\\Psi\\right)^2 +\\sum_{i_1+i_2+i_3+i_4\\leq 3}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} \\psi\\nabla^{i_4}\\psi\\right)^2.\n\\end{split}\n\\end{equation}\nIt is helpful to first note that \n$$\\sum_{i\\leq 2}||\\nabla^i\\Psi||_{L^2(D_{u,\\ub})}\\leq C\\epsilon^{\\frac 12}\\mathcal R,$$\nsince $\\nab^i\\Psi$ can be estimated either in $L^2(H_u)$ or $L^2(\\Hb_{\\ub})$.\nBy Sobolev Embedding in Propositions \\ref{L4} and \\ref{Linfty}, we thus have\n$$\\sum_{i\\leq 1}||\\nabla^i\\Psi||_{L^2_uL^2_{\\ub}L^4(S)}+||\\Psi||_{L^2_uL^2_{\\ub}L^\\infty(S)}\\leq C\\epsilon^{\\frac 12}\\mathcal R,$$\nA similar argument shows that\n$$\\sum_{i\\leq 3}||\\nabla^i\\psi||_{L^2(D_{u,\\ub})}+\\sum_{i\\leq 2}||\\nabla^i\\psi||_{L^2_uL^2_{\\ub}L^4(S)}+\\sum_{i\\leq 1}||\\nab^i\\psi||_{L^2_uL^2_{\\ub}L^\\infty(S)}\\leq C\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2},$$\nThus by Proposition \\ref{Riccielliptic}, we have\n$$\\sum_{i\\leq 3}||\\nabla^i\\psi||_{L^2(D_{u,\\ub})}+\\sum_{i\\leq 2}||\\nabla^i\\psi||_{L^2_uL^2_{\\ub}L^4(S)}+\\sum_{i\\leq 1}||\\nab^i\\psi||_{L^2_uL^2_{\\ub}L^\\infty(S)}\\leq C(\\mathcal O_0,\\mathcal R_0)\\epsilon^{\\frac 12}\\mathcal R.$$\nNow we estimate the first nonlinear term in (\\ref{eenonlinear}):\n\\begin{equation*}\n\\begin{split}\n\\sum_{i_1+i_2+i_3+i_4\\leq 2}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\right)^2. \\\\\n\\end{split}\n\\end{equation*}\nWe can estimate, by Proposition \\ref{Ricci},\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1,i_2\\leq 1,i_3\\leq 1}||\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})}\\\\\n\\leq &(\\sum_{i_1\\leq 2}\\sup_{u,\\ub}||\\psi||_{L^\\infty(S_{u,\\ub})}^{i_1})(\\sum_{i_2\\leq 2}\\sup_{u,\\ub}||\\nabla^{i_2}\\psi||_{L^2(S_{u,\\ub})})(\\sum_{i_3\\leq 2}||\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})})\\\\\n\\leq &\\epsilon^{\\frac 12} C(\\mathcal O_0)R.\n\\end{split}\n\\end{equation*}\nWe then estimate the term\n$$\\sum_{i_1+i_2+i_3\\leq 1}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\psi^{i_1}\\nabla^{i_2} K\\nabla^{i_3}\\Psi\\right)^2.$$\nWe have, by Propositions \\ref{Ricci} and \\ref{Kest},\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1}||\\psi^{i_1}\\nabla^{i_2} K\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})}\\\\\n\\leq &(\\sum_{i_1\\leq 2}\\sup_{u,\\ub}||\\psi||_{L^\\infty(S_{u,\\ub})}^{i_1})(\\sum_{i_2\\leq 1}\\sup_{u,\\ub}||\\nabla^{i_2}K||_{L^2(S_{u,\\ub})})(\\sum_{i_3\\leq 2}||\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})})\\\\\n\\leq &\\epsilon^{\\frac 12} C(\\mathcal O_0,\\mathcal R_0)R.\n\\end{split}\n\\end{equation*}\nFinally, we estimate\n$$\\sum_{i_1+i_2+i_3+i_4\\leq 3}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} \\psi\\nabla^{i_4}\\psi\\right)^2.$$\nBy Propositions \\ref{Ricci} and \\ref{Riccielliptic},\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3+i_4\\leq 3}||\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} \\psi\\nabla^{i_4}\\psi||_{L^2(D_{u,\\ub})}\\\\\n\\leq &(\\sum_{i_1\\leq 3}\\sup_{u,\\ub}||\\psi||_{L^\\infty(S_{u,\\ub})}^{i_1})(\\sum_{i_2\\leq 2}\\sup_{u,\\ub}||\\nabla^{i_2}\\psi||_{L^2(S_{u,\\ub})})(\\sum_{i_3\\leq 3}||\\nabla^{i_3}\\psi||_{L^2(D_{u,\\ub})})\\\\\n\\leq &\\epsilon^{\\frac 12} C(\\mathcal O_0,\\mathcal R_0)R.\n\\end{split}\n\\end{equation*}\nThus\n$$\\sum_{i=0}^{2}(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} (\\nabla^i\\Psi)^2+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\underline{H}_{\\underline{u}}} (\\nabla^i\\Psi)^2)\n\\leq C\\mathcal R_0^2+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0)+\\epsilon^{\\frac 12} C(\\mathcal O_0,\\mathcal R_0)R.$$\nBy choosing $\\epsilon$ sufficiently small we have\n$$\\mathcal R\\leq C(\\mathcal O_0,\\mathcal R_0).$$\nThis improves (\\ref{BA3}) for $\\Delta_2$ sufficiently large. Moreover, $\\Delta_2$ can be chosen to depend only on $\\mathcal O_0$ and $\\mathcal R_0$. Thus the choice of $\\epsilon$ depends only on $\\mathcal O_0$ and $\\mathcal R_0$.\n\\end{proof}\n\\subsection{Propagation of Regularity}\\label{Propregsec}\nUp to this point, we have been considering spacetimes with smooth characteristic initial data. In the context of an impulsive gravitational wave, our estimates in this Section apply to spacetimes arising from the smooth data approximating the given singular data. For these spacetimes, we can also prove that estimates for the higher order derivatives of the curvature. In view of the bounds that we have already obtained, this follows from standard arguments. In particular, energy estimates for the higher order derivatives can be derived as in Section \\ref{energyestimatessec}. In this case, since the equations are linear in the highest order derivatives, one can use the bounds from the previous Sections to obtain the desired estimates. More precisely, we have\n\\begin{proposition}\\label{propagationregularity}\nSuppose, in addition to the assumptions of Theorem \\ref{timeofexistence}, the bound \n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psi||_{L^2(\\Hb_0)}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^i \\nab_4^k\\nab^j\\Psi||_{L^2(H_0)})\\leq D_{I,J,K}. $$\nholds initially for some $I,J,K$. Then, the following bounds hold for $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$,\n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psi||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)})\\leq D'_{I,J,K}. $$\nfor $u_*,\\ub_*\\leq \\epsilon$, where $\\epsilon$ can be chosen as in Theorem \\ref{timeofexistence} and the constant $D'_{I,J,K}$ depends only on the size of the initial data norm in the assumption of Theorem \\ref{timeofexistence} and $D_{I,J,K}$.\n\\end{proposition}\n\\begin{proof}\nSince the arguments are similar to those in Section \\ref{energyestimatessec}, we will only provide a sketch. We will proceed with an induction argument in $i$, $j $ and $k$ and we will only consider the highest order terms, assuming appropriate control on the lower order terms. From \\cite{KN}, we have the commutation formula\n$$[\\nab_3,\\nab_4]f=-2\\omega \\nab_3 f+2\\omegab\\nab_4 f+4\\zeta\\cdot\\nab f$$\nfor all scalar functions $f$. Therefore, by commuting $\\nab_3^j\\nab_4^k\\nab^i$ with the equations \\eqref{rcsc} and \\eqref{bc}, we have\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nab_3^j\\nab_4^k\\nabla^i\\beta- \\nabla \\nab_3^j\\nab_4^k\\nabla^i\\rhoc-\\nabla\\nab_3^j\\nab_4^k\\nabla^i\\sigmac \\\\\n\\sim&\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)+\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi+\\mbox{terms with fewer derivatives},\\\\\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nab_3^j\\nab_4^k\\nabla^i\\rhoc- \\div\\nab_3^j\\nab_4^k\\nabla^i\\beta\\\\\n\\sim&\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)+\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi+\\mbox{terms with fewer derivatives},\\\\\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nab_3^j\\nab_4^k\\nabla^i\\sigmac- \\div^*\\nab_3^j\\nab_4^k\\nabla^i\\beta\\\\\n\\sim&\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)+\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi+\\mbox{terms with fewer derivatives}.\n\\end{split}\n\\end{equation*}\nNotice that since $\\alpha$ does not appear in \\eqref{rcsc} and \\eqref{bc}, we do not have the term $\\nab_3^j\\nab_4^k\\nab^i\\alpha$ in these commuted equations. Therefore, integrating by parts as in Section \\ref{energyestimatessec}, we have\n\\begin{equation*}\n\\begin{split}\n&||\\nab_3^j\\nab_4^k \\nab^i(\\rhoc,\\sigmac)||_{L^2(\\Hb_{\\ub})}^2+||\\nab_3^j \\nab_4^k\\nab^i\\beta||_{L^2(H_u)}^2\\\\\n\\leq &\\mbox{Data}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)\\|_{L^1(\\mathcal D_{u,\\ub})}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^1(\\mathcal D_{u,\\ub})}\\\\\n&+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\times(\\mbox{terms with fewer derivative})\\|_{L^1(\\mathcal D_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\nSimilarly, using the other null Bianchi equations in \\eqref{eq:null.Bianchi}, we also have\n\\begin{equation}\\label{hr.1}\n\\begin{split}\n&\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub})}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)}^2\\\\\n\\leq &\\mbox{Data}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab,\\alphab)\\|_{L^1(\\mathcal D_{u,\\ub})}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^1(\\mathcal D_{u,\\ub})}\\\\\n&+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\times(\\mbox{terms with fewer derivative})\\|_{L^1(\\mathcal D_{u,\\ub})}.\n\\end{split}\n\\end{equation}\nAs in Section \\ref{Ricciellipticsec}, the highest order derivatives of the Ricci coefficients $\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi$ can be controlled by the highest order derivatives of the curvature components using elliptic estimates:\n\\begin{equation}\\label{hr.2}\n\\begin{split}\n&\\sum_{\\psi\\in\\{\\trch,\\chih,\\omega,\\trchb,\\eta,\\etab\\}}\\|\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^2(H_u)}+\\sum_{\\psi\\in\\{\\trch,\\chih,\\omegab,\\trchb,\\eta,\\etab\\}}\\|\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^2(\\Hb_{\\ub})}\\\\\n\\leq &\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)}\\\\\n&+\\mbox{terms under control}.\n\\end{split}\n\\end{equation}\nUsing the boundedness of the $L^\\infty$ norm of $\\psi$ and the control of the terms with fewer derivatives that we have assumed as the induction hypothesis, \\eqref{hr.1} and \\eqref{hr.2} imply\n\\begin{equation}\\label{hr.3}\n\\begin{split}\n&\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub})}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)}^2\\\\\n\\leq &\\mbox{Data term}+\\mbox{Bounded terms}+\\int_0^{\\ub} \\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub_*})}^2 d\\ub_*\\\\\n&+\\int_0^u\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_{u_*})}^2 du_*\\\\\n&+\\int_0^{\\ub}\\int_0^u\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_{u_*})}\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub_*})} du_*d\\ub_*.\n\\end{split}\n\\end{equation}\nTo obtain the desired conclusion, introduce the bootstrap assumption\n$$\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub}(0,u))}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u(0,\\ub))}^2\\leq A e^{2A(u+\\ub)}$$\nfor $A$ sufficiently large. Then \\eqref{hr.3} implies that we have\n$$\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub}(0,u))}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u(0,\\ub))}^2\\leq C e^{2A(u+\\ub)}$$\nfor some constant $C$ depending on the data term and the bounded terms. Therefore, this improves the bootstrap assumption for $A$ sufficiently large. Finally, notice that since we can control the derivatives of the Ricci coefficients by the derivatives of the renormalized curvature components, we in fact have the bound for the curvature components themselves. This concludes the proof of the proposition.\n\\end{proof}\n\\begin{remark}\nNotice that for approximating solutions for an impulsive gravitational wave, the constants in the initial norms $D_{I,J,K}$ depend on $n$. However, the initial data satisfy uniform estimates for each $I\\geq 2,J\\geq 0$ with $K=0$. Therefore, the corresponding uniform estimates will hold in the region $0\\leq u\\leq u_*$ and $0\\leq\\ub\\leq \\ub_*$.\n\\end{remark}\n\n\n\\subsection{End of Proof of Theorem \\ref{timeofexistence}}\\label{EndofProof}\nOnce we have the estimates for all the higher derivatives, the proof of Theorem \\ref{timeofexistence} is standard. We refer, for example, to Section 6 of \\cite{L} for details.\n\n\\subsection{Additional Estimates for an Impulsive Gravitational Wave}\\label{AddEst}\nIn this Subsection, we focus our attention on the sequence of spacetimes $(\\mathcal M_n, g_n)$ with characteristic initial data converging to that of an impulsive gravitational wave as constructed in Section \\ref{initialcondition}. The conclusion of Theorem \\ref{timeofexistence} can be applied to $(\\mathcal M_n, g_n)$. The initial data of the sequence of spacetimes, however, possess an additional property: $\\alpha_n$ and its angular derivatives are bounded uniformly in $L^1_{\\ub}$. In this Subsection, we prove that in the constructed spacetimes $(\\mathcal M_n, g_n)$, the same estimates hold for all $u\\leq u_*$ with $u_*\\leq \\epsilon$. Moreover, we show that $\\alpha_n$ concentrates around $\\ub_s$ and if $\\alpha_n$ is initially more regular uniformly in $n$ away from $\\ub_s$, it is also uniformly more regular away from $\\ub_s$ for $0\\leq u\\leq u_*\\leq \\epsilon$.\n\nWe begin by proving energy estimates for $\\alpha_n$ in $L^2(H_u)$ and $\\beta_n$ in $L^2(\\Hb_{\\ub})$ using the pair of Bianchi equations for $(\\nab_3\\alpha_n,\\nab_4\\beta_n)$ in (\\ref{eq:null.Bianchi}).\n\\begin{proposition}\\label{alphaenergy}\nThe following estimate holds:\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\beta_n|^2_\\gamma  \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\beta|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi,\n\\end{split}\n\\end{equation*}\nwhere here, unlike in other places, we use $\\Psi$ to denote all possible curvature components, including $\\alpha_n$.\n\\end{proposition}\n\\begin{proof}\nConsider the following Bianchi equations:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\alpha+\\frac 12 tr\\chib \\alpha=\\nabla\\hot \\beta+ 4\\omegab\\alpha-3(\\chih\\rho+^*\\chih\\sigma)+\n(\\zeta+4\\eta)\\hot\\beta,\\\\\n&\\nabla_4\\beta+2tr\\chi\\beta = \\div\\alpha - 2\\omega\\beta +  \\eta \\alpha,\\\\\n\\end{split}\n\\end{equation*}\nUsing Proposition \\ref{intbypartssph},\n\\begin{equation*}\n\\begin{split}\n\\int <\\alpha_n,\\nabla_3\\alpha_n>_\\gamma =&\\int <\\alpha_n,\\nabla\\hot\\beta_n>_\\gamma+<\\alpha_n,\\psi\\Psi>_\\gamma \\\\\n=&\\int -<\\div\\alpha_n,\\beta_n>_\\gamma+<\\alpha_n,\\psi\\Psi>_\\gamma \\\\\n=&\\int -<\\nabla_4\\beta_n,\\beta_n>_\\gamma+<\\alpha_n,\\psi\\Psi>_\\gamma +<\\beta_n,\\psi\\Psi>_\\gamma\\\\\n\\end{split}\n\\end{equation*}\nIntegrate by parts using Proposition \\ref{intbyparts34} to get that for $u\\geq u'$, $\\ub\\geq \\ub'$,\n\\begin{equation*}\n\\begin{split}\n\\int_{H_u} |\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\beta_n|^2_\\gamma \\leq \\int_{H_{u'}} |\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\beta_n|^2_\\gamma+\\int_{D_{u,\\ub}}<(\\alpha_n,\\beta_n),\\psi\\Psi>_\\gamma\\\\\n\\end{split}\n\\end{equation*}\nWe use the commutation formula, and note that the special structure is preserved in the highest order:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nabla^i\\alpha- \\nabla\\hot \\nabla^i\\beta \\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\alpha \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\beta+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n&\\nabla_4\\nabla^i\\beta- \\div\\nabla^i\\alpha\\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\beta\\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\alpha+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nPerforming the integration by parts as before using Propositions \\ref{intbyparts34} and \\ref{intbypartssph}, we have\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\beta_n|^2_\\gamma  \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\beta_n|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi.\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\\subsubsection{Before the Impulse}\n\nWe first apply Proposition \\ref{alphaenergy} for the region $0\\leq \\ub\\leq \\ub_s$. Notice that the initial norm\n$$\\sum_{i\\leq 2}\\int_0^{\\ub_s} ||\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}^2 d\\ub\\leq C$$\nis bounded independent of $n$. Therefore,\n\\begin{proposition}\\label{beforeshock}\n$$\\sum_{i\\leq 2}(\\sup_{u\\leq \\epsilon}||\\nabla^i\\alpha_n||_{L^2(H_u(0,\\ub_s))}^2+\\sup_{\\ub\\leq \\ub_s}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0),$$\nindependent of $n$.\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{alphaenergy} and all the estimates in the previous section, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 2}(||\\nabla^i\\alpha_n||_{L^2(H_u(0,\\ub_s))}^2+\\sup_{\\ub\\leq \\ub_s}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})})\\\\\n \\leq &C(\\mathcal O_0,\\mathcal R_0)(1+||\\nabla^i\\alpha_n||_{L^2(H)}+\\int_0^u ||\\nabla^i\\alpha_n||^2_{L^2(H_{u'})} du').\n\\end{split}\n\\end{equation*}\nBy Gronwall's inequality, we have\n$$\\sum_{i\\leq 2}(\\sup_{u\\leq \\epsilon}||\\nabla^i\\alpha_n||_{L^2(H_u(0,\\ub_s))}^2+\\sup_{\\ub\\leq \\ub_s}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0),$$\nas claimed.\n\\end{proof} \n\n\\subsubsection{The Impulse Region}\n\nFor an impulsive gravitational wave, the initial data represented by the $\\alpha$ component of curvature and its derivative are not in $L^2$. By the construction of Section \\ref{initialcondition}, the approximating sequence $\\alpha_n$ satisfies initially \n$$\\sum_{i\\leq 2}\\int_{\\ub_s}^{\\ub_s+2^{-n}} ||\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}^2 d\\ub\\leq C2^{n}.$$\nIn the following Proposition, we show that this bound is propagated.\n\\begin{proposition}\\label{duringshock}\n$$\\sum_{i\\leq 2}(\\sup_{u\\leq \\epsilon}||\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s,\\ub_s+2^{-n}))}^2+\\sup_{\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n}}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0)2^{n},$$\nwhere $C(\\mathcal O_0,\\mathcal R_0)$ is independent of $n$.\n\\end{proposition}\n\\begin{proof}\nThe proof is the same as Proposition \\ref{beforeshock}, except that the initial data for $\\nabla^i\\alpha$ can only be bounded by $2^{\\frac n2}$ in $L^2(H_0(\\ub_s,\\ub_s+2^{-n}))$.\n\\end{proof}\n\n\\subsubsection{After the Impulse}\n\nFor $\\ub_s+2^{-n}\\leq \\ub\\leq \\ub_*$, since the initial data satisfy $$\\sum_{i\\leq 2}\\int_{\\ub_s+2^{-n}}^{\\ub_*} ||\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}^2 d\\ub\\leq C,$$\nindependent of $n$, we again have a better estimate. An extra challenge arises from the fact that the estimates derived in Proposition \\ref{duringshock} depends on $n$. This can be overcome if we allow a loss in derivative:\n\\begin{proposition}\\label{aftershock}\n$$\\sum_{i\\leq 1}(\\sup_{u\\leq u_*}||\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\ub_*))}^2+\\sup_{\\ub_s+2^{-n}\\leq\\ub\\leq \\ub_*}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0),$$\nindependent of $n$.\n\\end{proposition}\n\\begin{proof}\nTo bound the left hand side, we can apply the argument as in Proposition \\ref{beforeshock} to get\n$$\\sum_{i\\leq 1}(\\sup_{u\\leq u_*}||\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\ub_*))}^2+\\sup_{\\ub_s+2^{-n}\\leq\\ub\\leq \\ub_*}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0)2^{\\frac n2}.$$\nThe problem in proving Proposition \\ref{aftershock} with a bound independent of $n$ is that according to Proposition \\ref{duringshock}, $||\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s})}$ may grow in $n$. However, we can still obtain the desired bounds for $\\alpha_n$ and $\\nab\\alpha_n$ (note that the argument below will not apply to $\\nab^2\\alpha_n$ unless extra assumptions on the initial data are used) by using Proposition \\ref{RS}, which implies that\n$$\\sum_{i\\leq 1}\\sup_{u,\\ub}||\\nab^i\\beta_n||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0,\\mathcal R_0).$$\nTaking $\\ub=\\ub_s+2^{-n}$ and integrating, we thus have\n$$\\sum_{i\\leq 1}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C(\\mathcal O_0,\\mathcal R_0).$$\n\\end{proof}\n\nWe can also prove estimates for higher derivatives away from $\\ub_s$. The influence of a propagating curvature impulse results in a loss of derivatives in such estimates compared to the assumed regularity of the initial data. By the remark after Proposition \\ref{propagationregularity}, we have estimates, uniform in $n$, for the $\\nab_3$ and $\\nab$ derivatives for $\\psi$ and $\\Psi$. We now show that we can also have uniform in $n$ estimates for the $\\nab_4$ derivatives for $\\psi$ and $\\Psi$ for $\\ub\\geq\\ub_s+2^{-n}$. By the Bianchi equations (\\ref{eq:null.Bianchi}), this in turn can be reduced to showing estimates for $\\nab_4$ derivatives of $\\alpha$. This is proved in the following Proposition:\n\\begin{proposition}\\label{alphaapriori}\nFor $\\ub_s+2^{-n}\\leq\\ub\\leq\\epsilon$, if the initial data set satisfies\n$$\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_4^k \\nab^i\\Psi_n||_{L^2(\\Hb_0)}\\leq C_{I,K}. $$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I}\\sup_{\\ub}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}|| \\nab_4^k\\nab^i\\Psi_n||_{L^2(S_{0,\\ub})}\\leq C_{I,K}.$$\nthen\n$$\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}(\\sup_{\\ub}||\\nab_4^k\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\epsilon))}+\\sup_{\\ub_s+2^{-n}\\leq \\ub\\leq \\epsilon}||\\nab_4^k\\nabla^i\\beta_n||_{L^2(\\Hb_{\\ub})})\\leq C_{I,K}',$$\nwhere $C_{I,K}$ is independent of $n$.\n\\end{proposition}\n\\begin{proof}\nThe main step is to prove that on $\\Hb_{\\ub_s+2^{-n}}$, we have the estimate\n$$||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C$$\nindependent of $n$. For $k=0$, we can commute with angular derivatives and integrate along the $u$ direction using the Bianchi equation\n$$\\nab_3\\beta+\\trchb\\beta=\\nab\\rho+2\\omegab\\beta+^*\\nab\\sigma+2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma).$$\nSince the right hand side has one more angular derivative, this integration loses a derivative, allowing us only to prove\n$$\\sum_{i\\leq I-1}||\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C$$\n\nNow we consider also the $\\nab_4$ derivatives. Differentiate the equation by $\\nab_4^k\\nab_i$ and commute $[\\nab_3,\\nab_4^k\\nab_i]$ on the left hand side. Moreover, except for the component $\\alpha$, whenever we see $\\nab_4^{k_1}\\Psi$, we substitute an appropriate Bianchi equation from (\\ref{eq:null.Bianchi}). Then we get an equation\n$$\\nab_3\\nab_4^k\\nab^i\\beta=\\nab_4^{k-1}\\nab^{i+2}\\beta+...$$\nwhere $...$ denote terms that are lower order in terms of derivatives. We then integrate this along the $u$ direction. Thus\n$$||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C||\\nab_4^{k-1}\\nab^{i+2}\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}+...$$\nInducting in $k$, we get that\n$$||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C||\\nab^{i+2k}\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}+... $$\nNow we have reduced to the case where there are only angular derivatives falling on $\\beta_n$. By the above, we need $i+2k+1\\leq I$. Thus we can prove \n\\begin{equation}\\label{smoothnessdata}\n\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C.\n\\end{equation}\nThe conclusion thus follows from a standard energy estimate type argument as in Proposition \\ref{alphaenergy} and using (\\ref{smoothnessdata}) as the initial data on $\\Hb_{\\ub_s+2^{-n}}$\n\\end{proof}\n\nCombining the estimates in Propositions \\ref{beforeshock}, \\ref{duringshock}, \\ref{alphaapriori} and Cauchy-Schwarz, we have the following uniform $L^1_{\\ub}$ estimate.\n\\begin{proposition}\\label{totalvariation}\n$$\\sup_{u\\leq u_*}\\sum_{i\\leq I}\\int_0^{\\ub_*} ||\\nabla^i\\alpha_n||_{L^2(S_{u,\\ub})}d\\ub\\leq C_I,$$\nwhere $C_I$ is independent of $n$.\n\\end{proposition}\nProposition \\ref{totalvariation} is crucial in showing that the limiting spacetime will have $\\alpha$ defined as a finite measure with a singular atom at $\\ub_s$.\n\n\\section{Convergence}\\label{convergence}\nIn this Section, we show that a sequence of initial data satisfying uniform estimates of Theorem \\ref{rdthmv2} with converging initial data gives rise to a sequence of converging spacetimes. The convergence will be understood as follows: the spacetime will be identified in the system of double null coordinates $(u,\\ub,\\th^1,\\th^2)$ and convergence will be established for the sequence of the corresponding spacetime metrics. In view of the quasilinear nature of the Einstein equations, we can only hope to prove convergence of our approximating spacetimes in a norm with one derivative fewer than the a priori estimates that we established for them. \n\nTo get estimates of metric, we use the fact that the metric components satisfy inhomogeneous transport equations with right hand side expressed as Ricci coefficients. The estimates for the difference of Ricci coefficients and curvature components are derived by considering the system of difference equations obtained from the original system of transport, elliptic and Bianchi equations. As was the case for the a priori estimates, the challenge for the difference system is a lack of any a priori control of the difference of the singular curvature components $\\alpha$. For the approximating sequence of initial data, $\\alpha_n$ is not a Cauchy sequence in $L^2(H_0)$. In the proof of the a priori estimates, we handled the lack of information of the $\\alpha$ component of curvature via a renormalization procedure and the observation that the system satisfied by the Ricci coefficients and the renormalized curvature components can be closed without any reference to the $\\alpha$ component of curvature. Potentially, this property may fail when we consider the difference system. It is however a remarkable fact as we will show below that the difference equations still possess the same structure.\n\nThe following is the main Theorem in which we estimate the difference of the metrics, Ricci coefficients and curvature components of two spacetimes:\n\n\\begin{theorem}\\label{convergencethm}\nSuppose we have two sets of initial data $(1)$ and $(2)$ satisfying the conditions in Theorem \\ref{timeofexistence} with the same constants $C$ and $c$. By Theorem \\ref{timeofexistence}, we can solve for vacuum spacetimes $(\\mathcal M^{(1)}, g^{(1)})$ and $(\\mathcal M^{(2)}, g^{(2)})$ corresponding to the initial data sets $(1)$ and $(2)$ in the region $0\\leq u\\leq u_*$ and $0\\leq \\ub\\leq\\ub_*$ for $u_*,\\ub_*\\leq \\epsilon$ . Let $(u,\\ub,\\th^1,\\th^2)$ be the coordinate system introduced in Section \\ref{coordinates} such that the metrics take the form\n$$g^{(i)}=-2(\\Omega^{(i)})^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)})_{AB}(d\\th^A-(b^{(i)})^Adu)\\otimes (d\\th^B-(b^{(i)})^Bdu),$$ \nwhere $\\Omega=1$ and $b^{A}=0$ on $H_0$ and $\\Hb_0$. We can now identify the two spacetimes by identifying points with the same value of coordinate functions. Define $g'=g^{(1)}-g^{(2)}$, $\\psi'=\\psi^{(1)}-\\psi^{(2)}$ and $\\Psi'=\\Psi^{(1)}-\\Psi^{(2)}$ to be the difference of the metric, the difference of the Ricci coefficients and the difference of the curvature components respectively. If the data satisfy \n$$\\sup_u|(\\frac{\\partial}{\\partial\\th})^i\\gamma_{AB}'(\\ub=0)|+\\sup_{\\ub}|(\\frac{\\partial}{\\partial\\th})^i\\gamma_{AB}'(u=0)|\\leq a,$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}(\\sum_{i\\leq 1} \\sup_{u}||\\nabla^i\\psi'||_{L^2(S_{u,0})}+\\sum_{i\\leq 1} \\sup_{\\ub}||\\nabla^i\\psi'||_{L^2(S_{0,\\ub})})\\leq a,$$\n$$\\sum_{i\\leq 1} ||\\nabla^i(\\chih',\\omega')||_{L^{p_0}_{\\ub}L^2(S_{0,\\ub})}\\leq a\\quad\\mbox{for some fixed }2\\leq p_0<\\infty,$$\n$$||\\nabla^2(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_0)}+||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{0})}\\leq a,$$\n$$\\sup_{u} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{u,0})}+\\sup_{\\ub} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{0,\\ub})}\\leq a,$$\n$$\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}} ||\\nabla^i\\Psi'||_{L^{2}_{\\ub}L^2(S_{0,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}} ||\\nabla^i\\Psi'||_{L^{2}_{u}L^2(S_{u,0})}\\right)\\leq a,$$\nwhere the angular covariant derivative and all the norms are defined with respect to the spacetime $(1)$. Then the following estimates hold in $\\{0\\leq u\\leq u_*\\}\\cap\\{0\\leq\\ub\\leq\\ub_*\\}$:\n$$\\sup_{u,\\ub}|(\\frac{\\partial}{\\partial\\th})^i\\gamma_{AB}'|\\leq C'a,$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}\\sum_{i\\leq 1} \\sup_{u,\\ub}||\\nabla^i\\psi'||_{L^2(S_{u,\\ub})}\\leq C'a,$$\n$$\\sum_{i\\leq 1} \\sup_u ||\\nabla^i(\\chih',\\omega')||_{L^{p_0}_{\\ub}L^2(S_{u,\\ub})}\\leq C'a,$$\n$$\\sup_u ||\\nabla^2(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub} ||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{\\ub})}+\\sup_{u,\\ub} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{u,\\ub})}\\leq C'a,$$\n$$\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}} \\sup_u||\\nabla^i\\Psi'||_{L^2_{\\ub}L^2(S_{u,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}} \\sup_{\\ub}||\\nabla^i\\Psi'||_{L^2_{u}L^2(S_{u,\\ub})}\\right)\\leq C'a,$$\nfor some constant $C'$ depending only on $C$ and $c$ and independent of $a$.\n\\end{theorem}\nBy iterating Theorem \\ref{convergencethm}, we can reduce its proof to the region $0\\leq u\\leq \\delta$, $0\\leq \\ub\\leq \\delta$ for some sufficiently small $\\delta$ independent of $a$. Most of this Section will be devoted to a proof of Theorem \\ref{convergencethm} for $0\\leq u\\leq\\delta$ and $0\\leq \\ub\\leq\\delta$. The proof of Theorem \\ref{convergencethm} will be carried out in Sections \\ref{convsec1}, \\ref{convsec2}, \\ref{convsec3}, \\ref{convsec4} and \\ref{convsec5}.\n\nTheorem \\ref{convergencethm} implies the following convergence result:\n\\begin{theorem}\\label{convergencethm2}\nSuppose we have a sequence of initial data that coincide on $\\Hb_0$ and satisfy the assumptions of Theorem \\ref{timeofexistence} with uniform constants $C$ and $c$ on $H_0$ and $\\Hb_0$. By Theorem \\ref{timeofexistence}, for every initial data in the sequence, a unique smooth solution to the vacuum Einstein equations $(\\mathcal M_n, g_n)$ exists in $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$ for $u_*, \\ub_*\\leq \\epsilon$ and the metric takes the following form in the coordinate system $(u,\\ub,\\th^1,\\th^2)$:\n$$g_n=-2(\\Omega_n)^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma_n)_{AB}(d\\th^A-(b_n)^Adu)\\otimes (d\\th^B-(b_n)^Bdu),$$ \nwhere $\\Omega=1$ and $b^A=0$ on $H_0$ and $\\Hb_0$. Denote the Ricci coefficients and curvature components by $\\psi_n$ and $\\Psi_n$ respectively. Identify the spacetimes in the sequence by the value of the coordinate functions $(u,\\ub,\\th^1,\\th^2)$. Define also $g_n'=g_n-g_{n-1}$, $\\psi'_n=\\psi_n-\\psi_{n-1}$ and $\\Psi'_n=\\Psi_n-\\Psi_{n-1}$. If\n$$\\sup_u|(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB})'_n(\\ub=0)|\\leq a_n,$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}\\sum_{i\\leq 1} \\sup_{\\ub}||\\nabla^i\\psi'_n||_{L^2(S_{0,\\ub})}\\leq a_n,$$\n$$\\sum_{i\\leq 1} ||\\nabla^i(\\chih_n',\\omega_n')||_{L^{p_0}_{\\ub}L^2(S_{0,\\ub})}\\leq a_n\\quad\\mbox{for some fixed }2\\leq p_0<\\infty,$$\n$$||\\nabla^2(\\chih_n',\\omega_n',\\eta_n',\\etab_n')||_{L^2(H_0)}+\\sup_{\\ub} ||\\nabla^2(\\trch_n',\\trchb_n')||_{L^2(S_{0,\\ub})}\\leq a_n,$$\n$$\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}} ||\\nabla^i\\Psi_n'||_{L^{2}_{\\ub}L^2(S_{0,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}} ||\\nabla^i\\Psi_n'||_{L^{2}_{u}L^2(S_{u,0})}\\right)\\leq a_n,$$\nfor some $a_n$ such that $\\displaystyle\\sum a_n<\\infty$,\nthen the spacetime metrics converge uniformly to a continuous limiting spacetime metric $g_\\infty$\n$$g_\\infty=-2(\\Omega_\\infty)^2(du\\otimes d\\ub+d\\ub\\otimes du)+ (\\gamma_{\\infty})_{AB}(d\\th^A-(b_\\infty)^Adu)\\otimes(d\\th^B-(b_\\infty)^Bdu)$$\nin the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$.\nMoreover, \n$$(\\frac{\\partial}{\\partial \\th}g_n,\\frac{\\partial}{\\partial u}g_n)\\mbox{ converge to }(\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty)\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^4(S),$$\n$$(\\frac{\\partial^2}{\\partial \\th^2}g_n,\\frac{\\partial^2}{\\partial u\\partial\\th}g_n,\\frac{\\partial^2}{\\partial u^2}g_n)\\mbox{ converge to }(\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_{\\infty})\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^2(S),$$\n$$(\\frac{\\partial}{\\partial \\ub}g_n, \\frac{\\partial}{\\partial\\ub}((\\gamma_n^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_n)_{AB}))\\mbox{ converge to }(\\frac{\\partial}{\\partial \\ub}g_{\\infty}, \\frac{\\partial}{\\partial\\ub}((\\gamma_{\\infty}^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_{\\infty})_{AB}))\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^\\infty(S),$$\n$$(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_n,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_n,\\frac{\\partial^2}{\\partial\\ub^2}(b^A)_n)\\mbox{ converge to }(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial\\ub^2}(b^A)_\\infty)\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^4(S).$$\nAs a consequence, in the limiting spacetime,\n$$\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty\\in C^0_u C^0_{\\ub} L^4(S),$$\n$$\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_\\infty\\in C^0_u C^0_{\\ub} L^2(S),$$\n$$\\frac{\\partial}{\\partial \\ub}g_\\infty, \\frac{\\partial}{\\partial\\ub}((\\gamma_\\infty^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_\\infty)_{AB}) \\in L^\\infty_u L^\\infty_{\\ub} L^\\infty(S),$$\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial \\ub^2}(b^A)_\\infty\\in L^\\infty_u L^\\infty_{\\ub} L^4(S).$$\n\\end{theorem}\n\n\\begin{remark}\nNotice that in the case of initial data of the impulsive gravitational wave, we have $\\nabla^i\\chih_n$ converging in $L^p_{\\ub} L^2(S)$ for any $2\\leq p <\\infty$, but not for $p=\\infty$. We thus take a sequence of initial data that converges in a topology which is consistent with $\\nab^i\\chih_n$ converging in $L^p_{\\ub}L^2(S)$.\n\\end{remark}\n\nThe convergence of the Ricci coefficients and the acceptable curvature components follows from Theorem \\ref{convergencethm}. In Section \\ref{limit}, we will show that the convergence of the Ricci coefficients and the acceptable curvature components imply the asserted convergence of the metric and the regularity property of the limiting spacetime. The above convergence theorem is strong enough to show that the limiting spacetime is a solution to the Einstein equations:\n\\begin{theorem}\\label{Einstein}\nSuppose all the assumptions of Theorem \\ref{convergencethm2} hold. Then the limiting spacetime metric satisfies the Einstein equations in $L^\\infty_u L^\\infty_{\\ub}L^2(S)$.\n\\end{theorem}\nThis will also be proved in Section \\ref{limit}. Moreover, this limiting spacetime solution is the unique solution to the vacuum Einstein equations.\n\\begin{theorem}\\label{uniquenessthm}\nThe solution to the Einstein equations given by Theorems \\ref{convergencethm2} and \\ref{Einstein} is unique among spacetimes that arise as $C^0$ limit of smooth solutions to the vacuum Einstein equations.\n\\end{theorem}\nA more precise version of the uniqueness theorem is formulated as Proposition \\ref{uniquenessprop} and will be proved in Section \\ref{uniquenesssec}. Moreover, if the initial data is assumed to be more regular, the limit spacetime metric is more regular:\n\\begin{theorem}\\label{regularitythm}\nSuppose, in addition to the assumptions of Theorem \\ref{convergencethm2}, the bounds\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j\\nab^i\\Psi_n||_{L^2(\\Hb_0)}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^i \\nab^j\\Psi_n||_{L^2(H_0)})\\leq C $$\nhold uniformly independent of $n$.\nThen \n$$\\sum_{j\\leq J+2}\\sum_{i\\leq \\min\\{I,j-2\\}}(\\frac{\\partial}{\\partial u})^j(\\frac{\\partial}{\\partial \\th})^i g_\\infty\\in L^\\infty_u L^\\infty_{\\ub} L^2(S),$$\n$$\\sum_{j\\leq J+2}\\sum_{i\\leq \\min\\{I,j-2\\}}(\\frac{\\partial}{\\partial u})^j(\\frac{\\partial}{\\partial \\th})^i\\frac{\\partial}{ \\partial \\ub}g_\\infty\\in L^\\infty_u L^{p_0}_{\\ub} L^2(S).$$\n\\end{theorem}\nThis will be proved in Section \\ref{regularityp}. \n\nFor general initial data satisfying the assumptions of Theorem \\ref{rdthmv2}, using the construction in Section \\ref{initialcondition}, there exists an approximating sequence of smooth initial data satisfying the assumption of Theorem \\ref{convergencethm2}. Thus, the combination of Theorems \\ref{timeofexistence}, \\ref{convergencethm2}, \\ref{Einstein}, \\ref{regularitythm} and \\ref{uniquenessthm} together imply Theorem \\ref{rdthmv2}.\n\nThe remainder of this Section will be organized as follows: Theorem \\ref{convergencethm} is proved in Sections \\ref{convsec1}-\\ref{convsec5}. After the definition of the norms in Section \\ref{convsec1}, the proof is carried out in three steps:\\\\\n\n\\noindent{\\bf STEP 1} (Section \\ref{convsec2}): The difference of the metric components are estimated assuming the bounds for the difference of the Ricci coefficients.\\\\\n\n\\noindent{\\bf STEP 2} (Section \\ref{convsec4}): The difference of the Ricci coefficients are controlled assuming the estimates of the difference of the curvature components. This relies on the transport equations for the difference quantities derived in Section \\ref{convsec3}.\\\\\n\n\\noindent{\\bf STEP 3} (Section \\ref{convsec5}): Finally, the bounds for the difference of curvature components are obtained, closing all the estimates for Theorem \\ref{convergencethm}.\\\\\n\nIn Section \\ref{AddEstMetric}, additional estimates are derived for the metric components. These additional estimates will be used together with Theorem \\ref{convergencethm} to construct a limiting spacetime and to obtain Theorem \\ref{convergencethm2} in Section \\ref{limit}. In this Section, Theorem \\ref{Einstein} is also proved, showing that the limiting spacetime satisfies the Einstein equations. In Section \\ref{uniquenesssec}, we formulate and prove a precise version of Theorem \\ref{uniquenessthm}, establishing uniqueness of the limiting spacetime. In Section \\ref{regularityp}, Theorem \\ref{regularitythm} is proved, showing additional regularity in the spacetime with more regular initial data. Finally, in Section \\ref{limitgiw}, we return to the case of an impulsive gravitational wave and obtain extra regularity properties for these spacetimes.\n\n\\subsection{Norms}\\label{convsec1}\nWe begin the proof of Theorem \\ref{convergencethm}.\nDefine the following $L^2$ norms for the difference of the null curvature components or their renormalized versions:\n$$\\mathcal R'=\\sum_{i\\leq 1}\\sup_u||\\nabla^i(\\beta',\\rho',\\sigma',\\betab')||_{L^2(H_u)}+\\sum_{i\\leq 1}\\sup_{\\ub}||\\nabla^i(\\rhoc',\\sigmac',\\betab',\\alphab')||_{L^2(\\Hb_{\\ub})}$$\nDefine the following norm for difference of the the Ricci coefficients:\n$$\\mathcal O'=\\sum_{i\\leq 1}\\sup_{u}||\\nabla^i(\\hat{\\chi}',\\omega')||_{L^{p_0}_{\\ub}L^2(S_{u,\\ub})}+\\sum_{i\\leq 1}\\sup_{u,\\ub}||(\\nabla^i(tr\\chi',\\eta',\\underline{\\eta}',\\underline{\\omega}',\\underline{\\hat{\\chi}}',tr\\underline{\\chi}')||_{L^2(S_{u,\\ub})}.$$\nThese norms will be used to estimate the difference of the null Ricci coefficients except for those involving the highest derivatives, for which the estimates are weaker. We therefore also introduce the norms:\n\\begin{equation*}\n\\begin{split}\n\\tilde{\\mathcal O}'=\\sup_u ||\\nabla^2(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub} ||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{\\ub})}+\\sup_{u,\\ub} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{u,\\ub})},\n\\end{split}\n\\end{equation*}\nNotice that the norms $\\mathcal R'$, $\\mathcal O'$ and $\\tilde{\\mathcal O}'$ are difference counterparts of the norms $\\mathcal R$, $\\mathcal O$ and $\\tilde{\\mathcal O}$ for the difference quantities. Note, however, that the former provides control of one fewer derivatives than the latter. This is due to the fact that convergence will be proved in a norm which is one derivative weaker than the corresponding norms for the a priori estimates. Define also the following norms:\n\\begin{equation*}\n\\begin{split}\n\\mathcal O''=\\sup_u(||(\\chih,\\omega,\\eta,\\underline{\\eta},\\trch,\\trchb)'||_{L^2_{\\ub}L^\\infty(S_{u,\\ub})}+||\\nabla(\\chih,\\omega,\\eta,\\underline{\\eta},\\trch,\\trchb)'||_{L^2_{\\ub}L^4(S_{u,\\ub})}).\n\\end{split}\n\\end{equation*}\nIt follows by Sobolev Embedding in Propositions \\ref{L4} and \\ref{Linfty} that\n\\begin{proposition}\\label{OpSobolev}\n$$\\mathcal O''\\leq C(\\tilde{\\mathcal O}'+\\mathcal O').$$\n\\end{proposition}\n\n\\subsection{Estimates for the Difference of the Metrics}\\label{convsec2}\nIn this section, we show that the difference of the metrics and their coordinate angular derivatives can be controlled by the $\\mathcal O'$ and $\\tilde{\\mathcal O}'$ norms. Recall that the metrics take the form\n$$g^{(i)}=-2(\\Omega^{(i)})^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)})_{AB}(d\\th^A-(b^{(i)})^Adu)\\otimes (d\\th^B-(b^{(i)})^Bdu),$$ \nfor $i=1,2$.\n\\begin{proposition}\\label{Omegap}\n\\[\n\\sup_{u,\\ub}|(\\Omega',(\\Omega^{-1})')(u,\\ub)|\\leq C\\delta^{\\frac 12}\\mathcal O''.\n\\]\n\\end{proposition}\n\\begin{proof}\nRecall that\n\\[\n \\omega=-\\frac{1}{2}\\nabla_4\\log\\Omega=\\frac{1}{2}\\Omega\\nabla_4\\Omega^{-1}=\\frac{1}{2}\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}.\n\\]\nHence \n\\[\n \\frac{1}{2}\\frac{\\partial}{\\partial\\ub}(\\Omega^{-1})'=\\omega'.\n\\]\nBy integrating along the $\\ub$ direction, noticing that $\\Omega'=0$ on $\\Hb_0$, and using Cauchy-Schwarz we get\n$$|(\\Omega^{-1})'|\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\nIn order to get the estimate for $\\Omega'$, we note that\n\\[\n(\\Omega'+\\Omega^{(2)})^{-1}=(\\Omega^{(1)})^{-1}=(\\Omega^{-1})^{(2)}+(\\Omega^{-1})'.\n\\]\nTherefore,\n\\[\n \\Omega'=((\\Omega^{(2)})^{-1}+(\\Omega)^{-1})')^{-1}-\\Omega^{(2)}=\\frac{(\\Omega^{-1})'}{1+\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(2)}}}.\n\\]\nIn view of the upper and lower bounds of $\\Omega$ in Proposition \\ref{Omega},\n$$|\\Omega'|\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\n\n\\end{proof}\nUsing the estimates for $\\Omega'$, we also have estimates for $\\gamma'$.\n\\begin{proposition}\\label{gammap}\n$\\gamma'$ satisfies the following pointwise bounds:\n$$\\sup_{u,\\ub}|(\\gamma_{AB}',((\\gamma^{-1})^{AB})')(u,\\ub)|\\leq a+C\\delta^{\\frac 12}\\mathcal O''_u.$$\n\\end{proposition}\n\\begin{proof}\nThe components of $\\gamma$ solve the following ODE:\n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB}.$$\nThis implies\n$$\\frac{\\partial}{\\partial \\ub}\\log(\\det\\gamma)=\\Omega\\trch.$$\nFrom this we can derive an equation for $(\\det\\gamma)'$:\n$$\\frac{\\partial}{\\partial \\ub}(\\det\\gamma)'=\\frac{1}{(\\det\\gamma)^{(2)}}\\left(-(\\det\\gamma)'\\frac{\\partial}{\\partial \\ub}(\\det\\gamma)^{(1)}+(\\det\\gamma)^{(1)}(\\det\\gamma)^{(2)}(\\Omega\\trch)'\\right). $$\nBy Proposition \\ref{gamma}, we have uniform upper and lower bounds on $\\det\\gamma$ and uniform estimates for $\\Omega$ and $\\frac{\\partial}{\\partial \\ub}(\\det\\gamma)$. The previous Proposition gives $|\\Omega'|\\leq C\\delta^{\\frac 12}\\mathcal O''$. Moreover, $\\int |\\trch'| d\\ub'\\leq \\delta\\mathcal O''$ by definition. Thus,\n\\begin{equation}\\label{volumeformd}\n|(\\det\\gamma)'|\\leq C\\delta^{\\frac 12}\\mathcal O''.\n\\end{equation}\nWe can also derive an equation for $(\\gamma_{AB})'$:\n$$\\frac{\\partial}{\\partial \\ub}(\\gamma_{AB})'=2(\\Omega\\chi_{AB})'.$$\nThus\n$$|(\\gamma_{AB})'(u,\\ub)|\\leq |(\\gamma_{AB})'(u,0)|+C\\int_0^{\\ub}|(\\chi_{AB})'|d\\ub'+C\\delta^{\\frac 12}\\mathcal O'' |\\chi_{AB}|.$$\nBy assumption, $|(\\gamma_{AB})'(u,0)|\\leq a$. By Proposition \\ref{gamma}, matrices $\\gamma^{(1)}$ and $\\gamma^{(2)}$ are uniformly non-degenerate matrices. Hence, regardless of whether we define the $L^\\infty$ norm with respect to $\\gamma^{(1)}$ or $\\gamma^{(2)}$, we have\n$$\\sum_{A,B=1,2}|(\\chih_{AB})'|\\leq C\\sup_{u,\\ub}||\\chi'||_{L^\\infty(S_{u,\\ub})}.$$\nTherefore, \n\\begin{equation}\\label{metricd}\n|(\\gamma_{AB})'|\\leq a+C\\delta^{\\frac 12}\\mathcal O''.\n\\end{equation}\nNow, (\\ref{volumeformd}) and (\\ref{metricd}) together also imply the pointwise bound in coordinates for \n$$|((\\gamma^{-1})^{AB})'|\\leq a+C\\delta^{\\frac 12}\\mathcal O''.$$\n\\end{proof}\nThis estimate allows us to conclude that the $L^p$ norms defined with respect to either metric $(1)$ or $(2)$ differ only by $a+C\\delta^{\\frac 12}\\mathcal O''$.\n\\begin{proposition}\\label{normscomparable}\nGiven any tensor, we can define its $L^p(S)$ norms $||\\phi||^{(1)}_{L^p(S_{u,\\ub})}$ and $||\\phi||^{(2)}_{L^p(S_{u,\\ub})}$ with respect to the first and second metric. Suppose\n$$||\\phi||^{(1)}_{L^p(S_{u,\\ub})}<\\infty.$$\nThen\n$$|||\\phi||^{(1)}_{L^p(S_{u,\\ub})}-||\\phi||^{(2)}_{L^p(S_{u,\\ub})}|\\leq C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||^{(1)}_{L^p(S_{u,\\ub})}.$$\nIn particular, $\\phi$ is also in $L^p$ with respect to the second metric.\n\\end{proposition}\n\\begin{proof}\nThis follows from the pointwise control of $(\\gamma_{AB})'$, $((\\gamma^{-1})^{AB})'$ and $(\\det\\gamma)'$ in coordinates.\n\\end{proof}\nIn a similar manner, we can control the $(\\frac{\\partial}{\\partial\\th^A})$ derivatives of $\\gamma'$.\n\\begin{proposition}\\label{dgammap}\n$$\\sup_{u,\\ub}||(\\frac{\\partial}{\\partial\\th^C})\\gamma_{AB}'||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O'',$$\nwhere $L^4(S)$ is understood as the $L^4$ norm for a scalar function and by Proposition \\ref{normscomparable} can be defined with respect either to the metric $\\gamma^{(1)}$ or $\\gamma^{(2)}$.\n\\end{proposition}\n\\begin{proof}\nBy the equation \n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB}$$\nand the assumption, we have\n\\begin{equation}\\label{gammadd}\n\\begin{split}\n||(\\frac{\\partial}{\\partial\\th^C})^i\\gamma_{AB}'(u,\\ub)||_{L^4(S_{u,\\ub'})}\\leq& Ca+C\\int_0^{\\ub}\\left(\\|(\\frac{\\partial}{\\partial\\th^C})^i\\Omega'||_{L^4(S_{u,\\ub'})}+\\|(\\frac{\\partial}{\\partial\\th^C})^i\\chi'||_{L^4(S_{u,\\ub'})}\\right) d\\ub'.\n\\end{split}\n\\end{equation}\nSince\n$$\\nabla(\\log \\Omega)=\\frac 12 (\\eta+\\etab),$$\nwe have\n$$|(\\frac{\\partial}{\\partial\\th^C})\\Omega'|\\leq C|(\\eta,\\etab)'|+C|\\Gamma'||\\Omega|+C|\\Gamma||\\Omega'| \\leq C|(\\eta,\\etab)'|+C|\\frac{\\partial}{\\partial\\th^D}\\gamma_{AB}'|+C|\\Omega'|,$$\nwhere $\\Gamma$ is the connection coefficients on the spheres with respect to $\\gamma$.\nMoreover, by Cauchy-Schwarz,\n$$\\int ||(\\frac{\\partial}{\\partial\\th^C})\\chi'||_{L^4(S_{u,\\ub'})} d\\ub'\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\nThus by (\\ref{gammadd}), we have\n\\begin{equation*}\n\\begin{split}\n&||(\\frac{\\partial}{\\partial\\th^C})\\gamma_{AB}'(u,\\ub)||_{L^4(S_{u,\\ub})}\\\\\n\\leq&Ca+C\\delta^{\\frac 12}\\mathcal O''+C\\int_0^{\\ub} ||\\eta',\\etab',\\chih',\\frac{\\partial}{\\partial\\th^D}\\gamma_{AB}',\\Omega'||_{L^4(S_{u,\\ub'})}d\\ub'.\n\\end{split}\n\\end{equation*}\nUsing already established estimates, Gronwall's inequality and Cauchy-Schwarz, we thus have\n$$||(\\frac{\\partial}{\\partial\\th^C})\\gamma_{AB}'(u,\\ub)||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O''.$$\n\\end{proof}\nA consequence of the above Proposition is the estimates on the difference of the connection coefficients:\n\\begin{proposition}\\label{Gammap}\n$$\\sup_{u,\\ub}||\\Gamma'||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O''.$$\n\\end{proposition}\n\\begin{proof}\nThis follows from the fact that $\\Gamma'$ can be expressed as a linear combination $\\gamma'$ and its first angular derivative.\n\\end{proof}\nThis implies that $\\nabla^{(1)}$ and $\\nabla^{(2)}$ are comparable in the following sense:\n\\begin{proposition}\\label{connectionpL4}\nLet $p\\leq 4$. Suppose $\\phi$ is a tensor. \nThen\n$$||(\\nabla^{(1)})\\phi-(\\nabla^{(2)})\\phi||_{L^p(S_{u,\\ub})} \\leq C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||_{L^\\infty(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nSince we have estimates for $\\gamma_{AB}$ and $(\\gamma^{-1})^{AB}$. It suffice to estimate \n$$(\\nabla^{(1)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi-(\\nabla^{(2)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi.$$\nUsing Proposition \\ref{Gammap}, we have\n\\begin{equation*}\n\\begin{split}\n&||(\\nabla^{(1)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi-(\\nabla^{(2)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi||_{L^p(S_{u,\\ub})}\\\\\n\\leq &||\\Gamma^{(1)}\\phi-\\Gamma^{(2)}\\phi||_{L^p(S_{u,\\ub})}\\\\\n\\leq &C||\\Gamma'||_{L^4(S_{u,\\ub})}||\\phi||_{L^\\infty(S_{u,\\ub})}\\\\\n\\leq &C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||_{L^\\infty(S_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\n\\end{proof}\nWe now show that the difference between the angular covariant derivatives of two tensors is comparable to the angular covariant derivative of the difference:\n\\begin{proposition}\\label{angularpL4}\nLet $p\\leq 4$. Suppose $\\phi^{(1)}$ and $\\phi^{(2)}$ are tensors defined on spacetimes $(1)$ and $(2)$ respectively.\nThen\n$$||\\nabla^{(1)}(\\phi')-(\\nabla\\phi)'||_{L^p(S_{u,\\ub})}\\leq C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||_{L^\\infty(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nIt is easy to see that\n$$\\nabla^{(1)}(\\phi')-(\\nabla\\phi)'=-\\nabla^{(1)}\\phi^{(2)}+\\nabla^{(2)}\\phi^{(2)}.$$\nThe Proposition thus follows from Proposition \\ref{connectionpL4}.\n\\end{proof}\nWe now estimate the coordinate angular derivative of $\\Omega'$ in $L^4(S)$:\n\\begin{proposition}\\label{dOmegap}\nWe have\n\\[\n\\sup_{u,\\ub}||\\frac{\\partial}{\\partial \\th^A}\\Omega'||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O''.\n\\]\n\\end{proposition}\n\\begin{proof}\nRecall that\n\\[\n \\omega=-\\frac{1}{2}\\nabla_4\\log\\Omega=\\frac{1}{2}\\Omega\\nabla_4\\Omega^{-1}=\\frac{1}{2}\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}.\n\\]\nHence \n\\[\n \\frac{1}{2}\\frac{\\partial}{\\partial\\ub}(\\Omega^{-1})'=\\omega'.\n\\]\nThus, in order to estimate $\\frac{\\partial}{\\partial \\th^A}\\Omega'$, we need to estimate $\\frac{\\partial}{\\partial \\th^A}\\omega'$ in $L^1_{\\ub}L^4(S)$. Since by the a priori estimates in the previous section we have $||\\nabla\\omega||_{L^\\infty(S)}\\leq C$, it suffices by Proposition \\ref{angularpL4} to estimate \n$$\\sum_{i\\leq 1}\\int_0^{\\ub}||\\nabla^i\\omega'||_{L^4(S_{u,\\ub'})}d\\ub'$$\nBy the definition of the norm and Cauchy-Schwarz\n$$\\sum_{i\\leq 1}\\int_0^{\\ub}||\\nabla^i\\omega'||_{L^4(S_{u,\\ub'})}d\\ub'\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\nThe conclusion thus follows.\n\\end{proof}\nThe above Proposition allows us to estimate the second coordinate angular derivatives of $\\gamma'$ in $L^2(S)$.\n\\begin{proposition}\\label{ddgammap}\n$$\\sup_{u,\\ub}||(\\frac{\\partial^2}{\\partial\\th^C \\partial\\th^D})\\gamma_{AB}'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'),$$\nwhere $L^2(S)$ is understood as the $L^2$ norms for scalar functions.\n\\end{proposition}\n\\begin{proof}\nBy the equation \n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB},$$\nand the a priori estimates in the previous section, we have\n\\begin{equation}\\label{gamma2dd}\n\\begin{split}\n&||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub})}\\\\\n\\leq& Ca+C\\int_0^{\\ub}\\left(\\|(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\Omega'||_{L^2(S_{u,\\ub'})}+\\|(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\chi'||_{L^2(S_{u,\\ub'})}\\right) d\\ub'\\\\\n&+ C\\int_0^{\\ub}\\left(\\|(\\frac{\\partial}{\\partial\\th^C})\\Omega'||_{L^2(S_{u,\\ub'})}+\\|(\\frac{\\partial}{\\partial\\th^C})\\chi'||_{L^2(S_{u,\\ub'})}\\right) d\\ub'.\n\\end{split}\n\\end{equation}\nThe last term has already been estimated in the proof of Proposition \\ref{dgammap}:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\left(\\|(\\frac{\\partial}{\\partial\\th^C})\\Omega'||_{L^2(S_{u,\\ub'})}+\\|(\\frac{\\partial}{\\partial\\th^C})\\chi'||_{L^2(S_{u,\\ub'})}\\right) d\\ub'\\\\\n\\leq& Ca+C\\delta^{\\frac 12}\\mathcal O''.\n\\end{split}\n\\end{equation*}\nSince\n$$\\nabla(\\log \\Omega)=\\frac 12 (\\eta+\\etab),$$\nwe have\n\\begin{equation*}\n\\begin{split}\n&||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\Omega'||_{L^2(S_{u,\\ub'})}\\\\\n\\leq& C||\\frac{\\partial}{\\partial\\th}(\\eta,\\etab)'||_{L^2(S_{u,\\ub'})}+C||\\frac{\\partial}{\\partial\\th}\\Gamma'||_{L^2(S_{u,\\ub'})}||\\Omega||_{L^\\infty(S_{u,\\ub'})}+C||\\frac{\\partial}{\\partial\\th}\\Gamma||_{L^2(S_{u,\\ub'})}||\\Omega'|||_{L^\\infty(S_{u,\\ub'})}\\\\\n& +C||\\Gamma'||_{L^4(S_{u,\\ub'})}||\\frac{\\partial}{\\partial\\th}\\Omega||_{L^4(S_{u,\\ub'})}+C||\\Gamma||_{L^\\infty(S_{u,\\ub'})}||\\frac{\\partial}{\\partial\\th}\\Omega'||_{L^2(S_{u,\\ub'})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i(\\eta,\\etab)'||_{L^2(S_{u,\\ub'})}+C\\sum_{i\\leq 1}||(\\frac{\\partial}{\\partial\\th})^i\\Gamma'||_{L^2(S_{u,\\ub'})}+C\\sum_{i\\leq 1}||(\\frac{\\partial}{\\partial\\th})^i\\Omega'||_{L^2(S_{u,\\ub'})}.\n\\end{split}\n\\end{equation*}\nUsing Propositions \\ref{dOmegap} and the definition of the norms $\\mathcal O''$ and $\\tilde{\\mathcal O}'$, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}\\int_0^{\\ub}(||\\nabla^i(\\eta,\\etab)'||_{L^2(S_{u,\\ub'})}+||(\\frac{\\partial}{\\partial\\th})^i\\Gamma'||_{L^2(S_{u,\\ub'})}+||(\\frac{\\partial}{\\partial\\th})^i\\Omega'||_{L^2(S_{u,\\ub'})})d\\ub'\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'_u)+C\\int_0^{\\ub}||(\\frac{\\partial^2}{\\partial\\th^E\\partial\\th^F})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub'})}d\\ub').\n\\end{split}\n\\end{equation*}\nMoreover,\n$$\\int ||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\chi'||_{L^2(S_{u,\\ub'})} d\\ub'\\leq C(\\mathcal O''+\\tilde{\\mathcal O}').$$\nThus by (\\ref{gamma2dd}), we have\n\\begin{equation*}\n\\begin{split}\n&||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub})}\\\\\n\\leq&Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')+C\\int_0^{\\ub}||(\\frac{\\partial^2}{\\partial\\th^E\\partial\\th^F})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub'})}d\\ub').\n\\end{split}\n\\end{equation*}\nBy Gronwall's inequality, we thus have\n$$||(\\frac{\\partial^2}{\\partial\\th^C \\partial\\th^D})\\gamma_{AB}'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}').$$\n\\end{proof}\nThe proof of the previous Proposition also shows that\n\\begin{proposition}\\label{ddOmegap}\n$$\\sup_{u,\\ub}||(\\frac{\\partial^2}{\\partial\\th^C \\partial\\th^D})\\Omega'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'),$$\n\\end{proposition}\nProposition \\ref{ddgammap} implies the following estimates of $K'$, the difference of the Gauss curvatures:\n\\begin{proposition}\\label{Kp}\n$K'$ satisfies the following bounds:\n$$\\sup_{u,\\ub}||K'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}').$$\n\\end{proposition}\n\\begin{proof}\nThis follows from the estimates of the derivatives of $\\gamma'$ in Propositions \\ref{dgammap} and \\ref{ddgammap}.\n\\end{proof}\nMoreover, Proposition \\ref{ddgammap} allows us to conclude that the covariant second derivative with respect to connection $(1)$ and connection $(2)$ are comparable:\n\\begin{proposition}\\label{connectionpL2}\nLet $p\\leq 2$. Suppose $\\phi$ is a tensor. \nThen\n$$||(\\nabla^{(1)})^2\\phi-(\\nabla^{(2)})^2\\phi||_{L^p(S_{u,\\ub})} \\leq C(a+\\delta^{\\frac 12}\\mathcal O'')\\sum_{i\\leq 1}||\\nabla^i\\phi||_{L^4(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nBy Propositions \\ref{Gammap}, \\ref{ddgammap} and \\ref{Kp}.\n\\end{proof}\nThis implies that taking the covariant second derivatives of the difference is comparable to taking the difference of the covariant second derivatives:\n\\begin{proposition}\\label{angularpL2}\nLet $p\\leq 2$. Suppose $\\phi^{(1)}$ and $\\phi^{(2)}$ are tensors defined on spacetimes $(1)$ and $(2)$ respectively.\nThen\n$$||(\\nabla^{(1)})^2(\\phi')-(\\nabla^2\\phi)'||_{L^p(S_{u,\\ub})}\\leq C(a+\\delta^{\\frac 12}\\mathcal O'')\\sum_{i\\leq 1}||\\nabla^i\\phi||_{L^4(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nThe proof follows as that of Proposition \\ref{angularpL4}, using Proposition \\ref{connectionpL2} instead of Proposition \\ref{connectionpL4}.\n\\end{proof}\nWe now prove the estimates for $b'$:\n\\begin{proposition}\\label{bp}\n\\[\n\\sup_{u,\\ub}||(b^A)'||_{L^\\infty(S_{u,\\ub})}, \\sup_{u,\\ub}||\\frac{\\partial}{\\partial\\th^B}(b^A)'||_{L^4(S_{u,\\ub})},\\sup_{u,\\ub}||\\frac{\\partial^2}{\\partial \\th^C\\partial\\th^B}(b^A)'||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O''+\\tilde{\\mathcal O}').\n\\]\n\\end{proposition}\n\\begin{proof}\nRecall that $b$ satisfies the transport equation \n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A.$$\nThe Proposition thus follows from differentiating the equation by $\\frac{\\partial}{\\partial\\th}$, integrating in $\\ub$ and using the the bounds in Propositions \\ref{Omegap}, \\ref{dOmegap} and \\ref{ddOmegap} and the definitions of the norms.\n\\end{proof}\n\n\n\n\\subsection{Transport Equations for the Difference Quantities}\\label{convsec3}\nIn this Subsection, we show how to derive estimates for the difference $\\phi'$ of the quantities $\\phi^{(1)}$ and $\\phi^{(2)}$ satisfying transport equations in the $\\nab_4^{(1)}$, $\\nab_4^{(2)}$ or $\\nab_3^{(1)}$, $\\nab_3^{(2)}$ directions. This should be compared with the previous Subsection where we considered difference quantities satisfying transport equations, but unlike in this Subsection, the corresponding transport equations held with respect to the $\\frac{\\partial}{\\partial \\ub}$, $\\frac{\\partial}{\\partial u}$ derivatives rather than $\\nab_4$, $\\nab_3$, which themselves depend on the spacetimes under consideration.\n\nIn order to estimate the quantities $\\phi'$ associated to the difference between that of different spacetimes, we need to derive a transport equation for $\\phi'$ from the transport equation for $\\phi$. \n\\begin{proposition}\\label{peqn}\nConsider spacetimes that satisfy the hypotheses of Theorem \\ref{timeofexistence} and \\ref{convergencethm}. Let $\\phi$ be a $(0,r)$ S-tensor. Suppose $\\nabla_4\\phi=F$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_4^{(1)}\\phi'\\sim & F'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_4\\phi+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\gamma^{-1}\\chi\\phi+(\\gamma^{-1}\\chi)'\\phi.\n\\end{split}\n\\end{equation*}\nSimilarly, suppose $\\nabla_3\\phi=G$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_3^{(1)}\\phi'\\sim & G'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_3\\phi+\\Omega^{-1}(b^A)'\\nabla_{\\frac{\\partial}{\\partial \\th^A}}\\phi+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\gamma^{-1}\\chib\\phi+(\\gamma^{-1}\\chib)'\\phi+\\Omega^{-1}\\frac{\\partial b'}{\\partial\\th}\\phi.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nWe write in coordinates\n$$\\nabla_4\\phi_{A_1...A_r}=\\Omega^{-1}\\frac{\\partial}{\\partial \\ub}\\phi_{A_1...A_r}-(\\gamma^{-1})^{CD}\\chi_{A_iD}\\phi_{A_1...\\hat{A_i}C...A_r},$$\nwhere $\\hat{A_i}$ denotes that the original $A_i$ in the $i$-th slot of the tensor is removed. This equation holds in both spacetimes $(1)$ and $(2)$.\nThen, we derive the equations for $\\phi'$. We will write schematically without the exact constant depending only on $r$:\n\\begin{equation*}\n\\begin{split}\n&(\\nabla_4)^{(1)}\\phi' \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial \\ub}-(\\gamma^{-1})^{(1)}\\chi^{(1)}\\right)\\left(\\phi^{(1)}-\\phi^{(2)}\\right) \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial \\ub}-(\\gamma^{-1})^{(1)}\\chi^{(1)}\\right)\\phi^{(1)}-\\left((\\Omega^{-1})^{(2)}\\frac{\\partial}{\\partial \\ub}-(\\gamma^{-1})^{(2)}\\chi^{(2)}\\right)\\phi^{(2)}\\\\\n& -(\\Omega^{-1})'\\frac{\\partial}{\\partial \\ub}\\phi^{(2)}+(\\gamma^{-1}\\chi)'\\phi^{(2)}\\\\\n=&F'-(\\Omega^{-1})'\\frac{\\partial}{\\partial \\ub}\\phi^{(2)}+(\\gamma^{-1}\\chi)'\\phi^{(2)}\\\\\n=&F'-\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}(\\nabla_4)^{(1)}\\phi^{(2)}+\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}(\\gamma^{-1}\\chi)^{(1)}\\phi^{(2)}+(\\gamma^{-1}\\chi)'\\phi^{(2)}.\n\\end{split}\n\\end{equation*}\nFor the $\\nabla_3$ equations, we write in coordinates\n\\begin{equation*}\n\\begin{split}\n\\nabla_3\\phi_{A_1...A_r}=&\\Omega^{-1}\\frac{\\partial}{\\partial u}\\phi_{A_1...A_r}+\\Omega^{-1}b^B\\frac{\\partial}{\\partial \\th^B}\\phi_{A_1...A_r} \\\\\n&-(\\gamma^{-1})^{CD}\\chib_{A_i D}\\phi_{A_1...\\hat{A_i}C...A_r}+\\Omega^{-1}\\frac{\\partial b^B}{\\partial\\th^{A_i}}\\phi_{A_1...\\hat{A_i}C...A_r}.\n\\end{split}\n\\end{equation*}\nWe derive the difference equations as before. However, $e_3$ in the respective spacetimes are not parallel to each other and we therefore have extra terms.\n\\begin{equation*}\n\\begin{split}\n&(\\nabla_3)^{(1)}\\phi' \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial u}+(\\Omega^{-1})^{(1)}b^{(1)}\\frac{\\partial}{\\partial \\th} -(\\gamma^{-1})^{(1)}\\chib^{(1)}+(\\Omega^{-1})^{(1)}\\frac{\\partial b^{(1)}}{\\partial \\th}\\right)\\left(\\phi^{(1)}-\\phi^{(2)}\\right) \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial u}+(\\Omega^{-1})^{(1)}b^{(1)}\\frac{\\partial}{\\partial \\th} -(\\gamma^{-1})^{(1)}\\chib^{(1)}+(\\Omega^{-1})^{(1)}\\frac{\\partial b^{(1)}}{\\partial \\th}\\right)\\phi^{(1)} \\\\\n&-\\left((\\Omega^{-1})^{(2)}\\frac{\\partial}{\\partial u}+(\\Omega^{-1})^{(2)}b^{(2)}\\frac{\\partial}{\\partial \\th} -(\\gamma^{-1})^{(2)}\\chib^{(2)}+(\\Omega^{-1})^{(2)}\\frac{\\partial b^{(2)}}{\\partial \\th}\\right)\\phi^{(2)}\\\\\n& -(\\Omega^{-1})'\\frac{\\partial}{\\partial u}\\phi^{(2)}+(\\gamma^{-1}\\chib)'\\phi^{(2)}-(\\Omega^{-1}b)'\\frac{\\partial}{\\partial \\th}\\phi^{(2)}-(\\Omega^{-1}\\frac{\\partial b}{\\partial \\th})'\\phi^{(2)}\\\\\n=&G'-\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}\\nabla_3^{(1)}\\phi^{(1)}-(\\Omega^{-1})^{(2)}b'\\frac{\\partial}{\\partial\\th}\\phi^{(2)}-(\\Omega^{-1})^{(2)}\\frac{\\partial b'}{\\partial \\th}\\phi^{(2)}\\\\\n&+\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}(\\gamma^{-1}\\chib)^{(1)}\\phi^{(1)}+(\\gamma^{-1}\\chib)'\\phi^{(2)}.\n\\end{split}\n\\end{equation*}\n\\end{proof}\nThis, together with the commutation estimates in Proposition \\ref{commuteeqn}, gives the following estimates for $(\\nabla^i\\phi)'$.\n\\begin{proposition}\\label{pcommuteeqn}\nSuppose $\\nabla_4\\phi=F$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_4(\\nabla^i\\phi)'\\sim &\\sum_{i_1+i_2+i_3=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F)'+\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi)' \\\\\n&+(\\Omega^{-1})'\\left(\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\right)\\\\\n&+(\\gamma^{-1}\\chi)'\\nabla^i\\phi.\n\\end{split}\n\\end{equation*}\nSimilarly, suppose $\\nabla_3\\phi=G$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_3(\\nabla^i\\phi)'\\sim &\\sum_{i_1+i_2+i_3=i}(\\nabla^{i_1}(\\eta,\\etab\n)^{i_2}\\nabla^{i_3} G)'+\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chib\\nabla^{i_4} \\phi)' \\\\\n&+(\\Omega^{-1})'\\left(\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} G+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chib\\nabla^{i_4} \\phi\\right)\\\\\n&+b'\\nabla^{i+1}\\phi+((\\gamma^{-1}\\chi)'+\\frac{\\partial b'}{\\partial\\th})\\nabla^i\\phi.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nAccording to the already established estimates, we have $\\frac{1}{2}\\leq \\Omega\\leq 2$. The result now follows directly from Propositions \\ref{commuteeqn} and \\ref{peqn}.\n\\end{proof}\nWe now use Propositions \\ref{transport} and \\ref{pcommuteeqn} to obtain estimates from a transport equation.\n\\begin{proposition}\\label{transportp}\nSuppose $\\nabla_4\\phi=F$. Then\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,0})}+C\\int_0^{\\ub} \\sum_{i\\leq 1}||(\\nabla^iF)'||_{L^2(S_{u,\\ub'})} d\\ub'\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iF||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\n\nSimilarly, suppose $\\nabla_3\\phi=G$. Then\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{0,\\ub})}+C\\int_0^{\\ub} \\sum_{i\\leq 1}||(\\nabla^iG)'||_{L^2(S_{u,\\ub'})} d\\ub'\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iG||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nWe use the pointwise estimates for $\\gamma'$, $\\Omega'$ and $b'$ proved in the last section.\n\nBy Propositions \\ref{angularpL4}, it suffices to estimate \n$$\\sum_{i\\leq 1}||(\\nabla^i\\phi)'||_{L^2(S_{u,\\ub})}$$\nsince the difference can be estimated by\n$$\\sum_{i\\leq 1}||(\\nabla^i\\phi)'-\\nabla(\\phi')||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O''+\\tilde{\\mathcal O}')(||\\nabla\\phi||_{L^4(S_{u,\\ub})}+||\\phi||_{L^\\infty(S_{u,\\ub})}).$$\nWe can estimate $||(\\nabla^i\\phi)'||_{L^2(S_{u,\\ub})}$ using Proposition \\ref{transport} and the equation in Proposition \\ref{pcommuteeqn}. Recall the formula in Proposition \\ref{pcommuteeqn}\n\\begin{equation*}\n\\begin{split}\n\\nabla_4(\\nabla^i\\phi)'\\sim &\\sum_{i_1+i_2+i_3=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F)'+\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi)' \\\\\n&+(\\Omega^{-1})'\\left(\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\right)\\\\\n&+(\\gamma^{-1}\\chi)'\\nabla^i\\phi.\n\\end{split}\n\\end{equation*}\nWe estimate term by term:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i_1+i_2+i_3\\leq 1}||(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\int_0^{\\ub}(\\sum_{i_1\\leq 1}||(\\eta,\\etab)||^{i_1}_{L^\\infty(S_{u,\\ub'})})\\sum_{i_2\\leq 1}||(\\nabla^{i_2} F)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\sum_{i\\leq 1}||(\\nabla^{i} F)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\nThe second term can be estimated by\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i_1+i_2+i_3\\leq 1}||(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\int_0^{\\ub}(\\sum_{i_1\\leq 1}\\sum_{i_2\\leq 2}||\\nabla^{i_1}(\\eta,\\etab,\\trch,\\chih)||^{i_2}_{L^\\infty(S_{u,\\ub'})})\\sum_{i_3\\leq 1}||(\\nabla^{i_3} \\phi)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n&+C\\int_0^{\\ub}(\\sum_{i_1\\leq 1}||\\nab^{i_1}(\\eta',\\etab',\\trch',\\chih')||_{L^2(S_{u,\\ub'})})\\\\\n&\\quad\\quad\\quad\\quad\\times(\\sum_{i_2\\leq 1}\\sum_{i_3\\leq 1}||\\nabla^{i_1}(\\eta,\\etab,\\trch,\\chih)||^{i_3}_{L^\\infty(S_{u,\\ub'})})(\\sum_{i_4\\leq 1}||\\nabla^{i_4} \\phi||_{L^2(S_{u,\\ub'})})d\\ub'\\\\\n\\leq &C\\sum_{i\\leq 1}||(\\nabla^{i} \\phi)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')(||\\nabla\\phi||_{L^2(S_{u,\\ub})}+||\\phi||_{L^\\infty(S_{u,\\ub})}).\\\\\n\\end{split}\n\\end{equation*}\nThe third term is easier to estimate since by Proposition \\ref{Omegap}, $(\\Omega^{-1})'$ can be estimated in $L^\\infty$:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i_1+i_2+i_3\\leq 1}||(\\Omega^{-1})'\\left(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F+\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\right)||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}||\\nabla^iF||_{L^2(S_{u,\\ub'})}+\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})}).\n\\end{split}\n\\end{equation*}\nThe fourth term can be controlled in the same way as the third term, since $(\\gamma^{-1}\\chi)'$ can be estimated in $L^2_{\\ub}L^\\infty$:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i\\leq 1}||(\\gamma^{-1}\\chi)'\\nabla^i\\phi||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})}).\n\\end{split}\n\\end{equation*}\nPutting all these estimates together using Proposition \\ref{transport}, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{0,\\ub})}+C\\sum_{i\\leq 1}||(\\nabla^iF)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}+C\\delta||\\nabla^{i}\\phi'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 2}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iF||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\nTake $\\delta$ to be sufficiently small depending only on $C$ but not $a$, we can absorb the third term to the left hand side to get\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{0,\\ub})}+C\\sum_{i\\leq 1}||(\\nabla^iF)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 2}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iF||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\nThe proof for the $\\nabla_3$ equation is analogous.\n\\end{proof}\n\\subsection{Estimates for the Ricci Coefficient Difference}\\label{convsec4}\nThe results of the previous Subsection can now be used to estimate the norm ${\\mathcal O}'$ in terms of $\\mathcal R'$. \n\nIn this Subsection and the subsequent Section \\ref{convsec5}, in order to simplify notation, we will omit the superindices $(1)$ and $(2)$ in all the background quantities, for example, $\\nab_3, \\nab, \\Omega, \\beta...$\n\n\\begin{proposition}\\label{Riccip}\nThere exists $\\delta$ and $C$ depending only on the a priori estimates in Theorem \\ref{timeofexistence} (but independent of $a$) such that\n$$\\mathcal O'\\leq Ca+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'.$$\n\\end{proposition}\n\\begin{proof}\nWe control $\\mathcal O'$ using the null structure equations.\nFirst, we consider the Ricci coefficients $\\trchb, \\chibh, \\etab$. They all satisfy $\\nabla_3$ equations of the form\n$$\\nabla_3\\psi=\\psi\\psi+\\Psi$$\nsuch that $\\chih, \\omega$ do not appear in the $\\psi$ terms on the right hand side; and that $\\beta$ does not appear as $\\Psi$. This is important because $\\chih'$ and $\\omega'$ cannot be controlled by $C\\mathcal O'$ in the $L^2(S)$ norms and $\\beta'$ cannot be controlled by $C\\mathcal R'$ in $L^2(\\Hb)$. In order to estimate $\\psi$ using this equation, we need to estimate the curvature term:\n\\begin{equation*}\n\\begin{split}\n\\sum_{i\\leq 1}||(\\nabla^i\\Psi)'||_{L^1_uL^2(S)}\\leq C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')+C\\delta^{\\frac 12}\\mathcal R'.\n\\end{split}\n\\end{equation*}\nand the nonlinear terms:\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||(\\nabla^i(\\psi\\psi))'||_{L^1_u L^2(S_{u,\\ub})}\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')+C \\delta^{\\frac 12}\\sum_{i_1+i_2\\leq 1}\\sup_{u'\\leq u}||\\gamma'\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi||_{ L^2(S_{u',\\ub})}\\\\\n&+C\\delta^{\\frac 12}\\sum_{i_1+i_2\\leq 1}\\sup_{u'\\leq u}||\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi'||_{L^2(S_{u',\\ub})}\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation*}\nBy Proposition \\ref{transportp}, we thus have\n\\begin{equation}\\label{Ricci31}\n\\begin{split}\n&\\sum_{i\\leq 1}\\sup_{u}||\\nabla^i(\\trchb',\\chibh',\\etab')||_{L^2(S_{u,\\ub})}\\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^i(\\trchb',\\chibh',\\etab')||_{L^2(S_{0,\\ub})}+ C\\delta^{\\frac 12}\\sup_{u}(\\mathcal O''_{u}+\\tilde{\\mathcal O}'_{u}+\\mathcal O')+C\\delta^{\\frac 12}\\mathcal R'\\\\\n\\leq & a+ C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')+C\\delta^{\\frac 12}\\mathcal R'.\n\\end{split}\n\\end{equation}\nWe now consider the equation for $\\omega$, which schematically looks like\n$$\\nabla_3\\omega=\\omega\\psi+\\chih\\psi+\\psi\\psi+\\rhoc,$$\nwhere $\\psi\\neq \\chih,\\omega$ are the good components as above. The term $\\psi'\\omega$ or $\\psi'\\chih$ can be estimated since $\\psi'$ can be put in an appropriate $L^p(S)$ norm and be controlled by $C\\mathcal O'$. However, the most difficult terms are $\\omega'\\psi$ and $\\chih'\\psi$ since $\\omega'$ and $\\chih'$ cannot be bounded in $L^1_uL^p(S)$ by $C\\mathcal O'$. Thus, by Proposition \\ref{transportp}, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2(S_{u,\\ub})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2(S_{0,\\ub})}+ C\\sum_{i_1+i_2\\leq 1}||\\nabla^{i_1}\\psi\\nabla^{i_2}(\\chih',\\omega')||_{L^1_u L^p(S_{u,\\ub})}+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation*}\nSince this holds for every $\\ub$, we can integrate in $L^2$ in $\\ub$ to get\n\\begin{equation}\\label{Ricci32}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2_{\\ub}L^2(S_{0,\\ub})}+  \\int_0^u||\\nabla^{i_1}\\psi\\nabla^{i_2}(\\chih',\\omega')||_{L^2_{\\ub} L^2(S_{u',\\ub})} du'+C\\mathcal R'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')\\\\\n\\leq & a+C\\mathcal R'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O'),\n\\end{split}\n\\end{equation}\nsince we can control $\\nabla^i\\omega'$ by $C\\mathcal O'$ after integrating along the $\\ub$ direction.\nWe can estimate $\\chih$ in a similar manner as $\\omega$. $\\chih$ satisfies\n$$\\nabla_3\\chih=\\psi\\psi+\\psi\\chih+\\nabla\\eta,$$\nwhere $\\psi\\neq \\chih,\\omega$. Putting $\\nabla\\eta$ in the $\\tilde{\\mathcal O}'$ norm and using already obtained estimates, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2(S_{u,\\ub})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2(S_{0,\\ub})}+ C\\sum_{i_1+i_2\\leq 1}||\\nabla^{i_1}\\psi \\nabla^{i_2}\\chih'||_{L^1_u L^2(S_{u,\\ub})}+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation*}\nAs for $\\omega'$, we now integrate in $L^2$ in $\\ub$ to get\n\\begin{equation}\\label{Ricci33}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2_{\\ub}L^2(S_{0,\\ub})}+ C \\int_0^u||\\nabla^{i_1}\\psi \\nabla^{i_2}\\chih'||_{L^2_{\\ub} L^2(S_{u',\\ub})}du'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')\\\\\n\\leq &a+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation}\nIt remains to consider $\\trch,\\eta,\\omegab$. They satisfy $\\nabla_4$ equations of the form:\n$$\\nabla_4\\psi=\\psi\\psi+\\Psi,$$\nwhere $\\psi$ can be any Ricci coefficients and $\\Psi$ can be any curvature components $\\neq\\alpha,\\alphab$, i.e., all the $\\Psi$ terms can be controlled in $L^2(H)$ by $\\mathcal R'$. Moreover, since we are now integrating in $\\ub$, all terms $\\psi'$ can be estimated by $C\\mathcal O'$. By Proposition \\ref{transportp}, we have\n$$\\sum_{i\\leq 1}||\\nabla^i(\\trch',\\eta',\\omegab')||_{L^2(S_{u,\\ub})}\\leq \\sum_{i\\leq 1}||\\nabla^i(\\trch',\\eta',\\omegab')||_{L^2(S_{0,\\ub})}+ C\\mathcal R'+C\\int_0^{\\ub}\\mathcal O' d\\ub'.$$\nTherefore,\n$$\\sum_{i\\leq 1}||\\nabla^i(\\trch',\\eta',\\omegab')||_{L^2(S_{u,\\ub})}\\leq a+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}(\\sup_u\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').$$\nUsing $\\mathcal O''\\leq C(\\tilde{\\mathcal O}'+\\mathcal O')$ and all the above estimates, we have\n$$\\mathcal O'\\leq Ca+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal O').$$\nBy choosing $\\delta$ sufficiently small so that $C\\delta^{\\frac 12}\\leq \\frac 12$, we have\n$$\\mathcal O'\\leq Ca+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'.$$\nNotice that in particular $\\delta$ and $C$ depend only on the a priori estimates from Theorem \\ref{timeofexistence} and are independent of $a$.\n\\end{proof}\nWe now move on to estimate $\\tilde{\\mathcal O}'$ by $\\mathcal R'$. Recall from Propositions \\ref{Theta} and \\ref{ellipticTheta} that $\\tilde{\\mathcal O}$ was controlled using a combination of transport equations for $\\Theta$ and Hodge systems. The norm $\\tilde{\\mathcal O}'$ can be dealt with in a similar fashion. In order to perform this scheme, we need show that as was with the case of the norm $\\tilde{\\mathcal O}$ where we controlled $\\nab\\psi$ from $\\Theta$, the difference $\\psi'$ satisfies elliptic equations with $\\Theta'$ as a source. This is given by the following Proposition:\n\\begin{proposition}\\label{commuteelliptic}\nLet $\\phi$ be a $(0,r)$-tensorfield. Suppose \n$$\\div\\phi=f,\\quad\\curl\\phi=g,\\quad\\tr\\phi=h$$\non each of the spacetimes. Then\n$$\\div\\phi'\\sim f'+ (\\gamma^{-1})'\\nabla\\phi+\\Gamma'\\phi,$$\n$$\\curl\\phi'\\sim g'+ (\\eps)'\\nabla\\phi,$$\n$$\\tr\\phi' \\sim h' + (\\gamma^{-1})'\\phi.$$\n\\end{proposition}\n\\begin{proof}\nThis is a straightforward coordinate computation.\n\\end{proof}\nBased on the estimates we have on $\\gamma'$, $(\\gamma^{-1})'$ and $\\Gamma'$, and using the estimates in Proposition \\ref{ellipticthm}, we have\n\\begin{proposition}\\label{ellipticthmp}\nLet $\\phi^{(1)}$ and $\\phi^{(2)}$ be totally symmetric $r+1$ covariant tensorfields on 2-spheres $(\\mathbb S^2,\\gamma^{(1)})$, $(\\mathbb S^2,\\gamma^{(2)})$ respectively satisfying\n$$\\div\\phi^{(i)}=f^{(i)},\\quad \\curl\\phi^{(i)}=g^{(i)},\\quad \\mbox{tr}\\phi^{(i)}=h^{(i)},$$\nfor $i=1,2$. \nThen\n\\begin{equation*}\n\\begin{split}\n||\\nabla^{2}\\phi'||_{L^2(S)}\\leq &Ca\\sum_{i\\leq 1}(||\\nabla^{i}f||_{L^2(S)}+||\\nabla^{i}g||_{L^2(S)}+||\\nabla^{i}h||_{L^2(S)}+||\\phi||_{L^2(S)})\\\\\n&+C\\sum_{i\\leq 1}(||\\nabla^{i}f'||_{L^2(S)}+||\\nabla^{i}g'||_{L^2(S)}+||\\nabla^{i}h'||_{L^2(S)}+||\\phi'||_{L^2(S)})\n\\end{split}\n\\end{equation*}\n\n\\end{proposition}\n\\begin{proof}\nThis is a direct consequence of Propositions \\ref{commuteelliptic} and \\ref{ellipticthm}.\n\\end{proof}\n\n\n\nWe can now begin to estimate $\\tilde{\\mathcal O}'$. First, we define $\\Theta'$ and derive estimates for $\\Theta'$ using transport equations. Let $\\Theta'$ denote $(\\nabla\\trch)',(\\nabla\\trchb)',\\mu',\\mub',\\kappa',\\kappab'$. We have the following estimates:\n\\begin{proposition}\\label{Thetap}\n$$\\sum_{i\\leq 1}||\\nabla^i(\\nabla\\trch',\\nabla\\trchb',\\mu',\\mub')||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proposition}\n\\begin{proof}\nFrom the proof of Proposition \\ref{ellipticTheta}, we know that each of $\\nabla\\trch,\\nabla\\trchb,\\mu,\\mub$ satisfies either\n$$\\nab_3\\Theta=\\psi\\nab\\psi_{\\Hb}+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb},$$\nor\n$$\\nab_4\\Theta=\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_H,$$\nwhere as in Section \\ref{Ricciellipticsec}, we use the notation $\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}$, $\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}$, $\\Psi_H\\in\\{\\beta,\\rho,\\sigma,\\betab\\}$ and $\\Psi_{\\Hb}\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}$.\n\nWe focus on those $\\Theta$'s satisfying the $\\nab_4$ equation. The other case can be treated analogously. By Proposition \\ref{transportp}, we need to control\n$$\\sum_{i\\leq 1}||(\\nabla^i(\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_{H}))'||_{L^1_{\\ub}L^2(S_{u,\\ub})},$$\n$$\\sum_{i\\leq 1}||\\nabla^i(\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_{H})||_{L^2(S_{u,\\ub})},$$\nand \n$$\\sum_{i\\leq 1}||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})}.$$\nWe first estimate the $'$ terms. Among those we first look at the term with two derivatives on the Ricci coefficient. We have, using Proposition \\ref{Riccip}, \n$$||(\\psi\\nabla^2\\psi_H)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\leq C\\tilde{\\mathcal O}'+ C||\\psi'||_{L^1_{\\ub}L^\\infty(S_{u,\\ub})}+C\\delta^{\\frac 12}||\\nabla^2\\psi_H'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\leq C(a+\\tilde{\\mathcal O}'+\\delta^{\\frac 12}\\mathcal R').$$\nThe other term with Ricci coefficients can be estimated by $\\mathcal O'$ and thus by Proposition \\ref{Riccip}:\n$$\\sum_{i_1+i_2\\leq 1} ||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla\\psi)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\leq C(a+\\delta^{\\frac 12}\\mathcal R'+\\delta^{\\frac 12}\\tilde{\\mathcal O}').$$\nWe then move to the term with the curvature components:\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1} ||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi_{H})'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &C(a+\\delta^{\\frac 12}\\mathcal R'+\\delta^{\\frac 12}\\tilde{\\mathcal O}')+C\\delta^{\\frac 12}\\sum_{i\\leq 1}||\\nab^i\\Psi_{H}'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &C(a+\\delta^{\\frac 12}\\mathcal R'+\\delta^{\\frac 12}\\tilde{\\mathcal O}').\n\\end{split}\n\\end{equation*}\nThe terms without $'$ can be estimated using the a priori estimates in Theorem \\ref{timeofexistence}:\n$$\\sum_{i\\leq 1}(||\\nabla^i(\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_{H})||_{L^2(S_{u,\\ub})}+\n||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})})\\leq C.$$\nThus, by Proposition \\ref{transportp}, we have\n$$\\sum_{i\\leq 1}||\\nabla^i(\\nabla\\trch',\\nabla\\trchb',\\mu',\\mub')||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proof}\nWe now consider the difference quantities $\\kappa'$ and $\\kappab'$. As in Proposition \\ref{omegaelliptic}, they will only satisfy $L^2(H_u)$ and $L^2(\\Hb_{\\ub})$ estimates. As in the proof of Proposition \\ref{omegaelliptic}, in what follows, we allow $\\psi$ also to be $\\omega^\\dagger$ and $\\omegab^\\dagger$.\n\\begin{proposition}\\label{ellipticomegap}\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(H_u)}\\leq C\\delta^{\\frac 12}(a+\\tilde{\\mathcal O}'+\\mathcal R'),$$\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappab'||_{L^2(\\Hb_{\\ub})}\\leq C\\delta^{\\frac 12}(a+\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proposition}\n\\begin{proof}\nRecall equations (\\ref{kappaeqn}) and (\\ref{kappabeqn}):\n$$\\nab_3\\kappa=\\psi\\nab\\psi+\\psi\\psi\\psi+\\psi\\Psi,$$\nor\n$$\\nab_4\\kappab=\\psi\\nab\\psi+\\psi\\psi\\psi+\\psi\\Psi.$$\nWe focus on $\\kappa$. As $\\kappab$ is easier. The only terms that are new compared to the proof of Proposition \\ref{Thetap} \n$$\\sum_{i\\leq 1}||\\nab^i(\\psi\\beta)'||_{L^1_{u}L^2(S_{u,\\ub})}\\mbox{ and }\\sum_{i\\leq 2}||\\nab^i(\\chih',\\omega')||_{L^1_{u}L^2(S_{u,\\ub})}.$$\nThus using the estimates in the proof of Proposition \\ref{Thetap}, we have\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal R)+C\\delta^{\\frac 12}(\\sum_{i\\leq 1}||\\nab^i\\beta'||_{L^2_uL^2(S_{u,\\ub})}+\\sum_{i\\leq 2}||\\nab^i(\\chih',\\omega')||_{L^2_uL^2(S_{u,\\ub})}).$$\nIntegrating over $\\ub$ in $L^2$, we get\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(H_u)}\\\\\n\\leq &C\\delta^{\\frac 12}a+C\\delta(\\tilde{\\mathcal O}'+\\mathcal R')+C\\delta^{\\frac 12}(\\sum_{i\\leq 1}||\\nab^i\\beta'||_{L^2_{\\ub}L^2_uL^2(S_{u,\\ub})}+\\sum_{i\\leq 2}||\\nab^i(\\chih',\\omega')||_{L^2_{\\ub}L^2_uL^2(S_{u,\\ub})}).\n\\end{split}\n\\end{equation*}\nThus, using the definition of the norms $\\tilde{\\mathcal O}'$ and $\\mathcal R'$,\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(H_u)}\\leq C\\delta^{\\frac 12}a+C\\delta(\\tilde{\\mathcal O}'+\\mathcal R').$$\nSimilarly\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappab'||_{L^2(H_u)}\\leq C\\delta^{\\frac 12}a+C\\delta(\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proof}\nWe use Propositions \\ref{Thetap} and \\ref{ellipticomegap} and elliptic estimates to derive all the estimates for $\\nabla^{2}\\psi'$.\n\\begin{proposition}\\label{Ricciellipticp}\n$$\\tilde{\\mathcal O}'\\leq Ca+C\\mathcal R'.$$\n\\end{proposition}\n\\begin{proof}\nWe will first prove\n$$\\sup_u||\\nabla^{2}(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub}||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\mathcal R'.$$\nWe have the following div-curl systems:\n$$\\div(\\chih,\\eta,\\etab)=(\\nab\\trch,\\mu)+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\n$$\\curl(\\chih,\\eta,\\etab)=(\\nab\\trch,\\mu)+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\nand\n$$\\div(\\chibh,\\eta,\\etab)=(\\nab\\trchb,\\mub)+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab),$$\n$$\\curl(\\chibh,\\eta,\\etab)=(\\nab\\trchb,\\mub)+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab);$$\nas well as the div-curl system\n$$\\div(\\nab\\omega,\\nab\\omega^\\dagger)=\\nab\\kappa+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\n$$\\curl(\\nab\\omega,\\nab\\omega^\\dagger)=\\nab\\kappa+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\nand\n$$\\div(\\nab\\omegab,\\nab\\omegab^\\dagger)=\\nab\\kappab+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab),$$\n$$\\curl(\\nab\\omegab,\\nab\\omegab^\\dagger)=\\nab\\kappab+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab).$$\nBy Proposition \\ref{ellipticthmp}, we need to control the terms involving $\\Theta$\n$$\\sup_{u}\\sum_{i\\leq 1}||\\nabla^i(\\nab\\trch',\\mu',\\kappa')||_{L^2(H_u)}+\\sup_{\\ub}\\sum_{i\\leq 1}||\\nabla^i(\\nab\\trchb',\\mub',\\kappab')||_{L^2(\\Hb_{\\ub})},$$\nthe curvature terms\n$$\\sup_u\\sum_{i\\leq 1}||\\nabla^i(\\beta',\\rho',\\sigma')||_{L^2(H_u)}+\\sup_{\\ub}||\\nabla^i(\\rhoc',\\sigmac',\\betab')||_{L^2(\\Hb_{\\ub})}$$\nand the lower order terms involving the Ricci coefficients\n$$\\sum_{i\\leq 1}||\\nabla^i(\\psi'\\psi)||_{L^2(H_u)}+\\sum_{i\\leq 1}\\sum_{\\psi\\neq \\chih,\\omega}||\\nabla^i(\\psi'\\psi)||_{L^2(\\Hb_{\\ub})}.$$\nThe terms involving $\\Theta$ are controlled by the estimates in Proposition \\ref{Thetap} by\n$$C\\delta^{\\frac 12}(a+\\tilde{\\mathcal O}'+\\mathcal R').$$\nThe curvature terms can be estimated using the definition of $\\mathcal R'$ by\n$$C\\mathcal R'.$$\nThe remaining terms can be bounded by\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i(\\psi'\\psi)||_{L^2(H_u)}\\\\\n\\leq &C(||\\psi'||_{L^\\infty(S_{u,\\ub})}\\sum_{i\\leq 1}||\\nabla^{i}\\psi||_{L^2(S_{u,\\ub})}+\\sum_{i\\leq 1}||\\nabla^{i}\\psi'||_{L^2(S_{u,\\ub})}||\\nabla^{i_2}\\psi||_{L^\\infty(S_{u,\\ub})})\\\\\n\\leq &Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\delta^{\\frac 12}\\mathcal R'.\n\\end{split}\n\\end{equation*}\nThus we have\n$$\\sup_u||\\nabla^{2}(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub}||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{ub})}\\leq Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\mathcal R'.$$\nTherefore, together with the estimates for $\\nab^2\\trch'$ and $\\nab^2\\trchb'$ in Proposition \\ref{Thetap}, we have\n$$\\tilde{\\mathcal O}\\leq Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\mathcal R'.$$\nBy choosing $\\delta$ sufficiently small depending on $C$ so that $C\\delta^{\\frac 12}$, we have\n$$\\tilde{\\mathcal O}\\leq Ca+C\\mathcal R'.$$\nNotice that $C$ and $\\delta$ depend only on the a priori estimates in Theorem \\ref{timeofexistence} and are both independent of $a$.\n\\end{proof}\n\\subsection{Estimates for the Curvature Difference}\\label{convsec5}\nIn order to finish the proof of Theorem \\ref{convergencethm}, we need to estimate $\\mathcal R'$ by $Ca$. This will be proved using an energy-type estimate. As in the bounds for the curvature components themselves, we will derive the energy estimates directly from the Bianchi equations without using the Bel Robinson tensor. This method is especially useful in controlling the difference of the curvature components since these difference quantities do not arise from a Weyl field with respect to either of the background spacetime metrics.\n\\begin{proposition}\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} (\\nabla^i\\Psi')^2+\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\underline{H}_{\\underline{u}}} (\\nabla^i\\Psi')^2\\right)\\\\\n\\leq& Ca^2+C\\delta(\\mathcal R')^2+C||(K\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n&+C\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2+\\sum_{i_1+i_2+i_3\\leq 2}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nThe energy estimates for the curvature difference follows from the difference Bianchi equations. Once these are derived, the energy estimates will be obtained as in Section \\ref{estimates} by integration by parts in appropriate subsets of the difference Bianchi system. In the proof below, we concentrate on the most difficult case which involves renormalized curvature components. The estimates for the other components can be derived in a similar fashion.\n\nRecall that\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta+tr\\chib\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma),\\\\\n&\\nabla_4\\sigma+\\frac 32tr\\chi\\sigma=-\\div^*\\beta+\\frac 12\\chibh\\cdot ^*\\alpha-\\zeta\\cdot^*\\beta-2\\etab\\cdot\n^*\\beta,\\\\\n&\\nabla_4\\rho+\\frac 32tr\\chi\\rho=\\div\\beta-\\frac 12\\chibh\\cdot\\alpha+\\zeta\\cdot\\beta+2\\etab\\cdot\\beta,\\\\\n\\end{split}\n\\end{equation*}\nAs before, we renormalize the equations. Define\n$$\\sigmac:=\\sigma+\\frac{1}{2}\\chibh\\wedge\\chih,$$\n$$\\rhoc:=\\rho-\\frac{1}{2}\\chibh\\cdot\\chih.$$\nSchematically, we have\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta=\\nabla\\rhoc+^*\\nabla\\sigmac +\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,\\\\\n&\\nabla_4\\sigmac=-\\div^*\\beta+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,\\\\\n&\\nabla_4\\rhoc=\\div\\beta+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi.\n\\end{split}\n\\end{equation*}\nFrom these we can derive equations for $\\beta'$, $\\sigma'$ and $\\rhoc'$. By Proposition \\ref{peqn}, \n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta'=(\\nabla_3\\beta)'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_3\\beta+\\Omega^{-1}(b^A)'\\nabla_{\\frac{\\partial}{\\partial\\th^A}}\\beta+\\Omega^{-1}(\\gamma^{-1}\\chi)'\\beta+\\Omega^{-1}b\\Gamma'\\beta,\\\\\n&\\nabla_4\\sigmac=(\\nabla_4\\sigmac)'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_4\\sigmac+\\Omega^{-1}(\\gamma^{-1}\\chi)'\\sigmac,\\\\\n&\\nabla_4\\rhoc=(\\nabla_4\\rhoc)'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_4\\rhoc+\\Omega^{-1}(\\gamma^{-1}\\chi)'\\rhoc.\n\\end{split}\n\\end{equation*}\nMoreover, we can write schematically\n\\begin{equation*}\n\\begin{split}\n&(\\nab(\\rhoc,\\sigmac))'=\\nabla(\\rhoc',\\sigmac')+\\Gamma'(\\rhoc,\\sigmac)\\\\\n&(\\div\\beta)'=\\div\\beta'+\\gamma'\\nabla\\beta+\\Gamma'\\beta.\n\\end{split}\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta'=\\nabla\\rhoc'+^*\\nabla\\sigmac' +(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'+F_{\\beta},\\\\\n&\\nabla_4\\sigmac'=-\\div^*\\beta'+(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'+F_{\\sigmac},\\\\\n&\\nabla_4\\rhoc'=\\div\\beta'+(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'+F_{\\rhoc},\n\\end{split}\n\\end{equation*}\nwhere $F_{\\beta}$, $F_{\\sigmac}$ and $F_{\\rhoc}$ satisfy the bound\n$$\\sum_{i\\leq 1}||\\nab^iF||_{L^2_uL^2_{\\ub}L^2(S)}\\leq C\\delta^{\\frac 12}(a+\\mathcal R').$$\nTherefore, multiplying the first equation by $\\beta'$, integrating by parts and using the other equations, we have\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\beta'|^2+\\int_{\\Hb_{\\ub}} (|\\rhoc'|^2+|\\sigmac'|^2)\\\\\n\\leq& Ca^2 +||(\\beta',\\rhoc',\\sigma')(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'||_{L^1_u L^1_{\\ub}L^1(S)}+||(\\beta',\\rhoc',\\sigma')F_{\\beta,\\rhoc,\\sigmac}||_{L^1_u L^1_{\\ub}L^1(S)}\\\\\n\\leq &Ca^2 +||(\\beta',\\rhoc',\\sigma')||_{L^2_u L^2_{\\ub}L^2(S)}+||(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2++||F||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n\\leq &Ca^2 +C\\delta(\\mathcal R')^2+||(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2.\n\\end{split}\n\\end{equation*}\nWe now look at the equation the first covariant angular derivative for the \ncurvature components:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nabla\\beta=\\nabla^2\\rhoc+^*\\nabla^2\\sigmac+K(\\rhoc,\\sigmac) +\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+\\psi^2\\nabla\\psi+\\psi^4,\\\\\n&\\nabla_4\\nabla\\sigmac=-\\div^*\\nabla\\beta+K\\beta+\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+\\psi^2\\nabla\\psi+\\psi^4,\\\\\n&\\nabla_4\\nabla\\rhoc=\\div\\nabla\\beta+K\\beta+\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+\\psi^2\\nabla\\psi+\\psi^4.\n\\end{split}\n\\end{equation*}\nA similar argument as before involving writing the equations for $\\nabla(\\beta',\\rhoc',\\sigmac')$ and integrating by parts yield\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla\\beta'|^2+\\int_{\\Hb_{\\ub}} (|\\nabla\\rhoc'|^2+|\\nabla\\sigmac'|^2)\\\\\n\\leq &Ca^2 +C\\delta(\\mathcal R')^2\\\\\n&+||(K\\Psi+\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+(\\nabla\\psi)^2+\\psi^2\\nabla\\psi+\\psi^4)'||_{L^2_u L^2_{\\ub}L^2(S)}^2.\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nFrom this one can conclude\n\\begin{proposition}\\label{curvaturep}\n\\begin{equation*}\n\\begin{split}\n\\mathcal R'\\leq Ca.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nBy the previous Proposition, we need to estimate\n$$||(K\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2,$$\n$$\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2,$$\n$$\\sum_{i_1+i_2+i_3\\leq 2}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2$$\nBy Proposition \\ref{Kp} and the definition of the norm $\\mathcal R'$ and the a priori estimates from Theorem \\ref{timeofexistence}, we have\n$$||(K\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\leq ||K'\\Psi||_{L^2_u L^2_{\\ub}L^2(S)}^2+||K \\Psi'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\leq C\\delta^2(\\mathcal R')^2.$$\nTo estimate the remaining terms, we will repeatedly invoke Proposition \\ref{angularpL4} to exchange $(\\nabla\\Psi)'$ and $\\nabla(\\Psi')$ (or $(\\nabla\\psi)'$ and $\\nabla(\\psi')$ etc.), at the expense of an error term $\\delta^2(\\mathcal R')^2$.\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n\\leq &C\\delta^2(\\mathcal R')^2+C(\\sum_{i_1\\leq 1}\\sum_{i_2\\leq 2}||\\nabla^{i_1}\\psi||_{L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)}^{i_2})^2(\\sum_{i_3\\leq 1}||\\nabla^{i_3}\\Psi'||_{L^2_u L^2_{\\ub}L^2(S)}^2)\\\\\n&+C(\\sum_{i_1\\leq 1}||\\nabla^{i_1}\\psi||_{L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)}^2)(\\sum_{i_2\\leq 1}||\\nabla^{i_3}\\Psi||_{L^2_u L^2_{\\ub}L^4(S)}^2)(\\sum_{i_3\\leq 1}||\\nabla^{i_1}\\psi'||_{L^\\infty_u L^\\infty_{\\ub}L^4(S)}^2)\\\\\n\\leq &C\\delta(a^2+(\\mathcal R')^2),\n\\end{split}\n\\end{equation*}\nwhere the second term is estimated by Theorem \\ref{timeofexistence} and the definition of $\\mathcal R'$; and the last term is estimated by Theorem \\ref{timeofexistence} and Proposition \\ref{Riccip}. Finally, we estimate using Theorem \\ref{timeofexistence} and Proposition \\ref{Riccip}\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n\\leq &C\\delta^2(\\mathcal R')^2+C(\\sum_{i_1\\leq 1}\\sum_{i_2\\leq 2}||\\nabla^{i_1}\\psi||_{L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)}^{i_2})^2(\\sum_{i_3\\leq 2}||\\nabla^{i_3}\\psi'||_{L^2_u L^2_{\\ub}L^2(S)}^2)\\\\\n\\leq &C\\delta(a^2+(\\mathcal R')^2),\n\\end{split}\n\\end{equation*}\nPutting these together, we have\n$$(\\mathcal R')^2\\leq Ca^2+C\\delta(\\mathcal R')^2.$$\nThe conclusion follows by choosing $\\delta$ sufficiently small depending only on $C$.\n\\end{proof}\nThis implies all the quantities that we have estimated in this section can be estimated by $Ca$:\n\\begin{proposition}\\label{p}\n$$||g'||_{L^\\infty_u L^\\infty_{\\ub} L^\\infty(S)},||\\frac{\\partial}{\\partial\\th}g'||_{L^\\infty_u L^\\infty_{\\ub} L^4(S)},||\\frac{\\partial^2}{\\partial\\th^2}g'||_{L^\\infty_u L^\\infty_{\\ub} L^2(S)},\\mathcal O',\\tilde{\\mathcal O'},\\mathcal R'\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nThis is a direct consequence of Propositions \\ref{Omegap}, \\ref{gammap}, \\ref{dgammap}, \\ref{dOmegap}, \\ref{ddgammap}, \\ref{ddOmegap}, \\ref{bp}, \\ref{Riccip}, \\ref{Thetap}, \\ref{Ricciellipticp} and \\ref{curvaturep}.\n\\end{proof}\nThis concludes the proof of Theorem \\ref{convergencethm}.\n\n\\subsection{Additional Estimates on the Difference of the Metrics}\\label{AddEstMetric}\n\nIn this Subsection, we derive some additional estimates on the difference of the metrics. They follow directly from the bounds already derived. This will be used in Section \\ref{limit} to show that the limiting spacetime metric is in the space asserted in Theorem \\ref{convergencethm2} and satisfies the Einstein equations.\n\n\\begin{proposition}\\label{ugp}\n$$||\\frac{\\partial}{\\partial u}(\\gamma',\\Omega',b')||_{L^4(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\n$\\gamma$ satisfies a transport equation in the $\\Lb$ direction:\n$$\\Ls_{\\Lb}\\gamma=2\\Omega\\chib.$$\nFrom this we can derive an equation for $\\gamma'$:\n$$\\frac{\\partial}{\\partial u}\\gamma'=-b^A\\frac{\\partial}{\\partial\\th^A}\\gamma'-(b^A)'\\frac{\\partial}{\\partial\\th^A}\\gamma+2(\\Omega\\chi)'.$$\nEstimating the right hand side using Proposition \\ref{p}, we have\n$$||\\frac{\\partial}{\\partial u}\\gamma'||_{L^4(S_{u,\\ub})}\\leq Ca.$$\n$\\Omega$ also satisfies a transport equation in the $\\Lb$ direction:\n$$\\omegab=-\\frac 12\\nabla_3\\log\\Omega=\\frac 12\\Omega\\nab_3\\Omega^{-1}=\\frac 12(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial\\th^A})\\Omega^{-1}.$$\nAs for $\\gamma'$, we can derive an equation for $\\frac{\\partial}{\\partial u}(\\Omega^{-1})'$. By direct estimates using Proposition \\ref{p}, we have\n$$||\\frac{\\partial}{\\partial u}\\Omega'||_{L^4(S_{u,\\ub})}\\leq Ca.$$\nFinally, we move to $\\frac{\\partial}{\\partial u}b'$. $b$ does not satisfy a transport equation. We therefore resort to the equation\n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A.$$\nApplying $\\frac{\\partial}{\\partial u}$, we get\n$$\\frac{\\partial^2 b^A}{\\partial \\ub\\partial u}=-4\\frac{\\partial}{\\partial u}(\\Omega^2\\zeta^A).$$\nSince $(b')^A=0$ on $\\Hb_0$, $\\frac{\\partial b'}{\\partial u}=0$ on $\\Hb_0$.\nThus\n$$||\\frac{\\partial}{\\partial u}b'||_{L^4(S_{u,\\ub})}\\leq C\\int_0^{\\ub} ||\\frac{\\partial}{\\partial u}(\\Omega^2\\zeta^A)'||_{L^4(S_{u,\\ub'})}d\\ub'.$$\nBy Proposition \\ref{p}, the right hand side $\\leq Ca$.\n\\end{proof}\n\n\\begin{proposition}\\label{u2gp}\n$$||\\frac{\\partial^2}{\\partial u^2}(\\gamma',\\Omega',b')||_{L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nWe prove this Proposition by taking an extra $\\frac{\\partial}{\\partial u}$ derivative of the equations in the proof of Proposition \\ref{ugp}. By\n$$\\Ls_{\\Lb}\\gamma=2\\Omega\\chib,$$\nwe have\n$$\\frac{\\partial^2}{\\partial u^2}\\gamma=2\\frac{\\partial}{\\partial u}(\\Omega\\chib)-\\frac{\\partial}{\\partial u}(b^A\\frac{\\partial}{\\partial\\th^A}\\gamma).$$\nNotice that \n$$\\Omega\\frac{\\partial}{\\partial u}\\chib=\\nab_3\\chib-\\chib\\chib.$$\nThus using the null structure equation for $\\nab_3\\chib$, and estimating directly, we get\n$$||\\frac{\\partial^2}{\\partial u^2}\\gamma'||_{L^2(S_{u,\\ub})}\\leq Ca.$$\nTo estimate $\\frac{\\partial^2}{\\partial u^2}\\Omega$, we use\n$$\\omegab=\\frac 12(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial\\th^A})\\Omega^{-1}.$$\nDifferentiating with respect to $\\frac{\\partial}{\\partial u}$, using the estimates for $\\frac{\\partial}{\\partial u}b^A$ and the null structure equation for $\\nab_3\\omega$, we get\n$$||\\frac{\\partial^2}{\\partial u^2}\\Omega'||_{L^2(S_{u,\\ub})}\\leq Ca.$$\nFinally, differentiating \n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A$$\ntwice in the $u$ direction, we get\n$$\\frac{\\partial^3 b^A}{\\partial \\ub\\partial u^2}=-4\\frac{\\partial^2}{\\partial u^2}(\\Omega^2\\zeta^A).$$\nUsing the null structure equations as well as the Bianchi equation for $\\nab_3\\betab$, we see that all the terms on the right hand side can be bounded in $L^2(S_{u,\\ub})$. Thus we can integrate to get\n$$||\\frac{\\partial^2}{\\partial u^2}b'||_{L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proof}\nIt is also possible to prove estimates for the mixed second derivatives\n\\begin{proposition}\\label{utgp}\n$$||\\frac{\\partial^2}{\\partial u\\partial\\th^A}(\\gamma',\\Omega',b')||_{L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nThis can be proved in a similar way as Proposition \\ref{ugp}, taking $\\frac{\\partial}{\\partial\\th^A}$ instead of $\\frac{\\partial}{\\partial u}$ derivative.\n\\end{proof}\nOn the other hand, $\\ub$ derivatives can only be taken once and can only be estimated after taking the $L^{p_0}_{\\ub}$ norm:\n\\begin{proposition}\\label{ubgp}\n$$||\\frac{\\partial}{\\partial \\ub}(\\gamma',\\Omega',b')||_{L^{p_0}_{\\ub}L^\\infty(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nWe can directly estimate the right hand side of the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma'=2(\\Omega\\chi)',$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}(\\Omega^{-1})'=\\omega',$$\n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)'.$$\n\\end{proof}\nThe mixed $\\ub$ and $\\th$ derivatives can also be estimates:\n\\begin{proposition}\\label{ubtgp}\n$$||\\frac{\\partial^2}{\\partial \\ub \\partial \\th^A}(\\gamma',\\Omega',b')||_{L^{p_0}_{\\ub}L^4(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\\label{ubugp}\n\\begin{proof}\nTake $\\frac{\\partial}{\\partial \\th^A}$ derivatives of the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma'=2(\\Omega\\chi)',$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}(\\Omega^{-1})'=\\omega',$$\n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)',$$\nand apply the estimates in Proposition \\ref{p}.\n\\end{proof}\nSimilarly for the mixed $\\ub$ and $u$ derivatives:\n\\begin{proposition}\n$$||\\frac{\\partial^2}{\\partial \\ub \\partial u}(\\gamma',\\Omega',b')||_{L^{p_0}_{\\ub}L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nTake $\\frac{\\partial}{\\partial u}$ derivatives of the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma'=2(\\Omega\\chi)',$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}(\\Omega^{-1})'=\\omega',$$\n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)',$$\nand use the null structure equations.\n\\end{proof}\nWe do not have estimates for the second $\\ub$ derivatives of the metric, since they can only be understood as measures even for the initial data. However, the following substitute is useful in showing that the limit spacetime that we construct satisfies the Einstein equations, in particular, $R_{u\\ub}=R_{A\\ub}=R_{\\ub\\ub}=0$.\n\\begin{proposition}\\label{trchp}\n$$||\\frac{\\partial^2}{\\partial \\ub^2}(b^A)'||_{L^{p_0}_{\\ub}L^\\infty(S_{u,\\ub})}\\leq Ca$$\nand\n$$||\\frac{\\partial}{\\partial \\ub}((\\gamma^{-1})^{AB}\\frac{\\partial}{\\partial \\ub}\\gamma_{AB})'||_{L^{p_0}_{\\ub}L^4(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nThe first estimate follows directly from \n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)'$$\nand the null structure equations.\n\nFor the second estimate, notice that in coordinates,\n$$\\trch=(\\gamma^{-1})^{AB}\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}.$$\nThus we want to estimate $\\frac{\\partial}{\\partial\\ub}\\trch'$. Using the null structure equation\n$$\\nab_4\\trch+\\frac 12(\\trch)^2=-|\\chih|^2-2\\omega\\trch,$$\nwe have\n$$\\frac{\\partial}{\\partial\\ub}\\trch'=(-\\frac \\Omega 2(\\trch)^2-\\Omega|\\chih|^2-2\\Omega\\omega\\trch)'.$$\nThe conclusion follows from estimating the right hand side directly using Proposition \\ref{p}.\n\\end{proof}\n\n\n\\subsection{The Limiting Spacetime Metric}\\label{limit}\n\nIn this Subsection, we prove Theorems \\ref{convergencethm2} and \\ref{Einstein}. We therefore assume that the conditions of Theorem \\ref{convergencethm2} hold. Consider the sequence of metric\n$$g_n=-2(\\Omega_n)^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma_n)_{AB}(d\\th^A-(b_n)^Adu)\\otimes (d\\th^B-(b_n)^Bdu).$$ \nBy Proposition \\ref{p}, $\\gamma_n$, $b_n$ and $\\Omega_n$ converge uniformly to their limiting values $\\gamma_{\\infty}$, $b_\\infty$ and $\\Omega_\\infty$ and are therefore continuous functions of $(u,\\ub,\\th^1,\\th^2)$ and define a continuous limiting spacetime metric\n\\begin{equation}\\label{limitmetric}\ng_\\infty=-2(\\Omega_\\infty)^2(du\\otimes d\\ub+d\\ub\\otimes du)+ (\\gamma_{\\infty})_{AB}(d\\th^A-(b_\\infty)^Adu)\\otimes(d\\th^B-(b_\\infty)^Bdu).\n\\end{equation}\nTo complete the proof of Theorem \\ref{convergencethm2}, we need to demonstrate the desired regularity of the limiting spacetime. This follows easily from the estimates in the previous Subsections.\n\\begin{proposition}\n$$(\\frac{\\partial}{\\partial \\th}g_n,\\frac{\\partial}{\\partial u}g_n)\\mbox{ converge to }(\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty)\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^4(S),$$\n\\begin{equation*}\n\\begin{split}\n(\\frac{\\partial^2}{\\partial \\th^2}g_n,\\frac{\\partial^2}{\\partial u\\partial\\th}g_n,\\frac{\\partial^2}{\\partial u^2}g_n)\\mbox{ converge to }\\\\\n(\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_{\\infty})\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^2(S),\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n\\begin{split}\n(\\frac{\\partial}{\\partial \\ub}g_n, \\frac{\\partial}{\\partial\\ub}((\\gamma_n^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_n)_{AB}))\\mbox{ converge to }\\\\\n(\\frac{\\partial}{\\partial \\ub}g_{\\infty}, \\frac{\\partial}{\\partial\\ub}((\\gamma_{\\infty}^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_{\\infty})_{AB}))\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^\\infty(S),\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n\\begin{split}\n(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_n,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_n,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_n)\\mbox{ converge to }\\\\\n(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_{\\infty})\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^4(S).\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nThat\n$$\\frac{\\partial}{\\partial \\th}g_n,\\frac{\\partial}{\\partial u}g_n\\mbox{ converge in }L^\\infty_u L^\\infty_{\\ub} L^4(S)$$\nfollows from Propositions \\ref{p} and \\ref{ugp} respectively.\n\nThat\n$$\\frac{\\partial^2}{\\partial \\th^2}g_n,\\frac{\\partial^2}{\\partial u^2}g_n,\\frac{\\partial^2}{\\partial u\\partial\\th}g_n\\mbox{ converge in } L^\\infty_u L^\\infty_{\\ub} L^2(S)$$\nfollows from Propositions \\ref{p}, \\ref{u2gp} and \\ref{utgp} respectively.\n\nThat\n$$\\frac{\\partial}{\\partial \\ub}g_n, \\frac{\\partial}{\\partial\\ub}((\\gamma_n^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_n)_{AB})\\mbox{ converge in }L^\\infty_u L^{p_0}_{\\ub} L^\\infty(S)$$\nfollows from Propositions \\ref{ubgp} and \\ref{trchp} respectively.\n\nThat\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_n,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_n,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_n\\mbox{ converge in }L^\\infty_u L^{p_0}_{\\ub} L^4(S)$$\nfollows from Propositions \\ref{utgp}, \\ref{ubugp} and \\ref{trchp} respectively.\n\\end{proof}\n\\begin{proposition}\\label{gspace}\n$$\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty\\in C^0_u C^0_{\\ub} L^4(S),$$\n$$\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_\\infty\\in C^0_u C^0_{\\ub} L^2(S),$$\n$$\\frac{\\partial}{\\partial \\ub}g_\\infty, \\frac{\\partial}{\\partial\\ub}((\\gamma_\\infty^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_\\infty)_{AB}) \\in L^\\infty_u L^\\infty_{\\ub} L^\\infty(S),$$\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_{\\infty}\\in L^\\infty_u L^\\infty_{\\ub} L^4(S).$$\n\\end{proposition}\n\\begin{proof}\nTo prove the first two statements, notice that since $g_n$ are smooth, the convergence in $L^\\infty_u L^\\infty_{\\ub}L^p(S)$ implies that the limit is in $C^0_u C^0_{\\ub} L^p(S)$.\n\nFor the latter two statements, we use the fact that for $p_0<\\infty$, if $f_n\\to f$ in $L^{p_0}$ and $f_n$ is uniformly bounded in $L^\\infty$, then $f\\in L^\\infty$.\n\\end{proof}\n\n\nProposition \\ref{gspace} allows us to conclude that all first derivatives of $g_{\\infty}$ can be defined and belong to an appropriate space. These norms also allow us to conclude that the product of any two first derivatives of the metric belongs to $L^\\infty_u L^\\infty_{\\ub}L^2(S)$. For the second derivatives, Proposition \\ref{gspace} shows that all second derivatives of the metric except $\\frac{\\partial^2}{\\partial \\ub^2}(\\gamma,\\Omega)$ are defined as functions in $L^\\infty L^{p_0}_{\\ub}L^2(S)$. The second derivative $\\frac{\\partial^2}{\\partial \\ub^2}(\\gamma,\\Omega)$ is merely a distribution. As we will see below, its definition is not necessary to make sense of the Einstein equations. \n\nWe now show that all components of the curvature for the limiting spacetime, except for $R_{\\ub A\\ub B}$, $R_{\\ub u \\ub A}$ and $R_{\\ub u \\ub u}$, can be defined at worst as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. For this we use the coordinate definition of the Riemann curvature tensor:\n\\begin{equation}\\label{curvdef}\nR_{\\delta\\sigma\\mu\\nu}=g_{\\delta\\rho}(\\frac{\\partial}{\\partial x^{\\mu}}\\Gamma^\\rho_{\\nu\\sigma}-\\frac{\\partial}{\\partial x^{\\nu}}\\Gamma^\\rho_{\\mu\\sigma}+\\Gamma^{\\rho}_{\\mu\\lambda}\\Gamma^\\lambda_{\\nu\\sigma}-\\Gamma^\\rho_{\\nu\\lambda}\\Gamma^\\lambda_{\\mu\\sigma}).\n\\end{equation}\n\\begin{proposition}\nIn the limiting spacetime, all components of the Riemann curvature tensor, defined by the coordinate expression (\\ref{curvdef}) expect for $R_{\\ub A\\ub B}$, $R_{\\ub u \\ub A}$ and $R_{\\ub u \\ub u}$, can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Moreover, $R(L,\\Lb,L,e_A)$ and $R(L,\\Lb,L,\\Lb)$ can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$.\n\\end{proposition}\n\\begin{proof}\nUsing (\\ref{curvdef}), it is easy to see that if $\\delta,\\sigma,\\mu,\\nu\\neq \\ub$, the expression has at most one $\\ub$ derivative. Thus, by the regularity properties of $g_\\infty$, these components of the Riemann curvature tensor are well-defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions. \n\nIn the case that exactly one of $\\delta,\\sigma,\\mu,\\nu$ is equal to $\\ub$, we can assume, by the symmetry properties of the Riemann curvature tensor, that $\\delta=\\ub$. Then, the formula for the component of the Riemann curvature tensor has at most one $\\ub$ derivative and by the regularity properties, it is well-defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions. Notice, however, that strictly speaking, at the level of regularity that we have, the symmetry properties of the Riemann curvature tensor may not hold. Consider, for example, the component $R_{12\\ub 1}$. Using the formula above, we would have a term $g_{1A}(g^{A\\ub}g_{12,\\ub})_{,\\ub}$, which is not defined. However, the sum of \\emph{all} terms containing two $\\ub$ derivatives is equal to the expression\n$$g_{1\\alpha}\\frac{\\partial}{\\partial \\ub}(g^{\\ub\\alpha}\\frac{\\partial}{\\partial \\ub}g_{12}).$$\nwhich can be rearranged, up to terms with at most one $\\ub$ derivative, in the form\n$$\\frac{\\partial}{\\partial \\ub}(g_{1\\alpha}g^{\\ub\\alpha}\\frac{\\partial}{\\partial \\ub}g_{12}).$$\nThis expression vanishes since $g_{1\\alpha}g^{\\ub\\alpha}=0$. The above calculation is a reflection of the symmetry properties of the Riemann curvature tensor and provides an appropriate distributional definition of $R_{\\sigma\\delta\\mu\\nu}$ for which the symmetry properties formally hold.\n\nWe have therefore shown that all components of curvature with the exception of $R_{\\ub A\\ub B}$, $R_{\\ub u \\ub A}$ and $R_{\\ub u \\ub u}$ are defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions. However, replacing the double null coordinate system $(u,\\ub,\\th^1,\\th^2)$ by the null frame $(L,\\Lb,e^1,e^2)$, we can show that additional components of curvature are defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. To see this, we can write\n$$R(L,\\Lb,L,e_A)=R_{\\ub u \\ub A}+ b^B R_{\\ub B\\ub A},$$\nand\n$$R(L,\\Lb,L,\\Lb)=R_{\\ub u \\ub u}+ b^B R_{\\ub u \\ub B}+b^A R_{\\ub A \\ub u}+b^A b^B R_{\\ub A\\ub B}=R_{\\ub u \\ub u}+2b^A R_{\\ub A \\ub u}+b^A b^B R_{\\ub A\\ub B}.$$\nNote that\n$$R_{\\ub u \\ub u}=-\\frac 12 g_{uu,\\ub\\ub}+\\mbox{good terms}=-\\frac 12 b^A b^B (\\frac{\\partial}{\\partial \\ub})^2 g_{AB}+\\mbox{good terms},$$\n$$R_{\\ub u \\ub A}=-\\frac 12 g_{Au,\\ub\\ub}+\\mbox{good terms}=\\frac 12 b^B (\\frac{\\partial}{\\partial \\ub})^2 g_{AB}+\\mbox{good terms},$$\n$$R_{\\ub B\\ub A}=-\\frac 12 g_{AB,\\ub\\ub}+\\mbox{good terms},$$\nwhere good terms denote terms with at most one $\\ub$ derivative. Therefore, the terms with two $\\ub$ derivatives cancel in $R(L,\\Lb,L,A)$ and $R(L,\\Lb,L,\\Lb)$ and they can be defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions.\n\\end{proof}\nWe now verify that the limiting spacetime that we have constructed satisfies the vacuum Einstein equations in the sense that relative to the system of double null coordinates $(u,\\ub,\\th^1,\\th^2)$,\n$$R_{\\mu\\nu}=0\\quad\\mbox{in }L^\\infty_uL^\\infty_{\\ub}L^2(S).$$\nFor this, we use the coordinate definition of the Ricci curvature:\n\\begin{equation*}\nR_{\\mu\\nu}=\\frac{\\partial}{\\partial x^{\\rho}}\\Gamma^\\rho_{\\mu\\nu}-\\frac{\\partial}{\\partial x^{\\nu}}\\Gamma^\\rho_{\\rho\\mu}+\\Gamma^\\rho_{\\rho\\lambda}\\Gamma^{\\lambda}_{\\mu\\nu}-\\Gamma^\\rho_{\\mu\\lambda}\\Gamma^{\\lambda}_{\\rho\\nu}.\n\\end{equation*}\n\\begin{proposition}\nThe limiting spacetime satisfies the vacuum Einstein equations in the sense that relative to the system of double null coordinates $(u,\\ub,\\th^1,\\th^2)$,\n$$R_{\\mu\\nu}=0\\quad\\mbox{in }L^\\infty_uL^\\infty_{\\ub}L^2(S).$$.\n\\end{proposition}\n\\begin{proof}\nNow, notice that with the metric given by (\\ref{limitmetric}), we have\n$$(g_{\\infty}^{-1})^{\\ub\\ub}=(g_{\\infty}^{-1})^{uu}=(g_{\\infty}^{-1})^{u1}=(g_{\\infty}^{-1})^{u2}=0.$$\nTherefore,\n$\\Gamma^{\\ub}_{\\mu\\nu}$ does not contain a term with $g_{\\mu\\nu,\\ub}$. In other words, the expressions for \n$$R_{uu}, R_{uA}, R_{AB}\\quad\\mbox{for $A,B=1,2$}$$\ndo not contain two $\\ub$ derivatives. Therefore, we have \n$$R_{uu}= R_{uA}= R_{AB}= 0$$\nas functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. For the component $R_{u\\ub}$, there are terms involving two $\\ub$ derivatives of the metric. In particular, we have\n\\begin{equation*}\n\\begin{split}\nR_{u\\ub}=&\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{u\\ub}-\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub u}-\\frac{\\partial}{\\partial\\ub}\\Gamma^A_{Au}-\\frac{\\partial}{\\partial\\ub}\\Gamma^u_{uu}\\\\\n&+\\mbox{terms involving at most one $\\ub$ derivative of the metric}\\\\\n=&\\frac{-1}{4\\Omega^2}\\gamma_{AB}b^A\\frac{\\partial^2}{\\partial\\ub^2} b^B+\\mbox{terms involving at most one $\\ub$ derivative of the metric}.\n\\end{split}\n\\end{equation*}\nFrom this expression, we see that except for the second $\\ub$ derivatives of $b$, the terms involving two $\\ub$ derivatives of the metric cancel. Recall from Proposition \\ref{gspace} that $\\frac{\\partial^2}{\\partial\\ub^2} b^A$ is in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Thus we also have\n$$R_{u\\ub}=0$$\nas functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Similarly, for $R_{A\\ub}$, we have\n\\begin{equation*}\n\\begin{split}\nR_{A\\ub}=&\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{A\\ub}-\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub A}-\\frac{\\partial}{\\partial\\ub}\\Gamma^B_{BA}-\\frac{\\partial}{\\partial\\ub}\\Gamma^u_{uA}\\\\\n&+\\mbox{terms involving at most one $\\ub$ derivative of the metric}\\\\\n=&\\frac{1}{4\\Omega^2}\\gamma_{AB}\\frac{\\partial^2}{\\partial\\ub^2} b^B+\\mbox{terms involving at most one $\\ub$ derivative of the metric}.\n\\end{split}\n\\end{equation*}\nAs for $R_{u\\ub}$, because of the cancellation, we have\n$$R_{\\ub A}=0$$\nas functions in $L^\\infty_{u}L^\\infty_{\\ub}L^2(S)$.\n\nIt now remains to study $R_{\\ub\\ub}$. Indeed this term involves two $\\ub$ derivatives of the metric and there are no cancellations to remove this term. We have\n\\begin{equation*}\n\\begin{split}\nR_{\\ub\\ub}=&\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub\\ub}-\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub\\ub}+\\frac{\\partial}{\\partial \\th^A}\\Gamma^{A}_{\\ub\\ub}+\\frac{\\partial}{\\partial \\ub}\\Gamma^A_{A\\ub}\\\\\n&+\\mbox{terms involving at most one $\\ub$ derivative of the metric}.\n\\end{split}\n\\end{equation*}\nThe first two terms cancel each other. Note that\n$$\\Gamma^A_{\\ub\\ub}=0$$\nsince\n$$g_{\\ub\\ub}=g_{\\ub B}=0,\\quad g^{Au}=0.$$\nWe are thus left with\n$$\\frac{\\partial}{\\partial \\ub}\\Gamma^A_{A\\ub}=\\frac{\\partial}{\\partial \\ub}(g^{AB}\\frac{\\partial}{\\partial \\ub}g_{AB}).$$\nIt might seem that this term behaves like $\\frac{\\partial^2}{\\partial \\ub^2}\\gamma$ which we cannot control. However, this particular combination of derivatives, by Proposition \\ref{gspace}, is in fact in $L^\\infty_uL^\\infty_{\\ub}L^4(S)$. Thus $R_{\\ub\\ub}=0$ in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$.\n\\end{proof}\n\n\\subsection{Uniqueness}\\label{uniquenesssec}\n\nIn the Subsection, we show that the spacetime constructed in Theorem \\ref{convergencethm2} is the unique spacetime among the class of continuous spacetimes which arise as $C^0$ limits of smooth spacetime solutions to the vacuum Einstein equations. In view of Theorem \\ref{convergencethm}, uniqueness follows easily if the spacetime solution is assumed to be in the class of spacetimes satisfying the conclusions of \\ref{convergencethm}. Nevertheless, we prove uniqueness within a larger class of spacetimes for which it is not a priori assumed that the spacetime is Lipschitz, or that it is more regular in the $u$ and the angular directions.\n\nThe following is a precise formulation of the uniqueness theorem (Theorem \\ref{uniquenessthm}) and is the main result in this Subsection:\n\\begin{proposition}\\label{uniquenessprop}\nLet $(\\mathcal M^{(1)}, g^{(1)})$ and $(\\mathcal M^{(2)}, g^{(2)})$ be two $C^0$ Lorentzian spacetimes in double null coordinates, defined in $0\\leq u\\leq u_*, 0\\leq \\ub\\leq \\ub_*$ and taking the form\n$$g^{(i)}=-2(\\Omega^{(i)})^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)})_{AB}(d\\th^A-(b^{(i)})^Adu)\\otimes (d\\th^B-(b^{(i)})^Bdu)$$ \nsuch that\n\\begin{itemize}\n\\item $\\Omega=1$ and $b^A=0$ on $H_0$ and $\\Hb_0$ and $\\gamma^{(1)}|_{H_0}=\\gamma^{(2)}|_{H_0}$ and $\\gamma^{(1)}|_{\\Hb_0}=\\gamma^{(2)}|_{\\Hb_0}$\n\\item for $i=1,2$, there exists a sequence of smooth spacetimes $(\\mathcal M_n^{(i)}, g^{(i)}_n)$ such that for every $n$, $g^{(i)}_n$ is defined in $0\\leq u\\leq u_*$ and $0\\leq \\ub\\leq \\ub_*$ and takes the following form in double null coordinates:\n$$g^{(i)}_n=-2(\\Omega^{(i)}_n)^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)}_n)_{AB}(d\\th^A-(b^{(i)}_n)^Adu)\\otimes (d\\th^B-(b^{(i)}_N)^Bdu)$$ \nwith the property that\n\\begin{enumerate}\n\\item $$g^{(i)}_n \\to g^{(i)} \\quad\\mbox{in }C^0\\mbox{ in }0\\leq u\\leq u_*, 0\\leq \\ub\\leq \\ub_*,$$\n\n\\item The initial data for $g^{(i)}_n$ satisfy the assumptions of Theorem \\ref{timeofexistence} uniformly,\n\n\\item The initial data for $g^{(i)}_n$ converges to the initial data for $g^{(i)}$, i.e.,\n$$g^{(i)}_n|_{H_0} \\to g^{(i)}|_{H_0} \\quad\\mbox{and }g^{(i)}_n|_{\\Hb_0} \\to g^{(i)}|_{\\Hb_0}$$\nin the norms in the assumptions of Theorem \\ref{convergencethm}.\n\\end{enumerate}\n\\end{itemize}\nThen, if $u_*, \\ub_*\\leq \\epsilon$, where $\\epsilon$ is as given in Theorem \\ref{timeofexistence},\n$$g^{(1)}=g^{(2)}\\quad\\mbox{in    }0\\leq u\\leq u_*, 0\\leq \\ub\\leq \\ub_* .$$\n\\end{proposition}\n\\begin{proof}\nBy Property $(3)$ in the assumptions, for every $i$, there exists $n_i$ such that \n$$\\sup_u|(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB})_{n_i}'(\\ub=0)|+\\sup_{\\ub}||(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB})_{n_i}'(u=0)|\\leq 2^{-i},$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}(\\sum_{i\\leq 1} \\sup_{u}||\\nabla^i\\psi_{n_i}'||_{L^2(S_{u,0})}+\\sum_{i\\leq 1} \\sup_{\\ub}||\\nabla^i\\psi_{n_i}'||_{L^2(S_{0,\\ub})})\\leq 2^{-i},$$\n$$\\sum_{i\\leq 1} ||\\nabla^i(\\chih_{n_i}',\\omega_{n_i}')||_{L^{p_0}_{\\ub}L^2(S_{0,\\ub})}\\leq 2^{-i}\\quad\\mbox{for some fixed }2\\leq p_0<\\infty,$$\n$$||\\nabla^2(\\chih_{n_i}',\\omega_{n_i}',\\eta_{n_i}',\\etab_{n_i}')||_{L^2(H_0)}+\\sup_{\\ub} ||\\nabla^2(\\chibh_{n_i}',\\omegab_{n_i}',\\eta_{n_i}',\\etab_{n_i}')||_{L^2(\\Hb_{0})}\\leq 2^{-i},$$\n$$\\sup_{u} ||\\nabla^2(\\trch_{n_i}',\\trchb_{n_i}')||_{L^2(S_{u,0})}+\\sup_{\\ub} ||\\nabla^2(\\trch_{n_i}',\\trchb_{n_i}')||_{L^2(S_{0,\\ub})}\\leq 2^{-i},$$\n$$\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sum_{i\\leq 1} ||\\nabla^i\\Psi_{n_i}'||_{L^{2}_{\\ub}L^2(S_{0,\\ub})}\\leq 2^{-i}.$$\nBy Theorem \\ref{convergencethm}, there exists $C$ depending only on the uniform bounds in the assumption $(2)$ such that\n$$|g^{(1)}_{n_i}-g^{(2)}_{n_i}|\\leq C2^{-i}.$$\nTherefore, in $C^0$ norm,\n$$|g^{(1)}-g^{(2)}|\\leq C|g^{(1)}_{n_i}-g^{(2)}_{n_i}|\\leq C2^{-i}.$$\nSince this holds for every $i$, we have\n$$g^{(1)}=g^{(2)}.$$\n\\end{proof}\n\n\\subsection{Propagation of Regularity}\\label{regularityp}\nUsing Proposition \\ref{propagationregularity}, we show that if the initial data is more regular in the $\\nab_3$ and $\\nab$ directions, then so is the limiting spacetime.\n\\begin{proposition}\\label{regularityprop}\nSuppose, in addition to the assumptions of Theorem \\ref{convergencethm}, we have the following estimates for the initial data:\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j \\nab^i\\Psi||_{L^2(\\Hb_0)}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^i \\nab^j\\Psi||_{L^2(H_0)})\\leq C. $$\nThen\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_{\\ub}||\\nab_3^j \\nab^i\\Psi||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_u||\\nab_3^i \\nab^j\\Psi||_{L^2(H_u)})\\leq C'.$$\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{propagationregularity}, for an approximating sequence of spacetimes, the following bound holds independent of $n$:\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_{\\ub}||\\nab_3^j \\nab^i\\Psi_n||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_u||\\nab_3^i \\nab^j\\Psi_n||_{L^2(H_u)})\\leq C'.$$\nThe conclusion thus follows.\n\\end{proof}\nUsing the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma=2\\Omega\\chi,$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}=\\omega,$$\n$$\\frac{\\partial}{\\partial \\ub}b^A=-4\\Omega^2\\zeta^A,$$\ntogether with Proposition \\ref{regularityprop}, we can show that $g$ is in the desired space as indicated in Theorem \\ref{regularitythm}. The details are straightforward and will be omitted.\n\n\\subsection{The Limiting Spacetime for an Impulsive Gravitational Wave}\\label{limitgiw}\nIn the Subsection, we show that for the case of an impulsive gravitational wave, the spacetime is more regular than a general spacetime with merely bounded $\\chih$, thus concluding the proof of Theorem \\ref{giwthmv1}. In particular, we show that $\\alpha_\\infty$ is well defined as a measure which is the weak limit of $\\alpha_n$. All other curvature components can be defined in $L^\\infty$. Moreover, the singularity of $\\alpha_\\infty$ is supported on a null hypersurface and the constructed spacetime is smooth away from this null hypersurface.\n\n\\subsubsection{Limit of the Curvature Component $\\alpha$}\n\nWe first show that $\\alpha_n$ has a well defined limit as a measure.\n\\begin{proposition}\\label{alphalimit}\nIn the case of an impulsive gravitational wave, for every $u$ and every $\\vartheta=(\\th^1,\\th^2)\\in\\mathbb S^2$, $\\alpha_\\infty(u,\\vartheta)$ is a measure on $[0,\\ub_*]$, defined as the limit of $\\alpha_n(u,\\vartheta)$\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{totalvariation}, for every $u$ and $\\vartheta$, $\\alpha_n$ has uniformly bounded $L^1_{\\ub}$ norms. Thus to show convergence of $\\alpha_n$, we only need to show that for every $u$ and every $\\vartheta=(\\th^1,\\th^2)\\in\\mathbb S^2$, \n$$\\int_0^{\\ub} \\alpha_n(u,\\ub',\\vartheta) d\\ub' $$\nconverges to a limit $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ as $n\\to\\infty$ for all $\\ub$ such that $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ is continuous.\nBy the equation\n$$\\nab_4\\chih+\\trch\\chih=-2\\omega\\chih-\\alpha,$$\nwe have\n$$\\int_0^{\\ub} \\alpha_n(u,\\ub',\\vartheta) d\\ub'=\\int_0^{\\ub} (\\Omega^{-1}_n\\frac{\\partial}{\\partial\\ub}\\chih_n+\\trch_n\\chih_n+2\\omega_n\\chih_n)(u,\\ub',\\vartheta) d\\ub'.$$\nIntegrating by parts and using \n$$\\frac{\\partial}{\\partial\\ub}\\Omega^{-1}_n=2\\omega_n,$$\nwe derive\n\\begin{equation}\\label{intalpha}\n\\int_0^{\\ub} \\alpha_n(u,\\vartheta) d\\ub=(\\Omega^{-1}_n\\chih_n)(u,\\ub,\\vartheta)-(\\Omega^{-1}_n\\chih_n)(u,\\ub=0,\\vartheta)+\\int_0^{\\ub} (\\trch_n\\chih_n)(u,\\ub',\\vartheta) d\\ub'.\n\\end{equation}\nBy Proposition \\ref{p}, \n$$\\int_0^{\\ub} (\\trch_n\\chih_n)(u,\\ub',\\vartheta) d\\ub'$$\nconverges. Thus, in order to show weak convergence of $\\alpha_n$, we need to show pointwise convergence for $\\Omega^{-1}_n\\chih_n$ for all $\\ub$ such that $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ is continuous. By the construction of the data for an impulsive gravitational wave in Section \\ref{initialcondition}, we know that for the initial data\n$$(\\chih'_n)_{AB}(u=0,\\ub,\\vartheta)\\leq C2^{-n}\\quad\\mbox{for }\\ub\\leq\\ub_s\\mbox{ or }\\ub\\geq\\ub_s+2^{-n}.$$\nUsing the equation\n$$\\nabla_3\\chih+\\frac 12\\trchb\\chih=\\nabla\\hot\\eta+2\\omegab\\chih-\\frac 12\\trch\\chibh+\\eta\\hot\\eta,$$\none sees that \n$$(\\chih'_n)_{AB}(u,\\ub,\\vartheta)\\leq C2^{-n}\\quad\\mbox{for }\\ub\\leq\\ub_s\\mbox{ or }\\ub\\geq\\ub_s+2^{-n}.$$\nTherefore, $\\chih_n$ converges for all $\\ub\\neq\\ub_s$. By Proposition \\ref{convergencethm2}, $\\Omega^{-1}_n\\chih_n$ also converges for all $\\ub\\neq\\ub_s$.\n\nIt remains to show that the limit $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ is discontinuous at $\\ub=\\ub_s$. In view of (\\ref{intalpha}), it suffices to show that $\\chih$ has a jump discontinuity across $\\Hb_{\\ub_s}$. This follows from the equation\n$$\\nabla_3\\chih+\\frac 12\\trchb\\chih=\\nabla\\hot\\eta+2\\omegab\\chih-\\frac 12\\trch\\chibh+\\eta\\hot\\eta.$$\nRewriting in coordinates and expressing $\\chih$, $\\chibh$, $\\eta$, $\\etab$ in terms of the coordinate vector fields $\\frac{\\partial}{\\partial\\th^1}$ and $\\frac{\\partial}{\\partial\\th^2}$ on the spheres, we have\n\\begin{equation*}\n\\begin{split}\n&\\Omega^{-1}(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial\\th^A})\\chih-\\gamma^{-1}\\chib\\chih+\\frac 1 2 \\trchb \\chih\\\\\n=&\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta.\n\\end{split}\n\\end{equation*}\nConsider the coordinate system $(u,\\ub,\\tilde{\\th}^1,\\tilde{\\th}^2)$ such that \n$$\\frac{d}{d\\ub}\\tilde{\\th}^A(\\ub;u,\\th)=b^A(u,\\ub,\\tilde\\th^1,\\tilde\\th^2),$$\nwith initial data\n$$\\tilde{\\th}^A(0;u,\\th)=\\th^A.$$\nAccording to the proven estimates for $b$, this change of variable is $W^{2,\\infty}$. Therefore, the vector fields $\\frac{\\partial}{\\partial\\tilde{\\th}^1}$ and $\\frac{\\partial}{\\partial\\tilde{\\th}^2}$ associated to the new coordinate system are $W^{1,\\infty}$ with respect to the differentiable structure given by the original coordinate system $(u,\\ub,\\th^1,\\th^2)$. In the new coordinate system, we can rewrite the transport equation for $\\chih$ as\n$$\\Omega^{-1}\\frac{\\partial}{\\partial u}\\chih-\\gamma^{-1}\\chib\\chih+\\frac 1 2 \\trchb \\chih\n=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta,$$\nwhere $\\chih$, $\\chibh$, $\\eta$, $\\etab$ are now expressed in terms of the new coordinate vector fields $\\frac{\\partial}{\\partial\\tilde{\\th}^1}$ and $\\frac{\\partial}{\\partial\\tilde{\\th}^2}$.\nNotice that $\\chib$ is continuous and the expression\n$$\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta$$\nis also continuous. Since for the initial data, $\\chih(\\tilde{u}_0,\\ub,\\theta)$ has a jump discontinuity for $\\ub=\\ub_s$, $\\chih_{\\tilde{A}\\tilde{B}}$ also has a jump discontinuity across $\\ub=\\ub_s$. As noted before, the the vector fields $\\frac{\\partial}{\\partial\\tilde{\\th}^1}$ and $\\frac{\\partial}{\\partial\\tilde{\\th}^2}$ associated to the new coordinate system are $W^{1,\\infty}$ with respect to the differentiable structure given by the original coordinate system $(u,\\ub,\\th^1,\\th^2)$. Therefore, $\\chih_{AB}$ also has a jump discontinuity.\n\\end{proof}\n\n\\subsubsection{Control of the Curvature Components $\\beta,\\rho,\\sigma,\\betab,\\alphab$}\n\nExcept for the curvature component $\\alpha$ which can only be defined as a measure, all other curvature components can be defined as $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$ functions:\n\n\\begin{proposition}\n$\\beta,\\rho,\\sigma,\\betab,\\alphab$ can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$.\n\\end{proposition}\n\\begin{proof}\nThat $\\rho,\\sigma,\\betab,\\alphab$ can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$ follows from Proposition \\ref{regularityprop} and the Sobolev Embedding Theorem. To show that $\\beta$ can be defined as a $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$ function, we consider the following Bianchi equation:\n$$\\nabla_3\\beta=\\sum_{\\Psi\\in\\{\\rho,\\sigma\\}}\\nabla\\Psi+\\psi\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\Psi.$$\nSince $\\nab\\rho, \\psi, \\rho, \\sigma, \\betab$ can be bounded in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$, it follows from Proposition \\ref{transport} and Gronwall's inequality that $\\beta$ can be estimated in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$, as desired.\n\\end{proof}\n\n\n\\subsubsection{Smoothness of Spacetime away from $\\Hb_{\\ub_s}$}\n\nWe now show that the spacetime is smooth away from the null hypersurface $\\ub=\\ub_s$. In the region $\\ub<\\ub_s$ before the impulse, this follows from standard theory of local-wellposedness. We focus on the region after the impulse, where $\\ub>\\ub_s$. In the following Proposition, we prove uniform estimates for the $\\nab_4$ and $\\nab$ derivatives of $\\alpha_\\infty$ in the region $\\ub>\\ub_s$. The bounds for all derivatives of the curvature components follow from a combination of the estimates on $\\alpha_\\infty$, Theorem \\ref{regularitythm} and the Bianchi equations.\n\\begin{proposition}\\label{giwsmoothness}\nSuppose the data for the impulsive gravitational wave satisfy the estimates as given in Proposition \\ref{dataprop} for some $I\\geq 2$, $K\\geq 0$. Then\n$$\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}} \\limsup_{\\tilde{\\ub}\\to \\ub_s+}||\\nab_4^k\\nab^i\\alpha_{\\infty}||_{L^2(H_u(\\tilde{\\ub},\\ub_*))}\\leq C_{I,K}'.$$\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{alphaapriori},\n\\begin{equation}\\label{alphaaprioricon}\n\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}\\sup_{\\ub}||\\nab_4^k\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\epsilon))}\\leq C_{I,K}'.\n\\end{equation}\nFix $\\delta>0$. Take $n$ large enough such that $2^{-n}< \\delta$. Then (\\ref{alphaaprioricon}) implies that \n$$\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}\\sup_{\\ub}||\\nab_4^k\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+\\delta,\\epsilon))}\\leq C_{I,K}'.$$\nThe conclusion follows from the fact that this bound is uniform in $\\delta$.\n\\end{proof}\nAs a consequence, we show\n\\begin{proposition}\nThe limiting spacetime is smooth away from the the null hypersurface $\\Hb_{\\ub_s}$.\n\\end{proposition}\n\n\\begin{remark}\nIn Proposition \\ref{giwsmoothness}, we have only used the fact that for $\\ub>\\ub_s$, the initial data set is smooth. Therefore, suppose we have a initial data set satisfying the assumptions of Theorem \\ref{rdthmv2}, with the additional assumption that it is smooth for $\\ub>\\tilde{\\ub}$ for some $\\tilde{\\ub}$, we can also prove that the spacetime is smooth in $0\\leq u\\leq \\epsilon$, $\\tilde{\\ub} <\\ub\\leq\\epsilon$.\n\\end{remark}\n\n\\subsubsection{Decomposition of $\\alpha_\\infty$ into Singular and Regular Parts}\n\nFinally, we show that $\\alpha_\\infty$ has a delta singularity on the incoming null hypersurface $\\Hb_{\\ub_s}$.\n\\begin{proposition}\n$\\alpha_{\\infty}$ can be decomposed as\n$$\\alpha_{\\infty}=\\delta(\\ub_s)\\alpha_s+\\alpha_r,$$\nwhere $\\delta(\\ub_s)$ is the scalar delta function supported on the null hypersurface $\\Hb_{\\ub_s}$, $\\alpha_s=\\alpha_s(u,\\vartheta)\\neq 0$ belongs to $L^\\infty_uL^\\infty(S)$ and $\\alpha_r$ belongs to $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$.\n\\end{proposition}\n\\begin{proof}\nDefine \n$$\\alpha_s(u,\\vartheta):=\\lim_{\\ub\\to\\ub_s^+} \\Omega^{-1}\\chih (u,\\ub,\\vartheta)-\\lim_{\\ub\\to\\ub_s^-} \\Omega^{-1}\\chih (u,\\ub,\\vartheta),$$\nand \n$$\\alpha_r:=\\alpha_{\\infty}-\\delta(\\ub_s)\\alpha_s.$$\nWe now show that $\\alpha_s$ and $\\alpha_r$ have the desired property. By Theorem \\ref{convergencethm2}, $\\alpha_s$ belongs to $L^\\infty_uL^\\infty(S)$. That $\\alpha_s\\neq 0$ follows from the fact that $\\chih$ has a jump discontinuity across $\\ub=\\ub_s$, which is proved in Proposition \\ref{alphalimit}.\n\nIt remains to show that $\\alpha_r$ belongs to $L^\\infty_u L^\\infty_{\\ub} L^\\infty(S)$. To show this, we consider the measure of the half open interval $[0,\\ub)$ using the measure $\\alpha_r(u,\\vartheta)$:\n\\begin{equation*}\n\\begin{split}\n&(\\alpha_r(u,\\vartheta))([0,\\ub))\\\\\n=&(\\Omega^{-1}\\chih)(u,\\ub,\\vartheta)-\\lim_{\\tilde{\\ub}\\to\\ub_s^+}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub},\\vartheta)+\\lim_{\\tilde{\\ub}\\to\\ub_s^-}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub},\\vartheta)-(\\Omega^{-1}\\chih)(u,\\ub=0,\\vartheta)\\\\\n&+\\int_0^{\\ub} (\\trch\\chih)(u,\\tilde{\\ub},\\vartheta) d\\tilde{\\ub}\\\\\n=&\\lim_{\\tilde{\\ub}\\to\\ub_s^-}\\int_0^{\\tilde{\\ub}} \\frac{\\partial}{\\partial\\ub}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub}',\\vartheta)d\\tilde{\\ub}'+\\lim_{\\tilde{\\ub}\\to\\ub_s^+}\\int_{\\tilde{\\ub}}^{\\ub} \\frac{\\partial}{\\partial\\ub}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub}',\\vartheta)d\\tilde{\\ub}'\\\\\n&+\\int_0^{\\ub} (\\trch\\chih)(u,\\tilde{\\ub},\\vartheta) d\\tilde{\\ub}.\n\\end{split}\n\\end{equation*}\nBy Proposition \\ref{giwsmoothness}, $\\frac{\\partial}{\\partial\\ub}(\\Omega^{-1}\\chih)(u,\\ub,\\vartheta)$ is a bounded function away from the the hypersurface $\\Hb_{\\ub_s}$. Thus $(\\alpha_r(u,\\vartheta))([0,\\ub))$ can be expressed as an integral over $[0,\\ub)$ of an $L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)$ function, as desired.\n\\end{proof}\n\n\n\n\n\\bibliographystyle{hplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet an evolution equation $u_t=f[u]$ admits a symmetry $u_\\tau=g[t,u]$ (that is, vector fields $\\partial_t$ and $\\partial_\\tau$ commute), then the stationary equation $g[t,u]=0$ defines a constraint which is consistent with the dynamics in $t$. Moreover, if $\\partial_\\tau$ belongs to a commutative Lie subalgebra of the higher symmetries, then the stationary equation inherits this subalgebra. It is well known that many important classes of exact solutions satisfy such stationary equations, including the finite-gap, multisoliton and rational solutions. On the other hand, if $\\partial_\\tau$ contains some members of the additional noncommutative Lie subalgebra of symmetries then the constraint leads to a Painlev\\'e type equation. Such solutions are also considered in the literature for a long time, but, for the understandable reasons, they are studied much worse.\n\nIn this paper we study the solutions of the Volterra lattice $u_{n,t}=u_n(u_{n+1}-u_{n-1})$ which satisfy the stationary equation for the master-symmetry (plus lower order terms). An analogous, more simple constraint was studied in papers \\cite{Its_Kitaev_Fokas_1990,Fokas_Its_Kitaev_1991}, where the evolution in $n$ was governed by the discrete Painlev\\'e equation dP$_1$ and the evolution in $t$ was governed by the Painlev\\'e equation P$_4$. In our case, the respective equations are dP$_{34}$ and P$_5$ (or P$_3$, for degenerating parameters). The corresponding set of solutions is not so small and contains a family of solutions depending on three essential parameters, which are regular for all $n,t$. \n\nSection \\ref{s:constriants} contains the definition of the constraint under study, lowering of its order and reduction to the Painlev\\'e equations. In Section \\ref{s:regular}, we define a subclass of regular solutions. It is characterized by the special choice of initial data at the fixed singular point $t=0$ and by certain restrictions on the values of parameters, ensuring the absence of poles at $t\\ne0$. These solutions describe small-scale oscillations in a region that grows linearly with increase of $t$. Such behavior is typical for generic solutions outside the soliton sector with the initial data in the form of sharp spikes. The constraint which we study is of interest as an example of exact solution (in terms of the Painlev\\'e transcendents) for this mode. However, so far these solutions are investigated only numerically.\n\nSection \\ref{s:halfline} deals with the case when, in addition to the constraint equations, the condition $u_0=0$ is satisfied. Then the lattice equations admit a reduction on the half-line $n>0$, and the equations P$_5$ and P$_3$ are reduced, respectively, to the confluent hypergeometric equation and the Bessel equation. Section \\ref{s:asymp} contains asymptotic formulas for the regular solutions related to the hypergeometric equation.\n\nThe class of solutions in study contains, in particular, the solution with the most simple initial data in the form of the unit step $u_0=0$, $u_n(0)=1$ for $n>0$. It was constructed in our previous paper \\cite{Adler_Shabat_2018} by comparing of the Wronskian representation for the Volterra lattice solution \\cite{Leznov_1980} with the well known result from the combinatorics about the Hankel transform for the Catalan numbers \\cite{Aigner_1999, Stanley_1999, Layman_2001}. Now we are able to reverse this construction and to apply the constraint equations for computing of the Hankel transform. In Section \\ref{s:det}, this is done for the coefficients of the Taylor expansion of the Kummer function.\n\n\\section{Constraints for the Volterra lattice}\\label{s:constriants}\n\nThe Volterra lattice\n\\begin{equation}\\label{ut}\n u_{n,t}=u_n(u_{n+1}-u_{n-1})\n\\end{equation}\npossesses the symmetries\n\\begin{align*}\n u_{n,t_2}&=u_n(h_{n+1}-h_{n-1}),\\quad h_n:=u_n(u_{n+1}+u_n+u_{n-1}),\\\\\n u_{n,\\tau_0} &= tu_n(u_{n+1}-u_{n-1})+u_n,\\\\\n u_{n,\\tau_1} &= tu_n(h_{n+1}-h_{n-1})\n  +u_n\\bigl(\\bigl(n+\\tfrac{3}{2}\\bigr)u_{n+1}+u_n-\\bigl(n-\\tfrac{3}{2}\\bigr)u_{n-1}\\bigr).\n\\end{align*}\nThe flow $\\partial_{\\tau_0}$ corresponds to the scaling transformation and $\\partial_{\\tau_1}$ is the master-symmetry which generates the commutative Lie subalgebra of the symmetries by the formula $\\partial_{t_{k+1}}=[\\partial_{\\tau_1},\\partial_{t_k}]$, starting from $\\partial_{t_1}=\\partial_t$ \\cite{Cherdantsev_Yamilov_1995, Adler_Shabat_Yamilov_2000}. Both sequences $\\partial_{t_k}$ and $\\partial_{\\tau_k}$ are infinite (and the flows $\\partial_{\\tau_k}$ are non-local for $k>1$), but in this paper we will need only the above members of the whole hierarchy of the symmetries.\n\nThe stationary equation for any linear combination of the symmetries is a constraint compatible with (\\ref{ut}). The equations which correspond to the commutative symmetries only, bring to the algebro-geometric (in particular, multisoliton) solutions. The simplest example involving a noncommutative symmetry is given by equation\n\\[\n u_{n,t_2}+2u_{n,\\tau_0}=0\n\\]  \n(the coefficient at the second term is fixed by scaling and the term $u_{n,t}$ can be neglected, due to the shift $t\\to t-\\mathop{\\rm const}$). After dividing by $u_n$, the 5-point difference equation appears\n\\[\n h_{n+1}-h_{n-1} +2t(u_{n+1}-u_{n-1})+2=0.\n\\]\nIt can be easily reduced to the 3-point constraint\n\\begin{equation}\\label{IKF}\n u_n(u_{n+1}+u_n+u_{n-1})+2tu_n+n+(-1)^nb+c=0,\n\\end{equation}\nmoreover, a straightforward computation proves that it is consistent with (\\ref{ut}) if and only if the integration constants $b$ and $c$ do not depend on $t$. This equation, known as the discrete Painlev\\'e equation dP$_1$, turns the lattice equations (\\ref{ut}) into a coupled system for the variables $u_{n-1},u_n$ which is equivalent to the continuous Painlev\\'e equation P$_4$ for the function $y=u_n$  \\cite{Its_Kitaev_Fokas_1990, Fokas_Its_Kitaev_1991, Grammaticos_Ramani_1998, Grammaticos_Ramani_2014}:\n\\begin{equation}\\label{P4}\n y''=\\frac{(y')^2}{2y}+\\frac{3}{2}y^3+4ty^2+2(t^2-\\alpha)y+\\frac{\\beta}{2y},\n\\end{equation}\n\\[\n \\alpha=\\frac{1}{2}(n-3(-1)^nb+c),\\quad \\beta=-(n+(-1)^nb+c)^2.\n\\]\nThe mapping $(u_{n-1},u_n)\\mapsto(u_n,u_{n+1})$ defines one of the B\\\"acklund transformations for (\\ref{P4}).\n\nIn this paper, our main goal will be to investigate another, more complicated case\n\\[\n u_{n,\\tau_1} -4au_{n,\\tau_0} -du_{n,t}=0.\n\\]\nHere, the coefficient $a$ can be scaled either to 0 or to 1, and the shift of $t$ makes possible to remove the term $u_{n,t_2}$. Like in the previous example, this 5-point constraint can be reduced to a 3-point one, although it is less obvious in this case. First, we notice that the equation takes the following form, after dividing by $u_n$:\n\\[\n \\widetilde G_{n+1}+\\widetilde G_n=0 \\quad\\Rightarrow\\quad G_n=\\widetilde G_n+4(-1)^nb=0, \n\\]\nwhere we denote\n\\begin{equation}\\label{Gn}\n G_n=(q_{n+2}+q_{n+1})u_{n+1}-(q_n+q_{n-1})u_n-4a(q_{n+1}-q_n)+4(-1)^nb=0\n\\end{equation}\nand\n\\begin{equation}\\label{qn}\n q_n=2tu_n+n-d.\n\\end{equation}\nNext, we lower the order by use of the integrating factor:\n\\[\n (q_{n+1}+q_n)G_n=\\widetilde F_{n+1}-\\widetilde F_n=0 \\quad\\Rightarrow\\quad \n F_n=\\widetilde F_n-4c=0,\n\\]\nwhere $F_n$ is the left hand side of equation (\\ref{unn}) below.  \n\n\\begin{statement}\nThe lattice (\\ref{ut}) is consistent with the equation\n\\begin{equation}\\label{unn}\n F_n=(q_{n+1}+q_n)(q_n+q_{n-1})u_n-4(aq^2_n+(-1)^nbq_n+c)=0,\n\\end{equation}\nwhere $q_n=2tu_n+n-d$, for any constants $a,b,c,d$ and under assumption that $q_{n+1}+q_n\\ne0$ at some inital moment $t=t^*$, for all $n$.\n\\end{statement}\n\\begin{proof}\nA straightforward computation proves the identities\n\\begin{align*}\n G_{n,t}&=u_{n+1}(G_{n+1}+G_n)-u_n(G_n+G_{n-1}),\\\\ \n F_{n,t}&= u_n(q_n+q_{n-1})G_n+u_n(q_{n+1}+q_n)G_{n-1},\n\\end{align*}\nassuming, like for the constraint (\\ref{IKF}), that the integration constants $b$ and $c$ do not depend on $t$. Let equation $F_n=0$ is satisfied for $t=t^*$, then also $G_n=(F_{n+1}-F_n)\/(q_{n+1}+q_n)=0$ for $t=t^*$. Then it follows from the above identities that $G_n=F_n=0$ for all $t$ such that the solution $u_n(t)$ exists.\n\\end{proof}\n\n\\begin{remark}\\label{rem:confinement} \nThe stipulation that $q_{n+1}+q_n$ do not vanish at $t=t^*$ is not very essential. We need it only in order to provide $G_n(t^*)=0$. However, in the situation when $q_{n+1}+q_n=0$ for some $n$, we can directly require that the initial conditions satisfy the equality $G_n=0$, then the constraint (\\ref{unn}) will be preserved as before. In other words, if we consider equation (\\ref{unn}) as a mapping $(u_n,u_{n+1})\\mapsto(u_{n+1},u_{n+2})$ then the equality $q_{n+1}+q_n=0$ determines a singularity which is eliminated by use of the equation $G_n=0$.\n\\end{remark}\n\nSimilar to the case (\\ref{IKF}), the constraint (\\ref{unn}) turns the lattice equations (\\ref{ut}) into a  coupled system for the variables $u_{n-1},u_n$, and the shift $n\\mapsto n+1$ defines a B\\\"acklund transformation for the latter. This system is equivalent, after some additional changes, to the Painlev\\'e equations P$_5$ or P$_3$, depending on the value of the parameter $a$. The following statement is verified by straightforward substitution, for which it is convenient to completely rewrite equations (\\ref{ut}) and (\\ref{unn}) it terms of $q_n$:\n\\begin{gather}\n\\label{qt}\n q_{n,t}=\\frac{1}{2t}(q_n-n+d)(q_{n+1}-q_{n-1}),\\\\\n\\label{qnn}\n (q_{n+1}+q_n)(q_n+q_{n-1})= \\frac{8t(aq^2_n+(-1)^nbq_n+c)}{q_n-n+d}.\n\\end{gather}\nNotice, that (\\ref{qnn}) coincides, up to a scaling of parameters, with dP$_{34}$ equation \\cite{Grammaticos_Ramani_2014}. \n\n\\begin{statement}\nLet functions $q_n(t)$ satisfy equations (\\ref{qt}), (\\ref{qnn}). If $a\\ne0$ then functions \n\\[\n y_n(t)=1-\\frac{8at}{q_{n+1}(t)+q_n(t)}\n\\]\nsatisfy the P$_5$ equation  \n\\begin{equation}\\label{P5}\n y''=\\Bigl(\\frac{1}{2y}+\\frac{1}{y-1}\\Bigr)(y')^2-\\frac{y'}{t}\n    +\\frac{(y-1)^2}{t^2}\\Bigl(\\alpha y+\\frac{\\beta}{y}\\Bigr)\n    +\\gamma\\frac{y}{t}+\\delta\\frac{y(y+1)}{y-1},\n\\end{equation}\nwith parameters\n\\[\n \\alpha=\\frac{b^2-4ac}{8a^2},\\quad \\beta=-\\frac{(a+(-1)^nb)^2}{8a^2},\\quad\n \\gamma=-2a(2n-2d+1),\\quad \\delta=-8a^2.\n\\]\nIf $a=0$ then functions\n\\[\n y_n(z)=\\frac{1}{2z}(q_{n+1}(t)+q_n(t)),\\quad t=z^2,\n\\]\nsatisfy the P$_3$ equation: \n\\begin{equation}\\label{P3}\n y''=\\frac{(y')^2}{y}-\\frac{y'}{z} +\\frac{1}{z}(\\alpha y^2+\\beta) +\\gamma y^3+\\frac{\\delta}{y},\n\\end{equation}\n\\[\n \\alpha=-4n+4d-2,\\quad \\beta=-4(-1)^nb-8c,\\quad \\gamma=4,\\quad \\delta=-16b^2.\n\\]\n\\end{statement}\n\n\\section{Regular solutions}\\label{s:regular}\n\nIn general, solutions of equations (\\ref{ut}), (\\ref{unn}) may have singularities in $t$. Such solutions are of interest as well, but in this paper we restrict ourselves by consideration of a special family of solutions, such that functions $u_n(t)$ are continuously differentiable on the whole axis $t\\in\\mathbb{R}$, for all $n\\in\\mathbb{Z}$. This regularuty condition strictly fixes the initial data of the lattice at $t=0$, that is, at the singular point of the system (\\ref{ut}), (\\ref{unn}). Indeed, since the values $u_n(0)$ are finite for a regular solution, hence $q_{n+1}(0)+q_n(0)=2n-2d+1$ and we obtain from (\\ref{unn}) that, if $d$ is not a half-integer then\n\\begin{equation}\\label{und0}\n u_n(0)= a + \\frac{4(-1)^nb(n-d)+4c+a}{4(n-d)^2-1},\\quad d\\not\\in\\frac{1}{2}+\\mathbb{Z}.\n\\end{equation}\nTherefore, for the fixed values of $a,b,c$ and $d$, we are talking about one special solution of equations (\\ref{ut}) and (\\ref{unn}) (while the general solution is 2-parametric). In terms of the P$_5$ equation (\\ref{P5}), this solution corresponds to the functions $y_n(t)$ without singularity at $t=0$, with the initial data\n\\[\n y_n(0)=1,\\quad y'_n(0)=-\\frac{2\\delta}{\\gamma}=-\\frac{8a}{2n-2d+1}.\n\\]\nIn the case when $d=\\frac{1}{2}+k$ is half-integer, the integrating factor $q_{n+1}+q_n$ which we used for derivation of (\\ref{unn}) turns into 0 for $t=0$ and $n=k$, and we have to use equation (\\ref{Gn}) instead of (\\ref{unn}), as it was explained in Remark \\ref{rem:confinement}. For $t=0$, it takes the form\n\\[ \n (n-k+1)u_{n+1}(0)-(n-k-1)u_n(0)=2a-2(-1)^nb.\n\\]\nFrom here, all $u_n(0)$ are uniqueley defined, except for $u_k(0)$ and $u_{k+1}(0)$ which are chosen independently on the rest values in such a way that their sum is constant:\n\\begin{equation}\\label{unk0}\n\\begin{gathered} \n u_n(0)=a+b\\frac{(-1)^n(2n-2k-1)+(-1)^k}{2(n-k)(n-k-1)},~~ n\\ne k,k+1,\\\\\n u_{k+1}(0)+u_k(0)=2a-2(-1)^kb,\\quad d=\\frac{1}{2}+k,~ k\\in\\mathbb{Z}.\n\\end{gathered} \n\\end{equation}\nThese inital data can be viewed as a limiting case of (\\ref{und0}) with $c=(-1)^kb\/2-a\/4$, for $d\\to\\frac{1}{2}+k$.\n\nUnder certain relations between parameters, it is possible that the equality $u_m(0)=0$ is fulfilled which splits the lattice (\\ref{ut}) into two independent systems for $n<m$ and $n>m$. We consider this case in the rest sections in more details, and now we will assume that $u_n(0)\\ne0$ for all integer $n$.\n\nThe absence of singularity at $t=0$ does not guarantee that the solution is regular for all $t$, this requires an additional study. The numeric experiments show that, for the solution family under scrutiny, the crucial property is related with the signs of $u_n(0)$: is all $u_n(0)$ are of one and the same sign (positive, without loss of generality) then the solution is regular for all $n,t$; in contrast, if there are $u_n(0)$ with different signs then the solution acquires a singularity at a finte $t$.\n\n\\begin{remark}\nThe question about the regularity criterium fot the solutions of lattice equations (\\ref{ut}) with generic inital data is open. Regular solutions with different signs do exist: a simple explicit example is the solution\n\\begin{equation}\\label{expsol}\n u_{2n} = -\\frac{\\beta(n+\\delta)e^{2\\beta t}}{\\alpha+e^{2\\beta t}},\\quad \n u_{2n+1}=\\frac{\\beta(\\alpha n+\\gamma)}{\\alpha+e^{2\\beta t}},\n\\end{equation}\nwhich, apparently, does not have singularities for $\\alpha\\ge0$, and also the stationary solution $u_{2n}=\\alpha$, $u_{2n+1}=\\beta$ with constants of different signs. However, the nonalternating solutions are of primary interest. In many papers, this requirement is simply postulated; sometimes, the Volterra lattice is wrtten down in the variables $p_n=\\sqrt{u_n}$.\n\\end{remark}\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=70mm]{fig1a.pdf}\\quad\\includegraphics[width=70mm]{fig1b.pdf}}\n\\caption{Solution of the Volterra lattice with the initial data $u_n(0)=1-\\frac{2}{4n^2-1}$}\n\\label{fig:arrow}\n\\end{figure}\n\nFor $a=0$ and nonzero $b,c$, the initial data (\\ref{und0}), (\\ref{unk0}) always change the sign and there are no regular solutions. If $a\\ne0$ then one can set $a=1$ without loss of generality. In addition, taking into account the shift of $n$, one can assume that $d\\in(-\\frac{1}{2},\\frac{1}{2}]$. Then the condition of the positivity of the initial data reduces to inequalities\n\\[\n bd-c-d^2>0,\\quad b(d-1)+c+(d-1)^2>0,\\quad b(d+1)+c+(d+1)^2>0\n\\]\nwhich cut off a bounded region in the parameter space (for a fixed $d$ it is a triangle in the $b,c$ plane) and the corresponding solutions are regular. Fig.\\,\\ref{fig:arrow} demonstrates a typical solution from this family, corresponding to the values $a=1$, $b=0$, $c=-3\/4$ and $d=0$. The initial profile is shown on the left plot. It quickly collapses and generates a zone of small-scale oscillations (with the period comparable to the lattice spacing), which has an arrow-shaped profile and expands at a constant speed in both directions with increasing $t$ (for the negative $t$, the direction of the arrowhead changes). For nonzero $b$ and $d$, the initial data look a bit more complicated, but the general behavior of the solution remains the same. Moreover, the picture does not change much if we take the initial data that do not satisfy the constraint (\\ref{unn}), but are close to (\\ref{und0}). Apparently, this behavior is typical for solutions with generic initial data in the form of sharp spikes (as opposed to solutions of the soliton type, which are formed when the initial data are relatively gently sloping). Fig.\\,\\ref{fig:2arrow} shows a solution for initial data which differ from 1 at two points. Each spike generates oscillations of the type described, which form an interference pattern after fusion.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=70mm]{fig2a.pdf}\\quad\\includegraphics[width=70mm]{fig2b.pdf}}\\bigskip\n\n\\centerline{\\includegraphics[width=70mm]{fig2c.pdf}\\quad\\includegraphics[width=70mm]{fig2d.pdf}}\n\\caption{Solution of the Volterra lattice with the initial data\\\\ $u_{-40}(0)=0.25$, $u_{40}(0)=2$, $u_n(0)=1$, $n\\ne\\pm40$.}\n\\label{fig:2arrow}\n\\end{figure}\n\nThus, this is a fairly common mode in the Volterra lattice that deserves to be studied. It would be interesting to obtain its description from the point of view of the inverse scattering method. The constraint (\\ref{unn}), with positive initial data, is of interest as an exact solution example for this mode, in terms of the Painlev\\'e transcendents. More precisely, here we use not all solutions of P$_5$, but only one, which is distinguished by the regularity condition at $t=0$. However, this solution does not seem to be expressed in terms of classical special functions, at least for the general values of parameters.\n\n\\section{Solutions on the half-line}\\label{s:halfline}\n\nAssume that, in addition to the constraint (\\ref{unn}), the condition $u_0=0$ is fulfilled (which is also a constraint, consistent with the lattice equations (\\ref{ut})). In this case, the lattice splits into two unrelated subsystems for $n<0$ and $n>0$. It is enough to consider solutions on the half-line $n>0$. First of all, we notice that if $u_0=0$ then the system of ordinary differential equation for the variables $u_0,u_1$ (which is equivalent, in general, to a Painlev\\'e equation) reduces to the Riccati equation for $u_1$.\n\n\\begin{statement}\\label{st:Riccati}\nLet $u_n$ be a solution of the lattice (\\ref{ut}) governed by the constraints (\\ref{unn}) and $u_0=0$. Then the function $u_1(t)$ satisfies the equation\n\\begin{equation}\\label{u1'}\n u'_1+u^2_1-\\Bigl(4a+\\frac{2d-3}{2t}\\Bigr)u_1-\\frac{2(a-b)}{t}=0\n\\end{equation}\nwhich is linearizable via the substitution $u_1=f'\/f$:\n\\begin{equation}\\label{f''}\n tf''+\\Bigl(\\frac{3}{2}-d-4at\\Bigr)f'-2(a-b)f=0.\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nSubstituting of $u_0=0$ into equations (\\ref{unn}) for $n=0$ and $n=1$ gives\n\\begin{gather*}\n -ad^2+bd-c=0,\\\\\n \\frac{1}{4}(q_2+q_1)(q_1-d)u_1 -aq^2_1 +bq_1 -c =0,\n\\end{gather*}\nwhere $q_n=2tu_n+n-d$. Subtracting one equation from another and dividing by $q_1-d$, we obtain\n\\[\n \\frac{1}{4}(q_2+q_1)u_1-a(q_1+d)+b=0.\n\\]\nThis is equivalent to (\\ref{u1'}), taking into account the relation $u'_1=u_1u_2$ which follows from equation (\\ref{ut}). The passage to equation (\\ref{f''}) is standard.\n\\end{proof}\n\nThus, $u_0=0$ and $u_1$ is constructed by solving equation (\\ref{f''}), then all functions $u_{n+1}$ for $n=1,2,\\dots$ are constructed by recurrent relations, either by use of the lattice itself:\n\\begin{equation}\\label{recD}\n u_{n+1}=\\frac{u'_n}{u_n}+u_{n-1},\n\\end{equation}\nor, without using differentation, by relation\n\\begin{multline}\\label{rec0}\n\\quad u_{n+1}=-u_n-\\frac{1}{t}(n-d+1\/2)\\\\\n   +\\frac{a(2tu_n+n-d)^2+(-1)^nb(2tu_n+n-d)+bd-ad^2}{tu_n(tu_n+tu_{n-1}+n-d-1\/2)}\\qquad\n\\end{multline}\n(this is the constraint equation (\\ref{unn}) with $c=bd-ad^2$ resolved with respect to $u_{n+1}$). \n\nAs in the previous section, we will consider only regular solutions, and such that $u_n\\ne0$ for $n>0$ (since otherwise the solution is constrained to a finite interval). In this case, equations (\\ref{und0}), (\\ref{unk0}) are slightly refined and we arrive to the following statement. As before, it guarantees only regularity at $t=0$; the regularity for all $t$ is related with the constant sign property of the sequence (\\ref{und00}).\n\n\\begin{statement}\\label{st:un00}\nLet $u_0=0$ and let $u_n\\ne0$ be a solution of the lattice (\\ref{ut}) for $n>0$, satisfying the constraint (\\ref{unn}) and regular at $t=0$. Then\n\\begin{equation}\\label{und00}\n u_1(0)=\\frac{4(b-a)}{2d-3},~~ u_n(0)=\\frac{a(n-d)^2+(-1)^nb(n-d)+d(b-ad)}{(n-d)^2-1\/4},~~ n>1,\n\\end{equation}\nwhere\n\\begin{equation}\\label{bdnek}\n d\\ne\\frac{1}{2}+k,\\quad b\\ne a(2k-1),\\quad b\\ne 2a(d-k),\\quad k=1,2,3,\\dots\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nThe expression for $u_n(0)$, $n>1$, is obtained from (\\ref{und0}) by substituting $c=bd-ad^2$. Here, the values $d=\\frac{1}{2},-\\frac{1}{2},-\\frac{3}{2},\\dots$ are admissible, since the corresponding denominators do not vanish. If $d\\ne\\frac{1}{2}$ then the expression for $u_1(0)$ is also found from (\\ref{und0}) and if $d=\\frac{1}{2}$ then it is obtained from (\\ref{unk0}) for $k=0$ (or directly from equation (\\ref{u1'})). Solving the inequalities $u_{2k-1}(0)\\ne0$ for the obtained initial data brings to restrictions $b\\ne a(2k-1)$; the inequalities $u_{2k}(0)\\ne0$ yield $b\\ne 2a(d-k)$.\n\nFor $d=\\frac{1}{2}+k$, $k>0$, we use equations (\\ref{unk0}) instead of (\\ref{und0}). In this case, if follows from the equality $u_0=0$ that\n\\[\n a=b\\frac{2k+1-(-1)^k}{2k(k+1)}\n\\]\nand it is easy to check that then also $u_{2k+1}(0)=0$, so these values of $d$ are rejected.\n\\end{proof}\n\nIf $a=0$ and $b\\ne0$ then the change $z=2\\sqrt{2bt}$, $f(t)=t^{d\/2-1\/4}y(z)$ brings (\\ref{f''}) to the Bessel equation\n\\begin{equation}\\label{Bessel}\n z^2y''+zy'+\\bigl(z^2-(d-\\tfrac{1}{2})^2\\bigr)y=0.\n\\end{equation}\nThe corresponding initial data (\\ref{und00}) are alternating. Numeric experiments show that if $b<0$ (without loss of generality) then the solution acquires the poles at $t<0$, but it tends to 0 for $t>0$ (the corresponding function $y$ is the Bessel function of the imaginary argument). This gives an example of alternating solution which is bounded and regular in the quadrant $n,t>0$. However, this solution is very unstable with respect to the calculation errors and the perturbations of the initial data.\n\n\\section{Asymptotics in the case $a=1$, $u_0=0$}\\label{s:asymp}\n\nIf $a\\ne0$ then the scaling of the independent variable $z=4at$, $f(t)=y(z)$ brings (\\ref{f''}) to the confluent hypergeometric equation\n\\begin{equation}\\label{confl}\n zy''+(\\beta-z)y'-\\alpha y=0,\\quad \\alpha=\\frac{a-b}{2a},\\quad \\beta=\\frac{3}{2}-d.\n\\end{equation}\nSince the function $u_1=f'\/f$ must be regular at $t=0$, we should to choose as $y$ the Kummer function $M(\\alpha,\\beta,z)$ (or ${\\!}_1F_1(\\alpha;\\beta;z)$ in other notation), that is, $f(t)= M(\\alpha,\\beta,4at)$. \n\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[width=70mm]{fig3a.pdf}\\qquad\\includegraphics[width=70mm]{fig3b.pdf}}\n\\caption{Solution with the initial data $u_n(0)=1$, $n>0$. The dashed lines correspond to one or two terms of the asymptotic expansions.}\n\\label{fig:t25}\n\\end{figure}\n\nIn this section we set $a=1$ without loss of generality. If $b=0$ and $d=-\\frac{1}{2}$ then the initial data take especially simple form of the unit step: $u_n(0)=1$ for $n>0$. The corresponding solution, shown of fig.\\,\\ref{fig:t25}, was studied in \\cite{Adler_Shabat_2018} (notice, that we used there an equivalent representation of $f$ in terms of the modified Bessel function: $f(t)=e^{2t}(I_0(2t)-I'_0(2t))=M(\\frac{1}{2},2,4t)$; however, $f$ is not expressed through $I_n$ for the generic initial data (\\ref{und00})). \n\nThe initial data are changed for other admissible values of $b$ and $d$, but the general behaviour of the solution remains the same, exactly as for the case of solutions described in Section \\ref{s:regular}. Moreover, the numeric experiments demonstrate that this mode is stable with respect to small enough perturbations of the step-like initial data (of course, under the condition that the boundary value $u_0=0$ is not changed). Fig.\\,\\ref{fig:rand} shows the evolution of the initial data\n\\[\n u_n(0)=1+0.9\\exp(-0.001(n-30)^2)r_n,\\quad n>0,\n\\] \nwhere $r_n$ is a random value uniformly distributed in $[-1,1]$. This perturbation leads to appearance of soliton-like structures on the pure decay solution background, but the overall asymptotics does not change. \n\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[width=70mm]{fig4a.pdf}\\qquad\\includegraphics[width=70mm]{fig4b.pdf}}\n\\centerline{\\includegraphics[width=70mm]{fig4c.pdf}}\n\\caption{Solution with random perturbation of the initial unit step.}\n\\label{fig:rand}\n\\end{figure}\n\nTo determine the asymptotics, we use the formal expansion of the solution of the Riccati equation (\\ref{u1'}) \n\\[\n u_1=q_0+q_1t^{-1}+q_2t^{-2}+\\dots \n\\]\nSubstitution into equation proves that the leading term may take the values $q_0=4$ or $q_0=0$, while the subsequent coefficients are computed uniquely. This gives two series which correspond to the different asymptotics at $t\\to\\pm\\infty$, which is easy to see by comparing with known asymptotic formulas \\cite{AS}: \n\\[\n M(\\alpha,\\beta,z)=\\left\\{\n \\begin{aligned}\n  &\\frac{\\Gamma(\\beta)}{\\Gamma(\\alpha)}e^zz^{\\alpha-\\beta}\\bigl(1+O(|z|^{-1})\\bigr),  & \\mathop{\\rm Re} z>0,\\\\ \n  &\\frac{\\Gamma(\\beta)}{\\Gamma(\\beta-\\alpha)}(-z)^{-\\alpha}\\bigl(1+O(|z|^{-1})\\bigr), & \\mathop{\\rm Re} z<0. \n \\end{aligned}\\right.\n\\]\nFrom here it follows, for the function $u_1=f'\/f$ and for real $z=4t$, that $u_1\\sim 4-(\\alpha-\\beta)t^{-1}+\\dots$ at $t\\to+\\infty$ and $u_1\\sim 0+\\alpha t^{-1}+\\dots$ at $t\\to-\\infty$. Next, the expansions for all $u_n$ are obtained by intermediate use of equation (\\ref{u1'}) and the lattice equation (\\ref{ut}). We wil restrict ourselves by two first terms of the asymptotics.\n\n\\begin{statement}\\label{st:uinf}\nConsider the solution of the lattice (\\ref{ut}) with $u_0=0$ and the initial data $u_n(0)>0$ for $n>0$ defined by equations (\\ref{und00}), (\\ref{bdnek}) with $a=1$. Then the following asymptotic formulas are valid: for $t\\to+\\infty$\n\\[\n u_n=\n  \\left\\{\n  \\begin{aligned}\n   & \\frac{n(n-2d+b)}{16t^2}+\\frac{n(n-2d+b)(2n-2d+3b)}{128t^3}+O(t^{-4}),\\quad n=0,2,\\dots,\\\\\n   & 4-\\frac{2n-2d+b}{2t}\\\\\n   & \\qquad -\\frac{n^2-(2d-3b)n+b^2-2bd+1}{16t^2}+O(t^{-3}),\\quad n=1,3,\\dots\n  \\end{aligned}\n  \\right.\n\\]\nand for $t\\to-\\infty$\n\\[\n u_n=\n  \\left\\{\n  \\begin{aligned}\n   & -\\frac{n}{2t}+\\frac{(2d-b)n}{16t^2}+O(t^{-3}),\\quad n=0,2,\\dots,\\\\\n   & -\\frac{n-b}{2t}+\\frac{(2d-b)(n-b)}{16t^2}+O(t^{-3}),\\quad n=1,3,\\dots\n  \\end{aligned}\n  \\right.\n\\]\n\\end{statement}\n\\begin{proof}\nWe find from equation (\\ref{u1'}) that\n\\[\n u_1(t)=\\left\\{\n  \\begin{aligned} \n  & 4+\\frac{2d-b-2}{2t}+\\frac{(b+1)(2d-b-2)}{16t^2}+\\dots,\\quad t\\to+\\infty,\\\\\n  & \\frac{b-1}{2t}+\\frac{(b-1)(b-2d)}{16t^2}+\\dots,\\quad t\\to-\\infty.\n \\end{aligned}\\right.\n\\]\nIt is easy to prove by induction that expansions for $u_n$ at $t\\to+\\infty$ are of the following form, depending on the parity of $n$:\n\\[\n u_{2j}=\\frac{p_{j,2}}{t^2}+\\frac{p_{j,3}}{t^3}+\\dots,\\quad\n u_{2j+1}=4+\\frac{q_{j,1}}{t}+\\frac{q_{j,2}}{t^2}+\\dots\n\\]\nand that the substitution into (\\ref{ut}) gives the difference equations for the coefficients\n\\begin{gather*}\n q_{j,1}-q_{j-1,1}=-2,\\quad 4p_{j+1,2}-4p_{j,2}=-q_{j,1},\\\\\n p_{j,2}(q_{j,2}-q_{j-1,2})=-p_{j,3},\\quad 16p_{j+1,3}-16p_{j,3}+8q_{j,2}=q^2_{j,1}.\n\\end{gather*}\nSimilarly, the expansions at $t\\to-\\infty$ start from $t^{-1}$ for all $n$: \n\\[\n u_{2j}=p_{j,1}t^{-1}+p_{j,2}t^{-2}+\\dots,\\quad u_{2j+1}=q_{j,1}t^{-1}+q_{j,2}t^{-2}+\\dots\n\\] \nand the coefficients are governed by equations\n\\begin{gather*}\n q_{j,1}-q_{j-1,1}=-1,\\quad p_{j+1,1}-p_{j,1}=-1,\\\\ \n p_{j,1}(q_{j,2}-q_{j-1,2})=-p_{j,2},\\quad  q_{j,1}(p_{j+1,2}-p_{j,2})=-q_{j,2}.\n\\end{gather*}\nThe initial data for these equations are given by the coefficients $q_{0,1},q_{0,2}$ of the above series for $u_1$ and the values $p_{0,k}=0$ corresponding to $u_0=0$. In both cases, the solution is easily obtained as polynomials in $j$ \nand we obtain the required formulas by returning to the variable $n$.\n\\end{proof}\n\nFor $t>0$, one can obtain a rough estimate of the decay zone by constructing a triangular region bounded by the plots of one or two terms of the asymptotic expansions, as shown on figs.\\;\\ref{fig:t25} and \\ref{fig:rand}. In particular, an upper bound for the wedge point $n_0(t)$ of the decay zone can be obtained by solving the inequality $4-\\frac{2n_0-2d+b}{2t}>1$, which gives, apparently, $n_0<3t+\\mathop{\\rm const}$. More accurate estimates can be obtained by taking the next asymptotic terms. For the negative $t$, the solution is well approximated by the first term of the asymptotics.\n\nWe conclude this section with a note on the conservation laws of the Volterra lattice, that is, relations of the form\n\\begin{equation}\\label{conserv}\n \\frac{d}{dt}\\rho^{(k)}_n=\\sigma^{(k)}_{n+1}-\\sigma^{(k)}_n,\n\\end{equation}\nwhere $\\rho^{(k)}_n$ and $\\sigma^{(k)}_n$ depend on a finite number of variables $u_n$. Three simplest conservation laws are given by\n\\begin{gather*}\n \\rho^{(0)}_n=\\log u_n,~~ \\sigma^{(0)}_n=u_{n-1}+u_n,\\qquad\n \\rho^{(1)}_n=u_n,~~ \\sigma^{(1)}_n=u_{n-1}u_n,\\\\\n \\rho^{(2)}_n=\\frac{1}{2}u^2_n+u_nu_{n+1},~~ \\sigma^{(2)}_n=u_{n-1}u_n(u_n+u_{n+1}).\n\\end{gather*}\nIn the case of the problem on the whole line and with the initial data which have the same constant asymptotics for $n\\to\\pm\\infty$, it follows from (\\ref{conserv}) that the quantities $H_k=\\sum_n(\\rho^{(k)}_n-r^{(k)})$ are preserved, where the summation is taken over all integer $n$ and the constant $r^{(k)}$ is chosen so that the sum is well defined. For the lattice truncated by $u_0=0$, the analogous sums over $n>0$ do not preserve, since\n\\[\n \\frac{d}{dt}H_k=\\lim_{n\\to\\infty}\\sigma^{(k)}_n-\\sigma^{(k)}_1\\ne0.\n\\]\nFor solutions with the asymptotics $u_n\\to1$ for $n\\to\\infty$, the sums are regularized as follows\n\\[\n H_0=\\sum^\\infty_{n=1}\\log u_n,\\quad\n H_1=\\sum^\\infty_{n=1}(u_n-1),\\quad\n H_2=\\sum^\\infty_{n=1}\\Bigl(\\frac{1}{2}u^2_n+u_nu_{n+1}-\\frac{3}{2}\\Bigr),\n\\]\nand we have $\\sigma^{(0)}_\\infty=2$, $\\sigma^{(0)}_1=u_1$;\n$\\sigma^{(1)}_\\infty=1$, $\\sigma^{(1)}_1=0$ and \n$\\sigma^{(2)}_\\infty=2$, $\\sigma^{(2)}_1=0$. Then\n\\[\n \\frac{d}{dt}H_0=2-u_1,\\quad \\frac{d}{dt}H_1=1,\\quad \\frac{d}{dt}H_2=2,\n\\]\nand since all three sums are equal to 0 at $t=0$, hence\n\\[\n H_0= \\int^t_0(2-u_1(\\tau))d\\tau,\\quad H_1=t,\\quad H_2=2t.\n\\]\n\n\\section{Determinant identities}\\label{s:det}\n\nIn addition to the recurrent relations (\\ref{recD}), (\\ref{rec0}), there exists the Wronskian representation of the Volterra lattice solution on the half-line, which goes back to the Leznov paper \\cite{Leznov_1980}. It is not very convenient for a practical computing of solutions, but we will show that using it together with explicit expressions for $u_1(t)$ and $u_n(0)$ makes possible to get nontrivial identities for some number sequences.\n\n\\begin{statement}\\label{st:w}\nThe solution of the lattice (\\ref{ut}) on the half-line $n\\ge0$, such that $u_0=0$ and $u_1=f'\/f$, with an arbitrary infinitely differentiable function $f(t)$, is of the form\n\\begin{equation}\\label{uw}\n u_n=\\frac{w_{n-3}w_n}{w_{n-2}w_{n-1}},\\quad n=0,1,2,\\dots,\n\\end{equation}\nwhere $w_{-3}=0$, $w_{-2}=w_{-1}=1$ and, for $k\\ge0$,  \n\\begin{equation}\\label{ww}\n w_{2k}=\n \\left|\\begin{matrix}\n   f       & f'        & \\dots  & f^{(k)}  \\\\\n   f'      & f''       & \\dots  & f^{(k+1)}\\\\\n   \\vdots  & \\vdots    & \\ddots &\\vdots    \\\\\n   f^{(k)} & f^{(k+1)} & \\dots  & f^{(2k)}\n \\end{matrix}\\right|,\\quad\n w_{2k+1}=\n \\left|\\begin{matrix}\n   f'       & f''       & \\dots  & f^{(k+1)}  \\\\\n   f''      & f'''      & \\dots  & f^{(k+2)}\\\\\n   \\vdots   & \\vdots    & \\ddots &\\vdots    \\\\\n   f^{(k+1)}& f^{(k+2)} & \\dots  & f^{(2k+1)}\n \\end{matrix}\\right|.\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nWe will prove that $w_n$ satisfy equations\n\\begin{equation}\\label{wt}\n w_nw'_{n+1}-w'_nw_{n+1}=w_{n-1}w_{n+2},\\quad n=-2,-1,0,1,\\dots,\n\\end{equation}\nthen it is easy to check that substitution (\\ref{uw}) gives a (unique) solution of the lattice equations (\\ref{ut}) with $u_0=0$ and $u_1=f'\/f$. \n\nFor $n=-2,-1$, the relations (\\ref{wt}) are verified directly. For $n\\ge0$, let $W(A)$ denote the Wronskian of an arbitrary finite sequence $A$ of smooth functions (possibly empty). For $n=2k$ we set $A=f^{(1)},\\dots,f^{(k)}$, then\n\\begin{alignat*}{2}\n &w_{n-1}=W(A), &\\quad&     w_n=W(f^{(0)},A) =(-1)^kW(A,f^{(0)}),\\\\\n &w_{n+1}=W(A,f^{(k+1)}),&& w_{n+2}=W(f^{(0)},A,f^{(k+1)}) =(-1)^kW(A,f^{(0)},f^{(k+1)}).\n\\end{alignat*}\nSimilarly, for $n=2k+1$ we set $A=f^{(0)},\\dots,f^{(k)}$, then\n\\begin{alignat*}{2}\n &w_{n-1}=W(A), &\\quad&     w_n=W(1,A)=(-1)^{k+1}W(A,1),\\\\\n &w_{n+1}=W(A,f^{(k+1)}),&& w_{n+2}=W(1,A,f^{(k+1)})=(-1)^{k+1}W(A,1,f^{(k+1)}).\n\\end{alignat*}\nIn both cases, equation (\\ref{wt}) is satisfied due to the identity\n\\[\n W(A,b)\\frac{d}{dt}W(A,c)-\\frac{d}{dt}W(A,b)W(A,c)=W(A)W(A,b,c),\n\\]\nwhere $b,c$ are arbitrary smooth functions. In order to prove it, it is sufficient to consider both left and right hand sides as the differential operators with respect to $c$ and to compare their kernels and the coefficients at the highest derivative.\n\\end{proof}\n\nThe determinants of the form (\\ref{ww}) for the number sequences $f_0,f_1,f_2,\\dots$ are actively studied in combinatorics. Recall that the Hankel transformation for such a sequence is the sequence of determinants of size $n\\times n$ with $(i,j)$-th element equal to $f_{i+j-2}$. This mapping is not one-to-one, since the determinant of size $n$ involves $2n-1$ members of the sequence. To get a one-to-one mapping, one can use simultaneously Hankel transforms for the sequence itself and the sequence without the zero member, provided that all determinants do not vanish.\n\nFor instance, the Hankel transform of the Catalan numbers $1$, $1$, $2$, $5$, $14$, $42$, $132$, $429,\\dots$ is the sequence $1,1,1,\\dots$ and the same is true for the sequence of the Catalan numbers with the first member dropped \\cite{Aigner_1999, Stanley_1999, Layman_2001}. In \\cite{Adler_Shabat_2018}, we associated these identities of the Catalan numbers with the Wronskian representation of the solution with the initial data $u_0=0$, $u_n(0)=1$, $n>0$ and derived the Riccati equation for $u_1(t)$ from this.\n\nNow we are able to reverse and to generalize this result. Indeed, the Wronskians $w_{2k}$ and $w_{2k+1}$ define the Hankel transformations for the sequences $f,f',\\dots$ and $f',f'',\\dots$, respectively. By setting $t=0$ in (\\ref{ww}), we obtain the Hankel transformations for the coefficients of the Taylor expansion of the function $f(t)$ (here, the regularity for other values of $t$ is not necessary) and it remains to compare the result with the known initial data by substituting into equation (\\ref{uw}) at $t=0$.\n\n\\begin{statement}\\label{st:Hankel}\nLet a solution of the lattice equation (\\ref{ut}) on the half-line $n\\ge0$ be given by equations (\\ref{uw}), (\\ref{ww}) with the function $f$ given by the Taylor expansion\n\\[\n f(t)=f_0+f_1t+\\cdots+f_n\\frac{t^n}{n!}+\\cdots,\n\\]\nthen, for $k=0,1,2,\\dots$, \n\\begin{equation}\\label{wwu0}\n\\begin{gathered}\n h_{2k}=\\left|\\begin{matrix}\n   f_0    & \\dots  & f_k     \\\\\n   \\vdots & \\ddots &\\vdots   \\\\\n   f_k    & \\dots  & f_{2k}\n \\end{matrix}\\right|  =  \\prod^k_{j=1}(u_{2j-1}(0)u_{2j}(0))^{k+1-j},\\qquad\\\\\n h_{2k+1}=\\left|\\begin{matrix}\n   f_1     & \\dots  & f_{k+1}  \\\\\n   \\vdots  & \\ddots &\\vdots    \\\\\n   f_{k+1} & \\dots  & f_{2k+1}\n \\end{matrix}\\right|  =  u^{k+1}_1(0)\\prod^k_{j=1}(u_{2j}(0)u_{2j+1}(0))^{k+1-j}.\n\\end{gathered}\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nBy setting $h_n=w_n(0)$, we obtain $h_{-2}=h_{-1}=h_0=1$ and the recurrent relation\n\\[\n h_nh_{n-3}=u_n(0)h_{n-1}h_{n-2},\\quad n=1,2,\\dots, \n\\]\nwhich proves the statement by induction.\n\\end{proof}\n\nWe note that some nontrivial identities follows from here even for the explicit solution\n\\[\n u_{2k-1}=e^t,\\quad u_{2k}=k\n\\]\nwhich is a particular case of solution (\\ref{expsol}) at $\\alpha=0$, $\\beta=-1\/2$, $\\gamma=-2$ and $\\delta=0$. In this case we find, by solving equation $f'\/f=u_1=e^t$, that\n\\[\n f(t)=e^{e^t-1}= \\sum^\\infty_{n=0}B_n\\frac{t^n}{n!},\n\\]\nwhere $B_n$ are the Bell numbers $1,1,2,5,15,52,203,\\dots$ enumerating the partitions of a set of $n$ elements. Here $u_{2k-1}(0)=1$, $u_{2k}(0)=k$ and one obtains easily that $h_{2k}=h_{2k+1}=1!\\cdots k!$ (the superfactorial of $k$). This result is known since 1978~\\cite{Ehrenborg_2000}.\n\nNow, let us apply Statement \\ref{st:Hankel} to solutions desribed in the previous section (however, now we do not set $a=1$). Due to the known expansion of the Kummer function, we have\n\\begin{equation}\\label{Kummer}\n f(t)= M(\\alpha,\\beta,4at)\n     = 1+4a\\frac{\\alpha}{\\beta}t+(4a)^2\\frac{(\\alpha)_2}{(\\beta)_2}\\frac{t^2}{2!}+\\dots\n        +(4a)^n\\frac{(\\alpha)_n}{(\\beta)_n}\\frac{t^n}{n!}+\\dots,\n\\end{equation}\nwhere $(\\alpha)_n$ is the Pochhammer symbol \n\\[\n (\\alpha)_n=\\alpha(\\alpha+1)\\cdots(\\alpha+n-1),\\quad (\\alpha)_0=1. \n\\]\nIn addition, let us denote\n\\[\n ((\\alpha))_n= (\\alpha)_1\\cdots(\\alpha)_n= \\alpha^n(\\alpha+1)^{n-1}\\cdots(\\alpha+n-1)^1,\\quad ((\\alpha))_0=1; \n\\]\nin particular, the superfactorial is denoted as $((1))_n=1!\\cdots n!$.\n\n\\begin{statement}\\label{st:Poch-Hankel}\nLet\n\\begin{gather*}\n f_n=(4a)^n\\frac{(\\alpha)_n}{(\\beta)_n},\\quad n=0,1,2,\\dots,\\\\\n \\alpha\\ne -n,\\quad \\beta\\ne -n,\\quad \\alpha-\\beta\\ne n,\\quad n=0,1,2,\\dots,\n\\end{gather*}\nthen, for $k=0,1,2,\\dots$, \n\\begin{equation}\\label{Poch-Hankel}\n\\begin{gathered}\n h_{2k}=\\left|\\begin{matrix}\n   f_0    & \\dots  & f_k     \\\\\n   \\vdots & \\ddots &\\vdots   \\\\\n   f_k    & \\dots  & f_{2k}\n \\end{matrix}\\right|  = \\frac{((1))_k(4a)^{k(k+1)}((\\alpha))_k((\\beta-\\alpha))_k}{(\\beta)^{k+1}_k((\\beta+k))_k},\\\\\n h_{2k+1}=\\left|\\begin{matrix}\n   f_1     & \\dots  & f_{k+1}  \\\\\n   \\vdots  & \\ddots &\\vdots    \\\\\n   f_{k+1} & \\dots  & f_{2k+1}\n \\end{matrix}\\right|  = \\frac{((1))_k(4a)^{(k+1)^2}((\\alpha))_{k+1}((\\beta-\\alpha))_k}{(\\beta)^{k+1}_k((\\beta+k))_{k+1}},\n\\end{gathered}\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nSet \n\\[\n b=a(1-2\\alpha),\\quad d=\\frac{3}{2}-\\beta,\n\\]    \nthen the initial data (\\ref{und00}) take the form\n\\[\n \\begin{aligned}\n  u_1(0)=4a\\frac{\\alpha}{\\beta},\\quad\n   &u_{2k}(0)=4a\\frac{k(\\beta-\\alpha+k-1)}{(\\beta+2k-2)(\\beta+2k-1)},\\\\\n   &u_{2k+1}(0)=4a\\frac{(\\alpha+k)(\\beta+k-1)}{(\\beta+2k-1)(\\beta+2k)},\n \\end{aligned}\\quad k=1,2,3,\\dots,\n\\]\nand the inequalities for $\\alpha$ and $\\beta$ coincide exactly with the conditions (\\ref{bdnek}) which guarantee that all numerators and denominators are not 0. According to Statements \\ref{st:Riccati} and \\ref{st:w}, the corresponding regular (at $t=0$) solution of the Volterra lattice is defined by equations (\\ref{uw}), (\\ref{ww}) with function (\\ref{Kummer}) and we only have to substitute the initial data into (\\ref{wwu0}) and to arrange the factors. \n\\end{proof}\n\nIn particular, the example with the Catalan numbers $f_n=\\dfrac{(2n)!}{(n+1)!n!}$ corresponds to the choice $a=1$, $b=0$ and $d=-\\frac{1}{2}$ (or, $\\alpha=\\frac{1}{2}$ and $\\beta=2$). In this case the initial data are $u_n(0)=1$ for $n>0$ \nand instead of the general formula, it is easier to use the recurrence relation $h_nh_{n-3}=u_n(0)h_{n-1}h_{n-2}$ directly, which immediately gives that all $h_n=1$.\n\nSimilarly, the central binomial coefficients $f_n=\\dfrac{(2n)!}{(n!)^2}$ correspond to the choice $a=1$, $b=0$ and $d=\\frac{1}{2}$ (or, $\\alpha=\\frac{1}{2}$ and $\\beta=1$). In this case $u_1(0)=2$ and $u_n(0)=1$ for $n>1$ and the recurrent relation yields $h_{2k}=2^k$, $h_{2k+1}=2^{k+1}$. This example is also known in the combinatorics.\n\n\\subsubsection*{Acknowledgements}\n\nThis work was carried out under the State Assignment 0033-2019-0006 (Integrable systems of mathematical physics) of the Ministry of Science and Higher Education of the Russian Federation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Model 1: estimating the SNARC effect}\n\nThe model is a hierarchical Bayesian linear regression model, where $dRT$ is predicted by stimulus number.  A graphical representation of the model can be seen in Figure \\ref{fig:model}.  Formally, we define\n\n\\begin{equation}\\label{eq:model}\n  dRT_{ij} = \\alpha_i+\\beta_i(j) + \\varepsilon_{ij}\n\\end{equation}\n\n\\noindent\nwhere $i =$ subject number and $j=$ stimulus number.  The residuals $\\varepsilon_{ij}$ are assumed to be normally distributed with mean 0 and precision $1\/\\sigma^2$.  Thus, we can express the likelihood for the data as\n\n\\begin{equation}\\label{eq:likelihood}\n dRT_{ij} \\sim \\text{Normal}(\\mu_{ij}, 1\/\\sigma^2)\n\\end{equation}\n\n\\noindent\nwhere $\\mu_{ij}=\\alpha_i+\\beta_i(j)$.  The prior for each individual-level intercept $\\alpha_i$ is set to be uniformly distributed between -200 and 200.  The hierarchical structure is instantiated on slope.  First, I set the prior for each individual-level slope $\\beta_i$ to be normally distributed with two hyperparameters: mean $b$ and precision $1\/\\sigma_b^2$.  In turn, this requires priors on the hyperparameters -- the prior for $b$ is uniformly distributed between -20 and 20.  Note that this range is based on inspection of the slopes obtained in \\citet{viarouge2014}.  The prior for group-level slop precision $1\/\\sigma_b^2$ (as well as group-level residual precision $1\/\\sigma^2$) is set as a Gamma distribution with both shape and scale equal to 0.01. \n\n\\begin{figure}\n  \\centering\n\\begin{tikzpicture}\n\n \n  \\node[obs] (dRT) {$dRT_{ij}$};%\n  \\node[latent, above=of dRT] (mu) {$\\mu_{ij}$}; %\n  \\node[latent, above=of mu, xshift=-2cm] (beta) {$\\beta_i$};%\n  \\node[latent, above=of mu, xshift=2cm] (alpha) {$\\alpha_i$};%\n\n  \\node[latent, above=of beta, xshift=-1cm] (b) {$b$};%\n  \\node[latent, above=of beta, xshift=1cm] (sigma_b) {$\\sigma_b^2$};%\n\n  \\node[latent, left=of dRT, xshift=-2cm] (sigma) {$\\sigma^2$};%\n     \n    \n     \\edge {sigma,mu} {dRT}\n     \\edge {beta,alpha} {mu}\n     \\edge {b,sigma_b} {beta}\n\n    \n     \\plate[inner sep=10pt] {plate1} {(dRT)(mu)} {number $j$}; %\n     \\plate[inner sep=10pt] {plate2} {(plate1)(beta)(alpha)} {subject $i$}; %\n\n \n     \\node[text width=6cm, anchor=west, right] at (3,2)\n     {\n       \\begin{align*}\n         b & \\sim \\text{Uniform}(-20,20)\\\\\n         1\/\\sigma_b^2 & \\sim \\text{Gamma}(0.01,0.01)\\\\\n         \\alpha_i & \\sim \\text{Uniform}(-200,200)\\\\\n         \\beta_i & \\sim \\text{Gaussian}(b,\\sigma_b^2)\\\\\n         \\mu_{ij} &= \\alpha_i + \\beta_i(j)\\\\\n         1\/\\sigma^2 &\\sim \\text{Gamma}(0.01, 0.01)\\\\\n         dRT_{ij} & \\sim \\text{Gaussian}(\\mu_{ij},\\sigma^2)\n       \\end{align*}\n};\n \\end{tikzpicture}\n\n  \n  \\caption{Graphical model of the hierarchical Bayesian linear regression model for the SNARC effect. Following the convention of \\citet{lee2014}, nodes represent variables of interest (observed=shaded, latent=unshaded), with dependencies represented via the graph structure.}\n  \\label{fig:model}\n\\end{figure}\n\n\\subsection{Fitting the model}\n\n\\subsubsection{Data}\nI fit the model to data collected from 35 participants in a number parity task.  The numbers 1, 2, 8, and 9 were presented in the center of a computer screen, after which participants were asked to quickly indicate via a button press whether the number was even or odd.  The procedure mirrored that of Experiment 1 of \\citet{pinhas2014}.  Participants completed 112 trials of the task under two counterbalanced response rules (either even=left or even=right).  This resulted in a collection of 3,920 trials.  We removed 259 error trials and an additional 12 trials for which RT exceeded 3 seconds (a total of 6.9\\% of trials).  The remaining 3,649 trials were collapsed into $35 \\times 8 \\times 2$ cells by computing median $RT$ for each of the conditions defined by crossing the factors of subject, stimulus number, and response hand.  Then, $dRT$ was computed for each combination of subject and number by subtracting left-hand RT from right-hand RT.\n\n\\subsubsection{Results}\nThe regression model parameters were estimated using R \\citep{r} and JAGS \\citep{jags}.  Posterior sampling consisted of 3 MCMC chains, each containing 100,000 draws.  The first 5000 draws of each chain were discarded as ``burn-in'' samples, leaving 285,000 samples remaining.  These remaining samples were thinned by a factor of 10, leaving a final sample of 28,500 posterior draws for each parameter.  Visual inspection of trace plots indicated that all chains converged appropriately.  Additionally, the Gelman-Rubin statistic $\\hat{R}=1.001$ for each parameter, indicating that the Markov chains for each parameter converged to the appropriate stationary distribution \\citep{gelman1992,gelman2013}.\n\nSince the model is hierarchical, I was able to estimate posterior distributions for each $\\alpha_i$ and $\\beta_i$ (i.e., each participant's intercept and slope, respectively). Further, I estimated the posterior distribution of $b$, which is the group level mean slope. The flexibility of this model allows one to ask many questions about the SNARC effect, both at the group level and the individual level.  To illustrate, I will investigate whether there was an overall SNARC effect for the group \\citep[c.f.,][]{fias1996}. This can be answered by looking at the posterior distribution of the group-level $b$.\n\nThe posterior distribution of the group-level slope $b$ is depicted in Figure \\ref{fig:model1-posterior}.  As can be seen in the figure, the mass of the distribution is centered over the posterior mode $b=-11.5$.  Further, the 95\\% HPDI for the slope $b$ is $[-15.6, -7.3]$.  Finally, since the posterior distribution is a \\emph{probability distribution}, we can compute the probability that $b<0$ (that is, the probability that there is a non-zero SNARC effect).  This probability is greater than 0.999.  Other Bayesian tools can be applied to this model, such as computing a Bayes factor comparing a null-SNARC model to our obtained model via the Savage-Dickey density ratio \\citep{wagenmakers2010}. Briefly, this method amounts to computing the density of the value $b=0$ in the posterior distribution divided by the density of $b=0$ in the prior (i.e., $\\text{Uniform}(-20,20)$).  Doing this resulted in a Bayes factor of approximately 280,000 to 1 in favor of the alternative hypothesis, indicating overwhelming support for an overall SNARC effect.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{model1-posterior.jpeg}\n  \\caption{Posterior distribution of the group-level slope $b$ for Model 1, indicating a substantial SNARC effect across the group.}\n  \\label{fig:model1-posterior}\n\\end{figure}\n\n\n\\section{Model 2: estimating the numerical distance effect}\n\nAs with Model 1, this model is also a hierarchical Bayesian linear regression model, where $RT$ is predicted by numerical distance. For this specific instantiation of the model, I will index numerical distance via the \\emph{ratio} between compared numbers.  For example, \\citet{fazio2014} used four ratio bins as predictors of RT.  A graphical representation of the model can be seen in Figure \\ref{fig:model2}.  Formally, we define\n\n\\begin{equation}\\label{eq:model2}\n  RT_{ij} = \\alpha_i+\\beta_i(j) + \\varepsilon_{ij}\n\\end{equation}\n\n\\noindent\nwhere $i =$ participant number and $j=1,\\dots,4$ equals ratio bin number.  The residuals $\\varepsilon_{ij}$ are again assumed to be normally distributed with mean 0 and precision $1\/\\sigma^2$.  The likelihood for the data is\n\n\\begin{equation}\\label{eq:likelihood}\n RT_{ij} \\sim \\text{Normal}(\\mu_{ij}, 1\/\\sigma^2)\n\\end{equation}\n\n\\noindent\nwhere $\\mu_{ij}=\\alpha_i+\\beta_i(j)$.  The prior for each individual-level intercept $\\alpha_i$ is set to be uniformly distributed between 0 and 2000. The prior for each individual-level slope $\\beta_i$ to be normally distributed with two hyperparameters: mean $b$ (with uniform prior between -100 and 100) and precision $1\/\\sigma_b^2$ (with a $\\text{Gamma}(.01,.01)$ prior).  Finally, the prior for the group-level residual precision is $\\text{Gamma}(.01,.01)$.\n\n\\begin{figure}\n  \\centering\n\n\n\\begin{tikzpicture}\n\n \n  \\node[obs] (RT) {$RT_{ij}$};%\n  \\node[latent, above=of RT] (mu) {$\\mu_{ij}$}; %\n  \\node[latent, above=of mu, xshift=-2cm] (beta) {$\\beta_i$};%\n  \\node[latent, above=of mu, xshift=2cm] (alpha) {$\\alpha_i$};%\n\n  \\node[latent, above=of beta, xshift=-1cm] (b) {$b$};%\n  \\node[latent, above=of beta, xshift=1cm] (sigma_b) {$\\sigma_b^2$};%\n\n  \\node[latent, left=of dRT, xshift=-2cm] (sigma) {$\\sigma^2$};%\n     \n    \n     \\edge {sigma,mu} {dRT}\n     \\edge {beta,alpha} {mu}\n     \\edge {b,sigma_b} {beta}\n\n    \n     \\plate[inner sep=10pt] {plate1} {(dRT)(mu)} {number $j$}; %\n     \\plate[inner sep=10pt] {plate2} {(plate1)(beta)(alpha)} {subject $i$}; %\n\n \n     \\node[text width=6cm, anchor=west, right] at (3,2)\n     {\n       \\begin{align*}\n         b & \\sim \\text{Uniform}(-100,100)\\\\\n         1\/\\sigma_b^2 & \\sim \\text{Gamma}(0.01,0.01)\\\\\n         \\alpha_i & \\sim \\text{Uniform}(0,2000)\\\\\n         \\beta_i & \\sim \\text{Gaussian}(b,\\sigma_b^2)\\\\\n         \\mu_{ij} &= \\alpha_i + \\beta_i(j)\\\\\n         1\/\\sigma^2 &\\sim \\text{Gamma}(0.01, 0.01)\\\\\n         dRT_{ij} & \\sim \\text{Gaussian}(\\mu_{ij},\\sigma^2)\n       \\end{align*}\n};\n \\end{tikzpicture}\n\n  \n  \\caption{Graphical model of the hierarchical Bayesian linear regression model for the numerical distance effect. Following the convention of \\citet{lee2014}, nodes represent variables of interest (observed=shaded, latent=unshaded), with dependencies represented via the graph structure.}\n  \\label{fig:model2}\n\\end{figure}\n\n\\subsection{Fitting the model}\n\n\\subsubsection{Data}\nI fit the model to the RT data from \\citet{fazio2014}.  Fifty-five 5th graders completed a symbolic number comparison task, in which they were asked to choose which of two Arabic numerals was larger. Each child completed 40 trials using stimulus numbers ranging from 5 to 21.  Of the 40 trials, 10 came from each of 4 ratio bins: 1.15 - 1.28, 1.28 - 1.43, 1.48 - 1.65, and 2.46 - 2.71. The ratio for each stimulus pair was defined as the quotient obtained when dividing the larger number by the smaller number.  In all, participants completed 2,200 trials.  I removed 95 error trials and 6 additional trials for which RT exceeded 5 seconds (a total of 4.6\\% of trials). The remaining 2,099 trials were collapsed into $55 \\times 4$ cells by computing median $RT$ for each of the conditions defined by crossing the factors of subject and ratio bin.\n\n\\subsubsection{Results}\nThe regression model was fit using the same procedure as in Model 1.  As before, visual inspection of trace plots indicated that all chains converged appropriately, with $\\hat{R}=0.001$ for all parameters.\n\nThe posterior distribution of the group-level slope $b$ is depicted in Figure \\ref{fig:model2-posterior}.  As can be seen in the figure, the mass of the distribution is centered over the posterior mode $b=-65.9$, with 95\\% HPDI for the slope $b$ equal to $[-83.4, -50.7]$.  In addition to estimating the group-level slope, we can test the existence of the numerical distance effect via a Bayes factor, which as before I computed using the Savage-Dickey density ratio.  This Bayes factor was approximately 1.7 million to 1 in favor of the alternative hypothesis, indicating overwhelming support for an overall numerical distance effect.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{model2-posterior.jpeg}\n  \\caption{Posterior distribution of the group-level slope $b$ for Model 2, indicating a substantial numerical distance effect across the group.}\n  \\label{fig:model2-posterior}\n\\end{figure}\n\n\\section{Comparing the model to the classical approach}\nIn this section, I will demonstrate that the hierarchical Bayesian model developed in this paper can provide better measurement fidelity for the group-level regression slopes that are desired when assessing group-level SNARC effects or NDEs, particularly for small samples.  As mentioned earlier, the reason for this increased accuracy is because of a property that is unique to Bayesian inference -- namely the property of \\emph{shrinkage}.  When one specifies a group-level prior distribution for the slope (e.g., the $\\text{Uniform}(-20,20)$ distribution that I used in Model 1), this prior is then combined with the data likelihood via Bayes theorem by multiplication.  In our case, extreme individual slope estimates (those beyond -20 and 20) vanish by virtue of being multiplied by the prior probability of obtaining those estimates (i.e., probability = 0).  The resulting posterior distribution is then \\emph{shrunk} away from these parameter estimates.\n\nTo demonstrate this, I conducted a simulation.  I randomly generated $dRT$ values for a small samples of $n=15$ simulated participants as follows.  First, I generated 15 random slopes $b_i$, where $b_i\\sim \\text{Gaussian}(-10,1)$.  That is, I assumed that individual slopes are drawn from a normal distribution centered at -10, with standard deviation 1.  Similarly, I randomly generated 15 random intercepts $a_i$, where $a_i \\sim \\text{Uniform}(-200,200)$.  Then, I generated \n\n\\[\n  dRT_{ij} = a_i + b_i(j) + \\varepsilon_{ij}\n\\]\n\n\\noindent\nwhere $i=1,\\dots,15$, $j=1,2,8,9$, and $\\varepsilon_{ij} \\sim \\text{Gaussian}(0,100)$.  Then, I computed a classical linear regression for each participant $i=1,\\dots,15$, recording each estimated slope $\\hat{b}_i$.  Finally, I fit the hierarchical Bayesian model (Model 1) to the overall set of data to compute the posterior distribution of $b$.\n\nThe results can be seen in Figure \\ref{fig:simulation}.  Notice that while both the frequentist 95\\% confidence interval and the Bayesian 95\\% highest posterior density interval contain the population mean $b=-10$, the Bayesian estimate is much less variable.  This is because compared to the classical linear regression method (e.g., Lorch \\& Myers, 1990), the extreme parameter estimates are shrunk toward the posterior mean.  The same story is repeated with hypothesis testing, as well.  Indeed, performing a $t$-test on the individual regression slopes \\citep[as in][]{fias1996} results in a non-significant SNARC effect, $t(14)=-1.82$, $p=0.09$, which results in a Type II error.  However, a Bayesian hypothesis test (the Savage-Dickey method) yields a Bayes factor that favors the SNARC effect by a factor of 75 to 1. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{simulation.jpeg}\n  \\caption{Slope estimates from a simulated data set of $n=15$ participants.  The solid line represents the posterior density of $b$, the group-level slope estimate for the SNARC effect.  The histogram represents the slope estimates obtained from separate classical linear regressions for each individual.}\n  \\label{fig:simulation}\n\\end{figure}\n\n\\section{Discussion}\n\nThe purpose of this paper was to introduce a hierarchical Bayesian linear regression framework for measuring group-level and individual-level numerical representations. The models were then applied to two common markers of numerical representations: the SNARC effect, which indexes spatial-numerical associations, and the numerical distance effect (NDE), which indexes representations of numerical magnitude. The models developed here represent a straightforward Bayesian extension of the classical linear regression approach to measuring these effects that was introduced by \\citet{lorch1990} and first applied to the SNARC effect by \\citet{fias1996}.\n\nBoth models were built from a hierarchical Bayesian framework, which is advantageous for several reasons.  First, the models assume that the regression slopes (the primary object of interest in these analyses) are drawn from a group-level distribution. This hierarchical definition allows both group-level and individual-level slope estimates to be modeled simultaneously from the data. As such, we can answer questions at either the group-level (i.e., is there a SNARC effect?) or participant-level (i.e., are the slopes associated with ohter measures, such as math achievement?).  For example, one might use this framework as a stepping stone to a more complex model, where the group-level slopes $b$ are hypothesized to differ by some independent variable. A concrete instance of this comes from studies in which the authors found group differences in the SNARC effect \\citep[e.g.,][]{fischer2010,cipora2015}.  One could perform similar studies by building an additional linear model on these group-level parameters, with priors appropriate to such effects \\citep[e.g., a Cauchy prior, as in][]{rouder2009}. Importantly, since the inference would be done in a Bayesian framework, it is possible to measure evidence for null effects too \\citep{wagenmakers2007}, so the model could be used to test for \\emph{invariances} as well as differences. \n\nAnother advantage to using a hierarchical Bayesian model for numerical representations is that such models tend to have better measurement fidelity. Indeed, one of the advantages of the Bayesian framework in general is the notion of shrinkage, where extreme parameter estimates are ``shrunk'' toward the mean by virtue of the prior \\citep{gelman2013}. This property is particularly salient for small samples, where classical frequentist methods tend to perform poorly.  I demonstrated exactly this phenomenon in the simulation above: because of some extreme individual-level slope estimates, the frequentist confidence interval was quite wide, and as a result, we could not detect the SNARC effect.  However, the Bayesian highest posterior density interval provided a much more accurate estimate of the true population slope.  Critically, in this simulation, only the Bayesian method produced the correct inference.\n\nA final advantage to this modeling framework that I will mention is that all assumptions of the model are made explicit. This may be a new approach to some, especially those who are accustomed to classical inferential software packages for which the statistical assumptions are kept ``under the hood.''  However, I think the approach presented in this paper can be very useful to a wide variety of problems in cognitive psychology, as the researcher can take the models presented here and modify them for any desired context.  Indeed, the prior knowledge of a given field can be easily and coherently integrated into the model without too much work.\n\nIn summary, the models developed in this paper will provide a flexible tool that can be used to estimate group-level and individual-level numerical representations quickly and coherently.  Further, the methods developed are not specific to numerical cognition, so application to a wide variety of problems should not be too difficult. Regardless of the application, the hierarchical Bayesian linear regression framework presented here provides a powerful, coherent, and accurate measurement model for applied work in cognitive psychology.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nFew-shot learning \\cite{Erik,Brenden} is a classification framework from a very small amount of training data.\nThis framework is used in situations where there is a need to reduce the cost of adding annotations to a large amount of data or there is few data we can use. \nOne promising direction to few-shot learning is based on meta learning \\cite{Finn}, in which\nthe training data is separated into a support set for learning representations and a query set for prediction and computing the loss.\nThis separation unifies the process of learning from the support set and predicting the labels of the query set into a single task.\nThat is, the problem of few-shot learning is formulated as learning a representation of the support and query sets, called support and query vectors.\nThe model-agnostic meta learning (MAML) \\cite{J.Yoon} learns how to learn by optimizing initial parameters. The matching network (MatchNet) \\cite{Vinyals} learns how to add attention or weight from the support set and predicts query labels following the attention mechanism. \nThe prototypical network (ProtoNet) \\cite{Snell} consists of meta learning and metric learning.\nIt is simple to implement, highly extensible, and performs as well as complex models in few-shot learning.\nThe mean vectors of the support vectors are treated as representations for each class, and labels of query vectors are predicted by the distance to the class representations. \nThe task adaptive projection network (TapNet) \\cite{Sung} is based on ProtoNet and learns class-reference vectors  representing each class.\nIt also uses singular value decomposition (SVD) to find a subspace onto which the mean vectors and class-reference vectors are projected nearby. \n\nHowever, these existing methods still have several drawbacks, leading to undesired classification performance.\nProtoNet only takes into account the mean vectors; thus, it causes misclassification when the variance in the support set is relatively large.\nTapNet reduces misalignment of support vectors in the algorithm.\nIn few-shot learning, however, since the amount of data we can use is small, searching for the best features is difficult.\nThus, TapNet can not find better directions but only removes worse direction, which may result in weak feature extraction. \n\nIn this paper, we propose the use of local Fisher discriminant analysis (LFDA) \\cite{Sugiyama} to obtain a feature projection matrix in the feature extraction step in ProtoNet(Fig.\\,\\ref{figure: net}).\nIn LFDA, for samples in each class, first the local within-class covariance matrix and the local between-class covariance matrix are computed. Then, it finds \\emph{any} number of directions or features that minimize the local within-class covariance and maximize the local between-class covariance. Through this LFDA feature extraction step, we can choose a better subspace from the support set and compute the mean vector for each class after projecting \nthem by using the subspace. In the prediction of the query set, we also project query vectors embedded by the network using the subspace obtained in LFDA. By using the mean vectors of the support set and query vectors, we predict the query labels. \nCompared to ProtoNet and TapNet, the remarkable difference is explicitly searching for a better supspace in the algorithm, which leads to significantly better performance in classification.\n\n\\textbf{Contributions:}\nWe make three contributions in this work. \n\\begin{enumerate}\n\\setlength{\\parskip}{-0.1cm} \n\\item We propose a novel few-shot learning algorithm based on ProtoNet and LFDA, which we refer to as LFD-ProtoNet.\n\\item We provide an upper bound of the classification risk for LFDA-ProtoNets, which theoretically guarantees the performance of the proposed method. We analyze the effect of the shot number and feature projection matrix.\n\\item We experimentally show that loss decreases much faster and the accuracy is better than that of TapNet for small iteration complexity.\n\\end{enumerate}\n\nThe code is available online\\footnote{Code for LFD-ProtoNet is at \\url{https:\/\/github.com\/m8k8\/LFD_ProtoNet.git}. Note: there are some mistakes in our code and the results arer wrong and we'll do experiments again and show correct results.}.\n\n\\section{Problem formulation and notations}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=15cm]{network_proto.png}\n    \\caption{\\textbf{Prototypical few-shot learning algorithms.} The representation vectors and feature projection matrix are obtained from the support set $\\textbf{\\textrm{S}}$, and the query labels of the query set $\\textbf{\\textrm{Q}}$ are predicted by the squared euclidean distance (SED). The difference of networks is how to obtain the representation vectors and the feature projection matrix.}\n    \\label{figure: net}\n\\end{figure}\nIn few-shot learning, we need to prepare training data with a different configuration from ordinary machine learning \\cite{Erik,Brenden}.\nA task set is a training data consisting of meta-training data and meta-test data, referred to as a support set and a query set, respectively.\n\nLet $\\mathcal{X}$ be the input space and $\\mathcal{Y}$ be the output space.\nDenote by $x_{c,i}$ the $i$-th input of class $c$.\nThese samples are drawn independently from an unknown distribution $\\mathcal{D}$, i.e., $(x_{c,i},c)\\sim \\mathcal{D}$.\nWe define a task as a pair of the support set and query set, and we denote by $\\mathcal{T}$ the task distribution.\nThe task $T=\\{D_\\mathrm{s},D_\\mathrm{q}\\}\\sim \\mathcal{T}$ is composed of support set $D_{s}=\\{(x_{c,i}\\in \\mathcal{X},c\\in \\mathcal{Y})_{c,i}\\}$ and query set $D_{q}=\\{(x'_{c,i}\\in \\mathcal{X},c\\in \\mathcal{Y})_{c,i}\\}$. \nThe number of support sets is limited such that when we classify $C$ classes in $k$-shot learning, the number of the samples per class is $k$. \n\nDenote by $m$ the dimension of the embedded latent feature space and by $n$ that of the space projected by feature projection matrix.\nWe define the network $f_{\\theta}:\\mathcal{X}\\rightarrow \\mathbb{R}^m$ that embeds an input sample $x_{c,i}\\in \\mathcal{X}$ to a latent feature space space $\\mathbb{R}^m$ with the parameter $\\theta$.\nFor learning representations, we define the feature extractor $\\mathrm{FE}$ that is given some information such as embedded support vectors or support labels and returns a feature projection matrix $F\\in \\mathbb{R}^{m\\times n}$.\nIt is expected that $F$ extracts better statistical information from $m$-dimensional latent feature space to $n$-dimensional latent feature space.\nLet $\\overline{f_{\\theta}(x_{c,\\cdot})}\\in \\mathbb{R}^n$ be a representation vector for each class $c$ to predict the label of the query vector with the Euclidean distance.\n\nFinally, we define the loss function $g:\\mathbb{R}^n\\times \\mathbb{R}^n\\times \\mathbb{R}^{(k-1)\\times n}\\rightarrow \\mathbb{R}$ and a generalization loss of the network $f_{\\theta}$ as follows. \nLet $v_c$ be  the query vector that is embedded by $f_{\\theta}$ and projected by $F$.\n\\begin{align}\ng(v_c,\\overline{f_{\\theta}(x_{c,\\cdot})},\\{\\overline{f_{\\theta}(x_{s,\\cdot})}\\}_{s\\neq c}):=d(v_c,\\overline{f_{\\theta}(x_{c,\\cdot})})+\\log \\sum_{s\\neq c} \\exp (-d(v_c,\\overline{f_{\\theta}(x_{s,\\cdot})})),\n\\end{align}\nwhere  the distance function $d(\\cdot)$ is given by the L2 norm, i.e, $d(v_c,\\overline{f_{\\theta}(x_{s,\\cdot})})=\\|v_c-\\overline{f_{\\theta}(x_{s,\\cdot})}\\|_2^2$ and $\\{\\overline{f_{\\theta}(x_{s,\\cdot})}\\}_{s\\neq c}$ indicates all class representation vectors excluding class $c$. \n\nThe generalization loss $L_{\\mathcal{T}}(f)$ is defined by\n\\begin{align}\nL_{\\mathcal{T}}(f):=\\mathbb{E}_{T\\sim \\mathcal{T}}[g(Ff(x'_c),\\overline{f_{\\theta}(x_{c,\\cdot}))},\\{\\overline{f_{\\theta}(x_{s,\\cdot})}\\}_{s\\neq c}].\n\\end{align}\nThe empirical loss $\\widehat{L}_{\\mathcal{T}}(f)$ is also defined by\n\\begin{align}\n\\widehat{L}_{\\mathcal{T}}(f):=\\frac{1}{N}{\\displaystyle \\sum_{T_u\\sim \\mathcal{T},1\\leq u \\leq N}}\\frac{1}{CM}\\sum_{c=1}^C\\sum_{i=1}^Mg(Ff(x'_{c,i}),v_b,\\{\\overline{f_{\\theta}(x_{s,\\cdot})}\\}_{s\\neq b}),\n\\end{align}\nwhere $N$ is the number of tasks, $M$ is the number of query data per class, and $C$ is the number of classes.\n\nWe construct a hypothesis set $\\mathcal{H}=\\{f_{\\theta} | \\theta \\in \\Omega\\}$ and aim to solve the following optimization problem:\n\\begin{align}\nf^*=\\mathop{\\rm arg~min}\\limits_{f\\in \\mathcal{H}} L_{\\mathcal{T}}(f).\n\\end{align}\nHowever, since $L_{\\mathcal{T}}$ is not accessible, we minimize the empirical risk in practice:\n\\begin{align}\n\\widehat{f^*}=\\mathop{\\rm arg~min}\\limits_{f\\in \\mathcal{H}} \\widehat{L}_{\\mathcal{T}}(f).\n\\end{align}\nWe explain existing work on the basis of this problem formulation as follows. \n\n\n\\textbf{The prototypical network (ProtoNet)} \\cite{Snell} is a pioneering algorithm of combining meta learning and few-shot learning.\nA number of variants based on this algorithm have been proposed because of its simplicity and better performance \\cite{Sung,Oreshkin}.\nThe simplicity of ProtoNet lies in making the feature projection matrix $F_{\\textrm{Proto}}$ an identity matrix, i.e., $F_{\\textrm{Proto}}=I_m(m=n)$, in the feature extraction step. \nMoreover, what ProtoNet requires for obtaining the $c$-th class representation $\\overline{f_{\\theta}(x_{c,\\cdot})}$ is only to average support vectors, which are embedded by $f_{\\theta}$, belonging to class $c$. \nThat is, $\\overline{f_{\\theta}(x_{c,\\cdot})}$ is simply given by\n\\begin{align}\n\\overline{f_{\\theta}(x_{c,\\cdot})}=\\frac{1}{k} \\sum_{i=1}^k f_{\\theta}(x_{c,i}).\n\\end{align}\n\n\\textbf{The task adaptive projection network (TapNet)} \\cite{Sung} adds a feature projection function to ProtoNet. \nAs with ProtoNet, the training data is split into the support set and the query set.\nThen, a class reference vector $\\phi_c$ is introduced to represent each class $c$.\nIn the feature extraction step of TapNet, a matrix projecting the class reference vectors $\\phi_c$ and the mean vectors per class $\\frac{1}{k}\\sum_{i=1}^{k} f(x_{c,i})$ to the same location for each class is found where $k$ is the number of the data of class $c$ and $\\overline{f_{\\theta}(x_{c,\\cdot})}$ indicates that $x_c$ belongs to class $c$. \nWhen the norm of all vectors is ignored and only their directions are considered, the matrix $F_{\\textrm{SVD}}$ is obtained by the following equation with singular value decomposition:\n\\begin{align}\n\\forall c \\in \\{ 1,2,3\\ldots,C \\}.~F_{\\textrm{SVD}}\\frac{\\overline{f_{\\theta}(x_{c,\\cdot})}}{\\| \\overline{f_{\\theta}(x_{c,\\cdot})} \\|_2} =F_{\\textrm{SVD}}\\frac{\\phi_c}{\\| \\phi_c \\|_2}.\n\\end{align}\nThe query data is similarly embedded by $f_{\\theta}$ and projected by $F_{\\textrm{SVD}}$.\nFinally, their labels are estimated by the distances between $F_{\\textrm{SVD}} f(x_{c,i})$ and $F_{\\textrm{SVD}}\\phi_k$.\n\nFollowing Chen et al.\\cite{Chen}, the schematics of ProtoNet and TapNet are illustrated in Fig.\\,\\ref{figure: net}.\n\n\n\n\n\\section{Preliminaries}\n\\label{prelim}\n\\textbf{Fisher discriminant analysis (FDA)} \\cite{Fisher} is used for finding the subspace to make the within-class covariance $S_{\\mathrm{wit}}$ small and between-class covariance $S_{\\mathrm{bet}}$ large:\n\\begin{align}\nS_{\\mathrm{wit}} = \\frac{1}{kC} \\sum_{c=1}^C\\sum_{i=1}^k(x_{c,i} - \\mu_c)(x_{c,i}-\\mu_c)^{\\top},\n\\quad\nS_{\\mathrm{bet}} = \\frac{1}{C} \\sum_{c=1}^C(\\mu_c - \\mu)(\\mu_c -\\mu)^{\\top},\n\\end{align}\nwhere $k$ is the number of samples in each class, $\\mu_c$ is the mean vector of the class $c$, and $\\mu$ is the mean vector of all samples in $\\mathbb{R}^n$. We take $\\bm{w}$ as the directions to project samples and minimize the ratio of the within-class covariance of vectors projected by $\\bm{w}$ and their between-class covariance:\n\\begin{align}\n\\min_{\\bm{w}} \\frac{\\bm{w}^{\\top}S_{\\mathrm{wit}}\\bm{w}}{\\bm{w}^{\\top}S_{\\mathrm{bet}}\\bm{w}}.\n\\end{align}\nThe $\\bm{w}$ is the best direction that minimizes the ratio.\nWe can also find some directions that also make the ratio small if the problem is a multi-class case. To obtain these directions, we solve a modified problem\n\\begin{align}\nW_{\\mathrm{FDA}}=\\mathop{\\rm arg~min}\\limits_{W} \\mathrm{Tr}((W^{\\top}S_{\\mathrm{bet}}W)^{-1}W^{\\top}S_{\\mathrm{wit}}W),\n\\end{align}\nwhere $W$ is the $n\\times (C-1)$ matrix and $\\mathrm{Tr}(\\cdot)$ is the trace of the matrix. \nSince the solution is the eigenvectors of $S_{\\mathrm{bet}}^{-1}S_{\\mathrm{wit}}$, the rank of this matrix is just $C$, which is the number of the class.\nThat is, the total number of eigenvectors is also just $C$, which is a problem with few-shot learning described in Sec.\\ref{sec:prop}.\n\n\\textbf{Local Fisher discriminant analysis (LFDA)} \\cite{Sugiyama} is an extension of FDA.\nWhen samples in a class are multimodal, keeping local within-class scatter can be hard in FDA because multimodal samples should be merged into a single cluster.\nThis constraint results in less separate embedding due to less degree of freedom. \nTo solve this problem, LFDA combined FDA and \\textit{locality-preserving projection} \\cite{Niyogi} and construct the within-class covariance $S_{\\mathrm{wit}}^{A}$ and between-clss covariance $S_{\\mathrm{bet}}^{A}$ by using the affinity matrix $A$ whose elements are the similarities of samples in the same class, e.g.,\nthe squared exponential kernel $\\exp(-(||x_i-x_j||)^2)$ is used for $A_{i,j}$.\nThe details of $S_{\\mathrm{wit}}^{A}$ and $S_{\\mathrm{bet}}^{A}$ are in Appendix \\ref{Sec:LFDA} in the supplementary material.\nThe objective function to minimize is the same, i.e.,\n\\begin{align}\nW_{\\mathrm{LFDA}}=\\mathop{\\rm arg~min}\\limits_{W}\\mathrm{Tr}((W^{\\top}S_{\\mathrm{bet}}^{A}W)^{-1}W^{\\top}S_{\\mathrm{wit}}^{A}W).\n\\end{align}\nIn FDA, the rank of $(S_{\\mathrm{bet}})^{-1}S_{\\mathrm{wit}}$ is just the number of classes $C$; however, in LFDA the rank of $(S_{\\mathrm{bet}}^{A})^{-1}S_{\\mathrm{wit}}^{A}$ is the number of samples $kC$ because by adding the similarity terms, vectors with a linear dependency in FDA has a linear dependency.\n\n\n\\section{Proposed method}\n\\label{sec:prop}\nSince the feature projection matrix in ProtoNet is just an identity matrix $I_n$, ProtoNet uses no information about the support set.\nThat is, incorporating the support set can improve the classification performance of few-shot learning based on ProtoNet. \nThe feature projection matrix $F_{\\textrm{SVD}}$ of TapNet aims to reduce the misalignment of support vectors by eliminating worse directions up to $C$. \nThis means that if we have much more support data, we can reduce misalignment more; however, the more support data we obtain, the less features we can use. \nThis is counterintuitive because the ideal situation is that if we obtain more data for the support set, then we can obtain more features for each class. \nIn this section, we propose a novel feature projection matrix in accordance with the intuition that more data lead to more useful features.\n\n\\subsection{Algorithm}\n\\label{sec:alg}\nWe suppose that if we can make the local within-class covariance of the support set smaller and at the same time can make its local between-class covariance larger, then the classification performance is expected to be improved. \nThis concept was originally introduced in FDA \\cite{Fisher}; hence, using FDA for feature extraction is one option. \nThe dimension of the features extracted by FDA, however, is limited to the rank of the covariance matrix, i.e., $C-1$ as described in Sec.\\,\\ref{prelim}.\nThat is, the expression power of the FDA features is typically insufficient.\n To solve this problem, we propose to use LFDA, in which we can increase the dimension of the extracted features to $kC-1$  as described in Sec.\\,\\ref{prelim}.\n LFDA can usually extract any number of the features up to the number of the support set so we can use more features from the support set.\n \nBy using the feature projection matrix $ F_{\\textrm{LFDA}}^{c}$ in LFDA (see Eq.\\,\\eqref{eq:lfd} below), we formulate the representation vector $\\overline{f_{\\theta}(x_{c,\\cdot})}$ as the mean vectors of $F_{\\textrm{LFDA}}^{c}f_{\\theta}(x_{c,i})$ in terms of $i$, i.e.,\n\\begin{align}\n\\overline{f_{\\theta}(x_{c,\\cdot})}=\\frac{1}{k}\\sum_{i=1}^{k}F_{\\textrm{LFDA}}^{c}f_{\\theta}(x_{c,i}).\n\\end{align}\nThe query vectors are embedded by $f_{\\theta}$, projected by $F_{\\textrm{LFDA}}^{c}$, and predicted as the class $c$ that is the class of the nearest representation vector to the query vector.\nWe summarize the proposed algorithm in Algorithm\\,\\ref{fig: bo}.\n\\begin{algorithm}[t!]\n    \\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n    \\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n    \\caption{Few-shot learning (k-shot) algorithm framework based on ProtoNet}\n    \\label{fig: bo}\n    \\textbf{Notations:} Denote by $T$ a task, drawn from $\\mathcal{T}$, compased of support set $D_s$ and query set $D_q$. Denote by $L_{tr}$ a training loss, by $F_{\\mathrm{LFDA}}^{c}$ a feature projection matrix of local Fisher discriminant analysis, and by $f_{\\theta}$ an embedding function with parameters $\\theta$. In this algorithm $C$ means the number of classes in the task and $M$ means the number of query samples per class.\n    \\begin{algorithmic}[1]\n    \\REQUIRE training task $\\{T_u\\}_{u=1}^N \\sim \\mathcal{T}$ where $T_u=\\{D_s,D_q\\}$, $D_s=\\{(x_{c,i},c)\\}_{1\\leq c \\leq C,1\\leq i \\leq k}$ and $D_q=\\{(x'_{c,i},c)\\}_{1\\leq c \\leq C,1\\leq i \\leq M}$.\n    \\STATE $L_{tr}\\leftarrow 0$\n    \\FOR{$u$ in $u=0,1,\\ldots,N$}\n    \\STATE $(D_s,D_q)\\leftarrow T_u$\n    \\STATE $F_{\\mathrm{LFDA}}^{c}=\\mathop{\\rm arg~min}\\limits_{W} \\mathrm{Tr}((W^{\\top}\\Sigma_{F,c}W)^{-1}W^{\\top}\\Sigma_FW)$, $c=1,\\ldots,C$ using Eqs.\\,\\eqref{eq:sigmaf} and \\eqref{eq:sigmafc}.\n    \\STATE $\\{\\overline{f_{\\theta}(x_{c,\\cdot})}\\}_{c=1}^C=\\{\\frac{1}{k}\\sum_{i=1}^k F_{\\textrm{LFDA}}^{c}f_{\\theta}(x_{c,i})\\}_{c=1}^C$\n    \\STATE $L_{T_u}\\leftarrow 0$\n    \\FOR{$c$ in $c=0,1,\\ldots,C$}\n    \\FOR{$i$ in $i=0,1,\\ldots,M$}\n    \\STATE $L_{T_u} \\leftarrow L_{T_u} + g(F_{\\textrm{LFDA}}^{c}f_{\\theta}(x'_{c,i}),\\overline{f_{\\theta}(x_{c,\\cdot})},\\{\\overline{f_{\\theta}(x_{s,\\cdot})}\\}_{s\\neq c})$\n    \\ENDFOR\n    \\ENDFOR\n    \\STATE $L_{tr}\\leftarrow L_{tr} + \\frac{1}{CM}L_{T_u}$\n    \\ENDFOR\n    \\STATE $L_{tr}\\leftarrow \\frac{1}{N}L_{tr}$\n    \\STATE update $\\theta$ with $L_{tr}$\n    \\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Theoretical analysis}\n\\label{sec:theory}\nThe effect of $k$ in ProtoNet was analyzed by Cao et al. \\citep{Tianshi}. \nWe analyze our algorithm in line with their work and  show that how our algorithm is theoretically better than ProtoNet.\n\nAs in Cao et al. \\citep{Tianshi}, we first consider, for simplicity, the case where the query is the binary classification of class $\\mathrm{a}\\in \\mathcal{Y}$ or $\\mathrm{b}\\in \\mathcal{Y}$. The result can be easily generalized to the multi-class classification (see Appendix \\ref{ap:multi}).\nThese $\\mathrm{a}$ and $\\mathrm{b}$ are random variables from all class sets $\\mathcal{Y}$. The support set of $\\mathrm{a}$ is defined as $\\mathcal{S}_{\\mathrm{a}}=\\{x_{\\mathrm{a},i}\\}_{i=1}^k$, and that of $b$ is defined as $\\mathcal{S}_{\\mathrm{b}}=\\{x_{\\mathrm{b},i}\\}_{i=1}^k$. \nThe whole support set is $S=\\{\\mathcal{S}_{\\mathrm{a}},\\mathcal{S}_{\\mathrm{b}}\\}$. \nThe support sets $\\mathcal{S}_{\\mathrm{a}}$ and $\\mathcal{S}_{\\mathrm{b}}$ are embedded by the network $\\phi$ and a feature projection matrix $F$ is obtained by LFDA to make the local between-class covariance large and the local within-class covariance small. With LFDA, we get the representation vectors of classes $\\mathrm{a}$ and $\\mathrm{b}$. When we take $x\\in \\mathcal{X}$ from the query set, it is also embedded by $\\phi$, projected by $F$, and finally the distances to $\\mathrm{a}$ and $\\mathrm{b}$ are compared. In the analysis below, we assume that the query $x$ belongs to class $\\mathrm{a}$. \n\\begin{definition}[Representation vector of class]\nWe define $\\overline{\\phi(\\mathcal{S}_c)}$ as the mean vector of support vectors in class $c\\in \\{\\mathrm{a},\\mathrm{b}\\}$ so the representation vector of class $c$ is written as $\\overline{\\phi(\\mathcal{S}_c)}$. In the $k$-shot learning $|\\mathcal{S}_c|=k$, we can write $\\overline{\\phi(\\mathcal{S}_c)}$ as\n\\begin{align}\n\\overline{\\phi(\\mathcal{S}_c)}=\\frac{1}{k}\\sum_{i=1}^k\\phi(x_{a,i}).\n\\end{align}\nWe also define $\\overline{F\\phi(\\mathcal{S}_c)}$ as the mean vector of support vectors projected by feature projection matrix $F$ in class $c$.\nThe representation vector of class $c$ after the feature extraction step is written as\n\\begin{align}\n\\overline{F\\phi(\\mathcal{S}_c)}=\\frac{1}{k}\\sum_{i=1}^kF\\phi(x_{a,i}).\n\\end{align}\n\\end{definition}\n\n\\begin{definition}[Between-class covariance and within-class covariance]\nWe define between-class covariance matrix $\\Sigma$ and within-class covariance matrix of class $c$ $\\Sigma_c$ as\n\\begin{align}\n\\Sigma&=\\frac{1}{2}((\\overline{\\phi(\\mathcal{S}_a)}-\\overline{\\phi(\\mathcal{S})})(\\overline{\\phi(\\mathcal{S}_a)}-\\overline{\\phi(\\mathcal{S})})^{\\top}+(\\overline{\\phi(\\mathcal{S}_b)}-\\overline{\\phi(\\mathcal{S})})(\\overline{\\phi(\\mathcal{S}_b)}-\\overline{\\phi(\\mathcal{S})})^{\\top}),\\\\\n\\Sigma_c &=\\frac{1}{k}\\sum_{i=1}^k(\\phi(x_{c,i})-\\overline{\\phi(\\mathcal{S}_c)})(\\phi(x_{c,i})-\\overline{\\phi(\\mathcal{S}_c)})^{\\top}.\n\\end{align}\nWe also define $\\Sigma_{F}$ and $\\Sigma_{F,c}$ as both the between-class covariance matrix projected by $F$ and within-class covariance matrix projected by $F$ in class $c$ as follows.\n\\begin{align}\n\\Sigma_{F}=&\\frac{1}{2}\\Bigl\\{\\left(\\overline{F\\phi(\\mathcal{S}_a)}-\\overline{F\\phi(\\mathcal{S})}\\right)\\left(\\overline{F\\phi(\\mathcal{S}_a)}-\\overline{F\\phi(\\mathcal{S})}\\right)^{\\top}\\notag\\\\\n&+\\left(\\overline{F\\phi(\\mathcal{S}_b)}-\\overline{F\\phi(\\mathcal{S})}\\right)\\left(\\overline{F\\phi(\\mathcal{S}_b)}-\\overline{F\\phi(\\mathcal{S})}\\right)^{\\top}\\Bigr\\}\\label{eq:sigmaf},\\\\\n\\Sigma_{F,c} =&\\frac{1}{k}\\sum_{i=1}^k\\left(F\\phi(x_{c,i})-\\overline{F\\phi(\\mathcal{S}_c)}\\right)\\left(Ff(x_{c,i})-\\overline{F\\phi(\\mathcal{S}_c)}\\right)^{\\top}\\label{eq:sigmafc}.\n\\end{align}\n\\end{definition}\n\n\\begin{definition}[Task loss]\nThe task loss $\\ell_{\\mathrm{task}}(T)$ with $0$-$1$ loss $\\ell_{err}$ is defined as\n\\begin{align}\n\\ell_{\\mathrm{task}}(T) = \\frac{1}{MC}\\sum_{i=1}^{MC} \\ell_{\\mathrm{err}}(\\widehat{y_i},y_i),\n\\end{align}\nwhere $T=\\{D_s,D_q\\}\\sim \\mathcal{T}$, $D_\\mathrm{s}$ and $D_\\mathrm{q}$ are the support set and query set, and $\\widehat{y_i}$ and $y_i$ are the $i$-th estimated label and the $i$-th true label in the query set $D_q$.\n\\end{definition}\n\n\\begin{definition}[Empirical risk of $\\phi$]\nWe define the empirical risk of $\\phi$ using task loss $\\ell_{\\mathrm{task}}$ where $m$ tasks $T_u$ are drawn independently from the task distribution $\\mathcal{T}$, i.e.,\n\\begin{align}\n\\widehat{R}_{n,\\mathcal{T},c}(\\phi) = \\frac{1}{N}\\sum_{u=1}^N\\ell_{\\mathrm{task}}(T_u),\\quad T_u\\sim \\mathcal{T}~(u=1,\\ldots,N).\n\\end{align}\n\\end{definition}\n\n\\begin{definition}[Expected risk of $\\phi$]\n\\label{def:ex_risk}\nWe define the risk of $\\phi$ using the expectation in terms of the task distribution $\\mathcal{T}$ as\n\\begin{align}\nR_{\\mathcal{T},c}(\\phi)=\\mathbb{E}[\\widehat{R}_{n,\\mathcal{T},c}(\\phi)].\n\\end{align}\n\\end{definition}\n\n\\begin{thm}[Upper-bound of expected risk with LFDA]\\label{maintheorem}\nConsider $k$-shot learning. Under the same assumptions as Cao et al.\\cite{Tianshi}, in which $\\Sigma_a=\\Sigma_b$ and $p(\\phi(X)|Y(X)=c)$ is the Gaussian distribution with mean $\\mu_c$ and variance $\\Sigma_c$, i.e., $\\phi(X)|Y(X)=c \\sim \\mathcal{N}(\\mu_c,\\Sigma_c)$,  the expected risk of $\\phi$ with the $0$-$1$ loss is bounded as\n\\begin{align}\nR_{\\mathcal{T},c}(\\phi) \\leq 1 - \\frac{4\\mathrm{Tr}\\left(\\Sigma_F\\right)^2}{8\\left(1+\\frac{1}{k}\\right)^2\\mathrm{Tr}\\left(\\Sigma_{F,c}^2\\right)+16\\left(1+\\frac{1}{k}\\mathrm{Tr}(\\Sigma_F \\Sigma_{F,c})\\right) + \\mathbb{E}[\\left((\\mu_a-\\mu_b)^{\\top}F^{\\top}F(\\mu_a-\\mu_b)\\right)^2]}.\\label{eq:main}\n\\end{align}\n\\end{thm}\n\nA Proof of Theorem \\ref{maintheorem} is given in Appendix \\ref{ap:proof}.\nThe numerator $4\\mathrm{Tr}(\\Sigma_F)^2$ is $\\mathcal{O}(\\Sigma_F^2)$, the first term in the denominator $8(1+\\frac{1}{k})^2\\mathrm{Tr}(\\Sigma_{F,c}^2)$ is $\\mathcal{O}(\\Sigma_{F,c}^2)$, the second term $16(1+\\frac{1}{k}\\mathrm{Tr}(\\Sigma_F \\Sigma_{F,c}))$ is $\\mathcal{O}(\\Sigma_F\\Sigma_{F,c})$, and the last term $\\mathbb{E}[((\\mu_a-\\mu_b)^{\\top}F^{\\top}F(\\mu_a-\\mu_b))^2]$ is $\\mathcal{O}(\\Sigma_F^2)$. For the last term, if we assume that $F$ satisfies the conservation of the norm, it is clear that $(\\mu_a-\\mu_b)^{\\top}F^{\\top}F(\\mu_a-\\mu_b)$ becomes large when the between-class covariance is relatively large. Thus we can conclude that if $\\mathrm{Tr}(\\Sigma_{F}^{-1}\\Sigma_{F,c})$ is small, the right-hand side of the inequality in (\\ref{eq:main}) becomes small so that the risk will be close to zero. \nMoreover, LFDA tries to find the subspace that makes $\\mathrm{Tr}(\\Sigma_F^{-1}\\Sigma_{F,c})$ minimum, i.e.,\n\\begin{align}\n\\label{eq:lfd}\nF_{\\mathrm{LFDA}}^{c}=\\mathop{\\rm arg~min}\\limits_{W} \\mathrm{Tr}((W^{\\top}\\Sigma_{F}W)^{-1}W^{\\top}\\Sigma_{F,c}W).\n\\end{align}\nThus, we can expect that  LFD-ProtoNet performs better than ProtoNet since $F$ is an identity matrix in ProtoNet.\n\n\n\\section{Experiment}\nIn Section \\ref{sec:theory}, we showed that our algorithm improves the upper bound of the risk if $\\Sigma_F^{-1}\\Sigma_{F,c}$ is smaller than $\\Sigma^{-1} \\Sigma_c$.\nIn this experiment, we check how much better the performance of our algorithm compared to other few-shot methods.\nWe also did experiment for comparing the trace value of $\\Sigma_{F}^{-1}\\Sigma_{F,c}$ in LFD-ProtoNet and $\\Sigma^{-1} \\Sigma_c$ in ProtoNet.\n\\subsection{Dataset}\nWe used two benchmark datasets well-used in few-shot learning.\n\\paragraph{miniImageNet\\citep{Vinyals}}\nThis dataset is a subset of the ILSVRC-12 ImageNet datas\\citep{Russakovsky} with 100 classes and $600$ images per class. In this setting, the size of  the images is $84$ $\\times$ $84$. And the training data contains $64$ classes, the validation data contains $16$ classes, and the test data contains $20$ classes.\n\\paragraph{tieredImageNet\\citep{Ren}}\nThis dataset is a larger set than miniImageNet with $608$ classes and $779,165$ images. It has $34$ categories, and these categories are split into $20$ training, $6$ validation, and $8$ test categories.\n\n\\subsection{Implementation detail}\nWe performed an experiment on $5$-shot and $1$-shot case with miniImageNet and tieredImageNe. \nThe number of the iterations was $40,000$ and we used ResNet-$12$\\citep{K.He} as the network where the output dimension was $128$. We generated training data, validation data, and test data with a data generator. As the first step, the training data was split into the support set and query set. Then, we made many tasks that contain support data for feature extraction and query data for computing loss function. In this experiment, we used cross-entropy loss as the loss function. For each task, the loss was calculated, and the parameters of the network or embedding function were updated in the training step. In $5$-shot learning, we obtained $5$ samples for each class. That is, the number of all samples for feature extraction was $24$ in the $5$-class classification problem. As a property of the features, from $n$ samples, we can obtain at most $n-1$ features, which means that we can obtain by LFD at most $24$ features in this setting. Similarly in $1$-shot learning, we can obtain $4$ features from the support set.\n\nWe also performed an experiment with LFDA in the $5$-shot case of miniImageNet and compared the performance of the LFDA and FDA cases in respect of the number of features from the support set.\n\nAs we showed in Section \\ref{sec:theory}, small $\\mathrm{Tr}(\\Sigma_F^{-1}\\Sigma_{F,c})$ is preferable and we performed an experiment comparing the value $\\mathrm{Tr}(\\Sigma_F^{-1}\\Sigma_{F,c})$ with $\\mathrm{Tr}(\\Sigma^{-1}\\Sigma_c)$ of ProtoNet. This result is shown in the supplementary material due to the lack of space.\n\n\\subsection{Results}\nTable\\ref{tab:result:miniImage_tieredImage} shows that LFD-ProtoNet achieved $-\\%$ on the miniImageNet($1$-shot), $-\\%$ on the miniImageNet($5$-shot), $-\\%$ on the tieredImageNet($1$-shot), and $-\\%$ on the tieredImageNet($5$-shot). It is clear that the method with LFDA achieves the best performance of all other methods, and this is because searching for the best subspace to project positively is superior to reducing worse directions such as TapNet. In the case of $1$-shot learning, the number of samples for each class is exactly $1$ and this fact means that FDA can only consider the within-class covariance. However, it is sufficient for LFD-ProtoNet to outperform others only with the local within-class covariance and it also shows that searching for the better subspace makes sense. \n\nWe show the results of the case of FDA. In $5$-shot learning, when we use FDA as the feature extractor, the accuracy is only $-\\%$, which is the almost same as that of adaResNet. It can be considered that FDA returns features up to only the number of classes; thus, if total classes are $C$, we can extract only $C-1$ features, and it is insufficient for the training.  If we use LFDA, However, we can extract features up to the number of samples.\nTherefore, in the $k$-shot case, the number of samples is $kC-1$ in the training step and it is sufficient for the network to learn.\n\nWe measured loss decreasing speed of TapNet and LFD-ProtoNet. In TapNet, the training loss decreased slowly up to $30,000$ epochs. This can result in overfitting to training data. However, in LFD-ProtoNet, the training loss quickly decreased for $10,000$ epochs and this fact can be thought as fast adaptation without overfitting.\n\n\\begin{table}[!t]\n    \\caption{The result of the few-shot learning experiment with miniImageNet and tieredImageNet. The N\/A indicates ``not available in the original paper''.}\\label{tab:result:miniImage_tieredImage}\n    \\centering\n    \\renewcommand{\\arraystretch}{1.2}\n    \\begin{tabular}{l||c|c||c|c}\n    \\toprule\n    Method & \\multicolumn{2}{|c||}{miniImageNet} & \\multicolumn{2}{|c}{tieredImageNet}\\\\\n    \\hline\n    & $1$-shot & $5$-shot & $1$-shot & $5$-shot \\\\\n    \\midrule\n    Matching Nets \\citep{Vinyals} & $43.56 \\pm 0.84\\%$ & $55.31 \\pm 0.73\\%$ & N\/A & N\/A \\\\\n    MAML \\citep{Finn} & $48.70 \\pm 1.84\\%$ & $63.15\\pm 0.91\\%$ & $51.67 \\pm 1.81\\%$ & $70.30 \\pm 1.75\\%$\\\\\n    ProtoNet \\citep{Snell} & $49.42 \\pm 0.78\\%$ & $68.20 \\pm 0.66\\%$ & $53.31 \\pm 0.89\\%$ & $72.69 \\pm 0.74\\%$\\\\\n    SNAIL \\citep{Mishra} & $55.71 \\pm 0.99\\%$ & $68.88 \\pm 0.92\\%$ & N\/A & N\/A\\\\\n    adaResNet \\citep{Munkhdalai} & $56.88 \\pm 0.62\\%$ & $71.94 \\pm 0.57\\%$ & N\/A & N\/A\\\\\n    TPN \\citep{Liu} & $55.51 \\pm 0.86\\%$ & $69.86 \\pm 0.65\\%$ & $59.91 \\pm 0.94\\%$ & $73.30 \\pm 0.75\\%$ \\\\\n    TADAM-$\\alpha$ \\citep{Oreshkin} & $56.8 \\pm 0.3\\%$ & $75.7 \\pm 0.2\\%$ & N\/A & N\/A\\\\\n    TADAM-TC \\citep{Oreshkin} & $58.5 \\pm 0.3\\%$ & $76.7 \\pm 0.3\\%$ & N\/A & N\/A\\\\\n    Relation Nets \\citep{Yang} & N\/A & N\/A & $54.48 \\pm 0.93\\%$ & $71.31 \\pm 0.78\\%$\\\\\n    TapNet \\citep{Sung} & $61.65 \\pm 0.15\\%$ & $76.36 \\pm 0.10\\%$ & $63.08 \\pm 0.15\\%$ & $80.26 \\pm 0.12\\%$\\\\\n    \\textcolor{red}{LFD-ProtoNet(Ours)} & \\textcolor{red}{$- \\pm 0.32\\%$} & \\textcolor{red}{$76.5 \\pm 0.10\\%$} & \\textcolor{red}{$-\\pm 0.13\\%$} & \\textcolor{red}{$78.0 \\pm 0.10\\%$}\\\\\n    \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\nWe have proposed LFD-ProtoNet in few-shot learning problem settings.\nOur method focuses on the covariance and mean of the support set. Such a feature extraction method is realized with LFDA, and the accuracy of LFD-ProtoNet improves by $-\\%$ compared to TapNet which is the state-of-the- art variant of ProtoNet.\nMoreover, the speed of the loss decreasing is much faster than that of TapNet and these result shows that LFDA extracts sufficient information to describe each class.\nWe theoretically explained that our feature extraction can maximize the expected risk bound in the $k$-shot learning. \nAs in ProtoNet, LFD-ProtoNet is simple and easy to implement.\n\nAs our future work, we can add a pre-training step such as optimization of the initial parameters and we can consider the semi-supervised condition that we can also access some data without any annotation. These additional techniques are expected to further improve LFD-ProtoNet.\n\\newpage\n \n\\section*{Acknowledement}\nMS was supported by JST CREST Grant Number JPMJCR18A2.\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{sec:intro}\n\nIn an incomplete financial setting with noise governed by a continuous\nmartingale and in which the investor's preferences are modeled by a\nnegative power utility function, we provide a  second-order Taylor expansion\nof the investor's value function with respect to perturbations of the\nunderlying market price of risk process. We show that tractable models can\nbe used to approximate highly intractable ones as long as the latter can\nbe interpreted as perturbations of the former.  As a by-product of our  analysis we explicitly construct first-order approximations of both the primal and the dual optimizers. Finally, we apply our approximation in two numerical\nexamples.\n\n\nThere are two different ways of looking at our contribution: as a tool to \napproximate the value function and \nperform numerical computations, or as a stability result with applications to statistical estimation. Let us elaborate on these, and the\nrelated work,  in order.\n\n\\par{\\em An approximation interpretation.} The conditions for existence and uniqueness of the investor's utility optimizers are\nwell-established (see \\cite{KLSX} and  \\cite{KraSch99}). However, in\ngeneral settings, the numerical computation of the investor's value\nfunction remains a challenging problem. Various existing approaches\ninclude:\n\n\\begin{enumerate}\n\n\\item In Markovian settings, the value function can typically be\ncharacterized by a HJB-equation.  Its numerical implementation\nthrough a finite-grid approximation is naturally subject to the curse of\ndimensionality. Many authors (see \\cite{KimOmb96}, \\cite{Wac2002}, \\cite{ChaVic2005},\n\\cite{Kra05}, and \\cite{Liu2007}) opt for affine and \nquadratic models for\nwhich closed-form solutions exist. Going beyond these specifications\nin high-dimensional settings by using PDE-techniques seems to be\nvery hard computationally.\n\n\\item In general (i.e., not necessarily Markovian) complete models,\n\\cite{CviGouZap2003} and \\cite{DetGarRin2003} provide efficient Monte Carlo\nsimulation techniques based on the martingale method for complete markets\ndeveloped in \\cite{CoxHuang1} and \\cite{KLS}.\n\n\\item Other approximation methods are based on various Taylor-type\nexpansions. The authors of \n\\cite{Cam1993} and \\cite{CamVic1999} log-linearize the investor's budget constraint as well as the investor's first-order condition for optimality.  \\cite{KogUpp2000}  expand\nin the investor's risk-aversion coefficient around the log-investor (the myopic investor's problem is known to be tractable even in incomplete settings). \nWhen solving the HJB-equation numerically (using Longstaff-Schwartz type of techniques)\n\\cite{BraGoySanStr2005} expand the value function in the wealth variable to\na forth degree Taylor approximation.\n\n\\item Based on the duality results in \\cite{KLSX}, \\cite{HauKogWan2006} provide an upper bound on the error stemming from using sub-optimal strategies. \\cite{BicKraMun2013}\npropose a method based on minimizing over a subset of dual\nelements. This subset is chosen such that the corresponding dual\nutility can be computed explicitly and transformed into a feasible primal strategy.\n\n\\item \\revised{It is also important to mention the recent explosion in\nresearch in asymptotic methods in a variety of different ares in\nmathematical finance (transaction costs, pricing, etc.). Since we focus on\nmodel expansion in utility maximization in this paper, we simply point the\nreader to some of the most recent papers, namely \\cite{AltMuhSon15}, and\n\\cite{KalMuh15}, and the references therein, for further information.}\n\\end{enumerate}\n\nIn our work, no Markovian assumption is imposed and we deal with\ngeneral, possibly incomplete, markets with continuous price\nprocesses. We note that while our results apply only to $p<0$, it is possible to\nextend them to $p\\in(0,1)$ at the cost of imposing additional integrability\nrequirements. We do not pursue such an extension; the parameter range\n$p\\in (0,1)$ which we leave out seems to lie outside the typical\nrange of risk-aversion parameters observed in practice (see, e.g.,  \\cite{Szp86}).\nMoreover, we do not consider utility functions more general than the powers. While there are no significant additional mathematical difficulties in treating the general case under appropriate conditions on the relative risk-aversion coefficients, we do not believe that the added value justifies the corresponding \nnotational and technical overhead. For example, all our results would become dependent on the agent's initial wealth, and this dependence would permeate the entire analysis.\n\n\n\\par{\\em A stability interpretation.} As we mentioned above, our contribution can also be seen as a stability\nresult.  It is well-known (see, e.g., \\cite{Rogers}) that even in\nSamuelson's model, estimating the drift is far more challenging than\nestimating the volatility. \\cite{LarZit07} identify the kinds of\nperturbations of the market price of risk process under which the value\nfunction behaves continuously. In the present paper we take the stability\nanalysis one step further and provide a first-order Taylor expansion in \nan \ninfinite-dimensional space of the market price of risk processes. This way,\nwe not only identify the ``continuous'' directions, but also identify\nthose features of the market price of risk process that affect the\nsolution of the utility maximization problem the most (at least locally).  \nAny statistical procedure which is performed with utility maximization in\nmind should, therefore, focus on those, salient, features in order to\nuse the scarce data most efficiently. \n\nSimilar perturbations have been considered by \\cite{Monoyios},\nbut in a somewhat different setting. \\cite{Monoyios} is based on Malliavin\ncalculus and produces a first-order expansion for the\nutility-indifference price of an exponential investor in an It\\^o-process\ndriven market; some of the ideas used can be traced to the related\nwork  \\cite{Dav06}.  \n\n\n\n\\par{\\em Mathematical challenges.} From a mathematical point of view,\nour approach is founded on two ideas.  One of them is to extend the\ntechniques and results of \\cite{LarZit07}; indeed, the basic fact that the\noptimal dual minimizers converge when the market prices of risk process does is\nheavily exploited. It does not, however, suffice to get the full picture.\nFor that, one needs to work on the primal and the dual problems\nsimultaneously and use a pair of bounds. The ideas used there are related\nto and can be interpreted as a nonlinear version of the primal-dual\nsecond-order error estimation techniques first used in \\cite{Hen2002} in\nthe context of mathematical finance.  The first-order expansion in the\nquantity of the unspanned contingent claim developed in \\cite{Hen2002} was \ngeneralized in \n\\cite{KramkovSirbu2006b}\n(see also \\cite{KramkovSirbu2006a}). The arguments in these papers rely on convexity and\nconcavity properties in the expansion parameter (wealth and number of\nunspanned claims). This is not the case in the present paper; indeed, when seen as a function of the underlying\nmarket price of risk process, the investor's value function is neither\nconvex nor concave and a more delicate, local, analysis needs to be\nperformed.\n\n\\bigskip\n\n\\par{\\em Numerical examples.} \nIn Section \\ref{sec:examples} we use two examples to\nillustrate how our approximation performs under realistic conditions.\nFirst, we consider the  Kim-Omberg model (see \\cite{KimOmb96}) which is widely\nused in the financial literature. Under a calibrated set of parameters, \nwe find that our approximation is indeed very accurate when compared to the exact values. \n\nOur second example belongs to a class of extended\naffine models introduced in \\cite{CheFilKim07}. The authors  \nshow that this class of models has superior empirical properties when \ncompared to popular affine and quadratic specifications \n(such as those used, e.g.,  in \\cite{Liu2007}). \nThe resulting optimal investment problem for the extended affine models,\nunfortunately, does not seem to be explicitly solvable. Our approximation\ntechnique turns out to be easily applicable and our error bounds are quite\ntight in the relevant parameter ranges.\n\n\\section{A family of utility-maximization problems}\n\\label{sec:problem}\n\n\\subsection{The setup}\n\\label{sse:setup}\nWe work on a filtered probability space $(\\Omega,\\sF,\n{\\mathbb F}= \\prf{\\sF_t}, {\\mathbb P})$, \nwith the\nfinite time horizon $T>0$. We assume that the filtration ${\\mathbb F}$ is\nright-continuous and that the $\\sigma$-algebra $\\sF_0$ consists of all\n${\\mathbb P}$-trivial subsets of $\\sF$. \n\nLet $M$ be a continuous local martingale, and let $\\Re$, $\\varepsilon\\geq\n0$\nbe a family of continuous ${\\mathbb F}$-semimartingales given by\n\\begin{equation}\n\\label{equ:7283}\n\\begin{split}\n \\Re := M + \\int_0^{\\cdot} \\upeps{\\ld}_t\\, d\\ab{M}_t, \\text{ on } \n [0,T],\\text{ where } \\upeps{\\ld}: = \\lambda  + \\varepsilon \\lambda',\n\\end{split}\n\\end{equation}\nfor a pair $\\lambda,\\lambda'\\in\\sP_{M}^2$, where $\\sP_{M}^2$ denotes the collection of \n  all progressively\nmeasurable processes $\\pi$ with $\\int_0^T \\pi_t^2\\,\nd\\ab{M}_t<\\infty$.\nAs $\\upeps{S} := {\\mathcal E}(\\Re)$ (where ${\\mathcal E}$ denotes the stochastic exponential)\nwill be interpreted as the price process of a\nfinancial asset, the assumption that $\\upeps{\\ld}\\in\\sP_{M}^2$ can be taken \nas a minimal no-arbitrage-type condition.\nWe remark right away that further integrability conditions on $\\lambda$ and $\\lambda'$ will need to be\nimposed below for our main results to hold.\n\\subsection{The utility-maximization problem}\nGiven $x>0$ and $\\varepsilon \\in [0,\\infty)$, let $\\upeps{\\sX}(x)$ denote the set of all\nnonnegative \nwealth processes starting from initial wealth $x$ in the financial market consisting of \n$\\upeps{S} := {\\mathcal E}(\\Re)$ and a zero-interest bond, i.e., \n\\[ \\upeps{\\sX}(x) := \\Bsets{x {\\mathcal E}\\big(\\textstyle \\int_0^T \\pi_t\\, d\\Re_t\\big)}{  \\pi \\in\\ \\sP_{M}^2}.\\]\nHere, $\\pi$ is interpreted as the fraction of wealth invested in the risky asset $\\upeps{S}$.\nThe investor's preferences are modeled by a CRRA (power) utility function\nwith the risk-aversion parameter $p<0$:\n\\begin{equation}\\label{equ:U}\nU(x) := \\frac{x^p}{p}, \\quad x>0.\n\\end{equation}\nThe value function of the corresponding optimal-investment problem is defined\nby\n\\begin{equation}\\label{equ:ue}\n\\upeps{u}(x) := \\sup_{X\\in\\upeps{\\sX}(x)} {\\mathbb E}[ U(X_T)],\\ x>0.\n\\end{equation}\n\\subsection{The dual utility-maximization problem}\nAs is usual in the utility-ma\\-xi\\-mi\\-za\\-ti\\-on literature, a fuller\npicture is obtained if one also considers the appropriate version of the\noptimization problem dual to \\eqref{equ:ue}. For that, we need to examine\nthe no-arbitrage properties of the set of models introduced in Section\n\\ref{sse:setup} above.\n\n\nWe observe, first, that the assumptions we placed on the\nmarket price of risk processes $\\upeps{\\ld}$ above are  not sufficient to\nguarantee the existence of an equivalent martingale measure (NFLVR). They\ndo preclude so-called ``arbitrages of the first kind'' and imply the\nrelated condition NUBPR.  In particular, for all $x,y>0$ and $\\varepsilon\\geq 0$\nthere exists a (strictly) positive c\\` adl\\` ag supermartingale $Y$ with\nthe property that $Y_0=y$ and $YX$ is a supermartingale for each $X\\in\\upeps{\\sX}(x)$; we denote\nthe set of all such processes by $\\upeps{\\sY}(y)$. While this is a consequence of the\ncondition NUBPR in general, in this case an example of a process in\n$\\upeps{\\sY}(y)$ is given, explicitly, as $y\\upeps{Z}$, where $\\upeps{Z}$ is the \\emph{minimal} local martingale density:\n \\begin{equation}\n \\label{equ:Ze}\n \\begin{split}\n   \\upeps{Z} = {\\mathcal E}(-\\int_0^{\\cdot} \\upeps{\\ld}_t\\, dM_t). \n \\end{split}\n \\end{equation}\n\nHaving described the dual domain, we remind the reader that the\n\\emph{conjugate} utility function $V:(0,\\infty)\\to\\R$ is defined by\n\\begin{equation}\\label{equ:V}\nV(y) := \\sup_{x>0} \\left( U(x) - xy\\right) = \\frac{y^{-q}}{q},\\text{ where }\nq:=\\tfrac{p}{1-p}\\in(-1,0).\n\\end{equation}\nWe define the \\emph{dual value function} $\\upeps{v}:(0,\\infty)\\to \\R$ by\n\\begin{equation}\n\\label{equ:ve}\n\\begin{split}\n \\upeps{v}(y) := \\inf_{Y\\in \\upeps{\\sY}(y) } {\\mathbb E}[ V(Y_T)],\\ y>0,\\ \\varepsilon\\geq 0.\n\\end{split}\n\\end{equation}\nDue to negativity (and, a fortiori, finiteness) of the primal value\nfunction $\\upeps{u}$, the (abstract) Theorem 3.1 of \\cite{KraSch99} can now be applied\n(see also \\cite{Mostovyi2011}). Its \nmain assumption, namely the\nbipolar relationship between the primal and dual domains, holds due to the\nexistence of the num\\' eraire process, given explicitly by $1\/\\upeps{Z}$ (see\nTheorem 4.12~in \\cite{KarKar07}). One can also use \na simpler argument (see \\cite{Lar11}), which applies only to \nthe case of a CRRA utility with $p<0$, to obtain the following conclusions\nfor all $\\varepsilon\\geq 0$:\n\\begin{enumerate}\n  \\item  \nboth $\\upeps{u}$ and $\\upeps{v}$ are\nfinite \nand the following conjugacy\nrelationships hold \n \\begin{equation}\n \\label{equ:conj}\n \\begin{split}\n   \\upeps{v}(y) = \\sup_{x>0} \\Big( \\upeps{u}(x)-xy \\Big),\\text{ and }\n   \\upeps{u}(x) = \\inf_{y>0} \\Big( \\upeps{v}(y)+xy \\Big).\n \\end{split}\n \\end{equation}\n\\item For all $x,y>0$\nthere exist optimal solutions $\\upeps{\\hat{X}}(x)\\in\\upeps{\\sX}(x)$\nand $\\upeps{\\hat{Y}}(y)\\in\\upeps{\\sY}(y)$ of \\eqref{equ:ue} and \\eqref{equ:ve},\nrespectively,  and are related by\n\\[ U'(\\upeps{\\hat{X}}_T(x)) = \\upeps{\\hat{Y}}_T (\\upeps{y}(x))  \\text{ where } \\upeps{y}(x) = \\trn{}{x} \\upeps{u}(x) = p\nx^{p-1} \\upeps{u}(1). \\]\n\\item \\revised{The product $\\upeps{\\hat{X}}\\upeps{\\hat{Y}}$ is a uniformly-integrable martingale. \nIn particular \\[ {\\mathbb E}[ \\upeps{\\hat{X}}_T \\upeps{\\hat{Y}}_T]=xy.\\]} \n\\end{enumerate}\nThe homogeneity of the utility function $U$ and its conjugate\n$V$ transfers to the value functions $\\upeps{u}$ and $\\upeps{v}$ and the optimal\nsolutions $\\upeps{\\hat{X}}$ and $\\upeps{\\hat{Y}}$:\n \\begin{equation}\n \\label{equ:simpl}\n \\begin{split}\n    \\upeps{u}(x) &= x^p \\upeps{u},\\quad \\upeps{v}(y)=y^{-q} \\upeps{v},\\\\\n    \\upeps{\\hat{X}}(x)&=x \\upeps{\\hat{X}},\\quad \\upeps{\\hat{Y}}(y)=y \\upeps{\\hat{Y}},\n \\end{split}\n \\end{equation}\nwhere, to simplify the notation, we write $\\upeps{u}, \\upeps{v}, \\upeps{\\hat{X}}$ and $\\upeps{\\hat{Y}}$ for\n$\\upeps{u}(1)$,\n$\\upeps{v}(1)$, $\\upeps{\\hat{X}}(1)$ and $\\upeps{\\hat{Y}}(1)$, respectively. \n\\subsection{A change of measure}\nFor $\\varepsilon=0$ we denote by $\\upz{\\hat{\\pi}}$ the primal optimizer, i.e., the process\nin $\\sP_{M}^2$ such that\n\\[ \\upz{\\hat{X}} = {\\mathcal E}( \\int_0^{\\cdot} \\upz{\\hat{\\pi}}_u\\, d\\upz{R}_u).\\]\nWe define the probability measure $\\upz{\\tilde{\\PP}}$ \n by\n \\begin{equation}\n \\label{equ:Girsanov}\n \\begin{split}\n   \\rn{\\upz{\\tilde{\\PP}}}{{\\mathbb P}} = \\upz{\\hat{X}}_T \\upz{\\hat{Y}}_T\\, \\Big(= \\oo{\\upz{v}} V(\\upz{\\hat{Y}}_T) = \\oo{\\upz{u}}\nU(\\upz{\\hat{X}}_T)\\Big),\n \\end{split}\n \\end{equation}\n \\revised{where the last two equalities \n follow from the  identities $x U'(x) = p U(x)$ and $ y V'(y) = -q V(y)$, and the relations between the value functions outlined above.}\n\nThe measure $\\upz{\\tilde{\\PP}}$ has been in the mathematical finance literature for a while (see, e.g., p. 911-2 in \\cite{KraSch99}).\nThe explicit form of $\\upz{\\tilde{\\PP}}$ is not generally available, but, we note that, by Girsanov's Theorem (see \\eqref{equ:ZENL} and the discussion around it),\nthe process\n\\begin{equation}\n \\label{equ:tMp}\n \\begin{split}\n   \\tM^p := M + \\int_0^{\\cdot} \\Big( \\lambda_t - \\upz{\\hat{\\pi}}_t \\Big)\\, d\\ab{M}_t \n \\end{split}\n \\end{equation}\nis a $\\upz{\\tilde{\\PP}}$-local martingale; \\revised{this fact will be used below in the proof of Proposition \\ref{pro:u-down}.} \n\n\\section{The problem and the main results}\n\nWe first provide first-order expansions and error estimates of the primal and dual value functions. Secondly, we provide an expansion of the optimal controls in the Brownian setting.\n\n\\subsection{Value functions}\nAt the basic level, we are interested in the first-order properties\nof the convergence, as $\\varepsilon\\searrow 0$, of the value\nfunctions of the problems $\\upeps{u}$ and $\\upeps{v}$ to the value functions $\\upz{u}$ and\n$\\upz{v}$ of the\n``base'' model (corresponding to $\\varepsilon=0$). To familiarize ourselves with\nthe flavor of the results we can expect in the general case, we start by\nanalyzing a similar problem for the logarithmic utility. It has the\nadvantage that it admits a simple explicit solution. Let $\\ue_{\\log}(x)$ and\n$\\ve_{\\log}(y)$\ndenote the value function of the utility maximization problem as in \n\\eqref{equ:ue} and \\eqref{equ:ve} above, but with $U(x)=\\log(x)$ and\n$V(y) = \\sup_x (U(x)-xy)=-\\log(y) -1$.\nIt is a  classical result that, as long as ${\\mathbb E}[ \\int_0^T (\\lambda_t^2 + (\\lambda'_t)^2)\\, d\\ab{M}_t]<\\infty$,  we \nhave\n\\[ \\ue_{\\log}(x) = \\log(x)+\\tot {\\mathbb E}[ \\int_0^T (\\upeps{\\ld}_t)^2\\, d\\ab{M}_t] \\text{ and }\n\\ve_{\\log} = \\ue_{\\log} -1.\\] The (exact) second-order expansion in $\\varepsilon$ \nof $\\ue_{\\log}(x)$ is thus\ngiven by\n \\begin{equation*}\n \\label{equ:4AA5}\n \\begin{split}\n    \\ue_{\\log}(x) &= \\uz_{\\log}(x) + \\varepsilon {\\mathbb E}[ \\int_0^T \\lambda_t \\lambda'_t\\, d\\ab{M}_t] + \\tot \\varepsilon^2\n   {\\mathbb E}[ \\int_0^T (\\lambda'_t)^2\\, d\\ab{M}_t]\\\\\n    &=\\uz_{\\log}(x) + \\varepsilon {\\mathbb E}[ \\int_0^T \\lambda'_t\\, d\\upz{R}_t] + \\tot \\varepsilon^2\n   {\\mathbb E}[ \\int_0^T (\\lambda'_t)^2\\, d\\ab{M}_t],\\\\\n \\end{split}\n \\end{equation*}\n where $\\upz{R}$ is defined in (\\ref{equ:7283}).\n We cannot expect the value function to be a second order polynomial in\n $\\varepsilon$ in the case of a general power utility. We do obtain a formally similar\n first-order expansion in Theorem \\ref{thm:main1} below and an analogous\n error estimate in Theorem \\ref{thm:main2}. Section 5\n  is devoted to their proofs. We remind the reader of the homogeneity\n relationships in \\eqref{equ:simpl}; they allow us to assume from now on\n that $x=y=1$.\n\\begin{theorem}[The G\\^ateaux derivative]\n\\label{thm:main1}\nIn the setting of Section \\ref{sec:problem}, we assume that \n\\begin{equation}\n\\label{equ:A1}\n\\begin{split}\n\\int_0^T(\\lambda'_t)^2\\, d\\ab{M}_t \\in \\el^{1-p}({\\mathbb P}) \\text{ and } \\int_0^T \\lambda'_t\\, d\\upz{R}_t\n\\in \\cup_{s>(1-p)} \\el^s({\\mathbb P}).\n\\end{split}\n\\end{equation}\nThen, with\n$\\upz{\\Delta} := {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\textstyle \\int_0^T \\lambda'_t\\, d\\upz{R}_t]$, where $\\upz{\\tilde{\\PP}}$ is defined by \\eqref{equ:Girsanov}, we have\n\\begin{align}\n \\label{equ:main1-u}\n \\derep{\\upeps{u}}:=\\lim_{\\varepsilon \\searrow 0}  \\oo{\\varepsilon} \\Big( \\upeps{u} - \\upz{u} \\Big) \n &= p \\upz{u} \\upz{\\Delta}, \\text{ and } \\\\\n\\label{equ:main1-v}\n \\derep{\\upeps{v}}:=\\lim_{\\varepsilon \\searrow 0}  \\oo{\\varepsilon} \\Big( \\upeps{v} - \\upz{v} \\Big) \n &= q\\upz{v} \\upz{\\Delta}.\\qedhere\n\\end{align}\n\\end{theorem}\n\\begin{theorem}[An error estimate]\n\\label{thm:main2} \nIn the setting of Section \\ref{sec:problem}, we assume that \n\\begin{equation}\n\\label{equ:A2}\n\\begin{split}\n\\int_0^T(\\lambda'_t)^2\\, d\\ab{M}_t, \\int_0^T \\lambda'_t\\, d\\upz{R}_t\n\\in \\el^{2(1-p)}({\\mathbb P}) \\text{ and } \\Phi^2 e^{\\varepsilon_0 |p| \\Phi^-}\\in\\el^1(\\upz{\\tilde{\\PP}}),\n\\end{split}\n\\end{equation}\nfor some $\\varepsilon_0>0$, where $\\Phi := \\int_0^T \\upz{\\hat{\\pi}}_t \\lambda'_t\\, d\\ab{M}_t$. \nThen there exist constants $C>0$ and $\\varepsilon_0'\\in (0,\\varepsilon_0]$ such that for all $\\varepsilon\\in [0,\\varepsilon'_0]$ we have\n\\begin{align}\n\\label{equ:main2-u}\n\\Big| \\upeps{u} - \\upz{u} - \\varepsilon p  \\upz{u} \\upz{\\Delta} \\Big| & \\leq \n C \\varepsilon^2, \\text{ and }\n \\\\ \\label{equ:main2-v}\n\\Big| \n\\upeps{v} - \\upz{v} - \\varepsilon q \\upz{v} \\upz{\\Delta} \\Big| &\n \\leq C  \\varepsilon^2.\n \\end{align}\n\\end{theorem}\n\\begin{remark}\\ \n\\label{rem:761C}\n\\begin{enumerate}\n\\item\nIt is perhaps more informative to think of the results in Theorems\n\\ref{thm:main1} and \\ref{thm:main2} on the logarithmic scale. \nAs is evident from \\eqref{equ:main1-u} and\n\\eqref{equ:main1-v}, the functions $\\upeps{u}$ and $\\upeps{v}$ admit the right\n\\emph{logarithmic} derivative $p\\upz{\\Delta}$ and $q\\upz{\\Delta}$, respectively,  at\n$\\varepsilon=0$. Moreover, we have the following small-$\\varepsilon$ asymptotics:\n\\[ \n\\upeps{u} = \\upz{u} e^{\\varepsilon p \\upz{\\Delta} + O(\\varepsilon^2)} \\text{ and }\n\\upeps{v} = \\upz{v} e^{\\varepsilon q \\upz{\\Delta} + O(\\varepsilon^2)}.\\]\nIf one takes one step further and uses the \\emph{certainty\nequivalent} $\\upeps{\\mathrm{CE}}$, given by\n\\[ U(\\upeps{\\mathrm{CE}}) = \\upeps{u},\\]\nwe note that $\\upz{\\Delta}$ is precisely the infinitesimal growth-rate of $\\upeps{\\mathrm{CE}}$ at\n$\\varepsilon=0$ - an $\\varepsilon$-change of the market price of risk in the direction\n$\\lambda'$ yields to an $e^{\\varepsilon \\upz{\\Delta}}$-fold increase in the certainty-equivalent\nof the initial wealth. \n\\item \\label{rem:main2}\nA careful analysis of \nthe proof of Theorem \\ref{thm:main2} below reveals the following,\nadditional, information:\n\\begin{enumerate}\n  \\item The proof of Proposition \\ref{pro:u-down} reveals that\n  $\\upz{\\Delta}={\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\Phi]$. \n  \\item The condition involving $\\Phi$ in \\eqref{equ:A2} is needed only for\n  the upper bound in \\eqref{equ:main2-u} and the lower bound in\n  \\eqref{equ:main2-v}. The other two bounds hold for all \n  $\\varepsilon\\geq 0$ even \n  if \\eqref{equ:A2} holds with $\\varepsilon_0=0$. \n  \\item The constants $C$ and $\\varepsilon_0'$ depend - in a simple way - on\n  $\\varepsilon_0$, $p$ and the $\\el^{2(1-p)}(\\upz{\\tilde{\\PP}})$- and \n  $\\el^1(\\upz{\\tilde{\\PP}})$-bounds of the random variables in \\eqref{equ:A2}. \n  For two one-sided bounds, explicit formulas are given in Propositions \\ref{pro:v-up}\n  and \\ref{pro:u-down}. The other two bounds are somewhat less informative so we do not compute them explicitly. The reader will find an example of how this can be done in a specific setting in Subsection \\ref{sse:extaff}. \n   \\item Even though we cannot claim that the functions $\\upeps{u}$ and $\\upeps{v}$ are convex or concave, it is possible to show their local {\\it semiconcavity} in $\\varepsilon$  (see \\cite{Can04}).\n   This can be done via the techniques from the proof of  Theorem~\\ref{thm:main2}.\n\\end{enumerate} \n\\item\nThe assumption of constant risk aversion (power utility) \nallows us to incorporate many stochastic interest-rate models into our setting.\nIndeed, provided that $c:= \\ee{e^{p\\int_0^T\nr_tdt}} <\\infty$, we can introduce the probability measure ${\\mathbb P}^r$, defined by\n\\begin{align}\n\\rn{{\\mathbb P}^r}{{\\mathbb P}}:= c e^{p\\int_0^T r_t\\, dt},\n\\end{align}\non $\\sF_T$. For any admissible wealth process $X$ we then have\n\\[ {\\mathbb E}[ U(X_T) ] = c\\, {\\mathbb E}^{{\\mathbb P}^r}\\left[ U\\Big( X_T e^{-\\int_0^T r_u\\, du}\\Big) \\right].\\]\nThis way, the utility maximization under ${\\mathbb P}^r$ with a zero interest rate\nbecomes equivalent to the utility maximization problem under ${\\mathbb P}$ with\nthe interest rate process $\\prf{r_t}$. \\cite{Zitkovic2005} and \\cite{Mostovyi2011} consider the setting of utility maximization  with stochastic utility which embeds stochastic interest rates. \n\nPractical implementation of the above idea depends on how explicit one can\nbe about the Girsanov transformation associated with ${\\mathbb P}^r$. It turns out,\nfortunately, that many of the widely-used interest-rate models, \nsuch as Vasi\\v cek, CIR, or the quadratic normal models (see, e.g.,\n\\cite{Mun13} for a textbook discussion of these models)\nallow for a fully\nexplicit description (often due to their affine structure). For example, in\nthe Vasi\\v cek model, the Girsanov drift under ${\\mathbb P}^r$ can be computed\nquite explicitly, due to the underlying affine structure.\n Indeed, suppose that $r$ has the Ornstein-Uhlenbeck dynamics of\nthe form\n\\[ dr_t := \\kappa (\\theta-r_t)\\, dt + \\beta\\, dB_t,\\ r_0\\in\\R,\\]\nwhere $B$ is a Brownian motion and $\\kappa>0$, $\\theta,\\beta\\in\\R$.\nThen the process\n\\[ B^{(p)} := B - \\int_0^{\\cdot} b(T-t)\\, dt,\\text{ where }\nb(t) = \\tfrac{\\beta p}{\\kappa} (1-e^{-\\kappa t}),\\]\nis a ${\\mathbb P}^r$-Brownian motion. \n\\end{enumerate}\n\\end{remark}\n\n\\subsection{Optimal controls}\nThe estimates \\eqref{equ:main2-u} and \\eqref{equ:main2-v} are of type $O(\\varepsilon^2)$. A slight adjustment to the below proof of Proposition \\ref{pro:u-down}  shows that the wealth process $\\tilde{X} :=\\mathcal{E}\\big(\\int\\hat{\\pi}^{(0)}dR^{(\\varepsilon)})$ satisfies (see \\ref{lower_bound_est})\n\\begin{align*}\n\\big|{\\mathbb E}[U(\\tilde{X}_T)] - u^{(0)}(1+\\varepsilon p \\Delta^{(0)}) \\big|&\\le \\frac12p^2\\varepsilon^2 |u^{(0)}| {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\Phi^2e^{\\varepsilon |p| \\Phi^-}].\n\\end{align*}\n\nTherefore, under the conditions of Theorem \\ref{thm:main2}, $\\hat{\\pi}^{(0)}$ is an $O(\\varepsilon^2)$-optimal control for the $\\varepsilon$-model because the triangle inequality produces a constant $C>0$ such that\n$$\n\\big|{\\mathbb E}[U(\\tilde{X}_T)] -u^{(\\varepsilon)} \\big| \\le C \\varepsilon^2,\n$$\nfor all $\\varepsilon>0$ small enough. In this section we will provide a correction term to $\\hat{\\pi}^{(0)}$ such that the resulting wealth process upgrades the convergence  to $o(\\varepsilon^2)$. \n\nFor simplicity, we consider the (augmented) filtration generated by $(B,W)$ where $B\\in \\R$ and $W\\in\\R^d$, $d\\in\\mathbb{N}$, are two independent Brownian motions. In \\eqref{equ:7283} we take \n\\begin{align}\\label{dM_BM}\ndM_t := \\sigma_t dB_t,\\quad M_0:=0,\n\\end{align}\nfor a process $\\sigma\\in \\sP_{B}^2$ with $\\sigma\\neq0$. We define $\\upz{\\tilde{\\PP}}$ by\n  \\eqref{equ:Girsanov} and we denote by $(B^{\\upz{\\tilde{\\PP}}},W^{\\upz{\\tilde{\\PP}}})$ the\n  corresponding $\\upz{\\tilde{\\PP}}$-Brownian motions. Provided that $\\Phi :=\\int_0^T\n  \\upz{\\hat{\\pi}}_t \\lambda'_t\\sigma^2_t dt \\in \\el^{2}(\\upz{\\tilde{\\PP}})$, $\\Phi$ has the unique\n  martingale representation under $\\upz{\\tilde{\\PP}}$\n\\begin{align}\\label{Itorep}\n\\Phi = {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\Phi] + \\int_0^T \\gamma^B_{t}\\sigma_tdB^{\\upz{\\tilde{\\PP}}}_t  +  \\int_0^T \\gamma^W_{t} dW^{\\upz{\\tilde{\\PP}}}_t,\n\\end{align}\nwhere we have used $\\sigma\\neq0$. Because $\\Phi\\in \\el^{2}(\\upz{\\tilde{\\PP}})$ the two processes $\\gamma^W$ and $\\gamma^B$ in \\eqref{Itorep} satisfy the integrability conditions\n$$\n{\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\Big[\\int_0^T\\Big( (\\gamma^B_{t}\\sigma_t)^2 + (\\gamma^W_t)^2\\Big)dt\\Big]<\\infty.\n$$\nThese square integrability properties will be used in the proof of the next theorem.\n\n\\begin{theorem}[2nd order expansion] \\label{thm:main3}  In the above Brownian setting, we assume\n\\begin{align}\\label{2nd_int}\n\\int_0^T(\\lambda'_t)^2\\sigma^2_tdt \\in \\el^{1-p}({\\mathbb P})\\cap \\el^{1}(\\upz{\\tilde{\\PP}}) \n\\text{ and } \\int_0^T  \\hat{\\pi}_t^{(0)} \\lambda_t' \\sigma_t^2 dt \\in \\el^{2}(\\upz{\\tilde{\\PP}}),\n\\end{align}\nas well as the existence of a constant $\\varepsilon_0 >0$ such that $\\delta :=\\tfrac{\\lambda'+p\\gamma^B}{1-p}$ satisfies\n\\begin{align}\\label{exp_reg}\ne^{p\\int_0^T\\big (\\varepsilon\\hat{\\pi}^{(0)} \\lambda'+ \\varepsilon^2 (\\delta \\lambda'-\\frac12\\delta^2 ) \\big)\\sigma^2dt + p\\varepsilon \\int_0^T\\delta\\sigma dB_t^{\\upz{\\tilde{\\PP}}} }\\in \\el^{1}(\\upz{\\tilde{\\PP}}),\n\\end{align}\nfor all $\\varepsilon \\in (0,\\varepsilon_0)$. Then we have\n\\begin{align}\n\\label{equ:main3-u}\n \\upeps{u} - \\upz{u} - \\varepsilon p  \\upz{u} \\upz{\\Delta} - \\tfrac12 \\varepsilon^2p\\upz{u}\\left(\\Delta^{(00)}+p(\\Delta^{(0)})^2 \\right)& \\in O(\\varepsilon^3),\n \\\\ \\label{equ:main3-v}\n\\upeps{v} - \\upz{v} - \\varepsilon q \\upz{v} \\upz{\\Delta}  - \\tfrac12 \\varepsilon^2q\\upz{v}\\left(\\Delta^{(00)}+q(\\Delta^{(0)})^2 \\right) &\n\\in O(\\varepsilon^3),\n \\end{align}\nas $\\varepsilon\\searrow 0$. In \\eqref{equ:main3-u} and \\eqref{equ:main3-v} we have defined\n \\begin{align}\n\\Delta^{(00)} &:= {\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\left[ \\int_0^T\\left(p|\\gamma^W_{t}|^2 +\\frac{(\\lambda'_t)^2 + p\\gamma^B_t(\\gamma^B_t+2\\lambda'_t)}{1-p}\\sigma^2_t\\right)dt\\right],\\label{Delta00}\n\\end{align}\nwhere the processes $\\gamma^B$ and $\\gamma^W$ are given by the\nmartingale representation \\eqref{Itorep}.\n\\end{theorem}\n\n\n\\begin{remark}\\ \n\\label{rem:controls}\n\\begin{enumerate}\n\\item The below proof of Theorem \\ref{thm:main3} shows that the process\n\\begin{align}\\label{optimal_control_correction}\n\\tilde{\\pi} := \\hat{\\pi}^{(0)} + \\varepsilon \\tfrac{\\lambda'+p\\gamma^B}{1-p},\n\\end{align}\nis an $O(\\varepsilon^3)$-optimal control for the $\\varepsilon$-model in the sense that the wealth process $\\tilde{X} :=\\mathcal{E}\\big(\\int\\tilde{\\pi}dR^{(\\varepsilon)})$ satisfies\n$$\n{\\mathbb E}[U(\\tilde{X}_T)] -u^{(\\varepsilon)} \\in O(\\varepsilon^3)\\quad \\text{as}\\quad \\varepsilon \\searrow 0.\n$$\n\\item Because the filtration is generated by $(B,W)$, the optimizer $\\hat{H}^{(0)}$  for the dual problem \\eqref{equ:ve} can be written as $\\hat{H}^{(0)} = \\mathcal{E}(-\\int \\hat{\\nu}^{(0)}dW)$ for a $d$-dimensional process  $\\hat{\\nu}^{(0)}$ in $\\sP_{W}^2$. The below proof of Theorem \\ref{thm:main3} also shows that the process\n\\begin{align}\\label{optimal_dual_control_correction}\n\\tilde{\\nu} := \\hat{\\nu}^{(0)} - \\varepsilon p\\gamma^W,\n\\end{align}\nis an $O(\\varepsilon^3)$-optimal dual control in the $\\varepsilon$-model.\n\n\\item Throughout the paper we have considered $\\varepsilon=0$ as the base model. Because we can write \n$$\n\\lambda + (\\bar{\\varepsilon} +\\varepsilon)\\lambda' = \\lambda + \\bar{\\varepsilon}\\lambda' +\\varepsilon\\lambda',\n$$\nfor any $\\bar{\\varepsilon}\\in[\\varepsilon_L,\\varepsilon_U]$ with $\\varepsilon_L<\\varepsilon_U$, we can use Theorem \\ref{thm:main3} for the base model $\\lambda+ \\bar{\\varepsilon}\\lambda'$ to  provide a 2nd order Taylor expansion around any point $\\bar{\\varepsilon}$. Therefore, whenever $\\Delta^{(0)}$ and $\\Delta^{(00)}$ are bounded uniformly in $\\bar{\\varepsilon}\\in[\\varepsilon_L,\\varepsilon_U]$, Theorem 3 in \\cite{Oli54} ensures that $u^{(\\varepsilon)}$ is twice differentiable in $\\varepsilon$.\n\\item An easy way of eliminating the stochastic $B^{\\upz{\\tilde{\\PP}}}$-integral  \nappearing in \\eqref{exp_reg} is to use H\\\"older's inequality with the exponents $-1\/q$ and $(1-p)$; see Section \\ref{sse:extaff} below for an example.\n\\end{enumerate}\n\\end{remark}\n\n\n\\section{Proofs of the main theorems}\n\\label{sec:proofs}\nWe start the proof with a short discussion of the special structure \nthe dual domain $\\upeps{\\sY}$ has when the stock-price process $\\upeps{S} = \\mathcal{E}(\\Re)$ is\ncontinuous.  Indeed, it has been shown in \\cite{LarZit07}, Proposition 3.2, p.~1653,  that in that case the maximal\nelements in $\\upeps{\\sY}$ (in the pointwise order) are precisely \nlocal martingales of the form\n \\begin{equation*}\n \\begin{split}\n   Y = \\upeps{Z} H ,\\ H\\in\\sH,\n \\end{split}\n \\end{equation*}\n where $\\sH$ denotes the set of all $M$-orthogonal positive local\n martingales $H$ with $H_0=1$. \nWe remark that even though the results in\n\\cite{LarZit07} were written under the assumption NFLVR, a simple \nlocalization argument shows that they apply under the present conditions,\nas well.\n   Hence, we can write\n  \\[ \\upeps{v} = \\inf_{H\\in\\sH} {\\mathbb E}[ V(\\upeps{Z}_T H_T)],\\]\nand the minimizer $\\upeps{\\hat{Y}}$ always has the\nform \\begin{equation}\n \\label{equ:ZENL}\n \\begin{split}\n   \\upeps{\\hat{Y}} = \\upeps{Z} \\upeps{\\hat{H}},\\text{\nfor some $\\upeps{\\hat{H}}\\in\\sH$.}\n \\end{split}\n \\end{equation}\nFinally, we introduce two shortcuts for expressions that appear\nfrequently in the proof:\n\\begin{equation}\n\\label{equ:etaLambda}\n\\begin{split}\n \\eta := \\int_0^T \\lambda'_t\\, d\\upz{R}_t,\\ \\Lambda := \\int_0^T (\\lambda'_t)^2\\,\n d\\ab{M}_t,\n\\end{split}\n\\end{equation}\nand remind the reader that\n$\\Phi := \\int_0^T \\upz{\\hat{\\pi}}_t \\lambda'_t\\, d\\ab{M}_t$ and \n $\\Delta^{(0)} := {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[ \\eta]$. It will be useful\nto keep in mind that $(1-p)(1+q)=1$ and that $-1\/q$ and $1-p$ are\nconjugate exponents. \n\\subsection{A proof of Theorem \\ref{thm:main1}}\nThe proof is based on the stability results of \\cite{LarZit07} and the\nfollowing lemma:\n\\begin{lemma}\n\\label{lem:K-eps}\nLet $\\set{\\upeps{K}}_{\\varepsilon\\geq 0}$ be a family of \npositive random variables such that \n\\begin{enumerate}\n  \\item ${\\mathbb E}[ \\updel{Z}_T \\upeps{K}]\\leq 1$ for all $\\varepsilon,\\delta \\geq 0$, and \n  \\item $\\upeps{K}  \\to \\upz{K}$ in probability, as $\\varepsilon \\searrow 0$. \n\\end{enumerate}\nThen, under the conditions of Theorem \\ref{thm:main1}, we have\n\\[ \\lim_{\\varepsilon\\searrow 0} \\oo{\\varepsilon}\n{\\mathbb E}\\Big[ V(\\upeps{Z}_T \\upeps{K}) - V(\\upz{Z}_T \\upeps{K})\\Big] = \nq \\ee{ V(\\upz{Z}_T \\upz{K})\\eta}.\\]\n \\end{lemma} \n \\begin{proof}\nThe map\n$\\varepsilon\\mapsto \\upeps{Z}_T$ is almost surely\ncontinuously differentiable; \\revised{\nindeed, we have \n\\[\n\\log(\\upeps{Z}_T) \n= \\log(\\upz{Z}_T)\n  - \\varepsilon  \\int_0^T \\lambda'_t\\, d\\upz{R}_t \n  - \\tot \\varepsilon^2 \\int_0^T (\\lambda'_t)^2\\, d\\ab{M}_t,\n\\]\nand, so, \n\\[\n\\tfrac{d}{d \\varepsilon} \\upeps{Z}_T = -\\upeps{Z}_T \\Big( \\eta + \\varepsilon \\Lambda \\Big), \\text{ a.s.}\n\\]\nTherefore, }\n\\begin{align}\n\\label{equ:primal1}\nV(\\upeps{Z}_T K) -V(\\upz{Z}_T K)\n = \\int_0^\\varepsilon q V(\\updel{Z}_T K) (\\eta +\\delta \\Lambda)d\\delta,\n\\end{align} \nfor each $\\varepsilon$ and each positive random variable $K$. \nThus, \n\\begin{equation}\n\\label{equ:378}\n\\begin{split}\nV(\\upeps{Z}_T \\upeps{K}) & -V(\\upz{Z}_T \\upeps{K}) - \\varepsilon q V(\\upz{Z}_T \\upz{K})\\eta \n=\nA_{\\varepsilon}+B_{\\varepsilon},\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{equ:A-B}\n\\begin{split}\n  A_{\\varepsilon}&:=\\int_0^\\varepsilon\n  q \\Big( V(\\updel{Z}_T \\upeps{K}) - V(\\upz{Z}_T \\upz{K}) \\Big)\\eta\\, d\\delta, \\text{ and } \\\\\n  B_{\\varepsilon}&:= \\int_0^{\\varepsilon}q V(\\updel{Z}_T\\upeps{K}) \\Lambda\\,  \\delta\\, d\\delta. \n\\end{split}\n\\end{equation}\nH\\\" older's inequality implies that\n\\begin{equation}\n\\label{equ:Holder}\n\\begin{split}\n  {\\mathbb E}[ B_{\\varepsilon}] \\leq \\tot  \\varepsilon^2 \\sup_{\\delta\\in [0,\\varepsilon]} \\Big({\\mathbb E}[ \\updel{Z}_T \\upeps{K}]^{-q} {\\mathbb E}[\n  \\Lambda^{1-p}]^{1+q}\\Big)\n  \\leq \\tot \\varepsilon^2 {\\mathbb E}[ \\Lambda^{1-p}]^{1+q}.\n\\end{split}\n\\end{equation}\nThus, \nwe have\n$\\tfrac{1}{\\varepsilon} {\\mathbb E}[ B_{\\varepsilon}] \\to 0$, as $\\varepsilon\\searrow 0$. \nTo show that  $\\tfrac{1}{\\varepsilon} {\\mathbb E}[ A_{\\varepsilon}]\\to 0$, we note that \n${\\mathbb E}[ A_{\\varepsilon}] = \\int_0^{\\varepsilon} f(\\varepsilon,\\delta)\\, d\\delta$, where\nthe function\n$f:[0,\\infty)^2 \\to \\R$ is given by \n \\begin{equation}\n \\label{equ:function-f}\n \\begin{split}\n   f(\\varepsilon,\\delta) := q\\ee{ \\big(V(\\updel{Z}_T\\upeps{K}) - V(\\upz{Z}_T \\upz{K}) \\big)\\eta}.\n \\end{split}\n \\end{equation}\nSince $f(0,0)=0$, it will be enough to show that $f$ is continuous at $(0,0)$.  \nBy the assumptions of the lemma and the definition of $\\updel{Z}$, we have\n\\[ V(\\uppar{Z}{\\delta_n}_T \\uppar{K}{\\varepsilon_n}) \\to V(\\upz{Z}_T \\upz{K}),\\text{ in\nprobability},\\]\nfor each sequence $(\\varepsilon_n,\\delta_n)\\in [0,\\infty)^2$ such\n  $(\\varepsilon_n,\\delta_n)\\to (0,0)$.\n  Therefore, it suffices to\nestablish uniform integrability of the expression\ninside of the expectation in \\eqref{equ:function-f}. For that we\ncan use the theorem of de la Valle\\' e-Poussin, \n whose conditions hold\nthanks to an application H\\\"older's inequality as in \\eqref{equ:Holder}\nabove, remembering that not only $\\eta\\in\\el^{1-p}$, but\nalso in $\\el^s$, for some $s>(1-p)$. \n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm:main1}] \n  Thanks to the optimality of $\\upeps{Z}_T \\upeps{\\hat{H}}_T$, we have the upper estimate\n   \\begin{equation}\n   \\label{equ:upper}\n   \\begin{split}\n     \\oo{\\varepsilon}\\ee{ V(\\upeps{Z}_T \\upeps{\\hat{H}}_T) - V(\\upz{Z}_T \\upz{\\hat{H}}_T)} &\\leq\n     \\oo{\\varepsilon}\\ee{ V(\\upeps{Z}_T \\upz{\\hat{H}}_T) - V(\\upz{Z}_T \\upz{\\hat{H}}_T)}\n   \\end{split}\n   \\end{equation}\n   Similarly, we obtain the lower estimate\n   \\begin{equation}\n   \\label{equ:lower}\n   \\begin{split}\n     \\oo{\\varepsilon}\\ee{ V(\\upeps{Z}_T \\upeps{\\hat{H}}_T) - V(\\upz{Z}_T \\upz{\\hat{H}}_T)} &\\geq\n     \\oo{\\varepsilon}\\ee{ V(\\upeps{Z}_T \\upeps{\\hat{H}}_T) - V(\\upz{Z}_T \\upeps{\\hat{H}}_T)}.\n   \\end{split}\n   \\end{equation}\nOur next task is to prove that the limits of the right-hand sides of \n\\eqref{equ:upper} and \\eqref{equ:lower} exist and both coincide with the \nright-hand side of \\eqref{equ:main1-v}. In each case, Lemma \\ref{lem:K-eps}\ncan be applied; in the first with $\\upeps{K}= \\upz{\\hat{H}}_T$, and in the second with\n$\\upeps{K}=\\upeps{\\hat{H}}_T$. In both cases the assumption (1) of Lemma\n\\ref{lem:K-eps} follows directly from that fact that $\\upeps{Z}_T \\upeps{K} \\in \\upeps{\\sY}$.\nAs for the assumption (2), it trivially holds in\nthe first case. In the second case, we need to argue that\n$\\upeps{\\hat{H}}_T \\to \\upz{\\hat{H}}_T$ in probability, as $\\varepsilon\\searrow 0$. That, in turn,\nfollows easily from Lemma 3.10 in \\cite{LarZit07}; as mentioned above, the\nseemingly stronger assumption of NFLVR made in \\cite{LarZit07} is not necessary\nand its results hold under the weaker condition NUBPR. \n\nHaving proven \\eqref{equ:main1-v}, we turn to \\eqref{equ:main1-u}. \nThanks to \\eqref{equ:simpl}, the conjugacy relationship\n  \\eqref{equ:conj} takes the following, simple, form in our setting:\n  \\begin{equation}\n  \\label{equ:conj2}\n  \\begin{split}\n  p \\upeps{u} = (q\\upeps{v})^{1-p}.\n  \\end{split}\n  \\end{equation}\n  Therefore, $\\upeps{u}$ is right differentiable at $\\varepsilon=0$, and we have\n   \\begin{equation*}\n   \\label{equ:6630}\n   \\begin{split}\n      p \\derep{\\upeps{u}} \n     &= (1-p) (q\\upz{v})^{-p}\\, q^2 \\upz{v} \\upz{\\Delta} = p^2 \\upz{u} \\upz{\\Delta}. \\qedhere\n   \\end{split}\n   \\end{equation*}\n\\end{proof}\n\\subsection{Remaining proofs}\n\\begin{proposition} Suppose that $\\eta\\in\\el^{2(1-p)}$ and $\\Lambda, \\Lambda\\eta\\in \\el^{1-p}$. Then\n\\label{pro:v-up}\nfor all $\\varepsilon\\geq 0$ we have\n \\begin{equation}\n \\label{equ:ve-up}\n \\begin{split}\n  \\upeps{v} - \\upz{v} - \\varepsilon q \\upz{v} \\upz{\\Delta}  \\leq   \\tot C_v \\varepsilon^2 + \\tot C'_v \\varepsilon^3,\n \\end{split}\n \\end{equation} \n where $C_v= |q|\\|\\eta\\|_{\\el^{2(1-p)}}^{1\/2} + \\|\\Lambda\\|_{\\el^{1-p}}$ and\n $C'_v =|q|\\|\\eta \\Lambda\\|_{\\el^{1-p}}$. \n\\end{proposition}\n\\begin{proof}\nThe upper estimate \\eqref{equ:upper} and the representation\n\\eqref{equ:primal1} imply that\n\\[ \\ee{ V(\\upeps{Z}_T \\upeps{\\hat{H}}_T) - V(\\upz{Z}_T \\upz{\\hat{H}}_T) - \\varepsilon q V(\\upz{Z}_T \\upz{\\hat{H}}_T)\\eta}\n \\leq {\\mathbb E}[A_{\\varepsilon}] + {\\mathbb E}[B_{\\varepsilon}],\\]\n where $A_{\\varepsilon}$ and $B_{\\varepsilon}$ are defined by \\eqref{equ:A-B}, with\n $\\upeps{K}=\\upz{K}= \\upz{\\hat{H}}_T$.  As in \\eqref{equ:Holder}, we have\n \\[  {\\mathbb E}[ B_{\\varepsilon}]\\leq \\tot \\varepsilon^2 \\|\\Lambda\\|_{\\el^{1-p}}.\\] To deal with $A_{\\varepsilon}$ we note that its structure allows us to\n apply the representation from \\eqref{equ:primal1} once again to see\n \\[ \\oo{q^2} A_{\\varepsilon} = \\int_0^{\\varepsilon} \\int_0^{\\delta} V(\\uppar{Z}{\\beta}_T\n \\upz{\\hat{H}}_T) \\eta (\\eta+\\beta \\Lambda)\\, d\\beta\\, d\\delta. \\]\n This, in turn, can be estimated, via  H\\\" older inequality, as in\n \\eqref{equ:Holder}, as follows\n \\[ {\\mathbb E}[ A_{\\varepsilon}] \\leq \\tot  |q| \\varepsilon^2 \\sup_{ \\beta\\in [0,\\varepsilon] }   {\\mathbb E}[ (\\eta (\\eta+\\beta \\Lambda))^{1-p}]^{1+q}\n\\leq \\tot |q| \\varepsilon^2 \\Big(\n\\|\\eta^2\\|_{\\el^{1-p}}+\\varepsilon \\|\\eta \\Lambda\\|_{\\el^{1-p}}\\Big),\n\\]\n yielding the bound in \\eqref{equ:ve-up}.\n \\end{proof}\n Unfortunately, the same idea cannot be applied to obtain a similar lower\n bound. Instead, we turn to the primal problem and establish a lower bound\n for it. \n\\begin{proposition} \\label{pro:u-down} \nGiven $\\varepsilon_0>0$, assume that $\\Lambda\\in\\el^{1-p}$,\n and $\\Phi^2 e^{ \\varepsilon_0\\abs{p} \\Phi^-}\\in\\el^1(\\upz{\\tilde{\\PP}})$, where $\\upz{\\tilde{\\PP}}$ is defined by \\eqref{equ:Girsanov}. Then, \n\\[ \\upeps{u} - \\upz{u} - \\varepsilon p \\upz{u} \\upz{\\Delta} \\geq -  C_u(\\varepsilon) \\varepsilon^2 \\text{ for } \\varepsilon \\in\n[0,\\varepsilon_{0}],\\]\n where $C_u(\\varepsilon) := \\tot p^2 |\\upz{u}| {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[ \\Phi^2 e^{\\varepsilon \\abs{p} \\Phi^-}]$. \n\\end{proposition}\n\\begin{proof}\nFor $\\tX := {\\mathcal E}( \\int_0^{\\cdot} \\upz{\\hat{\\pi}}_t\\, d\\Re_t)$, we have \n$\\tX\\in\\upeps{\\sX}$ so that, by optimality, \n \\begin{equation}\n \\label{equ:first-u}\n \\begin{split}\n    \\upeps{u} - \\upz{u} - p\\varepsilon {\\mathbb E}[  U(\\upz{\\hat{X}}_T) \\Phi] &\\geq \n    {\\mathbb E}[   U(\\tX_T) - U(\\upz{\\hat{X}}_T) -  p \\varepsilon U(\\upz{\\hat{X}}_T)\\Phi].\n \\end{split}\n \\end{equation}\n Thanks to the form of $\\tX$, the right-hand side of \\eqref{equ:first-u}\n above can be written as ${\\mathbb E}[ U(\\upz{\\hat{X}}_T) D_{\\varepsilon}]$, where\n$D_{\\varepsilon} = \\exp( p \\varepsilon \\Phi) -1 - p\\varepsilon \\Phi = \n\\int_0^{\\varepsilon} \\int_0^{\\delta}  p^2 \\Phi^2 e^{p\\beta \\Phi}  \\, d\\beta\\, d\\delta$.\nThus, \n\\begin{align}\\label{lower_bound_est} \n\\begin{split}\n{\\mathbb E}[ U(\\upz{\\hat{X}}_T) D_{\\varepsilon}] &=  p^2 \\int_0^{\\varepsilon} \\int_0^{\\delta} {\\mathbb E}[ U(\\upz{\\hat{X}}_T) \\Phi^2 e^{ p \\beta \\Phi}]\\, d\\beta\\, d\\delta \\\\&\\geq \\tot p^2 \\varepsilon^2{\\mathbb E}[ U(\\upz{\\hat{X}}_T) \\Phi^2 e^{ \\varepsilon \\abs{p} \\Phi^-}].\n\\end{split}\n\\end{align}\nTherefore,  $\\upeps{u} - \\upz{u} - \\varepsilon p  {\\mathbb E}[ U(\\upz{\\hat{X}}_T) \\Phi] \\geq -  C_u(\\varepsilon) \\varepsilon^2$, for $\\varepsilon\\in [0,\\varepsilon_0]$ with $C_u$ as in the statement.\n \n It remains to show that \n ${\\mathbb E}[ U(\\upz{\\hat{X}}_T) \\Phi] =  {\\mathbb E}[ U(\\upz{\\hat{X}}_T) \\eta]$ which is equivalent to showing ${\\mathbb E}^{\\tilde{\\mathbb P}^{(0)}}[ \\Phi] =  {\\mathbb E}^{\\tilde{\\mathbb P}^{(0)}}[\\eta]$  by the definition of $\\tilde{\\mathbb P}^{(0)}$.\n We define the local $\\tilde{\\mathbb P}^{(0)}$-martingale $\\tM^p$ by \\eqref{equ:tMp}. Therefore, \n $N = \\int_0^{\\cdot} \\lambda'_t\\, d\\tM^p_t$ is also a local martingale.\nThe  desired\nequality\n is therefore equivalent\n to the equality ${\\mathbb E}^{\\upz{\\tilde{\\PP}}}[N_T]=0$ by the definition of $\\eta$ and $\\Phi$. In turn, it is sufficient to\n show that $N$ is an $\\sH^2$-martingale under $\\upz{\\tilde{\\PP}}$. Since $\\ab{N}_T =\n \\int_0^{T} (\\lambda'_t)^2\\,  d\\ab{M}_t = \\Lambda$ , H\\\" older's inequality implies\n that\n  \\begin{equation*}\n  \\label{equ:49B0}\n  \\begin{split}\n      {\\mathbb E}^{\\tilde{\\mathbb P}^{(0)}}[\\ab{N}_T] & = (q\\upz{v})^{-1} {\\mathbb E}[(\\upz{\\hat{Y}}_T)^{-q} \\Lambda]\n    \\leq (q\\upz{v})^{-1} {\\mathbb E}[\\Lambda^{1-p}]^{1+q}<\\infty.  \\qedhere\n  \\end{split}\n  \\end{equation*}\n\\end{proof}\n\\revised{\n\\begin{remark}\nIf one is interested in an error estimate which does not feature the optimal portfolio $\\upz{\\hat{\\pi}}$ (through $\\Phi$), one can adopt an alternative approach in the proof (and the statement) of Proposition \\ref{pro:u-down}. More specifically, \nby using $\\tX = \\upz{\\hat{X}}{\\mathcal E}( \\int_0^{\\cdot} \\varepsilon \\lambda' \\, d\\Re_t)$ as a test process (instead of ${\\mathcal E}( \\int_0^{\\cdot} \\upz{\\hat{\\pi}}_t\\, d\\Re_t)$), one obtains a constant $C_u(\\varepsilon)$ which depends only on the primal and dual optimizers $\\upz{\\hat{X}}$ and $\\upz{\\hat{Y}}$, in addition to $\\lambda'$, $\\eta$ and $\\Lambda$.\n\\end{remark}\n}\n\\begin{proof}[Proof of Theorem \\ref{thm:main2}]\nTwo of the four inequalities in Theorem \\ref{thm:main2} have been\nestablished in Propositions \\ref{pro:v-up} and \\ref{pro:u-down}. For the\nremaining two we use the special form \\eqref{equ:conj2}\nof the conjugacy relationship between $\\upeps{u}$ and\n$\\upeps{v}$. Thanks to Proposition \\ref{pro:u-down} \nand the positivity of $p\\upeps{u}$, $q\\upeps{v}$ and $1+q$, we have\n \\begin{equation*}\n \\label{equ:4797}\n \\begin{split}\n    q\\Big( \\upeps{v}-\n   \\upz{v} - \\varepsilon q \\upz{v} \\upz{\\Delta}\\Big) &= (p\\upeps{u})^{1+q} - (p\\upz{u})^{1+q} -\n    \\varepsilon q (p\\upz{u})^{1+q}\\upz{\\Delta}.\n \\end{split}\n \\end{equation*}\n The right-hand side above is further bounded from above,\n for $\\varepsilon$ in a (right) neighborhood of $0$, by\n\\[ F(\\varepsilon) := (p\\upz{u} + \\varepsilon p\\upz{u} \\upz{\\Delta} - p C\\varepsilon^2 )^{1+q} - (p\\upz{u})^{1+q} - \n\\varepsilon q  (p\\upz{u})^{1+q} \\upz{\\Delta},\\]\nwhere $C$ is the constant from Proposition \\ref{pro:u-down}. \n$F$ is a $C^2$-function in some neighborhood of $0$ with\n$F(0)=F'(0)=0$; hence, on each compact subset of that neighborhood it is bounded by a constant multiple of $\\varepsilon^2$. In particular, we have\n\\[ \\upeps{v} - \\upz{v} - \\varepsilon q \\upz{v} \\upz{\\Delta} \\geq -  C \\varepsilon^2,\\]\nfor some $C>0$ and $\\varepsilon$ in some (right) neighborhood of $0$. A similar argument, but based on\nProposition \\ref{pro:v-up}, shows that \\eqref{equ:main2-u} holds, as well. \n\\end{proof}\n\n \\proof[Proof of Theorem \\ref{thm:main3}] The first part of \\eqref{2nd_int} means that $\\Lambda \\in  \\el^{1-p}({\\mathbb P})$; hence, the second half of the proof of Proposition \\ref{pro:u-down} shows that \n${\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\Phi] = \\Delta^{(0)}$. Therefore, the martingale representation \\eqref{Itorep} can be written as\n\\begin{align}\\label{Itorep1}\n\\Phi = \\Delta^{(0)} + \\int_0^T \\gamma^B_{t}\\sigma_tdB^{\\upz{\\tilde{\\PP}}}_t  +  \\int_0^T \\gamma^W_{t} dW^{\\upz{\\tilde{\\PP}}}_t.\n\\end{align}\n\nBecause the filtration is generated by the  Brownian motions $(B,W)$ we can find $\\hat{\\nu}^{(0)} \\in \\sP_{W}^2$ such that the dual optimizer $\\hat{H}^{(0)}$ can be represented as\n$$\n\\hat{H}^{(0)} = \\mathcal{E}(-\\int \\hat{\\nu}^{(0)} dW).\n$$\nTherefore, Girsanov's Theorem ensures that under $\\upz{\\tilde{\\PP}}$, the processes\n$$\ndB^{\\upz{\\tilde{\\PP}}} := dB +(\\lambda -\\hat{\\pi}^{(0)})\\sigma dt, \\quad \\text{ and } \\quad dW^{\\upz{\\tilde{\\PP}}}:= dW +\\hat{\\nu}^{(0)}dt,\n$$\nare independent Brownian motions. We start with the primal problem and define $\\tilde{\\pi} := \\hat{\\pi}^{(0)} + \\varepsilon \\delta$  with $\\delta:=q\\gamma^B +\\frac{\\lambda'}{1-p}\\in \\sP_{B}^2$. Then we have\n\\begin{align*}\n(\\tilde{X})^{p} &:=\\mathcal{E}\\big(\\int\\tilde{\\pi}dR^{(\\varepsilon)})^p\\\\\n&= \\big(\\hat{X}^{(0)}\\big)^pe^{p\\int\\big (\\varepsilon\\hat{\\pi}^{(0)} \\lambda'+ \\varepsilon^2 (\\delta \\lambda'-\\frac12\\delta^2 ) \\big)\\sigma^2dt + p\\varepsilon \\int\\delta\\sigma dB^{\\upz{\\tilde{\\PP}}} }.\n\\end{align*}\nConsequently, by replacing $e^x$ with its Taylor expansion and using that the involved $\\upz{\\tilde{\\PP}}$-expectation is finite (here we use the integrability requirement \\ref{exp_reg}), we find a function $C_u(\\varepsilon) \\in O(\\varepsilon^3)$ such that\n\\begin{align}\\label{Cu}\n\\begin{split}\n{\\mathbb E}[U(\\tilde{X}_T)] \n&= u^{(0)} {\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\left[e^{p\\int_0^T \\big(\\varepsilon\\hat{\\pi}^{(0)} \\lambda'+ \\varepsilon^2 (\\delta \\lambda'-\\frac12\\delta^2)\\big)\\sigma^2dt +p\\varepsilon\\int_0^T \\delta\\sigma dB^{\\upz{\\tilde{\\PP}}} }\\right]\\\\\n& = u^{(0)}\\Big(1+p\\varepsilon\\Delta^{(0)}+ \\frac12 p\\varepsilon^2\\Big\\{p(\\Delta^{(0)})^2\n+\\Delta^{(00)}\\Big\\}\\Big)+ C_u(\\varepsilon).\n\\end{split}\n\\end{align}\n\nWe then turn to the dual problem. For the perturbed dual control $\\tilde{\\nu}:= \\hat{\\nu}^{(0)} -\\varepsilon p\\gamma^W\\in\\sP_{W}^2$ we have\n\\begin{align*}\n&\\Big(Z^{(\\varepsilon)} \\mathcal{E}(-\\int \\tilde{\\nu} dW)\\Big)^{-q}\\\\&= e^{q\\int ( \\lambda+\\varepsilon\\lambda' )\\sigma dB+q\\int ( \\hat{\\nu}^{(0)}-\\varepsilon p\\gamma^W )dW+q\\frac12\\int \\big(( \\lambda+\\varepsilon \\lambda' )\\sigma ^2 + | \\hat{\\nu}^{(0)}-\\varepsilon q\\gamma^W |^2 \\big)dt}\\\\\n&= (Z^{(0)}\\hat{H}^{(0)})^{-q}e^{\\varepsilon q\\int  \\lambda'\\sigma dB^{\\upz{\\tilde{\\PP}}}-\\varepsilon qp\\int \\gamma^W dW^{\\upz{\\tilde{\\PP}}}+q\\frac12\\int \\big( \\varepsilon^2 (\\lambda')^2\\sigma^2 +\\varepsilon^2 p^2|\\gamma^W|^2+2\\varepsilon\\lambda'\\pi^{(0)}\\sigma^2\\big)dt}.\n\\end{align*}\nSince $\\tilde{\\nu}$ is admissible in the $\\varepsilon$-problem we find \n\\begin{align*\n&v^{(\\varepsilon)} \\le \\frac1q{\\mathbb E}\\left[\\Big(Z_T^{(\\varepsilon)} \\mathcal{E}(-\\int_0^T \\tilde{\\nu} dW)\\Big)^{-q}\\right] \\\\&= v^{(0)} {\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\left[e^{\\varepsilon q\\int_0^T \\lambda'\\sigma dB^{\\upz{\\tilde{\\PP}}}-\\varepsilon qp\\int_0^T \\gamma^W dW^{\\upz{\\tilde{\\PP}}}+q\\frac12\\int_0^T \\big( \\varepsilon^2 (\\lambda')^2\\sigma^2 +\\varepsilon^2p^2 |\\gamma^W|^2+2\\varepsilon\\lambda'\\pi^{(0)}\\sigma^2\\big)dt}\\right].\n\\end{align*}\nFiniteness of $v^{(\\varepsilon)}$ ensures that the $\\upz{\\tilde{\\PP}}$-expectation appearing on the last line\nis also finite (recall that $q<0$). As in the primal problem, this allows us to replace $e^x$ with its Taylor series and in turn implies that we can find a function $C_v(\\varepsilon) \\in O(\\varepsilon^3)$ such that\n\\begin{align*}\n&v^{(0)} {\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\left[e^{\\varepsilon q\\int_0^T \\lambda'\\sigma dB^{\\upz{\\tilde{\\PP}}}-\\varepsilon qp\\int_0^T \\gamma^W dW^{\\upz{\\tilde{\\PP}}}+q\\frac12\\int_0^T \\big( \\varepsilon^2 (\\lambda')^2\\sigma^2 +\\varepsilon^2p^2 |\\gamma^W|^2+2\\varepsilon\\lambda'\\pi^{(0)}\\sigma^2\\big)dt}\\right]\\\\&=\nv^{(0)}\\Big(1+q\\varepsilon\\Delta^{(0)} + \\frac12q\\varepsilon^2\\Big\\{q(\\Delta^{(0)})^2 +\n\\Delta^{(00)}\\Big\\}\\Big)+ C_v(\\varepsilon).\n\\end{align*}\nBy combining this estimate and \\eqref{Cu} with the primal-dual relation \\eqref{equ:conj2} we find\n\\begin{align}\n\\label{fenchel}\n\\begin{split}\n& u^{(0)}\\Big(1+p\\varepsilon\\Delta^{(0)}+ \\frac12 p\\varepsilon^2\\Big\\{p(\\Delta^{(0)})^2\n+\\Delta^{(00)}\\Big\\}\\Big)+ C_u(\\varepsilon) \\\\&\\le u^{(\\epsilon)} \\\\&=\\frac1p (q v^{(\\epsilon)})^{1-p}\\\\& \\le \\frac1p\\Big(q v^{(0)}\\Big[1+q\\epsilon \\Delta^{(0)} + \\frac12q\\epsilon^2\\Big\\{q(\\Delta^{(0)})^2 +\n\\Delta^{(00)}\\Big\\}+C_v(\\epsilon)\\Big]\\Big)^{1-p}.\n\\end{split}\n\\end{align}\nThe function $x\\to x^{1-p}$ is real analytic on $(0,\\infty)$. Therefore, the fact that  $C_v\\in O(\\varepsilon^3)$ ensures that the last line of \\eqref{fenchel} agrees with the first line of \\eqref{fenchel} up to $O(\\varepsilon^3)$-terms. This establishes \\eqref{equ:main3-u}. \nA similar argument produces \\eqref{equ:main3-v}.\n\\endproof\n \n\n\n\\section{Examples}\n\\label{sec:examples}\n\\subsection{First examples} We start this section with a short list of\ntrivial and extreme cases. They are not here to illustrate the power of our\nmain results, but simply to help the reader understand them better. They\nalso tell a similar, qualitative, story: loosely speaking, the improvement\nin the utility (on the log scale) is proportional both to the base market\nprice of risk process and to the size of the deviation. Locally, around $\\lambda$, the \nvalue function of the utility maximization problem - parametrized by the\nmarket price of risk process $\\tilde{\\lambda}$ - is well approximated by\nan exponential function of the form\n \\begin{equation}\n \\label{equ:exp-approx}\n \\begin{split}\n    u(\\tilde{\\lambda}) \\approx u(\\lambda) e^{\\scl{\\tilde{\\lambda}-\\lambda}{\\upz{\\hat{\\pi}}}_{0}},\\text{ where }\n   \\scl{\\rho}{\\pi}_{0} =\n   {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[ \\int_0^T \\rho_t\\pi_t\\, dt],\n \\end{split}\n \\end{equation}\n where $u(\\tilde{\\lambda})$ and $u(\\lambda)$ denote the values of the utility-maximization problems with market price of risk processes $\\tilde{\\lambda}$ and $\\lambda$, respectively.\n\\begin{example}[Small market price of risk]\n\\label{exa:small}\nSuppose that $\\lambda\\equiv 0$ so that we can think of $\\upeps{S}$ as the stock price\nin a market with a ``small'' market price of risk. Since $\\upz{Z}\\equiv 1$, it\nis clearly the dual optimizer at $\\varepsilon=0$ and we have $\\upz{\\hat{\\pi}}\\equiv 0$.\nConsequently, under the assumptions of Theorem \\ref{thm:main2}, we have\n$\\upz{\\tilde{\\PP}}={\\mathbb P}$ and \n\\[ \\upz{\\Delta} = {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[ \\int_0^T \\lambda'_t\\, dM_t]=0.\\]\nIt follows that\n\\[  \\upeps{u} = \\upz{u} + O(\\varepsilon^2)\\text{ and } \\upeps{v}=\\upz{v} + O(\\varepsilon^2),\\]\nand the effects of $\\varepsilon \\lambda'$ are felt only in the second order,\nregardless of the risk-aversion coefficient $p<0$. \n\\end{example}\n\\begin{example}[Deviations from the Black-Scholes model]\n\\label{exa:Black-Scholes}\nSuppose that $M=B$ is an ${\\mathbb F}$-Brownian \nmotion and that $\\lambda\\ne 0$ is a constant process (we also use $\\lambda$ for the\nvalue of the constant). \nIn that case, it is classical that the dual minimizer in the base market\nis $\\upz{Z} = {\\mathcal E}( - \\lambda B)$ and, consequently, that \n$\\rn{\\upz{\\tilde{\\PP}}}{{\\mathbb P}} = {\\mathcal E}( q\\lambda B)$. It follows that\n\\[ \\upz{\\Delta} = \\tfrac{\\lambda}{1-p} {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[ \\int_0^T \\lambda'_t\\, dt].\\]\nAs we will see below, this form is especially convenient for computations.\n\\end{example}\n\\begin{example}[Uniform deviations] \n\\label{exa:constant}\nAnother special case where it is\nparticularly easy to compute the (logarithmic derivative) $\\upz{\\Delta}$ is when the\nperturbation $\\lambda'$ is a constant process (whose value is also\ndenoted by $\\lambda'$). Indeed, in that case\n \\begin{equation}\n \\label{equ:Dz-unif}\n \\begin{split}\n   \\upz{\\Delta} = \\lambda' {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[ \\int_0^T \\upz{\\hat{\\pi}}_t\\, dt].\n \\end{split}\n \\end{equation}\nIt is especially instructive to consider the case where the base model is\nBlack and Scholes' model since everything becomes explicit:\nthe optimal portfolio is\ngiven by the Merton proportion $\\upz{\\hat{\\pi}}_t = \\lambda\/(1-p)$, and the\nthe values $\\upeps{u}$ and $\\upeps{v}$ are given by\n\\[ p\\upz{u} = \\exp(\\tot q \\lambda^2 T) \\text{ and } q\\upz{v} = \\exp(\\tot \\tfrac{q}{1-p} \\lambda^2\nT). \\]\nUsing \\eqref{equ:Dz-unif} or by performing a \nstraightforward direct computation, we easily get\n\\[ p\\upz{\\Delta} = q \\lambda' \\lambda T, \\]\nmaking the approximation in \\eqref{equ:exp-approx} exact.\n\\end{example}\n\\subsection{The Kim-Omberg model}\nThe Kim-Omberg model (see \\cite{KimOmb96}) \nis one of the most widely used models for the\nmarket price of risk process. Because the Kim-Omberg model allows for explicit expressions for all quantities involved in CRRA utility maximization it serves as an excellent test\ncase for the practical implementation of our main results. \n\nWe assume that ${\\mathbb F}$ is the augmentation of the filtration generated by two independent one dimensional Brownian motions $B$ and $W$ and define $\\lambda^\\text{KO}$ be the Ornstein-Uhlenbeck\nprocess\n\\begin{align}\nd\\lambda_t^\\text{KO} &:= \\kappa (\\theta - \\lambda_t^\\text{KO})dt + \\beta dB_t + \\gamma dW_t, \\quad \\lambda^\\text{KO}_0\\in\\R\\label{OU},\n\\end{align}\nwhere $\\kappa, \\theta, \\beta$ and $\\gamma$ are constants. We define the volatility $M_t := B_t$ in what follows.\n\nThe following result summarizes the main properties in \\cite{KimOmb96}:\n\\begin{theorem}[Kim and Omberg 1996]\\label{thm:KO} Let the market price of risk process be defined by \\eqref{OU}, $M:=B$, and let $p<0$. Then there \nexist continuously differentiable functions $a,b,c:[0,\\infty)\\to \\R$ such that for $t\\in\n[0,T)$ we have\n\\begin{align*}\n-a'(t) & = \\alpha_1\\, b(t) \n+ \\tot \\alpha_3\\, c(t) - \n\\tot \\alpha_2\\, b^2(t), &\na(T)&=0, \\\\\n-b'(t) &= \\alpha_4\\, b(t)+ \\alpha_1\\, c(t)  - \\alpha_2\\, b(t) c(t),& b(T)&=0, \\\\\n-c'(t) &= -q + 2 \\alpha_4\\, c(t) - \\alpha_2\\, c^2(t), & c(T)&=0,\n\\end{align*}\nwhere $\\alpha_1:=\\theta\\kappa$, $\\alpha_2 := (1+q) \\beta^2+ \\gamma^2$,\n$\\alpha_3 := \\beta^2+\\gamma^2$ and $\\alpha_4 := q \\beta - \\kappa$. \nFurthermore, the primal  value function reads\n\\begin{align}\n\\label{equ:exact}\nu^\\text{KO}(x) = \\frac{x^p}{p}e^{-a(0) - b(0)\\lambda^\\text{KO}_0 - \\tfrac12 c(T)(\\lambda^\\text{KO}_0)^2},\\quad x>0,\n\\end{align}\nand the corresponding primal optimizer is given by\n\\begin{align}\\label{KO_primal}\n\\hat{\\pi}^\\text{KO}_t = \\frac{b(t)\\beta + \\big(c(t)\\beta-1\\big) \\lambda^\\text{KO}_t}{p-1},\\quad t\\in[0,T].\n\\end{align}\n\\end{theorem}\nFor $p<0$, the above Riccati equation describing $c$ has the ``normal\nnon-exploding solution'' as defined in the appendix of \\cite{KimOmb96}.\nTherefore, all three functions $a,b$, and $c$ are bounded on any finite\ntime-interval $[0,T]$ of $(0,\\infty)$.\n\nTo illustrate our approximation we think of the Kim-Omberg model as a\nperturbation of a base model. As base model we will consider the following model with ``totally-unhedgable-coefficients'' (see Example 7.4, p.~305, in \\cite{KarShr98}):\n\\begin{align}\\label{KO1}\nd\\lambda_t := \\kappa (\\theta-\\lambda_t) dt + \\gamma \\, dW_t,\\quad \n\\lambda_0:=\\lambda^\\text{KO}_0. \n\\end{align}\nThis way, $\\ld^\\text{KO} = \\lambda+\\varepsilon \\lambda'$, where $\\varepsilon = \\beta$ and\n\\begin{align}\\label{KO2}\nd\\lambda'_t := - \\kappa \\lambda'_t\\, dt + dB_t,\\quad \\lambda'_0:=0.\n\\end{align}\nThe following result provides closed-form expressions for our correction terms:\n\\begin{lemma}\\label{lem:KO} Let $(\\lambda,\\lambda')$ be defined by \\eqref{KO1}-\\eqref{KO2} and let $p<0$. For the $\\varepsilon =0$ model the primal and dual optimizers are given by\n\\begin{align}\\label{KO:base}\n\\hat{\\pi}^{(0)}_t = \\frac{\\lambda_t}{1-p},\\quad \\hat{\\nu}^{(0)}_t = \\gamma \\Big(b(t)+c(t)\\lambda_t\\Big),\\quad t\\in[0,T].\n\\end{align}\nFurthermore, the processes $(\\gamma^B,\\gamma^W)$ appearing in the\nmartingale representation \\eqref{Itorep} of $\\Phi$  are given by\n\\begin{align}\n\\gamma_t^B &=\\tfrac1{p-1}(C_2(t)+C_6(t)\\lambda_t),\\label{KOgammaB}\\\\\n\\gamma^W_t &= \\tfrac{\\gamma}{p-1}(C_4(t)+2C_5(t)\n\\lambda_t+C_6(t)\\lambda_t'),\\label{KOgammaW}\n\\end{align}\nwhere the functions $C_1,C_2,C_4, C_5$ and $C_6$ in \n\\eqref{KOgammaB}-\\eqref{KOgammaW} satisfy the ODEs\n\\begin{align*}\n-C_1'(t)&= \\tilde{b}(t) \\, C_4(t)+  \\gamma^2 \\, C_5(t),& C_1(T)&=0,\\\\\n-C_2'(t)&= \\tilde{b}(t) \\, C_6(t) - \\kappa \\, C_2(t),& C_2(T)&=0,\\\\\n-C_4'(t) &= q\\, C_2(t) - \\tilde{c}(t) \\, C_4(t) + 2\\tilde{b}(t) \\, C_5(t),\n& C_4(T)&=0,\\\\\n-C_5'(t) &= q\\, C_6(t) -2 \\tilde{c}(t) \\, C_5(t),& C_5(T)&=0,\\\\\n-C'_6(t) &= - (\\kappa +\\tilde{c}(t)) \\, C_6(t) -1,& C_6(T)&=0,\n\\end{align*}\non $[0,T)$, with\n$(a,b,c)$ as in Theorem \\ref{thm:KO} (with $\\beta:=0$),\n$\\tilde{b}(t):=\\kappa\\theta-\\gamma^2 b(t)$ and\n$\\tilde{c}(t) := \\kappa+\\gamma^2 c(t) $.\nFurthermore, for the measure $\\tilde{\\mathbb P}^{(0)}$ defined by \\eqref{equ:Girsanov} and for all $T>0$ we have\n\\begin{align}\\label{Delta0_KO}\n\\Delta^{(0)} &:= {\\mathbb E}^{\\tilde{\\mathbb P}^{(0)}}\\left[ \\int_0^T \\lambda'_s\\hat{\\pi}_s^{(0)} ds\\right]= -\\frac1{1-p}\\Big(C_1(T) + C_4(T) \\lambda_0 +C_5(T) \\lambda_0^2\\Big),\n\\end{align}\n\\end{lemma}\n\n\\proof The first part follows from Theorem \\ref{thm:KO} applied to the case\n$\\beta:=0$. To find the  martingale representation \\eqref{Itorep} we define the function \n$$\nf(t,x,\\lambda):= \\frac{x^p}{p}e^{-a(t) - b(t)\\lambda - \\tfrac12 c(t)\\lambda^2},\\quad t\\in[0,T],\\quad x>0,\\quad \\lambda\\in\\R,\n$$\nwhere the functions $(a,b,c)$ are as in Theorem \\ref{thm:KO}. The martingale properties of\n$f(t,\\hat{X}^{(0)}_t,\\lambda_t)$ and $\\hat{X}^{(0)}_t\\hat{Y}^{(0)}_t$ as well as the proportionality property $(\\hat{X}^{(0)}_T)^{p}\\propto\\hat{X}^{(0)}_T \\hat{Y}^{(0)}_T$ produce \n$$\npf(t, \\hat{X}^{(0)}_t,\\lambda_t)=p{\\mathbb E}[f(T, \\hat{X}^{(0)}_T,\\lambda_T)|\\sF_t]\\propto{\\mathbb E}[ \\hat{Y}^{(0)}_T\\hat{X}^{(0)}_T|\\sF_t] = \\hat{X}^{(0)}_t\\hat{Y}^{(0)}_t.\n$$\nBy computing the dynamics of the left-hand-side we see from Girsanov's Theorem that the two processes\n\\begin{align*}\ndB^{\\tilde{\\mathbb P}^{(0)}}_t &:= \n- q\\lambda_tdt + dB_t,\\\\\n\\quad dW^{\\tilde{\\mathbb P}^{(0)}}_t &:= \n\\Big(b(t)+c(t)\\lambda_t\\Big)\\gamma dt + dW_t,\n\\end{align*}\nare independent Brownian motions under $\\tilde{\\mathbb P}^{(0)}$. These dynamics and It\\^o's Lemma ensure that\n\\begin{align*}\nN_t := \\int_0^t &\\lambda'_s\\lambda_s ds -C_1(t) -C_2(t)\\lambda'_\n- C_4(t) \\lambda_t -C_5(t) \\lambda_t^2-C_6(t)\\lambda_t\\lambda'_t,\n\\end{align*}\nis a $\\tilde{\\mathbb P}^{(0)}$-local martingale.\n\nBecause the processes $(\\lambda,\\lambda')$ remain Ornstein-Uhlenbeck\nprocesses under $\\tilde{\\mathbb P}^{(0)}$ and the functions $C_1$-$C_6$ are bounded, $N$ is indeed a\n$\\tilde{\\mathbb P}^{(0)}$-martingale. Furthermore, thanks to the zero terminal conditions\nimposed on $C_1$-$C_6$, we see that\n\\begin{align*}\n\\Phi=\\tfrac1{1-p}\\int_0^T \\lambda_t \\lambda'_tdt &= \\tfrac1{1-p}N_T = \\tfrac1{1-p}N_0 +\\int_0^T \\gamma^B_t dB^{\\upz{\\tilde{\\PP}}}_t +\\int_0^T \\gamma^W_t dW^{\\upz{\\tilde{\\PP}}}_t,\n\\end{align*}\nfor $(\\gamma^B,\\gamma^W)$ defined by \\eqref{KOgammaB}-\\eqref{KOgammaW}. \n\\endproof\n\n\\subsubsection{Exact computations.} The proof of Lemma \\ref{lem:KO}  shows that \n$$\n\\Delta^{(0)} :={\\mathbb E}^{\\tilde{\\mathbb P}^{(0)}}\\left[ \\int_0^T \\lambda'_s\\hat{\\pi}_s^{(0)}\nds\\right]= \\frac1{p-1}\\Big(C_1(0) + C_4(0) \\lambda_0 +C_5(0)\n\\lambda_0^2\\Big).\n$$\nThis relation, a similar one (whose exact form and the derivation we omit) \nfor the second-order term $\\Delta^{(00)}$ of \\eqref{Delta00}, and the availability of the exact expression \\eqref{equ:exact} for the value function $u^{KO}$ \nallow for an efficient numerical computation of the zeroth-, first-, and second-order\napproximation, and their comparison with the exact values.\nThe model parameters used in the below Table 1 are the calibrated model parameters for the market portfolio reported in Section 4.2 in \\cite{LarMun2012} (we ignore\nthe constant interest rate and constant volatility used in Section 4 in\n\\cite{LarMun2012}). Moreover, we use negative values of $\\varepsilon$ because\nthe empirical covariation between excess return and the\nstock's return is typically negative (see, e.g., the discussion in Section 4.2 in\n\\cite{LarMun2012}).\n\nInstead of hard-to-interpret expected utility values, we report their certainty\nequivalents (i.e., their compositions with the\nfunction $CE:=U^{-1}$; see Remark \\ref{rem:761C}(1)).\nWe set $\\delta^{(0)} := p u^{(0)}  \\Delta^{(0)}$ and\n$\\delta^{(00)} :=p u^{(0)} \\big( \\Delta^{(00)} + p (\\Delta^{(0)})^2\\big)$. \n\n\\begin{center}\n\\begin{tabular}{cc||ccc|c}\n $\\varepsilon$ & \n $\\lambda_0$& \n $CE(u^{(0)}) $&\n $CE(u^{(0)}+ \\varepsilon \\delta^{(0)})$&\n $CE(u^{(0)}+ \\varepsilon \\delta^{(0)} + \\tfrac{\\varepsilon^2}{2} \\delta^{(00)})$&\n $CE(u^{(\\varepsilon)})$\\\\\n\\hline\\hline\n-0.01 & 0.1 & 1.046 & 1.047 & 1.048 & 1.048 \\\\\n- 0.05 & 0.1 & 1.046 & 1.054 & 1.081 & 1.084 \\\\\n- 0.10 & 0.1 & 1.046 & 1.063 & 1.181 & 1.206 \\\\\\hline\n- 0.01 & 0.5 & 1.614 & 1.647 & 1.648 & 1.649 \\\\\n- 0.05 & 0.5 & 1.614 & 1.794 & 1.850 & 1.846 \\\\\n- 0.10 & 0.5 & 1.614 & 2.020 & 2.339 & 2.272 \\\\\n\\hline\n\\end{tabular}\n\\footnotesize \n\\begin{quotation}\n \\textbf{Table 1.} Certainty equivalents for the zeroth-, first-, and second-order approximations and the exact values in the Kim-Omberg model with\n$\\beta:=\\varepsilon$ and unit initial wealth. The model parameters used are \n$\\gamma :=0.04395,\\,\\kappa := 0.0404,\\,\\theta := 0.117,\\,p:=-1$, and $T:=10$.\n\\end{quotation}\n\\end{center}\n\n\n\n\n\n\\subsubsection{Monte-Carlo-based computations}\nOne of the advantages of our approach is that it lends\nitself easily to computational methods based on Monte-Carlo (MC) simulation. For the Kim-Omberg model we use the standard explicit Euler scheme from MC simulation to compute the involved quantities of interest. In other words, we do not rely on the availability of exact expressions for the value\nfunctions or the correction terms $\\Delta^{(0)}$ and $\\Delta^{(00)}$.\n\nFor a portfolio $\\pi$ and the\nmodel-perturbation parameter $\\varepsilon$, the  constant\n$\\text{CE}^{(\\varepsilon)}(\\pi)\\in(0,\\infty)$ is uniquely defined by\n\\begin{align}\\label{approxCE}\nU\\left(\\text{CE}^{(\\varepsilon)}(\\pi)\\right) = {\\mathbb E}\\left[U\\left({\\mathcal E}\\big(\\int_0^T\n\\pi_t\\, d\\Re_t\\big)\\right)\\right].\n\\end{align}\nIn other words, $\\text{CE}^{(\\varepsilon)}{(\\pi)}$ is the dollar amount whose\nutility value matches that of the expected utility an investor would obtain\nin the $\\varepsilon$-model who uses the strategy $\\pi$. We remind the reader that\n$\\hat{\\pi}^{(0)}$ denotes the optimizer in the base ($\\varepsilon=0$)\nmodel, $\\tilde{\\pi}^{(\\varepsilon)}$ is the second-order improvement (as in\n\\ref{optimal_control_correction} above)\nof $\\hat{\\pi}^{(0)}$,\nand $\\hat{\\pi}^{(\\varepsilon)}$ is the exact optimizer in the $\\varepsilon$-model.\nBoth quantities \n$\\text{CE}^{(\\varepsilon)}(\\hat{\\pi}^{(0)})$ and \n$\\text{CE}^{(\\varepsilon)}(\\tilde{\\pi}^{(\\varepsilon)})$ serve as lower bounds for the exact\nvalue $\\text{CE}(u^{(\\varepsilon)})$. The second one, which we also denote by\n\\begin{align}\\label{def_LB}\n \\text{LB} := \\text{CE}^{(\\varepsilon)}(\\tilde{\\pi}^{(\\varepsilon)}),\n \\end{align}\nis second-order optimal and appears in our simulations. \nTo obtain a corresponding upper bound, we simulate the dynamics of the dual\nprocess, based on $\\eqref{equ:conj2}$ and the\nsecond-order optimal dual control $\\tilde{\\nu}$ defined by\n\\eqref{optimal_dual_control_correction}. We define\n\\begin{align}\\label{def_UB}\n\\text{UB} := U^{-1} \\left( \\frac1p{\\mathbb E}\\left[\\Big(Z_T^{(\\varepsilon)}\n\\mathcal{E}(-\\int_0^T \\tilde{\\nu}_u dW_u)\\Big)^{-q}\\right]^{1-p} \\right).\n\\end{align}\n\nTo quantify the simulation errors, we report the $95\\%$-confidence intervals based on MC simulated values of  $\\text{CE}^{(\\varepsilon)}(\\hat{\\pi}^{(0)})$, LB, and UB in the below Table 2. The value $\\text{CE}(u^{(\\varepsilon)})$,\ncomputed without MC simulation and included for comparison only, \nis exact to $3$ decimal places. \n\\begin{center}\n\\begin{tabular}{cc||ccc|c}\n $\\varepsilon$ & \n $\\lambda_0$& \n $\\text{CE}^{(\\varepsilon)}(\\hat{\\pi}^{(0)}) $ &\n $\\text{LB}$ &\n $\\text{UB}$ &\n $\\text{CE}(u^{(\\varepsilon)})$\\\\\n\\hline\\hline\n-0.01& \t0.10& $[1.047, 1.048]$& [1.048, 1.049]& [1.048, 1.049]& 1.048\\\\  \n-0.05& \t0.10& $[1.052, 1.053]$& [1.083, 1.084]& [1.083, 1.085]& 1.084\\\\  \n-0.10& \t0.10& $[1.057, 1.058]$& [1.200, 1.201]& [1.204, 1.208]& 1.206\\\\  \n\\hline\n-0.01& \t0.50& $[1.644, 1.649]$& [1.647, 1.653]& [1.646, 1.657]& 1.649\\\\  \n-0.05& \t0.50& $[1.760, 1.764]$& [1.844, 1.850]& [1.843, 1.857]& 1.846\\\\ \n-0.10& \t0.50& $[1.868, 1.871]$& [2.248, 2.256]& [2.266, 2.286]& 2.272\\\\  \n\\hline\n\\end{tabular}\n\\begin{quotation}\n\\footnotesize \\textbf{Table 2.} \n$95\\%$-confidence intervals for certainty equivalents for the upper and lower bounds as well as the base model optimizer $\\hat{\\pi}^{(0)}$ for the Kim-Omberg model. The true exact values for the $\\varepsilon$-model are included in the last column for comparison. Except for the last column, the numbers are based on MC simulation using Euler's scheme with one million paths each with time-step size $0.001$. The model parameters are the same as in Table 1.\n\\end{quotation}\n\\end{center}\n\nIn Table 2 we note the significant difference \nbetween the performance of the base-model optimizer\n$\\hat{\\pi}^{(0)}$ and its second-order improvement\n$\\tilde{\\pi}^{(\\varepsilon)}$; especially for larger values of $\\varepsilon$. Furthermore, the lower and upper bounds appear to be quite tight.\n\n\\subsection{Extended affine models} \n\\label{sse:extaff}\nWe turn to a class of models for which no closed-form expressions for the\nvalue functions $u$ and $v$ seem to be available. It constitutes the\nmain example of the class of so-called extended-affine specifications of the\nmarket price of risk models introduced by \\cite{CheFilKim07}. \n\nAs in the\nKim-Omberg model above we let the augmented filtration be generated by\ntwo independent Brownian motions $B$ and $W$. The\ncentral role is played by the following Feller process $F$\n\\begin{align}\ndF_t&:= \\kappa (\\theta - F_t)dt +  \\sqrt{F_t}\\, \\big( \\beta dB_t + \\gamma dW_t\\big), \\quad F_0>0,\\label{Feller}\n\\end{align}\nwhere $\\kappa, \\theta, \\beta$ and $\\gamma$ are strictly \npositive constants such that the (strict) Feller condition\n$2\\kappa\\theta > \\beta^2 +\\gamma^2$ holds. This ensures, in particular,\nthat $F$ is strictly positive on $[0,T]$, almost surely. \nUnlike in the Kim-Omberg model, the appropriate volatility normalization turns out to be \n$\\sqrt{F_t}$; that is, we define \n \\begin{equation}\n \\label{equ:M-F}\n \\begin{split}\n   M := \\int_0^{\\cdot} \\sqrt{F_t}\\, dB_t.\n \\end{split}\n \\end{equation}\nA particular extended affine specification of the market price of risk process considered in \\cite{CheFilKim07}  is given by\n\\begin{align}\\label{CFK}\n\\lambda^\\text{CFK}_t := \\frac{\\varepsilon}{F_t}+1,\n\\end{align}\nwhere $\\varepsilon$ is a (positive or negative) constant. Unless $\\varepsilon=0$,\nthere is \ncurrently no known closed-form \nsolution to the corresponding optimal investment problem (Theorem 4.5 in \\cite{GR15} expresses the corresponding value function as an infinite sum of weighted generalized Laguerre polynomials). However, for\n$\\varepsilon =0$, the resulting model is covered by the analysis in\n\\cite{Kra05}. Therefore, we choose the constant \nmarket price of risk process\n\\[ \\lambda_t:= 1\\] for the base model whereas we define the \nperturbation process $\\lambda'$ by\n\\begin{align}\\label{CV}\n\\lambda'_t := \\frac1{F_t}.\n\\end{align}\n\\begin{theorem}[Kraft 2005]\\label{thm:CV} For $p<0$ there exist continuously differentiable functions $a,b:[0,T)\\to \\R$ such\nthat \n\\begin{align*}\n- a'(t)&= \\alpha_1\\, b(t),& a(T)&=0, \\\\\n-b'(t) &= \\alpha_4 \\, b(t) - \\tot \\alpha_2\n\\, b^2(t) - \\tot q,& b(T)&=0,\n\\end{align*}\nwhere $\\alpha_1:=\\theta\\kappa$, $\\alpha_2 := (1+q) \\beta^2+ \\gamma^2$,\n and $\\alpha_4 := q \\beta - \\kappa$.  The value function of the utility-maximization problem with $\\lambda:= 1$ and $M$ as in \\eqref{equ:M-F} is given by \n$$\nu^{(0)}(x) = \\frac{x^p}{p}e^{-a(0) - b(0) F_0},\\quad x>0.\n$$\nThe corresponding primal and dual optimizers are given by\n\\begin{align}\\label{CV_primal}\n\\hat{\\pi}^{(0)}_t = \\frac{b(t)\\beta-1}{p-1},\\quad \\hat{\\nu}^{(0)}_t = b(t)\\gamma\\sqrt{F_t},\\quad  t\\in[0,T].\n\\end{align}\n\\end{theorem}\n\nTo check the conditions of our main\ntheorems, we use the explicit expression in \\cite{HurKuz2008}, Theorem\n3.1, for the Laplace transform \n\\[ L(a_1, a_2) := {\\mathbb E}[ \\exp( a_1 Q + a_2 \\Lambda)],\\quad Q:=\\int_0^T F_s\\, ds,\\quad \\Lambda:=\\int_0^T \\tfrac{1}{F_s}\\, ds.\\]\nIt is shown in \\cite{HurKuz2008} that $L$ is finite in some neighborhood\nof $0$ under the strict Feller condition $2\\kappa\\theta > \\beta^2\n+\\gamma^2$. This implies that both $\\Lambda$ and $Q$ have a finite exponential\nmoment. In particular, H\\\"older's inequality with exponents $-1\/q$ and $(1-p)$ implies that\n$$\n{\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\Lambda]=\\frac1{qv^{(0)}}{\\mathbb E}[(\\hat{Y}_T^{(0)})^{-q}\\Lambda]\\le \\frac1{qv^{(0)}}{\\mathbb E}[\\Lambda^{1-p}]^{\\frac{1}{1-p}}<\\infty.\n$$ Thanks to the deterministic behavior of $\\hat{\\pi}^{(0)}$ in\n\\eqref{CV_primal}, the  martingale representation \\eqref{Itorep} of $\\Phi$  holds\nwith $\\gamma^B = \\gamma^W=0$. Consequently, we have \n$$\n\\Phi :=  \\int_0^T  \\upz{\\hat{\\pi}}_s\\, ds =\\Delta^{(0)},\\quad \n\\Delta^{(00)} :=\\tfrac{1 }{1-p} {\\mathbb E}^{\\upz{\\tilde{\\PP}}}[\\Lambda].\n$$\nTo verify that \\eqref{exp_reg} holds, we can use H\\\"older's inequality (twice) with exponents $-1\/q$ and $(1-p)$ to see\n\\begin{align*}\n{\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\left[e^{-\\frac12\n\\varepsilon^2 \\frac{p}{(1-p)^2}\\Lambda + q\\varepsilon \\int_0^T \\frac1{\\sqrt{F_t}} dB^{\\upz{\\tilde{\\PP}}}_t}\\right]&\\l\n{\\mathbb E}^{\\upz{\\tilde{\\PP}}}\\left[e^{-\\frac12\n\\varepsilon^2 (p+q)\\Lambda}\\right]^{\\frac1{1-p}}\\\\\n&\\le\n\\frac1{qv^{(0)}}{\\mathbb E}\\left[e^{-\\frac12\n\\varepsilon^2 (1-p)(p+q)\\Lambda}\\right]^{\\frac1{(1-p)^2}},\n\\end{align*}\nwhich is finite for $\\varepsilon>0$ small enough. This allows Theorem\n\\ref{thm:main3} to be invoked for $\\varepsilon>0$ small enough.\nThe second-order optimal controls $(\\tilde{\\pi},\\tilde{\\nu})$  are then well defined\nby \\eqref{optimal_control_correction} and\n\\eqref{optimal_dual_control_correction}, and read\n\\begin{align}\n\\tilde{\\pi} := \\hat{\\pi}^{(0)} + \\varepsilon \\tfrac{\\lambda'}{1-p},\\quad\n\\tilde{\\nu} := \\hat{\\nu}^{(0)}.\n\\end{align}\n\n\n  \n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\nTable 3 is the analogue of Table 2 for the extended affine model with parameters taken from Figure 4 in Section 3.3 in \\cite{LarMun2012}.  The methodology and the simulated quantities are the same as for Table 2.\n\\newpage\n\\begin{center}\n  \\begin{tabular}{cc||ccc}\n$\\varepsilon$ & $F_0$ & $\\text{CE}^{(\\varepsilon)}(\\hat{\\pi}^{(0)})$  & LB &\n UB \\\\ \\hline\\hline\n$0.10$ & $0.01$ & $[1.724, 1.726]$& $[10.159, 10.399]$&$[10.226, 10.481]$  \\\\\n$0.05$& $0.01$ &  $[1.342, 1.343]$& $[2.141, 2.151]$&$[2.131, 2.149]$  \\\\\n$0.01$& $0.01$& $[1.097, 1.098]$& $[1.118, 1.119]$&$[1.117, 1.120]$  \\\\\n\\hline\n$0.10$& $0.05$& $[1.728, 1.729]$& $[9.660, 9.877]$&$[9.766, 10.000]$  \\\\\n$0.05$& $0.05$& $[1.344, 1.345]$& $[2.105, 2.115]$&$[2.102, 2.120 ]$  \\\\\n$0.01$& $0.05$& $[1.099, 1.100]$& $[1.119, 1.121]$&$[1.117, 1.121]$  \\\\\n\\hline\n\\end{tabular}\n\\begin{quotation}\n\\footnotesize \\textbf{Table 3.} $95\\%$-confidence intervals for certainty equivalents for the upper and lower bounds as well as the base model optimizer $\\hat{\\pi}^{(0)}$ for the extended affine model. The parameter values are \n$\\kappa := 5$, $\\theta := 0.0169$, $\\beta := -0.1$, $\\gamma :=0.1744$, $p:=-1$,\nand $T:=10$. The numbers are based on MC simulation using Euler's scheme with one million paths each with time-step size $0.001$.\n\\end{quotation}\n\n\\end{center}\n\nThe zeroth order approximation $\\text{CE}^{(0)}(\\hat{\\pi}^{(0)})$ produces the certainty equivalent values \n$$\n\\text{CE}^{(0)}(\\hat{\\pi}^{(0)}) = 1.043 \\; (F_0=0.01),\\quad\\text{and }\\quad   \\text{CE}^{(0)}(\\hat{\\pi}^{(0)}) = 1.045 \\;(F_0=0.05).\n$$ \nPerhaps even more than in the Kim-Omberg model, the numbers in Table 3 above illustrate the superiority of the second-order approximations (columns 4 and 5) over its first-order version (column 3) as well as the zeroth order values reported above. Again, the bounds in Table 3 appear quite tight when compared to the first-order approximations for moderate values of $\\varepsilon$. \n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAnalyzing the dynamics of quantum fields in curved black-hole\nspacetimes, Hawking \\cite{Haw} has reached the intriguing conclusion\nthat black holes have a well defined temperature \\cite{NoteBek,Bek}.\nIn particular, Hawking \\cite{Haw} has revealed that semi-classical\nblack holes are characterized by quantum emission spectra which have\ndistinct thermal features \\cite{Notegrey}. This remarkable\nprediction is certainly one of the most important outcomes of the\ninterplay between quantum field theory and classical general\nrelativity.\n\nIt has recently been shown \\cite{Vis} that the semi-classical\nHawking radiation flux out of a $(3+1)$-dimensional Schwarzschild\nblack hole is extremely sparse [see also \\cite{BekMuk,Mak,Hod1} for\nearlier discussions of this characteristic property of the\n$(3+1)$-dimensional Schwarzschild black hole]. In particular, the\nHawking emission of gravitons out of a $(3+1)$-dimensional\nSchwarzschild black hole is characterized by the dimensionless large\nratio \\cite{Vis}\n\\begin{equation}\\label{Eq1}\n\\eta\\equiv{{\\tau_{\\text{gap}}}\\over{\\tau_{\\text{emission}}}}\\gg1\\ ,\n\\end{equation}\nwhere $\\tau_{\\text{gap}}$ is the characteristic time gap between the\nemissions of successive black-hole gravitational quanta [see Eqs.\n(\\ref{Eq8}) and (\\ref{Eq9}) below], and $\\tau_{\\text{emission}}$ is\nthe characteristic timescale required for an individual Hawking\nquantum to be emitted from the black hole [see Eq. (\\ref{Eq10})\nbelow].\n\nThe dimensionless large ratio (\\ref{Eq1}) implies that the\nsemi-classical Hawking evaporation of $(3+1)$-dimensional\nSchwarzschild black holes is indeed sparse. That is, the\ncharacteristic time gap between the emissions of successive\ngravitational quanta out of an evaporating $(3+1)$-dimensional\nSchwarzschild black hole is very large on the natural timescale\n$2\\pi\/\\omega$ [see Eq. (\\ref{Eq10}) below] set by the characteristic\nenergy (frequency) of the emitted Hawking quanta. The characteristic\nlarge ratio (\\ref{Eq1}) therefore suggests a simple physical picture\nin which an evaporating $(3+1)$-dimensional Schwarzschild black hole\n\\cite{NoteHK,Hodkerr} typically emits Hawking quanta one at a time\n\\cite{Vis,BekMuk,Mak,Hod1}.\n\nOne naturally wonders whether the strong inequality (\\ref{Eq1}),\nwhich characterizes the Hawking evaporation process of the\n$(3+1)$-dimensional Schwarzschild black hole, is a generic feature\nof all $(D+1)$-dimensional Schwarzschild black-hole spacetimes? In\norder to answer this interesting question, we shall explore in this\npaper the semi-classical Hawking evaporation of higher-dimensional\nSchwarzschild black holes. Below we shall show that the\ndimensionless ratio\n$\\eta(D)\\equiv{{\\tau_{\\text{gap}}}\/{\\tau_{\\text{emission}}}}$, which\ncharacterizes the Hawking evaporation process of $(D+1)$-dimensional\nSchwarzschild black holes, is a {\\it decreasing} function of the\nspacetime dimension. In particular, we shall show that\nhigher-dimensional Schwarzschild black holes with $D\\gtrsim 10$ are\ncharacterized by the relation $\\eta(D)<1$. Thus, our analysis (to be\npresented below) reveals the fact that the Hawking cascades of\ngravitons from these higher-dimensional evaporating black holes have\na {\\it continuous} character.\n\n\\section{The Hawking evaporation process of $(D+1)$-dimensional Schwarzschild black\nholes}\n\nWe study the semi-classical Hawking emission of gravitational quanta\nby higher-dimensional Schwarzschild black holes. The characteristic\nBekenstein-Hawking temperature of an evaporating $(D+1)$-dimensional\nSchwarzschild black hole is given by \\cite{Noteunit}\n\\begin{equation}\\label{Eq2}\nT^{D}_{\\text{BH}}={{(D-2)\\hbar}\\over{4\\pi r_{\\text{H}}}}\\  ,\n\\end{equation}\nwhere $r_{\\text{H}}$ is the black-hole horizon radius\n\\cite{Noterh,SchTang,Notekk,Ober}.\n\nThe radiation flux (that is, the number of quanta emitted per unit\nof time) and the radiation power (that is, the energy emitted per\nunit of time) for one bosonic degree of freedom out of a\n$(D+1)$-dimensional Schwarzschild black hole are given respectively\nby the integral relations \\cite{Haw,Page,ZuKa}\n\\begin{equation}\\label{Eq3}\n{\\cal\nF}^{D}_{\\text{BH}}={{1}\\over{(2\\pi)^D}}\\sum_{j}{\\int_0^{\\infty}}dV_D(\\omega)\n{{\\Gamma}\\over{e^{\\hbar\\omega\/T_{\\text{BH}}}-1}}\\\n\\end{equation}\nand\n\\begin{equation}\\label{Eq4}\n{\\cal\nP}^{D}_{\\text{BH}}={{1}\\over{(2\\pi)^D}}\\sum_{j}{\\int_0^{\\infty}}dV_D(\\omega)\n{{\\Gamma\\hbar\\omega}\\over{e^{\\hbar\\omega\/T_{\\text{BH}}}-1}}\\ ,\n\\end{equation}\nwhere $j$ denotes the angular harmonic indices of the emitted field\nmode. The frequency-dependent coefficients\n$\\Gamma=\\Gamma(\\omega;j,D)$ are the dimensionless greybody factors\n\\cite{Page} which quantify the partial scattering of the emitted\nfield modes by the effective curvature potential that surrounds the\n$(D+1)$-dimensional Schwarzschild black hole. These factors are\ndetermined by the higher-dimensional version of the Regge-Wheeler\nequation \\cite{CKK}\n\\begin{equation}\\label{EqN1}\n\\Big({{d^2}\\over{dr^2_*}}+\\omega^2-V\\Big)\\phi=0\\  ,\n\\end{equation}\nwhere $r_*$ is a `tortoise' radial coordinate which is determined by\nthe relation $dr_*\/dr=[1-(r_{\\text{H}}\/r)^{D-3}]^{-1}$. The\neffective curvature potential in the Schr\\\"odinger-like wave\nequation (\\ref{EqN1}) is given by \\cite{CKK}\n\\begin{equation}\\label{EqN2}\nV(r;D)=\\Big[1-\\Big({{r_{\\text{H}}}\\over{r}}\\Big)^{D-3}\\Big]\\Big[{{l(l+D-2)+(D-1)(D-3)\/4}\\over{r^2}}\n+{{(1-p^2)(D-1)^2r^{D-2}_{\\text{H}}}\\over{4r^D}}\\Big]\\ ,\n\\end{equation}\nwhere $l$ is the angular harmonic index of the perturbation mode and\n$p=0,2$ for gravitational tensor perturbations and gravitational\nvector perturbations, respectively \\cite{CKK,Notecom}.\n\nSubstituting into (\\ref{Eq3}) and (\\ref{Eq4}) the expression\n\\begin{equation}\\label{Eq5}\ndV_D(\\omega)=[2\\pi^{D\/2}\/\\Gamma(D\/2)]\\omega^{D-1}d\\omega\\\n\\end{equation}\nfor the D-dimensional volume in frequency-space of the shell\n$(\\omega,\\omega+d\\omega)$, one finds\n\\begin{equation}\\label{Eq6}\n{\\cal\nF}^D_{\\text{BH}}={{1}\\over{2^{D-1}\\pi^{D\/2}\\Gamma(D\/2)}}\\sum_{j}{\\int_0^{\\infty}}\\Gamma\n{{\\omega^{D-1}}\\over{{e^{\\hbar\\omega\/T_{\\text{BH}}}-1}}} d\\omega\\\n\\end{equation}\nand\n\\begin{equation}\\label{Eq7}\n{\\cal\nP}^D_{\\text{BH}}={{\\hbar}\\over{2^{D-1}\\pi^{D\/2}\\Gamma(D\/2)}}\\sum_{j}{\\int_0^{\\infty}}\\Gamma\n{{\\omega^D}\\over{{e^{\\hbar\\omega\/T_{\\text{BH}}}-1}}} d\\omega\\\n\\end{equation}\nfor the semi-classical Hawking radiation flux and the semi-classical\nHawking radiation power which characterize the evaporating\n$(D+1)$-dimensional Schwarzschild black holes.\n\n\\section{The characteristic timescales of the Hawking evaporation\nprocess}\n\nAn important timescale which characterizes the Hawking evaporation\nprocess of the $(D+1)$-dimensional Schwarzschild black holes is\ngiven by the time gap between the emissions of successive Hawking\nquanta. There are several distinct (though closely related) ways to\nquantify this fundamental time scale. Here we shall use two natural\ndefinitions for this characteristic time gap:\n\\newline\n(1) One can use the reciprocal of the black-hole radiation flux\n(\\ref{Eq6}) in order to quantify the characteristic time gap between\nthe emissions of successive Hawking quanta. That is,\n\\begin{equation}\\label{Eq8}\n\\tau^{(1)}_{\\text{gap}}={{1}\\over{{\\cal F}^D_{\\text{BH}}}}\\  .\n\\end{equation}\n\\newline\n(2) One can also use the reciprocal of the black-hole radiation\npower (\\ref{Eq7}) and the characteristic peak frequency\n$\\omega_{\\text{peak}}$ of the semi-classical Hawking radiation\nspectrum in order to quantify the characteristic time gap between\nthe emissions of successive Hawking quanta. That is,\n\\begin{equation}\\label{Eq9}\n\\tau^{(2)}_{\\text{gap}}={{\\omega_{\\text{peak}}}\\over{{\\cal\nP}^D_{\\text{BH}}}}\\ .\n\\end{equation}\nBelow we shall show that the characteristic time gaps obtained from\nthese two definitions [Eqs. (\\ref{Eq8}) and (\\ref{Eq9})] are of the\nsame order of magnitude.\n\nA distinct timescale which characterizes the Hawking evaporation\nprocess of the $(D+1)$-dimensional Schwarzschild black holes is\ngiven by the time $\\tau_{\\text{emission}}$ required for an\nindividual Hawking quantum to be emitted from the evaporating black\nhole. This fundamental timescale can be bounded from below by the\ntime-period it takes to the characteristic wave field emitted from\nthe black hole to complete a full oscillation cycle\n\\cite{Vis,Hodkerr}. That is,\n\\begin{equation}\\label{Eq10}\n\\tau_{\\text{emission}}\\geq\\tau_{\\text{oscillation}}={{2\\pi}\\over{\\omega_{\\text{peak}}}}\\\n.\n\\end{equation}\n\nUsing the natural timescales (\\ref{Eq8}), (\\ref{Eq9}), and\n(\\ref{Eq10}), one can define the fundamental dimensionless ratio\n\\begin{equation}\\label{Eq11}\n\\eta^{(i)}\\equiv{{\\tau^{(i)}_{\\text{gap}}}\\over{\\tau_{\\text{emission}}}}\\\n\\end{equation}\nwhich provides important information about the Hawking evaporation\nprocess of the semi-classical black holes \\cite{Notepk}. In\nparticular, physical situations which are characterized by the\nrelation $\\eta\\gg1$ describe Hawking evaporation processes which are\nextremely {\\it sparse} (that is, the individual Hawking quanta\nemitted from the black hole are well separated in time), whereas\nphysical situations which are characterized by the relation\n$\\eta\\ll1$ describe Hawking evaporation processes which are\neffectively {\\it continuous}.\n\nIn the next sections we shall investigate the functional dependence\nof the dimensionless ratios\n$\\eta^{(i)}(D)\\equiv{{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{emission}}}}$\non the spacetime dimension $D+1$ of the evaporating black-hole\nspacetime.\n\n\\section{The $(3+1)$-dimensional case}\n\nFor the emission of gravitational Hawking quanta from a\n$(3+1)$-dimensional Schwarzschild black hole \\cite{Page} one finds\nthe characteristic dimensionless ratios\n\\begin{equation}\\label{Eq12}\n\\eta^{(1)}(D=3)=5175.8\\gg1\\ \\ \\ \\ \\ \\text{and}\\ \\ \\ \\ \\\n\\eta^{(2)}(D=3)=5371.5\\gg1\\ \\ .\n\\end{equation}\nAs emphasized earlier, these remarkably large ratios imply that the\nHawking cascade from the evaporating $(3+1)$-dimensional\nSchwarzschild black hole is extremely {\\it sparse}\n\\cite{Vis,BekMuk,Mak,Hod1}. In other words, the dimensionless large\nratios (\\ref{Eq12}) imply that, on average, an evaporating\n$(3+1)$-dimensional Schwarzschild black hole emits gravitational\nquanta which are well separated in time \\cite{Vis,BekMuk,Mak,Hod1}.\n\n\\section{Higher-dimensional black holes: Intermediate $D$-values}\n\nIn the previous section we have seen that the $(3+1)$-dimensional\nSchwarzschild black hole is characterized by large values of the\ndimensionless ratios $\\eta^{(i)}(D=3)$ [see Eq. (\\ref{Eq12})]. In\nthe present section we shall show that the dimensionless ratios\n$\\eta^{(i)}(D)$, which characterize the Hawking evaporation process\nof $(D+1)$-dimensional Schwarzschild black holes, are {\\it\ndecreasing} functions of the spacetime dimension.\n\nThe semi-classical Hawking emission of gravitons from\nhigher-dimensional Schwarzschild black holes was investigated\nnumerically in \\cite{CKK}. In Table \\ref{Table1} we display the\nnumerically computed dimensionless ratios\n${{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}$ which\ncharacterize the Hawking emission of gravitons from\nhigher-dimensional Schwarzschild black holes with intermediate\n$D$-values \\cite{Notebo}. One finds from Table \\ref{Table1} that the\ncharacteristic dimensionless ratios\n${{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}$ of the\n$(D+1)$-dimensional Schwarzschild black holes are {\\it decreasing}\nfunctions of the spacetime dimension $D+1$.\n\nIn particular, we find that Schwarzschild black holes in the regime\n$D=O(10)$ are characterized by the dimensionless ratios\n${{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}=O(1)$.\nThese higher-dimensional black holes therefore mark the boundary\nbetween sparse (that is, with $\\eta^{(i)}\\gg1$) Hawking cascades of\ngravitons and continuous (that is, with $\\eta^{(i)}\\ll1$) Hawking\ncascades of gravitons.\n\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline $D+1$ & \\ \\ 5 \\ \\ & \\ \\ 8 \\ \\ & \\ \\ 11 \\ \\ \\ \\\\\n\\hline \\ \\ ${{\\tau^{(1)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}$\n\\ \\ &\\ \\ \\ 120.9\\ \\ \\ \\ &\\ \\ \\ 3.42\\ \\ \\ \\ &\\ \\ \\ 0.57\\ \\ \\ \\ \\\\\n\\hline \\ \\ ${{\\tau^{(2)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}$\n\\ \\ &\\ \\ \\ 105.1\\ \\ \\ \\ &\\ \\ \\ 4.22\\ \\ \\ \\ &\\ \\ \\ 0.56\\ \\ \\ \\ \\\\\n\\hline\n\\end{tabular}\n\\caption{The dimensionless ratios\n${{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}$ which\ncharacterize the semi-classical Hawking emission of gravitons from\n$(D+1)$-dimensional Schwarzschild black holes \\cite{CKK}. Here\n$\\tau^{(i)}_{\\text{gap}}$ is the characteristic time gap between the\nemissions of successive Hawking quanta from the higher-dimensional\nblack hole [see Eqs. (\\ref{Eq8}) and (\\ref{Eq9})] and\n$\\tau_{\\text{oscillation}}$ is the characteristic oscillation period\nof the emitted wave field [see Eq. (\\ref{Eq10})]. One finds that the\ncharacteristic dimensionless ratios\n${{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{oscillation}}}}$ are {\\it\ndecreasing} functions of the spacetime dimension $D+1$.\n} \\label{Table1}\n\\end{table}\n\n\\section{Higher-dimensional black holes: The large-$D$ regime}\n\nThe D-dimensional frequency distribution\n$\\omega^{D}\/(e^{\\hbar\\omega\/T_{\\text{BH}}}-1)$ [see Eq. (\\ref{Eq7})]\nis characterized by the peak frequency\n\\begin{equation}\\label{Eq13}\n{{\\hbar\\omega_{\\text{peak}}}\\over{T^{D}_{\\text{BH}}}}=D+W(-De^{-D})\\\n,\n\\end{equation}\nwhere $W(x)$ is the Lambert function. Using the small argument\nrelation $W(x\\to0)\\to0$ and taking cognizance of the\nBekenstein-Hawking temperature (\\ref{Eq2}), one finds the\ncharacteristic strong inequality \\cite{Hoddd}\n\\begin{equation}\\label{Eq14}\n\\omega_{\\text{peak}}\\times\nr_{\\text{H}}={{D^2}\\over{4\\pi}}[1+O(D^{-1})]\\gg1\\\n\\end{equation}\nin the large $D\\gg1$ regime.\n\nThe strong inequality (\\ref{Eq14}) reflects the fact that, for\n$(D+1)$-dimensional Schwarzschild black holes in the large $D\\gg1$\nregime, the characteristic wavelengths in the semi-classical Hawking\nemission spectrum are very {\\it short} as compared to the curvature\nradius of the corresponding black-hole spacetime. As demonstrated in\n\\cite{Hoddd}, this fact implies that the semi-classical Hawking\nevaporation of these higher-dimensional black holes is described\nextremely well by the eikonal (geometric-optics) approximation\n\\cite{Notego}. In particular, one finds \\cite{Hoddd}\n\\begin{equation}\\label{Eq15}\n{\\cal F}^{D}_{\\text{BH}}\\times\nr_{\\text{H}}={{(D+1)(D-2)}\\over{2}}\\Big({{D-2}\\over{4\\pi}}\\Big)^{D}\\Big({{D}\\over{2}}\\Big)^{{D-1}\\over{D-2}}\n\\Big({{D}\\over{D-2}}\\Big)^{{D-1}\\over{2}}{{\\zeta(D)}\\over{\\pi}}\n\\end{equation}\nfor the semi-classical Hawking radiation flux, and \\cite{Hoddd}\n\\begin{equation}\\label{Eq16}\n{\\cal P}^{D}_{\\text{BH}}\\times\nr^2_{\\text{H}}={{(D+1)(D-2)}\\over{2}}\\Big({{D-2}\\over{4\\pi}}\\Big)^{D+1}\\Big({{D}\\over{2}}\\Big)^{{D-1}\\over{D-2}}\n\\Big({{D}\\over{D-2}}\\Big)^{{D-1}\\over{2}}{{D\\zeta(D+1)\\hbar}\\over{\\pi}}\n\\end{equation}\nfor the semi-classical Hawking radiation power in the large-D regime\n[which, as explained above, corresponds to the {\\it short}\nwavelengths (geometric-optics) approximation, see Eq. (\\ref{Eq14})].\nTaking the asymptotic large $D\\gg1$ limit in Eqs. (\\ref{Eq15}) and\n(\\ref{Eq16}), one obtains the compact asymptotic expressions\n\\begin{equation}\\label{Eq15N}\n{\\cal F}^{D}_{\\text{BH}}\\times\nr_{\\text{H}}={{(4\\pi)^2}\\over{e}}\\Big({{D}\\over{4\\pi}}\\Big)^{D+3}\\ \\\n\\ \\text{for}\\ \\ \\ D\\gg1\\\n\\end{equation}\nand\n\\begin{equation}\\label{Eq16N}\n{\\cal P}^{D}_{\\text{BH}}\\times\nr^2_{\\text{H}}=\\hbar{{(4\\pi)^3}\\over{e}}\\Big({{D}\\over{4\\pi}}\\Big)^{D+5}\\\n\\ \\ \\text{for}\\ \\ \\ D\\gg1\\  .\n\\end{equation}\n\nTaking cognizance of Eqs. (\\ref{Eq8}), (\\ref{Eq9}), (\\ref{Eq10}),\n(\\ref{Eq15N}), and (\\ref{Eq16N}), one finds the remarkably {\\it\nsmall} dimensionless ratios\n\\begin{equation}\\label{Eq17}\n\\eta^{(1)}(D\\gg1)=\\eta^{(2)}(D\\gg1)={{e}\\over{8\\pi^2}}\\Big({{4\\pi}\\over{D}}\\Big)^{D+1}\\ll1\n\\end{equation}\nwhich characterize the evaporating $(D+1)$-dimensional Schwarzschild\nblack holes in the large-D limit \\cite{Notemat}. These relations\n(and, in particular, the strong inequalities\n$\\tau^{(i)}_{\\text{gap}}\\ll\\tau_{\\text{emission}}$) imply that the\ncharacteristic Hawking cascades from these higher-dimensional black\nholes have a {\\it continuous} character.\n\n\\section{Summary}\n\nIt has long been known \\cite{Vis,BekMuk,Mak,Hod1} that an\nevaporating $(3+1)$-dimensional Schwarzschild black hole is\ncharacterized by an extremely {\\it sparse} Hawking radiation flux.\nIn particular, the dimensionless large ratios\n$\\eta^{(i)}\\equiv\\tau^{(i)}_{\\text{gap}}\/\\tau_{\\text{emission}}=O(10^3)$\nwhich characterize the $(3+1)$-dimensional Schwarzschild black hole\n[see Eq. (\\ref{Eq12})] imply a simple physical picture in which the\nevaporating black hole typically emits Hawking quanta one at a time\n\\cite{Vis,BekMuk,Mak,Hod1}.\n\nIn the present paper we have analyzed the Hawking emission rates of\nhigher-dimensional Schwarzschild black holes. It was shown that the\ndimensionless ratios\n$\\eta^{(i)}(D)\\equiv{{\\tau^{(i)}_{\\text{gap}}}\/{\\tau_{\\text{emission}}}}$,\nwhich characterize the semi-classical Hawking emission of gravitons\nfrom the $(D+1)$-dimensional Schwarzschild black holes, are {\\it\ndecreasing} functions of the spacetime dimension. In particular, we\nhave shown that higher-dimensional Schwarzschild black holes with\n$D\\gtrsim 10$ are characterized by the relation $\\eta^{(i)}(D)<1$.\nThis fact implies that the corresponding Hawking cascades from these\nhigher-dimensional black holes have a continuous character with the\nproperty $\\tau_{\\text{gap}}<\\tau_{\\text{emission}}$.\n\nMoreover, we have shown that the semi-classical Hawking emission\nspectra of higher-dimensional Schwarzschild black holes in the\nlarge-D regime are characterized by the strong inequality [see Eq.\n(\\ref{Eq17})]\n\\begin{equation}\\label{Eq18}\n\\tau_{\\text{gap}}(D\\gg1)\\ \\ll\\ \\tau_{\\text{emission}}(D\\gg1)\\  .\n\\end{equation}\nOur results therefore imply that, contrary to the\n$(3+1)$-dimensional case, the characteristic Hawking cascades from\nthese higher-dimensional black holes \\cite{Notetn} have a {\\it\ncontinuous} character.\n\n\\bigskip\n\\noindent {\\bf ACKNOWLEDGMENTS}\n\\bigskip\n\nThis research is supported by the Carmel Science Foundation. I thank\nYael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for\nstimulating discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \nThe sigma model approach to obtaining the effective action for the fields of the string has primarily been used for massless fields \\cite{L,CDMP,AS1,FT1,CG} although it was shown very early that it could be used for the massive modes (including closed string tachyon \\cite{DS}). In \\cite{BSPT} this technique was first applied to the open string and it was shown to all orders in perturbation theory that the equation of motion for the tachyon is proportional to the beta function, with the proportionality factor being the Zamolodchikov metric, something that was first argued on general grounds in \\cite{Poly}. Subsequently equations for higher spin (spin one and higher) fields were considered and contact was attempted with string field theory \\cite{HLP,BM,KPP,BSOS,ACNY,T,BSLV}. Interest in the tachyon has revived since the work on tachyon condensation in the open string \\cite{AS2,AS3}. Different approaches have been used to understand this and also to prove the Sen conjectures. Cubic String Field Theory \\cite{W} in particular has contributed significantly to this endeavour \\cite{FK}.\n\n    One parallel approach is the background independent formalism\nproposed by Witten\\cite{WiI,WiII,LW}. It seems well suited to the problem of tachyon potential (i.e zero momentum tachyons) as worked out in the original papers as well as in subsequent ones \\cite{ShI,ShII,KMM}. The advantage of this method is that it uses the sigma model formalism and is thus manifestly background independent unlike string field theory. \nAlso it is less tedious computationally. The space time geometry is manifest. The method seems to have some limitations \\cite{LW,ShII} in dealing with on shell tachyons. These  have to do with the problem of world sheet divergences which seem to clash with BRST symmetry. The divergences do have a physical meaning - they correspond to contributions to the equations\nof motion. \n\nThe loop variable approach is also based on the sigma model and is therefore manifestly background independent. Gauge invariance is present as a spacetime symmetry principle and does not depend on world sheet properties. World sheet divergences therefore can be regulated and RG techniques can freely be used. A gauge invariant action has been proposed. If one takes the limit $R>>a$ (where $a$ is the UV cutoff and $R$ the IR cutoff) the proposal reduces, for the tachyon, and also for gauge fixed fields, to the same expression as that of Witten's background independent scheme. It is interesting that the same expression is obtained from a completely different (at least superficially) approach. However the way gauge invariance is implemented is different. Furthermore there seems to be no problem extending this to on-shell fields as outlined in \\cite{BSA} and worked out in more detail here.\n \nThere is an arbitrariness in the RG procedure, which corresponds to off-shell field redefinitions. However this needs to be fixed in order to calculate for instance the height of the tachyon potential. Our approach for this problem is essentially the same as \\cite{WiI,WiII,KMM,ShI} and establishes that the height of the potential is the same as the D-brane tension but does not evaluate the tension in terms of $g_s$ and $\\al'$. This presumably needs some further calculation that fixes the remaining arbitrariness in the scheme. In Witten's approach the BRST  symmetry seems to fix this \\cite{GS}.\n\nThis paper is organized as follows: In section 2 we give a brief review of the proposal for the action.\nIn section 3 we calculate the quadratic (i.e) kinetic term and also the lowest order $\\beta$ function. This is done for generic momenta. For on shell fields more explicit results are obtained. These calculations are done first with bare fields (with a finite cutoff) and then after renormalization in terms of renormalized fields. The two give mutually consistent equations. In section 4 we give the cubic term in the action.\nHere also the calculation is done in the bare theory as well as in the renormalized theory. The equation of motion obtained from the action is shown to coincide with the $\\beta$ function. We also outline how the quartic and higher order terms work out as a fairly obvious extension of the cubic result. In section 5 for completeness we rederive the exact tree level potential of the constant tachyon (zero momentum) and fix also the leading derivative correction by considering the quadratic profile. Again it is done in the bare theory as well as the renormalized theory. This section reproduces earlier results based on Witten's work, but here, for uniformity with the rest of the paper, we work on the UHP with an IR cutoff, rather than on the unit disc.  Section 6 contains a summary and some conclusions.\n\n\n\\section{Gauge Invariant Action and Disk Partition Function}\n\nOur proposal for the gauge invariant action \\cite{BSA} starts with the following gauge fixed action:\n\n\\begin{equation}     \\label{action}\nS = K{\\p \\over \\p ln ~ a} \\{ {Z \\over \\int dz \\int dw \\langle O(z) O(w) \\rangle}\\}\n\\ee\n\nHere Z is the disk partition function with boundary interactions turned on.\\footnote{It is assumed to be normalized to $1$ when the boundary interactions are set to zero. Thus  $Z\\equiv {Z[\\phi]\\over Z[0]}$} These correspond to non trivial backgrounds for the open string fields. $O$ is a dimension-one (i.e. marginal) operator. The argument for this is as follows. \nFor three point and higher functions, the fully integrated partition function with on-shell backgrounds turned on, has divergences due to the integral over the non-compact SL(2,R) (Mobius) group that is a symmetry of the on shell amplitudes.These are only present for three and higher point functions. Thus for three and higher point functions one expects a divergence of the form $\\int dx \\int _{-R}^{x-a}dy {1\\over y^2} ln~{y\\over a} $ multiplying the S-matrix - with on-shell poles removed due to  regularization.\\footnote{This idea is an old one and has been used in many places. See for eg\\cite{BSPT,FT1,T}.}  The denominator is  $\\int dx \\int _{-R}^{x-a}dy {1\\over y^2} $. The numerator and denominater differ by terms proportional to $ln {R\\over a}$.  Thus when we divide by the denominator and take the derivative w.r.t $ln~a$, this gives something proportional to the action that reproduces the tree level S-matrix. \n\nThe denominator is just the integrated two point function of a dimension one operator (e.g.an on-shell tachyon). It is being used as an overall normalization factor.\nThe kinetic term comes from a two point function. If the particles are exactly on shell this is exactly of the form of the denominator. So the ratio is one and differentiation by $d\\over d~ln~a$ gives zero. This is the rationale for dividing by a term of this form.  The kinetic term should vanish on-shell, and this construction guarantees that. When the particle is off shell the two point function gives a different result - with a $ln ~y$ correction proportional to $p^2-m^2$ and gives a contribution of the same form as the three and higher point functions. Thus including this contribution we obtain the full action.  The constant $K$ is an overall normalization that needs to be specified. \n\nOne of the main motivations for constructing an action using the operation $d\\over d~ln~a$ is that this can be easily made gauge invariant (under space-time gauge transformations) using the loop variable formalism \\cite{BSLV,BSREV}. In this formalism $ln ~a$ is replaced first by $\\sigma$ (Liouville mode) and subsequently by a dressed up version of $\\sigma$ (called $\\Sigma$ in \\cite{BSLV,BSREV}) which is a function of $\\sigma$ and many extra variables introduced to paramterize gauge transformation. In this formalism $d\\over d\\Sigma$\nacting on the partition function (expressed in terms of loop variables) is automatically gauge invariant.\\footnote{In this sense it is reminiscent of the BRST operator}. \nThus this construction gives an action that reproduces the S-matrix, and is furthermore gauge invariant {\\em off shell}, it is a good candidate for the gauge invariant action. We do not describe the loop variable construction here, since in the case of the tachyon (interacting with itself) there is no issue of gauge invariance.\n\n\n\nThe idea that the partition function is closely related to the effective action has been suspected for a long time\\cite{FT1,T}. What we have given is one precise statement of the connection. As shown below, when $R>>a$ it reduces to a relation proposed by Witten in his work on background independent action for the open string\\cite{WiI,WiII,LW,ShI}. It is interesting that this should be the case, since the rationale for that construction seems to be, at least superficially, quite different. When two different approaches give the same result it increases our confidence in the result.\n\nExplicit calculation with on shell fields and also with zero momentum fields was outlined in \\cite{BSA}. The aim of this paper is to work out the details. Many aspects of the exact renormalization group(RG) \\cite{RG1,RG2,HS1,HS2} are probably relevant for a proper understanding of these issues. For instance one issue is that of a finite cutoff. It is true that starting with a bare theory on a lattice , one reaches a fixed point after only an infinte number of iterations. Therefore one can expect to have a finite cutoff for off shell fields. However since we are always an infinite number of rescalings away from the fixed point, one can further rescale the cutoff by an arbitrary large amount and thus approach the continuum arbitrarily closely. \nAnother way of saying this is that on the exact unique RG trajectory the equations don't depend on the cutoff scale\n\\cite{RG1,RG2} (modulo terms that vanish (exponentially) as ${R\\over a}\\rightarrow \\infty$ where $R$ is some IR cutoff length scale\\footnote{This was demonstrated by explicit evaluation of the exact RG for this boundary field theory in \\cite{BSERG}}.) so one can make it arbitrarily small. But actually this is true only for the exact RG. When we try to solve the equations  iteratively to obtain an equation for the marginal couplings, one can expect $ln{L\\over a}$ terms. This has to be dealt with by a renormalization.\n This is done in this paper.\n\nFinally it is worth briefly mentioning the connection between the notion of world sheet Renormalization Group as well as the notion of renormalization, with the space time action and space time fields. The exact RG equations can be thought of as the exact (cubic) string field equations. The exact RG is quadratic as is the string field equation. When we solve for say, the massive fields in terms of massless fields we get a higher order equation for massless fields. This is like writing down the \nbeta functions for marginal couplings by eliminating the irrelevant ones from the exact RG.\\footnote{See \\cite{Wil} for a clear description of this.} This is done iteratively and each iteration increases the degree of nonlinearity of the equation. This can be roughly compared with the level truncation in string field theory. As we iterate the RG and move towards the IR, it is equivalent to increases the level to arbitrary high values. When there are numerical coefficients in the equation that are of order $N$, the number of iterations, this shows up as factors of $ln {L\\over a}$ in the relation between bare and renormalized couplings, where $a$ is the cutoff and $L$ is the renormalization scale. If we want to take $a$ to zero we need to do an infinite renormalization. \n\n\\section{Action for Tachyon Felds with Generic Momenta: Quadratic Terms}\n\nWe shall discuss this first in terms of bare fields and then in terms of renormalized fields.\n\n\n\\subsection{Bare Fields}\n\n\\subsubsection{Partition Function,Z}\n\\begin{equation}\nZ = \\langle e^{-\\int _{\\p \\Gamma} {dz \\over a} \\phi _B (X(z))}\\rangle\n\\ee  \nWe can write $\\phi _B(X(z)) = \\int dk~\\phi _B(k)\\e$. The subscript \"B\" denotes that it is a bare coupling \nconstant of the two dimensional theory. $a$ is the world sheet cutoff.\n\n\nThe expectation value is calculated using the Polyakov measure\n\\[\n\\int {\\cal D} X e ^{- {1\\over \\al '}\\int _{\\Gamma} d^2z \\p X \\pb X }\n\\]\n\nWe are working with a Euclidean metric on the world sheet. $\\Gamma$ is\nthe upper half plane (UHP) because we are interested in the open\nstring and $\\p \\Gamma$ is the real axis. With this\nnormalization $\\langle X(z) X(w) \\rangle = - {\\al '\\over 2\\pi} (ln ~ |{z-w\\over 2R}|\n+ ln ~ |{z-\\bar w\\over 2R}|)$ for the upper half plane with Neumann boundary\nconditions on the real axis. The range of integration along the real axis is taken to be from $-R$ to $+R$. Thus $2R$ acts as an IR cutoff in the theory. Further, for a tachyon vertex operator\non the real axis at $z=x$, $e^{ik.X(x)} = :e^{ik.X(x)}: e^{{\\al '\n    k^2 \\over 2\\pi }ln ~ (a\/2R)}$. Thus if we set $\\al '=\\pi$, $k^2=2$ \nensures that the operator is marginal and this is\nthe mass shell condition for the open string tachyon. \n\nLet us evaluate $Z$ in powers of $\\phi _B$.\n\\begin{equation}   \nZ~=~\\langle 1 \\rangle ~+~\\langle-\\int _{-R}^{+R} {dz \\over a} \\phi _B\n(X(z))    \\rangle  ~+~\\langle \\int _{-R+a}^{+R}  {dz_1 \\over a}  \n\\phi _B (X(z_1))\\int _{-R}^{z_1-a} {dz_2 \\over a} \\phi _B (X(z_2))  \\rangle +...\n\\ee\n\nNote that the lower end in one of the integrations is $-R+a$. This ensures that the two vertex operators are always separated by a minimum distnce of $a$. This is sharp cutoff and is not good for an exact RG treatment because it does not provide a clean separation between modes in momentum space. However if we are near on-shell and $R\/a >>1$ this is not important. It has the advantage of being easy to calculate with. A better cutoff would be to use a cutoff propagator, eg. ${1\\over 2\\pi}ln~[(z-w)^2+a^2]$. \nYet another procedure is to use a momentum space cutoff: $G(k) = {e^{-\\eps k^2}\\over k^2}$ as used in \n\\cite{BSERG}. One can also cut off the momentum space integrals and we will use this last method in deriving the tachyon potential.\n \nAssume for the moment that $\\phi _B$ has non-zero space-time momentum so\nthat it is of the form $\\int dk~\\phi _B (k) \\e$ with $\\phi _B (0) =0$.\nThen the linear term vanishes by momentum conservation. The quadratic\nterm is \n\\[ \n\\int dp ~\\int dk~ \\int _{-R+a}^{R} {dz \\over a} ~\\int _{-R} ^{z-a}\n{du \\over a} ({z-u\\over 2R})^{k.p}\\phi _B (k)  \\phi _B (p) \\delta (k+p)({a\\over 2R})^{k^2 +p^2\\over 2}\n \\]\n\\begin{equation}   \\label{3.4}\n=~\\int dk \\int _{-R+a}^{R} {dz \\over a} ~\\int   _{-R} ^{z-a}{du \\over a}\n({z-u\\over 2R})^{-k^2}({a\\over 2R}) ^{k^2} \\phi _B (k)  \\phi _B (-k)\n\\ee\n Doing the $u$ integral:\n\\begin{equation}   \\label{3.5}\n=~-\\int dk \\int _{-R+a}^{R} dz \n{(z-u)^{-k^2+1}\\over -k^2+1}\\Big \\|^{z-a} _{-R}(a) ^{k^2-2} \\phi _B (k)  \\phi _B (-k)\n\\ee\nAnd the $z$-integral gives \n\\begin{equation}   \\label{3.6}\n-\\int dk~{\\{ {2R-a\\over a} - {[({2R\\over a})^{-k^2+2} -1]\\over -k^2+2} \\}\\over -k^2+1} \\phi _B (k)  \\phi _B (-k)\n\\ee\n\n\\subsubsection{\\bf \\boldmath $\\beta$ Beta-function:}\n\n We can now pause and calculate the $\\beta$- function to second order. One method is to use\n\\begin{equation}  \\label{3.7}\n{dZ\\over dln~a}=0 = {\\p Z\\over \\p ln~a} - \\int dk~\\beta _{\\phi _B(k)} {\\p Z \\over \\p \\phi _B(k)}\n\\ee\n\nTo lowest order we have \n\\begin{equation}\n\\langle ~~\\int {dx \\over a} ~\\phi _B (k) e^{{k^2\\over 2} ln~{a\\over 2R}}:\\e :~~\\rangle\n\\ee \n\nThis gives:\n\\begin{equation}\n\\beta _{\\phi _B(k)}=-{d\\phi _B (k)\\over d ln~a} = ({k^2\\over 2}-1)\\phi _B (k)\n\\ee\nWe have ignored for the moment the fact that momentum conservation forces $k=0$. (This lowest order result for $\\beta$ can be obtained by other methods as well.)\nSo the above method\nonly involves $\\beta _{\\phi _B (0)}$ to this order.\n\nKeeping terms to the next order (from (\\ref{3.6})):\n\n\\begin{equation}\nZ= -{2R\\over a} \\phi _B (0) - [{ ({2R\\over a}-1) - {(({2R\\over a})^{-k^2+2} -1)\\over -k^2+2}\\over -k^2+1}]\\phi _B(k)\\phi _B (-k)\n\\ee\n\n\n\nWe write $\\beta _{\\phi _B} = \\beta ^{(1)}_{\\phi _B} + \\beta ^{(2)}_{\\phi _B}= ({k^2\\over 2}-1)\\phi _{B} (k)+\\beta ^{(2)}_{\\phi _B} $ as the sum of the first order (linear in $\\phi$) and the second order (quadratic in $\\phi$) pieces. Then we get:\n\\[\n{dZ\\over dln~a}=0 \\Rightarrow {2R\\over a}\\beta _{\\phi _B(0)}+ {2R\\over a}\\phi _B (0) +\n\\]\n\\[\n{-1\\over -k^2+1}\\{ \n[-{2R\\over a} + ({2R\\over a})^{-k^2+2}]\\phi _B (k) \\phi _B (-k)\n\\]\n\\begin{equation}\n [({2R\\over a}-1) -{({2R\\over a})^{-k^2+2}-1\\over -k^2+2}][2-k^2]\\} (\\phi _B (k) \\phi _B (-k)) =0\n\\ee\n\\begin{equation}   \\label{betabare}\n\\Rightarrow  \\beta _{\\phi _B(0)} =-\\phi _B (0) +\\int dk~\\{(1-     {a\\over 2R}) (\\phi _B (k) \\phi _B (-k))  \\}\n\\ee\n\n{\\boldmath $k^2=2$:}\n\n\nWhen $k^2 = 2$ one has to be careful because of the pole. \n\\begin{equation}\nZ = -{2R\\over a} \\phi _B (0) + [{2R\\over a} -1-ln~{2R\\over a}]\\phi _B (k) \\phi _B (-k)\n\\ee \n\nSetting ${dZ\\over d ln~a}=0$ as before, we get the same expression for the beta function.  \n\n\\subsubsection{\\bf Action:}\n\n\nNow we turn to the action. Consider (\\ref{3.6}):\n\nThis expression is valid for general $k$. If we now specialize to $k^2-2 \\approx 0$ we get\n\\begin{equation}\n\\int dk~\\{ {2R\\over a} -1 - ln~({2R\\over a}) - {1\\over 2} (-k^2+2) ln^2 ({2R\\over a})\\}\\phi _B (k)  \\phi _B (-k)\n\\ee\nSetting $k^2-2=0$ in the above expression gives us the denominator of the formula (\\ref{action}) for the Action.\nAccording to (\\ref{action}) the action,S, is thus given by\n\\begin{equation}  \\label{S}\n S~=~K{\\p \\over \\p~ln~a}{-\\int dk~\\{ {2R\\over a} -1 - ln~({2R\\over a}) - {1\\over 2} (-k^2+2) ln^2 ({2R\\over a})\\}\\phi _B (k)  \\phi _B (-k)\\over\n  {2R\\over a} -1 - ln~({2R\\over a}) }           \n\\ee\nThus we can write, for near on shell tachyons: \n\\begin{equation}\nS (k^2\\approx 2)~=~ K h(R\/a) \\int dk~(2-k^2) \\phi _B (k) \\phi _B (-k)\n\\ee\nwhere \n\\begin{equation}  \\label{h}\nh(R\/a) =  {\\p \\over \\p~ln~a} [ {ln^2~({2R\\over a})\\over {2R\\over a}-1 - ln~({2R\\over a})}]\n\\ee\n\nThus even for near on-shell fields one cannot take $a \\rightarrow 0$, the continuum limit, because $h$ vanishes.\nThis may seem surprising. But there is no real need to take the continuum limit. We know that for off shell fields we need a finite cutoff. So we can treat $R\\over a$ as a parameter of the theory. What is required is that the\nS-matrix should not depend on this parameter. We will see later, that the coefficient of the cubic term is also\nexactly the same function. Thus we can absorb this ill defined factor into the overall factor $K$. The parameter $R\\over a$ will still show up in off-shell vertices. This is thus a parameter of the off-shell theory that does not affect the on-shell theory.  This parameter is analogous to the number $4\\over 3\\sqrt{3}$ in cubic string field theory.\\footnote{This is explained for instance in \\cite{BSFC}}\n\n\n\nThe more general expression for $S$ is:\n\n\\begin{equation}   \\label{S_B}\nS~=~K\n{\\p \\over \\p~ln~a} [{\n-\\int dk~{\\{ {2R\\over a} -1- {[({2R\\over a})^{-k^2+2} -1]\\over -k^2+2} \\}\\over -k^2+1} \\phi _B (k)  \\phi _B (-k)\n\\over {2R\\over a} -1 - ln~({2R\\over a}) }] \n\\ee\n\nIn the limit $R>>a$ we can approximate the denominator, which is $\\int dz~\\int dw~\\langle O(z)O(w)\\rangle$\nof (\\ref{action}), by $2R\\over a$. Then the expression for the action is\n\\begin{equation}   \\label{Sbeta}\nS = K {\\p \\over \\p ln~a}[{a\\over 2r} Z] = K {a\\over 2r}(1+\\int dk~\\beta _{\\phi_B (k)}{\\p \\over \\p \\phi _B (k)})Z\n\\ee  \nwhere we have used (\\ref{3.7}).\n\nThis is a useful form and can be applied to the partition function written in terms of renormalized variables.\n$\\beta _{\\phi _B}$ will be replaced by $\\beta _{\\phi _R}$. In this form it is the same as the expression derived by Witten (\\cite{WiI,WiII}) where the vector field is identified with the beta function (\\cite{ShI}).\n\n{\\bf Zero Momentum Tachyon:}\n\nThis expression is valid in particular for $k^2=0$, when it reduces to\n\\begin{equation}   \\label{S(0)}\nK{\\p \\over \\p ~ln ~ a}{\n[{-({2R\\over a}) +{1\\over 2} +{1\\over 2} ({2R\\over a})^2\\over {2R\\over a} -1 - ln~({2R\\over a})} ] \\phi _B (k) \\phi _B (-k)}\n\\ee\n\n\n\nAt zero momentum the tachyon coupling in the boundary action reduces to \\[\\Delta S_{world~sheet ~boundary} = \\int _R^R~{dz\\over a} \\phi _B(0) = {2R\\over a} \\phi _B (0)\\]. The contribution to the partition function is expected to be \n\\begin{equation}  \\label{naive}\ne^{-{2R\\over a} \\phi _B(0)}\n\\ee\n Consequently the quadratic term is expected to be $ {1\\over 2} ({2R\\over a})^2 \\phi _B(0)^2$. This does not quite match with the numerator of (\\ref{S(0)}). The discrepancy is due to the minimum distance of $a$ between vertex operators not being imposed in (\\ref{naive}). If we do impose that rule one gets\n\\begin{equation} \n\\int _{-R+a}^R {dz \\over a} \\int _R ^{z-a} {dw\\over a}  \\phi _B (0)^2 = \\frac{1}{2} (({2R\\over a}) -1)^2 \\phi _B (0)^2\n\\ee\n which agrees with the numerator.\n\nIn the limit $R\/a \\rightarrow \\infty$ this difference does not matter. Furthermore this would not be an issue in some other schemes such as when we use a cutoff propagator.\n\n{\\bf \\boldmath Tachyon at $k^2\\approx 0$}\n\nOne can turn now to the order $k^2$ term in an expansion about zero momentum. We consider the case where $R>>a$. This simplifies the algebra and moreover this is the case where (\\ref{action}) reduces to (\\ref{Sbeta}). This can then be compared with the result of the quadratic tachyon considered in section 5. The result is\n\\[\nS=K{\\p \\over \\p ln~a}[{{a\\over 2R}({2R\\over a})^{-k^2+2}\\over (1-k^2)(2-k^2)}]\n\\]\n\\begin{equation}\n={({2R\\over a})^{-k^2+1}\\over {2-k^2}} \\approx {R\\over a}(1+(\\frac{1}{2} -ln~{2R\\over a})k^2+...)\n\\ee\n\nWe see that the relative coefficient of the $1$ and $k^2$ terms is critically dependent on $2R\\over a$. We can choose it to match the derivative expansion obtained in section 5 but since this would not satisfy $R>>a$, we would have to go back to the general expression (\\ref{S_B}). The conclusion is that the coefficient of $k^2$ term is not universal and depends on the regularization scheme. Thus the tachyon fields that correspond to different regularization are expected to be related by field redefinitions, but these could depend on $k$ :\n\\[\n\\phi (k) = \\phi '(k) + A \\al' k^2 \\phi '(k) + ...~+O(\\phi '(k)^2)+...\n\\]\n\n{\\bf \\boldmath Normalization constant $K$ and Determination of $R\/a$:}\n\n The function $h$ that multiplies the {\\em on shell} kinetic term is universal in the sense that it multiplies the full action for {\\em on-shell} tachyons, including the cubic term that is calculated in sec 4, and also the quartic and higher order terms, as argued in sec 4.2. Normally the S-matrix is supposed to capture the physics of a theory and to that extant we have a well defined physical theory. \n\nHowever the height of the tachyon potential is also in some sense \"universal\"  since it encodes information about the D-brane tension \\cite{AS3}. This is related to the coefficient of the quadratic term of the {\\em zero momentum} tachyon. This is quite a different function of $R\/a$. Thus we can use this constraint in principle to determine the ration $R\/a$ {\\em within a regularization scheme}.\nAlso $K$ can be fixed once the string coupling is specified \\cite{GS}. However if there are other such universal quantities, that would be a problem because  there are no further free parameters to fix. In such a situation more minute details of the regularization would have to be specified. This seems to be an open question at this stage.\n\n\n\n\\subsection{Renormalized Fields: Intermediate Case}\n\nWorking with bare fields we see that the continuum limit cannot be taken. Although there is no problem with this, it is interesting to consider other possibilities. For instance it is possible to absorb the $a$ dependence by renormalizing the tachyon field. In that case we can take\nthe limit $a \\rightarrow 0$ without too much difficulty. In fact we can do it in two stages. We can define an intermediate field $\\phi _I$ that absorbs the $a$ dependence that arises from normal ordering the vertex operators. The calculation is then very similar to the S-matrix calculation. We can also define a fully renormalized field $\\phi _R$ in terms of which we can write the action without any $a$ dependence. In place of $a$ we have\na renormalization length scale $L$ entering the action. And the off-shell vertices will be functions of $L\\over 2R$.\n\n{\\bf \\boldmath $\\phi _I$ (Intermediate Field):}\n\nDefine $\\phi _I$ by:\n\\[\n\\int {dx\\over a} ~\\phi _B (k) \\e = \\int {dx \\over a}~ \\phi _B (k) ({a\\over 2R})^{k^2\\over 2}:\\e : = \n\\int {dx\\over L} ~Z_1 \\phi _I (k) \\e \n\\]\n\\begin{equation} \n=\\int {dx \\over L}~ \\phi _I (k) Z_1({a\\over 2R})^{k^2\\over 2}:\\e :\n\\ee\nChoose $Z_1=  ({L\\over a})^{k^2\\over 2}$.  Thus\n\\begin{equation}  \\label{phI}\n\\phi _B (k) = \\phi _I (k) ({L\\over a})^{{k^2\\over 2}-1}\n\\ee\nand\n\\begin{equation}\n\\int {dx\\over a} ~\\phi _B (k) \\e=\\int {dx\\over L}~\\phi _I (k) ({L\\over 2R})^{k^2\\over 2} :\\e :\n\\ee\nOff-shell, when $a\\rightarrow 0$ $\\phi _I$ is a good variable to work with because it is finite. In fact if this is the entire $a$ dependence in the theory (i.e. if we ignore the nonlinearities) we get using (\\ref{phI}),\n\\begin{equation}\n{d\\over d~ln~a} \\phi _I =0 = {d\\phi _B \\over d~ln~a} ({L\\over a})^{1-{k^2\\over 2}} + ({k^2\\over 2}-1) \\phi _B ({L\\over a})^{1-{k^2\\over 2}}\n\\ee\n\\begin{equation}  \\label{beta0}\n\\beta _{\\phi _B} = ({k^2\\over 2}-1) \\phi _B\n\\ee\n\nLet us use $\\phi _I$ to calculate the action. The quadratic term is the same as (\\ref{3.4}) with the replacements: $\\phi _B \\rightarrow \\phi _I$ and $a \\rightarrow L$.\n\\begin{equation}\n\\int dk \\int _{-R+a}^{R} {dz \\over L} ~\\int   _{-R} ^{z-a}{du \\over L}\n({z-u\\over 2R})^{-k^2}({L\\over 2R}) ^{k^2} \\phi _I (k)  \\phi _I (-k)\n\\ee\n\\begin{equation}\n=\\int dk~{({L\\over a})^{k^2-2}\\over k^2-1} [{2R\\over a}-1 - [ {({2R\\over a})^{-k^2+2}-1\\over -k^2+2}]]\\phi _I(k)\\phi _I(-k)\n\\ee\n\nFor near on shell fields with $k^2\\rightarrow 2$, this becomes:\n\\begin{equation}\n{1\\over k^2-1}[{2R\\over a}-1 -ln~{2R\\over a} -\\frac{1}{2} (2-k^2) ln^2~{2R\\over a}][1+(k^2-2) ln~{L\\over a}]\n\\ee\nFor $k^2=2$ this becomes as before ${2R\\over a} -1 -ln~{2R\\over a}$. \n\nThen the actions becomes\n\\begin{equation}\nS= K{\\p \\over \\p ln~a}\\int dk~[1+ (k^2-2) ln~{L\\over a} -\\frac{1}{2} {(2-k^2) ln^2~{2R\\over a} \\over {2R\\over a}-1 -ln~{2R\\over a}}]\n\\phi _I(k) \\phi _I (-k)\n\\ee\n\nThe limit $a \\rightarrow 0$ is well defined and gives\n\\begin{equation}\nS= K\\int dk~(2-k^2)\\phi _I(k)\\phi _I(-k)\n\\ee\nFor general $k^2$ we have \n\n\\begin{equation}   \\label{S_I}\nS=K \\int dk~{\\p\\over \\p~ln~a} {{({L\\over a})^{k^2-2}\\over k^2-1} [{2R\\over a}-1 - [ {({2R\\over a})^{-k^2+2}-1\\over -k^2+2}]]\\over {2R\\over a}-1 -ln~{2R\\over a}}\\phi _I(k)\\phi _I(-k)\n\\ee\nIn this expression, of course $a$ has to be kept finite, because we have not done a complete renormalization.\n\nThe connection between the two expressions can be seen as follows:\nLet $Z(a, \\phi _B)$ be the partition function in terms of bare fields. When we rewrite it in terms of\n$\\phi _I$ we get a function $Z_1$ that has a different {\\em explicit} '$a$' dependence.\n\\begin{equation} \nZ(a, \\phi _B) = Z_1 (a, \\phi _I (\\phi _B,a))\n\\ee\nThen\n\\begin{equation}\n{\\p Z\\over \\p~ln~a} = {\\p Z_1\\over \\p~ln~a} + {\\p Z_1 \\over \\p \\phi _I} {\\p \\phi _I \\over \\p ln~a}\n\\ee\nUsing ${d\\phi _I \\over d~ln ~ a}=0={\\p \\phi _I \\over \\p \\phi _B}\\beta _{\\phi _B} + {\\p \\phi _I \\over \\p ln~a}$\nwe get\n\\begin{equation}\n{\\p Z\\over \\p~ln~a}={\\p Z_1\\over \\p~ln~a} -{\\p Z_1 \\over \\p \\phi _I}{\\p \\phi _I \\over \\p \\phi _B}\\beta _{\\phi _B}\n\\ee\n\nWe see that on-shell, when $\\beta_{\\phi_B}=0$, the two actions are the same.\nUsing (\\ref{phI}) and (\\ref{beta0}) we get\n\\begin{equation}   \\label{329}\n{\\p Z\\over \\p~ln~a}={\\p Z_1\\over \\p~ln~a} -{\\p Z_1 \\over \\p \\phi _I}({k^2-2\\over 2})\\phi_I\n\\ee\n\nComparing (\\ref{S_B}) and (\\ref{S_I}), we can check that they are related by (\\ref{329}).\n\n\n\\subsection{Renormalized Fields: Full Renormalization}\n\n{\\bf \\boldmath $\\phi _R$ Renormalized Field:}\n\n\nLet us now turn to the fully renormalized calculation:\n\\begin{equation}\n\\int {dx\\over a} ~\\phi _B (k) \\e=\\int {dx\\over L}~\\phi _I (k) ({L\\over 2R})^{k^2\\over 2} :\\e :\n\\ee\nWe write $\\phi _I (k) = \\phi _R(k) + \\delta \\phi _R (k)$. Thus\n\\begin{equation}\n\\int {dx\\over a} ~\\phi _B (k) \\e=\\int {dx\\over L}~\\phi _R (k) ({L\\over 2R})^{k^2\\over 2} :\\e :+\\int {dx\\over L}~\\delta \\phi _R (k) ({L\\over 2R})^{k^2\\over 2} :\\e :\n\\ee\n\n\\subsubsection{Renormalization :}\n\n$\\delta \\phi _R$ is a counterterm that will be chosen order by order in such a way that correlation functions are finite in the limit $a\\rightarrow 0$. Of course this can only be achieved in some ranges of values of $k$ because in general there will be divergences corresponding to higher dimensional vertex operators. For general $k$ one will need all the massive mode operators as well which will be tantamount to an exact RG treatment of the problem of open strings.\n\nThe normal ordering divergences have been taken care of. So the first new divergence occurs in second order correction. Thus in any correlation function one can insert two powers of $\\int {dx\\over L}~\\phi _R (k) ({L\\over 2R})^{k^2\\over 2}:\\e :$:\n\\[\n\\langle O_1 O_2...\\{\\int dk_1~\\int dk_2~\\frac{1}{2} \\int {dx_1\\over L}~\\phi _R (k_1) ({L\\over 2R})^{k_1^2\\over 2}:e^{ik_1X(x_1)} :\\int {dx_2\\over L}~\\phi _R (k_2) ({L\\over 2R})^{k_2^2\\over 2}:e^{ik_2X(x_2)} :+ \n\\]\n\\begin{equation}\n\\int dk~\\int {dx\\over L}~\\delta \\phi _R (k) :e^{ikX(x)}:({L\\over 2R})^{k^2\\over 2} \\}\n.....O_N \\rangle\n\\ee\n$\\delta \\phi _R$ has to be chosen to make this finite. To quadratic order in $\\phi _R$ and to zeroeth order in all other fields we can do this by removing the divergence associated with the contraction between $X(x_1)$ and $X(x_2)$ and subsequent integration over the positions. This can be summarized by using the OPE:\n\\[\n\\langle O_1 O_2...\\{\\int dk_1~\\int dk_2~\\frac{1}{2} \\int {dx_1\\over L}~ \\int {dx_2\\over L}~\\phi _R (k_1)\\phi _R (k_2) ({L\\over 2R})^{k_1^2 +k_2^2\\over 2}\\Big\\|{x_1-x_2\\over 2R}\\Big\\|^{k_1.k_2}:e^{i(k_1X(x_1)+k_2X(x_2)} :-\n\\]\n\\begin{equation}    \\label{ope}\n\\int dk~\\int {dx\\over L}~\\delta \\phi _R (k) :e^{ikX(x)}:({L\\over 2R})^{k^2\\over 2} \\}\n.....O_N \\rangle\n\\ee\n\nTo proceed further we need to specify some range of momenta. We consider two cases:\n\n\\subsubsection{\\boldmath$ k_1 +k_2 = 0; k_1^2=k_2^2=2$} \n\nThis is actually the calculation done earlier - when\n$k_1+k_2$ is zero this is a contribution to the partition function. (i.e. without any operators $O_{1,2,3}$ in \n(\\ref{ope}). However we will be able to take $a\\rightarrow 0$  because the counterterms\nremove all divergences - in this momentum range. This simplifies the results somewhat. \nThus \\footnote{Note that we are not renormalizing the one-particle irreducible effective action. We are interested in the full partition function, which is to be made finite. Therefore we do not subtract from the correlator the term $1+ln~({x_1-x_2\\over 2R})$ as was done for instance in the original papers on the beta function method, for e.g. \\cite{DS}.}\n\\begin{equation}\n-\\int dk_1~\\frac{1}{2} \\int _{-\\infty}^\\infty {dx_1\\over L}~ \\int _{-\\infty}^\\infty {dx_2\\over L}~\\phi _R (k_1)\\phi _R (-k_1) ({L\\over 2R})^{k_1^2}\\big\\|{x_1-x_2\\over 2R}\\big\\|^{-k_1^2}\n\\ee\n\\begin{equation}\n=-\\int dk_1~\\int {dx_1\\over L}~ ~\\phi _R (k_1)\\phi _R (-k_1) (L)^{k_1^2-1}{(x_1-x_2)^{-k_1^2+1}\n\\big\\|_{-R}^{x_1-a}\\over -k_1^2+1}\n\\ee\n\\begin{equation}\n=-\\int dk_1~\\int _{-R+a}^{+R}{dx_1\\over L}~ ~\\phi _R (k_1)\\phi _R (-k_1) (L)^{k_1^2-1}\n{[a^{-k_1^2+1}- (x_1+R)^{-k_1^2+1}]\\over -k_1^2+1}\n\\ee\n\\begin{equation}\n= -\\int dk_1~{1\\over -k_1^2+1}[({2R-a\\over L})({a\\over L})^{-k_1^2+1} - \n[{(2R)^{-k_1^2 +2}-(a)^{-k_1^2+2}\\over -k_1^2+2}]L^{k_1^2-2}]\\phi _R (k_1)\\phi _R (-k_1)\n\\ee\n\\begin{equation}\n= -\\int dk_1~{1\\over -k_1^2+1}[({2R-a\\over L})({a\\over L})^{-k_1^2+1} - \n[{({2R\\over L})^{-k_1^2 +2}-({a\\over L})^{-k_1^2+2}\\over -k_1^2+2}]]\\phi _R (k_1)\\phi _R (-k_1)\n\\ee\n\nWe add a counterterm $\\int _{-R}^R {dx\\over L} \\delta \\phi _R(0)= {2R\\over L}\\delta \\phi _R(0)$, whose contribution to $Z$ is $\\langle -{2R\\over L} \\delta \\phi _R(0)\\rangle$.\nChoose\n\\[\n\\delta \\phi_R(0) = -\\int dk_1~{1\\over -k_1^2+1}[(1-{a\\over 2R})({a\\over L})^{-k_1^2+1} + \n[{{L\\over 2R}[ ({2R\\over L})^{-k^2+2}-({a\\over L})^{-k_1^2+2}]\\over -k_1^2+2}]] \\phi _R (k_1)\\phi _R (-k_1) +\n\\]\n\\begin{equation}\ng \\phi _R (k_1)\\phi _R (-k_1)\n\\ee\nWe choose $g$ such that $\\delta \\phi _R(0) =0$ when $L=a$. This fixes \n\\begin{equation}    \\label{g}\ng=\\{{1- {L\\over 2R}\\over -k_1^2+1} - ({L\\over 2R})[{[({2R\\over L})^{-k^2+2}-1]{1\\over -k_1^2+2}\\over -k_1^2+1}]\\} \n\\ee\n\nSo that the contribution to the partition function at this order is:\n\\begin{equation}\n-\\int dk_1~ {2R\\over L} g(k_1, {L\\over 2R})\\phi _R (k_1)\\phi _R (-k_1)\n\\ee\n\nIncluding the higher orders of the two point (i.e. two vertex operator)  functions has the effect of exponentiation: Thus $\\phi _R(0) \\rightarrow \\phi _R(0) + g\\phi _R (k_1)\\phi _R (-k_1)$. The cubic and higher order correlations will introduce further modification. Another way to see this is that we can define an effective Lagrangian with $\\phi _{eff}=\\phi _R(0) +g\\int dk_1~\\phi _R (k_1)\\phi _R (-k_1)$ as the coefficient of $1$. Thus \n\\begin{equation}   \\label{Z}\nZ = e^{-{2R\\over L} \\phi _R(0) - \\int dk_1~{2R\\over L} g(k_1, {L\\over 2R})\\phi _R (k_1)\\phi _R (-k_1) }\n\\ee\n\n{\\bf Beta function (\\boldmath $\\beta$):}\n\nLet us calculate the $\\beta $ function to this order:\n\\[\n{\\phi _B (0)\\over a} = {\\phi _R (0)\\over L} -{1\\over L}  \\{\\int dk_1~{1\\over -k_1^2+1}[(1-{a\\over 2R})({a\\over L})^{-k_1^2+1} +\n\\]\n\\begin{equation}\n[{{L\\over 2R}[({L\\over 2R})^{-k^2+2}-  ({a\\over L})^{-k_1^2+2}]\\over -k_1^2+2}]]\\phi _R (k_1)\\phi _R (-k_1)\\}\n +{g\\over L} \\phi_R(k)\\phi _R(-k)\n\\ee\nWe define $\\beta _{\\phi _R} = - {d\\over dln~L} \\phi _R$.  Note that the total power of $L$ in each part of the second order contribution is $k^2-2$. Thus operating with $\\p\\over \\p ln~L$ just gives an overall multiplicative  factor of $k^2-2$. When we act with $d\\over dln~L$ on $\\phi _R(k)\\phi _R (-k)$ we pick up a multiplicative factor\nof $2-k^2$. This is because the beta function to lowest order for $\\phi _R (k)$ is (as we will see below) \nis $\\beta _{\\phi _R(k)} = (k^2-2)\\phi _R (k)$. These two contributions thus exactly cancel and we are left with the contribution from the term $g\\phi _R(k) \\phi _R(-k)$. \n\\begin{equation}\n0=\n{-\\beta _{\\phi _R(0)} -\\phi _R(0)\\over L} +  \\int dk_1~{d\\over dln~L}[({g\\over L}) \\phi _R(k_1)\\phi _R(-k_1)] \n\\ee\n\nThus\n\\begin{equation}  \\label{3.53}\n\\beta _{\\phi _R (0)} = -\\phi _R (0) + \\int dk~[L{d\\over d ln ~L} ({g\\over L}) + g (2-k^2)]\\phi _R (k) \\phi _R(-k)\n\\ee\n\n\n\nNote that the $a$ dependent terms disappear from the $\\beta$ function.\n\nNote also that the finite part of the counterterm makes a non trivial contribution to the $\\beta$ function - just as it did to the partition function (\\ref{Z}). This is to be expected. The finite term modifies the renormalization scheme, which in turn corresponds to field\/coupling constant redefinitions. This is reflected in some redefinitions of the tachyon field also.  \n\nFinally if we set $g=\\{{1- {L\\over 2R}\\over -k_1^2+1} - ({L\\over 2R})[{[({2R\\over L})^{-k^2+2}-1]{1\\over -k_1^2+2}\\over -k_1^2+1}]\\} $ from (\\ref{g})\nwe find that the beta function becomes\n\\begin{equation}     \\label{betaR}\n\\beta _{\\phi _R(0)}=-\\phi _R(0) + \\int dk~ (1-{L\\over 2R} )\\phi _R (k) \\phi _R(-k)\n\\ee\n \\begin{equation}\n=\\beta _{\\phi _B (0)} ~~~~at~~L=a\n\\ee\nand agrees with (\\ref{betabare}) when $L=a$.\n- the beta function calculated in terms of bare fields. This is because the counterterms vanish when $L=a$, i.e. the two definitions of $\\phi$ agree when $L=a$. \n\n{\\boldmath $k^2=2$:}\n\nAgain for $k^2=2$ we have to be more careful. The partition function to this order is \n\\begin{equation}\nZ= -{2R\\over L} \\phi _R (0) + [{2R\\over a} -1 - ln {2R\\over a}] \\phi _R (k) \\phi _R(-k) -{2R\\over L} \\delta \\phi _R (0)\n\\ee\n\nWe let\n\\begin{equation}\n\\delta \\phi _R (0) = ({L\\over a} -{L\\over 2R}- {L\\over 2R} ln~{2R\\over a})\\phi _R (k) \\phi _R (-k) + g \\phi _R(k)\\phi _R(-k)\n\\ee\n\nNow choose $g({L\\over 2R})$ such that $\\delta \\phi _R(0)=0$ when $L=a$. Thus\n\n\\begin{equation}\ng({a\\over 2R})= -(1- {a\\over 2R}-{a\\over 2R} ln~{a\\over 2R}) \n\\ee\nThus \n\\begin{equation}\ng({L\\over 2R}) = -(1- {L\\over 2R}-{L\\over 2R} ln~{L\\over 2R}) \n\\ee\n\nThis gives for the partition function:\n\\begin{equation}\nZ=1 -{2R\\over L} \\phi _R (0) - \\int dk~{2R\\over L} g\\phi _R (k) \\phi _R (-k)\n\\ee\nor if we exponentiate due to the higher orders we get:\n\\begin{equation}\nZ=e^{-{2R\\over L} \\phi _R (0) - \\int dk~{2R\\over L} g\\phi _R (k) \\phi _R (-k)}\n\\ee\n\nWe can now calculate the beta function.\nAt $k^2=2$ since the leading contribution $({k^2\\over 2}-1)\\phi_R(k)$ vanishes, we get\n\\begin{equation}\n{d ({\\phi _B (0)\\over a})\\over dln~L}=0=-{\\beta _{\\phi _R (0)}\\over L} -{\\phi _R(0)\\over L} + {\\p \\over \\p ln~L}({g\\over L} )\\phi _R(k)\\phi _R(-k)\n\\ee\n which gives the same expression (\\ref{betaR}) for the beta function and also agrees with the bare beta function. \n\n\n\nWe now turn to the second case:\n\n\\subsubsection{\\boldmath$ (k_1 +k_2)^2 =  k_1^2=k_2^2=2$} \n\nIn this case the operators are marginal and there are log divergences. The limits of integration,$ \\pm R$ can be taken to be arbitrarily large and we can use translational invariance.\nIn this case $k_1.k_2 = -1$.  We go back to (\\ref{ope})\n\\[\n\\langle O_1 O_2...\\{\\int dk_1~\\int dk_2~\\frac{1}{2} \\int {dx_1\\over L}~ \\int {dx_2\\over L}~\\phi _R (k_1)\\phi _R (k_2)\n\\]\n\\[ ({L\\over 2R})^{k_1^2 +k_2^2\\over 2}\\Big\\|{x_1-x_2\\over 2R}\\Big\\|^{k_1.k_2}:e^{i(k_1X(x_1)+k_2X(x_2)} :- \n\\]\n\\begin{equation}    \n\\int dk~\\int {dx\\over L}~\\delta \\phi _R (k) :e^{ikX(x)}:({L\\over 2R})^{k^2\\over 2} \\}\n.....O_N \\rangle\n\\ee\nTo leading order $e^{i(k_1X(x_1)+k_2X(x_2)}) \\approx e^{i(k_1+k_2)X(x_1)}$. Using translational invariance and doing the integral over $x_2$ gives:\n\\begin{equation}\n\\int dk_1~\\int dk_2~ \\int {dx_1\\over L}~ ({L\\over 2R})^{k_1^2+k_2^2\\over 2} {1\\over L} ({1\\over 2R})^{k_1.k_2}{x^{k_1.k_2+1}\\over k_1.k_2+1}\\Big\\| _a^R \\phi _R (k_1)\\phi _R (k_2)\n\\ee\n\\begin{equation}\n=\\int dk_1~\\int dk_2~ \\int {dx_1\\over L}~ ({L\\over 2R})^{k_1^2+k_2^2\\over 2} {1\\over L} ({1\\over 2R})^{k_1.k_2}R^{k_1.k_2+1}{[1-({a\\over R})^{k_1.k_2 +1}]\\over k_1.k_2+1}\\Big\\| _a^R \\phi _R (k_1)\\phi _R (k_2)\n\\ee\nSince $k_1.k_2 +1\\approx 0$ we can expand  in powers of $(k_1.k_2+1)ln~{a\\over R}$. This gives (we set $k_1+k_2=k$)\\footnote{ In all expressions replace $k_2$ by $k-k_1$.}\n\\begin{equation}\n-\\int {dx_1\\over L}\\int dk~ \\int dk_1~ ({L\\over 2R})^{k^2\\over 2} ({L\\over R})^{-k_1.k_2 -1} ln~{a\\over R} :e^{ikX}: \n\\phi _R (k_1) \\phi _R (k -k_1)\n\\ee\n\nTo cancel the divergence we choose a counterterm:\n\\begin{equation}   \\label{CT}\n\\delta \\phi _R (k) = -\\int dk_1~ [({L\\over R})^{-k_1.k_2-1}ln~{a\\over L} + f(k,k_1,{L\\over R})] \\phi _R (k_1)\\phi _R (k-k_1)\n\\ee\nWe have added a finite term proportional to a function $f$. Thus the coefficient of the operator $e^{ikX}$ becomes modified by the addition of:\n\\begin{equation}\n-\\int {dx_1\\over L}\\int dk~ \\int dk_1~ ({L\\over 2R})^{k^2\\over 2}[ ({L\\over R})^{-k_1.k_2 -1} ln~{L\\over R} +f(k,k_1,{L\\over R})] :e^{ikX}: \n\\phi _R (k_1) \\phi _R (k -k_1)\n\\ee\n\nNote that as before if we want $\\delta \\phi _R(k)=0$ when $L=a$, $f$ is fixed to be zero.\n\n\n{\\bf Beta function\\boldmath ($\\beta$):}\n\nWe proceed now to calculate the $\\beta$ function. Start with:\n\n\\begin{equation}\n{\\phi _B (k) \\over a} ({a\\over 2R})^{k^2\\over 2}= {\\phi _R \\over L} ({L\\over 2R})^{k^2\\over 2} + {\\delta \\phi _R \\over L}({L\\over 2R})^{k^2\\over 2}\n\\ee\n\nApply ${d\\over dln~a}$ to both sides to get:\n\\begin{equation}   \\label{352}\n[-{\\beta _{\\phi _R(k)}\\over L} + ({k^2\\over 2}-1) {\\phi _R(k)\\over L}]({L\\over 2R})^{k^2\\over 2} +[({k^2\\over 2}-1){\\delta \\phi_R(k)\\over L} + {d\\delta \\phi_R(k)\\over dln~L}{1\\over L}]({L\\over 2R})^{k^2\\over 2}\n\\ee\nSince $\\delta \\phi _R(k)$ is $O(\\phi _R ^2)$, we have the lowest order result (which was used in (\\ref{3.53}):\n\\begin{equation}\n\\beta _{\\phi _R(k)} = ({k^2\\over 2}-1) \\phi _R(k)\n\\ee\nTo get the next order contribution we need to use (\\ref{CT}):\n\\[\n{d\\delta \\phi _R (k)\\over dln~L}=(1-{k^2\\over 2})\\delta \\phi_R(k) +\\int dk_1~({L\\over R})^{-k_1.k_2-1}\\phi _R (k_1)\\phi _R(k_2) +\n\\]\n\\begin{equation}   \\label{354}\n-\\int dk_1 \\{{df\\over dln~L}  + f[(1-{k_1^2\\over2}) + (1-{k_2^2\\over2})]\\}\\phi _R(k_1)\\phi _R(k_2)\n\\ee\n\nSubstituting (\\ref{354}) in (\\ref{352}), we see that the divergent terms proportional to $\\delta \\phi _R(k)$\ncancel leaving us with:\n\\[\n[-{\\beta _{\\phi _R(k)}\\over L} + ({k^2\\over 2}-1) {\\phi _R(k)\\over L}]({L\\over 2R})^{k^2\\over 2}  + \\int dk_1~({L\\over R})^{-k_1.k_2-1}({L\\over 2R})^{k^2\\over 2}\\phi _R (k_1)\\phi _R(k_2) +\n\\]\n\\begin{equation}   \\label{355}\n-\\int dk_1 \\{{df\\over dln~L}  + f[(1-{k_1^2\\over2}) + (1-{k_2^2\\over2})]\\}\\phi _R(k_1)\\phi _R(k_2)\n{1\\over L}]({L\\over 2R})^{k^2\\over 2}\n\\ee\nThus:\n\\[\n\\beta _{\\phi _R(k)}=({k^2\\over 2}-1) \\phi _R(k) +\\int dk_1~({L\\over R})^{-k_1.k_2-1}\\phi _R (k_1)\\phi _R(k_2) +\n\\]\n\\begin{equation}   \\label{beta}\n-\\int dk_1 \\{{df\\over dln~L}  + f[(1-{k_1^2\\over2}) + (1-{k_2^2\\over2})]\\}\\phi _R(k_1)\\phi _R(k_2)\n\\ee\nThe second line in (\\ref{beta}) represents the effect of the finite parts of the counterterm on the $\\beta$ function.\n\n\\subsubsection{\\bf Action}\n\nOnce we have $Z$ to second order, and also $\\beta$, we can use (\\ref{Sbeta}) to calculate the action. \n\n\n\\begin{equation}   \nS = K {\\p \\over \\p ln~a}[{a\\over 2r} Z] = K {a\\over 2r}(1+\\int dk~\\beta _{\\phi_B (k)}{\\p \\over \\p \\phi _B (k)})Z\n\\ee  \nwith\n\\begin{equation}   \nZ = e^{-{2R\\over L} \\phi _R(0) - \\int dk_1~{2R\\over L} g(k_1, {L\\over 2R})\\phi _R (k_1)\\phi _R (-k_1) }\n\\ee\nand\n\\[\n\\beta _{\\phi _R(k)}=({k^2\\over 2}-1) \\phi _R(k) +\\int dk_1~({L\\over R})^{-k_1.k_2-1}\\phi _R (k_1)\\phi _R(k_2) \n\\]\nFor concreteness  we take $g=-1$ (for on-shell fields, with $k^2=2$), and $f=0$. \n\\begin{equation}\nS = K (1+ \\int dk~ \\beta _{\\phi _R(k)}{\\p \\over \\p \\phi_R(k)})Z\n\\ee\n\nHere, the symbol $\\int dk$ stands for $\\int _{-\\infty}^{\\infty} {d^{26}k\\over (2\\pi )^{26}}$. \n\nUsing ${\\p Z\\over \\p \\phi _R(k)}={4R\\over L} \\phi_R(-k)$  we get:\n\\begin{equation}\nS= [1~+~{4R\\over L} \\phi_R(-k)({k^2\\over 2}-1) \\phi _R(k) + O(\\phi ^3)]Z\n\\ee\n\nThis gives the kinetic term as expected. There is also a contribution of $O(\\phi _R ^3)$ but since we need $Z$ to cubic order to get the full contribution we postpone this to the next section. The $\\phi _R(0)$ contribution will also be treated separately. \n\n\\section{Action for Tachyons with Generic Momenta:Cubic  Terms}\n\nWe now turn to a calculation of the cubic terms in the tachyon action. As in the last section we calculate it first using the bare Lagrangian. This can be done using the prescription (\\ref{action}). \nHaving calculated the action we can check whether the equation of motion is equal to the beta function to this (quadratic) order. Unfortunately the beta function has not been calculated for the bare theory to this order, only for the renormalized theory. However we have seen that for a \nparticular choice of the counterterm we expect these to be equal at $L=a$. For on shell fields, which is what we treat in this section, this choice is $f=0$. Thus for this choice we know the beta function equation. We will see that it does coincide exactly with the equation of motion obtained from the tachyon action.\n \nIn principle one can work all this out in the renormalized theory. We will do the following. We show that for a particular choice of the cubic term in $Z$ (this uses the freedom of counterterms) the tachyon action gives the same equation as before (but in terms of renormalized fields) and hence the equation of motion agrees with the beta function - with $f=0$. What could be done further is to show by actual renormalization of the cubic term that this choice for $f$ is consistent with the choice of the  cubic term in $Z$. This is not  done in this paper.\n\n\n\\subsection{Bare Fields}\n\nIn order to keep track of the combinatorics in a transparent way we specialize the on-shell tachyon field to the following form\n\\[\n\\phi _B(X) = \\phi_0(p) e^{ipX} +\\phi_0(-p) e^{-ipX} +\\phi_0(q) e^{iqX} +\n\\]\n\\begin{equation}\n\\phi_0(-q) e^{-iqX} +\n\\phi_0(r) e^{irX} +\\phi_0(-r) e^{-irX}\n\\ee\nwith some specific $p,q,r$ satisfying $ p=q+r$ and $p^2\\approx q^2\\approx r^2\\approx 2$. Momentum conservation ensures that \n$\\langle e^{ipX(z_1)} e^{-iqX(z_2)} e^{-irX(z_3)}\\rangle$ and its complex conjugate expression are the only non zero three point correlators.\n\\subsubsection{Quadratic Term}\n\nThis has been calculated earlier. We are reproducing it here for the particular choice of the tachyon field.\nThe quadratic term in the partition function, $Z_2$, is \n\\[\n{1\\over 2!}\\int dz_1~\\int dz_2~ [2\\phi _0(p)\\phi _0 (-p)\\langle e^{ipX(z_1)}e^{-ipX(z_2)}\\rangle +\n\\]\n\\begin{equation}\n2\\phi _0 (q)\\phi _0 (-q)\\langle e^{iqX(z_1)}e^{-iqX(z_2)}\\rangle+2\\phi _0 (r)\\phi _0 (-r)\\langle e^{irX(z_1)}e^{-irX(z_2)}\\rangle]\n\\ee\n\nWe use\n\\[\n\\int dz_1~\\int dz_2~ \\langle e^{ipX(z_1)}e^{-ipX(z_2)}\\rangle = 2 \\int _{-R+a}^{+R} dz_1~\\int _{-R}^{z_2-a} dz_2~ \\langle e^{ipX(z_1)}e^{-ipX(z_2)}\\rangle\n\\]\n\\begin{equation}\n=2[{2R\\over a} -1 -ln{2R\\over a} +({p^2\\over 2}-1) ln^2{2R\\over a}]\n\\ee\nThus\n\\begin{equation}\nZ_2 = [\\phi _0(p)\\phi _0 (-p)2({2R\\over a} -1 -ln{2R\\over a} +({p^2\\over 2}-1) ln^2{2R\\over a})]+\n[p\\leftrightarrow q] + [p\\leftrightarrow r]]\n\\ee\nIn terms of $\\phi (k)$ this would be written as\n\\begin{equation}\n\\frac{1}{2} \\int dk~ \\phi (k) \\phi (-k)2[{2R\\over a} -1 -ln{2R\\over a} +({k^2\\over 2}-1) ln^2{2R\\over a}]\n\\ee\n\nWe are using a somewhat loose notation. Since we are considering a discrete set of momenta the $\\int dk$ should actually be $\\Sigma _{k=p,-p,q,-q,r,-r}$. The combinatoric factor $\\frac{1}{2}$ outside should be noted.\n\n\\subsubsection{Cubic Term}\n\n\\begin{equation}\n{-1\\over 3!}\\int dz_1 ~\\int dz_2~\\int dz_3~ \\Sigma _{k_1,k_2,k_3}\\phi (k_1)\\phi (k_2) \\phi (k_3)\n\\langle e^{ik_1X(z_1)} e^{ik_2X(z_2)}e^{ik_3X(z_3)}\\rangle\n\\ee\n\\[\n=-\\int dz_1 ~\\int dz_2~\\int dz_3~[\\phi_0(p)\\phi_0(-q)\\phi _0(-r)\\langle e^{ipX(z_1)}e^{-iqX(z_2)}e^{-irX(z_3)}\\rangle +\n\\]\n\\begin{equation}\n\\phi_0(-p)\\phi_0(q)\\phi _0(r)\\langle e^{-ipX(z_1)}e^{iqX(z_2)}e^{irX(z_3)}\\rangle]\n\\ee\nWe can restrict $z_3 <z_1$ without loss of generality. This gives a factor of two (actually it permutes the momenta)\\footnote{The final answer for any term is independent of the momenta, so permuting momenta just gives a combinatoric factor} and then there are two inequivalent orderings: $z_1>z_2>z_3$ and $z_1>z_3>z_2$. Other orderings are related by Mobius transformations. So this gives another factor of two. \n\nThe integral that needs to be done is of the form ($k_1=-p,k_2=q,k_3=r$):\n\\begin{equation}\n-({a\\over 2R})^{k_1^2+k_2^2+k_3^2\\over 2}\\int _{-R+a}^{R} {dz_1\\over a}~\\int _{-R}^{z_1-a} {dz_3\\over a}~|{z_3-z_1\\over 2R}|^{k_1.k_3}\\int _{z_3+a}^{z_1-a} {dz_2\\over a}~|{z_2-z_3\\over 2R}|^{k_2.k_3}|{z_1-z_2\\over 2R}|^{k_1.k_2}\n\\ee\nAll the powers of $a,R$ cancel when the particles are on shell.\nMake the change of variables: $z_2-z_3=y$ and the $y'={y\\over z_1-z_3}$.\nWe get\n\\begin{equation}\n-\\int _{-R+a}^{R} dz_1~\\int _{-R}^{z_1-a} dz_3~|z_3-z_1|^{k_1.k_3+k_1.k_2+k_2.k_3+1}\n\\int _{a\\over z_1-z_3}^{1-{a\\over z_1-z_3}}dy'~|1-y'|^{k_1.k_2}y'^{k_2.k_3}\n\\ee\n\\[\n=-\\int _{-R+a}^{R} dz_1~\\int _{-R}^{z_1-a} dz_3~|z_3-z_1|^{k_1.k_3+k_1.k_2+k_2.k_3+1}\n\\]\n\\begin{equation}\n[B(k_1.k_2+1,k_2.k_3+1)-B_{a\\over z_1-z_3}(k_1.k_2+1,k_2.k_3+1)-B_{a\\over z_1-z_3}(k_2.k_3+1,k_1k_2+1)]\n\\ee\n\nMake the change of variables:$z_1-z_3=x$ and expand the incomplete Beta functions to get:\n\\[\n=-\\int _{-R+a}^{R} dz_1~\\int _{2a}^{z_1+R} dx~|x|^{k_1.k_3+k_1.k_2+k_2.k_3+1}\n\\]\n\\begin{equation}\n[{1\\over k_1.k_2+1}+ {1\\over k_2.k_3+1}- {({a\\over x})^{k_1.k_2+1}\\over k_1.k_2+1}\n-{({a\\over x})^{k_2.k_3+1}\\over k_2.k_3+1}]\n\\ee\nExpanding in logarithms:\n\\[\n=2\\int _{-R+a}^{R} dz_1~\\int _{a}^{z_1+R} dx~x^{-2}ln({x\\over a})\n\\]\n\\begin{equation}\n=2[{2R\\over a} -1 -ln{2R\\over a} -\\frac{1}{2} ln^2{2R\\over a}]\n\\ee\n\nNote that the precise choice of cutoff prescription reflected in the choice of the limits of integration affect the final result. This is as expected. This is equivalent to field redefinitions. \n\n\nThus we get\n\\begin{equation}\n4 \\int _{-R}^{R} dz_1 \\int _R^{z_1} dz_2~\\int _{-R}^{z_2}dz_3~[\\langle e^{-ipX(z_1)}e^{iqX(z_2)}e^{irX(z_3)}\\rangle \\phi_0(-p)\\phi_0(q)\\phi _0(r) +cc]\n\\ee\n\n\\begin{equation}\n=(4)2[{2R\\over a} -1 -ln{2R\\over a} -\\frac{1}{2} ln^2{2R\\over a}][\\phi_0(-p)\\phi_0(q)\\phi _0(r)+cc]\n\\ee\n\nNote that \n\\[\n\\phi_0(-p)\\phi_0(q)\\phi _0(r)+cc={1\\over 3!}\\int dk_1~\\int dk_2~ \\int dk_3 ~\\phi (k_1)\\phi (k_2)\\phi (k_3)\\delta(k_1+k_2+k_3)\n\\]\nThus\n\\begin{equation}\nZ_3=8[{2R\\over a} -1 -ln{2R\\over a} -\\frac{1}{2} ln^2{2R\\over a}][\\phi_0(-p)\\phi_0(q)\\phi _0(r)+cc]\n\\ee\n\nApplying the prescription for the Action we get\n\\begin{equation}\nS=K{\\p \\over \\p ln~a}[{Z_2+Z_3\\over {2R\\over a} -1 -ln{2R\\over a}}]\n\\ee\nWe take the limit $R>>a$ to simplify things:\n\\[\n=K\\underbrace{{\\p \\over \\p ln~a}  (2  {a\\over 2R}ln^2{2R\\over a})}_{2h({R\\over a})} [\\phi_0(p)({p^2\\over 2}-1)\\phi_0(-p) +p\\leftrightarrow q + p\\leftrightarrow r]+  \n\\]      \n\\begin{equation}\n  K \\underbrace{{\\p \\over \\p ln~a}(4{a\\over 2R}ln^2{2R\\over a})}_{4h({R\\over a})}[\\phi_0(p)\\phi_0(-q)\\phi_0(-r) +cc]\n\\ee\nThe function $h$ was introduced in Section 3.1.3.\n\nThe equation of motion obtained from $S$ is \n\\begin{equation}\n{\\p S \\over \\p \\phi _0(-p)} =K2h[ ({p^2\\over 2}-1)\\phi_0(p)+2 \\phi_0(q)\\phi_0(r)]=0\n\\ee\nThis is to be compared with (\\ref{beta}) that gives (for the choice $f=0$, which is when the bare and renormalized fields coincide at $L=a$, so we expect the beta functions to have the same form):\n\\[\n\\beta _{\\phi_R(p)}=({p^2\\over2}-1)\\phi_0(p) +\\int dk_1 \\phi_R(k_1)\\phi_R(p-k_1)\n\\]\n\\begin{equation}\n=({p^2\\over 2}-1)\\phi_0(p)+2 \\phi_0(q)\\phi_0(r)\n\\ee\n\n\n\\subsection{Quartic and Higher Order}\n\nThe calculation for higher order terms follows the same pattern. This has been known for a long time and has been used to obtain the tachyon equations to arbitrary order in perturbation theory for near on shell fields in \\cite{BSPT}. It is known that when the particles are on shell then all N-point amplitude correlators have an SL(2,R) Mobius invariance. The result of integrating over $N-3$  positions has the form $1\\over |x_1-x_2||x_2-x_3||x_3-x_1|$ $\\times$ [effective action that generates S-matrix {\\em minus} on shell poles].\nThe regularization of the integrals is what causes the subtraction of the divergent on-shell poles. Integrating over the remaining three is what gives the volume of the Mobius Group. (The cubic term we calculated above also has this form, when the particles are on shell.) Integrating over one of the three say $x_2$ gives the same integral as above:\n\\[\n\\int dx_1 ~\\int dx_3~ {1\\over |x_1-x_3|^2}ln {|x_1-x_3|\\over a} \n\\]\nThus the coefficient of the higher order terms in the action (for on shell fields) is the same as we found for the cubic term. Off shell it will be more of a more complicated form.\n\n\\subsection{Renormalized Fields}\n\nThe goal of this section is to outline how the calculation would be done in terms of renormalized fields. We have essentially all the ingredients necessary, except for the cubic term in the renormalized $Z$. For the purposes of this outline we will leave it as a free parameter. We show that it can be chosen by requirements of consistency. It needs to be checked by actual calculation that this term has this value.\n\nThe beta function has been calculated:\n\\begin{equation}\n\\beta _{\\phi _R (k)} = ({k^2\\over 2}-1) \\phi _R(k) + \\int dk_1~ \\phi _R(k_1) \\phi _R(k-k_1)\n\\ee\n\\begin{equation}\n\\Rightarrow \\beta _{\\phi_0(p)}=({p^2\\over 2}-1)\\phi_0(p) + 2 \\phi _0(q) \\phi _0(r)\n\\ee\n\\begin{equation}\nZ=[1+ 2(\\phi _0(p)\\phi_0(-p) +\\phi _0(q)\\phi_0(-q)+\\phi _0(r)\\phi_0(-r))+ b[\\phi_0(p)\\phi_0(-q)\\phi_0(-r) +cc]]\n\\ee\nThe coefficient $b$ is the one that needs to be fixed by an actual calculation.\n\nwe calculate $Z+\\Sigma _{k=p,q,r,-p,-q,-r} \\beta_{\\phi_0(k)}{\\p Z\\over \\p \\phi_0(k)}$.\n\nWe get\n\\[\nS=K\\{1+{8R\\over L}[\\phi_0(p)({p^2\\over 2}-1)\\phi_0(-p)+\\phi_0(q)({q^2\\over 2}-1)\\phi_0(-q)+\\phi_0(q)({q^2\\over 2}-1)\\phi_0(-q)]  +\n\\]\n\\begin{equation}\n {24R\\over L}[\\phi_0(p)\\phi_0(-q)\\phi_0(-r) +cc] +b[\\phi_0(p)\\phi_0(-q)\\phi_0(-r) +cc] + O(\\phi_0^4)\\}\n\\ee\n\nThe choice $b=-{8R\\over L}$ yields the same action as we obtained earlier using the bare theory. As checked there the equations of motion agree with the beta function.\n\nThis concludes our discussion of the tachyon action with generic  momenta. We have seen that in the on-shell case it yields the expected space-time action. In the next section we turn to the (near) zero momentum case treated in \\cite{WiI,WiII,LW,ShI,ShII,KMM}.\n\n\\section{Quadratic profile Tachyon and Tachyon Potential}\n\n\nIn this section we apply our prescription to the zero momentum tachyon, both in terms of bare fields as well as after renormalization. When $R>>a$ we have seen that this is the same as Witten's prescription so in this limit we recover the same results as \\cite{WiII,ShI,KMM}. We will continue to work in the UHP as in the earlier sections, rather than on the unit disc. The leading terms (which is all we are interested in) are the same. We present the results here mainly for completeness and to emphasize that our prescription for the action agrees with earlier results when these are available.\n\\subsection{Bare Fields}\n\nThe starting point is the evaluation of the bare partition function:\n\\begin{equation}\nZ= \\int {\\cal D}X e^{-{1\\over 2\\alpha '}\\int _\\Gamma d^2x \\p _\\al X^i \\p ^\\al X^i -\\int _{\\p \\Gamma} dx [{uX^2\\over 2\\al'}+\\phi _0]  }\n\\ee\nWe assume $\\Gamma$ is the Upper Half Plane (UHP) and $\\p \\Gamma$ is the boundary i.e. the Real axis. \nThen $\\langle X^i (x)X^j(u)\\rangle = -{\\al'\\over 2\\pi}(ln |{x-u\\over 2R}|- ln|{x-\\bar u\\over 2R}|)\\delta ^{ij}$ is the Green function\\footnote{The solution to $\\p_\\al \\p^\\al G(x,u)=\\delta ^2(x-u),~~G(x,0,u,v)=0$ is ${1\\over 2\\pi}(ln |{x-u\\over 2R}|- ln|{x-\\bar u\\over 2R}|)$.}\nthat satisfies Dirichlet boundary conditions $X(x,0)=0$.\nThe $\\phi_0$ dependence of the partition function is trivially obtained: $\\approx e^{-{2R\\over a}}\\phi_0$.\n In order to evaluate the $u$-dependence of the partition function we first evaluate the auxiliary quantity\n\\begin{equation}\nF[X_B(t)]=\\int {\\cal D}Xe^{-{1\\over 2\\alpha '}\\int _\\Gamma  d^2x \\p _\\al X^i \\p ^\\al X^i }\\delta[X|_{\\p \\Gamma}-X_B(t)]\n\\ee\nLet $X_c$ satisy  $\\pp X_c=0$ and $X_c|_{\\p\\Gamma}=X_B(t)$. Then we write $X=X_c +x$ to get\n\\[ \nF[X_B(t)]=e^{-{1\\over 2\\alpha '}\\int _\\Gamma  d^2x \\p _\\al X_c^i \\p ^\\al X_c^i }\\int {\\cal D}xe^{-{1\\over 2\\alpha '}\\int _\\Gamma  d^2x \\p _\\al x^i \\p ^\\al x^i }\\delta[x|_{\\p \\Gamma}]\n\\]\n\\[\n=e^{-{1\\over 2\\alpha '}\\int _\\Gamma  d^2x \\p _\\al X_c^i \\p ^\\al X_c^i }\nDet^{-D\/2}[\\pp ]\n\\]\n\\begin{equation}\ne^{-{1\\over 2\\alpha '}\\int _{\\p\\Gamma}  dx  X_c(x,0)^i \\p _y  X_c(x,0)^i }\nDet^{-D\/2}[\\pp ]\n\\ee\nFor the problem at hand we can use the Green's function defined above to obtain\n\\begin{equation}\nX_c(x,y)=\\int du \\p_v G(x,y,u,v=0)X_B(u)\n\\ee\nThus\n\\begin{equation}\nF[X_B(t)]= e^{-{1\\over 2\\al '\\pi} \\int dx du {1\\over (x-u)^2}X_B(x)X_B(u)} Det^{-D\/2}[\\pp ]\n\\ee\nIntroduce the Forier transform:\n\\[\n X_B(x) = 2R\\int {dp\\over 2\\pi}~ e^{ipx}\\tilde X_B(p)\n\\]\nThe factor of $2R$ has been added so that $X_B$ has the same scaling dimension (world sheet) as $X$.\n\\begin{equation}\nZ[u]=\\int {\\cal D}X_Be^{-{(2R)^2\\over 2\\al '} \\int {dp\\over 2\\pi}(2|p|R+{2R\\over a}u)\\tilde X_B(pP\\tilde X_B(-p)}\n\\ee\n\\begin{equation}\n=Det^{-D\/2}[2R(|p|+{u\\over a})] = e^{-{D\\over 2}Tr~ln~[2R(|p|+{u\\over a}]}\n\\ee\nNow as long as $u\\neq 0$ there is no infrared divergence. It is useful to separate out the zero mode.\nThus let us write\n\\[ \n\\int _{-\\infty} ^\\infty {dp\\over 2\\pi} ln~ [2R(|p|+ {u\\over a}]=ln {2Ru\\over a} + 2\\int _{2\\pi \\over 2R}^{2\\pi\\over a} ln~[2R(|p|+{u\\over a}]\n\\]\n\nThe zero mode gives a prefactor $({a\\over 2Ru})^{D\/2}$. In the exponent we do the integral over $p$ and keep only terms to linear order in $u$. We take the ratio $Z[u]\\over Z'[0]$, which gets rid of all the $u$-independent terms in Z[u]. The result is\n\\begin{equation}\n{Z[u,\\phi _0]\\over Z'[0]}=({a\\over 2Ru})^{D\/2} e^{-D{2R\\over a}{u\\over 2\\pi} ln~{2R\\over a}-{2R\\over a}\\phi_0}\n\\ee \nWe can also trivially generalize to the case where there are $D$ u's i.e. $\\Sigma _iu_i ({X^i})^2$ is the boundary Lagrangian density. We get:\n\\begin{equation}\n\\bar Z \\equiv {Z[u_i,\\phi _0]\\over Z'[0]}=[\\Pi _{i=1}^D (\\sqrt{a\\over 2Ru_i})] e^{-\\Sigma _{i=1}^D{2R\\over a}{u_i\\over 2\\pi} ln~{2R\\over a}-{2R\\over a}\\phi_0}\n\\ee \nNow we can apply our prescription for the action: \n\\[\nS =K {\\p \\over \\p ln~a}\\{ {a\\over 2R} \\bar Z \\}\n\\]\n\\begin{equation}\n=K[1+\\Sigma _i (\\frac{1}{2} + {2R\\over a}\\phi_0 + {2R\\over a} {u_i\\over 2\\pi} ln~{2R\\over a} + {2R\\over a} {u_i\\over 2\\pi}]\\Pi _{i=1}^D (\\sqrt{a\\over 2Ru_i})] e^{-\\Sigma _{i=1}^D{2R\\over a}{u_i\\over 2\\pi} ln~{2R\\over a}-{2R\\over a}\\phi_0}\n\\ee\nNote that $D$ is 26 here (i.e. for the bosonic string). Also if we happen to set $2R=a$ the expressions simplify. While we do want to keep $a$ finite in general, for some purposes it is better to have the flexibility to tak $a\\rightarrow 0$. In that case we need to renormalize the theory.\n\n\n\\subsection{Renormalized Fields}\n\nWe now renormalize the theory:\n\\[\nS=\\int dx~ [{\\phi _{0B}\\over a} + {u_B X^2\\over 2\\al' a}]\n\\]\n\\begin{equation}\n=\\int dx~[{\\phi _{0R}\\over L} + {u_R X^2\\over 2\\al' L}]+[{\\delta \\phi _{0R}\\over L} + {\\delta u_R X^2\\over 2\\al' L}]\n\\ee\n\nWe evaluate the partition function perturbatively using the usual Green's function on the UHP\n$\\langle X(x)X(u)\\rangle = -{\\al'\\over \\pi}ln~{|x-u|\\over 2R}$. \nThe first term is ${u_R \\over 2\\pi L}ln~{a\\over 2R}$. Choose $\\delta \\phi_{0R}={u_R \\over 2\\pi }ln~{a\\over L}$ to make this finite:\n\n\\[\\delta \\phi_{0R} -{u_R \\over 2\\pi }ln~{a\\over 2R}={u_R \\over 2\\pi }ln~{L\\over 2R}\n\\]\n\nAt this order $\\delta u_R =0$.\n\nThe relation between bare and renormalized parameters reads as:\n\\[\n{\\phi_{0B}\\over a}= {\\phi_{0R}\\over L}+{\\delta \\phi_{0R}\\over L}\n\\]\n\\[\n\\Rightarrow {\\phi_{0B}\\over a}= {\\phi_{0R}\\over L}-{u_R\\over 2\\pi L}ln~{L\\over a}\n\\]\nAlso ${u_B\\over a}={u_R\\over L}$.\nUsing ${d\\over d~ln~L}{u_B\\over a} =0$ we get\n\\[\n{-\\beta _{u_R}\\over L}-{u_R\\over L} =0\n\\]\n\\[\\Rightarrow \\beta _{u_R} =- u_R\\]\nAlso similarly\n\\[\n{-\\beta _{\\phi_0}\\over L}-{\\phi_{0R}\\over L} +{\\beta _{U_R}\\over 2\\pi L}ln~{L\\over a} + {u_R\\over 2\\pi L} ln~{L\\over a} - {u_R\\over 2\\pi L} =0\n\\]\nThis gives:\n\\[\n\\beta _{\\phi_{0R}}=-(\\phi_{0R}+{u_R\\over 2\\pi})\n\\]\nOne can now write the partition function in terms of the renormalized quantities to obtain (for simplicity we are assuming $D=1$):\n\\[\nZ=\\sqrt{L\\over u_R 2R} e^{-2R({\\phi_{0R}\\over L} + {u_R\\over 2\\pi L} ln~{2R\\over L})}\n\\]\nThe formula for the action:\n\\[\nS=K[1+\\beta _{u_R}{\\p\\over \\p u_R}+ \\beta _{\\phi_{0R}}{\\p \\over \\p \\phi_{0R}}]Z\n\\]\n\\[\n=K[{3\\over 2}+ {2R\\over L} (\\phi _{0R}+ {u_R\\over 2\\pi} + {u_R\\over 2\\pi} ln ~{2R\\over L})]\\sqrt{L\\over u_R2R}e^{-{2R\\over L} (\\phi_{0R} + {u_R\\over 2\\pi} ln~{2R\\over L})}\n\\]\n\n\nNote that if $u=0$ to begin with, we trivially get for the tachyon potential (setting $L=2R$)\n\\[\nV(\\phi_0) = -K(1+\\phi_0)e^{-\\phi_0}\n\\]\n\nThe rest of the contribution can be understood as being due to the derivative terms in the action.\nThus we can match it with an effective action as was done in \\cite{KMM}:\n\\[\nS=K[e^{-\\phi}[(1+\\phi ) +{\\al '\\over 2\\pi}\\p_\\mu \\phi \\p^\\mu \\phi]]\n\\] with $\\phi = (\\phi _0 + {uX^2\\over 2\\al '})$.\n\nThe potential term gives\n\\[\n\\int dX e^{-({uX^2\\over 2\\al '} + \\phi _0)}(1+\\phi _0 + {uX^2\\over 2\\al '})= e^{-\\phi_0}({3\\over 2}+\\phi_0) \\sqrt{2\\pi\\al '\\over u})\n\\]\nthe kinetic term gives\n\\[\ne^{-\\phi_0}{u\\over 2\\pi}\\sqrt{2\\pi\\al '\\over u})\n\\]\n\nThe potential has a minimum at $\\phi _0=\\infty$ and the height of the potential is clearly equal\nto the cosmological constant (or the brane tension). This is a partial verification of the Sen conjecture. Of course this does not fix the value of the brane tension in terms of the string coupling constant and the inverse string tension parameter $\\al'$. As discussed in Sec 3.1.3, this is because the coefficient of the on shell cubic term is not fixed in terms of the cosmological constant  depending as it does on the details of the renormalization scheme. In particular there is a parameter $2R\\over a$, even within this scheme.\n\n\\section{Summary and Conclusions}\n\nThe main aim of this paper is to work out the details of the prescription for a gauge invariant bosonic open string action that was presented in \\cite{BSA}. An issue we discussed is the issue of renormalization. At least in certain ranges of momenta, where we can restrict our attention to just the tachyon sector, it is possible to renormalize the theory so that the cutoff parameter does not appear. This means that the continuum limit can be taken. There may be some conceptual attraction in doing this, but from a computational viewpoint nothing is gained. One conceptual advantage is that in the continuum limit one can make contact with another expression for the action derived in a completely different way in the  background independent formalism of Witten.\n\n The result is that both for on shell tachyons as well as for zero momentum tachyons the action can be constructed. The on shell action clearly is consistent with the S-matrix. We also showed that the equation of motion agrees with the $\\beta$ function to this order.  For zero momentum tachyons the exact tree level potential can be calculated and agrees with earlier results using Witten's background independent formalism.\n\nThe normalization of the action depends on the renormalization scheme and introduces some arbitrariness. These have to be fixed by appealing to some universal quantities. One such is the \nS-matrix and another is the height of the tachyon potential. It is not clear whether there are  further constraints on the regularization scheme. This is an open question.\n\nThe issue of gauge invariance will arise when we deal with the vector and higher modes. The main obstacle in doing these calculations is the enormous amount of algebra. It would be useful if there are some clever field redefinitions that can be done to simplify things. \n\nThe loop variable technique has been applied to closed strings at the level of equations of motion. If the open string action can similarly be generalized to closed strings one can tackle the question of closed string tachyon potential. Finally the connection with the exact RG needs to be understood better (both on the world sheet and in space-time). We hope to turn to these questions soon. \n\n{\\bf Acknowledgements:} I thank Partha Mukhopadhyay for some useful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Additional data types}\n\\label{app:a}\n\n\\subsection*{Commutative Data Types}\n\n\\subsubsection*{Grow-only Counter (\\gctr)}\n\n\\begin{figure}[b]\n\\begin{eqnarray*}\n\\Sigma = \\mathds{N} && \\sigma_i^0 = 0 \\\\\n\\prepare_i(\\inc,n ) & = & \\inc \\\\\n\\effect_i(\\inc, n ) & = & n+1 \\\\\n\\eval_i(\\af{value},n ) & = & n\\\\\n\\end{eqnarray*}\n\\caption{Pure Grow-only Counter CRDT (\\gctr).}\n\\label{fig:gctr}\n\\end{figure}\n\n\n\nAs its name indicates, a \\gctr is a counter that can only be incremented. \nAmong its use-cases are implementing vector clocks in distributed systems \nor counting the number of clients visiting a web-site.\nIn \\gctr, the state is usually an integer $n\\in \\mathds{N}$ and the only mutating \noperation is $\\inc$,\nwhich reads the current state and increments it by $1$. Since \n$+$ is commutative over $\\mathds{N}$, \\gctr is a commutative data type.\n\nFigure~\\ref{fig:gctr} depicts a pure CRDT design of \\gctr. \nThe design is self explanatory: the state is an integer $n\\in \\mathds{N}$ \ninitialized to $0$; $\\prepare$ can only be $\\inc$ without any meta-data \n(and thus the design is pure).\nThe $\\effect$ function simply increments $n$ by $1$, whereas $\\eval$ invokes \nthe read operation $\\rd$ which returns $n$.\n\n\\subsection*{Non-Commutative Data Types}\n\n\\subsubsection*{Flags (\\ewfl and \\dwfl)}\nA flag is a datatype that retains a single boolean value indicating \nwhether it is ``enabled'' or ``disabled''. Flags~\\cite{riak-crdt:Cribbs:2012} have several uses in \npractice, e.g., showing if an email has been read.\nSince the two allowed mutating operations, i.e., $\\enable$ and $\\disable$,\ncan be concurrent and result in opposite outputs depending on the delivery\norder, a flag is obviously a non commutative CRDT. To ensure a consistent result on\nall replicas when $\\enable$ and $\\disable$ operations are concurrent, a design\ndecision is usually made to break this tie,\nthus yielding two flag versions:\nEnable-Wins Flag (\\ewfl) where two concurrent $\\enable$ and $\\disable$\noperations leave the flag ``enabled'', to the contrary of a \nDisable-Wins Flag (\\dwfl).\nThe formal concurrent semantics of \\ewfl and \\dwfl as well as their pure CRDT \ndesigns are depicted in Figures~\\ref{fig:ewflag-semantics},~\\ref{fig:ewflag-compact}, \nand~\\ref{fig:dwflag-semantics},~\\ref{fig:dwflag-compact}, respectively. \n\nThe \\ewfl semantics (in Figure~\\ref{fig:ewflag-semantics}) says that \na $\\af{read}$ operation on the\n\\polog must return ``enabled'' iff there is an $\\enable$ operation\nthat is not strictly followed by a $\\disable$ or $\\clear$ (i.e., a reset) \noperation in its causal future; meaning that, \na concurrent $\\disable$ or $\\clear$ will not ``disable'' the flag. \nConsequently, the first clause in the pure design in \nFigure~\\ref{fig:ewflag-compact} says that a newly delivered operation is always\nconsidered redundant, and never added to the \\polog, if it is $\\disable$ \nor $\\clear$. The second clause, however, compares\nthe existing operations in the \\polog to remove any operation $o$ that is strictly\nfollowed by another operation $o'$ in its causal future, i.e., if $t<t'$. \nSince only $\\enable$ operations are retained by the \\polog (as the first \nclause specifies), the second clause will actually remove any $\\enable$ \noperations in the \\polog with the $t<t'$ condition.\nConsequently, the output of the $\\af{read}$ operation,\ninvoked by $\\eval$, returns an existing $\\enable$ operation, or empty otherwise,\nyielding the same output of the $\\af{read}$ in the base-line semantics in \nFigure~\\ref{fig:ewflag-semantics}. Notice that $\\stabilize$ \nhas no effect in this datatype, and thus always returns the \\polog as is; \nwhich means that \n$\\stable$ only replaces the stable timestamps in the \\polog with $\\bot$. \n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n  \\eval_i(\\af{read}, s) & = & \\exists (t, \\enable) \\in  s \\cdot\n  \\forall (t', o')\\in s \\cdot \\\\&& (o'=\\disable \\mathrel{\\vee} \\clear) \\cdot t \\not< t'\n\\end{eqnarray*}\n\\caption{\\ewfl concurrent semantics}\n\\label{fig:ewflag-semantics}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\t(t,o) \\R s &=& o[0] = (\\disable \\mathrel{\\vee} \\clear) \\\\\n  (t',o') \\R_\\_ (t,o) &=& t' < t \\\\\n  \\stabilize_i(t,s) &=& s \\\\\n  \\eval_i(\\af{read}, s) &=& (\\_, \\enable) \\in s \\\\\n\\end{eqnarray*}\n\\caption{Compact \\ewfl Pure CRDT}\n\\label{fig:ewflag-compact}\n\\end{figure}\n\n\\begin{figure}[H]\n\\begin{eqnarray*}\n  \\eval_i(\\af{read}, s) & = & \\exists (t, \\enable) \\in  s \\cdot\n  \\forall (t',\\disable)\\in s \\cdot t' < t \\\\ && \\mathrel{\\wedge}\n  \\forall (t'',\\clear)\\in s \\cdot t \\not< t''\n\\end{eqnarray*}\n\\caption{\\dwfl concurrent semantics}\n\\label{fig:dwflag-semantics}\n\\end{figure}\n\\begin{figure}[H]\n\\begin{eqnarray*}\n\t(t,o) \\R s &=& o[0] = \\clear \\\\\n  (t',o') \\R_\\_ (t,o) &=& t' < t \\\\\n  \\stabilize_i(t,s) &=& s \\\\\n  \\eval_i(\\af{read}, s) &=& (\\_, \\enable) \\in s  \\mathrel{\\wedge} (\\_, \\disable) \\not\\in s \\\\\n\\end{eqnarray*}\n\\caption{Compact \\dwfl Pure CRDT}\n\\label{fig:dwflag-compact}\n\\end{figure}\n\nAs for \\dwfl, the concurrent specs (in Figure~\\ref{fig:dwflag-semantics})\nsays that the flag is enabled as long as there is at least one $\\enable$ operation\nwhere all existing $\\disable$ operations are in its strict causal past, \nand all existing $\\clear$ operations are either concurrent or in its causal past.\nConsequently, the first clause in the pure CRDT design, in \nFigure~\\ref{fig:dwflag-compact}, only removes newly delivered $\\clear$ operations.\nThe second clause however specifies that $\\disable$ and $\\enable$ operations are\nonly removed from the \\polog if they are succeeded by another operation with a \ngreater timestamp. To reflect this behavior, the $\\af{read}$ operation returns\nthe $\\enable$ operations such that no $\\disable$ operations exist in the\n\\polog (in which case a $\\disable$ should prevail), yielding an equivalent output\nto the base-line $\\af{read}$ operation in Figure~\\ref{fig:dwflag-semantics}. \nFinally, $\\stabilize$\nhas no effect in this datatype as well.\n\n\n\n\\section{Background}\n\\label{sec:back}\n\n\nBy tracking divergence, conflict-free replicated data-types allow local operation even when no communication is possible. We have seen that CRDTs can be approached by state-based\\footnote{For more information\n\tabout state-based CRDTs and the optimized variant Delta-State CRDTs,\n\tthe reader may refer to ~\\cite{rep:syn:sh138,syn:rep:sh143} \n\tand~\\cite{delta-states:Almeida:2016,deltas:Almeida:2015}.} or op-based frameworks, and both approaches share the same high-availability properties and resiliency to partitions. Onwards, this paper will focus on advancing the definition and possibilities of the operation-based approach.  \n\n\\subsection{Operation-Based CRDTs}\nIn op-based CRDTs~\\cite{rep:syn:sh138,syn:rep:sh143}, each replica\nmaintains a local state, and is subject to clients \\emph{query} and \\emph{update}\noperations that are executed locally as soon as they arrive. \nIn particular, update operations that are received through clients are \ndisseminated to other replicas, via a reliable causal broadcast middleware, \nin a form of operation name, arguments (if exist), and possibly some\nmeta-data~\\cite{rep:syn:sh138,syn:rep:sh143}.\nOnce all replicas receive and execute all update operations, they\neventually converge to a single state  as long as concurrent update operations commute, i.e. produce the same effect if delivered in different orders. \nOperation dissemination must be reliable, and most data-types require broadcast delivery to respect causal order. This also ensures that resulting states are causally consistent. \n\n\n\\begin{algorithm}[H]\n\\DontPrintSemicolon\n\\SetKwBlock{state}{state:}{}\n\\SetKwBlock{on}{on}{}\n\\state{\n  $\\sigma_i \\in \\Sigma$ \\;\n}\n\\on({$\\operation_i(o)$:}){\n\t$m:=\\prepare_i(o, \\sigma_i)$ \\;\\\\\n\t$\\cbcast_i(m)$ \\;\n}\n\\on({$\\cdeliver_i(m)$:}){\n  $\\sigma_i := \\effect_i(m, \\sigma_i)$ \\;\n}\n\\BlankLine\n\\caption{Distributed algorithm at node $i$ showing the interplay between a standard\nreliable causal broadcast API ($\\cbcast$ and $\\cdeliver$) and op-based CRDTs \n($\\prepare$ and $\\effect$).}\n\\label{algo:op-rcb}\n\\end{algorithm}\n\n\n\\begin{figure}[h]\n\t\\centering\n\\[{\\centering\n\t\\begin{array}{cl}\n\t\t\\Sigma&:\\text{\\small State type, $\\sigma_i$ is an instance}\\\\ \\prepare_i(o,\\sigma_i)&:\\text{\\small Prepares a message $m$  given an operation $o$}\\\\\n\t\t\\effect_i(m,\\sigma_i)&:\\text{\\small Applies a \\emph{prepared} message $m$ on a state}\\\\\n\t\t\\eval_i(q,\\sigma_i)&:\\text{\\small Read-only evaluation of query $q$ on a state}\n\t\\end{array}}\n\\]\n\t\\caption{The general scheme of an op-based CRDT.}\n\t\\label{fig:op-crdt}\n\\end{figure}\n\nMore specifically, the typical scheme of op-based CRDT is depicted \nin Figure~\\ref{fig:op-crdt}, while Algorithm~\\ref{algo:op-rcb} depicts the\ninterplay between the reliable causal broadcast middleware and the CRDT.\nWhen an update operation $o$ is issued at some node $i$ having state $\\sigma_i$, \nthe function $\\prepare_i(o,\\sigma_i)$ is called returning a message $m$. \nThis message $m$ is then broadcast by calling $\\cbcast_i(m)$ that is provided\nby the reliable causal broadcast API. \nOnce $m$ is delivered via $\\cdeliver_j(m)$ at each destination\nnode $j$ (including $i$ itself), $\\effect_j(m,\\sigma_j)$ is called, returning\na new replica state $\\sigma'_j$. For each node that broadcasts a given\noperation, the broadcast event, the corresponding local delivery, and the\n$\\effect$ on the local state are executed atomically. \nWhen a query operation $q$ is issued, $\\eval_i(q, \\sigma_i)$ is invoked, and no\ncorresponding broadcast occurs. $\\eval$ takes the query and the state as input \nand returns a result, leaving the state unchanged. \nWe further explain this through the following simple example.\n\n\\emph{Example on Op-Based Counters:}\nFigure~\\ref{fig:op-ctr} represents an op-based increment-only counter. The\nstate represents an integer that is initialized to $0$. The $\\prepare$ function\nonly returns $m=\\inc$ for an invoked $\\inc$ operation by a client. \nThe middleware disseminates $m$ to all replicas, whereas, $\\effect$ increments \nthe counter state at each replica. \nThe query operation $\\af{value}$ simply returns the value of the counter.\\\\\n\n\\subsection{The case for a new op-based CRDT model}\nThe above example on counters is useful to explain the op-based CRDT model, \nbut it falsely gives\nthe impression that the CRDT design is generally straightforward and lightweight.\nUnfortunately, this is not the case for more complex data types where operations\nare not commutative, e.g., in sets, maps, etc. \nFor instance, consider the \\emph{Add-Wins Set} design shown in Figure~\\ref{fig:op-set} \nand having the following semantics: an $\\add$ operation \nprevails if it is concurrent with a $\\remove$ (an alternative remove-wins design is shown later). \nSince $\\add$ and $\\remove$ are not commutative operations, the causal order of \ntheir execution must be ensured to guarantee that different replicas converge. \nCapturing this information leads to a more complex \ndesign through adding \nextra meta-data to the data type state (see $\\Sigma$) as well as to the exchanged messages \n(see the output of $\\prepare$). \n\nIndeed, after analyzing several op-based data types introduced \nin~\\cite{rep:syn:sh138,syn:rep:sh143}, we observed three general trends (we discuss\nthese in details in Section~\\ref{sec:discuss}): \nfirst, the entire design is data-type-dependent since all the components (i.e.,\n$\\Sigma, \\prepare, \\effect,$ and $\\eval$) can change per data type; \nsecond, designing (or understanding) an efficient data type becomes cumbersome;\nand third, this design induces a substantial communication overhead since \n$\\prepare$ returns additional meta-data to be sent (along with the operation \nname and parameters). The pure op-based CRDT model that we introduce in this\npaper tries to address these challenges.\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[b]{.35\\textwidth}\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\mathds{N} && \\sigma_i^0 = 0 \\\\\n\\prepare_i(\\inc,n ) & = & \\inc \\\\\n\\effect_i(\\inc, n ) & = & n+1 \\\\\n  \\eval_i(\\af{value},n ) & = & n\\\\\n\\end{eqnarray*}\n\\caption{Op-based increment-only counter.}\n\\label{fig:op-ctr}\n\\end{subfigure}\\\\\n\\begin{subfigure}[b]{.60\\textwidth}\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\mathds{N} \\times \\pow{I \\times \\mathds{N} \\times V} &&\n\\sigma_i^0 = (0,\\{\\})\\\\\n\\prepare_i([\\add,v],(n,s)) & = & [\\add,v,i,n+1] \\\\\n\\effect_i([\\add,v,i',n'], (n,s)) & = & (n' \\kw{if} i = i' \n\\kw{otherwise} n\\\\&&, s \\cup \\{(v,i',n')\\} )\\\\\n\\prepare_i[\\remove,v],(n,s)) & = & [\\remove,\\{ (v',i',n') \\in s | v'= v\\}]\\\\\n\\effect_i([\\remove,r],  (n,s )) & = & (n, s \\setminus r ) \\\\\n\\eval_i(\\elems,(n,s )) & = & \\{ v | (v,i',n') \\in s\\}\\\\\n\\end{eqnarray*}\n\\caption{Op-based Add-Wins Set (embeding causality)}\n\\label{fig:op-set}\n\\end{subfigure}\\\\\n\\caption{An example of two standard op-based CRDTs showing their \n\tdesign complexity (when causality is needed) and their\n\tdata-type-dependent nature.}\n\\label{fig:op-based}\n\\end{figure}\n\n\n\n\\section{Semantic \\polog Compaction}\n\\label{sec:compaction}\n\nIn this section, we show how \\polog based CRDTs can be made practical by\nperforming \\polog compaction, and thus extend the introduced framework.  The\npurpose of compaction is to reduce the space and computation overheads through\npruning the \\polog in a lossless manner. We do this through two main\nmechanisms: \\emph{causal redundancy} which once an operation is delivered, one \nmay prune that very operation or\/and an already existing \\polog operation,\nwhichever is deemed redundant according to the data type semantics; and\n\\emph{causal stabilization} which once some operation in the \\polog becomes\ncausally stable, it discards the operation timestamp and possibly removes it or\neven some other operations.\n\n\\subsection{Causal Redundancy}\n\nThe first mechanism prunes the \\polog once an operation is\ncausally delivered in the $\\effect$. The aim is to keep the smallest number of \\polog \noperations such that all queries return the same result as if the full \\polog was\npresent. In particular, this method discards operations from the \\polog if \nthey can be removed without impacting the output of query operations.\nSince this pruning is based on the semantics of a data type, it is obviously \ndata-type-dependent; however, we opt to call it causal redundancy\nsince pruning is mainly driven by causality.\n\nIn Fig.~\\ref{fig:compact-framework}, we extend the generic framework\nwhich now includes the \\polog, $\\prepare$, and a more sophisticated generic\n$\\effect$. The $\\effect$ can now prune the \\polog through three data-type-specific\nrelations that together define causal redundancy.\nThis has the advantage of keeping the framework generic, with the same\n$\\effect$ for all data types, while delegating data-type-specific behavior to\nmere relations (and not arbitrary procedures).\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\\Sigma = T \\mathrel{\\hookrightarrow} O &&\n\\sigma_i^0 = \\{\\}  \\\\\n\\prepare_i(o, s) & = & o \\\\\n\\effect_i(o, t, s) & = &\n  \\{(t,o) | (t,o) \\not\\R s\\} \\cup\n  \\{x \\in s | x \\not\\R_0 (t,o) \\\\ && \\mathrel{\\wedge} (t,o) \\R s \\mathrel{\\vee}\n              x \\not\\R_1 (t,o) \\mathrel{\\wedge} (t,o) \\not\\R s\\} \\\\\n\\R, \\R_0, \\R_1 & = & \\textrm{[datatype-specific redundancy relations]} \\\\\n\\eval_i(q, s ) & = & \\textrm{[datatype-specific query function]} \\\\\n\\end{eqnarray*}\n\\caption{Generic framework with \\polog compaction by causal redundancy}\n\\label{fig:compact-framework}\n\\end{figure}\n\nThe $\\R$ relation, in Figure~\\ref{fig:compact-framework}, defines whether the \ndelivered operation is itself redundant\nand does not need to be added itself to the \\polog. In most cases, as \nwe will see later, this can be decided by looking only at the delivered operation \nitself, regardless of the current \\polog and, thus, a unary $\\R$ would be enough;\nhowever, in other cases, this depends on the \\polog; therefore, $\\R$ is a binary \nrelation between the operation and the \\polog.\n\nThe other two relations, $\\R_0$ and $\\R_1$, define which operations in the\ncurrent \\polog become redundant given the delivery of the new operation. For\nsome data types this decision depends on whether the delivered operation is\nadded to the \\polog or whether it is itself redundant: we may be able to\ndiscard some operation if the new one is kept, but not if it is itself\ndiscarded. Having these two relations makes the framework expressive enough\nfor such data-types (which are fortunately not numerous): \n$\\R_0$ is used when the new arrival is discarded \nbeing redundant, and $\\R_1$ if it is added to the \\polog. \nFor most data types, these relations will be equal, i.e., $\\R_0 = \\R_1$, \nand we define a single $\\R_\\_$ relation to denote both $\\R_0$ and $\\R_1$.\nAs we will see, the \\polog compaction resulting from these relations will\nprovide invariants over the \\polog which allow simplified query functions that\ngive the same result over the compact \\polog as the original query functions\nover the full \\polog.\n\nHaving explained the roles of $\\R$, $\\R_0$, and $\\R_1$, the behavior of $\\effect$\nin Figure~\\ref{fig:compact-framework} becomes as follows: a newly delivered operation\n$(t,o)$ is added to the \\polog if it is not redundant by the \\polog operations,\nas the first clause says: $\\{(t,o) | (t,o) \\not\\R s\\}$. The remaining clause says\nthat an existing operation $x$ in the \\polog is removed if it is made redundant by $(t,o)$. \nSince, the behavior is sometimes different depending on whether $(t,o)$ has been discarded \nor not, the two clauses, separated by $\\mathrel{\\vee}$, distinguishes these two cases.\n\n\n\n\\subsubsection{Example CRDTs}\n\nWe apply this \\polog compaction to the previous examples presented in\nFigures~\\ref{fig:polog-orset} and~\\ref{fig:polog-mvreg}, extended with a $\\clear$ operation that removes all values; the corresponding\ncompact designs are presented in Figures~\\ref{fig:or-set-compacted} \nand~\\ref{fig:mvreg-compacted}, respectively. In the latter two, we\nonly mention the data-type-specific functions since $\\Sigma$, $\\prepare$, \nand $\\effect$ are generic. In these examples,\nthe left side of the relations $\\R$ and $\\R_\\_$ are made redundant by the\nright side. On the other hand, the notation $o[i]$ refers to the $i^{th}$\nelement in a list that comprises the operation name and subsequent arguments (analogous to\nthe $argv[]$ used in several programming languages); in particular, $o[0]$ \nrefers to the operation name.\n\nIn the \\awset design in Figure~\\ref{fig:or-set-compacted}, the first clause says\nthat $\\clear$ and $\\remove$ operations are always made redundant by\n(and thus not stored in) the \\polog. The second clause specifies that an existing\noperation $o'$ in the \\polog is redundant (and thus pruned) if the newly arrived \noperation $o$ is strictly in its causal future, provided that $o$ is either $\\clear$\nor otherwise (e.g., $\\add$) tries to modify the same value (i.e., set item) in the set (in \nwhich case it is clearly redundant). These relations will actually result in a \\polog\nthat is deprived of any $\\rmv$ or $\\clear$ operations; consequently, the $\\af{rd}$\noperation will simply return all the values in the \\polog.\n\nThe second example is the \\mvreg, presented in Figure~\\ref{fig:mvreg-compacted}.\nThe first clause specifies that a $\\clear$ operation is always redundant and must not be\nadded to \\polog. The second clause says that an existing operation $o'$ is made redundant\nby $o$ if the latter is strictly in its causal future; otherwise all concurrent $\\af{wr}$\noperations are kept in the \\polog. The $\\af{rd}$ operation is thus very simple: it returns\nall the present values in the \\polog.\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\t(t,o) \\R s &\\iff& o[0] = \\clear \\mathrel{\\vee} \\rmv \\\\\n\t(t',o') \\R_\\_ (t,o) &\\iff& t' < t \\mathrel{\\wedge} (o[0] = \\clear \\mathrel{\\vee} o[1] = o'[1]) \\\\\n  \\eval_i(\\elems, s) & = & \\{ v | (\\_, [\\add,v]) \\in s \\}\\\\\n\\end{eqnarray*}\n\\caption{Add-Wins set with \\polog compaction. $o[0]$ denotes the operation name\nand $o[1]$ denotes its first argument.}\n\\label{fig:or-set-compacted}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\t(t,o) \\R s &\\iff& o[0] = \\clear \\\\\n  (t', o') \\R_\\_ (t, o) &\\iff& t' < t \\\\\n  \\eval_i(\\af{rd}, s) & = & \\{ v | (\\_, [\\af{wr}, v]) \\in s \\}\\\\\n\\end{eqnarray*}\n\\caption{Multi-Value register with \\polog compaction}\n\\label{fig:mvreg-compacted}\n\\end{figure}\n\n\n\n\n\\subsection{Causal Stabilization}\n\n\\emph{Causal stabilization} is the second \\polog compaction mechanism; it\nexploits \\emph{causal stability} information to discard timestamp\nmeta-data for the elements that become \\emph{causally stable}, and may allow\nsome elements to be entirely removed from the \\polog.\n\n\\subsubsection{Discarding stable timestamps}\nThis mechanism discards the causally stable timestamps in the \\polog.\nThe motivation is that the $\\effect$ function makes use of the relations $\\R$, \n$\\R_0$, and $\\R_1$ to compare $(t',o')$ elements in the \\polog with newly \ndelivered operations, but never compares \\polog elements among themselves.\nOur observation is that most of the meta-data in the \\polog is not\nneeded when incoming operations are no longer concurrent.\nConsidering the definition of causal stability in Section~\\ref{sec:ercb}:\nif some pair $(t',o')$ is in the \\polog, with $t'$ a causally stable timestamp,\nall future deliveries $(t, o)$ used in $\\effect$ will be in its causal \nfuture, i.e., $t' < t$. This means that what is important is to know if this \ncomparison $t' < t$ is true or false regardless of the exact timestamp values.\nConsequently, we can replace an existing timestamp $t'$ that became stable with\nanother one that is always less than $<$ any other timestamp $t$ to be \ndelivered. (Notice that $t$ cannot be replaced now until it also becomes stable,\notherwise the concurrent semantics will break.) For instance, a\nstable timestamp $t'$ can be replaced by $\\bot$, i.e., the least element (a.k.a., \nbottom) of the timestamp domain, effectively discarding the timestamp; e.g.,\n$0$ is the $\\bot$ of a scalar timestamp in $\\mathds{N}$.\n\nHowever, to perform such a timestamp replacement, two necessary conditions \nhave to be met in the data-type-specific functions: queries should not be\naffected and the causal redundancy relations $\\R$, $\\R_0$, and $\\R_1$ must\nnot compare the existing \\polog timestamps together. It was a fortunate happenstance that\nthis was the case for all data types that we have considered in this paper, and\nthus all can be causally compacted using stabilization.\nThe explanation of this is that, in all these data types, after causal \nredundancy is applied, the resulting simplified query operations no longer \nneed to compare timestamps, and therefore, the queries will\ngive the same result as before even if stable timestamps are replaced by $\\bot$.\nOn the other hand, $\\R$, $\\R_0$ and $\\R_1$ never compare existing timestamps in the \n\\polog with each other, but only compare them with the newly arrived operations.\n\nThis optimization can have substantial gains in many data types as it \ngreatly diminishes the size of a replica state.\nIn practice, instead of having timestamps that are maps or\nvectors with linear size on the number of replicas, we can have a special\nmarker denoting $\\bot$ (e.g., a null pointer). Consequently, we designate two \ninteresting data type classes. \nThe first class is those CRDTs where values may take considerably less space \nthan timestamps, like sets of integers, and thus, stripping the timestamp from an \nelement can reduce the storage overhead several orders of magnitude.\nThe second class is those CRDTs whose state size can grow significantly;\nin such CRDTs, the percentage of elements in the \\polog that are not yet causally \nstable will be quite small, most of operations being already stabilized. \nTherefore, this optimization reduces the storage overhead of the element itself,\nand may impact most \\polog elements, as they become causally stable.\n\n\\emph{Further optimizations in practice:}\nIn actual implementations, the \\polog can be split in two components: \none that simply stores the set of stable operations and the other stores the \ntimestamped operations.\nFor instance, instead of being a map $T \\mathrel{\\hookrightarrow} O$, the state is split into\n\\[\\Sigma=\\pow{O} \\times (T \\mathrel{\\hookrightarrow} O)\\]\ndetaching into a plain set of all stable operations. Furthermore, for some\ndata types (such as \\awset) where only one kind of operations, i.e., $\\add$, \nis ever in the \\polog, the stable set of operations can become a plain\nset of elements (without timestamps), \ne.g., the traditional sequential set data type, with possibly\nspecialized implementations according to the domain (i.e., a bitmap for dense\nsets of integers). To keep the presentation clear and consistent, we leave \nsuch optimizations to actual implementations and we rather only consider \nusing a unified \\polog, containing both stable $(\\bot, o')$ and unstable operations $(t', o')$. \n\n\\subsubsection{Discarding stable operations}\nThis mechanism allows to discard stable operations, not only timestamps, if\nthey have no impact on the semantics of the data type.\nThe idea is that, for some data types like \\rwset (explained next), \nwe noticed that some operations\ncould not be considered redundant once delivered, but they become useless once\nother future operations become stable; and therefore,\nit may be possible to discard a set of operations at once. \\\\\n\nTo support both causal stabilization mechanisms, we extend the \nframework, presented in Figure~\\ref{fig:full-compaction}, to include a \ndata-type-specific $\\stabilize$ function. This function takes a stable timestamp \n$t$ (fed by the TCSB middleware) and the full \\polog $s$ as input, \nand returns a new \\polog (i.e., a map), possibly discarding a set of\noperations at once. The $\\stable$ handler, on the other hand, invokes $\\stabilize$\nand then strips the timestamp (if the operation has not been discarded by\n$\\stabilize$), by replacing a $(t',o')$ pair that is present in the returned \\polog by\n$(\\bot, o')$. In the following section, we show how the \\awset and \\mvreg, presented in \nFigures~\\ref{fig:or-set-compacted} and~\\ref{fig:mvreg-compacted},\nas well as other several data types, can be optimized with causal stabilization.\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\\Sigma = T \\mathrel{\\hookrightarrow} O &&\n\\sigma_i^0 = \\{\\}  \\\\\n\\prepare_i(o, s) & = & o \\\\\n\\effect_i(o, t, s) & = &\n  \\{(t,o) | (t,o) \\not\\R s\\} \\cup\n  \\{x \\in s | x \\not\\R_0 (t,o) \\\\ && \\mathrel{\\wedge} (t,o) \\R s \\mathrel{\\vee}\n              x \\not\\R_1 (t,o) \\mathrel{\\wedge} (t,o) \\not\\R s\\} \\\\\n  \\stable_i(t, s) &=& \\stabilize_i(t, s)[ (\\bot,o) \/ (t,o)] \\\\\n\\R, \\R_0, \\R_1 &=& \\textrm{[datatype-specific redundancy relations]} \\\\\n  \\stabilize_i(t,s) &=& \\textrm{[datatype-specific stabilization function]} \\\\\n\\eval_i(q, s ) & = & \\textrm{[datatype-specific query function]} \\\\\n\\end{eqnarray*}\n\\caption{Pure CRDTs framework including \\polog compaction: causal redundancy and stabilization.}\n\\label{fig:full-compaction}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nWe introduced Pure operation-based CRDTs: a novel generic and efficient\nop-based CRDT model that establishes a clear frontier with state-based models.\nThe model is ``pure'' in the sense that a disseminated message only contains \nthe operation name and its potential arguments. The causal information that\nis usually explicitly retained in the CRDT state and added to disseminated messages, \nin the classical op-based CRDT, are now provided by a  \nTagged Causal Stable Broadcast (TCSB) middleware that we proposed. Despite the useful design and\nperformance benefits it brings to CRDTs, we tried to introduce the TCSB in \na self-contained section as we believe it is an important extension that \ncan be exploited by the community to improve other concurrent designs as well.\nWe also believe that the TCSB deserves a dedicated practical analysis in the future\nto understand its characteristics and performance tradeoffs.\n\nOur paper distinguishes between two types of CRDTs: commutative and non commutative CRDTs.\nFor the non commutative CRDTs, we design the state as a partially ordered log (\\polog)\nthat can retain concurrent operations tagged with timestamps. The timestamps and\nsome operations can be discarded using causal redundancy and stabilization that are \nfeatures supported by the TCSB. \nOn the other hand, the commutative CRDTs are not based on \\polog \nbeing very simple. Indeed, they can be designed using the \\polog if desired, we however \ndon't recommend this due to the needless overhead and complexity of the \\polog in this case.\n\nWe tried to provide a catalog of Pure CRDTs addressing many useful data types in\npractice. We believe that these designs are now simpler, provided that the\ngeneric framework is well understood. Some data types, like Remove-Wins Sets \nare however more complex due to their non-trivial concurrent semantics especially\nwhen used with $\\clear$ operations that reset the state. We noticed that the framework can be easier if \nreset operations are not needed. \nWe did not address more composite data types like Maps since this will\nrequire extending the \\polog, probably to a Multi-\\polog, thus imposing more complexity\nto the paper, which we opt to avoid. Finally, we believe that a detailed \nempirical analysis that studies the different data access patterns to the CRDTs in\ndifferent models will help better understand the performance tradeoffs; we plan to \ndo such analysis in the future.\n\n\n\n\\section{Discussion}\n\\label{sec:discuss}\n\nWe recap this work by discussing the most substantial benefits the Pure \nCRDT model brings on the conceptual and performance levels. \nWe divide our discussion into the following three subsections. \n\n\\subsection{Making Op-based CRDTs Op-based}\nThe first incentive behind this work was to remove the confusion between\nop-based and state-based CRDTs~\\cite{crdt:shapiro:2011,delta-states:Almeida:2016}.\nIn fact, the nomenclature of these two approaches refers to the type of\nthe message being disseminated: the message is generally (supposed to be) the ``operation'' in\nthe operation-based model whereas a ``state'' in the state-based model.\nThis notion is, however, not reflected in the classical op-based design since\nthe disseminated message, i.e., the output of $\\prepare$, comprises other meta-data \nas we have seen in Figure~\\ref{fig:op-set}, Section~\\ref{sec:back}.\nIn many cases, this leads to other issues (discussed next) like ad-hoc \ndesigns and abuse of resources.\nThe Pure op-based CRDTs we presented are ``pure'' op-based designs that only \nrequire the operation (and potential parameters) to be disseminated; thus bringing \nback the essence of the ``op-based'' terminology. The tangible benefits of this\nconceptual feature is reflected in the following sections.\n\n\\subsection{Making Op-based CRDT Designs (Almost) Generic}\nThe Pure CRDT framework makes the design of op-based CRDTs ``almost'' generic.\nIn our experience, making them fully generic is impossible due to the native semantic \ndiscrepancy of the designed data types --- making them more generic will be \nimpractical as we've seen in the \\polog based CRDTs before compaction (presented in \nSection~\\ref{sec:pure}). \n\nThe classical op-based CRDT designs are data type dependent. Observing the \nseminal CRDT designs in~\\cite{crdt:shapiro:2011} shows that designing a new\nCRDT will require the designer\/developer to completely define every design \ncomponent, i.e., $\\Sigma$, $\\prepare$ for each operation type, $\\effect$ \nfor each operation type, and $\\eval$. In the Pure CRDT framework we introduced,\nthe basic components are generic: $\\Sigma$, $\\prepare$, $\\effect$, and $\\stable$. \nAlthough our approach requires additional data-type-specific components like \n$\\R$, $\\R_\\_$, and $\\stabilize$, we argue that their definitions are very simple;\nand as you may observe in the catalog of CRDTs in the previous section, these\nmere components can be implemented in a couple of conditionals or assertions.\nNevertheless, our recommendation was not to use the \\polog for commutative CRDTs,\nbeing simple and more efficient, even if it will be generic.\n\nFinally, introducing a generic framework for CRDTs brings notable benefits.\nFirst, it reduces the complexity of the design and allows the designer\/developer to \nfocus on the semantics of a data type rather than fiddle with\nother design details. Second,\nit reduces the implementation cost since the common components can now be\nimplemented once and used by all data types, avoiding repetitions. Third, it reduces \nthe likelihood of introducing new bugs and anomalies as the design limits the\nfreedom of the developer to change the rigorously thought-out generic components. \n\n\\subsection{Making Op-based CRDT Efficient}\nOur Pure CRDT model improves the efficiency of dissemination and storage.\nIn classical op-based CRDT designs, additional meta-data is usually used\nto express the causality information required by non commutative \ndata types~\\cite{crdt:shapiro:2011}.\nAn example we've seen in this paper is the \\awset, in Figure~\\ref{fig:op-set},\nwhere the state\n$\\Sigma = \\mathds{N} \\times \\pow{I \\times \\mathds{N} \\times V}$ comprises a set of\ntriples, and the output of $\\prepare$ for the $\\remove$ operation (which must be\nbroadcast to other replicas) contains also a set of triples. On the \ndissemination front, we believe that there is abuse of network resources since\nthe causality meta-data (mainly timestamps) is already being exchanged, though not used,\nin the underlying middleware. The introduction of Tagged Causal Stable Broadcast\nmiddleware (TCSB) exploits this information which allows designing pure CRDTs, i.e.,\nwhere the $\\prepare$ output (to be disseminated) is only the operation name and \npotential arguments. The dissemination overhead can be reduced several orders of\nmagnitude when the fraction of data to timestamps is small, e.g., when the data is\nan integer whereas the timestamp is a vector clock.\nTCSB has also other advantages on the storage front through\nthe causal stability information that helps pruning the useless meta-data in the \\polog,\nwhich as we've seen, turns the CRDT into a classical sequential data type in some cases.\nAs explained in Section~\\ref{sec:compaction}, the gain will be higher in data types \nthat exhibit large storage sizes (e.g,\nas set of millions of items) since most of the operations will already be causally \nstable and thus compacted. Measuring the real overhead across real data patterns is \nsubject future work.\n\n\n\\section{Tagged Causal Stable Broadcast (TCSB)}\n\\label{sec:ercb}\nReliable Causal Broadcast (RCB) is a prominent dissemination abstraction that is\noften used in AP systems (in the CAP~\\cite{cap:Gilbert:2002} context) to \nensure causal delivery of messages, being the strongest consistency \nmodel in always-available systems that eventually \nconverges~\\cite{syn:rep:1677,causal:Attiya:2017}.\nA common implementation strategy for a reliable causal broadcast \nservice~\\cite{Schmuck88theuse} is to assign a vector clock to each message\nbroadcast, and use the causality information in the vector clock to decide at\neach destination when a message can be delivered. If a message arrives at a\ngiven destination before causally preceding messages have been delivered,\nthe service delays delivery of that message until those messages arrive and\nare delivered. \n\nWhen messages are delivered to the application layer, in a sequence that is consistent with causality, there is some loss of information when using standard RCB middleware. Messages that the middleware knows be concurrent, are delivered in order and the application is unaware of that concurrency. Also the middleware can often determine that, for a given delivered message, that no more messages concurrent to it are to be delivered. This information, that can be used to optimize message processing by the application, is also not conveyed. Next we show how to expose this ordering information and how to provide a causal stability oracle that informs on stable messages, those with no further concurrent deliveries. \n\n\n\n\\subsection{Exposing Ordering Information}\nReliable Causal Broadcast (RCB) uses internally a logical timestamp $t$\n(e.g., a vector clock) to order messages. We propose to simply expose\nthis information to the upper layers. Technically, the \n$\\cdeliver(m)$ middleware API (used in Algo.~\\ref{algo:op-rcb}) will now \nlook like this: $\\tcdeliver(t, m)$; meaning that, the calling process\nwill receive the message $m$ as well as its associated timestamp $t$.\nHaving a timestamp provided by the middleware allows the application logic to access causality information and avoids duplicating it in the datatype logic, avoiding the complexity in Figure \\ref{fig:op-set} and simplifying the message payload. \n\n\n\\subsection{Causal Stability}\nIn concurrent data types, the order information provided by timestamps\nmay only be needed for an operation as long as its concurrent operations are being \ndelivered or expected. However, in our experience, this information is useless\nonce no concurrent operations are expected for a given operation, which we then call\na ``causally stable'' operation\nand thus, it makes sense to get rid of this extra meta-data.\nConsequently, we propose another extension to the RCB middleware in order to \nprovide \\emph{causal stability} information to the upper layers. We first\ndefine causal stability.\n\n\\begin{definition}[Causal Stability]\nA timestamp $\\tau$, and a corresponding message, is \\emph{causally stable} at node $i$\nwhen all messages subsequently delivered at $i$ will have timestamp $t > \\tau$.\n\\end{definition}\n\nThis implies that no message with a timestamp $t$ concurrent with $\\tau$ can\nbe delivered at $i$ when $\\tau$ is causally stable at $i$. This notion differs\nfrom the classical message stability, in~\\cite{Birman:1991:LCA:128738.128742},\nin which a message is considered stable if it has been received by all nodes.\nOur definition is stronger, as we also require that no further concurrent\nmessages may be delivered.\nOur definition is about delivered (as opposed to received) messages, and is\nper-node (some $m$ may be causally stable at some node $i$ but not at some\nnode $j$), not global. It is similar to the definition in\n\\cite{rep:syn:sh138}, but we consider important to use a different name, to\navoid confusion with classic message stability.\n\n\n\nThe TCSB middleware can offer this \\emph{causal stability} information through \nextending its API with $\\tcstable_i(\\tau)$ which informs the upper layers that\nmessage with timestamp $\\tau$ is now known to be causally stable.\nThis can be done by each node having a causal stability oracle which\nconservatively detects causal stability according to local knowledge.\nEach node $i$ can check if timestamp $\\tau$ is causally stable at\n$i$ by verifying if a message with timestamp $t > \\tau$ has already been\ndelivered at $i$ from every other node $j$ in the set of nodes $I$. More\nformally, if $\\deliv_i()$ returns the set of messages that have been delivered\nat node $i$ and $\\src(t)$ denotes the node from where the message\ncorresponding to $t$ has been issued:\n\\[\n  \\tcstable_i(\\tau)  \\kw{if} \\forall j \\in I \\setminus \\{i\\} \\cdot\n   \\exists t \\in \\deliv_i() \\cdot \\src(t) = j \\wedge \\tau < t.\n\\]\n\nOne possible implementation of a TCSB causal stability oracle uses a strategy\nsimilar to Roh's RGA tombstone deletion algorithm \\cite{syn:app:1649}.\nEach node $i$ keeps a local map $L_i$ (from $I$ to $T$) with the last vector\nclock timestamp, delivered locally, from each other node.\nWe define an auxiliary function $\\low$ that gives the greatest lower bound on messages\nissued at $j$ delivered at each other node:\n\n\\[\n  \\low(L,j) \\doteq \\min(\\{ L(k)(j) | k \\in \\dom(m) \\}).\n\\]\n\nFor instance, if $\\low(L_i,j)=4$, we know at node $i$ that all other nodes\nhave delivered at least message number 4 from node $j$.\nUsing this function we can now define a sufficient condition for the causal\nstability oracle.  A timestamp $\\tau$ is causally stable at node $i$ if\n\\[\n  \\tau(\\src(\\tau)) \\leq \\low(L_i,\\src(\\tau)).\n\\]\n\nThis ensures at $i$ that message $\\tau$ was already delivered at all nodes and\nthat each node issued a message after delivering $\\tau$, that is already\ndelivered at $i$. Since any messages concurrent to $\\tau$ from any node $k$\nmust have been issued prior to $\\tau$ being delivery at $k$ then, due to\ncausal delivery, they must have also been delivered at node $i$.\n\n\n\\section{Introduction}\n\n\nEventual consistency~\\cite{rep:syn:pan:1624} is a relaxed consistency model that\nis often adopted by large-scale distributed\nsystems~\\cite{riak+crdt,Bailis:2013:ECT:2460276.2462076,syn:optim:rep:1433,app:rep:optim:1606}\nwhere losing availability is normally not an option, whereas delayed consistency\nis acceptable. In eventually consistent systems, data replicas are allowed to\ntemporarily diverge, provided that they can eventually be reconciled into a\ncommon consistent state.  Reconciliation (or merging) used to be error-prone,\nbeing application-dependent, until new data-type-dependent models like the\nConflict-free Replicated Data Types~(CRDTs)~\\cite{rep:syn:sh138,syn:rep:sh143}\nwere introduced. CRDTs allow both researchers and practitioners\nto design correct replicated data types that are always available, and are\nguaranteed to eventually converge once all operations are known at replicas.\nThe concept of CRDT has been deployed in practice by industry, initially in Key-Value Stores~\\cite{riak+crdt}, and have since been ported to multiple languages and production platforms. \n\nCRDTs support two complementary designs: operation-based (or simply, op-based)\nand state-based. As the name suggests, the former are based on dissemination of operations and the later on shipping state that results from locally applied operations. \nIn op-based\ndesigns~\\cite{alg:rep:sh132,syn:rep:sh143}, the execution of an operation is\ndone in two phases: \\emph{prepare} and \\emph{effect}. The former is\nperformed only on the local replica and looks at the operation and current\nstate to produce a message that aims to represent the operation, which is then\nshipped to all replicas.  Once received, the representation of the\noperation is applied remotely using \\emph{effect}. Different replicas are\nguaranteed to converge as long as messages are disseminated through a\nreliable causal broadcast messaging\nmiddleware~\\cite{pro:pan:160}, and \\emph{effect} is designed to be commutative for concurrent\noperations. On the other hand, in a state-based\ndesign~\\cite{app:1639,syn:rep:sh143}, an operation is only executed on the\nlocal replica state. A replica propagates its local changes to other replicas\nthrough shipping its entire state. A received state is incorporated with the\nlocal state via a \\emph{merge} function that, deterministically, reconciles\nthe \\emph{merged} states. To maintain convergence, \\emph{merge} is defined as\na join: a least upper bound over a\njoin-semi-lattice~\\cite{app:1639,syn:rep:sh143}.\nTypically, state-based CRDTs support ad hoc dissemination of states and can\nhandle duplicate and out-of-order delivery of messages without breaking causal\nconsistency; however, they impose complex state designs and store extra\nmeta-data. \nOn the other hand, in the systems where the message dissemination\nlayer guarantees reliable causal broadcast, operation-based CRDTs have more\nadvantages as they can allow for simpler implementations, concise replica\nstate, and smaller messages.\n\n\nIn standard op-based CRDTs the designer is given much freedom in defining\n\\emph{prepare}, namely using the state in an arbitrary way.  This is needed to\nhave the \\emph{effect}s of concurrently invoked data-type operations commute,\nand thus provide replica convergence despite the absence of causality\ninformation in current causal delivery APIs. This forces current op-based\ndesigns to include causality information in the state to be used in\n\\emph{prepare}, sent in messages, and subsequently used in \\emph{effect}. The\ndesigner ends up intervening in many components (the state, \\emph{prepare},\n\\emph{effect}, and \\emph{query} functions) in an ad hoc way.  This can result\nin large complex state structures and also large messages.\n\nCurrently, a \\emph{prepare} not only builds messages that duplicate the\ninformation already present in the middleware (even if it is not currently\nmade available), but causality meta-data is often incorporated in the object\nstate, hence, reusing design choices similar to those used in state-based\napproaches. Such designs impose larger state size and do not fully exploit causal delivery information.\nThis freedom in current op-based designs is against the spirit of `sending\noperations', and leads to confusion with the state-based approach. Indeed, in\nthe current op-based framework, a \\emph{prepare} can return the full state,\nand an \\emph{effect} can do a full state-merge (which mimics a state-based\nCRDT)~\\cite{app:1639,syn:rep:sh143}.\n\n\n\n\nWe believe that the above weaknesses can be avoided if the causality meta-data\ncan be provided by the messaging middleware. Causal broadcast implementations\nalready possess that information internally, but it is not exposed to clients.\nIn this paper we propose and exploit such an extended API to achieve both\nsimplicity and efficiency in defining op-based CRDTs.\n\n\n\nWe introduce a \\emph{Pure} Op-Based CRDT framework, in which \\emph{prepare}\ncannot inspect the state, being limited to returning the operation (including\npotential parameters). The entire logic of executing the operation in each\nreplica is delegated to \\emph{effect}, which is also made generic (i.e., not\ndata type dependent).  For pure op-based CRDTs, we propose that the object\nstate is a \\emph{partially ordered log of operations -- a \\polog}.  Causality\ninformation is provided by an extended messaging API: \\emph{Tagged\nCausal Stable Broadcast} (TCSB). We use this information to preserve\nconvergence and also design compact and efficient CRDTs through a\n\\emph{semantically based \\polog compaction} framework, which makes use of a\ndata type-specific \\emph{obsolescence} relation, defined over\ntimestamp-operation pairs.\n\n\n\n\nFurthermore, we propose an extension that improves the design and implementation of\nop-based CRDTs through decomposing the state into two components: a \\polog (as\nbefore), and a causality-stripped-component which, in many cases, will be\nsimply a standard sequential data type.\nFor this, we introduce the notion of \\emph{causal stability}, to be provided\nby the TCSB middleware.\nThe idea is that operations are kept only transiently in the \\polog, but once\nthey become causally stable, causality meta-data is stripped, and the\noperations are stored in the sequential data type. This reduces the\nstorage overhead to a level that was never achieved before in CRDTs, neither\nstate-based nor op-based.\n\n\n\n\n\n\n\n\\section{System Model and Notations}\n\\label{sec:model}\n\nThe system is composed of a fixed set of nodes, each is associated with a \nglobally unique identifier in a set $I$. Nodes execute operations at different \nspeeds and communicate using asynchronous message passing.\nMessages can be lost, reordered, or duplicated, and the system can experience\narbitrary, but transient, network partitions. To ensure exactly-once delivery, \nmessage sending is abstracted by a\nreliable causal broadcast~\\cite{birman:1987:reliable,golding1992weak}.\nA node can fail by crashing and can recover later on. Upon recovery, the last\ndurable state of a node is assumed to be intact (not destroyed). We do not\nconsider Byzantine or malicious faults.\n\nFor presentation purposes, and without loss of generality, we consider a single\ndata type instance that is fully replicated at each node. All replicas (of a data type) are \ninitially equivalent. Once a data type operation is locally applied on a replica, \nthe latter can diverge from other replicas, but replicas may \neventually convergence as\noperations arrive everywhere. The local application of an operation is atomic\non each replica (i.e., upon a crash, no state `in the middle of an operation'\ncan be seen in the durable state).\n\n\n\\subsection{Definitions and Notations}\n$\\Sigma$ denotes the type of the state. $\\mathcal{P}(V)$ denotes a \\emph{power\nset} (the set of all subsets of $V$), where $V$ is a set of any type.\nThe initial state of a replica $i$ is denoted by $\\sigma^0_i\\in \\Sigma$.\nOperations belong to a set $O$ and can include arguments, in which case they\nare surrounded by square brackets, e.g., $\\inc$ and $[\\add,v]$.\nOn the other hand, the notation $o[j]$ refers to the $j^{th}$\nelement in a list that comprises the operation name and subsequent arguments (analogous to\nthe $argv[]$ used in several programming languages); in particular, $o[0]$ \nrefers to the operation name.\nWe use total functions $K \\to V$ and maps (partial functions) $K \\mathrel{\\hookrightarrow} V$\nfrom keys to values, both represented as sets of pairs $(k, v$). Given a\nfunction $m$, the notation $m\\{k \\mapsto v\\}$ maps a specific key $k$\nto $v$, and behaves like $m$ on other keys.\n\n\n\n\\section{Pure CRDTs Based on Partially Ordered Logs (\\polog)}\n\\label{sec:polog}\nWe introduce a new framework for designing pure op-based CRDTs of\nnon-commutative data types. Driven by the challenges discussed in Section~\\ref{sec:challenges},\nthe framework supports concurrent data types through the use of a Partial Ordered Log (\\polog \nfor short) which retains all invoked operations together with their timestamps. Being ``pure'',\ni.e., the $\\prepare$ in framework shall only have the name of the operation and potential arguments,\nthe timestamp that is stored in the \\polog is directly fed by the underlying TCSB middleware\nintroduced above. Figure~\\ref{fig:polog-reference} depicts a reference design of a pure\nCRDT, whereas Algorithm~\\ref{algo:ercb} describes its interaction with the TCSB middleware.\nWe describes the framework in this section showing that it is easy to understand\nand generic. The framework is however not practical without the optimizations we introduce\nin the subsequent sections --- due to the ever-growing \\polog.\n\nThe challenges leveraged in Section~\\ref{sec:challenges} can be summarized by \ntwo points. The first is that there is a real need to support data types that\nare concurrent in practice. Consequently, we choose to retain all operations in a partially \nordered log (\\polog) that can be implemented as a map from timestamps to \noperations: $\\Sigma:=T\\mathrel{\\hookrightarrow} O$ as shown in Figure~\\ref{fig:polog-reference}. \nThis brings two main benefits: (1) a \\polog can retain \nconcurrent operations, and (2) storing operations as black-boxes makes the \nstate type  generic across all data types.\nThe timestamps offered by the TCSB middleware through $\\tcdeliver$ (as in \nAlgorithm~\\ref{algo:ercb}) are crucial to keep information \nabout concurrent operations, not trying to impose a local total-order over \nthem, contrary to a classic sequential log.\n\nThe second challenge, i.e., how to design the $\\effect$, is interestingly made\neasy using the \\polog.\nIn fact, we can now obtain a universal data-type-independent definition of $\\effect$:\n\\[\n\\effect(o, t, s) = s \\union \\{(t, o)\\},\n\\]\nwhich associates a timestamp to the corresponding operation and joins it to the \\polog\nvia the set union $\\union$, which is commutative, therefore making the $\\effect$ commutative\nas needed.\n\nIn this novel framework, it is easy to notice that all the components, but one,\nof the data type design are the same regardless of the data type. This significantly \nreduces the burden of implementation on developers as well as the chances of\nintroducing new bugs.\nOnly the query functions, defined in $\\eval$, will need to be data-type-specific \naccording to the desired semantics; however, their definition over the \n\\polog will typically be a direct transposition of their specification, which \nis fairly easy to define as we show next in the examples.\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\\Sigma = T \\mathrel{\\hookrightarrow} O &&\n\\sigma_i^0 = \\{\\}  \\\\\n\\prepare_i(o, s) & = & o \\\\\n\\effect_i(o, t, s) & = &  s \\cup \\{(t,o)\\} \\\\\n\\eval_i(q, s ) & = & \\textrm{[datatype-specific query function]} \\\\\n\\end{eqnarray*}\n\\caption{\\polog based reference implementation for pure op-based CRDTs}\n\\label{fig:polog-reference}\n\\end{figure}\n\n\\begin{algorithm}[t]\n\\DontPrintSemicolon\n\\SetKwBlock{state}{state:}{}\n\\SetKwBlock{period}{periodically:}{}\n\\SetKwBlock{on}{on}{}\n\\state{\n  $\\sigma_i \\in \\Sigma$ \\;\n}\n\\on({$\\operation_i(o)$:}){\n\t$m:=\\prepare_i(o, \\sigma_i)$ \\;\\\\\n\t$\\tcbcast_i(m)$ \\;\n}\n\\on({$\\tcdeliver_i(m, t)$:}){\n  $\\sigma_i := \\effect_i(m, t, \\sigma_i)$ \\;\n}\n\\BlankLine\n\\on({$\\tcstable_i(t)$:}){\n  $\\sigma_i := \\stable_i(t, \\sigma_i)$ \\;\n}\n\\BlankLine\n\\caption{Distributed algorithm for node $i$ showing \n\tthe interaction between the Tagged\nCausal Stable Broadcast (TCSB) middleware and a Pure CRDT.}\n\\label{algo:ercb}\n\\end{algorithm}\n\n\n\\subsection{Example CRDTs}\nWe exemplify the use and simplicity of the above framework through two\nexamples: an Add-Wins Set (\\awset) and a Multi-Value\nRegister (\\mvreg). In a nutshell, the semantics of \\awset give priority for\n$\\add$ operation over a $\\remove$ if they are concurrent; whereas, an \\mvreg\nretains all concurrently written values and leaves the decision to the\napplication to choose the desired value. These semantics are explained in\ndetails in Section~\\ref{sec:catalog}.\n\nFigures~\\ref{fig:polog-orset} and~\\ref{fig:polog-mvreg} show two pure \\polog\ndesigns for \\awset and \\mvreg, respectively, using the framework presented\nin Fig.~\\ref{fig:polog-reference}. It is interesting to notice that the pure\nCRDT design of both data types is exactly the same considering the state $\\Sigma$,\n$\\prepare$, and also $\\effect$. The only data-type-specific function is the\n$\\eval$ query function which defines the behavior of the data types in both\nCRDTs. \nIn \\awset (Fig.~\\ref{fig:polog-orset}), the values reported to be in\nthe set are those corresponding to the $\\add$ operations in the \\polog \nhaving no $\\remove$ in their causal future. \nWhereas, in \\mvreg (Fig.~\\ref{fig:polog-mvreg}), a read\nreports the set of all distinct concurrently written values, via $\\af{wr}$, that have not\nbeen overwritten.\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\\Sigma = T \\mathrel{\\hookrightarrow} O &&\n\\sigma_i^0 = \\{\\}  \\\\\n\\prepare_i(o, s) & = & o\n\\qquad\\qquad\\qquad (o \\textrm{ is } [\\add, v] \\textrm{ or\n} [\\rmv, v])\\\\\n\\effect_i(o, t, s) & = &  s \\cup \\{(t,o)\\} \\\\\n\\eval_i(\\elems, s ) & = & \\{ v | (t, [\\add,v])\n\\in s \\mathrel{\\wedge} \\not\\exists (t',[\\rmv,v]) \\in s \\cdot t < t'\\}\\\\\n\\end{eqnarray*}\n\\caption{\\polog based Add-Wins Set (\\awset).}\n\\label{fig:polog-orset}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\\Sigma = T \\mathrel{\\hookrightarrow} O &&\n\\sigma_i^0 =  \\{\\}  \\\\\n\\prepare_i(o, s) & = & o\n\\qquad\\qquad\\qquad (o \\textrm{ is } [\\af{wr}, v])\\\\\n\\effect_i(o, t, s) & = &  s \\cup \\{(t,o)\\} \\\\\n\\eval_i(\\rd, s ) & = & \\{ v \\mid (t, [\\af{wr},v])\n\\in s \\mathrel{\\wedge} \\not\\exists (t',[\\af{wr},v']) \\in s \\cdot t < t' \\}\\\\\n\\end{eqnarray*}\n\\caption{\\polog based Multi-Value register (\\mvreg)}\n\\label{fig:polog-mvreg}\n\\end{figure}\n\nAlthough these designs are not realistic to be used in practice, since the\nstate size in each replica grows linearly with the number of operations, they\nrepresent a starting point from which actual efficient designs can be derived by\nsemantic \\polog compaction, as we show in the rest of the paper. In addition,\nthese designs are theoretically relevant as they provide a clear description\nof the concurrent semantics of the replicated data type, and therefore can be used\nas a base-line to compare with other designs. This is made possible\nthrough capturing the partial ordered set of all operations delivered to each\nreplica\\footnote{A similar approach to express data type semantics is found in\n\\cite{DBLP:conf\/popl\/BurckhardtGYZ14} when relating to the \\emph{visibility}\nrelation.}.\n\n\\section{Portfolio of Pure Op-based CRDTs}\n\\label{sec:catalog}\nIn this section, we provide a catalog of CRDT designs following the pure\nop-based framework introduced above. Our purpose it to make this framework and\nCRDTs easy to understand and implement by designer and developers.\nWe try to cover examples of the most \nused data types in practice, e.g., in Dynamo~\\cite{app:rep:optim:1606}, \nRiak~\\cite{riak-crdt:Cribbs:2012}, \nCassandra~\\cite{cassandra:2016}, etc. In particular, we address two \ncategories of data types: (1) Commutative CRDTs: Grow-only Counter (\\gctr),\nPositive-Negative Counter (\\pnctr), Grow-only Set (\\gset), and Two-Phase Set \n(\\twopset); and Non-commutative data types: Enable-Wins Flag (\\ewfl), Disable-Wins \nFlag (\\dwfl), Multi-Value Register (\\mvreg), Add-Wins Set (\\awset), and \nRemove-Wins Set (\\rwset). To limit redundancy, but still provide reference specifications, the \\gctr and both flags are collected in appendix.\n\nIn this catalog, our methodology is to define the data type\nand show how it can be used in practice. Then we describe its semantics in a \ngeneric way before presenting its pure op-based design. This analogy helps\nthe designer\/developer choose the desired semantics for his business logic as well as \nunderstand the design of the CRDT in the Pure framework.\nFinally, we discuss additional optimizations that are data type\nspecific when useful. (Recall that the notation used in this section are\ndescribed in Section~\\ref{sec:model}). \n\n\n\\subsection{Commutative Data Types}\nAccording to Section~\\ref{sec:pure}, a CRDT is said to be commutative if all \nits possible operations are commutative over the state.\nThe design of commutative CRDTs is datatype-specific, both in \nthe pure op-based as well as in classical op-based approaches. Indeed, it \nis possible to design them using the \\polog framework to be generic across all data types,\nas we do for other non-commutative data types; however, the tradeoff is to induce an \nextra unneeded overhead. We opt not to use the \\polog for such\ndata types and thus provide more practical designs. All the commutative designs\nassume the existence of a Reliable Causal Broadcast (RCB) middleware. The interaction\nbetween the presented CRDTs and the RCB are discussed in Algorithm~\\ref{algo:op-rcb},\nSection~\\ref{sec:back}.\n\n\n\\subsubsection{Positive-Negative Counter (\\pnctr)}\nA \\pnctr is a counter that supports increment and decrement operations. \nTherefore, it is very useful to represent quantities that can be added or removed, like\nthe number of available tickets in a system.\nIts state is usually implemented as $n\\in \\mathbb{Z}$ with two operations \n$\\inc$ ($+$ operator) and $\\dec$ ($-$ operator); thus, its value can go negative. \nSince $+$ and $-$ are commutative in $\\mathbb{Z}$, \\pnctr is a commutative data type.\n\nFigure~\\ref{fig:pnctr} conveys the pure CRDT design of \\pnctr. \nThe state is now defined as integer $z\\in \\mathbb{Z}$ that is initialized to $0$.\n$\\prepare$ can be either $\\inc$ or $\\dec$; however, in both cases, no meta-data is\nneeded (and thus it is pure). The $\\effect$ function increments or decrements $z$ \ndepending on $\\prepare$'s output. Finally, $\\eval$ invokes the read operation $\\af{value}$ and\nsimply returns $z$.\n\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\mathbb{Z}&&\n\\sigma_i^0 = 0 \\\\\n\\prepare_i(\\inc, z) & = & \\inc \\\\\n\\prepare_i(\\dec, z) & = & \\dec \\\\\n\\effect_i(\\inc, z ) & = & z+1 \\\\\n\\effect_i(\\dec, z ) & = & z-1 \\\\\n\\eval_i(\\af{value},z ) & = & z\\\\\n\\end{eqnarray*}\n\\caption{Pure Positive-Negative Counter CRDT (\\pnctr).}\n\\label{fig:pnctr}\n\\end{figure}\n\n\n\\subsubsection{Grow-only Set (\\gset)}\nA \\gset is a set where only adding items is supported. It is usually used when removals\nare not needed, as in storing the IPs of clients visiting a web-site.\nThe set comprises a collection of items, of some type, that are added via \n$\\add$ operation (a union $\\union$). Since $\\add$ is commutative over the set, \n\\gset is a commutative datatype. \n\nFigure~\\ref{fig:gset} shows the pure CRDT design of \\gset.\nThe state is represented by $s\\in \\pow{V}$ (a power set of $V$) and is originally empty. \nThe function $\\prepare$ can only be the $\\add$ operation with its parameter $v$, but\nwithout extra meta-data (which makes it pure). The function $\\effect$ adds $v$ to \nthe set using the set union operator: $\\union$. Finally, the $\\eval$ function invokes\nthe read operations $\\elems$ and $\\size$ to return all the elements of the set or \nits cardinality, respectively.\n\n\n\\subsubsection{Two-Phase Set (\\twopset)}\nThe \\twopset is a set where items can be added and removed, but never \nadded after being removed. It can be seen as a composition of two \n\\gset, one for additions and another for removals. \nIn practice, the \\twopset can be used to represent collections having\nitems with unique ids. The allowed mutating operations in \\twopset are \n$\\add$ and $\\rmv$. They are commutative since the presence of an item in\nthe removals set  prevents re-adding that element later;\nBoth $[\\add,a]$ followed by $[\\rmv,a]$, and $[\\rmv,a]$ followed by $[\\add,a]$\nwill have the same effect, i.e., $a$ not being present.\nTherefore --- contrary to what it looks like at a glance --- a \\twopset is a commutative CRDT.\n\nFigure~\\ref{fig:2pset} depicts the pure CRDT design of the \\twopset. The state\nis designed as two power sets $s,p\\in \\pow{V}$, initially empty, where $s$ retains \nthe added values and $p$ retains the removed ones. A $\\prepare$ can either be \n$\\add$ or $\\rmv$ together with a parameter $v$ and without additional meta-data\n(i.e., making the CRDT pure). When item $v$ is added, the $\\effect$\nunions $v$ to set $s$ only of it is not in the set of removals $p$ \n(i.e., $s \\union (\\{v\\} \\setminus p)$), which remains intact in this case. \nIf $v$ is to be removed via $\\rmv$, it is subtracted from $s$ and \nunioned to $p$. Notice that, set $s$ can grow and shrink in \nsize depending on the issued operations whereas $p$ can only grow in size as long \nas $\\rmv$ operations are issued. This imposes a space overhead, but ensures that removed\nitems are never added again. Finally, the $\\eval$ function invokes\nthe read operations $\\elems$ and $\\size$ to return all the elements in $s$ or \nits cardinality, respectively.\n\n\\begin{figure}[t]\n\\begin{subfigure}[b]{.50\\textwidth}\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\pow{V} &&\n\\sigma_i^0 = \\{\\} \\\\\n\\prepare_i([\\add, v], s) & = & [\\add, v] \\\\\n\\effect_i([\\add, v], s) & = & s \\union \\{v\\} \\\\\n\\eval_i(\\elems, s) & = & s\\\\\n\\eval_i(\\size, s) & = & \\setsize{s}\\\\\n\\end{eqnarray*}\n\\caption{GSet}\n\\label{fig:gset}\n\\end{subfigure}\n\\begin{subfigure}[b]{.50\\textwidth}\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\pow{V}\\times \\pow{V} &&\n  \\sigma_i^0 = (\\{\\},\\{\\}) \\\\\n\\prepare_i([\\add, v], \\sigma) & = & [\\add, v] \\\\\n\\prepare_i([\\rmv, v], \\sigma) & = & [\\rmv, v] \\\\\n \n  \\effect_i([\\add, v], (s,p)) & = & (s \\union (\\{v\\} \\setminus p),p) \\\\\n  \\effect_i([\\rmv, v], (s,p)) & = & (s\\setminus \\{v\\},p \\union \\{v\\}) \\\\\n  \\eval_i(\\elems, (s, p)) & = & s \\\\\n  \\eval_i(\\size, (s,p)) & = & \\setsize{s}\\\\\n\\end{eqnarray*}\n\\caption{2PSet}\n\\label{fig:2pset}\n\\end{subfigure}\n\\caption{Pure Grow-only Set and Two-Phase Set CRDTs.}\n\\label{fig:pure-set}\n\\end{figure}\n\n\\subsection{Non-Commutative Data Types}\nThese CRDTs accept operations that are not commutative. \nThe challenge is\nthen to design the $\\effect$ function to be commutative.\nThe design of such data types follows the framework presented in \nFigure~\\ref{fig:full-compaction} and supports using the \\polog as well as the\ncompaction mechanisms explained in the preceding sections. Consequently, the\npresented pure CRDTs have a common design for the state $\\Sigma$, \n$\\prepare$, $\\effect$, and $\\stable$ functions. \nAlthough the other functions like $\\R$, $\\stabilize$, and $\\eval$ are datatype-specific,\nthey are often fairly simple as shown next.\nAs explained earlier, non-commutative pure CRDTs assume the existence of an  \nTagged Causal Stable Broadcast middleware (TCSB), as presented in Section~\\ref{sec:ercb}.\nThe interplay between the TCSB and the pure CRDTs is described in Algorithm~\\ref{algo:ercb}.\n\nSince the semantics of non-commutative CRDTs is more complex than\ncommutative ones, being concurrent, we formally define their concurrent semantics assuming\nthat all operations are retained in a \\polog, where the output of the query operations\ndefines its semantics. This makes the definition very solid and stands as a base-line to verify \nfuture amendments across; i.e., the output of defined queries for a datatype must not change in \nany new design or optimization (which is true in all the designs we present).\n\n\\subsubsection{Multi-Value Register (\\mvreg)}\nContrary to a classical register that holds a single value at a time, \na \\mvreg can hold multiple values if their corresponding $\\af{write}$ \noperations are concurrent;\nthus letting the application choose the desired values returned.\nOne motivation for \\mvreg was to mimic the design of the Amazon Shopping Cart in\nDynamo DB~\\cite{app:rep:optim:1606}. Since the order of two $\\af{write}$ operations\non the \\mvreg can lead to different results if one of them is in the causal \nfuture of the other, the datatype is non commutative. \n\nThe concurrent\nsemantics of \\mvreg is depicted in Figure~\\ref{fig:mvreg-semantics} and says that a read $\\af{rd}$\nshould return all write $\\af{wr}$ operations that are not causally succeeded by any other \noperation (i.e., neither $\\af{wr}$ nor $\\clear$). Consequently,\nthe first clause in the Pure CRDT design, in Figure~\\ref{fig:mvreg-compact}, \nsays that $\\clear$ operations must not be added to the \\polog. The second \nclause then means that $\\af{wr}$ operations are removed from the \\polog if there\nis any $\\af{wr}$ or $\\clear$ operation in the causal future; otherwise, \nconcurrent $\\af{wr}$ operations will be kept. The $\\af{rd}$ operation, \ntherefore, simply returns all available $\\af{wr}$ operations in the \\polog.\nFinally, $\\stabilize$ has no effect on the \\polog, and thus the calling function\n$\\stable$ will simply replace causally stable timestamps with $\\bot$.\n\n\\begin{figure}[t]\n\t\\begin{eqnarray*}\n  \\eval_i(\\af{rd}, s) & = &\n  \\{ v | (t, [\\af{wr}, v]) \\in s \\mathrel{\\wedge} \\forall (t', \\_) \\in s \\cdot t \\not < t' \\}\n\\end{eqnarray*}\n\\caption{\\mvreg concurrent semantics}\n\\label{fig:mvreg-semantics}\n\\end{figure}\n\\begin{figure}[t\n\\begin{eqnarray*}\n\t(t,o) \\R s &\\iff& o[0] = \\clear \\\\\n  (t', o') \\R_\\_ (t, o) &\\iff& t' < t \\\\\n  \\stabilize_i(t,s) &=& s \\\\\n  \\eval_i(\\af{rd}, s) & = & \\{ v | (\\_, [\\af{wr}, v]) \\in s \\}\\\\\n\\end{eqnarray*}\n\\caption{Compact \\mvreg CRDT}\n\\label{fig:mvreg-compact}\n\\end{figure}\n\n\\subsubsection{Sets (\\awset and \\rwset)}\nA set is a very well known data structure that retains unordered items, being \nadded and removed through $\\add$ and $\\rmv$ operations, respectively. \nObviously, these operations are not commutative and thus reordering them\nwill yield different effects, making the datatype non commutative.\nSpecial consideration must be given when\nthese operations are concurrent. Typically, the designers explicitly define\nextra rules to make the behavior consistent across different replicas, thus\nleading to two variants: Add-Wins Set (\\awset) and Remove-Wins Set (\\rwset).\n\nIn \\awset, an $\\add$ operation dominates a concurrent $\\rmv$\nthat is mutating the same item in the set. \nThis is explained formally in the concurrent semantics in\nFigure~\\ref{fig:awset-semantics}: a $\\elems$ operation will return all the values\nassociated with $\\add$ such that no corresponding $\\rmv$ operation\nor $\\clear$ operation are found in the causal future of the $\\add$. \nConsequently, the first clause in the Pure CRDT design in \nFigure~\\ref{fig:awset-compact} specifies that newly delivered \n$\\rmv$ and $\\clear$ operations are never added to the \\polog; \nwhich leaves the \\polog with only $\\add$ operations.\nThe second clause however says that an existing $\\add$ in the \\polog is\nconsidered redundant if there is another $\\add$ or $\\rmv$ operations on \nthe same item (specified by $o[1]$ in Figure~\\ref{fig:awset-compact}) \nor a $\\clear$ operation such that they are in the \ncausal future of the candidate $\\add$. Therefore, the $\\elems$ operation\nwill simply return the items associated with an $\\add$ in the \\polog, since\nneither duplicate $\\add$s nor $\\rmv$ and $\\clear$ operations exist in the \\polog.\nFor the same reasons as previous data types, $\\stabilize$ has no effect and\nthe calling $\\stable$ function will only replace the causally stable timestamps \nwith $\\bot$.\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n  \\eval_i(\\elems, s) & = &\n  \\{ v | (t, [\\add, v]) \\in s \\\\&&\\mathrel{\\wedge} \n    \\forall (t', [\\rmv, v] \\mathrel{\\vee} \\clear) \\in s \\cdot t \\not < t' \n\\}\n\\end{eqnarray*}\n\\caption{\\awset concurrent semantics}\n\\label{fig:awset-semantics}\n\\end{figure}\n\\begin{figure}[t\n\\begin{eqnarray*}\n\t(t,o) \\R s &=& o[0] = (\\clear \\mathrel{\\vee} \\rmv) \\\\\n\t(t',o') \\R_\\_ (t,o) &=& t' < t \\mathrel{\\wedge} (o[0] = \\clear \\\\\n\t&&\\mathrel{\\vee} o[1] = o'[1]) \\\\\n  \\stabilize_i(t,s) &=& s \\\\\n  \\eval_i(\\elems, s) & = & \\{ v | (\\_, [\\add,v]) \\in s \\}\\\\\n\\end{eqnarray*}\n\\caption{Compact \\awset Pure CRDT}\n\\label{fig:awset-compact}\n\\end{figure}\n\nOn the other hand, in \\rwset, a $\\rmv$ operation dominates a \nconcurrent $\\add$ that is mutating the same item in the set, thus masking its\neffect. Therefore, the formal concurrent semantics in Figure~\\ref{fig:rwset-semantics}\nsays that a $\\elems$ operation on a \\polog returns all values associated\nwith an $\\add$ such that all $\\rmv$ operations are in its causal past, and\n$\\clear$ operations are not in the causal future. Notice that, contrary to \n\\awset, concurrent $\\add$ operations with a $\\rmv$ will be discarded (reflecting\nthe remove-wins semantics). The corresponding Pure CRDT design of \\rwset is\npresented in Figure~\\ref{fig:rwset-compact}. In particular, the first clause\nprevents any $\\clear$ operation from being added to \\polog. The second clause\nspecifies that a candidate operation on item $v$ is only removed if there is another\noperation on $v$ or a $\\clear$ in its causal future. This actually means that\n$\\rmv$ operations can still live in the \\polog, to the contrary of \\awset.\nIn fact, $\\rmv$ cannot be removed since a concurrent $\\add$ on the same\nitem may be delivered which breaks the remove-wins semantics. Even more,\nan $\\add$ operation $a$ cannot be removed despite the presence of concurrent \n$\\rmv$ operation $r$ (as one might think). \nThe reason is that $r$ can become stable at any time; if it happened to be followed\nby a $\\clear$ operation, i.e., in its causal future, all stable operations in \nthe \\polog will be deleted, $r$ among them. Since $a$ is not yet stable, and thus \nconcurrent with $\\clear$, the former will stay in the \\polog. This breaks the \nremove-wins semantics since $r$ and $a$ were originally concurrent and hence \n$r$ should win. These reasons lead to a more complex\n$\\stabilize$ function (explained next) that is in charge of removing the extra \noperations that were not safe to be removed previously through $\\R$ or $\\R_\\_$.\n\n\n\n\\def\\hspace{1.5pt}{\\hspace{1.5pt}}\n\\def\\s#1{\\setlength{\\fboxsep}{1.2pt}\\hspace{1.5pt}\\fbox{\\smash{#1}\\vphantom{x}}\\hspace{1.5pt}}\n\\def\\n#1{\\dot{\\s{#1}}}\n  \\def\\c#1{\\cancel{\\smash{#1}\\vphantom{x}}}\n\nWe explain the $\\stabilize$ function, depicted in Figure~\\ref{fig:rwset-compact},\nwith the help of a simulation of the \\polog state transitions upon stabilization in \nFigure~\\ref{fig:rwset-stabilization}. The $\\stabilize$ function tries to remove\nredundant operations once invoked by $\\stable$ when a timestamp $t$ becomes stable.\nThe removed operations are defined in three clauses, i.e., those removed from the \n\\polog using the \\emph{set minus} operator in Figure~\\ref{fig:rwset-compact}.\nThe first clause specifies that, once $t$ becomes stable, the associated $\\add$\noperations are removed from the \\polog as long as there is at least another \noperation in the \\polog with a different timestamp $t'\\neq t$ (otherwise the latter\noperation will also be stable). This clause corresponds to the first four cases in\nFigure~\\ref{fig:rwset-stabilization}: once an $\\add$ operation $a$ becomes \nstable, the existence\nof other $\\add$s will make it redundant, as they add the same item to the set; $a$ is\nalso removed if there exist any other $\\rmv$ operation $r$ on the same item since,\naccording to the remove-wins semantics,\n$r$ wins the concurrent $a$ which has just became stable, and thus, it is not expected \nto have another concurrent $\\add$ delivered (by definition of causal stability).\nThe second clause in $\\stabilize$ (Figure~\\ref{fig:rwset-compact}) deletes \nthe stable $\\rmv$ operations as long as the potential non stable operations on \nthe same item are not uniquely $\\add$ operations. This is explained in the simulation\nin Figure~\\ref{fig:rwset-stabilization} in cases (5) through (8): cases (7) and (8)\nare trivial since a $\\rmv$ is simply redundant by duplicate $\\rmv$s; case (5)\nremoves a stable $\\rmv$ if no other operation exist. The rational is that the stable\n$\\rmv$ being stable,\nno concurrent operations are expected, meaning that any delivered operations will be \nin its causal future, and thus, making stable $\\rmv$ useless. Case (6) does not\nlead to any removals. Indeed, if the stable $\\rmv$ operation $r$ is removed,\nany existing $\\add$ operation on the same item $a$ will win, which contradicts with\nthe remove-wins semantics since $r$ and $a$ are originally concurrent. This practically\nmeans the stable $\\rmv$ will be kept in the \\polog for a later stage, thus forking\nthe cases (9) and (10). When an $\\add$ operation becomes stable, i.e., all $\\add$ and\n$\\rmv$ operations on the same item are not stable, they can both be removed from the\n\\polog since $\\rmv$ wins, and it is not expected to have any concurrent operation\ndelivered any more. To the contrary, if there is at least another $\\add$ operation,\nthe stable $\\add$ will be removed while the stable $\\rmv$ will be kept for similar\nreasons as in point (6) explained above.\n\n\\begin{figure}[t]\n\\begin{eqnarray*}\n  \\eval_i(\\elems, s) & = &\n  \\{ v | (t, [\\add, v]) \\in s \\\\&&\\mathrel{\\wedge} \n  \\forall (t', [\\rmv, v]) \\in s \\cdot t'< t \\\\&& \\mathrel{\\wedge}\n  \\forall (t'', \\clear) \\in s \\cdot t \\not < t'' \n  \\}\n\\end{eqnarray*}\n\\caption{\\rwset concurrent semantics}\n\\label{fig:rwset-semantics}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{eqnarray*}\n\t(t,o) \\R s &=& o[0] = \\clear \\\\\n\t(t',o') \\R_\\_ (t,o) &=& t' < t \\mathrel{\\wedge} (o[0] = \\clear \\mathrel{\\vee} o[1] = o'[1])\\\\\n  \\stabilize_i(t,s)\n \n  &=& s \\setminus \\\\&& \\{(t,[\\add,e]) \\in s |\\exists (t',[\\_,e])\\in s \\cdot t\\neq t'\\} \\setminus \\\\\n \n  &&  \\{(t,[\\rmv,e])  \\in s | \\\\\n  &&  \\hphantom{s} \\{ op | (t', [op, e]) \\in s \\mathrel{\\wedge} t' \\neq t\\} \\neq\n  \\{ \\add \\} \\} \\setminus \\\\\n \n  && \\{(\\bot,[\\rmv,e]) \\in s| \\forall (t',[\\add,e]) \\in s \\cdot t' = t \\} \\\\\n  \\eval_i(\\elems, s) &=& \\{ v | (\\_, [\\add,v]) \\in s \\mathrel{\\wedge}\n                                 (\\_, [\\rmv,v]) \\not\\in s \\} \\\\\n\\end{eqnarray*}\n\\caption{Compact Pure \\rwset CRDT}\n\\label{fig:rwset-compact}\n\\end{figure}\n\nAfter invoking $\\stabilize$ and removing all redundant operations, the $\\stable$ function \nwill replace the stable timestamps with $\\bot$. Finally, the $\\elems$ operation will\nreturn all the items that are associated with $\\add$ if there is no corresponding $\\rmv$\nin the \\polog.\n\n\n\\begin{figure}[t]\n  \\[\n\\begin{array}{l@{\\quad\\rightarrow\\qquad}l}\n (1)\\quad \\n a & \\n a \\\\\n (2)\\quad \\n a a^+ & \\c{\\n a} a^+ \\\\\n (3) \\quad \\n a r^+ & \\c{\\n a} r^+ \\\\\n  (4) \\quad \\n a a^+ r^+ & \\c{\\n a} a^+ r^+ \\\\\n  (5)\\quad \\n r & \\c{\\n r} \\\\\n  (6)\\quad \\n r a^+ & \\n r a^+ \\\\\n  (7)\\quad \\n r r^+ & \\c{\\n r} r^+ \\\\\n  (8)\\quad \\n r a^+ r^+ & \\c{\\n r} a^+ r^+ \\\\\n  (9)\\quad \\s r \\n a & \\c{\\s r} \\c{\\n a} \\\\\n  (10)\\ \\ \\s r \\n a a^+ & \\s r \\c{\\n a} a^+ \\\\\n\\end{array}\n  \\]\n\\caption{Simulation of the possible transitions at stabilization in \\rwset.\nThe operations $\\add$ and $\\rmv$, i.e., $r$ and $a$ respectively, \nare assumed to mutate the same item. \nA boxed operation is stable. A dotted operation refers to the operation that has\njust became stable. The sign + is used to indicate the presence of duplicate \noperations.\nThe canceled boxed operations on the right side are the redundant operations \ndue to causal stabilization.\n}\n\\label{fig:rwset-stabilization}\n\\end{figure}\n\n\n\\section{Pure Op-based CRDTs}\n\\label{sec:pure}\n\nIn this section, we introduce \\emph{pure op-based CRDTs}. We show that\ndata types whose operations are commutative are natively ``pure''; whereas\nfor other data types to be pure, they need to leverage some information \nchanneled through the dissemination middleware.\n\n\\begin{definition}[Pure op-based CRDT]\nAn op-based CRDT is \\emph{pure} if disseminated messages contain only \nthe operation and its potential arguments. \n\\end{definition}\n\nTherefore, considering the general op-based CRDT scheme introduced \nin Figure~\\ref{fig:op-crdt}, and given an operation $o$ and a state $\\sigma$,\n$\\prepare$ is always defined as:\n\\[\n\\prepare(o, \\sigma) = o\n\\]\n\nIn addition to the generic nature of this property across different data types,\nat least two other benefits can be identified. \nFirst, $\\prepare$ cannot build an arbitrary \nmessage depending on the current state which avoids introducing design or \nimplementation bugs. Second, the operation can immediately\nbe broadcast without even reading the replica state $\\sigma$. \n(Indeed, the parameter $\\sigma$ of $\\prepare(o, \\sigma)$ can be omitted in\npure op-based CRDTs, but we keep it for consistency and clarity.). This has\ngood performance implications on the sender side as the access to persistent \nstorage and serialization\/deserialization are avoided. \n\nBack to the data type examples described in Figure~\\ref{fig:op-based}, we notice\nthat the counter data type is pure op-based since $\\prepare=\\inc$.\nOn the contrary, the Add-Wins Set in Figure~\\ref{fig:op-set} is not pure \nsince a $\\prepare$ builds a set of triples present in the current state, \nto be used by other replicas when performing $\\effect$ (either for $\\add$ \nor for $\\rmv$). To understand this difference, we make a distinction between\ntwo categories: commutative and non-commutative data types.\n\n\n\\begin{definition}[Commutative data type]\nA concurrent data type is commutative iff: \n\\begin{enumerate}\n\t\\item for any operations $f$ and $g$, their sequential \n\t\tinvocation commutes: $f(g(\\sigma))=g(f(\\sigma))$, and\n\t\\item all concurrent invocations are defined as equivalent to some linearization.\n\\end{enumerate}\n\\end{definition}\n\nWhile the first property removes the ordering overhead of operations (as no \nordering assumptions are required), the second property makes the pure op-based \ndesign of a commutative data type straightforward as we explain next.\n\n\\subsection{Pure designs for commutative data types}\nCommutative data types can natively be designed as pure op-based CRDTs on top\nof a standard reliable causal broadcast~\\cite{causal-multicast:Birman:1991}.\nThe reason is that commutative data types reflect a \\emph{principle of permutation\nequivalence}~\\cite{rep:opt:sh150}: if all sequential\npermutations of updates lead to equivalent states, then it should also hold\nthat concurrent executions of the updates lead to equivalent states.\nAs the extension to concurrent scenarios follows directly from their\nsequential definition (with no room for design choices), commutative data types\ncan have a standard sequential specification and design. Consequently,\na pure op-based CRDT design becomes trivial: since a message returned from\n$\\prepare$, and containing the operation name and arguments, will arrive \nexactly once at each replica (through the reliable causal broadcast), \nit is enough to have $\\effect$ only apply the received operation to the standard \nsequential data type state, i.e., defining for any operation $o$:\n\\[\n\\effect(o, \\sigma) = o(\\sigma)\n\\]\n\n\\emph{Examples on commutative data types:} Two examples are presented in\nFigure~\\ref{fig:pure-commutative}: a PN-Counter with $\\inc$ and $\\dec$\noperations, and a Grow-only Set (G-Set) with $\\add$ operation. Both cases use a\nstandard sequential data type for the replica state $\\Sigma$, \nand applying $\\effect$ is just invoking the corresponding standard \noperation ($+,-,\\cup$, etc.) to \nthe sequential data type. Both examples exploit commutativity \nand rely on the exactly-once delivery, leading to a trivial pure design. \n\n\\begin{figure}[t]\n\\begin{subfigure}[b]{.47\\textwidth}\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\mathds{N} &&\n\\sigma_i^0 = 0 \\\\\n\\prepare_i(o, \\sigma_i) & = & o \\text{\\quad($\\inc$ or $\\dec$)} \\\\\n\\effect_i(\\inc, n ) & = & n+1 \\\\\n\\effect_i(\\dec, n ) & = & n-1 \\\\\n\\eval_i(\\af{value},n ) & = & n\\\\\n\\end{eqnarray*}\n\\caption{Pure PN-Counter}\n\\label{fig:oppnctr}\n\\end{subfigure}\n\\begin{subfigure}[b]{.47\\textwidth}\n\\centering\n\\begin{eqnarray*}\n\\Sigma = \\pow{V} &&\n\\sigma_i^0 = \\{\\} \\\\\n\\prepare_i(o, \\sigma_i) & = & [\\add, v] \\\\\n\\effect_i([\\add, v], s) & = & s \\union \\{v\\} \\\\\n\\eval_i(\\elems, s) & = & s\\\\\n\\end{eqnarray*}\n\\caption{Pure Grow-Only Set}\n\\label{fig:opgset}\n\\end{subfigure}\n\\caption{Pure op-based CRDTs for commutative data types.}\n\\label{fig:pure-commutative}\n\\end{figure}\n\n\\subsection{The challenge of non-commutative data types}\n\\label{sec:challenges}\nIn the case where the operations of a data type are not commutative, \ne.g., $\\add$ and $\\rmv$ in a set ($\\add(v, \\rmv(v, s)) \\neq \\rmv(v, \\add(v,\ns))$), an $\\effect$ cannot simply apply the operations over a sequential \ndata type as done in the previous section: $o(\\sigma)$. This is due to\ntwo reasons, that we explain below, on which we base our extended model.\n\nThe first reason is that the messages corresponding to concurrent \noperations can be delivered in different orders on different \nreplicas. Since the operations do not commute (even if the \nsemantics of concurrent invocations can be defined as equivalent to some \nlinearization of those operations), simply applying them in different \norders on different replicas makes these replicas diverge.\nTherefore, \nconvergence is only guaranteed if $\\effect$ is commutative, and therefore, cannot be\ndefined directly as operations that are not commutative themselves (e.g., simple\n$\\add$ and $\\remove$ operations).\nTo address this point, $\\effect$ must have partial-ordering knowledge about operations\nand a corresponding logic in execution. While the latter suggests changing the\ndesign of $\\effect$, the former may require exchanging some ordering meta-data.\nHowever, since such information\nabout the partial order is already present in the meta-data of the causal delivery\nmiddleware, we propose to extend the API of the middleware (presented in \nSection~\\ref{sec:ercb}), which we call \n\\emph{Tagged Causal Stable Broadcast} (TCSB), to leverage this meta-data that can then be \nused in designing general non-commutative pure op-based CRDTs.\n\nThe other reason is that, indeed, there are useful concurrent data types in which\nthe outcomes of concurrent executions are not equivalent (on purpose) to some \nlinearization. The best example is the Multi-Value Register (MVRegister) that is used in \nAmazon's Dynamo system to model the shopping cart~\\cite{app:rep:optim:1606}. In MVRegister, invoking\na read operation following two concurrent writes in its causal past would return a \nset with \\emph{both} values written. This behavior does not arise under a sequential\nspecification of a register, and the state must retain concurrent operations.\nWe address this challenge by proposing (in Section~\\ref{sec:polog}) the type \nof a CRDT (i.e., $\\Sigma$) \nto be a partially-ordered log of operations provided that $\\effect$ understands \nhow to execute these operations, thanks to the TCSB meta-data.\n\n\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\emph{Causal broadcast}\nThe early definitions of \\emph{causality} were introduced by Lamport\nin~\\cite{con:rep:615}. Schiper et al. introduced an algorithm\nin~\\cite{Schiper:1989:NAI:645946.675010} to implement \\emph{causal ordering}\nbased on message delivery. The authors distinguished it from Lamports\ncausality definitions which they called \\emph{causal timestamping}. Then,\ncausal broadcast has been explored in the context of process group\ncommunication in~\\cite{pro:pan:160} to ensure delivery orders that do not\ncontradict potential causality \\cite{con:rep:615}. Implementations can\npiggyback to messages their causally preceding messages or tag it with a\nvector clock \\cite{Schmuck88theuse,alg:rep:738,alg:rep:738bis} and ensure that\npreceding messages are delivered first. \n\n\\emph{Weakly Consistent Replication.}\nThe design of replicated systems that are always available and eventually\nconverge can be traced back to historical designs\nin~\\cite{app:rep:optim:1501,db:rep:optim:1454,usenet:1986:NCN:6617.6618},\namong others.\nLazy Replication~\\cite{Ladin:1992:PHA:138873.138877} allows enforcing causal\nconsistency, but may apply concurrent operations in different orders in\ndifferent replicas, possibly leading to divergence if operations are not\ncommutative; TSAE~\\cite{pan:pro:optim:813} also either applies concurrent\noperations in possibly different orders, or allows enforcing a total order\ncompatible with causality, at the cost of delaying message delivery. Both\nthese systems use a message log, the former with complete causality\ninformation, but the log is \\emph{pre-delivery}, unseen by the application:\noperations are applied sequentially to the current state and queries use only\nthe state. In our framework the \\polog is \\emph{post-delivery}, being part of\nthe data type state, maintains causality information and is used in query\noperations.\n\n\\emph{Conflict-Free Replicated Data Types.}\nThe formalization of the commutativity requirements for concurrent operations\nin replicated data types was introduced in~\\cite{alg:rep:sh132,syn:app:1649},\nand that of the state based semi-lattices was presented in~\\cite{app:1639}.\nAfterwards, the integration of the two models with many extensions was\npresented in Conflict-free Replicated\nDatatypes~\\cite{rep:syn:sh138,syn:rep:sh143}. Currently, CRDTs have made their\nway into the industry through designing highly available scalable systems in\ncloud databases like RIAK~\\cite{riak+crdt}, and mobile gaming industry such as\nRovio~\\cite{rovio:2013}. \n\n\n\\emph{Message Stability.}\nThe notion of message stability was defined\nin~\\cite{Birman:1991:LCA:128738.128742} to represent a message that has been\nreceived by all recipients; each replica can discard any message it knows to\nbe stable after delivering it. Similar notions are used in Lazy\nReplication~\\cite{Ladin:1992:PHA:138873.138877} and\nTSAE~\\cite{pan:pro:optim:813}. In all these cases the aim is message\ngarbage collection. Our definition of \\emph{causal stability} is the stronger\nnotion that \\emph{no more concurrent messages will be delivered}; we\nuse it inside the Datatypes to discard causality information while keeping\nthe operation. Causal stability is close to what is used in the mechanics of\nReplicated Growable Arrays (RGA)~\\cite{syn:app:1649}, although no definition\nis presented there.\n\n\n\n\\emph{Message Obsolescence.}\nSemantically reliable multicast~\\cite{Pereira:2003:SRM:642778.642783} uses the\nconcept of \\emph{message obsolescence} to avoid delivering messages made\nredundant by some newly arrived message, where obsolescence is a strict\npartial order that is a subset of causality, possibly relating messages from the\nsame sender or totally ordered messages from different senders. Our\nobsolescence relation is more general, being defined on clock-operation pairs,\nand can relate concurrent messages. Also, it is defined per-data-type, being\nused inside each data type, post-delivery.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Properties of the operators $\\{\\Pi_{\\mu}\\}$} \nHere we show that the operators in Eq.~\\eqref{eq:pilambda} are mutually orthogonal projectors that add up to identity. To this end we use the \\emph{grand orthogonality theorem} (a consequence of Schur's lemma) of representation theory.\n\\begin{theorem}[Grand orthogonality theorem] (See \\cite[Sec. 2.2.]{serre1977linear})\n    Given irreducible representations $\\rho_\\mu$ and $\\rho_\\nu$, we have that: \n    \\begin{align}\n        \\sum_{g \\in S_n} \\rho_\\mu(g)^\\dag_{ij} \\rho_\\nu(g)_{k\\ell} &= \\frac{n!}{d_\\mu} \\delta_{ik} \\delta_{j \\ell} \\delta_{\\mu \\nu}. \n    \\end{align} \n\\label{thm:got}\n\\end{theorem}\nUsing Theorem \\ref{thm:got} we obtain the following identity for any fixed $h \\in S_n$.\n\\begin{align*}\n    \\sum_{g \\in S_n} \\chi_\\nu(g) \\chi_\\mu(g^{-1} h) &= \\sum_{g \\in S_n} \\sum_{i,j} \\rho_\\nu(g)_{ii} \\rho_\\mu(g^{-1}h)_{jj} \\\\ &= \\sum_{g \\in S_n} \\sum_{ijk}\\rho_\\nu(g)_{ii} \\rho_\\mu(g^{-1})_{jk} \\rho_\\mu(h)_{kj} \\\\ &=  \\sum_{ijk}  \\left( \\sum_{g \\in S_n} \\rho_\\nu(g)_{ii} \\rho_\\mu(g)^\\dag_{jk} \\right) \\rho_\\mu(h)_{kj} \\\\ \n    &= \\delta_{\\mu \\nu} \\sum_{ijk} \\delta_{ij} \\delta_{ik} \\frac{n!}{d_\\nu} \\rho_\\mu(h)_{kj} = \\delta_{\\mu \\nu} \\sum_i \\frac{n!}{d_\\nu} \\rho_\\mu(h)_{ii} = \\delta_{\\mu \\nu} \\frac{n!}{d_\\nu} \\chi_\\mu(h).\n    \\end{align*}\nIt follows that $\\{\\Pi_{\\mu}\\}$ are mutually orthogonal projectors. \n\\begin{align}\n    \\Pi_\\nu \\Pi_\\mu &= \\frac{d_\\mu d_\\nu}{(n!)^2} \\sum_{gh} \\chi_\\nu(g) \\chi_\\mu(h) \\rho(gh) = \\frac{d_\\mu d_\\nu}{(n!)^2} \\sum_{g \\ell} \\chi_\\nu(g) \\chi_\\mu(g^{-1} \\ell) \\rho(\\ell) = \\delta_{\\mu \\nu} \\frac{d_\\mu}{n!} \\sum_\\ell \\chi_\\mu(\\ell) \\rho(\\ell) = \\delta_{\\mu \\nu} \\Pi_\\mu.\n\\end{align}\nUsing the fact that $d_\\mu = \\chi_\\mu(e)$, where $e \\in S_n$ is the identity, we obtain\n\\begin{align}\n    \\sum_{\\mu \\vdash n} \\Pi_\\mu &= \\sum_{g \\in S_n} \\frac{1}{n!} \\rho(g) \\sum_{\\mu \\vdash n} d_\\mu \\chi_\\mu(g) = \\sum_{g \\in S_n} \\frac{1}{n!} \\rho(g) \\sum_{\\mu \\vdash n} \\chi_\\mu(e) \\chi_\\mu(g).\n\\label{eq:sumpi}\n\\end{align}\n\nThe column-orthogonality of characters (stated below) implies that\n\\begin{align}\n    \\sum_\\mu \\chi_\\mu(e) \\chi_\\mu(g) &= n! \\delta_{eg},\n\\end{align}\nand substituting this into Eq.~\\eqref{eq:sumpi} gives $\\sum_\\mu \\Pi_\\mu = I $.\n\n\n\n\\begin{lemma}[Column-orthogonality of characters] (See \\cite[Thm 16.4]{alperin2012groups})\nFor any $h,g\\in S_n$ we have\n    \\begin{align}\n    \\sum_{\\mu \\vdash n} \\chi_\\mu(h) \\chi_\\mu(g) &= \\begin{cases}\n    |Z(g)|  &\\text{ if $h$ and $g$ are conjugate.}\\\\ \n    0 & \\text{ otherwise. }\n    \\end{cases}\n\\end{align}\nwhere $Z(g)$ is the centralizer of $g$ in $S_n$ (the set of all group elements that commute with $g$).\n\\end{lemma}\n\n\n\n\\section{Beals' quantum Fourier transform over the symmetric group}\nFor completeness, here we review Beals' efficient algorithm for the quantum Fourier transform over the symmetric group \\cite{beals1997quantum} (see also \\cite{Moore03} for generalizations to other nonabelian groups).\n\n  Let $n\\geq 2$ be an integer and consider the symmetric group $S_n$ of permutations on $n$ objects. Also let $T=\\{\\tau_1,\\tau_2,\\ldots, \\tau_n\\}$ be a transversal of the left cosets of $S_{n-1}$ in $S_n$. Here $S_{n-1}$ is regarded as the subgroup of $S_n$ that does not permute the $n$-th object and transversality means that $S_n$ is a union of cosets $\\tau_1S_{n-1},\\ldots,\\tau_n S_{n-1}$.\n\nIt will be convenient to fix some notation and conventions concerning the function whose Fourier transform we are interested in. We shall fix a unitary matrix representation of each irreducible representation of $S_n$. In particular, we choose the so-called Young Orthogonal representation (more on this below) and write $\\rho_{\\lambda}(\\pi)$ for the matrix corresponding to the irreducible representation labeled by $\\lambda \\vdash n$ evaluated at $\\pi\\in S_n$, and $d_{\\lambda}$ for its dimension. \n\nFor a function $f:S_n \\rightarrow \\mathbb{C}$ we have\n \\begin{equation}\n \\mathrm{QFT}_n\\sum_{\\sigma\\in S_n} f(\\sigma)|\\sigma\\rangle=\\sum_{\\omega \\vdash n} \\sum_{i,j \\in d_\\omega} (\\hat{f}(\\omega))_{ij}  |\\omega, i,j\\rangle, \n\\label{eq:ft11}\n\\end{equation}\nwhere\n\\begin{equation}\n \\hat{f}(\\omega)=\\sqrt{\\frac{d_{\\omega}}{n!}} \\sum_{\\sigma\\in S_n} f(\\sigma)\\rho_\\omega(\\sigma).\n\\label{eq:ft22}\n \\end{equation}\nFor ease of notation in this section we include the normalization factor $\\sqrt{\\frac{d_{\\omega}}{n!}}$ in Eq.~(\\ref{eq:ft22}) rather than in Eq.~\\eqref{eq:ft11} which differs from our convention in the main text (Cf. Eqs.~(\\ref{eq:ft1},\\ref{eq:ft2})). However the quantum fourier transform $\\mathrm{QFT}_n$ is defined in exactly the same way.\n\nNote that the basis vectors in the Fourier basis can be labeled by triples from the set $\\Omega=\\{(\\omega, i,j): \\omega\\vdash n, 1\\leq i,j\\leq d_{\\omega}\\}$, since $|\\Omega|=\\sum_{\\omega\\vdash n} d_{\\omega}^2 =|S_n|$.\nSince the left cosets of $S_{n-1}$ partition $S_n$, we may write\n\\begin{equation}\nf(g)=\\sum_{j=1}^{n} F^{j}(g) \\qquad \\qquad F^{j}(g)\\equiv \\begin{cases} f(g) &, \\text{if }  g\\in \\tau_j S_{n-1}\\\\ 0 &, \\text{otherwise} \\end{cases}.\n\\label{eq:defF}\n\\end{equation}\nAlso define $f_j:S_{n-1}\\rightarrow \\mathbb{C}$ for $1\\leq j\\leq n$ by \n\\begin{equation}\nf_j(h)\\equiv f(\\tau_j h) \\qquad \\quad h\\in S_{n-1}.\n\\label{eq:deff}\n\\end{equation}\n\nYoung's orthogonal representation has a certain \\textit{adapted} property that allows us to express Eq.~\\eqref{eq:ft22} in terms of Fourier transforms of the functions $f_j$ for $1\\leq j\\leq n$ in a simple way. In particular, \n\\begin{equation}\n\\hat{f}(\\lambda)=\\sum_{k=1}^{n} \\rho_{\\lambda}(\\tau_k) \\bigoplus_{\\lambda^{-}\\in \\Phi(\\lambda) } \\sqrt{\\frac{d_{\\lambda}}{n\\cdot d_{\\lambda^{-}}}}\\cdot \\hat{f_k} (\\lambda^{-}),\n\\label{eq:dirsum}\n\\end{equation}\nwhere \n\\[\n\\Phi(\\lambda)=\\{\\lambda^{-}\\vdash (n-1) : \\lambda^{-}\\leq \\lambda\\}\n\\]\nis the set of partitions of $n-1$ whose diagram differs from $\\lambda$ in exactly one box. Note that the Fourier transforms $\\hat{f_k}$ appearing on the right hand side of Eq.~\\eqref{eq:dirsum} are over the group $S_{n-1}$.\n\n\n\nTo describe Beals' QFT algorithm it will be convenient to first define some unitary transformations that are used in the construction. In the following we shall assume that we have enough qubits to encode certain classical strings as computational basis states.\n\n\n Let $W$ denote the unitary which implements a classical reversible circuit that, given a  permutation $\\pi\\in S_n$, computes its factorization:\n\\[\nW|\\pi\\rangle=|\\tau_j\\rangle|h\\rangle,\n\\]\nwhere $j\\in [n]$ and $h\\in S_{n-1}$\nsatisfy $\\pi = \\tau_j h$. Note that this factorization is unique.\n\nFor $\\pi \\in S_n$ let $M(\\pi)$ be a unitary such that\n\\[\nM(\\pi) \\sum_{\\lambda\\vdash n}\\sum_{p,q=1}^{d_{\\lambda}} (A_{\\lambda})_{p,q}|\\lambda,p,q\\rangle= \\sum_{\\lambda\\vdash n} \\sum_{p,q=1}^{d_{\\lambda} }(\\rho_{\\lambda}(\\pi)A_{\\lambda})_{p,q}|\\lambda,p,q\\rangle\n\\]\nHere $(A_{\\lambda})_{p,q}$ are complex coefficients. Note that this can be implemented via a controlled-$\\rho_{\\lambda}(\\pi)$ operation on the $p$ register (where the control is $\\lambda$).\n The unitary $M(\\pi)$ acts trivially outside the subspace spanned by basis vectors $\\{|\\lambda,p,q\\rangle: \\lambda \\vdash n, p,q\\in [d_{\\lambda}]\\}$\n\nFor each irreducible representation $\\sigma$ of $S_{n-1}$ there is a set of irreps $\\lambda$ of $S_n$ such that the restriction of $\\rho_{\\lambda}$ to $S_{n-1}$ contains $\\sigma$. These correspond to the partitions $\\lambda$ that differ from $\\sigma$ by adding one box. Moreover, each matrix element $p,q$ of $\\rho_{\\sigma}$  (where $p,q\\leq d_{\\sigma}$) is identified with a unique matrix element $p',q'$ of $\\rho_{\\lambda}$ (Here $p',q'$ depend on $p,q$ as well as $\\lambda$, though our notation suppresses these dependencies).\n\nLet $U$ be a unitary such that for any $\\sigma \\vdash (n-1)$ and $p,q\\leq d_{\\sigma}$ we have\n\\begin{equation}\nU|0\\rangle|\\sigma,p,q\\rangle=|0\\rangle\\sum_{\\lambda \\vdash n: \\sigma \\leq \\lambda} \\sqrt{\\frac{d_{\\lambda}}{n\\cdot d_{\\sigma}}}|\\lambda,p',q'\\rangle \n\\label{eq:U}\n\\end{equation}\nand such that \n\\begin{equation}\nU|\\tau_i\\rangle|\\sigma,p,q\\rangle=|\\tau_i\\rangle|\\sigma,p,q\\rangle  \\quad \\qquad i\\in [n].\n\\label{eq:pkinvariant}\n\\end{equation}\n It is possible to construct such a unitary because the states appearing on the RHS of Eq.~\\eqref{eq:U} for different values of $\\sigma,p,q$ are orthonormal. However the action of $U$ in the rest of the Hilbert space is constrained by unitarity. The following lemma describes a property of $U$ that follows from this constraint.\n\\begin{lemma}\nLet $P$ project onto the subspace spanned by basis states than encode partitions of $n-1$, that is, \n\\[\n\\mathrm{span}\\{|w\\rangle|\\sigma,p,q\\rangle: w\\in \\{0,\\tau_1,\\ldots, \\tau_n\\}, \\sigma \\vdash (n-1), 1\\leq p,q\\leq d_{\\sigma}\\}.\n\\]\nSuppose $g:S_n\\rightarrow \\mathbb{C}$ is any function such that $g(\\pi)=0$ for all $\\pi\\in S_{n-1}$ (regarded as the subgroup of $S_n$ that fixes $n$). Then $PU^{\\dagger}|0\\rangle|\\hat{g}\\rangle=0$.\n\\label{lem:orthog}\n\\end{lemma}\n\\begin{proof}\nBelow we write $P=\\sum_{k=0}^{n}P_k$ where $P_0=\\mathrm{span}\\{|0\\rangle|\\sigma,p,q\\rangle: \\sigma \\vdash (n-1), 1\\leq p,q\\leq d_{\\sigma}\\}$ and $P_k=\\mathrm{span}\\{|\\tau_k\\rangle|\\sigma,p,q\\rangle: \\sigma \\vdash (n-1), 1\\leq p,q\\leq d_{\\sigma}\\}$ for $k\\in [n]$.\n\nFirst suppose that $\\hat{r}$ is the Fourier transform of some (arbitrary) function $r:S_{n-1}\\rightarrow \\mathbb{C}$. Define $R:S_n\\rightarrow \\mathbb{C}$ such that $R(\\pi)=r(\\pi)$ for all $\\pi\\in S_{n-1}$, and $R(\\pi)=0$ otherwise. Then from Eq.~\\eqref{eq:U} we have\n\\begin{equation}\n|0\\rangle\\sum_{\\sigma\\vdash n-1,p,q} (\\hat{r}(\\sigma))_{pq}|\\sigma,p,q\\rangle=U^{\\dagger}|0\\rangle\\sum_{\\lambda\\vdash n, p',q'} (\\hat{R}(\\lambda))_{p',q'}|\\lambda,p',q'\\rangle. \n\\label{eq:rR}\n\\end{equation}\nNow consider a function $g:S_n\\rightarrow \\mathbb{C}$ as in the statement of the Lemma. Since \n$g(\\pi)=0$ for all $\\pi\\in S_{n-1}$ we have $\\langle g|R\\rangle=0$ and therefore $\\langle \\hat{g}|\\hat{R}\\rangle=0$ and therefore \n$U^{\\dagger}|0\\rangle|\\hat{g}\\rangle$ is orthogonal to any state of the form on the LHS of Eq.~\\eqref{eq:rR}. Noting that the LHS of Eq.~\\eqref{eq:rR} is an arbitrary state in the image of $P_0$, we see that \n\\begin{equation}\nP_0U^{\\dagger}|0\\rangle|\\hat{g}\\rangle=0.\n\\label{eq:p0}\n\\end{equation}\n Then \n\\begin{align*}\nPU^{\\dagger}|0\\rangle|\\hat{g}\\rangle&=(\\sum_{k=1}^{n}P_k)U^{\\dagger}|0\\rangle|\\hat{g}\\rangle+P_0U^{\\dagger}|0\\rangle|\\hat{g}\\rangle\\\\\n&=(\\sum_{k=1}^{n}P_k)U^{\\dagger}|0\\rangle|\\hat{g}\\rangle\\\\&=(\\sum_{k=1}^{n}P_k)|0\\rangle|\\hat{g}\\rangle\\\\\n&=0\\\\\n\\end{align*}\nwhere in the first step we used Eq.~\\eqref{eq:p0} and in the following step we used Eq.~\\eqref{eq:pkinvariant}. \n\\end{proof}\n\nOur definitions of $U, f_k, F^k$ and $M(\\tau_k)$ are chosen so that the following holds.\n\\begin{claim}\nFor $1\\leq k\\leq n$ we have $M(\\tau_k)U|0\\rangle|\\hat{f_k}\\rangle=|0\\rangle|\\hat{F^k}\\rangle$.\n\\label{claim:Fkfk}\n\\end{claim}\n\\begin{proof}\nFollows by combining Eqs.~(\\ref{eq:dirsum}, \\ref{eq:U}, \\ref{eq:defF}, \\ref{eq:deff}).\n\\end{proof}\nFinally, for $k\\in [n]$ let $V_k$ be the unitary that acts as\n\\begin{align}\nV_k|0\\rangle|\\sigma,p,q\\rangle &=|\\tau_k\\rangle|\\sigma,p,q\\rangle \\\\\nV_k|\\tau_k\\rangle|\\sigma,p,q\\rangle &=|0\\rangle|\\sigma,p,q\\rangle,\n\\end{align}\nfor all $\\sigma\\vdash n-1$ and $1\\leq p,q\\leq d_{\\sigma}$,\nand which acts as the identity on all other computational basis states.\n\nThe quantum Fourier transform over the symmetric group is described in Algorithm \\ref{alg:qft}.\n\\begin{algorithm}[H]\n\\caption{Implements a unitary $\\mathrm{QFT_n}$\\label{alg:qft}}\n\\hspace*{\\algorithmicindent}  \\hspace{-22pt}\\textbf{Input:}  A state $|f\\rangle=\\sum_{g\\in S_n} f(g)|g\\rangle$.\\\\\n\\hspace*{\\algorithmicindent} \\hspace{-27pt} \\textbf{Output:} The state $|\\hat{f}\\rangle=\\mathrm{QFT}_n|f\\rangle$ corresponding to the Fourier transform of $f$ over $S_n$.\\\\\n\n\\begin{algorithmic}[1]\n\t\\State{$|\\phi\\rangle\\leftarrow W|\\phi\\rangle$} \\Comment{$|\\phi\\rangle=\\sum_{j=1}^{n} |\\tau_j\\rangle\\sum_{h\\in S_{n-1}} f_j(h)|h\\rangle$}.\n\t\\State{$|\\phi\\rangle \\leftarrow (I\\otimes \\mathrm{QFT}_{n-1})|\\phi\\rangle$} \\Comment{$|\\phi\\rangle=\\sum_{j=1}^{n} |\\tau_j\\rangle \\sum_{\\sigma \\vdash n-1} (\\hat{f_j}(\\sigma))_{pq} |\\sigma,p,q\\rangle$}\n\t\t\t\\For{$k=1$ to $n$}\n\t\t\\State{$|\\phi\\rangle\\leftarrow M(\\tau_k)^{-1}|\\phi\\rangle$ \n\t\t}\n\t\t\t\\State{$|\\phi\\rangle\\leftarrow UV_kU^{\\dagger}|\\phi\\rangle$}\n\t\t\\State{$|\\phi\\rangle\\leftarrow M(\\tau_k)|\\phi\\rangle$ } \n\t\t\t      \\EndFor \\Comment{$|\\phi\\rangle=|0\\rangle|\\hat{f}\\rangle$ (see Lemma \\ref{lem:forloop})} \n\\State{\\textbf{return} the second register $|\\hat{f}\\rangle$}.\n\t\t\\end{algorithmic}\n\\end{algorithm}\n\nThe following lemma shows that the algorithm performs the quantum Fourier transform, as claimed.\n\n\\begin{lemma}\nFor $0\\leq k\\leq n$, the state after the $k$th iteration of the for loop in Algorithm \\ref{alg:qft} is\n\\begin{equation}\n|\\phi_k\\rangle=|0\\rangle\\sum_{j=1}^{k}|\\hat{F^{j}}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle.\n\\label{eq:phik}\n\\end{equation}\nWhen $k=0$ this describes the state before the first iteration (in this case the first term is not present). For $k=n$ we have $|\\phi_n\\rangle=|0\\rangle\\sum_{j=1}^{n}|\\hat{F^{j}}\\rangle=|0\\rangle|\\hat{f}\\rangle$.\n\\label{lem:forloop}\n\\end{lemma}\n\\begin{proof}\nBy induction on $k$. The base case $k=0$ corresponds to the initial state $\\sum_{j=1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle$. \n\n Now suppose the state after the $(k-1)$-th iteration is given by $|\\phi_{k-1}\\rangle$ as described by Eq.~\\eqref{eq:phik}. Then after line 4 of Algorithm \\ref{alg:qft} the state is\n\\begin{align}\nM(\\tau_k^{-1})|\\phi_{k-1}\\rangle&=|0\\rangle\\sum_{j=1}^{k-1}M(\\tau_k^{-1})|\\hat{F^{j}}\\rangle+\\sum_{j=k}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle.\n\\end{align}\nFor each $j\\leq k-1$, the state $M(\\tau_k^{-1})|\\hat{F^j}\\rangle$ is the Fourier transform of a function \n$g(\\pi)=F^j(\\tau_k \\pi)$ which is zero on $S_{n-1}$.\nSo for $j\\leq k-1$, by Lemma \\ref{lem:orthog} we have $P U^{\\dagger}|0\\rangle M(\\tau_k^{-1})|\\hat{F^j}\\rangle =0$ where $P$ projects onto basis states that encode partitions of $n-1$. Therefore $V_k$ acts trivially on $U^{\\dagger}|0\\rangle M(\\tau_k^{-1})|\\hat{F^j}\\rangle$  for all $j\\leq k-1$, i.e., \n\\begin{align}\nV_kU^{\\dagger}M(\\tau_k^{-1})|\\phi_{k-1}\\rangle&=U^{\\dagger}|0\\rangle\\sum_{j=1}^{k-1}M(\\tau_k^{-1})|\\hat{F^j}\\rangle+\\sum_{j=k}^{n} V_k|\\tau_j\\rangle |\\hat{f_j}\\rangle\\\\\n&=U^{\\dagger}|0\\rangle\\sum_{j=1}^{k-1}M(\\tau_k^{-1})|\\hat{F^j}\\rangle+|0\\rangle|\\hat{f_k}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle.\n\\end{align}\nand, applying $M(\\tau_k)U$ to the above gives\n\\begin{align}\n|\\phi_k\\rangle&=|0\\rangle\\sum_{j=1}^{k-1}|\\hat{F^j}\\rangle+M(\\tau_k)U|0\\rangle|\\hat{f_k}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle\\\\\n&=|0\\rangle\\sum_{j=1}^{k}|\\hat{F^j}\\rangle+\\sum_{j=k+1}^{n} |\\tau_j\\rangle |\\hat{f_j}\\rangle\n\\end{align}\nwhere we used Claim \\ref{claim:Fkfk}. This completes the induction, and the proof.\n\\end{proof}\n\n\n\\section{ Generalized phase estimation}\nIn this section we will show that the circuits from Figure~\\ref{fig:gpe} implement weak Fourier sampling and generalized phase estimation respectively. The following Lemma shows that the circuit in Figure \\ref{fig:gpe}(a) implements weak Fourier sampling. We will make use of the inverse Fourier transform $\\mathrm{QFT}_n^{\\dagger}$ which acts as:\n\\begin{equation}\n\\mathrm{QFT}_n^{\\dagger} \\sum_{\\omega \\vdash n}\\sum_{1\\leq i,j\\leq d_{\\omega}} (c(\\omega))_{ij} |\\omega, i,j\\rangle=\\sum_{\\sigma\\in S_n}\\sum_{\\lambda \\vdash n} \\sqrt{\\frac{d_\\lambda}{n!}} \\trace{\\left(c(\\lambda) \\rho_{\\lambda}(\\sigma)^{\\dagger}\\right)}|\\sigma\\rangle.\n\\label{eq:invf}\n\\end{equation}\n\n \\begin{lemma}\n The POVM $M_L$ can be implemented by applying $\\mathrm{QFT}_n$, then performing a projective measurement of the representation label $\\omega$, then appplying $\\mathrm{QFT}_n^{\\dagger}$.\n \\label{Lemma:partition_measurement}\n \\end{lemma}\n \\begin{proof}\n Let a partition $\\lambda \\vdash n$ be given.  Let $P_{\\lambda}$ denote the projector such that $P_{\\lambda}|\\omega,i,j\\rangle=\\delta_{\\omega, \\lambda} |\\omega,i,j\\rangle.$ It suffices to show that for any state $|\\psi\\rangle\\in \\mathcal{H}$ we have $\\Pi^L_{\\lambda}|\\psi\\rangle=\\mathrm{QFT}_n^{\\dagger} P_{\\lambda} \\mathrm{QFT}_n|\\psi\\rangle$. To this end let $|\\psi\\rangle=\\sum_{\\sigma \\in S_n} f(\\sigma)|\\sigma\\rangle$. Using Eq.~\\eqref{eq:pilambda} we get\n \\begin{align}\n \\Pi_{\\lambda}|\\psi\\rangle& =\\frac{d_{\\lambda}}{n!} \\sum_{\\sigma\\in S_n}\\sum_{\\alpha\\in S_n}\\chi_{\\lambda}(\\alpha)f(\\sigma)|\\alpha \\sigma\\rangle\\\\\n &=\\frac{d_{\\lambda}}{n!}\\sum_{\\sigma,\\beta\\in S_n} \\chi_{\\lambda}(\\beta\\sigma^{-1})f(\\sigma)|\\beta\\rangle =\\frac{d_{\\lambda}}{n!}\\sum_{\\sigma,\\beta\\in S_n} \\chi_{\\lambda}(\\sigma \\beta^{-1})f(\\sigma)|\\beta\\rangle,\n \\label{eq:chif1}\n \\end{align}\n where in the last line we used the fact that $\\chi_\\lambda (g)=\\chi_{\\lambda}(g^{-1})$ for all $g\\in S_n$  (this can be seen for example using the fact that group characters are class functions and $g$ and $g^{-1}$ are always in the same conjugacy class since their cycle structure coincides). Now using Eq.~\\eqref{eq:chif1} and the fact that \n \\[\n \\chi_\\lambda(\\sigma \\beta^{-1})=\\trace{\\left(\\rho_\\lambda(\\sigma \\beta^{-1})\\right)}=\\trace{\\left(\\rho_\\lambda(\\sigma)\\rho_{\\lambda}( \\beta)^{\\dagger}\\right)}\n \\]\n gives\n \\begin{equation}\n \\Pi^L_{\\lambda}|\\psi\\rangle=\\sqrt{\\frac{d_{\\lambda}}{n!}}\\sum_{\\beta\\in S_n}\\trace{\\left(\\hat{f}(\\lambda)\\rho_{\\lambda}(\\beta)^{\\dagger}\\right)}|\\beta\\rangle\\label{eq:povm1}.\n \\end{equation}\n \n Below we show this is equal to $\\mathrm{QFT}_n^{\\dagger}P_\\lambda \\mathrm{QFT}_n|\\psi\\rangle$. We have:\n \\begin{align}\n \\mathrm{QFT}_n^{\\dagger}P_\\lambda \\mathrm{QFT}_n|\\psi\\rangle&= \\mathrm{QFT}_n^{\\dagger}P_\\lambda\\sum_{\\omega\\vdash n}\\sum_{i,j=1}^{d_{\\omega}} (\\hat{f}(\\omega))_{ij}|\\omega,i,j\\rangle \\\\\n &=\\mathrm{QFT}_n^{\\dagger}  \\sum_{i,j=1}^{d_{\\lambda}} (\\hat{f}(\\lambda))_{ij}|\\lambda,i,j\\rangle =\\sqrt{\\frac{d_{\\lambda}}{n!}}\\sum_{\\beta\\in S_n}\\trace{\\left(\\hat{f}(\\lambda)\\rho_{\\lambda}(\\beta)^{\\dagger}\\right)}|\\beta\\rangle,\n \\end{align}\n where we used Eq.~\\eqref{eq:invf} in the last line. This coincides with Eq.~\\eqref{eq:povm1} and completes the proof.\n \\end{proof}\nNext we show that the circuit in Fig.~\\ref{fig:gpe} (b) implements the generalized phase estimation measurement $M_{\\rho}$:\n\\begin{align}\n|\\tau,1,1\\rangle\\otimes |\\psi\\rangle  \\xrightarrow[]{\\mathrm{QFT}_n^{\\dagger}\\otimes I} \\frac{1}{\\sqrt{n!}}&\\sum_{\\alpha\\in S_n}|\\alpha\\rangle \\otimes |\\psi\\rangle \\\\\n\\xrightarrow[]{C-\\rho^{\\dagger}} \n&\\frac{1}{\\sqrt{n!}}\\sum_{\\alpha\\in S_n}|\\alpha\\rangle \\otimes \\rho^{\\dagger}(\\alpha)|\\psi\\rangle \\\\\n \\xrightarrow[]{\\text{Measure } M_L} \n&\\frac{1}{\\sqrt{n!}}\\sum_{\\alpha\\in S_n}\\Pi^L_{\\omega}|\\alpha\\rangle \\otimes \\rho^{\\dagger}(\\alpha)|\\psi\\rangle \\quad \\quad (\\text{conditioned on meas. outcome }\\omega)\\\\\n=&\\frac{d_{\\omega}}{(n!)^{3\/2}}\\sum_{\\alpha,\\sigma\\in S_n} \\chi_{\\omega}(\\sigma)|\\sigma\\alpha\\rangle \\otimes \\rho^{\\dagger}(\\alpha)|\\psi\\rangle\\\\\n\\xrightarrow[]{C-\\rho} & \\frac{d_{\\omega}}{(n!)^{3\/2}}\\sum_{\\alpha,\\sigma\\in S_n} \\chi_{\\omega}(\\sigma)|\\sigma\\alpha\\rangle \\otimes \\rho(\\sigma \\alpha)\\rho^{\\dagger}(\\alpha)|\\psi\\rangle.\\label{eq:outputstate}\n\\end{align}\nUsing the fact that $\\rho(\\sigma\\alpha)=\\rho(\\sigma)\\rho(\\alpha)$ and that $\\rho^{\\dagger}(\\alpha)=\\rho(\\alpha)^{-1}$ we see that the output state Eq.~\\eqref{eq:outputstate} is equal to\n\\begin{align}\n \\frac{d_{\\omega}}{(n!)^{3\/2}}\\sum_{\\alpha,\\sigma\\in S_n} \\chi_{\\omega}(\\sigma)|\\sigma\\alpha\\rangle \\otimes \\rho(\\sigma)|\\psi\\rangle\n=\\frac{1}{\\sqrt{n!}} \\sum_{\\beta\\in S_n}|\\beta\\rangle \\otimes \\frac{d_\\omega}{n!}\\sum_{\\sigma\\in S_n} \\chi_{\\omega}(\\sigma) \\rho(\\sigma)|\\psi\\rangle.\n\\end{align}\nOn the right hand side the state of the second register is $\\Pi_{\\omega}|\\psi\\rangle$, as desired.\n\n\\section{$\\QMA$, $\\#\\BQP$, and $\\QAPC$}\nRecall the definition of a verification circuit from the main text. In the following we write $p(\\psi)$ for the acceptance probability of a verification circuit with input state $\\psi$. \n\nA (promise) problem is in $\\QMA$ \\textit{with completeness $c$ and soundness $s$}, also denoted $\\QMA(c,s)$, if there exists a uniform polynomial-sized family of verification circuits $C_x$ labeled by instances $x$ of the problem, such that (A) If $x$ is a yes instance then there exists a witness $\\psi$ such that $p(\\psi)\\geq c$, and (B) If $x$ is a no instance then $p(\\psi)\\leq s$ for all  $\\psi$.  \n\nA standard convention is to define $\\QMA\\equiv \\QMA(2\/3,1\/3)$. The definition is not very sensitive to the choice of completeness $c$ and soundness $s$: it is known that $\\QMA(c,s)=\\QMA(2\/3,1\/3)$ whenever $0<s<c\\leq 1$ and $c-s=\\Omega(1\/\\mathrm{poly}(n))$.  In fact, one can reduce the error of any $\\QMA$ verifier exponentially in the input size $n$, so that, e.g.,  $\\QMA(2\/3,1\/3)=\\QMA(4^{-n},1-4^{-n})$ as well \\cite{kitaev2002classical, marriott2005quantum}. \n\nWe choose to define $\\#\\BQP$ as a class of relation problems rather than a class of functions, languages (decision problems), or promise problems. Recall from the main text that Eq.~\\eqref{eq:POVM} describes the POVM element corresponding to the accept outcome of a quantum verification circuit and $N_q$ denotes the number of eigenvalues of $A$ that are at least $q$. We define $\\#\\BQP$ as the class of problems that can be expressed as: given a quantum verification circuit with an $n$-qubit input register, total size $\\mathrm{poly}(n)$, and two thresholds $0<b<a\\leq 1$ and $a-b=\\Omega(1\/\\mathrm{poly}(n))$, compute an estimate $N$ such that $N_a\\leq N\\leq N_b$. This defines a relation problem because there may be more than one solution for a given instance (this happens whenever $N_a\\neq N_b$).\n\n Ref.~\\cite{brown2011computational} puts forward a slightly different definition of $\\#\\BQP$, as the class of promise problems obtained by augmenting and specializing the above definition with the promise that $N_a=N_b$. This has the pleasing feature that it results in a class of functions rather than relations, though at the expense of adding a promise.\n\nOne of the main results established in Ref.~\\cite{brown2011computational} is that any problem in  $\\#\\BQP$ (as they define it) can be efficiently reduced to computing a $\\#\\P$ function. We now describe how their proof  can be straightforwardly adapted to our setting, establishing the same result for the definition of $\\#\\BQP$ that we use in this paper.  Using the in-place error reduction for $\\QMA$ \\cite{marriott2005quantum} we may assume without loss of generality that $a=1-2^{-n-2}$ and $b=2^{-n-2}$ (see, e.g. the proof of Lemma 8 in the supplementary material of Ref.~\\cite{bravyi2022quantum}).\nThen \n\\begin{equation}\n\\mathrm{Tr}(A)\\leq N_b+2^{-n-2}\\cdot (2^n-N_b)\\leq N_b+1\/4,\n\\label{eq:trup}\n\\end{equation}\nand\n\\begin{equation}\n\\mathrm{Tr}(A)\\geq N_a (1-2^{-n-2})\\geq N_a-1\/4.\n\\label{eq:trlow}\n\\end{equation}\nEqs.~(\\ref{eq:trup},\\ref{eq:trlow}) show that the nearest integer $N$ to $\\mathrm{Tr}(A)$ solves the given $\\#\\BQP$ problem as it satisfies $N_a\\leq N\\leq N_b$. It then suffices to show that we can compute $\\mathrm{Tr}(A)$ to within an additive error of $1\/4$ using efficient classical computation along with a single call to a $\\#\\P$ oracle. This is established in Theorem 14 of Ref.~\\cite{brown2011computational} via a proof that is based on expressing $\\mathrm{Tr}(A)$ as a sum over paths, see Ref.~\\cite{brown2011computational} for details.\n\n\n\\section{Efficient sampling algorithm}\nNow let us describe the algorithm stated in Lemma \\ref{lem:samplekron}, which appears implicitly in Ref.~\\cite{Moore07}. \n\nThe completely mixed state over the image of the projector $\\Pi^L_{\\mu}$ can be prepared as follows.\n\\begin{align}\n    \\frac{\\Pi^L_\\mu}{d_\\mu^2} &= \\text{QFT}_n \\left( \\ket{\\mu} \\bra{\\mu} \\otimes I\/d_{\\mu} \\otimes  I\/d_{\\mu}\\right) \\text{QFT}_n^\\dag. \n\\end{align}\nStarting from $\\frac{\\Pi^L_\\mu \\otimes \\Pi^L_\\nu}{d_\\mu^2 d_\\nu^2}$ we then measure the projector $\\Pi^L_\\lambda$ using Weak Fourier sampling. We obtain outcome $\\lambda$ with probability: \n\\begin{align}\n    p(\\lambda) &= \\text{Tr} \\left(\\frac{\\Pi^L_\\mu \\otimes \\Pi^L_\\nu}{d_\\mu^2 d_\\nu^2} \\Pi_\\lambda \\right) = \\frac{d_\\lambda}{d_\\mu d_\\nu n!} \\sum_{g,h,\\ell} \\chi_\\mu(g) \\chi_\\nu(h) \\chi_\\lambda(\\ell)  = \\frac{d_\\lambda}{d_\\mu d_\\nu} g_{\\mu \\nu \\lambda}.\n\\end{align}\n\n\n\\section{Row sums}\n\n\\begin{lemma}\nThe multiplicity of an irreducible representation $\\rho_\\lambda$ in the conjugation representation $\\rho_c$ is given by the row sum $R_{\\lambda}$ defined in Eq.~\\eqref{eq:rowsum}.\n\n\\end{lemma}\n\\begin{proof}\nRecall that the centralizer $Z(\\pi)$ of an element $\\pi \\in S_n$ is defined as the set of elements $g \\in S_n$ that commute with $\\pi$. The conjugation representation acts on $\\mathbb{C}S_n$ as\n\\[\n\\rho_c(\\pi)|\\sigma\\rangle = |\\pi \\sigma \\pi^{-1}\\rangle \n\\]\nand therefore \n\\[\n\\chi_{c}(\\pi)\\equiv \\mathrm{Tr}(\\rho_c(\\pi))=|Z(\\pi)|,\n\\]\nwhere $Z(\\pi)$ is the centralizer of $\\pi \\in S_n$. Note that all elements $\\pi\\in S_n$ in the same conjugacy class $C$ have the same centralizer which we denote $Z(C)$.\n\nWe can decompose the conjugation representation into irreducibles as $\\rho_c \\simeq \\bigoplus_{\\lambda \\vdash n} m_{\\lambda} \\rho_{\\lambda}$ for some multiplicities $m_\\lambda$. Taking the trace gives \n\\[\n\\chi_c(\\pi)=\\sum_{\\lambda \\vdash n} m_{\\lambda}\\chi_{\\lambda}(\\pi)\n\\]\nfor all $\\pi\\in S_n$. Using orthogonality of the irreducible characters we arrive at\n\\begin{align}\n    m_\\lambda &= \\frac{1}{n!} \\sum_{g \\in G} \\chi_\\lambda(g) \\chi_{c}(g) = \\frac{1}{n!} \\sum_{C \\vdash n} \\chi_\\lambda(C) |C| |Z(C)| \\label{eq:rowsumC},\n\\end{align}\nwhere $|C|$ is the order of the conjugacy class $C$. By Lagrange's theorem, $n! \/ |C| = |Z(C)|$, and substituting this into Eq.~\\eqref{eq:rowsumC} shows that $m_{\\lambda}=R_{\\lambda}$ and completes the proof.\n\n\n\\end{proof}\n\\begin{lemma}\n$R_{\\lambda}=d_{\\lambda}^{-1}\\mathrm{Tr}(\\Pi_{\\lambda}^{c})$\n\\end{lemma}\n\\begin{proof}\nBy definition, \n\\[\n\\Pi_{\\lambda}^{c}=\\frac{d_{\\lambda}}{n!}\\sum_{g\\in S_n}\\chi_{\\lambda}(g)\\rho_c(g),\n\\]\nand therefore\n\\[\n\\mathrm{Tr}(\\Pi_{\\lambda}^{c})=\\frac{d_{\\lambda}}{n!}\\sum_{g\\in S_n}\\chi_{\\lambda}(g)\\chi_c(g) = d_\\lambda R_\\lambda,\n\\]\nby Eq. \\eqref{eq:rowsumC}\n\\end{proof}\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Sequential Patterns}\n\nLet $\\I$ be a finite set of {\\em items}. \nThe language of sequences  corresponds to $\\mathcal{L_I}=\\I^n$ where $n\\in\\mbox{$\\mathbb{N}$}^+$. \n\n\\begin{definition}[sequence, sequence database]\nA sequence $s$ over $\\mathcal{L_I}$ is an ordered list\n$\\langle s_1 s_2 \\ldots s_n \\rangle$, where $s_i$, $1 \\leq i \\leq n$,\nis an item. $n$ is called the length of the sequence $s$.\nA sequence database $\\SDB$ is a set of tuples $(sid, s)$,\nwhere $sid$ is a sequence identifier and $s$ a sequence. \n\\end{definition}\n\n\\begin{definition}[subsequence,  $\\mbox{$\\preceq$}$ relation]\nA sequence $\\alpha = \\langle \\alpha_1 \\ldots \\alpha_m \\rangle$ is a \n  subsequence of $s = \\langle s_1 \\ldots s_n \\rangle$,\ndenoted by ($\\alpha$ $\\mbox{$\\preceq$}$ $s$), if $m \\leq n$ and there exist integers \n$1 \\leq j_1 \\leq \\ldots \\leq j_m\\leq n$, such that \n$\\alpha_i=s_{j_i}$ for all $1 \\leq i \\leq m$.  We also say that $\\alpha$ is\ncontained in $s$ or $s$ is a super-sequence of $\\alpha$. \nFor example, the sequence $\\angx{BABC}$ is a super-sequence of\n$\\angx{AC}$ : $\\angx{AC}$ $\\mbox{$\\preceq$}$  $\\angx{BABC}$.\nA tuple $(sid, s)$  contains a sequence $\\alpha$, if $\\alpha$ $\\mbox{$\\preceq$}$ $s$. \n\\end{definition}\n\nThe cover of a sequence $p$ in $\\SDB$ is the set of all tuples in\n$\\SDB$ in which $p$ is contained. \nThe support of a sequence $p$ in $\\SDB$  is the number of \ntuples in $\\SDB$ which contain $p$. \n\n\\begin{definition}[coverage, support]\nLet $\\SDB$ be a sequence database and $p$ a sequence. \n$cover_{\\SDB}(p)$$=$$\\{(sid, s) \\in \\SDB \\, | \\, p \\,\\mbox{$\\preceq$}\\, s\\}$ and\n$sup_{\\SDB}(p)=\\# cover_{\\SDB}(p)$. \n\\end{definition}\n\n\\begin{definition}[sequential pattern]\nGiven a minimum support threshold $minsup$, every sequence $p$ \nsuch that $sup_{\\SDB}(p) \\geq minsup$ is called a sequential pattern~\\cite{DBLP:conf\/icde\/AgrawalS95}. \n$p$ is said to be frequent in $\\SDB$. \n\\end{definition}\n\n\\begin{table}[t]\n\t\\begin{center}\n\t\t\\scalebox{0.90}{\n\t\t\t\\begin{tabular}{|l|l|}\t\n\t\t\t\t\\hline\n                                ~sid~ &~Sequence~ \\\\\n\t\t\t\t\\hline\n\t\t\t\t $1$ & $\\angx{ABCBC}$  \\\\\n\t\t\t\t $2$ & $ \\angx{BABC}$ \\\\\n\t\t\t\t $3$ & $ \\angx{AB} $ \\\\\n\t\t\t\t $4$  & $\\angx{BCD}$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular} \n\t\t}  \n                        \\caption{$\\SDB_1$: a sequence database example.}\n\t\t\\label{tab:SDB}\n\t\\end{center}\n\\end{table}\t\n\n\\begin{example}\nTable~\\ref{tab:SDB} represents\n  a sequence database of four sequences \nwhere the set of items is $ \\I = \\{A, B, C, D\\}$. Let \nthe sequence $p= \\angx{A C}$. We have\n$cover_{{\\SDB}1}(p)=\\{(1, s_1), (2, s_2)\\}$. If we consider \n$minsup=2$, $p=\\angx{A C}$ is a sequential pattern because $sup_{\\SDB_1}(p) \\ge 2$. \n\\end{example}\n\n\\begin{definition}[sequential pattern mining (SPM)]\nGiven a sequence database $\\SDB$ and a minimum support threshold\n$minsup$. The problem of sequential pattern mining is to find all\npatterns $p$ such that $sup_{\\SDB}(p) \\geq minsup$. \n\\end{definition}\n\n\n\\subsection{SPM under Constraints}\n\\label{sec:local}\n\nIn this section, we define the problem of mining sequential \npatterns in a sequence database satisfying user-defined\nconstraints Then, we review the most usual constraints for\nthe sequential mining problem~\\cite{Pei2002}. \n\n\\noindent \n{\\bf Problem statement}. Given a constraint $C(p)$ on \npattern $p$ and a sequence database \\SDB, the problem\nof constraint-based pattern mining is to find the\ncomplete set of patterns satisfying $C(p)$.\nIn the following, we present different types of constraints\nthat we explicit in the context of sequence mining. All\nthese constraints will be handled by our concise encoding (see\nSections \\ref{consistency} and \\ref{pattern-encoding}). \n\n\\noindent\n- The minimum size constraint $size(p,\\ell_{min})$ states that \nthe number of items of $p$ must be greater than or equal to $\\ell_{min}$. \n\n\\noindent\n- The item constraint $item(p,t)$ states that an item $t$ must belong\n(or not) to a pattern $p$. \n\n\\noindent\n- The regular expression\nconstraint~\\cite{DBLP:journals\/tkde\/GarofalakisRS02}\n$reg(p,exp)$ states that a pattern $p$ must be \naccepted by the deterministic finite \nautomata associated to the regular expression $exp$. \n\n\n\\subsection{Projected Databases}\n\\label{PB-projected-databases}\nWe now present the necessary definitions related to the concept of\n{\\it projected databases}~\\cite{DBLP:conf\/icde\/PeiHPCDH01}.\n\n\\begin{definition}[prefix, projection, suffix]  \nLet $\\beta=\\angx{\\beta_1 \\ldots \\beta_n}$ and $\\alpha = \\angx{\\alpha_1\n  \\ldots \\alpha_m}$ be two sequences, where $m \\leq n$. \\\\ \n- Sequence $\\alpha$ is called the prefix of $\\beta$ iff $\\forall i \\in\n[1..m], \\alpha_i=\\beta_i$.   \\\\ \n- Sequence $\\beta=\\angx{\\beta_1 \\ldots \\beta_n}$ is called the \nprojection of some sequence $s$ w.r.t. $\\alpha$,  \niff (1) $\\beta\\, \\mbox{$\\preceq$}\\, s$, \n(2) $\\alpha$ is a prefix of $\\beta$ \nand (3) there exists no proper super-sequence $\\beta'$ of $\\beta$ such\nthat $\\beta' \\,\\mbox{$\\preceq$}\\,s$ and $\\beta'$ also has $\\alpha$ as prefix. \\\\\n- Sequence $\\gamma =\\angx{\\beta_{m+1} \\ldots \\beta_n}$ is called the suffix of $s$\nw.r.t. $\\alpha$. With the standard concatenation operator \"$concat$\",\nwe have $\\beta = concat(\\alpha, \\gamma)$.  \n\\end{definition}\n\n\\begin{definition}[projected database] \nLet  $\\SDB$ be a sequence database, \nthe $\\alpha$-projected database, denoted by $SDB|_\\alpha$, \nis the collection of suffixes of sequences in  $\\SDB$ w.r.t. prefix $\\alpha$. \n\\end{definition}\n\n\\cite{DBLP:conf\/icde\/PeiHPCDH01} have proposed an\nefficient algorithm, called \\code{PrefixSpan}, for mining sequential\npatterns based on the concept of {\\it projected databases}.   \nIt proceeds by dividing the initial database into smaller ones\nprojected on the frequent subsequences obtained so far; only their\ncorresponding suffixes are kept. Then, sequential patterns are mined \nin each projected database by exploring only locally  frequent\npatterns. \n\n\\begin{example}\n\\label{exp-projection}\nLet us consider the sequence database of Table~\\ref{tab:SDB} with\n$minsup=2$.  \n\\code{PrefixSpan} starts by scanning ${\\SDB}_1$ to find\nall the frequent items, each of them is used as\na prefix to get projected databases. \nFor ${\\SDB}_1$, we get $3$ disjoint subsets w.r.t. the prefixes\n$\\angx{A}$, $\\angx{B}$, and $\\angx{C}$.  \nFor instance, ${\\SDB}_1|_{\\angx{A}}$ consists of 3 suffix sequences:\n$\\{(1,\\angx{B C B C})$, $(2,\\angx{BC})$, $(3,\\angx{B})\\}$.   \nConsider the projected database ${\\SDB}_1|_{<A>}$, its\nlocally frequent items are $B$ and $C$. Thus, ${\\SDB}_1|_{<A>}$\ncan be recursively partitioned into 2 subsets w.r.t. the two \nprefixes $\\angx{A B}$ and $\\angx{A C}$. The $\\angx{A B}$- and $\\angx{A\n  C}$- projected databases can be constructed and recursively mined\nsimilarly. The processing of a \n$\\alpha$-projected database terminates  \nwhen no frequent subsequence can be generated. \n\\end{example}\n\nProposition~\\ref{prop-prefixspan} establishes the support count of a\nsequence $\\gamma$ in $\\SDB|_\\alpha$~\\cite{DBLP:conf\/icde\/PeiHPCDH01}: \n\\begin{proposition}[Support count]\\label{prop-prefixspan}\nFor any sequence $\\gamma$ in $\\SDB$ with prefix $\\alpha$ and suffix $\\beta$\ns.t. $\\gamma= concat(\\alpha,\\beta$), $sup_{\\SDB}(\\gamma) = sup_{\\SDB|_{\\alpha}}(\\beta)$. \n\\end{proposition}\n\nThis proposition ensures that only the sequences in $\\SDB$ grown from\n$\\alpha$ need to be considered for the support count of a sequence\n$\\gamma$. Furthermore, only those suffixes with prefix \n$\\alpha$ should be counted. \n\n\n\\subsection{CSP and Global Constraints}\n\n\\noindent\nA {\\it Constraint Satisfaction Problem} (CSP) consists of a \nset $X$ of $n$ variables, a domain $\\mathcal{D}$ mapping each variable \n$X_i \\in X$ to a finite set of values $D(X_i)$, and a set of constraints\n$\\mathcal{C}$. An assignment $\\sigma$ is a mapping from variables in $X$ to\nvalues in their domains: $\\forall X_i \\in X, \\sigma(X_i) \\in D(X_i)$. \nA constraint $c \\in \\mathcal{C}$ is a subset of the \ncartesian product of the domains of the variables that are in $c$. \nThe goal is to find an assignment \nsuch that all constraints are satisfied.  \n\n\\noindent \n\\textbf{Domain consistency (DC).} \nConstraint solvers typically use backtracking search to explore the\nspace of partial assignments. At each assignment, filtering\nalgorithms prune the search space by enforcing local consistency\nproperties like domain consistency. A constraint $c$ on $X$ is domain\nconsistent, if and only if, for every $X_i \\in X$ and for every\n$d_i \\in D(X_i)$, there is an assignment $\\sigma$ satisfying $c$ such\nthat $\\sigma(X_i) = d_i$. Such \nan assignment is called a support.  \n\n\\noindent \n\\textbf{Global constraints} \nprovide shorthands to often-used combinatorial substructures.   \nWe present two global constraints.\nLet $X=\\angx{X_1, X_2, ..., X_n}$ be a sequence of $n$ variables.\n\n\\noindent\nLet $V$ be a set of values, $l$ and $u$ be two integers s.t. $0\\leq l \\leq u \\leq n$,\nthe constraint \\texttt{Among}$(X, V ,l, u)$ states that each value $a \\in V$ \nshould occur at least $l$ times and at most $u$ times in $X$~\\cite{DBLP:journals\/jmcm\/Beldi94}.\nGiven a deterministic finite automaton $A$,\nthe constraint \\texttt{Regular}$(X, A)$ ensures that the sequence $X$\nis accepted by $A$~\\cite{regular}.  \n\n\n\n\\subsection{Ad hoc Methods for SPM}\n{\\tt GSP}~\\cite{DBLP:conf\/edbt\/SrikantA96} was the first algorithm\nproposed to extract sequential patterns. It uses a generate-and test\napproach. Later, two major classes of methods have been proposed: \n\n\\noindent -  Depth-first search based on a vertical database format\ne.g. \\texttt{cSpade}\\xspace incorporating contraints (max-gap, max-span,\nlength)~\\cite{DBLP:conf\/cikm\/Zaki00}, \n\\texttt{SPADE}\\xspace~\\cite{DBLP:journals\/ml\/Zaki01}   \n or {\\tt\n   SPAM}~\\cite{Ayres:2002:SPM:775047.775109}.\n\n\\noindent - Projected pattern growth such as {\\tt\n  PrefixSpan}~\\cite{DBLP:conf\/icde\/PeiHPCDH01}  \nand its extensions, e.g. \\texttt{CloSpan}\\xspace\\ for mining closed sequential\npatterns~\\cite{DBLP:conf\/sdm\/YanHA03} or \\texttt{Gap-BIDE}\\xspace~\\cite{Li2012}\ntackling the gap constraint. \n\nIn \\cite{DBLP:journals\/tkde\/GarofalakisRS02}, the authors\n  proposed {\\tt SPIRIT} based on {\\tt GSP}\n  for SPM with regular expressions. Later, \\cite{Bonchi:2008}\n  introduces Sequence Mining Automata (\\sma), a new approach\n  based on a specialized kind of Petri Net. Two variants of \\sma\n  were proposed:  \\smap (\\sma one pass) and \\texttt{SMA-FC}\\xspace (\\sma Full\n  Check).  \\smap processes by means\n  of the \\sma all sequences one by one, and enters all resulting valid\n  patterns in a hash table for support counting, while \\texttt{SMA-FC}\\xspace allows\n  frequency based pruning during the scan of the database. \nFinally, \\cite{Pei2002} provides a survey for other constraints \nsuch as regular expressions, length and aggregates. \nBut, \nall these proposals, though efficient, are ad hoc methods suffering\nfrom a lack of genericity. \nAdding new constraints often requires to develop new implementations.\n\n\n\\subsection{CP Methods for SPM}\n\\label{CP4SPM}\n\nFollowing the work of \\cite{DBLP:journals\/ai\/GunsNR11} for itemset mining,\nseveral methods have been proposed to mine sequential patterns using CP.\n\n\\noindent {\\bf Proposals.}\n\\cite{DBLP:conf\/ecai\/CoqueryJSS12} have\nproposed a first SAT-based model for discovering a special class of\npatterns with wildcards\\footnote{A wildcard is a special symbol that\n  matches any item of $\\I$ including itself.} in a single\nsequence under different types of \nconstraints (e.g. frequency, maximality, closedness). \n{\\cite{metivierLML13} have proposed a CSP model for SPM.\nEach sequence is encoded by an automaton capturing all subsequences that can occur in it.\n\\cite{DBLP:conf\/ictai\/KemmarULCLBC14} have proposed a CSP model for SPM with wildcards.\nThey show how some constraints dealing with local patterns (e.g. frequency, size, gap, regular expressions)\nand constraints defining more complex patterns such as relevant subgroups \\cite{DBLP:journals\/jmlr\/NovakLW09} \nand \\topk patterns can be modeled using a CSP. \n\\cite{NegrevergneCPIAOR15} have proposed two CP encodings for the SPM. \nThe first one uses a global constraint to encode the subsequence relation (denoted \\texttt{global-p.f}\\xspace),\nwhile the second one encodes explicitly this relation using additional\nvariables and constraints (denoted \\texttt{decomposed-p.f}\\xspace).\n\nAll \nthese proposals use {\\bf reified constraints} to encode the database.\nA reified constraint associates a boolean variable to a constraint\nreflecting whether the constraint is satisfied (value 1) or not (value\n0).   \nFor each sequence $s$ of $\\SDB$, a reified constraint, \nstating whether (or not) the unknown pattern $p$ is a subsequence of $s$,\nis imposed: $(S_s=1) \\Leftrightarrow (p \\preceq s)$.\nA great consequence is that the encoding of the frequency measure \nis straightforward: $freq(p) = \\sum_{s \\in \\SDB} S_s$.\nBut such an encoding has a major drawback since it\nrequires $(m=\\#\\SDB)$ reified constraints to encode the whole\ndatabase. This constitutes a strong limitation of the size of the\ndatabases that could be managed. \n\nMost\nof these proposals encode {\\bf the subsequence relation} $(p \\preceq s)$ \nusing variables $Pos_{s,j}$ $(s \\in  \\SDB$ and $1 \\le j \\le \\ell)$\nto determine a position where $p$ occurs in $s$.\nSuch an encoding requires a large number of additional \nvariables ($m$$\\times$$\\ell$) and makes the labeling computationally\nexpensive. \nIn order to address this drawback, \\cite{NegrevergneCPIAOR15} have\nproposed a global constraint \\texttt{exists-embedding}  \nto encode the subsequence relation,\nand used projected frequency within an ad hoc specific branching strategy\nto keep only frequent  items before branching over the variables of\nthe pattern. \nBut, \nthis encoding still relies on reified constraints and requires to\nimpose $m$ \\texttt{exists-embedding} global constraints. \n\n\nSo,\nwe propose in the next section the \\textsc{Prefix-Projection}\\xspace global constraint \nthat  fully exploits the principle of projected databases to encode\nboth the subsequence relation and the frequency constraint. \n \\textsc{Prefix-Projection}\\xspace does not require any reified constraints nor any extra\n variables to encode the subsequence relation. \nAs a consequence, usual SPM constraints (see Section~\\ref{sec:local}) \ncan be encoded in a straightforward way using directly the (global)\nconstraints of the CP solver. \n\n\n\n\n\n\\subsection{A Concise Encoding}\n\\label{sec:pattern}\n\nLet $P$ be the unknown pattern of size $\\ell$ we are looking for. \nThe symbol $\\vide$ stands for an empty item and denotes the end of a\nsequence.  \nThe unknown pattern $P$ is encoded with a sequence\nof $\\ell$ variables $\\angx{P_1,P_2,\\ldots,P_\\ell}$\ns.t. $\\forall i \\in [1\\ldots\\ell],  D(P_ i)= \\I\\cup\\{\\vide\\}$.  \nThere are two basic rules on the domains:\n\\begin{enumerate}\n\\vspace*{-.1cm}\\item \nTo avoid the empty sequence, the first item of $P$ must be non empty,\nso $(\\vide\\not\\in D_1)$.  \n\\item\nTo allow patterns with less than $\\ell$ items, we impose that $\\forall\ni \\in [1.. (\\ell$$-$$1)], (P_i=\\vide) \\rightarrow (P_{i+1} = \\vide)$. \n\\end{enumerate}\n\n\\subsection{Definition and Consistency Checking} \n\\label{consistency}\n\nThe global constraint \\textsc{Prefix-Projection}\\xspace ensures both subsequence relation and\nminimum frequency constraint.  \n\n\\begin{definition}[\\textsc{Prefix-Projection}\\xspace global constraint]\nLet $P = \\angx{P_1,P_2,\\ldots,P_\\ell}$ be a pattern of size $\\ell$.   \n$\\angx{d_1, ..., d_{\\ell}} \\in D(P_1)\\times \\ldots \\times D(P_\\ell)$\nis a solution of \\textsc{Prefix-Projection}\\xspace$(P, \\SDB, minsup)$ iff   \n$sup_{\\SDB}(\\angx{d_1, ..., d_{\\ell}})\n\\geq minsup$. \n\\end{definition}\n\n\\begin{proposition}\n\\label{prop-solution}\nA \\textsc{Prefix-Projection}\\xspace$(P, \\SDB,minsup)$ constraint has a solution if and only if  \nthere exists an assignment $\\sigma = \\angx{d_1, ..., d_{\\ell}}$ of\nvariables of $P$ s.t. $\\SDB|_{\\sigma}$ has at least $minsup$ suffixes of $\\sigma$: \n$\\#\\SDB|_{\\sigma}\\geq minsup$. \n\\end{proposition}\n\n{\\it Proof: }\nThis is a direct consequence of proposition \\ref{prop-prefixspan}. We\nhave straightforwardly   \n$sup_{\\SDB}(\\sigma) = sup_{\\SDB|_{\\sigma}}(\\angx{}) =\n\\#\\SDB|_{\\sigma}$. Thus, suffixes of $\\SDB|_{\\sigma}$ are supports  \nof $\\sigma$ in the constraint \\textsc{Prefix-Projection}\\xspace$(P, \\SDB,minsup)$,\nprovided that $\\#\\SDB|_{\\sigma}\\geq minsup$. \n$\\Box$\n\nThe following proposition characterizes values in the domain\nof unassigned (i.e. future) variable $P_{i+1}$ that are consistent with the \ncurrent assignment of variables $\\angx{P_1, ..., P_i}$.\n\n\\begin{proposition}\n\\label{prop-consistency}\nLet $\\sigma \\footnote{We indifferently denote $\\sigma$ by $\\angx{d_1,\n    \\dots, d_i}$ or by $\\angx{\\sigma(P_1), \\dots, \\sigma(P_{i})}$.} =\\angx{d_1, \\dots, d_i}$ be a current\nassignment of\nvariables $\\angx{P_1, \\dots, P_i}$, $P_{i+1} $ be a future variable. A\nvalue $d \\in D(P_{i+1})$ \nappears in a solution for \\textsc{Prefix-Projection}\\xspace$(P, \\SDB, minsup)$ if and only if\n$d$ is a frequent item in $\\SDB|_{\\sigma}$:  \n$$\\#\\{(sid,\\gamma) | (sid,\\gamma) \\in \\SDB|_{\\sigma} \\wedge \\angx{d}\\mbox{$\\preceq$}\n\\gamma\\} \\geq minsup$$\n\\end{proposition}\n\n\\noindent\n{\\it Proof: }\nSuppose that value $d \\in D(P_{i+1})$ occurs in $\\SDB|_{\\sigma}$\nmore than $minsup$. From proposition~\\ref{prop-prefixspan}, we have \n$sup_{\\SDB}(concat(\\sigma, \\angx{d})) =\nsup_{\\SDB|_{\\sigma}}(\\angx{d})$. Hence, the\nassignment $\\sigma \\cup \\angx{d}$ satisfies the constraint,\nso $d \\in D(P_{i+1})$ participates in a solution. \n$\\Box$\n\n\\noindent {\\bf Anti-monotonicity of the frequency measure.} \nIf a pattern $p$ is not frequent, then any pattern $p'$ satisfying \n$p \\,\\mbox{$\\preceq$}\\, p'$ is not frequent.   \nFrom proposition~\\ref{prop-consistency} and according \nto the {\\it anti-monotonicity property}, we can derive the following \npruning rule: \n\n\\begin{proposition}\n\\label{prop-filtering}\nLet $\\sigma=\\angx{d_1, \\dots, d_i}$ be a current assignment of\nvariables $\\angx{P_1, \\dots, P_i}$. All values $d \\in D(P_{i+1})$\nthat are locally not frequent in\n$\\SDB|_{\\sigma}$ can be pruned from the domain of variable\n$P_{i+1}$. Moreover, these values $d$ can also be pruned from the domains\nof variables $P_j$ with  $j \\in [i+2, \\dots, \\ell]$. \n\\end{proposition}\n\n\\noindent\n{\\it Proof: } \nLet $\\sigma=\\angx{d_1, \\dots, d_i}$ be a current assignment of\nvariables $\\angx{P_1, \\dots, P_i}$. Let $d \\in D(P_{i+1})$ s.t. \n$\\sigma' = {concat(\\sigma,\\angx{d})}$. \nSuppose that $d$ is not frequent in $\\SDB|_{\\sigma}$. \nAccording to proposition \\ref{prop-prefixspan},\n$sup_{\\SDB|_{\\sigma}}(\\angx{d}) = sup_{\\SDB}(\\sigma')<minsup$, thus $\\sigma'$\nis not frequent. So, $d$ can be pruned from the domain of $P_{i+1}$.\n\n\\noindent\nSuppose that the assignment $\\sigma$ has been extended to\n$concat(\\sigma, \\alpha)$, where $\\alpha$ corresponds to the assignment\nof variables $P_j$ (with $j > i$).  \nIf $d \\in D(P_{i+1})$ is not frequent, it is straightforward that\n$sup_{\\SDB|_{\\sigma}}(concat(\\alpha, \\angx{d}))\\leq\nsup_{\\SDB|_{\\sigma}}(\\angx{d}) < minsup$. Thus, if $d$ is not frequent\nin $\\SDB|_{\\sigma}$, it will be also not frequent in\n$\\SDB|_{concat(\\sigma,\\alpha)}$. \nSo, $d$ can be pruned from the domains of\n$P_j$ with $j \\in [i+2, \\dots, \\ell]$.\n$\\Box$\n\n\\begin{example}\nConsider the sequence database of Table~\\ref{tab:SDB} with\n$minsup=2$. Let $P = \\angx{P_1,P_2,P_3}$ with $D(P_1) = \\I$ and \n$D(P_2) = D(P_3) = \\I\\cup\\{\\vide\\}$. Suppose that $\\sigma(P_1) = A$, \n$\\textsc{Prefix-Projection}\\xspace(P, \\SDB, minsup)$ will remove values $A$ and\n$D$ from $D(P_2)$ and $D(P_3)$, since the only locally frequent items in\n${\\SDB}_1|_{<A>}$ are $B$ and $C$.\n\\end{example} \n\nProposition~\\ref{prop-filtering} guarantees \nthat any value (i.e. item)  \n$d \\in D(P_{i+1})$ present but not frequent in $\\SDB|_{\\sigma}$ does not need\nto be considered when extending $\\sigma$, thus avoiding searching over\nit. \nClearly, our global constraint encodes the anti-monotonicity\nof the frequency measure in a simple and elegant way, while \nCP methods for SPM have difficulties to handle this property. \nIn \\cite{NegrevergneCPIAOR15}, this is achieved by using very specific\npropagators and branching  \nstrategies, making the integration quite complex (see\n\\cite{NegrevergneCPIAOR15}). \n\n\n\\begin{algorithm}[t]\n\\begin{small}\n\\caption{\\small \\codex{ProjectSDB}($\\SDB$, $ProjSDB$, $\\alpha$)  \\label{algo:projection}}\n\\KwData{$\\SDB$: initial database; $ProjSDB$: projected sequences; $\\alpha$: prefix}\n\\Begin\n{\n\\lnl{p1}$\\SDB|_{\\alpha} \\leftarrow \\emptyset$ \\;\n\\lnl{p2}\\For{each pair $(sid,start) \\in ProjSDB$}{\n\\lnl{p3}$s \\leftarrow \\SDB[sid]$ \\; \n\\lnl{p4}$pos_{\\alpha}  \\leftarrow 1$;\\ $pos_s  \\leftarrow start$ \\;\n\\lnl{p6}\\While{$(pos_{\\alpha} \\leq \\#\\alpha$ $\\wedge$ $pos_s \\leq \\# s)$}{\n\\lnl{p7}\\If{$(\\alpha[pos_\\alpha] = s[pos_s])$}{\n\\lnl{p8}$pos_\\alpha \\leftarrow pos_\\alpha + 1$ \\;\n}\n\\lnl{p9}$pos_s \\leftarrow pos_s + 1$ \\;\n\n}\n\\lnl{p10}\\If{$(pos_\\alpha = \\#\\alpha + 1)$}{\n\\lnl{p11}$\\SDB|_{\\alpha} \\leftarrow \\SDB|_{\\alpha} \\cup \\{(sid, pos_s)\\}$\n}\n}\n\\lnl{p12}\\Return $\\SDB|_{\\alpha}$ \\;\n}\n\\end{small}\n\\end{algorithm}\n\n\n\\subsection{Building the projected databases.} \n\\label{sec-ProjectSDB}\nThe key issue of our approach lies in the construction of the \nprojected databases. \nWhen projecting a prefix, instead of storing the whole suffix as a\nprojected subsequence, one \ncan represent each suffix by a pair $(sid, start)$ where $sid$ is the\nsequence identifier and $start$ is the starting position of the\nprojected suffix in the \nsequence $sid$. For instance, let us consider the sequence database of Table~\\ref{tab:SDB}. As\nshown in example \\ref{exp-projection}, ${\\SDB}|_{\\angx{A}}$\nconsists of 3 suffix sequences: $\\{(1,\\angx{B C \n  B C})$, $(2,\\angx{BC})$, $(3,\\angx{B})\\}$. \nBy using the {\\it pseudo-projection}, ${\\SDB}|_{\\angx{A}}$ can be\nrepresented by the following three pairs: $\\{(1,2)$, $(2,3)$, $(3,2)\\}$. \nThis is the principle of {\\it pseudo-projection}, adopted in \\texttt{PrefixSpan}\\xspace, \nexploited during the filtering step of our \\textsc{Prefix-Projection}\\xspace global\nconstraint. Algorithm \\ref{algo:projection} details \nthis principle. It takes as input a set of projected sequences\n$ProjSDB$ and a prefix $\\alpha$.  \nThe algorithm processes all the pairs $(sid,start)$ of $ProjSDB$ one\nby one (line \\ref{p2}), and searches for the lowest location of\n$\\alpha$ in the sequence $s$ corresponding to the $sid$ of that\nsequence in $\\SDB$ (lines \\ref{p7}-\\ref{p9}). \n\nIn the worst case, \\codex{ProjectSDB} processes all the items of all\nsequences. So, the time complexity is $O(\\ell\\times m)$, with  \n$m=\\#\\SDB$ and $\\ell$ is the length of the longest sequence in \n$\\SDB$. \nThe worst case space complexity of pseudo-projection is $O(m)$, since\nwe need to store for each sequence only a pair ($sid, start$), while\nfor the standard projection the space complexity is $O(m\\times\n\\ell)$. Clearly, the pseudo-projection takes much less space than the standard projection.\n\n\\begin{algorithm}[!ht]\n\\begin{small}\n\\SetKwFunction{ProcProjection}{\\codex{Function ProjectSDB}}\n\\SetKwFunction{ProcGetFreqItems}{\\codex{Function getFreqItems}}\n\\caption{\\small \\codex{Filter-Prefix-Projection}($\\SDB$, $\\sigma$, $i$, $P$, $minsup$) \\label{algo:filter}}\n\\KwData{$\\SDB$: initial database; $\\sigma$: current prefix\n  $\\angx{\\sigma(P_1), \\ldots,\\sigma(P_i)}$; $minsup$: the minimum\n  support threshold; $\\mathcal{PSDB}$: internal data structure of\n  \\textsc{Prefix-Projection}\\xspace for storing pseudo-projected databases} \n\\Begin\n{\n\\lnl{pre1}\\If{$( i \\geq 2 \\wedge \\sigma(P_i) = \\vide)$}{ \n                  \\lnl{pre2}\\For{$j \\leftarrow i+1$ \\KwTo $\\ell$}{\n                                  \\lnl{pre3}$P_{j}  \\leftarrow \\vide$\\;\n                                 }\n                   \\lnl{pre4} \\Return True\\;\n                   }\n\\Else{\n          \\lnl{pre0} $\\mathcal{PSDB}_ {i} \\leftarrow\n          \\codex{ProjectSDB}(\\SDB, \\mathcal{PSDB}_{i-1},\n          \\angx{\\sigma(P_{i})})$\\;\n          \\lnl{proj5}\\If{$(\\#\\mathcal{PSDB}_{i} < minsup)$}{\n            \\lnl{proj6} \\Return False \\;\n          }\n          \\Else{\n                    \\lnl{proj7} $\\mathcal{FI} \\leftarrow\n                    \\codex{getFreqItems}(\\SDB, \\mathcal{PSDB}_{i}, minsup)$ \\;\n                     \\lnl{proj8}\\For{$j \\leftarrow i+1$ \\KwTo $\\ell$}{\n                                 \\lnl{proj9} \\ForEach{$a \\in D(P_j) \\, s.t. (a \\neq \\vide \\wedge a \\notin \\mathcal{FI} )$}{\n                                   \\lnl{proj10}$D(P_{j}) \\leftarrow\n                                   D(P_{j}) - \\{a\\}$\\;\n                                 }\n                                  }\n                       \\lnl{proj13} \\Return True\\;\n          }\n     }\n}\n\n\n\\ProcGetFreqItems($\\SDB$, $ProjSDB$, $minsup$) \\;\n \\KwData{$SDB$: the initial database; $ProjSDB$: pseudo-projected\n   database; $minsup$: the minimum support threshold; \n   $ExistsItem$, $SupCount$: internal data structures using a hash table\n   for support counting over items;}\n\\Begin\n{\n\\lnl{F1} $SupCount[ ] \\leftarrow \\{0, ..., 0\\}$;\\ \\ $F\\leftarrow \\emptyset$ \\;\n\\lnl{F2}\\For{each pair $(sid,start) \\in ProjSDB$}{\n\\lnl{F3} $ExistsItem[] \\leftarrow \\{false, ..., false\\}$; $s \\leftarrow SDB[sid]$ \\;\n\\lnl{F4}\\For{$i \\leftarrow start$ \\KwTo $\\# s$}{\n\\lnl{F5}$a \\leftarrow s[i]$ \\;\n\n\\lnl{F6}\\If{$(\\neg ExistsItem[a])$}{\n\\lnl{F7}$SupCount[a] \\leftarrow SupCount[a] + 1$ \\;\n \\lnl{F8}$ExistsItem[a] \\leftarrow true$\\;\n\\lnl{F10}\\If{$(SupCount[a] \\geq minsup)$}{\n\\lnl{F11}$F \\leftarrow F \\cup \\{a\\}$\\;\n}\n}\n}\n}\n\\lnl{F12}\\Return $F$\\;\n}\n\\end{small}\n\\end{algorithm}\n\n\n\\subsection{Filtering}\n\\label{Filtering}\n\nEnsuring DC on $\\textsc{Prefix-Projection}\\xspace(P, \\SDB, minsup)$ is equivalent to finding a sequential pattern of length $(\\ell-1)$ and \nthen checking whether this pattern remains a frequent pattern \nwhen extended to any item $d_{\\ell}$ in $D(P_{\\ell})$. \nThus, finding such an assignment (i.e. support) is as much as difficult than the original \nproblem of sequential pattern mining. \n\\cite{Guizhen-2006} has proved that the problem of \ncounting the number of maximal\\footnote{A sequential pattern $p$ is\nmaximal if there is no sequential pattern $q$ such that $p \\mbox{$\\preceq$} q$.}\nfrequent patterns in a database of sequences is \\#P-complete, thereby\nproving the NP-hardness of \nthe problem of mining maximal frequent sequences. \nThe difficulty is due to the exponential\nnumber of candidates that should \nbe parsed to find the frequent patterns. Thus, finding, for every\nvariable $P_i \\in\nP$ and for every $d_i \\in D(P_i)$, an assignment $\\sigma$ satisfying\n$\\textsc{Prefix-Projection}\\xspace(P, \\SDB, minsup)$ s.t. $\\sigma(P_i) = d_i$ is of \nexponential nature.  \n\nSo, the filtering of the \\textsc{Prefix-Projection}\\xspace constraint maintains a consistency lower than DC.\nThis consistency is based on specific properties of the projected\ndatabases (see Proposition~\\ref{prop-consistency}), and\nanti-monotonicity of the frequency constraint (see\nProposition~\\ref{prop-filtering}), \nand resembles forward-checking regarding Proposition~\\ref{prop-consistency}. \n\\textsc{Prefix-Projection}\\xspace is considered as a global constraint, since all variables\nshare the same internal data structures that awake and drive the\nfiltering. \n\nAlgorithm~\\ref{algo:filter} describes the  pseudo-code of the\nfiltering algorithm of the \\textsc{Prefix-Projection}\\xspace constraint.  \nIt is an incremental filtering algorithm\nthat should be run when some $i$ first variables are\nassigned according to the following lexicographic ordering $\\angx{P_1,\n  P_2, \\dots, P_\\ell}$ of variables of $P$. \nIt exploits internal data-structures enabling to enhance the\nfiltering algorithm. \nMore precisely, it uses an incremental data structure, denoted\n$\\mathcal{PSDB}$, that stores the intermediate pseudo-projections of\n$\\SDB$, where $\\mathcal{PSDB}_i$ ($i\\in [0, \\ldots, \\ell]$)\ncorresponds to the $\\sigma$-projected \ndatabase of the current partial assignment $\\sigma=\\angx{\\sigma(P_1),\n  \\ldots, \\sigma(P_i)}$ (also called prefix) of variables $\\angx{P_1,\n  \\dots, P_i}$, and $\\mathcal{PSDB}_0 = \\{(sid, 1)| (sid, s) \\in\n\\SDB\\}$ is the initial pseudo-projected database of $\\SDB$ \n(case where $\\sigma = \\angx{}$). \nIt also uses a hash table indexing the items $\\I$ into integers\n$(1 \\ldots \\#\\I)$ for an efficient support counting over items\n(see function {\\tt getFreqItems}).  \n\nAlgorithm~\\ref{algo:filter} takes as input the current partial\nassignment $\\sigma=\\angx{\\sigma(P_1), \\ldots, \\sigma(P_i)}$\nof variables $\\angx{P_1, \\dots, P_i}$, \nthe length $i$ of $\\sigma$ (i.e. position of the last assigned\nvariable in $P$) and the minimum support threshold\n$minsup$. \nIt starts by checking if the last assigned variable\n$P_i$ is instantiated to $\\vide$ (line \\ref{pre1}). In this case, the end\nof sequence is reached (since value $\\vide$ can only appear at the\nend) and the sequence $\\angx{\\sigma(P_1), \\dots, \\sigma(P_{i})}$\nconstitutes a frequent pattern in $\\SDB$; hence the algorithm sets the \nremaining $(\\ell - i)$ unassigned variables to $\\vide$ and returns\n{\\it true}  (lines \\ref{pre2}-\\ref{pre4}). Otherwise, the algorithm\ncomputes incrementally $\\mathcal{PSDB}_i$ from $\\mathcal{PSDB}_{i-1}$ \nby calling function \\codex{ProjectSDB} (see\nAlgorithm~\\ref{algo:projection}).  Then, it checks in line\n\\ref{proj5} whether the current assignment $\\sigma$ is a {\\it legal} prefix\nfor the constraint (see Proposition \\ref{prop-solution}). This is\ndone by computing the size of $\\mathcal{PSDB}_i$. If this size is less\nthan $minsup$, we stop growing $\\sigma$ and we return {\\it\n  false}. \nOtherwise, the algorithm computes the set of\n  locally frequent items $\\mathcal{F_I}$ in $\\mathcal{PSDB}_i$\nby calling function {\\tt getFreqItems} (line\n\\ref{proj7}). \n\n\nFunction  {\\tt getFreqItems} processes all the entries of the\npseudo-projected database \none by one, counts the number of first occurrences of items\n$a$ (i.e. $SupCount[a]$) in each entry $(sid, start)$, and keeps only\nthe frequent ones (lines \\ref{F1}-\\ref{F11}). \nThis is done by using $ExistsItem$ data structure. \nAfter the whole pseudo-projected database has been processed, the\nfrequent items are returned (line \\ref{F12}), and Algorithm~\\ref{algo:filter} \nupdates the current domains of variables $P_j$ with $j \\geq \n(i+1)$ by pruning inconsistent values, \nthus avoiding searching over not frequent items (lines\n\\ref{proj8}-\\ref{proj10}). \n\n\\begin{proposition}\\label{prop-complexity}\nIn the worst case, filtering with \\textsc{Prefix-Projection}\\xspace global constraint can be\nachieved in $O(m\\times\\ell + m\\times d + \\ell\\times d)$. The worst\ncase space complexity of \\textsc{Prefix-Projection}\\xspace is $O(m\\times \\ell)$. \n\\end{proposition}\n\n\\noindent\n{\\it Proof: }\nLet $\\ell$ be the length of the longest sequence in $\\SDB$, $m$ $=$\n$\\#\\SDB$, and $d$ $=$ $\\#\\I$. \nComputing the pseudo-projected database $\\mathcal{PSDB}_i$ can be\ndone in $O(m \\times \\ell)$: for \neach sequence $(sid,s)$ of $\\SDB$, checking if $\\sigma$ occurs in $s$\nis $O(\\ell)$ and there are $m$ sequences. The total complexity of\nfunction \\codex{GetFreqItems} is $O(m\\times(\\ell + d))$.  \nLines (\\ref{proj8}-\\ref{proj10}) can be achieved in $O(\\ell\\times d)$.\nSo, the whole complexity is $O(m \\times \\ell + m\\times(\\ell + d) +\n\\ell\\times d)$ $=$ $O(m\\times\\ell + m\\times d + \\ell\\times d)$. \nThe space complexity of the filtering algorithm lies in the storage of\nthe $\\mathcal{PSDB}$ internal data structure. In the worst case, we have to\nstore $\\ell$ pseudo-projected databases. Since each pseudo-projected database\nrequires $O(m)$, the worst case space complexity is $O(m\\times \\ell)$.  \n$\\Box$\n\n\n\\subsection{Encoding of SPM Constraints}\n\\label{pattern-encoding}\nUsual SPM constraints (see Section~\\ref{sec:local}) can be reformulated in a straightforward way.\nLet $P$ be the unknown pattern.\n \n\\noindent- {\\it Minimum size constraint}:\n$size(P,\\ell_{min})  \\equiv\n\\bigwedge_{i=1}^{i=\\ell_{min}} (P_i \\neq \\square)$\n\n\\noindent- {\\it Item constraint}: let $V$ be a subset of items, $l$ and $u$\ntwo integers  s.t. $0 \\leq l\\leq u \\leq \\ell$.  \n\\hspace*{-.07cm}$item(P,V) \\equiv \\bigwedge_{t \\in V} \\mbox{\\tt Among}(P,\\{t\\},l,u)$\nenforces that items of $V$ should occur at least $l$ times and at most $u$ times in $P$.\nTo forbid items of $V$ to occur in $P$, $l$ and $u$ must be set to $0$. \n\n\\noindent \n- {\\it Regular expression constraint}:\nlet $A_{reg}$ be the deterministic finite automaton encoding the regular\nexpression $exp$.\n$reg(P,exp) \\equiv {\\tt Regular}(P, A_{reg})$.\n\n\n\\subsection{Experimental protocol} \nThe implementation of our approach was carried out in the {\\tt Gecode}\nsolver\\footnote{\\url{http:\/\/www.gecode.org}}. \nAll experiments were conducted on a machine with a processor Intel X5670 and 24 GB of memory. \nA time limit of 1 hour has been used. \nFor each dataset, we varied the $minsup$ threshold until the\nmethods are not able to complete the extraction of all patterns\nwithin the time limit. \n$\\ell$ was set to the length of the longest\nsequence of $\\SDB$. The implementation and the datasets used in our\nexperiments are available\nonline\\footnote{\\url{https:\/\/sites.google.com\/site\/prefixprojection4cp\/}}. \nWe compare our approach (indicated by \\texttt{PP}\\xspace) with:\n\\begin{enumerate} \n\\item two CP encodings~\\cite{NegrevergneCPIAOR15}, the most efficient CP\n  methods for SPM: \\texttt{global-p.f}\\xspace and \n  \\texttt{decomposed-p.f}\\xspace; \n\\item state-of-the-art methods for SPM : \\texttt{PrefixSpan}\\xspace and \\texttt{cSpade}\\xspace; \n\\item \\sma~\\cite{Bonchi:2008} for SPM under regular expressions. \n\\end{enumerate}\n\nWe used the author's \\texttt{cSpade}\\xspace\nimplementation~\\footnote{\\url{http:\/\/www.cs.rpi.edu\/~zaki\/www-new\/pmwiki.php\/Software\/}} \nfor SPM, the publicly available implementations of \\texttt{PrefixSpan}\\xspace by\nY. Tabei~\\footnote{\\url{https:\/\/code.google.com\/p\/prefixspan\/}} and \nthe \\sma\nimplementation~\\footnote{\\url{http:\/\/www-kdd.isti.cnr.it\/SMA\/}} for\nSPM under regular expressions. The\nimplementation~\\footnote{\\url{https:\/\/dtai.cs.kuleuven.be\/CP4IM\/cpsm\/}}\nof the two CP encodings was carried out in the {\\tt Gecode} solver. \nAll methods have been executed on the same machine.\n\n\\begin{figure*}[t]\t\n{\\footnotesize\n\\begin{tabular}{ccc}\nBIBLE & Kosarak & PubMed \\\\\n\t\\includegraphics[width=3.8cm, height=2.8cm]{BIBLECPSM.pdf}\n&   \n\\includegraphics[width=3.8cm, height=2.8cm]{KosarakCPSM.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{PUBMEDCPSM.pdf}\n\\\\\n\\includegraphics[width=3.8cm, height=2.8cm]{numberBIBLE.pdf}\n&   \n\\includegraphics[width=3.8cm, height=2.8cm]{numberKOSARAK.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{numberPUBMED.pdf}\n\\\\\nFIFA & Leviathan & Protein \\\\\n\\includegraphics[width=3.8cm, height=2.8cm]{FIFACPSM.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{LeviathanCPSM.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{PROTEINCPSM.pdf}\n\\\\\n\\includegraphics[width=3.8cm, height=2.8cm]{numberFIFA.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{numberLEVIATHAN.pdf}\n&\n\\includegraphics[width=3.8cm, height=2.8cm]{numberPROTEIN.pdf}\n\\end{tabular}}\n\\vspace*{-.35cm}\n \n\\caption{\\small \\label{fig:FreqCPSM} Comparing \\texttt{PP}\\xspace with \\texttt{global-p.f}\\xspace\n  for SPM on real-life datasets: CPU times (top) and number of patterns (bottom).} \n\\end{figure*}\n\n\n\\subsection{Comparing with CP Methods for SPM}\nFirst we compare \\texttt{PP}\\xspace with the two CP encodings \n  \\texttt{global-p.f}\\xspace and \\texttt{decomposed-p.f}\\xspace (see Section\n  \\ref{CP4SPM}). \nFig.~\\ref{fig:FreqCPSM} shows the number of extracted sequential patterns and the CPU times\nto extract them (in logscale for BIBLE, Kosarak and PubMed) for the\nthree methods. \n\nFirst, as expected, the lower $minsup$ is, the larger the number of\nextracted sequential patterns. \nSecond, when comparing the CPU times, \\texttt{decomposed-p.f}\\xspace is the least performer\n  method. On all the datasets, it fails to complete the extraction within the time \n  limit for all values of $minsup$ we considered. \nThird, \\texttt{PP}\\xspace largely dominates \\texttt{global-p.f}\\xspace on all the datasets: \n\\texttt{PP}\\xspace is more than an order of magnitude faster than \\texttt{global-p.f}\\xspace. \nThe gains in terms of CPU times are greatly amplified for low values\nof $minsup$. On BIBLE (resp. PubMed), the speed-up is\n$84.4$ (resp. $33.5$) for $minsup$ equal to $1\\%$. \nAnother important observation that can be made is that, on most of the\ndatasets (except BIBLE and Kosarak), \\texttt{global-p.f}\\xspace is not able to mine for\npatterns at very low frequency within the time limit. \nFor example on FIFA, \\texttt{PP}\\xspace is able to\ncomplete the extraction for values of $minsup$ up to $6\\%$ in $1,457$ \nseconds, while \\texttt{global-p.f}\\xspace fails to complete the extraction for $minsup$\nless than $10\\%$. The same trend is also conformed on Leviathan, where\n\\texttt{global-p.f}\\xspace is not able to mine for patterns at $1\\%$ minimum frequency.  \n\n\nTo complement the results given by  Fig.~\\ref{fig:FreqCPSM},\nTable~\\ref{table:SolverStat} reports for different datasets and different\nvalues of $minsup$, the number of calls to the propagate routine of\n{\\tt Gecode} (column 5), and the number of nodes of the search tree (column\n6). \nFirst, \\texttt{PP}\\xspace explores less nodes than \\texttt{global-p.f}\\xspace. But, the difference is\nnot huge (gains of 45\\%  and 33\\% on FIFA and BIBLE\nrespectively). \nSecond, our approach is very\neffective in terms of number of propagations. For \\texttt{PP}\\xspace, the number of \npropagations remains small (in thousands for small values of $minsup$)\ncompared to  \\texttt{global-p.f}\\xspace (in millions). \nThis is due to \nthe huge number of reified constraints used in \\texttt{global-p.f}\\xspace to encode the\nsubsequence relation. On the contrary, our \\textsc{Prefix-Projection}\\xspace global constraint\ndoes not require any reified constraints nor any extra variables to\nencode the subsequence relation.\n\n\\begin{table*}[t] \\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|l|l|l|r|r|r|r|r|r|}\n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{$minsup$ (\\%)} &\n\\multirow{2}{*}{\\#PATTERNS} & \\multicolumn{2}{c|}{CPU times (s)} & \\multicolumn{2}{c|}{\\#PROPAGATIONS} & \\multicolumn{2}{c|}{\\#NODES}\\\\\n\\cline{4-9}\n&  &  & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace \\\\\n\\hline\n\\multirow{6}{*}{FIFA} &20 & 938 & {\\bf 8.16} & 129.54 & {\\bf 1884} & 11649290 & {\\bf 1025} & 1873 \\\\\n&18 & 1743 &{\\bf 13.39} & 222.68 & {\\bf 3502} & 19736442  & {\\bf 1922} & 3486 \\\\\n&16 & 3578 & {\\bf 24.39} & 396.11 &{\\bf 7181} & 35942314 & {\\bf 3923} & 7151\\\\\n&14 & 7313 & {\\bf 44.08} & 704 & {\\bf 14691} & 65522076 & {\\bf 8042} & 14616\\\\\n&12 & 16323 & {\\bf 86.46} & 1271.84 & {\\bf 32820} & 126187396 & {\\bf 18108} & 32604 \\\\\n&10 & 40642 & {\\bf 185.88} & 2761.47 & {\\bf 81767 } & 266635050  & {\\bf 45452} & 81181  \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{BIBLE} &10 & 174 &{\\bf 1.98}  & 105.01 & {\\bf 363}  & 4189140& {\\bf 235} & 348 \\\\\n&8 & 274 & {\\bf 2.47}  & 153.61 & {\\bf 575}  & 5637671 & {\\bf 362} & 548 \\\\\n&6 & 508 & {\\bf 3.45}  & 270.49 & {\\bf 1065} & 8592858 & {\\bf 669} & 1016 \\\\\n&4 & 1185 & {\\bf 5.7}  & 552.62 & {\\bf 2482} & 15379396 & {\\bf 1575}  & 2371 \\\\\n&2 & 5311 & {\\bf 15.05}  & 1470.45 & {\\bf 11104}  & 39797508  &  {\\bf 7048} & 10605  \\\\ \n&1 & 23340 & {\\bf 41.4} & 3494.27 & {\\bf 49057}  & 98676120 & {\\bf 31283} & 46557 \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{PubMed} &5 &  2312 & {\\bf 8.26} &  253.16 & {\\bf 4736}   & 15521327  & {\\bf 2833}  &  4619 \\\\ \n&4 & 3625 & {\\bf 11.17}  & 340.24 &{\\bf 7413} & 20643992 & {\\bf 4428}  & 7242 \\\\\n&3 & 6336 & {\\bf 16.51} & 536.96 &{\\bf 12988} &29940327  & {\\bf 7757} & 12643 \\\\\n&2 & 13998 & {\\bf 28.91} & 955.54 & {\\bf 28680} &50353208  & {\\bf 17145} & 27910  \\\\\n&1 & 53818 & {\\bf 77.01} & 2581.51 & {\\bf 110133} &  124197857 &  {\\bf 65587} &  107051 \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{Protein} &99.99 & 127 &{\\bf 165.31} & 219.69 & {\\bf 264} & 26731250 & {\\bf 172} & 221 \\\\\n&99.988 & 216 & {\\bf 262.12} & 411.83 & {\\bf 451} & 44575117& {\\bf 293} & 390\\\\\n&99.986 & 384 & {\\bf 467.9}6 & 909.47& {\\bf 805} &80859312 & {\\bf 514} & 679\\\\\n&99.984 & 631 & {\\bf 753.3} & 1443.92 & {\\bf 1322} & 132238827 & {\\bf 845} & 1119\\\\\n&99.982 & 964& {\\bf 1078.73} & 2615 & {\\bf 2014} & 201616651 &  {\\bf 1284} & 1749\\\\ \n&99.98 & 2143 & {\\bf 2315.65} & {\\bf $-$} & {\\bf 4485}  & {\\bf $-$} & {\\bf 2890} & {\\bf $-$} \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{Kosarak} &1 & 384 &{\\bf 2.59} & 137.95 & {\\bf 793} & 8741452 & {\\bf 482} & 769\\\\\n&0.5 & 1638 & {\\bf 7.42} & 491.11 & {\\bf 3350} & 26604840 & {\\bf 2087} & 3271\\\\\n&0.3 & 4943& {\\bf 19.25} & 1111.16& {\\bf 10103} & 56854431& {\\bf 6407} &  9836\\\\\n&0.28 & 6015 & {\\bf 22.83} & 1266.39 & {\\bf 12308} & 64003092 & {\\bf 7831} & 11954\\\\\n&0.24 & 9534 & {\\bf 36.54} & 1635.38 & {\\bf 19552} & 81485031 &  {\\bf 12667} & 18966 \\\\\n&0.2  & 15010& {\\bf 57.6} & 2428.23 & {\\bf 30893} & 111655799 &  {\\bf 20055} & 29713 \\\\\n\\hline\n\\hline\n\\multirow{6}{*}{Leviathan} &10 & 651 &{\\bf 1.78} & 12.56 & {\\bf 1366} &  2142870& {\\bf 849} & 1301\\\\\n&8 & 1133 & {\\bf 2.57} & 19.44 & {\\bf 2379} &3169615  & {\\bf 1487} & 2261\\\\\n&6 & 2300 & {\\bf 4.27} &32.85 & {\\bf 4824} &5212113 & {\\bf 3008} &4575 \\\\\n&4 & 6286 & {\\bf 9.08} &  66.31& {\\bf 13197} &10569654  & {\\bf 8227} &12500  \\\\\n&2 & 33387& {\\bf 32.27} & 190.45& {\\bf 70016} &  33832141&  {\\bf 43588} &66116 \\\\ \n&1 & 167189& {\\bf 121.89} & $-$ & {\\bf 350310} & $-$ &  {\\bf 217904} & $-$\\\\ \n\\hline\n\\end{tabular}\n}\n\\caption{\\texttt{PP}\\xspace vs. \\texttt{global-p.f}\\xspace.} \n\\label{table:SolverStat}\n\\end{table*}\n\\begin{figure*}[t]\n{\\footnotesize\n\\begin{tabular}{ccc}\nBIBLE & Kosarak & PubMed   \\\\\n\\includegraphics[width=4cm, height=3.0cm]{BIBLEC.pdf\n&   \n\\includegraphics[width=4cm, height=3.0cm]{KosarakC.pdf\n&\n\\includegraphics[width=4cm, height=3.0cm]{PUBMEDC.pdf\n\\\\\nFIFA & Leviathan & Protein \\\\\n\\includegraphics[width=4cm, height=3.0cm]{FIFAC.pdf\n&\n\\includegraphics[width=4cm, height=3.0cm]{LeviathanC.pdf\n&\n\\includegraphics[width=4cm, height=3.0cm]{PROTEINC.pdf}\n\\vspace*{-.35cm}\n\\end{tabular}}\n\n\\caption{\\small \\label{fig:FreqCLASSIC} Comparing \\textsc{Prefix-Projection}\\xspace with\n  state-of-the-art algorithms for SPM.} \n\\end{figure*}\n\n\\subsection{Comparing with ad hoc Methods for SPM}\nOur second experiment compares \\texttt{PP}\\xspace with state-of-the-art methods  \nfor SPM. Fig.~\\ref{fig:FreqCLASSIC} shows the CPU times of the three \nmethods.\nFirst, \\texttt{cSpade}\\xspace obtains the best\nperformance on all datasets (except on Protein).  \nHowever, \\texttt{PP}\\xspace exhibits a similar behavior as \\texttt{cSpade}\\xspace, but it is less\nfaster (not counting the highest values of $minsup$).  \nThe behavior of \\texttt{cSpade}\\xspace on Protein is due to the\n  vertical representation format that is not appropriated in the\n  case of databases having large sequences and small number of\n  distinct items, thus degrading the performance of the mining\n  process.\nSecond, \\texttt{PP}\\xspace which also uses the concept of projected databases,\nclearly outperforms \\texttt{PrefixSpan}\\xspace on all datasets. \nThis is due to\nour filtering algorithm combined together with incremental data\nstructures to manage the projected databases.  \nOn FIFA, \\texttt{PrefixSpan}\\xspace is not\nable to complete the extraction for $minsup$ less than $12\\%$, while\nour approach remains feasible until $6\\%$ within the time limit. \nOn Protein, \\texttt{PrefixSpan}\\xspace fails to complete the\n  extraction for all values of $minsup$ we considered. \nThese results clearly demonstrate that our approach competes well with\nstate-of-the-art methods for SPM on large datasets and achieves scalability \nwhile it is a major issue of existing CP approaches. \n\n\\subsection{SPM under size and item constraints}\nOur third experiment aims at assessing the interest of pushing\nsimultaneously different types of constraints. We impose on the PubMed\ndataset usual constraints such as {\\it the minimum frequency} and the\n{\\it minimum size} constraints and other useful constraints expressing\nsome linguistic knowledge such as {\\it the item\n  constraint}. The goal is to retain sequential patterns which convey\nlinguistic regularities (e.g., gene - rare disease relationships)~\\cite{BCCC2012cbms}.  \n{\\it The size constraint} allows to\nremove patterns that are too small w.r.t. the\nnumber of items (number of words) to be relevant patterns. We tested\nthis constraint with $\\ell_{min}$ set to 3.  \n{\\it The item constraint} imposes that the extracted patterns must\ncontain the item GENE and the item DISEASE. \nAs no ad hoc method exists for this combination of \nconstraints, we only compare \\texttt{PP}\\xspace with\n\\texttt{global-p.f}\\xspace. \nFig.~\\ref{fig:const} shows the CPU times and the number of\nsequential patterns extracted \nwith and without constraints. \nFirst, pushing simultaneously the two constraints enables to reduce\nsignificantly the number of patterns. Moreover, the CPU times for \\texttt{PP}\\xspace decrease \nslightly whereas for \\texttt{global-p.f}\\xspace (with and without\nconstraints), they are almost the same. \nThis\nis probably due to the weak communication between\nthe $m$ \\texttt{exists-embedding} reified global constraints and the\ntwo constraints. This reduces significantly the quality of the\nwhole filtering. \nSecond (see Table~\\ref{table:const:stat}), when considering the\ntwo constraints, \\texttt{PP}\\xspace clearly dominates \\texttt{global-p.f}\\xspace (speed-up value up to\n$51.5$). Moreover, the number of propagations performed by \\texttt{PP}\\xspace remains \nvery small as compared to \\texttt{global-p.f}\\xspace. \nFig.~\\ref{fig:const:prop} compares the two methods under \nthe minimum size constraint for different values of \n$\\ell_{min}$, with $minsup$ fixed to\n$1\\%$. Table~\\ref{table:const:stat} compares the two methods in terms\nof numbers of propagations (column $5$) and number of nodes of the\nsearch tree (column $6$). \nOnce again, \\texttt{PP}\\xspace is always the most performer method (speed-up\nvalue up to $53.1$). These results also confirm what we observed previously,\nnamely the weak communication between reified global constraints \nand constraints imposed on patterns (i.e., size and item constraints). \n\n\\begin{figure*}[t]\n\\centering\n\\subfloat[\\# of patterns\\label{fig:const:patterns}]{\n\\includegraphics[width=4cm, height=2.6cm]{numberPP-CPSM-const.pdf}\n}\n\\subfloat[CPU times (logscale) \\label{fig:const:time}]{\n\\includegraphics[width=4cm, height=2.8cm]{PP-CPSM-const.pdf}\n}\n\\subfloat[Minimum size constraint\\label{fig:const:prop}]{\n\\includegraphics[width=4cm, height=2.6cm]{PP-CPSM-size.pdf}\n}\n\\vspace*{-.35cm} \n\\caption{\\small Comparing \\texttt{PP}\\xspace with \\texttt{global-p.f}\\xspace under minimum size and item constraints on PubMed.} \\label{fig:const}\n\\end{figure*}\n\\begin{table*}[t] \\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|l|l|l|r|r|r|r|r|r|}\n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{$minsup$ (\\%)} &\n\\multirow{2}{*}{\\#PATTERNS} & \\multicolumn{2}{c|}{CPU times (s)} & \\multicolumn{2}{c|}{\\#PROPAGATIONS} & \\multicolumn{2}{c|}{\\#NODES}\\\\\n\\cline{4-9}\n&  &  & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace \\\\\n\\hline\n\\multirow{6}{*}{PubMed} & 5 & 279 & {\\bf 6.76} & 252.36 & {\\bf 7878} & 12234292  & {\\bf 2285} & 4619  \\\\ \n&4 & 445 & {\\bf 8.81}  & 339.09 & {\\bf 12091} & 16475953 & {\\bf 3618}  & 7242 \\\\\n&3 & 799 & {\\bf 12.35} & 535.32 & {\\bf 20268} & 24380096 & {\\bf 6271}   & 12643 \\\\\n&2& 1837 & {\\bf 20.41} & 953.32 & {\\bf 43088}   &  42055022 & {\\bf 13888}  & 27910  \\\\ \n&1 & 7187 & {\\bf 49.98} & 2574.42 & {\\bf 157899}  &  107978568 & {\\bf 52508}  &  107051 \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\\small {\\tt PP} vs. \\texttt{global-p.f}\\xspace under minimum size and item constraints.} \n\\label{table:const:stat}\n\\end{table*}\n\n\\begin{table*}[t] \\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|l|l|l|r|r|r|r|r|r|}\n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{$\\ell_{min}$} &\n\\multirow{2}{*}{\\#PATTERNS} & \\multicolumn{2}{c|}{CPU times (s)} & \\multicolumn{2}{c|}{\\#PROPAGATIONS} & \\multicolumn{2}{c|}{\\#NODES}\\\\\n\\cline{4-9}\n&  &  & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace & {\\tt PP} & \\texttt{global-p.f}\\xspace \\\\\n\\hline\n\\multirow{6}{*}{PubMed} & 8 & 12 & {\\bf 48.52} & 2577.09 & {\\bf 55523} & 105343528  & {\\bf 50264} &  107051\\\\ \n& 6 & 3596 & {\\bf 50.91} & 2576.9 & {\\bf 59144} & 106272419  & {\\bf 50486} & 107051 \\\\ \n& 4 & 40669 & {\\bf 70.61} & 2579.3 & {\\bf 96871} & 117781215  & {\\bf 59194} &  107051 \\\\ \n& 2 & 53486 & {\\bf 76.64} & 2580.41  & {\\bf 109801} &  123913176 & {\\bf 65334} & 107051 \\\\ \n& 1 & 53818 & {\\bf 78.49} & 2579.85  & {\\bf 110133} &  117208559 & {\\bf 65587} &  107051\\\\ \n\\hline\n\\end{tabular}\n}\n\\caption{\\small {\\tt PP} vs. \\texttt{global-p.f}\\xspace under minimum size constraint.} \n\\label{table:const:stat}\n\\end{table*}\n\n\\subsection{SPM under regular constraints}\nOur last experiment compares \\prefixCPREG against two variants of \\sma:\n\\smap (\\sma one pass) and\n\\texttt{SMA-FC}\\xspace (\\sma Full Check). \nTwo datasets are considered from~\\cite{Bonchi:2008}: one synthetic\ndataset ({data-200k}), and one real-life dataset (Protein). \nFor {data-200k}, we used two RE: \n\\begin{itemize}\n\\item $\\mathtt{RE}10 \\equiv A^*B(B|C)D^*EF^*(G|H)I^*$, \n\\item $\\mathtt{RE}14 \\equiv A^*(Q|BS^*(B|C))D^*E(I|S)^*(F|H)G^*R$. \n\\end{itemize}\n\nFor {Protein}, we used $\\mathtt{RE}2 \\equiv (S|T)$ $.$ $(R|K)$\nrepresenting {\\it Protein kinase C phosphorylation} \n(where $.$ represents any symbol).  \nFig.~\\ref{fig:reg} reports CPU-times comparison. \nOn the synthetic dataset, our approach is very effective. \nFor $\\mathtt{RE}14$, our\nmethod is more than an order of magnitude faster than \\sma. \nOn Protein, the gap between the $3$ methods\nshrinks, but our method remains effective. For the particular\ncase of $\\mathtt{RE}2$, the {\\tt Regular} constraint can  be substituted\nby restricting the domain of the first and third variables to\n$\\{S,T\\}$ and $\\{R,K\\}$ respectively (denoted as \\texttt{PP-SRE}\\xspace), thus\nimproving performances. \n\n\\begin{figure*}[t]\n{\\footnotesize\n\\begin{tabular}{ccc}\ndata-200k (RE10) & data-200k (RE14) & Protein (RE2) \\\\  \n\\includegraphics[width=4cm, height=3cm]{REG-data200k-RE10-col.pdf}\n&   \n\\includegraphics[width=4cm, height=3cm]{REG-data200k-RE14-col.pdf}\n&\n\\includegraphics[width=4cm, height=3cm]{REG-protein-col.pdf}\n\t\\end{tabular}}\n\\vspace*{-.35cm} \n\\caption{\\small \\label{fig:reg} Comparing \\textsc{Prefix-Projection}\\xspace with \\sma for SPM under RE constraint.}\n\\end{figure*}\n\n\n\n\\section{Introduction}\n\\label{section:introduction}\n\\input{1-introduction.tex}\n\n\\section{Preliminaries}\n\\label{section:preliminaries}\n\\input{2-context.tex}\n\n\\vspace*{-.15cm}\n\\section{Related works}\n\\label{PB-related-works}\n\\input{3-relatedworks.tex}\n\n\\section{\\textsc{Prefix-Projection}\\xspace Global Constraint}\n\\label{sec:model}\n\\input{4-projection.tex}\n\n\n\\section{Experimental Evaluation}\n\\label{section:experimentations}\n\\input{6-experiments.tex}\n\n\\section{Conclusion}\n\\label{section:conclusion}\n\\input{7-conclusion.tex}\n\n\\newpage\n\\bibliographystyle{splncs03}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum Hamiltonians modelling condensed matter typically exhibit a lattice structure. In the idealized setting one uses operators which are periodic { with respect to  } the lattice, while more realistic models, taking into account disorder present in the model, consist in random operators which are homogeneous and ergodic { with respect to  } lattice translations. In the framework of quantum mechanics the key  to understanding properties of the underlying physical system is the spectrum of the operator.  It is a fundamental fact in the theory of random operators that the spectrum of an ergodic or metrically-transitive ensemble of operators is almost surely non-random, in the sense that there exists a fixed set $\\Sigma \\subset \\RR$ such that for almost all elements in the ensemble, the spectrum of the operator coincides with $\\Sigma$.\nThis holds even in an abstract Hilbert space setting, cf. e.g. \\cite{KirschM-82a}.\n\n\nThe structure or shape of the spectrum as a subset of the real axis and its measure theoretic features are intimately connected to conductance and transport properties of the condensed matter.\nWith the emergence of disorder the spectrum of as a set grows. In order to understand this expansion in a quantitative way, one introduces a scalar parameter coupling the random part of the Hamiltonian to the periodic background operator.  In the simplest case the coupling is linear.  The question of interest in the present paper is: At what rate does the spectrum expand, as the scalar coupling increases? This is the first step in the analysis of more delicate spectral properties.\nIndeed, it is precisely the new portions of the spectrum generated by the random perturbation of the Hamiltonian where interesting phenomena (like a transition from point to continuous spectrum) are expected to occur. We consider here only the expansion of spectrum at its lowest edge, although it seems that a refinement of our methods together with substantially more technical effort would apply to general spectral edges as well.\n\n\nThe \\emph{motivation for the present work} is the following:\nThe most commonly encountered situation for weak-disorder random Schr\\\"odinger operators\nis that the spectrum expands linearly with the disorder --- provided that sign conspiracy does not prevent expansion altogether.\n(If $-\\Delta$ is perturbed by a non-negative potential, there is obviously no expansion of the spectrum.)\nHowever, there are examples where the expansion is quadratic in the disorder parameter. So it is natural to ask whether even\n\\emph{slower growth} (cubic, quartic, \\ldots) \\emph{is possible to occur?}\nIn the most important case, growth can occur only either with a linear or a quadratic rate,\nas we discuss in Corollary \\ref{c:1}  and Remark \\ref{r:1} below.\n\n\n\nMore generally, we derive in the present paper\nestimates on the expansion of the spectrum which are in line  with earlier research,\nin particular on the weak disorder regime, {e.~g.}\n\\cite{aizenman1994localization}, \\cite{wang2001},\n\\cite{klopp2002a}, \\cite{klopp2002b}, \\cite{elgart2009lifshitz}, \\cite{elgartetal2011locnonmono}, \\cite{caoelgart2012}, \\cite{hoeckerescuti2013}, \\cite{hoeckerescuti2014},\n\\cite{borisovveselic2011}, \\cite{borisovveselic2013}. Let us give some more details:\nIn \\cite{aizenman1994localization} the fractional moment method is developed to show localization for the Anderson model in the weak disorder regime,\n\\cite{wang2001} treats the Anderson model as well and gives an estimate on the size of the energy interval where localization occurs as a function of the disorder parameter.\nThis bound has been refined in \\cite{klopp2002a} and \\cite{elgart2009lifshitz}, while a similar result for random operator on $\\RR^d$ has been derived in\n\\cite{klopp2002b}.\nIn the later paper the single site potential is allowed to change sign, but needs to have a non-vanishing average.\nIn the context of the present paper, this means the spectral edge is shifted linearly as a function of the disorder parameter.\nDiscrete alloy type models on $\\ZZ^d$ have been analyzed in \\cite{elgartetal2011locnonmono},\nand for $\\ZZ^3$ an analogous result to \\cite{klopp2002b} has been derived in \\cite{caoelgart2012}. It covers also the case\nwhen the average of the single site potential is zero, which requires a more detailed analysis.\nIn particular,  this means the spectral edge is shifted quadratically as a function of the disorder parameter.\nA particular regime where one can prove Anderson localization is the Lifshitz tail region near the spectral bottom.\nUpper and lower bounds on the size of the region (again as a function of the disorder parameter) have been given in\n\\cite{hoeckerescuti2013} and \\cite{hoeckerescuti2014}. The results mentioned so far concern models where a random potential\nis coupled linearly to a kinetic energy term. In contrast to this \\cite{borisovveselic2011} and \\cite{borisovveselic2013} analyse\nlow lying eigenvalues of (long, finite segments of) randomly shifted and curved waveguides, respectively.\nIn this case the random perturbation term depends in a nonlinear way on the random variables and\nthe disorder parameter and consist of (lower order) differential operators.\n\nThe present paper\nprovides a fundamental analysis about the location of the spectrum which is a prerequisite for identifying the energy  region\nof Anderson localization.\nFor a broader discussion of the physical intuition and the relevance of our result to the general understanding of spectral properties of random Hamiltonians\nwe refer to the discussion in our previous work \\cite{BorisovHEV}.\nThere we have carried out an analogous analysis for discrete Hamiltonians, i.e. matrix operators over $\\ell^2(\\ZZ^d)$. In the discrete setting one has less technical questions to take care of, for instance properly defined domains or compactness properties, either because certain operators are automatically bounded, or because auxiliary Hilbert spaces are finite dimensional.\nNevertheless, our approach carries over to a very general class of operators in continuum space. In the present paper we chose not to present the most general model class (which we may do in some later manuscript) but to restrict ourselves to the case that the unperturbed periodic part of the Hamiltonian is a differential operator, and the perturbations satisfy certain relative boundedness conditions. This framework has the advantage that it on one hand covers a variety of physically relevant cases, but on the other hand is specific enough to avoid a long list of abstract hypotheses.\n\n\nWe close the section\nhighlighting  the general scope, the flexible versatility, and the technical advancements of our approach.\nIn comparison to mentioned previous work on spectral properties of random Schr\\\"odinger type operators in the weak-disorder regime\nthe present paper has the following new features\n\\begin{itemize}\n \\item The results are formulated for very general types of random operators (actually quadratic forms).\n\n \\item {The results require only weak assumptions on the random variables entering the model. In particular, the range of values need not be of fixed sign, nor an interval.}\n\n \\item {No monotonicity condition for the random perturbations is assumed.}\n\n \\item {We prove both upper and lower bounds for the bottom of the perturbed spectrum.}\n\\end{itemize}\nThis is motivated by the fact that the mechanism which determines the expansion of the spectrum is\nof quite universal nature. So it makes more sense to establish it once and for all models of practical importance,\nrather than to repeat the argument in each setting separately.\n\n\n\n\\section{Model and main results}\n\n\nLet $x=(x_1,\\ldots,x_d)$ be Cartesian coordinates in $\\mathds{R}^d$, $d\\geqslant d_1\\geqslant 1$, $e_1,\\ldots,e_{d_1}$ be linearly independent vectors in $\\mathds{R}^d$, and $\\G$ be the lattice $\\{x\\in\\mathds{R}^d:\\, x=z_1 e_1 +\\ldots + z_{d_1}e_{d_1},\\,z_i\\in \\mathds{Z}\\}$. By $\\Pi$ we denote an infinite domain in $\\mathds{R}^d$ which is invariant  with respect to   shifts along $\\G$, namely, for each $x\\in\\Pi$, $k\\in\\G$ we have $x-k\\in\\Pi$.\nLet $\\square$ be a periodicity cell of $\\Pi$, i.e., a minimal domain such that $\\Pi$ is the interior of $\\overline{\\bigcup\\limits_{k\\in\\G} \\square_k}$, $\\square_k:=\\{x\\in\\mathds{R}^d:\\, x-k\\in \\square \\}$.\n\nLet $m\\geqslant 1$ be a given natural number,\n$A_{\\a\\b}=A_{\\a\\b}(x)$, $\\a,\\b\\in\\mathds{Z}_+^d$, $|\\a|,|\\b|\\leqslant m$, be functions defined on $\\Pi$ satisfying the conditions:\n\\begin{equation}\\label{2.1}\n\\begin{aligned}\n&A_{\\a\\b}\\ \\text{are $\\square$-periodic},\\quad A_{\\a\\b}\\in C^{|\\a|}(\\overline{\\Pi}),\n\\quad \\overline{A_{\\b\\a}}=A_{\\a\\b}\n\\\\\n&\\sum\\limits_{\\genfrac{}{}{0 pt}{}{\\a,\\b\\in\\mathds{Z}_+^d}{|\\a|=|\\b|=m}} A_{\\a\\b}(x)\\xi^{\\a+\\b}\n\\geqslant c_\n|\\xi|^{2m},\\quad x\\in\\overline{\\Pi},\\quad \\xi\\in\\mathds{R}^d,\n\\end{aligned}\n\\end{equation}\nwhere for $\\xi=(\\xi_1,\\ldots,\\xi_d)$, $\\a+\\b=(\\a_1+\\b_1,\\ldots,\\a_2+\\b_2)$ we denote $\\xi^{\\a+\\b}:=\\xi_1^{\\a_1+\\b_1}\\cdot\\ldots\\cdot\\xi_d^{\\a_d+\\b_d}$,\nand $c_0$ is a fixed positive constant independent of $x$ and $\\xi$.\nThe boundary of the domain $\\Pi$ is assumed to be $C^{2m}$-smooth.\nIn $L_2(\\Pi)$ we consider the operator\n\\begin{equation}\\label{2.2}\n\\Op_0:=  \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|}\\p^\\a A_{\\a\\b} \\p^\\b,\\quad \\p^\\a:=\\frac{\\p^\\a\\,}{\\p x^\\a}\n\\end{equation}\nsubject to the Dirichlet condition\n\\begin{equation}\\label{2.3}\nu=0,\\quad \\frac{\\p^j u}{\\p\\nu^j}=0,\\quad \\text{on}\\quad \\p\\Pi,\\quad j=1,\\ldots,m-1,\n\\end{equation}\nwhere $\\nu$ is the outward normal to $\\p\\Pi$.\nThanks to  conditions (\\ref{2.1}), the operator $\\Op_0$ is self-adjoint, elliptic, and lower  semi-bounded\non the domain\n\\begin{equation*}\n\\Dom(\\Op_0):=\\{u\\in W_2^{2m}(\\Pi):\\, \\text{boundary conditions (\\ref{2.3}) are satisfied}\\}.\n\\end{equation*}\nThe associated sesquilinear form is\n\\begin{equation}\\label{2.4}\n\\fm_0(u,v)= \\sum\\limits_{ \\genfrac{}{}{0 pt}{}{\\a,\\b\\in\\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} (A_{\\a\\b} \\p^\\b u , \\p^\\a v )_{L_2(\\Pi)} \\quad\\text{in}\\quad L_2(\\Pi)\n\\end{equation}\non the domain\n\\begin{equation*}\n\\Dom(\\fm_0):=\\{u\\in W_2^{m}(\\Pi):\\, \\text{boundary conditions (\\ref{2.3}) are satisfied}\\}.\n\\end{equation*}\nThe above stated facts about $\\Op_0$ can be proven in the same way as for second order differential operators,\none should just  employ appropriate smoothness improving theorems, see \\cite[Ch. III, Sect. 6, Lm. 6.3]{Ber}.\n\n\nLet  $\\omega_k$, $k\\in\\G$, be a sequence  of bounded, non-trivial, independent, identically distributed random variables with distribution measure $\\mu$. We assume that\n\\begin{equation*}\n\\{s_-,s_+\\}\\in \\supp\\mu \\subseteq [s_-,s_+],\n\\end{equation*}\nwhere the\nnumbers $s_\\pm$ satisfy one of the following alternatives:\n\\begin{equation}\\label{2.7}\ns_-<0<s_+\n\\end{equation}\nor\n\\begin{equation}\\label{2.8}\n0\\leqslant s_-<s_+.\n\\end{equation}\nWithout loss of generality we suppose that each random variable $\\omega_k$ is defined on the probability space $(\\supp \\mu,\\mathfrak{B},\\mu)$, where $\\mathfrak{B}$ is the Borel $\\si$-algebra on $\\supp \\mu\\subset [s_-,s_+]$,\nand $\\omega_k$ are just the identity mappings on $\\supp \\mu$.\nThus the random field  $(\\omega_k)_{k\\in\\G}$\nis an element of the product probability space $(\\Om, \\bigotimes_\\G \\mathfrak{B}, \\PP)$ with\nmeasure $\\PP:=\\bigotimes_{k\\in\\G} \\mu$ and configuration space $\\Om:=\\times_{k\\in\\G}\\supp \\mu$.\n\nWe let $\\g_\\Pi:=\\p\\square\\cap\\p\\Pi$, $\\g_l:=\\p\\square\\setminus\\p\\Pi$.\nBy $\\Ho^{2m}(\\square,\\g_\\Pi)$ we denote the subspace of $W_2^{2m}(\\square)$ consisting of functions satisfying the boundary conditions\n\\begin{equation}\\label{2.5}\nu=0,\\quad \\frac{\\p^j u}{\\p\\nu^j}=0\\quad \\text{on}\\quad \\g_\\Pi,\\quad j=1,\\ldots,m-1.\n\\end{equation}\nLet $\\cL(t)$, $t\\in[-t_0,t_0]$, $t_0>0$, be a family of operators from $\\Ho^{2m}(\\square,\\g_\\Pi)$ into $L_2(\\square)$ defined as\n\\begin{equation}\\label{2.6}\n\\cL(t)=t \\cL_1 + t^2 \\cL_2 + t^3 \\cL_3(t),\n\\end{equation}\nwhere $\\cL_i$ are bounded symmetric operators from $\\Ho^{2m}(\\square,\\g_\\Pi)$ into $L_2(\\square)$;\nmoreover,  the operator $\\cL_3(t)$ is assumed to be bounded uniformly in $t$.\nWe also suppose that the map\n$$\nt \\mapsto \\left(\\cL_3(t): \\Ho^{2m}(\\square, \\g_\\Pi)\\to L_2(\\square)\\right)\n$$\nis continuous in $t\\in[-t_0,t_0]$.\n\nBy $\\cS(k)$, $k\\in\\G$, we denote the shift operator: $(\\cS(k)u)(x):=u(x-k)$.\nThe main operator of our study is\n\\begin{equation}\\label{2.8a}\n\\Op_\\e(\\om):=\\Op_0 + \\cL_\\e(\\om),\n\\quad \\om:=(\\omega_k)_{k\\in\\G},\\quad \\cL_\\e(\\om):=\\sum\\limits_{k\\in\\G} \\cS(k) \\cL(\\e\\omega_k) \\cS(-k),\n\\end{equation}\nin $L_2(\\Pi)$ on the domain $\\Dom(\\Op_\\e(\\om)):=\\Dom(\\Op_0)$. Here $\\e$ is a small positive parameter.\n\n\nLet us clarify the action of the operator $\\cL_\\e(\\om)$. Given $u\\in\\Dom(\\Op_0)$, it is clear that the restriction of $u$ on $\\square_k$\nbelongs to $\\Ho^{2m}(\\square_k,\\g_\\Pi^k)$, $\\g_\\Pi^k:=\\p\\square_k\\cap\\p\\Pi$. Then $(\\cS(-k)u)(x)=u(x+k)$ is an element of $\\Ho^{2m}(\\square,\\g_\\Pi)$ and the action of $\\cL(\\e\\omega_k)$ on $\\cS(-k)u$ is well-defined as an element of $L_2(\\square)$.\nBy applying $\\cS(k)$ to the result of the action, we just shift the function $\\cL(\\e\\omega_k)\\cS(-k)u$ to  the cell $\\square_k$.\nIn this way each operator $\\cS(k) \\cL(\\e\\omega_k) \\cS(-k)$ acts on $\\Ho^{2m}(\\square_k,\\g_\\Pi^k)$\nand the operator $\\cL_\\e(\\om)$ is a sum of such single cell actions.\nSince the operators $\\cL_i$ are $\\Op_0$-bounded and symmetric, by  the Kato-Rellich theorem  the operator $\\Op_\\e(\\om)$ is self-adjoint for sufficiently small $\\e$.\n\n\n\n\n\nOur main aim is to study the behavior of the spectrum $\\spec(\\Op_\\e(\\om))$ of the operator $\\Op_\\e(\\om)$.\nThe later results will consider the asymptotic behaviour  for (very) small $\\e>0$.\nFor our first main result we merely require that $\\e$ is sufficiently small so that for all $\\omega$ the\nperturbed operator $\\Op_\\e(\\om)$ is self-adjoint.\n\n\\begin{theorem}\\label{th2.1}\nFor all sufficiently small $\\e$ there exists a closed set $\\Sigma_\\e$ such that\n\\begin{equation}\\label{2.9}\n\\spec(\\Op_\\e(\\om))=\\Sigma_\\e\\quad\\PP-a.s.\n\\end{equation}\nThe set $\\Sigma_\\e$ is equal to the closure of the union of spectra of periodic realizations of $\\Op_\\e(\\om)$, explicitly,\n\\begin{equation}\\label{2.10}\n\\Sigma_\\e=\\overline{\\bigcup\\limits_{N\\in \\NN} \\quad \\bigcup\\limits_{\\xi \\ \\text{is $2^N \\G$-periodic}} \\spec\\big(\\Op_\\e(\\xi)\\big)},\n\\end{equation}\nwhere the second union is taken over all sequences $\\xi: \\G\\to\\supp \\mu $, which are periodic  with respect to   the sublattice $2^N \\G:=\\{2^N q:\\, q\\in\\G\\}$.\n\\end{theorem}\nHere we adopt the following convention: In statements which are deterministic, i.e.~valid for all configurations\n$\\xi\\colon \\G \\to \\supp\\mu$ we will denote the sequences by $\\xi$, in statements which are probabilistic, e.g.~hold only for almost all configurations, we will use the symbol $\\omega$ for the configuration $\\omega\\colon \\G \\to \\supp\\mu$.\n\nThe next part of our results is devoted to the position of $\\inf \\Sigma_\\e$.\nMore precisely, we shall describe how $\\inf \\spec(\\Op_0)$ is shifted by the perturbation $\\cL_\\e(\\om)$.\nFirst we describe the spectrum of $\\Op_0$. Since this operator is periodic, we can employ a Floquet-Bloch decomposition to find $\\spec(\\Op_0)$. We introduce the Brillouin zone\n\\begin{equation*}\n\\square^*:=\\{\\theta\\in\\la e_1,\\ldots,e_{d_1}\\ra:\\, \\theta=\\theta_1^* e_1^*+\\ldots +\\theta_{d_1}^* e_{d_1}^*,\\, \\theta_i^*\\in[0,1)\\},\n\\end{equation*}\nwhere $e_1^*,\\ldots,e_{d_1}^*$ are the vectors in the linear span $S:=\\la e_1,\\ldots,e_{d_1}\\ra$ defined by the conditions\n\\begin{equation*}\n(e_i,e_j^*)_{\\mathds{R}^d}=2\\pi\\d_{ij},\n\\end{equation*}\nand $\\d_{ij}$ is the Kronecker delta.\nOn $\\square$ we introduce the operator\n\\begin{equation}\\label{2.11}\n\\Op_0(\\theta):\n\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|} (\\p + i \\theta)^\\a A_{\\a\\b} (\\p + i \\theta)^\\b \n\\end{equation}\nsubject to boundary conditions (\\ref{2.5}) and to periodic boundary conditions on $\\g_l$.\nHere, $\\iu$ is the imaginary unit and for a given $\\a=(\\a_1,\\ldots,\\a_d)\\in \\mathds{Z}_+^d$, $\\theta=(\\theta_1,\\ldots,\\theta_d)\\in S$, the symbol $(\\p + i \\theta)^\\a$ stands for the differential expression\n\\begin{equation*}\n\\left(\\frac{\\p\\,}{\\p x_1}+ i\\theta_1\\right)^{\\a_1} \\left(\\frac{\\p\\,}{\\p x_2}+ i\\theta_2\\right)^{\\a_2}\\cdot\\dots\\cdot\\left(\\frac{\\p\\,}{\\p x_d}+ i\\theta_d\\right)^{\\a_d}.\n\\end{equation*}\nThe spectrum of $\\Op_0$ is given by the identity \\cite[Ch. 4, Sect. 4.5, Thm. 4.5.1]{K93}\n\\begin{equation}\\label{2.12}\n\\spec(\\Op_0)=\\bigcup\\limits_{\\theta\\in\\square^*} \\spec(\\Op_0(\\theta)).\n\\end{equation}\nWe assume that $\\inf\\spec(\\Op_0(\\theta))$ is a discrete eigenvalue for all $\\theta\\in\\square^*$.\nWe denote this eigenvalue by $E_0(\\theta)$.\nThe function $\\theta\\mapsto E_0(\\theta)$ is $\\square^*$-periodic and continuous in $\\overline{\\square^*}$, a compact set.\nConsequently it attains the global minimum at some point $\\theta_0\\in\\square^*$:\n\\begin{equation}\\label{2.14}\n\\L_0:=E_0(\\theta_0)=\\min\\limits_{\\square^*} E_0(\\theta).\n\\end{equation}\nThere is a positive constant  $c_1$ independent of $\\theta\\in\\square^*$ such that\n\\begin{equation}\\label{2.13}\n\\dist\\Big(E_0(\\theta),\\spec(\\Op_0(\\theta))\\setminus\\{E_0(\\theta)\\}\\Big)\\geqslant c_1,\n\\end{equation}\nAgain by compactness and formula (\\ref{2.12})\n\\begin{equation}\\label{2.15}\n\\inf\\spec(\\Op_0)=\\L_0.\n\\end{equation}\nThe eigenfunctions of $\\Op_0(\\theta_0)$ associated with $E_0(\\theta_0)$ and orthonormalized in $L_2(\\square)$ are denoted by $\\psi_0^{(j)}$, $j=1,\\ldots,n$.\nWe choose these eigenfunctions so that the matrix with the entries\n$(\\cL_1 e^{\\iu \\theta_0 x}\\psi_0^{(i)},e^{\\iu \\theta_0 x}\\psi_0^{(j)})_{L_2(\\square)}$ is diagonal.\nThis  is possible by the theorem on simultaneous diagonalization of two quadratic forms in a finite-dimensional space.\n\n\nConsider the $2n $ numbers\n\\begin{equation*}\ns_-(\\cL_1 \\psi_0^{(i)},\\psi_0^{(i)})_{L_2(\\square)},\\quad i=1,\\ldots,n, \\quad s_+(\\cL_1 \\psi_0^{(i)},\\psi_0^{(i)})_{L_2(\\square)},\\quad i=1,\\ldots,n.\n\\end{equation*}\nAssume that the minimal value is attained at an index $i=i_0$ and for $s_*\\in\\{s_-,s_+\\}$. We denote\n\\begin{equation}\\label{2.16}\n\\L_1:=(\\cL_1e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)},\\quad \\psi_0:=\\psi_0^{(i_0)}.\n\\end{equation}\nBy $\\psi_1$ we denote the unique solution to the equation\n\\begin{equation}\\label{2.17}\n(\\Op_0(\\theta_0)-\\L_0) \\psi_1=-e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x}\\psi_0+\\L_1\\psi_0\n\\quad\n\\perp \\ker  (\\Op_0(\\theta_0)-\\Lambda_0)\n\\end{equation}\nsuch that this solution is orthogonal to $\\psi_0^{(j)}$, $j=1,\\ldots,n$. We let\n\\begin{equation}\\label{2.18}\n\\L_2:=(\\cL_1e^{\\iu \\theta_0 x}\\psi_1,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)} +(\\cL_2e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}.\n\\end{equation}\nOur next main result reads as follows.\n\\begin{theorem}\\label{th2.2}\nFor any minimizing triple $\\theta_0\\in\\square^*$, $i_0\\in\\{1, \\ldots,n\\}$, $s_*\\in\\{s_-,s_+\\}$\nas defined above, there exist $C,\\e _0 \\in(0,\\infty)$ such that for all $\\e \\in [0, \\e _0]$\nand for all sequences $\\xi\\colon \\Gamma \\to [s_-,s_+]$\n\\begin{equation}\\label{2.19}\n\\spec(\\Op_\\e(\\xi))\\subseteq\\big\\{\\l\\in\\mathds{R}:\\, \\dist(\\l,\\spec(\\Op_0))\\leqslant C\\e(|\\l|+1)\\big\\}.\n\\end{equation}\nFor the bottom of the almost sure spectrum $\\Sigma_\\e$ the estimate\n\\begin{equation}\\label{2.20}\n\\Sigma_\\e\n\\leqslant \\L_0+\n\\e s_* \\L_1 + \\e^2 s_*^2 \\L_2 + \\frac{\\e^3 s_*^3 \\L_3(\\e s_*)}{1+\\e^2 s_*^2 \\|\\psi_1\\|_{L_2(\\square)}^2}\n\\end{equation}\nholds true, where $\\L_3 \\colon \\RR \\to \\RR$ is a continuous function, in particular, uniformly  bounded on compact intervals.\n\\end{theorem}\nIn fact, one can show the following explicit representation for $\\L_3(t)$:\n\\begin{equation}\\label{2.21}\n\\begin{aligned}\n\\L_3(t)=  &-(\\L_1+t\\L_2)\\|\\psi_1\\|_{L_2(\\square)}^2\n+2 \\RE ( \\cL_2 e^{\\iu \\theta_0 x}\\psi_0,  e^{\\iu \\theta_0 x}\\psi_1)_{L_2(\\square)}\\\\\n & + \\big( (\\cL_1+t\\cL_2) e^{\\iu \\theta_0 x}\\psi_1,  e^{\\iu \\theta_0 x}\\psi_1\\big)_{L_2(\\square)}\n\\\\\n&+\\big(\\cL_3(t) e^{\\iu \\theta_0 x}\\psi_0 (\\psi_0+t\\psi_1),e^{\\iu \\theta_0 x}(\\psi_0+t\\psi_1)\\big)_{L_2(\\square)}.\n\\end{aligned}\n\\end{equation}\n\n\n\\begin{remark}\\label{rm2.2}\nIf we have several minimizing triples $\\{\\tht_0,i_0,s_*\\}$, estimate (\\ref{2.20}) is valid for each of them.\nIn this situation one should choose a triple, for which $s_* \\L_1$ is minimal.\nIf all such quantities are same, then one should minimize $s_*^2\\L_2$.\n\\end{remark}\n\n\n\nFor many types of periodic operators $\\Op_0$ and generic perturbations $\\cL_1$ the coefficient $\\L_1$ does not vanish,\nand one sees a linear shift of $\\Sigma_\\e$. Let us consider the case $\\L_1=0$.\nWe will show that in this case, for many models,\nthe almost sure spectrum $\\Sigma_\\e$ must expand at least quadratically\n ---  i.e.~cubic, quartic, or weaker expansions cannot occur.\n\n\\begin{corollary}\\label{c:1}\nAssume that $\\L_1=0$ and $\\cL_2$ is a non-positive operator, cf.~(\\ref{2.6}) and (\\ref{2.16}).\nThen\n\\[\n\\L_2 \\leqslant - c_1 \\|\\psi_1\\|_{L_2(\\square)}^2\n\\]\nwhere $c_1$ is the strictly positive spectral gap in (\\ref{2.13}).\n\\end{corollary}\n\n\n\\begin{remark} \\label{r:1}\nThus the question arises, in what situations we can ensure that $\\psi_1$ does not vanish,\ni.e.~that $e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x}\\psi_0$ is not the zero vector.\nThis is for instance the case if $\\Op_0$ is the pure Laplacian and $\\cL_1$ a multiplication operator $V$.\nIn this case $\\theta_0=0$, $\\Op_0(0)$ is the Laplacian with periodic boundary conditions, $\\psi_0$\nis the normalized constant function, so that $V\\psi$ is only identically zero if $V$ itself is trivial.\nMore generally, $\\Op_0$ can be any operator satisfying a unique continuation property for eigenfunctions\nor the Harnack inequality for the ground state.\n\nLikewise, $\\psi_1$ cannot vanish if $\\Op_0$ satisfies the Harnack inequality and\n$e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x}$ is a positivity preserving operator.\\footnote{An $L_2$-function is called positive if $f$ is nonnegative almost everywhere and does not\nvanish identically. A self-adjoint operator $A$ on an $L_2$-space is called positivity preserving if $Af$ is positive\nwhenever $f$ is in the domain of $A$ and positive.}\n\\end{remark}\n\n\n\nOur next result provides a lower bound for $\\inf\\Sigma_\\e$.\nFirst we need to introduce additional notations and assumptions:\nGiven $u,v\\in \\Ho^{2m}(\\square,\\g_\\Pi)$, by integration by parts it is easy to convince oneself that\n\\begin{equation}\\label{2.23}\n\\begin{aligned}\n\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} & (-1)^{|\\a|} \\big( (\\p + \\iu \\theta_0)^\\a A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b  u,v\\big)_{L_2(\\square)}\n\\\\\n&=\\sum\\limits_{j=1}^{m} (\\cB_{2m-j}u,\\cB_{j-1} v)_{L_2(\\g_l)}\n+ \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} \\big( A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b  u, (\\p + \\iu \\theta_0)^\\a v\\big)_{L_2(\\square)},\n\\end{aligned}\n\\end{equation}\nwhere $\\cB_j=\\cB_j(x,\\p)$ are linear differential operators of order at most $j$ with continuous coefficients defined on $\\overline{\\g_l}$.\nSince the functions $A_{\\a\\b}$ are $\\square$-periodic, the above formula remains true if we replace $\\square$ by $\\square_k$.\nWe make the following assumption:\n\n\\begin{enumerate}\\def\\theenumi{(A\\arabic{enumi})}\n\\item \\label{A1} There are real-valued functions  $b_j\\in C(\\overline{\\p\\square})$ such that\n\\begin{equation}\\label{2.25}\nb_j=\\frac{\\cB_{2m-j}\\psi_0}{\\cB_{j-1}\\psi_0} \\chi_{\\{\\cB_{j-1}\\psi_0\\neq0\\}}\n\\quad \\text{ on} \\quad  \\g_l,\\quad j=1,\\ldots,m,\n\\end{equation}\nwhere $\\chi_\\natural$ is the characteristic function of a set $\\natural$.\nAssume also that the coefficients of the boundary operators\n$\\cB_{2m-j}-b_j\\cB_{j-1}$ belong to $C^j(\\g_l)$.\n\\end{enumerate}\n\nOn $\\Ho^m(\\square,\\g_l)$  we introduce the sesquilinear form\n\\begin{equation}\\label{2.26}\n\\widehat{\\fm}_0(u,v):= \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} \\big( A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b  u, (\\p + \\iu \\theta_0)^\\a v\\big)_{L_2(\\square)} + \\sum\\limits_{j=1}^{m} (b_j \\cB_{j-1}u,\\cB_{j-1}v)_{L_2(\\g_l)}.\n\\end{equation}\nThanks to conditions (\\ref{2.1}) and assumption \\ref{A1}, this form is symmetric, lower-semibounded and closed.\nBy $\\widehat{\\Op}_0$ we denote the self-adjoint operator in $L_2(\\square)$ associated with the form $\\widehat{\\fm}_0$.\nThis is the operator\n\\begin{equation}\\label{2.27}\n\\widehat{\\Op}_0= \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|}(\\p + \\iu \\theta_0)^\\a A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b,\n\\end{equation}\nin $L_2(\\square)$ subject to the Dirichlet condition on $\\g_\\Pi$ and to the condition\n\\begin{equation}\\label{2.28}\n(\\cB_{2m-j}-b_j \\cB_{j-1})u=0\\quad \\text{on}\\quad \\g_l.\n\\end{equation}\nThe domain of $\\widehat{\\Op}_0$ consists of the functions in $\\Ho^{2m}(\\square,\\g_{\\Pi})$\nsatisfying the above boundary conditions on $\\g_l$.\n\n\nIt is straightforward to check that $\\psi_0$ is an eigenfunction of $\\widehat{\\Op}_0$ associated with the eigenvalue $\\L_0$. Moreover, we make one more assumption\n\\begin{enumerate}\\def\\theenumi{(A\\arabic{enumi})}\\setcounter{enumi}{1}\n\\item \\label{A2} The bottom of the spectrum of $\\widehat{\\Op}_0$ is a simple eigenvalue  and equal to $\\L_0$.\n\\end{enumerate}\n\nLet $\\widehat{\\psi}_1\\in\\Dom(\\widehat{\\Op}_0)$ be the solution to the equation\n\\begin{equation*}\n\\big(\\widehat{\\Op}_0-\\L_0\\big)\\widehat{\\psi}_1=-e^{-\\iu \\theta_0 x}\\cL_1e^{\\iu \\theta_0 x}\\psi_0 + \\L_1\\psi_0\n\\end{equation*}\nsuch that this solution is orthogonal to $\\psi_0$ in $L_2(\\square)$. We denote\n\\begin{equation}\\label{2.29}\n\\widehat{\\L}_2:=(\\cL_2e^{ \\iu \\theta_0 x}\\psi_0,e^{ \\iu \\theta_0 x}\\psi_0)_{L_2(\\square)} + (\\cL_1 e^{ \\iu \\theta_0 x}\\widehat{\\psi}_1,e^{ \\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}.\n\\end{equation}\n\nThe main result concerning a lower bound is provided by the following theorem.\n\n\\begin{theorem}\\label{th2.3}\nAssume \\ref{A1} and \\ref{A2}. For all sufficiently small $\\e\\geqslant 0$ the estimate\n\\begin{equation}\\label{2.30}\n\\inf\\Sigma_\\e\\geqslant\n\\L_0+\\e s_* \\L_1+\\e^2 s_*^2\\widehat{\\L}_\n - C\\e^3\n\\end{equation}\nholds true, where $C$ is a non-negative constant independent of $\\e$.\n\\end{theorem}\n\nAn immediate corollary of Theorems~\\ref{th2.2} and \\ref{th2.3} is that under Assumptions~\\ref{A1},~\\ref{A2} we have the asymptotics\n\\begin{equation}\\label{2.31}\n\\inf\\spec(\\Op_\\e(\\om))=\\L_0+\\e s_*\\L_1+O(\\e^2)\\quad\\PP-a.s.\n\\end{equation}\n{Moreover, if $\\widehat{\\L}_2=\\L_2$ for at least one of the minimizing triples $(\\theta_0,i_0,s_*)$, the asymptotics is even more precise}\n\\begin{equation}\\label{2.32}\n\\inf\\spec(\\Op_\\e(\\om))=\\L_0+\\e s_*\\L_1+\\e^2 s_*^2\\L_2+O(\\e^3)\\quad\\PP-a.s.\n\\end{equation}\nNote that the additional assumptions~\\ref{A1},~\\ref{A2} required for the lower bound (\\ref{2.30}) are not very restrictive.\nA simple example is provided by the  operator $\\Op_0$ in $\\Pi=\\mathds{R}^d$ in the case $A_{\\a\\b}=0$ for $|\\a|<m$ or $|\\b|<m$. Then the ellipticity condition in (\\ref{2.1}) implies that $(\\Op_0 u,u)_{L_2(\\mathds{R}^d)}\\geqslant 0$.\nThe lowest eigenvalue of the operator $\\Op_0(\\theta)$ attains its minimum $\\L_0=0$ for $\\theta=0$; the associated eigenfunction $\\psi_0$ is just a constant: $\\psi_0=|\\square|^{-\\frac{1}{2}} \\mathbf{1}$, where $\\mathbf{1}$ stands for the function being $1$ everywhere in $\\square$.  This eigenvalue is simple.\nAll the associated functions $b_j$ vanish identically  and the form $\\widehat{\\fm}_0$ is given by the formula\n\\begin{equation*}\n\\widehat{\\fm}_0(u,v)= \\sum\\limits_{ \\genfrac{}{}{0 pt}{}{\\a,\\b\\in\\mathds{Z}_+^d}{|\\a|,|\\b|=m}} (A_{\\a\\b} \\p^\\b u, \\p^\\a v )_{L_2(\\square)}.\n\\end{equation*}\nThe quadratic form $\\widehat{\\fm}_0(u,u)$ is non-negative and its value vanishes only for the  function $\\psi_0$.\nIt means that in this case both Assumptions~\\ref{A1},~\\ref{A2} are satisfied.\nThe constant  $\\L_1$ is given by the formula\n\\begin{equation*}\n\\L_1=\\frac{1}{|\\square|} (\\cL_1\\mathbf{1},\\mathbf{1})_{L_2(\\square)}.\n\\end{equation*}\nThe number $s_*$ is fixed by the inequalities\n\\begin{equation*}\ns_*\\L_1\\leqslant s_+\\L_1,\\quad s_*\\L_1\\leqslant s_-\\L_1.\n\\end{equation*}\nThe asymptotics (\\ref{2.31}) holds true.\n\n\nIt should be also mentioned that we choose Dirichlet boundary condition (\\ref{2.5}) just for simplicity.\nThe type of  boundary condition is not used in our proofs and the Dirichlet condition can be replaced by\nany other boundary condition ensuring the self-adjointness of the operator $\\Op_0$ on an appropriate subspace of $W_2^{2m}(\\Pi)$.\nThe operators $\\cL_i$ are to be symmetric on functions in $W_2^{2m}(\\square)$ satisfying the boundary condition on $\\g_\\Pi$ involved in the definition of $\\Op_0$. Then the main results remain true.\n\n\n\n\\section{Examples}\n\nIn this section we discuss examples of unperturbed operators $\\Op_0$ and perturbations $\\cL_\\e$.\nFirst of all we stress that\nall the examples we discuss below are considered in an arbitrary periodic domain $\\Pi$.\nClassical examples of the differential operator $\\Op_0$ are Schr\\\"odinger operators\n\\begin{equation*}\n\\Op_0=-\\D+V(x),\n\\end{equation*}\nmagnetic Schr\\\"odinger operators\n\\begin{equation*}\n\\Op_0=(\\iu\\nabla+A)^2,\n\\end{equation*}\noperators with a variable metric\n\\begin{equation*}\n\\Op_0=-\\sum\\limits_{i,j=1}^{d} \\frac{\\p}{\\p x_i } A_{ij} \\frac{\\p}{\\p x_j},\n\\end{equation*}\nor the bi-Laplacian\n\\begin{equation*}\n\\Op_0=\\Delta^2.\n\\end{equation*}\nIn the above examples all the coefficients $V$, $A$, $A_{ij}$ are supposed to be periodic  with respect to   $\\G$ and smooth enough.\n\n\nOne more class of examples are Schr\\\"odinger operators with $\\d$-interaction on a periodic system of disjoint bounded manifolds. Namely, let $M_0\\subset \\square$ be a given $C^3$-manifold\n{of codimension one without boundary}.\nBy $M$ we denote the periodic system of manifolds obtained by the shifts of $M_0$ along $\\G$. We consider the operator\n\\begin{equation}\\label{5.1}\n\\Op_0=-\\D+V\\quad \\text{in}\\quad \\Pi\n\\end{equation}\nsubject to the Dirichlet boundary condition on $\\p \\Pi$ and to the boundary conditions\n\\begin{equation}\\label{5.2}\n[u]_M=0,\\quad \\left[\\frac{\\p u}{\\p\\nu}\\right]_M=b u|_{M}=0,\n\\end{equation}\nwhere $[\\cdot]_M$ denotes the jump of a function on $M$, $\\nu$ is the normal vector to $M$ and $b\\in C^2(M)$ is a given function periodic  with respect to   lattice $\\G$.\nOf course, the above introduced operator does not fit our assumptions\nsince its domain is not $\\Ho^2(\\Pi,{\\p\\Pi})$. But it was shown in \\cite[Sect. 8, Ex. 5]{Bo07-AHP}, \\cite[Sect. 8, Ex. 5]{Bo07-MPAG}\nthat there exists an explicit unitary transformation that reduces operator (\\ref{5.1}), (\\ref{5.2}) into a self-adjoint differential operator with domain $\\Ho^2(\\Pi,{\\p\\Pi})$.\nThis new operator has the same spectrum as the operator introduced in (\\ref{5.1}) and  (\\ref{5.2}).\n\n\nThe class of examples of operators $\\cL_i$  is wider than for the operator $\\Op_0$.\nThe reason is that the operators $\\cL_i$ are not necessarily differential ones.\nOf course, symmetric differential operators are standard examples for $\\cL_i$:\n\\begin{equation*}\n\\cL_i=\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} (-1)^{|\\a|}\\p^\\a B_{\\a\\b}\\p^b,\n\\end{equation*}\nwhere $B_{\\a\\b}$  are given functions  defined on $\\square$ satisfying the conditions:\n\\begin{equation*}\n B_{\\a\\b}\\in W_\\infty^{|\\a|}(\\square),\n\\quad \\overline{B_{\\b\\a}}=B_{\\a\\b}.\n\\end{equation*}\nThe values on $\\g_l$ are to be chosen so that by the integration by parts we get\n\\begin{equation*}\n \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} (-1)^{|\\a|} (\\p^\\a B_{\\a\\b}\\p^b u,v)_{L_2(\\square)}=\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}} ( B_{\\a\\b}\\p^b u, \\p^\\a v)_{L_2(\\square)},\\quad u,v\\in\\Ho^{2{m}}(\\square,\\g_\\Pi),\n\\end{equation*}\nand all the boundary integrals over $\\g_l$ vanish. In particular, this\nincludes potential and magnetic fields. In this case our results are applicable to the perturbation by a random potential and a random magnetic field.\n\nOur next example is an integral operator:\n\\begin{equation*}\n(\\cL_i u)(x)=\\int\\limits_{\\square} K(x,y)u(y)\\di y,\n\\end{equation*}\nwhere the kernel $K(x,y)$ is such that $\\cL_i$ is a symmetric operator. A sufficient condition is, for instance,\n\\begin{equation*}\nK(x,y)=\\overline{K(y,x)}\\quad\\text{and}\\quad \\int\\limits_{\\square\\times\\square} |K(x,y)|^2 \\di x\\di y<\\infty.\n\\end{equation*}\nAlso, more complicated examples like integro-differential operators or pseudodifferential operators are possible.\n\n\nThe next class of examples concerns the situation where a considered operator fits our assumptions\nafter certain transformations.\nHere the first example is the above mentioned $\\d$-interaction, see (\\ref{5.1}), (\\ref{5.2}). As $\\cL_i$, we can also choose a $\\d$-interaction of such type.\nOne just needs to replace the second boundary condition in (\\ref{5.2}) by\n\\begin{equation*}\n\\left[\\frac{\\p u}{\\p\\nu}\\right]_{M_k} + \\e \\omega_k b u\\big|_{M_k}=0,\n\\end{equation*}\nwhere $M_k$ is the shift of $M_0$ by $k\\in\\G$. Then we deal with a random $\\d$-potential.\nAnd again by applying the approach of \\cite[Sect. 8, Ex. 5]{Bo07-AHP}, \\cite[Sect. 8, Ex. 5]{Bo07-MPAG}\nwe can reduce the operator with  a $\\d$-interaction to a differential operator with the same spectrum.\nThis differential operator satisfies the representation (\\ref{2.8a}).\n\n\nThe next example is a small random deformation of a boundary. There are various ways how to introduce it, we dwell only on one of them.\nLet $\\nu$ be the outward normal to $\\p\\Pi$, $\\z$ be the distance measured along $\\nu$ and $\\rho\\in C^2(\\g_\\Pi)$ be a given function vanishing in the vicinity of $\\overline{\\g_l}\\cap\\p\\Pi$.\nWe introduce a domain $\\Pi_\\e(\\om)$ whose boundary is defined as\n\\begin{equation*}\n\\p\\Pi_\\e(\\om):=\\{x:\\, \\theta=\\e G(\\om,x)\\},\\quad G(\\om,x):=\\sum\\limits_{k\\in\\G} \\omega_k \\rho(x-k).\n\\end{equation*}\nIn the domain $\\Pi_\\e(\\om)$ we consider the operator\n\\begin{equation*}\n\\Op_\\e(\\om):=\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|}\\p^\\a A_{\\a\\b} \\p^\\b\n\\end{equation*}\nsubject to the Dirichlet condition with coefficients $A_{\\a\\b}$ satisfying (\\ref{2.1}). As in \\cite[Sect. 8, Ex. 4]{Bo07-MPAG},\none can construct a mapping of the domain $\\Pi_\\e(\\om)$  onto $\\Pi$ and a unitary transformation of $\\Op_\\e(\\om)$\nsuch that\nthe transformed  perturbed operator satisfies representation (\\ref{2.8a}) and has the same spectrum.\n\nOur next two examples are borrowed from \\cite{BKSh15}.\nLet $\\Wl=\\Wl(x)$ be a continuous compactly supported function defined on $\\mathds{R}^d$, and $\\Ws=\\Ws(x,\\z)$, $\\z=(\\z_1,\\ldots,\\z_d)$,\nstands for a function in $\\square\\times\\mathds{R}^{d}$\nbeing  1-periodic  with respect to   each of the variables $\\z_i$, $i=1,\\ldots,d$, having a zero mean\n\\begin{equation*}\n\\int\\limits_{(0,1)^{d}} \\Ws(x,\\xi)\\di\\xi=0\\quad \\text{for each} \\quad x\\in\\square,\n\\end{equation*}\nand compactly supported  with respect to   $x$:\n\\begin{equation*}\n\\supp \\Ws(\\cdot,\\xi)\\subseteq Q\\subset \\square \\quad\\text{for each}\\quad \\xi\\in\\mathds{R}^d,\n\\end{equation*}\nwhere $Q$ is a some fixed set. We assume the following smoothness for the function $\\Ws$:\n\\begin{equation*}\n\\frac{\\p^{|\\a|+|\\b|}\\Ws}{\\p x^\\a \\p\\xi^\\b} \\in C(\\square\\times\\mathds{R}^d),\\quad \\a,\\b\\in\\ZZ_+^d,\\quad |\\a|\\leqslant 3, \\quad |\\b|\\leqslant 1.\n\\end{equation*}\n We let\n\\begin{align*}\n&\\Wloc(x',\\e):=\\e^{-a} \\Wl\\left(\\frac{x'}{\\e}\\right), \\quad\n \\Wosc(x,\\e):=\\e^{-a} \\Ws\\left(x,\\frac{x}{\\e}\\right),\n \\\\\n & \\Wloc(x',0):=0,\\qquad \\Wosc(x',0):=0,\n\\end{align*}\nwhere $0\\leqslant a<1$ is a given number.\nThe potential $\\Wloc$ has {large} values due to the presence of the factor $\\e^{-a}$\nbut is localized on a set of diameter $\\e$. The potential $\\Wosc$ has also {large} values,\nis localized on set $Q$ and oscillates fast due to the presence of $\\frac{x}{\\e}$ in its definition.\n\n\nThe perturbed operators are introduced as\n\\begin{equation*}\n\\Op_\\e(\\om):=-\\D+ \\sum\\limits_{k\\in\\G} \\Wloc(\\cdot-k,\\e\\omega_k),\\quad \\Op_\\e(\\om):=-\\D+\\sum\\limits_{k\\in\\G} \\Wosc(\\cdot-k,\\e \\omega_k).\n\\end{equation*}\nIt was shown in  \\cite{BKSh15} that there exist unitary transformations reducing each of the above perturbed operators to operators satisfying representation (\\ref{2.8a}).\n\n\nWe also stress that any combination of the above perturbed operators is also covered by out general results. For instance, one can consider a combination of a random magnetic field with a random deformation of a boundary, a random integral operator and a $\\d$-interaction, a fast oscillating potential and a potential localized on a small set, etc.\n\n\n\n\n\\section{Proof of Theorem~\\ref{th2.1}}\nTo prove statement (\\ref{2.9}), we follow the ideas of \\cite{KirschM-82a}.\nIn accordance with Theorem~1 in that paper, it is sufficient to check the following conditions:\n\\begin{enumerate}\n\\item \\textsl{Measurability.}\n  For each $z\\in\\mathds{C}\\setminus\\mathds{R}$ the mapping $\\Om\\ni \\{\\xi_k\\}_{k\\in\\G}\\mapsto (\\Op_\\e(\\xi)-z)^{-1}$ defined on our product probability space $(\\Om,\\mathfrak{F},\\PP)$, where $\\mathfrak{F}:=\\bigotimes_\\G \\mathfrak{B}$ is the product $\\si$-algebra, is weakly measurable. Namely, for each $u,v\\in L_2(\\Pi)$ the mapping $\\{\\xi_k\\}_{k\\in\\G}\\mapsto \\big((\\Op_\\e(\\xi)-z)^{-1}u,v\\big)_{L_2(\\Pi)}$ is $\\PP$-measurable.\n\\item \\textsl{Homogeneity.}\n  There exists an index set $I$, a family of measure preserving transformations $T_i\\colon \\Om\\to\\Om$, $i\\in I$,\n  and a family of unitary operators $U_i$, $i\\in I$, on $L_2(\\Pi)$ satisfying for all $i \\in I$\n    \\begin{equation} \\label{eq:covariance}\n    \\Op_\\e( T_i \\xi)=U_i \\Op_\\e(\\xi) U_i^*.\n    \\end{equation}\n\\item \\textsl{Ergodicity.}\n    Any $A \\in \\mathfrak{F}$ such that $T_i^{-1}A=A$ for each $i\\in I$ satisfies  $\\PP(A)=0$ or $\\PP(A)=1$.\n\\end{enumerate}\nWe check the measurability following the argument of Proposition~6 in \\cite{KirschM-82a}.\nFirst we observe that by the continuity of $\\cL_3(t)$ in $t$, the scalar function $t\\mapsto (\\cL_3(t)u,v)_{L_2(\\square)}$ is measurable for each $u\\in\\Ho^{2m}(\\square,\\g_\\Pi)$, $v\\in L_2(\\square)$.\nThen we represent the resolvent of $\\Op_\\e(\\xi)$ as\n\\begin{align*}\n\\big(\\Op_\\e(\\xi)-z\\big)^{-1}=&(\\Op_0+\\cL_\\e(\\xi)-z)^{-1}\n\\\\\n=&(\\Op_0-z)^{-\\frac{1}{2}} \\big(I+(\\Op_0-z)^{-\\frac{1}{2}}\\cL_\\e(\\xi)(\\Op_0-z)^{-\\frac{1}{2}}\\big)^{-1} (\\Op_0-z)^{-\\frac{1}{2}}.\n\\end{align*}\n\nBy the lemma following Proposition~6 in \\cite{KirschM-82a} the operator $\\big(I+(\\Op_0-z)^{-\\frac{1}{2}}\\cL_\\e(\\xi)(\\Op_0-z)^{-\\frac{1}{2}}\\big)^{-1}$ is measurable and it implies the desired measurability of $\\Op_\\e(\\xi)$.\n\nTo show homogeneity, we let\n$I:=\\G$, $T_i(\\{\\xi_k\\}_{k\\in\\G}):=\\{\\xi_{k+i}\\}_{k\\in\\G}$, $U_i:=\\cS(i)$.\nIt is obvious that the operators $\\cS(i)$ are unitary and $\\cS^*(i)=\\cS^{-1}(i)=\\cS(-i)$. Employing definition (\\ref{2.8a}) of operator $\\Op_\\e(\\om)$ and the obvious identity $\\cS(k)\\cS(i)=\\cS(k+i)$, by straightforward calculations we see that\nCondition (\\ref{eq:covariance}) is satisfied.\nSince the marginal distributions of the product measure $\\PP$ are all equal, the mappings $T_i$ are measure preserving, namely, $\\PP(T_i^{-1}(A))=\\PP(A)$ for each $i\\in I$, $A\\in\\mathcal{F}$.\nMoreover, the product structure of $\\PP$ implies that the family $T_i, i\\in I$ is mixing, and in particular ergodic.\nThus there exists indeed a set $\\Sigma_\\e\\subseteq\\RR$ which is almost surely the spectrum of $\\Op_\\e(\\om)$.\nIn particular,  $\\Sigma_\\e$ is closed.\n\nIn order to prove (\\ref{2.10}), we adapt a similar proof in \\cite{KirschM-82b} (see Theorems~3,~4 and Lemma~2 in this paper).\nGiven $N\\in\\NN$, we denote\n\\begin{equation*}\n\\G_N:=\\{x\\in\\mathds{R}^d:\\, x=z_1 e_1 +\\ldots + z_{d_1}e_{d_1},\\,z_i\\in \\mathds{Z},\\, |z_i|\\leqslant N\\},\n\\end{equation*}\nand $\\Pi_N$ is the interior of $\\overline{\\bigcup\\limits_{k\\in\\G_N}\\square_k}$. For $\\xi,\\om\\in\\Omega$, $q\\in\\G$, we let\n\\begin{equation}\\label{3.0}\nD_N(\\xi,\\om,q):=\\left(\\sum\\limits_{k\\in\\G_N} \\|\\cL(\\e\\xi_k)-\\cS(q)\\cL(\\e\\omega_k) \\cS(-q)\\|_{\\Ho^{2m}(\\square_k,\\g_\\Pi^k)\\to L_2(\\square_k)}\\right)^{\\frac{1}{2}},\n\\end{equation}\nwhere $\\|\\cdot\\|_{X\\to Y}$ stands for the norm of an operator acting from Hilbert space $X$ into Hilbert space $Y$. We introduce the set\n\\begin{equation}\\label{3.4}\nB_\\xi:=\\left\\{\\om\\in\\Omega:\\, \\forall\\; N\\in \\NN \\  \\forall\\; p \\in \\NN \\\n\\exists\\;  q(N,p,\\xi)\\in\\G:\\, D_N(\\xi,\\om,q)<\\frac{1}{p} \\right\\}.\n\\end{equation}\n\n\\begin{lemma}\\label{lm3.1}\nFor each $\\xi\\in\\Om$ we have $\\PP(B_\\xi)=1$.\n\\end{lemma}\n\n\\begin{proof}\nWe follow the proof of Lemma~2 in \\cite{KirschM-82b}. We let\n\\begin{equation} \\label{eq:BxiNP}\nB_{\\xi,N,p}:=\\left\\{\\om\\in\\Omega:\\,\n\\exists\\; q(\\om)\\in\\G:\\, D_N(\\xi,\\om,q)<\\frac{1}{p}\\right\\}.\n\\end{equation}\nThe definition of $D_N$ implies $D_{N+1}(\\xi,\\om,p)\\geqslant D_N(\\xi,\\om,p)$ and hence $B_{\\xi,N+1,p}\\subseteq B_{\\xi,N,p}$.\nFurthermore $B_{\\xi,p+1}\\subseteq B_{\\xi,p}$.\nSet $B_{\\xi,p}:=\\bigcap\\limits_{N=1}^{\\infty} B_{\\xi,N,p}$\nand $B_\\xi:=\\bigcap\\limits_{p=1}^{\\infty} B_{\\xi,p}$.\nMonotone continuity of probability measures implies then\n\\begin{align*}\n& \\PP(B_{\\xi,p})=\\lim\\limits_{N\\to\\infty} \\PP(B_{\\xi,N,p}),\n\\quad \\PP(B_\\xi)=\n\\lim\\limits_{p\\to\\infty} \\PP(B_{\\xi,p}).\n\\end{align*}\nThus, it is sufficient to prove that $\\PP(B_{\\xi,N,p})=1$ for each $N,p\\in \\NN$.\nSince the set $B_{\\xi,N,p}$ is invariant under the shifts $\\{T_i\\}_{i\\in\\G}$,\nit is by ergodicity sufficient to prove that $\\PP(B_{\\xi,N,p})>0$.\nObviously\n\\begin{equation*}\nB_{\\xi,N,p}^*:=\\left\\{\\om\\in\\Om:\\, D_N(\\xi,\\om,0)<\\frac{1}{p}\\right\\}\n\\subseteq B_{\\xi,N,p}\n\\end{equation*}\n(just chose $q(\\omega)=0$).\nSo, it suffices to show $\\PP(B_{\\xi,N,p}^*)>0$ to complete the proof.\n\n\nThe definition of the operator $\\cL(t)$ yields\n\\begin{equation*}\n\\cL(t_1)-\\cL(t_2)=(t_1-t_2) \\cL_1 + (t_1^2-t_2^2) \\cL_2 + (t_1^3-t_2^3) \\cL_3(t_1) + t_2^3 \\big(\\cL_3(t_1)-\\cL_3(t_2)\\big).\n\\end{equation*}\nThis identity, definition (\\ref{3.0}) of $D_N$ and the continuity of operator $\\cL_3(t)$  in $t$ imply immediately\nthat $D_N(\\xi,\\om,0)$ tends to zero as $\\xi_i-\\omega_i\\to0$, $i\\in\\G_N$.\nHence, there exists $\\d=\\d(\\xi,p,N)$ such that $D_N(\\xi,\\om,0)<\\frac{1}{p}$ once $|\\xi_i-\\omega_i|<\\d$, $i\\in\\G_N$.\nIn other words,\n\\begin{equation*}\n\\left\\{\\om\\in\\Om:\\, |\\xi_i-\\omega_i|<\\d,\\, i\\in\\G_N\\right\\}\n\\subseteq B_{\\xi,N,p}^*\n\\end{equation*}\nSince  $\\G_N$ is finite, we see that the set on the right\nhas a positive measure, whenever all $\\xi_i, i \\in \\G_N$ are elements of $\\supp \\mu$.\nIt implies that $B_{\\xi,N,p}^*$ is of positive $\\PP$-measure.\n\\end{proof}\n\n\\begin{lemma}\\label{lm3.2}\nThe identity\n\\begin{equation*}\n\\Sigma_\\e=\\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))\n=\\overline{\\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))}\n\\end{equation*}\nholds true.\n\\end{lemma}\n\n\\begin{proof}\nWe adapt the proof of Theorem~3 in \\cite{KirschM-82b}. First of all, by (\\ref{2.9}) we have\n\\begin{equation*}\n\\Sigma_\\e\n\\subseteq \\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))\n\\end{equation*}\nLet us prove the opposite inclusion.\n\nWe fix $\\xi\\in\\Om$ and take $\\l\\in\\spec(\\Op_\\e(\\xi))$. By the Weyl criterion there exists a characteristic sequence $\\{\\vp_p\\}_{p\\in\\NN}\\subset \\Dom(\\Op_0)$ such that\n\\begin{equation}\\label{3.1}\n\\|\\vp_p\\|_{L_2(\\Pi)}=1,\\quad \\|(\\Op_\\e(\\xi)-\\l)\\vp_p\\|_{L_2(\\Pi)} \\leqslant \\frac 1 p, \\quad p\\in\\NN.\n\\end{equation}\nBy \\cite[Ch. III, Sect. 6, Lm. 6.3]{Ber} we have the estimate\n\\begin{equation*}\n\\|\\vp_p\\|_{W_2^{2m}(\\Pi)}\n\\leqslant C\\big (\\|(\\Op_0+\\iu)\\vp_p\\|_{L_2(\\Pi)}\n+|\\l|+1\\big  ),\n\\end{equation*}\nwhere $C$ is a constant independent of $p$ and $\\e$. Hence, the norm $\\|\\vp_p\\|_{W_2^{2m}(\\Pi)}$ is bounded uniformly in $p$, and the suprema\n\\begin{equation*}\n\\vp_p^*:=\\sup\\limits_{k\\in\\G} \\|\\vp_p\\|_{W_2^{2m}(\\square_k)},\n\\quad\n\\vp^*:=\\sup\\limits_{p\\in\\NN} \\vp_p^*\n\\end{equation*}\n{are} well-defined  and finite.\nPick $\\om$ from the full measure set\n$$\n\\Om^*:=B_\\xi\\cap \\{\\om\\in\\Om:\\, \\spec(\\Op_\\e(\\om))=\\Sigma_\\e\\}\n\\subseteq B_{\\xi,N_p,p}\n$$\nand\nlet us show that $\\psi_{p}:=\\cS(-q(N_p,p,\\om))\\vp_p$ is a characteristic sequence for $\\Op_\\e(\\om)$ at $\\l$.\nHere $N_p$ is a number fixed by the inequality\n\\begin{equation}\\label{3.2}\n\\|\\varphi_p\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{N_p})}\\leqslant \\frac{1}{p},\n\\end{equation}\nwhile  $q$ comes from definition (\\ref{eq:BxiNP}) of the set $B_{\\xi,N_p,p}$.\nEmploying definition (\\ref{3.4}) of the set $B_p$, (\\ref{3.1}), (\\ref{3.2}), by straightforward calculation we obtain\n\\begin{align*}\n\\|&(\\Op_\\e(\\om)-\\l)\\psi_{p}\\|_{L_2(\\Pi)} = \\|(\\Op_0 + \\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\l)\\vp_p\\|_{L_2(\\Pi)}\n\\\\\n&\\leqslant \\|(\\Op_\\e(\\xi)-\\l)\\vp_p\\|_{L_2(\\Pi)}  + \\|(\\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\cL_\\e(\\xi))\\vp_p\\|_{L_2(\\Pi)}\n\\\\\n&\\leqslant  \\frac{1}{p} + \\|(\\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\cL_\\e(\\xi))\\vp_p\\|_{L_2(\\Pi_{N_p})}\n\\\\\n&\\hphantom{\\leqslant}+ \\|(\\cS(-q)\\cL_\\e(\\om)\\cS(q)-\\cL_\\e(\\xi))\\vp_p\\|_{L_2(\\Pi\\setminus\\Pi_{N_p})}\n\\\\\n&\\leqslant \\frac{1}{p} + D_{N_p} (\\xi,\\om,q) \\vp^* + C\\|\\vp_p\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{N_p})}\n\\end{align*}\nwhere the constant $C$ is independent of $p$.\nSince the chosen $\\omega$ is in particular in $B_{\\xi,N_p,p}$ and by the choice (\\ref{3.2})\n\\begin{align*}\n\\frac{1}{p} + D_{N_p} (\\xi,\\om,q) \\vp^* + C\\|\\vp_p\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{N_p})}\n\\leqslant \\frac{1}{p} + \\frac{\\vp^* }{p} + \\frac{C}{p}.\n\\end{align*}\nHence, $\\psi_{p}$ is indeed a characteristic sequence for $\\Op_\\e(\\omega)-\\l$, i.e.~$\\l \\in\\spec(\\Op_\\e(\\omega))$\nfor any $\\omega \\in \\Omega^*$.\nTherefore, $\\spec(\\Op_\\e(\\xi))\\subseteq\\Sigma_\\e$ for each $\\xi\\in\\Om$. This completes the proof of the lemma.\n\\end{proof}\n\nIn view of the above lemma, we can obtain $\\Sigma_\\e$ as the union of the spectra $\\Op_\\e(\\xi)$ over all $\\xi\\in\\Om$.\nWe follow the proof of Theorem~4 in \\cite{KirschM-82b} to prove that this union can be taken over $2^k\\G$-periodic configurations $\\xi$ only.\n\nGiven $N\\in\\NN$, $\\xi\\in\\Om$, we introduce the $2^N\\G$-periodic sequence $\\xi^{(N)}\\in\\Om$ as follows: $\\xi_k^{(N)}=\\xi_k$, $k\\in\\G_{2^N}$, and for $k\\in\\G\\setminus\\G_{2^N}$ we define other terms of $\\xi^{(N)}$ by the periodic continuation. If we prove that $\\Op_\\e(\\xi^{(N)})$ converges to $\\Op_\\e(\\xi)$ as $N\\to\\infty$ in the strong resolvent sense, it will imply $\\spec(\\Op_\\e(\\xi))\\subseteq\\overline{\\bigcup\\limits_{N=1}^{\\infty} \\spec(\\Op_\\e(\\xi^{(N)}))}$, and hence,\n\\begin{equation*}\n\\overline{\\bigcup\\limits_{\\xi\\in\\Om} \\spec(\\Op_\\e(\\xi))}=\\overline{\\bigcup\\limits_{N\\in \\NN}\\bigcup\\limits_{\\xi \\ \\text{is $2^N \\G$-periodic}} \\spec\\big(\\Op_\\e(\\xi)\\big)},\n\\end{equation*}\nwhich will prove (\\ref{2.10}).\n\n\nThe desired strong resolvent convergence means that for each $f\\in L_2(\\Pi)$ the solution to the equation $\\big(\\Op_\\e(\\xi^{(N)})-\\iu\\big)u^{(N)}=f$ converges in $L_2(\\Pi)$ to the solution of the equation $\\big(\\Op_\\e(\\xi)-\\iu\\big)u=f$. These equations yield\n\\begin{equation*}\n\\big(\\Op_\\e(\\xi^{(N)})-\\iu\\big)(u-u^{(N)})=\\big(\\cL_\\e(\\xi^{(N)})-\\cL_\\e(\\xi)\\big)u.\n\\end{equation*}\nHence, by the definition of $\\xi^{(N)}$ and the estimate for the resolvent of a self-adjoint operator\n\\begin{equation*}\n\\|u-u^{(N)}\\|_{L_2(\\Pi)} \\leqslant \\big\\|\\big(\\cL_\\e(\\xi^{(N)})-\\cL_\\e(\\xi)\\big)u\\big\\|_{L_2(\\Pi)} \\leqslant C\\|u\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{2^N})},\n\\end{equation*}\nwhere $C$ is a constant independent of $u$ and $N$. Since $u\\in W_2^{2m}(\\Pi)$, we get $\\|u\\|_{W_2^{2m}(\\Pi\\setminus\\Pi_{2^N})}\\to0$ as $N\\to\\infty$\nand this completes the proof of Theorem~\\ref{th2.1}.\n\n\n\n\n\n\n\n\\section{Upper bound on $\\inf\\Sigma_\\e$: Proof of Theorem~\\ref{th2.2}.}\nIn order to prove (\\ref{2.19}), first  we estimate the resolvent of the unperturbed operator $\\Op_0$.\nSince this operator is self-adjoint, we have for any $\\l$ in the resolvent set\n\\begin{equation}\\label{4.1}\n\\|(\\Op_0-\\l)^{-1}\\|_{L_2(\\Pi)\\to L_2(\\Pi)}=\\frac{1}{\\dist(\\l,\\spec(\\Op_0))}.\n\\end{equation}\nWe rewrite the resolvent equation $(\\Op_0-\\l)u=f$ as\n\\begin{equation}\\label{4.2}\n(\\Op_0-\\iu)u=(\\l-\\iu)u+f.\n\\end{equation}\nSince the operator $(\\Op_0-\\iu)$ is invertible, by \\cite[Ch. I\\!I\\!I, Sect. 6, Lm. 6.3]{Ber} and (\\ref{4.1}), the solution of equation (\\ref{4.2}) satisfies the estimate\n\\begin{equation}\\label{4.3}\n\\|u\\|_{W_2^{2m}(\\Pi)} \\leqslant C\\|(\\l-\\iu)u+f\\|_{L_2(\\Pi)} \\leqslant \\frac{C(|\\l|+1)}{\\dist(\\l,\\spec(H_0))}\\|f\\|_{L_2(\\Pi)},\n\\end{equation}\nwhere $C$ is a positive constant independent of $\\l$ and $f$.\n\nConsider the resolvent equation for $\\Op_\\e(\\xi)$, $\\xi\\in\\Om$:\n\\begin{equation*}\n(\\Op_0+\\cL_\\e(\\xi)-\\l)u=f.\n\\end{equation*}\nWe can rewrite it as\n\\begin{equation*}\n\\big(I+\\cL_\\e(\\xi)(\\Op_0-\\l)^{-1}\\big)(\\Op_0-\\l)u=f,\n\\end{equation*}\nwhere $I$ is the identity mapping. Hence, once\n\\begin{equation}\\label{4.4}\n\\|\\cL_\\e(\\xi)(\\Op_0-\\l)^{-1}\\|<1,\n\\end{equation}\nthe resolvent of $\\Op_\\e$ is well-defined and  is given by the formula\n\\begin{equation*}\n(\\Op_\\e(\\xi)-\\l)^{-1}=(\\Op_0-\\l)^{-1} \\big(I+\\cL_\\e(\\xi)(\\Op_0\\l)^{-1}\\big)^{-1}.\n\\end{equation*}\nIn view of the definition of operators $\\cL_i$ and estimate (\\ref{4.3}), we see that inequality (\\ref{4.4}) is satisfied provided\n\n\\begin{equation}\\label{4.5} \\frac{C\\e(|\\l|+1)}{\\dist(\\l,\\spec(\\Op_0))}\\sup_{|t|\\leqslant t_0} \\|\\cL(t)\\|_{W_2^{2m}(\\square) \\to L_2(\\square)}<1,\n\\end{equation}\nwhere constant $C$ is the same as in (\\ref{4.3}). Hence, such $\\l$ are in the resolvent set of $\\Op_\\e(\\om)$.\nA contraposition of this statement yields (\\ref{2.19}).\n\n\nGiven $s\\in\\supp \\mu$, we denote by $\\xi^s$ the constant sequence $\\{s\\}_{k\\in\\G}$.\nIt follows from (\\ref{2.10}) that $\\Sigma_\\e\\supseteq\\spec(\\Op_\\e(\\xi^s))$, and hence, by the minimax principle\n\\begin{equation}\\label{4.6}\n\\inf\\Sigma_\\e\\leqslant \\inf\\spec(\\Op_\\e(\\xi^s))=\\inf\\limits_{\\genfrac{}{}{0 pt}{}{u\\in\\Dom(\\Op_\\e(\\xi^s))}{u\\not=0}} \\frac{\\big(\\Op_\\e(\\xi^s)u,u\\big)_{L_2(\\Pi)}}{\\|u\\|_{L_2(\\Pi)}^2}.\n\\end{equation}\nLet $\\phi^\\e(x,s):=\\psi_0(x)+\\e s\\psi_1(x)$. Since both functions $\\psi_0$ and $\\psi_1$ satisfy periodic boundary conditions on $\\g_l$, we extend $\\phi_s^\\e$ periodically to $\\Pi$ keeping the same notation for the extension.\nBy $\\chi_p\\colon \\Pi \\to [0,1]$ we denote an infinitely differentiable function being one in $\\Pi_p$, vanishing in $\\Pi\\setminus \\Pi_{p+2}$ and satisfying the estimates\n\\begin{equation}\\label{4.7}\n\\left|\\frac{\\p^\\a \\chi_p}{\\p x^\\a}\\right| \\leqslant C_\\a\n\\quad \\text{in}\\quad \\overline{\\Pi_{p+2}\\setminus\\Pi_p},\n\\end{equation}\nwhere $C_\\a$ is a positive constant independent of $p$ and $x$.\nWe also suppose that $\\chi_p$ in fact depends only on the $d_1$ coordinates in $S$.\nMore precisely, for each pair $x,\\, \\widetilde{x}\\in\\Pi$ such that $x-\\widetilde{x}$ is orthogonal to $S$, we have $\\chi_p(x)=\\chi_p(\\widetilde{x})$.\n\nIn view of the above described properties of $\\phi^\\e$ and $\\chi_p$, the function\n\\begin{equation*}\nu_p^\\e(x):=e^{\\iu\\theta_0x}\\phi^\\e(x,s)\\chi_p(x)\n\\end{equation*}\nbelongs to the domain of $\\Op_0$ and therefore, $u_p^\\e\\in \\Dom(\\Op_\\e(\\xi^s))$. We choose $u_p^\\e$ as the test function in (\\ref{4.6}) to obtain\n\\begin{equation}\\label{4.8}\n\\inf\\Sigma_\\e\\leqslant \\frac{\\big(\\Op_\\e(\\xi^s)u_p^\\e,u_p^\\e\\big)_{L_2(\\Pi)}}{\\|u_p^\\e\\|_{L_2(\\Pi)}^2}.\n\\end{equation}\n\n\nLet us calculate the right hand side of this inequality. It is clear that\n\\begin{equation}\\label{4.9}\n\\frac{\\big(\\Op_\\e(\\xi^s)u_p^\\e,u_p^\\e\\big)_{L_2(\\Pi)}}{\\|u_p^\\e\\|_{L_2(\\Pi)}^2}=\n\\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x})\\phi^\\e \\chi_p,\\phi^\\e \\chi_p\\big)_{L_2(\\Pi)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi)}^2}.\n\\end{equation}\n\nIt follows from the definition of $\\phi^\\e$ that for each $k\\in\\G$ the identities\n\\begin{equation}\\label{4.10}\n\\begin{aligned}\n&\\|\\phi^\\e\\|_{L_2(\\square_k)}^2=1+\\e^2s^2\\|\\psi_1\\|_{L_2(\\square)}^2,\n\\\\\n&\\|\\phi^\\e\\|_{W_2^{2m}(\\square_k)}^2=\\|\\psi_0\\|_{W_2^{2m}(\\square)}^2\n+2\\e s\\RE(\\psi_0,\\psi_1)_{W_2^{2m}(\\square)} + \\e^2 s^2 \\|\\psi_1\\|_{W_2^{2m}(\\square)}^2\n\\end{aligned}\n\\end{equation}\n{hold true.}\nThese identities and the above described properties of $\\chi_p$ imply\n\\begin{align}\n&\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi)}^2=\\|\\phi^\\e\\|_{L_2(\\Pi_p)}^2 + \\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)}^2,\\nonumber\n\\\\\n&\\|\\phi^\\e\\|_{L_2(\\Pi_p)}^2\\geqslant C p^{d_1},\n\\quad\n\\|\\phi^\\e\\|_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)}^2\\leqslant C p^{d_1-1}, \\label{4.11}\n\\end{align}\nwhere symbol $C$ stands for inessential constants independent of $p$, $\\e$ and $s$.\n\nThe boundedness of $\\cL_1, \\cL_1, \\cL_2, \\cL_3(t) : W_2^{2m}(\\square)\\to L_2(\\square)$ and (\\ref{4.7}) yield\n\\begin{equation}\\label{4.12}\n\\big\\|\\big(\\Op_0(\\theta_0) + e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x}\\big)\\phi^\\e \\chi_p \\big\\|_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)} \\leqslant C \\|\\phi^\\e\\|_{W_2^{2m}(\\Pi_{p+2}\\setminus\\Pi_p)}\\leqslant C p^{d_1-1},\n\\end{equation}\nwhere $C$ is a constant independent of $p$, $\\e$ and $s$.\n\n\nIt is clear that\n\\begin{align}\\label{4.13}\n& \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x})\\phi^\\e \\chi_p,\\phi^\\e \\chi_p\\big)_{L_2(\\Pi)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi)}^2}= T_p^1(\\e,s) + T_p^2(\\e, s),\n\\\\\n&T_p^1(\\e,s):= \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\theta_0 x})\\phi^\\e,\\phi^\\e \\big)_{L_2(\\Pi_p)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi_{p+2})}^2},\\nonumber\n\\\\\n&T_p^2(\\e,s):= \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\xi^s) e^{\\iu \\tau_0 x})\\phi^\\e\\chi_p,\\phi^\\e \\chi_p \\big)_{L_2(\\Pi_{p+2}\\setminus\\Pi_p)}}{\\|\\phi^\\e\\chi_p\\|_{L_2(\\Pi_{p+2})}^2}.\n\\end{align}\nBy (\\ref{4.11}), (\\ref{4.12}) we obtain\n\\begin{equation}\\label{4.14}\n|T_p^2(\\e,s)|\\leqslant C p^{-1},\n\\end{equation}\nwhere the constant $C$ is independent of $p$, $\\e$, and $s$, and\n\\begin{equation}\\label{4.15}\n\\lim\\limits_{p\\to+\\infty}  T_p^1(\\e,s) = \\frac{\\big((\\Op_0(\\tau_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\om^s) e^{\\iu \\theta_0 x})\\phi^\\e,\\phi^\\e \\big)_{L_2(\\square)}}{\\|\\phi^\\e\\|_{L_2(\\square)}^2}.\n\\end{equation}\nTogether with (\\ref{4.8}), (\\ref{4.9}) it yields\n\\begin{equation}\\label{4.16}\n\\inf \\Sigma_\\e\\leqslant \\frac{\\big((\\Op_0(\\theta_0)+e^{-\\iu \\theta_0 x} \\cL_\\e(\\om^s) e^{\\iu \\theta_0 x})\\phi^\\e,\\phi^\\e \\big)_{L_2(\\square)}}{\\|\\phi^\\e\\|_{L_2(\\square)}^2}.\n\\end{equation}\nEquation (\\ref{2.17}) and the eigenvalue equation for $\\psi_0$ imply that\n\\begin{align*}\n\\big(\\Op_0(\\theta_0) + e^{-\\iu \\theta_0 x} \\cL_\\e(\\om^s) e^{\\iu \\theta_0 x}\\big)\\phi^\\e=&\\L_0\\phi^\\e + \\e s \\L_1 \\psi_0\n\\\\\n&+ \\e^2 s^2 e^{-\\iu \\theta_0 x} (\\cL_2 + \\e s \\cL_3(\\e s) ) e^{\\iu \\theta_0 x} \\phi_\\e\n\\\\\n &+ \\e^2 s^2 e^{-\\iu \\theta_0 x} \\cL_1 e^{\\iu \\theta_0 x} \\psi_1.\n\\end{align*}\nSubstituting this identity into (\\ref{4.16}), we get\n\\begin{equation}\\label{4.17}\n\\begin{aligned}\n\\inf\\Sigma_\\e\\leqslant &\\L_0+ \\frac{1}{\\|\\phi^\\e\\|_{L_2(\\square)}^2} \\Big(\n\\e s\\L_1(\\psi_0,\\phi^\\e)_{L_2(\\square)} + \\e^2 s^2 (\\cL_2 e^{\\iu\\tht_0 x}\\phi^\\e, e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}\n\\\\\n&+\\e^2 s^2 (\\cL_1 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}\n+\\e^3 s^3 (\\cL_3(\\e s) e^{\\iu\\tht_0 x}\\phi^\\e,e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}.\n\\Big)\n\\end{aligned}\n\\end{equation}\nWe employ the identities\n\\begin{equation*}\n(\\psi_0,\\psi_1)_{L_2(\\square)}=0,\\quad \\|\\phi^\\e\\|_{L_2(\\square)}^2=1+\\e^2 s^2 \\|\\psi_1\\|_{L_2(\\square)}^2\n\\end{equation*}\nand definitions (\\ref{2.16}), (\\ref{2.18}) of $\\L_1$, $\\L_2$\nto check that\n\\begin{equation*}\n \\e s\\L_1 (\\psi_0,\\phi^\\e)_{L_2(\\square)}=\\e s \\L_1\\|\\phi^\\e\\|_{L_2(\\square)}^2 - \\e^3 s^3 \\L_1 \\|\\psi_1\\|_{L_2(\\square)}^2\n\\end{equation*}\nand\n\\begin{align*}\n(\\cL_2 e^{\\iu\\tht_0 x}\\phi^\\e,&e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)} + (\\cL_1 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\phi^\\e)_{L_2(\\square)}\n\\\\\n=& \\L_2\\big(\\|\\phi^\\e\\|_{L_2(\\square)}^2 - \\e^2 s^2 \\|\\psi_1\\|_{L_2}^2\\big)\n+ 2\\e s \\RE (\\cL_2 e^{\\iu\\tht_0 x}\\psi_0,e^{\\iu\\tht_0 x}\\psi_1)_{L_2(\\square)}\n\\\\\n&  + \\e^2 s^2 (\\cL_2 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\psi_1)_{L_2(\\square)} + \\e s (\\cL_1 e^{\\iu\\tht_0 x}\\psi_1,e^{\\iu\\tht_0 x}\\psi_1)_{L_2(\\square)}.\n\\end{align*}\nTogether with (\\ref{4.17}) it yields\n\\begin{equation*}\n\\inf \\Sigma_\\e \\leqslant \\L_0 + \\e s \\L_1 + \\e^2 s^2 \\L_2 + \\frac{\\e^3 s^3 \\L_3(\\e s)}{1+\\e^2 s^2 \\|\\psi_1\\|_{L_2(\\square)}^2}.\n\\end{equation*}\nAnd thanks to (\\ref{2.9}) it proves (\\ref{2.20}). The proof of Theorem~\\ref{th2.2} is complete.\n\n\\medskip\n\nIn this section we also prove Corollary~\\ref{c:1}.\n\n\\begin{proof}[Proof of Corollary \\ref{c:1}]\nSince {the function $e^{\\iu \\theta_0 x}\\psi_0(x)$}\nis in the domain of $\\cL_2$,\n\\begin{equation*}\n(\\cL_2e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)} \\leqslant 0.\n\\end{equation*}\nFor  $\\L_1=0$ {we have}\n\\begin{align*}\n(\\cL_1e^{\\iu \\theta_0 x}\\psi_1,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}\n= &-(\\Op_0(\\theta_0) \\psi_1,\\psi_1)_{L_2(\\square)}\n= -(E_2-\\L_0) \\|\\psi_1\\|^2_{L_2(\\square)}\n\\\\\n\\leqslant &-c_1 \\|\\psi_1\\|^2_{L_2(\\square)},\n\\end{align*}\nwhere $E_2=\\spec(\\Op_0(\\theta))\\setminus\\{E_0(\\theta)\\}$ and $c_1$ the spectral gap.\nThus,\n\\[\n\\L_2\n\\leqslant - c_1 \\|\\psi_1\\|^2 + (\\cL_2e^{\\iu \\theta_0 x}\\psi_0,e^{\\iu \\theta_0 x}\\psi_0)_{L_2(\\square)}\n\\leqslant - c_1 \\|\\psi_1\\|^2.\n\\]\n\\end{proof}\n\n\n\\section{Lower bound on $\\inf\\Sigma_\\e$: Proof of Theorem~\\ref{th2.3}}\nLet $\\g_l^\\pm$ be a pair of opposite faces in $\\g_l$, namely,\n\\begin{equation*}\n\\g_l^-:=\\p\\square\\cap\\p\\square_{-e_i},\\quad \\g_l^+:=\\p\\square\\cap\\p\\square_{e_i}\n\\end{equation*}\nfor some $1\\leqslant i\\leqslant d_1$.\nWe recall that $e_1,\\ldots,e_{d_1}$ is the basis of lattice $\\G$, and $\\square_{\\pm e_i}$ is just $\\square_k$ with $k=\\pm e_i$. Let us show that\n\\begin{equation}\\label{4.18}\nb_j\\big|_{\\g_l^-}=-b_j\\big|_{\\g_l^+}\n\\end{equation}\nfor any choice of $1\\leqslant i\\leqslant d_1$.\nWe first observe that by the periodicity of the functions $A_{\\a\\b}$ and $\\psi_0$ we have\n\\begin{equation}\\label{4.18a}\n\\Big|\\cB_j\\psi_0\\big|_{\\g_l^-}\\Big|=\\Big|\\cB_j\\psi_0\\big|_{\\g_l^+}\\Big|.\n\\end{equation}\nFor each $u\\in \\Dom(\\Op_0)$\nwith compact support we have\n\\begin{align*}\n \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} (-1)^{|\\a|} & \\int\\limits_{\\Pi} \\overline{\\psi_0}   (\\p + i \\theta)^\\a A_{\\a\\b} (\\p + i \\theta)^\\b u \\di x\n\\\\\n&= \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n}   \\int\\limits_{\\Pi} A_{\\a\\b} (\\p + i \\theta)^\\b u\\, \\overline{(\\p + i \\theta)^\\a\\psi_0}\\di x\n\\end{align*}\nand by (\\ref{2.23}) we also get\n\\begin{align*}\n\\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n} &(-1)^{|\\a|} \\int\\limits_{\\Pi} \\overline{\\psi_0}   (\\p + \\iu \\theta_0)^\\a A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b u \\di x\n\\\\\n=& \\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\psi_0)_{L_2(\\g_l)}\n \\\\\n &+ \\sum\\limits_{k\\in\\G}  \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n}  \\big( A_{\\a\\b} (\\p + \\iu \\theta_0)^\\b u, \\overline{(\\p + \\iu \\theta_0)^\\a\\psi_0}\\big)_{L_2(\\square_k)}\n\\\\\n=& \\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\psi_0)_{L_2(\\g_l)}\n \\\\\n &+   \\sum\\limits_{\n\\genfrac{}{}{0 pt}{}{\\a,\\b\\in \\mathds{Z}_+^d}{|\\a|,|\\b|\\leqslant m}\n}  \\int\\limits_{\\Pi} A_{\\a\\b} (\\p + i \\theta)^\\b u\\, \\overline{(\\p + i \\theta)^\\a\\psi_0}\\di x.\n\\end{align*}\nHence,\n\\begin{equation*}\n\\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\psi_0)_{L_2(\\g_l)}=0.\n\\end{equation*}\nSince $u$ is arbitrary and $\\psi_0$ is periodic, the above identity is possible only if for all $1\\leqslant i\\leqslant d_1$\n\\begin{align*}\n\\cB_{2m-j} \\cS(-k)u\\Big|_{\\g_l^-} \\overline{\\cB_j\\psi_0}\\Big|_{\\g_l^-}= & -\\cB_{2m-j} \\cS(-k-e_i)u\\Big|_{\\g_l^-} \\overline{\\cB_j\\cS(-e_i)\\psi_0}\\Big|_{\\g_l^-}\n\\\\\n=&-\\cB_{2m-j} \\cS(-k )u\\Big|_{\\g_l^+} \\overline{\\cB_j\\psi_0}\\Big|_{\\g_l^+}\n\\end{align*}\nfor each $k\\in\\G$. Dividing this identity by  $\\Big|\\cB_j\\psi_0\\Big|_{\\g_l^-}\\Big|^2=\\Big|\\cB_j\\psi_0\\Big|_{\\g_l^+}\\Big|^2$,\n cf.~(\\ref{4.18a}), and letting $u=\\psi_0$ in the vicinity of $\\square_k$, we arrive at (\\ref{4.18}).\n\n\nIt follows from (\\ref{4.18}) that for each\n{$u\\in\\Dom(\\Op_0)$}\n\\begin{equation*}\n\\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (\\cB_{2m-j}\\cS(-k)u,\\cB_j \\cS(-k) u)_{L_2(\\g_l)} = 0.\n\\end{equation*}\nEmploying this identity and the minimax principle, we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\inf\\spec\\big(\\Op_\\e(\\xi)\\big)& = \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}} \\frac{\\fm_0(u,u) + (\\cL_\\e(\\xi)u,u)_{L_2(\\Pi)}}{\\|u\\|_{L_2(\\Pi)}^2}\n\\\\\n &=  \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}} \\frac{\\fm_0(e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u) + (\\cL_\\e(\\xi)e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u)_{L_2(\\Pi)}}{\\|u\\|_{L_2(\\Pi)}^2}\n\\\\\n &=  \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}}\n\\frac{1}{\\|u\\|_{L_2(\\Pi)}^2} \\Big(\n \\fm_0(e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u) + ( \\cL_\\e(\\xi)e^{\\iu\\theta_0x}u,e^{\\iu\\theta_0x}u)_{L_2(\\Pi)}\n\\\\\n&\\hphantom{= \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}}\\frac{1}{\\|u\\|_{L_2(\\Pi)}^2} \\Big(}+ \\sum\\limits_{k\\in\\G} \\sum\\limits_{j=1}^{m} (b_j \\cB_{j-1}\\cS(-k)u,\\cB_{j-1}\\cS(-k)u)_{L_2(\\g_l)}  \\Big)\n\\\\\n&= \\inf\\limits_{\\genfrac{}{}{0 pt}{}{\n{u\\in\\Dom(\\Op_0)}\n}{u\\not=0}}\n\\frac{\\sum\\limits_{k\\in\\G} \\Big(\n\\widehat{\\fm}_0(u,u)  + \\big( \\cL(\\e\\xi_k)e^{\\iu\\theta_0x}\\cS(-k)u,e^{\\iu\\theta_0x}\\cS(-k)u\\big)_{L_2(\\square)}\n\\Big)}{\\sum\\limits_{k\\in\\G}\\|\\cS(-k)u\\|_{L_2(\\square)}^2}.\n\\end{aligned}\\label{4.20}\n\\end{equation}\nSince\n\\begin{equation*}\n\\Dom(\\Op_0)\n\\subseteq \\bigoplus\\limits_{k\\in\\G} \\Ho^{2m}(\\square_k,\\g_\\Pi^k):=\\left\\{u\\in L_2(\\Pi):\\, u\\big|_{\\square_k} \\in \\Ho^{2m}(\\square_k,\\g_\\Pi^k),\\,k\\in\\G\n\\right\\},\n\\end{equation*}\nby (\\ref{4.20}) we get\n\\begin{equation}\\label{4.21}\n\\begin{aligned}\n\\inf\\spec\\big(\\Op_\\e(\\xi)\\big)\n&\\geqslant  \\inf\\limits_{\\bigoplus\\limits_{k\\in\\G} \\Ho^{2m}(\\square_k,\\g_\\Pi^k)}\n\\frac{1}{\\sum\\limits_{k\\in\\G}\\|\\cS(-k)u\\|_{L_2(\\square)}^2}\n \\sum\\limits_{k\\in\\G}  \\Big(\n\\widehat{\\fm}_0(\\cS(-k)u, \\cS(-k)u)\n\\\\\n&\\hphantom{\\geqslant  \\inf\\limits_{\\bigoplus\\limits_{k\\in\\G}\\Ho^{2m}(\\square,\\g_\\Pi )} }+ \\big( \\cL(\\e\\xi_k)e^{\\iu\\theta_0x}\\cS(-k)u,e^{\\iu \\theta_0 x}\\cS(-k)u\\big)_{L_2(\\square)}\n\\Big)\n\\\\\n&\\geqslant \\inf\\limits_{[s_-,s_+]} \\l_\\e(s),\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n\\l_\\e(s):=\\inf\\limits_{\\Ho^{2m}(\\square,\\g_\\Pi) }    \\frac{\\widehat{\\fm}_0(u,u)  +\n\\big(\\cL(\\e s)e^{\\iu\\theta_0x} u, e^{\\iu\\theta_0x} u\\big)_{L_2(\\square)}\n}{\\|u\\|_{L_2(\\square)}^2}.\n\\end{equation*}\nBy the minimax principle, $\\l_\\e(s)$ is the bottom of the spectrum of the operator $$\\widehat{\\Op}_0 + e^{-\\iu\\theta_0 x}\\cL(\\e s)e^{\\iu\\theta_0 x}.$$ Assumption \\ref{A2} and the $\\widehat{\\Op}_0$-boundedness of $\\cL(\\e s)$ yield that $\\l_\\e(s)$ is a discrete eigenvalue of $\\widehat{\\Op}_0 + e^{-\\iu\\theta_0 x}\\cL(\\e s)e^{\\iu\\theta_0 x}$ and $\\l_\\e(s)\\to\\L_0$ as $\\e\\to+0$ uniformly in $s\\in[s_-,s_+]$. By regular perturbation theory one can easily construct the asymptotic expansion for $\\l_\\e(s)$:\n\\begin{equation}\\label{4.22}\n\\l_\\e(s)=\\L_0+\\e s\\widehat{\\L}_1 + \\e^2 s^2 \\widehat{\\L}_2 + O(\\e^3),\n\\end{equation}\nwhere the estimate for the error term is uniform in $s\\in[s_-,s_+]$, \\begin{equation*}\n\\widehat{\\L}_1=(e^{-\\iu\\theta_0 x}\\cL_1 e^{\\iu\\theta_0 x}\\psi_0,\\psi_0)_{L_2(\\square)}=\\L_1,\n\\end{equation*}\nand $\\widehat{\\L}_2$ is given by formula (\\ref{2.29}).\nThe asymptotics (\\ref{4.22}), estimate (\\ref{4.21}) and the definition of $\\l_\\e(s)$ imply\n(\\ref{2.30}).\nThe proof is complete.\n\n\n\\section*{Acknowledgments}\n\nThis work was initiated while the authors were at the Chair of Stochastics, Faculty of Mathematics, of the Technische Universit\\\"at Chemnitz. It was partially financially supported by the DFG through the project grant \\emph{Eindeutige-Fortsetzungsprinzipien und Gleichverteilungseigenschaften von Eigenfunktionen}. The research of D.B. was supported by the grant of Russian Science Foundation no. 14-11-00078.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:introduction}Introduction}\n\nFluid flow in porous media is studied for more than a century due to its high relevance to several engineering applications such as enhanced oil recovery~\\cite{green1998enhanced,farajzadeh2012foam,fraggedakis2015flow,sahimi2011flow}, filtration and separation~\\cite{herzig1970flow,tien1979advances,jaisi2008transport}, fermentation~\\cite{pandey2003solid,aufrecht2019pore}, soil sequestration~\\cite{schlesinger1999carbon}, energy storage~\\cite{duduta2011semi,sun2019hierarchical}, and food processing~\\cite{greenkorn1983flow}. In most cases, the fluids involved in these applications exhibit yield stress\/viscoplastic behavior~\\cite{bonn2009yield,balmforth2014yielding}. Therefore, understanding the conditions -- critical pressure drop and\/or stresses -- that lead to fluidization of yield-stress fluids in porous media can help boost the efficiency and lower the operational cost of several industrial applications.\n\nIn pressure-driven flows, the critical pressure drop $\\Delta P_c$ required to fluidize the yield-stress fluid and open the first channel~\\cite{chen2005flow,hewitt2016obstructed,waisbord2019anomalous} depends on the heterogeneous geometric characteristics and porosity $1-\\phi$ of the porous medium, where $\\phi$ is the volume fraction of the solid phase~\\cite{talon2013geometry,bauer2019experimental,chaparian2020complex}. Therefore, it is crucial to understand the relation between the yielding conditions and the structure of the porous medium, which will lead to predictive models for both the first open channel and $\\Delta P_c$.\n\nThe classical way to study yield-stress fluids is by solving the Cauchy momentum equations using viscoplastic constitutive relations, such as the Bingham and Herschel\u2013Bulkley models~\\cite{bird1987dynamics,huilgol2015fluid,saramito2016complex}. More recently, though, there is an increasing trend on using elastoviscoplastic~\\cite{saramito2007new} and kinematic hardening~\\cite{dimitriou2019canonical} models that originate from continuum mechanics~\\cite{gurtin2010mechanics,anand2020continuum}. The yield stress behavior, however, leads to an ill-defined problem that does not describe the stress distribution within the unyielded regions of the fluid~\\cite{balmforth2014yielding,saramito2017progress}. Common ways to resolve this problem is by using either optimization-~\\cite{hestenes1969multiplier,powell1978algorithms,glowinski2008lectures} or regularization-based methods~\\cite{papanastasiou1987flows,frigaard2005usage}. The former are accurate on predicting the yielded\/unyielded boundaries and the flow field~\\cite{dimakopoulos2013steady}, however, they are computationally expensive~\\cite{saramito2016damped} for conditions nearby the yield limit. Although the latter reduce the computational cost, they introduce non-physical parameters~\\cite{tsamopoulos2008steady,dimakopoulos2018pal} that lead to non-physical solutions, incorrect location of yield\/unyield boundaries, and inaccurate yield limits~\\cite{mitsoulis2017numerical,frigaard2005usage}. Here, we are interested in determining the statistics of the critical $\\Delta P_c$ for a yield-stress fluid in complex porous media. Thus, we need to use models that can predict accurately and efficiently $\\Delta P_c$ along with the first open channel.\n\nFluid flow in porous media is traditionally described through network models~\\cite{fatt1956network} that represent the complex geometric characteristics of the domain with spherical pore throats and cylindrical edges~\\cite{alim2017local,bryant1993network,blunt2001flow,blunt2013pore,stoop2019disorder}. In addition to their wide applicability in Newtonian fluids, network models have also been applied to describe $\\Delta P_c$ and the flow behavior with respect to the applied pressure drop in yield-stress fluids~\\cite{chen2005flow,frigaard2017bingham,liu2019darcy,talon2020effective}. When the relation between the local flow rate and the pressure drop is known, the network representation allows for the use of graph theoretic tools~\\cite{kharabaf1997invasion,chen2005flow,balhoff2012numerical,liu2019darcy} to quickly evaluate $\\Delta P_c$ and the flow response of the system. In general, though, the results of network viscoplastic models in complex porous media have been rarely compared and validated against those produced by solving the full fluid problem, and thus the conditions of their validity\/applicability are unknown. \n\nThe goal of the present work is to predict the first open channel for a yield-stress fluid in a complex porous medium along with the critical applied pressure drop required to open it. We develop a simple network model based on realistic porous media configurations, and use graph theoretical tools to study the statistics of yielding conditions in terms of the medium porosity. We validate our results against reported pressure-driven simulations of Bingham fluids in porous media. Finally, we discuss the relevance of our study to applications such as enhanced oil recovery and propose possible extensions.\n\n\\section{\\label{sec:modeling}Theory}\n\n\\begin{figure*}[!ht]\n    \\centering\n    \\hspace{0.08in}\\includegraphics[width=0.7\\textwidth]{fig1.pdf}\n    \\caption{(a) Typical configuration of a porous medium of length $L$ and porosity $1-\\phi$ that is made of non-overlapping monodisperse spheres of radius $R_s$, and its network representation. Across the domain, a macroscopic pressure gradient $\\Delta P\/L$ is applied to fluidize the yield-stress fluid. (b) Schematic of three network edges with different radii $r_i$ and lengths $l_i$ ($i=1,2,3$ ). The network representation includes the local geometric characteristics of the complex porous medium structure. (c) Characteristic pore-size distribution for the network shown in (a) as derived from the network model. (d) Typical shear stress response $\\tau$ as a function of the applied shear rate $\\dot{\\gamma}$ of a viscoplastic fluid with yield stress $\\tau_y$.}\n    \\label{fig:system}\n\\end{figure*}\n\n\\subsection{Network topology}\nWe are interested in the construction of realistic network models that capture the complex morphology of real porous media. The main scope of our work is to understand the statistics on the critical conditions that lead to fluidization in terms of the porosity $1-\\phi$ and topological characteristics of the medium. \n\nTo first approximation, we assume a porous medium that consists of monodisperse non-overlapping spheres\/disks of radius $R_s$ as shown in Fig.~\\ref{fig:system}(a). It is apparent that the structure of void space depends on the solid volume fraction defined as $\\phi=N_sV_s\/V_t$, where $N_s$ is the total number of spheres, $V_s$ is the volume of an individual sphere and $V_t$ the total volume of the system. We can use the given porous medium structure and create the network representation shown in the right panel of Fig.~\\ref{fig:system}(a).\n\nThe network consists of nodes and edges that span the entire medium, where its complex topological characteristics are encoded on the connectivity between them~\\cite{gostick2017versatile,khan2019dual}. Additionally, the local geometric characteristics of the porous medium are included on the length $l_i$ and radius $r_i$ of each individual edge, Fig.~\\ref{fig:system}(b). For demonstration, we show in Fig.~\\ref{fig:system}(c) the pore size distribution for the configuration of Fig.~\\ref{fig:system}(a). Details on the generation of porous media with monodisperse non-overlaping sphere, the construction of the network representation, and the choice of $r_i$ and $l_i$ are given in the Appendix of the paper.\n\n\\subsection{Yield-stress fluid in a network}\n\nYield-stress fluids are characterized by their solid-liquid transition when the Euclidean norm of the stress field exceeds the value of yield stress (Von Mises criterion)~\\cite{gurtin2010mechanics,hill1998mathematical}. The typical shear stress response in simple shear flow is shown in Fig.~\\ref{fig:system}(d), where for $\\dot{\\gamma}\\rightarrow0$ the shear stress reaches its critical value $\\tau\\rightarrow\\tau_y$. The most common constitutive relations used to describe yield-stress fluids are the Bingham and the Herschel-Bulkley models~\\cite{bird1987dynamics}. Both of them, though, have the same behavior for $\\dot{\\gamma}\\rightarrow0$. Therefore, it is sufficient to discuss only the Bingham model for a porous medium to understand the connection between the yielding conditions to the geometric and topological characteristics of the network.\n\nFor pressure-driven flows, the local flow rate $q_i$ of edge $i$ is described in terms of the local geometric properties $r_i,l_i$, the local pressure drop along the edge $\\Delta P_i$, and the yield stress $\\tau_y$ of the fluid as~\\cite{bird1987dynamics,liu2019darcy,frigaard2019background}\n\\begin{equation}\n\\label{eq:qflow}\nq_{i} = \\left\\{ \n\\begin{array}{cc}\n\\frac{r_i^4}{l_i} \\left(\\Delta P_{i}-\\frac{\\tau_y l_i}{r_i}\\right) &\\text{for} \\,\\, \\Delta P_{i}>\\frac{\\tau_y l_i}{r_i}\\\\\n 0   &\\text{ for} \\,\\, \\left\\lvert\\Delta P_{i}\\right\\rvert < \\frac{\\tau_y l_i}{r_i} \\\\\n\\frac{r_i^4}{l_i} \\left(\\Delta P_{i}+\\frac{\\tau_y l_i}{r_i}\\right) &\\text{ for} \\,\\, \\Delta P_{i}<-\\frac{\\tau_y l_i}{r_i}\\\\\n\\end{array} \n\\right.\n\\end{equation}\nNear the no-flow limit $q_i\\rightarrow0$, we see from Eq.~\\ref{eq:qflow} that $\\Delta P_i\\rightarrow \\tau_yl_i\/r_i$, and thus smaller in radius or longer in length channels require larger applied pressure drop to yield. Across the first open channel, we can calculate the total pressure drop across the medium to be $\\Delta P \\equiv \\sum_{i=1}^N \\Delta P_i = \\tau_y\\sum_{i=1}^N l_i\/r_i$, where $N$ is the total number of edges across the path. From this expression, we can see that the connectivity between the edges determines the first open channel in a real porous medium, and it corresponds to the path of `least resistance'. Thus, the problem of finding $\\Delta P_c$ can be formulated as finding the path of the minimum pressure drop as follows~\\cite{liu2019darcy}\n\\begin{equation}\n\\label{eq:least_res}\n    \\frac{\\Delta P_c}{\\tau_y} = \n    \\min_{C \\in \\mathcal{C}_{\\text{in-out}}}\n    \\sum_{i=1}^N \\frac{l_i}{r_i}\n\\end{equation}\nwhere $\\mathcal{C}_{\\text{in-out}}$ is the set of all paths between the corresponding boundaries. \n\nThe problem of Eq.~\\ref{eq:least_res} satisfies the principle of minimum dissipation rate and is valid near equilibrium~\\cite{kondepudi2014modern}. In particular, the entropy production for a pressure-driven flow is $\\sigma_D=q\\Delta P$~\\cite{de2013non}. Thus, for conditions near the solid-liquid transition where $q\\rightarrow0^+$, the minimum pressure drop path also minimizes $\\sigma_D$.\n\nTo solve Eq.~\\ref{eq:least_res}, we transform the generated network into a graph with edges that have weights equal to $l_i\/r_i$ and use the Dijkstra method~\\cite{dijkstra1959note} for directed graphs to determine the first open channel. This method is known to scale quadratically with the path length~\\cite{west1996introduction,bollobas2013modern}, and therefore for complex domains that lead to larger number of edges the computational cost increases. For a single porous medium configuration, however, the overall computational time to determine the first open channel is much lower (seconds to minutes) than that required to solve the full fluid flow problem using optimization methods (days to weeks)~\\cite{dimakopoulos2018pal,chaparian2020complex}. \n\n\\section{\\label{sec:results}Results}\n\n\\begin{figure}[!ht]\n    \\centering\n    \\hspace{0.08in}\\includegraphics[width=0.5\\textwidth]{fig2.pdf}\n    \\caption{Validation of the network model for the first open channel against simulations for a pressure-driven Bingham fluid in a complex porous medium with $R_s\/L=0.02$. (a) Simulation results of the open pathway for conditions near the critical pressure drop $\\Delta P_c$. The contour plot shows the magnitude of the local velocity, normalized with the maximum velocity across the channel. (b) Network model predictions for the first open channel. The cases of $\\phi=0.3$ and $\\phi=0.5$ are examined. It is clear that both the full Bingham fluid simulation and the network model predict the same location for the first open channel.}\n    \\label{fig:validation}\n\\end{figure}\n\n\\subsection{The first open channel}\nWhen the applied pressure drop approaches the critical value $\\Delta P \\rightarrow \\Delta P_c^+$, there exists a single open channel across the entire medium. Here, we test the validity of the proposed approach to determine the first open channel nearby when the solid-liquid transition occurs. For comparison, we solve the full flow field under pressure-driven conditions for a yield-stress fluid for the complex porous media shown in Fig.~\\ref{fig:validation}. We consider the cases of $\\phi=0.3$ and $\\phi=0.5$, respectively. All the lengths are normalized with the macroscopic length of the system $L$ and also $R_s\/L=0.02$. For simplicity, we use two-dimensional porous media, however, our approach is general and does not depend on the dimensionality of the problem.\n\nFigure~\\ref{fig:validation}(a) shows the normalized velocity magnitude that results from the solution of the Cauchy momentum equation for a Bingham fluid~\\cite{chaparian2020complex}. Also, Fig.~\\ref{fig:validation}(b) depicts the predictions for the first open channel after solving the minimization problem of Eq.~\\ref{eq:least_res}. In both cases, the network model is able to reproduce the results of the fluid problem for the first open path.\n\nNotably, in both cases of Fig.~\\ref{fig:validation}(a), where the full fluid flow problem is solved, we can see the existence of additional paths other than the one predicted by the network model. We justify this observation based on the fact that the fluid flow simulations are performed at extremely large, yet finite non-dimensional yield stress (i.e.~Bingham number; see \\cite{chaparian2020complex}). Indeed, we speculate that for simulations with $q\\rightarrow0^+$ (i.e.~infinite Bingham number), the secondary paths will eventually close and only the predicted one by the network model will be present. However, as it is clear from the Fig.~\\ref{fig:validation}(a), flow rate in those secondary paths are negligible and does not contribute to the leading order of the resistance.\n\n\\subsection{Critical pressure drop $\\Delta P_c$ and its statistics in complex porous media}\n\n\\begin{figure*}[!ht]\n    \\centering\n    \\hspace{0.08in}\\includegraphics[width=0.7\\textwidth]{fig3.pdf}\n    \\caption{Statistics on the predictions of the network model for 500 realizations per value of $\\phi$. (a) Critical pressure drop $\\Delta P_c$ as a function of the volume fraction $\\phi$ for different ratio of $R_s\/L$. $\\Delta P_c$ is normalized with the yield stress of the fluid $\\tau_y$. The error bars indicate the variance around the mean value. Inset -- Scaled critical pressure drop $\\Delta P_c R_s\/\\tau_y L$ in terms of $\\phi$. For non-overlapping spheres, the results for different $R_s\/L$ collapse in a master curve. (b) Probability density distributions for the normalized arc-length of the first open channel $L_c\/L-1$. The cases of $R_s\/L=0.02$ and $R_s\/L=0.08$ are shown, respectively, for $\\phi=0.1$ and $\\phi=0.5$. In general, increasing $\\phi$ leads to increase of $\\Delta P_c$ required to open the first channel, but also leads to a wide distribution of arc-lengths, which provide large uncertainty on the critical macroscopic pressure gradient $\\Delta P_c\/L_c$ required to yield the fluid in the porous medium.}\n    \\label{fig:result}\n\\end{figure*}\n\nIn addition to the first open path, the network model can predict the normalized critical pressure $\\Delta P_c$ required to yield the fluid in the porous medium. \n\n\\begin{table}[h]\n\\begin{ruledtabular}\n\\begin{tabular}{ c c c c}\n\\textrm{$\\phi$}&\n\\textrm{$\\frac{\\Delta P_c}{\\tau_y}$ Network}&\n\\textrm{$\\frac{\\Delta P_c}{\\tau_y}$ Simulations}&\n\\textrm{Rel. Error $\\%$}\\\\\n\\colrule\n$0.1$ & $22.16$ & $22.63$ & $1.87$\\\\\n$0.3$ & $61$ & $65.2$ & $6.43$\\\\\n$0.5$ & $142.16$ & $145.28$ & $2.05$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{\\label{tab:table_DPc}%\nPredicted normalized critical pressure drop $\\Delta P_c$ required to open the first open channel for the configurations shown in Fig.~\\ref{fig:validation}. We show both the predictions of the network model and the full simulation results, along with the relative error $\\frac{\\Delta P_{c}^{\\text{sim}}-\\Delta P_{c}^{\\text{net}}}{\\Delta P_{c}^{\\text{sim}}}$. The network model is adequate on predicting both the first open channel and the critical pressure drop required to open it.\n}\n\\end{table}\n\nIn table~\\ref{tab:table_DPc} we show the predictions of the normalized critical pressure drop $\\Delta P_c\/\\tau_y$ for both the network model and the full simulations. For $\\phi=0.3$ and $0.5$ we consider the configurations shown in Fig.~\\ref{fig:validation}. In all cases, it is clear that the relative difference of $\\Delta P_c$ between the two models never exceeds $6.5\\%$. The negligible difference can be justified by the fact that discussed above about the secondary paths, or\/and the simplification of the geometric characteristics of the medium by its network representation. We conclude, though, that network-based models are adequate to predict both the first open path and the critical pressure drop required to open it. \n\nThe validation of the model allows us to predict $\\Delta P_c$ as a function of $\\phi$ and the geometric characteristics of the system. For a porous medium made of non-overlapping disks, we control the microstructure characteristics by changing the ratio $R_s\/L$. For each combination of $\\phi$ and $R_s\/L$, we generate 500 realizations for gathering the statistics of $\\Delta P_c$. \n\nFigure~\\ref{fig:result}(a) shows the normalized critical pressure drop $\\Delta P_c\/\\tau_y$ in terms of $\\phi$. The colored lines indicate different value of $R_s\/L\\in[0.02,0.1]$, while the error bars correspond to the variance of the statistical sample. For all $R_s\/L$ the normalized pressure drop increases with increasing $\\phi$. This behavior is expected as the radius $r_i$ of each edge decreases monotonically as $\\phi$ increases~\\cite{torquato2002random}. Additionally, we observe that decreasing the ratio $R_s\/L$ leads to increase in $\\Delta P_c$, in qualitative agreement with experiments~\\cite{waisbord2019anomalous}. The reason for this behavior can be explained by examining the arc-length (defined as $L_c=\\sum_i^N l_i$) for the first open channel. \n\nFigure~\\ref{fig:result}(b) illustrates the histogram of $L_c$ for $\\phi=0.3$ and $0.5$ as well as $R_s\/L=0.02$ and $0.08$. In both cases we observe that increasing solid volume fraction leads to an increase in the total length of the first open path. While large $\\phi$ results in almost similar behavior for both $R_s\/L$, it is apparent that decreasing the size of the spheres results in more tortuous paths, even for low solid volume fraction. \n\nDimensional analysis on Eq.~\\ref{eq:least_res} indicates that $\\Delta P_c$ follows a simple scaling with $R_s\/L$. In particular, by rescaling the local edge radius $r_i$ with the sphere radius $R_s$, we find $\\Delta \\widetilde{P}_c=\\Delta P_c R_s\/\\tau_y L = \\sum_{i=1}^N \\left(l_i\/L\\right)\\left(R_s\/r_i\\right)$. The inset in Fig.~\\ref{fig:result}(a) shows the rescaled form of the critical pressure drop for all $R_s\/L$, were for non-overlapping disks, a master curve exists for all the examined values of $\\phi$. The dependence of $\\Delta \\widetilde{P}_c$ with $\\phi$, the effect of the pore microstructure, as well as the physical mechanism for the behavior of the tortuosity $\\tau=L_c\/L$ will be presented in a future work.\n\n\\section{\\label{sec:discussion}Discussion}\n\nWe find that network models provide an efficient and accurate way to model the fluidization conditions in porous media. They are also accurate in locating the first open channel, which is equivalent to the path of least resistance through the entire medium.\n\nOur results on the normalized critical pressure drop $\\Delta P_c\/\\tau_y$ can be used to design porous media systems with the desired flow properties. In applications such as semi-solid flow batteries~\\cite{duduta2011semi}, it is critical to keep the contact between the active material (electrode particles) and the conductive wiring (carbon nanoparticle network) intact during operation, otherwise there is a significant energy loss during cycling of the battery~\\cite{solomon2018enhancing,wei2015biphasic}. This can be achieved by immersing the active material and the electronically conductive agent in yield-stress fluids like Carbopol~\\cite{zhu2020high}. Therefore, we can use the predicted $\\Delta P_c\/\\tau_y$ to determine the size of the active particles to optimize the design and operation of semi-solid electrodes.\n\nThe present model can also provide insights on the design of porous media. By taking advantage of the computational efficiency of the proposed network model, we can perform on-the-fly optimization to construct porous media with optimal mixing and transport properties~\\cite{lester2013chaotic,kirkegaard2020optimal}. Such ideas have recently been implemented in elastic networks with optimal phonon band structures~\\cite{ronellenfitsch2019inverse,ronellenfitsch2019chiral}, and we believe they can also be used for designing porous media immersed in yield-stress fluids.\n\nGiven their inherent node\/edge structure, network models are fairly simple to be analyzed using graph theoretical tools. The unique property of yield-stress fluid, namely the fluidization conditions, allows us to use algorithms that can find the minimum resistance pathways with minimal efforts. In coarse-grained domains, however, where the microscopic geometric irregularities are encoded in the heterogeneous `permeability' tensor~\\cite{hewitt2016obstructed}, graph theory tools might not be the most suitable ones. An alternative way to calculate the first open channel in a continuum with spatially variable properties is through methodologies used in physical chemistry to identify reaction pathways~\\cite{henkelman2000climbing,swenson2018openpathsampling}. There, the first open channel corresponds to the path that passes through the minimum energy barrier, namely the transition state point.\n\nThe present model considers only the case of viscoplastic materials and disregards the fluid elasticity~\\cite{saramito2007new,cheddadi2012steady,fraggedakis2016yielding1,fraggedakis2016yielding2} prior to yielding. Therefore, further analysis is required to connect the yield criterion to the elastic modulus of the fluid, which can provide insights for engineering both the porous medium and the fluid itself. Additionally, the yielding and\/or stoppage conditions might be further affected by possible thixotropic~\\cite{mewis2012colloidal} and kinematic hardening~\\cite{dimitriou2019canonical,gurtin2010mechanics} phenomena.\n\n\\section{\\label{sec:conc}Summary}\nIn this work, we presented a network model for yield-stress fluids in porous media that describe the effects of porosity $1-\\phi$ and microstructure properties in complex geometries. We demonstrated the capabilities of the model to predict the critical applied pressure drop required to open the first channel in the medium. Also, we compared our results to direct numerical simulations of the full fluid problem for Bingham fluids and we showed the accuracy and computational efficiency of the network-based models to solid-liquid transition. Finally, we discussed the implications of our model on the optimization and design of porous media.\n\n\\section*{Contributions}\nD.F. conceptualized, designed, and performed the analysis in the present study. E.C. and O.T. provided the fluid flow simulation results. D.F. wrote the manuscript. All authors contributed to the final manuscript. \n\n\\section*{Acknowledgment}\nD.F. (aka dfrag) wants to thank T. Zhou and M. Mirzadeh for insightful discussions related to the validity of the network model. The authors declare no competing interests.\n\n\\section*{\\label{sec:appendix}Appendix}\n\\subsection*{Porous medium and network generation}\nFor the generation of porous media that consist of non-overlapping disks, we implemented the random sequential addition (RSA) algorithm~\\cite{zhang2013precise,cule1999generating,torquato2006random}. The procedure described in~\\cite{torquato2006random} allows for the fast generation of randomly packed disks with the desired volume fractions. Due to the constraint of non-overlapping disks, all generated microstructures never exceed $\\phi=0.52$ in two dimensions. \n\nFor the generation of the network model, we implemented the maximal ball algorithm as described in~\\cite{silin2006pore,dong2009pore,al2007network}. The procedure allows us to get the radius $r_i$ and length $l_i$ for each edge. The maximal ball algorithm represents `fits' a circle\/sphere within each pore~\\cite{alim2017local}, its radius of which represents the $r_i$ we use in Eq.~\\ref{eq:least_res}. Other choices of $r_i$ can be considered (e.g. equivalent radius etc.), however, our choice for the local radius works well in predicting both the first open channel and the critical pressure drop required to open it. The generated network was represented by a graph with vertices $V$ and edges $E$ using the open source library NetworkX~\\cite{hagberg2008exploring}.\n\nDetails on the numerical simulation of the fluid flow problem shown in Fig.~\\ref{fig:validation}(a) can be found in~\\cite{chaparian2020complex}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nVery recently IceCube Neutrino Observatory reported the detection of a high-energy muon neutrino event, IceCube-170922A of energy $\\sim 290$ TeV with a 56.5\\% probability of being a truly astrophysical neutrino \\cite{IceCube18a, IceCube18b}. The best-fit reconstructed arrival direction of the neutrino was consistent with the $0.1^0$ from the sky location of a flaring gamma ray blazar TXS 0506+056 \\cite{IceCube18a, Ansoldi18}. As a follow-up observation Fermi Large Area Telescope (LAT) Collaboration \\cite{Tanaka17} reported that the direction of origin of IceCube-170922A was consistent with the known gamma ray source TXS 0506+056 blazar which was in a state of enhanced emission with day scale variability \\cite{Keivani18} on 28 September 2017. The observed association of a high-energy neutrino with a blazar during a period of enhanced gamma-ray emission suggests that blazars may indeed be one of the long-sought sources of very-high-energy cosmic rays and hence these observations offers a unique possibility to explore the interrelate between energetic gamma rays, neutrinos, and cosmic rays.\n\nThe electromagnetic spectral energy distribution (SED) of the blazar TXS 0506+056 exhibits a double-hump structure which is a common feature of the non-thermal emission from blazars. The first hump, which peaking in the optical-ultraviolet range, is usually attributed to synchrotron radiation and the higher energy hump with peak energy in the GeV range, is often interpreted due to inverse Compton (IC) emission. An archival study of the time-dependent $\\gamma$-ray data over the last ten years or so reveal that the source was in quiescent stage most of the time, the flaring was noticed during the period July 2017 to September 2017. The averaged integrated flux above 0.1 GeV from TXS 0506+056 was found $(7.6 \\pm 0.2) \\times 10^{-8}$ $cm^{-2}s^{-1}$ during 2008 to 2017 from Fermi-LAT observations which in the week 4 to 11 July 2017 elevates to the level $(5.3 \\pm 0.6) \\times 10^{-7}$ $cm^{-2}s^{-1}$. The Astro-Rivelatore Gamma a Immagini Leggero (AGILE) gamma-ray telescope obtained flux of $(5.3 \\pm 2.1)\\times 10^{-7}$ $cm^{-2}s^{-1}$ during 10 to 23 September 2017. The Major Atmospheric Gamma Imaging Cherenkov (MAGIC) Telescopes detected a significant very-high-energy $\\gamma$-ray signal with observed energies up to about 400 GeV on 28 September 2017. Note that the Icecube observatory detected the neutrino event on 22 September 2017. It was found from optical to x-ray observations that the lower energy hump of the SED of the source does not show any noticeable time variation over the stated period of study.    \n\nSeveral efforts have been made so far to model the production of the detected neutrino event together with the electromagnetic (EM) observations from TXS 0506+056. Mainly two different production scenario, namely lepto-hadronic ($p\\gamma$) \\cite{Ansoldi18,Keivani18,Gao18,Cerruti19} and hadronic (pp) \\cite{Liu18,Sahakyan18} have been proposed in the literature to interpret the observations. A common feature of all the proposed models is that protons, like electrons, are also assumed to be accelerated to relativistic energies in the acceleration sites. Subsequently the accelerated protons interacting with low energy photons of blazar environment (lepto-hadronic interaction) and\/or with ambient matter produce high energy gamma rays and neutrinos.  \n\nAnsoldi et al. (2018) \\cite{Ansoldi18} showed that the measured neutrino event from the said blazar can be interpreted consistently with the EM observations by assuming a dense field of external low-energy photons originating outside of the jet as targets for photohadronic interactions. The lack of broadline signatures in the optical spectrum of TXS 0506+056 and other BL Lac objects suggest that such external photons emissions may be weak \\cite{Keivani18}. The model discussed in Ansoldi et al. (2018) \\cite{Ansoldi18}, however, does not invoke radiation from broadlines but, instead, assume the existence of soft radiation produced in a possible layer surrounding the jet. Therefore the lack of broadlines does not impact this specific scenario. In this context it is also to be noted that the BL Lac nature of TXS 0506+056 has been recently questioned by Padovani et al. 2019 \\cite{Padovani19}. Keivani et al. (2018) \\cite{Keivani18} considered a hybrid leptonic scenario of TXS 0506+056 where the production of high energy gamma rays was interpreted by external inverse-Compton processes and high-energy neutrinos via a radiatively sub-dominant hadronic component. \n\nFor efficient high energy $\\gamma-$ray production in AGN jet via $pp$ interaction demands high thermal plasma density; the thermal plasma in the jet should exceed $10^6$ cm$^{\u22123}$ in order to interpret the reported TeV flares of Markarian 501 by $pp$ interactions for any reasonable acceleration power of protons $L_p \\le 10^{45}$ erg\/s \\cite{Aharonian00}. The stated pure hadronic mechanism thus can be effectively realized in a scenario like ``relativistic jet meets target\" \\cite{Morrison84}, i.e. considering that $\\gamma-$radiation is produced in dense gas clouds that move across the jet \\cite{Dar97}. Recently, Liu et al. (2018) \\cite{Liu18} described the observed gamma ray \\& neutrino flux from the blazar TXS 0506+056 by assuming the presence of clouds in the vicinity of the super-massive black hole (SMBH) that provides targets for inelastic $pp$ collisions once they enter the jet. Liu et al. (2018) considered the synchrotron emission and inverse Compton emission of secondary electrons produced in cascade when high energy $\\gamma-$rays absorbed in $\\gamma\\gamma$ pair production with the emission region of the jet. However, the presence of broadline region (BLR) clouds in the vicinity of the SMBH for TXS 0506+056 is questionable due to the non-detection of the BLR emission from TXS 0506+056 and other BL Lac objects \\cite{Keivani18}. \n\nThe composition of bulk of the jet medium is not clearly known which makes difficulties to understand the interaction mechanism for gamma ray and neutrino production. But on average, jet plasma must be neutral to remain collimated \\cite{Hirotani05}. Therefore, the two main scenarios for their matter composition are suggested: a `pair plasma' consisting of only of relativistic electrons and positrons \\cite{Kino04} and a `normal plasma' consisting of (relativistic or non-relativistic) protons and relativistic electrons \\cite{Celotti93}. A useful quantity that can furnish some constraints on jet composition is the kinetic power of an AGN jet. By comparing the bulk kinetic energy of the parsec scale jet with the kinetic luminosities on extended scales \\cite{Rawlings91}, Celotti \\& Fabian (1993) \\cite{Celotti93} argued in favor of an electron-proton fluid. For high luminous blazars, to maintain the radiated power which would not exceed that carried by the jet, the proton component of plasma is necessary (see Ghisellini, 2010 \\cite{Ghisellini10}, and references therein). \n\nUnder the context, in the present work we exploit the main essences of proton blazar model \\cite{Protheroe01, Mucke01} to explain the observed higher-energy bump of the EM SED along with the neutrino from the blazar TXS 0506+056 at flaring stage. The detected lower-energy bump of EM SED from the blazar can be well interpreted with the synchrotron radiation of relativistic electrons present in jet plasma whereas the cold (non-relativistic) protons density that arose from charge neutrality condition can provide sufficient target matter (proton) for production of high energy gamma rays and neutrinos via the $pp$ interaction. For TXS 0506+056 such a scenario is more realistic then the scenario like the cloud-in-jet model \\cite{Aharonian17} as we argue later. We would also like to examine the maximum energy that a cosmic ray particle can attain in the blazar jet; the detected $\\sim 290$ TeV energy neutrino alone suggest that acceleration of protons in the jet of this object to energies of at least several times $10^{15}$ eV. \n\nThe organization of this paper is as follows: In the next section, we shall describe the methodology for evaluating the gamma-ray and neutrino  fluxes generated in interaction of cosmic rays with the ambient matter in the AGN jet under the framework of proton blazar model. The numerical results of the hadronically produced gamma-rays and neutrino fluxes from the AGN jet over the GeV to TeV energy range are shown in section III. The findings are compared with the observed gamma rays spectra and the neutrino event from the blazar and the results are discussed in the same section. Finally we conclude in Sec. IV.\n\n\\section{Methodology}\n\nThe overall jet composition of AGN is not properly known. In the adopted proton blazar inspired model it is assumed that the relativistic jet material is composed of relativistic protons (p) and electrons (e$^\u2212$). Some cold protons also exist, allowing charge neutrality to be fulfilled. The ratio of number of relativistic protons to electrons, the maximum energies attained by protons\/electrons in acceleration process and slope of their energy spectrum, luminosities of electrons and protons are adjustable parameters of the model. In this model flaring is produced due to high magnetic activities in the source (similar to the origin of flaring activities in the Sun). \n\nWe consider a spherical blob of size $R_b'$ (primed variables for jet frame) in the AGN jet which is the region responsible for the blazar emission. The blob is moving with a Doppler factor $\\delta = \\Gamma_j^{-1}(1-\\beta_j\\cos\\theta)^{-1}$ where $\\theta$ is the angle between the line of sight and the jet axis and $\\Gamma_j = 1\/\\sqrt{1-\\beta_j^2}$ is the bulk Lorentz factor \\cite{Petropoulou15}, and it contains a tangled magnetic field of strength $B'$.  \n\nIn the proton blazar framework the low-energy bump of the SED is explained by synchrotron radiation of accelerated relativistic electron in blazar jet having broken power law energy distribution as \\cite{Katarzynski01}\n\n\\begin{eqnarray}\nN_e'(\\gamma_e') = K_e \\gamma_e'^{-\\alpha_1} \\hspace{1.5cm} \\mbox{if}\\hspace{0.6cm} \\gamma_{e,min}' \\le \\gamma_e' \\le \\gamma_b' \\nonumber \\\\\n         = K_e \\gamma_b'^{\\alpha_2-\\alpha_1} \\gamma_e'^{-\\alpha_2} \\hspace{0.35cm} \\mbox{if}\\hspace{0.45cm} \\gamma_b' <\\gamma_e' \\le \\gamma_{e,max}'\\;\n\\end{eqnarray}\nwhere $\\gamma_e' = E_e'\/m_e c^2$ is  the  Lorentz  factor of electrons of energy $E_e'$, $\\alpha_1$ and $\\alpha_2$ are the spectral indices before and after the spectral break Lorentz factor $\\gamma_b'$ respectively. The normalization constant $k_e$ can be found from \\cite{Bottcher13}\n\\begin{equation}\nL_e' = \\pi R_b'^2 \\beta_j c \\int_{\\gamma_{e,min}'}^{\\gamma'_{e,max}}m_e c^2\\gamma_e' N_e(\\gamma_e') d\\gamma_e'\n\\end{equation}\nwhere $L_e'$ is the kinetic power in relativistic electrons in the blazar jet frame. The number density of highly relativistic (`hot') electrons is $n_{e,h}' = \\int N_e'(\\gamma_e') d\\gamma_e'$ and the corresponding energy density is $u_e' = 3p_e' = \\int m_e c^2\\gamma_e' N_e'(\\gamma_e') d\\gamma_e'$ where $p_e'$ is the radiation pressure due to relativistic electrons. Due to strong synchrotron and Inverse Compton cooling at relativistic energies, the acceleration efficiency of electrons in AGN jet is quite low and it can be assumed to be $\\chi_e \\approx 10^{-3}$ \\cite{Bykov96,Eichler05,Vazza15}. Hence total number can be determined as $n_{e}' = n_{e,h}'\/\\chi_e$. Thus the number density of non-relativistic (`cold') electrons is given by $n_{e,c}' = n_{e}' - n_{e,h}'$.\n\nThe emissivity of photons of energy $E_{s}'$ ($= m_e c^2 \\epsilon_{s}'$) due to the synchrotron emission of electrons which describe low energy component of EM SED of the blazar, can be written as \\cite{Bottcher13}\n\\begin{eqnarray}\nQ'_{s}(\\epsilon_{s}') = A_0 \\epsilon_{s}'^{-3\/2} \\int_1^\\infty d\\gamma_e' N_e'(\\gamma_e')\\gamma_e'^{-2\/3}e^{-\\epsilon_{s}'\/(b\\gamma_e'^2)}\n\\end{eqnarray}\nwith the normalization constant\n\\begin{eqnarray}\nA_0 = \\frac{c \\sigma_T B'^2}{6\\pi m_e c^2 \\Gamma(4\/3) b^{4\/3}}, \\nonumber\n\\end{eqnarray}\nwhere $\\sigma_T$ is the Thomson cross-section, $b = B'\/B_{crit}$ and $B_{crit} = 4.4\\times 10^{13}$ G. The magnetic field energy density is $u_B' = B'^2\/8\\pi = 3 p_B'$ where $p_B'$ is the corresponding pressure.\n\nThe emissivity of photons of energy $E_{c}'$ ($= m_e c^2 \\epsilon_{c}'$) due to the inverse compton scattering of primary accelerated electrons\nwith the seed photons co-moving with the AGN jet, which can describe lower part of high energy component of EM SED of the blazar, can be written as\\cite{Blumenthal70,Inoue96}\n\n\\begin{eqnarray}\nQ_{c}(\\epsilon_{c}') = \\int_0^\\infty d\\epsilon_{j}' n_j'(\\epsilon_{j}') \\int_{\\gamma_{e,0}'}^{\\gamma_{e,max}'} d\\gamma_e' N_e'(\\gamma_e') C(\\epsilon_{c}',\\gamma_e',\\epsilon_{j}'),\n\\end{eqnarray}\nwhere $\\gamma_{e,0}' = \\frac{1}{2}\\epsilon_{c}'\\left(1+\\sqrt{1+\\frac{1}{\\epsilon_{c}'\\epsilon_{j}'}}\\right)$ and the compton kernel $C(\\epsilon_{c}',\\gamma_e',\\epsilon_{j}')$ is given by Jones (1968) \\cite{Jones68} as\n\\begin{eqnarray}\nC(\\epsilon_{c}',\\gamma_e',\\epsilon_{j}') = \\frac{2\\pi r_e^2 c}{\\gamma_e'^2\\epsilon_{j}'} \\Big[2k \\ln(k) + (1+2k)(1-k)   \\big.  \\nonumber   \\\\\n\\big. + \\frac{(4\\epsilon_{j}'\\gamma_e' k)^2}{2(1+4\\epsilon_{j}'\\gamma_e'k)}(1-k)\\Big],\n\\end{eqnarray}\nwith $k = \\frac{\\epsilon_{c}'}{4\\epsilon_{j}'\\gamma_e'(\\gamma_e' - \\epsilon_{c}')}$ and $r_e$ is the classical electron radius. Here, $n_{j}'(\\epsilon_{j}')$ is the average number density of the seed photons of energy $\\epsilon_{j}'$( in $m_ec^2$) in the blob of AGN jet which can be directly related to observed photon flux $f_{\\epsilon_j}$ (in erg cm$^{-2}$ s$^{-1}$) from the blazar through \\cite{Dermer02}\n\\begin{eqnarray}\n\\epsilon_{j}'n_{j}'(\\epsilon_{j}') = \\frac{2 d_L^2}{c R_b'^2\\delta^2\\Gamma_j^2} \\frac{f_{\\epsilon_j}}{m_e c^2\\epsilon_{j}' }\n\\end{eqnarray}\nwhere $\\epsilon_{j} = \\delta \\epsilon_{j}'\/(1+z) $ \\cite{Atoyan03} relates photon energies in the observer and co-moving jet frame of red shift parameter $z$ respectively,  and $d_L$ is the luminosity distance of the AGN from the Earth. \n\nIn the proton blazar model the cosmic ray protons are also supposed to accelerate to very high energies $E_p' = m_p c^2 \\gamma_p'$ in the same region of blazar jet and the production  spectrum shall follow a power law \\cite{Malkov01,Cerruti15}:  \n\\begin{equation}\n N_p'(\\gamma'_p) =  K_p {\\gamma'_p}^{-\\alpha_p} .\n\\end{equation}\nwhere $\\alpha_p$ is the spectral index, $\\gamma_p'$ is the Lorentz factor of accelerated protons, $K_p$ denotes the proportionality constant which can be found from the same expression as eq (2) but for protons and $L_p'$ is the corresponding jet power in relativistic protons. The number density of relativistic protons is $n_p' = \\int N_p'(\\gamma_p') d\\gamma_p' $ and the corresponding energy density is $u_p' = 3p_p' = \\int m_p c^2\\gamma_p' N_p'(\\gamma_p') d\\gamma_p'$, where $p_p'$ is the radiation pressure due to relativistic protons. \n\nWe estimate the mechanical luminosity or total kinematic jet power of an AGN jet containing jet frame energy density $u'$ (sum of $u_e'$, $u_p'$ and $u_B'$), pressure $p'$ (sum of $p_e'$, $p_p'$ and $p_B'$) and matter density $\\rho'$ (including cold protons and electrons) from the following relation \\cite{Protheroe01}\n\\begin{eqnarray}\nL_{jet} = \\Gamma_j^2 \\beta_j c \\pi R_b'^2\\left[\\rho'c^2(\\Gamma_j-1)\/\\Gamma_j + u' + p' \\right].\n\\end{eqnarray}\nwhere we assume the Lorentz factor to be $\\Gamma \\approx \\delta\/2$ which is quite reasonable particularly for jets closely aligned to the line of sight of the observer. Applying charge conservation and considering that the number of relativistic electrons will be greater then the number of relativistic protons, the number of `cold' (non-relativistic) protons will be equal to the total number of electrons ($n_{e}'$) minus the number of hot protons ($n_p'$). Thus the cold matter density in protons and in electrons in the blob will be $\\rho_p' = (n_{e}'-n_p')m_p$ where $m_p$ is the rest mass of a proton and $\\rho_e' = n_{e,c}' m_e$ respectively. \n\nWhen the shock accelerated cosmic rays interact with the cold matter (protons) of density $n_{H} = \\rho_p'\/ m_p$ in the blob of AGN jet, the emissivity of produced secondary particles of energy $E_i' = m_e c^2 \\epsilon_i'$ in co-moving AGN jet frame is given by \\cite{Liu18,Anchordoqui07,Banik17a,Kelner06}\n\\begin{eqnarray}\nQ_{i,pp}'(\\epsilon'_{i}) = \\frac{c n_{H}m_e}{m_p} \\int_{\\frac{m_e\\epsilon'_{i}}{m_p}} \\sigma_{pp}(E'_{p}) N'_p(\\gamma'_{p})F_i\\Big(\\frac{E'_{i}}{E'_{p}},E'_{p}\\Big)\\frac{d\\gamma'_{p}}{\\gamma'_{p}}\n\\end{eqnarray}\nwhere $i$ could be $\\pi^{0}$ mesons, electrons (positrons) $e^{\\pm}$ or neutrinos $\\nu$ and $F_i$ is the spectrum of the corresponding secondary particles in a single $pp$ collision as given in Kelner et al. (2006) \\cite{Kelner06}.\n\nDue to decay of $\\pi^0$ mesons, the resulting gamma ray emissivity as a function of gamma ray energy $E_{\\gamma}'( = m_e c^2 \\epsilon_{\\gamma}')$ is given by \\cite{Banik17b} \n\n\\begin{eqnarray}\nQ_{\\gamma,pp}'(\\epsilon'_{\\gamma}) = 2\\int_{\\epsilon_{\\pi,min}'(\\epsilon'_{\\gamma})}^{\\epsilon_{\\pi,max}'}\\frac{Q_{\\pi,pp}'(\\epsilon'_{\\pi})}{\\left[{\\epsilon'_{\\pi}}^2-(\\frac{m_{\\pi}}{m_e})^2\\right]^{1\/2}}d\\epsilon'_{\\pi}\n\\end{eqnarray}\nwhere ${\\epsilon'_{\\pi,min}}(\\epsilon'_{\\gamma}) = \\epsilon'_{\\gamma} + (\\frac{m_{\\pi}}{m_e})^2\/(4\\epsilon'_{\\gamma})$ is the minimum energy of a pion required to produce a gamma ray photon of energy $\\epsilon'_{\\gamma}$ (in $m_e c^2$).\n\nWhen propagating through an isotropic source of low-frequency radiation, the TeV$-$PeV gamma-rays can be absorbed at photon-photon ($\\gamma\\gamma$) interactions \\cite{Aharonian08}. Thus, the emissivity of escaped gamma rays after $\\gamma\\gamma$-interaction can be written as \\cite{Bottcher13} \n\\begin{eqnarray}\nQ_{\\gamma,esc}'(\\epsilon_{\\gamma}') = Q_{\\gamma}'(\\epsilon_{\\gamma}') .\\left( \\frac{1-e^{-\\tau_{\\gamma \\gamma}}}{\\tau_{\\gamma \\gamma}} \\right).\n\\end{eqnarray}\nHere $\\tau_{\\gamma \\gamma}(\\epsilon_{\\gamma}')$ is the optical depth for the interaction and is given by \\cite{Aharonian08}\n\\begin{eqnarray}\n\\tau_{\\gamma \\gamma}(\\epsilon_{\\gamma}') = R_b' \\int \\sigma_{\\gamma\\gamma}(\\epsilon_{\\gamma}',\\epsilon_{j}') n_{j}'(\\epsilon_{j}')d\\epsilon_{j}'\n\\end{eqnarray}\nwhere $\\sigma_{\\gamma\\gamma}$ is the the total cross-section as given in Aharonian et al., 2008 \\cite{Aharonian08} and $n_{j}'(\\epsilon_{j}')$ describes the spectral distributions of target photons.  $n_{j}'(\\epsilon_{j}')$ is generally assumed to be the observed synchrotron radiation photons produced by the relativistic electron population in co-moving jet frame as given in eq.(6) because of the low luminosity of accretion disks in BL Lacs \\cite{Mucke01}.\n\nThe number of injected electrons (positrons) per unit volume and time in AGN blob with a Lorentz factor $\\gamma_e'$ coming from $\\gamma \\gamma$ pair production of high-energy photons as given by Aharonian,Atoian \\& Nagapetian (1983) \\cite{Aharonian83} reads\n\n\\begin{eqnarray}\nQ_{e,\\gamma\\gamma}'(\\gamma_e') = \\frac{3 \\sigma_T c}{32} \\int_{\\gamma_e'}^\\infty d\\epsilon_{\\gamma}' \\frac{n'_{\\gamma}(\\epsilon_{\\gamma}')}{\\epsilon_{\\gamma}'^3} \\int_{\\frac{\\epsilon_{\\gamma}'}{4\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}}^\\infty d\\epsilon_{j}' \\frac{n_{j}'(\\epsilon_{j}')}{\\epsilon_{j}'^2}  \\nonumber  \\\\\n\\times \\left[ \\frac{4\\epsilon_{\\gamma}'^2}{\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}\\ln\\left( \\frac{4\\gamma_e'\\epsilon_{\\gamma}'(\\epsilon_{\\gamma}'-\\gamma_e')}{\\epsilon_{\\gamma}'}\\right) - 8\\epsilon_{\\gamma}'\\epsilon_{j}' \\right.  \\nonumber   \\\\\n\\left. + \\frac{2\\epsilon_{\\gamma}'^2(\\epsilon_{\\gamma}'\\epsilon_{j}'-1)}{\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}-\\left( 1-\\frac{1}{\\epsilon_{\\gamma}'\\epsilon_{j}'}\\right)\\left(\\frac{\\epsilon_{\\gamma}'^2}{\\gamma_e'(\\epsilon_{\\gamma}'-\\gamma_e')}\\right)^2   \\right]  \\;\n\\end{eqnarray}\nwhere $n'_{\\gamma}(\\epsilon'_{\\gamma}) = (R_b'\/c)Q_{\\gamma,pp}'$ is the number density of photons of high energy $\\epsilon_{\\gamma}'$.\n\nThe high-energy injected electrons\/positrons ($Q_{e}'$) including both those ($Q_{e,\\gamma\\gamma}'$), produced in $\\gamma\\gamma$ pair production and those ($Q_{e,\\pi}'$), created directly due to the decay of $\\pi^{\\pm}$ mesons produced in $pp$ interaction (using eq.(9)) will initiate EM cascades in the AGN blob via the synchrotron radiation, the IC scattering.\n\nIn order to determine the stationary state of the population of produced electron distribution $N_e'(\\gamma_e')$, the injection function $Q_e'(\\gamma_e')$ has been used as a source term in the continuity equation for electrons as given by \\cite{Cerruti15}\n\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t} \\left[N_e'(\\gamma_e')\\right] = \\frac{\\partial}{\\partial \\gamma_e'}\\left[\\gamma_e'\\frac{N_e'(\\gamma_e')}{\\tau_c(\\gamma_e')}\\right] + Q_e'(\\gamma_e') - \\frac{N_e'(\\gamma_e')}{\\tau_{ad}},\n\\end{eqnarray}\nwhere we consider the adiabatic time-scale as $\\tau_{ad} = 2R_b'\/c$. The radiative cooling time, considering both inverse Compton losses and synchrotron losses is given by \\cite{Cerruti15}\n\n\\begin{eqnarray}\n\\tau_c(\\gamma_e') = \\frac{3m_e c}{4(u_B' + u_{ph}')\\sigma_T}\\frac{1}{\\gamma_e'}\n\\end{eqnarray}\nwhere $u_{ph}'$ is the energy density of photons in co-moving jet frame in equilibrium. \n\nUsing the integral expression given by Inoue \\& Takahara (1996) \\cite{Inoue96}, the solution of eq. (14) i.e, the cascade electron distribution in stationary state can be evaluated as  \n\n\\begin{eqnarray}\nN_e'(\\gamma_e') = e^{-\\gamma_e^*\/\\gamma_e'}\\frac{\\gamma_e^*\\tau_{ad}}{\\gamma_e'^2}\\int_{\\gamma_e'}^\\infty d\\zeta Q_e'(\\zeta)e^{+\\gamma_e^*\/\\zeta}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\gamma_e^* = \\frac{3m_e c^2}{8(u_B' + u_{ph}')\\sigma_T R_b'}\n\\end{eqnarray}\nindicating the Lorentz factor of electron when $\\tau_c(\\gamma_e') = \\tau_{ad}$. Once the equilibrium pair distribution $N_e'(\\gamma_e')$ is known, the associated stationary synchrotron emission is evaluated using Equation (3) and hence found the observable photon spectrum using eq. (11). \n\nLet $Q_{\\gamma,esc}'(\\epsilon_{\\gamma}')$ be the total gamma ray emissivity from the blob of AGN jet including all processes stated above i.e, the synchrotron and the IC radiation of relativistic electrons, the gamma rays produced in $pp$ interaction and also the synchrotron photons of EM cascade electrons. The observable differential flux of gamma rays reaching at the earth, therefore, can be written as   \n\n\\begin{eqnarray}\nE_{\\gamma}^2\\frac{d\\Phi_{\\gamma}}{dE_{\\gamma}} = \\frac{V'\\delta^2\\Gamma_j^2}{4\\pi d_L^{2}}\\frac{E_{\\gamma}'^2}{m_e c^2} Q_{\\gamma,esc}'(\\epsilon_{\\gamma}') . e^{-\\tau_{\\gamma\\gamma}^{EBL}}\n\\end{eqnarray}\nwhere $E_{\\gamma} = \\delta E_{\\gamma}'\/(1+z) $ \\cite{Atoyan03} relates photon energies in the observer and co-moving jet frame of red shift parameter $z$ respectively with $E_{\\gamma}' = m_e c^2 \\epsilon_{\\gamma}'$, $V' = \\frac{4}{3}\\pi R_b'^3$ is the volume of the emission region. Here we employ the Franceschini-Rodighiero-Vaccari (FRV) model \\cite{Franceschini08,website1} to find the optical depth $\\tau_{\\gamma\\gamma}^{EBL}(\\epsilon_{\\gamma}',z)$ for gamma-ray photons due to the absorption by the extragalactic background (EBL) light.\n\nThe corresponding flux of muon neutrinos reaching at the earth can be written as \n\\begin{eqnarray}\nE_{\\nu}^2\\frac{d\\Phi_{\\nu_{\\mu}}}{dE_{\\nu}} = \\xi.\\frac{V'\\delta^2\\Gamma_j^2}{4\\pi d_L^{2}} \\frac{E_{\\nu}'^2}{m_e c^2}Q_{\\nu,pp}'(\\epsilon_{\\nu}') \n\\end{eqnarray}\nwhere $E_{\\nu} = \\delta E_{\\nu}'\/(1+z) $ \\cite{Atoyan03} relates neutrino energies in the observer and co-moving jet frame respectively and the fraction $\\xi = 1\/3$ is considered due to neutrino oscillation.\n\n\\section{Numerical results and discussion}\nIn the third catalog of AGNs detected by Fermi-LAT listing 1773 objects \\cite{Ackermann15}, TXS 0506+056 is one of the most luminous objects with an average flux of $6.5 (\\pm 0.2)\\times 10^{-9}$ photons cm$^{-2}$ s$^{-1}$ between 1 GeV and 100 GeV. A high-energy neutrino-induced muon track IceCube-170922A, detected on 22 September 2017, was found to be positionally coincident with the flaring $\\gamma-$ray blazar, TXS 0506+056 \\cite{IceCube18a}. The coincidence detection probability by chance was found to be disfavored at a $3\\sigma$ confidence level mainly due to the precise determination of the direction of neutrino \\cite{IceCube18a} although no additional excess of neutrinos was found from the direction of TXS 0506+056 near the time of the alert. Assuming a spectral index of \u22122.13 (\u22122.0) for the diffuse astrophysical muon neutrino spectrum \\cite{Aartsen14}, the most probable energy of the neutrino event was found to be 290 TeV (311 TeV) with the 90\\% C.L. lower and upper limits being 183 TeV (200 TeV) and 4.3 PeV (7.5 PeV), respectively \\cite{IceCube18a, Ansoldi18}. \nExtensive follow-up observations by the Fermi-Large Area Telescope \\cite{Tanaka17} in GeV gamma-rays and by the Major Atmospheric Gamma-ray Imaging Cherenkov (MAGIC) \\cite{Mirzoyan17} telescopes in very-high-energy (VHE) gamma-rays above 100 GeV, revealed TXS 0506+056 to be active in all EM bands. The redshift of the blazar has been recently measured to be $z = 0.3365$ \\cite{Paiano18} and the luminosity distance, estimated with a consensus cosmology is $d_L \\sim 1750$ Mpc \\cite{Keivani18}.\n\nThe gamma ray variability time scale is found as $t_{ver} \\le 10^5$ s by analyzing the X-ray and gamma-ray light curves \\cite{Keivani18}. Consequently to describe the electromagnetic SED of TXS 0506+056 over the optical to gamma ray energy range we have chosen the size of emission region of $R_b' = 2.2\\times10^{16}$ cm with Doppler boosting factor $\\delta = 20$ and bulk Lorentz factor of AGN jet $\\Gamma_j = 10.4$  which are strongly consistent with the size inferred from the variability, namely $R_b' \\lesssim \\delta c t_{ver}\/(1+z) \\simeq 4.5\\times10^{16} (\\delta\/20) (t_{ver}\/10^5 s)$ cm \\cite{Keivani18}. \n\nThe low energy part of the experimental EM SED data can be explained well by synchrotron emission of primary relativistic electron's distribution obeying a broken power law as given by Eq.(1) with spectral indices $\\alpha_1 = 1.71$ and $\\alpha_2 = 4.3$ respectively before and after the spectral break Lorentz factor $\\gamma_b' = 8.5\\times 10^{3}$. The required kinematic power of relativistic electrons in blazar jet as given by Eq. (2) and the magnetic field to fit the observed data are $L_e' = 2.3\\times 10^{42}$ erg\/s and $B' = 0.38$ G respectively. Here we have not included the self-absorption of synchrotron photons spectrum. When the self-absorption mechanism \\cite{Katarzynski01} is included, the resultant spectrum will show slight mismatch with the observed photon flux at radio energies, particularly VLA and OVRO data.\n\nThe inverse Compton scattering of primary electrons with the target synchrotron photons (also including high energy photons) co-moving with the AGN jet as given by Eq. (4) are also found to produce lower part of high energy bump of EM spectrum, particularly from NuSTAR experimental data upto Fermi-LAT data.  The number of `hot' electrons in blob of the AGN jet are estimated to be $n_{e,h}' = 1.7\\times10^3$ particles\/cm$^{3}$ which is required to produce the EM SED due to both synchrotron and inverse Compton emission. But the acceleration efficiency of electrons in AGN jet may be quite low and it can be assumed to be $\\chi_e \\approx 10^{-3}$ \\cite{Bykov96,Eichler05,Vazza15} due to strong synchrotron and Inverse Compton cooling at relativistic energies and the total number of electrons including `cold' electrons are found out to be $n_e = 1.7\\times10^6$ particles\/cm$^{3}$. \n\nIn the original proton blazar model high energy gamma rays are produced through synchrotron radiation by high energy protons in strong magnetic field environment. However, due to low magnetic field strength of the source (obtained from the fitting of low energy hump of SED) the gamma ray spectrum of the source can not be modeled with the proton synchrotron radiation. The proton-photon interaction is also found inefficient in the present case  due to low amplitude of target synchrotron photon field. Instead required gamma rays are found to produce in interactions of relativistic protons with the ambient cold protons in the blob. The observed higher energy part of observed EM SED data, particularly those measured with Fermi-LAT and MAGIC observatory, can be reproduced well by the model as estimated following the best fit Eq. (18). The spectral index of the energy spectrum of AGN accelerated cosmic rays is taken as $\\alpha_p = - 2.13$ which is consistent with the best fit spectral slope of the observed astrophysical neutrinos of between 194 TeV and 7.8 PeV by IceCube observatory \\cite{Halzen17,Aartsen16}. The required accelerated primary proton injection luminosity is found to be $L_p' = 10^{46}$ erg\/s. The cold proton number density in jet turns out to be $1.68\\times 10^6$ particles\/cm$^3$ under charge neutrality condition which provides sufficient targets for hadronuclear interactions with accelerated relativistic protons. The estimated differential gamma-ray spectrum reaching at Earth from this AGN is shown in Fig. 1 along with the different space and ground based observations. It is clear from the figure that the observed spectrum is correctly reproduced by the model. The detection sensitivity of upcoming gamma-ray experiments like the Cherenkov telescope array (CTA) \\cite{Ong17} and the Large High Altitude Air Shower Observatory (LHAASO) \\cite{Liu17} are also shown in the figure which suggest that these experiments will be able to detect gamma rays up to nearly 100 TeV for any similar kind of event if detected in future and thereby will be able to provide a better understanding of the emission processes.\n\n\\begin{figure}[h]\n  \\begin{center}\n  \\includegraphics[width = 0.5\\textwidth,height = 0.45\\textwidth,angle=0]{txs_report_final_16_new.eps}\n\\end{center}\n\\caption{Estimated differential energy spectrum of gamma rays and neutrinos reaching at the Earth from the blazar TXS 0506+056. The pink small dashed line indicates the low energy component of EM spectrum due to synchrotron emission of relativistic electrons. The green long dash-double-dotted denotes the gamma ray flux produced due to Inverse Compton emission of relativistic electrons in seed photon distribution in co-moving jet frame. The red dotted line represents the gamma ray flux produced from neutral pion decay in $pp$-interaction together with the cascade emission of electron\/positron produced in (pionic) $\\gamma\\gamma$-absorption. The black continuous line represents the estimated overall differential multi wave-length EM SED. The blue small dash-single dotted line indicates the differential muon neutrino flux reaching at earth. The yellow dash-triple-dotted line and brown long dash-single dotted line denote the detection sensitivity of the CTA detector for 1000 h and the LHAASO detector for 1 year respectively. The cyan long dashed line indicates the expected level\\cite{Gao18} and energy range of the neutrino flux reaching at earth  to produce one muon neutrino in IceCube in 0.5 year, as observed.}  \n\\label{Fig:1}\n\\end{figure}\n\nHigh energy neutrinos are produced together with gamma rays in pp interactions. The high energy neutrino flux at the Earth from the blazar TXS 0506+056 in active state has also been estimated following Eq. (19) and also shown in Fig. 1 along with the concerned Icecube result. For estimation of the neutrino flux no additional adjustable parameters were available; the same parameters used to describe gamma ray spectrum lead the neutrino flux. The total mechanical jet power of the blazar in jet frame is found out to be $L_{jet}' = 1.2\\times 10^{47}$ erg\/s and physical jet power after Lorentz boost is $L_{jet} = 1.3\\times 10^{49}$ erg\/s. It is noticed that $\\eta'_p = L_p'\/L_{jet} = 8.5\\%$ under assumption of electron injection efficiency about $\\chi_e \\approx 10^{-3}$ i.e, cosmic ray protons carries 8.5\\% energy of total jet power in co-moving jet frame which is generally expected acceleration efficiency of cosmic rays in astrophysical sources \\cite{Banik17b,Sahakyan18}. \nThe total jet power in the form of magnetic field and relativistic electron and proton kinetic energy calculated as \\cite{Ansoldi18} $L^k_{jet} = \\Gamma_j^2 \\beta_j c \\pi R_b'^2\\left[u_e' + u_p' + u_B' \\right]$ and found out to be $10^{48}$ erg\/s. The estimated kinetic jet power of the blazar is consistent with the Eddington luminosity of $L_{edd}\\gtrsim 1.3\\times 10^{48}$ erg\/s if we assume a super-massive black hole of mass $M_{bh} \\gtrsim 10^{10} M_{\\odot}$, like blazar S5 $0014+813$ \\cite{Ghisellini09}. However, jet power may exceed the Eddington luminosity during outbursts or for a collimated outflow in a jet because in such situations the jet does not interfere with the accretion flow. Note that a moderate excess of jet power over the Eddington luminosity (within a factor of ten) seems physically viable \\cite{Gao18,Sadowski15}.\n\nThe MAGIC collaboration reported the prominent spectral steepening observed gamma ray spectra from the said blazar above $\\sim 100$ GeV which confirms the internal $\\gamma\\gamma$ absorption that is robustly expected as a consequence of $pp$ production of a $\\sim 290$ TeV neutrino and also restrict the $\\delta$ to a low value. The cascade emission of electron\/positron pairs induced by protons has been estimated following Eq. ($14-16$) where we include the contribution of high energy photons along with synchrotron photons in jet frame as target for internal $\\gamma\\gamma$ absorption of firstly produced gamma rays in $pp$ interaction. This mechanism also found to contribute significantly in the hard X-ray to VHE gamma-ray bands. A primary cosmic ray proton spectrum up to $E_{p,max}' = 10$ PeV in jet frame and magnetic fields of $B' = 0.38$ G which is mutually consistent with the synchrotron radiation of electrons for lower bump in EM SED, can somewhat describe the observed gamma-ray spectra. \n\nThe number of expected muon neutrino event in time $\\tau$ can be found from the relation $N_{\\nu_{\\mu}} = \\tau \\int_{\\epsilon_{\\nu,min}}^{\\epsilon_{\\nu,max}} A_{eff}(\\epsilon_{\\nu}). \\frac{d\\phi_{\\nu_{\\mu}}}{d\\epsilon_{\\nu}} d\\epsilon_{\\nu}$ where $A_{eff}$ be the IceCube detector effective area at the declination of the TXS 0506+056 in the sky \\cite{IceCube18b,Padovani18,Albert18}. We found that the expected muon neutrino event in IceCube detector from the blazar in 200 TeV and 7.5 PeV energy range are about $N_{\\nu_{\\mu}} = 1.007$ events in 0.5 years for the flaring VHE emission state with $E_{p,max}' = 10$ PeV. The expected muon neutrino event are about $N_{\\nu_{\\mu}} = 2.6$ for the same scenario but in the energy range of 32 TeV and 3.6 PeV with $E_{p,max}' = 10$ PeV which is in good agreement with the effective energy range \\cite{IceCube18b} of IceCube for astrophysical neutrinos. The model fitting parameters to match the EM SED as well as muon neutrino event are summarized in Table~\\ref{table1}.\n\n\\begin{table}[h]\n  \\begin{center}\n    \\caption{Model fitting parameters for TXS 0506+056 according to proton blazar model.}\n    \\label{table1}\n    \\begin{tabular}{c|c}\n      \\toprule\n      Parameters &  Values   \\\\ \\hline\n      $\\delta$              &  $20$ \\\\\n      $\\Gamma_j$              &   $10.4$  \\\\\n      $\\theta$              &   $1^{0}$  \\\\\n       $z$                  &  $0.3365$   \\\\\n      $R_b'$  (in cm)            &  $2.2\\times 10^{16}$ \\\\\n       $B$ (in G)          &  $0.38$  \\\\\n       $u_B$ (in erg\/cm$^{3}$)          &  $5.75\\times10^{-3}$  \\\\\n       $\\alpha_1$           &   $- 1.71$  \\\\ \n       $\\alpha_2$           &   $- 4.3$  \\\\\n       $\\gamma_b'$  &  $8.5\\times 10^{3}$ \\\\\n       $\\gamma_{e,min}'$  &  $1$ \\\\\n       $\\gamma_{e,max}'$  &  $1.5\\times 10^{5}$ \\\\\n       $u_e$ (in erg\/cm$^{3}$)          &  $4.5\\times10^{-2}$  \\\\\n       $L_e'$ (in erg\/s)    & $2.3\\times 10^{42}$  \\\\ \n       $n_H$  (in cm$^{-3}$) &  $1.68\\times 10^6$  \\\\\n       $\\alpha_p$           &   $- 2.13$  \\\\\n       $E_{p,max}'$ (in eV) &  $ 10^{16}$  \\\\ \n       $L_p'$ (in erg\/s)    &  $10^{46}$  \\\\\n       $L^k_{jet}$ (in erg\/s)   &  $ 10^{48}$ \\\\\n       $N_{\\nu_{\\mu}}$   &  1.007  \\\\ \\hline    \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n\nThe VHE gamma-ray flux from the blazar is found to be variable i.e, increasing by a factor of up to $\\sim 6$ within one day from low state (quiescent state) to the flaring state. The flux variability found mainly in high energy component but not in lower bump of EM spectra from the source disfavors the inverse-Compton origin for such variabilities. There may be two possible scenarios for such variabilities$-$ i) The VHE gamma-ray flux in low state is leptonic in origin, i.e, via inverse-Compton emission from electrons up-scattering synchrotron photons (synchrotron-self-Compton scenario, SSC \\cite{Maraschi92,Bloom96,Mastichiadis97}) or photons from the ambient fields (external inverse-Compton, EIC \\cite{Dermer92,Dermer93}) but consequently no neutrinos will produce. The higher flux of gamma rays in flaring state can be interpreted when the blazar jet meets with the external cloud \\cite{Aharonian17,Barkov12,Dar97} which will provide sufficient target matter (protons) for interaction with accelerated cosmic rays to produce observed high energy gamma rays and neutrinos efficiently. ii) The VHE gamma-ray flux in both low state and flaring state can be explained in a hadronic interaction model using a proton blazar model. In this scenario, observed gamma ray flux can be explained well with hadronic $pp$ interaction of accelerated cosmic rays of comparatively harder spectral slope ($\\sim 2.28$) and lowering the maximum energy of accelerated cosmic rays with ambient `cold' proton (in charge neutrality condition with co-accelerated electrons) in low state of the blazar compared to flaring state and subsequently produce neutrinos (of event $N_{\\nu_{\\mu}} = 0.13$ in 0.5 year) as well.  \n\n\\section{Conclusion}\nThe  coincident detection of the neutrino event, IceCube-170922A with the gamma ray flaring blazar, TXS 0506+056 provide support to the acceleration of cosmic rays in the blazar jet in diffusive shock acceleration process \\cite{IceCube18a}. In the framework of proton blazar model, our findings suggest that relative contributions to the total jet power of cold protons, accelerated protons, magnetic field, and accelerated electrons, obtained on the basis of charge neutrality, can explain both the low and high energy bump of the multi-wavelength EM SED and also the observed neutrino event, IceCube-170922A from the flaring blazar, TXS 0506+056 consistently. We find that maximum energy of cosmic ray particle achievable in the blazar is one order less then the ankle energy of cosmic ray energy spectrum or $2\\times 10^{17}$ eV in observer frame, is required to explain consistently the observed gamma ray and the neutrino signal from the source. The upcoming gamma-ray experiments like CTA \\cite{Ong17} and LHAASO \\cite{Liu17}, which are much sensitive up to 100 TeV energies, may provide clearer picture regarding the physical origin of gamma rays if more events like TXS 0506+056 are detected in future.\n\nThe gamma ray flux in the quiescent state of the source TXS 0506+056 is smaller by an order or so. Such a fact disfavor the cloud-jet interaction model as in the absence of cloud the gamma ray flux should decrease substantially. One may argue that the quiescent state gamma ray flux is due to inverse Compton process by relativistic electrons. But a fine tuning is needed to produce the exactly same kind of shape and same peak position of the second hump of EM SED in both enhanced and quiescent state if two different processes (hadronic and inverse Compton) are invoked to explain the observations. Recently, IceCube collaboration re-analyzed their historical data and reported significantly an evidence for a flare of 13 muon-neutrino events in the direction of TXS 0506+056 between September 2014 and March 2015 \\cite{IceCube18a}. Surprisingly, the blazar TXS 0506+056 was found to be in the quiescent state of both the radio and GeV emission at the arrival time window of such a neutrino flare \\cite{Padovani18}. Such an observation favors hadronic interaction mechanism for the production of observed high energy gamma rays and as well as neutrinos for both low and flaring state of the blazar. More elaborate studies are required to understand the production mechanism of the muon-neutrino events from TXS 0506+056 in the quiescent state.\n\n\\section*{Acknowledgments}\nThe authors would like to thank an anonymous reviewer for insightful comments and very useful suggestions that helped us to improve and correct the manuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\nContents\n\nINTRODUCTION\n\n**THE ABS DIET CHEAT SHEET**\n\nCHAPTER 1\n\n**YOUR WEIGHT IS REALLY NOT YOUR FAULT**\n\nCHAPTER 2\n\n**NEVER GO HUNGRY AGAIN**\n\nThe ABS DIET POWER 12\n\nCHAPTER 3\n\n**THE SIX STEPS TO LIFELONG LEANNESS**\n\nGuidelines for Easy Eating\n\nCHAPTER 4\n\n**SHOP 'TIL YOU DROP (POUNDS)**\n\nThe Complete Abs Diet Grocery List\n\nCHAPTER 5\n\n**LET'S GET IT STARTED**\n\n25 Abs Diet Breakfasts\n\nCHAPTER 6\n\n**LEAN IN THE MIDDLE**\n\n25 Abs Diet Lunches\n\nCHAPTER 7\n\n**START YOUR NIGHT RIGHT**\n\n30 Abs Diet Dinners\n\nCHAPTER 8\n\n**SHAKE THINGS UP**\n\n27 Abs Diet Smoothies and Snacks\n\nCHAPTER 9\n\n**EATING OUT, EATING RIGHT**\n\nThe Abs Diet Restaurant Survival Guide\n\nCHAPTER 10\n\n**INDULGE AND ENJOY**\n\nThe Abs Diet Holiday Survival Guide\n\nCHAPTER 11\n\n**CRANK UP THE FAT BURN**\n\nThe Abs Diet Workout\n\n**ABS DIET FAQ**\n\n**MEASURING YOUR PROGRESS ON THE ABS DIET**\nAcknowledgments\n\n**SEEING THE ABS DIET** come to fruition has been one of the great pleasures of my life. Seeing it make a real impact on the lives of tens of thousands of Americans has been one of the great rewards. For all of it, I have to thank a number of extraordinarily talented, hard-working, and dedicated people who continue to support, encourage and inspire me. In particular:\n\nSteve Murphy, whose courage and commitment to editorial quality has made Rodale Inc. the best publishing company in the world to work for.\n\nThe Rodale family, without whom none of this would be possible.\n\nJeremy Katz, executive editor of _Men's Health_ Books, whose wisdom and guidance has made The Abs Diet into an extraordinary success.\n\nBen Roter, whom I want to be when I grow up.\n\nSteve Perrine, who can make a silk purse out of just about anything.\n\nThe entire _Men's Health_ editorial staff, the smartest and hardest-working group of writers, editors, researchers, designers, and photo directors in the industry. Most important, a big shout out to Chris Krogermeier, Marilyn Hauptly, Jennifer Giandomenico, Erin Hobday, Phillip Rhodes, Brenda Miller, and everyone else who worked so hard and so fast to publish this book in record time.\n\nMy brother, Eric, whose relentless teasing shamed me into taking better care of myself. (Dude, you are sooo dead. . . .)\n\nMy mother, Janice, who raised two of us nearly single-handedly. Your strength and kindness guide my every action.\n\nMy dad, Bohdan, who left this world way to early. I wish you were still here.\n\nElaine Kaufman, who still lets me order off the menu.\n\nAnd special thanks also to: Mary Ann Bekkedahl, Michael Bruno, Jeff Csatari, Aimee Geller, Karen Mazzotta, Jon Hammond, Cathy Gruhn, Joe Heroun, Samantha Irwin, George Karabotsos, Charlene Lutz, Patrick McMullan, Peter Moore, Jeff Morgan, Myatt Murphy, Megan Phillips, Scott Quill, Cindy Ratzlaff, Leslie Schneider, Joyce Shirer, Bill Stump, Sara Vigneri, Bug and Fester, and my stepmother, Mickey.\n\nAnd to Rose. On the rollercoaster of life, you've taught me to let go of the safety bar and reach my hands into the air.\n\n# **INTRODUCTION**\n\n# Eat Right Every Time  \nThe Abs Diet Way to a Flat Belly\n\n**D IET IS A FOUR-LETTER** word.\n\nThat may sound like a strange sentence with which to start a diet book. But then again, this is no ordinary diet book.\n\nWe think of a diet as something we \"go on.\" A high school reunion looms, or a family vacation to beach territory is planned, or some other event that tells us it's time to bear down and get rid of that extra layer of flab comes up. And so we do\u2014we \"go on\" a diet. Then, once we've lost the desired weight, we \"go off\" the diet and go back to our habits of eating cold macaroni and cheese while standing over the kitchen sink. Soon enough, another important event looms, and we're back on another diet again. Up and down, up and down goes our weight, but mostly, over time, it goes up. That's because deprivation diets and fads like eating low-fat, eating low-carb, or eating nothing but grapefruit just don't work in the long run. In fact, they stress your body so much that your body responds by trying even harder to store fat, especially in the midsection.\n\nA recent study in the _American Journal of Preventive Medicine_ found that about 60 percent of Americans who try to lose weight do so by restricting their calorie intakes, with roughly one in 10 skipping meals in a desperate attempt to strip off the pounds. But study after study has shown that yo-yo dieting is one of the best ways to ensure your belly will get bigger and flabbier in the months and years ahead.\n\nWell, those days are over!\n\nThe Abs Diet is a revolutionary new way of eating, one that's swept America in the last year and helped tens of thousands of people lose hundreds of thousands of pounds. In fact, with the Abs Diet\u2014a simple, six-times-a-day eating plan that will never let you get hungry\u2014you can lose 10, 15, even 20 pounds, from your belly first, in 6 weeks or less.\n\nThe Abs Diet isn't a traditional diet, because you'll eat so much delicious food\u2014from steak to strawberries, bread to bacon, soup to nuts\u2014that you'll never want or need to stray from it. In fact, if you're ever hungry on the Abs Diet\u2014well, then, you're not on the Abs Diet.\n\nSee, I know how hard it is to lose weight using traditional methods. As a boy growing up in small-town Pennsylvania, I too struggled with a weight problem. I made bad choices\u2014choosing fast food over smart food, then trying to starve myself to get my body in the shape I wanted it. Sure enough, I'd get hungry, and there I'd be, barking my lunch order into a clown's mouth once again. My brother, Eric, used to invite his friends over to watch my dietary indiscretions: \"Don't disturb the big animal,\" he'd tell his buddies. \"It's feeding.\"\n\nBut all that's changed. As the editor-in-chief of _Men's Health_ , I've spent the past 10 years poring over cutting-edge research in nutrition, fitness, weight loss, and exercise. And what I've learned, I've distilled into the Abs Diet\u2014a program that's been proven time and time again to strip away belly fat and leave you looking and feeling better than ever.\n\n## **The Secret to Perfect Weight Control Is in Your Hands**\n\n**THE ABS DIET REVOLVES** around a dozen delicious, convenient foods I call the ABS DIET POWER 12. All you need to do is eat the acronym: **A** lmonds and other nuts, **B** eans and other legumes, **S** pinach and other green vegetables, **D** airy (low-fat), **I** nstant oatmeal, **E** ggs, **T** urkey and other lean meats, **P** eanut butter, **O** live oil, **W** hole-grain breads and cereals, **E** xtra-protein (whey) powder, **R** aspberries and other berries.\n\nYup, you read that right: You get to eat healthy protein, healthy fats, healthy carbs\u2014there's hardly anything you need to give up on this program. Ice cream? Sure. Grilled cheese sandwiches? Yup. Uncle Frank's famous chili? You bet. All I want you to do is eat more good food, more often, and to toy with the ingredients of your favorites to make them just as delicious\u2014and twice as nutritious. And as for the few foods I do want you to say goodbye to\u2014anything that stains your fingers orange, for example\u2014well, I've come up with some great, tasty, healthful alternatives.\n\nThe Abs Diet is so easy even Jessica Simpson could handle it. (And she'd get to eat Chicken of the Sea even if it really was chicken!) There's no measuring, no counting calories, no complicated equations, no hours-in-the-kitchen recipes. Just simple, smart, delicious food you'll enjoy all day long.\n\nBut even though the Abs Diet is simple, modern life is complicated. The grocery stores and chain restaurants are filled with foods that look healthy but really aren't; foods that are packed with high-calorie sweeteners that actually increase your appetite; foods that are greased up with unhealthy, chemically altered fats that clog up your plumbing; and foods that have had all their nutritional value stripped from them before they're wrapped in cellophane and set on the supermarket shelf. And even family life comes fraught with its own perils: In a survey of 274 single and married women by Texas Tech University Health Sciences Center, researchers found that almost 60 percent of married women were obese, compared to just 43 percent of singles. \"Women in larger households have four times the odds of being obese in comparison to women who live alone,\" says study author James E. Rohrer, Ph.D. He suggests that more people in the household translates into more food in the fridge and a risk of obesity that is four times greater than that of women who live solo.\n\nAnd that's why I've created this sequel\u2014 _The Abs Diet Eat Right Every Time Guide_ \u2014to help you make smart, healthy choices on the fly, whether you're cooking up the family dinner or ordering out on a romantic date. Wherever you may be, _The Abs Diet Eat Right Every Time Guide_ will show you how to eat more of the great foods out there and teach you to avoid the fat bombs that are looking to spoil your waistline.\n\nOur relationship with food is a complicated one, especially in America, the land of plenty. Sixty-five percent of American adults are obese or overweight, and for good reason: We've become a country that considers drive-throughs fine dining, that saves money if we order two pizzas instead of one, that loves bosses who treat the office to doughnuts, that's been snowed into thinking sausage is a diet food, and that builds its food pyramids with an order of 50 wings.\n\nWith _The Abs Diet Eat Right Every Time Guide_ , you'll be armed\u2014literally, to the teeth\u2014with the information you need to make the right choices. From Arby's to Zabar's, I've listed the best foods to buy\u2014and the worst foods to avoid\u2014in every shopping and dining situation. Slip this book inside your jacket pocket or into your purse and be ready whenever, wherever hunger strikes.\n\n## **Stay Lean\u2014Not Hungry**\n\n**I TRUST THAT** if you've read this far, you're inspired by the idea that you can rebuild your body. But every person who wants to lose fat is motivated by different factors\u2014whether it's because you want to feel better, live longer, look better in a bathing suit, run faster, or avoid having your kids mistake you for a Sea World attraction. No matter your motivation, I think building a diet plan that works into your lifestyle is the most important element for success.\n\nLike I said at the beginning of this book, diet is a four-letter word. It's been twisted around to mean eating less, not eating more or eating better. Say the word and all you see is a 6-month stint of celery sticks and rice cakes. Well, wrong. You should stop thinking about \"dieting\" and start thinking about building a healthy \"eating plan.\" Even the people you'll read about in this book\u2014people who have lost more than 20 pounds in 6 weeks\u2014found themselves forgetting about the word _diet._\n\nTake Jon Armond, who traded in his beer gut for a flat one with a 35-pound weight loss in 9 weeks. Jon says he doesn't even miss the foods he used to eat.\n\n\"I found myself not even consumed with dieting,\" Jon says. \"I've never been on a diet where you didn't have to think about being on the diet all the time.\" (See his story.) That's because the Abs Diet is a system that's easy to follow because it never leaves you hungry and gives you the nutritional balance to have all of your cravings fulfilled.\n\n## **Eat More\u2014and Weigh Less!**\n\n**MOST DIETS ARE ABOUT** losing\u2014losing meals, losing muscle mass, losing energy. The Abs Diet is about gaining\u2014gaining health and fitness, gaining time and energy, gaining delicious foods you can eat whenever you want. Just look at all you have to gain:\n\n**A longer life!** Belly fat is the most dangerous kind of fat. That's because belly fat often comprises both subcutaneous fat (fat that's under your skin) and visceral fat (fat that lies beneath your stomach muscles, snug up against your internal organs). It's this second type, visceral fat, which can cause you some long-term harm.\n\nHere's why: In a lean, healthy person, the liver uses both fat and glycogen (i.e., blood sugar) as energy. But if your belly is hard to the touch and protrudes out in front of you, it means your liver is basically encased in a layer of fat. With so much fat on hand, your liver gets lazy, forgets about glycogen, and just burns fat for energy. Now, normally, burning fat is good\u2014but not without glycogen to balance it. Your liver is like a big iron stove in your living room. You can fill it with clean, dry wood\u2014that's the glycogen. Or you can fill it with rotten fruit, egg shells, and styrofoam containers\u2014that's the fat. Both will burn, but one will burn cleanly, and the other will create a horrible, stinky mess.\n\nIn your body, that horrible stinky mess is excess cholesterol\u2014the byproduct of a liver that burns more fat than glycogen. Perhaps even worse, once your liver gets used to burning fat, it forgets how to properly manage blood sugar. In one report, researchers concluded that visceral fat is the single-best predictor of diabetes. So factor in the higher cholesterol, the increased risk of diabetes, and all the complications of the two, and you realize that losing that fat is just as important to your long-term health as losing that eyebrow ring is to your future employment. But that's really only the beginning of the story. Shedding fat from your frame is essential to living long and healthy. Consider:\n\n  A Canadian study of 8,000 people found that those with the weakest abdominal muscles (an indicator for fattier abdominal regions) had twice the death rate of those with stronger ones.\n\n  Many studies show that men with waists larger than about 35 inches have an increased risk of heart disease.\n\n  A Swedish study found that cancer rates are 33 percent higher in obese patients than in lean ones.\n\n  Overweight men are 50 percent more likely to develop heart diseases, 360 percent more likely to develop diabetes, and 16 percent more likely to die of a first heart attack.\n\nBottom line: Lose fat, gain years. You don't have to be George Steinbrenner to know who gets the better end of that trade.\n\n**More and better sex!** Sure, losing weight can turn you from a clown to a Clooney, from a Jujube to an Angelina Jolie. But besides the side effects of looking and feeling better (which gives you more confidence and makes you more attractive to others), losing fat helps with the mechanics of sex. Being overweight makes you 50 percent more likely to have erectile dysfunction. Of course, there are many factors that control sexual dysfunction in both men and women, but one of the major ones is a supply-and-demand issue. See, when you're sexually excited, your brain sends an all-points bulletin to your pants. _Wet bathing suit ahead, prepare for engagement._ With that message, your brain sends blood downward to cause an erection in a man or stimulate arousal-sensing and lubricating organs in a woman. But when you're overweight, the aftereffects of the afternoon's burrito supreme gum up your blood vessels and thus narrow the arteries that lead to Shangri La. Without a sufficient blood supply, nothing happens. If there's no gas in the tank, your car ain't going anywhere.\n\n**A pain-free and injury-proof life!** While the core goal of this book is to help you decipher the tricky menu mines you'll encounter through life, another important part of the Abs Diet is transforming your body with a modest amount of exercise. That will teach your body to grow and maintain lean muscle mass. I'll review the workout principles briefly in Chapter 11, but as part of the plan, you'll be doing some work to strengthen your abdominal region. That is, you'll teach your abdominal muscles to be strong. That's not just so you can score a modeling contract. That's because strong abdominal muscles are the infrastructure of your body. Your abs play a role in just about every physical movement you make. They help you run, lift, have sex, stretch, bend, pick up your kids, and shimmy through clothes racks. But most important, they act\u2014along with your lower back\u2014as your internal girdle to support you through the everyday rigors of life. One U.S. Army study showed that those people with the strongest abdominal muscles were the least likely to be injured (for all kinds of injuries, not just lower-back ones), and that points to the crucial role that strong abdominal muscles play. Strong abdominal muscles will help prevent and alleviate back pain\u2014one of the most debilitating and most common injuries for both men and women.\n\nIt helps to realize that abdominal muscles don't really work in isolation; they work as a cohesive group that crisscrosses your midsection and attaches to your spine. When abdominal muscles are weak, other muscles in your body have to pick up the slack. They end up overcompensating, and they end up causing back pain and strain or even more serious back problems.\n\n**Abs!** If you haven't figured it by now, Sherlock, then let me explain: This program is about living a healthier life. It'll help you lose weight, it'll help you gain control of what you eat, and it'll help turn your body into a fat-burning, muscle-building juggernaut. Ultimately, this book is about striving to meet your individual goals and looking the way you want to look, and that's where the abs come in. Abs are the byproduct of your new outlook on eating and exercising. They're the reward for following a plan that makes you healthier. Of course, if you're one bonbon away from a total couch collapse, it's going to take you a little longer than someone who's only 5, 10, or even 20 pounds over their ideal weight. (The typical standard for men is that you need to have a body-fat percentage of around 10 percent to have visible abs. For women, it's around 14 percent.) The fact is, everyone has abs, and everyone has the potential to see their abs. It's just a matter of banishing the fat off your gut so you can. Surely, some people are blessed with high-quality genetics, metabolism, or lipo docs, but the potential is there for everyone.\n\n## **Turn Fat into Muscle!**\n\n**MOST DIET PLANS DEPEND** on willpower for success. The Abs Diet, on the other hand, recruits an often untapped but highly potent ally in your search for slim\u2014your own body!\n\nSee, one of the most potent fat-burners around is lurking right below your own skin. It's muscle, and building more can turn your body into a fat-burning machine. For each pound of muscle you build, you'll burn an extra 50 calories a day\u2014just sitting still. If you were to build just 10 pounds of new muscle, you'd burn off enough extra calories to drop 50 pounds of fat in 1 year\u2014again, simply by sitting still.\n\nOf course, you can't build that muscle by sitting still. So I've included in this book a super-simple workout built around an easy weight-training program that anyone can do. It won't turn you into a gym rat, the Incredible Hulk, or the governor of California. But it will help you trade flab for lean, sexy muscle. (Feeling eager? Turn to this page for a quick peek at the Abs Diet Workout.)\n\n## **Take Back Control of Your Body\u2014And Your Life!**\n\n**YOU DON'T HAVE TO** just take my word for it. Listen to other people who've embraced the Abs Diet\u2014people like Linda Toomey, who lost 20 pounds in 6 weeks, while caring for four small children; or Jim Phillips, who was stuck in a weight rut for years until he found this revolutionary and easy plan; or Kyle Snay, who lost more than 20 pounds on the program and alleviated all of his back pain while doing so. As of now, there are more than 100,000 people taking the Abs Diet Challenge (you can find it on www.absdiet.com [inactive]). The stories of the many people who have tried and succeeded on the Abs Diet show the power of the plan.\n\nNow, with _The Abs Diet Eat Right Every Time Guide_ , you have a passport to success no matter where you and your stomach may wander. What I want is for you to use the Abs Diet Powerfoods as the guiding philosophy for the way you eat. With more than 100 recipes in this book, you'll cook up plenty of tasty ways to help you lose fat. But I also want you to use this guide to help you manage situations when you face more tough questions than Michael Jackson's publicist. It's one thing to eat right when you prepare the foods yourself; it's quite another when you're at the mercy of lard-slinging chefs.\n\nAbove all\u2014and this is really the foundation for _Eat Right Every Time_ \u2014I want you to feel one way while taking part in the Abs Diet: Satisfied.\n\nSatisfied . . . that you'll never be hungry.\n\nSatisfied . . . that you won't be denied great-tasting foods. Satisfied . . . that you have the flexibility to eat according to your lifestyle.\n\nSatisfied . . . that you'll have the knowledge to eat right anywhere you go.\n\nSatisfied . . . that the only new pants you'll ever have to buy will be smaller.\n\nSatisfied\u2014make that ecstatic\u2014with your new body!\n\n# **THE ABS DIET CHEAT SHEET**\n\n**T HIS AT-A-GLANCE GUIDE** summarizes the principles of the Abs Diet: the 6-week plan to flatten your stomach and keep you lean for life.\n\n**Number of meals**\n\nSix a day, spaced relatively evenly throughout the day. Eat snacks 2 hours before larger meals.\n\n**The _ABS DIET POWER 12_**\n\nBase most of your meals on these 12 groups of foods. Every meal should have at least two foods from the list.\n\n**A** lmonds and other nuts\n\n**B** eans and legumes\n\n**S** pinach and other green vegetables\n\n**D** airy (fat-free or low-fat milk, yogurt, cheese)\n\n**I** nstant oatmeal (unsweetened, unflavored)\n\n**E** ggs\n\n**T** urkey and other lean meats\n\n**P** eanut butter\n\n**O** live oil\n\n**W** hole-grain breads and cereals\n\n**E** xtra-protein (whey) powder\n\n**R** aspberries and other berries\n\n**Portion size**\n\nWhile many diets center around controlling portion size, the Abs Diet is designed to be self-controlling. The high-fiber, high-protein foods you'll encounter in this book will fill you up and keep you feeling full for hours. Your body will tell you when it's time to eat\u2014and when it's time to stop.\n\n**Secret weapons**\n\nEach of the ABS DIET POWER 12 has been chosen in part for its stealthy, healthy secret weapons\u2014the nutrients that will help power up your natural fat burners, protect you from illness and injury, and keep you lean and fit for life!\n\n**Nutritional ingredients to emphasize**\n\nProtein, monounsaturated and polyunsaturated fats, fiber, calcium.\n\n**Nutritional ingredients to limit**\n\nRefined carbohydrates (or carbs with high glycemic index), saturated fats, trans fats, high-fructose corn syrup.\n\n**Alcohol**\n\nLimit yourself to two or three drinks per week, to maximize the benefits of the Abs Diet plan.\n\n**Ultimate Powerfood**\n\nSmoothies. The combination of the calcium and protein in milk, yogurt, and whey powder, combined with the fiber in oatmeal and fruit, makes them one of the more filling and easy options.\n\n**Cheating**\n\nOne meal a week, eat anything you want.\n\n**Exercise program**\n\nOptional for the first 2 weeks. Weeks 3 through 6 incorporate a 20-minute, full-body workout 3 days a week. Emphasis is on strength training, brisk walking, and some abdominal work.\n\n**At-home workout**\n\nGym workouts and at-home workouts are both detailed to excuse-proof your fitness plan.\n\n**Abdominal workout**\n\nAt the beginning of two of your strength-training workouts. One exercise for each of the five different parts of your abs.\n\n### **ABS DIET SUCCESS STORY**\n\n## **INSPIRED TO LOSE, INSPIRED TO WIN**\n\n**Name:** Bret Freeman\n\n**Age:** 38\n\n**Height:** 5\u203210\u2033\n\n**Weight, Week 1:** 205\n\n**Weight, Week 6:** 183\n\n**Weight, Week 17:** 175\n\n\"I was just basically tired of being a fat slob,\" says Bret Freeman, who had been a lean, fit wrestler in high school but steadily gained weight over the years.\n\n\"I tried Atkins and all the fad diets out there. I'd go on and off, and it always seemed like I gained back more weight than what I started out with,\" says Freeman, who once reached an all-time high of 245 pounds. \"I tried The Zone diet, and that thing was so dang complicated; you had to put so much thought into what you ate, good sugars and bad sugars, what it did to your insulin levels. It was horrible.\"\n\nWhen Freeman heard about how simple the Abs Diet was, he couldn't wait to get started. So he ordered it, read it through in a day and a half and started the following Monday. At the same time, he bought a home gym and started the exercise program. The pounds started coming off immediately.\n\n\"I think the Abs Diet worked for me for a combination of reasons, but one was that it kept me eating delicious food all day long,\" he says. \"People would notice how much weight I lost, and they'd say, 'You must be starving yourself.' And I'd tell them, 'You wouldn't believe how much I eat.' \"\n\nThe change? Besides the pounds, Freeman lost inches. He started the diet in snug 38s, and now his 32s are a little loose. \"I think I'm ready to go down to a 30.\" Motivated by Lance Armstrong's story, he's now set to do a couple triathlons, and he's been asked to be the head coach for the high school junior varsity wrestling team. Plus, there's the added satisfaction he feels when he goes to his sons' football games, where he hasn't seen other parents since last season. Freeman says, \"One friend joked that they saw my wife, and they thought she got a new husband because I looked so different.\"\n\nAnd in a way, she did.\nChapter 1\n\n# **YOUR WEIGHT IS REALLY NOT YOUR FAULT**\n\n**E VERYONE HAS ENEMIES.** Batman has the Joker. Duke has UNC. Joan Rivers has gravity. You? Sure, your cranky neighbor may slice your perennials when he mows the lawn, and the boss's henchman may boil your blood, but to me, the biggest enemies you face are the profiteering junk-food merchants of the food industry.\n\nIn a time-strapped society and in your time-strapped life, these piranhas prey on consumers who need a quick fix\u2014a quick fix of fat, fries, and foods designed to hypnotize you with their taste and stretch you at your waist. As you surely can see at any interstate exit, mall food court, or stretch of commercialized suburbia, we live in a world of fast food. We have a world rich with hot dog vendors, pizza windows, order-at-the-counter restaurants, chain restaurants, and extra-value meals\u2014a world where, even when you want to eat healthy, it sometimes feels impossible to do so. These places pelt you with chips, suffocate your innards with ice cream, and shoot rapid-fire Goobers straight into your mouth.\n\nBut fast food doesn't need to be fat food. Convenient eating doesn't have to mean that fat conveniently claims squatter's rights on your gut. Easy eating doesn't mean greasy eating.\n\nThe secret of the Abs Diet's success is that you can follow it no matter where you are. If you just remember to eat the acronym\u2014the ABS DIET POWER 12\u2014you'll stay lean, fit, and healthy no matter where you are. You don't have to fall victim to kitchens that have more grease than a mechanic's fingernails, and you don't have to stop enjoying the taste of food to do so.\n\nYou can eat fast.\n\nYou can enjoy what you eat.\n\nAnd you can be free from hunger, deprivation, and culinary boredom.\n\nI want to arm you with the information you need to make smart decisions about eating on the fly or on a Friday family night out. Before I explain the Abs Diet plan and give you hundreds of choices for what to eat no matter what situation you're in, it's important to review your nutritional friends and enemies. You know, there are only a few things that you can do to fundamentally change the way your body's chemicals, hormones, and organs function, but the one major thing that you do that changes your body's internal systems is eat. Ultimately, your health is dictated by the nutrients that travel through your bloodstream once they pass through your Fritos hole. Here's a quick primer on the pitfalls of your nutritional world.\n\n## **Your Enemies**\n\n### **The Hunger-Booster: High-Fructose Corn Syrup (HFCS)**\n\nI believe there's one primary reason why many people fail on diets\u2014and why they eat poorly to begin with. They're so damn hungry.\n\nSound simple? Anatomically, it's a little more complicated. And the complication has come in the form of a little-known but shockingly ubiquitous food additive called High-Fructose Corn Syrup (HFCS).\n\nNever heard of HFCS before? Then you'll be even more shocked by this next fact: You, the average American, consume almost 63 pounds of it each year.\n\nSit and think about that for a moment. Sixty-two pounds of it each year\u2014or about 228 extra calories every day\u2014and you don't even know what it is.\n\nAnd you're not going to be any happier once you find out: See, up until the 1970s, most sweets were made with simple white sugar. Bad for you, but it was what it was. You sated your sweet tooth, your body absorbed the calories, and you got full pretty fast. But about 3 decades ago, food manufacturers discovered an easier way to make sodas, cereals, yogurts, and some 40,000 more manufactured foods taste sweeter. They developed HFCS, which is derived from corn and is many times cheaper\u2014and sweeter\u2014than simple sugar. Today, HFCS is added to a shocking number of foods, including foods you wouldn't equate with sweeteners: ketchup, pasta sauce, and even crackers. And it's screwing up America's metabolic system.\n\nMechanically, the system works like this: When you eat any type of carbohydrate (like bread or fruit), your body releases insulin to regulate your body weight\u2014pushing those carb calories into your muscles to be used as energy or storing them for later use. Then, like a shut-off mechanism on a gas pump, it suppresses your appetite. That's the signal that tells your body to stop filling; your tank is full.\n\nThe problem is that fructose doesn't stimulate insulin so your body doesn't register it the way it registers simple white sugar. That's why it's possible to see people drink 2-liter bottles of soft drinks in a single sitting. Thirty years ago, that would have been an impossible task, but today, HFCS makes the same number of calories go down and, incredibly, still leaves you hungry for more. Your muscles still need energy, so you crave more food and more sweets. And so you eat more foods containing HFCS, and the cycle continues. Less energy, more flab. HFCS could well stand for Here's Flab Coming to your Stomach.\n\nYou don't need to eliminate this artificial foodstuff from your diet entirely, but you need to ensure your meals don't revolve around it, as many Americans' do. So start checking the food labels: If HFCS is listed first or second on the ingredient list, look at the chart on the nutrition label to see how much sugar the food contains. If it's just a gram or two, it's fine. But if you see a food that has 8 or more grams of sugar, and HFCS is prominent on the list of ingredients, it's a sign that you should leave it on the shelf.\n\n**FOODS HIGH IN HFCS OR FRUCTOSE** | **REPLACE WITH**  \n---|---  \nRegular soft drinks | Unsweetened sparkling water or diet soda  \nCommercial candy (such as jelly beans) | Chocolate candy (check the label; some chocolate bars have HFCS)  \nPancake syrup | Real maple syrup  \nFrozen yogurt | Ice cream  \nFruit-flavored yogurt | Artificially sweetened or sugar-sweetened yogurt  \nHighly sweetened cereals | Sugar-free or low-sugar cereals  \nPasta sauce | Sugar-free pasta sauce\n\n### **The Artery Hardener: Trans Fat**\n\nIf HFCS is Bonnie, then trans fat is Clyde, because together, the duo is responsible for some seriously heinous crimes against your body.\n\nFirst, it's important to know that dietary fats are a little like college bands, minus the long hair and low pants. Some are actually good, and some are unbelievably bad. Trans fats falls into the latter category. Trans fat increases the amount of bad cholesterol in your body, for example, and has been linked to an increased risk of heart disease, diabetes, and a weakened immune system. Scientists have estimated that trans fat contributes to more than 30,000 premature deaths every year.\n\nSo what exactly is this stuff? Once again, it's a Frankenstein monster that's come lumbering out of the labs of the food industry.\n\nTrans fat is created by combining vegetable oil (a liquid) with hydrogen to create partially hydrogenated oil, or trans fatty acids. Once infused with hydrogen, liquid vegetable oil turns into a solid at room temperature, becoming what we recognize today as Crisco or margarine.\n\nTrans fats are beloved by the food and restaurant industries for several reasons. Number one, they're cheap. Number two, they can stick around seemingly forever without going bad. (How gross is that?) Number three, you can add them to myriad foods in a way you can't add regular oil\u2014a cookie with vegetable oil in it will ooze all over the supermarket shelf, but one with partially hydrogenated oil in it will stay crisp and tantalizing. So it's no surprise that food marketers, eager to deliver the sensuous flavor and mouth feel of fat to millions of unsuspecting consumers, now add partially hydrogenated oils to all sorts of things\u2014chips, frozen foods, fries, muffins, to name a few.\n\nBut think about what trans fats are: Fats that are supposed to be liquid but are turned into solids. Now think about what they do when they get inside you. Instead of melting like they would in their natural state, they try to revert to their waxy, solid nature. Once you understand that heart disease and stroke are caused in part by waxy buildups of fat solids in the circulatory system, it's easy to put two and two together. Turning oils into solids isn't doing us a favor\u2014not by a long run.\n\nThe good news is that the U.S. government is finally recognizing the dangers of trans fats. In 2003, food companies were required to list these trans fats for the first time. At least this gives you the power to see the enemy you need to fight (though food companies have several years to phase the new nutritional labels in). In the meantime, use these tactics to reduce your intake of trans fats:\n\n  Check the ingredient list for \"hydrogenated\" or \"partially hydrogenated.\" The higher these ingredients are on the label, the more trans fats they contain (with the exception of processed peanut butter, which contains trace amounts).\n\n  Decode the food label. For those products that don't list trans fats, add all the fat grams together that are listed on the label and then subtract that number from the total fat content. The number you're left with is the estimate for the amount of trans fat.\n\n  Snack on baked chips or chips fried in peanut oil instead of ones with vegetable shortening (check the ingredient list).\n\n  Pick high-protein breakfasts like eggs or Canadian bacon instead of waffles. If you have toast, use jelly instead of margarine.\n\n  At a restaurant, ask what kind of oil the chef uses. You want to hear olive oil\u2014not shortening.\n\n  When eating dinner out, avoid bread, which may be filled with trans fats. It's better to pick a baked potato, soup, or a salad.\n\n  Blot oil from your fries as quickly as possible. A napkin can absorb excess grease.\n\n**IF YOU WANT** | **PICK THIS TRANS FAT-FREE OPTION**  \n---|---  \nCandy bar | Dove dark chocolate bar  \nCereal | Kellogg's Frosted Mini-Wheats or Post Premium Raisin Bran  \nCookies | Archway fat-free cookies or Pamela's Products gourmet cookies  \nCrackers | Ryvita Multigrain crackers  \nFrench fries | McCain 5-Minute Shoestring French Fries  \nPotato chips | Ruffles Natural sea-salted, reduced-fat chips\n\n### **The Belly Buster: Saturated Fats**\n\nJust the name sounds threatening, doesn't it? Saturated fats, as in they're going to sink into your stomach and saturate your organs with soft little globs of putty. Bleeech!\n\nAnd the truth is just as gross as you imagine it to be. Your body likes to burn some kinds of fats\u2014polyunsaturated (from vegetables) and monounsaturated (from nuts and seeds) fats\u2014as energy. But your body would rather save saturated fats around your belly to use for future energy in case, I dunno, your plane crashes in the jungles of the Philippines or something. Assuming there's no Battan Death March in your immediate future, however, your body will continue to hold onto the fat it's stored, and the more you eat, the more you wear. Besides raising cholesterol rates, saturated fats have also been shown to increase your risk for heart disease and some types of cancer.\n\nSaturated fats are found primarily in meats and dairy products. \"But wait!\" you say. \"Aren't meats and dairy products part of the ABS DIET POWER 12?\" Yes, and that's why I emphasize lean meats like turkey, chicken, fish, and some cuts of beef and recommend that you look for low-fat dairy products like low-fat milk or low-fat yogurt whenever possible. The trick is to get the most nutrients\u2014muscle-building protein and fat-fighting calcium\u2014with the least amount of saturated fats.\n\n### **The Energy Sucker: Refined Carbohydrates**\n\nConsider the simple wheat stalk: There it stands, soaking up the sun and minding its own business, a single droplet in a vast sea of amber waves. Who'd have thought this humble grain would spark a controversy more complicated than the JFK assassination? And yet, the dietary landscape is wrought by forces arguing grain's place at the American dinner table. On this side, the USDA food guide pyramid, which calls for six to eleven servings of grains a day. And on that side, the Atkins addicts and other no-carb adherents who believe that simple stalk of wheat is evil incarnate.\n\nThe truth is . . . well, the truth is in neither corner, but somewhere more towards the middle.\n\nThe truth is that the human body can't survive without carbohydrates, because grains\u2014like fruits, vegetables, and other carbs\u2014provide crucial energy to feed the brain, the muscles, and the metabolism. Grains also provide crucial vitamins, minerals, and fiber, all of which the body needs to stay healthy. For example, a recent study at Brigham and Women's Hospital found that women who consumed 800 or more micrograms of folate a day had 29 percent less risk of high blood pressure than those who consumed less than 200 micrograms daily. Folate is just one of the many nutrients found in carbohydrates.\n\nBut the anti-carb movement does have its points to make, because most of the carbohydrates in the American diet aren't rich in all those great nutrients and fiber. They're \"refined carbohydrates,\" such as white sugar, white bread, bagels, waffles, et cetera. Most baked goods, in fact, are made from grains that have had all their great nutrients \"refined\" out of them.\n\nThis causes serious dietary problems for the carb lover. A slice of bread made with whole grains\u2014wheat, oats, or what have you\u2014is full of fiber. Fiber expands once it's in the belly, taking up space, slowing the digestive process, and keeping your energy and hunger levels even for several hours. Take out the fiber, though, and those carb calories go shooting through the digestive track faster than Bill Clinton at a sorority party. There's a rush of blood sugar as the carbs are quickly digested, a burst of energy, and then a letdown as insulin stores the blood sugar and your body cries out for more.\n\nThat's why the ABS DIET POWER 12 includes whole-grain breads and cereals, along with fruit, vegetables, and other carbohydrate sources. Carbs have the energy you need; you just have to choose wisely. Read the label; you want to see the words \"whole grain\" on your bread, cereal, and cracker boxes whenever possible.\n\n## **HOW METABOLISM WORKS\u2014AND HOW TO MAKE IT WORK FOR YOU**\n\nEven if you're lying in bed, lounging on the couch, or sitting on the toilet as you're reading this, your body is burning calories. It's burning calories to keep your heart beating, your lungs breathing, your brain dreaming about Cabo San Lucas. . . . Hey, wake up!\n\nThat calorie burn I'm referring to is your metabolism, and how high it's revving is what determines whether you're losing fat right now\u2014or gaining it. See, your body burns calories all the time, and it burns them in three different ways. One, you burn them when you eat, simply through the act of digestion (remember, it takes more energy to digest protein than it does carbs). Two, you burn calories by exercise and movement, whether you're running a marathon or just walking down the hall. And the last way you burn calories is when you're at rest; that's called your _basal metabolism_ , and it refers to the way your body uses fuel when you're not doing anything. Incredibly enough, this is when the majority of your calories are burned\u2014while you're doing nothing.\n\nThat explains, in part, why watching those calories tick away on the treadmill or the exercise bike is an exercise in frustration. If the majority of calories are burned during your non-exercising times, then it makes sense to boost your calorie burn during those times, and the way you do that is by adding muscle. In fact, for every pound of muscle you build, your body will need to burn off up to 50 extra calories a day, just sitting around doing nothing. Add 6 pounds of muscle, and you're burning up to 300 extra calories a day, just hanging out being you.\n\nLater on in this book, I'll walk you through the basics of the Abs Diet Workout, a muscle-building, fat-burning, 20-minute workout that will help you create the body you've always wanted.\n\n## **Your Friends**\n\n### ****The Muscle Builder: Protein****\n\nYou may think of protein as the staple ingredient for bodybuilders or the Atkins crowd, but protein has more super powers than the Hall of Justice. For one, protein helps kick-start your metabolism. It takes your body twice as much energy to break down protein as it does to break down carbohydrates, so when you eat a high-protein meal, you actually burn off additional calories at the dinner table. In one study, for example, people who ate a high-protein diet burned more than twice as many calories in the hours after their meals as people on a high-carbohydrate diet.\n\nProtein also flips your satisfaction switch. When you start your meal with protein\u2014say, downing a glass of fat-free milk before breakfast or ordering the shrimp cocktail appetizer at dinner\u2014your body registers its satiation level earlier on, and you wind up eating less. And that effect can carry on throughout the day. Some studies have shown that, if all calories are equal, people who eat a high-protein meal feel fuller and eat less at their next meal than those who don't. In another study, subjects who followed a high-protein diet lost an average of 20 pounds each, compared to just 11 pounds lost by participants who followed a low-protein diet. Amazingly, protein not only burned away fat, it burned away belly fat. The high-protein dieters lost twice as much abdominal fat as their low-protein dieting counterparts.\n\n### **The Cholesterol Cutters: Polyunsaturated and Monounsaturated Fats**\n\nTen years ago, if I had told you to eat more fat, I'd have been dragged from my office by the diet police and run out of town on a rail. Although it's generally a good idea to cut down on some kinds of fats\u2014like trans fats and saturated fats\u2014other kinds are actually good for you. Good? Heck, they're great.\n\nOur bodies need fat. We need it to deliver vitamins throughout our bodies. We need it to produce testosterone\u2014the hormone that leads to muscle growth. And we need it to keep satiated and full. In fact, one of the important things we've learned recently is that reducing your fat intake doesn't necessarily decrease your body-fat percentage over the long haul. In a recent study at Brigham and Women's Hospital in Boston, participants were put on either a low-fat or moderate-fat diet. After 6 months, the two groups lost about the same amount of weight. But when doctors checked in after 12 months, they discovered that the low-fat eaters had not only gained back what they lost, but they had added an average of 6 pounds! The dieters who were allowed to eat fats, however, lost an average of 9 pounds\u2014and kept it off.\n\nSo, as the song goes, Grease is the Word\u2014as long as it's the right kind of grease. The two kinds of fats that you'll incorporate into your eating plan are polyunsaturated and monounsaturated. Polyunsaturated fats include the famous omega-3's, which are found in fish like salmon and tuna and work to help clear your arteries. But more research shows that polyunsaturated fat also plays a role in helping speed your metabolism. Studies have shown that people who take omega-3's burned more calories throughout the day than those who don't. And a recent study of more than 35,000 women discovered that those who ate fish high in omega-3's had the lowest body mass indexes (BMIs)\u2014even lower than vegetarians. (And if you're a vegetarian, there's still no excuse: Flaxseed and flaxseed oil are also loaded with omega-3's, and you can find them in a health food store. Get the ground flaxseed so you can toss it on cereal or into smoothies.)\n\nThe other kind of good fat\u2014monounsaturated\u2014is found in nuts, olives, avocados, and olive and canola oils. Monounsaturated fats reduce cholesterol levels, as well, and they also help burn fat and keep you satiated. One study found that subjects who ate a meal with oil high in monounsaturated fats felt fuller than those who ate one cooked with the polyunsaturated kind. And that's a primary approach to how you need to eat: Balance your foods with the ingredients that keep your hunger in check. You want to keep your stomach satisfied\u2014not the folks at Big & Tall.\n\n### **The Appetite Suppressor: Fiber-Rich Carbohydrates**\n\nAs I stated earlier in this chapter, it's not the \"carbohydrate\" part you should be concerned about, it's the \"fiber\" part. Whole-grain breads and cereals, oatmeal, and berries\u2014these are the real weapons of mass destruction, and the mass they seek to destroy is the one collecting around your waist.\n\nThere are two types of fiber: soluble and insoluble. Let's see if I can define them both in one paragraph without making you nod off.\n\nSoluble fiber, like the kind you find in oatmeal, apples, and other fruits and grains, likes to hang out in your stomach. While it's hanging out, it does two things. First, it slows digestion, giving you longer-burning energy throughout the day. Second, it bonds with digestive acids, which happen to be made from cholesterol. When fiber splits, it takes the digestive acids with it, forcing your body to pull cholesterol from your bloodstream to make more. Presto, your cholesterol profile improves. Insoluble fiber, mean-while, does not like to hang around. It shoots through your plumbing like Dr no, picking up miscellaneous fats and whatever else happens to be loitering in your system and ushering them out the back. So both types of fiber help keep all your pipes and fittings in order, and if you eat them regularly, they should delay any need for professional plumbing services in the near future.\n\n### **The Weight-Loss Wonder: Calcium**\n\nIn the past few years, researchers have begun studying the effects of calcium-rich dairy foods on weight management. And quite frankly, it's hard not to be cowed by the evidence.\n\nFor example, researchers at Harvard Medical School showed that those who ate three servings of dairy a day\u2014to total the recommended 1,200 milligrams of calcium daily\u2014were 60 percent less likely to be overweight. But some of the most exciting research came from a study in which researchers put subjects on diets that were 500 calories a day less than what they were used to eating. The subjects lost weight\u2014about a pound of fat a week. But when researchers put another set of subjects on the same diet but added dairy to their meals, their fat loss doubled, to 2 pounds a week. Same calorie intake, double the fat loss. Calcium, it seems, is going to be one of the most exciting new areas of research about weight loss and metabolism, and that's why it's an important part of the ABS DIET POWER 12.\n\n### **ABS DIET SUCCESS STORY**\n\n## **SHE HAD THE GUTS TO LOSE HERS**\n\n**Name:** Linda Toomey\n\n**Age:** 35\n\n**Height:** 5\u20324\u2033\n\n**Weight, Week 1:** 145\n\n**Weight, Week 6:** 126\n\n**Body-Fat Percentage, Week 1:** 36\n\n**Body-Fat Percentage, Week 6:** 25\n\nWhen Linda Toomey had her fourth baby, she knew that she had to get the weight off. She was still carrying an extra 20 pounds she'd gained from her third child, and she wanted to act quickly, because she knew the longer she waited, the harder it would be. At 145 pounds and with four children under the age of 6, she knew that her own health\u2014and belly\u2014might take a backseat to everything else going on in her life. \"I'm the queen of excuses,\" she says.\n\nToomey also knew that she needed as much energy as possible\u2014especially considering she wasn't getting a full night's sleep anyway, caring for a newborn.\n\n\"At night, I expected to be tired,\" she says. \"But I was tired 2 hours after I woke up.\"\n\nHer goals: Get her body back, have more energy, and strengthen her back to be able to meet the demands of carrying larger-than-average children.\n\n\"I tried other diets, but being so crazy and busy, I didn't have a lot of time for exercise or food preparation. I needed something that was easy and fast to prepare,\" Toomey says.\n\nShe found it in the Abs Diet. \"It's not really a diet,\" she says. \"It's a lifelong eating plan. I think knowing that you can eat carbs and not resist cravings was one of the key factors. The eating plan was extremely easy to follow, and the whole family could enjoy the meals. I didn't have to prepare different foods for myself.\"\n\nToomey also included the 20-minute exercise plan and strengthened her abdominals and lower back to the point where she has no problem lifting her children.\n\nIn 6 weeks, Toomey dropped 19 pounds and went from 35-percent body fat to 25. And she also went from a size 14 dress to size 6.\n\n\"I'm hoping it motivates a lot of women,\" Toomey says. \"In the past after being pregnant, the waist was extremely hard for me. I may have lost inches from everywhere else in the past, but the waist was my real difficult area. It's amazing how it progressed in a short time.\"\nChapter 2\n\n# **NEVER GO HUNGRY AGAIN**\n\n# The ABS DIET POWER 12\n\n**T HE GOVERNMENT USES** acronyms: FBI, CIA, DOT. Media corporations use acronyms: ABC, CNN, BET. We love acronyms because they're easy to remember. (Do you even recall what CNN stands for after all these years?)\n\nThe Abs Diet comes with its own acronym\u2014the ABS DIET POWER 12. They are 12 foods you want to look for and fit into your eating plan whenever you can.\n\nI don't know about you, but if I had to consult a chart, formulate some kind of substitution plan, or calculate calories at every turn, I'd starve. Eating plans need to be simple, because life is complicated. And that's what the ABS DIET POWER 12 is\u2014simple. Figuratively, it stands for the principles of what this body-transforming diet is all about. Literally, it stands for:\n\n**A** lmonds and other nuts\n\n**B** eans and legumes\n\n**S** pinach and other green vegetables\n\n**D** airy (fat-free or low-fat milk, yogurt, cheese)\n\n**I** nstant oatmeal (unsweetened, unflavored)\n\n**E** ggs\n\n**T** urkey and other lean meats\n\n**P** eanut butter\n\n**O** live oil\n\n**W** hole-grain breads and cereals\n\n**E** xtra-protein (whey) powder\n\n**R** aspberries and other berries\n\nAs you can see, these 12 foods, or food groups, if you will, constitute a veritable cornucopia of dietary choices. You can use the acronym when you're shopping, when you're cooking at home, and when you're eating out.\n\nThe crux of the diet and _The Abs Diet Eat Right Every Time Guide_ is that if you can revolve your meals around these 12 Powerfoods, you'll know exactly what to order, what to shop for, and how to eat right every time. You'll have armed yourself with the \"friend\" category of nutrients that will help your body lose fat and gain lean muscle, and you'll have shunned the \"enemy\" category of junk foods that threaten to lay waste to your waist. Here's a quick overview of the ABS DIET POWER 12 and what they have to offer.\n\n_**How to read the key:**_ For at-a-glance scanning, I've included the following icons under the descriptions of each of the Abs Diet Powerfoods. Each icon demonstrates which important roles each food can help play in maintaining optimum health.\n\n  **Builds muscle:** Foods rich in muscle-building plant and animal proteins qualify for this seal of approval, as do foods rich in certain minerals linked to proper muscle maintenance, such as magnesium.\n\n  **Helps prevent weight gain:** Foods high in calcium and fiber (both of which protect against obesity) as well as foods that help build fat-busting muscle tissue earn this badge of respect.\n\n  **Strengthens bone:** Calcium and vitamin D are the most important bone builders, and they protect the body against osteoporosis. But beware: High levels of sodium can leach calcium out of bone tissue. Fortunately, all of the Powerfoods are naturally low in sodium.\n\n  **Lowers blood pressure:** Any food that's not high in sodium can help lower blood pressure\u2014and earn this designation\u2014if it has beneficial amounts of potassium, magnesium, or calcium.\n\n  **Fights cancer:** Research has shown that there is a lower risk of some types of cancer among people who maintain low-fat, high-fiber diets. You can also help foil cancer by eating foods that are high in calcium, beta-carotene, or vitamin C. In addition, all cruciferous (cabbage-type) and allium (onion-type) vegetables get the cancer protection symbol because research has shown they help prevent certain kinds of cancer.\n\n  **Improves immune function:** Vitamins A, E, B6, and C; folate; and the mineral zinc help to increase the body's immunity to certain types of disease. This icon indicates a Powerfood with high levels of one or more of these nutrients.\n\n  **Fights heart disease:** Artery-clogging cholesterol can lead to trouble if you eat foods that are predominant in saturated and trans fats, while foods that are high in monounsaturated or polyunsaturated fats will actually help protect your heart by keeping your cholesterol levels in check.\n\n## **#1: Almonds and Other Nuts**\n\n**Superpowers: builds muscle, fights cravings**\n\n**Secret weapons: protein, monounsaturated fats, vitamin E, fiber, magnesium, folate (peanuts), phosphorus**\n\n**Fights against: obesity, heart disease, muscle loss, wrinkles, cancer, high blood pressure**\n\n**Sidekicks: flaxseed, pumpkin seeds, sunflower seeds, avocados**\n\n**Imposters: salted or smoked nuts**\n\nThese days, you hear about good fats and bad fats the way you hear about good cops and bad cops. One's on your side, and one's gonna beat you silly. Oreos fall into the latter category, but nuts are clearly out to help you. They contain the monounsaturated fats that clear your arteries and help you feel full.\n\nAll nuts are high in protein and monounsaturated fat. But almonds are like Jack Nicholson in _One Flew over the Cuckoo's Nest_ : They're the king of the nuts. A handful of almonds provides half the amount of vitamin E you need in a day and 8 percent of the calcium. Almonds also contain 19 percent of your daily requirement of magnesium\u2014a key component for muscle building. In a Western Washington University study, people taking extra magnesium were able to lift 20 percent more weight and build more muscle than those who weren't. Eat as much as two handfuls of almonds a day. A Toronto University study found that men can eat this amount daily without gaining any extra weight. A Purdue University study showed that people who ate nuts high in monounsaturated fat felt full an hour and a half longer than those who ate fat-free food (rice cakes, in this instance). If you eat 2 ounces of almonds (about 24 of them), it should be enough to suppress your appetite\u2014especially if you wash them down with 8 ounces of water. The fluid helps expand the fiber in the nuts to help you feel fuller. Also, eat almonds with the nuts' nutrient-rich skins on them.\n\nHere are ways to seamlessly introduce almonds and other nuts into your diet.\n\n  Add chopped nuts to plain peanut butter.\n\n  Toss a handful of nuts on cereal, yogurt, or ice cream.\n\n  Put almond slivers in an omelet.\n\n  For a quick popcorn alternative, spray a handful of almonds with nonstick cooking spray and bake at 400 degrees for 5 to 10 minutes. Take them out of the oven and sprinkle them with a mixture of either brown sugar and cinnamon or cayenne pepper and thyme.\n\nOne caveat, before you get all nutty: Smoked and salted nuts don't make the cut here, because of their high sodium content. High sodium can mean high blood pressure.\n\nAlthough it's not technically a nut, I want you to consider adding ground flaxseed to your food. As I pointed out earlier, 1 tablespoon contains only 60 calories, but it packs in omega-3 fatty acids and has nearly 4 grams of fiber. It has a nutty flavor, so you can sprinkle it into a lot of different recipes, add some to your meat or beans, spoon it over cereal, or add a tablespoon to a smoothie.\n\n## **SUPPLEMENT SMARTS**\n\nYou can get nearly every nutrient in the ABS DIET POWER 12 from a bottle or a can. Health food stores are filled with multivitamins and dietary supplements that promise better health and nutrition. But I want you to be eating real, wholesome, delicious foods, not popping pills.\n\nOne main reason\u2014besides the fact that eating your meals in pill form makes you feel like you're on the set of _Logan's Run_ \u2014is that fruits, vegetables, whole grains and other Powerfoods offer not only the aforementioned vitamins, minerals, and fiber, but hundreds of \"micronutrients,\" as well. These micronutrients\u2014many of which are only now being studied and classified\u2014may have tremendous protective properties that scientists are still trying to discover. So the more real food you eat, the more real nutrition you get.\n\nThat said, there are two nutrients that aren't readily available through traditional food sources. If you want to supplement your healthy diet with some extra weight-loss power, consider:\n\n  Conjugated linoleic acid (CLA) is a fatty acid that's been shown to aid in weight loss. Researchers recently reported that subjects taking CLA for 1 year lost up to 8.7 percent of their body fat. The recommended dosage is 3 grams daily.\n\n  Pyruvate is an antioxidant that may help aid weight loss, as well. In two trials, people taking pyruvate in addition to a low-fat diet stepped up their weight loss. The recommended dosage is 25 grams daily.\n\n## **#2: Beans and Legumes**\n\n**Superpowers: builds muscle, helps burn fat, regulates digestion**\n\n**Secret weapons: fiber, protein, iron, folate**\n\n**Fights against: obesity, colon cancer, heart disease, high blood pressure**\n\n**Sidekicks: lentils, peas, bean dips, hummus, edamame**\n\n**Imposters: refried beans, which are high in saturated fats; baked beans, which are high in sugar**\n\nMost of us can trace our resistance to beans to some unfortunately timed intestinal upheaval (third-grade math class, a first date gone awry). But beans are, as the song says, good for your heart; the more you eat them, the more you'll be able to control your hunger. Black, lima, pinto, garbanzo\u2014you pick the bean (as long as it's not refried\u2014refried beans are loaded with fat). Beans are a low-calorie food packed with protein, fiber, and iron\u2014ingredients crucial for building muscle and losing weight. Gastrointestinal disadvantages notwithstanding, beans serve as one of the key members of the Abs Diet cabinet because of all their nutritional power. In fact, you can swap in a bean-heavy dish for a meat-heavy dish a couple of times per week; you'll be lopping a lot of saturated fat out of your diet and replacing it with higher amounts of fiber.\n\nThe best beans for your diet are:\n\n  Soybeans\n\n  Pinto beans\n\n  Chickpeas (garbanzo beans)\n\n  Navy beans\n\n  Black beans\n\n  White beans\n\n  Kidney beans\n\n  Lima beans\n\n## **#3: Spinach and Other Green Vegetables**\n\n**Superpowers: neutralizes free radicals, which are molecules that accelerate the aging process**\n\n**Secret weapons: vitamins including A, C, and K; folate; minerals including calcium and magnesium; fiber; beta-carotene**\n\n**Fights against: cancer, heart disease, stroke, obesity, osteoporosis**\n\n**Sidekicks: cruciferous vegetables like broccoli and brussels sprouts; green, yellow, red, and orange vegetables like asparagus, yellow beans, and peppers**\n\n**Imposters: none, as long as you don't fry them or smother them in fatty cheeses**\n\nYou know vegetables are packed with important nutrients, but they're also a critical part of your body-changing diet. I like spinach in particular because one serving supplies nearly a full day's worth of vitamin A and half of your vitamin C. It's also loaded with folate\u2014a vitamin that protects against heart disease, stroke, and colon cancer. To incorporate spinach into your diet, you can take the fresh stuff and use it as lettuce on a sandwich or try stir-frying it with a little fresh olive oil and garlic.\n\nAnother potent power vegetable is broccoli. It's high in fiber and more densely packed with vitamins and minerals than almost any other food. For instance, broccoli contains nearly 90 percent of the vitamin C of fresh orange juice and almost half as much calcium as milk. It is also a powerful defender against diseases like cancer because it increases the enzymes that help detoxify carcinogens. Tip: With broccoli, you can skip the stalks. The florets have three times as much beta-carotene as the stems, and they're also a great source of other antioxidants. Saut\u00e9 olive oil and garlic and douse them with hot sauce.\n\nIf you hate vegetables, you can learn to hide them but still reap the benefits. Try pureeing them and adding them to marinara sauce or chili. The more you chop and puree vegetables, the more invisible they become, and the easier it is for your body to absorb them.\n\n## **#4: Dairy (Fat-Free or Low-Fat Milk, Yogurt, Cheese)**\n\n**Superpowers: builds strong bones, fires up weight loss**\n\n**Secret weapons: calcium, vitamins A and B 12, riboflavin, phosphorus, potassium**\n\n**Fights against: osteoporosis, obesity, high blood pressure, cancer**\n\n**Sidekicks: cottage cheese, low-fat sour cream**\n\n**Imposters: whole milk, frozen yogurt**\n\nDairy is nutrition's version of a typecast actor. It gets so much attention for one thing it does well\u2014strengthening bones\u2014that it gets little or no attention for all the other stuff it does well. It's about time for dairy to accept a breakout role as a vehicle for weight loss. Just take a look at the mounting evidence: A University of Tennessee study found that dieters who consumed between 1,200 and 1,300 milligrams of calcium a day lost nearly twice as much weight as dieters getting less calcium. In a Purdue University study of 54 people, those who took in 1,000 milligrams of calcium a day (about 3 cups of fat-free milk) gained less weight over 2 years than those with low-calcium diets. Researchers think that calcium probably prevents weight gain by increasing the breakdown of body fat and hampering its formation. Low-fat yogurt, cheeses, and other dairy products can play an important role in your diet. But as your major source of calcium, I recommend milk for one primary reason: volume. Liquids can take up valuable room in your stomach and send the signal to your brain that you're full. Adding in a sprinkle of chocolate powder can also help curb sweet cravings while still providing nutritional power.\n\n## **#5: Instant Oatmeal (Unsweetened, Unflavored)**\n\n**Superpowers: boosts energy and sex drive, reduces cholesterol, maintains blood sugar levels**\n\n**Secret weapons: complex carbohydrates and fiber**\n\n**Fights against: heart disease, diabetes, colon cancer, obesity**\n\n**Sidekicks: high-fiber cereals like All-Bran and Fiber One**\n\n**Imposters: cereals with added sugar and high-fructose corn syrup**\n\nOatmeal is the Bo Derek of your pantry: It's a perfect 10. You can eat it at breakfast to propel you through sluggish mornings, a couple of hours before a workout to feel fully energized by the time you hit the weights, or at night to avoid a late-night binge. I recommend instant oatmeal for its convenience. But I want you to buy the unsweetened, unflavored variety and use other Powerfoods such as milk and berries to enhance the taste. Pre-flavored oatmeal often comes loaded with sugar calories.\n\nOatmeal contains soluble fiber, meaning that it attracts fluid and stays in your stomach longer than insoluble fiber (like vegetables). Soluble fiber is thought to reduce blood cholesterol by binding with digestive acids made from cholesterol and sending them out of your body. When this happens, your liver has to pull cholesterol from your blood to make more digestive acids, and your bad cholesterol levels drop.\n\nTrust me: You need more fiber, both soluble and insoluble. Doctors recommend we get between 25 and 35 grams of fiber per day, but most of us get half that. Fiber is like a bouncer for your body, kicking out troublemakers and showing them the door. Fiber protects you from heart disease, and it also protects you from colon cancer by sweeping carcinogens out of the intestines quickly.\n\nA Penn State study also showed that oatmeal sustains your blood sugar levels longer than many other foods, which keeps your insulin levels stable and ensures you won't be ravenous for the few hours that follow. That's good, because spikes in the production of insulin slow your metabolism and send a signal to the body that it's time to start storing fat. Since oatmeal breaks down slowly in the stomach, it causes less of a spike in insulin levels than foods like bagels. Include oatmeal in a smoothie or as your breakfast.(A U.S. Navy study showed that simply eating breakfast raised metabolism by 10 percent.)\n\nAnother cool fact about oatmeal: Preliminary studies indicate that oatmeal raises the levels of free testosterone in your body, enhancing your body's ability to build muscle and burn fat and boosting your sex drive.\n\n## **#6: Eggs**\n\n**Superpowers: builds muscle, burns fat**\n\n**Secret weapons: protein, vitamin B 12, vitamin A**\n\n**Fights against: obesity**\n\n**Sidekicks: none**\n\n**Imposters: none**\n\nFor a long time, eggs were considered pure evil, and doctors were more likely to recommend tossing eggs at passing cars than into omelet pans. That's because just two eggs contain enough cholesterol to put you over your daily recommended value. Though you can cut out some of the cholesterol by removing part of the yolk and using the whites, more and more research shows that eating an egg or two a day will not raise your cholesterol levels, as once previously believed. In fact, we've learned that most blood cholesterol is made by the body from dietary fat, not dietary cholesterol. And that's why you should take advantage of eggs and their powerful makeup of protein.\n\nThe protein found in eggs has the highest \"biological value\" of protein\u2014a measure of how well it supports your body's protein need\u2014of any food. In other words, the protein in eggs is more effective in building muscle than protein from other sources, even milk and beef. Eggs also contain vitamin B12, which is necessary for fat breakdown.\n\n## **#7: Turkey and Other Lean Meats**\n\n**Superpowers: builds muscle, improves the immune system**\n\n**Secret weapons: protein, iron, zinc, creatine (beef), omega-3 fatty acids (fish), vitamins B** **6** **(chicken and fish) and B** **12** **, phosphorus, potassium**\n\n**Fights against: obesity, various diseases**\n\n**Sidekicks: shellfish, Canadian bacon**\n\n**Imposters: sausage, bacon, cured meats, ham, fatty cuts of steak like T-bone and rib-eye**\n\nA classic muscle-building nutrient, protein is the base of any solid diet plan. You already know that it takes more energy for your body to digest the protein in meat than it does to digest carbohydrates or fat, so the more protein you eat, the more calories you burn. Many studies support the notion that high-protein diets promote weight loss. In one study, researchers in Denmark found that men who substituted protein for 20 percent of their carbs were able to increase their metabolism and increase the number of calories they burned every day by up to 5 percent.\n\nAmong meats, turkey is a rare bird. Turkey breast is one of the leanest meats you'll find, and it packs nearly one-third of your daily requirements of niacin and vitamin B6. Dark meat, if you prefer, has lots of zinc and iron. One caution, though: If you're roasting a whole turkey for a family feast, avoid self-basting birds, which have been injected with fat.\n\nBeef is another classic muscle-building protein. It's the top food source for creatine\u2014the substance your body uses when you lift weights. Beef does have a downside; it contains saturated fats, but some cuts have more than others. Look for rounds or loins (that's code for extra-lean); sirloins and New York strips are less fatty than prime ribs and T-bones. Wash down that steak with a glass of fat-free milk. Research shows that calcium (that magic bullet again!) may reduce the amount of saturated fat your body absorbs. Choose cuts on the left side of the chart below. They contain less fat but still pack high amounts of protein.\n\n**LEAN BEEF (55 calories and 2\u20133 grams of fat per 1-ounce serving)** | **MEDIUM-FAT BEEF (75 calories and 5 grams of fat per 1-ounce serving)**  \n---|---  \nFlank steak | Corned beef  \nGround beef (extra-lean or lean) | Ground beef (not marked as lean or extra-lean)  \nLondon broil |   \nRoast beef | Prime cut  \nTenderloin |\n\nTo cut down on saturated fats even more, concentrate on fish like tuna and salmon, because they contain a healthy dose of omega-3 fatty acids as well as protein. Those fatty acids lower levels of a hormone called leptin in your body. Several recent studies suggest that leptin directly influences your metabolism: The higher your leptin levels, the more readily your body stores calories as fat. Researchers at the University of Wisconsin found that mice with low leptin levels have faster metabolisms and are able to burn fat faster than animals with higher leptin levels. Mayo Clinic researchers studying the diets of two African tribes found that the tribe that ate fish frequently had leptin levels nearly five times lower than the tribe that primarily ate vegetables. A bonus benefit: Researchers in Stockholm studied the diets of more than 6,000 men and found that those who ate no fish had three times the risk of prostate cancer than those who ate it regularly. It's the omega-3's that inhibit prostate cancer growth.\n\n## **#8: Peanut Butter**\n\n**Superpowers: boosts testosterone, builds muscle, burns fat**\n\n**Secret weapons: protein, monounsaturated fat, vitamin E, niacin, magnesium**\n\n**Fights against: obesity, muscle loss, wrinkles, cardiovascular disease**\n\n**Sidekicks: cashew and almond butters**\n\n**Imposters: mass-produced sugary and trans fatty peanut butters**\n\nYes, PB has its disadvantages: It's high in calories, and it doesn't go over well when you order it in four-star restaurants. But it's packed with those heart-healthy monounsaturated fats that can increase your body's production of testosterone, which can help your muscles grow and your fat melt. In one 18-month experiment, people who integrated peanut butter into their diets maintained weight loss better than those on low-fat plans. A recent study from the University of Illinois showed that diners who had monounsaturated fats before a meal (in this case, it was olive oil) ate 25 percent fewer calories during that meal than those who didn't.\n\nPractically speaking, PB also works because it's a quick and versatile snack, and it tastes good. Since a diet that includes an indulgence like peanut butter doesn't leave you feeling deprived, it's easier to follow and won't make you fall prey to other cravings. Use it on an apple, on the go, or to add flavor to potentially bland smoothies. Two caveats: You can't gorge on it because of its fat content; limit yourself to about 3 tablespoons per day. And you should look for all-natural peanut butter, not the mass-produced brands that have added sugar and trans fat.\n\n## **#9: Olive Oil**\n\n**Superpowers: lowers cholesterol and boosts the immune system**\n\n**Secret weapons: monounsaturated fat, vitamin E**\n\n**Fights against: obesity, cancer, heart disease, high blood pressure**\n\n**Sidekicks: canola oil, peanut oil, sesame oil**\n\n**Imposters: vegetable and hydrogenated vegetable oils, trans fatty acids, margarine**\n\nYou read extensive information on the value of high-quality fats like olive oil in Chapter 1. But it's worth reiterating here: Olive oil and its brethren will help you eat less by controlling your food cravings; they'll also help you burn fat and keep your cholesterol in check. Do you need any more reason to pass the bottle?\n\n## **#10: Whole-Grain Breads and Cereals**\n\n**Superpower: prevents your body from storing fat**\n\n**Secret weapons: fiber, protein, thiamin, riboflavin, niacin, pyridoxine, vitamin E, magnesium, zinc, potassium, iron, calcium**\n\n**Fights against: obesity, cancer, high blood pressure, heart disease**\n\n**Sidekicks: brown rice, whole-wheat pastas, whole-wheat pretzels**\n\n**Imposters: processed bakery products like white bread, bagels, and doughnuts; breads labeled wheat instead of whole wheat**\n\nThere's only so long a person can survive on an all-protein diet or an all-salad diet or an all-anything diet. You will crave carbohydrates because your body needs carbohydrates. The key is to eat the ones that have been the least processed\u2014carbs that still have all their heart-healthy, belly-busting fiber intact.\n\nGrains like wheat, corn, oats, barley, and rye are seeds that come from grasses, and they're broken into three parts\u2014the germ, the bran, and the endosperm. Think of a kernel of corn. The biggest part of the kernel\u2014the part that blows up when you make popcorn\u2014is the endosperm. Nutritionally it's pretty much a big dud. It contains starch, a little protein, and some B vitamins. The germ is the smallest part of the grain; in the corn kernel, it's that little white seedlike thing. But while it's small, it packs the most nutritional power. It contains protein, oils, and the B vitamins thiamin, riboflavin, niacin, and pyridoxine. It also has vitamin E and the minerals magnesium, zinc, potassium, and iron. The bran is the third part of the grain and the part where all the fiber is stored. It's a coating around the endosperm that contains B vitamins, zinc, calcium, potassium, magnesium, and other minerals.\n\nSo what's the point of this little biology lesson? Well, get this: When food manufacturers process and refine grains, guess which two parts get tossed out? Yup, the bran, where all the fiber and minerals are, and the germ, where all the protein and vitamins are. And what they keep\u2014the nutritionally bankrupt endosperm (that is, starch)\u2014gets made into pasta, bagels, white bread, white rice, and just about every other wheat product and baked good you'll find. Crazy, right? But if you eat products made with all the parts of the grain\u2014whole-grain bread, whole-grain pasta, long-grain rice\u2014you get all the nutrition that food manufacturers are otherwise trying to cheat you out of.\n\nWhole-grain carbohydrates can play an important role in a healthy lifestyle. In an 11-year study of 16,000 middle-age people, researchers at the University of Minnesota found that consuming three daily servings of whole grains can reduce a person's mortality risk over the course of a decade by 23 percent. (Tell that to your buddy who's eating low-carb.) Whole-grain bread keeps insulin levels low, which keeps you from storing fat. In this diet, it's especially versatile because it'll supplement any kind of meal with little prep time. Toast for breakfast, sandwiches for lunch, bread with a dab of peanut butter for a snack. Don't believe the hype. Carbs\u2014the right kind of carbs\u2014are good for you.\n\nWarning: Food manufacturers are very sneaky. Sometimes, after refining away all the vitamins, fiber, and minerals from wheat, they'll add molasses to the bread, turning it brown, and put it on the grocery shelf with a label that says wheat bread. It's a trick! Truly nutritious breads and other products will say whole-wheat or whole-grain. Don't be fooled.\n\n## **CAN YOU STOMACH THIS?**\n\nWhen I tell people about the Abs Diet, I'm often asked about the issue of portion control. In most diets, portion control is a key element, but the Abs Diet comes with its own portion control built in. Because you'll be eating lots of fiber, protein, and healthy fats, the foods you eat on the Abs Diet will fill you up and give you long-running energy, so you won't feel the need to binge and won't have the urge to stuff your stomach like Santa's sack. But just for kicks, let's look at what happens to your body when you do overeat.\n\n  Your stomach, which is the size of your closed fist, doesn't crave more servings. When you're full, it is, too.\n\n  But if you're eating very quickly\u2014a common problem with junk food\u2014your stomach becomes distended and can swell to as much as three times its original size. It then pushes against your lungs and diaphragm.\n\n  If you really overeat, some of the food may not immediately reach your stomach, so it loiters in your esophagus, causing acid-reflux\u2013related belching and nausea.\n\n  Your liver works overtime to digest the surplus food by producing excess bile\u2014an emulsifier that helps fats and oils pass through your intestines. Your intestines also crank out extra digestive enzymes.\n\n  After all this stressful work, your GI system can't take it and rebels. Your body looks for a way to dump the excess cargo\u2014by any means necessary.\n\n## **#11: Extra-Protein (Whey) Powder**\n\n**Superpowers: builds muscle, burns fat**\n\n**Secret weapons: protein, cysteine, glutathione**\n\n**Fights against: obesity**\n\n**Sidekick: ricotta cheese**\n\n**Imposter: soy protein**\n\nProtein powder? What the heck is that? It's the only Abs Diet Powerfood that you may not be able to find at the supermarket, but it's the one that's worth the trip to a health food store. I'm talking about powdered whey protein, a type of animal protein that packs a muscle-building wallop. If you add whey powder to your meal\u2014in a smoothie, for instance\u2014you may very well have created the most powerful fat-burning meal possible. Whey protein is a high-quality protein that contains essential amino acids that build muscle and burn fat. But it's especially effective because it has the highest amount of protein for the fewest number of calories, making it fat's kryptonite. Smoothies with some whey powder can be most effective before a workout. A 2001 study at the University of Texas found that lifters who drank a shake containing amino acids and carbohydrates before working out increased their protein synthesis (their ability to build muscle) more than lifters who drank the same shake after exercising. Since exercise increases bloodflow to tissues, the theory goes that having whey protein in your system when you work out may lead to a greater uptake of amino acids\u2014the building blocks of muscle\u2014in your muscle.\n\nBut that's not all. Whey protein can help protect your body from prostate cancer. Whey is a good source of cysteine, which your body uses to build a prostate cancer\u2013fighting antioxidant called glutathione. Adding just a small amount may increase glutathione levels in your body by up to 60 percent.\n\nBy the way, the one great source of whey protein in your supermarket is ricotta cheese. Unlike other cheeses, which are made from milk curd, ricotta is made from whey\u2014a good reason to visit your local Italian eatery.\n\n## **#12: Raspberries and Other Berries**\n\n**Superpowers: protects your heart; enhances eyesight; improves balance, coordination, and short-term memory; prevents cravings**\n\n**Secret weapons: antioxidants, fiber, vitamin C, tannins (cranberries)**\n\n**Fights against: heart disease, cancer, obesity**\n\n**Sidekicks: most other fruits, especially apples and grapefruit**\n\n**Imposters: jellies, most of which eliminate fiber and add sugar**\n\nDepending on your taste, any berry will do (except Crunch Berries). I like raspberries as much for their power as for their taste. They carry powerful levels of antioxidants, all-purpose compounds that help your body fight heart disease and cancer. the berries' flavonoids may also help your eyesight, balance, coordination, and short-term memory. One cup of raspberries packs 6 grams of fiber and more than half of your daily requirement of vitamin C.\n\nBlueberries are also loaded with the soluble fiber that, like oatmeal, keeps you fuller longer. In fact, they're one of the most healthful foods you can eat. Blueberries beat out 39 other fruits and vegetables in the antioxidant power ratings. (One study also found that rats that ate blueberries were more coordinated and smarter than rats that didn't.)\n\nStrawberries contain another valuable form of fiber called pectin (as do grapefruits, peaches, apples, and oranges). In a study from the _Journal of the American College of Nutrition_ , subjects drank plain orange juice or juice spiked with pectin. The people who got the loaded juice felt fuller after drinking it than those who got the juice without the pectin. The difference lasted for an impressive 4 hours.\n\n## **MIDNIGHT MADNESS**\n\nIf you're dying for a midnight snack, chances are you're probably more bored, restless, or worried than you are hungry\u2014especially if you've eaten six meals already today. The key here is to do as little harm as possible and to get back to bed. Milk, oatmeal, and bananas contain trace amounts of melatonin, the send-me-to-sleep hormone. Combine them all in one bowl (one instant oatmeal packet made with milk and topped with half a sliced banana) or just have one alone. Backups: A small bowl (\u00bd cup or so) of whole-grain cereal with milk, 2 fig bars, or \u00bc cup ice cream.\n\n### **The Abs Diet Bull's Eye**\n\nVariety being the spice of life and all, you're often going to have the opportunity and the desire to mix in foods that don't fall squarely into the ABS DIET POWER 12. That's fine, as long as you keep your eye on the target and try to mix at least two Powerfoods into each meal and snack.\n\nTo help keep you focused on the ABS DIET POWER 12, I've created a little game I call the Abs Diet Bull's Eye. In the center of this chart are the foods you want to concentrate on. Around them, in concentric layers, are additional foods you'll no doubt run across every day.\n\nIndulge in them from time to time, but try to stay as focused on the center of the bull's eye as you can. You have the flexibility to eat what you want, but this bull's eye will help you focus on the center goal. The closer you stick to the inside, the healthier your diet will be and the better results you'll see. And if your dinnertime darts are always hitting outside the ring, then I think you may need to reevaluate your form.\n\n**Abs Diet Bull's Eye**\n\n**A** lmonds and other nuts\n\n**B** eans and legumes\n\n**S** pinach and other green vegetables\n\n**D** airy (fat-free or low-fat milk, yogurt, cheese)\n\n**I** nstant oatmeal (unsweetened, unflavored)\n\n**E** ggs\n\n**T** urkey and other lean meats\n\n**P** eanut butter\n\n**O** live oil\n\n**W** hole-grain breads and cereals\n\n**E** xtra-protein (whey) powder\n\n**R** aspberries and other berries\n\n**Eat Often**\n\n\u2022 Apples \u2022 Asparagus \u2022 Avocados \u2022 Bananas \u2022 Brown rice \u2022 Canadian bacon \u2022 Canola oil \u2022 Citrus fruits and juices \u2022 Corn \u2022 Cruciferous vegetables, such as broccoli and brussels sprouts \u2022 Edamame \u2022 Eggplant \u2022 Flaxseed \u2022 Fruit, dried \u2022 Fruit juice, no sugar added \u2022 Game, lean choices such as ostrich \u2022 Garlic \u2022 Lentils \u2022 Mushrooms \u2022 Melons \u2022 Nut butters, such as almond and cashew \u2022 Onions \u2022 Pasta,whole-wheat with tomato sauce and\/or vegetables \u2022 Peach \u2022 Peanut oil \u2022 Peas \u2022 Peppers \u2022 Pizza, plain orwith vegetables \u2022 Popcorn, fat-free \u2022 Pumpkin seeds \u2022 Prunes \u2022 Ricotta cheese, part-skim \u2022 Salsa \u2022 Sesame oil \u2022 Shellfish and bivalves, steamed or baked \u2022 Soup, broth-based \u2022 Sunflower seeds \u2022 Sweet potatoes \u2022 Tea \u2022 Tofu \u2022 Tomatoes \u2022 Tomato sauce \u2022 Vegetable juice \u2022 Veggie burgers \u2022 Wine, red\n\n**Eat Occasionally**\n\n\u2022 Applesauce \u2022 Baked beans \u2022 Baked potatoes,white \u2022 Beer, light \u2022 Butter, light \u2022 Chocolate \u2022 Coffee, with low-fat milk \u2022 French fries, baked \u2022 Graham crackers \u2022 Granola, low-fat \u2022 Granola or energy bars \u2022 Ham \u2022 Honey \u2022 Ice cream, low-fat \u2022 Jam or marmalade \u2022 Lamb \u2022 Lasagna \u2022 Lunchmeat \u2022 Macaroni \u2022 Margarine, trans fat-free \u2022 Nuts, salted or smoked \u2022 Pudding, with low-fat milk \u2022 Pork, tenderloin \u2022 Rice, white \u2022 Sauerkraut \u2022 Sorbet or sherbet \u2022 Veal \u2022 Wine,white \u2022 Yogurt, frozen\n\n**Eat Rarely**\n\n\u2022 Bacon \u2022 Bagels \u2022 Baked goods, such as cake and cookies \u2022 Beef, fatty cuts such as T-bone \u2022 Beer, regular \u2022 Breads, white-flour \u2022 Burgers, fast food \u2022 Candy \u2022 Cereals, sugared \u2022 Chicken, fried \u2022 Chicken wings \u2022 Chips \u2022 Dairy products, full-fat \u2022 Doughnuts \u2022 French fries \u2022 Game, fatty choices such as duck \u2022 Jellies \u2022 Margarine \u2022 Non-dairy creamer \u2022 Pasta, with creamy sauce \u2022 Pastries \u2022 Peanut butter, highly processed \u2022 Pizza, frozen or with extra cheese and meat toppings \u2022 Popcorn, buttery \u2022 Salad dressing, creamy \u2022 Sausage, pork \u2022 Seafood, fried \u2022 Soft drinks \u2022 Soup, creamy \u2022 Refried beans \u2022 Ribs \u2022 Vegetables, creamed or fried\n\n### **ABS DIET SUCCESS STORY**\n\n## **HE BEAT THE CURSE OF THE FAMILY GUT**\n\n**Name:** Jim Phillips\n\n**Age:** 37\n\n**Height:** 6\u20321\"\n\n**Weight, Week 1:** 190\n\n**Weight, Week 6:** 169\n\n**Body-Fat Percentage, Week 1:** 24\n\n**Body-Fat Percentage, Week 6:** 15\n\nEver since high school, Jim Phillips had been gaining weight\u2014not all at once, but just a few pounds every year. At one point, he found himself at 215 pounds on his 6-foot-1 frame and decided he needed to do something about it. So he took up running, started cutting down on his fat intake, and concentrated on a more vegetarian-like diet. He lost weight and got down to 190. And he stayed at 190\u2014for 5 years. But still, Phillips's gut lingered on.\n\n\"I think I kinda wore the weight well, but I knew I was definitely overweight\u2014even if other people didn't notice it,\" he says.\n\nOne of the places Phillips wore his weight was the infamous Phillips family gut (which he'd inherited through years of soda, pizza, and bagels). After reaching that plateau of 190, Phillips resigned himself to the fact that he was going to stay there, especially as he got older.\n\nThen, he found the Abs Diet.\n\n\"As soon as I saw the book, I really liked it because of the approach\u2014telling you the foods to eat, not what you can't,\" Phillips says. \"Psychologically, that was one of the biggest things. I like feeling good about what I'm eating.\"\n\nPhillips tried the Abs Diet\u2014eating six times a day (up from his habit of light meals early in the day and eating a big dinner) and following an exercise program that emphasizes strength-training and interval-training as the means to burning fat. He really feels that his change in exercise helped him. \"I was definitely into a lackadaisical routine the last 4 years. When I changed to a shorter routine and higher intensity, I noticed the difference.\"\n\nThe effect: He dropped 21 pounds in 6 weeks, 9 percentage points off his body-fat percentage, and 4 \u00bd inches off his waist. Plus, he has much better energy and fitness levels. Though he ran regularly when he was 190 pounds, he can really feel the difference now. \"I never felt as good running as I feel now. My knees aren't hurting. Hills are so much easier. I just did 13 miles yesterday,\" says Phillips, who has run marathons and is training for another. \"Before, I wouldn't have been able to walk the next day after doing that distance. Now it doesn't even bother me.\"\n\nWhile he still indulges in an occasional doughnut, he's made a change\u2014for good. \"After the fifth or sixth week on the diet, it hit me that I really didn't remember what my old habits were,\" Phillips says. \"The new ways just became natural and something I wanted to live with and how I wanted to eat. It wasn't going to be something I fell off of because I could eat the foods I like to eat.\"\nChapter 3\n\n# **THE SIX STEPS TO LIFELONG LEANNESS**\n\n# Guidelines for Easy Eating\n\n**L OTS OF GOOD THINGS** come in sixes. Besides the Brady kids and bottles of Bud, there's also that group of muscles that gets title billing on this book. And then there's the Abs Diet guidelines. Six of them. That's it.\n\nIt all goes back to the simple principle of, well, simplicity. No matter how nutritionally responsible popular diets are (some are and some aren't, by the way), most of them are more high-maintenance than an old Fiat. They require you to do complicated math, keep dietary journals, or stay \"in the zone\" (wherever the heck that is). Well, forget that. Instead, I've eliminated all the complex elements and boiled the Abs Diet down to six simple guidelines that don't require a degree in home economics or exotic ingredients you need to order from Madagascar.\n\nWhat you'll find are that these six general principles aren't really rules; they just serve as the map that shows you the way to your ultimate destination: a lean, strong, and healthy body.\n\n**Guideline 1: Eat six meals a day**\n\nWhen people are introduced to the Abs Diet, their first question is this: \"How can I lose weight if I'm eating twice as many meals?\" The explanation is simple\u2014and delicious.\n\nSee, nature doesn't much care if you have abs, which is why Cro Magnon man didn't have buffalo-skin Speedos. Our bodies evolved to do one thing: Stay alive long enough to pass our genes on to the next generation. When our ancestors roamed the jungles and plains, they encountered frequent times of deprivation and hardship. So just like bears in winter, our bodies became adept at storing calories, in the form of fat, to tide us over during times when food was scarce.\n\nThe more frequently the body is exposed to times of deprivation, the more it conspires to store fat and to jettison lean muscle tissue. That's because muscle requires more calories to maintain than fat does. The drought hits, the herds move on, and our ancestor starts living on his own muscle tissue. When food returns, his body reacts by bulking up with fat so he can make it through future lean times.\n\nNowadays, the herd never moves on; it's always there, 24 hours a day, at the drive-through. But people who go on calorie-restrictive diets create a modern version of starvation: They're not eating enough calories to maintain their lean muscle tissue. (In fact, most diets are not fat-loss diets, they're muscle-loss diets.) And as soon as people go off their diets, boom: The body instinctively begins storing fat. It doesn't know that it's being starved on purpose; it only knows it needs to survive the next diet.\n\nBut there's more bad news for the starvation-diet set. By burning away muscle, they're sacrificing the body's greatest weapon in the fight against flab. As I said above, muscle requires many more calories to maintain than fat does. In fact, for each pound of muscle you gain, your body burns up to 50 extra calories a day, just sitting still. Now let's say you go on a restrictive diet for a week, and you lose 4 pounds, and half of that is muscle. When you end your diet and return to eating as you normally do, you may be taking in the same number of calories you once did\u2014but your body actually requires 100 calories a day less, because of the muscle you lost. So where do those calories go? Right to your gut. It takes only 3,500 calories to build a pound of fat. So you've now programmed your body to gain a pound of fat every 35 days\u2014or more than 10 pounds in the coming year.\n\nSome diet plan, huh?\n\nAnd yet this is the crux of yo-yo dieting, and it's one of the worst things you can do for your figure and your health. But here's the thing: Most of us go on mini-starvation diets every single day. When we limit ourselves to three meals a day, we're asking our bodies to operate normally all the time, even though we're intermittently depriving it of fuel, then dumping big heaps of calories into it every 6 hours or so. That's why eating three squares will make you round. But if you break those meals into six of them, you're getting a steady stream of fuel throughout the day to balance what your body is doing.\n\nIn fact, the new catchphrase in obesity science is \"energy balance.\" Energy balance refers to taking in throughout the day a similar number of calories as you're burning off. As long as you're constantly fueling your body with small meals, you're constantly teaching it to shed fat. Ideally, you're trying to keep your hourly calorie surplus or deficit within 300 or 500 calories at all times, meaning that you're never going long periods without eating, and you're never overloading your system with too many calories at one sitting. As you burn calories throughout the day\u2014by exercising, by walking from the parking lot, by jumping up and down screaming, if that's your thing\u2014the calories you ingest should stay within similar levels of the calories you burn. That's balance\u2014never having too many calories and never being starved for more. In one study, researchers gave two sets of dieters the same daily calorie intake. But one group ate those calories in two big meals, while the other group ate them in six small meals spread throughout the day. At the end of the study, those who ate six meals a day lost an average of 11 pounds in two weeks; those who ate the same number of calories in just two meals a day also lost 11 pounds, but they lost 3 pounds more of muscle than men eating 6 meals who lost 3 more pounds of fat.\n\nSo the Abs Diet works in two ways: By maintaining your energy balance and by helping you build new muscle. The more lean muscle mass you have, the more energy it takes to fuel it\u2014meaning that calories go to your muscles to sustain them rather than being converted to fat. That's why I call muscle-building exercise the magic bullet in the chamber\u2014the secret weapon in your fight against fat.You'll read much more about the Abs Diet Workout in Chapter 11, and you'll learn how you can build new muscle that jump-starts your metabolism and turns your body into a fat-burning machine. In the meantime, I want you to keep this equation in mind:\n\nMORE FOOD = MORE MUSCLE = LESS FLAB\n\nThe alternative:\n\nLESS FOOD = LESS MUSCLE = MORE FLAB\n\nOh, and one more thing: If you eat six times a day, you never get hungry. If you never get hungry, you never get tired, cranky, or miserable. And if you never get hungry, tired, cranky, or miserable, you never feel the need to drown your hunger pangs or energy drain in a 32-ounce root beer float.\n\nTo keep the fat-burners firing, eat your six meals like this:\n\n  Alternate larger meals and smaller ones.\n\n  Eat your snacks roughly 2 hours before lunch, 2 hours before dinner, and roughly 2 hours after dinner.\n\n  Eat to fit your own schedule and lifestyle, but an ideal schedule would look something like this:\n\n**8 A.M.: Breakfast**\n\n**11 A.M.: Snack**\n\n**1 P.M.: Lunch**\n\n**4 P.M.: Snack**\n\n**6 P.M.: Dinner**\n\n**8 P.M.: Snack**\n\n**Guideline 2: Revolve your eating around the ABS DIET POWER 12**\n\nThe ABS DIET POWER 12 is your ticket to eating better anywhere, anytime. While every meal won't consist entirely of Powerfoods, the more Powerfoods you can squeeze into your day, the more quickly and effectively you'll change your body and your life.\n\nFor example, let's say you're trapped in some nutritional equivalent of Dante's Inferno, where the only food you have to eat comes out of a vending machine. You could choose the Coke and the tortilla chips (mmm . . . nacho cheese!) and those nifty Dunking Sticks for dessert. Or you could peer through the glass and look for Powerfoods. They're in there . . . the fat-free microwave popcorn, the unsalted peanuts, the low-fat chocolate milk.\n\nLet's see . . . soda, chips, and dunking sticks, or chocolate milk, popcorn, and peanuts. Both sound like junk-food meals, right? So which do you choose? Here's where the magic of the Powerfoods comes into play:\n\n**VENDING MACHINE MEAL #1:**\n\n_Can of Coke_ 155 calories, 0 g protein, 40 g carbohydrates, 0 g fat, 0 g fiber, 15 mg sodium\n\n_1.5-ounce bag of nacho cheese tortilla chips_ 212 calories, 3 g protein, 27 g carbohydrates, 11 g fat (2 g saturated), 2 g fiber, 301 mg sodium\n\n_2 Dunking Sticks_ 360 calories, 4 g protein, 44 g carbohydrates, 18 g fat (6 g saturated), 1 g fiber, 260 mg sodium\n\n**VENDING MACHINE MEAL #2:**\n\n_8 ounces low-fat chocolate milk_ 180 calories, 8 g protein, 28 g carbohydrates, 4.5 g fat (3 g saturated), 1 g fiber, 150 mg sodium\n\n_94% Fat-free microwave popcorn (3 cups)_ 120 calories, 4 g protein, 26 g carbohydrates, 2 g fat (0 g saturated), 4 g fiber, 380 mg sodium\n\n_1-ounce bag of unsalted peanuts_ 167 calories, 8 g protein, 5 g carbohydrates, 15 g fat (2 g saturated), 2.5 g fiber, 2 mg sodium\n\nWhat a difference choosing the Powerfoods can make: Meal #1 has about 50 percent more calories (727 versus 467), and pales in comparison nutritionally to Meal #2. By making your choices based on the Powerfoods, you get three times as much protein, more than twice the fiber, and significantly less fat and sodium.\n\nAnd that's something you can do no matter where you are, because every supermarket, every restaurant, every vending machine, and every movie-theater concession stand offers something that contains one or more Powerfoods. And that, my friends, will keep you losing weight\u2014without driving you nuts.\n\nWhile you're negotiating the nutrition wasteland, I'd like you to keep these three thoughts in mind:\n\n  Incorporate two to three Powerfoods into each of your three major meals and at least one or two of them into each of your three snacks.\n\n  Diversify your food at every meal to make sure you have a combination of protein, carbohydrates, and healthy fat.\n\n  Make sure you sneak a little bit of protein into each snack.\n\n**Guideline 3: Drink smoothies regularly**\n\nI'll never forget something that one of the first followers of the Abs Diet said. He lost some 20 pounds in the first 6 weeks, and he attributed most of that loss to one part of the diet: the smoothies. It was like getting to go to Dairy Queen every day, he said.\n\nSee, smoothies are more than just faux milk shakes packed with healthy ingredients. They're one of the keys to the Abs Diet because they pack your day with more energy than a busload of cheerleaders. In a recent study, researchers found that meal-replacement shakes work brilliantly as a weight-loss method. Subjects who replaced two daily meals with shakes or meal-replacement bars lost about 9 percent of their original body weight.\n\nMore important, smoothies feature everything you could possibly want out of a food, like:\n\n  Ease: They take less than 3 minutes to make\n\n  Taste: Like dessert, with one exception. They're guilt-free.\n\n  Satisfaction: Because they're so thick and packed with Powerfoods, they take up valuable room in your stomach to keep you full for hours.\n\n  Effectiveness: With the secret fat-fighter calcium, and satiating fiber and protein, they're some of the most potent meals or snacks you can make.\n\nWhile Chapter 8 gives you recipes for 27 different kinds of smoothies, I'd also encourage you to mix and match ingredients that fit your tastes. That's the great thing about smoothies. All you have to know is how to open a lid and press a button to make one. So dump in whatever Powerfoods you like, whether it's berries, peanut butter, or oatmeal, and experiment with your own flavors (I advise against the salmon). Then follow these guidelines.\n\n  Drink one or two 8-ounce smoothies a day, as a meal substitute or as a snack. In one study, researchers found that regularly drinking meal replacements increased a man's chance of losing weight and keeping it off for longer than a year.\n\n  Add ice cubes and blend your smoothie for at least 10 minutes. Thicker shakes incorporate a little more air and water but have an increased satiating effect. In one study, researchers found that people stayed fuller longer when they drank thick drinks than when they drank thin ones. Another study found that men who drank yogurt shakes that had been blended until they doubled in volume ate 96 fewer calories a day than men who drank shakes of normal thickness.\n\n  Don't skimp on the yogurt. A University of Tennessee study found that men who added three servings of yogurt a day to their diets lost 61 percent more body fat and 81 percent more stomach fat over 12 weeks than men who didn't eat yogurt. Calcium, baby!\n\n**Guideline 4: Stop counting**\n\nOn the Abs Diet, you'll count the number of pounds you've lost, you'll count the notches on your belt you've gained, you may even count your lucky stars. But one thing you will not count is calories. Counting calories is for suckers.\n\nFor one, most people don't have the time or discipline to record every pretzel stick they eat or weigh the three slices of turkey breast they carved. And two, all calories are not made equally when it comes to metabolism. Instead, the important thing is to eat the right foods. If you do, your body will essentially regulate your caloric intake all by itself. By including a good balance of fiber and protein, you'll be full throughout the day.\n\nWhat you want to do is think about\u2014not record\u2014the amount of food you eat by sticking to one helping of each food group you eat and keeping the total contents of each meal contained to the diameter of your plate (and please, no high-rise mashed potatoes).\n\n**Guideline 5: Watch what you drink**\n\nI think a diet with lots of restrictions is like a roller-coaster without hills or a zoo without animals. What's the point of having a diet plan if it's so regimented and boring that nobody's going to be able to follow it? That's why the Abs Diet is about what you can eat, not what you can't. But the one place you really have to be careful is in what you drink.\n\nAlcohol and soda, along with a lot of juice blends and prepackaged iced teas, contain empty, non-nutritious calories that will work against you in pursuit of a better body. While alcohol has some health benefits in moderation, it also encourages you to eat more, slows down fat-burning, and increases fat-storing. Drink water (eight 8-ounce glasses is ideal), diet soda, and low-fat milk. I encourage you to keep the booze in the cooler for the first 6 weeks of the plan, but if you drink, limit yourself to no more than two or three drinks a week.\n\n**Guideline 6: Go ahead, cheat**\n\nAmericans are rebels at heart\u2014inside each of us lies an inner James Dean or Mae West, an Ali or a Madonna, someone who wants to break the rules and make the world march to their own tune. Well, rebellion is built right into the Abs Diet.\n\nOn the Abs Diet, I want you to cheat\u2014with one whatever-you-want meal a week. That way, you can satisfy your cravings without derailing your commitment to the ABS DIET POWER 12. Of course, in the Abs Diet, there are enough great variations that you'll probably lose the urge for wings, Oreos, or Oreo-flavored wings pretty quickly. In fact, many people who've used the Abs Diet reported that they didn't even feel the urge to cheat, but they loved the fact that they had it as an option.\n\nNow, if you're going to cheat on me, I want you to do it conscientiously. When considering an adulterous encounter with an unhealthy food bear in mind the following:\n\n  Plan it. If you schedule your cheat meal, you'll be less likely to eat on impulse at other times during the week. Make it every Friday, when you meet the gang from work for beer and potato skins. Or every Sunday, when you treat yourself to cake and ice cream while watching Mike Wallace mysteriously not age.\n\n  Really, do it. While you don't have to cheat, it may help you in the long run. Researchers at the National Institutes of Health found that men who ate twice as many calories in a day as normal increased their metabolism by 9 percent in the 24-hour period that followed.\n\n  Stick to one a week. The cheat meal is like walking on a cliffside trail. It's entirely safe\u2014unless you step over the line. You need to view the cheat meal as satisfaction instead of temptation and remain focused on your weight-loss goals. You'll be better off for it.\n\n### **ABS DIET SUCCESS STORY**\n\n## **THE ABS DIET HEALS A BAD BACK**\n\n**Name:** Kyle Snay\n\n**Age:** 36\n\n**Height:** 6\u20325\u2033\n\n**Weight, Week 1:** 227\n\n**Weight, Week 6:** 205\n\n**Weight, Week 9:** 199\n\nKyle Snay knew he needed to get back into shape\u2014fast. Motivation to get in shape came in droves.\n\nReason 1: His dad. Several years earlier, Snay had made plans to visit his father\u2014whom he didn't know well because his parents had been divorced. But 6 months before Snay was to make his pilgrimage, his dad dropped dead of a heart attack. That's when it hit Snay that his health was something he needed better control of.\n\nReason 2: His kids. \"I was a walking Twinkie for years,\" Snay says. \"When my wife was pregnant with our second child, I said that I really needed to do something to get back in shape at least for my family. I was trying to keep up with our 2 year old, and it was exhausting.\"\n\nReason 3: His back. Snay, who's had a herniated disc for 10 years, says, \"My big flare-up came when I was bending down to pick up a Doritos chip.\" After years of twinges, spasms, and pain that laid him up for a couple days at a time, he was told by doctors his choices were surgery or physical therapy. Snay chose neither. He chose the Abs Diet.\n\nAfter deciding he was going to lose weight, Snay went searching for something hassle-free. \"I didn't want to count calories, or carbs, or fat. I read about the Abs Diet in _Men's Health_ and noticed results immediately.\"\n\nBefore, Snay never paid attention to what he ate: His staple foods included ice cream and \"Oreo soup\"\u2014a whole bunch of Oreos in a full bowl of milk. He immediately changed his diet and saw the pounds peel away. Plus, he got a bonus out of it: a better back. \"Since I started, I haven't had a twinge, a spasm, or anything\u2014nothing. My back is completely fine,\" he says. \"It's just unreal. That was a huge plus that I wasn't even expecting.\"\n\nSnay says he really stuck to the principles but appreciates the diet's flexibility. Eating six times a day, he says, makes you feel like you're never really \"on a diet.\"\n\nNow, almost 30 pounds lighter, Snay retrained his metabolism\u2014so much so that he lost his last 5 pounds even when he had to take a week off of exercise and sneak in a few extra cheat meals because he was away at a business conference. He also invents his own recipes following the ABS DIET POWER 12. \"I know people who are still trying to do Atkins, and they just pour meat down their stomach. In a couple of weeks, they're still walking around like a bowl of pudding because they're not doing any exercising, and they're taking too many shortcuts,\" he says.\n\n\"I can't ever see myself going back to who I was before,\" Snay says. \"I just can't wait until it gets warm again. I'm going to go out and wash my car\u2014with my shirt off for the first time in years. I'll even stay out there longer and wash my neighbor's car, too.\"\nChapter 4\n\n# **SHOP 'TIL YOU DROP (POUNDS)**\n\n# The Complete Abs Diet Grocery List\n\n**A S FAR AS I'M CONCERNED,** every time someone walks into a supermarket, they should play \"Welcome to the Jungle\" over the loudspeaker. Supermarkets are one of the most confusing and mind-numbing places on earth, right up there with the Balkans and your local DMV. Supermarkets have one goal: to sell you more calories than you actually came in to buy. The easiest foods to find are usually the worst for you (a special on doughnuts!), and in the checkout lane, eight rows of candy taunt you like the hecklers at a Knicks game. There's a reason why they're called supermarkets, you know; they're super marketers.\n\nSo next time you go food shopping, you should notice things, like . . . the cracker aisle never moves. Sure, the junk\u2014and it usually is junk\u2014on each aisle's end cap may shift, but the basic layout never changes. You can use this to your advantage. Make a list before you go. Be specific; instead of writing \"snacks\" and winding up with the entire Dolly Madison collection in your cart, write \"yogurt\" or \"sliced almonds.\" This way, you'll be able to make tactical strikes; quickly buy exactly what you need instead of the crap they're trying to foist off on you. Here's your guide to sneaking past the schlock and marching down the aisle in style.\n\n### **Produce: Work the Greens**\n\nMost produce is just as nutritious frozen as it is fresh, so be judicious. If you use it up slowly, pack your freezer. If you burn through greens like Vijay Singh, stick with fresh (taste and texture will be better).\n\n**Fresh strawberries and blueberries:** Don't think you'll finish them in 3 to 5 days? Buy frozen instead.\n\n**Bananas:** Avoid too-green ones. They'll add a strange tang to a smoothie.\n\n**Green onions:** Green onions are much easier to use than whole onions for small meals. Just chop the tops and toss the bulb\u2014no half-cut-up, plastic-wrapped onions stinking up your fridge.\n\n**Bagged baby spinach leaves:** Peer through the bottom of the bag. If you see any mush, choose another one.\n\n**Broccoli:** Look for tights buds or florets; they indicate a fresher find.\n\n**Cucumbers:** Choose English cucumbers. They're longer and thinner than ordinary ones and usually come wrapped individually in plastic. Buy them for the convenience factor; they have hardly any seeds.\n\n**Peppers:** Avoid peppers with wrinkled skin; it means they're starting to age.\n\n**Tomatoes:** When possible, buy vine-ripened over hot-house. You don't have to be a gourmand to notice the improvement in taste and texture.\n\n**Lemons:** Pick fruits that feel heavy; it means they have more juice.\n\n**Mixed-green salad blend:** The more colors, the more antioxidants. Look for one with red radicchio, pale green endive, and dark-green spinach.\n\n**Avocados:** The blacker, the better. Too green and they won't be ripe enough to eat.\n\n**Fresh cilantro:** Tear off a leaf and taste it to make sure you're buying cilantro and not flat-leaf parsley. Unless you don't like cilantro. In that case, buy flat-leaf parsley and use it instead.\n\n**Nuts:** Buy unroasted and unsalted loose nuts by the pound so you're not getting more sodium than you bargain for.\n\n### **Meat: Your Muscle Maker**\n\nWhile turkey is a top-shelf Powerfood, that doesn't mean other meats are off-limits. The key is getting the leanest protein for the least amount of saturated fat. Turkey does the job exceptionally well, but only if you buy breast meat. Mixed ground turkey, usually made from dark meat, can contain as much saturated fat as beef.\n\n**Fresh salmon:** Ask for wild-caught salmon; it has fewer chemicals.\n\n**Fresh turkey or chicken cutlets:** Check the label for sodium; some raw meats are plumped with a sodium solution\u2014added salt you don't need. Look for one that contains less than 10 percent broth solution.\n\n**Ground turkey:** If it's not made from ground turkey breast, it could have as much fat as beef.\n\n**Precooked chicken strips:** Try Perdue Short Cuts. No painted-on grill stripes or strange chicken-from-concentrate texture.\n\n**Deli slices:** Healthy Choice turkey or ham have next-to-nothing fat. For a little less sodium, chose plain or honey-roasted over smoked flavors.\n\n**Smoked salmon:** This is a high-sodium food, courtesy of the smoking. Pick the one with the lowest levels.\n\n**Canadian bacon:** Buy the thin-sliced varieties; a smaller piece will help you keep the sodium tab down.\n\n**Flank steak:** Pick the one with the bluest tinge. That means it's aged enough for extra flavor, but it's not too old.\n\n**Lean ground beef:** You want 95 percent lean; it's the one with the least amount of saturated fat.\n\n**Pork loin cuts:** Get the thin ones; they cook in half the time.\n\n### **Dairy: The Great White Help**\n\nDairy products play a key role in the Abs Diet\u2014as snacks, drinks, and as important ingredients in smoothies. They're your greatest source of calcium\u2014the mineral more and more studies point to as a key weight-loss ingredient\u2014and pack a potent protein punch as well. Think of the dairy section as fat-loss central. If you play the percentages.\n\n**Low-fat milk:** Horizon Organic. Cow antibiotics are for sick cows. Go organic.\n\n**Chocolate milk:** Choose Horizon Organic Reduced-Fat in stow-and-go boxes. They don't require refrigeration, so you can take a healthy source of calcium and protein with you anywhere.\n\n**Eggs:** Eggland's Best omega-3 fortified eggs provide an extra shot of heart-healthy omega-3 fatty acids.\n\n**Reduced-fat yogurt:** Stonyfield Farm 100 Percent organic has just-right creamy texture.\n\n**Shredded cheeses:** Sargento Reduced Fat shredded cheese blends offers the best taste for the least fat. Plus, they melt evenly, unlike most other reduced-fat cheese.\n\n**Part-skim ricotta cheese:** Get the big tub. It doesn't expire quickly, and you'll finish it long before it threatens to go bad. Ricotta is a great natural source of whey protein, one of nature's premium muscle-builders.\n\n**String cheese:** Sorrento Stringsters has 8 grams of muscle-building protein per stick, plus more calcium than other brands.\n\n**Cottage cheese:** Cottage cheese is a stealth-sodium food; choose Friendship All-Natural Low-Fat No Salt Added and save the salt for something else.\n\n**Pudding cups:** Swiss Miss. Ordinarily, you should watch out for added sugar. But no-sugar-added pudding is just nasty. This one only has 25 grams of sugary carbs, which won't kill you or blow the diet. As long as you just eat one. Occasionally.\n\n**Back-up smoothie:** Stonyfield Farm has no high-fructose corn syrup, which can't be said for many other bottled smoothies.\n\n## **THE BEST PREPARED FOOD YOU CAN BUY. PERIOD.**\n\nRotisserie chicken is an amazing thing. Not only does it look impressive, but it's an incredibly smart buy, both for the diet- and budget-conscious. First, always purchase a plain one; you're going to pick the skin off anyway, since that's where most of the fat hides. A breast and a leg will make a fine meal (check out the nutrition numbers below). Use the meat that's left to make a chicken salad or stuff it into a quesadilla. And then, toss the entire carcass into a stockpot full of water with chopped vegetables and spices of your choice, simmer for a couple of hours, and you'll wind up with the best chicken soup you've ever eaten. (Just fish out the bones before digging in.)\n\n**3 ounces breast meat (without skin)**\n\n**120 calories, 1.5 g fat (0.5 g saturated), 65 mg sodium**\n\n**3 ounces dark meat (without skin)**\n\n**130 calories, 4 g fat (1 g saturated), 80 mg sodium**\n\n### **Canned Foods: What's in Storage**\n\nCanned foods last will outlast your grandchildren, which is why they're used for currency in so many postapocalypse movies. Just watch out for sodium, the main preservative.\n\n**Tuna:** Star Kist Premium Chunk White Albacore Tuna in Water. Water cuts the fat. But the no-draining-needed bag seals the deal.\n\n**Salmon:** Tulip brand has more fat than others, but it also tastes like salmon and not canned tuna.\n\n**Garbanzos, black beans, kidney beans:** You can't go wrong. But before you cook with them, always rinse off the salty solution they soak in.\n\n**Chicken broth:** Swanson Fat-Free Reduced Sodium tastes like the chicken, not like the can it came in.\n\n**Marinara:** Muir Glen Italian Herb is preseasoned, so you don't have to spice it up. And it's not loaded with high-fructose corn syrup like other varieties.\n\n**Canned tomatoes:** Del Monte Diced Tomatoes, No Salt Added is low sodium and, unlike other brands, contains no high-fructose corn syrup.\n\n**Peaches:** Buy Dole sliced peaches, simply because they come in a resealable plastic jar. No fooling with a can opener or Tupperware.\n\n**Pumpkin:** Buy the in-store brand. Since you're just going to dump it in a smoothie anyway, might as well save a few cents.\n\n### **Sauces: Topping the List**\n\nNormally, sauces are like one-night stands. You enjoy them in the moment, but you feel pretty guilty afterwards. But I've uncovered some choice selections that might just make you interested in a lifelong commitment.\n\n**Olive oil:** Choose extra virgin, which means the goods haven't been damaged by mixing with other oils.\n\n**Balsamic or red wine vinegar:** There's no need to buy super-pricey vinegars shipped over from some forgotten enclave of the Mediterranean. Alessi is a quality, reasonably priced brand.\n\n**Soy sauce:** At 560 milligrams of sodium per tablespoon, La Choy Lite keeps you from getting buried in a salt mine.\n\n**Hot sauce:** Chohula is plenty potent, but it's not laced with extra sodium like many brands.\n\n**Peanut butter:** Crazy Richard's Natural tastes exactly like roast peanuts. Not surprising, since peanuts\u2014and a touch of salt\u2014are its only ingredients.\n\n**Salsa:** Walnut Acres Black Bean and Corn. Black beans provide a small protein boost, and corn contributes the antioxidant zeaxanthin.\n\n**Hummus:** Asmar's Roasted Pepper. The peppers give you a little extra flavor to go with the 3 grams of protein.\n\n**Mustard:** To me, deli-style mustards have more flavor than the sharp yellow kinds. Among them, Silver Springs Deli Style Horseradish has a nice extra kick, but barely any sodium (70 milligrams).\n\n## **THE LOW-FAT BUZZWORDS**\n\nRemember, food labeling isn't about education, it's about marketing. Labels can be confusing, misleading, and damaging if you don't know what you're getting into, especially when you're trying to figure out fat content. Here are some common phrases you'll see on food labels, and a look at what, exactly, they mean.\n\n**Regular:** Greater than 3 grams of fat per serving\n\n**Reduced fat:** 25 percent less fat than regular\n\n**Light or lite:** One-third fewer calories or 50 percent less fat than regular\n\n**Low fat:** No more than 3 grams of fat per serving\n\n**Nonfat or fat-free:** Less than 0.5 gram of fat per serving\n\n### **The Freezer: Cold Comfort**\n\nShop here last and you'll likely make it home with your ice cream intact. \"Ice cream?\" you say. Right. This plan is designed for human beings rather than robots.\n\n**Fish:** Gorton's Cajun Blackened Grilled Fillets are already spiced but not too salty.\n\n**Shrimp:** Since shrimp can be expensive, just pick the least expensive one. There's no real difference.\n\n**Waffles:** Van's Gourmet Flax Waffles have slightly sweet whole-wheat flavor along with 1.6 grams of heart-healthy omega-3 fatty acids.\n\n**Berries and fruit:** Choose Cascadian Farms organic. If you're buying frozen, it pays to go organic. Because they're delicate, berries and fruit often top the lists of high-pesticide produce.\n\n**Edamame:** Make things easy for yourself and buy the already-shelled kind.\n\n**Ice cream:** Edy's Grand Light is the hands-down best light ice cream on the market. It's sweet and creamy and has two-thirds less saturated fat than regular.\n\n**Condensed juice:** Go with Minute Maid. Off brands may have extra sweeteners.\n\n## **LOOK! DOWN THERE! IT'S A BARGAIN!**\n\nMore expensive items are typically found at eye level\u2014right where you'll see them and sweep them into the cart without thinking. Look down, though, and you'll likely find a lower-priced version of the same thing.\n\n**Frozen Dinners**\n\nFrozen dinners can fit into nearly any diet for one reason: They're perfectly portion-controlled. (Well, except for the aptly named Hungry Man XXL, a size designation you're trying to avoid anyway.) In a study at the University of Illinois, people who regularly ate dinners from the freezer section lost 31 percent more weight than their free-eating counterparts. Plus, many of these meals include a respectable amount of fiber. And the 10 frozen dinners in this section in particular balance reasonable amounts of saturated fat with not-too-high sodium (a hidden danger of the freezer aisle).\n\n**Ethnic Gourmet:**\n\n_Chicken Pad Thai_\n\n430 calories, 20 g protein, 71 g carbohydrate, 8 g fat (1.5 g saturated), 3 g fiber, 880 mg sodium\n\n**Healthy Choice:**\n\n_Grilled Chicken Marinara_\n\n270 calories, 22 g protein, 35 g carbohydrate, 4.5 g fat (1.5 g saturated), 5 g fiber, 580 mg sodium\n\n_Grilled Turkey Breast_\n\n250 calories, 18 g protein, 31 g carbohydrate, 5 g fat (2 g saturated), 5 g fiber, 600 mg sodium\n\n_Grilled Steak in Roasted_\n\n_Garlic Sauce_\n\n220 calories, 16 g protein, 22 g carbohydrate, 7 g fat (2.5 g saturated), 5 g fiber, 600 mg sodium\n\n**Lean Cuisine:**\n\n_Chicken & Vegetables_\n\n240 calories, 19 g protein, 30 g carbohydrate, 5 g fat (2 g saturated), 2 g fiber, 630 mg sodium\n\n_Skillet Sensations Herb Chicken_\n\n250 calories, 15 g protein, 38 g carbohydrate, 4 g fat (1 g saturated), 4 g fiber, 890 mg sodium\n\n_Roasted Turkey Breast_\n\n270 calories, 12 g protein, 51 g carbohydrate, 2.5 g fat (.5 g saturated), 3 g fiber, 690 mg sodium\n\n**Smart Ones:**\n\n_Beef Pot Roast_\n\n170 calories, 21 g protein, 10 g carbohydrate, 5 g fat (1.5 g saturated), 2 g fiber, 660 mg sodium\n\n_Stuffed Turkey Breast_\n\n270 calories, 13 g protein, 37 g carbohydrate, 7 g fat (2 g saturated), 5 g fiber, 720 mg sodium\n\n**Stouffer's:**\n\n_Steak Teriyaki Skillet Sensations_\n\n190 calories, 11 g protein, 29 g carbohydrate, 3 g fat (1.5 g saturated), 4 g fiber, 950 mg sodium\n\n## **LABELING LESSON**\n\nAny food label is like a store window. The front of it is designed to draw you in, and then once you're inside, you can figure out whether the product is worth it. With a food label, once you get past the pretty colors and the coquettish cartoon mermaid, you need to flip the can over and inspect the label. Here's what to look for.\n\n**Serving size:** How much you consume. A common trick is for something that appears to be one serving (a bottled drink) to actually be two or more. Don't be fooled.\n\n**Calories:** The measure of energy a food provides. Calories mean different things depending on what you eat, but in the end, extra calories that your body can't burn will get stored as fat.\n\n**Calories from fat:** Multiply that number by 3. If that number is close to the total calories, it could mean trouble.\n\n**% Daily value:** It's the percentage of the daily intake that the food supplies, based on a 2,000-calorie-a-day diet. A usually useless number.\n\n**Total fat:** It's the combined total of saturated, polyunsaturated, monoun-saturated, and trans fats. Look for a ratio that's at least 3 to 1, total to saturated. If monos and polys are listed, it's probably a healthy food. If it's more than 33 percent saturated or trans fat, consider another alternative.\n\n**Cholesterol:** It's a fatlike substance from animals. Your body manufactures most of your cholesterol; what food adds is only a small percentage. Don't worry about it and don't get drawn in by otherwise unhealthy foods that claim \"no cholesterol.\" As a nutritional selling point, it's worthless.\n\n**Sodium:** A mineral (salt, basically) added for flavor and to preserve foods. Unless you have high blood pressure or are sodium sensitive, use 2,000 milligrams as a reasonable daily target. More than that will not only raise your risk of hypertension, it will also make you retain water and look and feel heavier.\n\n**Total carbohydrate:** This includes all of the sugar, starch, and fiber. Total number isn't as important as the kind.\n\n**Dietary fiber:** The roughage that cleans your digestive and circulatory system. Insoluble fiber (in whole-grain foods, nuts, and beans) expands and takes up space in your belly, so it makes you feel full. Soluble fiber (in fruit and oats) keeps blood vessels lubed so cholesterol won't stick. Almost any food with at least 2 grams of fiber is good. Five is even better.\n\n**Protein:** Amino acids that build and maintain your entire body. It helps you feel satisfied. Men who exercise should shoot for 162 to 225 grams per day\u2014and women around 100. Thinner men need no more than 114 grams. Thinner women around 75.\n\n**Vitamin and mineral percentages:** The food's percentage of the minimum amounts of nutrients required to prevent various deficiency diseases. As long as you're eating the ABS DIET POWER 12, you should be getting all the vitamins and minerals you need, but it never hurts to take a daily multivitamin for extra insurance.\n\n**Ingredients:** Arranged in order by weight from most to least. Bad foods like high-fructose corn syrup and partially hydrogenated oils should be the fifth ingredient listed or lower. If not, move on.\n\n### **Bakery\/Grains: The Incredible Bulk**\n\nFiber is crucial to weight loss, and the best place to find it is in whole-grain baked goods. If the first ingredient isn't \"whole grain\" or \"whole wheat,\" keep looking.\n\n**Whole-wheat bread:** Pepperidge Farm or Milton's both offer a variety of high-fiber whole-wheat breads.\n\n**Tortillas:** MexAmerican or Tumaro Honey Wheat are flexible enough for a wrap, yet sturdy enough for a quesadilla. Bonus fiber, too.\n\n**Pitas:** Sahara Whole Wheat offers more fiber\u20145 grams\u2014than other brands.\n\n**English muffins:** Thomas' Whole-Wheat have a gram more fiber than the white-bread kinds.\n\n**Brown rice:** Kraft Minute Rice Instant Whole-Grain Brown is ready in less than 10 minutes.\n\n**Pasta:** De Cecco Whole Wheat is high in fiber, and this brand cooks well. It's not too tough or chewy.\n\n**Cereal:** Here's your best chance for fiber. My favorites include: Nature's Path Optimum (10 grams) and Kellogg's Complete Wheat Bran Flakes (5 grams).\n\n**Oats:** If your need is speed, choose Quick Quaker Oats. They cook in 1 minute and deliver 4 grams of fiber per half cup. If you have a little more time, buy Arrowhead Mills Steel Cut Oats. They take 7 to 9 minutes in the microwave, but they pack a potent 16 grams of fiber per half cup.\n\n### **Baking: Piece of Cake**\n\nYou'll have to navigate past the muffin mix and cake icing to find some important ingredients. Just grab your nuts and get outta there fast.\n\n**Nuts:** Choose sliced almonds, chopped pecans, crushed walnuts\u2014anything that'll slip into a bowl of oatmeal or smoothie and help save you time on the chopping block.\n\n**Honey:** Buy the in-store brand. You'll use only small amounts, so there's no point in buying the super-expensive ones.\n\n**Spices:** Here's another case where value rules. Despite what the label says, they'll last for ages, not just a couple of months. Get the cheap ones. You'll need basil, powdered ginger, cinnamon, cumin, and paprika.\n\n### **Cookies and Candies: No-Sweat Sweets**\n\nSpend too much time here and you'll get fatter just thinking about all the sugar, HFCS, and lard that are lurking in these products. You'll have plenty of opportunity to satisfy sweet cravings\u2014especially with smoothies that taste like shakes. But slip down this aisle for a few necessities.\n\n**Candy bars:** Pick up York Peppermint Patties and Snickers in the bite-sized bags. The key part is bite-sized. Enjoying one or two after dinner is a pleasure no one should be deprived of, diet or not. You just have to supply the willpower to keep it at one or two.\n\n**Crackers:** Triscuit Reduced Fat and Ryvita Crispbread are the best whole-wheat crackers for fiber at 3 grams per serving.\n\n**Cookies:** Fig bars\u2014they're sweet and satisfying, thanks to the fiber. Fig Newtons have 1 gram of fiber per 2-cookie serving; Bakery Barbara's Whole-Wheat Fig Bars have 1 gram per cookie.\n\n### **Drinks: Liquid assets**\n\nLoading up on soda or fruit juice is like reverse liposuction; it's pouring calories that'll be stored as fat back into your body. There's only one drink of choice, but you can spruce it up with a slice of lemon or lime.\n\n**Bottled water:** Buy an entire case. That way, you can't ignore it. Take a bottle everywhere.\n\n## **FAT CONTENT OF MEAT (4 OUNCES, RAW, WITHOUT SKIN OR BONE)**\n\n| **TOTAL (G)** | **SATURATED (G)**  \n---|---|---  \n**Skinless chicken breast** | 1.41 | 0.37  \n**Veal steak** | 2.45 | 0.74  \n**Wild rabbit** | 2.63 | 0.78  \n**Lean ground beef** | 4 | 1.50  \n**Cured ham** | 4.68 | 1.56  \n**Wild duck breast** | 4.82 | 1.50  \n**Chicken drumstick** | 5.05 | 1.34  \n**Lean pork tenderloin** | 5.06 | 1.79  \n**Beef sirloin steak** | 5.15 | 2  \n**Beefalo** | 5.44 | 2.31  \n**Turkey leg** | 7.62 | 2.34  \n**Turkey breast** | 7.96 | 2.17  \n**Lean beef tenderloin** | 8.02 | 3  \n**Lean pork chop** | 8.19 | 2.85  \n**Porterhouse steak** | 8.58 | 3  \n**Lean ground turkey** | 9.37 | 2.55  \n**Veal breast meat** | 9.73 | 3.80  \n**Rib-eye steak** | 18.03 | 7.30  \n**T-bone steak** | 19.63 | 7.69  \n**Ham** | 21.40 | 7.42  \n**Pork belly** | 60.11 | 21.92  \n**Cured pork** | 91.29 | 33.32\n\n### **Health food: Alternative routes**\n\nMost foods in the health food aisle or store taste like the box your computer came in\u2014only worse. (Rice cakes? No thanks.) But there are some great finds lurking in the aisles, and you don't have to be a dietary masochist or an unreformed hippie to shop for them.\n\n**Flaxseed:** Try Barleans's Forti-Flax. Buying your flax pre-ground saves time. And this one comes in a 16-ounce container that'll last for months. Look for it in the refrigerated section.\n\n**Whey protein:** Look for protein powder that also includes casein, another dairy-based muscle builder.\n\n### **Prepared Grocery-Store Foods**\n\nRemember when you were a kid, and all you had to do to score a heaping plateful of meat loaf and green beans was show up at the dinner table? The expanded deli sections of most upmarket grocery stores operate on the same principle. They cook it; you eat it. But this section of the grocery store is as littered with nutritional land mines as the rest (yeah, I mean you, crab cake). Here's how to chart a healthy path.\n\n### **Heat-and-Eat Entr\u00e9es**\n\n**ABS DIET ENDORSEMENT:**\n\n_Asian pan-seared salmon_ 250 calories, 12 g fat (1.5 g saturated), 350 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Homestyle meat loaf_ 230 calories, 11 g fat (4.5 g saturated), 470 mg sodium\n\n**Not that:** _Crab cake_ 500 calories, 32 g fat (5 g saturated), 990 mg sodium\n\n### **Heat-and-Eat Vegetables**\n\n**ABS DIET ENDORSEMENT:**\n\n_Asian-style bok choy_ 15 calories, 0 g fat, 190 mg sodium\n\n_Roasted asparagus and leeks_ 30 calories, 0 g fat, 130 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Pesto vegetables_ 70 calories, 3 g fat (0 g saturated), 190 mg sodium\n\n**Not that:** _Creamed spinach_ 90 calories, 4.5 g fat (2.5 g saturated), 460 mg sodium\n\n### **Cold Sides**\n\n**ABS DIET ENDORSEMENT:**\n\n_Roasted corn salad_ 140 calories, 8 g fat (0.5 g saturated), 390 mg sodium\n\n_Bean salad_ 150 calories, 9 g fat (0 g saturated), 390 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Orzo with vegetables_ 180 calories, 10 g fat (0.5 g saturated), 570 mg sodium\n\n**Not that:** _Tortellini salad_ 300 calories, 26 g fat (1 g saturated), 660 mg sodium\n\n### **ABS DIET SUCCESS STORY**\n\n## **\"IT DOESN'T EVEN FEEL LIKE A DIET\"**\n\n**Name:** Jon Armond\n\n**Height:** 6\u20324\u2033\n\n**Age:** 33\n\n**Weight, Week 1:** 254\n\n**Weight, Week 6:** 229\n\n**Weight, Week 9:** 219\n\n**Body-Fat Percentage, Week 1:** 27\n\n**Body-Fat Percentage, Week 9:** 18\n\nJon Armond's a big guy, and he's always carried his weight well. When he told people he was trying to lose weight, they told him he didn't need to. But Armond knew better. Carrying more than 250 pounds around everywhere he went was taking a serious toll on his body, and he knew he needed to drop some serious pounds.\n\nEven though he was active, Armond was one of those guys who rationalized that he could eat whatever he wanted because he had exercised.\n\nBut the fact was that he ate and drank too much, and he was always tired. (He frequently took naps in the middle of the day).\n\n\"I just didn't feel like I was 33,\" he says. \"I felt like I was 53.\" Then he found the Abs Diet.\n\n\"When I looked in _Men's Health_ and looked at the way the Abs Diet was set up, something clicked. This made perfect sense,\" he says. \"I knew I'd lose weight, but what I didn't expect was the total-body transformation\u2014to see the fat drop and the muscle gain at the same time. That's what I really appreciate about it.\"\n\nFor the first 6 weeks, Armond stuck to the principles without wavering, and he realized that it wasn't really a diet at all, but just a different approach to eating.\n\n\"I found myself not even consumed with it. I've never been on a diet that you didn't have to think about being on the diet all the time,\" he says. Instead, it was a plan that he simply incorporated into his life, and by doing so, he's traded in his \"huge beer gut.\"\n\n\"I've got some abs showing through now,\" Armond says. \"I was wearing size 38s, and I just bought 34s. I was wearing double-X shirts and now I'm wearing larges. I'm ready to go on the public-speaking circuit to talk about the Abs Diet.\"\n\n### **ABS DIET SUCCESS STORY**\n\n## **A FORMER A THLETE FINDS A BETTER BODY**\n\n**Name:** Mark Peterson\n\n**Age:** 25\n\n**Height:** 5\u20329\u2033\n\n**Weight, Week 1:** 187\n\n**Weight, Week 6:** 170\n\n**Weight, Week 12:** 164\n\nMark Peterson was a swimmer in high school and competed in triathlons. But then all that eating during college and law school hit him.\n\n\"Law school is the most unhealthy thing I've ever done in my life,\" he says. \"I was looking for something to get me back in shape.\"\n\nHe decided to train for the Chicago marathon and found a diet to support his training.\n\n\"The Abs Diet gave you a lot of different foods that you could have,\" he says. \"It just seemed like a diet you could totally stick to.\"\n\nHe lived on almonds as a snack, spinach salad as lunch, peanut butter on whole-wheat toast before his workout, and made chicken or fish for dinner. \"I didn't like oatmeal, so I substituted Kashi and Go Lean cereals.\"\n\nIn the fall, Peterson completed the marathon, and friends who hadn't seen him for months noticed the transformation. \"They all said that it looked like I didn't need to lose weight and that I was crazy, but it's totally different when they see me now. A lot of friends couldn't believe the weight I lost.\"\n\nNow that marathon training is over, he has another goal in mind. \"I swam like crazy in high school, and I focused a lot on running, but I've never been close to a six-pack,\" he says. \"This is the closest I've ever been.\"\nChapter 5\n\n# **LET'S GET IT STARTED**\n\n# 25 Abs Diet Breakfasts\n\n**B EFORE WORK,** there's a lot to do\u2014take a shower, check the weather, dress the kids, glance at the stocks and scores, brush your teeth, zip the pants. In that routine, there's not much you can sacrifice\u2014unless you want go to work in your skivvies. So many of us end up treating breakfast like it's a luxury\u2014something that we do only if we have time or if there's a stale Pop-Tart on the floor mats. But sacrificing breakfast is like arguing with the cops; there's absolutely nothing good that can come out of it. Without breakfast, you're operating on reserves, putting your body in a pseudostarvation mode that tells your body's metabolism to slow down to protect you. What you need to do is eat\u2014and eat heartily. Since I know you've apportioned about as much time to make breakfast as you have to slide on your socks, I've whipped up some recipes you can zip through faster than those artery-clogged lines at the drive-through.\n\nSo fire 'em up and start the day strong. What you eat the first hour you're awake will have a huge effect on what you eat for the next 16 or 17. I'm a firm believer that if every day of good eating is a race, then how well you start determines how well you finish.\n\n## **Quicker Oats**\n\n**USE PLAIN INSTANT OATMEAL** for the following recipes. Nothing personal, Quaker guy, but those powdery sugared packets are as antiquated as that hat. For each recipe, mix all ingredients in a microwave-safe bowl and nuke for 2 minutes, unless otherwise noted. All serve 1.\n\n### **Honey, I Shrunk My Gut (Powerfoods: 5)**\n\n1 cup 1% milk\n\n\u00be cup plain instant oatmeal\n\n\u00bd cup blue- or blackberries\n\n1 tablespoon chopped walnuts or pecans 1 teaspoon honey 1 teaspoon ground flaxseed\n\nDash of cinnamon\n\n_Per serving: 459 calories, 19 g protein, 70 g carbohydrates, 9 g fat (2 g saturated), 9 g fiber, 132 mg sodium_\n\n### **Flax Machine (Powerfoods: 4)**\n\n1 cup 1% milk\n\n\u00be cup plain instant oatmeal\n\n\u00bd banana, sliced\n\n1 tablespoon peanut butter\n\n1 teaspoon honey\n\n1 teaspoon ground flaxseed\n\nDash of brown sugar\n\n_Per serving: 515 calories, 22 g protein, 75 g carbohydrates, 13 g fat (3 g saturated), 9 g fiber, 172 mg sodium_\n\n### **You're Nuts (Powerfoods: 5)**\n\n1 cup 1% milk\n\n\u00be cup plain instant oatmeal\n\n1 tablespoon chopped pecans\n\n1 tablespoon chopped walnuts\n\n1\u00bd teaspoon honey\n\n1 teaspoon ground flaxseed\n\n1\u20448 teaspoon cinnamon\n\n_Per serving: 480 calories, 20 g protein, 63 g carbohydrates, 14 g fat (3 g saturated), 8 g fiber, 132 mg sodium_\n\n### **Ginger? Roger! (Powerfoods: 5)**\n\n1 cup 1% milk\n\n\u00be cup plain instant oatmeal\n\n1 tablespoon sliced almonds\n\n1 teaspoon honey\n\n1 teaspoon ground flaxseed\n\n\u00bd teaspoon powdered ginger\n\n1 tablespoon vanilla yogurt\n\n_Mix milk, oats, almonds, honey, flaxseed, and ginger in a microwave-safe bowl. Nuke for 2 minutes. Top with yogurt._\n\n_Per serving: 420 calories, 20 g protein, 62 g carbohydrates, 7 g fat (2 g saturated), 7 g fiber, 142 mg sodium_\n\n## **Instant Omelets**\n\nAt a diner, an omelet is just code for \"throw the leftovers in a pan with some eggs.\" At home, they're a quick source of protein\u2014and a chance to boost your Powerfood count with one pan only. For all recipes, stir 2 eggs with a fork until white and yolk are well blended. Add the remaining ingredients. Nuke for 2 minutes and 30 seconds or until the eggs are firmly set. All serve 1.\n\n### **Tom Tomelet (Powerfoods: 3)**\n\n2 eggs\n\n1 slice turkey, diced\n\n1 tablespoon shredded reduced-fat Mexican cheese blend\n\n_Per serving: 237 calories, 16 g protein, 3 g carbohydrates, 18 g fat (7 g saturated), 0 g fiber, 374 mg sodium_\n\n### **The Green and White (Powerfoods: 3)**\n\n2 eggs\n\n1 tablespoon shredded part-skim mozzarella cheese\n\n1\u20443 cup torn baby spinach leaves\n\n_Per serving: 221 calories, 15 g protein, 3 g carbohydrates, 16 g fat (6.5 g saturated),_ < _1 g fiber, 256 mg sodium_\n\n### **Lean Eggs and Ham (Powerfoods: 2)**\n\n2 eggs\n\n1 slice Canadian bacon, diced\n\n1 slice tomato, chopped\n\n_Per serving: 271 calories, 23 g protein, 4 g carbohydrates, 18 g fat (7 g saturated), 0 g fiber, 781 mg sodium_\n\n### **Bean Counter (Powerfoods: 2)**\n\n2 eggs\n\n2 tablespoons rinsed black beans\n\n1 teaspoon cilantro\n\n_After cooking, top with 1 tablespoon salsa._\n\n_Per serving: 230 calories, 15 g protein, 8 g carbohydrates, 15 g fat (6 g saturated), 1 g fiber, 361 mg sodium_\n\n### **Up in Smoke (Powerfoods: 3)**\n\n2 eggs\n\n1 ounce diced smoked salmon\n\n1\u20443 cup torn baby spinach leaves\n\n_Per serving: 236 calories, 18 g protein, 3 g carbohydrates, 16 g fat (6 g saturated), 0 g fiber, 790 mg sodium_\n\n### **The Mister Bean (Powerfoods: 4)**\n\n2 eggs\n\n1 slice turkey, diced\n\n1 tablespoon rinsed black or cannellini beans\n\n1 tablespoon shredded part-skim mozzarella cheese\n\n_Per serving: 240 calories, 17 g protein, 5 g carbohydrates, 17 g fat (7 g saturated),_ < _1 g fiber, 374 mg sodium_\n\n## **Breakfast Burritos**\n\nYou usually eat burritos for dinner, but a burrito is actually one of the easiest things you can make for breakfast. Just arrange all the ingredients on the tortilla, fold the ends, then neatly roll. For those recipes that call for nuked eggs, you can scramble them in 60 seconds. In a microwave-safe bowl, just stir the eggs with a fork until white and yolk are well blended and microwave for 1 minute per egg.\n\n### **Holy Guacamole! (Powerfoods: 5)**\n\n1 medium whole-wheat tortilla\n\n2 slices fat-free turkey deli slices\n\n2 nuked eggs\n\n\u00bd avocado, sliced\n\n2 tablespoon shredded reduced-fat Mexican-blend cheese\n\n_Per serving: 494 calories, 25 g protein, 31 g carbohydrates, 35 g fat (10 g saturated), 7 g fiber, 718 mg sodium_\n\n### **Yosemite Salmon (Powerfoods: 6)**\n\n2 tablespoons part-skim ricotta cheese\n\n1 medium whole-wheat tortilla\n\n1 ounce smoked salmon, torn into little pieces\n\n2 nuked eggs\n\n1 cup chopped baby spinach\n\n1 sliced green onion\n\n_Spread cheese on tortilla, then arrange remaining ingredients, fold ends in, and roll._\n\n_Per serving: 361 calories, 25 g protein, 26 g carbohydrates, 19 g fat (8 g saturated), 3 g fiber, 1026 mg sodium_\n\n### **Huevos Rancheros (Powerfoods: 4)**\n\n1 medium whole-wheat tortilla\n\n2 nuked eggs\n\n1 sliced green onion\n\n1 tablespoon diced cilantro\n\n2 tablespoons shredded reduced-fat Mexican-blend cheese\n\n2 tablespoons salsa\n\n_Per serving: 326 calories, 20 g protein, 25 g carbohydrates, 19 g fat (7 g saturated), 2 g fiber, 713 mg sodium_\n\n## **DAIRY DAIRY, QUITE CONTRARY**\n\nAt breakfast, put coffee in your milk instead of milk in your coffee. Fill your mug to the rim with fat-free milk first thing in the morning. Drink it down until all that's left is the amount you'd normally add to your coffee; then pour your java on top. You just took in 25 percent of the vitamin D you need every day and 30 percent of the calcium.\n\n## **Breakfast Sandwiches**\n\nOnce suitable only for the lunch box, today sandwiches might show up anywhere\u2014as fancy canap\u00e9s, as hearty dinner entr\u00e9es, as sexual fantasies. . . . Okay, let's not go there. Right now we're concentrating on breakfast, so let's take a look at some healthy choices you can serve up in the A.M. (Who you serve them to is your business.)\n\n### **Mex-illent Adventure (Powerfoods: 3)**\n\n1 tablespoons salsa\n\n1 toasted whole-wheat English muffin\n\n1 nuked egg\n\n1 teaspoon diced cilantro\n\n1 tablespoon shredded reduced-fat Mexican-blend cheese\n\n_Spread salsa on bottom half of muffin, top with egg, cilantro, and cheese. Toast in a toaster oven until cheese melts._\n\n_Per serving: 259 calories, 14 g protein, 29 g carbohydrates, 11 g fat (4 g saturated), 4 g fiber, 691 mg sodium_\n\n### **Foxy Lox (Powerfoods: 3)**\n\n2 tablespoons part-skim ricotta cheese\n\n1 toasted whole-wheat English muffin\n\n1 slice tomato\n\n1 ounce smoked salmon\n\n_Spread cheese over each muffin half, top with tomato and salmon._\n\n_Per serving: 214 calories, 15 g protein, 29 g carbohydrates, 5 g fat (2 g saturated), 5 g fiber, 1027 mg sodium_\n\n### **The Three-Country Breakfast (Powerfoods: 4)**\n\n1 nuked egg\n\n1 slice Canadian bacon\n\n1 slice tomato\n\n1 toasted whole-wheat English muffin\n\n1 tablespoon shredded reduced-fat Mexican cheese blend\n\n_Arrange egg, bacon, and tomato on one half of muffin. Top with cheese. Toast in a toaster oven until cheese melts._\n\n_Per serving: 326 calories, 24 g protein, 30 g carbohydrates, 13 g fat (5 g saturated), 5 g fiber, 1145 mg sodium_\n\n### **Apple Jacked (Powerfoods: 4)**\n\n1 toasted whole-wheat English muffin\n\n1 slice Canadian bacon\n\n2 slices apple\n\n2 tablespoons peanut butter\n\n_Per serving: 403 calories, 23 g protein, 38 g carbohydrates, 20 g fat (3.5 g saturated), 7 g fiber, 1069 mg sodium_\n\n### **Jam Session (Powerfoods: 3)**\n\n1 toasted whole-wheat English muffin\n\n2 tablespoons part-skim ricotta cheese\n\n1\u20443 cup slightly crushed blue-, black-, or raspberries (mash berries in small bowl with fork)\n\n_Per serving: 197 calories, 10 g protein, 33 g carbohydrates, 4 g fat (2 g saturated), 4 g fiber, 460 mg sodium_\n\n### **Wafflewich (Powerfoods: 3)**\n\n1 whole-wheat toaster waffle\n\n2 tablespoons peanut butter\n\n\u00bc cup slightly crushed blue-, black-, or raspberries\n\n_Prepare waffle according to package directions. Spread peanut butter on waffle. Cup waffle in half, add berries, then squeeze lightly. Think of it as a berry breakfast taco._\n\n_Per serving: 308 calories, 12 g protein, 24 g carbohydrates, 20 g fat (3.5 g saturated), 5 g fiber, 212 mg sodium_\n\n### **Turk before Work (Powerfoods: 3)**\n\n1 whole-wheat toaster waffle\n\n2 slices deli turkey\n\n2 slices apple\n\n_Prepare waffle according to package directions. Arrange turkey and apple on waffle, then fold lightly in half._\n\n_Per serving: 139 calories, 7 g protein, 18 g carbohydrates, 5 g fat (1.5 g saturated), 2 g fiber, 359 mg sodium_\n\n## **Turbocharged Yogurt**\n\nIn case you missed it (and I don't know how, since I've been jumping up and down and waving my arms about it for nearly 100 pages now), calcium is one of those ingredients that fortifies more than your skeleton. It seems to be a potent player in the weight-loss game. At breakfast, a cup of power-enhanced yogurt is the fastest power player since Mike Vick. For all, just mix the ingredients and eat.\n\n### **Bananarama (Powerfoods: 2)**\n\n1 cup vanilla yogurt\n\n1 banana, sliced\n\n1 tablespoon chopped walnuts\n\n_Per serving: 363 calories, 15 g protein, 62 g carbohydrates, 8 g fat (3 g saturated), 4 g fiber, 163 mg sodium_\n\n### **Berry Easy (Powerfoods: 3)**\n\n1 cup plain yogurt\n\n\u00bd cup mixed berries\n\n1 teaspoon ground flaxseed\n\n_Per serving: 204 calories, 14 g protein, 27 g carbohydrates, 5 g fat (2.5 g saturated), 4 g fiber, 173 mg sodium_\n\n### **The Rupert Pumpkin (Powerfoods: 2)**\n\n1 cup vanilla yogurt\n\n\u00bc cup canned pumpkin\n\n1 tablespoon chopped pecans\n\n_Per serving: 282 calories, 13 g protein, 40 g carbohydrates, 9 g fat (2.5 g saturated), 2.5 g fiber, 164 mg sodium_\n\n### **A Trip to the Peach (Powerfoods: 2)**\n\n1 cup vanilla yogurt\n\n1\u20443 cup frozen or canned peaches (drain the syrup)\n\n1 tablespoon sliced almonds\n\n_Per serving: 323 calories, 14 g protein, 55 g carbohydrates, 6 g fat (2 g saturated), 2 g fiber, 166 mg sodium_\n\n## **The Super Bowl of Breakfasts**\n\nEight power foods in one bowl? We haven't seen this many A-listers since Oscar night. Start mixing.\n\n### **The Ultimate Power Breakfast (Powerfoods:8)**\n\n1 egg\n\n1 cup 1% milk\n\n\u00be cup oatmeal\n\n\u00bd cup mixed berries\n\n1 tablespoon chopped pecans or sliced almonds\n\n1 teaspoon vanilla whey protein powder\n\n1 teaspoon ground flaxseed\n\n\u00bd banana, sliced\n\n1 tablespoon plain yogurt\n\n_In microwave-safe bowl, mix egg well, then add next 6 ingredients and nuke for 2 minutes. Remove, let cool for a minute or two. Top with sliced banana and yogurt._\n\n_Per serving: 587 calories, 30 g protein, 76 g carbohydrates, 15 g fat (5 g saturated), 13 g fiber, 254 mg sodium_\n\n### **ABS DIET SUCCESS STORY**\n\n## **HE LOST HIS POUNDS AND CURED HIS PAIN**\n\n**Name:** John Kelly\n\n**Age:** 40\n\n**Height:** 6\u20321\u2033\n\n**Weight, Week 1:** 215\n\n**Weight, Week 6:** 193\n\nWhen John Kelly was diagnosed with Barrett's disease\u2014a condition caused by chronic acid reflux that can lead to esophageal cancer\u2014he knew something had to change. He had a demanding job, three sons under the age of 9, and the stress of moving to a new house.\n\n\"It's hard to take care of yourself when there are so many other responsibilities that dominate your life,\" he says. Caught up in trying to be all things to all people, Kelly let his diet and exercise program slide, and, no surprise, he gained weight. \"I just thought that gaining weight and being tired were par for the course.\"\n\nBut what wasn't par for the course were the unexplained pains in his chest. The pain was building to the point where he'd feel nauseous three times a week, and it got so bad that he had to skip work sometimes. When his doctor diagnosed Barrett's disease, Kelly began searching for ways to change his lifestyle, and he found some in the Abs Diet.\n\nBecause of the limitations he had with his illness\u2014no fatty meats, spicy foods, or tomato sauce\u2014Kelly found that the Abs Diet worked for him.\n\n\"My condition meant no vices\u2014no alcohol or caffeine\u2014so the idea of a diet that featured daily smoothies was pretty appealing,\" Kelly says.\n\nWithin weeks, the fat began to melt, and the muscles appeared. But it wasn't just the fat\u2014or the fact that his energy levels had increased. Kelly wasn't feeling sick anymore\u2014and hasn't experienced any symptoms since starting the diet.\n\n\"The Abs Diet helped me return to a more normal life on so many levels that the weight loss and fitness gain are just the icing on the cake,\" Kelly says. \"Being able to contribute at my job and participate in my boys' lives more actively are blessings that I could have never imagined could come from a diet.\"\nChapter 6\n\n# **LEAN IN THE MIDDLE**\n\n# 25 Abs Diet Lunches\n\n**D EPENDING ON WHAT** you do during the day, lunch can often present you with the opportunity to blow your diet like a tropical storm through the Florida Keys. Lunch comes at the very time of day when forces outside of our control are vying for our time. If your job is demanding, lunch sometimes means you have 3 minutes to pop the crackers out of the machine and get back to your desk. If you work with the get-some-fresh-air types, you're on the road at 12 and ordering a burger and beer by 12:06. If you need to have fancy business lunches, you're susceptible to the onslaught of free bread, creamy sauces, and four-story desserts. If you're home with the kids, you're one temper tantrum away from a macaroni-and-cheese binge.\n\nThat's why I suggest that if there's one meal that you should plan every day, it's lunch. Preparing for it makes you less likely to fall victim to any one of the diet busters waiting to pounce. It doesn't matter whether you make your lunch the night before, in the morning, or 3 minutes before you're going to eat it. The point is: Preparation breeds motivation.\n\n## **Wraps**\n\nThe trend toward sandwich wraps over the past few years has made eating a healthy lunch easier than ever before. An easy-access tortilla cuts down on empty calories and eliminates the need for utensils. But there are some bum wraps out there, so I've rolled up and rolled out a handful of smart choices just packed with Powerfoods. All make 1 serving.\n\n### **Thai One On (Powerfoods: 4)**\n\n1\u00bd tablespoons peanut butter\n\n1 whole-wheat tortilla\n\n2\u20443 cup chopped precooked chicken\n\n\u00be cup mixed greens\n\n\u00bc cup matchstick carrots\n\n1 teaspoon diced cilantro\n\n_Spread peanut butter down center of tortilla. Add chicken and remaining ingredients. Fold outside edges in, then roll._\n\n_Per serving: 331 calories, 29 g protein, 30 g carbohydrates, 15 g fat (2 g saturated), 5 g fiber, 661 mg sodium_\n\n### **The Day after Thanksgiving (Powerfoods: 4)**\n\n2 tablespoons cranberry relish\n\n1 whole-wheat tortilla\n\n3 slices turkey\n\n1 slice Muenster cheese\n\n\u00be cup mixed greens\n\n_Spread cranberry relish down center of tortilla. Add turkey and remaining ingredients. Fold outside edges in, then roll._\n\n_Per serving: 311 calories, 24 g protein, 40 g carbohydrates, 11 g fat (6 g saturated), 3 g fiber, 1063 mg sodium_\n\n### **The Cow Tipper (Powerfoods: 3)**\n\n3 slices roast beef\n\n1 whole-wheat tortilla\n\n\u00be cup mixed greens\n\n\u00bc cup chopped tomato\n\n1 tablespoon Dijon mustard\n\n1 tablespoon blue cheese crumbles\n\n_Arrange beef slices down center of tortilla, then add remaining ingredients. Fold outside edges in, then roll._\n\n_Per serving: 208 calories, 22 g protein, 26 g carbohydrates, 5 g fat (2.5 g saturated), 3 g fiber, 1038 mg sodium_\n\n### **The Two Turks (Powerfoods: 6)**\n\n2 slices turkey\n\n1 whole-wheat tortilla\n\n2 slices cooked turkey bacon\n\n\u00be cup mixed greens\n\n3 slices avocado\n\n2 tablespoons shredded reduced-fat Mexican-blend cheese\n\n_Arrange turkey slices down center of tortilla, then add remaining ingredients. Fold outside edges in, then roll._\n\n_Per serving: 407 calories, 27 g protein, 32 g carbohydrates, 24 g fat (6 g saturated), 10 g fiber, 1376 mg sodium_\n\n### **Hot Curlers (Powerfoods: 4)**\n\n1 tablespoon Dijon mustard\n\n1 whole-wheat tortilla\n\n2\u20443 cup chopped precooked chicken\n\n\u00be mixed greens\n\n\u00bc cup diced tomato\n\nHot sauce to taste\n\n2 tablespoons shredded reduced-fat Mexican-blend cheese\n\n_Spread mustard down center of tortilla. Add chicken and remaining ingredients. Fold outside edges in, then roll._\n\n_Per serving: 244 calories, 30 g protein, 28 g carbohydrates, 6 g fat (2 g saturated), 3 g fiber, 1089 mg sodium_\n\n## **Salads**\n\nYour standard chain-restaurant salad bar is a nutritional minefield\u2014there are plenty of healthy, safe moves to make and plenty of fat bombs that can blow up your gut: potato salad, macaroni salad, croutons, those bacon bits that look like gravel. If you make your own salad, you can ensure that your salad has taste and power. All make 1 serving.\n\n### **The Olympiad (Powerfoods: 4)**\n\n2 \u00bd cups mixed greens\n\n1 Roma tomato, chopped\n\n\u00bc cup chopped cucumber\n\n1 tablespoon shredded part-skim mozzarella cheese\n\nPinch black pepper\n\n1 tablespoon balsamic vinegar or red wine\n\n1 teaspoon olive oil\n\n_Per serving: 109 calories, 5 g protein, 8 g carbohydrates, 6 g fat (1.6 g saturated), 4 g fiber, 77 mg sodium_\n\n## **DIY DRESSINGS**\n\nEver wondered how something that should by all laws of nature be refrigerated\u2014like blue cheese dressing\u2014manages to stay out on shelves? Trans fat and lots of preservatives do the work, making salad dressings some of the biggest belly-bloating foods out there.\n\nMaking your own salad dressing is just a little bit harder than boiling water, but it will ensure that you're eating 100 percent healthy (and it'll taste better, too). Add 1 part olive oil to 2 parts acid, such as lemon juice, orange juice, vinegar, or wine. Stir it together to blend or just pour each one on the salad.\n\n### **Sweet Cheeses! (Powerfoods: 5)**\n\n2 \u00bd cups mixed greens\n\n\u00bd cup berries (any type or a mix)\n\n1 tablespoon chopped onion\n\n1 tablespoon blue cheese crumbles\n\n1 tablespoon chopped pecans or walnuts\n\n1 teaspoon ground flaxseed\n\n1 tablespoon balsamic vinegar\n\n1 teaspoon olive oil\n\n_Per serving: 235 calories, 6 g protein, 24 g carbohydrates, 15 g fat (2.7 g saturated), 7 g fiber, 160 mg sodium_\n\n### **Orient Express (Powerfoods: 6)**\n\n2 \u00bd cups mixed greens\n\n\u00bd cup diced precooked chicken\n\n1 green onion, sliced\n\n\u00bc cup chopped green pepper\n\n1 tablespoon sliced almonds\n\n1 teaspoon diced cilantro\n\n1 tablespoon orange juice\n\n1 teaspoon olive oil\n\n_Per serving: 214 calories, 23 g protein, 10 g carbohydrates, 10 g fat (1 g saturated), 4 g fiber, 437 mg sodium_\n\n### **Kidney Punch (Powerfoods: 3)**\n\n2 \u00bd cups mixed greens\n\n\u00bc cup drained canned kidney beans\n\n1 tablespoon chopped onion\n\n1 tablespoon blue cheese crumbles\n\nPinch black pepper\n\n1 tablespoon red wine\n\n1 teaspoon olive oil\n\n_Per serving: 184 calories, 9 g protein, 20 g carbohydrates, 8 g fat (2 g saturated), 9 g fiber, 154 mg sodium_\n\n### **Potion of the Ocean (Powerfoods: 6)**\n\n2 \u00bd cups mixed greens\n\n6 frozen shrimp, defrosted\n\n1 green onion, sliced\n\n\u00bc cup chopped cucumber\n\n\u00bc cup chopped green pepper\n\n1 tablespoon lemon juice\n\n1 teaspoon olive oil\n\n_Per serving: 112 calories, 10 g protein, 8 g carbohydrates, 5 g fat (.78 g saturated), 4 g fiber, 112 mg sodium_\n\n### **El Tequila Ensalada (Powerfoods: 5)**\n\n2 \u00bd cups mixed greens\n\n\u00bc cup drained black beans\n\n1 Roma tomato, chopped\n\n1 green onion, sliced\n\n\u00bd sliced avocado\n\n1 teaspoon diced cilantro\n\n1 tablespoon tequila or, for the less stout of heart, lime juice\n\n1 teaspoon olive oil\n\n_Per serving: 325 calories, 8 g protein, 22 g carbohydrates, 21 g fat (3 g saturated), 13 g fiber, 250 mg sodium_\n\n## **WEEKLY GRIND**\n\nSeveral of these recipes call for lemon juice. If you have real lemons on hand, you can use the rind for zest. Zest adds flavor, and according to a University of Arizona study, a tablespoon of it each week can help cut the risk of developing skin cancer by 30 percent. Another handy tip: After using half a lemon for juice or zest, toss the rest down the drain and let it clean out your rank garbage disposal.\n\n## **Sandwiches**\n\nAs far as ease goes, nothing beats a sandwich. Bread, meat, eat. Still, as drive-throughs and the Hilton sisters prove, just because something's easy doesn't mean it's smart. There are plenty of sandwich fixings, from mile-high corned beef to do-nothing-for-you white bread, that simply don't stack up. Still, we love the sandwich. The trick is to incorporate as many Abs Diet Powerfoods into your concoctions as you can. That means some whole-wheat bread, some lean meat, and some leafy greens if you want. You want to build the ultimate power sandwich that's filling and more well rounded than Anna Nicole Smith (circa 2003). To shake things up, try making these substitutions without the extra layer of guilt.\n\n**Abs Diet BLT**\n\n**SUBSTITUTE THIS:** Turkey bacon\n\n**FOR THAT:** Regular bacon\n\n**ABS DIET ADVANTAGE:** 59 fewer calories\n\n**SUBSTITUTE THIS:** Reduced-fat sour cream\n\n**FOR THAT:** Mayo\n\n**ABS DIET ADVANTAGE:** 2 more grams protein\n\n**SUBSTITUTE THIS:** Baby spinach\n\n**FOR THIS:** Iceberg lettuce\n\n**ABS DIET ADVANTAGE:** 3 more grams fiber\n\n**SUBSTITUTE THIS:** Whole-wheat bread\n\n**FOR THIS:** White bread\n\n**ABS DIET ADVANTAGE:**\n\n**Abs Diet meatball sub**\n\n**SUBSTITUTE THIS:** Turkey meatballs\n\n**FOR THIS:** Beef meatballs\n\n**ABS DIET ADVANTAGE:** 194 fewer calories\n\n**SUBSTITUTE THIS:** Part-skim mozzarella\n\n**FOR THIS:** Full-fat mozzarella\n\n**ABS DIET ADVANTAGE:** 4 grams less saturated fat\n\n**SUBSTITUTE THIS:** Whole-wheat roll\n\n**FOR THIS:** White roll\n\n**ABS DIET ADVANTAGE:** 3.5 grams more fiber\n\n**Abs Diet po boy**\n\n**SUBSTITUTE THIS:** Defrosted frozen steamed shrimp\n\n**FOR THIS:** Deep-fried shrimp\n\n**ABS DIET ADVANTAGE:** 379 fewer calories\n\n**SUBSTITUTE THIS:** Reduced-fat sour cream\n\n**FOR THIS:** R\u00e9moulade\n\n**ABS DIET ADVANTAGE:** 3.5 grams less saturated fat\n\n**Abs Diet gyro**\n\n**SUBSTITUTE THIS:** Lean roast beef\n\n**FOR THIS:** Lamb\n\n**ABS DIET ADVANTAGE:** 14 fewer calories\n\n**SUBSTITUTE THIS:** Reduced-fat yogurt\n\n**FOR THIS:** Full-fat yogurt\n\n**ABS DIET ADVANTAGE:** 1.5 grams less saturated fat\n\n**SUBSTITUTE THIS:** Whole-wheat pita\n\n**FOR THIS:** White pita\n\n**ABS DIET ADVANTAGE:** 3 grams more fiber\n\n**Abs Diet PB&J**\n\n**SUBSTITUTE THIS:** Natural peanut butter\n\n**FOR THIS:** Trans-fatty kind\n\n**ABS DIET ADVANTAGE:** 74 fewer calories\n\n**SUBSTITUTE THIS:** Slightly crushed fresh berries\n\n**FOR THIS:** Sugary jelly\n\n**ABS DIET ADVANTAGE:** 1 gram less saturated fat\n\n**SUBSTITUTE THIS:** Whole-wheat bread\n\n**FOR THIS:** White bread\n\n**ABS DIET ADVANTAGE:** 4 grams more fiber\n\n### **Fill-in-the-Blank-Salad Sandwich (Powerfoods: 4)**\n\n6-ounces salmon or tuna or 2\u20443 cup chopped precooked chicken\n\n1 green onion, sliced\n\n\u00bc cup finely diced cucumber\n\n1 tablespoon reduced fat sour cream\n\n2 teaspoons Dijon mustard\n\n1 teaspoon lemon juice\n\n1\u20448 teaspoon lemon zest\n\nSalt and pepper to taste\n\n_Mix everything together in a big bowl, stirring well to blend in the lemon juice and sour cream. Eat with mixed greens or whole-wheat crackers_.\n\n_Salmon: Per serving: 149 calories, 19 g protein, 2 g carbohydrates, 7 g fat (2 g saturated), 0 g fiber, 661 mg sodium_\n\n_Tuna: Per serving: 127 calories, 22 g protein, 2 g carbohydrates, 3 g fat (1 g saturated), 0 g fiber, 523 mg sodium_\n\n_Chicken: Per serving: 118 calories, 20 g protein, 3 g carbohydrates, 3 g fat (.5 g saturated), 0 g fiber, 603 mg sodium_\n\n## **Soups**\n\nSoup is not the kind of food you think of as fitting into a busy lifestyle, it's hard to stash in your briefcase, tough to eat on the train, and generally frowned upon at business meetings. But soup does offer one big convenience. You can whip up a batch, store it in the fridge, then nuke it the next day (and the next, and the next) for a hearty, healthy, power-packed meal.\n\n### **Macho Gazpacho (Powerfoods: 3)**\n\n3 large tomatoes\n\n1 cup peeled chopped cucumber\n\n\u00bd cup reduced-fat plain yogurt\n\n1 teaspoon balsamic vinegar\n\n1 teaspoon olive oil\n\n1 teaspoon lemon juice\n\n\u00bc teaspoon salt\n\n_Seed the tomatoes (cut a crosshatch in the bottom, hold over the sink and squeeze\u2014most of the seeds will come out of the bottom). Core and chop them into rough chunks and chuck them, along with everything else, into the blender. Puree until smooth. Serve cold_.\n\n_Serves 2_\n\n_Per serving: 118 calories, 6 g protein, 17 g carbohydrates, 4 g fat (1 g saturated), 4 g fiber, 349 mg sodium_\n\n### **The Beaning of Life (Powerfoods: 3)**\n\n1\u00bd cans drained or no salt added black beans\n\n\u00bd cup reduced-fat, low-sodium chicken broth\n\n1 chopped Roma tomato\n\n1 sliced green onion\n\n1 tablespoon lime juice\n\n1 teaspoon hot sauce\n\n1 teaspoon olive oil\n\n1 tablespoon diced cilantro\n\n\u00bd teaspoon cumin\n\nSalt and pepper to taste\n\n1 tablespoon shredded reduced-fat Mexican-blend cheese\n\n_Dump everything except cheese into the blender. Pur\u00e9e until smooth, scraping the sides if needed. Pour into bowls and microwave for 2 or 3 minutes, stirring occasionally. Top with a pinch of cheese._\n\n_Serves 4_\n\n_Per serving: 365 calories, 24 g protein, 60 g carbohydrates, 5 g fat (1 g saturated), 22 g fiber, 475 mg sodium_\n\n## **Pita Pizzas**\n\nContrary to popular belief, pizza _is_ a health food. The tomato sauce provides vitamin C and the anticancer nutrient lycopene; the cheese gives you a hit of calcium and protein; and any vegetables you toss on top bring extra helpings of vitamins, minerals, and fiber.\n\nUnfortunately, most commercial pizzas are corrupted by excess oil, fatty pepperoni, and horrific mutations like \"cheese-filled crust.\" Talk about turning a good thing bad!\n\nOn this page, I've ranked the national pizza chains from healthiest to unhealthiest. But when you want a quick hit of pie, try these home versions for the taste without suffocating your organs in pepperoni. For all of them, spread sauce evenly over pita, then top with remaining ingredients. Bake in an oven pre-heated to 475\u00b0F for 4 to 6 minutes. All serve 1.\n\n### **The Whitey Ford (Powerfoods: 5)**\n\n1 small whole-wheat pita\n\nWhite sauce (stir together \u00bc cup part-skim ricotta cheese, 1 teaspoon olive oil, \u00bc teaspoon dried basil or oregano)\n\n1 green onion, sliced\n\n1 tablespoon part-skim mozzarella cheese\n\n\u00bc cup diced precooked chicken; 1 ounce smoked salmon, chopped; or three slices tomato\n\n_Chicken: Per serving: 254 calories, 18 g protein, 20 g carbohydrates, 12 g fat (4.5 g saturated), 2 g fiber, 393 mg sodium_\n\n_Salmon: Per serving: 254 calories, 17 g protein, 19 g carbohydrates, 13 g fat (5 g saturated), 2 g fiber, 827 mg sodium_\n\n_Tomato: Per serving: 231 calories, 12 g protein, 22 g carbohydrates, 11 g fat (4.5 g saturated), 3 g fiber, 263 mg sodium_\n\n### **The Red Auerbach (Powerfoods: 3)**\n\n1 small whole-wheat pita\n\n\u00bc cup marinara sauce\n\n1 green onion, sliced\n\n2 tablespoons part-skim mozzarella cheese\n\n3 frozen turkey meatballs; 2 turkey slices, chopped; or \u00bc cup chopped green pepper\n\n_Meatball: Per serving: 292 calories, 24 g protein, 27 g carbohydrates, 10 g fat (3 g saturated), 3.5 g fiber, 557 mg sodium_\n\n_Turkey: Per serving: 169 calories, 10 g protein, 21 g carbohydrates, 5 g fat (2 g saturated), 3 g fiber, 676 mg sodium_\n\n_Pepper: Per serving: 156 calories, 7 g protein, 22 g carbohydrates, 5 g fat (2 g saturated), 4 g fiber, 451 mg sodium_\n\n### **The Pancho Villa (Powerfoods: 2)**\n\n1 small whole-wheat pita\n\n\u00bc cup salsa, with a little of the liquid drained\n\n1 teaspoon diced cilantro\n\n2 tablespoons shredded reduced-fat Mexican-blend cheese\n\n\u00bc cup diced precooked chicken; \u00bc cup chopped green or red pepper; or \u00bc cup chopped avocado\n\n_Chicken: Per serving: 166 calories, 14 g protein, 20 g carbohydrates, 5 g fat (2 g saturated), 3 g fiber, 664 mg sodium_\n\n_Pepper: Per serving: 139 calories, 8 g protein, 21 g carbohydrates, 4 g fat (2 g saturated), 4 g fiber, 430 mg sodium_\n\n_Avocado: Per serving: 193 calories, 8 g protein, 23 g carbohydrates, 10 g fat (3 g saturated), 5 g fiber, 434 mg sodium_\nChapter 7\n\n# **START YOUR NIGHT RIGHT**\n\n# 30 Abs Diet Dinners\n\n**T YPICAL SCENARIO:** For the first 10 or 12 hours of your day, your life is somebody else's, whether you're working for a boss or catering to your kids. And just about the time you finally finish feeding everybody else's needs, your stomach starts sending you signals that it's tired of being ignored.\n\nWhen you work hard all day\u2014especially if you're too busy to eat right\u2014it makes sense that you'd want to reward yourself with a big plate of Whatever the Hell You Want, maybe washed down with a Manhattan and a couple glasses of wine. And dinners should be a celebration of sorts\u2014there's nothing wrong with rewarding yourself for forging through another demanding day and taking time to surround yourself with family, friends\u2014and food.\n\nBut you can have your reward without sacrificing your health or your waistline. How well you eat at dinner is in great part determined by what you ate earlier in the day. If you fuel your body throughout the day with four smart, sensible meals and snacks, you'll find you aren't ravenously hungry the minute you walk in the door or craving the half-a-pig special at O'Bloaty's Tavern. You can, instead, enjoy a hearty\u2014and healthy\u2014meal, either at your favorite restaurant or at home with one of these simple recipes. Instead of using dinner as an opportunity to get drunk on fat and sweets, use it as an opportunity to pack in as many of the POWER 12 as you can.\n\n## **Burgers**\n\nBurgers don't have to resemble deep-fried hockey pucks. Make yours with lean beef or turkey, grill or broil it just the way you want it, and enjoy a protein blast that will fire up your fat-burners and stimulate muscle growth. Most grocery stores now carry whole-wheat hamburger buns, so your classic \"junk-food\" dinner can morph into a perfect diet food\u2014without sacrificing taste.\n\n### **The Official Abs Diet Burger (Powerfoods: 5)**\n\n1 egg\n\n1 pound lean ground beef\n\n\u00bd cup oats\n\n1\u20443 cup diced onion\n\n\u00bd cup chopped spinach\n\n2 tablespoons reduced-fat shredded Mexican-blend cheese\n\nSalt and pepper\n\n_In a large bowl, whisk egg. Add everything else, mixing it\u2014your hands are the best tool\u2014until well blended. Form into four patties. Place burgers in a grill pan or nonstick skillet that's heated over medium-high. Cook 6 minutes per side or to desired level of doneness._\n\n_Serves 4 (Wrap any extra burgers in plastic and freeze them for later.)_\n\n_Per serving: 263 calories, 27 g protein, 8 g carbohydrates, 13 g fat (5 g saturated), 1 g fiber, 416 mg sodium_\n\n### **Alaskan Burger (Powerfoods: 4)**\n\n1 egg (use omega-3 eggs to up the good-fat count)\n\n1 can salmon, drained\n\n2 diced pieces of whole-wheat toast\n\n1 tablespoon ground flaxseed\n\nSalt and pepper\n\n_In a large bowl, break open egg and stir. Add everything else, mixing it with your hands until well blended. Form into four patties. Bake in oven that's preheated to 375\u00b0F for 20 minutes, turning once._\n\n_Serves 4_\n\n_Per serving: 230 calories, 23 g protein, 12 g carbohydrates, 10 g fat (2 g saturated), 2 g fiber, 590 mg sodium_\n\n### **Ciao Down Burger (Powerfoods: 2)**\n\n1 egg\n\n1 pound ground turkey breast\n\n2 crushed cloves garlic\n\n1\u20443 cup drained canned diced tomatoes\n\n1 teaspoon dried basil\n\n\u00bc teaspoon salt\n\n_In a large bowl, whisk egg. Add everything else, mixing it with your hands until well blended. Form into four patties. Place patties in a grill pan or nonstick skillet that's heated over medium-high. Cook 6 minutes per side or to desired level of doneness._\n\n_Serves 4_\n\n_Per serving: 147 calories, 30 g protein, 2 g carbohydrates, 3 g fat (.5 g saturated), .5 g fiber, 235 mg sodium_\n\n## **Chicken and Turkey**\n\nAsk a culinary adventurer to describe the flavor of any exotic dish, from frog's legs to turtle soup to medallions of rattlesnake, and chances are you'll hear it \"tastes like chicken.\"\n\nThat's not a lot of respect given to the humble chicken and its larger cousin, the turkey. But these two birds are powerful sources of lean protein and great go-to options whenever you're stuck for something to eat. Try these simple concoctions out for size, and the answer to the eternal question, \"What does it taste like?\" will always be, \"Tastes like I want more.\"\n\n### **Faux Fried Chicken (Powerfoods: 3)**\n\n1 cup of high-fiber bran flakes\n\n1 egg\n\nHot sauce\n\n2 boneless, skinless chicken breasts\n\n_Put cereal in a zip-top bag, seal, and pound the hell out of it. In a large bowl, whisk egg and desired amount of hot sauce together. Dip chicken breasts in egg mixture, then roll in crushed cereal. Bake in an oven preheated to 350\u00b0F for 20 minutes._\n\n_Serves 2_\n\n_Per serving: 211 calories, 31 g protein, 6 g carbohydrates, 6 g fat (2 g saturated), 1 g fiber, 243 mg sodium_\n\n### **The Dijon Lennon (Powerfoods: 3)**\n\n2 boneless, skinless turkey cutlets\n\n1 teaspoon olive oil\n\n1\u20443 cup reduced-fat low-sodium chicken broth\n\n1 tablespoon Dijon mustard\n\n2 sliced green onions\n\nSalt and pepper to taste\n\n_Pound turkey cutlets to an even thickness. Heat oil in nonstick skillet over medium heat. Brown each side of turkey cutlets, about 3 minutes each. Add broth, mustard, onions, salt, and pepper, stirring well. Reduce heat to low. Simmer for 10 to12 minutes._\n\n_Serves 2_\n\n_Per serving: 140 calories, 27 g protein, 2 g carbohydrates, 3 g fat (.4 g saturated), 0 g fiber, 364 mg sodium_\n\n### **Chicken a l'Orange (Powerfoods: 4)**\n\n1 teaspoon olive oil\n\n2 boneless, skinless chicken breasts\n\n1 tablespoon orange juice concentrate\n\n2 tablespoons low-sodium soy sauce\n\n\u00bc teaspoon powdered ginger\n\n1 sliced green onion\n\n1 tablespoon sliced almonds\n\n1 teaspoon diced cilantro\n\n_Heat oil in a nonstick skillet over medium heat. Add chicken and sear for 3 minutes per side. Add orange juice concentrate, soy sauce, and ginger, stirring to mix. Reduce heat and simmer for 5 to 6 minutes. Before serving, top with green onion, almonds, and cilantro._\n\n_Serves 2_\n\n_Per serving: 179 calories, 25 g protein, 7 g carbohydrates, 5 g fat (1 g saturated),_ < _1 g fiber, 572 mg sodium_\n\n## **Precooked Chicken**\n\nPrecooked chicken\u2014low in saturated fat and high in protein\u2014is ideal for busy people. You can eat it plain, chop it up and toss it onto a salad for a quick protein boost, or cook with it. Just beware, though\u2014in order to preserve the cuts for your ease of use, these chicken pieces are saltier than fresh, uncooked chicken. So add as little extra sodium as possible to your meal.\n\n### **Que Sera Quesadilla (Powerfoods: 4)**\n\n2 medium whole-wheat tortillas\n\n\u00bd cup reduced-fat grated Mexican-blend cheese, divided\n\n\u00bd cup diced precooked grilled chicken\n\n1 heaping tablespoon diced fresh cilantro\n\n1 sliced green onion\n\nSalsa\n\n_Place 1 tortilla in a nonstick pan that's preheated over medium-low heat, and top with \u00bd_ _of the cheese and the chicken, cilantro, and onion. Add remaining cheese and top tortilla, pressing down to flatten. Cook 3 minutes, then flip and cook 3 minutes more. Serve with salsa._\n\n_Serves 1_\n\n_Per serving: 425 calories, 42 g protein, 43 g carbohydrates, 15 g fat (7 g saturated), 4 g fiber, 800 mg sodium_\n\n### **Gonzo Chicken (Powerfoods: 8)**\n\n2 cups bagged mixed-green salad mix\n\n1 cup baby spinach leaves\n\n1\u20443 cup rinsed garbanzo beans\n\n\u00bd cup diced precooked chicken\n\n1 tablespoon chopped pecans\n\n1 sliced green onion\n\n3 slices avocado\n\n2 teaspoons olive oil\n\n1\u00bd tablespoons balsamic or red wine vinegar\n\nSalt and pepper to taste\n\n_Mix greens, beans, chicken, nuts, and onion together in a bowl. Top with avocado, oil, and vinegar._\n\n_Serves 1_\n\n_Per serving: 420 calories, 30 g protein, 27 g carbohydrates, 23 g fat (2 g saturated), 9 g fiber, 799 mg sodium_\n\n### **It Takes Stew, Baby (Powerfoods: 5)**\n\n2 cups roughly torn baby spinach leaves\n\n\u00bd cup diced red pepper\n\n1 crushed garlic clove\n\n1 teaspoon olive oil\n\n1 can (10.5 ounces) cannellini beans, drained and rinsed\n\n1 cup cubed precooked chicken\n\n\u00bd cup fat-free reduced-sodium chicken broth\n\nSalt and pepper\n\n_In a nonstick skillet heated over medium-low heat, saut\u00e9 spinach, peppers and garlic in oil 2 minutes, turning frequently. Add beans and chicken. Saut\u00e9 1 minute more. Add broth. Simmer for 10 minutes. Add salt and pepper to taste._\n\n_Serves 2_\n\n_Per serving: 311 calories, 27 g protein, 43 g carbohydrates, 4 g fat (.5 g saturated), 11 g fiber, 420.5 mg sodium_\n\n## **Steak**\n\nRed meat pulsates with amino acids, the cinder blocks of your body's architecture. In fact, steak is the best natural source of creatine, an enzyme that helps stimulate muscle growth. So unleash your inner carnivore, and you'll unleash your abs as well.\n\n### **Fruit of Your Loins (Powerfoods: 3)**\n\n2 beef tenderloin steaks\n\n1 teaspoon olive oil\n\n2 cloves garlic\n\n\u00bc teaspoon pepper\n\n\u00bd cup raspberries or blackberries\n\n1\u20443 cup red wine\n\nSalt\n\n_Place steaks in a skillet heated over medium-high heat. Cook 2 minutes per side. Remove from skillet. Add oil, saut\u00e9 garlic for 30 seconds, then add pepper, berries, and wine. Return steaks to pan and cook 4 to 6 minutes more (or until desired level of doneness). Add salt to taste_.\n\n_Serves 2_\n\n_Per serving: 341 calories, 19 g protein, 5 g carbohydrates, 24 g fat (9 g saturated), 0 g fiber, 130 mg sodium_\n\n### **Sergeant Pepper (Powerfoods: 4)**\n\n6 ounces flank steak (about half of one)\n\n\u00bd green or red pepper, cut lengthwise into strips\n\n1\u20443 cup cashew pieces\n\n2 sliced green onions\n\n3 tablespoons reduced-sodium soy sauce\n\nHot sauce, to taste\n\n1 teaspoon sugar\n\n_Cut meat diagonally and across the grain into thin strips (freezing it for 20 minutes first helps a lot). Place in large zip-top plastic bag with all other ingredients. Shake well to combine. Place into a skillet that's preheated over medium-high heat. Cook for 5 to 6 minutes or until meat reaches desired doneness, turning frequently._\n\n_Serves 2_\n\n_Per serving: 363 calories, 29 g protein, 14 g carbohydrates, 22 g fat (7 g saturated), 1 g fiber, 873 mg sodium_\n\n### **Steak Fa-heat-as (Powerfoods: 4)**\n\n6 ounces flank steak (the other half of the one called for above)\n\n1 small onion, cut into eighths\n\n1 green or red pepper, cut lengthwise into strips\n\n1 small jalape\u00f1o pepper, cut into rings\n\n1 teaspoon olive oil\n\n1 tablespoon diced cilantro\n\n1\u20448 teaspoon cinnamon\n\n\u00bc teaspoon cumin\n\nSalt and pepper to taste\n\n_Cut meat diagonally and across the grain into thin strips. Place in large zip-top plastic bag with all other ingredients. Shake well to combine. Place into a skillet that's preheated over medium-high heat. Cook for 5 to 6 minutes, or until meat reaches desired doneness, turning frequently. Serve with four whole-wheat tortillas and salsa_.\n\n_Serves 2_\n\n_Per serving: 310 calories, 24 g protein, 46 g carbohydrates, 8 g fat (2.5 g saturated), 5 g fiber, 458 mg sodium_\n\n### **Mighty Muffins (Powerfoods: 3)**\n\n1 egg\n\n1 pound lean ground beef\n\n2 tablespoons balsamic vinegar\n\n\u00bd cup oats\n\n\u00bc cup minced onion\n\nSalt and pepper to taste\n\n_In a large bowl, whisk egg. Add everything else, mixing it with your hand until well blended. Divide mixture evenly into a 6-cup nonstick muffin pan. Preheat the oven to 375\u00b0F and bake for 25 minutes_.\n\n_Serves 3 (Wrap extras in plastic and freeze them for later.)_\n\n_Per serving: 349 calories, 35 g protein, 13 g carbohydrates, 16 g fat (6 g saturated), 1.5 g fiber, 329 mg sodium_\n\n### **Aztec Casserole (Powerfoods: 3)**\n\n6 ounces lean ground beef\n\n1\u20443 cup diced onion\n\n1 crushed clove garlic\n\n\u00bc teaspoon cumin\n\n1\u00be cup cooked brown rice\n\n\u00be cup salsa\n\n1 tablespoon diced cilantro\n\n2 tablespoons reduced-fat shredded Mexican blend cheese\n\n_Brown beef in a large nonstick skillet preheated over medium heat (about 3 to 4 minutes.) Add onion and garlic and saut\u00e9 3 to 5 minutes or until soft. Drain fat. Add cumin, rice, salsa, and cilantro, stirring to mix well. Reduce heat to low and simmer for 6 to 8 minutes. Top each serving with a tablespoon of cheese._\n\n_Serves 2_\n\n_Per serving: 397 calories, 25 g protein, 50 g carbohydrates, 11 g fat (4 g saturated), 5 g fiber, 537 mg sodium_\n\n## **One-Pot Dishes**\n\nMost of us don't mind the cooking so much as we mind the cleaning, which is where these recipes come in. Everything gets cooked in one pot, which leaves you more time for _nip\/tuck_ reruns.\n\n### **Three Amigos Chili (Powerfoods: 5)**\n\n1 tablespoon olive oil\n\n1 small onion, diced\n\n1 pound ground turkey breast\n\n1 can diced tomatoes with jalape\u00f1os\n\n1 can (10.5 ounces) each garbanzos, black beans, and kidney beans, drained\n\n1 can (14 ounces) low-sodium chicken broth\n\n\u00bc teaspoon each salt and cumin\n\n1\u20448 teaspoon cinnamon\n\nHot sauce to taste\n\n_Heat oil on medium-low. Add onion and saut\u00e9 until soft (about 3 to 5 minutes). Add turkey and brown (about 5 minutes.) Add tomatoes with juice, beans, broth, and spices. Stir and bring to a boil, then reduce heat and simmer 20 minutes._\n\n_Serves 6 (Freeze the leftovers and save $5 by taking them for lunch.)_\n\n_Per serving: 293 calories, 31 g protein, 32 g carbohydrates, 5 g fat (0 g saturated), 11 g fiber, 788 mg sodium_\n\n### **Hot-Headed Chicken (Powerfoods: 5)**\n\n1 teaspoon olive oil\n\n\u00bd cup diced onion\n\n1\u20443 cup diced red pepper\n\n1 egg\n\n1 tablespoon reduced-sodium soy sauce\n\nHot sauce to taste\n\n1\u00be cup cooked brown rice\n\n1\u00bd cups diced precooked chicken\n\n_Heat oil in nonstick skillet over medium heat. Add onion and red pepper. Saut\u00e9 for 3 to 5 minutes, until onion softens. Add egg, stirring frequently. Cook for 2 to 4 minutes more until egg scrambles. Add soy sauce, hot sauce, rice, and chicken. Stir and cook about 3 minutes more, until dish is well blended and evenly heated._\n\n_Serves 2_\n\n_Per serving: 413 calories, 33 g protein, 49 g carbohydrates, 9 g fat (1.4 g saturated), 5 g fiber, 846 mg sodium_\n\n### **Pot Luck of the Irish (Powerfoods: 3)**\n\n2 small pork tenderloins\n\n\u00bc teaspoon olive oil\n\n\u00bd cup diced onion\n\n\u00bd cup red pepper\n\n1 cup packaged shredded cabbage\n\n1 small chopped tomato\n\n1\u20443 cup low-fat, reduced-sodium chicken broth\n\n1\u20448 teaspoon paprika\n\nSalt and pepper to taste\n\n_In a nonstick skillet, brown pork 2 to 3 minutes per side over medium-high heat. Remove pork. Add oil, onion, and red pepper, saut\u00e9ing for 3 to 5 minutes or until onion softens. Reduce heat to medium. Stir in cabbage, tomato, broth, and spices. Add chops to skillet. Cook 10 minutes more, stirring occasionally._\n\n_Serves 2_\n\n_Per serving: 272 calories, 23 g protein, 10 g carbohydrates, 16 g fat (5 g saturated), 2 g fiber, 60 mg sodium_\n\n## **Seafood**\n\nTo quote the culinarily confused sharks from _Finding Nemo_ , fish are our friends. Besides delivering plenty of lean protein, most fish are packed with the omega-3 fatty acids that help control cholesterol and your appetite at the same time. To be as sleek and energetic as a dolphin, try these no-hassle recipes.\n\n### **No-Scrimp Shrimp (Powerfoods: 3)**\n\n24 large frozen shrimp, peeled and deveined\n\n\u00bd cup chopped baby spinach leaves\n\n1 teaspoon olive oil\n\n1 crushed garlic clove\n\n\u00bd teaspoon dried basil\n\nHot sauce to taste\n\n_Mix all ingredients in a large microwave-safe bowl, tossing well to coat shrimp. Microwave 1 minute. Remove and toss again. Microwave 1\u00bc minutes more._\n\n_Serves 2_\n\n_Per serving: 91 calories, 14 g protein, 1 g carbohydrates, 3 g fat (.5 g saturated), 0 g fiber, 165 mg sodium_\n\n### **The Aqua Man (Powerfoods: 3)**\n\n\u00bd cup trimmed asparagus\n\n\u00bd cup matchstick carrots\n\n1 teaspoon olive oil\n\nJuice of 1 lemon\n\n\u00bd teaspoon lemon rind\n\n1 crushed clove garlic\n\nSalt and pepper to taste\n\n2 tilapia fillets\n\n_In a small bowl, mix vegetables with oil, lemon juice, rind, garlic, and salt and pepper. Arrange fish in a small, shallow microwave-safe baking dish. Pour vegetable mixture over each fillet. Wrap dish tightly in plastic wrap, pricking a couple of times with a fork or toothpick. Microwave for 3 to 4 minutes or until fish flakes lightly with a fork._\n\n_Serves 2_\n\n_Per serving: 136 calories, 22 g protein, 6 g carbohydrates, 3 g fat (1 g saturated), 1 g fiber, 220 mg sodium_\n\n### **Fish Tacos (Powerfoods: 3)**\n\n2 frozen grilled fish fillets, Cajun- or blackened-style\n\n4 small corn tortillas\n\n1 cup chopped baby spinach\n\n4 tablespoons reduced-fat grated Mexican-blend cheese\n\nSalsa\n\n_Microwave fish according to package instructions. Slice fillets into strips, then divide evenly among four tortillas. Top with spinach, 1 tablespoon of cheese, and salsa to taste._\n\n_Serves 2_\n\n_Per serving: 270 calories, 25 g protein, 29 g carbohydrates, 7 g fat (2 g saturated), 4 g fiber, 674 mg sodium_\n\n### **Hot Pink (Powerfoods: 3)**\n\n2 3-ounce cuts of fresh salmon\n\n1\u00bd tablespoons reduced-sodium soy sauce\n\n1 teaspoon olive oil\n\nHot sauce to taste\n\n\u00bc teaspoon powdered ginger\n\n1 sliced green onion\n\n1 teaspoon chopped cilantro\n\n_On a foil-lined pan, place fish on top oven rack under preheated broiler. Broil 4 to 5 minutes or until fish flakes with a fork. Mix soy sauce, olive oil, hot sauce, ginger, onion, and cilantro in a small microwave-safe bowl. Microwave on high for 5 minutes, stirring once. Remove fish from oven. Transfer to plate, then pour half of the soy sauce mixture over each piece of fish._\n\n_Serves 2_\n\n_Per serving: 203 calories, 20 g protein, 1 g carbohydrates, 13 g fat (2.5 g saturated), 0 g fiber, 460 mg sodium_\n\n### **The Perfect Storm (Powerfoods: 3)**\n\n2 tilapia fillets\n\n2 tablespoons mustard\n\n1 egg\n\n\u00bd cup finely chopped pecans\n\nHoney\n\n_Spread each fillet with about 1 tablespoon of mustard, then dip in beaten egg. Roll in chopped nuts. Bake in an oven preheated to 350\u00b0F for 10 to 12 minutes or until fish flakes. When done, drizzle each fillet lightly with honey._\n\n_Serves 2_\n\n_Per serving: 371 calories, 30 g protein, 13 g carbohydrates, 25 g fat (3 g saturated), 3 g fiber, 513 mg sodium_\n\n## **Pasta**\n\nLegend has it that Martin Scorsese took Robert De Niro to Little Italy and directed him to eat pasta in order to balloon up for the second act of _Raging Bull_. But that doesn't mean you can't eat noodles and still look fighting trim. Whenever possible, opt for the whole-wheat versions; they'll give you gut-filling fiber and help take out a hit on your cholesterol. But with all pasta dishes, the key is to avoid fatty sauces and pile your plate high with Powerfoods.\n\n### **The You-Can Noodle (Powerfoods: 3)**\n\n10 or 12 frozen turkey meatballs\n\n2 cups diced tomatoes with sauce\n\n\u00bd teaspoon dried basil\n\n1 crushed clove garlic\n\n4 ounces whole-wheat spaghetti\n\n2 tablespoons shredded part-skim mozzarella cheese\n\n_Defrost meatballs according to package directions. Toss with tomatoes, basil, and garlic. Microwave 2 minutes,_ _stirring once. Toss with cooked pasta and top with cheese._\n\n_Serves 2_\n\n_Per serving: 410 calories, 39 g protein, 39 g carbohydrates, 11 g fat (4 g saturated), 8 g fiber, 557 mg sodium_\n\n### **\"Alfredo, I Know It Was You . . . \"(Powerfoods: 6)**\n\n1 teaspoon olive oil\n\n\u00bd cup part-skim ricotta cheese\n\n\u00bc cup 1% milk\n\n1 packed cup canned salmon, drained\n\n4 ounces whole-wheat spaghetti\n\n2 tablespoons shredded part-skim mozzarella cheese\n\nSalt and pepper\n\n_In a saut\u00e9 pan or skillet, heat oil and garlic over low-medium heat for 1 minute. Add ricotta and milk, stir, then add salmon and simmer 5 to 6 minutes. Thin with additional milk if needed. Pour over cooked pasta and top with cheese. Add salt and pepper to taste._\n\n_Serves 2_\n\n_Per serving: 311 calories, 27 g protein, 20 g carbohydrates, 14 g fat (6 g saturated), 3 g fiber, 543 mg sodium_\n\n### **The Pesto R\u00e9sistance (Powerfoods: 5)**\n\n1 tablespoon olive oil\n\n\u00bd cup walnut pieces\n\n1 crushed clove garlic\n\n2 cups torn baby spinach leaves\n\n1 teaspoon dried basil\n\nSalt and pepper to taste\n\n4 ounces whole-wheat spaghetti\n\n2 tablespoons shredded part-skim mozzarella cheese\n\n_Heat oil in nonstick skillet over medium-low heat. Add nuts and toast 3 to 4 minutes, stirring frequently. Add garlic, spinach, basil, salt, and pepper. Cook 3 to 5 minutes more, turning frequently. Toss with cooked pasta and top with cheese._\n\n_Serves 2_\n\n_Per serving: 335 calories, 9 g protein, 17 g carbohydrates, 28 g fat (4 g saturated), 5 g fiber, 160 mg sodium_\n\n## **Side Dishes**\n\nNo matter what your main course, it's always smart to build your side dishes from the ABS DIET POWER 12\u2014whether it's brown rice, spinach, salad, or beans. But you have many other options, as well.\n\n### **The Breathalyzer (Powerfoods: 2)**\n\n1 teaspoon olive oil\n\n1 crushed clove garlic\n\n3 cups roughly torn baby spinach leaves\n\n_Heat olive oil in a skillet on medium-high. Add garlic and saut\u00e9 2 minutes. Add spinach, saut\u00e9ing 3 to 5 minutes until all leaves are wilted, turning frequently with tongs._\n\n_Serves 2_\n\n_Per serving: 37 calories, 1 g protein, 4 g carbohydrates, 2 g fat (0 g saturated), 2 g fiber, 136 mg sodium_\n\n### **The Green Party (Powerfoods: 2)**\n\n2 cups fresh green beans trimmed into inch-long pieces\n\n1 heaping tablespoon sliced almonds\n\n\u00bc teaspoon lemon rind\n\nSalt to taste\n\nJuice of \u00bd lemon\n\n_Arrange beans, almonds, rind, and salt (in that order) in steamer. Squeeze juice over the top. Steam to desired texture (3 to 5 minutes for firm beans). Add more salt to taste._\n\n_Serves 2_\n\n_Per serving: 54 calories, 2 g protein, 8 g carbohydrates, 1.5 g fat (0 g saturated), 4 g fiber, 290 mg sodium_\n\n### **Nuclear Orange Spud Missiles (Powerfoods: 1)**\n\n2 medium sweet potatoes\n\n2 tablespoons finely chopped pecans\n\n2 tablespoons raisins\n\n2 teaspoons whipped butter\n\n_Pierce potatoes with a fork. Microwave on high for 6 to 8 minutes, turning once. Cut them open and top each with 1 tablespoon pecans, 1 tablespoon raisins, and 1 teaspoon butter._\n\n_Serves 2_\n\n_Per serving: 192 calories, 3 g protein, 32 g carbohydrates, 7 g fat (1 g saturated), 5 g fiber, 57 mg sodium_\n\n### **El El Bean (Powerfoods: 3)**\n\n\u00bd cup canned black beans, drained and rinsed\n\n1 cup canned corn kernels, drained and rinsed\n\n1 sliced green onion\n\n1 teaspoon diced cilantro\n\n1 teaspoon olive oil\n\n\u00bc teaspoon red pepper flakes\n\nSalt and pepper to taste\n\n_Mix ingredients together in a bowl._\n\n_Serves 2_\n\n_Per serving: 150 calories, 5 g protein, 24 g carbohydrates, 4 g fat (0 g saturated), 6 g fiber, 550 mg sodium_\n\n### **Jerry's Rice (Powerfoods: 2)**\n\n1 packet quick-cooking brown rice\n\nReduced-fat, low-sodium chicken broth\n\n1 cup broccoli florets, cut uniformly to thumb tip\u2013sized pieces\n\n_Follow packaged rice directions, substituting chicken broth for the recommended amount of water and adding broccoli after broth has been brought to a boil._\n\n_Serves 4_\n\n_Per serving: 136 calories, 4 g protein, 27 g carbohydrates, 1 g fat (0 g saturated), 2 g fiber, 80 mg sodium_\nChapter 8\n\n# **SHAKE THINGS UP**\n\n# 27 Abs Diet Smoothies and Snacks\n\n**W E LIKE TO DRINK** \u2014whether it's beer, Gatorade, or shots off the bartender's belly. But all drinks are not made alike\u2014and some drinks can really sabotage your diet (or your relationship, in the case of the belly shots).\n\nWhile some drinks, like presweetened iced teas, sports drinks, and Bud, are laden with empty calories, the right drinks can jump-start your metabolism and top off your tank with a healthy dose of Powerfoods. If you've got 3 minutes to spare, you can mix up a Powerfood smoothie. Dump the ingredients in a blender, push the button, and whip up a frenzy of belly-busting nutrition.\n\nThe best thing about smoothies is their versatility. You can down a smoothie at breakfast, use it as a meal replacement at lunch, make it your late-afternoon snack to take the edge off before dinner, or have it at night as your dessert. For all recipes, first blend together any liquid ingredients (milk, yogurt, juice, etc.) and protein powder; this will help break down the grainy powder and make sure it's evenly distributed. Next, add mushy ingredients, like precooked oatmeal and fruit, then add ice at the end. For a thicker shake, you can toss in more ice cubes; you'll add volume without the calories.\n\n### **Check Your Blackberry (Powerfoods: 5)**\n\n\u00be cup instant oatmeal nuked in water\n\n1\u20443 cup blackberries\n\n2 tablespoons low-fat plain yogurt\n\n2 teaspoons vanilla whey protein powder\n\n1 teaspoon ground flaxseed\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 149 calories, 8 g protein, 23 g carbohydrates, 1 g fat (0 g saturated), 3 g fiber, 17 mg sodium_\n\n### **Choco-nana (Powerfoods: 4)**\n\n1 cup 1% chocolate milk\n\n1 banana\n\n2 tablespoons low-fat vanilla yogurt\n\n1 tablespoon chopped walnuts\n\n2 teaspoons chocolate whey protein powder\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 185 calories, 9 g protein, 31 g carbohydrates, 4 g fat (1 g saturated), 2 g fiber, 106 mg sodium_\n\n### **The Orangeman (Powerfoods: 3)**\n\n1 cup 1% milk\n\n\u00bd cup frozen orange juice concentrate\n\n2 tablespoons low-fat plain yogurt\n\n1 banana\n\n2 teaspoons whey protein powder\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 241 calories, 10 g protein, 48 g carbohydrates, 2 g fat (1 g saturated), 2 g fiber, 84 mg sodium_\n\n### **Tirami-Smooth (Powerfoods: 5)**\n\n\u00be cup part-skim ricotta cheese\n\n2 tablespoons low-fat plain yogurt\n\n1 tablespoon slivered almonds\n\n2 teaspoons chocolate whey protein powder\n\n2 teaspoons ground flaxseed\n\n\u00bd teaspoon finely ground coffee\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 207 calories, 15 g protein, 15 g carbohydrates, 10 g fat (5 g saturated), 2 g fiber, 134 mg sodium_\n\n### **The Endless Summer (Powerfoods: 4)**\n\n\u00bc cup 1% milk\n\n\u00be cup ready-to-eat cubed seedless watermelon pieces\n\n\u00bd cup strawberries\n\n\u00bd cup low-fat plain yogurt\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 92 calories, 7 g protein, 13 g carbohydrates, 2 g fat (1 g saturated), 1 g fiber, 66 mg sodium_\n\n### **Punk'd Pie (Powerfoods: 5)**\n\n\u00bd cup canned pumpkin\n\n\u00be cup instant oatmeal nuked in water\n\n\u00bc cup chopped pecans\n\n2 tablespoons low-fat vanilla yogurt\n\n2 teaspoons vanilla whey protein powder\n\n1 teaspoon ground flaxseed\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 270 calories, 9 g protein, 29 g carbohydrates, 11 g fat (1 g saturated), 6 g fiber, 19 mg sodium_\n\n### **Lime Dancing (Powerfoods: 3)**\n\n\u00bd cup frozen lime juice concentrate\n\n1 cup 1% milk\n\n2 tablespoons low-fat vanilla yogurt\n\n1 banana\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 274 calories, 8 g protein, 58 g carbohydrates, 2 g fat (1 g saturated), 2 g fiber, 88 mg sodium_\n\n### **Honey-Pecan Smoothie (Powerfoods: 5)**\n\n\u00bd cup 1% milk\n\n\u00bd cup low-fat vanilla yogurt\n\n\u00bc cup chopped pecans\n\n2 teaspoons whey protein powder\n\n1 teaspoon honey\n\n2 teaspoons ground flaxseed\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 224 calories, 11 g protein, 18 g carbohydrates, 13 g fat (2 g saturated), 2 g fiber, 78 mg sodium_\n\n## **PACK SNACKS**\n\nFor snacks, you can eat leftovers, smoothies, or smaller portions of the ABS DIET POWER 12. Make sure each snack contains one or two Powerfoods, one of which must be protein. (Note: Dairy options count as protein, too.)\n\n**Protein options**\n\n2 teaspoons peanut butter\n\n1 ounce almonds, pecans, walnuts, or peanuts\n\n3 slices low-sodium deli cold cuts\n\n\u00bd cup shelled edamame\n\n**Dairy options**\n\n8 ounces low-fat yogurt\n\n8 ounces 1% milk or chocolate milk\n\n1\u00bd slices low-fat cheese\n\n1 stick string cheese\n\nLow-fat, no-salt-added cottage cheese\n\nLow-fat yogurt smoothie\n\n**Fruit or vegetable options**\n\n1 ounce raisins\n\nRaw vegetables (celery, baby carrots, broccoli) in unlimited quantity\n\n1\u00bd cups berries\n\n4 ounces cantaloupe\n\n1 large orange\n\n**Whole-grain options**\n\n1 or 2 slices whole-grain bread\n\n1 bowl oatmeal or high-fiber cereal\n\n3 whole-wheat crackers\n\n3 cups fat-free popcorn\n\n1 granola bar\n\n**Dessert options**\n\nAs long as you're pairing them with Powerfoods (like a glass of milk), these indulgences will add a taste of decadence to a healthy snack.\n\n1 chocolate pudding cup\n\n3 mini York Peppermint Patties\n\n3 bite-size Snickers\n\n\u00bd cup reduced-fat ice cream\n\n**Desk-drawer snacks**\n\nThese complete snacks balance complex carbs and protein for a oneand-done fast snack.\n\n* Clif Bar (5 g fiber, 10 g protein)\n\n* Erin Baker's breakfast cookies (6 g fiber, 7 g protein)\n\n* Nile Spice Dried bean soups (11 g fiber, 11 g protein)\n\n### **Mango Tango (Powerfoods: 4)**\n\n\u00bd cup ready-to-eat cubed mango\n\n1\u20443 cup blueberries\n\n\u00bd banana\n\n\u00bd cup 1% milk\n\n\u00bd cup low-fat vanilla yogurt\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 158 calories, 8 g protein, 29 g carbohydrates, 2 g fat (1 g saturated), 2 g fiber, 80 mg sodium_\n\n### **Honey-Nut Cheery Oat (Powerfoods: 5)**\n\n\u00be cup instant oatmeal nuked in water\n\n\u00bc cup 1% milk\n\n1 tablespoon peanut butter\n\n2 teaspoons whey protein powder\n\n1 teaspoon honey\n\n1 teaspoon ground flaxseed\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 206 calories, 11 g protein, 26 g carbohydrates, 6 g fat (1 g saturated), 4 g fiber, 51 mg sodium_\n\n### **Blue Velvet (Powerfoods: 5)**\n\n\u00bd cup 1% chocolate milk\n\n\u00bd cup low-fat vanilla yogurt\n\n\u00bd cup blueberries\n\n2 teaspoons chocolate whey protein powder\n\n2 teaspoons ground flaxseed\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 140 calories, 8 g protein, 22 g carbohydrates, 3 g fat (1 g saturated), 2 g fiber, 92 mg sodium_\n\n### **The Peachy Keen (Powerfoods: 4)**\n\n1 cup 1% milk\n\n2 tablespoons low-fat vanilla yogurt\n\n\u00bd cup frozen peaches\n\n\u00bd cup strawberries\n\n1\u20448 teaspoon powdered ginger\n\n2 teaspoons whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 152 calories, 9 g protein, 27 g carbohydrates, 2 g fat (1 g saturated), 2 g fiber, 83 mg sodium_\n\n### **Cheesecake in a Cup (Powerfoods: 4)**\n\n\u00be cup part-skim ricotta cheese\n\n\u00bc cup 1% milk\n\n\u00bd cup blueberries\n\n\u00bd banana\n\n2 teaspoons vanilla whey protein powder\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 200 calories, 15 g protein, 19 g carbohydrates, 8 g fat (5 g saturated), 2 g fiber, 139 mg sodium_\n\n### **Juicy Fruit Juice (Powerfoods: 4)**\n\n1 banana\n\n\u00bd cup strawberries\n\n\u00bd cup 1% milk\n\n2 tablespoons low-fat vanilla yogurt\n\n1\u20443 cup orange juice\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 137 calories, 6 g protein, 27 g carbohydrates, 1 g fat (.5 g saturated), 3 g fiber, 49 mg sodium_\n\n### **Chocolate Factory (Powerfoods: 5)**\n\n1 cup 1% chocolate milk\n\n2 tablespoons low-fat plain yogurt\n\n2 tablespoons peanut butter\n\n2 teaspoons chocolate whey protein powder\n\n2 teaspoons ground flaxseed\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 219 calories, 11 g protein, 20 g carbohydrates, 11 g fat (2 g saturated), 2 g fiber, 167 mg sodium_\n\n### **The Chocolate Latte (Powerfoods: 4)**\n\n\u00bd cup 1% chocolate milk\n\n\u00bd cup low-fat plain yogurt\n\n\u00bd banana\n\n1 tablespoon peanut butter\n\n\u00bd teaspoon finely ground coffee\n\n2 teaspoons chocolate whey protein powder\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 171 calories, 10 g protein, 20 g carbohydrates, 6 g fat (2 g saturated), 2 g fiber, 116 mg sodium_\n\n### **Abs Diet Trail Mix (Powerfoods: 4)**\n\nA ready-made power snack is an effective one: It gives you something to reach for when you're hungry, and it gives you good ingredients without guilt. Make this trail mix on the weekend, then keep stashes on hand at work and at home.\n\n1 cup whole almonds\n\n1 cup pecan halves\n\n1\u20443 cup plain oats\n\nCooking spray\n\n2 tablespoons honey\n\n\u00bd teaspoon cinnamon\n\n1 cup orange-flavored dried cranberries (Craisins)\n\n1 large packet dried apricots\n\n_Preheat oven to 350\u00b0F. Place nuts and oats in a bowl and spray evenly with cooking spray (about 3 shots). Drizzle with 1 tablespoon honey and add cinnamon, stirring to coat. Spread nuts evenly on a pan and toast for 15 minutes, stirring occasionally. Once they've cooled, stir in the second tablespoon of honey and mix in the dried fruit. Divide into \u00bc cup servings and place each into a zip-top bag._\n\n_Makes 9 to 10 servings_\n\n_Per serving: 170 calories, 3 g protein, 18 g carbohydrates, 9 g fat (0 g saturated), 3 g fiber, 0 mg sodium_\n\n### **The Cinnamon Girl (Powerfoods: 3)**\n\n\u00be cup instant oatmeal nuked in water\n\n2 tablespoons low-fat vanilla yogurt\n\n1 teaspoon honey\n\n1 teaspoon ground flaxseed\n\n1\u20448 teaspoon cinnamon\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 142 calories, 5 g protein, 25 g carbohydrates, 1 g fat (0 g saturated), 3 g fiber, 11 mg sodium_\n\n### **Peach Vacation (Powerfoods: 3)**\n\n\u00be cup 1% milk\n\n\u00be cup frozen or canned peaches (drain the syrup)\n\n2 tablespoons low-fat plain yogurt\n\n2 teaspoons vanilla whey protein powder\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 150 calories, 7 g protein, 29 g carbohydrates, 1.5 g fat (1 g saturated), 2 g fiber, 71 mg sodium_\n\n### **Mint Chocolate Morning (Powerfoods: 3)**\n\n\u00be cup 1% chocolate milk\n\n\u00bd cup low-fat vanilla yogurt\n\n2 teaspoons chocolate whey protein powder\n\n1 frozen Peppermint Pattie\n\n2 teaspoons ground flaxseed\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 168 calories, 9 g protein, 26 g carbohydrates, 3 g fat (1.5 g saturated), 1 g fiber, 116 mg sodium_\n\n### **The Hawaiian Five-O (Powerfoods: 4)**\n\n\u00bd cup 1% milk\n\n2 tablespoons low-fat plain yogurt\n\n\u00bc cup frozen orange juice concentrate\n\n\u00bd banana\n\n\u00bc cup strawberries\n\n\u00bc cup cubed ripe mango\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 154 calories, 7 g protein, 31 g carbohydrates, 1 g fat (.5 g saturated), 2 g fiber, 50 mg sodium_\n\n### **The New Zealander (Powerfoods: 4)**\n\n1 cup 1% milk\n\n2 tablespoons low-fat plain yogurt\n\n1 medium peeled kiwifruit\n\n\u00bd cup strawberries\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 110 calories, 8 g protein, 16 g carbohydrates, 2 g fat (1 g saturated), 2 g fiber, 83 mg sodium_\n\n### **The Almond Hammer (Powerfoods: 5)**\n\n\u00bd cup 1% milk\n\n\u00bd cup low-fat vanilla yogurt\n\n\u00bc cup sliced almonds\n\n2 teaspoons chocolate whey protein powder\n\n1 teaspoon honey\n\n1 teaspoon ground flaxseed\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 179 calories, 10 g protein, 18 g carbohydrates, 8 g fat (1.5 g saturated), 2 g fiber, 79 mg sodium_\n\n### **The Nutty Professor (Powerfoods: 5)**\n\n1 cup 1% chocolate milk\n\n2 tablespoons low-fat vanilla yogurt\n\n1 tablespoon frozen orange juice concentrate\n\n\u00bd banana\n\n1 tablespoon sliced almonds\n\n2 teaspoons chocolate whey protein powder\n\n2 teaspoons ground flaxseed\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 181 calories, 9 g protein, 29 g carbohydrates, 4 g fat (1 g saturated), 2 g fiber, 107 mg sodium_\n\n### **The Neapolitan (Powerfoods: 5)**\n\n\u00be cup 1% chocolate milk\n\n\u00bd cup low-fat vanilla yogurt\n\n\u00be cup sliced strawberries\n\n1 teaspoon ground flaxseed\n\n2 teaspoons vanilla whey protein powder\n\n3 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 154 calories, 9 g protein, 25 g carbohydrates, 2 g fat (1 g saturated), 2 g fiber, 114 mg sodium_\n\n### **The Whey-Too-Good Smoothie (Powerfoods: 6)**\n\n\u00be cup part-skim ricotta cheese\n\n\u00be cup 1% chocolate milk\n\n\u00bc cup chopped pecans\n\n\u00bd banana\n\n2 tablespoons low-fat vanilla yogurt\n\n2 teaspoons ground flaxseed\n\n2 teaspoons chocolate whey protein powder\n\n6 ice cubes\n\n_Makes 2 8-ounce servings_\n\n_Per serving: 340 calories, 16 g protein, 30 g carbohydrates, 18 g fat (5 g saturated), 3 g fiber, 184 mg sodium_\n\n### **Abs Diet Sundae Parfait (Powerfoods: 3)**\n\nYep, it's still a diet. The ice cream gives you calcium and protein with only a modest amount of fat and refined sugar. Indulge and enjoy!\n\n\u00bd cup reduced-fat chocolate or vanilla ice cream\n\n\u00bc cup berries of your choice, slightly crushed\n\n1 tablespoon chopped nuts of your choice\n\n_Starting with the ice cream, layer the ingredients in a small bowl_.\n\n_Makes 1 serving_\n\n_Per serving: 162 calories, 4.5 g protein, 25 g carbohydrates, 5 g fat (2 g saturated), 1 g fiber, 56 mg sodium_\n\nChapter 9\n\n# **EATING OUT, EATING RIGHT**\n\n# The Abs Diet Restaurant Survival Guide\n\nEating out in a restaurant can be exciting, tempting, arousing . . . dangerous. Here's this seductive menu in front of you\u2014pictures and descriptions of creamy pasta, juicy burgers, cheese-drenched potato skins. They're looking at you. They're daring you. They're whispering in your ear with soft, slow, raspy voices: \"Order meeeeeee.\"\n\nSo you\u2014a person of dignity and restraint, a person with serious goals and a desire to change your body\u2014now have to decide. Do you give in to temptation? You know what's right\u2014what's good for you\u2014but the temptations are crying out to you: the smells, the specials, the free bread, the \"save any room for Chocolate Death?\" sales pitches. Oh, just this once won't hurt. . . .\n\nNo, it won't. But you're going to find yourself in restaurants again and again, still facing the same choices, still being lured by the same temptations. What do you do? It's simple: Look for the Powerfoods. And pull out your copy of _The Abs Diet Eat Right Every Time Guide._\n\nSee, you don't have to sacrifice flavor to eat healthy, and you don't have to deny yourself a night out on the town. You just have to know which menu options deliver the most nutrition with the least number of empty calories and bad-for-you ingredients. To help you through the decision-making process, I've compiled a tireless compendium of nearly every major chain restaurant in the country, with the best and the worst choices each of them has to offer.\n\n## **Fast-Food Joints**\n\nYou've probably already figured out that the grilled-chicken sandwich is the default option when you don't know what else to order. But even the greasiest of junk food Valhallas offers more healthy choices than you might suspect. And even if nothing on the menu qualifies as a health food, you can still cut down on empty calories and saturated fat by ordering smartly. The goal is to stick to the plan, not bore yourself off it. One fast-food burger now and then won't kill you. Just beware of special sauces, dips, and salad dressings. They're often loaded with calories and fat. Stick to ketchup, mustard, or barbecue sauce. And when given the choice, a second burger is usually better than the side of fries.\n\n### **Arby's**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Hot Ham & Swiss Melt_ 270 calories, 8 g fat (3.5 g saturated), 1140 mg sodium\n\n_Grilled Chicken Deluxe_ 380 calories, 12 g fat (2 g saturated), 920 mg sodium\n\n_Arby-Q sandwich_ 360 calories, 11 g fat (4.5 g saturated), 1210 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Arby baked potato with sour cream_ 320 calories, 12 g fat (7 g saturated), 45 mg sodium\n\n**Not that:** _Medium fries_ 380 calories, 16 g fat (2.5 g saturated), 690 mg sodium\n\n**Eat that:** _Martha's Vineyard Salad with almonds and raspberry vinaigrette_ 501 calories, 27 g fat (6 g saturated), 830 mg sodium\n\n**Not that:** _Market Fresh Roast Beef & Swiss_ 780 calories, 39 g fat (12 g saturated), 1740 mg sodium\n\n### **Baja Fresh**\n\n**ABS DIET ENDORSEMENTS:**\n\n_2 Baja-Style Charbroiled Chicken Tacos_ 360 calories, 8 g fat (1 g saturated), 460 mg sodium\n\n_Shrimp Ensalada_ 180 calories, 4 g fat (1.5 g saturated), 1010 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _2 Baja-Style Wild Gulf Shrimp Tacos_ 380 calories, 10 g fat (1 g saturated), 700 mg sodium\n\n**Not that:** _Chicken Enchilada plate_ 770 calories, 24 g fat (10 g saturated), 2270 mg sodium\n\n**Eat that:** _Charbroiled Chicken Baja Ensalada with salsa_ 325 calories, 7 g fat (2 g saturated), 1150 mg sodium\n\n**Not that:** _Burrito Ultimo with Charbroiled Steak_ 930 calories, 37 g fat (16 g saturated), 2010 mg sodium\n\n### **Burger King\/Breakfast**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Croissan'wich with egg_ 270 calories, 2 g fat (1.5 g saturated), 175 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Croissan'wich with sausage_ 380 calories, 27 g fat (9 g saturated), 630 mg sodium\n\n**Not that:** _Croissan'wich with sausage, egg, and cheese_ 520 calories, 39 g fat (14 g saturated), 1090 mg sodium\n\n### **Burger King\/Lunch & Dinner**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Chicken Tenders (5 piece)_ 210 calories, 12 g fat (3.5 g saturated), 920 mg sodium (Add 35 calories for BBQ sauce or 90 for honey.)\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Medium Onion Rings_ 320 calories, 16 g fat (4 g saturated), 460 mg sodium\n\n**Not that:** _Medium French Fries_ 360 calories, 18 g fat (5 g saturated), 640 mg sodium\n\n**Eat that:** _Chicken Whopper_ 570 calories, 25 g fat (4.5 g saturated), 1410 mg sodium\n\n**Not that:** _Whopper_ 700 calories, 42 g fat (13 g saturated), 1020 mg sodium\n\n### **Chick-fil-A**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Chargrilled Chicken Sandwich_ 270 calories, 3.5 g fat (1 g saturated), 940 mg sodium\n\n_Fresh Fruit Cup_ 60 calories, 0 g fat (0 g saturated), 0 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Carrot and Raisin Salad_ 170 calories, 6 g fat (1 g saturated), 110 mg sodium\n\n**Not that:** _Small waffle fries_ 280 calories, 14 g fat (5 g saturated), 105 mg sodium\n\n**Eat that:** _Chick-n-strips (4 count) (fried)_ 290 calories, 13 g fat (2.5 g saturated), 730 mg sodium\n\n**Not that:** _Chargrilled Chicken Club_ 380 calories, 11 g fat (5 g saturated), 1240 mg sodium\n\n### **Chipotle**\n\n**ABS DIET ENDORSEMENT:**\n\n_Burrito with black beans, vegetables, lettuce, and salsa_ 600 calories, 18 g fat (3.5 g saturated), 2378 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Burrito with black beans, vegetables, lettuce, and guacamole_ 770 calories, 33 g fat (6 g saturated), 2248 mg sodium\n\n**Not that:** _Burrito with barbecue, rice, lettuce, salsa, cheese, sour cream_ 1120 calories, 51 g fat (20 g saturated), 2920 mg sodium\n\n### **Dairy Queen**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Grilled Chicken Sandwich_ 340 calories, 16 g fat (2.5 g saturated), 1000 mg sodium\n\n_Chocolate Soft Serve_ (\u00bd _cup)_ 150 calories, 5 g fat (3.5 g saturated), 75 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _2 Hot Dogs_ 480 calories, 28 g fat (10 g saturated), 1460 mg sodium\n\n**Not that:** _Chicken Strip Basket with gravy_ 1000 calories, 50 g fat (13 g saturated), 2510 mg sodium\n\n**Eat that:** _Small Chocolate Sundae_ 280 calories, 7 g fat (4.5 g saturated), 140 mg sodium\n\n**Not that:** _Medium Chocolate Malt_ 870 calories, 22 g fat (14 g saturated), 450 mg sodium\n\n### **Hardee's**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Country Ham Biscuit_ 440 calories, 26 g fat (6 g saturated), 1710 mg sodium\n\n_Slammer_ 240 calories, 12 g fat (5 g saturated), 300 mg sodium\n\n_Hot Ham 'n Cheese_ 287 calories, 13 g fat (6 g saturated), 1100 mg sodium\n\n_Mashed potatoes_ 90 calories, 2 g fat, 410 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Frisco Breakfast Sandwich_ 410 calories, 17 g fat (7 g saturated), 870 mg sodium\n\n**Not that:** _Loaded Omelet Biscuit_ 640 calories, 44 g fat (14 g saturated), 1510 mg sodium\n\n**Eat this:** _Cole Slaw_ 170 calories, 10 g fat (2 g saturated), 140 mg sodium\n\n**Not that:** _Medium Crispy Curls_ 410 calories, 20 g fat (5 g saturated), 1020 mg sodium\n\n**Eat that:** _Regular Roast Beef Sandwich_ 330 calories, 16 g fat (7 g saturated), 1220 mg sodium\n\n**Not that:** _Half Pound Six Dollar Burger_ 1060 calories, 72 g fat (30 g saturated), 1860 mg sodium\n\n### **Jack in the Box\/Breakfast**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Breakfast Jack_ 305 calories, 14 g fat (4 g saturated), 715 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Sourdough Breakfast Sandwich_ 445 calories, 26 g fat (8 g saturated), 875 mg sodium\n\n**Not that:** _Extreme Sausage Sandwich_ 690 calories, 50 g fat (17 g saturated), 1265 mg sodium\n\n### **Jack in the Box\/Lunch & Dinner**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Hamburger_ 310 calories, 14 g fat (5 g saturated), 590 mg sodium\n\n_Side salad with low-fat balsamic dressing_ 195 calories, 9.5 g fat (4.5 g saturated), 880 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Egg Roll (1)_ 175 calories, 6 g fat (2 g saturated), 470 mg sodium\n\n**Not that:** _Stuffed Jalape\u00f1os (3 pieces)_ 230 calories, 13 g fat (6 g saturated), 690 mg sodium\n\n**Eat that:** _Chicken Fajita Pita_ 315 calories, 9 g fat (4 g saturated), 1080 mg sodium\n\n**Not that:** _Sourdough Jack_ 715 calories, 51 g fat (18 g saturated), 1165 mg sodium\n\n### **KFC**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Original Recipe Chicken Breast (with skin and breading removed)_ 140 calories, 3 g fat (1 g saturated), 410 mg sodium\n\n_Mashed Potatoes with Gravy_ 110 calories, 4 g fat (1 g saturated), 260 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _BBQ Baked Beans_ 230 calories, 1 g fat (1 g saturated), 720 mg sodium\n\n**Not that:** _Potato Wedges_ 240 calories, 12 g fat (3 g saturated), 830 mg sodium\n\n**Eat that:** _Honey Barbecue Sandwich_ 300 calories, 6 g fat (1.5 g saturated), 640 mg sodium\n\n**Not that:** _Original Recipe Breast_ 380 calories, 19 g fat (6 g saturated), 1150 mg sodium\n\n### **Long John Silver's**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Baked Cod with cocktail sauce_ 145 calories, 2.5 g fat (1 g saturated), 490 mg sodium\n\n_Corn Cobette_ 95 calories, 3 g fat (0.5 g saturated), 0 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Chicken Plank (fried)_ 140 calories, 8 g fat (2.5 g saturated), 400 mg sodium\n\n**Not that:** _Ultimate Fish Sandwich_ 500 calories, 25 g fat (8 g saturated), 1310 mg sodium\n\n**Eat that:** _Pineapple Cream Pie_ 290 calories, 13 g fat (7 g saturated), 210 mg sodium\n\n**Not that:** _Chocolate Cream Pie_ 310 calories, 22 g fat (14 g saturated), 170 mg sodium\n\n### **McDonald's\/Breakfast**\n\n**ABS DIET ENDORSEMENT:**\n\n_Egg McMuffin_ 290 calories, 11 g fat (4.5 g saturated), 850 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Sausage Burrito_ 300 calories, 16 g fat (6 g saturated), 760 mg sodium\n\n**Not that:** _Sausage McMuffin with Egg_ 450 calories, 26 g fat (10 g saturated), 930 mg sodium\n\n### **McDonald's\/Lunch & Dinner**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Chicken McGrill_ 400 calories, 16 g fat (3 g saturated), 1010 mg sodium\n\n_Side Salad with Low-Fat Balsamic Vinaigrette_ 55 calories, 3 g fat (0 g saturated), 740 mg sodium\n\n_Fiesta salad with salsa_ 390 calories, 22 g fat (10 g saturated), 870 mg sodium\n\n_Fruit 'n Yogurt Parfait (with granola)_ 160 calories, 2 g fat (1 g saturated), 85 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Quarter Pounder_ 420 calories, 18 g fat (7 g saturated), 730 mg sodium\n\n**Not that:** _Big Mac_ 560 calories, 30 g fat (10 g saturated), 1010 mg sodium\n\n**Eat this:** _Chicken Selects (3 piece) (Add 60 calories for buffalo sauce, 70 for honey mustard)_ 380 calories, 20 g fat (3.5 g saturated), 930 mg sodium\n\n**Not that:** _Crispy Chicken Bacon Ranch Salad with dressing_ 620 calories, 31 g fat (8 g saturated), 1560 mg sodium\n\n**Eat that:** _Apple Dippers with Low-Fat Caramel Dip_ 100 calories, 1 g fat (0.5 g saturated), 35 mg sodium\n\n**Not that:** _Baked Apple Pie_ 250 calories, 11 g fat (3 g saturated), 150 mg sodium\n\n### **Taco Bell**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Chicken Burrito, Fiesta style_ 370 calories, 12 g fat (3.5 g saturated), 1090 mg sodium\n\n_2 Ranchero Chicken Soft Tacos_ 540 calories, 14 g fat (4 g saturated), 1710 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Bean Burrito_ 370 calories, 10 g fat (3.5 g saturated), 1200 mg sodium\n\n**Not that:** _Fiesta Taco Salad (without shell)_ 500 calories, 27 g fat (12 g saturated), 1520 mg sodium\n\n### **Wendy's**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Ultimate Chicken Grill sandwich_ 360 calories, 7 g fat (1.5 g saturated), 1100 mg sodium\n\n_Side Salad with Low-Fat Honey Mustard Dressing_ 145 calories, 3 g fat (0 g saturated), 360 mg sodium\n\n_Chili (small)_ 200 calories, 5 g fat (2 g saturated), 870 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Baked potato with sour cream_ 340 calories, 6 g fat (3.5 g saturated), 40 mg sodium\n\n**Not that:** _Biggie fries_ 440 calories, 19 g fat (3.5 g saturated), 380 mg sodium\n\n**Eat that:** _Jr. Cheeseburger_ 310 calories, 12 g fat (6 g saturated), 820 mg sodium\n\n**Not that:** _Spicy Chicken Fillet sandwich_ 510 calories, 19 g fat (3.5 g saturated), 1480 mg sodium\n\n## **Sandwich Shops**\n\nAlways start with building the right base: whole-wheat bread if they have it (rye is also a good choice because it has nearly as much fiber). Then make smart choices\u2014low-fat cheese (go easy on it if all they have is full-fat cheese); no fatty, salty cured cold cuts like pepperoni or salami; and top it with tomatoes and all the greens that bun can hold. And don't always trust the salads; many are loaded with saturated fats hiding in fried noodles, butter-soaked croutons, and goopy dressings.\n\n### **Atlanta Bread Company**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Roast beef sandwich_ 450 calories, 5 g fat (n\/a g saturated), 980 mg sodium\n\n_Turkey breast sandwich_ 420 calories, 4.5 g fat (n\/a g saturated), 1450 mg sodium\n\n_Greek chicken salad_ 210 calories, 10 g fat (n\/a g saturated), 690 mg sodium\n\n_Garden veggie soup (cup)_ 80 calories, 1 g fat (n\/a g saturated), 820 mg sodium\n\n_Country bean soup (cup)_ 140 calories, 1.5 g fat (n\/a g saturated), 1040 mg sodium\n\n_Fruit salad_ 140 calories, 1 g fat (n\/a g saturated), 15 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Chicken Caesar salad_ 310 calories, 11 g fat (N\/A g saturated), 580 mg sodium\n\n**Not that:** _Turkey Club Panini_ 780 calories, 32 g fat (N\/A g saturated), 2000 mg sodium\n\n### **Au Bon Pain**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Grilled Salmon Salad Ficelle_ 280 calories, 7 g fat (2 g saturated), 490 mg sodium\n\n_Turkey Tenderloin Ficelle_ 310 calories, 9 g fat (1.5 g saturated), 860 mg sodium\n\n_Small Garden Salad_ 50 calories, 1 g fat (0 g saturated), 10 mg sodium\n\n_Blueberry yogurt with granola and fresh fruit (small)_ 310 calories, 6 g fat (2 g saturated), 130 mg sodium\n\n_Chargrilled Salmon Filet with Yellow Peppers salad_ 300 calories, 7 g fat (5 g saturated), 420 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Dijon Albacore Tuna sandwich_ 450 calories, 10 g fat (4.5 g saturated), 800 mg sodium\n\n**Not that:** _Clam Chowder in a bread bowl_ 880 calories, 19 g fat (7 g saturated), 2500 mg sodium\n\n### **Blimpie**\n\n**ABS DIET ENDORSEMENTS:**\n\n_6-inch Turkey sub_ 344 calories, 5 g fat (0.5 g saturated), 1551 mg sodium\n\n_6-inch Grilled Chicken sub_ 373 calories, 9 g fat (2.5 g saturated), 836 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Vegi-Max sub_ 395 calories, 7 g fat (1.5 g saturated), 982 mg sodium\n\n**Not that:** _Buffalo Chicken salad_ 390 calories, 28 g fat (7 g saturated), 1300 mg sodium\n\n### **Einstein Bros. Bagels**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Harvest Chicken Salad on Challah_ 400 calories, 9 g fat (1 g saturated), 520 mg sodium\n\n_100% Albacore Tuna on Artisan Wheat_ 400 calories, 9 g fat (1 g saturated), 520 mg sodium\n\n_Tortilla soup (cup)_ 170 calories, 2 g fat (0 g saturated), 1150 mg sodium\n\n_Jamaican Jerk Entr\u00e9e Salad_ 340 calories, 10 g fat (1 g saturated), 650 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Calypso Chicken Salad Sandwich_ 460 calories, 9 g fat (1 g saturated), 1100 mg sodium\n\n**Not that:** _Club Mex on Challah Sandwich_ 620 calories, 27 g fat (10 g saturated), 1660 mg sodium\n\n**Eat that:** _Low-Fat Minestrone (cup)_ 120 carlories, 3.5 g fat (0 g saturated), 940 mg sodium)\n\n**Not that:** _Caribbean Crab Chowder (cup)_ 220 calories, 14 g fat (12 g saturated), 790 mg sodium)\n\n### **Panera Bread**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Smoked Turkey on Artisan Bread_ 590 calories, 16 g fat (1.5 g saturated), 2320 mg sodium\n\n_Asian Sesame Chicken Salad_ 330 calories, 17 g fat (2 g saturated), 1170 mg sodium\n\n_Low-Fat Vegetarian Black Bean soup (cup)_ 160 calories, 1 g fat (0 g saturated), 820 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** \u00bd _Chicken Caesar Salad\/Low-Fat Vegetarian Autumn Tomato Bisque combo_ 360 calories, 20 g fat (4 g saturated) 1675 mg sodium\n\n**Not that:** _Pepperblue Steak Sandwich_ 780 calories, 38 g fat (8 g saturated), 2070 mg sodium _Broccoli Cheddar soup (cup)_ 230 calories, 16 g fat (9 g saturated), 1000 mg sodium\n\n### **Schlotzsky's Deli**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Dijon Chicken Sandwich (small)_ 329 calories, 4 g fat (N\/A g saturated), 1456 mg sodium\n\n_Fresh Fruit salad (small)_ 86 calories, 1 g fat (N\/A g saturated), 22 mg sodium\n\n_California Pasta salad (small)_ 58 calories, 3 g fat (N\/A g saturated), 250 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Albacore Tuna Sandwich (small)_ 334 calories, 7 g fat (N\/A g saturated), 1230 mg sodium\n\n**Not that:** _The Original sandwich (small)_ 525 calories, 24 g fat (N\/A g saturated), 1781 mg sodium\n\n### **Subway**\n\n**ABS DIET ENDORSEMENTS:**\n\n_6-inch roast beef sub_ 290 calories, 5 g fat (2 g saturated), 910 mg sodium\n\n_6-inch savory turkey sub_ 280 calories, 4.5 g fat (1.5 g saturated), 1010 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _6-inch roast beef sub with provolone_ 340 calories, 9 g fat (4 g saturated), 1035 mg sodium\n\n**Not that:** _Meatball marinara sub with provolone_ 550 calories, 26 g fat (13 g saturated), 1305 mg sodium\n\n**Eat this:** _6-inch savory turkey with provolone_ 330 calories, 8.5 g fat (3.5 g saturated), 1135 mg sodium\n\n**Not that:** _Atkins-Friendly Chicken Bacon Ranch Wrap_ 440 calories, 26 g fat (9 g saturated), 1550 mg sodium\n\n**Eat that:** _Oatmeal raisin cookie_ 200 calories, 8 g fat (2.5 g saturated), 170 mg sodium\n\n**Not that:** _Peanut butter cookie_ 220 calories, 12 g fat (4 g saturated), 200 mg sodium\n\n### **Zabar's and other NYC-style delis**\n\n**ABS DIET ENDORSEMENT:**\n\n_Turkey or ham on rye with lettuce, tomato, provolone, and mustard_ 370 calories, 12 g fat (6 g saturated), 1900 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Corned beef on rye with slaw and mustard_ 380 calories, 20 g fat (5 g saturated), 1500 mg sodium\n\n**Not that:** _Egg salad on white with lettuce and tomato_ 750 calories, 60 g fat (11 g saturated), 900 mg sodium\n\n## **Breakfast Places**\n\nI can't emphasize enough the importance of eating the moment you wake up. They call it \"breakfast\" for a reason\u2014you've been fasting for the past 8 to 10 hours, and your body needs fuel. Eating immediately jump-starts your metabolism and starts you on your daily quest to turn fat into muscle. On the other hand, if you skip breakfast, even for a few hours, you signal your body to begin breaking down muscle for fuel. That's right: Every minute you wait between waking and eating is a minute more of muscle loss.\n\nThat said, sometimes the fastest way to add fuel to the fire is to swing by one of these joints (especially on those mornings where you wake up somewhere . . . unusual). As a rule, make sure they don't put full-fat milk in your coffee concoction, go with reduced-fat vegetable spread on your bagel, and never buy a pastry that's bigger than your head. In fact, if there's one food category I would ban if I could, it's pastries and doughnuts\u2014or, as they're known by their technical, scientific names, \"empty sugar calories fried in lard.\"\n\n### **Breakfast Diner (such as Bob Evans, Denny's, IHOP, Perkins, or Shoney's)**\n\n**ABS DIET ENDORSEMENTS:**\n\n_2 poached eggs_ 148 calories, 10 g fat (3 g saturated), 294 mg sodium\n\n_Plain whole-wheat toast (per slice)_ 128 calories, 2.5 g fat (.5 g saturated), 160 mg sodium\n\n_Canadian bacon_ 44 calories, 2 g fat (.5 g saturated), 364 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _2 scrambled eggs_ 170 calories, 11 g fat (3 g saturated), 482 mg sodium\n\n**Not that:** _Western omelet with eggs_ 520 calories, 39 g fat (13 g saturated), 1280 mg sodium\n\n**Eat this:** _Plain English muffin_ 139 calories, 1 g fat (0 g saturated), 229 mg sodium\n\n**Not that:** _Plain biscuit_ 280 calories, 12 g fat (3 g saturated), 760 mg sodium\n\n**Eat this:** _Bacon (2 slices)_ 72 calories, 4 g fat (2 g saturated), 54 mg sodium\n\n**Not that:** _Sausage (2 links)_ 250 calories, 22 g fat (6 g saturated), 370 mg sodium\n\n### **Coffee Shop (such as Caribou Coffee, Dunkin' Donuts, Peet's Coffee, or Starbucks)**\n\n**ABS DIET ENDORSEMENT:**\n\n_Cappuccino with fat-free milk (12 ounces)_ 80 calories, 0 g fat (0 g saturated), 105 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Latte with fat-free milk_ (12 ounces) 120 calories, 0 g fat (0 g saturated), 170 mg sodium\n\n**Not that:** _Latte with whole milk_ (12 ounces) 200 calories, 11 g fat (7 g saturated), 160 mg sodium\n\n## **NUTRITIONAL DETECTIVE WORK**\n\nFood labels are all well and good when you're cooking for yourself, but what about those occasional nights (okay, endless stream of nights) when dinner is dished up by some pimply-faced kid with a paper hat and a bored expression? How do you know what, exactly, is in that paper-and-Styrofoam-wrapped monstrosity you're eating? In the land of \"special sauces\" and \"secret recipes,\" sussing out the nutritional realities of your favorite entrees can take a little detective work.\n\nIf the greasy floor beneath your feet is that of a national fast-food joint, the answers may be pretty easy to come by. Most fast-food restaurants post complete nutritional information on their Web sites. Many will also have posters hanging in some dim corner of the restaurant, if you're willing to search. Be aware of two caveats: First, these menus change all the time, so you need to make sure what you're ordering is what they've posted. Second, watch out for serving size and extra ingredients: At some chains, the numbers for the \"healthy\" salad don't include the fried Chinese noodles that come on top. Others give you the calorie count for a serving of their dressing, but the package that comes with the salad can be two or even three servings, not one.\n\nChain restaurants that feature actual menus and waiters are a mixed bag. Some deserve applause: Panera Bread Company and Don Pablo's Mexican Kitchen provide complete nutritional information on their Web sites. Others deserve a round of jeers: T.G.I. Friday's for example, refused to give us a nutritional breakdown of some of their foods, even when I called their corporate headquarters and harassed them. Applebee's will tell you the calorie count of their Weight Watchers items but not their regular fare. What's up with that?\n\nSo check your favorite chain's Web site. Even if they only list total calories, that's enough to make an informed decision\u2014when you read that half of an onion-blossom appetizer has 1600 calories, you can pretty much figure the meal should only be administered under a cardiologist's care. If you can't find the information you want on the Web, call their customer service line. They might be able to e-mail or fax it to you. And if that's not an option, well, tell them you want it to be.\n\n## **Bagel Shops**\n\n### **Bruegger's Bagels**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Honey Grain bagel_ 330 calories, 3 g fat (0 g saturated), 500 mg sodium\n\n. . . _with Light garden veggie cream cheese_ 60 calories, 4 g fat (2.5 g saturated), 75 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Pumpernickel bagel_ 320 calories, 3 g fat (0 g saturated), 600 mg sodium\n\n**Not that:** _Salt bagel_ 300 calories, 2 g fat (0 g saturated), 1540 mg sodium\n\n**Eat this:** . . . _with Light plain cream cheese_ 70 calories, 4.5 g fat (2 g saturated), 90 mg sodium\n\n**Not this:** . . . _with Light plain cream cheese_ 100 calories, 8 g fat (5 g saturated), 105 mg sodium\n\n### **Manhattan Bagel Co.**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Spinach bagel_ 270 calories, <1 g fat (0 g saturated), 580 mg sodium\n\n. . . _with Light vegetable cream cheese_ 50 calories, 5 g fat (2.5 g saturated), 100 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Rye bagel_ 260 calories, 1 g fat (0 g saturated), 560 mg sodium\n\n**Not that:** _Chocolate chip bagel_ 290 calories, 2.5 g fat (1.5 g saturated), 530 mg sodium **Not that:** . . . _with French vanilla cream cheese_ 100 calories, 7 g fat (4.5 g saturated), 85 mg sodium\n\n**Eat that:** . . . _with Light raisin walnut cream cheese_ 80 calories, 5 g fat (3 g saturated), 75 mg sodium\n\n**Not that:** . . . _with French vanilla cream cheese_ 100 calories, 7 g fat (4.5 g saturated), 85 mg sodium\n\n### **Bakery (such as Au Bon Pain, Atlanta Bread Company, Bruegger's Bagels, Caribou Coffee, Einstein Bros. Bagels, Panera, or Starbucks)**\n\n**ABS DIET ENDORSEMENT:**\n\n_Biscotti_ 110 calories, 5 g fat (1.5 g saturated), 75 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Fruit or Raisin Scone_ 410 calories, 14 g fat (8 g saturated), 350 mg sodium\n\n**Not that:** _Coffee cake_ 570 calories, 28 g fat (10 g saturated), 310 mg sodium\n\n## **Doughnut Shop**\n\n### **Dunkin' Donuts**\n\n**ABS DIET ENDORSEMENT:**\n\n_Plain glazed donut_ 180 calories, 8 g fat (1.5 g saturated), 250 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Jelly-filled donut_ 210 calories, 8 g fat (1.5 g saturated), 280 mg sodium\n\n**Not that:** _Old Fashioned cake donut_ 300 calories, 19 g fat (5 g saturated), 330 mg sodium\n\n### **Krispy Kreme**\n\n**ABS DIET ENDORSEMENT:**\n\n_Original glazed donut_ 200 calories, 12 g fat (3 g saturated), 95 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Maple iced glazed donut_ 240 calories, 12 g fat (3 g saturated), 100 mg sodium\n\n**Not that:** _Dulce de leche_ 290 calories, 18 g fat (4.5 g saturated), 160 mg sodium\n\n## **Sit-Down Restaurants**\n\nHere's the big secret about those casual-dinner chain restaurants: They're really all the same place. Sure, there are some slight variations\u2014Ruby Tuesday may give its spinach dip the pedestrian name while T.G.I. Friday's dubs theirs \"Tuscan\"\u2014but both are the same vat of once-healthy vegetables sunk in saturated fat. The same principle applies to steak houses\u2014a sirloin is a sirloin is a sirloin; maybe one place offers it in a 10-ounce cut while another in 12 ounces, but the meat remains the same. So even though these recommendations are not keyed to a specific restaurant, you can use them anywhere. At the steak house, look for cuts with \"loin\" in its name.\n\n### **Steak House**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Filet, aka tenderloin (9 ounces)_ 360 calories, 19 g fat (10 g saturated), 330 mg sodium\n\n_Barbecued chicken breast (10 ounces)_ 280 calories, 5 g fat (2 g saturated), 860 mg sodium\n\n_Steamed mixed vegetables_ 70 calories, 1 g fat (0 g saturated), 900 mg sodium\n\n_Side salad with light dressing_ 200 calories, 9 g fat (1 g saturated), 51 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Sirloin (12 ounces)_ 410 calories, 18 g fat (9 g saturated), 470 mg sodium\n\n**Not that:** _Prime rib (16 ounces)_ 1280 calories, 94 g fat (52 g saturated), 620 mg sodium\n\n**Eat this:** _Saut\u00e9ed mushrooms_ 115 calories, 9 g fat (2 g saturated), 600 mg sodium\n\n**Not that:** _Steak fries_ 590 calories, 31 g fat (12 g saturated), 460 mg sodium\n\n**Eat that:** _Baked potato with sour cream_ 280 calories, 3 g fat (2 g saturated), 200 mg sodium\n\n**Not that:** _Side Caesar salad_ 310 calories, 26 g fat (7 g saturated), 620 mg sodium\n\n### **Casual Sit-Down (such as Applebee's, Bennigan's, Cheesecake Factory, Friendly's, Perkins, O'Charley's, Ruby Tuesday, T.G.I. Friday's, or pretty much any restaurant with an apostrophe s)**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Vegetable Soup (cup)_ 100 calories, 1 g fat (0 g saturated), 610 mg sodium\n\n_Grilled chicken (6 ounces)_ 270 calories, 8 g fat (3 g saturated), 650 mg sodium\n\n_Grilled salmon (8 ounces)_ 420 calories, 21 g fat (4 g saturated), 340 mg sodium\n\n_Vegetable of the day_ 60 calories, 3 g fat (1 g saturated), 150 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Steak fajitas with salsa_ 860 calories, 31 g fat (12 g saturated), 1660 mg sodium\n\n**Not that:** _Rack of ribs (16 ounces)_ 770 calories, 54 g fat (21 g saturated), 770 mg sodium\n\n**Eat this:** _Baked potato with sour cream_ 280 calories, 3 g fat (2 g saturated), 30 mg sodium\n\n**Not that:** _Loaded baked potato_ 620 calories, 31 g fat (19 g saturated), 570 mg sodium\n\n**Eat this:** _Apple pie_ 540 calories, 28 g fat (13 g saturated), 440 mg sodium\n\n**Not that:** _Fudge cake or brownie sundae_ 1130 calories, 57 g fat (30 g saturated), 400 mg sodium\n\n### **Diner**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Pot roast_ 370 calories, 16 g fat (7 g saturated), 570 mg sodium\n\n_Grilled chicken_ 476 calories, 14 g fat (3 g saturated), 1494 mg sodium\n\n_Vegetable of the day_ 60 calories, 3 g fat (1 g saturated), 150 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Roast turkey with stuffing_ 500 calories, 19 g fat (9 g saturated), 2190 mg sodium\n\n**Not that:** _Chicken pot pie_ 680 calories, 37 g fat (17 g saturated), 1590 mg sodium\n\n**Eat this:** _Cole slaw_ 170 calories, 14 g fat (2 g saturated), 380 mg sodium\n\n**Not that:** _Fries_ 600 calories, 30 g fat (12 g saturated), 460 mg sodium\n\n**Eat that:** _Tuna melt_ 537 calories, 28 g fat (13 g saturated), 1518 mg sodium\n\n**Not that:** _Patty melt_ 770 calories, 50 g fat (25 g saturated), 1130 mg sodium\n\n### **Boston Market**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Hand-carved rotisserie turkey_ 170 calories, 1 g fat (0 g saturated), 850 mg sodium\n\n_Fresh steamed broccoli_ 30 calories, 0 g fat (0 g saturated), 30 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** \u00bc _sweet garlic rotisserie chicken (white meat, no skin or wing)_ 170 calories, 4 g fat (1 g saturated), 480 mg sodium\n\n**Not that:** _Chipotle meat loaf and chipotle gravy_ 860 calories, 55 g fat (23 g saturated), 1750 mg sodium\n\n**Eat that:** _Green beans_ 70 calories, 4 g fat (0.5 g saturated), 250 mg sodium\n\n**Not that:** _Squash casserole_ 330 calories, 24 g fat (13 g saturated), 1110 mg sodium\n\n### **Italian Restaurant (such as Olive Garden, Carrabba's, Macaroni Grill, or Fazoli's)**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Minestrone soup (cup)_ 100 calories, 1 g fat (0 g saturated), 610 mg sodium\n\n_Chicken Marsala_ 460 calories, 25 g fat (7 g saturated), 790 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Pasta with marinara sauce_ 850 calories, 17 g fat (4 g saturated), 1450 mg sodium\n\n**Not that:** _Pasta with Alfredo sauce_ 1500 calories, 97 g fat (48 g saturated), 1030 mg sodium\n\n**Eat this:** _Cheese ravioli with tomato sauce_ 620 calories, 26 g fat (11 g saturated), 1290 mg sodium\n\n**Not that:** _Lasagna_ 960 calories, 53 g fat (21 g saturated), 2060 mg sodium\n\n**Eat that:** _Soft breadsticks (2)_ 280 calories, 12 g fat (2 g saturated), 640 mg sodium\n\n**Not that:** _Antipasto (half an order)_ 315 calories, 24 g fat (8 g saturated), 1480 mg sodium\n\n### **Mexican Restaurant (such as Don Pablo's or Chili's)**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Chicken fajitas with lettuce and pico de gallo_ 850 calories, 30 g fat (6 g saturated), 2100 mg sodium\n\n_Side of stewed black, kidney, or pinto beans_ 120 calories, 2 g fat (0 g saturated), 400 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Enchiladas combo (beef and chicken)_ 615 calories, 35 g fat (15 g saturated), 1800 mg sodium\n\n**Not that:** _Chicken chimichanga_ 1100 calories, 50 g fat (15 g saturated), 3300 mg sodium\n\n**Eat that:** _Chips (12) with guacamole (2.5 ounces)_ 380 calories, 22 g fat (4 g saturated), 300 mg sodium\n\n**Not that:** _Chips (12) with cheese dip (2 ounces)_ 440 calories, 25 g fat (7 g saturated), 920 mg sodium\n\n### **Chinese Restaurant (such as Panda Express or P.F. Chang's)**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Egg drop soup_ 60 calories, 3 g fat (1 g saturated), 1000 mg sodium _Stir-fried vegetables_ 750 calories, 19 g fat (3 g saturated), 2150 mg sodium\n\n_Szechuan shrimp_ 950 calories, 20 g fat (2 g saturated), 2460 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Shrimp with garlic sauce_ 950 calories, 30 g fat (4 g saturated), 2950 mg sodium\n\n**Not that:** _Beef with broccoli_ 1180 calories, 46 g fat (9 g saturated), 3150 mg sodium\n\n**Eat this:** _Chicken chow mein_ 1000 calories, 32 g fat (10 g saturated), 2450 mg sodium\n\n**Not that:** _General Tso's chicken_ 1600 calories, 60 g fat (10 g saturated), 3200 mg sodium\n\n**Eat that:** _Vegetarian spring roll_ 80 calories, 2.5 g fat (0 g saturated), 270 mg sodium\n\n**Not that:** _Chicken\/pork egg roll_ 200 calories, 10 g fat (1 g saturated), 450 mg sodium\n\n### **Sushi Bar**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Edamame (2 ounces)_ 86 calories, 4 g fat (0 g saturated), 8 mg sodium\n\n_Miso soup (cup)_ 50 calories, 2 g fat (n\/a g saturated), 310 mg sodium\n\n_Tuna roll (12 pieces)_ 260 calories, 1.5 g fat (0 g saturated), 270 mg sodium\n\n_Nigiri, salmon (3 pieces)_ 192 calories, 2 g fat (0 g saturated), 126 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _California roll (12 pieces)_ 290 calories, 5 g fat (1 g saturated), 380 mg sodium\n\n**Not that:** _Crunchy shrimp roll (12 pieces)_ 650 calories, 19 g fat (2 g saturated), 1247 mg sodium\n\n**Eat that:** _Nigiri, shrimp (3 pieces)_ 267 calories, 2.5 g fat (0 g saturated), 273 mg sodium\n\n**Not that:** _Nigiri, eel (3 pieces)_ 258 calories, 7 g fat (1.5 g saturated), 432 mg sodium\n\n### **Seafood Restaurant**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Shrimp cocktail (3 ounces)_ 80 calories, 1 g fat (0 g saturated), 190 mg sodium\n\n_Grilled or broiled white fish (6 ounces)_ 210 calories, 5 g fat (1 g saturated), 360 mg sodium\n\n_Grilled or broiled salmon (8 ounces)_ 420 calories, 21 g fat (4 g saturated), 340 mg sodium\n\n_Salad with light dressing_ 90 calories, 7 g fat (1 g saturated), 490 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Broiled salmon (8 ounces)_ 420 calories, 21 g fat (4 g saturated), 340 mg sodium\n\n**Not that:** _Fried white fish (9 ounces)_ 520 calories, 24 g fat (8 g saturated), 840 mg sodium\n\n**Eat this:** _Cup of Manhattan-style clam chowder_ 153 calories, 1 g fat (0 g saturated), 680 mg sodium\n\n**Not that:** _Cup of New England\u2013style clam chowder_ 250 calories, 7 g fat (2 g saturated), 1400 mg sodium\n\n## **Pizza Places**\n\n### ****Domino's****\n\n**ABS DIET ENDORSEMENT:**\n\n_2 slices medium thin-crust Vegi-Feast cheese pizza_ 336 calories, 19 g fat (7 g saturated), 793 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _2 slices medium thin-crust Hawaiian Feast pizza_ 348 calories, 19 g fat (7 g saturated), 908 mg sodium\n\n**Not that:** _2 slices medium hand-tossed Bacon Cheeseburger Feast pizza_ 546 calories, 26 g fat (11 g saturated), 1268 mg sodium\n\n**Eat that:** _Hot buffalo wings (2 pieces)_ 90 calories, 5 g fat (1 g saturated), 508 mg sodium\n\n**Not that:** _Cheesy Bread (2 pieces)_ 246 calories, 13 g fat (4 saturated), 324 mg sodium\n\n## **DANGER AT YOUR DOORSTEP**\n\nThe TV ads are tempting, but which delivery pizza will turn your Blockbuster night into your very own _Return of the Blob_?\n\n**Domino's** _2 slices medium (12\u2033) cheese pizza_ 372 calories, 11 g fat (4 g saturated), 770 mg sodium\n\n**Little Caesar's** _2 slices medium (12\u2033) round cheese pizza_ 320 calories, 12 g fat (5 g saturated), 640 mg sodium\n\n**Pizza Hut** _2 slices medium (12\u2033) hand-tossed crust cheese pizza_ 480 calories, 16 g fat (9 g saturated), 1040 mg sodium\n\n**Papa John's** _2 slices medium (12\u2033) original-crust cheese pizza_ 420 calories, 16 g fat (5 g saturated), 1060 mg sodium\n\n### **Pizza Hut**\n\n**ABS DIET ENDORSEMENT:**\n\n_2 slices medium Fit 'n' Delicious Ham, Pineapple, and Diced Tomato pizza_ 320 calories, 8 g fat (4 g saturated), 940 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _2 slices medium Thin 'n' Crispy Chicken Supreme pizza_ 400 calories, 14 g fat (7 g saturated), 1040 mg sodium\n\n**Not that:** _2 slices medium Pepperoni Lover's Pan pizza_ 660 calories, 36 g fat (14 g saturated), 1340 mg sodium\n\n**Eat that:** _Hot wings (2 pieces)_ 110 calories, 6 g fat (2 g saturated), 450 mg sodium\n\n**Not that:** _Cheese Breadsticks (2 pieces)_ 400 calories, 20 g fat (7 g saturated), 680 mg sodium\n\n### **Papa John's**\n\n**ABS DIET ENDORSEMENT:**\n\n_2 slices medium original-crust Garden Fresh pizza_ 380 calories, 12 g fat (3 g saturated), 940 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _2 slices medium original-crust Grilled Chicken Alfredo pizza_ 420 calories, 14 g fat (5 g saturated), 1020 mg sodium\n\n**Not that:** _2 slices medium original-crust The Meats pizza_ 540 calories, 26 g fat (8 g saturated), 1420 mg sodium\n\n**Eat this:** _Papa's Chicken Strips (2 pieces)_ 160 calories, 8 g fat (2 g saturated), 350 mg sodium\n\n**Not that:** _Cheesesticks (2 pieces)_ 360 calories, 16 g fat (4.5 g saturated), 830 mg sodium\n\n## **DANGER IN YOUR FREEZER**\n\nDelivery pizza might seem like an extravagance when you could just pop one of those frozen pies into the oven, but ordering delivery can actually save you on calories and sodium as well. That's because pizza joints use mostly fresh ingredients. Here's a rundown of the most popular brands:\n\n**Tony's** _2 slices cheese (Cut into_ _1 \u20446ths)_ 370 calories, 17 g fat (6 g saturated), 780 mg sodium\n\n**Freschetta** _2 slices cheese (Cut into 1\u20448ths)_ 463 calories, 16 g fat (7.5 g saturated), 1288 mg sodium\n\n**DiGiorno** _2 slices cheese (Cut into_ _1 \u20448ths)_ 465 calories, 16.5 g fat (7.5 g saturated), 1275 mg sodium\n\n## **Smoothie Stands**\n\nI prefer you make your own smoothie, but sometimes you need one on the go. If so, always get yours with extra protein (you'd be surprised how many of these shakes are just fruit, water, and air). Then, make sure the shake contains a gram or two of fat, otherwise your body won't absorb many of the fat-soluble nutrients from all that fruit. Simply asking it to be made with low-fat milk or yogurt will do the trick. But beware fat bombs. Here's a clue: If it sounds more like a dessert than a beverage for someone with an active lifestyle, skip it.\n\n### **Jamba Juice**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Berry Fulfilling (16 ounces)_ 160 calories, 0.5 g fat (0 g saturated), 230 mg sodium\n\n_Orange Divine (16 ounces)_ 160 calories, 0.5 g fat (0 g saturated), 230 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Banana Berry (16 ounces)_ 310 calories, 0.5 g fat (0 g saturated), 75 mg sodium\n\n**Not that:** _Chocolate Moo'd (16 ounces)_ 500 calories, 6 g fat (4 g saturated), 260 mg sodium\n\n### **Smoothie King**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Slim-N-Trim vanilla_ 227 calories, 1 g fat (0 g saturated), 150 mg sodium\n\n_Blueberry Heaven_ 260 calories, 1 g fat (0 g saturated), 200 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _The Activator, Chocolate_ 429 calories, 1 g fat (0 g saturated), 260 mg sodium\n\n**Not that:** _Pina Colada Island_ 550 calories, 11 g fat (9 g saturated), 300 mg sodium\n\n## **Food Courts at Malls**\n\nSpend more time eating than you do shopping, and you'll end up having to go back\u2014for bigger clothes. Don't even think about stopping at Cinnabon; eat one of their cinnamon-buns-on-steroids, and you'll rack up 813 calories and 8 grams of saturated fat. Here are the best stops so you won't drop while you shop.\n\n### **Auntie Anne's**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Jalape\u00f1o pretzel (without butter) with marinara sauce_ 280 calories, 1 g fat (0 g saturated), 960 mg sodium\n\n_Whole wheat pretzel (without butter)_ 350 calories, 1.5 g fat (0 g saturated), 1100 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Garlic pretzel (without butter) with sweet mustard_ 380 calories, 2.5 g fat (1 g saturated), 950 mg sodium\n\n**Not that:** _Original pretzel (without butter) with cheese sauce_ 440 calories, 9 g fat (4 g saturated), 1410 mg sodium\n\n### **Mrs. Fields**\n\n**ABS DIET ENDORSEMENT:**\n\n_Debra's Special Nibbler Cookie_ 100 calories, 4.5 g fat (2 g saturated), 80 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Peanut Butter Bite-Size Nibbler Cookie_ 110 calories, 6 g fat (2.5 g saturated), 95 mg sodium\n\n**Not that:** _Milk Chocolate Chip Cookie_ 280 calories, 13 g fat (8 g saturated), 180 mg sodium\n\n### **Sbarro**\n\n**ABS DIET ENDORSEMENTS:**\n\n_1 slice fresh tomato pizza_ 450 calories, 14 g fat (n\/a g saturated), 1040 mg sodium\n\n_1 slice mushroom pizza_ 460 calories, 14 g fat (n\/a g saturated), 1310 mg sodium\n\n_Greek salad (side)_ 60 calories, 5 g fat (n\/a g saturated), 130 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _1 slice chicken vegetable pizza_ 530 calories, 17 g fat (n\/a g saturated), 1260 mg sodium\n\n**Not that:** _1 slice pepperoni pizza_ 730 calories, 37 g fat (n\/a g saturated), 2200 mg sodium\n\n## **Ballparks**\n\nI'm a big fan of smuggling in your own food: It makes healthy eating easier, it makes you feel like a little bit of a rebel, and it avoids the humiliation of paying $453 for one hot dog. But if you can't or won't smuggle trail mix in your underwear, you can still grab a handful of Powerfoods. If you have to pile something on, make it insults to the other team, not nacho cheese.\n\n**ABS DIET ENDORSEMENTS:**\n\n_Roasted peanuts in shell, unsalted (3 ounces)_ 510 calories, 37 g fat (7.5 g saturated), 0 mg sodium\n\n_Light beer (12 ounces)_ 103 calories, 0 g fat (0 g saturated), 0 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Nachos (12\u201316 nachos)_ 692 calories, 38 g fat (16 g saturated), 1632 mg sodium\n\n**Not that:** _Popcorn popped in oil (medium, 16-cup serving)_ 880 calories, 50 g fat (9 g saturated), 1556 mg sodium\n\n**Eat this:** _Regular beer (12 ounces)_ 139 calories, 0 g fat (0 g saturated), 14 mg sodium\n\n**Not that:** _Cola (16 ounces)_ 155 calories, 0 g fat (0 g saturated), 15 mg sodium\n\n## **Sports Bars**\n\nAlcohol will weaken your defenses; so will the fact that 380-pound offensive tackles are considered pro athletes. So I'll make you a deal: If you are paid millions of dollars to shove enormous men in helmets each Sunday, then order whatever you want. If not, then follow the guidelines below.\n\n**ABS DIET ENDORSEMENT:**\n\n_Salted nuts (per ounce)_ 168 calories, 15 g fat (2 g saturated), 190 mg sodium\n\n_Hot wings (4\u20135 wings)_ 350 calories, 24 g fat (8 g saturated), 510 mg sodium\n\n_Fried mozzarella sticks (3 pieces)_ 375 calories, 21 g fat (7 saturated), 780 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Chicken fingers (5 pieces)_ 620 calories, 34 g fat (13 g saturated), 1450 mg sodium\n\n**Not that:** _Cheese fries with ranch dressing (2 cups)_ 1500 calories, 108 g fat (46 g saturated), 1300 mg sodium\n\n**Eat that:** _Burger (Lettuce, tomato, onion, mustard)_ 660 calories, 36 g fat (17 g saturated), 810 mg sodium\n\n**Not that:** _Mushroom cheese-burger_ 900 calories, 57 g fat (28 g saturated), 1070 mg sodium\n\n## **Ice Cream Shops**\n\nFirst, come here directly after a meal. You'll be less likely to pig out. Second, don't stress too much about it; ice cream may be sugary and high in calories, but it also provides protein and calcium. As indulgences go, it's hardly a high crime. So don't waste your time on nasty no-sugar-added ice creams; no sense eating food you don't enjoy. Instead, look for reduced-fat options and add Powerfoods (nuts and fruit) whenever possible. And if you completely ignore all that advice, take this tip: Order the ice cream you absolutely love . . . in a single-scoop cup.\n\n### **Baskin-Robbins**\n\n**ABS DIET ENDORSEMENTS:**\n\n_Espresso 'n Cream low-fat ice cream (1 scoop)_ 180 calories, 4 g fat (1.5 g saturated), 120 mg sodium\n\n_Perils of Praline low-fat yogurt (1 scoop)_ 190 calories, 3.5 g fat (1.5 g saturated), 170 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Very Berry Strawberry ice cream (1 scoop)_ 220 calories, 11 g fat (7 g saturated), 70 mg sodium\n\n**Not that:** _Fudge Brownie ice cream (2 scoops)_ 600 calories, 38 g fat (22 g saturated), 280 mg sodium\n\n### **Cold Stone Creamery**\n\n**ABS DIET ENDORSEMENT:**\n\n_Low-fat chocolate ice cream with blueberries or strawberries (\"Like it\" size)_ 240 calories, 1.5 g fat (1 g saturated), 140 mg sodium\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Any fruit-flavored ice cream (strawberry, raspberry, banana, etc.) with blueberries (\"Like it\" size)_ 210 calories, 22 g fat (14 g saturated), 90 mg sodium\n\n**Not that:** _Any fruit-flavored ice cream (strawberry, raspberry, banana, etc.) with brownie pieces in a cone (\"Love it\" size)_ 670 calories, 46 g fat (26 g saturated), 400 mg sodium\n\n### **ABS DIET SUCCESS STORY**\n\n## **HE WANTED TO CHASE AFTER HIS KIDS, SO HE CHASED AFTER A GOAL**\n\n**Name:** William Salerno\n\n**Age:** 38\n\n**Height:** 5\u20329\u2033\n\n**Weight, Week 1:** 222\n\n**Weight, Week 6:** 198\n\nHaving twins 2 years ago meant that William Salerno was going to be busier than ever. He was in a job that required him to routinely work 12-hour days, plus now he had to help care for two young children. And that's when it hit him.\n\n\"I feel like I'm an older dad and that I gotta stay ahead of these guys,\" Salerno says. \"I want to be like my father was with me\u2014always active. And that got me thinking.\"\n\nThat's when his business manager told him he had to read _The Abs Diet_.\n\n\"When I read it, something clicked. I'm amazed by it because I usually don't go for those kinds of books. Usually they don't do anything for me,\" he says. \"I started to read it in my office one day and read it for a couple hours. I carried it everywhere. Now I talk about trans fats and high-fructose corn syrup to anybody who'll listen to me.\"\n\nHe reads food ingredients on everything\u2014and remains pumped up about the plan.\n\n\"Almonds are a god,\" he says.\n\nA side benefit is that he no longer experiences any symptoms with gout\u2014a disorder that causes pain in the joints that's hampered Salerno in the past.\n\n\"My theory is that if I'm 220, then I could very well have been 230\u2014I was in that range for a couple of years,\" Salerno says. \"People see me now and they say, where'd it all go? They think I'm starving myself, and I'm not. It's so refreshing and unbelievable.\"\n\nNow, he can't stop talking about it.\n\n\"I quoted passages to my wife so often that she began calling me 'the _Abs Diet_ zealot.'\"\nChapter 10\n\n# **INDULGE AND ENJOY**\n\n# The Abs Diet Holiday Survival Guide\n\n**I F A DIET IS AN ANT,** then the holidays are the bottom of a shoe, because they have the power to squash you every time. Between the cookies, meat loaf, more cookies, eggnog, fruitcake, mashed potatoes and gravy (oh, the gravy), many of us cram 6 months of eating into 6 weeks. The result: Come January, the only thing you're trying to cram is 4 extra inches into your jeans.\n\nAs you know, the Abs Diet isn't about deprivation, so I want you to have the occasional tussle with a kilo-high mound of stuffing. If you do that for 6 weeks straight, you'll have succeeded\u2014on the Flabs Diet. Since you have one cheat meal per week, I suggest you play your card around the biggest bomb\u2014whether it's Thanksgiving dinner, the Christmas party, or a date with Aunt Matilda's can't-resist chocolate cheesecake\u2014and then sidestep the dietary land mines through the rest of the week. To eat right throughout the holidays, these choices will help you avoid turning your diet into a disaster.\n\n### **Halloween**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Bit-O-Honey_ Per 6 pieces: 186 calories, 4 g fat (0 g saturated), 124 mg sodium\n\n**Not that:** _Whoppers_ Per 9 pieces: 125 calories, 7 g fat (4 g saturated), 38 mg sodium\n\n**Eat this:** _Kit Kat_ Per miniature bar: 51 calories, 3 g fat (1.7 g saturated), 6 mg sodium\n\n**Not that:** _Reese's Pieces_ About 30 pieces: 120 calories, 6 g fat (4 g saturated), 47 mg sodium\n\n**Eat this:** _Milky Way_ Per fun-size piece: 76 calories, 3 g fat (1 g saturated), 43 mg sodium\n\n**Not that:** _Butterfinger_ Per fun-size piece: 100 calories, 4 g fat (2 g saturated), 45 mg sodium\n\n### **Thanksgiving Dinner**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:**\n\n_1 serving turkey breast_\n\n_2 \u20443 cup mashed potatoes_\n\n_1 \u20443 cup turkey gravy_\n\n_1 dinner roll_\n\n_1 cup corn_\n\n_1 slice jellied cranberry sauce_\n\n_1 slice pumpkin pie_\n\nTOTAL: 1233 calories, 39 g fat (11 g saturated), 1493 mg sodium\n\n**Not that:**\n\n_1 serving dark turkey meat_\n\n_1 cup stuffing_\n\n_2 \u20443 cup sweet potatoes with marshmallow topping_\n\n_1 cup green-bean casserole_\n\n_1 cup homemade cranberry sauce_\n\n_1 slice pecan pie_\n\nTOTAL: 1944 calories, 90 g fat (26 g saturated), 2512 mg sodium\n\n### **Thanksgiving Leftovers**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:**\n\n_1 serving turkey breast on 2 pieces whole grain bread with lettuce, cranberry sauce_\n\n\u00bd _cup green-bean casserole_\n\nTOTAL: 530 calories, 20 g fat (7 g saturated), 1015 mg sodium\n\n**Not that:**\n\n_1 serving turkey on kaiser roll with mayo  \n2\u20443 cup mashed potatoes_\n\n_1 \u20443 cup turkey gravy_\n\nTOTAL: 665 calories, 30 g fat (7 g saturated), 1300 mg sodium\n\n### **Black Friday at the Mall**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _1 Auntie Anne's Cinnamon-Sugar pretzel (without butter)_ 350 calories, 2 g fat (0 g saturated), 430 mg sodium\n\n**Not that:** _1 Cinnabon cinnamon bun_ 813 calories, 32 g fat (8 g saturated, 5 g trans fat), 801 mg sodium\n\n**Eat this:** _TCBY frozen yogurt (small)_ 110 calories, 0 g fat, 60 mg sodium\n\n**Not that:** _Orange Julius (small)_ 220 calories, 1 g fat (0 g saturated), 10 mg sodium\n\n### **At the Office in December**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _1 handful roasted pistachios_ 175 calories, 14 g fat (1.7 g saturated), 125 mg sodium\n\n**Not that:** _1 handful_ (\u00bc _cup) holiday M &M's_ 256 calories, 11 g fat (7 g saturated), 32 mg sodium\n\n**Eat this:** _1 brownie (2\u2033 square)_ 165 calories, 7 g fat (2 g saturated), 82 mg sodium\n\n**Not that:** _A few chunks of peanut brittle_ 274 calories, 11 g fat (2 g saturated), 252 mg sodium\n\n**Eat this:** _1 Hershey's Kiss_ 26 calories, 1.5 g fat (1 g saturated), 4 mg sodium\n\n**Not that:** _1 1-ounce candy cane_ 110 calories, 0 g fat, 11 mg sodium\n\n### **Christmas Dinner**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:**\n\n_8 ounces turkey breast_\n\n_2 \u20443 cup stuffing_\n\n_1 cup sweet potatoes_\n\n_1 slice cranberry sauce_\n\n_2 \u20443 cup creamed corn_\n\n_Sliver pecan pie_\n\n_Sliver pumpkin pie_\n\n_Glass of red wine_\n\nTOTAL: 1653 calories, 50 g fat (12 g saturated), 1444 mg sodium\n\n**Not that:**\n\n_8 ounces ham_\n\n_Dinner roll with 2 pats butter_\n\n_1 cup mashed potatoes_\n\n\u00bd _cup gravy_\n\n_1 cup green-bean casserole_\n\n_Slice pumpkin pie with heavy whipped cream_\n\n_Glass of beer_\n\nTOTAL: 1670 calories, 87 g fat (41 g saturated), 5671 mg sodium\n\n### **New Year's Eve**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _12 large shrimp with 2 tablespoons cocktail sauce_ 125 calories, 1 g fat (0 g saturated), 948 mg sodium\n\n**Not that:** _1 small crab cake_ 285 calories, 12 g fat (2 g saturated), 807 mg sodium\n\n**Drink this:** _Champagne_ 105 calories, 0 g fat, 0 mg sodium\n\n**Not that:** _Vodka tonic_ 125 calories, 0 g fat, 7 mg sodium\n\n**Eat this:** _8 melon balls wrapped in prosciutto_ 220 calories, 11 g fat (4 g saturated), 1650 mg sodium\n\n**Not that:** _1 potato pancake with smoked salmon and caviar_ 313 calories, 17 g fat (4 g saturated), 1760 mg sodium\n\n## **Feasts, Not Famine**\n\nAt the end of the year, you're tempted everywhere\u2014at meals, at the secretary's candy jar, under the mistletoe. If you can make it through the biggest fat wave of the year, the other 10 months will be smoother than a supermodel's freshly shaved legs. Sure, you'll hit the occasional holiday feast. Go ahead and enjoy them, but know that you can always make decisions to satisfy your tastes and cravings without drowning yourself in fat.\n\n### **Super Bowl Party**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:**\n\n_2 handfuls assorted nuts_\n\n_Baked tortilla chips with salsa_\n\n_2 slices turkey or roast beef hoagie with cheese_\n\nTOTAL: 848 calories, 36 g fat (10 g saturated), 2372 mg sodium\n\n**Not that:**\n\n_Celery with onion dip_\n\n_Nacho chips with 7-layer bean and cheese dip_\n\n_15 hot wings_\n\nTOTAL:1264 calories, 86 g fat (26 g saturated), 2229 mg sodium\n\n### **St. Patrick's Day**\n\n**THE LESSER OF TWO EVILS**\n\n**Drink this:** _Guinness (12 ounces)_ 126 calories; _Beamish Irish Cr\u00e8me Stout (12 ounces)_ 146 calories; _Amstel Light (12 ounces)_ 99 calories\n\n**Not that:** _Beck's Dark (12 ounces)_ 146 calories; _Sam Adams Cream Stout (12 ounces)_ 195 calories; _Michelob Light (12 ounces)_ 134 calories\n\n### **Easter Dinner**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:**\n\n_1 8-ounce serving lamb_\n\n_1 cup roasted potatoes with olive oil_\n\n_1 cup steamed spinach_\n\n_Salad with oil and balsamic vinegar_\n\nTOTAL: 958 calories, 64 g fat (18 g saturated), 502 mg sodium\n\n**Not that:**\n\n_1 8-ounce serving ham_\n\n_1 cup potato salad_\n\n_1 cup corn_\n\n_Caesar salad with Caesar dressing_\n\nTOTAL: 1245 calories, 72 g fat (18 g saturated), 1818 mg sodium\n\n### **Easter Basket**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _Jelly beans (about 20 pieces)_ 82 calories, 0 g fat, 11 mg sodium\n\n**Not that:** _Werther's Original_ Per 6 pieces: 120 calories, 2 g fat (2 g saturated fat), 120 mg sodium\n\n**Eat this:** _2 marshmallow peeps_ 64 calories, 0 g fat, 6 mg sodium\n\n**Not that:** _1 Cadbury Cr\u00e8me Egg_ 170 calories, 6 g fat (3.5 g saturated), 25 mg sodium\n\n### **Fourth of July**\n\n**THE LESSER OF TWO EVILS**\n\n**Eat this:** _1 quarter-pound grilled burger with ketchup, mustard, lettuce, tomato, slice of cheese 1 cup homemade coleslaw_\n\nTOTAL: 548 calories, 24 g fat (9 g saturated), 806 mg sodium\n\n**Not that:** _2 fried chicken legs \u00bd cup macaroni salad_\n\nTOTAL: 1093 calories, 70 g fat (16 g saturated), 1061 mg sodium\n\nChapter 11\n\n# **CRANK UP THE FAT BURN**\n\n# The Abs Diet Workout\n\n**B ETWEEN YOUR JOB,** family, home, and Google addiction, you probably feel pushed and pulled in more directions than a piece of Silly Putty. Me, too.\n\nThe clock seems to move faster and faster, and we're all running to keep up. We crave less stress, more time, and 30 uninterrupted seconds during which nobody nags us about meeting deadlines, caulking cracks, or playing a 14th consecutive game of Candy Land.\n\nWe're all busy. But we're not too busy to exercise.\n\nHere's the proof: The average American spends 28 hours a week watching television. If you watch only one-half as much TV, that still means you have more than a solid half a day of quality time you're spending with O'Reilly, the Donald, or the Three Stooges (Simon, Paula, and Randy) every week.\n\nGive up just one of those hours each week\u2014just one\u2014and you can change your body and your life forever.\n\nI said at the very beginning of this book that the Abs Diet was no ordinary diet, and I meant it. To me, one of the biggest errors in most mainstream diets is that they treat exercise the way the prom queen treats the chess champ\u2014with indifference. They operate on ho-hum exercise principles: Great if you do it, so what if you don't.\n\nBut most plans overlook one simple truth: The best way to lose fat is to build muscle and let that muscle eat away flab from the inside out. Each pound of muscle you build means your body will burn an extra up to 50 calories a day just sitting still. Add 5 pounds of muscle\u2014something you can easily do over the course of 6 weeks\u2014and you're now burning an extra up to 1,750 calories a week! When it comes down to it, exercise can do just as much to reshape your body as any broccoli floret can.\n\nOf course, the ABS DIET POWER 12 gives you all the weapons you need to fight fat: protein, healthy fat, whole grains, and fiber to keep you full, to feed your body nutrients, and to ward off potentially devastating hunger pangs. But you'll accelerate all of your gains with the secret weapon in your dietary artillery: muscle.\n\nI'm not talking bar-bouncer, piano-lifting muscle. I'm talking about lean muscle mass, which works as your body's natural metabolism booster. To make the Abs Diet as effective as possible, you need to add a muscle-building and strength-training workout to your program.\n\nNow, if the only dumbbells you know live four doors down, then the prospect of taking up strength training may seem intimidating. It shouldn't be. Like the Abs Diet itself, the Abs Diet Workout is designed to be fast, simple, effective, and convenient. I know you're not going to spend hours a day in the gym, I know you're not going to enjoy a workout program that's all pain and no gain, and I know you're not going to stick with it if you don't see impressive results, fast. That's why I've created a workout program that will build muscle, melt away fat, and reintroduce you to your abs\u2014all in just 20 minutes a day, 3 days a week.\n\nHow do I know it works? Besides the countless success stories of people who reshaped their bodies with minimal time at the gym, one study found that you can put on 6 pounds of muscle and lose 15 pounds of fat in 6 weeks by following the exercise principles used in the Abs Diet Workout. We're talking just 20 minutes a day, just 3 days a week.\n\nDon't believe it? Think you need to quit your job and spend your life in the gym to see results? Well, check this out: Scientists at the University of Glamorgan in Wales studied 16 weight lifters doing either one or three sets of upper-body exercises three times a week. Those who did one set gained just as much muscle\u2014and burned twice as much fat\u2014as the three-set group. In other words, the less time you spend, the better your results!\n\nWhat's great about this program is that you can do it in a gym or in your house, you don't need fancy equipment, and you can finish it before _ER_ is even halfway over. (In fact, I've even included a no-weight workout that you can do in your backyard, in a hotel room, in a prison cell if need be. No excuses, remember?) One hour a week. That's all I want, and that's all you need.\n\n## **Your 3-Days-a-Week Muscle-Building, Fat-Burning Program**\n\nThink of muscle as your body's python and fat as a quivering mouse. In the battle between the two, the python will always win. And the bigger your python, the more mice it eats.\n\nWhat's great about weight training is that it burns fat in three ways. First, there are the calories you burn off breaking a sweat. Second, there's the fact that new muscle eats up calories, making your body more efficient at burning fat. And third, there's the \"afterburn\"\u2014the additional calories burned off in the hours immediately following your workout. All kinds of exercise raise your metabolism and give you an afterburn. But the effects of weight training far outstrip those of aerobic exercise. In one study, researchers found that the increased calorie burn of aerobics (that is, steady-state cardiovascular exercise like running or cycling) lasted only 30 minutes to an hour after a workout. In subjects who trained with weights, the increased metabolism lasted as long as 48 hours. That's 2 days during which your body burns fat after the fact.\n\nAnd you don't need to push iron like a Nebraska offensive tackle to achieve the afterburn effect. All you have to do is employ the two major Abs Diet components of muscle building: circuit training and compound exercises.\n\n_**Circuit training.**_ There are many different ways to lift weights. A lot of people employ the lift-rest-lift-rest approach to weight training, and that's fine. But my guess is that you're interested in building the most muscle and burning the most fat in the least amount of time possible. So I've built the Abs Diet Workout around circuit training. In this type of training, there is no rest phase, no part of the workout where you're standing around the water fountain looking lost. Circuit training means that your body is constantly working, constantly improving.\n\n## **BONUS: ADVANCED ABS!**\n\nIf you're starting to feel as if your abs are tighter than the jeans you just pulled out of the dryer, then you're probably ready for a more challenging abdominal exercise. Try this one. Lie on your back on a Swiss ball, with your knees bent at 90 degrees, your feet flat, and your hands behind your ears. Keeping your right foot planted, lift your left foot off the floor and bring it toward you as you curl your torso up and to the left so that you right elbow meets your left knee. It's like the classic bicycle maneuver, and it works your entire core at once. Do 12 repetitions. Then plant your left foot on the floor and curl toward your right knee for another 12 reps.\n\nSimply put, circuit training involves moving from one exercise to the next with little rest in between. Once you complete the circuit of 9 or 11 exercises, you rest for 2 minutes and then start again. In your time-compressed life, circuit training works because it means you're working the most muscles in the least amount of time; plus, with little rest, your heart rate stays elevated to give you an additional calorie burn. A recent Ohio University study found that a short-but-hard workout was effective in burning fat. Using a circuit of three exercises in a row for 31 minutes, the subjects were still burning more calories than normal 38 hours after the workout.\n\n_**Compound exercises.**_ Just like compound interest, compound exercises give you more back than you put in. Compound exercises hit many muscles during a single move (the squat engages 256 muscles at once!), so that you're gaining the most benefit from your workout. Compound exercises also ensure that you work the largest muscles in your body (like your chest, legs, and back). Larger muscles take more calories to maintain than smaller ones, making your workout that much more effective.\n\nTo put it all together, all you need is a gym membership or, if you're exercising at home, a set of dumbbells and a bench. (For dumbbells, I recommend you invest in an adjustable pair so that you can change weights for different exercises and up the resistance as you grow stronger.) Though it doesn't matter where you do the circuit, it does matter when you do it. Stick to 3 days a week with at least 1 day a week of rest in between to allow your muscles to recover and grow.\n\n## **ULTIMATE NO-WEIGHT WORKOUT**\n\nSometimes, you can't get to the gym or even access the workout gear you have stashed in your basement. Maybe you're stuck in some god-forsaken Motel 6. Maybe you're trapped at your in-laws' house. Maybe the Feds discovered you gave stock advice to Martha Stewart. Well, that's still no excuse. This full-body routine requires no equipment and only 8 minutes of your time. Perform each exercise below for 30 seconds. Do one move after another without rest and repeat the sequence without rest for a total of four times, if you can.\n\nThe intensity of this workout conditions your muscular and cardiovascular systems, helps improve flexibility, and melts fat. You'll burn hundreds of calories.\n\n**Jumping Jack.** Just like grade school. Start with your hands on your hips and your feet together. Raise your hands out to your sides and up overhead as you move your feet out to the sides. Then bring your feet back together and lower your hands to your sides.\n\n**Split Hop.** Stand with your hands on your hips and your feet together. Move your left foot 6 inches forward and your right foot 6 inches back. Now jump up and switch leg positions so that your right foot is forward.\n\n**Squat Thrust with Pushup.** Stand with your arms at your sides. Bend your knees and lower your hands to the floor. Kick your legs behind you so that you're in a Pushup position. Now do a Pushup. Thrust your knees to your chest so that your feet are back underneath you and stand back up.\n\n**Mountain Climber.** Get back in the Pushup position and kick your knees to your chest, one leg at a time. Alternate thrusting your knees forward, like you're running, so that one leg is extended when one knee is forward.\n\n## **The Abs Diet Circuit**\n\nPerform each exercise once, then move immediately to the next exercise with only 30 seconds of rest in between. When you reach the end of the circuit, rest for 2 minutes and then repeat. Beginners can start with light weights and one circuit. More advanced lifters can do two or three circuits with weights that they can comfortably handle for at least 8 repetitions but no more than 12 repetitions.\n\n**EXERCISE** | **REPETITIONS** | **REST** | **SETS**  \n---|---|---|---  \nSquat | 10\u201312 | 30 seconds | 2  \nBench Press | 10 | 30 seconds | 2  \nPulldown | 10 | 30 seconds | 2  \nMilitary Press | 10 | 30 seconds | 2  \nUpright Row | 10 | 30 seconds | 2  \nTriceps Pushdown | 10\u201312 | 30 seconds | 2  \nLeg Extension | 10\u201312 | 30 seconds | 2  \nBiceps Curl | 10 | 30 seconds | 2  \nLeg Curl | 10\u201312 | 30 seconds | 2\n\n_Note:_ One day a week, add the following two exercises (do the Traveling Lunge after the Pulldown, and the Step-Up after the Upright Row). Because your legs contain your body's largest muscles, the fat-burning potential increases with a little extra time spent working your legs muscles.\n\n**EXERCISE** | **REPETITIONS** | **REST** | **SETS**  \n---|---|---|---  \nTraveling Lunge | 10\u201312 | 30 seconds | 2  \nStep-Up | 10\u201312 | 30 seconds | 2\n\n**Squat.** Hold a barbell with an overhand grip so that it rests comfortably on your upper back. Set your feet shoulder width apart, and keep your knees slightly bent, back straight, and eyes focused straight ahead. Slowly lower your body as if you were sitting back into a chair, keeping your back in its natural alignment and your lower legs perpendicular to the floor. When your thighs are parallel to the floor, pause, then return to the starting position.\n\n_**Home variation:**_ Same, but with one dumbbell in each hand, your palms facing your outer thighs.\n\n**Bench Press.** Lie on your back on a flat bench with your feet on the floor. Grab the barbell with an overhand grip, your hands just beyond shoulder width apart. Lift the bar off the uprights, and hold it at arm's length over your chest. Slowly lower the bar to your chest. Pause, then push the bar back to the starting position.\n\n_**Home variation:**_ Pushups. Get in a Pushup position with your hands shoulder width apart. Bend at the elbows while keeping your back straight, until your chin almost touches the floor, then push back up.\n\n**Pulldown.** Stand facing a lat pulldown machine. Reach up and grasp the bar with an overhand grip that's 4 to 6 inches wider than your shoulders. Sit on the seat, letting the resistance of the bar extend your arms above your head. When you're in position, pull the bar down until it touches your upper chest. Hold the position for a second, then return to the starting position.\n\n_**Home variation:**_ Bent-Over Row. Stand with your knees slightly bent and shoulder width apart. Bend over so that your back is almost parallel to the floor. Holding a dumbbell in each hand, let your arms hang toward the floor. With your palms facing in, pull the dumbbells toward you until they touch the outside of your chest. Pause, then return to the starting position.\n\n**Military Press.** Sitting on an exercise bench, hold a barbell at shoulder height with your hands shoulder width apart. Press the weight straight overhead so that your arms are almost fully extended, hold for a count of one, then bring it down to the front of your shoulders.\n\n_**Home variation:**_ Sitting on a sturdy chair instead of a bench, hold one dumbbell in each hand, about level with your ears. Push the dumbbells straight overhead so that your arms are almost fully extended, hold for a count of one, then return to the starting position.\n\n**Upright Row.** Grab a barbell with an overhand grip and stand with your feet shoulder width apart and your knees slightly bent. Let the barbell hang at arm's length on top of your thighs, thumbs pointed toward each other. Bending your elbows, lift your upper arms straight out to the sides, and pull the barbell straight up until your upper arms are parallel to the floor and the bar is just below chin level. Pause, then return to the starting position.\n\n_**Home variation:**_ Same, using one dumbbell in each hand.\n\n**Triceps Pushdown.** While standing, grip a bar attached to a high pulley cable or lat machine with your hands 6 inches apart. With your elbows tucked against your sides, bring the bar down until it is directly in front of you. With your forearms parallel to the floor (the starting position), push the bar down until your arms are extended straight down with the bar near your thigh. Don't lock your elbows. Return to the starting position.\n\n_**Home variation:**_ Triceps Kickback. Holding a light dumbbell in each hand, stand with your knees slightly bent and shoulder width apart. Bend over so that your back is almost parallel to the ground. Bend your elbows to about 90-degree angles, raising them to just above the level of your back. This is the starting position. Extend your forearms backward, keeping your upper arms stationary. When they're fully extended, your arms should be parallel to the ground. Pause, then return to the starting position.\n\n## **WHAT'S MY MOTIVATION?**\n\nNo matter how dedicated you are to your weight-loss goals, there are some days when you don't feel like getting out of bed, much less getting to the gym. How do you keep making progress? Here are a few tricks to keep you from falling off the exercise wagon.\n\n**Make a bet.** Challenge a coworker to a contest\u2014the first to drop 10 pounds, best of seven in one-on-one, and so forth. Competition is the ultimate motivator.\n\n**Switch training partners.** Working out with a partner who will hold you accountable for showing up at the gym works well\u2014for a while. But the longer you know him, the easier it is to back out of a workout. Find a new one every few months.\n\n**Strike an agreement with your family.** The rule: You get 1 hour to yourself every day, provided that you use it for exercise (and reciprocate the favor).\n\n**Schedule a body-composition test every 2 months.** The short-term end date will keep you focused to keep moving forward.\n\n**Leg Extension.** Sitting on a leg extension machine with your feet under the footpads, lean back slightly, and lift the pads with your feet until your legs are extended.\n\n_**Home variation:**_ Stand with your back flat against a wall. Squat down so that your thighs are parallel to the ground. Hold that position for as long as you can. That consists of one set. Aim for 20 seconds to start and work your way up to 45 seconds.\n\n**Biceps Curl.** Stand while holding a barbell in front of you, palms facing out, with your hands shoulder width apart and your arms hanging in front of you. Curl the weight toward your shoulders, hold for a second, then return to the starting position.\n\n_**Home variation:**_ Same, only use a set of dumbbells instead.\n\n**Leg Curl.** Lie facedown on a leg curl machine and hook your ankles under the padded bar. Keeping your stomach and pelvis against the bench, slowly raise your feet toward your butt, curling up the weight. Come up so that your feet nearly touch your butt, and slowly return to the starting position.\n\n_**Home variation:**_ Lie down with your stomach on the floor. Put a light dumbbell between your feet (so that the top end of the dumbbell rests on the bottom of your feet). Squeeze your feet together and curl them up toward your butt.\n\n**Traveling Lunge.** Rest a barbell against your upper back. Stand, with your feet hip width apart, at one end of the room; you need room to walk about 20 steps. Step forward with your left foot, and lower your body so that your left thigh is parallel to the floor and your right thigh is perpendicular to the floor (your right knee should bend and almost touch the floor). Stand and bring your right foot up next to your left, then repeat with the right leg lunging forward.\n\n_**Home variation:**_ Use dumbbells, holding one in each hand with your arms at your sides. If you don't have enough space, do the move in one place, alternating your lead foot with each lunge.\n\n**Step-Up.** Use a step or bench that's 18 inches off the ground. Place your left foot on the step so that your knee is bent at 90 degrees. Your knee should not advance past the toes of your left foot. Push off with your left foot and bring your right foot onto the step, keeping your back straight. Now step down with the left foot, followed by the right. Alternate the leading foot, or do all of the repetitions leading with one foot and then alternating. Once you're comfortable, add dumbbells.\n\n_**Home variation:**_ Same, only use a staircase instead of a step (if you don't have one).\n\n## **Abdominal Exercises: Before Your Circuit**\n\nYou'd think with a name like the Abs Diet, I'd be asking you to spend more time working your abs than J.Lo spends working the counters at Tiffany & Co. But finding your abs isn't as much about abdominal exercises as it is about changing your body composition to eliminate the belly fat that's covering them. Once you do that, then you can concentrate on building the muscle. It's sort of like a construction company trying to build a housing development. If you don't clear the lot of sand, debris, and trees, you aren't going to have any room to build the house. But if you clear the lot and lay the foundation, then you can build a house that everyone will see.\n\nTo work your abs, you'll want to hit all five parts of your abdominal region (as outlined below). At the beginning of your strength circuit (2 or 3 days a week), do these five exercises in a circuit with little rest in between each. Start with one set, but work up to doing two or three sets. To vary your exercises, see _The Abs Diet,_ which has 50 variations of abdominal moves.\n\n**Traditional Crunch** (works the upper part of the rectus abdominis, which is the six-pack muscle that helps you maintain good posture). Lie on your back with your knees bent and your hands behind your ears. Slowly crunch up, bringing your shoulder blades off the ground. Do 12 to 15 repetitions.\n\n**Flutter Kick** (works the lower part of the rectus abdominis). Lie on your back, raise both feet off the ground, and scissor-kick one leg over the other. Do 20 repetitions.\n\n**Saxon Side Bend** (works the external and internal obliques, which extend diagonally down the sides of your waist and rotate the torso). Hold a pair of lightweight dumbbells over your head, in line with your shoulders, with your elbows slightly bent. Keep your back straight, and slowly bend directly to your left side as far as possible without twisting your upper body. Pause, return to an upright position, then bend to your right side as far as possible. Do 6 to 10 repetitions.\n\n**Bridge** (works the transverse abdominis, which is known as the girdle because it compresses the abdomen). Start to get in a Pushup position, but bend your elbows and rest your weight on your forearms instead. Your body should form a straight line. Pull your abdominals in. Hold for 20 seconds, breathing steadily. Do 1 to 2 repetitions.\n\n**Superman** (works the lower back, which anchors all of the abdominal muscles). Lie with your stomach on the ground and your arms in front of you and legs behind you (in Superman-flying position). Lift your arms and legs about 6 inches off the ground and hold for as long as you can. Repeat three times.\n\n## **Interval Training: 1 Day a Week**\n\nBack in the '80s, the only thing more popular than big hair and shoulder pads was running. Weight-loss experts touted long, steady aerobic exercise as nearly the best way to burn fat, build endurance, and keep the heart pumping.\n\nAnd aerobic exercise (running, cycling, swimming) is good for you. I've run the New York City Marathon twice myself, and I can attest to the fact that cardiovascular exercise strengthens your heart, burns calories, and decreases stress.\n\nBut cardio has two significant drawbacks. First, it only burns calories while you're doing it, not afterward. And second, it does nothing to build muscle. Unless . . . unless you try interval training.\n\nInterval training refers to a shorter, more intense method of working out. Instead of long, slow, boring runs or rides, interval training intersperses short bursts of high-intensity exertion with periods of slow, more restful exercise. In a Canadian study from Laval University, researchers measured differences in fat loss between two groups of exercisers following two different workout programs. The first group rode stationary bikes at a steady pace four or five times a week and burned 300 to 400 calories per 30- to 45-minute session. The second group did the same, but only one or two times a week, and they filled the rest of their sessions with short intervals of high-intensity cycling. They hopped on their stationary bikes and pedaled as quickly as they could for 30 to 90 seconds, rested, and then repeated the process several times per exercise session. As a result, they burned 225 to 250 calories while cycling, but they burned more fat at the end of the study than the workers in group one. In fact, even though they exercised less, their fat loss was nine times greater. Researchers said that the majority of the fat-burning took place after the workout.\n\nSo instead of asking you to spend 30 minutes on a stairclimber or a stationary bike every day, I want you to add one simple interval workout per week to complement your strength training. Your mode of transportation isn't important, so pick whatever activity you prefer. What's important is making sure you change gears. You can vary it in whatever time frames you want (1-minute high-intensity, 1-minute low-intensity, or maybe build up with 30 seconds of high, rest, then 45 seconds, then rest, and so on). Always warm up and cool down for at least 5 minutes at the beginning and end of each interval workout.\n\n## **ULTIMATE TOTAL-BODY DUMBBELL EXERCISE**\n\nDumbbells are one of the smartest fitness investments you can make. They're portable and affordable, and they condition your body even more effectively than standard barbells, because they teach your body to balance and allow for a greater range of motion.\n\nI've created here a super-simple, super-fast dumbbell workout that will hit your whole body at once. Do one set of 12 repetitions, and you're set for the day.\n\n  Grab a pair of light dumbbells and get into a Pushup position, with your arms straight and directly beneath your shoulders.\n\n  Do a Pushup. Then bring your feet underneath you, one foot at a time.\n\n  Keeping your back flat, stand up. (This is a deadlift.)\n\n  From the standing position, curl the weights up to your shoulders.\n\n  Swing your elbows out to your sides so that the weights are above your shoulders.\n\n  Lower your body until your thighs are parallel to the floor. Pause, then stand up as you press the weights overhead.\n\n## **Sample Workout Schedule**\n\nYou'll do circuit training 3 days a week, making sure you have at least 1 day of rest in between days. Additionally, do an interval workout 1 day a week. You also have the option of doing light cardiovascular exercise (cycling, walking, swimming, tennis, or golf without the cart) on your off days. On these days, make having fun your main goal\u2014don't think about calorie burns, afterburns, George Burns, or any other kind of burns.\n\n**Monday: Circuit Training**\n\n**Tuesday (optional): Walking at brisk pace or light cardiovascular exercise**\n\n**Wednesday: Circuit Training**\n\n**Thursday (optional): Walking at brisk pace or light cardiovascular exercise**\n\n**Friday: Circuit Training**\n\n**Saturday: Interval Workout**\n\n**Sunday: Off**\n\n## **The Abs Diet Workout Worksheet**\n\nLike naked bodies, all workouts are not made alike. If you're currently not exercising, then anything is better than a nightly chips festival on the love seat. But for the most powerful workouts for burning fat and adding muscle, follow the Abs Diet Workout Worksheet.\n\nYour goal here is to amass a total of 40 points per week. For each activity, you get full credit for doing the activity continuously for 20 minutes. Exercises in the Abs Diet Workout give you the highest number of points, but if you can't keep to that workout schedule, you can sneak in a few extra points here and there to make your workout work. Don't try to go over 40 points a week, though\u2014there's no extra credit for total exhaustion.\n\n**10 POINTS (FOR EACH 20 MINUTES SPENT)**\n\nAbs Diet Circuit\n\nAbdominals Circuit\n\nInterval Training (solo sport: running, swimming, cycling, or machine)\n\n**6 POINTS (FOR EACH 30 MINUTES SPENT)**\n\nAbs-specific classes\n\nBasketball (full-court)\n\nBoot Camp classes\n\nBoxing\n\nBull running, Pamplona\n\nCalisthenics: pushups, pullups, situps\n\nCross-country skiing, hilly\n\nHiking, hilly\n\nHockey, inline or ice\n\nMountain biking, intermediate to advanced, hilly course\n\nPilates, advanced\n\nPower lifting\n\nSnowshoeing, hilly\n\nSpinning classes\n\nSports-conditioning classes\n\nStairclimbing, stadium stairs\n\nStrength training, noncircuit\n\nVolleyball, beach, competitive\n\n**4 POINTS (FOR EACH 30 MINUTES SPENT)**\n\nBasketball, (half-court)\n\nBOSU classes\n\nDodgeball\n\nDownhill skiing, intermediate to advanced\n\nInline skating, steady\n\nKickboxing classes\n\nMartial arts\n\nPilates, beginner\n\nRacquetball\n\nRowing machine, steady pace\n\nRugby\n\nSoccer\n\nStep classes\n\nStrength-training, resistance bands\n\nStrength-training, ultra-light weights\n\nSurfing\n\nSwimming, steady pace\n\nTennis, competitive\n\nUltimate Frisbee\n\nVolleyball, indoor\n\nYoga, advanced or power\n\n**3 POINTS (FOR EVERY 30 MINUTES SPENT)**\n\nAdventure racing\n\nCross-country skiing, flat\n\nCycling, road, steady pace, flat\n\nDance-aerobic classes\n\nDownhill skiing, easy\n\nFishing, big-league tuna\n\nGolf, without cart\n\nHiking, flat\n\nJogging, steady pace\n\nKayaking\/canoeing\n\nRock climbing\n\nSnowboarding\n\nStairclimbing machine, steady pace\n\nTennis, recreational\n\nUrban Rebounding classes\n\nVolleyball, beach, recreational\n\nWalking, brisk\n\nYoga, beginner\n\n**2 POINTS (FOR EVERY 30 MINUTES SPENT)**\n\nBasketball, solo\n\nBowling\n\nFishing, recreational\n\nFrisbee golf\n\nGardening\n\nGolf, driving range\n\nGolf, with cart\n\nIce skating, recreational\n\nSoftball\n\nStretching, general\n\nWalking, slow\n\nWater aerobics classes\n\nWrestling, with kids\n\nYoga, meditative\n\n### **ABS DIET SUCCESS STORY**\n\n## **MORE ENERGY TO WORK, TO PLAY, AND TO CARE FOR HIS FAMILY**\n\n**Name:** Pete Hemmer\n\n**Age:** 40\n\n**Weight, Week 1:** 244\n\n**Weight, Week 6:** 231\n\n**Weight, Week 10:** 217\n\nPete Hemmer knew he had to get his body back in order. He had reached 286 pounds and was trying to care for premature triplets. He was unhappy\u2014and out of shape, out of breath, out of energy. \"When I stepped on the scale and saw 286, I said at that moment, 'I'm going to be 300 pounds. If I keep going where I'm going, I'll be dead by the time I'm 40,'\" he says. That's when he knew he needed to change, so he just tried adjusting portion sizes and making smart choices about eating. Over the course of about a year and a half, he whittled himself down to 244 pounds\u2014still far off from where he wanted to be. Then he read about the Abs Diet and told his wife, Krista, \"That's the diet plan I need.\n\n\"I went out that weekend and bought the book, read it in a weekend from cover to cover, and my wife said I wouldn't shut up talking about it,\" he says. Then Krista asked him if it was something she could try, too. On Monday, they made their plan, and on Tuesday, they started.\n\n\"As I was pushing the cart down the aisle, I noticed there weren't any boxes in my cart. It was fresh vegetables, dairy and meat, and there were no frozen pizzas. All the stuff that had becomes staples of our diet were missing,\" he says. \"It was just something that struck me\u2014that even though we had been doing well for a year, there was a lot of room for improvement.\"\n\nHemmer, whose goal was to be in better shape by his 40th birthday than he was at his 30th, instantly saw results\u2014in fat loss and in energy levels. \"I noticed when I was going up the steps to a loading dock, I went up two steps at a time and I didn't even blink, and I realized I had never done that. I just felt lighter.\" After the first 2 weeks, Krista even made the comment that they had eaten better the past 2 weeks than they ever had before in their 12 years of marriage.\n\nThe rewards kept coming: Krista has lost 15 pounds and a dress size, Pete's down from a nearly 44-size pants to almost a 34, it's easier to take care of the kids, friends who haven't seen them in months don't even recognize them, and they're even planning a sea kayaking trip. \"That wouldn't have happened before,\" Hemmer says. \"For us, sitting on the sofa and watching a pay-per-view movie was the extent of our physical activity.\"\n\nWhen he went to the gym to have his body composition checked after 6 weeks, the woman who ran the machine looked at Hemmer's 6-week improvement and asked him what he was doing. \"I told her briefly about that the Abs Diet, and she said that even the gym's weight-loss program doesn't get such good results,\" Hemmer says. \"She also said that it appeared that my body had become efficient at burning fat. I told her politely that it was sort of the whole point.\"\n\nAnd, of course, there's also one other side effect: \"Our sex life has improved dramatically,\" Pete says. \"Krista says that if I get in any better shape, she's going to need smelling salts.\"\n\n# **ABS DIET FAQ**\n\n**How does the Abs Diet differ from other diets?**\n\nFollowing most popular diet books is like sending your body to prison\u2014they're all about deprivation and restriction. These diets may work in the short term by shackling you to keep you away from certain foods and calories. But once you get a sniff of freedom (in the form of fries and shakes), you're more likely to try to escape. And you know the severe penalties for trying to escape\u2014a life sentenced to a fat gut.\n\nThe Abs Diet is about gaining, not restricting. You gain a leaner, firmer body, more control over your life, and even more opportunities to enjoy the foods you love. By eating six times a day, you'll keep your energy levels high and your hunger always at bay, while training your body to burn calories and make itself lean. You'll feed yourself the foods that force your body to lose weight, to burn more calories.\n\n**What about really popular ones, like Atkins and South Beach?**\n\nThe Atkins Diet eliminates practically all carbohydrates for the first part of the plan, leaving you with only foods that contain protein and fat. It's a gimmick that works\u2014in the short term. By restricting the foods you eat to only a handful of them, you'll automatically drop pounds because you've dramatically reduced your total calories. But you'll also dramatically reduce your intake of vitamins, minerals, and fiber, while upping your intake of artery-clogging saturated fats. The one good thing about Atkins: It eliminates empty carb calories like doughnuts and Doritos. But on the Abs Diet, you dump the lousy carbs and keep the healthy and delicious ones (whole-grain bread, cereal, fruit, vegetables), which will keep you full, fuel your body, and dampen your cravings.\n\nThe South Beach Diet is based on sound principles of healthy nutrition, and it's a pretty good choice for pure weight loss. But the Abs Diet isn't just about losing weight\u2014it's about turning unsightly flab into lean, sexy muscle. That's why, unlike South Beach, the Abs Diet features muscle-building exercise as part of its foundation. Muscle exponentially speeds up the fat-burning process\u20141 pound of muscle requires your body to burn up to 50 calories a day just to maintain that muscle. This plan combines exercise with the foods that most promote muscle growth.\n\nThere are many diets out there today, and as long as people have the opportunity to choose a course for better health, that's what ultimately counts. Frankly, I hate the word _diet,_ because diet implies that you have to eliminate and restrict, but to call this the Abs Nutritional Superiority Plan seemed a bit much.\n\n**How much weight will I lose?**\n\nI can't give you a firm number, because people's bodies have more variables than a calculus textbook. I can tell you that some people who have used the Abs Diet have lost 25 pounds in 6 weeks. That doesn't necessarily mean that you will, but it means that you might or that you could. Much of your success depends on so many factors\u2014including your intensity of exercise and your starting weight. But we've seen a lot of people who've lost somewhere between 10 and 15 pounds in the first 6 weeks while also gaining a few pounds of muscle, which helps them keep burning fat after that initial 6-week surge.\n\n**What percentage of body fat do you have to get to so that you can see your abs?**\n\nFinding your abs might feel as daunting as digging through sand to find a buried treasure. But if you dig diligently enough\u2014and by that I mean burning off that extra, unwanted flab\u2014you will find the treasure. As far as the specific body-fat percentage, it varies. I've seen both men and women with body-fat percentages in the teens with defined stomach muscles. For most men, you need to get to around 10 percent. The good news is that if you're exercising regularly, then at least 80 percent of every pound you lose will be fat.\n\n**Will it work for women?**\n\nAbsolutely. It doesn't take a Peeping Tom to know that men and women have some distinct physiological differences, but in this case, what works for a man will also work for a woman. This eating plan revolves around eating foods that are good for everyone, and the exercise plan is about building lean muscle mass, not line-backer muscle mass. The key is not to count calories but to eat six meals a day to keep you properly fueled. A woman's portion sizes can be a little smaller than a man's, but by eating well-balanced meals with the ABS DIET POWER 12, anyone can lose fat.\n\n**So the ABS DIET POWER 12 foods are the only ones I can eat?**\n\nAbsolutely not. You build your diet around the ABS DIET POWER 12\u2014and make sure you have several of them at every meal and snack. You can supplement with other foods, but if your meals are centered on the Powerfoods, you'll ensure yourself a well-balanced diet that keeps you satiated and provides you with the ingredients that help keep your body properly fueled and turn it into a fat-burning machine.\n\n**What about portion control? Can I eat any amount I want?**\n\nIf you're eating these Powerfoods, they should take care of your hunger so that you don't have the physical urge to eat a lot. By doing things like eating six times a day and making sure you get enough fiber, you'll decrease the chance of shoveling enormous amounts of food into your mouth. That said, it's always smart to be aware of how much you eat\u2014especially when you're starting out. You can cover your plate with the ABS DIET POWER 12, but there's no need to pile the food so high that you need a city council permit to construct your plate. Let's just say that a height restriction is in effect.\n\n**The Abs Diet includes a weekly cheat meal. Does that mean I really can eat anything?**\n\nYes. One time a week, you can eat anything. You want a Bavarian cream? Eat it. You want a plate of wings? Eat it. You want fried alligator? Eat it. The cheat meal is designed to reward you for good work throughout the week and help keep cravings in check. And what we found is that once people start seeing results, many don't even want the cheat meal\u2014because the ABS DIET POWER 12 satisfy so many different kinds of cravings. The only caveat is that you must limit yourself to just one cheat meal a week. Do it any more, and you might as well be on the Doughnut Diet.\n\n**Can I make food substitutions for the ABS DIET POWER 12?**\n\nMany of the 12 categories are broad enough to include many different dietary choices. Low-fat dairy includes milk, yogurt, cheese, and lean meat can include turkey, beef, fish, and chicken, for example. We do offer some substitution examples for different kinds of foods. Even if you're allergic to one category of food\u2014nuts, for example\u2014you can still figure out what ingredient you're missing by not eating that food and make up for it somewhere else. So having avocado or pumpkin seeds can give you the monounsaturated fats found in nuts\u2014without the danger of an allergic reaction. (That said, if you do suffer from food allergies, consult your doctor before trying this or any other diet plan.)\n\n**Some of the smoothie recipes have cooked instant oatmeal. That sounds gross. How does it taste?**\n\nRecipes are a little like CD collections\u2014what works for me might not work for you. But from the smoothies I've had, I can't even taste the oatmeal after it's blended\u2014especially when there are berries or chocolate whey powder in them. So be brave and give it a go: Oatmeal adds bulk to the smoothie as well as the all-important satiating fiber.\n\n**What about this issue of targeting body fat? I thought you couldn't spot-reduce.**\n\nWell, you can't spot-reduce per se. That's a myth. But when you lose weight on the Abs Diet\u2014particularly if you're doing some moderate exercise at the same time\u2014you'll lower your body fat and will likely notice much of that loss around your midsection, since that's where most of the fat accumulates. And if you are doing ab workouts and strengthening those muscles, then as the fat peels away, you'll begin to see the washboard.\n\n### **ABS DIET SUCCESS STORY**\n\n## **HE INSPIRED HIS WIFE\u2014AND HIS SON**\n\n**Name:** Bill and Kathy Bartz\n\n**Bill's age:** 56\n\n**Height:** 5\u20329\u2033\n\n**Weight, Week 1:** 187\n\n**Weight, Week 6:** 170\n\n**Weight, Week 12:** 166\n\n**Body-fat percentage, Week 1:** 18.7\n\n**Body-fat percentage, Week 12:** 10.2\n\n**Kathy's height:** 5\u20326\u2033\n\n**Weight, Week 1:** 126\n\n**Weight, Week 12:** 118\n\nLast year, Bill Bartz made a New Year's resolution: He wanted to see his six-pack before the end of the year. He had been lifting weights regularly for 15 years, so he had a lot of bulky muscle mass, but he was missing the definition.\n\n\"I saw the author on _Good Morning America_ on the Fourth of July, and I thought, hmmmmm,\" Bartz says. \"So I decided to go to Border's and get the book. I just started reading it and it all made sense. And what I really liked was how it took other diets and blew them apart.\"\n\nSo Bartz decided to jump on the Abs Diet, and the pounds jumped off.\n\n\"One day, as I was getting out of the shower with a towel wrapped around me, my wife said, 'Whoa, that thing is working.'\" So Bartz's wife, Kathy, joined in, too. With not much weight to lose, Kathy went from 126 pounds to 118.\n\nBartz took ab-specific classes 3 days a week at the gym, continued lifting weights, and incorporated interval training into his workouts.\n\nBut maybe the best outcome of the program is what it's done for his overall health. He's taking part in a clinical study in which his blood pressure is measured every 6 months. Since he started, his BP dropped from 130\/90 to around 115\/70. \"When I went there, they were blown away,\" he says. \"Right away, they noticed a difference in weight and they noticed a drastic drop in blood pressure. The nurse even wanted the name of the book so she could get one for her husband.\"\n\nBartz has been spreading the word about the Abs Diet\u2014now, he's in competition with his 25-year-old son in a race for the first one to get a six-pack. His son isn't on the Abs Diet\u2014and he' s losing out. Bartz says, \"Last Saturday, I sent him a picture and he said, 'Dad, that's not you. You found that picture on the Internet.'\n\n\"I'm not at a six-pack yet, but I'm darn close,\" Bartz says, \"and I know I'll have that six-pack for the end of the year.\"\n\n# **MEASURING YOUR PROGRESS ON THE ABS DIET**\n\n**E VERY PERSON HAS HIS** or her own way of measuring success on a diet. Some measure it by the way their pants feel. Some measure it by the compliments they receive. Some measure it by the fact that they no longer have to buy two seats when they fly cross-country. But if you want harder numbers to know where you stand, here are some pretty good ways to gauge your progress.\n\n**Weight.** Pounds lost help give you some idea, but it's an incomplete number because it doesn't take into account the amount of muscle you're going to develop over the course of a plan. Muscle weighs more than fat, so even a dramatic fat loss may not translate into a dramatic drop in body weight.\n\n**Body mass index.** The BMI is a formula that takes into consideration your height and your weight, and gives you an indication of whether you're overweight, obese, or in good shape. To calculate your BMI, the easiest thing to do is use an online calculator, like the one at MensHealth.com\/BMI. A BMI between 25 and 30 indicates that you're overweight. Over 30 signifies obesity. BMI has its flaws as well (it doesn't take into account muscle mass, and it also leaves out another important factor\u2014weight distribution, i.e. where most of the fat on your body resides). But for most people, it's a better indicator of progress than simply body weight itself.\n\n**Waist-to-hip ratio.** Researchers have recently started using waist size and its relationship to hip size as a more definitive way to determine your health risk. It's considered more important than BMI because visceral fat\u2014the fat that pushes your waist out in front of you\u2014is a leading indicator for diabetes and heart disease. British researchers recently reported that men with waists of 40 or more inches, and women with waists of 35 or more, are at substantially higher risk for these diseases\u2014up to four times higher.\n\nThat's why I want you to concentrate on lowering your waist-to-hip ratio. To figure out yours, measure your waist at your belly button and your hips at the widest point (around your butt). Divide your waist by your hips. For example, if your hips measure 40 inches and your waist at belly-button level measures 38 inches, your waist-to-hip ratio is 0.95. You want a waist-to-hip ratio of 0.92 or lower.\n\n**Body-fat percentage.** Though this is the most difficult for the average person to measure because it requires a bit of technology, it's the most useful. That's because it doesn't just take into consideration weight, but also how much of your weight is fat and how much is muscle. Many gyms offer body-fat measurements, or you can try an at-home body-fat calculator. If you want a simple low-tech test (and this isn't as accurate as what the electronic versions will give you), try this: Sit in a chair with your knees and your feet flat on the floor. Using your thumb and index finger, gently pinch the skin on top of your right thigh. Measure the thickness of the pinched skin with a ruler. If it's \u00be inch or less, you have about 14 percent body fat\u2014ideal for a guy, quite fit for a woman. If it's an inch, you're probably closer to 18 percent fat, which is a tad high for a man, but desirable for a woman. If you pinch more than an inch, you're at increased risk for diabetes and heart disease.\n\nThis last measurement can be the most significant because it'll really help give you a sense of how well you're sticking to a plan. As you see your body-fat percentage decrease, you'll see an increase in the amount of visible muscle. Experts say that in order for your abs to show, your body fat needs to be between 7 and 10 percent. For the average slightly overweight man, that means cutting body fat in about half. If you want to see your progress, take measurements every 2 weeks or so\u2014and certainly not every day.\n\n**MEASUREMENT** | **START** | **END OF WEEK 2** | **END OF WEEK 4** | **END OF WEEK 6**  \n---|---|---|---|---  \nWeight |  |  |  |   \nBMI |  |  |  |   \nWaist-to-hip ratio |  |  |  |   \nBody-fat percentage* |  |  |  |\n\n_*Make sure to have the same person administer body-fat readings using the same method to ensure consistency._\n\n### **ABS DIET SUCCESS STORY**\n\n## **A FORMER KICKBOXER KICKS IT INTO GEAR**\n\n**Name:** Ray and Kim Welborn\n\n**Ray's age:** 44\n\n**Height:** 5\u20328\u2033\n\n**Weight, week 1:** 188\n\n**Weight, week 6:** 169\n\nWhen he was in his twenties, Ray Welborn weighed in at about 130 pounds\u2014it was his fighting weight as a professional kickboxer. But after he stopped, he started putting on the pounds. One day recently, he and his wife, Kim, were in the bookstore and saw the _Abs Diet_. They immediately took to it\u2014Ray for what he thought it could do for his body, and Kim because it reminded her of the way her parents brought her up to eat healthy.\n\nRay, who's lost nearly 20 pounds, and Kim, who's lost 7, follow the weekly meal planner in the original Abs Diet. \"That's our bible,\" Ray says. \"And we stick to it. We even find that we can't eat as much on our cheat meal because of our shrinking bellies.\"\n\nWith the meals, they adjusted recipes just a bit\u2014so that Kim would eat 4\u20445 of a portion. That's when she started seeing a change. \"I like it because of the healthy food,\" Kim says. \"I used to do Weight Watchers, and they all concentrated just on points\u2014it didn't matter whether it had high-fructose corn syrup in it or not. With Weight Watchers, I was always hungry. With this, I'm never hungry.\"\n\nBoth Ray and Kim have lost the weight from their bellies, and Ray is seeing muscles he hasn't seen in years, including ones right around his rib cage. \"She really feels it when she cuddles up to me,\" he says.\n\nMore importantly, Ray _feels_ healthier. \"I feel lighter. My knees don't hurt as much anymore. With the excess weight off, it doesn't hurt to go downstairs like it used to,\" he says. \"And I can tell you. I can bend over and tie my shoes and breathe at the same time. Seriously.\"\n\n# **INDEX**\n\nUnderscored page references indicate sidebars and charts. **Boldface** references indicate illustrations.\n\n**A**\n\nAbdominal exercises,\n\nadvanced,\n\nbridge,\n\nflutter kick, 183\u201384\n\nsaxon side bend,\n\nsuperman,\n\ntraditional crunch,\n\nAbdominal muscles\n\nbenefits of strengthening,\n\nbody-fat percentage for seeing, , 192\u201393,\n\nAbdominal workout, in Abs Diet,\n\nAbs Diet\n\nbenefits of, , 5\u20136\n\nbetter sex,\n\nhealthier life, 8\u20139\n\nlonger life,\n\npain and injury prevention, 7\u20138\n\nBull's Eye, **48\u201349**\n\ncheat sheet, 11\u201312\n\ncompared with other diets, 191\u201392\n\nexpected weight loss from,\n\nfood choices in, ,  ( _see also_ ABS DIET POWER 12)\n\nguidelines\n\nABS DIET POWER 12 foods, 57\u201359\n\ncheating, 61\u201362\n\ndrinks,\n\nno calorie counting, 60\u201361\n\nsix meals a day, 53\u201357\n\nsmoothies, 59\u201360\n\nmeasuring progress on, 197\u2013200\n\norigin of,\n\nportions in, , , , 193\u201394\n\nsuccess with ( _see_ Success stories)\n\nfor women,\n\nAbs Diet Challenge,\n\n_Abs Diet Eat Right Every Time Guide, The_\n\nhow to use, ,\n\npurpose of,\n\nAbs Diet FAQ, 191\u201396\n\nABS DIET POWER , , , , , ,\n\nfood substitutions in, 195\u201396\n\nguidelines for, 57\u201359\n\nmeaning of icons in, 29\u201330\n\noverview of foods in, , 28\u201329\n\nalmonds and other nuts, 31\u201333\n\nbeans and legumes, 33\u201334\n\ndairy products, 35\u201336\n\neggs, 38\u201339\n\nextra-protein (whey) powder, 45\u201346\n\ninstant oatmeal, 36\u201338\n\nolive oil,\n\npeanut butter,\n\nraspberries and other berries, 46\u201347\n\nspinach and other green vegetables, 34\u201335\n\nturkey and other lean meats, 39\u201340\n\nwhole-grain breads and cereals, 42\u201345\n\nstaying focused on, **48\u201349**\n\nsupplementing, with other foods,\n\nAbs Diet Sundae Parfait,\n\nAbs Diet Trail Mix,\n\nAbs Diet Workout,\n\nabdominal exercises in,\n\nadvanced exercise,\n\nbridge,\n\nflutter kick, 183\u201384\n\nsaxon side bend,\n\nsuperman,\n\ntraditional crunch,\n\ncalorie burning from, 175\u201376\n\ncircuit training in, 176\u201377, 177\u201378,\n\nbench press, 179\u201380\n\nbent-over row,\n\nbiceps curl,\n\nleg curl,\n\nleg extension,\n\nmilitary press,\n\npulldown,\n\npushups,\n\nsquat,\n\nstep-up, 182\u201383\n\ntraveling lunge,\n\ntriceps kickback,\n\ntriceps pushdown,\n\nupright row,\n\ncompound exercises in,\n\nfat burning from, , , 175\u201376\n\nflexibility of,\n\ninterval training in, 185\u201388\n\nmuscle building from, 174\u201376\n\nsample schedule for,\n\nAbs Diet Workout Worksheet, 189\u201390\n\nAerobic exercise, , 184\u201385\n\nAlcohol, ,\n\nAllium vegetables, for cancer prevention,\n\nAlmonds, 31\u201333\n\nApplebee's, 154\u201355\n\nArby's, 137\u201338\n\nAtkins Diet,\n\nAtlanta Bread Company, ,\n\nAu Bon Pain, 145\u201346,\n\nAuntie Anne's,\n\nAvocados,\n\n**B**\n\nBacon, Canadian,\n\nBagel shops, best choices at, 151\u201352\n\nBaja Fresh,\n\nBakeries, chain,\n\nBallparks, best food choices at, 163\u201364\n\nBalsamic vinegar,\n\nBananas, ,\n\nBaskin-Robbins,\n\nBeans, 33\u201334,\n\nBeef, 39\u201340, ,\n\nBelly fat\n\ndangers of, 6\u20137\n\nprotein burning,\n\nBench press, 179\u201380\n\nBennigan's, 154\u201355\n\nBent-over row,\n\nBerries, 46\u201347, . _See also specific berries_\n\nBeta-carotene, for cancer prevention,\n\nBiceps curl,\n\nBlack Friday at the mall,\n\nBlimpie,\n\nBlood-pressure-lowering icon,\n\nBlood sugar control, from oatmeal,\n\nBlueberries, 46\u201347,\n\nBob Evans, 149\u201350\n\nBody fat\n\ndangers of, 6\u20137\n\ntargeting,\n\nBody-fat percentage\n\nfor defined abs, , 192\u201393,\n\nfor measuring progress, 199\u2013200\n\nBody mass index, for measuring progress,\n\nBone-strengthening icon,\n\nBoston Market,\n\nBreads, whole-grain, 42\u201345,\n\nBreakfast, 80\u201381\n\nimportance of, 80\u201381\n\nrecipes\n\nburritos, 84\u201385\n\noats, 81\u201382\n\nomelets, 82\u201384\n\nsandwiches, 86\u201389\n\nUltimate Power\n\nBreakfast, 90\u201391\n\nyogurt, 89\u201390\n\nBreakfast places, best choices at, 149\u201350\n\nBridge,\n\nBroccoli, ,\n\nBrown rice,\n\nBruegger's Bagels, 151\u201352\n\nBull's Eye, Abs Diet, **48\u201349**\n\nBurger King, 138\u201339\n\nBurgers\n\nAlaskan Burger,\n\nCiao Down Burger,\n\nOfficial Abs Diet Burger,\n\nBurritos, breakfast,\n\nHoly Guacamole!,\n\nHuevos Rancheros,\n\nYosemite Salmon,\n\n**C**\n\nCalcium\n\nin almonds,\n\nfor bone strength,\n\nin broccoli,\n\nfor cancer prevention,\n\nfor lowering blood pressure,\n\nsaturated fat and,\n\nweight loss from, , , ,\n\nCalorie burning\n\nfrom metabolism,\n\nfrom weight training, 175\u201376\n\nCalorie counting, lack of, in Abs Diet,\n\nCanadian bacon,\n\nCancer-fighting icon,\n\nCancer prevention,\n\nfish for,\n\nvegetables for,\n\nwhey protein for,\n\nCandy bars, 74\u201375\n\nCanned foods, shopping for, 67\u201368\n\nCarbohydrates\n\nin Abs Diet,\n\nfiber-rich, 26\u201327\n\nrefined,\n\nwhole-grain, 43\u201345\n\nCaribou Coffee, ,\n\nCarrabba's,\n\nCereals, whole-grain, 42\u201345,\n\nCheating on Abs Diet, , 61\u201362, 194\u201395\n\nCheat sheet, 11\u201312\n\nCheesecake Factory, 154\u201355\n\nCheeses,\n\nChicken,\n\nChicken a l'Orange,\n\nFaux Fried Chicken,\n\nprecooked, , 107\u20138\n\nGonzo Chicken, 108\u20139\n\nHot-Headed Chicken,\n\nIt Takes Stew, Baby,\n\nQue Sera Quesadilla,\n\nrotisserie,\n\nChicken broth,\n\nChick-fil-A,\n\nChili's,\n\nChinese restaurants,\n\nChipotle, 139\u201340\n\nChocolate milk,\n\nCholesterol, blood\n\nfats increasing, ,\n\nfiber lowering,\n\nobesity and,\n\nCholesterol, dietary, in eggs,\n\nChristmas dinner,\n\nCilantro,\n\nCircuit training, 176\u201377, 177\u201383\n\nCLA, recommended dosage of,\n\nCoffee, milk in,\n\nCoffee shops,\n\nCold Stone Creamery,\n\nCompound exercises,\n\nConjugated linoleic acid (CLA), recommended dosage of,\n\nConvenience foods, as unhealthy,\n\nCookies,\n\nCottage cheese,\n\nCrackers, 75\u201376\n\nCreatine, in beef, 39\u201340\n\nCruciferous vegetables, for cancer prevention,\n\nCrunch, traditional,\n\nCucumbers,\n\n**D**\n\nDairy products, 35\u201336\n\nin ABS DIET POWER ,\n\nshopping for, 66\u201367\n\nweight loss from, ,\n\nDairy Queen,\n\nDeli slices,\n\nDenny's, 149\u201350\n\nDiabetes, visceral fat and, ,\n\nDietary fat\n\nbad\n\nsaturated fat, , ,\n\ntrans fats, 17\u201319, , ,\n\ngood\n\nmonounsaturated fats, , ,\n\npolyunsaturated fats, 25\u201326,\n\nin meat,\n\nneed for, 23\u201324\n\nDiets\n\nvs. eating plan,\n\npopular, compared with Abs Diet, 191\u201392\n\nproblems with, 1\u20132\n\nDiGiorno pizza,\n\nDiners,\n\nbreakfast, 149\u201350\n\nDinner, 103\u20134\n\nrecipes\n\nburgers, 104\u20135\n\nchicken and turkey, 106\u20137\n\none-pot dishes, 112\u201314\n\npasta, 117\u201319\n\nprecooked chicken, 107\u20139\n\nseafood, 114\u201317\n\nside dishes, 119\u201321\n\nsteak, 109\u201312\n\nDomino's, ,\n\nDon Pablo's,\n\nDoughnut shop, best choices at,\n\nDrinks, , ,\n\nsmoothies ( _see_ Smoothies)\n\nDumbbell exercise, ultimate total-body,\n\nDunkin' Donuts, ,\n\n**E**\n\nEaster basket,\n\nEaster dinner,\n\nEating out. _See_ Restaurants\n\nEating plan, vs. diets,\n\nEating schedule, for six meals a day,\n\nEdamame,\n\nEggs, 38\u201339, . _See also_ Omelet(s)\n\nEinstein Bros. Bagels, 146\u201347,\n\nEnergy balance,\n\nEnglish muffins,\n\nErectile dysfunction,\n\nEthnic Gourmet frozen dinners,\n\nExercise\n\naerobic, , 184\u201385\n\nlack of, in diets,\n\nmotivation for,\n\nExercise program, in Abs Diet,  ( _see also_ Abs Diet Workout)\n\n**F**\n\nFast food, 14\u201315\n\nbest choices of, 137\u201344\n\nFat. _See_ Body fat; Dietary fat\n\nFat burning\n\nfrom muscle,\n\nfrom whey protein,\n\nFazoli's,\n\nFiber\n\nfor cancer prevention,\n\nin oatmeal,\n\nfor preventing weight gain,\n\nsoluble and insoluble, 26\u201327\n\nin whole grains,\n\nFish, 25\u201326, , . _See also_ Seafood\n\nFlank steak,\n\nFlaxseed, , ,\n\nFlaxseed oil,\n\nFlutter kick, 183\u201384\n\nFolate\n\nin carbohydrates,\n\nfor improving immune function,\n\nin spinach,\n\nFood courts, best choices at, 162\u201363\n\nFood labels\n\nlow-fat phrases on,\n\nunderstanding, 72\u201373\n\nFood shopping, for bargain items,\n\nFood shopping list\n\ncanned foods, 67\u201368\n\ncookies and candies, 74\u201376\n\ndairy, 66\u201367\n\ndrinks,\n\nfrozen foods, 69\u201372\n\ngrains, 72\u201374\n\nhealth food,\n\nmeat, 65\u201366,\n\nnuts, honey, and spices,\n\nprepared foods, 78\u201379\n\nproduce, 64\u201365\n\nsauces, 68\u201369\n\nFourth of July,\n\nFrequently asked questions, 191\u201396\n\nFreschetta pizza,\n\nFriendly's, 154\u201355\n\nFrozen dinners, 70\u201372\n\nFrozen foods, shopping for, 69\u201370\n\nFruit\n\nfrozen,\n\nshopping for,\n\n**G**\n\nGrains\n\nbenefits of, 21\u201322\n\nparts of,\n\nwhole, , 42\u201345\n\nshopping for, 72\u201374\n\nGreen onions,\n\n**H**\n\nHalloween,\n\nHardee's, 140\u201341\n\nHealth food, shopping for,\n\nHealthy Choice frozen dinners,\n\nHeart-disease-fighting icon,\n\nHigh-fructose corn syrup, 15\u201317,\n\nHoliday eating, 167\u201372\n\nHoney,\n\nHot sauce,\n\nHummus,\n\nHunger, lack of, from six meals a day,\n\n**I**\n\nIce cream,\n\nAbs Diet Sundae Parfait,\n\nIce cream shops, best choices at, 164\u201365\n\nIcons, in ABS DIET POWER , 29\u201330\n\nIHOP, 149\u201350\n\nImmune-function-boosting icon,\n\nInsulin, , ,\n\nInterval training, 185\u201388\n\nIron, in turkey,\n\nItalian restaurants,\n\n**J**\n\nJack in the Box,\n\nJamba Juice,\n\nJuice, condensed,\n\nJumping jacks,\n\n**K**\n\nKFC,\n\nKrispy Kreme,\n\n**L**\n\nLabels, food. _See_ Food labels\n\nLean Cuisine frozen dinners,\n\nLeg curl,\n\nLeg extension,\n\nLemons, ,\n\nLeptin,\n\nLittle Caesar's,\n\nLiver, effect of belly fat on,\n\nLong John Silver's,\n\nLow-fat diets\n\nfor cancer prevention,\n\ndrawbacks of,\n\nLow-fat products, labeling on,\n\nLunch,\n\nrecipes\n\npita pizzas, 101\u20132\n\nsalads, 95\u201397\n\nsandwiches, 98\u201399\n\nsoups, 99\u2013101\n\nwraps, 93\u201395\n\n**M**\n\nMacaroni Grill,\n\nMagnesium\n\nin almonds,\n\nfor lowering blood pressure,\n\nfor muscle building, ,\n\nManhattan Bagel Co.,\n\nMarinara sauce,\n\nMcDonald's, 142\u201343\n\nMeals, number of, in Abs Diet,\n\nMeats. _See also specific meats_\n\nin ABS DIET POWER , , 39\u201340\n\nfat content of,\n\nshopping for, 65\u201366,\n\nMetabolism,\n\nleptin influencing,\n\nmuscle increasing,\n\nMexican restaurants,\n\nMicronutrients,\n\nMidnight snacks,\n\nMilitary press,\n\nMilk\n\nas calcium source,\n\nchoosing,\n\nwith coffee,\n\nfor midnight snack,\n\nsaturated fat and,\n\nMonounsaturated fats, ,\n\nlowering cholesterol, ,\n\nin nuts,\n\nin peanut butter,\n\nMotivation for exercise,\n\nMountain climber,\n\nMrs. Fields,\n\nMuscle\n\ncalorie burning from,\n\nfat burning from,\n\nMuscle building\n\nfrom Abs Diet Workout, 174\u201376\n\nfor fat loss, 56\u201357\n\nfrom protein, ,\n\nfrom whey protein, 45\u201346\n\nMuscle-building icon, 29\u201330\n\nMuscle loss\n\nfrom starvation diets, 53\u201355\n\nstudy on,\n\nMustard,\n\n**N**\n\nNew Year's Eve,\n\nNiacin, in turkey,\n\nNo-weight workout,\n\nNutritional information\n\non commercial pizza,\n\non frozen pizza,\n\nfrom restaurants, 150\u201351\n\nNutritional ingredients to emphasize and limit,\n\nNuts, 31\u201333, ,\n\n**O**\n\nOatmeal, 36\u201338,\n\nfor midnight snack,\n\nrecipes\n\nFlax Machine,\n\nGinger _Roger!,_\n\nHoney, I Shrunk My Gut,\n\nYou're Nuts,\n\nin smoothies,\n\nObesity, ,\n\nO'Charley's, 154\u201355\n\nOffice party in December,\n\nOlive Garden,\n\nOlive oil, ,\n\nOmega-3 fatty acids, 25\u201326, ,\n\nOmelet(s), 82\u201383\n\nBean Counter,\n\nGreen and White,\n\nLean Eggs and Ham,\n\nMister Bean,\n\nTom Tomelet,\n\nUp in Smoke,\n\nOne-pot dishes\n\nHot-Headed Chicken,\n\nPot Luck of the Irish,\n\nThree-Amigos Chili, 112\u201313\n\nOnions, green,\n\nOvereating, effects of,\n\nOverweight,\n\n**P**\n\nPanda Express,\n\nPanera Bread, ,\n\nPapa John's, , 160\u201361\n\nPasta, ,\n\n\"Alfredo, I Know It Was\n\nYou . . . ,\"\n\nPesto R\u00e9sistance, 118\u201319\n\nYou-Can Noodle, 117\u201318\n\nPeaches,\n\nPeanut butter, ,\n\nPectin,\n\nPeet's Coffee,\n\nPeppers,\n\nPerkins, 149\u201350, 154\u201355\n\nP.F. Chang's,\n\nPita pizzas,\n\nPancho Villa,\n\nRed Auerbach,\n\nWhitey Ford,\n\nPitas,\n\nPizza\n\ncommercial, ,\n\nfrozen,\n\npita ( _see_ Pita pizzas)\n\nPizza Hut, ,\n\nPizza places, best choices at, 159\u201361\n\nPolyunsaturated fats, 25\u201326,\n\nPork\n\nPot Luck of the Irish,\n\nPortions, in Abs Diet, , , , 193\u201394\n\nPotassium, for lowering blood pressure,\n\nPrepared grocery-store foods, 78\u201379\n\nProduce, shopping for, 64\u201365\n\nProgress on Abs Diet, measuring, 197\u2013200\n\nProstate cancer, preventing, ,\n\nProtein\n\nbenefits of, 22\u201323\n\nin eggs, 38\u201339\n\nfor fat burning, ,\n\nfor muscle building, 29\u201330, ,\n\nfor weight loss,\n\nProtein powder, 45\u201346\n\nPudding cups,\n\nPulldown,\n\nPumpkin,\n\nPushups,\n\nPyruvate, recommended dosage of,\n\n**R**\n\nRaspberries,\n\nRed wine vinegar,\n\nRestaurants, 136\u201337\n\nbest food choices at\n\nbagel shops, 151\u201352\n\nballparks, 163\u201364\n\nbreakfast places, 149\u201350\n\ndoughnut shop,\n\nfast-food places, 137\u201344\n\nfood courts at malls, 162\u201363\n\nice cream shops, 164\u201365\n\npizza places, 159\u201361\n\nsandwich shops, 144\u201348\n\nsit-down restaurants, 153\u201359\n\nsmoothie stands, 161\u201362\n\nsports bars,\n\nnutritional information from, 150\u201351\n\nRice, brown,\n\nRicotta cheese, ,\n\nRotisserie chicken,\n\nRuby Tuesday, 154\u201355\n\n**S**\n\nSalad blends,\n\nSalad dressing, homemade,\n\nSalads\n\nEl Tequila Ensalada,\n\nKidney Punch, 96\u201397\n\nOlympiad,\n\nOrient Express,\n\nPotion of the Ocean,\n\nSweet Cheeses!,\n\nSalmon, , ,\n\nSalsa,\n\nSandwiches\n\nbreakfast\n\nApple Jacked,\n\nFoxy Lox,\n\nJam Session,\n\nMex-illent Adventure, 86\u201387\n\nThree-Country Breakfast,\n\nTurk before Work,\n\nWafflewich, 88\u201389\n\nlunch, , 98\u201399\n\nFill-in-the-Blank-Salad Sandwich,\n\nSandwich shops, best choices at, 144\u201348\n\nSaturated fats, , ,\n\nSauces, shopping for, 68\u201369\n\nSaxon side bend,\n\nSbarro,\n\nSchlotzsky's Deli, 147\u201348\n\nSeafood. _See also_ Fish\n\nAqua Man,\n\nFish Tacos,\n\nHot Pink,\n\nNo-Scrimp Shrimp, 114\u201315\n\nPerfect Storm,\n\nSeafood restaurants, 158\u201359\n\nSecret weapons, in Abs Diet,\n\nSex, improved, after Abs Diet,\n\nSex drive, oatmeal boosting,\n\nShoney's, 149\u201350\n\nShrimp,\n\nSide dishes\n\nBreathalyzer,\n\nEl El Bean, 120\u201321\n\nGreen Party, 119\u201320\n\nJerry's Rice,\n\nNuclear Orange Spud Missiles,\n\nSit-down restaurants, best choices at, 153\u201359\n\nSix meals a day, as Abs Diet guideline, 53\u201357\n\nSmart Ones frozen dinners,\n\nSmoothie King, 161\u201362\n\nSmoothies, 122\u201323\n\nin Abs Diet,\n\nadvantages of, 59\u201360\n\nbottled,\n\nguidelines for,\n\noatmeal in,\n\nrecipes\n\nAlmond Hammer,\n\nBlue Velvet, 128\u201329\n\nCheck Your Blackberry,\n\nCheesecake in a Cup,\n\nChocolate Factory,\n\nChocolate Latte, 130\u201331\n\nChoco-nana,\n\nCinnamon Girl,\n\nEndless Summer, 124\u201325\n\nHawaiian Five-O,\n\nHoney-Nut Cheery Oat,\n\nHoney-Pecan Smoothie, 126\u201327\n\nJuicy Fruit Juice,\n\nLime Dancing,\n\nMango Tango,\n\nMint Chocolate Morning, 132\u201333\n\nNeapolitan, 134\u201335\n\nNew Zealander,\n\nNutty Professor,\n\nOrangeman,\n\nPeach Vacation,\n\nPeachy Keen,\n\nPunk'd Pie,\n\nTirami-Smooth,\n\nWhey-Too-Good Smoothie,\n\nwhey protein in,\n\nSmoothie stands, best choices at, 161\u201362\n\nSnacks, 126\u201327\n\nAbs Diet Trail Mix,\n\nmidnight,\n\nSoda,\n\nSodium, ,\n\nSoups, 99\u2013100\n\nBeaning of Life, 100\u2013101\n\nMacho Gazpacho,\n\nSouth Beach Diet,\n\nSoy sauce,\n\nSpices,\n\nSpinach, 34\u201335,\n\nSplit hop,\n\nSports bars, best choices at,\n\nSpot-reducing, myth of,\n\nSquat,\n\nSquat thrust with pushup,\n\nSt. Patrick's Day,\n\nStarbucks, ,\n\nStarvation diets, muscle loss from, 53\u201355\n\nSteak,\n\nAztec Casserole,\n\nflank,\n\nFruit of Your Loins,\n\nMighty Muffins,\n\nSergeant Pepper,\n\nSteak Fa-heat-as,\n\nSteak houses,\n\nStep-up, 182\u201383\n\nStouffer's frozen dinners,\n\nStrawberries, ,\n\nString cheese,\n\nSubway,\n\nSuccess stories, , 9\u201310, , 24\u201325, 50\u201351, 54\u201355, 76\u201377, 78\u201379, 86\u201387, , 186\u201387, 194\u201395, 198\u201399\n\nSundae Parfait, Abs Diet,\n\nSuper Bowl party,\n\nSuperman,\n\nSupermarkets\n\nbargain items in,\n\npitfalls of, 63\u201364\n\nSupplements, vs. food,\n\nSushi bar,\n\n**T**\n\nTaco Bell, 143\u201344\n\nT.G.I. Friday's, 154\u201355\n\nThanksgiving,\n\nThree meals a day, drawbacks of, 55\u201356\n\nTomatoes, ,\n\nTony's pizza,\n\nTortillas,\n\nTrail Mix, Abs Diet,\n\nTrans fats, 17\u201319, ,\n\nTraveling lunge,\n\nTriceps kickback,\n\nTriceps pushdown,\n\nTuna, , 67\u201368\n\nTurkey, ,\n\nDijon Lennon, 106\u20137\n\nThree Amigos Chili, 112\u201313\n\n**U**\n\nUltimate Power Breakfast, 90\u201391\n\nUpright row,\n\n**V**\n\nVegetables, 34\u201335. _See also_ Side dishes\n\nshopping for, 64\u201365\n\nVending machine meals, comparison of,\n\nVinegars,\n\nVisceral fat, dangers of, 6\u20137,\n\nVitamin A\n\nfor improving immune function,\n\nin spinach,\n\nVitamin B6\n\nfor improving immune function,\n\nin turkey,\n\nVitamin B12, in eggs,\n\nVitamin C\n\nin broccoli,\n\nfor cancer prevention,\n\nfor improving immune function,\n\nin spinach,\n\nVitamin D, for bone strength,\n\nVitamin E\n\nin almonds,\n\nfor improving immune function,\n\n**W**\n\nWaffles,\n\nWaist-to-hip ratio, for measuring progress, 198\u201399\n\nWater, bottled,\n\nWeight, for measuring progress,\n\nWeight gain prevention icon,\n\nWeight loss, expected, from Abs Diet,\n\nWeight training. _See also_ Abs Diet Workout, circuit training in\n\ncalorie burning from, 175\u201376\n\nWendy's,\n\nWhey protein, 45\u201346,\n\nWhole grains, , 42\u201345\n\nshopping for, 72\u201374\n\nWhole-wheat bread, 42\u201345,\n\nWomen\n\nAbs Diet for,\n\nobesity in,\n\nWorkouts\n\nin Abs Diet,  ( _see also_ Abs Diet Workout)\n\nwhey protein before, 45\u201346\n\nWraps\n\nCow Tipper,\n\nDay after Thanksgiving,\n\nHot Curlers, 94\u201395\n\nThai One On,\n\nTwo Turks,\n\n**Y**\n\nYogurt,\n\nin breakfast recipes\n\nBananarama,\n\nBerry Easy,\n\nRupert Pumpkin,\n\nTrip to the Peach,\n\nin smoothies,\n\n**Z**\n\nZabar's,\n\nZinc\n\nfor improving immune function,\n\nin turkey,\n\n* * *\n\n_Note on sources: Whenever possible, I have used published data from major national chain restaurants. In cases of similar foods, I examined the nutritional profiles of each and selected one that seemed most representative of an industry-wide standard. In cases where published material was not available, I have used independent tests from the following sources: The USDA and The Center for Science in the Public Interest (Restaurant Confidential)._\n\n* * *\n\n**Notice**\n\nThis book is intended as a reference volume only, not as a medical manual. The information given here is designed to help you make informed decisions about your health. It is not intended as a substitute for any treatment that may have been prescribed by your doctor. If you suspect that you have a medical problem, we urge you to seek competent medical help. Mention of specific companies, organizations, or authorities in this book does not imply endorsement by the publisher, nor does mention of specific companies, organizations, or authorities imply that they endorse this book. Internet addresses and telephone numbers given in this book were accurate at the time it went to press.\n\nSales of this book without a front cover may be unauthorized. If this book is coverless, it may have been reported to the publisher as \"unsold or destroyed\" and neither the author nor the publisher may have received payment for it.\n\n\u00a9 2005 by Rodale Inc.\n\nAll rights reserved. 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+{"text":"\n\n\n\nProduced by KD Weeks, Colin Bell and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n                          Transcriber's Note:\n\nThis version of the text cannot represent certain typographical effects.\nItalics are delimited with the '_' character as _italic_. Bold font is\ndelimited by the '=' character. Superscripted and subscripted characters\nare shown as '^2' and '_{2}' respectively. Emphasized words in Greek\nare, by convention, printed using wider spacing (_gesperrt_). The '_'\ncharacter is used as a delimiter in this version, e.g. (\u03bc\u03b9\u03bc\u03b7\u03c4\u03b1\u03af _\u03bc\u03bf\u03c5\n\u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b5_).\n\nThe original text includes annotations on two Greek texts, the Epistle\nto the Colossians, and an Epistle to Philemon. On each page, several\nlines of Greek are accompanied by a double column of notations on key\nwords. It was not possible to follow that convention in this version,\ngiven the nature of our medium.\n\nAny hyphenations in the Greek text that occurred on page breaks have\nbeen removed, and the word's end has been moved to the previous page. On\nmany occasions, a note appears on an earlier page than the text it\nglosses. In this version, the notes have been arranged so that each\n_follows_ the text to which it refers.\n\nThere are Greek inscriptions printed in an uncial font, and using a\nlunate sigma (\u03f2). These will appear here as ==\u03bc\u03b9\u03bc\u03b7\u03c4\u03b1\u03af \u03bc\u03bf\u03c5 \u03b3\u03af\u03bd\u03b5\u03f2\u03b8\u03b5==.\nThe occasional blackletter font appears here as '=blackletter\ntext='.\n\nFootnotes have been moved to follow each paragraph, and are resequenced\nto be unique across the text. Any internal references to those notes\nhave been modified as well.\n\nThe section entitled AD LAODICENSES presents the text of an apocryphal\nletter from St Paul. Given the brevity of the text, these notes are\ngathered after the letter.\n\nThe index includes references to both pages and to the verses of the two\nEpistles included here. Those references to a verse may refer to either\nthe Greek itself, or to any of the notes on that verse.\n\nPlease consult the note at the end of this text for any other issues\nthat arose during its presentation.\n\n\n\n\n                        THE EPISTLES OF ST PAUL.\n\n                                  III.\n\n                       THE FIRST ROMAN CAPTIVITY.\n\n                                   2.\n\n                       EPISTLE TO THE COLOSSIANS.\n\n                                   3.\n\n                          EPISTLE TO PHILEMON.\n\n[Illustration]\n\n                          =Cambridge:=\n                      PRINTED BY C. J. CLAY, M.A.\n                        AT THE UNIVERSITY PRESS.\n\n                               ST PAUL'S\n                       EPISTLES TO THE COLOSSIANS\n                                 AND TO\n                               PHILEMON.\n\n                            _A REVISED TEXT_\n\n                                  WITH\n\n                INTRODUCTIONS, NOTES, AND DISSERTATIONS.\n\n\n\n\n                                   BY\n\n                         J. B. LIGHTFOOT, D.D.\n\n                          CANON OF ST PAUL'S;\n                     HULSEAN PROFESSOR OF DIVINITY,\n                                  AND\n             HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE.\n\n\n\n\n                            =London:=\n                          _MACMILLAN AND CO._\n                                 1875.\n\n                        [_All Rights reserved._]\n\n------------------------------------------------------------------------\n\n\n\n\n         ==\u03bc\u03b9\u03bc\u03b7\u03c4\u03b1\u03af \u03bc\u03bf\u03c5 \u03b3\u03af\u03bd\u03b5\u03f2\u03b8\u03b5 \u03ba\u03b1\u03b8\u1f7c\u03f2 \u03ba\u1f00\u03b3\u1f7c \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u1fe6.==\n\n                         ---------------------\n\n              \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf\u03c2 \u1f51\u03c0\u03bf\u03b3\u03c1\u03b1\u03bc\u03bc\u03cc\u03c2.\n\n                                                    CLEMENT.\n\n                               ----------\n\n       \u039f\u1f50\u03c7 \u1f61\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03b4\u03b9\u03b1\u03c4\u03ac\u03c3\u03c3\u03bf\u03bc\u03b1\u03b9 \u1f51\u03bc\u1fd6\u03bd\u00b7 \u1f10\u03ba\u03b5\u1fd6\u03bd\u03bf\u03c2 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03c2,\n       \u1f10\u03b3\u1f7c \u03ba\u03b1\u03c4\u03ac\u03ba\u03c1\u03b9\u03c4\u03bf\u03c2\u00b7 \u1f10\u03ba\u03b5\u1fd6\u03bd\u03bf\u03c2 \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2, \u1f10\u03b3\u1f7c \u03b4\u1f72 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03bd\u1fe6\u03bd\n          \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2.\n\n                                                   IGNATIUS.\n\n                               ----------\n\n     \u039f\u1f54\u03c4\u03b5 \u1f10\u03b3\u1f7c \u03bf\u1f54\u03c4\u03b5 \u1f04\u03bb\u03bb\u03bf\u03c2 \u1f45\u03bc\u03bf\u03b9\u03bf\u03c2 \u1f10\u03bc\u03bf\u03af \u03b4\u03cd\u03bd\u03b1\u03c4\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u1fc6\u03c3\u03b1\u03b9\n            \u03c4\u1fc7 \u03c3\u03bf\u03c6\u03af\u1fb3 \u03c4\u03bf\u1fe6 \u03bc\u03b1\u03ba\u03b1\u03c1\u03af\u03bf\u03c5 \u03ba\u03b1\u1f76 \u1f10\u03bd\u03b4\u03cc\u03be\u03bf\u03c5 \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5.\n\n                                                   POLYCARP.\n\n------------------------------------------------------------------------\n\n\n\n\n                                PREFACE.\n\n\nOn the completion of another volume of my commentary, I wish again to\nrenew my thanks for the assistance received from previous labourers in\nthe same field. Such obligations must always be great; but it is not\neasy in a few words to apportion them fairly, and I shall not make the\nattempt. I have not consciously neglected any aid which might render\nthis volume more complete; but at the same time I venture to hope that\nmy previous commentaries have established my claim to be regarded as an\nindependent worker, and in the present instance more especially I have\nfound myself obliged to diverge widely from the treatment of my\npredecessors, and to draw largely from other materials than those which\nthey have collected.\n\nIn the preface to a previous volume I expressed an intention of\nappending to my commentary on the Colossian Epistle an essay on\n'Christianity and Gnosis.' This intention has not been fulfilled in the\nletter; but the subject enters largely into the investigation of the\nColossian heresy, where it receives as much attention as, at all events\nfor the present, it seems to require. It will necessarily come under\ndiscussion again, when the Pastoral Epistles are taken in hand.\n\nThe question of the genuineness of the two epistles contained in this\nvolume has been deliberately deferred. It could not be discussed with\nany advantage apart from the Epistle to the Ephesians, for the three\nletters are inseparably bound together. Meanwhile however the doctrinal\nand historical discussions will, if I mistake not, have furnished\nanswers to the main objections which have been urged; while the\ncommentary will have shown how thoroughly natural the language and\nthoughts are, if conceived as arising out of an immediate emergency.\nMore especially it will have been made apparent that the Epistle to the\nColossians hangs together as a whole, and that the phenomena are\naltogether adverse to any theory of interpolation such as that recently\nput forward by Professor Holtzmann.\n\nIn the commentary, as well as in the introduction, it has been a chief\naim to illustrate and develope the theological conception of the Person\nof Christ, which underlies the Epistle to the Colossians. The Colossian\nheresy for instance owes its importance mainly to the fact that it\nthrows out this conception into bolder relief. To this portion of the\nsubject therefore I venture to direct special attention.\n\nI cannot conclude without offering my thanks to Mr A. A. VanSittart who,\nas on former occasions, has given his aid in correcting the proof sheets\nof this volume; and to the Rev. J. J. Scott, of Trinity College, who has\nprepared the index. I wish also to express my obligations to Dr\nSchiller-Szinessy, of whose Talmudical learning I have freely availed\nmyself in verifying Frankel's quotations and in other ways. I should add\nhowever that he is not in any degree responsible for my conclusions and\nhas not even seen what I have written.\n\n    TRINITY COLLEGE,\n        _April 30, 1875_.\n\n------------------------------------------------------------------------\n\n\n\n\n                               CONTENTS.\n\n                       EPISTLE TO THE COLOSSIANS.\n\n     _INTRODUCTION._\n\n                                                              PAGE\n\n          I.   _The Churches of the Lycus_                    1\u201372\n\n          II.  _The Colossian Heresy_                       73\u2013113\n\n               _On some points connected with the\n               Essenes._\n\n               1. _The name Essene_                        114\u2013119\n\n               2. _Origin and Affinities of the            119\u2013157\n               Essenes_\n\n               3. _Essenism and Christianity_              158\u2013179\n\n          III. _Character and Contents of the Epistle_     180\u2013194\n\n     _TEXT AND NOTES_                                      197\u2013311\n\n          _On some Various Readings in the Epistle_        312\u2013322\n\n          _On the meaning of \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1_                      323\u2013339\n\n          _The Epistle from Laodicea_                      340\u2013366\n\n                          EPISTLE TO PHILEMON.\n\n     _INTRODUCTION_                                        369\u2013395\n\n     _TEXT AND NOTES_                                      399\u2013412\n\n     _INDEX_                                               415\u2013424\n\n\n\n\n                                   I.\n                       THE CHURCHES OF THE LYCUS.\n\n\n[Sidenote: Situation of the three cities.]\n\nLying in, or overhanging, the valley of the Lycus, a tributary of the\nM\u00e6ander, were three neighbouring towns, Laodicea, Hierapolis, and\nColoss\u00e6[1]. The river flows, roughly speaking, from east to west; but at\nthis point, which is some few miles above its junction with the M\u00e6ander,\nits direction is more nearly from south-east to north-west[2]. Laodicea\nand Hierapolis stand face to face, being situated respectively on the\nsouthern and northern sides of the valley, at a distance of six\nmiles[3], and within sight of each other, the river lying in the open\nplain between the two. The site of Coloss\u00e6 is somewhat higher up the\nstream, at a distance of perhaps ten or twelve miles[4] from the point\nwhere the road between Laodicea and Hierapolis crosses the Lycus. Unlike\nLaodicea and Hierapolis, which overhang the valley on opposite sides,\nColoss\u00e6 stands immediately on the river-bank, the two parts of the town\nbeing divided by the stream. The three cities lie so near to each other,\nthat it would be quite possible to visit them all in the course of a\nsingle day.\n\nFootnote 1:\n\n  The following are among the most important books of travel relating to\n  this district; Pococke _Description of the East and Some Other\n  Countries_, Vol. II, Part II, London 1745; Chandler _Travels in Asia\n  Minor_ etc., Oxford 1775; Leake _Tour in Asia Minor_, London 1824;\n  Arundell _Discoveries in Asia Minor_, London 1834; Hamilton\n  _Researches in Asia Minor, Pontus, and Armenia_, London 1842; Fellows\n  _Asia Minor_, London 1839, _Discoveries in Lycia_, London 1840; de\n  Tchihatcheff _Asie Mineure, Description Physique, Statistique et\n  Arch\u00e9ologique_, Paris 1853 etc., with the accompanying Atlas (1860);\n  de Laborde _Voyage de l'Asie Mineure_ (the expedition itself took\n  place in 1826, but the date on the title-page is 1838, and the\n  introduction was written in 1861); Le Bas _Voyage Arch\u00e9ologique en\n  Gr\u00e8ce et en Asie Mineure_, continued by Waddington and not yet\n  completed; Texier _Description de l'Asie Mineure_, Vol. I (1839). It\n  is hardly necessary to add the smaller works of Texier and Le Bas on\n  _Asie Mineure_ (Paris 1862, 1863) in Didot's series _L'Univers_, as\n  these have only a secondary value. Of the books enumerated, Hamilton's\n  work is the most important for the topography, etc.; Tchihatcheff's\n  for the physical features; and Le Bas and Waddington's for the\n  inscriptions, etc. The best maps are those of Hamilton and\n  Tchihatcheff; to which should be added the _Karte von Klein-Asien_ by\n  v. Vincke and others, published by Schropp, Berlin 1844.\n\n  Besides books on Asia Minor generally, some works relating especially\n  to the Seven Churches may be mentioned. Smith's _Survey of the Seven\n  Churches of Asia_ (1678) is a work of great merit for the time, and\n  contains the earliest description of the sites of these Phrygian\n  cities. It was published in Latin first, and translated by its author\n  afterwards. Arundell's _Seven Churches_ (1828) is a well-known book.\n  Allom and Walsh's _Constantinople and the Scenery of the Seven\n  Churches of Asia Minor illustrated_ (1850) gives some views of this\n  district. Svoboda's _Seven Churches of Asia_ (1869) contains 20\n  photographs and an introduction by the Rev. H. B. Tristram. This is a\n  selection from a larger series of Svoboda's photographs, published\n  separately.\n\nFootnote 2:\n\n  The maps differ very considerably in this respect, nor do the\n  statements of travellers always agree. The direction of the river, as\n  given in the text, accords with the maps of Hamilton and Tchihatcheff,\n  and with the accounts of the most accurate writers.\n\nFootnote 3:\n\n  _Anton. Itin._ p. 337 (Wesseling) gives the distance as 6 miles. See\n  also Fellows _Asia Minor_ p. 283, Hamilton I. p. 514. The relative\n  position of the two cities appears in Laborde's view, pl. xxxix.\n\nFootnote 4:\n\n  I do not find any distinct notice of the distance; but, to judge from\n  the maps and itineraries of modern travellers, this estimate will\n  probably be found not very far wrong.\n\n[Sidenote: Their neighbourhood and intercourse.]\n\nThus situated, they would necessarily hold constant intercourse with\neach other. We are not surprised therefore to find them so closely\nconnected in the earliest ages of Christianity. It was the consequence\nof their position that they owed their knowledge of the Gospel to the\nsame evangelist, that the same phases of thought prevailed in them, and\nthat they were exposed to the same temptations, moral as well as\nintellectual.\n\n[Sidenote: Physical forces at work.]\n\nThe physical features of the neighbourhood are very striking. Two potent\nforces of nature are actively at work to change the face of the country,\nthe one destroying old land-marks, the other creating fresh ground.\n\n[Sidenote: Frequent earthquakes.]\n\nOn the one hand, the valley of the Lycus was and is especially liable to\nviolent earthquakes. The same danger indeed extends over large portions\nof Asia Minor, but this district is singled out by ancient writers[5]\n(and the testimony of modern travellers confirms the statement[6]), as\nthe chief theatre of these catastrophes. Not once or twice only in the\nhistory of Laodicea do we read of such visitations laying waste the city\nitself or some flourishing town in the neighbourhood[7]. Though the\nexterior surface of the earth shows no traces of recent volcanoes, still\nthe cavernous nature of the soil and the hot springs and mephitic\nvapours abounding here indicate the presence of those subterranean\nfires, which from time to time have manifested themselves in this work\nof destruction.\n\nFootnote 5:\n\n  See especially Strabo xii. 8. 16 (p. 578) \u03c4\u1f78 \u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03b7\u03c4\u03bf\u03bd \u03c4\u1fc6\u03c2 \u03c7\u03ce\u03c1\u03b1\u03c2 \u03ba\u03b1\u1f76\n  _\u03c4\u1f78 \u03b5\u1f54\u03c3\u03b5\u03b9\u03c3\u03c4\u03bf\u03bd_\u00b7 \u03b5\u1f30 \u03b3\u03ac\u03c1 \u03c4\u03b9\u03c2 \u1f04\u03bb\u03bb\u03b7, \u03ba\u03b1\u1f76 \u1f21 \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1 \u03b5\u1f54\u03c3\u03b5\u03b9\u03c3\u03c4\u03bf\u03c2, \u03ba\u03b1\u1f76\n  \u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b7\u03c3\u03b9\u03bf\u03c7\u03ce\u03c1\u03bf\u03c5 \u03b4\u1f72 \u039a\u03ac\u03c1\u03bf\u03c5\u03c1\u03b1.\n\nFootnote 6:\n\n  Thus Pococke (p. 71) in 1745 writes of Denizli, which is close to\n  Laodicea, 'The old town was destroyed about 25 years past by an\n  earthquake, in which 12,000 people perished.'\n\nFootnote 7:\n\n  See below p. 38.\n\n[Sidenote: Deposits of travertine.]\n\nBut, while the crust of the earth is constantly broken up by these\nforces from beneath, another agency is actively employed above ground in\nlaying a new surface. If fire has its fitful outbursts of devastation,\nwater is only less powerful in its gradual work of reconstruction. The\nlateral streams which swell the waters of the Lycus are thickly\nimpregnated with calcareous matter, which they deposit in their course.\nThe travertine formations of this valley are among the most remarkable\nin the world, surpassing even the striking phenomena of Tivoli and\nClermont[8]. Ancient monuments are buried, fertile lands overlaid,\nriver-beds choked up and streams diverted, fantastic grottos and\ncascades and archways of stone formed, by this strange capricious power,\nat once destructive and creative, working silently and relentlessly\nthrough long ages. Fatal to vegetation, these incrustations spread like\na stony shroud over the ground. Gleaming like glaciers on the hill-side\nthey attract the eye of the traveller at a distance of twenty miles[9],\nand form a singularly striking feature in scenery of more than common\nbeauty and impressiveness.\n\nFootnote 8:\n\n  Tchihatcheff P. I. _Geogr. Phys. Comp._ p. 344 sq., esp. p. 353. See\n  the references below, pp. 9 sq., 15.\n\nFootnote 9:\n\n  Fellows _Asia Minor_ p. 283.\n\n[Sidenote: Produce and manufactures of the district.]\n\nAt the same time, along with these destructive agencies, the fertility\nof the district was and is unusually great. Its rich pastures fed large\nflocks of sheep, whose fleeces were of a superior quality; and the trade\nin dyed woollen goods was the chief source of prosperity to these towns.\nFor the bounty of nature was not confined to the production of the\nmaterial, but extended also to the preparation of the fabric. The\nmineral streams had chemical qualities, which were highly valued by the\ndyer[10]. Hence we find that all the three towns, with which we are\nconcerned, were famous in this branch of trade. At Hierapolis, as at\nThyatira, the guild of the dyers appears in the inscriptions as an\nimportant and influential body[11]. Their colours vied in brilliancy\nwith the richest scarlets and purples of the farther east[12]. Laodicea\nagain was famous for the colour of its fleeces, probably a glossy black,\nwhich was much esteemed[13]. Here also we read of a guild of dyers[14].\nAnd lastly, Coloss\u00e6 gave its name to a peculiar dye, which seems to have\nbeen some shade of purple, and from which it derived a considerable\nrevenue[15].\n\nFootnote 10:\n\n  See note 13.\n\nFootnote 11:\n\n  Boeckh _Corp. Inscr._ no. 3924 (at Hierapolis) \u03c4\u03bf\u1fe6\u03c4\u03bf \u03c4\u1f78 \u1f21\u03c1\u1ff7\u03bf\u03bd \u03c3\u03c4\u03b5\u03c6\u03b1\u03bd\u03bf\u1fd6\n  _\u1f21 \u1f10\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03c4\u1ff6\u03bd \u03b2\u03b1\u03c6\u03b5\u03ce\u03bd_. See Laborde's view, pl. xxxv. In another\n  inscription too (Le Bas and Waddington, no. 1687) there is mention of\n  the purple-dyers, \u03c0\u03bf\u03c1\u03c6\u03c5\u03c1\u03b1\u03b2\u03b1\u03c6\u03b5\u1fd6\u03c2.\n\nFootnote 12:\n\n  Strabo xiii. 4. 14 (p. 630) \u1f14\u03c3\u03c4\u03b9 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03b2\u03b1\u03c6\u1f74\u03bd \u1f10\u03c1\u03af\u03c9\u03bd \u03b8\u03b1\u03c5\u03bc\u03b1\u03c3\u03c4\u1ff6\u03c2\n  \u03c3\u03cd\u03bc\u03bc\u03b5\u03c4\u03c1\u03bf\u03bd \u03c4\u1f78 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u1f39\u03b5\u03c1\u1f70\u03bd \u03c0\u03cc\u03bb\u03b9\u03bd \u1f55\u03b4\u03c9\u03c1, \u1f64\u03c3\u03c4\u03b5 \u03c4\u1f70 \u1f10\u03ba \u03c4\u1ff6\u03bd \u1fe5\u03b9\u03b6\u1ff6\u03bd \u03b2\u03b1\u03c0\u03c4\u03cc\u03bc\u03b5\u03bd\u03b1\n  \u1f10\u03bd\u03ac\u03bc\u03b9\u03bb\u03bb\u03b1 \u03b5\u1f34\u03bd\u03b1\u03b9 \u03c4\u03bf\u1fd6\u03c2 \u1f10\u03ba \u03c4\u1fc6\u03c2 \u03ba\u03cc\u03ba\u03ba\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u1f01\u03bb\u03bf\u03c5\u03c1\u03b3\u03ad\u03c3\u03b9\u03bd.\n\nFootnote 13:\n\n  Strabo xii. 8. 16 (p. 578) \u03c6\u03ad\u03c1\u03b5\u03b9 \u03b4' \u1f41 \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f74\u03bd \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1\u03bd \u03c4\u03cc\u03c0\u03bf\u03c2\n  \u03c0\u03c1\u03bf\u03b2\u03ac\u03c4\u03c9\u03bd \u1f00\u03c1\u03b5\u03c4\u1f70\u03c2 \u03bf\u1f50\u03ba \u03b5\u1f30\u03c2 \u03bc\u03b1\u03bb\u03b1\u03ba\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bc\u03cc\u03bd\u03bf\u03bd \u03c4\u1ff6\u03bd \u1f10\u03c1\u03af\u03c9\u03bd, \u1f96 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u039c\u03b9\u03bb\u03b7\u03c3\u03af\u03c9\u03bd\n  \u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03ba\u03bf\u03c1\u03b1\u03be\u1f74\u03bd \u03c7\u03c1\u03cc\u03b1\u03bd, \u1f65\u03c3\u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c3\u03bf\u03b4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9\n  \u03bb\u03b1\u03bc\u03c0\u03c1\u1ff6\u03c2 \u1f00\u03c0' \u03b1\u1f50\u03c4\u1ff6\u03bd, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03ba\u03b1\u1f76 \u03bf\u1f31 \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b7\u03bd\u03bf\u1f76 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u1f41\u03bc\u03c9\u03bd\u03cd\u03bc\u03bf\u03c5 \u03c7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2,\n  \u03c0\u03bb\u03b7\u03c3\u03af\u03bf\u03bd \u03bf\u1f30\u03ba\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2. For this strange adjective \u03ba\u03bf\u03c1\u03b1\u03be\u03cc\u03c2 (which seems to\n  be derived from \u03ba\u03cc\u03c1\u03b1\u03be and to mean 'raven-black') see the passages in\n  Hase and Dindorf's _Steph. Thes._ In Latin we find the form\n  _coracinus_, Vitruv. viii. 3 \u00a7 14 'Aliis coracino colore,' Laodicea\n  being mentioned in the context. Vitruvius represents this as the\n  natural colour of the fleeces, and attributes it to the water drunk by\n  the sheep. See also Plin. _N. H._ viii. 48 \u00a7 73. So too Hieron. _adv.\n  Jovin._ ii. 21 (II. p. 358) 'Laodice\u00e6 indumentis ornatus incedis.' The\n  ancient accounts of the natural colour of the fleeces in this\n  neighbourhood are partially confirmed by modern travellers; e.g.\n  Pococke p. 74, Chandler p. 228.\n\nFootnote 14:\n\n  Boeckh _Corp. Inscr._ 3938 [\u1f21 \u1f10\u03c1\u03b3\u03b1\u03c3\u03af\u03b1] \u03c4\u1ff6\u03bd \u03b3\u03bd\u03b1\u03c6\u03ad[\u03c9\u03bd \u03ba\u03b1\u1f76 \u03b2\u03b1\u03c6\u03ad\u03c9\u03bd \u03c4\u1ff6\u03bd]\n  (\u03b1\u03bb\u03bf\u03c5\u03c1\u03b3[\u1ff6]\u03bd.\n\nFootnote 15:\n\n  See the passage of Strabo quoted p. 4, note 13. The place gives its\n  name to the colour, and not conversely, as stated in Blakesley's\n  Herod. vii. 113. See also Plin. _N. H._ xxi. 9 \u00a7 27, 'In vepribus\n  nascitur cyclaminum ... flos ejus _colossinus_ in coronas admittitur,'\n  a passage which assists in determining the colour.\n\n[Sidenote: 1. LAODICEA.]\n\n[Sidenote: Its name and history.]\n\n1. Of these three towns LAODICEA, as the most important, deserves to be\nconsidered first. Laodice was a common name among the ladies of the\nroyal house of the Seleucid\u00e6, as Antiochus was among the princes. Hence\nAntiochia and Laodicea occur frequently as the designations of cities\nwithin the dominions of the Syrian kings. Laodicea on the Lycus[16], as\nit was surnamed to distinguish it from other towns so called, and more\nespecially perhaps from its near neighbour Laodicea Catacecaumene, had\nborne in succession the names of Diospolis and Rhoas[17]; but when\nrefounded by Antiochus Theos (B.C. 261\u2013246), it was newly designated\nafter his wife Laodice[18]. It is situated[19] on an undulating hill, or\ngroup of hills, which overhangs the valley on the south, being washed on\neither side by the streams of the Asopus and the Caprus, tributaries of\nthe Lycus[20]. Behind it rise the snow-capped heights of Cadmus, the\nlofty mountain barrier which shuts in the south side of the main\nvalley[21]. [Sidenote: Its growing prosperity.] A place of no great\nimportance at first, it made rapid strides in the last days of the\nrepublic and under the earliest C\u00e6sars, and had become, two or three\ngenerations before St Paul wrote, a populous and thriving city[22].\nAmong its famous inhabitants are mentioned the names of some\nphilosophers, sophists, and rhetoricians, men renowned in their day but\nforgotten or almost forgotten now[23]. More to our purpose, as\nillustrating the boasted wealth and prosperity of the city, which\nappeared as a reproach and a stumblingblock in an Apostle's eyes[24],\nare the facts, that one of its citizens, Polemo, became a king and a\nfather of kings, and that another, Hiero, having accumulated enormous\nwealth, bequeathed all his property to the people and adorned the city\nwith costly gifts[25]. To the good fortune of her principal sons, as\nwell as to the fertility of the country around, the geographer Strabo\nascribes the increase and prosperity of Laodicea. The ruins of public\nbuildings still bear testimony by their number and magnificence to the\npast greatness of the city[26].\n\nFootnote 16:\n\n  \u1f10\u03c0\u1f76 \u039b\u03cd\u03ba\u1ff3, Boeckh _Corp. Inscr._ no. 3938, Ptol. _Geogr._ v. 2, Tab.\n  Peut. 'laudicium pilycum'; \u03c0\u03c1\u1f78\u03c2 [\u03c4\u1ff7] \u039b\u03cd\u03ba\u1ff3, Eckhel _Num. Vet._ III. p.\n  166, Strabo l.c., Boeckh _Corp. Inscr._ 5881, 5893; \u03c0\u03c1\u1f78\u03c2 \u039b\u03cd\u03ba\u03bf\u03bd, Boeckh\n  _Corp. Inscr._ 6478. A citizen was styled \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u1f7a\u03c2 \u1f00\u03c0\u1f78 \u039b\u03cd\u03ba\u03bf\u03c5, Diog.\n  Laert. ix. 12 \u00a7 116.\n\nFootnote 17:\n\n  Plin. _N. H._ v. 29.\n\nFootnote 18:\n\n  Steph. Byz. s.v., who quotes the oracle in obedience to which (\u1f61\u03c2\n  \u1f10\u03ba\u03ad\u03bb\u03b5\u03c5\u03c3\u03b5 \u0396\u03b5\u1f7a\u03c2 \u1f51\u03c8\u03b9\u03b2\u03c1\u03b5\u03bc\u03ad\u03c4\u03b7\u03c2) it was founded.\n\nFootnote 19:\n\n  For descriptions of Laodicea see Smith p. 250 sq., Pococke p. 71 sq.,\n  Chandler p. 224 sq., Arundell _Seven Churches_ p. 84 sq., _Asia Minor_\n  II. p. 180 sq., Fellows _Asia Minor_ 280 sq., Hamilton I. p. 514 sq.,\n  Tchihatcheff P. I. p. 252 sq., 258 sq. See also the views in Laborde,\n  pl. xxxix, Allom and Walsh II. p. 86, and Svoboda phot. 36\u201338.\n\n  The modern Turkish name is Eskihissar, 'the Old Castle,' corresponding\n  to the modern Greek, Pale\u00f3kastro, a common name for the sites of\n  ancient cities; Leake p. 251. On the ancient site itself there is no\n  town or village; the modern city Denizli is a few miles off.\n\nFootnote 20:\n\n  The position of Laodicea with respect to the neighbouring streams is\n  accurately described by Pliny _N.H._ v. 29 'Imposita est Lyco flumini,\n  latera affluentibus Asopo et Capro'; see Tchihatcheff P. I. p. 258.\n  Strabo xii. (l.c.) is more careless in his description (for it can\n  hardly be, as Tchihatcheff assumes, that he has mistaken one of these\n  two tributaries for the Lycus itself), \u1f10\u03bd\u03c4\u03b1\u1fe6\u03b8\u03b1 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u1f41 \u039a\u03ac\u03c0\u03c1\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f41\n  \u039b\u03cd\u03ba\u03bf\u03c2 \u03c3\u03c5\u03bc\u03b2\u03ac\u03bb\u03bb\u03b5 \u03c4\u1ff7 \u039c\u03b1\u03b9\u03ac\u03bd\u03b4\u03c1\u1ff3 \u03c0\u03bf\u03c4\u03b1\u03bc\u1ff7 \u03c0\u03bf\u03c4\u03b1\u03bc\u1f78\u03c2 \u03b5\u1f50\u03bc\u03b5\u03b3\u03ad\u03b8\u03b7\u03c2, where \u1f10\u03bd\u03c4\u03b1\u1fe6\u03b8\u03b1\n  refers to \u1f41 \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f74\u03bd \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1\u03bd \u03c4\u03cc\u03c0\u03bf\u03c2, and where by the junction of\n  the stream with the M\u00e6ander must be intended the junction of the\n  _combined_ stream of the Lycus and Caprus. On the coins of Laodicea\n  (Eckhel III. p. 166, Mionnet IV. p. 330, ib. Suppl. VII. p. 587, 589)\n  the Lycus and Caprus appear together, being sometimes represented as a\n  wolf and a wild-boar. The Asopus is omitted, either as being a less\n  important stream or as being less capable of symbolical\n  representation. Of modern travellers, Smith (p. 250), and after him\n  Pococke (p. 72), have correctly described the position of the streams.\n  Chandler (p. 227), misled by Strabo, mistakes the Caprus for the Lycus\n  and the Lycus for the M\u00e6ander. The modern name of the Lycus is Tchoruk\n  S\u00fa.\n\nFootnote 21:\n\n  The modern name of Cadmus is Baba-Dagh, 'The father of mountains.'\n\nFootnote 22:\n\n  Strabo xii. l.c. \u1f21 \u03b4\u1f72 \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1 \u03bc\u03b9\u03ba\u03c1\u1f70 \u03c0\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u1f40\u1fe6\u03c3\u03b1 \u1f00\u03cd\u03be\u03b7\u03c3\u03b9\u03bd \u1f14\u03bb\u03b1\u03b2\u03b5\u03bd \u1f10\u03c6'\n  \u1f21\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f21\u03bc\u03b5\u03c4\u03ad\u03c1\u03c9\u03bd \u03c0\u03b1\u03c4\u03ad\u03c1\u03c9\u03bd, \u03ba\u03b1\u03af\u03c4\u03bf\u03b9 \u03ba\u03b1\u03ba\u03c9\u03b8\u03b5\u1fd6\u03c3\u03b1 \u1f10\u03ba \u03c0\u03bf\u03bb\u03b9\u03bf\u03c1\u03ba\u03af\u03b1\u03c2 \u1f10\u03c0\u1f76\n  \u039c\u03b9\u03b8\u03c1\u03b9\u03b4\u03ac\u03c4\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u1f18\u03c5\u03c0\u03ac\u03c4\u03bf\u03c1\u03bf\u03c2. Strabo flourished in the time of Augustus\n  and the earlier years of Tiberius. The growing importance of Laodicea\n  dates from before the age of Cicero: see p. 7.\n\nFootnote 23:\n\n  Strabo l.c.; Diog. Laert. ix. 11 \u00a7 106, 12 \u00a7 116; Philostr. _Vit.\n  Soph._ i. 25; Eckhel _Doctr. Num. Vet._ III. p. 162, 163 sq.\n\nFootnote 24:\n\n  Rev. iii. 17; see below p. 43.\n\nFootnote 25:\n\n  Strabo l.c.\n\nFootnote 26:\n\n  The ruins of Laodicea have formed the quarry out of which the modern\n  town of Denizli is built. Yet notwithstanding these depredations they\n  are still very extensive, comprising an amphitheatre, two or three\n  theatres, an aqueduct, etc. The amphitheatre was built by the\n  munificence of a citizen of Laodicea only a few years after St Paul\n  wrote, as the inscription testifies; Boeckh _Corp. Inscr._ no. 3935.\n  See especially Hamilton I. p. 515 sq., who describes these ruins as\n  'bearing the stamp of Roman extravagance and luxury, rather than of\n  the stern and massive solidity of the Greeks.'\n\n[Sidenote: Its political rank, as the capital of a _conventus_.]\n\nNot less important, as throwing light on the Apostolic history, is the\npolitical status of Laodicea. Asia Minor under the Romans was divided\ninto districts, each comprising several towns and having its chief city,\nin which the courts were held from time to time by the proconsul or\nlegate of the province, and where the taxes from the subordinate towns\nwere collected[27]. Each of these political aggregates was styled in\nLatin _conventus_, in Greek \u03b4\u03b9\u03bf\u03af\u03ba\u03b7\u03c3\u03b9\u03c2\u2014a term afterwards borrowed by the\nChristian Church, being applied to a similar ecclesiastical aggregate,\nand thus naturalised in the languages of Christendom as _diocese_. At\nthe head of the most important of these political dioceses, the\n'Cibyratic convention' or 'jurisdiction,' as it was called, comprising\nnot less than twenty-five towns, stood Laodicea[28]. Here in times past\nCicero, as proconsul of Cilicia, had held his court[29]; hither at\nstated seasons flocked suitors, advocates, clerks, sheriffs'-officers,\ntax-collectors, pleasure-seekers, courtiers\u2014all those crowds whom\nbusiness or leisure or policy or curiosity would draw together from a\nwealthy and populous district, when the representative of the laws and\nthe majesty of Rome appeared to receive homage and to hold his\nassize[30]. To this position as the chief city of the Cibyratic union\nthe inscriptions probably refer, when they style Laodicea the\n'metropolis[31].' And in its metropolitan rank we see an explanation of\nthe fact, that to Laodicea, as to the centre of a Christian diocese\nalso, whence their letters would readily be circulated among the\nneighbouring brotherhoods, two Apostles address themselves in\nsuccession, the one writing from his captivity in Rome[32], the other\nfrom his exile at Patmos[33].\n\nFootnote 27:\n\n  See Becker and Marquardt _R\u00f6m. Alterth._ III. 1. p. 136 sq.\n\nFootnote 28:\n\n  See Cic. _ad Att._ v. 21,'Idibus Februariis ... forum institueram\n  agere Laodice\u00e6 Cibyraticum,'with the references in the next note:\n  comp. also Plin. _N.H._ v. 29 'Una (jurisdictio) appellatur\n  Cibyratica. Ipsum (i.e. Cibyra) oppidum Phrygi\u00e6 est. Conveniunt eo xxv\n  civitates, celeberrima urbe Laodicea.'\n\n  Besides these passages, testimony is borne to the importance of the\n  Cibyratic 'conventus' by Strabo, xiii. 4 \u00a7 17 (p. 631), \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2\n  \u03bc\u03b5\u03b3\u03af\u03c3\u03c4\u03b1\u03b9\u03c2 \u1f10\u03be\u03b5\u03c4\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03bf\u03b9\u03ba\u03ae\u03c3\u03b5\u03c3\u03b9 \u03c4\u1fc6\u03c2 \u1f08\u03c3\u03af\u03b1\u03c2 \u1f21 \u039a\u03b9\u03b2\u03c5\u03c1\u03b1\u03c4\u03b9\u03ba\u03ae. It will be\n  remembered also that Horace singles out the _Cibyratica negotia_\n  (_Epist._ i. 6. 33) to represent Oriental trade generally. The\n  importance of Laodicea may be inferred from the fact that, though the\n  union was named after Cibyra, its head-quarters were from the first\n  fixed at or soon afterwards transferred to Laodicea.\n\nFootnote 29:\n\n  See _ad Fam._ ii. 17, iii. 5, 7, 8, ix. 25, xiii. 54, 67, xv. 4; _ad\n  Att._ v. 16, 17, 20, 21, vi. 1, 2, 3, 7. He visited Laodicea on\n  several occasions, sometimes making a long stay there, and not a few\n  of his letters are written thence. See especially his account of his\n  work there, _ad Att._ vi. 2, 'Hoc foro quod egi ex Idibus Februariis\n  Laodice\u00e6 ad Kalendas Maias omnium dioecesium, pr\u00e6ter Cilici\u00e6,\n  mirabilia qu\u00e6dam efficimus; ita mult\u00e6 civitates, etc.' Altogether\n  Laodicea seems to have been second in importance to none of the cities\n  in his province, except perhaps Tarsus. See also the notice, _in\n  Verr._ Act. ii. I. c. 30.\n\nFootnote 30:\n\n  The description which Dion Chrysostom gives in his eulogy of Cel\u00e6n\u00e6\n  (Apamea Cibotus), the metropolis of a neighbouring 'dioecesis,'\n  enables us to realise the concourse which gathered together on these\n  occasions: _Orat._ XXXV (II. p. 69) \u03be\u03c5\u03bd\u03ac\u03b3\u03b5\u03c4\u03b1\u03b9 \u03c0\u03bb\u1fc6\u03b8\u03bf\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd\n  \u03b4\u03b9\u03ba\u03b1\u03b6\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd, \u03b4\u03b9\u03ba\u03b1\u03b6\u03cc\u03bd\u03c4\u03c9\u03bd, \u1f21\u03b3\u03b5\u03bc\u03cc\u03bd\u03c9\u03bd, \u1f51\u03c0\u03b7\u03c1\u03b5\u03c4\u1ff6\u03bd, \u03bf\u1f30\u03ba\u03b5\u03c4\u1ff6\u03bd, \u03ba.\u03c4.\u03bb.\n\nFootnote 31:\n\n  On this word see Becker and Marquardt l.c. p. 138 sq. It had lost its\n  original sense, as the mother city of a colony. Laodicea is styled\n  'metropolis' on the coins, Mionnet IV. p. 321.\n\nFootnote 32:\n\n  Col. iv. 16 with the notes. See also below p. 37, and the introduction\n  to the Epistle to the Ephesians.\n\nFootnote 33:\n\n  Rev. iii. 14.\n\n[Sidenote: Its religious\n           worship.]\n\nOn the religious worship of Laodicea very little special information\nexists. Its tutelary deity was Zeus, whose guardianship had been\nrecognised in Diospolis, the older name of the city, and who, having\n(according to the legend) commanded its rebuilding, was commemorated on\nits coins with the surname Laodicenus[34]. Occasionally he is also\ncalled Aseis, a title which perhaps reproduces a Syrian epithet of this\ndeity, 'the mighty.' If this interpretation be correct, we have a link\nof connexion between Laodicea and the religions of the farther East\u2014a\nconnexion far from improbable, considering that Laodicea was refounded\nby a Syrian king and is not unlikely to have adopted some features of\nSyrian worship[35].\n\nFootnote 34:\n\n  See Eckhel III. p. 159 sq. (passim), Mionnet IV. p. 315 sq., ib.\n  Suppl. VII. p. 578 sq. (passim). In the coins commemorating an\n  alliance with some other city Laodicea is represented by Zeus; e.g.\n  Mionnet IV. pp. 320, 324, 331 sq., Suppl. VII. pp. 586, 589.\n\nFootnote 35:\n\n  ==\u03b1\u03f2\u03b5\u03b9\u03f2== or ==\u03b1\u03f2\u03b5\u03b9\u03f2 \u03bb\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03c9\u03bd==. See Waddington _Voyage en Asie\n  Mineure au point de vue Numismatique_ (Paris 1853) pp. 25, 26 sq. Mr\n  Waddington adopts a suggestion communicated to him by M. de Longp\u00e9rier\n  that this word represents the Aramaic \u05e2\u05d6\u05d9\u05d6\u05d0 'the strong, mighty,'\n  which appears also in the Arabic 'Aziz.' This view gains some\n  confirmation from the fact, not mentioned by Mr Waddington, that\n  \u1f0c\u03b6\u03b9\u03b6\u03bf\u03c2 was an epithet of the Ares of Edessa: Julian _Orat._ iv; comp.\n  Cureton _Spic. Syr._ p. 80, and see de Lagarde _Gesamm. Abhandl._ p.\n  16. On the other hand this Shemitic word elsewhere, when adopted into\n  Greek or Latin, is written \u1f0c\u03b6\u03b9\u03b6\u03bf\u03c2 or Azizus: see Garrucci in the\n  _Arch\u00e6ologia_ XLIII. p. 45 'Tyrio Septimio Azizo,' and Boeckh _Corp.\n  Inscr._ 9893 \u1f0c\u03b6\u03b9\u03b6\u03bf\u03c2 \u1f08\u03b3\u03c1\u03af\u03c0\u03b1 \u03a3\u03cd\u03c1\u03bf\u03c2. M. de Longp\u00e9rier offers the\n  alternative that \u03b1\u03c3\u03b5\u03b9\u03f2, _i.e._ \u1f08\u03c3\u03af\u03c2, is equivalent to \u1f08\u03c3\u03b9\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2. An\n  objection to this view, stronger than those urged by Mr Waddington, is\n  the fact that \u1f08\u03c3\u03af\u03c2 seems only to be used as a feminine adjective. M.\n  Renan points to the fact that this \u03b6\u03b5\u03c5\u03c2 \u03b1\u03c3\u03b5\u03b9\u03c2 is represented with his\n  hand on the horns of a goat, and on the strength of this coincidence\n  would identify him with 'the Azazel of the Semites' (_Saint Paul_, p.\n  359), though tradition and orthography alike point to some other\n  derivation of Azazel (\u05e2\u05d6\u05d0\u05d6\u05dc).\n\n[Sidenote: 2. HIERAPOLIS.]\n\n[Sidenote: Its situation.]\n\n2. On the north of the valley, opposite to the sloping hills which mark\nthe site of Laodicea, is a broad level terrace jutting out from the\nmountain side and overhanging the plain with almost precipitous sides.\nOn this plateau are scattered the vast ruins of HIERAPOLIS[36]. The\nmountains upon which it abuts occupy the wedge of ground between the\nM\u00e6ander and the Lycus; but, as the M\u00e6ander above its junction with the\nLycus passes through a narrow ravine, they blend, when seen from a\ndistance, with the loftier range of the Mesogis which overhangs the\nright bank of the M\u00e6ander almost from its source to its embouchure, and\nform with it the northern barrier to the view, as the Cadmus range does\nthe southern, the broad valley stretching between. Thus Hierapolis may\nbe said to lie over against Mesogis, as Laodicea lies over against\nCadmus[37].\n\nFootnote 36:\n\n  For descriptions of Hierapolis, see Smith p. 245 sq., Pococke p. 75\n  sq., Chandler 229 sq., Arundell _Seven Churches_ p. 79 sq., Hamilton\n  p. 517 sq., Fellows _Asia Minor_ p. 283 sq. For the travertine\n  deposits see especially the description and plates in Tchihatcheff P.\n  I. p. 345, together with the views in Laborde (pl. xxxii-xxxviii), and\n  Svoboda (photogr. 41\u201347). Tchihatcheff repeatedly calls the place\n  Hieropolis; but this form, though commonly used of other towns (see\n  Steph. Byz. s.v. \u1f39\u03b5\u03c1\u03b1\u03c0\u03cc\u03bb\u03b9\u03c2, Leake _Num. Hell._ p. 67), appears not to\n  occur as a designation of the Phrygian city, which seems always to be\n  written Hierapolis. The citizens however are sometimes called\n  \u1f39\u03b5\u03c1\u03bf\u03c0\u03bf\u03bb\u1fd6\u03c4\u03b1\u03b9 on the coins.\n\n  The modern name is given differently by travellers. It is generally\n  called Pambouk-Kalessi, i.e. 'cotton-castle,' supposed to allude to\n  the appearance of the petrifactions, though cotton is grown in the\n  neighbourhood (Hamilton I. p. 517). So Smith, Pococke, Chandler,\n  Arundell, Tchihatcheff, Waddington, and others. M. Renan says\n  '_Tambouk_, et non _Pambouk, Kalessi_' (_S. Paul_ p. 357). Laborde\n  gives the word _Tambouk_ in some places and _Pambouk_ in others; and\n  Leake says 'Hierapolis, now called _Tab\u00fak-Kale_ or _Pambuk-Kale_' (p.\n  252).\n\nFootnote 37:\n\n  Strabo xiii. 4. 14 (p. 629) says \u1f51\u03c0\u03b5\u03c1\u03b2\u03b1\u03bb\u03bf\u1fe6\u03c3\u03b9 \u03b4\u1f72 \u03c4\u1f74\u03bd \u039c\u03b5\u03c3\u03c9\u03b3\u03af\u03b4\u03b1 ...\n  \u03c0\u03cc\u03bb\u03b5\u03b9\u03c2 \u03b5\u1f30\u03c3\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03bc\u1f72\u03bd \u03c4\u1fc7 \u039c\u03b5\u03c3\u03c9\u03b3\u1f76\u03b4\u03b9 \u03ba\u03b1\u03c4\u03b1\u03bd\u03c4\u03b9\u03ba\u03c1\u1f7a \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 \u1f39\u03b5\u03c1\u1f70 \u03c0\u03cc\u03bb\u03b9\u03c2,\n  \u03ba.\u03c4.\u03bb. He cannot mean that Hierapolis was situated immediately in or\n  by the Mesogis (for the name does not seem ever to be applied to the\n  mountains between the Lycus and M\u00e6ander), but that with respect to\n  Laodicea it stood over against the Mesogis, as I have explained it in\n  the text. The view in Laborde (pl. xxxix) shows the appearance of\n  Hierapolis from Laodicea. Strabo had himself visited the place and\n  must have known how it was situated. Some modern travellers however\n  (e.g. Chandler and Arundell) speak of the plateau of Hierapolis as\n  part of the Mesogis. Steiger (_Kolosser_ p. 33) gets over the\n  difficulty by translating Strabo's words, 'near the Mesogis but on the\n  opposite side (i.e. of the M\u00e6ander) is the Laodicean Hierapolis' (to\n  distinguish it from others of the name); but \u03ba\u03b1\u03c4\u03b1\u03bd\u03c4\u03b9\u03ba\u03c1\u1f7a cannot be\n  separated from \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 without violence.\n\n[Sidenote: Remarkable physical features.]\n\nIt is at Hierapolis that the remarkable physical features which\ndistinguish the valley of the Lycus display themselves in the fullest\nperfection. Over the steep cliffs which support the plateau of the city,\ntumble cascades of pure white stone, the deposit of calcareous matter\nfrom the streams which, after traversing this upper level, are\nprecipitated over the ledge into the plain beneath and assume the most\nfantastic shapes in their descent. At one time overhanging in cornices\nfringed with stalactites, at another hollowed out into basins or broken\nup with ridges, they mark the site of the city at a distance, glistening\non the mountain-side like foaming cataracts frozen in the fall.\n\n[Sidenote: Their relation to the Apostolic history.]\n\nBut for the immediate history of St Paul's Epistles the striking beauty\nof the scenery has no value. It is not probable that he had visited this\ndistrict when the letters to the Colossians and Laodiceans were written.\nWere it otherwise, we can hardly suppose, that educated under widely\ndifferent influences and occupied with deeper and more absorbing\nthoughts, he would have shared the enthusiasm which this scenery\ninspires in the modern traveller. Still it will give a reality to our\nconceptions, if we try to picture to ourselves the external features of\nthat city, which was destined before long to become the adopted home of\nApostles and other personal disciples of the Lord, and to play a\nconspicuous part\u2014second perhaps only to Ephesus\u2014in the history of the\nChurch during the ages immediately succeeding the Apostles.\n\n[Sidenote: Hierapolis a famous watering-place.]\n\nLike Laodicea, Hierapolis was at this time an important and a growing\ncity, though not like Laodicea holding metropolitan rank[38]. Besides\nthe trade in dyed wools, which it shared in common with the neighbouring\ntowns, it had another source of wealth and prosperity peculiar to\nitself. The streams to which the scenery owes the remarkable features\nalready described, are endowed with valuable medicinal qualities, while\nat the same time they are so copious that the ancient city is described\nas full of self-made baths[39]. An inscription, still legible among the\nruins, celebrates their virtues in heroic verse, thus apostrophizing the\ncity:\n\n             Hail, fairest soil in all broad Asia's realm;\n             Hail, golden city, nymph divine, bedeck'd\n             With flowing rills, thy jewels[40].\n\nCoins of Hierapolis too are extant of various types, on which \u00c6sculapius\nand Hygeia appear either singly or together[41]. To this fashionable\nwatering-place, thus favoured by nature, seekers of pleasure and seekers\nof health alike were drawn.\n\nFootnote 38:\n\n  On its _ecclesiastical_ title of metropolis, see below, p. 70, note\n  277.\n\nFootnote 39:\n\n  Strabo l.c. \u03bf\u1f55\u03c4\u03c9 \u03b4' \u1f10\u03c3\u03c4\u1f76\u03bd \u1f04\u03c6\u03b8\u03bf\u03bd\u03bf\u03bd \u03c4\u1f78 \u03c0\u03bb\u1fc6\u03b8\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u1f55\u03b4\u03b1\u03c4\u03bf\u03c2 \u1f65\u03c3\u03c4\u03b5 \u1f21 \u03c0\u03cc\u03bb\u03b9\u03c2\n  \u03bc\u03b5\u03c3\u03c4\u1f74 \u03c4\u1ff6\u03bd \u03b1\u1f50\u03c4\u03bf\u03bc\u03ac\u03c4\u03c9\u03bd \u03b2\u03b1\u03bb\u03b1\u03bd\u03b5\u03af\u03c9\u03bd \u1f10\u03c3\u03c4\u03af.\n\nFootnote 40:\n\n  Boeckh _Corp. Inscr._ 3909, \u1f08\u03c3\u03af\u03b4\u03bf\u03c2 \u03b5\u1f50\u03c1\u03b5\u03af\u03b7\u03c2 \u03c0\u03c1\u03bf\u03c6\u03b5\u03c1\u03ad\u03c3\u03c4\u03b1\u03c4\u03bf\u03bd \u03bf\u1f56\u03b4\u03b1\u03c2\n  \u1f01\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd, \u03c7\u03b1\u03af\u03c1\u03bf\u03b9\u03c2, \u03c7\u03c1\u03c5\u03c3\u03cc\u03c0\u03bf\u03bb\u03b9 \u1f39\u03b5\u03c1\u03ac\u03c0\u03bf\u03bb\u03b9, \u03c0\u03cc\u03c4\u03bd\u03b9\u03b1 \u039d\u03c5\u03bc\u03c6\u1ff6\u03bd, \u03bd\u03ac\u03bc\u03b1\u03c3\u03b9\u03bd,\n  \u1f00\u03b3\u03bb\u03b1\u0390\u1fc3\u03c3\u03b9, \u03ba\u03b5\u03ba\u03b1\u03c3\u03bc\u03ad\u03bd\u03b7.\n\nFootnote 41:\n\n  Mionnet IV. p. 297, 306, 307, ib. Suppl. VII. p. 567; Waddington\n  _Voyage_ etc. p. 24.\n\n[Sidenote: The magnificence of its ruins.]\n\nTo the ancient magnificence of Hierapolis its extant ruins bear ample\ntestimony. More favoured than Laodicea, it has not in its immediate\nneighbourhood any modern town or village of importance, whose\ninhabitants have been tempted to quarry materials for their houses out\nof the memorials of its former greatness. Hence the whole plateau is\ncovered with ruins, of which the extent and the good taste are equally\nremarkable; and of these the pal\u00e6stra and the therm\u00e6, as might be\nexpected, are among the more prominent.\n\n[Sidenote: Its religious worship.]\n\nA city, which combined the pursuit of health and of gaiety, had fitly\nchosen as its patron deity Apollo, the god alike of medicine and of\nfestivity, here worshipped especially as 'Archegetes,' the Founder[42].\nBut more important, as illustrating the religious temper of this\nPhrygian city, is another fact connected with it. [Sidenote: The\nPlutonium.]In Hierapolis was a spot called the Plutonium, a hot well or\nspring, from whose narrow mouth issued a mephitic vapour immediately\nfatal to those who stood over the opening and inhaled its fumes. To the\nmutilated priests of Cybele alone (so it was believed) an immunity was\ngiven from heaven, which freed them from its deadly effects[43]. Indeed\nthis city appears to have been a chief centre of the passionate mystical\ndevotion of ancient Phrygia. But indications are not wanting, that in\naddition to this older worship religious rites were borrowed also from\nother parts of the East, more especially from Egypt[44]. By the\nmultitude of her temples Hierapolis established her right to the title\nof the 'sacred city,' which she bore[45].\n\nFootnote 42:\n\n  Boeckh _Corp. Inscr._ 3905, 3906; Mionnet iv. pp. 297, 301, 307, ib.\n  Suppl. vii. p. 568, 569, 570. In coins struck to commemorate alliances\n  with other cities, Hierapolis is represented by Apollo Archegetes:\n  Mionnet IV. p. 303, ib. Suppl. VII. 572, 573, 574; Waddington _Voyage_\n  etc. p. 25; and see Eckhel III. p. 156. On the meaning of\n  _Archegetes_, under which name Apollo was worshipped by other cities\n  also, who regarded him as their founder, see Spanheim on Callim.\n  _Hymn. Apoll._ 57.\n\nFootnote 43:\n\n  Strabo l.c. He himself had seen the phenomenon and was doubtful how to\n  account for the immunity of these priests, \u03b5\u1f34\u03c4\u03b5 \u03b8\u03b5\u03af\u1fb3 \u03c0\u03c1\u03bf\u03bd\u03bf\u03af\u1fb3 ... \u03b5\u1f34\u03c4\u03b5\n  \u1f00\u03bd\u03c4\u03b9\u03b4\u03cc\u03c4\u03bf\u03b9\u03c2 \u03c4\u03b9\u03c3\u1f76 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c3\u03b9 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03c3\u03c5\u03bc\u03b2\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03bf\u03c2. See also Plin. _N. H._\n  ii. 93 \u00a7 95 'locum ... matris tantum magn\u00e6 sacerdoti innoxium.' Dion\n  Cass. (Xiphil.) lxviii. 27, who also witnessed the phenomenon, adds \u03bf\u1f50\n  \u03bc\u1f74\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03b1\u1f30\u03c4\u03af\u03b1\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c3\u03c5\u03bd\u03bd\u03bf\u1fc6\u03c3\u03b1\u03b9 \u1f14\u03c7\u03c9, \u03bb\u03ad\u03b3\u03c9 \u03b4\u1f72 \u1f05 \u03c4\u03b5 \u03b5\u1f36\u03b4\u03bf\u03bd \u1f61\u03c2 \u03b5\u1f36\u03b4\u03bf\u03bd\n  \u03ba\u03b1\u1f76 \u1f03 \u1f24\u03ba\u03bf\u03c5\u03c3\u03b1 \u1f61\u03c2 \u1f24\u03ba\u03bf\u03c5\u03c3\u03b1. Ammian. Marc. xxiii. 6. 18 also mentions this\n  marvel, but speaks cautiously, 'ut asserunt quidam,' and adds 'quod\n  qua causa eveniat, rationibus physicis permittatur.' Comp. Anthol.\n  VII. p. 190 \u0395\u1f34 \u03c4\u03b9\u03c2 \u1f00\u03c0\u03ac\u03b3\u03be\u03b1\u03c3\u03b8\u03b1\u03b9 \u03bc\u1f72\u03bd \u1f40\u03ba\u03bd\u03b5\u1fd6 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5 \u03b4' \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03b5\u1fd6, \u1f10\u03be \u1f39\u03b5\u03c1\u1fb6\u03c2\n  \u03c0\u03cc\u03bb\u03b5\u03c9\u03c2 \u03c8\u03c5\u03c7\u03c1\u1f78\u03bd \u1f55\u03b4\u03c9\u03c1 \u03c0\u03b9\u03ad\u03c4\u03c9; Stob\u00e6us _Ecl._ i. 34, p. 680. Laborde states\n  (p. 83) that he discovered by experiment that the waters are sometimes\n  fatal to animal life and sometimes perfectly harmless; and if this be\n  substantiated, we have a solution of the marvel. Other modern\n  travellers, who have visited the Plutonium, are Cockerell (Leake p.\n  342), and Svoboda. In Svoboda's work a chemical analysis of the waters\n  is given.\n\nFootnote 44:\n\n  On a coin of Hierapolis, Pluto-Serapis appears seated, while before\n  him stands Isis with a sistrum in her hand; Waddington _Voyage_ etc.\n  p. 24. See also Mionnet IV. pp. 296, 305; Leake _Num. Hell._ p. 66.\n\n  The worship of Serapis appears elsewhere in this neighbourhood. At\n  Chon\u00e6 (Coloss\u00e6) is an inscription recording a vow to this deity; Le\n  Bas _Asie Mineure_ inscr. 1693 b.\n\nFootnote 45:\n\n  Steph. Byz. s.v. \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u1f31\u03b5\u03c1\u1f70 \u03c0\u03bf\u03bb\u03bb\u1f70 \u1f14\u03c7\u03b5\u03b9\u03bd.\n\n[Sidenote: The birth-place of Epictetus.]\n\nThough at this time we have no record of famous citizens at Hierapolis,\nsuch as graced the annals of Laodicea, yet a generation or two later she\nnumbered among her sons one nobler far than the rhetoricians and\nsophists, the millionaires and princes, of whom her neighbour could\nboast. The lame slave Epictetus, the loftiest of heathen moralists, must\nhave been growing up to manhood when the first rumours of the Gospel\nreached his native city. Did any chance throw him across the path of\nEpaphras, who first announced the glad-tidings there? [Sidenote:\nEpictetus and Christianity.]Did he ever meet the great Apostle himself,\nwhile dragging out his long captivity at Rome, or when after his release\nhe paid his long-promised visit to the valley of the Lycus? We should be\nglad to think that these two men met together face to face\u2014the greatest\nof Christian, and the greatest of heathen preachers. Such a meeting\nwould solve more than one riddle. A Christian Epictetus certainly was\nnot; his Stoic doctrine and his Stoic morality are alike apparent: but\nnevertheless his language presents some strange coincidences with the\nApostolic writings, which would thus receive an explanation[46]. It must\nbe confessed however, that of any outward intercourse between the\nApostle and the philosopher history furnishes no hint.\n\nFootnote 46:\n\n  See _Philippians_, pp. 312, 313.\n\n[Sidenote: 3. COLOSS\u00c6.]\n\n3. While the sites of Laodicea and Hierapolis are conspicuous, so that\nthey were early identified by their ruins, the same is not the case with\nCOLOSS\u00c6. [Sidenote: Difficulty of determining its site.]Only within the\npresent generation has the position of this once famous city been\nascertained, and even now it lacks the confirmation of any inscription\nfound _in situ_ and giving the name[47]. [Sidenote: Subterranean channel\nof the Lycus.]Herodotus states that in Coloss\u00e6 the river Lycus\ndisappears in a subterranean cave, emerging again at a distance of about\nfive stades[48]; and this very singular landmark--the underground\npassage of a stream for half a mile\u2014might be thought to have placed the\nsite of the city beyond the reach of controversy. But this is not the\ncase. In the immediate neighbourhood of the only ruins which can\npossibly be identified with Coloss\u00e6, no such subterranean channel has\nbeen discovered. But on the other hand the appearance of the river at\nthis point suggests that at one time the narrow gorge through which it\nruns, as it traverses the ruins, was overarched for some distance with\nincrustations of travertine, and that this natural bridge was broken up\nafterwards by an earthquake, so as to expose the channel of the\nstream[49]. This explanation seems satisfactory. If it be rejected, we\nmust look for the underground channel, not within the city itself, as\nthe words of Herodotus strictly interpreted require, but at some point\nhigher up the stream. In either case there can be little doubt that\nthese are the ruins of Coloss\u00e6. [Sidenote: Petrifying stream.]The fact\nmentioned by Pliny[50], that there is in this city a river which turns\nbrick into stone, is satisfied by a side stream flowing into the Lycus\nfrom the north, and laying large deposits of calcareous matter; though\nin this region, as we have seen, such a phenomenon is very far from\nrare. The site of Coloss\u00e6 then, as determined by these considerations,\nlies two or three miles north of the present town of Chonos, the\nmedi\u00e6val Chon\u00e6, and some twelve miles east of Laodicea. The Lycus\ntraverses the site of the ruins, dividing the city into two parts, the\nnecropolis standing on the right or northern bank, and the town itself\non the left.\n\nFootnote 47:\n\n  See however a mutilated inscription (Boeckh _Corp. Inscr._ 3956) with\n  the letters ...==\u03b7\u03bd\u03c9\u03bd==, found near Chon\u00e6.\n\nFootnote 48:\n\n  Herod. vii. 30 \u1f00\u03c0\u03af\u03ba\u03b5\u03c4\u03bf \u1f10\u03c2 \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03ac\u03c2, \u03c0\u03cc\u03bb\u03b9\u03bd \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7\u03bd \u03a6\u03c1\u03c5\u03b3\u03af\u03b7\u03c2, \u1f10\u03bd \u03c4\u1fc7 \u039b\u03cd\u03ba\u03bf\u03c2\n  \u03c0\u03bf\u03c4\u03b1\u03bc\u1f78\u03c2 \u1f10\u03c2 \u03c7\u03ac\u03c3\u03bc\u03b1 \u03b3\u1fc6\u03c2 \u1f10\u03c3\u03b2\u03ac\u03bb\u03bb\u03c9\u03bd \u1f00\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9, \u1f14\u03c0\u03b5\u03b9\u03c4\u03b1 \u03b4\u03b9\u1f70 \u03c3\u03c4\u03b1\u03b4\u03af\u03c9\u03bd \u1f61\u03c2 \u03c0\u03ad\u03bd\u03c4\u03b5\n  \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03ac \u03ba\u1fc3 \u1f00\u03bd\u03b1\u03c6\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f10\u03ba\u03b4\u03b9\u03b4\u03bf\u1fd6 \u03ba\u03b1\u1f76 \u03bf\u1f56\u03c4\u03bf\u03c2 \u1f10\u03c2 \u03c4\u1f78\u03bd \u039c\u03b1\u03af\u03b1\u03bd\u03b4\u03c1\u03bf\u03bd.\n\nFootnote 49:\n\n  This is the explanation of Hamilton (I. p. 509 sq.), who (with the\n  doubtful exception of Laborde) has the merit of having first\n  identified and described the site of Coloss\u00e6. It stands on the Tchoruk\n  S\u00fa (Lycus) at the point where it is joined by two other streams, the\n  Bounar Bashi S\u00fa and the Ak-S\u00fa. In confirmation of his opinion,\n  Hamilton found a tradition in the neighbourhood that the river had\n  once been covered over at this spot (p. 522). He followed the course\n  of the Lycus for some distance without finding any subterranean\n  channel (p. 521 sq.).\n\n  It is difficult to say whether the following account in Strabo xii. 8\n  \u00a7 16 (p. 578) refers to the Lycus or not; \u1f44\u03c1\u03bf\u03c2 \u039a\u03ac\u03b4\u03bc\u03bf\u03c2 \u1f10\u03be \u03bf\u1f53 \u03ba\u03b1\u1f76 \u1f41\n  \u039b\u03cd\u03ba\u03bf\u03c2 \u1fe5\u03b5\u1fd6 \u03ba\u03b1\u1f76 \u1f04\u03bb\u03bb\u03bf\u03c2 \u1f41\u03bc\u03ce\u03bd\u03c5\u03bc\u03bf\u03c2 \u03c4\u1ff7 \u1f44\u03c1\u03b5\u03b9\u00b7 \u03c4\u1f78 \u03c0\u03bb\u03ad\u03bf\u03bd \u03b4' \u03bf\u1f57\u03c4\u03bf\u03c2 \u1f51\u03c0\u1f78 \u03b3\u1fc6\u03c2 \u1fe5\u03c5\u1f72\u03b9\u03c2\n  \u03b5\u1f36\u03c4' \u1f00\u03bd\u03b1\u03ba\u03cd\u03c8\u03b1\u03c2 \u03c3\u03c5\u03bd\u03ad\u03c0\u03b5\u03c3\u03b5\u03bd \u03b5\u1f30\u03c2 \u03c4\u03b1\u1f50\u03c4\u1f78 \u03c4\u03bf\u1fd6\u03c2 \u1f04\u03bb\u03bb\u03bf\u03b9\u03c2 \u03c0\u03bf\u03c4\u03b1\u03bc\u03bf\u1fd6\u03c2, \u1f10\u03bc\u03c6\u03b1\u03af\u03bd\u03c9\u03bd \u1f05\u03bc\u03b1\n  \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03b7\u03c4\u03bf\u03bd \u03c4\u1fc6\u03c2 \u03c7\u03ce\u03c1\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03b5\u1f54\u03c3\u03b5\u03b9\u03c3\u03c4\u03bf\u03bd. If the Lycus is meant,\n  may not \u03c3\u03c5\u03bd\u03ad\u03c0\u03b5\u03c3\u03b5\u03bd imply that this remarkable feature had changed\n  before Strabo wrote?\n\n  Laborde (p. 103), who visited the place before Hamilton, though his\n  account was apparently not published till later, fixes on the same\n  site for Coloss\u00e6, but thinks that he has discovered the subterranean\n  course of the Lycus, to which Herodotus refers, much higher up a\n  stream, close to its source ('\u00e0 dix pas de cette source'), which he\n  describes as '\u00e0 deux lieues au nord de Coloss\u00e6.' Yet in the same\n  paragraph he says 'Or il [H\u00e9rodote, exact cicerone] savait que _le\n  Lycus dispara\u00eet pr\u00e8s de Coloss\u00e6, ville consid\u00e9rable de la Phrygie_'\n  (the italics are his own). He apparently does not see the vast\n  difference between his _pr\u00e8s de Coloss\u00e6_ thus widely interpreted and\n  the precise \u1f10\u03bd \u03c4\u1fc7 of Herodotus himself. Obviously no great reliance\n  can be placed on the accuracy of a writer, who treats his authorities\n  thus. The subterranean stream which Laborde saw, and of which he gives\n  a view (pl. xl), may possibly be the phenomenon to which Herodotus\n  alludes; but if so, Herodotus has expressed himself very carelessly.\n  On the whole Hamilton's solution seems much more probable.\n\n  Arundell's account (_Seven Churches_ p. 98 sq., _Asia Minor_ p. 160\n  sq.) is very confused, and it is not clear whether he has fixed on the\n  right site for Coloss\u00e6; but it bears testimony to the existence of two\n  subterranean courses of rivers, though neither of them is close enough\n  to the city to satisfy Herodotus' description.\n\nFootnote 50:\n\n  Plin. _N.H._ xxxi. 2 \u00a7 20. This is the Ak-S\u00fa, which has strongly\n  petrifying qualities.\n\n[Sidenote: Its ancient greatness]\n\nCommanding the approaches to a pass in the Cadmus range, and standing on\na great high-way communicating between Eastern and Western Asia, Coloss\u00e6\nat an early date appears as a very important place. Here the mighty host\nof Xerxes halted on its march against Greece; it is mentioned on this\noccasion as 'a great city of Phrygia[51].' Here too Cyrus remained seven\ndays on his daring enterprise which terminated so fatally; the Greek\ncaptain, who records the expedition, speaks of it as 'a populous city,\nprosperous and great[52].' But after this time its glory seems to wane.\nThe political supremacy [Sidenote: and later decline.]of Laodicea and\nthe growing popularity of Hierapolis gradually drain its strength; and\nStrabo, writing about two generations before St Paul, describes it as a\n'small town[53]' in the district of which Laodicea was the capital. We\nshall therefore be prepared to find that, while Laodicea and Hierapolis\nboth hold important places in the early records of the Church, Coloss\u00e6\ndisappears wholly from the pages of history. Its comparative\ninsignificance is still attested by its ruins, which are few and\nmeagre[54], while the vast remains of temples, baths, theatres,\naqueducts, gymnasia, and sepulchres, strewing the extensive sites of its\nmore fortunate neighbours, still bear witness to their ancient\nprosperity and magnificence. It is not even mentioned by Ptolemy, though\nhis enumeration of towns includes several inconsiderable places[55].\nWithout doubt Coloss\u00e6 was the least important Church, to which any\nepistle of St Paul was addressed.\n\nFootnote 51:\n\n  Herod. vii. 30. See p. 14, note 48.\n\nFootnote 52:\n\n  Xen. _Anab._ i. 2. 6 \u1f10\u03be\u03b5\u03bb\u03b1\u03cd\u03bd\u03b5\u03b9 \u03b4\u03b9\u1f70 \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03c2 ... \u03b5\u1f30\u03c2 \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03ac\u03c2, \u03c0\u03cc\u03bb\u03b9\u03bd\n  \u03bf\u1f30\u03ba\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03bd, \u03b5\u1f50\u03b4\u03b1\u03af\u03bc\u03bf\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7\u03bd.\n\nFootnote 53:\n\n  \u03c0\u03cc\u03bb\u03b9\u03c3\u03bc\u03b1, Strabo xii. 8. 13 (p. 576). Plin. _N. H._ v. 32 \u00a7 41 writes\n  'Phrygia ... oppida ibi celeberrima pr\u00e6ter jam dicta, Ancyra, Andria,\n  Cel\u00e6n\u00e6, Coloss\u00e6,' etc. The commentators, referring to this passage,\n  overlook the words 'pr\u00e6ter jam dicta,' and represent Pliny as calling\n  Coloss\u00e6 'oppidum celeberrimum.' Not unnaturally they find it difficult\n  to reconcile this expression with Strabo's statement. But in fact\n  Pliny has already exhausted all the considerable towns, Hierapolis,\n  Laodicea, Apamea, etc., and even much less important places than these\n  (see v. 28, 29 \u00a7 29), so that only decayed and third-rate towns\n  remain. The Ancyra here mentioned is not the capital of Galatia, but a\n  much smaller Phrygian town.\n\nFootnote 54:\n\n  Laborde p. 102 'De cette grande c\u00e9l\u00e9brit\u00e9 de Coloss\u00e6 il ne reste\n  presque rien: ce sont des substructions sans suite, des fragments sans\n  grandeur; les restes d'un th\u00e9\u00e2tre de m\u00e9diocre dimension, une acropole\n  sans hardiesse,' etc.\n\nFootnote 55:\n\n  _Geogr._ v. 2.\n\n[Sidenote: Uncertain orthography of the name.]\n\nAnd perhaps also we may regard the variation in the orthography of the\nname as another indication of its comparative obscurity and its early\nextinction. Are we to write _Coloss\u00e6_ or _Colass\u00e6_? So far as the\nevidence goes, the conclusion would seem to be that, while Coloss\u00e6 alone\noccurs during the classical period and in St Paul's time, it was\nafterwards supplanted by Colass\u00e6, when the town itself had either\ndisappeared altogether or was already passing out of notice[56].\n\nFootnote 56:\n\n  All Greek writers till some centuries after the Christian era write it\n  \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u03af: so Herod. vii. 30, Xen. _Anab._ i. 2. 6, Strabo xii. 8. 13,\n  Diod. xiv. 80, Poly\u00e6n. _Strat._ vii. 16. 1; though in one or more MSS\n  of some of these authors it is written \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03af, showing the tendency\n  of later scribes. _Coloss\u00e6_ is also the universal form in Latin\n  writers. The coins moreover, even as late as the reign of Gordian\n  (A.D. 238\u2013244) when they ceased to be struck, universally have\n  ==\u03ba\u03bf\u03bb\u03bf\u03f2\u03f2\u03b7\u03bd\u03bf\u03b9== (or ==\u03ba\u03bf\u03bb\u03bf\u03f2\u03b7\u03bd\u03bf\u03b9==); Mionnet IV. p. 267 sq.: see\n  Babington _Numismatic Chronicle_ New series III. p. 1 sq., 6. In\n  Hierocles (_Synecd._ p. 666, Wessel.) and in the _Apostolic\n  Constitutions_ (vii. 46) \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03af seems to be the original reading of\n  the text, and in later Byzantine writers this form is common. If Prof.\n  Babington (p. 3) were right in supposing that it is connected with\n  \u03ba\u03bf\u03bb\u03bf\u03c3\u03c3\u03cc\u03c2, the question of the correct spelling might be regarded as\n  settled; but in a Phrygian city over which so many Eastern nations\n  swept in succession, who shall say to what language the name belonged,\n  or what are its affinities?\n\n  Thus, judging from classical usage, we should say that \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u03af was\n  the old form and that \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03af did not supplant it till some time\n  after St Paul's age. This view is confirmed by a review of the\n  authorities for the different readings in the New Testament.\n\n  In the opening of the epistle (i. 1) the authorities for \u1f10\u03bd \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u1fd6\u03c2\n  are overwhelming. It is read by \u05d0BDFGL (A is obliterated here and C is\n  wanting); and in the Old Latin, Vulgate, and Armenian Versions. On the\n  other hand \u1f10\u03bd \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u1fd6\u03c2 is read by KP. 17. 37. 47, and among the\n  versions by the Memphitic and the Philoxenian Syriac (\u0729\u0718\u0720\u0710\u0723\u0718\u0724, though\n  the marg. gives ==\u03ba\u03bf\u03bb\u03f2\u03f2\u03b1\u03b9\u03f2==). In the Peshito also the present\n  reading represents \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u1fd6\u03c2, but as the vowel was not expressed\n  originally and depends on the later pointing, its authority can hardly\n  be quoted. The Thebaic is wanting here.\n\n  In the _heading_ of the epistle however there is considerably more\n  authority for the form in \u03b1. \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u03b9\u03c2 is the reading of AB^* KP. 37\n  (\u039a\u03bf\u03bb\u03b1\u03c3\u03b1\u03b5\u03b9\u03c2). 47. C is wanting here, but has \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u03b9\u03c2 in the\n  subscription. On the other hand \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u03b5\u03b9\u03c2 (or \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u03b9\u03c2) appears in\n  \u05d0B^1 (according to Tregelles, but B^3 Tisch.; see his introd. p.\n  xxxxviii) DFG (but G has left \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u03b9\u03c2 in the heading of one page,\n  and \u039a\u03bf\u03bb\u03b1\u03bf\u03c3\u03b1\u03b5\u03b9\u03c2 in another) L. 17 (\u039a\u03bf\u03bb\u03bf\u03c3\u03b1\u03b5\u03b9\u03c2), in the Latin Version,\n  and in the margin of the Philoxenian Syriac. The readings of both\n  Peshito and Philoxenian (text) here depend on the vocalisation; and\n  those of other versions are not recorded. In the _subscription_ the\n  preponderance of authority is even more favourable to \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u03b9\u03c2.\n\n  Taking into account the obvious tendency which there would be in\n  scribes to make the title \u03c0\u03c1\u1f78\u03c2 \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u03b5\u1fd6\u03c2 or \u03c0\u03c1\u1f78\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u1fd6\u03c2 conform\n  to the opening \u1f10\u03bd \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u1fd6\u03c2 or \u1f10\u03bd \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u1fd6\u03c2, as shown in G, we seem to\n  arrive at the conclusion that, while \u1f10\u03bd \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u1fd6\u03c2 was indisputably the\n  original reading in the opening, \u03c0\u03c1\u1f78\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u1fd6\u03c2 was probably the\n  earlier reading in the title. If so, the title must have been added at\n  a somewhat later date; which is not improbable.\n\n  Connected with this question is the variation in the adjectival form,\n  -\u03b7\u03bd\u03cc\u03c2 or -\u03b1\u03b5\u03cd\u03c2. Parallels to this double termination occur in other\n  words; e.g. \u0394\u03bf\u03ba\u03b9\u03bc\u03b7\u03bd\u03cc\u03c2, \u0394\u03bf\u03ba\u03b9\u03bc\u03b5\u03cd\u03c2; \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b7\u03bd\u03cc\u03c2, \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03cd\u03c2; \u039d\u03b9\u03ba\u03b1\u03b7\u03bd\u03cc\u03c2,\n  \u039d\u03b9\u03ba\u03b1\u03b5\u03cd\u03c2; \u03a3\u03b1\u03b3\u03b1\u03bb\u03b1\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, \u03a3\u03b1\u03b3\u03b1\u03bb\u03b1\u03c3\u03c3\u03b5\u03cd\u03c2, etc. The coins, while they\n  universally exhibit the form in \u03bf, are equally persistent in the\n  termination -\u03b7\u03bd\u03cc\u03c2, ==\u03ba\u03bf\u03bb\u03bf\u03f2\u03f2\u03b7\u03bd\u03c9\u03bd==; and it is curious that to the\n  form \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b7\u03bd\u03bf\u03af in Strabo xii. 8 \u00a7 16 (p. 578) there is a various\n  reading \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u1fd6\u03c2. Thus, though there is no necessary connexion\n  between the two, the termination -\u03b7\u03bd\u03cc\u03c2 seems to go with the \u03bf form,\n  and the termination -\u03b1\u03b5\u03cd\u03c2 with the \u03b1 form.\n\n  For the above reasons I have written confidently \u1f10\u03bd \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u1fd6\u03c2 in the\n  text, and with more hesitation \u03c0\u03c1\u1f78\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u1fd6\u03c2 in the superscription.\n\n[Sidenote: Ethnological relations of the three cities.]\n\nConsidered ethnologically, these three cities are generally regarded as\nbelonging to Phrygia. But as they are situated on the western border of\nPhrygia, and as the frontier line separating Phrygia from Lydia and\nCaria was not distinctly traced, this designation is not persistent[57].\nThus Laodicea is sometimes assigned to Caria, more rarely to Lydia[58];\nand again, Hierapolis is described as half Lydian, half Phrygian[59]. On\nthe other hand I have not observed that Coloss\u00e6 is ever regarded as\nother than Phrygian[60], partly perhaps because the notices relating to\nit belong to an earlier date when these several names denoted political\nas well as ethnological divisions, and their limits were definitely\nmarked in consequence, but chiefly because it lies some miles to the\neast of the other cities, and therefore farther from the doubtful border\nland.\n\nFootnote 57:\n\n  Strabo, xiii. 4. 12 (p. 628) \u03c4\u1f70 \u03b4' \u1f11\u03be\u1fc6\u03c2 \u1f10\u03c0\u1f76 \u03c4\u1f70 \u03bd\u03cc\u03c4\u03b9\u03b1 \u03bc\u03ad\u03c1\u03b7 \u03c4\u03bf\u1fd6\u03c2 \u03c4\u03cc\u03c0\u03bf\u03b9\u03c2\n  \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u1f10\u03bc\u03c0\u03bb\u03bf\u03ba\u1f70\u03c2 \u1f14\u03c7\u03b5\u03b9 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03a4\u03b1\u1fe6\u03c1\u03bf\u03bd, \u1f65\u03c3\u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03a6\u03c1\u03cd\u03b3\u03b9\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f70\n  \u039a\u03b1\u03c1\u03b9\u03ba\u1f70 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u039b\u03cd\u03b4\u03b9\u03b1 \u03ba\u03b1\u1f76 \u1f14\u03c4\u03b9 \u03c4\u1f70 \u03c4\u1ff6\u03bd \u039c\u03c5\u03c3\u1ff6\u03bd \u03b4\u03c5\u03c3\u03b4\u03b9\u03ac\u03ba\u03c1\u03b9\u03c4\u03b1 \u03b5\u1f36\u03bd\u03b1\u03b9\n  \u03c0\u03b1\u03c1\u03b1\u03c0\u03af\u03c0\u03c4\u03bf\u03bd\u03c4\u03b1 \u03b5\u1f30\u03c2 \u1f04\u03bb\u03bb\u03b7\u03bb\u03b1\u00b7 \u03b5\u1f30\u03c2 \u03b4\u1f72 \u03c4\u1f74\u03bd \u03c3\u03cd\u03b3\u03c7\u03c5\u03c3\u03b9\u03bd \u03c4\u03b1\u03cd\u03c4\u03b7\u03bd \u03bf\u1f50 \u03bc\u03b9\u03ba\u03c1\u1f70\n  \u03c3\u03c5\u03bb\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9 \u03c4\u1f78 \u03c4\u03bf\u1f7a\u03c2 \u1fec\u03c9\u03bc\u03b1\u03af\u03bf\u03c5\u03c2 \u03bc\u1f74 \u03ba\u03b1\u03c4\u1f70 \u03c6\u1fe6\u03bb\u03b1 \u03b4\u03b9\u03b5\u03bb\u03b5\u1fd6\u03bd \u03b1\u1f50\u03c4\u03bf\u03cd\u03c2 \u03ba.\u03c4.\u03bb.\n\nFootnote 58:\n\n  To _Phrygia_, Strabo xii. 8. 13 (p. 576), Polyb. v. 57, and so\n  generally; to _Caria, Orac. Sibyll._ iii. 472 \u039a\u03b1\u03c1\u1ff6\u03bd \u1f00\u03b3\u03bb\u03b1\u1f78\u03bd \u1f04\u03c3\u03c4\u03c5, Ptol.\n  v. 2, Philostr. _Vit. Soph._ i. 25 (though in the context Philostratus\n  adds that at one time \u03c4\u1fc7 \u03a6\u03c1\u03c5\u03b3\u03af\u1fb3 \u03be\u03c5\u03bd\u03b5\u03c4\u03ac\u03c4\u03c4\u03b5\u03c4\u03bf); to _Lydia_, Steph. Byz.\n  s.v. On the coins the city is sometimes represented as seated between\n  two female figures ==\u03c6\u03c1\u03c5\u03b3\u03b9\u03b1== and ==\u03ba\u03b1\u03c1\u03b9\u03b1==; Eckhel III. p. 160,\n  comp. Mionnet IV. p. 329. From its situation on the confines of the\n  three countries Laodicea seems to have obtained the surname\n  _Trimitaria_ or _Trimetaria_, by which it is sometimes designated in\n  later times: see below, p. 65, note 205, and comp. Wesseling, _Itin._\n  p. 665.\n\nFootnote 59:\n\n  Steph. Byz. s.v. says \u03bc\u03b5\u03c4\u03b1\u03be\u1f7a \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u039b\u03c5\u03b4\u03af\u03b1\u03c2 \u03c0\u03cc\u03bb\u03b9\u03c2. But generally\n  Hierapolis is assigned to Phrygia: e.g. Ptol. v. 2, Vitruv. viii. 3 \u00a7\n  10.\n\nFootnote 60:\n\n  Coloss\u00e6 is assigned to Phrygia in Herod. vii. 30, Xen. _Anab._ i. 2.\n  6, Strabo xii. 8. 13, Diod. xiv. 80, Plin. _N. H._ v. 32 \u00a7 41, Poly\u00e6n.\n  _Strat._ vii. 16. 1.\n\n[Sidenote: Their political relations.]\n\nPhrygia however ceased to have any political significance, when this\ncountry came under the dominion of the Romans. Politically speaking, the\nthree cities with the rest of the Cibyratic union belonged at this time\nto Asia, the proconsular province[61]. As an _Asiatic_ Church\naccordingly Laodicea is addressed in the Apocalyptic letter. To this\nprovince they had been assigned in the first instance; then they were\nhanded over to Cilicia[62]; afterwards they were transferred and\nre-transferred from the one to the other; till finally, before the\nChristian era, they became a permanent part of Asia, their original\nprovince. Here they remained, until the close of the fourth century,\nwhen a new distribution of the Roman empire was made, and the province\nof Phrygia Pacatiana created with Laodicea as its capital[63].\n\nFootnote 61:\n\n  After the year B.C. 49 they seem to have been permanently attached to\n  'Asia': before that time they are bandied about between Asia and\n  Cilicia. These alternations are traced by Bergmann _de Asia provincia_\n  (Berlin, 1846) and in _Philologus_ II. 4 (1847) p. 641 sq. See Becker\n  and Marquardt _R\u00f6m. Alterth._ III. I. p. 130 sq. Laodicea is assigned\n  to 'Asia' in Boeckh _Corp. Inscr._ 6512, 6541, 6626.\n\n  The name 'Asia' will be used throughout this chapter in its political\n  sense, as applying to the Roman province.\n\nFootnote 62:\n\n  Cic. _ad Fam._ xiii. 67 'ex provincia mea Ciliciensi, cui scis \u03c4\u03c1\u03b5\u1fd6\u03c2\n  \u03b4\u03b9\u03bf\u03b9\u03ba\u03ae\u03c3\u03b5\u03b9\u03c2 Asiaticas [i.e. Cibyraticam, Apamensem, Synnadensem]\n  attributas fuisse'; _ad Att._ v. 21 'mea expectatio Asi\u00e6 nostrarum\n  di\u0153cesium' and 'in hac mea Asia.' See also above p. 7, notes 2, 3.\n\nFootnote 63:\n\n  3 Hierocles _Synecd._ p. 664 sq. (Wessel.): see below p. 69.\n\n[Sidenote: Important Jewish settlement in this neighbourhood.]\n\nThe Epistle to the Colossians supposes a powerful Jewish colony in\nLaodicea and the neighbourhood. We are not however left to draw this\ninference from the epistle alone, but the fact is established by ample\nindependent testimony. When, with the insolent licence characteristic\nof Oriental kings, Antiochus the Great transplanted two thousand\nJewish families from Babylonia and Mesopotamia into Lydia and\nPhrygia[64],[Sidenote: Colony of Antiochus the Great.] we can hardly\ndoubt that among the principal stations of these new colonists would\nbe the two most thriving cities of Phrygia, which were also the two\nmost important settlements of the Syrian kings, Apamea and Laodicea,\nthe one founded by his grandfather Antiochus the First, the other by\nhis father Antiochus the Second. If the commercial importance of\nApamea at this time was greater (for somewhat later it was reckoned\nsecond only to Ephesus among the cities of Asia Minor as a centre of\ntrade), the political rank of Laodicea stood higher[65]. When mention\nis made of Lydia and Phrygia[66], this latter city especially is\npointed out by its position, for it stood near the frontier of the two\ncountries. A Jewish settlement once established, the influx of their\nfellow-countrymen would be rapid and continuous. Accordingly under the\nRoman domination we find them gathered here in very large\nnumbers.[Sidenote: Confiscations of Flaccus.] When Flaccus the\npropr\u00e6tor of Asia (B.C. 62), who was afterwards accused of\nmaladministration in his province and defended by Cicero, forbade the\ncontributions of the Jews to the temple-worship and the consequent\nexportation of money to Palestine, he seized as contraband not less\nthan twenty pounds weight in gold in the single district of which\nLaodicea was the capital[67]. Calculated at the rate of a half-shekel\nfor each man, this sum represents a population of more than eleven\nthousand adult freemen[68]; for women, children, and slaves were\nexempted. It must be remembered however, that this is only the sum\nwhich the Roman officers succeeded in detecting and confiscating; and\nthat therefore the whole Jewish population would probably be much\nlarger than this partial estimate implies. The amount seized at\nApamea, the other great Phrygian centre, was five times as large as\nthis[69]. [Sidenote: Other evidence.]Somewhat later we have a document\npurporting to be a decree of the Laodiceans, in which they thank the\nRoman Consul for a measure granting to Jews the liberty of observing\ntheir sabbaths and practising other rites of their religion[70]; and\nthough this decree is probably spurious, yet it serves equally well to\nshow that at this time Laodicea was regarded as an important centre of\nthe dispersion in Asia Minor. To the same effect may be quoted the\nextravagant hyperbole in the Talmud, that when on a certain occasion\nan insurrection of the Jews broke out in C\u00e6sarea the metropolis of\nCappadocia, which brought down upon their heads the cruel vengeance of\nking Sapor and led to a massacre of 12,000, 'the wall of Laodicea was\ncloven with the sound of the harpstrings' in the fatal and premature\nmerriment of the insurgents[71]. This place was doubtless singled out,\nbecause it had a peculiar interest for the Jews, as one of their chief\nsettlements[72]. It will be remembered also, that Phrygia is\nespecially mentioned among those countries which furnished their quota\nof worshippers at Jerusalem, and were thus represented at the baptism\nof the Christian Church on the great day of Pentecost[73].\n\nFootnote 64:\n\n  Joseph. _Antiq._ xii. 3, 4.\n\nFootnote 65:\n\n  Strabo xii. 8. 13 (p. 576) \u03b5\u1f36\u03c4\u03b1 \u1f08\u03c0\u03ac\u03bc\u03b5\u03b9\u03b1 \u1f21 \u039a\u03b9\u03b2\u03c9\u03c4\u1f78\u03c2 \u03bb\u03b5\u03b3\u03bf\u03bc\u03ad\u03bd\u03b7 \u03ba\u03b1\u1f76\n  \u039b\u03b1\u03bf\u03b4\u1f76\u03ba\u03b5\u03b9\u03b1 \u03b1\u1f35\u03c0\u03b5\u03c1 \u03b5\u1f30\u03c3\u1f76 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd \u03c0\u03cc\u03bb\u03b5\u03c9\u03bd. Below \u00a7 15\n  (p. 577) he says \u1f08\u03c0\u03ac\u03bc\u03b5\u03b9\u03b1 \u03b4' \u1f10\u03c3\u03c4\u1f76\u03bd \u1f10\u03bc\u03c0\u03cc\u03c1\u03b9\u03bf\u03bd \u03bc\u03ad\u03b3\u03b1 \u03c4\u1fc6\u03c2 \u1f30\u03b4\u03af\u03c9\u03c2 \u03bb\u03b5\u03b3\u03bf\u03bc\u03ad\u03bd\u03b7\u03c2\n  \u1f08\u03c3\u03af\u03b1\u03c2 \u03b4\u03b5\u03c5\u03c4\u03b5\u03c1\u03b5\u1fe6\u03bf\u03bd \u03bc\u03b5\u03c4\u1f70 \u03c4\u1f74\u03bd \u1f1c\u03c6\u03b5\u03c3\u03bf\u03bd. The relative importance of Apamea\n  and Laodicea two or three generations earlier than St Paul may be\n  inferred from the notices in Cicero; but there is reason for thinking\n  that Laodicea afterwards grew more rapidly than Apamea.\n\nFootnote 66:\n\n  In Josephus l.c. the words are \u03c4\u1f70 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd \u03ba\u03b1\u1f76 \u039b\u03c5\u03b4\u03af\u03b1\u03bd, the two\n  names being under the vinculum of the one article: while immediately\n  afterwards Lydia is dropped and Phrygia alone named, \u03c0\u03ad\u03bc\u03c8\u03b1\u03b9 \u03c4\u03b9\u03bd\u1f70\u03c2 ...\n  \u03b5\u1f30\u03c2 \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd.\n\nFootnote 67:\n\n  Cic. _pro Flacc._ 28 'Sequitur auri illa invidia Judaici.... Quum\n  aurum Jud\u00e6orum nomine quotannis ex Italia et ex omnibus provinciis\n  Hierosolyma exportari soleret, Flaccus sanxit edicto ne ex Asia\n  exportari liceret ... multitudinem Jud\u00e6orum, flagrantem nonnumquam in\n  concionibus, pro republica contemnere gravitatis summ\u00e6 fuit.... Apame\u00e6\n  manifesto comprehensum ante pedes pr\u00e6toris in foro expensum est auri\n  pondo centum paullo minus ... Laodice\u00e6 viginti pondo paullo amplius.'\n\n  Josephus (_Antiq._ xiv. 7. 2), quoting the words of Strabo, \u03c0\u03ad\u03bc\u03c8\u03b1\u03c2 \u03b4\u1f72\n  \u039c\u03b9\u03b8\u03c1\u03b9\u03b4\u03ac\u03c4\u03b7\u03c2 \u03b5\u1f30\u03c2 \u039a\u1ff6 \u1f14\u03bb\u03b1\u03b2\u03b5 ... \u03c4\u1f70 \u03c4\u1ff6\u03bd \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd \u1f40\u03ba\u03c4\u03b1\u03ba\u03cc\u03c3\u03b9\u03b1 \u03c4\u03ac\u03bb\u03b1\u03bd\u03c4\u03b1,\n  explains this enormous sum as composed of the temple-offerings of the\n  Jews which they sent to Cos for safety out of the way of Mithridates.\n\nFootnote 68:\n\n  This calculation supposes (1) That the half-shekel weighs 110 gr; (2)\n  That the Roman pound is 5050 gr; (3) That the relation of gold to\n  silver was at this time as 12 : 1. This last estimate is possibly\n  somewhat too high.\n\nFootnote 69:\n\n  The coinage of Apamea affords a striking example of Judaic influence\n  at a later date. On coins struck at this place in the reigns of\n  Severus, Macrinus, and the elder Philip, an ark is represented\n  floating on the waters. Within are a man and a woman: on the roof a\n  bird is perched; while in the air another bird approaches bearing an\n  olive-branch in its claws. The ark bears the inscription \u03bd\u03c9\u03b5. Outside\n  are two standing figures, a man and a woman (apparently the same two\n  who have been represented within the ark), with their hands raised as\n  in the attitude of prayer. The connexion of the ark of Noah with\n  Apamea is explained by a passage in one of the Sibylline Oracles (i.\n  261 sq.), where the mountain overhanging Apamea is identified with\n  Ararat, and the ark (\u03ba\u03b9\u03b2\u03c9\u03c4\u03cc\u03c2) is stated to have rested there. Whether\n  this Apamea obtained its distinctive surname of Cibotus, the Ark or\n  Chest, from its physical features, or from its position as the centre\n  of taxation and finance for the district, or from some other cause, it\n  is difficult to say. In any case this surname might naturally suggest\n  to those acquainted with the Old Testament a connexion with the deluge\n  of Noah; but the idea would not have been adopted in the coinage of\n  the place without the pressure of strong Jewish influences. On these\n  coins see Eckhel _Doctr. Num. Vet._ III. p. 132 sq., and the paper of\n  Sir F. Madden in the _Numismatic Chronicle_ N. S. VI. p. 173 sq.\n  (1866), where they are figured.\n\nFootnote 70:\n\n  Joseph. _Ant._ xiv. 10. 21.\n\nFootnote 71:\n\n  Talm. Babl. _Mo\u00ebd Katon_ 26a, quoted by Neubauer, _La G\u00e9ographie du\n  Talmud_ p. 319, though he seems to have misunderstood the expression\n  quoted in the text, of which he gives the sense, 'Cette ville\n  tremblait au bruit des fl\u00e8ches qu'on avait tir\u00e9es.'\n\n  It is probably this same Laodicea which is meant in another Talmudical\n  passage, Talm. Babl. _Baba Metziah_ 84a (also quoted by Neubauer, p.\n  311), in which Elijah appearing to R. Ishmael ben R. Jose, says 'Thy\n  father fled to Asia; flee thou to Laodicea,' where Asia is supposed to\n  mean Sardis.\n\nFootnote 72:\n\n  An inscription found at Rome in the Jewish cemetery at the Porta\n  Portuensis (Boeckh _Corp. Inscr._ 9916) runs thus; ==\u03b5\u03bd\u03b8\u03b1 . \u03ba\u03b9\u03c4\u03b5 .\n  \u03b1\u03bc\u03bc\u03b9\u03b1 . [\u03b5]\u03b9\u03bf\u03c5\u03b4\u03b5\u03b1 . \u03b1\u03c0\u03bf . \u03bb\u03b1\u03b4\u03b9\u03ba\u03b9\u03b1\u03f2.== \u03ba.\u03c4.\u03bb. i.e. \u1f14\u03bd\u03b8\u03b1 \u03ba\u03b5\u1fd6\u03c4\u03b1\u03b9 \u1f08\u03bc\u03bc\u03af\u03b1\n  \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03b1 \u1f00\u03c0\u1f78 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2. Probably Laodicea on the Lycus is meant.\n  Perhaps also we may refer another inscription (6478), which mentions\n  one Trypho from Laodicea on the Lycus, to a Jewish source.\n\nFootnote 73:\n\n  Acts ii. 10.\n\n[Sidenote: Special attractions of Hierapolis.]\n\nMention has already been made of the traffic in dyed wools, which formed\nthe staple of commerce in the valley of the Lycus[74]. It may be\ninferred from other notices that this branch of trade had a peculiar\nattraction for the Jews[75]. If so, their commercial instincts would\nconstantly bring fresh recruits to a colony which was already very\nconsiderable. But the neighbourhood held out other inducements besides\nthis. Hierapolis, the gay watering place, the pleasant resort of idlers,\nhad charms for them, as well as Laodicea the busy commercial city. At\nleast such was the complaint of stricter patriots at home. 'The wines\nand the baths of Phrygia,' writes a Talmudist bitterly, 'have separated\nthe ten tribes from Israel[76].'\n\nFootnote 74:\n\n  See p. 4.\n\nFootnote 75:\n\n  Acts xvi. 14. Is there an allusion to this branch of trade in the\n  message to the Church of Laodicea, Rev. iii. 17 \u03bf\u1f50\u03ba \u03bf\u1f36\u03b4\u03b1\u03c2 \u1f45\u03c4\u03b9 \u03c3\u03cd \u03b5\u1f36 \u1f41\n  ... \u03b3\u03c5\u03bc\u03bd\u03cc\u03c2\u00b7 \u03c3\u03c5\u03bc\u03b2\u03bf\u03c5\u03bb\u03b5\u03cd\u03c9 \u03c3\u03bf\u03b9 \u1f00\u03b3\u03bf\u03c1\u03ac\u03c3\u03b1\u03b9 ... \u1f31\u03bc\u03ac\u03c4\u03b9\u03b1 \u03bb\u03b5\u03c5\u03ba\u1f70 \u1f35\u03bd\u03b1 \u03c0\u03b5\u03c1\u03b9\u03b2\u03ac\u03bb\u1fc3,\n  \u03ba.\u03c4.\u03bb.? The only other of the seven messages, which contains an\n  allusion to the white garments, is addressed to the Church of Sardis,\n  where again there might be a reference to the \u03b2\u03ac\u03bc\u03bc\u03b1 \u03a3\u03b1\u03c1\u03b4\u03b9\u03b1\u03bd\u03b9\u03ba\u03cc\u03bd\n  (Arist. _Pax_ 1174, _Acharn._ 112) and the \u03c6\u03bf\u03b9\u03bd\u03b9\u03ba\u03af\u03b4\u03b5\u03c2 \u03a3\u03b1\u03c1\u03b4\u03b9\u03b1\u03bd\u03b9\u03ba\u03b1\u03af\n  (Plato Com. in Athen. II. p. 48 E) of the comic poets.\n\nFootnote 76:\n\n  Talm. Babl. _Sabbath_ 147 b, quoted by Neubauer _La G\u00e9ographie du\n  Talmud_ p. 317: see Wiesner _Schol. zum Babyl. Talm._ p. 259 sq., and\n  p. 207 sq. On the word translated 'baths,' see Rapoport's _Erech\n  Millin_ p. 113, col. 1.\n\n[Sidenote: St Paul had not visited the district when he wrote.]\n\nThere is no ground for supposing that, when St Paul wrote his Epistle to\nthe Colossians, he had ever visited the church in which he evinces so\ndeep an interest. Whether we examine the narrative in the Acts, or\nwhether we gather up the notices in the epistle itself, we find no hint\nthat he had ever been in this neighbourhood; but on the contrary some\nexpressions indirectly exclude the supposition of a visit to the\ndistrict.\n\n[Sidenote: What is meant by _Phrygia_ in St Luke?]\n\nIt is true that St Luke more than once mentions Phrygia as lying on St\nPaul's route or as witnessing his labours. But Phrygia was a vague and\ncomprehensive term; nor can we assume that the valley of the Lycus was\nintended, unless the direction of his route or the context of the\nnarrative distinctly points to this south-western corner of Phrygia. In\nneither of the two passages, where St Paul is stated to have travelled\nthrough Phrygia, is this the case.\n\n[Sidenote: 1. St Paul's visit to Phrygia on his second missionary\n           journey.]\n\n1. On his second missionary journey, after he has revisited and\nconfirmed the churches of Pisidia and Lycaonia founded on his first\nvisit, he passes through 'the Phrygian and Galatian country[77].' I have\npointed out elsewhere that this expression must be used to denote the\nregion which might be called indifferently Phrygia or Galatia\u2014the land\nwhich had originally belonged to the Phrygians and had afterwards been\ncolonised by the Gauls; or the parts of either country which lay in the\nimmediate neighbourhood of this debatable ground[78]. This region lies\nconsiderably north and east of the valley of the Lycus. Assuming that\nthe last of the Lycaonian and Pisidian towns at which St Paul halted was\nAntioch, he would not on any probable supposition approach nearer to\nColoss\u00e6 than Apamea Cibotus on his way to 'the Phrygian and Galatian\ncountry', nor indeed need he have gone nearly so far westward as this.\nAnd again on his departure from this region he journeys by Mysia to\nTroas, leaving 'Asia' on his left hand and Bithynia on his right. Thus\nthe notices of his route conspire to show that his path on this occasion\nlay far away from the valley of the Lycus.\n\nFootnote 77:\n\n  Acts xvi. 6 \u03c4\u1f74\u03bd \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd \u03ba\u03b1\u1f76 \u0393\u03b1\u03bb\u03b1\u03c4\u03b9\u03ba\u1f74\u03bd \u03c7\u03ce\u03c1\u03b1\u03bd, the correct reading. For\n  this use of \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd as an adjective comp. Mark i. 5 \u03c0\u1fb6\u03c3\u03b1 \u1f21 \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03b1\n  \u03c7\u03ce\u03c1\u03b1, Joh. iii. 22 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03b1\u03bd \u03b3\u1fc6\u03bd, Luke iii. 1 \u03c4\u1fc6\u03c2 \u1f38\u03c4\u03bf\u03c5\u03c1\u03b1\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76\n  \u03a4\u03c1\u03b1\u03c7\u03c9\u03bd\u03af\u03c4\u03b9\u03b4\u03bf\u03c2 \u03c7\u03ce\u03c1\u03b1\u03c2, Acts xiii. 14 \u1f08\u03bd\u03c4\u03b9\u03cc\u03c7\u03b5\u03b9\u03b1\u03bd \u03c4\u1f74\u03bd \u03a0\u03b9\u03c3\u03b9\u03b4\u03af\u03b1\u03bd (the correct\n  reading).\n\nFootnote 78:\n\n  See _Galatians_, p. 18 sq., 22.\n\n[Sidenote: 2. His visit on his third missionary journey.]\n\n2. But if he was not brought into the neighbourhood of Coloss\u00e6 on his\nsecond missionary journey, it is equally improbable that he visited it\non his third. So far as regards Asia Minor, he seems to have confined\nhimself to revisiting the churches already founded; the new ground which\nhe broke was in Macedonia and Greece. Thus when we are told that during\nthis third journey St Paul after leaving Antioch 'passed in order\nthrough the Galatian country and Phrygia, confirming all the\ndisciples,'[79] we can hardly doubt that 'the Galatian country and\nPhrygia' in this latter passage denotes essentially the same region as\n'the Phrygian and Galatian country' in the former. The slight change of\nexpression is explained by the altered direction of his route. In the\nfirst instance his course, as determined by its extreme limits\u2014Antioch\nin Pisidia its starting point, and Alexandria Troas its\ntermination\u2014would be northward for the first part of the way, and thus\nwould lie on the border land of Phrygia and Galatia; whereas on this\nsecond occasion, when he was travelling from Antioch in Syria to\nEphesus, its direction would be generally from east to west, and the\nmore strictly Galatian district would be traversed before the Phrygian.\nIf we suppose him to leave Galatia at Pessinus on its western border, he\nwould pass along the great highway\u2014formerly a Persian and at this time a\nRoman road\u2014by Synnada and Sardis to Ephesus, traversing the heart of\nPhrygia, but following the valleys of the Hermus and Cayster, and\nseparated from the M\u00e6ander and Lycus by the high mountain ranges which\nbound these latter to the north[80].\n\nFootnote 79:\n\n  Acts xviii. 23.\n\nFootnote 80:\n\n  M. Renan (_Saint Paul_ pp. 51 sq., 126, 313) maintains that the\n  Galatia of St Paul and St Luke is not the country properly so called,\n  but that they are speaking of the Churches of Pisidian Antioch,\n  Iconium, Lystra, and Derbe, which lay within the _Roman province_ of\n  Galatia. This interpretation of Galatia necessarily affects his view\n  of St Paul's routes (pp. 126 sq., 331 sq.); and he supposes the\n  Apostle on his third missionary journey to have passed through the\n  valley of the Lycus, without however remaining to preach the Gospel\n  there (pp. 331 sq., 356 sq., 362). As Antioch in Pisidia would on this\n  hypothesis be the farthest church in 'Galatia and Phrygia' which St\n  Paul visited, his direct route from that city to Ephesus (Acts xviii.\n  23, xix. 1) would naturally lie by this valley. I have already\n  (_Galatians_ pp. 18 sq., 22) stated the serious objections to which\n  this interpretation of 'Galatia' is open, and (if I mistake not) have\n  answered most of M. Renan's arguments by anticipation. But, as this\n  interpretation nearly affects an important point in the history of St\n  Paul's dealings with the Colossians, it is necessary to subject it to\n  a closer examination.\n\n  Without stopping to enquire whether this view is reconcilable with St\n  Paul's assertion (Col. ii. 1) that these churches in the Lycus valley\n  'had not seen his face in the flesh,' it will appear (I think) that M.\n  Renan's arguments are in some cases untenable and in others may be\n  turned against himself. The three heads under which they may be\n  conveniently considered are: (i) The use of the name 'Galatia'; (ii)\n  The itinerary of St Paul's travels; (iii) The historical notices in\n  the Epistle to the Galatians.\n\n  (i) On the first point, M. Renan states that St Paul was in the habit\n  of using the _official_ name for each district and therefore called\n  the country which extends from Antioch in Pisidia to Derbe 'Galatia,'\n  supporting this view by the Apostle's use of Asia, Macedonia, and\n  Achaia (p. 51). The answer is that the names of these elder provinces\n  had very generally superseded the local names, but this was not the\n  case with the other districts of Asia Minor where the provinces had\n  been formed at a comparatively late date. The usage of St Luke is a\n  good criterion. He also speaks of Asia, Macedonia, and Achaia; but at\n  the same time his narrative abounds in historical or ethnographical\n  names which have no official import; e.g. Lycaonia, Mysia, Pamphylia,\n  Pisidia, Phrygia. Where we have no evidence, it is reasonable to\n  assume that St Paul's usage was conformable to St Luke's. And again,\n  if we consider St Luke's account alone, how insuperable are the\n  difficulties which this view of Galatia creates. The part of Asia\n  Minor, with which we are immediately concerned, was comprised\n  officially in the provinces of Asia and Galatia. On M. Renan's\n  showing, St Luke, after calling Antioch a city of Pisidia (xiii. 14)\n  and Lystra and Derbe cities of Lycaonia (xiv. 6), treats all the\n  three, together with the intermediate Iconium, as belonging to Galatia\n  (xvi. 6, xviii. 23). He explains the inconsistency by saying that in\n  the former case the narrative proceeds in detail, in the latter in\n  masses. But if so, why should he combine a historical and ethnological\n  name Phrygia with an official name Galatia in the same breath, when\n  the two are different in kind and cannot be mutually exclusive?\n  'Galatia and Asia,' would be intelligible on this supposition, but not\n  'Galatia and Phrygia.' Moreover the very form of the expression in\n  xvi. 6, 'the Phrygian and Galatian country' (according to the correct\n  reading which M. Renan neglects) appears in its studied vagueness to\n  exclude the idea that St Luke means the province of Galatia, whose\n  boundaries were precisely marked. And even granting that the Christian\n  communities of Lycaonia and Pisidia could by a straining of language\n  be called Churches of Galatia, is it possible that St Paul would\n  address them personally as 'ye foolish Galatians' (Gal. iii. 1)? Such\n  language would be no more appropriate than if a modern preacher in a\n  familiar address were to appeal to the Poles of Warsaw as 'ye\n  Russians,' or the Hungarians of Pesth as 'ye Austrians,' or the Irish\n  of Cork as 'ye Englishmen.'\n\n  (ii) In the itinerary of St Paul several points require consideration.\n  (_a_) M. Renan lays stress on the fact that in Acts xvi. 6, xviii. 23,\n  the order in which the names of Phrygia and Galatia occur is inverted.\n  I seem to myself to have explained this satisfactorily in the text. He\n  appears to be unaware of the correct reading in xvi. 6, \u03c4\u1f74\u03bd \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd\n  \u03ba\u03b1\u1f76 \u0393\u03b1\u03bb\u03b1\u03c4\u03b9\u03ba\u1f74\u03bd \u03c7\u03ce\u03c1\u03b1\u03bd (see _Galatians_ p. 22), though it has an\n  important bearing on St Paul's probable route. (_b_) He states that\n  Troas was St Paul's aim ('l'objectif de Saint Paul') in the one case\n  (xvi. 6), and Ephesus in the other (xviii. 23): consequently he argues\n  that Galatia, properly so called, is inconceivable, as there was no\n  reason why he should have made 'this strange detour towards the\n  north.' The answer is that Troas was not his 'objectif' in the first\n  instance, nor Ephesus in the second. On the first occasion St Luke\n  states that the Apostle set out on his journey with quite different\n  intentions, but that after he had got well to the north of Asia Minor\n  he was driven by a series of divine intimations to proceed first to\n  Troas and thence to cross over into Europe (see _Philippians_ p. 48).\n  This narrative seems to me to imply that he starts for his further\n  travels from some point in the western part of Galatia proper. When he\n  comes to the borders of Mysia, he designs bearing to the left and\n  preaching in Asia; but a divine voice forbids him. He then purposes\n  diverging to the right and delivering his message in Bithynia; but the\n  same unseen power checks him again. Thus he is driven forward, and\n  passes by Mysia to the coast at Troas (Acts xvi. 6\u20138). Here all is\n  plain. But if we suppose him to start, not from some town in Galatia\n  proper such as Pessinus, but from Antioch in Pisidia, why should\n  Bithynia, which would be far out of the way, be mentioned at all? On\n  the second occasion, St Paul's primary object is to revisit the\n  Galatian Churches which he had planted on the former journey (xviii.\n  23), and it is not till after he has fulfilled this intention that he\n  goes to Ephesus. (_c_) M. Renan also calls attention to the difficulty\n  of traversing 'the central steppe' of Asia Minor. 'There was\n  probably,' he says, 'at this epoch no route from Iconium to Ancyra,'\n  and in justification of this statement he refers to Perrot, _de Gal.\n  Rom. prov._ p. 102, 103. Even so, there were regular roads from either\n  Iconium or Antioch to Pessinus; and this route would serve equally\n  well. Moreover the Apostle, who was accustomed to 'perils of rivers,\n  perils of robbers, perils in the wilderness' (2 Cor. xi. 26), and who\n  preferred walking from Troas to Assos (Acts xx. 13) while his\n  companions sailed, would not be deterred by any rough or unfrequented\n  paths. But the facts adduced by Perrot do not lend themselves to any\n  such inference, nor does he himself draw it. He cites an inscription\n  of the year A.D. 82 which speaks of A. C\u00e6sennius Gallus, the legate of\n  Domitian, as a great road-maker throughout the Eastern provinces of\n  Asia Minor, and he suggests that the existing remains of a road\n  between Ancyra and Iconium may be part of this governor's work. Even\n  if the suggestion be adopted, it is highly improbable that no road\n  should have existed previously, when we consider the comparative\n  facility of constructing a way along this line of country (Perrot p.\n  103) and the importance of such a direct route. (_d_) 'In the\n  conception of the author of the Acts,' writes M. Renan, 'the two\n  journeys across Asia Minor are journeys of confirmation and not of\n  conversion (Acts xv. 36, 41, xvi. 5, 6, xviii. 23).' This statement\n  seems to me to be only partially true. In both cases St Paul _begins_\n  his tour by confirming churches already established, but in both he\n  advances beyond this and breaks new ground. In the former he starts\n  with the existing churches of Lycaonia and Pisidia and extends his\n  labours to Galatia: in the latter he starts with the then existing\n  churches of Galatia, and carries the Gospel into Macedonia and Achaia.\n  This, so far as I can discover, was his general rule.\n\n  (iii) The notices in the Galatian Epistle, which appear to M. Renan to\n  favour his view, are these: (_a_) St Paul appears to have 'had\n  intimate relations with the Galatian Church, at least as intimate as\n  with the Corinthians and Thessalonians,' whereas St Luke disposes of\n  the Apostle's preaching in Galatia very summarily, unless the\n  communities of Lycaonia and Pisidia be included. But the Galatian\n  Epistle by no means evinces the same close and varied personal\n  relations which we find in the letters to these other churches, more\n  especially to the Corinthians. And again; St Luke's history is more or\n  less fragmentary. Whole years are sometimes dismissed in a few verses.\n  The stay in Arabia which made so deep an impression on St Paul himself\n  is not even mentioned: the three months' sojourn in Greece, though\n  doubtless full of stirring events, only occupies a single verse in the\n  narrative (Acts xx. 3). St Luke appears to have joined St Paul after\n  his visit to Galatia (xvi. 10); and there is no reason why he should\n  have dwelt on incidents with which he had no direct acquaintance.\n  (_b_) M. Renan sees in the presence of emissaries from Jerusalem in\n  the Galatian Churches an indication that Galatia proper is not meant.\n  'It is improbable that they would have made such a journey.' But why\n  so? There were important Jewish settlements in Galatia proper\n  (_Galatians_ p. 9 sq.); there was a good road through Syria and\n  Cilicia to Ancyra (_Itin. Anton._ p. 205 sq., _Itin. Hierosol._ p. 575\n  sq. ed. Wessel.); and if we find such emissaries as far away from\n  Jerusalem as Corinth (2 Cor. xi. 13, etc.), there is at least no\n  improbability that they should have reached Galatia. (_c_) Lastly; M.\n  Renan thinks that the mention of Barnabas (Gal. ii. 1, 9, 13) implies\n  that he was personally known to the churches addressed, and therefore\n  points to Lycaonia and Pisidia. But are we to infer on the same\n  grounds that he was personally known to the Corinthians (1 Cor. ix.\n  6), and to the Colossians (Col. iv. 10)? In fact the name of Barnabas,\n  as a famous Apostle and an older disciple even than St Paul himself,\n  would not fail to be well known in all the churches. On the other hand\n  one or two notices in the Galatian Epistle present serious obstacles\n  to M. Renan's view. What are we to say for instance to St Paul's\n  statement, that he preached the Gospel in Galatia \u03b4\u03b9' \u1f00\u03c3\u03b8\u03ad\u03bd\u03b5\u03b9\u03b1\u03bd \u03c4\u1fc6\u03c2\n  \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 (iv. 13), i.e. because he was detained by sickness (see\n  _Galatians_ pp. 23 sq., 172), whereas his journey to Lycaonia and\n  Pisidia is distinctly planned with a view to missionary work? Why\n  again is there no mention of Timothy, who was much in St Paul's\n  company about this time, and who on this showing was himself a\n  Galatian? Some mention would seem to be especially suggested where St\n  Paul is justifying his conduct respecting the attempt to compel Titus\n  to be circumcised.\n\n[Sidenote: The inference from]\n\nThus St Luke's narrative seems to exclude any visit of the Apostle to\nthe Churches of the Lycus before his first [Sidenote: St Luke's\nnarrative]Roman captivity. And this inference is confirmed by St Paul's\nown language to the Colossians.\n\n[Sidenote: borne out by St Paul's own language.]\n\nHe represents his knowledge of their continued progress, and even of\ntheir first initiation, in the truths of the Gospel, as derived from the\nreport of others. He describes himself as _hearing_ of their faith in\nChrist and their love to the saints[81]. He recalls the day when he\nfirst _heard_ of their Christian profession and zeal[82]. .[Sidenote:\nSilence of St Paul.]Though opportunities occur again and again where he\nwould naturally have referred to his direct personal relations with\nthem, if he had been their evangelist, he abstains from any such\nreference. He speaks of their being instructed in the Gospel, of his own\npreaching the Gospel, several times in the course of the letter, but he\nnever places the two in any direct connexion, though the one reference\nstands in the immediate neighbourhood of the other[83]. Moreover, if he\nhad actually visited Coloss\u00e6, it must appear strange that he should not\nonce allude to any incident occurring during his sojourn there, for this\nepistle would then be the single exception to his ordinary practice. And\nlastly; in one passage at least, if interpreted in its natural sense, he\ndeclares that the Colossians were personally unknown to him: 'I would\nhave you know,' he writes, 'how great a conflict I have for you and them\nthat are in Laodicea and as many as have not seen my face in the\nflesh'[84].\n\nFootnote 81:\n\n  Col. i. 4.\n\nFootnote 82:\n\n  i. 9 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2, \u1f00\u03c6' \u1f25\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03bc\u03b5\u03bd, \u03bf\u1f50 \u03c0\u03b1\u03c5\u03cc\u03bc\u03b5\u03b8\u03b1, \u03ba.\u03c4.\u03bb.\n  This corresponds to ver. 6 \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, \u1f00\u03c6' \u1f27\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5\n  \u03ba\u03b1\u1f76 \u1f10\u03c0\u03ad\u03b3\u03bd\u03c9\u03c4\u03b5 \u03c4\u1f74\u03bd \u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3. The day when they first\n  heard the preaching of the Gospel, and the day when he first heard the\n  tidings of this fact, are set against each other.\n\nFootnote 83:\n\n  e.g. i. 5\u20138, 21\u201323, 25, 28, 29. ii. 5, 6.\n\nFootnote 84:\n\n  ii. 1 \u03b8\u03ad\u03bb\u03c9 \u03b3\u1f70\u03c1 \u1f51\u03bc\u1fb6\u03c2 \u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9 \u1f21\u03bb\u03af\u03ba\u03bf\u03bd \u1f00\u03b3\u1ff6\u03bd\u03b1 \u1f14\u03c7\u03c9 \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f10\u03bd\n  \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u1fb3 \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 \u03bf\u1f50\u03c7 \u1f11\u03ce\u03c1\u03b1\u03ba\u03b1\u03bd \u03c4\u1f78 \u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03cc\u03bd \u03bc\u03bf\u03c5 \u1f10\u03bd \u03c3\u03b1\u03c1\u03ba\u03af, \u1f35\u03bd\u03b1\n  \u03c0\u03b1\u03c1\u03b1\u03ba\u03bb\u03b7\u03b8\u1ff6\u03c3\u03b9\u03bd \u03b1\u1f31 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9 \u03b1\u1f50\u03c4\u1ff6\u03bd, \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03c3\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2 \u03ba.\u03c4.\u03bb. The question of\n  interpretation is whether the people of Coloss\u00e6 and Laodicea belong to\n  the same category with the \u1f45\u03c3\u03bf\u03b9, or not. The latter view is taken by\n  one or two ancient interpreters (e.g. Theodoret in his introduction to\n  the epistle), and has been adopted by several modern critics. Yet it\n  is opposed alike to grammatical and logical considerations. (1) The\n  grammatical form is unfavourable; for the preposition \u1f51\u03c0\u1f72\u03c1 is not\n  repeated, so that all the persons mentioned are included under a\n  vinculum. (2) No adequate sense can be extracted from the passage, so\n  interpreted. For in this case what is the drift of the enumeration? If\n  intended to be exhaustive, it does not fulfil the purpose; for nothing\n  is said of others whom he had seen beside the Colossians and\n  Laodiceans. If not intended to be exhaustive, it is meaningless; for\n  there is no reason why the Colossians and Laodiceans especially should\n  be set off against those whom he had not seen, or indeed why in this\n  connexion those whom he had not seen should be mentioned at all. The\n  whole context shows that the Apostle is dwelling on his spiritual\n  communion with and interest in those with whom he has had no personal\n  communications. St Jerome (_Ep._ cxxx. ad Demetr. \u00a7 2) has rightly\n  caught the spirit of the passage; 'Ignoti ad ignotam scribimus,\n  dumtaxat juxta faciem corporalem. Alioquin interior <DW25> pulcre sibi\n  cognitus est illa notitia qua et Paulus apostolus Colossenses\n  multosque credentium noverat quos ante non viderat.' For parallels to\n  this use of \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9, see the note on the passage.\n\n[Sidenote: Epaphras was the evangelist of this district.]\n\nBut, if he was not directly their evangelist, yet to him they were\nindirectly indebted for their knowledge of the truth. Epaphras had been\nhis delegate to them, his representative in Christ. By Epaphras they had\nbeen converted to the Gospel. This is the evident meaning of a passage\nin the opening of the epistle, which has been much obscured by\nmisreading and mistranslation, and which may be paraphrased thus: 'The\nGospel, which has spread and borne fruit throughout the rest of the\nworld, has been equally successful among yourselves. This fertile growth\nhas been manifested in you from the first day when the message of God's\ngrace was preached to you, and accepted by you\u2014preached not as now with\nadulterations by these false teachers, but in its genuine simplicity by\nEpaphras our beloved fellowservant; he has been a faithful minister of\nChrist and a faithful representative of us, and from him we have\nreceived tidings of your love in the Spirit'[85].\n\nFootnote 85:\n\n  i. 6 \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u1ff7 \u03ba\u03cc\u03c3\u03bc\u1ff3 \u1f14\u03c3\u03c4\u03b9\u03bd \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd, \u03ba\u03b1\u03b8\u1f7c\u03c2\n  \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, \u1f00\u03c6' \u1f24\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03ad\u03b3\u03bd\u03c9\u03c4\u03b5 \u03c4\u1f74\u03bd \u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u1f10\u03bd\n  \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 \u1f00\u03c0\u1f78 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6 \u03c4\u03bf\u1fe6 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03bf\u1fe6 \u03c3\u03c5\u03bd\u03b4\u03bf\u03cd\u03bb\u03bf\u03c5 \u1f21\u03bc\u1ff6\u03bd, \u1f45\u03c2\n  \u1f10\u03c3\u03c4\u03b9\u03bd \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f51\u03c0\u1f72\u03c1 \u1f21\u03bc\u1ff6\u03bd \u03b4\u03b9\u03ac\u03ba\u03bf\u03bd\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, \u1f41 \u03ba\u03b1\u1f76 \u03b4\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2 \u1f21\u03bc\u1fd6\u03bd \u03c4\u1f74\u03bd\n  \u1f51\u03bc\u1ff6\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u1f10\u03bd \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9.\n\n  The various readings which obscure the meaning are these. (i) The\n  received text for \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 has \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5. With this\n  reading the passage suggests that the instructions of Epaphras were\n  _superadded to_, and so distinct from, the original evangelization of\n  Coloss\u00e6; whereas the correct text identifies them. (ii) For \u1f51\u03c0\u1f72\u03c1 \u1f21\u03bc\u1ff6\u03bd\n  the received reading is \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd. Thus the fact that St Paul did not\n  preach at Coloss\u00e6 in person, but _through his representative_, is\n  obliterated. In both cases the authority for the readings which I have\n  adopted against the received text is overwhelming.\n\n  The obscurity of rendering is in \u03ba\u03b1\u03b8\u1f7c\u03c2 [\u03ba\u03b1\u1f76] \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 \u1f00\u03c0\u1f78 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6,\n  translated in our English Version by the ambiguous expression, 'as ye\n  also learned of Epaphras.' The true force of the words is, 'according\n  as ye were taught by Epaphras,' being an explanation of \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3.\n  See the notes on the passage.]\n\n[Sidenote: St Paul's residence at Ephesus instrumental in their\n           conversion.]\n\nHow or when the conversion of the Colossians took place, we have no\ndirect information. Yet it can hardly be wrong to connect the event with\nSt Paul's long sojourn at Ephesus. Here he remained preaching for three\nwhole years. It is possible indeed that during this period he paid short\nvisits to other neighbouring cities of Asia: [Sidenote: A.D. 54\u201357.] but\nif so, the notices in the Acts oblige us to suppose these interruptions\nto his residence in Ephesus to have been slight and infrequent[86]. Yet,\nthough the Apostle himself was stationary in the capital, the Apostle's\ninfluence and teaching spread far beyond the limits of the city and its\nimmediate neighbourhood. It was hardly an exaggeration when Demetrius\ndeclared that 'almost throughout all Asia this Paul had persuaded and\nturned away much people'[87]. The sacred historian himself uses equally\nstrong language in describing the effects of the Apostle's preaching;\n'All they which dwelt in Asia heard the word of the Lord, both Jews and\nGreeks'[88]. In accordance with these notices, the Apostle himself in an\nepistle written during this sojourn sends salutations to Corinth, not\nfrom the Church of Ephesus specially, as might have been anticipated,\nbut from the 'Churches of Asia' generally[89]. St Luke, it should be\nobserved, ascribes this dissemination of the Gospel, not to journeys\nundertaken by the Apostle, but to his preaching at Ephesus itself[90].\nThither, as to the metropolis of Western Asia, would flock crowds from\nall the towns and villages far and near. Thence they would carry away,\neach to his own neighbourhood, the spiritual treasure which they had so\nunexpectedly found.\n\nFootnote 86:\n\n  See especially xx. 18 'Ye know, from the first day when I set foot on\n  Asia, how I was with you _all the time_', and ver. 31 'For three years\n  _night and day I ceased not_ warning every one with tears.' As it\n  seems necessary to allow for a brief visit to Corinth (2 Cor. xii. 14,\n  xiii. 1) during this period, other interruptions of long duration\n  should not be postulated.\n\nFootnote 87:\n\n  Acts xix. 26.\n\nFootnote 88:\n\n  Acts xix. 10.\n\nFootnote 89:\n\n  1 Cor. xvi. 19 \u1f00\u03c3\u03c0\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2 \u03b1\u1f31 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03b9 \u03c4\u1fc6\u03c2 \u1f08\u03c3\u03af\u03b1\u03c2. In accordance\n  with these facts it should be noticed that St Paul himself alluding to\n  this period speaks of 'Asia', as the scene of his ministry (2 Cor. i.\n  8, Rom. xvi. 5).\n\nFootnote 90:\n\n  Acts xix. 10 'disputing daily in the School of Tyrannus; and this\n  continued for two years, so that all they which dwelt in Asia, etc.'\n\n[Sidenote: Close alliance of these cities with Ephesus.]\n\nAmong the places thus represented at the Asiatic metropolis would\ndoubtless be the cities lying in the valley of the Lycus. The bonds of\namity between these places and Ephesus appear to have been unusually\nstrong. The _Concord of the Laodiceans and Ephesians_, the _Concord of\nthe Hierapolitans and Ephesians_, are repeatedly commemorated on medals\nstruck for the purpose[91]. [Sidenote: The work of Philemon and\nNymphas,] Thus the Colossians, Epaphras and Philemon, the latter with\nhis household[92], and perhaps also the Laodicean Nymphas[93], would\nfall in with the Apostle of the Gentiles and hear from his lips the\nfirst tidings of a heavenly life.\n\nFootnote 91:\n\n  \u03bb\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03c9\u03bd . \u03b5\u03c6\u03b5\u03f2\u03b9\u03c9\u03bd . \u03bf\u03bc\u03bf\u03bd\u03bf\u03b9\u03b1, Eckhel III. p. 165, Mionnet IV. p. 324,\n  325, 331, 332, _Suppl._ VII. p. 583, 586, 589; \u03b9\u03b5\u03c1\u03b1\u03c0\u03bf\u03bb\u03b5\u03b9\u03c4\u03c9\u03bd . \u03b5\u03c6\u03b5\u03c3\u03b9\u03c9\u03bd\n  . \u03bf\u03bc\u03bf\u03bd\u03bf\u03b9\u03b1, Eckhel III. p. 155, 157, Mionnet IV. p. 299, 300, 307,\n  _Suppl._ VII. p. 569, 571, 572, 574, 575. See Steiger _Kolosser_ p.\n  50, and comp. Krause _Civitat. Neocor._ \u00a7 20.\n\nFootnote 92:\n\n  Philem. 1, 2, 19.\n\nFootnote 93:\n\n  Col. iv. 15. On the question whether the name is _Nymphas_ or\n  _Nympha_, see the notes there.\n\n[Sidenote: but especially Epaphras.]\n\nBut, whatever service may have been rendered by Philemon at Coloss\u00e6, or\nby Nymphas at Laodicea, it was to Epaphras especially that all the three\ncities were indebted for their knowledge of the Gospel. Though he was a\nColossian by birth, the fervency of his prayers and the energy of his\nlove are represented as extending equally to Laodicea and\nHierapolis[94]. It is obvious that he looked upon himself as responsible\nfor the spiritual well-being of all alike.\n\nFootnote 94:\n\n  iv. 12, 13.\n\n[Sidenote: St Paul still a stranger to this district.]\n\nWe pass over a period of five or six years. St Paul's first captivity in\nRome is now drawing to a close. During this interval he has not once\nvisited the valley of the Lycus. He has, it is true, skirted the coast\nand called at Miletus, which lies near the mouth of the M\u00e6ander; but,\nthough the elders of Ephesus were summoned to meet him there[95], no\nmention is made of any representatives from these more distant towns.\n\nFootnote 95:\n\n  Acts xx. 16, 17.\n\n[Sidenote: His imprisonment at Rome.]\n\nI have elsewhere described the Apostle's circumstances during his\nresidence in Rome, so far as they are known to us[96]. It is sufficient\nto say here, that though he is still a prisoner, friends new and old\nminister freely to his wants. Meanwhile the alienation of the Judaic\nChristians is complete. Three only, remaining faithful to him, are\ncommemorated as honourable exceptions in the general desertion[97].\n\nFootnote 96:\n\n  See _Philippians_ p. 6 sq.\n\nFootnote 97:\n\n  Col. iv. 10, 11. See _Philippians_ p. 17 sq.\n\n[Sidenote: Coloss\u00e6 brought before his notice by two incidents.]\n\nWe have seen that Coloss\u00e6 was an unimportant place, and that it had no\ndirect personal claims on the Apostle. We might therefore feel surprise\nthat, thus doubly disqualified, it should nevertheless attract his\nspecial attention at a critical moment, when severe personal trials were\nsuperadded to 'the care of all the churches.' But two circumstances, the\none affecting his public duties, the other private and personal,\nhappening at this time, conspired to bring Coloss\u00e6 prominently before\nhis notice.\n\n[Sidenote: 1. The mission of EPAPHRAS.]\n\n1. He had received a visit from EPAPHRAS. The dangerous condition of the\nColossian and neighbouring churches had filled the mind of their\nevangelist with alarm. A strange form of heresy had broken out in these\nbrotherhoods\u2014a combination of Judaic formalism with Oriental mystic\nspeculation\u2014and was already spreading rapidly. His distress was extreme.\nHe gratefully acknowledged and reported their faith in Christ and their\nworks of love[98]. But this only quickened his anxiety. He had 'much\ntoil for them'; he was 'ever wrestling in his prayers on their behalf,'\nthat they might stand fast and not abandon the simplicity of their\nearlier faith[99]. He came to Rome, we may suppose, for the express\npurpose of laying this state of things before the Apostle and seeking\nhis counsel and assistance.\n\nFootnote 98:\n\n  i. 4, 8.\n\nFootnote 99:\n\n  iv. 12, 13.\n\n[Sidenote: 2. ONESIMUS a fugitive in Rome.]\n\n2. But at the time when Epaphras paid this visit, St Paul was also in\ncommunication with another Colossian, who had visited Rome under very\ndifferent circumstances. ONESIMUS, the runaway slave, had sought the\nmetropolis, the common sink of all nations[100], probably as a\nconvenient hiding place, where he might escape detection among its\ncrowds and make a livelihood as best he could. Here, perhaps\naccidentally, perhaps through the intervention of Epaphras, he fell in\nwith his master's old friend. The Apostle interested himself in his\ncase, instructed him in the Gospel, and transformed him from a\ngood-for-nothing slave[101] into a 'faithful and beloved brother[102].'\n\nFootnote 100:\n\n  Tac. _An._ xv. 44.\n\nFootnote 101:\n\n  Philem. 11 \u03c4\u03cc\u03bd \u03c0\u03bf\u03c4\u03ad \u03c3\u03bf\u03b9 \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03bd \u03ba.\u03c4.\u03bb.\n\nFootnote 102:\n\n  Col. iv. 9; comp. Philem. 16.\n\n[Sidenote: The Apostle despatches three letters simultaneously.]\n\nThis combination of circumstances called the Apostle's attention to the\nChurches of the Lycus, and more especially to Coloss\u00e6. His letters,\nwhich had been found 'weighty and powerful' in other cases, might not be\nunavailing now; and in this hope he took up his pen. Three epistles were\nwritten and despatched at the same time to this district.\n\n[Sidenote: 1. The EPISTLE TO THE COLOSSIANS.]\n\n1. He addresses a special letter to the COLOSSIANS, written in the joint\nnames of himself and Timothy, warning them against the errors of the\nfalse teachers. He gratefully acknowledges the report which he has\nreceived of their love and zeal[103]. He assures them of the conflict\nwhich agitates him on their behalf[104]. He warns them to be on their\nguard against the delusive logic of enticing words, against the vain\ndeceit of a false philosophy[105]. [Sidenote: The theological and the\npractical error of the Colossians.]The purity of their Christianity is\nendangered by two errors, recommended to them by their heretical\nleaders\u2014the one theological, the other practical\u2014but both alike\nspringing from the same source, the conception of matter as the origin\nand abode of evil. Thus, regarding God and matter as directly\nantagonistic and therefore apart from and having no communication with\neach other, they sought to explain the creation and government of the\nworld by interposing a series of intermediate beings, emanations or\nangels, to whom accordingly they offered worship. At the same time,\nsince they held that evil resided, not in the rebellious spirit of man,\nbut in the innate properties of matter, they sought to overcome it by a\nrigid ascetic discipline, which failed after all to touch the springs of\naction. [Sidenote: The proper corrective to both lies in the Christ of\nthe Gospel.]As both errors flowed from the same source, they must be\ncorrected by the application of the same remedy, the Christ of the\nGospel. In the Person of Christ, the one mediator between heaven and\nearth, is the true solution of the theological difficulty. Through the\nLife in Christ, the purification of the heart through faith and love, is\nthe effectual triumph over moral evil[106]. [Sidenote: References to\nEpaphras.]St Paul therefore prescribes to the Colossians the true\nteaching of the Gospel, as the best antidote to the twofold danger which\nthreatens at once their theological creed and their moral principles;\nwhile at the same time he enforces his lesson by the claims of personal\naffection, appealing to the devotion of their evangelist Epaphras on\ntheir behalf[107].\n\nFootnote 103:\n\n  i. 3\u20139, 21 sq.\n\nFootnote 104:\n\n  ii. 1 sq.\n\nFootnote 105:\n\n  ii. 4, 8, 18.\n\nFootnote 106:\n\n  i. 1\u201320, ii. 9, iii. 4. The two threads are closely interwoven in St\n  Paul's refutation, as these references will show. The connexion of the\n  two errors, as arising from the same false principle, will be\n  considered more in detail in the next chapter.\n\nFootnote 107:\n\n  i. 7, iv. 12.\n\nOf Epaphras himself we know nothing beyond the few but significant\nnotices which connect him with Coloss\u00e6[108]. He did not return to\nColoss\u00e6 as the bearer of the letter, but remained behind with St\nPaul[109]. As St Paul in a contemporary epistle designates him his\nfellow-prisoner[110], it may be inferred that his zeal and affection had\ninvolved him in the Apostle's captivity, and that his continuance in\nRome was enforced. But however this may be, the letter was placed in the\nhands of Tychicus, a native of proconsular Asia, probably of\nEphesus[111],[Sidenote: Tychicus and Onesimus accompany the letter.] who\nwas entrusted with a wider mission at this time, and in its discharge\nwould be obliged to visit the valley of the Lycus[112]. At the same time\nhe was accompanied by Onesimus, whom the Colossians had only known\nhitherto as a worthless slave, but who now returns to them with the\nstamp of the Apostle's warm approval. St Paul says very little about\nhimself, because Tychicus and Onesimus would be able by word of mouth to\ncommunicate all information to the Colossians[113]. [Sidenote: The\nsalutations.]But he sends one or two salutations which deserve a few\nwords of explanation. Epaphras of course greets his fellow-townsmen and\nchildren in the faith. Other names are those of Aristarchus the\nThessalonian, who had been with the Apostle at Ephesus[114] and may\npossibly have formed some personal connexion with the Colossians at that\ntime: Mark, against whom apparently the Apostle fears that a prejudice\nmay be entertained (perhaps the fact of his earlier desertion, and of St\nPaul's dissatisfaction in consequence[115], may have been widely known),\nand for whom therefore he asks a favourable reception at his approaching\nvisit to Coloss\u00e6, according to instructions which they had already\nreceived; and Jesus the Just, of whose relations with the Colossians we\nknow nothing, and whose only claim to a mention may have been his\nsingular fidelity to the Apostle at a critical juncture. Salutations\nmoreover are added from Luke and from Demas; and here again their close\ncompanionship with the Apostle is, so far as we know, the sole cause of\ntheir names appearing[116].\n\nFootnote 108:\n\n  For the reasons why Epaphras cannot be identified with Epaphroditus,\n  who is mentioned in the Philippian letter, see _Philippians_ p. 60,\n  note 4. The later tradition, which makes him bishop of Coloss\u00e6, is\n  doubtless an inference from St Paul's language and has no independent\n  value. The further statement of the martyrologies, that he suffered\n  martyrdom for his flock, can hardly be held to deserve any higher\n  credit. His day is the 19th of July in the Western Calendar. His body\n  is said to lie in the Church of S. Maria Maggiore at Rome.\n\nFootnote 109:\n\n  Col. iv. 12.\n\nFootnote 110:\n\n  Philem. 23 \u1f41 \u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03cc\u03c2 \u03bc\u03bf\u03c5. The word may possibly have a\n  metaphorical sense (see _Philippians_ p. 11); but the literal meaning\n  is more probable. St Jerome on Philem. 23 (VII. p. 762) gives the\n  story that St Paul's parents were natives of Giscala and, when the\n  Romans invaded and wasted Jud\u00e6a, were banished thence with their son\n  to Tarsus. He adds that Epaphras may have been St Paul's\n  fellow-prisoner at this time, and have been removed with his parents\n  to Coloss\u00e6. It is not quite clear whether this statement respecting\n  Epaphras is part of the tradition, or Jerome's own conjecture appended\n  to it.\n\nFootnote 111:\n\n  Acts xx. 4, 2 Tim. iv. 12.\n\nFootnote 112:\n\n  See below, p. 37.\n\nFootnote 113:\n\n  Col. iv. 7\u20139.\n\nFootnote 114:\n\n  Acts xix. 29.\n\nFootnote 115:\n\n  Acts xiii. 13, xv. 37\u201339.\n\nFootnote 116:\n\n  Col. iv. 10\u201314.\n\n[Sidenote: Charge respecting Laodicea.]\n\nLastly, the Laodiceans were closely connected with the Colossians by\nlocal and spiritual ties. To the Church of Laodicea therefore, and to\nthe household of one Nymphas who was a prominent member of it, he sends\ngreeting. At the same time he directs them to interchange letters with\nthe Laodiceans; for to Laodicea also he had written. And he closes his\nsalutations with a message to Archippus, a resident either at Coloss\u00e6 or\nat Laodicea (for on this point we are left to conjecture), who held some\nimportant office in the Church, and respecting whose zeal he seems to\nhave entertained a misgiving[117].\n\nFootnote 117:\n\n  iv. 15\u201317.\n\n[Sidenote: 2. The LETTER TO PHILEMON.]\n\n2. But, while providing for the spiritual welfare of the whole Colossian\nChurch, he did not forget the temporal interests of its humblest member.\nHaving attended to the solicitations of the evangelist Epaphras, he\naddressed himself to the troubles of the runaway slave Onesimus. The\nmission of Tychicus to Coloss\u00e6 was a favourable opportunity of restoring\nhim to Philemon; for Tychicus, well known as the Apostle's friend and\nfellow-labourer, might throw the shield of his protection over him and\navert the worst consequences of Philemon's anger. But, not content with\nthis measure of precaution, the Apostle himself writes to PHILEMON on\nthe offender's behalf, recommending him as a changed man[118], and\nclaiming forgiveness for him as a return due from Philemon to himself as\nto his spiritual father[119].\n\nFootnote 118:\n\n  Philem. 11, 16.\n\nFootnote 119:\n\n  ver. 19.\n\nThe salutations in this letter are the same as those in the Epistle to\nthe Colossians with the exception of Jesus Justus, whose name is\nomitted[120]. Towards the close St Paul declares his hope of release and\nintention of visiting Coloss\u00e6, and asks Philemon to 'prepare a lodging'\nfor him[121].\n\nFootnote 120:\n\n  vv. 23, 24.\n\nFootnote 121:\n\n  ver. 22.\n\n[Sidenote: 3. The CIRCULAR LETTER, of which a copy is sent to LAODICEA.]\n\n3. But at the same time with the two letters destined especially for\nColoss\u00e6, the Apostle despatched a third, which had a wider scope. It has\nbeen already mentioned that Tychicus was charged with a mission to the\nAsiatic Churches. It has been noticed also that the Colossians were\ndirected to procure and read a letter in the possession of the\nLaodiceans. These two facts are closely connected. The Apostle wrote at\nthis time a circular letter to the Asiatic Churches, which got its\nultimate designation from the metropolitan city and is consequently\nknown to us as the Epistle to the EPHESIANS[122]. It was the immediate\nobject of Tychicus' journey to deliver copies of this letter at all the\nprincipal centres of Christianity in the district, and at the same time\nto communicate by word of mouth the Apostle's special messages to\neach[123]. Among these centres was Laodicea. Thus his mission brought\nhim into the immediate neighbourhood of Coloss\u00e6. But he was not charged\nto deliver another copy of the circular letter at Coloss\u00e6 itself, for\nthis Church would be regarded only as a dependency of Laodicea; and\nbesides he was the bearer of a special letter from the Apostle to them.\nIt was sufficient therefore to provide that the Laodicean copy should be\ncirculated and read at Coloss\u00e6.\n\nFootnote 122:\n\n  See the introduction to the epistle.\n\nFootnote 123:\n\n  Ephes. vi. 21, 22.\n\n[Sidenote: Personal links connecting the three letters.]\n\nThus the three letters are closely related. Tychicus is the personal\nlink of connexion between the Epistles to the Ephesians and to the\nColossians; Onesimus between those to the Colossians and to Philemon.\n\nFor reasons given elsewhere[124], it would appear that these three\nletters were written and despatched towards the close of [Sidenote:\nEarthquake in the Lycus Valley.]the Apostle's captivity, about the year\n63. At some time not very distant from this date, a great catastrophe\novertook the cities of the Lycus valley. An earthquake was no uncommon\noccurrence in this region[125]. But on this occasion the shock had been\nunusually violent, and Laodicea, the flourishing and populous, was laid\nin ruins. Tacitus, who is our earliest authority for this fact, places\nit in the year 60 and is silent about the neighbouring towns[126].\nEusebius however makes it subsequent [Sidenote: Its probable date.]to\nthe burning of Rome (A.D. 64), and mentions Hierapolis and Coloss\u00e6 also\nas involved in the disaster[127]; while later writers, adopting the date\nof Eusebius and including the three cities with him, represent it as one\nof a series of divine judgments on the heathen world for the persecution\nof the Christians which followed on the fire[128]. Having no direct\nknowledge of the source from which Eusebius derived his information, we\nshould naturally be disposed to accept the authority of Tacitus for the\ndate, as more trustworthy. But, as indications occur elsewhere that\nEusebius followed unusually good authorities in recording these\nearthquakes[129], it is far from improbable that he [Sidenote: Bearing\non the chronology of these letters.] gives the correct date[130]. In\nthis case the catastrophe was subsequent to the writing of these\nletters. If on the other hand the year named by Tacitus be adopted, we\ngain a subsidiary confirmation of the comparatively late date which I\nhave ventured to assign to these epistles on independent grounds; for,\nif they had been written two years earlier, when the blow was recent, we\nmight reasonably have expected to find some reference to a disaster\nwhich had devastated Laodicea and from which Coloss\u00e6 cannot have escaped\naltogether without injury. The additional fact mentioned by the Roman\nhistorian, that Laodicea was rebuilt from her own resources without the\nusual assistance from Rome[131], is valuable as illustrating a later\nnotice in the Apostolic writings[132].\n\nFootnote 124:\n\n  See _Philippians_ p. 29 sq.; where reasons are given for placing the\n  Philippian Epistle at an earlier, and the others at a later stage in\n  the Apostle's captivity.\n\nFootnote 125:\n\n  See above, p. 3. Laodicea was visited by the following earthquakes in\n  the ages preceding and subsequent to the Christian era.\n\n  (1) Before about B.C. 125, _Orac. Sibyll._ iii. 471, if the date now\n  commonly assigned to this Sibylline Oracle be correct, and if the\n  passage is to be regarded as a prophecy after the event. In iii. 347\n  Hierapolis is also mentioned as suffering in the same way; but it may\n  be questioned whether the Phrygian city is meant.\n\n  (2) About B.C. 12, Strabo xii. 8, p. 579, Dion Cass. liv. 30. Strabo\n  names only Laodicea and Tralles, but Dion Cassius says \u1f21 \u1f08\u03c3\u03af\u03b1 \u03c4\u1f78 \u1f14\u03b8\u03bd\u03bf\u03c2\n  \u1f10\u03c0\u03b9\u03ba\u03bf\u03c5\u03c1\u03af\u03b1\u03c2 \u03c4\u03b9\u03bd\u1f78\u03c2 \u03b4\u03b9\u1f70 \u03c3\u03b5\u03b9\u03c3\u03bc\u03bf\u1f7a\u03c2 \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u1f10\u03b4\u03b5\u1fd6\u03c4\u03bf.\n\n  (3) A.D. 60 according to Tacitus (_Ann._ xiv. 27); A.D. 64 or 65\n  according to Eusebius (_Chron._ s.a.), who includes also Hierapolis\n  and Coloss\u00e6. To this earthquake allusion is made in a Sibylline Oracle\n  written not many years after the event; _Orac. Sibyll._ iv. 107 (see\n  also v. 289, vii. 23).\n\n  (4) Between A.D. 222 and A.D. 235, in the reign of Alexander Severus,\n  as we learn from another Sibylline Oracle (xii. 280). On this occasion\n  Hierapolis also suffered.\n\n  This list will probably be found not to have exhausted all these\n  catastrophes on record.\n\n  The following earthquakes also are mentioned as happening in the\n  neighbouring towns or in the district generally: the date uncertain,\n  _Carura_ (Strabo xii. 8, p. 578); A.D. 17 the _twelve cities_,\n  _Sardis_ being the worst sufferer (Tac. _Ann._ ii. 7, Plin. _N.H._ ii.\n  86, Dion Cass. lvii. 17, Strabo xii. 8, p. 579); A.D. 23 _Cibyra_\n  (Tac. _Ann._ iv. 13); A.D. 53 _Apamea_ (Tac. _Ann._ xii. 58): about\n  A.D. 155, under Antoninus Pius, 'Rhodiorum et _Asi\u00e6_ oppida' (Capitol.\n  _Anton. Pius_ 9); A.D. 178, under M. Aurelius, _Smyrna_ and other\n  cities (_Chron. Pasch._ I. p. 489, ed. Dind., Aristid. _Or._ xx, xxi,\n  xli; see Clinton _Fast. Rom._ I. p. 176 sq., Hertzberg _Griechenland_\n  etc. II. pp. 371, 410); A.D. 262, under Gallienus II (Trebell.\n  _Gallien._ 5 'Malum tristius in _Asi\u00e6_ urbibus fuit ... hiatus terr\u00e6\n  plurimis in locis fuerunt, cum aqua salsa in fossis appareret,' ib. 6\n  'vastatam _Asiam_ ... elementorum concussionibus'). Strabo says (p.\n  579) that _Philadelphia_ is more or less shaken daily (\u03ba\u03b1\u03b8' \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd),\n  and that _Apamea_ has suffered from numerous earthquakes.\n\nFootnote 126:\n\n  Tac. _Ann._ xiv. 27 'Eodem anno ex inlustribus Asi\u00e6 urbibus Laodicea,\n  tremore terr\u00e6 prolapsa, nullo a nobis remedio propriis opibus\n  revaluit.' The year is given 'Nerone iv, Corn. Cosso consulibus' (xiv.\n  20). Two different writers, in _Smith's Dictionary of Geography_ and\n  _Smith's Dictionary of the Bible_, s.v. Laodicea, place the\n  destruction of Laodicea in the reign of Tiberius, confusing this\n  earthquake with an earlier one (_Ann._ ii. 47). By this earlier\n  earthquake 'duodecim celebres Asi\u00e6 urbes conlaps\u00e6,' but their names\n  are given, and not one is situated in the valley of the Lycus.\n\nFootnote 127:\n\n  Euseb. _Chron._ Ol. 210 (II. p. 154 sq., ed. Sch\u00f6ne) 'In Asia tres\n  urbes terr\u00e6 motu conciderunt Laodicea Hierapolis Coloss\u00e6.' The\n  Armenian version and Jerome agree in placing it the next event in\n  order after the fire at Rome (A.D. 64), though there is a difference\n  of a year in the two texts. If the Sibylline Oracle, v. 317, refers to\n  this earthquake, as seems probable, we have independent testimony that\n  Hierapolis was involved in the catastrophe; comp. _ib._ v. 289.\n\nFootnote 128:\n\n  This is evidently the idea of Orosius, vii. 7.\n\nFootnote 129:\n\n  I draw this inference from his account of the earthquake in the reign\n  of Tiberius. Tacitus (_Ann._ ii. 47) states that _twelve_ cities were\n  ruined in one night, and records their names. Pliny also, who mentions\n  this earthquake as 'the greatest within the memory of man' (_N.H._ ii.\n  86), gives the same number. Eusebius however, _Chron._ Ol. 198 (II. p.\n  146 sq., ed. Sch\u00f6ne), names _thirteen_ cities, coinciding with Tacitus\n  as far as he goes, but including Ephesus also. Now a monument was\n  found at Puteoli (see Gronov. _Thes._ _Gr\u00e6c. Ant._ VII. p. 433 sq.),\n  and is now in the Museum at Naples (_Museo Borbonico_ XV, Tav. iv, v),\n  dedicated to Tiberius and representing _fourteen_ female figures with\n  the names of fourteen Asiatic cities underneath; these names being the\n  same as those mentioned by Tacitus with the addition of Ephesus and\n  Cibyra. There can be no doubt that this was one of those monuments\n  mentioned by Apollonius quoted in Phlegon (_Fragm._ 42, M\u00fcller's\n  _Fragm. Hist. Gr\u00e6c._ III. p. 621) as erected to commemorate the\n  liberality of Tiberius in contributing to the restoration of the\n  ruined cities (see Eckhel _Doct. Num. Vet._ VI. 192 sq.). But no\n  earthquake at Ephesus is mentioned by Tacitus. He does indeed speak of\n  such a catastrophe as happening at Cibyra (_Ann._ iv. 13) six years\n  later than the one which ruined the twelve cities, and of the relief\n  which Tiberius afforded on this latter occasion as on the former. But\n  we owe to Eusebius alone the fact that Ephesus also was seriously\n  injured by an earthquake in the same year\u2014perhaps not on the same\n  night\u2014with the twelve cities: and this fact is necessary to explain\n  the monument. It should be added that Nipperdey (on Tac. _Ann._ ii.\n  47) supposes the earthquake at Ephesus to have been recorded in the\n  lost portion of the fifth book of the _Annals_ which comprised the\n  years A.D. 29\u201331; but this bare hypothesis cannot outweigh the direct\n  testimony of Eusebius.\n\nFootnote 130:\n\n  Hertzberg (_Geschichte Griechenlands unter der Herrschaft der R\u00f6mer_,\n  II. p. 96) supposes that Tacitus and Eusebius refer to two different\n  events, and that Laodicea was visited by earthquakes twice within a\n  few years, A.D. 60 and A.D. 65.\n\nFootnote 131:\n\n  Tac. _Ann._ xiv. 27, quoted above, p. 38, note 126. To this fact\n  allusion is made in the feigned prediction of the Sibyllines, iv. 107\n  \u03a4\u03bb\u1fc6\u03bc\u03bf\u03bd \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1, \u03c3\u1f72 \u03b4\u1f72 \u03c4\u03c1\u03ce\u03c3\u03b5\u03b9 \u03c0\u03bf\u03c4\u1f72 \u03c3\u03b5\u03b9\u03c3\u03bc\u1f78\u03c2 \u03c0\u03c1\u03b7\u03bd\u03af\u03be\u03b1\u03c2, \u03c3\u03c4\u03ae\u03c3\u03b5\u03b9 \u03b4\u1f72 \u03c0\u03ac\u03bb\u03b9\u03bd\n  \u03c0\u03cc\u03bb\u03b9\u03bd \u03b5\u1f50\u03c1\u03c5\u03ac\u03b3\u03c5\u03b9\u03b1\u03bd, where \u03c3\u03c4\u03ae\u03c3\u03b5\u03b9 must be the 2nd person, 'Thou wilt\n  rebuild thy city with its broad streets.' This Sibylline poem was\n  written about the year 80. The building of the amphitheatre mentioned\n  above (p. 6, note 6), would form part of this work of reconstruction.\n\nFootnote 132:\n\n  See below, p. 43.\n\n[Sidenote: St Mark's intended visit.]\n\nIt has been seen that, when these letters were written, St Mark was\nintending shortly to visit Coloss\u00e6, and that the Apostle himself,\nlooking forward to his release, hoped at length to make a personal\nacquaintance with these Churches, which hitherto he knew only through\nthe report of others. Whether St Mark's visit was ever paid or not, we\nhave no means of determining[133]. Of St Paul himself it is reasonable\nto assume, [Sidenote: St Paul probably visits Coloss\u00e6.] that in the\ninterval between his first and second Roman captivity he found some\nopportunity of carrying out his design. At all events we find him at\nMiletus, near to the mouth of the M\u00e6ander[134]; and the journey between\nthis place and Laodicea is neither long nor difficult.\n\nFootnote 133:\n\n  Two notices however imply that St Mark had some personal connexion\n  with Asia Minor in the years immediately succeeding the date of this\n  reference: (1) St Peter, writing to the Churches of Asia Minor, sends\n  a salutation from St Mark (1 Pet. v. 13); (2) St Paul gives charge to\n  Timothy, who appears to be still residing at Ephesus, to take up Mark\n  and bring him to Rome (2 Tim. iv. 11 \u039c\u03ac\u03c1\u03ba\u03bf\u03bd \u1f00\u03bd\u03b1\u03bb\u03b1\u03b2\u1f7c\u03bd \u1f04\u03b3\u03b5 \u03bc\u03b5\u03c4\u1f70\n  \u03c3\u03b5\u03b1\u03c5\u03c4\u03bf\u1fe6). Thus it seems fairly probable that St Mark's projected visit\n  to Coloss\u00e6 was paid.\n\nFootnote 134:\n\n  2 Tim. iv. 20. By a strange error Lequien (_Oriens Christ._ I. p. 833)\n  substitutes Hierapolis for Nicopolis in Tit. iii. 12, and argues from\n  the passage that the Church of Hierapolis was founded by St Paul.\n\n[Sidenote: St John in Asia Minor.]\n\nAt the time of this visit\u2014the first and last, we may suppose, which he\npaid to the valley of the Lycus\u2014St Paul's direction of the Asiatic\nChurches is drawing to a close. With his death they pass into the hands\nof St John[135], who takes up his abode in Asia Minor. Of Coloss\u00e6 and\nHierapolis we hear nothing more in the New Testament: but from his exile\nin[Sidenote: The message to Laodicea.] Patmos the beloved disciple\ndelivers his Lord's message to the Church of Laodicea[136]; a message\ndoubtless intended to be communicated also to the two subordinate\nChurches, to which it would apply almost equally well.\n\nFootnote 135:\n\n  It was apparently during the interval between St Paul's first\n  captivity at Rome and his death, that St Peter wrote to the Churches\n  of Asia Minor (1 Pet. i. 1). Whether in this interval he also visited\n  personally the districts evangelized directly or indirectly by St\n  Paul, we have no means of deciding. Such a visit is far from unlikely,\n  but it can hardly have been of long duration. A copy of his letters\n  would probably be sent to Laodicea, as a principal centre of\n  Christianity in Proconsular Asia, which is among the provinces\n  mentioned in the address of the First Epistle.\n\nFootnote 136:\n\n  Rev. iii. 14\u201321.\n\n[Sidenote: Correspondences between the Apocalypse and St Paul's\n           Epistles.]\n\nThe message communicated by St John to Laodicea prolongs the note which\nwas struck by St Paul in the letter to Coloss\u00e6. An interval of a very\nfew years has not materially altered the character of these Churches.\nObviously the same temper prevails, the same errors are rife, the same\ncorrection must be applied.\n\n[Sidenote: 1. The doctrine of the Person of Christ,]\n\n1. Thus, while St Paul finds it necessary to enforce the truth that\nChrist is the image of the invisible God, that in Him all the divine\nfulness dwells, that He existed before all things, that through Him all\nthings were created and in Him all things are sustained, that He is the\nprimary source (\u1f00\u03c1\u03c7\u03ae) and has the pre-eminence in all things[137]; so in\nalmost identical language St John, speaking in the person of our Lord,\ndeclares that He is the Amen, the faithful and true witness, the primary\nsource (\u1f00\u03c1\u03c7\u03ae) of the creation of God[138]. Some lingering shreds of the\nold heresy, we may suppose, still hung about these Churches, and instead\nof 'holding fast the Head' they were even yet prone to substitute\nintermediate agencies, angelic mediators, as links in the chain which\nshould bind man to God. They still failed to realise the majesty and\nsignificance, the _completeness_, of the Person of Christ.\n\nFootnote 137:\n\n  Col. i. 15\u201318.\n\nFootnote 138:\n\n  Rev. iii. 14. It should be observed that this designation of our Lord\n  (\u1f21 \u1f00\u03c1\u03c7\u1f74 \u03c4\u1fc6\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6), which so closely resembles the language\n  of the Colossian Epistle, does not occur in the messages to the other\n  six Churches, nor do we there find anything resembling it.\n\n[Sidenote: and practical duties which follow upon it.]\n\nAnd the practical duty also, which follows from the recognition of the\ntheological truth, is enforced by both Apostles in very similar\nlanguage. If St Paul entreats the Colossians to seek those things which\nare above, where Christ is seated on the right hand of God[139], and in\nthe companion epistle, which also he directs them to read, reminds the\nChurches that God raised them with Christ and seated them with him in\nheavenly places in Christ Jesus[140]; in like manner St John gives this\npromise to the Laodiceans in the name of his Lord: 'He that overcometh,\nI will grant to him to sit with me in my throne, even as I also overcame\nand did sit with my Father in His throne[141]'.\n\nFootnote 139:\n\n  Col. iii. 1.\n\nFootnote 140:\n\n  Ephes. ii. 6 \u03c3\u03c5\u03bd\u03ae\u03b3\u03b5\u03b9\u03c1\u03b5\u03bd \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b5\u03ba\u03ac\u03b8\u03b9\u03c3\u03b5\u03bd \u03ba.\u03c4.\u03bb.\n\nFootnote 141:\n\n  Rev. iii. 21 \u03b4\u03ce\u03c3\u03c9 \u03b1\u1f50\u03c4\u1ff7 \u03ba\u03b1\u03b8\u03af\u03c3\u03b1\u03b9 \u03bc\u03b5\u03c4' \u1f10\u03bc\u03bf\u1fe6, \u03ba.\u03c4.\u03bb. Here again it must be\n  noticed that there is no such resemblance in the language of the\n  promises to the faithful in the other six Churches. This double\n  coincidence, affecting the two ideas which may be said to cover the\n  whole ground in the Epistle to the Colossians, can hardly, I think, be\n  fortuitous, and suggests an acquaintance with and recognition of the\n  earlier Apostle's teaching on the part of St John.\n\n[Sidenote: 2. Warning against lukewarmness.]\n\n2. But again; after a parting salutation to the Church of Laodicea St\nPaul closes with a warning to Archippus, apparently its chief pastor, to\ntake heed to his ministry[142]. Some signs of slackened zeal seem to\nhave called forth this rebuke. It may be an accidental coincidence, but\nit is at least worthy of notice, that lukewarmness is the special sin\ndenounced in the angel of the Laodiceans, and that the necessity of\ngreater earnestness is the burden of the message to that Church[143]. As\nwith the people, so is it with the priest. The community takes its\ncolour from and communicates its colour to its spiritual rulers. The 'be\nzealous' of St John is the counterpart to the 'take heed' of St Paul.\n\nFootnote 142:\n\n  Col. iv. 17.\n\nFootnote 143:\n\n  Rev. iii. 19. If the common view, that by the angel of the Church its\n  chief pastor is meant, were correct, and if Archippus (as is very\n  probable) had been living when St John wrote, the coincidence would be\n  still more striking; see Trench's _Epistles to the Seven Churches in\n  Asia_, p. 180. But for reasons given elsewhere (Philippians p. 197\n  sq.), this interpretation of the angels seems to me incorrect.\n\n[Sidenote: 3. The pride of wealth denounced.]\n\n3. Lastly; in the Apocalyptic message the pride of wealth is sternly\ncondemned in the Laodicean Church: 'For that thou sayest I am rich and\nhave gotten me riches and have need of nothing, and knowest not that\nthou art utterly wretched and miserable and beggarly and blind and\nnaked, I counsel thee to buy gold of me refined with fire, that thou\nmayest have riches[144].' This proud vaunt receives its best\nillustration from a recent occurrence at Laodicea, to which allusion has\nalready been made. Only a very few years before this date an earthquake\nhad laid the city in ruins. Yet from this catastrophe she rose again\nwith more than her former splendour. [Sidenote: The vaunt of\nLaodicea.]This however was not her chief title to respect. While other\ncities, prostrated by a like visitation, had sought relief from the\nconcessions of the Roman senate or the liberality of the emperor's\npurse, it was the glory of Laodicea that she alone neither courted nor\nobtained assistance, but recovered by her own resources. 'Nullo a nobis\nremedio,' says the Roman historian, 'propriis opibus revaluit[145].'\nThus she had asserted a proud independence, to which neither far-famed\nmetropolitan Ephesus, nor old imperial Sardis, nor her prosperous\ncommercial neighbours, Apamea and Cibyra, could lay claim[146]. No one\nwould dispute her boast that she 'had gotten riches and had need of\nnothing.'\n\nFootnote 144:\n\n  Rev. iii. 17, 18, where the correct reading with the repetition of the\n  definite articles, \u1f41 \u03c4\u03b1\u03bb\u03b1\u03af\u03c0\u03c9\u03c1\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f41 \u1f10\u03bb\u03b5\u03b9\u03bd\u03cc\u03c2, signifies the type, the\n  embodiment of wretchedness, etc.\n\nFootnote 145:\n\n  Tac. Ann. xiv. 27.\n\nFootnote 146:\n\n  In all the other cases of earthquake which Tacitus records as\n  happening in these Asiatic cities, Ann. ii. 47 (the twelve cities),\n  iv. 13 (Cibyra), xii. 58 (Apamea), he mentions the fact of their\n  obtaining relief from the Senate or the Emperor. On an earlier\n  occasion Laodicea herself had not disdained under similar\n  circumstances to receive assistance from Augustus: Strabo, xii. p.\n  579.\n\n[Sidenote: Pride of intellectual wealth.]\n\nBut is there not a second and subsidiary idea underlying the Apocalyptic\nrebuke? The pride of intellectual wealth, we may well suspect, was a\ntemptation at Laodicea hardly less strong than the pride of material\nresources. When St Paul wrote, the theology of the Gospel and the\ncomprehension of the Church were alike endangered by a spirit of\nintellectual exclusiveness[147] in these cities. He warned them against\na vain philosophy, against a show of wisdom, against an intrusive mystic\nspeculation, which vainly puffed up the fleshly mind[148]. He tacitly\ncontrasted with this false intellectual wealth 'the riches of the glory\nof God's mystery revealed in Christ[149],' the riches of the full\nassurance of understanding, the genuine treasures of wisdom and\nknowledge[150]. May not the same contrast be discerned in the language\nof St John? The Laodiceans boast of their enlightenment, but they are\nblind, and to cure their blindness they must seek eye-salve from the\nhands of the great Physician. They vaunt their wealth of knowledge, but\nthey are wretched paupers, and must beg the refined gold of the Gospel\nto relieve their wants[151].\n\nFootnote 147:\n\n  See the next chapter of this introduction.\n\nFootnote 148:\n\n  Col. ii. 8, 18, 23.\n\nFootnote 149:\n\n  i. 27.\n\nFootnote 150:\n\n  ii. 2, 3.\n\nFootnote 151:\n\n  Comp. Eph. i. 18 'The _eyes of your understanding being enlightened_,\n  that ye may know what is the hope of his calling, what the _riches of\n  the glory of his inheritance_ in the saints.'\n\nThis is the last notice in the Apostolic records relating to the\nChurches in the valley of the Lycus; but during the succeeding ages the\nChristian communities of this district play a conspicuous part in the\nstruggles and the development of the Church. [Sidenote: The early\ndisciples settle in proconsular Asia]When after the destruction of\nJerusalem St John fixed his abode at Ephesus, it would appear that not a\nfew of the oldest surviving members of the Palestinian Church\naccompanied him into 'Asia,' which henceforward became the head-quarters\nof Apostolic authority. In this body of emigrants Andrew[152] and Philip\namong the Twelve, Aristion and John the presbyter[153] among other\npersonal disciples of the Lord, are especially mentioned.\n\nFootnote 152:\n\n  _Canon Murator._ fol. 1, l. 14 (p. 17, ed. Tregelles), Cureton's\n  _Ancient Syriac Documents_ pp. 32, 34. Comp. Papias in Euseb. _H.E._\n  iii. 39.\n\nFootnote 153:\n\n  Papias in Euseb. _H.E._ iii. 39.\n\n[Sidenote: and especially at Hierapolis.]\n\nAmong the chief settlements of this Christian dispersion was Hierapolis.\nThis fact explains how these Phrygian Churches assumed a prominence in\nthe ecclesiastical history of the second century, for which we are\nhardly prepared by their antecedents as they appear in connexion with St\nPaul, and which they failed to maintain in the history of the later\nChurch.\n\nHere at all events was settled Philip of Bethsaida[154], the [Sidenote:\nPhilip the Apostle with his daughters.] early friend and fellow-townsman\nof St John, and the first Apostle who is recorded to have held\ncommunication with the Gentiles[155]. Here he died and was buried; and\nhere after his decease lived his two virgin daughters, who survived to a\nvery advanced age and thus handed down to the second century the\ntraditions of the earliest days of the Church. A third daughter, who was\nmarried, had settled in Ephesus, where her body rested[156]. [Sidenote:\nTheir traditions collected by Papias.]It was from the two daughters who\nresided at Hierapolis, that Papias heard several stories of the first\npreachers of the Gospel, which he transmitted to posterity in his\nwork[157].\n\nFootnote 154:\n\n  Polycrates in Euseb. _H.E._ iii. 31, v. 24 \u03a6\u03af\u03bb\u03b9\u03c0\u03c0\u03bf\u03bd [\u03c4\u1f78\u03bd] _\u03c4\u1ff6\u03bd\n  \u03b4\u03ce\u03b4\u03b5\u03ba\u03b1 \u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03c9\u03bd_, \u1f43\u03c2 \u03ba\u03b5\u03ba\u03bf\u03af\u03bc\u03b7\u03c4\u03b1\u03b9 \u1f10\u03bd \u1f39\u03b5\u03c1\u03b1\u03c0\u03cc\u03bb\u03b5\u03b9, \u03ba\u03b1\u1f76 \u03b4\u03cd\u03bf \u03b8\u03c5\u03b3\u03b1\u03c4\u03ad\u03c1\u03b5\u03c2\n  \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b3\u03b5\u03b3\u03b7\u03c1\u03b1\u03ba\u03c5\u1fd6\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b8\u03ad\u03bd\u03bf\u03b9, \u03ba\u03b1\u1f76 \u1f20 \u1f11\u03c4\u03ad\u03c1\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b8\u03c5\u03b3\u03ac\u03c4\u03b7\u03c1 \u1f10\u03bd \u1f01\u03b3\u03af\u1ff3 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9\n  \u03c0\u03bf\u03bb\u03b9\u03c4\u03b5\u03c5\u03c3\u03b1\u03bc\u03ad\u03bd\u03b7, \u1f23 \u1f10\u03bd \u1f18\u03c6\u03ad\u03c3\u1ff3 \u1f00\u03bd\u03b1\u03c0\u03b1\u03cd\u03b5\u03c4\u03b1\u03b9. To this third daughter the\n  statement of Clement of Alexandria must refer, though by a common\n  looseness of expression he uses the plural number (Euseb. _H.E._ iii.\n  30), \u1f23 \u03ba\u03b1\u1f76 _\u03c4\u03bf\u1f7a\u03c2 \u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03bf\u03c5\u03c2_ \u1f00\u03c0\u03bf\u03b4\u03bf\u03ba\u03b9\u03bc\u03ac\u03c3\u03bf\u03c5\u03c3\u03b9\u00b7 \u03a0\u03ad\u03c4\u03c1\u03bf\u03c2 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76\n  \u03a6\u03af\u03bb\u03b9\u03c0\u03c0\u03bf\u03c2 \u1f10\u03c0\u03b1\u03b9\u03b4\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03b1\u03bd\u03c4\u03bf, \u03a6\u03af\u03bb\u03b9\u03c0\u03c0\u03bf\u03c2 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1f70\u03c2 \u03b8\u03c5\u03b3\u03b1\u03c4\u03ad\u03c1\u03b1\u03c2 \u1f00\u03bd\u03b4\u03c1\u03ac\u03c3\u03b9\u03bd\n  \u1f10\u03be\u03ad\u03b4\u03c9\u03ba\u03b5. On the other hand in the _Dialogue between Caius and\n  Proclus_, Philip the Evangelist was represented as residing at\n  Hierapolis (Euseb. _H.E._ iii. 31) \u03bc\u03b5\u03c4\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03b4\u1f72 \u03c0\u03c1\u03bf\u03c6\u03ae\u03c4\u03b9\u03b4\u03b5\u03c2 \u03c4\u03ad\u03c3\u03c3\u03b1\u03c1\u03b5\u03c2\n  \u03b1\u1f31 \u03a6\u03af\u03bb\u03b9\u03c0\u03c0\u03bf\u03c5 \u03b3\u03b5\u03b3\u03ad\u03bd\u03b7\u03bd\u03c4\u03b1\u03b9 \u1f10\u03bd \u1f39\u03b5\u03c1\u03b1\u03c0\u03cc\u03bb\u03b5\u03b9 \u03c4\u1fc7 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u1f08\u03c3\u03af\u03b1\u03bd\u00b7 \u1f41 \u03c4\u03ac\u03c6\u03bf\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd\n  \u1f10\u03c3\u03c4\u1f76\u03bd \u1f10\u03ba\u03b5\u1fd6, \u03ba\u03b1\u1f76 \u1f41 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd, where the mention of the _four_\n  daughters _prophesying_ identifies the person meant (see Acts xxi. 8).\n  Nothing can be clearer than that St Luke distinguishes Philip the\n  Evangelist from Philip the Apostle; for (1) When the Seven are\n  appointed, he distinctly states that this new office is created to\n  relieve the Twelve of some onerous duties (Acts vi. 2\u20135). (2) After\n  Philip the Evangelist has preached in Samaria, two of the Twelve are\n  sent thither to convey the gifts of the Spirit, which required the\n  presence of an Apostle (viii. 14\u201317). (3) When St Paul and his\n  companions visit Philip at C\u00e6sarea, he is carefully described as 'the\n  Evangelist, being one of the Seven' (xxi. 8). As St Luke was a member\n  of the Apostle's company when this visit was paid, and stayed 'many\n  days' in Philip's house, the accuracy of his information cannot be\n  questioned. Yet Eusebius (_H.E._ iii. 31) assumes the identity of the\n  Apostle with the Evangelist, and describes the notice in the _Dialogue\n  of Caius and Proclus_ as being 'in harmony with (\u03c3\u03c5\u03bd\u1fb4\u03b4\u03c9\u03bd)' the\n  language of Polycrates. And accordingly in another passage (_H.E._\n  iii. 39), when he has occasion to mention the conversations of Papias\n  with Philip's daughters at Hierapolis, he again supposes them to be\n  the same who are mentioned in the Acts.\n\n  My reasons for believing that the Philip who lived at Hierapolis was\n  not the Evangelist, but the Apostle, are as follows. (1) This is\n  distinctly stated by the earliest witness, Polycrates, who was bishop\n  of Ephesus at the close of the second century, and who besides claimed\n  to have and probably had special opportunities of knowing early\n  traditions. It is confirmed moreover by the notice in Clement of\n  Alexandria, who is the next in order of time, and whose means of\n  information also were good, for one of his earliest teachers was an\n  Ionian Greek (_Strom._ I. 1, p. 322). (2) The other view depends\n  solely on the authority of the _Dialogue of Caius and Proclus_. I have\n  given reasons elsewhere for questioning the separate existence of the\n  Roman presbyter Caius, and for supposing that this dialogue was\n  written by Hippolytus bishop of Portus (_Journal of Philology_ I. p.\n  98 sq., Cambridge, 1868). But however this may be, its author was a\n  Roman ecclesiastic, and probably wrote some quarter of a century at\n  least after Polycrates. In all respects therefore his authority is\n  inferior. Moreover it is suspicious in form. It mentions four\n  daughters instead of three, makes them all virgins, and represents\n  them as prophetesses, thus showing a distinct aim of reproducing the\n  particulars as given in Acts xxi. 9; whereas the account of Polycrates\n  is divergent in all three respects. (3) A life-long friendship would\n  naturally draw Philip the Apostle of Bethsaida after John, as it also\n  drew Andrew. And, when we turn to St John's Gospel, we can hardly\n  resist the impression that incidents relating to Andrew and Philip had\n  a special interest, not only for the writer of the Gospel, but also\n  for his hearers (John i. 40, 43\u201346, vi. 5\u20138, xii. 20\u201322, xiv. 8, 9).\n  Moreover the Apostles Andrew and Philip appear in this Gospel as\n  inseparable companions. (4) Lastly; when Papias mentions collecting\n  the sayings of the Twelve and of other early disciples from those who\n  heard them, he gives a prominent place to these two Apostles \u03c4\u03af\n  \u1f08\u03bd\u03b4\u03c1\u03ad\u03b1\u03c2 ... \u03b5\u1f36\u03c0\u03b5\u03bd \u1f22 \u03c4\u03af \u03a6\u03af\u03bb\u03b9\u03c0\u03c0\u03bf\u03c2, but there is no reference to Philip\n  the Evangelist. When therefore we read later that he conversed with\n  the daughters of Philip, it seems natural to infer that the Philip\n  intended is the same person whom he has mentioned previously. It\n  should be added, though no great value can be assigned to such\n  channels of information, that the Acts of Philip place the Apostle at\n  Hierapolis; Tischendorf, _Act. Apost. Apocr._ p. 75 sq.\n\n  On the other hand, those who suppose that the Evangelist, and not the\n  Apostle, resided at Hierapolis, account for the other form of the\n  tradition by the natural desire of the Asiatic Churches to trace their\n  spiritual descent directly from the Twelve. This solution of the\n  phenomenon might have been accepted, if the authorities in favour of\n  Philip the Evangelist had been prior in time and superior in quality.\n  There is no improbability in supposing that both the Philips were\n  married and had daughters.\n\nFootnote 155:\n\n  John xii. 20.\n\nFootnote 156:\n\n  See above p. 45, note 154.\n\nFootnote 157:\n\n  Euseb. _H.E._ iii. 39. This is the general reference for all those\n  particulars respecting Papias which are derived from Eusebius.\n\nThis Papias had conversed not only with the daughters of Philip, but\nalso with at least two personal disciples of the Lord, Aristion and John\nthe presbyter. He made it his business to gather traditions respecting\nthe sayings of the Saviour and His Apostles; and he published a work in\nfive books, entitled _An Exposition of the Oracles of the Lord_, using\nthe information thus collected to illustrate the discourses, and perhaps\nthe doings, of Christ as recorded in the Gospels[158]. Among other\nstories he related, apparently on the authority of these daughters of\nPhilip, how a certain dead man had been restored to life in his own day,\nand how Justus Barsabas, who is mentioned in the Acts, had drunk a\ndeadly poison and miraculously escaped from any evil effects[159].\n\nFootnote 158:\n\n  See Westcott, _Canon_ p. 63. On the opinions of Papias and on the\n  nature of his work, I may perhaps be allowed to refer to an article in\n  the _Contemporary Review_ Aug. 1867, where I have collected and\n  investigated all the notices of this father. The object of Papias'\n  work was not to construct a Gospel narrative, but to interpret and\n  illustrate those already existing. I ought to add that on two minor\n  points, the martyrdom of Papias and the identity of Philip with the\n  Evangelist, I have been led to modify my views since the article was\n  written.\n\nFootnote 159:\n\n  Euseb. l.c. \u1f61\u03c2 \u03b4\u1f72 \u03ba\u03b1\u03c4\u1f70 \u03c4\u03bf\u1f7a\u03c2 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 \u1f41 \u03a0\u03b1\u03c0\u03af\u03b1\u03c2 \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b4\u03b9\u03ae\u03b3\u03b7\u03c3\u03b9\u03bd\n  \u03c0\u03b1\u03c1\u03b5\u03b9\u03bb\u03b7\u03c6\u03ad\u03bd\u03b1\u03b9 \u03b8\u03b1\u03c5\u03bc\u03b1\u03c3\u03af\u03b1\u03bd \u1f51\u03c0\u1f78 [\u1f00\u03c0\u03bf]; \u03c4\u1ff6\u03bd \u03c4\u03bf\u1fe6 \u03a6\u03b9\u03bb\u03af\u03c0\u03c0\u03bf\u03c5 \u03b8\u03c5\u03b3\u03b1\u03c4\u03ad\u03c1\u03c9\u03bd\n  \u03bc\u03bd\u03b7\u03bc\u03bf\u03bd\u03b5\u03cd\u03b5\u03b9, \u03c4\u1f70 \u03bd\u1fe6\u03bd \u03c3\u03b7\u03bc\u03b5\u03b9\u03c9\u03c4\u03ad\u03bf\u03bd\u00b7 \u03bd\u03b5\u03ba\u03c1\u03bf\u1fe6 \u03b3\u1f70\u03c1 \u1f00\u03bd\u03ac\u03c3\u03c4\u03b1\u03c3\u03b9\u03bd \u03ba\u03b1\u03c4' \u03b1\u1f50\u03c4\u1f78\u03bd\n  \u03b3\u03b5\u03b3\u03bf\u03bd\u03c5\u1fd6\u03b1\u03bd \u1f31\u03c3\u03c4\u03bf\u03c1\u03b5\u1fd6, \u03ba\u03b1\u1f76 \u03b1\u1f56 \u03c0\u03ac\u03bb\u03b9\u03bd \u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03be\u03bf\u03bd \u03c0\u03b5\u03c1\u1f76 \u1f38\u03bf\u1fe6\u03c3\u03c4\u03bf\u03bd \u03c4\u1f78\u03bd\n  \u1f10\u03c0\u03b9\u03ba\u03bb\u03b7\u03b8\u03ad\u03bd\u03c4\u03b1 \u0392\u03b1\u03c1\u03c3\u03b1\u03b2\u1fb6\u03bd \u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c2 \u03ba.\u03c4.\u03bb. The information respecting the\n  raising of the dead man might have come from the daughters of Philip,\n  as the context seems certainly to imply, while yet the event happened\n  in Papias' own time (\u03ba\u03b1\u03c4' \u03b1\u1f50\u03c4\u03cc\u03bd). It will be remembered that even\n  Iren\u00e6us mentions similar miracles as occurring in his own age (_H\u00e6r._\n  ii. 32. 4). Eusebius does not say that the miraculous preservation of\n  Justus Barsabas also occurred in the time of Papias.\n\n[Sidenote: Life and teaching of PAPIAS.]\n\nIf we may judge by his name, PAPIAS was a native of Phrygia, probably of\nHierapolis[160], of which he afterwards became bishop, and must have\ngrown up to youth or early manhood before the close of the first\ncentury. He is said to have suffered martyrdom at Pergamum about the\nyear 165; but there is good reason for distrusting this statement,\nindependently of any chronological difficulty which it involves[161].\nOtherwise [Sidenote: Account of Eusebius.]he must have lived to a very\nadvanced age. Eusebius, to whom chiefly we owe our information\nrespecting him, was repelled by his millennarian views, and describes\nhim as a man of mean intelligence[162], accusing him of misunderstanding\nthe Apostolic sayings respecting the kingdom of Christ and thus\ninterpreting in a material sense expressions which were intended to be\nmystical and symbolical. This disparaging account, though one-sided, was\nindeed not altogether undeserved, for his love of the marvellous seems\nto have overpowered his faculty of discrimination. But the adverse\nverdict of Eusebius must be corrected by the more sympathetic language\nof Iren\u00e6us[163], who possibly may have known him personally, and who\ncertainly must have been well acquainted with his reputation and\ncharacter.\n\nFootnote 160:\n\n  Papias, or (as it is very frequently written in inscriptions) Pappias,\n  is a common Phrygian name. It is found several times at Hierapolis,\n  not only in inscriptions (Boeckh _Corp. Inscr._ no. 3930, 3912 a add.)\n  but even on coins (Mionnet IV. p. 301). This is explained by the fact\n  that it was an epithet of the Hierapolitan Zeus (Boeckh 3817 \u03a0\u03b1\u03c0\u03af\u1fb3 \u0394\u03b9\u1fd2\n  \u03c3\u03c9\u03c4\u1fc6\u03c1\u03b9), just as in Bithynia this same god was called \u03a0\u03ac\u03c0\u03b1\u03c2 (Lobeck\n  _Aglaoph._ p. 1048; see Boeckh _Corp. Inscr._ III. p. 1051). Hence as\n  the name of a mortal it is equivalent to the Greek Diogenes; e.g.\n  Boeckh no. 3912 a add., \u03a0\u03b1\u03c0\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03a3\u03c4\u03c1\u03ac\u03c4\u03c9\u03bd\u03bf\u03c2 \u1f41 \u03ba\u03b1\u03bb\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u0394\u03b9\u03bf\u03b3\u03ad\u03bd\u03b7\u03c2. In\n  an inscription at Trajanopolis we meet with it in a curious\n  conjunction with other familiar names (Boeckh no. 3865 i add.) \u03a0\u03b1\u03c0\u03c0\u03af\u03b1\u03c2\n  \u03a4\u03c1\u03bf\u03c6\u03af\u03bc\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03a4\u03c5\u03c7\u03b9\u03ba\u1fc6\u03c2 \u03ba.\u03c4.\u03bb. (see Waddington on Le Bas, Inscr. no.\n  718). This last belongs to the year A.D. 199. Other analogous Phrygian\n  names are Ammias, Tatias (with the corresponding feminines), which\n  with Latin terminations become Ammianus, Tatianus.\n\n  Thus at Hierapolis the name Papias is derived from heathen mythology,\n  and accordingly the persons bearing it on the inscriptions and coins\n  are all heathens. It may therefore be presumed that our Papias was of\n  Gentile origin. The inference however is not absolutely certain, since\n  elsewhere it is found borne by Jews; see the Talmudical references in\n  Zunz _Namen der Juden_ p. 16.\n\nFootnote 161:\n\n  _Chron. Pasch._ sub ann. 163 \u03c3\u1f7a\u03bd \u03c4\u1ff7 \u1f01\u03b3\u03af\u1ff3 \u03b4\u1f72 \u03a0\u03bf\u03bb\u03c5\u03ba\u03ac\u03c1\u03c0\u1ff3 \u03ba\u03b1\u1f76 \u1f04\u03bb\u03bb\u03bf\u03b9 \u03b8' \u1f00\u03c0\u1f78\n  \u03a6\u03b9\u03bb\u03b1\u03b4\u03b5\u03bb\u03c6\u03b5\u03af\u03b1\u03c2 \u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03bf\u1fe6\u03c3\u03b9\u03bd \u1f10\u03bd \u03a3\u03bc\u03cd\u03c1\u03bd\u1fc3\u00b7 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03a0\u03b5\u03c1\u03b3\u03ac\u03bc\u1ff3 \u03b4\u1f72 \u1f15\u03c4\u03b5\u03c1\u03bf\u03b9, \u1f10\u03bd \u03bf\u1f37\u03c2\n  \u1f27\u03bd \u03ba\u03b1\u1f76 \u03a0\u03b1\u03c0\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f04\u03bb\u03bb\u03bf\u03b9 \u03c0\u03bf\u03bb\u03bb\u03bf\u03af, \u1f67\u03bd \u03ba\u03b1\u1f76 \u1f14\u03b3\u03b3\u03c1\u03b1\u03c6\u03b1 \u03c6\u03ad\u03c1\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c4\u1f70 \u03bc\u03b1\u03c1\u03c4\u03cd\u03c1\u03b9\u03b1.\n  See also the Syrian epitome of Euseb. _Chron._ (II. p. 216 ed. Sch\u00f6ne)\n  'Cum persecutio in Asia esset, Polycarpos martyrium subiit et Papias,\n  quorum martyria in libro (scripta) extant,' but the Armenian version\n  of the _Chronicon_ mentions only Polycarp, while Jerome says\n  'Polycarpus et Pionius fecere martyrium.' In his history (iv. 15)\n  Eusebius, after quoting the _Martyrdom of Polycarp_ at length, adds \u1f10\u03bd\n  \u03c4\u1fc7 \u03b1\u1f50\u03c4\u1fc7 \u03b4\u1f72 \u03c0\u03b5\u03c1\u1f76 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b3\u03c1\u03b1\u03c6\u1fc7 \u03ba\u03b1\u1f76 _\u1f04\u03bb\u03bb\u03b1 \u03bc\u03b1\u03c1\u03c4\u03cd\u03c1\u03b9\u03b1 \u03c3\u03c5\u03bd\u1fc6\u03c0\u03c4\u03bf_ ... \u03bc\u03b5\u03b8'\n  \u1f67\u03bd \u03ba\u03b1\u1f76 \u039c\u03b7\u03c4\u03c1\u03cc\u03b4\u03c9\u03c1\u03bf\u03c2 ... \u1f00\u03bd\u03ae\u03c1\u03b7\u03c4\u03b1\u03b9\u00b7 \u03c4\u1ff6\u03bd \u03b3\u03b5 \u03bc\u1f74\u03bd \u03c4\u1f78\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03b2\u03bf\u03ae\u03c4\u03c9\u03bd \u03bc\u03b1\u03c1\u03c4\u03cd\u03c1\u03c9\u03bd\n  \u03b5\u1f37\u03c2 \u03c4\u03b9\u03c2 \u1f10\u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03b5\u03c4\u03bf _\u03a0\u03b9\u03cc\u03bd\u03b9\u03bf\u03c2_ ... \u1f11\u03be\u1fc6\u03c2 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u1f04\u03bb\u03bb\u03c9\u03bd \u1f10\u03bd \u03a0\u03b5\u03c1\u03b3\u03ac\u03bc\u1ff3\n  \u03c0\u03cc\u03bb\u03b5\u03b9 \u03c4\u1fc6\u03c2 \u1f08\u03c3\u03af\u03b1\u03c2 \u1f51\u03c0\u03bf\u03bc\u03bd\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03b5\u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03b7\u03ba\u03cc\u03c4\u03c9\u03bd \u03c6\u03ad\u03c1\u03b5\u03c4\u03b1\u03b9, \u039a\u03ac\u03c1\u03c0\u03bf\u03c5 \u03ba\u03b1\u1f76\n  _\u03a0\u03b1\u03c0\u03cd\u03bb\u03bf\u03c5_ \u03ba\u03b1\u1f76 \u03b3\u03c5\u03bd\u03b1\u03b9\u03ba\u1f78\u03c2 \u1f08\u03b3\u03b1\u03b8\u03bf\u03bd\u03af\u03ba\u03b7\u03c2 \u03ba.\u03c4.\u03bb. He here falls into the\n  serious error of imagining that Metrodorus, Pionius, Carpus, Papylus,\n  and the others were martyred under M. Aurelius, whereas we know from\n  their extant Acts that they suffered in the Decian persecution. For\n  the martyrdoms of Pionius and Metrodorus see _Act. SS. Bolland._ Feb.\n  1; for those of Carpus, Papylus, and Agathonica, _ib._ April 13. The\n  Acts of the former, which are included in Ruinart (_Act. Sinc. Mart._\n  p. 120 sq., 1689) are apparently the same which were seen by Eusebius.\n  Those of the latter are a late compilation of the Metaphrast, but were\n  probably founded on the earlier document. At all events the tradition\n  of the persecution in which they suffered could hardly have been\n  perverted or lost. Eusebius seems to have found their Acts bound up in\n  the same volume with those of Polycarp, and without reading them\n  through, to have drawn the hasty inference that they suffered at the\n  same time. But notwithstanding the error, or perhaps owing to it, this\n  passage in the Ecclesiastical History, by a confusion of the names\n  Papias and Papylus, seems to have given rise to the statement\n  respecting Papias in the Chronicon Paschale and in the Syrian epitome,\n  as it obviously has misled Jerome respecting Pionius. If so, the\n  martyrdom of Papias is a fiction, and he may have died a natural death\n  at an earlier date; so that the not very serious difficulty of his\n  longevity will disappear. The time of Polycarp's martyrdom is fixed by\n  various data as Easter A.D. 166 (see Clinton's _Fast. Rom._ I. p.\n  157).\n\nFootnote 162:\n\n  _H E._ iii. 39 \u03c3\u03c6\u03cc\u03b4\u03c1\u03b1 \u03c3\u03bc\u03b9\u03ba\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03bd\u03bf\u1fe6\u03bd. In another passage (iii. 36),\n  as commonly read, Eusebius makes partial amends to Papias by calling\n  him \u1f00\u03bd\u1f74\u03c1 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f45\u03c4\u03b9 \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03bb\u03bf\u03b3\u03b9\u03ce\u03c4\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u03c1\u03b1\u03c6\u1fc6\u03c2 \u03b5\u1f30\u03b4\u03ae\u03bc\u03c9\u03bd, but\n  this passage is found to be a spurious interpolation (see\n  _Contemporary Review_ l.c. p. 12), and was probably added by some one\n  who was acquainted with the work of Papias and desired to do him\n  justice.\n\nFootnote 163:\n\n  Iren. v. 33. 3, 4.\n\nMuch has been written respecting the relation of this writer to the\nCanonical Gospels, but the discussion has no very direct bearing on our\nspecial subject, and may be dismissed here[164]. One question however,\nwhich has a real importance as affecting the progress of the Gospel in\nthese parts, has been raised by modern criticism and must not be passed\nover in silence.\n\nFootnote 164:\n\n  See on this subject Westcott _Canon_ p. 64 sq.; _Contemporary Review_\n  l.c. p. 12 sq.\n\n[Sidenote: A modern hypothesis respecting Christianity in Asia Minor\n           stated and discussed.]\n\nIt has been supposed that there was an entire dislocation and\ndiscontinuity in the history of Christianity in Asia Minor at a certain\nepoch; that the Apostle of the Gentiles was ignored and his teaching\nrepudiated, if not anathematized; and that on its ruins was erected the\nstandard of Judaism, around which with a marvellous unanimity deserters\nfrom the Pauline Gospel rallied. Of this retrograde faith St John is\nsupposed to have been the great champion, and Papias a typical and\nimportant representative[165].\n\nFootnote 165:\n\n  The theory of the T\u00fcbingen school may be studied in Baur's\n  _Christliche Kirche der drei ersten Jahrhunderte_ or in Schwegler's\n  _Nachapostolisches Zeitalter_. It has been reproduced (at least as far\n  as regards the Asiatic Churches) by Renan _Saint Paul_ p. 366 sq.\n\nThe subject, as a whole, is too wide for a full investigation here. I\nmust content myself with occupying a limited area, showing not only the\nhistorical baselessness, but the strong inherent improbability of the\ntheory, as applied to Hierapolis and the neighbouring churches. As this\ndistrict is its chief strong-hold, a repulse at this point must involve\nits ultimate defeat along the whole line.\n\n[Sidenote: The position of St John]\n\nOf St John himself I have already spoken[166]. It has been shown that\nhis language addressed to these Churches is not only not opposed to St\nPaul's teaching, but presents remarkable coincidences with it. So far at\nleast the theory finds no support; and, when from St John we turn to\nPapias, the case is not different. [Sidenote: and of Papias.]The\nadvocates of the hypothesis in question lay the chief stress of their\nargument on the silence of Papias, or rather of Eusebius. Eusebius\nquotes a passage from Papias, in which the bishop of Hierapolis mentions\ncollecting from trustworthy sources the sayings of certain Apostles and\nearly disciples; but St Paul is not named among them. He also gives\nshort extracts from Papias referring to the Gospels of St Matthew and St\nMark, and mentions that this writer made use of the first Epistle of St\nJohn and the first Epistle of St Peter; but here again there is no\nallusion to St Paul's writings. Whether referring to the personal\ntestimony or to the Canonical writings of the Apostles, Papias, we are\nreminded, is equally silent about St Paul.\n\nFootnote 166:\n\n  See above p. 41 sq.\n\nOn both these points a satisfactory answer can be given; but the two\ncases are essentially different, and must be considered apart.\n\n[Sidenote: 1. The traditions collected by Papias.]\n\n(1) The range of _personal testimony_ which Papias would be able to\ncollect depended on his opportunities. Before he had grown up to\nmanhood, the personal reminiscences of St Paul would have almost died\nout. The Apostle of the Gentiles had not resided more than three years\neven at Ephesus, and seems to have paid only one brief visit to the\nvalley of the Lycus, even if he visited it at all. Such recollections of\nSt Paul as might once have lingered here would certainly be overshadowed\nby and forgotten in the later sojourn of St John, which, beginning where\nthey ceased, extended over more than a quarter of a century. To St John,\nand to those personal disciples of Christ who surrounded him, Papias and\nhis contemporaries would naturally and almost inevitably look for the\ntraditions which they so eagerly collected. This is the case with the\nleading representative of the Asiatic school in the next generation,\nIren\u00e6us, whose traditions are almost wholly derived from St John and his\ncompanions, while at the same time he evinces an entire sympathy with\nthe work and teaching of St Paul. But indeed, even if it had been\notherwise, the object which Papias had directly in view did not suggest\nany appeal to St Paul's authority. He was writing an 'Exposition of the\nOracles of the Lord,' and he sought to supplement and interpret these by\ntraditions of our Lord's life, such as eyewitnesses only could give. St\nPaul could have no place among those personal disciples of Christ, of\nwhom alone he is speaking in this preface to his work, which Eusebius\nquotes.\n\n[Sidenote: 2. His references to the Canonical writings.]\n\n(2) But, though we have no right to expect any mention of St Paul where\nthe appeal is to personal testimony, yet with quotations from or\nreferences to the _Canonical writings_ the case, it may be argued, is\ndifferent. Here at all events we might look for some recognition of St\nPaul. To this argument it would perhaps be a sufficient reply, that St\nPaul's Epistles do not furnish any matter which must necessarily have\nbeen introduced into a work such as Papias composed. But the complete\nand decisive answer is this; that the silence of Eusebius, so far from\ncarrying with it the silence of Papias, does not [Sidenote: No weight to\nbe attached to the silence of Eusebius.] even afford a presumption in\nthis direction. Papias may have quoted St Paul again and again, and yet\nEusebius would see no reason to chronicle the fact. His usage in other\ncases is decisive on this point. The Epistle of Polycarp which was read\nby Eusebius is the same which we still possess. Not only does it teem\nwith the most obvious quotations from St Paul, but in one passage it\ndirectly mentions his writing to the Philippians[167]. Yet the\nhistorian, describing its relation to the Canonical Scriptures, contents\nhimself with saying that it 'employs some testimonies from the former\nEpistle of Peter[168].' Exactly similar is his language respecting\nIren\u00e6us also. Iren\u00e6us, as is well known, cites by name almost every one\nof St Paul's Epistles; yet the description which Eusebius gives under\nthis same head, after quoting this writer's notices respecting the\nhistory of the Gospels and the Apocalypse, is that 'he mentions also the\nfirst Epistle of John, alleging very many testimonies from it, and in\nlike manner also the former Epistle of Peter[169].' There is every\nreason therefore to suppose that Eusebius would deal with Papias as he\nhas dealt with Polycarp and Iren\u00e6us, and that, unless Papias had\nintroduced some curious fact relating to St Paul, it would not have\noccurred to him to record mere quotations from or references to this\nApostle's letters. It may be supposed that Eusebius records with a fair\namount of attention references to the Catholic Epistles in early\nwriters, because the limits of the Canon in this part were not\naccurately fixed. On the other hand the Epistles of St Paul were\nuniversally received and therefore did not need to be accredited by any\nsuch testimony. But whatever may be the explanation, the fact is patent,\nand it furnishes a complete answer to the argument drawn from his\nsilence in the case of Papias[170].\n\nFootnote 167:\n\n  \u00a7 3.\n\nFootnote 168:\n\n  _H.E._ iv. 14 \u1f41 \u03b3\u03ad \u03c4\u03bf\u03b9 \u03a0\u03bf\u03bb\u03cd\u03ba\u03b1\u03c1\u03c0\u03bf\u03c2 \u1f10\u03bd \u03c4\u1fc7 \u03b4\u03b7\u03bb\u03c9\u03b8\u03b5\u03af\u03c3\u1fc3 \u03c0\u03c1\u1f78\u03c2 \u03a6\u03b9\u03bb\u03b9\u03c0\u03c0\u03b7\u03c3\u03af\u03bf\u03c5\u03c2\n  \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b3\u03c1\u03b1\u03c6\u1fc7 \u03c6\u03b5\u03c1\u03bf\u03bc\u03ad\u03bd\u1fc3 \u03b5\u1f30\u03c2 \u03b4\u03b5\u1fe6\u03c1\u03bf \u03ba\u03ad\u03c7\u03c1\u03b7\u03c4\u03b1\u03af \u03c4\u03b9\u03c3\u03b9 \u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03af\u03b1\u03b9\u03c2 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03a0\u03ad\u03c4\u03c1\u03bf\u03c5\n  \u03c0\u03c1\u03bf\u03c4\u03ad\u03c1\u03b1\u03c2 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1fc6\u03c2. This is all that Eusebius says with reference to\n  Polycarp's knowledge of the Canonical writings. It so happens that in\n  an earlier passage (iii. 36) he has given an extract from Polycarp, in\n  which St Paul's name is mentioned; but the quotation is brought to\n  illustrate the life of Ignatius, and the mention of the Apostle there\n  is purely accidental.\n\nFootnote 169:\n\n  _H.E._ v. 8 \u03bc\u03ad\u03bc\u03bd\u03b7\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u1f38\u03c9\u03ac\u03bd\u03bd\u03bf\u03c5 \u03c0\u03c1\u03ce\u03c4\u03b7\u03c2 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1fc6\u03c2, \u03bc\u03b1\u03c1\u03c4\u03cd\u03c1\u03b9\u03b1 \u1f10\u03be\n  \u03b1\u1f50\u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b5\u1fd6\u03c3\u03c4\u03b1 \u03b5\u1f30\u03c3\u03c6\u03ad\u03c1\u03c9\u03bd, \u1f41\u03bc\u03bf\u03af\u03c9\u03c2 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03a0\u03ad\u03c4\u03c1\u03bf\u03c5 \u03c0\u03c1\u03bf\u03c4\u03ad\u03c1\u03b1\u03c2.\n\nFootnote 170:\n\n  It is necessary to press this argument, because though it has never\n  been answered and (so far as I can see) is quite unanswerable, yet\n  thoughtful men, who have no sympathy with the T\u00fcbingen views of early\n  Christian history, still continue to argue from the silence of\n  Eusebius, as though it had some real significance. To illustrate the\n  omissions of Eusebius I have given only the instances of Polycarp and\n  Iren\u00e6us, because they are historically connected with Papias; but his\n  silence is even more remarkable in other cases. Thus, when speaking of\n  the epistle of the Roman Clement (_H.E._ iii. 38), he alludes to the\n  coincidences with the Epistle to the Hebrews, but omits to mention the\n  direct references to St Paul's First Epistle to the Corinthians which\n  is referred to by name, and is even silent about the numerous and\n  patent quotations from the Epistle of St James.\n\n[Sidenote: The views of Papias inferred from his associates.]\n\nBut, if the assumption has been proved to be baseless, have we any\ngrounds for saying that it is also highly improbable? Here it seems fair\nto argue from the well-known to the unknown. Of the opinions of Papias\nrespecting St Paul we know absolutely nothing; of the opinions of\nPolycarp and Iren\u00e6us ample evidence lies before us. _Noscitur a sociis_\nis a sound maxim to apply in such a case. Papias was a companion of\nPolycarp, and he is quoted with deference by Iren\u00e6us[171]. Is it\nprobable that his opinions should be diametrically opposed to those of\nhis friend and contemporary on a cardinal point affecting the very\nconception of Christianity (for the rejection of St Paul must be\nconsidered in this light)? or that this vital heterodoxy, if it existed,\nshould have escaped an intelligent critic of the next generation who had\nthe five books of his work before him, who himself had passed his early\nlife in Asia Minor, and who yet appeals to Papias as preserving the\ndoctrinal tradition which had been handed down from the Apostles\nthemselves to his own time? I say nothing of Eusebius himself, who, with\na distinct prejudice against Papias, accuses him of no worse heresy in\nhis writings than entertaining millennarian views.\n\nFootnote 171:\n\n  Iren. _H\u00e6r._ v. 33. 4.\n\n[Sidenote: Millennarian views consistent with the recognition of St\n           Paul.]\n\nIt may indeed be confessed that a man like Papias, whose natural bent,\nassisted by his Phrygian education, was towards sensuous views of\nreligion, would not be likely to appreciate the essentially spiritual\nteaching of St Paul; but this proves nothing. The difference between\nunconscious want of sympathy and conscious rejection is all important\nfor the matter in hand. The same charge might be brought against\nnumberless theologians, whether in the middle ages or in more modern\ntimes, into whose minds it never entered to question the authority of\nthe Apostle and who quote his writings with the utmost reverence.\nNeither in the primitive days of Christianity nor in its later stages\nhas the profession of Chiliastic views been found inconsistent with the\nfullest recognition of St Paul's Apostolic claims. In the early Church\nIren\u00e6us and Tertullian are notable instances of this combination; and in\nour own age and country a tendency to millennarian speculations has been\ncommonly associated with the staunchest adherence to the fundamental\ndoctrines of St Paul.\n\n[Sidenote: ABERCIUS probably his successor.]\n\nAs the successor of Papias and the predecessor of Claudius Apollinaris\nin the see of Hierapolis, we may perhaps name ABERCIUS or AVIRCIUS[172].\nHis legendary Acts assign his episcopate to the reign of Marcus\nAurelius; and, though they are disfigured by extravagant fictions, yet\nthe date may perhaps be accepted, as it seems to be confirmed by other\nevidence. An inscription on his tombstone recorded how he had paid one\n[Sidenote: His journeys.] visit to the city of Rome, and another to the\nbanks of the Euphrates. These long journeys are not without parallels in\nthe lives of contemporary bishops. Polycarp of Smyrna visited Rome,\nhoping to adjust the Paschal controversy; Melito of Sardis went as far\nas Palestine, desiring to ascertain on the spot the facts relating to\nthe Canon of the Old Testament Scriptures. These or similar motives may\nhave influenced Abercius to undertake his distant journeys. If we may\nassume the identification of this bishop with one Avircius Marcellus who\nis mentioned in a contemporary document, he took an active interest in\nthe Montanist controversy, as from his position he was likely to do.\n\nFootnote 172:\n\n  The life of this Abercius is printed in the Bollandist _Acta\n  Sanctorum_ Oct. 22. It may safely be pronounced spurious. Among other\n  incidents, the saint goes to Rome and casts out a demon from Lucilla,\n  the daughter of M. Aurelius and Faustina, at the same time compelling\n  the demon to take up an altar from Rome and transport it through the\n  air to Hierapolis. But these Acts, though legendary themselves,\n  contain an epitaph which has the ring of genuineness and which seems\n  to have suggested the story to the pious forger who invented the Acts.\n  This very interesting memorial is given and discussed at length by\n  Pitra, _Spicil. Solesm._ III. p. 532 sq. It is inscribed by one\n  Abercius of Hierapolis on his tomb, which he erected during his\n  life-time. He declares himself a disciple of the good shepherd, who\n  taught him trustworthy writings (\u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c4\u03b1 \u03c0\u03b9\u03c3\u03c4\u03ac) and sent him to visit\n  queenly Rome, where he saw a people sealed with the bright seal [of\n  baptism]. He recounts also a journey to Syria and the East, when he\n  crossed the Euphrates. He says that faith served up to him as a\n  banquet the \u03b9\u03c7\u03b8\u03c5\u03c2 from the fountain, giving him bread and wine. He\n  states that he has reached his 72nd year. And he closes by threatening\n  with severe penalties those who disturb his tomb. The resemblance of\n  this inscription to others found _in situ_ in the cemetery at\n  Hierapolis, after allowance made for the Christian element, is very\n  striking. The commencement \u1f18\u03ba\u03bb\u03b5\u03ba\u03c4\u1fc6\u03c2 \u03c0\u03cc\u03bb\u03b5\u03c9\u03c2 closely resembles the form\n  of another Hierapolitan inscription, Boeckh _Corp. Inscr._ 3906; the\n  enumeration of foreign tours has a counterpart in the monument of one\n  Flavius Zeuxis which states that the deceased had made 72 voyages\n  round the promontory of Malea to Italy (_ib._ 3920); and lastly, the\n  prohibition against putting another grave upon his, and the imposition\n  of fines to be paid to the treasury and the city if this injunction is\n  violated, are echos of language which occurs again and again on\n  tombstones in this city (_ib._ 3915, 3916, 3922, 3923, etc.). Out of\n  this epitaph, which he found probably at Hierapolis, and which, as he\n  himself tells us (\u00a7 41), was in a much mutilated condition, the\n  legend-writer apparently created his story, interpreting the queen, by\n  which Abercius himself probably meant the city of Rome, to be the\n  empress Faustina, with whom the saint is represented as having an\n  interview, M. Aurelius himself being absent at the time on his German\n  campaign. This view, that the epitaph is genuine and gave rise to the\n  Acts, is also maintained by Garrucci (_Civilt\u00e0 Cattolica_ 1856, I. p.\n  683, II. p. 84, quoted in the _Acta Sanct._ l.c.), whose criticisms\n  however are not always sound; and indeed as a whole it bears every\n  mark of authenticity, though possibly it may contain some\n  interpolations, which its mutilated condition would encourage.\n\n  The inscription itself however does not tell us what office Abercius\n  held or when he lived. There was a person of this name bishop of\n  Hierapolis present at the Council of Chalcedon A.D. 451 (Labb. _Conc._\n  IV. 862, 1204, 1341, 1392, 1496, 1744, ed. Coleti). But a chief pastor\n  of the Church at this late date would have declared his office\n  plainly; and the inscription points to a more primitive age, for the\n  expressions are archaic and the writer seems to veil his profession of\n  Christianity under language studiously obscure. The open profession of\n  Christianity on inscriptions occurs at an earlier date in these parts\n  than elsewhere. Already the word \u03c7\u03c1\u03b9\u03f2\u03c4\u03b9\u03b1\u03bd\u03bf\u03f2 or \u03c7\u03c1\u03b7\u03f2\u03c4\u03b9\u03b1\u03bd\u03bf\u03f2 is found on\n  tombstones of the third century; Boeckh _Corp. Inscr._ 3857 g, 3857 p,\n  3865 l; see Renan _Saint Paul_ p. 363. Thus we are entirely at fault\n  unless we accept the statement in the Acts.\n\n  And it is not unreasonable to suppose that, so far as regards the date\n  and office of Abercius, the writer of these Acts followed some\n  adequate historical tradition. Nor indeed is his statement altogether\n  without confirmation. We have evidence that a person bearing this name\n  lived in these parts of Asia Minor, somewhere about this time. An\n  unknown writer of a polemical tract against Montanism dedicates his\n  work to one Avircius Marcellus, at whose instigation it was written.\n  Eusebius (_H.E._ v. 16), who is our authority for this fact, relates\n  that Montanism found a determined and formidable opponent in\n  Apollinaris at Hierapolis and 'several other learned men of that day\n  with him,' who left large materials for a history of the movement. He\n  then goes on to say; \u1f00\u03c1\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b3\u03bf\u1fe6\u03bd \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03c4' \u03b1\u1f50\u03c4\u1ff6\u03bd \u03b3\u03c1\u03b1\u03c6\u1fc6\u03c2 _\u03c4\u1ff6\u03bd\n  \u03b5\u1f30\u03c1\u03b7\u03bc\u03ad\u03bd\u03c9\u03bd \u03b4\u03ae \u03c4\u03b9\u03c2_ ... \u03c0\u03c1\u03bf\u03bf\u03b9\u03bc\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 ... \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03c4\u1f78\u03bd \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd\u00b7 \u1f18\u03ba\n  \u03c0\u03bb\u03b5\u03af\u03c3\u03c4\u03bf\u03c5 \u1f45\u03c3\u03bf\u03c5 \u03ba\u03b1\u1f76 \u1f31\u03ba\u03b1\u03bd\u03c9\u03c4\u03ac\u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5, \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1f72 \u1f08\u03bf\u03c5\u03af\u03c1\u03ba\u03b9\u03b5 \u039c\u03ac\u03c1\u03ba\u03b5\u03bb\u03bb\u03b5,\n  \u1f10\u03c0\u03b9\u03c4\u03b1\u03c7\u03b8\u03b5\u1f76\u03c2 \u1f51\u03c0\u1f78 \u03c3\u03bf\u1fe6 \u03c3\u03c5\u03b3\u03b3\u03c1\u03ac\u03c8\u03b1\u03b9 \u03c4\u03b9\u03bd\u1f70 \u03bb\u03cc\u03b3\u03bf\u03bd \u03ba.\u03c4.\u03bb., i.e. 'One of the\n  aforesaid writers at the commencement of his treatise against them\n  (the Montanists) etc.' May not the person here addressed be the\n  Abercius of the epitaph?\n\n  But if so, who is the writer that addresses him, and when did he live?\n  Some MSS omit \u03b4\u03ae \u03c4\u03b9\u03c2, and others substitute \u1f24\u03b4\u03b7, thus making\n  Apollinaris himself the writer. But the words seem certainly to have\n  been part of the original text, as the sense requires them; for if\n  they are omitted, \u03c4\u1ff6\u03bd \u03b5\u1f30\u03c1\u03b7\u03bc\u03ad\u03bd\u03c9\u03bd must be connected with \u03ba\u03b1\u03c4' \u03b1\u1f50\u03c4\u1ff6\u03bd,\n  where it is not wanted. Thus Eusebius quotes the writer anonymously;\n  and those who assign the treatise to Apollinaris cannot plead the\n  authority of the original text of the historian himself.\n\n  But after all may it not have been written by Apollinaris, though\n  Eusebius was uncertain about the authorship? He quotes in succession\n  three \u03c3\u03c5\u03b3\u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c4\u03b1 or treatises, speaking of them as though they\n  emanated from the same author. The first of these, from which the\n  address to Avircius Marcellus is quoted, might very well have been\n  composed soon after the Montanist controversy broke out (as Eusebius\n  himself elsewhere states was the case with the work of Apollinaris,\n  iv. 27 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1fc6\u03c2 \u03c4\u1ff6\u03bd \u03a6\u03c1\u03c5\u03b3\u1ff6\u03bd \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03c2 ... \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f10\u03ba\u03c6\u03cd\u03b5\u03b9\u03bd \u1f00\u03c1\u03c7\u03bf\u03bc\u03ad\u03bd\u03b7\u03c2); but\n  the second and third distinctly state that they were written some time\n  after the death of Montanus. May not Eusebius have had before him a\n  volume containing a collection of tracts against Montanism 'by\n  Claudius Apollinaris and others,' in which the authorship of the\n  several tracts was not distinctly marked? This hypothesis would\n  explain the words with which he prefaces his extracts, and would also\n  account for his vague manner of quotation. It would also explain the\n  omission of \u03b4\u03ae \u03c4\u03b9\u03c2 in some texts (the ancient Syriac version boldly\n  substitutes the name of Apollinaris), and would explain how Rufinus,\n  Nicephorus, and others, who might have had independent information,\n  ascribed the treatise to this father. I have already pointed out how\n  Eusebius was led into a similar error of connecting together several\n  martyrologies and treating them as contemporaneous, because they were\n  collected in the same volume (p. 48, note 161). Elsewhere too I have\n  endeavoured to show that he mistook the authorship of a tract which\n  was bound up with others, owing to the absence of a title (_Caius or\n  Hippolytus?_ in the _Journal of Philology_ I. p. 98 sq.).\n\n  On this hypothesis, Claudius Apollinaris would very probably be the\n  author of the first of these treatises. If so, it would appear to have\n  been written while he was still a presbyter, at the instigation of his\n  bishop Avircius Marcellus whom he succeeded not long after in the see\n  of Hierapolis.\n\n  If on the other hand Eusebius has correctly assigned the first\n  treatise to the same writer as the second and third, who must have\n  written after the beginning of the third century, Avircius Marcellus\n  to whom it is addressed cannot have held the see of Hierapolis during\n  the reign of M. Aurelius (A.D. 161\u2013180); and, if he was ever bishop of\n  this city, must have been a successor, not a predecessor, of Claudius\n  Apollinaris. In this case we have the alternative of abandoning the\n  identification of this Avircius with the Hierapolitan bishop of the\n  same name, or of rejecting the statement of the Acts which places his\n  episcopate in this reign.\n\n  The occurrence of the name Abercius in the later history of the see of\n  Hierapolis (see p. 55) is no argument against the existence of this\n  earlier bishop. It was no uncommon practice for the later occupants of\n  sees to assume the name of some famous predecessor who lived in\n  primitive or early times. The case of Ignatius at Antioch is only one\n  of several examples which might be produced.\n\n  There is some ground for supposing that, like Papias and Apollinaris,\n  Abercius earned a place in literary history. Baronio had in his hands\n  an epistle to M. Aurelius, purporting to have been written by this\n  Abercius, which he obviously considered genuine and which he describes\n  as 'apostolicum redolens spiritum,' promising to publish it in his\n  Annals (_Martyr. Rom._ Oct. 22). To his great grief however he\n  afterwards lost it ('doluimus vehementer e manibus nostris elapsam\n  nescio quomodo'), and was therefore unable to fulfil his promise\n  (_Annal._ s. a. 163, n. 15). A \u03b2\u03af\u03b2\u03bb\u03bf\u03c2 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2 by Abercius is\n  mentioned in the Acts (\u00a7 39); but this, if it ever existed, was\n  doubtless spurious.\n\n[Sidenote: CLAUDIUS APOLLINARIS bishop of Hierapolis.]\n\nThe literary character of the see of Hierapolis, which had been\ninaugurated by Papias, was ably sustained by CLAUDIUS APOLLINARIS. His\nsurname, which seems to have been common in these parts[173], may have\nbeen derived from the patron deity of Hierapolis[174] and suggests a\nGentile origin. His intimate acquaintance with heathen literature, which\nis mentioned by more than one ancient writer, points in the same\ndirection. During the reign of M. Aurelius he had already made himself a\nname by his writings, and seems to have been promoted to the see of\nHierapolis before the death of that emperor[175].\n\nFootnote 173:\n\n  Some of the family, as we may infer from the monuments, held a high\n  position in another Phrygian town. On a tablet at \u00c6zani, on which is\n  inscribed a letter from the emperor Septimius Severus in reply to the\n  congratulations of the people at the elevation of Caracalla to the\n  rank of Augustus (A.D. 198), we find the name of ==\u03ba\u03bb\u03b1\u03c5\u03b4\u03b9\u03bf\u03f2 .\n  \u03b1\u03c0\u03bf\u03bb\u03bb\u03b9\u03bd\u03b1\u03c1\u03b9\u03bf\u03f2 . \u03b1\u03c5\u03c1\u03b7\u03bb\u03b9\u03b1\u03bd\u03bf\u03f2==, Boeckh 3837 (see III. p. 1066 add.). In\n  another inscription at the same place, the same or another member of\n  the family is commemorated as holding the office of pr\u00e6tor for the\n  second time, ==\u03f2\u03c4\u03c1\u03b1\u03c4\u03b7\u03b3\u03bf\u03c5\u03bd\u03c4\u03bf\u03f2 . \u03c4\u03bf . \u03b2 . \u03ba\u03bb . \u03b1\u03c0\u03bf\u03bb\u03bb\u03b9\u03bd\u03b1\u03c1\u03b9\u03bf\u03c5==; Boeckh\n  3840, _ib._ p. 1067. See also the inscriptions 3842 c, 3846 z (_ib._\n  pp. 1069, 1078) at the same place, where again the name Apollinarius\n  occurs. It is found also at Appia no. 3857 b (_ib._ p. 1086). At an\n  earlier date one Claudius Apollinaris appears in command of the Roman\n  fleet at Misenum (Tac. _Hist._ iii. 57, 76, 77). The name occurs also\n  at Hierapolis itself, Boeckh, no. 3915. ==\u03c0 . \u03b1\u03b9\u03bb\u03b9\u03bf\u03f2 . \u03c0 . \u03b1\u03b9\u03bb\u03b9\u03bf\u03c5 .\n  \u03b1\u03c0\u03bf\u03bb\u03bb\u03b9\u03bd\u03b1\u03c1\u03b9\u03bf\u03c5 . \u03b9\u03bf\u03c5\u03bb\u03b9\u03b1\u03bd\u03bf[\u03c5] . \u03c5\u03b9\u03bf\u03f2 . \u03f2\u03b5[...] . \u03b1\u03c0\u03bf\u03bb\u03bb\u03b9\u03bd\u03b1\u03c1\u03b9\u03f2 . \u03bc\u03b1\u03ba\u03b5\u03b4\u03c9\u03bd\n  .== \u03ba.\u03c4.\u03bb., which shows that both the forms, _Apollinaris_ and\n  _Apollinarius_, by which the bishop of Hierapolis is designated, are\n  legitimate. The former however is the correct Latin form, the latter\n  being the Greek adaptation.\n\n  More than a generation later than our Apollinaris, Origen in his\n  letter to Africanus (_Op._ I. 30, Delarue) sends greeting to a bishop\n  bearing this name (\u03c4\u1f78\u03bd \u03ba\u03b1\u03bb\u1f78\u03bd \u1f21\u03bc\u1ff6\u03bd \u03c0\u03ac\u03c0\u03b1\u03bd \u1f08\u03c0\u03bf\u03bb\u03b9\u03bd\u03ac\u03c1\u03b9\u03bf\u03bd), of whom nothing\n  more is known.\n\nFootnote 174:\n\n  Apollo Archegetes; see above p. 12, note 42.\n\nFootnote 175:\n\n  Euseb. _H.E._ iv. 26, _Chron._ s. a. 171, 172, 'Apollinaris Asianus,\n  Hierapolitanus episcopus, insignis habetur.'\n\n[Sidenote: His literary works.]\n\nOf his works, which were very numerous, only a few scanty fragments have\nsurvived[176]. The imperfect lists however, which have reached us, bear\nample testimony both to the literary activity of the man, and to the\nprominence of the Church, over which he presided, in the great\ntheological and ecclesiastical controversies of the age.\n\nFootnote 176:\n\n  Collected in Routh's _Reliqui\u00e6 Sacr\u00e6_ I. p. 159 sq., and quite\n  recently in Otto's _Corp. Apol. Christ._ IX. p. 479 sq.\n\n[Sidenote: He takes part in the two chief controversies of the day.]\n\nThe two questions, which especially agitated the Churches of Asia Minor\nduring the last thirty years of the first century, were the celebration\nof the Easter festival and the pretensions of the Montanist prophets. In\nboth disputes Claudius Apollinaris took an active and conspicuous part.\n\n[Sidenote: 1. The Paschal question.]\n\n1. The Paschal controversy, after smouldering long both here and\nelsewhere, first burst into flames in the neighbouring Church of\nLaodicea[178]. An able bishop of Hierapolis therefore must necessarily\nhave been involved in the dispute, even if he had been desirous of\navoiding it. What side Apollinaris took in the controversy the extant\nfragments of his work do not by themselves enable us to decide; for they\ndeal merely with a subsidiary question which does not seriously affect\nthe main issue[179]. But we can hardly doubt that with Polycarp of\nSmyrna and Melito of Sardis and Polycrates of Ephesus he defended the\npractice which was universal in Asia[180], observing the Paschal\nanniversary on the 14th Nisan whether it fell on a Friday or not, and\ninvoking the authority of St John at Ephesus, and of St Philip at his\nown Hierapolis[181], against the divergent usage of Alexandria and\nPalestine and the West.\n\nFootnote 178:\n\n  See below, p. 63.\n\nFootnote 179:\n\n  The main point at issue was whether the exact day of the month should\n  be observed, as the Quartodecimans maintained, irrespective of the day\n  of the week. The fragment of Apollinaris (preserved in the _Chron.\n  Pasch._ p. 13) relates to a discrepancy which some had found in the\n  accounts of St Matthew and St John.\n\nFootnote 180:\n\n  Eusebius represents the dioceses of 'Asia' and the neighbourhood, as\n  absolutely unanimous; _H.E._ v. 23 \u03c4\u1fc6\u03c2 \u0391\u03c3\u03af\u03b1\u03c2 \u1f01\u03c0\u03ac\u03c3\u03b7\u03c2 \u03b1\u1f31 \u03c0\u03b1\u03c1\u03bf\u03b9\u03ba\u03af\u03b1\u03b9, v.\n  24 \u03c4\u1fc6\u03c2 \u0391\u03c3\u03af\u03b1\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u1f05\u03bc\u03b1 \u03c4\u03b1\u1fd6\u03c2 \u1f41\u03bc\u03cc\u03c1\u03bf\u03b9\u03c2 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03b9\u03c2 \u03c4\u1f70\u03c2 \u03c0\u03b1\u03c1\u03bf\u03b9\u03ba\u03af\u03b1\u03c2. 'Asia'\n  includes all this district, as appears from Polycrates, _ib._\n\nFootnote 181:\n\n  See Polycrates of Ephesus in Euseb. _H.E._ v. 24.\n\n[Sidenote: 2. Montanism.]\n\n2. His writings on the Montanist controversy were still more famous, and\nare recommended as an authority on the subject by Serapion of Antioch a\nfew years after the author's death[182]. Though later than many of his\nworks[183], they were written soon after Montanus had divulged the\nextravagance of his pretensions and before Montanism had attained its\ncomplete development. If a later notice may be trusted, Apollinaris was\nnot satisfied with attacking Montanism in writing, but summoned at\nHierapolis a council of twenty-six bishops besides himself, where this\nheresy was condemned and sentence of excommunication pronounced against\nMontanus together with his adherent the pretended prophetess\nMaximilla[184].\n\nFootnote 182:\n\n  In Euseb. _H.E._ v. 19.\n\nFootnote 183:\n\n  Eusebius (_H.E._ iv. 27) at the close of his list of the works of\n  Apollinarius gives \u03ba\u03b1\u1f76 _\u1f03 \u03bc\u03b5\u03c4\u1f70 \u03c4\u03b1\u1fe6\u03c4\u03b1_ \u03c3\u03c5\u03bd\u03ad\u03b3\u03c1\u03b1\u03c8\u03b5 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1fc6\u03c2 [\u03c4\u1ff6\u03bd]\n  \u03a6\u03c1\u03c5\u03b3\u1ff6\u03bd \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03c2 \u03bc\u03b5\u03c4' \u03bf\u1f50 \u03c0\u03bf\u03bb\u1f7a\u03bd \u03ba\u03b1\u03b9\u03bd\u03bf\u03c4\u03bf\u03bc\u03b7\u03b8\u03b5\u03af\u03c3\u03b7\u03c2 \u03c7\u03c1\u03cc\u03bd\u03bf\u03bd, \u03c4\u03cc\u03c4\u03b5 \u03b3\u03b5 \u03bc\u1f74\u03bd\n  \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f10\u03ba\u03c6\u03cd\u03b5\u03b9\u03bd \u1f00\u03c1\u03c7\u03bf\u03bc\u03ad\u03bd\u03b7\u03c2, \u1f14\u03c4\u03b9 \u03c4\u03bf\u1fe6 \u039c\u03bf\u03bd\u03c4\u03b1\u03bd\u03bf\u1fe6 \u1f05\u03bc\u03b1 \u03c4\u03b1\u1fd6\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6\n  \u03c8\u03b5\u03c5\u03b4\u03bf\u03c0\u03c1\u03bf\u03c6\u03ae\u03c4\u03b9\u03c3\u03b9\u03bd \u1f00\u03c1\u03c7\u1f70\u03c2 \u03c4\u1fc6\u03c2 \u03c0\u03b1\u03c1\u03b5\u03ba\u03c4\u03c1\u03bf\u03c0\u1fc6\u03c2 \u03c0\u03bf\u03b9\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5, _i.e._ the vagaries\n  of Montanus and his followers had already begun when Apollinaris\n  wrote, but Montanism assumed a new phase shortly after.\n\nFootnote 184:\n\n  Included in the _Libellus Synodicus_ published by Pappus; see Labb.\n  _Conc._ I. 615, ed. Coleti. Though this council is not mentioned\n  elsewhere, there is no sufficient ground for questioning its\n  authenticity. The important part taken by Apollinaris against the\n  Montanists is recognised by Eusebius _H.E._ v. 16, \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f74\u03bd \u03bb\u03b5\u03b3\u03bf\u03bc\u03ad\u03bd\u03b7\u03bd\n  \u03ba\u03b1\u03c4\u1f70 \u03a6\u03c1\u03cd\u03b3\u03b1\u03c2 \u1f05\u03b9\u03c1\u03b5\u03c3\u03b9\u03bd \u1f45\u03c0\u03bb\u03bf\u03bd \u1f30\u03c3\u03c7\u03c5\u03c1\u1f78\u03bd \u03ba\u03b1\u1f76 \u1f00\u03ba\u03b1\u03c4\u03b1\u03b3\u03ce\u03bd\u03b9\u03c3\u03c4\u03bf\u03bd \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u1f39\u03b5\u03c1\u03b1\u03c0\u03cc\u03bb\u03b5\u03c9\u03c2\n  \u03c4\u1f78\u03bd \u1f08\u03c0\u03bf\u03bb\u03b9\u03bd\u03ac\u03c1\u03b9\u03bf\u03bd.\n\n  After mentioning the council the compiler of this Synodicon speaks\n  thus of the false prophets; \u03bf\u1f33 \u03ba\u03b1\u1f76 \u03b2\u03bb\u03b1\u03c3\u03c6\u03ae\u03bc\u03c9\u03c2, \u1f24\u03c4\u03bf\u03b9 \u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u1ff6\u03bd\u03c4\u03b5\u03c2, \u03ba\u03b1\u03b8\u03ce\u03c2\n  \u03c6\u03b7\u03c3\u03b9\u03bd \u1f41 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03c0\u03b1\u03c4\u03ae\u03c1 [_i.e._ \u1f08\u03c0\u03bf\u03bb\u03b9\u03bd\u03ac\u03c1\u03b9\u03bf\u03c2], \u03c4\u1f78\u03bd \u03b2\u03af\u03bf\u03bd \u03ba\u03b1\u03c4\u03ad\u03c3\u03c4\u03c1\u03b5\u03c8\u03b1\u03bd, \u03c3\u1f7a\u03bd\n  \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03b4\u1f72 \u03ba\u03b1\u03c4\u03ad\u03ba\u03c1\u03b9\u03bd\u03b5 \u03ba\u03b1\u1f76 \u0398\u03b5\u03cc\u03b4\u03bf\u03c4\u03bf\u03bd \u03c4\u1f78\u03bd \u03c3\u03ba\u03c5\u03c4\u03ad\u03b1. He evidently has before\n  him the fragments of the anonymous treatises quoted by Eusebius\n  (_H.E._ v. 16), as the following parallels taken from these fragments\n  shew: \u1f61\u03c2 \u1f10\u03c0\u1f76 \u1f10\u03bd\u03b5\u03c1\u03b3\u03bf\u03c5\u03bc\u03ad\u03bd\u1ff3 \u03ba\u03b1\u1f76 _\u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u1ff6\u03bd\u03c4\u03b9_ ... _\u03b2\u03bb\u03b1\u03c3\u03c6\u03b7\u03bc\u03b5\u1fd6\u03bd_\n  \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u1f00\u03c0\u03b7\u03c5\u03b8\u03b1\u03b4\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 ... _\u03c4\u1f78\u03bd \u03b2\u03b9\u1f78\u03bd\n  \u03ba\u03b1\u03c4\u03b1\u03c3\u03c4\u03c1\u03ad\u03c8\u03b1\u03b9_ \u1f38\u03bf\u03cd\u03b4\u03b1 \u03c0\u03c1\u03bf\u03b4\u03cc\u03c4\u03bf\u03c5 \u03b4\u03af\u03ba\u03b7\u03bd ... \u03bf\u1f36\u03bf\u03bd \u1f10\u03c0\u03af\u03c4\u03c1\u03bf\u03c0\u03cc\u03bd \u03c4\u03b9\u03bd\u03b1\n  _\u0398\u03b5\u03cc\u03b4\u03bf\u03c4\u03bf\u03bd_ \u03c0\u03bf\u03bb\u1f7a\u03c2 \u03b1\u1f31\u03c1\u03b5\u1fd6 \u03bb\u03cc\u03b3\u03bf\u03c2 ... \u03c4\u03b5\u03c4\u03b5\u03bb\u03b5\u03c5\u03c4\u03ae\u03ba\u03b1\u03c3\u03b9 \u039c\u03bf\u03bd\u03c4\u03b1\u03bd\u03cc\u03c2 \u03c4\u03b5 \u03ba\u03b1\u1f76\n  _\u0398\u03b5\u03cc\u03b4\u03bf\u03c4\u03bf\u03c2_ \u03ba\u03b1\u03b9 \u1f21 \u03c0\u03c1\u03bf\u03b5\u03b9\u03c1\u03b7\u03bc\u03ad\u03bd\u03b7 \u03b3\u03c5\u03bd\u03ae. Thus he must have had before\n  him a text of Eusebius which omitted the words \u03b4\u03ae \u03c4\u03b9\u03c2 at the\n  commencement, as they are omitted in some existing MSS (see above, p.\n  56, note); and accordingly he ascribed all the treatises to\n  Apollinaris. The parallels are taken from the first and second\n  treatises; the first might have been written by Apollinaris, but the\n  second was certainly not by his hand, as it refers to much later\n  events (see above, p. 56).\n\n  Hefele (_Conciliengeschichte_ I. p. 71) places the date of this\n  council before A.D. 150. But if the testimony of Eusebius is worth\n  anything, this is impossible; for he states that the writings of\n  Claudius Apollinaris against the Montanists were later than his\n  Apology to M. Aurelius (see p. 59, note 183), and this Apology was not\n  written till after A.D. 174 (see p. 61, note 187). The chronology of\n  Montanism is very perplexing, but Hefele's dates appear to be much too\n  early. The _Chronicon_ of Eusebius gives the rise of Montanism under\n  A.D. 172 or 173, and this statement is consistent with the notices in\n  his History. But if this date be correct, it most probably refers to\n  Montanism as a distinct system; and the fires had probably been\n  smouldering within the Church for some time before they broke out.\n\n  It will be observed that the writer of the Synodicon identifies\n  Theodotus the Montanist (see Euseb. _H.E._ v. 3) with Theodotus the\n  leather-seller who was a Monarchian. There is no authority for this\n  identification in Eusebius.\n\n[Sidenote: His other h\u00e6resiological writings.]\n\nNor were his controversial writings confined to these two topics. In one\nplace he refuted the Encratites[185]; in another he upheld the orthodox\nteaching respecting the true humanity of Christ[186]. It is plain that\nhe did not confine himself to questions especially affecting Asia Minor;\nbut that the doctrine and the practice of the Church generally found in\nhim a vigorous advocate, who was equally opposed to the novelties of\nheretical teaching and the rigours of overstrained asceticism.\n\nFootnote 185:\n\n  Theodoret. _H.E._ i. 21.\n\nFootnote 186:\n\n  Socr. _H.E._ iii. 7.\n\nNor again did Apollinaris restrict himself to controversies carried on\nbetween Christian and Christian. He appears alike as the champion of the\nGospel against attacks from without, and as the promoter of Christian\nlife and devotion within the pale of the Church. [Sidenote: His\napologetic] On the one hand he was the author of an apology addressed to\nM. Aurelius[187], of a controversial treatise in five books against the\nGreeks, and of a second in two books against the Jews[188]; on the other\nwe find mentioned among his [Sidenote: and didactic works.] writings a\nwork in two books _on Truth_, and a second _on Piety_, besides several\nof which the titles have not come down to us[189]. He seems indeed to\nhave written on almost every subject which interested the Church of his\nage. He was not only well versed in the Scriptures, but showed a wide\nacquaintance with secular literature also[190]. His style is praised by\na competent judge[191], and his orthodoxy was such as to satisfy the\ndogmatic precision of the post-Nicene age[192].\n\nFootnote 187:\n\n  Euseb. _H.E._ iv. 26, 27. He referred in this Apology to the incident\n  of the so-called Thundering Legion, which happened A.D. 174; and as\n  reported by Eusebius (_H.E._ v. 5), he stated that the legion was thus\n  named by the emperor in commemoration of this miraculous thunderstorm.\n  As a contemporary however, he must probably have known that the title\n  _Legio Fulminata_ existed long before; and we may conjecture that he\n  used some ambiguous expression implying that it was fitly so named\n  (_e.g._ \u1f10\u03c0\u03ce\u03bd\u03c5\u03bc\u03bf\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03bd\u03c4\u03c5\u03c7\u03af\u03b1\u03c2), which Eusebius and later writers\n  misunderstood; just as Eusebius himself (v. 24) speaks of Iren\u00e6us as\n  \u03c6\u03b5\u03c1\u03ce\u03bd\u03c5\u03bc\u03cc\u03c2 \u03c4\u03b9\u03c2 \u1f62\u03bd \u03c4\u1fc7 \u03c0\u03c1\u03bf\u03c3\u03b7\u03b3\u03bf\u03c1\u03af\u1fb3 \u03b1\u1f50\u03c4\u1ff7 \u03c4\u03b5 \u03c4\u1ff7 \u03c4\u03c1\u03cc\u03c0\u1ff3 \u03b5\u1f30\u03c1\u03b7\u03bd\u03bf\u03c0\u03bf\u03af\u03bf\u03c2. Of the\n  words used by Eusebius, \u03bf\u1f30\u03ba\u03b5\u03af\u03b1\u03bd \u03c4\u1ff7 \u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c4\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1fe6 \u03b2\u03b1\u03c3\u03b9\u03bb\u03ad\u03c9\u03c2\n  \u03b5\u1f30\u03bb\u03b7\u03c6\u03ad\u03bd\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b7\u03b3\u03bf\u03c1\u03af\u03b1\u03bd, we may suspect that \u03bf\u1f30\u03ba\u03b5\u03af\u03b1\u03bd \u03c4\u1ff7 \u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c4\u03b9\n  \u03c0\u03c1\u03bf\u03c3\u03b7\u03b3\u03bf\u03c1\u03af\u03b1\u03bd is an expression borrowed from Apollinaris himself, while\n  \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1fe6 \u03b2\u03b1\u03c3\u03b9\u03bb\u03ad\u03c9\u03c2 \u03b5\u1f30\u03bb\u03b7\u03c6\u03ad\u03bd\u03b1\u03b9 gives Eusebius' own erroneous\n  interpretation of his author's meaning.\n\n  The name of this legion was _Fulminata_, not _Fulminatrix_, as it is\n  often carelessly written out, where the inscriptions have merely\n  FVLM.; see Becker and Marquardt _R\u00f6m. Alterth._ III. 2, p. 353.\n\nFootnote 188:\n\n  The words \u03ba\u03b1\u1f76 \u03c0\u03c1\u1f78\u03c2 \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03bf\u03c5\u03c2 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03bf\u03bd are omitted in some\n  MSS and by Rufinus. They are found however in the very ancient Syriac\n  version, and are doubtless genuine. Their omission is due to the\n  hom\u0153oteleuton, as they are immediately preceded by \u03ba\u03b1\u1f76 \u03c0\u03b5\u03c1\u1f76\n  \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03bf\u03bd.\n\nFootnote 189:\n\n  A list of his works is given by Eusebius (_H.E._ iv. 27), who explains\n  that there were many others which he had not seen. This list omits the\n  work on the Paschal Feast, which is quoted in the _Chronicon Paschale_\n  p. 13 (ed. Dind.), and the treatise _on Piety_, of which we know from\n  Photius _Bibl._ 14.\n\nFootnote 190:\n\n  Theodoret. _H\u00e6r. Fab._ iii. 2 \u1f00\u03bd\u1f74\u03c1 \u1f00\u03be\u03b9\u1f72\u03c0\u03b1\u03b9\u03bd\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1fc7 \u03b3\u03bd\u03ce\u03c3\u03b5\u03b9 \u03c4\u1ff6\u03bd\n  \u03b8\u03b5\u03af\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f14\u03be\u03c9\u03b8\u03b5\u03bd \u03c0\u03b1\u03b9\u03b4\u03b5\u03af\u03b1\u03bd \u03c0\u03c1\u03bf\u03c3\u03b5\u03b9\u03bb\u03b7\u03c6\u03ce\u03c2. So too Jerome, _Ep._ 70 (I.\n  p. 428, ed. Vallarsi), names him among those who were equally versed\n  in sacred and profane literature.\n\nFootnote 191:\n\n  Photius l.c., \u1f00\u03be\u03b9\u03cc\u03bb\u03bf\u03b3\u03bf\u03c2 \u03b4\u1f72 \u1f41 \u1f00\u03bd\u1f74\u03c1 \u03ba\u03b1\u1f76 \u03c6\u03c1\u03ac\u03c3\u03b5\u03b9 \u1f00\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u1ff3 \u03ba\u03b5\u03c7\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03c2.\n\nFootnote 192:\n\n  Euseb. _H.E._ iv. 21, Jerome l.c., Theodoret. l.c., Socr. _H.E._ iii.\n  7.\n\n[Sidenote: Important bearing of these facts on the history of\n           Christianity.]\n\nThese facts are not unimportant in their bearing on the question which\nhas been already discussed in relation to Papias. If there had been such\na discontinuity of doctrine and practice in the Church of Hierapolis as\nthe theory in question assumes, if the Pauline Gospel was repudiated in\nthe later years of the first century and rank Judaism adopted in its\nstead, how can we explain the position of Apollinaris? Obviously a\ncounter-revolution must have taken place, which undid the effects of the\nformer. One dislocation must have been compensated by another. And yet\nIren\u00e6us knows nothing of these religious convulsions which must have\nshaken the doctrine of the Church to its foundations, but represents the\ntradition as one, continuous, unbroken, reaching back through the elders\nof the Asiatic Churches, through Papias and Polycarp, to St John\nhimself\u2014Iren\u00e6us who received his Christian education in Asia Minor, who\nthroughout life was in communication with the churches there, and who\nhad already reached middle age when this second revolution is supposed\nto have occurred. The demands on our credulity, which this theory makes,\nare enormous. And its improbability becomes only the more glaring, as we\nextend our view.[Sidenote: Solidarity of the Church in the second\ncentury.] For the _solidarity_ of the Church is the one striking fact\nunmistakably revealed to us, as here and there the veil which shrouds\nthe history of the second century is lifted. Anicetus and Soter and\nEleutherus and Victor at Rome, Pant\u00e6nus and Clement at Alexandria,\nPolycrates at Ephesus, Papias and Apollinaris at Hierapolis, Polycarp at\nSmyrna, Melito at Sardis, Ignatius and Serapion at Antioch, Primus and\nDionysius at Corinth, Pothinus and Iren\u00e6us in Gaul, Philippus and\nPinytus in Crete, Hegesippus and Narcissus in Palestine, all are bound\ntogether by the ties of a common organization and the sympathy of a\ncommon creed. The Paschal controversy is especially valuable, as showing\nthe limits of divergence consistent with the unity of the Church. The\nstudy of this controversy teaches us to appreciate with ever increasing\nforce the pregnant saying of Iren\u00e6us that the difference of the usage\nestablishes the harmony of the faith[193].\n\nFootnote 193:\n\n  Iren. in Euseb. _H.E._ v. 24 \u1f21 \u03b4\u03b9\u03b1\u03c6\u03c9\u03bd\u03af\u03b1 \u03c4\u1fc6\u03c2 \u03bd\u03b7\u03c3\u03c4\u03b5\u03af\u03b1\u03c2 (the fast which\n  preceded the Paschal festival) \u03c4\u1f74\u03bd \u1f41\u03bc\u03cc\u03bd\u03bf\u03b9\u03b1\u03bd \u03c4\u1fc6\u03c2 \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03c3\u03c5\u03bd\u03af\u03c3\u03c4\u03b7\u03c3\u03b9.\n\n[Sidenote: Activity of Laodicea.]\n\nThough Laodicea cannot show the same intellectual activity as Hierapolis\nduring the second century, yet in practical energy she is not wanting.\n\n[Sidenote: Martyrdom of Sagaris. c. A.D. 165.]\n\nThe same persecution, which, permitted if not encouraged by the imperial\nStoic, was fatal to Polycarp at Smyrna, deprived Laodicea also of her\nbishop Sagaris[194]. The exact year in which he fell a martyr is not\nknown; but we can hardly be wrong in assuming that his death was nearly\ncoincident with those of Polycarp and his companions. His name appears\nto have been held in great honour[195].\n\nFootnote 194:\n\n  Melito in Euseb. _H.E._ iv. 26 \u1f10\u03c0\u1f76 \u03a3\u03b5\u03c1\u03bf\u03c5\u03b9\u03bb\u03bb\u03af\u03bf\u03c5 \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5 \u1f00\u03bd\u03b8\u03c5\u03c0\u03ac\u03c4\u03bf\u03c5 \u03c4\u1fc6\u03c2\n  \u1f08\u03c3\u03af\u03b1\u03c2, \u1fa7 \u03a3\u03ac\u03b3\u03b1\u03c1\u03b9\u03c2 \u03ba\u03b1\u03b9\u03c1\u1ff7 \u1f10\u03bc\u03b1\u03c1\u03c4\u03cd\u03c1\u03b7\u03c3\u03b5\u03bd, \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u03b6\u03ae\u03c4\u03b7\u03c3\u03b9\u03c2 \u03c0\u03bf\u03bb\u03bb\u1f74 \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u1fb3\n  \u03c0\u03b5\u03c1\u1f76 \u03c4\u03bf\u1fe6 \u03c0\u03ac\u03c3\u03c7\u03b1 \u1f10\u03bc\u03c0\u03b5\u03c3\u03cc\u03bd\u03c4\u03bf\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03ba\u03b1\u03b9\u03c1\u1f78\u03bd \u1f10\u03bd \u1f10\u03ba\u03b5\u03af\u03bd\u03b1\u03b9\u03c2 \u03c4\u03b1\u1fd6\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03b9\u03c2, \u03ba\u03b1\u1f76\n  \u1f10\u03b3\u03c1\u03ac\u03c6\u03b7 \u03c4\u03b1\u1fe6\u03c4\u03b1 (_i.e._ Melito's own treatise on the Paschal festival).\n\nFootnote 195:\n\n  Besides Melito (l.c.), Polycrates of Ephesus refers to him with\n  respect; Euseb. _H.E._ v. 24, \u03c4\u1f76 \u03b4\u1f72 \u03b4\u03b5\u1fd6 \u03bb\u03ad\u03b3\u03b5\u03b9\u03bd \u03a3\u03ac\u03b3\u03b1\u03c1\u03b9\u03bd \u1f10\u03c0\u03af\u03c3\u03ba\u03bf\u03c0\u03bf\u03bd \u03ba\u03b1\u1f76\n  \u03bc\u03ac\u03c1\u03c4\u03c5\u03c1\u03b1, \u1f45\u03c2 \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u1fb3 \u03ba\u03b5\u03ba\u03bf\u03af\u03bc\u03b7\u03c4\u03b1\u03b9.\n\n[Sidenote: Outbreak of the Paschal controversy.]\n\nBut while the Church of Laodicea was thus contending against foes\nwithout, she was also torn asunder by feuds within. Coincident with the\nmartyrdom of Sagaris was the outburst of the Paschal controversy, of\nwhich mention has been already made, and which for more than a century\nand a half disturbed the peace of the Church, until it was finally laid\nat rest by the Council of Nic\u00e6a. The Laodiceans would naturally regulate\ntheir festival by the Asiatic or Quartodeciman usage, strictly observing\nthe day of the month and disregarding the day of the week. But a great\ncommercial centre like Laodicea must have attracted large crowds of\nforeign Christians from Palestine or Egypt or Rome or Gaul, who were\naccustomed to commemorate the Passion always on a Friday and the\nResurrection on a Sunday according to the western practice; and in this\nway probably the dispute arose. The treatise _on the Paschal Festival_\nby Melito of Sardis was written on this occasion to defend the Asiatic\npractice. The fact that Laodicea became the head-quarters of the\ncontroversy is a speaking testimony to the prominence of this Church in\nthe latter half of the second century.\n\n[Sidenote: Hierapolis and Laodicea in later history.]\n\nAt a later date the influence of both Hierapolis and Laodicea has\nsensibly declined. In the great controversies of the fourth s and fifth\ncenturies they take no conspicuous part. Among their bishops there is\nnot one who has left his mark on history. And yet their names appear at\nmost of the great Councils, in which they bear a silent part. [Sidenote:\nThe _Arian_ heresy. NIC\u00c6A A.D. 325.]At Nic\u00e6a Hierapolis was represented\nby Flaccus[196], Laodicea by Nunechius[197]. They both acquiesced in its\ndecrees, and the latter as metropolitan published them throughout the\nPhrygian Churches[198]. Soon after, both sees lapsed into Arianism.\n[Sidenote: Philippopolis A.D. 347.] At the synod of Philippopolis,\ncomposed of bishops who had seceded from the Council of Sardica, the\nrepresentatives of these two sees were present and joined in the\ncondemnation of the Athanasians. On this occasion Hierapolis was still\nrepresented by Flaccus, who had thus turned traitor to his former\nfaith[199]. On the other hand Laodicea had changed its bishop twice\nmeanwhile. Cecropius had won the imperial favour by his abuse of the\northodox party, and was first promoted to Laodicea, whence he was\ntranslated to Nicomedia[200]. He was succeeded by Nonnius, who signed\nthe Arian decree at Philippopolis[201]. When these sees recovered their\northodoxy we [Sidenote: CONSTANTINOPLE. A.D. 381.]] do not know; but it\nis perhaps a significant fact, that neither is represented at the second\ngeneral Council, held at Constantinople [Sidenote: The _Nestorian_ and\n_Eutychian_ heresies. EPHESUS. A.D. 431.] (A.D. 381)[202]. At the third\ngeneral Council, which met at Ephesus, Laodicea is represented by\nAristonicus, Hierapolis by Venantius[203]. Both bishops sign the decrees\ncondemning Nestorius. Again in the next Christological controversy which\nagitated the Church the two sees bear their part. At the notorious\n[Sidenote: Latrocinium. A.D. 449.] Robbers' Synod, held also at Ephesus,\nLaodicea was represented by another Nunechius, Hierapolis by Stephanus.\nBoth bishops committed themselves to the policy of Dioscorus and the\nopinions of the heretic Eutyches[204]. Yet with the fickleness which\ncharacterized these sees at an earlier date during the Arian\ncontroversy, we find their representatives two years [Sidenote:\nCHALCEDON. A.D. 451.] later at the Council of Chalcedon siding with the\northodox party and condemning the Eutychian heresy which they had so\nlately supported[205]. Nunechius is still bishop of Laodicea, and\nreverses his former vote. Stephanus has been succeeded at Hierapolis by\nAbercius, whose orthodoxy, so far as we know, had not been compromised\nby any previous expression of opinion[206].\n\nFootnote 196:\n\n  Labb. _Conc._ II. 57, 62, ed. Coleti; Cowper's _Syriac Miscellanies_\n  p. 11, 28. It is remarkable that after Papias all the early bishops of\n  Hierapolis of whom any notice is preserved, have Roman names; Avircius\n  Marcellus (?), Claudius Apollinaris, Flaccus, Lucius, Venantius.\n\nFootnote 197:\n\n  Labb. _Conc._ II. 57, 62; Cowper's _Syriac Miscellanies_ pp. 11, 28,\n  34. He had also been present at the Synod of Ancyra held about A.D.\n  314 (see _Galatians_ p. 34); _ib._ p. 41.\n\nFootnote 198:\n\n  Labb. _Conc._ II. 236.\n\nFootnote 199:\n\n  _ib._ 744.\n\nFootnote 200:\n\n  Athanas. _ad Episc. \u00c6gypt._ 8 (_Op._ I. p. 219), _Hist. Arian. ad\n  Mon._ 74 (_ib._ p. 307).\n\nFootnote 201:\n\n  Labb. _Conc._ II. 744.\n\nFootnote 202:\n\n  Cowper's _Syriac Miscell._ p. 39.\n\nFootnote 203:\n\n  Labb. _Conc._ III. 1085, 1222, Mans. _Conc._ IV. 1367. The name of\n  this bishop of Hierapolis is variously written, but Venantius seems to\n  be the true orthography. For some unexplained reason, though present\n  in person he signs by deputy. He had before subscribed the protest to\n  Cyril against commencing the proceedings before the arrival of John of\n  Antioch (Mans. _Conc._ V. 767), and perhaps his acquiescence in the\n  decisions of the Council was not very hearty.\n\nFootnote 204:\n\n  Labb. _Conc._ IV. 892, 925, 928, 1107, 1170, 1171, 1185. In the Acts\n  of this heretical council, as occasionally in those of the Council of\n  Chalcedon, Laodicea is surnamed _Trimitaria_ (see above, p. 18, note\n  2). Following Le Quien (_Or. Christ._ I. p. 835), I have assumed the\n  Stephanus who was present at the _Latrocinium_ to have been bishop of\n  the _Phrygian_ Hierapolis, though I have not found any decisive\n  indication which Hierapolis is meant. On the other hand the bishop of\n  the _Syrian_ Hierapolis at this time certainly bore the name Stephanus\n  (Labb. _Conc._ IV. 727, 1506, [1550], 1644, 1836, V. 46); and the\n  synod held under Stephanus A.D. 445, which Wiltsch (_Geography and\n  Statistics of the Church_ I. p. 170, Eng. Trans.) assigns to our\n  Hierapolis, belongs to the Syrian city of the same name, as the\n  connexion with Perrha shews: Labb. _Conc._ IV. 727, 1644.\n\nFootnote 205:\n\n  Labb. _Conc._ IV. 853, 862, 1195, 1204, 1241, 1312, 1337, 1383, 1392,\n  1444, 1445, 1463, 1480, 1481, 1496, 1501, 1505, 1716, 1732, 1736,\n  1744, 1746, 1751.\n\nFootnote 206:\n\n  The bishops of both sees are addressed by the Emperor Leo in his\n  letter respecting the Council of Chalcedon: but their replies are not\n  preserved. Nunechius is still bishop of Laodicea; but Hierapolis has\n  again changed hands, and Philippus has succeeded Abercius (Labb.\n  _Conc._ IV. 1836 sq.). Nunechius of Laodicea was one of those who\n  signed the decree against simony at the Council of Constantinople\n  (A.D. 459): _Conc._ V. 50.\n\n[Sidenote: Later vacillation of these sees.]\n\nThe history of these churches at a later date is such as might have been\nanticipated from their attitude during the period of the first Four\nGeneral Councils. The sees of Laodicea and Hierapolis, one or both, are\nrepresented at all the more important assemblies of the Church; and the\nsame vacillation and infirmity of purpose, which had characterized their\nholders in the earlier councils, marks the proceedings of their later\nsuccessors[207].\n\nFootnote 207:\n\n  See for instance the tergiversation of Theodorus of Laodicea and\n  Ignatius of Hierapolis in the matter of Photius and the 8th General\n  Council.\n\n[Sidenote: Their comparative unimportance.]\n\nBut, though the two sees thus continue to bear witness to their\nexistence by the repeated presence of their occupants at councils and\nsynods, yet the real influence of Laodicea and Hierapolis on the Church\nat large has terminated with the close of the second century. On one\noccasion only did either [Sidenote: COUNCIL OF LAODICEA an exception.]\ncommunity assume a position of prominence. About the middle of the\nfourth century a council was held at Laodicea[208]. It [Sidenote: Its\ndecree on the Canon.] was convened more especially to settle some points\nof ecclesiastical discipline; but incidentally the assembled bishops\nwere led to make an order respecting the Canon of Scripture[209]. As\nthis was the first occasion in which the subject had been brought\nformally before the notice of an ecclesiastical assembly this Council of\nLaodicea secured a notoriety which it would not otherwise have obtained,\nand to which it was hardly entitled by its constitution or its\nproceedings. Its decrees were confirmed and adopted by later councils\nboth in the East and in the West[210].\n\nFootnote 208:\n\n  This council cannot have been held earlier than the year 344, as the\n  7th canon makes mention of the Photinians, and Photinus did not\n  attract notice before that year: see Hefele, _Conciliengesch._ I. p.\n  722 sq. In the ancient lists of Councils it stands after that of\n  Antioch (A.D. 341), and before that of Constantinople (A.D. 381). Dr\n  Westcott (_History of the Canon_, p. 400) is inclined to place it\n  about A.D. 363, and this is the time very generally adopted.\n\n  Here however a difficulty presents itself, which has not been noticed\n  hitherto. In the Syriac MS _Brit. Mus._ Add. 14,528, are lists of the\n  bishops present at the earlier councils, including Laodicea (see\n  Wright's _Catalogue of the Syriac MSS in the British Museum_, DCCCVI,\n  p. 1030 sq.). These lists have been published by Cowper (_Syriac\n  Miscell._ p. 42 sq., _Analecta Nic\u00e6na_ p. 36), who however has\n  transposed the lists of Antioch and Laodicea, so that he ascribes to\n  the Antiochian Synod the names which really belong to the Laodicean.\n  This is determined (as I am informed by Prof. Wright) by the position\n  of the lists.\n\n  The Laodicean list then, which seems to be imperfect, contains twenty\n  names; and, when examined, it yields these results. (1) At least\n  three-fourths of the names can be identified with bishops who sat at\n  Nic\u00e6a, and probably the exceptions would be fewer, if in some cases\n  they had not been obscured by transcription into Syriac and by the\n  errors of copyists. (2) When identified, they are found to belong in\n  almost every instance to C\u0153lesyria, Ph\u0153nicia, Palestine,\n  Cilicia, and Isauria, whereas apparently not one comes from Phrygia,\n  Lydia, or the other western districts of Asia Minor.\n\n  Supposing that this is a genuine Laodicean list, we are led by the\n  first result to place it as near in time as possible to the Council of\n  Nic\u00e6a; and by the second to question whether after all the Syrian\n  Laodicea may not have been meant instead of the Phrygian. On the other\n  hand tradition is unanimous in placing this synod in the Phrygian\n  town, and in this very Syriac MS the heading of the canons begins 'Of\n  the Synod of Laodicea of Phrygia.' On the whole it appears probable\n  that this supposed list of bishops who met at Laodicea belongs to some\n  other Council. The Laodicean Synod seems to have been, as Dr Westcott\n  describes it (l.c.), 'A small gathering of clergy from parts of Lydia\n  and Phrygia.'\n\n  In a large mosaic in the Church at Bethlehem, in which all the more\n  important Councils are represented, we find the following inscription;\n  [\u1f29] \u1f01\u03b3\u03af\u03b1 \u03c3\u03cd\u03bd\u03bf\u03b4\u03bf\u03c2 \u1f21 \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u1fb3 \u03c4\u1fc6\u03c2 \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03c2 \u03c4\u03c9\u03bd \u03ba\u1f72 \u1f10\u03c0\u03b9\u03c3\u03ba\u03cc\u03c0\u03c9\u03bd \u03b3\u03ad\u03b3\u03bf\u03bd\u03b5\u03bd\n  \u03b4\u03b9\u1f70 \u039c\u03bf\u03bd\u03c4\u03b1\u03bd\u1f78\u03bd \u03ba\u1f72 [\u03c4]\u1f70[\u03c2] \u03bb\u03bf\u03b9\u03c0\u1f70\u03c2 \u1f11\u03c1\u03ad\u03c3\u03b5\u03b9\u03c2\u00b7 \u03c4\u03bf\u03cd[\u03c4\u03bf\u03c5\u03c2] \u1f61\u03c2 \u03b1\u1f31\u03c1\u03b5\u03c4\u03b9\u03ba\u03bf\u1f7a\u03c2 \u03ba\u03b1\u1f76\n  \u1f10\u03c7\u03b8\u03c1\u03bf\u1f7a\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b5\u03b8\u03b5\u03af\u03b1\u03c2 \u1f21 \u1f01\u03b3\u03af\u03b1 \u03c3\u03cd\u03bd\u03bf\u03b4\u03bf\u03c2 \u1f00\u03bd\u03b5\u03b8\u03b5\u03bc\u03ac\u03c4\u03b9\u03c3\u03b5\u03bd (Ciampini _de Sacr.\n  \u00c6dif. a Constant. constr._ p. 156; comp. Boeckh _Corp. Inscr._ 8953).\n  From its position we might infer that the synod to which this\n  inscription refers was supposed to have taken place before the Council\n  of Nic\u00e6a; and if so, it may have been one of those Asiatic synods held\n  against Montanism at the end of the second or beginning of the third\n  century. But, inasmuch as no record of any such synod is preserved\n  elsewhere, we must probably refer it to the well-known Council of\n  Laodicea in the fourth century. In this case however the description\n  is not very correct, for though Montanism is incidentally condemned in\n  the eighth canon, yet this condemnation was not the main object of the\n  council and occupies a very subordinate place. The Bethlehem mosaics\n  were completed A.D. 1169: see Boeckh _C. I._ 8736.\n\nFootnote 209:\n\n  The canons of this Council, 59 in number, will be found in Labb.\n  _Conc._ I. 1530 sq., ed. Coleti. The last of these forbids the reading\n  of any but 'the Canonical books of the New and Old Testament.' To this\n  is often appended (sometimes as a 60th canon) a list of the Canonical\n  books; but Dr Westcott has shown that this list is a later addition\n  and does not belong to the original decrees of the council (_Canon_ p.\n  400 sq.).\n\nFootnote 210:\n\n  By the Quinisextine Council (A.D. 692) in the East (Labb. _Conc._ VII.\n  1345), and by the Synod of Aix-la-Chapelle (A.D. 789) in the West\n  (_Conc._ IX. 10 sq.).\n\n[Sidenote: Its decrees illustrate the Epistle to the Colossians.]\n\nMore important however for my special purpose, than the influence of\nthis synod on the Church at large, is the light which its canons throw\non the heretical tendencies of this district, and on the warnings of St\nPaul in the Colossian Epistle. To illustrate this fact it will only be\nnecessary to write out some of these canons at length:\n\n[Sidenote: Col. ii. 14, 16, 17.]\n\n29. 'It is not right for Christians to Judaize and abstain from labour\non the sabbath, but to work on this same day. They should pay respect\nrather to the Lord's day, and, if possible, abstain from labour on it as\nChristians. But if they should be found Judaizers, let them be anathema\nin the sight of Christ.'\n\n[Sidenote: Col. ii. 18.]\n\n35. 'It is not right for Christians to abandon the Church of God and go\naway and invoke angels (\u1f00\u03b3\u03b3\u03ad\u03bb\u03bf\u03c5\u03c2 \u1f40\u03bd\u03bf\u03bc\u03ac\u03b6\u03b5\u03b9\u03bd)[211] and hold conventicles\n(\u03c3\u03c5\u03bd\u03ac\u03be\u03b5\u03b9\u03c2 \u03c0\u03bf\u03b9\u03b5\u1fd6\u03bd); for these things are forbidden. If therefore any one\nis found devoting himself to this secret idolatry, let him be anathema,\nbecause he abandoned our Lord Jesus Christ and went after idolatry.'\n\nFootnote 211:\n\n  Theodoret about a century after the Laodicean Council, commenting on\n  Col. ii. 18, states that this disease (\u03c4\u1f78 \u03c0\u03ac\u03b8\u03bf\u03c2) which St Paul\n  denounces 'long remained in Phrygia and Pisidia.' 'For this reason\n  also,' he adds, 'a synod convened in Laodicea of Phrygia forbad by a\n  decree the offering prayer to angels; and even to the present time\n  oratories of the holy Michael may be seen among them and their\n  neighbours.' See also below p. 71, note 219. A curious inscription,\n  found in the theatre at Miletus (Boeckh _C. I._ 2895), illustrates\n  this tendency. It is an inscription in seven columns, each having a\n  different planetary symbol, and a different permutation of the vowels\n  with the same invocation ==\u03b1\u03b3\u03b9\u03b5 . \u03c6\u03c5\u03bb\u03b1\u03c4\u03bf\u03bd . \u03c4\u03b7\u03bd . \u03c0\u03bf\u03bb\u03b9\u03bd . \u03bc\u03b9\u03bb\u03b7\u03f2\u03b9\u03c9\u03bd .\n  \u03ba\u03b1\u03b9 . \u03c0\u03b1\u03bd\u03c4\u03b1\u03f2 . \u03c4\u03bf\u03c5\u03f2 . \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03bf\u03c5\u03bd\u03c4\u03b1\u03f2==, while at the common base is\n  written ==\u03b1\u03c1\u03c7\u03b1\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9 . \u03c6\u03c5\u03bb\u03b1\u03f2\u03f2\u03b5\u03c4\u03b1\u03b9 . \u03b7 . \u03c0\u03bf\u03bb\u03b9\u03f2 . \u03bc\u03b9\u03bb\u03b7\u03f2\u03b9\u03c9\u03bd . \u03ba\u03b1\u03b9 .\n  \u03c0\u03b1\u03bd\u03c4\u03b5\u03f2 . \u03bf\u03b9 . \u03ba\u03b1\u03c4...== Boeckh writes, 'Etsi hic titulus Gnosticorum\n  et Basilidianorum commentis prorsus congruus est, tamen potuit ab\n  ethnicis Milesiis scriptus esse; quare nolui eum inter Christianos\n  rejicere, quum pr\u00e6sertim public\u00e6 Milesiorum superstitionis documentum\n  insigne sit.' The idea of the seven h\u03ac\u03b3\u03b9\u03bf\u03b9, combined in the one\n  \u03b1\u03c1\u03c7\u03ac\u03b3\u03b3\u03b5\u03bb\u03bf\u03c2, seems certainly to point to Jewish, if not Christian,\n  influences: Rev. i. 4, iii. 1, iv. 5, v. 6.\n\n36. 'It is not right for priests or clergy to be magicians or enchanters\nor mathematicians or astrologers[212], or to make safe-guards\n(\u03c6\u03c5\u03bb\u03b1\u03ba\u03c4\u03ae\u03c1\u03b9\u03b1) as they are called, for such things are prisons\n(\u03b4\u03b5\u03c3\u03bc\u03c9\u03c4\u03ae\u03c1\u03b9\u03b1) of their souls[213]: and we have enjoined that they which\nwear them be cast out of the Church.'\n\nFootnote 212:\n\n  Though there is no direct mention of 'magic' in the letter to the\n  Colossians, yet it was a characteristic tendency of this part of Asia:\n  Acts xix. 19, 2 Tim. iii. 8, 13. See the note on Gal. v. 20. The term\n  \u03bc\u03b1\u03b8\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u1f76 is used in this decree in its ordinary sense of\n  astrologers, soothsayers.\n\nFootnote 213:\n\n  A Play on the double sense of \u03c6\u03c5\u03bb\u03b1\u03ba\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd (1) a safeguard or amulet,\n  (2) a guard-house.\n\n37. 'It is not right to receive from Jews or heretics the festive\nofferings which they send about, nor to join in their festivals.'\n\n38. 'It is not right to receive unleavened bread from the Jews or to\nparticipate in their impieties.'\n\nIt is strange, at this late date, to find still lingering in these\nchurches the same readiness to be 'judged in respect of an holiday or a\nnew moon or a sabbath,' with the same tendency to relinquish the hold of\nthe Head and to substitute 'a voluntary humility and worshipping of\nangels,' which three centuries before had called forth the Apostle's\nrebuke and warning in the Epistle to the Colossians.\n\n[Sidenote: Ecclesiastical status of Laodicea and Hierapolis.]\n\nDuring the flourishing period of the Eastern Church, Laodicea appears as\nthe metropolis of the province of Phrygia Pacatiana, counting among its\nsuffragan bishoprics the see of Coloss\u00e6[214]. On the other hand\nHierapolis, though only six miles distant, belonged to the neighbouring\nprovince of Phrygia Salutaris[215], whose metropolis was Synnada, and of\nwhich it was one of the most important sees. The stream of the Lycus\nseems to have formed the boundary line between the two ecclesiastical\nprovinces. At a later date Hierapolis itself was raised to metropolitan\nrank[216].\n\nFootnote 214:\n\n  A list of the bishoprics belonging to this province at the time of the\n  Council of Chalcedon is given, Labb. _Conc._ IV. 1501, 1716.\n\nFootnote 215:\n\n  _Conc._ IV. 1716, 1744.\n\nFootnote 216:\n\n  At the 5th and 6th General Councils (A.D. 553 and A.D. 680) Hierapolis\n  is styled a metropolis (Labb. _Conc._ VI. 220, VII. 1068, 1097, 1117);\n  and in the latter case it is designated metropolis of _Phrygia\n  Pacatiana_, though this same designation is still given to Laodicea.\n  Synnada retains its position as metropolis of _Phrygia Salutaris_.\n\n  From this time forward Hierapolis seems always to hold metropolitan\n  rank. But no notice is preserved of the circumstances under which the\n  change was made. In the _Notiti\u00e6_ it generally occurs twice\u2014first as a\n  suffragan see of Phrygia Salutaris, and secondly as metropolis of\n  another Phrygia Pacatiana (distinct from that which has Laodicea for\n  its metropolis): _Hieroclis Synecdemus et Notiti\u00e6_ (ed. Parthey) Not.\n  1, pp. 56, 57, 69, 73; Not. 3, pp. 114, 124; Not. 7, pp. 152, 161;\n  Not. 8, pp. 164, 176, 180; Not. 9, pp. 193, 197; Not. 10, pp. 212,\n  220. In this latter position it is placed quite out of the proper\n  geographical order, thus showing that its metropolitan jurisdiction\n  was created comparatively late. The number of dioceses in the province\n  is generally given as 9; Nilus _ib._ p. 301. The name of the province\n  is variously corrupted from \u03a0\u03b1\u03ba\u03b1\u03c4\u03b9\u03b1\u03bd\u1fc6\u03c2, e.g. \u039a\u03b1\u03c0\u03c0\u03b1\u03c4\u03b9\u03b1\u03bd\u1fc6\u03c2, \u039a\u03b1\u03c0\u03c0\u03b1\u03b4\u03bf\u03ba\u03af\u03b1\u03c2.\n  Unless the ecclesiastical position of Hierapolis was altogether\n  anomalous, as a province within a province, its double mention in the\n  _Notiti\u00e6_ must be explained by a confusion of its earlier and later\n  status.\n\n[Sidenote: Obscurity of Coloss\u00e6.]\n\nBut while Laodicea and Hierapolis held the foremost place in the records\nof the early Church, and continued to bear an active, though\ninconspicuous part, in later Christian history, Coloss\u00e6 was from the\nvery first a cipher. The town itself, as we have seen, was already\nwaning in importance, when the Apostle wrote; and its subsequent decline\nseems to have been rapid. Not a single event in Christian history is\nconnected with its name; and its very existence is only rescued from\noblivion, when at long intervals some bishop of Coloss\u00e6 attaches his\nsignature to the decree of an ecclesiastical synod. The city ceased to\nstrike coins in the reign of Gordian (A.D. [Sidenote: It is supplanted\nby Chon\u00e6.] 238\u2013244)[217]. It fell gradually into decay, being supplanted\nby the neighbouring town Chon\u00e6, the modern Chonos, so called from the\nnatural funnels by which the streams here disappear in underground\nchannels formed by the incrustations of travertine[218]. We may\nconjecture also that its ruin was hastened by a renewed assault of its\nancient enemy, the earthquake[219]. It is commonly said that Chon\u00e6 is\nbuilt on the site of the ancient Coloss\u00e6; but the later town stands at\nsome distance from the earlier, as Salisbury does from Old Sarum. The\nepiscopal see necessarily followed the population; though for some time\nafter its removal to the new town the bishop still continued to use the\nolder title, with or without the addition of Chon\u00e6 by way of\nexplanation, till at length the name of this primitive Apostolic Church\npasses wholly out of sight[220].\n\nFootnote 217:\n\n  See Mionnet IV. p. 269, Leake _Numism. Hellen._ p. 45.\n\nFootnote 218:\n\n  Joannes Curopalata p. 686 (ed. Bonn.) \u03c6\u03ae\u03bc\u03b7 ... \u03c4\u03bf\u1f7a\u03c2 \u03a4\u03bf\u03cd\u03c1\u03ba\u03bf\u03c5\u03c2\n  \u1f00\u03c0\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03bf\u03c5\u03c3\u03b1 \u03c4\u1f74\u03bd \u1f10\u03bd \u03a7\u03ce\u03bd\u03b1\u03b9\u03c2 \u03c0\u03bf\u03bb\u03b9\u03c4\u03b5\u03af\u03b1\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03bd \u03c4\u1f78\u03bd \u03c0\u03b5\u03c1\u03b9\u03b2\u03cc\u03b7\u03c4\u03bf\u03bd \u1f10\u03bd\n  \u03b8\u03b1\u03cd\u03bc\u03b1\u03c3\u03b9 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03b1\u03b8\u03ae\u03bc\u03b1\u03c3\u03b9 \u03c4\u03bf\u1fe6 \u1f00\u03c1\u03c7\u03b9\u03c3\u03c4\u03c1\u03b1\u03c4\u03ae\u03b3\u03bf\u03c5 \u03bd\u03b1\u1f78\u03bd \u03ba\u03b1\u03c4\u03b1\u03bb\u03b1\u03b2\u03b5\u1fd6\u03bd \u1f10\u03bd \u03bc\u03b1\u03c7\u03b1\u03af\u03c1\u1fb3 ...\n  \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03b4\u1f74 \u03c3\u03c7\u03b5\u03c4\u03bb\u03b9\u03ce\u03c4\u03b5\u03c1\u03bf\u03bd, \u03bc\u03b7\u03b4\u1f72 \u03c4\u1f70\u03c2 \u03c4\u03bf\u1fe6 \u03c7\u03ac\u03c3\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c3\u03ae\u03c1\u03b1\u03b3\u03b3\u03b1\u03c2 \u1f10\u03bd \u1fa7\u03c0\u03b5\u03c1 \u03bf\u1f31\n  \u03c0\u03b1\u03c1\u03b1\u03c1\u03c1\u03ad\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c0\u03bf\u03c4\u03b1\u03bc\u03bf\u1f76 \u1f10\u03ba\u03b5\u1fd6\u03c3\u03b5 _\u03c7\u03c9\u03bd\u03b5\u03c5\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9_ \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u03c4\u03bf\u1fe6\n  \u1f00\u03c1\u03c7\u03b9\u03c3\u03c4\u03c1\u03b1\u03c4\u03ae\u03b3\u03bf\u03c5 \u03c0\u03b1\u03bb\u03b1\u03b9\u1fb6\u03c2 \u1f10\u03c0\u03b9\u03b4\u03b7\u03bc\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b8\u03b5\u03bf\u03c3\u03b7\u03bc\u03af\u03b1\u03c2 \u1f61\u03c2 \u03b4\u03b9\u1f70 \u03c0\u03c1\u03b1\u03bd\u03bf\u1fe6\u03c2 \u1f00\u03c3\u03c4\u03b1\u03c4\u03bf\u1fe6\u03bd\n  \u03c4\u1f78 \u1fe5\u03b5\u1fe6\u03bc\u03b1 \u03ba\u03b1\u1f76 \u03bb\u03b9\u1f70\u03bd \u03b5\u1f50\u03b4\u03c1\u03bf\u03bc\u03bf\u1fe6\u03bd \u1f14\u03c7\u03bf\u03c5\u03c3\u03b9, \u03c4\u03bf\u1f7a\u03c2 \u03ba\u03b1\u03c4\u03b1\u03c0\u03b5\u03c6\u03b5\u03c5\u03b3\u03cc\u03c4\u03b1\u03c2 \u03b4\u03b9\u03b1\u03c4\u03b7\u03c1\u1fc6\u03c3\u03b1\u03b9,\n  \u03ba.\u03c4.\u03bb. The 'worship of angels' is curiously connected with the\n  physical features of the country in the legend to which Curopalata\n  refers. The people were in imminent danger from a sudden inundation of\n  the Lycus, when the archangel Michael appeared and opened a chasm in\n  the earth through which the waters flowed away harmlessly: Hartley's\n  _Researches in Greece_ p. 53. See another legend, or another version\n  of the legend, in which the archangel interposes, in Laborde p. 103.\n\n  It was the birth-place of Nicetas Choniates, one of the most important\n  of the Byzantine historians, who thus speaks of it (_de Manuel_ vi. 2,\n  p. 230, ed. Bonn.); \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1\u03bd \u03b4\u03b9\u03b5\u03bb\u03b8\u1f7c\u03bd \u1f00\u03c6\u03b9\u03ba\u03bd\u03b5\u1fd6\u03c4\u03b1\u03b9 \u1f10\u03c2\n  \u03a7\u03ce\u03bd\u03b1\u03c2, \u03c0\u03cc\u03bb\u03b9\u03bd \u03b5\u1f50\u03b4\u03b1\u03af\u03bc\u03bf\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7\u03bd, \u03c0\u03ac\u03bb\u03b1\u03b9 \u03c4\u1f70\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03ac\u03c2, \u03c4\u1f72\u03bd \u1f10\u03bc\u03bf\u1fe6 \u03c4\u03bf\u1fe6\n  \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03ad\u03c9\u03c2 \u03c0\u03b1\u03c4\u03c1\u03af\u03b4\u03b1, \u03ba\u03b1\u1f76 \u03c4\u1f78\u03bd \u1f00\u03c1\u03c7\u03b1\u03b3\u03b3\u03b5\u03bb\u03b9\u03ba\u1f78\u03bd \u03bd\u03b1\u1f78\u03bd \u03b5\u1f30\u03c3\u03b9\u1f7c\u03bd \u03bc\u03b5\u03b3\u03ad\u03b8\u03b5\u03b9 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf\u03bd\n  \u03ba\u03b1\u1f76 \u03ba\u03ac\u03bb\u03bb\u03b5\u03b9 \u03ba\u03ac\u03bb\u03bb\u03b9\u03c3\u03c4\u03bf\u03bd \u1f44\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03b8\u03b1\u03c5\u03bc\u03b1\u03c3\u03af\u03b1\u03c2 \u03c7\u03b5\u03b9\u03c1\u1f78\u03c2 \u1f05\u03c0\u03b1\u03bd\u03c4\u03b1 \u1f14\u03c1\u03b3\u03bf\u03bd \u03ba.\u03c4.\u03bb.,\n  where a corrupt reading \u03a0\u03b1\u03bb\u03b1\u03c3\u03c3\u1f70\u03c2 for \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u1f70\u03c2 has misled some. It will\n  be remembered that the words \u03c0\u03cc\u03bb\u03b9\u03bd \u03b5\u1f50\u03b4\u03b1\u03af\u03bc\u03bf\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7\u03bd are borrowed\n  from Xenophon's description of Coloss\u00e6 (_Anab._ i. 2. 6): see above,\n  p. 15, note 52.\n\n  He again alludes to his native place, _de Isaac._ ii. 2, pp. 52, 3\n  \u03c4\u03bf\u1f7a\u03c2 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u1fd6\u03c2 \u03b4\u1f72 \u03a6\u03c1\u03cd\u03b3\u03b1\u03c2 \u03bc\u03c5\u03c1\u03b9\u03b1\u03c7\u1ff6\u03c2 \u1f10\u03ba\u03ac\u03ba\u03c9\u03c3\u03b5\u03bd, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1f7a\u03c2 \u03c4\u1ff6\u03bd \u03a7\u03c9\u03bd\u1ff6\u03bd\n  \u03c4\u1ff6\u03bd \u1f10\u03bc\u1ff6\u03bd \u03bf\u1f30\u03ba\u03ae\u03c4\u03bf\u03c1\u03b1\u03c2, and _Urbs Capta_ 16, p. 842, \u03c4\u1f78 \u03b4\u1f72 \u1f24\u03bd \u1f10\u03bc\u03bf\u1fe6 \u03c4\u03bf\u1fe6\n  \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03ad\u03c9\u03c2 \u039d\u03b9\u03ba\u03ae\u03c4\u03b1 \u03c0\u03b1\u03c4\u03c1\u1f76\u03c2 \u03b1\u1f31 \u03a7\u1ff6\u03bd\u03b1\u03b9 \u03ba\u03b1\u1f76 \u1f21 \u1f00\u03b3\u03c7\u03b9\u03c4\u03ad\u03c1\u03bc\u03c9\u03bd \u03c4\u03b1\u03cd\u03c4\u1fc3 \u03a6\u03c1\u03c5\u03b3\u03b9\u03ba\u1f74\n  \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1.\n\nFootnote 219:\n\n  Thus Hamilton (I. p. 514) reports that an earthquake which occurred at\n  Denizli about a hundred years ago caused the inhabitants to remove\n  their residences to a different locality, where they have remained\n  ever since. Earthquakes have been largely instrumental in changing the\n  sites of cities situated within the range of their influence.\n\nFootnote 220:\n\n  At the Council of Chalcedon (A.D. 451) Nunechius of Laodicea\n  subscribes 'for the absent bishops under him,' among whom is mentioned\n  \u1f18\u03c0\u03b9\u03c6\u03b1\u03bd\u03af\u03bf\u03c5 \u03c0\u03cc\u03bb\u03b5\u03c9\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u1ff6\u03bd (Labb. _Conc._ IV. 1501, ed. Coleti; comp.\n  _ib._ 1745). At the Quinisextine Council (A.D. 692) occurs the\n  signature of \u039a\u03bf\u03c3\u03bc\u1fb6\u03c2 \u1f10\u03c0\u03af\u03c3\u03ba\u03bf\u03c0\u03bf\u03c2 \u03c0\u03cc\u03bb\u03b5\u03c9\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u1fc6\u03c2 (_sic_) \u03a0\u03b1\u03ba\u03b1\u03c4\u03b9\u03b1\u03bd\u1fc6\u03c2\n  (_Conc._ VII. 1408). At the 2nd Council of Nic\u00e6a (A.D. 787) the name\n  of the see is in a transition state; the bishop Theodosius (or\n  Dositheus) signs himself sometimes \u03a7\u03c9\u03bd\u1ff6\u03bd \u1f24\u03c4\u03bf\u03b9 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u1ff6\u03bd, sometimes\n  \u03a7\u03c9\u03bd\u1ff6\u03bd simply (_Conc._ VIII. 689, 796, 988, 1200, 1222, 1357, 1378,\n  1432, 1523, 1533, in many of which passages the word \u03a7\u03c9\u03bd\u1ff6\u03bd is grossly\n  corrupted). At later Councils the see is called \u03a7\u1ff6\u03bd\u03b1\u03b9; and this is the\n  name which it bears in the _Notiti\u00e6_ (pp. 97, 127, 199, 222, 303, ed.\n  Parthey).\n\n[Sidenote: Turkish conquest.]\n\nThe Turkish conquest pressed with more than common severity on these\ndistricts. When the day of visitation came, the Church was taken by\nsurprise. Occupied with ignoble quarrels and selfish interests, she had\nno ear for the voice of Him who demanded admission. The door was barred\nand the knock unheeded. The long-impending doom overtook her, and the\ngolden candlestick was removed for ever from the Eternal Presence[221].\n\nFootnote 221:\n\n  For the remains of Christian churches at Laodicea see Fellows _Asia\n  Minor_ p. 282, Pococke p. 74. A description of the three fine churches\n  at Hierapolis is given in Fergusson's _Illustrated Handbook of\n  Architecture_ II. p. 967 sq.; comp. Texier _Asie Mineure_ I. p. 143.\n\n\n\n\n                                  II.\n                         THE COLOSSIAN HERESY.\n\n\n[Sidenote: Two elements in the Colossian heresy.]\n\nFrom the language of St Paul, addressed to the Church of Coloss\u00e6, we may\ninfer the presence of two disturbing elements which threatened the\npurity of Christian faith and practice in this community. These elements\nare distinguishable in themselves, though it does not follow that they\npresent the teaching of two distinct parties.\n\n[Sidenote: 1. JUDAIC.]\n\n1. A mere glance at the epistle suffices to detect the presence of\nJUDAISM in the teaching which the Apostle combats. The observance of\nsabbaths and new moons is decisive in this respect. The distinction of\nmeats and drinks points in the same direction[222]. Even the enforcement\nof the initiatory rite of Judaism may be inferred from the contrast\nimplied in St Paul's recommendation of the spiritual circumcision[223].\n\n[Sidenote: 2. GNOSTIC.]\n\n2. On the other hand a closer examination of its language shows that\nthese Judaic features do not exhaust the portraiture of the heresy or\nheresies against which the epistle is directed. We discern an element of\ntheosophic speculation, which is alien to the spirit of Judaism proper.\nWe are confronted with a shadowy mysticism, which loses itself in the\ncontemplation of the unseen world. We discover a tendency to interpose\ncertain spiritual agencies, intermediate beings, between God and man, as\nthe instruments of communication and the objects of worship[224].\nAnticipating the result which will appear more clearly hereafter, we may\nsay that along with its Judaism there was a GNOSTIC element in the false\nteaching which prevailed at Coloss\u00e6.\n\nFootnote 222:\n\n  Col. ii. 16, 17, 21 sq.\n\nFootnote 223:\n\n  ii. 11.\n\nFootnote 224:\n\n  ii. 4, 8, 18, 23.\n\n[Sidenote: Are these combined or separate?]\n\nHave we then two heresies here, or one only? Were these elements\ndistinct, or were they fused into the same system? In other words, Is St\nPaul controverting a phase of Judaism on the one hand, and a phase of\nGnosticism on the other; or did he find himself in conflict with a\nJud\u00e6o-Gnostic heresy which combined the two[225]?\n\nFootnote 225:\n\n  The Colossian heresy has been made the subject of special\n  dissertations by SCHNECKENBURGER _Beitr\u00e4ge zur Einleitung ins N. T._\n  (Stuttgart 1832), and _Ueber das Alter der j\u00fcdischen Proselyten-Taufe,\n  nebst einer Beilage \u00fcber die Irrlehrer zu Coloss\u00e4_ (Berlin 1828); by\n  OSIANDER _Ueber die Colossischen Irrlehrer_ (_T\u00fcbinger Zeitschrift_\n  for 1834, III. p. 96 sq.); and by RHEINWALD _De Pseudodoctoribus\n  Colossensibus_ (Bonn 1834). But more valuable contributions to the\n  subject will often be found in introductions to the commentaries on\n  the epistle. Those of BLEEK, DAVIES, MEYER, OLSHAUSEN, STEIGER, and DE\n  WETTE may be mentioned. Among other works which may be consulted are\n  BAUR _Der Apostel Paulus_ p. 417 sq.; BOEHMER _Isagoge in Epistolam ad\n  Colossenses_, Berlin 1829, p. 56 sq., p. 277 sq.; BURTON _Inquiry into\n  the Heresies of the Apostolic Age_, Lectures IV, V; EWALD _Die\n  Sendschreiben des Apostels Paulus_ p. 462 sq.; HILGENFELD _Der\n  Gnosticismus u. das Neue Testament_ in the _Zeitschr. f. Wissensch.\n  Theol._ XIII. p. 233 sq.; R. A. LIPSIUS in _Schenkels Bibel-Lexikon_,\n  s.v. Gnosis; MAYERHOFF _Der Brief an die Colosser_ p. 107 sq.; NEANDER\n  _Planting of the Christian Church_ I. p. 319 sq. (Eng. Trans.); DE\n  PRESSENS\u00c9 _Trois Premiers Si\u00e8cles_ II. p. 194 sq.; STORR _Opuscula_\n  II. p. 149 sq.; THIERSCH _Die Kirche im Apostolischen Zeitalter_ p.\n  146 sq. Of all the accounts of these Colossian false teachers, I have\n  found none more satisfactory than that of Neander, whose opinions are\n  followed in the main by the most sober of later writers.\n\n  In the investigation which follows I have assumed that the Colossian\n  false teachers were Christians in some sense. The views maintained by\n  some earlier critics, who regarded them as (1) Jews, or (2) Greek\n  philosophers, or (3) Chaldean magi, have found no favour and do not\n  need serious consideration. See Meyer's introduction for an\n  enumeration of such views. A refutation of them will be found in\n  Bleek's _Vorlesungen_ p. 12 sq.\n\n[Sidenote: General reasons for supposing one heresy only, in which they\n           are fused.]\n\nOn closer examination we find ourselves compelled to adopt the latter\nalternative. The epistle itself contains no hint that the Apostle has\nmore than one set of antagonists in view; and the needless\nmultiplication of persons or events is always to be deprecated in\nhistorical criticism. Nor indeed does the hypothesis of a single complex\nheresy present any real difficulty. If the two elements seem\nirreconcileable, or at least incongruous, at first sight, the\nincongruity disappears on further examination. It will be shown in the\ncourse of this investigation, that some special tendencies of religious\nthought among the Jews themselves before and about this time prepared\nthe way for such a combination in a Christian community like the Church\nof Coloss\u00e6[226]. Moreover we shall find that the Christian heresies of\nthe next succeeding ages exhibit in a more developed form the same\ncomplex type, which here appears in its nascent state[227]; this later\ndevelopment not only showing that the combination was historically\npossible in itself, but likewise presupposing some earlier stage of its\nexistence such as confronts us at Coloss\u00e6.\n\nFootnote 226:\n\n  See below, p. 83 sq.\n\nFootnote 227:\n\n  See below, p. 107 sq.\n\n[Sidenote: S. Paul's language is decisive on this point.]\n\nBut in fact the Apostle's language hardly leaves the question open. The\ntwo elements are so closely interwoven in his refutation, that it is\nimpossible to separate them. He passes backwards and forwards from the\none to the other in such a way as to show that they are only parts of\none complex whole. On this point the logical connexion of the sentences\nis decisive: 'Beware lest any man make spoil of you through philosophy\nand vain deceit after the tradition of men, after the rudiments of the\nworld.... Ye were circumcised with a circumcision not made with\nhands.... And you ... did He quicken, ... blotting out the handwriting\nof ordinances which was against you.... Let no man therefore judge you\nin meat or drink, or in respect of a holy day or a new moon or a\nsabbath.... Let no man beguile you of your prize in a self-imposed\nhumility and service of angels.... If ye died with Christ from the\nrudiments of the world, why ... are ye subject to ordinances ... which\nthings have a show of wisdom in self-imposed service and humility and\nhard treatment of the body, but are of no value against indulgence of\nthe flesh[228].' Here the superior wisdom, the speculative element which\nis characteristic of Gnosticism, and the ritual observance, the\npractical element which was supplied by Judaism, are regarded not only\nas springing from the same stem, but also as intertwined in their\ngrowth. And the more carefully we examine the sequence of the Apostle's\nthoughts, the more intimate will the connexion appear.\n\nFootnote 228:\n\n  Col. ii. 8\u201323. Hilgenfeld (_Der Gnosticismus_ etc. p. 250 sq.)\n  contends strenuously for the separation of the two elements. He argues\n  that 'these two tendencies are related to one another as fire and\n  water, and nothing stands in the way of allowing the author after the\n  first side-glance at the Gnostics to pass over with ver. 11 to the\n  Judaizers, with whom Col. ii. 16 sq. is exclusively concerned.' He\n  supposes therefore that ii. 8\u201310 refers to 'pure Gnostics,' and ii.\n  16\u201323 to 'pure Judaizers.' To this it is sufficient to answer (1)\n  That, if the two elements be so antagonistic, they managed\n  nevertheless to reconcile their differences; for we find them united\n  in several Jud\u00e6o-Gnostic heresies in the first half of the second\n  century, \u03be\u03c5\u03bd\u03ce\u03bc\u03bf\u03c3\u03b1\u03bd \u03b3\u03ac\u03c1, \u1f44\u03bd\u03c4\u03b5\u03c2 \u1f14\u03c7\u03b8\u03b9\u03c3\u03c4\u03bf\u03b9 \u03c4\u1f78 \u03c0\u03c1\u03af\u03bd, \u03c0\u1fe6\u03c1 \u03ba\u03b1\u1f76 \u03b8\u03ac\u03bb\u03b1\u03c3\u03c3\u03b1, \u03ba\u03b1\u1f76\n  \u03c4\u1f70 \u03c0\u03af\u03c3\u03c4' \u1f10\u03b4\u03b5\u03b9\u03be\u03ac\u03c4\u03b7\u03bd; (2) That the two passages are directly connected\n  together by \u03c4\u1f70 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5, which occurs in both vv. 8, 20;\n  (3) That it is not a simple transition once for all from the Gnostic\n  to the Judaic element, but the epistle passes to and fro several times\n  from the one to the other; while no hint is given that two separate\n  heresies are attacked, but on the contrary the sentences are connected\n  in a logical sequence (e.g. ver. 9 \u1f45\u03c4\u03b9, 10 \u1f43\u03c2, 11 \u1f10\u03bd \u1fa7, 12 \u1f10\u03bd \u1fa7, 13\n  \u03ba\u03b1\u1f76, 16 \u03bf\u1f56\u03bd). I hope to make this point clear in my notes on the\n  passage.\n\n  The hypothesis of more than one heresy is maintained also by Heinrichs\n  (Koppe N. T. VII. Part 2, 1803). At an earlier date it seems to be\n  favoured by Grotius (notes on ii. 16, 21); but his language is not\n  very explicit. And earlier still Calvin in his argument to the epistle\n  writes, 'Putant aliqui duo fuisse hominum genera, qui abducere\n  tentarent Colossenses ab evangelii puritate,' but rejects this view as\n  uncalled for.\n\n  The same question is raised with regard to the heretical teachers of\n  the Pastoral Epistles, and should probably be answered in the same\n  way.\n\n[Sidenote: Gnosticism must be defined and described.]\n\nHaving described the speculative element in this complex heresy\nprovisionally as Gnostic, I purpose enquiring in the first place, how\nfar Judaism prior to and independently of Christianity had allied itself\nwith Gnostic modes of thought; and afterwards, whether the description\nof the Colossian heresy is such as to justify us in thus classing it as\na species of Gnosticism. But, as a preliminary to these enquiries, some\ndefinition of the word, or at least some conception of the leading ideas\nwhich it involves, will be necessary. With its complex varieties and\nelaborate developments we have no concern here: for, if Gnosticism can\nbe found at all in the records of the Apostolic age, it will obviously\nappear in a simple and elementary form. Divested of its accessories and\npresented in its barest outline, it is not difficult of\ndelineation[229].\n\nFootnote 229:\n\n  The chief authorities for the history of Gnosticism are NEANDER\n  _Church History_ II. p. 1 sq.; BAUR _Die Christliche Gnosis_\n  (T\u00fcbingen, 1835); MATTER _Histoire Critique du Gnosticisme_ (2nd ed.,\n  Strasbourg and Paris, 1843); R. A. LIPSIUS _Gnosticismus_ in Ersch u.\n  Gruber _s.v._ (Leipzig, 1860); and for Gnostic art, KING _Gnostics and\n  their Remains_ (London 1864).\n\n[Sidenote: 1. Intellectual exclusiveness of Gnosticism.]\n\n1. As the name attests[230], Gnosticism implies the possession of a\nsuperior wisdom, which is hidden from others. It makes a distinction\nbetween the select few who have this higher gift, and the vulgar many\nwho are without it. Faith, blind faith, suffices the latter, while\nknowledge is the exclusive possession of the former. Thus it recognises\na separation of intellectual _caste_ in religion, introducing the\ndistinction of an esoteric and an exoteric doctrine, and interposing an\ninitiation of some kind or other between the two classes. In short it is\nanimated by the exclusive _aristocratic_ spirit[231], which\ndistinguishes the ancient religions, and from which it was a main\nfunction of Christianity to deliver mankind.\n\nFootnote 230:\n\n  See esp. Iren. i. 6. 1 sq., Clem. Alex. Strom. ii. p. 433 sq.\n  (Potter). On the words \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03b9, \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03af, by which they designated\n  the possessors of this higher _gnosis_, see the notes on Col. i. 28,\n  and Phil. iii. 15.\n\nFootnote 231:\n\n  See Neander l.c. p. 1 sq., from whom the epithet is borrowed.\n\n[Sidenote: Speculative tenets of Gnosticism.]\n\n2. This was its spirit; and the intellectual questions, on which its\nenergies were concentrated and to which it professed to hold the key,\nwere mainly twofold. How can the work of creation be explained? and, How\nare we to account for the existence of evil[232]? To reconcile the\ncreation of the world and [Sidenote: Creation of the world, and\nexistence of evil.] the existence of evil with the conception of God as\nthe absolute Being, was the problem which all the Gnostic systems set\nthemselves to solve. It will be seen that the two questions cannot be\ntreated independently but have a very close and intimate connexion with\neach other.\n\nFootnote 232:\n\n  The fathers speak of this as the main question about which the\n  Gnostics busy themselves; _Unde malum?_ \u03c0\u03cc\u03b8\u03b5\u03bd \u1f21 \u03ba\u03b1\u03ba\u03af\u03b1; Tertull. _de\n  Pr\u00e6scr._ 7, _adv. Marc._ I. 2, Eus. _H.E._ v. 27; passages quoted by\n  Baur _Christliche Gnosis_ p. 19. On the leading conceptions of\n  Gnosticism see especially Neander, l.c. p. 9 sq.\n\n[Sidenote: Existence of evil, how to be explained?]\n\nThe Gnostic argument ran as follows: Did God create the world out of\nnothing, evolve it from Himself? Then, God being perfectly good and\ncreation having resulted from His sole act without any opposing or\nmodifying influence, evil would have been impossible; for otherwise we\nare driven to the conclusion that God created evil.\n\n[Sidenote: Matter the abode of evil.]\n\nThis solution being rejected as impossible, the Gnostic was obliged to\npostulate some antagonistic principle independent of God, by which His\ncreative energy was thwarted and limited. This opposing principle, the\nkingdom of evil, he conceived to be the world of matter. The precise\nidea of its mode of operation varies in different Gnostic systems. It is\nsometimes regarded as a dead passive resistance, sometimes as a\nturbulent active power. But, though the exact point of view may shift,\nthe object contemplated is always the same. In some way or other evil is\nregarded as residing in the material, sensible world. Thus Gnostic\nspeculation on the existence of evil ends in a dualism.\n\n[Sidenote: Creation, how to be explained?]\n\nThis point being conceded, the ulterior question arises: How then is\ncreation possible? How can the Infinite communicate with the Finite, the\nGood with the Evil? How can God act upon matter? God is perfect,\nabsolute, incomprehensible.\n\nThis, the Gnostic went on to argue, could only have been possible by\nsome self-limitation on the part of God. God must express Himself in\nsome way. There must be some evolution, some effluence, of Deity.\n[Sidenote: Doctrine of emanations.] Thus the Divine Being germinates, as\nit were; and the first germination again evolves a second from itself in\nlike manner. In this way we obtain a series of successive emanations,\nwhich may be more or fewer, as the requirements of any particular system\ndemand. In each successive evolution the Divine element is feebler. They\nsink gradually lower and lower in the scale, as they are farther removed\nfrom their source; until at length contact with matter is possible, and\ncreation ensues. These are the emanations, \u00e6ons, spirits, or angels, of\nGnosticism, conceived as more or less concrete and personal according to\nthe different aspects in which they are regarded in different systems.\n\n[Sidenote: 3. Practical errors of Gnosticism.]\n\n3. Such is the bare outline (and nothing more is needed for my immediate\npurpose) of the speculative views of Gnosticism. But it is obvious that\nthese views must have exerted a powerful influence on the ethical\nsystems of their advocates, and thus they would involve important\npractical consequences. If matter is the principle of evil, it is of\ninfinite moment for a man to know how he can avoid its baneful influence\nand thus keep his higher nature unclogged and unsullied.\n\n[Sidenote: Two opposite ethical rules.]\n\nTo this practical question two directly opposite answers were\ngiven[233]:\n\nFootnote 233:\n\n  On this point see Clem. _Strom._ iii. 5 (p. 529) \u03b5\u1f30\u03c2 \u03b4\u03cd\u03bf \u03b4\u03b9\u03b5\u03bb\u03cc\u03bd\u03c4\u03b5\u03c2\n  \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b1 \u1f01\u03c0\u03ac\u03c3\u03b1\u03c2 \u03c4\u1f70\u03c2 \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03b9\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03b9\u03bd\u03ce\u03bc\u03b5\u03b8\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2\u00b7 \u1f22 \u03b3\u03ac\u03c1 \u03c4\u03bf\u03b9 \u1f00\u03b4\u03b9\u03b1\u03c6\u03cc\u03c1\u03c9\u03c2\n  \u03b6\u1fc6\u03bd \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03c5\u03c3\u03b9\u03bd, \u1f22 \u03c4\u1f78 \u1f51\u03c0\u03ad\u03c1\u03c4\u03bf\u03bd\u03bf\u03bd \u1f04\u03b3\u03bf\u03c5\u03c3\u03b1\u03b9 \u1f10\u03b3\u03ba\u03c1\u03ac\u03c4\u03b5\u03b9\u03b1\u03bd \u03b4\u03b9\u1f70 \u03b4\u03c5\u03c3\u03c3\u03b5\u03b2\u03b5\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76\n  \u03c6\u03b9\u03bb\u03b1\u03c0\u03b5\u03c7\u03b8\u03b7\u03bc\u03bf\u03c3\u03cd\u03bd\u03b7\u03c2 \u03ba\u03b1\u03c4\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03bf\u03c5\u03c3\u03b9, with the whole passage which follows.\n  As examples of the one extreme may be instanced the Carpocratians and\n  Cainites: of the other the Encratites.\n\n[Sidenote: (i) Rigid asceticism.]\n\n(i) On the one hand, it was contended that the desired end might best be\nattained by a rigorous abstinence. Thus communication with matter, if it\ncould not be entirely avoided, might be reduced to a minimum. Its\ngrosser defilements at all events would be escaped. The material part of\nman would be subdued and mortified, if it could not be annihilated; and\nthe spirit, thus set free, would be sublimated, and rise to its proper\nlevel. Thus the ethics of Gnosticism pointed in the first instance to a\nstrict _asceticism_.\n\n[Sidenote: (ii) Unrestrained license.]\n\n(ii) But obviously the results thus attained are very slight and\ninadequate. Matter is about us everywhere. We do but touch the skirts of\nthe evil, when we endeavour to fence ourselves about by prohibitive\nordinances, as for instance, when we enjoin a spare diet or forbid\nmarriage. Some more comprehensive rule is wanted, which shall apply to\nevery contingency and every moment of our lives. Arguing in this way,\nother Gnostic teachers arrived at an ethical rule directly opposed to\nthe former. 'Cultivate an entire indifference,' they said, 'to the world\nof sense. Do not give it a thought one way or the other, but follow your\nown impulses. The ascetic principle assigns a certain importance to\nmatter. The ascetic fails in consequence to assert his own independence.\nThe true rule of life is to treat matter as something alien to you,\ntowards which you have no duties or obligations and which you can use or\nleave unused as you like[234].' In this way the reaction from rigid\nasceticism led to the opposite extreme of unrestrained _licentiousness_,\nboth alike springing from the same false conception of matter as the\nprinciple of evil.\n\nFootnote 234:\n\n  See for instance the description of the Carpocratians in Iren. i. 25.\n  3 sq., ii. 32. 1 sq., Hippol. _H\u00e6r._ vii. 32, Epiphan. _H\u00e6r._ xxvii. 2\n  sq.; from which passages it appears that they justified their moral\n  profligacy on the principle that the highest perfection consists in\n  the most complete contempt of mundane things.\n\n[Sidenote: Original independence of Gnosticism and its subsequent\n           connexion with Christianity.]\n\nGnosticism, as defined by these characteristic features, has obviously\nno necessary connexion with Christianity[235]. Christianity would\nnaturally arouse it to unwonted activity, by leading men to dwell more\nearnestly on the nature and power of evil, and thus stimulating more\nsystematic thought on the theological questions which had already\narrested attention. After no long time Gnosticism would absorb into its\nsystem more or fewer Christian elements, or Christianity in some of its\nforms would receive a tinge from Gnosticism. But the thing itself had an\nindependent root, and seems to have been prior in time. The\nprobabilities of the case, and the scanty traditions of history, alike\npoint to this independence of the two[236]. If so, it is a matter of\nlittle moment at what precise time the name 'Gnostic' was adopted,\nwhether before or after contact with Christianity; for we are concerned\nonly with the growth and direction of thought which the name\nrepresents[237].\n\nFootnote 235:\n\n  It will be seen from the description in the text, that Gnosticism (as\n  I have defined it) presupposes only a belief in one God, the absolute\n  Being, as against the vulgar polytheism. All its _essential_ features,\n  as a speculative system, may be explained from this simple element of\n  belief, without any intervention of specially Christian or even Jewish\n  doctrine. Christianity added two new elements to it; (1) the idea of\n  _Redemption_, (2) the person of _Christ_. To explain the former, and\n  to find a place for the latter, henceforth become prominent questions\n  which press for solution; and Gnosticism in its several developments\n  undergoes various modifications in the endeavour to solve them.\n  Redemption must be set in some relation to the fundamental Gnostic\n  conception of the antagonism between God and matter; and Christ must\n  have some place found for Him in the fundamental Gnostic doctrine of\n  emanations.\n\n  If it be urged that there is no authority for the name 'Gnostic' as\n  applied to these pre-Christian theosophists, I am not concerned to\n  prove the contrary, as my main position is not affected thereby. The\n  term 'Gnostic' is here used, only because no other is so convenient,\n  or so appropriate. See note 239, p. 81.\n\nFootnote 236:\n\n  This question will require closer investigation when I come to discuss\n  the genuineness of the Epistle to the Colossians. Meanwhile I content\n  myself with referring to Baur _Christliche Gnosis_ p. 29 sq. and\n  Lipsius _Gnosticismus_ p. 230 sq. Both these writers concede, and\n  indeed insist upon, the non-Christian basis of Gnosticism, at least so\n  far as I have maintained it in the text. Thus for instance Baur says\n  (p. 52), 'Though Christian gnosis is the completion of gnosis, yet the\n  Christian element in gnosis is not so essential as that gnosis cannot\n  still be gnosis even without this element. But just as we can abstract\n  it from the Christian element, so can we also go still further and\n  regard even the Jewish as not strictly an essential element of\n  gnosis.' In another work (_Die drei ersten Jahrhunderte_, p. 167, 1st\n  ed.) he expresses himself still more strongly to the same effect, but\n  the expressions are modified in the second edition.\n\nFootnote 237:\n\n  We may perhaps gather from the notices which are preserved that,\n  though the substantive \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 was used with more or less precision\n  even before contact with Christianity to designate the superior\n  illumination of these opinions, the adjective \u03b3\u03bd\u03c9\u03c3\u03c4\u03b9\u03ba\u03bf\u03af was not\n  distinctly applied to those who maintained them till somewhat later.\n  Still it is possible that pre-Christian Gnostics already so designated\n  themselves. Hippolytus speaks of the Naassenes or Ophites as giving\n  themselves this name; _H\u00e6r._ v. 6 \u03bc\u03b5\u03c4\u1f70 \u03b4\u1f72 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u1f10\u03c0\u03b5\u03ba\u03ac\u03bb\u03b5\u03c3\u03b1\u03bd \u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2\n  \u03b3\u03bd\u03c9\u03c3\u03c4\u03b9\u03ba\u03bf\u1f7a\u03c2, \u03c6\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03bc\u03cc\u03bd\u03bf\u03b9 \u03c4\u1f70 \u03b2\u03ac\u03b8\u03b7 \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd; comp. \u00a7\u00a7 8, 11. His\n  language seems to imply (though it is not explicit) that they were the\n  first to adopt the name. The Ophites were plainly among the earliest\n  Gnostic sects, as the heathen element is still predominant in their\n  teaching, and their Christianity seems to have been a later graft on\n  their pagan theosophy; but at what stage in their development they\n  adopted the name \u03b3\u03bd\u03c9\u03c3\u03c4\u03b9\u03ba\u03bf\u03af does not appear. Iren\u00e6us (_H\u00e6r._ i. 25. 6)\n  speaks of the name as affected especially by the Carpocratians. For\n  the use of the substantive \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 see 1 Cor. viii. 1, xiii. 2, 8, 1\n  Tim. vi. 20, and the note on Col. ii. 3: comp. Rev. ii. 24 \u1f44\u03b9\u03c4\u03b9\u03bd\u03b5\u03c2 \u03bf\u1f50\u03ba\n  \u1f14\u03b3\u03bd\u03c9\u03c3\u03b1\u03bd \u03c4\u1f70 \u03b2\u03b1\u03b8\u03ad\u03b1 \u03c4\u03bf\u1fe6 \u03a3\u03b1\u03c4\u03b1\u03bd\u1fb6, \u1f61\u03c2 \u03bb\u03ad\u03b3\u03bf\u03c5\u03c3\u03b9\u03bd (as explained by the passage\n  already quoted from Hippol. _H\u00e6r._ v. 6; see _Galatians_, p. 298, note\n  3).\n\n                  *       *       *       *       *\n\n[Sidenote: Its alliance with Judaism before Christianity.]\n\nIf then Gnosticism was not an offspring of Christianity, but a direction\nof religious speculation which existed independently, we are at liberty\nto entertain the question whether it did not form an alliance with\nJudaism, contemporaneously with or prior to its alliance with\nChristianity. There is at least no obstacle which bars such an\ninvestigation at the outset. If this should prove to be the case, then\nwe have a combination which prepares the way for the otherwise strange\nphenomena presented in the Epistle to the Colossians.\n\n[Sidenote: The three sects of the Jews.]\n\nThose, who have sought analogies to the three Jewish sects among the\nphilosophical schools of Greece and Rome, have compared the Sadducees to\nthe Epicureans, the Pharisees to the Stoics, and the Essenes to the\nPythagoreans. Like all historical parallels, this comparison is open to\nmisapprehension: but, carefully guarded, the illustration is pertinent\nand instructive.\n\n[Sidenote: Sadduceeism, purely negative.]\n\nWith the Sadducees we have no concern here. Whatever respect may be due\nto their attitude in the earlier stages of their history, at the\nChristian era at least they have ceased to deserve our sympathy; for\ntheir position has become mainly _negative_. They take their stand on\ndenials\u2014the denial of the existence of angels, the denial of the\nresurrection of the dead, the denial of a progressive development in the\nJewish Church. In these negative tendencies, in the materialistic\nteaching of the sect, and in the moral consequences to which it led, a\nvery rough resemblance to the Epicureans will appear[238].\n\nFootnote 238:\n\n  The name _Epicureans_ seems to be applied to them even in the Talmud;\n  see Eisenmenger's _Entdecktes Judenthum_ i. pp. 95, 694 sq.; comp.\n  Keim _Geschichte Jesu von Nazara_ i. p. 281.\n\n[Sidenote: Phariseeism and Essenism compared.]\n\nThe two _positive_ sects were the Pharisees and the Essenes. Both alike\nwere strict observers of the ritual law; but, while the Pharisee was\nessentially _practical_, the tendency of the Essene was to _mysticism_;\nwhile the Pharisee was a man of the world, the Essene was a member of a\nbrotherhood. In this respect the Stoic and the Pythagorean were the\nnearest counterparts which the history of Greek philosophy and social\nlife could offer. These analogies indeed are suggested by Josephus\nhimself[239].\n\nFootnote 239:\n\n  For the Pharisees see Vit. 2 \u03c0\u03b1\u03c1\u03b1\u03c0\u03bb\u03ae\u03c3\u03b9\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9 \u03c4\u1fc7 \u03c0\u03b1\u03c1' \u1f1d\u03bb\u03bb\u03b7\u03c3\u03b9 \u03a3\u03c4\u03c9\u03ca\u03ba\u1fc7\n  \u03bb\u03b5\u03b3\u03bf\u03bc\u03ad\u03bd\u1fc3: for the Essenes, _Ant._ xv. 10. 4 \u03b4\u03b9\u03b1\u03af\u03c4\u1fc3 \u03c7\u03c1\u03ce\u03bc\u03b5\u03bd\u03bf\u03bd \u03c4\u1fc7 \u03c0\u03b1\u03c1'\n  \u1f1d\u03bb\u03bb\u03b7\u03c3\u03b9\u03bd \u1f51\u03c0\u1f78 \u03a0\u03c5\u03b8\u03b1\u03b3\u03cc\u03c1\u03bf\u03c5 \u03ba\u03b1\u03c4\u03b1\u03b4\u03b5\u03b4\u03b5\u03b9\u03b3\u03bc\u03ad\u03bd\u1fc3.\n\n[Sidenote: Elusive features of Essenism.]\n\nWhile the portrait of the Pharisee is distinctly traced and easily\nrecognised, this is not the case with the Essene. The Essene is the\ngreat enigma of Hebrew history. Admired alike by Jew, by Heathen, and by\nChristian, he yet remains a dim vague outline, on which the highest\nsubtlety of successive critics has been employed to supply a substantial\nform and an adequate colouring. An ascetic mystical dreamy recluse, he\nseems too far removed from the hard experience of life to be capable of\nrealisation.\n\n[Sidenote: A sufficiently distinct portrait of the sect attainable.]\n\nAnd yet by careful use of the existing materials the portrait of this\nsect may be so far restored, as to establish with a reasonable amount of\nprobability the point with which alone we are here concerned. It will\nappear from the delineations of ancient writers, more especially of\nPhilo and Josephus, that the characteristic feature of Essenism was a\nparticular direction of mystic speculation, involving a rigid asceticism\nas its practical consequence. Following the definition of Gnosticism\nwhich has been already given, we may not unfitly call this tendency\n_Gnostic_.\n\n[Sidenote: Main features of Essenism.]\n\nHaving anticipated the results in this statement, I shall now endeavour\nto develope the main features of Essenism; and, while doing so, I will\nask my readers to bear in mind the portrait of the Colossian heresy in\nSt Paul, and to mark the resemblances, as the enquiry proceeds[240].\n\nFootnote 240:\n\n   The really important contemporary sources of information respecting\n  the Essenes are JOSEPHUS, _Bell. Jud._ ii. 8. 2\u201313, _Ant._ xiii. 5. 9,\n  xviii. 1. 5, _Vit._ 2 (with notices of individual Essenes _Bell. Jud._\n  i. 3. 5, ii. 7. 3, ii. 20. 4, iii. 2. 1, _Ant._ xiii. 11. 2, xv. 10.\n  4, 5); and PHILO, _Quod omnis probus liber_ \u00a7 12 sq. (II. p. 457 sq.),\n  _Apol. pro Jud._ (II. p. 632 sq., a fragment quoted by Eusebius _Pr\u00e6p.\n  Evang._ viii. 11 ). The account of the Therapeutes by the latter\n  writer, _de Vita Contemplativa_ (II. p. 471 sq.), must also be\n  consulted, as describing a closely allied sect. To these should be\n  added the short notice of PLINY, _N.H._ v. 15. 17, as expressing the\n  views of a Roman writer. His account, we may conjecture, was taken\n  from Alexander Polyhistor, a contemporary of Sulla, whom he mentions\n  in his prefatory elenchus as one of his authorities for this 5th book,\n  and who wrote a work _On the Jews_ (Clem. Alex. _Strom._ i. 21, p.\n  396, Euseb. _Pr\u00e6p. Ev._ ix. 17). Significant mention of the Essenes is\n  found also in the Christian HEGESIPPUS (Euseb. _H.E._ iv. 22) and in\n  the heathen DION CHRYSOSTOM (Synesius _Dion_ 3, p. 39). EPIPHANIUS\n  (_H\u00e6r._ pp. 28 sq., 40 sq.) discusses two separate sects, which he\n  calls _Essenes_ and _Oss\u00e6ans_ respectively. These are doubtless\n  different names of the same persons. His account is, as usual,\n  confused and inaccurate, but has a certain value. All other\n  authorities are secondary. HIPPOLYTUS, _H\u00e6r._ ix. 18\u201328, follows\n  Josephus (_Bell. Jud._ ii. 8. 2 sq.) almost exclusively. PORPHYRY also\n  (_de Abstinentia_, iv. II sq.) copies this same passage of Josephus,\n  with a few unimportant exceptions probably taken from a lost work by\n  the same author, \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f19\u03bb\u03bb\u03b7\u03bd\u03b1\u03c2, which he mentions by name.\n  EUSEBIUS (_Pr\u00e6p. Evang._ viii. II sq., ix. 3) contents himself with\n  quoting Philo and Porphyry. SOLINUS (_Polyh._ xxxv. 9 sq.) merely\n  abstracts Pliny. TALMUDICAL and RABBINICAL passages, supposed to\n  contain references to the Essenes, are collected by Frankel in the\n  articles mentioned in a later paragraph; but the allusions are most\n  uncertain (see the appendix to this chapter). The authorities for the\n  history of the Essenes are the subject of an article by W. Clemens in\n  the _Zeitschr. f. Wiss. Theol._ 1869, p. 328 sq.\n\n  The attack on the genuineness of Philo's treatise _De Vita\n  Contemplativa_ made by Gr\u00e4tz (III. p. 463 sq.) has been met by Zeller\n  (_Philosophie_, III. ii. p. 255 sq.), whose refutation is complete.\n  The attack of the same writer (III. p. 464) on the genuineness of the\n  treatise _Quod omnis probus liber_> Zeller considers too frivolous to\n  need refuting (_ib._ p. 235). A refutation will be found in the\n  above-mentioned article of W. Clemens (p. 340 sq.).\n\n  Of modern writings relating to the Essenes the following may be\n  especially mentioned; BELLERMANN _Ueber Ess\u00e4er u. Therapeuten_, Berlin\n  1821; GFR\u00d6RER _Philo_ II. p. 299 sq.; D\u00c4HNE _Ersch u. Gruber's\n  Encyklop\u00e4die_ s.v.; FRANKEL _Zeitschrift f\u00fcr die religi\u00f6sen Interessen\n  des Judenthums_ 1846 p. 441 sq., _Monatschrift f\u00fcr Geschichte u.\n  Wissenschaft des Judenthums_ 1853 p. 30 sq., 61 sq.; B\u00d6TTGER_Ueber den\n  Orden der Ess\u00e4er_, Dresden 1849; EWALD _Geschichte des Volkes Israel_\n  IV. p. 420 sq., VII. p. 153 sq.; RITSCHL _Entstehung der\n  Altkatholischen Kirche_ p. 179 sq. (ed. 2, 1857), and _Theologische\n  Jahrb\u00fccher_ 1855, p. 315 sq.; JOST _Geschichte des Judenthums_ I. p.\n  207 sq.; GRAETZ _Geschichte der Juden_ III. p. 79 sq., 463 sq. (ed. 2,\n  1863); HILGENFELD _J\u00fcdische Apocalyptik_ p. 245 sq., and _Zeitschr. f.\n  Wiss. Theol._ X. p. 97 sq., XI. p. 343 sq., XIV. p. 30 sq.; WESTCOTT\n  _Smith's Dictionary of the Bible_ s.v.; GINSBURG _The Essenes_, London\n  1864, and in _Kitto's Cyclop\u00e6dia_ s.v.; DERENBOURG _L'Histoire et la\n  G\u00e9ographie de la Palestine_ p. 166 sq., 460 sq.; KEIM _Geschichte Jesu\n  von Nazara_ I. p. 282 sq.; HAUSRATH _Neutestamentliche Zeitgeschichte_\n  I. p. 133 sq.; LIPSIUS _Schenkel's Bibel Lexikon_ s.v.; HERZFELD\n  _Geschichte des Volkes Israel_ II. 368 sq., 388 sq., 509 sq. (ed. 2,\n  1863); ZELLER _Philosophie der Griechen_ III. 2. p. 234 sq. (ed. 2,\n  1868); LANGEN _Judenthum in Pal\u00e4stina_ p. 190 sq.; L\u00d6WY\n  _Kritisch-talmudisches Lexicon_ s.v. (Wien 1863); WEISS _Zur\n  Geschichte der j\u00fcdischen Tradition_ p. 120 sq. (Wien).\n\nThe Judaic element is especially prominent in the life and teaching of\nthe sect. The Essene was exceptionally rigorous in his observance of the\nMosaic ritual. In his strict abstinence [Sidenote: Observance of the\nMosaic law.] from work on the sabbath he far surpassed all the other\nJews. He would not light a fire, would not move a vessel, would not\nperform even the most ordinary functions of life[241]. The whole day was\ngiven up to religious exercises and to exposition of the\nScriptures[242]. His respect for the law extended also to the law-giver.\nAfter God, the name of Moses was held in the highest reverence. He who\nblasphemed his name was punished with death[243]. In all these points\nthe Essene was an exaggeration, almost a caricature, of the Pharisee.\n\nFootnote 241:\n\n  _B.J._ ii. 8. 9 \u03c6\u03c5\u03bb\u03ac\u03c3\u03c3\u03bf\u03bd\u03c4\u03b1\u03b9 ... \u03c4\u03b1\u1fd6\u03c2 \u1f11\u03b2\u03b4\u03cc\u03bc\u03b1\u03c3\u03b9\u03bd \u1f14\u03c1\u03b3\u03c9\u03bd \u1f10\u03c6\u03ac\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9\n  \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ce\u03c4\u03b1\u03c4\u03b1 \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd \u1f01\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd\u00b7 \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u03b3\u1f70\u03c1 \u03c4\u03c1\u03bf\u03c6\u1f70\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 \u03c0\u03c1\u1f78 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2\n  \u03bc\u03b9\u1fb6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c3\u03ba\u03b5\u03c5\u03ac\u03b6\u03bf\u03c5\u03c3\u03b9\u03bd, \u1f61\u03c2 \u03bc\u03b7\u03b4\u1f72 \u03c0\u1fe6\u03c1 \u1f10\u03bd\u03b1\u03cd\u03bf\u03b9\u03b5\u03bd \u1f10\u03ba\u03b5\u03af\u03bd\u1fc3 \u03c4\u1fc7 \u1f21\u03bc\u03ad\u03c1\u1fb3, \u1f00\u03bb\u03bb' \u03bf\u1f50\u03b4\u1f72\n  \u03c3\u03ba\u03b5\u1fe6\u03cc\u03c2 \u03c4\u03b9 \u03bc\u03b5\u03c4\u03b1\u03ba\u03b9\u03bd\u1fc6\u03c3\u03b1\u03b9 \u03b8\u03b1\u03c1\u1fe5\u03bf\u1fe6\u03c3\u03b9\u03bd \u03ba.\u03c4.\u03bb. Hippolytus (_H\u00e6r._ ix. 25) adds\n  that some of them do not so much as leave their beds on this day.\n\nFootnote 242:\n\n  Philo _Quod omn. prob. lib._ \u00a7 12. Of the Therapeutes see Philo _Vit.\n  Cont._ \u00a7 3, 4.\n\nFootnote 243:\n\n  _B.J._ l.c. \u00a7 9 \u03c3\u03ad\u03b2\u03b1\u03c2 \u03b4\u1f72 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03bc\u03b5\u03c4\u1f70 \u03c4\u1f78\u03bd \u0398\u03b5\u1f78\u03bd \u03c4\u1f78 \u1f44\u03bd\u03bf\u03bc\u03b1\n  \u03c4\u03bf\u1fe6 \u03bd\u03bf\u03bc\u03bf\u03b8\u03ad\u03c4\u03bf\u03c5, \u03ba\u1f02\u03bd \u03b2\u03bb\u03b1\u03c3\u03c6\u03b7\u03bc\u03ae\u03c3\u1fc3 \u03c4\u03b9\u03c2 \u03b5\u1f30\u03c2 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd (i.e. \u03c4\u1f78\u03bd \u03bd\u03bf\u03bc\u03bf\u03b8\u03ad\u03c4\u03b7\u03bd),\n  \u03ba\u03bf\u03bb\u03ac\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9 \u03b8\u03b1\u03bd\u03ac\u03c4\u1ff3: comp. \u00a7 10.\n\n[Sidenote: External elements superadded.]\n\nSo far the Essene has not departed from the principles of normal\nJudaism; but here the divergence begins. In three main points we trace\nthe working of influences, which must have been derived from external\nsources.\n\n[Sidenote: 1. Rigid asceticism, in respect to]\n\n1. To the legalism of the Pharisee, the Essene added an asceticism,\nwhich was peculiarly his own, and which in many respects contradicted\nthe tenets of the other sect. The honourable, and even exaggerated,\nestimate of marriage, which was characteristic of the Jew, and of the\nPharisee as the typical Jew, found no favour with the Essene[244].\n[Sidenote: marriage,]Marriage was to him an abomination. Those Essenes\nwho lived together as members of an order, and in whom the principles of\nthe sect were carried to their logical consequences, eschewed it\naltogether. To secure the continuance of their brotherhood they adopted\nchildren, whom they brought up in the doctrines and practices of the\ncommunity. There were others however who took a different view. They\naccepted marriage, as necessary for the preservation of the race. Yet\neven with them it seems to have been regarded only as an inevitable\nevil. They fenced it off by stringent rules, demanding a three years'\nprobation and enjoining various purificatory rites[245]. The conception\nof marriage, as quickening and educating the affections and thus\nexalting and refining human life, was wholly foreign to their minds.\nWoman was a mere instrument of temptation in their eyes, deceitful,\nfaithless, selfish, jealous, misled and misleading by her passions.\n\nFootnote 244:\n\n  _B.J._ l.c. \u00a7 2 \u03b3\u03ac\u03bc\u03bf\u03c5 \u03bc\u1f72\u03bd \u1f51\u03c0\u03b5\u03c1\u03bf\u03c8\u03af\u03b1 \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 ... \u03c4\u1f70\u03c2 \u03c4\u1ff6\u03bd \u03b3\u03c5\u03bd\u03b1\u03af\u03ba\u03c9\u03bd\n  \u1f00\u03c3\u03b5\u03bb\u03b3\u03b5\u03af\u03b1\u03c2 \u03c6\u03c5\u03bb\u03b1\u03c3\u03c3\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03bc\u03b7\u03b4\u03b5\u03bc\u03af\u03b1\u03bd \u03c4\u03b7\u03c1\u03b5\u1fd6\u03bd \u03c0\u03b5\u03c0\u03b5\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf\u03b9 \u03c4\u1f74\u03bd \u03c0\u03c1\u1f78\u03c2 \u1f15\u03bd\u03b1\n  \u03c0\u03af\u03c3\u03c4\u03b9\u03bd, _Ant._ xviii. 1. 5; Philo _Fragm._ p. 633 \u03b3\u03ac\u03bc\u03bf\u03bd \u03c0\u03b1\u03c1\u1fc3\u03c4\u03ae\u03c3\u03b1\u03bd\u03c4\u03bf\n  \u03bc\u03b5\u03c4\u1f70 \u03c4\u03bf\u1fe6 \u03b4\u03b9\u03b1\u03c6\u03b5\u03c1\u03cc\u03bd\u03c4\u03c9\u03c2 \u1f00\u03c3\u03ba\u03b5\u1fd6\u03bd \u1f10\u03b3\u03ba\u03c1\u03ac\u03c4\u03b5\u03b9\u03b1\u03bd\u00b7 \u1f18\u03c3\u03c3\u03b1\u03af\u03c9\u03bd \u03b3\u1f70\u03c1 \u03bf\u1f50\u03b4\u03b5\u03b9\u03c2 \u1f04\u03b3\u03b5\u03c4\u03b1\u03b9\n  \u03b3\u03c5\u03bd\u03b1\u1fd6\u03ba\u03b1, \u03b4\u03af\u03bf\u03c4\u03b9 \u03c6\u03af\u03bb\u03b1\u03c5\u03c4\u03bf\u03bd \u1f21 \u03b3\u03c5\u03bd\u1f74 \u03ba\u03b1\u1f76 \u03b6\u03b7\u03bb\u03cc\u03c4\u03c5\u03c0\u03bf\u03bd \u03bf\u1f50 \u03bc\u03b5\u03c4\u03c1\u03af\u03c9\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03b5\u03b9\u03bd\u1f78\u03bd\n  \u1f00\u03bd\u03b4\u03c1\u1f78\u03c2 \u1f24\u03b8\u03b7 \u03c0\u03b1\u03c1\u03b1\u03c3\u03b1\u03bb\u03b5\u1fe6\u03c3\u03b1\u03b9, with more to the same purpose. This\n  peculiarity astonished the heathen Pliny, _N.H._ v. 15, 'gens sola et\n  in toto orbe pr\u00e6ter ceteros mira, sine ulla femina, venere\n  abdicata.... In diem ex \u00e6quo convenarum turba renascitur large\n  frequentantibus.... Ita per s\u00e6culorum millia (incredibile dictu) gens\n  \u00e6terna est, in qua nemo nascitur. Tam f\u0153cunda illis aliorum vit\u00e6\n  p\u0153nitentia est.'\n\nFootnote 245:\n\n  _B.J._ l.c. \u00a7 13. Josephus speaks of these as \u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u1f18\u03c3\u03c3\u03b7\u03bd\u1ff6\u03bd \u03c4\u03ac\u03b3\u03bc\u03b1, \u1f43\n  \u03b4\u03af\u03b1\u03b9\u03c4\u03b1\u03bd \u03bc\u1f72\u03bd \u03ba\u03b1\u1f76 \u1f14\u03b8\u03b7 \u03ba\u03b1\u1f76 \u03bd\u03cc\u03bc\u03b9\u03bc\u03b1 \u03c4\u03bf\u1fd6\u03c2 \u1f04\u03bb\u03bb\u03bf\u03b9\u03c2 \u1f41\u03bc\u03bf\u03c6\u03c1\u03bf\u03bd\u03bf\u1fe6\u03bd, \u03b4\u03b9\u03b5\u03c3\u03c4\u1f78\u03c2 \u03b4\u1f72 \u03c4\u1fc7\n  \u03ba\u03b1\u03c4\u1f70 \u03b3\u03ac\u03bc\u03bf\u03bd \u03b4\u03cc\u03be\u1fc3. We may suppose that they corresponded to the third\n  order of a Benedictine or Franciscan brotherhood; so that, living in\n  the world, they would observe the rule up to a certain point, but\n  would not be bound by vows of celibacy or subject to the more rigorous\n  discipline of the sect.\n\n[Sidenote: meats and drinks,]\n\nBut their ascetic tendencies did not stop here. The Pharisee was very\ncareful to observe the distinction of meats lawful and unlawful, as laid\ndown by the Mosaic code, and even rendered these ordinances vexatious by\nminute definitions of his own. But the Essene went far beyond him. He\ndrank no wine, he did not touch animal food. His meal consisted of a\npiece of bread and a single mess of vegetables. Even this simple fare\nwas prepared for him by special officers consecrated for the purpose,\nthat it might be free from all contamination[246]. Nay, so stringent\nwere the rules of the order on this point, that when an Essene was\nexcommunicated, he often died of starvation, being bound by his oath not\nto take food prepared by defiled hands, and thus being reduced to eat\nthe very grass of the field[247].\n\nFootnote 246:\n\n  _B.J._ l.c. \u00a7 5; see Philo's account of the Therapeutes, _Vit. Cont._\n  \u00a7 4 \u03c3\u03b9\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03c0\u03bf\u03bb\u03c5\u03c4\u03b5\u03bb\u1f72\u03c2 \u03bf\u1f50\u03b4\u03ad\u03bd, \u1f00\u03bb\u03bb\u1f70 \u1f04\u03c1\u03c4\u03bf\u03bd \u03b5\u1f50\u03c4\u03b5\u03bb\u1fc6\u00b7 \u03ba\u03b1\u1f76 \u1f44\u03c8\u03bf\u03bd \u1f05\u03bb\u03b5\u03c2,\n  \u03bf\u1f53\u03c2 \u03bf\u1f31 \u1f00\u03b2\u03c1\u03bf\u03b4\u03b9\u03b1\u03b9\u03c4\u03cc\u03c4\u03b1\u03c4\u03bf\u03b9 \u03c0\u03b1\u03c1\u03b1\u03c1\u03c4\u03cd\u03bf\u03c5\u03c3\u03b9\u03bd \u1f51\u03c3\u03c3\u03ce\u03c0\u1ff3\u00b7 \u03c0\u03bf\u03c4\u1f78\u03bd \u1f55\u03b4\u03c9\u03c1 \u03bd\u03b1\u03bc\u03b1\u03c4\u03b9\u03b1\u1fd6\u03bf\u03bd\n  \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd; and again more to the same effect in \u00a7 9: and compare\n  the Essene story of St James in Hegesippus (Euseb. _H.E._ ii. 23)\n  \u03bf\u1f36\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c3\u03af\u03ba\u03b5\u03c1\u03b1 \u03bf\u1f50\u03ba \u1f14\u03c0\u03b9\u03b5\u03bd, \u03bf\u1f50\u03b4\u1f72 \u1f14\u03bc\u03c8\u03c5\u03c7\u03bf\u03bd \u1f14\u03c6\u03b1\u03b3\u03b5. Their abstention from\n  animal food accounts for Porphyry's giving them so prominent a place\n  in his treatise: see Zeller, p. 243.\n\nFootnote 247:\n\n  _B.J._ l.c. \u00a7 8.\n\n[Sidenote: and oil for anointing.]\n\nAgain, in hot climates oil for anointing the body is almost a necessary\nof life. From this too the Essenes strictly abstained. Even if they were\naccidentally smeared, they were careful at once to wash themselves,\nholding the mere touch to be a contamination[248].\n\nFootnote 248:\n\n  _B.J._ l.c. \u00a7 3 \u03ba\u03b7\u03bb\u1fd6\u03b4\u03b1 \u03b4\u1f72 \u1f51\u03c0\u03bf\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03c5\u03c3\u03b9 \u03c4\u1f78 \u1f14\u03bb\u03b1\u03b9\u03bf\u03bd \u03ba.\u03c4.\u03bb.; Hegesippus\n  l.c. \u1f14\u03bb\u03b1\u03b9\u03bf\u03bd \u03bf\u1f50\u03ba \u1f20\u03bb\u03b5\u03af\u03c8\u03b1\u03c4\u03bf.\n\n[Sidenote: Underlying principle of this asceticism.]\n\nFrom these facts it seems clear that Essene abstinence was something\nmore than the mere exaggeration of Pharisaic principles. The rigour of\nthe Pharisee was based on his obligation of obedience to an absolute\nexternal law. The Essene introduced a new principle. He condemned in any\nform the gratification of the natural cravings, nor would he consent to\nregard it as moral or immoral only according to the motive which\nsuggested it or the consequences which flowed from it. It was in itself\nan absolute evil. He sought to disengage himself, as far as possible,\nfrom the conditions of physical life. In short, in the asceticism of the\nEssene we seem to see the germ of that Gnostic dualism which regards\nmatter as the principle, or at least the abode, of evil.\n\n[Sidenote: 2. Speculative tenets.]\n\n2. And, when we come to investigate the speculative tenets of the sect,\nwe shall find that the Essenes have diverged appreciably from the common\ntype of Jewish orthodoxy.\n\n[Sidenote: (i) Tendency to sun-worship.]\n\n(i) Attention was directed above to their respect for Moses and the\nMosaic law, which they shared in common with the Pharisee. But there was\nanother side to their theological teaching. Though our information is\nsomewhat defective, still in the scanty notices which are preserved we\nfind sufficient indications that they had absorbed some foreign elements\nof religious thought into their system. Thus at day-break they addressed\ncertain prayers, which had been handed down from their forefathers, to\nthe Sun, 'as if entreating him to rise[249].' They were careful also to\nconceal and bury all polluting substances, so as not 'to insult the rays\nof the god[250].' We cannot indeed suppose that they regarded the sun as\nmore than a symbol of the unseen power who gives light and life; but\ntheir outward demonstrations of reverence were sufficiently prominent to\nattach to them, or to a sect derived from them, the epithet of\n'Sun-worshippers[251],' and some connexion with the characteristic\nfeature of Parsee devotion at once suggests itself. The practice at all\nevents stands in strong contrast to the denunciations of worship paid to\nthe 'hosts of heaven' in the Hebrew prophets.\n\nFootnote 249:\n\n  _B.J._ l.c. \u00a7 5 \u03c0\u03c1\u03cc\u03c2 \u03b3\u03b5 \u03bc\u1f74\u03bd \u03c4\u1f78 \u03b8\u03b5\u1fd6\u03bf\u03bd \u1f30\u03b4\u03af\u03c9\u03c2 \u03b5\u1f50\u03c3\u03b5\u03b2\u03b5\u1fd6\u03c2\u00b7 \u03c0\u03c1\u1f76\u03bd \u03b3\u1f70\u03c1 \u1f00\u03bd\u03b1\u03c3\u03c7\u03b5\u1fd6\u03bd\n  \u03c4\u1f78\u03bd \u1f25\u03bb\u03b9\u03bf\u03bd \u03bf\u1f50\u03b4\u1f72\u03bd \u03c6\u03b8\u03ad\u03b3\u03b3\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03b2\u03b5\u03b2\u03ae\u03bb\u03c9\u03bd, \u03c0\u03b1\u03c4\u03c1\u03af\u03bf\u03c5\u03c2 \u03b4\u03b5\u03ce\u03c9 \u03c4\u03b9\u03bd\u03b1\u03c2 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd\n  \u03b5\u1f50\u03c7\u03ac\u03c2, \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f31\u03ba\u03b5\u03c4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f00\u03bd\u03b1\u03c4\u03b5\u1fd6\u03bb\u03b1\u03b9. Compare what Philo says of the\n  Therapeutes, _Vit. Cont._ \u00a7 3 \u1f21\u03bb\u03af\u03bf\u03c5 \u03bc\u1f72\u03bd \u1f00\u03bd\u03af\u03c3\u03c7\u03bf\u03bd\u03c4\u03bf\u03c2 \u03b5\u1f50\u03b7\u03bc\u03b5\u03c1\u03af\u03b1\u03bd\n  \u03b1\u1f30\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f74\u03bd \u1f44\u03bd\u03c4\u03c9\u03c2 \u03b5\u1f50\u03b7\u03bc\u03b5\u03c1\u03af\u03b1\u03bd, \u03c6\u03c9\u03c4\u1f78\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03af\u03bf\u03c5 \u03c4\u03b7\u03bd \u03b4\u03af\u03b1\u03bd\u03bf\u03b9\u03b1\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd\n  \u1f00\u03bd\u03b1\u03c0\u03bb\u03b7\u03c3\u03b8\u1fc6\u03bd\u03b1\u03b9, and _ib._ \u00a7 11. On the attempt of Frankel (_Zeitschr._\n  p. 458) to resolve this worship, which Josephus states to be offered\n  to the sun (\u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd), into the ordinary prayers of the Pharisaic Jew\n  at day-break, see the appendix to this chapter.\n\nFootnote 250:\n\n  _B.J._ l.c. \u00a7 9 \u1f61\u03c2 \u03bc\u1f74 \u03c4\u1f70\u03c2 \u03b1\u1f50\u03b3\u1f70\u03c2 \u1f51\u03b2\u03c1\u03af\u03b6\u03bf\u03b9\u03b5\u03bd \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6. There can be no\n  doubt, I think, that by \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 is meant the 'sun-god'; comp. Eur.\n  _Heracl._ 749 \u03b8\u03b5\u03bf\u1fe6 \u03c6\u03b1\u03b5\u03c3\u03af\u03bc\u03b2\u03c1\u03bf\u03c4\u03bf\u03b9 \u03b1\u1f50\u03b3\u03b1\u03af, _Alc._ 722 \u03c4\u1f78 \u03c6\u03ad\u03b3\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03c4\u03bf\u1fe6\n  \u03b8\u03b5\u03bf\u1fe6, Appian _Pr\u00e6f._ 9 \u03b4\u03c5\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6, _Lib._ 113 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 \u03c0\u03b5\u03c1\u1f76\n  \u03b4\u03b5\u03af\u03bb\u03b7\u03bd \u1f11\u03c3\u03c0\u03ad\u03c1\u03b1\u03bd \u1f44\u03bd\u03c4\u03bf\u03c2, _Civ._ iv. 79 \u03b4\u03cd\u03bd\u03bf\u03bd\u03c4\u03bf\u03c2 \u1f04\u03c1\u03c4\u03b9 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6: comp.\n  Herod. ii. 24. Dr Ginsburg has obliterated this very important touch\n  by translating \u03c4\u1f70\u03c2 \u03b1\u1f50\u03b3\u1f70\u03c2 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 'the Divine rays' (_Essenes_ p. 47).\n  It is a significant fact that Hippolytus (_H\u00e6r._ ix. 25) omits the\n  words \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6, evidently regarding them as a stumbling-block. How\n  Josephus expressed himself in the original Hebrew of the _Bellum\n  Judaicum_, it is vain to speculate: but the Greek translation was\n  authorised, if not made, by him.\n\nFootnote 251:\n\n  Epiphan. _H\u00e6r._ xix. 2, xx. 3 \u1f48\u03c3\u03c3\u03b7\u03bd\u03bf\u1f76 \u03b4\u1f72 \u03bc\u03b5\u03c4\u03ad\u03c3\u03c4\u03b7\u03c3\u03b1\u03bd \u1f00\u03c0\u1f78 \u1f38\u03bf\u03c5\u03b4\u03b1\u03ca\u03c3\u03bc\u03bf\u1fe6 \u03b5\u1f30\u03c2\n  \u03c4\u1f74\u03bd \u03c4\u1ff6\u03bd \u03a3\u03b1\u03bc\u03c8\u03b1\u03af\u03c9\u03bd \u03b1\u1f35\u03c1\u03b5\u03c3\u03b9\u03bd, liii. 1, 2 \u03a3\u03b1\u03bc\u03c8\u03b1\u1fd6\u03bf\u03b9 \u03b3\u1f70\u03c1 \u1f11\u03c1\u03bc\u03b7\u03bd\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9\n  \u1f29\u03bb\u03b9\u03b1\u03ba\u03bf\u03af, from the Hebrew \u05e9\u05de\u05e9 'the sun.' The historical connexion of\n  the Samps\u00e6ans with the Essenes is evident from these passages: though\n  it is difficult to say what their precise relations to each other\n  were. See the appendix.\n\n[Sidenote: (ii) Resurrection of the body denied.]\n\n(ii) Nor again is it an insignificant fact that, while the Pharisee\nmaintained the resurrection of the body as a cardinal article of his\nfaith, the Essene restricted himself to a belief in the immortality of\nthe soul. The soul, he maintained, was confined in the flesh, as in a\nprison-house. Only when disengaged from these fetters would it be truly\nfree. Then it would soar aloft, rejoicing in its newly attained\nliberty[252]. This doctrine accords with the fundamental conception of\nthe malignity of matter. To those who held this conception a\nresurrection of the body would be repulsive, as involving a perpetuation\nof evil.\n\nFootnote 252:\n\n  _B.J._ l.c. \u00a7 11 \u03ba\u03b1\u1f76 \u03b3\u1f70\u03c1 \u1f15\u03c1\u03c1\u03c9\u03c4\u03b1\u03b9 \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f25\u03b4\u03b5 \u1f21 \u03b4\u03cc\u03be\u03b1, \u03c6\u03b8\u03b1\u03c1\u03c4\u1f70 \u03bc\u1f72\u03bd\n  \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1f70 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f55\u03bb\u03b7\u03bd \u03bf\u1f50 \u03bc\u03cc\u03bd\u03b9\u03bc\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2, \u03c4\u1f70\u03c2 \u03b4\u1f72 \u03c8\u03c5\u03c7\u1f70\u03c2 \u1f00\u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5\u03c2\n  \u1f00\u03b5\u1f76 \u03b4\u03b9\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9\u03bd ... \u1f10\u03c0\u03b5\u03b9\u03b4\u1f70\u03bd \u03b4\u1f72 \u1f00\u03bd\u03b5\u03b8\u1ff6\u03c3\u03b9 \u03c4\u1ff6\u03bd \u03ba\u03b1\u03c4\u1f70 \u03c3\u03ac\u03c1\u03ba\u03b1 \u03b4\u03b5\u03c3\u03bc\u1ff6\u03bd, \u03bf\u1f37\u03b1 \u03b4\u1f74\n  \u03bc\u03b1\u03ba\u03c1\u1fb6\u03c2 \u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u03b1\u03c2 \u1f00\u03c0\u03b7\u03bb\u03bb\u03b1\u03b3\u03bc\u03ad\u03bd\u03b1\u03c2, \u03c4\u03cc\u03c4\u03b5 \u03c7\u03b1\u03af\u03c1\u03b5\u03b9\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b5\u03c4\u03b5\u1ff6\u03c1\u03bf\u03c5\u03c2 \u03c6\u03ad\u03c1\u03b5\u03c3\u03b8\u03b1\u03b9\n  \u03ba.\u03c4.\u03bb. To this doctrine the teaching of the Pharisees stands in direct\n  contrast; _ib._ \u00a7 13: comp. also _Ant._ xviii. 1. 3, 5.\n\n  Nothing can be more explicit than the language of Josephus. On the\n  other hand Hippolytus (_H\u00e6r._ ix. 27) says of them \u1f41\u03bc\u03bf\u03bb\u03bf\u03b3\u03bf\u1fe6\u03c3\u03b9 \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76\n  \u03c4\u1f74\u03bd \u03c3\u03ac\u03c1\u03ba\u03b1 \u1f00\u03bd\u03b1\u03c3\u03c4\u03ae\u03c3\u03b5\u03c3\u03b8\u03b1\u03b9 \u03ba\u03b1\u1f76 \u1f14\u03c3\u03b5\u03c3\u03b8\u03b1\u03b9 \u1f00\u03b8\u03ac\u03bd\u03b1\u03c4\u03bf\u03bd \u1f43\u03bd \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd \u1f24\u03b4\u03b7 \u1f00\u03b8\u03ac\u03bd\u03b1\u03c4\u03cc\u03c2\n  \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03c8\u03c5\u03c7\u03ae \u03ba.\u03c4.\u03bb.; but his authority is worthless on this point, as\n  he can have had no personal knowledge of the facts: see Zeller p. 251,\n  note 2. Hilgenfeld takes a different view; _Zeitschr._ XIV. p. 49.\n\n[Sidenote: (iii) Prohibition of sacrifices.]\n\n(iii) But they also separated themselves from the religious belief of\nthe orthodox Jew in another respect, which would provoke more notice.\nWhile they sent gifts to the temple at Jerusalem, they refused to offer\nsacrifices there[253]. It would appear that the slaughter of animals was\naltogether forbidden by their creed[254]. It is certain that they were\nafraid of contracting some ceremonial impurity by offering victims in\nthe temple. Meanwhile they had sacrifices, bloodless sacrifices, of\ntheir own. They regarded their simple meals with their accompanying\nprayers and thanksgiving, not only as devotional but even as sacrificial\nrites. Those who prepared and presided over these meals were their\nconsecrated priests[255].\n\nFootnote 253:\n\n  _Ant._ xviii. 1. 5 \u03b5\u1f30\u03c2 \u03b4\u1f72 \u03c4\u1f78 \u1f31\u03b5\u03c1\u1f78\u03bd \u1f00\u03bd\u03b1\u03b8\u03ae\u03bc\u03b1\u03c4\u03ac \u03c4\u03b5 \u03c3\u03c4\u03ad\u03bb\u03bb\u03bf\u03bd\u03c4\u03b5\u03c2 \u03b8\u03c5\u03c3\u03af\u03b1\u03c2 \u03bf\u1f50\u03ba\n  \u1f10\u03c0\u03b9\u03c4\u03b5\u03bb\u03bf\u1fe6\u03c3\u03b9 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03cc\u03c4\u03b7\u03c4\u03b9 \u1f01\u03b3\u03bd\u03b5\u03b9\u1ff6\u03bd, \u1f03\u03c2 \u03bd\u03bf\u03bc\u03af\u03b6\u03bf\u03b9\u03b5\u03bd, \u03ba\u03b1\u1f76 \u03b4\u03b9' \u03b1\u1f50\u03c4\u1f78 \u03b5\u1f30\u03c1\u03b3\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9\n  \u03c4\u03bf\u1fe6 \u03ba\u03bf\u03b9\u03bd\u03bf\u1fe6 \u03c4\u03b5\u03bc\u03b5\u03bd\u03af\u03c3\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f10\u03c6' \u03b1\u1f51\u03c4\u1ff6\u03bd \u03c4\u1f70\u03c2 \u03b8\u03c5\u03c3\u03af\u03b1\u03c2 \u1f10\u03c0\u03b9\u03c4\u03b5\u03bb\u03bf\u1fe6\u03c3\u03b9. So Philo\n  _Quod omn. prob. lib._ \u00a7 12 describes them as \u03bf\u1f50 \u03b6\u1ff7\u03b1 \u03ba\u03b1\u03c4\u03b1\u03b8\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f00\u03bb\u03bb'\n  \u1f31\u03b5\u03c1\u03bf\u03c0\u03c1\u03b5\u03c0\u03b5\u1fd6\u03c2 \u03c4\u1f70\u03c2 \u1f11\u03b1\u03c5\u03c4\u1ff6\u03bd \u03b4\u03b9\u03b1\u03bd\u03bf\u03af\u03b1\u03c2 \u03ba\u03b1\u03c4\u03b1\u03c3\u03ba\u03b5\u03c5\u03ac\u03b6\u03b5\u03b9\u03bd \u1f00\u03be\u03b9\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2.\n\nFootnote 254:\n\n  The following considerations show that their abstention should\n  probably be explained in this way: (1) Though the language of Josephus\n  may be ambiguous, that of Philo is unequivocal on this point; (2)\n  Their abstention from the temple-sacrifices cannot be considered apart\n  from the fact that they ate no animal food: see above p. 86, note 246.\n  (3) The Christianized Essenes, or Ebionites, though strong Judaizers\n  in many respects, yet distinctly protested against the sacrifice of\n  animals; see Clem. _Hom._ iii. 45, 52, and comp. Ritschl p. 224. On\n  this subject see also Zeller p. 242 sq., and the appendix to this\n  chapter.\n\nFootnote 255:\n\n  _Ant._ xviii. 1. 5 \u1f31\u03b5\u03c1\u03b5\u1fd6\u03c2 \u03c4\u03b5 [\u03c7\u03b5\u03b9\u03c1\u03bf\u03c4\u03bf\u03bd\u03bf\u1fe6\u03c3\u03b9] \u03b4\u03b9\u1f70 \u03c0\u03bf\u03af\u03b7\u03c3\u03b9\u03bd \u03c3\u03af\u03c4\u03bf\u03c5 \u03c4\u03b5 \u03ba\u03b1\u1f76\n  \u03b2\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd, _B.J._ ii. 8. 5 \u03c0\u03c1\u03bf\u03ba\u03b1\u03c4\u03b5\u03cd\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u1f41 \u1f31\u03b5\u03c1\u03b5\u1f7a\u03c2 \u03c4\u1fc6\u03c2 \u03c4\u03c1\u03bf\u03c6\u1fc6\u03c2 \u03ba.\u03c4.\u03bb.;\n  see Ritschl p. 181.\n\n[Sidenote: (iv) Esoteric doctrine of angels.]\n\n(iv) In what other respects they may have departed from, or added to,\nthe normal creed of Judaism, we do not know. But it is expressly stated\nthat, when a novice after passing through the probationary stages was\nadmitted to the full privileges of the order, the oath of admission\nbound him 'to conceal nothing from the members of the sect, and to\nreport nothing concerning them to others, even though threatened with\ndeath; not to communicate any of their doctrines to anyone otherwise\nthan as he himself had received them; but to abstain from robbery, and\nin like manner to guard carefully the books of their sect, and _the\nnames of the angels_[256].' It may be reasonably supposed that more\nlurks under this last expression than meets the ear. This esoteric\ndoctrine, relating to angelic beings, may have been another link which\nattached Essenism to the religion of Zoroaster[257]. At all events we\nseem to be justified in connecting it with the self-imposed service and\nworshipping of angels at Coloss\u00e6: and we may well suspect that we have\nhere a germ which was developed into the Gnostic doctrine of \u00e6ons or\nemanations.\n\nFootnote 256:\n\n  _B.J._ l.c. \u00a7 7 \u1f45\u03c1\u03ba\u03bf\u03c5\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f44\u03bc\u03bd\u03c5\u03c3\u03b9 \u03c6\u03c1\u03b9\u03ba\u03ce\u03b4\u03b5\u03b9\u03c2 ... \u03bc\u03ae\u03c4\u03b5 \u03ba\u03c1\u03cd\u03c8\u03b5\u03b9\u03bd \u03c4\u03b9\n  \u03c4\u03bf\u1f7a\u03c2 \u03b1\u1f31\u03c1\u03b5\u03c4\u03b9\u03c3\u03c4\u1f70\u03c2 \u03bc\u03ae\u03c4\u03b5 \u1f11\u03c4\u03ad\u03c1\u03bf\u03b9\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c4\u03b9 \u03bc\u03b7\u03bd\u03cd\u03c3\u03b5\u03b9\u03bd, \u03ba\u03b1\u1f76 \u1f02\u03bd \u03bc\u03ad\u03c7\u03c1\u03b9 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5\n  \u03c4\u1f76\u03c2 \u03b2\u03b9\u03ac\u03b6\u03b7\u03c4\u03b1\u03b9. \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u1f40\u03bc\u03bd\u03cd\u03bf\u03c5\u03c3\u03b9 \u03bc\u03b7\u03b4\u03b5\u03bd\u1f76 \u03bc\u1f72\u03bd \u03bc\u03b5\u03c4\u03b1\u03b4\u03bf\u1fe6\u03bd\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03b4\u03bf\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd\n  \u1f11\u03c4\u03ad\u03c1\u03c9\u03c2 \u1f22 \u1f61\u03c2 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03bc\u03b5\u03c4\u03ad\u03bb\u03b1\u03b2\u03b5\u03bd\u00b7 \u1f00\u03c6\u03ad\u03be\u03b5\u03c3\u03b8\u03b1\u03b9 \u03b4\u1f72 \u03bb\u1fc3\u03c3\u03c4\u03b5\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03c4\u03b7\u03c1\u03ae\u03c3\u03b5\u03b9\u03bd\n  \u1f41\u03bc\u03bf\u03af\u03c9\u03c2 \u03c4\u1f70 \u03c4\u03b5 \u03c4\u1fc6\u03c2 \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03b2\u03b9\u03b2\u03bb\u03af\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b1.\n  With this notice should be compared the Ebionite \u03b4\u03b9\u03b1\u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03af\u03b1, or\n  protest of initiation, prefixed to the _Clementine Homilies_, which\n  shows how closely the Christian Essenes followed the practice of their\n  Jewish predecessors in this respect. See Zeller p. 254.\n\nFootnote 257:\n\n  See below, in the appendix.\n\n[Sidenote: (v) Speculations on God and Creation.]\n\n(v) If so, it is not unconnected with another notice relating to Essene\npeculiarities. The Gnostic doctrine of intermediate beings between God\nand the world, as we have seen, was intimately connected with\nspeculations respecting creation. Now we are specially informed that the\nEssenes, while leaving physical studies in general to speculative idlers\n(\u03bc\u03b5\u03c4\u03b5\u03c9\u03c1\u03bf\u03bb\u03ad\u03c3\u03c7\u03b1\u03b9\u03c2), as being beyond the reach of human nature, yet\nexcepted from their general condemnation that philosophy which treats of\nthe existence of God and the generation of the universe[258].\n\nFootnote 258:\n\n  Philo _Omn. prob. lib._ \u00a7 12 (p. 458) \u03c4\u1f78 \u03b4\u1f72 \u03c6\u03c5\u03c3\u03b9\u03ba\u1f78\u03bd \u1f61\u03c2 \u03bc\u03b5\u1fd6\u03b6\u03bf\u03bd \u1f22 \u03ba\u03b1\u03c4\u1f70\n  \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03af\u03bd\u03b7\u03bd \u03c6\u03cd\u03c3\u03b9\u03bd \u03bc\u03b5\u03c4\u03b5\u03c9\u03c1\u03bf\u03bb\u03ad\u03c3\u03c7\u03b1\u03b9\u03c2 \u1f00\u03c0\u03bf\u03bb\u03b9\u03c0\u03cc\u03bd\u03c4\u03b5\u03c2, \u03c0\u03bb\u1f74\u03bd \u1f45\u03c3\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c0\u03b5\u03c1\u1f76\n  \u1f51\u03c0\u03ac\u03c1\u03be\u03b5\u03c9\u03c2 \u0398\u03b5\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u03b3\u03b5\u03bd\u03ad\u03c3\u03b5\u03c9\u03c2 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03b5\u1fd6\u03c4\u03b1\u03b9.\n\n[Sidenote: (vi) Magical charms.]\n\n(vi) Mention has been made incidentally of certain secret books peculiar\nto the sect. The existence of such an apocryphal literature was a sure\ntoken of some abnormal development in doctrine[259]. In the passage\nquoted it is mentioned in relation to some form of angelology. Elsewhere\ntheir skill in prediction, for which they were especially famous, is\nconnected with the perusal of certain 'sacred books,' which however are\nnot described[260]. But more especially, we are told that the Essenes\nstudied with extraordinary diligence the writings of the ancients,\nselecting those especially which could be turned to profit for soul and\nbody, and that from these they learnt the qualities of roots and the\nproperties of stones[261]. This expression, as illustrated by other\nnotices, points clearly to the study of occult sciences, and recalls the\nalliance with the practice of magical arts, which was a distinguishing\nfeature of Gnosticism, and is condemned by Christian teachers even in\nthe heresies of the Apostolic age.\n\nFootnote 259:\n\n  The word _Apocrypha_ was used originally to designate the secret books\n  which contained the esoteric doctrine of a sect. The secondary sense\n  'spurious' was derived from the general character of these writings,\n  which were heretical, generally Gnostic, forgeries. See Prof.\n  Plumptre's article _Apocrypha_ in Smith's _Dictionary of the Bible_,\n  and the note on \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9 below, ii. 3.\n\nFootnote 260:\n\n  _B.J._ ii. 8. 12 \u03b5\u1f30\u03c3\u1f76 \u03b4\u1f72 \u1f10\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f33 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03bc\u03ad\u03bb\u03bb\u03bf\u03bd\u03c4\u03b1 \u03c0\u03c1\u03bf\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd\n  \u1f51\u03c0\u03b9\u03c3\u03c7\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9, \u03b2\u03af\u03b2\u03bb\u03bf\u03b9\u03c2 \u1f31\u03b5\u03c1\u03b1\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03c6\u03cc\u03c1\u03bf\u03b9\u03c2 \u1f01\u03b3\u03bd\u03b5\u03af\u03b1\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c6\u03b7\u03c4\u1ff6\u03bd\n  \u1f00\u03c0\u03bf\u03c6\u03b8\u03ad\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd \u1f10\u03bc\u03c0\u03b1\u03b9\u03b4\u03bf\u03c4\u03c1\u03b9\u03b2\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9\u00b7 \u03c3\u03c0\u03ac\u03bd\u03b9\u03bf\u03bd \u03b4\u1f72, \u03b5\u1f34\u03c0\u03bf\u03c4\u03b5, \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2\n  \u03c0\u03c1\u03bf\u03b1\u03b3\u03bf\u03c1\u03b5\u03cd\u03c3\u03b5\u03c3\u03b9\u03bd \u1f00\u03c3\u03c4\u03bf\u03c7\u03ae\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd. Dr Ginsburg (p. 49) translates \u03b2\u03af\u03b2\u03bb\u03bf\u03b9\u03c2\n  \u1f31\u03b5\u03c1\u03b1\u1fd6\u03c2 '_the_ sacred Scripture,' and \u03c0\u03c1\u03bf\u03c6\u03b7\u03c4\u1ff6\u03bd \u1f00\u03c0\u03bf\u03c6\u03b8\u03ad\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd '_the_\n  sayings of _the_ prophets'; but as the definite articles are wanting,\n  the expressions cannot be so rendered, nor does there seem to be any\n  reference to the Canonical writings.\n\n  We learn from an anecdote in _Ant._ xiii. II. 2, that the teachers of\n  this sect communicated the art of prediction to their disciples by\n  instruction. We may therefore conjecture that with the Essenes this\n  acquisition was connected with magic or astrology. At all events it is\n  not treated as a direct inspiration.\n\nFootnote 261:\n\n  _B.J._ ii. 8. 6 \u03c3\u03c0\u03bf\u03c5\u03b4\u03ac\u03b6\u03bf\u03c5\u03c3\u03b9 \u03b4\u1f72 \u1f10\u03ba\u03c4\u03cc\u03c0\u03c9\u03c2 \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f70 \u03c4\u1ff6\u03bd \u03c0\u03b1\u03bb\u03b1\u03b9\u1ff6\u03bd\n  \u03c3\u03c5\u03b3\u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c4\u03b1, \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03c4\u1f70 \u03c0\u03c1\u1f78\u03c2 \u1f60\u03c6\u03ad\u03bb\u03b5\u03b9\u03b1\u03bd \u03c8\u03c5\u03c7\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f10\u03ba\u03bb\u03ad\u03b3\u03bf\u03bd\u03c4\u03b5\u03c2\u00b7\n  \u1f14\u03bd\u03b8\u03b5\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03af\u03b1\u03bd \u03c0\u03b1\u03b8\u1ff6\u03bd \u1fe5\u03af\u03b6\u03b1\u03b9 \u03c4\u03b5 \u1f00\u03bb\u03b5\u03be\u03b9\u03c4\u03ae\u03c1\u03b9\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03bb\u03af\u03b8\u03c9\u03bd\n  \u1f30\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u1f00\u03bd\u03b5\u03c1\u03b5\u03c5\u03bd\u1ff6\u03bd\u03c4\u03b1\u03b9. This passage might seem at first sight to\n  refer simply to the _medicinal_ qualities of vegetable and mineral\n  substances; but a comparison with another notice in Josephus invests\n  it with a different meaning. In _Ant._ viii. 2. 5 he states that\n  Solomon, having received by divine inspiration the art of defeating\n  demons for the advantage and healing of man (\u03b5\u1f30\u03c2 \u1f60\u03c6\u03ad\u03bb\u03b5\u03b9\u03b1\u03bd \u03ba\u03b1\u1f76\n  \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03af\u03b1\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03bd\u03b8\u03c1\u1ff6\u03c0\u03bf\u03b9\u03c2), composed and left behind him charms\n  (\u1f10\u03c0\u1ff3\u03b4\u03ac\u03c2) by which diseases were allayed, and diverse kinds of\n  exorcisms (\u03c4\u03c1\u03cc\u03c0\u03bf\u03c5\u03c2 \u1f10\u03be\u03bf\u03c1\u03ba\u03ce\u03c3\u03b5\u03c9\u03bd) by which demons were cast out. 'This\n  mode of healing,' he adds, 'is very powerful even to the present day';\n  and he then relates how, as he was credibly informed (\u1f31\u03c3\u03c4\u03cc\u03c1\u03b7\u03c3\u03b1), one\n  of his countrymen, Eleazar by name, had healed several persons\n  possessed by demons in the presence of Vespasian and his sons and a\n  number of officers and common soldiers. This he did by applying to the\n  nose of the possessed his ring, which had concealed in it one of the\n  _roots_ which Solomon had directed to be used, and thus drawing out\n  the demon through the nostrils of the person smelling it. At the same\n  time he adjured the evil spirit not to return, 'making mention of\n  Solomon and repeating the charms composed by him.' On one occasion\n  this Eleazar gave ocular proof that the demon was exorcized; and thus,\n  adds Josephus, \u03c3\u03b1\u03c6\u1f74\u03c2 \u1f21 \u03a3\u03bf\u03bb\u03bf\u03bc\u1ff6\u03bd\u03bf\u03c2 \u03ba\u03b1\u03b8\u03af\u03c3\u03c4\u03b1\u03c4\u03bf \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03bf\u03c6\u03af\u03b1. On these\n  books relating to the occult arts and ascribed to Solomon see\n  Fabricius _Cod. Pseud. Vet. Test._ I. p. 1036 sq., where many curious\n  notices are gathered together. Comp. especially Origen, _In Matth.\n  Comm._ xxxv. \u00a7 110 (III. p. 910), Pseudo-Just. _Qu\u00e6st._ 55.\n\n  This interpretation explains all the expressions in the passage. The\n  \u03bb\u03af\u03b8\u03c9\u03bd \u1f30\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 naturally points to the use of charms or amulets, as\n  may be seen e.g. from the treatise, Damigeron _de Lapidibus_, printed\n  in the _Spicil. Solemn._ III. p. 324 sq.: comp. King _Antique Gems_\n  Sect. IV, _Gnostics and their Remains_. The reference to 'the books of\n  the ancients' thus finds an adequate explanation. On the other hand\n  the only expression which seemed to militate against this view,\n  \u1f00\u03bb\u03b5\u03be\u03b9\u03c4\u03ae\u03c1\u03b9\u03bf\u03b9 \u1fe5\u03af\u03b6\u03b1\u03b9, is justified by the story in the _Antiquities_. It\n  should be added also that Hippolytus (_H\u00e6r._ ix. 22) paraphrases the\n  language of Josephus so as to give it this sense; \u03c0\u03ac\u03bd\u03c5 \u03b4\u1f72\n  _\u03c0\u03b5\u03c1\u03b9\u03ad\u03c1\u03b3\u03c9\u03c2_ \u1f14\u03c7\u03bf\u03c5\u03c3\u03b9 \u03c0\u03b5\u03c1\u1f76 \u03b2\u03bf\u03c4\u03ac\u03bd\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03bb\u03af\u03b8\u03bf\u03c5\u03c2, _\u03c0\u03b5\u03c1\u03b9\u03b5\u03c1\u03b3\u03cc\u03c4\u03b5\u03c1\u03bf\u03b9_\n  \u1f44\u03bd\u03c4\u03b5\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70\u03c2 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u03af\u03b1\u03c2, \u03c6\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03bc\u1f74 \u03bc\u03ac\u03c4\u03b7\u03bd \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03b3\u03b5\u03bd\u03bf\u03bd\u03ad\u03bd\u03b1\u03b9.\n  The sense which \u03c0\u03b5\u03c1\u03af\u03b5\u03c1\u03b3\u03bf\u03c2 ('curiosus') bears in Acts xix. 19 and\n  elsewhere, referring to magical arts, illustrates its use here.\n\n  Thus these Essenes were dealers in charms, rather than physicians. And\n  yet it is quite possible that along with this practice of the occult\n  sciences they studied the healing art in its nobler forms. The works\n  of Alexander of Tralles, an eminent ancient physician, constantly\n  recommend the use of such charms, of which some obviously come from a\n  Jewish source and not improbably may have been taken from these\n  Solomonian books to which Josephus refers. A number of passages from\n  this and other writers, specifying charms of various kinds, are given\n  in Becker and Marquardt _Rom. Alterth._ IV. p. 116 sq. See also\n  Spencer's note on Orig. _c. Cels._ p. 17 sq.\n\n[Sidenote: 3. Exclusive spirit of Essenism.]\n\n3. But the notice to which I have just alluded suggests a broader\naffinity with Gnosticism. Not only did the theological speculations of\nthe Essenes take a Gnostic turn, but they guarded their peculiar tenets\nwith Gnostic reserve. They too had their esoteric doctrine which they\nlooked upon as the exclusive possession of the privileged few; their\n'mysteries' which it was a grievous offence to communicate to the\nuninitiated. This doctrine was contained, as we have seen, in an\napocryphal literature. Their whole organisation was arranged so as to\nprevent the divulgence of its secrets to those without. The long period\nof noviciate, the careful rites of initiation, the distinction of the\nseveral orders[262] in the community, the solemn oaths by which they\nbound their members, were so many safeguards against a betrayal of this\nprecious deposit, which they held to be restricted to the inmost circle\nof the brotherhood.\n\nFootnote 262:\n\n  See especially _B.J._ ii. 8. 7, 10.\n\n[Sidenote: The three notes of Gnosticism found in the Essenes.]\n\nIn selecting these details I have not attempted to give a finished\nportrait of Essenism. From this point of view the delineation would be\nimperfect and misleading: for I have left out of sight the nobler\nfeatures of the sect, their courageous endurance, their simple piety,\ntheir brotherly love. My object was solely to call attention to those\nfeatures which distinguish it from the normal type of Judaism, and seem\nto justify the attribution of Gnostic influences. And here it has been\nseen that the three characteristics, which were singled out above as\ndistinctive of Gnosticism, reappear in the Essenes; though it has been\nconvenient to consider them in the reversed order. This Jewish sect\nexhibits the same exclusiveness in the communication of its doctrines.\nIts theological speculations take the same direction, dwelling on the\nmysteries of creation, regarding matter as the abode of evil, and\npostulating certain intermediate spiritual agencies as necessary links\nof communication between heaven and earth. And lastly, its speculative\nopinions involve the same ethical conclusions, and lead in like manner\nto a rigid asceticism. If the notices relating to these points do not\nalways explain themselves, yet read in the light of the heresies of the\nApostolic age and in that of subsequent Jud\u00e6o-Gnostic Christianity,\ntheir bearing seems to be distinct enough; so that we should not be far\nwrong, if we were to designate Essenism as Gnostic Judaism[263].\n\nFootnote 263:\n\n  I have said nothing of the Cabbala, as a development of Jewish thought\n  illustrating the Colossian heresy: because the books containing the\n  Cabbalistic speculations are comparatively recent, and if they contain\n  ancient elements, it seems impossible to separate these from later\n  additions or to assign to them even an approximate date. The\n  Cabbalistic doctrine however will serve to show to what extent Judaism\n  may be developed in the direction of speculative mysticism.\n\n[Sidenote: How widely were the Essenes dispersed?]\n\nBut the Essenes of whom historical notices are preserved were\ninhabitants of the Holy Land. Their monasteries were situated on the\nshores of the Dead Sea. We are told indeed, that the sect was not\nconfined to any one place, and that members of the order were found in\ngreat numbers in divers cities and villages[264]. But Jud\u00e6a in one\nnotice, Palestine and Syria in another, are especially named as the\nlocalities of the Essene settlements[265]. Have we any reason to suppose\nthat they were represented among the Jews of the Dispersion? In Egypt\nindeed we find ourselves confronted with a similar ascetic sect, the\nTherapeutes, who may perhaps have had an independent origin, but who\nnevertheless exhibit substantially the same type of Jewish thought and\npractice[266]. But the Dispersion of Egypt, it may be argued, was\nexceptional; and we might expect to find here organisations and\ndevelopments of Judaism hardly less marked and various than in the\nmother country. [Sidenote: Do they appear in Asia Minor?] What ground\nhave we for assuming the existence of this type in Asia Minor? Do we\nmeet with any traces of it in the cities of the Lycus, or in proconsular\nAsia generally, which would justify the opinion that it might make its\ninfluence felt in the Christian communities of that district?\n\nFootnote 264:\n\n  Philo _Fragm._ p. 632 \u03bf\u1f30\u03ba\u03bf\u1fe6\u03c3\u03b9 \u03b4\u1f72 \u03c0\u03bf\u03bb\u03bb\u1f70\u03c2 \u03bc\u1f72\u03bd \u03c0\u03cc\u03bb\u03b5\u03b9\u03c2 \u03c4\u1fc6\u03c2 \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03b1\u03c2,\n  \u03c0\u03bf\u03bb\u03bb\u1f70\u03c2 \u03b4\u1f72 \u03ba\u03ce\u03bc\u03b1\u03c2, \u03ba\u03b1\u1f76 \u03bc\u03b5\u03b3\u03ac\u03bb\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03bf\u03bb\u03c5\u03b1\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03c5\u03c2 \u1f41\u03bc\u03af\u03bb\u03bf\u03c5\u03c2; Joseph.\n  _B.J._ ii. 8. 4 \u03bc\u03af\u03b1 \u03b4\u1f72 \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c0\u03cc\u03bb\u03b9\u03c2, \u1f00\u03bb\u03bb' \u1f10\u03bd \u1f11\u03ba\u03ac\u03c3\u03c4\u1fc3\n  \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03bf\u1fe6\u03c3\u03b9 \u03c0\u03bf\u03bb\u03bb\u03bf\u03af. On the notices of the settlements and dispersion of\n  the Essenes see Zeller p. 239.\n\nFootnote 265:\n\n  Philo names _Jud\u00e6a_ in _Fragm._ p. 632; _Palestine_ and _Syria_ in\n  _Quod omn. prob. lib._ 12 p. 457. Their chief settlements were in the\n  neighbourhood of the Dead Sea. This fact is mentioned by the heathen\n  writers Pliny (_N.H._ v. 15) and Dion Chrysostom (Synesius _Dio_ 3).\n  The name of the 'Essene gate' at Jerusalem (_B.J._ v. 4. 2) seems to\n  point to some establishment of the order close to the walls of that\n  city.\n\nFootnote 266:\n\n  They are only known to us from Philo's treatise _de Vita\n  Contemplativa_. Their settlements were on the shores of the Mareotic\n  lake near Alexandria. Unlike the Essenes, they were not gathered\n  together in convents as members of a fraternity, but lived apart as\n  anchorites, though in the same neighbourhood. In other respects their\n  tenets and practices are very similar to those of the Essenes.\n\n[Sidenote: How the term _Essene_ is to be understood.]\n\nNow it has been shown that the colonies of the Jews in this\nneighbourhood were populous and influential[267]; and it might be argued\nwith great probability that among these large numbers Essene Judaism\ncould not be unrepresented. But indeed throughout this investigation,\nwhen I speak of the Judaism in the Colossian Church as Essene, I do not\nassume a precise identity of origin, but only an essential affinity of\ntype, with the Essenes of the mother country. As a matter of history, it\nmay or may not have sprung from the colonies on the shores of the Dead\nSea; but as this can neither be proved nor disproved, so also it is\nimmaterial to my main [Sidenote: Probabilities of the case.] purpose.\nAll along its frontier, wherever Judaism became enamoured of and was\nwedded to Oriental mysticism, the same union would produce substantially\nthe same results. In a country where Phrygia, Persia, Syria, all in turn\nhad moulded religious thought, it would be strange indeed if Judaism\nentirely escaped these influences. Nor, as a matter of fact, are\nindications wanting to show that it was not unaffected [Sidenote: Direct\nindications.] by them. If the traces are few, they are at least as\nnumerous and as clear as with our defective information on the whole\nsubject we have any right to expect in this particular instance.\n\nFootnote 267:\n\n  See above, p. 19 sq.\n\n[Sidenote: St Paul at Ephesus A.D. 54\u201357.]\n\nWhen St Paul visits Ephesus, he comes in contact with certain strolling\nJews, exorcists, who attempt to cast out evil spirits[268]. Connecting\nthis fact with the notices of Josephus, from which we infer that\nexorcisms of this kind were especially [Sidenote: Exorcisms and]\npractised by the Essenes[269], we seem to have an indication of their\npresence in the capital of proconsular Asia. If so, it is a significant\nfact that in their exorcisms they employed the name of our Lord: for\nthen we must regard this as the earliest notice of those overtures of\nalliance on the part of Essenism, which involved such important\nconsequences in the subsequent history of the Church[270]. It is also\nworth observing, that the next incident in St Luke's narrative is the\nburning [Sidenote: magical books.] of their magical books by those whom\nSt Paul converted on this occasion[271]. As Jews are especially\nmentioned among these converts, and as books of charms are ascribed to\nthe Essenes by Josephus, the two incidents, standing in this close\nconnexion, throw great light on the type of Judaism which thus appears\nat Ephesus[272].\n\nFootnote 268:\n\n  Acts xix. 13 \u03c4\u1ff6\u03bd \u03c0\u03b5\u03c1\u03b9\u03b5\u03c1\u03c7\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd \u1f10\u03be\u03bf\u03c1\u03ba\u03b9\u03c3\u03c4\u1ff6\u03bd.\n\nFootnote 269:\n\n  See above p. 91, note 261.\n\nFootnote 270:\n\n  On the later contact of Essenism with Christianity, see the appendix,\n  and _Galatians_ p. 310 sq.\n\nFootnote 271:\n\n  There is doubtless a reference to the charms called \u1f18\u03c6\u03ad\u03c3\u03b9\u03b1 \u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c4\u03b1 in\n  this passage: see Wetstein ad loc., and the references in Becker and\n  Marquardt _R\u00f6m. Alterth._ IV. p. 123 sq. But this supposition does not\n  exclude the Jews from a share in these magical arts, while the context\n  points to some such participation.\n\nFootnote 272:\n\n  I can only regard it as an accidental coincidence that the epulones of\n  the Ephesian Artemis were called _Essenes_, Pausan. viii. 13. 1 \u03c4\u03bf\u1f7a\u03c2\n  \u03c4\u1fc7 \u1f08\u03c1\u03c4\u03ad\u03bc\u03b9\u03b4\u03b9 \u1f31\u03c3\u03c4\u03b9\u03ac\u03c4\u03bf\u03c1\u03b1\u03c2 \u03c4\u1fc7 \u1f18\u03c6\u03b5\u03c3\u03af\u1fb3 \u03b3\u03b9\u03bd\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2, \u03ba\u03b1\u03bb\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u03b4\u1f72 \u1f51\u03c0\u1f78 \u03c4\u1ff6\u03bd\n  \u03c0\u03bf\u03bb\u03b9\u03c4\u1ff6\u03bd \u1f18\u03c3\u03c3\u1fc6\u03bd\u03b1\u03c2: see Guhl _Ephesiaca_ 106 sq. The _Etymol. Magn._ has\n  \u1f18\u03c3\u03c3\u03ae\u03bd: \u1f41 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u1f7a\u03c2 \u03ba\u03b1\u03c4\u1f70 \u1f18\u03c6\u03b5\u03c3\u03af\u03bf\u03c5\u03c2, and adds several absurd derivations\n  of the word. In the sense of 'a king' it is used by Callimachus _Hymn.\n  Jov. 66_ \u03bf\u1f54 \u03c3\u03b5 \u03b8\u03b5\u1ff6\u03bd \u1f10\u03c3\u03c3\u1fc6\u03bd\u03b1 \u03c0\u03ac\u03bb\u03b9\u03bd \u03b8\u03ad\u03c3\u03b1\u03bd. It is probably not a Greek\n  word, as other terms connected with the worship of the Ephesian\n  Artemis (e.g. \u03bc\u03b5\u03b3\u03ac\u03b2\u03c5\u03b6\u03bf\u03c2, a Persian word) point to an oriental or at\n  least a non-Greek origin; and some have derived it from the Aramaic\n  \u05d4\u05e1\u05d9\u05df _chasin_ 'strong' or 'powerful.' But there is no sufficient\n  ground for connecting it directly with the name of the sect \u1f18\u03c3\u03c3\u03b7\u03bd\u03bf\u03af or\n  \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9, as some writers are disposed to do (e.g. Spanheim on\n  _Callim._ l.c., Creuzer _Symbolik_ IV. pp. 347, 349); though this view\n  is favoured by the fact that certain ascetic practices were enjoined\n  on these pagan 'Essenes.'\n\n[Sidenote: Sibylline Oracle A.D. 80.]\n\nSomewhat later we have another notice which bears in the same direction.\nThe Sibylline Oracle, which forms the fourth book in the existing\ncollection, is discovered by internal evidence to have been written\nabout A.D. 80[273]. It is plainly a product of Judaism, but its Judaism\ndoes not belong to the normal Pharisaic type. With Essenism it rejects\nsacrifices, even regarding the shedding of blood as a pollution[274],\nand with Essenism also it inculcates the duty of frequent washings[275].\nYet from other indications we are led to the conclusion, that this poem\nwas not written in the interests of Essenism properly so called, but\nrepresents some allied though independent development of Judaism. In\nsome respects at all events its language seems quite inconsistent with\nthe purer type of Essenism[276]. But its general tendency is clear: and\nof its locality there can hardly be a doubt. The affairs of Asia Minor\noccupy a disproportionate space in the poet's description of the past\nand vision of the future. The cities of the M\u00e6ander and its\nneighbourhood, among these Laodicea, are mentioned with emphasis[277].\n\nFootnote 273:\n\n  Its date is fixed by the following allusions. The temple at Jerusalem\n  has been destroyed by Titus (vv. 122 sq.), and the cities of Campania\n  have been overwhelmed in fire and ashes (vv. 127 sq.). Nero has\n  disappeared and his disappearance has been followed by bloody contests\n  in Rome (vv. 116 sq.); but his return is still expected (vv. 134 sq.).\n\nFootnote 274:\n\n  See vv. 27\u201330 \u03bf\u1f33 \u03bd\u03b7\u03bf\u1f7a\u03c2 \u03bc\u1f72\u03bd \u1f05\u03c0\u03b1\u03bd\u03c4\u03b1\u03c2 \u1f00\u03c0\u03bf\u03c3\u03c4\u03c1\u03ad\u03c8\u03bf\u03c5\u03c3\u03b9\u03bd \u1f30\u03b4\u03cc\u03bd\u03c4\u03b5\u03c2, \u03ba\u03b1\u1f76 \u03b2\u03c9\u03bc\u03bf\u1f7a\u03c2,\n  \u03b5\u1f30\u03ba\u03b1\u1fd6\u03b1 \u03bb\u03af\u03b8\u03c9\u03bd \u1f31\u03b4\u03c1\u03cd\u03bc\u03b1\u03c4\u03b1 \u03ba\u03c9\u03c6\u1ff6\u03bd \u1f05\u03b9\u03bc\u03b1\u03c3\u03b9\u03bd \u1f10\u03bc\u03c8\u03cd\u03c7\u03c9\u03bd \u03bc\u03b5\u03bc\u03b9\u03b1\u03c3\u03bc\u03ad\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03b8\u03c5\u03c3\u03af\u1fc3\u03c3\u03b9\n  \u03c4\u03b5\u03c4\u03c1\u03b1\u03c0\u03cc\u03b4\u03c9\u03bd \u03ba.\u03c4.\u03bb. In an earlier passage vv. 8 sq. it is said of God,\n  \u03bf\u1f54\u03c4\u03b5 \u03b3\u1f70\u03c1 \u03bf\u1f34\u03ba\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9 \u03bd\u03b1\u1ff7 \u03bb\u03af\u03b8\u03bf\u03bd \u1f31\u03b4\u03c1\u03c5\u03b8\u03ad\u03bd\u03c4\u03b1 \u03ba\u03c9\u03c6\u03cc\u03c4\u03b1\u03c4\u03bf\u03bd \u03bd\u03c9\u03b4\u03cc\u03bd \u03c4\u03b5, \u03b2\u03c1\u03bf\u03c4\u1ff6\u03bd\n  \u03c0\u03bf\u03bb\u03c5\u03b1\u03bb\u03b3\u03ad\u03b1 \u03bb\u03ce\u03b2\u03b7\u03bd.\n\nFootnote 275:\n\n  ver. 160 \u1f10\u03bd \u03c0\u03bf\u03c4\u03b1\u03bc\u03bf\u1fd6\u03c2 \u03bb\u03bf\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u1f45\u03bb\u03bf\u03bd \u03b4\u03ad\u03bc\u03b1\u03c2 \u03b1\u1f10\u03bd\u03ac\u03bf\u03b9\u03c3\u03b9. Another point of\n  contact with the Essenes is the great stress on prayers before meals,\n  ver. 26 \u03b5\u1f50\u03bb\u03bf\u03b3\u03ad\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c0\u03c1\u1f76\u03bd \u03c0\u03b9\u03ad\u03b5\u03b9\u03bd \u03c6\u03b1\u03b3\u03ad\u03b5\u03b9\u03bd \u03c4\u03b5. Ewald (_Sibyll. B\u00fccher_ p.\n  46) points also to the prominence of the words \u03b5\u1f50\u03c3\u03b5\u03b2\u03b5\u1fd6\u03bd, \u03b5\u1f50\u03c3\u03b5\u03b2\u03ae\u03c2,\n  \u03b5\u1f50\u03c3\u03b5\u03b2\u03af\u03b1 (vv. 26, 35, 42, 45, 133, 148, 151, 162, 165, 181, 183) to\n  designate the elect of God, as tending in the same direction. The\n  force of this latter argument will depend mainly on the derivation\n  which is given to the name _Essene_. See the appendix.\n\nFootnote 276:\n\n  Thus for instance, Ewald (l.c., p. 47) points to the tacit approval of\n  marriage in ver. 33. I hardly think however that this passage, which\n  merely condemns adultery, can be taken to imply so much. More\n  irreconcilable with pure Essenism is the belief in the resurrection of\n  the body and the future life on earth, which is maintained in vv. 176\n  sq.; though Hilgenfeld (_Zeitschr._ XIV. p. 49) does not recognise the\n  difficulty. See above p. 88. This Sibylline writer was perhaps rather\n  a Hemerobaptist than an Essene. On the relation of the Hemerobaptists\n  and Essenes see the appendix. Alexandre, _Orac. Sibyll._ (II. p. 323),\n  says of this Sibylline Oracle, 'Ipse liber haud dubie Christianus\n  est,' but there is nothing distinctly Christian in its teaching.\n\nFootnote 277:\n\n  vv. 106 sq., 145 sq.; see above p. 40, note 131. It begins \u03ba\u03bb\u1fe6\u03b8\u03b9 \u03bb\u03b5\u1f7c\u03c2\n  \u1f08\u03c3\u03af\u03b7\u03c2 \u03bc\u03b5\u03b3\u03b1\u03bb\u03b1\u03c5\u03c7\u03ad\u03bf\u03c2 \u0395\u1f50\u03c1\u03ce\u03c0\u03b7\u03c2 \u03c4\u03b5.\n\n[Sidenote: Phrygia and Asia congenial to this type of religion.]\n\nAnd certainly the moral and intellectual atmosphere would not be\nunfavourable to the growth of such a plant. The same district, which in\nspeculative philosophy had produced a Thales and a Heraclitus[278], had\ndeveloped in popular religion the worship of the Phrygian Cybele and\nSabazius and of the Ephesian Artemis[279]. Cosmological speculation,\nmystic theosophy, religious fanaticism, all had their home here.\nAssociated with Judaism or with Christianity the natural temperament and\nthe intellectual bias of the people would take a new direction; but the\nold type would not be altogether obliterated. Phrygia reared the hybrid\nmonstrosities of Ophitism[280]. She was the mother of Montanist\nenthusiasm[281], and the foster-mother of Novatian rigorism[282]. The\nsyncretist, the mystic, the devotee, the puritan, would find a congenial\nclimate in these regions of Asia Minor.\n\nFootnote 278:\n\n  The exceptional activity of the forces of nature in these districts of\n  Asia Minor may have directed the speculations of the Ionic school\n  towards physics, and more especially towards cosmogony. In Heraclitus\n  there is also a strong mystical element. But besides such broader\n  affinities, I venture to call attention to special dicta of the two\n  philosophers mentioned in the text, which curiously recall the tenets\n  of the Jud\u00e6o-Gnostic teachers. Thales declared (Diog. Laert. i. 27)\n  \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u1f14\u03bc\u03c8\u03c5\u03c7\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b1\u03b9\u03bc\u03cc\u03bd\u03c9\u03bd \u03c0\u03bb\u03ae\u03c1\u03b7, or, as reported by Aristotle\n  (_de An._ i. 5, p. 411), \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c0\u03bb\u03ae\u03c1\u03b7 \u03b8\u03b5\u1ff6\u03bd \u03b5\u1f30^\u03bd\u03b1\u03b9. In a recorded\n  saying of Heraclitus we have the very language of a Gnostic teacher;\n  Clem. Alex. _Strom._ v. 13, p. 699, _\u03c4\u1f70 \u03bc\u1f72\u03bd \u03c4\u1fc6\u03c2 \u03b3\u03bd\u03ce\u03c3\u03b9\u03bf\u03c2 \u03b2\u03ac\u03b8\u03b7\n  \u03ba\u03c1\u03cd\u03c0\u03c4\u03b5\u03b9\u03bd_ \u1f00\u03c0\u03b9\u03c3\u03c4\u03af\u03b7 \u1f00\u03b3\u03b1\u03b8\u03ae, \u03ba\u03b1\u03b8' \u1f29\u03b7\u03c1\u03ac\u03ba\u03bb\u03b5\u03b9\u03c4\u03bf\u03bd\u00b7 \u1f00\u03c0\u03b9\u03c3\u03c4\u03af\u03b7 \u03b3\u1f70\u03c1 \u03b4\u03b9\u03b1\u03c6\u03c5\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\n  \u03c4\u1f78 \u03bc\u1f74 \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03c3\u03b8\u03b1\u03b9. See above pp. 77, 92.\n\nFootnote 279:\n\n  For the characteristic features of Phrygian religious worship see\n  Steiger _Kolosser_ p. 70 sq.\n\nFootnote 280:\n\n  The prominence, which the Phrygian mysteries and Phrygian rites held\n  in the syncretism of the Ophites, is clear from the account of\n  Hippolytus _H\u00e6r._ v. 7 sq. Indeed Phrygia appears to have been the\n  proper home of Ophitism. Yet the admixture of Judaic elements is not\n  less obvious, as their name _Naassene_, derived from the Hebrew word\n  for a serpent, shows.\n\nFootnote 281:\n\n  The name, by which the Montanists were commonly known in the early\n  ages, was the sect of the 'Phrygians'; Clem. _Strom._ vii. 17, p. 900\n  \u03b1\u1f31 \u03b4\u1f72 [\u03c4\u1ff6\u03bd \u03b1\u1f31\u03c1\u03b5\u03c3\u03ad\u03c9\u03bd] \u1f00\u03c0\u1f78 \u1f14\u03b8\u03bd\u03bf\u03c5\u03c2 [\u03c0\u03c1\u03bf\u03c3\u03b1\u03b3\u03bf\u03c1\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9], \u1f61\u03c2 \u1f21 \u03c4\u1ff6\u03bd \u03a6\u03c1\u03c5\u03b3\u1ff6\u03bd\n  (comp. Eus. _H.E._ iv. 27, v. 16, Hipp. _H\u00e6r._ viii. 19, x. 25). From\n  \u03bf\u1f31 (or \u1f21) \u03ba\u03b1\u03c4\u1f70 \u03a6\u03c1\u03c5\u03b3\u03ac\u03c2 (Eus. _H.E._ ii. 25, v. 16, 18, vi. 20) comes\n  the sol\u0153cistic Latin name _Cataphryges_.\n\nFootnote 282:\n\n  Socrates (iv. 28) accounts for the spread of Novatianism in Phrygia by\n  the \u03c3\u03c9\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7 of the Phrygian temper. If so, it is a striking\n  testimony to the power of Christianity, that under its influence the\n  religious enthusiasm of the Phrygians should have taken this\n  direction, and that they should have exchanged the fanatical orgiasm\n  of their heathen worship for the rigid puritanism of the Novatianist.\n\n                  *       *       *       *       *\n\n[Sidenote: Previous results summed up.]\n\nIt has thus been shown _first_, that Essene Judaism was Gnostic in its\ncharacter; and _secondly_, that this type of Jewish thought and practice\nhad established itself in the Apostolic age in those parts of Asia Minor\nwith which we are more directly concerned. It now remains to examine the\nheresy of the [Sidenote: Is the Colossian heresy Gnostic?] Colossian\nChurch more nearly, and to see whether it deserves the name, which\nprovisionally was given to it, of Gnostic Judaism. Its Judaism all will\nallow. Its claim to be regarded as Gnostic will require a closer\nscrutiny. And in conducting [Sidenote: Three notes of Gnosticism.] this\nexamination, it will be convenient to take the three notes of Gnosticism\nwhich have been already laid down, and to enquire how far it satisfies\nthese tests.\n\n[Sidenote: 1. Intellectual exclusiveness.]\n\n1. It has been pointed out that Gnosticism strove to establish, or\nrather to preserve, an _intellectual oligarchy_ in religion. It had its\nhidden wisdom, its exclusive mysteries, its privileged class.\n\nNow I think it will be evident, that St Paul in this epistle [Sidenote:\nSt Paul contends for the universality of the Gospel,] feels himself\nchallenged to contend for the _universality_ of the Gospel. This indeed\nis a characteristic feature of the Apostle's teaching at all times, and\nholds an equally prominent place in the epistles of an earlier date. But\nthe point to be observed is, that the Apostle, in maintaining this\ndoctrine, has changed the mode of his defence; and this fact suggests\nthat there has been a change in the direction of the attack. It is no\nlonger against national exclusiveness, but against intellectual\nexclusiveness, that he contends. His adversaries do not now plead\nceremonial restrictions, or at least do not plead these alone: but they\nerect an artificial barrier of spiritual privilege, even more fatal to\nthe universal claims of the Gospel, because more specious and more\ninsidious. It is not now against Jew as such, but against the Jew become\nGnostic, that he fights the battle of liberty. In other words; it is not\nagainst Christian Pharisaism but against Christian Essenism that he\ndefends his position. Only in the light of such an antagonism can we\nunderstand the emphatic iteration with which he claims to 'warn _every_\nman and teach _every_ man in _every_ wisdom, that he may present\n[Sidenote: against the pretentions of an aristocracy of intellect.]\n_every_ man perfect in Christ Jesus[283].' It will be remembered that\n'wisdom' in Gnostic teaching was the exclusive possession of the few; it\nwill not be forgotten that 'perfection' was the term especially applied\nin their language to this privileged minority, as contradistinguished\nfrom the common herd of believers; and thus it will be readily\nunderstood why St Paul should go on to say that this universality of the\nGospel is the one object of his contention, to which all the energies of\nhis life are directed, and having done so, should express his intense\nanxiety for the Churches of Coloss\u00e6 and the neighbourhood, lest they\nshould be led astray by a spurious wisdom to desert the true\nknowledge[284]. This danger also will enable us to appreciate a novel\nfeature in another passage of the epistle. While dwelling on the\nobliteration of all distinctions in Christ, he repeats his earlier\ncontrasts, 'Greek and Jew,' 'circumcision and uncircumcision,'\n'bondslave and free'; but to these he adds new words which at once give\na wider scope and a more immediate application to the lesson. In Christ\nthe existence of 'barbarian' and even 'Scythian,' the lowest type of\nbarbarian, is extinguished[285]. As culture, civilisation, philosophy,\nknowledge, are no conditions of acceptance, so neither is their absence\nany disqualification in the believer. The aristocracy of intellectual\ndiscernment, which Gnosticism upheld in religion, is abhorrent to the\nfirst principles of the Gospel.\n\nFootnote 283:\n\n  i. 28 \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u03cd\u03bd\u03c4\u03b5\u03c2 _\u03c0\u03ac\u03bd\u03c4\u03b1_ \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 _\u03c0\u03ac\u03bd\u03c4\u03b1_\n  \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u1f10\u03bd _\u03c0\u03ac\u03c3\u1fc3_ \u03c3\u03bf\u03c6\u03af\u1fb3 \u1f35\u03bd\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ae\u03c3\u03c9\u03bc\u03b5\u03bd _\u03c0\u03ac\u03bd\u03c4\u03b1_ \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd\n  _\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd_ \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03ba.\u03c4.\u03bb. The reiteration has offended the\n  scribes; and the first \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd is omitted in some copies, the\n  second in others. For \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd see the note on the passage.\n\nFootnote 284:\n\n  The connexion of the sentences should be carefully observed. After the\n  passage quoted in the last note comes the asseveration that this is\n  the one object of the Apostle's preaching (i. 29) \u03b5\u1f30\u03c2 \u1f43 \u03ba\u03b1\u1f76 \u03ba\u03bf\u03c0\u03b9\u1ff6\n  \u03ba.\u03c4.\u03bb.; then the expression of concern on behalf of the Colossians\n  (ii. 1) \u03b8\u03ad\u03bb\u03c9 \u03b3\u1f70\u03c1 \u1f51\u03bc\u1fb6\u03c2 \u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9 \u1f21\u03bb\u03af\u03ba\u03bf\u03bd \u1f00\u03b3\u1ff6\u03bd\u03b1 \u1f14\u03c7\u03c9 _\u1f51\u03c0\u1f72\u03c1_ \u1f51\u03bc\u1ff6\u03bd\n  \u03ba.\u03c4.\u03bb.; then the desire that they may be brought (ii. 2) \u03b5\u1f30\u03c2\n  _\u03c0\u1fb6\u03bd_ \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 _\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03c2_ \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03bd\u03ad\u03c3\u03b5\u03c9\u03c2, \u03b5\u1f30\u03c2\n  _\u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd_ \u03c4\u03bf\u1fe6 \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6; then the definition of this\n  mystery (ii. 2, 3), \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f10\u03bd \u1fa7 \u03b5\u1f30\u03c3\u1f76\u03bd _\u03c0\u03ac\u03bd\u03c4\u03b5\u03c2_ \u03bf\u1f31 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u1f76\n  \u03ba.\u03c4.\u03bb.; then the warning against the false teachers (ii. 4) \u03c4\u03bf\u1fe6\u03c4\u03bf \u03bb\u03ad\u03b3\u03c9\n  \u1f35\u03bd\u03b1 \u03bc\u03b7\u03b4\u03b5\u1f76\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bb\u03bf\u03b3\u03af\u03b6\u03b7\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb.\n\nFootnote 285:\n\n  Col. iii. 11 after \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1f74 \u03ba\u03b1\u1f76 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u03b1 the Apostle adds \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2,\n  \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2. There is nothing corresponding to this in the parallel\n  passage, Gal. iii. 28.\n\n[Sidenote: He contrasts the true wisdom with the false,]\n\nHence also must be explained the frequent occurrence of the words\n'wisdom' (\u03c3\u03bf\u03c6\u03af\u03b1), 'intelligence' (\u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2), 'knowledge' (\u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2),\n'perfect knowledge' (\u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2), in this epistle[286]. St Paul takes up\nthe language of his opponents, and translates it into a higher sphere.\nThe false teachers put forward a 'philosophy,' but it was only an empty\ndeceit, only a plausible display of false-reasoning[287]. They pretended\n'wisdom,' but it was merely the profession, not the reality[288].\nAgainst these pretentions the Apostle sets the true wisdom of the\nGospel. On its wealth, its fulness, its perfection, he is never tired of\ndwelling[289]. The true wisdom, he would argue, is essentially spiritual\nand yet essentially definite; while the false is argumentative, is\nspeculative, [Sidenote: and dwells on the veritable mystery.] is vague\nand dreamy[290]. Again they had their rites of initiation. St Paul\ncontrasts with these the one universal, comprehensive mystery[291], the\nknowledge of God in Christ. This mystery is complete in itself: it\ncontains 'all the treasures of wisdom and of knowledge hidden' in\nit[292]. Moreover it is offered to all without distinction: though once\nhidden, its revelation is unrestricted, except by the waywardness and\ndisobedience of men. The esoteric spirit of Gnosticism finds no\ncountenance in the Apostle's teaching.\n\nFootnote 286:\n\n  For \u03c3\u03bf\u03c6\u03af\u03b1 see i. 9, 28, ii. 3, iii. 16, iv. 5; for \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 i. 9, ii.\n  2; for \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 ii. 3; for \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2 i. 9, 10, ii. 2, iii. 10.\n\nFootnote 287:\n\n  ii. 4 \u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03af\u03b1, ii. 8 \u03ba\u03b5\u03bd\u1f74 \u1f00\u03c0\u03ac\u03c4\u03b7.\n\nFootnote 288:\n\n  ii. 23 \u03bb\u03cc\u03b3\u03bf\u03bd \u03bc\u1f72\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2, where the \u03bc\u1f72\u03bd suggests the contrast of\n  the suppressed clause.\n\nFootnote 289:\n\n  e.g. i. 9, 28, iii. 16 \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3; ii. 2 \u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03c2. For the\n  'wealth' of this knowledge compare i. 27, ii. 2, iii. 16; and see\n  above p. 44.\n\nFootnote 290:\n\n  ii. 4, 18.\n\nFootnote 291:\n\n  i. 26, 27, ii. 2, iv. 3.\n\nFootnote 292:\n\n  ii. 2 \u1f10\u03bd \u1fa7 \u03b5\u1f30\u03c3\u1f76\u03bd \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03bf\u1f31 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u1f76 \u03c4\u1fc6\u03c2 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2\n  \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9. For the meaning of \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9 see above p. 90, and the note\n  on the passage.\n\n[Sidenote: 2. Speculative tenets.\n           Cosmogony and theology.]\n\n2. From the informing spirit of Gnosticism we turn to the speculative\ntenets\u2014the cosmogony and the theology of the Gnostic.\n\nAnd here too the affinities to Gnosticism reveal themselves in the\nColossian heresy. We cannot fail to observe that the [Sidenote: St Paul\nattacks the doctrine of angelic mediators,] Apostle has in view the\ndoctrine of intermediate agencies, regarded as instruments in the\ncreation and government of the world. Though this tenet is not\ndistinctly mentioned, it is tacitly assumed in the teaching which St\nPaul opposes to it. Against the philosophy of successive evolutions from\nthe Divine nature, angelic mediators forming the successive links in the\nchain which binds the finite to the Infinite, he sets the doctrine\n[Sidenote: setting against it the doctrine of the Word Incarnate,] of\nthe one Eternal Son, the Word of God begotten before the worlds[293].\nThe angelology of the heretics had a twofold bearing; it was intimately\nconnected at once with cosmogony and with religion. Correspondingly St\nPaul represents the mediatorial function of Christ as twofold: it is\nexercised in the natural creation, and it is exercised in the spiritual\ncreation. In both these spheres His initiative is absolute, His control\nis universal, His action is complete. By His agency the world of matter\nwas created and is sustained. He is at once the beginning and the\n[Sidenote: as the reconciler of heaven and earth.] end of the material\nuniverse; 'All things have been created through Him and unto Him.' Nor\nis His office in the spiritual world less complete. In the Church, as in\nthe Universe, He is sole, absolute, supreme; the primary source from\nwhich all life proceeds and the ultimate arbiter in whom all feuds are\nreconciled.\n\nFootnote 293:\n\n  The two great Christological passages are i. 15\u201320, ii. 9\u201315. They\n  will be found to justify the statements in this and the following\n  paragraphs of the text. For the meaning of individual expressions see\n  the notes on the passages.\n\n[Sidenote: His relations to (1) Deity; as God manifested.]\n\nOn the one hand, in relation to Deity, He is the visible image of the\ninvisible God. He is not only the chief manifestation of the Divine\nnature: He exhausts the Godhead manifested. In Him resides the totality\nof the Divine powers and attributes. For this totality Gnostic teachers\nhad a technical [Sidenote: The _pleroma_ resides in Him.] term, the\n_pleroma_ or _plenitude_[294]. From the pleroma they supposed that all\nthose agencies issued, through which God has at any time exerted His\npower in creation, or manifested His will through revelation. These\nmediatorial beings would retain more or less of its influence, according\nas they claimed direct parentage from it or traced their descent through\nsuccessive evolutions. But in all cases this pleroma was distributed,\ndiluted, transformed and darkened by foreign admixture. They were only\npartial and blurred images, often deceptive caricatures, of their\noriginal, broken lights of the great central Light. It is not improbable\nthat, like later speculators of the same school, they found a place\nsomewhere or other in their genealogy of spiritual beings for the\nChrist. If so, St Paul's language becomes doubly significant. But this\nhypothesis is not needed to explain its reference. In contrast to their\ndoctrine, he asserts and repeats the assertion, that the pleroma abides\nabsolutely and wholly in Christ as the Word of God[295]. The entire\nlight is concentrated in Him.\n\nFootnote 294:\n\n  See the detached note on \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1.\n\nFootnote 295:\n\n  i. 19 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9, ii. 9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7\n  \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2.\n\n[Sidenote: (2) Created things; as absolute Lord.]\n\nHence it follows that, as regards created things, His supremacy must be\nabsolute. In heaven as in earth, over things immaterial as over things\nmaterial, He is king. Speculations on the nature of intermediate\nspiritual agencies\u2014their names, their ranks, their offices\u2014were rife in\nthe schools of Jud\u00e6o-Gnostic thought. 'Thrones, dominations, princedoms,\nvirtues, powers'\u2013these formed part of the spiritual nomenclature which\nthey had invented to describe different grades of angelic mediators.\nWithout entering into these speculations, the Apostle asserts that\nChrist is Lord of all, the highest and the lowest, whatever rank they\nmay hold and by whatever name they are called[296], for they are parts\nof creation and He is the source of creation. Through Him they became,\nand unto Him they tend.\n\nFootnote 296:\n\n  See especially i. 16 \u03b5\u1f34\u03c4\u03b5 \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9 \u03b5\u1f34\u03c4\u03b5 \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03b5\u1f34\u03c4\u03b5 \u1f00\u03c1\u03c7\u03b1\u1f76 \u03b5\u1f34\u03c4\u03b5\n  \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9 \u03ba.\u03c4.\u03bb., compared with the parallel passage in Eph. i. 21\n  \u1f51\u03c0\u03b5\u03c1\u03ac\u03bd\u03c9 \u03c0\u03ac\u03c3\u03b7\u03c2 \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03b5\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c9\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76\n  \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f40\u03bd\u03bf\u03bc\u03b1\u03b6\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 \u03ba.\u03c4.\u03bb. Compare also ii. 10 \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74 \u03c0\u03ac\u03c3\u03b7\u03c2\n  \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2, and ii. 15 \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f70\u03c2\n  \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u03ba.\u03c4.\u03bb.\n\n[Sidenote: Angelolatry is therefore condemned]\n\nHence the worship of angels, which the false teachers inculcated, was\nutterly wrong in principle. The motive of this angelolatry it is not\ndifficult to imagine. There was a show of humility[297], for there was a\nconfession of weakness, in this subservience to inferior mediatorial\nagencies. It was held feasible to grasp at the lower links of the chain\nwhich bound earth to heaven, when heaven itself seemed far beyond the\nreach of man. The successive grades of intermediate beings were as\nsuccessive steps, by which man might mount the ladder leading up to the\nthrone of God. This carefully woven web of sophistry the Apostle tears\nto shreds. The doctrine of the false teachers was based on confident\nassumptions respecting angelic beings of whom they could know nothing.\nIt was moreover a denial of Christ's twofold personality and His\n[Sidenote: as a denial of His perfect mediation.] mediatorial office. It\nfollows from the true conception of Christ's Person, that He and He\nalone can bridge over the chasm between earth and heaven; for He is at\nonce the lowest and the highest. He raises up man to God, for He brings\ndown God to man. Thus the chain is reduced to a single link, this link\nbeing the Word made flesh. As the _pleroma_ resides in Him, so is it\ncommunicated to us through Him[298]. To substitute allegiance to any\nother spiritual mediator is to sever the connexion of the limbs with the\nHead, which is the centre of life and the mainspring of all energy\nthroughout the body[300].\n\nFootnote 297:\n\n  ii. 18 \u03b8\u03ad\u03bb\u03c9\u03bd \u1f10\u03bd \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3 \u03ba\u03b1\u1f76 \u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u1fb3 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd \u03ba.\u03c4.\u03bb.\n\nFootnote 298:\n\n  ii. 10; comp. i. 9.\n\nFootnote 300:\n\n  ii. 18.\n\n[Sidenote: The Apostle's practical inference.]\n\nHence follows the practical conclusion, that, whatever is done, must be\ndone in the name of the Lord[301]. Wives must submit to their husbands\n'in the Lord': children must obey their parents 'in the Lord': servants\nmust work for the masters as working 'unto the Lord[302].' This\niteration, 'in the Lord,' 'unto the Lord,' is not an irrelevant form of\nwords; but arises as an immediate inference from the main idea which\nunderlies the doctrinal portion of the epistle.\n\nFootnote 301:\n\n  iii. 17.\n\nFootnote 302:\n\n  iii. 18, 20, 23.\n\n[Sidenote: 3. Moral results of Gnostic doctrine.]\n\n3. It has been shown that the speculative tenets of Gnosticism might\nlead (and as a matter of fact we know that they did lead) to either of\ntwo practical extremes, to rigid asceticism or to unbridled license. The\nlatter alternative appears to some extent in the heresy of the Pastoral\nEpistles[303], and still more plainly in those of the Catholic\nEpistles[304] and the Apocalypse[305]. It is constantly urged by\nCatholic writers as a reproach against later Gnostic sects[306].\n\nFootnote 303:\n\n  At least in 2 Tim. iii. 1\u20137, where, though the most monstrous\n  developments of the evil were still future, the Apostle's language\n  implies that it had already begun. On the other hand in the picture of\n  the heresy in 1 Tim. iv. 2 the ascetic tendency still predominates.\n\nFootnote 304:\n\n  2 Pet. ii. 10 sq., Jude 8.\n\nFootnote 305:\n\n  Apoc. ii. 14, 20\u201322.\n\nFootnote 306:\n\n  See the notes on Clem. Rom. _Ep._ ii. \u00a7 9.\n\n[Sidenote: Asceticism of the Colossian heresy]\n\nBut the former and nobler extreme was the first impulse of the Gnostic.\nTo escape from the infection of evil by escaping from the domination of\nmatter was his chief anxiety. This appears very plainly in the Colossian\nheresy. Though the prohibitions to which the Apostle alludes might be\nexplained in part by the ordinances of the Mosaic ritual, this\nexplanation will not cover all the facts. Thus for instance drinks are\nmentioned as well as meats[307], though on the former the law of Moses\nis silent. Thus again the rigorous denunciation, 'Touch not, taste not,\nhandle not[308],' seems to go very far beyond the Levitical enactments.\nAnd moreover the _motive_ of these prohibitions [Sidenote: not explained\nby its Judaism.] is Essene rather than Pharisaic, Gnostic rather than\nJewish. These severities of discipline were intended 'to check\nindulgence of the flesh[309].' They professed to treat the body with\nentire disregard, to ignore its cravings and to deny its wants. In\nshort; they betray a strong _ascetic_ tendency[310], of which normal\nJudaism, as represented by the Pharisee, offers no explanation.\n\nFootnote 307:\n\n  ii. 16.\n\nFootnote 308:\n\n  ii. 21.\n\nFootnote 309:\n\n  ii. 23.\n\nFootnote 310:\n\n  Asceticism is of two kinds. There is the asceticism of dualism\n  (whether conscious or unconscious), which springs from a false\n  principle; and there is the asceticism of self-discipline, which is\n  the training of the Christian athlete (1 Cor. ix. 27). I need not say\n  that the remarks in the text apply only to the former.\n\n[Sidenote: St Paul's reply shows its Gnostic bearing.]\n\nAnd St Paul's answer points to the same inference. The difference will\nappear more plainly, if we compare it with his treatment of Pharisaic\nJudaism in the Galatian Church. This epistle offers nothing at all\ncorresponding to his language on that occasion; 'If righteousness be by\nlaw, then Christ died in vain'; 'If ye be circumcised, Christ shall\nprofit you nothing'; 'Christ is nullified for you, whosoever are\njustified by law; ye are fallen from grace[311].' The point of view in\nfact is wholly changed. With these Essene or Gnostic Judaizers the\nMosaic law was neither the motive nor the standard, it was only the\nstarting point, of their austerities. Hence in replying the [Sidenote:\nIt is no longer the contrast of law and grace.] Apostle no longer deals\nwith law, as law; he no longer points the contrast of grace and works;\nbut he enters upon the _moral_ aspects of these ascetic practices. He\ndenounces them, as concentrating the thoughts on earthly and perishable\nthings[312]. He points out that they fail in their purpose, and are\nfound valueless against carnal indulgences[313]. In their place he\noffers the true and only remedy against sin\u2014the elevation of the inner\nlife in Christ, the transference of the affections into a higher\nsphere[314], where the temptations of the flesh are powerless. Thus\ndying with Christ, they will kill all their earthly members[315]. Thus\nrising with Christ, they will be renewed in the image of God their\nCreator[316].\n\nFootnote 311:\n\n  Gal. ii. 21, v. 2, 4.\n\nFootnote 312:\n\n  ii. 8, 20\u201322.\n\nFootnote 313:\n\n  ii. 23 \u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2: see the note on\n  these words.\n\nFootnote 314:\n\n  iii. 1, 2.\n\nFootnote 315:\n\n  iii. 3, 5.\n\nFootnote 316:\n\n  iii. 10.\n\n[Sidenote: The truth of the above result tested by]\n\nIn attempting to draw a complete portrait of the Colossian heresy from a\nfew features accidentally exhibited in St Paul's epistle, it has been\nnecessary to supply certain links; and some assurance may not\nunreasonably be required that this has not been done arbitrarily. Nor is\nthis security wanting. In all such cases the test will be twofold. The\nresult must be consistent with itself: and it must do no violence to the\nhistorical conditions under which the phenomena arose.\n\n[Sidenote: (1) Its inherent consistency and symmetry.]\n\n1. In the present instance the former of these tests is fully satisfied.\nThe consistency and the symmetry of the result is its great\nrecommendation. The postulate of a Gnostic type brings the separate\nparts of the representation into direct connexion. The speculative\nopinions and the practical tendencies of the heresy thus explain, and\nare explained by, each other. It is analogous to the hypothesis of the\ncomparative anatomist, who by referring the fossil remains to their\nproper type restores the whole skeleton of some unknown animal from a\nfew bones belonging to different extremities of the body, and without\nthe intermediate and connecting parts. In the one case, as in the other,\nthe result is the justification of the postulate.\n\n[Sidenote: (2) Its place in a historical sequence.]\n\n2. And again; the historical conditions of the problem are carefully\nobserved. It has been shown already, that Judaism in the preceding age\nhad in one of its developments assumed a form which was the natural\nprecursor of the Colossian heresy. In order to complete the argument it\nwill be necessary to show that Christianity in the generation next\nsucceeding exhibited a perverted type, which was its natural outgrowth.\nIf this can be done, the Colossian heresy will take its proper place in\na regular historical sequence.\n\n                  *       *       *       *       *\n\n[Sidenote: Continuance of this type of Jud\u00e6o-Gnosticism in the\n           district.]\n\nI have already pointed out, that the language of St John in the\nApocalypse, which was probably written within a few years of this\nepistle, seems to imply the continuance in this district of the same\ntype of heresy which is here denounced by St Paul[317]. But the notices\nin this book are not more definite than those of the Epistle to the\nColossians itself; and we are led to look outside the Canonical writings\nfor some more explicit evidence. Has early Christian history then\npreserved any record of a distinctly Gnostic school existing on the\nconfines of the Apostolic age, which may be considered a legitimate\ndevelopment of the phase of religious speculation that confronts us\nhere?\n\nFootnote 317:\n\n  See above p. 41 sq.\n\n[Sidenote: Heresy of Cerinthus.]\n\n[Sidenote: His date and place.]\n\nWe find exactly the phenomenon which we are seeking in the heresy of\nCerinthus[318]. The time, the place, the circumstances, all agree. This\nheresiarch is said to have been originally a native of Alexandria[319];\nbut proconsular Asia is allowed on all hands to have been the scene of\nhis activity as a teacher[320]. He lived and taught at the close of the\nApostolic age, that is, in the latest decade of the first century. Some\nwriters indeed make him an antagonist of St Peter and St Paul[321], but\ntheir authority is not trustworthy, nor is this very early date at all\nprobable. But there can be no reasonable doubt that he was a\ncontemporary of St John, who was related by Polycarp to have denounced\nhim face to face on one memorable occasion[322], and is moreover said by\nIren\u00e6us to have written his Gospel with the direct object of confuting\nhis errors[323].\n\nFootnote 318:\n\n  The relation of Cerinthus to the Colossian heresy is briefly indicated\n  by Neander _Planting of Christianity_ I. p. 325 sq. (Eng. Trans.). It\n  has been remarked by other writers also, both earlier and later. The\n  subject appeared to me to deserve a fuller investigation than it has\n  yet received.\n\nFootnote 319:\n\n  Hippol. _H\u00e6r._ vii. 33 \u0391\u1f30\u03b3\u03c5\u03c0\u03c4\u03af\u03c9\u03bd \u03c0\u03b1\u03b9\u03b4\u03b5\u03af\u1fb3 \u1f00\u03c3\u03ba\u03b7\u03b8\u03b5\u03af\u03c2, x. 21 \u1f41 \u1f10\u03bd \u0391\u1f30\u03b3\u03cd\u03c0\u03c4\u1ff3\n  \u1f00\u03c3\u03ba\u03b7\u03b8\u03b5\u03af\u03c2, Theodoret. _H\u00e6r. Fab._ ii. 3 \u1f10\u03bd \u0391\u1f30\u03b3\u03cd\u03c0\u03c4\u1ff3 \u03c0\u03bb\u03b5\u00ee\u03c3\u03c4\u03bf\u03bd \u03b4\u03b9\u03b1\u03c4\u03c1\u03af\u03c8\u03b1\u03c2\n  \u03c7\u03c1\u03cc\u03bd\u03bf\u03bd.\n\nFootnote 320:\n\n  Iren. i. 26. 1 'et Cerinthus autem quidam ... in Asia docuit,'\n  Epiphan. _H\u00e6r._ xxviii. 1 \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u03b4\u1f72 \u03bf\u1f57\u03c4\u03bf\u03c2 \u1f41 \u039a\u03ae\u03c1\u03b9\u03bd\u03b8\u03bf\u03c2 \u1f10\u03bd \u03c4\u1fc7 \u1f08\u03c3\u03af\u1fb3\n  \u03b4\u03b9\u03b1\u03c4\u03c1\u03af\u03b2\u03c9\u03bd, \u03ba\u1f00\u03ba\u03b5\u1fd6\u03c3\u03b5 \u03c4\u03bf\u1fe6 \u03ba\u03b7\u03c1\u03cd\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1f74\u03bd \u1f00\u03c1\u03c7\u1f74\u03bd \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03bc\u03ad\u03bd\u03bf\u03c2, Theodoret. 1.\n  c. \u1f55\u03c3\u03c4\u03b5\u03c1\u03bf\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u1f08\u03c3\u03af\u03b1\u03bd \u1f00\u03c6\u03af\u03ba\u03b5\u03c4\u03bf. The scene of his encounter with St\n  John in the bath is placed at Ephesus: see below, note 322.\n\nFootnote 321:\n\n  Epiphanius (xxviii. 2 sq.) represents him as the ringleader of the\n  Judaizing opponents of the Apostles in the Acts and Epistles to the\n  Corinthians and Galatians. Philastrius (_H\u00e6r._ 36) takes the same\n  line.]\n\nFootnote 322:\n\n  The well-known story of the encounter between St John and Cerinthus in\n  the bath is related by Iren\u00e6us (iii. 3. 4) on the authority of\n  Polycarp, who appears from the sequence of Iren\u00e6us' narrative to have\n  told it at Rome, when he paid his visit to Anicetus; \u1f43\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03c0\u1f76\n  \u1f08\u03bd\u03b9\u03ba\u03ae\u03c4\u03bf\u03c5 \u1f10\u03c0\u03b9\u03b4\u03b7\u03bc\u03ae\u03c3\u03b1\u03c2 \u03c4\u1fc4 \u1fec\u03ce\u03bc\u1fc3 \u03c0\u03bf\u03bb\u03bb\u03bf\u1f7a\u03c2 \u1f00\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03c0\u03c1\u03bf\u03b5\u03b9\u03c1\u03b7\u03bc\u03ad\u03bd\u03c9\u03bd \u03b1\u1f31\u03c1\u03b5\u03c4\u03b9\u03ba\u1ff6\u03bd\n  \u1f10\u03c0\u03ad\u03c3\u03c4\u03c1\u03b5\u03c8\u03b5\u03bd ... \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c3\u1f76\u03bd \u03bf\u1f31 \u1f00\u03ba\u03b7\u03ba\u03bf\u03cc\u03c4\u03b5\u03c2 \u03b1\u1f50\u03c4\u03bf\u00fb \u1f45\u03c4\u03b9 \u1f38\u03c9\u03ac\u03bd\u03bd\u03b7\u03c2 \u03ba.\u03c4.\u03bb.\n\nFootnote 323:\n\n  Iren. iii. II. 1.\n\n[Sidenote: Cerinthus a link between Judaism and Gnosticism.]\n\n'Cerinthus,' writes Neander, 'is best entitled to be considered as the\nintermediate link between the Judaizing and the Gnostic sects.' 'Even\namong the ancients,' he adds, 'opposite reports respecting his doctrines\nhave been given from opposite points of view, according as the Gnostic\nor the Judaizing element was exclusively insisted upon: and the dispute\non this point has been kept up even to modern times. In point of\nchronology too Cerinthus may be regarded as representing the principle\nin its transition from Judaism to Gnosticism[324].'\n\nFootnote 324:\n\n  _Church History_ II. p. 42 (Bohn's Trans.).\n\n[Sidenote: Judaism still prominent in his system]\n\nOf his Judaism no doubt has been or can be entertained. The gross\nChiliastic doctrine ascribed to him[325], even though it may have been\nexaggerated in the representations of adverse writers, can only be\nexplained by a Jewish origin. His conception of the Person of Christ was\nEbionite, that is Judaic, in its main features[326]. He is said moreover\nto have enforced the rite of circumcision and to have inculcated the\nobservance of sabbaths[327]. It is related also that the Cerinthians,\nlike the Ebionites, accepted the Gospel of St Matthew alone[328].'\n\nFootnote 325:\n\n  See the _Dialogue of Caius and Proclus_ in Euseb. _H.E._ iii. 28,\n  Dionysius of Alexandria, ib. vii. 25, Theodoret. l.c., Augustin.\n  _H\u00e6r._ 8.\n\nFootnote 326:\n\n  See below p. 111.\n\nFootnote 327:\n\n  Epiphan. _H\u00e6r._ xxviii. 4, 5, Philastr. _H\u00e6r._ 36, Augustin. l.c. The\n  statements of these writers would not carry much weight in themselves;\n  but in this instance they are rendered highly probable by the known\n  Judaism of Cerinthus.\n\nFootnote 328:\n\n  Epiphan. _H\u00e6r._ xxviii. 5, xxx. 14, Philastr. _H\u00e6r._ 36.\n\n[Sidenote: though Gnosticism is already aggressive.]\n\nAt the same time, it is said by an ancient writer that his adherence to\nJudaism was only partial[329]. This limitation is doubtless correct. As\nGnostic principles asserted themselves more distinctly, pure Judaism\nnecessarily suffered. All or nearly all the early Gnostic heresies were\nJudaic; and for a time a compromise was effected which involved more or\nless concession on either side. But the ultimate incompatibility of the\ntwo at length became evident, and a precarious alliance was exchanged\nfor an open antagonism. This final result however was not reached till\nthe middle of the second century: and meanwhile it was a question to\nwhat extent Judaism was prepared to make concessions for the sake of\nthis new ally. Even the Jewish Essenes, as we have seen, departed from\nthe orthodox position in the matter of sacrifices; and if we possessed\nfuller information, we should probably find that they made still larger\nconcessions than this. Of the Colossian heretics we can only form a\nconjecture, but the angelology and angelolatry attributed to them point\nto a further step in the same direction. As we pass from them to\nCerinthus we are [Sidenote: Gnostic element in his teaching.] no longer\nleft in doubt; for the Gnostic element has clearly gained the ascendant,\nthough it has not yet driven its rival out of the field. Two\ncharacteristic features in his teaching especially deserve\nconsideration, both as evincing the tendency of his speculations and as\nthrowing back light on the notices in the Colossian Epistle.\n\nFootnote 329:\n\n  Epiphan. _H\u00e6r._ xxviii. 1 \u03c0\u03c1\u03bf\u03c3\u03ad\u03c7\u03b5\u03b9\u03bd \u03c4\u1ff7 \u1f38\u03bf\u03c5\u03b4\u03b1\u03ca\u03c3\u03bc\u1ff7 \u1f00\u03c0\u1f78 \u03bc\u03ad\u03c1\u03bf\u03c5\u03c2.\n\n[Sidenote: 1. His Gnostic Cosmogony]\n\n1. His cosmogony is essentially Gnostic. The great problem of creation\npresented itself to him in the same aspect; and the solution which he\noffered was generically the same. The world, he asserted, was not made\nby the highest God, but by an angel or power far removed from, and\nignorant of, this supreme Being[330]. Other authorities describing his\nsystem speak not of a single power, but of powers, as creating the\nuniverse[331]; but all alike represent this demiurge, or these\ndemiurges, as ignorant of the absolute God. It is moreover stated that\nhe held the Mosaic law to have been given not by the supreme God\nHimself, but by this angel, or one of these angels, who created the\nworld[332].\n\nFootnote 330:\n\n  Iren. i. 26. 1 'Non a primo Deo factum esse mundum docuit, sed a\n  virtute quadam valde separata et distante ab ea principalitate qu\u00e6 est\n  super universa, et ignorante eum qui est super omnia Deum'; Hippol.\n  _H\u00e6r._ vii. 33 \u1f14\u03bb\u03b5\u03b3\u03b5\u03bd \u03bf\u1f50\u03c7 \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03c0\u03c1\u03ce\u03c4\u03bf\u03c5 \u0398\u03b5\u03bf\u1fe6 \u03b3\u03b5\u03b3\u03bf\u03bd\u03ad\u03bd\u03b1\u03b9 \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd,\n  \u1f00\u03bb\u03bb' \u1f51\u03c0\u1f78 \u03b4\u03c5\u03bd\u03ac\u03bc\u03ad\u03c9\u03c2 \u03c4\u03b9\u03bd\u03bf\u03c2 \u03ba\u03b5\u03c7\u03c9\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03b7\u03c2 \u03c4\u1fc6\u03c2 \u1f51\u03c0\u1f72\u03c1 \u03c4\u1f70 \u1f45\u03bb\u03b1 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76\n  \u1f00\u03b3\u03bd\u03bf\u03bf\u1fe6\u03c3\u03b7\u03c2 \u03c4\u1f78\u03bd \u1f51\u03c0\u1f72\u03c1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u0398\u03b5\u03cc\u03bd, x. 21 \u1f51\u03c0\u1f78 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03ce\u03c2 \u03c4\u03b9\u03bd\u03bf\u03c2 \u1f00\u03b3\u03b3\u03b5\u03bb\u03b9\u03ba\u1fc6\u03c2,\n  \u03c0\u03bf\u03bb\u1f7a \u03ba\u03b5\u03c7\u03c9\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b5\u03c3\u03c4\u03ce\u03c3\u03b7\u03c2 \u03c4\u1fc6\u03c2 \u1f51\u03c0\u1f72\u03c1 \u03c4\u1f70 \u1f45\u03bb\u03b1 \u03b1\u1f50\u03b8\u03b5\u03bd\u03c4\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76\n  \u1f00\u03b3\u03bd\u03bf\u03bf\u03c5\u03c3\u03b7\u03c2 \u03c4\u1f78\u03bd \u1f51\u03c0\u1f72\u03c1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u0398\u03b5\u03cc\u03bd.\n\nFootnote 331:\n\n  Pseudo-Tertull. _H\u00e6r._ 3 'Carpocrates pr\u00e6terea hanc tulit sectam: Unam\n  esse dicit virtutem in superioribus principalem, ex hac prolatos\n  angelos atque virtutes, quos distantes longe a superioribus virtutibus\n  mundum istum in inferioribus partibus condidisse.... Post hunc\n  Cerinthus h\u00e6reticus erupit, similia docens. Nam et ipse mundum\n  institutum esse ab illis dicit'; Epiphan. _H\u00e6r._ xxviii. 1 \u1f15\u03bd\u03b1 \u03b5\u1f36\u03bd\u03b1\u03b9\n  \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03ba\u03cc\u03c4\u03c9\u03bd; Theodoret. H. F. ii. 3 \u1f15\u03bd\u03b1 \u03bc\u1f72\u03bd\n  \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1f78\u03bd \u03c4\u1ff6\u03bd \u1f45\u03bb\u03c9\u03bd \u0398\u03b5\u03cc\u03bd, \u03bf\u1f50\u03ba \u03b1\u1f50\u03c4\u1f78\u03bd \u03b4\u1f72 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03cc\u03bd,\n  \u1f00\u03bb\u03bb\u1f70 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u03b9\u03bd\u1f70\u03c2 \u03ba\u03b5\u03c7\u03c9\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03bd\u03c4\u03b5\u03bb\u1ff6\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f00\u03b3\u03bd\u03bf\u03bf\u03cd\u03c3\u03b1\u03c2;\n  Augustin. _H\u00e6r._ 8. The one statement is quite reconcilable with the\n  other. Among those angels by whose instrumentality the world was\n  created, Cerinthus appears to have assigned a position of preeminence\n  to one, whom he regarded as the demiurge in a special sense and under\n  whom the others worked; see Neander _Church History_ II. p. 43.\n\nFootnote 332:\n\n  Pseudo-Tertull. l.c.; Epiphan. _H\u00e6r._ xxviii. 4 \u03c4\u1f78\u03bd \u03b4\u03b5\u03b4\u03c9\u03ba\u03cc\u03c4\u03b1 \u03bd\u03cc\u03bc\u03bf\u03bd \u1f15\u03bd\u03b1\n  \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03ba\u03cc\u03c4\u03c9\u03bd.\n\n[Sidenote: and consequent angelology.]\n\nFrom these notices it is plain that angelology had an important place in\nhis speculations; and that he employed it to explain the existence of\nevil supposed to be inherent in the physical world, as well as to\naccount for the imperfections of the old dispensation. The 'remote\ndistance' of his angelic demiurge from the supreme God can hardly be\nexplained except on the hypothesis of _successive_ generations of these\nintermediate agencies. Thus his solution is thoroughly Gnostic. At the\nsame time, as contrasted with later and more sharply defined Gnostic\nsystems, the Judaic origin and complexion of his cosmogony is obvious.\nHis intermediate agencies still retain the name and the personality of\nangels, and have not yet given way to those vague idealities which, as\nemanations [Sidenote: Angels of earlier and \u00e6ons of later Gnostics.] or\n\u00e6ons, took their place in later speculations. Thus his theory is linked\non to the angelology of later Judaism founded on the angelic appearances\nrecorded in the Old Testament narrative. And again: while later Gnostics\nrepresent the demiurge and giver of the law as antagonistic to the\nsupreme and good God, Cerinthus does not go beyond postulating his\nignorance. He went as far as he could without breaking entirely with the\nOld Testament and abandoning his Judaic standing-ground.\n\n[Sidenote: Cerinthus a link between the Colossian heresy and later\n           Gnosticism.]\n\nIn these respects Cerinthus is the proper link between the incipient\ngnosis of the Colossian heretics and the mature gnosis of the second\ncentury. In the Colossian epistle we still breathe the atmosphere of\nJewish angelology, nor is there any trace of the _\u00e6on_ of later\nGnosticism[333]; while yet speculation is so far advanced that the\nangels have an important function in explaining the mysteries of the\ncreation and government of the world. On the other hand it has not\nreached the point at which we find it in Cerinthus. Gnostic conceptions\nrespecting the relation of the demiurgic agency to the supreme God would\nappear to have passed through three stages. This relation was\nrepresented first, as imperfect appreciation; next, as entire ignorance;\nlastly, as direct antagonism. The second and third are the standing\npoints of Cerinthus and of the later Gnostic teachers respectively. The\nfirst was probably the position of the Colossian false teachers. The\nimperfections of the natural world, they would urge, were due to the\nlimited capacities of these angels to whom the demiurgic work was\ncommitted, and to their imperfect sympathy with the supreme God; but at\nthe same time they might fitly receive worship as mediators between God\nand man; and indeed humanity seemed in its weakness to need the\nintervention of some such beings less remote from itself than the\nhighest heaven.\n\nFootnote 333:\n\n  I am quite unable to see any reference to the Gnostic conception of an\n  _\u00e6on_ in the passages of the New Testament, which are sometimes quoted\n  in support of this view, e.g., by Baur Paulus p. 428, Burton\n  _Lectures_ p. 111 sq.\n\n[Sidenote: 2. His Christology.]\n\n2. Again the Christology of Cerinthus deserves attention from this point\nof view. Here all our authorities are agreed. As a Judaizer Cerinthus\nheld with the Ebionites that Jesus was only the son of Joseph and Mary,\nborn in the natural way. As a Gnostic he maintained that the Christ\nfirst descended in the form of a dove on the carpenter's son at his\nbaptism; that He revealed to him the unknown Father, and worked miracles\nthrough him: and that at length He took His flight and left him, so that\nJesus alone suffered and rose, while the Christ remained\nimpassible[334]. It would appear also, though this is not certain, that\nhe described this re-ascension of the Christ, as a return 'to His own\n_pleroma_[335].'\n\nFootnote 334:\n\n  Iren. i. 26. 1, Hippol. _H\u00e6r._ vii. 33, x. 21, Epiphan. _H\u00e6r._ xxviii.\n  1, Theodoret. _H. F._ ii. 3. The arguments by which Lipsius\n  (_Gnosticismus_ pp. 245, 258, in Ersch u. Gruber; _Quellenkritik des\n  Epiphanios_ p. 118 sq.) attempts to show that Cerinthus did not\n  separate the Christ from Jesus, and that Iren\u00e6us (and subsequent\n  authors copying him) have wrongly attributed to this heretic the\n  theories of later Gnostics, seem insufficient to outweigh these direct\n  statements. It is more probable that the system of Cerinthus should\n  have admitted some foreign elements not very consistent with his\n  Judaic standing point, than that these writers should have been\n  misinformed. Inconsistency was a necessary condition of Judaic\n  Gnosticism. The point however is comparatively unimportant as\n  affecting my main purpose.\n\nFootnote 335:\n\n  Iren\u00e6us (iii. 11. 1), after speaking of Cerinthus and the Nicolaitans,\n  proceeds 'non, quemadmodum illi dicunt, alterum quidem fabricatorem\n  (i.e. demiurgum), alium autem Patrem Domini: et alium quidem\n  fabricatoris filium, alterum vero de superioribus Christum, quem et\n  impassibilem perseverasse, descendentem in Jesum filium fabricatoris,\n  et iterum _revolasse in suum pleroma_.' The doctrine is precisely that\n  which he has before ascribed to Cerinthus (i. 26. 1), but the mode of\n  statement may have been borrowed from the Nicolaitans or from some\n  later Gnostics. There is however no improbability in the supposition\n  that Cerinthus used the word pleroma in this way; see the detached\n  note on \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 below.\n\n[Sidenote: Approach towards Cerinthian Christology in the Colossian\n           heresy.]\n\nNow it is not clear from St Paul's language what opinions the Colossian\nheretics held respecting the person of our Lord; but we may safely\nassume that he regarded them as inadequate and derogatory. The emphasis,\nwith which he asserts the eternal being and absolute sovereignty of\nChrist, can hardly be explained in any other way. But individual\nexpressions tempt us to conjecture that the same ideas were already\nfloating in the air, which ultimately took form and consistency in the\ntenets of Cerinthus. Thus, when he reiterates the statement that the\n_whole_ pleroma abides _permanently_ in Christ[336], he would appear to\nbe tacitly refuting some opinion which maintained only mutable and\nimperfect relations between the two. When again he speaks of the true\ngospel first taught to the Colossians as the doctrine of 'the Christ,\n_even_ Jesus the Lord[337],' his language might seem to be directed\nagainst the tendency to separate the heavenly Christ from the earthly\nJesus, as though the connexion were only transient. When lastly he\ndwells on the work of reconciliation, as wrought 'through the blood of\nChrist's cross,' 'in the body of His flesh through death[338],' we may\nperhaps infer that he already discerned a disposition to put aside\nChrist's passion as a stumbling-block in the way of philosophical\nreligion. Thus regarded, the Apostle's language gains force and point;\nthough no stress can be laid on explanations which are so largely\nconjectural.\n\nFootnote 336:\n\n  i. 19, ii. 9. See above p. 102, note 295. On the force of \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6\u03bd\n  see the note on the earlier of the two passages.\n\nFootnote 337:\n\n  ii. 6 \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 \u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd, \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd.\n\nFootnote 338:\n\n  i. 20, 22.\n\n[Sidenote: The Gnosticism of the Colossians being vague and\n           undeveloped.]\n\nBut if so, the very generality of his language shows that these\nspeculations were still vague and fluctuating. The difference which\nseparates these heretics from Cerinthus may be measured by the greater\nprecision and directness in the Apostolic counter-statement, as we turn\nfrom the Epistle to the Colossians to the Gospel of St John. In this\ninterval, extending over nearly a quarter of a century, speculation had\ntaken a definite shape. The elements of Gnostic theory, which were\nbefore held in solution, had meanwhile crystallized around the facts of\nthe Gospel. Yet still we seem justified, even at the earlier date, in\nspeaking of these general ideas as Gnostic, guarding ourselves at the\nsame time against misunderstanding with the twofold caution, that we\nhere employ the term to express the simplest and most elementary\nconceptions of this tendency of thought, and that we do not postulate\nits use as a distinct designation of any sect or sects at this early\ndate. Thus limited, the view that the writer of this epistle is\ncombating a Gnostic heresy seems free from all objections, while it\nappears necessary to explain his language; and certainly it does not, as\nis sometimes imagined, place any weapon in the hands of those who would\nassail the early date and Apostolic authorship of the epistle.\n\n------------------------------------------------------------------------\n\n\n              _On some points connected with the Essenes._\n\n\n                                   I.\n                            THE NAME ESSENE.\n\n\n[Sidenote: Various forms of the name in Greek.]\n\nThe name is variously written in Greek;\n\n1. \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2: Joseph. _Ant._ xiii. 5. 9, xiii. 10. 6, xv. 10. 5, xviii. 1.\n  2, 5, _B.J._ ii. 8. 2, 13, _Vit._ 2; Plin. _N.H._ v. 15. 17 (Essenus);\n  Dion Chrys. in Synes. _Dion_ 3; Hippol. _H\u00e6r._ ix. 18, 28 (MS \u1f10\u03c3\u03b7\u03bd\u03cc\u03c2);\n  Epiphan. _H\u00e6r._ p. 28 sq, 127 (ed. Pet.).\n\n2. \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2: Philo II. pp. 457, 471, 632 (ed. Mang.); Hegesippus in\n  Euseb. _H.E._ iv. 22; Porphyr. _de Abstin._ iv. 11. So too Joseph.\n  _B.J._ ii. 7. 3, ii. 20. 4, iii. 2. 1; _Ant._ xv. 10. 4; though in the\n  immediate context of this last passage he writes \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, if the\n  common texts may be trusted.\n\n3. \u1f48\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2: Epiphan. _H\u00e6r._ pp. 40 sq., 125, 462. The common texts very\n  frequently make him write \u1f48\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, but see Dindorf's notes, Epiphan.\n  _Op._ 1. pp. 380, 425. With Epiphanius the Essenes are a Samaritan,\n  the Oss\u00e6ans a Judaic sect. He has evidently got his information from\n  two distinct sources, and does not see that the same persons are\n  intended.\n\n4. \u0399\u1f10\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2, Epiphan. _H\u00e6r._ p. 117. From the connexion the same sect\n  again seems to be meant: but owing to the form Epiphanius conjectures\n  (\u03bf\u1f36\u03bc\u03b1\u03b9) that the name is derived from Jesse, the father of David.\n\n[Sidenote: All etymologies to be rejected which derive the name.]\n\nIf any certain example could be produced where the name occurs in any\nearly Hebrew or Aramaic writing, the question of its derivation would\nprobably be settled; but in the absence of a single decisive instance a\nwide field is opened for conjecture, and critics have not been backward\nin availing themselves of the license. In discussing the claims of the\ndifferent etymologies proposed we may reject:\n\n[Sidenote: (i) From the Greek;]\n\n_First_: derivations from the Greek. Thus Philo connects the word with\n\u1f45\u03c3\u03b9\u03bf\u03c2 'holy': _Quod omn. prob._ 12, p. 457 \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9 ... \u03b4\u03b9\u03b1\u03bb\u03ad\u03ba\u03c4\u03bf\u03c5\n\u1f11\u03bb\u03bb\u03b7\u03bd\u03b9\u03ba\u1fc6\u03c2 \u03c0\u03b1\u03c1\u03ce\u03bd\u03c5\u03bc\u03bf\u03b9 \u1f41\u03c3\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2, \u00a7 13, p. 459 \u03c4\u1ff6\u03bd \u1f18\u03c3\u03c3\u03b1\u03af\u03c9\u03bd \u1f22 \u1f41\u03c3\u03af\u03c9\u03bd,\n_Fragm._ p. 632 \u03ba\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u03bc\u1f72\u03bd \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9, \u03c0\u03b1\u03c1\u1f70 \u03c4\u1f74\u03bd \u1f41\u03c3\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1, \u03bc\u03bf\u1f76 \u03b4\u03bf\u03ba\u1ff6\n[\u03b4\u03bf\u03ba\u03b5\u1fd6;], \u03c4\u1fc6\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b7\u03b3\u03bf\u03c1\u03af\u03b1\u03c2 \u1f00\u03be\u03b9\u03c9\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2. It is not quite clear whether\nPhilo is here playing with words after the manner of his master Plato,\nor whether he holds a pre-established harmony to exist among different\nlanguages by which similar sounds represent similar things, or whether\nlastly he seriously means that the name was directly derived from the\nGreek word \u1f45\u03c3\u03b9\u03bf\u03c2. The last supposition is the least probable; but he\ncertainly does not reject this derivation 'as incorrect' (Ginsburg\n_Essenes_ p. 27), nor can \u03c0\u03b1\u03c1\u03ce\u03bd\u03c5\u03bc\u03bf\u03b9 \u1f41\u03c3\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 be rendered 'from an\nincorrect derivation from the Greek homonym _hosiotes_' (ib. p. 32),\nsince the word \u03c0\u03b1\u03c1\u03ce\u03bd\u03c5\u03bc\u03bf\u03c2 never involves the notion of _false_ etymology.\nThe amount of truth which probably underlies Philo's statement will be\nconsidered hereafter. Another Greek derivation is \u1f34\u03c3\u03bf\u03c2, 'companion,\nassociate,' suggested by Rapoport, _Erech Millin_ p. 41. Several others\nagain are suggested by L\u00f6wy, s.v. Ess\u00e4er, e.g. \u1f14\u03c3\u03c9 from their esoteric\ndoctrine, or \u03b1\u1f36\u03c3\u03b1 from their fatalism. All such may be rejected as\ninstances of ingenious trifling, if indeed they deserve to be called\ningenious.\n\n[Sidenote: (ii) From names of persons or places;]\n\n_Secondly_: derivations from proper names whether of persons or of\nplaces. Thus the word has been derived from _Jesse_ the father of David\n(Epiphan. l.c.), or from one \u05d9\u05e9\u05d9 Isai, the disciple of R. Joshua ben\nPerachia who migrated to Egypt in the time of Alexander Jann\u00e6us (L\u00f6w in\n_Ben Chananja_ i. p. 352). Again it has been referred to the town _Essa_\n(a doubtful reading in Joseph. _Ant._ xiii. 15. 3) beyond the Jordan.\nAnd other similar derivations have been suggested.\n\n[Sidenote: From Hebrew roots not supplying the right consonants,]\n\n_Thirdly_: etymologies from the Hebrew or Aramaic, which do not supply\nthe right consonants, or do not supply them in the right order. Under\nthis head several must be rejected;\n\n\u05d0\u05e1\u05e8 _\u0101sar_ 'to bind,' Adler _Volkslehrer_ VI. p. 50, referred to by\n  Ginsburg _Essenes_ p. 29.\n\n\u05d7\u05e1\u05d9\u05d3 _ch\u0101s\u012bd_ 'pious,' which is represented by \u1f08\u03c3\u03b9\u03b4\u03b1\u1fd6\u03bf\u03c2 (1 Macc. ii. 42\n  (v. l.), vii. 13, 2 Macc. xiv. 6), and could not possibly assume the\n  form \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 or \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2. Yet this derivation appears in Josippon ben\n  Gorion (iv. 6, 7, v. 24, pp. 274, 278, 451), who substitutes\n  _Chasidim_ in narratives where the Essenes are mentioned in the\n  original of Josephus; and it has been adopted by many more recent\n  writers.\n\n\u05e1\u05d7\u05d0 _s'\u0101ch_ 'to bathe,' from which with an _Aleph_ prefixed we might get\n  \u05d0\u05e1\u05d4\u05d0\u05d9 _as'chai_ 'bathers' (a word however which does not occur): Gr\u00e4tz\n  _Gesch. der Juden_ iii. pp. 82, 468.\n\n\u05e6\u05e0\u05d5\u05e2 _tsan\u016ba\u0639_ 'retired, modest,' adopted by Frankel (_Zeitschrift_\n  1846, p. 449, _Monatschrift_ II[. p. 32) after a suggestion by L\u00f6w.\n\n[Sidenote: such as those which make _n_ part of the root.]\n\nTo this category must be assigned those etymologies which contain a \u05d5 as\nthe third consonant of the root; since the comparison of the parallel\nforms \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 and \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2 shows that in the latter word the \u03bd is only\nformative. On this ground we must reject:\n\n\u05d7\u05e1\u05d9\u05df _ch\u0101s\u012bn_; see below under \u05e2\u05e9\u05d9\u05df.\n\n\u05d7\u05e6\u05df _ch\u014dtsen_ 'a fold' of a garment, and so supposed to signify the\n\u03c0\u03b5\u03c1\u03af\u03b6\u03c9\u03bc\u03b1 or 'apron', which was given to every neophyte among the Essenes\n(Joseph. _B.J._ ii. 8. 5, 7): suggested by Jellinek _Ben Chananja_ IV.\np. 374.\n\n\u05e2\u05e9\u05d9\u05df _\u0639\u0101sh\u012bn_ 'strong': see Cohn in Frankel's _Monatschrift_ VII. p.\n271. This etymology is suggested to explain Epiphanius _H\u00e6r._ p. 40\n\u03c4\u03bf\u1fe6\u03c4\u03bf \u03b4\u1f72 \u03c4\u1f78 \u03b3\u03ad\u03bd\u03bf\u03c2 \u03c4\u1ff6\u03bd \u1f48\u03c3\u03c3\u03b7\u03bd\u1ff6\u03bd \u1f11\u03c1\u03bc\u03b7\u03bd\u03b5\u03cd\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f10\u03ba\u03b4\u03cc\u03c3\u03b5\u03c9\u03c2 \u03c4\u03bf\u1fe6 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2\n\u03c3\u03c4\u03b9\u03b2\u03b1\u03c1\u1f78\u03bd \u03b3\u03ad\u03bd\u03bf\u03c2 ('a sturdy race'). The name 'Essene' is so interpreted\nalso in Makrisi (de Sacy, _Chrestom. Arab._ I. p. 114, 306); but, as he\nhimself writes it with _Elif_ and not _Ain_, it is plain that he got\nthis interpretation from some one else, probably from Epiphanius. The\ncorrect reading however in Epiphanius is \u1f48\u03c3\u03c3\u03b1\u03af\u03c9\u03bd, not \u1f48\u03c3\u03c3\u03b7\u03bd\u1ff6\u03bd; and it\nwould therefore appear that this father or his informant derived the\nword from the Hebrew root \u05e2\u05df\u05d5 rather than from the Aramaic \u05e2\u05e9\u05df. The\n\u1f48\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9 would then be the \u05e2\u05d5\u05d9\u05e1, and this is so far a possible\nderivation, that the _n_ does not enter into the root. Another word\nsuggested to explain the etymology of Epiphanius is the Aramaic \u05d7\u05e1\u05d9\u05df\n_ch\u0101s\u012bn_ 'powerful, strong' (from \u05d4\u05e1\u05df); but this is open to the same\nobjections as \u05e2\u05e9\u05d9\u05df.\n\n[Sidenote: Other derivations considered:]\n\nWhen all such derivations are eliminated as untenable or improbable,\nconsiderable uncertainty still remains. The 1st and 3rd radicals might\nbe any of the gutturals \u05d0,\u05d4,\u05d7,\u05e2; and the Greek \u03c2, as the 2nd radical,\nmight represent any one of several Shemitic sibilants.\n\nThus we have the choice of the following etymologies, which have found\nmore or less favour.\n\n[Sidenote: (1) \u05d0\u05e1\u05d9\u05d0 'a physician';]\n\n(1) \u05d0\u05e1\u05d0 _\u0103s\u0101_ 'to heal,' whence \u05d0\u05e1\u05d9\u05d0 _asy\u0101_, 'a physician.' The Essenes\nare supposed to be so called because Josephus states (_B.J._ ii. 8. 6)\nthat they paid great attention to the qualities of herbs and minerals\nwith a view to the healing of diseases (\u03c0\u03c1\u1f78\u03c2 \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03af\u03b1\u03bd \u03c0\u03b1\u03b8\u1ff6\u03bd). This\netymology is supported likewise by an appeal to the name \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03c5\u03c4\u03b1\u03af,\nwhich Philo gives to an allied sect in Egypt (_de Vit. Cont._ \u00a7 1, II.\np. 471). It seems highly improbable however, that the ordinary name of\nthe Essenes should have been derived from a pursuit which was merely\nsecondary and incidental; while the supposed analogy of the Therapeut\u00e6\nrests on a wrong interpretation of the word. Philo indeed (l.c.), bent\nupon extracting from it as much moral significance as possible, says,\n\u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03c5\u03c4\u03b1\u1f76 \u03ba\u03b1\u1f76 \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03c5\u03c4\u03c1\u03af\u03b4\u03b5\u03c2 \u03ba\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9, \u1f24\u03c4\u03bf\u03b9 \u03c0\u03b1\u03c1' \u1f45\u03c3\u03bf\u03bd \u1f30\u03b1\u03c4\u03c1\u03b9\u03ba\u1f74\u03bd\n\u1f10\u03c0\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03bf\u03bd\u03c4\u03b1\u03b9 \u03ba\u03c1\u03b5\u03af\u03c3\u03c3\u03bf\u03bd\u03b1 \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c0\u03cc\u03bb\u03b5\u03b9\u03c2 \u1f21 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b1 \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03cd\u03b5\u03b9\n\u03bc\u03cc\u03bd\u03bf\u03bd, \u1f10\u03ba\u03b5\u03af\u03bd\u03b7 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c8\u03c5\u03c7\u1f70\u03c2 \u03ba.\u03c4.\u03bb.) \u1f22 \u03c0\u03b1\u03c1' \u1f45\u03c3\u03bf\u03bd \u1f10\u03ba \u03c6\u03cd\u03c3\u03b5\u03c9\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f31\u03b5\u03c1\u1ff6\u03bd\n\u03bd\u03cc\u03bc\u03c9\u03bd \u1f10\u03c0\u03b1\u03b9\u03b4\u03b5\u03cd\u03b8\u03b7\u03c3\u03b1\u03bd \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03cd\u03b5\u03b9\u03bd \u03c4\u1f78 \u1f43\u03bd \u03ba.\u03c4.\u03bb.: but the latter meaning alone\naccords with the usage of the word; for \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03c5\u03c4\u03ae\u03c2, used absolutely,\nsignifies 'a worshipper, devotee,' not 'a physician, healer.' This\netymology of \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 is ascribed, though wrongly, to Philo by Asaria di\nRossi (_Meor Enayim_ 3, fol. 33 _a_) and has been very widely received.\nAmong more recent writers, who have adopted or favoured it, are\nBellermann (_Ueber Ess\u00e4er u. Therapeuten_ p. 7), Gfr\u00f6rer (_Philo_ II. p.\n341), D\u00e4hne (_Ersch u. Gruber_, s.v.), Baur (_Christl. Kirche der drei\nerst. Jahrh._ p. 20), Herzfeld (_Gesch. des Judenthums_ II. p. 371, 395,\n397 sq.), Geiger (_Urschrift_ p. 126), Derenbourg (_L'Histoire et la\nG\u00e9ographie de la Palestine_ pp. 170, 175, notes), Keim (_Jesus von\nNazara_ I. p. 284 sq.), and Hamburger (_Real-Encyclop\u00e4die f\u00fcr Bibel u.\nTalmud_, s.v.). Several of these writers identify the Essenes with the\nBaithusians (\u05d1\u05d9\u05d4\u05d5\u05e1\u05d9\u05df) of the Talmud, though in the Talmud the\nBaithusians are connected with the Sadducees. This identification was\nsuggested by di Rossi (l.c. fol. 33 _b_), who interprets 'Baithusians'\nas 'the school of the Essenes' (\u05d1\u05d9\u05d7 \u05d0\u05d9\u05e1\u05d9\u05d0): while subsequent writers,\ngoing a step further, have explained it 'the school of the physicians'\n(\u05d1\u05d9\u05d7 \u05d0\u05d9\u05e1\u05d9\u05d0).\n\n[Sidenote: (2) \u05d7\u05d6\u05d9\u05d0 'a seer';]\n\n(2) \u05d7\u05d6\u05d0 _ch\u0103z\u0101_ 'to see', whence \u05d7\u05d6\u05d9\u05d0 _chazy\u0101_ 'a seer', in reference to\nthe prophetic powers which the Essenes claimed, as the result of ascetic\ncontemplation: Joseph. _B.J._ ii. 8. 12 \u03b5\u1f30\u03c3\u1f76 \u03b4\u1f72 \u1f10\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f43\u03b9 \u03ba\u03b1\u1f76 \u03c4\u1f70\n\u03bc\u03ad\u03bb\u03bb\u03bf\u03bd\u03c4\u03b1 \u03c0\u03c1\u03bf\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd \u1f51\u03c0\u03b9\u03c3\u03c7\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb. For instances of such Essene\nprophets see _Ant._ xiii. II. 2, xv. 10. 5, _B.J._ I. 3. 5, ii. 7. 3.\nSuidas, s.v. \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9, says: \u03b8\u03b5\u03c9\u03c1\u03af\u1fb3 \u03c4\u1f70 \u03c0\u03bf\u03bb\u03bb\u1f70 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03bf\u03c5\u03c3\u03b9\u03bd, \u1f14\u03bd\u03b8\u03b5\u03bd \u03ba\u03b1\u1f76\n\u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9 \u03ba\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9, \u03c4\u03bf\u1fe6\u03c4\u03bf \u03b4\u03b7\u03bb\u03bf\u1fe6\u03bd\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2, \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9, \u03b8\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03bf\u03af.\nFor this derivation, which was suggested by Baumgarten (see Bellermann\np. 10) and is adopted by Hilgenfeld (_J\u00fcd. Apocal._ p. 278), there is\nsomething to be said: but \u05d7\u05d6\u05d0 is rather \u1f41\u03c1\u1fb6\u03bd than \u03b8\u03b5\u03c9\u03c1\u03b5\u1fd6\u03bd; and thus it\nmust denote the result rather than the process, the _vision_ which was\nthe privilege of the few rather than the _contemplation_ which was the\nduty of all. Indeed in a later paper (_Zeitschr_. XI. p. 346, 1868)\nHilgenfeld expresses himself doubtfully about this derivation, feeling\nthe difficulty of explaining the \u03c3\u03c2 from the \u05d6. This is a real\nobjection. In the transliteration of the LXX the \u05d6 is persistently\nrepresented by \u03b6, and the \u05e6 by \u03c2. The exceptions to this rule, where the\nmanuscript authority is beyond question, are very few, and in every case\nthey seem capable of explanation by peculiar circumstances.\n\n[Sidenote: (3) \u05e2\u05e9\u05d4 'to do';]\n\n(3) _\u0639\u0101s\u0101h_ 'to do,' so that \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9 would signify 'the doers, the\nobservers of the law,' thus referring to the strictness of Essene\npractices: see Oppenheim in Frankel's _Monatschrift_ VII. p. 272 sq. It\nhas been suggested also that, as the Pharisees were especially\ndesignated the teachers, the Essenes were called the 'doers' by a sort\nof antithesis: see an article in Jost's _Annalen_ 1839, p. 145. Thus the\ntalmudic phrase \u05d0\u05e0\u05e9\u05d9 \u05de\u05e2\u05e9\u05d4, interpreted 'men of practice, of good deeds,'\nis supposed to refer to the Essenes (see Frankel's _Zeitschrift_ III. p.\n458, _Monatschrift_ II. p. 70). In some passages indeed (see Surenhuis\n_Mishna_ III. p. 313) it may possibly mean 'workers of miracles' (as\n\u1f14\u03c1\u03b3\u03bf\u03bd Joh. v. 20, vii. 21, x. 25, etc.); but in this sense also it might\nbe explained of the thaumaturgic powers claimed by the Essenes. (See\nbelow, p. 126.) On the use which has been made of a passage in the\n_Aboth_ of R. Nathan c. 37, as supporting this derivation, I shall have\nto speak hereafter. Altogether this etymology has little or nothing to\nrecommend it.\n\nI have reserved to the last the two derivations which seem to deserve\nmost consideration.\n\n[Sidenote: (4) _chasyo_ 'pious';]\n\n(4) \u071a\u0724\u071d _chasi_ (\u071a\u0723\u0710 _ch's\u0113_) or \u071a\u0723\u071d\u0710 _chasyo_, 'pious,' in Syriac. This\nderivation, which is also given by de Sacy (_Chrestom. Arab._ I. p.\n347), is adopted by Ewald (_Gesch. des V. Isr._ IV. p. 484, ed. 3, 1864,\nVII. pp. 154, 477, ed. 2, 1859), who abandons in its favour another\netymology (\u05d4\u05d6\u05df _chazzan_ 'watcher, worshipper' = \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03c5\u03c4\u03ae\u03c2) which he\nhad suggested in an earlier edition of his fourth volume (p. 420). It is\nrecommended by the fact that it resembles not only in sound, but in\nmeaning, the Greek \u1f45\u03c3\u03b9\u03bf\u03c2, of which it is a common rendering in the\nPeshito (Acts ii. 27, xiii. 35, Tit. i. 8). Thus it explains the\nderivation given by Philo (see above, p. 115), and it also accounts for\nthe tendency to write \u1f48\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 for \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 in Greek. Ewald moreover\npoints out how an Essenizing Sibylline poem (_Orac. Sib._ iv; see above,\np. 96) dwells on the Greek equivalents, \u03b5\u1f50\u03c3\u03b5\u03b2\u03ae\u03c2, \u03b5\u1f50\u03c3\u03b5\u03b2\u03af\u03b7, etc. (vv. 26,\n35, 42 sq., 148 sq., 162, 165 sq., 178 sq., ed. Alexandre), as if they\nhad a special value for the writer: see _Gesch._ VII. p. 154, _Sibyll.\nB\u00fccher_ p. 46. Lipsius (Schenkel's _Bibel-Lexikon_, _s.v._) also\nconsiders this the most probable etymology.\n\n[Sidenote: (5) \u05d7\u05e9\u05d0\u05d9\u05dd 'silent ones.']\n\n(5) \u05d7\u05e9\u05d0 _ch\u0101sh\u0101_ (also \u05d7\u05e9\u05d4) Heb., 'to be silent'; whence \u05d7\u05e9\u05d0\u05d9\u05dd\n_chashsh\u0101\u012bm_ 'the silent ones,' who meditate on mysteries. Jost (_Gesch.\nd. Judenth._ I. p. 207) believes that this was the derivation accepted\nby Josephus, since he elsewhere (_Ant._ iii. 7. 5, iii. 8. 9) writes out\n\u05d7\u05e9\u05df, _ch\u014dshen_ 'the high-priest's breast-plate' (Exod. xxviii. 15 sq),\n\u1f10\u03c3\u03c3\u03ae\u03bd or \u1f10\u03c3\u03c3\u03ae\u03bd\u03b7\u03c2 in Greek, and explains it \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9, \u03c4\u03bf\u1fe6\u03c4\u03bf \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd\n\u1f19\u03bb\u03bb\u03ae\u03bd\u03c9\u03bd \u03b3\u03bb\u1ff6\u03c4\u03c4\u03b1\u03bd \u03bb\u03bf\u03b3\u03b5\u1fd6\u03bf\u03bd (i.e. the 'place of oracles' or 'of reason':\ncomp. Philo _de Mon._ ii. \u00a7 5, II. p. 226 \u03ba\u03b1\u03bb\u03b5\u1fd6\u03c4\u03b1\u03b9 \u03bb\u03bf\u03b3\u03b5\u1fd6\u03bf\u03bd \u1f10\u03c4\u03cd\u03bc\u03c9\u03c2,\n\u1f10\u03c0\u03b5\u03b9\u03b4\u1f74 \u03c4\u1f70 \u1f10\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1ff7 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03bb\u03cc\u03b3\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03b1\u03bb\u03bf\u03b3\u03af\u03b1\u03b9\u03c2 \u03b4\u03b5\u03b4\u03b7\u03bc\u03b9\u03bf\u03cd\u03c1\u03b3\u03b7\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb.),\nas it is translated in the LXX. Even though modern critics should be\nright in connecting \u05d7\u05e9\u05df with the Arab. \ufea3\u0633\u0646 'pulcher fuit, ornavit' (see\nGesen. _Thes._ p. 535, s.v.), the other derivation may have prevailed in\nJosephus' time. We may illustrate this derivation by Josephus'\ndescription of the Essenes, _B.J._ ii. 8. 5 \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03be\u03c9\u03b8\u03b5\u03bd \u1f61\u03c2 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03cc\u03bd \u03c4\u03b9\n\u03c6\u03c1\u03b9\u03ba\u03c4\u1f78\u03bd \u1f21 \u03c4\u1ff6\u03bd \u1f14\u03bd\u03b4\u03bf\u03bd \u03c3\u03b9\u03c9\u03c0\u1f74 \u03ba\u03b1\u03c4\u03b1\u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9; and perhaps this will also\nexplain the Greek equivalent \u03b8\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03bf\u03af, which Suidas gives for \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03b9.\nThe use of the Hebrew word \u05d7\u05e9\u05d0\u05d9\u05dd in Mishna _Shekalim_ v. 6, though we\nneed not suppose that the Essenes are there meant, will serve to show\nhow it might be adopted as the name of the sect. On this word see Levy\n_Chald\u00e4isches W\u00f6rterbuch_ p. 287. On the whole this seems the most\nprobable etymology of any, though it has not found so much favour as the\nlast. At all events the rules of transliteration are entirely satisfied,\nand this can hardly be said of the other derivations which come into\ncompetition with it.\n\n\n                                   2.\n                 ORIGIN AND AFFINITIES OF THE ESSENES.\n\n[Sidenote: The principle of the restoration.]\n\nThe ruling principle of the Restoration under Ezra was the isolation of\nthe Jewish people from all influences of the surrounding nations. Only\nby the rigorous application of this principle was it possible to guard\nthe nationality of the Hebrews, and thus to preserve the sacred deposit\nof religious truth of which this nationality was the husk. Hence the\nstrictest attention was paid to the Levitical ordinances, and more\nespecially to those which aimed at ceremonial purity. The principle,\nwhich was thus distinctly asserted at the period of the national\nrevival, gained force and concentration at a later date from the active\nantagonism to which the patriotic Jews were driven by the religious and\npolitical aggressions of the Syrian kings. During the Maccab\u00e6an wars we\nread of a party or sect [Sidenote: Rise of the Asid\u00e6ans.] called the\n_Chasidim_ or _Asid\u00e6ans_ (\u1f08\u03c3\u03b9\u03b4\u03b1\u1fd6\u03bf\u03b9), the 'pious' or 'devout,' who\nzealous in their observance of the ceremonial law stoutly resisted any\nconcession to the practices of Hellenism, and took their place in the\nvan of the struggle with their national enemies, the Antiochene monarchs\n(1 Macc. ii. 42, vii. 13, 2 Macc. xiv. 6). But, though their names\nappear now for the first time, they are not mentioned as a newly formed\nparty; and it is probable that they had their origin at a much earlier\ndate.\n\nThe subsequent history of this tendency to exclusiveness and isolation\nis wrapt in the same obscurity. At a somewhat later date [Sidenote:\nPharisaism and Essenism traced to the same principle.] it is exhibited\nin the _Pharisees_ and the _Essenes_; but whether these were\nhistorically connected with the Chasidim as divergent offshoots of the\noriginal sect, or whether they represent independent developments of the\nsame principle, we are without the proper data for deciding. The\nprinciple itself appears in the name of the Pharisees, which, as\ndenoting 'separation,' points to the avoidance of all foreign and\ncontaminating influences. On the other hand the meaning of the name\n_Essene_ is uncertain, for the attempt to derive it directly from\n_Chasidim_ must be abandoned; but the tendency of the sect is\nunmistakeable. If with the Pharisees ceremonial purity was a principal\naim, with the Essenes it was an absorbing passion. It was enforced and\nguarded moreover by a special organization. While the Pharisees were a\nsect, the Essenes were an order. Like the Pythagoreans in Magna Gr\u00e6cia\nand the Buddhists in India before them, like the Christian monks of the\nEgyptian and Syrian deserts after them, they were formed into a\nreligious brotherhood, fenced about by minute and rigid rules, and\ncarefully guarded from any contamination with the outer world.\n\n[Sidenote: Foreign elements in Essenism.]\n\nThus the sect may have arisen in the heart of Judaism. The idea of\nceremonial purity was essentially Judaic. But still, when we turn to the\nrepresentations of Philo and Josephus, it is impossible to overlook\nother traits which betoken foreign affinities. Whatever the Essenes may\nhave been in their origin, at the Christian era at least and in the\nApostolic age they no longer represented the current type of religious\nthought and practice among the Jews. This foreign element has been\nderived by some from the Pythagoreans, by others from the Syrians or\nPersians or even from the farther East; but, whether Greek or Oriental,\nits existence has until lately been almost universally allowed.\n\n[Sidenote: Frankel's theory well received,]\n\nThe investigations of Frankel, published first in 1846 in his\n_Zeitschrift_, and continued in 1853 in his _Monatschrift_, have given a\ndifferent direction to current opinion. Frankel maintains that Essenism\nwas a purely indigenous growth, that it is only Pharisaism in an\nexaggerated form, and that it has nothing distinctive and owes nothing,\nor next to nothing, to foreign influences. To establish this point, he\ndisparages the representations of Philo and Josephus as  to suit\nthe tastes of their heathen readers, while in their place he brings\nforward as authorities a number of passages from talmudical and\nrabbinical writings, in which he discovers references to this sect. In\nthis view he is followed implicitly by some later writers, and has\nlargely influenced the opinions of others; while nearly all speak of his\ninvestigations as throwing great light on the subject.\n\n[Sidenote: but groundless and misleading.]\n\nIt is perhaps dangerous to dissent from a view which has found so much\nfavour; but nevertheless I am obliged to confess my belief that,\nwhatever value Frankel's investigations may have as contributions to our\nknowledge of Jewish religious thought and practice, they throw little or\nno light on the Essenes specially; and that the blind acceptance of his\nresults by later writers has greatly obscured the distinctive features\nof this sect. I cannot but think that any one, who will investigate\nFrankel's references and test his results step by step, will arrive at\nthe conclusion to which I myself have been led, that his talmudical\nresearches have left our knowledge of this sect where it was before, and\nthat we must still refer to Josephus and Philo for any precise\ninformation respecting them.\n\n[Sidenote: His double derivation of the name.]\n\nFrankel starts from the etymology of the name. He supposes that \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2,\n\u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, represent two different Hebrew words, the former \u05d7\u05e1\u05d9\u05d3 _ch\u0101s\u012bd_,\nthe latter \u05e6\u05e0\u05d5\u05e2 _tsan\u016ba\u0639_, both clothed in suitable Greek dresses[339].\nWherever therefore either of these words occurs, there is, or there may\nbe, a direct reference to the Essenes.\n\nFootnote 339:\n\n  _Zeitschrift_ p. 449 'F\u00fcr _Ess\u00e4er_ liegt, wie schon von anderen Seiten\n  bemerkt wurde, das Hebr. \u05d7\u05e1\u05d9\u05d3, f\u00fcr _Essener_, nach einer Bemerkung des\n  Herrn L. L\u00f6w im _Orient_, das Hebr. \u05e6\u05e0\u05d5\u05e2 nahe'; see also pp. 454, 455;\n  _Monatschrift_ p. 32.\n\n[Sidenote: Fatal objections to it.]\n\nIt is not too much to say that these etymologies are impossible; and\nthis for several reasons. (1) The two words \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2, \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, are\nplainly duplicate forms of the same Hebrew or Aramaic original, like\n\u03a3\u03b1\u03bc\u03c8\u03b1\u1fd6\u03bf\u03c2 and \u03a3\u03b1\u03bc\u03c8\u03b7\u03bd\u03cc\u03c2 (Epiphan. _H\u00e6r._ pp. 40, 47, 127; and even\n\u03a3\u03b1\u03bc\u03c8\u03af\u03c4\u03b7\u03c2 p. 46), \u039d\u03b1\u03b6\u03c9\u03c1\u03b1\u1fd6\u03bf\u03c2 and \u039d\u03b1\u03b6\u03b1\u03c1\u03b7\u03bd\u03cc\u03c2, \u0393\u03b9\u03c4\u03c4\u03b1\u1fd6\u03bf\u03c2 and \u0393\u03b9\u03c4\u03c4\u03b7\u03bd\u03cc\u03c2 (Steph.\nByz. s.v., Hippol. _H\u00e6r._ vi. 7), with which we may compare \u0392\u03bf\u03c3\u03c4\u03c1\u03b1\u1fd6\u03bf\u03c2\nand \u0392\u03bf\u03c3\u03c4\u03c1\u03b7\u03bd\u03cc\u03c2, \u039c\u03b5\u03bb\u03b9\u03c4\u03b1\u1fd6\u03bf\u03c2 and \u039c\u03b5\u03bb\u03b9\u03c4\u03b7\u03bd\u03cc\u03c2, and numberless other examples.\n(2) Again; when we consider either word singly, the derivation offered\nis attended with the most serious difficulties. There is no reason why\nin \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 the _d_ should have disappeared from _chasid_, while it is\nhardly possible to conceive that tsanua\u0639 should have taken such an\nincongruous form as \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2. (3) And lastly; the more important of the\ntwo words, _chasid_, had already a recognised Greek equivalent in\n\u1f08\u03c3\u03b9\u03b4\u03b1\u1fd6\u03bf\u03c2; and it seems highly improbable that a form so divergent as\n\u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 should have taken its place.\n\n[Sidenote: Dependence of the theory on the derivation.]\n\nIndeed Frankel's derivations are generally, if not universally,\nabandoned by later writers; and yet these same writers repeat his\nquotations and accept his results, as if the references were equally\nvalid, though the name of the sect has disappeared. They seem to be\nsatisfied with the stability of the edifice, even when the foundation is\nundermined. Thus for instance Gr\u00e4tz not only maintains after Frankel\nthat the Essenes 'were properly nothing more than stationary or, more\nstrictly speaking, consistently logical (consequente) _Chasidim_,' and\n'that therefore they were not so far removed from the Pharisees that\nthey can be regarded as a separate sect,' and 'accepts entirely these\nresults' which, as he says, 'rest on critical investigation' (III. p.\n463), but even boldly translates _chasiduth_ 'the Essene mode of life'\n(ib. 84), though he himself gives a wholly different derivation of the\nword 'Essene,' making it signify 'washers' or 'baptists' (see above, p.\n116). And even those who do not go to this length of inconsistency, yet\navail themselves freely of the passages where _chasid_ occurs, and\ninterpret it of the Essenes, while distinctly repudiating the\netymology[340].\n\nFootnote 340:\n\n  e.g. Keim (p. 286) and Derenbourg (p. 166, 461 sq.), who both derive\n  Essene from \u05d0\u05e1\u05d9\u05d0 'a physician.'\n\n[Sidenote: The term _chasid_ not applied specially to the Essenes.]\n\nBut, although \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 or \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2 is not a Greek form of _chasid_, it\nmight still happen that this word was applied to them as an epithet,\nthough not as a proper name. Only in this case the reference ought to be\nunmistakeable, before any conclusions are based upon it. But in fact,\nafter going through all the passages which Frankel gives, it is\nimpossible to feel satisfied that in a single instance there is a direct\nallusion to the Essenes. Sometimes the word seems to refer to the old\nsect of the _Chasidim_ or _Asid\u00e6ans_, as for instance when Jose ben\nJoezer, who lived during the Maccab\u00e6an war, is called a _chasid_[341].\nAt all events this R. Jose is known to have been a married man, for he\nis stated to have disinherited his children (_Baba Bathra_ 133 _b_); and\ntherefore he cannot have belonged to the stricter order of Essenes.\nSometimes it is employed quite generally to denote pious observers of\nthe ceremonial law, as for instance when it is said that with the death\nof certain famous teachers the Chasidim ceased[342]. In this latter\nsense the expression \u05d7\u05e1\u05d9\u05d3\u05d9\u05dd \u05d4\u05e8\u05d0\u05e9\u05d5\u05e0\u05d9\u05dd, 'the ancient or primitive\nChasidim' (_Monatschr._ pp. 31, 62), is perhaps used; for these\nprimitive Chasidim again are mentioned as having wives and\nchildren[343], and it appears also that they were scrupulously exact in\nbringing their sacrificial offerings[344]. Thus it is impossible to\nidentify them with the Essenes, as described by Josephus and Philo. Even\nin those passages of which most has been made, the reference is more\nthan doubtful. Thus great stress is laid on the saying of R. Joshua ben\nChananiah in Mishna _Sotah_ iii. 4, 'The foolish _chasid_ and the clever\nvillain (\u05d7\u05e1\u05d9\u05d3 \u05e9\u05d5\u05d8\u05d4 \u05d5\u05e8\u05e9\u05e2 \u05e2\u05e8\u05d5\u05dd), etc., are the ruin of the world.' But the\nconnexion points to a much more general meaning of _chasid_, and the\nrendering in Surenhuis, '<DW25> pius qui insipiens, improbus qui astutus,'\ngives the correct antithesis. So we might say that there is no one more\nmischievous than the wrong-headed conscientious man. It is true that the\nGemaras illustrate the expression by examples of those who allow an\nover-punctilious regard for external forms to stand in the way of deeds\nof mercy. And perhaps rightly. But there is no reference to any\ndistinctive Essene practices in the illustrations given. Again; the\nsaying in Mishna _Pirke Aboth_ v. 10, 'He who says Mine is thine and\nthine is thine is [a] _chasid_ (\u05e9\u05dc\u05d9 \u05e9\u05dc\u05da \u05d5\u05e9\u05dc\u05da \u05e9\u05dc\u05da \u05d4\u05e1\u05d9\u05d3),' is quoted by\nseveral writers as though it referred to the Essene community of\ngoods[345]. But in the first place the idea of community of goods would\nrequire 'Mine is thine and thine is mine': and in the second place, the\nwhole context, and especially the clause which immediately follows (and\nwhich these writers do not give), 'He who says Thine is mine and mine is\nmine is wicked (\u05e8\u05e9\u05e2),' show plainly that \u05d7\u05e1\u05d9\u05d3 must be taken in its\ngeneral sense 'pious,' and the whole expression implies not reciprocal\ninterchange but individual self-denial.\n\nFootnote 341:\n\n  Mishna _Chagigah_ ii. 7; _Zeitschr._ p. 454, _Monatschr._ pp. 33, 62.\n  See Frankel's own account of this R. Jose in an earlier volume,\n  _Monatschr._ I. p. 405 sq.\n\nFootnote 342:\n\n  _Zeitschr._ p. 457, _Monatschr._ p. 69 sq.; see below, p. 126.\n\nFootnote 343:\n\n  _Niddah_ 38 _a_; see L\u00f6wy s.v. Ess\u00e4er.\n\nFootnote 344:\n\n  Mishna _Kerithuth_ vi. 3, _Nedarim_ 10 a; see _Monatschr._ p. 65.\n\nFootnote 345:\n\n  Thus Gr\u00e4tz (III. p. 81) speaking of the community of goods among the\n  Essenes writes, 'From this view springs the proverb; Every Chassid\n  says; _Mine and thine belong to thee_ (not _me_)' thus giving a turn\n  to the expression which in its original connexion it does not at all\n  justify. Of the existence of such a proverb I have found no traces. It\n  certainly is not suggested in the passage of _Pirke Aboth_. Later in\n  the volume (p. 467) Gr\u00e4tz tacitly alters the words to make them\n  express reciprocation or community of goods, substituting 'Thine is\n  mine' for 'Thine is thine' in the second clause; 'The Chassid must\n  have no property of his own, but must treat it as belonging to the\n  Society (\u05e9\u05dc\u05d9 \u05e9\u05dc\u05da \u05e9\u05dc\u05da \u05e9\u05dc\u05d9 \u05d7\u05e1\u05d9\u05d3).' At least, as he gives no reference, I\n  suppose that he refers to the same passage. In this loose way he\n  treats the whole subject. Keim (p. 294) quotes the passage correctly,\n  but refers it nevertheless to Essene communism.\n\n[Sidenote: Possible connexion of\n           _chasid_ and _chasyo_\n           discussed.]\n\nIt might indeed be urged, though this is not Frankel's plea, that\nsupposing the true etymology of the word \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2, \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, to be the\nSyriac \u071a\u0723\u0710, \u071a\u0723\u071d\u0710, _ch's\u0113_, _chasyo_ (a possible derivation), _chasid_\nmight have been its Hebrew equivalent as being similar in sound and\nmeaning, and perhaps ultimately connected in derivation, the exactly\ncorresponding triliteral root \u05d7\u05e1\u05d0 (comp. \u05d7\u05d5\u05dd) not being in use in\nHebrew[346]. But before we accept this explanation we have a right to\ndemand some evidence which, if not demonstrative, is at least\ncircumstantial, that _chasid_ is used of the Essenes: and this we have\nseen is not forthcoming. Moreover, if the Essenes had thus inherited the\nname of the _Chasidim_, we should have expected that its old Greek\nequivalent \u1f08\u03c3\u03b9\u03b4\u03b1\u1fd6\u03bf\u03b9, which is still used later than the Maccab\u00e6an era,\nwould also have gone with it; rather than that a new Greek word \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2\n(or \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2) should have been invented to take its place. But indeed the\nSyriac Version of the Old Testament furnishes an argument against this\nconvertibility of the Hebrew _chasid_ and the Syriac _chasyo_, which\nmust be regarded as [Sidenote: Usage is unfavourable to this view.]\nalmost decisive. The numerous passages in the Psalms, where the\nexpressions 'My _chasidim_,' 'His _chasidim_,' occur (xxx. 5, xxxi. 24,\nxxxvii. 28, lii. 11, lxxix. 2, lxxxv. 9, xcvii. 10, cxvi. 15, cxxxii. 9,\ncxlix. 9: comp. xxxii. 6, cxlix. 1, 5) seem to have suggested the\nassumption of the name to the original Asid\u00e6ans. But in such passages\n\u05d7\u05e1\u05d9\u05d3 is commonly, if not universally, rendered in the Peshito not by\n\u071a\u0723\u0710, \u071a\u0723\u071d\u0710, but by a wholly different word \u0719\u0715\u071d\u0729 _zad\u012bk_. And again, in\nthe Books of Maccabees the Syriac rendering for the name \u1f08\u03c3\u03b9\u03b4\u03b1\u1fd6\u03bf\u03b9,\n_Chasidim_, is a word derived from another quite distinct root. These\nfacts show that the Hebrew _chasid_ and the Syriac _chasyo_ were not\npractically equivalents, so that the one would suggest the other; and\nthus all presumption in favour of a connexion between \u1f08\u03c3\u03b9\u03b4\u03b1\u1fd6\u03bf\u03c2 and\n\u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u03c2 is removed.\n\nFootnote 346:\n\n  This is Hitzig's view (_Geschichte des Volkes Israel_ p. 427). He\n  maintains that \"they were called '_Hasidim_' by the later Jews because\n  the Syrian _Essenes_ means exactly the same as '_Hasidim_.'\"\n\n[Sidenote: Frankel's second derivation]\n\nFrankel's other derivation \u05e6\u05e0\u05d5\u05e2, _tsan\u016ba\u0639_, suggested as an equivalent\nto \u1f18\u03c3\u03c3\u03b7\u03bd\u03cc\u03c2, has found no favour with later writers, and indeed is too\nfar removed from the Greek form to be tenable. [Sidenote: _tsanua\u0639_\nconsidered.]\n\nNor do the passages quoted by him[347] require or suggest any allusion\nto this sect. Thus in Mishna _Demai_, vi. 6, we are told that the school\nof Hillel permits a certain license in a particular matter, but it is\nadded, 'The \u05e6\u05e0\u05d5\u05e2\u05d9 of the school of Hillel followed the precept of the\nschool of Shammai.' Here, as Frankel himself confesses, the Jerusalem\nTalmud knows nothing about Essenes, but explains the word by \u05d1\u05e9\u05d3\u05d9, i.e.\n'upright, worthy[348]'; while elsewhere, as he allows[349], it must have\nthis general sense. Indeed the mention of the 'school of Hillel' here\nseems to exclude the Essenes. In its comprehensive meaning it will most\nnaturally be taken also in the other passage quoted by Frankel,\n_Kiddushin_ 71 _a_, where it is stated that the pronunciation of the\nsacred name, which formerly was known to all, is now only to be divulged\nto the \u05e6\u05e0\u05d5\u05e2\u05d9\u05dd, i.e. the discreet, among the priests; and in fact it\noccurs in reference to the communication of the same mystery in the\nimmediate context also, where it could not possibly be treated as a\nproper name; \u05e9\u05e6\u05e0\u05d5\u05e2 \u05d5\u05e2\u05e0\u05d9\u05d5 \u05d5\u05e2\u05d5\u05de\u05d3 \u05d1\u05d7\u05e6\u05d9 \u05d9\u05de\u05d9\u05d5, 'who is _discreet_ and meek\nand has reached middle age,' etc.\n\nFootnote 347:\n\n  _Zeitschr_. pp. 455, 457; _Monatschr._ p. 32.\n\nFootnote 348:\n\n  _Monatschr._ p. 32.\n\nFootnote 349:\n\n  _Zeitschr._ p. 455.\n\n[Sidenote: Other supposed etymologies in the Talmud. (1) _Asya_ 'a\n           physician,']\n\nOf other etymologies, which have been suggested, and through which it\nmight be supposed the Essenes are mentioned by name in the Talmud, \u05d0\u05d9\u05e1\u05d0,\n_asya_, 'a physician,' is the one which has found most favour. For the\nreasons given above (p. 117) this derivation seems highly improbable,\nand the passages quoted are quite insufficient to overcome the\nobjections. Of these the strongest is in the Talm. Jerus. _Yoma_ iii. 7,\nwhere we are told that a certain physician\n\n[Sidenote: not supported by the passages quoted in its behalf.]\n\n(\u05d0\u05e1\u05d9) offered to communicate the sacred name to R. Pinchas the son of\nChama, and the latter refused on the ground that he ate of the\ntithes\u2014this being regarded as a disqualification, apparently because it\nwas inconsistent with the highest degree of ceremonial purity[350]. The\nsame story is told with some modifications in Midrash _Qoheleth_ iii.\n11[351]. Here Frankel, though himself (as we have seen) adopting a\ndifferent derivation of the word 'Essene,' yet supposes that this\nparticular physician belonged to the sect, on the sole ground that\nceremonial purity is represented as a qualification for the initiation\ninto the mystery of the Sacred Name. L\u00f6wy (l.c.) denies that the\nallusion to the tithes is rightly interpreted: but even supposing it to\nbe correct, the passage is quite an inadequate basis either for\nFrankel's conclusion that this particular physician was an Essene, or\nfor the derivation of the word Essene which others maintain. Again, in\nthe statement of Talm. Jerus. _Kethuboth_ ii. 3, that correct\nmanuscripts were called books of \u05d0\u05e1\u05d9[352], the word _Asi_ is generally\ntaken as a proper name. But even if this interpretation be false, there\nis absolutely nothing in the context which suggests any allusion to the\nEssenes[353]. In like manner the passage from _Sanhedrin_ 99 _b_, where\na physician is mentioned[354], supports no such inference. Indeed, as\nthis last passage relates to the family of the _Asi_, he obviously can\nhave had no connexion with the celibate Essenes.\n\nFootnote 350:\n\n  Frankel _Monatschr._ p. 71: comp. Derenbourg p. 170 sq.\n\nFootnote 351:\n\n  See L\u00f6wy _Krit.-Talm. Lex._ s.v. Ess\u00e4er.\n\nFootnote 352:\n\n  Urged in favour of this derivation by Herzfeld II. p. 398.\n\nFootnote 353:\n\n  The oath taken by the Essenes (Joseph. _B.J._ ii. 8. 7) \u03c3\u03c5\u03bd\u03c4\u03b7\u03c1\u03ae\u03c3\u03b5\u03b9\u03bd\n  ... \u03c4\u1f70 \u03c4\u1fc6\u03c2 \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03b2\u03b9\u03b2\u03bb\u03af\u03b1 can have nothing to do with accuracy\n  in transcribing copies, as Herzfeld (II. pp. 398, 407) seems to think.\n  The natural meaning of \u03c3\u03c5\u03bd\u03c4\u03b7\u03c1\u03b5\u1fd6\u03bd, 'to keep safe or close' and so 'not\n  to divulge' (e.g. Polyb. xxxi. 6. 5 \u03bf\u1f50\u03ba \u1f10\u03be\u03ad\u03c6\u03b1\u03b9\u03bd\u03b5 \u03c4\u1f74\u03bd \u1f11\u03b1\u03c5\u03c4\u1fc6\u03c2 \u03b3\u03bd\u03ce\u03bc\u03b7\u03bd\n  \u1f00\u03bb\u03bb\u1f70 \u03c3\u03c5\u03bd\u03b5\u03c4\u03ae\u03c1\u03b5\u03b9 \u03c0\u03b1\u03c1' \u1f11\u03b1\u03c5\u03c4\u1fc7), is also the meaning suggested here by the\n  context.\n\nFootnote 354:\n\n  The passage is adduced in support of this derivation by Derenbourg p.\n  175.\n\n[Sidenote: (2) _\u0639asah_ 'to do.']\n\nHitherto our search for the name in the Talmud has been unsuccessful.\nOne possibility however still remains. The talmudical writers speak of\ncertain \u05d0\u05e0\u05e9\u05d9 \u05de\u05e2\u05e9\u05d4 'men of deeds'; and if (as some suppose) the name\nEssene is derived from \u05e2\u05e9\u05d4, have we not here the mention which we are\nseeking? Frankel rejects the etymology, but presses the\nidentification[355]. The expression, he urges, is often used in\nconnexion with _chasidim_. It signifies 'miracle workers,' and therefore\naptly describes the supernatural powers supposed to be exercised by the\nEssenes[356]. Thus we are informed in Mishna _Sotah_ ix. 15, that 'When\nR. Chaninah ben Dosa died, the men of deeds ceased; when R. Jose Ketinta\ndied, the chasidim ceased.' In the Jerusalem Talmud however this mishna\nis read, 'With the death of R. Chaninah ben Dosa and R. Jose Ketinta the\nchasidim ceased'; while the Gemara there explains R. Chaninah to have\nbeen one of the \u05de\u05e2\u05e9\u05d4 \u05d0\u05e0\u05e9\u05d9. Thus, Frankel concludes, 'the identity of\nthese with \u05d4\u05e1\u05d9\u05d3\u05d9\u05dd becomes still more plain.' Now it seems clear that\nthis expression \u05d0\u05e0\u05e9\u05d9 \u05de\u05e2\u05e9\u05d4 in some places cannot refer to miraculous\npowers, but must mean 'men of practical goodness,' as for instance in\n_Succah_ 51_a_, 53_a_; and being a general term expressive of moral\nexcellence, it is naturally connected with _chasidim_, which is likewise\na general term expressive of piety and goodness. Nor is there any reason\nwhy it should not always be taken in this sense. It is true that stories\nare told elsewhere of this R. Chaninah, which ascribe miraculous powers\nto him[357], and hence there is a temptation to translate it\n'wonder-worker,' as applied to him. But the reason is quite\ninsufficient. Moreover it must be observed that R. Chaninah's wife is a\nprominent person in the legends of his miracles reported in _Taanith_ 24\n_b_; and thus we need hardly stop to discuss the possible meanings of\n\u05d0\u05e0\u05e9\u05d9 \u05de\u05e2\u05e9\u05d4, since his claims to being considered an Essene are barred at\nthe outset by this fact[358].\n\nFootnote 355:\n\n  See _Zeitschr._ p 438, _Monatschr._ pp. 68\u201370.\n\nFootnote 356:\n\n  See above, p. 118.\n\nFootnote 357:\n\n  _Taanith_ 24_b_, _Yoma_ 53_b_; see Surenhuis _Mishna_ III. p. 313.\n\nFootnote 358:\n\n  In this and similar cases it is unnecessary to consider whether the\n  persons mentioned might have belonged to those looser disciples of\n  Essenism, who married (see above, p. 85): because the identification\n  is meaningless unless they belonged to the strict order itself.\n\nIt has been asserted indeed by a recent author, that one very ancient\nJewish writer distinctly adopts this derivation, and as distinctly\nstates that the Essenes were a class of Pharisees[359]. If this were the\ncase, Frankel's theory, though not his etymology, would receive a\nstriking confirmation: and it is therefore important to enquire on what\nfoundation the assertion rests.\n\nFootnote 359:\n\n  Ginsburg in Kitto's _Cyclop\u00e6dia_ s.v., I. p. 829: comp. _Essenes_ pp.\n  22, 28.\n\n[Sidenote: The authority for this derivation traced to an error.]\n\nDr Ginsburg's authority for this statement is a passage from the _Aboth_\nof Rabbi Nathan, c. 37, which, as he gives it, appears conclusive;\n'There are eight kinds of Pharisees ... and those Pharisees who live in\ncelibacy are Essenes.' But what are the facts of the case? _First_; This\nbook was certainly not written by its reputed author, the R. Nathan who\nwas vice-president under the younger Gamaliel about A. D. 140. It may\npossibly have been founded on an earlier treatise by that famous\nteacher, though even this is very doubtful: but in its present form it\nis a comparatively modern work. On this point all or almost all recent\nwriters on Hebrew literature are agreed[360]. _Secondly_; Dr Ginsburg\nhas taken the reading \u05de\u05d7\u05d5\u05e4\u05ea\u05d5 \u05e2\u05e9\u05d0\u05e0\u05d9, without even mentioning any\nalternative. Whether the words so read are capable of the meaning which\nhe has assigned to them, may be highly questionable; but at all events\nthis cannot have been the original reading, as the parallel passages,\nBabl. Sotah fol. 22_b_, Jerus. _Sotah_ v. 5, Jerus. _Berakhoth_ ix. 5,\n(quoted by Buxtorf and Levy, s.v. \u05e4\u05e8\u05d9\u05e9), distinctly prove. In Babl.\n_Sotah_ l.c., the corresponding expression is \u05de\u05d4 \u05d4\u05d5\u05d1\u05ea\u05d9 \u05d5\u05d0\u05e2\u05e9\u05e0\u05d4 'What is\nmy duty, and I will do it,' and the passage in Jerus. _Berakhoth_ l.c.\nis to the same effect. These parallels show that the reading \u05de\u05d4 \u05d4\u05d5\u05d1\u05ea\u05d9\n\u05d5\u05d0\u05e2\u05e9\u05e0\u05d4 must be taken also in _Aboth_ c. 37, so that the passage will be\nrendered, 'The Pharisee _who says_, What is my duty, and I will do it.'\nThus the Essenes and celibacy disappear together. _Lastly_; Inasmuch as\nDr Ginsburg himself takes a wholly different view of the name Essene,\nconnecting it either with \u05d7\u05e6\u05df 'an apron,' or with \u05d4\u05e1\u05d9\u05dd 'pious[361],' it\nis difficult to see how he could translate \u05e2\u05e9\u05d0\u05e0\u05d9 'Essene' (from \u05e2\u05e9\u05d0 'to\ndo') in this passage, except on the supposition that R. Nathan was\nentirely ignorant of the orthography and derivation of the word Essene.\nYet, if such ignorance were conceivable in so ancient a writer, his\nauthority on this question would be absolutely worthless. But indeed Dr\nGinsburg would appear to have adopted this reference to R. Nathan, with\nthe reading of the passage and the interpretation of the name, from some\nother writer[362]. At all events it is quite inconsistent with his own\nopinion as expressed previously.\n\nFootnote 360:\n\n  e.g. Geiger _Zeitschrift f. J\u00fcdische Theologie_ VI. p. 20 sq.; Zunz\n  _Gottesdienstliche Vortr\u00e4ge_ p. 108 sq.: comp. Steinschneider _Catal.\n  Heb. Bibl. Bodl._ col. 2032 sq. These two last references are given by\n  Dr Ginsburg himself.\n\nFootnote 361:\n\n  _Essenes_ p. 30; comp. _Kitto's Cyclop\u00e6dia_, s.v. Essenes.\n\nFootnote 362:\n\n  It is given by Landsberg in the _Allgemeine Zeitung des Judenthums_\n  1862, no. 33, p. 459, a reference pointed out to me by a friend.\n\n                  *       *       *       *       *\n\n[Sidenote: Are the Essenes alluded to, though not named, in the Talmud?]\n\nBut, though we have not succeeded in finding any direct mention of this\nsect by name in the Talmud, and all the identifications of the word\nEssene with diverse expressions occurring there have failed us on\nexamination, it might still happen that allusions to them were so\nfrequent as to leave no doubt about the persons meant. Their\norganisation or their practices or their tenets might be precisely\ndescribed, though their name was suppressed. Such allusions Frankel\nfinds scattered up and down the Talmud in great profusion.\n\n[Sidenote: (1) The _chaber_ or Associate.]\n\n(1) He sees a reference to the Essenes in the \u05d7\u05d1\u05d5\u05e8\u05d0 _ch\u0103b\u016bra_ or\n'Society,' which is mentioned several times in talmudical writers[363].\nThe _ch\u0101ber_ (\u05d4\u05d1\u05e8) or 'Associate' is, he supposes, a member of this\nbrotherhood. He is obliged to confess that the word cannot always have\nthis sense, but still he considers this to be a common designation of\nthe Essenes. The chaber was bound to observe certain rules of ceremonial\npurity, and a period of probation was imposed upon him before he was\nadmitted. With this fact Frankel connects the passage in Mishna\n_Chagigah_ ii. 5, 6, where several degrees of ceremonial purity are\nspecified. Having done this, he considers that he has the explanation of\nthe statement in Josephus (_B.J._ ii. 8. 7, 10), that the Essenes were\ndivided into four different grades or orders according to the time of\ntheir continuance in the ascetic practices demanded by the sect.\n\nFootnote 363:\n\n  _Zeitschr_. p. 450 sq., _Monatschr._ pp. 31, 70.\n\n[Sidenote: A passage in _Chagigah_ considered.]\n\nBut in the first place there is no reference direct or indirect to the\nchaber, or indeed to any organisation of any kind, in the passage of\n_Chagigah_. It simply contemplates different degrees of purification as\nqualifying for the performance of certain Levitical rites in an\nascending scale. There is no indication that these lustrations are more\nthan temporary and immediate in their application; and not the faintest\nhint is given of distinct orders of men, each separated from the other\nby formal barriers and each demanding a period of probation before\nadmission from the order below, as was the case with the grades of the\nEssene brotherhood described by Josephus. Moreover the orders in\nJosephus are four in number[364], while the degrees of ceremonial purity\nin _Chagigah_ are five. Frankel indeed is inclined to maintain that only\nfour degrees are intended in _Chagigah_, though this interpretation is\nopposed to the plain sense of the passage. But, even if he should be\nobliged to grant that the number of degrees is five[365], he will not\nsurrender the allusion to the Essenes, but meets the difficulty by\nsupposing (it is a pure hypothesis) that there was a fifth and highest\ndegree of purity among the Essenes, to which very few attained, and\nwhich, as I understand him, is not mentioned by Josephus on this\naccount. But enough has already been said to show, that this passage in\n_Chagigah_ can have no connexion with the Essenes and gives no\ncountenance to Frankel's views.\n\nFootnote 364:\n\n  As the notices in Josephus (_B.J._ ii. 8) relating to this point have\n  been frequently misunderstood, it may be well once for all to explain\n  his meaning. The grades of the Essene order are mentioned in two\n  separate notices, apparently, though not really, discordant. (1) In \u00a7\n  10 he says that they are 'divided into four sections according to the\n  duration of their discipline' (\u03b4\u03b9\u1fc4\u03c1\u03b7\u03bd\u03c4\u03b1\u03b9 \u03ba\u03b1\u03c4\u1f70 \u03c7\u03c1\u03cc\u03bd\u03bf\u03bd \u03c4\u1fc6\u03c2 \u1f00\u03c3\u03ba\u03ae\u03c3\u03b5\u03c9\u03c2 \u03b5\u1f30\u03c2\n  \u03bc\u03bf\u03af\u03c1\u03b1\u03c2 \u03c4\u03ad\u03c3\u03c3\u03b1\u03c1\u03b1\u03c2), adding that the older members are considered to be\n  defiled by contact with the younger, i.e. each superior grade by\n  contact with the inferior. So far his meaning is clear. (2) In \u00a7 8 he\n  states that one who is anxious to become a member of the sect\n  undergoes a year's probation, submitting to discipline but 'remaining\n  outside.' Then, 'after he has given evidence of his perseverance (\u03bc\u03b5\u03c4\u1f70\n  \u03c4\u1f74\u03bd \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03c1\u03c4\u03b5\u03c1\u03af\u03b1\u03c2 \u1f10\u03c0\u03af\u03b4\u03b5\u03b9\u03be\u03b9\u03bd), his character is tested for two years\n  more; and, if found worthy, he is accordingly admitted into the\n  society.' A comparison with the other passage shows that these two\n  years comprise the period spent in the second and third grades, each\n  extending over a year. After passing through these three stages in\n  three successive years, he enters upon the fourth and highest grade,\n  thus becoming a perfect member.\n\n  It is stated by Dr Ginsburg (_Essenes_ p. 12 sq., comp. Kitto's\n  _Cyclop\u00e6dia_ s.v. p. 828) that the Essenes passed through eight stages\n  'from the beginning of the noviciate to the achievement of the highest\n  spiritual state,' this last stage qualifying them, like Elias, to be\n  forerunners of the Messiah. But it is a pure hypothesis that the\n  Talmudical notices thus combined have anything to do with the Essenes;\n  and, as I shall have occasion to point out afterwards, there is no\n  ground for ascribing to this sect any Messianic expectations whatever.\n\nFootnote 365:\n\n  _Zeitschr._ p. 452, note.\n\n[Sidenote: Difference between]\n\nAs this artificial combination has failed, we are compelled to fall back\non the notices relating to the chaber, and to ask whether [Sidenote: the\nchaber and the Essene.] these suggest any connexion with the account of\nthe Essenes in Josephus. And the facts oblige us to answer this question\nin the negative. Not only do they not suggest such a connexion, but they\nare wholly irreconcilable with the account in the Jewish historian. This\nassociation or confraternity (if indeed the term is applicable to an\norganisation so loose and so comprehensive) was maintained for the sake\nof securing a more accurate study and a better observance of the\nceremonial law. Two grades of purity are mentioned in connexion with it,\ndesignated by different names and presenting some difficulties[366],\ninto which it is not necessary to enter here. A chaber, it would appear,\nwas one who had entered upon the second or higher stage. For this a\nperiod of a year's probation was necessary. The chaber enrolled himself\nin the presence of three others who were already members of the\nassociation. This apparently was all the formality necessary: and in the\ncase of a teacher even this was dispensed with, for being presumably\nacquainted with the law of things clean and unclean he was regarded as\n_ex officio_ a chaber. The chaber was bound to keep himself from\nceremonial defilements, and was thus distinguished from the _[\u0639am\nhaarets_ or common people[367]; but he was under no external\nsurveillance and decided for himself as to his own purity. Moreover he\nwas, or might be a married man: for the doctors disputed whether the\nwives and children of an associate were not themselves to be regarded as\nassociates[368]. In one passage, _Sanhedrin_ 41_a_, it is even assumed,\nas a matter of course, that a woman may be an associate (\u05d7\u05d1\u05e8\u05d4). In\nanother (_Niddah_ 33_b_)[369] there is mention of a Sadducee and even of\na Samaritan as a chaber. An organisation so flexible as this has\nobviously only the most superficial resemblances with the rigid rules of\nthe Essene order; and in many points it presents a direct contrast to\nthe characteristic tenets of that sect.\n\nFootnote 366:\n\n  The entrance into lower grade was described as 'taking \u05d1\u05e0\u05e4\u05d9\u05dd' or\n  'wings.' The meaning of this expression has been the subject of much\n  discussion; see e.g. Herzfeld II. p. 390 sq., Frankel _Monatschr._ p.\n  33 sq.\n\nFootnote 367:\n\n  The contempt with which a chaber would look down upon the vulgar herd,\n  the _\u0639am haarets_, finds expression in the language of the Pharisees,\n  Joh. vii. 49 \u1f41 \u1f44\u03c7\u03bb\u03bf\u03c2 \u03bf\u1f57\u03c4\u03bf\u03c2 \u1f41 \u03bc\u1f74 \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03c9\u03bd \u03c4\u1f78\u03bd \u03bd\u03cc\u03bc\u03bf\u03bd \u1f10\u03c0\u03ac\u03c1\u03b1\u03c4\u03bf\u03af \u03b5\u1f30\u03c3\u03b9\u03bd.\n  Again in Acts iv. 13, where the Apostles are described as \u1f30\u03b4\u03b9\u1ff6\u03c4\u03b1\u03b9, the\n  expression is equivalent to _\u0639am haarets._ See the passages quoted in\n  Buxtorf, _Lex._ p. 1626.\n\nFootnote 368:\n\n  All these particulars and others may be gathered from _Bekhoroth_ 30\n  _b_, Mishna _Demai_ ii. 2, 3, Jerus. _Demai_ ii. 3, v. 1, Tosifta\n  _Demai_ 2, _Aboth R. Nathan_ c. 41.\n\nFootnote 369:\n\n  See Herzfeld II. p. 386.\n\n[Sidenote: (2) The _Bene hakkeneseth_.]\n\n(2) Having discussed Frankel's hypothesis respecting the chaber, I need\nhardly follow his speculations on the _B\u0115n\u0113-hakk\u0115neseth_, \u05d1\u05e0\u05d9 \u05d4\u05db\u05e0\u05e1\u05d7,\n'sons of the congregation' (_Zabim_ iii. 2), in which expression\nprobably few would discover the reference, which he finds, to the lowest\nof the Essene orders[370].\n\nFootnote 370:\n\n  _Monatschr._ p. 35.\n\n[Sidenote: (3) The 'holy congregation at Jerusalem']\n\n(3) But mention is also made of a 'holy congregation' or 'assembly' (\u05e2\u05d3\u05d4\n\u05e7\u05d3\u05d9\u05e9\u05d4 \u05e7\u05d4\u05dc\u05d0 \u05e7\u05d3\u05d9\u05e9\u05d0) 'in Jerusalem'; and, following Rapoport, Frankel sees\nin this expression also an allusion to the Essenes[371]. The grounds for\nthis identification are, that in one passage (_Berakhoth_ 9_b_) they are\nmentioned in connexion with prayer at day break, and in another (Midrash\n_Qoheleth_ ix. 9) two persons are stated to belong to this 'holy\ncongregation,' because they divided their day into three parts, devoting\none-third to learning, another to prayer, and another to work. The first\nnotice would suit the Essenes very well, though the practice mentioned\nwas not so distinctively Essene as to afford any safe ground for this\nhypothesis. Of the second it should be observed, that no such division\nof the day is recorded of the Essenes, and indeed both Josephus (_B.J._\nii. 8. 5) and Philo (_Fragm._ p. 633) describe them as working from\nmorning till night with the single interruption of their mid-day\nmeal[372]. But in fact the identification is beset with other and more\nserious difficulties. For this 'holy congregation' at Jerusalem is\nmentioned long [Sidenote: not an Essene community.] after the second\ndestruction of the city under Hadrian[373], when on Frankel's own\nshowing[374] the Essene society had in all probability ceased to exist.\nAnd again certain members of it, e.g. Jose ben Meshullam (Mishna\n_Bekhoroth_ iii. 3, vi. 1), are represented as uttering precepts\nrespecting animals fit for sacrifice, though we have it on the authority\nof Josephus and Philo that the Essenes avoided the temple sacrifices\naltogether. The probability therefore seems to be that this 'holy\ncongregation' was an assemblage of devout Jews who were drawn to the\nneighbourhood of the sanctuary after the destruction of the nation, and\nwhose practices were regarded with peculiar reverence by the later\nJews[375].\n\nFootnote 371:\n\n  _Zeitschr._ pp. 458, 461, _Monatschr._ pp. 32, 36.\n\nFootnote 372:\n\n  It is added however in Midrash _Qoheleth_ ix. 9 'Some say that they\n  (the holy congregation) devoted the whole of the winter to studying\n  the Scriptures and the summer to work.'\n\nFootnote 373:\n\n  _Monatschr._ p. 32.\n\nFootnote 374:\n\n  _Ib._ p. 70.\n\nFootnote 375:\n\n  See Derenbourg p. 175.\n\n[Sidenote: (4) The _Vethikin_.]\n\n(4) Neither can we with Frankel[376] discern any reference to the\nEssenes in those \u05d5\u05ea\u05d9\u05e7\u05d9\u05d5 _Vethikin_, 'pious' or 'learned' men (whatever\nmay be the exact sense of the word), who are mentioned in _Berakhoth_\n9_b_ as praying before sunrise; because the word itself seems quite\ngeneral, and the practice, though enforced among the Essenes, as we know\nfrom Josephus (_B.J._ ii. 8. 5), would be common to all devout and\nearnest Jews. If we are not justified in saying that these \u05d5\u05ea\u05d9\u05e7\u05d9\u05d5 were\nnot Essenes, we have no sufficient grounds for maintaining that they\nwere.\n\nFootnote 376:\n\n  _Monatschr._ p. 32.\n\n[Sidenote: (5) The 'primitive elders.']\n\n(5) Nor again can we find any such reference in the \u05d6\u05e7\u05e0\u05d9\u05dd or 'primitive\nelders[377].' It may readily be granted that this term is used\nsynonymously, or nearly so, with \u05d4\u05e8\u05d0\u05e9\u05d5\u05e0\u05d9\u05dd \u05d4\u05e1\u05d9\u05d3\u05d9\u05dd 'the primitive\nchasidim'; but, as we failed to see anything more than a general\nexpression in the one, so we are naturally led to take the other in the\nsame sense. The passages where the expression occurs (e.g. _Shabbath_\n64_b_) simply refer to the stricter observances of early times, and do\nnot indicate any reference to a particular society or body of men.\n\nFootnote 377:\n\n  _Monatschr._ pp. 32, 68.\n\n[Sidenote: (6) The 'morning bathers.']\n\n(6) Again Frankel finds another reference to this sect in the \u05d8\u05d1\u05dc\u05d9 \u05e9\u05d7\u05e8\u05d9\u05ea\n_T\u014dbl\u0113-shach\u0103r\u012bth_, or 'morning-bathers,' mentioned in Tosifta _Yadayim_\nc. 2[378]. The identity of these with the \u1f21\u03bc\u03b5\u03c1\u03bf\u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03c4\u03b1\u1f76 of Greek\nwriters seems highly probable. The latter however, though they may have\nhad some affinities with Essene practices and tenets, are nevertheless\ndistinguished from this sect wherever they are mentioned[379]. But the\npoint to be observed is that, even though we should identify these\nToble-shacharith with the Essenes, the passage in Tosifta _Yadayim_, so\nfar from favouring, runs directly counter to Frankel's view which\nregards the Essenes as only a branch of Pharisees: for the two are here\nrepresented as in direct antagonism. The Toble-shacharith say, 'We\ngrieve over you, Pharisees, because you pronounce the (sacred) Name in\nthe morning without having bathed.' The Pharisees retort, 'We grieve\nover you, Toble-shacharith, because you pronounce the Name from this\nbody in which is impurity.'\n\nFootnote 378:\n\n  _Ib._ p. 67.\n\nFootnote 379:\n\n  See below, p. 166.\n\n[Sidenote: (7) The _Banaim_.]\n\n(7) In connexion with the Toble-shacharith we may consider another name,\n_Ban\u0101\u012bm_ (\u05d1\u05e0\u05d0\u05d9\u05dd), in which also Frankel discovers an allusion to the\nEssenes[380]. In Mishna _Mikvaoth_ ix. 6 the word is opposed to \u05d1\u05d5\u05e8\n_b\u014dr_, 'an ignorant or stupid person'; and this points to its proper\nmeaning 'the builders,' i.e. the edifiers or teachers, according to the\ncommon metaphor in Biblical language. The word is discussed in\n_Shabbath_ 114 and explained to mean 'learned.' But, because in\n_Mikvaoth_ it is mentioned in connexion with ceremonial purity, and\nbecause in Josephus the Essenes are stated to have carried an 'axe and\nshovel' (_B.J._ ii. 8. 7, 9), and because moreover the Jewish historian\nin another place (_Vit._ 2) mentions having spent some time with one\nBanus a dweller in the wilderness, who lived on vegetables and fruits\nand bathed often day and night for the sake of purity, and who is\ngenerally considered to have been an Essene; therefore Frankel holds\nthese Banaim to have been Essenes. This is a specimen of the misplaced\ningenuity which distinguishes Frankel's learned speculations on the\nEssenes. Josephus does [Sidenote: Josephus misinterpreted.] not mention\nan 'axe _and_ shovel,' but an axe only (\u00a7 7 \u1f00\u03be\u03b9\u03bd\u03ac\u03c1\u03b9\u03bf\u03bd), which he\nafterwards defines more accurately as a spade (\u00a7 9 \u03c4\u1fc7 \u03c3\u03ba\u03b1\u03bb\u03af\u03b4\u03b9, \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03bd\n\u03b3\u03ac\u03c1 \u1f10\u03c3\u03c4\u03b9 \u03c4\u1f78 \u03b4\u03b9\u03b4\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u1f51\u03c0' \u03b1\u1f50\u03c4\u1ff6\u03bd \u1f00\u03be\u03b9\u03bd\u03af\u03b4\u03b9\u03bf\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bd\u03b5\u03bf\u03c3\u03c5\u03c3\u03c4\u03ac\u03c4\u03bf\u03b9\u03c2) and which,\nas he distinctly states, was given them for the purpose of burying\nimpurities out of sight (comp. Deut. xxiii. 12\u201314). Thus it has no\nconnexion whatever with any 'building' implement. And again, it is true\nthat Banus has frequently been regarded as an Essene, but there is\nabsolutely no ground for this supposition. On the contrary the narrative\nof Josephus in his _Life_ seems to [Sidenote: Another derivation of\nBanaim.] exclude it, as I shall have occasion to show hereafter[381]. I\nshould add that Sachs interprets Banaim 'the bathers,' regarding the\nexplanation in _Shabbath_ l.c. as a 'later accommodation[382].' This\nseems to me very improbable; but, if it were conceded, the Banaim would\nthen apparently be connected not with the Essenes, but with the\nHemerobaptists.\n\nFootnote 380:\n\n  _Zeitschr._ p. 455.\n\nFootnote 381:\n\n  See below, p. 161.\n\nFootnote 382:\n\n  _Beitr\u00e4ge_ II. p. 199. In this derivation he is followed by Graetz\n  (III. p. 82, 468) and Derenbourg (p. 166).\n\n                  *       *       *       *       *\n\n[Sidenote: Results of this investigation.]\n\nFrom the preceding investigation it will have appeared how little\nFrankel has succeeded in establishing his thesis that 'the talmudical\nsources are acquainted with the Essenes and make mention of them\nconstantly[383].' We have seen not only that no instance of the name\nEssene has been produced, but that all those passages which are supposed\nto refer to them under other designations, or to describe their\npractices or tenets, fail us on closer examination. In no case can we\nfeel sure that there is any direct reference to this sect, while in most\ncases such reference seems to be excluded by the language or the\nattendant circumstances[384]. Thus we are [Sidenote: Philo and Josephus\nour main authorities.] obliged to fall back upon the representations of\nPhilo and Josephus. Their accounts are penned by eye-witnesses. They are\ndirect and explicit, if not so precise or so full as we could have\nwished. The writers obviously consider that they are describing a\ndistinct and exceptional phenomenon. And it would be a reversal of all\nestablished rules of historical criticism to desert the solid\nstanding-ground of contemporary history for the artificial combinations\nand shadowy hypotheses, which Frankel would substitute in its place.\n\nFootnote 383:\n\n  _Monatschr._ p. 31.\n\nFootnote 384:\n\n  'The attempt to point out the Essenes in our patristic (i.e.\n  rabbinical) literature,' says Herzfeld truly (II. p. 397), 'has led to\n  a splendid hypothesis-hunt (_einer stattlichen Hypothesenjagd_).'\n\n[Sidenote: Frankel's depreciation of them is unreasonable, and explains\n           nothing.]\n\nBut here we are confronted with Frankel's depreciation of these ancient\nwriters, which has been echoed by several later critics. They were\ninterested, it is argued, in making their accounts attractive to their\nheathen contemporaries, and they  them highly for this\npurpose[385]. We may readily allow that they would not be uninfluenced\nby such a motive, but the concession does not touch the main points at\nissue. This aim might have led Josephus, for example, to throw into bold\nrelief the coincidences between the Essenes and Pythagoreans; it might\neven have induced him to give a semi-pagan tinge to the Essene doctrine\nof the future state of the blessed (_B.J._ ii. 8. 11). But it entirely\nfails to explain those peculiarities of the sect, which marked them off\nby a sharp line from orthodox Judaism, and which fully justify the term\n'separatists' as applied to them by a recent writer. In three main\nfeatures especially the portrait of the Essenes retains its distinctive\ncharacter unaffected by this consideration.\n\nFootnote 385:\n\n  _Monatschr._ p. 31.\n\n[Sidenote: (i) The avoidance of sacrifices is not accounted for.]\n\n(i) How, for instance, could this principle of accommodation have led\nboth Philo and Josephus to lay so much stress on their divergence from\nJudaic orthodoxy in the matter of sacrifices? Yet this is perhaps the\nmost crucial note of heresy which is recorded of the Essenes. What was\nthe law to the orthodox Pharisee without the sacrifices, the\ntemple-worship, the hierarchy? Yet the Essene declined to take any part\nin the sacrifices; he had priests of his own independently of the\nLevitical priesthood. On Frankel's hypothesis that Essenism is merely an\nexaggeration of pure Pharisaism, no explanation of this abnormal\nphenomenon can be given. Frankel does indeed attempt to meet the case by\nsome speculations respecting the red-heifer[386], which are so obviously\ninadequate that they have not been repeated by later writers and may\nsafely be passed over in silence here. On this point indeed the language\nof Josephus is not [Sidenote: The notices of Josephus and Philo\nconsidered.] quite explicit. He says (Ant. xviii. 1. 5) that, though\nthey send offerings (\u1f00\u03bd\u03b1\u03b8\u1fc6\u03bc\u03b1\u03c4\u03b1) to the temple, they perform no\nsacrifices, and he assigns as the reason their greater strictness as\nregards ceremonial purity (\u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03cc\u03c4\u03b7\u03c4\u03b9 \u1f01\u03b3\u03bd\u03b5\u03b9\u1ff6\u03bd \u1f03\u03c2 \u03bd\u03bf\u03bc\u03af\u03b6\u03bf\u03b9\u03b5\u03bd), adding\nthat 'for this reason being excluded from the common sanctuary\n(\u03c4\u03b5\u03bc\u03b5\u03bd\u03af\u03c3\u03bc\u03b1\u03c4\u03bf\u03c2) they perform their sacrifices by themselves (\u1f10\u03c6' \u03b1\u1f51\u03c4\u1ff6\u03bd\n\u03c4\u1f70\u03c2 \u03b8\u03c5\u03c3\u03af\u03b1\u03c2 \u1f10\u03c0\u03b9\u03c4\u03b5\u03bb\u03bf\u1fe6\u03c3\u03b9).' Frankel therefore supposes that their only\nreason for abstaining from the temple sacrifices was that according to\ntheir severe notions the temple itself was profaned and therefore unfit\nfor sacrificial worship. But if so, why should it not vitiate the\nofferings, as well as the sacrifices, and make them also unlawful? And\nindeed, where Josephus is vague, Philo is explicit. Philo (II. p. 457)\ndistinctly states that the Essenes being more scrupulous than any in the\nworship of God (\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03c5\u03c4\u03b1\u1f76 \u03b8\u03b5\u03bf\u1fe6) do not sacrifice\nanimals (\u03bf\u1f50 \u03b6\u1ff6\u03b1 \u03ba\u03b1\u03c4\u03b1\u03b8\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2), but hold it right to dedicate their own\nhearts as a worthy offering (\u1f00\u03bb\u03bb' \u1f31\u03b5\u03c1\u03bf\u03c0\u03c1\u03b5\u03c0\u03b5\u1fd6\u03c2 \u03c4\u1f70\u03c2 \u1f11\u03b1\u03c5\u03c4\u1ff6\u03bd \u03b4\u03b9\u03b1\u03bd\u03bf\u03af\u03b1\u03c2\n\u03ba\u03b1\u03c4\u03b1\u03c3\u03ba\u03b5\u03c5\u03ac\u03b6\u03b5\u03b9\u03bd \u1f00\u03be\u03b9\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2). Thus the greater strictness, which Josephus\nascribes to them, consists in the abstention from shedding blood, as a\npollution in itself. And, when he speaks of their substituting private\nsacrifices, his own qualifications show that he does not mean the word\nto be taken literally. Their simple meals are their sacrifices; their\nrefectory is their sanctuary; their president is their priest[387]. It\nshould be added also that, though we once hear of an Essene apparently\nwithin the temple precincts (_B.J._ i. 3. 5, _Ant._ xiii. II. 2)[388],\nno mention is ever made of one offering sacrifices. Thus it is clear\nthat with the Essene it was the sacrifices which polluted the temple,\nand not the [Sidenote: Their statements confirmed by the doctrine of\nChristian Essenes.] temple which polluted the sacrifices. And this view\nis further recommended by the fact that it alone will explain the\nposition of their descendants, the Christianized Essenes, who condemned\nthe slaughter of victims on grounds very different from those alleged in\nthe Epistle to the Hebrews, not because they have been superseded by the\nAtonement, but because they are in their very nature repulsive to God;\nnot because they have ceased to be right, but because they never were\nright from the beginning.\n\nFootnote 386:\n\n  _Monatschr._ p. 64.\n\nFootnote 387:\n\n  _B.J._ ii. 8. 5 \u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1 \u03b5\u1f30\u03c2 \u1f05\u03b3\u03b9\u03cc\u03bd \u03c4\u03b9 \u03c4\u03ad\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c0\u03b1\u03c1\u03b1\u03b3\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c4\u1f78\n  \u03b4\u03b5\u03b9\u03c0\u03bd\u03b7\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd: see also the passages quoted above p. 89, note 255.\n\nFootnote 388:\n\n  See below, p. 142.\n\nIt may be said indeed, that such a view could not be maintained without\nimpugning the authority, or at least disputing the integrity, of the Old\nTestament writings. The sacrificial system is so bound up with the\nMosaic law, that it can only be rejected by the most arbitrary excision.\nThis violent process however, uncritical as it is, was very likely to\nhave been adopted by the Essenes[389]. As a matter of fact, it did\nrecommend itself to those Judaizing Christians who reproduced many of\nthe Essene tenets, and who both theologically and historically may be\nregarded as the lineal [Sidenote: The Clementine Homilies justify this\ndoctrine by arbitrary excision of the Scriptures.] descendants of this\nJudaic sect[390]. Thus in the _Clementine Homilies_, an Ebionite work\nwhich exhibits many Essene features, the chief spokesman St Peter is\nrepresented as laying great stress on the duty of distinguishing the\ntrue and the false elements in the current Scriptures (ii. 38, 51, iii.\n4, 5, 10, 42, 47, 49, 50, comp. xviii. 19). The saying traditionally\nascribed to our Lord, 'Show yourselves approved money-changers' (\u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b5\n\u03c4\u03c1\u03b1\u03c0\u03b5\u03b6\u1fd6\u03c4\u03b1\u03b9 \u03b4\u03cc\u03ba\u03b9\u03bc\u03bf\u03b9), is more than once quoted by the Apostle as\nenforcing this duty (ii. 51, iii. 50, xviii. 20). Among these false\nelements he places all those passages which represent God as enjoining\nsacrifices (iii. 45, xviii. 19). It is plain, so he argues, that God did\nnot desire sacrifices, for did He not kill those who lusted after the\ntaste of flesh in the wilderness? and, if the slaughter of animals was\nthus displeasing to Him, how could He possibly have commanded victims to\nbe offered to Himself (iii. 45)? It is equally clear from other\nconsiderations that this was no part of God's genuine law. For instance,\nChrist declared that He came to fulfil every tittle of the Law; yet\nChrist abolished sacrifices (iii. 51). And again, the saying 'I will\nhave mercy and not sacrifice' is a condemnation of this practice (iii.\n56). The true prophet 'hates sacrifices, bloodshed, libations'; he\n'extinguishes the fire of altars' (iii. 26). The frenzy of the lying\nsoothsayer is a mere intoxication produced by the reeking fumes of\nsacrifice (iii. 13). When in the immediate context of these\ndenunciations we find it reckoned among the highest achievements of man\n'to know the _names of angels_, to drive away demons, to endeavour to\nheal diseases by charms (\u03c6\u03b1\u03c1\u03bc\u03b1\u03ba\u03af\u03b1\u03b9\u03c2), [Sidenote: Essene features in this\nwork.] and to find incantations (\u1f10\u03c0\u03b1\u03bf\u03b9\u03b4\u03ac\u03c2) against venomous serpents\n(iii. 36)'; when again St Peter is made to condemn as false those\nscriptures which speak of God swearing, and to set against them Christ's\ncommand 'Let your yea be yea' (iii. 55); we feel how thoroughly this\nstrange production of Ebionite Christianity is saturated with Essene\nideas[391].\n\nFootnote 389:\n\n  Herzfeld (II. p. 403) is unable to reconcile any rejection of the Old\n  Testament Scriptures with the reverence paid to Moses by the Essenes\n  (_B.J._ ii. 8. 9, 10). The Christian Essenes however did combine both\n  these incongruous tenets by the expedient which is explained in the\n  text. Herzfeld himself suggests that allegorical interpretation may\n  have been employed to justify this abstention from the temple\n  sacrifices.\n\nFootnote 390:\n\n  See _Galatians_, p. 310 sq.\n\nFootnote 391:\n\n  Epiphanius (_H\u00e6r._ xviii. I, p. 38) again describes, as the account\n  was handed down to him (\u1f61\u03c2 \u1f41 \u03b5\u1f30\u03c2 \u1f21\u03bc\u1fb6\u03c2 \u1f10\u03bb\u03b8\u1f7c\u03bd \u03c0\u03b5\u03c1\u03b9\u03ad\u03c7\u03b5\u03b9 \u03bb\u03cc\u03b3\u03bf\u03c2), the\n  tenets of a Jewish sect which he calls the Nasareans, \u03b1\u1f50\u03c4\u1f74\u03bd \u03b4\u1f72 \u03bf\u1f50\n  \u03c0\u03b1\u03c1\u03b5\u03b4\u03ad\u03c7\u03b5\u03c4\u03bf \u03c4\u1f74\u03bd \u03c0\u03b5\u03bd\u03c4\u03ac\u03c4\u03b5\u03c5\u03c7\u03bf\u03bd, \u1f00\u03bb\u03bb\u1f70 \u1f61\u03bc\u03bf\u03bb\u03cc\u03b3\u03b5\u03b9, \u03bc\u1f72\u03bd \u03c4\u1f78\u03bd \u039c\u03c9\u03cb\u03c3\u03ad\u03b1, \u03ba\u03b1\u1f76 \u1f45\u03c4\u03b9\n  \u1f10\u03b4\u03ad\u03be\u03b1\u03c4\u03bf \u03bd\u03bf\u03bc\u03bf\u03b8\u03b5\u03c3\u03af\u03b1\u03bd \u1f10\u03c0\u03af\u03c3\u03c4\u03b5\u03c5\u03b5\u03bd, \u03bf\u1f50 \u03c4\u03b1\u03cd\u03c4\u03b7\u03bd \u03b4\u03ad \u03c6\u03b7\u03c3\u03b9\u03bd, \u1f00\u03bb\u03bb' \u1f11\u03c4\u03ad\u03c1\u03b1\u03bd. \u1f45\u03b8\u03b5\u03bd \u03c4\u1f70\n  \u03bc\u1f72\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c6\u03c5\u03bb\u03ac\u03c4\u03c4\u03bf\u03c5\u03c3\u03b9 \u03c4\u1ff6\u03bd \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9v \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03b9 \u1f44\u03bd\u03c4\u03b5\u03c2, \u03b8\u03c5\u03c3\u03af\u03b1\u03bd \u03b4\u1f72 \u03bf\u1f50\u03ba \u1f14\u03b8\u03c5\u03bf\u03bd\n  \u03bf\u1f54\u03c4\u03b5 \u1f10\u03bc\u03c8\u03cd\u03c7\u03c9\u03bd \u03bc\u03b5\u03c4\u03b5\u1fd6\u03c7\u03bf\u03bd, \u1f00\u03bb\u03bb\u1f70 \u03b8\u03ad\u03bc\u03b9\u03c4\u03bf\u03bd \u1f26\u03bd \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c4\u1f78 \u03ba\u03c1\u03b5\u1ff6\u03bd\n  \u03bc\u03b5\u03c4\u03b1\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9\u03bd \u1f22 \u03b8\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9\u03bd \u03b1\u1f50\u03c4\u03bf\u03cd\u03c2. \u1f14\u03c6\u03b1\u03c3\u03ba\u03bf\u03bd \u03b3\u1f70\u03c1 \u03c0\u03b5\u03c0\u03bb\u03ac\u03c3\u03b8\u03b1\u03b9 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c4\u1f70\n  \u03b2\u03b9\u03b2\u03bb\u03af\u03b1 \u03ba\u03b1\u1f76 \u03bc\u03b7\u03b4\u1f72\u03bd \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u1f51\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03c0\u03b1\u03c4\u03ad\u03c1\u03c9\u03bd \u03b3\u03b5\u03b3\u03b5\u03bd\u1fc6\u03c3\u03b8\u03b1\u03b9. Here we have in\n  combination all the features which we are seeking. The cradle of this\n  sect is placed by him in Gilead and Bashan and 'the regions beyond the\n  Jordan.' He uses similar language also (xxx. 18, p. 142) in describing\n  the Ebionites, whom he places in much the same localities (naming Moab\n  also), and whose Essene features are unmistakeable: \u03bf\u1f54\u03c4\u03b5 \u03b3\u1f70\u03c1 \u03b4\u03ad\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9\n  \u03c4\u1f74\u03bd \u03c0\u03b5\u03bd\u03c4\u03ac\u03c4\u03b5\u03c5\u03c7\u03bf\u03bd \u039c\u03c9\u03cb\u03c3\u03ad\u03c9\u03c2 \u1f45\u03bb\u03b7\u03bd \u1f00\u03bb\u03bb\u03ac \u03c4\u03b9\u03bd\u03b1 \u1fe5\u03ae\u03bc\u03b1\u03c4\u03b1 \u1f00\u03c0\u03bf\u03b2\u03ac\u03bb\u03bb\u03bf\u03c5\u03c3\u03b9\u03bd. \u1f45\u03c4\u03b1\u03bd \u03b4\u1f72\n  \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03b5\u1f34\u03c0\u1fc3\u03c2 \u03c0\u03b5\u03c1\u1f76 \u1f10\u03bc\u03c8\u03cd\u03c7\u03c9\u03bd \u03b2\u03c1\u03ce\u03c3\u03b5\u03c9\u03c2 \u03ba.\u03c4.\u03bb. These parallels will speak\n  for themselves.\n\n[Sidenote: (ii) The Essene worship of the Sun cannot be explained away.]\n\n(ii) Nor again is Frankel successful in explaining the Essene prayers to\nthe sun by rabbinical practices[392]. Following Rapoport, he supposes\nthat Josephus and Philo refer to the beautiful hymn of praise for the\ncreation of light and the return of day, which forms part of the\nmorning-prayer of the Jews to the present time[393], and which seems to\nbe enjoined in the Mishna itself[394]; and this view has been adopted by\nmany subsequent writers. But the language of Josephus is not satisfied\nby this explanation. For he says plainly (_B.J._ ii. 8. 5) that they\naddressed prayers to the sun[395], and it is difficult to suppose that\nhe has wantonly introduced a dash of paganism into his picture; nor\nindeed was there any adequate motive for his doing so. Similarly Philo\nrelates of the Therapeutes (_Vit. Cont._ II, II. p. 485), that they\n'stand with their faces and their whole body towards the East, and when\nthey see that the sun is risen, holding out their hands to heaven they\npray for a happy day (\u03b5\u1f50\u03b7\u03bc\u03b5\u03c1\u03af\u03b1\u03bd) and for truth and for keen vision of\nreason (\u1f40\u03be\u03c5\u03c9\u03c0\u03af\u03b1\u03bd \u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03bf\u1fe6).' And here again it is impossible to overlook\nthe confirmation which these accounts receive from the history of\ncertain Christian heretics deriving their descent from this Judaic sect.\n[Sidenote: The Samps\u00e6ans are an Essene sect,] Epiphanius (_H\u00e6r._ xix. 2,\nxx. 3, pp. 40 sq., 47) speaks of a sect called the Samps\u00e6ans or\n'Sun-worshippers[396],' as existing in his own time in Per\u00e6a on the\nborders of Moab and on the shores of the Dead Sea. He describes them as\na remnant of the Ossenes (i.e. Essenes), who have accepted a spurious\nform of Christianity and are neither Jews nor Christians. This debased\nChristianity which they adopted is embodied, he tells us, in the\npretended revelation of the Book of Elchasai, and dates from the time of\nTrajan[397]. Elsewhere (xxx. 3, p. 127) he seems to use the terms\nSamps\u00e6an, Ossene, and Elchasaite as synonymous (\u03c0\u03b1\u03c1\u1f70 \u03c4\u03bf\u1fd6\u03c2 \u03a3\u03b1\u03bc\u03c8\u03b7\u03bd\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76\n\u1f48\u03c3\u03c3\u03b7\u03bd\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u1f18\u03bb\u03ba\u03b5\u03c3\u03c3\u03b1\u03af\u03bf\u03b9\u03c2 \u03ba\u03b1\u03bb\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2). Now we happen to know something\nof this book of Elchasai, not only from Epiphanius himself (xix. 1 sq.,\np. 40 sq., xxx. 17, p. 141), but also from Hippolytus [Sidenote: as\nappears from their sacred book of Elchesai.] (_H\u00e6r._ ix. 13 sq.) who\ndescribes it at considerable length. From these accounts it appears that\nthe principal feature in the book was the injunction of frequent\nbathings for the remission of sins (Hipp. _H\u00e6r._ ix. 13, 15 sq.). We are\nlikewise told that it 'anathematizes immolations and sacrifices (\u03b8\u03c5\u03c3\u03af\u03b1\u03c2\n\u03ba\u03b1\u1f76 (\u03b9\u03b5\u03c1\u03bf\u03c5\u03c1\u03b3\u03af\u03b1\u03c2) as being alien to God and certainly not offered to God\nby tradition from (\u1f10\u03ba) the fathers and the law,' while at the same time\nit 'says that men ought to pray there at Jerusalem, where the altar was\nand the sacrifices (were offered), prohibiting the eating of flesh which\nexists among the Jews, and the rest (of their customs), and the altar\nand the fire, as being alien to God' (Epiphan. xix. 3, p. 42).\nNotwithstanding, [Sidenote: Its Essene peculiarities.] we are informed\nthat the sect retained the rite of circumcision, the observance of the\nsabbath, and other practices of the Mosaic law (Hipp. _H\u00e6r._ ix. 14;\nEpiph. _H\u00e6r._ xix. 5, p. 43, comp. xxx. 17, p. 141). This inconsistency\nis explained by a further notice in Epiphanius (l.c.) that they treated\nthe Scriptures in the same way as the Nasar\u00e6ans[398]; that is, they\nsubmitted them to a process of arbitrary excision, as recommended in the\nClementine Homilies, and thus rejected as falsifications all statements\nwhich did not square with their own theory. Hippolytus also speaks of\nthe Elchasaites as studying astrology and magic, and as practising\ncharms and incantations on the sick and the demoniacs (\u00a7 14). Moreover\nin two formularies, one of expiation, another of purification, which\nthis father has extracted from the book, invocation is made to 'the holy\nspirits and the angels of prayer' (\u00a7 15, comp. Epiph. xix. 1). It should\nbe added that the word Elchasai probably signifies the 'hidden\npower'[399]; while the book itself directed that its mysteries should be\nguarded as precious pearls, and should not be communicated to the world\nat large, but only to the faithful few (Hipp. ix. 15, 17). It is hardly\nnecessary to call attention to the number of Essene features which are\nhere combined[400]. I would only remark that the value of the notice is\nnot at all diminished, but rather enhanced, by the uncritical character\nof Epiphanius' work; for this very fact prevents us from ascribing the\ncoincidences, which here reveal themselves, to this father's own\ninvention.\n\nFootnote 392:\n\n  _Zeitschr._ p. 458.\n\nFootnote 393:\n\n  See Ginsburg _Essenes_ p. 69 sq.\n\nFootnote 394:\n\n  _Berakhoth_ i. 4; see Derenbourg, p. 169 sq.\n\nFootnote 395:\n\n  See above, p. 87, note 249.\n\nFootnote 396:\n\n  See above, p. 83.\n\nFootnote 397:\n\n  _Galatians_ p. 311 sq. See also below, p. 167.\n\nFootnote 398:\n\n  See p. 136, note 391.\n\nFootnote 399:\n\n  _Galatians_ p. 312, note 1. For another derivation see below, p. 167.\n\nFootnote 400:\n\n  Celibacy however is not one of these: comp. Epiphan. _H\u00e6r._ xix. 1 (p.\n  40) \u1f00\u03c0\u03b5\u03c7\u03b8\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03c4\u1fc7 \u03c0\u03b1\u03c1\u03b8\u03b5\u03bd\u03af\u1fb3, \u03bc\u03b9\u03c3\u03b5\u1fd6 \u03b4\u1f72 \u03c4\u1f74\u03bd \u1f10\u03b3\u03ba\u03c1\u03ac\u03c4\u03b5\u03b9\u03b1\u03bd, \u1f00\u03bd\u03b1\u03b3\u03ba\u03ac\u03b6\u03b5\u03b9 \u03b4\u1f72\n  \u03b3\u03ac\u03bc\u03bf\u03bd. In this respect they departed from the original principles of\n  Essenism, alleging, as it would appear, a special revelation (\u1f61\u03c2 \u03b4\u1fc6\u03b8\u03b5\u03bd\n  \u1f00\u03c0\u03bf\u03ba\u03b1\u03bb\u03cd\u03c8\u03b5\u03c9\u03c2) in justification. In like manner marriage is commended in\n  the Clementine Homilies.\n\n[Sidenote: Doubtful bearing of this Sun-worship.]\n\nIn this heresy we have plainly the dregs of Essenism, which has only\nbeen corrupted from its earlier and nobler type by the admixture of a\nspurious Christianity. But how came the Essenes to be called Samps\u00e6ans?\nWhat was the original meaning of this outward reverence which they paid\nto the sun? Did they regard it merely as the symbol of Divine\nillumination, just as Philo frequently treats it as a type of God, the\ncentre of all light (e.g. _de Somn._ i. 13 sq., I. p. 631 sq.), and even\ncalls the heavenly bodies 'visible and sensible gods' (_de Mund. Op._ 7,\nI. p. 6)[401]? Or did they honour the light, as the pure ethereal\nelement in contrast to gross terrestrial matter, according to a\nsuggestion of a recent writer[402]? [Sidenote: The practice repugnant to\nJewish orthodoxy.]Whatever may have been the motive of this reverence,\nit is strangely repugnant to the spirit of orthodox Judaism. In Ezek.\nviii. 16 it is denounced as an abomination, that men shall turn towards\nthe east and worship the sun; and accordingly in _Berakhoth_ 7_a_, a\nsaying of R. Meir is reported to the effect that God is angry when the\nsun appears and the kings of the East and the West prostrate themselves\nbefore this luminary[403]. We cannot fail therefore to recognise the\naction of some foreign influence in this Essene practice\u2014whether Greek\nor Syrian or Persian, it will be time to consider hereafter.\n\nFootnote 401:\n\n  The important place which the heavenly bodies held in the system of\n  Philo, who regarded them as animated beings, may be seen from\n  Gfr\u00f6rer's _Philo_ I. p. 349 sq.\n\nFootnote 402:\n\n  _Keim_ I. p. 289.\n\nFootnote 403:\n\n  See Wiesner Schol. zum _Babyl. Talm._ I. pp. 18, 20.\n\n[Sidenote: (iii) The depreciation of marriage not accounted for.]\n\n(iii) On the subject of marriage again, talmudical and rabbinical\nnotices contribute nothing towards elucidating the practices of this\nsect. Least of all do they point to any affinity between the Essenes and\nthe Pharisees. The nearest resemblance, which Frankel can produce, to\nany approximation in this respect is an injunction in Mishna _Kethuboth_\nv. 8 respecting the duties of the husband in providing for the wife in\ncase of his separating from her, and this he ascribes to Essene\ninfluences[404]; but this mishna does not express any approval of such a\nseparation. The direction seems to be framed entirely in the interests\nof the wife: nor can I see that it is at all inconsistent, as Frankel\nurges, with Mishna _Kethuboth_ vii. 1 which allows her to claim a\ndivorce under such circumstances. But however this may be, Essene and\nPharisaic opinion stand generally in the sharpest contrast to each other\nwith respect to marriage. The talmudic writings teem with passages\nimplying not only the superior sanctity, but even the imperative duty,\nof marriage. The words 'Be fruitful and multiply' (Gen. i. 28) were\nregarded not merely as a promise, but as a command, which was binding on\nall. It is a maxim of the Talmud that 'Any Jew who has not a wife is no\nman' (\u05d0\u05d9\u05e0\u05d5 \u05d0\u05d3\u05dd,) _Yebamoth_ 63_a_. The fact indeed is so patent, that\nany accumulation of examples would be superfluous, and I shall content\nmyself with referring to _Pesachim_ 113_a_, _b_, as fairly illustrating\nthe doctrine of orthodox Judaism on this point[405]. As this question\naffects the whole framework not only of religious, but also of social\nlife, the antagonism between the Essene and the Pharisee in a matter so\nvital could not be overlooked.\n\nFootnote 404:\n\n  _Monatschr._ p. 37.\n\nFootnote 405:\n\n  Justin Martyr more than once taunts the Jewish rabbis with their\n  reckless encouragement of polygamy. See _Dial._ 134, p. 363 D, \u03c4\u03bf\u1fd6\u03c2\n  \u1f00\u03c3\u03c5\u03bd\u03ad\u03c4\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03c5\u03c6\u03bb\u03bf\u1fd6\u03c2 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03ac\u03bb\u03bf\u03b9\u03c2 \u1f51\u03bc\u1ff6\u03bd, \u1f41\u03af\u03c4\u03b9\u03bd\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03bd\u1fe6\u03bd \u03ba\u03b1\u1f76\n  \u03c4\u03ad\u03c3\u03c3\u03b1\u03c1\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03ad\u03bd\u03c4\u03b5 \u1f14\u03c7\u03b5\u03b9\u03bd \u1f51\u03bc\u1fb6\u03c2 \u03b3\u03c5\u03bd\u03b1\u1fd6\u03ba\u03b1\u03c2 \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03bd \u03c3\u03c5\u03b3\u03c7\u03c9\u03c1\u03bf\u1fe6\u03c3\u03b9\u00b7 \u03ba\u03b1\u1f76 \u1f10\u1f70\u03bd\n  \u03b5\u1f54\u03bc\u03bf\u03c1\u03c6\u03cc\u03bd \u03c4\u03b9\u03c2 \u1f30\u03b4\u1f7c\u03bd \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03ae\u03c3\u1fc3 \u03b1\u1f50\u03c4\u1fc6\u03c2 \u03ba.\u03c4.\u03bb., _ib._ 141, p. 371 A, B,\n  \u1f41\u03c0\u03bf\u1fd6\u03bf\u03bd \u03c0\u03c1\u03ac\u03c4\u03c4\u03bf\u03c5\u03c3\u03b9\u03bd \u03bf\u1f31 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03b3\u03ad\u03bd\u03bf\u03c5\u03c2 \u1f51\u03bc\u1ff6\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03b9, \u03ba\u03b1\u03c4\u1f70 \u03c0\u1fb6\u03c3\u03b1\u03bd \u03b3\u1fc6\u03bd \u1f14\u03bd\u03b8\u03b1\n  \u1f02\u03bd \u1f10\u03c0\u03b9\u03b4\u03b7\u03bc\u03ae\u03c3\u03c9\u03c3\u03b9\u03bd \u1f22 \u03c0\u03c1\u03bf\u03c3\u03c0\u03b5\u03bc\u03c6\u03b8\u1ff6\u03c3\u03b9\u03bd \u1f00\u03b3\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b9 \u03b3\u03ac\u03bc\u03bf\u03c5 \u03b3\u03c5\u03bd\u03b1\u1fd6\u03ba\u03b1\u03c2\n  \u03ba.\u03c4.\u03bb., with Otto's note on the first passage.\n\n[Sidenote: (iv) The Essene practice of magic still a difficulty.]\n\n(iv) Nor again is it probable that the magical rites and incantations\nwhich are so prominent in the practice of the Essenes would, as a rule,\nhave been received with any favour by the Pharisaic Jew. In Mishna\n_Pesachim_ iv. 9 (comp. _Berakhoth_ 10_b_) it is mentioned with approval\nthat Hezekiah put away a 'book of healings'; where doubtless the author\nof the tradition had in view some volume of charms ascribed to Solomon,\nlike those which apparently formed part of the esoteric literature of\nthe Essenes[406]. In the same spirit in Mishna _Sanhedrin_ xi. 1 R.\nAkiba shuts out from the hope of eternal life any 'who read profane or\nforeign (i.e. perhaps, apocryphal) books, and who mutter over a wound'\nthe words of Exod. xv. 26. On this point of difference however no great\nstress can be laid. Though the nobler teachers among the orthodox Jews\nset themselves steadfastly against the introduction of magic, they were\nunable to resist the inpouring tide of superstition. In the middle of\nthe second century Justin Martyr alludes to exorcists and magicians\namong the Jews, as though they were neither few nor obscure[407].\nWhether these were a remnant of Essene Judaism, or whether such\npractices had by this time spread throughout the whole body, it is\nimpossible to say; but the fact of their existence prevents us from\nfounding an argument on the use of magic, as an absolutely distinctive\nfeature of Essenism.\n\nFootnote 406:\n\n  See above, p. 91, note 261.\n\nFootnote 407:\n\n  _Dial._ 85, p. 311 C, \u1f24\u03b4\u03b7 \u03bc\u03ad\u03bd\u03c4\u03bf\u03b9 \u03bf\u1f31 \u1f10\u03be \u1f51\u03bc\u1ff6\u03bd \u1f10\u03c0\u03bf\u03c1\u03ba\u03b9\u03c3\u03c4\u03b1\u1f76 \u03c4\u1fc7 \u03c4\u03ad\u03c7\u03bd\u1fc3, \u1f65\u03c3\u03c0\u03b5\u03c1\n  \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f14\u03b8\u03bd\u03b7, \u03c7\u03c1\u03ce\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f10\u03be\u03bf\u03c1\u03ba\u03af\u03b6\u03bf\u03c5\u03c3\u03b9 \u03ba\u03b1\u1f76 \u03b8\u03c5\u03bc\u03b9\u03ac\u03bc\u03b1\u03c3\u03b9 \u03ba\u03b1\u1f76 \u03ba\u03b1\u03c4\u03b1\u03b4\u03ad\u03c3\u03bc\u03bf\u03b9\u03c2\n  \u03c7\u03c1\u1ff6\u03bd\u03c4\u03b1\u03b9.\n\n[Sidenote: General result.]\n\nOther divergences also have been enumerated[408]; but, as these do not\nfor the most part involve any great principles, and refer only to\npractical details in which much fluctuation was possible, they cannot\nunder any circumstances be taken as crucial tests, and I have not\nthought it worth while to discuss them. But the antagonisms on which I\nhave dwelt will tell their own tale. In three respects more especially,\nin the avoidance of marriage, in the abstention from the temple\nsacrifices, and (if the view which I have adopted be correct) in the\noutward reverence paid to the sun, we have seen that there is an\nimpassable gulf between the Essenes and the Pharisees. No known\ninfluences within the sphere of Judaism proper will serve to account for\nthe position of the Essenes in these respects; and we are obliged to\nlook elsewhere for an explanation.\n\nFootnote 408:\n\n  Herzfeld, II. p. 392 sq.\n\n                  *       *       *       *       *\n\n[Sidenote: Frankel has failed in establishing his point.]\n\nIt was shown above that the investigations of Frankel and others failed\nto discover in the talmudical writings a single reference to the\nEssenes, which is at once direct and indisputable. It has now appeared\nthat they have also failed (and this is the really important point) in\nshowing that the ideas and practices generally considered characteristic\nof the Essenes are recognised and incorporated in these representative\nbooks of Jewish orthodoxy; and thus the hypothesis that Essenism was\nmerely a type, though an exaggerated type, of pure Judaism falls to the\nground.\n\n[Sidenote: Affinities between Essenes and Pharisees confined to the\n           Judaic side.]\n\nSome affinities indeed have been made out by Frankel and by those who\nhave anticipated or followed him. But these are exactly such as we might\nhave expected. Two distinct features combine to make up the portrait of\nthe Essene. The Judaic element is quite as prominent in this sect as the\nnon-Judaic. It could not be more strongly emphasized than in the\ndescription given by Josephus himself. In everything therefore which\nrelates to the strictly Judaic side of their tenets and practices, we\nshould expect to discover not only affinities, but even close\naffinities, in talmudic and rabbinic authorities. And this is exactly\nwhat, as a matter of fact, we do find. The Essene rules respecting the\nobservance of the sabbath, the rites of lustration, and the like, have\noften very exact parallels in the writings of more orthodox Judaism. But\nI have not thought it necessary to dwell on these coincidences, because\nthey may well be taken for granted and my immediate purpose did not\nrequire me to emphasize them.\n\n[Sidenote: The divergence of the Essenes from the Pharisees gradual.]\n\nAnd again; it must be remembered that the separation between Pharisee\nand Essene cannot always have been so great as it appears in the\nApostolic age. Both sects apparently arose out of one great movement, of\nwhich the motive was the avoidance of pollution[409]. The divergence\ntherefore must have been gradual. At the same time, it does not seem a\nvery profitable task to write a hypothetical history of the growth of\nEssenism, where the data are wanting; and I shall therefore abstain from\nthe attempt. Frankel indeed has not been deterred by this difficulty;\nbut he has been obliged to assume his data by postulating that such and\nsuch a person, of whom notices are preserved, was an Essene, and thence\ninferring the character of Essenism at the period in question from his\nrecorded sayings or doings. But without attempting any such\nreconstruction of history, we may fairly allow that there must have been\na gradual development; and consequently in the earlier stages of its\ngrowth we should not expect to find that sharp antagonism between the\ntwo sects, which the principles of the Essenes when fully matured would\ninvolve. [Sidenote: Hence the possibility of their appearing in the\nrecords of orthodox Judaism.] If therefore it should be shown that the\ntalmudical and rabbinical writings here and there preserve with approval\nthe sayings of certain Essenes, this fact would present no difficulty.\nAt present however no decisive example has been produced; and the\ndiscoveries of Jellinek for instance[410], who traces the influence of\nthis sect in almost every page of _Pirke Aboth_, can only be regarded as\nanother illustration of the extravagance with which the whole subject\nhas been treated by a large section of modern Jewish writers. More to\nthe point is a notice of an earlier Essene preserved in Josephus\nhimself. We learn from this historian that one Judas, a member of the\nsect, who had prophesied the death of Antigonus, saw this prince\n'passing by through the temple[411],' when his prophecy was on the point\nof fulfilment (about B.C. 110). At this moment Judas is represented as\nsitting in the midst of his disciples, instructing them in the science\nof prediction. The expression quoted would seem to imply that he was\nactually teaching within the temple area. Thus he would appear not only\nas mixing in the ordinary life of the Jews, but also as frequenting the\nnational sanctuary. But even supposing this to be the right explanation\nof the passage, it will not present any serious difficulty. Even at a\nlater date, when (as we may suppose) the principles of the sect had\nstiffened, the scruples of the Essene were directed, if I have rightly\ninterpreted the account of Josephus, rather against the sacrifices than\nagainst the locality[412]. The temple itself, independently of its\naccompaniments, would not suggest any offence to his conscience.\n\nFootnote 409:\n\n  See above, p. 120.\n\nFootnote 410:\n\n  _Orient_ 1849, pp. 489, 537, 553.\n\nFootnote 411:\n\n  _B.J._ i. 3. 5 \u03c0\u03b1\u03c1\u03b9\u03cc\u03bd\u03c4\u03b1 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u1f31\u03b5\u03c1\u03bf\u1fe6. In the parallel narrative,\n  _Ant._ xiii. II. 2, the expression is \u03c0\u03b1\u03c1\u03b9\u03cc\u03bd\u03c4\u03b1 \u03c4\u1f78 \u1f31\u03b5\u03c1\u03cc\u03bd, which does\n  not imply so much; but the less precise notice must be interpreted by\n  the more precise. Even then however it is not directly stated that\n  Judas himself was within the temple area.\n\nFootnote 412:\n\n  See above, pp. 89, 134 sq.\n\n[Sidenote: The approbation of Philo and Josephus is no evidence of\n           orthodoxy.]\n\nNor again, is it any obstacle to the view which is here maintained, that\nthe Essenes are regarded with so much sympathy by Philo and Josephus\nthemselves. Even though the purity of Judaism might have been somewhat\nsullied in this sect by the admixture of foreign elements, this fact\nwould attract rather than repel an eclectic like Philo, and a\nlatitudinarian like Josephus. The former, as an Alexandrian, absorbed\ninto his system many and diverse elements of heathen philosophy,\nPlatonic, Stoic, and Pythagorean. The latter, though professedly a\nPharisee, lost no opportunity of ingratiating himself with his heathen\nconquerors, and would not be unwilling to gratify their curiosity\nrespecting a society with whose fame, as we infer from the notice of\nPliny, they were already acquainted.\n\n                  *       *       *       *       *\n\n[Sidenote: What was the foreign element in Essenism?]\n\nBut if Essenism owed the features which distinguished it from Pharisaic\nJudaism to an alien admixture, whence were these foreign influences\nderived? From the philosophers of Greece or from the religious mystics\nof the East? On this point recent writers are divided.\n\n[Sidenote: Theory of Neopythagorean influence.]\n\nThose who trace the distinctive characteristics of the sect to Greece,\nregard it as an offshoot of the Neopythagorean School grafted on the\nstem of Judaism. This solution is suggested by the statement of\nJosephus, that 'they practise the mode of life which among the Greeks\nwas introduced (\u03ba\u03b1\u03c4\u03b1\u03b4\u03b5\u03b4\u03b5\u03b9\u03b3\u03bc\u03ad\u03bd\u1fc3) by Pythagoras[413].' It is thought to be\nconfirmed by the strong resemblances which as a matter of fact are found\nto exist between the institutions and practices of the two.\n\n[Sidenote: Statement of the theory by Zeller.]\n\nThis theory, which is maintained also by other writers, as for instance\nby Baur and Herzfeld, has found its ablest and most persistent advocate\nin Zeller, who draws out the parallels with great force and precision.\n'The Essenes,' he writes, 'like the Pythagoreans, desire to attain a\nhigher sanctity by an ascetic life; and the abstentions, which they\nimpose on themselves for this end, are the same with both. They reject\nanimal food and bloody sacrifices; they avoid wine, warm baths, and oil\nfor anointing; they set a high value on celibate life: or, so far as\nthey allow marriage, they require that it be restricted to the one\nobject of procreating children. Both wear only white garments and\nconsider linen purer than wool. Washings and purifications are\nprescribed by both, though for the Essenes they have a yet higher\nsignificance as religious acts. Both prohibit oaths and (what is more)\non the same grounds. Both find their social ideal in those institutions,\nwhich indeed the Essenes alone set themselves to realise\u2014in a corporate\nlife with entire community of goods, in sharply defined orders of rank,\nin the unconditional submission of all the members to their superiors,\nin a society carefully barred from without, into which new members are\nreceived only after a severe probation of several years, and from which\nthe unworthy are inexorably excluded. Both require a strict initiation,\nboth desire [Sidenote: Zeller's theory.] to maintain a traditional\ndoctrine inviolable; both pay the highest respect to the men from whom\nit was derived, as instruments of the deity: yet both also love\nfigurative clothing for their doctrines, and treat the old traditions as\nsymbols of deeper truths, which they must extract from them by means of\nallegorical explanation. In order to prove the later form of teaching\noriginal, newly-composed writings were unhesitatingly forged by the one\nas by the other, and fathered upon illustrious names of the past. Both\nparties pay honour to divine powers in the elements, both invoke the\nrising sun, both seek to withdraw everything unclean from his sight, and\nwith this view give special directions, in which they agree as well with\neach other as with older Greek superstition, in a remarkable way. For\nboth the belief in intermediate beings between God and the world has an\nimportance which is higher in proportion as their own conception of God\nis purer; both appear not to have disdained magic; yet both regard the\ngift of prophecy as the highest fruit of wisdom and piety, which they\npique themselves on possessing in their most distinguished members.\nFinally, both agree (along with the dualistic character of their whole\nconception of the world ...) in their tenets respecting the origin of\nthe soul, its relation to the body, and the life after death[414]....'\n\nFootnote 413:\n\n  _Ant._ xv. 10. 4.\n\nFootnote 414:\n\n  Zeller _Philosophie der Griechen_, Th. III. Abth. 2, p. 281.\n\n[Sidenote: Absence of distinctive Pythagorean features in the Essenes.]\n\nThis array of coincidences is formidable, and thus skilfully marshalled\nmight appear at first sight invincible. But a closer examination\ndetracts from its value. In the first place the two distinctive\ncharacteristics of the Pythagorean philosophy are wanting to the\nEssenes. The Jewish sect did not believe in the transmigration of souls;\nand the doctrine of numbers, at least so far as our information goes,\nhad no place in their system. Yet these constitute the very essence of\nthe Pythagorean teaching. In the next place several of the coincidences\nare more apparent than real. Thus [Sidenote: The coincidences are in\nsome cases only apparent,] for instance the demons who in the\nPythagorean system held an intermediate place between the Supreme God\nand man, and were the result of a compromise between polytheism and\nphilosophy, have no near relation to the angelology of the Essenes,\nwhich arose out of a wholly different motive. Nor again can we find\ndistinct traces among the Pythagoreans of any such reverence for the sun\nas is ascribed to the Essenes, the only notice which is adduced having\nno prominence whatever in its own context, and referring to a rule which\nwould be dictated by natural decency and certainly was not peculiar to\nthe Pythagoreans[415]. When these imperfect and (for the purpose)\nvalueless resemblances have been subtracted, the only basis on which the\ntheory of a direct affiliation can rest is withdrawn. All the remaining\ncoincidences are unimportant. Thus the respect paid to founders is not\nconfined to any one sect or any one age. The reverence of the Essenes\nfor Moses, and the reverence of the Pythagoreans for Pythagoras, are\nindications of a common humanity, but not of a common philosophy. And\nagain the forgery of supposititious documents is unhappily not the badge\nof any one school. The Solomonian books of the Essenes, so far as we can\njudge from the extant notices, were about as unlike the tracts ascribed\nto Pythagoras and his disciples by the Neopythagoreans as two such\nforgeries could well be. All or nearly all that remains in common to the\nGreek school and the Jewish sect after these deductions is [Sidenote:\nand in others do not suggest any historical connexion.] a certain\nsimilarity in the type of life. But granted that two bodies of men each\nheld an esoteric teaching of their own, they would secure it\nindependently in a similar way, by a recognised process of initiation,\nby a solemn form of oath, by a rigid distinction of orders. Granted\nalso, that they both maintained the excellence of an ascetic life, their\nasceticism would naturally take the same form; they would avoid wine and\nflesh; they would abstain from anointing themselves with oil; they would\ndepreciate, and perhaps altogether prohibit, marriage. Unless therefore\nthe historical conditions are themselves favourable to a direct and\nimmediate connexion between the Pythagoreans and the Essenes, this\ntheory of affiliation has little to recommend it.\n\nFootnote 415:\n\n  Diog. Laert. viii. 17; see Zeller l.c. p. 282, note 5. The precept in\n  question occurs among a number of insignificant details, and has no\n  special prominence given to it. In the _Life of Apollonius_ by\n  Philostratus (e.g. vi. 10) considerable stress is laid on the worship\n  of the sun (Zeller l.c. p. 137, note 6); but the syncretism of this\n  late work detracts from its value as representing Pythagorean\n  doctrine.\n\n[Sidenote: Twofold objection to this theory.]\n\nAnd a closer examination must pronounce them to be most unfavourable.\nChronology and geography alike present serious obstacles to any solution\nwhich derives the peculiarities of the Essenes from the Pythagoreans.\n\n[Sidenote: (i) Chronological facts are adverse.]\n\n(i) The priority of time, if it can be pleaded on either side, must be\nurged in favour of the Essenes. The Pythagoreans as a philosophical\nschool entirely disappear from history before the middle of the fourth\ncentury before Christ. The last Pythagoreans were scholars of Philolaus\nand Eurytus, the contemporaries of Socrates and Plato[416]. For nearly\ntwo centuries after their extinction we hear [Sidenote: Disappearance of\nthe Pythagoreans.] nothing of them. Here and there persons like Diodorus\nof Aspendus are satirised by the Attic poets of the middle comedy as\n'pythagorizers,' in other words, as total abstainers and\nvegetarians[417]; but the philosophy had wholly died or was fast dying\nout. This is the universal testimony of ancient writers. It is not till\nthe first century before Christ, that we meet with any distinct traces\nof a revival. In Alexander Polyhistor[418], a younger contemporary of\nSulla, for the first time we find references to certain writings, which\nwould seem to have emanated from this incipient Neopythagoreanism,\nrather than from the elder school of Pythagoreans. And a little later\nCicero commends his friend Nigidius Figulus as one specially raised up\nto revive the extinct philosophy[419]. But so slow or so chequered was\nits progress, that a whole century after Seneca can still speak of the\n[Sidenote: Priority of Essenism to Neopythagoreanism.] school as\npractically defunct[420]. Yet long before this the Essenes formed a\ncompact, well-organized, numerous society with a peculiar system of\ndoctrine and a definite rule of life. We have seen that Pliny the elder\nspeaks of this celibate society as having existed 'through thousands of\nages[421].' This is a gross exaggeration, but it must at least be taken\nto imply that in Pliny's time the origin of the Essenes was lost in the\nobscurity of the past, or at least seemed so to those who had not access\nto special sources of information. If, as I have given reasons for\nsupposing[422], Pliny's authority in this passage is the same Alexander\nPolyhistor to whom I have just referred, and if this particular\nstatement, however exaggerated in expression, is derived from him, the\nfact becomes still more significant. But on any showing the priority in\ntime is distinctly in favour of the Essenes as against the\nNeopythagoreans.\n\nFootnote 416:\n\n  Zeller l.c. p. 68 (comp. I. p. 242). While disputing Zeller's\n  position, I have freely made use of his references. It is impossible\n  not to admire the mastery of detail and clearness of exposition in\n  this work, even when the conclusions seem questionable.\n\nFootnote 417:\n\n  Athen. iv. p. 161, Diog. Laert. viii. 37. See the index to Meineke\n  _Fragm. Com._ s. vv. \u03c0\u03c5\u03b8\u03b1\u03b3\u03bf\u03c1\u03b9\u03ba\u03cc\u03c2, etc. The words commonly used by\n  these satirists are \u03c0\u03c5\u03b8\u03b1\u03b3\u03bf\u03c1\u03af\u03b6\u03b5\u03b9\u03bd, \u03c0\u03c5\u03b8\u03b1\u03b3\u03bf\u03c1\u03b9\u03c3\u03c4\u03ae\u03c2, \u03c0\u03c5\u03b8\u03b1\u03b3\u03bf\u03c1\u03b9\u03c3\u03bc\u03cc\u03c2. The\n  persons so satirized were probably in many cases not more Pythagoreans\n  than modern teetotallers are Rechabites.\n\nFootnote 418:\n\n  Diog. Laert. viii. 24 sq.; see Zeller l.c. p. 74\u201378.\n\nFootnote 419:\n\n  Cic. _Tim._ I 'sic judico, post illos nobiles Pythagoreos quorum\n  disciplina _extincta est_ quodammodo, cum aliquot s\u00e6cula in Italia\n  Siciliaque viguisset, hunc exstitisse qui illam _renovaret_.'\n\nFootnote 420:\n\n  Sen. _N.Q._ vii. 32 'Pythagorica illa invidiosa turb\u00e6 schola\n  pr\u00e6ceptorem non invenit.'\n\nFootnote 421:\n\n  _N.H._ v. 15. The passage is at which Josephus thinks it necessary to\n  insert an account of the Essenes as already flourishing (_Ant._ xiii.\n  5. 9), is prior to the revival of the Neopythagorean school. How much\n  earlier the Jewish sect arose, we are without data for determining.\n\nFootnote 422:\n\n  See p. 83, note 240.\n\n[Sidenote: The Essene tenets more developed than the Neopythagorean.]\n\nAnd accordingly we find that what is only a tendency in the\nNeopythagoreans is with the Essenes an avowed principle and a definite\nrule of life. Such for instance is the case with celibacy, of which\nPliny says that it has existed as an institution among the Essenes _per\ns\u00e6culorum millia_, and which is a chief corner-stone of their practical\nsystem. The Pythagorean notices (whether truly or not, it is unimportant\nfor my purpose to enquire) speak of Pythagoras as having a wife and a\ndaughter[423]. Only at a late date do we find the attempt to represent\ntheir founder in another light; and if virginity is ascribed to\nApollonius of Tyana, the great Pythagorean of the first Christian\ncentury, in the fictitious biography of Philostratus[424], this\nrepresentation is plainly due to the general plan of the novelist, whose\nhero is intended to rival the Founder of Christianity, and whose work is\nsaturated with Christian ideas. In fact virginity can never be said to\nhave been a Pythagorean principle, though it may have been an exalted\nideal of some not very early adherents of the school. And the same\nremark applies to other resemblances between the Essene and\nNeopythagorean teaching. The clearness of conception and the\ndefiniteness of practice are in almost every instance on the side of the\nEssenes; so that, looking to the comparative chronology of the two, it\nwill appear almost inconceivable that they can have derived their\nprinciples from the Neopythagoreans.\n\nFootnote 423:\n\n  Diog. Laert. viii. 42.\n\nFootnote 424:\n\n  _Vit. Apoll._ i. 15 sq. At the same time Philostratus informs us that\n  the conduct of his hero in this respect had been differently\n  represented by others.\n\n[Sidenote: (ii) Geographical difficulties in the theory.]\n\n(ii) But the geographical difficulty also, which this theory of\naffiliation involves, must be added to the chronological. The home of\nthe Essene sect is allowed on all hands to have been on the eastern\nborders of Palestine, the shores of the Dead Sea, a region least of all\nexposed to the influences of Greek philosophy. It is true that we find\nnear Alexandria a closely allied school of Jewish recluses, the\nTherapeutes; and, as Alexandria may have been the home of\nNeopythagoreanism, a possible link of connexion is here disclosed. But,\nas Zeller himself has pointed out, it is not among the Therapeutes, but\namong the Essenes, that the principles in question appear fully\ndeveloped and consistently carried out[425]; and therefore, if there be\na relation of paternity between Essene and Therapeute, the latter must\nbe derived from the former and not conversely. How then can we suppose\nthis influence of Neopythagoreanism brought to bear on a Jewish\ncommunity in the south-eastern border of Palestine? Zeller's answer is\nas follows[426]. Jud\u00e6a was for more than a hundred and fifty years\nbefore the Maccabean period under the sovereignty first of the Egyptian\nand then of the Syrian Greeks. We know that at this time Hellenizing\ninfluences did infuse themselves largely into Judaism: and what more\nnatural than that among these the Pythagorean philosophy and discipline\nshould have recommended itself to a section of the Jewish people? It may\nbe said in reply, that at all events the special locality of the Essenes\nis the least favourable to such a solution: but, without pressing this\nfact, Zeller's hypothesis is open to two serious objections which\ncombined seem fatal to it, unsupported as it is by any historical\nnotice. First, this influence of Pythagoreanism is assumed to have taken\nplace at the very time when the Pythagorean school was practically\nextinct: and secondly, it is supposed to have acted upon that very\nsection of the Jewish community, which was the most vigorous advocate of\nnational exclusiveness and the most averse to Hellenizing influences.\n\nFootnote 425:\n\n  l.c. p. 288 sq.\n\nFootnote 426:\n\n  l.c. p. 290 sq.\n\n[Sidenote: The foreign element of Essenism to be sought in the East,]\n\nIt is not therefore to Greek but to Oriental influences that\nconsiderations of time and place, as well as of internal character, lead\nus to look for an explanation of the alien elements in Essene Judaism.\nAnd have we not here also the account of any real coincidences which may\nexist between Essenism and Neopythagoreanism? We should perhaps be\nhardly more justified in tracing Neopythagoreanism directly to Essenism\nthan conversely (though, if we had no other alternative, this would\nappear to be the more probable solution of the two): but were not both\nalike due to substantially the same influences acting in different\ndegrees? [Sidenote: to which also Pythagoreanism may have been\nindebted.]I think it will hardly be denied that the characteristic\nfeatures of Pythagoreanism, and especially of Neopythagoreanism, which\ndistinguish it from other schools of Greek philosophy, are much more\nOriental in type, than Hellenic. The asceticism, the magic, the\nmysticism, of the sect all point in the same direction. And history\nmoreover contains indications that such was the case. There seems to be\nsufficient ground for the statement that Pythagoras himself was indebted\nto intercourse with the Egyptians, if not with more strictly Oriental\nnations, for some leading ideas of his system. But, however this may be,\nthe fact that in the legendary accounts, which the Neopythagoreans\ninvented to do honour to the founder of the school, he is represented as\ntaking lessons from the Chaldeans, Persians, Brahmins, and others, may\nbe taken as an evidence that their own philosophy at all events was\npartially derived from eastern sources[427].\n\nFootnote 427:\n\n  See the references in Zeller I. p. 218 sq.; comp. III. 2, p. 67.\n\n                  *       *       *       *       *\n\nBut, if the alien elements of Essenism were borrowed not so much from\nGreek philosophy as from Oriental mysticism, to what nation or what\nreligion was it chiefly indebted? To this question it is difficult, with\nour very imperfect knowledge of the East at the Christian era, to reply\nwith any confidence. [Sidenote: Resemblances to Parsism.]Yet there is\none system to which we naturally look, as furnishing the most probable\nanswer. The Medo-Persian religion supplies just those elements which\ndistinguish the tenets and practices of the Essenes from the normal type\nof Judaism. [Sidenote: (i) Dualism.](1) First; we have here a very\ndefinite form of dualism, which exercised the greatest influence on\nsubsequent Gnostic sects, and of which Manicheism, the most mature\ndevelopment of dualistic doctrine in connexion with Christianity, was\nthe ultimate fruit. For though dualism may not represent the oldest\ntheology of the Zend-Avesta in its unadulterated form, yet long before\nthe era of which we are speaking it had become the fundamental principle\nof the Persian religion. [Sidenote: (ii) Sun-worship.](2) Again; the\nZoroastrian symbolism of light, and consequent worship of the sun as the\nfountain of light, will explain those anomalous notices of the Essenes\nin which they are represented as paying reverence to this luminary[428].\n[Sidenote: (iii) Angelolatry.](3) Moreover; the 'worship of angels' in\nthe Essene system has a striking parallel in the invocations of spirits,\nwhich form a very prominent feature in the ritual of the Zend-Avesta.\nAnd altogether their angelology is illustrated, and not improbably was\nsuggested, by the doctrine of intermediate beings concerned in the\ngovernment of nature and of man, such as the Amshaspands, which is an\nintegral part of the Zoroastrian system[429]. [Sidenote: (iv) Magic.](4)\nAnd once more; the magic, which was so attractive to the Essene, may\nhave received its impulse from the priestly caste of Persia, to whose\nworld-wide fame this form of superstition is indebted for its name.\n[Sidenote: (v) Striving after purity.](5) If to these parallels I\nventure also to add the intense striving after purity, which is the\nnoblest feature in the Persian religion, I do so, not because the\nEssenes might not have derived this impulse from a higher source, but\nbecause this feature was very likely to recommend the Zoroastrian system\nto their favourable notice, and because also the particular form which\nthe zeal for purity took among them was at all events congenial to the\nteaching of the Zend-Avesta, and may not have been altogether free from\nits influences.\n\nFootnote 428:\n\n  Keim (_Geschichte Jesu von Nazara_ I.<\/sc> p. 303) refers to Tac.\n  Hist. iii. 24 'Undique clamor; et _orientem solem_ (ita in Syria mos\n  est) tertiani salutavere,' as illustrating this Essene practice. The\n  commentators on Tacitus quote a similar notice of the Parthians in\n  Herodian iv. 15 _\u1f05\u03bc\u03b1 \u03b4\u1f72 \u1f21\u03bb\u03af\u1ff3 \u1f00\u03bd\u03af\u03c3\u03c7\u03bf\u03bd\u03c4\u03b9 \u1f10\u03c6\u03ac\u03bd\u03b7 \u1f08\u03c1\u03c4\u03ac\u03b2\u03b1\u03bd\u03bf\u03c2 \u03c3\u1f7a\u03bd \u03bc\u03b5\u03b3\u03af\u03c3\u03c4\u1ff3\n  \u03c0\u03bb\u03ae\u03b8\u03b5\u03b9 \u03c3\u03c4\u03c1\u03b1\u03c4\u03bf\u1fe6\u00b7 \u1f00\u03c3\u03c0\u03b1\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03b4\u1f72 \u03c4\u1f78\u03bd \u1f24\u03bb\u03b9\u03bf\u03bd, \u1f61\u03c2 \u1f14\u03b8\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2, \u03bf\u1f31 \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03b9\n  \u03ba.\u03c4.\u03bb._\n\nFootnote 429:\n\n  See e.g. _Vendidad_ Farg. xix; and the liturgical portions of the book\n  are largely taken up with invocations of these intermediate beings.\n  Some extracts are given in Davies' _Colossians_ p. 146 sq.\n\n[Sidenote: Other coincidences accidental.]\n\nI have preferred dwelling on these broader resemblances, because they\nare much more significant than any mere coincidence of details, which\nmay or may not have been accidental. Thus for instance the magi, like\nthe Essenes, wore white garments, and eschewed gold and ornaments; they\npractised frequent lustrations; they avoided flesh, living on bread and\ncheese or on herbs and fruits; they had different orders in their\nsociety; and the like[430]. All these, as I have already remarked, may\nbe the independent out-growth of the same temper and direction of\nconduct, and need not imply any direct historical connexion. Nor is\nthere any temptation to press such resemblances; for even without their\naid the general connexion seems to be sufficiently established[431].\n\nFootnote 430:\n\n  Hilgenfeld (_Zeitschrift_ x. p. 99 sq.) finds coincidences even more\n  special than these. He is answered by Zeller (III. 2. p. 276), but\n  defends his position again (_Zeitschrift_ xi. p. 347 sq.), though with\n  no great success. Among other points of coincidence Hilgenfeld remarks\n  on the axe (Jos. _B.J._ ii. 8. 7) which was given to the novices among\n  the Essenes, and connects it with the \u1f00\u03be\u03b9\u03bd\u03bf\u03bc\u03b1\u03bd\u03c4\u03b5\u03af\u03b1 (Plin. _N.H._\n  xxxvi. 19) of the magi. Zeller contents himself with replying that the\n  use of the axe among the Essenes for purposes of divination is a pure\n  conjecture, not resting on any known fact. He might have answered with\n  much more effect that Josephus elsewhere (\u00a7 9) defines it as a spade\n  or shovel, and assigns to it a very different use. Hilgenfeld has\n  damaged his cause by laying stress on these accidental resemblances.\n  So far as regards minor coincidences, Zeller makes out as good a case\n  for his Pythagoreans, as Hilgenfeld for his magians.\n\nFootnote 431:\n\n  Those who allow any foreign Oriental element in Essenism most commonly\n  ascribe it to Persia: e.g. among the more recent writers, Hilgenfeld\n  (l.c.) and Lipsius _Schenkel's Bibel-Lexikon_ s.v. Ess\u00e4er p. 189.\n\n[Sidenote: The destruction of the Persian empire not adverse]\n\nBut it is said, that the history of Persia does not favour the\nhypothesis of such an influence as is here assumed. The destruction of\nthe Persian empire by Alexander, argues Zeller[432], and the subsequent\nerection of the Parthian domination on its ruins, must have been fatal\nto the spread of Zoroastrianism. From the middle of the third century\nbefore Christ, when the Parthian empire was established, till towards\nthe middle of the third century of our era, when the Persian monarchy\nand religion were once more restored[433], its influence must have been\nreduced within the narrowest limits. [Sidenote: but favourable to the\nspread of Parsism.] But does analogy really suggest such an inference?\nDoes not the history of the Jews themselves show that the religious\ninfluence of a people on the world at large may begin just where its\nnational life ends? The very dispersion of Zoroastrianism, consequent on\nthe fall of the empire, would impregnate the atmosphere far and wide;\nand the germs of new religious developments would thus be implanted in\nalien soils. For in tracing Essenism to Persian influences I have not\nwished to imply that this Jewish sect consciously incorporated the\nZoroastrian philosophy and religion as such, but only that Zoroastrian\nideas were infused into its system by more or less direct contact. And,\nas a matter of fact, it seems quite certain that Persian ideas were\nwidely spread during this very interval, when the Persian nationality\nwas eclipsed. [Sidenote: Indications of its influence during this\nperiod.]It was then that Hermippus gave to the Greeks the most detailed\naccount of this religion which had ever been laid before them[434]. It\nwas then that its tenets suggested or moulded the speculations of the\nvarious Gnostic sects. It was then that the worship of the Persian\nMithras spread throughout the Roman Empire. It was then, if not earlier,\nthat the magian system took root in Asia Minor, making for itself (as it\nwere) a second home in Cappadocia[435]. It was then, if not earlier,\nthat the Zoroastrian demonology stamped itself so deeply on the\napocryphal literature of the Jews themselves, which borrowed even the\nnames of evil spirits[436] from the Persians. There are indeed abundant\nindications that Palestine was surrounded by Persian influences during\nthis period, when the Persian empire was in abeyance.\n\nFootnote 432:\n\n  l.c. p. 275.\n\nFootnote 433:\n\n  See Gibbon _Decline and Fall_ c. viii, Milman _History of\n  Christianity_ II. p. 247 sq. The latter speaks of this restoration of\n  Zoroastrianism, as 'perhaps the only instance of the vigorous revival\n  of a Pagan religion.' It was far purer and less Pagan than the system\n  which it superseded; and this may account for its renewed life.\n\nFootnote 434:\n\n  See M\u00fcller _Fragm. Hist. Gr\u00e6c._ III. p. 53 sq. for this work of\n  Hermippus \u03c0\u03b5\u03c1\u1f76 \u039c\u03ac\u03b3\u03c9\u03bd. He flourished about B.C. 200. See Max M\u00fcller\n  _Lectures on the Science of Language_ 1st ser. p. 86.\n\nFootnote 435:\n\n  Strabo xv. 3. 15 (p. 733) \u1f18\u03bd \u03b4\u1f72 \u03c4\u1fc7 \u039a\u03b1\u03c0\u03c0\u03b1\u03b4\u03bf\u03ba\u03af\u1fb3 (\u03c0\u03bf\u03bb\u1f7a \u03b3\u1f70\u03c1 \u1f10\u03ba\u03b5\u1fd6 \u03c4\u1f78 \u03c4\u1ff6\u03bd\n  \u039c\u03ac\u03b3\u03c9\u03bd \u03c6\u1fe6\u03bb\u03bf\u03bd, \u03bf\u1f33 \u03ba\u03b1\u1f76 \u03c0\u03cd\u03c1\u03b1\u03b9\u03b8\u03bf\u03b9 \u03ba\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9\u00b7 \u03c0\u03bf\u03bb\u03bb\u1f70 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u03a0\u03b5\u03c1\u03c3\u03b9\u03ba\u1ff6\u03bd \u03b8\u03b5\u1ff6\u03bd\n  \u1f31\u03b5\u03c1\u03ac) \u03ba.\u03c4.\u03bb.\n\nFootnote 436:\n\n  At least in one instance, Asmodeus (Tob. iii. 17); see M. M\u00fcller\n  _Chips from a German Workshop_ I. p. 148 sq. For the different dates\n  assigned to the book of Tobit see Dr Westcott's article _Tobit_ in\n  Smith's _Dictionary of the Bible_ p. 1525.\n\nThus we seem to have ample ground for the view that certain alien\nfeatures in Essene Judaism were derived from the Zoroastrian religion.\n[Sidenote: Are Buddhist influences also perceptible?]But are we\njustified in going a step further, and attributing other elements in\nthis eclectic system to the more distant East? The monasticism of the\nBuddhist will naturally occur to our minds, as a precursor of the\ncenobitic life among the Essenes; and Hilgenfeld accordingly has not\nhesitated to ascribe this characteristic of Essenism directly to\nBuddhist influences[437]. But at the outset we are obliged to ask\nwhether history gives any such indication of the presence of Buddhism in\nthe West as this hypothesis requires. Hilgenfeld answers this question\nin the affirmative. [Sidenote: Supposed Buddhist establishment at\nAlexandria.]He points triumphantly to the fact that as early as the\nmiddle of the second century before Christ the Buddhist records speak of\ntheir faith as flourishing in Alasanda the chief city of the land of\nYavana. The place intended, he conceives, can be none other than the\ngreat Alexandria, the most famous of the many places bearing the\nname[438]. [Sidenote: The authority misinterpreted] In this opinion\nhowever he stands quite alone. Neither K\u00f6ppen[439], who is his authority\nfor this statement, nor any other Indian scholar[440], so far as I am\naware, for a moment contemplates this identification. Yavana, or Yona,\nwas the common Indian name for the Gr\u00e6co-Bactrian kingdom and its\ndependencies[441]; and to this region we naturally turn. The Alasanda or\nAlasadda therefore, which is here mentioned, will be one of several\nEastern cities bearing the name of the great conqueror, most probably\n_Alexandria ad Caucasum_. But indeed I hardly think that, if Hilgenfeld\nhad referred to the original authority for the statement, the great\nBuddhist history _Mahawanso_, he would have ventured to lay any stress\nat all on this notice, as supporting his theory. [Sidenote: and wholly\nuntrustworthy in itself.]The historian, or rather fabulist (for such he\nis in this earlier part of his chronicle), is relating the foundation of\nthe Mah\u00e1 th\u00fapo, or great tope, at Ruanwelli by the king Dutthag\u00e1mini in\nthe year B.C. 157. Beyond the fact that this tope was erected by this\nking the rest is plainly legendary. All the materials for the\nconstruction of the building, we are told, appeared spontaneously as by\nmiracle\u2014the bricks, the metals, the precious stones. The dewos, or\ndemons, lent their aid in the erection. In fact\n\n                                  the fabric huge\n                      Rose like an exhalation.\n\nPriests gathered in enormous numbers from all the great Buddhist\nmonasteries to do honour to the festival of the foundation. One place\nalone sent not less than 96,000. Among the rest it is mentioned that\n'Maha Dhammarakkito, th\u00e9ro (_i.e._ senior priest) of Y\u00f3na, accompanied\nby 30,000 priests from the vicinity of Alasadd\u00e1, the capital of the Y\u00f3na\ncountry, attended[442].' It is obvious that no weight can be attached to\na statement occurring as part of a story of which the other details are\nso manifestly false. An establishment of 30,000 Buddhist priests at\nAlexandria would indeed be a phenomenon of which historians have shown a\nstrange neglect.\n\nFootnote 437:\n\n  _Zeitschrift_ X. p. 103 sq.; comp. XI. p. 351. M. Renan also (_Langues\n  S\u00e9mitiques_ III. iv. 1, _Vie de J\u00e9sus_ p. 98) suggests that Buddhist\n  influences operated in Palestine.\n\nFootnote 438:\n\n  X. p. 105 'was schon an sich, zumal in dieser Zeit, schwerlich\n  Alexandria ad Caucasum, sondern nur Alexandrien in Aegypten bedeuten\n  kann.' Comp. XI. p. 351, where he repeats the same argument in reply\n  to Zeller. This is a very natural inference from a western point of\n  view; but, when we place ourselves in the position of a Buddhist\n  writer to whom Bactria was Greece, the relative proportions of things\n  are wholly changed.\n\nFootnote 439:\n\n  _Die Religion des Buddha_ I. p. 193.\n\nFootnote 440:\n\n  Comp. e.g. Weber _Die Verbindungen Indiens mit den L\u00e4ndern im Westen_\n  p.675 in the _Allgem. Monatschr. f. Wissensch. u. Literatur_,\n  Braunschweig 1853; Lassen _Indische Alterthumskunde_ II. p. 236; Hardy\n  _Manual of Budhism_ p. 516.\n\nFootnote 441:\n\n  For its geographical meaning in older Indian writers see K\u00f6ppen _l.c._\n  Since then it has entirely departed from its original signification,\n  and Yavana is now a common term used by the Hindoos to designate the\n  Mohammedans. Thus the Greek name has come to be applied to a people\n  which of all others is most unlike the Greeks. This change of meaning\n  admirably illustrates the use of \u1f19\u03bb\u03bb\u03b7\u03bd among the Jews, which in like\n  manner, from being the name of an alien nation, became the name of an\n  alien religion, irrespective of nationality: see the note on Gal. ii.\n  3.\n\nFootnote 442:\n\n  _Mahawanso_ p. 171, Turnour's translation.\n\n[Sidenote: General ignorance of Buddhism in the West.]\n\nNor is the presence of any Buddhist establishment even on a much smaller\nscale in this important centre of western civilization at all\nreconcilable with the ignorance of this religion, which the Greeks and\nRomans betray at a much later date[443]. For some centuries after the\nChristian era we find that the information possessed by western writers\nwas most shadowy and confused; and in almost every instance we are able\nto trace it to some other cause than the actual presence of Buddhists in\nthe Roman Empire[444]. [Sidenote: Strabo.]Thus Strabo, who wrote under\nAugustus and Tiberius, apparently mentions the Buddhist priests, the\nsramanas, under the designation _sarman\u00e6_, (\u03a3\u03b1\u03c1\u03bc\u03ac\u03bd\u03b1\u03c2)[445]; but he\navowedly obtains his information from Megasthenes, who travelled in\nIndia somewhere about the year 300 B.C. and wrote a book on Indian\naffairs. [Sidenote: Bardesanes.]Thus too Bardesanes at a much later date\ngives an account of these Buddhist ascetics, without however naming the\nfounder of the religion; but he was indebted for his knowledge of them\nto conversations with certain Indian ambassadors who visited Syria on\ntheir way westward in the reign of one of the Antonines[446]. [Sidenote:\nClement of Alexandria.] Clement of Alexandria, writing in the latest\nyears of the second century or the earliest of the third, for the first\ntime[447] mentions Buddha by name; and even he betrays a strange\nignorance of this Eastern religion[448].\n\nFootnote 443:\n\n  How for instance, if any such establishment had ever existed at\n  Alexandria, could Strabo have used the language which is quoted in the\n  next note?\n\nFootnote 444:\n\n  Consistently with this view, we may allow that single Indians would\n  visit Alexandria from time to time for purposes of trade or for other\n  reasons, and not more than this is required by the rhetorical passage\n  in Dion Chrysost. Or. xxxii (p. 373) \u1f41\u03c1\u1ff6 \u03b3\u1f70\u03c1 \u1f14\u03b3\u03c9\u03b3\u03b5 \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f1d\u03bb\u03bb\u03b7\u03bd\u03b1\u03c2\n  \u03c0\u03b1\u03c1' \u1f51\u03bc\u1fd6\u03bd ... \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u0392\u03b1\u03ba\u03c4\u03c1\u03af\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u03a3\u03ba\u03cd\u03b8\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03a0\u03ad\u03c1\u03c3\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f38\u03bd\u03b4\u1ff6\u03bd\n  \u03c4\u03b9\u03bd\u03ac\u03c2. The qualifying \u03c4\u03b9\u03bd\u03b1\u03c2 shows how very slight was the\n  communication between India and Alexandria. The mission of Pant\u00e6nus\n  may have been suggested by the presence of such stray visitors. Jerome\n  (_Vir. Ill._ 36) says that he went 'rogatus ab illius gentis legatis.'\n  It must remain doubtful however, whether some other region than\n  Hindostan, such as \u00c6thiopia for instance, is not meant, when Pant\u00e6nus\n  is said to have gone to India: see Cave's _Lives of the Primitive\n  Fathers_ p. 188 sq.\n\n  How very slight the communication was between India and the West in\n  the early years of the Christian era, appears from this passage of\n  Strabo XV. 1. 4 (p. 686); \u03ba\u03b1\u1f76 \u03bf\u1f31 \u03bd\u1fe6\u03bd \u03b4\u1f72 \u1f10\u03be \u0391\u1f30\u03b3\u03cd\u03c0\u03c4\u03bf\u03c5 \u03c0\u03bb\u03ad\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bc\u03c0\u03bf\u03c1\u03b9\u03ba\u03bf\u1f76\n  \u03c4\u1ff7 \u039d\u03b5\u03af\u03bb\u1ff3 \u03ba\u03b1\u1f76 \u03c4\u1ff7 \u1f08\u03c1\u03b1\u03b2\u03af\u1ff3 \u03ba\u03cc\u03bb\u03c0\u1ff3 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u1fc6\u03c2 \u1f38\u03bd\u03b4\u03b9\u03ba\u1fc6\u03c2 \u03c3\u03c0\u03ac\u03bd\u03b9\u03bf\u03b9 \u03bc\u1f72\u03bd \u03ba\u03b1\u1f76\n  \u03c0\u03b5\u03c1\u03b9\u03c0\u03b5\u03c0\u03bb\u03b5\u03cd\u03ba\u03b1\u03c3\u03b9 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03bf\u1fe6 \u0393\u03ac\u03b3\u03b3\u03bf\u03c5, \u03ba\u03b1\u1f76 \u03bf\u1f57\u03c4\u03bf\u03b9 \u03b4' \u1f30\u03b4\u03b9\u1ff6\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03bf\u1f50\u03b4\u1f72\u03bd \u03c0\u03c1\u1f78\u03c2\n  \u1f31\u03c3\u03c4\u03bf\u03c1\u03af\u03b1\u03bd \u03c4\u1ff6\u03bd \u03c4\u03cc\u03c0\u03c9\u03bd \u03c7\u03c1\u1fc6\u03c3\u03b9\u03bc\u03bf\u03b9, after which he goes on to say that the\n  only instance of Indian travellers in the West was the embassy sent to\n  Augustus (see below p. 155), which came \u1f00\u03c6' \u1f11\u03bd\u1f78\u03c2 \u03c4\u03cc\u03c0\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1' \u1f11\u03bd\u1f78\u03c2\n  \u03b2\u03b1\u03c3\u03b9\u03bb\u03ad\u03c9\u03c2.\n\n  The communications between India and the West are investigated by two\n  recent writers, Reinaud _Relations Politiques et Commerciales de\n  l'Empire Romain avec l'Asie Centrale_, Paris 1863, and Priaulx _The\n  Indian Travels of Apollonius of Tyana and the Indian Embassies to\n  Rome_, 1873. The latter work, which is very thorough and satisfactory,\n  would have saved me much labour of independent investigation, if I had\n  seen it in time.\n\nFootnote 445:\n\n  Strabo XV. 1. 59, p. 712. In the MSS it is written \u0393\u03b1\u03c1\u03bc\u03ac\u03bd\u03b1\u03c2, but this\n  must be an error either introduced by Strabo's transcribers or found\n  in the copy of Megasthenes which this author used. This is plain not\n  only from the Indian word itself, but also from the parallel passage\n  in Clement of Alexandria (_Strom._ i. 15). From the coincidences of\n  language it is clear that Clement also derived his information from\n  Megasthenes, whose name he mentions just below. The fragments of\n  Megasthenes relating to the Indian philosophers will be found in\n  M\u00fcller _Fragm. Hist. Gr\u00e6c._ II. p. 437. They were previously edited by\n  Schwanbeck, _Megasthenis Indica_ (Bonn\u00e6 1846).\n\n  For \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9 we also find the form \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9 in other writers; e.g.\n  Clem. Alex. l.c., Bardesanes in Porphyr. _de Abstin._ iv. 17, Orig.\n  _c. Cels._ i. 19 (I. p. 342). This divergence is explained by the fact\n  that the Pali word _sammana_ corresponds to the Sanskrit _sramana_.\n  See Schwanbeck, l.c. p. 17, quoted by M\u00fcller p. 437.\n\n  It should be borne in mind however, that several eminent Indian\n  scholars believe Megasthenes to have meant not Buddhists but Brahmins\n  by his \u03a3\u03b1\u03c1\u03bc\u03ac\u03bd\u03b1\u03c2. So for instance Lassen _Rhein. Mus._ 1833, p. 180\n  sq., _Ind. Alterth._ II. p. 700: and Prof. Max M\u00fcller (Pref. to\n  Rogers's _Translation of Buddhaghosha's Parables_, London 1870, p.\n  lii) says; 'That Lassen is right in taking the \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9, mentioned by\n  Megasthenes, for Brahmanic, not for Buddhist ascetics, might be proved\n  also by their dress. Dresses made of the bark of trees are not\n  Buddhistic.' If this opinion be correct, the earlier notices of\n  Buddhism in Greek writers entirely disappear, and my position is\n  strengthened. But for the following reasons the other view appears to\n  me more probable: (1) The term _sramana_ is the common term for the\n  Buddhist ascetic, whereas it is very seldom used of the Brahmin.\n\n  (2) The \u0396\u03ac\u03c1\u03bc\u03b1\u03bd\u03bf\u03c2 (another form of _sramana_), mentioned below p. 156,\n  note 450, appears to have been a Buddhist. This view is taken even by\n  Lassen, _Ind. Alterth._ III. p. 60.\n\n  (3) The distinction of \u0392\u03c1\u03b1\u03c7\u03bc\u1fb6\u03bd\u03b5\u03c2 and \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9 in Megasthenes or the\n  writers following him corresponds to the distinction of \u0392\u03c1\u03b1\u03c7\u03bc\u1fb6\u03bd\u03b5\u03c2 and\n  \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9 in Bardesanes, Origen, and others; and, as Schwanbeck has\n  shown (l.c.), the account of the \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9 in Megasthenes for the most\n  part is a close parallel to the account of the \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9 in Bardesanes\n  (or at least in Porphyry's report of Bardesanes). It seems more\n  probable therefore that Megasthenes has been guilty of confusion in\n  describing the dress of the \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9, than that Brahmins are intended\n  by the term.\n\n  The Pali form, \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9, as a designation of the Buddhists, first\n  occurs in Clement of Alexandria or Bardesanes, whichever may be the\n  earlier writer. It is generally ascribed to Alexander Polyhistor, who\n  flourished B.C. 80\u201360, because his authority is quoted by Cyril of\n  Alexandria (c. _Julian_. iv. p. 133) in the same context in which the\n  \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9 are mentioned. This inference is drawn by Schwanbeck, Max\n  M\u00fcller, Lassen, and others. An examination of Cyril's language however\n  shows that the statement for which he quotes the authority of\n  Alexander Polyhistor does not extend to the mention of the Saman\u00e6i.\n  Indeed all the facts given in this passage of Cyril (including the\n  reference to Polyhistor) are taken from Clement of Alexandria\n  (_Strom._ i. 15; see the next note), whose account Cyril has abridged.\n  It is possible indeed that Clement himself derived the statement from\n  Polyhistor, but nothing in Clement's own language points to this.\n\nFootnote 446:\n\n  The narrative of Bardesanes is given by Porphyry _de Abst._ iv. 17.\n  The Buddhist ascetics are there called \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9 (see the last note).\n  The work of Bardesanes, recounting his conversations with these Indian\n  ambassadors, is quoted again by Porphyry in a fragment preserved by\n  Stob\u00e6us _Ecl._ iii. 56 (p. 141). In this last passage the embassy is\n  said to have arrived \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u1f08\u03bd\u03c4\u03c9\u03bd\u03af\u03bd\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u1f10\u03be \u1f18\u03bc\u03b9\u03c3\u1ff6\u03bd, by\n  which, if the words be correct, must be meant Elagabalus (A.D.\n  218\u2013222), the spurious Antonine (see Hilgenfeld _Bardesanes_ p. 12\n  sq.). Other ancient authorities however place Bardesanes in the reign\n  of one of the older Antonines; and, as the context is somewhat\n  corrupt, we cannot feel quite certain about the date. Bardesanes gives\n  by far the most accurate account of the Buddhists to be found in any\n  ancient Greek writer; but even here the monstrous stories, which the\n  Indian ambassadors related to him, show how little trustworthy such\n  sources of information were.\n\nFootnote 447:\n\n  Except possibly Arrian, _Ind._ viii. 1, who mentions an ancient Indian\n  king, Budyas (\u0392\u03bf\u03c5\u03b4\u03cd\u03b1\u03c2) by name; but what he relates of him is quite\n  inconsistent with the history of Buddha, and probably some one else is\n  intended.\n\nFootnote 448:\n\n  In this passage (_Strom._ i. 15, p. 359) Clement apparently mentions\n  these same persons three times, supposing that he is describing three\n  different schools of Oriental philosophers. (1) He speaks of \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9\n  \u0392\u03ac\u03ba\u03c4\u03c1\u03c9\u03bd (comp. Cyrill. Alex. l.c.); (2) He distinguishes two classes\n  of Indian gymnosophists, whom he calls \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9 and \u0392\u03c1\u03b1\u03c7\u03bc\u1fb6\u03bd\u03b1\u03b9. These\n  are Buddhists and Brahmins respectively (see p. 153, note 445); (3) He\n  says afterwards \u03b5\u1f30\u03c3\u1f76 \u03b4\u1f72 \u03c4\u1ff6\u03bd \u1f38\u03bd\u03b4\u1ff6\u03bd \u03bf\u1f31 \u03c4\u03bf\u1fd6\u03c2 \u0392\u03bf\u1fe6\u03c4\u03c4\u03b1 \u03c0\u03b5\u03b9\u03b8\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9\n  \u03c0\u03b1\u03c1\u03b1\u03b3\u03b3\u03ad\u03bb\u03bc\u03b1\u03c3\u03b9\u03bd, \u1f43\u03bd \u03b4\u03b9' \u1f51\u03c0\u03b5\u03c1\u03b2\u03bf\u03bb\u1f74\u03bd \u03c3\u03b5\u03bc\u03bd\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03b5\u1f30\u03c2 [\u1f61\u03c2?] \u03b8\u03b5\u1f78\u03bd \u03c4\u03b5\u03c4\u03b9\u03bc\u03ae\u03ba\u03b1\u03c3\u03b9.\n  Schwanbeck indeed maintains that Clement here intends to describe the\n  same persons whom he has just mentioned as \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9; but this is not\n  the natural interpretation of his language, which must mean 'There are\n  also among the Indians those who obey the precepts of Buddha.'\n  Probably Schwanbeck is right in identifying the \u03a3\u03b1\u03c1\u03bc\u1fb6\u03bd\u03b1\u03b9 with the\n  Buddhist ascetics, but Clement appears not to have known this. In fact\n  he has obtained his information from different sources, and so\n  repeated himself without being aware of it. Where he got the first\n  fact it is impossible to say. The second, as we saw, was derived from\n  Megasthenes. The third, relating to Buddha, came, as we may\n  conjecture, either from Pant\u00e6nus (if indeed Hindostan is really meant\n  by the India of his missionary labours) or from some chance Indian\n  visitor at Alexandria.\n\n  In another passage (_Strom._ iii. 7, p. 539) Clement speaks of certain\n  Indian celibates and ascetics, who are called \u03a3\u03b5\u03bc\u03bd\u03bf\u03af. As he\n  distinguishes them from the gymnosophists, and mentions the pyramid as\n  a sacred building with them, the identification with the Buddhists can\n  hardly be doubted. Here therefore \u03a3\u03b5\u03bc\u03bd\u03bf\u03af is a Grecized form of\n  \u03a3\u03b1\u03bc\u03b1\u03bd\u03b1\u1fd6\u03bf\u03b9; and this modification of the word would occur naturally to\n  Clement, because \u03c3\u03b5\u03bc\u03bd\u03bf\u03af, \u03c3\u03b5\u03bc\u03bd\u03b5\u1fd6\u03bf\u03bd, were already used of the ascetic\n  life: e.g. Philo _de Vit. Cont._ 3 (p. 475 M) \u1f31\u03b5\u03c1\u1f78\u03bd \u1f43 \u03ba\u03b1\u03bb\u03b5\u1fd6\u03c4\u03b1\u03b9\n  \u03c3\u03b5\u03bc\u03bd\u03b5\u1fd6\u03bf\u03bd \u03ba\u03b1\u1f76 \u03bc\u03bf\u03bd\u03b1\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd \u1f10\u03bd \u1fa7 \u03bc\u03bf\u03bd\u03bf\u1fe6\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f70 \u03c4\u03bf\u1fe6 \u03c3\u03b5\u03bc\u03bd\u03bf\u1fe6 \u03b2\u03af\u03bf\u03c5 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03b1\n  \u03c4\u03b5\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9.\n\n[Sidenote: Hippolytus.]\n\nStill later than this, Hippolytus, while he gives a fairly intelligent,\nthough brief, account of the Brahmins[449], says not a word about the\nBuddhists, though, if he had been acquainted with their teaching, he\nwould assuredly have seen in them a fresh support to his theory of the\naffinity between Christian heresies and pre-existing heathen\nphilosophies. [Sidenote: A Buddhist at Athens.]With one doubtful\nexception\u2014an Indian fanatic attached to an embassy sent by king Porus to\nAugustus, who astonished the Greeks and Romans by burning himself alive\nat Athens[450]\u2013there is apparently no notice in either heathen or\nChristian writers, which points to the presence of a Buddhist within the\nlimits of the Roman Empire, till long after the Essenes had ceased to\nexist[451].\n\nFootnote 449:\n\n  _H\u00e6r._ i. 24.\n\nFootnote 450:\n\n  The chief authority is Nicolaus of Damascus in Strabo xv. i. 73 (p.\n  270). The incident is mentioned also in Dion Cass. liv. 9. Nicolaus\n  had met these ambassadors at Antioch, and gives an interesting account\n  of the motley company and their strange presents. This fanatic, who\n  was one of the number, immolated himself in the presence of an\n  astonished crowd, and perhaps of the emperor himself, at Athens. He\n  anointed himself and then leapt smiling on the pyre. The inscription\n  on his tomb was \u0396\u03b1\u03c1\u03bc\u03b1\u03bd\u03bf\u03c7\u03b7\u03b3\u1f70\u03c2 \u1f38\u03bd\u03b4\u1f78\u03c2 \u1f00\u03c0\u1f78 \u0392\u03b1\u03c1\u03b3\u03cc\u03c3\u03b7\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70 \u03c0\u03ac\u03c4\u03c1\u03b9\u03b1 \u1f38\u03bd\u03b4\u1ff6\u03bd\n  \u1f14\u03b8\u03b7 \u1f11\u03b1\u03c5\u03c4\u1f78\u03bd \u1f00\u03c0\u03b1\u03b8\u03b1\u03bd\u03b1\u03c4\u03af\u03c3\u03b1\u03c2 \u03ba\u03b5\u1fd6\u03c4\u03b1\u03b9. The tomb was visible at least as late\n  as the age of Plutarch, who recording the self-immolation of Calanus\n  before Alexander (_Vit. Alex._ 69) says, \u03c4\u03bf\u1fe6\u03c4\u03bf \u03c0\u03bf\u03bb\u03bb\u03bf\u1fd6\u03c2 \u1f14\u03c4\u03b5\u03c3\u03b9\u03bd \u1f55\u03c3\u03c4\u03b5\u03c1\u03bf\u03bd\n  \u1f04\u03bb\u03bb\u03bf\u03c2 \u1f38\u03bd\u03b4\u1f78\u03c2 \u1f10\u03bd \u1f08\u03b8\u03ae\u03bd\u03b1\u03b9\u03c2 \u039a\u03b1\u03af\u03c3\u03b1\u03c1\u03af \u03c3\u03c5\u03bd\u1f7c\u03bd \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5, \u03ba\u03b1\u1f76 \u03b4\u03b5\u03af\u03ba\u03bd\u03c5\u03c4\u03b1\u03b9 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03bd\u1fe6\u03bd\n  \u03c4\u1f78 \u03bc\u03bd\u03b7\u03bc\u03b5\u1fd6\u03bf\u03bd \u1f38\u03bd\u03b4\u03bf\u1fe6 \u03c0\u03c1\u03bf\u03c3\u03b1\u03b3\u03bf\u03c1\u03b5\u03c5\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd. Strabo also places the two\n  incidents in conjunction in another passage in which he refers to this\n  person, xv. 1. 4 (p. 686) \u1f41 \u03ba\u03b1\u03c4\u03b1\u03ba\u03b1\u03cd\u03c3\u03b1\u03c2 \u1f11\u03b1\u03c5\u03c4\u1f78\u03bd \u1f08\u03b8\u03ae\u03bd\u03b7\u03c3\u03b9 \u03c3\u03bf\u03c6\u03b9\u03c3\u03c4\u1f74\u03c2 \u1f38\u03bd\u03b4\u03cc\u03c2,\n  \u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1 \u03ba\u03b1\u1f76 \u1f41 \u039a\u03ac\u03bb\u03b1\u03bd\u03bf\u03c2 \u03ba.\u03c4.\u03bb.\n\n  The reasons for supposing this person to have been a Buddhist, rather\n  than a Brahmin, are: (1) The name \u0396\u03b1\u03c1\u03bc\u03b1\u03bd\u03bf\u03c7\u03b7\u03b3\u1f70\u03c2 (which appears with\n  some variations in the MSS of Strabo), being apparently the Indian\n  _sramanakarja_, i.e. 'teacher of the ascetics,' in other words, a\n  Buddhist priest; (2) The place Bargosa, i.e. Barygaza, where Buddhism\n  flourished in that age. See Priaulx p. 78 sq. In Dion Cassius it is\n  written \u0396\u03ac\u03c1\u03bc\u03b1\u03c1\u03bf\u03c2.\n\n  And have we not here an explanation of 1 Cor. xiii. 3, if \u1f35\u03bd\u03b1\n  \u03ba\u03b1\u03c5\u03b8\u03ae\u03c3\u03c9\u03bc\u03b1\u03b9 be the right reading? The passage, being written before the\n  fires of the Neronian persecution, requires explanation. Now it is\n  clear from Plutarch that the 'Tomb of the Indian' was one of the\n  sights shown to strangers at Athens: and the Apostle, who observed the\n  altar ==\u03b1\u03b3\u03bd\u03c9\u03f2\u03c4\u03c9\u03b9 \u03b8\u03b5\u03c9\u03b9==, was not likely to overlook the sepulchre\n  with the strange inscription ==\u03b5\u03b1\u03c5\u03c4\u03bf\u03bd \u03b1\u03c0\u03b1\u03b8\u03b1\u03bd\u03b1\u03c4\u03b9\u03f2\u03b1\u03f2 \u03ba\u03b5\u03b9\u03c4\u03b1\u03b9==. Indeed\n  the incident would probably be pressed on his notice in his\n  discussions with Stoics and Epicureans, and he would be forced to\n  declare himself as to the value of these Indian self-immolations, when\n  he preached the doctrine of self-sacrifice. We may well imagine\n  therefore that the fate of this poor Buddhist fanatic was present to\n  his mind when he penned the words \u03ba\u03b1\u1f76 \u1f10\u1f70\u03bd \u03c0\u03b1\u03c1\u03b1\u03b4\u1ff6 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03ac \u03bc\u03bf\u03c5 ...\n  \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u03b4\u1f72 \u03bc\u1f74 \u1f14\u03c7\u03c9, \u03bf\u1f50\u03b4\u1f72\u03bd \u1f60\u03c6\u03b5\u03bb\u03bf\u1fe6\u03bc\u03b1\u03b9. Indeed it would furnish an almost\n  equally good illustration of the text, whether we read \u1f35\u03bd \u03ba\u03b1\u03c5\u03b8\u03ae\u03c3\u03c9\u03bc\u03b1\u03b9\n  or \u1f35\u03bd\u03b1 \u03ba\u03b1\u03c5\u03c7\u03ae\u03c3\u03c9\u03bc\u03b1\u03b9. Dion Cassius (l.c.) suggests that the deed was done\n  \u1f51\u03c0\u1f78 \u03c6\u03b9\u03bb\u03bf\u03c4\u03b9\u03bc\u03af\u03b1\u03c2 or \u03b5\u1f30\u03c2 \u1f10\u03c0\u03af\u03b4\u03b5\u03b9\u03b6\u03b9\u03bd. How much attention these religious\n  suicides of the Indians attracted in the Apostolic age (doubtless\n  because the act of this Buddhist priest had brought the subject\n  vividly before men's minds in the West), we may infer from the speech\n  which Josephus puts in the mouth of Eleazar (_B.J._ vii. 8. 7),\n  \u03b2\u03bb\u03ad\u03c8\u03c9\u03bc\u03b5\u03bd \u03b5\u1f30\u03c2 \u1f38\u03bd\u03b4\u03bf\u1f7a\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u1f00\u03c3\u03ba\u03ad\u03b9\u03bd \u1f51\u03c0\u03b9\u03c3\u03c7\u03bd\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 ... \u03bf\u1f31 \u03b4\u1f72 ...\n  _\u03c0\u03c5\u03c1\u1f76 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c0\u03b1\u03c1\u03b1\u03b4\u03cc\u03bd\u03c4\u03b5\u03c2_, \u1f45\u03c0\u03c9\u03c2 \u03b4\u1f74 \u03ba\u03b1\u1f76 _\u03ba\u03b1\u03b8\u03b1\u03c1\u03c9\u03c4\u03ac\u03c4\u03b7\u03bd_\n  \u1f00\u03c0\u03bf\u03ba\u03c1\u03af\u03bd\u03c9\u03c3\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1f74\u03bd \u03c8\u03c5\u03c7\u03ae\u03bd, \u1f51\u03bc\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u03b5\u03bb\u03b5\u03c5\u03c4\u1ff6\u03c3\u03b9 ... \u1f06\u03c1' \u03bf\u1f56\u03bd \u03bf\u1f50\u03ba\n  \u03b1\u1f30\u03b4\u03bf\u03cd\u03bc\u03b5\u03b8\u03b1 \u03c7\u03b5\u1fd6\u03c1\u03bf\u03bd \u1f38\u03bd\u03b4\u1ff6\u03bd \u03c6\u03c1\u03bf\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2;\n\nFootnote 451:\n\n  In the reign of Claudius an embassy arrived from Taprobane (Ceylon);\n  and from these ambassadors Pliny derived his information regarding the\n  island, _N.H._ vi. 24. Respecting their religion however he says only\n  two words 'coli Herculem,' by whom probably Rama is meant (Priaulx p.\n  116). From this and other statements it appears that they were Tamils\n  and not Singalese, and thus belonged to the non-Buddhist part of the\n  island; see Priaulx p. 91 sq.\n\n[Sidenote: The alleged coincidences prove nothing.]\n\nAnd, if so, the coincidences must be very precise, before we are\njustified in attributing any peculiarities of Essenism to Buddhist\ninfluences. This however is far from being the case. They both exhibit a\nwell-organized monastic society: but the monasticism of the Buddhist\npriests, with its systematized mendicancy, has little [Sidenote:\nMonasticism.] in common with the monasticism of the Essene recluse,\nwhose life was largely spent in manual labour. [Sidenote:\nAsceticism.]They both enjoin celibacy, both prohibit the use of flesh\nand of wine, both abstain from the slaughter of animals. But, as we have\nalready seen, such resemblances prove nothing, for they may be explained\nby the independent development of the same religious principles. One\ncoincidence, and one only, is noticed by Hilgenfeld, which at first\nsight seems more striking and might suggest a historical connexion.\n[Sidenote: Four orders and four steps.]He observes that the four orders\nof the Essene community are derived from the four steps of Buddhism.\nAgainst this it might fairly be argued that such coincidences of numbers\nare often purely accidental, and that in the present instance there is\nno more reason for connecting the four steps of Buddhism with the four\norders of Essenism than there would be for connecting the ten precepts\nof Buddha with the Ten Commandments of Moses. But indeed a nearer\nexamination will show that the two have nothing whatever in common\nexcept the number. The four steps or paths of Buddhism are not four\ngrades of an external order, but four degrees of spiritual progress on\nthe way to nirvana or annihilation, the ultimate goal of the Buddhist's\nreligious aspirations. They are wholly unconnected with the Buddhist\nmonastic system, as an organization. A reference to the Buddhist notices\ncollected in Hardy's _Eastern Monachism_ (p. 280 sq.) will at once\ndispel any suspicion of a resemblance. A man may attain to the highest\nof these four stages of Buddhist illumination instantaneously. He does\nnot need to have passed through the lower grades, but may even be a\nlayman at the time. Some merit obtained in a previous state of existence\nmay raise him _per saltum_ to the elevation of a rahat, when all earthly\ndesires are crushed and no future birth stands between him and nirvana.\n[Sidenote: Buddhist influences seen first in Manicheism.]There remains\ntherefore no coincidence which would suggest any historical connexion\nbetween Essenism and Buddhism. Indeed it is not till some centuries\nlater, when Manicheism starts into being, that we find for the first\ntime any traces of the influence of Buddhism on the religions of the\nWest[452].\n\nFootnote 452:\n\n  Even its influence on Manicheism however is disputed in a learned\n  article in the _Home and Foreign Review_ III. p. 143 sq. (1863), by Mr\n  P. Le Page Renouf (see _Academy_ 1873, p. 399).\n\n------------------------------------------------------------------------\n\n                                   3.\n                       ESSENISM AND CHRISTIANITY.\n\n\n[Sidenote: The theory which explains Christianity as an outgrowth of\n           Essenism,]\n\nIt has become a common practice with a certain class of writers to\ncall Essenism to their aid in accounting for any distinctive features\nof Christianity, which they are unable to explain in any other way.\nWherever some external power is needed to solve a perplexity, here is\nthe _deus ex machina_ whose aid they most readily invoke. Constant\nrepetition is sure to produce its effect, and probably not a few\npersons, who want either the leisure or the opportunity to investigate\nthe subject for themselves, have a lurking suspicion that the Founder\nof Christianity may have been an Essene, or at all events that\nChristianity was largely indebted to Essenism for its doctrinal and\nethical teaching[453]. Indeed, when very confident and sweeping\nassertions are made, it is natural to presume that they rest on a\nsubstantial basis of fact. Thus for instance we are told by one writer\nthat Christianity is 'Essenism alloyed with foreign elements'[454]:\nwhile another, who however approaches the subject in a different\nspirit, says; 'It will hardly be doubted that our Saviour Himself\nbelonged to this holy brotherhood. This will especially be apparent,\nwhen we remember that _the whole Jewish community_ at the advent of\nChrist was divided into three parties, the Pharisees, the Sadducees,\nand the Essenes, and that _every Jew had to belong to one of these\nsects_. Jesus who in all things conformed to the Jewish law, and who\nwas holy, harmless, undefiled, and separate from sinners, would\ntherefore naturally associate Himself with that order of Judaism which\nwas most congenial to his nature'.[455][Sidenote: tested by facts.]I\npurpose testing these strong assertions by an appeal to facts.\n\nFootnote 453:\n\n  De Quincey's attempt to prove that the Essenes were actually\n  Christians (_Works_ VI p. 270 sq., IX p. 253 sq.), who used the\n  machinery of an esoteric society to inculcate their doctrines 'for\n  fear of the Jews,' is conceived in a wholly different spirit from the\n  theories of the writers mentioned in the text; but it is even more\n  untenable and does not deserve serious refutation.\n\nFootnote 454:\n\n  Gr\u00e4tz III p. 217.\n\nFootnote 455:\n\n  Ginsburg _Essenes_ p. 24.\n\n[Sidenote: Our Lord need not have belonged to any sect.]\n\nFor the statements involved in those words of the last extract which I\nhave underlined, no authority is given by the writer himself; nor have I\nbeen able to find confirmation of them in any quarter. On the contrary\nthe frequent allusions which we find to the vulgar herd, the \u03b9\u03b4\u03b9\u1ff6\u03c4\u03b1\u03b9,\nthe _\u0639am haarets_, who are distinguished from the disciples of the\nschools[456], suggest that a large proportion of the people was\nunattached to any sect. If it had been otherwise, we might reasonably\npresume that our Lord, as one who 'in all things conformed to the Jewish\nlaw,' would have preferred attaching Himself to the Pharisees who 'sat\nin Moses' seat' and whose precepts He recommended His disciples to\nobey[457], rather than to the Essenes who in one important respect at\nleast\u2014the repudiation of the temple sacrifices\u2014acted in flagrant\nviolation of the Mosaic ordinances.\n\nFootnote 456:\n\n  See above, p. 130.\n\nFootnote 457:\n\n  Matt. xxiii. 2, 3.\n\n[Sidenote: The argument from the silence of the New Testament answered.]\n\nThis preliminary barrier being removed, we are free to investigate the\nevidence for their presumed connexion. And here we are met first with a\nnegative argument, which obviously has great weight with many persons.\nWhy, it is asked, does Jesus, who so unsparingly denounces the vices and\nthe falsehoods of Pharisees and Sadducees, never once mention the\nEssenes by way of condemnation, or indeed mention them by name at all?\nWhy, except that He himself belonged to this sect and looked favourably\non their teaching? This question is best answered by another. How can we\nexplain the fact, that throughout the enormous mass of talmudical and\nearly rabbinical literature this sect is not once mentioned by name, and\nthat even the supposed allusions to them, which have been discovered for\nthe first time in the present century, turn out on investigation to be\nhypothetical and illusory? The difficulty is much greater in this latter\ninstance; but the answer is the same in both cases. The silence is\nexplained by the comparative insignificance of the sect, their small\nnumbers and their retired habits. Their settlements were far removed\nfrom the great centres of political and religious life. Their recluse\nhabits, as a rule, prevented them from interfering in the common\nbusiness of the world. Philo and Josephus have given prominence to them,\nbecause their ascetic practices invested them with the character of\nphilosophers and interested the Greeks and Romans in their history; but\nin the national life of the Jews they bore a very insignificant\npart[458]. If the Sadducees, who held the highest offices in the\nhierarchy, are only mentioned directly on three occasions in the\nGospels[459], it can be no surprise that the Essenes are not named at\nall.\n\nFootnote 458:\n\n  This fact is fully recognised by several recent writers, who will not\n  be suspected of any undue bias towards traditional views of Christian\n  history. Thus Lipsius writes (p. 190), 'In the general development of\n  Jewish life Essenism occupies a far more subordinate place than is\n  commonly ascribed to it.' And Keim expresses himself to the same\n  effect (I. p. 305). Derenbourg also, after using similar language,\n  adds this wise caution, 'In any case, in the present state of our\n  acquaintance with the Essenes, which is so imperfect and has no chance\n  of being extended, the greatest prudence is required of science, if\n  she prefers to be true rather than adventurous, if she has at heart\n  rather to enlighten than to surprise' (p. 461). Even Gr\u00e4tz in one\n  passage can write soberly on this subject: 'The Essenes had throughout\n  no influence on political movements, from which they held aloof as far\n  as possible' (III. p. 86).\n\nFootnote 459:\n\n  These are (1) Matt. iii. 7; (2) Matt. xvi. 1 sq.; (3) Matt. xxii. 23\n  sq., Mark xii. 18, Luke xx. 27.\n\n[Sidenote: The positive arguments for a connexion may be twofold.]\n\nAs no stress therefore can be laid on the argument for silence, any\nhypothesis of connexion between Essenism and Christianity must make good\nits claims by establishing one or both of these two points: _first_,\nthat there is direct historical evidence of close intercourse between\nthe two; and _secondly_, that the resemblances of doctrine and practice\nare so striking as to oblige, or at least to warrant, the belief in such\na connexion. If both these lines of argument fail, the case must be\nconsidered to have broken down.\n\n[Sidenote: 1. Absence of direct historical evidence of a connexion.]\n\n1. On the former point it must be premised that the Gospel narrative\ndoes not suggest any hint of a connexion. Indeed its general tenor is\ndirectly adverse to such a supposition. From first to last Jesus and his\ndisciples move about freely, taking part in the common business, even in\nthe common recreations, of Jewish life. The recluse ascetic brotherhood,\nwhich was gathered about the shores of the Dead Sea, does not once\nappear above the Evangelists' horizon. Of this close society, as such,\nthere is not the faintest indication. [Sidenote: Two individual cases\nalleged.] But two individuals have been singled out, as holding an\nimportant place either in the Evangelical narrative or in the Apostolic\nChurch, who, it is contended, form direct and personal links of\ncommunication with this sect. These are John the Baptist and James the\nLord's brother. The one is the forerunner of the Gospel, the first\nherald of the Kingdom; the other is the most prominent figure in the\nearly Church of Jerusalem.\n\n[Sidenote: (i) John the Baptist]\n\n(i) John the Baptist was an ascetic. His abode was the desert; his\nclothing was rough; his food was spare; he baptized his penitents.\nTherefore, it is argued, he was an Essene. Between the premisses and the\nconclusion however there is a broad gulf, which cannot very easily be\nbridged over. [Sidenote: not an Essene.]The solitary independent life,\nwhich John led, presents a type wholly different from the cenobitic\nestablishments of the Essenes, who had common property, common meals,\ncommon hours of labour and of prayer. It may even be questioned whether\nhis food of locusts would have been permitted by the Essenes, if they\nreally ate nothing which had life (\u1f14\u03bc\u03c8\u03c5\u03c7\u03bf\u03bd[460]). And again; his baptism\nas narrated by the Evangelists, and their lustrations as described in\nJosephus, have nothing in common except the use of water for a religious\npurpose. When therefore we are told confidently that 'his manner of life\nwas altogether after the Essene pattern[461],' and that 'he without\ndoubt baptized his converts into the Essene order,' we know what value\nto attach to this bold assertion. If positive statements are allowable,\nit would be more true to fact to say that he could not possibly have\nbeen an Essene. The rule of his life was _isolation_; the principle of\ntheirs, _community_[462].\n\nFootnote 460:\n\n  See above p. 86.\n\nFootnote 461:\n\n  Gr\u00e4tz III. p. 220.\n\nFootnote 462:\n\n  \u03c4\u1f78 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03b7\u03c4\u03b9\u03ba\u03cc\u03bd, Joseph. _B.J._ ii. 8. 3. See also Philo _Fragm._ 632\n  \u1f51\u03c0\u1f72\u03c1 \u03c4\u03bf\u1fe6 \u03ba\u03bf\u03b9\u03bd\u03c9\u03c6\u03b5\u03bb\u03bf\u1fe6\u03c2, and the context.\n\n[Sidenote: External resemblances to John in Banus,]\n\nIn this mode of life John was not singular. It would appear that not a\nfew devout Jews at this time retired from the world and buried\nthemselves in the wilderness, that they might devote themselves\nunmolested to ascetic discipline and religious meditation. One such\ninstance at all events we have in Banus the master of Josephus, with\nwhom the Jewish historian, when a youth, spent three years in the\ndesert. This anchorite was clothed in garments made of bark or of\nleaves; his food was the natural produce of the earth; he bathed day and\nnight in cold water for purposes of purification. To the careless\nobserver doubtless John and Banus would appear to be men of the same\nstamp. In their outward mode of life there was perhaps not very much\ndifference[463]. The consciousness of a divine mission, the gift of a\nprophetic insight, in John was the real and all-important distinction\nbetween the two. [Sidenote: who was not an Essene.]But here also the\nsame mistake is made; and we not uncommonly find Banus described as an\nEssene. It is not too much to say however, that the whole tenor of\nJosephus' narrative is opposed to this supposition[464]. He says that\nwhen sixteen years old he desired to acquire a knowledge of the three\nsects of the Jews before making his choice of one; that accordingly he\nwent through (\u03b4\u03b9\u1fc6\u03bb\u03b8\u03bf\u03bd) all the three at the cost of much rough\ndiscipline and toil; that he was not satisfied with the experience thus\ngained, and hearing of this Banus he attached himself to him as his\nzealous disciple (\u03b6\u03b7\u03bb\u03c9\u03c4\u1f74\u03c2 \u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b7\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6); that having remained three\nyears with him he returned to Jerusalem; and that then, being nineteen\nyears old, he gave in his adhesion to the sect of the Pharisees. Thus\nthere is no more reason for connecting this Banus with the Essenes than\nwith the Pharisees. The only natural interpretation of the narrative is\nthat he did not belong to any of the three sects, but represented a\ndistinct type of religious life, of which Josephus was anxious to gain\nexperience. And his hermit life seems to demand this solution, which the\nsequence of the narrative suggests.\n\nFootnote 463:\n\n  Ewald (VI. p. 649) regards this Banus as representing an extravagant\n  development of the school of John, and thus supplying a link between\n  the real teaching of the Baptist and the doctrine of the\n  Hemerobaptists professing to be derived from him.\n\nFootnote 464:\n\n  The passage is so important that I give it in full; Joseph. _Vit._ 2\n  \u03c0\u03b5\u03c1\u1f76 \u1f11\u03ba\u03ba\u03b1\u03af\u03b4\u03b5\u03ba\u03b1 \u03b4\u1f72 \u1f14\u03c4\u03b7 \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f10\u03b2\u03bf\u03c5\u03bb\u03ae\u03b8\u03b7\u03bd \u03c4\u1ff6\u03bd \u03c0\u03b1\u03c1' \u1f21\u03bc\u1fd6\u03bd \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03bd\n  \u1f10\u03bc\u03c0\u03b5\u03b9\u03c1\u03af\u03b1\u03bd \u03bb\u03b1\u03b2\u03b5\u1fd6\u03bd. \u03c4\u03c1\u03b5\u1fd6\u03c2 \u03b4' \u03b5\u1f30\u03c3\u1f76\u03bd \u03b1\u1f57\u03c4\u03b1\u03b9\u00b7 \u03a6\u03b1\u03c1\u03b9\u03c3\u03b1\u03af\u03c9\u03bd \u03bc\u1f72\u03bd \u1f21 \u03c0\u03c1\u03ce\u03c4\u03b7, \u03ba\u03b1\u1f76\n  \u03a3\u03b1\u03b4\u03b4\u03bf\u03c5\u03ba\u03b1\u03af\u03c9\u03bd \u1f21 \u03b4\u03b5\u03c5\u03c4\u03ad\u03c1\u03b1, \u03c4\u03c1\u03af\u03c4\u03b7 \u03b4\u1f72 \u1f21 \u1f18\u03c3\u03c3\u03b7\u03bd\u1ff6\u03bd, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03c0\u03bf\u03bb\u03bb\u03ac\u03ba\u03b9\u03c2 \u03b5\u1f34\u03c0\u03b1\u03bc\u03b5\u03bd.\n  \u03bf\u1f55\u03c4\u03c9\u03c2 \u03b3\u1f70\u03c1 \u1fa0\u03cc\u03bc\u03b7\u03bd \u03b1\u1f31\u03c1\u03ae\u03c3\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f74\u03bd \u1f00\u03c1\u03af\u03c3\u03c4\u03b7\u03bd, \u03b5\u1f30 \u03c0\u03ac\u03c3\u03b1\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bc\u03ac\u03b8\u03bf\u03b9\u03bc\u03b9.\n  \u03c3\u03ba\u03bb\u03b7\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae\u03c3\u03b1\u03c2 \u03b3\u03bf\u1fe6\u03bd \u1f10\u03bc\u03b1\u03c5\u03c4\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c0\u03bf\u03bb\u03bb\u1f70 \u03c0\u03bf\u03bd\u03b7\u03b8\u03b5\u1f76\u03c2 \u03c4\u1f70\u03c2 \u03c4\u03c1\u03b5\u1fd6\u03c2 \u03b4\u03b9\u1fc6\u03bb\u03b8\u03bf\u03bd. \u03ba\u03b1\u1f76\n  \u03bc\u03b7\u03b4\u1f72 \u03c4\u1f74\u03bd \u1f10\u03bd\u03c4\u03b5\u1fe6\u03b8\u03b5\u03bd \u1f10\u03bc\u03c0\u03b5\u03b9\u03c1\u03af\u03b1\u03bd \u1f31\u03ba\u03b1\u03bd\u1f74\u03bd \u1f10\u03bc\u03b1\u03c5\u03c4\u1ff7 \u03bd\u03bf\u03bc\u03af\u03c3\u03b1\u03c2 \u03b5\u1f36\u03bd\u03b1\u03b9, \u03c0\u03c5\u03b8\u03cc\u03bc\u03b5\u03bd\u03cc\u03c2\n  \u03c4\u03b9\u03bd\u03b1 \u0392\u03b1\u03bd\u03bf\u1fe6\u03bd \u1f44\u03bd\u03bf\u03bc\u03b1 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u1f10\u03c1\u03b7\u03bc\u03af\u03b1\u03bd \u03b4\u03b9\u03b1\u03c4\u03c1\u03af\u03b2\u03b5\u03b9\u03bd, \u1f10\u03c3\u03b8\u1fc6\u03c4\u03b9 \u03bc\u1f72\u03bd \u1f00\u03c0\u1f78 \u03b4\u03ad\u03bd\u03b4\u03c1\u03c9\u03bd\n  \u03c7\u03c1\u03ce\u03bc\u03b5\u03bd\u03bf\u03bd, \u03c4\u03c1\u03bf\u03c6\u1f74\u03bd \u03b4\u1f72 \u03c4\u1f74\u03bd \u03b1\u1f50\u03c4\u03bf\u03bc\u03ac\u03c4\u03c9\u03c2 \u03c6\u03c5\u03bf\u03bc\u03ad\u03bd\u03b7\u03bd \u03c0\u03c1\u03bf\u03c3\u03c6\u03b5\u03c1\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd, \u03c8\u03c5\u03c7\u03c1\u1ff7 \u03b4\u1f72\n  \u1f55\u03b4\u03b1\u03c4\u03b9 \u03c4\u1f74\u03bd \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03bd\u03cd\u03ba\u03c4\u03b1 \u03c0\u03bf\u03bb\u03bb\u03ac\u03ba\u03b9\u03c2 \u03bb\u03bf\u03c5\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u03c0\u03c1\u1f78\u03c2 \u1f01\u03b3\u03bd\u03b5\u03af\u03b1\u03bd,\n  \u03b6\u03b7\u03bb\u03c9\u03c4\u1f74\u03c2 \u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b7\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6. \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03c4\u03c1\u03af\u03c8\u03b1\u03c2 \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u1ff7 \u1f10\u03bd\u03b9\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u03c4\u03c1\u03b5\u1fd6\u03c2 \u03ba\u03b1\u1f76\n  \u03c4\u1f74\u03bd \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03bd \u03c4\u03b5\u03bb\u03b5\u03b9\u03ce\u03c3\u03b1\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03c0\u03cc\u03bb\u03b9\u03bd \u1f51\u03c0\u03ad\u03c3\u03c4\u03c1\u03b5\u03c6\u03bf\u03bd. \u1f10\u03bd\u03bd\u03b5\u03b1\u03ba\u03b1\u03af\u03b4\u03b5\u03ba\u03b1 \u03b4' \u1f14\u03c4\u03b7\n  \u1f14\u03c7\u03c9\u03bd \u1f20\u03c1\u03be\u03ac\u03bc\u03b7\u03bd \u03c4\u03b5 \u03c0\u03bf\u03bb\u03b9\u03c4\u03b5\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c4\u1fc7 \u03a6\u03b1\u03c1\u03b9\u03c3\u03b1\u03af\u03c9\u03bd \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03b9 \u03ba\u03b1\u03c4\u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u1ff6\u03bd \u03ba.\u03c4.\u03bb.\n\n[Sidenote: General result.]\n\nOf John himself therefore no traits are handed down which suggest that\nhe was a member of the Essene community. He was an ascetic, and the\nEssenes were ascetics; but this is plainly an inadequate basis for any\nsuch inference. Nor indeed is the relation of his asceticism to theirs a\nquestion of much moment for the matter in hand; since this was the very\npoint in which Christ's mode of life was so essentially different from\nJohn's as to provoke criticism and to point a contrast[465]. But the\nlater history of his real or supposed disciples has, or may seem to\nhave, some bearing on this investigation. [Sidenote: The\nHemerobaptists.]Towards the close of the first and the beginning of the\nsecond century we meet with a body of sectarians called in Greek\n_Hemerobaptists_[466], in Hebrew _Toble-shacharith_[467], 'day' or\n'morning bathers.' What were their relations to John the Baptist on the\none hand, and to the Essenes on the other? Owing to the scantiness of\nour information the whole subject is wrapped in obscurity, and any\nrestoration of their history must be more or less hypothetical; but it\nwill be possible at all events to suggest an account which is not\nimprobable in itself, and which does no violence to the extant notices\nof the sect.\n\nFootnote 465:\n\n  Matt. ix. 14 sq., xi. 17 sq., Mark ii. 18 sq., Luke v. 33, vii. 31 sq.\n\nFootnote 466:\n\n  The word \u1f21\u03bc\u03b5\u03c1\u03bf\u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03c4\u03b1\u1f76 is generally taken to mean 'daily-bathers,'\n  and this meaning is suggested by Apost. Const. vi. 6 \u03bf\u1f35\u03c4\u03b9\u03bd\u03b5\u03c2, \u03ba\u03b1\u03b8'\n  (\u03b5\u03ba\u03ac\u03c3\u03c4\u03b7\u03bd \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd \u1f10\u1f70\u03bd \u03bc\u1f74 \u03b2\u03b1\u03c0\u03c4\u03af\u03c3\u03c9\u03bd\u03c4\u03b1\u03b9, \u03bf\u1f50\u03ba \u1f10\u03c3\u03b8\u03af\u03bf\u03c5\u03c3\u03b9\u03bd, _ib._ 23 \u1f00\u03bd\u03c4\u1f76\n  \u03ba\u03b1\u03b8\u03b7\u03bc\u03b5\u03c1\u03b9\u03bd\u03bf\u1fe6 \u1f13\u03bd \u03bc\u03cc\u03bd\u03bf\u03bd \u03b4\u03bf\u1fe6\u03c2 \u03b2\u03ac\u03c0\u03c4\u03b9\u03c3\u03bc\u03b1, Epiphan. _H\u00e6r._ xvii. 1 (p. 37) \u03b5\u1f30\n  \u03bc\u03ae \u03c4\u03b9 \u1f04\u03c0\u03b1 \u03ba\u03b1\u03b8' \u1f11\u03ba\u03ac\u03c3\u03c4\u03b7\u03bd \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd \u03b2\u03b1\u03c0\u03c4\u03af\u03b6\u03bf\u03b9\u03c4\u03cc \u03c4\u03b9\u03c2 \u1f10\u03bd \u1f55\u03b4\u03b1\u03c4\u03b9. But, if the\n  word is intended as a translation of _Toble-shacharith_ 'morning\n  bathers,' as it seems to be, it must signify rather 'day-bathers'; and\n  this is more in accordance with the analogy of other compounds from\n  \u1f21\u03bc\u03ad\u03c1\u03b1, as \u1f21\u03bc\u03b5\u03c1\u03cc\u03b2\u03b9\u03bf\u03c2, \u1f21\u03bc\u03b5\u03c1\u03bf\u03b4\u03c1\u03cc\u03bc\u03bf\u03c2, \u1f21\u03bc\u03b5\u03c1\u03bf\u03c3\u03ba\u03cc\u03c0\u03bf\u03c2, etc.\n\n  Josephus (_B.J._ ii. 8. 5) represents the Essenes as bathing, not at\n  dawn, but at the fifth hour, just before their meal. This is hardly\n  consistent either with the name of the _Toble-shacharith_, or with the\n  Talmudical anecdote of them quoted above p. 132. Of Banus he reports\n  (_Vit._ 2) that he 'bathed often day and night in cold water.'\n\nFootnote 467:\n\n  See above p. 132.\n\n[Sidenote: (_a_) Their relation to John the Baptist.]\n\n(_a_) We must not hastily conclude, when we meet with certain persons at\nEphesus about the years A.D. 53, 54, who are described as 'knowing only\nthe baptism of John,' or as having been 'baptized unto John's\nbaptism[468],' that we have here some early representatives of the\nHemerobaptist sect. [Sidenote: John's disciples at Ephesus.]These were\nChristians, though imperfectly informed Christians. Of Apollos, who was\nmore fully instructed by Aquila and Priscilla, this is stated in the\nmost explicit terms[469]. Of the rest, who owed their fuller knowledge\nof the Gospel to St Paul, the same appears to be implied, though the\nlanguage is not free from ambiguity[470]. But these notices have an\nimportant bearing on our subject; for they show how profoundly the\neffect of John's preaching was felt in districts as remote as\nproconsular Asia, even after a lapse of a quarter of a century. With\nthese disciples it was the initial impulse towards Christianity; but to\nothers it represented a widely different form of belief and practice.\n[Sidenote: Professed followers at a later date.]The Gospel of St John\nwas written, according to all tradition, at Ephesus in the later years\nof the first century. Again and again the Evangelist impresses on his\nreaders, either directly by his own comments or indirectly by the course\nof the narrative, the transient and subordinate character of John's\nministry. He was not the light, says the Evangelist, but came to bear\nwitness of the light[471]. He was not the sun in the heavens: he was\nonly the waning lamp, which shines when kindled from without and burns\nitself away in shining. His light might well gladden the Jews while it\nlasted, but this was only 'for a season[472].' John himself lost no\nopportunity of bearing his testimony to the loftier claims of\nJesus[473]. From such notices it is plain that in the interval between\nthe preaching of St Paul and the Gospel of St John the memory of the\nBaptist at Ephesus had assumed a new attitude towards Christianity. His\nname is no longer the sign of imperfect appreciation, but the watchword\nof direct antagonism. John had been set up as a rival Messiah to Jesus.\nIn other words, this Gospel indicates the spread of Hemerobaptist\nprinciples, if not the presence of a Hemerobaptist community, in\nproconsular Asia, when it was written. In two respects these\nHemerobaptists distorted the facts of history. [Sidenote: The facts of\nhistory distorted by them.]They perverted John's teaching, and they\nmisrepresented his office. His baptism was no more a single rite, once\nperformed and initiating an amendment of life; it was a daily recurrence\natoning for sin and sanctifying the person[474]. He himself was no\nlonger the forerunner of the Messiah; he was the very Messiah[475].\n[Sidenote: Spread of Hemerobaptist principles.]In the latter half of the\nfirst century, it would seem, there was a great movement among large\nnumbers of the Jews in favour of frequent baptism, as the one\npurificatory rite essential to salvation. Of this superstition we have\nhad an instance already in the anchorite Banus to whom Josephus attached\nhimself as a disciple. Its presence in the western districts of Asia\nMinor is shown by a Sibylline poem, dating about A.D. 80, which I have\nalready had occasion to quote[476]. Some years earlier these sectarians\nare mentioned by name as opposing James the Lord's brother and the\nTwelve at Jerusalem[477]. Nor is there any reason for questioning their\nexistence as a sect in Palestine during the later years of the Apostolic\nage, though the source from which our information comes is legendary,\nand the story itself a fabrication. But when or how they first connected\nthemselves with the name of John the Baptist, and whether this\nassumption was made by all alike or only by one section of them, we do\nnot know. Such a connexion, however false to history, was obvious and\nnatural; nor would it be difficult to accumulate parallels to this false\nappropriation of an honoured name. Baptism was the fundamental article\nof their creed; and John was the Baptist of world-wide fame. [Sidenote:\nA wrong use made of John's name.]Nothing more than this was needed for\nthe choice of an eponym. From St John's Gospel it seems clear that this\nappropriation was already contemplated, if not completed, at Ephesus\nbefore the first century had drawn to a close. In the second century the\nassumption is recognised as a characteristic of these Hemerobaptists, or\nBaptists, as they are once called[478], alike by those who allow and\nthose who deny its justice[479]. Even in our age the name of 'John's\ndisciples' has been given, though wrongly given, to an obscure sect in\nBabylonia, the Mandeans, whose doctrine and practice have some\naffinities to the older sect, and of whom perhaps they are the\ncollateral, if not the direct, descendants[480].\n\nFootnote 468:\n\n  The former expression is used of Apollos, Acts xviii. 24; the latter\n  of 'certain disciples,' Acts xix. 1.\n\nFootnote 469:\n\n  This appears from the whole narrative, but is distinctly stated in\n  ver. 25, as correctly read, \u1f10\u03b4\u03af\u03b4\u03b1\u03c3\u03ba\u03b5\u03bd \u1f00\u03ba\u03c1\u03b9\u03b2\u1ff6\u03c2 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6, not\n  \u03c4\u03bf\u1fe6 \u03ba\u03c5\u03c1\u03af\u03bf\u03c5 as in the received text.\n\nFootnote 470:\n\n  The \u03c0\u03b9\u03c3\u03c4\u03b5\u1fe6\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 in xix. 1 is slightly ambiguous, and some expressions\n  in the passage might suggest the opposite: but \u03bc\u03b1\u03b8\u03b7\u03c4\u1f70\u03c2 seems decisive,\n  for the word would not be used absolutely except of Christian\n  disciples; comp. vi. 1, 2, 7, ix. 10, 19, 26, 38, and frequently.\n\nFootnote 471:\n\n  John i. 8.\n\nFootnote 472:\n\n  John v. 35 \u1f10\u03ba\u03b5\u1fd6\u03bd\u03bf\u03c2 \u1f27\u03bd \u1f41 \u03bb\u03cd\u03c7\u03bd\u03bf\u03c2 \u1f41 \u03ba\u03b1\u03b9\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c6\u03b1\u03af\u03bd\u03c9\u03bd \u03ba.\u03c4.\u03bb. The word\n  \u03ba\u03b1\u03af\u03b5\u03b9\u03bd is not only 'to burn,' but not unfrequently also 'to kindle, to\n  set on fire,' as e.g. Xen. _Anab._ iv. 4. 12 \u03bf\u1f31 \u1f04\u03bb\u03bb\u03bf\u03b9 \u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03bd\u03c4\u03b5\u03c2 \u03c0\u1fe6\u03c1\n  \u1f14\u03ba\u03b1\u03b9\u03bf\u03bd, so that \u1f41 \u03ba\u03b1\u03b9\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 may mean either 'which burns away' or\n  'which is lighted.' With the former meaning it would denote the\n  _transitoriness_, with the latter the _derivative character_, of\n  John's ministry. There seems no reason for excluding either idea here.\n  Thus the whole expression would mean 'the lamp which is kindled and\n  burns away, and (only so) gives light.' For an example of two verbs or\n  participles joined together, where the second describes a result\n  conditional upon the first, see 1 Pet. ii. 20 \u03b5\u1f30 \u1f01\u03bc\u03b1\u03c1\u03c4\u03ac\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76\n  \u03ba\u03bf\u03bb\u03b1\u03c6\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f51\u03c0\u03bf\u03bc\u03b5\u03bd\u03b5\u1fd6\u03c4\u03b5 ... \u03b5\u1f30 \u1f00\u03b3\u03b1\u03b8\u03bf\u03c0\u03bf\u03b9\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03ac\u03c3\u03c7\u03bf\u03bd\u03c4\u03b5\u03c2\n  \u1f51\u03c0\u03bf\u03bc\u03b5\u03bd\u03b5\u1fd6\u03c4\u03b5, 1 Thess. iv. 1 \u03c0\u1ff6\u03c2 \u03b4\u03b5\u1fd6 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03bd \u03ba\u03b1\u1f76 \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03bd \u0398\u03b5\u1ff7.\n\nFootnote 473:\n\n  See John i. 15\u201334, iii. 23\u201330, v. 33 sq.: comp. x. 41, 42. This aspect\n  of St John's Gospel has been brought out by Ewald _Jahrb. der Bibl.\n  Wissensch._ III. p. 156 sq.; see also _Geschichte_ VII. p. 152 sq.,\n  _die Johanneischen Schriften_ p. 13. There is perhaps an allusion to\n  these 'disciples of John' in 1 Joh. v. 6 \u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u1ff7 \u1f55\u03b4\u03b1\u03c4\u03b9 \u03bc\u03cc\u03bd\u03bf\u03bd, \u1f00\u03bb\u03bb'\n  \u1f10\u03bd \u03c4\u1ff7 \u1f55\u03b4\u03b1\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03c4\u1ff7 \u1f05\u03b9\u03bc\u03b1\u03c4\u03b9\u00b7 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c0\u03bd\u03b5\u1fe6\u03bc\u03b1 \u03ba.\u03c4.\u03bb.; comp. Acts i. 5,\n  xi. 16, xix. 4.\n\nFootnote 474:\n\n  _Apost. Const._ vi. 6; comp. \u00a7 23. See p. 162, note 2.\n\nFootnote 475:\n\n  _Clem. Recogn._ i. 54 'ex discipulis Johannis, qui ... magistrum suum\n  veluti Christum praedicarunt,' _ib._ \u00a7 60 'Ecce unus ex discipulis\n  Johannis adfirmabat Christum Johannem fuisse, et non Jesum; in tantum,\n  inquit, ut et ipse Jesus omnibus hominibus et prophetis majorem esse\n  pronuntiaverit Johannem etc.'; see also \u00a7 63.\n\nFootnote 476:\n\n  See above p. 96.\n\nFootnote 477:\n\n  _Clem. Recogn._ l.c. This portion of the Clementine Recognitions is\n  apparently taken from an older Judaizing romance, the _Ascents of\n  James_ (see _Galatians_ pp. 316, 349). Hegesippus also (in Euseb.\n  _H.E._ iv. 22) mentions the Hemerobaptists in his list of Jewish\n  sects; and it is not improbable that this list was given as an\n  introduction to his account of the labours and martyrdom of St James\n  (see Euseb. _H.E._ ii. 23). If so, it was probably derived from the\n  same source as the notice in the Recognitions.\n\nFootnote 478:\n\n  They are called Baptists by Justin Mart. _Dial._ 10, p. 307 A. He\n  mentions them among other Jewish sects, without however alluding to\n  John.\n\nFootnote 479:\n\n  By the author of the _Recognitions_ (l.c.) who denies the claim; and\n  by the author of the _Homilies_ (see below p. 166, note 482), who\n  allows it.\n\nFootnote 480:\n\n  These Mandeans are a rapidly diminishing sect living in the region\n  about the Tigris and the Euphrates, south of Bagdad. Our most exact\n  knowledge of them is derived from Petermann (_Herzog's\n  Real-Encyklop\u00e4die_ s. vv. Mend\u00e4er, Zabier, and _Deutsche Zeitschrift_\n  1854 p. 181 sq. 1856 p. 331 sq., 342 sq., 363 sq., 386 sq.) who has\n  had personal intercourse with them; and from Chwolson (_die Ssabier u.\n  der Ssabismus_ I. p. 100 sq.) who has investigated the Arabic\n  authorities for their earlier history. The names by which they are\n  known are (1) _Mendeans_, or more properly _Mandeans_, \u05de\u05e0\u05d3\u05d9\u05d9\u05d0\n  _Mand\u0101y\u0113_, contracted from \u05de\u05e0\u05d3\u05d0 \u05d3\u05d7\u05d9\u05d9\u05d0 _Mand\u0101 d\u0115ch\u0101y\u0113_ 'the word of\n  life.' This is their own name among themselves, and points to their\n  Gnostic pretentions. (2) _Sabeans, Tsabiyun_, possibly from the root\n  \u05e6\u05d1\u05e2 'to dip' on account of their frequent lustrations (Chwolson I. p.\n  110; but see _Galatians_ p. 312), though this is not the derivation of\n  the word which they themselves adopt, and other etymologies have found\n  favour with some recent writers (see Petermann _Herzog's Real-Encykl_.\n  Suppl. XVIII. p. 342 s.v. Zabier). This is the name by which they are\n  known in the Koran and in Arabic writers, and by which they call\n  themselves when speaking to others. (3) _Nasoreans_, \u05e0\u05e6\u05d5\u05e8\u05d9\u05d9\u05d0\n  _N\u0101ts\u014dr\u0101y\u0113_. This term is at present confined to those among them who\n  are distinguished in knowledge or in business. (4) 'Christians of St\n  John, or Disciples of St John' (i.e. the Baptist). This name is not\n  known among themselves, and was incorrectly given to them by European\n  travellers and missionaries. At the same time John the Baptist has a\n  very prominent place in their theological system, as the one true\n  prophet. On the other hand they are not Christians in any sense.\n\n  These Mandeans, the true Sabeans, must not be confused with the false\n  Sabeans, polytheists and star-worshippers, whose locality is Northern\n  Mesopotamia. Chwolson (I. p. 139 sq.) has shown that these last\n  adopted the name in the 9th century to escape persecution from the\n  Mohammedans, because in the Koran the Sabeans, as monotheists, are\n  ranged with the Jews and Christians, and viewed in a more favourable\n  light than polytheists. The name however has generally been applied in\n  modern times to the false rather than to the true Sabeans.\n\n[Sidenote: (_b_) Their relation to the Essenes.]\n\n(_b_) Of the connexion between this sect and John the Baptist we have\nbeen able to give a probable, though necessarily hypothetical account.\nBut when we attempt to determine its relation to the Essenes, we find\nourselves entangled in a hopeless mesh of perplexities. The notices are\nso confused, the affinities so subtle, the ramifications so numerous,\nthat it becomes a desperate task to distinguish and classify these\nabnormal Jewish and Judaizing heresies. [Sidenote: They were at first\ndistinct, if not antagonistic.] One fact however seems clear that,\nwhatever affinities they may have had originally, and whatever relations\nthey may have contracted afterwards with one another, the\nHemerobaptists, properly speaking, were not Essenes. The Sibylline poem\nwhich may be regarded as in some respects a Hemerobaptist manifesto\ncontains, as we saw, many traits inconsistent with pure Essenism[481].\nIn two several accounts, the memoirs of Hegesippus and the Apostolic\nConstitutions, the Hemerobaptists are expressly distinguished from the\nEssenes[482]. In an early production of Judaic Christianity, whose\nJudaism has a strong Essene tinge, the Clementine Homilies, they and\ntheir eponym are condemned in the strongest language. The system of\nsyzygies, or pairs of opposites, is a favourite doctrine of this work,\nand in these John stands contrasted to Jesus, as Simon Magus to Simon\nPeter, as the false to the true; for according to this author's\nphilosophy of history the manifestation of the false always precedes the\nmanifestation of the true[483]. And again, Epiphanius speaks of them as\nagreeing substantially in their doctrines, not with the Essenes, but\nwith the Scribes and Pharisees[484]. His authority on such a point may\nbe worth very little; but connected with other notices, it should not be\npassed over in silence. Yet, whatever may have been their differences,\nthe Hemerobaptists and the Essenes had one point of direct contact,\ntheir belief in the moral efficacy of lustrations. When the temple and\npolity were destroyed, the shock vibrated through the whole fabric of\nJudaism, loosening and breaking up existing societies, and preparing the\nway for new combinations. [Sidenote: But after the destruction of the\nTemple]More especially the cessation of the sacrificial rites must have\nproduced a profound effect equally on those who, like the Essenes, had\ncondemned them already, and on those who, as possibly was the case with\nthe Hemerobaptists, had hitherto remained true to the orthodox ritual.\n[Sidenote: there may have been a fusion.]One grave obstacle to friendly\novertures was thus removed; and a fusion, more or less complete, may\nhave been the consequence. At all events the relations of the Jewish\nsects must have been materially affected by this great national crisis,\nas indeed we know to have been the case. In the confusion which follows,\nit is impossible to attain any clear view of their history. At the\nbeginning of the second century however this pseudo-baptist movement\nreceived a fresh impulse from the pretended revelation of Elchesai,\nwhich came from the farther East[485]. Henceforth Elchesai is the\nprominent name in the history of those Jewish and Judaizing sects whose\nproper home is east of the Jordan[486], and who appear to have\nreproduced, with various modifications derived from Christian and\nHeathen sources, the Gnostic theology and the pseudo-baptist ritual of\ntheir Essene predecessors. It is still preserved in the records of the\nonly extant people who have any claim to be regarded as the religious\nheirs of the Essenes. Elchesai is regarded as the founder of the sect of\nMandeans[487].\n\nFootnote 481:\n\n  See p. 96 sq.\n\nFootnote 482:\n\n  Hegesipp. in Euseb. _H.E._ iv. 22, _Apost. Const._ vi. 6. So also the\n  Pseudo-Hieronymus in the _Indiculus de H\u00e6resibus_ (_Corp. H\u00e6res._ I.\n  p. 283, ed. Oehler).\n\nFootnote 483:\n\n  Clem. Hom. ii. 23 \u1f38\u03c9\u03ac\u03bd\u03bd\u03b7\u03c2 \u03c4\u03b9\u03c2 \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u1f21\u03bc\u03b5\u03c1\u03bf\u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03c4\u03ae\u03c2, \u1f43\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6\n  \u03ba\u03c5\u03c1\u03af\u03bf\u03c5 \u1f21\u03bc\u1ff6\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03b6\u03c5\u03b3\u03af\u03b1\u03c2 \u03bb\u03cc\u03b3\u03bf\u03bd \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u03c0\u03c1\u03cc\u03bf\u03b4\u03bf\u03c2. It is\n  then stated that, as Christ had twelve leading disciples, so John had\n  thirty. This, it is argued, was a providential dispensation\u2014the one\n  number represents the solar, the other the lunar period; and so they\n  illustrate another point in this writer's theory, that in the syzygies\n  the true and the false are the male and female principle respectively.\n  Among these 30 disciples he places Simon Magus. With this the doctrine\n  of the Mandeans stands in direct opposition. They too have their\n  syzygies, but John with them represents the true principle.\n\nFootnote 484:\n\n  _H\u00e6r._ xvii. 1 (p. 37) \u1f34\u03c3\u03b1 \u03c4\u1ff6\u03bd \u03b3\u03c1\u03b1\u03bc\u03bc\u03b1\u03c4\u03ad\u03c9\u03bd \u03ba\u03b1\u1f76 \u03a6\u03b1\u03c1\u03b9\u03c3\u03b1\u03af\u03c9\u03bd \u03c6\u03c1\u03bf\u03bd\u03bf\u1fe6\u03c3\u03b1. But\n  he adds that they resemble the Sadducees 'not only in the matter of\n  the resurrection of the dead, but also in their unbelief and in the\n  other points.'\n\nFootnote 485:\n\n  See _Galatians_ p. 311 sq. on this Book of Elchesai.\n\nFootnote 486:\n\n  See above p. 137.\n\nFootnote 487:\n\n  See Chwolson I. p. 112 sq., II. p. 543 sq. The Arabic writer En-Nedim,\n  who lived towards the close of the tenth century, says that the\n  founder of the Sabeans (i.e. Mandeans) was _El-chasaich_ (\u0627\u0644\u062d\u0633\u064a\u062d) the\n  doctrine of two coordinate principles, the male and female. This\n  notice, as far as it goes, agrees with the account of Elchesai or\n  Elxai in Hippolytus (_H\u00e6r._ ix. 13 sq.) and Epiphanius (_H\u00e6r._ xix. 1\n  sq.). But the derivation of the name Elchesai given by Epiphanius\n  (_H\u00e6r._ xix. 2) \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 \u03ba\u03b5\u03ba\u03b1\u03bb\u03c5\u03bc\u03bc\u03ad\u03bd\u03b7 (\u05d7\u05d9\u05dc \u05db\u05e1\u05d9) is different and\n  probably correct (see _Galatians_ p. 312).\n\n[Sidenote: (ii) James the Lord's Brother]\n\n(ii) But, if great weight has been attached to the supposed connexion of\nJohn the Baptist with the Essenes, the case of James the Lord's brother\nhas been alleged with still more confidence. Here, it is said, we have\nan indisputable Essene connected by the closest family ties with the\nFounder of Christianity. [Sidenote: invested with Essene\ncharacteristics.]James is reported to have been holy from his birth; to\nhave drunk no wine nor strong drink; to have eaten no flesh; to have\nallowed no razor to touch his head, no oil to anoint his body; to have\nabstained from using the bath; and lastly to have worn no wool, but only\nfine linen[488]. Here we have a description of Nazarite practices at\nleast and (must it not be granted) of Essene tendencies also.\n\nBut what is our authority for this description? The writer, from whom\nthe account is immediately taken, is the Jewish-Christian historian\nHegesippus, who flourished about A.D. 170. He cannot therefore have been\nan eye-witness of the facts which he relates. [Sidenote: But the account\ncomes from untrustworthy sources.]And his whole narrative betrays its\nlegendary character. Thus his account of James's death, which follows\nimmediately on this description, is highly improbable and melodramatic\nin itself, and directly contradicts the contemporary notice of Josephus\nin its main facts[489]. From whatever source therefore Hegesippus may\nhave derived his information, it is wholly untrustworthy. Nor can we\ndoubt that he was indebted to one of those romances with which the\nJudaizing Christians of Essene tendencies loved to gratify the natural\ncuriosity of their disciples respecting the first founders of the\nChurch[490]. In like manner Essene portraits are elsewhere preserved of\nthe Apostles Peter[491] and Matthew[492], which represent them as living\non a spare diet of herbs and berries. I believe also that I have\nelsewhere pointed out the true source of this description in Hegesippus,\nand that it is taken from the 'Ascents of James[493],' a Jud\u00e6o-Christian\nwork stamped, as we happen to know, with the most distinctive Essene\nfeatures[494]. But if we turn from these religious novels of Judaic\nChristianity to earlier and more trustworthy sources of information\u2014to\nthe Gospels or the Acts or the Epistles of St Paul\u2014we fail to discover\nthe faintest traces of Essenism in James. [Sidenote: No Essene features\nin the true portraits of James or of the earliest disciples.]'The\nhistorical James,' says a recent writer, 'shows Pharisaic but not Essene\nsympathies[495].' This is true of James, as it is true of the early\ndisciples in the mother Church of Jerusalem generally. The\ntemple-ritual, the daily-sacrifices, suggested no scruples to them. The\nonly distinction of meats, which they recognised, was the distinction of\nanimals clean and unclean as laid down by the Mosaic law. The only\nsacrificial victims, which they abhorred, were victims offered to idols.\nThey took their part in the religious offices, and mixed freely in the\ncommon life, of their fellow-Israelites, distinguished from them only in\nthis, that to their Hebrew inheritance they superadded the knowledge of\na higher truth and the joy of a better hope. It was altogether within\nthe sphere of orthodox Judaism that the Jewish element in the Christian\nbrotherhood found its scope. Essene peculiarities are the objects\nneither of sympathy nor of antipathy. In the history of the infant\nChurch for the first quarter of a century Essenism is as though it were\nnot.\n\nFootnote 488:\n\n  Hegesippus in Euseb. _H.E._ ii. 23.\n\nFootnote 489:\n\n  See _Galatians_ p. 348 sq.\n\nFootnote 490:\n\n  See _Galatians_ p. 311.\n\nFootnote 491:\n\n  _Clem. Hom._ xii. 6, where St Peter is made to say \u1f04\u03c1\u03c4\u1ff3 \u03bc\u03cc\u03bd\u1ff3 \u03ba\u03b1\u1f76\n  \u1f10\u03bb\u03b1\u03af\u03b1\u03b9\u03c2 \u03c7\u03c1\u1ff6\u03bc\u03b1\u03b9, \u03ba\u03b1\u1f76 \u03c3\u03c0\u03b1\u03bd\u03af\u03c9\u03c2 \u03bb\u03b1\u03c7\u03ac\u03bd\u03bf\u03b9\u03c2; comp. xv. 7 \u1f55\u03b4\u03b1\u03c4\u03bf\u03c2 \u03bc\u03cc\u03bd\u03bf\u03c5 \u03ba\u03b1\u1f76\n  \u1f04\u03c1\u03c4\u03bf\u03c5.\n\nFootnote 492:\n\n  Clem. Alex. _P\u00e6dag._ ii. 1 (p. 174) \u03c3\u03c0\u03b5\u03c1\u03bc\u03ac\u03c4\u03c9\u03bd \u03ba\u03b1\u1f76 \u1f00\u03ba\u03c1\u03bf\u03b4\u03c1\u03cd\u03c9\u03bd \u03ba\u03b1\u1f76\n  \u03bb\u03b1\u03c7\u03ac\u03bd\u03c9\u03bd \u1f04\u03bd\u03b5\u03c5 \u03ba\u03c1\u03b5\u1ff6\u03bd \u03bc\u03b5\u03c4\u03b5\u03bb\u03b1.\u03bc\u03b2\u03b1\u03bd\u03b5\u03bd.\n\nFootnote 493:\n\n  See _Galatians_ p. 349, note.\n\nFootnote 494:\n\n  Epiphanius (_H\u00e6r._ xxx. 16) mentions two points especially, in which\n  the character of this work is shown: (1) It represented James as\n  condemning the sacrifices and the fire on the altar (see above pp.\n  134\u2013136): (2) It published the most unfounded calumnies against St\n  Paul.\n\nFootnote 495:\n\n  Lipsius, _Schenkel's Bibel-Lexicon_, p. 191.\n\n[Sidenote: Essene influences visible before the close of the Apostolic\n           age.]\n\nBut a time came, when all this was changed. Even as early as the year\n58, when St Paul wrote to the Romans, we detect practices in the\nChristian community of the metropolis, which may possibly have been due\nto Essene influences[496]. Five or six years later, the heretical\nteaching which threatened the integrity of the Gospel at Coloss\u00e6 shows\nthat this type of Judaism was already strong enough within the Church to\nexert a dangerous influence on its doctrinal purity. Then came the great\nconvulsion\u2014the overthrow of the Jewish polity and nation. This was the\nturning-point in the relations between Essenism and Christianity, at\nleast in Palestine. [Sidenote: Consequences of the Jewish war.]The\nEssenes were extreme sufferers in the Roman war of extermination. It\nseems probable that their organization was entirely broken up. Thus cast\nadrift, they were free to enter into other combinations, while the shock\nof the recent catastrophe would naturally turn their thoughts into new\nchannels. At the same time the nearer proximity of the Christians, who\nhad migrated to Per\u00e6a during the war, would bring them into close\ncontact with the new faith and subject them to its influences, as they\nhad never been subjected before[497]. But, whatever may be the\nexplanation, the fact seems certain, that after the destruction of\nJerusalem the Christian body was largely reinforced from their ranks.\nThe Judaizing tendencies among the Hebrew Christians, which hitherto had\nbeen wholly Pharisaic, are henceforth largely Essene.\n\nFootnote 496:\n\n  Rom. xiv. 2, 21.\n\nFootnote 497:\n\n  See _Galatians_ p. 310 sq.\n\n[Sidenote: 2. Do the resemblances favour the theory of a connexion?]\n\n2. If then history fails to reveal any such external connexion with\nEssenism in Christ and His Apostles as to justify the opinion that\nEssene influences contributed largely to the characteristic features of\nthe Gospel, such a view, if tenable at all, must find its support in\nsome striking coincidence between the doctrines and practices of the\nEssenes and those which its Founder stamped upon Christianity. This\nindeed is the really important point; for without it the external\nconnexion, even if proved, would be valueless. The question is not\nwhether Christianity arose amid such and such circumstances, but how far\nit was created and moulded by those circumstances.\n\n[Sidenote: (i) Observance of the sabbath.]\n\n(i) Now one point which especially strikes us in the Jewish historian's\naccount of the Essenes, is their strict observance of certain points in\nthe Mosaic ceremonial law, more especially the ultra-Pharisaic rigour\nwith which they kept the sabbath. How far their conduct in this respect\nwas consistent with the teaching and practice of Christ may be seen from\nthe passages quoted in the parallel columns which follow:\n\n'Jesus went on the sabbath-day through the corn fields; and his\ndisciples began to pluck the ears of corn and to eat[498].... But when\nthe Pharisees saw it, they said unto him, 'Behold, thy disciples do that\nwhich it is not lawful to do upon the sabbath-day. But he said unto\nthem, Have ye not read what David did.... The sabbath was made for man,\nand not man for the sabbath. Therefore the Son of Man is Lord even of\nthe sabbath-day....'\n\n'It is lawful to do well on the sabbath-days' (Matt. xii. 1\u201312; Mark ii.\n23.-iii. 6; Luke vi. 1\u201311, xiv. 1\u20136. See also a similar incident in Luke\nxiii. 10\u201317). 'The Jews therefore said unto him that was cured; It is\nthe sabbath-day; it is not lawful for thee to carry thy bed. But he\nanswered them, He that made me whole, the same said unto me, Take up thy\nbed and walk.... Therefore the Jews did persecute Jesus and sought to\nslay him, because he did these things on the sabbath-day. But Jesus\nanswered them, My Father worketh hitherto, and I work, etc. (John v.\n10\u201318; comp. vii. 22, 23).' 'And it was the sabbath-day when Jesus made\nthe clay, and opened his eyes.... Therefore said some of the Pharisees,\nThis man is not of God, because he keepeth not the sabbath-day (John ix.\n14, 16).'\n\n'And they avoid ... touching any work (\u1f10\u03c6\u03ac\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 \u1f14\u03c1\u03b3\u03c9\u03bd) on the\nsabbath-day more scrupulously than any of the Jews (\u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ce\u03c4\u03b1\u03c4\u03b1 \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd\n\u1f01\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd); for they do not venture so much as to move a vessel[499], nor\nto perform the most necessary offices of life (_B.J._ ii. 8. 9).'\n\nFootnote 498:\n\n  Gr\u00e4tz (III. p. 233) considers this narrative an interpolation\n  made from a Pauline point of view ('eine paulinistische\n  Tendenz-interpolation'). This theory of interpolation,\n  interposing wherever the evidence is unfavourable, cuts up all\n  argument by the roots. In this instance however Gr\u00e4tz is\n  consistently carrying out a principle, which he broadly lays\n  down elsewhere. He regards it as the great merit of Baur and his\n  school, that they explained the origin of the Gospels by the\n  conflict of two opposing camps, the Ebionite and the Pauline.\n  'By this master-key,' he adds, 'criticism was first put in a\n  position to test what is historical in the Gospels, and what\n  bears the stamp of a polemical tendency (was einen tendenti\u00f6sen\n  polemischen Charakter hat). Indeed by this means the element of\n  trustworthy history in the Gospels melts down to a minimum'\n  (III. p. 224). In other words the judgment is not to be\n  pronounced upon the evidence, but the evidence must be mutilated\n  to suit the judgment. The method is not new. The sectarians of\n  the second century, whether Judaic or anti-Judaic, had severally\n  their 'master-key.' The master-key of Marcion was a conflict\n  also\u2014the antagonism of the Old and New Testaments. Under his\n  hands the historical element in the New Testament dissolved\n  rapidly. The master-key of the anti-Marcionite writer of the\n  Clementine Homilies was likewise a conflict, though of another\n  kind\u2014the conflict of fire and water, of the sacrificial and the\n  baptismal systems. Wherever sacrifice was mentioned with\n  approval, there was a 'Tendenz-interpolation' (see above p.\n  136). In this manner again the genuine element in the Old\n  Testament melted down to a minimum.\n\nFootnote 499:\n\n  Gr\u00e4tz however (III. p. 228) sees a coincidence between Christ's\n  teaching and Essenism in this notice. Not to do him injustice, I will\n  translate his own words (correcting however several misprints in the\n  Greek): 'For the connexion of Jesus with the Essenes compare moreover\n  Mark xi. 16 \u03ba\u03b1\u1f76 \u03bf\u1f50\u03ba \u1f24\u03c6\u03b9\u03b5\u03bd \u1f41 \u1f38\u03b7\u03c3\u03bf\u1fe6\u03c2 \u1f35\u03bd\u03b1 \u03c4\u03b9\u03c2 \u03b4\u03b9\u03b5\u03bd\u03ad\u03b3\u03ba\u1fc3 \u03c3\u03ba\u03b5\u1fe6\u03bf\u03c2 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6\n  \u1f31\u03b5\u03c1\u03bf\u1fe6 with Josephus _B.J._ ii. 8. 9 \u1f00\u03bb\u03bb' \u03bf\u1f50\u03b4\u1f72 \u03c3\u03ba\u03b5\u1fe6\u03cc\u03c2 \u03c4\u03b9 \u03bc\u03b5\u03c4\u03b1\u03ba\u03b9\u03bd\u1fc6\u03c3\u03b1\u03b9\n  \u03b8\u03b1\u03c1\u03c1\u03bf\u1fe6\u03c3\u03b9\u03bd \u03bf\u1f35 \u1f18\u03c3\u03c3\u03b1\u1fd6\u03bf\u1f30.' He does not explain what this notice, which\n  refers solely to the scrupulous observance of the sabbath, has to do\n  with the profanation of the temple, with which the passage in the\n  Gospel is alone concerned. I have seen Gr\u00e4tz's history described as a\n  'masterly' work. The first requisites in a historian are accuracy in\n  stating facts and sobriety in drawing inferences. Without these, it is\n  difficult to see what claims a history can have to this honourable\n  epithet: and in those portions of his work, which I have consulted, I\n  have not found either.\n\n[Sidenote: (ii) Lustrations and other ceremonial observances.]\n\n(ii) But there were other points of ceremonial observance, in which the\nEssenes superadded to the law. Of these the most remarkable was their\npractice of constant lustrations. In this respect the Pharisee was\nsufficiently minute and scrupulous in his observances; but with the\nEssene these ablutions were the predominant feature of his religious\nritual. Here again it will be instructive to compare the practice of\nChrist and His disciples with the practice of the Essenes.\n\n'And when they saw some of his disciples eat bread with defiled (that is\nto say, unwashen) hands; for the Pharisees and all the Jews, except they\nwash their hands oft (\u03c0\u03c5\u03b3\u03bc\u1fc7), eat not...The Pharisees and scribes asked\nhim, Why walk not thy disciples according to the tradition of the\nelders.... But he answered ... Ye hypocrites, laying aside the\ncommandment of God, ye hold the tradition of men....'\n\n'Not that which goeth into the mouth defileth the man; but that which\ncometh out of the mouth, this defileth the man.... Let them alone, they\nbe blind leaders of the blind....'\n\n'To eat with unwashen hands defileth not the man (Matt. xv. 1\u201320, Mark\nvii. 1\u201323).'\n\n'So they wash their whole body (\u1f00\u03c0\u03bf\u03bb\u03bf\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1) in cold water; and\nafter this purification (\u1f01\u03b3\u03bd\u03b5\u03af\u03b1\u03bd) ... being clean (\u03ba\u03b1\u03b8\u03b1\u03c1\u03bf\u1f76) they come to\nthe refectory (to dine).... And when they have returned (from their\nday's work) they sup in like manner (_B.J._ ii. 8. 5).'\n\n'After a year's probation (the novice) is admitted to closer intercourse\n(\u03c0\u03c1\u03cc\u03c3\u03b5\u03b9\u03c3\u03b9\u03bd \u1f14\u03b3\u03b3\u03b9\u03bf\u03bd \u03c4\u1fc7 \u03b4\u03b9\u03b1\u03af\u03c4\u1fc3), and the lustral waters in which he\nparticipates have a higher degree of purity (\u03ba\u03b1\u1f76 \u03ba\u03b1\u03b8\u03b1\u03c1\u03c9\u03c4\u03ad\u03c1\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c0\u03c1\u1f78\u03c2\n\u1f01\u03b3\u03bd\u03b5\u03af\u03b1\u03bd \u1f51\u03b4\u03ac\u03c4\u03c9\u03bd \u03bc\u03b5\u03c4\u03b1\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9, \u00a7 7).'\n\n'It is a custom to wash after it, as if polluted by it (\u00a7 9).'\n\n'And when the Pharisee saw it, he marvelled that he had not first washed\nbefore dinner (\u03c4\u03bf\u1fe6 \u1f00\u03c1\u03af\u03c3\u03c4\u03bf\u03c5). And the Lord said unto him: Now do ye\nPharisees make clean the outside of the cup and the platter.... Ye fools\n... behold all things are clean unto you (Luke xi. 38\u201341).'\n\n'Racked and dislocated, burnt and crushed, and subjected to every\ninstrument of torture ... to make them eat strange food (\u03c4\u03b9 \u03c4\u1ff6\u03bd\n\u1f00\u03c3\u03c5\u03bd\u03ae\u03b8\u03c9\u03bd) ... they were not induced to submit (\u00a7 10).'\n\n'Exercising themselves in ... divers lustrations (\u03b4\u03b9\u03b1\u03c6\u03cc\u03c1\u03bf\u03b9\u03c2 \u1f01\u03b3\u03bd\u03b5\u03af\u03b1\u03b9\u03c2 ...\n\u1f10\u03bc\u03c0\u03b1\u03b9\u03b4\u03bf\u03c4\u03c1\u03b9\u03b2\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9, \u00a7 12).'\n\n[Sidenote: Avoidance of strangers.]\n\nConnected with this idea of external purity is the avoidance of contact\nwith strangers, as persons who would communicate ceremonial defilement.\nAnd here too the Essene went much beyond the Pharisee. The Pharisee\navoided Gentiles or aliens, or those whose profession or character\nplaced them in the category of 'sinners'; but the Essene shrunk even\nfrom the probationers and inferior grades of his own exclusive\ncommunity. Here again we may profitably compare the sayings and doings\nof Christ with the principles of this sect.\n\n'And when the scribes and Pharisees saw him eat with the publicans and\nsinners they said unto the disciples, Why eateth your Master with the\npublicans and the sinners....' (Mark ii. 15 sq.; Matth. ix. 10 sq., Luke\nv. 30 sq.)\n\n'They say ... a friend of publicans and sinners (Matth. xi. 19).'\n\n'The Pharisees and the scribes murmured, saying, This man receiveth\nsinners and eateth with them (Luke xv. 2).'\n\n'They all murmured saying that he was gone to be a guest with a man that\nis a sinner (Luke xix. 7).'\n\n'Behold, a woman in the city that was a sinner ... began to wash his\nfeet with her tears, and did wipe them with the hairs of her head and\nkissed his feet.... Now when the Pharisee which had bidden him saw it,\nhe spake within himself saying, This man, if he had been a prophet,\nwould have known who and what manner of woman this is that toucheth him;\nfor she is a sinner (Luke vii. 37 sq.).'\n\n'And after this purification they assemble in a private room, where no\nperson of a different belief (\u03c4\u1ff6\u03bd \u1f11\u03c4\u03b5\u03c1\u03bf\u03b4\u03cc\u03be\u03c9\u03bd, i.e. not an Essene) is\npermitted to enter; and (so) being by themselves and clean (\u03b1\u1f50\u03c4\u03bf\u1f76\n\u03ba\u03b1\u03b8\u03b1\u03c1\u03bf\u1f76) they present themselves at the refectory (\u03b4\u03b5\u03b9\u03c0\u03bd\u03b7\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd), as if\nit were a sacred precinct (\u00a7 5).'\n\n'And they are divided into four grades according to the time passed\nunder the discipline: and the juniors are regarded as so far inferior to\nthe seniors, that, if they touch them, the latter wash their bodies\nclean (\u1f00\u03c0\u03bf\u03bb\u03bf\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9), as if they had come in contact with a foreigner\n(\u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1 \u1f00\u03bb\u03bb\u03bf\u03c6\u03cd\u03bb\u1ff3 \u03c3\u03c5\u03bc\u03c6\u03c5\u03c1\u03ad\u03bd\u03c4\u03b1\u03c2, \u00a7 10).'\n\nIn all these minute scruples relating to ceremonial observances, the\ndenunciations which are hurled against the Pharisees in the Gospels\nwould apply with tenfold force to the Essenes.\n\n[Sidenote: (iii) Asceticism.]\n\n(iii) If the lustrations of the Essenes far outstripped the enactments\nof the Mosaic law, so also did their asceticism. I have given reasons\nabove for believing that this asceticism was founded on a false\nprinciple, which postulates the malignity of matter and is wholly\ninconsistent with the teaching of the Gospel[500]. But without pressing\nthis point, of which no absolutely demonstrative proof can be given, it\nwill be sufficient to call attention to the trenchant contrast in\npractice which Essene habits present to the life of Christ. He who 'came\neating and drinking' and was denounced in consequence as 'a glutton and\na wine-bibber'[501], [Sidenote: Eating and drinking.]He whose first\nexercise of power is recorded to have been the multiplication of wine at\na festive entertainment, and whose last meal was attended with the\ndrinking of wine and the eating of flesh, could only have excited the\npity, if not the indignation, of these rigid abstainers. And again,\nattention should be directed to another kind of abstinence, where the\ncontrast is all the more speaking, because the matter is so trivial and\nthe scruple so minute.\n\nFootnote 500:\n\n  See above p. 87.\n\nFootnote 501:\n\n  Matt. xi. 19, Luke vii. 34.\n\n'My head with oil thou didst not anoint (Luke vii. 46).'\n\n'Thou, when thou fastest, anoint thy head (Matt. vi. 17).'\n\n'And they consider oil a pollution (\u03ba\u03b7\u03bb\u1fd6\u03b4\u03b1), and though one is smeared\ninvoluntarily, he rubs his body clean (\u03c3\u03bc\u03ae\u03c7\u03b5\u03c4\u03b1\u03b9 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1, \u00a7 3).'\n\nAnd yet it has been stated that 'the Saviour of the world ... showed\nwhat is required for a holy life in the Sermon on the Mount by a\ndescription of the Essenes[502].'\n\nFootnote 502:\n\n  Ginsburg _Essenes_ p. 14.\n\n[Sidenote: Celibacy.]\n\nBut much stress has been laid on the celibacy of the Essenes; and our\nLord's saying in Matt. xix. 12 is quoted to establish an identity of\ndoctrine. Yet there is nothing special in the language there used. Nor\nis there any close affinity between the stern invectives against\nmarriage which Josephus and Philo attribute to the Essene, and the\ngentle concession 'He that is able to receive it, let him receive it.'\nThe best comment on our Lord's meaning here is the advice of St\nPaul[503], who was educated not in the Essene, but in the Pharisaic\nschool. Moreover this saying must be balanced by the general tenour of\nthe Gospel narrative. When we find Christ discussing the relations of\nman and wife, gracing the marriage festival by His presence, again and\nagain employing wedding banquets and wedded life as apt symbols of the\nhighest theological truths, without a word of disparagement or rebuke,\nwe see plainly that we are confronted with a spirit very different from\nthe narrow rigour of the Essenes.\n\nFootnote 503:\n\n  1 Cor. vii. 26\u201331.\n\n[Sidenote: (iv) Avoidance of the Temple sacrifices.]\n\n(iv) But not only where the Essenes superadded to the ceremonial law,\ndoes their teaching present a direct contrast to the phenomena of the\nGospel narrative. The same is true also of those points in which they\nfell short of the Mosaic enactments. I have already discussed at some\nlength the Essene abstention from the temple sacrifices[504]. There can,\nI think, be little doubt that they objected to the slaughter of\nsacrificial victims altogether. But for my present purpose it matters\nnothing whether they avoided the temple on account of the sacrifices, or\nthe sacrifices on account of the temple. Christ did neither. Certainly\nHe could not have regarded the temple as unholy; for his whole time\nduring his sojourns at Jerusalem was spent within its precincts. It was\nthe scene of His miracles, of His ministrations, of His daily\nteaching[505]. And in like manner it is the common rendezvous of His\ndisciples after Him[506]. Nor again does He evince any abhorrence of the\nsacrifices. On the contrary He says that the altar consecrates the\ngifts[507]; He charges the cleansed lepers to go and fulfil the Mosaic\nordinance and offer the sacrificial offerings to the priests[508].\n[Sidenote: Practice of Christ and His disciples.]And His practice also\nis conformable to His teaching. He comes to Jerusalem regularly to\nattend the great festivals, where sacrifices formed the most striking\npart of the ceremonial, and He himself enjoins preparation to be made\nfor the sacrifice of the Paschal lamb. If He repeats the inspired\nwarning of the older prophets, that mercy is better than sacrifice[509],\nthis very qualification shows approval of the practice in itself. Nor is\nHis silence less eloquent than His utterances or His actions. Throughout\nthe Gospels there is not one word which can be construed as condemning\nthe sacrificial system or as implying a desire for its cessation until\neverything is fulfilled.\n\nFootnote 504:\n\n  See p. 134 sq.\n\nFootnote 505:\n\n  Matt. xxi. 12 sq., 23 sq., xxiv. 1 sq., xxvi. 55, Mark xi. 11, 15 sq.,\n  27, xii. 35, xiii. 1 sq., xiv. 49, Luke ii. 46, xix. 45, xx. 1 sq.,\n  xxi. 37 sq., xxii. 53, John ii. 14 sq., v. 14, vii. 14, viii. 2, 20,\n  59, x. 23, xi. 56, xviii. 20.\n\nFootnote 506:\n\n  Luke xxiv. 53, Acts ii. 46, iii. 1 sq., v. 20 sq., 42.\n\nFootnote 507:\n\n  Matt. xxiii. 18 sq.: comp. v. 23, 24.\n\nFootnote 508:\n\n  Matt. viii. 4, Mark i. 44, Luke v. 14.\n\nFootnote 509:\n\n  Matt. ix. 13, xii. 7.\n\n[Sidenote: (v) Denial of the resurrection of the body.]\n\n(v) This last contrast refers to the ceremonial law. But not wide is the\ndivergence on an important point of doctrine. The resurrection of the\nbody is a fundamental article in the belief of the early disciples. This\nwas distinctly denied by the Essenes[510]. However gross and sensuous\nmay have been the conceptions of the Pharisees on this point, still they\nso far agreed with the teaching of Christianity, as against the Essenes,\nin that the risen man could not, as they held, be pure soul or spirit,\nbut must necessarily be body and soul conjoint.\n\nFootnote 510:\n\n  See above p. 88.\n\n[Sidenote: Some supposed coincidences considered.]\n\nThus at whatever point we test the teaching and practice of our Lord by\nthe characteristic tenets of Essenism, the theory of affinity fails.\nThere are indeed several coincidences on which much stress has been\nlaid, but they cannot be placed in the category of distinctive features.\nThey are either exemplifications of a higher morality, which may indeed\nhave been honourably illustrated in the Essenes, but is in no sense\nconfined to them, being the natural outgrowth of the moral sense of\nmankind whenever circumstances are favourable. Or they are more special,\nbut still independent developments, which owe their similarity to the\nsame influences of climate and soil, though they do not spring from the\nsame root. To this latter class belong such manifestations as are due to\nthe social conditions of the age or nation, whether they result from\nsympathy with, or from repulsion to, those conditions.\n\n[Sidenote: Simplicity and brotherly love.]\n\nThus, for instance, much stress has been laid on the aversion to war and\nwarlike pursuits, on the simplicity of living, and on the feeling of\nbrotherhood which distinguished Christians and Essenes alike. But what\nis gained by all this? It is quite plain that Christ would have approved\nwhatever was pure and lovely in the morality of the Essenes, just as He\napproved whatever was true in the doctrine of the Pharisees, if any\noccasion had presented itself when His approval was called for. But it\nis the merest assumption to postulate direct obligation on such grounds.\nIt is said however, that the moral resemblances are more particular than\nthis. [Sidenote: Prohibition of oaths.]There is for instance Christ's\nprecept 'Swear not at all ... but let your communication be Yea, yea,\nNay, nay.' Have we not here, it is urged, the very counterpart to the\nEssene prohibition of oaths[511]? Yet it would surely be quite as\nreasonable to say that both alike enforce that simplicity and\ntruthfulness in conversation which is its own credential and does not\nrequire the support of adjuration, both having the same reason for\nlaying stress on this duty, because the leaders of religious opinion\nmade artificial distinctions between oath and oath, as regards their\nbinding force, and thus sapped the foundations of public and private\nhonesty[512]. And indeed this avoidance of oaths is anything but a\nspecial badge of the Essenes. It was inculcated by Pythagoreans, by\nStoics, by philosophers and moralists of all schools[513]. When Josephus\nand Philo called the attention of Greeks and Romans to this feature in\nthe Essenes, they were simply asking them to admire in these practical\nphilosophers among the 'barbarians' the realisation of an ideal which\ntheir own great men had laid down. Even within the circles of Pharisaism\nlanguage is occasionally heard, which meets the Essene principle\nhalf-way[514].\n\nFootnote 511:\n\n  Jos. _B.J._ ii. 8. 6 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u1fe5\u03b7\u03b8\u1f72\u03bd \u1f51\u03c0' \u03b1\u1f50\u03c4\u1ff6\u03bd \u1f30\u03c3\u03c7\u03c5\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u1f45\u03c1\u03ba\u03bf\u03c5\u00b7 \u03c4\u1f78 \u03b4\u1f72\n  \u1f40\u03bc\u03bd\u03cd\u03b5\u03b9\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c0\u03b5\u03c1\u03b9\u0390\u03c3\u03c4\u03b1\u03c4\u03b1\u03b9, \u03c7\u03b5\u1fd6\u03c1\u03cc\u03bd \u03c4\u03b9 \u03c4\u1fc6\u03c2 \u1f10\u03c0\u03b9\u03bf\u03c1\u03ba\u03af\u03b1\u03c2 \u1f51\u03c0\u03bf\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2\u00b7\n  \u1f24\u03b4\u03b7 \u03b3\u1f70\u03c1 \u03ba\u03b1\u03c4\u03b5\u03b3\u03bd\u1ff6\u03c3\u03b8\u03b1\u03af \u03c6\u03b1\u03c3\u03b9 \u03c4\u1f78\u03bd \u1f00\u03c0\u03b9\u03c3\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03b4\u03af\u03c7\u03b1 \u03b8\u03b5\u03bf\u1fe6, Philo _Omn. prob.\n  lib._ 12 (II. p. 458) \u03c4\u03bf\u1fe6 \u03c6\u03b9\u03bb\u03bf\u03b8\u03ad\u03bf\u03c5 \u03b4\u03b5\u03af\u03b3\u03bc\u03b1\u03c4\u03b1 \u03c0\u03b1\u03c1\u03ad\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9 \u03bc\u03c5\u03c1\u03af\u03b1 ... \u03c4\u1f78\n  \u1f00\u03bd\u03ce\u03bc\u03bf\u03c4\u03bf\u03bd \u03ba.\u03c4.\u03bb. Accordingly Josephus relates (_Ant._ xv. 10. 4) that\n  Herod the Great excused the Essenes from taking the oath of allegiance\n  to him. Yet they were not altogether true to their principles; for\n  Josephus says (_B.J._ ii. 8. 7), that on initiation into the sect the\n  members were bound by fearful oaths (\u1f45\u03c1\u03ba\u03bf\u03c5\u03c2 \u03c6\u03c1\u03b9\u03ba\u03ce\u03b4\u03b5\u03b9\u03c2) to fulfil\n  certain conditions; and he twice again in the same passage mentions\n  oaths (\u1f40\u03bc\u03bd\u03cd\u03bf\u03c5\u03c3\u03b9, \u03c4\u03bf\u03b9\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u1f45\u03c1\u03ba\u03bf\u03b9\u03c2) in this connexion.\n\nFootnote 512:\n\n  On the distinctions which the Jewish doctors made between the validity\n  of different kinds of oaths, see the passages quoted in Lightfoot and\n  Sch\u00f6ttgen on Matt. v. 33 sq. The Talmudical tract _Shebhuoth_ tells\n  its own tale, and is the best comment on the precepts in the Sermon on\n  the Mount.\n\nFootnote 513:\n\n  See e.g. the passages in Wetstein on Matt. v. 37.\n\nFootnote 514:\n\n  _Baba Metsia_ 49 _a_. See also Lightfoot on Matt. v. 34.\n\n[Sidenote: Community of goods.]\n\nAnd again; attention has been called to the community of goods in the\ninfant Church of Christ, as though this were a legacy of Essenism. But\nhere too the reasonable explanation is, that we have an independent\nattempt to realise the idea of brotherhood\u2014an attempt which naturally\nsuggested itself without any direct imitation, but which was soon\nabandoned under the pressure of circumstances. Indeed the communism of\nthe Christians was from the first wholly unlike the communism of the\nEssenes. The surrender of property with the Christians was not a\nnecessary condition of entrance into an order; it was a purely voluntary\nact, which might be withheld without foregoing the privileges of the\nbrotherhood[515]. And the common life too was obviously different in\nkind, at once more free and more sociable, unfettered by rigid\nordinances, respecting individual liberty, and altogether unlike a\nmonastic rule.\n\nFootnote 515:\n\n  514: Acts v. 4.\n\nNot less irrelevant is the stress, which has been laid on another point\nof supposed coincidence in the social doctrines of the two communities.\n[Sidenote: Prohibition of slavery.]The prohibition of slavery was indeed\na highly honourable feature in the Essene order[516], but it affords no\nindication of a direct connexion with Christianity. It is true that this\nsocial institution of antiquity was not less antagonistic to the spirit\nof the Gospel, than it was abhorrent to the feelings of the Essene; and\nultimately the influence of Christianity has triumphed over it. But the\nimmediate treatment of the question was altogether different in the two\ncases. The Essene brothers proscribed slavery wholly; they produced no\nappreciable results by the proscription. The Christian Apostles, without\nattempting an immediate and violent revolution in society, proclaimed\nthe great principle that all men are equal in Christ, and left it to\nwork. It did work, like leaven, silently but surely, till the whole lump\nwas leavened. In the matter of slavery the resemblance to the Stoic is\nmuch closer than to the Essene[517]. The Stoic however began and ended\nin barren declamation, and no practical fruits were reaped from his\ndoctrine.\n\nFootnote 516:\n\n  Philo _Omn. prob. lib._ \u00a7 12 (II. p. 458) \u03b4\u03bf\u1fe6\u03bb\u03cc\u03c2 \u03c4\u03b5 \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f30\u03b4\u1f72\n  \u03b5\u1f36\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u03bb\u03bb' \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03b9 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03ba.\u03c4.\u03bb., _Fragm._ II. p. 632 \u03bf\u1f50\u03ba\n  \u1f00\u03bd\u03b4\u03c1\u03ac\u03c0\u03bf\u03b4\u03bf\u03bd, Jos. _Ant._ xviii. I. 5 \u03bf\u1f54\u03c4\u03b5 \u03b4\u03bf\u03cd\u03bb\u03c9\u03bd \u1f10\u03c0\u03b9\u03c4\u03b7\u03b4\u03b5\u03cd\u03bf\u03c5\u03c3\u03b9 \u03ba\u03c4\u1fc6\u03c3\u03b9\u03bd.\n\nFootnote 517:\n\n  See for instance the passages from Seneca quoted in _Philippians_ p.\n  305.\n\n[Sidenote: Respect paid to poverty.]\n\nMoreover prominence has been given to the fact, that riches are decried,\nand a preference is given to the poor, in the teaching of our Lord and\nHis Apostles. Here again, it is urged, we have a distinctly Essene\nfeature. We need not stop to enquire with what limitations this\nprerogative of poverty, which appears in the Gospels, must be\ninterpreted; but, quite independently of this question, we may fairly\ndecline to lay any stress on such a coincidence, where all other\nindications of a direct connexion have failed. The Essenes, pursuing a\nsimple and ascetic life, made it their chief aim to reduce their\nmaterial wants as far as possible, and in doing so they necessarily\nexalted poverty. Ascetic philosophers in Greece and Rome had done the\nsame. Christianity was entrusted with the mission of proclaiming the\nequal rights of all men before God, of setting a truer standard of human\nworth than the outward conventions of the world, of protesting against\nthe tyranny of the strong and the luxury of the rich, of redressing\nsocial inequalities, if not always by a present compensation, at least\nby a future hope. The needy and oppressed were the special charge of its\npreachers. It was the characteristic feature of the 'Kingdom of Heaven,'\nas described by the prophet whose words gave the keynote to the\nMessianic hopes of the nation, that the glad-tidings should be preached\nto the poor[518]. The exaltation of poverty therefore was an absolute\ncondition of the Gospel.\n\nFootnote 518:\n\n  Is. lxi. I. \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 \u03c0\u03c4\u03c9\u03c7\u03bf\u1fd6\u03c2, quoted in Luke iv. 18. There are\n  references to this particular part of the prophecy again in Matt. xi.\n  5, Luke vii. 22, and probably also in the beatitude \u03bc\u03b1\u03ba\u03ac\u03c1\u03b9\u03bf\u03b9 \u03bf\u1f31 \u03c0\u03c4\u03c9\u03c7\u03bf\u03af\n  \u03ba.\u03c4.\u03bb., Matt. v. 3, Luke vi. 20.\n\n[Sidenote: The preaching of the Kingdom wrongly ascribed to the\n           Essenes.]\n\nThe mention of the kingdom of heaven leads to the last point on which it\nwill be necessary to touch before leaving this subject. 'The whole\nascetic life of the Essenes,' it has been said, 'aimed only at\nfurthering the _Kingdom of Heaven_ and the _Coming Age_.' Thus John the\nBaptist was the proper representative of this sect. 'From the Essenes\nwent forth the first call that the Messiah must shortly appear, _The\nkingdom of heaven is at_ _hand_'[519]. 'The announcement of the kingdom\nof heaven unquestionably went forth from the Essenes'[520]. For this\nconfident assertion there is absolutely no foundation in fact; and, as a\nconjectural hypothesis, the assumption is highly improbable.\n\nFootnote 519:\n\n  Gr\u00e4tz _Gesch._ III. p. 219.\n\nFootnote 520:\n\n  _ib._ p. 470.\n\n[Sidenote: The Essenes not prophets, but fortune-tellers.]\n\nAs fortune-tellers or soothsayers, the Essenes might be called prophets;\nbut as preachers of righteousness, as heralds of the kingdom, they had\nno claim to the title. Throughout the notices in Josephus and Philo we\ncannot trace the faintest indication of Messianic hopes. Nor indeed was\ntheir position at all likely to foster such hopes[521]. The Messianic\nidea was built on a belief in the resurrection of the body. The Essenes\nentirely denied this doctrine. The Messianic idea was intimately bound\nup with the national hopes and sufferings, with the national life, of\nthe Jews. The Essenes had no interest in the Jewish polity; they\nseparated themselves almost entirely from public affairs. [Sidenote:\nThey had no vivid Messianic expectations.]The deliverance of the\nindividual is the shipwreck of the whole, it has been well said, was the\nplain watchword of Essenism[522]. How entirely the conception of a\nMessiah might be obliterated, where Judaism was regarded only from the\nside of a mystic philosophy, we see from the case of Philo. Throughout\nthe works of this voluminous writer only one or two faint and doubtful\nallusions to a personal Messiah are found[523]. The philosophical tenets\nof the Essenes no doubt differed widely from those of Philo; but in the\nsubstitution of the individual and contemplative aspect of religion for\nthe national and practical they were united; and the effect in obscuring\nthe Messianic idea would be the same. When therefore it is said that the\nprominence given to the proclamation of the Messiah's kingdom is a main\nlink which connects Essenism and Christianity, we may dismiss the\nstatement as a mere hypothesis, unsupported by evidence and improbable\nin itself.\n\nFootnote 521:\n\n  Lipsius Schenkel's _Bibel-Lexikon_ s.v. Ess\u00e4er p. 190, Keim _Jesus von\n  Nazara_ I. p. 305. Both these writers express themselves very\n  decidedly against the view maintained by Gr\u00e4tz. 'The Essene art of\n  soothsaying,' writes Lipsius, 'has absolutely nothing to do with the\n  Messianic prophecy. 'Of all this,' says Keim,'there is no trace.'\n\nFootnote 522:\n\n  Keim _l.c._\n\nFootnote 523:\n\n  How little can be made out of Philo's Messianic utterances by one who\n  is anxious to make the most possible out of them, may be seen from\n  Gfr\u00f6rer's treatment of the subject, Philo I. p. 486 sq. The treatises\n  which bear on this topic are the _de Pr\u00e6miis et P\u0153nis_ (I. p. 408,\n  ed. Mangey) and the _de Execrationibus_ (I. p. 429). They deserve to\n  be read, if only for the negative results which they yield.\n\n------------------------------------------------------------------------\n\n\n\n\n                                  III.\n                 CHARACTER AND CONTENTS OF THE EPISTLE.\n\n\n[Sidenote: The understanding of the heresy necessary.]\n\nWithout the preceding investigation the teaching of this epistle would\nbe very imperfectly understood; for its direction was necessarily\ndetermined by the occasion which gave rise to it. Only when we have once\ngrasped the nature of the doctrine which St Paul is combating, do we\nperceive that every sentence is instinct with life and meaning.\n\n[Sidenote: The errors though twofold sprang from one root.]\n\nWe have seen that the error of the heretical teachers was twofold. They\nhad a false conception in theology, and they had a false basis of\nmorals. It has been pointed out also, that these two were closely\nconnected together, and had their root in the same fundamental error,\nthe idea of matter as the abode of evil and thus antagonistic to God.\n\n[Sidenote: So the answer to both is in the same truth.]\n\nAs the two elements of the heretical doctrine were derived from the same\nsource, so the reply to both was sought by the Apostle in the same idea,\nthe conception of the Person of Christ as the one absolute mediator\nbetween God and man, the true and only reconciler of heaven and earth.\n\nBut though they are thus ultimately connected, yet it will be necessary\nfor the fuller understanding of St Paul's position to take them apart,\nand to consider first the theological and then the ethical teaching of\nthe epistle.\n\n[Sidenote: 1. The _theological_ teaching of the heretics.]\n\n1. This Colossian heresy was no coarse and vulgar development of\nfalsehood. It soared far above the Pharisaic Judaism which St Paul\nrefutes in the Epistle to the Galatians. The questions in which it was\ninterested lie at the very root of our religious consciousness.\n[Sidenote: Its lofty motive,]The impulse was given to its speculations\nby an overwhelming sense of the unapproachable majesty of God, by an\ninstinctive recognition of the chasm which separates God from man, from\nthe world, from matter. Its energy was sustained by the intense yearning\nafter some mediation which might bridge over this chasm, might establish\ninter-communion between the finite and the Infinite. Up to this point it\nwas deeply religious in the best sense of the term.\n\n[Sidenote: but complete failure.]\n\nThe answer which it gave to these questions we have already seen. In two\nrespects this answer failed signally. On the one hand it was drawn from\nthe atmosphere of mystical speculation. It had no foundation in history,\nand made no appeal to experience. On the other hand, notwithstanding its\ncomplexity, it was unsatisfactory in its results; for in this plurality\nof mediators none was competent to meet the requirements of the case.\nGod here and man there\u2014no angel or spirit, whether one or more, being\nneither God nor man, could truly reconcile the two. Thus as regards\ncredentials it was without a guarantee; while as regards efficiency it\nwas wholly inadequate.\n\n[Sidenote: The Apostle's answer is in the Person of Christ.]\n\nThe Apostle pointed out to the Colossians a more excellent way. It was\nthe one purpose of Christianity to satisfy those very yearnings which\nwere working in their hearts, to solve that very problem which had\nexercised their minds. In Christ they would find the answer which they\nsought. His life\u2014His cross and resurrection\u2014was the guarantee;\n[Sidenote: The mediator in the world and in the Church.]His Person\u2014the\nWord Incarnate\u2014was the solution. He alone filled up, He alone could fill\nup, the void which lay between God and man, could span the gulf which\nseparated the Creator and creation. This solution offered by the Gospel\nis as simple as it is adequate. To their cosmical speculations, and to\ntheir religious yearnings alike, Jesus Christ is the true answer. In the\nWorld, as in the Church, He is the one only mediator, the one only\nreconciler. This two-fold idea runs like a double thread through the\nfabric of the Apostle's teaching in those passages of the epistle where\nhe is describing the Person of Christ.\n\nIt will be convenient for the better understanding of St Paul's teaching\nto consider these two aspects of Christ's mediation apart\u2014its function\nin the natural and in the spiritual order respectively.\n\n[Sidenote: (i) In the _Universe_.]\n\n(i) The heresy of the Colossian teachers took its rise, as we saw, in\ntheir cosmical speculations. It was therefore natural that the Apostle\nin replying should lay stress on the function of the Word in the\ncreation and government of the world. This is the aspect of His work\nmost prominent in the first of the two distinctly Christological\npassages. The Apostle there predicates of the Word, not only prior, but\nabsolute existence. All things were created through Him, are sustained\nin Him, are tending towards Him. Thus He is the beginning, middle, and\nend, of creation. This He is, because He is the very _image_ of the\nInvisible God, because in Him dwells the _plenitude_ of Deity.\n\n[Sidenote: Importance of this aspect of the Person of Christ,]\n\nThis creative and administrative work of Christ the Word in the natural\norder of things is always emphasized in the writings of the Apostles,\nwhen they touch upon the doctrine of His Person. It stands in the\nforefront of the prologue to St John's Gospel: it is hardly less\nprominent in the opening of the Epistle to the Hebrews. His mediatorial\nfunction in the Church is represented as flowing from His mediatorial\nfunction in the world. With ourselves this idea has retired very much\ninto the background. Though in the creed common to all the Churches we\nprofess our belief in Him, as the Being 'through whom all things were\ncreated,' yet in reality this confession seems to exercise very little\ninfluence on our thoughts. And the loss is serious. How much our\ntheological conceptions suffer in breadth and fulness by the neglect, a\nmoment's reflexion will show. How much more hearty would be the sympathy\nof theologians with the revelations of science and the developments of\nhistory, if they habitually connected them with the operation of the\nsame Divine Word who is the centre of all their religious aspirations,\nit is needless to say. Through the recognition of this idea with all the\nconsequences which flow from it, as a living influence, more than in any\nother way, may we hope to strike the chords of that 'vaster music,'\nwhich results only from the harmony of knowledge and faith, of reverence\nand research.\n\n[Sidenote: notwithstanding difficulties yet unsolved.]\n\nIt will be said indeed, that this conception leaves untouched the\nphilosophical difficulties which beset the subject; that creation still\nremains as much a mystery as before. This may be allowed. But is there\nany reason to think that with our present limited capacities the veil\nwhich shrouds it ever will be or can be removed? The metaphysical\nspeculations of twenty-five centuries have done nothing to raise it. The\nphysical investigations of our own age from their very nature can do\nnothing; for, busied with the evolution of phenomena, they lie wholly\noutside this question, and do not even touch the fringe of the\ndifficulty. But meanwhile revelation has interposed and thrown out the\nidea, which, if it leaves many questions unsolved, gives a breadth and\nunity to our conceptions, at once satisfying our religious needs and\nlinking our scientific instincts with our theological beliefs.\n\n[Sidenote: (ii) In the _Church_.]\n\n(ii) But, if Christ's mediatorial office in the physical creation was\nthe starting point of the Apostle's teaching, His mediatorial office in\nthe spiritual creation is its principal theme. The cosmogonies of the\nfalse teachers were framed not so much in the interests of philosophy as\nin the interests of religion; and the Apostle replies to them in the\nsame spirit and with the same motive. If the function of Christ is\nunique in the Universe, so is it also in the Church. [Sidenote: Its\nabsolute character.]He is the sole and absolute link between God and\nhumanity. Nothing short of His personality would suffice as a medium of\nreconciliation between the two. Nothing short of His life and work in\nthe flesh, as consummated in His passion, would serve as an assurance of\nGod's love and pardon. His cross is the atonement of mankind with God.\nHe is the Head with whom all the living members of the body are in\ndirect and immediate communication, who suggests their manifold\nactivities to each, who directs their several functions in subordination\nto the healthy working of the whole, from whom they individually receive\ntheir inspiration and their strength.\n\n[Sidenote: Hence angelic mediations are fundamentally wrong.]\n\nAnd being all this He cannot consent to share His prerogative with\nothers. He absorbs in Himself the whole function of mediation. Through\nHim alone, without any interposing link of communication, the human soul\nhas access to the Father. Here was the true answer to those deep\nyearnings after spiritual communion with God, which sought, and could\nnot find, satisfaction in the manifold and fantastic creations of a\ndreamy mysticism. The worship of angels might have the semblance of\nhumility; but it was in fact a contemptuous defiance of the fundamental\nidea of the Gospel, a flat denial of the absolute character of Christ's\nPerson and office. It was a severance of the proper connexion with the\nHead, an amputation of the disordered limb, which was thus disjoined\nfrom the source of life and left to perish for want of spiritual\nnourishment.\n\n[Sidenote: Christ's mediation in the Church justified by His mediation\n           in the World.]\n\nThe language of the New Testament writers is beset with difficulties, so\nlong as we conceive of our Lord only in connexion with the Gospel\nrevelation: but, when with the Apostles we realise in Him the same\nDivine Lord who is and ever has been the light of the whole world, who\nbefore Christianity wrought first in mankind at large through the\navenues of the conscience, and afterwards more particularly in the Jews\nthrough a special though still imperfect revelation, then all these\ndifficulties fall away. Then we understand the significance, and we\nrecognise the truth, of such passages as these: 'No man cometh unto the\nFather, but by me': 'There is no salvation in any other'; 'He that\ndisbelieveth the Son shall not see life, but the wrath of God abideth\nupon him[524].' The exclusive claims advanced in Christ's name have\ntheir full and perfect justification in the doctrine of the Eternal\nWord.\n\nFootnote 524:\n\n  Joh. xiv. 6, Acts iv. 12, Joh. iii. 36.\n\n[Sidenote: Relation of the doctrine of the Word]\n\nThe old dispensation is primarily the revelation of the absolute\nsovereignty of God. It vindicates this truth against two opposing forms\nof error, which in their extreme types are represented by Pantheism and\nManicheism respectively. [Sidenote: to the monotheism of the Old\nTestament.]The Pantheist identifies God with the world: the Manichee\nattributes to the world an absolute existence, independent of God. With\nthe Pantheist sin ceases to have any existence: for it is only one form\nof God's working. With the Manichee sin is inherent in matter, which is\nantagonistic to God. The teaching of the Old Testament, of which the\nkey-note is struck in the opening chapters of Genesis, is a refutation\nof both these errors. God is distinct from the world, and He is the\nCreator of the world. Evil is not inherent in God, but neither is it\ninherent in the material world. Sin is the disobedience of intelligent\nbeings whom He has created, and whom He has endowed with a free-will,\nwhich they can use or misuse.\n\n[Sidenote: The New Testament is complementary to the Old.]\n\nThe revelation of the New Testament is the proper complement to the\nrevelation of the Old. It holds this position in two main respects. If\nthe Old Testament sets forth the absolute unity of God\u2014His distinctness\nfrom and sovereignty over His creatures\u2014the New Testament points out how\nHe holds communion with the world and with humanity, how man becomes one\nwith Him. And again, if the Old Testament shows the true character of\nsin, the New Testament teaches the appointed means of redemption. On the\none hand the monotheism of the Old Testament is supplemented by the\ntheanthropism[525] of the New. Thus the _theology_ of revelation is\ncompleted. On the other hand, the hamartiology of the Old Testament has\nits counterpart in the soteriology of the New. Thus the _economy_ of\nrevelation is perfected.\n\nFootnote 525:\n\n  I am indebted for the term _theanthropism_, as describing the\n  substance of the new dispensation, to an article by Prof. Westcott in\n  the _Contemporary Review_ IV. p. 417 (December, 1867); but it has been\n  used independently, though in very rare instances, by other writers.\n  The value of terms such as I have employed here in fixing ideas is\n  enhanced by their strangeness, and will excuse any appearance of\n  affectation.\n\n  In applying the terms _theanthropism_ and _soteriology_ to the New\n  Testament, as distinguished from the Old, it is not meant to suggest\n  that the ideas involved in them were wholly wanting in the Old, but\n  only to indicate that the conceptions, which were inchoate and\n  tentative and subsidiary in the one, attain the most prominent\n  position and are distinctly realised in the other.\n\n[Sidenote: 2. The _ethical_ error of the heretics.]\n\n2. When we turn from the theology of these Colossian heretics to their\nethical teaching, we find it characterised by the same earnestness. Of\nthem it might indeed be said that they did 'hunger and thirst after\nrighteousness.' [Sidenote: Their practical earnestness,]Escape from\nimpurity, immunity from evil, was a passion with them. But it was no\nless true that notwithstanding all their sincerity they 'went astray in\nthe wilderness'; 'hungry and thirsty, their soul fainted within them.'\nBy their fatal transference of the abode of sin from the human heart\nwithin to the material world without, they had incapacitated themselves\nfrom finding the true antidote. [Sidenote: but fundamental misconception\nand consequent failure.] Where they placed the evil, there they\nnecessarily sought the remedy. Hence they attempted to fence themselves\nabout, and to purify their lives by a code of rigorous prohibitions.\nTheir energy was expended on battling with the physical conditions of\nhuman life. Their whole mind was absorbed in the struggle with imaginary\nforms of evil. Necessarily their character was moulded by the thoughts\nwhich habitually engaged them. Where the 'elements of the world,' the\n'things which perish in the using[526],' engrossed all their attention,\nit could not fail but that they should be dragged down from the serene\nheights of the spiritual life into the cloudy atmosphere which shrouds\nthis lower earth.\n\nFootnote 526:\n\n  ii. 20, 22.\n\n[Sidenote: St Paul substitutes a principle for ordinances.]\n\nSt Paul sets himself to combat this false tendency. For negative\nprohibitions he substitutes a positive principle; for special\nenactments, a comprehensive motive. He tells them that all their\nscrupulous restrictions are vain, because they fail to touch the springs\nof action. If they would overcome the evil, they must strike at the root\nof the evil. Their point of view must be entirely changed. They must\ntransfer themselves into a wholly new sphere of energy. This\ntransference is nothing less than a migration from earth to heaven\u2014from\nthe region of the external and transitory to the region of the spiritual\nand eternal[527]. For a code of rules they must substitute a principle\nof life, which is one in its essence but infinite in its application,\nwhich will meet every emergency, will control every action, will resist\nevery form of evil.\n\nFootnote 527:\n\n  iii. 1 sq.\n\n[Sidenote: This principle is the heavenly life in Christ.]\n\nThis principle they have in Christ. With Him they have died to the\nworld; with Him they have risen to God. Christ, the revelation of God's\nholiness, of God's righteousness, of God's love, is light, is life, is\nheaven. With Him they have been translated into a higher sphere, have\nbeen brought face to face with the Eternal Presence. Let them only\nrealise this translation. It involves new insight, new motives, new\nenergies. They will no more waste themselves upon vexatious special\nrestrictions: for they will be furnished with a higher inspiration which\nwill cover all the minute details of action. They will not exhaust their\nenergies in crushing this or that rising desire but they will kill the\nwhole body[528] of their earthly passions through the strong arm of this\npersonal communion with God in Christ.\n\nFootnote 528:\n\n  ii. 11 \u1f10\u03bd \u03c4\u1fc7 \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03c3\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, iii. 5 \u03bd\u03b5\u03ba\u03c1\u03ce\u03c3\u03b1\u03c4\u03b5 \u03bf\u1f56\u03bd \u03c4\u1f70\n  \u03bc\u03ad\u03bb\u03b7 with ver. 8 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03cc\u03b8\u03b5\u03c3\u03b8\u03b5 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, and ver. 9\n  \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f78\u03bd \u03c0\u03b1\u03bb\u03b1\u03b9\u1f78\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd. See the notes on the several\n  passages.\n\n[Sidenote: St Paul's doctrine of faith and works considered in the light\n           of this principle.]\n\nWhen we once grasp this idea, which lies at the root of St Paul's\nethical teaching, the moral difficulty which is supposed to attach to\nhis doctrine of faith and works has vanished. It is simply an\nimpossibility that faith should exist without works. Though in form he\nstates his doctrine as a relation of contrast between the two, in\nsubstance it resolves itself into a question of precedence. Faith and\nworks are related as principle and practice. Faith\u2014the repose in the\nunseen, the recognition of eternal principles of truth and right, the\nsense of personal obligations to an Eternal Being who vindicates these\nprinciples\u2014must come first. Faith is not an intellectual assent, nor a\nsympathetic sentiment merely. It is the absolute surrender of self to\nthe will of a Being who has a right to command this surrender. It is\nthis which places men in personal relation to God, which (in St Paul's\nlanguage) justifies them before God. For it touches the springs of their\nactions; it fastens not on this or that detail of conduct, but extends\nthroughout the whole sphere of moral activity; and thus it determines\ntheir character as responsible beings in the sight of God.\n\n                  *       *       *       *       *\n\n[Sidenote: The Christology of this epistle]\n\nFrom the above account it will have appeared that the distinctive\nfeature of this epistle is its Christology. The doctrine of the Person\nof Christ is here stated with greater precision and fulness than in any\nother of St Paul's epistles. It is therefore pertinent to ask (even\nthough the answer must necessarily be brief) what relation this\nstatement bears to certain other enunciations of the same doctrine;\n[Sidenote: considered in relation to]to those for instance which occur\nelsewhere in St Paul's own letters, to those which are found in other\nApostolic writings, and to those which appear in the fathers of the\nsucceeding generations.\n\n[Sidenote: 1. The Christology of St Paul's earlier epistles]\n\n1. The Christology of the Colossian Epistle is in no way different from\nthat of the Apostle's earlier letters. It may indeed be called a\ndevelopment of his former teaching, but only as exhibiting the doctrine\nin fresh relations, as drawing new deductions from it, as defining what\nhad hitherto been left undefined, not as superadding any foreign element\nto it. The doctrine is practically involved in the opening and closing\nwords of his earliest extant epistle: 'The Church which is in God the\nFather and the Lord Jesus Christ'; 'The grace of our Lord Jesus Christ\nbe with you[529].' The main conception of the Person of Christ, as\nenforced in the Colossian Epistle, alone justifies and explains this\nlanguage, which otherwise would be emptied of all significance. And\nagain; it had been enunciated by the Apostle explicitly, though briefly,\nin the earliest directly doctrinal passage which bears on the subject;\n'One Lord Jesus Christ, through whom are all things and we through\nHim[530].' [Sidenote: the same in substance but]The absolute universal\nmediation of the Son is declared as unreservedly in this passage from\nthe First Epistle to the Corinthians, as in any later statement of the\nApostle: and, [Sidenote: less fully developed]if all the doctrinal and\npractical inferences which it implicitly involves were not directly\nemphasized at this early date, it was because the circumstances did not\nyet require explicitness on these points. New forms of error bring into\nprominence new aspects of the truth. The heresies of Laodicea and\nColoss\u00e6 have been invaluable to the later Church in this respect. The\nApostle himself, it is not too much to say, realised with ever\nincreasing force the manifoldness, the adaptability, the completeness of\nthe Christian idea, notwithstanding its simplicity, as he opposed it to\neach successive development of error. The Person of Christ proved the\ncomplete answer to false speculations at Coloss\u00e6, as it had been found\nthe sovereign antidote to false practices at Corinth. All these\nunforeseen harmonies must have appeared to him, as they will appear to\nus, fresh evidences of its truth.\n\nFootnote 529:\n\n  1 Thess. i. 1, v. 28.\n\nFootnote 530:\n\n  1 Cor. viii. 6 \u03b4\u03b9' \u03bf\u1f57 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6. The expression \u03b4\u03b9'\n  \u03bf\u1f57 implies the conception of the Logos, even where the term itself is\n  not used. See the dissertation on the doctrine of the Logos in the\n  Apostolic writers.\n\n[Sidenote: 2. The Christology of other Apostolic writings.]\n\n[Sidenote: Their fundamental identity.]\n\n2. And when we turn from St Paul to the other Apostolic writings which\ndwell on the Person of Christ from a doctrinal point of view, we find\nthem enunciating it in language which implies the same fundamental\nconception, though they may not always present it in exactly the same\naspect. More especially in the Epistle to the Hebrews first, and in the\nGospel of St John afterwards, the form of expression is identical with\nthe statement of St Paul. In both these writings the universe is said to\nhave been created or to exist _by_ or _through_ Him. This is the crucial\nexpression, which involves in itself all the higher conceptions of the\nPerson of Christ[531]. The Epistle to the Hebrews seems to have been\nwritten by a disciple of St Paul immediately after the Apostle's death,\nand therefore within some five or six years from the date which has been\nassigned to the Colossian letter. The Gospel of St John, if the\ntraditional report may be accepted, dates about a quarter of a century\nlater; but it is linked with our epistle by the fact that the readers\nfor whom it was primarily intended belonged to the neighbouring\ndistricts of Proconsular Asia. Thus it illustrates, and is illustrated\nby, the teaching of St Paul in this letter. More especially by the\nemphatic use of the term _Logos_, which St Paul for some reason has\nsuppressed, it supplies the centre round which the ideas gather, and\nthus gives unity and directness to the conception.\n\nFootnote 531:\n\n  Joh. i. 3 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u03ba.\u03c4.\u03bb., Heb. i. 2 \u03b4\u03b9' \u03bf\u1f57 \u03ba\u03b1\u1f76\n  \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u03c4\u03bf\u1f7a\u03c2 \u03b1\u1f30\u1ff6\u03bd\u03b1\u03c2.\n\n[Sidenote: Firmness of the apostolic idea.]\n\nIn the Christology of these Apostolic writings there is a firmness and\nprecision which leaves no doubt about the main conception present to the\nmind of the writers. The idea of Christ as an intermediate being,\nneither God nor man, is absolutely and expressly excluded. On the one\nhand His humanity is distinctly emphasized. On the other He is\nrepresented as existing from eternity, as the perfect manifestation of\nthe Father, as the absolute mediator in the creation and government of\nthe world.\n\n[Sidenote: 3. The Christology of the succeeding ages.]\n\n3. But, when we turn from these Apostolic statements to the writings of\nsucceeding generations, we are struck with the contrast[532]. A\nvagueness, a flaccidity, of conception betrays itself in their language.\n\nFootnote 532:\n\n  The remarks on the theology of the Apostolic Fathers, as compared with\n  the Apostles, in Dorner's _Lehre von der Person Christi_ I. p. 130 sq.\n  seem to me perfectly just and highly significant. See also de\n  Pressens\u00e9 _Trois Premiers Si\u00e8cles_ II. p. 406 sq. on the unsystematic\n  spirit of the Apostolic Fathers.\n\n[Sidenote: Its looseness of conception.]\n\nIn the Apostolic Fathers and in the earlier Apologists we find indeed\nfor the most part a _practical_ appreciation of the Person of Christ,\nwhich leaves nothing to be desired; but as soon as they venture upon any\ndirectly dogmatic statement, we miss at once the firmness of grasp and\nclearness of conception which mark the writings of the Apostles. If they\ndesire to emphasize the majesty of His Person, they not unfrequently\nfall into language which savours of patripassianism[533]. If on the\nother hand they wish to present Him in His mediatorial capacity, they\nuse words which seem to imply some divine being, who is God and yet not\nquite God, neither Creator nor creature[534].\n\nFootnote 533:\n\n  See for instance the passages quoted in the note on Clem. Rom. 2 \u03c4\u1f70\n  \u03c0\u03b1\u03b8\u03ae\u03bc\u03b1\u03c4\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6.\n\nFootnote 534:\n\n  The unguarded language of Justin for instance illustrates the\n  statement in the text. On the one hand Petavius, _Theol. Dogm._ de\n  Trin. ii. 3. 2, distinctly accuses him of Arianism: on the other Bull,\n  _Def. Fid. Nic._ ii. 4. 1 sq., indignantly repudiates the charge and\n  claims him as strictly orthodox. Petavius indeed approaches the\n  subject from the point of view of later Western theology and, unable\n  to appreciate Justin's doctrine of the Logos, does less than justice\n  to this father; but nevertheless Justin's language is occasionally\n  such as no Athanasian could have used. The treatment of this father by\n  Dorner (_Lehre_ I. p. 414 sq.) is just and avoids both extremes.\n\n[Sidenote: The Apostolic idea applied in later ages.]\n\nThe Church needed a long education, before she was fitted to be the\nexpositor of the true Apostolic doctrine. A conflict of more than two\ncenturies with Gnostics, Ebionites, Sabellians, Arians, supplied the\nnecessary discipline. The true successors of the Apostles in this\nrespect are not the fathers of the second century, but the fathers of\nthe third and fourth centuries. In the expositors of the Nicene age we\nfind indeed technical terms and systematic definitions, which we do not\nfind in the Apostles themselves; but, unless I have wholly misconceived\nthe nature of the heretical teaching at Coloss\u00e6 and the purport of St\nPaul's reply, the main idea of Christ's Person, with which he here\nconfronts this Gnostic Judaism, is essentially the same as that which\nthe fathers of these later centuries opposed to the Sabellianism and the\nArianism of their own age. If I mistake not, the more distinctly we\nrealise the nature of the heresy, the more evident will it become that\nany conception short of the perfect deity and perfect humanity of Christ\nwould not have furnished a satisfactory answer; and this is the reason\nwhy I have dwelt at such length on the character of the Colossian false\nteaching, and why I venture to call especial attention to this part of\nmy subject.\n\n                  *       *       *       *       *\n\n[Sidenote: Style of this epistle.]\n\nOf the style of the letter to the Colossians I shall have occasion to\nspeak hereafter, when I come to discuss its genuineness. It is\nsufficient to say here, that while the hand of St Paul is unmistakable\nthroughout this epistle, we miss the flow and the versatility of the\nApostle's earlier letters.\n\n[Sidenote: Its ruggedness and compression,]\n\nA comparison with the Epistles to the Corinthians and to the Philippians\nwill show the difference. It is distinguished from them by a certain\nruggedness of expression, a 'want of finish' often bordering on\nobscurity. What account should be given of this characteristic, it is\nimpossible to say. The divergence of style is not greater than will\nappear in the letters of any active-minded man, written at different\ntimes and under different circumstances. The epistles which I have\nselected for contrast suggest that the absence of all personal connexion\nwith the Colossian Church will partially, if not wholly, explain the\ndiminished fluency of this letter. [Sidenote: but essential vigour.]At\nthe same time no epistle of St Paul is more vigorous in conception or\nmore instinct with meaning. It is the very compression of the thoughts\nwhich creates the difficulty. If there is a want of fluency, there is no\nwant of force. Feebleness is the last charge which can be brought\nagainst this epistle.\n\n                  *       *       *       *       *\n\n[Sidenote: Analysis.]\n\nThe following is an analysis of the epistle:\n\nI. INTRODUCTORY (i. 1\u201313).\n\n  (1) i. 1, 2. Opening salutation.\n\n  (2) i. 3\u20138. Thanksgiving for the progress of the Colossians hitherto.\n\n  (3) i. 9\u201313. Prayer for their future advance in knowledge and\n          well-doing through Christ.\n\n        [This leads the Apostle to speak of Christ as the only path of\n          progress.]\n\nII. DOCTRINAL (i. 13-ii. 3).\n\n    The Person and Office of Christ.\n\n  (1) i. 13, 14. Through the Son we have our deliverance, our\n      redemption.\n\n  (2) i. 15\u201319. The _Preeminence_ of the Son;\n\n      (i) As the Head of the natural Creation, the Universe (i. 15\u201317);\n\n      (ii) As the Head of the new moral Creation, the Church (i. 18).\n\n      Thus He is first in all things; and this, because the pleroma has\n        its abode in Him (i. 19).\n\n  (3) i. 20-ii. 3. The _Work_ of the Son\u2014a work of reconciliation;\n\n      (i) Described generally (i. 20).\n\n      (ii) Applied specially to the Colossians (i. 21\u201323).\n\n      (iii) St Paul's own part in carrying out this work. His sufferings\n        and preaching. The 'mystery' with which he is charged (i.\n        24\u201327).\n\n        His anxiety on behalf of all (i. 28, 29): and more especially of\n          the Colossian and neighbouring Churches (ii. 1\u20133).\n\n        [This expression of anxiety leads him by a direct path to the\n          next division of the epistle.]\n\nIII. POLEMICAL (ii. 4-iii. 4).\n\n  Warning against errors.\n\n  (1) ii. 4\u20138. The Colossians charged to abide in the truth of the\n      Gospel as they received it at first, and not to be led astray by a\n      strange philosophy which the new teachers offer.\n\n  (2) ii. 9\u201315. The truth stated first positively and then negatively.\n\n    [In the passage which follows (ii. 9\u201323) it will be observed how St\n      Paul vibrates between the theological and practical bearings of\n      the truth, marked \u03b1, \u03b2, respectively.]\n\n    (i) _Positively._\n\n      (\u03b1) The _pleroma_ dwells wholly in Christ and is communicated\n        through Him (ii. 9, 10).\n\n      (\u03b2) The true circumcision is a spiritual circumcision (ii. 11,\n        12).\n\n    (ii) _Negatively_. Christ has\n\n      (\u03b2) annulled the law of ordinances (ii. 14);\n\n      (\u03b1) triumphed over all spiritual agencies, however powerful (ii.\n        15).\n\n  (3) ii. 16-iii. 4. Obligations following thereupon.\n\n    (i) Consequently the Colossians must not\n\n      (\u03b2) either submit to ritual prohibitions (ii. 16, 17),\n\n      (\u03b1) or substitute the worship of inferior beings for allegiance to\n        the Head (ii. 18, 19).\n\n    (ii) On the contrary this must henceforth be their rule:\n\n      1. They have _died_ with Christ; and with Him they have died to\n        their old life, to earthly _ordinances_ (ii. 20\u201323).\n\n      2. They have _risen_ with Christ; and with Him they have risen to\n        a new life, to heavenly _principles_ (iii. 1\u20134).\n\nIV. HORTATORY (iii. 5-iv. 6).\n\n  Practical application of this death and this resurrection.\n\n  (1) iii. 5\u201312. _Comprehensive_ rules.\n\n      (i) What vices are to be put off, being mortified in this death\n        (iii. 5\u201311).\n\n      (ii) What graces are to be put on, being quickened through this\n        resurrection (iii. 12\u201317).\n\n  (2) iii. 13-iv. 6. _Special_ precepts.\n\n      (_a_) The obligations\n\n          Of wives and husbands (iii. 18, 19);\n\n          Of children and parents (iii. 20, 21);\n\n          Of slaves and masters (iii. 22-iv. 1).\n\n      (_b_) The duty of prayer and thanksgiving; with special\n          intercession on the Apostle's behalf (iv. 2\u20134).\n\n      (_c_) The duty of propriety in behaviour towards the unconverted\n          (iv. 5, 6).\n\nV. PERSONAL (iv. 7\u201318).\n\n  (1) iv. 7\u20139. Explanations relating to the letter itself.\n\n  (2) iv. 10\u201314. Salutations from divers persons.\n\n  (3) iv. 15\u201317. Salutations to divers persons. A message relating to\n      Laodicea.\n\n  (4) iv. 18. Farewell.\n\n------------------------------------------------------------------------\n\n\n\n\n                            \u03a0\u03a1\u039f\u03a3 \u039a\u039f\u039b\u0391\u03a3\u03a3\u0391\u0395\u0399\u03a3.\n\n\n\n\n------------------------------------------------------------------------\n\n\n\n\n              WE SPEAK WISDOM AMONG THEM THAT ARE PERFECT.\n\n                   YET NOT THE WISDOM OF THIS WORLD.\n\n              BUT WE SPEAK THE WISDOM OF GOD IN A MYSTERY.\n\n\n                             --------------\n\n                      _Iste vas electionis\n                      Vires omnes rationis\n                        Human\u00e6 transgreditur:\n                      Super choros angelorum\n                      Raptus, c\u0153li secretorum\n                        Doctrinis imbuitur._\n\n                      _De hoc vase tam fecundo,\n                      Tam electo et tam mundo,\n                        Tu nos, Christe, complue;\n                      Nos de luto, nos de f\u00e6ce,\n                      Tua sancta purga prece,\n                        Regno tuo statue._\n\n------------------------------------------------------------------------\n\n\n\n\n                            \u03a0\u03a1\u039f\u03a3 \u039a\u039f\u039b\u0391\u03a3\u03a3\u0391\u0395\u0399\u03a3.\n\n------------------------------------------------------------------------\n\nI. 1, 2]\n\n\u03a0\u0391\u03a5\u039b\u039f\u03a3 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03b4\u03b9\u1f70 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u0398\u03b5\u03bf\u1fe6, \u03ba\u03b1\u1f76 \u03a4\u03b9\u03bc\u03cc\u03b8\u03b5\u03bf\u03c2 \u1f41\n\u1f00\u03b4\u03b5\u03bb\u03c6\u03cc\u03c2, ^2\u03c4\u03bf\u1fd6\u03c2 \u1f10\u03bd \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u1fd6\u03c2 >\n\n\n1, 2. 'PAUL, an apostle of Christ Jesus by no personal merit but by\nGod's gracious will alone, and TIMOTHY, our brother in the faith, to the\nconsecrated people of God in Coloss\u00e6, the brethren who are stedfast in\ntheir allegiance and faithful in Christ. May grace the well-spring of\nall mercies, and peace the crown of all blessings, be bestowed upon you\nfrom God our Father.'\n\n1. \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03c2] On the exceptional omission of this title in some of St\nPaul's epistles see Phil. i. 1. Though there is no reason for supposing\nthat his authority was directly impugned in the Colossian Church, yet he\ninterposes by virtue of his Apostolic commission and therefore uses his\nauthoritative title.\n\n\u03b4\u03b9\u1f70 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u0398\u03b5\u03bf\u1fe6] As in 1 Cor. i. 1, 2 Cor. i. 1, Ephes. i. 1, 2 Tim.\ni. 1. These passages show that the words cannot have a polemical\nbearing. If they had been directed against those who questioned his\nApostleship, they would probably have taken a stronger form. The\nexpression must therefore be regarded as a renunciation of all personal\nworth, and a declaration of God's unmerited grace; comp. Rom. ix. 16 \u1f04\u03c1\u03b1\n\u03bf\u1f56\u03bd \u03bf\u1f50 \u03c4\u03bf\u1fe6 \u03b8\u03ad\u03bb\u03bf\u03bd\u03c4\u03bf\u03c2 \u03bf\u1f50\u03b4\u1f72 \u03c4\u03bf\u1fe6 \u03c4\u03c1\u03ad\u03c7\u03bf\u03bd\u03c4\u03bf\u03c2 \u1f00\u03bb\u03bb\u1f70 \u03c4\u03bf\u1fe6 \u1f10\u03bb\u03b5\u1ff6\u03bd\u03c4\u03bf\u03c2 \u03b8\u03b5\u03bf\u1fe6. The same\nwords \u03b4\u03b9\u1f70 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b8\u03b5\u03bf\u1fe6 are used in other connexions in Rom. xv. 32, 2\nCor. viii. 5, where no polemical reference is possible.\n\n\u03a4\u03b9\u03bc\u03cc\u03b8\u03b5\u03bf\u03c2] The name of this disciple is attached to the Apostle's own in\nthe heading of the Philippian letter, which was probably written at an\nearlier stage in his Roman captivity. It appears also in the same\nconnexion in the Epistle to Philemon, but not in the Epistle to the\nEphesians, though these two letters were contemporaneous with one\nanother and with the Colossian letter. For an explanation of the\nomission, see the introduction to that epistle.\n\nIn the Epistles to the Philippians and to Philemon the presence of\nTimothy is forgotten at once (see Phil. i. 1). In this epistle the\nplural is maintained throughout the thanksgiving (vv. 3, 4, 7, 8, 9),\nbut afterwards dropped, when the Apostle begins to speak in his own\nperson (i. 23, 24), and so he continues to the end. The exceptions (i.\n28, iv. 3) are rather apparent than real.\n\n\u1f41 \u1f00\u03b4\u03b5\u03bb\u03c6\u03cc\u03c2] Timothy is again designated simply 'the brother' in 2 Cor. i.\n1, Philem. 1, but not in Heb. xiii. 23, where the right reading is \u03c4\u1f78\u03bd\n\u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03bd \u1f21\u03bc\u1ff6\u03bd. The same designation is used of Quartus (Rom. xvi. 23), of\nSosthenes (1 Cor. i. 1), of Apollos (1 Cor. xvi. 12); comp. 2 Cor. viii.\n18, ix. 3, 5, xii. 18. As some designation seemed to be required, and as\nTimothy could not be called an Apostle (see _Galatians_, p. 96, note 2),\nthis, as the simplest title, would naturally suggest itself.\n\n2. \u039a\u03bf\u03bb\u03bf\u03c3\u03c3\u03b1\u1fd6\u03c2] For the reasons why this form is preferred here, while\n\u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u1fd6\u03c2 is adopted in the heading of the epistle, see above, p. 16\nsq.\n\n------------------------------------------------------------------------\n\nI. 3]\n\n\u1f01\u03b3\u03af\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7\u00b7 \u03c7\u03ac\u03c1\u03b9\u03c2 \u1f51\u03bc\u1fd6\u03bd \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u1f00\u03c0\u1f78 \u0398\u03b5\u03bf\u1fe6\n\u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u1f21\u03bc\u1ff6\u03bd.\n\n^3\u0395\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bc\u03b5\u03bd \u03c4\u1ff7 \u0398\u03b5\u1ff7 [\u03ba\u03b1\u1f76] \u03c0\u03b1\u03c4\u03c1\u1f76 \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5\n\n\n\u1f01\u03b3\u03af\u03bf\u03b9\u03c2] '_saints_,' i.e. the people consecrated to God, the Israel of\nthe new covenant; see the note on Phil. i. 1. This mode of address marks\nthe later epistles of St Paul. In his earlier letters (1, 2 Thess., 1, 2\nCor., Gal.) he writes \u03c4\u1fc7 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3, \u03c4\u03b1\u1fd6\u03c2 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03b9\u03c2. The change begins\nwith the Epistle to the Romans, and from that time forward the Apostle\nalways uses \u1f01\u03b3\u03af\u03bf\u03b9\u03c2 in various combinations in addressing Churches (Rom.,\nPhil., Col., Ephes.). For a similar phenomenon, serving as a\nchronological mark, see the note on \u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2, iv. 18. The word \u1f01\u03b3\u03af\u03bf\u03b9\u03c2\nmust here be treated as a substantive in accordance with its usage in\nparallel passages, and not as an adjective connected with \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2. See\nthe next note.\n\n\u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2] This unusual addition is full of meaning. Some\nmembers of the Colossian Church were shaken in their allegiance, even if\nthey had not fallen from it. The Apostle therefore wishes it to be\nunderstood that, when he speaks of the saints, he means the true and\nstedfast members of the brotherhood. In this way he obliquely hints at\nthe defection. Thus the words \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2 are a supplementary\nexplanation of \u03c4\u03bf\u1fd6\u03c2 \u1f01\u03b3\u03af\u03bf\u03b9\u03c2. He does not directly exclude any, but he\nindirectly warns all. The epithet \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 cannot mean simply 'believing';\nfor then it would add nothing which is not already contained in \u1f01\u03b3\u03af\u03bf\u03b9\u03c2\nand \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2. Its passive sense, 'trustworthy, stedfast, unswerving,'\nmust be prominent here, as in Acts xvi. 15 \u03b5\u1f30 \u03ba\u03b5\u03ba\u03c1\u03af\u03ba\u03b1\u03c4\u03ad \u03bc\u03b5 \u03c0\u03b9\u03c3\u03c4\u1f74\u03bd \u03c4\u1ff7\n\u039a\u03c5\u03c1\u03af\u1ff3 \u03b5\u1f36\u03bd\u03b1\u03b9. See _Galatians_ p. 155.\n\n\u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7] most naturally connected with both words \u03c0\u03b9\u03c3\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2,\nthough referring chiefly to \u03c0\u03b9\u03c3\u03c4\u03bf\u1fd6\u03c2; comp. Ephes. vi. 21 \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2\n\u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3, 1 Tim. i. 2 \u03b3\u03bd\u03b7\u03c3\u03af\u1ff3 \u03c4\u03ad\u03ba\u03bd\u1ff3 \u1f10\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03b9. For the expression \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2\n\u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7, \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3, see also 1 Cor. iv. 17, Ephes. i. 1. The Apostle\nassumes that the Colossian brethren are 'stedfast in Christ.' Their\nstate thus contrasts with the description of the heretical teacher, who\n(ii. 19) \u03bf\u1f50 \u03ba\u03c1\u03b1\u03c4\u03b5\u1fd6 \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae\u03bd.\n\n\u03c7\u03ac\u03c1\u03b9\u03c2 \u03ba.\u03c4.\u03bb.] On this form of salutation see the note to 1 Thess. i. 1.\n\n\u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u1f21\u03bc\u1ff6\u03bd] The only instance in St Paul's epistles, where the name of\nthe Father stands alone in the opening benediction without the addition\nof Jesus Christ. The omission was noticed by Origen (_Rom._ 1. \u00a7 8, IV.\np. 467), and by Chrysostom (_ad loc._ XI. p. 324, _Hom. in 2 Cor._ XXX,\nx. p. 651). But transcribers naturally aimed at uniformity, and so in\nmany copies we find the addition \u03ba\u03b1\u1f76 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. The only\nother exception to the Apostle's usual form is in 1 Thessalonians, where\nthe benediction is shorter still, \u03c7\u03ac\u03c1\u03b9\u03c2 \u1f51\u03bc\u1fd6\u03bd \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7, and where\nlikewise the copyists have supplied words to lengthen it out in\naccordance with St Paul's common practice.\n\n3\u20138. 'We never cease to pour forth our thanksgiving to God the Father of\nour Lord Jesus Christ on your account, whensoever we pray to Him. We are\nfull of thankfulness for the tidings of the _faith_ which ye have in\nChrist Jesus, and the _love_ which ye show towards all the people of\nGod, while ye look forward to the _hope_ which is stored up for you in\nheaven as a treasure for the life to come. This hope was communicated to\nyou in those earlier lessons, when the Gospel was preached to you in its\npurity and integrity\u2014the one universal unchangeable Gospel, which was\nmade known to you, even as it was carried throughout the world,\napproving itself by its fruits wheresoever it is planted. For, as\nelsewhere, so also in you, these fruits were manifested from the first\nday when ye received your lessons in, and apprehended the power of, the\ngenuine Gospel, which is not a law of ordinances but a dispensation of\ngrace, not a device of men but a truth of God. Such was the word\npreached to you by Epaphras, our beloved fellow-servant in our Master's\nhousehold, who in our absence and on our behalf has ministered to you\nthe Gospel of Christ, and who now brings back to us the welcome tidings\nof the love which ye show in the Spirit.'\n\n3. \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bc\u03b5\u03bd] See the notes on 1 Thess. i. 2.\n\n\u03c0\u03b1\u03c4\u03c1\u1f76] If the \u03ba\u03b1\u1f76 be omitted, as the balance of authorities appears to\nsuggest, the form of words here is quite exceptional. Elsewhere it runs\n\u1f41 \u03b8\u03b5\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c4\u1f74\u03c1 \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5, Rom. XV. 6, 2 Cor. i. 3, xi. 31, Ephes. i.\n3 (v.l.), 1 Pet. i. 3; comp. Rev. i. 6: and in analogous cases, such as\n\u1f41 \u03b8\u03b5\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c4\u1f74\u03c1 \u1f21\u03bc\u1ff6\u03bd, the rule is the same. See the note on Clem. Rom.\n\u00a7 7. In iii. 17 however we have \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af, where the evidence is more\ndecisive and the expression quite as unusual. On the authorities for the\nvarious readings here see the detached note.\n\n------------------------------------------------------------------------\n\nI. 4, 5]\n\n\u1f21\u03bc\u1ff6\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9\u00b7 ^4 \u1f00\u03ba\u03bf\u03cd\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 \u03c4\u1f74\u03bd\n\u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u1f51\u03bc\u1ff6\u03bd \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6, \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd [\u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5] \u03b5\u1f30\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1f7a\u03c2\n\u1f01\u03b3\u03af\u03bf\u03c5\u03c2, ^5 \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd\n\n\n\u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] We here meet the same difficulty about the connexion of\nthe clauses, which confronts us in several of St Paul's opening\nthanksgivings. The words \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 and \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd must clearly be taken\ntogether, because the emphasis of \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd would be inexplicable, if it\nstood at the beginning of a clause. But are they to be attached to the\npreceding or to the following sentence? The connexion with the previous\nwords is favoured by St Paul's usual conjunction of \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03b5\u1fd6\u03bd \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5\n(see the note on Phil. i. 3), and by the parallel passage \u03bf\u1f50 \u03c0\u03b1\u03cd\u03bf\u03bc\u03b1\u03b9\n\u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6\u03bd \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd in Ephes. i. 16. Thus the words will mean '_We give\nthanks for you always in our prayers_.' For this absolute use of\n\u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 see Matt. vi. 7, Acts xvi. 25.\n\n4. \u1f00\u03ba\u03bf\u03cd\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2] '_having heard_' from Epaphras (ver. 8); for the Apostle\nhad no direct personal knowledge of the Colossian Church: see the\nintroduction, p. 27 sq.\n\n\u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6] to be connected with \u03c4\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u1f51\u03bc\u1ff6\u03bd. The strict\nclassical language would require \u03c4\u1f74\u03bd \u1f10\u03bd \u03a7. \u1f38., but the omission of the\narticle is common to the New Testament (e.g. ver. 8); see the note on 1\nThess. i. 1, and Winer \u00a7 xx. p. 169 (ed. Moulton). The preposition \u1f10\u03bd\nhere and in the parallel passage, Ephes. i. 15, denotes the sphere in\nwhich their faith moves, rather than the object to which it is directed\n(comp. 1 Cor. iii. 5); for, if the object had been meant, the natural\npreposition would have been \u1f10\u03c0\u1f76 or \u03b5\u1f30\u03c2 (e.g. ii. 5). This is probably\nthe case also in the passages where at first sight it might seem\notherwise, e.g. 1 Tim. iii. 13, 2 Tim. iii. 15; for compare 2 Tim. i. 13\n\u1f10\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03b9 \u03ba\u03b1\u1f76 \u1f00\u03b3\u03ac\u03c0\u1fc3 \u03c4\u1fc7 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6, where the meaning is\nunambiguous. There is however authority in the LXX for the use of \u1f10\u03bd\nwith \u03c0\u03af\u03c3\u03c4\u03b9\u03c2, \u03c0\u03b9\u03c3\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd, to denote the object, in Jer. xii. 6, Ps.\nlxxviii. 22, and perhaps in Mark i. 15, Rom. iii. 25, and (more\ndoubtfully still) in Joh. iii. 15.\n\n\u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5] See the detached note on the various readings.\n\n5. \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u1f10\u03bb\u03c0\u03af\u03b4\u03b1] '_for the hope_,' i.e. looking to the hope. The\nfollowing reasons seem decisive in favour of connecting \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u1f10\u03bb\u03c0\u03af\u03b4\u03b1,\nnot with \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bc\u03b5\u03bd, but with \u03c4\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u03ba.\u03c4.\u03bb., whether \u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5 be\nretained or not. (1) The great distance of \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bc\u03b5\u03bd is against the\nformer connexion; (2) The following clause, \u1f23\u03bd \u03c0\u03c1\u03bf\u03b7\u03ba\u03bf\u1fe6\u03c3\u03b1\u03c4\u03b5 \u03ba.\u03c4.\u03bb.,\nsuggests that the words \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u1f10\u03bb\u03c0\u03af\u03b4\u03b1 describe the motives of the\nColossians for well-doing, rather than the reasons of the Apostle for\nthanksgiving: (3) The triad of Christian graces, which St Paul delights\nto associate together, would otherwise be broken up. This last argument\nseems conclusive; see especially the corresponding thanksgiving in 1\nThess. i. 3, \u03bc\u03bd\u03b7\u03bc\u03bf\u03bd\u03b5\u1fe6\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03c4\u03bf\u1fe6 \u1f14\u03c1\u03b3\u03bf\u03c5 \u03c4\u1fc6\u03c2 _\u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2_ \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6\n\u03ba\u03cc\u03c0\u03bf\u03c5 \u03c4\u1fc6\u03c2 _\u1f00\u03b3\u03ac\u03c0\u03b7\u03c2_ \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u1fc6\u03c2 \u03c4\u1fc6\u03c2 _\u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2_ \u03ba.\u03c4.\u03bb., with\nthe note there. The order is the same here, as there; and it is the\nnatural sequence. Faith rests on the past; love works in the present;\nhope looks to the future. They may be regarded as the efficient,\nmaterial, and final causes respectively of the spiritual life. Compare\nPolycarp _Phil._ 3 \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u1f25\u03c4\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76 \u03bc\u03ae\u03c4\u03b7\u03c1 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u1f21\u03bc\u1ff6\u03bd, \u1f10\u03c0\u03b1\u03ba\u03bf\u03bb\u03bf\u03c5\u03b8\u03bf\u03cd\u03c3\u03b7\u03c2\n\u03c4\u1fc6\u03c2 \u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2, \u03c0\u03c1\u03bf\u03b1\u03b3\u03bf\u03cd\u03c3\u03b7\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03b3\u03ac\u03c0\u03b7\u03c2.\n\nThe hope here is identified with the object of the hope: see the\npassages quoted on Gal. v. 5. The sense of \u1f10\u03bb\u03c0\u03af\u03c2, as of the\ncorresponding words in any language, oscillates between the subjective\nfeeling and the objective realisation; comp. Rom. viii. 24 \u03c4\u1fc7 \u03b3\u1f70\u03c1 \u1f10\u03bb\u03c0\u03af\u03b4\u03b9\n\u1f10\u03c3\u03ce\u03b8\u03b7\u03bc\u03b5\u03bd\u00b7 \u1f10\u03bb\u03c0\u1f76\u03c2 \u03b4\u1f72 \u03b2\u03bb\u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03b7 \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9\u03bd \u1f10\u03bb\u03c0\u03af\u03c2\u00b7 \u1f43 \u03b3\u1f70\u03c1 \u03b2\u03bb\u03ad\u03c0\u03b5\u03b9 \u03c4\u03b9\u03c2 \u03ba.\u03c4.\u03bb.,\nwhere it passes abruptly from the one to the other.\n\n------------------------------------------------------------------------\n\nI. 6]\n\n\u1f10\u03bb\u03c0\u03af\u03b4\u03b1 \u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03ba\u03b5\u03b9\u03bc\u03ad\u03bd\u03b7\u03bd \u1f51\u03bc\u1fd6\u03bd \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2, \u1f23\u03bd \u03c0\u03c1\u03bf\u03b7\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5 \u1f10\u03bd \u03c4\u1ff7 \u03bb\u03cc\u03b3\u1ff3\n\u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5, ^6 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c1\u03cc\u03bd\u03c4\u03bf\u03c2 \u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd\n\u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u1ff7 \u03ba\u03cc\u03c3\u03bc\u1ff3 \u1f10\u03c3\u03c4\u03b9\u03bd\n\n\n\u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03ba\u03b5\u03b9\u03bc\u03ad\u03bd\u03b7\u03bd] '_which is stored up_.' It is the \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u1f78\u03c2 \u1f10\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1ff7 of\nthe Gospels (Matt. vi. 20, 21, Luke xii. 34, xviii. 22).\n\n\u03c0\u03c1\u03bf\u03b7\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5] '_of which ye were told in time past_.' The preposition\nseems intended to contrast their earlier with their later lessons\u2014the\ntrue Gospel of Epaphras with the false gospel of their recent teachers\n(see the next note). The expression would gain force, if we might\nsuppose that the heretical teachers obscured or perverted the doctrine\nof the resurrection (comp. 2 Tim. ii. 18); and their speculative tenets\nwere not unlikely to lead to such a result. But this is not necessary;\nfor under any circumstances the false doctrine, as leading them astray,\ntended to cheat them of their hope; see ver. 23. The common\ninterpretations, which explain \u03c0\u03c1\u03bf- as meaning either 'before its\nfulfilment' or 'before my writing to you,' seem neither so natural in\nthemselves nor so appropriate to the context.\n\n\u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5] '_the truth of the Gospel_,' i.e. the true\nand genuine Gospel as taught by Epaphras, and not the spurious\nsubstitute of these later pretenders: comp. ver. 6 \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3. See also\nGal. ii. 5, 14, where a similar contrast is implied in the use of \u1f21\n\u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5.\n\n6. \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c1\u03cc\u03bd\u03c4\u03bf\u03c2 \u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2 '_which reached you_.' The expression \u03c0\u03b1\u03c1\u03b5\u1fd6\u03bd\u03b1\u03b9\n\u03b5\u1f30\u03c2 is not uncommon in classical writers; comp. \u03c0\u03b1\u03c1\u03b5\u1fd6\u03bd\u03b1\u03b9 \u03c0\u03c1\u1f78\u03c2 in Acts\nxii. 20, Gal. iv. 18, 20. So also \u03b5\u1f51\u03c1\u03b5\u03b8\u1fc6\u03bd\u03b1\u03b9 \u03b5\u1f30\u03c2 (Acts viii. 40),\n\u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9 \u03b5\u1f30\u03c2 (e.g. Acts xxv. 15), and even \u03b5\u1f36\u03bd\u03b1\u03b9 \u03b5\u1f30\u03c2 (Luke xi. 7). See\nWiner \u00a7 l. p. 516 sq.\n\n\u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u1ff7 \u03ba\u03cc\u03c3\u03bc\u1ff3] For a similar hyperbole see Rom. i. 8 \u1f10\u03bd \u1f45\u03bb\u1ff3 \u03c4\u1ff7\n\u03ba\u03cc\u03c3\u03bc\u1ff3; comp. 1 Thess. i. 8, 2 Cor. ii. 14, \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u03cc\u03c0\u1ff3. More lurks\nunder these words than appears on the surface. The true Gospel, the\nApostle seems to say, proclaims its truth by its universality. The false\ngospels are the outgrowths of local circumstances, of special\nidiosyncrasies; the true Gospel is the same everywhere. The false\ngospels address themselves to limited circles; the true Gospel proclaims\nitself boldly throughout the world. Heresies are at best ethnic: truth\nis essentially catholic. See ver. 23 \u03bc\u1f74 \u03bc\u03b5\u03c4\u03b1\u03ba\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2\n\u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5 \u1f41\u1fe6 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5, _\u03c4\u03bf\u1fe6_ \u03ba\u03b7\u03c1\u03c5\u03c7\u03b8\u03ad\u03bd\u03c4\u03bf\u03c2 _\u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03ba\u03c4\u03af\u03c3\u03b5\u03b9_\n\u03c4\u1fc7 \u1f51\u03c0\u1f78 \u03c4\u1f78\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u03cc\u03bd.\n\n------------------------------------------------------------------------\n\nI. 6]\n\n\u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, \u1f00\u03c6' \u1f27\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2\n\u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03ad\u03b3\u03bd\u03c9\u03c4\u03b5 \u03c4\u1f74\u03bd \u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u03bf\u1fe6\n\n\n\u1f10\u03c3\u03c4\u1f76\u03bd \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd] '_is constantly bearing fruit_.' The fruit, which\nthe Gospel bears without fail in all soils and under every climate, is\nits credential, its verification, as against the pretensions of spurious\ncounterfeits. The substantive verb should here be taken with the\nparticiple, so as to express _continuity_ of present action; as in 2\nCor. ix. 12 \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f10\u03c3\u03c4\u1f76\u03bd \u03c0\u03c1\u03bf\u03c3\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03c3\u03b1 \u03ba.\u03c4.\u03bb., Phil. ii. 26 \u1f10\u03c0\u03b9\u03c0\u03bf\u03b8\u1ff6\u03bd\n\u1f26\u03bd. It is less common in St Paul than in some of the Canonical writers,\ne.g. St Mark and St Luke; but probably only because he deals less in\nnarrative.\n\nOf the middle \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 no other instance has been found. The voice\nis partially illustrated by \u03ba\u03c9\u03b4\u03c9\u03bd\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, \u03c3\u03b9\u03b4\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9,\n\u03c4\u03c5\u03bc\u03c0\u03b1\u03bd\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, though, as involving a different sense of -\u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9\n'to wear,' these words are not exact parallels. Here the use of the\nmiddle is the more marked, inasmuch as the active occurs just below\n(ver. 10) in the same connexion, \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9. This\nfact however points to the force of the word here. The middle is\n_intensive_, the active _extensive_. The middle denotes the inherent\nenergy, the active the external diffusion. The Gospel is essentially a\nreproductive organism, a plant whose 'seed is in itself.' For this\n'dynamic' middle see Moulton's note on Winer \u00a7 xxxviii. p. 319.\n\n\u03ba\u03b1\u1f76 \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd] The Gospel is not like those plants which exhaust\nthemselves in bearing fruit and wither away. The external growth keeps\npace with the reproductive energy. While \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd describes the\ninner working, \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd gives the outward extension of the Gospel.\nThe words \u03ba\u03b1\u1f76 \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd are not found in the received text, but the\nauthority in their favour is overwhelming.\n\n\u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd The comparison is thus doubled back, as it were, on\nitself. This irregularity disappears in the received text, \u03ba\u03b1\u1f76 \u1f10\u03c3\u03c4\u1f76\u03bd\n\u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, where the insertion of \u03ba\u03b1\u1f76 before\n\u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd straightens the construction. For a similar irregularity\nsee 1 Thess. iv. 1 \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03bf\u1fe6\u03bc\u03b5\u03bd \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u1f35\u03bd\u03b1, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5\n\u03c0\u03b1\u03c1' \u1f21\u03bc\u1ff6\u03bd \u03c4\u1f78 \u03c0\u1ff6\u03c2 \u03b4\u03b5\u1fd6 \u1f51\u03bc\u1fb6\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03bd \u03ba\u03b1\u1f76 \u1f00\u03c0\u03ad\u03c3\u03ba\u03b5\u03b9\u03bd \u03b8\u03b5\u1ff7, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76\n\u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5, \u1f35\u03bd\u03b1 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03b5\u1fe6\u03b7\u03c4\u03b5 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd, where again the received text\nsimplifies the construction, though in a different way, by omitting the\nfirst \u1f35\u03bd\u03b1 and the words \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5. In both cases the\nexplanation of the irregularity is much the same; the clause\nreciprocating the comparison (here \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, there \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76\n\u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5) is an afterthought springing out of the Apostle's anxiety\nnot to withhold praise where praise can be given.\n\nFor the appearance of \u03ba\u03b1\u1f76 in both members of the comparison, \u03ba\u03b1\u1f76 \u1f10\u03bd\n\u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u1ff7 \u03ba\u03cc\u03c3\u03bc\u1ff3 ... \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76, comp. Rom. i. 13 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd\n\u03c4\u03bf\u1fd6\u03c2 \u03bb\u03bf\u03b9\u03c0\u03bf\u1fd6\u03c2 \u1f15\u03b8\u03bd\u03b5\u03c3\u03b9\u03bd; and in the reversed order below, iii. 13 \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76\n\u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 \u1f10\u03c7\u03b1\u03c1\u03af\u03c3\u03b1\u03c4\u03bf \u1f51\u03bc\u1fd6\u03bd, \u1f45\u03c5\u03c4\u03c9\u03c2 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 (with the note): see also Winer\nliii. p. 549 (ed. Moulton). The correlation of the clauses is thus\nrendered closer, and the comparison emphasized.\n\n\u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03ad\u03b3\u03bd\u03c9\u03c4\u03b5] The accusative is governed by both verbs equally,\n'Ye were instructed in and fully apprehended the grace of God.' For this\nsense of \u1f00\u03ba\u03bf\u03cd\u03b5\u03b9\u03bd see below, ver. 23. For \u1f10\u03c0\u03b9\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd as denoting\n'advanced knowledge, thorough appreciation,' see the note on \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2,\nver. 9.\n\n\u03c4\u1f74\u03bd \u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6] St Paul's synonyme for the Gospel. In Acts xx. 24 he\ndescribes it as his mission to preach \u03c4\u1f78 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03bd\u03c4\u03b7\u03c2 \u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6.\nThe true Gospel as taught by Epaphras was an offer of free grace, a\nmessage from God; the false gospel, as superposed by the heretical\nteachers, was a code of rigorous prohibitions, a system of human\ndevising. It was not \u03c7\u03ac\u03c1\u03b9\u03c2 but \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1 (ii. 14); not \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 but \u03c4\u03bf\u1fe6\n\u03ba\u03cc\u03c3\u03bc\u03bf\u03c5, \u03c4\u1ff6\u03c5 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd (ii. 8, 20, 22). For God's power and goodness it\nsubstituted self-mortification and self-exaltation. The Gospel is called\n\u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 again in 2 Cor. vi. 1, viii. 9, with reference to the\nsame leading characteristic which the Apostle delights to dwell upon\n(e.g. Rom. iii. 24, v. 15, Eph. ii. 5, 8), and which he here tacitly\ncontrasts with the doctrine of the later intruders. The false teachers\nof Coloss\u00e6, like those of Galatia, would lead their hearers \u1f00\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd \u03c4\u1f74\u03bd\n\u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 (Gal. ii. 21); to accept their doctrine was \u1f10\u03ba\u03c0\u03af\u03c0\u03c4\u03b5\u03b9\u03bd \u03c4\u1fc6\u03c2\n\u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2 (Gal. v. 4).\n\n------------------------------------------------------------------------\n\nI. 7, 8]\n\n\u0398\u03b5\u03bf\u1fe6 \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3, ^7\u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 \u1f00\u03c0\u1f78 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6 \u03c4\u03bf\u1fe6 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03bf\u1fe6 \u03c3\u03c5\u03bd\u03b4\u03bf\u03cd\u03bb\u03bf\u03c5\n\u1f21\u03bc\u1ff6\u03bd, \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f51\u03c0\u1f72\u03c1 \u1f21\u03bc\u1ff6\u03bd \u03b4\u03b9\u03ac\u03ba\u03bf\u03bd\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, ^8 \u1f41 \u03ba\u03b1\u1f76 \u03b4\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2\n\u1f21\u03bc\u1fd6\u03bd \u03c4\u1f74\u03bd \u1f51\u03bc\u1ff6\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u1f10\u03bd \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9.\n\n\n\u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3 i.e. 'in its genuine simplicity, without adulteration': see\nthe note on \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5, ver. 5.\n\n7. \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5] '_even as ye were instructed_ in it,' the clause being\nan explanation of the preceding \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3; comp. ii. 7 \u03ba\u03b1\u03b8\u1f7c\u03c2\n\u1f10\u03b4\u03b9\u03b4\u03ac\u03c7\u03b8\u03b7\u03c4\u03b5. On the insertion of \u03ba\u03b1\u1f76 before \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 in the received text,\nand the consequent obscuration of the sense, see above, p. 29 sq. The\ninsertion however was very natural, inasmuch as \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 is an ordinary\ncollocation of particles and has occurred twice in the preceding verse.\n\n\u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6] On the notices of Epaphras, and on his work as the evangelist of\nthe Colossians, see above, p. 29 sq., p. 34 sq., and the note on iv. 12.\n\n\u03c3\u03c5\u03bd\u03b4\u03bf\u03cd\u03bb\u03bf\u03c5] See iv. 7. The word does not occur elsewhere in St Paul.\n\n\u1f51\u03c0\u1f72\u03c1 \u1f21\u03bc\u1ff6\u03bd] As the evangelist of Coloss\u00e6, Epaphras had _represented_ St\nPaul there and preached in his stead; see above, p. 30. The other\nreading \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd might be interpreted in two ways: either (1) It might\ndescribe the personal ministrations of Epaphras to St Paul as the\nrepresentative of the Colossians (see a similar case in Phil. ii. 25,\niv. 18), and so it might be compared with Philem. 13 \u1f35\u03bd\u03b1 \u1f51\u03c0\u1f72\u03c1 \u03c3\u03bf\u1fe6 \u03bc\u03bf\u03b9\n\u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u1fc7; but this interpretation is hardly consistent with \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.\nOr (2) It might refer to the preaching of Epaphras for the good of the\nColossians; but the natural construction in this case would hardly be\n\u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd (of which there is no direct example), but either \u1f51\u03bc\u1ff6\u03bd (Rom.\nxv. 8) or \u1f51\u03bc\u1fd6\u03bd (1 Pet. i. 12). The balance of external authority however\nis against it. Partly by the accidental interchange of similar sounds,\npartly by the recurrence of \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd in the context (vv. 3, 9), and\npartly also from ignorance of the historical circumstances, \u1f51\u03bc\u1ff6\u03bd would\nreadily be substituted for \u1f21\u03bc\u1ff6\u03bd. See the detached note on various\nreadings.\n\n8. \u1f41 \u03ba\u03b1\u1f76 \u03b4\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2] 'As he preached to you from us, so _also_ he brought\nback to us from you the tidings, etc.'\n\n\u1f10\u03bd \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9] to be connected with \u03c4\u1f74\u03bd \u1f51\u03bc\u1ff6\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd. 'The fruit of the\nSpirit is love,' Gal. v. 22. For the omission of the article, \u03c4\u1f74\u03bd \u1f10\u03bd\n\u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9, see the note on ver. 4.\n\n------------------------------------------------------------------------\n\nI. 9]\n\n^9\u0394\u03b9\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2, \u1f00\u03c6' \u1f27\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03bc\u03b5\u03bd, \u03bf\u1f50 \u03c0\u03b1\u03c5\u03cc\u03bc\u03b5\u03b8\u03b1 \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd\n\u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03b1\u1f30\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f35\u03bd\u03b1 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03c4\u03b5 \u03c4\u1f74\u03bd \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2\n\u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f10\u03bd\n\n\n9\u201314. 'Hearing then that ye thus abound in works of faith and love, we\non our part have not ceased, from the day when we received the happy\ntidings, to pray on your behalf. And this is the purport of our\npetitions; that ye may grow more and more in knowledge, till ye attain\nto the perfect understanding of God's will, being endowed with all\nwisdom to apprehend His verities and all intelligence to follow His\nprocesses, living in the mind of the Spirit\u2014to the end that knowledge\nmay manifest itself in practice, that your conduct in life may be worthy\nof your profession in the Lord, so as in all ways to win for you the\ngracious favour of God your King. Thus, while ye bear fruit in every\ngood work, ye will also grow as the tree grows, being watered and\nrefreshed by this knowledge, as by the dew of heaven: thus will ye be\nstrengthened in all strength, according to that power which centres in\nand spreads from His glorious manifestation of Himself, and nerved to\nall endurance under affliction and all long-suffering under provocation,\nnot only without complaining, but even with joy: thus finally (for this\nis the crown of all), so rejoicing ye will pour forth your thanksgiving\nto the Universal Father, who prepared and fitted us all\u2014you and us\nalike\u2014to take possession of the portion which His goodness has allotted\nto us among the saints in the kingdom of light. Yea, by a strong arm He\nrescued us from the lawless tyranny of Darkness, removed us from the\nland of our bondage, and settled us as free citizens in our new and\nglorious home, where His Son, the offspring and the representative of\nHis love, is King; even the same, who paid our ransom and thus procured\nour redemption from captivity\u2014our redemption, which (be assured) is\nnothing else than the remission of our sins.'\n\n9. \u0394\u03b9\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf] '_for this cause_,' i.e. 'by reason of your progressive\nfaith and love,' referring not solely to \u1f41 \u03ba\u03b1\u1f76 \u03b4\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2 \u03ba.\u03c4.\u03bb. but to the\nwhole of the preceding description. For \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2 in an\nexactly similar connexion, see 1 Thess. ii. 13; comp. Ephes. i. 15 \u03b4\u03b9\u1f70\n\u03c4\u03bf\u1fe6\u03c4\u03bf \u03ba\u1f00\u03b3\u1f7c \u03ba.\u03c4.\u03bb. In all these cases the \u03ba\u03b1\u1f76 denotes the _response_ of\nthe Apostle's personal feeling to the favourable character of the news;\n'we on our part.' This idea of correspondence is still further\nemphasized by the repetition of the same words: \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd \u1f00\u03c6' \u1f27\u03c2\n\u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5 (ver. 6), \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2 \u1f00\u03c6' \u1f27\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03bc\u03b5\u03bd (ver. 9).\n\n\u03ba\u03b1\u1f76 \u03b1\u1f30\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9] The words have an exact parallel in Mark xi. 24 (as\ncorrectly read) \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f45\u03c3\u03b1 \u03c0\u03c1\u03bf\u03c3\u03b5\u03cd\u03c7\u03b5\u03c3\u03b8\u03b5 \u03ba\u03b1\u1f76 \u03b1\u1f30\u03c4\u03b5\u1fd6\u03c3\u03b8\u03b5.\n\n\u1f35\u03bd\u03b1] With words like \u03c0\u03c1\u03bf\u03c3\u03b5\u03cd\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9, \u03b1\u1f30\u03c4\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, etc., the earlier and\nstronger force of \u1f35\u03bd\u03b1, implying _design_, glides imperceptibly into its\nlater and weaker use, signifying merely _purport_ or _result_, so that\nthe two are hardly separable, unless one or other is directly indicated\nby something in the context. See the notes on Phil. i. 9, and comp.\nWiner \u00a7 xliv. p. 420 sq.\n\n\u03c4\u1f74\u03bd \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd] A favourite word in the later epistles of St Paul; see\nthe note on Phil. i. 9. In all the four epistles of the first Roman\ncaptivity it is an element in the Apostle's opening prayer for his\ncorrespondents' well-being (Phil. i. 9, Ephes. i. 17, Philem. 6, and\nhere). The greater stress which is thus laid on the contemplative\naspects of the Gospel may be explained partly by St Paul's personal\ncircumstances, partly by the requirements of the Church. His enforced\nretirement and comparative leisure would lead his own thoughts in this\ndirection, while at the same time the fresh dangers threatening the\ntruth from the side of mystic speculation required to be confronted by\nan exposition of the Gospel from a corresponding point of view.\n\nThe compound \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2 is an advance upon \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2, denoting a larger and\nmore thorough knowledge. So Chrysostom here, \u1f14\u03b3\u03bd\u03c9\u03c4\u03b5, \u1f00\u03bb\u03bb\u1f70 \u03b4\u03b5\u1fd6 \u03c4\u03b9 \u03ba\u03b1\u1f76\n\u1f10\u03c0\u03b9\u03b3\u03bd\u1ff6\u03bd\u03b1\u03b9. Comp. Justin Mart. _Dial._ 3. p. 221 A, \u1f21 \u03c0\u03b1\u03c1\u03ad\u03c7\u03bf\u03c5\u03c3\u03b1 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c4\u1ff6\u03bd\n\u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03af\u03bd\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u03b8\u03b5\u03af\u03c9\u03bd _\u03b3\u03bd\u1ff6\u03c3\u03b9\u03bd_, \u1f14\u03c0\u03b5\u03b9\u03c4\u03b1 \u03c4\u1fc6\u03c2 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03b8\u03b5\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76\n\u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7\u03c2 _\u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd_. So too St Paul himself contrasts\n\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd, \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2, with \u1f10\u03c0\u03b9\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd, \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2, as the partial with the\ncomplete, in two passages, Rom. i. 21, 38, 1 Cor. xiii. 12. With this\nlast passage (\u1f04\u03c1\u03c4\u03b9 \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03c9 \u1f10\u03ba \u03bc\u03ad\u03c1\u03bf\u03c5\u03c2, \u03c4\u03cc\u03c4\u03b5 \u03b4\u1f72 \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03bf\u03bc\u03b1\u03b9) compare Clem.\nAlex. _Strom._ i. 17, p. 369, \u03c0\u03b1\u03c1\u1f70 \u03c4\u1ff6\u03bd \u1f19\u03b2\u03c1\u03b1\u03ca\u03ba\u1ff6\u03bd \u03c0\u03c1\u03bf\u03c6\u03b7\u03c4\u1ff6\u03bd _\u03bc\u03ad\u03c1\u03b7_ \u03c4\u1fc6\u03c2\n\u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03bf\u1f50 \u03ba\u03b1\u03c4' _\u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd_ \u03bb\u03b1\u03b2\u03cc\u03bd\u03c4\u03b5\u03c2, where \u03ba\u03b1\u03c4' \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd is\ncommonly but wrongly translated 'without proper recognition' (comp.\nTatian _ad Gr\u00e6c._ 40). Hence also \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2 is used especially of the\nknowledge of God and of Christ, as being the perfection of knowledge:\ne.g. Prov. ii. 5, Hos. iv. 1, vi. 6, Ephes. i. 17, iv. 13, 2 Pet. i. 2,\n8, ii. 20, Clem. Alex. _P\u00e6d._ ii. 1, p. 173.\n\n------------------------------------------------------------------------\n\nI. 10]\n\n\u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03ad\u03c3\u03b5\u03b9 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u1fc7, ^{10}\u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u1fc6\u03c3\u03b1\u03b9 \u1f00\u03be\u03af\u03c9\u03c2 \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u03b5\u1f30\u03c2\n\u03c0\u1fb6\u03c3\u03b1\u03bd \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1\u03bd\u00b7 \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u1f14\u03c1\u03b3\u1ff3\n\n\n\u03c3\u03bf\u03c6\u03af\u1fb3 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03ad\u03c3\u03b5\u03b9] '_wisdom and intelligence_.' The two words are\nfrequently found together: e.g. Exod. xxxi. 3, Deut. iv. 6, 1 Chron.\nxxii. 12, 2 Chron. i. 10 sq., Is. xi. 2, xxix. 14, Dan. ii. 20, Baruch\niii. 23, 1 Cor. i. 19, Clem. Rom. 32. So too \u03c3\u03bf\u03c6\u03bf\u1f76 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b5\u03c4\u03bf\u03af, Prov.\nxvi. 21, Matt. xi. 25, and elsewhere. In the parallel passage, Eph. i.\n8, the words are \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3 \u03ba\u03b1\u1f76 \u03c6\u03c1\u03bf\u03bd\u03ae\u03c3\u03b5\u03b9, and the substitution of\n\u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2 for \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 there is instructive. The three words are mentioned\ntogether, Arist. _Eth. Nic._ i. 13, as constituting the intellectual\n(\u03b4\u03b9\u03b1\u03bd\u03bf\u03b7\u03c4\u03b9\u03ba\u03b1\u1f76) virtues. \u03a3\u03bf\u03c6\u03af\u03b1 is mental excellence in its highest and\nfullest sense; Arist. _Eth. Nic._ vi. 7 \u1f21 \u1f00\u03ba\u03c1\u03b9\u03b2\u03b5\u03c3\u03c4\u03ac\u03c4\u03b7 \u03c4\u1ff6\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03b7\u03bc\u1ff6\u03bd ...\n\u1f65\u03c3\u03c0\u03b5\u03c1 \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74\u03bd \u1f14\u03c7\u03bf\u03c5\u03c3\u03b1 \u1f10\u03c0\u03b9\u03c3\u03c4\u03ae\u03bc\u03b7 \u03c4\u1ff6\u03bd \u03c4\u03b9\u03bc\u03b9\u03c9\u03c4\u03ac\u03c4\u03c9\u03bd (see Waitz on Arist.\n_Organ._ II. p. 295 sq.), Cicero _de Off._ i. 43 'princeps omnium\nvirtutum,' Clem. Alex. _P\u00e6d._ ii. 2, p. 181, \u03c4\u03b5\u03bb\u03b5\u03af\u03b1 ... \u1f10\u03bc\u03c0\u03b5\u03c1\u03b9\u03bb\u03b1\u03b2\u03bf\u1fe6\u03c3\u03b1 \u03c4\u1f70\n\u1f45\u03bb\u03b1. The Stoic definition of \u03c3\u03bf\u03c6\u03af\u03b1, as \u1f10\u03c0\u03b9\u03c3\u03c4\u03ae\u03bc\u03b7 \u03b8\u03b5\u03af\u03c9\u03bd \u03ba\u03b1\u1f76 \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03af\u03bd\u03c9\u03bd \u03ba\u03b1\u1f76\n\u03c4\u1ff6\u03bd \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03b1\u1f30\u03c4\u03b9\u1ff6\u03bd, is repeated by various writers: e.g. Cic. _de Off._\nii. 5, Philo. _Congr. erud. grat._ 14, p. 530, [Joseph.] _Macc._ 2,\nClem. Alex. _P\u00e6d._ ii. 2, p. 181, _Strom._ i. 5, p. 333, Aristob. in\nEus. _Pr\u00e6p. Ev._ xiii. 12 p. 667). And the glorification of \u03c3\u03bf\u03c6\u03af\u03b1 by\nheathen writers was even surpassed by its apotheosis in the Proverbs and\nin the Wisdom of Solomon. While \u03c3\u03bf\u03c6\u03af\u03b1 'wisdom' is thus primary and\nabsolute (_Eth. Nic._ vi. 7 \u03bc\u1f74 \u03bc\u03cc\u03bd\u03bf\u03bd \u03c4\u1f70 \u1f10\u03ba \u03c4\u1ff6\u03bd \u1f00\u03c1\u03c7\u1ff6\u03bd \u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9 \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76\n\u03c0\u03b5\u03c1\u1f76 \u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03cd\u03b5\u03b9\u03bd), both \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 'intelligence' and \u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2\n'prudence' are derivative and special (_Eth. Nic._ vi. 12 \u03c4\u1ff6\u03bd \u1f10\u03c3\u03c7\u03ac\u03c4\u03c9\u03bd\n\u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u03ba\u03b1\u03b8' \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03bd). They are both applications of \u03c3\u03bf\u03c6\u03af\u03b1 to details,\nbut they work on different lines; for, while \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 is critical,\n\u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2 is practical; while \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 apprehends the bearings of things,\n\u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2 suggests lines of action: see Arist. _Eth. Nic._ vi. 11 \u1f21 \u03bc\u1f72\u03bd\n\u03b3\u1f70\u03c1 \u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2 \u1f10\u03c0\u03b9\u03c4\u03b1\u03ba\u03c4\u03b9\u03ba\u03ae \u1f10\u03c3\u03c4\u03b9\u03bd ... \u1f21 \u03b4\u1f72 \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 \u03ba\u03c1\u03b9\u03c4\u03b9\u03ba\u03ae. For \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 see\n2 Tim. ii. 7 _\u03bd\u03cc\u03b5\u03b9_ \u1f43 \u03bb\u03ad\u03b3\u03c9, \u03b4\u03ce\u03c3\u03b5\u03b9 \u03b3\u03ac\u03c1 \u03c3\u03bf\u03b9 \u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 _\u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03bd_\n\u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd. This relation of \u03c3\u03bf\u03c6\u03af\u03b1 to \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 explains why in almost every\ncase \u03c3\u03bf\u03c6\u03af\u03b1 (\u03c3\u03bf\u03c6\u03cc\u03c2) precedes \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 (\u03c3\u03c5\u03bd\u03b5\u03c4\u03cc\u03c2), where they are found\ntogether, and also why in Baruch iii. 23 \u03bf\u1f31 \u1f10\u03ba\u03b6\u03b7\u03c4\u03b7\u03c4\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03bd\u03ad\u03c3\u03b5\u03c9\u03c2, \u1f41\u03b4\u1f78\u03bd\n\u03b4\u1f72 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03bf\u1f50\u03ba \u1f14\u03b3\u03bd\u03c9\u03c3\u03b1\u03bd, we find \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 implying a tentative, partial,\napproach to \u03c3\u03bf\u03c6\u03af\u03b1. The relation of \u03c3\u03bf\u03c6\u03af\u03b1 to \u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2 will be considered\nmore at length in the note on the parallel passage, Ephes. i. 8.\n\n\u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u1fc7] The word is emphatic from its position. The false teachers\nalso offered a \u03c3\u03bf\u03c6\u03af\u03b1, but it had only a show of wisdom (ii. 23); it was\nan empty counterfeit calling itself philosophy (ii. 8); it was the\noffspring of vanity nurtured by the mind of the _flesh_ (ii. 18). See 2\nCor. i. 12 \u03bf\u1f50\u03ba \u1f10\u03bd \u03c3\u03bf\u03c6\u03af\u1fb3 \u03c3\u03b1\u03c1\u03ba\u03b9\u03ba\u1fc7, where a similar contrast is implied,\nand 1 Cor. i. 20, ii. 5, 6, 13, iii. 19, where it is directly expressed\nby \u03c3\u03bf\u03c6\u03af\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5, \u03c3\u03bf\u03c6\u03af\u03b1 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd, \u03c3\u03bf\u03c6\u03af\u03b1 \u03c4\u03bf\u1fe6 \u03b1\u1f30\u1ff6\u03bd\u03bf\u03c2 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5, \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03af\u03bd\u03b7\n\u03c3\u03bf\u03c6\u03af\u03b1, etc.\n\n10. \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u1fc6\u03c3\u03b1\u03b9 \u1f00\u03be\u03af\u03c9\u03c2 \u03ba.\u03c4.\u03bb.] So 1 Thess. ii. 12, Ephes. iv. i; comp.\nPhil. i. 27. The infinitive here denotes the consequence (not\nnecessarily the purpose) of the spiritual enlightenment described in \u1f35\u03bd\u03b1\n\u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03c4\u03b5 \u03ba.\u03c4.\u03bb.; see Winer \u00a7 xliv. p. 399 sq. With the received text\n_\u03c4\u03bf\u1fe6_ \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u1fc6\u03c3\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2 \u1f00\u03be\u03af\u03c9\u03c2 \u03ba.\u03c4.\u03bb. the connexion might be\ndoubtful; but this reading is condemned by external evidence. The\nemphasis of the sentence would be marred by the insertion of \u1f51\u03bc\u1fb6\u03c2. The\nend of all knowledge, the Apostle would say, is conduct.\n\n\u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5] i.e. 'of Christ.' In 1 Thess. ii. 12 indeed we have\n\u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03bd \u1f00\u03be\u03af\u03c9\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6; but St Paul's common, and apparently\nuniversal, usage requires us to understand \u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 of Christ.\n\n\u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1\u03bd] i.e. 'to please _God_ in all ways'; comp. 1 Thess. iv. 1 \u03c0\u1ff6\u03c2\n\u03b4\u03b5\u1fd6 \u1f51\u03bc\u1fb6\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03bd \u03ba\u03b1\u1f76 \u03b1\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03bd \u0398\u03b5\u1ff7. As this word was commonly used to\ndescribe the proper attitude of men towards God, the addition of \u03c4\u03bf\u1fe6\n\u0398\u03b5\u03bf\u1fe6 would not be necessary: Philo _Quis rer. <DW37>. her._ 24 (I. p. 490)\n\u1f61\u03c2 \u1f00\u03c0\u03bf\u03b4\u03b5\u03c7\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 (\u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1f56 \u03c4\u1f70\u03c2 \u03c8\u03c5\u03c7\u1fc6\u03c2 \u1f11\u03ba\u03bf\u03c5\u03c3\u03af\u03bf\u03c5 \u1f00\u03c1\u03b5\u03c3\u03ba\u03b5\u03af\u03b1\u03c2, _de Abrah._ 25\n(II. p. 20) \u03c4\u1f70\u03c2 \u03c0\u03c1\u1f78\u03c2 \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1\u03bd \u1f41\u03c1\u03bc\u03ac\u03c2, _de Vict. Off._ 8 (II. p. 257) \u03b4\u03b9\u1f70\n\u03c0\u03b1\u03c3\u1ff6\u03bd \u1f30\u03ad\u03bd\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03b5\u1f30\u03c2 \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1\u03bd \u1f41\u03b4\u1ff6\u03bd, with other passages quoted by\nLoesner. Otherwise it is used especially of ingratiating oneself with a\nsovereign or potentate, e.g. Polyb. vi. 2. 12; and perhaps in the higher\nconnexion, in which it occurs in the text, the idea of a king is still\nprominent, as e.g. Philo _de Mund. Op._ 50 (I. p. 34) \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03bb\u03ad\u03b3\u03b5\u03b9\u03bd\n\u03ba\u03b1\u1f76 \u03c0\u03c1\u03ac\u03c4\u03c4\u03b5\u03b9\u03bd \u1f10\u03c3\u03c0\u03bf\u03cd\u03b4\u03b1\u03b6\u03b5\u03bd \u03b5\u1f30\u03c2 \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03b2\u03b1\u03c3\u03b9\u03bb\u03ad\u03c9\u03c2. Towards\nmen this complaisance is always dangerous and most commonly vicious;\nhence \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1 is a bad quality in Aristotle [?] (_Eth. Eud._ ii. 3 \u03c4\u1f78\n\u03bb\u03af\u03b1\u03bd \u03c0\u03c1\u1f78\u03c2 \u1f21\u03b4\u03bf\u03bd\u03ae\u03bd) as also in Theophrastus (_Char._ 5 \u03bf\u1f50\u03ba \u1f10\u03c0\u1f76 \u03c4\u1ff7 \u03b2\u03b5\u03bb\u03c4\u03af\u03c3\u03c4\u1ff3\n\u1f21\u03b4\u03bf\u03bd\u1fc6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c3\u03ba\u03b5\u03c5\u03b1\u03c3\u03c4\u03b9\u03ba\u03ae), but towards the King of kings no obsequiousness\ncan be excessive. The \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1 of Aristotle and Theophrastus presents\nthe same moral contrast to the \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1 here, as \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2 \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03bd to\n\u0398\u03b5\u1ff7 \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03bd in such passages as 1 Thess. ii. 4, Gal. i. 10. Opposed to\nthe \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1 commended here is \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03b1\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1 condemned below, iii. 22.\n\n\u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03ba.\u03c4.\u03bb.] i.e. 'not only showing the fruits of your faith before\nmen (Matt. vii. 16), but yourselves growing meanwhile in moral stature\n(Eph. iv. 13).'\n\n------------------------------------------------------------------------\n\nI. 11]\n\n\u1f00\u03b3\u03b1\u03b8\u1ff7 \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03be\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1fc7 \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6\u00b7 ^{11}\u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3\n\u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9 \u03b4\u03c5\u03bd\u03b1\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78\n\n\n\u03c4\u1fc7 \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03b5\u03b9] '_by the knowledge_.' The other readings, \u1f10\u03bd \u03c4\u1fc7 \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03b5\u03b9,\n\u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd, are unsuccessful attempts to define the construction.\nThe simple instrumental dative represents the knowledge of God as the\ndew or the rain which nurtures the growth of the plant; Deut. xxxii. 2,\nHos. xiv. 5.\n\n11. \u03b4\u03c5\u03bd\u03b1\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9] A word found more than once in the Greek versions of\nthe Old Testament, Ps. lxvii (lxviii). 29 (LXX), Eccles. x. 10 (LXX),\nDan. ix. 27 (Theod.), Ps. lxiv (lxv). 4 (Aq.), Job xxxvi. 9 (Aq.), but\nnot occurring elsewhere in the New Testament, except in Heb. xi. 34 and\nas a various reading in Ephes. vi. 10. The compound \u1f10\u03bd\u03b4\u03c5\u03bd\u03b1\u03bc\u03bf\u1fe6\u03bd however\nappears several times in St Paul and elsewhere.\n\n\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78 \u03ba\u03c1\u03ac\u03c4\u03bf\u03c2] The power communicated to the faithful corresponds to,\nand is a function of, the Divine might whence it comes. Unlike \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2\nor \u1f30\u03c3\u03c7\u1f7a\u03c2, the word \u03ba\u03c1\u03ac\u03c4\u03bf\u03c2 in the New Testament is applied solely to God.\n\n------------------------------------------------------------------------\n\nI. 12]\n\n\u03ba\u03c1\u03ac\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b5\u1f30\u03c2 \u03c0\u1fb6\u03c3\u03b1\u03bd \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u1f74\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1\u03bd \u03bc\u03b5\u03c4\u1f70 \u03c7\u03b1\u03c1\u1fb6\u03c2\u00b7\n^{12} \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c4\u1ff7 \u03c0\u03b1\u03c4\u03c1\u1f76 \u03c4\u1ff7 \u1f31\u03ba\u03b1\u03bd\u03ce\u03c3\u03b1\u03bd\u03c4\u03b9\n\n                     12 \u03c4\u1ff7 \u1f31\u03ba\u03b1\u03bd\u03ce\u03c3\u03b1\u03bd\u03c4\u03b9 _\u1f51\u03bc\u1fb6\u03c2_.\n\n\n\u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6] The 'glory' here, as frequently, stands for the majesty\nor the power or the goodness of God, as _manifested_ to men; e.g. Eph.\ni. 6, 12, 17, iii. 16; comp. ver. 27, below. The \u03b4\u03cc\u03be\u03b1, the bright light\nover the mercy-seat (Rom. ix. 4), was a symbol of such manifestations.\nGod's revelation of Himself to us, however this revelation may be made,\nis the one source of all our highest strength (\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78 \u03ba\u03c1\u03ac\u03c4\u03bf\u03c2 \u03ba.\u03c4.\u03bb.).\n\n\u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u1f74\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1\u03bd] '_endurance and long-suffering_.' The two\nwords occur in the same context in 2 Cor. vi. 4, 6, 2 Tim. iii. 10,\nJames v. 10, 11, Clem. Rom. 58, Ign. _Ephes._ 3. They are distinguished\nin Trench _Synon._ \u00a7 liii. p. 184 sq. The difference of meaning is best\nseen in their opposites. While \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u1f74 is the temper which does not\neasily succumb under suffering, \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1 is the self-restraint which\ndoes not hastily retaliate a wrong. The one is opposed to _cowardice_ or\n_despondency_, the other to _wrath_ or _revenge_ (Prov. xv. 18, xvi. 32;\nsee also the note on iii. 12). While \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u1f74 is closely allied to _hope_\n(1 Thess. i. 3), \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1 is commonly connected with _mercy_ (e.g.\nExod. xxxiv. 6). This distinction however, though it applies generally,\nis not true without exception. Thus in Is. lvii. 15 \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1 is\nopposed to \u1f40\u03bb\u03b9\u03b3\u03bf\u03c8\u03c5\u03c7\u03af\u03b1, where we should rather have expected \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u03ae; and\n\u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03b5\u1fd6\u03bd is used similarly in James v. 7.\n\n\u03bc\u03b5\u03c4\u1f70 \u03c7\u03b1\u03c1\u1fb6\u03c2] So James i. 2, 3, \u03c0\u1fb6\u03c3\u03b1\u03bd _\u03c7\u03b1\u03c1\u1f70\u03bd_ \u1f21\u03b3\u03ae\u03c3\u03b1\u03c3\u03b8\u03b5 ... \u1f45\u03c4\u03b1\u03bd\n\u03c0\u03b5\u03b9\u03c1\u03b1\u03c3\u03bc\u03bf\u1fd6\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c0\u03ad\u03c3\u03b7\u03c4\u03b5 \u03c0\u03bf\u03b9\u03ba\u03af\u03bb\u03bf\u03b9\u03c2, \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f45\u03c4\u03b9 \u03c4\u1f78 \u03b4\u03bf\u03ba\u03af\u03bc\u03b9\u03bf\u03bd \u1f51\u03bc\u1ff6\u03bd \u03c4\u1fc6\u03c2\n\u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03ba\u03b1\u03c4\u03b5\u03c1\u03b3\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u03ad\u03bd \u03ba.\u03c4.\u03bb.: comp. 1 Pet. iv. 13, and see below\ni. 24. This parallel points to the proper connexion of \u03bc\u03b5\u03c4\u1f70 \u03c7\u03b1\u03c1\u1fb6\u03c2, which\nshould be attached to the preceding words. On the other hand some would\nconnect it with \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 for the sake of preserving the balance of\nthe three clauses, \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u1f14\u03c1\u03b3\u1ff3 \u1f00\u03b3\u03b1\u03b8\u1ff7 \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2, \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\n\u03b4\u03c5\u03bd\u03b1\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9, \u03bc\u03b5\u03c4\u1f70 \u03c7\u03b1\u03c1\u1fb6\u03c2 \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2; and this seems to be favoured\nby Phil. i. 4 \u03bc\u03b5\u03c4\u1f70 \u03c7\u03b1\u03c1\u1fb6\u03c2 \u03c4\u1f74\u03bd \u03b4\u03ad\u03b7\u03c3\u03b9\u03bd \u03c0\u03bf\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2: but when it is so\nconnected, the emphatic position of \u03bc\u03b5\u03c4\u1f70 \u03c7\u03b1\u03c1\u1fb6\u03c2 cannot be explained; nor\nindeed would these words be needed at all, for \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03af\u03b1 is in itself\nan act of rejoicing.\n\n12. \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2] most naturally coordinated with the preceding\nparticiples and referred to the Colossians. The duty of thanksgiving is\nmore than once enforced upon them below, ii. 7, iii. 17, iv. 2; comp. 1\nThess. v. 18. On the other hand the first person \u1f21\u03bc\u1fb6\u03c2, which follows,\nhas led others to connect \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 with the primary verb of the\nsentence, \u03bf\u1f50 \u03c0\u03b1\u03c5\u03cc\u03bc\u03b5\u03b8\u03b1 ver. 9. But the sudden transition from the second\nto the first person is quite after St Paul's manner (see the note on ii.\n13, 14, \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u1f51\u03bc\u1fb6\u03c2 ... \u03c7\u03b1\u03c1\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f21\u03bc\u1fd6\u03bd), and cannot create any\ndifficulty.\n\n\u03c4\u1ff7 \u1f31\u03ba\u03b1\u03bd\u03ce\u03c3\u03b1\u03bd\u03c4\u03b9] '_who made us competent_'; comp. 2 Cor. iii. 6. On the\nvarious readings see the detached note.\n\n------------------------------------------------------------------------\n\nI. 13]\n\n\u1f21\u03bc\u1fb6\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03bc\u03b5\u03c1\u03af\u03b4\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03bb\u03ae\u03c1\u03bf\u03c5 \u03c4\u1ff6\u03bd \u1f01\u03b3\u03af\u03c9\u03bd \u1f10\u03bd \u03c4\u1ff7 \u03c6\u03c9\u03c4\u03af\u00b7 ^{13}\u1f43\u03c2 \u1f10\u03c1\u03cd\u03c3\u03b1\u03c4\u03bf \u1f21\u03bc\u1fb6\u03c2\n\u1f10\u03ba \u03c4\u1fc6\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6\n\n\n\u03c4\u1f74\u03bd \u03bc\u03b5\u03c1\u03af\u03b4\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03bb\u03ae\u03c1\u03bf\u03c5] '_the parcel of the lot_,' 'the portion which\nconsists in the lot,' \u03c4\u03bf\u1fe6 \u03ba\u03bb\u03ae\u03c1\u03bf\u03c5 being the genitive of apposition: see\nWiner \u00a7 lix. p. 666 sq., and comp. Ps. xv (xvi). 5 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 \u03bc\u03b5\u03c1\u1f76\u03c2 \u03c4\u1fc6\u03c2\n\u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03c2 \u03bc\u03bf\u03c5. In Acts viii. 21 \u03bc\u03b5\u03c1\u1f76\u03c2 and \u03ba\u03bb\u1fc6\u03c1\u03bf\u03c2 are coordinated; in\nGen. xxxi. 14, Num. xviii. 20, Is. lvii. 6, \u03bc\u03b5\u03c1\u1f76\u03c2 and \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03bf\u03bc\u03af\u03b1. The\ninheritance of Canaan, the allotment of the promised land, here presents\nan analogy to, and supplies a metaphor for, the higher hopes of the new\ndispensation, as in Heb. iii. 7-iv. 11. See also below, iii. 24 \u03c4\u1f74\u03bd\n\u1f00\u03bd\u03c4\u03b1\u03c0\u03cc\u03b4\u03bf\u03c3\u03b9\u03bd \u03c4\u1fc6\u03c2 \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03c2, and Ephes. i. 18. St Chrysostom writes, \u03b4\u03b9\u1f70\n\u03c4\u03af _\u03ba\u03bb\u1fc6\u03c1\u03bf\u03bd_ \u03ba\u03b1\u03bb\u03b5\u1fd6\u00b7 \u03b4\u03b5\u03b9\u03ba\u03bd\u1f7a\u03c2 \u1f45\u03c4\u03b9 \u03bf\u1f50\u03b4\u1f72\u03b9\u03c2 \u1f00\u03c0\u1f78 \u03ba\u03b1\u03c4\u03bf\u03c1\u03b8\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u03bf\u1f30\u03ba\u03b5\u03af\u03c9\u03bd\n\u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03c2 \u03c4\u03c5\u03b3\u03c7\u03ac\u03bd\u03b5\u03b9, referring to Luke xvii. 10. It is not won by us, but\nallotted to us.\n\n\u1f10\u03bd \u03c4\u1ff7 \u03c6\u03c9\u03c4\u03af] best taken with the expression \u03c4\u1f74\u03bd \u03bc\u03b5\u03c1\u03af\u03b4\u03b1 \u03ba.\u03c4.\u03bb. For the\nomission of the definite article, [\u03c4\u1f74\u03bd] \u1f10\u03bd \u03c4\u1ff7 \u03c6\u03c9\u03c4\u1f76, see above, vv. 2, 4,\n8. The portion of the saints is situated in the kingdom of light. For\nthe whole context compare St Paul's narrative in Acts xxvi. 18 \u03c4\u03bf\u1fe6\n\u1f10\u03c0\u03b9\u03c3\u03c4\u03c1\u03ad\u03c8\u03b1\u03b9 _\u1f00\u03c0\u1f78 \u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2 \u03b5\u1f30\u03c2 \u03c6\u1ff6\u03c2_ \u03ba\u03b1\u1f76 _\u03c4\u1fc6\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2_ \u03c4\u03bf\u1fe6 \u03a3\u03b1\u03c4\u03b1\u03bd\u1fb6\n\u1f10\u03c0\u1f76 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd, \u03c4\u03bf\u1fe6 \u03bb\u03b1\u03b2\u03b5\u1fd6\u03bd \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 _\u1f04\u03c6\u03b5\u03c3\u03b9\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03b9\u1ff6\u03bd_ \u03ba\u03b1\u1f76 _\u03ba\u03bb\u1fc6\u03c1\u03bf\u03bd \u1f10\u03bd\n\u03c4\u03bf\u1fd6\u03c2 \u1f21\u03b3\u03b9\u03b1\u03c3\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2_, where all the ideas and many of the expressions\nrecur. See also Acts xx. 32, in another of St Paul's later speeches. As\na classical parallel, Plato _Resp._ vii. p. 518 A, \u1f14\u03ba \u03c4\u03b5 \u03c6\u03c9\u03c4\u1f78\u03c2 \u03b5\u1f30\u03c2\n\u03c3\u03ba\u03cc\u03c4\u03bf\u03c2 \u03bc\u03b5\u03b8\u03b9\u03c3\u03c4\u03b1\u03bc\u03ad\u03bd\u03c9\u03bd \u03ba\u03b1\u1f76 \u1f10\u03ba \u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2 \u03b5\u1f30\u03c2 \u03c6\u1ff6\u03c2, is quoted.\n\n13. 'We were slaves in the land of darkness. God rescued us from this\nthraldom. He transplanted us thence, and settled us as free colonists\nand citizens in the kingdom of His Son, in the realms of light.'\n\n\u1f10\u03c1\u03cd\u03c3\u03b1\u03c4\u03bf] '_rescued, delivered us_' by His strong arm, as a mighty\nconqueror: comp. ii. 15 \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03c3\u03b1\u03c2. On the form \u1f10\u03c1\u03cd\u03c3\u03b1\u03c4\u03bf see A.\nButtmann, p. 29: comp. Clem. Rom. 55, and see the note on \u1f10\u03be\u03b5\u03c1\u03af\u03b6\u03c9\u03c3\u03b5\u03bd,\n_ib._ 6.\n\n\u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2] here 'arbitrary power, tyranny.' The word \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1 properly\nsignifies 'liberty of action' (\u1f14\u03be\u03b5\u03c3\u03c4\u03b9), and thence, like the\ncorresponding English word 'license,' invokes two secondary ideas, of\nwhich either may be so prominent as to eclipse the other; (1)\n'authority,' '_delegated_ power' (e.g. Luke xx. 2); or (2) 'tyranny,'\n'lawlessness,' '_unrestrained_ or _arbitrary_ power.' For this second\nsense comp. e.g. Demosth. _F.L._ p. 428 \u03c4\u1f74\u03bd \u1f04\u03b3\u03b1\u03bd \u03c4\u03b1\u03cd\u03c4\u03b7\u03bd \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03bd,\nXenoph. _Hiero_ 5 \u03c4\u1fc6\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f78 \u03c0\u03b1\u03c1\u1f78\u03bd \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u1f15\u03bd\u03b5\u03ba\u03b1 (speaking of tyrants),\nPlut. _Vit. Eum._ 13 \u1f00\u03bd\u03ac\u03b3\u03c9\u03b3\u03bf\u03b9 \u03c4\u03b1\u1fd6\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03b1\u03bb\u03b1\u03ba\u03bf\u1f76 \u03c4\u03b1\u1fd6\u03c2 \u03b4\u03b9\u03b1\u03af\u03c4\u03b1\u03b9\u03c2,\n_Vit. Alex._ 33 \u03c4\u1f74\u03bd \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78\u03bd \u1f44\u03b3\u03ba\u03bf\u03bd \u03c4\u1fc6\u03c2 \u1f08\u03bb\u03b5\u03be\u03ac\u03bd\u03b4\u03c1\u03bf\u03c5 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c9\u03c2,\nHerodian ii. 4 \u03ba\u03b1\u03b8\u03b1\u03af\u03c1\u03b5\u03c3\u03b9\u03bd \u03c4\u1fc6\u03c2 \u1f00\u03bd\u03ad\u03c4\u03bf\u03c5 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2. This latter idea of a\ncapricious unruly rule is prominent here. The expression \u1f21 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1 \u03c4\u03bf\u1fe6\n\u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2 occurs also in Luke xxii. 53, where again the idea of disorder\nis involved. The transference from darkness to light is here represented\nas a transference from an arbitrary tyranny, an \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1, to a\nwell-ordered sovereignty, a \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1. This seems also to be St\nChrysostom's idea; for he explains \u03c4\u1fc6\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 by \u03c4\u1fc6\u03c2 \u03c4\u03c5\u03c1\u03b1\u03bd\u03bd\u03af\u03b4\u03bf\u03c2,\nadding \u03c7\u03b1\u03bb\u03b5\u03c0\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f00\u03c0\u03bb\u1ff6\u03c2 \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f51\u03c0\u1f78 \u03c4\u1ff7 \u03b4\u03b9\u03b1\u03b2\u03cc\u03bb\u1ff3\u00b7 \u03c4\u1f78 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03c4'\n\u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2, \u03c4\u03bf\u1fe6\u03c4\u03bf \u03c7\u03b1\u03bb\u03b5\u03c0\u03ce\u03c4\u03b5\u03c1\u03bf\u03bd.\n\n\u03bc\u03b5\u03c4\u03ad\u03c3\u03c4\u03b7\u03c3\u03b5\u03bd] '_removed_,' when they were baptized, when they accepted\nChrist. The image of \u03bc\u03b5\u03c4\u03ad\u03c3\u03c4\u03b7\u03c3\u03b5\u03bd is supplied by the wholesale\ntransportation of peoples (\u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03c4\u03bf\u03c5\u03c2 or \u1f00\u03bd\u03b1\u03c3\u03c0\u03ac\u03c3\u03c4\u03bf\u03c5\u03c2 \u03c0\u03bf\u03b9\u03b5\u1fd6\u03bd), of which\nthe history of oriental monarchies supplied so many examples. See\nJoseph. _Ant._ ix. 11. 1 \u03c4\u03bf\u1f7a\u03c2 \u03bf\u1f30\u03ba\u03ae\u03c4\u03bf\u03c1\u03b1\u03c2 \u03b1\u1f30\u03c7\u03bc\u03b1\u03bb\u03c9\u03c4\u03af\u03c3\u03b1\u03c2 \u03bc\u03b5\u03c4\u03ad\u03c3\u03c4\u03b7\u03c3\u03b5\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd\n\u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03bd, speaking of Tiglath-Pileser and the Transjordanic\ntribes.\n\n------------------------------------------------------------------------\n\nI. 13]\n\n\u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2, \u03ba\u03b1\u1f76 \u03bc\u03b5\u03c4\u03ad\u03c3\u03c4\u03b7\u03c3\u03b5\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03bd \u03c4\u03bf\u1fe6 \u03c5\u1f31\u03bf\u1fe6 \u03c4\u1fc6\u03c2\n\n\n\u03c4\u03bf\u1fe6 \u03c5\u1f31\u03bf\u1fe6] Not of inferior angels, as the false teachers would have it\n(ii. 18), but of His own Son. The same contrast between a dispensation\nof angels and a dispensation of the Son underlies the words here, which\nis explicitly brought out in Heb. i. 1-ii. 8; see especially i. 2\n\u1f10\u03bb\u03ac\u03bb\u03b7\u03c3\u03b5\u03bd \u1f21\u03bc\u1fd6\u03bd \u1f10\u03bd \u03c5\u1f31\u1ff7, compared with ii. 5 \u03bf\u1f50 \u03b3\u1f70\u03c1 \u1f00\u03b3\u03b3\u03ad\u03bb\u03bf\u03b9\u03c2 \u1f51\u03c0\u03ad\u03c4\u03b1\u03be\u03b5\u03bd \u03c4\u1f74\u03bd\n\u03bf\u1f30\u03ba\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03bd \u03c4\u1f74\u03bd \u03bc\u03ad\u03bb\u03bb\u03bf\u03c5\u03c3\u03b1\u03bd. Severianus has rightly caught the idea\nunderlying \u03c4\u03bf\u1fe6 \u03c5\u1f31\u03bf\u1fe6 here; \u1f51\u03c0\u1f78 \u03c4\u1f78\u03bd \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03cc\u03bc\u03bf\u03bd \u1f10\u03c3\u03bc\u03ad\u03bd, \u03bf\u1f50\u03c7 \u1f51\u03c0\u1f78 \u03c4\u03bf\u1f7a\u03c2\n\u03bf\u1f30\u03ba\u03ad\u03c4\u03b1\u03c2.\n\n\u03c4\u1fc6\u03c2 \u1f00\u03b3\u03ac\u03c0\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6] '_of His love_.' As love is the essence of the Father\n(1 Joh. iv. 8, 16), so is it also of the Son. The mission of the Son is\nthe revelation of the Father's love; for as He is the \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u03ae\u03c2, the\nFather's love is perfectly represented in Him (see 1 Joh. iv. 9). St\nAugustine has rightly interpreted St Paul's words here, _de Trin._ XV.\n19 (VIII. p. 993) 'Caritas quippe Patris ... nihil est quam ejus ipsa\nnatura atque substantia ... ac per hoc filius caritatis ejus nullus est\nalius quam qui de ejus substantia est genitus.' Thus these words are\nintimately connected with the expressions which follow, \u03b5\u1f30\u03ba\u1f7c\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6\n\u03c4\u03bf\u1fe6 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5 (ver. 15), and \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9\n(ver. 19). The loose interpretation, which makes \u03c4\u03bf\u1fe6 \u03c5\u1f31\u03bf\u1fe6 \u03c4\u1fc6\u03c2 \u1f00\u03b3\u03ac\u03c0\u03b7\u03c2\nequivalent to \u03c4\u03bf\u1fe6 \u03c5\u1f31\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03c5, destroys the whole force of the\nexpression.\n\nIn the preceding verses we have a striking illustration of St Paul's\nteaching in two important respects. _First._ The reign of Christ has\nalready begun. His kingdom is a present kingdom. Whatever therefore is\n_essential_ in the kingdom of Christ must be capable of realisation now.\nThere may be some exceptional manifestation in the world to come, but\nthis cannot alter its inherent character. In other words the sovereignty\nof Christ is essentially a moral and spiritual sovereignty, which has\nbegun now and will only be perfected hereafter. _Secondly._\nCorresponding to this, and equally significant, is his language in\nspeaking of individual Christians. He regards them as already rescued\nfrom the power of darkness, as already put in possession of their\ninheritance as saints. They are _potentially_ saved, because the\nknowledge of God is itself salvation, and this knowledge is within their\nreach. Such is St Paul's constant mode of speaking. He uses the language\nnot of exclusion, but of comprehension. He prefers to dwell on their\npotential advantages, rather than on their actual attainments. He hopes\nto make them saints by dwelling on their calling as saints. See\nespecially Ephes. ii. 6 \u03c3\u03c5\u03bd\u03ae\u03b3\u03b5\u03b9\u03c1\u03b5\u03bd \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b5\u03ba\u03ac\u03b8\u03b9\u03c3\u03b5\u03bd \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f10\u03c0\u03bf\u03c5\u03c1\u03b1\u03bd\u03af\u03bf\u03b9\u03c2\n\u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03ba.\u03c4.\u03bb.\n\n------------------------------------------------------------------------\n\nI. 14]\n\n\u1f00\u03b3\u03ac\u03c0\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, ^14\u1f10\u03bd \u1fa7 \u1f14\u03c7\u03bf\u03bc\u03b5\u03bd \u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd, \u03c4\u1f74\u03bd \u1f04\u03c6\u03b5\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03b9\u1ff6\u03bd\u00b7\n\n                        14 \u1f10\u03bd \u1fa7 _\u1f14\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd_.\n\n\n14. \u1f14\u03c7\u03bf\u03bc\u03b5\u03bd] For the reading \u1f14\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd, which is possibly correct here, and\nwhich carries out the idea enforced in the last note, see the detached\nnote on the various readings. In the parallel passage, Ephes. i. 7,\nthere is the same variation of reading.\n\n\u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd] '_ransom, redemption_.' The image of a captive and\nenslaved people is still continued: Philo _Omn. prob. lib._ 17 (II. p.\n463) \u03b1\u1f30\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03bf\u03c2 \u1f00\u03c0\u03ae\u03c7\u03b8\u03b7 ... \u1f00\u03c0\u03bf\u03b3\u03bd\u03bf\u1f7a\u03c2 \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd, Plut. _Vit. Pomp._ 24\n\u03c0\u03cc\u03bb\u03b5\u03c9\u03bd \u03b1\u1f30\u03c7\u03bc\u03b1\u03bb\u03ce\u03c4\u03c9\u03bd \u1f00\u03c0\u03bf\u03bb\u03c5\u03c4\u03c1\u03ce\u03c3\u03b5\u03b9\u03c2. The metaphor however has changed from\nthe victor who rescues the captive by force of arms (ver. 13 \u1f10\u03c1\u03cd\u03c3\u03b1\u03c4\u03bf) to\nthe philanthropist who releases him by the payment of a ransom. The\nclause which follows in the received text, \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u1f01\u03af\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, is\ninterpolated from the parallel passage, Ephes. i. 7.\n\n\u03c4\u1f74\u03bd \u1f04\u03c6\u03b5\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03b9\u1ff6\u03bd] So in the parallel passage Ephes. i. 7 the\nApostle defines \u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd as \u03c4\u1f74\u03bd \u1f04\u03c6\u03b5\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd \u03c0\u03b1\u03c1\u03b1\u03c0\u03c4\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd. May not\nthis studied precision point to some false conception of \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03c2 put\nforward by the heretical teachers? Later Gnostics certainly perverted\nthe meaning of the term, applying it to their own formularies of\ninitiation. This is related of the Marcosians by Iren\u00e6us i. 13. 6 \u03b4\u03b9\u1f70\n\u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd \u1f00\u03ba\u03c1\u03b1\u03c4\u03ae\u03c4\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5\u03c2 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c4\u1ff7 \u03ba\u03c1\u03b9\u03c4\u1fc7 \u03ba.\u03c4.\u03bb., i. 21.\n1 \u1f45\u03c3\u03bf\u03b9 \u03b3\u03ac\u03c1 \u03b5\u1f30\u03c3\u03b9 \u03c4\u03b1\u03cd\u03c4\u03b7\u03c2 \u03c4\u1fc6\u03c2 \u03b3\u03bd\u03ce\u03bc\u03b7\u03c2 \u03bc\u03c5\u03c3\u03c4\u03b1\u03b3\u03c9\u03b3\u03bf\u03af, \u03c4\u03bf\u03c3\u03b1\u1fe6\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u1f00\u03c0\u03bf\u03bb\u03c5\u03c4\u03c1\u03ce\u03c3\u03b5\u03b9\u03c2,\n_ib._ \u00a7 4 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03b4\u1f72 \u03c4\u03b5\u03bb\u03b5\u03af\u03b1\u03bd \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd \u03b1\u1f50\u03c4\u1f74\u03bd \u03c4\u1f74\u03bd \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u1f00\u03c1\u03c1\u03ae\u03c4\u03bf\u03c5\n\u03bc\u03b5\u03b3\u03ad\u03b8\u03bf\u03c5\u03c2 (with the whole context), and Hippolytus _H\u00e6r._ vi. 41 \u03bb\u03ad\u03b3\u03bf\u03c5\u03c3\u03af\n\u03c4\u03b9 \u03c6\u03c9\u03bd\u1fc7 \u1f00\u03c1\u03c1\u03ae\u03c4\u1ff3, \u1f10\u03c0\u03b9\u03c4\u03b9\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2 \u03c7\u03b5\u1fd6\u03c1\u03b1 \u03c4\u1ff7 \u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd \u03bb\u03b1\u03b2\u03cc\u03bd\u03c4\u03b9 \u03ba.\u03c4.\u03bb.\n(comp. ix. 13). In support of their nomenclature they perverted such\npassages as the text, Iren. i. 21. 2 \u03c4\u1f78\u03bd \u03a0\u03b1\u1fe6\u03bb\u03bf\u03bd \u1fe5\u03b7\u03c4\u1ff6\u03c2 \u03c6\u03ac\u03c3\u03ba\u03bf\u03c5\u03c3\u03b9 \u03c4\u1f74\u03bd \u1f10\u03bd\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd _\u03c0\u03bf\u03bb\u03bb\u03ac\u03ba\u03b9\u03c2_ \u03bc\u03b5\u03bc\u03b7\u03bd\u03c5\u03ba\u03ad\u03bd\u03b1\u03b9. It seems not\nimprobable that the communication of similar mystical secrets, perhaps\nconnected with their angelology (ii. 18), was put forward by these\nColossian false teachers as an \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03c2. Compare the words in the\nbaptismal formula of the Marcosians as given in Iren. i. 21. 3 (comp.\nTheodt. _H\u00e6r. Fab._ i. 9) \u03b5\u1f30\u03c2 \u1f15\u03bd\u03c9\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1\u03bd \u03c4\u1ff6\u03bd\n\u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c9\u03bd, where the last words (which have been differently interpreted)\nmust surely mean 'communion with the (spiritual) powers.' Thus it is a\nparallel to \u03b5\u1f30\u03c2 \u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd \u1f00\u03b3\u03b3\u03b5\u03bb\u03b9\u03ba\u03ae\u03bd, which appears in an alternative\nformula of these heretics given likewise by Iren\u00e6us in the context; for\nthis latter is explained in Clem. Alex. _Exc. Theod._ p. 974, \u03b5\u1f30\u03c2\n\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03bd \u1f00\u03b3\u03b3\u03b5\u03bb\u03b9\u03ba\u03ae\u03bd, \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9\u03bd, \u1f23\u03bd \u03ba\u03b1\u1f76 \u1f04\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9 \u1f14\u03c7\u03bf\u03c5\u03c3\u03b9\u03bd. Any direct\nhistorical connexion between the Colossian heretics and these later\nGnostics of the Valentinian school is very improbable; but the passages\nquoted will serve to show how a false idea of \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03c2 would\nnaturally be associated with an esoteric doctrine of angelic powers. See\nthe note on i. 28 \u1f35\u03bd\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ae\u03c3\u03c9\u03bc\u03b5\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd.\n\n------------------------------------------------------------------------\n\nI. 15]\n\n^{15}\u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03b5\u1f30\u03ba\u1f7c\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2\n\n\n15 sq. In the passage which follows St Paul defines the Person of\nChrist, claiming for Him the absolute supremacy,\n\n  (1) In relation to the _Universe_, the _Natural_ Creation (vv. 15\u201317);\n\n  (2) In relation to the _Church_, the new _Moral_ Creation (ver. 18);\n\nand he then combines the two, \u1f35\u03bd\u03b1 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 _\u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd_ \u03b1\u1f50\u03c4\u1f78\u03c2\n\u03c0\u03c1\u03c9\u03c4\u03b5\u03cd\u03c9\u03bd, explaining this twofold sovereignty by the absolute indwelling\nof the _pleroma_ in Christ, and showing how, as a consequence, the\nreconciliation and harmony of all things must be effected in Him (vv.\n19, 20).\n\nAs the idea of the _Logos_ underlies the whole of this passage, though\nthe term itself does not appear, a few words explanatory of this term\nwill be necessary by way of preface. The word \u03bb\u03cc\u03b3\u03bf\u03c2 then, denoting both\n'reason' and 'speech,' was a philosophical term adopted by Alexandrian\nJudaism before St Paul wrote, to express the _manifestation_ of the\nUnseen God, the Absolute Being, in the creation and government of the\nWorld. It included all modes by which God makes himself known to man. As\nHis _reason_, it denoted His purpose or design; as His _speech_, it\nimplied His revelation. Whether this \u03bb\u03cc\u03b3\u03bf\u03c2 was conceived merely as the\ndivine energy personified, or whether the conception took a more\nconcrete form, I need not stop now to enquire. A fuller account of the\nmatter will be found in the dissertation at the end of this volume. It\nis sufficient for the understanding of what follows to say that\nChristian teachers, when they adopted this term, exalted and fixed its\nmeaning by attaching to it two precise and definite ideas: (1) 'The Word\nis a Divine Person,' \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f26\u03bd \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd \u03ba\u03b1\u1f76 \u0398\u03b5\u1f78\u03c2 \u1f26\u03bd \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2; and\n(2) 'The Word became incarnate in Jesus Christ,' \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c3\u1f70\u03c1\u03be \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf.\nIt is obvious that these two propositions must have altered materially\nthe significance of all the subordinate terms connected with the idea of\nthe \u03bb\u03cc\u03b3\u03bf\u03c2; and that therefore their use in Alexandrian writers, such as\nPhilo, cannot be taken to _define_, though it may be brought to\n_illustrate_, their meaning in St Paul and St John. With these cautions\nthe Alexandrian phraseology, as a providential preparation for the\nteaching of the Gospel, will afford important aid in the understanding\nof the Apostolic writings.\n\n15\u201317. 'He is the perfect image, the visible representation, of the\nunseen God. He is the Firstborn, the absolute Heir of the Father,\nbegotten before the ages; the Lord of the Universe by virtue of\nprimogeniture, and by virtue also of creative agency. For in and through\nHim the whole world was created, things in heaven and things on earth,\nthings visible to the outward eye and things cognisable by the inward\nperception. His supremacy is absolute and universal. All powers in\nheaven and earth are subject to Him. This subjection extends even to the\nmost exalted and most potent of angelic beings, whether they be called\nThrones or Dominations or Princedoms or Powers, or whatever title of\ndignity men may confer upon them. Yes: He is first and He is last.\nThrough Him, as the mediatorial Word, the universe has been created; and\nunto Him, as the final goal, it is tending. In Him is no before or\nafter. He is pre-existent and self-existent before all the worlds. And\nin Him, as the binding and sustaining power, universal nature coheres\nand consists.'\n\n15. \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03ba.\u03c4.\u03bb.] The Person of Christ is described _first_ in\nrelation more especially to Deity, as \u03b5\u1f30\u03ba\u1f7c\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5, and\n_secondly_ in relation more especially to created things, as \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2\n\u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2. The fundamental conception of the Logos involves the idea\nof _mediation_ between God and creation. A perverted view respecting the\nnature of the mediation between the two lay, as we have seen, at the\nroot of the heretical teaching at Coloss\u00e6 (p. 34, p. 101 sq., p. 181\nsq.), and required to be met by the true doctrine of Christ as the\nEternal Logos.\n\n\u03b5\u1f30\u03ba\u03ce\u03bd] '_the image_.' This expression is used repeatedly by Philo, as a\ndescription of the Logos; _de Mund. Op._ 8 (I. p. 6) \u03c4\u1f78\u03bd \u1f00\u03cc\u03c1\u03b1\u03c4\u03bf\u03bd \u03ba\u03b1\u1f76\n\u03bd\u03bf\u03b7\u03c4\u1f78\u03bd \u03b8\u03b5\u1fd6\u03bf\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u03bb\u03ad\u03b3\u03b5\u03b9 \u0398\u03b5\u03bf\u1fe6, _de Confus. ling._ 20 (I. p. 419)\n\u03c4\u1f74\u03bd \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6, \u03c4\u1f78\u03bd \u1f31\u03b5\u03c1\u03ce\u03c4\u03b1\u03c4\u03bf\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd, _ib._ \u00a7 28 (I. p. 427) \u03c4\u1fc6\u03c2 \u1f00\u03ca\u03b4\u03af\u03bf\u03c5\n\u03b5\u1f30\u03ba\u03cc\u03bd\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03bb\u03cc\u03b3\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u1f31\u03b5\u03c1\u03c9\u03c4\u03ac\u03c4\u03bf\u03c5 \u03ba.\u03c4.\u03bb., _de Profug._ 19 (I. p. 561) \u1f41\n\u1f51\u03c0\u03b5\u03c1\u03ac\u03bd\u03c9 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03bb\u03cc\u03b3\u03bf\u03c2 \u03b8\u03b5\u1fd6\u03bf\u03c2 ... \u03b1\u1f50\u03c4\u1f78\u03c2 \u03b5\u1f30\u03ba\u1f7c\u03bd \u1f51\u03c0\u03ac\u03c1\u03c7\u03c9\u03bd \u0398\u03b5\u03bf\u1fe6, _de Monarch._\nii. 5 (II. p. 225) \u03bb\u03cc\u03b3\u03bf\u03c2 \u03b4\u03ad \u1f10\u03c3\u03c4\u03b9\u03bd \u03b5\u1f30\u03ba\u1f7c\u03bd \u0398\u03b5\u03bf\u1fe6 \u03b4\u03b9' \u1f41\u1fe6 \u03c3\u03cd\u03bc\u03c0\u03b1\u03c2 \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2\n\u1f10\u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03b5\u1fd6\u03c4\u03bf, _de Somn._ i. 41 (I. p. 656), etc. For the use which\nPhilo made of the text Gen. i. 26, 27, \u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u1f21\u03bc\u03b5\u03c4\u03ad\u03c1\u03b1\u03bd, \u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1\n\u0398\u03b5\u03bf\u1fe6, see the note on iii. 10. Still earlier than Philo, before the idea\nof the \u03bb\u03cc\u03b3\u03bf\u03c2 had assumed such a definite form, the term was used of the\nDivine \u03c3\u03bf\u03c6\u03af\u03b1 personified in Wisd. vii. 26 \u1f00\u03c0\u03b1\u03cd\u03b3\u03b1\u03c3\u03bc\u03b1 \u03b3\u03ac\u03c1 \u1f10\u03c3\u03c4\u03b9 \u03c6\u03c9\u03c4\u1f78\u03c2\n\u1f00\u03ca\u03b4\u03af\u03bf\u03c5 ... \u03ba\u03b1\u1f76 \u03b5\u1f30\u03ba\u1f7c\u03bd \u03c4\u1fc6\u03c2 \u1f00\u03b3\u03b1\u03b8\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6. St Paul himself applies the\nterm to our Lord in an earlier epistle, 2 Cor. iv. 4 \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 \u03c4\u03bf\u1fe6\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03b5\u1f30\u03ba\u1f7c\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 (comp. iii. 18 \u03c4\u1f74\u03bd \u03b1\u1f50\u03c4\u1f74\u03bd\u03b5\u1f30\u03ba\u03cc\u03bd\u03b1\n\u03bc\u03b5\u03c4\u03b1\u03bc\u03bf\u03c1\u03c6\u03bf\u03cd\u03bc\u03b5\u03b8\u03b1). Closely allied to \u03b5\u1f76\u03ba\u1f7c\u03bd also is \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03ae\u03c1, which appears\nin the same connexion in Heb. i. 3 \u1f62\u03bd \u1f00\u03c0\u03b1\u03cd\u03b3\u03b1\u03c3\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u1f74\u03c1\n\u03c4\u1fc6\u03c2 \u1f51\u03c0\u03bf\u03c3\u03c4\u03ac\u03c3\u03b5\u03c9\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, a passage illustrated by Philo _de Plant._ 5 (I.\np. 332) \u03c3\u03c6\u03c1\u03b1\u03b3\u1fd6\u03b4\u03b9 \u0398\u03b5\u03bf\u1fe6 \u1f27\u03c2 \u1f41 \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03ae\u03c1 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u0390\u03b4\u03b9\u03bf\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c2. See also Phil.\nii. 6 \u1f10\u03bd \u03bc\u03bf\u03c1\u03c6\u1fc7 \u0398\u03b5\u03bf\u1fe6 \u1f51\u03c0\u03ac\u03c1\u03c7\u03c9\u03bd.\n\nBeyond the very obvious notion of _likeness_, the word \u03b5\u1f30\u03ba\u03ce\u03bd involves\ntwo other ideas;\n\n(1) _Representation._ In this respect it is allied to \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03ae\u03c1, and\ndiffers from (\u03bf\u03bc\u03bf\u03af\u03c9\u03bc\u03b1. In \u1f41\u03bc\u03bf\u03af\u03c9\u03bc\u03b1 the resemblance may be accidental, as\none egg is like another; but \u03b5\u1f31\u03ba\u03ce\u03bd implies an archetype of which it is a\n_copy_, as Greg. Naz. _Orat._ 30 (I. p. 554) says \u1f05\u03c5\u03c4\u03b7 \u03b3\u1f70\u03c1 \u03b5\u1f30\u03ba\u03cc\u03bd\u03bf\u03c2 \u03c6\u03cd\u03c3\u03b9\u03c2\n_\u03bc\u03af\u03bc\u03b7\u03bc\u03b1_ \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u03bf\u1fe6 _\u1f00\u03c1\u03c7\u03b5\u03c4\u03cd\u03c0\u03bf\u03c5_. So too Io. Damasc. _de Imag._\ni. 9 (I. p. 311) \u03b5\u1f30\u03ba\u03ce\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f41\u03bc\u03bf\u03af\u03c9\u03bc\u03b1 _\u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03b7\u03c1\u03af\u03b6\u03bf\u03bd_ \u03c4\u1f78\n_\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03c5\u03c0\u03bf\u03bd_; comp. Philo _de Mund. Op. 23_ (I. p. 16). On this\ndifference see Trench _N. T. Synon._ \u00a7 xv. p. 47. The \u03b5\u1f30\u03ba\u1f7c\u03bd might be the\nresult of direct imitation (\u03bc\u03b9\u03bc\u03b7\u03c4\u03b9\u03ba\u03ae) like the head of a sovereign on a\ncoin, or it might be due to natural causes (\u03c6\u03c5\u03c3\u03b9\u03ba\u03ae) like the parental\nfeatures in the child, but in any case it was _derived_ from its\nprototype: see Basil. _de Spir. Sanct._ 18 \u00a7 45 (III. p. 38). The word\nitself however does not necessarily imply _perfect_ representation. Thus\nman is said to be the image of God; 1 Cor. xi. 7 \u03b5\u1f30\u03ba\u1f7c\u03bd \u03ba\u03b1\u1f76 \u03b4\u03cc\u03be\u03b1 \u0398\u03b5\u03bf\u1fe6\n\u1f51\u03c0\u03ac\u03c1\u03c7\u03c9\u03bd, Clem. Rom. 33 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd ... \u03c4\u1fc6\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 \u03b5\u1f30\u03ba\u03cc\u03bd\u03bf\u03c2 \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u1fc6\u03c1\u03b1. Thus\nagain an early Jud\u00e6o-Christian writer so designates the duly appointed\nbishop, as the representative of the divine authority; _Clem. Hom._ iii.\n62 \u1f61\u03c2 \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u0398\u03b5\u03bf\u1fe6 \u03c0\u03c1\u03bf\u03c4\u03b9\u03bc\u1ff6\u03bd\u03c4\u03b1\u03c2. The idea of _perfection_ does not lie in\nthe word itself, but must be sought from the context (e.g. \u03c0\u1fb6\u03bd \u03c4\u1f78\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 ver. 19). The use which was made of this expression, and\nespecially of this passage, in the Christological controversies of the\nfourth and fifth centuries may be seen from the patristic quotations in\nPetav. _Theol. Dogm._ de Trin. ii. 11. 9 sq., vi. 5. 6.\n\n(2) _Manifestation._ This idea comes from the implied contrast to \u03c4\u03bf\u1fe6\n_\u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5_ \u0398\u03b5\u03bf\u1fe6. St Chrysostom indeed maintains the direct opposite,\narguing that, as the archetype is invisible, so the image must be\ninvisible also, \u1f21 \u03c4\u03bf\u1fe6 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5 \u03b5\u1f30\u03ba\u1f7c\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f74 \u1f00\u03cc\u03c1\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f41\u03bc\u03bf\u03af\u03c9\u03c2 \u1f00\u03cc\u03c1\u03b1\u03c4\u03bf\u03c2.\nSo too Hilary _c. Const. Imp._ 21 (II. p. 378) 'ut imago invisibilis\nDei, etiam per id quod ipse invisibilis est, invisibilis Dei imago\nesset.' And this was the view of the Nicene and post-Nicene fathers\ngenerally. But the underlying idea of the \u03b5\u1f30\u03ba\u03ce\u03bd, and indeed of the \u03bb\u03cc\u03b3\u03bf\u03c2\ngenerally, is the manifestation of the hidden: comp. Philo _de Vit.\nMoys._ ii. 12 (II. p. 144) \u03b5\u1f30\u03ba\u1f7c\u03bd \u03c4\u1fc6\u03c2 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5 \u03c6\u03cd\u03c3\u03b5\u03c9\u03c2 \u1f10\u03bc\u03c6\u03b1\u03bd\u03ae\u03c2. And adopted\ninto Christian theology, the doctrine of the \u03bb\u03cc\u03b3\u03bf\u03c2 expresses this\nconception still more prominently by reason of the Incarnation; comp.\nTertull. _adv. Marc._ v. 19 'Scientes filium semper retro visum, si\nquibus visus est in Dei nomine, ut imaginem ipsius,' Hippol. _c. Noet._\n7 \u03b4\u03b9\u1f70 \u03b3\u1f70\u03c1 \u03c4\u1fc6\u03c2 \u03b5\u1f30\u03ba\u03cc\u03bd\u03bf\u03c2 \u1f41\u03bc\u03bf\u03af\u03b1\u03c2 \u03c4\u03c5\u03b3\u03c7\u03b1\u03bd\u03bf\u03cd\u03c3\u03b7\u03c2 _\u03b5\u1f54\u03b3\u03bd\u03c9\u03c3\u03c4\u03bf\u03c2_ \u1f41 \u03c0\u03b1\u03c4\u1f74\u03c1\n\u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9, _ib._ \u00a7 12, 13, Orig. _in Ioann._ vi. \u00a7 2 (IV. p. 104). Among\nthe post-Nicene fathers too St Basil has caught the right idea, _Epist._\nxxxviii. 8 (III. p. 121) \u1f41 \u03c4\u1fc6\u03c2 \u03b5\u1f30\u03ba\u03cc\u03bd\u03bf\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03ae\u03c3\u03b1\u03c2 \u03ba\u03ac\u03bb\u03bb\u03bf\u03c2 \u1f10\u03bd \u03c0\u03b5\u03c1\u03b9\u03bd\u03bf\u03af\u1fb3 \u03c4\u03bf\u1fe6\n\u1f00\u03c1\u03c7\u03b5\u03c4\u03cd\u03c0\u03bf\u03c5 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 ... \u03b2\u03bb\u03ad\u03c0\u03b5\u03b9\u03bd \u03b4\u03b9\u1f70 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u1f10\u03ba\u03b5\u1fd6\u03bd\u03bf\u03bd ... \u03c4\u1f78 \u1f00\u03b3\u03ad\u03bd\u03bd\u03b7\u03c4\u03bf\u03bd \u03ba\u03ac\u03bb\u03bb\u03bf\u03c2\n\u1f10\u03bd \u03c4\u1ff7 \u03b3\u03b5\u03bd\u03bd\u03b7\u03c4\u1ff7 \u03ba\u03b1\u03c4\u03bf\u03c0\u03c4\u03b5\u03cd\u03c3\u03b1\u03c2. The Word, whether pre-incarnate or incarnate,\nis the revelation of the unseen Father: comp. John i. 18 \u0398\u03b5\u1f78\u03bd _\u03bf\u1f50\u03b4\u03b5\u1f76\u03c2\n\u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd_ \u03c0\u03ce\u03c0\u03bf\u03c4\u03b5\u00b7 \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u1f74\u03c2 \u0398\u03b5\u03cc\u03c2, \u1f41 \u1f62\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u03ba\u03cc\u03bb\u03c0\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u03cc\u03c2,\n\u1f10\u03ba\u03b5\u1fd6\u03bd\u03bf\u03c2 _\u1f10\u03be\u03b7\u03b3\u03ae\u03c3\u03b1\u03c4\u03bf_, xiv. 9, 10 _\u1f41 \u1f11\u03c9\u03c1\u03ac\u03ba\u03c9\u03c2 \u1f10\u03bc\u1f72 \u1f11\u03ce\u03c1\u03b1\u03ba\u03ad\u03bd \u03c4\u1f78\u03bd\n\u03c0\u03b1\u03c4\u03ad\u03c1\u03b1_\u00b7 \u03c0\u1ff6\u03c2 \u03c3\u1f7a \u03bb\u03ad\u03b3\u03b5\u03b9\u03c2, _\u0394\u03b5\u1fd6\u03be\u03bf\u03bd_ \u1f21\u03bc\u1fd6\u03bd \u03c4\u1f78\u03bd \u03c0\u03b1\u03c4\u03ad\u03c1\u03b1; (compared with\nvi. 46 \u03bf\u1f50\u03c7 \u1f45\u03c4\u03b9 \u03c4\u1f78\u03bd \u03c0\u03b1\u03c4\u03ad\u03c1\u03b1 \u1f11\u03ce\u03c1\u03b1\u03ba\u03ad\u03bd \u03c4\u03b9\u03c2 \u03ba.\u03c4.\u03bb.). The epithet \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5\nhowever must not be confined to the apprehension of the bodily senses,\nbut will include the cognisance of the inward eye also.\n\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2] '_the First-born of all creation_.' The word\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 has a twofold parentage:\n\n(1) Like \u03b5\u1f30\u03ba\u03ce\u03bd it is closely connected with and taken from the\nAlexandrian vocabulary of the Logos. The word however which Philo\napplies to the \u03bb\u03cc\u03b3\u03bf\u03c2 is not \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 but \u03c0\u03c1\u03c9\u03c4\u03cc\u03b3\u03bf\u03bd\u03bf\u03c2: _de Agric._ 12\n(I. p. 308) \u03c0\u03c1\u03bf\u03c3\u03c4\u03b7\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c4\u1f78\u03bd \u1f40\u03c1\u03b8\u1f78\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03bb\u03cc\u03b3\u03bf\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03b3\u03bf\u03bd\u03bf\u03bd \u1f51\u03af\u03bf\u03bd, _de\nSomn._ i. 37 (I. p. 653) \u1f41 \u03c0\u03c1\u03c9\u03c4\u03cc\u03b3\u03bf\u03bd\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b8\u03b5\u1fd6\u03bf\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c2, _de Confus.\nling._ i. 28 (I. p. 427) \u03c3\u03c0\u03bf\u03c5\u03b4\u03b1\u03b6\u03ad\u03c4\u03c9 \u03ba\u03bf\u03c3\u03bc\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03b3\u03bf\u03bd\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6\n\u03bb\u03cc\u03b3\u03bf\u03bd: comp. _ib._ i. 14 (I. p. 414) \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b1\u03c4\u03bf\u03bd \u03c5\u1f31\u1f78\u03bd \u1f41 \u03c4\u1ff6\u03bd \u1f44\u03bd\u03c4\u03c9\u03bd\n\u1f00\u03bd\u03ad\u03c4\u03b5\u03b9\u03bb\u03b5 \u03c0\u03b1\u03c4\u03ae\u03c1, \u1f43\u03bd \u1f11\u03c4\u03ad\u03c1\u03c9\u03b8\u03b9 \u03c0\u03c1\u03c9\u03c4\u03cc\u03b3\u03bf\u03bd\u03bf\u03bd \u1f60\u03bd\u03cc\u03bc\u03b1\u03c3\u03b5: and this designation\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b1\u03c4\u03bf\u03c2 \u03c5\u1f31\u1f78\u03c2 is several times applied to the \u03bb\u03cc\u03b3\u03bf\u03c2. Again in _Quis\nrer. <DW37>. her._ \u00a7 24 (I. p. 489) the language of Exod. xiii. 2 \u1f01\u03b3\u03af\u03b1\u03c3\u03cc\u03bd\n\u03bc\u03bf\u03b9 \u03c0\u1fb6\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03c0\u03c1\u03c9\u03c4\u03bf\u03b3\u03b5\u03bd\u03ad\u03c2 \u03ba.\u03c4.\u03bb. is so interpreted as to apply to\nthe Divine Word. These appellations, 'the first-begotten, the eldest\nson,' are given to the Logos by Philo, because in his philosophy it\nincludes the original conception, the archetypal idea, of creation,\nwhich was afterwards realised in the material world. Among the early\nChristian fathers Justin Martyr again and again recognises the\napplication of the term \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 to the Word; _Apol._ i. 23 (p. 68)\n\u03bb\u03cc\u03b3\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f51\u03c0\u03ac\u03c1\u03c7\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2, _ib._ \u00a7 46 (p. 83) \u03c4\u1f78\u03bd\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03b5\u1f36\u03bd\u03b1\u03b9 ... \u03bb\u03cc\u03b3\u03bf\u03bd \u1f44\u03bd\u03c4\u03b1 \u03bf\u1f57 \u03c0\u1fb6\u03bd \u03b3\u03ad\u03bd\u03bf\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd\n\u03bc\u03b5\u03c4\u03ad\u03c3\u03c7\u03b5, _ib._ \u00a7 33 (p. 75 C) \u03c4\u1f78\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u1f43\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c4\u1ff7 \u0398\u03b5\u1ff7 \u1f10\u03c3\u03c4\u03b9.\nSo too Theophilus _ad Antol._ ii. 22 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03c4\u1f78\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u1f10\u03b3\u03ad\u03bd\u03bd\u03b7\u03c3\u03b5\u03bd\n\u03c0\u03c1\u03bf\u03c6\u03bf\u03c1\u03b9\u03ba\u03cc\u03bd, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2.\n\n(2) The word \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 had also another not less important link of\nconnexion with the past. The Messianic reference of Ps. lxxxix. 28, \u1f10\u03b3\u1f7c\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03b8\u03ae\u03c3\u03bf\u03bc\u03b1\u03b9 \u03b1\u1f50\u03c4\u1f78\u03bd \u03ba.\u03c4.\u03bb., seems to have been generally allowed.\nSo at least it is interpreted by R. Nathan in _Shemoth Rabba_ 19, fol.\n118. 4, 'God said, As I made Jacob a first-born (Exod. iv. 22), so also\nwill I make king Messiah a first-born (Ps. lxxxix. 28).' Hence 'the\nfirst-born' \u1f41 \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 (\u05d1\u05db\u05d5\u05e8), used absolutely, became a recognised\ntitle of Messiah. The way had been paved for this Messianic reference of\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 by its prior application to the Israelites, as the\nprerogative race, Exod. iv. 22 'Israel is my son, my first-born': comp.\nPsalm. Salom. xviii. 4 \u1f21 \u03c0\u03b1\u03b9\u03b4\u03b5\u03af\u03b1 \u03c3\u03bf\u03c5 \u1f10\u03c6' \u1f21\u03bc\u1fb6\u03c2 \u1f61\u03c2 \u03c5\u1f31\u1f78\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd\n\u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u1fc6, 4 Esdr. vi. 58 'nos populus tuus, quem vocasti primogenitum,\nunigenitum,' where the combination of the two titles applied in the New\nTestament to the Son is striking. Here, as elsewhere (see the note on\nGal. iii. 16 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u03c3\u03c0\u03ad\u03c1\u03bc\u03b1\u03c3\u03b9\u03bd \u03ba.\u03c4.\u03bb.), the terms are transferred from\nthe race to the Messiah, as the representative, the embodiment, of the\nrace.\n\nAs the Person of Christ was the Divine response alike to the\nphilosophical questionings of the Alexandrian Jew and to the patriotic\nhopes of the Palestinian, these two currents of thought meet in the term\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 as applied to our Lord, who is both the true Logos and the\ntrue Messiah. For this reason, we may suppose, as well as for others,\nthe Christian Apostles preferred \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 to \u03c0\u03c1\u03c9\u03c4\u03cc\u03b3\u03bf\u03bd\u03bf\u03c2, which (as we\nmay infer from Philo) was the favourite term with the Alexandrians,\nbecause the former alone would include the Messianic reference as well.\n\nThe main ideas then which the word involves are twofold; the one more\ndirectly connected with the Alexandrian conception of the Logos, the\nother more nearly allied to the Palestinian conception of the Messiah.\n\n(1) _Priority_ to all creation. In other words it declares the absolute\npre-existence of the Son. At first sight it might seem that Christ is\nhere regarded as one, though the earliest, of created things. This\ninterpretation however is not required by the expression itself. The\nfathers of the fourth century rightly called attention to the fact that\nthe Apostle writes not \u03c0\u03c1\u03c9\u03c4\u03cc\u03ba\u03c4\u03b9\u03c3\u03c4\u03bf\u03c2, but \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2; e.g. Basil, _c.\nEunom._ iv (p. I. p. 292). Much earlier, in Clem. Alex. _Exc. Theod._ 10\n(p. 970), though without any direct reference to this passage, the\n\u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u1f74\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 is contrasted with the \u03c0\u03c1\u03c9\u03c4\u03cc\u03ba\u03c4\u03b9\u03c3\u03c4\u03bf\u03b9, the\nhighest order of angelic beings; and the word \u03c0\u03c1\u03c9\u03c4\u03cc\u03ba\u03c4\u03b9\u03c3\u03c4\u03bf\u03c2 occurs more\nthan once elsewhere in his writings (e.g. _Strom._ v. 14, p. 699). Nor\nagain does the genitive case necessarily imply that the \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2\nHimself belonged to the \u03ba\u03c4\u03af\u03c3\u03b9\u03c2, as will be shown presently. And if this\nsense is not required by the words themselves, it is directly excluded\nby the context. It is inconsistent alike with the universal agency in\ncreation which is ascribed to Him in the words following, \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7\n\u1f10\u03ba\u03c4\u03af\u03c3\u03b8\u03b7 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, and with the absolute pre-existence and self-existence\nwhich is claimed for Him just below, \u03b1\u1f50\u03c4\u1f78\u03c2 \u1f14\u03c3\u03c4\u03b9\u03bd \u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd. We may add\nalso that it is irreconcileable with other passages in the Apostolic\nwritings, while it contradicts the fundamental idea of the Christian\nconsciousness. More especially the description \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2\nmust be interpreted in such a way that it is not inconsistent with His\nother title of \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u03ae\u03c2, _unicus_, alone of His kind and therefore\ndistinct from created things. The two words express the same eternal\nfact; but while \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u03ae\u03c2 states it in itself, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 places it in\nrelation to the Universe. The correct interpretation is supplied by\nJustin Martyr, _Dial._ \u00a7 100 (p. 326 D) \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c0\u03c1\u1f78\n\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03c4\u1ff6\u03bd \u03ba\u03c4\u03b9\u03c3\u03bc\u03ac\u03c4\u03c9\u03bd. He does not indeed mention this passage, but it\nwas doubtless in his mind, for he elsewhere uses the very expression\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2, _Dial._ \u00a7 85 (p. 311 B), \u00a7 138 (p. 367 D);\ncomp. also \u00a7 84 (p. 310 B), where the words \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c4\u1ff6\u03bd \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd\n\u03c0\u03bf\u03b9\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd occur.\n\n(2) _Sovereignty_ over all creation. God's 'first-born' is the natural\nruler, the acknowledged head, of God's household. The right of\nprimogeniture appertains to Messiah over all created things. Thus in Ps.\nlxxxix. 28 after \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03b8\u03ae\u03c3\u03bf\u03bc\u03b1\u03b9 \u03b1\u1f50\u03c4\u1f78\u03bd the explanation is added,\n\u1f51\u03c8\u03b7\u03bb\u1f78\u03bd \u03c0\u03b1\u03c1\u1f70 \u03c4\u03bf\u1fd6\u03c2 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u1fe6\u03c3\u03b9\u03bd \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2, i.e. (as the original implies)\n'above all the kings of the earth.' In its Messianic reference this\nsecondary idea of sovereignty predominated in the word \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2, so\nthat from this point of view \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 would mean\n'Sovereign Lord over all creation by virtue of primogeniture.' The\n\u1f14\u03b8\u03b7\u03ba\u03b5\u03bd \u03ba\u03bb\u03b7\u03c1\u03cc\u03bd\u03bf\u03bc\u03bf\u03bd \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd of the Apostolic writer (Heb. i. 2) exactly\ncorresponds to the \u03b8\u03ae\u03c3\u03bf\u03bc\u03b1\u03b9 \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd of the Psalmist (lxxxix. 28), and\ndoubtless was tacitly intended as a paraphrase and application of this\nMessianic passage. So again in Heb. xii. 23, \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3 \u03c0\u03c1\u03c9\u03c4\u03bf\u03c4\u03cc\u03ba\u03c9\u03bd, the\nmost probable explanation of the word is that which makes it equivalent\nto 'heirs of the kingdom,' all faithful Christians being _ipso facto_\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03b9, because all are kings. Nay, so completely might this idea of\ndominion by virtue of priority eclipse the primary sense of the term\n'first-born' in some of its uses, that it is given as a title to God\nHimself by R. Bechai on the Pentateuch, fol. 124. 4, 'Who is\n_primogenitus mundi_,' \u05e9\u05d4\u05d5\u05d0 \u05d1\u05db\u05d5\u05e8\u05d9 \u05e9\u05dc \u05e2\u05d5\u05dc\u05dd, i.e. \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u03ba\u03cc\u03c3\u03bc\u03bf\u03c5, as it would be rendered in Greek. In this same work again, fol.\n74. 4, Exod. xiii. 2 is falsely interpreted so that God is represented\nas calling Himself 'primogenitus': see Sch\u00f6ttgen p. 922. For other\ninstances of secondary uses of \u05d1\u05db\u05d5\u05e8 in the Old Testament, where the idea\nof 'priority of birth' is over-shadowed by and lost in the idea of\n'pre-eminence,' see Job xviii. 13 'the first-born of death,' Is. xiv. 30\n'the first-born of the poor'.\n\n\u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 '_of all creation_,' rather than '_of every created\nthing_.' The three senses of \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 in the New Testament; are (1)\ncreation, as the act of creating, e.g. Rom. i. 20 \u1f00\u03c0\u1f78 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5:\n(2) creation, as the aggregate of created things, Mark xiii. 19 \u1f00\u03c0'\n\u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 \u1f23\u03bd \u1f14\u03ba\u03c4\u03b9\u03c3\u03b5\u03bd \u1f41 \u0398\u03b5\u03cc\u03c2 (where the parallel passage, Matt. xxiv.\n21, has \u1f00\u03c0' \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5), Rom. viii. 22 \u03c0\u1fb6\u03c3\u03b1 \u1f21 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 \u03c3\u03c5\u03c3\u03c4\u03b5\u03bd\u03ac\u03b6\u03b5\u03b9: (3) a\ncreation, a single created thing, a creature, e.g. Rom. viii. 39 \u03bf\u1f54\u03c4\u03b5\n\u03c4\u03b9\u03c2 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 \u1f11\u03c4\u03ad\u03c1\u03b1, Heb. iv. 13 \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9\u03bd \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 \u1f00\u03c6\u03b1\u03bd\u03ae\u03c2. As \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 without\nthe definite article is sometimes used of the created world generally\n(e.g. Mark xiii. 19), and indeed belongs to the category of anarthrous\nnouns like \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2, \u03b3\u1fc6, \u03bf\u1f50\u03c1\u03b1\u03bd\u03cc\u03c2, etc. (see Winer \u00a7 xix. p. 149 sq.), it\nis best taken so here. Indeed \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2, in the sense of \u03c0\u03ac\u03bd\u03c4\u03bf\u03c2\n\u03ba\u03c4\u03af\u03c3\u03bc\u03b1\u03c4\u03bf\u03c2, would be awkward in this connexion; for \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 seems to\nrequire either a collective noun, or a plural \u03c0\u03b1\u03c3\u1ff6\u03bd \u03c4\u1ff6\u03bd \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03bd. In ver.\n23 the case is different (see the note there). The anarthrous \u03c0\u1fb6\u03c3\u03b1\n\u03ba\u03c4\u03af\u03c3\u03b9\u03c2 is found in Judith ix. 12 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u1fe6 \u03c0\u03b1\u03c3\u1fc6\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03ce\u03c2 \u03c3\u03bf\u03c5, while \u03c0\u1fb6\u03c3\u03b1 \u1f21\n\u03ba\u03c4\u03af\u03c3\u03b9\u03c2 occurs in Judith xvi. 14, Mark xvi. 15, Rom. viii. 22, Clem. Rom.\n19, _Mart. Polyc._ 14. For \u03c0\u1fb6\u03c2, signifying '_all_,' and not '_every_,'\nwhen attached to this class of nouns, see Winer \u00a7 xviii. p. 137.\n\nThe genitive case must be interpreted so as to include the full meaning\nof \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2, as already explained. It will therefore signify: 'He\nstands in the relation of \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 to all creation,' i.e. 'He is the\nFirstborn, and, as the Firstborn, the absolute Heir and sovereign Lord,\nof all creation.' The connexion is the same as in the passage of R.\nBechai already quoted, where God is called _primogenitus mundi_. Another\nexplanation which would connect the genitive with the first part of the\ncompound alone (\u03c0\u03c1\u03c9\u03c4\u03cc-), comparing Joh. i. 15, 30, \u03c0\u03c1\u1ff6\u03c4\u03cc\u03c2 \u03bc\u03bf\u03c5 \u1f26\u03bd, unduly\nstrains the grammar, while it excludes the idea of 'heirship,\nsovereignty.'\n\nThe history of the patristic exegesis of this expression is not without\na painful interest. All the fathers of the second and third centuries\nwithout exception, so far as I have noticed, correctly refer it to the\nEternal Word and not to the Incarnate Christ, to the Deity and not to\nthe humanity of our Lord. So Justin _l.c._, Theophilus _l.c._, Clement\nof Alexandria _Exc. Theod._ 7, 8, 19 (pp. 967, 973), Tertullian _adv.\nPrax._ 7, _adv. Marc._ v. 19, Hippolytus _H\u00e6r._ x. 33, Origen _c. Cels._\nvi. 47, 63, 64, _in Ioann._ i. \u00a7 22 (IV. p. 21), xix. \u00a7 5 (p. 305),\nxxviii. \u00a7 14 (p. 392), Cyprian _Test._ ii. 1, Novatian _de Trin._ 16,\nand the Synod of Antioch (Routh's _Rel. Sacr._ III. pp. 290, 293). The\nArian controversy however gave a different turn to the exegesis of the\npassage. The Arians fastened upon the expression \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2\n\u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2, and drew from it the inference that the Son was a created\nbeing. The great use which they made of the text appears from the\ndocument in Hilary, _Fragm. Hist._ Op. II. p. 644. The right answer to\nthis false interpretation we have already seen. Many orthodox fathers\nhowever, not satisfied with this, transferred the expression into a new\nsphere, and maintained that \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 describes the\nIncarnate Christ. By so doing they thought to cut up the Arian argument\nby the roots. As a consequence of this interpretation, they were obliged\nto understand the \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 and the \u03ba\u03c4\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9 in the context of the new\nspiritual creation, the \u03ba\u03b1\u03b9\u03bd\u1f74 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 of 2 Cor. v. 17, Gal. vi. 15. Thus\ninterpreted, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 here becomes nearly equivalent to\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u1f10\u03bd \u03c0\u03bf\u03bb\u03bb\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03c2 in Rom. viii. 29. The arguments alleged\nin favour of this interpretation are mainly twofold: (1) That, if\napplied to the Divine nature, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 would contradict \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u1f74\u03c2\nwhich elsewhere describes the nature of the Eternal Son. But those who\nmaintained, and rightly maintained, that \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 (Luke ii. 7) did not\nnecessarily imply that the Lord's mother had other sons, ought not to\nhave been led away by this fallacy. (2) That \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 in other\npassages (e.g. Rom. viii. 29, Rev. i. 5, and just below, ver. 18) is\napplied to the humanity of Christ. But elsewhere, in Heb. i. 6 \u1f45\u03c4\u03b1\u03bd \u03b4\u1f72\n\u03c0\u03ac\u03bb\u03b9\u03bd \u03b5\u1f30\u03c3\u03b1\u03b3\u03ac\u03b3\u1fc3 \u03c4\u1f78\u03bd \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03bd \u03ba.\u03c4.\u03bb., the term must almost necessarily\nrefer to the pre-existence of the Son; and moreover the very point of\nthe Apostle's language in the text (as will be seen presently) is the\nparallelism in the two relations of our Lord\u2014His relation to the natural\ncreation, as the Eternal Word, and His relation to the spiritual\ncreation, as the Head of the Church\u2014so that the same word (\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2\n\u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 ver. 15, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u1f10\u03ba \u03c4\u1ff6\u03bd \u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd ver. 18) is studiously\nused of both. A false exegesis is sure to bring a nemesis on itself.\nLogical consistency required that this interpretation should be carried\nfarther; and Marcellus, who was never deterred by any considerations of\nprudence, took this bold step. He extended the principle to the whole\ncontext, including even \u03b5\u1f76\u03ba\u1f7c\u03bd \u03c4\u03bf\u1fe6 \u1f00\u03bf\u03c1\u03ac\u03c4\u03bf\u03c5 \u0398\u03b5\u03bf\u1fe6, which likewise he\ninterpreted of our Lord's humanity. In this way a most important\nChristological passage was transferred into an alien sphere; and the\nstrongest argument against Arianism melted away in the attempt to combat\nArianism on false grounds. The criticisms of Eusebius on Marcellus are\nperfectly just: _Eccl. Theol._ i. 20 (p. 96) \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c0\u03b5\u03c1\u1f76 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u03c5\u1f31\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, \u03ba\u1f02\u03bd \u03bc\u1f74 \u039c\u03b1\u03c1\u03ba\u03ad\u03bb\u03bb\u1ff3 \u03b4\u03bf\u03ba\u1fc7, \u03b5\u1f34\u03c1\u03b7\u03c4\u03b1\u03b9\u00b7 \u03bf\u1f50 \u03b3\u1f70\u03c1 \u1f02\u03bd \u03c0\u03b5\u03c1\u1f76 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2\n\u1f02\u03bd \u03b5\u1f36\u03c0\u03b5\u03bd \u03c4\u03bf\u03c3\u03b1\u1fe6\u03c4\u03b1 \u1f41 \u03b8\u03b5\u1fd6\u03bf\u03c2 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03c2 \u03ba.\u03c4.\u03bb.; comp. _ib._ ii. 9 (p. 67),\niii. 6 sq. (p. 175), _c. Marcell._ i. 1 (p. 6), i. 2 (p. 12), ii. 3 (pp.\n43, 46 sq., 48). The objections to this interpretation are threefold:\n(1) It disregards the history of the terms in their connexion with the\npre-Christian speculations of Alexandrian Judaism. These however, though\ndirectly or indirectly they were present to the minds of the earlier\nfathers and kept them in the right exegetical path, might very easily\nhave escaped a writer in the fourth century. (2) It shatters the\ncontext. To suppose that such expressions as \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u1f10\u03ba\u03c4\u03af\u03c3\u03b8\u03b7 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1\n[\u03c4\u1f70] \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 [[\u03c4\u1f70] \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2, or \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 ...\n\u1f14\u03ba\u03c4\u03b9\u03c3\u03c4\u03b1\u03b9, or \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5\u03bd, refer to the work of the\nIncarnation, is to strain language in a way which would reduce all\ntheological exegesis to chaos; and yet this, as Marcellus truly saw, is\na strictly logical consequence of the interpretation which refers\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 to Christ's humanity. (3) It takes no account\nof the cosmogony and angelology of the false teachers against which the\nApostle's exposition here is directed (see above, pp. 101 sq., 110 sq.,\n181 sq.). This interpretation is given by St Athanasius _c. Arian._ ii.\n62 sq. (I. p. 419 sq.) and appears again in Greg. Nyss. _c. Eunom._ ii\n(II. pp. 451\u2013453, 492), _ib._ iii (II. p. 540\u2013545), _de Perf._ (III. p.\n290 sq.), Cyril Alex. _Thes._ 25, p. 236 sq., _de Trin. Dial._ iv. p.\n517 sq., vi. p. 625 sq., Anon. _Chrysost. Op._ VIII. p. 223, appx.\n(quoted as Chrysostom by Photius _Bibl._ 277). So too Cyril expresses\nhimself at the Council of Ephesus, Labb. _Conc._ III. p. 652 (ed.\nColet.). St Athanasius indeed does not confine the expression to the\ncondescension (\u03c3\u03c5\u03b3\u03ba\u03b1\u03c4\u03ac\u03b2\u03b1\u03c3\u03b9\u03c2) of the Word in the Incarnation, but\nincludes also a prior condescension in the Creation of the world (see\nBull _Def. Fid. Nic._ iii. 9. \u00a7 1, with the remarks of Newman _Select\nTreatises of S. Athanasius_ I. pp. 278, 368 sq.). This double reference\nhowever only confuses the exegesis of the passage still further, while\ntheologically it might lead to very serious difficulties. In another\nwork, _Expos. Fid._ 3 (I. p. 80), he seems to take a truer view of its\nmeaning. St Basil, who to an equally clear appreciation of doctrine\ngenerally unites a sounder exegesis than St Athanasius, while mentioning\nthe interpretation which refers the expression to Christ's human nature,\nhimself prefers explaining it of the Eternal Word; _c. Eunom._ iv (I. p.\n292). Of the Greek commentators on this passage, Chrysostom's view is\nnot clear; Severianus (Cram. _Cat._ p. 303) and Theodoret understand it\nrightly of the Eternal Word; while Theodore of Mopsuestia (Cram. _Cat._\npp. 306, 308, 309, Rab. Maur. _Op._ VI. p. 511 sq. ed. Migne) expresses\nhimself very strongly on the opposite side. Like Marcellus, he carries\nthe interpretation consistently into the whole context, explaining \u1f10\u03bd\n\u03b1\u1f50\u03c4\u1ff7 to refer not to the original creation (\u03ba\u03c4\u03af\u03c3\u03b9\u03c2) but to the moral\nre-creation (\u1f00\u03bd\u03ac\u03ba\u03c4\u03b9\u03c3\u03b9\u03c2), and referring \u03b5\u1f30\u03ba\u03ce\u03bd to the Incarnation in the\nsame way. At a later date, when the pressure of an immediate controversy\nhas passed away, the Greek writers generally concur in the earlier and\ntruer interpretation of the expression. Thus John Damascene (_de Orthod.\nFid._ iv. 8, I. p. 258 sq.), Theophylact (_ad loc._), and [OE]cumenius\n(_ad loc._), all explain it of Christ's Divine Nature. Among Latin\nwriters, there is more diversity of interpretation. While Marius\nVictorinus (_adv. Arium_ i. 24, p. 1058, ed. Migne), Hilary of Poictiers\n(_Tract. in ii Ps._ \u00a7 28 sq. I. p. 47 sq. _de Trin._ viii. 50, II. p.\n248 sq.), and Hilary the commentator (_ad loc._), take it of the Divine\nNature, Augustine (_Expos. ad Rom._ 56, III. p. 914) and Pelagius (_ad\nloc._) understand it of the Incarnate Christ. This sketch of the history\nof the interpretation of the expression would not be complete without a\nreference to another very different explanation. Isidore of Pelusium,\n_Epist._ iii. 31 (p. 268), would strike out a new path of interpretation\naltogether (\u03b5\u1f30 \u03ba\u03b1\u1f76 \u03b4\u03cc\u03be\u03b1\u03b9\u03bc\u03af \u03c4\u03b9\u03c3\u03b9 \u03ba\u03b1\u03b9\u03bd\u03bf\u03c4\u03ad\u03c1\u03b1\u03bd \u1f11\u03c1\u03bc\u03b7\u03bd\u03af\u03b1\u03c2 \u1f00\u03bd\u03b1\u03c4\u03ad\u03bc\u03bd\u03b5\u03b9\u03bd \u1f41\u03b4\u03cc\u03bd),\nand for the passive \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 suggests reading the active \u03c0\u03c1\u03c9\u03c4\u03bf\u03c4\u03cc\u03ba\u03bf\u03c2,\nalluding to the use of this latter word in Homer (_Il._ xvii. 5 \u03bc\u03ae\u03c4\u03b7\u03c1\n\u03c0\u03c1\u03c9\u03c4\u03bf\u03c4\u03cc\u03ba\u03bf\u03c2 ... \u03bf\u1f50 \u03c0\u03c1\u1f76\u03bd \u03b5\u1f30\u03b4\u03c5\u1fd6\u03b1 \u03c4\u03cc\u03ba\u03bf\u03b9\u03bf: comp. Plat. _The\u00e6t._ 151 C \u1f65\u03c3\u03c0\u03b5\u03c1\n\u03b1\u1f31 \u03c0\u03c1\u03c9\u03c4\u03bf\u03c4\u03cc\u03ba\u03bf\u03b9). Thus St Paul is made to say that Christ \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd\n\u03c4\u03b5\u03c4\u03bf\u03ba\u03ad\u03bd\u03b1\u03b9, \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9, \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03ba\u03ad\u03bd\u03b1\u03b9 \u03c4\u1f74\u03bd \u03ba\u03c4\u03af\u03c3\u03b9\u03bd.\n\n------------------------------------------------------------------------\n\nI. 16]\n\n\u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2\u00b7 ^{16} \u1f45\u03c4\u03b9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u1f10\u03ba\u03c4\u03af\u03c3\u03b8\u03b7 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, [\u03c4\u1f70]\n\n\n16. \u1f45\u03c4\u03b9 \u03ba.\u03c4.\u03bb.] We have in this sentence the justification of the title\ngiven to the Son in the preceding clause, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2. It\nmust therefore be taken to explain the sense in which this title is\nused. Thus connected, it shows that the \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 Himself is not\nincluded in \u03c0\u1fb6\u03c3\u03b1 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2; for the expression used is not \u03c4\u1f70 \u1f04\u03bb\u03bb\u03b1 or \u03c4\u1f70\n\u03bb\u03bf\u03b9\u03c0\u03ac, but \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03ba\u03c4\u03af\u03c3\u03b8\u03b7\u2013words which are absolute and comprehensive,\nand will admit no exception.\n\n\u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7] '_in Him_,' as below ver. 17 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5\u03bd. For the\npreposition comp. Acts xvii, 28 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03b3\u1f70\u03c1 \u03b6\u1ff6\u03bc\u03b5\u03bd \u03ba\u03b1\u1f76 \u03ba\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03b8\u03b1 \u03ba\u03b1\u03af\n\u1f10\u03c3\u03bc\u03b5\u03bd. All the laws and purposes which guide the creation and government\nof the Universe reside in Him, the Eternal Word, as their meeting-point.\nThe Apostolic doctrine of the Logos teaches us to regard the Eternal\nWord as holding the same relation to the Universe which the Incarnate\nChrist holds to the Church. He is the source of its life, the centre of\nall its developments, the mainspring of all its motions. The use of \u1f10\u03bd\nto describe His relations to the Church abounds in St Paul (e.g. Rom.\nviii. 1, 2, xii. 5, xvi. 3, 7, 9, etc., 1 Cor. i. 30, iv. 15, 17, vii.\n39, xv. 18, 22, etc.), and more especially in the Epistles to the\nColossians and Ephesians (e.g. below ii. 7, 10). In the present passage,\nas in ver. 17, the same preposition is applied also to His relations to\nthe Universe; comp. Joh. i. 4 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03b6\u03c9\u1f74 \u1f26\u03bd (more especially if we\nconnect the preceding \u1f43 \u03b3\u03ad\u03b3\u03bf\u03bd\u03b5\u03bd with it).\n\nThus it is part of the parallelism which runs through the whole passage,\nand to which the occurrence of \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 in both relations gives the\nkey. The Jud\u00e6o-Alexandrian teachers represented the Logos, which in\ntheir view was nothing more than the Divine mind energizing, as the\n\u03c4\u03cc\u03c0\u03bf\u03c2 where the eternal ideas, the \u03bd\u03bf\u03b7\u03c4\u1f78\u03c2 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2, had their abode; Philo\n_de Mund. Op._ 4 (I. p. 4) \u1f45\u03c3\u03b1\u03c0\u03b5\u03c1 \u1f10\u03bd \u1f10\u03ba\u03b5\u03af\u03bd\u1ff3 \u03bd\u03cc\u03b7\u03c4\u03b1, _ib._ \u00a7 5 (p. 4) \u03bf\u1f50\u03b4\u1f72\n\u1f41 \u1f10\u03ba \u03c4\u1ff6\u03bd \u1f30\u03b4\u03b5\u1ff6\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 \u1f04\u03bb\u03bb\u03bf\u03bd \u1f02\u03bd \u1f14\u03c7\u03bf\u03b9 \u03c4\u03cc\u03c0\u03bf\u03bd \u1f23 \u03c4\u1f78\u03bd \u03b8\u03b5\u1fd6\u03bf\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03c4\u1f78\u03bd \u03c4\u03b1\u1fe6\u03c4\u03b1\n\u03b4\u03b9\u03b1\u03ba\u03bf\u03c3\u03bc\u03ae\u03c3\u03b1\u03bd\u03c4\u03b1, _ib._ \u00a7 10 (p. 8) \u1f41 \u1f00\u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 ... \u1f31\u03b4\u03c1\u03c5\u03b8\u03b5\u1f76\u03c2 \u1f10\u03bd \u03c4\u1ff7\n\u03b8\u03b5\u03af\u1ff3 \u03bb\u03cc\u03b3\u1ff3; and see especially _de Migr. Abr._ I. p. 437) \u03bf\u1f36\u03ba\u03bf\u03c2 \u1f10\u03bd \u1fa7\n\u03b4\u03b9\u03b1\u03b9\u03c4\u1fb6\u03c4\u03b1\u03b9 ... \u1f45\u03c3\u03b1 \u1f02\u03bd \u1f10\u03bd\u03b8\u03c5\u03bc\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c4\u03ad\u03ba\u03b7, \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f10\u03bd \u03bf\u1f34\u03ba\u1ff3 \u03c4\u1ff7 \u03bb\u03cc\u03b3\u1ff3 \u03b4\u03b9\u03b1\u03b8\u03b5\u03af\u03c2. The\nApostolic teaching is an enlargement of this conception, inasmuch as the\nLogos is no longer a philosophical abstraction but a Divine Person: see\nHippol. _H\u00e6r._ x. 33 \u1f04\u03b9\u03c4\u03b9\u03bf\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03b3\u03b9\u03bd\u03bf\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2 \u039b\u03cc\u03b3\u03bf\u03c2 \u1f26\u03bd, \u1f10\u03bd \u1f11\u03b1\u03c5\u03c4\u1ff7 \u03c6\u03ad\u03c1\u03c9\u03bd \u03c4\u1f78\n\u03b8\u03ad\u03bb\u03b5\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03b3\u03b5\u03b3\u03b5\u03bd\u03bd\u03b7\u03ba\u03cc\u03c4\u03bf\u03c2 ... \u1f14\u03c7\u03b5\u03b9 \u1f10\u03bd \u1f11\u03b1\u03c5\u03c4\u1ff7 \u03c4\u1f70\u03c2 \u1f10\u03bd \u03c4\u1ff7 \u03c0\u03b1\u03c4\u03c1\u1f76 \u03c0\u03c1\u03bf\u03b5\u03bd\u03bd\u03bf\u03b7\u03b8\u03b5\u03af\u03c3\u03b1\u03c2\n\u1f30\u03b4\u03ad\u03b1\u03c2 \u1f45\u03b8\u03b5\u03bd \u03ba\u03b5\u03bb\u03b5\u03cd\u03bf\u03bd\u03c4\u03bf\u03c2 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u03c4\u1f78 \u03ba\u03b1\u03c4\u1f70 \u1f12\u03bd \u039b\u03cc\u03b3\u03bf\u03c2 \u1f00\u03c0\u03b5\u03c4\u03b5\u03bb\u03b5\u1fd6\u03c4\u03bf\n\u1f00\u03c1\u03ad\u03c3\u03ba\u03c9\u03bd \u0398\u03b5\u1ff7: comp. Orig. _in Ioann._ i. \u00a7 22, IV. p. 21.\n\n\u1f10\u03ba\u03c4\u03af\u03c3\u03b8\u03b7] The aorist is used here; the perfect below. \u1f18\u03ba\u03c4\u03af\u03c3\u03b8\u03b7 describes\nthe definite historical act of creation; \u1f14\u03ba\u03c4\u03b9\u03c3\u03c4\u03b1\u03b9 the continuous and\npresent relations of creation to the Creator: comp. Joh. i. 3 \u03c7\u03c9\u03c1\u1f76\u03c2\n\u03b1\u1f50\u03c4\u03bf\u1fe6 _\u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf_ \u03bf\u1f50\u03b4\u1f72 \u1f14\u03bd with _ib._ _\u1f43 \u03b5\u03b3\u03ad\u03bf\u03bd\u03b5\u03bd_, 1 Cor. ix. 22\n\u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b7\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03c3\u03b8\u03ad\u03bd\u03b5\u03c3\u03b9\u03bd \u1f00\u03c3\u03b8\u03b5\u03bd\u03ae\u03c2 with ib. \u03c4\u03bf\u1fd6\u03c2 \u03c0\u1fb6\u03c3\u03b9\u03bd \u03b3\u03ad\u03b3\u03bf\u03bd\u03b1 \u03c0\u03ac\u03bd\u03c4\u03b1, 2 Cor.\nxii. 17 \u03bc\u03ae \u03c4\u03b9\u03bd\u03b1 \u1f67\u03bd \u1f00\u03c0\u03ad\u03c3\u03c4\u03b1\u03bb\u03ba\u03b1 with ver. 18 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b1\u03c0\u03ad\u03c3\u03c4\u03b5\u03b9\u03bb\u03b1 \u03c4\u1f78\u03bd \u1f00\u03b4\u03b5\u03bb\u03c6\u03cc\u03bd,\n1 Joh. iv. 9 \u03c4\u1f78\u03bd \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u1fc6 _\u1f00\u03c0\u03ad\u03c3\u03c4\u03b1\u03bb\u03ba\u03b5\u03bd_ \u1f41 \u0398\u03b5\u1f78\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u1f35\u03bd\u03b1\n\u03b6\u03ae\u03c3\u03c9\u03bc\u03b5\u03bd \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 with ver. 10 \u1f45\u03c4\u03b9 \u03b1\u1f50\u03c4\u1f78\u03c2 \u1f20\u03b3\u03ac\u03c0\u03b7\u03c3\u03b5\u03bd \u1f21\u03bc\u1fb6\u03c2 \u03ba\u03b1\u1f76\n_\u1f00\u03c0\u03ad\u03c3\u03c4\u03b5\u03b9\u03bb\u03b5\u03bd_ \u03c4\u1f78\u03bd \u1f51\u03b9\u1f78\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6.\n\n\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1] '_the universe of things_,' not \u03c0\u03ac\u03bd\u03c4\u03b1 'all things severally,'\nbut \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 'all things collectively.' With very few exceptions,\nwherever this phrase occurs elsewhere, it stands in a similar connexion;\nsee below, vv. 17, 20, iii. 11, Rom. xi. 36, 1 Cor. viii. 6, xi. 12,\nxii. 6, xv. 27, 28, 2 Cor. v. 18, Eph. i. 10, 11, 23, iv. 10, Heb. i. 3,\nii. 8, Rev. iv. 11. Compare Rom. viii. 32 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f21\u03bc\u1fd6\u03bd \u03c7\u03b1\u03c1\u03af\u03c3\u03b5\u03c4\u03b1\u03b9, 2\nCor. iv. 15 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u1f51\u03bc\u1fb6\u03c2, with 1 Cor. iii. 22 \u1f14\u03b9\u03c4\u03b5 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 ... \u1f51\u03bc\u1ff6\u03bd;\nand Phil. iii. 8 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b5\u03b6\u03b7\u03bc\u03b9\u03ce\u03b8\u03b7\u03bd with Matt. xvi. 26 \u1f10\u1f70 \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd\n\u1f45\u03bb\u03bf\u03bd \u03ba\u03b5\u03c1\u03b4\u03ae\u03c3\u1fc3. Thus it will appear that \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 is nearly equivalent to\n'the universe.' It stands midway between \u03c0\u03ac\u03bd\u03c4\u03b1 and \u03c4\u1f78 \u03c0\u1fb6\u03bd. The last\nhowever is not a scriptural phrase; for, while with \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 it involves\nthe idea of connexion, it suggests also the unscriptural idea of\n_self-contained_ unity, the great world-soul of the Stoic pantheist.\n\n------------------------------------------------------------------------\n\nI. 16]\n\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 [\u03c4\u1f70] \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2, \u03c4\u1f70 \u1f41\u03c1\u03b1\u03c4\u1f70 \u03ba\u03b1\u1f76 \u03c4\u03b1\n\n\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2, \u03ba.\u03c4.\u03bb.] This division of the universe is not the same\nwith the following, as if [\u03c4\u1f70] \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2 were equivalent to \u03c4\u1f70\n\u1f00\u03cc\u03c1\u03b1\u03c4\u03b1 and [\u03c4\u1f70] \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2 to \u03c4\u1f70 \u1f41\u03c1\u03b1\u03c4\u03ac. It should rather be compared\nwith Gen. i. 1 \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u1f41 \u0398\u03b5\u1f78\u03c2 \u03c4\u1f78\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03b3\u1fc6\u03bd, ii. 1\n\u03c3\u03c5\u03bd\u03b5\u03c4\u03b5\u03bb\u03ad\u03c3\u03b8\u03b7\u03c3\u03b1\u03bd \u1f41 \u03bf\u1f50\u03c1\u03b1\u03bd\u1f78\u03c2 \u03ba\u03b1\u1f76 \u1f21 \u03b3\u1fc6 \u03ba\u03b1\u1f76 \u03c0\u1fb6\u03c2 \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd, xiv. 19 \u1f43\u03c2\n\u1f14\u03ba\u03c4\u03b9\u03c3\u03b5\u03bd \u03c4\u1f78\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03b3\u1fc6\u03bd, Rev. x. 6 \u1f43\u03c2 \u1f14\u03ba\u03c4\u03b9\u03c3\u03b5\u03bd \u03c4\u1f78\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f70\n\u1f10\u03bd \u03b1\u1f50\u03c4\u1ff3 \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03b3\u1fc6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f10\u03bd \u03b1\u1f50\u03c4\u1fc3. It is a classification by\n_locality_, as the other is a classification by _essences_. Heaven and\nearth together comprehend all space; and all things whether material or\nimmaterial are conceived for the purposes of the classification as\nhaving their abode in space. Thus the sun and the moon would belong to\n\u1f41\u03c1\u03b1\u03c4\u03ac, but they would be \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2; while the human soul would be\nclassed among \u1f00\u03cc\u03c1\u03b1\u03c4\u03b1 but would be regarded as \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2; see below\nver. 20.\n\nIt is difficult to say whether \u03c4\u1f70 ... \u03c4\u03b1 should be expunged or retained.\nThe elements in the decision are; (1) The facility either of omission or\nof addition in the first clause, owing to the termination of \u03c0\u03ac\u03bd\u03c4\u03b1: (2)\nThe much greater authority for the omission in the first clause than in\nthe second. These two combined suggest that \u03c4\u1f70 was omitted accidentally\nin the first clause, and then expunged purposely in the second for the\nsake of uniformity. On the other hand there is (3) The possibility of\ninsertion in both cases either for the sake of grammatical completeness\nor owing to the parallel passages, ver. 20, Ephes. i. 10. On the whole\nthe reasons for their omission preponderate. At all events we can hardly\nretain the one without the other.\n\n\u03c4\u1f70 \u1f40\u03c1\u03b1\u03c4\u1f70 \u03ba.\u03c4.\u03bb.] 'Things material and immaterial,' or, according to the\nlanguage of philosophy, \u03c6\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03b1 and \u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03b1: comp. Plato _Ph\u00e6d._ 79 A\n\u03b8\u1ff6\u03bc\u03b5\u03bd \u03bf\u1f56\u03bd, \u03b5\u1f30 \u03b2\u03bf\u03cd\u03bb\u03b5\u03b9, \u1f14\u03c6\u03b7, \u03b4\u03cd\u03bf \u03b5\u1f34\u03b4\u03b7 \u03c4\u1ff6\u03bd \u1f44\u03bd\u03c4\u03c9\u03bd, \u03c4\u1f78 \u03bc\u1f72\u03bd \u1f41\u03c1\u03b1\u03c4\u03cc\u03bd, \u03c4\u1f78 \u03b4\u1f72\n\u1f00\u03b5\u03b9\u03b4\u03ad\u03c2, \u03ba.\u03c4.\u03bb.\n\n------------------------------------------------------------------------\n\n\n\u1f00\u03cc\u03c1\u03b1\u03c4\u03b1, \u03b5\u1f34\u03c4\u03b5 \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9 \u03b5\u1f34\u03c4\u03b5 \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, \u03b5\u1f34\u03c4\u03b5 \u1f00\u03c1\u03c7\u03b1\u1f76 \u03b5\u1f34\u03c4\u03b5\n\n\n\u03b5\u1f34\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] '_whether they be thrones or lordships, etc._' The\nsubdivision is no longer exhaustive. The Apostle singles out those\ncreated beings that from their superior rank had been or might be set in\nrivalry with the Son.\n\nA comparison with the parallel passage Ephes. i. 21, \u1f51\u03c0\u03b5\u03c1\u03ac\u03bd\u03c9 \u03c0\u03ac\u03c3\u03b7\u03c2 \u1f00\u03c1\u03c7\u1fc6\u03c2\n\u03ba\u03b1\u1f76 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c9\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u03ba.\u03c4.\u03bb., brings out\nthe following points:\n\n(1) No stress can be laid on the sequence of the names, as though St\nPaul were enunciating with authority some precise doctrine respecting\nthe grades of the celestial hierarchy. The names themselves are not the\nsame in the two passages. While \u1f00\u03c1\u03c7\u03ae, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1, \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c2, are common to\nboth, \u03b8\u03c1\u03cc\u03bd\u03bf\u03c2 is peculiar to the one and \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 to the other. Nor again\nis there any correspondence in the sequence. Neither does \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 take\nthe place of \u03b8\u03c1\u03cc\u03bd\u03bf\u03c2, nor do the three words common to both appear in the\nsame order, the sequence being \u1f00\u03c1\u03c7. \u1f10\u03be. [\u03b4\u03cd\u03bd.] \u03ba\u03c5\u03c1. in Eph. i. 21, and\n[\u03b8\u03c1\u03cc\u03bd.] \u03ba\u03c5\u03c1. \u1f00\u03c1\u03c7. \u1f10\u03be. here.\n\n(2) An expression in Eph. i. 21 shows the Apostle's _motive_ in\nintroducing these lists of names: for he there adds \u03ba\u03b1\u1f76 \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2\n\u1f40\u03bd\u03bf\u03bc\u03b1\u03b6\u03bf\u03bc\u03ad\u03bd\u03bf\u03bd \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f10\u03bd \u03c4\u1ff7 \u03b1\u1f30\u1ff6\u03bd\u03b9 \u03c4\u03bf\u03cd\u03c4\u1ff3 \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03c4\u1ff7 \u03bc\u03ad\u03bb\u03bb\u03bf\u03bd\u03c4\u03b9, i.e.\n'of every dignity or title (whether real or imaginary) which is\nreverenced,' etc.; for this is the force of \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f40\u03bd\u03bf\u03bc\u03b1\u03b6\u03bf\u03bc\u03ad\u03bd\u03bf\u03bd\n(see the notes on Phil. ii. 9, and Eph. _l.c._). Hence it appears that\nin this catalogue St Paul does not profess to describe objective\nrealities, but contents himself with repeating subjective opinions. He\nbrushes away all these speculations without enquiring how much or how\nlittle truth there may be in them, because they are altogether beside\nthe question. His language here shows the same spirit of impatience with\nthis elaborate angelology, as in ii. 18.\n\n(3) Some commentators have referred the terms used here solely to\nearthly potentates and dignities. There can be little doubt however that\ntheir chief and primary reference is to the orders of the celestial\nhierarchy, as conceived by these Gnostic Judaizers. This appears from\nthe context; for the words \u03c4\u1f70 \u1f00\u03cc\u03c1\u03b1\u03c4\u03b1 immediately precede this list of\nterms, while in the mention of \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 and in other expressions\nthe Apostle clearly contemplates the rivalry of _spiritual_ powers with\nChrist. It is also demanded by the whole design and purport of the\nletter, which is written to combat the worship paid to angels. The names\ntoo, more especially \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, are especially connected with the\nspeculations of Jewish angelology. But when this is granted, two\nquestions still remain. First; are evil as well as good spirits\nincluded, demons as well as angels? And next; though the primary\nreference is to spiritual powers, is it not possible that the expression\nwas intended to be comprehensive and to include earthly dignities as\nwell? The clause added in the parallel passage, \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f10\u03bd \u03c4\u1ff7 \u03b1\u1f30\u1ff6\u03bd\u03b9\n\u03c4\u03bf\u03cd\u03c4\u1ff3 \u03ba.\u03c4.\u03bb., encourages us thus to extend the Apostle's meaning; and we\nare led in the same direction by the comprehensive words which have\npreceded here, [\u03c4\u1f70] \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2 \u03ba.\u03c4.\u03bb. Nor is there anything in the\nterms themselves which bars such an extension; for, as will be seen, the\ncombination \u1f00\u03c0\u03c7\u03b1\u1f76 \u03ba\u03b1\u1f76 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9 is applied not only to good angels but to\nbad, not only to spiritual powers but to earthly. Compare Ignat.\n_Smyrn._ 6 \u03c4\u1f70 \u1f10\u03c0\u03bf\u03c5\u03c1\u03ac\u03bd\u03b9\u03b1 \u03ba\u03b1\u1f76 \u1f21 \u03b4\u03cc\u03be\u03b1 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd \u03ba\u03b1\u1f76 \u03bf\u1f31 \u1f04\u03c1\u03c7\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f40\u03c1\u03b1\u03c4\u03bf\u03af \u03c4\u03b5\n\u03ba\u03b1\u1f76 \u1f00\u03cc\u03c1\u03b1\u03c4\u03bf\u03b9.\n\nThus guided, we may paraphrase the Apostle's meaning as follows: 'You\ndispute much about the successive grades of angels; you distinguish each\ngrade by its special title; you can tell how each order was generated\nfrom the preceding; you assign to each its proper degree of worship.\nMeanwhile you have ignored or you have degraded Christ. I tell you, it\nis not so. He is first and foremost, Lord of heaven and earth, far above\nall thrones or dominations, all princedoms or powers, far above every\ndignity and every potentate\u2014whether earthly or heavenly\u2014whether angel or\ndemon or man\u2014that evokes your reverence or excites your fear.' See\nabove, pp. 103 sq.\n\nJewish and Jud\u00e6o-Christian speculations respecting the grades of the\ncelestial hierarchy took various forms. In the _Testaments of the Twelve\nPatriarchs_ (Levi 3), which as coming near to the Apostolic age supplies\na valuable illustration (see _Galatians_ p. 307 sq.), these orders are\narranged as follows: (1) \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9, these two in the highest or\nseventh heaven; (2) \u03bf\u1f31 \u1f04\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9 \u03bf\u1f31 \u03c6\u03ad\u03c1\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c4\u1f70\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03af\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b3\u03b3\u03ad\u03bb\u03bf\u03b9\u03c2\n\u03c4\u03bf\u1fe6 \u03c0\u03c1\u03bf\u03c3\u03ce\u03c0\u03bf\u03c5 in the sixth heaven; (3) \u03bf\u1f31 \u1f04\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9 \u03c4\u03bf\u1fe6 \u03c0\u03c1\u03bf\u03c3\u03ce\u03c0\u03bf\u03c5 in the\nfifth heaven; (4) \u03bf\u1f31 \u1f04\u03b3\u03b9\u03bf\u03b9 in the fourth heaven; (5) \u03b1\u1f31 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03c4\u1ff6\u03bd\n\u03c0\u03b1\u03c1\u03b5\u03bc\u03b2\u03bf\u03bb\u1ff6\u03bd in the third heaven; (6) \u03c4\u1f70 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b1 \u03c4\u1ff6\u03bd \u1f10\u03c0\u03b1\u03b3\u03c9\u03b3\u1ff6\u03bd (i.e. of\nvisitations, retributions) in the second heaven: or perhaps the denizens\nof the sixth and fifth heavens, (2) and (3), should be transposed. The\nlowest heaven is not peopled by any spirits. In Origen _de Princ._ i. 5.\n3, _ib._ i. 6. 2, I. pp. 66, 70 (comp. i. 8. 1, _ib._ p. 74), we have\nfive classes, which are given in an ascending scale in this order; (1)\nangels (_sancti angeli_, \u03c4\u03ac\u03be\u03b9\u03c2 \u1f00\u03b3\u03b3\u03b5\u03bb\u03b9\u03ba\u03ae); (2) princedoms (_principatus_,\n\u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 \u1f00\u03c1\u03c7\u03b9\u03ba\u03ae, \u1f00\u03c1\u03c7\u03b1\u03af); (3) powers (_potestates_, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9); (4) thrones\n(_throni vel sedes_, \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9); (5) dominations (_dominationes_,\n\u03ba\u03c5\u03c1\u03af\u03bf\u03c4\u03b7\u03c4\u03b5\u03c2); though elsewhere, _in Ioann._ i. \u00a7 34, IV. p. 34, he seems\nto have a somewhat different classification in view. In Ephrem Syrus\n_Op. Syr._ I. p. 270 (where the translation of Benedetti is altogether\nfaulty and misleading) the ranks are these: (1) \u03b8\u03b5\u03bf\u03af, \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9,\n\u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2; (2) \u1f00\u03c1\u03c7\u03ac\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9, \u1f00\u03c1\u03c7\u03b1\u03af, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9; (3) \u1f04\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9, \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2,\n\u03c7\u03b5\u03c1\u03bf\u03c5\u03b2\u03af\u03bc, \u03c3\u03b5\u03c1\u03b1\u03c6\u03af\u03bc; these three great divisions being represented by the\n\u03c7\u03b9\u03bb\u03af\u03b1\u03c1\u03c7\u03bf\u03b9, the \u1f10\u03ba\u03b1\u03c4\u03cc\u03bd\u03c4\u03b1\u03c1\u03c7\u03bf\u03b9, and the \u03c0\u03b5\u03bd\u03c4\u03b7\u03ba\u03cc\u03bd\u03c4\u03b1\u03c1\u03c7\u03bf\u03b9 respectively in\nDeut. i. 15, on which passage he is commenting. The general agreement\nbetween these will be seen at once. This grouping also seems to underlie\nthe conception of Basil of Seleucia _Orat._ 39 (p. 207), who mentions\nthem in this order; \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, \u1f00\u03c1\u03c7\u03b1\u03af, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9, \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2,\n\u03c7\u03b5\u03c1\u03bf\u03c5\u03b2\u03af\u03bc, \u03c3\u03b5\u03c1\u03b1\u03c6\u03af\u03bc. On the other hand the arrangement of the\npseudo-Dionysius, who so largely influenced subsequent speculations, is\nquite different and probably later (Dion. Areop. _Op._ I. p. 75, ed.\nCord.); (1) \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, \u03c7\u03b5\u03c1\u03bf\u03c5\u03b2\u03af\u03bc, \u03c3\u03b5\u03c1\u03b1\u03c6\u03af\u03bc; (2) \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9, \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2,\n\u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2; (3) \u1f04\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9, \u1f00\u03c1\u03c7\u03ac\u03b3\u03b3\u03b5\u03bb\u03bf\u03b9, \u1f00\u03c1\u03c7\u03b1\u03b9. But the earlier lists for the\nmost part seem to suggest as their common foundation a classification in\nwhich \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, belonged to the highest order, and \u1f00\u03c1\u03c7\u03b1\u03af,\n\u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9 to the next below. Thus it would appear that the Apostle takes\nas an illustration the titles assigned to the two highest grades in a\nsystem of the celestial hierarchy which he found current, and which\nprobably was adopted by these Gnostic Judaizers. See also the note on\nii. 18.\n\n\u03b8\u03c1\u03cc\u03bd\u03bf\u03b9] In all systems alike these 'thrones' belong to the highest grade\nof angelic beings, whose place is in the immediate presence of God. The\nmeaning of the name however is doubtful: (1) It may signify the\n_occupants of thrones_ which surround the throne of God; as in the\nimagery of Rev. iv. 4 \u03ba\u03cd\u03ba\u03bb\u03bf\u03b8\u03b5\u03bd \u03c4\u03bf\u1fe6 \u03b8\u03c1\u03cc\u03bd\u03bf\u03c5 \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9 \u03b5\u1f34\u03ba\u03bf\u03c3\u03b9 \u03c4\u03ad\u03c3\u03c3\u03b1\u03c1\u03b5\u03c2 (comp.\nxi. 16, xx. 4). The imagery is there taken from the court of an earthly\nking: see Jer. lii. 32. This is the interpretation given by Origen _de\nPrinc._ i. 5. 3 (p. 66), i. 6. 2 (p. 70) 'judicandi vel regendi ...\nhabentes officium.' Or (2) They were so called, as _supporting or\nforming the throne of God_; just as the chariot-seat of the Almighty is\nrepresented as resting on the cherubim in Ezek. i. 26, ix. 3, x. 1 sq.,\nxi. 22, Ps. xviii. 10, 1 Chron. xxviii. 18. So apparently Clem. Alex.\n_Proph. Ecl._ 57 (p. 1003) \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9 \u1f02\u03bd \u03b5\u1f36\u03b5\u03bd ... \u03b4\u03b9\u1f70 \u03c4\u1f78 \u1f00\u03bd\u03b1\u03c0\u03b1\u03cd\u03c3\u03b8\u03b1\u03b9 \u1f10\u03bd\n\u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd. From this same imagery of the prophet the later\nmysticism of the Kabbala derived its name 'wheels,' which it gave to one\nof its ten orders of Sephiroth. Adopting this interpretation, several\nfathers identify the 'thrones' with the cherubim: e.g. Greg. Nyss. _ad\nEunom._ i (II. p. 349 sq.), Chrysost. _de Incompr. Nat._ iii. 5 (I. p.\n467), Theodoret (_ad loc._), August. _in Psalm._ xcviii. \u00a7 3 (iv. p.\n1061). This explanation was adopted also by the pseudo-Dionysius _de\nC\u0153l. Hier._ 7 (I. p. 80), without however identifying them with the\ncherubim; and through his writings it came to be generally adopted. The\nformer interpretation however is more probable; for (1) This highly\nsymbolical nomenclature accords better with a later stage of mystic\nspeculation, like the Kabbala; and (2) It seems natural to treat \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9\nas belonging to the same category with \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, \u1f00\u03c1\u03c7\u03b1\u03af, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9,\nwhich are concrete words borrowed from different grades of human rank\nand power. As implying _regal_ dignity, \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9 naturally stands at the\nhead of the list.\n\n\u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2] '_dominations_,' as Ephes. i. 21. These appear to have been\nregarded as belonging to the first grade, and standing next in dignity\nto the \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9. This indeed would be suggested by their name.\n\n\u1f00\u03c1\u03c7\u03b1\u03af, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9 as Ephes. i. 21. These two words occur very frequently\ntogether. In some places they refer to human dignities, as Luke xii. 11,\nTit. iii. 1 (comp. Luke xx. 20); in others to a spiritual hierarchy. And\nhere again there are two different uses: sometimes they designate good\nangels, e.g. below ii. 10, Ephes. iii. 10; sometimes evil spirits, e.g.\nii. 15, Ephes. vi. 12: while in one passage at least (1 Cor. xv. 24)\nboth may be included. In Rom. viii. 38 we have \u1f00\u03c1\u03c7\u03b1\u1f76 without \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9\n(except as a v. l.), and in 1 Pet. iii. 22 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9 without \u1f00\u03c1\u03c7\u03b1\u03af, in\nconnexion with the angelic orders.\n\n------------------------------------------------------------------------\n\nI. 16]\n\n\u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03b9\u00b7 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f14\u03ba\u03c4\u03b9\u03c3\u03c4\u03b1\u03b9\u00b7\n\n\n\u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba.\u03c4.\u03bb.] 'As all creation passed out from Him, so does it all\nconverge again towards Him.' For the combination of prepositions see\nRom. xi. 36 \u1f10\u03be \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1. He is not\nonly the ==\u03b1== but also the ==\u03c9==, not only the \u1f00\u03c1\u03c7\u03ae but also the\n\u03c4\u03ad\u03bb\u03bf\u03c2 of creation, not only the first but also the last in the history\nof the Universe: Rev. xxii. 13. For this double relation of Christ to\nthe Universe, as both the initial and the final cause, see Heb. ii. 10\n\u03b4\u03b9' \u1f43\u03bd \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03b4\u03b9' \u03bf\u1f57 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, where \u03b4\u03b9' \u1f43\u03bd is nearly equivalent\nto \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd of the text.\n\nIn the Judaic philosophy of Alexandria the preposition \u03b4\u03b9\u1f70 with the\ngenitive was commonly used to describe the function of the Logos in the\ncreation and government of the world; e.g. _de Cherub._ 35 (I. p. 162)\nwhere Philo, enumerating the causes which combine in the work of\nCreation, describes God as \u1f51\u03c6' \u03bf\u1f57, matter as \u1f10\u03be \u03bf\u1f57, and the Word as \u03b4\u03b9'\n\u03bf\u1f57; comp. _de Mon._ ii. 5 (II. p. 225) \u03bb\u03cc\u03b3\u03bf\u03c2 ... \u03b4\u03b9' \u03bf\u1f57 \u03c3\u03cd\u03bc\u03c0\u03b1\u03c2 \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2\n\u1f10\u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03b5\u1fd6\u03c4\u03bf. The Christian Apostles accepted this use of \u03b4\u03b9\u1f70 to\ndescribe the mediatorial function of the Word in creation; e.g. John i.\n3 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u03ba.\u03c4.\u03bb., _ib._ ver. 10 \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6\n\u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf, Heb. i. 2 \u03b4\u03b9' \u03bf\u1f57 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u03c4\u03bf\u1f7a\u03c2 \u03b1\u1f30\u1ff6\u03bd\u03b1\u03c2. This mediatorial\nfunction however has entirely changed its character. To the Alexandrian\nJew it was the work of a passive tool or instrument (_de Cherub._ l.c.\n\u03b4\u03b9' \u03bf\u1f57, _\u03c4\u1f78 \u1f10\u03c1\u03b3\u03b1\u03bb\u03b5\u1fd6\u03bf\u03bd_, \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd ... \u03b4\u03b9' \u03bf\u1f57); but to the Christian\nApostle it represented a cooperating agent. Hence the Alexandrian Jew\nfrequently and consistently used the simple instrumental dative \u1fa7 to\ndescribe the relation of the Word to the Creator, e.g. _Quod Deus\nimmut._ 12 (I. p. 281) \u1fa7 \u03ba\u03b1\u1f76 \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u03b5\u1f30\u03c1\u03b3\u03ac\u03b6\u03b5\u03c4\u03bf, _Leg. All._ i. 9 (I.\np. 47) \u03c4\u1ff7 \u03c0\u03b5\u03c1\u03b9\u03c6\u03b1\u03bd\u03b5\u03c3\u03c4\u03ac\u03c4\u1ff3 \u03ba\u03b1\u1f76 \u03c4\u03b7\u03bb\u03b1\u03c5\u03b3\u03b5\u03c3\u03c4\u03ac\u03c4\u1ff3 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 \u03bb\u03cc\u03b3\u1ff3 \u1fe5\u03ae\u03bc\u03b1\u03c4\u03b9 \u1f41 \u0398\u03b5\u1f78\u03c2\n\u1f00\u03bc\u03c6\u03cc\u03c4\u03b5\u03c1\u03b1 \u03c0\u03bf\u03b9\u03b5\u1fd6, comp. _ib._ iii. 31 (I. p. 106) \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 ... \u1fa7 \u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1\n\u1f40\u03c1\u03b3\u03ac\u03bd\u1ff3 \u03c0\u03c1\u03bf\u03c3\u03c7\u03c1\u03b7\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2. This mode of speaking is not found in the New\nTestament.\n\n\u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd] '_unto Him_.' As of the Father it is said elsewhere, 1 Cor.\nviii. 6 \u1f10\u03be \u1f41\u1fe6 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd, so here of the Son we read\n\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd. All things must find their\nmeeting-point, their reconciliation, at length in Him from whom they\ntook their rise\u2014in the Word as the mediatorial agent, and through the\nWord in the Father as the primary source. The Word is the final cause as\nwell as the creative agent of the Universe. This ultimate goal of the\npresent dispensation in time is similarly stated in several passages.\nSometimes it is represented as the birth-throe and deliverance of all\ncreation through Christ; as Rom. viii. 19 sq. \u03b1\u1f50\u03c4\u1f74 \u1f21 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2\n\u1f10\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03c9\u03b8\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9, \u03c0\u1fb6\u03c3\u03b1 \u1f21 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 ... \u03c3\u03c5\u03bd\u03c9\u03b4\u03af\u03bd\u03b5\u03b9. Sometimes it is the\nabsolute and final subjection of universal nature to Him; as 1 Cor. xv.\n28 \u1f45\u03c4\u03b1\u03bd \u1f51\u03c0\u03bf\u03c4\u03b1\u03b3\u1fc7 \u03b1\u1f50\u03c4\u1ff7 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1. Sometimes it is the reconciliation of all\nthings through Him; as below, ver. 20 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u03b1\u03b9 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1.\nSometimes it is the recapitulation, the gathering up in one head, of the\nUniverse in Him; as Ephes. i. 10 \u1f00\u03bd\u03b1\u03ba\u03b5\u03c6\u03b1\u03bb\u03b1\u03b9\u03ce\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03c4\u1ff7\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7. The image involved in this last passage best illustrates the\nparticular expression in the text \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd ... \u1f14\u03ba\u03c4\u03b9\u03c3\u03c4\u03b1\u03b9; but all alike\nenunciate the same truth in different terms. The Eternal Word is the\ngoal of the Universe, as He was the starting-point. It must end in\nunity, as it proceeded from unity: and the centre of this unity is\nChrist. This expression has no parallel, and could have none, in the\nAlexandrian phraseology and doctrine.\n\n------------------------------------------------------------------------\n\nI. 17]\n\n^{17}\u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2 \u1f14\u03c3\u03c4\u03b9\u03bd \u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd, \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7\n\n\n17. \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u03bf\u03c2 \u03ba.\u03c4.\u03bb.] '_and HE IS before all things_': comp. Joh. viii.\n58 \u03c0\u03c1\u1f76\u03bd \u1f08\u03b2\u03c1\u03b1\u1f70\u03bc \u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9, _\u1f10\u03b3\u1f7c \u03b5\u1f30\u03bc\u1f76_ (and perhaps also viii. 24, 28,\nxiii. 19). The imperfect \u1f26\u03bd might have sufficed (comp. Joh. i. 1), but\nthe present \u1f14\u03c3\u03c4\u03b9\u03bd declares that this pre-existence is absolute\nexistence. The ==\u03b1\u03c5\u03c4\u03bf\u03f2 \u03b5\u03f2\u03c4\u03b9\u03bd== here corresponds exactly to the ==\u03b5\u03b3\u03c9\n\u03b5\u03b9\u03bc\u03b9== in St John, and this again is illustrated by Exod. iii. 14. The\nverb therefore is not an enclitic, but should be accentuated \u1f14\u03c3\u03c4\u03b9\u03bd. See\nBasil _adv. Eunom._ iv (I. p. 294) \u1f41 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03c2 \u03b5\u1f30\u03c0\u03ce\u03bd, \u03a0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6\n\u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f14\u03ba\u03c4\u03b9\u03c3\u03c4\u03b1\u03b9, \u1f64\u03c6\u03b5\u03b9\u03bb\u03b5\u03bd \u03b5\u1f30\u03c0\u03b5\u1fd6\u03bd, \u039a\u1f70\u03b9 \u03b1\u1f50\u03c4\u1f78\u03c2 _\u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf_ \u03c0\u03c1\u1f78\n\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd, \u03b5\u1f30\u03c0\u1f7c\u03bd \u03b4\u1f72, \u039a\u03b1\u03af \u03b1\u1f50\u03c4\u1f78\u03c2 _\u1f14\u03c3\u03c4\u03b9_ \u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd, \u1f14\u03b4\u03b5\u03b9\u03be\u03b5 \u03c4\u03bf\u03bd \u03bc\u1f72\u03bd \u1f00\u03b5\u1f76\n\u1f44\u03bd\u03c4\u03b1 \u03c4\u1f74\u03bd \u03b4\u1f72 \u03ba\u03c4\u03af\u03c3\u03b9\u03bd \u03b3\u03b5\u03bd\u03bf\u03bc\u03ad\u03bd\u03b7\u03bd. The \u03b1\u1f50\u03c4\u03cc\u03c2 is as necessary for the\ncompleteness of the meaning, as the \u1f14\u03c3\u03c4\u03b9\u03bd. The one emphasizes the\n_personality_, as the other declares the _pre-existence_. For this\nemphatic \u03b1\u1f50\u03c4\u03cc\u03c2 see again ver. 18; comp. Ephes. ii. 14, iv. 10, 11, 1\nJoh. ii. 2, and esp. Rev. xix. 15 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03c0\u03bf\u03b9\u03bc\u03b1\u03bd\u03b5\u1fd6 ... \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2\n\u03c0\u03b1\u03c4\u03b5\u1fd6. The other interpretation which explains \u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd of superiority\nin rank, and not of priority in time, is untenable for several reasons.\n(1) This would most naturally be expressed otherwise in Biblical\nlanguage, as \u1f10\u03c0\u1f76 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd (e.g. Rom. ix. 5, Eph. iv. 6), or \u1f51\u03c0\u1f72\u03c1 \u03c0\u03ac\u03bd\u03c4\u03b1\n(Eph. i. 22), or \u1f51\u03c0\u03b5\u03c1\u03ac\u03bd\u03c9 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd (Eph. i. 21, iv. 10). (2) The key to the\ninterpretation is given by the analogous words in the context, esp.\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2, vv. 15, 18. (3) Nothing short of this declaration of\nabsolute pre-existence would be adequate to introduce the statement\nwhich follows, \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5\u03bd.\n\n\u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd] '_before all things_.' In the Latin it was translated 'ante\nomnes,' i.e. thronos, dominationes, etc.; and so Tertullian _adv. Marc._\nv. 19 'Quomodo enim ante omnes, si non ante omnia? Quomodo ante omnia,\nsi non primogenitus conditionis?' But the neuter \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, standing in\nthe context before and after, requires the neuter here also.\n\n------------------------------------------------------------------------\n\nI. 18]\n\n\u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5\u03bd. ^{18} \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2,\n\n\n\u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5\u03bd] '_hold together, cohere_.' He is the principle of cohesion\nin the universe. He impresses upon creation that unity and solidarity\nwhich makes it a cosmos instead of a chaos. Thus (to take one instance)\nthe action of gravitation, which keeps in their places things fixed and\nregulates the motions of things moving, is an expression of His mind.\nSimilarly in Heb. i. 3 Christ the Logos is described as \u03c6\u03ad\u03c1\u03c9\u03bd \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1\n(_sustaining_ the universe) \u03c4\u1ff7 \u1fe5\u03ae\u03bc\u03b1\u03c4\u03b9 \u03c4\u1fc6\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c9\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6. Here again the\nChristian Apostles accept the language of Alexandrian Judaism, which\ndescribes the Logos as the \u03b4\u03b5\u03c3\u03bc\u1f78\u03c2 of the Universe; e.g. Philo _de\nProfug._ 20 (I. p. 562) \u1f45 \u03c4\u03b5 \u03b3\u1f70\u03c1 \u03c4\u03bf\u1fe6 \u1f44\u03bd\u03c4\u03bf\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c2 _\u03b4\u03b5\u03c3\u03bc\u1f78\u03c2 \u1f62\u03bd \u03c4\u1ff6\u03bd\n\u1f01\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd_ ... \u03ba\u03b1\u1f76 _\u03c3\u03cd\u03bd\u03b5\u03c7\u03b5\u03b9_ \u03c4\u1f70 \u03bc\u03ad\u03c1\u03b7 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03c3\u03c6\u03af\u03b3\u03b3\u03b5\u03b9 \u03ba\u03b1\u1f76 \u03ba\u03c9\u03bb\u03cd\u03b5\u03b9\n\u03b1\u1f50\u03c4\u1f70 \u03b4\u03b9\u03b1\u03bb\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03c1\u03c4\u1fb6\u03c3\u03b8\u03b1\u03b9, _de Plant._ 2 (I. p. 331) _\u03c3\u03c5\u03bd\u03ac\u03b3\u03c9\u03bd_\n\u03c4\u1f70 \u03bc\u03ad\u03c1\u03b7 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03c3\u03c6\u03af\u03b3\u03b3\u03c9\u03bd\u00b7 _\u03b4\u03b5\u03c3\u03bc\u1f78\u03bd_ \u03b3\u1f70\u03c1 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f04\u03c1\u03c1\u03b7\u03ba\u03c4\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u1f41\n\u03b3\u03b5\u03bd\u03bd\u03ae\u03c3\u03b1\u03c2 \u1f10\u03c0\u03bf\u03af\u03b5\u03b9 \u03c0\u03b1\u03c4\u03ae\u03c1, _Quis rer. <DW37>. her._ 38 (I. p. 507) \u03bb\u03cc\u03b3\u1ff3\n\u03c3\u03c6\u03af\u03b3\u03b3\u03b5\u03c4\u03b1\u03b9 \u03b8\u03b5\u03af\u1ff3\u00b7 \u03ba\u03cc\u03bb\u03bb\u03b1 \u03b3\u03ac\u03c1 \u1f10\u03c3\u03c4\u03b9 \u03ba\u03b1\u1f76 _\u03b4\u03b5\u03c3\u03bc\u1f78\u03c2_ \u03bf\u1f57\u03c4\u03bf\u03c2 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1fc6\u03c2\n\u03bf\u1f50\u03c3\u03af\u03b1\u03c2 \u1f10\u03ba\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03ba\u03ce\u03c2: and for the word itself see _Quis rer. <DW37>. her._\n12 (I. p. 481) _\u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5_ \u03ba\u03b1\u1f76 \u03b6\u03c9\u03c0\u03c5\u03c1\u03b5\u1fd6\u03c4\u03b1\u03b9 \u03c0\u03c1\u03bf\u03bd\u03bf\u03af\u1fb3 \u0398\u03b5\u03bf\u1fe6, Clem. Rom.\n27 \u1f10\u03bd \u03bb\u03cc\u03b3\u1ff3 \u03c4\u1fc6\u03c2 \u03bc\u03b5\u03b3\u03b1\u03bb\u03c9\u03c3\u03cd\u03bd\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c3\u03c5\u03bd\u03b5\u03c3\u03c4\u03ae\u03c3\u03b1\u03c4\u03bf \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1. In the same\nconnexion \u03c3\u03cd\u03b3\u03ba\u03b5\u03b9\u03c4\u03b1\u03b9 is used, Ecclus. xliii. 26. The indices to Plato and\nAristotle amply illustrate this use of \u03c3\u03c5\u03bd\u03ad\u03c3\u03c4\u03b7\u03ba\u03b5\u03bd. This mode of\nexpression was common also with the Stoics.\n\n18. 'And not only does He hold this position of absolute priority and\nsovereignty over the Universe\u2014the natural creation. He stands also in\nthe same relation to the Church\u2014the new spiritual creation. He is its\nhead, and it is His body. This is His prerogative, because He is the\nsource and the beginning of its life, being the First-born from the\ndead. Thus in all things\u2014in the spiritual order as in the natural\u2014in the\nChurch as in the World\u2014He is found to have the pre-eminence.'\n\nThe elevating influence of this teaching on the choicest spirits of the\nsubapostolic age will be seen from a noble passage in the noblest of\nearly Christian writings, _Epist. ad Diogn._ \u00a7 7 \u03c4\u1f78\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03c4\u1f78\u03bd \u1f05\u03b3\u03b9\u03bf\u03bd ...\n\u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2 \u1f10\u03bd\u03af\u03b4\u03c1\u03c5\u03c3\u03b5 ... \u03bf\u1f50, \u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1 \u1f04\u03bd \u03c4\u03b9\u03c2 \u03b5\u1f30\u03ba\u03ac\u03c3\u03b5\u03b9\u03b5\u03bd, \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2 \u1f51\u03c0\u03b7\u03c1\u03ad\u03c4\u03b7\u03bd\n\u03c4\u03b9\u03bd\u1f70 \u03c0\u03ad\u03bc\u03c8\u03b1\u03c2 \u1f22 \u1f04\u03b3\u03b3\u03b5\u03bb\u03bf\u03bd \u1f22 \u1f04\u03c1\u03c7\u03bf\u03bd\u03c4\u03b1 \u1f24 \u03c4\u03b9\u03bd\u03b1 \u03c4\u1ff6\u03bd \u03b4\u03b9\u03b5\u03c0\u03cc\u03bd\u03c4\u03c9\u03bd \u03c4\u1f70 \u1f10\u03c0\u03af\u03b3\u03b5\u03b9\u03b1 \u1f24 \u03c4\u03b9\u03bd\u03b1\n\u03c4\u1ff6\u03bd \u03c0\u03b5\u03c0\u03b9\u03c3\u03c4\u03b5\u03c5\u03bc\u03ad\u03bd\u03c9\u03bd \u03c4\u1f70\u03c2 \u1f10\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2 \u03b4\u03b9\u03bf\u03ba\u03ae\u03c3\u03b5\u03b9\u03c2, \u1f00\u03bb\u03bb' \u03b1\u1f50\u03c4\u1f78\u03bd \u03c4\u1f78\u03bd \u03c4\u03b5\u03c7\u03bd\u03af\u03c4\u03b7\u03bd \u03ba\u03b1\u1f76\n\u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u1f78\u03bd \u03c4\u1ff6\u03bd \u1f45\u03bb\u03c9\u03bd ... \u1fa7 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c4\u03ad\u03c4\u03b1\u03ba\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03ce\u03c1\u03b9\u03c3\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76\n\u1f51\u03c0\u03bf\u03c4\u03ad\u03c4\u03b1\u03ba\u03c4\u03b1\u03b9, \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1f76 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2, \u03b3\u1fc6 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f10\u03bd \u03c4\u1fc7 \u03b3\u1fc7 \u03ba.\u03c4.\u03bb.\nSee the whole context.\n\n\u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2] '_and He_,' repeated from the preceding verse, to emphasize\nthe identity of the Person who unites in Himself these prerogatives: see\non ver. 17, and comp. ver. 18 \u03b1\u1f50\u03c4\u03cc\u03c2, ver. 19 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6. The Creator of\nthe World is also the Head of the Church. There is no blind ignorance,\nno imperfect sympathy, no latent conflict, in the relation of the\ndemiurgic power to the Gospel dispensation, as the heretical teachers\nwere disposed consciously or unconsciously to assume (see above, p. 101\nsq., p. 110 sq.), but an absolute unity of origin.\n\n\u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae] '_the head_,' the inspiring, ruling, guiding, combining,\nsustaining power, the mainspring of its activity, the centre of its\nunity, and the seat of its life. In his earlier epistles the relations\nof the Church to Christ are described under the same image (1 Cor. xii.\n12\u201327; comp. vi. 15, x. 17, Rom. xii. 4 sq.); but the Apostle there\ntakes as his starting-point the various functions of the members, and\nnot, as in these later epistles, the originating and controlling power\nof the Head. Comp. i. 24, ii. 19, Eph. i. 22 sq., ii. 16, iv. 4, 12, 15\nsq., v. 23, 30.\n\n------------------------------------------------------------------------\n\nI. 18]\n\n\u03c4\u1fc6\u03c2 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03c2\u00b7 \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u03c1\u03c7\u03ae, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2\n\n\n\u03c4\u1fc6\u03c2 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03c2] in apposition with \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2: comp. i. 24 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2\n\u03b1\u1f50\u03c4\u03bf\u1fe6, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1, Eph. i. 23.\n\n\u1f00\u03c1\u03c7\u03ae] '_the origin_, _the beginning_.' The term is here applied to the\nIncarnate Christ in relation to the Church, because it is applicable to\nthe Eternal Word in relation to the Universe, Rev. iii. 14 \u1f21 \u1f00\u03c1\u03c7\u1f74 \u03c4\u1fc6\u03c2\n\u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6. The parallelism of the two relations is kept in view\nthroughout. The word \u1f00\u03c1\u03c7\u03ae here involves two ideas: (1) Priority in time;\nChrist was the first-fruits of the dead, \u1f00\u03c0\u03b1\u03c1\u03c7\u03ae (1 Cor. xv. 20, 23): (2)\nOriginating power; Christ was also the source of life, Acts iii. 14 \u1f41\n\u1f00\u03c1\u03c7\u03b7\u03b3\u1f78\u03c2 \u03c4\u1fc6\u03c2 \u03b6\u03c9\u1fc6\u03c2; comp. Acts v. 31, Heb. ii. 10. He is not merely the\n_principium principiatum_ but the _principium principians_ (see Trench\n_Epistles to the Seven Churches_ p. 183 sq.). He rose first from the\ndead, that others might rise through Him.\n\nThe word \u1f00\u03c1\u03c7\u03ae, like \u03c0\u03c1\u1ff6\u03c4\u03bf\u03c2 (see the note on Phil. i. 5), being absolute\nin itself, does not require the definite article. Indeed the article is\nmost commonly omitted where \u1f00\u03c1\u03c7\u03ae occurs as a predicate, as will appear\nfrom several examples to be gathered from the extracts in Plut. _Mor._\np. 875 sq., Stob. _Ecl. Phys._ i. 10. 12 sq. Comp. also Aristot. _Met._\nx. 7, p. 1064, \u03c4\u1f78 \u03b8\u03b5\u1fd6\u03bf\u03bd ... \u1f02\u03bd \u03b5\u1f34\u03b7 \u03c0\u03c1\u03ce\u03c4\u03b7 \u03ba\u03b1\u1f76 \u03ba\u03c5\u03c1\u03b9\u03c9\u03c4\u03ac\u03c4\u03b7 \u1f00\u03c1\u03c7\u03ae, Onatas in\nStob. _Ecl. Phys._ i. 2. 39 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03b3\u1f70\u03c1 [\u03b8\u03b5\u1f78\u03c2] \u1f00\u03c1\u03c7\u1f70 \u03ba\u03b1\u1f76 \u03c0\u03c1\u1fb6\u03c4\u03bf\u03bd, Tatian.\n_ad Gr\u00e6c._ 4 \u0398\u03b5\u1f78\u03c2 ... \u03bc\u03cc\u03bd\u03bf\u03c2 \u1f04\u03bd\u03b1\u03c1\u03c7\u03bf\u03c2 \u1f62\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2 \u1f51\u03c0\u03ac\u03c1\u03c7\u03c9\u03bd \u03c4\u1ff6\u03bd \u1f45\u03bb\u03c9\u03bd \u1f00\u03c1\u03c7\u03ae,\nClem. Alex. _Strom._ iv. 25, p. 638, \u1f41 \u0398\u03b5\u1f78\u03c2 \u03b4\u1f72 \u1f04\u03bd\u03b1\u03c1\u03c7\u03bf\u03c2, \u1f00\u03c1\u03c7\u1f74 \u03c4\u1ff6\u03bd \u1f45\u03bb\u03c9\u03bd\n\u03c0\u03b1\u03bd\u03c4\u03b5\u03bb\u03ae\u03c2, \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03c0\u03bf\u03b9\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2, Method. _de Creat._ 3 (p. 100, ed. Jahn)\n\u03c0\u03ac\u03c3\u03b7\u03c2 \u1f00\u03c1\u03b5\u03c4\u1fc6\u03c2 \u1f00\u03c1\u03c7\u1f74\u03bd \u03ba\u03b1\u1f76 \u03c0\u03b7\u03b3\u1f74\u03bd ... \u1f21\u03b3\u1fc7 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd, pseudo-Dionys. _de <DW37>.\nNom._ v. \u00a7 6 \u1f00\u03c1\u03c7\u1f74 \u03b3\u03ac\u03c1 \u1f10\u03c3\u03c4\u03b9 \u03c4\u1ff6\u03bd \u1f44\u03bd\u03c4\u03c9\u03bd, \u00a7 10 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03bf\u1f56\u03bd \u1f00\u03c1\u03c7\u1f74 \u03ba\u03b1\u1f76 \u03c4\u03b5\u03bb\u03b5\u03c5\u03c4\u1f74\n\u03c4\u1ff6\u03bd \u1f44\u03bd\u03c4\u03c9\u03bd \u1f41 \u03c0\u03c1\u03bf\u03ce\u03bd.\n\nThe text is read with the definite article, \u1f21 \u1f00\u03c1\u03c7\u03ae, in one or two\nexcellent authorities at least; but the obvious motive which would lead\na scribe to aim at greater distinctness renders the reading suspicious.\n\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2] Comp. Rev. i. 5 \u1f41 \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c4\u1ff6\u03bd \u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd \u03ba\u03b1\u1f76 \u1f41 \u1f04\u03c1\u03c7\u03c9\u03bd \u03c4\u1ff6\u03bd\n\u03b2\u03b1\u03c3\u03b9\u03bb\u03ad\u03c9\u03bd \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2. His resurrection from the dead is His title to the\nheadship of the Church; for 'the power of His resurrection' (Phil. iii.\n10) is the life of the Church. Such passages as Gen. xlix. 3, Deut. xxi.\n17, where the \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 is called \u1f00\u03c1\u03c7\u1f74 \u03c4\u03ad\u03ba\u03bd\u03c9\u03bd and superior privileges\nare claimed for him as such, must necessarily be only very faint and\npartial illustrations of the connexion between \u1f00\u03c1\u03c7\u1f74 and \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 here,\nwhere the subject-matter and the whole context point to a fuller meaning\nof the words. The words \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u1f10\u03ba \u03c4\u1ff6\u03bd \u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd here correspond to\n\u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03ba\u03c4\u03af\u03c3\u03b5\u03c9\u03c2 ver. 15, so that the parallelism between\nChrist's relations to the Universe and to the Church is thus emphasized.\n\n------------------------------------------------------------------------\n\nI. 19]\n\n\u1f10\u03ba \u03c4\u1ff6\u03bd \u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd, \u1f35\u03bd\u03b1 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd \u03b1\u1f50\u03c4\u1f78\u03c2 \u03c0\u03c1\u03c9\u03c4\u03b5\u03cd\u03c9\u03bd\u00b7 ^{19} \u1f45\u03c4\u03b9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7\n\u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9,\n\n\n\u1f35\u03bd\u03b1 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb.] As He _is_ first with respect to the Universe, so it\nwas ordained that He should _become_ first with respect to the Church as\nwell. The _\u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9_ here answers in a manner to the _\u1f14\u03c3\u03c4\u03b9\u03bd_ of\nver. 17. Thus \u1f14\u03c3\u03c4\u03b9\u03bd and \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 are contrasted as the absolute being and\nthe historical manifestation. The relation between Christ's headship of\nthe Universe by virtue of His Eternal Godhead and His headship of the\nChurch by virtue of His Incarnation and Passion and Resurrection is\nsomewhat similarly represented in Phil. ii. 6 sq. \u1f10\u03bd \u03bc\u03bf\u03c1\u03c6\u1fc7 \u0398\u03b5\u03bf\u1fe6 \u1f55\u03b1\u03c1\u03c7\u03c9\u03bd\n... \u03bc\u03bf\u03c1\u03c6\u1f74\u03bd \u03b4\u03bf\u03cd\u03bb\u03bf\u03c5 \u03bb\u03b1\u03b2\u03ce\u03bd ... \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f51\u03c0\u03ae\u03ba\u03bf\u03bf\u03c2 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5 ...\n_\u03b4\u03b9\u1f78_ \u03ba\u03b1\u1f76 \u1f41 \u0398\u03b5\u1f78\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f51\u03c0\u03b5\u03c1\u03cd\u03c8\u03c9\u03c3\u03b5\u03bd \u03ba.\u03c4.\u03bb.\n\n\u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd] '_in all things_' not in the Universe only but in the Church\nalso. \u039a\u03b1\u1f76 \u03b3\u03ac\u03c1, writes Theodoret, \u1f61\u03c2 \u0398\u03b5\u1f78\u03c2, \u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u1f10\u03c3\u03c4\u1f76 \u03ba\u03b1\u1f76 \u03c3\u1f7a\u03bd \u03c4\u1ff7\n\u03c0\u03b1\u03c4\u03c1\u03af \u1f10\u03c3\u03c4\u03b9, \u03ba\u03b1\u1f76 \u1f61\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2, \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2 \u1f10\u03ba \u03c4\u1ff6\u03bd \u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2\n\u03ba\u03b5\u03c6\u03b1\u03bb\u03ae. Thus \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd is neuter and not masculine, as it is sometimes\ntaken. Either construction is grammatically correct, but the context\npoints to the former interpretation here; and this is the common use of\n\u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd, e.g. iii. 11, Eph. i. 23, Phil. iv. 12. For the neuter compare\nPlut. _Mor._ p. 9 \u03c3\u03c0\u03b5\u03cd\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03c0\u03b1\u1fd6\u03b4\u03b1\u03c2 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9 \u03c4\u03ac\u03c7\u03b9\u03bf\u03bd \u03c0\u03c1\u03c9\u03c4\u03b5\u1fe6\u03c3\u03b1\u03b9. On\nthe other hand in [Demosth.] _Amat._ p. 1416 \u03ba\u03c1\u03ac\u03c4\u03b9\u03c3\u03c4\u03bf\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1f78\n\u03c0\u03c1\u03c9\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd \u1f10\u03bd \u1f05\u03c0\u03b1\u03c3\u03b9 the context shows that \u1f05\u03c0\u03b1\u03c3\u03b9 is masculine.\n\n\u03b1\u1f50\u03c4\u1f78\u03c2] '_He Himself_'; see the note on \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2 above.\n\n19, 20. 'And this absolute supremacy is His, because it was the Father's\ngood pleasure that in Him all the plenitude of Deity should have its\nhome; because He willed through Him to reconcile the Universe once more\nto Himself. It was God's purpose to effect peace and harmony through the\nblood of Christ's cross, and so to restore all things, whatsoever and\nwheresoever they be, whether on the earth or in the heavens.'\n\n19. \u1f45\u03c4\u03b9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03ba.\u03c4.\u03bb.] The eternal indwelling of the Godhead explains\nthe headship of the Church, not less than the headship of the Universe.\nThe resurrection of Christ, whereby He became the \u1f00\u03c1\u03c7\u1f74 of the Church,\nwas the result of and the testimony to His deity; Rom. i. 4 \u03c4\u03bf\u1fe6\n\u1f41\u03c1\u03b9\u03c3\u03b8\u03ad\u03bd\u03c4\u03bf\u03c2 \u03c5\u1f31\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 ... \u1f10\u03be \u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03c9\u03c2 \u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd.\n\n\u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd] sc. \u1f41 \u0398\u03b5\u03cc\u03c2, the nominative being understood; see Winer \u00a7\nlviii. p. 655 sq., \u00a7 lxiv. p. 735 sq.; comp. James i. 12 (the right\nreading), iv. 6. Here the omission is the more easy, because \u03b5\u1f50\u03b4\u03bf\u03ba\u03af\u03b1,\n\u03b5\u1f50\u03b4\u03bf\u03ba\u03b5\u1fd6\u03bd etc. (like \u03b8\u03ad\u03bb\u03b7\u03bc\u03b1) are used absolutely of God's good purpose,\ne.g. Luke ii. 14 \u1f10\u03bd \u1f00\u03bd\u03b8\u03c1\u1ff6\u03c0\u03bf\u03b9\u03c2 \u03b5\u1f50\u03b4\u03bf\u03ba\u03af\u03b1\u03c2 (or \u03b5\u1f50\u03b4\u03bf\u03ba\u03af\u03b1), Phil. ii. 13 \u1f51\u03c0\u1f72\u03c1\n\u03c4\u1fc6\u03c2 \u03b5\u1f50\u03b4\u03bf\u03ba\u03af\u03b1\u03c2, Clem. Rom. \u00a7 40 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1f70 \u03b3\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03b1 \u1f10\u03bd \u03b5\u1f50\u03b4\u03bf\u03ba\u03ae\u03c3\u03b5\u03b9; see the\nnote in Clem. Rom. \u00a7 2. For the expression generally comp. 2 Macc. xiv.\n35 \u03c3\u03cd, \u039a\u03cd\u03c1\u03b9\u03b5, \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b1\u03c2 \u03bd\u03b1\u1f78\u03bd \u03c4\u1fc6\u03c2 \u03c3\u1fc6\u03c2 \u03ba\u03b1\u03c4\u03b1\u03c3\u03ba\u03b7\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2 \u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd \u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9. The\nalternative is to consider \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 personified as the nominative;\nbut it is difficult to conceive St Paul so speaking, more especially as\nwith \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd personification would suggest personality. The \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1\nindeed is personified in Clem. Alex. _Exc. Theod._ 43 (p. 979)\n\u03c3\u03c5\u03bd\u03b1\u03b9\u03bd\u03ad\u03c3\u03b1\u03bd\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2, and in Iren. i. 2. 6 \u03b2\u03bf\u03c5\u03bb\u1fc7 \u03bc\u03b9\u1fb7 \u03ba\u03b1\u1f76\n\u03b3\u03bd\u03ce\u03bc\u1fc3 \u03c4\u1f78 \u03c0\u1fb6\u03bd \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1ff6\u03bd \u03b1\u1f30\u03ce\u03bd\u03c9\u03bd \u03ba.\u03c4.\u03bb., i. 12. 4 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1\n\u03b7\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd [\u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b4\u03bf\u03be\u03ac\u03c3\u03b1\u03b9 \u03c4\u1f78\u03bd \u03c0\u03ac\u03c4\u03b5\u03c1\u03b1]; but the phraseology of the\nValentinians, to which these passages refer, cannot be taken as an\nindication of St Paul's usage, since their view of the \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 was\nwholly different. A third interpretation is found in Tertullian _adv.\nMarc._ v. 19, who translates \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 _in semetipso_, taking \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 as\nthe nominative to \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd: and this construction is followed by some\nmodern critics. But, though grammatically possible, it confuses the\ntheology of the passage hopelessly.\n\n\u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1] '_the plenitude_,' a recognised technical term in theology,\ndenoting the totality of the Divine powers and attributes; comp. ii. 9.\nSee the detached note on \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1. On the relation of this statement to\nthe speculations of the false teachers at Coloss\u00e6 see the introduction,\npp. 102, 112. Another interpretation, which explains \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 as\nreferring to the Church (comp. Ephes. i. 22), though adopted by several\nfathers, is unsuited to the context and has nothing to recommend it.\n\n\u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9] '_should have its permanent abode_.' The word occurs again\nin the same connexion, ii. 9. The false teachers probably, like their\nlater counterparts, maintained only a partial and transient connexion of\nthe \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 with the Lord. Hence St Paul declares in these two passages\nthat it is not a \u03c0\u03b1\u03c1\u03bf\u03b9\u03ba\u03af\u03b1 but a \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03af\u03b1. The two words \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6\u03bd,\n\u03c0\u03b1\u03c1\u03bf\u03b9\u03ba\u03b5\u1fd6\u03bd, occur in the LXX as the common renderings of \u05d9\u05e9\u05d1 and \u05e0\u05d5\u05e8\nrespectively, and are distinguished as the _permanent_ and the\n_transitory_; e.g. Gen. xxxvi. 44 (xxxvii. 1) \u03ba\u03b1\u03c4\u1ff7\u03ba\u03b5\u03b9 \u03b4\u1f72 \u1f38\u03b1\u03ba\u1f7c\u03b2 \u1f10\u03bd \u03c4\u1fc7 \u03b3\u1fc7\n\u03bf\u1f57 \u03c0\u03b1\u03c1\u1ff4\u03ba\u03b7\u03c3\u03b5\u03bd \u1f41 \u03c0\u03b1\u03c4\u03ae\u03c1 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f10\u03bd \u03b3\u1fc7 \u03a7\u03b1\u03bd\u03b1\u03ac\u03bd (comp. Hos. x. 5), Philo _Sacr.\nAb. et Ca._ 10 (I. p. 170 M) \u1f41 \u03c4\u03bf\u1fd6\u03c2 \u1f10\u03b3\u03ba\u03c5\u03ba\u03bb\u03af\u03bf\u03b9\u03c2 \u03bc\u03cc\u03bd\u03bf\u03b9\u03c2 \u1f10\u03c0\u03b1\u03bd\u03ad\u03c7\u03c9\u03bd \u03c0\u03b1\u03c1\u03bf\u03b9\u03ba\u03b5\u1fd6\n\u03c3\u03bf\u03c6\u03af\u1fb3, \u03bf\u1f50 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6, Greg. Naz. _Orat._ xiv. (I. p. 271 ed. Caillau) \u03c4\u03af\u03c2\n\u03c4\u1f74\u03bd \u03ba\u03ac\u03c4\u03c9 \u03c3\u03ba\u03b7\u03bd\u1f74\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f04\u03bd\u03c9 \u03c0\u03cc\u03bb\u03b9\u03bd; \u03c4\u03af\u03c2 \u03c0\u03b1\u03c1\u03bf\u03b9\u03ba\u03af\u03b1\u03bd \u03ba\u03b1\u1f76 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03af\u03b1\u03bd; comp.\n_Orat._ vii. (I. p. 200). See also the notes on Ephes. ii. 19, and on\nClem. Rom. \u00a7 1.\n\n------------------------------------------------------------------------\n\nI. 20]\n\n^{20}\u03ba\u03b1\u1f76 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u03b1\u03b9 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b5\u1f30\u03c2\n\n\n20. The false teachers aimed at effecting a partial reconciliation\nbetween God and man through the interposition of angelic mediators. The\nApostle speaks of an absolute and complete reconciliation of universal\nnature to God, effected through the mediation of the Incarnate Word.\nTheir mediators were ineffective, because they were neither human nor\ndivine. The true mediator must be both human and divine. It was\nnecessary that in Him all the plenitude of the Godhead should dwell. It\nwas necessary also that He should be born into the world and should\nsuffer as a man.\n\n\u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6] i.e. \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, as appears from the preceding \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7, and\nthe following \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u03b1\u1f35\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03b1\u03c5\u03c1\u03bf\u1fe6 \u03b1\u1f50\u03c4\u03bf\u1fe6, [\u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6]. This\nexpression \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 has been already applied to the Preincarnate Word\nin relation to the Universe (ver. 16); it is now used of the Incarnate\nWord in relation to the Church.\n\n\u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u03b1\u03b9] sc. \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd \u1f41 \u0398\u03b5\u03cc\u03c2. The personal pronoun \u03b1\u1f50\u03c4\u03cc\u03bd, instead\nof the reflexive \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd, is no real obstacle to this way of connecting\nthe words (see the next note). The alternative would be to take \u03c4\u1f78\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 as governing \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u03b1\u03b9, but this mode of expression is harsh\nand improbable.\n\nThe same double compound \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03c3\u03c3\u03b5\u03b9\u03bd is used below, ver. 21 and\nEphes. ii. 16, in place of the usual \u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03c3\u03c3\u03b5\u03b9\u03bd. It may be compared\nwith \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b9\u03c2, Acts iii. 21. Tertullian, arguing against the\ndualism of Marcion who maintained an antagonism between the demiurge and\nthe Christ, lays stress on the compound, _adv. Marc._ v. 19\n'_conciliari_ extraneo possent, _reconciliari_ vero non alii quam suo.'\nThe word \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03c3\u03c3\u03b5\u03b9\u03bd corresponds to \u1f00\u03c0\u03b7\u03bb\u03bb\u03bf\u03c4\u03c1\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 here and in\nEphes. ii. 16, implying a _restitution_ to a state from which they had\nfallen, or which was potentially theirs, or for which they were\ndestined. Similarly St Augustine on Gal. iv. 5 remarks that the word\nused of the \u03c5\u1f31\u03bf\u03b8\u03b5\u03c3\u03af\u03b1 is not _accipere_ (\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9\u03bd) but _recipere_\n(\u1f00\u03c0\u03bf\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9\u03bd). See the note there.\n\n\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1] The whole universe of things, material as well as spiritual,\nshall be restored to harmony with God. How far this restoration of\nuniversal nature maybe subjective, as involved in the changed\nperceptions of man thus brought into harmony with God, and how far it\nmay have an objective and independent existence, it were vain to\nspeculate.\n\n------------------------------------------------------------------------\n\nI. 21]\n\n\u03b1\u1f50\u03c4\u03cc\u03bd, \u03b5\u1f30\u03c1\u03b7\u03bd\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03b1\u03c2 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u1f05\u03b9\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03b1\u03c5\u03c1\u03bf\u1fe6 \u03b1\u1f50\u03c4\u03bf\u1fe6, \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f14\u03b9\u03c4\u03b5\n\u03c4\u1f70 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2 \u1f14\u03b9\u03c4\u03b5 \u03c4\u1f70 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2, ^{21}\u03ba\u03b1\u1f76 \u1f51\u03bc\u1fb6\u03c2 \u03c0\u03bf\u03c4\u1f72 \u1f44\u03bd\u03c4\u03b1\u03c2\n\u1f00\u03c0\u03ae\u03bb\u03bb\u03bf\u03c4\u03c1\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 >\n\n\n\u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd] '_to Him_,' i.e. 'to Himself.' The reconciliation is always\nrepresented as made to the Father. The reconciler is sometimes the\nFather Himself (2 Cor. v. 18, 19 \u1f10\u03ba \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u03b1\u03bd\u03c4\u03bf\u03c2 \u1f21\u03bc\u1fb6\u03c2\n\u1f11\u03b1\u03c5\u03c4\u1ff7 \u03b4\u03b9\u1f70 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 ... \u0398\u03b5\u1f78\u03c2 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd \u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03c3\u03c3\u03c9\u03bd \u1f11\u03b1\u03c5\u03c4\u1ff7),\nsometimes the Son (Ephes. ii. 16: comp. Rom. v. 10, 11). Excellent\nreasons are given (Bleek _Hebr._ II. p. 69, A. Buttmann _Gramm._ p. 97)\nfor supposing that the reflexive pronoun \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 etc. is never contracted\ninto \u03b1\u1f50\u03c4\u03bf\u1fe6 etc. in the Greek Testament. But at the same time it is quite\nclear that the oblique cases of the personal pronoun \u03b1\u1f50\u03c4\u03cc\u03c2 are there\nused very widely, and in cases where we should commonly find the\nreflexive pronoun in classical authors: e.g. Ephes. i. 4, 5 \u1f10\u03be\u03b5\u03bb\u03ad\u03be\u03b1\u03c4\u03bf\n\u1f21\u03bc\u1fb6\u03c2 ... \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f21\u03bc\u1fb6\u03c2 \u1f01\u03b3\u03af\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bc\u03ce\u03bc\u03bf\u03c5\u03c2 \u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd _\u03b1\u1f50\u03c4\u03bf\u1fe6_ ...\n\u03c0\u03c1\u03bf\u03bf\u03c1\u03af\u03c3\u03b1\u03c2 \u1f21\u03bc\u1fb6\u03c2 \u03b5\u1f30\u03c2 \u1f51\u03b9\u03bf\u03b8\u03b5\u03c3\u03af\u03b1\u03bd \u03b4\u03b9\u1f70 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03b5\u1f30\u03c2 _\u03b1\u1f50\u03c4\u03cc\u03bd_. See\nalso the instances given in A. Buttmann p. 98. It would seem indeed that\n\u03b1\u1f50\u03c4\u03bf\u1fe6 etc. may be used for \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 etc. in almost every connexion, except\nwhere it is the direct object of the verb.\n\n\u03b5\u1f30\u03c1\u03b7\u03bd\u03bf\u03c0\u03bf\u03b9\u03ae\u03c3\u03b1\u03c2] The word occurs in the LXX, Prov. x. 10, and in Hermes in\nStob. _Ecl. Phys._ xli. 45. The substantive \u03b5\u1f30\u03c1\u03b7\u03bd\u03bf\u03c0\u03bf\u03b9\u03cc\u03c2 (see Matt. v. 9)\nis found several times in classical writers.\n\n\u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6] The external authority for and against these words is nearly\nevenly balanced: but there would obviously be a tendency to reject them\nas superfluous. They are a resumption of the previous \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6. For\nother examples see ii. 13 \u1f51\u03bc\u1fb6\u03c2, Rom. viii. 23 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u03bf\u1f76, Gal. ii. 15, 16\n\u1f21\u03bc\u03b5\u1fd6\u03c2, Ephes. i. 13 \u1f10\u03bd \u1fa7 \u03ba\u03b1\u03af, iii. 1, 14 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03c7\u03ac\u03c1\u03b9\u03bd, where words are\nsimilarly repeated for the sake of emphasis or distinctness. In 2 Cor.\nxii. 7 there is a repetition of \u1f35\u03bd\u03b1 \u03bc\u1f74 \u1f51\u03c0\u03b5\u03c1\u03b1\u03af\u03c1\u03c9\u03bc\u03b1\u03b9, where again it is\nomitted in several excellent authorities.\n\n21\u201323. 'And ye too\u2014ye Gentiles\u2014are included in the terms of this peace.\nIn times past ye had estranged yourselves from God. Your hearts were\nhostile to Him, while ye lived on in your evil deeds. But now, in\nChrist's body, in Christ's flesh which died on the Cross for your\natonement, ye are reconciled to Him again. He will present you a living\nsacrifice, an acceptable offering unto Himself, free from blemish and\nfree even from censure, that ye may stand the piercing glance of Him\nwhose scrutiny no defect can escape. But this can only be, if ye remain\ntrue to your old allegiance, if ye hold fast (as I trust ye are holding\nfast) by the teaching of Epaphras, if the edifice of your faith is built\non solid foundations and not reared carelessly on the sands, if ye\nsuffer not yourselves to be shifted or shaken but rest firmly on the\nhope which ye have found in the Gospel\u2014the one universal unchangeable\nGospel, which was proclaimed to every creature under heaven, of which I\nPaul, unworthy as I am, was called to be a minister.'\n\n21. \u1f00\u03c0\u03b7\u03bb\u03bb\u03bf\u03c4\u03c1\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2] '_estranged_,' not \u1f00\u03bb\u03bb\u03bf\u03c4\u03c1\u03af\u03bf\u03c5\u03c2, '_strangers_';\ncomp. Ephes. ii. 12, iv. 18. See the note on \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u03b1\u03b9 ver. 20.\n\n------------------------------------------------------------------------\n\nI. 22]\n\n\u1f10\u03c7\u03b8\u03c1\u03bf\u1f7a\u03c2 \u03c4\u1fc7 \u03b4\u03b9\u03b1\u03bd\u03bf\u03af\u1fb3 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03c1\u03b3\u03bf\u03b9\u03c2 \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03bf\u03bd\u03b7\u03c1\u03bf\u1fd6\u03c2, \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5\n^{22}\u1f10\u03bd \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b4\u03b9\u1f70\n\n                    21 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 _\u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd._\n\n\n\u1f10\u03c7\u03b8\u03c1\u03bf\u03cd\u03c2] '_hostile_ to God,' as the consequence of \u1f00\u03c0\u03b7\u03bb\u03bb\u03bf\u03c4\u03c1\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2, not\n'_hateful_ to God,' as it is taken by some. The active rather than the\npassive sense of \u1f10\u03c7\u03b8\u03c1\u03bf\u03cd\u03c2 is required by the context, which (as commonly\nin the New Testament) speaks of the sinner as reconciled to God, not of\nGod as reconciled to the sinner: comp. Rom. v. 10 \u03b5\u1f30 \u03b3\u1f70\u03c1 \u1f10\u03c7\u03b8\u03c1\u03bf\u1f76 \u1f44\u03bd\u03c4\u03b5\u03c2\n\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03bc\u03b5\u03bd \u03c4\u1ff7 \u0398\u03b5\u1ff7 \u03ba.\u03c4.\u03bb. It is the mind of man, not the mind of God,\nwhich must undergo a change, that a reunion may be effected.\n\n\u03c4\u1fc7 \u03b4\u03b9\u03b1\u03bd\u03bf\u03b9|\u1fb3] '_in your mind, intent_.' For the dative of the part\naffected compare Ephes. iv. 18 \u1f10\u03c3\u03ba\u03bf\u03c4\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 \u03c4\u1fc7 \u03b4\u03b9\u03b1\u03bd\u03bf\u03af\u1fb3, Luke i. 51\n\u1f51\u03c0\u03b5\u03c1\u03b7\u03c6\u03ac\u03bd\u03bf\u03c5\u03c2 \u03b4\u03b9\u03b1\u03bd\u03bf\u03af\u1fb3 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd. So \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3, \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2, Matt. v. 8, xi.\n29, Acts vii. 51, 2 Cor. ix. 7, 1 Thess. ii. 17; \u03c6\u03c1\u03b5\u03c3\u03af\u03bd, 1 Cor. xiv. 20.\n\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03c1\u03b3\u03bf\u03b9\u03c2 \u03ba.\u03c4.\u03bb.] '_in_ the midst of, _in_ the performance of _your\nwicked works_'; the same use of the preposition as e.g. ii. 23, iv. 2.\n\n\u03bd\u03c5\u03bd\u03af] Here, as frequently, \u03bd\u1fe6\u03bd (\u03bd\u03c5\u03bd\u03af) admits an aorist, because it\ndenotes not 'at the present _moment_,' but 'in the present\n_dispensation_, the present _order of things_': comp. e.g. ver. 26, Rom.\nv. 11, vii. 6, xi. 30, 31, xvi. 26, Ephes. ii. 13, iii. 5, 2 Tim. i. 10,\n1 Pet. i. 12, ii. 10, 25. In all these passages there is a direct\ncontrast between the old dispensation and the new, more especially as\naffecting the relation of the Gentiles to God. The aorist is found also\nin Classical writers, where a similar contrast is involved; e.g. Plato\n_Symp._ 193 A \u03c0\u03c1\u1f78 \u03c4\u03bf\u1fe6, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03bb\u03ad\u03b3\u03c9, \u1f13\u03bd \u1f26\u03bc\u03b5\u03bd\u00b7 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u1f00\u03b4\u03b9\u03ba\u03af\u03b1\u03bd\n\u03b4\u03b9\u1ff3\u03ba\u03af\u03c3\u03b8\u03b7\u03bc\u03b5\u03bd \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6, Is\u00e6us _de Cleon. her._ 20 \u03c4\u03cc\u03c4\u03b5 \u03bc\u1f72\u03bd ... \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72\n... \u1f10\u03b2\u03bf\u03c5\u03bb\u03ae\u03b8\u03b7.\n\n\u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5] The reasons for preferring this reading, though the\ndirect authority for it is so slight, are given in the detached note on\nthe various readings. But, whether \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 or \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd be\npreferred, the construction requires explanation. If \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd be\nadopted, it is perhaps best to treat \u03b4\u1f72 as introducing the apodosis, the\nforegoing participial clause serving as the protasis: '_And you, though\nye were once estranged ... yet now hath he reconciled_,' in which case\nthe first \u1f51\u03bc\u1fb6\u03c2 will be governed directly by \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd; see Winer\n_Gramm._ \u00a7 liii. p. 553. If this construction be adopted, \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9\n\u1f51\u03bc\u1fb6\u03c2 will describe the result of \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd, 'so as _to present\nyou_'; but \u1f41 \u0398\u03b5\u1f78\u03c2 will still be the nominative to \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd as in 2\nCor. v. 19. If on the other hand \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 be taken, it is best to\nregard \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 as a direct indicative clause substituted\nfor the more regular participial form \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03b1\u03b3\u03ad\u03bd\u03c4\u03b1\u03c2 for the\nsake of greater emphasis: see the note on ver. 26 \u03c4\u1f78 \u1f00\u03c0\u03bf\u03ba\u03b5\u03ba\u03c1\u03c5\u03bc\u03bc\u03ad\u03bd\u03bf\u03bd ...\n\u03bd\u1fe6\u03bd \u03b4\u1f72 \u1f10\u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03b8\u03b7. In this case \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9 will be governed directly by\n\u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd, and will itself govern \u1f51\u03bc\u03b1^\u03c2 \u03c0\u03cc\u03c4\u03b5 \u1f44\u03bd\u03c4\u03b1\u03c2 \u03ba.\u03c4.\u03bb., the second\n\u1f51\u03bc\u1fb6\u03c2 being a repetition of the first; '_And you who once were estranged\n... but now ye have been reconciled ... to present you_, I say, _holy\nand without blemish_.' For the repetition of \u1f51\u03bc\u1fb6\u03c2, which was needed to\ndisentangle the construction, see the note on \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6 ver. 20.\n\n22. \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6] It has been supposed that St Paul added these\nwords, which are evidently emphatic, with a polemical aim either; (1) To\ncombat docetism. Of this form of error however there is no direct\nevidence till a somewhat later date: or (2) To combat a false\nspiritualism which took offence at the doctrine of an atoning sacrifice.\nBut for this purpose they would not have been adequate, because not\nexplicit enough. It seems simpler therefore to suppose that they were\nadded for the sake of greater clearness, to distinguish the natural body\nof Christ intended here from the mystical body mentioned just above ver.\n18. Similarly in Ephes. ii. 14 \u1f10\u03bd \u03c4\u1fc7 \u03c3\u03b1\u03c1\u03ba\u1f76 \u03b1\u1f50\u03c4\u03bf\u1fe6 is used rather than \u1f10\u03bd\n\u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u03b1\u1f50\u03c4\u03bf\u1fe6, because \u03c3\u1ff6\u03bc\u03b1 occurs in the context (ver. 16) of\nChrist's mystical body. The same expression, \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, which\nwe have here, occurs also below, ii. 11, but with a different emphasis\nand meaning. There the emphasis is on \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1, the contrast lying\nbetween the whole _body_ and a single _member_ (see the note); whereas\nhere \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 is the emphatic part of the expression, the antithesis\nbeing between the _material_ and the _spiritual_. Compare also Ecclus.\nxxiii. 16 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2 \u03c0\u03cc\u03c1\u03bd\u03bf\u03c2 \u1f10\u03bd \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6.\n\nMarcion omitted \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 as inconsistent with his views, and explained\n\u1f10\u03bd \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 to mean the Church. Hence the comment of Tertullian _adv.\nMarc._ v. 19, 'utique in eo corpore, in quo mori potuit per carnem,\nmortuus est, non per ecclesiam sed propter ecclesiam, corpus commutando\npro corpore, carnale pro spiritali.'\n\n------------------------------------------------------------------------\n\nI. 23]\n\n\u03c4\u03bf\u1fe6 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5 [\u03b1\u1f50\u03c4\u03bf\u1fe6], \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2 \u1f01\u03b3\u03af\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bc\u03ce\u03bc\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03b5\u03b3\u03ba\u03bb\u03ae\u03c4\u03bf\u03c5\u03c2\n\u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6, ^{23} \u03b5\u1f34 \u03b3\u03b5 \u1f10\u03c0\u03b9\u03bc\u03ad\u03bd\u03b5\u03c4\u03b5 \u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9 \u03c4\u03b5\u03b8\u03b5\u03bc\u03b5\u03bb\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76\n\u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03bc\u1f74 \u03bc\u03b5\u03c4\u03b1\u03ba\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9\n\n\n\u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9] If the construction which I have adopted be correct, this is\nsaid of God Himself, as in 2 Cor. iv. 14 \u1f41 \u1f10\u03b3\u03ad\u03b9\u03c1\u03b1\u03c2 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03ba\u03b1\u1f76\n\u1f21\u03bc\u1fb6\u03c2 \u03c3\u1f7a\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6 \u1f10\u03b3\u03b5\u03c1\u03b5\u1fd6 \u03ba\u03b1\u1f76 _\u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ae\u03c3\u03b5\u03b9_ \u03c3\u1f7a\u03bd \u1f51\u03bc\u1fd6\u03bd. This construction\nseems in all respects preferable to connecting \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9 directly with\n\u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 and interpreting the words, _'Ye have been reconciled so\nthat ye should present yourselves_ (\u1f51\u03bc\u1fb6\u03c2) ... _before Him_.' This latter\ninterpretation leaves the \u03ba\u03b1\u1f76 \u1f51\u03bc\u1fb6\u03c2 \u03c0\u03bf\u03c4\u1f72 \u1f44\u03bd\u03c4\u03b1\u03c2 \u03ba.\u03c4.\u03bb. without a\ngovernment, and it gives to the second \u1f51\u03bc\u1fb6\u03c2 a reflexive sense (as if\n\u1f51\u03bc\u1fb6\u03c2 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 or \u1f11\u03b1\u03c5\u03c4\u03bf\u03cd\u03c2), which is at least harsh.\n\n\u1f00\u03bc\u03ce\u03bc\u03bf\u03c5\u03c2] _'without blemish_' rather than '_without blame_,' in the\nlanguage of the New Testament; see the noteon Ephes. i. 4. It is a\nsacrificial word, like \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, \u1f40\u03bb\u03cc\u03ba\u03bb\u03b7\u03c1\u03bf\u03c2, etc. The verb \u03c0\u03b1\u03c1\u03b9\u03c3\u03c4\u03ac\u03bd\u03b1\u03b9 also\nis used of presenting a sacrifice in Rom. xii. 1 _\u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9_ \u03c4\u1f70\n\u03c3\u03ce\u03bc\u03b1\u03c4\u03b1 \u1f51\u03bc\u1ff6\u03bd \u03b8\u03c5\u03c3\u03af\u03b1\u03bd \u03b6\u1ff6\u03c3\u03b1\u03bd \u1f01\u03b3\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb., Lev. xvi. 7 (v. l.): comp. Luke\nii. 22.\n\n\u1f00\u03bd\u03b5\u03b3\u03ba\u03bb\u03ae\u03c4\u03bf\u03c5\u03c2] an advance upon \u1f00\u03bc\u03ce\u03bc\u03bf\u03c5\u03c2, 'in whom not only no blemish is\nfound, but against whom no charge is brought': comp. 1 Tim. vi. 14\n\u1f04\u03c3\u03c0\u03b9\u03bb\u03bf\u03bd, \u1f00\u03bd\u03b5\u03c0\u03af\u03bb\u03b7\u03bc\u03c0\u03c4\u03bf\u03bd. The word \u1f00\u03bd\u03ad\u03b3\u03ba\u03bb\u03b7\u03c4\u03bf\u03c2 occurs again in 1 Cor. i. 8,\n1 Tim. iii. 10, Tit. i. 6, 7.\n\n\u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6] '_before Him_,' i.e. 'Himself,' as in the parallel\npassage, Ephes. i. 4; if the construction here adopted be correct. For\nthis use of the personal pronoun instead of the reflexive see the note\non \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd, ver. 20. But does \u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 refer to God's future\njudgment or His present approbation? The latter seems more probable,\nboth because the expression certainly has this meaning in the parallel\npassage, Ephes. i. 4, and because \u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03b1\u03bd, \u1f10\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd, \u03ba\u03b1\u03c4\u03ad\u03bd\u03b1\u03bd\u03c4\u03b9, etc.,\nare commonly so used; e.g. Rom. xiv. 22, 1 Cor. i. 29, 2 Cor. ii. 17,\niv. 2, vii. 12, xii. 19, etc. On the other hand, where the future\njudgment is intended, a different expression is found, 2 Cor. v. 10\n\u1f14\u03bc\u03c0\u03c1\u03bf\u03c3\u03b8\u03b5\u03bd \u03c4\u03bf\u1fe6 \u03b2\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. Thus God is here regarded, not as the\njudge who tries the accused, but as the \u03bc\u03c9\u03bc\u03bf\u03c3\u03ba\u03cc\u03c0\u03bf\u03c2 who examines the\nvictims (Polyc. _Phil._ 4, see the note on Ephes. i. 4). Compare Heb.\niv. 12, 13 for a closely allied metaphor. The passage in Jude 24, \u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9\n\u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f00\u03bc\u03ce\u03bc\u03bf\u03c5\u03c2 \u1f10\u03bd \u1f00\u03b3\u03b1\u03bb\u03bb\u03b9\u03ac\u03c3\u03b5\u03b9, though perhaps\nreferring to final approval, is too different in expression to influence\nthe interpretation of Paul's language here.\n\n23. \u03b5\u1f34 \u03b3\u03b5] On the force of these particles see Gal. iii. 4. They express\na pure hypothesis in themselves, but the indicative mood following\nconverts the hypothesis into a hope.\n\n\u1f10\u03c0\u03b9\u03bc\u03ad\u03bd\u03b5\u03c4\u03b5] '_ye abide by, ye adhere to_,' with a dative; the common\nconstruction of \u1f10\u03c0\u03b9\u03bc\u03ad\u03bd\u03b5\u03b9\u03bd in St Paul: see the note on Phil. i. 24. In\nthis connexion \u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9 is perhaps '_your_ faith,' rather than '_the_\nfaith.'\n\n\u03c4\u03b5\u03b8\u03b5\u03bc\u03b5\u03bb\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 \u03ba.\u03c4.\u03bb.] '_built on a foundation and so firm_'; not like\nthe house of the foolish man in the parable who built \u03c7\u03c9\u03c1\u1f76\u03c2 \u03b8\u03b5\u03bc\u03b5\u03bb\u03af\u03bf\u03c5,\nLuke vi. 49. For \u03c4\u03b5\u03b8\u03b5\u03bc\u03b5\u03bb\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 comp. Ephes. iii. 17. The consequence of\n\u03c4\u03b5\u03b8\u03b5\u03bc\u03b5\u03bb\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 is \u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03b9: Clem. Rom. 33 _\u1f25\u03b4\u03c1\u03b1\u03c3\u03b5\u03bd_ \u1f10\u03c0\u1f76 \u03c4\u1f78\u03bd \u1f00\u03c3\u03c6\u03b1\u03bb\u1fc6\n\u03c4\u03bf\u1fe6 \u1f30\u03b4\u03af\u03bf\u03c5 \u03b2\u03bf\u03c5\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 _\u03b8\u03b5\u03bc\u03ad\u03bb\u03b9\u03bf\u03bd_. The words \u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03c2, \u1f11\u03b4\u03c1\u03ac\u03b6\u03c9, etc.,\nare not uncommonly applied to buildings, e.g. \u1f11\u03b4\u03c1\u03b1\u03af\u03c9\u03bc\u03b1 1 Tim. iii. 15.\nComp. Ign. _Ephes._ 10 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03b9 \u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9.\n\n\u03bc\u1f74 \u03bc\u03b5\u03c4\u03b1\u03ba\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9] '_not constantly shifting_,' a present tense; the\nsame idea as \u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03b9 expressed from the negative side, as in 1 Cor. xv.\n58 \u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03b9 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b5, \u1f00\u03bc\u03b5\u03c4\u03b1\u03ba\u03af\u03bd\u03b7\u03c4\u03bf\u03b9, Polyc. _Phil._ 10 'firmi in fide et\nimmutabiles.'\n\n------------------------------------------------------------------------\n\nI. 23]\n\n\u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5 \u03bf\u1f57 \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5, \u03c4\u03bf\u1fe6 \u03ba\u03b7\u03c1\u03c5\u03c7\u03b8\u03ad\u03bd\u03c4\u03bf\u03c2 \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3\n\u03ba\u03c4\u03af\u03c3\u03b5\u03b9 \u03c4\u1fc7 \u1f51\u03c0\u1f78 \u03c4\u1f78\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u03cc\u03bd, \u03bf\u1f57 \u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b7\u03bd \u1f10\u03b3\u1f7c \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03b4\u03b9\u03ac\u03ba\u03bf\u03bd\u03bf\u03c2.\n\n\n\u03c4\u1fc6\u03c2 \u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2 \u03ba.\u03c4.\u03bb.] '_the hope held out by the Gospel_,' \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5\nbeing a subjective genitive, as in Ephes. i. 18 \u1f21 \u1f10\u03bb\u03c0\u1f76\u03c2 \u03c4\u1fc6\u03c2 \u03ba\u03bb\u03ae\u03c3\u03b5\u03c9\u03c2\n(comp. iv. 4).\n\n\u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03ba\u03c4\u03af\u03c3\u03b5\u03b9] '_among every creature_,' in fulfilment of the Lord's\nlast command, Mark xvi. 15 \u03ba\u03b7\u03c1\u03cd\u03be\u03b1\u03c4\u03b5 \u03c4\u1f78 \u03b5\u03c5\u03b1\u03b3\u03b3\u03ad\u03bb\u03b9\u03bf\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c4\u1fc7 \u03ba\u03c4\u03af\u03c3\u03b5\u03b9. Here\nhowever the definitive article, though found in the received text, \u1f10\u03bd\n\u03c0\u03ac\u03c3\u1fc3 \u03c4\u1fc7 \u03ba\u03c4\u03af\u03c3\u03b5\u03b9, must be omitted in accordance with the best authorities.\nFor the meanings of \u03c0\u1fb6\u03c3\u03b1 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2, \u03c0\u1fb6\u03c3\u03b1 \u1f21 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2, see the note on ver. 15.\nThe expression \u03c0\u1fb6\u03c3\u03b1 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2 must not be limited to man. The statement is\ngiven in the broadest form, all creation animate and inanimate being\nincluded, as in Rev. v. 13 _\u03c0\u1fb6\u03bd \u03ba\u03c4\u03af\u03c3\u03bc\u03b1_ ... \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f10\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2\n_\u03c0\u03ac\u03bd\u03c4\u03b1_ \u1f25\u03ba\u03bf\u03c5\u03c3\u03b1 \u03bb\u03ad\u03b3\u03bf\u03bd\u03c4\u03b1 \u03ba.\u03c4.\u03bb. For the hyperbole \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03ba\u03c4\u03af\u03c3\u03b5\u03b9\ncompare 1 Thess. i. 8 \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u03cc\u03c0\u1ff3. To demand statistical exactness in\nsuch a context would be to require what is never required in similar\ncases. The motive of the Apostle here is at once to emphasize the\nuniversality of the genuine Gospel, which has been offered without\nreserve to all alike, and to appeal to its publicity, as the credential\nand guarantee of its truth: see the notes on ver. 6 \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03c4\u1ff7 \u03ba\u03cc\u03c3\u03bc\u1ff3\nand on ver. 28 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd.\n\n\u03bf\u1f56 \u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b7\u03bd \u03ba.\u03c4.\u03bb.] Why does St Paul introduce this mention of himself\nso abruptly? His motive can hardly be the assertion of his Apostolic\nauthority, for it does not appear that this was questioned; otherwise he\nwould have declared his commission in stronger terms. We can only answer\nthat impressed with the dignity of his office, as involving the offer of\ngrace to the Gentiles, he cannot refrain from magnifying it. At the same\ntime this mention enables him to link himself in bonds of closer\nsympathy with the Colossians, and he passes on at once to his relations\nwith them: comp. Ephes. iii. 2\u20139, 1 Tim. i. 11 sq., in which latter\npassage the introduction of his own name is equally abrupt.\n\n\u1f10\u03b3\u1f7c \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2] i.e. 'weak and unworthy as I am': comp. Ephes. iii. 8 \u1f10\u03bc\u03bf\u1f76\n\u03c4\u1ff7 \u1f10\u03bb\u03b1\u03c7\u03b9\u03c3\u03c4\u03bf\u03c4\u03ad\u03bc\u03c9 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u1f01\u03b3\u03af\u03c9\u03bd.\n\n------------------------------------------------------------------------\n\nI. 24]\n\n^{24}\u039d\u1fe6\u03bd \u03c7\u03b1\u03af\u03c1\u03c9 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03b1\u03b8\u03ae\u03bc\u03b1\u03c3\u03b9\u03bd \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd, \u03ba\u03b1\u1f76\n\n\n24\u201327. '_Now_ when I see the full extent of God's mercy, _now_ when I\nponder over His mighty work of reconciliation, I cannot choose but\nrejoice in my sufferings. Yes, I Paul the persecutor, I Paul the feeble\nand sinful, am permitted to supplement\u2014I do not shrink from the word\u2014to\nsupplement the afflictions of Christ. Despite all that He underwent, He\nthe Master has left something still for me the servant to undergo. And\nso my flesh is privileged to suffer for His body\u2014His spiritual body, the\nChurch. I was appointed a minister of the Church, a steward in God's\nhousehold, for this very purpose, that I might administer my office on\nyour behalf, might dispense to you Gentiles the stores which His\nbountiful grace has provided. Thus I was charged to preach without\nreserve the whole Gospel of God, to proclaim the great mystery which had\nremained a secret through all the ages and all the generations from the\nbeginning, but which now in these last times was revealed to His holy\npeople. For such was His good pleasure. God willed to make known to\nthem, in all its inexhaustible wealth thus displayed through the call of\nthe Gentiles, the glorious revelation of this mystery\u2014Christ not the\nSaviour of the Jews only, but Christ dwelling in _you_, Christ become to\n_you_ the hope of glory.'\n\n24. \u039d\u1fe6\u03bd \u03c7\u03b1\u03af\u03c1\u03c9] A sudden outburst of thanksgiving, that he, who was less\nthan the least, who was not worthy to be called an Apostle, should be\nallowed to share and even to supplement the sufferings of Christ. The\nrelative \u1f45\u03c2, which is found in some authorities, is doubtless the\nrepetition of the final syllable of \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2; but its insertion would be\nassisted by the anxiety of scribes to supply a connecting link between\nthe sentences. The genuine reading is more characteristic of St Paul.\nThe abruptness, which dispenses with a connecting particle, has a\nparallel in Tim. i. 12 \u03c7\u03ac\u03c1\u03b9\u03bd \u1f14\u03c7\u03c9 \u03c4\u1ff7 \u1f10\u03bd\u03b4\u03c5\u03bd\u03b1\u03bc\u03ce\u03c3\u03b1\u03bd\u03c4\u03af \u03bc\u03b5 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03ba.\u03c4.\u03bb.,\nwhere also the common text inserts a link of connexion, \u03ba\u03b1\u1f76 \u03c7\u03ac\u03c1\u03b9\u03bd \u1f14\u03c7\u03c9\n\u03ba.\u03c4.\u03bb. Compare also 2 Cor. vii. 9 \u03bd\u1fe6\u03bd \u03c7\u03b1\u03af\u03c1\u03c9, \u03bf\u1f50\u03c7 \u1f45\u03c4\u03b9 \u03ba.\u03c4.\u03bb., where again\nthere is no connecting particle.\n\nThe thought underlying \u03bd\u1fe6\u03bd seems to be this: 'If ever I have been\ndisposed to repine at my lot, if ever I have felt my cross almost too\nheavy to bear, yet _now_\u2013now, when I contemplate the lavish wealth of\nGod's mercy\u2014now when I see all the glory of bearing a part in this\nmagnificent work\u2014my sorrow is turned to joy.'\n\n------------------------------------------------------------------------\n\nI. 24]\n\n\u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6 \u03c4\u1f70 \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c4\u1ff6\u03bd \u03b8\u03bb\u03af\u03c8\u03b5\u03c9\u03bd \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\n\n\n\u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6] '_I fill up on my part_', '_I supplement_.' The single\ncompound \u1f00\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd occurs several times (e.g. 1 Cor. xiv. 16, xvi. 17,\nGal. vi. 2); another double compound \u03c0\u03c1\u03bf\u03c3\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd twice (2 Cor. ix.\n12, xi. 9; comp. Wisd. xix. 4, v.l.); but \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd only here in the\nLXX or New Testament. For this verb compare Demosth. _de Symm._ p. 182\n\u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c3\u03c5\u03bc\u03bc\u03bf\u03c1\u03b9\u1ff6\u03bd \u1f11\u03ba\u03ac\u03c3\u03c4\u03b7\u03bd \u03b4\u03b9\u03b5\u03bb\u03b5\u1fd6\u03bd \u03ba\u03b5\u03bb\u03b5\u03cd\u03c9 \u03c0\u03ad\u03bd\u03c4\u03b5 \u03bc\u03ad\u03c1\u03b7 \u03ba\u03b1\u03c4\u1f70 \u03b4\u03ce\u03b4\u03b5\u03ba\u03b1\n\u1f04\u03bd\u03b4\u03c1\u03b1\u03c2, \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b1\u03c2 _\u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03b5\u1f50\u03c0\u03bf\u03c1\u03ce\u03c4\u03b1\u03c4\u03bf\u03bd_ \u1f00\u03b5\u1f76 \u03c4\u03bf\u1f7a\u03c2\n\u1f00\u03c0\u03bf\u03c1\u03c9\u03c4\u03ac\u03c4\u03bf\u03c5\u03c2 (where \u03c4\u03bf\u1f7a\u03c2 \u1f00\u03c0\u03bf\u03c1\u03c9\u03c4\u03ac\u03c4\u03bf\u03c5\u03c2 should be taken as the subject to\n\u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b1\u03c2), Dion Cass. xliv. 48 \u1f35\u03bd' \u1f45\u03c3\u03bf\u03bd ... \u1f10\u03bd\u03ad\u03b4\u03b5\u03b9, \u03c4\u03bf\u1fe6\u03c4\u03bf \u1f10\u03ba \u03c4\u1fc6\u03c2\n_\u03c0\u03b1\u03c1\u1f70 \u03c4\u1ff6\u03bd \u1f04\u03bb\u03bb\u03c9\u03bd_ \u03c3\u03c5\u03bd\u03c4\u03b5\u03bb\u03b5\u03af\u03b1\u03c2 \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc7, Clem. Alex. _Strom._\nvii. 12 p. 878 \u03bf\u1f57\u03c4\u03bf\u03c2 ... \u03c4\u1f74\u03bd _\u1f00\u03c0\u03bf\u03c3\u03c4\u03bf\u03bb\u03b9\u03ba\u1f74\u03bd \u1f00\u03c0\u03bf\u03c5\u03c3\u03af\u03b1\u03bd_ \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fd6,\nApollon. _Constr. Or._ i. 3 (p. 13 sq.) \u1f21 \u1f00\u03bd\u03c4\u03c9\u03bd\u03c5\u03bc\u03af\u03b1 \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03c3\u03b1 \u03ba\u03b1\u1f76\n\u03c4\u1f74\u03bd \u03b8\u03ad\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c4\u03ac\u03be\u03b9\u03bd \u03c4\u03bf\u1fe6 \u1fe5\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2, Ptol. _Math. Comp._\nvi. 9 (I. p. 435 ed. Halma) \u1f10\u03c0\u03b5\u1f76 \u03b4' \u1f21 _\u03bc\u1f72\u03bd \u1f10\u03bb\u03bb\u03b5\u03af\u03c0\u03b5\u03b9\u03bd_ \u1f10\u03c0\u03bf\u03af\u03b5\u03b9 \u03c4\u1f74\u03bd\n\u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b9\u03bd \u1f21 _\u03b4\u1f72 \u03c0\u03bb\u03b5\u03bf\u03bd\u03ac\u03b6\u03b5\u03b9\u03bd_ \u03ba\u03b1\u03c4\u03ac \u03c4\u03b9\u03bd\u03b1 \u03c3\u03c5\u03bd\u03c4\u03c5\u03c7\u03af\u03b1\u03bd \u1f23\u03bd \u1f34\u03c3\u03c9\u03c2 \u03ba\u03b1\u1f76 \u1f41\n\u1f3d\u03c0\u03c0\u03b1\u03c1\u03c7\u03bf\u03c2 \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03bd \u03c0\u03c9\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bd\u03b5\u03bd\u03bf\u03ae\u03ba\u03b5\u03b9 \u03ba.\u03c4.\u03bb. The substantive\n\u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03ae\u03c1\u03c9\u03c3\u03b9\u03c2 occurs in Diog. Laert. x. 48. So too \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03ae\u03b8\u03b5\u03b9\u03bd Xen.\n_Hell._ ii. 4. 11, 12 \u03be\u03c5\u03bd\u03b5\u03c4\u03ac\u03be\u03b1\u03bd\u03c4\u03bf, \u1f65\u03c3\u03c4\u03b5 _\u1f11\u03bc\u03c0\u03bb\u1fc6\u03c3\u03b1\u03b9 \u03c4\u1f74\u03bd \u1f41\u03b4\u03cc\u03bd_ ... \u03bf\u1f31\n\u03b4\u1f72 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03c6\u03c5\u03bb\u1fc6\u03c2 \u1f00\u03bd\u03c4\u03b1\u03bd\u03ad\u03c0\u03bb\u03b7\u03c3\u03b1\u03bd ... \u03c4\u1f74\u03bd \u1f41\u03b4\u03cc\u03bd. Compare also \u1f00\u03bd\u03c4\u03b1\u03bd\u03b9\u03c3\u03bf\u1fe6\u03bd\nThemist. _Paraphr. Arist._ 43 B \u03bf\u1f50\u03b4\u1f72\u03bd \u03ba\u03c9\u03bb\u03cd\u03b5\u03b9 \u03ba\u03b1\u03c4\u1f70 \u03c4\u03b1\u1f50\u03c4\u1f78\u03bd _\u1f04\u03bb\u03bb\u03bf\u03b8\u03af \u03c0\u03bf\u03c5\n\u03bc\u03b5\u03c4\u03b1\u03b2\u03ac\u03bb\u03bb\u03b5\u03b9\u03bd_ \u1f00\u03ad\u03c1\u03b1 \u03b5\u1f30\u03c2 \u1f55\u03b4\u03c9\u03c1 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03c4\u03b1\u03bd\u03b9\u03c3\u03bf\u1fe6\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f78\u03bd \u03c3\u03cd\u03bc\u03c0\u03b1\u03bd\u03c4\u03b1 \u1f44\u03b3\u03ba\u03bf\u03bd, and\n\u1f00\u03bd\u03c4\u03b1\u03bd\u03af\u03c3\u03c9\u03bc\u03b1 Joseph. _Ant._ xviii. 9. 7. The meaning of \u1f00\u03bd\u03c4\u1f76 in this\ncompound will be plain from the passages quoted. It signifies that the\nsupply comes _from an opposite quarter_ to the deficiency. This idea is\nmore or less definitely expressed in the context of all the passages, in\nthe words which are spaced. The force of \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd in St Paul is\noften explained as denoting simply that the supply _corresponds in\nextent_ to the deficiency. This interpretation practically deprives \u1f00\u03bd\u03c4\u03af\nof any meaning, for \u1f00\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd alone would denote as much. If indeed the\nsupply had been the subject of the verb, and the sentence had run \u03c4\u1f70\n\u03c0\u03b1\u03b8\u03ae\u03bc\u03b1\u03c4\u03ac \u03bc\u03bf\u03c5 \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fd6 \u03c4\u1f70 \u1f51\u03c3\u03c4\u03b7\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1 \u03ba.\u03c4.\u03bb., this idea might perhaps\nbe reached without sacrificing the sense of \u1f00\u03bd\u03c4\u03af; but in such a passage\nas this, where one personal agent is mentioned in connexion with the\nsupply and another in connexion with the deficiency, the one forming the\nsubject and the other being involved in the object of the verb, the \u1f00\u03bd\u03c4\u1f76\ncan only describe the correspondence of these personal agents. So\ninterpreted, it is eminently expressive here. The point of the Apostle's\nboast is that Christ the sinless Master should have _left_ something for\nPaul the unworthy servant to suffer. The right idea has been seized and\nis well expressed by Photius _Amphil._ 121 (I. p. 709 Migne) \u03bf\u1f50 \u03b3\u1f70\u03c1\n\u1f01\u03c0\u03bb\u1ff6\u03c2 \u03c6\u03b7\u03c3\u03b9\u03bd \u1f08\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6, \u1f00\u03bb\u03bb' \u1f08\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6, \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9\u03bd, \u1f08\u03bd\u03c4\u1f76 \u03b4\u03b5\u03c3\u03c0\u03cc\u03c4\u03bf\u03c5 \u03ba\u03b1\u1f76\n\u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03ac\u03bb\u03bf\u03c5 \u1f41 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u1f10\u03b3\u1f7c \u03ba\u03b1\u1f76 \u03bc\u03b1\u03b8\u03b7\u03c4\u1f74\u03c2 \u03ba.\u03c4.\u03bb. Similar in meaning, though\nnot identical, is the expression in 2 Cor. i. 5, where the sufferings of\nChrist are said to 'overflow' (\u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03b5\u03cd\u03b5\u03b9\u03bd) upon the Apostle. The\ntheological difficulty which this plain and natural interpretation of\n\u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd is supposed to involve will be considered in the note on\n\u03c4\u1ff6\u03bd \u03b8\u03bb\u03af\u03c8\u03b5\u03c9\u03bd.\n\n\u03c4\u1f70 \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1] '_the things lacking_.' This same word \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1\n'deficiency' occurs with \u1f00\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd 1 Cor. xvi. 17, Phil. ii. 30, and\nwith \u03c0\u03c1\u03bf\u03c3\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd 2 Cor. ix. 12, xi. 9. Its direct opposite is\n\u03c0\u03b5\u03c1\u03af\u03c3\u03c3\u03b5\u03c5\u03bc\u03b1 'abundance, superfluity,' 2 Cor. viii. 13, 14; comp. Luke\nxxi. 4. Another interpretation, which makes \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1 an antithesis to\n\u03c0\u03c1\u03bf\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1, explaining it as 'the later' as opposed to the earlier\n'sufferings of Christ,' is neither supported by the usage of the word\nnor consistent with \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6.\n\n\u03c4\u1ff6\u03bd \u03b8\u03bb\u03af\u03c8\u03b5\u03c9\u03bd \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6] '_of the afflictions of Christ_,' i.e. which\nChrist endured. This seems to be the only natural interpretation of the\nwords. Others have explained them as meaning 'the afflictions imposed by\nChrist,' or 'the afflictions endured for Christ's sake,' or 'the\nafflictions which resemble those of Christ.' All such interpretations\nput a more or less forced meaning on the genitive. All alike ignore the\nmeaning of \u1f00\u03bd\u03c4\u1f76 in \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6 which points to a _distinction_ of\npersons suffering. Others again suppose the words to describe St Paul's\nown afflictions regarded as Christ's, because Christ suffers in His\nsuffering Church; e.g. Augustine _in Psalm._ cxlii. \u00a7 3 (IV. p. 1590)\n'Patitur, inquit, adhuc Christus pressuram, non in carne sua in qua\nascendit in c\u00e6lum, sed in carne mea qu\u00e6 adhuc laborat in terra,' quoting\nGal. ii. 20. This last is a very favourite explanation, and has much to\nrecommend it. It cannot be charged with wresting the meaning of \u03b1\u1f31\n\u03b8\u03bb\u03af\u03c8\u03b5\u03b9\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. Moreover it harmonizes with St Paul's mode of\nspeaking elsewhere. But, like the others, it is open to the fatal\nobjection that it empties the first preposition in \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6 of any\nforce. The central idea in this interpretation is the _identification_\nof the suffering Apostle with the suffering Christ, whereas \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6\nemphasizes the _distinction_ between the two. It is therefore\ninconsistent with this context, however important may be the truth which\nit expresses.\n\nThe theological difficulty, which these and similar explanations are\nintended to remove, is imaginary and not real. There is a sense in which\nit is quite legitimate to speak of Christ's afflictions as _incomplete_,\na sense in which they may be, and indeed must be, _supplemented_. For\nthe sufferings of Christ may be considered from two different points of\nview. They are either _satisfactori\u00e6_ or _\u00e6dificatori\u00e6_. They have their\nsacrificial efficacy, and they have their ministerial utility. (1) From\nthe former point of view the Passion of Christ was the one full perfect\nand sufficient sacrifice, oblation, and satisfaction for the sins of the\nwhole world. In this sense there could be no \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1 of Christ's\nsufferings; for, Christ's sufferings being different _in kind_ from\nthose of His servants, the two are incommensurable. But in this sense\nthe Apostle would surely have used some other expression such as \u03c4\u03bf\u1fe6\n\u03c3\u03c4\u03b1\u03c5\u03c1\u03bf\u1fe6 (i. 20, Eph. ii. 16 etc.), or \u03c4\u03bf\u1fe6 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5 (i. 22, Rom. v. 10,\nHeb. ii. 14, etc.), but hardly \u03c4\u1ff6\u03bd \u03b8\u03bb\u03af\u03c8\u03b5\u03c9\u03bd. Indeed \u03b8\u03bb\u03af\u03c8\u03b9\u03c2, 'affliction,'\nis not elsewhere applied in the New Testament in any sense to Christ's\nsufferings, and certainly would not suggest a sacrificial act. (2) From\nthe latter point of view it is a simple matter of fact that the\nafflictions of every saint and martyr do supplement the afflictions of\nChrist. The Church is built up by repeated acts of self-denial in\nsuccessive individuals and successive generations. They continue the\nwork which Christ began. They bear their part in the sufferings of\nChrist (2 Cor. i. 7 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03bf\u1f76 \u03c4\u1ff6\u03bd \u03c0\u03b1\u03b8\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd, Phil. iii. 10 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1\u03bd \u03c4\u1ff6\u03bd\n\u03c0\u03b1\u03b8\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd); but St Paul would have been the last to say that they bear\ntheir part in the atoning sacrifice of Christ. This being so, St Paul\ndoes not mean to say that his own sufferings filled up all the\n\u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1, but only that they _went towards_ filling them up. The\npresent tense \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6 denotes an inchoate, and not a complete act.\nThese \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1 will never be fully supplemented, until the struggle of\nthe Church with sin and unbelief is brought to a close.\n\nThus the idea of expiation or satisfaction is wholly absent from this\npassage; and with it is removed the twofold temptation which has beset\ntheologians of opposite schools. (1) On the one hand Protestant\ncommentators, rightly feeling that any interpretation which infringed\nthe completeness of the work wrought by Christ's death must be wrong,\nbecause it would make St Paul contradict himself on a cardinal point of\nhis teaching, have been tempted to wrest the sense of the words. They\nhave emptied \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u1ff6 of its proper force; or they have assigned a\nfalse meaning to \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1; or they have attached a non-natural sense\nto the genitive \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. (2) On the other hand Romanist\ncommentators, while protesting (as they had a right to do) against these\nmethods of interpretation, have fallen into the opposite error. They\nhave found in this passage an assertion of the merits of the saints, and\n(as a necessary consequence) of the doctrine of indulgences. They have\nnot observed that, if the idea of vicarious satisfaction comes into the\npassage at all, the satisfaction of St Paul is represented here as the\nsame in kind with the satisfaction of Christ, however different it may\nbe in degree; and thus they have truly exposed themselves to the\nreproach which Estius indignantly repudiates on their behalf, 'quasi\nChristus non satis passus sit ad redemptionem nostram, ideoque\nsupplemento martyrum opus habeat; quod impium est sentire, quodque\nCatholicos dicere non minus impie calumniantur h\u00e6retici.' It is no part\nof a commentator here to enquire generally whether the Roman doctrine of\nthe satisfaction of the saints can in any way be reconciled with St\nPaul's doctrine of the satisfaction of Christ. It is sufficient to say\nthat, so far as regards this particular passage, the Roman doctrine can\nonly be imported into it at the cost of a contradiction to the Pauline\ndoctrine. It is only fair to add however that Estius himself says, 'qu\u00e6\nquidem doctrina, etsi Catholica et Apostolica sit, atque aliunde satis\nprobetur, ex hoc tamen Apostoli loco nobis non videtur admodum solide\nstatui posse.' But Roman Catholic commentators generally find this\nmeaning in the text, as may be seen from the notes of \u00e0 Lapide.\n\n------------------------------------------------------------------------\n\nI. 25]\n\n \u1f10\u03bd \u03c4\u1fc7 \u03c3\u03b1\u03c1\u03ba\u03af \u03bc\u03bf\u03c5 \u1f51\u03c0\u1f72\u03c1 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u00b7 ^{25}\u1f27\u03c2\n\u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b7\u03bd \u1f10\u03b3\u1f7c \u03b4\u03b9\u03ac\u03ba\u03bf\u03bd\u03bf\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd\n\n\n\u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 An antithesis of the Apostle's own flesh and Christ's\nbody. This antithetical form of expression obliges St Paul to explain\nwhat he means by the body of Christ, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1; comp. ver. 18.\nContrast the explanation in ver. 22 \u1f10\u03bd \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, and\nsee the note there.\n\n25. \u03c4\u1f74\u03bd \u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb.] '_stewardship in the house of God_.' The word\n\u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1 seems to have two senses: (1) 'The actual administration of a\nhousehold'; (2) 'The office of the administrator.' For the former\nmeaning see the note on Ephes. i. 10; for the latter sense, which it has\nhere, compare 1 Cor. ix. 17 \u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03bd \u03c0\u03b5\u03c0\u03af\u03c3\u03c4\u03b5\u03c5\u03bc\u03b1\u03b9, Luke xvi. 2\u20134,\nIsaiah xxii. 19, 21. So the Apostles and ministers of the Church are\ncalled \u03bf\u1f30\u03ba\u03bf\u03bd\u03cc\u03bc\u03bf\u03b9, 1 Cor. iv. 1, 2, Tit. i. 7: comp. 1 Pet. iv. 10.\n\n------------------------------------------------------------------------\n\nI.26]\n\n\u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u1f74\u03bd \u03b4\u03bf\u03b8\u03b5\u1fd6\u03c3\u03ac\u03bd \u03bc\u03bf\u03b9 \u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2, \u03c0\u03bb\u03b7\u03c1\u1ff6\u03c3\u03b1\u03b9 \u03c4\u1f78\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03c4\u03bf\u1fe6\n\u0398\u03b5\u03bf\u1fe6, ^{26} \u03c4\u1f78 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd \u03c4\u1f78 \u1f00\u03c0\u03bf\u03ba\u03b5\u03ba\u03c1\u03c5\u03bc\u03bc\u03ad\u03bd\u03bf\u03bd\n\n\n\u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2] '_to youward_,' i.e. 'for the benefit of you, the Gentiles';\n\u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2 being connected with \u03c4\u1f74\u03bd \u03b4\u03bf\u03b8\u03b5\u1fd6\u03c3\u03ac\u03bd \u03bc\u03bf\u03b9, as in Ephes. iii. 2 \u03c4\u1f74\u03bd\n\u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03bd \u03c4\u1fc6\u03c2 \u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u1fc6\u03c2 \u03b4\u03bf\u03b8\u03b5\u03af\u03c3\u03b7\u03c2 \u03bc\u03bf\u03b9 \u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2; comp. Rom.\nxv. 16 \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u1f74\u03bd \u03b4\u03bf\u03b8\u03b5\u1fd6\u03c3\u03ac\u03bd \u03bc\u03bf\u03b9 \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03b5\u1f30\u03c2 \u03c4\u1f78 \u03b5\u1f36\u03bd\u03b1\u03af \u03bc\u03b5\n\u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 _\u03b5\u1f30\u03c2 \u03c4\u1f70 \u1f14\u03b8\u03bd\u03b7_.\n\n\u03c0\u03bb\u03b7\u03c1\u1ff6\u03c3\u03b1\u03b9] '_to fulfil_,' i.e. 'to preach fully,' 'to give its complete\ndevelopment to'; as Rom. xv. 19 \u1f65\u03c3\u03c4\u03b5 \u03bc\u03b5 \u1f00\u03c0\u1f78 \u1f38\u03b5\u03c1\u03bf\u03c5\u03c3\u03b1\u03bb\u1f74\u03bc \u03ba\u03b1\u1f76 \u03ba\u03cd\u03ba\u03bb\u1ff3 \u03bc\u03ad\u03c7\u03c1\u03b9\n\u03c4\u03bf\u1fe6 \u1f38\u03bb\u03bb\u03c5\u03c1\u03b9\u03ba\u03bf\u1fe6 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03ba\u03ad\u03bd\u03b1\u03b9 \u03c4\u1f78 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. Thus 'the word of\nGod' here is 'the Gospel,' as in most places (1 Cor. xiv. 36, 2 Cor. ii.\n17, iv. 2, etc.), though not always (e.g. Rom. ix. 6), in St Paul, as\nalso in the Acts. The other interpretation, 'to accomplish the promise\nof God,' though suggested by such passages as 1 Kings ii. 27 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03bd\u03b1\u03b9\n\u03c4\u1f78 \u1fe5\u1fc6\u03bc\u03b1 \u039a\u03c5\u03c1\u03af\u03bf\u03c5, 2 Chron. xxxvi. 21 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03bd\u03b1\u03b9 \u03bb\u03cc\u03b3\u03bf\u03bd \u039a\u03c5\u03c1\u03af\u03bf\u03c5, etc., is\nalien to the context here.\n\n26. \u03c4\u1f78 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd] This is not the only term borrowed from the ancient\nmysteries, which St Paul employs to describe the teaching of the Gospel.\nThe word \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd just below, ver. 28, seems to be an extension of the\nsame metaphor. In Phil. iv. 12 again we have the verb \u03bc\u03b5\u03bc\u03cd\u03b7\u03bc\u03b1\u03b9: and in\nEphes. i. 14 \u03c3\u03c6\u03c1\u03b1\u03b3\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9 is perhaps an image derived from the same\nsource. So too the Ephesians are addressed as \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5 \u03c3\u03c5\u03bc\u03bc\u03cd\u03c3\u03c4\u03b1\u03b9 in Ign.\n_Ephes._ 12. The Christian teacher is thus regarded as a \u1f31\u03b5\u03c1\u03bf\u03c6\u03ac\u03bd\u03c4\u03b7\u03c2 (see\nEpict. iii. 21. 13 sq.) who initiates his disciples into the rites.\nThere is this difference however; that, whereas the heathen mysteries\nwere strictly confined to a narrow circle, the Christian mysteries are\nfreely communicated to all. There is therefore an intentional paradox in\nthe employment of the image by St Paul. See the notes on \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd\n\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd below.\n\nThus the idea of _secresy_ or _reserve_ disappears when \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd is\nadopted into the Christian vocabulary by St Paul: and the word signifies\nsimply 'a truth which was once hidden but now is revealed,' 'a truth\nwhich without special revelation would have been unknown.' Of the nature\nof the truth itself the word says nothing. It may be transcendental,\nincomprehensible, mystical, mysterious, in the modern sense of the term\n(1 Cor. xv. 51, Eph. v. 32): but this idea is quite accidental, and must\nbe gathered from the special circumstances of the case, for it cannot be\ninferred from the word itself. Hence \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd is almost universally\nfound in connexion with words denoting revelation or publication; e.g.\n\u1f00\u03c0\u03bf\u03ba\u03b1\u03bb\u03cd\u03c0\u03c4\u03b5\u03b9\u03bd, \u1f00\u03c0\u03bf\u03ba\u03ac\u03bb\u03c5\u03c8\u03b9\u03c2, Rom. xvi. 25, Ephes. iii. 3, 5, 2 Thess. ii.\n7; \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03b5\u03b9\u03bd Rom. xvi. 26, Ephes. i. 9, iii. 3, 10, vi. 19; \u03c6\u03b1\u03bd\u03b5\u03c1\u03bf\u1fe6\u03bd\nCol. iv. 3, Rom. xvi. 26, 1 Tim. iii. 16; \u03bb\u03b1\u03bb\u03b5\u1fd6\u03bd iv. 3, 1 Cor. ii. 7,\nxiv. 2; \u03bb\u03ad\u03b3\u03b5\u03b9\u03bd, 1 Cor. xv. 51.\n\nBut the one special 'mystery' which absorbs St Paul's thoughts in the\nEpistles to the Colossians and Ephesians is the free admission of the\nGentiles on equal terms to the privileges of the covenant. For this he\nis a prisoner; this he is bound to proclaim fearlessly (iv. 3, Ephes.\nvi. 19); this, though hidden from all time, was communicated to him by a\nspecial revelation (Ephes. iii. 3 sq.); in this had God most signally\ndisplayed the lavish wealth of His goodness (ver. 27, ii. 2 sq., Ephes.\ni. 6 sq., iii. 8 sq.). In one passage only throughout these two epistles\nis \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd applied to anything else, Ephes. v. 32. The same idea of\nthe \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd appears very prominently also in the thanksgiving (added\napparently later than the rest of the letter) at the end of the Epistle\nto the Romans, xvi. 25 sq. \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5 ... \u03b5\u1f30\u03c2 \u1f51\u03c0\u03b1\u03ba\u03bf\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03b5\u1f30\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1\n\u03c4\u1f70 \u1f14\u03b8\u03bd\u03b7 \u03b3\u03bd\u03c9\u03c1\u03b9\u03c3\u03b8\u03ad\u03bd\u03c4\u03bf\u03c2.\n\n------------------------------------------------------------------------\n\nI. 27]\n\n\u1f00\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03b1\u1f34\u03c9\u03bd\u03c9\u03bd \u03ba\u03b1\u1f76 \u1f00\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03b3\u03b5\u03bd\u03b5\u1ff6\u03bd, \u03bd\u1fe6\u03bd \u03b4\u1f72 \u1f10\u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03b8\u03b7 \u03c4\u03bf\u1fd6\u03c2 \u1f01\u03b3\u03af\u03bf\u03b9\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6,\n^{27}\u03bf\u1f37\u03c2 \u1f20\u03b8\u03ad\u03bb\u03b7\u03c3\u03b5\u03bd \u1f41 \u0398\u03b5\u1f78\u03c2 \u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03b1\u03b9 \u03c4\u03af \u03c4\u1f78 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 \u03c4\u03bf\u1fe6 \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5\n\u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2\n\n\n\u1f00\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03b1\u1f34\u03c9\u03bd\u03c9\u03bd \u03ba.\u03c4.\u03bb.] The preposition is doubtless temporal here, being\nopposed to \u03bd\u1fe6\u03bd, as in the parallel passage, Ephes. iii. 9: comp. Rom.\nxvi. 25 \u03ba\u03b1\u03c4\u1f70 \u1f00\u03c0\u03bf\u03ba\u03ac\u03bb\u03c5\u03c8\u03b9\u03bd \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03c9\u03c5 _\u03c7\u03c1\u03cc\u03bd\u03bf\u03b9\u03c2 \u03b1\u1f30\u03c9\u03bd\u03af\u03bf\u03b9\u03c2_ \u03c3\u03b5\u03c3\u03b9\u03b3\u03b7\u03bc\u03ad\u03bd\u03bf\u03c5, 1\nCor. ii. 7 \u0398\u03b5\u03bf\u1fe6 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u1f10\u03bd \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u1ff3 \u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03ba\u03b5\u03ba\u03c1\u03c5\u03bc\u03bc\u03ad\u03bd\u03b7\u03bd \u1f23\u03bd \u03c0\u03c1\u03bf\u03ce\u03c1\u03b9\u03c3\u03b5\u03bd \u1f41\n\u0398\u03b5\u1f78\u03c2 _\u03c0\u03c1\u1f78 \u03c4\u1ff6\u03bd \u03b1\u1f30\u03ce\u03bd\u03c9\u03bd_. So too \u1f00\u03c0' \u03b1\u1f30\u1ff6\u03bd\u03bf\u03c2, Acts iii. 21, xv. 18, Ps.\nxcii. 3, etc.; \u1f00\u03c0\u1f78 \u03ba\u03b1\u03c4\u03b1\u03b2\u03bf\u03bb\u1fc6\u03c2 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5, Matt. xiii. 35, xxv. 34, etc.\n\n\u03c4\u1ff6\u03bd \u03b3\u03b5\u03bd\u03b5\u1ff6\u03bd] An \u03b1\u1f32\u03c9\u03bd is made up of many \u03b3\u03b5\u03bd\u03b5\u03b1\u03af; comp. Ephes. iii. 21 \u03b5\u1f30\u03c2\n\u03c0\u03ac\u03c3\u03b1\u03c2 \u03c4\u1f70\u03c2 \u03b3\u03b5\u03bd\u03b5\u1f70\u03c2 \u03c4\u03bf\u1fe6 \u03b1\u1f30\u1ff6\u03bd\u03bf\u03c2 \u03c4\u1ff6\u03bd \u03b1\u1f30\u03ce\u03bd\u03c9\u03bd, Is. li. 9 \u1f61\u03c2 \u03b3\u03b5\u03bd\u03b5\u1f70 \u03b1\u1f30\u1ff6\u03bd\u03bf\u03c2 (where\nthe Hebrew has the plural 'generations'). Hence the order here. Not only\nwas this mystery unknown in remote periods of antiquity, but even in\nrecent generations. It came upon the world as a sudden surprise. The\nmoment of its revelation was the moment of its fulfilment.\n\n27. \u1f20\u03b8\u03ad\u03bb\u03b7\u03c3\u03b5\u03bd] '_willed_,' '_was pleased_.' It was God's grace: it was no\nmerit of their own. See the note on i. 1 \u03b4\u03b9\u1f70 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u0398\u03b5\u03bf\u1fe6.\n\n\u03bd\u1fe6\u03bd \u03b4\u1f72 \u03ba.\u03c4.\u03bb.] An indicative clause is substituted for a participial,\nwhich would otherwise have been more natural, for the sake of\nemphasizing the statement; comp. ver. 22 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5, and see\nWiner \u00a7 lxiii. p. 717.\n\n\u03c4\u1f78 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2] The '_wealth_ of God,' as manifested in His dispensation of\ngrace, is a prominent idea in these epistles; comp. ii. 2, Ephes. i. 7,\n18, iii. 8, 16; comp. Rom. xi. 33. See above p. 43 sq. St Paul uses the\nneuter and the masculine forms indifferently in these epistles (e.g. \u03c4\u1f78\n\u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 Ephes. i. 7, \u1f41 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 Ephes. i. 18), as in his other letters\n(e.g. \u03c4\u1f78 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 2 Cor. viii. 2, \u1f41 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 Rom. ix. 23). In most\npassages however there are various readings. On the neuter forms \u03c4\u1f78\n\u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2, \u03c4\u1f78 \u03b6\u1fc6\u03bb\u03bf\u03c2, etc., see Winer \u00a7 ix. p. 76.\n\n\u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2] i.e. 'of the glorious manifestation.' This word in\nHellenistic Greek is frequently used of a bright light; e.g. Luke ii. 9\n\u03c0\u03b5\u03c1\u03af\u03b5\u03bb\u03b1\u03bc\u03c8\u03b5\u03bd, Acts xxii. 11 \u03c4\u03bf\u1fe6 \u03c6\u03c9\u03c4\u03cc\u03c2, 1 Cor. xv. 41 \u1f21\u03bb\u03af\u03bf\u03c5, \u03c3\u03b5\u03bb\u03ae\u03bd\u03b7\u03c2, etc.\n2 Cor. iii. 7 \u03c4\u03bf\u1fe6 \u03c0\u03c1\u03bf\u03c3\u03ce\u03c0\u03bf\u03c5 [\u039c\u03c9\u03c5\u03c3\u03ad\u03c9\u03c2]. Hence it is applied generally to a\ndivine _manifestation_, even where there is no physical accompaniment of\nlight; and more especially to the revelation of God in Christ (e.g. Joh.\ni. 14, 2 Cor. iv. 4, etc.). The expression \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2 occurs\nagain, Rom. ix. 23, Ephes. i. 18, iii. 16. See above ver. 11 with the\nnote.\n\n------------------------------------------------------------------------\n\nI. 28]\n\n\u1f14\u03b8\u03bd\u03b5\u03c3\u03b9\u03bd, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, \u1f21 \u1f10\u03bb\u03c0\u1f76\u03c2 \u03c4\u1fc6\u03c2 \u03b4\u03cc\u03be\u03b7\u03c2\u00b7 ^{28} \u1f43\u03bd \u1f21\u03bc\u03b5\u1fd6\u03c2\n\u03ba\u03b1\u03c4\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03bf\u03bc\u03b5\u03bd \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd\n\n                             ^{27} \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd.\n\n\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03b8\u03bd\u03b5\u03c3\u03b9\u03bd] i.e. 'as exhibited among the Gentiles.' It was just\nhere that this 'mystery,' this dispensation of grace, achieved its\ngreatest triumphs and displayed its transcendant glory; \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1\n\u03ba\u03b1\u1f76 \u1f10\u03bd \u1f11\u03c4\u03b5\u03c1\u03bf\u03b9\u03c2, writes Chrysostom, \u03c0\u03bf\u03bb\u03bb\u1ff7 \u03b4\u1f72 \u03c0\u03bb\u03ad\u03bf\u03bd \u1f10\u03bd \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u1f21 \u03c0\u03bf\u03bb\u03bb\u1f74 \u03c4\u03bf\u1fe6\n\u03bc\u03c5\u03c3\u03c4\u03b5\u03c1\u03af\u03bf\u03c5 \u03b4\u03cc\u03be\u03b1. Here too was its _wealth_; for it overflowed all\nbarriers of caste or race. Judaism was 'beggarly' (Gal. iv. 9) in\ncomparison, since its treasures sufficed only for a few.\n\n\u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd] The antecedent is probably \u03c4\u03bf\u1fe6 \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5; comp. ii. 2 \u03c4\u03bf\u1fe6\n\u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f10\u03bd \u1fa7 \u03b5\u1f30\u03c3\u03b9\u03bd \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03ba.\u03c4.\u03bb.\n\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd] '_Christ in you_,' i.e. 'you Gentiles.' Not Christ, but\nChrist given freely to the Gentiles, is the 'mystery' of which St Paul\nspeaks; see the note on \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd above. Thus the various reading, \u1f43\u03c2\nfor \u1f45, though highly supported, interferes with the sense. With \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2\n\u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd compare \u03bc\u03b5\u03b8' \u1f21\u03bc\u1ff6\u03bd \u0398\u03b5\u03cc\u03c2 Matt. i. 23. It may be a question\nhowever, whether \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd means '_within you_' or '_among you_.' The\nformer is perhaps the more probable interpretation, as suggested by Rom.\nviii. 10, 2 Cor. xiii. 5, Gal. iv. 19; comp. Ephes. iii. 17 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9\n\u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 \u1f51\u03bc\u1ff6\u03bd.\n\n\u1f21 \u1f10\u03bb\u03c0\u03af\u03c2] comp. 1 Tim. i. 2; so \u1f21 [\u03ba\u03bf\u03b9\u03bd\u1f74] \u1f10\u03bb\u03c0\u1f76\u03c2 \u1f21\u03bc\u1ff6\u03bd Ign. _Eph._ 21,\n_Magn. Philad._ 5, etc., applied to our Lord.\n\n28, 29. 'This Christ we, the Apostles and Evangelists, proclaim without\ndistinction and without reserve. We know no restriction either of\npersons or of topics. We admonish every man and instruct every man. We\ninitiate every man in all the mysteries of wisdom. It is our single aim\nto present every man fully and perfectly taught in Christ. For this end\nI train myself in the discipline of self-denial; for this end I commit\nmyself to the arena of suffering and toil, putting forth in the conflict\nall that energy which He inspires, and which works in me so powerfully.'\n\n28. \u1f21\u03bc\u03b5\u1fd6\u03c2] '_we_,' the preachers; the same opposition as in 1 Cor. iv.\n8, 10, ix. 11, 2 Cor. xiii. 5 sq., 1 Thess. ii. 13 sq., etc. The Apostle\nhastens, as usual, to speak of the part which he was privileged to bear\nin this glorious dispensation. He is constrained to magnify his office.\nSee the next note, and comp. ver. 23.\n\n------------------------------------------------------------------------\n\nI. 28]\n\n\u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3, \u1f35\u03bd\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ae\u03c3\u03c9\u03bc\u03b5\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1\n\u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7\u00b7\n\n\n\u1f43\u03bd \u1f21\u03bc\u03b5\u1fd6\u03c2 \u03ba.\u03c4.\u03bb.] as in St Paul's own language at Thessalonica, Acts\nxvii. 3 \u1f43\u03bd \u1f10\u03b3\u1f7c \u03ba\u03b1\u03c4\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03c9 \u1f51\u03bc\u1fd6\u03bd, and at Athens, Acts xvii. 23 \u03c4\u03bf\u1fe6\u03c4\u03bf \u1f10\u03b3\u1f7c\n\u03ba\u03b1\u03c4\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03c9 \u1f51\u03bc\u1fd6\u03bd, in both which passages, as here, emphasis is laid on\nthe person of the preacher.\n\n\u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2] '_admonishing_.' The two words \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd and \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b5\u03b9\u03bd\npresent complementary aspects of the preacher's duty, and are related\nthe one to the other, as \u03bc\u03b5\u03c4\u03ac\u03bd\u03bf\u03b9\u03b1 to \u03c0\u03af\u03c3\u03c4\u03b9\u03c2, '_warning_ to repent,\n_instructing_ in the faith.' For the relation of \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd to \u03bc\u03b5\u03c4\u03ac\u03bd\u03bf\u03b9\u03b1\nSee Plut. _Mor._ p. 68 \u1f15\u03bd\u03b5\u03c3\u03c4\u03b9 \u03c4\u1f78 \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b5\u03c4\u03ac\u03bd\u03bf\u03b9\u03b1\u03bd \u1f10\u03bc\u03c0\u03bf\u03b9\u03bf\u1fe6\u03bd, p.\n452 \u1f21 \u03bd\u03bf\u03c5\u03b8\u03b5\u03c3\u03af\u03b1 \u03ba\u03b1\u1f76 \u1f41 \u03c8\u03cc\u03b3\u03bf\u03c2 \u1f10\u03bc\u03c0\u03bf\u03b9\u03b5\u1fd6 \u03bc\u03b5\u03c4\u03ac\u03bd\u03bf\u03b9\u03b1\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f30\u03c3\u03c7\u03cd\u03bd\u03b7\u03bd. The two verbs\n\u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd and \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b5\u03b9\u03bd are connected in Plato _Protag._ 323 D, _Legg._\n845 B, Plut. _Mor._ p. 46 (comp. p. 39), Dion Chrys. _Or._ xxxiii. p.\n369; the substantives \u03b4\u03b9\u03b4\u03b1\u03c7\u1f74 and \u03bd\u03bf\u03c5\u03b8\u03ad\u03c4\u03b7\u03c3\u03b9\u03c2 in Plato _Resp._ 399 B.\nSimilarly \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd and \u03c0\u03b5\u03af\u03b8\u03b5\u03b9\u03bd occur together in Arist. _Rhet._ ii. 18.\nFor the two functions of the preacher's office, corresponding\nrespectively to the two words, see St Paul's own language in Acts xx. 21\n\u03b4\u03b9\u03b1\u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 ... \u03c4\u1f74\u03bd \u03b5\u1f30\u03c2 \u0398\u03b5\u1f78\u03bd \u03bc\u03b5\u03c4\u03ac\u03bd\u03bf\u03b9\u03b1\u03bd \u03ba\u03b1\u1f76 \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd\n\u1f21\u03bc\u1ff6\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd.\n\n\u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd] three times repeated for the sake of emphasizing the\n_universality_ of the Gospel. This great truth, for which St Paul gave\nhis life, was now again endangered by the doctrine of an intellectual\nexclusiveness taught by the Gnosticizers at Coloss\u00e6, as before it had\nbeen endangered by the doctrine of a ceremonial exclusiveness taught by\nthe Judaizers in Galatia. See above pp. 77, 92, 98 sq. For the\nrepetition of \u03c0\u03ac\u03bd\u03c4\u03b1 compare especially 1 Cor. x. 1 sq., where \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 is\nfive times, and _ib._ xii. 29, 30, where it is seven times repeated; see\nalso Rom. ix. 6, 7, xi. 32, 1 Cor. xii. 13, xiii. 7, xiv. 31, etc.\nTranscribers have been offended at this characteristic repetition here,\nand consequently have omitted \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd in one place or other.\n\n\u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3] The Gnostic spoke of a blind faith for the many, of a\nhigher \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 for the few. St Paul declares that the fullest wisdom is\noffered to all alike. The character of the teaching is as free from\nrestriction, as are the qualifications of the recipients. Comp. ii. 2, 3\n\u03c0\u1fb6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03bd\u03ad\u03c3\u03b5\u03c9\u03c2 ... \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03bf\u1f31 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u1f76 \u03c4\u1fc6\u03c2\n\u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b3\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2.\n\n\u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ae\u03c3\u03c9\u03bc\u03b5\u03bd] See the note on \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9, ver. 22.\n\n\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd] So 1 Cor. ii. 6, 7 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u03b4\u1f72 \u03bb\u03b1\u03bb\u03bf\u1fe6\u03bc\u03b5\u03bd \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03c4\u03b5\u03bb\u03b5\u03af\u03bf\u03b9\u03c2 ... \u0398\u03b5\u03bf\u1fe6\n\u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u1f10\u03bd \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u1ff3 \u03c4\u1f74\u03bd \u1f00\u03c0\u03bf\u03ba\u03b5\u03ba\u03c1\u03c5\u03bc\u03bc\u03ad\u03bd\u03b7\u03bd. In both these passages the\nepithet \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2 is probably a metaphor borrowed from the ancient\nmysteries, where it seems to have been applied to the fully instructed,\nas opposed to the novices: comp. Plato _Ph\u00e6dr._ 249 C \u03c4\u03b5\u03bb\u03ad\u03bf\u03c5\u03c2 \u1f00\u03b5\u1f76\n\u03c4\u03b5\u03bb\u03b5\u03c4\u1f70\u03c2 \u03c4\u03b5\u03bb\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c4\u03ad\u03bb\u03b5\u03bf\u03c2 \u1f44\u03bd\u03c4\u03c9\u03c2 \u03bc\u03cc\u03bd\u03bf\u03c2 \u03b3\u03af\u03b3\u03bd\u03b5\u03c4\u03b1\u03b9... 250 B, C \u03b5\u1f36\u03b4\u03cc\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76\n\u1f10\u03c4\u03b5\u03bb\u03bf\u1fe6\u03bd\u03c4\u03bf \u03c4\u03b5\u03bb\u03b5\u03c4\u1ff6\u03bd \u1f23\u03bd \u03b8\u03ad\u03bc\u03b9\u03c2 \u03bb\u03ad\u03b3\u03b5\u03b9\u03bd \u03bc\u03b1\u03ba\u03b1\u03c1\u03b9\u03c9\u03c4\u03ac\u03c4\u03b7\u03bd ... \u03bc\u03c5\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03af \u03c4\u03b5 \u03ba\u03b1\u1f76\n\u1f10\u03c0\u03bf\u03c0\u03c4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03b1\u1f52\u03b3\u1fc7 \u03ba\u03b1\u03b8\u03b1\u03c1\u1fb7, _Symp._ 209 E \u03c4\u03b1\u1fe6\u03c4\u03b1 ... \u03ba\u1f02\u03bd \u03c3\u1f7a \u03bc\u03c5\u03b7\u03b8\u03b5\u03af\u03b7\u03c2\u00b7 \u03c4\u1f70\n\u03b4\u1f72 \u03c4\u03ad\u03bb\u03b5\u03b1 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03bf\u03c0\u03c4\u03b9\u03ba\u03ac ... \u03bf\u1f50\u03ba \u03bf\u1f37 \u03b4' \u03b5\u1f30 \u03bf\u1f37\u03cc\u03c2 \u03c4' \u1f02\u03bd \u03b5\u1f34\u03b7\u03c2, Plut. _Fragm. de\nAn._ vi. 2 (v. p. 726 Wyttenb.) \u1f41 \u03c0\u03b1\u03bd\u03c4\u03b5\u03bb\u1f74\u03c2 \u1f24\u03b4\u03b7 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03bc\u03c5\u03b7\u03bc\u03ad\u03bd\u03bf\u03c2 (with the\ncontext), Dion Chrys. _Or._ xii. p. 203 \u03c4\u1f74\u03bd \u1f41\u03bb\u03cc\u03ba\u03bb\u03b7\u03c1\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c4\u1ff7 \u1f44\u03bd\u03c4\u03b9\n\u03c4\u03b5\u03bb\u03b5\u03af\u03b1\u03bd \u03c4\u03b5\u03bb\u03b5\u03c4\u1f74\u03bd \u03bc\u03c5\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd; see Valcknaer on Eurip. _Hippol._ 25, and\nLobeck _Aglaoph._ p. 33 sq., p. 126 sq. Somewhat similarly in the LXX 1\nChron. xxv. 8 \u03c4\u03b5\u03bb\u03ad\u03b9\u03c9\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b1\u03bd\u03b8\u03b1\u03bd\u03cc\u03bd\u03c4\u03c9\u03bd stands for 'the teachers (or the\nwise) and the scholars.' So also in 2 Pet. i. 16 _\u1f10\u03c0\u03cc\u03c0\u03c4\u03b1\u03b9_\n\u03b3\u03b5\u03bd\u03b7\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2 \u03c4\u1fc6\u03c2 \u1f10\u03ba\u03b5\u03af\u03bd\u03bf\u03c5 \u03bc\u03b5\u03b3\u03b1\u03bb\u03b5\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 we seem to have the same metaphor.\nAs an illustration it may be mentioned that Plato and Aristotle called\nthe higher philosophy \u1f10\u03c0\u03bf\u03c0\u03c4\u03b9\u03ba\u03cc\u03bd, because those who have transcended the\nbounds of the material, \u03bf\u1f37\u03bf\u03bd \u1f10\u03bd\u03c4\u03b5\u03bb\u1fc6 [l. \u1f10\u03bd \u03c4\u03b5\u03bb\u03b5\u03c4\u1fc7] \u03c4\u03ad\u03bb\u03bf\u03c2 \u1f14\u03c7\u03b5\u03b9\u03bd\n\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03bd [\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2] \u03bd\u03bf\u03bc\u03af\u03b6\u03bf\u03c5\u03c3\u03b9, Plut. _Mor._ 382 D, E. For other\nmetaphorical expressions in St Paul, derived from the mysteries, see\nabove on \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd ver. 26. Influenced probably by this heathen use of\n\u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, the early Christians applied it to the baptized, as opposed to\nthe catechumens: e.g. Justin _Dial._ 8 (p. 225 C) \u03c0\u03ac\u03c1\u03b5\u03c3\u03c4\u03b9\u03bd \u1f10\u03c0\u03b9\u03b3\u03bd\u03cc\u03bd\u03c4\u03b9 \u03c3\u03bf\u03b9\n\u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c4\u03b5\u03bb\u03b5\u03af\u1ff3 \u03b3\u03b5\u03bd\u03bf\u03bc\u03ad\u03bd\u1ff3 \u03b5\u1f50\u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u03b5\u1fd6\u03bd, _Clem. Hom._ iii.\n29 \u1f51\u03c0\u03bf\u03c7\u03c9\u03c1\u03b5\u1fd6\u03bd \u03bc\u03bf\u03b9 \u03ba\u03b5\u03bb\u03b5\u03cd\u03c3\u03b1\u03c2, \u1f61\u03c2 \u03bc\u03ae\u03c0\u03c9 \u03b5\u1f30\u03bb\u03b7\u03c6\u03cc\u03c4\u03b9 \u03c4\u1f78 \u03c0\u03c1\u1f78\u03c2 \u03c3\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03bd \u03b2\u03ac\u03c0\u03c4\u03b9\u03c3\u03bc\u03b1,\n\u03c4\u03bf\u1fd6\u03c2 \u1f24\u03b4\u03b7 \u03c4\u03b5\u03bb\u03b5\u03af\u03bf\u03b9\u03c2 \u1f14\u03c6\u03b7 \u03ba.\u03c4.\u03bb., xi. 36 \u03b2\u03b1\u03c0\u03c4\u03af\u03c3\u03b1\u03c2 ... \u1f24\u03b4\u03b7 \u03bb\u03bf\u03b9\u03c0\u1f78\u03bd \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd\n\u1f44\u03bd\u03c4\u03b1 \u03ba.\u03c4.\u03bb.; and for later writers see Suicer _Thes._ s. vv. \u03c4\u03b5\u03bb\u03b5\u03b9\u03cc\u03c9,\n\u03c4\u03b5\u03bb\u03b5\u03af\u03c9\u03c3\u03b9\u03c2. At all events we may ascribe to its connexion with the\nmysteries the fact that it was adopted by Gnostics at a later date, and\nmost probably by the Gnosticizers at this time, to distinguish the\npossessors of the higher \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 from the vulgar herd of believers: see\nthe passages quoted in the note on Phil. iii. 15. While employing the\nfavourite Gnostic term, the Apostle strikes at the root of the Gnostic\ndoctrine. The language descriptive of the heathen mysteries is\ntransferred by him to the Christian dispensation, that he may thus more\neffectively contrast the things signified. The true Gospel also has its\nmysteries, its hierophants, its initiation: but these are open to all\nalike. In Christ every believer is \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, for he has been admitted as\n\u1f10\u03c0\u03cc\u03c0\u03c4\u03b7\u03c2 of its most profound, most awful, secrets. See again the note on\n\u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9, ii. 3.\n\n------------------------------------------------------------------------\n\nI. 29]\n\n^{29}\u03b5\u1f30\u03c2 \u1f43 \u03ba\u03b1\u1f76 \u03ba\u03bf\u03c0\u03b9\u1ff6 \u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u1f10\u03bd\u03ad\u03c1\u03b3\u03b5\u03b9\u03b1\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c4\u1f74\u03bd\n\u1f10\u03bd\u03b5\u03c1\u03b3\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03bd \u1f10\u03bd \u1f11\u03bc\u03bf\u1f76 \u1f10\u03bd \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9.\n\n\n29. \u03b5\u1f30\u03c2 \u1f43] i.e. \u03b5\u1f30\u03c2 \u03c4\u1f78 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u1fc6\u03c3\u03b1\u03b9 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd, 'that I may\ninitiate all mankind in the fulness of this mystery,' 'that I may preach\nthe Gospel to all without reserve.' If St Paul had been content to\npreach an exclusive Gospel, he might have saved himself from more than\nhalf the troubles of his life.\n\n\u03ba\u03bf\u03c0\u03b9\u1ff6] This word is used especially of the labour undergone by the\nathlete in his training, and therefore fitly introduces the metaphor of\n\u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2: comp. 1 Tim. iv. 10 \u03b5\u1f30\u03c2 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03b3\u1f70\u03c1 \u03ba\u03bf\u03c0\u03b9\u1ff6\u03bc\u03b5\u03bd \u03ba\u03b1\u1f76 \u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03b8\u03b1\n(the correct reading), and see the passages quoted on Phil. ii. 16.\n\n\u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2] '_contending in the lists_,' the metaphor being continued\nin the next verse (ii. 1), \u1f21\u03bb\u03af\u03ba\u03bf\u03bd \u1f00\u03b3\u1ff6\u03bd\u03b1; comp. iv. 12. These words \u1f00\u03b3\u03ce\u03bd,\n\u1f00\u03b3\u03c9\u03bd\u03af\u03b1, \u1f00\u03b3\u03c9\u03bd\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, are only found in St Paul and the Pauline writings\n(Luke, Hebrews) in the New Testament. They occur in every group of St\nPaul's Epistles. The use here most resembles 1 Thess. ii. 2 \u03bb\u03b1\u03bb\u1fc6\u03c3\u03b1\u03b9 \u03c0\u03c1\u1f78\u03c2\n\u1f51\u03bc\u1fb6\u03c2 \u03c4\u1f78 \u03b5\u1f50\u03b1\u03b3\u03b3\u03ad\u03bb\u03b9\u03bf\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u1f10\u03bd \u03c0\u03bf\u03bb\u03bb\u1ff7 \u1f00\u03b3\u1ff6\u03bd\u03b9.\n\n\u1f10\u03bd\u03b5\u03c1\u03b3\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03bd] Comp. Eph. iii. 20. For the difference between \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u1fd6\u03bd\nand \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 see the note on Gal. v. 6.\n\n------------------------------------------------------------------------\n\nII. 1, 2]\n\nII. ^1 \u0398\u03ad\u03bb\u03c9 \u03b3\u1f70\u03c1 \u1f51\u03bc\u1fb6\u03c2 \u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9, \u1f21\u03bb\u03af\u03ba\u03bf\u03bd \u1f00\u03b3\u1ff6\u03bd\u03b1 \u1f14\u03c7\u03c9 \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f10\u03bd\n\u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u1fb3 \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 \u03bf\u1f50\u03c7 \u1f11\u03ce\u03c1\u03b1\u03ba\u03b1\u03bd \u03c4\u1f78 \u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03cc\u03bd \u03bc\u03bf\u03c5 \u1f10\u03bd \u03c3\u03b1\u03c1\u03ba\u03af, ^2 \u1f35\u03bd\u03b1\n\u03c0\u03b1\u03c1\u03b1\u03ba\u03bb\u03b7\u03b8\u1ff6\u03c3\u03b9\u03bd \u03b1\u1f31 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\n\n\nII. 1\u20133. 'I spoke of an _arena_ and a _conflict_ in describing my\napostolic labours. The image was not lightly chosen. I would have you\nknow that my care is not confined to my own direct and personal\ndisciples. I wish you to understand the magnitude of the struggle, which\nmy anxiety for you costs me\u2014for you and for your neighbours of Laodicea\nand for all who, like yourselves, have never met me face to face in the\nflesh. I am constantly wrestling in spirit, that the hearts of all such\nmay be confirmed and strengthened in the faith; that they may be united\nin love; that they may attain to all the unspeakable wealth which comes\nfrom the firm conviction of an understanding mind, may be brought to the\nperfect knowledge of God's mystery, which is nothing else than\nChrist\u2014Christ containing in Himself all the treasures of wisdom and\nknowledge hidden away.'\n\n1. \u0398\u03ad\u03bb\u03c9 \u03ba.\u03c4.\u03bb.] as in 1 Cor. xi. 3. The corresponding negative form, \u03bf\u1f50\n\u03b8\u03ad\u03bb\u03c9 [\u03b8\u03ad\u03bb\u03bf\u03bc\u03b5\u03bd] \u1f51\u03bc\u1fb6\u03c2 \u1f00\u03b3\u03bd\u03bf\u03b5\u1fd6\u03bd, is the more common expression in St Paul;\nRom. i. 13, xi. 25, 1 Cor. x. 1, xii. 1, 2 Cor. i. 8, 1 Thess. iv. 13.\n\n\u1f00\u03b3\u1ff6\u03bd\u03b1] The arena of the contest to which \u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 in the preceding\nverse refers may be either outward or inward. It will include the\n'fightings without,' as well as the 'fears within.' Here however the\ninward struggle, the wrestling in prayer, is the predominant idea, as in\niv. 12 \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u03c9\u03bd \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03b1\u1fd6\u03c2 \u1f35\u03bd\u03b1 \u03c3\u03c4\u03b1\u03b8\u1fc6\u03c4\u03b5\n\u03ba.\u03c4.\u03bb.\n\n\u03c4\u1ff6\u03bd \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u1fb3] The Laodiceans were exposed to the same doctrinal\nperils as the Colossians: see above pp. 2, 41 sq. The Hierapolitans are\ndoubtless included in \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 \u03ba.\u03c4.\u03bb. (comp. iv. 13), but are not\nmentioned here by name, probably because they were less closely\nconnected with Coloss\u00e6 (see iv. 15 sq.), and perhaps also because the\ndanger was less threatening there.\n\n\u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 \u03ba.\u03c4.\u03bb.] '_and all who_, like yourselves, _have not seen_,\netc.'; where the \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 introduces the whole class to which the\npersons previously enumerated belong; so Acts iv. 6 \u1f0c\u03bd\u03bd\u03b1\u03c2 \u1f41 \u1f00\u03c1\u03c7\u03b9\u03b5\u03c1\u1f72\u03c5\u03c2\n\u03ba\u03b1\u1f76 \u039a\u03b1\u0390\u03b1\u03c6\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f3c\u03c9\u03b1\u03bd\u03bd\u03b7\u03c2 \u03ba\u03b1\u1f76 \u1f08\u03bb\u03ad\u03be\u03b1\u03bd\u03b4\u03c1\u03bf\u03c2 _\u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9_ \u1f26\u03c3\u03b1\u03bd \u1f10\u03ba \u03b3\u03ad\u03bd\u03bf\u03c5\u03c2\n\u1f00\u03c1\u03c7\u03b9\u03b5\u03c1\u03b1\u03c4\u03b9\u03ba\u03bf\u1fe6, Rev. xviii. 17 \u03ba\u03b1\u1f76 \u03c0\u1fb6\u03c2 \u03ba\u03c5\u03b2\u03b5\u03c1\u03bd\u03ae\u03c4\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03c0\u1fb6\u03c2 \u1f41 \u1f10\u03c0\u1f76 \u03c4\u03cc\u03c0\u03bf\u03bd\n\u03c0\u03bb\u03ad\u03c9\u03bd \u03ba\u03b1\u1f76 \u03bd\u03b1\u1fe6\u03c4\u03b1\u03b9 _\u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9_ \u03c4\u1f74\u03bd \u03b8\u03ac\u03bb\u03b1\u03c3\u03c3\u03b1\u03bd \u1f10\u03c1\u03b3\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9. Even a simple\n\u03ba\u03b1\u1f76 will sometimes introduce the general after the particular, e.g. Acts\nv. 29 \u1f41 \u03a0\u03ad\u03c4\u03c1\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03bf\u1f31 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03b9, Ar. _Nub._ 413 \u1f10\u03bd \u1f08\u03b8\u03b7\u03bd\u03b1\u03af\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fd6\u03c2\n\u1f1d\u03bb\u03bb\u03b7\u03c3\u03b9, etc.; see K\u00fchner _Gramm._ \u00a7 521, II. p. 791. On the other hand\n\u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9, occurring in an enumeration, sometimes introduces a different\nclass from those previously mentioned, as e.g. in Herod, vii. 185. As a\npure grammatical question therefore it is uncertain whether St Paul's\nlanguage here implies his personal acquaintance with his correspondents\nor the contrary. But in all such cases the sense of the context must be\nour guide. In the present instance \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 is quite out of place,\nunless the Colossians and Laodiceans also were personally unknown to the\nApostle. There would be no meaning in singling out _individuals_ who\nwere known to him, and then mentioning comprehensively _all_ who were\nunknown to him: see above p. 28, note 84. Hence we may infer from the\nexpression here, that St Paul had never visited Coloss\u00e6\u2013an inference\nwhich has been already shown (p. 23 sq.) to accord both with the\nincidental language of this epistle elsewhere and with the direct\nhistorical narrative of the Acts.\n\n\u1f11\u03ce\u03c1\u03b1\u03ba\u03b1\u03bd] For this ending of the 3rd pers. plur. perfect in -\u03b1\u03bd see Winer\n\u00a7 xiii. p. 90. The received text reads \u1f11\u03c9\u03c1\u03ac\u03ba\u03b1\u03c3\u03b9. In this passage the \u03c9\nform has the higher support; but below in ver. 18 the preponderance of\nauthority favours \u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd rather than \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd. On the use of the form in\n\u03bf see Buttmann _Ausf. Griech. Sprachl._ \u00a7 84, I. p. 325.\n\n2. \u03c0\u03b1\u03c1\u03b1\u03ba\u03bb\u03b7\u03b8\u1ff6\u03c3\u03b9\u03bd] '_encouraged_, _confirmed_,' i.e. 'comforted' in the\nolder and wider meaning of the word, ('confortati'), but not with its\nmodern and restricted sense: see \u03c0\u03b1\u03c1\u03ac\u03ba\u03bb\u03b7\u03c3\u03b9\u03c2 Phil. ii. 1. For \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03b5\u1fd6\u03bd\n\u03c4\u1f70\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2 comp. iv. 8, Ephes. vi. 22, 2 Thess. ii. 17.\n\n------------------------------------------------------------------------\n\nII. 3]\n\n\u03b1\u1f50\u03c4\u1ff6\u03bd, \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03c3\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u1f00\u03b3\u03ac\u03c0\u1fc3 \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03c0\u1fb6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2\n\u03c3\u03c5\u03bd\u03ad\u03c3\u03b5\u03c9\u03c2, \u03b5\u1f30\u03c2 \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 ^3\u1f10\u03bd \u1fa7 \u03b5\u1f30\u03c3\u1f76\u03bd\n\u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03bf\u1f31 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u1f76\n\n\n\u03b1\u1f31 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9] They met the Apostle heart to heart, though not face to\nface. We have here the same opposition of \u03ba\u03b1\u03c1\u03b4\u03af\u03b1 and \u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03bf\u03bd as in 1\nThess. ii. 17, though less directly expressed; see ver. 5.\n\n\u03b1\u1f50\u03c4\u1ff6\u03bd] where we should expect \u1f51\u03bc\u1ff6\u03bd, but the substitution of the third\nperson for the second is suggested by the immediately preceding \u03ba\u03b1\u1f76\n\u1f45\u03c3\u03bf\u03b9. This substitution confirms the interpretation of \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9 already\ngiven. Unless the Colossians are included in \u1f45\u03c3\u03bf\u03b9, they must be excluded\nby \u03b1\u1f50\u03c4\u1ff6\u03bd. Yet this exclusion is hardly conceivable in such a context.\n\n\u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03c3\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2] '_they being united_, _compacted_,' for \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03ac\u03b6\u03b5\u03b9\u03bd must\nhere have its common meaning, as it has elsewhere in this and the\ncompanion epistle: ver. 19 \u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd \u1f01\u03c6\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03bc\u03c9\u03bd ...\n\u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd, Ephes. iv. 16 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c3\u03c5\u03bd\u03b1\u03c1\u03bc\u03bf\u03bb\u03bf\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76\n\u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd. Otherwise we might be disposed to assign to this verb\nhere the sense which it always bears in the LXX (e.g. in Is. xl. 13, 14,\nquoted in 1 Cor. ii. 16), 'instructed, taught,' as it is rendered in the\nVulgate. Its usage in the Acts is connected with this latter sense; e.g.\nix. 22 \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03ac\u03b6\u03c9\u03bd 'proving,' xvi. 10 \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03ac\u03b6\u03bf\u03bd\u03c4\u03b5\u03c2 'concluding'; and so\nin xix. 33 \u03c3\u03c5\u03bd\u03b5\u03b2\u03af\u03b2\u03b1\u03c3\u03b1\u03bd \u1f08\u03bb\u03ad\u03be\u03b1\u03bd\u03b4\u03c1\u03bf\u03bd (the best supported reading) can only\nmean 'instructed Alexander.' For the different sense of the nominative\nabsolute see the note on iii. 16. The received text substitutes\n\u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03c3\u03b8\u03ad\u03bd\u03c4\u03c9\u03bd here. ` \u1f10\u03bd \u1f00\u03b3\u03ac\u03c0\u1fc3] for love is the \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2 (iii. 14) of\nperfection.\n\n\u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2] '_and brought unto_,' the thought being supplied from the\npreceding \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03c3\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2, which involves an idea of motion, comp. Joh.\nxx. 7 \u1f10\u03bd\u03c4\u03b5\u03c4\u03c5\u03bb\u03b9\u03b3\u03bc\u03ad\u03bd\u03bf\u03bd \u03b5\u1f30\u03c2 \u1f15\u03bd\u03b1 \u03c4\u03cc\u03c0\u03bf\u03bd.\n\n\u03c0\u1fb6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2] This reading is better supported than either \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2\nor \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03bd, while, as the intermediate reading, it also explains\nthe other two.\n\n\u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03c2] '_the full assurance_,' for such seems to be the\nmeaning of the substantive wherever it occurs in the New Testament; 1\nThess. i. 5 \u1f10\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u1fb3 \u03c0\u03bf\u03bb\u03bb\u1fc7, Heb. vi. 11 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f74\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03bd \u03c4\u1fc6\u03c2\n\u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2, x. 22 \u1f10\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u1fb3 \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2, comp. Clem. Rom. 42 \u03bc\u03b5\u03c4\u1f70\n\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1\u03c2 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f01\u03b3\u03af\u03bf\u03c5. With the exception of 1 Thess. i. 5\nhowever, all the Biblical passages might bear the other sense 'fulness':\nsee Bleek on Heb. vi. 11. For the verb see the note on \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9\nbelow, iv. 12.\n\n\u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd] See the note on i. 9.\n\n\u03c4\u03bf\u1fe6 \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5 \u03ba.\u03c4.\u03bb.] '_the mystery of God_, even _Christ in whom_,\netc.,' \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 being in apposition with \u03c4\u03bf\u1fe6 \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5; comp. i. 27 \u03c4\u03bf\u1fe6\n\u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u03bf\u03c5 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 ... \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, 1 Tim. iii. 16 \u03c4\u1f78 \u03c4\u1fc6\u03c2\n\u03b5\u1f50\u03c3\u03b5\u03b2\u03b5\u03af\u03b1\u03c2 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd, \u1f4d\u03c2 \u1f10\u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03b8\u03b7 \u03ba.\u03c4.\u03bb. The reasons for adopting the\nreading \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 are given in the detached note on various\nreadings. Other interpretations of this reading are; (1) 'the God\nChrist,' taking \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 in apposition with \u0398\u03b5\u03bf\u1fe6; or (2) 'the God of\nChrist,' making it the genitive after \u0398\u03b5\u03bf\u1fe6: but both expressions are\nwithout a parallel in St Paul. The mystery here is not 'Christ,' but\n'Christ as containing in Himself all the treasures of wisdom'; see the\nnote on i. 27 \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd. For the form of the sentence comp. Ephes.\niv. 15, 16 \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae, \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f10\u03be \u03bf\u1f57 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03ba.\u03c4.\u03bb.\n\n3. \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2] So \u03c0\u1fb6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 ver. 2, \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3 ii. 28. These repetitions\nserve to emphasize the character of the Gospel, which is as complete in\nitself, as it is universal in its application.\n\n------------------------------------------------------------------------\n\nII. 4]\n\n\u03c4\u1fc6\u03c2 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b3\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2 \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9. ^4\u03c4\u03bf\u1fe6\u03c4\u03bf\n\n\n\u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b3\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2] The two words occur together again Rom. xi. 33 \u1f66\n\u03b2\u03ac\u03b8\u03bf\u03c2 \u03c0\u03bb\u03bf\u03cd\u03c4\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03b3\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2 \u0398\u03b5\u03bf\u1fe6, 1 Cor. xii. 8. They are found\nin conjunction also several times in the LXX of Eccles. i. 7, 16, 18,\nii. 21, 26, ix. 10, where \u05d7\u05d1\u05db\u05d5\u05d4 is represented by \u03c3\u03bf\u03c6\u03af\u03b1 and \u05d3\u05e2\u05ea by\n\u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2. While \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 is simply _intuitive_, \u03c3\u03bf\u03c6\u03af\u03b1 is _ratiocinative_\nalso. While \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 applies chiefly to the apprehension of truths, \u03c3\u03bf\u03c6\u03af\u03b1\nsuperadds the power of reasoning about them and tracing their relations.\nWhen Bengel on 1 Cor. xii. 8 sq. says, 'Cognitio [\u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2] est quasi\nvisus; sapientia [\u03c3\u03bf\u03c6\u03af\u03b1] visus cum sapore,' he is so far right; but when\nhe adds, 'cognitio, rerum agendarum; sapientia, rerum aeternarum,' he is\nquite wide of the mark. Substantially the same, and equally wrong, is St\nAugustine's distinction _de Trin._ xii. 20, 25 (VIII. pp. 923, 926)\n'intelligendum est ad contemplationem sapientiam [\u03c3\u03bf\u03c6\u03af\u03b1\u03bd], ad actionem\nscientiam [\u03b3\u03bd\u1ff6\u03c3\u03b9\u03bd] pertinere ... quod alia [\u03c3\u03bf\u03c6\u03af\u03b1] sit intellectualis\ncognitio aeternarum rerum, alia [\u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2] rationalis temporalium' (comp.\nxiv. 3, p. 948), and again _de <DW37>. Qu\u00e6st. ad Simpl._> ii. 2 \u00a7 3 (VI. p.\n114) 'ita discerni probabiliter solent, ut sapientia pertineat ad\nintellectum \u00e6ternorum, scientia vero ad ea qu\u00e6 sensibus corporis\nexperimur.' This is directly opposed to usage. In Aristotle _Eth. Nic._\ni. 1 \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 is opposed to \u03c0\u03c1\u1fb6\u03be\u03b9\u03c2. In St Paul it is connected with the\napprehension of eternal mysteries, 1 Cor. xiii. 2 \u03b5\u1f30\u03b4\u1ff6 \u03c4\u1f70 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03b1 \u03c0\u03ac\u03bd\u03c4\u03b1\n\u03ba\u03b1\u1f76 \u03c0\u1fb6\u03c3\u03b1\u03bd \u03c4\u1f74\u03bd \u03b3\u03bd\u1ff6\u03c3\u03b9\u03bd. On the relation of \u03c3\u03bf\u03c6\u03af\u03b1 to \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2 see above, i.\n9.\n\n\u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9] So 1 Cor. ii. 7 \u03bb\u03b1\u03bb\u03bf\u1fe6\u03bc\u03b5\u03bd \u0398\u03b5\u03bf\u1fe6 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u1f10\u03bd \u03bc\u03c5\u03c3\u03c4\u03b7\u03c1\u03af\u1ff3, \u03c4\u1f74\u03bd\n\u1f00\u03c0\u03bf\u03ba\u03b5\u03ba\u03c1\u03c5\u03bc\u03bc\u03ad\u03bd\u03b7\u03bd. As before in \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2 (i. 28), so here again in \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9\nthe Apostle adopts a favourite term of the Gnostic teachers, only that\nhe may refute a favourite doctrine. The word _apocrypha_ was especially\napplied to those esoteric writings, for which such sectarians claimed an\n_auctoritas secreta_ (Aug. _c. Faust._ xi. 2, VIII. p. 219) and which\nthey carefully guarded from publication after the manner of their Jewish\nprototypes the Essenes (see above p. 89 sq.): comp. Iren. i. 20. 1\n\u1f00\u03bc\u03cd\u03b8\u03b7\u03c4\u03bf\u03bd \u03c0\u03bb\u1fc6\u03b8\u03bf\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03c9\u03bd \u03ba\u03b1\u1f76 \u03bd\u03cc\u03b8\u03c9\u03bd \u03b3\u03c1\u03b1\u03c6\u1ff6\u03bd, Clem. Alex. _Strom._ i. 15\n(p. 357) \u03b2\u03af\u03b2\u03bb\u03bf\u03c5\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03bf\u03c5\u03c2 \u03c4\u1f00\u03bd\u03b4\u03c1\u1f78\u03c2 \u03c4\u03bf\u1fe6\u03b4\u03b5 \u03bf\u1f31 \u03c4\u1f74\u03bd \u03a0\u03c1\u03bf\u03b4\u03af\u03ba\u03bf\u03c5 \u03bc\u03b5\u03c4\u03b9\u03cc\u03bd\u03c4\u03b5\u03c2\n\u1f05\u03b9\u03c1\u03b5\u03c3\u03b9\u03bd _\u03b1\u1f50\u03c7\u03bf\u1fe6\u03c3\u03b9_ \u03ba\u03b5\u03ba\u03c4\u1fc6\u03c3\u03b8\u03b1\u03b9, _ib._ iii. 4 (p. 524) \u1f10\u03c1\u03c1\u03cd\u03b7 \u03b4\u1f72 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2\n\u03c4\u1f78 \u03b4\u03cc\u03b3\u03bc\u03b1 \u1f14\u03ba \u03c4\u03b9\u03bd\u03bf\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03bf\u03c5. See also the application of the text Prov.\nix. 17 \u1f04\u03c1\u03c4\u03c9\u03bd \u03ba\u03c1\u03c5\u03c6\u03af\u03c9\u03bd \u1f21\u03b4\u03ad\u03c9\u03c2 \u1f05\u03c8\u03b1\u03c3\u03b8\u03b5 to these heretics in _Strom._ i. 19\n(p. 375). Thus the word _apocrypha_ in the first instance was an\nhonourable appellation applied by the heretics themselves to their\nesoteric doctrine and their secret books; but owing to the general\ncharacter of these works the term, as adopted by orthodox writers, got\nto signify 'false,' 'spurious.' The early fathers never apply it, as it\nis now applied, to _deutero-canonical_ writings, but confine it to\n_supposititious_ and _heretical_ works: see Smith's _Dictionary of the\nBible_ s.v. In the text St Paul uses it \u03ba\u03b1\u03c4\u03b1\u03c7\u03c1\u03b7\u03c3\u03c4\u03b9\u03ba\u1ff6\u03c2, as he uses\n\u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd. 'All the richest treasures of that secret wisdom,' he would\nsay, 'on which you lay so much stress, are buried in Christ, and being\nburied there are accessible to all alike who seek Him.' But, while the\nterm \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03c2 is adopted because it was used to designate the secret\ndoctrine and writings of the heretics, it is also entirely in keeping\nwith the metaphor of the 'treasure'; e.g. Is. xlv. 3 \u03b4\u03ce\u03c3\u03c9 \u03c3\u03bf\u03b9 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u1f7a\u03c2\n\u03c3\u03ba\u03bf\u03c4\u03b5\u03b9\u03bd\u03bf\u1f7a\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03bf\u03c5\u03c2, 1 Macc. i. 23 \u1f14\u03bb\u03b1\u03b2\u03b5 \u03c4\u03bf\u1f7a\u03c2 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u1f7a\u03c2 \u03c4\u03bf\u1f7a\u03c2\n\u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03bf\u03c5\u03c2, Dan. xi. 43 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03bf\u03b9\u03c2 \u03c4\u03bf\u1fe6 \u03c7\u03c1\u03c5\u03c3\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u03b1\u03c1\u03b3\u03cd\u03c1\u03bf\u03c5:\ncomp. Matt. xiii. 44.\n\nThe stress thus laid on \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9 will explain its position. It is not\nconnected with \u03b5\u1f30\u03c3\u03b9\u03bd, but must be taken apart as a secondary predicate:\ncomp. ver. 10 \u1f10\u03c3\u03c4\u1f72 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9, iii. 1 \u03bf\u1f57 \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f10\u03bd\n\u03b4\u03b5\u03be\u03b9\u1fb7 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03ba\u03b1\u03b8\u03ae\u03bc\u03b5\u03bd\u03bf\u03c2, James i. 17 \u03c0\u1fb6\u03bd \u03b4\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd \u1f04\u03bd\u03c9\u03b8\u03ad\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd,\n\u03ba\u03b1\u03c4\u03b1\u03b2\u03b1\u1fd6\u03bd\u03bf\u03bd \u03ba.\u03c4.\u03bb.\n\n4\u20137. 'I do not say this without a purpose. I wish to warn you against\nany who would lead you astray by specious argument and persuasive\nrhetoric. For I am not an indifferent spectator of your doings. Although\nI am absent from you in my flesh, yet I am present with you in my\nspirit. I rejoice to behold the orderly array and the solid phalanx\nwhich your faith towards Christ presents against the assaults of the\nfoe. I entreat you therefore not to abandon the Christ, as you learnt\nfrom Epaphras to know Him, even Jesus the Lord, but to walk still in Him\nas heretofore. I would have you firmly rooted once for all in Him. I\ndesire to see you built up higher in Him day by day, to see you growing\never stronger and stronger through your faith, while you remain true to\nthe lessons taught you of old, so that you may abound in it, and thus\nabounding may pour forth your hearts in gratitude to God the giver of\nall.'\n\n------------------------------------------------------------------------\n\nII. 5]\n\n#\u03bb\u03ad\u03b3\u03c9, \u1f35\u03bd\u03b1 \u03bc\u03b7\u03b4\u03b5\u1f76\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bb\u03bf\u03b3\u03af\u03b6\u03b7\u03c4\u03b1\u03b9 \u1f10\u03bd \u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03af\u1fb3\u00b7 ^5\u03b5\u1f30 \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76 \u03c4\u1fc7\n\u03c3\u03b1\u03c1\u03ba\u1f76 \u1f04\u03c0\u03b5\u03b9\u03bc\u03b9, \u1f00\u03bb\u03bb\u1f70 \u03c4\u1ff7 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9 \u03c3\u1f7a\u03bd\n\n\n4. \u03c4\u03bf\u1fe6\u03c4\u03bf \u03bb\u03ad\u03b3\u03c9 \u03ba.\u03c4.\u03bb.] 'I say all this to you, lest you should be led\nastray by those false teachers who speak of another knowledge, of other\nmysteries.' In other connexions \u03c4\u03bf\u1fe6\u03c4\u03bf \u03bb\u03ad\u03b3\u03c9 will frequently refer to the\nwords following (e.g. Gal. iii. 17, 1 Cor. i. 12); but with \u1f35\u03bd\u03b1 it\npoints to what has gone before, as in Joh. v. 34 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03bb\u03ad\u03b3\u03c9 \u1f35\u03bd\u03b1 \u1f51\u03bc\u03b5\u1fd6\u03c2\n\u03c3\u03c9\u03b8\u1fc6\u03c4\u03b5.\n\nThe reference in \u03c4\u03bf\u1fe6\u03c4\u03bf \u03bb\u03ad\u03b3\u03c9 extends over vv. 1\u20133, and involves two\nstatements; (1) The declaration that all knowledge is comprehended in\nChrist, vv. 2, 3; (2) The expression of his own personal anxiety that\nthey should remain stedfast in this conviction, vv. 1, 2. This last\npoint explains the language which follows, \u03b5\u1f30 \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76 \u03c4\u1fc7 \u03c3\u03b1\u03c1\u03ba\u1f76 \u03ba.\u03c4.\u03bb.\n\n\u03c0\u03b1\u03c1\u03b1\u03bb\u03bf\u03b3\u03af\u03b6\u03b7\u03c4\u03b1\u03b9] '_lead you astray by false reasoning_', as in Daniel xiv.\n7 \u03bc\u03b7\u03b4\u03ad\u03b9\u03c2 \u03c3\u03b5 \u03c0\u03b1\u03c1\u03b1\u03bb\u03bf\u03b3\u03b9\u03b6\u03ad\u03c3\u03b8\u03c9 (LXX): comp. James i. 22, Ign. _Magn._ 3. It\nis not an uncommon word either in the LXX or in classical writers. The\nsystem against which St Paul here contends professed to be a \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\n(ver. 8) and had a \u03bb\u03cc\u03b3\u03bf\u03bd \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 (ver. 23).\n\n\u1f10\u03bd \u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03af\u1fb3] The words \u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03b5\u1fd6\u03bd (Arist. _Eth. Nic._ i. 1),\n\u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 (Plat. _The\u00e6t._ 162 E), \u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03b9\u03ba\u03cc\u03c2 (Epictet. i. 8. 7),\noccur occasionally in classical writers, but do not bear a bad sense,\nbeing most frequently opposed to \u1f00\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b9\u03c2, as probable argument to\nstrict mathematical demonstration. This contrast probably suggested St\nPaul's language in 1 Cor. ii. 4 \u03bf\u1f50\u03ba \u1f10\u03bd _\u03c0\u03b5\u03b9\u03b8\u03bf\u1fd6\u03c2_ \u03c3\u03bf\u03c6\u03af\u03b1\u03c2\n_\u03bb\u03cc\u03b3\u03bf\u03b9\u03c2_ \u1f00\u03bb\u03bb' \u1f10\u03bd _\u1f00\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03b9_ \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba.\u03c4.\u03bb., and may\npossibly have been present to his mind here.\n\n5. \u1f00\u03bb\u03bb\u1f70] frequently introduces the apodosis after \u03b5\u1f30 or \u03b5\u1f30 \u03ba\u03b1\u1f76 in St\nPaul; e.g. Rom. vi. 5, 1 Cor. ix. 2, 2 Cor. iv. 16, v. 16, xi. 6, xiii.\n4 (v. l.).\n\n\u03c4\u1ff7 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9] '_in my spirit_', not '_by the Spirit_'. We have here the\ncommon antithesis of flesh and spirit, or body and spirit: comp. 1 Cor.\nv. 3 \u1f00\u03c0\u1f7c\u03bd \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9, \u03c0\u03b1\u03c1\u1f7c\u03bd \u03b4\u1f72 \u03c4\u1ff7 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9. St Paul elsewhere uses\nanother antithesis, \u03c0\u03c1\u03bf\u03c3\u03ce\u03c0\u1ff3 and \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3, to express this same thing; 1\nThess. ii. 17.\n\n\u03c7\u03b1\u03af\u03c1\u03c9\u03bd \u03ba\u03b1\u1f76 \u03b2\u03bb\u03ad\u03c0\u03c9\u03bd] '_rejoicing and beholding_'. This must not be\nregarded as a logical inversion. The contemplation of their orderly\narray, though it might have been first the cause, was afterwards the\nconsequence, of the Apostle's rejoicing. He looked, because it gave him\nsatisfaction to look.\n\n------------------------------------------------------------------------\n\nII. 6]\n\n\u1f51\u03bc\u1fd6\u03bd \u03b5\u1f30\u03bc\u03af, \u03c7\u03b1\u03af\u03c1\u03c9\u03bd \u03ba\u03b1\u1f76 \u03b2\u03bb\u03ad\u03c0\u03c9\u03bd \u1f51\u03bc\u1ff6\u03bd \u03c4\u1f74\u03bd \u03c4\u03ac\u03be\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c3\u03c4\u03b5\u03c1\u03ad\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b5\u1f30\u03c2\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u1f51\u03bc\u1ff6\u03bd. ^6\u1f61\u03c2 \u03bf\u1f56\u03bd \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 \u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd, \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03c4\u1f78\u03bd\n\u039a\u03cd\u03c1\u03b9\u03bf\u03bd, \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5,\n\n\n\u03c4\u1f74\u03bd \u03c4\u03ac\u03be\u03b9\u03bd] '_your orderly array_', a military metaphor: comp. e.g. Xen.\n_Anab._ i. 2. 18 \u1f30\u03b4\u03bf\u1fe6\u03c3\u03b1 \u03c4\u1f74\u03bd \u03bb\u03b1\u03bc\u03c0\u03c1\u03cc\u03c4\u03b7\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c4\u03ac\u03be\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03c1\u03b1\u03c4\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2\n\u1f10\u03b8\u03b1\u03cd\u03bc\u03b1\u03c3\u03b5, Plut. _Vit. Pyrrh._ 16 \u03ba\u03b1\u03c4\u03b9\u03b4\u1f7c\u03bd \u03c4\u03ac\u03be\u03b9\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c6\u03c5\u03bb\u03b1\u03ba\u1f70\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd\n\u03b1\u1f50\u03c4\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c3\u03c7\u1fc6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03c4\u03c1\u03b1\u03c4\u03bf\u03c0\u03b5\u03b4\u03b5\u03af\u03b1\u03c2 \u1f10\u03b8\u03b1\u03cd\u03bc\u03b1\u03c3\u03b5. The enforced\ncompanionship of St Paul with the soldiers of the pr\u00e6torian guard at\nthis time (Phil. i. 13) might have suggested this image. At all events\nin the contemporary epistle (Ephes. vi. 14 sq.) we have an elaborate\nmetaphor from the armour of a soldier.\n\n\u03c4\u1f78 \u03c3\u03c4\u03b5\u03c1\u03ad\u03c9\u03bc\u03b1] '_solid front_, _close phalanx_', a continuation of the\nmetaphor: comp. 1 Macc. ix. 14 \u03b5\u1f36\u03b4\u03b5\u03bd \u1f38\u03bf\u03cd\u03b4\u03b1\u03c2 \u1f45\u03c4\u03b9 \u0392\u03b1\u03ba\u03c7\u03af\u03b4\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c3\u03c4\u03b5\u03c1\u03ad\u03c9\u03bc\u03b1\n\u03c4\u1fc6\u03c2 \u03c0\u03b1\u03c1\u03b5\u03bc\u03b2\u03bf\u03bb\u1fc6\u03c2 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03be\u03b9\u03bf\u1fd6\u03c2. Somewhat similar are the expressions\n\u03c3\u03c4\u03b5\u03c1\u03b5\u03bf\u1fe6\u03bd \u03c4\u1f78\u03bd \u03c0\u03cc\u03bb\u03b5\u03bc\u03bf\u03bd 1 Macc. x. 50, \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u03c3\u03c4\u03b5\u03c1\u03ad\u03c9\u03c3\u03b9\u03bd \u03c4\u1fc6\u03c2 \u03bc\u03ac\u03c7\u03b7\u03c2 Ecclus.\nxxviii. 10. For the connexion here compare 1 Pet. v. 9 \u1f00\u03bd\u03c4\u03af\u03c3\u03c4\u03b7\u03c4\u03b5 \u03c3\u03c4\u03b5\u03c1\u03b5\u03bf\u1f76\n\u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9, Acts xvi. 5 \u1f10\u03c3\u03c4\u03b5\u03c1\u03b5\u03bf\u1fe6\u03bd\u03c4\u03bf \u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9.\n\n6. \u1f61\u03c2 \u03bf\u1f56\u03bd \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] i.e. 'Let your conviction and conduct be in\nperfect accordance with the doctrines and precepts of the Gospel as it\nwas taught to you'. For this use of \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 'ye received from your\nteachers, were instructed in', comp. 1 Cor. xv. 1, 3, Gal. i. 9, Phil.\niv. 9, 1 Thess. ii. 13, iv. 1, 2 Thess. iii. 6. The word \u03c0\u03b1\u03c1\u03b1\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9\u03bd\nimplies either 'to receive as transmitted', or 'to receive for\ntransmission': see the note on Gal. i. 12. The \u1f61\u03c2 of the protasis\nsuggests a \u03bf\u1f55\u03c4\u03c9\u03c2 in the apodosis, which in this case is unexpressed but\nmust be understood. The meaning of \u1f61\u03c2 \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 here is explained by\nthe \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 \u1f00\u03c0\u1f78 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6 in i. 7; see the note there, and comp.\nbelow ver. 7 \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03b4\u03b9\u03b4\u03ac\u03c7\u03b8\u03b7\u03c4\u03b5.\n\n\u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd] '_the Christ_', rather than 'the Gospel', because the\ncentral point in the Colossian heresy was the subversion of the true\nidea of the Christ.\n\n\u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd] 'even _Jesus the Lord_', in whom the true conception\nof the Christ is realised: comp. Ephes. iv. 20, 21, \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03b4\u1f72 \u03bf\u1f50\u03c7 \u03bf\u1f55\u03c4\u03c9\u03c2\n\u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 _\u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd_, \u03b5\u1f34\u03b3\u03b5 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f20\u03ba\u03bf\u03cd\u03c3\u03b1\u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u1f10\u03b4\u03b9\u03b4\u03ac\u03c7\u03b8\u03b7\u03c4\u03b5,\n_\u03ba\u03b1\u03b8\u03ce\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u03bb\u03ae\u03b8\u03b5\u03b9\u03b1 \u1f10\u03bd \u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6_, where the same idea is more\ndirectly expressed. The genuine doctrine of the Christ consists in (1)\nthe recognition of the historical person _Jesus_, and (2) the acceptance\nof Him as _the Lord_. This doctrine was seriously endangered by the\nmystic theosophy of the false teachers. The same order which we have\nhere occurs also in Ephes. iii. 11 \u1f10\u03bd \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3 \u1f21\u03bc\u1ff6\u03bd (the\ncorrect reading).\n\n7. \u1f10\u03c1\u03c1\u03b9\u03b6\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9] Two points may be noticed here; (1) The expressive\nchange of tenses; \u1f10\u03c1\u03c1\u03b9\u03b6\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 'firmly rooted' _once for all_,\n\u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9, \u03b2\u03b5\u03b2\u03b1\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9, 'built up and strengthened' _from hour\nto hour_. (2) The rapid transition of metaphor, \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5,\n\u1f10\u03c1\u1fe5\u03b9\u03b6\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9, \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9, the path, the tree, the building: comp.\nEphes. iii. 17 \u1f10\u03c1\u03c1\u03b9\u03b6\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03c4\u03b5\u03b8\u03b5\u03bc\u03b5\u03bb\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9. The metaphors of the\nplant and the building occur together in 1 Cor. iii. 9 \u0398\u03b5\u03bf\u1fe6 \u03b3\u03b5\u03ce\u03c1\u03b3\u03b9\u03bf\u03bd,\n\u0398\u03b5\u03bf\u1fe6 \u03bf\u1f30\u03ba\u03bf\u03b4\u03bf\u03bc\u03ae. The transition in this passage is made easier by the fact\nthat \u1fe5\u03b9\u03b6\u03bf\u1fe6\u03bd (Plut. _Mor._ 321 D), \u1f10\u03ba\u03c1\u03b9\u03b6\u03bf\u1fe6\u03bd (Jer. i. 10, 1 Macc. v. 51),\n\u03c0\u03c1\u03cc\u03c1\u03c1\u03b9\u03b6\u03bf\u03c2 (Jos. _B.J._ vii. 8. 7), etc., are not uncommonly used of\ncities and buildings.\n\n------------------------------------------------------------------------\n\nII. 7]\n\n^7\u1f10\u03c1\u03c1\u03b9\u03b6\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03ba\u03b1\u1f76 \u03b2\u03b5\u03b2\u03b1\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9,\n\u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03b4\u03b9\u03b4\u03ac\u03c7\u03b8\u03b7\u03c4\u03b5, \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03b1\u1f50\u03c4\u1fc7 \u1f10\u03bd \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03af\u1fb3.\n\n\n\u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9] '_being built up_,' as in 1 Cor. iii. 10\u201314. After\nthis verb we might have expected \u1f10\u03c0' \u03b1\u1f50\u03c4\u1ff7 or \u1f10\u03c0' \u03b1\u1f50\u03c4\u03cc\u03bd (1 Cor. iii. 12)\nrather than \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7; but in this and the companion epistle Christ is\nrepresented rather as the binding element than as the foundation of the\nbuilding: e.g. Ephes. ii. 20 \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03b7\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2 \u1f10\u03c0\u1f76 \u03c4\u1ff7 \u03b8\u03b5\u03bc\u03b5\u03bb\u03af\u1ff3 \u03c4\u1ff6\u03bd\n\u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c6\u03b7\u03c4\u1ff6\u03bd, \u1f44\u03bd\u03c4\u03bf\u03c2 _\u1f00\u03ba\u03c1\u03bf\u03b3\u03c9\u03bd\u03b9\u03b1\u03af\u03bf\u03c5_ \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6,\n_\u1f10\u03bd_ \u1fa7 \u03c0\u1fb6\u03c3\u03b1 [\u1f21] \u03bf\u1f30\u03ba\u03bf\u03b4\u03bf\u03bc\u1f74 \u03b1\u1f54\u03be\u03b5\u03b9 \u03b5\u1f30\u03c2 \u03bd\u03b1\u1f78\u03bd \u1f05\u03b3\u03b9\u03bf\u03bd _\u1f10\u03bd_ \u039a\u03c5\u03c1\u03af\u1ff3,\n_\u1f10\u03bd_ \u1fa7 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03c3\u03c5\u03bd\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03b5\u1fd6\u03c3\u03b8\u03b5. The \u1f10\u03c0\u1f76 in \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03b5\u1fd6\u03bd does not\nnecessarily refer to the original foundation, but may point to the\ncontinued progress of the building by successive layers, as e.g.\n[Aristot.] _Rhet. ad Alex._ 4 (p. 1426) \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03bf\u1fe6\u03bd\u03c4\u03b1 \u03c4\u1f78 \u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u1f61\u03c2 \u1f10\u03c0\u1f76\n\u03c4\u1f78 \u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u03b1\u1f54\u03be\u03b5\u03b9\u03bd. Hence \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03b5\u1fd6\u03bd is frequently used absolutely, '_to\nbuild up_' (e.g. Jude 20, Polyb. iii. 27, 4), as here. The repetition of\n\u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 emphasizes the main idea of the passage, and indeed of the whole\nepistle.\n\n\u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9] '_by your faith_', the dative of the instrument; comp. Heb.\nxiii. 9 \u03ba\u03b1\u03bb\u1f78\u03bd \u03b3\u1f70\u03c1 \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9 \u03b2\u03b5\u03b2\u03b1\u03b9\u03bf\u1fe6\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f74\u03bd \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03bd. Faith is, as it were,\nthe cement of the building: comp. Clem. Rom. 22 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b2\u03b5\u03b2\u03b1\u03b9\u03bf\u1fd6 \u1f21 \u1f10\u03bd\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03c0\u03af\u03c3\u03c4\u03b9\u03c2.\n\n\u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03b4\u03b9\u03b4\u03ac\u03c7\u03b8\u03b7\u03c4\u03b5] i.e. 'remaining true to the lessons which you received\nfrom Epaphras, and not led astray by any later pretenders': comp. i. 6,\n7 \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03bc\u03ac\u03b8\u03b5\u03c4\u03b5 \u1f00\u03c0\u1f78 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6.\n\n\u1f10\u03bd \u03b1\u1f50\u03c4\u1fc7 \u03ba.\u03c4.\u03bb.] The same ending occurs in iv. 2. Thanksgiving is the end\nof all human conduct, whether exhibited in words or in works. For the\nstress laid on thanksgiving in St Paul's epistles generally, see the\nnote on Phil. iv. 6. The words \u03b5\u1f50\u03c7\u03ac\u03c1\u03b9\u03c3\u03c4\u03bf\u03c2, \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03b5\u1fd6\u03bd, \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03af\u03b1,\noccur in St Paul's writings alone of the Apostolic epistles. In this\nepistle especially the duty of thanksgiving assumes a peculiar\nprominence by being made a refrain, as here and in iii. 15, 17, iv. 2:\nsee also i. 12.\n\n------------------------------------------------------------------------\n\nII. 8]\n\n^8 \u0392\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03bc\u03ae \u03c4\u03b9\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u1f14\u03c3\u03c4\u03b1\u03b9 \u1f41 \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u1ff6\u03bd \u03b4\u03b9\u1f70\n\n                      8. \u03bc\u03ae \u03c4\u03b9\u03c2 _\u1f14\u03c3\u03c4\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2_.\n\n\n8\u201315. 'Be on your guard; do not suffer yourselves to fall a prey to\ncertain persons who would lead you captive by a hollow and deceitful\nsystem, which they call philosophy. They substitute the traditions of\nmen for the truth of God. They enforce an elementary discipline of\nmundane ordinances fit only for children. Theirs is not the Gospel of\nChrist. In Christ the entire fulness of the Godhead abides for ever,\nhaving united itself with man by taking a human body. And so in Him\u2014not\nin any inferior mediators\u2014ye have your life, your being, for ye are\nfilled from His fulness. He, I say, is the Head over all spiritual\nbeings\u2014call them principalities or powers or what you will. In Him too\nye have the true circumcision\u2014the circumcision which is not made with\nhands but wrought by the Spirit\u2014the circumcision which divests not of a\npart only but of the whole carnal body\u2014the circumcision which is not of\nMoses but of Christ. This circumcision ye have, because ye were buried\nwith Christ to your old selves beneath the baptismal waters, and were\nraised with Him from those same waters to a new and regenerate life,\nthrough your faith in the powerful working of God who raised Him from\nthe dead. Yes, you\u2014you Gentiles who before were dead, when ye walked in\nyour transgressions and in the uncircumcision of your unchastened carnal\nheathen heart\u2014even you did\n\nGod quicken into life together with Christ; then and there freely\nforgiving all of us\u2014Jews and Gentiles alike\u2014all our transgressions; then\nand there cancelling the bond which stood valid against us (for it bore\nour own signature), the bond which engaged us to fulfil all the law of\nordinances, which was our stern pitiless tyrant. Ay, this very bond hath\nChrist put out of sight for ever, nailing it to His cross and rending it\nwith His body and killing it in His death. Taking upon Him our human\nnature, He stripped off and cast aside all the powers of evil which\nclung to it like a poisonous garment. As a mighty conqueror He displayed\nthese His fallen enemies to an astonished world, leading them in triumph\non His cross.'\n\n8. \u0392\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] The form of the sentence is a measure of the\nimminence of the peril. The usual construction with \u03b2\u03bb\u03ad\u03c0\u03b5\u03b9\u03bd \u03bc\u1f74 is a\nconjunctive; e.g. in Luke xxi. 8 \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03bc\u1f74 \u03c0\u03bb\u03b1\u03bd\u03b7\u03b8\u1fc6\u03c4\u03b5. Here the\nsubstitution of an indicative shows that the danger is real; comp. Heb.\niii. 12 \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03bc\u03ae\u03c0\u03bf\u03c4\u03b5 \u1f14\u03c3\u03c4\u03b1\u03b9 \u1f15\u03bd \u03c4\u03b9\u03bd\u03b9 \u1f51\u03bc\u1ff6\u03bd \u03ba\u03b1\u03c1\u03b4\u03af\u03b1 \u03c0\u03bf\u03bd\u03b7\u03c1\u1f70 \u1f00\u03c0\u03b9\u03c3\u03c4\u03af\u03b1\u03c2. For\nother instances of \u03bc\u1f74 with a future indicative comp. Mark xiv. 2 \u03bc\u03ae\u03c0\u03bf\u03c4\u03b5\n\u1f14\u03c3\u03c4\u03b1\u03b9 \u03b8\u03cc\u03c1\u03c5\u03b2\u03bf\u03c2, Rom. xi. 21 \u03bc\u03ae\u03c0\u03c9\u03c2 \u03bf\u1f50\u03b4\u1f72 \u03c3\u03bf\u1fe6 \u03c6\u03b5\u03af\u03c3\u03b5\u03c4\u03b1\u03b9; and see Winer \u00a7 lvi.\np. 631 sq.\n\n\u03c4\u03b9\u03c2] This indefinite \u03c4\u03b9\u03c2 is frequently used by St Paul, when speaking of\nopponents whom he knows well enough but does not care to name: see the\nnote on Gal. i. 7. Comp. Ign. _Smyrn._ 5 \u1f45\u03bd _\u03c4\u03b9\u03bd\u03b5\u03c2_ \u1f00\u03b3\u03bd\u03bf\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2\n\u1f00\u03c1\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 ... \u03c4\u1f70 \u03b4\u1f72 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b1 \u03b1\u1f50\u03c4\u1ff6\u03bd, \u1f44\u03bd\u03c4\u03b1 \u1f04\u03c0\u03b9\u03c3\u03c4\u03b1, \u03bf\u1f50\u03ba \u1f14\u03b4\u03bf\u03be\u03ad \u03bc\u03bf\u03b9 \u1f10\u03b3\u03b3\u03c1\u03ac\u03c8\u03b1\u03b9.\n\n\u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u1ff6\u03bd] '_makes you his prey_, carries you off body and soul'. The\nword appears not to occur before St Paul, nor after him, independently\nof this passage, till a late date: e.g. Heliod. _Aeth._ x. 35 \u03bf\u1f57\u03c4\u03cc\u03c2\n\u1f10\u03c3\u03c4\u03b9\u03bd \u1f41 \u03c4\u1f74\u03bd \u1f10\u03bc\u1f74\u03bd \u03b8\u03c5\u03b3\u03ac\u03c4\u03b5\u03c1\u03b1 \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u03ae\u03c3\u03b1\u03c2. In Tatian _ad Gr\u00e6c._ 22 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03b4\u1f72\n\u1f51\u03c0\u1f78 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u03b5\u1fd6\u03c3\u03b8\u03b5 it seems to be a reminiscence of St Paul. Its\nfull and proper meaning, as appears from the passages quoted, is not 'to\ndespoil,' but 'to carry off as spoil', in accordance with the analogous\ncompounds, \u03b4\u03bf\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u03b5\u1fd6\u03bd, \u03c3\u03ba\u03b5\u03c5\u03b1\u03b3\u03c9\u03b3\u03b5\u1fd6\u03bd. So too the closely allied word\n\u03bb\u03b1\u03c6\u03c5\u03c1\u03b1\u03b3\u03c9\u03b3\u03b5\u1fd6\u03bd in Plut. _Mor._ p. 5 \u03c0\u03cc\u03bb\u03b5\u03bc\u03bf\u03c2 \u03b3\u1f70\u03c1 \u03bf\u1f50 \u03bb\u03b1\u03c6\u03c5\u03c1\u03b1\u03b3\u03c9\u03b3\u03b5\u1fd6 \u1f00\u03c1\u03b5\u03c4\u03ae\u03bd,\n_Vit. Galb._ 5 \u03c4\u1f70 \u03bc\u1f72\u03bd \u0393\u03b1\u03bb\u03b1\u03c4\u1ff6\u03bd, \u1f45\u03c4\u03b1\u03bd \u1f51\u03c0\u03bf\u03c7\u03b5\u03af\u03c1\u03b9\u03bf\u03b9 \u03b3\u03ad\u03bd\u03c9\u03bd\u03c4\u03b1\u03b9,\n\u03bb\u03b1\u03c6\u03c5\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae\u03c3\u03b5\u03c3\u03b8\u03b1\u03b9. The Colossians had been rescued from the bondage of\ndarkness; they had been transferred to the kingdom of light; they had\nbeen settled there as free citizens (i. 12, 13); and now there was\ndanger that they should fall into a state worse than their former\nslavery, that they should be carried off as so much booty. Comp. 2 Tim.\niii. 6 \u03b1\u1f30\u03c7\u03bc\u03b1\u03bb\u03c9\u03c4\u03af\u03b6\u03bf\u03bd\u03c4\u03b5\u03c2 \u03b3\u03c5\u03bd\u03b1\u03b9\u03ba\u03ac\u03c1\u03b9\u03b1.\n\nFor the construction \u1f14\u03c3\u03c4\u03b1\u03b9 \u1f41 \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u1ff6\u03bd see the notes on Gal. i. 7, iii.\n21. The former passage is a close parallel to the words here, \u03b5\u1f30 \u03bc\u03ae\n\u03c4\u03b9\u03bd\u03ad\u03c2 \u03b5\u1f30\u03c3\u03b9\u03bd \u03bf\u1f31 \u03c4\u03b1\u03c1\u03ac\u03c3\u03c3\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u03ba.\u03c4.\u03bb. The expression \u1f41 \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u1ff6\u03bd gives\na directness and individuality to the reference, which would have been\nwanting to the more natural construction \u1f43\u03c2 \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u03ae\u03c3\u03b5\u03b9.\n\n\u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba.\u03c4.\u03bb.] '_through his philosophy which is an empty\ndeceit_'. The absence of both preposition and article in the second\nclause shows that \u03ba\u03b5\u03bd\u1fc6\u03c2 \u1f00\u03c0\u03ac\u03c4\u03b7\u03c2 describes and qualifies \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2.\nClement therefore (_Strom._ vi. 8, p. 771) had a right to contend that\nSt Paul does not here condemn 'philosophy' absolutely. The \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1 \u03ba\u03b1\u1f76\n\u03ba\u03b5\u03bd\u1f74 \u1f00\u03c0\u03ac\u03c4\u03b7 of this passage corresponds to the \u03c8\u03b5\u03c5\u03b4\u03ce\u03bd\u03c5\u03bc\u03bf\u03c2 \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 of 1\nTim. vi. 20.\n\nBut though 'philosophy' is not condemned, it is disparaged by the\nconnexion in which it is placed. St Chrysostom's comment is not\naltogether wrong, \u1f10\u03c0\u03b5\u03b9\u03b4\u1f74 \u03b4\u03bf\u03ba\u03b5\u1fd6 \u03c3\u03b5\u03bc\u03bd\u1f78\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1f78 \u03c4\u1fc6\u03c2 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2, \u03c0\u03c1\u03bf\u03c3\u03ad\u03b8\u03b7\u03ba\u03b5\n\u03ba\u03b1\u1f76 \u03ba\u03b5\u03bd\u1fc6\u03c2 \u1f00\u03c0\u03ac\u03c4\u03b7\u03c2. The term was doubtless used by the false teachers\nthemselves to describe their system. Though essentially Greek as a name\nand as an idea, it had found its way into Jewish circles. Philo speaks\nof the Hebrew religion and Mosaic law as \u1f21 \u03c0\u03ac\u03c4\u03c1\u03b9\u03bf\u03c2 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1 (_Leg. ad\nCai._ 23, II. p. 568, _de Somn._ ii. 18, I. p. 675) or \u1f21 \u1f38\u03bf\u03c5\u03b4\u03b1\u03ca\u03ba\u1f74\n\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1 (_Leg. ad Cai._ 33, II. p. 582) or \u1f21 \u03ba\u03b1\u03c4\u1f70 \u039c\u03c9\"\u03c5\u03c3\u1fc6\u03bd \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\n(_de Mut. Nom._ 39, I. p. 612). The system of the Essenes, the probable\nprogenitors of the false teachers at Coloss\u00e6, he describes as \u1f21 \u03b4\u03af\u03c7\u03b1\n\u03c0\u03b5\u03c1\u03b9\u03b5\u03c1\u03b3\u03b5\u03af\u03b1\u03c2 \u1f19\u03bb\u03bb\u03b7\u03bd\u03b9\u03ba\u1ff6\u03bd \u1f40\u03bd\u03bf\u03bc\u03ac\u03c4\u03c9\u03bd \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1 (_Omn. prob. lib._ 13, II. p.\n459). So too Josephus speaks of the three Jewish sects as \u03c4\u03c1\u03b5\u1fd6\u03c2\n\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03b9 (_Ant._ xviii. 1. 2, comp. _B.J._ ii. 8. 2). It should be\nremembered also, that in this later age, owing to Roman influence, the\nterm was used to describe practical not less than speculative systems,\nso that it would cover the ascetic life as well as the mystic theosophy\nof these Colossian heretics. Hence the Apostle is here flinging back at\nthese false teachers a favourite term of their own, 'their vaunted\n_philosophy_, which is hollow and misleading'.\n\nThe word indeed could claim a truly noble origin; for it is said to have\narisen out of the humility of Pythagoras, who called himself 'a lover of\nwisdom', \u03bc\u03b7\u03b4\u03ad\u03bd\u03b1 \u03b3\u1f70\u03c1 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c3\u03bf\u03c6\u1f78\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u1f00\u03bb\u03bb' \u1f22 \u0398\u03b5\u03cc\u03bd (Diog. Laert.\nPro\u0153m. \u00a7 12; comp. Cic. _Tusc._ v. 3). In such a sense the term would\nentirely accord with the spirit and teaching of St Paul; for it bore\ntestimony to the insufficiency of the human intellect and the need of a\nrevelation. But in his age it had come to be associated generally with\nthe idea of subtle dialectics and profitless speculation; while in this\nparticular instance it was combined with a mystic cosmogony and\nangelology which contributed a fresh element of danger. As contrasted\nwith the power and fulness and certainty of revelation, all such\nphilosophy was 'foolishness' (1 Cor. i. 20). It is worth observing that\nthis word, which to the Greeks denoted the highest effort of the\nintellect, occurs here alone in St Paul, just as he uses \u1f00\u03c1\u03b5\u03c4\u03ae, which\nwas their term to express the highest moral excellence, in a single\npassage only (Phil. iv. 8; see the note there). The reason is much the\nsame in both cases. The Gospel had deposed the terms as inadequate to\nthe higher standard, whether of knowledge or of practice, which it had\nintroduced.\n\nOn the attitude of the fathers towards philosophy, while philosophy was\na living thing, see Smith's _Dictionary of the Bible_ s.v. Clement, who\nwas followed in the main by the earlier Alexandrian fathers, regards\nGreek philosophy not only as a preliminary training (\u03c0\u03c1\u03bf\u03c0\u03b1\u03b9\u03b4\u03b5\u03af\u03b1) for the\nGospel, but even as in some sense a covenant (\u03b4\u03b9\u03b1\u03b8\u03ae\u03ba\u03b7) given by God to\nthe Greeks (_Strom._ i. 5, p. 331, vi. 5, p. 761, _ib._ \u00a7 8, p. 771\nsq.). Others, who were the great majority and of whom Tertullian may be\ntaken as an extreme type, set their faces directly against it, seeing in\nit only the parent of all heretical teaching: e.g. _de Anim._ 2, 3,\n_Apol._ 46, 47. In the first passage, referring to this text, he says,\n'Ab apostolo jam tunc philosophia concussio veritatis providebatur'; in\nthe second he asks, 'Quid simile philosophus et Christianus?' St Paul's\nspeech at Athens, on the only occasion when he is known to have been\nbrought into direct personal contact with Greek philosophers (Acts xvii.\n18), shows that his sympathies would have been at least as strong with\nClement's representations as with Tertullian's.\n\n------------------------------------------------------------------------\n\nII. 8]\n\n\u03c4\u1fc6\u03c2 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03b5\u03bd\u1fc6\u03c2 \u1f00\u03c0\u03ac\u03c4\u03b7\u03c2, \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03c3\u03b9\u03bd\n\n\n\u03ba\u03b1\u03c4\u1f70 \u03ba.\u03c4.\u03bb.] The false teaching is described (1) As regards its\nsource\u2013'the traditions of _men_'; (2) As regards its subject matter\u2013'the\nrudiments of _the world_'.\n\n\u03c4\u1f74\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03c3\u03b9\u03bd \u03ba.\u03c4.\u03bb.] Other systems, as for instance the ceremonial\nmishna of the Pharisees, might fitly be described in this way (Matt.\nxv. 2 sq., Mark vii. 3 sq.): but such a description was peculiarly\nappropriate to a mystic theosophy like this of the Colossian false\nteachers. The teaching might be oral or written, but it was\nessentially esoteric, essentially traditional. It could not appeal to\nsacred books which had been before all the world for centuries. The\nEssenes, the immediate spiritual progenitors of these Colossian\nheretics, distinctly claimed to possess such a source of knowledge,\nwhich they carefully guarded from divulgence; _B.J._ ii. 8. 7\n\u03c3\u03c5\u03bd\u03c4\u03b7\u03c1\u03ae\u03c3\u03b5\u03b9\u03bd \u1f41\u03bc\u03bf\u03af\u03c9\u03c2 \u03c4\u03ac \u03c4\u03b5 \u03c4\u1fc6\u03c2 \u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03c2 \u03b1\u03c5\u03c4\u1ff6\u03bd \u03b2\u03b9\u03b2\u03bb\u03af\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd\n\u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b1 (see above pp. 89, 90 sq., 95). The various Gnostic sects,\ntheir direct or collateral spiritual descendants, almost without\nexception traced their doctrines to a similar source: e.g. Hippol.\n_H\u00e6r._ v. 7 \u1f03 \u03c6\u03b7\u03c3\u1f76 _\u03c0\u03b1\u03c1\u03b1\u03b4\u03b5\u03b4\u03c9\u03ba\u03ad\u03bd\u03b1\u03b9_ \u039c\u03b1\u03c1\u03b9\u03ac\u03bc\u03bd\u1fc3 \u03c4\u1f78\u03bd \u1f38\u03ac\u03ba\u03c9\u03b2\u03bf\u03bd \u03c4\u03bf\u1fe6\n\u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u03c4\u1f78\u03bd \u1f00\u03b4\u03b5\u03bb\u03c6\u03cc\u03bd, vii. 20 \u03c6\u03b1\u03c3\u1f76\u03bd \u03b5\u1f30\u03c1\u03b7\u03ba\u03ad\u03bd\u03b1\u03b9 \u039c\u03b1\u03c4\u03b8\u03af\u03b1\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c5\u03c2\n\u1f00\u03c0\u03bf\u03ba\u03c1\u03cd\u03c6\u03bf\u03c5\u03c2 \u03bf\u1f53\u03c2 \u1f24\u03ba\u03bf\u03c5\u03c3\u03b5 \u03c0\u03b1\u03c1\u1f70 \u03c4\u03bf\u1fe6 \u03c3\u03c9\u03c4\u1fc6\u03c1\u03bf\u03c2, Clem. Alex. _Strom._ vii. 17\n(p. 898) \u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1 \u1f41 \u0392\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b4\u03b7\u03c2, \u03ba\u1f02\u03bd \u0393\u03bb\u03b1\u03c5\u03ba\u03af\u03b1\u03bd \u1f10\u03c0\u03b9\u03b3\u03c1\u03ac\u03c6\u03b7\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b1\u03bb\u03bf\u03bd, \u1f61\u03c2\n\u03b1\u1f50\u03c7\u03bf\u1fe6\u03c3\u03b9\u03bd \u03b1\u1f50\u03c4\u03bf\u03af, \u03c4\u1f78\u03bd \u03a0\u03ad\u03c4\u03c1\u03bf\u03c5 \u1f11\u03c1\u03bc\u03b7\u03bd\u03ad\u03b1\u00b7 \u1f61\u03c3\u03b1\u03cd\u03c4\u03c9\u03c2 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u039f\u1f50\u03b1\u03bb\u03b5\u03bd\u03c4\u1fd6\u03bd\u03bf\u03bd \u0398\u03b5\u03bf\u03b4\u1fb6\n\u03b4\u03b9\u03b1\u03ba\u03b7\u03ba\u03bf\u03ad\u03bd\u03b1\u03b9 \u03c6\u03ad\u03c1\u03bf\u03c5\u03c3\u03b9\u03bd, \u03b3\u03bd\u03ce\u03c1\u03b9\u03bc\u03bf\u03c2 \u03b4\u1f72 \u03bf\u1f57\u03c4\u03bf\u03c2 \u1f10\u03b3\u03b5\u03b3\u03cc\u03bd\u03b5\u03b9 \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5. So too a\nlater mystic theology of the Jews, which had many affinities with the\nteaching of the Christianized Essenes at Coloss\u00e6, was self-designated\n_Kabbala_ or 'tradition', professing to have been handed down orally\nfrom the patriarchs. See the note on \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9, ii. 3.\n\n------------------------------------------------------------------------\n\nII. 8]\n\n\u03c4\u1ff6\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd, \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5,\n\n\n\u03c4\u1f70 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1] '_the rudiments_, _the elementary teaching_'; comp. ver.\n20. The same phrase occurs again Gal. iv. 3 (comp. ver. 9). As \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1\nsignifies primarily 'the letters of the alphabet', so as a secondary\nmeaning it denotes 'rudimentary instruction'. Accordingly it is\ncorrectly interpreted by Clement _Strom._ vi. 8 (p. 771) \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 ... \u03bf\u1f50\u03ba\n\u1f14\u03c4\u03b9 \u03c0\u03b1\u03bb\u03b9\u03bd\u03b4\u03c1\u03bf\u03bc\u03b5\u1fd6\u03bd \u1f00\u03be\u03b9\u03bf\u1fd6 \u1f10\u03c0\u1f76 \u03c4\u1f74\u03bd \u1f19\u03bb\u03bb\u03b7\u03bd\u03b9\u03ba\u1f74\u03bd \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03bd, \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5\n\u03c4\u03ac\u03c5\u03c4\u03b7\u03bd \u1f00\u03bb\u03bb\u03b7\u03b3\u03bf\u03c1\u1ff6\u03bd, \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03b9\u03c9\u03c4\u03b9\u03ba\u03ae\u03bd \u03c4\u03b9\u03bd\u03b1 \u03bf\u1f56\u03c3\u03b1\u03bd (i.e. elementary) \u03ba\u03b1\u1f76\n\u03c0\u03c1\u03bf\u03c0\u03b1\u03b9\u03b4\u03b5\u03af\u03b1\u03bd \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 (comp. _ib._ vi. 15, p. 799), and by Tertullian\n_adv. Marc._ v. 19 '_secundum elementa mundi_, non secundum c\u00e6lum et\nterram dicens, sed secundum literas seculares'. A large number of the\nfathers however explained the expression to refer to the heavenly bodies\n(called \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1), as marking the seasons, so that the observance of\n'festivals and new-moons and sabbaths' was a sort of bondage to them. It\nwould appear from Tertullian's language that Marcion also had so\ninterpreted the words. On this false interpretation see the note on Gal.\niv. 3. It is quite out of place here: for (1) The context suggests some\n_mode of instruction_, e.g. \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd here, and\n\u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c3\u03b8\u03b5 in ver. 20; (2) The keeping of days and seasons is quite\nsubordinate to other external observances. The rite of circumcision\n(ver. 11), and the distinction of meats (ver. 21) respectively, are\nplaced in close and immediate connexion with \u03c4\u1f70 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5 in\nthe two places where it occurs, whereas the observance of days and\nseasons (ver. 16) stands apart from either.\n\n\u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5] '_of the world_', that is, 'belonging to the sphere of\nmaterial and external things'. See the notes on Gal. iv. 3, vi. 14.\n\n'In Christ', so the Apostle seems to say, 'you have attained the liberty\nand the intelligence of manhood; do not submit yourselves again to a\nrudimentary discipline fit only for children (\u03c4\u1f70 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1). In Christ\nyou have been exalted into the sphere of the Spirit: do not plunge\nyourselves again into the atmosphere of material and sensuous things\n(\u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5).'\n\n------------------------------------------------------------------------\n\nII. 9, 10]\n\n\u03ba\u03b1\u1f76 \u03bf\u1f50 \u03ba\u03b1\u03c4\u1f70 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd\u00b7 ^9\u1f45\u03c4\u03b9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2\n\u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2, ^{10}\u03ba\u03b1\u1f76 \u1f10\u03c3\u03c4\u1f72 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7\n\n\n\u03bf\u1f50 \u03ba\u03b1\u03c4\u1f70 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd] '_not after Christ_'. This expression is wide in\nitself, and should be interpreted so as to supply the negative to both\nthe preceding clauses; 'Christ is neither the author nor the substance\nof their teaching: not the author, for they listen to human traditions\n(\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd); not the substance, for they replace\nHim by formal ordinances (\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5) and by angelic\nmediators'.\n\n9 sq. In explaining the true doctrine which is 'after Christ', St Paul\ncondemns the two false principles, which lay at the root of this\nheretical teaching; (1) The _theological_ error of substituting inferior\nand created beings angelic mediators for the divine Head Himself (vv. 9,\n10); and (2) The _practical_ error of insisting upon ritual and ascetic\nobservances, as the foundation of their moral teaching (vv. 11\u201314).\nTheir theological speculations and their ethical code alike were at\nfault. On the intimate connexion between these two errors, as springing\nout of a common root, the Gnostic dualism of these false teachers, see\nthe introduction, pp. 33 sq., 79, 87, 180 sq.\n\n\u1f45\u03c4\u03b9 \u03ba.\u03c4.\u03bb.] The Apostle justifies the foregoing charge that this\ndoctrine was not \u03ba\u03b1\u03c4\u1f70 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd; 'In Christ dwells the whole pleroma, the\nentire fulness of the Godhead, whereas they represent it to you as\ndispersed among several spiritual agencies. Christ is the one\nfountain-head of all spiritual life, whereas they teach you to seek it\nin communion with inferior creatures.' The same truths have been stated\nbefore (i. 14 sq.) more generally and they are now restated with direct\nand immediate reference to the heretical teaching.\n\n\u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6] '_has its fixed abode_'. On the force of this compound in\nrelation to the false teaching, see the note on i. 19.\n\n\u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1] '_all the plenitude_', 'the totality of the divine\npowers and attributes'. On this theological term see i. 19, and the\ndetached note at the end of the epistle.\n\n\u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2] '_of the Godhead_'. 'Non modo divin\u00e6 virtutes, sed ipsa\ndivina natura', writes Bengel. For the difference between \u03b8\u03ad\u03bf\u03c4\u03b7\u03c2\n'_deitas_', the essence, and \u03b8\u03b5\u03b9\u03cc\u03c4\u03b7\u03c2 '_divinitas_', the quality, see\nTrench _N. T. Syn._ \u00a7 ii. p. 6. The different force of the two words may\nbe seen by a comparison of two passages in Plutarch, _Mor._ p. 857 A\n\u03c0\u1fb6\u03c3\u03b9\u03bd \u0391\u1f30\u03b3\u03c5\u03c0\u03c4\u03af\u03bf\u03b9\u03c2 \u03b8\u03b5\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c0\u03bf\u03bb\u03bb\u1f74\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7\u03bd \u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03ae\u03c3\u03b1\u03c2 (where it\nmeans a divine inspiration or faculty, and where no one would have used\n\u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03b1), and _Mor._ 415 C \u1f10\u03ba \u03b4\u1f72 \u1f21\u03c1\u03ce\u03c9\u03bd \u03b5\u1f30\u03c2 \u03b4\u03b1\u03af\u03bc\u03bf\u03bd\u03b1\u03c2 \u03b1\u1f31 \u03b2\u03b5\u03bb\u03c4\u03af\u03bf\u03bd\u03b5\u03c2 \u03c8\u03c5\u03c7\u03b1\u1f76\n\u03c4\u1f74\u03bd \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u1f74\u03bd \u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03c5\u03c3\u03b9\u03bd, \u1f10\u03ba \u03b4\u1f72 \u03b4\u03b1\u03b9\u03bc\u03cc\u03bd\u03c9\u03bd \u1f40\u03bb\u03af\u03b3\u03b1\u03b9 \u03bc\u1f72\u03bd \u1f14\u03c4\u03b9 \u03c7\u03c1\u03cc\u03bd\u1ff3 \u03c0\u03bf\u03bb\u03bb\u1ff7 \u03b4\u03b9'\n\u1f00\u03c1\u03b5\u03c4\u1fc6\u03c2 \u03ba\u03b1\u03b8\u03b1\u03c1\u03b8\u03b5\u1fd6\u03c3\u03b1\u03b9 \u03c0\u03b1\u03bd\u03c4\u03ac\u03c0\u03b1\u03c3\u03b9 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03bc\u03b5\u03c4\u03ad\u03c3\u03c7\u03bf\u03bd (where \u03b8\u03b5\u03b9\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 would be\nquite out of place, because all \u03b4\u03b1\u03af\u03bc\u03bf\u03bd\u03b5\u03c2 without exception were \u03b8\u03b5\u1fd6\u03bf\u03b9,\nthough they only became \u03b8\u03b5\u03bf\u03af in rare instances and after long probation\nand discipline). In the New Testament the one word occurs here alone,\nthe other in Rom. i. 20 alone. So also \u03c4\u1f78 \u03b8\u03b5\u1fd6\u03bf\u03bd, a very favourite\nexpression in Greek philosophy, is found once only, in Acts xvii. 29,\nwhere it is used with singular propriety; for the Apostle is there\nmeeting the heathen philosophers on their own ground and arguing with\nthem in their own language. Elsewhere he instinctively avoids a term\nwhich tends to obscure the idea of a personal God. In the Latin\nversions, owing to the poverty of the language, both \u03b8\u03ad\u03bf\u03c4\u03b7\u03c2 and \u03b8\u03b5\u03af\u03bf\u03c4\u03b7\u03c2\nare translated by the same term _divinitas_; but this was felt to be\ninadequate, and the word _deitas_ was coined at a later date to\nrepresent \u03b8\u03ad\u03bf\u03c4\u03b7\u03c2: August. _de Civ. Dei_ vii. \u00a7 1, VII. p. 162 (quoted in\nTrench) 'Hanc divinitatem vel, ut sic dixerim, _deitatem_: nam et hoc\nverbo uti jam nostros non piget, ut de Gr\u00e6co expressius transferant id\nquod illi \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03b1 appellant etc.'\n\n\u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2] '_bodily-wise_', '_corporeally_', i.e. 'assuming a bodily\nform, becoming incarnate'. This is an addition to the previous statement\nin i. 19 \u1f10\u03bd \u03b1\u03c5\u03c4\u1ff7 \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9. The indwelling of\nthe pleroma refers to the Eternal Word, and not to the Incarnate Christ;\nbut \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 is added to show that the Word, in whom the pleroma thus\nhad its abode from all eternity, crowned His work by the Incarnation.\nThus while the main statement \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 of St\nPaul corresponds to the opening sentence \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f26\u03bd \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u0398\u1f72\u03bf\u03bd \u03ba\u03b1\u1f76\n\u0398\u03b5\u1f78\u03c2 \u1f26\u03bd \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 of St John, the subsidiary adverb \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 of St Paul\nhas its counterpart in the additional statement \u03ba\u03b1\u1f76 \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c3\u1f70\u03c1\u03be \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf\nof St John. All other meanings which have been assigned to \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2\nhere, as 'wholly' (Hieron. _in Is._ xi. 1 sq., IV. p. 156, 'nequaquam\nper partes, ut in ceteris sanctis'), or 'really' (Aug. _Epist._ cxlix,\nII. p. 513 'Ideo corporaliter dixit, quia illi umbratiliter\nseducebant'), or 'essentially' (Hilar. _de Trin._ viii. 54, II. p. 252\n'Dei ex Deo significat veritatem etc.', Cyril. Alex. in Theodoret. _Op._\nV. p. 34 \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9\u03bd, \u03bf\u1f50 \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u1ff6\u03c2, Isid. Pelus. _Ep._ iv. 166 \u1f00\u03bd\u03c4\u1f76 \u03c4\u03bf\u1fe6\n\u03bf\u1f50\u03c3\u03b9\u03c9\u03b4\u1ff6\u03c2), are unsupported by usage. Nor again can the body be\nunderstood of anything else but Christ's human body; as for instance of\nthe created World (Theod. Mops. in Rab. _Op._ VI. p. 522) or of the\nChurch (Anon. in Chrysost. _ad loc._). According to these two last\ninterpretations \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03bf\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 is taken to mean the Universe\n('universam naturam repletam ab eo') and the Church (\u03c4\u1f74\u03bd \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03bd\n\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03b7\u03bd \u1f51\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, see Ephes. i. 23) respectively,\nbecause either of these may be said to reside in Him, as the source of\nits life, and to stand to Him in the relation of the body to the head\n(\u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2). But these forced interpretations have nothing to recommend\nthem.\n\nSt Paul's language is carefully guarded. He does not say \u1f10\u03bd \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9, for\nthe Godhead cannot be confined to any limits of space; nor \u03c3\u03c9\u03bc\u03b1\u03c4\u03bf\u03b5\u03b9\u03b4\u1ff6\u03c2,\nfor this might suggest the unreality of Christ's human body; but\n\u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2, 'in bodily wise', 'with a bodily manifestation'. The relation\nof \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 to the clause which it qualifies will depend on the\ncircumstances of the case: comp. e.g. Plut. _Mor._ p. 424 E \u03bb\u03ad\u03b9\u03c0\u03b5\u03c4\u03b1\u03b9\n\u03c4\u03bf\u03af\u03bd\u03c5\u03bd \u03c4\u1f78 \u03bc\u03ad\u03c3\u03bf\u03bd \u03bf\u1f50 \u03c4\u03bf\u03c0\u03b9\u03ba\u1ff6\u03c2 \u1f00\u03bb\u03bb\u1f70 \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 \u03bb\u03ad\u03b3\u03b5\u03c3\u03b8\u03b1\u03b9, i.e. 'ratione\ncorporis habita', Athan. _Exp. Fid._ 4 (I. p. 81) \u1f11\u03ba\u03ac\u03c4\u03b5\u03c1\u03b1 \u03c4\u03bf\u03af\u03bd\u03c5\u03bd \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76\n\u03c4\u1f78 \u03ba\u03c4\u03af\u03c3\u03bc\u03b1 \u1fe5\u03b7\u03c4\u1f70 \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03b3\u03ad\u03b3\u03c1\u03b1\u03c0\u03c4\u03b1\u03b9, i.e. 'secundum\ncorpus', Ptolem. in Epiphan. _H\u00e6r._ xxxiii. 5 \u03ba\u03b1\u03c4\u1f70 \u03bc\u1f72\u03bd \u03c4\u1f78 \u03c6\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76\n\u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 \u1f10\u03ba\u03c4\u03b5\u03bb\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u1f00\u03bd\u1fc3\u03c1\u03ad\u03b8\u03b7.\n\n10. \u03ba\u03b1\u1f76 \u1f10\u03c3\u03c4\u1f72 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7] '_and ye are in Him_', where \u1f10\u03c3\u03c4\u1f72 should be\nseparated from the following \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9; comp. John xvii. 21, Acts\nxvii. 28. True life consists in union with Him, and not in dependence on\nany inferior being; comp. ver. 19 \u03bf\u1f50 \u03ba\u03c1\u03b1\u03c4\u1ff6\u03bd \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae\u03bd, \u1f10\u03be \u1f41\u1fe6 \u03ba.\u03c4.\u03bb.\n\n------------------------------------------------------------------------\n\nII. 10]\n\n\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9, \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74 \u03c0\u03ac\u03c3\u03b7\u03c2 \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03b1\u1f76\n\n\n\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9] '_being fulfilled_', with a direct reference to the\npreceding \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1; 'Your fulness comes from His fulness; His \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 is\ntransfused into you by virtue of your incorporation in Him'. So too John\ni. 16 \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f21\u03bc\u03b5\u1fd6\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bb\u03ac\u03b2\u03bf\u03bc\u03b5\u03bd, Ephes. iii. 19 \u1f35\u03bd\u03b1\n\u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03c4\u03b5 \u03b5\u1f30\u03c2 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, iv. 13 \u03b5\u1f30\u03c2 \u03bc\u03ad\u03c4\u03c1\u03bf\u03bd \u1f21\u03bb\u03b9\u03ba\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6\n\u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, comp. Ign. _Ephes._ init. \u03c4\u1fc7 \u03b5\u1f50\u03bb\u03bf\u03b3\u03b7\u03bc\u03ad\u03bd\u1fc3 \u1f10\u03bd\n\u03bc\u03b5\u03b3\u03ad\u03b8\u03b5\u03b9 \u0398\u03b5\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9. Hence also the Church, as ideally\nregarded, is called the \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 of Christ, because all His graces and\nenergies are communicated to her; Ephes. i. 23 \u1f25\u03c4\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6,\n\u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5.\n\n\u1f45\u03c2] For the various reading \u1f45 see the detached note. It was perhaps a\ncorrection made on the false supposition that \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 referred to the\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1. At all events it must be regarded as an impossible reading; for\nthe image would be altogether confused and lost, if the \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 were\nrepresented as the head. And again \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74 is persistently said\nelsewhere of Christ; i. 18, ii. 19, Ephes. i. 22, iv. 15, v. 23. Hilary\n_de Trin._ ix. 8 (II. p. 264) explains the \u1f45 as referring to the whole\nsentence \u03c4\u1f78 \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2, but this also is an\ninconceivable sense. Again it has been suggested that \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd (like\n\u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9\u03bd) may be taken as equivalent to _scilicet_ (comp. Clem. _Hom._\nviii. 22); but this would require \u03c4\u1fc7 \u03ba\u03b5\u03c6\u03b1\u03bb\u1fc7, even if it were otherwise\nadmissible here.\n\n------------------------------------------------------------------------\n\nII. 11]\n\n\u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2\u00b7 ^{11}\u1f10\u03bd \u1fa7 \u03ba\u03b1\u1f76 \u03c0\u03b5\u03c1\u03b9\u03b5\u03c4\u03bc\u03ae\u03b8\u03b7\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc7 \u1f00\u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03b9\u03ae\u03c4\u1ff3,\n\n\n\u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74] The image expresses much more than the idea of sovereignty:\nthe head is also the centre of vital force, the source of all energy and\nlife: see the note on ver. 19.\n\n\u03c0\u03ac\u03c3\u03b7\u03c2 \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba.\u03c4.\u03bb.] '_of every principality and power_', and therefore\nof those angelic beings whom the false teachers adopted as mediators,\nthus transferring to the inferior members the allegiance due to the\nHead: comp. ver. 18 sq. For \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2, see the note on i. 16.\n\n11. The previous verses have dealt with the theological tenets of the\nfalse teachers. The Apostle now turns to their practical errors; 'You do\nnot need the circumcision of the flesh; for you have received the\ncircumcision of the heart. The distinguishing features of this higher\ncircumcision are threefold. (1) It is not external but inward, not made\nwith hands but wrought by the Spirit. (2) It divests not of a part only\nof the flesh, but of the whole body of carnal affections. (3) It is the\ncircumcision not of Moses or of the patriarchs, but of Christ'. Thus it\nis distinguished, as regards _first_ its character, _secondly_ its\nextent, and _thirdly_ its author.\n\n\u03c0\u03b5\u03c1\u03b9\u03b5\u03c4\u03bc\u03ae\u03b8\u03b7\u03c4\u03b5] The moment at which this is conceived as taking place is\ndefined by the other aorists, \u03c3\u03c5\u03bd\u03c4\u03b1\u03c6\u03ad\u03bd\u03c4\u03b5\u03c2, \u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5, etc., as the\ntime of their baptism, when they 'put on Christ'.\n\n\u1f00\u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03b9\u03ae\u03c4\u1ff3] i.e. 'immaterial', 'spiritual', as Mark xiv. 58, 2 Cor. v.\n1. So \u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03af\u03b7\u03c4\u03bf\u03c2, which is used in the N. T. of material temples and\ntheir furniture (Acts vii. 48, xvii. 24, Heb. ix. 11, 24, comp. Mark\n_l.c._), and of the material circumcision (Ephes. ii. 11 \u03c4\u1fc6\u03c2 \u03bb\u03b5\u03b3\u03bf\u03bc\u03ad\u03bd\u03b7\u03c2\n\u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc6\u03c2 \u1f10\u03bd \u03c3\u03b1\u03c1\u03ba\u1f76 \u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03b9\u03ae\u03c4\u03bf\u03c5). In the LXX \u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03af\u03b7\u03c4\u03b1 occurs\nexclusively as a rendering of idols (\u05d0\u05dc\u05d9\u05dc\u05dd, e.g. Lev. xxvi. 1, Is. ii.\n18, etc.), false gods (\u05d0\u05dc\u05d4\u05d9\u05dd Is. xxi. 9, where perhaps they read\n\u05d0\u05dc\u05d9\u05dc\u05d9\u05dd), or images (\u05d7\u05de\u05e0\u05d9\u05dd Lev. xxvi. 30), except in one passage, Is.\nxvi. 12, where it is applied to an idol's sanctuary. Owing to this\nassociation of the word the application which we find in the New\nTestament would sound much more depreciatory to Jewish ears than it does\nto our own; e.g. \u1f10\u03bd \u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03b9\u03ae\u03c4\u03bf\u03b9\u03c2 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6 in St Stephen's speech, where\nthe force of the passage is broken in the received text by the\ninterpolation of \u03bd\u03b1\u03bf\u1fd6\u03c2.\n\nFor illustrations of the typical significance of circumcision, as a\nsymbol of purity, see the note on Phil. iii. 3.\n\n\u1f10\u03bd \u03c4\u1fc7 \u03ba.\u03c4.\u03bb.] The words are chosen to express the _completeness_ of the\nspiritual change. (1) It is not an \u1f14\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2 nor an \u1f00\u03c0\u03cc\u03b4\u03c5\u03c3\u03b9\u03c2, but an\n\u1f00\u03c0\u03ad\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2. The word \u1f00\u03c0\u03ad\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2 is extremely rare, and no earlier\ninstances of it are produced; see the note on ver. 15 \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2. (2)\nIt is not a single member but the whole body, which is thus cast aside;\nsee the next note. Thus the idea of completeness is brought out both in\nthe energy of the action and in the extent of its operation, as in iii.\n9 \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f78\u03bd \u03c0\u03b1\u03bb\u03b1\u03b9\u1f78\u03bd _\u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd_.\n\n------------------------------------------------------------------------\n\nII. 12]\n\n\u1f10\u03bd \u03c4\u1fc7 \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03c3\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, \u1f10\u03bd \u03c4\u1fc7 \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc7 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6,\n^{12}\u03c3\u03c5\u03bd\u03c4\u03b1\u03c6\u03ad\u03bd\u03c4\u03b5\u03c2 \u03b1\u1f50\u03c4\u1ff7 \u1f10\u03bd\n\n\n\u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba.\u03c4.\u03bb.] '_the whole body_ which consists _of the flesh_',\ni.e. 'the body with all its corrupt and carnal affections'; as iii. 5\n\u03bd\u03b5\u03ba\u03c1\u03ce\u03c3\u03b1\u03c4\u03b5 \u03bf\u1f56\u03bd _\u03c4\u1f70 \u03bc\u03ad\u03bb\u03b7_. For illustrations of the expression see\nRom. vi. 6 \u1f35\u03bd\u03b1 \u03ba\u03b1\u03c4\u03b1\u03c1\u03b3\u03b7\u03b8\u1fc7 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u03b1\u03c2, vii. 24 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5, Phil. iii. 21 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2 \u1f21\u03bc\u1ff6\u03bd. Thus \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1\n\u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 here means 'the fleshly body' and not 'the entire mass of the\nflesh'; but the contrast between the whole and the part still remains.\nIn i. 22 the same expression \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 occurs, but with a\ndifferent emphasis and meaning: see the note there.\n\nThe words \u03c4\u1ff6\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03b9\u1ff6\u03bd, inserted between \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 and \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 in\nthe received text, are clearly a gloss, and must be omitted with the\nvast majority of ancient authorities.\n\n12. Baptism is the grave of the old man, and the birth of the new. As he\nsinks beneath the baptismal waters, the believer buries there all his\ncorrupt affections and past sins; as he emerges thence, he rises\nregenerate, quickened to new hopes and a new life. This it is, because\nit is not only the crowning act of his own faith but also the seal of\nGod's adoption and the earnest of God's Spirit. Thus baptism is an image\nof his participation both in the death and in the resurrection of\nChrist. See _Apost. Const._ iii. 17 \u1f21 \u03ba\u03b1\u03c4\u03ac\u03b4\u03c5\u03c3\u03b9\u03c2 \u03c4\u1f78 \u03c3\u03c5\u03bd\u03b1\u03c0\u03bf\u03b8\u03b1\u03bd\u03b5\u1fd6\u03bd, \u1f21\n\u1f00\u03bd\u03ac\u03b4\u03c5\u03c3\u03b9\u03c2 \u03c4\u1f78 \u03c3\u03c5\u03bd\u03b1\u03bd\u03b1\u03c3\u03c4\u1fc6\u03bd\u03b1\u03b9. For this twofold image, as it presents itself\nto St Paul, see especially Rom. vi. 3 sq.\n\n\u1f10\u03bd \u03c4\u1ff7 \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff7] '_in the act of baptism_'. A distinction seems to be\nobserved elsewhere in the New Testament between \u03b2\u03ac\u03c0\u03c4\u03b9\u03c3\u03bc\u03b1 'baptism'\nproperly so called, and \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2 'lustration' or 'washing' of divers\nkinds, e.g. of vessels (Mark vii. 4, [8,] Heb. ix. 10). Even Heb. vi. 2\n\u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff6\u03bd \u03b4\u03b9\u03b4\u03b1\u03c7\u1fc6\u03c2, which at first sight might seem to be an exception to\nthis rule, is perhaps not really so (Bleek _ad loc._). Here however,\nwhere the various readings \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff7 and \u03b2\u03b1\u03c0\u03c4\u03af\u03c3\u03bc\u03b1\u03c4\u03b9 appear in\ncompetition, the preference ought probably to be given to \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff7 as\nbeing highly supported in itself (see the detached note on various\nreadings) and as the less usual word in this sense. There is no _a\npriori_ reason why St Paul should not have used \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2 with this\nmeaning, for it is so found in Josephus _Ant._ xviii. 5. 2 \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff7\n\u03c3\u03c5\u03bd\u03b9\u03ad\u03bd\u03b1\u03b9 (of John the Baptist). Doubtless the form \u03b2\u03ac\u03c0\u03c4\u03b9\u03c3\u03bc\u03b1 was more\nappropriate to describe the one final and complete act of Christian\nbaptism, and it very soon obtained exclusive possession of the ground in\nGreek; but in St Paul's age the other form \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2 may not yet have\nbeen banished. In the Latin Version _baptisma_ and _baptismus_ are used\nindiscriminately: and this is the case also with the Latin fathers. The\nsubstantive 'baptism' occurs so rarely in any sense in St Paul (only\nRom. vi. 4, Eph. iv. 5, besides this passage), or indeed elsewhere in\nthe N. T. of Christian baptism (only in 1 Pet. iii. 21), that we have\nnot sufficient data for a sound induction. So far as the two words have\nany inherent difference of meaning, \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2 denotes rather the act in\nprocess and \u03b2\u03ac\u03c0\u03c4\u03b9\u03c3\u03bc\u03b1 the result.\n\n------------------------------------------------------------------------\n\nII. 12]\n\n\u03c4\u1ff7 \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff7, \u1f10\u03bd \u1fa7 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03c4\u1fc6\u03c2 \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6\n\u03c4\u03bf\u1fe6 \u1f10\u03b3\u03b5\u03af\u03c1\u03b1\u03bd\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f10\u03ba [\u03c4\u1ff6\u03bd]\n\n                           12. \u03c4\u1ff7 \u03b2\u03b1\u03c0\u03c4\u03af\u03c3\u03bc\u03b1\u03c4\u03b9.\n\n\n\u1f10\u03bd \u1fa7] i.e. \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u1ff7. Others would understand \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 for the sake of\nthe parallelism with ver. 11 \u1f10\u03bd h\u1ff7 \u03ba\u03b1\u1f76 ... \u03b5\u03bd \u1fa7 \u03ba\u03b1\u03af. But this\nparallelism is not suggested by the sense: while on the other hand\nthere is obviously a very close connexion between \u03c3\u03c5\u03bd\u03c4\u03b1\u03c6\u03ad\u03bd\u03c4\u03b5\u03c2 and\n\u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5 as the two complementary aspects of baptism; comp. Rom.\nvi. 4 sq. _\u03c3\u03c5\u03bd\u03b5\u03c4\u03ac\u03c6\u03b7\u03bc\u03b5\u03bd_ \u03b1\u1f50\u03c4\u1ff7 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u03b2\u03b1\u03c0\u03c4\u03af\u03c3\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f35\u03bd\u03b1 \u1f65\u03c3\u03c0\u03b5\u03c1\n_\u1f10\u03b3\u03ad\u03c1\u03b8\u03b7_ \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 ... \u1f45\u03c5\u03c4\u03c9\u03c2 \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2 ... \u03b5\u1f30 \u03b3\u1f70\u03c1 \u03c3\u03cd\u03bc\u03c6\u03c5\u03c4\u03bf\u03b9\n\u03b3\u03b5\u03b3\u03cc\u03bd\u03b1\u03bc\u03b5\u03bd \u03c4\u1ff7 \u1f41\u03bc\u03bf\u03af\u03c9\u03bc\u03b1\u03c4\u03b9 _\u03c4\u03bf\u1fe6 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5_ \u03b1\u1f50\u03c4\u03bf\u1fe6, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 _\u03c4\u1fc6\u03c2\n\u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03c9\u03c2_ \u1f10\u03c3\u03cc\u03bc\u03b5\u03b8\u03b1, 2 Tim. ii. 11 \u03b5\u1f30 \u03b3\u1f70\u03c1 _\u03c3\u03c5\u03bd\u03b1\u03c0\u03b5\u03b8\u03ac\u03bd\u03bf\u03bc\u03b5\u03bd_, \u03ba\u03b1\u1f76\n_\u03c3\u03c5\u03bd\u03b6\u03ae\u03c3\u03bf\u03bc\u03b5\u03bd_. In fact the idea of \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 must be reserved for\n\u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5 where it is wanted, '_ye were raised together with Him_'.\n\n\u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03ba.\u03c4.\u03bb.] '_through your faith in the operation_,'\n\u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u03af\u03b1\u03c2 being the objective genitive. So St Chrysostom, \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u1f45\u03bb\u03bf\u03bd\n\u1f10\u03c3\u03c4\u03af\u03bd\u00b7 \u1f10\u03c0\u03b9\u03c3\u03c4\u03b5\u03cd\u03c3\u03b1\u03c4\u03b5 \u1f45\u03c4\u03b9 \u03b4\u03cd\u03bd\u03b1\u03c4\u03b1\u03b9 \u1f41 \u0398\u03b5\u1f78\u03c2 \u1f10\u03b3\u03b5\u1fd6\u03c1\u03b1\u03b9, \u03ba\u03b1\u1f76 \u03bf\u1f55\u03c4\u03c9\u03c2 \u1f20\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5. Only\nby a belief in the resurrection are the benefits of the resurrection\nobtained, because only so are its moral effects produced. Hence St Paul\nprays that he may 'know the _power_ of Christ's resurrection' (Phil.\niii. 10). Hence too he makes this the cardinal article in the\nChristian's creed, 'If thou ... believest in thy heart that God raised\nHim from the dead, thou shalt be saved' (Rom. x. 9). For the influence\nof Christ's resurrection on the moral and spiritual being, see the note\non Phil. _l.c._ Others take \u03c4\u1fc6\u03c2 \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u03af\u03b1\u03c2 as the subjective genitive,\n'faith which comes from the operation etc.', arguing from a mistaken\ninterpretation of the parallel passage Ephes. i. 19 (where \u03ba\u03b1\u03c4\u1f70 \u03c4\u1fc6\u03bd\n\u1f10\u03bd\u03ad\u03c1\u03b3\u03b5\u03b9\u03b1\u03bd should be connected, not with \u03c4\u03bf\u1f7a\u03c2 \u03c0\u03b9\u03c3\u03c4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03c2, but with \u03c4\u03af \u03c4\u1f78\n\u1f51\u03c0\u03b5\u03c1\u03b2\u03ac\u03bb\u03bb\u03bf\u03bd \u03bc\u03ad\u03b3\u03b5\u03b8\u03bf\u03c2 \u03ba.\u03c4.\u03bb.). The former explanation however yields a\nbetter sense, and the genitive after \u03c0\u03af\u03c3\u03c4\u03b9\u03c2 far more commonly describes\nthe object than the source of the faith, e.g. Rom. iii. 22, 26, Gal.\niii. 22, Ephes. iii. 12, Phil. i. 27, iii. 9, 2 Thess. ii. 13.\n\n13. In the sentence which follows it seems necessary to assume a change\nof subject. There can be little doubt that \u1f41 \u0398\u03b5\u1f78\u03c2 is the nominative to\n\u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd: for (1) The parallel passage Ephes. ii. 4, 5 directly\nsuggests this. (2) This is uniformly St Paul's mode of speaking\nelsewhere. It is always God who \u1f10\u03b3\u03ad\u03b9\u03c1\u03b5\u03b9, \u03c3\u03c5\u03bd\u03b5\u03b3\u03ad\u03b9\u03c1\u03b5\u03b9, \u03b6\u03c9\u03bf\u03c0\u03bf\u03b9\u03b5\u1fd6,\n\u03c3\u03c5\u03bd\u03b6\u03c9\u03bf\u03c0\u03bf\u03b9\u03b5\u1fd6, etc., with or in or through Christ. (3) Though it might be\npossible to assign \u03c3\u1f7a\u03bd \u03b1\u1f50\u03c4\u1ff7 to the subject of \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd (see the\nnote on i. 20), yet a reference to some other person is more natural.\nThese reasons seem to decide the subject of \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd. But at the\nsame time it appears quite impossible to continue the same subject, \u1f41\n\u0398\u03b5\u03cc\u03c2, to the end of the sentence. No grammatical meaning can be assigned\nto \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2, by which it could be understood of God the Father. We\nmust suppose therefore that a new subject, \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2, is introduced\nmeanwhile, either with \u1f26\u03c1\u03ba\u03b5\u03bd or with \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 itself; and of the\ntwo the former seems the easier point of transition. For a similar\ninstance of abrupt transition, which is the more natural owing to the\nintimate connexion of the work of the Son with the work of the Father,\nsee e.g. i. 17 sq.\n\n\u03ba\u03b1\u1f76 \u1f51\u03bc\u1fb6\u03c2] i.e. 'you Gentiles'. This will appear from a study of the\nparallel passages iii. 7, 8, Ephes. i. 13, ii. 1 sq., 11, 13, 17, 22,\niii. 2, iv. 17; see the notes on Ephes. i. 13, and on \u03c4\u1fc7 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u1fb3 just\nbelow.\n\n------------------------------------------------------------------------\n\nII. 13]\n\n\u03bd\u03b5\u03ba\u03c1\u1ff6\u03bd\u00b7 ^{13}\u03ba\u03b1\u1f76 \u1f51\u03bc\u1fb6\u03c2 \u03bd\u03b5\u03ba\u03c1\u03bf\u1f7a\u03c2 \u1f44\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c0\u03c4\u03ce\u03bc\u03b1\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1fc7 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u1fb3\n\u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u1f51\u03bc\u1ff6\u03bd, \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd\n\n\n\u03c4\u03bf\u1fd6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c0\u03c4\u03ce\u03bc\u03b1\u03c3\u03b9\u03bd \u03ba.\u03c4.\u03bb.] '_by reason of your transgressions etc._' The\n\u03c0\u03b1\u03c1\u03b1\u03c0\u03c4\u03ce\u03bc\u03b1\u03c4\u03b1 are the actual definite transgressions, while the \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u03b1\n\u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 is the impure carnal disposition which prompts to them. For\nthe dative comp. Ephes. ii. 1, 5, where the same expression occurs; see\nWiner _Gramm._ \u00a7 xxxi. p. 270. On the other hand in Rom. vi. 11 \u03bd\u03b5\u03ba\u03c1\u03bf\u1f7a\u03c2\n\u03bc\u1f72\u03bd \u03c4\u1fc7 \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u1fb3, \u03b6\u1ff6\u03bd\u03c4\u03b1\u03c2 \u03b4\u1f72 \u03c4\u1ff7 \u0398\u03b5\u1ff7, the dative has a wholly different\nmeaning, as the context shows. The \u1f10\u03bd of the received text, though\nhighly supported, is doubtless an interpolation for the sake of\ngrammatical clearness.\n\n\u03c4\u1fc7 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u1fb3 \u03ba.\u03c4.\u03bb.] The external fact is here mentioned, not for its\nown sake but for its symbolical meaning. The outward uncircumcision of\nthe Gentiles is a type of their unchastened carnal mind. In other words,\nthough the literal meaning is not excluded, the spiritual reference is\nmost prominent, as appears from ver. 11 \u1f10\u03bd \u03c4\u1fc6\u03b9 \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03c3\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2.\nHence Theodore's comment, \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u03b1\u03bd (\u1f10\u03ba\u03ac\u03bb\u03b5\u03c3\u03b5\u03bd) \u03c4\u1f78 \u03c0\u03b5\u03c1\u03b9\u03ba\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u1f14\u03c4\u03b9 \u03c4\u1f74\u03bd\n\u03b8\u03bd\u03b7\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1. At the same time the choice of the expression shows that the\nColossian converts addressed by St Paul were mainly Gentiles.\n\n\u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd] It has been questioned whether the life here spoken of\nshould be understood in a spiritual sense of the regeneration of the\nmoral being, or in a literal sense of the future life of immortality\nregarded as conferred on the Christian potentially now, though only to\nbe realised hereafter. But is not such an issue altogether superfluous?\nIs there any reason to think that St Paul would have separated these two\nideas of life? To him the future glorified life is only the continuation\nof the present moral and spiritual life. The two are the same in\nessence, however the accidents may differ. Moral and spiritual\nregeneration is salvation, is life.\n\n\u1f51\u03bc\u1fb6\u03c2] The pronoun is repeated for the sake of emphasis. The omission in\nsome good copies is doubly explained; (1) By the desire to simplify the\ngrammar; (2) By the wish to relieve the awkwardness of the close\nproximity between \u1f51\u03bc\u1fb6\u03c2 and \u1f21\u03bc\u1fd6\u03bd. This latter consideration has led a few\ngood authorities to substitute \u1f21\u03bc\u1fb6\u03c2 for \u1f51\u03bc\u1fb6\u03c2, and others to substitute\n\u1f51\u03bc\u1fd6\u03bd for \u1f21\u03bc\u1fd6\u03bd. For instances of those emphatic repetitions in St Paul\nsee the note on i. 20 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6.\n\n\u03c3\u1f7a\u03bd \u03b1\u1f50\u03c4\u1ff3] 'with Christ', as in Ephes. ii. 5 \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7. On\nthe inadmissibility of the reading \u1f01\u03c5\u03c4\u1ff7 see the note on \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd i. 20.\n\n\u03c7\u03b1\u03c1\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2] '_having forgiven_', as in Luke vii. 42 sq., 2 Cor. ii. 7,\n10, xii. 13, Ephes. iv. 32; see also the note on iii. 13 below. The idea\nof sin as a debt incurred to God (Matt. vi. 12 \u03c4\u1f70 \u1f40\u03c6\u03b5\u03b9\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 \u1f21\u03bc\u1ff6\u03bd, comp.\nLuke xi. 4) underlies this expression, as it does also the commoner term\nfor pardon, \u1f04\u03c6\u03b5\u03c3\u03b9\u03c2 'remission'. The image is carried out in the\ncancelled bond, ver. 14.\n\n\u1f21\u03bc\u1fd6\u03bd] The person is changed; 'not to you Gentiles only, but to us all\nalike'. St Paul is eager to claim his share in the transgression, that\nhe may claim it also in the forgiveness. For other examples of the\nchange from the second to the first person, see i. 10\u201313, iii. 3, 4,\nEphes. ii. 2, 3, 13, 14, iv. 31, 32, v. 2 (the correct reading), 1\nThess. v. 5, where the motive of the change is similar. See also Gal.\niii. 25, 26, iv. 5, 6, where there is the converse transition.\n\n------------------------------------------------------------------------\n\nII. 14]\n\n\u1f51\u03bc\u1fb6\u03c2 \u03c3\u1f7a\u03bd \u03b1\u1f50\u03c4\u1ff7, \u03c7\u03b1\u03c1\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f21\u03bc\u1fd6\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1f70 \u03c0\u03b1\u03c1\u03b1\u03c0\u03c4\u03ce\u03bc\u03b1\u03c4\u03b1, ^{14}\u1f10\u03be\u03b1\u03bb\u03b5\u03af\u03c8\u03b1\u03c2 \u03c4\u1f78\n\u03ba\u03b1\u03b8' \u1f21\u03bc\u1ff6\u03bd \u03c7\u03b5\u03b9\u03c1\u03cc\u03b3\u03c1\u03b1\u03c6\u03bf\u03bd \u03c4\u03bf\u1fd6\u03c2\n\n\n14. \u1f10\u03be\u03b1\u03bb\u03ad\u03b9\u03c8\u03b1\u03c2] '_having cancelled_'. The word \u1f10\u03be\u03b1\u03bb\u03ad\u03b9\u03c6\u03b5\u03b9\u03bd, like\n\u03b4\u03b9\u03b1\u03b3\u03c1\u03ac\u03c6\u03b5\u03b9\u03bd, signifying 'to blot out, to erase', is commonly opposed to\n\u1f10\u03b3\u03b3\u03c1\u03ac\u03c6\u03b5\u03b9\u03bd 'to enter a name, etc.'; e.g. Arist. _Pax_ 1181, Lysias _c.\nNicom._ p. 183, Plato _Resp._ vi. p. 501 B. More especially is it so\nused in reference to an _item_ in an account, e.g. Demosth. _c.\nAristog._ i. p. 791 \u1f10\u03b3\u03b3\u03c1\u03ac\u03c6\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03bf\u1f31 \u1f40\u03c6\u03bb\u03b9\u03c3\u03ba\u03ac\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2 ... \u1f10\u03be\u03b1\u03bb\u03ae\u03bb\u03b9\u03c0\u03c4\u03b1\u03b9\n\u03c4\u1f78 \u1f44\u03c6\u03bb\u03b7\u03bc\u03b1.\n\n\u03c4\u1f78 \u03ba\u03b1\u03b8' \u1f21\u03bc\u1ff6\u03bd \u03ba.\u03c4.\u03bb.] '_the bond standing against us_'. The word\n\u03c7\u03b5\u03b9\u03c1\u03cc\u03b3\u03c1\u03b1\u03c6\u03bf\u03bd, which means properly an autograph of any kind, is used\nalmost exclusively for a note of hand, a bond or obligation, as having\nthe 'sign-manual' of the debtor or contractor: e.g. Tobit v. 3 (comp.\nix. 5) \u1f14\u03b4\u03c9\u03ba\u03b5\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c4\u1f78 \u03c7\u03b5\u03b9\u03c1\u03cc\u03b3\u03c1\u03b1\u03c6\u03bf\u03bd, Plut. _Mor._ p. 829 A \u03c4\u1ff6\u03bd \u03c7\u03b5\u03b9\u03c1\u03bf\u03b3\u03c1\u03ac\u03c6\u03c9\u03bd\n\u03ba\u03b1\u1f76 \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03b1\u03af\u03c9\u03bd. It is more common in Latin than in Greek, e.g. Cic.\n_Fam._ vii. 18 'Misi cautionem chirographi mei', Juv. _Sat._ xvi. 41\n'Debitor aut sumptos pergit non reddere nummos, vana supervacui dicens\nchirographa ligni' (comp. xiii. 137). Hence chirographum,\nchirographarius, are frequent terms in the Roman law-books; see Hesse\n_Handlexicon zu den Quellen des r\u00f6mischen Rechts_ s.v. p. 74.\n\nIn the case before us the Jewish people might be said to have signed the\ncontract when they bound themselves by a curse to observe all the\nenactments of the law (Deut. xxvii. 14\u201326; comp. Exod. xxiv. 3); and the\nprimary reference would be to them. But \u1f21\u03bc\u1fd6\u03bd, \u1f21\u03bc\u1ff6\u03bd, seem to include\nGentiles as well as Jews, so that a wider reference must be given to the\nexpression. The \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1 therefore, though referring primarily to the\nMosaic ordinances, will include all forms of positive decrees in which\nmoral or social principles are embodied or religious duties defined; and\nthe 'bond' is the moral assent of the conscience, which (as it were)\nsigns and seals the obligation. The Gentiles, though 'not having a law,\nare a law to themselves', \u1f45\u03b9\u03c4\u03b9\u03bd\u03b5\u03c2 \u1f10\u03bd\u03b4\u03b5\u03af\u03ba\u03bd\u03c5\u03bd\u03c4\u03b1\u03b9 \u03c4\u1f78 \u1f14\u03c1\u03b3\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03bd\u03cc\u03bc\u03bf\u03c5\n_\u03b3\u03c1\u03b1\u03c0\u03c4\u1f78\u03bd_ \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd, _\u03c3\u03c5\u03bc\u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03bf\u03cd\u03c3\u03b7\u03c2_ \u03b1\u1f50\u03c4\u03c9\u03bd \u03c4\u1fc6\u03c2\n\u03c3\u03c5\u03bd\u03b5\u03b9\u03b4\u03ae\u03c3\u03b5\u03c9\u03c2, Rom. ii. 14, 15. See the notes on Gal. ii. 19, iv. 11.\nComp. Orig. _Hom. in Gen._ xiii. 4 (II. p. 96).\n\n\u03c4\u03bf\u1fd6\u03c2 \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd] '_consisting in ordinances_': comp. Ephes. ii. 15 \u03c4\u1f78\u03bd\n\u03bd\u03cc\u03bc\u03bf\u03bd \u03c4\u1ff6\u03bd \u1f10\u03bd\u03c4\u03bf\u03bb\u1ff6\u03bd \u1f10\u03bd \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd. The word \u03b4\u03cc\u03b3\u03bc\u03b1 is here used in its proper\nsense of a 'decree', 'ordinance', corresponding to \u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c3\u03b8\u03b5 below,\nver. 20. This is its only sense in the N. T.; e.g. Luke ii. 1, Acts\nxvii. 7, of the Emperor's decrees; Acts xvi. 4 of the Apostolic\nordinances. Here it refers especially to the Mosaic law, as in Joseph.\n_Ant._ xv. 5. 3 \u03c4\u1f70 \u03ba\u03ac\u03bb\u03bb\u03b9\u03c3\u03c4\u03b1 \u03c4\u1ff6\u03bd \u03b4\u03bf\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f41\u03c3\u03b9\u03ce\u03c4\u03b1\u03c4\u03b1 \u03c4\u1ff6\u03bd \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2\n\u03bd\u03cc\u03bc\u03bf\u03b9\u03c2, Philo _Leg. All._ i. 16 (I. p. 54) \u03b4\u03b9\u03b1\u03c4\u03ae\u03c1\u03b7\u03c3\u03b9\u03c2 \u03c4\u1ff6\u03bd \u1f01\u03b3\u03af\u03c9\u03bd\n\u03b4\u03bf\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd, 3 Macc. i. 3 \u03c4\u1ff6\u03bd \u03c0\u03b1\u03c4\u03c1\u03af\u03c9\u03bd \u03b4\u03bf\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd. Comp. Iren. _Fragm._ 38 (p.\n855 Stieren) where, immediately after a reference to our text, \u03c4\u03bf\u1fd6\u03c2 \u03c4\u1ff6\u03bd\n\u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9 \u03c0\u03c1\u03bf\u03c3\u03ad\u03c1\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9 is opposed to \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2 \u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u03b5\u1fd6\u03bd. In\nthe parallel passage, Ephes. ii. 15, this is the exclusive reference;\nbut here (for reasons explained in the last note) it seems best to give\nthe term a secondary and more extensive application.\n\nThe dative is perhaps best explained as governed by the idea of\n\u03b3\u03b5\u03b3\u03c1\u03b1\u03bc\u03bc\u03ad\u03bd\u03bf\u03bd involved in \u03c7\u03b5\u03b9\u03c1\u03cc\u03b3\u03c1\u03b1\u03c6\u03bf\u03bd (comp. Plat. _Ep._ vii. p. 243 A \u03c4\u1f70\n\u03b3\u03b5\u03b3\u03c1\u03b1\u03bc\u03bc\u03ad\u03bd\u03b1 \u03c4\u03cd\u03c0\u03bf\u03b9\u03c2); as in 1 Tim. ii. 6 \u03c4\u1f78 \u03bc\u03b1\u03c1\u03c4\u03cd\u03c1\u03b9\u03bf\u03bd \u03ba\u03b1\u03b9\u03c1\u03bf\u1fd6\u03c2 \u1f30\u03b4\u03af\u03bf\u03b9\u03c2,\nwhere \u03ba\u03b1\u03b9\u03c1\u03bf\u1fd6\u03c2 depends on an implied \u03bc\u03b5\u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd. Otherwise it is\ntaken as closely connected with \u03ba\u03b1\u03b8' \u1f21\u03bc\u1ff6\u03bd, 'the bond which was in force\nagainst us by reason of the ordinances': see Winer \u00a7 xxxi. p. 273, A.\nButtmann p. 80. Possibly an \u1f10\u03bd has dropped out of the text before \u03c4\u03bf\u1fd6\u03c2\n\u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd, owing to the similar ending ==\u03c7\u03b5\u03b9\u03c1\u03bf\u03b3\u03c1\u03b1\u03c6\u03bf\u03bd\u03b5\u03bd== (comp. Ephes.\nii. 15); but, if so, the omission must date from the earliest age, since\nno existing authorities exhibit any traces of such a reading; see the\nnote on ver. 18 \u1f03 \u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd, and comp. Phil. ii. 1 \u03b5\u1f34 \u03c4\u03b9\u03c2 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1.\n\nA wholly different interpretation however prevails universally among\nGreek commentators both here and in Ephes. ii. 15. They take \u03c4\u03bf\u1fd6\u03c2\n\u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd, \u1f10\u03bd \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd, to mean the 'doctrines or precepts of the\nGospel', and so to describe the instrument by which the abrogation of\nthe law was effected. So Chrysostom, Severianus, Theodore of Mopsuestia,\nand Theodoret, followed by the later commentators [OE]cumenius and\nTheophylact. Strangely enough they do not allude to the correct\ninterpretation; nor (with the exception of the passage ascribed to\nIren\u00e6us which is quoted above) have I found any distinct traces of it in\nany Greek father. The grammatical difficulty would be taken to favour\nthis interpretation, which moreover was characteristic of the age when\nthe battle of creeds was fought. But it has been universally abandoned\nby modern interpreters, as plainly inappropriate to the context and also\nas severing the substantive \u03b4\u03cc\u03b3\u03bc\u03b1 here from the verb \u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03b9\u03bd in ver.\n20. The Latin fathers, who had either _decretis_ or _sententiis_ in\ntheir version, were saved from this false interpretation; e.g. Hilar.\n_de Trin._ i. 12 (II. p. 10), ix. 10 (II. p. 265 sq.), Ambros. _Apol.\nDav._ 13 (I. p. 698), _de Fid._ iii. 2 (II. p. 499), August. _de Pecc.\nMer._ i. 47 (X. p. 26): though they very commonly took \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd, \u1f10\u03bd\n\u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd, to refer to the decree of condemnation. Jerome however on\nEphes. ii. 15 (VII. p. 581) follows the Greeks. The later Christian\nsense of \u03b4\u03cc\u03b3\u03bc\u03b1, meaning 'doctrine', came from its secondary classical\nuse, where it was applied to the authoritative and categorical\n'sentences' of the philosophers: comp. Just. Mart. _Apol._ i. 7 (p. 56\nD) \u03bf\u1f31 \u1f10\u03bd \u1f1d\u03bb\u03bb\u03b7\u03c3\u03b9 \u03c4\u1f70 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03c1\u03b5\u03c3\u03c4\u1f70 \u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 \u1f10\u03ba \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u03c4\u1ff7 \u1f11\u03bd\u1f76 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b9\n\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b1\u03b3\u03bf\u03c1\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9, \u03ba\u03b1\u03af\u03c0\u03b5\u03c1 \u03c4\u1ff6\u03bd \u03b4\u03bf\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03c9\u03bd \u1f44\u03bd\u03c4\u03c9\u03bd, Cic.\n_Acad._ ii. 9 'de suis decretis qu\u00e6 philosophi vocant \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1', Senec.\n_Epist._ xcv. 10 'Nulla ars contemplativa sine decretis suis est, qu\u00e6\nGr\u00e6ci vocant _dogmata_, nobis vel _decreta_ licet adpellare vel _scita_\nvel _placita_'. See the indices to Plutarch, Epictetus, etc., for\nillustrations of the use of the term. There is an approach towards the\necclesiastical meaning in Ignat. _Magn._ 13 \u03b2\u03b5\u03b2\u03b1\u03b9\u03c9\u03b8\u1fc6\u03bd\u03b1\u03b9 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd\n\u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03c9\u03bd, Barnab. \u00a7 1 \u03c4\u03c1\u03af\u03b1 \u03bf\u1f56\u03bd \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03ac \u1f10\u03c3\u03c4\u03b9\u03bd \u039a\u03c5\u03c1\u03af\u03bf\u03c5\n(comp. \u00a7 9, 10).\n\n------------------------------------------------------------------------\n\nII. 14]\n\n\u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd, \u1f43 \u1f26\u03bd \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03bd \u1f21\u03bc\u1fd6\u03bd\u00b7 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78 \u1f26\u03c1\u03ba\u03b5\u03bd \u1f10\u03ba\n\n\n\u1f43 \u1f26\u03bd \u03ba.\u03c4.\u03bb.] '_which was directly opposed to us_'. The former\nexpression, \u03c4\u1f78 \u03ba\u03b1\u03b8' \u1f21\u03bc\u1ff6\u03bd, referred to the _validity_ of the bond; the\npresent, \u1f43 \u1f26\u03bd \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03bd \u1f21\u03bc\u1fd6\u03bd, describes its _active hostility_. It is\nquite a mistake to suppose that the first preposition in \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2\nmitigates its force, as in \u1f51\u03c0\u03bf\u03b4\u03ae\u03bb\u03c9\u03c3\u03b9\u03c2, \u1f51\u03c0\u03cc\u03bb\u03b5\u03c5\u03ba\u03bf\u03c2, \u1f51\u03c0\u03bf\u03bc\u03b1\u03af\u03bd\u03bf\u03bc\u03b1\u03b9,\n\u1f51\u03c0\u03bf\u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9\u03bd, etc. Neither in classical writers nor in the LXX has the\nword any shade of this meaning. It is very commonly used for instance,\nof things which are directly antagonistic and mutually exclusive: e.g.\nAristot. _de Gen. et Corr._ i. 7 (p. 323) \u0394\u03b7\u03bc\u03cc\u03ba\u03c1\u03b9\u03c4\u03bf\u03c2 ... \u03c6\u03b7\u03c3\u1f76 ... \u03c4\u1f78\n\u03b1\u1f50\u03c4\u1f78 \u03ba\u03b1\u1f76 \u1f45\u03bc\u03bf\u03b9\u03bf\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u03cc \u03c4\u03b5 \u03c0\u03bf\u03b9\u03bf\u1fe6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c0\u03ac\u03c3\u03c7\u03bf\u03bd ... \u1f10\u03bf\u03af\u03ba\u03b1\u03c3\u03b9 \u03b4\u1f72 \u03bf\u1f31\n\u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03c4\u1f78\u03bd \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd \u03bb\u03ad\u03b3\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03b1 (i.e. self-contradictory) \u03c6\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9\n\u03bb\u03ad\u03b3\u03b5\u03b9\u03bd\u00b7 \u03b1\u1f34\u03c4\u03b9\u03bf\u03bd \u03b4\u1f72 \u03c4\u1fc6\u03c2 \u1f10\u03bd\u03b1\u03bd\u03c4\u03b9\u03bf\u03bb\u03bf\u03b3\u03af\u03b1\u03c2 \u03ba.\u03c4.\u03bb., [Plato] _Alcib. Sec._ 138 C\n\u03a3\u03a9. \u03a4\u1f78 \u03bc\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 \u1f06\u03c1\u03b1 \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03bd \u03c3\u03bf\u03b9 \u03b4\u03bf\u03ba\u03b5\u1fd6 \u03c4\u1ff7 \u03c6\u03c1\u03bf\u03bd\u03b5\u1fd6\u03bd; \u0391\u039b. \u03a0\u03ac\u03bd\u03c5 \u03bc\u1f72\u03bd\n\u03bf\u1f56\u03bd.... 139 B \u03a3\u03a9. \u039a\u03b1\u1f76 \u03bc\u1f74\u03bd \u03b4\u03cd\u03bf \u03b3\u03b5 \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03b1 \u1f11\u03bd\u1f76 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b9 \u03c0\u1ff6\u03c2 \u1f02\u03bd \u03b5\u1f34\u03b7;\n(i.e. how can one thing have two direct opposites?), where the whole\nargument depends on this sense of \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2. In compounds with \u1f51\u03c0\u1f78 the\nforce of the preposition will generally be determined by the meaning of\nthe other element in the compound; and, as \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2 (\u1f14\u03bd\u03b1\u03bd\u03c4\u03b9) implies\nlocality, a local sense is communicated to \u1f51\u03c0\u03cc. Thus \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2 may be\ncompared with \u1f51\u03c0\u03b1\u03bb\u03bb\u03ac\u03c3\u03c3\u03b5\u03b9\u03bd, \u1f51\u03c0\u03b1\u03bd\u03c4\u1fb6\u03bd, \u1f51\u03c0\u03b1\u03bd\u03c4\u03b9\u03ac\u03b6\u03b5\u03b9\u03bd, \u1f51\u03c0\u03bf\u03c4\u03c1\u03ad\u03c7\u03b5\u03b9\u03bd (Xen.\n_Cyrop._ i. 2. 12 \u03bb\u03b7\u03c3\u03c4\u1f70\u03c2 \u1f51\u03c0\u03bf\u03b4\u03c1\u03b1\u03bc\u03b5\u1fd6\u03bd, 'to hunt down'), \u1f51\u03c0\u03b5\u03bb\u03ac\u03c5\u03bd\u03b5\u03b9\u03bd (Xen.\n_Anab._ i. 8. 15 \u1f51\u03c0\u03b5\u03bb\u03ac\u03c3\u03b1\u03c2 \u1f61\u03c2 \u03c3\u03c5\u03bd\u03b1\u03bd\u03c4\u1fc6\u03c3\u03b1\u03b9, 'riding up'), \u1f51\u03c6\u03b9\u03c3\u03c4\u03ac\u03bd\u03b1\u03b9 (Polyb.\ni. 50. 6 \u1f51\u03c0\u03ad\u03c3\u03c4\u03b7\u03c3\u03b5 \u03c4\u1f74\u03bd \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 \u03bd\u03b1\u1fe6\u03bd \u1f00\u03bd\u03c4\u03af\u03c0\u03c1\u03c9\u03c1\u03bf\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03bf\u03bb\u03b5\u03bc\u03af\u03bf\u03b9\u03c2, 'he brought\nup' his own ship). With this meaning, 'over against,' 'close in upon,'\nthe preposition does not weaken but enhance the force of \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2, so\nthat the compound will denote 'direct,' 'close,' or 'persistent\nopposition.'\n\n\u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78 \u1f26\u03c1\u03ba\u03b5\u03bd \u03ba.\u03c4.\u03bb.] '_and He_, i.e. Christ, _hath taken it away_'.\nThere is a double change in this clause: (1) The participles\n(\u03c7\u03b1\u03c1\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2, \u1f10\u03be\u03b1\u03bb\u03ad\u03b9\u03c8\u03b1\u03c2) are replaced by a finite verb. (2) The aorists\n(\u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd, \u03c7\u03b1\u03c1\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2, \u1f10\u03be\u03b1\u03bb\u03ad\u03b9\u03c8\u03b1\u03c2) are replaced by a perfect. The\nsubstitution of \u1f96\u03c1\u03b5\u03bd for \u1f26\u03c1\u03ba\u03b5\u03bd in some copies betrays a consciousness on\nthe part of the scribes of the dislocation produced by the new tense. As\na new subject, \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2, must be introduced somewhere (see the note on\nver. 13), the severance thus created suggests this as the best point of\ntransition. The perfect \u1f26\u03c1\u03ba\u03b5\u03bd, 'He hath removed it', is suggested by the\nfeeling of relief and thanksgiving, which rises up in the Apostle's mind\nat this point. For the strong expression \u1f04\u03b9\u03c1\u03b5\u03b9\u03bd \u1f10\u03ba [\u03c4\u03bf\u1fe6] \u03bc\u03ad\u03c3\u03bf\u03c5, 'to\nremove and put out of sight', comp. LXX Is. lvii. 2, Epictet. iii. 3.\n15, Plut. _Mor._ p. 519 D; so 2 Thess. ii. 7 \u1f10\u03ba \u03bc\u03ad\u03c3\u03bf\u03c5 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9.\n\n------------------------------------------------------------------------\n\nII. 15]\n\n\u03c4\u03bf\u1fe6 \u03bc\u03ad\u03c3\u03bf\u03c5, \u03c0\u03c1\u03bf\u03c3\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2 \u03b1\u1f50\u03c4\u1f78 \u03c4\u1ff7 \u03c3\u03c4\u03b1\u03c5\u03c1\u1ff7\u00b7 ^{15}\u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2\n\n\n\u03c0\u03c1\u03bf\u03c3\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2 \u03ba.\u03c4.\u03bb.] 'The abrogation was even more emphatic. Not only was\nthe writing erased, but the document itself was torn up and cast aside.'\nBy \u03c0\u03c1\u03bf\u03c3\u03b7\u03bb\u03ce\u03c3\u03b1\u03c2 is meant that the law of ordinances was nailed to the\ncross, rent with Christ's body, and destroyed with His death: see the\nnotes on Gal. vi. 14 \u03b4\u03b9' \u03bf\u1f57 [\u03c4\u03bf\u1fe6] \u03c3\u03c4\u03b1\u03c5\u03c1\u03bf\u1fe6 \u1f11\u03bc\u03bf\u1f76 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 (the world, the\nsphere of material ordinances) \u1f10\u03c3\u03c4\u03b1\u03cd\u03c1\u03c9\u03c4\u03b1\u03b9 \u03ba\u1f00\u03b3\u1f7c \u03ba\u03cc\u03c3\u03bc\u1ff3, where the idea is\nthe same. It has been supposed that in some cities the abrogation of a\ndecree was signified by running a nail through it and hanging it up in\npublic. The image would thus gain force, but there is no distinct\nevidence of such a custom.\n\n15. \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba.\u03c4.\u03bb.] This word appears not to occur at all before\nSt Paul, and rarely if ever after his time, except in writers who may be\nsupposed to have his language before them; e.g. Hippol. _H\u00e6r._ i. 24\n\u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u1f43 \u03c0\u03b5\u03c1\u03b9\u03ba\u03b5\u1fd6\u03c4\u03b1\u03b9. In Joseph. _Ant._ vi. 14. 2 \u1f00\u03c0\u03b5\u03ba\u03b4\u1f7a\u03c2\nis only a variation for \u03bc\u03b5\u03c4\u03b5\u03ba\u03b4\u1f7a\u03c2 which seems to be the correct reading.\nThe word also appears in some texts of Babrius _Fab._ xviii. 3, but it\nis merely a conjectural emendation. Thus the occurrence of \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9\nhere and in iii. 9, and of \u1f00\u03c0\u03ad\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2 above in ver. 11, is remarkable;\nand the choice of an unusual, if not a wholly new, word must have been\nprompted by the desire to emphasize the _completeness_ of the action.\nThe force of the double compound may be inferred from a passage of\nLysias, where the two words \u1f00\u03c0\u03bf\u03b4\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 and \u1f10\u03ba\u03b4\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 occur together; _c.\nTheomn._ i. 10 (p. 117) \u03c6\u03ac\u03c3\u03ba\u03c9\u03bd \u03b8\u03bf\u03b9\u03bc\u03ac\u03c4\u03b9\u03bf\u03bd \u1f00\u03c0\u03bf\u03b4\u03b5\u03b4\u03cd\u03c3\u03b8\u03b1\u03b9 \u1f22 \u03c4\u1f78\u03bd \u03c7\u03b9\u03c4\u03c9\u03bd\u03af\u03c3\u03ba\u03bf\u03bd\n\u1f10\u03ba\u03b4\u03b5\u03b4\u03cd\u03c3\u03b8\u03b1\u03b9. Here however the sense of \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 is difficult. The\nmeaning generally assigned to it, 'having spoiled, stripped of their\narms', disregards the middle voice. St Jerome is chiefly responsible for\nthis common error of interpretation: for in place of the Old Latin\n'exuens se', which was grammatically correct, he substituted\n'exspolians' in his revised version. In his interpretation however he\nwas anticipated by the commentator Hilary, who read 'exuens' for 'exuens\nse' in his text. Discarding this sense, as inconsistent with the voice,\nwe have the choice of two interpretations.\n\n(1) The common interpretation of the Latin fathers, '_putting off_ the\nbody', thus separating \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 from \u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba.\u03c4.\u03bb. and\nunderstanding \u03c4\u1f74\u03bd \u03c3\u03ac\u03c1\u03ba\u03b1 or \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 with it; comp. 2 Cor. v. 3\n\u1f10\u03bd\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9. So Novat. _de Trin._ 16 'exutus carnem'; Ambros. _Expos.\nLuc._ v. \u00a7 107 (I. p. 1381) 'exuens se carnem', comp. _de Fid._ iii. 2\n(II. p. 499); Hilar. _de Trin._ i. 13 (II. p. 10) 'exutus carnem' (comp.\nix. 10, p. 265), x. 48 (p. 355) 'spolians se carne' (comp. ix. 11, p.\n266); Augustin. _Epist._ 149 (II. p. 513) 'exuens se carne', etc. This\nappears to have been the sense adopted much earlier in a Docetic work\nquoted by Hippol. _H\u00e6r._ viii. 10 \u03c8\u03c5\u03c7\u1f74 \u1f10\u03ba\u03ad\u03b9\u03bd\u03b7 \u1f10\u03bd \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u03c4\u03c1\u03b1\u03c6\u03b5\u1fd6\u03c3\u03b1,\n\u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03b1\u03bc\u03ad\u03bd\u03b7 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c3\u03b7\u03bb\u03ce\u03c3\u03b1\u03c3\u03b1 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78 \u03be\u03cd\u03bb\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03c3\u03b1\u03c3\u03b1\n\u03ba.\u03c4.\u03bb. It is so paraphrased likewise in the Peshito Syriac and the\nGothic. The reading \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c4\u1f74\u03bd \u03c3\u03ac\u03c1\u03ba\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f70\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 (omitting\n\u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba\u03b1\u1f76), found in some ancient authorities, must be a corruption\nfrom an earlier text, which had inserted the gloss \u03c4\u1f74\u03bd \u03c3\u03ac\u03c1\u03ba\u03b1 after\n\u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2, while retaining \u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba\u03b1\u1f76, and which seems to have\nbeen in the hands of some of the Latin fathers already quoted. This\ninterpretation has been connected with a common metaphorical use of\n\u1f00\u03c0\u03bf\u03b4\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9, signifying 'to strip' and so 'to prepare for a contest';\ne.g. Plut. _Mor._ 811 E \u03c0\u03c1\u1f78\u03c2 \u03c0\u1fb6\u03c3\u03b1\u03bd \u1f00\u03c0\u03bf\u03b4\u03cd\u03bf\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f74\u03bd \u03c0\u03bf\u03bb\u03b9\u03c4\u03b9\u03ba\u1f74\u03bd \u03c0\u03c1\u1fb6\u03be\u03b9\u03bd,\nDiod. Sic. ii. 29 \u1f10\u03c0\u1f76 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u1f00\u03c0\u03bf\u03b4\u03cd\u03bd\u03c4\u03b5\u03c2. The serious objection to\nthis rendering is, that it introduces an isolated metaphor which is not\nexplained or suggested by anything in the context.\n\n(2) The common interpretation of the Greek fathers; '_having stripped\noff and put away the powers_ of evil', making \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 govern \u03c4\u1f70\u03c2\n\u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba.\u03c4.\u03bb. So Chrysostom, Severianus, Theodore of Mopsuestia, and\nTheodoret. This also appears to have been the interpretation of Origen,\n_in Matt._ xii. \u00a7 25 (III. p. 544), _ib._ \u00a7 40 (p. 560), _in Ioann._ vi.\n\u00a7 37 (IV. p. 155), _ib._ xx. \u00a7 29 (p. 356), though his language is not\nexplicit, and though his translators, e.g. _in Libr. Ies. Hom._ vii. \u00a7 3\n(II. p. 413), make him say otherwise. The meaning then will be as\nfollows. Christ took upon Himself our human nature with all its\ntemptations (Heb. iv. 15). The powers of evil gathered about Him. Again\nand again they assailed Him; but each fresh assault ended in a new\ndefeat. In the wilderness He was tempted by Satan; but Satan retired for\nthe time baffled and defeated (Luke iv. 13 \u1f00\u03c0\u03ad\u03c3\u03c4\u03b7 \u1f00\u03c0' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f04\u03c7\u03c1\u03b9\n\u03ba\u03b1\u03b9\u03c1\u03bf\u1fe6). Through the voice of His chief disciple the temptation was\nrenewed, and He was entreated to decline His appointed sufferings and\ndeath. Satan was again driven off (Matt. xvi. 23 \u1f55\u03c0\u03b1\u03b3\u03b5 \u1f40\u03c0\u03af\u03c3\u03c9 \u03bc\u03bf\u03c5,\n\u03a3\u03b1\u03c4\u03b1\u03bd\u1fb6, \u03c3\u03ba\u03ac\u03bd\u03b4\u03b1\u03bb\u03bf\u03bd \u03b5\u1f36 \u1f10\u03bc\u03bf\u1fe6: comp. Matt. viii. 31). Then the last hour\ncame. This was the great crisis of all, when 'the power of darkness'\nmade itself felt (Luke xxii. 53 _\u1f21 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1_ \u03c4\u03bf\u1fe6 \u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2; see above\ni. 13), when the prince of the world asserted his tyranny (Joh. xii. 30\n_\u1f41 \u1f04\u03c1\u03c7\u03c9\u03bd_ \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5). The final act in the conflict began with the\nagony of Gethsemane; it ended with the cross of Calvary. The victory was\ncomplete. The enemy of man was defeated. The powers of evil, which had\nclung like a Nessus robe about His humanity, were torn off and cast\naside for ever. And the victory of mankind is involved in the victory of\nChrist. In His cross we too are divested of the poisonous clinging\ngarments of temptation and sin and death; \u03c4\u1ff7 \u1f00\u03c0\u03bf\u03b8\u03ad\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f74\u03bd \u03b8\u03bd\u03b7\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1,\nsays Theodore, \u1f23\u03bd \u1f51\u03c0\u1f72\u03c1 \u03c4\u1fc6\u03c2 \u03ba\u03bf\u03b9\u03bd\u1fc6\u03c2 \u1f00\u03c6\u03b5\u1fd6\u03bb\u03b5\u03bd \u03b5\u1f50\u03b5\u03c1\u03b3\u03b5\u03c3\u03af\u03b1\u03c2, \u1f00\u03c0\u03b5\u03b4\u03cd\u03c3\u03b1\u03c4\u03bf \u03ba\u1f00\u03ba\u03b5\u03af\u03bd\u03c9\u03bd\n(i.e. \u03c4\u1ff6\u03bd \u1f00\u03bd\u03c4\u03b9\u03ba\u03b5\u03b9\u03bc\u03ad\u03bd\u03c9\u03bd \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03c9\u03bd) \u03c4\u1f74\u03bd \u03b1\u1f50\u03b8\u03b5\u03bd\u03c4\u03b5\u03af\u03b1\u03bd \u1f97\u03c0\u03b5\u03c1 \u1f10\u03ba\u03ad\u03c7\u03c1\u03b7\u03bd\u03c4\u03bf \u03ba\u03b1\u03b8'\n\u1f21\u03bc\u1ff6\u03bd. For the image of the garments comp. Is. lxiv. 6, but especially\nZech. iii. 1 sq., 'And he showed me Joshua the high-priest standing\nbefore the angel of the Lord and _Satan standing at his right hand to\nresist him_. And the Lord said unto Satan, The Lord rebuke thee, O\nSatan.... Now Joshua was clothed with filthy garments.... And He\nanswered and spake unto those that stood before Him saying _Take away\nthe filthy garments_ from him. And unto him He said Behold, _I have\ncaused thine iniquity to pass from thee_'. In this prophetic passage the\nimage is used of His type and namesake, the Jesus of the Restoration,\nnot in his own person, but as the high-priest and representative of a\nguilty but cleansed and forgiven people, with whom he is identified. For\nthe metaphor of \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 more especially, see Philo _Quod det. pot.\nins._ 13 (I. p. 199) \u1f10\u03be\u03b1\u03bd\u03b1\u03c3\u03c4\u03ac\u03bd\u03c4\u03b5\u03c2 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b5\u03c1\u03b5\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f70\u03c2 \u1f10\u03bd\u03c4\u03ad\u03c7\u03bd\u03bf\u03c5\u03c2\n\u03b1\u1f50\u03c4\u1ff6\u03bd \u03c0\u03b5\u03c1\u03b9\u03c0\u03bb\u03bf\u03ba\u1f70\u03c2 \u03b5\u1f50\u03bc\u03b1\u03c1\u1ff6\u03c2 _\u1f10\u03ba\u03b4\u03c5\u03c3\u03cc\u03bc\u03b5\u03b8\u03b1_, where the image in the\ncontext is that of a wrestling bout.\n\nThis interpretation is grammatical; it accords with St Paul's teaching;\nand it is commended by the parallel uses of the substantive in ver. 11\n\u1f10\u03bd \u03c4\u1fc7 \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03c3\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, and of the verb in iii. 9\n\u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f78\u03bd \u03c0\u03ac\u03bb\u03b1\u03b9\u03bf\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03ba.\u03c4.\u03bb. The \u1f00\u03c0\u03ad\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2 accomplished in\nus when we are baptized into His death is a counterpart to the \u1f00\u03c0\u03ad\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2\nwhich He accomplished by His death. With Him indeed it was only the\ntemptation, with us it is the sin as well as temptation; but otherwise\nthe parallel is complete. In both cases it is a divestiture of the\npowers of evil, a liberation from the dominion of the flesh. On the\nother hand the common explanation 'spoiling' is not less a violation of\nSt Paul's usage (iii. 9) than of grammatical rule.\n\n------------------------------------------------------------------------\n\nII. 15]\n\n\u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f70\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2 \u1f10\u03b4\u03b5\u03b9\u03b3\u03bc\u03ac\u03c4\u03b9\u03c3\u03b5\u03bd\n\n\n\u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u1f70\u03c2 \u03ba.\u03c4.\u03bb.] What powers are especially meant here will appear from\nEphes. vi. 12 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u03ac\u03c2, \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70\u03c2 \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1\u03c2, \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03ba\u03bf\u03c3\u03bc\u03bf\u03ba\u03c1\u03ac\u03c4\u03bf\u03c1\u03b1\u03c2\n\u03c4\u03bf\u1fe6 \u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5, \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u1f70 \u03c4\u1fc6\u03c2 \u03c0\u03bf\u03bd\u03b7\u03c1\u03af\u03b1\u03c2 \u03ba.\u03c4.\u03bb. See the note\non i. 16.\n\n\u1f10\u03b4\u03b5\u03b9\u03b3\u03bc\u03ac\u03c4\u03b9\u03c3\u03b5\u03bd] '_displayed_', as a victor displays his captives or\ntrophies in a triumphal procession: Hor. _Epist._ i. 17. 33 'captos\nostendere civibus hostes'. The word is extremely rare; Matt. i. 19 \u03bc\u1f74\n\u03b8\u03ad\u03bb\u03c9\u03bd \u03b1\u1f50\u03c4\u1f74\u03bd \u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03b9 (where it ought probably to be read for the more\ncommon word \u03c0\u03b1\u03c1\u03b1\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03b9), _Act. Paul. et Petr._ 33 \u1f14\u03bb\u03b5\u03b3\u03b5 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd\n\u03bb\u03b1\u1f78\u03bd \u1f35\u03bd\u03b1 \u03bc\u1f74 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03c4\u03bf\u1fe6 \u03a3\u03af\u03bc\u03c9\u03bd\u03bf\u03c2 \u1f00\u03c0\u03ac\u03c4\u03b7\u03c2 \u03c6\u03cd\u03b3\u03c9\u03c3\u03b9\u03bd \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76\n\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd \u03b1\u1f50\u03c4\u03cc\u03bd. Nowhere does the word convey the idea of 'making\nan example' (\u03c0\u03b1\u03c1\u03b1\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03b9) but signifies simply 'to display, publish,\nproclaim'. In the context of the last passage we have as the\nconsequence, \u1f65\u03c3\u03c4\u03b5 \u03c0\u03ac\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03b5\u1f50\u03bb\u03b1\u03b2\u03b5\u1fd6\u03c2 \u1f04\u03bd\u03b4\u03c1\u03b1\u03c2 \u03b2\u03b4\u03b5\u03bb\u03cd\u03c4\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 \u03a3\u03af\u03bc\u03c9\u03bd\u03b1 \u03c4\u1f78\u03bd\n\u03bc\u03ac\u03b3\u03bf\u03bd \u03ba\u03b1\u1f76 \u1f00\u03bd\u03cc\u03c3\u03b9\u03bf\u03bd \u03b1\u1f50\u03c4\u1f78\u03bd _\u03ba\u03b1\u03c4\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03b5\u03b9\u03bd_, i.e. to _proclaim_ his\nimpieties. The substantive occurs on the Rosetta stone l. 30 (Boeckh,\n_C. I._ 4697) \u03c4\u1ff6\u03bd \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b5\u03bb\u03b5\u03c3\u03bc\u03ad\u03bd\u03c9\u03bd \u03c4\u1f70 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u1f78\u03bd \u03b4\u03b9\u03ac\u03c6\u03bf\u03c1\u03b1.\n\n------------------------------------------------------------------------\n\nII. 15]\n\n\u1f10\u03bd \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u1fb3, \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03c3\u03b1\u03c2 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7.\n\n\n\u1f10\u03bd \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u1fb3] '_boldly_', not '_publicly_'. As \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u03b1 is\n'unreservedness, plainness of speech' (\u03c0\u03b1\u03bd-\u03c1\u03b7\u03c3\u03af\u03b1, its opposite being\n\u1f00\u03c1\u03c1\u03b7\u03c3\u03af\u03b1 'silence'), so while applied still to language, it may be\nopposed either (1) to 'fear', as John vii. 13, Acts iv. 29, or (2) to\n'ambiguity, reserve', Joh. xi. 14, xvi. 25, 29; but 'misgiving,\napprehension' in some form or other seems to be always the correlative\nidea. Hence, when it is transferred from words to actions, it appears\nalways to retain the idea of 'confidence, boldness'; e.g. 1 Macc. iv. 18\n\u03bb\u03ae\u03c8\u03b5\u03c4\u03b5 \u03c4\u1f70 \u03c3\u03ba^\u03c5\u03bb\u03b1 \u03bc\u03b5\u03c4\u1f70 \u03c0\u03b1\u03c1\u1fe5\u03b7\u03c3\u03af\u03b1\u03c2, _Test. xii. Patr._ Rub. 4 \u03bf\u1f50\u03ba \u03b5\u1f36\u03c7\u03bf\u03bd\n\u03c0\u03b1\u03c1\u1fe5\u03b7\u03c3\u03af\u03b1\u03bd \u1f00\u03c4\u03b5\u03bd\u03af\u03c3\u03b1\u03b9 \u03b5\u1f30\u03c2 \u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03bf\u03bd \u1f38\u03b1\u03ba\u03ce\u03b2, Jos. _Ant._ ix. 1O. 4 \u1f51\u03c0'\n\u03b1\u1f30\u03c3\u03c7\u03cd\u03bd\u03b7\u03c2 \u03c4\u03b5 \u03c4\u03bf\u1fe6 \u03c3\u03c5\u03bc\u03b2\u03b5\u03b2\u03b7\u03ba\u03cc\u03c4\u03bf\u03c2 \u03b4\u03b5\u03b9\u03bd\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u03bc\u03b7\u03ba\u03ad\u03c4' \u03b1\u1f50\u03c4\u1ff7 \u03c0\u03b1\u03c1\u1fe5\u03b7\u03c3\u03af\u03b1\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9.\nThe idea of publicity may sometimes be connected with the word as a\nsecondary notion, e.g. in Joh. vii. 4, where \u1f10\u03bd \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u1fb3 \u03b5\u1f36\u03bd\u03b1\u03b9 'to\nassume a bold attitude' is opposed to \u1f10\u03bd \u03ba\u03c1\u03c5\u03c0\u03c4\u1ff7 \u03c0\u03bf\u03b9\u03b5\u1fd6\u03bd (comp. xviii.\n20); but it does not displace the primary sense.\n\n------------------------------------------------------------------------\n\nII. 16]\n\n^{16}\u039c\u1f74 \u03bf\u1f56\u03bd \u03c4\u03b9\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u03ba\u03c1\u03b9\u03bd\u03ad\u03c4\u03c9 \u1f10\u03bd \u03b2\u03c1\u03ce\u03c3\u03b5\u03b9 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03c0\u03cc\u03c3\u03b5\u03b9 \u1f22 >\n\n                            16. \u1f22 \u1f10\u03bd \u03c0\u03cc\u03c3\u03b5\u03b9.\n\n\n\u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03c3\u03b1\u03c2] '_leading them in triumph_', the same metaphor as in 2 Cor.\nii. 14 \u03c4\u1ff7 \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03bf\u03bd\u03c4\u03b9 \u1f21\u03bc\u1fb6\u03c2 \u1f10\u03bd \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03ba.\u03c4.\u03bb., where it is\nwrongly translated in the A. V. 'causeth us to triumph'. Here however it\nis the defeated powers of evil, there the subjugated persons of men, who\nare led in public, chained to the triumphal car of Christ. This is the\nproper meaning and construction of \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd, as found elsewhere. This\nverb takes an accusative (1) of the person over whom the triumph is\ncelebrated, e.g. Plut. _Vit. Arat._ 54 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u0391\u1f30\u03bc\u03af\u03bb\u03b9\u03bf\u03c2 \u1f10\u03b8\u03c1\u03b9\u03ac\u03bc\u03b2\u03b5\u03c5\u03c3\u03b5,\n_Thes. et Rom. Comp._ 4 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u1fd6\u03c2 \u1f10\u03b8\u03c1\u03b9\u03ac\u03bc\u03b2\u03b5\u03c5\u03c3\u03b5: (2) of the spoils\nexhibited in the triumph, e.g. Tatian _c. Gr\u00e6c._ 26 \u03c0\u03b1\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u03bb\u03cc\u03b3\u03bf\u03c5\u03c2\n\u1f00\u03bb\u03bb\u03bf\u03c4\u03c1\u03af\u03bf\u03c5\u03c2 \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u03af, \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f41 \u03ba\u03bf\u03bb\u03bf\u03b9\u03cc\u03c2, \u03bf\u1f50\u03ba \u1f30\u03b4\u03af\u03bf\u03b9\u03c2 \u1f10\u03c0\u03b9\u03ba\u03bf\u03c3\u03bc\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9\n\u03c0\u03c4\u03b5\u03c1\u03bf\u1fd6\u03c2: (3) more rarely of the substance of the triumph, e.g. _Vit.\nCamill._ 30 \u1f41 \u03b4\u1f72 \u039a\u03ac\u03bc\u03b9\u03bb\u03bb\u03bf\u03c2 \u1f10\u03b8\u03c1\u03b9\u03ac\u03bc\u03b2\u03b5\u03c5\u03c3\u03b5 ... \u03c4\u1f78\u03bd \u1f00\u03c0\u03bf\u03bb\u03c9\u03bb\u03c5\u03af\u03b1\u03c2 \u03c3\u03c9\u03c4\u1fc6\u03c1\u03b1 \u03c0\u03b1\u03c4\u03c1\u03af\u03b4\u03bf\u03c2\n\u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd, i.e. 'in the character of his country's saviour'. The passive\n\u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 is 'to be led in triumph', 'to be triumphed over', e.g.\n_Vit. C. Marc._ 35. So the Latins say 'triumphare aliquem' and\n'triumphari'.\n\n\u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7] i.e. \u03c4\u1ff7 \u03c3\u03c4\u03b1\u03c5\u03c1\u1ff7: comp. Ephes. ii. 16 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03be\u1fc3 \u03c4\u03bf\u1f7a\u03c2\n\u1f00\u03bc\u03c6\u03bf\u03c4\u03ad\u03c1\u03bf\u03c5\u03c2 ... \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03b1\u03c5\u03c1\u03bf\u1fe6. The violence of the metaphor is its\njustification. The paradox of the crucifixion is thus placed in the\nstrongest light\u2014triumph in helplessness and glory in shame. The\nconvict's gibbet is the victor's car.\n\n16\u201319. 'Seeing then that the bond is cancelled, that the law of\nordinances is repealed, beware of subjecting yourselves to its tyranny\nagain. Suffer no man to call you to account in the matter of eating or\ndrinking, or again of the observance of a festival or a new moon or a\nsabbath. These are only shadows thrown in advance, only types of things\nto come. The substance, the reality, in every case belongs to the Gospel\nof Christ. The prize is now fairly within your reach. Do not suffer\nyourselves to be robbed of it by any stratagem of the false teachers.\nTheir religion is an officious humility which displays itself in the\nworship of angels. They make a parade of their visions, but they are\nfollowing an empty phantom. They profess humility, but they are puffed\nup with their vaunted wisdom, which is after all only the mind of the\nflesh. Meanwhile they have substituted inferior spiritual agencies for\nthe One true Mediator, the Eternal Word. Clinging to these lower\nintelligences, they have lost their hold of the Head; they have severed\ntheir connexion with Him, on whom the whole body depends; from whom it\nderives its vitality, and to whom it owes its unity, being supplied with\nnourishment and knit together in one by means of the several joints and\nattachments, so that it grows with a growth which comes from God\nHimself.'\n\n16 sq. The two main tendencies of the Colossian heresy are discernible\nin this warning (vv. 16\u201319), as they were in the previous statement (vv.\n9\u201315). Here however the order is reversed. The practical error, an\nexcessive ritualism and ascetic rigour, is first dealt with (vv. 16,\n17); the theological error, the interposition of angelic mediators,\nfollows after (vv. 18, 19). The first is the substitution of a shadow\nfor the substance; the second is the preference of an inferior member to\nthe head. The reversal of order is owing to the connexion of the\nparagraphs; the opening subject in the second paragraph being a\ncontinuation of the concluding subject in the first, by the figure\ncalled chiasm: comp. Gal. iv. 5.\n\n\u03ba\u03c1\u03b9\u03bd\u03ad\u03c4\u03c9] not '_condemn you_', but '_take you to task_'; as e.g. Rom.\nxiv. 3 sq. The judgment may or may not end in an acquittal; but in any\ncase it is wrong, since these matters ought not to be taken as the basis\nof a judgment.\n\n\u1f10\u03bd \u03b2\u03c1\u03ce\u03c3\u03b5\u03b9 \u03ba.\u03c4.\u03bb.] '_in eating and in drinking_'; Rom. xiv. 17 \u03bf\u1f50 \u03b3\u03ac\u03c1\n\u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03b2\u03c1\u1ff6\u03c3\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03cc\u03c3\u03b9\u03c2, \u1f00\u03bb\u03bb\u1f70 \u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7 \u03ba.\u03c4.\u03bb., Heb.\nix. 10 \u1f10\u03c0\u1f76 \u03b2\u03c1\u03ce\u03bc\u03b1\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c0\u03cc\u03bc\u03b1\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03c6\u03cc\u03c1\u03bf\u03b9\u03c2 \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u03bf\u1fd6\u03c2, \u03b4\u03b9\u03ba\u03b1\u03af\u03c9\u03bc\u03b1\u03c4\u03b1\n\u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, comp. 1 Cor. viii. 8 \u03b2\u03c1\u1ff6\u03bc\u03b1 \u03b4\u1f72 \u1f21\u03bc\u1fb6\u03c2 \u03bf\u1f50 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c4\u03ae\u03c3\u03b5\u03b9 \u03c4\u1ff7 \u0398\u03b5\u1ff7 \u03ba.\u03c4.\u03bb.\nThe first indication that the Mosaic distinctions of things clean and\nunclean should be abolished is given by our Lord Himself: Mark vii. 14\nsq. (the correct reading in ver. 19 being \u03ba\u03b1\u03b8\u03b1\u03c1\u03af\u03b6\u03c9\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1f70 \u03b2\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1).\nThey were afterwards formally annulled by the vision which appeared to\nSt Peter: Acts x. 11 sq. The ordinances of the Mosaic law applied almost\nexclusively to meats. It contained no prohibitions respecting drinks\nexcept in a very few cases; e.g. of the priests ministering in the\ntabernacle (Lev. x. 9), of liquids contained in unclean vessels etc.\n(Lev. xi. 34, 36), and of Nazarite vows (Num. vi. 3). These directions,\ntaken in connexion with the rigid observances which the later Jews had\ngrafted on them (Matt. xxiii. 24), would be sufficient to explain the\nexpression, when applied to the Mosaic law by itself, as in Heb. _l.c._\nThe rigour of the Colossian false teachers however, like that of their\nJewish prototypes the Essenes, doubtless went far beyond the injunctions\nof the law. It is probable that they forbad wine and animal food\naltogether: see the introduction pp. 86, 104 sq. For allusions in St\nPaul to similar observances not required by the law, see Rom. xiv. 2 \u1f41\n\u03b4\u1f72 \u1f00\u03c3\u03b8\u03b5\u03bd\u1ff6\u03bd \u03bb\u03ac\u03c7\u03b1\u03bd\u03b1 \u1f10\u03c3\u03b8\u03af\u03b5\u03b9, ver. 21 \u03ba\u03b1\u03bb\u1f78\u03bd \u03c4\u1f78 \u03bc\u1f74 \u03c6\u03b1\u03b3\u03b5\u1fd6\u03bd \u03ba\u03c1\u03ad\u03b1 \u03bc\u03b7\u03b4\u1f72 \u03c0\u03b9\u03b5\u1fd6\u03bd\n\u03bf\u1f36\u03bd\u03bf\u03bd \u03ba.\u03c4.\u03bb., 1 Tim. iv. 2, 3 \u03ba\u03c9\u03bb\u03c5\u03cc\u03bd\u03c4\u03c9\u03bd ... \u1f00\u03c0\u03ad\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9 \u03b2\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u1f03 \u1f41 \u0398\u03b5\u1f78\u03c2\n\u1f14\u03ba\u03c4\u03b9\u03c3\u03b5\u03bd \u03ba.\u03c4.\u03bb., Tit. i. 14 \u03bc\u1f74 \u03c0\u03c1\u03bf\u03c3\u03ad\u03c7\u03bf\u03bd\u03c4\u03b5\u03c2 ... \u1f10\u03bd\u03c4\u03bf\u03bb\u03b1\u1fd6\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd ...\n\u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03b1\u03b8\u03b1\u03c1\u1f70 \u03c4\u03bf\u1fd6\u03c2 \u03ba\u03b1\u03b8\u03b1\u03c1\u03bf\u1fd6\u03c2. The correct reading seems to be _\u03ba\u03b1\u1f76_\n\u1f10\u03bd \u03c0\u03cc\u03c3\u03b5\u03b9, thus connecting together the words between which there is a\nnatural affinity. Comp. Philo _Vit. Moys._ i. \u00a7 33 (II. p. 110)\n\u03b4\u03b5\u03c3\u03c0\u03bf\u03af\u03bd\u03b1\u03b9\u03c2 \u03c7\u03b1\u03bb\u03b5\u03c0\u03b1\u1fd6\u03c2 \u03c3\u03c5\u03bd\u03b5\u03b6\u03b5\u03c5\u03b3\u03bc\u03ad\u03bd\u03bf\u03c5 \u03b2\u03c1\u03ce\u03c3\u03b5\u03b9 \u03ba\u03b1\u1f76 \u03c0\u03cc\u03c3\u03b5\u03b9, Ign. _Trall._ 2 \u03bf\u1f50\n\u03b3\u1f70\u03c1 \u03b2\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c0\u03bf\u03c4\u1ff6\u03bd \u03b5\u1f30\u03c3\u1f76\u03bd \u03b4\u03b9\u03ac\u03ba\u03bf\u03bd\u03bf\u03b9.\n\n------------------------------------------------------------------------\n\nII. 17]\n\n\u1f10\u03bd \u03bc\u03ad\u03c1\u03b5\u03b9 \u1f11\u03bf\u03c1\u03c4\u1fc6\u03c2 \u1f22 \u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1\u03c2 \u1f22 \u03c3\u03b1\u03b2\u03b2\u03ac\u03c4\u03c9\u03bd, ^{17}\u1f05 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c3\u03ba\u1f76\u03b1\n\n                            17 \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c3\u03ba\u03b9\u1f70.\n\n\n\u1f10\u03bd \u03bc\u03ad\u03c1\u03b5\u03b9] '_in the matter of_,' etc.; comp. 2 Cor. iii. 10, ix. 3 \u1f10\u03bd \u03c4\u1ff7\n\u03bc\u03ad\u03c1\u03b5\u03b9 \u03c4\u03bf\u03cd\u03c4\u1ff3. The expression seems originally to mean 'in the division or\ncategory', and in classical writers most commonly occurs in connexion\nwith such words as \u03c4\u03b9\u03b8\u03ad\u03bd\u03b1\u03b9, \u03c0\u03bf\u03b9\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, \u1f00\u03c1\u03b9\u03b8\u03bc\u03b5\u1fd6\u03bd, etc.: comp. Demosth.\n_c. Aristocr._ \u00a7 148 \u1f45\u03c3\u03b1 ... \u03c3\u03c4\u03c1\u03b1\u03c4\u03af\u03c9\u03c4\u03b7\u03c2 \u1f62\u03bd \u1f10\u03bd \u03c3\u03c6\u03b5\u03bd\u03b4\u03bf\u03bd\u03ae\u03c4\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c8\u03b9\u03bb\u03bf\u1fe6\n\u03bc\u03ad\u03c1\u03b5\u03b9 ... \u1f10\u03c3\u03c4\u03c1\u03ac\u03c4\u03b5\u03c5\u03c4\u03b1\u03b9, i.e. 'in the capacity of.' Hence it gets to\nsignify more widely, as here, 'with respect to', 'by reason of': comp.\nPhilo _Quod det. pot. ins._ \u00a7 2 (I. p. 192) \u1f10\u03bd \u03bc\u03ad\u03c1\u03b5\u03b9 \u03bb\u03cc\u03b3\u03bf\u03c5 \u03c4\u03bf\u1fe6\n\u03c0\u03c1\u03bf\u03ba\u03cc\u03c0\u03c4\u03bf\u03bd\u03c4\u03bf\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78\u03bd \u03c0\u03ac\u03c4\u03b5\u03c1\u03b1 \u03ba\u03bf\u03c3\u03bc\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9, _in Flacc._ 20 (II. p. 542) \u1f45\u03c3\u03b1\n\u1f10\u03bd \u03bc\u03ad\u03c1\u03b5\u03b9 \u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03c9\u03c1\u03b5\u1fb6\u03c2 \u1f14\u03bb\u03b1\u03b2\u03bf\u03bd. But \u00c6lian _V. H._ viii. 3 \u03ba\u03c1\u03af\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2\n\u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03bd \u1f10\u03bd \u03c4\u1ff7 \u03bc\u03ad\u03c1\u03b5\u03b9 \u03c6\u03cc\u03bd\u03bf\u03c5, quoted by the commentators, is a false\nparallel: for \u03c6\u03cc\u03bd\u03bf\u03c5 is there governed by \u03ba\u03c1\u03af\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2 and \u1f10\u03bd \u03c4\u1ff7 \u03bc\u03ad\u03c1\u03b5\u03b9 means\n'in his turn'.\n\n\u1f11\u03bf\u03c1\u03c4\u1fc6\u03c2 \u03ba.\u03c4.\u03bb.] The same three words occur together, as an exhaustive\nenumeration of the sacred times among the Jews, in 1 Chron. xxiii. 31, 2\nChron. ii. 4, xxxi. 3, Ezek. xlv. 17, Hos. ii. 11, Justin _Dial._ 8, p.\n226; comp. Is. i. 13, 14. See also Gal. iv. 10 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c4\u03b7\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b5 \u03ba\u03b1\u1f76\n\u03bc\u1fc6\u03bd\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03b1\u03b9\u03c1\u03bf\u1f7a\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd\u03b9\u03b1\u03c5\u03c4\u03bf\u03cd\u03c2, where the first three words correspond\nto the three words used here, though the order is reversed. The \u1f11\u03bf\u03c1\u03c4\u03ae\nhere, like the \u03ba\u03b1\u03b9\u03c1\u03bf\u03af there, refers chiefly to the _annual_ festivals,\nthe passover, pentecost, etc. The \u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1 here describes more precisely\nthe _monthly_ festival, which is there designated more vaguely as \u03bc\u1fc6\u03bd\u03b5\u03c2.\nThe \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1 here gives by name the _weekly_ holy-day, which is there\nindicated more generally by \u1f21\u03bc\u03ad\u03c1\u03b1\u03b9.\n\n\u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1\u03c2] See Num. xxviii. 11 sq. The forms \u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1 and \u03bd\u03bf\u03c5\u03bc\u03b7\u03bd\u03af\u03b1 seem\nto be used indifferently in the common dialect, though the latter is\nmore common. In the Attic \u03bd\u03bf\u03c5\u03bc\u03b7\u03bd\u03af\u03b1 alone was held to be correct; see\nLobeck _Phryn._ p. 148. On the whole the preference should perhaps be\ngiven to \u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1\u03c2 here, as supported by some authorities which are\ngenerally trustworthy in matters of orthography, and as being the less\nusual form in itself.\n\n\u03c3\u03b1\u03b2\u03b2\u03ac\u03c4\u03c9\u03bd] '_a sabbath-day_', not, as the A.V., '_sabbath days_'; for the\ncoordinated words \u1f11\u03bf\u03c1\u03c4\u1fc6\u03c2, \u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1\u03c2, are in the singular. The word\n\u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1 is derived from the Aramaic (as distinguished from the Hebrew)\nform \u05e9\u05d1\u05ea\u05d0, and accordingly preserves the Aramaic termination in \u03b1. Hence\nit was naturally declined as a plural noun, \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1, \u03c3\u03b1\u03b2\u03b2\u03ac\u03c4\u03c9\u03bd. The\ngeneral use of \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1, when a single sabbath-day was meant, will appear\nfrom such passages as Jos. _Ant._ i. 1. 1 \u1f04\u03b3\u03bf\u03bc\u03b5\u03bd \u03c4\u1f74\u03bd \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd,\n\u03c0\u03c1\u03bf\u03c3\u03b1\u03b3\u03bf\u03c1\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u03b1\u1f50\u03c4\u1f74\u03bd \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1, _ib._ iii. 10. 1 \u1f11\u03b2\u03b4\u03cc\u03bc\u03b7\u03bd \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd \u1f25\u03c4\u03b9\u03c2\n\u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1 \u03ba\u03b1\u03bb\u03b5\u1fd6\u03c4\u03b1\u03b9, Plut. _Mor._ 169 C \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03b9 \u03c3\u03b1\u03b2\u03b2\u03ac\u03c4\u03c9\u03bd \u1f44\u03bd\u03c4\u03c9\u03bd \u1f10\u03bd\n\u1f00\u03b3\u03bd\u03ac\u03bc\u03c0\u03c4\u03bf\u03b9\u03c2 \u03ba\u03b1\u03b8\u03b5\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9, _ib._ 671 F \u03bf\u1f36\u03bc\u03b1\u03b9 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c4\u1ff6\u03bd \u03c3\u03b1\u03b2\u03b2\u03ac\u03c4\u03c9\u03bd \u1f11\u03bf\u03c1\u03c4\u1f74\u03bd\n\u03bc\u1f74 \u03c0\u03b1\u03bd\u03c4\u03ac\u03c0\u03b1\u03c3\u03b9\u03bd \u1f00\u03c0\u03c1\u03bf\u03c3\u03b4\u03af\u03bf\u03bd\u03c5\u03c3\u03bf\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9, Hor. _Sat._ i. 9. 69 'hodie tricesima\nsabbata'. In the New Testament \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1 is only once used distinctly of\nmore than a single day, and there the plurality of meaning is brought\nout by the attached numeral; Acts xvii. 2 \u1f10\u03c0\u1f76 \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1 \u03c4\u03c1\u03af\u03b1.\n\nOn the observance of days and seasons see again Gal. iv. 10, Rom. xiv.\n5, 6. A strong anti-Judaic view on the subject is expressed in the\n_Epist. ad Diogn._ \u00a7 4. Origen _c. Cels._ viii. 21, 22, after referring\nto Thucyd. i. 70 \u03bc\u03ae\u03c4\u03b5 \u1f11\u03bf\u03c1\u03c4\u1f74\u03bd \u1f04\u03bb\u03bb\u03bf \u03c4\u03b9 \u1f21\u03b3\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u1f22 \u03c4\u1f78 \u03c4\u1f70 \u03b4\u03ad\u03bf\u03bd\u03c4\u03b1 \u03c0\u03c1\u1fb6\u03be\u03b1\u03b9,\nsays \u1f41 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, \u1f00\u03b5\u1f76 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bb\u03cc\u03b3\u03bf\u03b9\u03c2 \u1f62\u03bd \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03c1\u03b3\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fd6\u03c2\n\u03b4\u03b9\u03b1\u03bd\u03bf\u03ae\u03bc\u03b1\u03c3\u03b9 \u03c4\u03bf\u1fe6 \u03c4\u1fc7 \u03c6\u03cd\u03c3\u03b5\u03b9 \u03ba\u03c5\u03c1\u03af\u03bf\u03c5 \u03bb\u03cc\u03b3\u03bf\u03c5 \u0398\u03b5\u03bf\u1fe6, \u1f00\u03b5\u03af \u1f10\u03c3\u03c4\u03b9\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2\n\u1f21\u03bc\u03ad\u03c1\u03b1\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03b5\u1f76 \u1f04\u03b3\u03b5\u03b9 \u03ba\u03c5\u03c1\u03b9\u03b1\u03ba\u1f70\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2, and he then goes on to explain\nwhat is the \u03c0\u03b1\u03c1\u03b1\u03c3\u03ba\u03b5\u03c5\u03ae, the \u03c0\u03ac\u03c3\u03c7\u03b1, the \u03c0\u03b5\u03bd\u03c4\u03b5\u03ba\u03bf\u03c3\u03c4\u03ae, of such a man. The\nobservance of sacred times was an integral part of the old dispensation.\nUnder the new they have ceased to have any value, except as a means to\nan end. The great principle that 'the sabbath was made for man and not\nman for the sabbath', though underlying the Mosaic ordinances, was first\ndistinctly pronounced by our Lord. The setting apart of special days for\nthe service of God is a confession of our imperfect state, an avowal\nthat we cannot or do not devote our whole time to Him. Sabbaths will\nthen ultimately be superseded, when our life becomes one eternal\nsabbath. Meanwhile the Apostle's rebuke warns us against attributing to\nany holy days whatever a meaning and an importance which is alien to the\nspirit of the New Covenant. Bengel on the text writes, 'Sabbatum non\nlaudatur, non imperatur; dominica memoratur, non pr\u00e6cipitur. Qui\nprofundius in mundi negotiis h\u00e6rent, his utilis et necessarius est dies\ndefinitus: qui semper sabbatizant, majori libertate gaudent'. Yes: but\nthese last are just they who will most scrupulously restrict their\nliberty, so as \u1f00\u03c0\u03c1\u03cc\u03c3\u03ba\u03bf\u03c0\u03bf\u03b9 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9.\n\n17. Two ideas are prominent in this image. (1) The contrast between the\nordinances of the Law and the teaching of the Gospel, as the shadow and\nthe substance respectively; Philo _de Conf. ling._ 37 (I. p. 434)\n\u03bd\u03bf\u03bc\u03af\u03c3\u03b1\u03bd\u03c4\u03b1\u03c2 \u03c4\u1f70 \u03bc\u1f72\u03bd \u1fe4\u03b7\u03c4\u1f70 \u03c4\u1ff6\u03bd \u03c7\u03c1\u03b7\u03c3\u03bc\u1ff6\u03bd \u03c3\u03ba\u03b9\u03ac\u03c2 \u03c4\u03b9\u03bd\u03b1\u03c2 \u1f61\u03c3\u03b1\u03bd\u03b5\u1f76 _\u03c3\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd_\n\u03b5\u1f36\u03bd\u03b1\u03b9, Joseph. _B.J._ ii. 2. 5 \u03c3\u03ba\u03b9\u1f70\u03bd \u03b1\u1f30\u03c4\u03b7\u03c3\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03c2 \u1f27\u03c2 \u1f25\u03c1\u03c0\u03b1\u03c3\u03b5\u03bd\n\u1f11\u03b1\u03c5\u03c4\u1ff7 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1; comp. Philo _in Flacc._ 19 (II. p. 541) \u03c3\u03ba\u03b9\u1f70 \u03c0\u03c1\u03b1\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd\n\u1f04\u03c1' \u1f26\u03c3\u03b1\u03bd, \u03bf\u1f50 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b1. (2) The conception of the shadow as thrown before\nthe substance (\u1f21 \u03b4\u1f72 \u03c3\u03ba\u03b9\u1f70 \u03c0\u03c1\u03bf\u03c4\u03c1\u03ad\u03c7\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2, says a Greek\ncommentator), so that the Law was a type and presage of the Gospel; Heb.\nx. 1 \u03c3\u03ba\u03b9\u1f70\u03bd \u1f14\u03c7\u03c9\u03bd \u1f41 \u03bd\u03cc\u03bc\u03bf\u03c2 \u03c4\u1ff6\u03bd \u03bc\u03b5\u03bb\u03bb\u03cc\u03bd\u03c4\u03c9\u03bd \u1f00\u03b3\u03b1\u03b8\u1ff6\u03bd (comp. viii. 5). Thus it\nimplies both the _unsubstantiality_ and the _supersession_ of the Mosaic\nritual.\n\n\u1f05] '_which things_', whether distinctions of meats or observances of\ntimes. If the other reading \u1f45 be taken, it will refer to the preceding\nsentence generally, as if the antecedent were 'the whole system of\nordinances'.\n\n------------------------------------------------------------------------\n\nII. 18]\n\n\u03c4\u1ff6\u03bd \u03bc\u03b5\u03bb\u03bb\u03cc\u03bd\u03c4\u03c9\u03bd, \u03c4\u1f78 \u03b4\u1f72 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. ^{18}\u03bc\u03b7\u03b4\u03b5\u1f76\u03c2\n\n\n\u03c4\u1f78 \u03b4\u1f72 \u03c3\u1ff6\u03bc\u03b1 \u03ba.\u03c4.\u03bb.] As the shadow belonged to Moses, so '_the substance\nbelongs to Christ_'; i.e. the reality, the antitype, in each case is\nfound in the Christian dispensation. Thus the passover typifies the\natoning sacrifice; the unleavened bread, the purity and sincerity of the\ntrue believer; the pentecostal feast, the ingathering of the first\nfruits; the sabbath, the rest of God's people; etc.\n\n18. The Christian's career is the contest of the stadium (\u03b4\u03c1\u03cc\u03bc\u03bf\u03c2, Acts\nxx. 24, 2 Tim. iv. 7); Christ is the umpire, the dispenser of the\nrewards (2 Tim. iv. 8); life eternal is the bay wreath, the victor's\nprize (\u03b2\u03c1\u03b1\u03b2\u03b5\u1fd6\u03bf\u03bd, 1 Cor. ix. 24, Phil. iii. 14). The Colossians were in a\nfair way to win this prize; they had entered the lists duly; they were\nrunning bravely: but the false teachers, thrusting themselves in the\nway, attempted to trip them up or otherwise impede them in the race, and\nthus to rob them of their just reward. For the idea of \u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03c4\u03c9\ncompare especially Gal. v. 7 \u1f10\u03c4\u03c1\u03ad\u03c7\u03b5\u03c4\u03b5 \u03ba\u03b1\u03bb\u1ff6\u03c2\u00b7 \u03c4\u03af\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u1f10\u03bd\u03ad\u03ba\u03bf\u03c8\u03b5\u03bd \u03ba.\u03c4.\u03bb.\n\n------------------------------------------------------------------------\n\nII. 18]\n\n\u1f51\u03bc\u1fb6\u03c2 \u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03c5\u03ad\u03c4\u03c9 \u03b8\u03ad\u03bb\u03c9\u03bd \u1f10\u03bd \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3 \u03ba\u03b1\u1f76\n\n\n\u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03c5\u03ad\u03c4\u03c9] '_rob of the prize, the_ \u03b2\u03c1\u03b1\u03b2\u03b5\u1fd6\u03bf\u03bd'; comp. Demosth. _Mid._\np. 544 (one of the documents) \u1f10\u03c0\u03b9\u03c3\u03c4\u03ac\u03bc\u03b5\u03b8\u03b1 \u03a3\u03c4\u03c1\u03ac\u03c4\u03c9\u03bd\u03b1 \u1f51\u03c0\u1f78 \u039c\u03b5\u03b9\u03b4\u03af\u03bf\u03c5\n_\u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03c5\u03b8\u03ad\u03bd\u03c4\u03b1_ \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1f70 \u03b4\u03af\u03ba\u03b1\u03b9\u03b1 \u1f00\u03c4\u03b9\u03bc\u03c9\u03b8\u03ad\u03bd\u03c4\u03b1, which\npresents a close parallel to the use of \u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd here. See also\nEustath. _in Il._ i. 403 sq. (p. 43) \u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9 \u03b1\u1f50\u03c4\u03cc\u03bd, \u1f65\u03c2 \u03c6\u03b1\u03c3\u03b9\u03bd \u03bf\u1f31\n\u03c0\u03ac\u03bb\u03b1\u03b9\u03bf\u03b9, ib. _Opusc._ 277, etc. The false teachers at Coloss\u00e6 are not\nregarded as umpires nor as successful rivals, but simply as persons\nfrustrating those who otherwise would have won the prize. The word\n\u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd is wide enough to include such. The two compounds\n\u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd and \u03c0\u03b1\u03c1\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd (Plut. _Mor._ p. 535 C \u03bf\u1f31\n\u03c0\u03b1\u03c1\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b3\u1ff6\u03c3\u03b9) only differ in this respect, that\n_deprivation_ is the prominent idea in the former word and _trickery_ in\nthe latter. Jerome, _Epist._ cxxi. _ad Algas._ (I. p. 879), sets down\nthis word, which he wrongly interprets 'bravium accipiat adversum vos',\nas one of St Paul's Cilicisms. The passages quoted (whether the document\nin the Midias be authentic or not) are sufficient to show that this\nstatement is groundless.\n\n\u03b8\u03ad\u03bb\u03c9\u03bd \u1f10\u03bd] '_taking delight in_', '_devoting himself to_'. The expression\nis common in the LXX, most frequently as a translation of \u05d7\u05e4\u05e5 \u05d1\u05f4, 1\nSam.\u05d7\u05e4\u05e5 xviii. 22, 2 Sam. xv. 26, 1 Kings x. 9, 2 Chron. ix. 8, Ps. cxi.\n1, cxlvi. 10, but in one passage of \u05e8\u05e6\u05d4 \u05d1\u05f4, 1 Chron. xxviii. 4. So too\n_Test. xii. Patr._ Asher 1 \u1f10\u1f70\u03bd \u03bf\u1f56\u03bd \u1f21 \u03c8\u03c5\u03c7\u1f74 \u03b8\u03ad\u03bb\u1fc3 \u1f10\u03bd \u03ba\u03b1\u03bb\u1ff7. Comp. also 1\nMacc. iv. 42 \u03b8\u03b5\u03bb\u03b7\u03c4\u1f70\u03c2 \u03bd\u03cc\u03bc\u03bf\u03c5, and see \u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1 below. Against this\nconstruction no valid objection has been urged. Otherwise \u03b8\u03ad\u03bb\u03c9\u03bd is taken\nabsolutely, and various senses have been assigned to it, such as\n'imperiously' or 'designedly' or 'wilfully' or 'gladly, readily'; but\nthese are either unsupported by usage or inappropriate to the context.\nLeclerc (_ad loc._) and Bentley (_Crit. Sacr._ p. 59) conjectured\n\u03b8\u03ad\u03bb\u03b3\u03c9\u03bd; Toup (_Emend. in Suid._ II. p. 63) more plausibly \u1f10\u03bb\u03b8\u03ce\u03bd; but the\npassages quoted show that no correction is needed.\n\n\u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3] Humility is a vice with heathen moralists, but a virtue\nwith Christian Apostles; see the note on Phil. ii. 3. In this passage,\nwhich (with ver. 23) forms the sole exception to the general language of\nthe Apostles, the divergence is rather apparent than real. The\ndisparagement is in the accompaniments and not in the word itself.\nHumility, when it becomes self-conscious, ceases to have any value; and\nself-consciousness at least, if not affectation, is implied by \u03b8\u03ad\u03bb\u03c9\u03bd \u1f10\u03bd.\nMoreover the character of the \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7 in this case is further\ndefined as \u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd, which was altogether a perversion of\nthe truth.\n\n------------------------------------------------------------------------\n\nII. 18]\n\n\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u1fb3 \u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd, \u1f03 \u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd \u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03c9\u03bd, \u03b5\u1f30\u03ba\u1fc7 \u03c6\u03c5\u03c3\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2\n\n\n\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u1fb3] This word is closely connected with the preceding by the\nvinculum of the same preposition. There was an officious parade of\nhumility in selecting these lower beings as intercessors, rather than\nappealing directly to the throne of grace. The word refers properly to\nthe external rites of religion, and so gets to signify an\nover-scrupulous devotion to external forms; as in Philo _Quod det. pot.\nins._ 7 (i. p. 195) \u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1\u03bd \u1f00\u03bd\u03c4\u1f76 \u1f41\u03c3\u03af\u03bf\u03c4\u03b7\u03c4\u03bf\u03c2 \u1f21\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2, Plut. _Vit.\nAlex._ 2 \u03b4\u03bf\u03ba\u03b5\u1fd6 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03cd\u03b5\u03b9\u03bd \u1f44\u03bd\u03bf\u03bc\u03b1 \u03c4\u03b1\u1fd6\u03c2 _\u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03bf\u03b9\u03c2_ \u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9\n\u03ba\u03b1\u1f76 _\u03c0\u03b5\u03c1\u03b9\u03ad\u03c1\u03b3\u03bf\u03b9\u03c2_ \u1f31\u03b5\u03c1\u03bf\u03c5\u03c1\u03b3\u03af\u03b1\u03b9\u03c2: comp. Acts xxvi. 5, and see the\nwell-known remarks of Coleridge on James i. 26, 27, in _Aids to\nReflection_ p. 14. In the LXX \u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03cd\u03b5\u03b9\u03bd, \u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1, together occur four\ntimes (Wisd. xi. 16, xiv. 16, 18, 27), and in all these examples the\nreference is to idolatrous or false worship. Indeed generally the usage\nof the word exhibits a tendency to a bad sense.\n\n\u03c4\u1ff6\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd] For the angelology and angelolatry of these Colossian false\nteachers, more especially in its connexion with Essene teaching, see the\nintroduction, pp. 89 sq., 101 sq., 110, 181 sq. For the prominence which\nwas given to angelology in the speculations of the Jews generally, see\nthe _Preaching of Peter_ quoted in Clem. Alex. _Strom._ vi. 5 (p. 760)\n\u03bc\u03b7\u03b4\u1f72 \u03ba\u03b1\u03c4\u1f70 \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03bf\u03c5\u03c2 \u03c3\u03ad\u03b2\u03b5\u03c3\u03b8\u03b5, \u03ba\u03b1\u1f76 \u03b3\u1f70\u03c1 \u03b5\u03ba\u03b5\u1fd6\u03bd\u03bf\u03b9 ... \u03bf\u1f50\u03ba \u1f10\u03c0\u03af\u03c3\u03c4\u03b1\u03bd\u03c4\u03b1\u03b9\n\u03bb\u03b1\u03c4\u03c1\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f00\u03b3\u03b3\u03ad\u03bb\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03c1\u03c7\u03b1\u03b3\u03b3\u03ad\u03bb\u03bf\u03b9\u03c2, Celsus in Orig. _c. Cels._ v. 6\n(i. p. 580) \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03bf\u1f56\u03bd \u03c4\u1ff6\u03bd \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd \u03b8\u03b1\u03c5\u03bc\u03ac\u03b6\u03b5\u03b9\u03bd \u1f04\u03be\u03b9\u03bf\u03bd, \u03b5\u1f30 \u03c4\u1f78\u03bd \u03bc\u1f72\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1f78\u03bd\n\u03ba\u03b1\u1f76 \u03c4\u03bf\u1f7a\u03c2 \u1f10\u03bd \u03c4\u1ff7\u03b4\u03b5 \u1f00\u03b3\u03b3\u03ad\u03bb\u03bf\u03c5\u03c2 \u03c3\u03ad\u03b2\u03bf\u03c5\u03c3\u03b9 \u03ba.\u03c4.\u03bb., comp. _ib._ i. 26 (p. 344).\nFrom Jews it naturally spread to Judaizing Christians; e.g. _Clem. Hom._\niii. 36 \u1f00\u03b3\u03b3\u03ad\u03bb\u03c9\u03bd \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b1 \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03b5\u03b9\u03bd, viii. 12 sq., _Test. xii. Patr._ Levi\n3 (quoted above on i. 16). The interest however extended to more\northodox circles, as appears from the strange passage in Ignat. _Trall._\n5 \u03bc\u1f74 \u03bf\u1f50 \u03b4\u03cd\u03bd\u03b1\u03bc\u03b1\u03b9 \u03c4\u1f70 \u1f10\u03c0\u03bf\u03c5\u03c1\u03ac\u03bd\u03b9\u03b1 \u03b3\u03c1\u03ac\u03c8\u03b1\u03b9; ... \u03b4\u03cd\u03bd\u03b1\u03bc\u03b1\u03b9 \u03bd\u03bf\u03b5\u1fd6\u03bd \u03c4\u1f70 \u1f10\u03c0\u03bf\u03c5\u03c1\u03ac\u03bd\u03b9\u03b1 \u03ba\u03b1\u1f76\n\u03c4\u1f70\u03c2 \u03c4\u03bf\u03c0\u03bf\u03b8\u03b5\u03c3\u03af\u03b1\u03c2 \u03c4\u1f70\u03c2 \u1f00\u03b3\u03b3\u03b5\u03bb\u03b9\u03ba\u1f70\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f70\u03c2 \u03c3\u03c5\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2 \u03c4\u1f70\u03c2 \u1f00\u03c1\u03c7\u03bf\u03bd\u03c4\u03b9\u03ba\u03ac\u03c2 \u03ba.\u03c4.\u03bb. Of\nangelology among Gnostic sects see Iren. ii. 30. 6, ii. 32. 5, Orig. _c.\nCels._ vi. 30 sq. (I. p. 653), Clem. Alex. _Exc. Theod._ p. 970 sq.,\n_Pistis Sophia_ pp. 2, 19, 23, etc.\n\n\u1f03 \u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd \u03ba.\u03c4.\u03bb.] literally '_invading what he has seen,_' which is\ngenerally explained to mean 'parading' or 'poring over his visions'. For\nthis sense of \u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd, which takes either a genitive or a dative or\nan accusative, comp. Philo _de Plant. Noe_ ii. 19 (i. p. 341) \u03bf\u1f31\n\u03c0\u03c1\u03bf\u03c3\u03c9\u03c4\u03ad\u03c1\u03c9 \u03c7\u03c9\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c4\u1ff6\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03b7\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u1f10\u03c0\u1f76 \u03c0\u03bb\u03ad\u03bf\u03bd \u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03c2 \u03b1\u1f50\u03c4\u03b1\u1fd6\u03c2, 2\nMacc. ii. 30 \u03c4\u1f78 \u03bc\u1f72\u03bd \u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd \u03ba\u03b1\u03af \u03c0\u03b5\u03c1\u1f76 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03c0\u03bf\u03b9\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u03bb\u03cc\u03b3\u03bf\u03bd \u03ba\u03b1\u1f76\n\u03c0\u03bf\u03bb\u03c5\u03c0\u03c1\u03b1\u03b3\u03bc\u03bf\u03bd\u03b5\u1fd6\u03bd \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03bc\u03ad\u03c1\u03bf\u03c2. At a later date this sense becomes\ncommon, e.g. Nemesius _de Nat. Hom._ p. 64 (ed. Matth\u00e6i) \u03bf\u1f50\u03c1\u03b1\u03bd\u1f78\u03bd\n\u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03b9 \u03c4\u1fc7 \u03b8\u03b5\u03c9\u03c1\u03af\u1fb3. In Xen. _Symp._ iv. 27 \u1f10\u03bd \u03c4\u1ff7 \u03b1\u1f50\u03c4\u1ff7 \u03b2\u03b9\u03b2\u03bb\u03af\u1ff7 \u1f00\u03bc\u03c6\u03cc\u03c4\u03b5\u03c1\u03bf\u03b9\n\u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03c4\u03ad \u03c4\u03b9, the reading may be doubtful. But though \u1f03 \u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd singly\nmight mean 'his visions', and \u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03c9\u03bd 'busying himself with', the\ncombination 'invading what he has seen', thus interpreted, is so harsh\nand incongruous as to be hardly possible; and there was perhaps some\ncorruption in the text prior to all existing authorities (see the note\non Phil. ii. 1 for a parallel case). Did the Apostle write )\u03ad\u03c9\u03c1\u1fb3 (or\n\u03b1\u1f34\u03c9\u03c1\u1fb3) \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03c9\u03bd? In this case the existing text ==\u03b1\u03b5\u03c9\u03c1\u03b1\u03ba\u03b5\u03bd\u03b5\u03bc\n\u03b2\u03b1\u03c4\u03b5\u03c5\u03c9\u03bd== might be explained partly by an attempt to correct the form\n)\u03b5\u03ce\u03c1\u1fb3 into \u03b1\u1f30\u03ce\u03c1\u1fb3 or conversely, and partly by the perplexity of\ntranscribers when confronted with such unusual words. This reading had\nsuggested itself to me independently without the knowledge that, so far\nas regards the latter word, it had been anticipated by others in the\nconjecture \u1f03 \u1f11\u03ce\u03c1\u03b1 (or \u1f03 \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd) \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03c9\u03bd. The word \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u1fd6\u03bd 'to\nwalk on emptiness', 'to tread the air', and so metaphorically (like\n\u03b1\u1f10\u03c1\u03bf\u03b2\u03b1\u03c4\u03b5\u1fd6\u03bd, \u03b1\u1f30\u03b8\u03b5\u03c1\u03bf\u03b2\u03b1\u03c4\u03b5\u1fd6\u03bd, \u03b1\u1f30\u03b8\u03b5\u03c1\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u1fd6\u03bd, etc.) 'to indulge in vain\nspeculations', is not an uncommon word. For its metaphorical sense\nespecially see Plut. _Mor._ p. 336 F \u03bf\u1f55\u03c4\u03c9\u03c2 \u1f10\u03c1\u03ad\u03bc\u03b2\u03b5\u03c4\u03bf \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03bf\u1fe6\u03bd \u03ba\u03b1\u1f76\n\u03c3\u03c6\u03b1\u03bb\u03bb\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u1f51\u03c0' \u1f00\u03bd\u03b1\u03c1\u03c7\u03af\u03b1\u03c2 \u03c4\u1f78 \u03bc\u03ad\u03b3\u03b5\u03b8\u03bf\u03c2 \u03b1\u1f50\u03c4\u1fc6\u03c2, Basil. _Op._ I. p. 135 \u03c4\u1f78\u03bd\n\u03bd\u03bf\u1fe6\u03bd ... \u03bc\u03c5\u03c1\u03af\u03b1 \u03c0\u03bb\u03b1\u03bd\u03b7\u03b8\u03ad\u03bd\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03c0\u03bf\u03bb\u03bb\u1f70 \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03ae\u03c3\u03b1\u03bd\u03c4\u03b1 \u03ba.\u03c4.\u03bb., _ib._ I. p.\n596 \u03c3\u03bf\u1fe6 \u03b4\u1f72 \u03bc\u1f74 \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03ad\u03b9\u03c4\u03c9 \u1f41 \u03bd\u03bf\u1fe6\u03c2, Synes. _de Insomn._ p. 156 \u03bf\u1f54\u03c4\u03b5 \u03b3\u1f70\u03c1\n\u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c5\u03c2 \u1f10\u03be\u03ae\u03bd\u03b5\u03b3\u03ba\u03b1\u03bd. Though the precise form\n\u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd does not occur, yet it is unobjectionable in itself. For\nthe other word which I have ventured to suggest, \u1f10\u03ce\u03c1\u1fb3 or \u03b1\u1f30\u03ce\u03c1\u1fb3, see\nPhilo _de Somn._ ii. 6 (I. p. 665) _\u1f51\u03c0\u03bf\u03c4\u03c5\u03c6\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2_ \u1f51\u03c0' \u03b1\u1f30\u03ce\u03c1\u03b1\u03c2\n\u03c6\u03c1\u03b5\u03bd\u1ff6\u03bd \u03ba\u03b1\u1f76 _\u03ba\u03b5\u03bd\u03bf\u1fe6_ \u03c6\u03c5\u03c3\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2, _ib._ \u00a7 9 (p. 667) \u03c4\u1f74\u03bd \u1f10\u03c0'\n_\u03b1\u1f30\u03ce\u03c1\u03b1\u03c2_ \u03c6\u03bf\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03bd _\u03ba\u03b5\u03bd\u1f74\u03bd_ \u03b4\u03cc\u03be\u03b1\u03bd, _Quod Deus immut._ \u00a7 36 (I.\np. 298) \u1f61\u03c3\u03c0\u03b5\u03c1 \u1f10\u03c0' \u03b1\u1f30\u03ce\u03c1\u03b1\u03c2 \u03c4\u03b9\u03bd\u1f78\u03c2 \u03c8\u03b5\u03c5\u03b4\u03bf\u1fe6\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03b2\u03b5\u03b2\u03b1\u03af\u03bf\u03c5 \u03b4\u03cc\u03be\u03b7\u03c2 \u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9\n_\u03ba\u03b1\u03c4\u1f70 \u03ba\u03b5\u03bd\u03bf\u1fe6 \u03b2\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1_. The first and last passages more especially\npresent striking parallels, and show how germane to St Paul's subject\nthese ideas of 'suspension or balancing in the air' (\u1f10\u03ce\u03c1\u03b1 or \u03b1\u1f30\u03ce\u03c1\u03b1) and\n'treading the void' (\u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd) would be, as expressing at once the\nspiritual pride and the emptiness of these speculative mystics; see also\n_de Somn._ ii. 2 (p. 661) \u1f10\u03bc\u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c4\u1fc6\u03c2 _\u03ba\u03b5\u03bd\u1fc6\u03c2_ \u03b4\u03cc\u03be\u03b7\u03c2, \u1f10\u03c6'\n\u1f23\u03bd, \u1f61\u03c2 \u1f10\u03c6' \u1f05\u03c1\u03bc\u03b1, \u03b4\u03b9\u1f70 \u03c4\u1f78 \u03ba\u03bf\u1fe6\u03c6\u03bf\u03bd _\u1f00\u03bd\u03b1\u03b2\u03b1\u03af\u03bd\u03b5\u03b9, \u03c6\u03c5\u03c3\u03ce\u03bc\u03b5\u03bd\u03bf\u03c2_ \u03ba\u03b1\u1f76 \u03bc\u03b5\u03c4\u03ad\u03c9\u03c1\u03bf\u03bd\n_\u1f90\u03c9\u03c1\u03b7\u03ba\u1f7c\u03c2_ \u1f11\u03b1\u03c5\u03c4\u03cc\u03bd. The substantive, \u1f10\u03ce\u03c1\u03b1 or \u03b1\u1f30\u03ce\u03c1\u03b1, is used sometimes\nof the instrument for suspending, sometimes of the position of\nsuspension. In this last sense it describes the poising of a bird, the\nfloating of a boat on the waters, the balancing on a rope, and the like.\nHence its expressiveness when used as a metaphor.\n\nIn the received text a negative is inserted, \u1f03 \u03bc\u1f74 \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd \u1f10\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03c9\u03bd.\nThis gives a very adequate sense '_intruding into those things which he\nhas not seen_'; \u03bf\u1f50 \u03b3\u1f70\u03c1 \u03b5\u1f36\u03b4\u03b5\u03bd \u1f00\u03b3\u03b3\u03ad\u03bb\u03bf\u03c5\u03c2, says Chrysostom, \u03ba\u03b1\u1f76 \u03bf\u1f55\u03c4\u03c9\n\u03b4\u03b9\u03ac\u03ba\u03b5\u03b9\u03c4\u03b1\u03b9 \u1f61\u03c2 \u1f30\u03b4\u03ce\u03bd: comp. Ezek. xiii. 3 \u03bf\u1f50\u1f70\u03b9 \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03c1\u03bf\u03c6\u03b7\u03c4\u03b5\u03cd\u03bf\u03c5\u03c3\u03b9\u03bd \u1f00\u03c0\u1f78\n\u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03ba\u03b1\u03b8\u03cc\u03bb\u03bf\u03c5 \u03bc\u1f74 \u03b2\u03bb\u03ad\u03c0\u03bf\u03c5\u03c3\u03b9\u03bd. But, though the difficulty is\nthus overcome, this cannot be regarded as the original reading of the\ntext, the authorities showing that the negative was an after insertion.\nSee the detached note on various readings.\n\nFor the form \u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd, which is better supported here than \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd, see\nthe note on ii. 1.\n\n------------------------------------------------------------------------\n\nII. 19]\n\n\u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03bd\u03bf\u1f78\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, ^{19}\u03ba\u03b1\u1f76 \u03bf\u1f50\n\n\n\u03b5\u1f30\u03ba\u1fc7 \u03c6\u03c5\u03c3\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2] '_vainly puffed up_.' Their profession of humility was\na cloke for excessive pride: for, as St Paul says elsewhere (1 Cor.\nviii. 1), \u1f21 \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2 \u03c6\u03c5\u03c3\u03b9\u03bf\u1fd6. It may be questioned whether \u03b5\u1f30\u03ba\u1fc7 should be\nconnected with the preceding or the following words. Its usual position\nin St Paul, before the words which it qualifies (Rom. xiii. 4, 1 Cor.\nxv. 2, Gal. iv. 11; there is an exceptional reason for the exceptional\nposition in Gal. iii. 4), points to the latter construction.\n\n\u03c4\u03bf\u1fe6 \u03bd\u03bf\u1f78\u03c2 \u03ba.\u03c4.\u03bb.] '_the mind of his flesh_', i.e. unenlightened by the\nSpirit; comp. Rom. viii. 7 \u03c4\u1f78 \u03c6\u03c1\u03cc\u03bd\u03b7\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2. It would seem that the\nApostle is here taking up some watchword of the false teachers. They\ndoubtless boasted that they were directed \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03bd\u03cc\u03bf\u03c2. Yes, he answers,\nbut it is \u1f41 \u03bd\u03bf\u1fe6\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u1f51\u03bc\u1ff6\u03bd. Compare Rev. ii. 24, where the\nfavourite Gnostic boast \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd \u03c4\u1f70 \u03b2\u03b1\u03b8\u03ad\u03b1 is characterized by the\naddition of \u03c4\u03bf\u1fe6 \u03a3\u03b1\u03c4\u03b1\u03bd\u1fb6 (see _Galatians_ p. 298 note 3). Comp. August.\n_Conf._ x. 67 'Quem invenirem qui me reconciliaret tibi? Ambiendum mihi\nfuit ad angelos? Qua prece? quibus sacramentis? Multi conantes ad te\nredire, neque per se ipsos valentes, sicut audio, tentaverunt h\u00e6c et\ninciderunt in desiderium curiosarum visionum et digni habiti sunt\nillusionibus. Elati enim te qu\u00e6rebant doctrin\u00e6 fastu, etc.'\n\n19. \u03bf\u1f50 \u03ba\u03c1\u03b1\u03c4\u1ff6\u03bd] '_not holding fast_.' This is the most common\nconstruction and meaning of \u03ba\u03c1\u03b1\u03c4\u03b5\u1fd6\u03bd in the New Testament; e.g. Mark vii.\n8 _\u1f00\u03c6\u03ad\u03bd\u03c4\u03b5\u03c2_ \u03c4\u1f74\u03bd \u1f10\u03bd\u03c4\u03bf\u03bb\u1f74\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 _\u03ba\u03c1\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5_ \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd\n\u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd; comp. Cant. iii. 4 \u1f11\u1fe6\u03c1\u03bf\u03bd \u1f43\u03bd \u1f20\u03b3\u03ac\u03c0\u03b7\u03c3\u03b5\u03bd \u1f21 \u03c8\u03c5\u03c7\u03ae \u03bc\u03bf\u03c5, \u1f10\u03ba\u03c1\u03ac\u03c4\u03b7\u03c3\u03b1\n\u03b1\u1f50\u03c4\u1f78\u03bd \u03ba\u03b1\u1f76 \u03bf\u1f50\u03ba \u1f00\u03c6\u1fc6\u03ba\u03b1 \u03b1\u1f50\u03c4\u03cc\u03bd.\n\n------------------------------------------------------------------------\n\nII. 19]\n\n\u03ba\u03c1\u03b1\u03c4\u1ff6\u03bd \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae\u03bd, \u1f10\u03be \u03bf\u1f57 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd \u1f01\u03c6\u1ff6\u03bd\n\n\n\u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae\u03bd] '_the Head_' regarded as a title, so that a person is at\nonce suggested, and the relative which follows is masculine, \u1f10\u03be \u03bf\u1f57;\ncomp. the parallel passage, Ephes. iv. 16 \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae, \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u1f10\u03be\n\u03bf\u1f57 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03ba.\u03c4.\u03bb. The supplication and worship of angels is a\nsubstitution of inferior members for the Head, which is the only source\nof spiritual life and energy. See the introduction pp. 34, 78, 101 sq.,\n181 sq.\n\n\u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd \u1f01\u03c6\u1ff6\u03bd \u03ba.\u03c4.\u03bb.] '_through the junctures and ligaments_.' Galen,\nwhen describing the structure of the human frame, more than once\nspecifies the elements of union as twofold: the body owes its\ncompactness partly to the _articulation_, partly to the _attachment_;\ne.g. _Op._ II. p. 734 (ed. K\u00fchn) \u1f14\u03c3\u03c4\u03b9 \u03b4\u1f72 \u1f41 \u03c4\u03c1\u03cc\u03c0\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03bd\u03b8\u03ad\u03c3\u03b5\u03c9\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd\n\u03b4\u03b9\u03c4\u03c4\u1f78\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03b3\u03ad\u03bd\u03bf\u03c2, \u1f41 \u03bc\u1f72\u03bd \u1f15\u03c4\u03b5\u03c1\u03bf\u03c2 \u03ba\u03b1\u03c4\u1f70 _\u1f04\u03c1\u03b8\u03c1\u03bf\u03bd_, \u1f41 \u03b4\u1f72 \u1f15\u03c4\u03b5\u03c1\u03bf\u03c2 \u03ba\u03b1\u03c4\u1f70\n_\u03c3\u03cd\u03bc\u03c6\u03c5\u03c3\u03b9\u03bd_. Similarly, though with a more general reference,\nAristotle speaks of two kinds of union, which he describes as \u1f01\u03c6\u03ae\n'contact' and \u03c3\u03cd\u03bc\u03c6\u03c5\u03c3\u03b9\u03c2 'cohesion' respectively; _Metaph._ iv. 4 (p.\n1014) \u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03b4\u1f72 _\u03c3\u03cd\u03bc\u03c6\u03c5\u03c3\u03b9\u03c2 \u1f01\u03c6\u1fc6\u03c2_\u00b7 \u1f15\u03bd\u03b8\u03b1 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u03bf\u1f50\u03b8\u1f72\u03bd \u03c0\u03b1\u03c1\u1f70 \u03c4\u1f74\u03bd \u1f01\u03c6\u1f74\u03bd\n\u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u1f00\u03bd\u03ac\u03b3\u03ba\u03b7 \u03b5\u1f36\u03bd\u03b1\u03b9, \u1f10\u03bd \u03b4\u1f72 \u03c4\u03bf\u1fd6\u03c2 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c6\u03cd\u03ba\u03bf\u03c3\u03b9\u03bd \u1f10\u03c3\u03c4\u03af \u03c4\u03b9 \u1f13\u03bd \u03c4\u1f78 \u03b1\u1f50\u03c4\u1f78 \u1f10\u03bd\n\u1f00\u03bc\u03c6\u03bf\u1fd6\u03bd \u1f43 \u03c0\u03bf\u03b9\u03b5\u1fd6 \u1f00\u03bd\u03c4\u1f76 \u03c4\u03bf\u1fe6 \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c6\u03c5\u03ba\u03ad\u03bd\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f13\u03bd \u03ba.\u03c4.\u03bb.,\n_Phys. Ausc._ iv. 6 (p. 213) \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u1f01\u03c6\u03ae \u1f10\u03c3\u03c4\u03b9\u03bd\u00b7 \u03c3\u03cd\u03bc\u03c6\u03c5\u03c3\u03b9\u03c2 \u03b4\u03ad, \u1f45\u03c4\u03b1\u03bd \u1f04\u03bc\u03c6\u03c9\n\u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u03af\u1fb3 \u1f13\u03bd \u03b3\u03ad\u03bd\u03c9\u03bd\u03c4\u03b1\u03b9 (comp. _ib._ v. 3, p. 227), _Metaph._ x. 3 (p.\n1071) \u1f45\u03c3\u03b1 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f01\u03c6\u1fc7 \u03ba\u03b1\u1f76 \u03bc\u1f74 \u03c3\u03c5\u03bc\u03c6\u03cd\u03c3\u03b5\u03b9. The relation of contiguous surfaces\nand the connexion of different parts together effect structural unity.\nThis same distinction appears in the Apostle's language here. Contact\nand attachment are the primary ideas in \u1f01\u03c6\u03b1\u03af and \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9 respectively.\n\nOf the function of \u1f01\u03c6\u03ae, 'contact', in physiology (\u03c0\u03b5\u03c1\u1f76 \u1f01\u03c6\u1fc6\u03c2 \u03c4\u1fc6\u03c2 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2\n\u03c6\u03c5\u03c3\u03b9\u03ba\u03bf\u1fd6\u03c2) Aristotle speaks at some length in one passage, _de Gen. et\nCorr._ i. 6 (p. 322 sq.). It may be mentioned, as illustrating St Paul's\nimage, that Aristotle in this passage lays great stress on the mutual\nsympathy and influence of the parts in contact, describing them as\n\u03c0\u03b1\u03b8\u03b7\u03c4\u03b9\u03ba\u1f70 \u03ba\u03b1\u1f76 \u03c0\u03bf\u03b9\u03b7\u03c4\u03b9\u03ba\u03ac and as \u03ba\u03b9\u03bd\u03b7\u03c4\u03b9\u03ba\u1f70 \u03ba\u03b1\u1f76 \u03ba\u03b9\u03bd\u03b7\u03c4\u1f70 \u1f51\u03c0' \u1f00\u03bb\u03bb\u03b7\u03bb\u1ff6\u03bd. Elsewhere,\nlike St Paul here, he uses the plural \u03b1\u1f31 \u1f01\u03c6\u03b1\u03af; _de C\u00e6lo_ i. 11 (p. 280)\n\u03c4\u1f78 \u1f04\u03bd\u03b5\u03c5 \u03c6\u03b8\u03bf\u03c1\u1fb6\u03c2 \u1f41\u03c4\u1f72 \u03bc\u1f72\u03bd \u1f42\u03bd \u1f41\u03c4\u1f72 \u03b4\u1f72 \u03bc\u1f74 \u1f44\u03bd, \u03bf\u1f37\u03bf\u03bd \u03c4\u1f70\u03c2 \u1f01\u03c6\u03ac\u03c2, \u1f45\u03c4\u03b9 \u1f04\u03bd\u03b5\u03c5 \u03c4\u03bf\u1fe6\n\u03c6\u03b8\u03b5\u03af\u03c1\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c0\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u03bf\u1f56\u03c3\u03b1\u03b9 \u1f55\u03c3\u03c4\u03b5\u03c1\u03bf\u03bd \u03bf\u1f50\u03ba \u03b5\u1f30\u03c3\u03af\u03bd, _de Gen. et Corr._ i. 8 (p.\n326) \u1f44\u03c5\u03c4\u03b5 \u03b3\u1f70\u03c1 _\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70\u03c2 \u1f01\u03c6\u1f70\u03c2_ \u1f10\u03bd\u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b9\u03ad\u03bd\u03b1\u03b9 \u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd \u03b4\u03b9\u03b1\u03c6\u03b1\u03bd\u1ff6\u03bd\n\u1f44\u03c5\u03c4\u03b5 \u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd _\u03c0\u03cc\u03c1\u03c9\u03bd_, _ib._ \u00a7 9 (p. 327) \u03b5\u1f30 \u03b3\u1f70\u03c1 \u03b4\u03b9\u03b1\u03ba\u03c1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9\n\u03b4\u03cd\u03bd\u03b1\u03c4\u03b1\u03b9\n\n------------------------------------------------------------------------\n\nII. 19]\n\n\u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03bc\u03c9\u03bd \u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd\n\n\n\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70\u03c2 _\u1f01\u03c6\u03ac\u03c2_, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03c6\u03b1\u03c3\u03af \u03c4\u03b9\u03bd\u03b5\u03c2, \u03ba\u1f02\u03bd \u03bc\u03ae\u03c0\u03c9 \u1f96 \u03b4\u03b9\u03b7\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd, \u1f14\u03c3\u03c4\u03b1\u03b9\n\u03b4\u03b9\u03b7\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd\u00b7 \u03b4\u03c5\u03bd\u03b1\u03c4\u1f78\u03bd \u03b3\u1f70\u03c1 \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03b8\u1fc6\u03bd\u03b1\u03b9: comp. [Plat.] _Axioch._ p. 365 A\n\u03c3\u03c5\u03bd\u03b5\u03b9\u03bb\u03b5\u03b3\u03bc\u03ad\u03bd\u03bf\u03bd \u03c4\u1f70\u03c2 \u1f01\u03c6\u1f70\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u1fe5\u03c9\u03bc\u03b1\u03bb\u03ad\u03bf\u03bd. It is quite clear from\nthese passages of Aristotle, more especially from the distinction of\n\u1f01\u03c6\u03b1\u03af and \u03c0\u03cc\u03c1\u03bf\u03b9, that \u03b1\u1f31 \u1f01\u03c6\u03b1\u03af are the joinings, the junctures. When\napplied to the human body they would be 'joints,' provided that we use\nthe word accurately of the relations between contiguous limbs, and not\nloosely (as it is often used) of the parts of the limbs themselves in\nthe neighbourhood of the contact. Hippocrates indeed used \u1f01\u03c6\u03b1\u03af as a\nphysiological term in a different sense, employing it as a synonyme for\n\u1f05\u03bc\u03bc\u03b1\u03c4\u03b1 i.e. the fasciculi of muscles (see Galen _Op._ XIX. p. 87), but\nthis use was quite exceptional and can have no place here. Thus \u03b1\u1f31 \u1f01\u03c6\u03b1\u03af\nwill be almost a synonyme for \u03c4\u1f70 \u1f04\u03c1\u03b8\u03c1\u03b1, differing however (1) as being\nmore wide and comprehensive, and (2) as not emphasizing so strongly the\n_adaptation_ of the contiguous parts.\n\nThe considerations just urged seem decisive as to the meaning of the\nword. Some eminent modern critics however explain \u03b1\u1f31 \u1f01\u03c6\u03b1\u03af to be 'the\nsenses', following Theodoret on Ephes. iv. 16 \u1f01\u03c6\u1f74\u03bd \u03b4\u1f72 \u03c4\u1f74\u03bd \u1f04\u03b9\u03c3\u03b8\u03b7\u03c3\u03b9\u03bd\n\u03c0\u03c1\u03bf\u03c3\u03b7\u03b3\u03cc\u03c1\u03b5\u03c5\u03c3\u03b5\u03bd, \u1f10\u03c0\u03b5\u03b9\u03b4\u1f74 \u03ba\u03b1\u1f76 \u03b1\u1f55\u03c4\u03b7 \u03bc\u03af\u03b1 \u03c4\u1ff6\u03bd \u03c0\u03ad\u03bd\u03c4\u03b5 \u03b1\u1f30\u03c3\u03b8\u03ae\u03c3\u03b5\u03c9\u03bd, \u03ba\u03b1\u1f76 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6\n\u03bc\u03ad\u03c1\u03bf\u03c5\u03c2 \u03c4\u1f78 \u03c0\u1fb6\u03bd \u1f60\u03bd\u03cc\u03bc\u03b1\u03c3\u03b5. St Chrysostom had led the way to this\ninterpretation, though his language is less explicit than Theodoret's.\nTo such a meaning however there are fatal objections. (1) This sense of\n\u1f01\u03c6\u03ae is wholly unsupported. It is true that touch lies at the root of all\nsensations, and that this fact was recognised by ancient physiologists:\ne.g. Aristot. _de Anim._ i. 13 (p. 435) \u1f04\u03bd\u03b5\u03c5 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u1f01\u03c6\u1fc6\u03c2 \u03bf\u1f50\u03b4\u03b5\u03bc\u03af\u03b1\u03bd\n\u1f10\u03bd\u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u1f04\u03bb\u03bb\u03b7\u03bd \u1f04\u03b9\u03c3\u03b8\u03b7\u03c3\u03b9\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd. But here the connexion ends; and unless\nmore cogent examples not hitherto adduced are forthcoming, we are\njustified in saying that \u03b1\u1f31 \u1f01\u03c6\u03b1\u03af could no more be used for \u03b1\u1f31 \u03b1\u1f30\u03c3\u03b8\u03ae\u03c3\u03b5\u03b9\u03c2,\nthan in English 'the touches' could be taken as a synonyme for 'the\nsenses.' (2) The image would be seriously marred by such a meaning. The\n\u1f01\u03c6\u03b1\u03af and \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9 would no longer be an exhaustive description of the\nelements of union in the anatomical structure; the conjunction of things\nso incongruous under the vinculum of the same article and preposition,\n_\u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd_ \u1f01\u03c6\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03bc\u03c9\u03bd, would be unnatural; and the intrusion\nof the 'senses' would be out of place, where the result specified is the\nsupply of nourishment (\u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd) and the compacting of the parts\n(\u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd). (3) All the oldest versions, the Latin, the Syriac,\nand the Memphitic, explain it otherwise, so as to refer in some way to\nthe connexion of the parts of the body; e.g. in the Old Latin it is\nrendered _nexus_ here and _junctura_ in Ephes. iv. 16.\n\n\u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03bc\u03c9\u03bd] '_bands_,' '_ligaments_.' The Greek \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2, like the\nEnglish 'ligament,' has a general and a special sense. In its general\nand comprehensive meaning it denotes any of the connecting bands which\nstrap the body together, such as muscles or tendons or ligaments\nproperly so called; in its special and restricted use it is a 'ligament'\nin the technical sense; comp. Galen _Op._ IV. p. 369 \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2 \u03b3\u03ac\u03c1\n\u1f10\u03c3\u03c4\u03b9\u03bd, \u1f41 \u03b3\u03bf\u1fe6\u03bd \u1f30\u03b4\u03af\u03c9\u03c2, \u03bf\u1f50 \u03ba\u03bf\u03b9\u03bd\u1ff6\u03c2 \u1f40\u03bd\u03bf\u03bc\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2, \u03c3\u1ff6\u03bc\u03b1 \u03bd\u03b5\u03c5\u03c1\u1ff6\u03b4\u03b5\u03c2 \u1f10\u03be \u1f40\u03c3\u03c4\u03bf\u1fe6 \u03bc\u1f72\u03bd\n\u1f41\u03c1\u03bc\u03ce\u03bc\u03b5\u03bd\u03bf\u03bd \u03c0\u03ac\u03bd\u03c4\u03c9\u03c2 \u03b4\u03b9\u03b1\u03c0\u03b5\u03c6\u03c5\u03ba\u1f78\u03c2 \u03b4\u1f72 \u1f22 \u03b5\u1f30\u03c2 \u1f40\u03c3\u03c4\u03bf\u1fe6\u03bd \u1f22 \u03b5\u1f30\u03c2 \u03bc\u1fe6\u03bd. Of the \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9\nor ligaments properly so called Galen describes at length the several\nfunctions and uses, more especially as binding and holding together the\n\u03b4\u03b9\u03b1\u03c1\u03b8\u03c1\u03ce\u03c3\u03b5\u03b9\u03c2; _Op._ I. 236, II. 268, 739, III. 149, IV. 2, etc., comp.\nTim. Locr. _de An. Mund._ p. 557 \u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03bc\u03bf\u03b9\u03c2 \u03c0\u03bf\u03c4\u03c4\u1f70\u03bd \u03ba\u03af\u03bd\u03b1\u03c3\u03b9\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bd\u03b5\u03cd\u03c1\u03bf\u03b9\u03c2\n\u03c3\u03c5\u03bd\u1fb6\u03c8\u03b5 \u03c4\u1f70 \u1f04\u03c1\u03b8\u03c1\u03b1 (_Opusc. Mythol._ etc. ed. Gale). In our text indeed\n\u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9 must be taken in its comprehensive sense; but the relation of\nthe \u1f01\u03c6\u03b1\u03af to the \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9 in St Paul still remains the same as that of\nthe \u03b4\u03b9\u03b1\u03c1\u03b8\u03c1\u03ce\u03c3\u03b5\u03b9\u03c2 to the \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9 in Galen.\n\n\u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba.\u03c4.\u03bb.] The two functions performed by the \u1f01\u03c6\u03b1\u03af and\n\u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9 are _first_ the supply of nutriment etc. (\u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd),\nand _secondly_ the compacting of the frame (\u03c3\u03c5\u03bd\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd). In other\nwords they are the communication of life and energy, and the\npreservation of unity and order. The _source_ of all (\u1f10\u03be \u03bf\u1f57) is Christ\nHimself the Head; but the _channels_ of communication (\u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd \u03ba.\u03c4.\u03bb.)\nare the different members of His body, in their relation one to another.\nFor \u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd 'bountifully furnished' see the note on Gal. iii. 5.\nSomewhat similarly Aristotle speaks of \u03c3\u1ff6\u03bc\u03b1 \u03ba\u03ac\u03bb\u03bb\u03b9\u03c3\u03c4\u03b1 \u03c0\u03b5\u03c6\u03c5\u03ba\u1f78\u03c2 \u03ba\u03b1\u1f76\n\u03ba\u03b5\u03c7\u03bf\u03c1\u03b7\u03b3\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd, _Pol._ iv. 1 (p. 1288). For examples of \u03c7\u03bf\u03c1\u03b7\u03b3\u03af\u03b1 applied\nto functions of the bodily organs, see Galen _Op._ III. p. 617 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2\n\u03b5\u1f30\u03c3\u03c0\u03bd\u03bf\u03b1\u1fd6\u03c2 \u03c7\u03bf\u03c1\u03b7\u03b3\u03af\u1fb3 \u03c8\u03c5\u03c7\u03c1\u1fb6\u03c2 \u03c0\u03bf\u03af\u03bf\u03c4\u03b7\u03c4\u03bf\u03c2, Alex. _Probl._ i. 81 \u03c4\u1f78 \u03c0\u03bb\u03b5\u1fd6\u03c3\u03c4\u03bf\u03bd \u03c4\u1fc6\u03c2\n\u03c4\u03c1\u03bf\u03c6\u1fc6\u03c2 \u1f10\u03be\u03c5\u03b4\u03b1\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03c7\u03bf\u03c1\u03b7\u03b3\u03b5\u1fd6\u03c4\u03b1\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03b3\u03ad\u03bd\u03b5\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03ac\u03b8\u03bf\u03c5\u03c2. For\n\u03c3\u03c5\u03bd\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd, 'joined together, compacted', see the note on ii. 2. In\nthe parallel passage, Ephes. iv. 16, this part of the image is more\ndistinctly emphasized, \u03c3\u03c5\u03bd\u03b1\u03c1\u03bc\u03bf\u03bb\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b2\u03b9\u03b2\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd. The\ndifference corresponds to the different aims of the two epistles. In the\nColossian letter the vital connexion with the Head is the main theme; in\nthe Ephesian, the unity in diversity among the members.\n\n------------------------------------------------------------------------\n\nII. 20]\n\n\u03b1\u1f54\u03be\u03b5\u03b9 \u03c4\u1f74\u03bd \u03b1\u1f54\u03c5\u03be\u03b7\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6. ^{20}\u03b5\u1f30 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5 \u03c3\u1f7a\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7\n\n\n\u03b1\u1f54\u03be\u03b5\u03b9 \u03c4\u1f74\u03bd \u03b1\u1f55\u03be\u03b7\u03c3\u03b9\u03bd \u03ba.\u03c4.\u03bb.] By the two-fold means of contact and\nattachment nutriment has been diffused and structural unity has been\nattained, but these are not the ultimate result; they are only\nintermediate processes; the end is _growth_. Comp. Arist. _Metaph._ iv.\n4 (p. 1014) _\u03b1\u1f54\u03be\u03b7\u03c3\u03b9\u03bd \u1f14\u03c7\u03b5\u03b9\u03b4'_ \u1f11\u03c4\u03ad\u03c1\u03bf\u03c5 \u03c4\u1ff7 _\u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9_ \u03ba\u03b1\u1f76\n_\u03c3\u03c5\u03bc\u03c0\u03b5\u03c6\u03c5\u03ba\u03ad\u03bd\u03b1\u03b9_ ... \u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03b4\u1f72 \u03c3\u03cd\u03bc\u03c6\u03c5\u03c3\u03b9\u03c2 \u1f01\u03c6\u1fc6\u03c2, where growth is\nattributed to the same two physiological conditions as here.\n\n\u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6] i.e. 'which partakes of God, which belongs to God, which has\nits abode in God.' Thus the finite is truly united with the Infinite;\nthe end which the false teachers strove in vain to compass is attained;\nthe Gospel vindicates itself as the true theanthropism, after which the\nhuman heart is yearning and the human intellect is feeling. See above p.\n183 sq. With this conclusion of the sentence contrast the parallel\npassage Ephes. iv. 16 \u03c4\u1f74\u03bd \u03b1\u1f54\u03be\u03b7\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 _\u03c0\u03bf\u03b9\u03b5\u1fd6\u03c4\u03b1\u03b9 \u03b5\u1f30\u03c2 \u03bf\u1f30\u03ba\u03bf\u03b4\u03bf\u03bc\u1f74\u03bd\n\u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 \u1f10\u03bd \u1f00\u03b3\u03ac\u03c0\u1fc3_, where again the different endings are determined by\nthe different motives of the two epistles.\n\nThe discoveries of modern physiology have invested the Apostle's\nlanguage with far greater distinctness and force than it can have worn\nto his own contemporaries. Any exposition of the nervous system more\nespecially reads like a commentary on his image of the relations between\nthe body and the head. At every turn we meet with some fresh\nillustration which kindles it with a flood of light. The volition\ncommunicated from the brain to the limbs, the sensations of the\nextremities telegraphed back to the brain, the absolute mutual sympathy\nbetween the head and the members, the instantaneous paralysis ensuing on\nthe interruption of continuity, all these add to the completeness and\nlife of the image. But the following passages will show how even ancient\nscientific speculation was feeling after those physiological truths\nwhich the image involves; Hippocr. _de Morb. Sacr._ p. 309 (ed Foese)\n\u03ba\u03b1\u03c4\u1f70 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03bd\u03bf\u03bc\u03af\u03b6\u03c9 \u03c4\u1f78\u03bd \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03bd \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03bd \u03c0\u03bb\u03b5\u03af\u03c3\u03c4\u03b7\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd \u1f10\u03bd \u03c4\u1ff7 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u1ff3 ...\n\u03bf\u1f31 \u03b4\u1f72 \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u1f76 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03bf\u1f54\u03b1\u03c4\u03b1 \u03ba\u03b1\u1f76 \u1f21 \u03b3\u03bb\u1ff6\u03c3\u03c3\u03b1 \u03ba\u03b1\u1f76 \u03b1\u1f31 \u03c7\u03b5\u1fd6\u03c1\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03bf\u1f31 \u03c0\u03cc\u03b4\u03b5\u03c2, \u03bf\u1f37\u03b1\n\u1f02\u03bd \u1f41 \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03c2 \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u1fc3, \u03c4\u03bf\u03b9\u03b1\u1fe6\u03c4\u03b1 \u1f51\u03c0\u03b7\u03c1\u03b5\u03c4\u03bf\u1fe6\u03c3\u03b9 ... \u1f10\u03c2 \u03b4\u1f72 \u03c4\u1f74\u03bd \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03bd \u1f41\n_\u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03c2_ \u1f10\u03c3\u03c4\u1f76\u03bd \u1f41 \u03b4\u03b9\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03c9\u03bd ... \u03b4\u03b9\u03cc\u03c4\u03b9 \u03c6\u03b7\u03bc\u1f76 \u03c4\u1f78\u03bd \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9\n\u03c4\u1f78\u03bd \u1f11\u03c1\u03bc\u03b7\u03bd\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1 \u03c4\u1f74\u03bd \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03bd, \u03b1\u1f31 \u03b4\u1f72 _\u03c6\u03c1\u03ad\u03bd\u03b5\u03c2_ \u1f04\u03bb\u03bb\u03c9\u03c2 \u1f44\u03bd\u03bf\u03bc\u03b1 \u1f14\u03c7\u03bf\u03c5\u03c3\u03b9 \u03c4\u1fc7\n\u03c4\u03cd\u03c7\u1fc3 \u03ba\u03b5\u03ba\u03c4\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd ... \u03bb\u03ad\u03b3\u03bf\u03c5\u03c3\u03b9 \u03b4\u03ad \u03c4\u03b9\u03bd\u03b5\u03c2 \u1f61\u03c2 \u03c6\u03c1\u03bf\u03bd\u03ad\u03bf\u03bc\u03b5\u03bd \u03c4\u1fc7 _\u03ba\u03b1\u03c1\u03b4\u03af\u1fc3_ \u03ba\u03b1\u1f76\n\u03c4\u1f78 \u1f00\u03bd\u03af\u03c9\u03bc\u03b5\u03bd\u03bf\u03bd \u03c4\u03bf\u1fe6\u03c4\u03bf \u1f10\u03c3\u03c4\u03b9 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c6\u03c1\u03bf\u03bd\u03c4\u03af\u03b6\u03bf\u03bd\u00b7 \u03c4\u1f78 \u03b4\u1f72 \u03bf\u1f50\u03c7 \u03bf\u1f55\u03c4\u03c9\u03c2 \u1f14\u03c7\u03b5\u03b9 ... \u03c4\u1fc6\u03c2\n... \u03c6\u03c1\u03bf\u03bd\u03ae\u03c3\u03b9\u03bf\u03c2 \u03bf\u1f50\u03b4\u03b5\u03c4\u03ad\u03c1\u1ff3 \u03bc\u03ad\u03c4\u03b5\u03c3\u03c4\u03b9\u03bd \u1f00\u03bb\u03bb\u1f70 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03c4\u03bf\u03c5\u03c4\u03ad\u03c9\u03bd \u1f41 \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03c2 \u03b1\u1f34\u03c4\u03b9\u03cc\u03c2\n\u1f10\u03c3\u03c4\u03b9\u03bd ... \u03c0\u03c1\u1ff6\u03c4\u03bf\u03c2 \u03b1\u1f30\u03c3\u03b8\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9 \u1f41 \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03c2 \u03c4\u1ff6\u03bd \u1f10\u03bd \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u1f10\u03bd\u03b5\u03cc\u03bd\u03c4\u03c9\u03bd (where\nthe theory is mixed up with some curious physiological speculations),\nGalen _Op._ I. 235 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03b4\u1f72 \u1f41 \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03c2 \u1f45\u03c4\u03b9 \u03bc\u1f72\u03bd \u1f00\u03c1\u03c7\u1f74 \u03c4\u03bf\u1fd6\u03c2 \u03bd\u03b5\u03cd\u03c1\u03bf\u03b9\u03c2 \u1f05\u03c0\u03b1\u03c3\u03b9\n\u03c4\u1fc6\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03ce\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd, \u1f10\u03bd\u03b1\u03c1\u03b3\u1ff6\u03c2 \u1f10\u03bc\u03ac\u03b8\u03bf\u03bc\u03b5\u03bd ... \u03c0\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u03b4\u1f72 \u1f61\u03c2 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03c4\u03bf\u1fd6\u03c2\n\u03bd\u03b5\u03cd\u03c1\u03bf\u03b9\u03c2, \u03bf\u1f55\u03c4\u03c9 \u1f10\u03ba\u03ad\u03b9\u03bd\u1ff3 \u03c0\u03ac\u03bb\u03b9\u03bd \u1f15\u03c4\u03b5\u03c1\u03cc\u03bd \u03c4\u03b9 \u03bc\u03cc\u03c1\u03b9\u03bf\u03bd \u1f10\u03c0\u03b9\u03c0\u03ad\u03bc\u03c0\u03b5\u03b9, \u1f22 \u03c0\u03b7\u03b3\u03ae \u03c4\u03b9\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd\n\u1f10\u03c3\u03c4\u03af\u03bd, \u1f14\u03c4' \u1f04\u03b4\u03b7\u03bb\u03bf\u03bd, _ib._ IV. p. 11 \u1f00\u03c1\u03c7\u1f74 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u03b1\u1f50\u03c4\u1ff6\u03bd (i.e. \u03c4\u1ff6\u03bd \u03bd\u03b5\u03cd\u03c1\u03c9\u03bd)\n\u1f41 \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9, \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03c0\u03ac\u03b8\u03b7 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u03c6\u03ad\u03c1\u03b5\u03b9, \u03bf\u1f37\u03bf\u03bd \u03b5\u1f30\u03c2 \u1f04\u03c1\u03bf\u03c5\u03c1\u03ac\u03bd \u03c4\u03b9\u03bd\u03b1 \u03c4\u1fc6\u03c2\n\u03bb\u03bf\u03b3\u03b9\u03c3\u03c4\u03b9\u03ba\u1fc6\u03c2 \u03c8\u03c5\u03c7\u1fc6\u03c2\u00b7 \u1f14\u03ba\u03c6\u03c5\u03c3\u03b9\u03c2 \u03b4' \u1f10\u03bd\u03c4\u03b5\u1fe6\u03b8\u03b5\u03bd, \u03bf\u1f37\u03bf\u03bd \u03c0\u03c1\u03ad\u03bc\u03bd\u03bf\u03c5 \u03c4\u03b9\u03bd\u1f78\u03c2 \u03b5\u1f30\u03c2 \u03b4\u03ad\u03bd\u03b4\u03c1\u03bf\u03bd\n\u1f00\u03bd\u03ae\u03ba\u03bf\u03bd\u03c4\u03bf\u03c2 \u03bc\u03ad\u03b3\u03b1, \u1f41 \u03bd\u03c9\u03c4\u03b9\u03b1\u1fd6\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9 \u03bc\u03c5\u03b5\u03bb\u1f78\u03c2 ... \u03c3\u03cd\u03bc\u03c0\u03b1\u03bd \u03b4' \u03bf\u1f55\u03c4\u03c9 \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1\n\u03bc\u03b5\u03c4\u03b1\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9 \u03b4\u03b9' \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c0\u03c1\u03ce\u03c4\u03b7\u03c2 \u03bc\u1f72\u03bd \u03ba\u03b1\u1f76 \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03ba\u03b9\u03bd\u03ae\u03c3\u03b5\u03c9\u03c2, \u1f10\u03c0\u1f76 \u03c4\u03ac\u03c5\u03c4\u1fc3 \u03b4'\n\u03b1\u1f30\u03c3\u03b8\u03ae\u03c3\u03b5\u03c9\u03c2, XIV. p. 313 h\u03ac\u03c5\u03c4\u03b7 \u03b3\u1f70\u03c1 (i.e. \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae) \u03ba\u03b1\u03b8\u03ac\u03c0\u03b5\u03c1 \u03c4\u03b9\u03c2 \u1f00\u03ba\u03c1\u03cc\u03c0\u03bf\u03bb\u03af\u03c2\n\u1f10\u03c3\u03c4\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u03c4\u03b9\u03bc\u03b9\u03c9\u03c4\u03ac\u03c4\u03c9\u03bd \u03ba\u03b1\u1f76 \u1f00\u03bd\u03b1\u03b3\u03ba\u03b1\u03b9\u03bf\u03c4\u03ac\u03c4\u03c9\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2\n\u03b1\u1f30\u03c3\u03b8\u03ae\u03c3\u03b5\u03c9\u03bd \u03bf\u1f30\u03ba\u03b7\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd. Plato had made the head the central organ of the\nreason (_Tim._ 69 sq.: see Grote's _Plato_ III. pp. 272, 287,\n_Aristotle_ II. p. 179 sq.), if indeed the speculations of the Tim\u00e6us\nmay be regarded as giving his serious physiological views; but he had\npostulated other centres of the emotions and appetites, the heart and\nthe abdomen. Aristotle, while rightly refusing to localize the mind as\nmind, had taken a retrograde step physiologically, when he transferred\nthe centre of sensation from the brain to the heart; e.g. _de Part.\nAnim._ ii. 10 (p. 656). Galen, criticizing his predecessors, says of\nAristotle \u03b4\u1fc6\u03bb\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9 \u03ba\u03b1\u03c4\u03b5\u03b3\u03bd\u03c9\u03ba\u1f7c\u03c2 \u03bc\u1f72\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 (i.e. \u03c4\u03bf\u1fe6 \u1f10\u03b3\u03ba\u03b5\u03c6\u03ac\u03bb\u03bf\u1f50 \u03c4\u03b5\u03bb\u03ad\u03b1\u03bd\n\u1f00\u03c7\u03c1\u03b7\u03c3\u03c4\u03af\u03b1\u03bd, \u03c6\u03b1\u03bd\u03b5\u03c1\u1ff6\u03c2 \u03b4' \u1f41\u03bc\u03bf\u03bb\u03bf\u03b3\u03b5\u1fd6\u03bd \u03b1\u1f30\u03b4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 (_Op._ III. p. 625). The\nStoics however (\u0396\u03ae\u03bd\u03c9\u03bd \u03ba\u03b1\u1f76 \u03a7\u03c1\u03cd\u03c3\u03b9\u03c0\u03c0\u03bf\u03c2 \u1f05\u03bc\u03b1 \u03c4\u1ff7 \u03c3\u03c6\u03b5\u03c4\u03ad\u03c1\u1ff3 \u03c7\u03bf\u03c1\u1ff7 \u03c0\u03b1\u03bd\u03c4\u03af) were even\nworse offenders; and in reply to them more especially Galen elsewhere\ndiscusses the question \u03c0\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u1f10\u03b3\u03ba\u03ad\u03c6\u03b1\u03bb\u03bf\u03c2 \u1f22 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1 \u03c4\u1f74\u03bd \u1f00\u03c1\u03c7\u1f74\u03bd \u1f14\u03c7\u03b5\u03b9, _Op._\nV. p. 213 sq. Bearing in mind all this diversity of opinion among\nancient physiologists, we cannot fail to be struck in the text not only\nwith the correctness of the image but also with the propriety of the\nterms; and we are forcibly reminded that among the Apostle's most\nintimate companions at this time was one whom he calls 'the beloved\nphysician' (iv. 14).\n\n20\u201323. 'You died with Christ to your old life. All mundane relations\nhave ceased for you. Why then do you\u2014you who have attained your\nspiritual manhood\u2014submit still to the rudimentary discipline of\nchildren? Why do you\u2014you who are citizens of heaven\u2014bow your necks\nafresh to the tyranny of material ordinances, as though you were still\nliving in the world? It is the same old story again; the same round of\nhard, meaningless, vexatious prohibitions, 'Handle not,' 'Taste not,'\n'Touch not.' What folly! When all these things\u2014these meats and drinks\nand the like\u2014are earthly, perishable, wholly trivial and unimportant!\nThey are used, and there is an end of them. What is this, but to draw\ndown upon yourselves the denunciations uttered by the prophet of old?\nWhat is this but to abandon God's word for precepts which are issued by\nhuman authority and inculcated by human teachers? All such things have a\nshow of wisdom, I grant. There is an officious parade of religious\ndevotion, an eager affectation of humility; there is a stern ascetic\nrigour, which ill-treats the body; but there is nothing of any real\nvalue to check indulgence of the flesh.'\n\n20. From the theological tenets of the false teachers the Apostle turns\nto the ethical\u2014from the objects of their worship to the principles of\ntheir conduct. The baptism into Christ, he argues, is death to the\nworld. The Christian has passed away to another sphere of existence.\nMundane ordinances have ceased to have any value for him, because his\nmundane life has ended. They belong to the category of the perishable;\nhe has been translated to the region of the eternal. It is therefore a\ndenial of his Christianity to subject himself again to their tyranny, to\nreturn once more to the dominion of the world. See again the note on\niii. 1.\n\n\u03b5\u1f30 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5] '_if ye died_, when ye were baptized into Christ.' For\nthis connexion between baptism and death see the notes on ii. 11, iii.\n3. This death has many aspects in St Paul's teaching. It is not only a\ndying _with_ Christ, 2 Tim. ii. 11 \u03b5\u1f30 \u03b3\u1f70\u03c1 \u03c3\u03c5\u03bd\u03b1\u03c0\u03b5\u03b8\u03ac\u03bd\u03bf\u03bc\u03b5\u03bd; but it is also\na dying to or from something. This is sometimes represented as _sin_,\nRom. vi. 2 \u1f41\u03af\u03c4\u03b9\u03bd\u03b5\u03c2 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03bf\u03bc\u03b5\u03bd \u03c4\u1fc7 _\u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u1fb3_ (comp. vv. 7, 8);\nsometimes as _self_, 2 Cor. v. 14, 15 \u1f04\u03c1\u03b1 \u03bf\u1f31 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f00\u03c0\u03ad\u03b8\u03b1\u03bd\u03bf\u03bd ... \u1f35\u03bd\u03b1 \u03bf\u1f31\n\u03b6\u1ff6\u03bd\u03c4\u03b5\u03c2 \u03bc\u03ae\u03ba\u03b5\u03c4\u03b9 _\u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2_ \u03b6\u1ff6\u03c3\u03b9\u03bd; sometimes as the _law_, Rom. vii. 6\n\u03ba\u03b1\u03c4\u03b7\u03c1\u03b3\u03ae\u03b8\u03b7\u03bc\u03b5\u03bd \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03bd\u03cc\u03bc\u03bf\u03c5 \u1f00\u03c0\u03bf\u03b8\u03b1\u03bd\u03cc\u03bd\u03c4\u03b5\u03c2, Gal. ii. 19 \u03b4\u03b9\u1f70 \u03bd\u03cc\u03bc\u03bf\u03c5 \u03bd\u03cc\u03bc\u1ff3\n\u1f00\u03c0\u03ad\u03b8\u03b1\u03bd\u03bf\u03bd; sometimes still more widely as the _world_, regarded as the\nsphere of all material rules and all mundane interests, so here and iii.\n3 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5 \u03b3\u03ac\u03c1. In all cases St Paul uses the aorist \u1f00\u03c0\u03ad\u03b8\u03b1\u03bd\u03bf\u03bd, never\nthe perfect \u03c4\u03ad\u03b8\u03bd\u03b7\u03ba\u03b1; for he wishes to emphasize the one absolute\n_crisis_, which was marked by the change of changes. When the aorist is\nwanted, the compound verb \u1f00\u03c0\u03bf\u03b8\u03bd\u03ae\u03c3\u03ba\u03b5\u03b9\u03bd is used; when the perfect, the\nsimple verb \u03b8\u03ae\u03c3\u03ba\u03b5\u03b9\u03bd; see Buttmann _Ausf. Gramm._ \u00a7 114. This rule holds\nuniversally in the Greek Testament.\n\n------------------------------------------------------------------------\n\nII. 20]\n\n\u1f00\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03c3\u03c4\u03bf\u03b9\u03c7\u03ad\u03b9\u03c9\u03bd \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5, \u03c4\u03af \u1f61\u03c2 \u03b6\u1ff6\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03ba\u03cc\u03c3\u03bc\u1ff3\n\n\n\u1f00\u03c0\u1f78 \u03c4\u1ff6\u03bd \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03c9\u03bd \u03ba.\u03c4.\u03bb.] i.e. 'from the rudimentary, disciplinary,\nordinances, whose sphere is the mundane and sensuous': see the note on\nver. 8. For the pregnant expression \u1f00\u03c0\u03bf\u03b8\u03b1\u03bd\u03b5\u1fd6\u03bd \u1f00\u03c0\u1f78 comp. Gal. v. 4\n\u03ba\u03b1\u03c4\u03b7\u03c1\u03b3\u03ae\u03b8\u03b7\u03c4\u03b5 \u1f00\u03c0\u1f78 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 (so too Rom. vii. 2, 6), 2 Cor. xi. 3 \u03c6\u03b8\u03b1\u03c1\u1fc7 ...\n\u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u1f01\u03c0\u03bb\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2, and see A. Buttmann p. 277 note.\n\n------------------------------------------------------------------------\n\nII. 21, 22]\n\n\u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c3\u03b8\u03b5; ^{21}\u039c\u1f74 \u1f05\u03c8\u1fc3 \u03bc\u03b7\u03b4\u1f72 \u03b3\u03b5\u03cd\u03c3\u1fc3 \u03bc\u03b7\u03b4\u1f72 \u03b8\u03af\u03b3\u1fc3\u03c2 ^{22}\u1f05\n\n\n\u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c3\u03b8\u03b5] '_are ye overridden with precepts, ordinances_.' In the\nLXX the verb \u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03b9\u03bd is used several times, meaning 'to issue a\ndecree,' Esth. iii. 9, 1 Esdr. vi. 33, 2 Macc. x. 8, xv. 36, 3 Macc. iv.\n11. Elsewhere it is applied most commonly to the precepts of\nphilosophers; e.g. Justin _Apol._ i. 7 \u03bf\u1f31 \u1f10\u03bd \u1f1d\u03bb\u03bb\u03b7\u03c3\u03b9 \u03c4\u1f70 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03c1\u03b5\u03c3\u03c4\u1f70\n_\u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2_ \u1f10\u03ba \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u03c4\u1ff7 \u1f11\u03bd\u1f76 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b9 _\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2_\n\u03c0\u03c1\u03bf\u03c3\u03b1\u03b3\u03bf\u03c1\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9 (comp. \u00a7 4), Epict. iii. 7. 17 sq. \u03b5\u1f30 \u03b8\u03ad\u03bb\u03b5\u03b9\u03c2 \u03b5\u1f36\u03bd\u03b1\u03b9\n\u03c6\u03b9\u03bb\u03cc\u03c3\u03bf\u03c6\u03bf\u03c2 ... \u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03c9\u03bd \u03c4\u1f70 \u03b1\u1f30\u03c3\u03c7\u03c1\u03ac. Here it would include alike the\n\u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1 of the Mosaic law (ver. 14) and the \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1 of the 'philosophy'\ndenounced above (ver. 8). Both are condemned; the one as superseded\nthough once authoritative, the other as wholly vexatious and\nunwarrantable. Examples are given in the following verse, \u03bc\u1f74 \u1f05\u03c8\u1fc3 \u03ba.\u03c4.\u03bb.\nFor the construction here, where the more remote object, which would\nstand in the dative with the active voice (2 Macc. x. 8 \u1f10\u03b4\u03bf\u03b3\u03bc\u03ac\u03c4\u03b9\u03c3\u03b1\u03bd ...\n\u03c4\u1ff7 \u03c4\u1ff6\u03bd \u1f38\u03bf\u03c5\u03b4\u03b1\u03af\u03c9\u03bd \u1f14\u03b8\u03bd\u03b5\u03b9), becomes the nominative of the passive, compare\n\u03c7\u03c1\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9 Matt. ii. 12, 22, \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 Mark x. 45, and see Winer \u00a7\nxxxix. p. 326, A. Buttmann p. 163, K\u00fchner \u00a7 378, II. p. 109.\n\n21. \u039c\u1f74 \u1f05\u03c8\u1fc3 \u03ba.\u03c4.\u03bb.] The Apostle disparagingly repeats the prohibitions of\nthe false teachers in their own words, 'Handle not, neither taste,\nneither touch.' The rabbinical passages quoted in Sch\u00f6ttgen show how\nexactly St Paul's language reproduces, not only the spirit, but even the\nform, of these injunctions. The Latin commentators, Hilary and Pelagius,\nsuppose these prohibitions to be the Apostle's own, thus making a\ncomplete shipwreck of the sense. So too St Ambrose _de Noe et Arca_ 25\n(I. p. 267), _de Abr._ i. 6 (I. p. 300). We may infer from the language\nof St Augustine who argues against it, that this was the popular\ninterpretation in his day: _Epist._ cxix. (II. p. 512) 'tanquam\npr\u00e6ceptum putatur apostoli, nescio quid tangere, gustare, attaminare,\nprohibentis.' The ascetic tendency of the age thus fastened upon a\nslight obscurity in the Greek and made the Apostle recommend the very\npractices which he disparaged. For a somewhat similar instance of a\nmisinterpretation commonly received see the note on \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03cc\u03b3\u03bc\u03b1\u03c3\u03b9\u03bd ver.\n14. Jerome however (I. p. 878) had rightly interpreted the passage,\nillustrating it by the precepts of the Talmud. At a still earlier date\nTertullian, _Adv. Marc._ v. 19, gives the correct interpretation.\n\nThese prohibitions relate to defilement contracted in divers ways by\ncontact with impure objects. Some were doubtless reenactments of the\nMosaic law; while others would be exaggerations or additions of a\nrigorous asceticism, such as we find among the Essene prototypes of\nthese Colossian heretics, e.g. the avoidance of oil, of wine, or of\nflesh-meat, the shunning of contact with a stranger or a religious\ninferior, and the like; see pp. 85 sq. For the religious bearing of this\nasceticism, as springing from the _dualism_ of these heretical teachers,\nsee above pp. 79, 104 sq.\n\n------------------------------------------------------------------------\n\nII. 22]\n\n\u1f10\u03c3\u03c4\u03b9\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b5\u1f30\u03c2 \u03c6\u03b8\u03bf\u03c1\u1f70\u03bd \u03c4\u1fc7 \u1f00\u03c0\u03bf\u03c7\u03c1\u03ae\u03c3\u03b5\u1f30, \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70\n\n\n\u1f05\u03c8\u1fc3] The difference between \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 and \u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\u03bd is not great, and in\nsome passages where they occur together, it is hard to distinguish them:\ne.g. Exod. xix. 12 \u03c0\u03c1\u03bf\u03c3\u03ad\u03c7\u03b5\u03c4\u03b5 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 \u03c4\u03bf\u1fe6 \u1f00\u03bd\u03b1\u03b2\u1fc6\u03bd\u03b1\u03b9 \u03b5\u1f30\u03c2 \u03c4\u1f78 \u1f44\u03c1\u03bf\u03c2 \u03ba\u03b1\u1f76\n_\u03b8\u03b9\u03b3\u03b5\u1fd6\u03bd_ \u03c4\u03b9 \u03b1\u1f50\u03c4\u03bf\u1fe6\u00b7 \u03c0\u1fb6\u03c2 \u1f41 _\u1f01\u03c8\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2_ \u03c4\u03bf\u1fe6 \u1f44\u03c1\u03bf\u03c5\u03c2 \u03b8\u03b1\u03bd\u03ac\u03c4\u1ff3\n\u03c4\u03b5\u03bb\u03b5\u03c5\u03c4\u03ae\u03c3\u03b5\u03b9, Eur. _Bacch._ 617 \u03bf\u1f54\u03c4' _\u1f14\u03b8\u03b9\u03b3\u03b5\u03bd_ \u03bf\u1f54\u03b8' _\u1f23\u03c8\u03b1\u03b8'_ \u1f21\u03bc\u1ff6\u03bd,\nArist. _de Gen. et Corr._ i. 8 (p. 326) \u03b4\u03b9\u1f70 \u03c4\u03af \u03bf\u1f50 \u03b3\u03af\u03b3\u03bd\u03b5\u03c4\u03b1\u03b9\n_\u1f01\u03c8\u03ac\u03bc\u03b5\u03bd\u03b1_ \u1f15\u03bd, \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f55\u03b4\u03c9\u03c1 \u1f55\u03b4\u03b1\u03c4\u03bf\u03c2 \u1f45\u03c4\u03b1\u03bd _\u03b8\u03af\u03b3\u03b7_|; Dion Chrys.\n_Or._ xxxiv. (II. p. 50) \u03bf\u1f31 \u03b4' \u1f10\u03ba \u03c0\u03b1\u03c1\u03ad\u03c1\u03b3\u03bf\u03c5 \u03c0\u03c1\u03bf\u03c3\u03af\u03b1\u03c3\u03b9\u03bd \u1f01\u03c0\u03c4\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03bc\u03cc\u03bd\u03bf\u03bd \u03c4\u03bf\u1fe6\n\u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03bf\u1f31 \u03c3\u03c0\u03bf\u03bd\u03b4\u1fc6\u03c2 \u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2, Themist. _Paraphr. Arist._ 95\n\u03c4\u1f74\u03bd \u03b4\u1f72 \u1f01\u03c6\u1f74\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03b1\u1f30\u03c3\u03b8\u03b7\u03c4\u1ff6\u03bd _\u1f00\u03bd\u03b1\u03b3\u03ba\u03b1\u1fd6\u03bf\u03bd_\u00b7 \u03ba\u03b1\u1f76 \u03b3\u1f70\u03c1\n\u03c4\u03bf\u1f54\u03bd\u03bf\u03bc\u03b1 \u03b1\u1f50\u03c4\u1fc6\u03c2 \u1f10\u03ba \u03c4\u03bf\u1fe6 _\u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9_ \u03ba\u03b1\u1f76 _\u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\u03bd_. But \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9\nis the stronger word of the two. This arises from the fact that it\nfrequently suggests, though it does not necessarily involve, the idea of\na voluntary or conscious effort, 'to take hold of'\u2013a suggestion which is\nentirely wanting to the colourless word \u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\u03bd; comp. Themist.\n_Paraphr. Arist._ 94 \u1f21 \u03c4\u1ff6\u03bd \u03b6\u03ce\u03c9\u03bd _\u1f01\u03c6\u1f74_ \u03ba\u03c1\u03af\u03c3\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03c4\u03af\u03bb\u03b7\u03c8\u03b9\u03c2 \u03c4\u03bf\u1fe6\n_\u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03bf\u03bd\u03c4\u03bf\u03c2_. Hence in Xen. _Cyrop._ i. 3. 5 \u1f45\u03c4\u03b9 \u03c3\u03b5, \u03c6\u03ac\u03bd\u03b1\u03b9, \u1f41\u03c1\u1ff6,\n\u1f45\u03c4\u03b1\u03bd \u03bc\u1f72\u03bd \u03c4\u03bf\u1fe6 \u1f04\u03c1\u03c4\u03bf\u03c5 \u1f05\u03c8\u1fc3, \u03b5\u1f30\u03c2 \u03bf\u1f50\u03b4\u1f72\u03bd \u03c4\u1f74\u03bd \u03c7\u03b5\u1fd6\u03c1\u03b1 \u1f00\u03c0\u03bf\u03c8\u03ce\u03bc\u03b5\u03bd\u03bf\u03bd, \u1f45\u03c4\u03b1\u03bd \u03b4\u1f72 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd\n\u03c4\u03b9\u03bd\u1f78\u03c2 _\u03b8\u03af\u03b3\u1fc3\u03c2_, \u03b5\u1f50\u03b8\u1f7a\u03c2 \u1f00\u03c0\u03bf\u03ba\u03b1\u03b8\u03b1\u03af\u03c1\u03b5\u03b9 \u03c4\u1f74\u03bd \u03c7\u03b5\u1fd6\u03c1\u03b1 \u03b5\u1f30\u03c2 \u03c4\u1f70 \u03c7\u03b5\u03b9\u03c1\u03cc\u03bc\u03b1\u03ba\u03c4\u03c1\u03b1\n\u03ba.\u03c4.\u03bb. Thus the words chosen in the Latin Versions, _tangere_ for\n\u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 and _attaminare_ or _contrectare_ for \u03b8\u03b9\u03b3\u03b5\u1fd6\u03bd, are unfortunate,\nand ought to be transposed. Our English Version, probably influenced by\nthe Latin, has erred in the same direction, translating \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 by\n'touch' and \u03b8\u03b9\u03b3\u03b5\u1fd6\u03bd by 'handle'. Here again they must be transposed.\n'Handle' is too strong a word for either; though in default of a better\nit may stand for \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9, which it more nearly represents. Thus the two\nwords \u1f05\u03c8\u1fc3 and \u03b8\u03af\u03b3\u1fc3\u03c2 being separate in meaning, \u03b3\u03b5\u03cd\u03c3\u1fc3 may well interpose;\nand the three together will form a descending series, so that, as Beza\n(quoted in Trench _N. T. Syn._ \u00a7 xvii. p. 57) well expresses it,\n'decrescente semper oratione, intelligatur crescere superstitio'.\n\nOn the other hand \u1f05\u03c8\u1fc3 has been interpreted here as referring to the\nrelation of husband and wife, as e.g. in 1 Cor. vii. 1 \u03b3\u03c5\u03bd\u03b1\u03b9\u03ba\u1f78\u03c2 \u03bc\u1f74\n\u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9; and the prohibition would then be illustrated by the teaching\nof the heretics in 1 Tim. iv. 3 \u03ba\u03c9\u03bb\u03cd\u03bf\u03bd\u03c4\u03c9\u03bd \u03b3\u03b1\u03bc\u03b5\u1fd6\u03bd. But, whatever\nlikelihood there may be that the Colossian false teachers also held this\ndoctrine (see above p. 85 sq.), it nowhere appears in the context, and\nwe should not expect so important a topic to be dismissed thus\ncursorily. Moreover \u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\u03bd is used as commonly in this meaning as\n\u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 (see Gataker _Op. Crit._ p. 79, and examples might be\nmultiplied); so that all ground for assigning it to \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 especially\nis removed. Both \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9 and \u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\u03bd refer to defilement incurred\nthrough the sense of touch, though in different degrees; 'Handle not,\nnor yet taste, nor even touch.'\n\n22. 'Only consider what is the real import of this scrupulous avoidance.\nWhy, you are attributing an inherent value to things which are fleeting;\nyou yourselves are citizens of eternity, and yet your thoughts are\nabsorbed in the perishable'.\n\n\u1f05] '_which things_', i.e. the meats and drinks and other material\nobjects, regarded as impure to the touch. The antecedent to \u1f05 is\nimplicitly involved in the prohibitions \u03bc\u1f74 \u1f05\u03c8\u1fc3 \u03ba.\u03c4.\u03bb.\n\n\u1f10\u03c3\u03c4\u03b9\u03bd \u03b5\u1f30\u03c2 \u03c6\u03b8\u03bf\u03c1\u1f70\u03bd] '_are destined for corruption_'. For similar\nexpressions see Acts viii. 20 \u1f14\u03b9\u03b7 \u03b5\u1f30\u03c2 \u1f00\u03c0\u03c9\u03bb\u03b5\u03af\u03b1\u03bd (comp. ver. 23 \u03b5\u1f30\u03c2 \u03c7\u03bf\u03bb\u1f74\u03bd\n\u03c0\u03b9\u03ba\u03c1\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03bd \u1f00\u03b4\u03b9\u03ba\u03af\u03b1\u03c2 ... \u1f44\u03bd\u03c4\u1f00, 2 \u03a0\u03b5\u03c4. \u03b9\u03b9. 12 [\u0393\u03c1\u03b5\u03b5\u03ba:\n\u03b3\u03b5\u03b3\u03b5\u03bd\u03bd\u03b7\u03bc\u03ad\u03bd\u03b1 ... \u03b5\u1f30\u03c2 \u1f05\u03bb\u03c9\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c6\u03b8\u03bf\u03c1\u03ac\u03bd. For the word \u03c6\u03b8\u03bf\u03c1\u03ac, involving the\nidea of 'decomposition', see the note on Gal. vi. 8. The expression here\ncorresponds to \u03b5\u1f30\u03c2 \u1f00\u03c6\u03b5\u03b4\u03c1\u1ff6\u03bd\u03b1 \u1f10\u03ba\u03b2\u03ac\u03bb\u03bb\u03b5\u03c4\u03b1\u03b9 (\u1f10\u03ba\u03c0\u03bf\u03c1\u03b5\u03cd\u03b5\u03c4\u03b1\u1f30, Matt. xv. 17, Mark\nvii. 19.\n\n\u03c4\u1fc7 \u1f00\u03c0\u03bf\u03c7\u03c1\u03ae\u03c3\u03b5\u03b9] '_in the consuming_'. While the verb \u1f00\u03c0\u03bf\u03c7\u03c1\u1ff6\u03bc\u03b1\u03b9 is common,\nthe substantive \u1f00\u03c0\u03cc\u03c7\u03c1\u03b7\u03c3\u03b9\u03c2 is extremely rare: Plut. _Mor._ p. 267 F\n\u03c7\u03b1\u03af\u03c1\u03b5\u03b9\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03c4\u03bf\u03b9\u03ac\u03c5\u03c4\u03b1\u03b9\u03c2 \u1f00\u03c0\u03bf\u03c7\u03c1\u03ae\u03c3\u03b5\u03c3\u03b9 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03c3\u03c4\u03bf\u03bb\u03b1\u1fd6\u03c2 \u03c4\u1ff6\u03bd \u03c0\u03b5\u03c1\u03b9\u03c4\u03c4\u1ff6\u03bd (i.e. 'by\nsuch modes of consuming and abridging superfluities'), Dion. Hal. _A.\nR._ i. 58 \u1f10\u03bd \u1f00\u03c0\u03bf\u03c7\u03c1\u03ae\u03c3\u03b5\u03b9 \u03b3\u1fc6\u03c2 \u03bc\u03bf\u03af\u03c1\u03b1\u03c2. The unusual word was chosen for its\nexpressiveness: the \u03c7\u03c1\u1fc6\u03c3\u03b9\u03c2 here was an \u1f00\u03c0\u03cc\u03c7\u03c1\u03b7\u03c3\u03b9\u03c2; the things could not\nbe used without rendering them unfit for further use. The subtlety of\nthe expression in the original cannot be reproduced in any translation.\n\nOn the other hand the clause is sometimes interpreted as a continuation\nof the language of the ascetic teachers; 'Touch not things which all\nlead to ruin by their abuse'. This interpretation however has nothing to\nrecommend it. It loses the point of the Apostle's argument; while it\nputs upon \u03b5\u1f36\u03bd\u03b1\u03b9 \u03b5\u1f30\u03c2 \u03c6\u03b8\u03bf\u03c1\u03ac\u03bd a meaning which is at least not natural.\n\n\u03ba\u03b1\u03c4\u1f70 \u03ba.\u03c4.\u03bb.] connected directly with vv. 20, 21, so that the words \u1f05\n\u1f10\u03c3\u03c4\u03b9\u03bd ... \u03c4\u1fc7 \u1f00\u03c0\u03bf\u03c7\u03c1\u03ae\u03c3\u03b5\u03b9 are a parenthetical comment.\n\n------------------------------------------------------------------------\n\nII. 22]\n\n\u1f10\u03bd\u03c4\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2 \u03c4\u1ff6\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd\u00b7\n\n\n\u03c4\u1f70 \u1f10\u03bd\u03c4\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03ba.\u03c4.\u03bb.] The absence of both preposition and article before\n\u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2 shows that the two words are closely connected. They are\nplaced here in their proper order; for \u1f10\u03bd\u03c4\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 describes the source of\nauthority and \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2 the medium of communication. The expression is\ntaken ultimately from Isaiah xxix. 13, where the words run in the LXX,\n\u03bc\u03ac\u03c4\u03b7\u03bd \u03b4\u1f72 \u03c3\u03ad\u03b2\u03bf\u03bd\u03c4\u03b1\u03af \u03bc\u03b5, \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd\u03c4\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2.\nThe Evangelists (Matt. xv. 9, Mark vii. 7), quoting the passage,\nsubstitute in the latter clause \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2 \u1f10\u03bd\u03c4\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1\n\u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd.\n\nThe coincidences in St Paul's language here with our Lord's words as\nrelated in the Gospels (Matt. xv. 1\u201320, Mark vii. 1\u201323) are striking,\nand suggest that the Apostle had this discourse in his mind. (1) Both\nalike argue against these vexatious ordinances from the _perishableness_\nof meats. (2) Both insist upon the indifference of such things in\nthemselves. In Mark vii. 19 the Evangelist emphasizes the importance of\nour Lord's words on this occasion, as practically abolishing the Mosaic\ndistinction of meats by declaring all alike to be clean (\u03ba\u03b1\u03b8\u03b1\u03c1\u03af\u03b6\u03c9\u03bd; see\nthe note on ver. 16). (3) Both alike connect such ordinances with the\npractices condemned in the prophetic denunciation of Isaiah.\n\n------------------------------------------------------------------------\n\nII. 23]\n\n^{23}\u1f05\u03c4\u03b9\u03bd\u03ac \u1f10\u03c3\u03c4\u03b9\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03bc\u1f72\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u1f10\u03bd \u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u1fb3\n\n\n23. 'All such teaching is worthless. It may bear the semblance of\nwisdom; but it wants the reality. It may make an officious parade of\nreligious service; it may vaunt its humility; it may treat the body with\nmerciless rigour; but it entirely fails in its chief aim. It is\npowerless to check indulgence of the flesh.'\n\n\u1f05\u03c4\u03b9\u03bd\u03b1] '_which sort of things_'. Not only these particular precepts, \u03bc\u1f74\n\u1f05\u03c8\u1fc3 \u03ba.\u03c4.\u03bb., but all precepts falling under the same category are\ncondemned. For this force of \u1f05\u03c4\u03b9\u03bd\u03b1 as distinguished from \u1f05, see the\nnotes on Gal. iv. 24, v. 19, Phil. iv. 3. The antecedent here is not\n\u1f10\u03bd\u03c4\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03c2 \u03ba.\u03c4.\u03bb., but the prohibitions given in ver. 21.\n\n\u03bb\u03cc\u03b3\u03bf\u03bd \u03bc\u1f72\u03bd \u03ba.\u03c4.\u03bb.] '_having a reputation for wisdom_', but not the\nreality. The corresponding member, which should be introduced by \u03b4\u03ad, is\nsuppressed; the oppositive clause being postponed and appearing later in\na new form, \u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u03b9 \u03ba.\u03c4.\u03bb. Such suppressions are common in\nclassical writers, more especially in Plato; see K\u00fchner \u00a7 531, II. p.\n813 sq., Jelf \u00a7 766, and comp. Winer \u00a7 lxiii. p. 719 sq. St Jerome\ntherefore is not warranted in attributing St Paul's language here to\n'imperitia artis grammatic\u00e6' (_Epist._ cxxi, _Op._ II. p. 884). On the\ncontrary it is just the license which an adept in a language would be\nmore likely to take than a novice.\n\nIn this sentence \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 is best taken as a single\npredicate, so that \u1f10\u03c3\u03c4\u03b9\u03bd is disconnected from \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1. Otherwise the\nconstruction \u1f10\u03c3\u03c4\u03b9\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 (for \u1f14\u03c7\u03b5\u03b9) would be supported by many\nparallels in the Greek Testament; see Winer \u00a7 xlv. p. 437.\n\nThe phrase \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd \u03c4\u03b9\u03bd\u03bf\u03c2, so far as I have observed, has four\nmeanings. (A) Two as applied to the _thinking subject_. (i) 'To take\naccount of, to hold in account, to pay respect to': e.g. \u00c6sch. _Prom._\n231 \u03b2\u03c1\u03bf\u03c4\u1ff6\u03bd \u03b4\u1f72 \u03c4\u1ff6\u03bd \u03c4\u03b1\u03bb\u03b1\u03b9\u03c0\u03ce\u03c1\u03c9\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03bf\u1f50\u03ba \u1f14\u03c3\u03c7\u03b5\u03bd \u03bf\u1f50\u03b4\u03ad\u03bd\u03b1, Demosth. _de\nCoron._ \u00a7 199 )\u03ad\u03b9\u03c0\u03b5\u03c1 \u1f22 \u03b4\u03cc\u03be\u03b7\u03c2 \u1f22 \u03c0\u03c1\u03bf\u03b3\u03cc\u03bd\u03c9\u03bd \u1f22 \u03c4\u03bf\u1fe6 \u03bc\u03ad\u03bb\u03bb\u03bf\u03bd\u03c4\u03bf\u03c2 \u03b1\u1f30\u1ff6\u03bd\u03bf\u03c2 \u03b5\u1f36\u03c7\u03b5\n\u03bb\u03cc\u03b3\u03bf\u03bd, Plut. _Vit. Philop._ 18 \u03c0\u1ff6\u03c2 \u1f04\u03be\u03b9\u03bf\u03bd \u1f10\u03ba\u03ad\u03b9\u03bd\u03bf\u03c5 \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd \u03c4\u03bf\u1fe6 \u1f00\u03bd\u03b4\u03c1\u1f78\u03c2\n\u03ba.\u03c4.\u03bb. (ii) 'To possess the reason or account or definition of', 'to\nhave a scientific knowledge of'; Plato _Gorg._ p. 465 A \u03c4\u03ad\u03c7\u03bd\u03b7\u03bd \u03b4\u1f72 \u03b1\u1f50\u03c4\u1f74\u03bd\n\u03bf\u1f54 \u03c6\u03b7\u03bc\u03b9 \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f00\u03bb\u03bb' \u1f10\u03bc\u03c0\u03b5\u03b9\u03c1\u03af\u03b1\u03bd, \u1f45\u03c4\u03b9 \u03bf\u1f50\u03ba \u1f14\u03c7\u03b5\u03b9 \u03bb\u03cc\u03b3\u03bf\u03bd \u03bf\u1f50\u03b4\u03ad\u03bd\u03b1 h\u1ff6\u03bd \u03c0\u03c1\u03bf\u03c3\u03c6\u03ad\u03c1\u03b5\u03b9,\n\u1f41\u03c0\u03bf\u1fd6\u03b1 \u1f04\u03c4\u03c4\u03b1 \u03c4\u1f74\u03bd \u03c6\u03cd\u03c3\u03b9\u03bd \u1f10\u03c3\u03c4\u03af\u03bd, and so frequently. These two senses are\nrecognised by Aristotle, _Eth. Nic._ i. 13 (p. 1102), where he\ndistinguishes the meaning of the expressions \u1f14\u03c7\u03b5\u03b9\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u1f22\n\u03c4\u1ff6\u03bd \u03c6\u03af\u03bb\u03c9\u03bd and \u1f14\u03c7\u03b5\u03b9\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u03c4\u1ff6\u03bd \u03bc\u03b1\u03b8\u03b7\u03c4\u03b9\u03ba\u1ff6\u03bd. (B) Two as applied to the\n_object of thought_. (iii) 'To have the credit or reputation of', as\nhere. This sense of \u1f14\u03c7\u03b5\u03b9\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd, 'to be reputed', is more commonly found\nwith an infinitive: e.g. Plato _Epin._ 987 B \u03b1\u1f51\u03c4\u1f78\u03c2 \u1f08\u03c6\u03c1\u03bf\u03b4\u03af\u03c4\u03b7\u03c2 \u03b5\u1f36\u03bd\u03b1\u03b9\n\u03c3\u03c7\u03ad\u03b4\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9 \u03bb\u03cc\u03b3\u03bf\u03bd. (iv) 'To fulfil the definition of, to possess the\ncharacteristics, to have the nature of'; e.g. Philo _Vit. Cont._ 4 (II.\np. 477) \u1f11\u03ba\u03ac\u03c4\u03b5\u03c1\u03bf\u03bd \u03b4\u1f72 \u03c0\u03b7\u03b3\u1fc6\u03c2 \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03bf\u03bd, Plut. _Mor._ p. 637 D \u03c4\u1f78 \u03b4\u1f72 \u1f62\u03bf\u03bd\n\u03bf\u1f54\u03c4\u03b5 \u1f00\u03c1\u03c7\u1fc6\u03c2 \u1f14\u03c7\u03b5\u03b9 \u03bb\u03cc\u03b3\u03bf\u03bd, \u03bf\u1f50 \u03b3\u1f70\u03c1 \u1f51\u03c6\u03af\u03c3\u03c4\u03b1\u03c4\u03b1\u03b9 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd, \u03bf\u1f54\u03c4\u03b5 \u1f45\u03bb\u03bf\u03c5 \u03c6\u03cd\u03c3\u03b9\u03bd, \u1f00\u03c4\u03b5\u03bb\u1f72\u03c2\n\u03b3\u03ac\u03c1 \u1f10\u03c3\u03c4\u03b9\u03bd, ib. 640 F \u03b4\u03b5\u1fd6 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78 \u1f10\u03bc\u03c6\u03c5\u03c4\u03b5\u03cd\u03bf\u03bc\u03b5\u03bd\u03bf\u03bd \u03c7\u03ce\u03c1\u03b1\u03c2 \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd \u03c4\u1f78\n\u03b4\u03b5\u03be\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd. The senses of \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd with other constructions, or as\nused absolutely, are very various, e.g. 'to be reasonable', 'to hold\ndiscourse', 'to bear a ratio', etc., but do not come under consideration\nhere. Nor again does such an expression as Plut. _Mor._ p. 550 C \u03bc\u03ae\u03c4\u03b5\n\u03c4\u1f78\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03c9\u03bd \u03c4\u03bf\u1fe6 \u03bd\u03bf\u03bc\u03bf\u03b8\u03ad\u03c4\u03bf\u03c5, 'not being in possession of, not knowing,\nthe intention of the legislator'; for the definite article removes it\nfrom the category of the cases considered.\n\n------------------------------------------------------------------------\n\n\n\u03ba\u03b1\u1f76 \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3 [\u03ba\u03b1\u1f76] \u1f00\u03c6\u03b5\u03b9\u03b4\u03b5\u03af\u1fb3 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2, \u03bf\u1f50\u03ba\n\n\n\u1f10\u03bd \u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u1fb3] '_in volunteered_, self-imposed, officious,\nsupererogatory _service_'. One or both of these two ideas, (i)\n'excessive readiness, officious zeal,' (ii) 'affectation, unreality,'\nare involved in this and similar compounds; e.g. \u1f10\u03b8\u03b5\u03bb\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u03b1,\n\u1f10\u03b8\u03b5\u03bb\u03bf\u03ba\u03ac\u03ba\u03b7\u03c3\u03b9\u03c2, \u1f10\u03b8\u03b5\u03bb\u03bf\u03ba\u03af\u03bd\u03b4\u03c5\u03bd\u03bf\u03c2, \u1f10\u03b8\u03b5\u03bb\u03bf\u03ba\u03c9\u03c6\u03ad\u03b9\u03bd, \u1f10\u03b8\u03b5\u03bb\u03bf\u03c1\u03ae\u03c4\u03c9\u03c1, \u1f10\u03b8\u03b5\u03bb\u03bf\u03c0\u03c1\u03cc\u03be\u03b5\u03bd\u03bf\u03c2:\nthese compounds being used most frequently, though not always (as this\nlast word shows), in a bad sense. This mode of expression was\nnaturalised in Latin, as appears from Augustine _Epist._ cxlix. 27 (II.\np. 514) 'Sic enim et vulgo dicitur qui divitem affectat thelodives, et\nqui sapientem thelosapiens, et cetera hujusmodi'. Epiphanius, when\nwriting of the Pharisees, not content with the word here supplied by St\nPaul, coins a double compound \u1f10\u03b8\u03b5\u03bb\u03bf\u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1, _H\u00e6r._ i. 16 (p.\n34).\n\n\u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3] The word is here disparaged by its connexion, as in ver.\n18 (see the note there). The force of \u1f10\u03b8\u03b5\u03bb\u03bf- may be regarded as carried\non to it. Real genuine \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7 is commended below; iii. 12.\n\n\u1f00\u03c6\u03b5\u03b9\u03b4\u03b5\u03af\u1fb3 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2] '_hard treatment of the body_'. The expression\n\u1f00\u03c6\u03b5\u03b9\u03b4\u03b5\u1fd6\u03bd \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 is not uncommon, being used most frequently, not as\nhere of ascetic discipline, but rather of courageous exposure to\nhardship and danger in war, e.g. Lysias _Or. Fun._ 25, Joseph. _B.J._\niii. 7. 18, Lucian _Anach._ 24, Plut. _Vit. Pericl._ 10; in Plut. _Mor._\np. 137 C however of a student's toil, and _ib._ p. 135 E, more generally\nof the rigorous demands made by the soul on the body. The substantive\n\u1f00\u03c6\u03ad\u03b9\u03b4\u03b5\u03b9\u03b1 or \u1f00\u03c6\u03b5\u03b9\u03b4\u03af\u03b1 does not often occur. On the forms in -\u03b5\u03b9\u03b1 and -\u03af\u03b1\nderived from adjectives in -\u03b7\u03c2 see Buttmann _Ausf. Gramm._ \u00a7 119, II. p.\n416 sq. The great preponderance of manuscript authority favours the form\n\u1f00\u03c6\u03b5\u03b9\u03b4\u03b5\u03af\u1fb3 here: but in such questions of orthography the fact carries\nless weight than in other matters. The \u03ba\u03b1\u1f76 before \u1f00\u03c6\u03b5\u03b9\u03b4\u03b5\u03af\u1fb3 should\nprobably be omitted; in which case \u1f00\u03c6\u03b5\u03b9\u03b4\u03b5\u03af\u1fb3 becomes an instrumental\ndative, explaining \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2. While the insertion would\nnaturally occur to scribes, the omission gives more point to the\nsentence. The \u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1 \u03ba\u03b1\u1f76 \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7 as the religious elements\nare thus separated from the \u1f00\u03c6\u03b5\u03af\u03b4\u03b5\u03b9\u03b1 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 as the practical rule.\n\n------------------------------------------------------------------------\n\n\n\u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2.\n\n\n\u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03ba.\u03c4.\u03bb.] 'yet _not_ really _of any value to remedy indulgence\nof the flesh_.' So interpreted the words supply the oppositive clause to\n\u03bb\u03cc\u03b3\u03bf\u03bd \u03bc\u1f72\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2, as the presence of the negative \u03bf\u1f50\u03ba naturally\nsuggests. If the sentence had been undisturbed, this oppositive clause\nwould naturally have been introduced by \u03b4\u03ad, but the interposition of \u1f10\u03bd\n\u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u1fb3 \u03ba.\u03c4.\u03bb. has changed its form by a sort of attraction. For\nthis sense of \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 comp. Lucian _Merc. cond._ 17 \u03c4\u1f70 \u03ba\u03b1\u03b9\u03bd\u1f70 \u03c4\u1ff6\u03bd\n\u1f51\u03c0\u03bf\u03b4\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u03b9 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03b9\u03bc\u03b5\u03bb\u03b5\u03af\u1fb3 \u1f10\u03c3\u03c4\u03af\u03bd: similarly Hom. _Il._ ix.\n319 \u1f10\u03bd \u03b4\u1f72 \u1f30\u1fc7 \u03c4\u03b9\u03bc\u1fc7 \u03ba.\u03c4.\u03bb. The preposition \u03c0\u03c1\u03cc\u03c2, like our English '_for_',\nwhen used after words denoting utility, value, sufficiency, etc., not\nuncommonly introduces the object to _check_ or _prevent_ or _cure_ which\nthe thing is to be employed. And even though utility may not be directly\nexpressed in words, yet if the idea of a something to be _remedied_ is\npresent, this preposition is freely used notwithstanding. See Isocr.\n_Phil._ 16 (p. 85) \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03b2\u03b1\u03c1\u03b2\u03ac\u03c1\u03bf\u03c5\u03c2 \u03c7\u03c1\u03ae\u03c3\u03b9\u03bc\u03bf\u03bd, Arist. _H. A._ iii. 21\n(p. 522) \u03c3\u03c5\u03bc\u03c6\u03ad\u03c1\u03b5\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70\u03c2 \u03b4\u03b9\u03b1\u03c1\u1fe5\u03bf\u03af\u03b1\u03c2 \u1f21 \u03c4\u03bf\u03b9\u03ac\u03c5\u03c4\u03b7 \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1, _de Respir._ 8\n(p. 474) \u1f00\u03bd\u03ac\u03b3\u03ba\u03b7 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 \u03ba\u03b1\u03c4\u03ac\u03c8\u03c5\u03be\u03b9\u03bd, \u03b5\u1f30 \u03bc\u03ad\u03bb\u03bb\u03b5\u03b9 \u03c4\u03b5\u03cd\u03be\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c3\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03c2\u00b7 \u03c4\u03bf\u1fe6\u03c4\u03bf\n\u03b3\u1f70\u03c1 \u03b2\u03bf\u03b7\u03b8\u03b5\u1fd6 \u03c0\u03c1\u1f78\u03c2 \u03c4\u03ac\u03c5\u03c4\u03b7\u03bd \u03c4\u1f74\u03bd \u03c6\u03b8\u03bf\u03c1\u03ac\u03bd, Lucian _Pisc._ 27 \u03c7\u03c1\u03ae\u03c3\u03b9\u03bc\u03bf\u03bd \u03b3\u03bf\u1fe6\u03bd \u03ba\u03b1\u1f76\n\u03c0\u03c1\u1f78\u03c2 \u1f10\u03ba\u03ad\u03b9\u03bd\u03bf\u03c5\u03c2 \u03c4\u1f78 \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03bd, Galen _Op._ XII. p. 399 \u03c7\u03c1\u03c9\u03bc\u03ad\u03bd\u1ff3 \u03b3\u03b5 \u03c4\u03af\u03bd\u03b9 \u03c0\u03c1\u1f78\u03c2\n\u03c4\u1f78 \u03c0\u03ac\u03b8\u03bf\u03c2 \u1f00\u03c1\u03ba\u03c4\u03ad\u03b9\u1ff3 \u03c3\u03c4\u03ad\u03b1\u03c4\u03b9, \u03c0. 420 [\u0393\u03c1\u03b5\u03b5\u03ba: \u03c4\u03bf\u1fe6 \u03b4\u03cc\u03bd\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u1f70 \u03c0\u03c1\u1f78\u03c2 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1\u03c2\n\u03c6\u03b1\u03bb\u03b1\u03ba\u03c1\u03ce\u03c3\u03b5\u03b9\u03c2 \u03ba.\u03c4.\u03bb., p. 430 \u03c3\u03c5\u03bd\u03ad\u03b8\u03b7\u03ba\u03b1\u03bd ... \u03c6\u03ac\u03c1\u03bc\u03b1\u03ba\u03b1 \u03c0\u03c1\u1f78\u03c2 \u1fe5\u03b5\u03bf\u03cd\u03c3\u03b1\u03c2 \u03c4\u03c1\u03af\u03c7\u03b1\u03c2, p.\n476 \u03b2\u03c1\u03b1\u03c7\u03c5\u03c4\u03ac\u03c4\u03b7\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b9 \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03bd \u1f61\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78 \u03c0\u03c1\u03bf\u03ba\u03ad\u03b9\u03bc\u03b5\u03bd\u03bf\u03bd \u03c3\u03cd\u03bc\u03c0\u03c4\u03c9\u03bc\u03b1, p. 482\n\u03c4\u03bf\u1fe6\u03c4\u03bf \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70 \u1f10\u03bd \u1f45\u03bb\u1ff3 \u03c4\u1ff7 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u1f10\u03be\u03b1\u03bd\u03b8\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03c6\u03cc\u03b4\u03c1\u03b1 \u03c7\u03c1\u03ae\u03c3\u03b9\u03bc\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd,\np. 514 \u03c7\u03c1\u03b7\u03c3\u03c4\u03ad\u03bf\u03bd \u03b4\u1f72 \u03c0\u1fb6\u03c3\u03b9 \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03bd\u03b1\u03b3\u03b5\u03b3\u03c1\u03b1\u03bc\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2 \u03b2\u03bf\u03b7\u03b8\u03ae\u03bc\u03b1\u03c3\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70\u03c2\n\u03b3\u03b9\u03bd\u03bf\u03bc\u03ad\u03bd\u03b1\u03c2 \u03b4\u03b9' \u1f15\u03b3\u03ba\u03b1\u03c5\u03c3\u03b9\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u03b1\u03bb\u03b3\u03af\u03b1\u03c2, p. 601 \u03ba\u03ac\u03bb\u03bb\u03b9\u03c3\u03c4\u03bf\u03bd \u03c0\u03c1\u1f78\u03c2 \u03b1\u1f50\u03c4\u1f74\u03bd \u03c6\u03ac\u03c1\u03bc\u03b1\u03ba\u03bf\u03bd\n\u1f10\u03b3\u03c7\u03ad\u03bf\u03bc\u03b5\u03bd\u03bf\u03bd \u03bd\u03ac\u03c1\u03b4\u03b9\u03bd\u03bf\u03bd \u03bc\u03cd\u03c1\u03bf\u03bd. These examples from Galen are only a few out\nof probably some hundreds, which might be collected from the treatise in\nwhich they occur, the _de Compositione Medicamentorum_.\n\nThe language, which the Colossian false teachers would use, may be\ninferred from the account given by Philo of a Judaic sect of mystic\nascetics, who may be regarded, not indeed as their direct, but as their\ncollateral ancestors (see p. 86, note 246, p. 94), the Therapeutes of\nEgypt; _de Vit. Cont._ \u00a7 4 (II. p. 476 sq.) \u03c4\u03c1\u03c5\u03c6\u1ff6\u03c3\u03b9\u03bd \u1f51\u03c0\u1f78 _\u03c3\u03bf\u03c6\u03af\u03b1\u03c2_\n\u1f11\u03c3\u03c4\u03af\u03c9\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c0\u03bb\u03bf\u03c5\u03c3\u03af\u03c9\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03c6\u03b8\u03cc\u03bd\u03c9\u03c2 _\u03c4\u1f70 \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1_ \u03c7\u03bf\u03c1\u03b7\u03b3\u03bf\u03cd\u03c3\u03b7\u03c2, \u1f61\u03c2 \u03ba\u03b1\u1f76 ...\n_\u03bc\u03cc\u03bb\u03b9\u03c2_ \u03b4\u03b9' \u1f13\u03be \u1f21\u03bc\u03b5\u03c1\u1ff6\u03bd _\u1f00\u03c0\u03bf\u03b3\u03b5\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9_ \u03c4\u03c1\u03bf\u03c6\u1fc6\u03c2 \u1f00\u03bd\u03b1\u03b3\u03ba\u03b1\u03af\u03b1\u03c2 ...\n\u03c3\u03b9\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u03b4\u1f72 ... \u1f04\u03c1\u03c4\u03bf\u03bd \u03b5\u1f50\u03c4\u03b5\u03bb\u1fc6, \u03ba\u03b1\u1f76 \u1f44\u03c8\u03bf\u03bd \u1f05\u03bb\u03b5\u03c2 ... \u03c0\u03cc\u03c4\u03bf\u03bd \u1f55\u03b4\u03c9\u03c1 \u03bd\u03b1\u03bc\u03b1\u03c4\u03b9\u03b1\u1fd6\u03bf\u03bd\n\u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f10\u03c3\u03c4\u03af\u03bd ... _\u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd_ \u1f61\u03c2 \u1f10\u03c7\u03b8\u03c1\u03cc\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03af\u03b2\u03bf\u03c5\u03bb\u03bf\u03bd\n\u1f10\u03ba\u03c4\u03c1\u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c8\u03c5\u03c7\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2. St Paul apparently has before him some\nsimilar exposition of the views of the Colossian heretics, either in\nwriting or (more probably) by report from Epaphras. In reply he\naltogether denies the claims of this system to the title of \u03c3\u03bf\u03c6\u03af\u03b1; he\ndisputes the value of these \u03b4\u03cc\u03b3\u03bc\u03b1\u03c4\u03b1; he allows that this \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u03ae is the\ngreat evil to be checked, the fatal disease to be cured; but he will not\nadmit that the remedies prescribed have any substantial and lasting\nefficacy.\n\nThe interpretation here offered is not new, but it has been strangely\noverlooked or despised. The passages adduced will I trust show the\ngroundlessness of objections which have been brought against it owing to\nthe use of the preposition; and in all other respects it seems to be far\npreferable to any rival explanation which has been suggested. The\nfavourite interpretations in ancient or modern times divide themselves\ninto two classes, according to the meaning assigned to \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd\n\u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2. (1) It is explained in a good sense: 'to satisfy the\nreasonable wants of the body'. In this case \u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u03af is\ngenerally interpreted, '_not_ holding it (the body) _in any honour_'. So\nthe majority of the fathers, Greek and Latin. This has the advantage of\npreserving the continuity of the words \u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd\n\u03ba.\u03c4.\u03bb.: but it assigns an impossible sense to \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2. For\n\u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u03ae always denotes 'repletion', 'surfeiting', 'excessive\nindulgence', and cannot be used of a reasonable attention to the\nphysical cravings of nature; as Galen says, _Op._ XV. p. 113 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd\n\u03b5\u1f30\u03c9\u03b8\u03cc\u03c4\u03c9\u03bd \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f30\u03b1\u03c4\u03c1\u1ff6\u03bd \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f04\u03bb\u03bb\u03c9\u03bd \u1f19\u03bb\u03bb\u03ae\u03bd\u03c9\u03bd \u03c4\u1f78 \u03c4\u1fc6\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1fc6\u03c2\n\u1f44\u03bd\u03bf\u03bc\u03b1 \u03bc\u1fb6\u03bb\u03bb\u03cc\u03bd \u03c0\u03c9\u03c2 \u1f10\u03c0\u03b9\u03c6\u03ad\u03c1\u03b5\u03b9\u03bd _\u03c4\u03b1\u1fd6\u03c2 \u1f51\u03c0\u03b5\u03c1\u03b2\u03bf\u03bb\u03b1\u1fd6\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03c5\u03bc\u03bc\u03ad\u03c4\u03c1\u03bf\u03c5\n\u03c0\u03bf\u03c3\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2_: and certainly neither the Apostle nor the Colossian\nascetics were likely to depart from this universal rule. To the long\nlist of passages quoted in Wetstein may be added such references as\nPhilo _Leg. ad. Cai._ \u00a7 1 (II. p. 546), _Clem. Hom._ viii. 15, Justin\n_Dial._ 126, Dion. Alex. in Euseb. _H.E._ vii. 25; but they might be\nincreased to any extent. (2) A bad sense is attached to \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u03ae, as\nusage demands. And here two divergent interpretations have been put\nforward. (i) The proper continuity of the sentence is preserved, and the\nwords \u03bf\u1f50\u03ba \u1f10\u03bd \u03c4\u03b9\u03bc\u1fc7 \u03c4\u03b9\u03bd\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 are regarded as an\nexposition of the doctrine of the false teachers from _their own point\nof view_. So Theodore of Mopsuestia, \u03bf\u1f50 \u03c4\u03af\u03bc\u03b9\u03bf\u03bd \u03bd\u03bf\u03bc\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03c2 \u03c4\u1f78 \u03b4\u03b9\u1f70 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd\n\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd \u03c4\u1f74\u03bd \u03c3\u03ac\u03c1\u03ba\u03b1, \u1f00\u03bb\u03bb\u1f70 \u03b3\u1f70\u03c1 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03b1\u1f31\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u1f00\u03c0\u03ad\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03c0\u03bf\u03bb\u03bb\u1ff6\u03bd \u03b4\u03b9\u1f70\n\u03c4\u1f74\u03bd \u03c4\u03bf\u1fe6 \u03bd\u03cc\u03bc\u03bf\u03c5 \u03c0\u03b1\u03c1\u03ac\u03b4\u03bf\u03c3\u03b9\u03bd. This able expositor however is evidently\ndissatisfied, for he introduces his explanation with the words \u1f00\u03c3\u03b1\u03c6\u1f72\u03c2\n\u03bc\u03ad\u03bd \u1f10\u03c3\u03c4\u03b9, \u03b2\u03bf\u03cd\u03bb\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03b5\u1f30\u03c0\u03b5\u1fd6\u03bd \u03ba.\u03c4.\u03bb.; and his explanation has not been\nadopted by others. Either the sentence, so interpreted, becomes flat and\nunmeaning, though it is obviously intended to clinch the whole matter;\nor the Apostle is made to confirm the value of the very doctrines which\nhe is combating. (ii) The sentence is regarded as discontinuous; and it\nis interpreted, '_not of any real value_' (or '_not_ consisting _in\nanything commendable_', or '_not_ holding the body _in any honour_') but\n'_tending to gratify the carnal_ desires' (or 'mind'). This in some form\nor other is almost universally adopted by modern interpreters, and among\nthe ancients is found in the commentator Hilary. The objections to it\nare serious. (\u03b1) The dislocation of the sentence is inexplicable. There\nis no indication either in the grammar or in the vocabulary that a\nseparate and oppositive clause begins with \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd \u03ba.\u03c4.\u03bb., but on\nthe contrary everything points to an unbroken continuity. (\u03b2) The sense\nwhich it attaches to \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 is either forced and unnatural,\nor it makes the Apostle say what he could not have said. If \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74 \u03c4\u1fc6\u03c2\n\u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 could have the sense which Hilary assigns to it, 'sagina carnalis\nsensus traditio humana est', or indeed if it could mean 'the _mind_ of\nthe flesh' in any sense (as it is generally taken by modern\ncommentators), this is what St Paul might well have said. But obviously\n\u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 conveys a very different idea from such expressions\nas \u03c4\u1f78 \u03c6\u03c5\u03c3\u03b9\u03bf\u1fe6\u03c3\u03b8\u03b1\u03b9 \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03bd\u03bf\u1f78\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 (ver. 18) or \u03c4\u1f78 \u03c6\u03c1\u03cc\u03bd\u03b7\u03bc\u03b1 \u03c4\u1fc6\u03c2\n\u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 (Rom. viii. 6, 7), which include pride, self-sufficiency, strife,\nhatred, bigotry, and generally everything that is earth-bound and\nselfish. On the other hand, if \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2 be taken in its\nnatural meaning, as applying to coarse sensual indulgences, then St Paul\ncould not have said without qualification, that this rigorous asceticism\nconduced \u03c0\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u1f74\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2. Such language would defeat its own\nobject by its extravagance.\n\n------------------------------------------------------------------------\n\nIII. 1]\n\nIII. ^1\u0395\u1f30 \u03bf\u1f56\u03bd \u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5 \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7, \u03c4\u1f70 \u1f04\u03bd\u03c9 \u03b6\u03b7\u03c4\u03b5\u1fd6\u03c4\u03b5, \u03bf\u1f57 \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd\n\u1f10\u03bd \u03b4\u03b5\u03be\u03b9\u1fb7 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03ba\u03b1\u03b8\u03ae\u03bc\u03b5\u03bd\u03bf\u03c2\u00b7\n\n\nIII. 1\u20134. 'If this be so; if ye were raised with Christ, if ye were\ntranslated into heaven, what follows? Why you must realise the change.\nAll your aims must centre in heaven, where reigns the Christ who has\nthus exalted you, enthroned on God's right hand. All your thoughts must\nabide in heaven, not on the earth. For, I say it once again, you have\nnothing to do with mundane things: you _died_, died once for all to the\nworld: you are living another life. This life indeed is hidden now: it\nhas no outward splendour as men count splendour; for it is a life with\nChrist, a life in God. But the veil will not always shroud it. Christ,\nour life, shall be manifested hereafter; then ye also shall be\nmanifested with Him and the world shall see your glory'.\n\n1. \u03b5\u1f30 \u03bf\u1f56\u03bd \u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] '_If then ye were raised_', not '_have\nbeen raised_'. The aorist \u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5, like \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5 (ii. 20), refers\nto their baptism; and the \u03b5\u1f30 \u03bf\u1f56\u03bd here is a resumption of the \u03b5\u1f30 in ii.\n20. The sacrament of baptism, as administered in the Apostolic age,\ninvolved a twofold symbolism, a death or burial and a resurrection: see\nthe note on ii. 12. In the rite itself these were represented by two\ndistinct acts, the disappearance beneath the water and the emergence\nfrom the water: but in the change typified by the rite they are two\naspects of the same thing, 'like the concave and convex in a circle', to\nuse an old simile. The negative side\u2014the death and burial\u2014implies the\npositive side\u2014the resurrection. Hence the form of the Apostle's\nresumption, \u03b5\u1f30 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5, \u03b5\u1f30 _\u03bf\u1f56\u03bd_ \u03c3\u03c5\u03bd\u03b7\u03b3\u03ad\u03c1\u03b8\u03b7\u03c4\u03b5.\n\nThe change involved in baptism, if truly realised, must pervade a man's\nwhole nature. It affects not only his practical conduct, but his\nintellectual conceptions also. It is nothing less than a removal into a\nnew sphere of being. He is translated from earth to heaven; and with\nthis translation his point of view is altered, his standard of judgment\nis wholly changed. Matter is to him no longer the great enemy; his\nposition towards it is one of absolute neutrality. Ascetic rules, ritual\nordinances, have ceased to have any absolute value, irrespective of\ntheir effects. All these things are of the earth, earthy. The material,\nthe transitory, the mundane, has given place to the moral, the eternal,\nthe heavenly.\n\n\u03c4\u1f70 \u1f04\u03bd\u03c9 \u03b6\u03b7\u03c4\u03b5\u1fd6\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] 'Cease to concentrate your energies, your\nthoughts, on mundane ordinances, and realise your new and heavenly life,\nof which Christ is the pole-star'.\n\n\u1f10\u03bd \u03b4\u03b5\u03be\u03b9\u1fb7 \u03ba.\u03c4.\u03bb.] '_being seated on the right hand of God_', where\n\u03ba\u03b1\u03b8\u03ae\u03bc\u03b5\u03bd\u03bf\u03c2 must not be connected with \u1f10\u03c3\u03c4\u03b9\u03bd; see the note on \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03b9,\nii. 3. This participial clause is pertinent and emphatic, for the\nsession of Christ implies the session of the believer also; Ephes. ii.\n4\u20136 \u1f41 \u03b4\u1f72 \u0398\u03b5\u03cc\u03c2 ... \u1f21\u03bc\u1fb6\u03c2 ... \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd ... \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03ae\u03b3\u03b5\u03b9\u03c1\u03b5\u03bd \u03ba\u03b1\u1f76\n_\u03c3\u03c5\u03bd\u03b5\u03ba\u03ac\u03b8\u03b9\u03c3\u03b5\u03bd_ \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f10\u03c0\u03bf\u03c5\u03c1\u03b1\u03bd\u03af\u03bf\u03b9\u03c2 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03ba.\u03c4.\u03bb.; comp.\nRev. iii. 21 \u1f41 \u03bd\u03b9\u03ba\u1ff6\u03bd, \u03b4\u03ce\u03c3\u03c9 \u03b1\u1f50\u03c4\u1ff7 \u03ba\u03b1\u03b8\u03af\u03c3\u03b1\u03b9 \u03bc\u03b5\u03c4' \u1f10\u03bc\u03bf\u1fe6 \u1f10\u03bd \u03c4\u1ff7 \u03b8\u03c1\u03cc\u03bd\u1ff3 \u03bc\u03bf\u03c5, \u1f61\u03c2\n\u03ba\u1f00\u03b3\u1f7c \u1f10\u03bd\u03af\u03ba\u03b7\u03c3\u03b1 \u03ba\u03b1\u1f76 \u1f10\u03ba\u03ac\u03b8\u03b9\u03c3\u03b1 \u03bc\u03b5\u03c4\u1f70 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u03cc\u03c2 \u03bc\u03bf\u03c5 \u1f10\u03bd \u03c4\u1ff7 \u03b8\u03c1\u03cc\u03bd\u1ff3 \u03b1\u1f50\u03c4\u03bf\u1fe6, in the\nmessage addressed to the principal church of this district: see above p.\n42. \u0392\u03b1\u03b2\u03b1\u03af, says Chrysostom, \u03c0\u03bf\u1fe6 \u03c4\u1f78\u03bd \u03bd\u03bf\u1fe6\u03bd \u1f00\u03c0\u03ae\u03b3\u03b1\u03b3\u03b5 \u03c4\u1f78\u03bd \u1f21\u03bc\u03ad\u03c4\u03b5\u03c1\u03bf\u03bd; \u03c0\u1ff6\u03c2\n\u03c6\u03c1\u03bf\u03bd\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 \u1f10\u03c0\u03bb\u03ae\u03c1\u03c9\u03c3\u03b5 \u03bc\u03b5\u03b3\u03ac\u03bb\u03bf\u03c5; \u03bf\u1f50\u03ba \u1f24\u03c1\u03ba\u03b5\u03b9 \u03a4\u1f70 \u1f04\u03bd\u03c9 \u03b5\u1f30\u03c0\u03b5\u1fd6\u03bd, \u03bf\u1f50\u03b4\u1f72, \u039f\u1f57 \u1f41\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd, \u1f00\u03bb\u03bb\u1f70 \u03c4\u03af; \u1f18\u03bd \u03b4\u03b5\u03be\u03b9\u1fb7 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03ba\u03b1\u03b8\u03ae\u03bc\u03b5\u03bd\u03bf\u03c2. \u1f10\u03ba\u03b5\u1fd6\u03b8\u03b5\u03bd \u03bb\u03bf\u03b9\u03c0\u1f78\u03bd \u03c4\u1f74\u03bd\n\u03b3\u1fc6\u03bd \u1f41\u03c1\u1fb6\u03bd \u03c0\u03b1\u03c1\u03b5\u03c3\u03ba\u03b5\u03cd\u03b1\u03b6\u03b5.\n\n------------------------------------------------------------------------\n\nIII. 2, 3]\n\n^2\u03c4\u1f70 \u1f04\u03bd\u03c9 \u03c6\u03c1\u03bf\u03bd\u03b5\u1fd6\u03c4\u03b5, \u03bc\u1f74 \u03c4\u1f70 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2. ^3\u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5 \u03b3\u03ac\u03c1, \u03ba\u03b1\u1f76 \u1f21 \u03b6\u03c9\u1f74 \u1f51\u03bc\u1ff6\u03bd\n\u03ba\u03ad\u03ba\u03c1\u03c5\u03c0\u03c4\u03b1\u03b9 \u03c3\u1f7a\u03bd \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f10\u03bd \u03c4\u1ff7 \u0398\u03b5\u1ff7\u00b7\n\n\n2. \u03c4\u1f70 \u1f04\u03bd\u03c9] The same expression repeated for emphasis; 'You must not only\n_seek_ heaven; you must also _think_ heaven.' For the opposition of \u03c4\u1f70\n\u1f04\u03bd\u03c9 and \u03c4\u1f70 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2 in connexion with \u03c6\u03c1\u03bf\u03bd\u03b5\u1fd6\u03bd, comp. Phil. iii. 19,\n20 \u03bf\u1f31 \u03c4\u1f70 _\u1f10\u03c0\u03af\u03b3\u03b5\u03b9\u03b1 \u03c6\u03c1\u03bf\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2_, \u1f21\u03bc\u1ff6\u03bd \u03b3\u1f70\u03c1 \u03c4\u1f78 \u03c0\u03bf\u03bb\u03af\u03c4\u03b5\u03c5\u03bc\u03b1 _\u1f10\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fd6\u03c2\n\u1f51\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9_; see also Theoph. _ad Autol._ ii. 17. Extremes meet. Here\nthe Apostle points the antithesis to controvert a Gnostic asceticism: in\nthe Philippian letter he uses the same contrast to denounce an Epicurean\nsensualism. Both alike are guilty of the same fundamental error; both\nalike concentrate their thoughts on material, mundane things.\n\n3. \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5] '_ye died_' in baptism. The aorist \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5 denotes the\npast act; the perfect \u03ba\u03ad\u03ba\u03c1\u03c5\u03c0\u03c4\u03b1\u03b9 the permanent effects. For \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5 see\nthe notes on ii. 12, 20.\n\n\u03ba\u03ad\u03ba\u03c1\u03c5\u03c0\u03c4\u03b1\u03b9] '_is hidden_, is buried out of sight, to the world'. The\nApostle's argument is this: 'When you sank under the baptismal water,\nyou disappeared for ever to the world. You rose again, it is true, but\nyou rose only to God. The world henceforth knows nothing of your new\nlife, and (as a consequence) your new life must know nothing of the\nworld.' 'Neque Christum', says Bengel, 'neque Christianos novit mundus;\nac ne Christiani quidem plane seipsos'; comp. Joh. xiv. 17\u201319 \u03c4\u1f78 \u03c0\u03bd\u03b5\u1fe6\u03bc\u03b1\n\u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u1f43 \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 \u03bf\u1f50 \u03b4\u03cd\u03bd\u03b1\u03c4\u03b1\u03b9 \u03bb\u03b1\u03b2\u03b5\u1fd6\u03bd, \u1f45\u03c4\u03b9 _\u03bf\u1f50 \u03b8\u03b5\u03c9\u03c1\u03b5\u1fd6 \u03b1\u1f50\u03c4\u1f78 \u03bf\u1f50\u03b4\u1f72\n\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9_ \u03b1\u1f50\u03c4\u1f78, \u1f51\u03bc\u03b5\u1fd6\u03c2 [\u03b4\u1f72] \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03c4\u03b5 \u03b1\u1f50\u03c4\u03cc ... \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 \u03bc\u03b5 \u03bf\u1f50\u03ba \u1f14\u03c4\u03b9\n\u03b8\u03b5\u03c9\u03c1\u03b5\u1fd6 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03b4\u1f72 \u03b8\u03b5\u03c9\u03c1\u03b5\u1fd6\u03c4\u03ad \u03bc\u03b5\u00b7 _\u1f45\u03c4\u03b9 \u1f10\u03b3\u1f7c \u03b6\u1ff6, \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03b6\u03ae\u03c3\u03b5\u03c4\u03b5_.\n\n------------------------------------------------------------------------\n\nIII. 4]\n\n^4\u1f45\u03c4\u03b1\u03bd \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u03c6\u03b1\u03bd\u03b5\u03c1\u03c9\u03b8\u1fc7, \u1f21 \u03b6\u03b5\u03b9\u03c9\u1f74 \u1f21\u03bc\u1ff6\u03bd, \u03c4\u03cc\u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03c3\u1f7a\u03bd \u03b1\u1f50\u03c4\u1ff7\n\u03c6\u03b1\u03bd\u03b5\u03c1\u03c9\u03b8\u03ae\u03c3\u03b5\u03c3\u03b8\u03b5 \u1f10\u03bd \u03b4\u03cc\u03be\u1fc3.\n\n                          4 \u1f21 \u03b6\u03c9\u1f74 _\u1f51\u03bc\u1ff6\u03bd_.\n\n\n4. \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2] A fourth occurrence of the name of Christ in this context;\ncomp. ver. 2 \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7, \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2, ver. 3 \u03c3\u1f7a\u03bd \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7. A pronoun would\nhave been more natural, but less emphatic.\n\n\u1f21 \u03b6\u03c9\u1f74 \u1f21\u03bc\u1ff6\u03bd] This is an advance on the previous statement, \u1f21 \u03b6\u03c9\u1f74 \u1f51\u03bc\u1ff6\u03bd\n\u03ba\u03ad\u03ba\u03c1\u03c5\u03c0\u03c4\u03b1\u03b9 \u03c3\u1f7a\u03bd \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7, in two respects: (1) It is not enough to have\nsaid that the life is shared _with_ Christ. The Apostle declares that\nthe life _is_ Christ. Comp. 1 Joh. v. 12 \u1f41 \u1f14\u03c7\u03c9\u03bd \u03c4\u1f78\u03bd \u1f51\u1f76\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9 \u03c4\u1f74\u03bd \u03b6\u03ce\u03b7\u03bd,\nIgn. _Ephes._ 7 \u1f10\u03bd \u03b8\u03b1\u03bd\u03ac\u03c4\u1ff3 \u03b6\u03c9\u1f74 \u1f00\u03bb\u03b7\u03b8\u03b9\u03bd\u03ae (of Christ), _Smyrn._ 4 \u1f38\u03b7\u03c3\u03bf\u1fe6\u03c2\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u03c4\u1f78 \u1f00\u03bb\u03b7\u03b8\u03b9\u03bd\u1f78\u03bd \u1f21\u03bc\u1ff6\u03bd \u03b6\u1fc6\u03bd, _Ephes._ 3 \u1f38\u03b7\u03c3\u03bf\u1fe6\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 \u03c4\u1f78 \u1f00\u03b4\u03b9\u03ac\u03ba\u03c1\u03b9\u03c4\u03bf\u03bd\n\u1f21\u03bc\u1ff6\u03bd \u03b6\u1fc6\u03bd, _Magn._ 1 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u03b4\u03b9\u03b1\u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u1f21\u03bc\u1ff6\u03bd \u03b6\u1fc6\u03bd. (2) For \u1f51\u03bc\u1ff6\u03bd\nis substituted \u1f21\u03bc\u1ff6\u03bd. The Apostle hastens to include himself among the\nrecipients of the bounty. For this characteristic transition from the\nsecond person to the first see the note on ii. 13. The reading \u1f51\u03bc\u1ff6\u03bd here\nhas very high support, and on this account I have given it as an\nalternative; but it is most probably a transcriber's correction, for the\nsake of uniformity with the preceding.\n\n\u03c4\u03cc\u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03ba.\u03c4.\u03bb.] 'The veil which now shrouds your higher life from\nothers, and even partly from yourselves, will then be withdrawn. The\nworld which persecutes, despises, ignores now, will then be blinded,\nwith the dazzling glory of the revelation'. Comp. 1 Joh. iii. 1, 2 \u1f41\n\u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 \u03bf\u1f50 \u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9 \u1f21\u03bc\u1fb6\u03c2, \u1f45\u03c4\u03b9 \u03bf\u1f50\u03ba \u1f14\u03b3\u03bd\u03c9 \u03b1\u1f50\u03c4\u03cc\u03bd. \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03bf\u03af, \u03bd\u1fe6\u03bd \u03c4\u03ad\u03ba\u03bd\u03b1 \u0398\u03b5\u03bf\u1fe6\n\u1f10\u03c3\u03bc\u03ad\u03bd, \u03ba\u03b1\u1f76 \u03bf\u1f54\u03c0\u03c9 \u1f10\u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03b8\u03b7 \u03c4\u03af \u1f10\u03c3\u03bf\u03bc\u03b5\u03b8\u03b1\u00b7 \u03bf\u1f35\u03b4\u03b1\u03bc\u03b5\u03bd \u1f45\u03c4\u03b9 \u1f10\u1f70\u03bd \u03c6\u03b1\u03bd\u03b5\u03c1\u03c9\u03b8\u1fc7, \u1f45\u03bc\u03bf\u03b9\u03bf\u03b9\n\u03b1\u1f50\u03c4\u1ff7 \u1f10\u03c3\u03cc\u03bc\u03b5\u03b8\u03b1 \u03ba.\u03c4.\u03bb., Clem. Rom. 50 \u03bf\u1f33 \u03c6\u03b1\u03bd\u03b5\u03c1\u03c9\u03b8\u03ae\u03c3\u03bf\u03bd\u03c4\u03b1\u03b9 (or \u03c6\u03b1\u03bd\u03b5\u03c1\u03bf\u1f76\n\u1f14\u03c3\u03bf\u03bd\u03c4\u03b1\u03b9) \u1f10\u03bd \u03c4\u1fc7 \u1f10\u03c0\u03b9\u03c3\u03ba\u03bf\u03c0\u1fc7 \u03c4\u1fc6\u03c2 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.\n\n\u1f10\u03bd \u03b4\u03cc\u03be\u1fc3] Joh. xvii. 22 \u03c4\u1f74\u03bd \u03b4\u03cc\u03be\u03b1\u03bd \u1f23\u03bd \u03b4\u03ad\u03b4\u03c9\u03ba\u03ac\u03c2 \u03bc\u03bf\u03b9, \u03b4\u03ad\u03b4\u03c9\u03ba\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2, Rom.\nviii. 17 \u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b4\u03bf\u03be\u03b1\u03c3\u03b8\u1ff6\u03bc\u03b5\u03bd.\n\n------------------------------------------------------------------------\n\nIII. 5]\n\n^5\u039d\u03b5\u03ba\u03c1\u03ce\u03c3\u03b1\u03c4\u03b5 \u03bf\u1f56\u03bd \u03c4\u1f70 \u03bc\u03ad\u03bb\u03b7 \u03c4\u1f70 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2\u00b7 \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1\u03bd, \u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1\u03bd, \u03c0\u03ac\u03b8\u03bf\u03c2,\n\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03bd \u03ba\u03b1\u03ba\u03ae\u03bd, \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1\u03bd,\n\n\n5\u201311. 'So then realise this death to the world; kill all your earthly\nmembers. Is it fornication, impurity of whatever kind, passion, evil\ndesire? Or again, is it that covetousness which makes a religion, an\nidolatry, of greed? Do not deceive yourselves. For all these things\nGod's wrath will surely come. In these sins ye, like other Gentiles,\nindulged in times past, when your life was spent amidst them. But now\neverything is changed. Now you also must put away not this or that\ndesire, but all sins whatsoever. Anger, wrath, malice, slander, filthy\nabuse; banish it from your lips. Be not false one to another in word or\ndeed; but cast off for ever the old man with his actions, and put on the\nnew, who is renewed from day to day, growing unto perfect knowledge and\nrefashioned after the image of his Creator. In this new life, in this\nregenerate man, there is not, there cannot be, any distinction of Greek\nor Jew, of circumcision or uncircumcision; there is no room for\nbarbarian, for Scythian, for bond or free. Christ has displaced, has\nannihilated, all these; Christ is Himself all things and in all things'.\n\n5. The false doctrine of the Gnostics had failed to check sensual\nindulgence (ii. 23). The true doctrine of the Apostle has power to kill\nthe whole carnal man. The substitution of a comprehensive principle for\nspecial precepts\u2014of the heavenly life in Christ for a code of minute\nordinances\u2014at length attains the end after which the Gnostic teachers\nhave striven, and striven in vain.\n\n\u03bd\u03b5\u03ba\u03c1\u03ce\u03c3\u03b1\u03c4\u03b5 \u03bf\u1f56\u03bd] i.e. 'Carry out this principle of _death_ to the world\n(ii. 20 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5, iii. 3 \u1f00\u03c0\u03b5\u03b8\u03ac\u03bd\u03b5\u03c4\u03b5), and kill everything that is\nmundane and carnal in your being'.\n\n\u03c4\u1f70 \u03bc\u03ad\u03bb\u03b7 \u03ba.\u03c4.\u03bb.] Each person has a twofold moral personality. There is in\nhim the 'old man', and there is in him also 'the new' (vv. 9, 10). The\nold man with all his members must be pitilessly slain. It is plain that\n\u03c4\u1f70 \u03bc\u03ad\u03bb\u03b7 here is used, like \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2 in ver. 9, not physically, but\nmorally. Our actual limbs may be either \u03c4\u1f70 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2 or \u03c4\u1f70 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2\n\u03bf\u1f50\u03c1\u03ac\u03bd\u03bf\u03b9\u03c2, according as they are made instruments for the world or for\nChrist: just as we\u2014our whole being\u2014may identify ourselves with the\n\u03c0\u03b1\u03bb\u03b1\u03b9\u1f78\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2 or with the \u03bd\u03ad\u03bf\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2 of our twofold potentiality.\nFor this use of the physical, as a symbol of the moral of which it is\nthe potential instrument, compare Matt. v. 29 sq. \u03b5\u1f30 \u03b4\u1f72 \u1f41 \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03cc\u03c2 \u03c3\u03bf\u03c5 \u1f41\n\u03b4\u03b5\u03be\u03b9\u1f78\u03c2 \u03c3\u03ba\u03b1\u03bd\u03b4\u03b1\u03bb\u03af\u03b6\u03b5\u03b9 \u03c3\u03b5, \u1f14\u03be\u03b5\u03bb\u03b5 \u03b1\u1f50\u03c4\u1f78\u03bd \u03ba.\u03c4.\u03bb.\n\nI have ventured to punctuate after \u03c4\u1f70 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2. Thus \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb.\nare prospective accusatives, which should be governed directly by some\nsuch word as \u1f00\u03c0\u03cc\u03b8\u03b5\u03c3\u03b8\u03b5. But several dependent clauses interpose; the last\nof these incidentally suggests a contrast between the past and the\npresent; and this contrast, predominating in the Apostle's mind, leads\nto an abrupt recasting of the sentence, _\u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72_ \u1f00\u03c0\u03cc\u03b8\u03b5\u03c3\u03b8\u03b5 \u03ba\u03b1\u1f76\n\u1f51\u03bc\u03b5\u1fd6\u03c2 _\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1_ in disregard of the original construction. This\nopposition of \u03c0\u03bf\u03c4\u03ad and \u03bd\u1fe6\u03bd has a tendency to dislocate the construction\nin St Paul, as in i. 22 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 (or \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd), i. 26\n\u03bd\u1fe6\u03bd \u03b4\u1f72 \u1f10\u03c6\u03b1\u03bd\u03ad\u03c1\u03c9\u03b8\u03b7: see the note on this latter passage. For the whole run\nof the sentence (the parenthetic relative clauses, the contrast of past\nand present, and the broken construction) compare Ephes. ii. 1\u20135 \u03ba\u03b1\u1f76\n\u1f51\u03bc\u1fb6\u03c2 ... \u1f10\u03bd \u03b1\u1f37\u03c2 \u03c0\u03bf\u03c4\u03ad ... \u1f10\u03bd \u03bf\u1f37\u03c2 \u03ba\u03b1\u1f76 ... \u03c0\u03bf\u03c4\u03b5 ... \u1f41 \u03b4\u1f72 \u0398\u03b5\u03cc\u03c2 ... \u03ba\u03b1\u1f76 \u1f44\u03bd\u03c4\u03b1\u03c2\n\u1f21\u03bc\u1fb6\u03c2 \u03c3\u03c5\u03bd\u03b5\u03b6\u03c9\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd.\n\nWith the common punctuation the interpretation is equally awkward,\nwhether we treat \u03c4\u1f70 \u03bc\u03ad\u03bb\u03b7 and \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb. as in direct apposition, or\nas double accusatives, or in any other way. The case is best put by\nSeverianus, \u03c3\u03ac\u03c1\u03ba\u03b1 \u03ba\u03b1\u03bb\u03b5\u1fd6 \u03c4\u1f74\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u03b1\u03bd, \u1f27\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03bc\u03ad\u03bb\u03b7 \u03ba\u03b1\u03c4\u03b1\u03c1\u03b9\u03b8\u03bc\u03b5\u1fd6 ... \u1f41\n\u03c0\u03b1\u03bb\u03b1\u03b9\u1f78\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c4\u1f78 \u03c6\u03c1\u03cc\u03bd\u03b7\u03bc\u03b1 \u03c4\u1f78 \u03c4\u1fc6\u03c2 \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u03b1\u03c2, \u03bc\u03ad\u03bb\u03b7 \u03b4\u1f72 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b1\u1f31\n\u03c0\u03c1\u03ac\u03be\u03b5\u03b9\u03c2 \u03c4\u1ff6\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd; but this is an evasion of the difficulty, which\nconsists in the direct apposition of the instruments and the activities,\nfrom whatever point they are viewed.\n\n\u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb.] The general order is from the less comprehensive to the\nmore comprehensive. Thus \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1 is a special kind of uncleanness, while\n\u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1 is uncleanness in any form, Ephes. v. 3 \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1 \u03b4\u1f72 \u03ba\u03b1\u1f76\n\u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1 \u03c0\u1fb6\u03c3\u03b1; comp. Gal. v. 19 \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1, \u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1, \u1f00\u03c3\u03ad\u03bb\u03b3\u03b5\u03b9\u03b1, with the\nnote there. Thus again \u03c0\u03ac\u03b8\u03bf\u03c2, though frequently referring to this class\nof sins (Rom. i. 26, 1 Thess. iv. 5), would include other base passions\nwhich do not fall under the category of \u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1, as for instance\ngluttony and intemperance.\n\n\u03c0\u03ac\u03b8\u03bf\u03c2, \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03bd] The two words occur together in 1 Thess. iv. 5 \u03bc\u1f74 \u1f10\u03bd\n\u03c0\u03ac\u03b8\u03b5\u03b9 \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03c2. So in a passage closely resembling the text, Gal. v. 24\n\u03bf\u1f31 \u03b4\u1f72 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03c4\u1f74\u03bd \u03c3\u03ac\u03c1\u03ba\u03b1 \u1f10\u03c3\u03c4\u03ac\u03c5\u03c1\u03c9\u03c3\u03b1\u03bd \u03c3\u1f7a\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03b1\u03b8\u03ae\u03bc\u03b1\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u03b1\u1fd6\u03c2\n\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03b9\u03c2. The same vice may be viewed as a \u03c0\u03ac\u03b8\u03bf\u03c2 from its passive and\nan \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 from its active side. The word \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 is not used here in\nthe restricted sense which it has e.g. in Arist. _Eth. Nic._ ii. 4,\nwhere it ranges with anger, fear, etc., being related to \u03c0\u03ac\u03b8\u03bf\u03c2 as the\nspecies to the genus (see Gal. l.c. note). In the Greek Testament\n\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 has a much more comprehensive sense; e.g. Joh. viii. 44 \u03c4\u1f70\u03c2\n\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03b8\u03ad\u03bb\u03b5\u03c4\u03b5 \u03c0\u03bf\u03b9\u03b5\u1fd6\u03bd. Here, if anything, \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 is\nwider than \u03c0\u03ac\u03b8\u03bf\u03c2. While \u03c0\u03ac\u03b8\u03bf\u03c2 includes all ungovernable affections,\n\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 \u03ba\u03b1\u03ba\u03ae reaches to all evil longings. \u1f38\u03b4\u03bf\u03cd, says Chrysostom,\n\u03b3\u03b5\u03bd\u03b9\u03ba\u1ff6\u03c2 \u03c4\u1f78 \u03c0\u1fb6\u03bd \u03b5\u1f36\u03c0\u03b5\u00b7 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b3\u1f70\u03c1 \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 \u03ba\u03b1\u03ba\u03ae, \u03b2\u03b1\u03c3\u03ba\u03b1\u03bd\u03af\u03b1, \u1f40\u03c1\u03b3\u03ae, \u03bb\u03cd\u03c0\u03b7. The\nepithet is added because \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1 is capable of a good sense: comp. 1\nCor. x. 6 \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03b7\u03c4\u1f70\u03c2 \u03ba\u03b1\u03ba\u1ff6\u03bd.\n\n\u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1\u03bd] '_and_ especially _covetousness_'. Impurity and\ncovetousness may be said to divide between them nearly the whole domain\nof human selfishness and vice; 'Si avaritia prostrata est, exsurgit\nlibido' (Cypr. _de Mort._ 3). The one has been already dealt with; the\nother needs now to be specially denounced; comp. Ephes. v. 3 \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1 \u03b4\u1f72\n\u03ba\u03b1\u1f76 \u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1 \u03c0\u1fb6\u03c3\u03b1 \u1f22 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1. '<DW25> extra Deum', says Bengel (on Rom.\ni. 29), 'qu\u00e6rit pabulum in creatura materiali vel per voluptatem vel per\navaritiam.' Comp. _Test. xii Patr._ Jud. 18 \u03c6\u03c5\u03bb\u03ac\u03be\u03b1\u03c3\u03b8\u03b5 \u03bf\u1f56\u03bd, \u03c4\u03ad\u03ba\u03bd\u03b1 \u03bc\u03bf\u03c5,\n\u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03c6\u03b9\u03bb\u03b1\u03c1\u03b3\u03c5\u03c1\u03af\u03b1\u03c2 ... \u1f45\u03c4\u03b9 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u1f00\u03c6\u03b9\u03c3\u03c4\u1fb7 \u03bd\u03cc\u03bc\u03bf\u03c5 \u0398\u03b5\u03bf\u1fe6.\nSimilarly Lysis Pythag. 4 (_Epistol. Gr\u00e6c._ p. 602, ed. Hercher)\n\u1f40\u03bd\u03bf\u03bc\u03ac\u03be\u03b1\u03b9\u03bc\u03b9 \u03b4' \u1f02\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd [i.e. the vices] \u03c0\u03c1\u1fb6\u03c4\u03bf\u03bd \u1f10\u03c0\u03b5\u03bb\u03b8\u1f7c\u03bd \u03c4\u1f70\u03c2 \u03bc\u03b1\u03c4\u03ad\u03c1\u03b1\u03c2,\n\u1f00\u03ba\u03c1\u03b1\u03c3\u03af\u03b1\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1\u03bd\u00b7 \u1f04\u03bc\u03c6\u03c9 \u03b4\u1f72 \u03c0\u03bf\u03bb\u03cd\u03b3\u03bf\u03bd\u03bf\u03b9 \u03c0\u03b5\u03c6\u03cd\u03ba\u03b1\u03bd\u03c4\u03b9. It must be\nremembered that \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1 is much wider than \u03c6\u03b9\u03bb\u03b1\u03c1\u03b3\u03c5\u03c1\u03af\u03b1 (see Trench _N.\nT. Syn._ \u00a7 xxiv, p. 77 sq.), which itself is called \u1fe5\u03af\u03b6\u03b1 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03c4\u1ff6\u03bd\n\u03ba\u03b1\u03ba\u1ff6\u03bd (1 Tim. vi. 10).\n\nThe attempt to give \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1 here and in other passages the sense of\n'impurity' (see e.g. Hammond on Rom. i. 29) is founded on a\nmisconception. The words \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03ba\u03c4\u03b5\u1fd6\u03bd, \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1, will sometimes be used\nin relation to sins of uncleanness, because such may be acts of\ninjustice also. Thus adultery is not only impurity, but it is robbery\nalso: hence 1 Thess. iv. 6 \u03c4\u1f78 \u03bc\u1f74 \u1f51\u03c0\u03b5\u03c1\u03b2\u03b1\u03af\u03bd\u03b5\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03ba\u03c4\u03b5\u1fd6\u03bd \u1f10\u03bd \u03c4\u1ff7\n\u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03b9 \u03c4\u1f78\u03bd \u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 (see the note there). In other passages again\nthere will be an accidental connexion; e.g. Ephes. iv. 19 \u03b5\u1f30\u03c2 \u1f10\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03bd\n\u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u1f10\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u1fb3, i.e. 'with greediness', 'with entire\ndisregard for the rights of others'. But no where do the words in\nthemselves suggest this meaning. Here the particles \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd show that a\nnew type of sin is introduced with \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1\u03bd: and in the parallel\npassage Ephes. v. 3 (quoted above) the same distinction is indicated by\nthe change from the conjunctive particle \u03ba\u03b1\u03af to the disjunctive \u1f24. It is\nan error to suppose that this sense of \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1 is supported by Clem.\nAlex. _Strom._ iii. 12 (p. 551 sq.) \u1f61\u03c2 \u03b3\u1f70\u03c1 \u1f21 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1 \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1 \u03bb\u03ad\u03b3\u03b5\u03c4\u03b1\u03b9,\n\u03c4\u1fc7 \u03b1\u1f50\u03c4\u03b1\u03c1\u03ba\u03b5\u03af\u1fb3 \u1f10\u03bd\u03b1\u03bd\u03c4\u03b9\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7. On the converse error of explaining \u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1\nto mean 'greediness', 'covetousness', see the note on 1 Thess. ii. 3.\n\n------------------------------------------------------------------------\n\nIII. 6]\n\n\u1f25\u03c4\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76\u03bd \u03b5\u1f30\u03b4\u03c9\u03bb\u03bf\u03bb\u03b1\u03c4\u03c1\u03b5\u03af\u03b1, ^6\u03b4\u03b9' \u1f03 \u1f10\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u1f21 \u1f40\u03c1\u03b3\u1f74\n\n\n\u1f25\u03c4\u03b9\u03c2 \u03ba.\u03c4.\u03bb.] '_for it is idolatry_': comp. Ephes. v. 5 \u03c0\u03bb\u03b5\u03bf\u03bd\u03ad\u03ba\u03c4\u03b7\u03c2, \u1f45 (or\n\u1f45\u03c2) \u1f10\u03c3\u03c4\u03b9\u03bd \u03b5\u1f30\u03b4\u03c9\u03bb\u03bf\u03bb\u03ac\u03c4\u03c1\u03b7\u03c2, Polyc. _Phil._ 11 'Si quis non abstinuerit se ab\navaritia, ab idololatria coinquinabitur' (see _Philippians_ p. 63 on the\nmisunderstanding of this passage). The covetous man sets up another\nobject of worship besides God. There is a sort of religious purpose, a\ndevotion of the soul, to greed, which makes the sin of the miser so\nhateful. The idea of avarice as a _religion_ may have been suggested to\nSt Paul by our Lord's words, Matt. vi. 24 \u03bf\u1f50 \u03b4\u03cd\u03bd\u03b1\u03c3\u03b8\u03b5 \u0398\u03b5\u1ff7 \u03b4\u03bf\u03c5\u03bb\u03b5\u03cd\u03b5\u03b9\u03bd \u03ba\u03b1\u1f76\n\u03bc\u03b1\u03bc\u03c9\u03bd\u1fb7, though it is a mistake to suppose that Mammon was the name of a\nSyrian deity. It appears however elsewhere in Jewish writers of this and\nlater ages: e.g. Philo _de Mon._ i. 2 (II. p. 214 sq.) \u03c0\u03b1\u03bd\u03c4\u03b1\u03c7\u03cc\u03b8\u03b5\u03bd \u03bc\u1f72\u03bd\n\u1f00\u03c1\u03b3\u03cd\u03c1\u03b9\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c7\u03c1\u03c5\u03c3\u03af\u03bf\u03bd \u1f10\u03ba\u03c0\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03c3\u03b9, \u03c4\u1f78 \u03b4\u1f72 \u03c0\u03bf\u03c1\u03b9\u03c3\u03b8\u1f72\u03bd \u1f61\u03c2 \u1f04\u03b3\u03b1\u03bb\u03bc\u03b1 \u03b8\u03b5\u1fd6\u03bf\u03bd \u1f10\u03bd\n\u1f00\u03b4\u03cd\u03c4\u03bf\u03b9\u03c2 \u03b8\u03b7\u03c3\u03b1\u03c5\u03c1\u03bf\u03c6\u03c5\u03bb\u03b1\u03ba\u03bf\u1fe6\u03c3\u03b9\u03bd (with the whole context), and _Shemoth Rabba_\nfol. 121. 3 'Qui opes suas multiplicat per f\u0153nus, ille est\nidololatra' (with other passages quoted by Wetstein and Sch\u00f6ttgen on\nEphes. v. 5). St Chrysostom, _Hom. in Johann. lxv_ (VIII. p. 392 sq.),\nenlarges on the cult of wealth\u2014the consecration of it, the worship paid\nto it, the sacrifices demanded by it: \u1f21 \u03b4\u1f72 \u03c6\u03b9\u03bb\u03b1\u03c1\u03b3\u03c5\u03c1\u03af\u03b1 \u03bb\u03ad\u03b3\u03b5\u03b9, \u0398\u1fe6\u03c3\u03cc\u03bd \u03bc\u03bf\u03b9\n\u03c4\u1f74\u03bd \u03c3\u03b1\u03c5\u03c4\u03bf\u1fe6 \u03c8\u03c5\u03c7\u03ae\u03bd, \u03ba\u03b1\u1f76 \u03c0\u03b5\u03af\u03b8\u03b5\u03b9\u00b7 \u1f41\u03c1\u1fb7\u03c2 \u1f45\u03b9\u03bf\u03c5\u03c2 \u1f14\u03c7\u03b5\u03b9 \u03b2\u03c9\u03bc\u03bf\u03cd\u03c2, \u03bf\u1f37\u03b1 \u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b8\u03cd\u03bc\u03b1\u03c4\u03b1\n(p. 393). The passage in _Test. xii Patr._ Jud. 18 \u1f21 \u03c6\u03b9\u03bb\u03b1\u03c1\u03b3\u03c5\u03c1\u03af\u03b1 \u03c0\u03c1\u1f78\u03c2\n\u1f14\u03b9\u03b4\u03c9\u03bb\u03b1 \u1f41\u03b4\u03b7\u03b3\u03b5\u1fd6 is no real parallel to St Paul's language, though at first\nsight it seems to resemble it. For \u1f25\u03c4\u03b9\u03c2, 'seeing that it', see the note\non Phil. iv. 3.\n\n6, 7. \u03b4\u03b9' \u1f05 \u03ba.\u03c4.\u03bb.] The received text requires correction in two points.\n(1) It inserts the words \u1f10\u03c0\u1f76 \u03c4\u03bf\u1f7a\u03c2 \u1f51\u03b9\u03bf\u1f7a\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03c0\u03b5\u03b9\u03b8\u03b5\u03af\u03b1\u03c2 after \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6.\nThough this insertion has preponderating support, yet the words are\nevidently interpolated from the parallel passage, Ephes. v. 6 \u03b4\u03b9\u1f70 \u03c4\u03b1\u1fe6\u03c4\u03b1\n\u03b3\u1f70\u03c1 \u1f10\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u1f21 \u1f40\u03c1\u03b3\u1f74 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u1f10\u03c0\u1f76 \u03c4\u03bf\u1f7a\u03c2 \u1f51\u03b9\u03bf\u1f7a\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03c0\u03b5\u03b9\u03b8\u03b5\u03af\u03b1\u03c2. We are\ntherefore justified in rejecting them with other authorities, few in\nnumber but excellent in character. See the detached note on various\nreadings. When the sentence is thus corrected, the parallelism of \u03b4\u03b9' \u1f05\n... \u03b5\u03bd \u03bf\u1f37\u03c2 \u03ba\u03b1\u03af ... may be compared with Ephes. i. 11 \u1f10\u03bd h\u1ff7 \u03ba\u03b1\u1f76\n\u1f10\u03ba\u03bb\u03b7\u03c1\u03ce\u03b8\u03b7\u03bc\u03b5\u03bd ... \u1f10\u03bd h\u1ff7 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 ... \u1f10\u03bd h\u1ff7 \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u03b5\u03cd\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 \u1f10\u03c3\u03c6\u03c1\u03b1\u03b3\u03af\u03c3\u03b8\u03b7\u03c4\u03b5,\nand ii. 21, 22 \u1f10\u03bd h\u1ff7 \u03c0\u1fb6\u03c3\u03b1 [\u1f21] \u03bf\u1f30\u03ba\u03bf\u03b4\u03bf\u03bc\u1f74 ... \u1f10\u03bd \u1fa7 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2\n\u03c3\u03c5\u03bd\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03b5\u1fd6\u03c3\u03b8\u03b5. (2) The vast preponderance of authority obliges us to\nsubstitute \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 for \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2.\n\n6. \u1f10\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9] This may refer either to the present and continuous\ndispensation, or to the future and final judgment. The present \u1f10\u03c1\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9\nis frequently used to denote the _certainty_ of a future event, e.g.\nMatt. xvii. 11, Joh. iv. 21, xiv. 3, whence \u1f41 \u1f10\u03c1\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 is a designation\nof the Messiah: see Winer \u00a7 xl. p. 332.\n\n------------------------------------------------------------------------\n\nIII. 7, 8]\n\n\u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6\u00b7 ^7\u1f10\u03bd \u03bf\u1f37\u03c2 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03c0\u03b5\u03c1\u03b9\u03b5\u03c0\u03b1\u03c4\u03ae\u03c3\u03b1\u03c4\u03ad \u03c0\u03bf\u03c4\u03b5, \u1f45\u03c4\u03b5 \u1f10\u03b6\u1fc6\u03c4\u03b5 \u1f10\u03bd \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2\u00b7\n^8\u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u1f00\u03c0\u03cc\u03b8\u03b5\u03c3\u03b8\u03b5 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1,\n\n\n\u1f10\u03bd \u03bf\u1f37\u03c2 \u03ba.\u03c4.\u03bb.] The clause \u1f10\u03c0\u1f76 \u03c4\u03bf\u1f7a\u03c2 \u1f51\u03b9\u03bf\u1f7a\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03c0\u03b5\u03b9\u03b8\u03b5\u03af\u03b1\u03c2 having been\nstruck out, \u1f10\u03bd \u03bf\u1f37\u03c2 must necessarily be neuter and refer to the same as\n\u03b4\u03b9' \u1f05. Independently of the rejection of the clause, this neuter seems\nmore probable in itself than the masculine: for (1) The expression\n\u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03bd \u1f10\u03bd is most commonly used of things, not of persons,\nespecially in this and the companion epistle; iv. 5, Ephes. ii. 2, 10,\niv. 17, v. 2; (2) The Apostle would hardly denounce it as a sin in his\nColossian converts that they 'walked _among_ the sons of disobedience';\nfor the Christian, though not of the world, is necessarily in the world:\ncomp. 1 Cor. v. 10. The apparent parallel, Ephes. ii. 3 \u1f10\u03bd \u03bf\u1f37\u03c2 \u03ba\u03b1\u1f76 \u1f21\u03bc\u03b5\u1fd6\u03c2\n\u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f00\u03bd\u03b5\u03c3\u03c4\u03c1\u03ac\u03c6\u03b7\u03bc\u03ad\u03bd \u03c0\u03bf\u03c4\u03b5 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03b9\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u1f21\u03bc\u1ff6\u03bd (where \u03bf\u1f37\u03c2\nseems to be masculine), does not hold, because the addition \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2\n\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03b9\u03c2 \u03ba.\u03c4.\u03bb. makes all the difference. Thus the rejection of the\nclause, which was decided by textual considerations, is confirmed by\nexegetical reasons.\n\n7. \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2] '_ye_, like the other heathen' (i. 6 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd), but in\nthe next verse \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 is rather 'ye yourselves', 'ye notwithstanding\nyour former lives'.\n\n\u1f45\u03c4\u03b5 \u1f10\u03b6\u1fc6\u03c4\u03b5 \u03ba.\u03c4.\u03bb.] 'When _ye_ lived in this atmosphere of sin, when ye\nhad not yet died to the world'.\n\n\u1f10\u03bd \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2] '_in these things_.' We should have expected \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2, but\n\u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 is substituted as more emphatic and condemnatory: comp. Ephes.\nv. 6 \u03b4\u03b9\u1f70 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03b3\u1f70\u03c1 \u1f10\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb. The two expressions \u03b6\u1fc6\u03bd \u1f10\u03bd and\n\u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03bd \u1f10\u03bd involve two distinct ideas, denoting the condition of\ntheir life and the character of their practice respectively. Their\nconduct was conformable to their circumstances. Comp. Gal. v. 25 \u03b5\u1f30\n\u03b6\u1ff6\u03bc\u03b5\u03bd \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9, \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9 \u03ba\u03b1\u1f76 \u03c3\u03c4\u03bf\u03b9\u03c7\u1ff6\u03bc\u03b5\u03bd.\n\n8. The errors of the past suggest the obligations of the present. Thus\nthe Apostle returns to the topic with which the sentence commenced. But\nthe violence of the contrast has broken up the grammar of the sentence:\nsee the note on ver. 5.\n\n\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1] 'not only those vices which have been specially named before\n(ver. 5), but _all_ of whatever kind.' The Apostle accordingly goes on\nto specify sins of a wholly different type from those already mentioned,\nsins of uncharitableness, such as anger, detraction, malice, and the\nlike.\n\n------------------------------------------------------------------------\n\nIII. 9]\n\n\u1f40\u03c1\u03b3\u03ae\u03bd, \u03b8\u03c5\u03bc\u03cc\u03bd, \u03ba\u03b1\u03ba\u03af\u03b1\u03bd, \u03b2\u03bb\u03b1\u03c3\u03c6\u03b7\u03bc\u03af\u03b1\u03bd, \u03b1\u1f30\u03c3\u03c7\u03c1\u03bf\u03bb\u03bf\u03b3\u03af\u03b1\u03bd \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f51\u03bc\u1ff6\u03bd\u00b7\n^9\u03bc\u1f74 \u03c8\u03b5\u03cd\u03b4\u03b5\u03c3\u03b8\u03b5 \u03b5\u1f30\u03c2 \u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2\u00b7 \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9\n\n\n\u1f40\u03c1\u03b3\u03ae\u03bd, \u03b8\u03c5\u03bc\u03cc\u03bd] '_anger_, _wrath_'. The one denotes a more or less settled\nfeeling of hatred, the other a tumultuous outburst of passion. This\ndistinction of the two words was fixed chiefly by the definitions of the\nStoics: Diog. Laert. vii. 114 \u1f41 \u03b4\u1f72 \u03b8\u03c5\u03bc\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f40\u03c1\u03b3\u1f74 \u1f00\u03c1\u03c7\u03bf\u03bc\u03ad\u03bd\u03b7. So\nAmmianus \u03b8\u1f7a\u03bc\u03bf\u03c2 \u03bc\u03ad\u03bd \u1f10\u03c3\u03c4\u03b9 \u03c0\u03c1\u03cc\u03c3\u03ba\u03b1\u03b9\u03c1\u03bf\u03c2, \u1f40\u03c1\u03b3\u1f74 \u03b4\u1f72 \u03c0\u03bf\u03bb\u03c5\u03c7\u03c1\u03cc\u03bd\u03b9\u03bf\u03c2 \u03bc\u03bd\u03b7\u03c3\u03b9\u03ba\u03b1\u03ba\u03af\u03b1,\nGreg. Naz. _Carm._ 34 (II. p. 612) \u03b8\u03c5\u03bc\u1f78\u03c2 \u03bc\u03ad\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u03b8\u03c1\u03cc\u03bf\u03c2 \u03b6\u03ad\u03c3\u03b9\u03c2 \u03c6\u03c1\u03b5\u03bd\u03cc\u03c2,\n\u1f40\u03c1\u03b3\u1f74 \u03b4\u1f72 \u03b8\u03c5\u03bc\u1f78\u03c2 \u1f10\u03bc\u03bc\u03ad\u03bd\u03c9\u03bd. They may be represented in Latin by _ira_ and\n_furor_; Senec. _de Ira_ ii. 36 'Ajacem in mortem egit furor, in furorem\nira', and Jerome in Ephes. iv. 31 'Furor incipiens ira est': see Trench\n_N. T. Syn._ \u00a7 xxxvii, p. 123 sq. On other synonymes connected with\n\u03b8\u03c5\u03bc\u03cc\u03c2 and \u1f40\u03c1\u03b3\u03ae see the note on Ephes. iv. 31.\n\n\u03ba\u03b1\u03ba\u03af\u03b1\u03bd] '_malice_', or '_malignity_', as it may be translated in default\nof a better word. It is not (at least in the New Testament) vice\ngenerally, but the vicious nature which is bent on _doing harm to\nothers_, and is well defined by Calvin (on Ephes. iv. 31) 'animi\npravitas, qu\u00e6 _humanitati et \u00e6quitati_ est opposita'. This will be\nevident from the connexion in which it appears, e.g. Rom. i. 29, Eph.\niv. 31, Tit. iii. 3. Thus \u03ba\u03b1\u03ba\u03af\u03b1 and \u03c0\u03bf\u03bd\u03b7\u03c1\u03af\u03b1 (which frequently occur\ntogether, e.g. 1 Cor. v. 8) only differ in so far as the one denotes\nrather the vicious disposition, the other the active exercise of it. The\nword is carefully investigated in Trench _N. T. Syn._ \u00a7 xi. p. 35 sq.\n\n\u03b2\u03bb\u03b1\u03c3\u03c6\u03b7\u03bc\u03af\u03b1\u03bd] '_evil speaking_, _railing_, _slandering_', as frequently,\ne.g. Rom. iii. 8, xiv. 16, 1 Cor. iv. 13 (v.l.), x. 30, Ephes. iv. 31,\nTit. iii. 2. The word has the same twofold sense, 'evil speaking' and\n'blasphemy', in classical writers, which it has in the New Testament.\n\n\u03b1\u1f30\u03c3\u03c7\u03c1\u03bf\u03bb\u03bf\u03b3\u03af\u03b1\u03bd] '_foul-mouthed abuse_'. The word, as used elsewhere, has\ntwo meanings: (1) '_Filthy-talking_', as defined in Clem. Alex. _P\u00e6d._\nii. 6 (p. 189 sq.), where it is denounced at length: comp. Arist. _Pol._\nvii. 17, Epict. _Man._ 33, Plut. _Mor._ 9, and so commonly; (2)\n'_Abusive language_', as e.g. Polyb. viii. 13. 8, xii. 13. 3, xxxi. 10.\n4. If the two senses of the word had been quite distinct, we might have\nhad some difficulty in choosing between them here. The former sense is\nsuggested by the parallel passage Ephes. v. 4 \u03b1\u1f30\u03c3\u03c7\u03c1\u03cc\u03c4\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03c9\u03c1\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 \u1f24\n\u03b5\u1f50\u03c4\u03c1\u03b1\u03c0\u03b5\u03bb\u03af\u03b1; the second by the connexion with \u03b2\u03bb\u03b1\u03c3\u03c6\u03b7\u03bc\u03af\u03b1 here. But the\nsecond sense is derived from the first. The word can only mean 'abuse',\nwhen the abuse is 'foul-mouthed'. And thus we may suppose that both\nideas, 'filthiness' and 'evil-speaking', are included here.\n\n9. \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03ba.\u03c4.\u03bb.] '_putting off_'. Do these aorist participles\ndescribe an action coincident with or prior to the \u03c8\u03b5\u03cd\u03b4\u03b5\u03c3\u03b8\u03b5? In other\nwords are they part of the command, or do they assign the reason for the\ncommand? Must they be rendered 'putting off', or 'seeing that ye did (at\nyour baptism) put off'? The former seems the more probable\ninterpretation: for (1) Though both ideas are found in St Paul, the\nimperative is the more usual; e.g. Rom. xiii. 12 sq. \u1f00\u03c0\u03bf\u03b8\u03ce\u03bc\u03b5\u03b8\u03b1 \u03bf\u1f56\u03bd \u03c4\u1f70\n\u1f10\u03c1\u03b3\u03b1 \u03c4\u03bf\u1fe6 \u03c3\u03ba\u03cc\u03c4\u03bf\u03c5\u03c2, \u1f10\u03bd\u03b4\u03c5\u03c3\u03ce\u03bc\u03b5\u03b8\u03b1 \u03b4\u1f72 \u03c4\u1f70 \u1f45\u03c0\u03bb\u03b1 \u03c4\u03bf\u1fe6 \u03c6\u03c9\u03c4\u03cc\u03c2 ... \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u03c4\u1f78\u03bd\n\u039a\u03cd\u03c1\u03b9\u03bf\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd, Ephes. vi. 11 \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u03c4\u1f74\u03bd \u03c0\u03b1\u03bd\u03bf\u03c0\u03bb\u03af\u03b1\u03bd with ver.\n14 \u03c3\u03c4\u1fc6\u03c4\u03b5 \u03bf\u1f56\u03bd ... \u1f10\u03bd\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03ba.\u03c4.\u03bb., 1 Thess. V. 8 \u03bd\u03ae\u03c6\u03c9\u03bc\u03b5\u03bd \u1f10\u03bd\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9\n\u03ba.\u03c4.\u03bb. The one exception is Gal. iii. 27 \u1f45\u03c3\u03bf\u03b9 \u03b3\u1f70\u03c1 \u03b5\u1f30\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd\n\u1f10\u03b2\u03b1\u03c0\u03c4\u03af\u03c3\u03b8\u03b7\u03c4\u03b5, \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u1f10\u03bd\u03b5\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5. (2) The 'putting on' in the parallel\npassage, Ephes. iv. 24, is imperative, not affirmative, whether we read\n\u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 or \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5. (3) The participles here are followed\nimmediately by an imperative in the context, ver. 12 \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u03bf\u1f56\u03bd,\nwhere the idea seems to be the same. For the synchronous aorist\nparticiple see Winer \u00a7 xlv. p. 430. St Paul uses \u1f00\u03c0\u03b5\u03ba\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9,\n\u1f10\u03bd\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 (not \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03bf\u03bc\u03b5\u03bd\u03bf\u03b9, \u1f10\u03bd\u03b4\u03cd\u03bf\u03bc\u03b5\u03bd\u03bf\u03b9), for the same reason for\nwhich he uses \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 (not \u1f10\u03bd\u03b4\u03cd\u03b5\u03c3\u03b8\u03b5), because it is a thing to be done\n_once for all_. For the double compound \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 see the notes on ii.\n11, 15.\n\n------------------------------------------------------------------------\n\nIII. 10, 11]\n\n\u03c4\u1f78\u03bd \u03c0\u03b1\u03bb\u03b1\u1f76\u03bf\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03c3\u1f7a\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03c0\u03c1\u03ac\u03be\u03b5\u03c3\u03b9\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6, ^{10}\u03ba\u03b1\u1f76 \u1f10\u03bd\u03b4\u03c5\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f78\u03bd\n\u03bd\u03ad\u03bf\u03bd, \u03c4\u1f78\u03bd \u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03b5\u1f30\u03c2 \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd \u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd\u00b7\n^{11}\u1f45\u03c0\u03bf\u03c5\n\n\n\u03c0\u03b1\u03bb\u03b1\u1f76\u03bf\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd] as Rom. vi. 6, Ephes. iv. 22. With this expression\ncompare \u1f41 \u1f14\u03be\u03c9, \u1f41 \u1f14\u03c3\u03c9 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2, Rom. vii. 22, 2 Cor. iv. 16, Ephes. iii.\n16; \u1f41 \u03ba\u03c1\u03c5\u03c0\u03c4\u1f78\u03c2 \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2, 1 Pet. iii. 4; \u1f41 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c2 \u03bc\u03bf\u03c5\n\u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2, 'my insignificance', Polycr. in Euseb. _H.E._ v. 24.\n\n10. \u03c4\u1f78\u03bd \u03bd\u03ad\u03bf\u03bd \u03ba.\u03c4.\u03bb.] In Ephes. iv. 24 it is \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 _\u03c4\u1f78\u03bd \u03ba\u03b1\u03b9\u03bd\u1f78\u03bd_\n\u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd. Of the two words \u03bd\u03ad\u03bf\u03c2 and \u03ba\u03b1\u03b9\u03bd\u03cc\u03c2, the former refers solely to\ntime, the other denotes quality also; the one is new as being _young_,\nthe other new as being _fresh_: the one is opposed to long duration, the\nother to effeteness; see Trench _N. T. Syn._ \u00a7 lx. p. 206. Here the idea\nwhich is wanting to \u03bd\u03ad\u03bf\u03c2, and which \u03ba\u03b1\u03b9\u03bd\u1f78\u03c2 gives in the parallel\npassage, is more than supplied by the addition \u03c4\u1f78\u03bd \u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba.\u03c4.\u03bb.\n\nThe \u03bd\u03ad\u03bf\u03c2 or \u03ba\u03b1\u03b9\u03bd\u1f78\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2 in these passages is not Christ Himself, as\nthe parallel expression \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 might suggest, and as it is\nactually used in Ign. _Ephes._ 20 \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u03ba\u03b1\u03b9\u03bd\u1f78\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd, but the regenerate man formed after Christ. The idea here is\nthe same as in \u03ba\u03b1\u03b9\u03bd\u1f74 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2, 2 Cor. v. 17, Gal. vi. 15: comp. Rom. vi. 4\n\u03ba\u03b1\u03b9\u03bd\u03cc\u03c4\u03b7\u03c2 \u03b6\u03c9\u1fc6\u03c2, Barnab. 16 \u1f10\u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03b8\u03b1 \u03ba\u03b1\u03b9\u03bd\u03bf\u03af, \u03c0\u03ac\u03bb\u03b9\u03bd \u1f10\u03be \u1f00\u03c1\u03c7\u1fc6\u03c2 \u03ba\u03c4\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9.\n\n\u03c4\u1f78\u03bd \u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd] '_which is_ ever _being renewed_'. The force of\nthe present tense is explained by 2 Cor. iv. 16 \u1f41 \u1f14\u03c3\u03c9 \u1f21\u03bc\u1ff6\u03bd [\u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2]\n\u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03bf\u1fe6\u03c4\u03b1\u03b9 _\u1f21\u03bc\u03ad\u03c1\u1fb3 \u03ba\u03b1\u1f76 \u1f21\u03bc\u03ad\u03c1\u1fb3_. Compare also the use of the\ntenses in the parallel passage, Ephes. iv. 22 sq. \u1f00\u03c0\u03bf\u03b8\u03ad\u03c3\u03b8\u03b1\u03b9,\n_\u1f00\u03bd\u03b1\u03bd\u03b5\u03bf\u1fe6\u03c3\u03b8\u03b1\u03b9_, \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9. For the opposite see Ephes. iv. 22 \u03c4\u1f78\u03bd\n\u03c0\u03b1\u03bb\u03b1\u1f76\u03bf\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03c4\u1f78\u03bd _\u03c6\u03b8\u03b5\u03b9\u03c1\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd_ \u03ba.\u03c4.\u03bb.\n\n\u03b5\u1f30\u03c2 \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd] '_unto perfect knowledge_', the true knowledge in Christ,\nas opposed to the false knowledge of the heretical teachers. For the\nimplied contrast see above pp. 44, 99 sq. (see the notes on i. 9, ii.\n3), and for the word \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2 the note on i. 9. The words here are to\nbe connected closely with \u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd: comp. Heb. vi. 6 \u03c0\u03ac\u03bb\u03b9\u03bd\n_\u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03af\u03b6\u03b5\u03b9\u03bd_ \u03b5\u1f30\u03c2 \u03bc\u03b5\u03c4\u03ac\u03bd\u03bf\u03b9\u03b1\u03bd.\n\n\u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u03ba.\u03c4.\u03bb.] The reference is to Gen. i. 26 \u03ba\u03b1\u1f76 \u03b5\u1f36\u03c0\u03b5\u03bd \u1f41 \u0398\u03b5\u03cc\u03c2\n\u03a0\u03bf\u03af\u03b7\u03c3\u03c9\u03bc\u03b5\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u1f21\u03bc\u03b5\u03c4\u03ad\u03c1\u03b1\u03bd \u03ba.\u03c4.\u03bb.; comp. ver. 28 \u03ba\u03b1\u03c4'\n\u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u0398\u03b5\u03bf\u1fe6 \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u03b1\u1f50\u03c4\u03cc\u03bd. See also Ephes. iv. 24 \u03c4\u1f78\u03bd \u03ba\u03b1\u03b9\u03bd\u1f78\u03bd \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd\n\u03c4\u1f78\u03bd \u03ba\u03b1\u03c4\u1f70 \u0398\u1f72\u03bf\u03bd \u03ba\u03c4\u03b9\u03c3\u03b8\u03ad\u03bd\u03c4\u03b1. This reference however does not imply an\nidentity of the creation here mentioned with the creation of Genesis,\nbut only an analogy between\n\nthe two. The spiritual man in each believer's heart, like the primal man\nin the beginning of the world, was created after God's image. The \u03ba\u03b1\u03b9\u03bd\u1f74\n\u03ba\u03c4\u03af\u03c3\u03b9\u03c2 in this respect resembles the \u1f00\u03c1\u03c7\u03b1\u03af\u03b1 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2. The pronoun \u03b1\u1f50\u03c4\u1f78\u03bd\ncannot be referred to anything else but the \u03bd\u03ad\u03bf\u03c2 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2, the\nregenerate man; and the aorist \u03ba\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03bf\u03c2 (compare \u03ba\u03c4\u03b9\u03c3\u03b8\u03ad\u03bd\u03c4\u03b1 in the\nparallel passage Ephes. iv. 24) refers to the time of this \u1f00\u03bd\u03b1\u03b3\u03ad\u03bd\u03bd\u03b7\u03c3\u03b9\u03c2\nin Christ. See Barnab. 6 _\u1f00\u03bd\u03b1\u03ba\u03b1\u03b9\u03bd\u03af\u03c3\u03b1\u03c2_ \u1f21\u03bc\u1fb6\u03c2 \u1f10\u03bd \u03c4\u1fc7 \u1f00\u03c6\u03ad\u03c3\u03b5\u03b9 \u03c4\u1ff6\u03bd\n\u1f01\u03bc\u03b1\u03c1\u03c4\u03b9\u1ff6\u03bd \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u1f21\u03bc\u1fb6\u03c2 \u1f04\u03bb\u03bb\u03bf\u03bd \u03c4\u03cd\u03c0\u03bf\u03bd ... \u1f61\u03c3\u1f70\u03bd \u03b4\u1f74 _\u1f00\u03bd\u03b1\u03c0\u03bb\u03ac\u03c3\u03c3\u03bf\u03bd\u03c4\u03bf\u03c2_\n\u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f21\u03bc\u1fb6\u03c2, after which Gen. i. 26 is quoted. The new birth was a\nrecreation in God's image; the subsequent life must be a deepening of\nthis image thus stamped upon the man.\n\nThe allusion to Genesis therefore requires us to understand \u03c4\u03bf\u1fe6\n\u03ba\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03bf\u03c2 of God, and not of Christ, as it is taken by St Chrysostom and\nothers; and this seems to be demanded also by the common use of \u1f41\n\u03ba\u03c4\u03af\u03c3\u03b1\u03c2. But if Christ is not \u1f41 \u03ba\u03c4\u03af\u03c3\u03b1\u03c2, may He not be intended by the\n\u03b5\u1f30\u03ba\u1ff6\u03bd \u03c4\u03bf\u1fe6 \u03ba\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03bf\u03c2? In favour of this interpretation it may be urged\n(1) That Christ elsewhere is called the \u03b5\u1f30\u03ba\u1f7c\u03bd of God, i. 15, 2 Cor. iv.\n4; (2) That the Alexandrian school interpreted the term in Gen. i. 26 as\ndenoting the Logos; thus Philo _de Mund._ Op. 6 (I. p. 5 M) \u03c4\u1f78 \u1f00\u03c1\u03c7\u03ad\u03c4\u03c5\u03c0\u03bf\u03bd\n\u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1, \u1f30\u03b4\u03ad\u03b1 \u03c4\u1ff6\u03bd \u1f30\u03b4\u03b5\u1ff6\u03bd \u1f41 \u0398\u03b5\u03bf\u1fe6 \u03bb\u03cc\u03b3\u03bf\u03c2 (comp. ib. \u00a7\u00a7 7, 23, 24, 48),\n_Fragm._ II. p. 625 M \u03b8\u03bd\u03b7\u03c4\u1f78\u03bd \u03b3\u1f70\u03c1 \u03bf\u1f50\u03b4\u1f72\u03bd \u1f00\u03c0\u03b5\u03b9\u03ba\u03bf\u03bd\u03b9\u03c3\u03b8\u1fc6\u03bd\u03b1\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u1f00\u03bd\u03c9\u03c4\u03ac\u03c4\u03c9\n\u03ba\u03b1\u1f76 \u03c0\u03b1\u03c4\u03ad\u03c1\u03b1 \u03c4\u1ff6\u03bd \u1f45\u03bb\u03c9\u03bd \u1f10\u03b4\u03cd\u03bd\u03b1\u03c4\u03bf, \u1f00\u03bb\u03bb\u1f70 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03bf\u03bd \u0398\u1f72\u03bf\u03bd \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd\n\u1f10\u03ba\u03ad\u03b9\u03bd\u03bf\u03c5 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03ba.\u03c4.\u03bb. _Leg. Alleg._ i. 31, 32 (I. p. 106 sq.). Hence\nPhilo speaks of the first man as \u03b5\u1f30\u03ba\u1f7c\u03bd \u03b5\u1f30\u03ba\u03cc\u03bd\u03bf\u03c2 (_de Mund. Op._ 6), and\nas \u03c0\u03b1\u03b3\u03ba\u03ac\u03bb\u03bf\u03c5 \u03c0\u03b1\u03c1\u03b1\u03b4\u03ad\u03b9\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c0\u03ac\u03b3\u03ba\u03b1\u03bb\u03bf\u03bd \u03bc\u03af\u03bc\u03b7\u03bc\u03b1 (ib. \u00a7 48). A pregnant meaning\nis thus given to \u03ba\u03b1\u03c4\u1f70, and \u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 is rendered 'after the fashion\n(or pattern) of the Image'. But this interpretation seems very\nimprobable in St Paul; for (1) In the parallel passage Ephes. iv. 24 the\nexpression is simply \u03ba\u03b1\u03c4\u1f70 \u0398\u03b5\u03cc\u03bd, which may be regarded as equivalent to\n\u03ba\u03b1\u03c4' \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03c4\u03af\u03c3\u03b1\u03bd\u03c4\u03bf\u03c2 here; (2) The Alexandrian explanation of Gen.\ni. 26 just quoted is very closely allied to the Platonic doctrine of\nideas (for the \u03b5\u1f30\u03ba\u03ce\u03bd, so interpreted, is the archetype or ideal pattern\nof the sensible world), and thus it lies outside the range of those\nconceptions which specially recommended the Alexandrian terminology of\nthe Logos to the Apostles, as a fit vehicle for communicating the truths\nof Christianity.\n\n11. \u1f45\u03c0\u03bf\u03c5] i.e. 'in this regenerate life, in this spiritual region into\nwhich the believer is transferred in Christ.'\n\n------------------------------------------------------------------------\n\nIII. 11]\n\n\u03bf\u1f50\u03ba \u1f15\u03bd\u03b9 \u1f1d\u03bb\u03bb\u03b7\u03bd \u03ba\u03b1\u1f76 \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2, \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1f74 \u03ba\u03b1\u1f76 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u03b1,\n\n\n\u03bf\u1f50\u03ba \u1f15\u03bd\u03b9] 'Not only does the distinction not exist, but it _cannot_\nexist.' It is a mundane distinction, and therefore it has disappeared.\nFor the sense of \u1f15\u03bd\u03b9, negativing not merely the fact but the\npossibility, see the note on Gal. iii. 28.\n\n\u1f1d\u03bb\u03bb\u03b7\u03bd \u03ba.\u03c4.\u03bb.] Comparing the enumeration here with the parallel passage\nGal. iii. 28, we mark this difference. In Galatians the abolition of all\ndistinctions is stated in the broadest way by the selection of three\ntypical instances; religious prerogative (\u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2, \u1f1d\u03bb\u03bb\u03b7\u03bd), social caste\n(\u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2, \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2), natural sex (\u1f04\u03c1\u03c3\u03b5\u03bd, \u03b8\u1fc6\u03bb\u03c5). Here on the other hand\nthe examples are chosen with special reference to the immediate\ncircumstances of the Colossian Church. (1) The Judaism of the Colossian\nheretics is met by \u1f1d\u03bb\u03bb\u03b7\u03bd \u03ba\u03b1\u1f76 \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2, and as it manifested itself\nespecially in enforcing circumcision, this is further emphasized by\n\u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1f74 \u03ba\u03b1\u1f76 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u03b1 (see above, p. 73). (2) Their Gnosticism again\nis met by \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2. They laid special stress on intelligence,\npenetration, gnosis. The Apostle offers the full privileges of the\nGospel to barbarians and even barbarians of the lowest type (see p. 99\nsq.). In Rom. i. 14, the division \u1f1d\u03bb\u03bb\u03b7\u03c3\u03af\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03b2\u03b1\u03c1\u03b2\u03b1\u03c1\u03bf\u1fd6\u03c2 is almost\nsynonymous with \u03c3\u03bf\u03c6\u03bf\u1fd6\u03c2 \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03cc\u03b7\u03c4\u03bf\u03b9\u03c2. (3) Special circumstances,\nconnected with an eminent member of the Church of Coloss\u00e6, had directed\nhis attention at this moment to the relation of masters and slaves.\nHence he cannot leave the subject without adding \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2, \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2,\nthough this has no special bearing on the Colossian heresy. See above p.\n33, and the note on iii. 22, together with the introduction to the\nEpistle to Philemon.\n\n\u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1f74 \u03ba.\u03c4.\u03bb.] Enforcing and extending the lesson of the previous\nclause. This abolition of distinctions applies to religious privilege,\nnot only as inherited by birth (\u1f1d\u03bb\u03bb\u03b7\u03bd \u03ba\u03b1\u1f76 \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2), but also as assumed\nby adoption (\u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1f74 \u03ba\u03b1\u1f76 \u1f00\u03ba\u03c1\u03bf\u03b2\u03c5\u03c3\u03c4\u03af\u03b1). If it is no advantage to be born\na Jew, it is none to become as a Jew; comp. 1 Cor. vii. 19, Gal. v. 6,\nvi. 15.\n\n------------------------------------------------------------------------\n\nIII. 11]\n\n\u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2, \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2, \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2, \u1f00\u03bb\u03bb\u1f70 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1\n\n\n\u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2] To the Jew the whole world was divided into \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03b9 and\n\u1f1d\u03bb\u03bb\u03b7\u03bd\u03b5\u03c2, the privileged and unprivileged portions of mankind, religious\nprerogative being taken as the line of demarcation (see notes Gal. ii.\n3). To the Greek and Roman it was similarly divided into \u1f1d\u03bb\u03bb\u03b7\u03bd\u03b5\u03c2 and\n\u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03b9, again the privileged and unprivileged portion of the human\nrace, civilization and culture being now the criterion of distinction.\nThus from the one point of view the \u1f1d\u03bb\u03bb\u03b7\u03bd is contrasted\ndisadvantageously with the \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2, while from the other he is\ncontrasted advantageously with the \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2. Both distinctions are\nequally antagonistic to the Spirit of the Gospel. The Apostle declares\nboth alike null and void in Christ. The twofold character of the\nColossian heresy enables him to strike at these two opposite forms of\nerror with one blow.\n\nThe word \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2 properly denoted one who spoke an inarticulate,\nstammering, unintelligible language; see Max M\u00fcller _Lectures on the\nScience of Language_ 1st ser. p. 81 sq., 114 sq., Farrar _Families of\nSpeech_ p. 21: comp. 1 Cor. xiv. 11. Hence it was adopted by Greek\nexclusiveness and pride to stigmatize the rest of mankind, a feeling\nembodied in the proverb \u03c0\u1fb6\u03c2 \u03bc\u1f74 \u1f1d\u03bb\u03bb\u03b7\u03bd \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2 (Servius on Verg. _\u00c6n._\nii. 504); comp. Plato _Polit._ 262 E \u03c4\u1f78 \u03bc\u1f72\u03bd \u1f19\u03bb\u03bb\u03b7\u03bd\u03b9\u03ba\u1f78\u03bd \u1f61\u03c2 h\u1f72\u03bd \u1f00\u03c0\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd\n\u1f00\u03c6\u03b1\u03b9\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c7\u03c9\u03c1\u03af\u03c2, \u03c3\u03cd\u03bc\u03c0\u03b1\u03c3\u03b9 \u03b4\u1f72 \u03c4\u03bf\u1fd6\u03c2 \u1f04\u03bb\u03bb\u03bf\u03b9\u03c2 \u03b3\u03ad\u03bd\u03b5\u03c3\u03b9\u03bd ... \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03bd \u03bc\u03b9\u1fb7\n\u03ba\u03bb\u03ae\u03c3\u03b5\u03b9 \u03c0\u03c1\u03bf\u03c3\u03ad\u03b9\u03c0\u03bf\u03bd\u03c4\u03b5\u03c2 \u03b1\u1f50\u03c4\u1f78 \u03ba.\u03c4.\u03bb., Dionys. Hal. _Rhet._ xi. 5 \u03b4\u03b9\u03c0\u03bb\u03bf\u1fe6\u03bd \u03b4\u1f72\n\u03c4\u1f78 \u1f14\u03b8\u03bd\u03bf\u03c2, \u1f1d\u03bb\u03bb\u03b7\u03bd \u1f22 \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2 \u03ba.\u03c4.\u03bb. So Philo _Vit. Moys._ ii. 5 (II. p.\n138) speaks of \u03c4\u1f78 \u1f25\u03bc\u03b9\u03c3\u03c5 \u03c4\u03bc\u1fc6\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd \u03b3\u03ad\u03bd\u03bf\u03c5\u03c2, \u03c4\u1f78 \u03b2\u03b1\u03c1\u03b2\u03b1\u03c1\u03b9\u03ba\u03cc\u03bd, as\nopposed to \u03c4\u1f78 \u1f19\u03bb\u03bb\u03b7\u03bd\u03b9\u03ba\u03cc\u03bd. It is not necessary to suppose that they\nadopted it from the Egyptians, who seem to have called non-Egyptian\npeoples _berber_ (see Sir G. Wilkinson in Rawlinson's Herod. ii. 158);\nfor the onomatop\u0153ia will explain its origin independently, Strabo\nxiv. 2. 28 (p. 662) \u03bf\u1f36\u03bc\u03b1\u03b9 \u03b4\u1f72 \u03c4\u1f78 \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03bd \u03ba\u03b1\u03c4' \u1f00\u03c1\u03c7\u1f70\u03c2 \u1f10\u03ba\u03c0\u03b5\u03c6\u03c9\u03bd\u1fc6\u03c3\u03b8\u03b1\u03b9 \u03bf\u1f55\u03c4\u03c9\u03c2\n\u03ba\u03b1\u03c4' \u1f40\u03bd\u03bf\u03bc\u03b1\u03c4\u03bf\u03c0\u03bf\u03b9\u03af\u03b1\u03bd \u1f10\u03c0\u1f76 \u03c4\u1ff6\u03bd \u03b4\u03c5\u03c3\u03b5\u03ba\u03c6\u03cc\u03c1\u03c9\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03ba\u03bb\u03b7\u03c1\u1ff6\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03c1\u03b1\u03c7\u03ad\u03c9\u03c2 \u03bb\u03b1\u03bb\u03bf\u03cd\u03bd\u03c4\u03c9\u03bd,\n\u1f61\u03c2 \u03c4\u1f78 \u03b2\u03b1\u03c4\u03c4\u03b1\u03c1\u03af\u03b6\u03b5\u03b9\u03bd \u03ba.\u03c4.\u03bb. The Latins, adopting the Greek culture, adopted\nthe Greek distinction also, e.g. Cic. _de Fin._ ii. 15 'Non solum Gr\u00e6cia\net Italia, sed etiam omnis barbaria': and accordingly Dionysius, _Ant.\nRom._ i. 69, classes the Romans with the Greeks as distinguished from\nthe 'barbarians'\u2014this twofold division of the human race being taken for\ngranted as absolute and final. So too in v. 8, having mentioned the\nRomans, he goes on to speak of \u03bf\u1f31 \u1f04\u03bb\u03bb\u03bf\u03b9 \u1f1d\u03bb\u03bb\u03b7\u03bd\u03b5\u03c2. The older Roman poets\nhowever, writing from a Greek point of view, (more than half in irony)\nspeak of themselves as _barbari_ and of their country as _barbaria_;\ne.g. Plaut. _Mil. Glor._ ii. 2. 58 'poeta barbaro' (of N\u00e6vius), _Asin._\nProl. II. 'Maccus vortit barbare', _P\u0153n._ iii. 2. 21 'in barbaria\nboves'.\n\nIn this classification the Jews necessarily ranked as 'barbarians'. At\ntimes Philo seems tacitly to accept this designation (_Vit. Moys._\nl.c.); but elsewhere he resents it, _Leg. ad Cai._ 31 (II. p. 578) \u1f51\u03c0\u1f78\n\u03c6\u03c1\u03bf\u03bd\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2, \u1f61\u03c2 \u03bc\u1f72\u03bd \u1f15\u03bd\u03b9\u03bf\u03b9 \u03c4\u1ff6\u03bd \u03b4\u03b9\u03b1\u03b2\u03b1\u03bb\u03bb\u03cc\u03bd\u03c4\u03c9\u03bd \u1f14\u03b9\u03c0\u03bf\u03b9\u03b5\u03bd \u1f02\u03bd, \u03b2\u03b1\u03c1\u03b2\u03b1\u03c1\u03b9\u03ba\u03bf\u1fe6, \u1f61\u03c2 \u03b4'\n\u1f14\u03c7\u03b5\u03b9 \u03c4\u1f78 \u1f00\u03bb\u03b7\u03b8\u03ad\u03c2, \u1f10\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03af\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03b5\u1f50\u03b3\u03b5\u03bd\u03bf\u1fe6\u03c2. On the other hand the Christian\nApologists with a true instinct glory in the 'barbarous' origin of their\nreligion: Justin _Apol._ i. 5 (p. 56 A) \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03b2\u03b1\u03c1\u03b2\u03ac\u03c1\u03bf\u03b9\u03c2 \u1f51\u03c0' \u03b1\u1f50\u03c4\u03bf\u1fe6\n\u03c4\u03bf\u1fe6 \u039b\u03cc\u03b3\u03bf\u03c5 \u03bc\u03bf\u03c1\u03c6\u03c9\u03b8\u03ad\u03bd\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03c5 \u03b3\u03b5\u03bd\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5, ib. \u00a7 46 (p. 83 D) \u1f10\u03bd\n\u03b2\u03b1\u03c1\u03b2\u03ac\u03c1\u03bf\u03b9\u03c2 \u03b4\u1f72 \u1f08\u03b2\u03c1\u03ac\u03b1\u03bc \u03ba.\u03c4.\u03bb., Tatian. _ad Gr\u00e6c._ 29 \u03b3\u03c1\u03b1\u03c6\u03b1\u1fd6\u03c2 \u03c4\u03b9\u03c3\u1f76\u03bd \u1f10\u03bd\u03c4\u03c5\u03c7\u03b5\u1fd6\u03bd\n\u03b2\u03b1\u03c1\u03b2\u03b1\u03c1\u03b9\u03ba\u03b1\u1fd6\u03c2, ib. 31 \u03c4\u1f78\u03bd \u03b4\u1f72 (\u039c\u03c9\u03c5\u03c3\u1fc6\u03bd) \u03c0\u03ac\u03c3\u03b7\u03c2 \u03b2\u03b1\u03c1\u03b2\u03ac\u03c1\u03bf\u03c5 \u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u1f00\u03c1\u03c7\u03b7\u03b3\u03cc\u03bd, ib.\n35 \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03b8' \u1f21\u03bc\u1fb6\u03c2 \u03b2\u03b1\u03c1\u03b2\u03ac\u03c1\u03bf\u03c5 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2. By glorying in the name they gave\na practical comment on the Apostle's declaration, that the distinction\nof Greek and barbarian was abolished in Christ. In a similar spirit\nClem. Alex. _Strom._ i. 16 (p. 361) endeavours to prove that \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd\n\u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2 \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03c3\u03c7\u03b5\u03b4\u1f78\u03bd \u03c4\u03ad\u03c7\u03bd\u03b7\u03c2 \u03b5\u1f51\u03c1\u03b5\u03c4\u1f70\u03b9 \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03b9.\n\n'Not till that word _barbarian_', writes Prof. Max M\u00fcller (l.c. p. 118),\n'was struck out of the dictionary of mankind and replaced by _brother_,\nnot till the right of all nations of the world to be classed as members\nof one genus or kind was recognised, can we look even for the first\nbeginnings of our science. This change was effected by Christianity....\n_Humanity_ is a word which you look for in vain in Plato or Aristotle;\nthe idea of mankind as one family, as the children of one God, is an\nidea of Christian growth: and the science of mankind, and of the\nlanguages of mankind, is a science which, without Christianity, would\nnever have sprung into life. When people had been taught to look upon\nall men as brethren, then and then only, did the variety of human speech\npresent itself as a problem that called for a solution in the eyes of\nthoughtful observers: and I therefore date the real beginning of the\nscience of language from the first day of Pentecost.... The common\norigin of mankind, the differences of race and language, the\nsusceptibility of all nations of the highest mental culture, these\nbecome, in the new world in which we live, problems of scientific,\nbecause of more than scientific interest'. St Paul was the great\nexponent of the fundamental principle in the Christian Church which was\nsymbolized on the day of Pentecost, when he declared, as here, that in\nChrist there is neither \u1f1d\u03bb\u03bb\u03b7\u03bd nor \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, or as in Rom. i. 14 that he\nhimself was a debtor equally \u1f1d\u03bb\u03bb\u03b7\u03c3\u03af\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03b2\u03b1\u03c1\u03b2\u03ac\u03c1\u03bf\u03b9\u03c2.\n\nThe only other passage in the New Testament (besides those quoted) in\nwhich \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2 occurs is Acts xxviii. 2, 4, where it is used of the\npeople of Melita. If this Melita be Malta, they would be of Ph\u0153nician\ndescent.\n\n\u03a3\u03ba\u03cd\u03b8\u03b7\u03c2] the lowest type of barbarian. There is the same collocation of\nwords in Dionys. Halic. _Rhet._ xi. 5, 6 \u03c0\u03b1\u03c4\u03ae\u03c1, \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2, \u03bd\u03ad\u03bf\u03c2,\n\u00c6sch. _c. Ctes._ 172 \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2, \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, \u1f11\u03bb\u03bb\u03b7\u03bd\u03af\u03b6\u03c9\u03bd \u03c4\u1fc7 \u03c6\u03c9\u03bd\u1fc7 (of\nDemosthenes). The savageness of the Scythians was proverbial. The\nearlier Greek writers indeed, to whom _omne ignotum_ was _pro\nmagnifico_, had frequently spoken of them otherwise (see Strabo vii. 3.\n7 sq., p. 300 sq.). \u00c6schylus for instance called them \u1f14\u03c5\u03bd\u03bf\u03bc\u03bf\u03b9 \u03a3\u03ba\u03cd\u03b8\u03b1\u03b9,\n_Fragm._ 189 (comp. _Eum._ 703). Like the other Hyperboreans, they were\na simple, righteous people, living beyond the vices and the miseries of\ncivilisation. But the common estimate was far different, and probably\nfar more true: e.g. 3 Macc. vii. 5 \u03bd\u03cc\u03bc\u03bf\u03c5 \u03a3\u03ba\u03c5\u03b8\u1ff6\u03bd \u1f00\u03b3\u03c1\u03b9\u03c9\u03c4\u03ad\u03c1\u03b1\u03bd ... \u1f60\u03bc\u03cc\u03c4\u03b7\u03c4\u03b1\n(comp. 2 Macc. iv. 47), Joseph. _c. Ap._ ii. 37 \u03a3\u03ba\u03cd\u03b8\u03b1\u03b9 ... \u03b2\u03c1\u03b1\u03c7\u1f7a \u03c4\u1ff6\u03bd\n\u03b8\u03b7\u03c1\u03af\u03c9\u03bd \u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03bf\u03bd\u03c4\u03b5\u03c2, Philo _Leg. ad Cai._ 2 (II p. 547) \u03a3\u03b1\u03c1\u03bc\u03b1\u03c4\u1ff6\u03bd \u03b3\u03ad\u03bd\u03b7 \u03ba\u03b1\u1f76\n\u03a3\u03ba\u03c5\u03b8\u1ff6\u03bd, \u1f05\u03c0\u03b5\u03c1 \u03bf\u1f50\u03c7 \u1f27\u03c4\u03c4\u03bf\u03bd \u1f10\u03be\u03b7\u03b3\u03c1\u03af\u03c9\u03c4\u03b1\u03b9 \u03c4\u1ff6\u03bd \u0393\u03b5\u03c1\u03bc\u03b1\u03bd\u03b9\u03ba\u1ff6\u03bd, Tertull. _adv. Marc._\ni. 1 'Scytha tetrior'. In _Vit. Moys._ ii. 4 (I. p. 137) Philo seems to\nplace the Egyptians and the Scythians at the two extremes in the scale\nof barbarian nations. The passages given in Wetstein from classical\nwriters are hardly less strong in the same direction. Anacharsis the\nScythian is said to have retorted \u1f11\u03bc\u03bf\u1f76 \u03b4\u1f72 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f1d\u03bb\u03bb\u03b7\u03bd\u03b5\u03c2 \u03c3\u03ba\u03c5\u03b8\u03af\u03b6\u03bf\u03c5\u03c3\u03b9\u03bd,\nClem. _Strom._ i. 16 (p. 364).\n\nThe Jews had a special reason for their unfavourable estimate of the\nScythians. In the reign of Josiah hordes of these northern barbarians\nhad deluged Palestine and a great part of Western Asia (Herod. i.\n103\u2013106). The incident indeed is passed over in silence in the\nhistorical books; but the terror inspired by these invaders has found\nexpression in the prophets (Ezek. xxxviii, xxxix, Jer. i. 13 sq., vi. 1\nsq.), and they left behind them a memorial in the Greek name of\nBeth-shean, \u03a3\u03ba\u03c5\u03b8\u1ff6\u03bd \u03c0\u03cc\u03bb\u03b9\u03c2 (Judith iii. 10, 2 Macc. xii. 29: comp. Judges\ni. 27 LXX) or \u03a3\u03ba\u03c5\u03b8\u03cc\u03c0\u03bf\u03bb\u03b9\u03c2, which seems to have been derived from a\nsettlement on this occasion (Plin. _N.H._ v. 16; see Ewald. _Gesch._\nIII. p. 689 sq., Grove s.v. _Scythopolis_ in Smith's _Bibl. Dict._).\n\nHence Justin, _Dial._ \u00a7 28 (p. 246 A), describing the largeness of the\nnew dispensation, says \u03ba\u1f02\u03bd \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2 \u1f96 \u03c4\u03b9\u03c2 \u1f22 \u03a0\u03ad\u03c1\u03c3\u03b7\u03c2, \u1f14\u03c7\u03b5\u03b9 \u03b4\u1f72 \u03c4\u1f74\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6\n\u03b3\u03bd\u1ff6\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c6\u03c5\u03bb\u03ac\u03c3\u03c3\u03b5\u03b9 \u03c4\u1f70 \u03b1\u1f34\u03c9\u03bd\u03b9\u03b1 \u03b4\u03af\u03ba\u03b1\u03b9\u03b1 ... \u03c6\u03af\u03bb\u03bf\u03c2\n\u1f10\u03c3\u03c4\u1f76 \u03c4\u1ff7 \u0398\u03b5\u1ff7, where he singles out two different but equally low types of\nbarbarians, the Scythians being notorious for their ferocity, the\nPersians for their licentiousness (Clem. Alex. _P\u00e6d._ i. 7, p. 131,\n_Strom._ iii. 2, p. 515, and the Apologists generally). So too the\nPseudo-Lucian, _Philopatris_ 17, satirising Christianity, \u039a\u03a1. \u03c4\u03cc\u03b4\u03b5 \u03b5\u1f36\u03c0\u03b5,\n\u03b5\u1f30 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03c4\u1ff6\u03bd \u03a3\u03ba\u03c5\u03b8\u1ff6\u03bd \u1f10\u03bd \u03c4\u1ff7 \u03bf\u1f50\u03c1\u03b1\u03bd\u1ff7 \u1f10\u03b3\u03c7\u03b1\u03c1\u03ac\u03c4\u03bf\u03c5\u03c3\u03b9. \u03a4\u03a1. \u03c0\u03ac\u03bd\u03c4\u03b1, \u03b5\u1f30 \u03c4\u03cd\u03c7\u03bf\u03b9 \u03b3\u03b5\n\u03c7\u03c1\u03b7\u03c3\u03c4\u1f78\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03bd \u1f14\u03b8\u03bd\u03b5\u03c3\u03b9. From a misconception of this passage in the\nColossians, heresiologers distinguished four main forms of heresy in the\npre-Christian world, \u03b2\u03b1\u03c1\u03b2\u03b1\u03c1\u03b9\u03c3\u03bc\u03cc\u03c2, \u03c3\u03ba\u03c5\u03b8\u03b9\u03c3\u03bc\u03cc\u03c2, \u1f11\u03bb\u03bb\u03b7\u03bd\u03b9\u03c3\u03bc\u03cc\u03c2, \u1f30\u03bf\u03c5\u03b4\u03b1\u03ca\u03c3\u03bc\u03cc\u03c2; so\nEpiphan. _Epist. ad. Acac._ 2 \u03c3\u03b1\u03c6\u1ff6\u03c2 \u03b3\u1f70\u03c1 \u03c0\u03b5\u03c1\u1f76 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c4\u03b5\u03c3\u03c3\u03ac\u03c1\u03c9\u03bd\n\u03b1\u1f31\u03c1\u03ad\u03c3\u03b5\u03c9\u03bd \u1f41 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03c2 \u1f10\u03c0\u03b9\u03c4\u03b5\u03bc\u1f7c\u03bd \u1f14\u03c6\u03b7, \u1f18\u03bd \u03b3\u1f70\u03c1 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03bf\u1f50 \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, \u03bf\u1f50\n\u03a3\u03ba\u03cd\u03b8\u03b7\u03c2, \u03bf\u1f50\u03c7 \u1f1d\u03bb\u03bb\u03b7\u03bd, \u03bf\u1f50\u03ba \u1f38\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u03b9\u03bd\u1f74 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2: comp. _H\u00e6r._ i. 4, 7\nsq., I. p. 5, 8 sq., _Anaceph._ II. pp. 127, 129 sq.\n\n\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u1f70 \u03ba.\u03c4.\u03bb.] '_Christ is all things and in all things._' Christ has\ndispossessed and obliterated all distinctions of religious prerogative\nand intellectual preeminence and social caste; Christ has substituted\nHimself for all these; Christ occupies the whole sphere of human life\nand permeates all its developments; comp. Ephes. i. 23 \u03c4\u03bf\u1fe6 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd\n\u03c0\u1fb6\u03c3\u03b9\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5. For \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, which is stronger than \u03bf\u1f31 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2, see\nGal. iii. 22 \u03c3\u03c5\u03bd\u03ad\u03ba\u03bb\u03b5\u03b9\u03c3\u03b5\u03bd \u1f21 \u03b3\u03c1\u03b1\u03c6\u1f74 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f51\u03c0\u1f78 \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u03b1\u03bd with the note. In\nthis passage \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd is probably neuter, as in 2 Cor. xi. 6, Phil. iv.\n12, 1 Tim. iii. II, 2 Tim. ii. 7, iv. 5, Ephes. iv. 6, vi. 16.\n\nIn the parallel passage Gal. iii. 28 the corresponding clause is \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2\n\u1f51\u03bc\u03b5\u1fd6\u03c2 \u1f11\u1fd6\u03c2 \u1f10\u03c3\u03c4\u1f72 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6. The inversion here accords with a chief\nmotive of the epistle, which is to assert the absolute and universal\nsupremacy of Christ; comp. i. 17 sq., ii. 10 sq., 19. The two parts of\nthe antithesis are combined in our Lord's saying, Joh. xiv. 20 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u1f10\u03bd\n\u1f10\u03bc\u03bf\u03af, \u03ba\u1f00\u03b3\u1f7c \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd.\n\n------------------------------------------------------------------------\n\nIII. 12]\n\n\u03ba\u03b1\u1f76 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2. ^{12}\u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u03bf\u1f56\u03bd, \u1f61\u03c2 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1f76\n\n\n12\u201315. 'Therefore, as the elect of God, as a people consecrated to His\nservice and specially endowed with His love, array yourselves in hearts\nof compassion, in kindliness and humility, in a gentle and yielding\nspirit. Bear with one another: forgive freely among yourselves. As your\nMaster forgave you His servants, so ought ye to forgive your\nfellow-servants. And over all these robe yourselves in love; for this is\nthe garment which binds together all the graces of perfection. And let\nthe one supreme umpire in your hearts, the one referee amidst all your\ndifficulties, be the peace of Christ, which is the destined goal of your\nChristian calling, in which is realised the unity belonging to members\nof one body. Lastly of all; show your gratitude by your thanksgiving.'\n\n12. \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5 \u03bf\u1f56\u03bd] '_Put on therefore_', as men to whom Christ has\nbecome all in all. The incidental mention of Christ as superseding all\nother relations gives occasion to this argumentative \u03bf\u1f56\u03bd: comp. iii. 1,\n5.\n\n\u1f61\u03c2 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1f76 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6] '_as elect ones of God._' Comp. Rom. viii. 33,\nTit. i. 1. In the Gospels \u03ba\u03bb\u03b7\u03c4\u03bf\u03af and \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af are distinguished as an\nouter and an inner circle (Matt. xxii. 14 \u03c0\u03bf\u03bb\u03bb\u03bf\u1f76 \u03b3\u03ac\u03c1 \u03b5\u1f30\u03c3\u03b9\u03bd \u03ba\u03bb\u03b7\u03c4\u03bf\u03af,\n\u1f40\u03bb\u03af\u03b3\u03bf\u03b9 \u03b4\u1f72 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af), \u03ba\u03bb\u03b7\u03c4\u03bf\u03af being those summoned to the privileges of\nthe Gospel and \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af those appointed to final salvation (Matt. xxiv.\n22, 24, 31, Mark xiii. 20, 22, 27, Luke xviii. 7). But in St Paul no\nsuch distinction can be traced. With him the two terms seem to be\ncoextensive, as two aspects of the same process, \u03ba\u03bb\u03b7\u03c4\u03bf\u03af having special\nreference to the goal and \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af to the starting-point. The same\npersons are 'called' to Christ, and 'chosen out' from the world. Thus in\n1 Thess. i. 4 \u03b5\u1f30\u03b4\u03cc\u03c4\u03b5\u03c2 \u03c4\u1f74\u03bd \u1f10\u03ba\u03bb\u03bf\u03b3\u1f74\u03bd \u1f51\u03bc\u1ff6\u03bd \u03ba.\u03c4.\u03bb. the word clearly denotes\nelection to Church-membership. Thus also in 2 Tim. ii. 10, where St Paul\nsays that he endures all things \u03b4\u03b9\u1f70 \u03c4\u03bf\u1f7a\u03c2 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03cd\u03c2, adding \u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03b9\n\u03c3\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03c2 \u03c4\u03cd\u03c7\u03c9\u03c3\u03b9\u03bd \u03ba.\u03c4.\u03bb., the uncertainty implied in these last words\nclearly shows that election to final salvation is not meant. In the same\nsense he speaks of an individual Christian as 'elect', Rom. xvi. 13. And\nagain in 1 Cor. i. 26, 27 \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03c4\u1f74\u03bd \u03ba\u03bb\u1fc6\u03c3\u03b9\u03bd \u1f51\u03bc\u1ff6\u03bd ... \u03c4\u1f70 \u03bc\u1ff6\u03c1\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5\n\u1f10\u03be\u03b5\u03bb\u03ad\u03be\u03b1\u03c4\u03bf, the words appear as synonymes. The same is also the usage of\nSt Peter. Thus in an opening salutation he addresses whole Christian\ncommunities as \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af (1 Pet. i. 1; comp. v. 13 \u1f21 \u03c3\u03c5\u03bd\u03b5\u03ba\u03bb\u03b5\u03ba\u03c4\u1f74 \u1f10\u03bd\n\u0392\u03b1\u03b2\u03c5\u03bb\u1ff6\u03bd\u03b9, i.e. probably \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1), as St Paul under similar\ncircumstances (Rom. i. 6, 7, 1 Cor. i. 2) designates them \u03ba\u03bb\u03b7\u03c4\u03bf\u03af; and in\nanother passage (2 Pet. i. 10) he appeals to his readers to make their\n\u03ba\u03bb\u1fc6\u03c3\u03b9\u03c2 and \u1f10\u03ba\u03bb\u03bf\u03b3\u03ae sure. The use of \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03cc\u03c2 in 2 Joh. 1. 13 is\napparently the same; and in Apoc. xvii. 14 \u03bf\u1f31 \u03bc\u03b5\u03c4' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03ba\u03bb\u03b7\u03c4\u03bf\u1f76 \u03ba\u03b1\u1f76\n\u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1f76 \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u03bf\u03af this is also the case, as we may infer from the\naddition of \u03c0\u03b9\u03c3\u03c4\u03bf\u03af, which points to those who have been _true_ to their\n'calling and election'. Thus the Gospels stand alone in this respect. In\nfact \u1f10\u03ba\u03bb\u03bf\u03b3\u03ae denotes election by God not only to final salvation, but to\n_any_ special privilege or work, whether it be (1) Church-membership, as\nin the passages cited from the epistles; or (2) The work of preaching,\nas when St Paul (Acts ix. 15) is called \u03c3\u03ba\u03b5\u1fe6\u03bf\u03c2 \u1f10\u03ba\u03bb\u03bf\u03b3\u1fc6\u03c2, the object of\nthe 'election' being defined in the words following, \u03c4\u03bf\u1fe6 \u03b2\u03b1\u03c3\u03c4\u03ac\u03c3\u03b1\u03b9 \u03c4\u1f78\n\u1f44\u03bd\u03bf\u03bc\u03ac \u03bc\u03bf\u03c5 \u1f10\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd [\u03c4\u1ff6\u03bd] \u1f10\u03b8\u03bd\u1ff6\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03b2\u03b1\u03c3\u03b9\u03bb\u03ad\u03c9\u03bd \u03ba.\u03c4.\u03bb.; or (3) The\nMessiahship, 1 Pet. ii. 4, 6; or (4) The fatherhood of the chosen\npeople, as in the case of Isaac and Jacob, Rom. ix. 11; or (5) The\nfaithful remnant under the theocracy, Rom. xi. 5, 7, 28. This last\napplication presents the closest analogy to the idea of final salvation:\nbut even here St Paul treats \u03ba\u03bb\u1fc6\u03c3\u03b9\u03c2 and \u1f10\u03ba\u03bb\u03bf\u03b3\u03ae as coextensive, Rom. xi.\n28, 29 \u03ba\u03b1\u03c4\u1f70 \u03b4\u1f72 \u03c4\u1f74\u03bd \u1f10\u03ba\u03bb\u03bf\u03b3\u1f74\u03bd \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03bf\u1f76 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1f7a\u03c2 \u03c0\u03b1\u03c4\u03ad\u03c1\u03b1\u03c2\u00b7 \u1f00\u03bc\u03b5\u03c4\u03b1\u03bc\u03ad\u03bb\u03b7\u03c4\u03b1 \u03b3\u1f70\u03c1 \u03c4\u1f70\n\u03c7\u03b1\u03c1\u03af\u03c3\u03bc\u03b1\u03c4\u03b1 \u03ba\u03b1\u1f76 \u1f21 \u03ba\u03bb\u1fc6\u03c3\u03b9\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6.\n\n------------------------------------------------------------------------\n\nIII. 12]\n\n\u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, \u1f05\u03b3\u03b9\u03bf\u03b9 [\u03ba\u03b1\u1f76] \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9, \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 \u03bf\u1f30\u03ba\u03c4\u03b9\u03c1\u03bc\u03bf\u1fe6,\n\n\n\u1f05\u03b3\u03b9\u03bf\u03b9 \u03ba.\u03c4.\u03bb.] These are not to be taken as vocatives, but as predicates\nfurther defining the meaning of \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af. All the three terms \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u03af,\n\u1f05\u03b3\u03b9\u03bf\u03b9, \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9, are transferred from the Old Covenant to the New,\nfrom the Israel after the flesh to the Israel after the Spirit. For the\ntwo former comp. 1 Pet. ii. 9 \u03b3\u03ad\u03bd\u03bf\u03c2 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03cc\u03bd ... \u1f14\u03b8\u03bd\u03bf\u03c2 \u1f05\u03b3\u03b9\u03bf\u03bd; and for\nthe sense of \u1f05\u03b3\u03b9\u03bf\u03b9, 'the consecrated people of God', see the note on\nPhil. i. 1. For the third word, \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9, see Is. v. 1 \u1f0c\u03c3\u03c9 \u03b4\u1f74 \u03c4\u1ff7\n\u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u1ff3 \u03ba.\u03c4.\u03bb., Hos. ii. 25 \u03c4\u1f74\u03bd \u03bf\u1f50\u03ba \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03b7\u03bd \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03b7\u03bd (as quoted\nin Rom. ix. 25). In the New Testament it seems to be used always of the\nobjects of _God's_ love: e.g. 1 Thess. i. 4 \u03b5\u1f30\u03b4\u03bf\u03c4\u03ad\u03c2, \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1f76 \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9\n\u1f51\u03c0\u1f78 \u0398\u03b5\u03bf\u1fe6, \u03c4\u1f74\u03bd \u1f10\u03ba\u03bb\u03bf\u03b3\u1f74\u03bd \u1f51\u03bc\u1ff6\u03bd, 2 Thess. ii. 13 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1f76 \u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9 \u1f51\u03c0\u1f78\n\u039a\u03c5\u03c1\u03af\u03bf\u03c5 (comp. Jude 1); and so probably Rev. xx. 9 \u03c4\u1f74\u03bd \u03c0\u03cc\u03bb\u03b9\u03bd \u03c4\u1f74\u03bd\n\u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03b7\u03bd. For the connexion of God's election and God's love see Rom.\nxi. 28 (quoted above), 1 Thess. _l.c._ The \u03ba\u03b1\u1f76 is omitted in one or two\nexcellent copies (though it has the great preponderance of authorities\nin its favour), and it is impossible not to feel how much the sentence\ngains in force by the omission, \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1f76 \u0398\u03b5\u03bf\u1fe6, \u1f05\u03b3\u03b9\u03bf\u03b9, )\u03b7\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9; comp.\n1 Pet. ii. 6.\n\n\u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 \u03bf\u1f30\u03ba\u03c4\u03b9\u03c1\u03bc\u03bf\u1fe6] '_a heart of pity_'. For the meaning of \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 see\nthe note on Phil. i. 8, and for the whole expression comp. \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1\n\u1f10\u03bb\u03ad\u03bf\u03c5\u03c2 Luke i. 78, _Test. xii Patr._ Zab. 7, 8.\n\n------------------------------------------------------------------------\n\nIII. 12]\n\n\u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1, \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7\u03bd, \u03c0\u03c1\u03ac\"\u03c5\u03c4\u03b7\u03c4\u03b1, \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1\u03bd\u00b7\n\n\n\u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u03ba.\u03c4.\u03bb.] The two words \u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c4\u03b7\u03c2 and \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7,\n'kindliness' and 'humility', describe the Christian _temper of mind_\ngenerally, and this in two aspects, as it affects either (1) our\nrelation to others (\u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c4\u03b7\u03c2), or (2) our estimate of self\n(\u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7). For \u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c4\u03b7\u03c2 see the note on Gal. v. 22; for\n\u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7, the note on Phil. ii. 3.\n\n\u03c0\u03c1\u03ac\"\u03c5\u03c4\u03b7\u03c4\u03b1 \u03ba.\u03c4.\u03bb.] These next two words, \u03c0\u03c1\u03ac\"\u03c5\u03c4\u03b7\u03c2 and \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1, denote\nthe _exercise_ of the Christian temper in its outward bearing towards\nothers. They are best distinguished by their opposites. \u03c0\u03c1\u03ac\"\u03c5\u03c4\u03b7\u03c2 is\nopposed to 'rudeness, harshness', \u1f00\u03b3\u03c1\u03af\u03bf\u03c4\u03b7\u03c2 (Plato _Symp._ 197 D),\n\u03c7\u03b1\u03bb\u03b5\u03c0\u03cc\u03c4\u03b7\u03c2 (Arist. _H. A._ ix. i); \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1 to 'resentment, revenge,\nwrath,' \u1f40\u03c1\u03b3\u03ae (Prov. xvi. 32), \u1f40\u03be\u03c5\u03c7\u03bf\u03bb\u03af\u03b1 (Herm. _Mand._ v. 1, 2). For the\nmeaning of \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1 see above, on i. 11; for the form of \u03c0\u03c1\u03ac\"\u03c5\u03c4\u03b7\u03c2\n(\u03c0\u03c1\u03ac\u03bf\u03c4\u03b7\u03c2), on Gal. v. 23. The words are discussed in Trench _N. T. Syn._\n\u00a7 xlii. p. 140 sq., \u00a7 xliii. p. 145 sq., \u00a7 liii. p. 184 sq. They appear\nin connexion Ephes. iv. 2, Ign. _Polyc._ 6 \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03ae\u03c3\u03b1\u03c4\u03b5 \u03bf\u1f56\u03bd \u03bc\u03b5\u03c4'\n\u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd \u1f10\u03bd \u03c0\u03c1\u03ac\"\u03c5\u03c4\u03b7\u03c4\u03b9.\n\n------------------------------------------------------------------------\n\nIII. 13]\n\n^{13}\u1f00\u03bd\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd, \u03ba\u03b1\u1f76 \u03c7\u03b1\u03c1\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2,\n\n\n13. \u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd, \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2] The pronoun is varied, as in Ephes. iv. 32 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b5\n\u03b5\u1f30\u03c2 _\u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2_ \u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03af ... \u03c7\u03b1\u03c1\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 _\u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2_ \u03ba.\u03c4.\u03bb., 1\nPet. iv. 8\u201310 \u03c4\u1f74\u03bd \u03b5\u1f30\u03c2 _\u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2_ \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u1f10\u03ba\u03c4\u03b5\u03bd\u1fc6 \u1f14\u03c7\u03bf\u03bd\u03c4\u03b5\u03c2 ... \u03c6\u03b9\u03bb\u03cc\u03be\u03b5\u03bd\u03bf\u03b9\n\u03b5\u1f30\u03c2 _\u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2_ ... \u03b5\u1f30\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u03b1\u1f50\u03c4\u1f78 [\u03c4\u1f78 \u03c7\u03ac\u03c1\u03b9\u03c3\u03bc\u03b1] \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2. The\nreciprocal \u1f11\u03b1\u03c5\u03c4\u1ff6\u03bd differs from the reciprocal \u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd in emphasizing the\nidea of _corporate unity_: hence it is more appropriate here (comp.\nEphes. iv. 2, 32) with \u03c7\u03b1\u03c1\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 than with \u1f00\u03bd\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9; comp. Xen.\n_Mem._ iii. 5. 16 \u1f00\u03bd\u03c4\u1f76 \u03bc\u1f72\u03bd \u03c4\u03bf\u1fe6 \u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u03b5\u1fd6\u03bd _\u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2_ \u03c4\u1f70 \u03c3\u03c5\u03bc\u03c6\u03ad\u03c1\u03bf\u03bd\u03c4\u03b1,\n\u1f10\u03c0\u03b7\u03c1\u03ad\u03b1\u03b6\u03bf\u03c5\u03c3\u03b9\u03bd _\u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03b9\u03c2_, \u03ba\u03b1\u1f76 \u03c6\u03b8\u03bf\u03bd\u03bf\u1fe6\u03c3\u03b9\u03bd _\u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2_ \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u1f22 \u03c4\u03bf\u1fd6\u03c2\n\u1f04\u03bb\u03bb\u03bf\u03b9\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2 ... \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03b1\u03b9\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03bf\u1f55\u03c4\u03c9 \u03ba\u03b5\u03c1\u03b4\u03b1\u03af\u03bd\u03b5\u03b9\u03bd \u1f00\u03c0'\n_\u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd_ \u1f22 \u03c3\u03c5\u03bd\u03c9\u03c6\u03b5\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 _\u03b1\u1f51\u03c4\u03bf\u03cd\u03c2_, where the propriety of the\ntwo words in their respective places will be evident; and ib. ii. 7. 12\n\u1f00\u03bd\u03c4\u1f76 \u1f51\u03c6\u03bf\u03c1\u03c9\u03bc\u03ad\u03bd\u03c9\u03bd \u1f11\u03b1\u03c5\u03c4\u1f70\u03c2 \u1f21\u03b4\u03ad\u03c9\u03c2 _\u1f00\u03bb\u03bb\u03ae\u03bb\u03b1\u03c2_ h\u03ad\u03c9\u03c1\u03c9\u03bd, where the variation\nis more subtle but not less appropriate. For instances of this use of\n\u1f11\u03b1\u03c5\u03c4\u1ff6\u03bd see Bleek _Hebr\u00e4erbrief_ iii. 13 (p. 453 sq.), K\u00fchner _>Griech.\nGramm._ \u00a7 455 (II. p. 497 sq.).\n\n\u03c7\u03b1\u03c1\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9] i.e. 'forgiving'; see the note on ii. 13. An _a fortiori_\nargument lurks under the use of \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 (rather than \u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03b9\u03c2): if\nChrist forgave them, much more should they forgive _themselves_.\n\n------------------------------------------------------------------------\n\nIII. 14]\n\n\u1f10\u03ac\u03bd \u03c4\u03b9\u03c2 \u03c0\u03c1\u03cc\u03c2 \u03c4\u03b9\u03bd\u03b1 \u1f14\u03c7\u1fc3 \u03bc\u03bf\u03bc\u03c6\u03ae\u03bd\u00b7 \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 \u1f10\u03c7\u03b1\u03c1\u03af\u03c3\u03b1\u03c4\u03bf \u1f51\u03bc\u1fd6\u03bd, \u03bf\u1f55\u03c4\u03c9\u03c2\n\u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2\u00b7 ^{14}\u1f10\u03c0\u1f76 \u03c0\u1fb6\u03c3\u03b9\u03bd \u03b4\u1f72 \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2\n\n\n\u03bc\u03bf\u03bc\u03c6\u03ae\u03bd] '_a complaint_'. As \u03bc\u03ad\u03bc\u03c6\u03b5\u03c3\u03b8\u03b1\u03b9 is 'to find _fault_ with',\nreferring most commonly to errors of omission, so \u03bc\u03bf\u03bc\u03c6\u03ae here is regarded\nas a _debt_, which needs to be remitted. The rendering of the A. V. 'a\nquarrel' (= querela) is only wrong as being an archaism. The phrase\n\u03bc\u03bf\u03bc\u03c6\u1f74\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd occurs several times in classical Greek, but generally in\npoetry: e.g. Eur. _Orest._ 1069, Arist. _Pax_ 664.\n\n\u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u03ba.\u03c4.\u03bb.] This must not be connected with the preceding words,\nbut treated as an independent sentence, the \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u03af being answered by\nthe \u03bf\u1f55\u03c4\u03c9\u03c2 \u03ba\u03b1\u03af. For the presence of \u03ba\u03b1\u03af in both clauses of the comparison\nsee the note on i. 6. The phenomenon is common in the best classical\nwriters, e.g. Xen. _Mem._ i. 6. 3 \u1f65\u03c3\u03c0\u03b5\u03c1 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f04\u03bb\u03bb\u03c9\u03bd \u1f10\u03c1\u03b3\u03c9\u03bd \u03bf\u1f31\n\u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b1\u03bb\u03bf\u03b9 ... \u03bf\u1f55\u03c4\u03c9 \u03ba\u03b1\u1f76 \u03c3\u03cd \u03ba.\u03c4.\u03bb.; see the references in Heindorf on\nPlato _Ph\u00e6do_ 64 C, _Sophist._ 217 B, and K\u00fchner _Griech. Gramm._ \u00a7 524\n(II. p. 799).\n\n\u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2] This reading, which is better supported than \u1f41 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2, is\nalso more expressive. It recalls more directly the lesson of the parable\nwhich enforces the duty of fellow-servant to fellow-servant; Matt.\nxviii. 27 \u03c3\u03c0\u03bb\u03b1\u03b3\u03c7\u03bd\u03b9\u03c3\u03b8\u1f72\u03b9\u03c2 \u03b4\u1f72 \u1f41 \u03ba\u03cd\u03c1\u03b9\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03b4\u03bf\u03cd\u03bb\u03bf\u03c5 \u1f10\u03ba\u03ad\u03b9\u03bd\u03bf\u03c5 \u1f00\u03c0\u03ad\u03bb\u03c5\u03c3\u03b5\u03bd \u03b1\u1f50\u03c4\u1f78\u03bd\n\u03ba\u03b1\u1f76 \u03c4\u1f78 \u03b4\u03ac\u03bd\u03b5\u03b9\u03bf\u03bd \u1f00\u03c6\u1fc6\u03ba\u03b5\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03ba.\u03c4.\u03bb.: comp. below iv. 1 \u03b5\u1f30\u03b4\u03cc\u03c4\u03b5\u03c2 \u1f45\u03c4\u03b9 \u03ba\u03b1\u1f76\n\u1f51\u03bc\u03b5\u1fd6\u03c2 \u1f14\u03c7\u03b5\u03c4\u03b5 \u03ba\u03cd\u03c1\u03b9\u03bf\u03bd \u1f10\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1ff7. The reading \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03c2 perhaps comes from the\nparallel passage Ephes. iv. 32 \u03c7\u03b1\u03c1\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u1f76 \u1f41 \u0398\u03b5\u1f78\u03c2 \u1f10\u03bd\n\u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f10\u03c7\u03b1\u03c1\u03af\u03c3\u03b1\u03c4\u03bf \u1f21\u03bc\u1fd6\u03bd (or \u1f51\u03bc\u1fd6\u03bd).\n\n\u03bf\u1f55\u03c4\u03c9\u03c2 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2] sc. \u03c7\u03b1\u03c1\u03af\u03b6\u03b5\u03c3\u03b8\u03b5 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2.\n\n14. \u1f10\u03c0\u1f76 \u03c0\u1fb6\u03c3\u03b9\u03bd] '_over and above all these_', comp. Luke iii. 20\n\u03c0\u03c1\u03bf\u03c3\u03ad\u03b8\u03b7\u03ba\u03b5\u03bd \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6\u03c4\u03bf \u1f10\u03c0\u1f76 \u03c0\u1fb6\u03c3\u03b9\u03bd. In Luke xvi. 26, Ephes. vi. 16, the\ncorrect reading is probably \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd. Love is the outer garment which\nholds the others in their places.\n\n------------------------------------------------------------------------\n\nIII. 15]\n\n\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c4\u03b5\u03bb\u03b5\u03af\u03bf\u03c4\u03b7\u03c4\u03bf\u03c2. ^{15}\u03ba\u03b1\u1f76 \u1f21 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u03c4\u03bf\u1fe6\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03c4\u03c9 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 \u1f51\u03bc\u1ff6\u03bd, \u03b5\u1f30\u03c2 \u1f23\u03bd \u03ba\u03b1\u1f76 \u1f10\u03ba\u03bb\u03ae\u03b8\u03b7\u03c4\u03b5 \u1f10\u03bd \u1f11\u03bd\u1f76\n\u03c3\u03ce\u03bc\u03b1\u03c4\u03b9. \u03ba\u03b1\u1f76 \u03b5\u1f50\u03c7\u03ac\u03c1\u03b9\u03c3\u03c4\u03bf\u03b9\n\n\n\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd] sc. \u1f10\u03bd\u03b4\u03cd\u03c3\u03b1\u03c3\u03b8\u03b5, from ver. 12.\n\n\u1f45] '_which thing_', i.e. 'love'; comp. Ephes. v. 5 \u03c0\u03bb\u03b5\u03bf\u03bd\u03ad\u03ba\u03c4\u03b7\u03c2, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd\n\u03b5\u1f30\u03b4\u03c9\u03bb\u03bf\u03bb\u03ac\u03c4\u03c1\u03b7\u03c2, Ign. _Rom._ 7 \u1f05\u03c1\u03c4\u03bf\u03bd \u0398\u03b5\u03bf\u1fe6 \u03b8\u03ad\u03bb\u03c9, \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c3\u1f70\u03c1\u03be \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6,\n_Magn._ 10 \u03bc\u03b5\u03c4\u03ac\u03b2\u03ac\u03bb\u03b5\u03c3\u03b8\u03b5 \u03b5\u1f30\u03c2 \u03bd\u03ad\u03b1\u03bd \u03b6\u03cd\u03bc\u03b7\u03bd \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2, _Trall._ 7\n\u1f00\u03bd\u03b1\u03ba\u03c4\u03ae\u03c3\u03b1\u03c3\u03b8\u03b5 \u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u1f10\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03b9 \u1f45 \u1f10\u03c3\u03c4\u03b9\u03bd \u03c3\u1f70\u03c1\u03be \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5. Though there are\nvarious readings in the passages of the Ignatian Epistles, the \u1f45 seems\nto be right in every case. These instances will show that \u1f45 may be\nreferred to \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd alone. Otherwise we might suppose the antecedent\nto be \u03c4\u1f78 \u1f10\u03bd\u03b4\u03c5\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd, but this hardly suits the sense. The\ncommon reading \u1f25\u03c4\u03b9\u03c2 is obviously a scribe's correction.\n\n\u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2 \u03ba.\u03c4.\u03bb.] '_the bond of perfection_', i.e. the power, which\nunites and holds together all those graces and virtues, which together\nmake up perfection. \u03a0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03ba\u03b5\u1fd6\u03bd\u03b1, says Chrysostom, h\u03ac\u03c5\u03c4\u03b7 \u03c3\u03c5\u03c3\u03c6\u03af\u03b3\u03b3\u03b5\u03b9\u00b7 \u1f45\u03c0\u03b5\u03c1\n\u1f02\u03bd \u03b5\u1f34\u03c0\u1fc3\u03c2 \u1f00\u03b3\u03b1\u03b8\u03cc\u03bd, \u03c4\u03ac\u03c5\u03c4\u03b7\u03c2 \u1f00\u03c0\u03bf\u03cd\u03c3\u03b7\u03c2 \u03bf\u1f50\u03b4\u03ad\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u03bb\u03bb\u1f70 \u03b4\u03b9\u03b1\u03c1\u03c1\u03b5\u1fd6: comp. Clem.\nRom. 49 \u03c4\u1f78\u03bd \u03b4\u03b5\u03c3\u03bc\u1f78\u03bd \u03c4\u1fc6\u03c2 \u1f00\u03b3\u03b1\u03c0\u1fc6\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u03af\u03c2 \u03b4\u03cd\u03bd\u03b1\u03c4\u03b1\u03b9 \u1f10\u03be\u03b7\u03b3\u03ae\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9; Thus the\nPythagoreans (Simplic. _in Epictet._ p. 208 A) \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u1ff6\u03c2 \u03c4\u1ff6\u03bd \u1f04\u03bb\u03bb\u03c9\u03bd \u1f00\u03c1\u03b5\u03c4\u1ff6\u03bd\n\u03c4\u1f74\u03bd \u03c6\u03b9\u03bb\u03af\u03b1\u03bd \u1f10\u03c4\u03af\u03bc\u03c9\u03bd \u03ba\u03b1\u1f76 \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03bd \u03b1\u1f50\u03c4\u1f74\u03bd \u03c0\u03b1\u03c3\u1ff6\u03bd \u03c4\u1ff6\u03bd \u1f00\u03c1\u03b5\u03c4\u1ff6\u03bd \u1f14\u03bb\u03b5\u03b3\u03bf\u03bd. So too\nThemist. _Orat._ i. (p. 5 C) \u03b2\u03b1\u03c3\u03b9\u03bb\u03b9\u03ba\u1f74 (\u1f00\u03c1\u03b5\u03c4\u1f74) \u03c0\u03b1\u03c1\u1f70 \u03c4\u1f70\u03c2 \u1f04\u03bb\u03bb\u03b1\u03c2 \u03b5\u1f30\u03c2 \u1f23\u03bd\n\u03be\u03c5\u03bd\u03b4\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03b1\u1f31 \u03bb\u03bf\u03b9\u03c0\u03b1\u03af, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03b5\u1f30\u03c2 \u03bc\u03af\u03b1\u03bd \u03ba\u03bf\u03c1\u03c5\u03c6\u1f74\u03bd \u1f00\u03bd\u03b7\u03bc\u03bc\u03ad\u03bd\u03b1\u03b9. The word\nwill take a genitive either of the object bound or of the binding force:\ne.g. Plato _Polit._ 310 A \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03b8\u03b5\u03b9\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1f78\u03bd \u03be\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03bd \u1f00\u03c1\u03b5\u03c4\u1fc6\u03c2\n\u03bc\u03b5\u03c1\u1ff6\u03bd \u03c6\u03cd\u03c3\u03b5\u03c9\u03c2 \u1f00\u03bd\u03cc\u03bc\u03bf\u03b9\u03c9\u03bd \u03ba\u03b1\u1f76 \u1f10\u03c0\u1f76 \u03c4\u1f00\u03bd\u03b1\u03bd\u03c4\u03af\u03b1 \u03c6\u03b5\u03c1\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd, where the \u1f00\u03c1\u03b5\u03c4\u1f74 \u03be\u03c5\u03bd\u03b4\u03b5\u1fd6\nand the \u03bc\u03ad\u03c1\u03b7 \u03c6\u03cd\u03c3\u03b5\u03c9\u03c2 \u03be\u03c5\u03bd\u03b4\u03b5\u1fd6\u03c4\u03b1\u03b9. We have an instance of the one genitive\n(the objective) here, of the other (the subjective) in Ephes. iv. 3 \u1f10\u03bd\n\u03c4\u1ff7 \u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03bc\u1ff3 \u03c4\u1fc6\u03c2 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7\u03c2 (see the note there).\n\nAnother explanation makes \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2 = \u03c3\u03cd\u03bd\u03b8\u03b5\u03c3\u03b9\u03c2 here, 'the bundle, the\ntotality', as e.g. Herodian. iv. 12 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1f78\u03bd \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03bd \u03c4\u1ff6\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1ff6\u03bd\n(comp. Ign. _Trall._ 3 \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03bd \u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03c9\u03bd); but this unusual metaphor\nis highly improbable and inappropriate here, not to mention that we\nshould expect the definite article \u1f41 \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2 in this case. With either\ninterpretation, the function assigned to \u1f00\u03b3\u03ac\u03c0\u03b7 here is the same as when\nit is declared to be \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03bd\u03cc\u03bc\u03bf\u03c5, Rom. xiii. 10 (comp. Gal. v. 14).\nSee also the all-embracing office which is assigned to it in 1 Cor.\nxiii.\n\n15. \u1f21 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6] '_Christ's peace_', which He left as a legacy\nto His disciples: Joh. xiv. 27 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7\u03bd \u1f00\u03c6\u03af\u03b7\u03bc\u03b9 \u1f51\u03bc\u1fd6\u03bd, \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7\u03bd _\u03c4\u1f74\u03bd\n\u1f10\u03bc\u1f74\u03bd_ \u03b4\u03af\u03b4\u03c9\u03bc\u03b9 \u1f51\u03bc\u1fd6\u03bd; comp. Ephes. ii. 14 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03b3\u03ac\u03c1 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u1f21\u03bc\u1ff6\u03bd\nwith the context. The common reading \u1f21 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 has a parallel in\nPhil. iv. 7.\n\n\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03c4\u03c9] '_be umpire_', for the idea of a contest is only less\nprominent here, than in \u03b2\u03c1\u03b1\u03b2\u03b5\u1fd6\u03bf\u03bd 1 Cor. ix. 24, Phil. iii. 14 (see the\nnote there). \u03a3\u03c4\u03ac\u03b4\u03b9\u03bf\u03bd \u1f15\u03bd\u03b4\u03bf\u03bd \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03bf\u1fd6\u03c2, writes\nChrysostom, \u03ba\u03b1\u1f76 \u1f00\u03b3\u1ff6\u03bd\u03b1 \u03ba\u03b1\u1f76 \u1f04\u03b8\u03bb\u03b7\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03b2\u03c1\u03b1\u03b2\u03b5\u03c5\u03c4\u03ae\u03bd. Wherever there is a\nconflict of motives or impulses or reasons, the peace of Christ must\nstep in and decide which is to prevail; \u039c\u1f74 \u03b8\u03c5\u03bc\u1f78\u03c2 \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03c4\u03c9, says\nChrysostom again, \u03bc\u1f74 \u03c6\u03b9\u03bb\u03bf\u03bd\u03b5\u03b9\u03ba\u03af\u03b1, \u03bc\u1f74 \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03af\u03bd\u03b7 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7\u00b7 \u1f21 \u03b3\u1f70\u03c1 \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03af\u03bd\u03b7\n\u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u1f10\u03ba \u03c4\u03bf\u1fe6 \u1f00\u03bc\u03cd\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9, \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03bc\u03b7\u03b4\u1f72\u03bd \u03c0\u03ac\u03c3\u03c7\u03b5\u03b9\u03bd \u03b4\u03b5\u03b9\u03bd\u03cc\u03bd.\n\nFor this metaphor of some one paramount consideration acting as umpire,\nwhere there is a conflict of internal motives, see Polyb. ii. 35. 3 \u1f05\u03c0\u03b1\u03bd\n\u03c4\u1f78 \u03b3\u03b9\u03b3\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u1f51\u03c0\u1f78 \u03c4\u1ff6\u03bd \u0393\u03b1\u03bb\u03ac\u03c4\u03c9\u03bd \u03b8\u03c5\u03bc\u1ff7 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u1f22 _\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u1ff7_ \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9,\nPhilo _de Migr. Abr._ 12 (I[. p. 446) \u03c0\u03bf\u03c1\u03b5\u03cd\u03b5\u03c4\u03b1\u03b9 \u1f41 \u1f04\u03c6\u03c1\u03c9\u03bd \u03b4\u03b9' \u1f00\u03bc\u03c6\u03bf\u03c4\u03ad\u03c1\u03c9\u03bd\n\u03b8\u03c5\u03bc\u03bf\u1fe6 \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1\u03c2 )\u03b1\u1f72\u03b9 ... \u03c4\u1f78\u03bd \u1f21\u03bd\u03af\u03bf\u03c7\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b2\u03c1\u03b1\u03b2\u03b5\u03c5\u03c4\u1f74\u03bd \u03bb\u03cc\u03b3\u03bf\u03bd \u1f00\u03c0\u03bf\u03b2\u03b1\u03bb\u03ce\u03bd\n(comp. _de Ebriet._ 19, I. p. 368), Jos. B. J. vi. 2. 6 \u1f10\u03b2\u03c1\u03ac\u03b2\u03b5\u03c5\u03b5 \u03c4\u1f70\u03c2\n\u03c4\u03cc\u03bb\u03bc\u03b1\u03c2 \u1f41 ... \u03c6\u03cc\u03b2\u03bf\u03c2. Somewhat similarly \u03c4\u03cd\u03c7\u03b7 (Polyb. xxvii. 14. 4) or\n\u03c6\u03cd\u03c3\u03b9\u03c2 (Athen. xv. p. 670 A) are made \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd. In other passages, where\n\u1f41 \u0398\u03b5\u1f78\u03c2 or \u03c4\u1f78 \u03b8\u03b5\u1fd6\u03bf\u03bd is said \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd, this implies that, while man\nproposes, God _disposes_. In Philo \u1f00\u03bb\u03ae\u03b8\u03b5\u03b9\u03b1 \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03bf\u03c5\u03c3\u03b1 (_Qui rer. <DW37>.\nher._ 19, I. p. 486) is a rough synonyme for \u1f00\u03bb\u03ae\u03b8\u03b5\u03b9\u03b1 \u03b4\u03b9\u03ba\u03ac\u03b6\u03bf\u03c5\u03c3\u03b1 (_de\nAbrah._ 14, II. p. 10, etc.): and in Josephus (_Ant._ vi. 3. 1) \u03b4\u03b9\u03ba\u03ac\u03b6\u03b5\u03b9\u03bd\nand \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd are used together of the same action. In all such cases it\nappears that the idea of a _decision_ and an _award_ is prominent in the\nword, and that it must not be taken to denote simply _rule_ or _power_.\n\n\u03b5\u1f30\u03c2 \u1f23\u03bd \u03ba.\u03c4.\u03bb.] Comp. 1 Cor. vii. 15 \u1f10\u03bd \u03b4\u1f72 \u03b5\u1f30\u03c1\u03ae\u03bd\u1fc3 \u03ba\u03ad\u03ba\u03bb\u03b7\u03ba\u03b5\u03bd \u1f21\u03bc\u1fb6\u03c2 \u1f41 \u0398\u03b5\u03cc\u03c2.\n\n\u1f10\u03bd \u1f11\u03bd\u1f76 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9] 'As ye were called as members of one body, so let there\nbe one spirit animating that body': Ephes. iv. 4 h\u1f72\u03bd \u03c3\u1ff6\u03bc\u03b1 \u03ba\u03b1\u1f76 h\u1f72\u03bd\n\u03c0\u03bd\u03b5\u1fe6\u03bc\u03b1. This passage strikes the keynote of the companion Epistle to the\nEphesians (see esp. ii. 16 sq., iv. 3 sq.).\n\n\u03b5\u1f50\u03c7\u03ac\u03c1\u03b9\u03c3\u03c4\u03bf\u03b9] 'And to crown all forget yourselves in thanksgiving towards\nGod': see the notes on i. 12, ii. 7. The adjective \u03b5\u1f50\u03c7\u03ac\u03c1\u03b9\u03c3\u03c4\u03bf\u03c2, though\nnot occurring elsewhere in the Greek Bible, is not uncommon in classical\nwriters, and like the English 'grateful', has two meanings; either (1)\n'pleasurable' (e.g. Xen. _Cyr._ ii. 2. 1); or (2) 'thankful' (e.g.\nBoeckh _C. I._ no. 1625), as here.\n\n------------------------------------------------------------------------\n\nIII. 16]\n\n\u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b5. ^{16}\u1f49 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f10\u03bd\u03bf\u03b9\u03ba\u03ad\u03b9\u03c4\u03c9 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd \u03c0\u03bb\u03bf\u03c5\u03c3\u03af\u03c9\u03c2 \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3\n\u03c3\u03bf\u03c6\u03af\u1fb3\u00b7 \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2\n\n\n16, 17. 'Let the inspiring word of Christ dwell in your hearts,\nenriching you with its boundless wealth and endowing you with all\nwisdom. Teach and admonish one another with psalms, with hymns of\npraise, with spiritual songs of all kinds. Only let them be pervaded\nwith grace from heaven. Sing to God in your hearts and not with your\nlips only. And generally; whatever ye do, whether in word or in deed,\nlet everything be done in the name of Jesus Christ. And (again I repeat\nit) pour out your thanksgiving to God the Father through Him'.\n\n16. \u1f49 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6] '_the word of Christ_', \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 being the\nsubjective genitive, so that Christ is the speaker. Though \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u0398\u03b5\u03bf\u1fe6 and \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 occur frequently, \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 is\nfound here only. There seems to be no direct reference in this\nexpression to any definite body of truths either written or oral, but \u1f41\n\u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 denotes the presence of Christ in the heart, as an\ninward monitor: comp. 1 Joh. ii. 14 \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd \u03bc\u03ad\u03bd\u03b5\u03b9, with\n_ib._ i. 10 \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9\u03bd \u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd, and so perhaps Acts xviii.\n5 \u03c3\u03c5\u03bd\u03ad\u03b9\u03c7\u03b5\u03c4\u03bf \u03c4\u1ff7 \u03bb\u03cc\u03b3\u1ff3 (the correct reading).\n\n\u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd] '_in your hearts_', not '_among you_'; comp. Rom. viii. 9, 11\n\u03c4\u1f78 \u1f10\u03bd\u03bf\u03b9\u03ba\u03bf\u1fe6\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c0\u03bd\u03b5\u1fe6\u03bc\u03b1 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, 2 Tim. i. 5, 14, and Lev. xxvi. 12, as\nquoted in 2 Cor. vi. 16, \u1f10\u03bd\u03bf\u03b9\u03ba\u03ae\u03c3\u03c9 \u1f10\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2.\n\n\u03c0\u03bb\u03bf\u03c5\u03c3\u03af\u03c9\u03c2] See above p. 43 sq., and the note on i. 27.\n\n\u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3] '_in every kind of wisdom_'. It seems best to take these\nwords with the preceding clause, though Clem. Alex. _P\u00e6d._ ii. 4 (p.\n194) attaches them to what follows. For this position of \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3,\nat the end of the sentence to which it refers, comp. i. 9, Ephes. i. 8.\nThe connexion here adopted is also favoured by the parallel passage\nEphes. v. 18, 19 (see the note below). Another passage i. 28\n\u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03bd \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3\nhas a double bearing: while the _connexion_ favours our taking \u1f10\u03bd \u03c0\u03ac\u03c3\u1fc3\n\u03c3\u03bf\u03c6\u03af\u1fb3 here with the following words, the _order_ suggests their being\nattached to the preceding clause.\n\n\u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba.\u03c4.\u03bb.] The participles are here used for imperatives, as\nfrequently in hortatory passages, e.g. Rom. xii. 9 sq., 16 sq., Ephes.\niv. 2, 3, Hebr. xiii. 5, 1 Pet. ii. 12[?], iii. 1, 7, 9, 15, 16. It is\nnot, as some insist, that the participle itself has any imperatival\nforce; nor, as maintained by others, that the construction should be\nexplained by the hypothesis of a preceding parenthesis or of a verb\nsubstantive understood or by any other expedient to obtain a regular\ngrammatical structure (see Winer, \u00a7 xlv. p. 441 sq., \u00a7 lxii. p. 707, \u00a7\nlxiii. p. 716, \u00a7 lxiv. p. 732). But the absolute participle, being (so\nfar as regards mood) neutral in itself, takes its colour from the\ngeneral complexion of the sentence. Thus it is sometimes indicative\n(e.g. 2 Cor. vii. 5, and frequently), sometimes imperative (as in the\npassages quoted), sometimes optative (as above, ii. 2, 2 Cor. ix. 11,\ncomp. Ephes. iii. 17). On the distinction of \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b5\u03b9\u03bd and \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd see\nthe note on i. 28; they describe respectively the positive and the\nnegative side of instruction. On the reciprocal \u1f11\u03b1\u03c5\u03c4\u03bf\u03cd\u03c2 see the note on\niii. 13.\n\n------------------------------------------------------------------------\n\nIII. 16]\n\n\u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 \u1fa0\u03b4\u03b1\u1fd6\u03c2 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2 \u1f10\u03bd \u03c4\u1fc7\n\n\n\u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u03ba.\u03c4.\u03bb.] to be connected with the preceding sentence, as\nsuggested by Ephes. v. 18 sq. \u1f00\u03bb\u03bb\u1f70 \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03c3\u03b8\u03b5 \u1f10\u03bd \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9, \u03bb\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2\n\u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 [\u1f10\u03bd] \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03bd\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u1fa0\u03b4\u03b1\u1fd6\u03c2 [\u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2], \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76\n\u03c8\u03ac\u03bb\u03bb\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u1f51\u03bc\u1ff6\u03bd \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3. The datives describe the instruments\nof the \u03b4\u03b9\u03b4\u03b1\u03c7\u03ae and \u03bd\u03bf\u03c5\u03b8\u03b5\u03c3\u03af\u03b1.\n\nThe three words \u03c8\u03b1\u03bb\u03bc\u03cc\u03c2, \u1f55\u03bc\u03bd\u03bf\u03c2, \u1fa0\u03b4\u03ae, are distinguished, so far as they\nare distinguishable, in Trench _N.T. Syn._ \u00a7 lxxviii. p. 279 sq. They\nare correctly defined by Gregory Nyssen _in Psalm._ c. iii (I. p. 295)\n\u03c8\u03b1\u03bb\u03bc\u1f78\u03c2 \u03bc\u03ad\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u1f40\u03c1\u03b3\u03ac\u03bd\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u03bc\u03bf\u03c5\u03c3\u03b9\u03ba\u03bf\u1fe6 \u03bc\u03b5\u03bb\u03c9\u03b4\u03af\u03b1, \u1fa0\u03b4\u1f74 \u03b4\u1f72 \u1f21 \u03b4\u03b9\u1f70\n\u03c3\u03c4\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b3\u03b5\u03bd\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c4\u03bf\u1fe6 \u03bc\u03ad\u03bb\u03bf\u03c5\u03c2 \u03bc\u03b5\u03c4\u1f70 \u1fe5\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd \u1f10\u03c0\u03b9\u03c6\u03ce\u03bd\u03b7\u03c3\u03b9\u03c2 ... \u1f55\u03bc\u03bd\u03bf\u03c2 \u03b4\u1f72 \u1f21 \u1f10\u03c0\u1f76\n\u03c4\u03bf\u1fd6\u03c2 \u1f51\u03c0\u03ac\u03c1\u03c7\u03bf\u03c5\u03c3\u03b9\u03bd \u1f21\u03bc\u1fd6\u03bd \u1f00\u03b3\u03b1\u03b8\u03bf\u1fd6\u03c2 \u1f00\u03bd\u03b1\u03c4\u03b9\u03b8\u03b5\u03bc\u03ad\u03bd\u03b7 \u03c4\u1ff7 \u0398\u03b5\u1ff7 \u03b5\u1f50\u03c6\u03b7\u03bc\u03af\u03b1; see also\nHippol. p. 191 sq. (ed. de Lagarde). In other words, while the leading\nidea of \u03c8\u03b1\u03bb\u03bc\u03cc\u03c2 is a musical accompaniment and that of \u1f55\u03bc\u03bd\u03bf\u03c2 praise to\nGod, \u1fa0\u03b4\u03ae is the general word for a song, whether accompanied or\nunaccompanied, whether of praise or on any other subject. Thus it was\nquite possible for the same song to be at once \u03c8\u03b1\u03bb\u03bc\u03cc\u03c2, \u1f55\u03bc\u03bd\u03bf\u03c2, and \u1fa0\u03b4\u03ae.\nIn the text the reference in \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2, we may suppose, is specially,\nthough not exclusively (1 Cor. xiv. 26), to the Psalms of David, which\nwould early form part of the religious worship of the Christian\nbrotherhood. On the other hand \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 would more appropriately designate\nthose hymns of praise which were composed by the Christians themselves\non distinctly Christian themes, being either set forms of words or\nspontaneous effusions of the moment. The third word \u1fa0\u03b4\u03b1\u1fd6\u03c2 gathers up the\nother two, and extends the precept to all forms of song, with the\nlimitation however that they must be \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u03af. St Chrysostom treats\n\u1f55\u03bc\u03bd\u03bf\u03b9 here as an advance upon \u03c8\u03b1\u03bb\u03bc\u03bf\u03af, which in one aspect they are; \u03bf\u1f31\n\u03c8\u03b1\u03bb\u03bc\u03bf\u03af, he says, \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f14\u03c7\u03bf\u03c5\u03c3\u03b9\u03bd, \u1f41\u03b9 \u03b4\u1f72 \u1f55\u03bc\u03bd\u03bf\u03b9 \u03c0\u03ac\u03bb\u03b9\u03bd \u03bf\u1f50\u03b4\u1f72\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03b9\u03bd\u03bf\u03bd\u00b7 \u1f45\u03c4\u03b1\u03bd\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u03bc\u03ac\u03b8\u1fc3, _\u03c4\u03cc\u03c4\u03b5_ \u03ba\u03b1\u1f76 \u1f55\u03bc\u03bd\u03bf\u03c5\u03c2 \u03b5\u1f34\u03c3\u03b5\u03c4\u03b1\u03b9, \u1f05\u03c4\u03b5 \u03b8\u03b5\u03af\u03bf\u03c4\u03b5\u03c1\u03bf\u03bd\n\u03c0\u03c1\u1fb6\u03b3\u03bc\u03b1.\n\nPsalmody and hymnody were highly developed in the religious services of\nthe Jews at this time: see Philo _in Flacc._ 14 (II. p. 535) \u03c0\u03ac\u03bd\u03bd\u03c5\u03c7\u03bf\u03b9 \u03b4\u1f72\n\u03b4\u03b9\u03b1\u03c4\u03b5\u03bb\u03ad\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1fa0\u03b4\u03b1\u1fd6\u03c2, _de Vit. Cont._ \u00a7 3 (II. p. 476)\n\u03c0\u03bf\u03b9\u03bf\u1fe6\u03c3\u03b9\u03bd \u1f84\u03c3\u03bc\u03b1\u03c4\u03b1 \u03ba\u03b1\u1f76 \u1f55\u03bc\u03bd\u03bf\u03c5\u03c2 \u03b5\u1f30\u03c2 \u0398\u03b5\u1f78\u03bd \u03b4\u03b9\u1f70 \u03c0\u03b1\u03bd\u03c4\u03bf\u03af\u03c9\u03bd \u03bc\u03ad\u03c4\u03c1\u03c9\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b5\u03bb\u1ff6\u03bd, \u1f03\n\u1fe5\u03c5\u03b8\u03bc\u03bf\u1fd6\u03c2 \u03c3\u03b5\u03bc\u03bd\u03bf\u03c4\u03ad\u03c1\u03bf\u03b9\u03c2 \u1f00\u03bd\u03b1\u03b3\u03ba\u03b1\u03af\u03c9\u03c2 \u03c7\u03b1\u03c1\u03ac\u03c4\u03c4\u03bf\u03c5\u03c3\u03b9, \u00a7 10 (p. 484) \u1f41 \u1f00\u03bd\u03b1\u03c3\u03c4\u1f70\u03c2 \u1f55\u03bc\u03bd\u03bf\u03bd\n\u1f84\u03b4\u03b5\u03b9 \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd, \u1f22 \u03ba\u03b1\u03b9\u03bd\u1f78\u03bd \u03b1\u1f50\u03c4\u1f78\u03c2 \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03ba\u1f7c\u03c2 \u1f22 \u1f00\u03c1\u03c7\u03b1\u1fd6\u03cc\u03bd \u03c4\u03b9\u03bd\u03b1\n\u03c4\u1ff6\u03bd \u03c0\u03ac\u03bb\u03b1\u03b9 \u03c0\u03bf\u03b9\u03b7\u03c4\u1ff6\u03bd\u00b7 \u03bc\u03ad\u03c4\u03c1\u03b1 \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76 \u03bc\u03ad\u03bb\u03b7 \u03ba\u03b1\u03c4\u03b1\u03bb\u03b5\u03bb\u03bf\u03af\u03c0\u03b1\u03c3\u03b9 \u03c0\u03bf\u03bb\u03bb\u1f70 \u1f10\u03c0\u1ff6\u03bd\n\u03c4\u03c1\u03b9\u03bc\u03ad\u03c4\u03c1\u03c9\u03bd, \u03c0\u03c1\u03bf\u03c3\u03bf\u03b4\u03af\u03c9\u03bd, \u1f55\u03bc\u03bd\u03c9\u03bd, \u03c0\u03b1\u03c1\u03b1\u03c3\u03c0\u03bf\u03bd\u03b4\u03b5\u03af\u03c9\u03bd, \u03c0\u03b1\u03c1\u03b1\u03b2\u03c9\u03bc\u03af\u03c9\u03bd, \u03c3\u03c4\u03b1\u03c3\u03af\u03bc\u03c9\u03bd,\n\u03c7\u03bf\u03c1\u03b9\u03ba\u1ff6\u03bd, \u03c3\u03c4\u03c1\u03bf\u03c6\u03b1\u1fd6\u03c2 \u03c0\u03bf\u03bb\u03c5\u03c3\u03c4\u03c1\u03cc\u03c6\u03bf\u03b9\u03c2 \u03b5\u1f56 \u03b4\u03b9\u03b1\u03bc\u03b5\u03bc\u03b5\u03c4\u03c1\u03b7\u03bc\u03ad\u03bd\u03c9\u03bd \u03ba.\u03c4.\u03bb., \u00a7 11 (p. 485)\n\u1f84\u03b4\u03bf\u03c5\u03c3\u03b9 \u03c0\u03b5\u03c0\u03bf\u03b9\u03b7\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u0398\u03b5\u1f78\u03bd \u1f55\u03bc\u03bd\u03bf\u03c5\u03c2 \u03c0\u03bf\u03bb\u03bb\u03bf\u1fd6\u03c2 \u03bc\u03ad\u03c4\u03c1\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03ad\u03bb\u03b5\u03c3\u03b9\n\u03ba.\u03c4.\u03bb., with the whole context. They would thus find their way into the\nChristian Church from the very beginning. For instances of singing hymns\nor psalms in the Apostolic age see Acts iv. 24, xvi. 25, 1 Cor. xiv. 15,\n26. Hence even in St Paul's epistles, more especially his later\nepistles, fragments of such hymns appear to be quoted; e.g. Ephes. v. 14\n(see the note there). For the use of hymnody in the early Church of the\nsucceeding generations see Plin. _Epist._ x. 97 'Ante lucem convenire,\ncarmenque Christo quasi Deo dicere secum invicem,' Anon. [Hippolytus] in\nEuseb. _H.E._ v. 28 \u03c8\u03b1\u03bb\u03bc\u03bf\u1f76 \u03b4\u1f72 \u1f45\u03c3\u03bf\u03b9 \u03ba\u03b1\u1f76 \u1fa0\u03b4\u1f70\u03b9 \u1f00\u03b4\u03b5\u03bb\u03c6\u1ff6\u03bd \u1f00\u03c0' _)\u03b1\u03c1\u03c7\u1fc6\u03c2_\n\u1f51\u03c0\u1f78 \u03c0\u03b9\u03c3\u03c4\u1ff6\u03bd \u03b3\u03c1\u03b1\u03c6\u03b5\u1fd6\u03c3\u03b1\u03b9 \u03c4\u1f78\u03bd \u039b\u03cc\u03b3\u03bf\u03bd \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03c4\u1f78\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u1f51\u03bc\u03bd\u03bf\u1fe6\u03c3\u03b9\n\u03b8\u03b5\u03bf\u03bb\u03bf\u03b3\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2. The reference in the text is not solely or chiefly to\npublic worship as such. Clem. Alex. _P\u00e6d._ ii. 4 (p. 194) treats it as\napplying to social gatherings; and again Tertullian says of the agape,\n_Apol._ 39 'Ut quisque de scripturis sanctis vel de proprio ingenio\npotest, provocatur in medium Deo canere,' and of the society of husband\nand wife, _Ad Uxor._ ii. 8 'Sonant inter duos psalmi et hymni, et mutuo\nprovocant quis melius Domino suo cantet.' On the psalmody etc. of the\nearly Christians see Bingham _Antiq._ xiv. c. 1, and especially Probst\n_Lehre und Gebet_ p. 256 sq.\n\n\u1f10\u03bd \u03c4\u1fc7 \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9] '_in_ God's _grace_'; comp. 2 Cor. i. 12 \u03bf\u1f50\u03ba \u1f10\u03bd \u03c3\u03bf\u03c6\u03af\u03b1\n\u03c3\u03b1\u03c1\u03ba\u03b9\u03ba\u1fc7 \u1f00\u03bb\u03bb' \u1f10\u03bd \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9 \u0398\u03b5\u03bf\u1fe6. These words are perhaps best connected with\nthe preceding clause, as by Chrysostom. Thus the parallelism with \u1f10\u03bd\n\u03c0\u03ac\u03c3\u1fc3 \u03c3\u03bf\u03c6\u03af\u1fb3 is preserved. The correct reading is \u1f10\u03bd \u03c4\u1fc7 \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9, not \u1f10\u03bd\n\u03c7\u03ac\u03c1\u03b9\u03c4\u03b9. For \u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2, 'divine grace', see Phil. i. 7 \u03c3\u03c5\u03b3\u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03bf\u03cd\u03c2 \u03bc\u03bf\u03c5 \u03c4\u1fc6\u03c2\n\u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2 with the note. The definite article seems to exclude all lower\nsenses of \u03c7\u03ac\u03c1\u03b9\u03c2 here, such as 'acceptableness', 'sweetness' (see iv. 6).\nThe interpretation 'with gratitude', if otherwise tenable (comp. 1 Cor.\nx. 30), seems inappropriate here, because the idea of thanksgiving is\nintroduced in the following verse.\n\n------------------------------------------------------------------------\n\nIII. 17, 18]\n\n\u03c7\u03ac\u03c1\u03b9\u03c4\u03b9, \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03c4\u1ff7 \u0398\u03b5\u1ff7\u00b7 ^{17}\u03ba\u03b1\u1f76 \u03c0\u1fb6\u03bd \u1f45 \u03c4\u03b9 \u1f10\u1f70\u03bd\n\u03c0\u03bf\u03b9\u1fc6\u03c4\u03b5 \u1f10\u03bd \u03bb\u03cc\u03b3\u1ff3 \u1f22 \u1f10\u03bd \u1f14\u03c1\u03b3\u1ff3, \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b9 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f38\u03b7\u03c3\u03bf\u1fe6, \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2\n\u03c4\u1ff7 \u0398\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u1f76 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6.\n\n^{18}\u0391\u1f31 \u03b3\u03c5\u03bd\u03b1\u1fd6\u03ba\u03b5\u03c2, \u1f51\u03c0\u03bf\u03c4\u03ac\u03c3\u03c3\u03b5\u03c3\u03b8\u03b5 \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03bd\u03b4\u03c1\u03ac\u03c3\u03b9\u03bd, \u1f61\u03c2 \u1f00\u03bd\u1fc6\u03ba\u03b5\u03bd\n\n\n\u1f04\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba.\u03c4.\u03bb.] This external manifestation must be accompanied by the\ninward emotion. There must be the thanksgiving of the heart, as well as\nof the lips; comp. Ephes. v. 19 \u1f04\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03c8\u03ac\u03bb\u03bb\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3\n(probably the correct reading), where \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 '_with_ the heart'\nbrings out the sense more distinctly.\n\n17. \u03c0\u1fb6\u03bd \u1f45 \u03c4\u03b9 \u03ba.\u03c4.\u03bb.] This is probably a nominative absolute, as Matt. x.\n32 \u03c0\u1fb6\u03c2 \u03bf\u1f56\u03bd \u1f45\u03c3\u03c4\u03b9\u03c2 \u1f41\u03bc\u03bf\u03bb\u03bf\u03b3\u03ae\u03c3\u03b5\u03b9 ... \u1f41\u03bc\u03bf\u03bb\u03bf\u03b3\u03ae\u03c3\u03c9 \u03ba\u1f00\u03b3\u1f7c \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 (comp. Luke xii.\n8), Luke xii. 10 \u03c0\u1fb6\u03c2 \u1f43\u03c2 \u1f10\u03c1\u03b5\u1fd6 \u03bb\u03cc\u03b3\u03bf\u03bd ... \u1f00\u03c6\u03b5\u03b8\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9 \u03b1\u1f50\u03c4\u1ff7, John xvii. 2 \u03c0\u1fb6\u03bd\n\u1f43 \u03b4\u03ad\u03b4\u03c9\u03ba\u03b1\u03c2 \u03b1\u1f50\u03c4\u1ff7, \u03b4\u03ce\u03c3\u1fc3 _\u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2_ \u03ba.\u03c4.\u03bb.; comp. Matt. vii. 24 (v.l.).\n\n\u03c0\u03ac\u03bd\u03c4\u03b1] sc. \u03c0\u03bf\u03b9\u03b5\u1fd6\u03c4\u03b5, as the following \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 suggests; comp. ver.\n23.\n\n\u1f10\u03bd \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03b9 \u03ba.\u03c4.\u03bb.] This is the great practical lesson which flows from\nthe theological teaching of the epistle. Hence the reiteration of \u039a\u03c5\u03c1\u03af\u1ff3,\n\u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3, etc., vv. 18, 20, 22, 23, 24. See above p. 104.\n\n\u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2] On this refrain see the notes on i. 12, ii. 7.\n\n\u03c4\u1ff7 \u0398\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u1f76] This, which is quite the best authenticated reading, gives\na very unusual, if not unique, collocation of words, the usual form\nbeing either \u1f41 \u0398\u03b5\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c4\u03ae\u03c1 or \u0398\u03b5\u1f78\u03c2 \u03c0\u03b1\u03c4\u03ae\u03c1. The \u03ba\u03b1\u03af before \u03c0\u03b1\u03c4\u03c1\u03af in the\nreceived text is an obvious emendation. See the note on i. 3, and the\nappendix on various readings.\n\n18\u201321. 'Ye wives, be subject to your husbands, for so it becomes you in\nChrist. Ye husbands, love and cherish your wives, and use no harshness\ntowards them. Ye children, be obedient to your parents in all things;\nfor this is commendable and lovely in Christ. Ye parents, vex not your\nchildren, lest they lose heart and grow sullen'.\n\n18 sq. These precepts, providing for the conduct of Christians in\nprivate households, should be compared with Ephes. v. 22\u2013vi. 9, 1 Pet.\nii. 18\u2013iii. 7, Tit. ii. 1 sq.; see also Clem. Rom. 1, Polyc. _Phil._ 4\nsq.\n\n\u0391\u1f31 \u03b3\u03c5\u03bd\u03b1\u1fd6\u03ba\u03b5\u03c2] '_Ye wives_', the nominative with the definite article\nbeing used for a vocative, as frequently in the New Testament, e.g.\nMatt. xi. 26, Mark v. 41, Luke viii. 54; see Winer \u00a7 xxix. p. 227 sq.\nThe frequency of this use is doubtless due to the fact that it is a\nreproduction of the Hebrew idiom. In the instances quoted from classical\nwriters (see Bernhardy _Syntax_ p. 67) the address is not so directly\nvocative, the nominative being used rather to _define_ or _select_ than\nto _summon_ the person in question.\n\n\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03bd\u03b4\u03c1\u03ac\u03c3\u03b9\u03bd] The \u1f30\u03b4\u03af\u03bf\u03b9\u03c2 of the received text may have been inserted\n(as it is inserted also in Ephes. v. 24) from Ephes. v. 22, Tit. ii. 5,\n1 Pet. iii. 1, 5, in all which passages this same injunction occurs. The\nscribes however show a general fondness for this adjective; e.g. Mark\nxv. 20, Luke ii. 3, Acts i. 19, Ephes. iv. 28, 1 Thess. ii. 15, iv. 11.\n\n------------------------------------------------------------------------\n\nIII. 19\u201322]\n\n\u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3. ^{19}\u039f\u1f31 \u1f04\u03bd\u03b4\u03c1\u03b5\u03c2, \u1f00\u03b3\u03b1\u03c0\u1fb6\u03c4\u03b5 \u03c4\u1f70\u03c2 \u03b3\u03c5\u03bd\u03b1\u1fd6\u03ba\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03bc\u1f74 \u03c0\u03b9\u03ba\u03c1\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b5 \u03c0\u03c1\u1f78\u03c2\n\u03b1\u1f50\u03c4\u03ac\u03c2. ^{20}\u03a4\u1f70 \u03c4\u03ad\u03ba\u03bd\u03b1, \u1f51\u03c0\u03b1\u03ba\u03bf\u03cd\u03b5\u03c4\u03b5 \u03c4\u03bf\u1fd6\u03c2 \u03b3\u03bf\u03bd\u03b5\u1fe6\u03c3\u03b9\u03bd \u03ba\u03b1\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1\u00b7 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03b3\u1f70\u03c1\n\u03b5\u1f50\u03ac\u03c1\u03b5\u03c3\u03c4\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3. ^{21}\u039f\u1f31 \u03c0\u03b1\u03c4\u03ad\u03c1\u03b5\u03c2, \u03bc\u1f74 \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5 \u03c4\u1f70 \u03c4\u03ad\u03ba\u03bd\u03b1 \u1f51\u03bc\u1ff6\u03bd,\n\u1f35\u03bd\u03b1 \u03bc\u1f74 \u1f00\u03b8\u03c5\u03bc\u1ff6\u03c3\u03b9\u03bd. ^{22}\u039f\u1f31 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03b9, \u1f51\u03c0\u03b1\u03ba\u03bf\u03cd\u03b5\u03c4\u03b5 \u03ba\u03b1\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1\n\n\n\u1f00\u03bd\u1fc6\u03ba\u03b5\u03bd] The imperfect, as Ephes. v. 4 \u1f03 \u03bf\u1f50\u03ba \u1f00\u03bd\u1fc6\u03ba\u03b5\u03bd (the correct\nreading); comp. _Clem. Hom._ Contest. 3 \u03c4\u03bf\u1fe6\u03b4\u03b5 \u03bc\u1f74 \u03bc\u03b5\u03c4\u03b1\u03b4\u03bf\u1fe6\u03bd\u03b1\u03b9 \u03c7\u03ac\u03c1\u03b9\u03bd, \u1f61\u03c2 \u03bf\u1f50\n\u03c0\u03c1\u03bf\u03c3\u1fc6\u03ba\u03b5\u03bd, Xen. _de Re Equestr._ xii. 14 \u1f03 \u1f31\u03c0\u03c0\u03ac\u03c1\u03c7\u1ff3 \u03c0\u03c1\u03bf\u03c3\u1fc6\u03ba\u03b5\u03bd \u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9 \u03c4\u03b5\n\u03ba\u03b1\u1f76 \u03c0\u03c1\u03ac\u03c4\u03c4\u03b5\u03b9\u03bd; and see D'Orville on Charito viii. 2 (p. 699 sq.). The\ncommon uses of the imperfect \u1f14\u03b4\u03b5\u03b9, \u1f14\u03c0\u03c1\u03b5\u03c0\u03b5\u03bd, etc., in classical writers\ndo not present a very exact parallel; for they imply that the thing\nwhich ought to have been done has been left undone. And so we might\ninterpret Acts xxii. 22 \u03bf\u1f50 \u03b3\u1f70\u03c1 \u03ba\u03b1\u03b8\u1fc6\u03ba\u03b5\u03bd \u03b1\u1f50\u03c4\u1f78\u03bd \u03b6\u1fc6\u03bd (the correct reading).\nHere however there can hardly be any such reference; and the best\nillustration is the English past tense 'ought' (= 'owed'), which is used\nin the same way. The past tense perhaps implies an essential _\u00e0 priori_\nobligation. The use of \u03c7\u03c1\u1fc6\u03bd, \u1f14\u03c7\u03c1\u03b7\u03bd, occasionally approximates to this;\ne.g. Eur. _Andr._ 423.\n\nThe idea of 'propriety' is the link which connects the primary meaning\nof such words as \u1f00\u03bd\u03ae\u03ba\u03b5\u03b9\u03bd, \u03c0\u03c1\u03bf\u03c3\u03ae\u03ba\u03b5\u03b9\u03bd, \u03ba\u03b1\u03b8\u03ae\u03ba\u03b5\u03b9\u03bd, 'aiming at or pertaining\nto', with their ultimate meaning of moral obligation. The word \u1f00\u03bd\u03ae\u03ba\u03b5\u03b9\u03bd\noccurs in the New Testament only here and in the contemporary epistles,\nEphes. v. 4, Philem. 8.\n\n\u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3] probably to be connected with \u1f61\u03c2 \u1f00\u03bd\u1fc6\u03ba\u03b5\u03bd, rather than with\n\u1f51\u03c0\u03bf\u03c4\u03ac\u03c3\u03c3\u03b5\u03c3\u03b8\u03b5; comp. ver. 20 \u03b5\u1f50\u03ac\u03c1\u03b5\u03c3\u03c4\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3.\n\n19. \u03bc\u1f74 \u03c0\u03b9\u03ba\u03c1\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b5 \u03ba.\u03c4.\u03bb.] '_show no bitterness, behave not harshly_';\ncomp. Lynceus in Athen. vi. p. 242 C \u03c0\u03b9\u03ba\u03c1\u03b1\u03bd\u03b8\u03b5\u03af\u03b7 \u03c0\u03c1\u03cc\u03c2 \u03c4\u03b9\u03bd\u03b1 \u03c4\u1ff6\u03bd \u03c3\u03c5\u03b6\u03ce\u03bd\u03c4\u03c9\u03bd,\nJoseph. _Ant._ v. 7. I \u03b4\u03b5\u03b9\u03bd\u1ff6\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03c4\u03bf\u1fe6 \u03b4\u03b9\u03ba\u03b1\u03af\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ca\u03c3\u03c4\u03b1\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2\n\u1f10\u03ba\u03c0\u03b9\u03ba\u03c1\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2, Plut. _Mor._ p. 457 A \u03c0\u03c1\u1f78\u03c2 \u03b3\u03cd\u03bd\u03b1\u03b9\u03b1 \u03b4\u03b9\u03b1\u03c0\u03b9\u03ba\u03c1\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9. So\nalso \u03c0\u03b9\u03ba\u03c1\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 \u1f10\u03c0\u03af \u03c4\u03b9\u03bd\u03b1 in the LXX, Jerem. xliv (xxxvii). 15, 3 Esdr.\niv. 31. This verb \u03c0\u03b9\u03ba\u03c1\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9 and its compounds occur frequently in\nclassical writers.\n\n20. \u03ba\u03b1\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1] as ver. 22. The rule is stated absolutely, because the\nexceptions are so few that they may be disregarded.\n\n\u03b5\u1f50\u03ac\u03c1\u03b5\u03c3\u03c4\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd] '_is well pleasing, commendable_'. The received text\nsupplies this adjective with a dative of reference \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3 (from Ephes.\nv. 10), but \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 is unquestionably the right reading. With the\nreading thus corrected \u03b5\u1f50\u03ac\u03c1\u03b5\u03c3\u03c4\u03bf\u03bd, like \u1f00\u03bd\u1fc6\u03ba\u03b5\u03bd ver. 18, must be taken\nabsolutely, as perhaps in Rom. xii. 2 \u03c4\u1f78 \u03b8\u03ad\u03bb\u03b7\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u0398\u1f11\u03bf\u1fe6 \u03c4\u1f78 \u1f00\u03b3\u03b1\u03b8\u1f78\u03bd \u03ba\u03b1\u1f76\n\u03b5\u1f50\u03ac\u03c1\u03b5\u03c3\u03c4\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd: comp. Phil. iv. 8 \u1f45\u03c3\u03b1 \u03c3\u03b5\u03bc\u03bd\u03ac ... \u1f45\u03c3\u03b1 \u03c0\u03c1\u03bf\u03c3\u03c6\u03b9\u03bb\u1fc6. The\nqualification \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 implies 'as judged by a Christian standard', 'as\njudged by those who are members of Christ's body.'\n\n21. \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5] '_provoke, irritate_'. The other reading \u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03c4\u03b5 has\nhigher support, but is doubtless taken from the parallel passage, Ephes.\nvi. 4. 'Irritation' is the first consequence of being too exacting with\nchildren, and irritation leads to moroseness (\u1f00\u03b8\u03c5\u03bc\u03af\u03b1). In 2 Cor. IX. 2\n\u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03b9\u03bd is used in a good sense and produces the opposite result, not\ndespondency but energy.\n\n\u1f00\u03b8\u03c5\u03bc\u1ff6\u03c3\u03b9\u03bd] '_lose heart, become spiritless_', i.e. 'go about their task\nin a listless, moody, sullen frame of mind'. '_Fractus animus_', says\nBengel, 'pestis juventutis'. In Xen. _Cyr._ i. 6. 13 \u1f00\u03b8\u03c5\u03bc\u03af\u03b1 is opposed\nto \u03c0\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1, and in Thuc. ii. 88 and elsewhere \u1f00\u03b8\u03c5\u03bc\u03b5\u1fd6\u03bd is opposed to\n\u03b8\u03b1\u03c1\u03c3\u03b5\u1fd6\u03bd.\n\n------------------------------------------------------------------------\n\nIII. 23]\n\n\u03c4\u03bf\u1fd6\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c3\u03ac\u03c1\u03ba\u03b1 \u03ba\u03c5\u03c1\u03af\u03bf\u03b9\u03c2, \u03bc\u1f74 \u1f10\u03bd \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u1fb3 \u1f61\u03c2 \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03ac\u03c1\u03b5\u03c3\u03ba\u03bf\u03b9, \u1f00\u03bb\u03bb' \u1f10\u03bd\n\u1f01\u03c0\u03bb\u03cc\u03c4\u03b7\u03c4\u03b9 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2, \u03c6\u03bf\u03b2\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd. ^{23}\u1f43 \u1f10\u1f70\u03bd \u03c0\u03bf\u03b9\u1fc6\u03c4\u03b5, \u1f10\u03ba \u03c8\u03c5\u03c7\u1fc6\u03c2\n\u1f10\u03c1\u03b3\u03ac\u03b6\u03b5\u03c3\u03b8\u03b5 \u1f61\u03c2\n\n                        22. \u1f10\u03bd \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u03b1\u03b9\u03c2.\n\n\n22.\u2013iv. 1. 'Ye slaves, be obedient in all things to the masters set over\nyou in the flesh, not rendering them service only when their eyes are\nupon you, as aiming merely to please men, but serving in all sincerity\nof heart, as living in the sight of God and standing in awe of Him. And\nin every thing that ye do, work faithfully and with all your soul, as\nlabouring not for men, but for the great Lord and Master Himself;\nknowing that ye have a Master, from whom ye will receive the glorious\ninheritance as your recompense, whether or not ye may be defrauded of\nyour due by men. Yes, Christ is your Master and ye are his slaves. He\nthat does a wrong shall be requited for his wrong-doing. I say not this\nof slaves only, but of masters also. There is no partiality, no respect\nof persons, in God's distribution of rewards and punishments. Therefore,\nye masters, do ye also on your part deal justly and equitably by your\nslaves, knowing that ye too have a Master in heaven'.\n\n22. \u039f\u1f31 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03b9] The relations of masters and slaves, both here and in the\ncompanion epistle (Ephes. vi. 5\u20139), are treated at greater length than\nis usual with St Paul. Here especially the expansion of this topic,\ncompared with the brief space assigned to the duties of wives and\nhusbands (vv. 18, 19), or of children and parents (vv. 20, 21), deserves\nto be noticed. The fact is explained by a contemporary incident in the\nApostle's private life. His intercourse with Onesimus had turned his\nthoughts in this direction. See above, p. 33, and the introduction to\nthe Epistle to Philemon: comp. also the note on ver. 11.\n\n\u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u1fb3] '_eye-service_', as Ephes. vi. 6: comp. _Apost. Const._\niv. 12 \u03bc\u1f74 \u1f61\u03c2 \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03cc\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2 \u1f00\u03bb\u03bb' \u1f61\u03c2 \u03c6\u03b9\u03bb\u03bf\u03b4\u03ad\u03c3\u03c0\u03bf\u03c4\u03bf\u03c2. This happy expression\nwould seem to be the Apostle's own coinage. At least there are no traces\nof it earlier. Compare \u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1 ii. 23. The reading \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u1fb3\nis better supported than \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u03b1\u03b9\u03c2, though the plural is rendered\nslightly more probable in itself by its greater difficulty.\n\n\u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03ac\u03c1\u03b5\u03c3\u03ba\u03bf\u03b9] again in Ephes. vi. 6. It is a LXX word, Ps. lii. 6,\nwhere the Greek entirely departs from the Hebrew: comp. also\n\u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03b1\u03c1\u03b5\u03c3\u03ba\u03b5\u1fd6\u03bd Ign. _Rom._ 2, \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03b1\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1 Justin _Apol._ i. 2 (p. 53\nE). So \u1f40\u03c7\u03bb\u03bf\u03b1\u03c1\u03ad\u03c3\u03ba\u03b7\u03c2 or \u1f40\u03c7\u03bb\u03cc\u03b1\u03c1\u03b5\u03c3\u03ba\u03bf\u03c2, Timo Phlias. in Diog. Laert. iv. 42\n(vv. 11.).\n\n\u1f01\u03c0\u03bb\u03cc\u03c4\u03b7\u03c4\u03b9 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2] as in Ephes. vi. 5, i.e. 'with _undivided_ service'; a\nLXX expression, 1 Chron. xxix. 17, Wisd. i. 1.\n\n\u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd] '_the_ one _Lord_ and Master', as contrasted with \u03c4\u03bf\u1fd6\u03c2 \u03ba\u03b1\u03c4\u1f70\n\u03c3\u03ac\u03c1\u03ba\u03b1 \u03ba\u03c5\u03c1\u03af\u03bf\u03b9\u03c2: the idea being carried out in the following verses. The\nreceived text, by substituting \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd, blunts the edge of the\ncontrast.\n\n23. \u1f10\u03c1\u03b3\u03ac\u03b6\u03b5\u03c3\u03b8\u03b5] i.e. 'do it diligently', an advance upon \u03c0\u03bf\u03b9\u03b7\u03c4\u03b5.\n\n\u03bf\u1f50\u03ba \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2] For the use of \u03bf\u1f50 rather than \u03bc\u1f74 in antitheses, see Winer\n\u00a7 lv. p. 601 sq. The negative here is wholly unconnected with the\nimperative, and refers solely to \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3.\n\n------------------------------------------------------------------------\n\nIII. 24, 25]\n\n\u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3, \u03ba\u03b1\u1f76 \u03bf\u1f50\u03ba \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2,] ^{24}\u03b5\u1f30\u03b4\u03cc\u03c4\u03b5\u03c2 \u1f45\u03c4\u03b9 \u1f00\u03c0\u1f78 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f00\u03c0\u03bf\u03bb\u03ae\u03bc\u03c8\u03b5\u03c3\u03b8\u03b5 \u03c4\u1f74\u03bd\n\u1f00\u03bd\u03c4\u03b1\u03c0\u03cc\u03b4\u03bf\u03c3\u03b9\u03bd \u03c4\u1fc6\u03c2 \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03c2\u00b7 \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3 \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03b4\u03bf\u03c5\u03bb\u03b5\u03cd\u03b5\u03c4\u03b5\u00b7 ^{25}\u1f41 \u03b3\u1f70\u03c1 \u1f00\u03b4\u03b9\u03ba\u1ff6\u03bd\n\u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c4\u03b1\u03b9 \u1f43\n\n\n24. \u1f00\u03c0\u1f78 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 'However you may be treated by your earthly masters, you\nhave still _a_ Master who will recompense you.' The absence of the\ndefinite article here (comp. iv. 1) is the more remarkable, because it\nis studiously inserted in the context, vv. 22\u201324, \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd, \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3,\n\u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3. In the parallel passage Ephes. vi. 8 it is \u03c0\u03b1\u03c1\u1f70 \u039a\u03c5\u03c1\u03af\u03bf\u03c5: for\nthe difference between the two see Gal. i. 12.\n\n\u03c4\u1f74\u03bd \u1f00\u03bd\u03c4\u03b1\u03c0\u03cc\u03b4\u03bf\u03c3\u03b9\u03bd] '_the just recompense_', a common word both in the lxx\nand in classical writers, though not occurring elsewhere in the New\nTestament; comp. \u1f00\u03bd\u03c4\u03b1\u03c0\u03cc\u03b4\u03bf\u03bc\u03b1 Luke xiv. 12, Rom. xi. 9. The double\ncompound involves the idea of 'exact requital'.\n\n\u03c4\u1fc6\u03c2 \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03c2] '_which consists in the inheritance_', the genitive of\napposition: see the note on \u03c4\u1f74\u03bd \u03bc\u03b5\u03c1\u03af\u03b4\u03b1 \u03c4\u03bf\u1fe6 \u03ba\u03bb\u03ae\u03c1\u03bf\u03c5, i. 12. There is a\nparadox involved in this word: elsewhere the \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 and the \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03cc\u03bc\u03bf\u03c2\nare contrasted (Matt. xxi. 35\u201338, etc., Rom. viii. 15\u201317, Gal. iv. 1,\n7), but here the \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 is the \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03cc\u03bc\u03bf\u03c2. This he is because, though\n\u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd, he is \u1f00\u03c0\u03b5\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 (1 Cor. vii. 22) and thus\n\u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03cc\u03bc\u03bf\u03c2 \u03b4\u03b9\u1f70 \u0398\u03b5\u03bf\u1fe6 (Gal. iv. 7); comp. Hermas _Sim._ v. 2 \u1f35\u03bd\u03b1\n\u03c3\u03c5\u03b3\u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03cc\u03bc\u03bf\u03c2 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 \u1f41 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u03c4\u1ff7 \u1f51\u03b9\u1ff7 (with the context).\n\n\u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3 \u03ba.\u03c4.\u03bb.] i.e. '_you serve_ as your master _the_ great _Master\nChrist_.' This clause is added to explain who is meant by the preceding\n\u1f00\u03c0\u1f78 \u039a\u03c5\u03c1\u03af\u03bf\u03c5. For this application of \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 compare (besides the parallel\npassage, Ephes. vi. 6\u20139) 1 Cor. vii. 22 \u1f41 \u03b3\u1f70\u03c1 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 \u03ba\u03bb\u03b7\u03b8\u1f72\u03b9\u03c2 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2\n\u1f00\u03c0\u03b5\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f10\u03c3\u03c4\u03af\u03bd \u03ba.\u03c4.\u03bb. It seems best to take \u03b4\u03bf\u03c5\u03bb\u03b5\u03cd\u03b5\u03c4\u03b5 here as\nan indicative, rather than as an imperative; for (1) The indicative is\nwanted to explain the previous \u1f00\u03c0\u1f78 \u039a\u03c5\u03c1\u03af\u03bf\u03c5; (2) The imperative would seem\nto require \u1f61\u03c2 \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3, as in Ephes. vi. 7 (the correct text). On the\nother hand see Rom. xii. 11.\n\n25. \u1f41 \u03b3\u1f70\u03c1 \u1f00\u03b4\u03b9\u03ba\u1ff6\u03bd \u03ba.\u03c4.\u03bb.] Who is this unrighteous person ? The slave who\ndefrauds his master of his service, or the master who defrauds his slave\nof his reward? Some interpreters confine it exclusively to the former;\nothers to the latter. It seems best to suppose that both are included.\nThe connexion of the sentence \u1f41 \u03b3\u1f70\u03c1 \u1f00\u03b4\u03b9\u03ba\u1ff6\u03bd (where \u03b3\u03ac\u03c1, not \u03b4\u03ad, is\ncertainly the right reading) points to the slave. On the other hand the\nexpression which follows, \u03c4\u1f78 \u03b4\u03af\u03ba\u03b1\u03b9\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03ba.\u03c4.\u03bb., suggests\nthe master. Thus there seems to be a twofold reference; the warning is\nsuggested by the case of the slave, but it is extended to the case of\nthe master; and this accords with the parallel passage, Ephes. vi. 8\n\u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03c2 \u1f43 \u1f02\u03bd \u03c0\u03bf\u03af\u03b7\u03c3\u1fc3 \u1f00\u03b3\u03b1\u03b8\u1f78\u03bd \u03c4\u03bf\u1fe6\u03c4\u03bf \u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c4\u03b1\u03b9 \u03c0\u03b1\u03c1\u1f70 \u039a\u03c5\u03c1\u03af\u03bf\u03c5, _\u03b5\u1f34\u03c4\u03b5 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2\n\u03b5\u1f34\u03c4\u03b5 \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2_.\n\nThe recent fault of Onesimus would make the Apostle doubly anxious to\nemphasize the duties of the slave towards the master, lest in his love\nfor the offender he should seem to condone the offence. This same word\n\u1f20\u03b4\u03af\u03ba\u03b7\u03c3\u03b5\u03bd is used by St Paul to describe the crime of Onesimus in Philem.\n18. But on the other hand it is the Apostle's business to show that\njustice has a double edge. There must be a _reciprocity_ between the\nmaster and the slave. The philosophers of Greece taught, and the laws of\nRome assumed, that the slave was a chattel. But a chattel could have no\nrights. It would be absurd to talk of treating a chattel with justice.\nSt Paul places the relations of the master and the slave in a wholly\ndifferent light. Justice and equity are the expression of the Divine\nmind: and with God there is no \u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1. With Him the claims of the\nslave are as real as the claims of the master.\n\n\u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c4\u03b1\u03b9] For this sense of the middle, 'to recover', 'to get back',\nand so (with an accusative of the thing to be recompensed), 'to be\nrequited for', see e.g. Lev. xx. 17 \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u03b1\u03bd \u03ba\u03bf\u03bc\u03b9\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9, 2 Cor. v. 10\n\u03ba\u03bf\u03bc\u03af\u03c3\u03b7\u03c4\u03b1\u03b9 \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03c2 \u03c4\u1f70 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2; comp. Barnab.\n\n------------------------------------------------------------------------\n\nIV. 1]\n\n\u1f20\u03b4\u03af\u03ba\u03b7\u03c3\u03b5\u03bd, \u03ba\u03b1\u1f76 \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9\u03bd \u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1. IV. ^1\u039f\u1f31 \u03ba\u03cd\u03c1\u03b9\u03bf\u03b9, \u03c4\u1f78 \u03b4\u03af\u03ba\u03b1\u03b9\u03bf\u03bd \u03ba\u03b1\u1f76\n\u03c4\u1f74\u03bd \u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03bf\u03cd\u03bb\u03bf\u03b9\u03c2 \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03c3\u03b8\u03b5, \u03b5\u1f30\u03b4\u03cc\u03c4\u03b5\u03c2 \u1f45\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u1f14\u03c7\u03b5\u03c4\u03b5 \u039a\u03cd\u03c1\u03b9\u03bf\u03bd\n\u1f10\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u1ff7.\n\n\n\u00a7 4 \u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 \u1f00\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03ae\u03bc\u03c0\u03c4\u03c9\u03c2 \u03ba\u03c1\u03b9\u03bd\u03b5\u1fd6 \u03c4\u1f78\u03bd \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5\u00b7 \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03c2, \u03ba\u03b1\u03b8\u1f7c\u03c2 \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd,\n\u03ba\u03bf\u03bc\u03b9\u03b5\u1fd6\u03c4\u03b1\u03b9. In the parallel passage Ephes. vi. 8, the form is certainly\n\u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c4\u03b1\u03b9: here it is more doubtful, the authorities being more equally\ndivided between \u03ba\u03bf\u03bc\u03b9\u03b5\u1fd6\u03c4\u03b1\u03b9 and \u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c4\u03b1\u03b9. See however the note on\n\u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd iv. 9.\n\n\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1] On this word see the note Gal. ii. 6. This \u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1,\nthough generally found on the side of rank and power, may also be\nexercised in favour of the opposite; Levit. xix. 15 \u03bf\u1f50 \u03bb\u03ae\u03c8\u1fc3 \u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03bf\u03bd\n\u03c0\u03c4\u03c9\u03c7\u03bf\u1fe6 \u03bf\u1f50\u03b4\u1f72 \u03bc\u1f74 \u03b8\u03b1\u03c5\u03bc\u03ac\u03c3\u1fc3\u03c2 \u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03bf\u03bd \u03b4\u03c5\u03bd\u03ac\u03c3\u03c4\u03bf\u03c5. There would be a tendency in\nthe mind of the slave to assume that, because the \u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1 of man\nwas on the side of the master, there must be a corresponding\n\u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1 of God on the side of the slave. This assumption is\ncorrected by St Paul.\n\nIV. 1. \u03c4\u1f74\u03bd \u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1] '_equity_', '_fairness_'; comp. Plut. _Sol. et Popl.\nComp._ 3 \u03bd\u03cc\u03bc\u03c9\u03bd \u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c0\u03b1\u03c1\u03b5\u03c7\u03cc\u03bd\u03c4\u03c9\u03bd. Somewhat similarly Lysias _Or. Fun._\n77 (speaking of death) \u03bf\u1f54\u03c4\u03b5 \u03b3\u1f70\u03c1 \u03c4\u03bf\u1f7a\u03c2 \u03c0\u03bf\u03bd\u03b7\u03c1\u03bf\u1f7a\u03c2 \u1f51\u03c0\u03b5\u03c1\u03bf\u03c1\u1fb7 \u03bf\u1f54\u03c4\u03b5 \u03c4\u03bf\u1f7a\u03c2 \u1f00\u03b3\u03b1\u03b8\u03bf\u1f7a\u03c2\n\u03b8\u03b1\u03c5\u03bc\u03ac\u03b6\u03b5\u03b9, \u1f00\u03bb\u03bb' _\u1f34\u03c3\u03bf\u03bd \u1f11\u03b1\u03c5\u03c4\u1f78\u03bd \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03b9_ \u03c0\u1fb6\u03c3\u03b9\u03bd. It seems a mistake to\nsuppose that \u1f30\u03c3\u03cc\u03c4\u03b7\u03c2 here has anything to do with the treatment of slaves\n_as equals_ (comp. Philem. 16). When connected with \u03c4\u1f78 \u03b4\u03af\u03ba\u03b1\u03b9\u03bf\u03bd, the word\nnaturally suggests an even-handed, impartial treatment, and is\nequivalent to the Latin _\u00e6quitas_: comp. Arist. _Top._ vi. 5 (p. 143) \u1f41\n\u03c4\u1f74\u03bd \u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7\u03bd (\u03bb\u03ad\u03b3\u03c9\u03bd) \u1f15\u03be\u03b9\u03bd \u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03c0\u03bf\u03b9\u03b7\u03c4\u03b9\u03ba\u1f74\u03bd \u1f22 \u03b4\u03b9\u03b1\u03bd\u03b5\u03bc\u03b7\u03c4\u03b9\u03ba\u1f74\u03bd \u03c4\u03bf\u1fe6 \u1f34\u03c3\u03bf\u03c5,\nPhilo _de Creat. Princ._ 14 (II. p. 373) \u1f14\u03c3\u03c4\u03b9 \u03b3\u1f70\u03c1 \u1f30\u03c3\u03cc\u03c4\u03b7\u03c2 ... \u03bc\u03ae\u03c4\u03b7\u03c1\n\u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7\u03c2, Clem. Alex. _Strom._ vi. 6 (p. 764) \u03bc\u03b5\u03c4\u1f70 \u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7\u03c2 \u03ba\u03b1\u1f76\n\u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03c4\u1fc6\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u03b5\u03c0\u03b9\u03c3\u03c4\u03c1\u03ad\u03c6\u03bf\u03bd\u03c4\u03b1\u03c2. Thus in Arist. _Eth. Nic._ v. 1 \u03c4\u1f78\n\u03b4\u03af\u03ba\u03b1\u03b9\u03bf\u03bd and \u03c4\u1f78 \u1f34\u03c3\u03bf\u03bd are regarded as synonymes, and in Plut. _Mor._ p.\n719 the relation of \u1f30\u03c3\u03cc\u03c4\u03b7\u03c2 to \u03b4\u03b9\u03ba\u03b1\u03af\u03bf\u03c4\u03b7\u03c2 is discussed. The word here is\nused in the same sense in which the adjective occurs in the common\nexpressions \u1f34\u03c3\u03bf\u03c2 \u03b4\u03b9\u03ba\u03b1\u03c3\u03c4\u03ae\u03c2, \u1f34\u03c3\u03bf\u03c2 \u1f00\u03ba\u03c1\u03bf\u03b1\u03c4\u03ae\u03c2, etc. Philo, describing the\nEssene condemnation of slavery, says, _Omn. prob. lib._ 12 (II. p. 457)\n\u03ba\u03b1\u03c4\u03b1\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03bf\u03c5\u03c3\u03af \u03c4\u03b5 \u03c4\u1ff6\u03bd \u03b4\u03b5\u03c3\u03c0\u03bf\u03c4\u1ff6\u03bd, \u03bf\u1f50 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f61\u03c2 \u1f00\u03b4\u03af\u03ba\u03c9\u03bd, \u1f30\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1\n\u03bb\u03c5\u03bc\u03b1\u03b9\u03bd\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u1f61\u03c2 \u1f00\u03c3\u03b5\u03b2\u1ff6\u03bd \u03ba.\u03c4.\u03bb., but he possibly does mean\n'equality' rather than 'equity.'\n\n\u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03c3\u03b8\u03b5] '_exhibit on your part_.' The middle \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9, 'to afford\nfrom oneself,' will take different shades of meaning according to the\ncontext, as 'to furnish one's _quota_' (e.g. Herod. viii. 1, 2) or 'to\nput forward one's _representative_' (esp. of witnesses, e.g. Plato\n_Apol._ 19 D). Here the idea is 'reciprocation', the master's duty as\ncorresponding to the slave's.\n\n\u1f14\u03c7\u03b5\u03c4\u03b5 \u039a\u03cd\u03c1\u03b9\u03bf\u03bd] as Ephes. vi. 9; comp. 1 Cor. vii. 22 \u1f41 \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2 \u03ba\u03bb\u03b7\u03b8\u1f72\u03b9\u03c2\n\u03b4\u03bf\u1fe6\u03bb\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.\n\n------------------------------------------------------------------------\n\nIV. 2\u20134]\n\n^2\u03a4\u1fc7 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u1fc7 \u03c0\u03c1\u03bf\u03c3\u03ba\u03b1\u03c1\u03c4\u03b5\u03c1\u03b5\u1fd6\u03c4\u03b5, \u03b3\u03c1\u03b7\u03b3\u03bf\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03b1\u1f50\u03c4\u1fc7 \u1f10\u03bd \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03af\u1fb3\u00b7\n^3\u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u1f05\u03bc\u03b1 \u03ba\u03b1\u1f76 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd, \u1f35\u03bd\u03b1 \u1f41 \u0398\u03b5\u1f78\u03c2 \u1f00\u03bd\u03bf\u03af\u03be\u1fc3 \u1f21\u03bc\u1fd6\u03bd \u03b8\u03cd\u03c1\u03b1\u03bd \u03c4\u03bf\u1fe6\n\u03bb\u03cc\u03b3\u03bf\u03c5, \u03bb\u03b1\u03bb\u1fc6\u03c3\u03b1\u03b9 \u03c4\u1f78 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, \u03b4\u03b9' \u1f43 \u03ba\u03b1\u1f76 \u03b4\u03ad\u03b4\u03b5\u03bc\u03b1\u03b9\u00b7 ^4\u1f35\u03bd\u03b1\n\u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03c3\u03c9\n\n\n2\u20136. 'Be earnest and unceasing in prayer; keep your hearts and minds\nawake while praying: remember also (as I have so often told you) that\nthanksgiving is the goal and crown of prayer. Meanwhile in your\npetitions forget not us\u2014myself Paul\u2014my fellow-labourer Timothy\u2014your\nevangelist Epaphras\u2014all the teachers of the Gospel; but pray that God\nmay open a door for the preaching of the word, to the end that we may\nproclaim the free offer of grace to the Gentiles\u2014that great mystery of\nChrist for which I am now a prisoner in bonds. So shall I declare it\nfearlessly, as I am bound to proclaim it. Walk wisely and discreetly in\nall your dealings with unbelievers; allow no opportunity to slip through\nyour hands, but buy up every passing moment. Let your language be always\npervaded with grace and seasoned with salt. So will you know how to give\na fit answer to each man, as the occasion demands.'\n\n2. \u03c0\u03c1\u03bf\u03c3\u03ba\u03b1\u03c1\u03c4\u03b5\u03c1\u03b5\u1fd6\u03c4\u03b5] '_cling closely to_', '_remain constant to_' (comp.\nMark iii. 9, Acts viii. 13, x. 7), and so 'continue stedfast in'. This\nword occurs again with \u03c4\u1fc7 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u1fc7, \u03c4\u03b1\u1fd6\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03b1\u1fd6\u03c2, Acts i. 14, ii. 42,\nvi. 4, Rom. xii. 12. The construction is with a simple dative both in\nthe New Testament (ll. cc.) and in classical writers, except where it\nstands absolutely (Acts ii. 46, Rom. xiii. 6). The injunction here\ncorresponds to the \u1f00\u03b4\u03b9\u03b1\u03bb\u03ad\u03b9\u03c0\u03c4\u03c9\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03cd\u03c7\u03b5\u03c3\u03b8\u03b5 of 1 Thess. v. 17.\n\n\u03b3\u03c1\u03b7\u03b3\u03bf\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2] Long continuance in prayer is apt to produce listlessness.\nHence the additional charge that the heart must be _awake_, if the\nprayer is to have any value. The word is not to be taken literally here,\nbut metaphorically. In Matt. xxvi. 41 etc., \u03b3\u03c1\u03b7\u03b3\u03bf\u03c1\u03b5\u1fd6\u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c3\u03b5\u03cd\u03c7\u03b5\u03c3\u03b8\u03b5,\nthe idea is not quite the same.\n\n\u1f10\u03bd \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03af\u1fb3] as the crown of all prayer; see the notes on i. 12, ii.\n7.\n\n3. \u1f21\u03bc\u1ff6\u03bd] '_us_', 'the Apostles and preachers of the Gospel', with\nreference more especially to Timothy (i. 1) and Epaphras (iv. 12, 13).\nWhere the Apostle speaks of himself alone, he uses the singular (ver. 3,\n4 \u03b4\u03ad\u03b4\u03b5\u03bc\u03b1\u03b9, \u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03c3\u03c9). Indeed there is no reason to think that St Paul\never uses an 'epistolary' plural, referring to himself solely: see the\nnote on 1 Thess. iii. 1.\n\n\u1f35\u03bd\u03b1 \u03ba.\u03c4.\u03bb.] On the sense of \u1f35\u03bd\u03b1 after \u03c0\u03c1\u03bf\u03c3\u03b5\u03cd\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9 etc., see the note on\ni. 9.\n\n\u03b8\u03cd\u03c1\u03b1\u03bd \u03c4\u03bf\u1fe6 \u03bb\u03cc\u03b3\u03bf\u03c5] '_a door_ of admission _for the word_', i.e. 'an\nopportunity of preaching the Gospel', as 1 Cor. xvi. 9 \u03b8\u03cd\u03c1\u03b1 \u03b3\u03ac\u03c1 \u03bc\u03bf\u03b9\n\u1f00\u03bd\u03ad\u1ff3\u03b3\u03b5\u03bd \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7 \u03ba\u03b1\u1f76 \u1f10\u03bd\u03b5\u03c1\u03b3\u03ae\u03c2, 2 Cor. ii. 12 \u03b8\u03cd\u03c1\u03b1\u03c2 \u03bc\u03bf\u03b9 \u1f00\u03bd\u03b5\u1ff3\u03b3\u03bc\u03ad\u03bd\u03b7\u03c2 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3:\ncomp. Plut. _Mor._ p. 674 D \u1f65\u03c3\u03c0\u03b5\u03c1 \u03c0\u03cd\u03bb\u03b7\u03c2 \u1f00\u03bd\u03bf\u03b9\u03c7\u03b8\u03ad\u03b9\u03c3\u03b7\u03c2, \u03bf\u1f50\u03ba \u1f00\u03bd\u03c4\u03ad\u03c3\u03c7\u03bf\u03bd ...\n\u03c3\u03c5\u03bd\u03b5\u03b9\u03c3\u03b9\u03bf\u1fe6\u03c3\u03b9 \u03c0\u03b1\u03bd\u03c4\u03bf\u03b4\u03b1\u03c0\u03bf\u1fd6\u03c2 \u1f00\u03ba\u03c1\u03cc\u03b1\u03bc\u03b1\u03c3\u03b9\u03bd. Similarly )\u03ad\u03b9\u03c3\u03bf\u03b4\u03bf\u03c2 is used in 1\nThess. i. 9, ii. 1. The converse application of the metaphor appears in\nActs xiv. 27 \u1f24\u03bd\u03bf\u03b9\u03be\u03b5\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03b8\u03bd\u03b5\u03c3\u03b9\u03bd \u03b8\u03cd\u03c1\u03b1\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2, where the door is\nopened not to the teachers, but to the recipients of the Gospel.\nAccording to another interpretation (suggested by Ephes. vi. 19 \u1f35\u03bd\u03b1 \u03bc\u03bf\u03b9\n\u03b4\u03bf\u03b8\u1fc7 \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f10\u03bd \u1f00\u03bd\u03bf\u03af\u03b6\u03b5\u03b9 \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u03bc\u03bf\u03c5) it is explained 'the door of our\nspeech', i.e. 'our mouth': comp. Ps. cxli (cxl). 3, Mic. vii. 5, Ecclus.\nxxviii. 25. But the parallel passages do not favour this sense, nor will\nthe words themselves admit it. In that case for \u1f21\u03bc\u1fd6\u03bd \u03b8\u03cd\u03c1\u03b1\u03bd \u03c4\u03bf\u1fe6 \u03bb\u03cc\u03b3\u03bf\u03c5 we\nshould require \u03c4\u1f74\u03bd \u03b8\u03cd\u03c1\u03b1\u03bd \u03c4\u1ff6\u03bd \u03bb\u03cc\u03b3\u03c9\u03c5 [\u1f21\u03bc\u1ff6\u03bd]. 'The word' here is 'the\nGospel', as frequently.\n\n\u03bb\u03b1\u03bb\u1fc6\u03c3\u03b1\u03b9 '_so as to speak_', the infinitive of the consequence, like\n\u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9 ver. 6 ; see Winer \u00a7 xliv, p. 400.\n\n\u03c4\u1f78 \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd \u03ba.\u03c4.\u03bb. i.e. the doctrine of the free admission of the\nGentiles. For the leading idea which St Paul in these epistles attaches\nto 'the mystery' of the Gospel, see the note on i. 26.\n\n\u03b4\u03b9' \u1f45 St Paul might have been still at large, if he had been content to\npreach a Judaic Gospel. It was because he contended for Gentile liberty,\nand thus offended Jewish prejudices, that he found himself a prisoner.\nSee Acts xxi. 28, xxii. 21, 22, xxiv. 5, 6, xxv. 6, 8. The other\nreading, \u03b4\u03b9' \u1f45\u03bd, destroys the point of the sentence.\n\n\u03ba\u03b1\u1f76 \u03b4\u03ad\u03b4\u03b5\u03bc\u03b1\u03b9 2 Tim. ii. 9 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03b4\u03b5\u03c3\u03bc\u1ff6\u03bd, Philem. 9 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2.\n\n4. \u1f35\u03bd\u03b1 \u03c6\u03b1\u03bd\u03b5\u03c1\u03ce\u03c3\u03c9 \u03ba.\u03c4.\u03bb. This is best taken as dependent on the previous\nclause \u1f35\u03bd\u03b1 \u1f41 \u0398\u03b5\u03cc\u03c2 ... \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6. For instances of a double \u1f35\u03bd\u03b1, where\nthe second is not coordinated with, but subordinated to, the first, see\nthe note on Gal. iii. 14. The immediate purport of the Colossians'\nprayers must be that the Apostle should have all opportunities of\npreaching the Gospel: the ulterior object, that he should use these\nopportunities boldly.\n\n------------------------------------------------------------------------\n\nIV. 5, 6]\n\n\u03b1\u1f50\u03c4\u03cc, \u1f61\u03c2 \u03b4\u03b5\u1fd6 \u03bc\u03b5 \u03bb\u03b1\u03bb\u1fc6\u03c3\u03b1\u03b9. ^5\u1f10\u03bd \u03c3\u03bf\u03c6\u03af\u1fb3 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5 \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f14\u03be\u03c9, \u03c4\u1f78\u03bd\n\u03ba\u03b1\u03b9\u03c1\u1f78\u03bd \u1f10\u03be\u03b1\u03b3\u03bf\u03c1\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9\u00b7 ^6\u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2\n\n\n5. \u1f10\u03bd \u03c3\u03bf\u03c6\u03af\u1fb3] Matt. x. 16 \u03b3\u03af\u03bd\u03b5\u03c3\u03b8\u03b5 \u03bf\u1f56\u03bd \u03c6\u03c1\u03cc\u03bd\u03b9\u03bc\u03bf\u03b9 \u1f61\u03c2 \u03bf\u1f31 \u1f44\u03c6\u03b5\u03b9\u03c2.\n\n\u03c4\u03bf\u1f7a\u03c2 \u1f14\u03be\u03c9] '_those without the pale_' of the Church, the unbelievers; as\nin 1 Cor. v. 12, 13, 1 Thess. iv. 12. So \u03bf\u1f31 \u1f14\u03be\u03c9\u03b8\u03b5\u03bd, 1 Tim. iii. 7. The\nbelievers on the other hand are \u03bf\u1f31 \u1f14\u03c3\u03c9, 1 Cor. v. 12. This mode of\nspeaking was derived from the Jews, who called the heathen \u05d4\u05d7\u05d9\u05e6\u05d5\u05e0\u05d9\u05dd\n(Sch\u00f6ttgen on 1 Cor. _l.c._), translated \u03bf\u1f31 \u1f10\u03ba\u03c4\u03cc\u03c2 Ecclus. Prol. and \u03bf\u1f31\n\u1f14\u03be\u03c9\u03b8\u03b5\u03bd Joseph. _Ant._ xv. 9. 2.\n\n\u1f10\u03be\u03b1\u03b3\u03bf\u03c1\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03ba.\u03c4.\u03bb. '_buying up the opportunity for yourselves_,\nletting no opportunity slip you, of saying and doing what may further\nthe cause of God': comp. Ephes. v. 16. The expression occurs also in\nDan. ii. 8 \u03bf\u1f36\u03b4\u03b1 \u1f45\u03c4\u03b9 \u03ba\u03b1\u03b9\u03c1\u1f78\u03bd \u1f51\u03bc\u03b5\u1fd6\u03c2 \u1f10\u03be\u03b1\u03b3\u03bf\u03c1\u03ac\u03b6\u03b5\u03c4\u03b5, i.e. 'are eager to gain\ntime'. Somewhat similar are the phrases \u03c4\u1f78\u03bd \u03c7\u03c1\u03cc\u03bd\u03bf\u03bd \u03ba\u03b5\u03c1\u03b4\u03b1\u03af\u03bd\u03b5\u03b9\u03bd, \u03c4\u1f78 \u03c0\u03b1\u03c1\u1f78\u03bd\n\u03ba\u03b5\u03c1\u03b4\u03b1\u03af\u03bd\u03b5\u03b9\u03bd. In much the same sense Ignatius says, _Polyc._ 3 \u03c4\u03bf\u1f7a\u03c2\n\u03ba\u03b1\u03b9\u03c1\u1f78\u03c5\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bc\u03ac\u03bd\u03b8\u03b1\u03bd\u03b5. For this sense of \u1f10\u03be\u03b1\u03b3\u03bf\u03c1\u03ac\u03b6\u03c9 'coemo' (closely allied\nin meaning to \u03c3\u03c5\u03bd\u03b1\u03b3\u03bf\u03c1\u03ac\u03b6\u03c9), see Polyb. iii. 42. 2 \u1f10\u03be\u03b7\u03b3\u03cc\u03c1\u03b1\u03c3\u03b5 \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c4\u03ac\n\u03c4\u03b5 \u03bc\u03bf\u03bd\u03cc\u03be\u03c5\u03bb\u03b1 \u03c0\u03bb\u03bf\u1fd6\u03b1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba.\u03c4.\u03bb., Plut. _Vit. Crass._ 2. More commonly the\nword signifies 'to redeem' (see the note on Gal. iii. 13), and some\nwould assign this sense to it here; but no appropriate meaning is thus\nobtained. In Mart. _Polyc._ 2 \u03b4\u03b9\u1f70 \u03bc\u03b9\u1fb6\u03c2 \u1f65\u03c1\u03b1\u03c2 \u03c4\u1f74\u03bd \u03b1\u1f34\u03c9\u03bd\u03b9\u03bf\u03bd \u03ba\u03cc\u03bb\u03b1\u03c3\u03b9\u03bd\n\u1f10\u03be\u03b1\u03b3\u03bf\u03c1\u03b1\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 it means 'buying off', a sense in which \u1f10\u03be\u03c9\u03bd\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 occurs\nseveral times. The reason for the injunction is added in Ephes. v. 16,\n\u1f45\u03c4\u03b9 \u03b1\u1f31 \u1f21\u03bc\u03ad\u03c1\u03b1\u03b9 \u03c0\u03bf\u03bd\u03b7\u03c1\u03b1\u03af \u03b5\u1f30\u03c3\u03b9\u03bd: the prevailing evil of the times makes the\nopportunities for good more precious.\n\n6. \u1f10\u03bd \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9] '_with grace, favour_', i.e. 'acceptableness',\n'pleasingness'; comp. Eccles. x. 12 \u03bb\u03cc\u03b3\u03bf\u03b9 \u03c3\u03c4\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c3\u03bf\u03c6\u03bf\u1fe6 \u03c7\u03ac\u03c1\u03b9\u03c2, Ps. xliv\n(xlv). 3 \u1f10\u03be\u03b5\u03c7\u03cd\u03b8\u03b7 \u03c7\u03ac\u03c1\u03b9\u03c2 \u1f10\u03bd \u03c7\u03ad\u03b9\u03bb\u03b5\u03c3\u03af \u03c3\u03bf\u03c5, Eccles. xxi. 16 \u1f10\u03c0\u1f76 \u03c7\u03ad\u03b9\u03bb\u03bf\u03c5\u03c2\n\u03c3\u03c5\u03bd\u03b5\u03c4\u03bf\u1fe6 \u1f11\u03c5\u03c1\u03b5\u03b8\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9 \u03c7\u03ac\u03c1\u03b9\u03c2. In classical writers \u03c7\u03ac\u03c1\u03b9\u03c2 \u03bb\u03cc\u03b3\u03c9\u03bd is a still\nmore common connexion; e.g. Demosth. _c. Phil._ i. 38, Dionys. Hal. _de\nLys._ \u00a7\u00a7 10, 11, Plut. _Vit. Mar._ 44.\n\n------------------------------------------------------------------------\n\nIV. 7]\n\n\u1f51\u03bc\u1ff6\u03bd \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u1f10\u03bd \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9, \u1f05\u03bb\u03b1\u03c4\u03b9 \u1f20\u03c1\u03c4\u03c5\u03bc\u03ad\u03bd\u03bf\u03c2, \u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9, \u1f51\u03bc\u1fb6\u03c2 \u03c0\u1ff6\u03c2 \u03b4\u03b5\u1fd6 \u1f11\u03bd\u1f76\n\u1f11\u03ba\u03ac\u03c3\u03c4\u1ff3 \u1f00\u03c0\u03bf\u03ba\u03c1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9.\n\n^7\u03a4\u1f70 \u03ba\u03b1\u03c4 \u1f10\u03bc\u1f72 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03b5\u03b9 \u1f51\u03bc\u1fd6\u03bd \u03a4\u03cd\u03c7\u03b9\u03ba\u03bf\u03c2 \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1f78\u03c2\n\n\n\u1f05\u03bb\u03b1\u03c4\u03b9] comp. Mark ix. 50 \u1f10\u1f70\u03bd \u03b4\u1f72 \u03c4\u1f78 \u1f05\u03bb\u03b1\u03c2 \u1f04\u03bd\u03b1\u03bb\u03bf\u03bd \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9, \u1f10\u03bd \u03c4\u03af\u03bd\u03b9 \u03b1\u1f50\u03c4\u1f78\n_)\u03b1\u03c1\u03c4\u03c5\u03c3\u03b5\u03c4\u03b5_\u00b7 \u1f14\u03c7\u03b5\u03c4\u03b5 \u1f10\u03bd \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 \u1f05\u03bb\u03b1. The salt has a twofold purpose.\n(1) It gives a flavour to the discourse and recommends it to the palate:\ncomp. Job vi. 6 \u03b5\u1f30 \u03b2\u03c1\u03c9\u03b8\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9 \u1f04\u03c1\u03c4\u03bf\u03c2 \u1f04\u03bd\u03b5\u03c5 \u1f01\u03bb\u03cc\u03c2\u00b7 \u03b5\u1f30 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u1f14\u03c3\u03c4\u03b9 \u03b3\u03b5\u1fe6\u03bc\u03b1 \u1f10\u03bd\n\u1fe5\u03ae\u03bc\u03b1\u03c3\u03b9 \u03ba\u03b5\u03bd\u03bf\u1fd6\u03c2; in which passage the first clause was rendered by\nSymmachus \u03bc\u03ae\u03c4\u03b9 \u03b2\u03c1\u03c9\u03b8\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9 _\u1f00\u03bd\u03ac\u03c1\u03c4\u03c5\u03c4\u03bf\u03bd_ \u03c4\u1ff7 \u03bc\u1f74 \u1f14\u03c7\u03b5\u03b9\u03bd \u1f05\u03bb\u03b1; This is the\nprimary idea of the metaphor here, as the word \u1f20\u03c1\u03c4\u03c5\u03bc\u03ad\u03bd\u03bf\u03c2 seems to show.\n(2) It preserves from corruption and renders wholesome; Ign. _Magn._ 10\n\u1f01\u03bb\u03af\u03c3\u03b8\u03b7\u03c4\u03b5 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u1f35\u03bd\u03b1 \u03bc\u1f74 \u03b4\u03b9\u03b1\u03c6\u03b8\u03b1\u03c1\u1fc7 \u03c4\u03b9\u03c2 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, \u1f10\u03c0\u1f72\u03b9 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u1f40\u03c3\u03bc\u1fc6\u03c2\n\u1f10\u03bb\u03b5\u03b3\u03c7\u03b8\u03ae\u03c3\u03b5\u03c3\u03b8\u03b5. Hence the Pythagorean saying, Diog. Laert. viii. I. 35 \u03bf\u1f31\n\u1f05\u03bb\u03b5\u03c2 \u03c0\u1fb6\u03bd \u03c3\u03ce\u03b6\u03bf\u03c5\u03c3\u03b9\u03bd \u1f45 \u03c4\u03b9 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03b1\u03bb\u03ac\u03b2\u03c9\u03c3\u03b9. It may be inferred that this\nsecondary application of the metaphor was present to the Apostle's mind\nhere, because in the parallel epistle, Ephes. iv. 29, he says \u03c0\u1fb6\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c2\n_\u03c3\u03b1\u03c0\u03c1\u1f78\u03c2_ \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c3\u03c4\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03bc\u1f74 \u1f10\u03ba\u03c0\u03bf\u03c1\u03b5\u03cd\u03b5\u03c3\u03b8\u03c9 \u03ba.\u03c4.\u03bb. In the first\napplication the opposite to \u1f05\u03bb\u03b1\u03c4\u03b9 \u1f20\u03c1\u03c4\u03c5\u03bc\u03ad\u03bd\u03bf\u03c2 would be \u03bc\u03c9\u03c1\u03cc\u03c2 '_insipid_'\n(Luke xiv. 34); in the second, \u03c3\u03b1\u03c0\u03c1\u03cc\u03c2 '_corrupt_'.\n\nHeathen writers also insisted that discourse should be 'seasoned with\nsalt'; e.g. Cic. _de Orat._ i. 34 'facetiarum quidam lepos quo, tanquam\nsale, perspergatur omnis oratio'. They likewise dwelt on the connexion\nbetween \u03c7\u03ac\u03c1\u03b9\u03c2 and \u1f05\u03bb\u03b5\u03c2; e.g. Plut. _Mor._ p. 514 F \u03c7\u03ac\u03c1\u03b9\u03bd \u03c4\u03b9\u03bd\u1f70\n\u03c0\u03b1\u03c1\u03b1\u03c3\u03ba\u03b5\u03c5\u03ac\u03b6\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f00\u03bb\u03bb\u03ae\u03bb\u03bf\u03b9\u03c2, \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f01\u03bb\u03c3\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u03bb\u03cc\u03b3\u03bf\u03b9\u03c2 \u1f10\u03c6\u03b7\u03b4\u03cd\u03bd\u03bf\u03c5\u03c3\u03b9 \u03c4\u1f74\u03bd\n\u03b4\u03b9\u03b1\u03c4\u03c1\u03b9\u03b2\u03ae\u03bd, p. 697 D (comp. p. 685 A) \u03bf\u1f31 \u03c0\u03bf\u03bb\u03bb\u03bf\u1f76 \u03c7\u03ac\u03c1\u03b9\u03c4\u03b1\u03c2 \u03ba\u03b1\u03bb\u03bf\u1fe6\u03c3\u03b9\u03bd [\u03c4\u1f78\u03bd\n\u1f05\u03bb\u03b1], \u1f45\u03c4\u03b9 \u1f10\u03c0\u1f76 \u03c4\u1f70 \u03c0\u03bb\u03b5\u1fd6\u03c3\u03c4\u03b1 \u03bc\u03b9\u03b3\u03bd\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b5\u1f50\u03ac\u03c1\u03bc\u03bf\u03c3\u03c4\u03b1 \u03c4\u1fc7 \u03b3\u03b5\u03cd\u03c3\u03b5\u03b9 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c3\u03c6\u03b9\u03bb\u1fc6\n\u03c0\u03bf\u03b9\u03b5\u1fd6 \u03ba\u03b1\u1f76 \u03ba\u03b5\u03c7\u03b1\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03b1, p. 669 A \u1f21 \u03b4\u1f72 \u03c4\u1ff6\u03bd \u1f01\u03bb\u1ff6\u03bd \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 ... \u03c7\u03ac\u03c1\u03b9\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03ba\u03b1\u1f76\n\u1f21\u03b4\u03bf\u03bd\u1f74\u03bd \u03c0\u03c1\u03bf\u03c3\u03c4\u03af\u03b8\u03b7\u03c3\u03b9, Dion. Chrys. _Or._ xviii. \u00a7 13. Their notion of\n'salt' however was wit, and generally the kind of wit which degenerated\ninto the \u03b5\u1f50\u03c4\u03c1\u03b1\u03c0\u03b5\u03bb\u03af\u03b1 denounced by St Paul in Ephes. v. 4 (see the note\nthere).\n\nThe form \u1f05\u03bb\u03b1\u03c2 is common in the LXX and Greek Testament. Otherwise it is\nrare: see Buttmann _Gramm._ I. p. 220, and comp. Plut. _Mor._ 668 F.\n\n\u03b5\u1f30\u03b4\u03ad\u03bd\u03b1\u03b9] '_so as to know_'; see the note on \u03bb\u03b1\u03bb\u1fc6\u03c3\u03b1\u03b9 ver. 3.\n\n\u1f11\u03bd\u1f76 \u1f11\u03ba\u03ac\u03c3\u03c4\u1ff3] 'Not only must your conversation be opportune as regards the\ntime; it must also be appropriate as regards the person'. The Apostle's\nprecept was enforced by his own example, for he made it a rule to become\n\u03c4\u03bf\u1fd6\u03c2 \u03c0\u1fb6\u03c3\u03b9\u03bd \u03c0\u03ac\u03bd\u03c4\u03b1, \u1f35\u03bd\u03b1 \u03c0\u03ac\u03bd\u03c4\u03c9\u03c2 \u03c4\u03b9\u03bd\u1f70\u03c2 \u03c3\u03ce\u03c3\u1fc3 (1 Cor. ix. 22).\n\n7\u20139. 'You will learn everything about me from Tychicus, the beloved\nbrother who has ministered to me and served with me faithfully in the\nLord. This indeed was my purpose in sending him to you: that you might\nbe informed how matters stand with me, and that he might cheer your\nhearts and strengthen your resolves by the tidings. Onesimus will\naccompany him\u2014a faithful and beloved brother, who is one of yourselves,\na Colossian. These two will inform you of all that is going on here.'\n\n7. \u03a4\u1f70 \u03ba\u03b1\u03c4' \u1f10\u03bc\u1f72 \u03c0\u03ac\u03bd\u03c4\u03b1] '_all that relates to me_'; see the note on Phil.\ni. 12, and comp. Bion in Diog. Laert. iv. 47. So Acts xxv. 14 \u03c4\u1f70 \u03ba\u03b1\u03c4\u1f70\n\u03c4\u1f78\u03bd \u03a0\u03b1\u1fe6\u03bb\u03bf\u03bd.\n\n\u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03b5\u03b9] On this word see the note Phil. i. 22.\n\n\u03a4\u03cd\u03c7\u03b9\u03ba\u03bf\u03c2] Tychicus was charged by St Paul at this same time with a more\nextended mission. He was entrusted with copies of the circular letter,\nwhich he was enjoined to deliver in the principal churches of\nproconsular Asia (see above, p. 37, and the introduction to the Epistle\nto the Ephesians). This mission would bring him to Laodicea, which was\none of these great centres of Christianity (see p. 8); and, as Coloss\u00e6\nwas only a few miles distant, the Apostle would naturally engage him to\npay a visit to the Colossians. At the same time the presence of an\nauthorised delegate of St Paul, as Tychicus was known to be, would serve\nto recommend Onesimus, who owing to his former conduct stood in every\nneed of such a recommendation. The two names \u03a4\u03cd\u03c7\u03b9\u03ba\u03bf\u03c2 and \u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03c2 occur\nin proximity in Phrygian inscriptions found at Altentash (Bennisoa?)\nBoeckh 3857r sqq. appx.\n\nTychicus was a native of proconsular Asia (Acts xx. 4) and perhaps of\nEphesus (2 Tim. iv. 12: see _Philippians_ p. 11). He is found with St\nPaul at three different epochs in his life. (1) He accompanied him when\non his way eastward at the close of the third missionary journey A.D. 58\n(Acts xx. 4), and probably like Trophimus (Acts xxi. 29) went with him\nto Jerusalem (for the words \u1f04\u03c7\u03c1\u03b9 \u03c4\u1fc6\u03c2 \u1f08\u03c3\u03af\u03b1\u03c2 must be struck out in Acts\nxx. 4). It is probable indeed that Tychicus, together with others\nmentioned among the Apostle's numerous retinue on this occasion, was a\ndelegate appointed by his own church according to the Apostle's\ninjunctions (1 Cor. xvi. 3, 4) to bear the contributions of his brethren\nto the poor Christians of Jud\u00e6a; and if so, he may possibly be the\nperson commended as the brother \u03bf\u1f57 \u1f41 \u1f14\u03c0\u03b1\u03b9\u03bd\u03bf\u03c2 \u1f10\u03bd \u03c4\u1ff7 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u1ff3 \u03b4\u03b9\u1f70 \u03c0\u03b1\u03c3\u1ff6\u03bd\n\u03c4\u1ff6\u03bd \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03b9\u1ff6\u03bd (2 Cor. viii. 18): but this will depend on the\ninterpretation of the best supported reading in Acts xx. 5 \u03bf\u1f57\u03c4\u03bf\u03b9 \u03b4\u1f72\n\u03c0\u03c1\u03bf\u03c3\u03b5\u03bb\u03b8\u03cc\u03bd\u03c4\u03b5\u03c2 \u1f14\u03bc\u03b5\u03bd\u03bf\u03bd \u1f21\u03bc\u1fb6\u03c2 \u1f10\u03bd \u03a4\u03c1\u03c9\u03ac\u03b4\u03b9. (2) We find Tychicus again in St\nPaul's company at the time with which we are immediately concerned, when\nthis epistle was written, probably towards the end of the first Roman\ncaptivity A.D. 62, 63 (see _Philippians_ p. 31 sq.). (3) Once more, at\nthe close of St Paul's life (about A.D. 67), he appears again to have\nassociated himself with the Apostle, when his name is mentioned in\nconnexion with a mission to Crete (Tit. iii. 12) and another to Ephesus\n(2 Tim. iv. 12). For the legends respecting him, which are slight and\ninsignificant, see _Act. Sanct. Boll._ April. 29 (III. p. 619).\n\nTychicus is not so common a name as some others which occur in the New\nTestament, e.g. Onesimus, Trophimus; but it is found occasionally in\ninscriptions belonging to Asia Minor, e.g. Boeckh _C. I._ 2918, 3665,\n[3857 c], 3857 r, (comp. 3865 i, etc.); and persons bearing it are\ncommemorated on the coins of both Magnesia ad M\u00e6andrum (Mionnet III. p.\n153 sq., _Suppl._ VI. p. 236) and Magnesia ad Sipylum (_ib._ IV. p. 70).\nThe name occurs also in Roman inscriptions; e.g. Muratori, pp.\nDCCCCXVII, MCCCXCIV, MMLV. Along with several other proper names\nsimilarly formed, this word is commonly accentuated \u03a4\u03c5\u03c7\u03b9\u03ba\u03cc\u03c2 (Chandler\n_Greek Accentuation_ \u00a7 255), and so it stands in all the critical\nSeditions, though according to rule (Winer \u00a7 vi. p. 58) it should be\n\u03a4\u03cd\u03c7\u03b9\u03ba\u03bf\u03c2.\n\n------------------------------------------------------------------------\n\nIV. 8]\n\n\u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3\u00b7 ^8\u1f43\u03bd \u1f14\u03c0\u03b5\u03bc\u03c8\u03b1 \u03c0\u03c1\u1f78\u03c2\n\u1f51\u03bc\u1fb6\u03c2 \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78 \u03c4\u03bf\u1fe6\u03c4\u03bf, \u1f35\u03bd\u03b1\n\n\n\u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u03ba.\u03c4.\u03bb.] The connexion of the words is not quite obvious. It\nseems best however to take \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 as referring to the whole clause\n\u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2 rather than to \u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2 alone: for (1)\nThe two substantives are thus bound together by the preceding \u03c0\u03b9\u03c3\u03c4\u03cc\u03c2 and\nthe following \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 in a natural way: (2) The attachment of \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3\nto \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2 is suggested by the parallel passage Ephes. vi. 21\n\u03a4\u03cd\u03c7\u03b9\u03ba\u03bf\u03c2 \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1f78\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c3\u03c4\u1f78\u03c2 \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3. The question of\nconnecting \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 with \u1f00\u03b4\u03b5\u03bb\u03c0\u1f45\u03c2 as well need not be entertained, since\nthe idea of \u1f00\u03b4\u03b5\u03bb\u03c0\u1f45\u03c2, 'a Christian brother', is complete in itself: see\nthe note on Phil. i. 14. The adjective \u03c0\u03b9\u03c3\u03c4\u03cc\u03c2 will here have its passive\nsense, 'trustworthy, stedfast', as also in ver. 9: see _Galatians_ p.\n154 sq.\n\n\u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2] '_minister_', but to whom? To the churches, or to St Paul\nhimself? The following \u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2 suggests the latter as the prominent\nidea here. So in Acts xix. 22 Timothy and Erastus are described as \u03b4\u03cd\u03bf\n\u03c4\u1ff6\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03bf\u03cd\u03bd\u03c4\u03c9\u03bd \u03b1\u1f50\u03c4\u1ff7. Tychicus himself also was one of several who\nministered to St Paul about that same time (Acts xx. 4). It is not\nprobable however, that \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03c2 has here its strict official sense, 'a\ndeacon', as in Rom. xvi. 1, Phil. i. 1, 1 Tim. iii. 8, 12.\n\n\u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2] The word does not occur elsewhere in St Paul, except in i. 7,\nwhere it is said of Epaphras. It is probably owing to the fact of St\nPaul's applying the term in both these passages to persons whom he calls\n\u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03b9, that \u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2 seems to have been adopted as a customary form\nof address in the early Church on the part of a bishop, when speaking of\na deacon. In Ignatian letters for instance, the term is never used\nexcept of deacons; _Ephes._ 2, _Magn._ 2, _Philad._ 4, _Smyrn._ 12.\nWhere the martyr has occasion to speak of a bishop or a presbyter some\nother designation is used instead.\n\n8. \u1f14\u03c0\u03b5\u03bc\u03c8\u03b1] '_I send_,' or '_I have sent_,' \u1f14\u03c0\u03b5\u03bc\u03c8\u03b1 being the epistolary\naorist; see the note on \u1f14\u03b3\u03c1\u03b1\u03c8\u03b1, Gal. vi. 11. Tychicus appears to have\naccompanied the letter itself. For similar instances of the epistolary\n\u1f14\u03c0\u03b5\u03bc\u03c8\u03b1, \u1f10\u03c0\u03ad\u03c3\u03c4\u03b5\u03b9\u03bb\u03b1, etc., see 2 Cor. viii. 18, 22, ix. 3, Ephes. vi. 22,\nPhil. ii. 25, 28, Philem. 11, Hebr. xiii. 22, Polyc. _Phil._ 13.\n\n------------------------------------------------------------------------\n\nIV. 9]\n\n\u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03ad\u03c3\u1fc3 \u03c4\u1f70\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2 \u1f51\u03bc\u1ff6\u03bd, ^9\u03c3\u1f7a\u03bd \u1f48\u03bd\u03b7\u03c3\u03af\u03bc\u1ff3 \u03c4\u1ff7\n\u03c0\u03b9\u03c3\u03c4\u1ff7 \u03ba\u03b1\u1f76 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1ff7 \u1f00\u03b4\u03b5\u03bb\u03c6\u1ff7, \u1f45\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f10\u03be \u1f51\u03bc\u1ff6\u03bd. \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f51\u03bc\u1fd6\u03bd \u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd \u03c4\u1f70\n\u1f67\u03b4\u03b5.\n\n\n\u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd] This must be preferred to the received reading, \u03b3\u03bd\u1ff7\n\u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd, for two independent reasons. (1) The preponderance of\nancient authority is decidedly in its favour. (2) The emphatic \u03b5\u1f30\u03c2 \u03b1\u1f50\u03c4\u1f78\n\u03c4\u03bf\u1fe6\u03c4\u03bf \u1f35\u03bd\u03b1 seems imperatively to demand it. St Paul in the context twice\nstates the object of Tychicus' visit to be that the Colossians might be\ninformed about the Apostle's own doings, \u03c4\u1f70 \u03ba\u03b1\u03c4' \u1f10\u03bc\u1f72 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03b5\u03b9 \u1f51\u03bc\u1fd6\u03bd\n(ver. 7), and \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f51\u03bc\u1fd6\u03bd \u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd \u03c4\u1f70 \u1f67\u03b4\u03b5. He could hardly therefore\nhave described 'the very purpose' of his mission in the same breath as\nsomething quite different.\n\nIt is urged indeed, that this is a scribe's alteration to bring the\npassage into accordance with Ephes. vi. 21. But against this it may\nfairly be argued that, on any hypothesis as regards the authorship and\nrelation of the two letters, this strange variation from \u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76\n\u1f21\u03bc\u1ff6\u03bd to \u03b3\u03bd\u1ff7 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd in the author himself is improbable. On the\nother hand a transcriber was under a great temptation to substitute \u03b3\u03bd\u1ff7\nfor \u03b3\u03bd\u1ff6\u03c4\u03b5 owing to the following \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03ad\u03c3\u03b7, and this temptation would\nbecome almost irresistible, if by any chance \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd had been written\nfor \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd in the copy before him, as we find to be the case in some\nMSS. See the detached note on various readings.\n\n\u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03ad\u03c3\u1fc3 \u03ba.\u03c4.\u03bb.] i.e. 'encourage you to persevere by his tidings and\nexhortations'. The phrase occurs again, Ephes. vi. 22, 2 Thess, ii. 17:\nsee above ii. 2. The prominent idea in all these passages is not comfort\nor consolation but perseverance in the right way.\n\n9. \u03c3\u1f7a\u03bd \u1f48\u03bd\u03b7\u03c3\u03af\u03bc\u1ff3] See above, p. 33, and the introduction to the Epistle to\nPhilemon.\n\n\u03c4\u1ff7 \u03c0\u03b9\u03c3\u03c4\u1ff7 \u03ba.\u03c4.\u03bb.] The man whom the Colossians had only known hitherto, if\nthey knew him at all, as a worthless runaway slave, is thus commended to\nthem as no more a slave but a brother, no more dishonest and faithless\nbut trustworthy, no more an object of contempt but of love; comp.\nPhilem. 11, 16.\n\n\u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd] This form has rather better support from the MSS than\n\u03b3\u03bd\u03c9\u03c1\u03b9\u03bf\u1fe6\u03c3\u03b9\u03bd: see also above, iii. 25. On the Attic future from verbs in\n-\u03b9\u03b6\u03c9 in the Greek Testament generally see Winer \u00a7 xiii. p. 88, A.\nButtmann p. 32 sq. Is there any decisive instance of these Attic forms\nin St Paul, except in quotations from the LXX (e.g. Rom. x. 19, xv. 12)?\n\n10\u201314. 'I send you greeting from Aristarchus who is a fellow-prisoner\nwith me; from Marcus, Barnabas' cousin, concerning whom I have already\nsent you directions, that you welcome him heartily, if he pays you a\nvisit; and from Jesus, surnamed Justus; all three Hebrew converts. They\nalone of their fellow-countrymen have worked loyally with me in\nspreading the kingdom of God; and their stedfastness has indeed been a\ncomfort to me in the hour of trial. Greeting also from Epaphras, your\nfellow-townsman, a true servant of Christ, who is ever wrestling in his\nprayers on your behalf, that ye may stand firm in the faith, perfectly\ninstructed and fully convinced in every will and purpose of God. I bear\ntestimony to the earnestness with which he labours for you and the\nbrethren of Laodicea and those of Hierapolis. Greeting also from Luke\nthe physician, my very dear friend, and from Demas.'\n\n------------------------------------------------------------------------\n\nIV. 10]\n\n^{10}\u1f08\u03c3\u03c0\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2 \u1f08\u03c1\u03af\u03c3\u03c4\u03b1\u03c1\u03c7\u03bf\u03c2 \u1f41 \u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03cc\u03c2\n\n\n10. The salutations to Philemon are sent from the same persons as to the\nColossians, except that in the former case the name of Jesus Justus is\nomitted.\n\n\u1f08\u03c1\u03af\u03c3\u03c4\u03b1\u03c1\u03c7\u03bf\u03c2] the Thessalonian. He had started with St Paul on his voyage\nfrom Jerusalem to Rome, but probably had parted from the Apostle at Myra\n(see _Philippians_ p. 33 sq.). If so, he must have rejoined him at Rome\nat a later date. On this Aristarchus see _Philippians_ p. 10 and the\nintroduction to the Epistles to the Thessalonians. He would be well\nknown in proconsular Asia, which he had visited from time to time; Acts\nxix. 29, xx. 4, xxvii. 2.\n\n\u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03cc\u03c2 \u03bc\u03bf\u03c5] In Philem. 23 this honourable title is withheld from\nAristarchus and given to Epaphras. In Rom. xvi. 7 St Paul's kinsmen,\nAndronicus and Junias, are so called. On the possibility of its\nreferring to a spiritual captivity or subjection see _Philippians_ p.\n11. In favour of this meaning it may be urged, that, though St Paul as a\nprisoner was truly a \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2, he was not strictly an \u03b1\u1f30\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03bf\u03c2 'a\nprisoner of war'; nor could he have called himself so, except by a\nconfusion of the actual and metaphorical. If on the other hand\n\u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03bf\u03c2 refers to a physical captivity, it cannot easily be\nexplained by any known fact. The incident in Acts xix. 29 is hardly\nadequate. The most probable solution would be, that his relations with\nSt Paul in Rome excited suspicion and led to a temporary confinement.\nAnother possible hypothesis is that he voluntarily shared the Apostle's\ncaptivity by living with him.\n\n\u039c\u03ac\u03c1\u03ba\u03bf\u03c2] doubtless John Mark, who had been associated with St Paul in his\nearlier missionary work; Acts xii. 25, xv. 37 sq. This commendatory\nnotice is especially interesting as being the first mention of him since\nthe separation some twelve years before, Acts xv. 39. In the later years\nof the Apostle's life he entirely effaced the unfavourable impression\nleft by his earlier desertion; 2 Tim. iv. 11 \u1f14\u03c3\u03c4\u03b9\u03bd \u03b3\u03ac\u03c1 \u03bc\u03bf\u03b9 \u03b5\u1f54\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c2 \u03b5\u1f30\u03c2\n\u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd.\n\nThis notice is likewise important in two other respects. (1) Mark\nappears here as commended to a church of proconsular _Asia_, and\nintending to visit those parts. To the churches of this same region he\nsends a salutation in 1 Pet. v. 13; and in this district apparently also\nhe is found some few years later than the present time, 2 Tim. iv. 11.\n(2) Mark is now residing at _Rome_. His connexion with the metropolis\nappears also from 1 Pet. v. 13, if \u0392\u03b1\u03b2\u03c5\u03bb\u1ff6\u03bd there (as seems most\nprobable) be rightly interpreted of Rome; and early tradition speaks of\nhis Gospel as having been written for the Romans (Iren. iii. 1. 1; comp.\nPapias in Euseb. _H.E._ iii. 39).\n\n------------------------------------------------------------------------\n\nIV. 10]\n\n\u03bc\u03bf\u03c5, \u03ba\u03b1\u1f76 \u039c\u03ac\u03c1\u03ba\u03bf\u03c2 \u1f41 \u1f00\u03bd\u03b5\u03c8\u03b9\u1f78\u03c2 \u0392\u03b1\u03c1\u03bd\u03ac\u03b2\u03b1, \u03c0\u03b5\u03c1\u1f76 \u03bf\u1f57 \u1f10\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5\n\n\n\u1f41 \u1f00\u03bd\u03b5\u03c8\u03b9\u1f78\u03c2] '_the cousin_'. The term \u1f00\u03bd\u03b5\u03c8\u03b9\u03bf\u03af is applied to cousins\ngerman, the children whether of two brothers or of two sisters or of a\nbrother and sister, as it is carefully defined in Pollux iii. 28. This\nwriter adds that \u03b1\u1f50\u03c4\u03b1\u03bd\u03ad\u03c8\u03b9\u03bf\u03b9 means neither more nor less than \u1f00\u03bd\u03b5\u03c8\u03b9\u03bf\u03af. As\na synonyme we find \u1f10\u03be\u03ac\u03b4\u03b5\u03bb\u03c6\u03bf\u03c2, which however is condemned as a vulgarism;\nPhryn. p. 306 (ed. Lobeck). Many instances of \u1f00\u03bd\u03b5\u03c8\u03b9\u03bf\u03af are found in\ndifferent authors of various ages (e.g. Herod, vii. 5, 82, ix. 10,\nThucyd. i. 132, Plato _Charm._ 154 B, _Gorg._ 471 B, Andoc. _de Myst._ \u00a7\n47, Is\u00e6us _Hagn. Her._ \u00a7 8 sq., Demosth. _c. Macart._ \u00a7 24, 27, etc.,\nDion. Hal. _A. R._ i. 79, Plut. _Vit. Thes._ 7, _Vit. C\u00e6s._ 1, _Vit.\nBrut._ 13, Lucian _Dial. Mort._ xxix. 1, Hegesipp. in Euseb. _H.E._ iv.\n22), where the relationship is directly defined or already known, and\nthere is no wavering as to the meaning. This sense also it has in the\nLXX, Num. xxxvi. 11. In very late writers however (e.g. Io. Malalas\n_Chron._ xvii. p. 424, Io. Damasc. _adv. Const. Cab._ 12, II. p. 621;\nbut in Theodt. _H.E._ v. 39, which is also quoted by E. A. Sophocles\n_Gr. Lex._ s.v. for this meaning, the text is doubtful) the word comes\nto be used for a nephew, properly \u1f00\u03b4\u03b5\u03bb\u03c6\u03b9\u03b4\u03bf\u1fe6\u03c2; and to this later use the\nrendering of our English versions must be traced. The German\ntranslations also (Luther and the Z\u00fcrich) have 'Neffe'. The earliest of\nthe ancient versions (Latin, Syriac, Egyptian) seem all to translate it\ncorrectly; not so in every case apparently the later. There is no reason\nto suppose that St Paul would or could have used it in any other than\nits proper sense. St Mark's relationship with Barnabas may have been\nthrough his mother Mary, who is mentioned Acts xii. 12. The incidental\nnotice here explains why Barnabas should have taken a more favourable\nview of Mark's defection than St Paul, Acts xv. 37\u201339. The notices in\nthis passage and in 2 Tim. iv. 11 show that Mark had recovered the\nApostle's good opinion. The studious recommendation of St Mark in both\npassages indicates a desire to efface the unfavourable impression of the\npast.\n\nThe name of Mark occurs in five different relations, as (1) The early\ndisciple, John Mark, Acts xii. 12, 25, xv. 39; (2) The later companion\nof St Paul, here and Philem. 24, 2 Tim. iv. 11; (3) The companion and\n'son' of St Peter, 1 Pet. v. 13; (4) The evangelist; (5) The bishop of\nAlexandria. Out of these notices some writers get three or even four\ndistinct persons (see the note of Cotelier on _Apost. Const._ ii. 57).\nEven Tillemont (_Mem. Eccl._ II. p. 89 sq., 503 sq.) assumes two Marks,\nsupposing (1) (2) to refer to one person, and (3) (4) (5) to another.\nHis main reason is that he cannot reconcile the notices of the first\nwith the tradition (Euseb. _H.E._ ii. 15, 16) that St Mark the\nevangelist accompanied St Peter to Rome in A. D. 43, having first\npreached the Gospel in Alexandria (p. 515). To most persons however this\nearly date of St Peter's visit to Rome will appear quite irreconcilable\nwith the notices in the Apostolic writings, and therefore with them\nTillemont's argument will carry no weight. But in fact Eusebius does not\nsay, either that St Mark went _with_ St Peter to Rome, or that he had\npreached in Alexandria _before_ this. The Scriptural notices suggest\nthat the same Mark is intended in all the occurrences of the name, for\nthey are connected together by personal links (Peter, Paul, Barnabas);\nand the earliest forms of tradition likewise identify them.\n\n\u0392\u03b1\u03c1\u03bd\u03ac\u03b2\u03b1] On the affectionate tone of St Paul's language, whenever he\nmentions Barnabas after the collision at Antioch (Gal. ii. 11 sq.) and\nthe separation of missionary spheres (Acts xv. 39), see the note on Gal.\nii. 13. It has been inferred from the reference here, that inasmuch as\nMark has rejoined St Paul, Barnabas must have died before this epistle\nwas written (about A. D. 63); and this has been used as an argument\nagainst the genuineness of the letter bearing his name (Hefele\n_Sendschr. d. Apost. Barnab._ p. 29 sq.); but this argument is somewhat\nprecarious. From 1 Cor. ix. 6 we may infer that he was still living, A.\nD. 57. The notices bearing on the biography of Barnabas are collected\nand discussed by Hefele, p. 1 sq.\n\n\u1f10\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5 \u1f10\u03bd\u03c4\u03bf\u03bb\u03ac\u03c2] These injunctions must have been communicated\npreviously either by letter or by word of mouth: for it cannot be a\nquestion here of an epistolary aorist. The natural inference is, that\nthey were sent by St Paul himself, and not by any one else, e.g. by St\nPeter or St Barnabas, as some have suggested. Thus the notice points to\nearlier communications between the Apostle and Coloss\u00e6.\n\nBut what was their tenour? It seems best to suppose that this is given\nin the next clause \u1f10\u1f70\u03bd \u1f14\u03bb\u03b8\u1fc3 \u03ba.\u03c4.\u03bb. By an abrupt change to the _oratio\nrecta_ the injunction is repeated as it was delivered; comp. Ps. cv\n(civ). 15 \u1f24\u03bb\u03b5\u03b3\u03be\u03b5\u03bd \u1f51\u03c0\u1f72\u03c1 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u1fd6\u03c2\u00b7 \u03bc\u1f74 \u1f05\u03c8\u03b7\u03c3\u03b8\u03b5 \u03ba.\u03c4.\u03bb. After verbs\nsignifying 'to command, charge, etc.', there is a tendency to pass from\nthe oblique to the direct; e.g. Luke v. 14, Acts i. 4, xxiii. 22. The\nreading \u03b4\u03ad\u03be\u03b1\u03c3\u03b8\u03b1\u03b9 gives the right sense, but can hardly be correct. If\nthis construction be not accepted, it is vain to speculate what may have\nbeen the tenour of the injunction.\n\n------------------------------------------------------------------------\n\nIV. 11]\n\n\u1f10\u03bd\u03c4\u03bf\u03bb\u03ac\u03c2, \u1f18\u1f70\u03bd \u1f14\u03bb\u03b8\u1fc3 \u03c0\u03c1\u1f78\u03c2 \u1f51\u03bc\u1fb6\u03c2, \u03b4\u03ad\u03be\u03b1\u03c3\u03b8\u03b5 \u03b1\u1f50\u03c4\u03cc\u03bd, ^{11}\u03ba\u03b1\u1f76 \u1f38\u03b7\u03c3\u03bf\u1fe6\u03c2 \u1f41 \u03bb\u03b5\u03b3\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2\n\u1f38\u03bf\u1fe6\u03c3\u03c4\u03bf\u03c2, \u03bf\u1f31 \u1f44\u03bd\u03c4\u03b5\u03c2 \u1f10\u03ba \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc6\u03c2\u00b7 \u03bf\u1f57\u03c4\u03bf\u03b9 \u03bc\u03cc\u03bd\u03bf\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u03bf\u1f76 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03bd\n\u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, \u03bf\u1f35\u03c4\u03b9\u03bd\u03b5\u03c2\n\n\n11. \u03ba\u03b1\u1f76 \u1f38\u03b7\u03c3\u03bf\u1fe6\u03c2] He is not mentioned elsewhere. Even in the Epistle to\nPhilemon his name is omitted. Probably he was not a man of any\nprominence in the Church, but his personal devotion to the Apostle\nprompted this honourable mention. For the story which makes him bishop\nof Eleutheropolis in Palestine, see Le Quien _Oriens Christ._ in III. p.\n633.\n\n\u1f38\u03bf\u1fe6\u03c3\u03c4\u03bf\u03c2] A common name or surname of Jews and proselytes, denoting\nobedience and devotion to the law. It is applied to two persons in the\nNew Testament, besides this Jesus; (1) Joseph Barsabbas, Acts i. 23; (2)\nA proselyte at Corinth, Acts xviii. 7. It occurs twice in the list of\nearly Jewish Christian bishops of Jerusalem, in Euseb. _H.E._ iii. 35,\niv. 5. It was borne by a Jew of Tiberias who wrote the history of the\nJewish war (Joseph. _Vit._ \u00a7\u00a7 9, 65), and by a son of the historian\nJosephus himself (_ib._ \u00a7 1). It occurs in the rabbinical writings\n(\u05d9\u05d5\u05e1\u05d8\u05d0 or \u05d9\u05d5\u05e1\u05d8\u05d9, Sch\u00f6ttgen on Acts. i. 23, Zunz _Judennamen_ p. 20), and\nin monumental inscriptions from Jewish cemeteries in various places\n(Boeckh _C. I._ no. 9922, 9925; _Revue Arch\u00e9ologique_ 1860, II. p. 348;\nGarrucci _Dissertazioni Archeologiche_ II. p. 182). So also the\ncorresponding female name Justa (Garrucci _l.c._ p. 180). In _Clem.\nHom._ ii. 19, iii. 73, iv. 1, xiii. 7, the Syroph\u0153nician woman of the\nGospels is named \u1f38\u03bf\u1fe6\u03c3\u03c4\u03b1, doubtless because she is represented in this\nJudaizing romance as a proselytess \u03c0\u03c1\u03bf\u03c3\u03ae\u03bb\u03c5\u03c4\u03bf\u03c2 xiii. 7) who strictly\nobserves the Mosaic ordinances ( \u03c4\u1f74\u03bd \u03bd\u03cc\u03bc\u03b9\u03bc\u03bf\u03bd \u1f00\u03bd\u03b1\u03b4\u03b5\u03be\u03b1\u03bc\u03ad\u03bd\u03b7 \u03c0\u03bf\u03bb\u03b9\u03c4\u03b5\u03af\u03b1\u03bd ii.\n20), and is contrasted with the heathen 'dogs' ( \u03c4\u1f70 \u1f14\u03b8\u03bd\u03b7 \u1f10\u03bf\u03b9\u03ba\u03cc\u03c4\u03b1 \u03ba\u03c5\u03c3\u03af\u03bd\nii. 19) who disregard them. In some cases Justus might be the only name\nof the person, as a Latin rendering of the Hebrew Zadok; while in\nothers, as here and in Acts i. 23, it is a surname. Its Greek\nequivalent, \u1f41 \u03b4\u03af\u03ba\u03b1\u03b9\u03bf\u03c2, is the recognised epithet of James the Lord's\nbrother: see _Galatians_, p. 348.\n\n\u03bf\u1f31 \u1f44\u03bd\u03c4\u03b5\u03c2 \u03ba.\u03c4.\u03bb.] i.e. 'converts from Judaism' (see the note Gal. ii.\n12), or perhaps 'belonging to the Circumcision'; but in this latter case\n\u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc6\u03c2, though without the article, must be used in a concrete sense,\nlike \u03c4\u1fc6\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc6\u03c2, for 'the Jews'. Of Mark and of Jesus the fact is\nplain from their name or their connexions. Of Aristarchus we could not\nhave inferred a Jewish origin, independently of this direct statement.\n\n\u03bc\u03cc\u03bd\u03bf\u03b9] i.e. of the Jewish Christians in Rome. On this antagonism of the\nconverts from the Circumcision in the metropolis, see _Philippians_ p.\n16 sq. The words however must not be closely pressed, as if absolutely\nno Jewish Christian besides had remained friendly; they will only imply\nthat among the more prominent members of the body the Apostle can only\nname these three as stedfast in their allegiance: comp. Phil. ii. 20\n\u03bf\u1f50\u03b4\u03ad\u03bd\u03b1 \u1f14\u03c7\u03c9 \u1f30\u03c3\u03cc\u03c8\u03c5\u03c7\u03bf\u03bd ... \u03bf\u1f31 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03b3\u1f70\u03c1 \u03ba.\u03c4.\u03bb. (with the note).\n\n------------------------------------------------------------------------\n\nIV. 12]\n\n\u1f10\u03b3\u03b5\u03bd\u03ae\u03b8\u03b7\u03c3\u03ac\u03bd \u03bc\u03bf\u03b9 \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u03b1. ^{12}\u1f00\u03c3\u03c0\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6\u03c2 \u1f41 \u1f10\u03be \u1f51\u03bc\u1ff6\u03bd, \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6, \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f51\u03c0\u1f72\u03c1 \u1f51\u03bc\u1ff6\u03bd \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03b1\u1fd6\u03c2, \u1f35\u03bd\u03b1\n\u03c3\u03c4\u03b1\u03b8\u1fc6\u03c4\u03b5 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03b9\n\n\n\u03c4\u1f74\u03bd \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb.] See the note on i. 13.\n\n\u03bf\u1f35\u03c4\u03b9\u03bd\u03b5\u03c2 \u03ba.\u03c4.\u03bb.] '_men whom I found_ etc.'; comp. Acts xxviii. 15 \u03bf\u1f53\u03c2\n\u1f30\u03b4\u1f7c\u03bd \u1f41 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03ae\u03c3\u03b1\u03c2 \u03c4\u1ff7 \u0398\u03b5\u1ff7 \u1f14\u03bb\u03b1\u03b2\u03b5\u03bd \u03b8\u03ac\u03c1\u03c3\u03bf\u03c2, and see _Philippians_\np. 17. For \u03bf\u1f35\u03c4\u03b9\u03bd\u03b5\u03c2, not specifying the individuals, but referring them\nto their class characteristics, see the notes on Gal. iv. 24, v. 19,\nPhil. iii. 7, iv. 3.\n\n\u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u03b1] '_encouragement_', '_comfort_'. The range of meaning in this\nword is even wider than in \u03c0\u03b1\u03c1\u03b1\u03bc\u03c5\u03b8\u03af\u03b1 or \u03c0\u03b1\u03c1\u03ac\u03ba\u03bb\u03b7\u03c3\u03b9\u03c2 (see the note Phil.\nii. 1). The verb \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03b5\u1fd6\u03bd denotes either (1) 'to exhort, encourage'\n(Herod. v. 104, Apoll. Rhod. ii. 64); (2) 'to dissuade' (Herod. ix. 54,\n55); (3) 'to appease', 'quiet' (Plut. _Vit. Pomp._ 13, _Mor._ p. 737 C);\nor (4) 'to console, comfort' (\u00c6sch. _Eum._ 507). The word however, and\nits derivates \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u03b1, \u03c0\u03b1\u03c1\u03b7\u03b3\u03cc\u03c1\u03b7\u03bc\u03b1, \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03b9\u03ba\u03cc\u03c2, \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03b7\u03c4\u03b9\u03ba\u03cc\u03c2, were\nused especially as medical terms, in the sense of 'assuaging',\n'alleviating'; e.g. Hippocr. pp. 392, 393, 394, Galen XIV. p. 335, 446,\nPlut. _Mor._ pp. 43 D, 142 D; and perhaps owing to this usage, the idea\nof consolation, comfort, is on the whole predominant in the word; e.g.\nPlut. _Mor._ p. 56 A \u03c4\u1f70\u03c2 \u1f10\u03c0\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u1f00\u03c4\u03c5\u03c7\u03ae\u03bc\u03b1\u03c3\u03b9 \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c0\u03af\u03b1\u03c2, p. 118 A \u03c4\u03bf\u1fd6\u03c2\n\u1f00\u03c6\u03b1\u03b9\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2 \u03c4\u1f70\u03c2 \u03bb\u03cd\u03c0\u03b1\u03c2 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u03b3\u03b5\u03bd\u03bd\u03b1\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c3\u03b5\u03bc\u03bd\u1fc6\u03c2 \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u03b1\u03c2, _Vit.\nCim._ 4 \u1f10\u03c0\u1f76 \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u1fb3 \u03c4\u03bf\u1fe6 \u03c0\u03ad\u03bd\u03b8\u03bf\u03c5\u03c2. In Plut. _Mor._ p. 599 B \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u03b1\nand \u03c3\u03c5\u03bd\u03b7\u03b3\u03bf\u03c1\u03af\u03b1 are contrasted, as the right and wrong method of dealing\nwith the sorrows of the exile; and the former is said to be the part of\nmen \u03c0\u03b1\u03c1\u1fe5\u03b7\u03c3\u03b9\u03b1\u03b6\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03cc\u03bd\u03c4\u03c9\u03bd \u1f45\u03c4\u03b9 \u03c4\u1f78 \u03bb\u03c5\u03c0\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u1fe6\u03bd\n\u1f11\u03b1\u03c5\u03c4\u1f78\u03bd \u1f10\u03c0\u1f76 \u03c0\u03b1\u03bd\u03c4\u1f76 \u03bc\u1f72\u03bd \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9 \u03ba.\u03c4.\u03bb.\n\n12. \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6\u03c2] His full name would be Epaphroditus, but he is always\ncalled by the shortened form Epaphras, and must not be confused with the\nPhilippian Epaphroditus (see _Philippians_ p. 60), who also was with St\nPaul at one period of his Roman captivity. Of Epaphras, as the\nevangelist of Coloss\u00e6, and perhaps of the neighbouring towns, see above,\npp. 29 sq., 34 sq.\n\n\u1f41 \u1f10\u03be \u1f51\u03bc\u1ff6\u03bd] '_who belongs to you_', '_who is one of you_', i.e. a native,\nor at least an inhabitant, of Coloss\u00e6, as in the case of Onesimus ver.\n9; comp. Acts iv. 6, xxi. 8, Rom. xvi. 10, 11, 1 Cor. xii. 16, Phil. iv.\n22, etc.\n\n\u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u03a7. \u1f38.] This title, which the Apostle uses several times of\nhimself, is not elsewhere conferred on any other individual, except once\non Timothy (Phil. i. 1), and probably points to exceptional services in\nthe cause of the Gospel on the part of Epaphras.\n\n\u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2] '_wrestling_'; comp. Rom. xv. 30 \u03c3\u03c5\u03bd\u03b1\u03b3\u03c9\u03bd\u03af\u03c3\u03b1\u03c3\u03b8\u03b1\u03af \u03bc\u03bf\u03b9 \u1f10\u03bd\n\u03c4\u03b1\u1fd6\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03b1\u1fd6\u03c2. See also the great \u1f00\u03b3\u03c9\u03bd\u03af\u03b1 of prayer in Luke xxii. 44.\nComp. Justin _Apol._ ii. 13 (p. 51 B) \u03ba\u03b1\u1f76 \u03b5\u1f50\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03bc\u03bc\u03ac\u03c7\u03c9\u03c2\n\u1f00\u03b3\u03c9\u03bd\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2. See also i. 29, ii. 1, with the notes.\n\n\u03c3\u03c4\u03b1\u03b8\u1fc6\u03c4\u03b5] '_stand fast_', doubtless the correct reading rather than \u03c3\u03c4\u1fc6\u03c4\u03b5\nwhich the received text has; comp. Matt. ii. 9, xxvii. 11, where also\nthe received text substitutes the weaker word.\n\n------------------------------------------------------------------------\n\nIV. 13]\n\n\u03ba\u03b1\u1f76 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9 \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03b9 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6. ^{13}\u03bc\u03b1\u03c1\u03c4\u03c5\u03c1\u1ff6 \u03b3\u1f70\u03c1 \u03b1\u1f50\u03c4\u1ff7\n\u1f45\u03c4\u03b9 \u1f14\u03c7\u03b5\u03b9 \u03c0\u03bf\u03bb\u1f7a\u03bd \u03c0\u03cc\u03bd\u03bf\u03bd \u1f51\u03c0\u1f72\u03c1\n\n\n\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9] '_fully persuaded_'. The verb \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03bd has several\nsenses. (1) 'To fulfil, accomplish'; 2 Tim. iv. 5 \u03c4\u1f74\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd \u03c3\u03bf\u03c5\n\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03cc\u03c1\u03b7\u03c3\u03bf\u03bd, _ib._ ver. 17 \u03c4\u1f78 \u03ba\u03ae\u03c1\u03c5\u03b3\u03bc\u03b1 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03b8\u1fc7, _Clem. Hom._ xix. 24\n\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd \u03bd\u1fe6\u03bd \u1f24\u03b4\u03b7 \u03c4\u03c1\u03b9\u1ff6\u03bd \u1f21\u03bc\u03b5\u03c1\u1ff6\u03bd. So perhaps Hermas _Sim._ 2\n\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03bf\u1fe6\u03c3\u03b9 \u03c4\u1f78\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd ... \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03bf\u1fe6\u03c3\u03b9 \u03c4\u1f70\u03c2 \u03c8\u03c5\u03c7\u1f70\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd, though\nit is a little difficult to carry the same sense into the latter clause,\nwhere the word seems to signify rather 'to satisfy'. (2)'To persuade\nfully, to convince'; Rom. iv. 21 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03b8\u1f72\u03b9\u03c2 \u1f45\u03c4\u03b9 \u1f43 \u1f10\u03c0\u03ae\u03b3\u03b3\u03b5\u03bb\u03c4\u03b1\u03b9 \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc\u03c2\n\u1f10\u03c3\u03c4\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c0\u03bf\u03b9\u1fc6\u03c3\u03b1\u03b9, xiv. 5 \u1f10\u03bd \u03c4\u1ff7 \u1f30\u03b4\u03af\u1ff3 \u03bd\u03bf\u1f76 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03ad\u03b9\u03c3\u03b8\u03c9, Clem. Rom. 42\n\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03b8\u03ad\u03c5\u03c4\u03b5\u03c2 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03c9\u03c2 \u03ba.\u03c4.\u03bb., Ign. _Magn._ 8 \u03b5\u1f30\u03c2 \u03c4\u1f78\n\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03b8\u1fc6\u03bd\u03b1\u03b9 \u03c4\u03bf\u1f7a\u03c2 \u1f00\u03c0\u03b5\u03b9\u03b8\u03bf\u1fe6\u03bd\u03c4\u03b1\u03c2, _ib._ 11 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u1fc6\u03c3\u03b8\u03b1\u03b9 \u1f10\u03bd \u03c4\u1fc7 \u03b3\u03b5\u03bd\u03bd\u1f74\u03c3\u03b5\u03b9\n\u03ba.\u03c4.\u03bb., _Philad._ inscr. \u1f10\u03bd \u03c4\u1fc7 \u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03b7 \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76\n\u1f10\u03bb\u03ad\u03b5\u03b9, _Smyrn._ 1 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd \u1f21\u03bc\u1ff6\u03bd, _Mart. Ign._ 7\n\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u1fc6\u03c3\u03b1\u03b9 \u03c4\u03bf\u1f7a\u03c2 \u1f00\u03c3\u03b8\u03b5\u03bd\u03b5\u1fd6\u03c2 \u1f21\u03bc\u1fb6\u03c2 \u1f10\u03c0\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03c1\u03bf\u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c3\u03b9\u03bd, _Clem. Hom._ Ep.\nad Iac. 10 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03c2 \u1f45\u03c4\u03b9 \u1f10\u03ba \u0398\u03b5\u03bf\u1fe6 \u03b4\u03b9\u03ba\u03b1\u03af\u03bf\u03c5, _ib._ xix. 24\n\u03c3\u03c5\u03bd\u03b5\u03c4\u03b9\u03b8\u03ad\u03bc\u03b7\u03bd \u1f61\u03c2 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2. So too LXX Eccles. viii. 11 \u1f10\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03ae\u03b8\u03b7\n\u03ba\u03b1\u03c1\u03b4\u03af\u03b1 \u03c4\u03bf\u1fe6 \u03c0\u03bf\u03b9\u1fc6\u03c3\u03b1\u03b9 \u03c4\u1f78 \u03c0\u03bf\u03bd\u03b7\u03c1\u03cc\u03bd. (3) 'To fill'; Rom. xv. 13 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03ae\u03c3\u03b1\u03b9\n\u1f51\u03bc\u1fb6\u03c2 \u03c0\u03ac\u03c3\u03b7\u03c2 \u03c7\u03b1\u03c1\u1fb6\u03c2 (a doubtful v. l.), Clem. Rom. 54 \u03c4\u03af\u03c2 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03c2\n\u1f00\u03b3\u03ac\u03c0\u03b7\u03c2; _Test. xii Patr._ Dan 2 \u03c4\u1fc7 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u1fb3 \u1f10\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03ae\u03b8\u03b7\u03bd \u03c4\u1fc6\u03c2\n\u1f00\u03bd\u03b1\u03b9\u03c1\u03ad\u03c3\u03b5\u03c9\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, where it means 'I was filled with', i.e. 'I was fully\nbent on', a sense closely allied to the last. From this account it will\nbe seen that there is in the usage of the word no justification for\ntranslating it 'most surely believed' in Luke i. 1 \u03c4\u1ff6\u03bd \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03c9\u03bd\n\u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd \u03c0\u03c1\u03b1\u03b3\u03bc\u03ac\u03c4\u03c9\u03bd, and it should therefore be rendered 'fulfilled,\naccomplished'. The word is almost exclusively biblical and\necclesiastical; and it seems clear that the passage from Ctesias in\nPhotius (_Bibl._ 72) \u03c0\u03bf\u03bb\u03bb\u03bf\u1fd6\u03c2 \u03bb\u03cc\u03b3\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1f45\u03c1\u03ba\u03bf\u03b9\u03c2 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03ae\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 \u039c\u03b5\u03b3\u03ac\u03b2\u03c5\u03b6\u03bf\u03bd\nis not quoted with verbal exactness. In Isocr. _Trapez._ \u00a7 8 the word is\nnow expunged from the text on the authority of the MSS. For the\nsubstantive \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 see the note on ii. 2 above. The reading of the\nreceived text here, \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9, must be rejected as of inferior\nauthority.\n\n\u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03ba.\u03c4.\u03bb.] '_in every thing willed by God_'; comp. 1 Kings ix. 11.\nSo the plural \u03c4\u1f70 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 in Acts xiii. 22, Ephes. ii. 3, and several\ntimes in the LXX. The words are best connected directly with\n\u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9. The passages quoted in the last note amply illustrate\nthis construction. The preposition may denote (1) The abode of the\nconviction, as Rom. xiv. 5 \u1f10\u03bd \u03c4\u1ff7 \u1f30\u03b4\u03af\u1ff3 \u03bd\u03bf\u0390; or (2) The object of the\nconviction, as Ign. _Magn._ II \u1f10\u03bd \u03c4\u1fc7 \u03b3\u03b5\u03bd\u03bd\u03ae\u03c3\u03b5\u03b9, _Philad._ inscr. \u1f10\u03bd \u03c4\u1fc7\n\u1f00\u03bd\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9; or (3) The atmosphere, the surroundings, of the conviction,\nas _Philad._ inscr. \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 )\u03b5\u03bb\u03ad\u03b5\u03b9. This last seems to be its sense\nhere. The connexion \u03c3\u03c4\u03b1\u03b8\u1fc6\u03c4\u03b5 ... \u1f10\u03bd, though legitimate in itself (Rom. v.\n2, 1 Cor. xv. 1), is not favoured by the order of the words here.\n\n13. \u03c0\u03bf\u03bb\u1f7a\u03bd \u03c0\u03cc\u03bd\u03bf\u03bd] '_much toil_', both inward and outward, though from the\nconnexion the former notion seems to predominate, as in \u1f00\u03b3\u1ff6\u03bd\u03b1 ii. 1;\ncomp. Plat. _Ph\u00e6dr._ p. 247 B \u03c0\u03cc\u03bd\u03bf\u03c2 \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f00\u03b3\u1f7c\u03bd \u1f14\u03c3\u03c7\u03b1\u03c4\u03bf\u03c2 \u03c8\u03c5\u03c7\u1fc7 \u03c0\u03c1\u03cc\u03ba\u03b5\u03b9\u03c4\u03b1\u03b9.\nOf the two variations which transcribers have substituted for the\ncorrect reading, \u03b6\u1fc6\u03bb\u03bf\u03bd emphasizes the former idea and \u03ba\u03cc\u03c0\u03bf\u03bd the latter.\nThe true reading is more expressive than either. The word \u03c0\u03cc\u03bd\u03bf\u03c2 however\nis very rare in the New Testament (occurring only Rev. xvi. 10, 11, xxi.\n4, besides this passage), and was therefore liable to be changed.\n\n\u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u03ba.\u03c4.\u03bb.] The neighbouring cities are taken in their geographical\norder, commencing from Coloss\u00e6; see above, p. 2. Epaphras, though a\nColossian, may have been the evangelist of the two larger cities also.\n\n------------------------------------------------------------------------\n\nIV. 14]\n\n\u1f51\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u1fb3 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u1f10\u03bd \u1f39\u03b5\u03c1\u03b1\u03c0\u03cc\u03bb\u03b5\u03b9. ^{14}\u1f00\u03c3\u03c0\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03c2 \u039b\u03bf\u03c5\u03ba\u1fb6\u03c2\n\u1f41 \u1f30\u03b1\u03c4\u03c1\u1f78\u03c2 \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2, \u03ba\u03b1\u1f76 \u0394\u03b7\u03bc\u1fb6\u03c2.\n\n\n\u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u1fb3] This form has not the same overwhelming preponderance of\nauthority in its favour here and in vv. 15, 16, as in ii. 1, but is\nprobably correct in all these places. It is quite possible however, that\nthe same person would write \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b9\u03b1 and \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03b9\u03b1 indifferently. Even\nthe form \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b7\u03b1 is found in Mionnet, Suppl. VII. p. 581. Another\nvariation is the contraction of \u039b\u03b1\u03bf\u03b4- into \u039b\u03b1\u03b4-; e.g. \u039b\u03b1\u03b4\u03b9\u03ba\u03b7\u03bd\u03cc\u03c2, which\noccurs frequently in the edict of Diocletian.\n\n14. \u039b\u03bf\u03c5\u03ba\u1fb6\u03c2] St Luke had travelled with St Paul on his last journey to\nJerusalem (Acts xxi. 1 sq.). He had also accompanied him two years later\nfrom Jerusalem to Rome (Acts xxvii. 2 sq.). And now again, probably\nafter another interval of two years (see _Philippians_ p. 31 sq.), we\nfind him in the Apostle's company. It is not probable that he remained\nwith St Paul in the meanwhile (_Philippians_ p. 35), and this will\naccount for his name not occurring in the Epistle to the Philippians. He\nwas at the Apostle's side again in his second captivity (2 Tim. iv. 11).\n\nLucas is doubtless a contraction of Lucanus. Several Old Latin MSS write\nout the name _Lucanus_ in the superscription and subscription to the\nGospel, just as elsewhere Apollos is written in full Apollonius. On the\nfrequent occurrence of this name Lucanus in inscriptions see _Ephem.\nEpigr._ II. p. 28 (1874). The shortened form Lucas however seems to be\nrare. He is here distinguished from \u03bf\u1f31 \u1f44\u03bd\u03c4\u03b5\u03c2 \u1f10\u03ba \u03c0\u03b5\u03c1\u03b9\u03c4\u03bf\u03bc\u1fc6\u03c2 (ver. 11).\nThis alone is fatal to his identification (mentioned as a tradition by\nOrigen _ad loc._) with the Lucius, St Paul's 'kinsman' (i.e. a Jew; see\n_Philippians_ pp. 17, 171, 173), who sends a salutation from Corinth to\nRome (Rom. xvi. 21). It is equally fatal to the somewhat later tradition\nthat he was one of the seventy (_Dial. c. Marc._ \u00a7 1 in Orig. _Op._ I.\np. 806, ed. De la Rue; Epiphan. _H\u00e6r._ li. 11). The identification with\nLucius of Cyrene (Acts xiii. 13) is possible but not probable. Though\nthe example of Patrobius for Patrobas (Rom. xvi. 14) shows that such a\ncontraction is not out of the question, yet probability and testimony\nalike point to Lucanus, as the longer form of the Evangelist's name.\n\n\u1f41 \u1f30\u03b1\u03c4\u03c1\u1f78\u03c2] Indications of medical knowledge have been traced both in the\nthird Gospel and in the Acts; see on this point Smith's _Voyage and\nShipwreck of St Paul_ p. 6 sq. (ed. 2). It has been observed also, that\nSt Luke's first appearance in company with St Paul (Acts xvi. 10) nearly\nsynchronizes with an attack of the Apostle's constitutional malady (Gal.\niv. 13, 14); so that he may have joined him partly in a professional\ncapacity. This conjecture is perhaps borne out by the personal feeling\nwhich breathes in the following \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2. But whatever may be thought\nof these points, there is no ground for questioning the ancient belief\n(Iren. iii. 14. 1 sq.) that the physician is also the Evangelist. St\nPaul's motive in specifying him as the physician may not have been to\ndistinguish him from any other bearing the same name, but to emphasize\nhis own obligations to his medical knowledge. The name in this form does\nnot appear to have been common. The tradition that St Luke was a painter\nis quite late (Niceph. Call. ii. 43). It is worthy of notice that the\ntwo Evangelists are mentioned together in this context, as also in\nPhilem. 24, 2 Tim. iv. 11.\n\n\u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2] '_the beloved one_', not to be closely connected with \u1f41\n\u1f30\u03b1\u03c4\u03c1\u03cc\u03c2, for \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2 is complete in itself; comp. Philem. 1, Rom. xvi.\n12 (comp. vv. 5, 8, 9), 3 Joh. 1. For the form compare the expression in\nthe Gospels, Matt. iii. 17, etc. \u1f41 \u1f51\u03af\u03bf\u03c2 \u03bc\u03bf\u03c5, \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2 \u03ba.\u03c4.\u03bb.; where a\ncomparison of Is. xlii. 1, as quoted in Matt. xii. 18, seems to show\nthat \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2 \u03ba.\u03c4.\u03bb. forms a distinct clause from \u1f41 \u1f51\u03af\u03bf\u03c2 \u03bc\u03bf\u03c5.\n\n\u0394\u03b7\u03bc\u1fb6\u03c2] On the probability that this person was a Thessalonian (2 Tim.\niv. 10) and that his name was Demetrius, see the introduction to the\nEpistles to the Thessalonians. He appears in close connexion with St\nLuke in Philem. 24, as here. In 2 Tim. iv. 10 their conduct is placed in\ndirect contrast, \u0394\u03b7\u03bc\u1fb6\u03c2 \u03bc\u03b5 \u1f10\u03b3\u03ba\u03b1\u03c4\u03ad\u03bb\u03b9\u03c0\u03b5\u03bd ... \u039b\u03bf\u1fe6\u03ba\u03b1\u03c2 \u1f10\u03c3\u03c4\u1f76\u03bd \u03bc\u03cc\u03bd\u03bf\u03c2 \u03bc\u03b5\u03c4' \u1f10\u03bc\u03bf\u1fe6.\nThere is perhaps a foreshadowing of this contrast in the language here.\nWhile Luke is described with special tenderness as \u1f41 \u1f30\u03b1\u03c4\u03c1\u03cc\u03c2, \u1f41 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03c2,\nDemas alone is dismissed with a bare mention and without any epithet of\ncommendation.\n\n------------------------------------------------------------------------\n\nIV. 15, 16]\n\n^{15}\u1f08\u03c3\u03c0\u03ac\u03c3\u03b1\u03c3\u03b8\u03b5 \u03c4\u03bf\u1f7a\u03c2 \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u1fb3 \u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03c5\u03c2 \u03ba\u03b1\u1f76 \u039d\u03c5\u03bc\u03c6\u1fb6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03bf\u03bd\n\u03b1\u1f50\u03c4\u1ff6\u03bd \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03bd. ^{16}\u039a\u03b1\u1f76 \u1f45\u03c4\u03b1\u03bd\n\n\n15\u201317. 'Greet from me the brethren who are in Laodicea, especially\nNymphas, and the church which assembles in their house. And when this\nletter has been read among you, take care that it is read also in the\nChurch of the Laodiceans, and be sure that ye also read the letter which\nI have sent to Laodicea, and which ye will get from them. Moreover give\nthis message from me to Archippus; Take heed to the ministry which thou\nhast received from me in Christ, and discharge it fully and faithfully.'\n\n15. \u039d\u03c5\u03bc\u03c6\u1fb6\u03bd] as the context shows, an inhabitant of Laodicea. The name in\nfull would probably be Nymphodorus, as Artemas (Tit. iii. 12) for\nArtemidorus, Zenas (Tit. iii. 13) for Zenodorus, Theudas (Acts v. 36)\nfor Theodorus, Olympas (Rom. xvi. 15) for Olympiodorus, and probably\nHermas (Rom. xvi. 14) for Hermodorus (see _Philippians_ p. 174). Other\nnames in \u03b1\u03c2 occurring in the New Testament and representing different\nterminations are Amplias (Ampliatus, a _v. l._), Antipas (Antipater),\nDemas (Demetrius?), Epaphras (Epaphroditus), Lucas (Lucanus), Parmenas\n(Parmenides), Patrobas (Patrobius), Silas (Sylvanus), Stephanas\n(Stephanephorus), and perhaps Junias (Junianus, Rom. xvi. 7). For a\ncollection of names with this contraction, found in different places,\nsee Chandler _Greek Accentuation_ \u00a7 34; comp. Lobeck _Pathol._ p. 505\nsq. Some remarkable instances are found in the inscriptions; e.g.\n\u1f08\u03c3\u03ba\u03bb\u1fb6\u03c2, \u0394\u03b7\u03bc\u03bf\u03c3\u03b8\u1fb6\u03c2, \u0394\u03b9\u03bf\u03bc\u1fb6\u03c2, \u1f19\u03c1\u03bc\u03bf\u03b3\u1fb6\u03c2, \u039d\u03b9\u03ba\u03bf\u03bc\u1fb6\u03c2, \u1f48\u03bd\u03b7\u03c3\u1fb6\u03c2, \u03a4\u03c1\u03bf\u03c6\u1fb6\u03c2, etc.; see\nesp. Boeckh _C. I._ III. pp. 1072, 1097. The name Nymphodorus is found\nnot unfrequently; e.g. Herod. vii. 137, Thuc. ii. 29, Athen. i. p. 19 F,\nvi. p. 265 C, Mionnet _Suppl._ VI. p. 88, Boeckh _C. I._ no. 158, etc.\nThe contracted form \u039d\u03c5\u03bc\u03c6\u1fb6\u03c2 however is very rare, though it appears to\noccur in a Spartan inscription, Boeckh _C. I._ no. 1240 )\u0388\u03c5\u03c4\u03c5\u03c7\u03bf\u03c2 \u039d\u03c5\u03bd\u03c6\u1fb6.\nIn Murat. MDXXXV. 6, is an inscription to one _Nu. Aquilius Nymphas_, a\nfreedman, where the dative is _Nymphadi_. Other names from which Nymphas\nmight be contracted are Nymphius, Nymphicus, Nymphidius, Nymphodotus,\nthe first and last being the most common.\n\nThose, who read \u03b1\u1f50\u03c4\u1fc6\u03c2 in the following clause, take it as a woman's name\n(\u039d\u03cd\u03bc\u03c6\u03b1\u03bd, not \u039d\u03c5\u03bc\u03c6\u1fb6\u03bd); and the name Nymphe, Nympha, Nympa, etc., occurs\nfrom time to time in Latin inscriptions; e.g. _Inscr. Hisp._ 1099, 1783,\n3763, _Inscr. As. Prov. etc._ 525, Murator. CMXXIV. 1, MCLIX. 8, MCCXCV.\n9, MDXCI. 3. But a Doric form of the Greek name here seems in the\nhighest degree improbable.\n\n\u03c4\u1f74\u03bd \u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03bf\u03bd \u03ba.\u03c4.\u03bb.] The same expression is used of Prisca and Aquila\nboth at Rome (Rom. xvi. 5) and at Ephesus (1 Cor. xvi. 19), and also of\nPhilemon, whether at Coloss\u00e6 or at Laodicea is somewhat uncertain\n(Philem. 2); comp. Acts xii. 12 \u03c4\u1f74\u03bd \u03bf\u1f30\u03ba\u03af\u03b1\u03bd \u03c4\u1fc6\u03c2 \u039c\u03b1\u03c1\u03af\u03b1\u03c2 ... \u1f41\u1fe6 \u1f26\u03c3\u03b1\u03bd \u1f31\u03ba\u03b1\u03bd\u03bf\u1f76\n\u03c3\u03c5\u03bd\u03b7\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03ad\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9, and see _Philippians_ p. 56. Perhaps\nsimilar gatherings may be implied by the expressions in Rom. xvi. 14, 15\n\u03c4\u03bf\u1f7a\u03c2 \u03c3\u1f7a\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u03cd\u03c2, \u03c4\u03bf\u1f7a\u03c2 \u03c3\u1f7a\u03bd \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1\u03c2 \u1f01\u03b3\u03af\u03bf\u03c5\u03c2 (Probst\n_Kirchliche Disciplin_ p. 182, 1873). See also _Act. Mart. Justin._ \u00a7 3\n(II. p. 262 ed. Otto), _Clem. Recogn._ x. 71 'Theophilus ... domus su\u00e6\ningentem basilicam ecclesi\u00e6 nomine consecraret' (where the word\n'basilica' was probably introduced by the translator Ruffinus). Of the\nsame kind must have been the 'collegium quod est in domo Sergi\u00e6 Paulin\u00e6'\n(de Rossi _Roma Sotteranea_ I. p. 209); for the Christians were first\nrecognised by the Roman government as 'collegia' or burial clubs, and\nprotected by this recognition doubtless held their meetings for\nreligious worship. There is no clear example of a separate building set\napart for Christian worship within the limits of the Roman empire before\nthe third century, though apartments in private houses might be\nspecially devoted to this purpose. This, I think, appears as a negative\nresult from the passages collected in Bingham VIII. I. 13 and Probst p.\n181 sq. with a different view. Hence the places of Christian assembly\nwere not commonly called \u03bd\u03b1\u03bf\u03af till quite late (Ignat. _Magn._ 7 is not\nreally an exception), but \u03bf\u1f36\u03ba\u03bf\u03b9 \u0398\u03b5\u03bf\u1fe6, \u03bf\u1f36\u03ba\u03bf\u03b9 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03b9\u1ff6\u03bd, \u03bf\u1f36\u03ba\u03bf\u03b9 \u03b5\u1f50\u03ba\u03c4\u03ae\u03c1\u03b9\u03bf\u03b9,\nand the like (Euseb. _H.E._ vii. 30, viii. 13, ix. 9, etc.).\n\n\u03b1\u1f50\u03c4\u1ff6\u03bd] The difficulty of this reading has led to the two corrections,\n\u03b1\u1f50\u03c4\u03bf\u1fe6 and \u03b1\u1f50\u03c4\u1fc6\u03c2, of which the former appears in the received text and\nthe latter is supported by one or two very ancient authorities. Of these\nalternative readings however, \u03b1\u1f50\u03c4\u03bf\u1fe6 is condemned by its simplicity, and\n\u03b1\u1f50\u03c4\u1fc6\u03c2 has arisen from the form \u039d\u03c5\u03bc\u03c6\u03b1\u03bd, which _prima facie_ would look\nlike a woman's name, and yet hardly can be so. We should require to know\nmore of the circumstances to feel any confidence in explaining \u03b1\u1f50\u03c4\u1ff6\u03bd. A\nsimple explanation is that \u03b1\u1f50\u03c4\u1ff6\u03bd denotes 'Nymphas and his friends', by a\ntransition which is common in classical writers; e.g. Xen. _Anab._ iii.\n3. 7 \u03c0\u03c1\u03bf\u03c3\u1fc4\u03b5\u03b9 \u03bc\u1f72\u03bd (\u039c\u03b9\u03b8\u03c1\u03b9\u03b4\u03ac\u03c4\u03b7\u03c2) ... \u03c0\u03c1\u1f78\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f1d\u03bb\u03bb\u03b7\u03bd\u03b1\u03c2\u00b7 \u1f10\u03c0\u1f72\u03b9 \u03b4' \u1f10\u03b3\u03b3\u1f7a\u03c2\n_\u1f10\u03b3\u03ad\u03bd\u03bf\u03bd\u03c4\u03bf_ \u03ba.\u03c4.\u03bb., iv. 5. 33 \u1f10\u03c0\u1f72\u03b9 \u03b4' \u1f26\u03bb\u03b8\u03bf\u03bd \u03c0\u03c1\u1f78\u03c2 \u03a7\u03b5\u03b9\u03c1\u03af\u03c3\u03bf\u03c6\u03bf\u03bd,\n\u03ba\u03b1\u03c4\u03b5\u03bb\u03ac\u03bc\u03b2\u03b1\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 _)\u03b5\u03ba\u03ad\u03b9\u03bd\u03bf\u03c5\u03c2_ \u03c3\u03ba\u03b7\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b1\u03c2: see also K\u00fchner _Gramm._ \u00a7\n371 (II. p. 77), Bernhardy _Syntax_ p. 288. Or perhaps \u03c4\u03bf\u1f7a\u03c2 \u1f10\u03bd \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u1fb3\n\u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u03cd\u03c2 may refer not to the whole body of the Laodicean Church, but to\na family of Colossian Christians established in Laodicea. Under any\ncircumstances this \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1 is only a section of \u1f21 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03ad\u03c9\u03bd \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\nmentioned in ver. 16. On the authorities for the various readings see\nthe detached note.\n\n------------------------------------------------------------------------\n\nIV. 16]\n\n\u1f00\u03bd\u03b1\u03b3\u03bd\u03c9\u03c3\u03b8\u1fc7 \u03c0\u03b1\u03c1' \u1f51\u03bc\u1fd6\u03bd \u1f21 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae, \u03c0\u03bf\u03b9\u03ae\u03c3\u03b1\u03c4\u03b5 \u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76\n\n\n16. \u1f21 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae] '_the letter_', which has just been concluded, for these\nsalutations have the character of a postscript; comp. Rom. xvi. 22\n\u03a4\u03ad\u03c1\u03c4\u03b9\u03bf\u03c2 \u1f41 \u03b3\u03c1\u03ac\u03c8\u03b1\u03c2 \u03c4\u1f74\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae\u03bd, 2 Thess. iii. 14 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1fc6\u03c2,\n_Mart. Polyc._ 20 \u03c4\u1f74\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1f74\u03bd \u03b4\u03b9\u03b1\u03c0\u03ad\u03bc\u03c8\u03b1\u03c3\u03b8\u03b5. Such examples however do\nnot countenance the explanation which refers \u1f14\u03b3\u03c1\u03b1\u03c8\u03b1 \u1f51\u03bc\u1fd6\u03bd \u1f10\u03bd \u03c4\u1fc7 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1fc7\nin 1 Cor. v. 9 to the First Epistle itself, occurring (as it does) in\nthe middle of the letter (comp. 2 Cor. vii. 8).\n\n\u03c0\u03bf\u03b9\u03ae\u03c3\u03b1\u03c4\u03b5 \u1f35\u03bd\u03b1] '_cause that_'; so John xi. 37, Apoc. xiii. 15. In such\ncases the \u1f35\u03bd\u03b1 is passing away from its earlier sense of _design_ to its\nlater sense of _result_. A corresponding classical expression is \u03c0\u03bf\u03b9\u03b5\u1fd6\u03bd\n\u1f61\u03c2 or \u1f45\u03c0\u03c9\u03c2, e.g. Xen. _Cyr._ vi. 3. 18.\n\nA similar charge is given in 1 Thess. v. 27. The precaution here is\nprobably suggested by the distastefulness of the Apostle's warnings,\nwhich might lead to the suppression of the letter.\n\n------------------------------------------------------------------------\n\nIV. 17]\n\n\u1f10\u03bd \u03c4\u1fc7 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03ad\u03c9\u03bd \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3 \u1f00\u03bd\u03b1\u03b3\u03bd\u03c9\u03c3\u03b8\u1fc7, \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u03b1\u03c2 \u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2\n\u1f00\u03bd\u03b1\u03b3\u03bd\u1ff6\u03c4\u03b5. ^{17}\u039a\u03b1\u1f76 \u03b5\u1f34\u03c0\u03b1\u03c4\u03b5 \u1f08\u03c1\u03c7\u03af\u03c0\u03c0\u1ff3, \u0392\u03bb\u03ad\u03c0\u03b5 \u03c4\u1f74\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd \u1f23\u03bd \u03c0\u03b1\u03c1\u03ad\u03bb\u03b1\u03b2\u03b5\u03c2 \u1f10\u03bd\n\u039a\u03c5\u03c1\u03af\u1ff3, \u1f35\u03bd\u03b1 \u03b1\u1f50\u03c4\u1f74\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u1fd6\u03c2.\n\n\n\u03c4\u1f74\u03bd \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u03b1\u03c2] i.e. 'the letter left at Laodicea, which you will\nprocure thence'. For this abridged expression compare Luke xi. 13 \u1f41\n\u03c0\u03b1\u03c4\u1f74\u03c1 _\u1f41 \u1f10\u03be_ \u03bf\u1f50\u03c1\u03b1\u03bd\u03bf\u1fe6 \u03b4\u03ce\u03c3\u03b5\u03b9 \u03c0\u03bd\u03b5\u1fe6\u03bc\u03b1 \u1f05\u03b3\u03b9\u03bf\u03bd, xvi. 26 (v. l.) \u03bc\u03b7\u03b4\u1f72 _\u1f41\u03b9\n\u1f10\u03ba\u03b5\u1fd6\u03b8\u03b5\u03bd_ \u03c0\u03c1\u1f78\u03c2 \u1f21\u03bc\u1fb6\u03c2 \u03b4\u03b9\u03b1\u03c0\u03b5\u03c1\u1ff6\u03c3\u03b9\u03bd, Susann. 26 \u1f61\u03c2 \u03b4\u1f72 \u1f24\u03ba\u03bf\u03c5\u03c3\u03b1\u03bd \u03c4\u1f74\u03bd \u03ba\u03c1\u03b1\u03c5\u03b3\u1f74\u03bd\n\u1f10\u03bd \u03c4\u1ff7 \u03c0\u03b1\u03c1\u03b1\u03b4\u03ad\u03b9\u03c3\u1ff3 _\u1f41\u03b9 \u1f10\u03ba_ \u03c4\u1fc6\u03c2 \u03bf\u1f30\u03ba\u03af\u03b1\u03c2, \u03b5\u1f30\u03c3\u03b5\u03c0\u03ae\u03b4\u03b7\u03c3\u03b1\u03bd \u03ba.\u03c4.\u03bb. For\ninstances of this proleptic use of the preposition in classical writers,\nwhere it is extremely common, see K\u00fchner _Gr._ \u00a7 448 (II. p. 474), Jelf\n_Gr._ \u00a7 647, Matthi\u00e6 _Gr._ \u00a7 596: e.g. Plat. _Apol._ 32 B \u03c4\u03bf\u1f7a\u03c2 \u03bf\u1f50\u03ba\n\u1f00\u03bd\u03b5\u03bb\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f10\u03ba \u03c4\u1fc6\u03c2 \u03bd\u03b1\u03c5\u03bc\u03b1\u03c7\u03af\u03b1\u03c2, Xen. _Cyr._ vii. 2. 5 \u1f01\u03c1\u03c0\u03b1\u03c3\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f70\n\u1f10\u03ba \u03c4\u1ff6\u03bd \u03bf\u1f30\u03ba\u03b9\u1ff6\u03bd, Isocr. _Paneg._ \u00a7 187 \u03c4\u1f74\u03bd \u03b5\u1f50\u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u03af\u03b1\u03bd \u03c4\u1f74\u03bd \u1f10\u03ba \u03c4\u1fc6\u03c2 \u1f08\u03c3\u03af\u03b1\u03c2\n\u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd )\u0395\u03c5\u03c1\u03ce\u03c0\u03b7\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bc\u03af\u03c3\u03b1\u03b9\u03bc\u03b5\u03bd. There are good reasons for the belief\nthat St Paul here alludes to the so-called Epistle to the Ephesians,\nwhich was in fact a circular letter addressed to the principal churches\nof proconsular Asia (see above p. 37, and the introduction to the\nEpistle to the Ephesians). Tychicus was obliged to pass through Laodicea\non his way to Coloss\u00e6, and would leave a copy there, before the\nColossian letter was delivered. For other opinions respecting this\n'letter from Laodicea' see the detached note.\n\n\u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2 \u03ba.\u03c4.\u03bb.] 'see _that ye also read_'. At first sight it might\nseem as though this \u1f35\u03bd\u03b1 also were governed by \u03c0\u03bf\u03af\u03b7\u03c3\u03b1\u03c4\u03b5, like the former;\nbut, inasmuch as \u03c0\u03bf\u03af\u03b7\u03c3\u03b1\u03c4\u03b5 would be somewhat awkward in this connexion,\nit is perhaps better to treat the second clause as independent and\nelliptical, (\u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u1f10 \u1f35\u03bd\u03b1 \u03ba.\u03c4.\u03bb. This is suggested also by the position\nof \u03c4\u1f74\u03bd \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u03b1\u03c2 before \u1f35\u03bd\u03b1; comp. Gal. ii. 10 \u03bc\u03cc\u03bd\u03bf\u03bd \u03c4\u1ff6\u03bd \u03c0\u03c4\u03c9\u03c7\u1ff6\u03bd \u1f35\u03bd\u03b1\n\u03bc\u03bd\u03b7\u03bc\u03bf\u03bd\u03b5\u03cd\u03c9\u03bc\u03b5\u03bd (with the note). Ellipses before \u1f35\u03bd\u03b1 are frequent; e.g.\nJohn ix. 3, 2 Cor. viii. 13, 2 Thess. iii. 9, 1 Joh. ii. 19.\n\n17. \u039a\u1f70\u03b9 )\u03ad\u03b9\u03c0\u03b1\u03c4\u03b5] Why does not the Apostle address himself directly to\nArchippus? It might be answered that he probably thought the warning\nwould come with greater emphasis, when delivered by the voice of the\nChurch. Or the simpler explanation perhaps is, that Archippus was not\nresident at Coloss\u00e6 but at Laodicea: see the introduction to the Epistle\nto Philemon. On this warning itself see above, p. 42.\n\n\u0392\u03bb\u03ad\u03c0\u03b5] '_look to_', as 2 Joh. 8 \u03b2\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u1f35\u03bd\u03b1 \u03bc\u1f74 \u03ba.\u03c4.\u03bb. More\ncommonly it has the accusative of the thing to be avoided; see Phil.\niii. 2 (with the note).\n\n\u03c4\u1f74\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd] From the stress which is laid upon it, the \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1 here\nwould seem to refer, as in the case of Timothy cited below, to some\nhigher function than the diaconate properly so called. In Acts xii. 25\nthe same phrase, \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd \u03c4\u1f74\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd, is used of a temporary\nministration, the collection and conveyance of the alms for the poor of\nJerusalem (Acts xi. 29); but the solemnity of the warning here points to\na continuous office, rather than an immediate service.\n\n\u03c0\u03b1\u03c1\u03ad\u03bb\u03b1\u03b2\u03b5\u03c2] i.e. probably \u03c0\u03b1\u03c1' \u1f10\u03bc\u03bf\u1fe6. The word suggests, though it does\nnot necessarily imply, a mediate rather than a direct reception: see the\nnote Gal. i. 12. Archippus received the charge immediately from St Paul,\nthough ultimately from Christ. 'Non enim sequitur', writes Bengel, '_a\nDomino_ (1 Cor. xi. 23), sed _in Domino_'.\n\n\u03c0\u03bb\u03b7\u03c1\u03bf\u1fd6\u03c2] '_fulfil_', i.e. '_discharge fully_'; comp. 2 Tim. iv. 5 \u03c4\u1f74\u03bd\n\u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd \u03c3\u03bf\u03c5 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03cc\u03c1\u03b7\u03c3\u03bf\u03bd.\n\n------------------------------------------------------------------------\n\nIV. 18]\n\n^{18}\u1f49 \u1f00\u03c3\u03c0\u03b1\u03c3\u03bc\u1f78\u03c2 \u03c4\u1fc7 \u1f10\u03bc\u1fc7 \u03c7\u03b5\u03b9\u03c1\u1f76 \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5. \u039c\u03bd\u03b7\u03bc\u03bf\u03bd\u03b5\u03cd\u03b5\u03c4\u03ad \u03bc\u03bf\u03c5 \u03c4\u1ff6\u03bd \u03b4\u03b5\u03c3\u03bc\u1ff6\u03bd. \u1f29 \u03c7\u03ac\u03c1\u03b9\u03c2\n\u03bc\u03b5\u03b8' \u1f51\u03bc\u1ff6\u03bd.\n\n\n18. 'I add this salutation with my own hand, signing it with my name\nPaul. Be mindful of my bonds. God's grace be with you.'\n\n\u1f49 \u1f00\u03c3\u03c0\u03b1\u03c3\u03bc\u1f78\u03c2 \u03ba.\u03c4.\u03bb.] The letter was evidently written by an amanuensis\n(comp. Rom. xvi. 22). The final salutation alone, with the accompanying\nsentence \u03bc\u03bd\u03b7\u03bc\u03bf\u03bd\u03b5\u03cd\u03b5\u03c4\u03b5 \u03ba.\u03c4.\u03bb., was in the Apostle's own handwriting. This\nseems to have been the Apostle's general practice, even where he does\nnot call attention to his own signature. In 2 Thess. iii. 17 sq., 1 Cor.\nxvi. 21, as here, he directs his readers' notice to the fact, but in\nother epistles he is silent. In some cases however he writes much more\nthan the final sentence. Thus the whole letter to Philemon is apparently\nin his own handwriting (see ver. 19), and in the Epistle to the\nGalatians he writes a long paragraph at the close (see the note on vi.\n11).\n\n\u03c4\u1fc7 \u1f10\u03bc\u1fc7 \u03c7\u03b5\u03b9\u03c1\u1f76 \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5] The same phrase occurs in 2 Thess. iii. 17, 1 Cor.\nxvi. 21. For the construction comp. e.g. Philo _Leg. ad Cai._ 8 (II. p.\n554) \u1f10\u03bc\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9 \u03c4\u03bf\u1fe6 \u039c\u03ac\u03ba\u03c1\u03c9\u03bd\u03bf\u03c2 \u1f10\u03c1\u03b3\u03bf\u03bd \u0393\u03ac\u03ca\u03bf\u03c2, and see K\u00fchner \u00a7 406 (II. p.\n242), Jelf \u00a7 467.\n\n\u03c4\u1ff6\u03bd \u03b4\u03b5\u03c3\u03bc\u1ff6\u03bd] His bonds establish an additional claim to a hearing. He who\nis suffering for Christ has a right to speak on behalf of Christ. The\nappeal is similar in Ephes. iii. 1 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03c7\u03ac\u03c1\u03b9\u03bd \u1f10\u03b3\u1f7c \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u1f41 \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u03a7. \u1f38., which is resumed again (after a long digression) in iv. 1\n\u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u1ff6 \u03bf\u1f56\u03bd \u1f51\u03bc\u1fb6\u03c2 \u1f10\u03b3\u1f7c \u1f41 \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 \u1f00\u03be\u03af\u03c9\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u1fc6\u03c3\u03b1\u03b9 \u03ba.\u03c4.\u03bb. (comp.\nvi. 20 \u1f51\u03c0\u1f72\u03c1 \u1f41\u1fe6 \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03cd\u03c9 \u1f10\u03bd \u1f01\u03bb\u03cd\u03c3\u03b5\u03b9). So too Philem. 9 \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f62\u03bd \u1f61\u03c2\n\u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 ... \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6. These passages seem to show that the\nappeal here is not for himself, but for his teaching\u2014not for sympathy\nwith his sufferings but for obedience to the Gospel. His bonds were not\nhis own; they were \u03c4\u1f70 \u03b4\u03b5\u03c3\u03bc\u1f70 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5 (Philem. 13). In Heb. x. 34\nthe right reading is not \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2 \u03bc\u03bf\u03c5, but \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03c3\u03bc\u03af\u03bf\u03b9\u03c2\n\u03c3\u03c5\u03bd\u03b5\u03c0\u03b1\u03b8\u03ae\u03c3\u03b1\u03c4\u03b5 (comp. xiii. 3). Somewhat similar is the appeal to his\n\u03c3\u03c4\u03af\u03b3\u03bc\u03b1\u03c4\u03b1 in Gal. vi. 17, 'Henceforth let no man trouble me.' See the\nnotes on Philem. 10, 13.\n\n\u1f29 \u03c7\u03ac\u03c1\u03b9\u03c2 \u03ba.\u03c4.\u03bb.] This very short form of the final benediction appears\nonly here and in 1 Tim. vi. 21, 2 Tim. iv. 22. In Tit. iii. 15 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd is\ninserted, and so in Heb. xiii. 25. In Ephes. vi. 24 the form so far\nagrees with the examples quoted, that \u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2 is used absolutely, though\nthe end is lengthened out. In all the earlier epistles \u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2 is\ndefined by the addition of \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 [\u1f21\u03bc\u1ff6\u03bd] \u1f38\u03b7\u03c3\u03bf\u1fe6 [\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6]; 1 Thess.\nv. 28, 2 Thess. iii. 18, 1 Cor. xvi. 23, 2 Cor. xiii. 13, Gal. vi. 18,\nRom. xvi. 20, [24], Phil. iv. 23. Thus the absolute \u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2 in the final\nbenediction may be taken as a chronological note. A similar phenomenon\nhas been already observed (\u03c4\u1fc7 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3, \u03c4\u03b1\u1fd6\u03c2 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03b9\u03c2) in the opening\naddresses: see the note on i. 2.\n\n\n\n\n            _On some Various Readings in the Epistle[535]._\n\n\n[Sidenote: Harmonistic readings.]\n\nIn one respect the letters to the Ephesians and Colossians hold a unique\nposition among the Epistles of St Paul, as regards textual criticism.\nThey alone have been exposed, or exposed in any considerable degree, to\nthose harmonizing tendencies in transcribers, which have had so great an\ninfluence on the text of the Synoptic Gospels.\n\n[Sidenote: Preponderant evidence (1) for the correct reading;]\n\nIn such cases there is sometimes no difficulty in ascertaining the\ncorrect reading. The harmonistic change is condemned by the majority of\nthe oldest and best authorities; or there is at least a nearly even\nbalance of external testimony, and the suspicious character of the\nreading is quite sufficient to turn the scale. Thus we cannot hesitate\nfor a moment about such readings as i. 14 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u1f05\u03b9\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 (from\nEphes. i. 7), or iii. 16 \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 _\u03ba\u03b1\u1f76_ \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 _\u03ba\u03b1\u1f76_ \u1fa0\u03b4\u03b1\u1fd6\u03c2\n\u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2, and \u03c4\u1ff7 _\u039a\u03c5\u03c1\u03af\u1ff3_ (for \u03c4\u1ff7 \u0398\u03b5\u1ff7) in the same verse (both\nfrom Ephes. v. 19).\n\n[Sidenote: (2) against the correct reading.]\n\nIn other instances again there can hardly be any doubt about the text,\neven though the vast preponderance of authority is in favour of the\nharmonistic reading; and these are especially valuable because they\nenable us to test the worth of our authorities. Such examples are:\n\nFootnote 535:\n\n  The references to the patristic quotations in the following pages have\n  all been verified. I have also consulted the Egyptian and Syriac\n  Versions in every case, and the Armenian and Latin in some instances,\n  before giving the readings. As regards the MSS, I have contented\n  myself with the collations as given in Tregelles and Tischendorf, not\n  verifying them unless I had reason to suspect an error.\n\n  The readings of the Memphitic Version are very incorrectly given even\n  by the principal editors, such as Tregelles and Tischendorf; the\n  translation of Wilkins being commonly adopted, though full of errors,\n  and no attention being paid to the various readings of Boetticher's\n  text. Besides the errors corrected in the following pages, I have also\n  observed these places where the text of this version is incorrectly\n  reported; ii. 7 \u1f10\u03bd \u03b1\u1f50\u03c4\u1fc7 not omitted; ii. 13 the second \u1f51\u03bc\u1fb6\u03c2 not\n  omitted; ii. 17 the singular (\u1f45), not the plural (\u1f05); iii. 4 \u1f51\u03bc\u1ff6\u03bd, not\n  \u1f21\u03bc\u1ff6\u03bd; iii. 16 \u03c4\u1ff7 \u0398\u03b5\u1ff7, not \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3; iii. 22 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd, not \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd;\n  iv. 3 doubtful whether \u03b4\u03b9' \u1f45 or \u03b4\u03b9' \u1f45\u03bd; and probably there are others.\n\n\n[Sidenote: Examples.]\n\n[Sidenote: iii. 6, words inserted.]iii. 6. The omission of the words \u1f10\u03c0\u1f76\n\u03c4\u03bf\u1f7a\u03c2 \u1f51\u03b9\u1f78\u03c5\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03c0\u03b5\u03b9\u03b8\u03b5\u03af\u03b1\u03c2 (taken from Ephes. v. 6). Apparently the only\nextant MS in favour of the omission is B. In D however they are written\n(though by the first hand) in smaller letters and extend beyond the line\n(in both Greek and Latin), whence we may infer that they were not found\nin a copy which was before the transcriber. They are wanting also in the\nThebaic Version and in one form of the \u00c6thiopic (Polyglott). They were\nalso absent from copies used by Clement of Alexandria (_P\u00e6d._ iii. 11,\np. 295, where however they are inserted in the printed texts; _Strom._\niii. 5, p. 531), by Cyprian (_Epist._ lv. 27, p. 645 ed. Hartel), by an\nunknown writer (_de Sing. Cler._> 39, in Cypr. _Op._ III. p. 215), by\nthe Ambrosian Hilary (_ad loc._), and by Jerome (_Epist._ xiv. 5, I. p.\n32), though now found apparently in all the Latin MSS.\n\n[Sidenote: iii. 21. \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5.]\n\niii. 21. \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5 is only found in B K and in later hands of D (with\nits transcript E) among the uncial MSS. All the other uncials read\n\u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03c4\u03b5, which is taken from Ephes. vi. 4. In this case however the\nreading of B is supported by the greater number of cursives, and it\naccordingly has a place in the received text. The versions (so far as we\ncan safely infer their readings) go almost entirely with the majority of\nuncials. [Sidenote: Syriac version misrepresented.]The true readings of\nthe Syriac Versions are just the reverse of those assigned to them even\nby the chief critical editors, Tregelles and Tischendorf. Thus in the\nPeshito the word used is the Aphel of \u072a\u0713\u0719, the same mood of the same\nverb being employed to translate \u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03b9\u03bd, not only in Rom. x. 19,\nbut even in the parallel passage Ephes. vi. 4. The word in the text of\nthe Harclean is the same, \u072c\u072a\u0713\u071b\u0719\u0718\u0722\u0722, but in the margin the alternative\n\u072c\u0713\u072a\u0713\u0718\u0722 is given. White interprets this as saying that the text is\n\u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5 and the margin \u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03c4\u03b5, and he is followed by Tregelles\nand Tischendorf. But in this version, as in the Peshito, the former word\ntranslates \u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03b9\u03bd in Rom. x. 19, Ephes. vi. 4; while in the Peshito\nthe latter word is adopted to render \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03b9\u03bd in 2 Cor. ix. 2 (the only\nother passage in the N. T. where \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03b9\u03bd occurs). In the Harclean of 2\nCor. ix. 2 a different word from either, \u071a\u072c\u071a\u072c, is used. It seems\ntolerably clear therefore that \u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03c4\u03b5 was read in the text of both\nPeshito and Harclean here, while \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5 was given in the margin of\nthe latter. [Sidenote: Latin versions.]The Latin Versions seem also to\nhave read \u03c0\u03b1\u03c1\u03bf\u03c1\u03b3\u03af\u03b6\u03b5\u03c4\u03b5; for the Old Latin has '_ad iram_ (or _in iram_ or\n_ad iracundiam_) _provocare_, and the Vulgate _ad indignationem\nprovocare_' here, while both have _ad iracundiam provocare_ in Ephes.\nvi. 4. The Memphitic too has the same rendering \u03ef\u03eb\u03c9\u2c9b\u03c4 in both passages.\nOf the earlier Greek fathers Clement, _Strom._ iv. 8 (p. 593), reads\n\u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03c4\u03b5: and it is found in Chrysostom and some later writers.\n\n[Sidenote: Great value of B.]\n\nThese examples show how singularly free B is from this passion for\nharmonizing, and may even embolden us to place reliance on its authority\nin extreme cases.\n\n[Sidenote: Parallel passages.]\n\n[Sidenote: Col. iii. 16, Eph. v. 19.]\n\nFor instance, the parallel passages Ephes. v. 19 and Col. iii. 16 stand\nthus in the received text:\n\n                               EPHESIANS.\n\n\u03bb\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1fa0\u03b4\u03b1\u1fd6\u03c2 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2 \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76\n\u03c8\u03ac\u03bb\u03bb\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u1f51\u03bc\u1ff6\u03bd \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3.\n\n                              COLOSSIANS.\n\n\u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1fa0\u03b4\u03b1\u1fd6\u03c2\n\u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2 \u1f10\u03bd \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9 \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u1f51\u03bc\u1ff6\u03bd \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3.\n\nAnd A carries the harmonizing tendency still further by inserting \u1f10\u03bd\n\u03c7\u03ac\u03c1\u03b9\u03c4\u03b9 before \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 in Ephes. from the parallel passage.\n\nIn B they are read as follows:\n\n\u03bb\u03b1\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1fd6\u03c2 \u1f10\u03bd \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u03ba\u03b1\u1f76 \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1fa0\u03b4\u03b1\u1fd6\u03c2 \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03c8\u03ac\u03bb\u03bb\u03bf\u03bd\u03c4\u03b5\u03c2\n\u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u1f51\u03bc\u1ff6\u03bd \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3.\n\n\u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f11\u03b1\u03c5\u03c4\u03bf\u1f7a\u03c2 \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 \u1f55\u03bc\u03bd\u03bf\u03b9\u03c2 \u1fa0\u03b4\u03b1\u1fd6\u03c2 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2\n\u1f10\u03bd \u03c4\u1fc7 \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9 \u1f84\u03b4\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03c4\u1ff7 \u0398\u03b5\u1ff7.\n\n[Sidenote: Alterations for the sake of harmonizing.] Here are seven\ndivergences from the received text. (1) The insertion of \u1f10\u03bd before\n\u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 in Ephes.; (2) The omission of \u03ba\u03b1\u03af, \u03ba\u03b1\u03af, attaching \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2,\n\u1f55\u03bc\u03bd\u03bf\u03b9\u03c2, \u1fa0\u03b4\u03b1\u1fd6\u03c2 in Col.; (3) The omission of \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2 in Ephes.; (4)\nThe insertion of \u03c4\u1fc7 before \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9 in Col.; (5) The omission of \u1f10\u03bd before\n\u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 in Ephes.; (6) The substitution of \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 for \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3\nin Col.; (7) The substitution of \u03c4\u1ff7 \u0398\u03b5\u1ff7 for \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3 in Col.\n\nOf these seven divergences the fourth alone does not affect the\nquestion: of the remaining six, the readings of B in (2), (6), (7) are\nsupported by the great preponderance of the best authorities, and are\nunquestionably right. In (1), (3), (5) however the case stands thus:\n\n[Sidenote: \u1f10\u03bd \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2.]\n\n  (1) \u1f10\u03bd \u03c8\u03b1\u03bb\u03bc\u03bf\u1fd6\u03c2 B, P, with the cursives 17, 67^{**}, 73, 116, 118, and\n          in Latin, d, e, vulg., with the Latin commentators Victorinus,\n          Hilary and Jerome. Of these however it is clear that the Latin\n          authorities can have little weight in such a case, as the\n          preposition might have been introduced by the translator. All\n          the other Greek MSS with several Greek fathers omit \u1f10\u03bd.\n\n[Sidenote: \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2.]\n\n  (3) \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2 omitted in B, d, e. Of the Ambrosian Hilary\n          Tischendorf says 'fluct. lectio'; but his comment 'In quo enim\n          est spiritus, semper spiritualia meditatur' seems certainly to\n          recognise the word. It appears to be found in every other\n          authority.\n\n[Sidenote: \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3.]\n\n  (5) \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u05d0^* B with Origen in Cramer's _Catena_, p. 201.\n\n          \u1f10\u03bd \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 K L, and the vast majority of later MSS, the\n            Armenian and \u00c6thiopic Versions, Euthalius (Tischendorf's\n            MS), Theodoret, and others. The Harclean Syriac (text) is\n            quoted by Tischendorf and Tregelles in favour of \u1f10\u03bd \u03c4\u1fc7\n            \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3, but it is impossible to say whether the translator\n            had or had not the preposition.\n\n          \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2 \u05d0^c A D F G P, 47, 8^{pe}; the Old Latin,\n            Vulgate, Memphitic, Peshito Syriac, and Gothic Versions,\n            together with the margin of the Harclean Syriac; the fathers\n            Basil (II. p. 464), Victorinus (probably), Theodore of\n            Mopsuestia, the Ambrosian Hilary, Jerome, and others.\n            Chrysostom (as read in the existing texts) wavers between \u1f10\u03bd\n            \u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 and \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03b9\u03c2. This form of the reading is\n            an attempt to bring Ephes. into harmony with Col., just as\n            (6) is an attempt to bring Col. into harmony with Ephes.\n\nIt will be seen how slenderly B is supported; and yet we can hardly\nresist the impression that it has the right reading in all three cases.\nIn the omission of \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03b1\u1fd6\u03c2 more especially, where the support is\nweakest, this impression must, I think, be very strong.\n\n[Sidenote: Excellence of B elsewhere.]\n\nThis highly favourable estimate of B is our starting-point; and on the\nwhole it will be enhanced as we proceed. Thus for instance in i. 22 and\nii. 2 we shall find this MS alone (with one important Latin father)\nretaining the correct text; in the latter case amidst a great\ncomplication of various readings. And when again, as in iv. 8, we find B\nfor once on the side of a reading which might otherwise be suspected as\na harmonistic change, this support alone will weigh heavily in its\nfavour. Other cases in which B (with more or less support) preserves the\ncorrect reading against the mass of authorities are ii. 2 \u03c0\u1fb6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2,\nii. 7 \u03c4\u1fc7 \u03c0\u03af\u03c3\u03c4\u03b5\u03b9, ii. 13 \u03c4\u03bf\u1fd6\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c0\u03c4\u03ce\u03bc\u03b1\u03c3\u03b9\u03bd (omitting \u1f10\u03bd), v. 12 \u03c3\u03c4\u03b1\u03b8\u1fc6\u03c4\u03b5,\ntogether with several instances which will appear in the course of the\nfollowing investigation. On the other hand its value must not be\noverestimated. [Sidenote: False readings in B.]Thus in iv. 3 \u03c4\u1f78\n\u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03b4\u03b9' \u1f43 \u03ba\u03b1\u1f76 \u03b4\u03ad\u03b4\u03b5\u03bc\u03b1\u03b9[536] there can be little doubt\nthat the great majority of ancient authorities correctly read \u03b4\u03b9' \u1f45,\nthough B F G have \u03b4\u03b9' \u1f45\u03bd: but the variation is easily explained. A\nsingle stroke, whether accidental or deliberate, alone would be\nnecessary to turn the neuter into a masculine and make the relative\nagree with the substantive nearest to it in position. Again in ii. 10 \u1f45\u03c2\n\u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae, the reading of B which substitutes \u1f45 for \u1f45\u03c2 is plainly\nwrong, though supported in this instance by D F G 47^*, by the Latin\ntext d, and by Hilary in one passage (_de Trin._ ix. 8, II. p. 263),\nthough elsewhere (ib. i. 13, I. p. 10) he reads \u1f45. But here again we\nhave only an instance of a very common interchange. Whether for\ngrammatical reasons or from diplomatic confusion or from some other\ncause, five other instances of this interchange occur in this short\nepistle alone; i. 15 \u1f45 for \u1f45\u03c2 F G; i. 18 \u1f45 for \u1f45\u03c2 F G; i. 24 \u1f45\u03c2 for \u1f45 C\nD^* etc.; i. 27 \u1f45\u03c2 for \u1f45 \u05d0 C D K L etc.; iii. 14 \u1f45\u03c2 for \u1f45 \u05d0^* D. Such\nreadings again as the omission of \u03ba\u03b1\u1f76 \u03b1\u1f30\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03b9 i. 9 by B K, or of \u03b4\u03b9'\n\u03b1\u1f50\u03c4\u03bf\u1fe6 in i. 20 by B D^* F G etc., or of \u1f21 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae in iv. 16 by B\nalone, need not be considered, since the motive for the omission is\nobvious, and the authority of B will not carry as great weight as it\nwould in other cases. Similarly the insertion of \u1f21 in i. 18, \u1f21 \u1f00\u03c1\u03c7\u03ae, by\nB, 47, 67^{**}, b^{scr}, and of \u03ba\u03b1\u03af in ii. 15, \u03ba\u03b1\u1f76 \u1f10\u03b4\u03b5\u03b9\u03b3\u03bc\u03ac\u03c4\u03b9\u03c3\u03b5\u03bd, by B\nalone, do not appear to deserve consideration, because in both instances\nthese readings would suggest themselves as obvious improvements. In\nother cases, as in the omission of \u03c4\u1fc6\u03c2 before \u03b3\u1fc6\u03c2 (i. 20), and of \u1f11\u03bd\u03af in\n\u1f10\u03bd \u1f11\u03bd\u03af \u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 (iii. 15), the scribe of B has erred as any scribe might\nerr.\n\nFootnote 536:\n\n  In this passage B (with some few other authorities) has \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 for\n  \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, thus substituting a commoner expression (ii. 2, 1 Cor.\n  iv. 1, Rev. x. 7; comp. 1 Cor. ii. 1, v.l.) for a less common (Ephes.\n  iii. 4).\n\n                  *       *       *       *       *\n\nThe various readings in this epistle are more perplexing than perhaps in\nany portion of St Paul's Epistles of the same length. The following\ndeserve special consideration.\n\n                       i. 3  ==\u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af.==\n\n[Sidenote: i. 3 \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af,]]\n\nOn this very unusual collocation I have already remarked in the notes\n(p. 199). The authorities stand as follows:\n\n            (1) \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af B C^*.\n\n            (2) \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c4\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af D^* F G Chrysostom.\n\nOne or other is also the reading of the Old Latin (d, e, g, harl.^{**}),\nof the Memphitic, the two Syriac (Peshito and Harclean), the \u00c6thiopic,\nand the Arabic (Erpenius, Bedwell, Leipzig) Versions; and of Augustine\n(_de Unit. Eccl._ 45, IX. p. 368) and Cassiodorus (II. p. 1351, Migne).\n\n            (3) \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c4\u03c1\u03af \u05d0 A C^2 D^c K L P and apparently all\nthe other MSS; the Vulgate and Armenian Versions; Euthalius\n(Tischendorf's MS), Theodore of Mopsuestia (transl.), Theodoret, the\nAmbrosian Hilary, and others.\n\nA comparison of these authorities seems to show pretty clearly that \u03c4\u1ff7\n\u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af was the original reading. The other two were expedients for\ngetting rid of a very unusual collocation of words. [Sidenote: compared\nwith iii. 17,]The scribes have felt the same difficulty again in iii. 17\n\u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u1f76 \u03b4\u03b9' \u03b1\u1f50\u03c4\u03bf\u1fe6, and there again we find \u03ba\u03b1\u03af\ninserted before \u03c0\u03b1\u03c4\u03c1\u03af. In this latter instance however the great\npreponderance of ancient authority is in favour of the unusual form \u03c4\u1ff7\n\u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af.\n\n[Sidenote: and i. 12.]\n\nIt is worth observing also that in i. 12, where \u03c4\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af has the highest\nsupport, there is sufficient authority for \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af to create a\nsuspicion that there too it may be possibly the correct reading. Thus \u03c4\u1ff7\n\u03b8\u03b5\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af is read in \u05d0 37, while \u03b8\u03b5\u1ff7 \u03c4\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af stands in F G. One or\nother must have been the reading of some Old Latin and Vulgate texts (f,\ng, m, fuld.), of the Peshito Syriac, of the Memphitic (in some texts;\nfor others read \u03c4\u1ff7 \u03c0\u03b1\u03c4\u03c1\u03af simply), of the Arabic (Bedwell), of the\nArmenian (Uscan), and of Origen (II. p. 451, the Latin translator);\nwhile several other authorities, Greek and Latin, read \u03c4\u1ff7 \u03b8\u03b5\u1ff7 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c4\u03c1\u03af.\n\n[Sidenote: Unique collocation.]\n\nThere is no other instance of this collocation of words, \u1f41 \u0398\u03b5\u1f78\u03c2 \u03c0\u03b1\u03c4\u03ae\u03c1,\nin the Greek Testament, so far as I remember; and it must be regarded as\npeculiar to this epistle.\n\n                   i. 4 ==\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd [\u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5]==.\n\n[Sidenote: i. 4 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd [\u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5].]\n\nHere the various readings are;\n\n          (1) \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd B.\n\n          (2) \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5 A \u05d0 C D^* F G P 17, 37, 47; the Old\n              Latin and Vulgate, Memphitic (apparently), and Harclean\n              Syriac Versions; the Ambrosian Hilary, Theodore of\n              Mopsuestia (transl.), and others.\n\n          (3) \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u03c4\u03ae\u03bd. D^c K L; the Peshito Syriac (apparently),\n              and Armenian (apparently) Versions; Chrysostom, Theodoret\n              and others.\n\nIf the question were to be decided by external authority alone, we could\nnot hesitate. It is important however to observe that (2) conforms to\nthe parallel passage Philem. 5 \u1f00\u03ba\u03bf\u03cd\u03c9\u03bd \u03c3\u03bf\u03c5 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u1f23\u03bd\n\u1f14\u03c7\u03b5\u03b9\u03c2, while (3) conforms to the other parallel passage Ephes. i. 15 \u03ba\u03b1\u1f76\n[\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd] \u03c4\u1f74\u03bd \u03b5\u1f30\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f05\u03b3\u03b9\u03bf\u03c5\u03c2. Thus, though \u1f23\u03bd \u1f14\u03c7\u03b5\u03c4\u03b5 is so\nhighly supported and though it helps out the sense, it is open to\nsuspicion. Still the omission in B may be an instance of that impatience\nof apparently superfluous words, which sometimes appears in this MS.\n\n                    i. 7 ==\u1f51\u03c0\u1f72\u03c1 \u1f21\u03bc\u1ff6\u03bd \u03b4\u03af\u03b1\u03ba\u03bf\u03bd\u03bf\u03f2==.\n\n[Sidenote: i. 7 \u1f51\u03c0\u1f72\u03c1 \u1f21\u03bc\u1ff6\u03bd.]\n\nHere there is a conflict between MSS and Versions.\n\n        (1) \u1f21\u03bc\u1ff6\u03bd A B \u05d0^* D^* F G, 3, 13, 33, 43, 52, 80, 91, 109. This\n            must also have been the reading of the Ambrosian Hilary\n            (though the editors make him write 'pro _vobis_'), for he\n            explains it 'qui eis ministravit gratiam Christi _vice\n            Apostoli_.'\n\n        (2) \u1f51\u03bc\u1ff6\u03bd \u05d0^c C D^b K L P, 17, 37, 47, and many others; the\n            Vulgate, the Peshito and Harclean Syriac, the Memphitic,\n            Gothic, and Armenian Versions; Chrysostom, Theodore of\n            Mopsuestia (transl.), and Theodoret (in their respective\n            texts, for with the exception of Chrysostom there is nothing\n            decisive in their comments), with others.\n\nThe Old Latin is doubtful; d, e having _vobis_ and g _nobis_.\n\nThough the common confusion between these two words even in the best MSS\nis a caution against speaking with absolute certainty, yet such a\ncombination of the highest authorities as we have here for \u1f21\u03bc\u1ff6\u03bd does not\nleave much room for doubt: and considerations of internal criticism\npoint in the same direction. See the note on the passage.\n\n                      i. 12 ==\u03c4\u1ff7 \u1f31\u03ba\u03b1\u03bd\u03ce\u03f2\u03b1\u03bd\u03c4\u03b9==.\n\n[Sidenote: i. 12 \u1f31\u03ba\u03b1\u03bd\u03ce\u03c3\u03b1\u03bd\u03c4\u03b9.]\n\nAgainst this, which is the reading of all the other ancient authorities,\nwe have\n\n        (2) \u03c4\u1ff7 \u03ba\u03b1\u03bb\u03ad\u03c3\u03b1\u03bd\u03c4\u03b9 D^* F G, 17, 80, with the Latin authorities d,\n            e, f, g, m, and the Gothic, Armenian, and \u00c6thiopic Versions.\n            It is so read also by the Ambrosian Hilary, by Didymus _de\n            Trin._ iii. 4 (p. 346), and by Vigilius Thapsensis _c.\n            Varim._ i. 50 (p. 409).\n\n        (3) \u03c4\u1ff7 \u03ba\u03b1\u03bb\u03ad\u03c3\u03b1\u03bd\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f31\u03ba\u03b1\u03bd\u03ce\u03c3\u03b1\u03bd\u03c4\u03b9, found in B alone.\n\nHere the confusion between ==\u03c4\u03c9\u03b9\u03b9\u03ba\u03b1\u03bd\u03c9\u03f2\u03b1\u03bd\u03c4\u03b9== and ==\u03c4\u03c9\u03b9\u03ba\u03b1\u03bb\u03b5\u03f2\u03b1\u03bd\u03c4\u03b9==\nwould be easy, more especially at a period prior to the earliest\nexisting MSS, when the iota adscript was still written; while at the\nsame time \u03ba\u03b1\u03bb\u03ad\u03c3\u03b1\u03bd\u03c4\u03b9 would suggest itself to scribes as the obvious word\nin such a connexion. It is a Western reading.\n\nThe text of B obviously presents a combination of both readings.\n\n                       i. 14 ==\u1f10\u03bd \u1fa7 \u1f14\u03c7\u03bf\u03bc\u03b5\u03bd==.\n\n[Sidenote: i. 14 \u1f14\u03c7\u03bf\u03bc\u03b5\u03bd or \u1f14\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd?]\n\nFor \u1f14\u03c7\u03bf\u03bc\u03b5\u03bd B, the Memphitic Version, and the Arabic (Bedwell, Leipzig),\nread \u1f14\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd. This is possibly the correct reading. In the parallel\npassage, Ephes. i. 7, several authorities (\u05d0^* D^*, the Memphitic and\n\u00c6thiopic Versions, and the translator of Iren\u00e6us v. 14. 3) similarly\nread \u1f14\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd for \u1f14\u03c7\u03bf\u03bc\u03b5\u03bd. It may be conjectured that \u1f14\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd in these\nauthorities was a harmonistic change in Ephes. i. 7, to conform to the\ntext which they or their predecessors had in Col. i. 14. Tischendorf on\nEphes. l.c. says 'aut utroque loco \u03b5\u03c7\u03bf\u03bc\u03b5\u03bd aut \u03b5\u03c3\u03c7\u03bf\u03bc\u03b5\u03bd Paulum scripsisse\nputo'; but if any inference can be drawn from the phenomena of the MSS,\nthey point rather to a different tense in the two passages.\n\n                      i. 22 ==\u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5==.\n\n[Sidenote: i. 22 \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5.]\n\nThis reading is perhaps the highest testimony of all to the great value\nof B.\n\nThe variations are;\n\n        (1) \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 B. This also seems to be the reading of\n            Hilary of Poitiers _In xci Psalm._ 9 (I. p. 270), who\n            transfers the Apostle's language into the first person, 'cum\n            aliquando essemus alienati et inimici sensus ejus in factis\n            malis, nunc autem reconciliati sumus corpore carnis ejus.'\n\n        (2) \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03ba\u03b7\u03c4\u03b1\u03b9 17.\n\n        (3) \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03b1\u03b3\u03ad\u03bd\u03c4\u03b5\u03c2 D^* F G, and the Latin authorities d, e,\n            g, m, the Gothic Version, the translator of Iren\u00e6us (v. 14.\n            3), and others.\n\n        (4) \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd, all the other authorities.\n\nOf these (2) is obviously a corruption of (1) from similarity of sound;\nand (3) is an emendation, though a careless emendation, of (1) for the\nsake of the grammar. It should have been \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03b1\u03b3\u03ad\u03bd\u03c4\u03b1\u03c2. The reading\ntherefore must lie between \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 and \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd. This latter\nhowever is probably a grammatical correction to straighten the syntax.\nIn the Memphitic a single letter \u2c81\u03eb for \u2c81\u03e5 would make the difference\nbetween \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b7\u03bb\u03bb\u03ac\u03b3\u03b7\u03c4\u03b5 and \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ae\u03bb\u03bb\u03b1\u03be\u03b5\u03bd; but no variation from the\nlatter is recorded.\n\n                    ii. 2 ==\u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6, \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u1fe6==.\n\n[Sidenote: ii. 2 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.]\n\nThe various readings here are very numerous and at first sight\nperplexing; but the result of an investigation into their several claims\nis far from unsatisfactory. The reading which explains all the rest may\nsafely be adopted as the original.\n\n[Sidenote: Original reading.]\n\n(1) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==.\n\nThis is the reading of B and of Hilary of Poitiers, _de Trin._ ix. 62\n(I. p. 306), who quotes the passage _sacramenti Dei Christi in quo_\netc., and wrongly explains it 'Deus Christus sacramentum est'.\n\n[Sidenote: Variations;]\n\nAll the other variations are derived from this, either by explanation or\nby omission or by amplification.\n\n[Sidenote: (_a_) by interpretation,]\n\nBy explanation we get;\n\n(2) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03bf \u03b5\u03f2\u03c4\u03b9\u03bd \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03f2==,\n\nthe reading of D, with the Latin authorities d, e, which have _Dei quod\nest Christus_. So it is quoted by Vigilius Thapsensis _c. Varim._ i. 20\n(p. 380), and in a slightly longer form by Augustine _de Trin._ xiii. 24\n(VIII. p. 944) _mysterium Dei quod est Christus Jesus_.\n\n(3) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03b5\u03bd \u03c7\u03c1\u03b9\u03f2\u03c4\u03c9==.\n\nSo it is twice quoted by Clement of Alexandria _Strom._ v. 10 (p. 683),\n_ib._ 12 (p. 694); or\n\n            ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03c4\u03bf\u03c5 \u03b5\u03bd \u03c7\u03c1\u03b9\u03f2\u03c4\u03c9==,\n\nthe reading of 17.\n\nSo the Ambrosian Hilary (both text and commentary) has _Dei in Christo_.\nAnd the Armenian has the same lengthened out, _Dei in Christo Jesu_\n(Zohrab) or _Dei patris in Christo Jesu_ (Uscan).\n\n(4) _Domini quod de Christo_\n\nis the \u00c6thiopic rendering. Whether this represents another various\nreading in the Greek or whether the paraphrase is the translator's own,\nit is impossible to say.\n\n[Sidenote: (_b_) by omission,]\n\nThe two following variations strive to overcome the difficulty by\nomission;\n\n(5) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5==,\n\nthe reading of D by a second hand, of P, 37, 67^{**}, 71, 80, 116.\n\n(6) ==\u03c4\u03bf\u03c5 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==,\n\nthe reading of Euthalius in Tischendorf's MS; but Tischendorf adds the\ncaution 'sed non satis apparet'.\n\n[Sidenote: (_c_) by amplification;]\n\nAll the remaining readings are attempts to remedy the test by\namplification. They fall into two classes; those which insert \u03c0\u03b1\u03c4\u03c1\u03cc\u03c2 so\nas to make \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 dependent on it, (7), (8), and those which separate\n\u0398\u03b5\u03bf\u1fe6 from \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 by the interposition of a \u03ba\u03b1\u03af, (9), (10), (11).\n\n[Sidenote: (i) by inserting \u03c0\u03b1\u03c4\u03c1\u03cc\u03c2 to govern \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6;]\n\n(7) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03c0\u03b1\u03c4\u03c1\u03bf\u03f2 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==,\n\nthe reading of \u05d0 (by the first hand). Tischendorf also adds b^{scr*} and\no^{scr}; but I read Scrivener's collations differently (_Cod. Aug._ p.\n506): or\n\n            ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03c0\u03b1\u03c4\u03c1\u03bf\u03f2 \u03c4\u03bf\u03c5 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==,\n\nthe reading of A C, 4.\n\nOne or other is the reading of the Thebaic Version (given by Griesbach)\nand of the Arabic (Leipz.).\n\nA lengthened form of the same, _Dei patris Christi Jesu_, appears in the\noldest MSS of the Vulgate, am. fuld. f: and the same is also the reading\nof the Memphitic (Boetticher).\n\n(8) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c0\u03b1\u03c4\u03c1\u03bf\u03f2 \u03c4\u03bf\u03c5 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==.\n\nSo \u05d0 (the third hand), b^{scr*}, o^{scr}, and a corrector in the Harclean\nSyriac.\n\n[Sidenote: (ii) by separating \u0398\u03b5\u03bf\u1fe6 from \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 by a conjunction.]\n\n(9) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==,\n\nthe simplest form of the other class of emendations by amplification. It\nis found in Cyril _Thes._ p. 287.\n\n(10) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03c0\u03b1\u03c4\u03c1\u03bf\u03f2 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==.\n\nSo 47, 73, the Peshito Syriac (ed. princeps and Schaaf). And so it\nstands in the commentators Chrysostom (but with various readings) and\nTheodore of Mopsuestia (_Spicil. Solesm._ I. p. 131 _Dei patris et\nChristi_, but in Rab. Maur. _Op._ VI. p. 521 _Dei patris Christi Jesu_).\n\nPelagius has _Dei patris et Christi Jesu_>, and so the Memphitic\n(Wilkins).\n\n[Sidenote: The common text the latest development.]\n\n(11) ==\u03c4\u03bf\u03c5 \u03b8\u03b5\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03c0\u03b1\u03c4\u03c1\u03bf\u03f2 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u03c5==.\n\nThis, which may be regarded as the latest development, is the reading of\nthe received text. It is found in D (third hand) K L, and in the great\nmajority of cursives; in the text of the Harclean Syriac, and in\nTheodoret and others.\n\nBesides these readings some copies of the Vulgate exhibit other\nvariations; e.g. demid. _Dei patris et Domini nostri Christi Jesu_,\ntolet. _Dei Christi Jesu patris et Domini_.\n\nIt is not necessary to add any remarks. The justification of \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 as the original reading will have appeared in the variations to\nwhich it has given rise. The passage is altogether an instructive lesson\nin textual criticism.\n\n                 ii. 16 ==\u1f10\u03bd \u03b2\u03c1\u03ce\u03f2\u03b5\u03b9 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03c0\u03cc\u03f2\u03b5\u03b9.==\n\n[Sidenote: ii. 16 \u03ba\u03b1\u03af or \u1f24?]\n\nIn this reading B stands alone among the MSS; but it is supported by the\nPeshito Syriac and Memphitic Versions, by Tertullian (_adv. Marc._ v.\n19), and by Origen (_in Ioann._ x. \u00a7 11, IV. p. 174). The testimony of\nTertullian however is invalidated by the fact that he uses _et_ as the\nconnecting particle throughout the passage; and the Peshito Syriac also\nhas 'and' for \u1f24 in the two last clauses, though not in the second.\n\nThe rest have \u1f10\u03bd \u03b2\u03c1\u03ce\u03c3\u03b5\u03b9 \u1f22 \u1f10\u03bd \u03c0\u03cc\u03c3\u03b5\u03b9. This may be explained as a very\nobvious, though not very intelligent, alteration of scribes to conform\nto the disjunctive particles in the context, \u1f22 \u1f10\u03bd \u03bc\u03ad\u03c1\u03b5\u03b9 \u1f11\u03bf\u03c1\u03c4\u1fc6\u03c2 \u1f22\n\u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1\u03c2 \u1f22 \u03c3\u03b1\u03b2\u03b2\u03ac\u03c4\u03c9\u03bd.\n\nIn this same context it is probable that B retains the right form\n\u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1\u03c2 (supported here by F G and others) as against the Attic\n\u03bd\u03bf\u03c5\u03bc\u03b7\u03bd\u03af\u03b1\u03c2. In the same way in iii. 25 \u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c4\u03b1\u03b9 and iv. 9 \u03b3\u03bd\u03c9\u03c1\u03af\u03c3\u03bf\u03c5\u03c3\u03b9\u03bd B\n(with some others) has resisted the tendency to Attic forms.\n\n                        ii. 18 ==\u1f03 \u1f15\u03bf\u03c1\u03b1\u03ba\u03b5\u03bd==.\n\n[Sidenote: ii. 18, the omission of the negative.]\n\nThat this is the oldest reading which the existing texts exhibit, will\nappear from the following comparison of authorities.\n\n      (1) \u1f03 \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd) A B \u05d0^* D^*, 17^*, 28, 67^{**}; the Old\n            Latin authorities d, e, m; the Memphitic, \u00c6thiopic, and\n            Arabic (Leipz.) Versions; Tertull. _c. Marc._ v. 19 ('ex\n            visionibus angelicis'; and apparently Marcion himself also);\n            Origen (_c. Cels._ v. 8, I. p. 583, though the negative is\n            here inserted by De la Rue, and _in Cant._ ii, III. p. 63,\n            _in his qu\u00e6 videt_); Lucifer (_De non conv. c. h\u00e6r._ p. 782\n            Migne); the Ambrosian Hilary (_ad loc._ explaining it\n            'Inflantur motum pervidentes stellarum, quas angelos\n            vocat'). So too the unknown author of _Qu\u00e6st. ex N. T._ ii.\n            62 in August. _Op._ III. Appx. p. 156. Jerome (_Epist. cxxi\n            ad Alg._ \u00a7 10, I. p. 880) mentions both readings (with and\n            without the negative) as found in the Greek text: and\n            Augustine (_Epist._ 149, II. p. 514), while giving the\n            preference to _qu\u00e6 non vidit_>, says that some MSS have _qu\u00e6\n            vidit_>.\n\n      (2) \u1f03 \u03bc\u1f74 \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd (\u1f11\u03cc\u03c1\u03b1\u03ba\u03b5\u03bd) \u05d0^c C D^{bc} K L P, and the great\n            majority of cursives;\n\n      (3) \u1f03 \u03bf\u1f50\u03ba \u1f11\u03ce\u03c1\u03b1\u03ba\u03b5\u03bd F G.\n\nThe negative is also read in g; in the Vulgate, the Gothic, both the\nSyriac, and the Armenian Versions; in the translator of Origen _In Rom._\nix. \u00a7 42 (IV. p. 665), in Ambrose _In Psalm. cxviii Exp._ xx (I. p.\n1222), and in the commentators Pelagius, Chrysostom, Theodore of\nMopsuestia (_Spic. Solesm._ I. p. 132 'qu\u00e6 nec sciunt'), Theodoret, and\nothers.\n\nFrom a review of these authorities we infer that the insertion of the\nnegative was a later correction, and that \u1f03 \u1f15\u03c9\u03c1\u03b1\u03ba\u03b5\u03bd (or \u1f15\u03bf\u03c1\u03b1\u03ba\u03b5\u03bd)\nrepresents the prior reading. In my note I have expressed my suspicion\nthat \u1f03 \u1f15\u03c9\u03c1\u03b1\u03ba\u03b5\u03bd (or \u1f15\u03bf\u03c1\u03b1\u03ba\u03b5\u03bd) is itself corrupt, and that the original\nreading is lost.\n\n[Sidenote: The form \u1f15\u03bf\u03c1\u03b1\u03ba\u03b5\u03bd.]\n\nThe unusual form \u1f15\u03bf\u03c1\u03b1\u03ba\u03b5\u03bd is found in \u05d0 B^* C D P, and is therefore to\nbe preferred to \u1f15\u03c9\u03c1\u03b1\u03ba\u03b5\u03bd.\n\n                  ii. 23 ==[\u03ba\u03b1\u1f76] \u1f00\u03c6\u03b5\u03b9\u03b4\u03af\u1fb3 \u03f2\u03ce\u03bc\u03b1\u03c4\u03bf\u03f2==.\n\n[Sidenote: ii. 23. Is \u03ba\u03b1\u03af to be omitted?]\n\nHere \u03ba\u03b1\u03af is found in all the Greek copies except B, but is omitted in\nthese Latin authorities, m, the translator of Origen (_In Rom._ ix. \u00a7\n42, IV. p. 665), Hilary of Poitiers (_Tract. in xiv Ps._ \u00a7 7, p. 73),\nthe Ambrosian Hilary, Ambrose (_de Noe_ 25, p. 267), and Paulinus\n(_Epist._ 50, p. 292 sq.). We have more than once found B and Hilary\nalone in supporting the correct reading (i. 22, ii. 2); and this fact\ngives weight to their joint authority here. The omission also seems to\nexplain the impossible reading of d, e, which have _in religione et\nhumilitate sensus et vexationem corporis_, where for _et vexationem_, we\nshould probably read _ad vexationem_, as in the Ambrosian Hilary. There\nwas every temptation for a scribe to insert the \u03ba\u03b1\u03af so as to make\n\u1f00\u03c6\u03b5\u03b9\u03b4\u03af\u1fb3 range with the other datives: while on the other hand a finer\nappreciation of the bearing of the passage suggests that St Paul would\nhave dissociated it, so as to give it a special prominence.\n\nA similar instance occurs in iii. 12 \u1f61\u03c2 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1f76 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, \u1f05\u03b3\u03b9\u03bf\u03b9 \u03ba\u03b1\u1f76\n\u1f20\u03b3\u03b1\u03c0\u03b7\u03bc\u03ad\u03bd\u03bf\u03b9, where B omits the \u03ba\u03b1\u03af with 17 and the Thebaic Version[537].\nIn 219 \u03ba\u03b1\u1f76 \u1f05\u03b3\u03b9\u03bf\u03b9 is read for \u1f05\u03b3\u03b9\u03bf\u03b9 \u03ba\u03b1\u03af. The great gain in force leads to\nthe suspicion that this omission may be correct, notwithstanding the\nenormous preponderance of authority on the other side.\n\nFootnote 537:\n\n  It is true that in the text (_Spicil. Solesm._ I. p. 123, Rab. Maur.\n  Op. VII. p. 539, Migne) he is credited with the later Latin reading\n  _ut cognoscat qu\u00e6 circa vos sunt_, but his comment implies the other;\n  'Quoniam omnia vobis nota faciet Tychicus illa qu\u00e6 erga me sunt,\n  propterea a me directus est cum Onesimo fratre qui a vobis venerat, ut\n  nota vobis faciant qu\u00e6 erga nos sunt [= \u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd] et\n  oblectent vos per suum adventum [= \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03ad\u03c3\u1fc3 \u03c4\u1f70\u03c2 \u03ba\u03b1\u03c1\u03b4\u03af\u03b1\u03c2 \u1f51\u03bc\u1ff6\u03bd],\n  omnia qu\u00e6 hic aguntur manifesta facientes vobis.' See _Spicil.\n  Solesm._ l.c.; the comment is mutilated in Rab. Maur. _Op._ l.c.\n\n                   iv. 8. ==\u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd==.\n\n[Sidenote: iv. 8 \u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd.]\n\nOf the various readings of this passage I have already spoken (p. 29\nsq., note 1, p. 301).\n\nThe authorities are as follows:\n\n      (1) \u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd A B D^* F G P, 10, 17, 33, 35, 37, 44, 47,\n            71, 111, 116, 137; d, e, g; the Armenian and \u00c6thiopic\n            Versions; Theodore of Mopsuestia[537], Theodoret[538],\n            Jerome (on Ephes. vi. 21 sq., VII. p. 682), and Euthalius\n            (Tischendorf's MS). This is also the reading of \u05d0^*, except\n            that it has \u1f51\u03bc\u1ff6\u03bd for \u1f21\u03bc\u1ff6\u03bd.\n\n      (2) \u03b3\u03bd\u1ff7 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd \u05d0^c C D^{bc} K L and the majority of\n            cursives; the Memphitic, Gothic, Vulgate, and both Syriac\n            Versions; the Ambrosian Hilary, Jerome (on Philem. I, VII.\n            p. 748), Chrysostom (expressly), and others.\n\nFootnote 538:\n\n  In the text; but in the commentary he is made to write \u1f35\u03bd\u03b1 \u03b3\u03bd\u1ff7 \u03b3\u03ac\u03c1,\n  \u03c6\u03b7\u03c3\u03af, \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd, an impossible reading.\n\n[Sidenote: The various readings accounted for.]\n\nThe internal evidence is considered in the note on the passage, and\nfound to accord with the vast preponderance of external authority in\nfavour of \u03b3\u03bd\u1ff6\u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f21\u03bc\u1ff6\u03bd. The reading of \u05d0 by the first hand\nexhibits a transitional stage. It would appear as though the transcriber\nintended it to be read \u03b3\u03bd\u1ff7 \u03c4\u03b5 \u03c4\u1f70 \u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd. At all events this is the\nreading of III and of Io. Damasc. _Op._ II. p. 214. The variation \u03b3\u03bd\u1ff7 \u03c4\u1f70\n\u03c0\u03b5\u03c1\u1f76 \u1f51\u03bc\u1ff6\u03bd is thus easily explained. (1) \u1f21\u03bc\u1ff6\u03bd would be accidentally\nsubstituted for \u1f51\u03bc\u1ff6\u03bd; (2) \u03b3\u03bd\u1ff6\u03c4\u03b5 would then be read \u03b3\u03bd\u1ff7 \u03c4\u03b5; (3) the\nawkward and superfluous \u03c4\u03b5 would be omitted. In illustration of the\ntendency to conform the persons of the two verbs \u03b3\u03bd\u1ff7, \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03ad\u03c3\u1fc3, (see\np. 301) it may be mentioned that 17 reads \u03b3\u03bd\u1ff6\u03c4\u03b5, \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03ad\u03c3\u03b7\u03c4\u03b5, both here\nand in Ephes. vi. 22.\n\n                    iv. 15. ==\u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03bf\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd==.\n\n[Sidenote: iv. 15 \u03b1\u1f50\u03c4\u1ff6\u03bd.]\n\nThe readings here are:\n\n      (1) \u03b1\u1f50\u03c4\u1ff6\u03bd \u05d0 A C P, 5, 9, 17, 23, 34, 39, 47, 73; together with the\n            Memphitic Version, the Arabic (Leipz.), and Euthalius\n            (Tischendorf's MS). The Memphitic Version is commonly but\n            wrongly quoted in favour of \u03b1\u1f50\u03c4\u03bf\u1fe6, owing to a mistranslation\n            of Wilkins. But both Wilkins and Boetticher give without any\n            various reading \u03a0\u039f\u03a5\u0397\u0399, i.e. \u03bf\u1f36\u03ba\u03bf\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd. This seems also to\n            be the reading of Theodore of Mopsuestia (_Spic. Solesm._ I.\n            p. 133) _qu\u00e6 in domo eorum est ecclesia_; though in Rab.\n            Maur. _Op._ VI. p. 540 his text runs _qu\u00e6 in domo ejus est\n            ecclesiam_, and he is made to say _Nympham cum omnibus suis\n            qui in domo ejus sunt_.\n\n      (2) \u03b1\u1f50\u03c4\u1fc6\u03c2 B 67^{**}.\n\n      (3) \u03b1\u1f50\u03c4\u03bf\u1fe6 D F G K L and the great majority of cursives; and so the\n            Gothic Version, Chrysostom, and Theodoret (the latter\n            distinctly).\n\n[Sidenote: _Nymphas_ or _Nympha_?]]\n\nThe singular, whether \u03b1\u1f50\u03c4\u03bf\u1fe6 or \u03b1\u1f50\u03c4\u1fc6\u03c2, is the reading of the old Latin\nand Vulgate, which have _ejus_, and of the Armenian. The pronoun is also\nsingular in the Peshito and Harclean Syriac. In this language the same\nconsonants express masculine and feminine alike, the difference lying in\nthe pointing and vocalisation. And here the copies are inconsistent with\nthemselves. [Sidenote: The Syriac versions.]In the Peshito (both the\neditio princeps and Schaaf) the proper name is vocalised as a feminine\n_Numph\u0113_ (= \u039d\u03cd\u03bc\u03c6\u03b7), and yet \u0712\u0712\u071d\u072c\u0717 is treated as having a masculine affix\n\u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6. In the text of the Harclean \u0715\u071d\u0720\u0717\u073f is pointed thus, as\na feminine \u03b1\u1f50\u03c4\u1fc6\u03c2; while the margin gives the alternative reading \u0715\u071d\u0720\u0717\n(without the point) = \u03b1\u1f50\u03c4\u03bf\u1fe6. The name itself is written Nympha, which\naccording to the transliteration of this version might stand either for\na masculine (as _Barnaba_, _Luka_, in the context, for \u0392\u03b1\u03c1\u03bd\u03b1\u03b2\u03b1\u03c2, \u039b\u03bf\u03c5\u03ba\u1fb6\u03c2)\nor for a feminine (since _Demas_, _Epaphras_, are written with an\n_s_)[539]. [Sidenote: The Latin authorities.]The Latin _ejus_ leaving\nthe gender undetermined, the Latin commentators were free to take either\nNymphas or Nympha; and, as Nympha was common Latin form of \u039d\u03cd\u03bc\u03c6\u03b7, they\nwould naturally adopt the female name. So the commentator Hilary\ndistinctly.\n\nFootnote 539:\n\n  More probably the latter. In Rom. xvi the terminations -\u03b1 and \u1fb6\u03c2 for\n  the feminine and masculine names respectively are carefully reproduced\n  in the Harclean Version. In ver. 15 indeed we have _Julias_, but the\n  translator doubtless considered the name to be a contraction for\n  _Julianus_. The proper Syriac termination _-a_ seems only to be\n  employed for the Greek -\u03b1\u03c2 in very familiar names such as _Barnaba_,\n  _Luka_.\n\nIt should be added that the word is accentuated as a masculine \u03bd\u03c5\u03bc\u03c6\u1fb6\u03bd in\nD^c L P, and as a feminine \u03bd\u03cd\u03bc\u03c6\u03b1\u03bd in B^c and Euthalius (Tischendorf's\nMS.).\n\n\n                      _On the meaning of \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1._\n\n[Sidenote: The meaning of the verb \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd.]\n\nThe verb \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd has two senses. It signifies either (1) 'To fill', e.g.\nActs ii. 2 \u1f10\u03c0\u03bb\u03ae\u03c1\u03c9\u03c3\u03b5\u03bd \u1f45\u03bb\u03bf\u03bd \u03c4\u1f78\u03bd \u03bf\u1f36\u03ba\u03bf\u03bd; or (2) 'To fulfil, complete,\nperfect, accomplish', e.g. Matt. xxvi. 56 \u1f35\u03bd\u03b1 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1ff6\u03c3\u03b9\u03bd \u03b1\u1f31 \u03b3\u03c1\u03b1\u03c6\u03b1\u03af, Rom.\nxiii. 8 \u03bd\u03cc\u03bc\u03bf\u03bd \u03c0\u03b5\u03c0\u03bb\u03ae\u03c1\u03c9\u03ba\u03b5\u03bd, Acts xii. 25 \u03c0\u03bb\u03b7\u03c1\u03ce\u03c3\u03b1\u03bd\u03c4\u03b5\u03c2 \u03c4\u1f74\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd. The\nlatter sense indeed is derived from the former, but practically it has\nbecome separate from it. The word occurs altogether about a hundred\ntimes in the New Testament, and for every one instance of the former\nsense there are at least four of the latter.\n\n[Sidenote: False issue raised respecting \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1]\n\n[Sidenote: resulting in theological confusion]\n\nIn the investigations which have hitherto been made into the\nsignification of the derived substantive \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1, as it occurs in the\nNew Testament, an almost exclusive prominence has been given to the\nformer meaning of the verb; and much confusion has arisen in\nconsequence. The question has been discussed whether \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 has an\nactive or a passive sense, whether it describes the filling substance or\nthe filled receptacle: and not unfrequently critics have arrived at the\nresult that different grammatical senses must be attached to it in\ndifferent passages, even within the limits of the same epistle. Thus it\nhas been maintained that the word has a passive sense 'id quod impletur'\nin Ephes. i. 23 \u03c4\u1fc7 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3 \u1f25\u03c4\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6, \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03c4\u1f70\n\u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5, and an active sense 'id quod implet' in\nEphes. iii. 19 \u1f35\u03bd\u03b1 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03c4\u03b5 \u03b5\u1f30\u03c2 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6. Indeed so long\nas we see in \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd only the sense 'to fill', and refuse to contemplate\nthe sense 'to complete', it seems impossible to escape from the\ndifficulties which meet us at every turn, otherwise than by assigning to\nits derivative \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 both an active and a passive sense; but the\ngreatest violence is thus done to the connexion of theological ideas.\n\n[Sidenote: and disregard of grammar.]\n\nMoreover the disregard of lexical rules is not less violent[540].\nSubstantives in -\u03bc\u03b1, formed from the perfect passive, appear always to\nhave a passive sense. They may denote an abstract notion or a concrete\nthing; they may signify the action itself regarded as complete, or the\nproduct of[Sidenote: Meaning of substantives in -\u03bc\u03b1.] the action; but in\nany case they give the _result_ of the agency involved in the\ncorresponding verb. Such for example are \u1f04\u03b3\u03b3\u03b5\u03bb\u03bc\u03b1 'a message', \u1f05\u03bc\u03bc\u03b1 'a\nknot', \u1f00\u03c1\u03b3\u03cd\u03c1\u03c9\u03bc\u03b1 'a silver-made vessel', \u03b2\u03bf\u03cd\u03bb\u03b5\u03c5\u03bc\u03b1 'a plan', \u03b4\u03b9\u03ba\u03b1\u03af\u03c9\u03bc\u03b1 'a\nrighteous deed' or 'an ordinance', \u03b6\u03ae\u03c4\u03b7\u03bc\u03b1 'an investigation', \u03ba\u03ae\u03c1\u03c5\u03b3\u03bc\u03b1 'a\nproclamation', \u03ba\u03ce\u03bb\u03c5\u03bc\u03b1 'a hindrance', \u1f41\u03bc\u03bf\u03af\u03c9\u03bc\u03b1 'a likeness', \u1f45\u03c1\u03b1\u03bc\u03b1 'a\nvision', \u03c3\u03c4\u03c1\u1ff6\u03bc\u03b1 'a carpet', \u03c3\u03c6\u03b1\u03af\u03c1\u03c9\u03bc\u03b1 'a round thing', etc. In many cases\nthe same word will have two meanings, both however passive; it will\ndenote both the completed action and the result or object of the action:\ne.g. \u1f05\u03c1\u03c0\u03b1\u03b3\u03bc\u03b1 the 'robbery' or the 'booty', \u1f00\u03bd\u03c4\u03ac\u03bb\u03bb\u03b1\u03b3\u03bc\u03b1 the 'exchange' or\nthe 'thing given or taken in exchange', \u03b8\u03ae\u03c1\u03b5\u03c5\u03bc\u03b1 the 'hunt' or the\n'prey', \u03c0\u03ac\u03c4\u03b7\u03bc\u03b1 the 'tread' or the 'carpet', and the like. But in all\ncases the word is strictly passive; it describes that which might have\nstood after the active verb, either as the direct object or as the\ncognate notion. [Sidenote: Apparent exceptions.]The apparent exceptions\nare only apparent. Sometimes this deceptive appearance is in the word\nitself. Thus \u03ba\u03ac\u03bb\u03c5\u03bc\u03bc\u03b1 'a veil' seems to denote 'that which _covers_', but\nit is really derived from another sense and construction of \u03ba\u03b1\u03bb\u03cd\u03c0\u03c4\u03b5\u03b9\u03bd,\nnot 'to hide', but 'to wrap round' (e.g. Hom. _Il._ v. 315 \u03c0\u03c1\u03cc\u03c3\u03b8\u03b5 \u03b4\u03ad \u03bf\u1f31\n\u03c0\u03ad\u03c0\u03bb\u03bf\u03b9\u03bf \u03c6\u03b1\u03b5\u03b9\u03bd\u03bf\u1fe6 \u03c0\u03c4\u03cd\u03b3\u03bc' \u1f10\u03ba\u03ac\u03bb\u03c5\u03c8\u03b5\u03bd, xxi. 321 \u03c4\u03cc\u03c3\u03c3\u03b7\u03bd \u03bf\u1f31 \u1f04\u03c3\u03b9\u03bd \u03ba\u03b1\u03b8\u03cd\u03c0\u03b5\u03c1\u03b8\u03b5\n\u03ba\u03b1\u03bb\u03cd\u03c8\u03c9), and therefore is strictly passive. Sometimes again we may be\nled astray by the apparent connexion with the following genitive. Thus\nin Plut. _Mor._ 78 E \u03b4\u03ae\u03bb\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03c0\u03c1\u03bf\u03ba\u03cc\u03c0\u03c4\u03b5\u03b9\u03bd the word does not mean, as\nmight appear at first sight, 'a thing showing', but 'a thing shown', 'a\ndemonstration given'; nor in 2 Thess. i. 5 \u1f10\u03bd\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b4\u03b9\u03ba\u03b1\u03af\u03b1\u03c2 \u03ba\u03c1\u03af\u03c3\u03b5\u03c9\u03c2\nmust we explain \u1f15\u03bd\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 'a thing proving', but 'a thing proved', 'a\nproof'. And the same is probably the case also with such expressions as\n\u03c3\u03c5\u03bc\u03c0\u03bf\u03c3\u03af\u03c9\u03bd \u1f10\u03c1\u03ad\u03b8\u03b9\u03c3\u03bc\u03b1 (Critias in Athen. xiii. p. 600 D), \u03c4\u03cc\u03be\u03bf\u03c5 \u1fe5\u1fe6\u03bc\u03b1 (\u00c6sch.\n_Pers._ 147), and the like; where the substantives in -\u03bc\u03b1 are no more\ndeprived of their passive sense by the connexion, than they are in\n\u1f51\u03c0\u03cc\u03b4\u03b7\u03bc\u03b1 \u03c0\u03bf\u03b4\u1ff6\u03bd or \u03c3\u03c4\u03c1\u1ff6\u03bc\u03b1 \u03ba\u03bb\u03af\u03bd\u03b7\u03c2; though in such instances the license of\npoetical construction may often lead to a false inference. Analogous to\nthis last class of cases is Eur. _Troad._ 824 \u0396\u03b7\u03bd\u1f78\u03c2 \u1f14\u03c7\u03b5\u03b9\u03c2 \u03ba\u03c5\u03bb\u03af\u03ba\u03c9\u03bd\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03ba\u03b1\u03bb\u03bb\u03af\u03c3\u03c4\u03b1\u03bd \u03bb\u03b1\u03c4\u03c1\u03b5\u03af\u03b1\u03bd, not 'the filling', but 'the fulness of the\ncups, the brimming cups, of Zeus.'\n\nFootnote 540:\n\n  The meaning of this word \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 is the subject of a paper _De vocis\n  \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 vario sensu in N. T._ in Storr's _Opusc. Acad._ I. p. 144 sq.,\n  and of an elaborate note in Fritzsche's _Rom._ II. p. 469 sq. Storr\n  attempts to show that it always has an active sense 'id quod implet'\n  in the New Testament. Fritzsche rightly objects to assigning a\n  persistently active sense to a word which has a directly passive\n  termination: and he himself attributes to it two main senses, 'id quod\n  impletur' and 'id quo res impletur', the latter being the more common.\n  He apparently considers that he has surmounted the difficulties\n  involved in Storr's view, for he speaks of this last as a passive\n  sense, though in fact it is nothing more than 'id quod implet'\n  expressed in other words. In Rom. xiii. 10 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03bd\u03cc\u03bc\u03bf\u03c5 he concedes\n  an active sense 'legis completio', h. e. 'observatio'.\n\n[Sidenote: \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 connected with the second sense of \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd.]]\n\nNow if we confine ourselves to the second of the two senses above\nascribed to \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd, it seems possible to explain \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 in the same\nway, at all events in all the theological passages of St Paul and St\nJohn, without doing any violence to the grammatical form. As \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd is\n'to complete', so \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 is 'that which is completed', i.e. the\ncomplement[541], the full tale, the entire number or quantity, the\nplenitude, the perfection.\n\nFootnote 541:\n\n  The English word complement has two distinct senses. It is either (i)\n  the complete set, the entire quantity or number, which satisfies a\n  given standard or cadre, as e.g. the complement of a regiment; or (ii)\n  the number or quantity which, when added to a preexisting number or\n  quantity, produces completeness; as e.g. the complement of an angle,\n  i.e. the angle by which it falls short of being a complete right\n  angle. In other words, it is either the whole or the part. As a\n  theological term, \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 corresponds to the first of these two\n  senses; and with this meaning alone the word 'complement' will be used\n  in the following dissertation.\n\n[Sidenote: Its uses in Classical writers.]\n\nThis indeed is the primary sense to which its commonest usages in\nclassical Greek can be most conveniently referred. Thus it signifies (1)\n[Sidenote: (1) 'A ship's crew.'] 'A ship's crew': e.g. Xen. _Hell._ i.\n6. 16 \u03b4\u03b9\u1f70 \u03c4\u1f78 \u1f10\u03ba \u03c0\u03bf\u03bb\u03bb\u1ff6\u03bd \u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u1f10\u03c2 \u1f40\u03bb\u03af\u03b3\u03b1\u03c2 (\u03bd\u03b1\u1fe6\u03c2) \u1f10\u03ba\u03bb\u03b5\u03bb\u03ad\u03c7\u03b8\u03b1\u03b9 \u03c4\u03bf\u1f7a\u03c2\n\u1f00\u03c1\u03af\u03c3\u03c4\u03bf\u03c5\u03c2 \u1f10\u03c1\u03ad\u03c4\u03b1\u03c2. In this sense, which is very frequent, it is generally\nexplained as having an active force, 'that which fills the ships'; and\nthis very obvious explanation is recommended by the fact that \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd\n\u03bd\u03b1\u1fe6\u03bd is a recognized expression for 'manning a ship', e.g. Xen. _Hell._\ni. 6. 24. But \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 is used not only of the crew which mans a ship,\nbut also of the ship which is manned with a crew; e.g. Polyb. i. 49. 4,\n5 \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03af\u03b1\u03bd \u03c4\u1ff6\u03bd \u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd ... \u03c4\u1f70 \u03c0\u03c1\u03bf\u03c3\u03c6\u03ac\u03c4\u03c9\u03c2 \u03c0\u03b1\u03c1\u03b1\u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c4\u03b1 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1,\nLucian _Ver. Hist._ ii. 37, 38, \u1f00\u03c0\u1f78 \u03b4\u03cd\u03bf \u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u1f10\u03bc\u03ac\u03c7\u03bf\u03bd\u03c4\u03bf ... \u03c0\u03ad\u03bd\u03c4\u03b5\n\u03b3\u1f70\u03c1 \u03b5\u1f36\u03c7\u03bf\u03bd \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1; and it is difficult to see how the word could be\ntransferred from the crew to the ship as a whole, if the common\nexplanation were correct. Fritzsche (_Rom._ II. p. 469 sq.), to whom I\nam chiefly indebted for the passages quoted in this paragraph, has\nboldly given the word two directly opposite senses in the two cases,\nexplaining it in the one 'ea quibus naves complentur, _h. e._ vel socii\nnavales vel milites classiarii vel utrique', and in the other 'id quod\ncompletur, _v. c._ navigium'; but this severance of meaning can hardly\nbe maintained. On the other hand, if we suppose that the crew is so\ncalled as 'the complement,' (i.e. 'not that which fills the ship,' but\n'that which is itself full or complete in respect of the ship'), we\npreserve the passive sense of the word, while at the same time the\ntransference to the fully equipped and manned vessel itself becomes\nnatural. In this sense 'a complement' we have the word used again of an\narmy, [Sidenote: (2) 'Population.']Aristid. _Or._ I. p. 381 \u03bc\u03ae\u03c4\u03b5\n\u03b1\u1f50\u03c4\u03ac\u03c1\u03ba\u03b5\u03b9\u03c2 \u03b5\u03c3\u1f14\u03c3\u03b8\u03b1\u03b9 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u1f11\u03bd\u1f78\u03c2 \u03bf\u1f30\u03ba\u03ad\u03b9\u03bf\u03c5 \u03c3\u03c4\u03c1\u03b1\u03c4\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c7\u03ad\u03c3\u03b8\u03b1\u03b9. (2) It\nsometimes signifies 'the population of a city', Arist. _Pol._ iii. 13\n(p. 1284) \u03bc\u1f74 \u03bc\u03ad\u03bd\u03c4\u03bf\u03b9 \u03b4\u03c5\u03bd\u03b1\u03c4\u03bf\u1f76 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c3\u03c7\u03ad\u03c3\u03b8\u03b1\u03b9 \u03c0\u03cc\u03bb\u03b5\u03c9\u03c2 (comp. iv. 4, p.\n1291). Clearly the same idea of completeness underlies this meaning of\nthe word, so that here again it signifies 'the complement': comp. Dion.\nHal. _A. R._ vi. 51 \u03c4\u03bf\u1fe6 \u03b4' \u1f40\u03bb\u03af\u03b3\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03bf\u1f50\u03ba \u1f00\u03be\u03b9\u03bf\u03bc\u03ac\u03c7\u03bf\u03c5 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1f78 \u03c0\u03bb\u03b5\u1fd6\u03cc\u03bd\n\u1f10\u03c3\u03c4\u03b9 \u03b4\u03b7\u03bc\u03bf\u03c4\u03b9\u03ba\u03cc\u03bd \u03ba.\u03c4.\u03bb., Eur. _Ion_ 663 \u03c4\u1ff6\u03bd \u03c6\u03af\u03bb\u03c9\u03bd \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc' \u1f00\u03b8\u03c1\u03bf\u03af\u03c3\u03b1\u03c2 'the\nwhole body of his friends'. [Sidenote: (3) 'Total amount.'](3) 'The\nentire sum', Arist. _Vesp._ 660 \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03ac\u03bb\u03b1\u03bd\u03c4' \u1f10\u03b3\u03b3\u1f7a\u03c2 \u03b4\u03b9\u03c3\u03c7\u03af\u03bb\u03b9\u03b1\n\u03b3\u03af\u03b3\u03bd\u03b5\u03c4\u03b1\u03b9 \u1f21\u03bc\u1fd6\u03bd, 'From these sources a total of nearly two thousand\ntalents accrues to us'. [Sidenote: (4) 'Entire term.'](4) 'The full\nterm', Herod. iii. 22 \u1f40\u03b3\u03b4\u03ce\u03ba\u03bf\u03bd\u03c4\u03b1 \u03b4' \u1f14\u03c4\u03b5\u03b1 \u03b6\u03cc\u03b7\u03c2 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u1f00\u03bd\u03b4\u03c1\u1f76 \u03bc\u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03bf\u03bd\n\u03c0\u03c1\u03bf\u03ba\u03ad\u03b5\u03c3\u03b8\u03b1\u03b9. [Sidenote: (5) 'Fulfilment.'](5) 'The perfect attainment',\n'the full accomplishment', e.g. Philo _de Abr._ 46 (II. p. 39) \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1\n\u03c7\u03c1\u03b7\u03c3\u03c4\u1ff6\u03bd \u1f10\u03bb\u03c0\u03af\u03b4\u03c9\u03bd. In short the fundamental meaning of the word generally,\nthough perhaps not universally, is neither 'the filling material', nor\n'the vessel filled'; but 'that which is complete in itself', or in other\nwords 'plenitude, fulness, totality, abundance'.\n\n[Sidenote: Use of \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 in the Gospels.\n           Matt. ix. 16.]\n\nIn the Gospels the uses of the word present some difficulty. (1) In\nMatt. ix. 16 \u03b1\u1f34\u03c1\u03b5\u03b9 \u03b3\u1f70\u03c1 \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u1f31\u03bc\u03b1\u03c4\u03af\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c7\u03b5\u1fd6\u03c1\u03bf\u03bd\n\u03c3\u03c7\u03af\u03c3\u03bc\u03b1 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9, it refers to the \u1f10\u03c0\u03af\u03b2\u03bb\u03b7\u03bc\u03b1 \u1fe5\u03ac\u03ba\u03bf\u03c5\u03c2 \u1f00\u03b3\u03bd\u03ac\u03c6\u03bf\u03c5 which has gone\nbefore; but \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 need not therefore be equivalent to \u1f10\u03c0\u03af\u03b2\u03bb\u03b7\u03bc\u03b1 so as\nto mean the patch itself, as is often assumed. The following pronoun\n\u03b1\u1f50\u03c4\u03bf\u1fe6 is most naturally referred to \u1f10\u03c0\u03af\u03b2\u03bb\u03b7\u03bc\u03b1; and if so \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1\ndescribes 'the completeness', which results from the patch. The\nstatement is thus thrown into the form of a direct paradox, the very\ncompleteness making the garment more imperfect than before. [Sidenote:\nMark ii. 21.]In the parallel passage Mark ii. 21 the variations are\nnumerous, but the right reading seems certainly to be \u03b1\u1f34\u03c1\u03b5\u03b9 \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1\n\u1f00\u03c0' \u03b1\u1f50\u03c4\u03bf\u1fe6, \u03c4\u1f78 \u03ba\u03b1\u03b9\u03bd\u1f78\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03bb\u03b1\u03b9\u03bf\u1fe6 \u03ba.\u03c4.\u03bb. The received text omits the\npreposition before \u03b1\u1f50\u03c4\u03bf\u1fe6, but a glance at the authorities is convincing\nin favour of its insertion. In this case the construction will be \u03b1\u1f34\u03c1\u03b5\u03b9\n\u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 (nom.) \u1f00\u03c0' \u03b1\u1f50\u03c4\u03bf\u1fe6 (i.e. \u03c4\u03bf\u1fe6 \u1f31\u03bc\u03b1\u03c4\u03af\u03bf\u03c5, which has been mentioned\nimmediately before), \u03c4\u1f78 \u03ba\u03b1\u03b9\u03bd\u1f78\u03bd (\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u1f00 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03bb\u03b1\u03b9\u03bf\u1fe6 (\u1f31\u03bc\u03b1\u03c4\u03af\u03bf\u1f50; 'The\ncompleteness takes away from the garment, the new _completeness_ of the\nold _garment_', where the paradox is put still more emphatically.\n[Sidenote: Mark vi. 43.] (2) In Mark vi. 43 the right reading is \u03ba\u03b1\u1f76\n\u1f26\u03c1\u03b1\u03bd \u03ba\u03bb\u03b1\u03c3\u03bc\u03ac\u03c4\u03c9\u03bd \u03b4\u03ce\u03b4\u03b5\u03ba\u03b1 \u03ba\u03bf\u03c6\u03af\u03bd\u03bf\u03c5\u03c2 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1, i.e. 'full' or 'complete\nmeasures', where the apposition to \u03ba\u03bf\u03c6\u03af\u03bd\u03bf\u03c5\u03c2 obviates the temptation to\nexplain \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1 as 'ea qu\u00e6 implent'. [Sidenote: Mark viii. 20.]On the\nother hand in Mark viii. 20 \u03c0\u03cc\u03c3\u03c9\u03bd \u03c3\u03c0\u03c5\u03c1\u03af\u03b4\u03c9\u03bd \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1 \u03ba\u03bb\u03b1\u03c3\u03bc\u03ac\u03c4\u03c9\u03bd \u1f24\u03c1\u03b1\u03c4\u03b5;\nthis would be the _prima facie_ explanation; comp. Eccles. iv. 6 \u1f00\u03b3\u03b1\u03b8\u03cc\u03bd\n\u1f10\u03c3\u03c4\u03b9 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b4\u03c1\u03b1\u03ba\u1f78\u03c2 \u1f00\u03bd\u03b1\u03c0\u03b1\u03cd\u03c3\u03b5\u03c9\u03c2 \u1f51\u03c0\u1f72\u03c1 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1 \u03b4\u03cd\u03bf \u03b4\u03c1\u03b1\u03ba\u1ff6\u03bd \u03bc\u03cc\u03c7\u03b8\u03bf\u03c5. But it\nis objectionable to give an active sense to \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 under any\ncircumstances; and if in such passages the patch itself is meant, it\nmust still be so called, not because it fills the hole, but because it\nis itself fulness or full measure as regards the defect which needs\nsupplying.\n\n[Sidenote: Usage in St. Paul's Epistles.]\n\nFrom the Gospels we pass to the Epistles of St Paul, whose usage bears\nmore directly on our subject. And here the evidence seems all to tend in\nthe same direction. [Sidenote: 1 Cor. x. 26.](1) In 1 Cor. x. 26 \u03c4\u03bf\u1fe6\n\u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u03b3\u1f70\u03c1 \u1f21 \u03b3\u1fc6 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b1\u1f50\u03c4\u1fc6\u03c2 it occurs in a quotation from Ps.\nxxiv (xxiii). 1. The expressions \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2, \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2\n\u03b8\u03b1\u03bb\u03ac\u03c3\u03c3\u03b7\u03c2, occur several times in the LXX (e.g. Ps. xcvi (xcv). 11, Jer.\nviii. 16), where \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 is a translation of \u05de\u05dc\u05d0, a word denoting\nprimarily 'fulness', but having in its secondary uses a considerable\nlatitude of meaning ranging between 'contents' and 'abundance'. This\nlast sense seems to predominate in its Greek rendering \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1, and\nindeed the other is excluded altogether in some passages, e.g. Cant. v.\n13 \u1f10\u03c0\u1f76 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1 \u1f51\u03b4\u03ac\u03c4\u03c9\u03bd. [Sidenote: Rom. xiii. 10.](2) In Rom. xiii. 10\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03bd\u03cc\u03bc\u03bf\u03c5 \u1f21 \u1f00\u03b3\u03ac\u03c0\u03b7, the best comment on the meaning of the word is\nthe context, ver. 8 \u1f41 \u1f00\u03b3\u03b1\u03c0\u1ff6\u03bd \u03c4\u1f78\u03bd \u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u03bd\u03cc\u03bc\u03bf\u03bd \u03c0\u03b5\u03c0\u03bb\u03ae\u03c1\u03c9\u03ba\u03b5\u03bd, so that\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 here means the 'completeness' and so 'fulfilment,\naccomplishment': see the note on Gal. v. 14. [Sidenote: Rom. xv. 29.](3)\nIn Rom. xv. 29 \u1f10\u03bd \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9 \u03b5\u1f50\u03bb\u03bf\u03b3\u03af\u03b1\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f10\u03bb\u03b5\u03cd\u03c3\u03bf\u03bc\u03b1\u03b9, it plainly has\nthe sense of 'fulness, abundance'. [Sidenote: Gal. iv. 4.](4) In Gal.\niv. 4 \u1f45\u03c4\u03b5 \u03b4\u1f72 \u1f26\u03bb\u03b8\u03b5\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03c7\u03c1\u03bf\u03bd\u03bf\u1fe6 and [Sidenote: Eph. i.\n10.]Ephes. i. 10 \u03b5\u1f30\u03c2 \u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1ff6\u03bd \u03ba\u03b1\u03b9\u03c1\u1ff6\u03bd, its force is\nillustrated by such passages as Mark i. 15 \u03c0\u03b5\u03c0\u03bb\u03ae\u03c1\u03c9\u03c4\u03b1\u03b9 \u1f41 \u03ba\u03b1\u03b9\u03c1\u1f78\u03c2 \u03ba\u03b1\u1f76\n\u1f24\u03b3\u03b3\u03b9\u03ba\u03b5\u03bd \u1f21 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b1 \u03ba.\u03c4.\u03bb., Luke xxi. 24 \u1f04\u03c7\u03c1\u03b9 \u03bf\u1f57 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1ff6\u03c3\u03b9\u03bd \u03ba\u03b1\u03b9\u03c1\u03bf\u1f76 \u1f10\u03b8\u03bd\u1ff6\u03bd\n(comp. Acts ii. 1, vii. 23, 30, ix. 23, xxiv. 27), so that the\nexpressions will mean 'the full measure of the time, the full tale of\nthe seasons'. [Sidenote: Rom. xi. 25.](5) In Rom. xi. 25 \u03c0\u03ce\u03c1\u03c9\u03c3\u03b9\u03c2 \u1f00\u03c0\u1f78\n\u03bc\u03ad\u03c1\u03bf\u03c5\u03c2 \u03c4\u1ff7 \u1f38\u03c3\u03c1\u03b1\u1f74\u03bb \u03b3\u03ad\u03b3\u03bf\u03bd\u03b5\u03bd \u1f04\u03c7\u03c1\u03b9\u03c2 \u1f41\u1fe6 \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1ff6\u03bd \u1f10\u03b8\u03bd\u1ff6\u03bd \u03b5\u1f30\u03c3\u03ad\u03bb\u03b8\u1fc3, it seems\nto mean 'the full number', 'the whole body', (whether the whole\nabsolutely, or the whole relatively to God's purpose), of whom only a\npart had hitherto been gathered into the Church. [Sidenote: Rom. xi.\n12.](6) In an earlier passage in this chapter the same expression occurs\nof the Jews, xi. 12 \u03b5\u1f30 \u03b4\u1f72 \u03c4\u1f78 \u03c0\u03b1\u03c1\u03ac\u03c0\u03c4\u03c9\u03bc\u03b1 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c4\u1f78\n\u1f25\u03c4\u03c4\u03b7\u03bc\u03b1 \u03b1\u1f50\u03c4\u1ff6\u03bd \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f10\u03b8\u03bd\u1ff6\u03bd, \u03c0\u03cc\u03c3\u1ff3 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b1\u1f50\u03c4\u1ff6\u03bd. Here the\nantithesis between \u1f25\u03c4\u03c4\u03b7\u03bc\u03b1 and \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1, 'failure' and 'fulness', is not\nsufficiently direct to fix the sense of \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1; and (in the absence of\nanything to guide us in the context) we may fairly assume that it is\nused in the same sense of the Jews here, as of the Gentiles in ver. 25.\n\n[Sidenote: General result.]\n\nThus, whatever hesitation may be felt about the exact force of the word\nas it occurs in the Gospels, yet substantially one meaning runs through\nall the passages hitherto quoted from St Paul. In these \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 has its\nproper passive force, as a derivative from \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd 'to make complete'.\nIt is 'the full complement, the entire measure, the plenitude, the\nfulness'. There is therefore a presumption in favour of this meaning in\nother passages where it occurs in this Apostle's writings.\n\n[Sidenote: Theological passages in]\n\nWe now come to those theological passages in the Epistles to the\nColossians and Ephesians and in the Gospel of St John, for the sake of\nwhich this investigation has been undertaken. They are as follows;\n\n[Sidenote: Colossians and Ephesians.]\n\nCol. i. 19 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u1fc6\u03c3\u03b1\u03b9.\n\nCol. ii. 9 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2 \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2, \u03ba\u03b1\u1f76\n\u1f10\u03c3\u03c4\u1f72 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c0\u03b5\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9.\n\nEphes. i. 23 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f14\u03b4\u03c9\u03ba\u03b5\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74\u03bd \u1f51\u03c0\u1f72\u03c1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1fc7 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3, \u1f25\u03c4\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76\u03bd \u03c4\u1f78\n\u03c3\u1ff6\u03bc\u03b1 \u03b1\u1f50\u03c4\u03bf\u1fe6, \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5.\n\nEphes. iii. 19 \u1f35\u03bd\u03b1 \u03c0\u03bb\u03b7\u03c1\u03c9\u03b8\u1fc6\u03c4\u03b5 \u03b5\u1f30\u03c2 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6.\n\nEphes. iv. 13 \u03b5\u1f30\u03c2 \u1f04\u03bd\u03b4\u03c1\u03b1 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd, \u03b5\u1f30\u03c2 \u03bc\u03ad\u03c4\u03c1\u03bf\u03bd \u1f21\u03bb\u03b9\u03ba\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.\n\n[Sidenote: St. John.]\n\nJohn i. 14, 16 \u03ba\u03b1\u1f76 \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u03c3\u1f70\u03c1\u03be \u1f10\u03b3\u03ad\u03bd\u03b5\u03c4\u03bf \u03ba\u03b1\u1f76 \u1f10\u03c3\u03ba\u03ae\u03bd\u03c9\u03c3\u03b5\u03bd \u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd (\u03ba\u03b1\u1f76\n\u1f10\u03b8\u03b5\u03b1\u03c3\u03ac\u03bc\u03b5\u03b8\u03b1 \u03c4\u1f74\u03bd \u03b4\u03cc\u03be\u03b1\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6, \u03b4\u03cc\u03be\u03b1\u03bd \u1f61\u03c2 \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u03bf\u1fe6\u03c2 \u03c0\u03b1\u03c1\u1f70 \u03c0\u03b1\u03c4\u03c1\u03cc\u03c2) \u03c0\u03bb\u03ae\u03c1\u03b7\u03c2\n\u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 ... \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f21\u03bc\u03b5\u1fd6\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f10\u03bb\u03ac\u03b2\u03bf\u03bc\u03b5\u03bd\n\u03ba\u03b1\u1f76 \u03c7\u03ac\u03c1\u03b9\u03bd \u1f00\u03bd\u03c4\u1f76 \u03c7\u03ac\u03c1\u03b9\u03c4\u03bf\u03c2.\n\n[Sidenote: Ignatius.]]\n\nTo these should be added two passages from the Ignatian Epistles[542],\nwhich as belonging to the confines of the Apostolic age afford valuable\nillustration of the Apostolic language.\n\n_Ephes._ inscr. \u1f38\u03b3\u03bd\u03ac\u03c4\u03b9\u03bf\u03c2, \u1f41 \u03ba\u03b1\u1f76 \u0398\u03b5\u03bf\u03c6\u03cc\u03c1\u03bf\u03c2, \u03c4\u1fc7 \u03b5\u1f50\u03bb\u03bf\u03b3\u03b7\u03bc\u03ad\u03bd\u1fc3 \u1f10\u03bd \u03bc\u03b5\u03b3\u03ad\u03b8\u03b5\u03b9 \u0398\u03b5\u03bf\u1fe6\n\u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9[543] ... \u03c4\u1fc7 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3 \u03c4\u1fc7 \u1f00\u03be\u03b9\u03bf\u03bc\u03b1\u03ba\u03b1\u03c1\u03af\u03c3\u03c4\u1ff3 \u03c4\u1fc7 \u03bf\u1f54\u03c3\u1fc3 \u1f10\u03bd \u1f18\u03c6\u03ad\u03c3\u1ff3\n\u03ba.\u03c4.\u03bb.\n\n_Trall._ inscr. \u1f38\u03b3\u03bd\u03ac\u03c4\u03b9\u03bf\u03c2, \u1f41 \u03ba\u03b1\u1f76 \u0398\u03b5\u03bf\u03c6\u03cc\u03c1\u03bf\u03c2 ... \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3 \u1f01\u03b3\u03af\u1fb3 \u03c4\u1fc7 \u03bf\u1f54\u03c3\u1fc3 \u1f10\u03bd\n\u03a4\u03c1\u03ac\u03bb\u03bb\u03b5\u03c3\u03b9\u03bd ... \u1f23\u03bd \u03ba\u03b1\u1f76 )\u03b1\u03c3\u03c0\u03ac\u03b6\u03bf\u03bc\u03b1\u03b9 \u1f10\u03bd \u03c4\u1ff7 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9, \u1f10\u03bd \u1f00\u03c0\u03bf\u03c3\u03c4\u03bf\u03bb\u03b9\u03ba\u1ff7\n\u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u1fc6\u03c1\u03b9.\n\nFootnote 542:\n\n  The first of the two passages is contained in the short Syriac\n  recension of the Ignatian Epistles, though loosely translated; the\n  other is wanting there. I need not stop to enquire whether the second\n  was written by St Ignatius himself or by an interpolator. The\n  interpolated epistles, if they be interpolated, can hardly be later\n  than the middle of the second century and are therefore early enough\n  to afford valuable illustrations of the Apostles' language.\n\nFootnote 543:\n\n  The common texts read \u03ba\u03b1\u1f76 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9, but there can be little doubt\n  (from a comparison of the authorities) that \u03ba\u03b1\u1f76 should be struck out.\n  The present Syriac text has _et perfect\u00e6_ for \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9; but there is\n  no reason for supposing that the Syriac translator had another reading\n  before him. A slight change in the Syriac, \u0712\u072b\u0718\u0721\u0720\u071d\u0710 for \u0718\u0721\u072b\u0721\u0720\u071d\u0710, would\n  bring this Version into entire accordance with the Greek; and the\n  confusion was the more easy, because the latter word occurs in the\n  immediate context. Or the translator may have indulged in a paraphrase\n  according to his wont; just as in the longer Latin Version \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9\n  here is translated _replet\u00e6_.\n\n[Sidenote: The term has a recognised value]\n\nIt will be evident, I think, from the passages in St Paul, that the word\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 'fulness, plenitude', must have had a more or less definite\ntheological value when he wrote. This inference, which is suggested by\nthe frequency of the word, seems almost inevitable when we consider the\nform of the expression in the first passage quoted, Col. i. 19. The\nabsolute use of the word, \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 'all _the_ fulness', would\notherwise be unintelligible, for it does not explain itself. In my notes\nI have taken \u1f41 \u0398\u03b5\u03cc\u03c2 to be the nominative to \u03b5\u1f50\u03b4\u03cc\u03ba\u03b7\u03c3\u03b5\u03bd, but if the\nsubject of the verb were \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1, as some suppose, the inference\nwould be still more necessary. The word however, regarded as a\ntheological term, does not appear to have been adopted, like so many\nother expressions in the Apostolic writers[544], from the nomenclature\nof Alexandrian Judaism. [Sidenote: derived from Palestine and not\nAlexandria.]At least no instance of its occurrence in this sense is\nproduced from Philo. We may therefore conjecture that it had a\nPalestinian origin, and that the Essene Judaizers of Coloss\u00e6, whom St\nPaul is confronting, derived it from this source. In this case it would\nrepresent the Hebrew \u05de\u05dc\u05d0, of which it is a translation in the LXX, and\nthe Aramaic \u0721\u0718\u0720\u0719\u0710 or some other derivative of the same root, such being\nits common rendering in the Peshito.\n\nFootnote 544:\n\n  See the notes on Col. i. 15 sq.\n\n[Sidenote: It denotes the totality of the Divine powers, etc. in the\n           Colossian letter.]\n\nThe sense in which St Paul employs this term was doubtless the sense\nwhich he found already attached to it. He means, as he explicitly states\nin the second Christological passage of the Colossian Epistle (ii. 9),\nthe pleroma, the plenitude of 'the Godhead' or 'of Deity'. In the first\npassage (i. 19), though the word stands without the addition \u03c4\u1fc6\u03c2\n\u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03bf\u03c2, the signification required by the context is the same. The\ntrue doctrine of the one Christ, who is the absolute mediator in the\ncreation and government of the world, is opposed to the false doctrine\nof a plurality of mediators, 'thrones, dominions, principalities,\npowers'. An absolute and unique position is claimed for Him, because in\nHim resides 'all the pleroma', i.e. the full complement, the aggregate\nof the Divine attributes, virtues, energies. This is another way of\nexpressing the fact that He is the Logos, for the Logos is the synthesis\nof all the various \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2, in and by which God manifests Himself\nwhether in the kingdom of nature or in the kingdom of grace.\n\n[Sidenote: Analogy to its usage elsewhere: e.g.]\n\nThis application is in entire harmony with the fundamental meaning of\nthe word. The term has been transferred to the region of theology, but\nin itself it conveys exactly the same idea as before. It implies that\nall the several elements which are required to realise the conception\nspecified are present, and that each appears in its full proportions.\n[Sidenote: in Philo of the family]Thus Philo, describing the ideal state\nof prosperity which will result from absolute obedience to God's law,\nmentions among other blessings the perfect development of the family:\n'Men shall be fathers and fathers too of goodly sons, and women shall be\nmothers of goodly children, so that each household shall be the\n_pleroma_ of a numerous kindred, where no part or name is wanting of all\nthose which are used to designate relations, whether in the ascending\nline, as parents, uncles, grandfathers, or again in the descending line\nin like manner, as brothers, nephews, sons' sons, daughters' sons,\ncousins, cousins' sons, kinsmen of all degrees[545].' [Sidenote: and in\nAristotle, of the state.]So again Aristotle, criticizing the _Republic_\nof Plato, writes; 'Socrates says that a city (or state) is composed of\nfour classes, as its indispensable elements (\u03c4\u1ff6\u03bd \u1f00\u03bd\u03b1\u03b3\u03ba\u03b1\u03b9\u03bf\u03c4\u03ac\u03c4\u03c9\u03bd): by\nthese he means the weaver, the husbandman, the shoemaker, and the\nbuilder; and again, because these are not sufficient by themselves, he\nadds the smith and persons to look after the necessary cattle, and\nbesides them the merchant and the retail dealer: these together make up\nthe _pleroma_ of a city in its simplest form (\u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c0\u03c1\u03ce\u03c4\u03b7\u03c2 \u03c0\u03cc\u03bb\u03b5\u03c9\u03c2); thus he assumes that a city is formed to\nsupply the bare necessities of life (\u03c4\u1ff6\u03bd \u1f00\u03bd\u03b1\u03b3\u03ba\u03c9\u03bd \u03c7\u03ac\u03c1\u03b9\u03bd) etc.'[546]. From\nthese passages it will be seen that the adequacy implied by the word, as\nso used, consists not less in the variety of the elements than in the\nfulness of the entire quantity or number.\n\nFootnote 545:\n\n  _de Pr\u00e6m. et P\u00e6n._ 18 (II. p. 425). The important words are \u1f61\u03c2 \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03bd\n  \u03bf\u1f36\u03ba\u03bf\u03bd \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c0\u03bf\u03bb\u03c5\u03b1\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03c5 \u03c3\u03c5\u03b3\u03b3\u03b5\u03bd\u03b5\u03af\u03b1\u03c2, \u03bc\u03b7\u03b4\u03b5\u03bd\u1f78\u03c2 \u1f10\u03bb\u03bb\u03b5\u03b9\u03c6\u03b8\u03ad\u03bd\u03c4\u03bf\u03c2 \u1f22\n  \u03bc\u03ad\u03c1\u03bf\u03c5\u03c2 \u1f22 \u1f40\u03bd\u03cc\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1ff6\u03bd \u1f45\u03c3\u03b1 \u1f10\u03c0\u03b9\u03c6\u03b7\u03bc\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb. The construction of the\n  subsequent part of the sentence is obscure; and for \u1f41\u03bc\u03bf\u03af\u03bf\u03c5\u03c2 we should\n  probably read \u1f41\u03bc\u03bf\u03af\u03c9\u03c2.\n\nFootnote 546:\n\n  Arist. _Pol._ iv. 4 (p. 1291).\n\n[Sidenote: Transition from Colossians to Ephesians.]\n\nSo far the explanation seems clear. But when we turn from the Colossian\nletter to the Ephesian, it is necessary to bear in mind the different\naims of the two epistles. While in the former the Apostle's main object\nis to assert the supremacy of the Person of Christ, in the latter his\nprincipal theme is the life and energy of the Church, as dependent on\nChrist[547]. So the pleroma residing in Christ is viewed from a\ndifferent aspect, no longer in relation to God, so much as in relation\nto the Church. [Sidenote: Corresponding application of \u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03b1 to the\nChurch.]It is that plenitude of Divine graces and virtues which is\ncommunicated through Christ to the Church as His body. The Church, as\n_ideally_ regarded, the bride 'without spot or wrinkle or any such\nthing', becomes in a manner identified with Him[548]. All the Divine\ngraces which reside in Him are imparted to her; His 'fulness' is\ncommunicated to her: and thus she may be said to be His pleroma (i. 23).\nThis is the ideal Church. The actual militant Church must be ever\nadvancing, ever struggling towards the attainment of this ideal. Hence\nthe Apostle describes the end of all offices and administrations in the\nChurch to be that the collective body may attain its full and mature\ngrowth, or (in other words) may grow up to the complete stature of\nChrist's fulness[549]. But Christ's fulness is God's fulness. Hence in\nanother passage he prays that the brethren may by the indwelling of\nChrist be fulfilled till they attain to the _pleroma_ of God (iii. 19).\nIt is another way of expressing the continuous aspiration and effort\nafter holiness which is enjoined in our Lord's precept, 'Ye shall be\nperfect as your heavenly Father is perfect'[550].\n\nFootnote 547:\n\n  See the notes on Col. ii. 19 (p. 266).\n\nFootnote 548:\n\n  Ephes. v. 27 sq.\n\nFootnote 549:\n\n  The Apostle in this passage (Ephes. iv. 13) is evidently contemplating\n  the collective body, and not the individual believers. He writes \u03bf\u1f31\n  \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2, not \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2, and \u1f04\u03bd\u03b4\u03c1\u03b1 \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd, not \u1f04\u03bd\u03b4\u03c1\u03b1\u03c2 \u03c4\u03b5\u03bb\u03ad\u03b9\u03bf\u03c5\u03c2. As he has\n  said before _\u1f11\u03bd\u1f76 \u1f11\u03ba\u03ac\u03c3\u03c4\u1ff3_ \u1f21\u03bc\u1ff6\u03bd \u1f10\u03b4\u03cc\u03b8\u03b7 [\u1f21] \u03c7\u03ac\u03c1\u03b9\u03c2 _\u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78\n  \u03bc\u03ad\u03c4\u03c1\u03bf\u03bd_ \u03c4\u1fc6\u03c2 \u03b4\u03c9\u03c1\u03b5\u1fb6\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, so now he describes the result of\n  these various partial graces bestowed on _individuals_ to be the unity\n  and mature growth of the _whole_, 'the building up of the _body_',\n  \u03bc\u03b5\u03c7\u03c1\u1f76 \u03ba\u03b1\u03c4\u03b1\u03bd\u03c4\u03ae\u03c3\u03c9\u03bc\u03b5\u03bd _\u03bf\u1f31 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2_ \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd _\u1f11\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1_ ... \u03b5\u1f30\u03c2\n  _\u1f04\u03bd\u03b4\u03c1\u03b1_ \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03bd, \u03b5\u1f30\u03c2 \u03bc\u03ad\u03c4\u03c1\u03bf\u03bd \u1f21\u03bb\u03b9\u03ba\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.\n  This corporate being must grow up into the one colossal Man, the\n  standard of whose spiritual and moral stature is nothing less than the\n  pleroma of Christ Himself.\n\nFootnote 550:\n\n  Matt. v. 48.\n\n[Sidenote: Gospel of St. John.]\n\nThe Gospel of St John, written in the first instance for the same\nChurches to which the Epistle to the Ephesians was sent, has numerous\nand striking points of resemblance with St Paul's letter. This is the\ncase here. As St Paul tells the Ephesians that the ideal Church is the\npleroma of Christ and that the militant Church must strive to become the\npleroma of Christ, so St John (i. 14 sq.) after describing our Lord as\n\u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u03ae\u03c2, i.e. the unique and absolute representative of the Father,\nand as such 'full (\u03c0\u03bb\u03ae\u03c1\u03b7\u03c2) of grace and of truth', says that they, the\ndisciples, had 'received out of His pleroma' ever fresh accessions of\ngrace. Each individual believer in his degree receives a fraction of\nthat pleroma which is communicated whole to the ideal Church.\n\n[Sidenote: Ignatian letters.]\n\nThe use of the word is not very different in the Ignatian letters. St\nIgnatius greets this same Ephesian Church, to which St Paul and St John\nsuccessively here addressed the language already quoted, as 'blessed in\ngreatness by the pleroma of God the Father,' i.e. by graces imparted\nfrom the pleroma. To the Trallians again he sends a greeting 'in the\npleroma', where the word denotes the sphere of Divine gifts and\noperations, so that \u1f10\u03bd \u03c4\u1ff7 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b9 is almost equivalent to \u1f10\u03bd \u03c4\u1ff7 \u039a\u03c5\u03c1\u03af\u1ff3\nor \u1f10\u03bd \u03c4\u1ff7 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9.\n\n[Sidenote: Gnostic sects.]\n\nWhen we turn from Catholic Christianity to the Gnostic sects we find\nthis term used, though (with one important exception) not in great\nfrequency. Probably however, if the writings of the earlier Gnostics had\nbeen preserved, we should have found that it occupied a more important\nplace than at present appears. One class of early Gnostics separated the\nspiritual being Christ from the man Jesus; they supposed that the Christ\nentered Jesus at the time of His baptism and left him at the moment of\nHis crucifixion. Thus the Christ was neither born as a man nor suffered\nas a man. In this way they obviated the difficulty, insuperable to the\nGnostic mind, of conceiving the connexion between the highest spiritual\nagency and gross corporeal matter which was involved in the Catholic\ndoctrine of the Incarnation and Passion, and which Gnostics of another\ntype more effectually set aside by the theory of docetism, i.e. by\nassuming that the human body of our Lord was only a phantom body and not\nreal flesh and blood. Iren\u00e6us represents the former class as teaching\nthat 'Jesus was the receptacle of the Christ', and that the Christ\n'descended upon him from heaven in the form of a dove and after He had\ndeclared (to mankind) the nameless Father, entered (again) into the\npleroma imperceptibly and invisibly'[551]. [Sidenote: The\nCerinthians.]Here no names are given. But in another passage he ascribes\nprecisely the same doctrine, without however naming the pleroma, to\nCerinthus[552]. And in a third passage, which links together the other\ntwo, this same father, after mentioning this heresiarch, again alludes\nto the doctrine which maintained that the Christ, having descended on\nJesus at his baptism, 'flew back again into His own pleroma'[553]. In\nthis last passage indeed the opinions of Cerinthus are mentioned in\nconnexion with those of other Gnostics, more especially the\nValentinians, so that we cannot with any certainty attribute this\nexpression to Cerinthus himself. But in the first passage the unnamed\nheretics who maintained this return of the Christ 'into the pleroma' are\nexpressly distinguished from the Valentinians; and presumably therefore\nthe allusion is to the Cerinthians, to whom the doctrine, though not the\nexpression, is ascribed in the second passage. [Sidenote: Connexion of\nthis use with St Paul and with the Colossian heretics.]Thus there seems\nto be sufficient reason for attributing the use of the term to\nCerinthus[554]. This indeed is probable on other grounds. The term\n_pleroma_, we may presume, was common to St Paul and the Colossian\nheretics whom he controverts. To both alike it conveyed the same idea,\nthe totality of the divine powers or attributes or agencies or\nmanifestations. But after this the divergence begins. They maintained\nthat a single divine power, a fraction of the pleroma, resided in our\nLord: the Apostle urges on the contrary, that the whole pleroma has its\nabode in Him[555]. The doctrine of Cerinthus was a development of the\nColossian heresy, as I have endeavoured to show above[556]. He would\ntherefore inherit the term _pleroma_ from it. [Sidenote: The _pleroma_\nlocalized.]At the same time he seems to have given a poetical colouring\nto his doctrine, and so doing to have treated the pleroma as a\n_locality_, a higher spiritual region, from which this divine power,\ntypified by the dove-like form, issued forth as on wings, and to which,\ntaking flight again, it reascended before the Passion. If so, his\nlanguage would prepare the way for the still more elaborate poetic\nimagery of the Valentinians, in which the pleroma, conceived as a\nlocality, a region, an _abode_ of the divine powers, is conspicuous.\n\nFootnote 551:\n\n  iii. 16. 1 'Quoniam autem sunt qui dicunt Iesum quidem receptaculum\n  Christi fuisse, in quem desuper quasi columbam descendisse, et quum\n  indicasset innominabilem Patrem, incomprehensibiliter et invisibiliter\n  _intrasse in pleroma_'.\n\nFootnote 552:\n\n  i. 26. 1 'post baptismum descendisse in eum ab ea principalitate, qu\u00e6\n  est super omnia, Christum figura columb\u00e6; et tunc annuntiasse\n  incognitum Patrem et virtutes perfecisse: in fine autem _revolasse\n  iterum_ Christum de Iesu et Iesum passum esse et resurrexisse, etc.'\n\nFootnote 553:\n\n  iii. 11. 1 'iterum revolasse in suum pleroma'. This expression is the\n  connecting link between the other two passages. This third passage is\n  quoted more at length, above, p. 112: but I ought to have stated there\n  that _illi_ is referred by several critics to the Valentinians, and\n  that certainly some characteristic errors of the Valentinian teaching\n  are specified immediately after. The probable explanation seems to be\n  that _illi_ is intended to include the Gnostics generally, and that\n  Iren\u00e6us mentions in illustration the principal errors of Gnostic\n  teaching, irrespective of the schools to which they belong. He goes on\n  to say that St John in his Gospel desired to exclude 'omnia talia'.\n\nFootnote 554:\n\n  I have not been able however to verify the statement in Harvey's\n  _Iren\u00e6us_ I. p. lxxiii that 'The Valentinian notion of a spiritual\n  marriage between the souls of the elect and the angels of the Pleroma\n  originated with Cerinthus'.\n\nFootnote 555:\n\n  See p. 101 sq., and the notes on i. 19.\n\nFootnote 556:\n\n  p. 107 sq.\n\n[Sidenote: The term avoided by Basilides,]\n\nThe attitude of later Gnostics towards this term is widely divergent.\nThe word is not, so far as I am aware, once mentioned in connexion with\nthe system of Basilides. Indeed the nomenclature of this heresiarch\nbelongs to a wholly different type; and, as he altogether repudiated the\ndoctrine of emanations[557], it is not probable that he would have any\nfondness for a term which was almost inextricably entangled with this\ndoctrine.\n\nFootnote 557:\n\n  Hippol. _R. H._ vii. 22 \u03c6\u03b5\u03cd\u03b3\u03b5\u03b9 \u03b3\u1f70\u03c1 \u03c0\u03ac\u03bd\u03c5 \u03ba\u03b1\u1f76 \u03b4\u03ad\u03b4\u03bf\u03b9\u03ba\u03b5 \u03c4\u1f70\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c0\u03c1\u03bf\u03b2\u03bf\u03bb\u1f74\u03bd\n  \u03c4\u1ff6\u03bd \u03b3\u03b5\u03b3\u03bf\u03bd\u03cc\u03c4\u03c9\u03bd \u03bf\u1f50\u03c3\u03af\u03b1\u03c2 \u1f41 \u0392\u03b1\u03c3\u03b9\u03bb\u03b5\u03af\u03b4\u03b7\u03c2. Basilides asked why the absolute\n  First Cause should be likened to a spider spinning threads from\n  itself, or a smith or carpenter working up his materials. The later\n  Basilideans, apparently influenced by Valentinianism, superadded to\n  the teaching of their founder in this respect; but the strong language\n  quoted by Hippolytus leaves no doubt about the mind of Basilides\n  himself.\n\n[Sidenote: but prominent in Valentinianism.]\n\nOn the other hand with Valentinus and the Valentinians the doctrine of\nthe _pleroma_ was the very key-stone of their system; and, since at\nfirst sight it is somewhat difficult to connect their use of the term\nwith St Paul's, a few words on this subject may not be out of place.\n\n[Sidenote: Poetic teaching of Valentinus.]\n\nValentinus then dressed his system in a poetic imagery not unlike the\nmyths of his master Plato. But a myth or story involves action, and\naction requires a scene of action. Hence the mysteries of theology and\ncosmogony and redemption call for a _topographical_ representation, and\nthe pleroma appears not as an abstract idea, but as a locality.\n\n[Sidenote: _Topographical_ conception of the pleroma.]\n\nThe Valentinian system accordingly maps out the universe of things into\ntwo great regions, called respectively the _pleroma_ and the _kenoma_,\nthe 'fulness' and the 'void'. From a Christian point of view these may\nbe described as the kingdoms of light and of darkness respectively.\n[Sidenote: Antithesis of _pleroma_ and _kenoma_.]From the side of\nPlatonism, they are the regions of real and of phenomenal existences\u2014the\nworld of eternal archetypes or ideas, and the world of material and\nsensible things. The identification of these two antitheses was rendered\neasy for the Gnostic; because with him knowledge was one with morality\nand with salvation, and because also matter was absolutely bound up with\nevil. It is difficult to say whether the Platonism or the Christianity\npredominates in the Valentinian theology; but the former at all events\nis especially prominent in their conception of the relations between the\npleroma and the kenoma.\n\n[Sidenote: _Pleroma_ the abode of the \u00c6ons.]\n\nThe pleroma is the abode of the \u00c6ons, who are thirty in number. These\n\u00c6ons are successive emanations, of which the first pair sprang\nimmediately from the preexistent Bythus or Depth. This Bythus is deity\nin itself, the absolute first principle, as the name suggests; the\nprofound, unfathomable, limitless, of whom or of which nothing can be\npredicated and nothing known. Here again we have something like a\n_local_ representation. The \u00c6ons or emanations are plainly the\nattributes and energies of deity; they are, or they comprise, the\neternal ideas or archetypes of the Platonic philosophy. In short they\nare deity relative, deity under self-imposed limitations, deity derived\nand divided up, as it were, so as at length to be conceivable.\n\n[Sidenote: Different forms of Valentinianism.]\n\nThe topographical relation of Bythus to the derived \u00c6ons was differently\ngiven in different developments of the Valentinian teaching. According\nto one representation he was outside the pleroma; others placed his\nabode within it, but even in this case he was separated from the rest by\nHorus (\u1f4d\u03c1\u03bf\u03c2), a personified Boundary or Fence, whom none, not even the\n\u00c6ons themselves, could pass[558]. The former mode of representation\nmight be thought to accord better with the imagery, at the same time\nthat it is more accurate if regarded as the embodiment of a\nphilosophical conception. Nevertheless the latter was the favourite mode\nof delineation; and it had at least this recommendation, that it\ncombined in one all that is real, as opposed to all that is phenomenal.\nIn this pleroma every existence which is suprasensual and therefore true\nhas its abode.\n\nFootnote 558:\n\n  For the various modes in which the relation of the absolute first\n  principle to the pleroma was represented in different Valentinian\n  schools, see Iren. i. 1. 1, i. 2. 4, i. 11. 1, 3, 5, i. 12. 1, etc.\n  The main distinction is that stated in the text: the first principle\n  was represented in two ways; either (i) as a monad, outside the\n  pleroma; or (ii) as a dyad, a syzygy, most commonly under the\n  designation of \u0392\u03c5\u03b8\u03cc\u03c2 and \u03a3\u03b9\u03b3\u03ae, included within the pleroma but fenced\n  off from the other \u00e6ons. The Valentinian doctrine as given by\n  Hippolytus (vi. 29 sq.) represents the former type. There are good,\n  though perhaps not absolutely decisive, reasons for supposing that\n  this father gives the original teaching of Valentinus himself. For (1)\n  this very doctrine of the monad seems to point to an earlier date. It\n  is the link which connects the system of Valentinus not only with\n  Pythagoreanism to which (as Hippolytus points out) he was so largely\n  indebted, but also with the teaching of the earlier heresiarch\n  Basilides, whose first principle likewise was a monad, the absolute\n  nothing, the non-existent God. The conception of the first principle\n  as a dyad seems to have been a later, and not very happy, modification\n  of the doctrine of the founder, being in fact an extension of the\n  principle of syzygies which Valentinus with a truer philosophical\n  conception had restricted to the derived essences. (2) The exposition\n  of Hippolytus throughout exhibits a system at once more consistent and\n  more simple, than the luxuriant developments of the later\n  Valentinians, such as Ptolem\u00e6us and Marcus. (3) The sequence of his\n  statement points to the same conclusion. He gives a consecutive\n  account of some one system, turning aside from time to time to notice\n  the variations of different Valentinian schools from this standard and\n  again resuming the main thread of his exposition. It seems most\n  natural therefore that he should have taken the system of the founder\n  as his basis. On the other hand Iren\u00e6us (i. 11. 1) states that\n  Valentinus represented the first principle as a dyad (\u1f0c\u03c1\u03c1\u03b7\u03c4\u03bf\u03c2 or\n  \u0392\u03c5\u03b8\u1f41\u03c2, and \u03a3\u03b9\u03b3\u03ae): but there is no evidence that he had any direct or\n  indirect knowledge of the writings of Valentinus himself, and his\n  information was derived from the later disciples of the school, more\n  especially from the Ptolem\u00e6ans.\n\n[Sidenote: _Kenoma_, the region of phenomena.]\n\nSeparated from this celestial region by Horus, another Horus or\nBoundary, which, or who, like the former is impassable, lies the\n'kenoma' or 'void'\u2014the kingdom of this world, the region of matter and\nmaterial things, the land of shadow and darkness[559]. Here is the\nempire of the Demiurge or Creator, who is not a celestial \u00c6on at all,\nbut was born in this very void over which he reigns. Here reside all\nthose phenomenal, deceptive, transitory things, of which the eternal\ncounterparts are found only in the pleroma.\n\nFootnote 559:\n\n  Iren. i. 4. 1, 2, ii. 3. 1, ii. 4. 1, 3, ii. 5. 1, ii. 8. 1\u20133, ii. 14.\n  3, iii. 25. 6, 7, etc.\n\n[Sidenote: Platonism of this antithesis.]\n\nIt is in this antithesis that the Platonism of the Valentinian theory\nreaches its climax. All things are set off one against another in these\ntwo regions[560]: just as\n\n                    The swan on still St Mary's lake\n                    Floats double, swan and shadow.\n\nNot only have the thirty \u00c6ons their terrestrial counterparts; but their\nsubdivisions also are represented in this lower region. The kenoma too\nhas its ogdoad, its decad, its dodecad, like the pleroma[561]. There is\none Sophia in the supramundane region, and another in the mundane; there\nis one Christ who redeems the \u00c6ons in the spiritual world, and a second\nChrist who redeems mankind, or rather a portion of mankind, in the\nsensible world. There is an \u00c6on Man and another \u00c6on Ecclesia in the\ncelestial kingdom, the ideal counterparts to the Human Race and the\nChristian Church of the terrestrial. Even individual men and women, as\nwe shall see presently, have their archetypes in this higher sphere of\nintelligible being.\n\nFootnote 560:\n\n  Iren. i. 6. 3, i. 7. 1 sq., ii. 14. 3, ii. 15. 3 sq., ii. 20. 5, ii.\n  30. 3, etc.\n\nFootnote 561:\n\n  Iren. i. 5. 2, ii. 14. 3; comp. Hippol. vi. 34.\n\n[Sidenote: The localization of the _pleroma_ carried out in detail.]\n\nThe topographical conception of the pleroma moreover is carried out in\nthe details of the imagery. The second Sophia, called also Achamoth, is\nthe desire, the offspring, of her elder namesake, separated from her\nmother, cast out of the pleroma, and left 'stranded' in the void\nbeyond[562], being prevented from returning by the inexorable Horus who\nguards the frontier of the supramundane kingdom. The second Christ\u2014a\nbeing compounded of elements contributed by all the \u00c6ons[563]\u2014was sent\ndown from the pleroma, first of all at the eve of creation to infuse\nsomething like order and to provide for a spiritual element in this\nlower world; and secondly, when He united Himself with the man Jesus for\nthe sake of redeeming those who were capable of redemption[564]. At the\nend of all things Sophia Achamoth, and with her the spiritual portion of\nmankind, shall be redeemed and received up into the pleroma, while the\npsychical portion will be left outside to form another kingdom under the\ndominion of their father the Demiurge. This redemption and ascension of\nAchamoth (by a perversion of a scriptural image) was represented as her\nespousals with the Saviour, the second Christ; and the pleroma, the\nscene of this happy union, was called the bridal-chamber[565]. Indeed\nthe localization of the pleroma is as complete as language can make it.\nThe constant repetition of the words 'within' and 'without', 'above' and\n'beneath', in the development of this philosophical and religious myth\nstill further impresses this local sense on the term[566].\n\nFootnote 562:\n\n  Iren. i. 4. 1 \u03bb\u03ad\u03b3\u03bf\u03c5\u03c3\u03b9\u03bd \u1f10\u03bd \u03c3\u03ba\u03b9\u03b1\u1fd6\u03c2 [\u03c3\u03ba\u03b9\u1fb6\u03c2] \u03ba\u03b1\u1f76 \u03ba\u03b5\u03bd\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03cc\u03c0\u03bf\u03b9\u03c2\n  _)\u03b5\u03ba\u03b2\u03b5\u03b2\u03c1\u03ac\u03c3\u03b8\u03b1\u03b9_ \u03ba.\u03c4.\u03bb. The Greek MS reads \u03ba\u03b1\u1f76 \u03c3\u03ba\u03b7\u03bd\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2, but the\n  rendering of the early Latin translation 'in umbr\u00e6 [et?] vacuitatis\n  locis' leaves no doubt about the word in the original text. Tertullian\n  says of this Achamoth (_adv. Valent._ 14) 'explosa est in loca luminis\n  aliena ... in vacuum atque inane illud Epicuri'. See note 567.\n\nFootnote 563:\n\n  Iren. i. 2. 6, Hippol. vi. 32.\n\nFootnote 564:\n\n  They quoted, as referring to this descent of the second Christ into\n  the kenoma, the words of St Paul, Phil. ii. 7 \u1f11\u03b1\u03c5\u03c4\u1f78\u03bd \u1f10\u03ba\u03ad\u03bd\u03c9\u03c3\u03b5\u03bd; Clem.\n  Alex. _Exc. Theod._ 35 (p. 978).\n\nFootnote 565:\n\n  Iren. i. 7. 1 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6\u03c4\u03bf \u03b5\u1f36\u03bd\u03b1\u03b9 \u03bd\u03c5\u03bc\u03c6\u03af\u03bf\u03bd \u03ba\u03b1\u1f76 \u03bd\u03cd\u03bc\u03c6\u03b7\u03bd, _\u03bd\u03c5\u03bc\u03c6\u1ff6\u03bd\u03b1_ \u03b4\u1f72 \u03c4\u1f78\n  \u03c0\u1fb6\u03bd \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1: comp. Hippol. vi. 34 \u1f41 \u03bd\u03c5\u03bc\u03c6\u03af\u03bf\u03c2 \u03b1\u1f50\u03c4\u1fc6\u03c2.\n\nFootnote 566:\n\n  This language is so frequent that special references are needless. In\n  Iren. ii. 5. 3 we have a still stronger expression, 'in ventre\n  pleromatis'.\n\n[Sidenote: The connexion with St. Paul's use of the term obscured,]\n\nIn this topographical representation the connexion of meaning in the\nword _pleroma_ as employed by St Paul and by Valentinus respectively\nseems at first sight to be entirely lost. When we read of the contrast\nbetween the pleroma and the kenoma, the fulness and the void, we are\nnaturally reminded of the _plenum_ and the _vacuum_ of physical\nspeculations. The sense of pleroma, as expressing completeness and so\ndenoting the aggregate or totality of the Divine powers, seems\naltogether to have disappeared. [Sidenote: owing partly to the false\nantithesis \u03ba\u03ad\u03bd\u03c9\u03bc\u03b1]But in fact this antithesis of \u03ba\u03ad\u03bd\u03c9\u03bc\u03b1 was, so far as\nwe can make out, a mere after-thought, and appears to have been\nborrowed, as Iren\u00e6us states, from the physical theories of Democritus\nand Epicurus[567]. It would naturally suggest itself both because the\nopposition of \u03c0\u03bb\u03ae\u03c1\u03b7\u03c2 and \u03ba\u03b5\u03bd\u1f78\u03c2 was obvious, and because the word \u03ba\u03ad\u03bd\u03c9\u03bc\u03b1\nmaterially assisted the imagery as a description of the kingdom of waste\nand shadow. But in itself it is a false antithesis. [Sidenote: borrowed\nfrom physical philosophers;]The true antithesis appears in another, and\nprobably an earlier, term used to describe the mundane kingdom. In this\nearlier representation, which there is good reason for ascribing to\nValentinus himself, it is called not \u03ba\u03ad\u03bd\u03c9\u03bc\u03b1 'the void', but \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1\n'the deficiency, incompleteness'[568]. [Sidenote: but reappears in their\ncommon phraseology.]Moreover the common phraseology of the Valentinian\nschools shows that the idea suggested by this opposition to \u03ba\u03ad\u03bd\u03c9\u03bc\u03b1 was\nnot the original idea of the term. They speak of \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1ff6\u03bd \u03b1\u1f34\u03c9\u03bd\u03c9\u03bd,\n\u03c4\u1f78 \u03c0\u1fb6\u03bd \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03c4\u1ff6\u03bd \u03b1\u1f34\u03c9\u03bd\u03c9\u03bd, 'the whole aggregate of the \u00c6ons'[569]. And\nthis (making allowance for the personification of the \u00c6ons) corresponds\nexactly to its use in St Paul.\n\nFootnote 567:\n\n  Iren. ii. 14. 3 'Umbram autem et vacuum ipsorum a Democrito et Epicuro\n  sumentes sibimetipsis _aptaverunt_, quum illi primum multum sermonem\n  fecerint de vacuo et de atomis'.\n\nFootnote 568:\n\n  Hippol. vi. 31 \u03ba\u03b1\u03bb\u03b5\u1fd6\u03c4\u03b1\u03b9 \u03b4\u1f72 \u1f45\u03c1\u03bf\u03c2 \u03bc\u1f72\u03bd \u1f41\u1fe6\u03c4\u03bf\u03c2 \u1f45\u03c4\u03b9 \u1f00\u03c6\u03bf\u03c1\u03af\u03b6\u03b5\u03b9 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6\n  \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f14\u03be\u03c9 \u03c4\u1f78 \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1\u00b7 \u03bc\u03b5\u03c4\u03bf\u03c7\u1f72\u03c5\u03c2 \u03b4\u03b5 \u1f45\u03c4\u03b9 \u03bc\u03b5\u03c4\u03ad\u03c7\u03b5\u03b9 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6\n  \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 (i.e. as standing between the \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 and \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1)\u00b7\n  \u03c3\u03c4\u03b1\u03c5\u03c1\u1f78\u03c2 \u03b4\u03ad, \u1f45\u03c4\u03b9 \u03c0\u03ad\u03c0\u03b7\u03b3\u03b5\u03bd \u1f00\u03ba\u03bb\u03b9\u03bd\u1ff6\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bc\u03b5\u03c4\u03b1\u03bd\u03cc\u03b7\u03c4\u03c9\u03c2, \u1f61\u03c2 \u03bc\u1f74 \u03b4\u03cd\u03bd\u03b1\u03c3\u03b8\u03b1\u03b9 \u03bc\u03b7\u03b4\u1f72\u03bd\n  \u03c4\u03bf\u1fe6 \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u03c4\u03b1\u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9 \u1f10\u03b3\u03b3\u1f7a\u03c2 \u03c4\u1ff6\u03bd \u1f10\u03bd\u03c4\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b1\u1f34\u03c9\u03bd\u03c9\u03bd.\n  Iren\u00e6us represents the Marcosians as designating the Demiurge \u03ba\u03b1\u03c1\u03c0\u1f78\u03c2\n  \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 i. 17. 2, i. 19. 1, ii. pr\u00e6f. 1, ii. 1. 1 (comp. i. 14.\n  1). This was perhaps intended originally as an antithesis to the name\n  of the Christ, who was \u03ba\u03b1\u03c1\u03c0\u1f78\u03c2 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2. The Marcosians however\n  apparently meant Sophia Achamoth by this \u1f51\u03c3\u03c4\u03b5\u03c1\u03b7\u03bc\u03b1. This transference\n  from the whole to the part would be in strict accordance with their\n  terminology: for as they called the supramundane \u00e6ons \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03b1 (Iren.\n  i. 14. 2, 5; quoted in Hippol. vi. 43, 46), so also by analogy they\n  might designate the mundane Powers \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1 (comp. Iren. i. 16. 3).\n  The term, as it occurs in the document used by Hippolytus, plainly\n  denotes the _whole_ mundane region.\n\n  Hippolytus does not use the word \u03ba\u03ad\u03bd\u03c9\u03bc\u03b1, though so common in Iren\u00e6us.\n  This fact seems to point to the earlier date of the Valentinian\n  document which he uses, and so to bear out the result arrived at in a\n  previous note (p. 332) that we have here a work of Valentinus himself.\n  The word \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1 appears also in _Exc. Theod._ 22 (p. 974).\n\nFootnote 569:\n\n  e.g. Hippol. vi. 34, Iren. i. 2. 6. See especially Iren. ii. 7. 3\n  'Quoniam enim pleroma ipsorum triginta Aeones sunt, ipsi testantur'.\n\n[Sidenote: The original meaning shown by other uses.]\n\nAgain the teaching of the Valentinian schools supplies other uses which\nserve to illustrate its meaning. Not only does the supramundane kingdom\nas a whole bear this name, but each separate \u00c6on, of which that kingdom\nis the aggregation, is likewise called a pleroma[570]. This designation\nis given to an \u00c6on, because it is the fulness, the perfection, of which\nits mundane counterpart is only a shadowy and defective copy. Nor does\nthe narrowing of the term stop here. There likewise dwells in this\nhigher region a pleroma, or eternal archetype, not only of every\ncomprehensive mundane power, but of each individual man; and to wed\nhimself with this heavenly partner, this Divine ideal of himself, must\nbe the study of his life. [Sidenote: Interpretation of John iv. 17,\n18.]The profound moral significance, which underlies the exaggerated\nPlatonism and perverse exegesis of this conception, will be at once\napparent. But the manner in which the theory was carried out is\ncuriously illustrated by the commentary of the Valentinian Heracleon on\nour Lord's discourse with the Samaritan woman[571]. This woman, such is\nhis explanation, belongs to the spiritual portion of mankind. But she\nhad had six[572] husbands, or in other words she had entangled herself\nwith the material world, had defiled herself with sensuous things. The\nhusband however, whom she now has, is not her husband; herein she has\nspoken rightly: the Saviour in fact means 'her partner from the\npleroma'. Hence she is bidden to go and call him; that is, she must find\n'her pleroma, that coming to the Saviour with him (or it), she may be\nable to obtain from Him the power and the union and the combination with\nher pleroma' (\u03c4\u1f74\u03bd \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f15\u03bd\u03c9\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f00\u03bd\u03ac\u03ba\u03c1\u03b1\u03c3\u03b9\u03bd \u03c4\u1f74\u03bd \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\n\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b1\u1f50\u03c4\u1fc6\u03c2). 'For', adds Heracleon, 'He did not speak of a mundane\n(\u03ba\u03bf\u03c3\u03bc\u03b9\u03ba\u03bf\u1fe6) husband when He told her to call him, since He was not\nignorant that she had no lawful husband'.\n\nFootnote 570:\n\n  See the passages from Iren\u00e6us quoted above, note 568; comp. _Exc.\n  Theod._ 32, 33 (p. 977). Similarly \u03bb\u03cc\u03b3\u03bf\u03b9 is a synonym for the \u00c6ons,\n  \u1f41\u03bc\u03c9\u03bd\u03cd\u03bc\u03c9\u03c2 \u03c4\u1ff7 \u039b\u03cc\u03b3\u1ff3, _Exc. Theod._ 25 (p. 975).\n\nFootnote 571:\n\n  Heracleon in Orig. _in Ioann._ xiii, iv. p. 205 sq. The passages are\n  collected in Stieren's Iren\u00e6us p. 947 sq. See especially p. 950 \u1f44\u03b9\u03b5\u03c4\u03b1\u03b9\n  [\u1f41 \u1f29\u03c1\u03b1\u03ba\u03bb\u03ad\u03c9\u03bd] \u03c4\u1fc6\u03c2 \u03a3\u03b1\u03bc\u03b1\u03c1\u03b5\u03af\u03c4\u03b9\u03b4\u03bf\u03c2 \u03c4\u1f78\u03bd \u03bb\u03b5\u03b3\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03c3\u03c9\u03c4\u1fc6\u03c1\u03bf\u03c2 \u1f04\u03bd\u03b4\u03c1\u03b1 \u03c4\u1f78\n  _\u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03b1\u1f50\u03c4\u1fc6\u03c2_, \u1f35\u03bd\u03b1 \u03c3\u1f7a\u03bd \u1f10\u03ba\u03b5\u03af\u03bd\u1ff3 \u03b3\u03b5\u03bd\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03c3\u03c9\u03c4\u1fc6\u03c1\u03b1\n  \u03ba\u03bf\u03bc\u03af\u03c3\u03b5\u03c3\u03b8\u03b1\u03b9 \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c4\u1f74\u03bd \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f15\u03bd\u03c9\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f00\u03bd\u03ac\u03ba\u03c1\u03b1\u03c3\u03b9\u03bd \u03c4\u1f74\u03bd\n  \u03c0\u03c1\u1f78\u03c2 _\u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u03b1\u1f51\u03c4\u1fc6\u03c2_ \u03b4\u03c5\u03bd\u03b7\u03b8\u1fc7\u00b7 \u03bf\u1f50 \u03b3\u1f70\u03c1 \u03c0\u03b5\u03c1\u1f76 \u1f00\u03bd\u03b4\u03c1\u03cc\u03c2, \u03c6\u03b7\u03c3\u03af,\n  \u03ba\u03bf\u03c3\u03bc\u03b9\u03ba\u03bf\u1fe6 \u1f14\u03bb\u03b5\u03b3\u03b5\u03bd ... \u03bb\u03ad\u03b3\u03c9\u03bd \u03b1\u1f50\u03c4\u1fc7 \u03c4\u1f78\u03bd \u03c3\u03c9\u03c4\u1fc6\u03c1\u03b1 \u03b5\u1f30\u03c1\u03b7\u03ba\u03ad\u03bd\u03b1\u03b9, \u03a6\u03ce\u03bd\u03b7\u03c3\u03cc\u03bd \u03c3\u03bf\u03c5 \u03c4\u1f78\u03bd\n  \u1f04\u03bd\u03b4\u03c1\u03b1 \u03ba\u03b1\u1f76 \u1f10\u03bb\u03b8\u1f72 \u1f10\u03bd\u03b8\u03ac\u03b4\u03b5\u00b7 \u03b4\u03b7\u03bb\u03bf\u1fe6\u03bd\u03c4\u03b1 _\u03c4\u1f78\u03bd \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c3\u03cd\u03b6\u03c5\u03b3\u03bf\u03bd_.\n  Lower down Heracleon says \u1f26\u03bd \u03b1\u1f50\u03c4\u1fc6\u03c2 \u1f41 \u1f00\u03bd\u1f74\u03c1 \u1f10\u03bd \u03c4\u1ff7 \u0391\u1f30\u1ff6\u03bd\u03b9. By this last\n  expression I suppose he means that the great \u00e6on Man of the Ogdoad,\n  the eternal archetype of mankind, comprises in itself archetypes\n  corresponding to each individual man and woman, not indeed of the\n  whole human race (for the Valentinian would exclude the psychical and\n  carnal portion from any participation in this higher region) but of\n  the spiritual portion thereof.\n\nFootnote 572:\n\n  Origen expressly states that Heracleon read \u1f15\u03be for \u03c0\u03ad\u03bd\u03c4\u03b5. The number\n  six was supposed to symbolize the material creature: see Heracleon on\n  'the forty and _six_ years' of John ii. 20 (Stieren p. 947). There is\n  no reason to think that Heracleon falsified the text here; he appears\n  to have found this various reading already in his copy.\n\n[Sidenote: Valentinians accept St Paul and St John,]\n\nImpossible as it seems to us to reconcile the Valentinian system with\nthe teaching of the Apostles, the Valentinians themselves felt no such\ndifficulty. They intended their philosophy not to supersede or\ncontradict the Apostolic doctrine, but to supplement it and to explain\nit on philosophical principles. Hence the Canon of the Valentinians\ncomprehended the Canon of Catholic Christianity in all its essential\nparts, though some Valentinian schools at all events supplemented it\nwith Apocryphal writings. More particularly the Gospel of St John and\nthe Epistles to the Colossians and Ephesians were regarded with especial\nfavour; and those passages which speak of the pleroma are quoted more\nthan once in their writings to illustrate their teaching. [Sidenote: and\nquote them in support of their views.]By isolating a few words from the\ncontext and interpreting them wholly without reference to their setting,\nthey had no difficulty in finding a confirmation of their views, where\nwe see only an incongruity or even a contradiction. For instance, their\nsecond Christ\u2014the redeemer of the spiritual element in the mundane\nworld\u2014was, as we saw, compacted of gifts contributed by all the \u00c6ons of\nthe pleroma. Hence he was called 'the common fruit of the pleroma', 'the\nfruit of all the pleroma'[573], 'the most perfect beauty and\nconstellation of the pleroma'[574]; hence also he was designated 'All'\n(\u03c0\u1fb6\u03bd) and 'All things' (\u03c0\u03ac\u03bd\u03c4\u03b1)[575]. Accordingly, to this second Christ,\nnot to the first, they applied these texts; Col. iii. 11 'And He is all\nthings', Rom. xi. 36 'All things are unto Him and from Him are all\nthings', Col. ii. 9 'In Him dwelleth all the fulness of the Godhead',\nEphes. i. 10 'To gather together in one all things in Christ through\nGod'[576]. So too they styled him )\u0395\u03c5\u03b4\u03cc\u03ba\u03b7\u03c4\u03bf\u03c2, with a reference to Col.\ni. 19, because 'all the pleroma was pleased through Him to glorify the\nFather'[577]. And inasmuch as this second Christ was according to the\nValentinian theory instrumental in the creation of the mundane powers,\nthey quoted, or rather misquoted, as referring to this participation in\nthe work of the Demiurge, the passage Col. i. 16 'In Him were created\nall things, visible and invisible, thrones, deities, dominions'[578].\nIndeed it seems clear that these adaptations were not always\nafterthoughts, but that in several instances at least their nomenclature\nwas originally chosen for the sake of fitting the theory to isolated\nphrases and expressions in the Apostolic writings, however much it might\nconflict with the Apostolic doctrine in its main lines[579].\n\nFootnote 573:\n\n  The expression is \u1f41 \u03ba\u03bf\u03b9\u03bd\u1f78\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u03c1\u03c0\u1f78\u03c2 in Hippolytus vi. 32,\n  34, 36 (pp. 190, 191, 192, 193, 196). In Iren\u00e6us i. 8. 5 it is \u03ba\u03b1\u03c1\u03c0\u1f78\u03c2\n  \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2.\n\nFootnote 574:\n\n  Iren. i. 2. 6 \u03c4\u03b5\u03bb\u03b5\u03b9\u03cc\u03c4\u03b1\u03c4\u03bf\u03bd \u03ba\u03ac\u03bb\u03bb\u03bf\u03c2 \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f04\u03c3\u03c4\u03c1\u03bf\u03bd \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2.\n\nFootnote 575:\n\n  Iren. i. 2. 6, i. 3. 4.\n\nFootnote 576:\n\n  Iren. i. 3. 4. The passages are given in the text as they are quoted\n  by Iren\u00e6us from the Valentinians. Three out of the four are incorrect.\n\nFootnote 577:\n\n  Iren. i. 12. 4; comp. _Exc. Theod._ 31 (p. 977) \u03b5\u1f30 \u1f41 \u03ba\u03b1\u03c4\u03b5\u03bb\u03b8\u1f7c\u03bd \u03b5\u1f50\u03b4\u03bf\u03ba\u03af\u03b1\n  \u03c4\u03bf\u1fe6 \u1f45\u03bb\u03bf\u03c5 \u1f26\u03bd\u00b7 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03b3\u1f70\u03c1 \u03c0\u1fb6\u03bd \u03c4\u1f78 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 \u1f26\u03bd \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2.\n\nFootnote 578:\n\n  Iren. i. 4. 5 \u1f45\u03c0\u03c9\u03c2 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba\u03c4\u03b9\u03c3\u03b8\u1fc7, \u03c4\u1f70 \u1f41\u03c1\u03b1\u03c4\u1f70 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f00\u03cc\u03c1\u03b1\u03c4\u03b1,\n  \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, where the misquotation is remarkable. In\n  _Exc. Theod._ 43 (p. 979) the words run \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b3\u1f70\u03c1 \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7 \u1f10\u03ba\u03c4\u03af\u03c3\u03b8\u03b7 \u03c4\u1f70\n  \u1f41\u03c1\u03b1\u03c4\u1f70 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f00\u03cc\u03c1\u03b1\u03c4\u03b1, \u03b8\u03c1\u03cc\u03bd\u03bf\u03b9, \u03ba\u03c5\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2, \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u1fd6\u03b1\u03b9, \u03b8\u03b5\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2,\n  \u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u03af\u03b1\u03b9\u00b7 \u03b4\u1f76\u03bf \u03ba\u03b1\u1f76 \u1f41 \u0398\u03b5\u1f78\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f51\u03c0\u03b5\u03c1\u03cd\u03c8\u03c9\u03c3\u03b5\u03bd \u03ba.\u03c4.\u03bb. (the last words\n  being taken from Phil. ii. 9 sq.).\n\nFootnote 579:\n\n  Thus they interpreted Ephes. iii. 21 \u03b5\u1f30\u03c2 \u03c0\u03ac\u03c3\u03b1\u03c2 \u03c4\u1f70\u03c2 \u03b3\u03b5\u03bd\u1f72\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03b1\u1f30\u1ff6\u03bd\u03bf\u03c2\n  \u03c4\u1ff6\u03bd \u03b1\u1f30\u03ce\u03bd\u03c9\u03bd as referring to their generated \u00e6ons: Iren. i. 3. 1.\n  Similar is the use which they made of expressions in the opening\n  chapter of St John, where they found their first Ogdoad described:\n  _ib._ i. 8. 5.\n\n[Sidenote: Use of the term by the Docet\u00e6]\n\nThe heretics called Docet\u00e6 by Hippolytus have no connexion with\ndocetism, as it is generally understood, i.e. the tenet that Christ's\nbody was not real flesh and blood, but merely a phantom body. Their\nviews on this point, as represented by this father, are wholly\ndifferent[580]. Of their system generally nothing need be said here,\nexcept that it is largely saturated with Valentinian ideas and phrases.\nFrom the Valentinians they evidently borrowed their conception of the\npleroma, by which they understood the aggregate, or (as localized) the\nabode, of the \u00c6ons. With them, as with the Valentinians, the Saviour is\nthe common product of all the \u00c6ons[581]; and in speaking of him they\necho a common Valentinian phrase 'the pleroma of the entire \u00c6ons'[582].\n\nFootnote 580:\n\n  _R. H._ viii. 10 (p. 267).\n\nFootnote 581:\n\n  _ib._ viii. 9.\n\nFootnote 582:\n\n  _ib._ viii. 10 (p. 266).\n\n[Sidenote: and by two Ophite sects.]\n\nThe Ophite heresy, Proteus-like, assumes so many various forms, that the\nskill of critics has been taxed to the utmost to bind it with cords and\nextract its story from it. It appears however from the notices of\nHippolytus, that the term _pleroma_ was used in a definite theological\nsense by at least two branches of the sect, whom he calls Naassenes and\nPerat\u00e6.\n\n[Sidenote: (i) Naassenes.]\n\nOf the Naassenes Hippolytus tells us that among other images borrowed\nfrom the Christian and Jewish Scriptures, as well as from heathen\npoetry, they described the region of true knowledge\u2014their kingdom of\nheaven, which was entered by initiation into their mysteries\u2014as the land\nflowing with milk and honey, 'which when the perfect (the true Gnostics,\nthe fully initiated) have tasted, they are freed from subjection to\nkings (\u1f00\u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03cd\u03c4\u03bf\u03c5\u03c2) and partake of the pleroma.' Here is a plain\nallusion to Joh. i. 16. 'This', the anonymous Naassene writer goes on to\nsay, 'is the pleroma, through which all created things coming into being\nare produced and fulfilled (\u03c0\u03b5\u03c0\u03bb\u03ae\u03c1\u03c9\u03c4\u03b1\u03b9) from the Uncreated'[583]. Here\nagain, as in the Valentinian system, the conception of the pleroma is\nstrongly tinged with Platonism. The pleroma is the region of ideas, of\narchetypes, which intervenes between the author of creation and the\nmaterial world, and communicates their specific forms to the phenomenal\nexistences of the latter.\n\nFootnote 583:\n\n  _R. H._ v. 8.\n\n[Sidenote: (ii) Perat\u00e6.]\n\nThe theology of the second Ophite sect, the Perat\u00e6, as described by\nHippolytus, is a strange phenomenon. [Sidenote: Their theology]They\ndivided the universe into three regions, the uncreate, the self-create,\nand the created. Again the middle region may be said to correspond\nroughly to the Platonic kingdom of ideas. But their conception of deity\nis entirely their own. They postulate three of every being; three Gods,\nthree Words, three Minds (i.e. as we may suppose three Spirits), three\nMen. Thus there is a God for each region, just as there is a Man. In\nfull accordance with this perverse and abnormal theology is their\napplication of St Paul's language. [Sidenote: and corresponding\napplication of \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1.]Their Christ has three natures, belonging to\nthese three kingdoms respectively; and this completeness of His being is\nimplied by St Paul in Col. i. 19, ii. 9, which passages are combined in\ntheir loose quotation or paraphrase, 'All the pleroma was pleased to\ndwell in him bodily, and there is in him all the godhead', i.e. (as\nHippolytus adds in explanation) 'of this their triple division (\u03c4\u1fc6\u03c2 \u03bf\u1f55\u03c4\u03c9\n\u03b4\u03b9\u1fc3\u03c1\u03b7\u03bc\u03ad\u03bd\u03b7\u03c2 \u03c4\u03c1\u03af\u03b1\u03b4\u03bf\u03c2)'[584]. This application is altogether arbitrary,\nhaving no relation whatever to the theological meaning of the term in St\nPaul. It is also an entire departure from the conception of the\nCerinthians, Valentinians, and Naassenes, in which this meaning, however\nobscured, was not altogether lost. These three heresies took a\nhorizontal section of the universe, so to speak, and applied the term as\ncoextensive with the supramundane stratum. The Perat\u00e6 on the other hand\ndivided it vertically, and the pleroma, in their interpretation of the\ntext, denoted the whole extent of this vertical section. There is\nnothing in common between the two applications beyond the fundamental\nmeaning of the word, 'completeness, totality'.\n\nFootnote 584:\n\n  _R. H._ v. 12.\n\n[Sidenote: Pistis Sophia.]\n\nThe extant Gnostic work, called Pistis Sophia, was attributed at one\ntime on insufficient grounds to Valentinus. It appears however to\nexhibit a late development of Ophitism[585], far more Christian and less\nheathen in its character than those already considered. In this work the\nword pleroma occurs with tolerable frequency; but its meaning is not\neasily fixed. [Sidenote: Frequent use of the term.]Early in the treatise\nit is said that the disciples supposed a certain 'mystery', of which\nJesus spoke, to be 'the end of all the ends' and 'the head (\u03ba\u03b5\u03c6\u03b1\u03bb\u03ae\u03bd) of\nthe Universe' and 'the whole pleroma'[586]. Here we seem to have an\nallusion to the Platonic kingdom of ideas, i.e. of intelligible being,\nof absolute truth, as reproduced in the Valentinian pleroma. And the\nword is used sometimes in connexion with the completeness of revelation\nor the perfection of knowledge. Thus our Lord is represented as saying\nto His disciples, 'I will tell you the whole mystery and the whole\npleroma, and I will conceal nothing from you from this hour; and in\nperfection will I perfect you in every pleroma and in every perfection\nand in every mystery, which things are the perfection of all the\nperfections and the pleroma of all the pleromas'[587]. Elsewhere however\nMary, to whom Jesus is represented as making some of his chief\nrevelations, is thus addressed by Him; 'Blessed art thou above (\u03c0\u03b1\u03c1\u1f70)\nall women that are on the earth, for thou shalt be pleroma of all the\npleromas and perfection of all the perfections'[588], where the word\nmust be used in a more general sense.\n\nFootnote 585:\n\n  See K\u00f6stlin in _Theolog. Jahrb._ T\u00fcbingen 1854, p. 185.\n\nFootnote 586:\n\n  _Pistis Sophia_ p. 3 sq.\n\nFootnote 587:\n\n  _ib._ p. 15 sq.: comp. pp. 4, 60, 75, 187, 275.\n\nFootnote 588:\n\n  _ib._ p. 28 sq.: comp. p. 56. On p. 7 \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1 is opposed to \u1f00\u03c1\u03c7\u03ae,\n  apparently in the sense of 'completion'.\n\n[Sidenote: Monoimus the Arabian.]\n\nOne heresy still remains to be noticed in connexion with this word.\nHippolytus has preserved an account of the teaching of Monoimus the\nArabian, of whom previously to the discovery of this father's treatise\nwe knew little more than the name. In this strange form of heresy the\nabsolute first principle is the uncreate, imperishable, eternal Man. I\nneed not stop to enquire what this statement means. It is sufficient for\nthe present purpose to add that this eternal Man is symbolized by the\nletter \u0399, the 'one iota', the 'one tittle' of the Gospel[589]; and this\n\u0399, as representing the number ten, includes in itself all the units from\none to nine. 'This', added Monoimus, 'is (meant by) the saying (of\nscripture) _All the pleroma was pleased to dwell upon the Son of Man_\n_bodily_'[590]. Here the original idea of the word as denoting\ncompleteness, totality, is still preserved.\n\nFootnote 589:\n\n  Matt. v. 18.\n\nFootnote 590:\n\n  _R. H._ viii. 13.\n\n                  _The Epistle from_ _Laodicea_[591].\n\n[Sidenote: Different theories classified.]\n\nThe different opinions respecting the epistle thus designated by St\nPaul, which have been held in ancient or modern times, will be seen from\nthe following table;\n\n1. An _Epistle written by the Laodiceans_; to\n\n      (\u03b1) St Paul;\n\n      (\u03b2) Epaphras;\n\n      (\u03b3) Coloss\u00e6.\n\n2. An _Epistle written by St Paul from Laodicea_.\n\n      (\u03b1) 1 Timothy;\n\n      (\u03b2) 1 Thessalonians;\n\n      (\u03b3) 2 Thessalonians;\n\n      (\u03b4) Galatians.\n\n3. An _Epistle addressed to the Laodiceans_ by\n\n      (_a_) St John (the First Epistle);\n\n      (_b_) Some companion of St Paul (Epaphras or Luke);\n\n      (_c_) St Paul himself;\n\n          (i) A lost Epistle.\n\n          (ii) One of the Canonical Epistles.\n\n              (\u03b1) Hebrews;\n\n              (\u03b2) Philemon;\n\n              (\u03b3) Ephesians.\n\n          (iii) The Apocryphal Epistle.\n\nFootnote 591:\n\n  The work of Anger, _Ueber den Laodicenerbrief_ (Leipzig 1843), is very\n  complete. He enumerates and discusses very thoroughly the opinions of\n  his predecessors, omitting hardly anything relating to the literature\n  of the subject which was accessible at the time when he wrote. His\n  exposition of his own view, though not less elaborate, is less\n  satisfactory. A later monograph by A. Sartori, _Ueber den\n  Laodicenserbrief_ (L\u00fcbeck 1853), is much slighter and contributes\n  nothing new.\n\nIn this maze of conflicting hypotheses we might perhaps be tempted to\ndespair of finding our way and give up the search as hopeless. Yet I\nventure to think that the true identification of the epistle in question\nis not, or at least ought not to be, doubtful.\n\n[Sidenote: 1. An epistle written by the Laodiceans.]\n\n1. The opinion that the epistle was addressed by the Laodiceans to St\nPaul, and not conversely, found much support in the age of the Greek\ncommentators. It is mentioned by St Chrysostom as held by 'some\npersons', though he himself does not pronounce a definite opinion on the\nsubject[592]. [Sidenote: Advocates of this theory.]It is eagerly\nadvocated by Theodore of Mopsuestia. He supposes that the letter of the\nLaodiceans contained some reflexions on the Colossian Church, and that\nSt Paul thought it good for the Colossians to hear what their neighbours\nsaid of them[593]. Theodoret, though not mentioning Theodore by name,\nfollows in his footsteps[594]. The same opinion is also expressed in a\nnote ascribed to Photius in the [OE]cumenian Catena. This view seems to\nhave been very widely entertained in ancient times. It possibly\nunderlies the Latin Version 'ea qu\u00e6 Laodicensium est'[595]: it is\ndistinctly expressed in the rendering of the Peshito, 'that which was\nwritten by the Laodiceans'[596]. At a more recent date too it found\ngreat favour. It was adopted on the one hand by Calvin[597] and\nBeza[598] and Davenant and Lightfoot[599], on the other by Baronius[600]\nand \u00e0 Lapide and Estius, besides other very considerable names[601].\nLatterly its popularity has declined, but it has secured the support of\none or two commentators even in the present century.\n\nFootnote 592:\n\n  _ad loc._ \u03c4\u03b9\u03bd\u1f72\u03c2 \u03bb\u03ad\u03b3\u03bf\u03c5\u03c3\u03b9\u03bd \u1f45\u03c4\u03b9 \u03bf\u1f50\u03c7\u1f76 \u03c4\u1f74\u03bd \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5 \u03c0\u03c1\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 \u1f00\u03c0\u03b5\u03c3\u03c4\u03b1\u03bb\u03bc\u03ad\u03bd\u03b7\u03bd,\n  \u1f00\u03bb\u03bb\u1f70 \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u1ff6\u03bd \u03a0\u03ac\u03c5\u03bb\u1ff3\u00b7 \u03bf\u1f50 \u03b3\u1f70\u03c1 \u03b5\u1f36\u03c0\u03b5 \u03c4\u1f74\u03bd \u03c0\u03c1\u1f78\u03c2 _\u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03ad\u03b1\u03c2_ \u1f00\u03bb\u03bb\u1f70\n  \u03c4\u1f74\u03bd \u1f10\u03ba _\u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2_.\n\nFootnote 593:\n\n  Rab. Maur. _Op._ VI. p. 540 (Migne) 'Non quia ad Laodicenses scribit.\n  Unde quidam falsam epistolam ad Laodicenses ex nomine beati Pauli\n  confingendam esse existimaverunt; nec enim erat vera epistola.\n  \u00c6stimaverunt autem quidam illam esse, qu\u00e6 in hoc loco est significata.\n  Apostolus vero non [_ad_] _Laodicenses_ dicit sed _ex Laodicea_; quam\n  illi scripserunt ad apostolum, in quam aliqua reprehensionis digna\n  inferebantur, quam etiam hac de causa jussit apud eos legi, ut ipsi\n  reprehendant seipsos discentes qu\u00e6 de ipsis erant dicta' (see _Spic.\n  Solesm._ I. p. 133) etc.\n\nFootnote 594:\n\n  After repeating the argument based on the expression \u03c4\u1f74\u03bd \u1f10\u03ba\n  \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2, Theodoret says \u03b5\u1f30\u03ba\u1f78\u03c2 \u03b4\u1f72 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 \u1f22 \u03c4\u1f70 \u1f10\u03bd \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u1fd6\u03c2 \u03b3\u03b5\u03bd\u03cc\u03bc\u03b5\u03bd\u03b1\n  \u03b1\u1f30\u03c4\u03b9\u03ac\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 \u1f22 \u03c4\u1f70 \u03b1\u1f50\u03c4\u1f70 \u03c4\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u03bd\u03b5\u03bd\u03bf\u03c3\u03b7\u03ba\u03ad\u03bd\u03b1\u03b9.\n\nFootnote 595:\n\n  This however may be questioned. On the other hand Beza (_ad loc._),\n  Whitaker (_Disputation on Scripture_ pp. 108, 303, 468 sq., 526, 531,\n  Parker Society's ed.), and others, who explain the passage in this\n  way, urge that it is required by the Greek \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2, and complain\n  that the other interpretation depends on the erroneous Latin\n  rendering.\n\nFootnote 596:\n\n  Or, 'that which was written from Laodicea.' The difference depends on\n  the vocalisation of \u0720\u0715\u071d\u0729\u071d\u0710 which may be either (1) 'Laodicea,' as in\n  vv. 13, 15, or (2) 'the Laodiceans,' as in the previous clause in this\n  same ver. 16.\n\nFootnote 597:\n\n  Calvin is very positive; 'Bis hallucinati sunt qui Paulum arbitrati\n  sunt ad Laodicenses scripsisse. Non dubito quin epistola fuerit ad\n  Paulum missa.... Impostura autem nimis crassa fuit, quod nebulo nescio\n  quis hoc pr\u00e6textu epistolam supponere ausus est adeo insulsam, ut\n  nihil a Pauli spiritu magis alienum fingi queat.' The last sentence\n  reveals the motive which unconsciously led so many to adopt this\n  unnatural interpretation of St Paul's language.\n\nFootnote 598:\n\n  _ad loc._ 'Multo f\u0153dius errarunt qui ex hoc loco suspicati sunt\n  quandam fuisse epistolam Pauli ad Laodicenses ... quum potius\n  significet Paulus epistolam aliquam ad se missam Laodicea, aut potius\n  qua responsuri essent Laodicenses Colossensibus.'\n\nFootnote 599:\n\n  _Works_ II. p. 326.\n\nFootnote 600:\n\n  _Ann. Eccl._ s. a. 60, \u00a7 xiii.\n\nFootnote 601:\n\n  e.g. Tillemont _Mem. Eccl._ p. 576.\n\n[Sidenote: Reasons for it.]\n\nThe underlying motive of this interpretation was to withdraw the support\nwhich the apocryphal epistle seemed to derive from this reference,\nwithout being obliged at the same time to postulate a lost epistle of St\nPaul. The critical argument adduced in its support was the form of\nexpression, \u03c4\u1f74\u03bd \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2. [Sidenote: Objections to it.]The whole\ncontext however points to a different explanation. The Colossian and\nLaodicean Epistles are obviously regarded as in some sense companion\nepistles, of which the Apostle directs an interchange between the two\nChurches. And again, if the letter in question had been written by the\nLaodiceans to St Paul, why should he enjoin the Colossians to get it\nfrom Laodicea? How could he assume that a copy had been kept by the\nLaodiceans; or, if kept, would be given up when required? Indeed the\ndifficulties in this hypothesis are so great, that nothing but the most\nimperious requirements of the Greek language would justify its\nacceptance. But the expression in the original makes no such demand. It\nis equally competent for us to explain \u03c4\u1f74\u03bd \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 either 'the\nletter written from Laodicea', or 'the letter to be procured from\nLaodicea', as the context may suggest. The latter accords at least as\nwell with Greek usage as the former[602].\n\nFootnote 602:\n\n  See the note on iv. 16.\n\n[Sidenote: Views respecting the person addressed.]\n\nThe vast majority of those who interpret the expression in this way\nassume that the letter was written to (\u03b1) St Paul. The modifications of\nthis view, which suppose it addressed to some one else, need hardly be\nconsidered. The theory for instance, which addresses it to (\u03b2)\nEpaphras[603], removes none of the objections brought against the\nsimpler hypothesis. Another opinion, which takes (\u03b3) the Colossians\nthemselves to have been the recipients[604], does indeed dispose of one\ndifficulty, the necessity of assuming a copy kept by the Laodiceans, but\nit is even more irreconcileable with the language of the context. Why\nthen should St Paul so studiously charge them to see that they read it?\nWhy above all should he say \u03ba\u03b1\u1f76 \u1f51\u03bc\u03b5\u1fd6\u03c2, '_ye also_', when they were the\nonly persons who would read it as a matter of course?\n\nFootnote 603:\n\n  e.g. Storr _Opusc._ II. p. 124 sq.\n\nFootnote 604:\n\n  So for instance Corn. \u00e0 Lapide, as an alternative, 'vel certe ad ipsos\n  Colossenses, ut vult Theodor.'; but I do not find anything of the kind\n  in Theodoret. This view also commends itself to Beza.\n\n[Sidenote: 2. A letter written from Laodicea by St Paul.]\n\n2. A second class of identifications rests on the supposition that it\nwas a letter written _from_ Laodicea, though not by the Laodiceans\nthemselves. The considerations which recommend this hypothesis for\nacceptance are the same as in the last case. It withdraws all support\nfrom the apocryphal Epistle to the Laodiceans, and it refrains from\npostulating a lost Apostolic epistle. It is not exposed to all the\nobjections of the other theory, but it introduces new difficulties still\nmore serious. Here a choice of several epistles is offered to us.\n[Sidenote: 1 Timothy.](\u03b1) _The First Epistle to Timothy._ This view is\ndistinctly maintained by John Damascene[605] and by Theophylact[606];\nbut it took its rise much earlier. It appears in the margin of the\nPhiloxenian Syriac[607], and it seems to have suggested the\nsubscriptions found in many authorities at the close of that epistle.\nThe words \u1f10\u03b3\u03c1\u03ac\u03c6\u03b7 \u1f00\u03c0\u1f78 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 are found in AKL 47 etc., and many of\nthese define the place meant by the addition \u1f25\u03c4\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76 \u03bc\u03b7\u03c4\u03c1\u03cc\u03c0\u03bf\u03bb\u03b9\u03c2\n\u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u03a0\u03b1\u03ba\u03b1\u03c4\u03b9\u03b1\u03bd\u1fc6\u03c2. A similar note is found in some Latin MSS. It is\nquite possible that this subscription was prior to the theory respecting\nthe interpretation of Col. iv. 16, and gave rise to it; but the converse\nis more probable, and in some MSS (a^{scr}, 74) the bearing of this\nsubscription on Col. iv. 16 is emphasized, \u1f30\u03b4\u1f78\u03c5 \u03b4\u1f74 \u03ba\u03b1\u1f76 \u1f21 \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2.\nThis identification has not been altogether without support in later\ntimes[608]. [Sidenote: 1 Thessalonians.](\u03b2) The _First Epistle to the\nThessalonians_. A final colophon in the Philoxenian Syriac asserts that\nit was 'written from Laodicea': and the same is stated in a later hand\nof d, 'scribens a Laodicea'. Again an \u00c6thiopic MS, though giving Athens\nas the place of writing, adds that it was 'sent with Timotheus,\n_Tychicus_, and _Onesimus_[609].' This identification was perhaps\nsuggested by the fact that 1 Thessalonians follows next after Colossians\nin the common order of St Paul's Epistles. [Sidenote: 2\nThessalonians.](\u03b3) The _Second Epistle to the Thessalonians_. In the\nPeshito (as given by Schaaf[609]) there is a final colophon stating\nthat this epistle 'was written from Laodicea of Pisidia and was sent by\nthe hand of Tychicus[610]'. Though the addition of Pisidia wrongly\ndefines the place as _Laodicea Combusta_, instead of _Laodicea ad\nLycum_, yet the mention of the messenger's name shows plainly that the\nidentification with the missing epistle of Col. iv. 16 was contemplated.\nSo too the Memphitic 'per Silvanum et _Tychicum_', and a Latin prologue\n'per Titum et _Onesimum_'. Again an \u00c6thiopic MS points to the same\nidentification, though strangely confused in its statements. In the\nsuperscription we are told that this epistle was written when the\nApostle was at Laodicea, but in the subscription that it 'was written at\nAthens to Laodicea and sent by Tychicus'; while the prolegomena state\nthat it was written and left at Laodicea, and that afterwards, when St\nPaul wrote his letter to the Colossians from Rome, he gave directions\nthat it should be transmitted to the Thessalonians by the\nColossians[611]. [Sidenote: Galatians.](\u03b4) The _Epistle to the\nGalatians_[612]. This might have been chosen, partly because it affords\nno internal data for deciding where it was written, partly because like\nthe Colossian Epistle it is directed against a form of Judaism, and the\nadvocates of this hypothesis might not be careful to distinguish the two\ntypes, though very distinct in themselves. I find no support for it in\nthe subscriptions, except the notice 'per _Tychicum_' in some Slavonic\nMSS.\n\nFootnote 605:\n\n  _Op._ II. p. 214 (ed. Lequien) \u03c4\u1f74\u03bd \u03c0\u03c1\u1f78\u03c2 \u03a4\u03b9\u03bc\u03cc\u03b8\u03b5\u03bf\u03bd \u03c0\u03c1\u03ce\u03c4\u03b7\u03bd \u03bb\u03ad\u03b3\u03b5\u03b9. But he\n  adds \u03c4\u03b9\u03bd\u1f72\u03c2 \u03c6\u03b1\u03c3\u1f76\u03bd \u1f45\u03c4\u03b9 \u03bf\u1f50\u03c7\u1f76 \u03c4\u1f74\u03bd \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5 \u03c0\u03c1\u1f78\u03c2 \u03b1\u1f50\u03c4\u03bf\u1f7a\u03c2 \u1f10\u03c0\u03b5\u03c3\u03c4\u03b1\u03bb\u03bc\u03ad\u03bd\u03b7\u03bd ... \u1f00\u03bb\u03bb\u1f70\n  \u03c4\u1f74\u03bd \u03c0\u03b1\u03c1' \u03b1\u1f50\u03c4\u1ff6\u03bd \u03a0\u03ac\u03c5\u03bb\u1ff3 \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 \u03b3\u03c1\u03b1\u03c6\u03b5\u1fd6\u03c3\u03b1\u03bd.\n\nFootnote 606:\n\n  _ad loc._ \u03c4\u03af\u03c2 \u03b4\u1f72 \u1f26\u03bd \u1f21 \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2; \u1f21 \u03c0\u03c1\u1f78\u03c2 \u03a4\u03b9\u03bc\u03cc\u03b8\u03b5\u03bf\u03bd \u03c0\u03c1\u03ce\u03c4\u03b7\u00b7 \u03b1\u1f55\u03c4\u03b7 \u03b3\u1f70\u03c1\n  \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 \u1f10\u03b3\u03c1\u03ac\u03c6\u03b7. \u03a4\u03b9\u03bd\u1f72\u03c2 \u03b4\u03ad \u03c6\u03b1\u03c3\u03b9\u03bd \u1f45\u03c4\u03b9 \u1f23\u03bd \u03bf\u1f31 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u1fd6\u03c2 \u03a0\u03b1\u03cd\u03bb\u1ff3\n  \u1f10\u03c0\u03ad\u03c3\u03c4\u03b5\u03b9\u03bb\u03b1\u03bd, \u1f00\u03bb\u03bb' \u03bf\u1f50\u03ba \u03bf\u1f36\u03b4\u03b1 \u03c4\u03af \u1f02\u03bd \u1f10\u03ba\u03b5\u03af\u03bd\u1fc3\u03c2 \u1f14\u03b4\u03b5\u03b9 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03b2\u03b5\u03bb\u03c4\u03af\u03c9\u03c3\u03b9\u03bd.\n\nFootnote 607:\n\n  _ad loc._ 'Propter eam qu\u00e6 est ad Timotheum dixit.'\n\nFootnote 608:\n\n  It is adopted by Erasmus in his paraphrase; 'vicissim vos legatis\n  epistolam qu\u00e6 Timotheo scripta fuit ex Laodicensium urbe': but in his\n  commentary he does not commit himself to it. For other names see Anger\n  p. 17, note k.\n\nFootnote 609:\n\n  _Catal. Bibl. Bodl. Cod. \u00c6thiop._ p. 23.\n\nFootnote 610:\n\n  In the editio princeps (Vienna 1555) the latter part of this colophon,\n  'and was sent by the hand of Tychicus,' is wanting.\n\nFootnote 611:\n\n  _Catal. Bibl. Bodl. Cod. \u00c6thiop._ p. 23.\n\nFootnote 612:\n\n  Bloch, quoted in Anger p. 17 note l.\n\n[Sidenote: Objections to these solutions.]\n\nThe special difficulties attending this class of solutions are manifold.\n(1) It does not appear that St Paul had ever been at Laodicea when he\nwrote the letter to the Colossians. (2) All the epistles thus singled\nout are separated from the Colossian letter by an interval of some years\nat least. (3) In every case they can with a high degree of probability\nbe shown to have been written elsewhere than at Laodicea. Indeed, as St\nPaul had been long a prisoner either at C\u00e6sarea or at Rome, when he\nwrote to Coloss\u00e6, he could not have despatched a letter recently from\nLaodicea.\n\n[Sidenote: 3. A letter to the Laodiceans written by (_a_) St John. (_b_)\n           A companion of St Paul. (_c_) St Paul.]\n\n3. Thus we are thrown back on some form of the solution which makes it a\nletter written _to the Laodiceans_. And here we may at once reject the\nhypothesis that the writer was (_a_) St John[613]. The First Epistle of\nSt John, which has been selected, was written (as is allowed on all\nhands) much later than this date. Nor again does St Paul's language\nfavour the alternative, which others have maintained, that the letter in\nquestion was written by (_b_) one of St Paul's companions, e.g. Epaphras\nor Luke[614]. The writer must therefore have been (_c_) St Paul himself.\n\nFootnote 613:\n\n  A conjecture of Lightfoot's (_Works_ II. pp. 326, 339, London 1684),\n  but he does not lay much stress on it. He offers it 'rather then\n  conceive that any epistle of Paul is lost.' See also Anger p. 17, note\n  m.\n\nFootnote 614:\n\n  Baumgarten _Comm._ ad loc., quoted by Anger p. 25, note g.\n\nOn this assumption three alternatives offer themselves.\n\n[Sidenote: (i) A lost letter.]]\n\n(i). We may suppose that the epistle in question has been lost. It has\nbeen pointed out elsewhere that the Apostle must have written many\nletters which are not preserved in our Canon[615]. Thus there is no _a\npriori_ objection to this solution; and, being easy and obvious in\nitself, it has found common support in recent times. If therefore we had\nno positive reasons for identifying the Laodicean letter with one of the\nextant epistles of our Canon, we might at once close with this account\nof the matter. But such reasons do exist. And moreover, as we are\nobliged to suppose that at least three letters\u2014the Epistles to the\nColossians, to the Ephesians, and to Philemon\u2014were despatched by St Paul\nto Asia Minor at the same time, it is best not to postulate a fourth,\nunless we are obliged to do so.\n\nFootnote 615:\n\n  _Philippians_ p. 136 sq.\n\n[Sidenote: (ii) A Canonical epistle.]\n\n(ii). But, if it was not a lost letter, with which of the Canonical\nEpistles of St Paul can we identify it with most probability? Was it\n\n[Sidenote: (\u03b1) Hebrews. Philastrius.]\n\n(\u03b1) _The Epistle to the Hebrews?_ The supporters of this hypothesis are\nable to produce ancient evidence of a certain kind, though not such as\ncarries any real weight. Philastrius, writing about the close of the\nfourth century, says that some persons ascribed the authorship of the\nEpistle to the Hebrews to Luke the Evangelist, and adds that it was\nasserted (apparently by these same persons, though this is not quite\nclear) to have been written to the Laodiceans[616]. [Sidenote: Supposed\ntestimony of MS G.]Again in the Gr\u00e6co-Latin MS G of St Paul's Epistles,\nthe _Codex Boernerianus_, probably written in the ninth century, after\nthe Epistle to Philemon, which breaks off abruptly at ver. 20, a vacant\nspace is left, as if for the conclusion of this epistle: and then\nfollows a fresh title\n\n             ad      laudicenses      incipit      epistola\n   ==\u03c0\u03c1\u03bf\u03f2\\ \\ \\ \\ \u03bb\u03b1\u03bf\u03c5\u03b4\u03b1\u03ba\u03b7\u03f2\u03b1\u03f2 \\ \\ \\ \\ \u03b1\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \\ \\ \\ \\   \u03b5\u03c0\u03b9\u03f2\u03c4\u03bf\u03bb\u03b7==\n\nThis is evidently intended as the heading to another epistle. No other\nepistle however succeeds, but the leaf containing this title is followed\nby several leaves, which were originally left blank, but were filled at\na later date with extraneous matter. What then was this Epistle to the\nLaodiceans, which was intended to follow, but which the scribe was\nprevented from transcribing? As the Epistle to the Hebrews is not found\nin this MS, and as in the common order of the Pauline Epistles it would\nfollow the Epistle to Philemon, the title has frequently been supposed\nto refer to it. This opinion however does not appear at all probable.\nAnger[617] indeed argues in its favour on the ground that in the\ncompanion MS F, the _Codex Augiensis_, which (so far as regards the\nGreek text) must have been derived immediately from the same\narchetype[618], the Epistle to the Hebrews does really follow. But what\nare the facts? [Sidenote: Relation of G to F.]It is plain that the Greek\ntexts of G and F came from the same original: but it is equally plain\nthat the two scribes had different Latin texts before them\u2014that of G\nbeing the Old Latin, and that of F Jerome's revised Vulgate. No argument\ntherefore derived from the Latin text holds good for the Greek. But the\nphenomena of both MSS alike[619] show that the Greek text of their\ncommon archetype ended abruptly at Philem. 20 (probably owing to the\nloss of the final leaves of the volume). The two scribes therefore were\nleft severally to the resources of their respective Latin MSS. The\nscribe of F, whose Greek and Latin texts are in parallel columns,\nconcluded the Epistle to Philemon in Latin, though he could not match it\nwith its proper Greek; and after this he added the Epistle to the\nHebrews in Latin, no longer however leaving a blank column, as he had\ndone for the last few verses of Philemon. On the other hand the Latin\ntext in G is interlinear, the Latin words being written above the Greek\nto interpret them. When therefore the Greek text came to an end the\nscribe's work was done, for he could no longer interlineate. But he left\na blank space for the remainder of Philemon, hoping doubtless hereafter\nto find a Greek MS from which he could fill it in; and he likewise gave\nthe title of the epistle which he found next in his Latin copy, in Greek\nas well as in Latin. The Greek title however he had to supply for\nhimself. This is clear from the form, which shows it to have been\ntranslated from the Latin by a person who had the very smallest\nknowledge of Greek. No Greek in the most barbarous age would have\nwritten ==\u03bb\u03b1\u03bf\u03c5\u03b4\u03b1\u03ba\u03b7\u03f2\u03b1\u03f2== for ==\u03bb\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03b1\u03f2== or ==\u03bb\u03b1\u03bf\u03b4\u03b9\u03ba\u03b7\u03bd\u03bf\u03c5\u03f2==.\nThe \u03b1\u03bf\u03c5 is a Latin corruption _au_ for _ao_, and the termination\n==\u03b1\u03f2== is a Latin's notion of the Greek accusative. Thus the whole\nword is a reproduction of the Latin 'Laudicenses,' the _en_ being\nrepresented as usual by the Greek \u03b5[620]. [Sidenote: The spurious\nLaodicean Epistle intended.]If so, we have only to ask what writing\nwould probably appear as _Epistola ad Laudicenses_ in a Latin copy; and\nto this question there can be only one answer. The apocryphal Epistle to\nthe Laodiceans occurs frequently in the Latin Bibles, being found at\nleast two or three centuries before the MS G was written. Though it does\nnot usually follow the Epistle to Philemon, yet its place varies very\nconsiderably in different Latin copies, and an instance will be given\nbelow[621] where it actually occurs in this position.\n\nFootnote 616:\n\n  _H\u00e6r._ lxxxix 'Sunt alii quoque qui epistolam Pauli ad Hebr\u00e6os non\n  adserunt esse ipsius, sed dicunt aut Barnab\u00e6 esse apostoli aut\n  Clementis de urbe Roma episcopi; alii autem Luc\u00e6 evangelist\u00e6 aiunt\n  epistolam etiam ad Laodicenses scriptam. Et quia addiderunt in ea\n  qu\u00e6dam non bene sentientes, inde non legitur in ecclesia; et si\n  legitur a quibusdam, non tamen in ecclesia legitur populo, nisi\n  tredecim epistol\u00e6 ipsius, et ad Hebr\u00e6os interdum. Et in ea quia\n  rhetorice scripsit, sermone plausibili, inde non putant esse ejusdem\n  apostoli; et quia factum Christum dicit in ea [Heb. iii. 2], inde non\n  legitur; de p\u0153nitentia autem [Heb. vi. 4, x. 26] propter Novatianos\n  \u00e6que. Cum ergo factum dicit Christum, corpore, non divinitate, dicit\n  factum, cum doceat ibidem quod divin\u00e6 sit et patern\u00e6 substanti\u00e6\n  filius, _Qui est splendor glori\u00e6_, inquit, _et imago substanti\u00e6 ejus_\n  [Heb. i. 3]' etc. Oehler punctuates the sentence with which we are\n  concerned thus: 'alii autem Luc\u00e6 evangelist\u00e6. Aiunt epistolam etiam ad\n  Laodicenses scriptam,' and in his note he adds 'videlicet Pauli esse\n  apostoli.' Thus he supposes the clause to refer to the apocryphal\n  Epistle to the Laodiceans: and Fabricius explains the reference\n  similarly. Such a reference however would be quite out of place here.\n  The whole paragraph before and after is taken up with discussing the\n  Epistle to the Hebrews; and the interposition of just six words,\n  referring to a wholly different matter, is inconceivable. We must\n  therefore punctuate either 'alii autem Luc\u00e6 evangelist\u00e6 aiunt\n  epistolam, etiam ad Laodicenses scriptam', or 'alii autem Luc\u00e6\n  evangelist\u00e6 aiunt; epistolam etiam ad Laodicenses scriptam.' In either\n  case it will mean that some persons supposed the Epistle to the\n  Hebrews to have been written to the Laodiceans.\n\nFootnote 617:\n\n  _Laodicenerbrief_ p. 29 sq.\n\nFootnote 618:\n\n  If indeed the Greek text of F was not copied immediately from G, as\n  has been recently maintained by Mr Hort in the _Journal of Philology_\n  III. p. 67. The divergent phenomena of the two Latin texts seem to me\n  unfavourable to this hypothesis; but it ought not to be hastily\n  rejected.\n\nFootnote 619:\n\n  Volkmar, the editor of Credner's _Geschichte des Neutestamentlichen\n  Kanon_ p. 299, with strange carelessness speaks of 'the appearance\n  (das Vorkommen) of the Laodicean Epistle in both the _Codices\n  Augiensis_ and _Boernerianus_ which in other respects are closely\n  allied.' There is no mention of it in the Codex Augiensis.\n\nFootnote 620:\n\n  It is curious that this MS, which was written by an Irish scribe,\n  should give the same corrupt form, Laud_ac_- for Laod_ac_-, which we\n  find in the Book of Armagh; see below p. 348.\n\nFootnote 621:\n\n  See p. 352. It occurs also in this position in the list of Aelfric\n  (see below p. 362), where the order of the Pauline Epistles is ...\n  Col., Hebr., 1, 2 Tim., Tit., Philem., Laod.\n\n[Sidenote: This identification unsatisfactory.]\n\nThus beyond the notice in Philastrius there is no ancient support for\nthe identification of the missing letter of Col. iv. 16 with the Epistle\nto the Hebrews; and doubtless the persons to whom Philastrius alludes\nhad no more authority for their opinion than their modern successors.\nCritical conjecture, not historical tradition, led them to this result.\nThe theory therefore must stand or fall by its own merits. It has been\nmaintained by one or two modern writers[622], chiefly on the ground of\nsome partial coincidences between the Epistles to the Hebrews and the\nColossians; but the general character and purport of the two is wholly\ndissimilar, and they obviously deal with antagonists of a very different\ntype. The insuperable difficulty of supposing that two epistles so\nunlike in style were written by the same person to the same\nneighbourhood at or about the same time would still remain, even though\nthe Pauline authorship of the Epistle to the Hebrews should be for a\nmoment granted.\n\nFootnote 622:\n\n  See especially Schneckenburger _Beitr\u00e4ge_ p. 153 sq.\n\n[Sidenote: (\u03b2) Philemon.]\n\n(\u03b2) The _Epistle to Philemon_ has been strongly advocated by\nWieseler[623], as the letter to which St Paul refers in this passage.\nFor this identification it is necessary to establish two points; (1)\nthat Philemon lived not at Coloss\u00e6, but at Laodicea; and (2) that the\nletter is addressed not to a private individual, but to a whole church.\nFor the first point there is something to be said. Though for reasons\nexplained elsewhere the abode of Philemon himself appears to have been\nat Coloss\u00e6, wherever Archippus may have resided[624], still two opinions\nmay very fairly be held on this point. But Wieseler's arguments entirely\nfail to establish his other position. [Sidenote: This epistle does not\nanswer the conditions.]The theme, the treatment, the whole tenour of the\nletter, mark it as private: and the mere fact that the Apostle's\ncourtesy leads him to include in the opening salutation the Christians\nwho met at Philemon's house is powerless to change its character. Why\nshould a letter, containing such intimate confidences, be read publicly\nin the Church, not only at Laodicea but at Coloss\u00e6, by the express order\nof the Apostle? The tact and delicacy of the Apostle's pleading for\nOnesimus would be nullified at one stroke by the demand for publication.\n\nFootnote 623:\n\n  Some earlier writers who maintained this view are mentioned by Anger,\n  p. 25, note f. It has since been more fully developed and more\n  vigorously urged by Wieseler, first in a programme _Commentat. de\n  Epist. Laodicena quam vulgo perditam putant_ 1844, and afterwards in\n  his well known work _Chronol. des Apostol. Zeit._ p. 450 sq. It may\n  therefore be identified with his name. He speaks of it with much\n  confidence as 'scarcely open to a doubt,' but he has not succeeded in\n  convincing others.\n\nFootnote 624:\n\n  See the introduction to the Epistle to Philemon.\n\n[Sidenote: (\u03b3) Ephesians.]\n\n(\u03b3) But may we not identify the letter in question with the _Epistle to\nthe Ephesians_, which also is known to have been despatched at the same\ntime with the Epistle to the Colossians? Unlike the Epistle to Philemon,\nit was addressed not to a private person but to a church or churches. If\ntherefore it can be shown that the Laodiceans were the recipients,\neither alone or with others, we have found the object of our search.\n[Sidenote: This is the true solution.]The arguments in favour of this\nsolution are reserved for the introduction to that epistle. Meanwhile it\nis sufficient to say that educated opinion is tending, though slowly, in\nthis direction, and to express the belief that ultimately this view will\nbe generally received[625].\n\nFootnote 625:\n\n  See above p. 37.\n\n[Sidenote: (iii) The extant un-canonical Epistle to the Laodiceans.]\n\n(iii) Another wholly different identification remains to be mentioned.\nIt was neither a lost epistle nor a Canonical epistle, thought some, but\nthe writing which is extant under the title of the 'Epistle to the\nLaodiceans,' though not generally received by the Church. Of the various\nopinions held respecting this apocryphal letter I shall have to speak\npresently. It is sufficient here to say that the advocates of its\ngenuineness fall into two classes. Either they assign to it a place in\nthe Canon with the other Epistles of St Paul, or they acquiesce in its\nexclusion, holding that the Church has authority to pronounce for or\nagainst the Canonicity even of Apostolic writings.\n\n                  *       *       *       *       *\n\n[Sidenote: General character of the spurious epistle.]\n\nThe apocryphal Epistle to the Laodiceans is a cento of Pauline phrases\nstrung together without any definite connexion or any clear object. They\nare taken chiefly from the Epistle to the Philippians, but here and\nthere one is borrowed elsewhere, e.g. from the Epistle to the Galatians.\nOf course it closes with an injunction to the Laodiceans to exchange\nepistles with the Colossians. The Apostle's injunction in Col. iv. 16\nsuggested the forgery, and such currency as it ever attained was due to\nthe support which that passage was supposed to give to it. Unlike most\nforgeries, it had no ulterior aim. It was not framed to advance any\nparticular opinions, whether heterodox or orthodox. It has no doctrinal\npeculiarities. Thus it is quite harmless, so far as falsity and\nstupidity combined can ever be regarded as harmless.\n\n                  *       *       *       *       *\n\nAmong the more important MSS which contain this epistle are the\nfollowing. The letters in brackets [ ] give the designations adopted in\nthe apparatus of various readings which follows.\n\n1. _Fuldensis_ [F]. The famous MS of the Vulgate N. T. written for\nVictor Bishop of Capua, by whom it was read and corrected in the years\n546, 547; edited by Ern. Ranke, _Marburgi et Lipsi\u00e6_ 1868. The Laodicean\nEpistle occurs between Col. and 1 Tim. without any indication of\ndoubtful authenticity, except that it has no argument or table of\ncontents, like the other epistles. The scribe however has erroneously\ninterpolated part of the argument belonging to 1 Tim. between the title\nand the epistle; see p. 291 sq. of Ranke's edition.\n\n2. _Cavensis._ A MS of the whole Latin Bible, at the Monastery of La\nCava near Salerno, ascribed to the 6th or 7th or 8th century. See\nVercellone _Var. Lect. Vulg. Lat. Bibl._ I. p. lxxxviii. Unfortunately\nwe have no account of the readings in the Laodicean Epistle (for which\nit would be the most important authority after the Codex Fuldensis),\nexcept the last sentence quoted by Mai _Nov. Patr. Bibl._ I. 2. p. 63,\n'Et facite legi Colossensium vobis.' Laod. here occurs between Col. and\n1 Thess. (Mai p. 62). Dr Westcott (Smith's _Dict. of the Bible s.v.\nVulgate_, p. 1713) has remarked that the two oldest authorities for the\ninterpolation of the three heavenly witnesses in 1 Joh. v. 7, this La\nCava MS and the _Speculum_ published by Mai, also support the Laodicean\nEpistle (see Mai l.c. pp. 7, 62 sq.). The two phenomena are combined in\nanother very ancient MS, Brit. Mus. _Add._ 11,852, described below.\n\n3. _Armachanus_ [A]. A MS of the N. T., now belonging to Trinity\nCollege, Dublin, and known as the 'Book of Armagh.' It was written in\nthe year 807, as ascertained by Bp. Graves; see the _Proceedings of the\nRoyal Irish Academy_ III. pp. 316, 356. The Laodicean Epistle follows\nColossians on fol. 138, but with the warning that Jerome denies its\ngenuineness. The text of the Laodicean Epistle in this MS is not so pure\nas might have been anticipated from its antiquity. I owe the collation\nof readings which is given below to the kindness of Dr Reeves, who is\nengaged in editing the MS.\n\n4. _Darmstadiensis_ [D]. A fol. MS of the whole Bible, defective from\nApoc. xxii. 12 to the end, now in the Grand-ducal library at Darmstadt,\nbut formerly belonging to the Cathedral Library at Cologne; presented by\nHermann Pius, Archbishop of Cologne from A.D. 890\u2013925. Laod. follows\nCol. A collation was made for Anger, from whom (p. 144) this account is\ntaken.\n\n5. _Bernensis_ no. 334 [B]. A 4to MS of miscellaneous contents, ending\nwith the Pauline Epistles, the last being the Epistle to the Laodiceans,\nwritten in the 9th cent. The Laodicean Epistle is a fragment, ending\nwith 'Gaudete in Christo et pr\u00e6cavete sordibus in lucro' (ver. 13). This\naccount is taken by Anger from Sinner _Catal. Cod. MSS. Bibl. Bern._ I.\np. 28. In his Addenda (p. 179) Anger gives a collation of this MS.\n\n6. _Toletanus_ [T] A MS of the Latin Bible belonging to the Cathedral\nLibrary at Toledo, and written about the 8th century: see Westcott in\nSmith's _Dict. of the Bible_, s.v. _Vulgate_ p. 1710, Vercellone _Var.\nLect._ I. p. lxxxiv. sq. The readings in the Laodicean Epistle are taken\nfrom Joh. Mariana _Schol. in Vet. et Nov. Test._, where it is printed in\nfull. The edition which I have used is dated Paris 1620 (p. 831). The\ntext however cannot be assumed to be strictly accurate, as Mariana had a\nprinted copy of the epistle before him, from which at all events he\nsupplied in brackets words wanting in the MS (see Anger p. 144), and\nwhich may have influenced his readings in other ways. In this MS Laod.\nfollows Col.\n\n7. _Parisiensis_ Reg. Lat. 3 (formerly 3562)[626] [P_{1}]. A Latin Bible,\nin one volume fol., called after Anowaretha by whom it was given to the\nmonastery of Glanfeuille (St Maur), and ascribed in the printed\nCatalogue to the 9th cent. Laod. follows Col. on fol. 379.\n\nFootnote 626:\n\n  So at least I find the number given in my notes. But in _Bentl. Crit.\n  Sacr._ p. xxxvii it is 3561.\n\n8. _Parisiensis_ Reg. Lat. 6 [P_{2}]. A MS of the Latin Bible in 4 vols.\nfol., according to the Catalogue probably written in the 10th cent. [?].\nIt belonged formerly to the Duc de Noailles. Laod. follows Col. It\ncontains numerous corrections in a later hand either between the lines\nor in the margin. The two hands are distinguished as P_{2}*, P_{2}**.\n\n9. _Parisiensis_ Reg. Lat. 250 (formerly 3572) [P_{3}]. A fol. MS of the\nN. T., described in the Catalogue as probably belonging to end of the\n9th cent. Laod. follows Col. It has a few corrections in a later hand.\nThe two hands are distinguished as P_{3}*, P_{3}**.\n\nThese three Parisian MSS I collated myself, but I had not time to\nexamine them as carefully as I could have wished.\n\n10. _Brit. Mus._ Add. 11,852. [G]. An important MS of St Paul's Epistles\nwritten in the 9th cent. It formerly belonged to the monastery of St\nGall, being one of the books with which the library there was enriched\nby Hartmot who was Abbot from A.D. 872 to 884 or 885. Laod. follows Heb.\nand has no capitula like the other epistles.\n\n11. _Brit. Mus._ Add. 10,546 [C]. A fol. MS of the Vulgate, commonly\nknown as 'Charlemagne's Bible,' but probably belonging to the age of\nCharles the Bald (\u2020 877). Laod. stands between Heb. and Apoc. It has no\nargument or capitula.\n\n12. _Brit. Mus._ Reg. 1. E. vii, viii [R]. An English MS of the Latin\nBible from Christ Church, Canterbury, written about the middle of the\n10th cent. Laod. follows Heb. This is the most ancient MS, so far as I\nam aware, in which the epistle has capitulations. It is here given in\nits fullest form, and thus presents the earliest example of what may be\ncalled the modern recension.\n\n13. _Brit. Mus._ Harl. 2833, 2834 [H_{1}]. A MS of the 13th cent. written\nfor the Cathedral of Angers. Laod. follows Apoc.\n\nThe readings of the four preceding MSS are taken from the collations in\nWestcott _Canon_ Appx. E p. 572 sq. (ed. 4).\n\n14. _Brit. Mus._ Harl. 3131 [H_{2}]. A smallish 4to. of the 12th cent.,\nsaid to be of German origin, with marginal and interlinear glosses in\nsome parts. Laod. stands between Philem. and Heb. It has no heading but\nonly a red initial letter P. At the end is 'Expl. Ep\u0305la\u0305 ad Laodicenses.\nPrologus ad Ebreos.'\n\n15. _Brit. Mus._ Sloane 539 [S]. A small fol. of the 12th cent., said to\nbe German. It contains St Paul's Epistles with glosses. The gloss on\nCol. iv. 16 'et ea qu\u00e6 est Laodicensium etc.' runs 'quam ego eis misi ut\nipsi michi ut videatis hic esse responsum.' Laod. follows Heb., and has\nno glosses.\n\nThe two last MSS I collated myself.\n\n16. _Bodl._ Laud. Lat. 13 (formerly 810) [L_{1}]. A 4to MS in double\ncolumns of the 13th cent. containing the Latin Bible. See _Catal. Bibl.\nLaud. Cod. Lat._ p. 10. Laod. follows Col. Notwithstanding the date of\nthe MS, it gives a very ancient text of this epistle.\n\n17. _Bodl._ Laud. Lat. 8 (formerly 757) [L_{2}]. A fol. MS of the Latin\nBible, belonging to the end of the 12th cent. See _Catal. Bibl. Laud.\nCod. Lat._ p. 9. This is the same MS, which Anger describes (p. 145) as\n115C (its original mark), and of which he gives a collation. Laod.\nstands between 2 Thess. and 1 Tim.\n\nI am indebted for collations of these two Laudian MSS to the kindness of\nthe Rev. J. Wordsworth, Fellow of Brasenose College.\n\n18. _Vindob._ 287 [V]. The Pauline Epp., written by Marianus Scotus\n(i.e. the Irishman), A.D. 1079. See Alter _Nov. Test. ad Cod. Vindob.\nGr\u00e6ce Expressum_ II. p. 1040 sq., Denis _Cod. MSS Lat. Bibl. Vindob._ I.\nno. lviii, Zeuss _Grammatica Celtica_ p. xviii (ed. 2). The Epistle to\nthe Laodiceans is transcribed from this MS by Alter l.c. p. 1067 sq. It\nfollows Col.\n\n19. _Trin. Coll. Cantabr._ B. 5. 1 [X]. A fol. MS of the Latin Bible,\nwritten probably in the 12th century. Laod. follows Col. I have given a\ncollation of this MS, because (like Brit. Mus. Reg. 1. E. viii) it is an\nearly example of the completed form. The epistle is preceded by\ncapitula, as follows.\n\n              INCIPIUNT CAPITULA EPISTOLE AD LAODICENSES.\n\n1. Paulus Apostolus pro Laodicensibus domino gratias refert et hortatur\neos ne a seductoribus decipiantur.\n\n2. De manifestis vinculis apostoli in quibus letatur et gaudet.\n\n3. Monet Laodicenses apostolus ut sicut sui audierunt praesentia ita\nretineant et sine retractu faciant.\n\n4. Hortatur apostolus Laodicenses ut fide sint firmi et qu\u00e6 integra et\nvera et deo placita sunt faciant. et salutatio fratrum. EXPLICIUNT\nCAPITULA. INCIPIT EPISTOLA BEATI PAULI APOSTOLI AD LAODICENSES.\n\nThese capitulations may be compared with those given by Dr Westcott from\nReg. 1. E. viii, with which they are nearly identical.\n\nBesides these nineteen MSS, of which (with the exception of _Cavensis_)\ncollations are given below, it may be worth while recording the\nfollowing, as containing this epistle.\n\nAmong the Lambeth MSS are (i) no. 4, large folio, 12th or 13th cent.\nLaod. stands between Col. and 1 Thess. (ii) no. 90, small folio, 13th or\n14th cent. Laod. stands between Col. and 1 Thess. without title or\nheading of any kind. Apparently a good text. (iii) no. 348, 4to, 15th\ncent. Laod. stands between Col. and 1 Thess., without heading etc. (iv)\nno. 544, 8vo, 15th cent. Laod. stands between Col. and 1 Thess., without\nheading etc. (v) no. 1152, 4to, 13th or 14th cent. Laod. occupies the\nsame position as in the four preceding MSS and has no heading or title.\nThe first and last of these five MSS are collated by Dr Westcott\n(_Canon_ p. 572 sq.). I inspected them all.\n\nIn the Bodleian Library at Oxford belonging to the Canonici collection\nare (i) Canon. Bibl. 82 (see _Catal._ p. 277), very small 4to, 13th\ncent., containing parts of the N. T. St Paul's Epp. are at the end of\nthe volume, following Apoc. Laod. intervenes between Tit. and Philem.,\nbeginning 'Explicit epistola ad titum. Incipit ad laud.', and ending\n'Explicit epistola ad laudicenses. Incipit ad phylemonem'. (ii) Canon.\nBibl. 7 (see _Catal._ p. 251), small 4to, beginning of 14th cent.,\ncontaining Evv., Acts, Cath. Epp., Apoc., Paul. Epp. Laod. is at the\nend. (iii) Canon. Bibl. 16 (_Catal._ p. 256), small 4to, containing the\nN. T., 15th cent., written by the hand 'Stephani de Tautaldis'. Laod.\nfollows Col. (iv) Canon. Bibl. 25 (_Catal._ p. 258), very small 4to,\nmutilated, early part of 15th cent. It contains a part of St Paul's Epp.\n(beginning in the middle of Gal.) and the Apocalypse. Laod. follows Col.\nFor information respecting these MSS I am indebted to the Rev. J.\nWordsworth.\n\nIn the University Library, Cambridge, I have observed the Epistle to the\nLaodiceans in the following MSS. (i) Dd. 5. 52 (see _Catal._ I. p. 273),\n4to, double columns, 14th cent. Laod. is between Col. and 1 Thess. (ii)\nEe. I. 9 (see _Catal._ II. p. 10), 4to, double columns, very small neat\nhand, 15th cent. It belonged to St Alban's. Laod. is between Col. and 1\nThess. (iii) Mm. 3. 2 (see _Catal._ IV. p. 174), fol., Latin Bible,\ndouble columns, 13th cent. Laod. is between Col. and 1 Thess., but the\nheading is 'Explicit epistola ad Colocenses, et hic incipit ad\nthesalocenses', after which Laod. follows immediately. At the top of the\npage is 'Ad Laudonenses'. (iv) Ee. I. 16 (see _Catal._ II. p. 16), 4to,\ndouble columns, Latin Bible, 13th or 14th cent. The order of the N. T.\nis Evv., Acts, Cath. Epp., Paul Epp., Apoc. Here Laod. is between Heb.\nand Rev.; it is treated like the other books, except that it has no\nprologue.\n\nIn the College Libraries at Cambridge I have accidentally noticed the\nfollowing MSS as containing the epistle; for I have not undertaken any\nsystematic search. (i) St Peter's, O. 4. 6, fol., 2 columns, 13th cent.,\nLatin Bible. The order of the N. T. is Evv., Acts, Cath. Epp., Paul\nEpp., Apoc. The Epistle to the Laodiceans is between Heb. and Apoc. (ii)\nSidney \u0394. 5. 11, fol., 2 columns, Latin Bible, 13th cent. The order of\nthe N. T. is Evv., Paul. Epp., Acts, Cath. Epp., Apoc.; and Laod. is\nbetween 2 Thess. and 1 Tim. (iii) Emman. 2. 1. 6, large fol., Latin\nBible, early 14th cent. The order of the N. T. is different from the\nlast, being Evv., Acts, Cath. Epp., Paul. Epp.; Apoc.; but Laod. is in\nthe same position, between 2 Thess. and 1 Tim.\n\nNotice of a few other MSS, in which this epistle occurs, will be found\nin Hody _de Bibl. Text. Orig._ p. 664, and in Anger p. 145 sq.\n\nThis list, slight and partial as it is, will serve to show the wide\ncirculation of the Laodicean Epistle. At the same time it will have been\nobserved that its position varies very considerably in different copies.\n\n(i) The most common position is immediately after Colossians, as the\nnotice in Col. iv. 16 would suggest. This is its place in the most\nancient authorities, e.g. the Fulda, La Cava, and Toledo MSS, and the\nBook of Armagh.\n\n(ii) Another position is after 2 Thess. So Laud. Lat. 8, Sidn. [Delta].\n5. 11, Emman. 2. 1. 6: see also MSS in Hody _Bibl. Text. Orig._ p. 664.\nIt must be remembered that in the Latin Bibles the Epistles to the\nThessalonians sometimes precede and sometimes follow the Epistle to the\nColossians. Hence we get three arrangements in different MSS; (1) 1, 2\nThess., Col., Laod.; (2) Col., Laod., 1, 2 Thess.; (3) Col., 1, 2\nThess., Laod.\n\n(iii) It occurs at least in one instance between Titus and Philemon;\nOxon. Bodl. Canon. 82. Mai also (_Nov. Patr. Bibl._ I. 2. p. 63)\nmentions a 'very ancient MS', in which it stands between Titus and 1\nJohn; but he does not say how Titus and 1 John appear in such close\nneighbourhood.\n\n(iv) Again it follows Philemon in Brit. Mus. Harl. 3131. This also must\nhave been its position in the Latin MS which the scribe of the Codex\nBoernerianus had before him: see above p. 346.\n\n(v) Another and somewhat common position is after Hebrews; e.g. Brit.\nMus. Add. 11,852, Add. 10,546, Reg. I. E. viii, Sloane 539, Camb. Univ.\nEe. I. 16, Pet. O. 4. 6. See also Hody l.c.\n\n(vi) It is frequently placed at the end of the New Testament, and so\nafter the Apocalypse when the Apocalypse comes last, e.g. Harl. 2833.\nSometimes the Pauline Epistles follow the Apocalypse, so that Laod.\noccurs at the end at once of the Pauline Epistles and of the N. T.; e.g.\nBodl. Canon. Lat. 7.\n\nOther exceptional positions, e.g. after Galatians or after 3 John, are\nfound in versions and printed texts (see Anger p. 143); but no authority\nof Latin MSS is quoted for them.\n\nThe _Codex Fuldensis_, besides being the oldest MS, is also by far the\nmost trustworthy. In some instances indeed a true reading may be\npreserved in later MSS, where it has a false one; but such cases are\nrare. The text however was already corrupt in several places at this\ntime; and the variations in the later MSS are most frequently attempts\nof the scribes to render it intelligible by alteration or amplification.\nSuch for instance is the case with the mutilated reading 'quod est'\n(ver. 13), which is amplified, even as early as the Book of Armagh, into\n'quodcunque optimum est', though there can be little doubt that the\nexpression represents \u03c4\u1f78 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd of Phil. iii. 2, and the missing word\ntherefore is 'reliquum'. The greatest contrast to F is presented by such\nMSS as RX, where the epistle has not only been filled out to the amplest\nproportions, but also supplied with a complete set of capitulations like\nthe Canonical books. Though for this reason these two MSS have no great\nvalue, yet they are interesting as being among the oldest which give the\namplified text, and I have therefore added a collation of them. On the\nother hand some much later MSS, especially L_{1}, preserve a very ancient\ntext, which closely resembles that of F.[627]\n\nFootnote 627:\n\n  The epistle has been critically edited by Anger _Laodicenerbrief_ p.\n  155 sq. and Westcott _Canon_ App. E. p. 572. I have already expressed\n  my obligations to both these writers for their collations of MSS.\n\n  In the apparatus of various readings, which is subjoined to the\n  epistle, I have not attempted to give such minute differences of\n  spelling as _e_ and _ae_, or _c_ and _t_ (_Laodicia_, _Laoditia_), nor\n  is the punctuation of the MSS noted.\n\n\n                            AD LAODICENSES.\n\n[Sidenote: Text of the epistle.]\n\nPaulus Apostolus non ab hominibus neque per hominem sed per Ihesum\nChristum, fratribus qui sunt Laodiciae. ^2Gratia vobis et pax a Deo\npatre et Domino Ihesu Christo.\n\n^3Gratias ago Christo per omnem orationem meam, quod permanentes estis\nin eo et perseverantes in operibus eius, promissum expectantes in diem\niudicii. ^4Neque destituant vos quorundam vaniloquia insinuantium, ut\nvos avertant a veritate evangelii quod a me praedicatur. ^5Et nunc\nfaciet Deus ut qui sunt ex me ad profectum veritatis evangelii\ndeservientes et facientes benignitatem operum quae salutis vitae\naeternae.\n\n^6Et nunc palam sunt vincula mea quae patior in Christo; quibus laetor\net gaudeo. ^7Et hoc mihi est ad salutem perpetuam; quod ipsum factum\norationibus vestris et administrante Spiritu sancto, sive per vitam sive\nper mortem. ^8Est enim mihi vivere in Christo et mori gaudium. ^9Et id\nipsum in vobis faciet misericordia sua, ut eandem dilectionem habeatis\net sitis unianimes.\n\n^{10}Ergo, dilectissimi, ut audistis praesentia mei, ita retinete et\nfacite in timore Dei, et erit vobis vita in aeternum: ^{11}Est enim Deus\nqui operatur in vos. ^{12}Et facite sine retractu quaecumque facitis.\n\n#^{13}al_13#Et quod est [reliquum], dilectissimi, gaudete in Christo; et\npraecavete sordidos in lucro. ^{14}Omnes sint petitiones vestrae palam\napud Deum; et estote firmi in sensu Christi. ^{15}Et quae integra et vera\net pudica et iusta et amabilia, facite. ^{16}Et quae audistis et\naccepistis in corde retinete; et erit vobis pax.\n\n^{18}Salutant vos sancti.\n\n^{19}Gratia Domini Ihesu cum spiritu vestro.\n\n^{20}Et facite legi Colosensibus et Colosensium vobis.\n\nInc. ad laodicenses F; Incipit epistola ad laodicenses (laudicenses\nP_{2}R) BDP_{1}P_{2}P_{3}CRH_{2}SV; Epistola ad laodicenses TM (_if this\nheading be not due to the editors themselves_); Incipit epistola pauli\nad laodicenses GH_{1}; Incipit epistola beati pauli ad laodicenses X;\nIncipit aepistola ad laudicenses sed hirunimus eam negat esse pauli A:\n_no heading_ in L_{1}L_{2}H_{2}.\n\napostolus] _om._ TM. hominibus] homine G.      ihesum christum] christum\nihesum T.      christum] _add._ 'et deum patrem omnipotentem qui\nsuscitavit eum a mortuis' RX.      fratribus qui sunt] his qui sunt\nfratribus A. _For_ fratribus B _has_ fratres.      laodiciae] laodicae\nT; ladoicie L; laudaciae A; laudiciae R; laodiceae B.\n\n2. patre] et patre nostro L_{1}; patre nostro H_{1}H_{2}SM; nostro\nA.     domino] add. nostro P_{2}P_{3}RGL_{2}.\n\n3. christo] deo meo DP_{1}P_{2}P_{3}CL_{1}; deo meo et christo ihesu\nRX.      meam] memoriam M.      permanentes estis] estis permanentes\nAGR.      in operibus eius] in operibus bonis H_{1}H_{2}S; om\nBDTP_{1}P_{2}P_{3}CM.      promissum expectantes] promissa expectantes T;\net promissum expectantes M; promissionem expectantes V; sperantes\npromissionem AG; sperantes promissum RX.      diem] die\nBTDP_{1}P_{3}GCRH_{1}H_{2}SL_{1}VMX.      iudicii] iudicationis GRX.\n\n4. neque] _add._ enim R.      destituant] distituant A; destituunt H_{1};\ndestituat M, Spec.; destituit DP_{1}P_{3}CM; distituit B; destitui P_{2};\ndisturbat T.       vaniloquia] vaniloquentia BDTP_{1}P_{2}P_{3}GCVM;\nvaneloquentia, Spec.      insinuantium] insinuantium se GM; insanientium\nH_{1}S;      insimulantium T. ut] sed ut BA; sed peto ne R; seductorum ne\nX.      vos] _om._ T. avertant] Spec.; evertant FML_{2}; evertent\nB.      evangelii] aevanguelii A (_and so below_).\n\n5. et nunc ... veritatis evangelii] _om._ L. faciet deus] deus faciet\nAG.       ut] _add._ sint G.      qui] que (_altered from_ qui) P_{3}*\n(or P_{3}**).      me] _add._ perveniant TM; _add._ proficiant V.      ad\nprofectum] imperfectum A; ad perfectum R; in profectum G.      veritatis\nevangelii] evangelii veritatis V.      deservientes] _add._ sint\nP_{2}**P_{3}**H_{1}H_{2}S. _For_ deservientes RX _have_ dei\nservientes.      et facientes] _repeated in_ L_{1}. _For_ facientes\nbenignitatem operum T _has_ benignitatem operum facientes.\noperum]      eorum RX; opera L_{2}.      quae] _om._ M; _add._ sunt\nAP_{2}**GCRH_{1}H_{2}SVX. _It is impossible to say in many cases whether a\nscribe intended_ operum quae _or_ operumque. _Ranke prints_ operumque\n_in_ F.       salutis] _add._ et L_{1}.\n\n6. nunc] n\u014d = non L_{2}.      palam sunt] sunt palam G; sunt\n(_om._ palam) A.       Christo] _add._ Ihesu (iesu)\nDP_{1}P_{2}P_{3}CVX.      quibus] in quibus TRMP_{2}. et] ut C.\n\n7. mihi] michi H_{1}S (_and so below_); enim (_for_ mihi)\nM.      factum] fletum L_{2}M; factum est TP_{3}**H_{1}S. orationibus]\noperationibus B.      vestris] meis DP_{1}.      et] est M: _om._\nTGRL_{1}X.      administrante spiritu sancto] administrantem spiritum\nsanctum FBL_{2}; amministrante spiritum sanctum DCP_{1}P_{2}* (_but there\nis an erasure in_ P_{1}). _For_ administrante L_{1}X _have_\namministrante; _and for_ spiritu sancto G _transposes and reads_\nsancto spiritu.      per mortem] mortem (_om._ per) H_{1}.\n\n8. est enim] etenim T.      mihi] _om._ M.      vivere] vivere vita\nDTP_{1}P_{2}P_{3}CVH_{1}H_{2}S; vere vita FL_{1}RMX; vera vita B;      vere\n(_altered into_ vivere _prima manu_) vita L_{2}.      gaudium] lucrum et\ngaudium A; gaudium ut lucrum H_{2}P_{2}**; gaudium vel lucrum H_{1}S.\n\n9. et] _om._ T; qui (_om._ et) V. id ipsum] in ipsum FBL_{2};      in\nidipsum L_{1}V; ipsum P_{2}GM; ipse TAH_{1}H_{2}SRX.      in vobis] vobis\nP_{2}; in nobis H_{2}.      misericordia sua] misericordiam suam\nFBDAP_{1}P_{2}P_{3}CH_{1}H_{2}RSVL_{1}XL_{2} (_but written_ misericordi\u0101 su\u0101\n_in several cases_).      et] _om._ L_{1}; ut V.      unianimes] unanimes\nBDTP_{1}P_{2}P_{3}GCH_{1}RL_{1}L_{2}VMSX.\n\n10. ergo] ego H_{2}. ut] et L_{2}.      praesentia mei] praesentiam ei DP;\npraesentiam G**; in praesentia mei P_{3}**; praesentiam mihi M;\npresenciam eius L_{2}; praesentiam dei A; pr\u00e6sentiam domini (dn\u012b)\nP_{2}**H_{1}H_{2}S.      ita] _om._ DP_{1}P_{2}**P_{3}CX.      retinete]\nretinere A; sentite T.      in] cum TM; _om._ B.       timore] timorem\nAB.      dei] domini H_{1}S.      vita] pax et vita RX. in aeternum] in\naeterno A; in aeterna G*; aeterna (eterna) G**PL_{1}.\n\n11. Est enim ... vos] _om._ (?) T.      enim] _om._ B.      vos] vobis\nGAH_{1}H_{2}SRVP_{2}** (_or_ P_{2}*) P_{3}**MX.\n\n12. retractu] retractatu BP_{2}RL_{2}; retractatione AGV; tractu T;\nreatu H_{1}S. _In_ P_{2}** ut peccato _is added_; _in_ H_{2} t\npeccato.      quaecumque] quodcumque TM.\n\n13. quod est reliquum] quod est FBTDP_{1}P_{2}*P_{3}*RCL_{1}L_{2}MX; quod est\noptimum GH_{1}H_{2}SV; quodcunque optimum est A; quodcunque est obtimum\nP_{2}**; quod bonum est P_{3}**: see p. 356. dilectissimi] dilectissime\nB.      christo] domino DP_{1}P_{2}P_{3}CX.      sordidos] _add._ omnes\nP_{2}**H_{1}H_{2}S; _add._ homines A. in] ut L_{1}.      lucro] lucrum RX.\n\n14. omnes] in omnibus G; homines (_attached to the preceding sentence_)\nTM.      sint] _omitted here and placed after_ palam H_{1}S.      apud]\naput F; ante AG.      deum] dominum A.      firmi in sensu christi]\nsensu firmi in christo ihesu R.\n\n15. quae] _add._ sunt R.      integra] intigra; _add._ sunt\nT.      vera] _add._ sunt DP_1P_2P_3CVX.      pudica et iusta] iusta et\npudica R.      iusta] iusta et casta AGV; casta et iusta\nP_2**H_1H_2S.      amabilia] _add._ sunt TH_1H_2SM; _add._ et sancta RX.\n\n16. audistis] _add._ et vidistis L_2.      accepistis] accipistis\nA.      pax] _add. ver._ 17, salutate omnes fratres (sanctos _for_\nfratres GV) in osculo sancto AGP_2**H_1H_2SRVX.\n\n18. sancti] omnes sancti AGRH_1SVX; sancti omnes H_2; _add._ in christo\nihesu RX.\n\n19. domini ihesu] domini nostri ihesu (iesu) christi\nDTAP_1P_2P_3GCH_1H_2SVMRX.\n\n20. et] _add._ hanc H_1H_2SP_2**.      legi] _add._ epistolam\nL_1P_3**      colosensibus et] _om._ FTDP_1P_2*P_3CVL_1L_2. _They are\nalso omitted in the La Cava_ MS; _see above_ p. 348.      colosensium]\n_add._ epistolam L_2. _The words_ colosensibus, colosensium, _are\ncommonly written with a single_ s, _more especially in the oldest MSS.\nIn L_1 the form is_ cholosensium.\n\n_The last sentence_ et facite etc. _is entirely omitted in_ M. _In_ RX\n_it is expanded into_ et facite legi colosensibus hanc epistolam et\ncolosensium (colosensibus R) vos legite. deus autem et pater domini\nnostri ihesu christi custodiat vos immaculatos in christo ihesu cui est\nhonor et gloria in secula seculorum. amen.\n\n_Subscriptions._ Explicit P_2P_3H_1; Exp. ad laodicenses F; Explicit\nepistola ad laodicenses (laudicenses R) DP_1GCH_2SRVX. _There is no\nsubscription in_ AL_1L_2, _and none is given for_ TM.\n\n[Sidenote: Notes on the epistle.]\n\nThe following notes are added for the sake of elucidating one or two\npoints of difficulty in the text or interpretation of the epistle.\n\n4 Neque] This is the passage quoted in the _Speculum_ \u00a7 50 published by\nMai _Nov. Patr. Bibl._ I. 2. p. 62 sq., 'Item ad Laodicenses: Neque\ndestituat vos quorundam vaneloquentia (_sic_) insinuantium, ut vos\navertant a veritate evangelii quod a me praedicatur'. We ought possibly\nto adopt the reading 'destituat ... vaniloquentia' of this and other old\nmss in preference to the 'destituant ... vaniloquia' of F. 'Vaniloquium'\nhowever is the rendering of \u03bc\u03b1\u03c4\u03b1\u03b9\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 1 Tim. i. 6, and is supported by\nsuch analogies as inaniloquium, maliloquium, multiloquium,\nstultiloquium, etc.; see Hagen _Sprachl. Er\u00f6rter. zur Vulgata_ p. 74,\nRoensch _Das Neue Testament Tertullians_ p. 710.\n\ndestituant] Properly '_leave in the lurch_' and so '_cheat_',\n'_beguile_', e.g. Cic. _pro Rosc. Am._ 40 'induxit, decepit, destituit,\nadversariis tradidit, omni fraude et perfidia fefellit.' In Heb. ix. 26\n\u03b5\u1f30\u03c2 \u1f00\u03b8\u03ad\u03c4\u03b7\u03c3\u03b9\u03bd \u03c4\u1fc6\u03c2 \u1f01\u03bc\u03b1\u03c1\u03c4\u03af\u03b1\u03c2 is translated 'ad destitutionem peccati'. The\noriginal here may have been \u1f10\u03be\u03b1\u03c0\u03b1\u03c4\u03ae\u03c3\u03c9\u03c3\u03b9\u03bd or \u1f00\u03b8\u03b5\u03c4\u03ae\u03c3\u03c9\u03c3\u03b9\u03bd.    insinuantium]\nIn late Latin this word means little more than 'to communicate', 'to\ninculcate', 'to teach': see the references in Roensch _Itala u. Vulgata_\np. 387, Heumann _Handlexicon des r\u00f6mischen Rechts_ s.v., Ducange\n_Glossarium_ s.v. So too 'insinuator' Tertull. _ad Nat._ ii. 1,\n'insinuatrix' August. _Ep._ 110 (II. p. 317). In Acts xvii. 3 it is the\nrendering of \u03c0\u03b1\u03c1\u03b1\u03c4\u03b9\u03b8\u03ad\u03bc\u03b5\u03bd\u03bf\u03c2.\n\n5 ut qui sunt etc.] The passage, as it stands, is obviously corrupt; and\na comparison with Phil. i. 12 \u03c4\u1f70 \u03ba\u03b1\u03c4' \u1f10\u03bc\u1f72 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03b5\u1f30\u03c2 \u03c0\u03c1\u03bf\u03ba\u03bf\u03c0\u1f74\u03bd \u03c4\u03bf\u1fe6\n\u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5 \u1f10\u03bb\u03ae\u03bb\u03c5\u03b8\u03b5\u03bd seems to reveal the nature of the corruption. (1)\nFor 'qui' we should probably read 'qu\u00e6', which indeed is found in some\nlate MSS of no authority. (2) There is a lacuna somewhere in the\nsentence, probably after 'evangelii'. The original therefore would run\nin this form 'ut qu\u00e6 sunt ex me ad profectum veritatis [eveniant] ...\ndeservientes etc.,' the participles belonging to a separate sentence of\nwhich the beginning is lost. The supplements 'perveniant', 'proficiant',\nfound in some MSS give the right sense, though perhaps they are\nconjectural. The Vulgate of Phil. i. 12 is 'qu\u00e6 circa me sunt magis ad\nprofectum venerunt evangelii'. In the latter part of the verse it is\nimpossible in many cases to say whether a MS intends 'operum qu\u00e6' or\n'operumque'; but the former is probably correct, as representing \u1f10\u03c1\u03b3\u03c9\u03bd\n\u03c4\u1ff6\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03c2: unless indeed this sentence also is corrupt or\nmutilated.\n\n7 administrante etc.] Considering the diversity of readings here, we may\nperhaps venture on the emendation 'administratione spiritus sancti', as\nthis more closely resembles the passage on which our text is founded,\nPhil. i. 19 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03b4\u03b5\u03ae\u03c3\u03b5\u03c9\u03c2 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03af\u03b1\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba.\u03c4.\u03bb.\n\n12 retractu] '_wavering_', '_hesitation_'. For this sense of\n'retractare', 'to rehandle, discuss', and so 'to question, hesitate',\nand even 'to shirk, decline', see Oehler _Tertullian_, index p. cxciii,\nRoensch _N. T. Tertullian's_ p. 669, Ducange _Glossarium_ s.v.: comp.\ne.g. Iren. v. 11. 1 'ne relinqueretur qu\u00e6stio his qui infideliter\nretractant de eo'. So 'retractator' is equivalent to 'detractator' in\nTert. _de Jejun._ 15 'retractatores hujus officii' (see Oehler's note);\nand in 1 Sam. xiv. 39 'absque retractatione morietur' is the rendering\nof 'dying he shall die', \u03b8\u03b1\u03bd\u03ac\u03c4\u1ff3 \u1f00\u03c0\u03bf\u03b8\u03b1\u03bd\u03b5\u1fd6\u03c4\u03b1\u03b9. Here the expression\nprobably represents \u03c7\u03c9\u03c1\u1f76\u03c2 ... \u03b4\u03b9\u03b1\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u1ff6\u03bd of Phil. ii. 14, which in the\nOld Latin is 'sine ... detractionibus'. All three forms occur, retractus\n(Tert. _Scorp._ 1), retractatus (Tert. _Apol._ 4, _adv. Marc._ i. 1, v.\n3, _adv. Prax._ 2, and frequently), retractatio (Cic. _Tusc._ v. 29,\n'sine retractatione' and so frequently; 1 Sam. l. c). Here 'retractus'\nmust be preferred, both as being the least common form and as having the\nhighest MS authority. In Tert. _Scorp._ 1 however it is not used in this\nsame sense.\n\n13 quod est reliquum] I have already spoken of this passage, p. 352, and\nshall have to speak of it again, p. 357. The oldest and most trustworthy\nMSS have simply 'quod est'. The word 'reliquum' must be supplied, as\nAnger truly discerned (p. 163); for the passage is taken from Phil. iii.\n1 \u03c4\u1f78 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd, \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u03af \u03bc\u03bf\u03c5, \u03c7\u03b1\u03af\u03c1\u03b5\u03c4\u03b5 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3. See the Vulgate translation\nof \u03c4\u1f78 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd in 1 Cor. vii. 29. Later and less trustworthy authorities\nsupply 'optimum' or 'bonum'.\n\n14 in sensu Christi] '_in the mind of Christ_': for in 1 Cor. ii. 16\n\u03bd\u03bf\u1fe6\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 is rendered 'sensum Christi'.\n\n20 facite legi etc.] Though the words 'Colosensibus et' are wanting in\nvery many of the authorities which are elsewhere most trustworthy, yet I\nhave felt justified in retaining them with other respectable copies,\nbecause (1) The hom\u0153oteleuton would account for their omission even\nin very ancient MSS; (2) The parallelism with Col. iv. 16 requires their\ninsertion; (3) The insertion is not like the device of a Latin scribe,\nwho would hardly have manipulated the sentence into a form which savours\nso strongly of a Greek original.\n\n                  *       *       *       *       *\n\n[Sidenote: Theory of a Greek original discussed.]\n\nIt is the general, though not universal, opinion that this epistle was\naltogether a forgery of the Western Church[628]; and consequently that\nthe Latin is not a translation from a lost Greek original, but preserves\nthe earliest form of the epistle. Though the forgery doubtless attained\nits widest circulation in the West, there are, I venture to think,\nstrong reasons for dissenting from this opinion.\n\nFootnote 628:\n\n  e.g. Anger _Laodicenerbrief_ p. 142 sq., Westcott _Canon_ p. 454 sq.\n  (ed. 4). Erasmus asks boldly, 'Qui factum est ut h\u00e6c epistola apud\n  Latinos extet, cum nullus sit apud Gr\u00e6cos, ne veterum quidem, qui\n  testetur eam a se lectam?' The accuracy of this statement will be\n  tested presently.\n\n[Sidenote: Frequent Grecisms in the epistle.]\n\nIf we read the epistle in its most authentic form, divested of the\nadditions contributed by the later MSS, we are struck with its cramped\nstyle. Altogether it has not the run of a Latin original. And, when we\ncome to examine it in detail, we find that this constraint is due very\nlargely to the fetters imposed by close adherence to Greek idiom. Thus\nfor instance we have ver. 5 '_qui_ [or _qu\u00e6_] _sunt ex me_', \u03bf\u1f31 [or \u03c4\u1f70]\n\u1f10\u03be \u1f10\u03bc\u03bf\u1fe6; _operum qu\u00e6 salutis_, \u1f10\u03c1\u03b3\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c4\u1fc6\u03c2 \u03c3\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03c2; ver. 6 _palam\nvincula mea qu\u00e6 patior_, \u03c6\u03b1\u03bd\u03b5\u03c1\u03bf\u1f76 \u03bf\u1f31 \u03b4\u03b5\u03c3\u03bc\u03bf\u03af \u03bc\u03bf\u03c5 \u1f43\u03c5\u03c2 \u1f51\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9; ver. 13\n_sordidos in lucro_, \u03b1\u1f30\u03c3\u03c7\u03c1\u03bf\u03ba\u03b5\u03c1\u03b4\u03b5\u1fd6\u03c2; ver. 20 _et facite legi Colosensibus\net Colosensium vobis_, \u03ba\u03b1\u1f76 \u03c0\u03bf\u03b9\u03ae\u03c3\u03b1\u03c4\u03b5 \u1f35\u03bd\u03b1 \u03c4\u03bf\u1fd6\u03c2 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03b5\u1fe6\u03c3\u03b9\u03bd \u1f00\u03bd\u03b1\u03b3\u03bd\u03c9\u03c3\u03b8\u1fc7 \u03ba\u03b1\u1f76\n\u1f21 \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u03ad\u03c9\u03bd \u1f35\u03bd\u03b1 [\u03ba\u03b1\u1f76] \u1f51\u03bc\u1fd6\u03bd. It is quite possible indeed that parallels\nfor some of these anomalies may be found in Latin writers. Thus Tert.\n_c. Marc._ i. 23 'redundantia justiti\u00e6 super _scribarum et Pharis\u00e6orum_'\nis quoted to illustrate the genitive 'Colossensium' ver. 20.[629] The\nGreek cast however is not confined to one or two expressions but extends\nto the whole letter.\n\nFootnote 629:\n\n  Anger, p. 165.\n\n[Sidenote: It differs widely from the Old Latin and Vulgate Versions.]\n\nBut a yet stronger argument in favour of a Greek original remains. This\nepistle, as we saw, is a cento of passages from St Paul. If it had been\nwritten originally in Latin, we should expect to find that the passages\nwere taken directly from the Latin versions. This however is not the\ncase. Thus compare ver. 6 '_palam_ sunt vincula mea' with Phil. i. 13\n'ut vincula mea _manifesta_ fierent': ver. 7 '_orationibus vestris et\nadministrante_ spiritu sancto' [_administratione_ spiritus sancti'?]\nwith Phil. i. 19 '_per vestram obsecrationem_ (V. _orationem_) et\n_subministrationem _ spiritus sancti'; ver. 9 'ut eandem _dilectionem_\nhabeatis et sitis unianimes' with Phil. ii. 2 'eandem _caritatem_\nhabentes, unanimes'; ver. 10 '_ergo_, dilectissimi, ut _audistis_\npr\u00e6sentia mei ... _facite in_ timore' with Phil. ii. 12 '_Propter quod_\n(V. _Itaque_) dilectissimi mihi (V. _charissimi_ mei) sicut semper\n_obaudistis_ (V. _obedistis_) ... pr\u00e6sentia (V. in pr\u00e6sentia) mei ...\n_cum_ timore (V. _metu_) ... _operamini_'; ver. 11, 12 '_Est enim Deus_\nqui operatur in vos (v. 1. vobis). Et facite sine _retractu_ qu\u00e6cumque\nfacitis' with Phil. ii. 13, 14 _Deus enim est_ qui operatur in vobis ...\nOmnia autem facite sine ... _detractionibus_ (V. _h\u00e6sitationibus_)';\nver. 13 _quod est_ [_reliquum_], dilectissimi, gaudete in Christo et\n_pr\u00e6cavete_' with Phil. iii. 1, 2 '_de c\u00e6tero_, fratres mei, gaudete in\nDomino ... _Videte_'; ib. '_sordidos in lucro_' with the Latin\nrenderings of \u03b1\u1f30\u03c3\u03c7\u03c1\u03bf\u03ba\u03b5\u03c1\u03b4\u03b5\u1fd6\u03c2 1 Tim. iii. 8 '_turpilucros_' (V. '_turpe\nlucrum sectantes_'), \u03b1\u1f30\u03c3\u03c7\u03c1\u03bf\u03ba\u03b5\u03c1\u03b4\u1fc6 Tit. i. 7 _turpilucrum_ (V. '_turpis\nlucri cupidum_'); ver. 14 '_sint petitiones_ vestr\u00e6 _palam_ apud Deum'\nwith Phil. iv. 6 '_postulationes_ (V. petitiones) vestr\u00e6 _innotescant_\napud Deum'; ver. 20 'facite _legi_ Colosensibus et _Colosensium vobis_'\nwith Col. iv. 16 'facite _ut_ et in Laodicensium ecclesia _legatur_ et\n_eam qu\u00e6 Laodicensium_ (MSS Laodiciam) _est ut_ (om. V.) _vos legatis_'.\nThese examples tell their own tale. [Sidenote: Thus internal evidence\nfavours a Greek original.]The occasional resemblances to the Latin\nVersion are easily explained on the ground that reminiscences of this\nversion would naturally occur to the translator of the epistle. The\nhabitual divergences from it are only accounted for on the hypothesis\nthat the original compiler was better acquainted with the New Testament\nin Greek than in Latin, and therefore presumably that he wrote in Greek.\n\n[Sidenote: External testimony to the same effect.]\n\nAnd, if we are led to this conclusion by an examination of the epistle\nitself, we shall find it confirmed by an appeal to external testimony.\nThere is ample evidence that a spurious Epistle to the Laodiceans was\nknown to Greek writers, as well as Latin, at a sufficiently early date.\n[Sidenote: [Muratorian Fragment].]A mention of such an epistle occurs as\nearly as the Muratorian Fragment on the Canon (about A.D. 170), where\nthe writer speaks of two letters, one to the Laodiceans and another to\nthe Alexandrians, as circulated under the name of Paul[630]. The bearing\nof the words however is uncertain. He may be referring to the Marcionite\nrecension of the canonical Epistle to the Ephesians, which was entitled\nby that heretic an Epistle to the Laodiceans[631]. Or, if this\nexplanation of his words be not correct (as perhaps it is not), still we\nshould not feel justified in assuming that he is referring to the extant\napocryphal epistle. Indeed we should hardly expect that an epistle of\nthis character would be written and circulated at so early a date. The\nreference in Col. iv. 16 offered a strong temptation to the forger, and\nprobably more than one unscrupulous person was induced by it to try his\nhand at falsification[632]. But, however this may be, it seems clear\nthat before the close of the fourth century our epistle was largely\ncirculated in the East and West alike. [Sidenote: Jerome.]'Certain\npersons', writes Jerome in his account of St Paul, 'read also an Epistle\nto the Laodiceans, but it is rejected by all[633]'. No doubt is\nentertained, that this father refers to our epistle. [Sidenote:\nTheodore.]If then we find that about the same time Theodore of\nMopsuestia also mentions an Epistle to the Laodiceans, which he condemns\nas spurious[634], it is a reasonable inference that the same writing is\nmeant. [Sidenote: Theodoret.]In this he is followed by Theodoret[635];\nand indeed the interpretations of Col. iv. 16 given by the Greek Fathers\nof this age were largely influenced as we have seen, by the presence of\na spurious epistle which they were anxious to discredit[636]. [Sidenote:\n2nd Council of Nic\u00e6a.]Even two or three centuries later the epistle\nseems to have been read in the East. At the Second Council of Nic\u00e6a\n(A.D. 787) it was found necessary to warn people against 'a forged\nEpistle to the Laodiceans' which was 'circulated, having a place in some\ncopies of the Apostle[637].'\n\nFootnote 630:\n\n  _Canon Murat._ p. 47 (ed. Tregelles). The passage stands in the MS,\n  'Fertur etiam ad Laudecenses alia ad Alexandrinos Pauli nomine fincte\n  ad heresem Marcionis et alia plura qu\u00e6 in catholicam eclesiam recepi\n  non potest.' There is obviously some corruption in the text. One very\n  simple emendation is the repetition of 'alia', so that the words would\n  run 'ad Laudicenses alia, alia ad Alexandrinos'. In this case fincte\n  (= finct\u00e6) might refer to the two epistles first mentioned, and the\n  Latin would construe intelligibly. The writing described as 'ad\n  Laodicenses alia' might then be the Epistle to the Ephesians under its\n  Marcionite title, the writer probably not having any personal\n  knowledge of it, but supposing from its name that it was a different\n  and a forged writing. But what can then be the meaning of 'alia ad\n  Alexandrinos'? Is it, as some have thought, the Epistle to the\n  Hebrews? But this could not under any circumstances be described as\n  'fincta ad h\u00e6resem Marcionis', even though we should strain the\n  meaning of the preposition and interpret the words '_against_ the\n  heresy of Marcion'. And again our knowledge of Marcion's Canon is far\n  too full to admit the hypothesis that it included a spurious Epistle\n  to the Alexandrians, of which no notice is elsewhere preserved. We are\n  therefore driven to the conclusion that there is a hiatus here, as in\n  other places of this fragment, probably after 'Pauli nomine'; and\n  'finct\u00e6' will then refer not to the two epistles named before, but to\n  the mutilated epistles of Marcion's Canon which he had 'tampered with\n  to adapt them to his heresy'. In this case the letter 'ad Laudicenses'\n  may refer to our apocryphal epistle or to some earlier forgery.\n\nFootnote 631:\n\n  See the introduction to the Epistle to the Ephesians.\n\nFootnote 632:\n\n  Timotheus, who became Patriarch of Constantinople in 511, while still\n  a presbyter, includes in a list of apocryphal works forged by the\n  Manicheans \u1f21 \u03c0\u03b5\u03bd\u03c4\u03b5\u03ba\u03b1\u03b9\u03b4\u03b5\u03ba\u03ac\u03c4\u03b7 [i.e. \u03c4\u03bf\u1fe6 \u03a0\u03b1\u03cd\u03bb\u03bf\u03c5] \u03c0\u03c1\u1f78\u03c2 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u1fd6\u03c2 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae,\n  Meurse p. 117 (quoted by Fabricius, _Cod. Apocr. N. T._ I.. p. 139).\n  Anger (p. 27) suggests that there is a confusion of the Marcionites\n  and Manicheans here. I am disposed to think that Timotheus recklessly\n  credits the Manicheans with several forgeries of which they were\n  innocent, among others with our apocryphal Epistle to the Laodiceans.\n  Still it is possible that there was another Laodicean Epistle forged\n  by these heretics to support their peculiar tenets.\n\nFootnote 633:\n\n  _Vir. Ill._ 5 (II. p. 840) 'Legunt quidam et ad Laodicenses, sed ab\n  omnibus exploditur'.\n\nFootnote 634:\n\n  The passage is quoted above, p. 341, note 593.\n\nFootnote 635:\n\n  \u03c4\u03b9\u03bd\u1f72\u03c2 \u1f51\u03c0\u03ad\u03bb\u03b1\u03b2\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c0\u03c1\u1f78\u03c2 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03ad\u03b1\u03c2 \u03b1\u1f50\u03c4\u1f78\u03bd \u03b3\u03b5\u03b3\u03c1\u03b1\u03c6\u03ad\u03bd\u03b1\u03b9\u00b7 \u03b1\u1f50\u03c4\u03af\u03ba\u03b1 \u03c4\u03bf\u03af\u03bd\u03c5\u03bd \u03ba\u03b1\u1f76\n  \u03c0\u03c1\u03bf\u03c3\u03c6\u03ad\u03c1\u03bf\u03c5\u03c3\u03b9 \u03c0\u03b5\u03c0\u03bb\u03b1\u03c3\u03bc\u03ad\u03bd\u03b7\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae\u03bd.\n\nFootnote 636:\n\n  Anger (p. 143) argues against a Greek original on the ground that the\n  Eastern Church, unlike the Latin, did not generally interpret Col. iv.\n  16 as meaning an epistle written _to_ the Laodiceans. The fact is\n  true, but the inference is wrong, as the language of the Greek\n  commentators themselves shows.\n\nFootnote 637:\n\n  Act. vi. Tom. v (Labbe viii. p. 1125 ed. Colet.) \u03ba\u03b1\u1f76 \u03b3\u1f70\u03c1 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03af\u03bf\u03c5\n  \u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03bf\u03c5 \u03c0\u03c1\u1f78\u03c2 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u1fd6\u03c2 \u03c6\u03ad\u03c1\u03b5\u03c4\u03b1\u03b9 \u03c0\u03bb\u03b1\u03c3\u03c4\u1f74 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1f74 \u1f15\u03bd \u03c4\u03b9\u03c3\u03b9 \u03b2\u03af\u03b2\u03bb\u03bf\u03b9\u03c2 \u03c4\u03bf\u1fe6\n  \u1f00\u03c0\u03bf\u03c3\u03c4\u03cc\u03bb\u03bf\u03c5 \u1f10\u03b3\u03ba\u03b5\u03b9\u03bc\u03ad\u03bd\u03b7, \u1f23\u03bd \u03bf\u1f31 \u03c0\u03ac\u03c4\u03b5\u03c1\u03b5\u03c2 \u1f21\u03bc\u1ff6\u03bd \u1f00\u03c0\u03b5\u03b4\u03bf\u03ba\u03af\u03bc\u03b1\u03c3\u03b1\u03bd \u1f61\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6\n  \u1f00\u03bb\u03bb\u03bf\u03c4\u03c1\u03af\u03b1\u03bd.\n\n                  *       *       *       *       *\n\n[Sidenote: The Greek restored.]\n\nThe Epistle to the Laodiceans then in the original Greek would run\nsomewhat as follows[638]:\n\n                         ==\u03a0\u03a1\u039f\u03a3 \u039b\u0391\u039f\u0394\u0399\u039a\u0395\u0391\u03a3.==\n\n[Sidenote: ^a Gal. i. 1.]\n\n^a==\u03a0\u0391\u03a5\u039b\u039f\u03a3 \u1f00\u03c0\u03cc\u03c3\u03c4\u03bf\u03bb\u03bf\u03f2 \u03bf\u1f50\u03ba \u1f00\u03c0' \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd \u03bf\u1f50\u03b4\u1f72 \u03b4\u03b9' \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03c5 \u03b1\u03bb\u03bb\u1f70 \u03b4\u03b9\u1f70 \u1f30\u03b7\u03f2\u03bf\u1fe6\n\u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u1fe6, \u03c4\u03bf\u1fd6\u03f2 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fd6\u03f2 \u03c4\u03bf\u1fd6\u03f2 \u03bf\u1f56\u03f2\u03b9\u03bd \u03b5\u03bd \u03bb\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u1fb3.== [Sidenote: ^b Gal.\ni. 3.; Phil. i. 2.]^2^b==\u03c7\u03ac\u03c1\u03b9\u03f2 \u1f51\u03bc\u1fd6\u03bd \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u1f00\u03c0\u1f78 \u03b8\u03b5\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03f2 \u03ba\u03b1\u1f76\n\u03ba\u03c5\u03c1\u03af\u03bf\u03c5 \u1f30\u03b7\u03f2\u03bf\u1fe6 \u03c7\u03c1\u03b9\u03f2\u03c4\u03bf\u1fe6.==\n\n[Sidenote: ^c Phil. i. 3.]\n\n^3^c==\u0395\u1f50\u03c7\u03b1\u03c1\u03b9\u03f2\u03c4\u1ff6 \u03c4\u1ff7 \u03a7\u03c1\u03b9\u03f2\u03c4\u1ff7 \u1f10\u03bd \u03c0\u03ac\u03f2\u1fc3 \u03b4\u03b5\u03ae\u03f2\u03b5\u03b9 \u03bc\u03bf\u03c5, \u1f45\u03c4\u03b9 \u1f10\u03f2\u03c4\u1f72 \u03b5\u03bd \u03b1\u1f50\u03c4\u1ff7\n\u03bc\u03ad\u03bd\u03bf\u03bd\u03c4\u03b5\u03f2 \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03f2\u03ba\u03b1\u03c1\u03c4\u03b5\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03f2 \u03c4\u03bf\u1fd6\u03f2 \u1f10\u03c1\u03b3\u03bf\u03b9\u03f2 \u03b1\u1f50\u03c4\u03bf\u1fe6,== [Sidenote: ^d Gal.\nv. 5.]^d==\u1f00\u03c0\u03b5\u03ba\u03b4\u03ad\u03c7\u03bf\u03bc\u03b5\u03bd\u03bf\u03b9 \u03c4\u1f74\u03bd \u1f10\u03c0\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03b1\u03bd== [Sidenote: ^e 2 Pet. ii.\n9; iii. 7; cf. Phil. ii. 16.]^e==\u03b5\u1f30\u03f2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03bd \u03ba\u03c1\u03af\u03f2\u03b5\u03c9\u03f2.==\n\n^4==\u039c\u03b7\u03b4\u1f72 \u1f51\u03bc\u1fb6\u03f2 \u1f10\u03be\u03b1\u03c0\u03b1\u03c4\u03ae\u03f2\u03c9\u03f2\u03b9\u03bd== [Sidenote: ^f 1 Tim. i.\n6.]^f==\u03bc\u03b1\u03c4\u03b1\u03b9\u03bf\u03bb\u03bf\u03b3\u03af\u03b1\u03b9 \u03c4\u03b9\u03bd\u1ff6\u03bd \u03b4\u03b9\u03b4\u03b1\u03f2\u03ba\u03cc\u03bd\u03c4\u03c9\u03bd \u1f35\u03bd\u03b1== [Sidenote: ^g 2 Tim.\niv. 4.]^g==\u1f00\u03c0\u03bf\u03f2\u03c4\u03c1\u03ad\u03c0\u03f2\u03c9\u03f2\u03b9\u03bd \u03c5\u03bc\u1fb6\u03f2 \u1f00\u03c0\u1f78== [Sidenote: ^h Col. i. 5; Gal.\nii. 5, 14.]^h==\u03c4\u1fc6\u03f2 \u03b1\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03f2== [Sidenote: ^i Gal. i. 11 (cf. i.\n8).]^i==\u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03b9\u03f2\u03b8\u03ad\u03bd\u03c4\u03bf\u03f2 \u1f51\u03c0' \u1f10\u03bc\u03bf\u1fe6.== ^5==\u03ba\u03b1\u1f76 \u03bd\u1fe6\u03bd\n\u03c0\u03bf\u03b9\u03ae\u03f2\u03b5\u03b9 \u1f41 \u0398\u03b5\u1f78\u03f2 \u1f35\u03bd\u03b1== [Sidenote: ^k Phil. i. 12.]^k==\u03c4\u03b1 \u1f10\u03be \u1f10\u03bc\u03bf\u1fe6 \u03b5\u1f30\u03f2\n\u03c0\u03c1\u03bf\u03ba\u03bf\u03c0\u1f74\u03bd \u03c4\u1fc6\u03f2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03f2 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5 * * * \u03bb\u03b1\u03c4\u03c1\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03f2 \u03ba\u03b1\u1f76 \u03c0\u03bf\u03b9\u03bf\u1fe6\u03bd\u03c4\u03b5\u03f2\n\u03c7\u03c1\u03b7\u03f2\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u1f10\u03c1\u03b3\u03c9\u03bd \u03c4\u1ff6\u03bd \u03c4\u1fc6\u03f2 \u03f2\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03f2 [\u03ba\u03b1\u1f76] \u03c4\u1fc6\u03f2 \u03b1\u1f30\u03c9\u03bd\u03af\u03bf\u03c5 \u03b6\u03c9\u1fc6\u03f2.== ^6==\u03ba\u03b1\u1f76\n\u03bd\u1fe6\u03bd== [Sidenote: ^l Phil. i. 13.]^l==\u03c6\u03b1\u03bd\u03b5\u03c1\u03bf\u1f76 \u03bf\u1f31 \u03b4\u03b5\u03f2\u03bc\u03bf\u03af \u03bc\u03bf\u03c5, \u03bf\u1f53\u03f2\n\u1f51\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9 \u1f10\u03bd \u03a7\u03c1\u03b9\u03f2\u03c4\u1ff7, \u1f10\u03bd \u03bf\u1f37\u03f2== [Sidenote: ^m Matt. v. 12; cf. Phil. i.\n18]^m==\u03c7\u03c1\u03c9 \u03ba\u03b1\u1f76 \u1f00\u03b3\u03b1\u03bb\u03bb\u03b9\u1ff6\u03bc\u03b1\u03b9.== ^7==\u03ba\u03b1\u1f76== [Sidenote: ^n Phil. i.\n19.]^n==\u03c4\u03bf\u1fe6\u03c4\u03cc \u1f10\u03f2\u03c4\u03af\u03bd \u03bc\u03bf\u03b9 \u03b5\u1f30\u03f2 \u03f2\u03c9\u03c4\u03b7\u03c1\u03af\u03b1\u03bd \u1f00\u0390\u03b4\u03b9\u03bf\u03bd, \u1f43 \u03ba\u03b1\u1f76 \u1f00\u03c0\u03ad\u03b2\u03b7 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03f2 \u1f51\u03bc\u1ff6\u03bd\n\u03b4\u03ad\u03b7\u03f2\u03b5\u03c9\u03f2 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03af\u03b1\u03f2 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03f2 \u1f01\u03b3\u03af\u03bf\u03c5,== [Sidenote: ^o Phil. i.\n20.]^o==\u03b5\u1f34\u03c4\u03b5 \u03b4\u03b9\u1f70 \u03b6\u03c9\u1fc6\u03f2 )\u03ad\u03b9\u03c4\u03b5 \u03b4\u03b9\u1f70 \u03b8\u03b1\u03bd\u03ac\u03c4\u03bf\u03c5.== [Sidenote: ^p Phil. i.\n21.]^8^p==\u1f11\u03bc\u03bf\u1f76 \u03b3\u1f70\u03c1 \u03c4\u1f78 \u03b6\u1fc6\u03bd \u1f10\u03bd \u03a7\u03c1\u03b9\u03f2\u03c4\u1ff7 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f00\u03c0\u03bf\u03b8\u03b1\u03bd\u03b5\u1fd6\u03bd \u03c7\u03b1\u03c1\u03ac.==\n^9==\u03ba\u03b1\u1f76 \u03c4\u1f78 \u03b1\u1f50\u03c4\u1f78 \u03c0\u03bf\u03af\u03b7\u03f2\u03b5\u03b9 [\u03ba\u03b1\u1f76] \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6 \u1f10\u03bb\u03ad\u03bf\u03c5\u03f2 \u03b1\u1f50\u03c4\u03bf\u1fe6, \u1f35\u03bd\u03b1==\n[Sidenote: ^q Phil. ii. 2.]^q==\u03c4\u1f74\u03bd \u03b1\u1f50\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u1f14\u03c7\u03b7\u03c4\u03b5, \u03f2\u03cd\u03bc\u03c0\u03f2\u03c5\u03c7\u03bf\u03b9\n\u1f44\u03bd\u03c4\u03b5\u03f2.== [Sidenote: ^r Phil. ii. 12.]^{10}^r==\u1f65\u03f2\u03c4\u03b5, \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03bf\u03af, \u03ba\u03b1\u03b8\u1f7c\u03f2\n\u1f51\u03c0\u03b7\u03ba\u03bf\u03cd\u03f2\u03b1\u03c4\u03b5 \u1f10\u03bd \u03c4\u1fc7 \u03c0\u03b1\u03c1\u03bf\u03c5\u03f2\u03af\u1fb3 \u03bc\u03bf\u03c5, \u03bf\u1f55\u03c4\u03c9\u03f2== [Sidenote: ^s 2 Thess. ii. 5\n(see vulg.).]^s==\u03bc\u03bd\u03b7\u03bc\u03bf\u03bd\u03b5\u03cd\u03bf\u03bd\u03c4\u03b5\u03f2 \u03bc\u03b5\u03c4\u1f70 \u03c6\u03cc\u03b2\u03bf\u03c5 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f10\u03c1\u03b3\u03ac\u03b6\u03b5\u03f2\u03b8\u03b5, \u03ba\u03b1\u1f76 \u1f14\u03f2\u03c4\u03b1\u03b9\n\u1f51\u03bc\u1fd6\u03bd \u03b6\u03c9\u1f74 \u03b5\u1f30\u03f2 \u03c4\u1f78\u03bd \u03b1\u1f30\u1ff6\u03bd\u03b1\u00b7== [Sidenote: ^t Phil. ii. 13.]^{11}^t==\u0398\u03b5\u1f78\u03f2\n\u03b3\u03ac\u03c1 \u1f10\u03f2\u03c4\u03b9\u03bd \u1f41 \u1f10\u03bd\u03b5\u03c1\u03b3\u1ff6\u03bd \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd.== [Sidenote: ^u Phil. ii.\n14.]^{12}==\u03ba\u03b1\u1f76== ^u==\u03c0\u03bf\u03b9\u03b5\u1fd6\u03c4\u03b5 \u03c7\u03c9\u03c1\u1f76\u03f2 \u03b4\u03b9\u03b1\u03bb\u03bf\u03b3\u03b9\u03f2\u03bc\u1ff6\u03bd== [Sidenote: ^x\nCol. iii. 17, 23.]^x==\u1f45 \u03c4\u03b9 \u1f10\u1f70\u03bd \u03c0\u03bf\u03b9\u1fc6\u03c4\u03b5.==\n\n[Sidenote: ^y Phil. iii. 1.]^{13}==\u039a\u03b1\u1f76== ^y==\u03c4\u1f78 \u03bb\u03bf\u03b9\u03c0\u03cc\u03bd, \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03bf\u03af,\n\u03c7\u03b1\u03af\u03c1\u03b5\u03c4\u03b5 \u1f10\u03bd \u03a7\u03c1\u03b9\u03f2\u03c4\u1ff7. \u0392\u03bb\u03ad\u03c0\u03b5\u03c4\u03b5 \u03b4\u1f72 \u03c4\u03bf\u1f7a\u03f2== [Sidenote: ^z 1 Tim. iii. 8;\nTit. i. 7.]^z==\u03b1\u1f30\u03f2\u03c7\u03c1\u03bf\u03ba\u03b5\u03c1\u03b4\u03b5\u1fd6\u03f2.== [Sidenote: ^a Phil. iv.\n6.]^{14}^a==\u03c0\u03ac\u03bd\u03c4\u03b1 \u03c4\u1f70 \u03b1\u1f30\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u1f51\u03bc\u1ff6\u03bd \u03b3\u03bd\u03c9\u03c1\u03b9\u03b6\u03ad\u03f2\u03b8\u03c9 \u03c0\u03c1\u1f78\u03f2 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd. \u03ba\u03b1\u1f76==\n[Sidenote: ^b 1 Cor. xv. 58.]^b==\u1f11\u03b4\u03c1\u03b1\u1fd6\u03bf\u03b9 \u03b3\u03af\u03bd\u03b5\u03f2\u03b8\u03b5 \u1f10\u03bd== [Sidenote:\n^c 1 Cor. ii. 16.]^c==\u03c4\u1ff7 \u03bd\u03bf\u0390 \u03c4\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03f2\u03c4\u03bf\u1fe6.== [Sidenote: ^d Phil.\niv. 8, 9.]^{15}^d==\u1f45\u03f2\u03b1 \u03c4\u03b5 \u1f41\u03bb\u03cc\u03ba\u03bb\u03b7\u03c1\u03b1 \u03ba\u03b1\u1f76 \u1f00\u03bb\u03b7\u03b8\u1fc6 \u03ba\u03b1\u1f76 \u03f2\u03b5\u03bc\u03bd\u1f70 \u03ba\u03b1\u1f76 \u03b4\u03af\u03ba\u03b1\u03b9\u03b1 \u03ba\u03b1\u1f76\n\u03c0\u03c1\u03bf\u03f2\u03c6\u03b9\u03bb\u1fc6, \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c0\u03c1\u03ac\u03f2\u03f2\u03b5\u03c4\u03b5.== ^{16}==\u1f03 \u03ba\u03b1\u1f76 \u1f20\u03ba\u03bf\u03cd\u03f2\u03b1\u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03b5\u03bb\u03ac\u03b2\u03b5\u03c4\u03b5, \u1f10\u03bd\n\u03c4\u1fc7 \u03ba\u03b1\u03c1\u03b4\u03af\u1fb3 \u03ba\u03c1\u03b1\u03c4\u03b5\u1fd6\u03c4\u03b5, \u03ba\u03b1\u1f76 \u1f21 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u1f14\u03f2\u03c4\u03b1\u03b9 \u03bc\u03b5\u03b8' \u1f51\u03bc\u1ff6\u03bd.==\n\n[Sidenote: ^e Phil. iv. 22.]\n\n^{18}^e==\u1f08\u03f2\u03c0\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 \u1f51\u03bc\u1fb6\u03f2 \u03bf\u1f31 \u1f05\u03b3\u03b9\u03bf\u03b9.==\n\n[Sidenote: ^f Phil. iv. 23.]\n\n^{19}^f==\u1f29 \u03c7\u03ac\u03c1\u03b9\u03f2 \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f38\u03b7\u03f2\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03f2\u03c4\u03bf\u1fe6 \u03bc\u03b5\u03c4\u1f70 \u03c4\u03bf\u1fe6 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03f2 \u1f51\u03bc\u1ff6\u03bd.==\n\n[Sidenote: ^g Col. iv. 16.]\n\n^{20}^g==\u03ba\u03b1\u1f76 \u03c0\u03bf\u03b9\u03ae\u03f2\u03b1\u03c4\u03b5 \u1f35\u03bd\u03b1 \u03c4\u03bf\u1fd6\u03f2 \u039a\u03bf\u03bb\u03b1\u03f2\u03f2\u03b1\u03b5\u1fe6\u03f2\u03b9\u03bd \u03b1\u03bd\u03b1\u03b3\u03bd\u03c9\u03f2\u03b8\u1fc7, \u03ba\u03b1\u1f76 \u03b7 \u03c4\u1ff6\u03bd\n\u039a\u03bf\u03bb\u03b1\u03f2\u03f2\u03b1\u03ad\u03c9\u03bd \u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76 \u1f51\u03bc\u1fd6\u03bd.==\n\nFootnote 638:\n\n  A Greek version is given in Elias Hutter's Polyglott New Testament\n  (Noreb. 1599); see Anger p. 147 note g. But I have retranslated the\n  epistle anew, introducing the Pauline passages, of which it is almost\n  entirely made up, as they stand in the Greek Testament. The references\n  are given in the margin.\n\n                  *       *       *       *       *\n\n[Sidenote: Scanty circulation in the East,]\n\nBut, though written originally in Greek, it was not among Greek\nChristians that this epistle attained its widest circulation. In the\nlatter part of the 8th century indeed, when the Second Council of Nic\u00e6a\nmet, it had found its way into some copies of St Paul's Epistles[639].\nBut the denunciation of this Council seems to have been effective in\nsecuring its ultimate exclusion. We discover no traces of it in any\nextant Greek MS, with the very doubtful exception which has already been\nconsidered[640]. [Sidenote: but wide diffusion in the West.]But in the\nLatin Church the case was different. St Jerome, as we saw, had\npronounced very decidedly against it. Yet even his authority was not\nsufficient to stamp it out. At least as early as the sixth century it\nfound a place in some copies of the Latin Bibles: and before the close\nof that century its genuineness was affirmed by perhaps the most\ninfluential theologian whom the Latin Church produced during the eleven\ncenturies which elapsed between the age of Jerome and Augustine and the\nera of the Reformation. [Sidenote: Gregory the Great.]Gregory the Great\ndid not indeed affirm its canonicity. He pronounced that the Church had\nrestricted the canonical Epistles of St Paul to fourteen, and he found a\nmystical explanation of this limitation in the number itself, which was\nattained by adding the number of the Commandments to the number of the\nGospels and thus fitly represented the teaching of the Apostle which\ncombines the two[641]. But at the same time he states that the Apostle\nwrote fifteen; and, though he does not mention the Epistle to the\nLaodiceans by name, there can be little doubt that he intended to\ninclude this as his fifteenth epistle, and that his words were rightly\nunderstood by subsequent writers as affirming its Pauline authorship.\nThe influence of this great name is perceptible in the statements of\nlater writers. [Sidenote: Haymo of Halberstadt.]Haymo of Halberstadt,\nwho died A.D. 853, commenting on Col. iv. 16, says, The Apostle 'enjoins\nthe Laodicean Epistle to be read to the Colossians, because though it is\nvery short and is not reckoned in the Canon, yet still it has some\nuse[642]'.[Sidenote: Hervey of Dole.] And between two or three centuries\nlater Hervey of Dole (c. A.D. 1130), if it be not Anselm of Laon[643],\ncommenting on this same passage, says: 'Although the Apostle wrote this\nepistle also as his fifteenth or sixteenth[644], and it is established\nby Apostolic authority like the rest, yet holy Church does not reckon\nmore than fourteen,' and he proceeds to justify this limitation of the\nCanon with the arguments and in the language of Gregory[645]. Others\nhowever did not confine themselves to the qualified recognition given to\nthe epistle by the great Bishop of Rome. Gregory had carefully\ndistinguished between genuineness and canonicity; but this important\ndistinction was not seldom disregarded by later writers. [Sidenote:\nEnglish Church. Aelfric.]In the English Church more especially it was\nforgotten. Thus Aelfric abbot of Cerne, who wrote during the closing\nyears of the tenth century, speaks as follows of St Paul: 'Fifteen\nepistles wrote this one Apostle to the nations by him converted unto the\nfaith: which are large books in the Bible and make much for our\namendment, if we follow his doctrine that was teacher of the Gentiles'.\nHe then gives a list of the Apostle's writings, which closes with 'one\nto Philemon and one to the Laodiceans; fifteen in all as loud as thunder\nto faithful people[646]'. [Sidenote: John of Salisbury.]Again, nearly\ntwo centuries later John of Salisbury, likewise writing on the Canon,\nreckons 'Fifteen epistles of Paul included in one volume, though it be\nthe wide-spread and common opinion of nearly all that there are only\nfourteen; ten to churches and four to individuals: supposing that the\none addressed to the Hebrews is to be reckoned among the Epistles of\nPaul, as Jerome the doctor of doctors seems to lay down in his preface,\nwhere he refuteth the cavils of those who contended that it was not\nPaul's. But the fifteenth is that which is addressed to the Church of\nthe Laodiceans; and though, as Jerome saith, it be rejected by all,\nnevertheless was it written by the Apostle. Nor is this opinion assumed\non the conjecture of others, but it is confirmed by the testimony of the\nApostle himself: for he maketh mention of it in the Epistle to the\nColossians in these words, _When this epistle shall have been read among\nyou_, etc. (Col. iv. 16)[647]'. Aelfric and John are the typical\ntheologians of the Church in this country in their respective ages. The\nConquest effected a revolution in ecclesiastical and theological\nmatters. The Old English Church was separated from the Anglo-Norman\nChurch in not a few points both of doctrine and of discipline. Yet here\nwe find the representative men of learning in both agreed on this one\npoint\u2014the authorship and canonicity of the Epistle to the Laodiceans.\nFrom the language of John of Salisbury however it appears that such was\nnot the common verdict at least in his age, and that on this point the\ninstinct of the many was more sound than the learning of the few. Nor\nindeed was it the undisputed opinion even of the learned in this country\nduring this interval. [Sidenote: The epistle repudiated by Lanfranc.]The\nfirst Norman Archbishop, Lanfranc, an Italian by birth and education,\nexplains the passage in the Colossian Epistle as referring to a letter\nwritten by the Laodiceans to the Apostle, and adds that otherwise 'there\nwould be more than thirteen Epistles of Paul[648]'. Thus he tacitly\nignores the Epistle to the Laodiceans, with which he can hardly have\nbeen unacquainted.\n\nFootnote 639:\n\n  Quoted above, p. 359, note 637.\n\nFootnote 640:\n\n  See above, p. 315 sq.\n\nFootnote 641:\n\n  Greg. Magn. _Mor. in Iob._ xxxv. \u00a7 25 (III. p. 433, ed. Gallicc.)\n  'Recte vita ecclesi\u00e6 multiplicata per decem et quattuor computatur;\n  quia utrumque testamentum custodiens, et tam secundum Legis decalogum\n  quam secundum quattuor Evangelii libros vivens, usque ad perfectionis\n  culmen extenditur. Unde et Paulus apostolus quamvis epistolas\n  quindecim scripserit, sancta tamen ecclesia non amplius quam\n  quatuordecim tenet, ut ex ipso epistolarum numero ostenderet quod\n  doctor egregius Legis et Evangelii secreta rimasset'.\n\nFootnote 642:\n\n  _Patrol. Lat._ CXVII. p. 765 (ed. Migne) 'Et eam qu\u00e6 erat Laodicensium\n  ideo pr\u00e6cipit Colossensibus legi, quia, licet perparva sit et in\n  Canone non habeatur, aliquid tamen utilitatis habet'. He uses the\n  expression 'eam qu\u00e6 erat Laodicensium', because \u03c4\u1f74\u03bd \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 was\n  translated in the Latin Bible 'eam qu\u00e6 Laodicensium est'.\n\nFootnote 643:\n\n  See _Galatians_ p. 232 on the authorship of this commentary.\n\nFootnote 644:\n\n  A third Epistle to the Corinthians being perhaps reckoned as the 15th;\n  see Fabric. _Cod. Apocr. Nov. Test._ II p. 866.\n\nFootnote 645:\n\n  _Patrol. Lat._ CLXXXI. p. 1355 sq. (ed. Migne) '_et ea_ similiter\n  epistola, _qu\u00e6 Laodicensium_ est, i.e. quam ego Laodicensibus misi,\n  legatur vobis. Quamvis et hanc epistolam quintamdecimam vel\n  sextamdecimam apostolus scripserit, et auctoritas eam apostolica sicut\n  c\u00e6tera firmavit, sancta tamen ecclesia non amplius quam quatuordecim\n  tenet, ut ex ipso epistolarum numero ostenderet etc.' At the end of\n  the notes to the Colossians he adds 'Hucusque protenditur epistola qu\u00e6\n  missa est ad Colossenses. Congruum autem videtur ut propter notitiam\n  legentium subjiciamus eam qu\u00e6 est ad Laodicenses directa; quam, ut\n  diximus, in usu non habet ecclesia. Est ergo talis.' Then follows the\n  text of the Laodicean Epistle, but it is not annotated.\n\nFootnote 646:\n\n  _A Saxon Treatise concerning the Old and New Testament_ by \u00c6lfricus\n  Abbas, p. 28 (ed. W. L'Isle, London 1623).\n\nFootnote 647:\n\n  Ioann. Sarisb. _Epist._ 143 (I. p. 210 ed. Giles) 'Epistol\u00e6 Pauli\n  quindecim uno volumine comprehens\u00e6, licet sit vulgata et fere omnium\n  communis opinio non esse nisi quatuordecim, decem ad ecclesias,\n  quatuor ad personas; si tamen illa qu\u00e6 ad Hebr\u00e6os est connumeranda est\n  epistolis Pauli, quod in pr\u00e6fatione ejus astruere videtur doctorum\n  doctor Hieronymus, illorum dissolvens argutias qui eam Pauli non esse\n  contendebant. C\u00e6terum quintadecima est illa qu\u00e6 ecclesi\u00e6 Laodicensium\n  scribitur; et licet, ut ait Hieronymus, ab omnibus explodatur, tamen\n  ab apostolo scripta est: neque sententia h\u00e6c de aliorum pr\u00e6sumitur\n  opinione sed ipsius apostoli testimonio roboratur. Meminit enim ipsius\n  in epistola ad Colossenses his verbis, _Quum lecta fuerit apud vos h\u00e6c\n  epistola, etc._'\n\nFootnote 648:\n\n  _Patrol. Lat._ CL. p. 331 (ed. Migne) on Col. iv. 16 'H\u00e6c si esset\n  apostoli, _ad Laodicenses_ diceret, non _Laodicensium_; et plusquam\n  tredecim essent epistol\u00e6 Pauli'. We should perhaps read xiiii for\n  xiii, 'quatuordecim' for 'tredecim', as Lanfranc is not likely to have\n  questioned the Pauline authorship of the Epistle to the Hebrews.\n\n[Sidenote: Occurrence in MSS of all ages and countries.]\n\nIndeed the safest criterion of the extent to which this opinion\nprevailed, is to be found in the manuscripts. At all ages from the sixth\nto the fifteenth century we have examples of its occurrence among the\nPauline Epistles and most frequently without any marks which imply doubt\nrespecting its canonicity. These instances are more common in proportion\nto the number of extant MSS in the earlier epoch than in the later[649].\nIn one of the three or four extant authorities for the Old Latin Version\nof the Pauline Epistles it has a place[650]. In one of the two most\nancient copies of Jerome's revised Vulgate it is found[651]. Among the\nfirst class MSS of this latter Version its insertion is almost as common\nas its omission. This phenomenon moreover is not confined to any one\ncountry. Italy, Spain, France, Ireland, England, Germany,\nSwitzerland\u2014all the great nations of Latin Christendom\u2014contribute\nexamples of early manuscripts in which this epistle has a place[652].\n\nFootnote 649:\n\n  The proportion however is very different in different collections. In\n  the Cambridge University Library I found the epistle in four only out\n  of some thirty MSS Which I inspected; whereas in the Lambeth Library\n  the proportion was far greater.\n\nFootnote 650:\n\n  The _Speculum_ of Mai, see above, p. 348.\n\nFootnote 651:\n\n  The Codex Fuldensis, which was written within a few years of the Codex\n  Amiatinus.\n\nFootnote 652:\n\n  The list of MSS given above p. 348 sq. will substantiate this\n  statement.\n\n[Sidenote: Versions.]\n\nAnd, when the Scriptures came to be translated into the vernacular\nlanguages of modern Europe, this epistle was not uncommonly included.\n[Sidenote: Albigensian.]Thus we meet with an Albigensian version, which\nis said to belong to the thirteenth century[653]. [Sidenote:\nBohemian.]Thus too it is found in the Bohemian language, both in\nmanuscript and in the early printed Bibles, in various recensions[654].\n[Sidenote: German.]And again an old German translation is extant, which,\njudging from linguistic peculiarities, cannot be assigned to a later\ndate than about the fourteenth century, and was printed in not less than\nfourteen editions of the German Bible at the close of the fifteenth and\nthe beginning of the sixteenth centuries, before Luther's version\nappeared[655]. [Sidenote: English.]In the early English Bibles too it\nhas a place. Though it was excluded by both Wycliffe and Purvey, yet it\ndid not long remain untranslated and appears in two different and quite\nindependent versions, in MSS written before the middle of the fifteenth\ncentury[656]. The prologue prefixed to the commoner of the two forms\nruns as follows:\n\n[Sidenote: English prologue.]\n\n'Laodicensis ben also Colocenses, as tweye townes and oo peple in\nmaners. These ben of Asie, and among hem hadden be false apostlis, and\ndisceyuede manye. Therfore the postle bringith hem to mynde of his\nconuersacion and trewe preching of the gospel, and excitith hem to be\nstidfast in the trewe witt and loue of Crist, and to be of oo wil. But\nthis pistil is not in comyn Latyn bookis, and therfor it was but late\ntranslatid into Englisch tunge[657].'\n\nFootnote 653:\n\n  An account of this MS, which is at Lyons, is given by Reuss in the\n  _Revue de Th\u00e9ologie_ v. p. 334 (Strassb. 1852). He ascribes the\n  translation of the New Testament to the 13th century, and dates the MS\n  a little later.\n\nFootnote 654:\n\n  This version is printed by Anger, p. 170 sq.\n\nFootnote 655:\n\n  See Anger, p. 149 sq., p. 166 sq.\n\nFootnote 656:\n\n  These two versions are printed in Lewis's New Testament translated by\n  J. Wiclif (1731) p. 99 sq., and in Forshall and Madden's _Wycliffite\n  Versions of the Holy Bible_ (1850) IV. p. 438 sq. They are also given\n  by Anger p. 168 sq. (1843), who takes the rarer form from Lewis and\n  the other from a Dresden MS. Dr Westcott also has printed the commoner\n  version in his _Canon_, p. 457 (ed. 4), from Forshall and Madden.\n\n  Of one of these two versions Forshall and Madden give a collation of\n  several MSS; the other is taken from a single MS (I. p. xxxii). Lewis\n  does not state whence he derived the rarer of these two versions, but\n  there can be little doubt that it came from the same MS _Pepys._ 2073\n  (belonging to Magd. Coll. Cambridge) from which it was taken by\n  Forshall and Madden (I. p. lvii); since he elsewhere mentions using\n  this MS (p. 104). The version is not known to exist in any other.\n  Forshall and Madden give the date of the MS as about 1440.\n\nFootnote 657:\n\n  From Forshall and Madden, IV. p. 438. The earliest MSS which contain\n  the common version of the Laodicean Epistle (to which this prologue is\n  prefixed) date about A.D. 1430.\n\n[Sidenote: Two Versions of the epistle.]\n\nThe two forms of the epistle in its English dress are as follows[658].\nThe version on the left hand is extant only in a single MS; the other,\nwhich occupies the right column, is comparatively common.\n\n'Poul, apostle, not of men, ne bi man, but bi Jhesu Crist, to the\nbritheren that ben of Laodice, grace to \u021dou, and pees of God the fadir,\nand of the Lord Jhesu Crist. Gracis I do to Crist bi al myn orisoun,\nthat \u021de be dwellinge in him and lastinge, bi the biheest abidinge in the\ndai of doom. Ne he vnordeynede vs of sum veyn speche feynynge, that vs\nouerturne fro the sothfastnesse of the gospel that of me is prechid.\nAlso now schal God do hem leuynge, and doynge of blessdnesse of werkis,\nwhich heelthe of lyf is. And now openli ben my boondis, whiche I suffre\nin Crist Jhesu, in whiche I glad and ioie. And that is to me heelthe\neuerlastynge, that that I dide with oure preieris, and mynystringe the\nHoly Spirit, bi lijf or bi deeth. It is forsothe to me lijf into Crist,\nand to die ioie withouten eende. In vs he schal do his merci, that \u021de\nhaue the same louynge, and that \u021de be of o wil. Therfore, derlyngis, as\n\u021de han herd in presence of me, hold \u021de, and do \u021de in drede of God; and\nit schal be to \u021dou lijf withouten eend. It is forsothe God that worchith\nin vs. And do \u021de withouten ony withdrawinge, what soeuere \u021de doon. And\nthat it is, derlyngis, ioie \u021de in Crist, and flee \u021de maad foul in clay.\nAlle \u021doure axingis ben open anentis God, and be \u021de fastned in the witt\nof Crist. And whiche been hool, and sooth, and chast, and rightwijs, and\nlouable, do \u021de; and whiche herden and take in herte, hold \u021de; and it\nschal be to \u021dou pees. Holi men greeten \u021dou weel, in the grace of oure\nLord Jhesu Crist, with the Holi Goost. And do \u021de that pistil of\nColosensis to be red to \u021dou. Amen.\n\n'Poul, apostle, not of men, ne by man, but bi Jhesu Crist, to the\nbritheren that ben at Laodice, grace to \u021dou, and pees of God the fadir,\nand of the Lord Jhesu Crist. I do thankyngis to my God bi al my preier,\nthat \u021de be dwelling and lastyng in him, abiding the biheest in the day\nof doom. For neithir the veyn spekyng of summe vnwise men hath lettide\n\u021dou, the whiche wolden turne \u021dou fro the treuthe of the gospel, that is\nprechid of me. And now hem that ben of me, to the profi\u021dt of truthe of\nthe gospel, God schal make disseruyng, and doyng benygnyte of werkis,\nand helthe of euerlasting lijf. And now my boondis ben open, which Y\nsuffre in Crist Jhesu, in whiche Y glade and ioie. And that is to me to\neuerlastyng helthe, that this same thing be doon by \u021doure preiers, and\nmynystryng of the Holi Goost, either bi lijf, either bi deeth. Forsothe\nto me it is lijf to lyue in Crist, and to die ioie. And his mercy schal\ndo in \u021dou the same thing, that \u021de moun haue the same loue, and that \u021de\nbe of oo will. Therfore, \u021de weel biloued britheren, holde \u021de, and do \u021de\nin the dreede of God, as \u021de han herde the presence of me; and lijf schal\nbe to \u021dou withouten eende. Sotheli it is God that worchith in \u021dou. And,\nmy weel biloued britheren, do \u021de without eny withdrawyng what euer\nthingis \u021de don. Joie \u021de in Crist, and eschewe \u021de men defoulid in lucre,\n_either foul wynnyng_. Be alle \u021doure askyngis open anentis God, and be\n\u021de stidefast in the witt of Crist. And do \u021de tho thingis that ben hool,\nand trewe, and chaast, and iust, and able to be loued; and kepe \u021de in\nherte tho thingis that \u021de haue herd and take; and pees schal be to \u021dou.\nAlle holi men greten \u021dou weel. The grace of oure Lord Jhesu Crist be\nwith \u021doure spirit. And do \u021de that pistil of Colocensis to be red to \u021dou.\n\nFootnote 658:\n\n  Printed from Forshall and Madden l.c. I am assured by those who are\n  thoroughly conversant with old English, that they can discern no\n  difference of date in these two versions, and that they both belong\n  probably to the early years of the 15th century. The rarer version is\n  taken from a better Latin text than the other.\n\n[Sidenote: Revival of learning and condemnation of the epistle.]\n\nThus for more than nine centuries this forged epistle hovered about the\ndoors of the sacred Canon, without either finding admission or being\nperemptorily excluded. At length the revival of learning dealt its\ndeath-blow to this as to so many other spurious pretensions. As a rule,\nRoman Catholics and Reformers were equally strong in their condemnation\nof its worthlessness. The language of Erasmus more especially is worth\nquoting for its own sake, and must not be diluted by translation:\n\n[Sidenote: Strictures of Erasmus.]\n\n'Nihil habet Pauli pr\u00e6ter voculas aliquot ex c\u00e6teris ejus epistolis\nmendicatas.... Non est cujusvis hominis Paulinum pectus effingere.\nTonat, fulgurat, meras flammas loquitur Paulus. At h\u00e6c, pr\u00e6terquam quod\nbrevissima est, quam friget, quam jacet!... Quanquam quid attinet\nargumentari? Legat, qui volet, epistolam.... Nullum argumentum\nefficacius persuaserit eam non esse Pauli quam ipsa epistola. Et si quid\nmihi naris est, ejusdem est opificis qui n\u00e6niis suis omnium veterum\ntheologorum omnia scripta contaminavit, conspurcavit, perdidit, ac\npr\u00e6cipue ejus qui pr\u00e6 c\u00e6teris indignus erat ea contumelia, nempe D.\nHieronymi[659]'.\n\nFootnote 659:\n\n  On Col. iv. 16. Erasmus is too hard upon the writer of this letter,\n  when he charges him with such a mass of forgeries. He does not explain\n  how this hypothesis is consistent with the condemnation of the Epistle\n  to the Laodiceans in Hieron. _Vir. Ill._ 5 (quoted above p. 359).\n\n[Sidenote: Exceptions.]\n\nBut some eccentric spirits on both sides were still found to maintain\nits genuineness. [Sidenote: Pr\u00e6torius.]Thus on the one hand the Lutheran\nSteph. Pr\u00e6torius prefaces his edition of this epistle (A.D. 1595) with\nthe statement that he 'restores it to the Christian Church'; he gives\nhis opinion that it was written 'either by the Apostle himself or by\nsome other Apostolic man': he declares that to himself it is 'redolent\nof the spirit and grace of the most divine Paul'; and he recommends\nyounger teachers of the Gospel to 'try their strength in explaining it',\nthat thus 'accustoming themselves gradually to the Apostolic doctrine\nthey may extract thence a flavour sweeter than ambrosia and\nnectar[660].' [Sidenote: Stapleton.]On the other hand the Jesuit\nStapleton was not less eager in his advocacy of this miserable cento. To\nhim its genuineness had a controversial value. Along with several other\napocryphal writings which he accepted in like manner, it was important\nin his eyes as showing that the Church had authority to exclude even\nApostolic writings from the Canon, if she judged fit[661]. But such\nphenomena were quite abnormal. The dawn of the Reformation epoch had\neffectually scared away this ghost of a Pauline epistle, which (we may\nconfidently hope) has been laid for ever and will not again be suffered\nto haunt the mind of the Church.\n\nFootnote 660:\n\n  _Pauli Apostoli ad Laodicenses Epistola, Latine et Germanice_,\n  Hamburg, 1595, of which the preface is given in Fabricius _Cod. Apocr.\n  Nov. Test._ II. p. 867. It is curious that the only two arguments\n  against its genuineness which he thinks worthy of notice are (1) Its\n  brevity; which he answers by appealing to the Epistle to Philemon; and\n  (2) Its recommendation of works ('quod scripsit opera esse facienda\n  qu\u00e6 sunt salutis \u00e6tern\u00e6'); which he explains to refer to works that\n  proceed of faith.\n\nFootnote 661:\n\n  See Bp. Davenant on Col. iv. 16: 'Detestanda Stapletonis opinio, qui\n  ipsius Pauli epistolam esse statuit, quam omnes patres ut adulterinam\n  et insulsam repudiarunt: nec sanior conclusio, quam inde deducere\n  voluit, posse nimirum ecclesiam germanam et veram apostoli Pauli\n  epistolam pro sua authoritate e Canone excludere'. So also Whitaker\n  _Disputation on Scripture_ passim (see the references given above, p.\n  341, note 595).\n\n------------------------------------------------------------------------\n\n\n\n\n                          EPISTLE TO PHILEMON.\n\n                      INTRODUCTION TO THE EPISTLE.\n\n[Sidenote: Unique character of the epistle.]\n\n\nThe Epistle to Philemon holds a unique place among the Apostle's\nwritings. It is the only strictly private letter which has been\npreserved. The Pastoral Epistles indeed are addressed to individuals,\nbut they discuss important matters of Church discipline and government.\nEvidently they were intended to be read by others besides those to whom\nthey are immediately addressed. On the other hand the letter before us\ndoes not once touch upon any question of public interest. It is\naddressed apparently to a layman. It is wholly occupied with an incident\nof domestic life. The occasion which called it forth was altogether\ncommon-place. It is only one sample of numberless letters which must\nhave been written to his many friends and disciples by one of St Paul's\neager temperament and warm affections, in the course of a long and\nchequered life. [Sidenote: Its value.]Yet to ourselves this fragment,\nwhich has been rescued, we know not how, from the wreck of a large and\nvaried correspondence, is infinitely precious. Nowhere is the social\ninfluence of the Gospel more strikingly exerted; nowhere does the\nnobility of the Apostle's character receive a more vivid illustration\nthan in this accidental pleading on behalf of a runaway slave.\n\n[Sidenote: The persons addressed.]\n\nThe letter introduces us to an ordinary household in a small town of\nPhrygia. Four members of it are mentioned by name, the father, the\nmother, the son, and the slave.\n\n[Sidenote: 1. Philemon.]\n\n1. The head of the family bears a name which, for good or for evil, was\nnot unknown in connexion with Phrygian story. [Sidenote: Occurrence of\nthe name in Phrygia.]The legend of Philemon and Baucis, the aged\npeasants who entertained not angels but gods unawares, and were rewarded\nby their divine guests for their homely hospitality and their conjugal\nlove[662], is one of the most attractive in Greek mythology, and\ncontrasts favourably with many a revolting tale in which the powers of\nOlympus are represented as visiting this lower earth. It has a special\ninterest too for the Apostolic history, because it suggests an\nexplanation of the scene at Lystra, when the barbarians would have\nsacrificed to the Apostles, imagining that the same two gods, Zeus and\nHermes, had once again deigned to visit, in the likeness of men, those\nregions which they had graced of old by their presence[663]. Again, in\nhistorical times we read of one Philemon who obtained an unenviable\nnotoriety at Athens by assuming the rights of Athenian citizenship,\nthough a Phrygian and apparently a slave[664]. Otherwise the name is not\ndistinctively Phrygian. It does not occur with any special frequency in\nthe inscriptions belonging to this country; and though several persons\nbearing this name rose to eminence in literary history, not one, so far\nas we know, was a Phrygian.\n\nFootnote 662:\n\n  Ovid. _Met._ vii. 626 sq. 'Jupiter huc, _specie mortali_, cumque\n  parente Venit Atlantiades positis caducifer alis' etc.\n\nFootnote 663:\n\n  Acts xiv. 11 \u03bf\u1f35 \u03b8\u03b5\u03bf\u1f76 _\u1f41\u03bc\u03bf\u03b9\u03c9\u03b8\u03ad\u03bd\u03c4\u03b5\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2_ \u03ba\u03b1\u03c4\u03ad\u03b2\u03b7\u03c3\u03b1\u03bd \u03c0\u03c1\u1f78\u03c2 \u1f21\u03bc\u1fb6\u03c2\n  \u03ba.\u03c4.\u03bb. There are two points worth observing in the Phrygian legend, as\n  illustrating the Apostolic history. (1) It is a miracle, which opens\n  the eyes of the peasant couple to the divinity of their guests thus\n  disguised; (2) The immediate effect of this miracle is their attempt\n  to sacrifice to their divine visitors, 'dis hospitibus mactare\n  parabant'. The familiarity with this beautiful story may have\n  suggested to the barbarians of Lystra, whose 'Lycaonian speech' was\n  not improbably a dialect of Phrygian, that the same two gods, Zeus and\n  Hermes, had again visited this region on an errand at once of\n  beneficence and of vengeance, while at the same time it would prompt\n  them to conciliate the deities by a similar mode of propitiation,\n  \u1f24\u03b8\u03b5\u03bb\u03bf\u03bd \u03b8\u03cd\u03b5\u03b9\u03bd.\n\nFootnote 664:\n\n  Aristoph. _Av._ 762 \u03b5\u1f30 \u03b4\u1f72 \u03c4\u03c5\u03b3\u03c7\u03ac\u03bd\u03b5\u03b9 \u03c4\u03b9\u03c2 \u1f62\u03bd \u03a6\u03c1\u1f7a\u03be ... \u03c6\u03c1\u03c5\u03b3\u03af\u03bb\u03bf\u03c2 \u1f44\u03c1\u03bd\u03b9\u03c2\n  \u1f10\u03bd\u03b8\u03ac\u03b4' \u1f14\u03c3\u03c4\u03b1\u03b9, \u03c4\u03bf\u1fe6 \u03a6\u03b9\u03bb\u03ae\u03bc\u03bf\u03bd\u03bf\u03c2 \u03b3\u03ad\u03bd\u03bf\u03c5\u03c2.\n\n[Sidenote: This Philemon a Colossian]\n\nThe Philemon with whom we are concerned was a native, or at least an\ninhabitant, of Coloss\u00e6. This appears from the fact that his slave is\nmentioned as belonging to that place. It may be added also, in\nconfirmation of this view, that in one of two epistles written and\ndespatched at the same time St Paul announces the restoration of\nOnesimus to his master, while in the other he speaks of this same person\nas revisiting Coloss\u00e6[665]. On the other hand it would not be safe to\nlay any stress on the statement of Theodoret, that Philemon's house was\nstill standing at Coloss\u00e6 when he wrote[666], for traditions of this\nkind have seldom any historical worth.\n\nFootnote 665:\n\n  Compare Col. iv. 9 with Philem. 11 sq.\n\nFootnote 666:\n\n  Theodoret in his preface to the epistle says \u03c0\u03cc\u03bb\u03b9\u03bd \u03b4\u03b5 \u03b5\u1f36\u03c7\u03b5 [[\u1f41\n  \u03a6\u03b9\u03bb\u03ae\u03bc\u03c9\u03bd] \u03c4\u1f70\u03c2 \u039a\u03bf\u03bb\u03ac\u03c3\u03c3\u03b1\u03c2\u00b7 \u03ba\u03b1\u1f76 \u1f21 \u03bf\u1f30\u03ba\u03af\u03b1 \u03b4\u1f72 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03bc\u03ad\u03c7\u03c1\u03b9 \u03c4\u03bf\u1fe6 \u03c0\u03b1\u03c1\u03cc\u03bd\u03c4\u03bf\u03c2\n  \u03bc\u03b5\u03bc\u03ad\u03bd\u03b7\u03ba\u03b5. This is generally taken to mean that Philemon's house was\n  still standing, when Theodoret wrote. This may be the correct\n  interpretation, but the language is not quite explicit.\n\n[Sidenote: converted by St Paul.]\n\nPhilemon had been converted by St Paul himself[667]. At what time or\nunder what circumstances he received his first lessons in the Gospel, we\ndo not know: but the Apostle's long residence at Ephesus naturally\nsuggests itself as the period when he was most likely to have become\nacquainted with a citizen of Coloss\u00e6[668].\n\nFootnote 667:\n\n  ver. 19.\n\nFootnote 668:\n\n  See above, p. 30 sq.\n\n[Sidenote: His evangelical zeal,]\n\nPhilemon proved not unworthy of his spiritual parentage. Though to\nEpaphras belongs the chief glory of preaching the Gospel at\nColoss\u00e6[669], his labours were well seconded by Philemon. The title of\n'fellow-labourer,' conferred upon him by the Apostle[670], is a noble\ntestimony to his evangelical zeal. Like Nymphas in the neighbouring\nChurch of Laodicea[671], Philemon had placed his house at the disposal\nof the Christians at Coloss\u00e6 for their religious and social\ngatherings[672]. Like Gaius[673], to whom the only other private letter\nin the Apostolic Canon is addressed[674], he was generous in his\nhospitalities. [Sidenote: and wide hospitality.]All those with whom he\ncame in contact spoke with gratitude of his kindly attentions[675]. Of\nhis subsequent career we have no certain knowledge. [Sidenote: Legendary\nmartyrdom.]Legendary story indeed promotes him to the bishopric of\nColoss\u00e6[676], and records how he was martyred in his native city under\nNero[677]. But this tradition or fiction is not entitled to any credit.\nAll that we really know of Philemon is contained within this epistle\nitself.\n\nFootnote 669:\n\n  See above, p. 31 sq.\n\nFootnote 670:\n\n  ver. 1 \u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u1ff7 \u1f21\u03bc\u1ff6\u03bd.\n\nFootnote 671:\n\n  Col. iv. 15.\n\nFootnote 672:\n\n  ver. 2 \u03c4\u1fc7 \u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03cc\u03bd \u03c3\u03bf\u03c5 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3. The Greek commentators, Chrysostom\n  and Theodoret, suppose that St Paul designates Philemon's own family\n  (including his slaves) by this honourable title of \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1, in order\n  to interest them in his petition. This is plainly wrong. See the note\n  on Col. iv. 15.\n\nFootnote 673:\n\n  3 Joh. 5 sq.\n\nFootnote 674:\n\n  I take the view that the \u03ba\u03c5\u03c1\u03af\u03b1 addressed in the Second Epistle of St\n  John is some church personified, as indeed the whole tenour of the\n  epistle seems to imply: see esp. vv. 4, 7 sq. The salutation to the\n  'elect lady' (ver. 1) from her 'elect sister' (ver. 15) will then be a\n  greeting sent to one church from another; just as in 1 Peter, the\n  letter is addressed at the outset \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1fd6\u03c2 \u03a0\u03cc\u03bd\u03c4\u03bf\u03c5 \u03ba.\u03c4.\u03bb. (i. 1) and\n  contains at the close a salutation from \u1f21 \u1f10\u03bd \u0392\u03b1\u03b2\u03c5\u03bb\u1ff6\u03bd\u03b9 \u03c3\u03c5\u03bd\u03b5\u03ba\u03bb\u03b5\u03ba\u03c4\u03ae (v.\n  13).\n\nFootnote 675:\n\n  vv. 5, 7.\n\nFootnote 676:\n\n  _Apost. Const._ vii. 46 \u03c4\u1fc6\u03c2 \u03b4\u1f72 \u1f10\u03bd \u03a6\u03c1\u03c5\u03b3\u03af\u1fb3 \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 [\u1f10\u03c0\u03af\u03c3\u03ba\u03bf\u03c0\u03bf\u03c2]\n  \u1f0c\u03c1\u03c7\u03b9\u03c0\u03c0\u03bf\u03c2, \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03ac\u03b5\u03c9\u03bd \u03b4\u1f72 \u03a6\u03b9\u03bb\u03ae\u03bc\u03c9\u03bd, \u0392\u03b5\u03c1\u03bf\u03af\u03b1\u03c2 \u03b4\u1f72 \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03c4\u1f70 \u039c\u03b1\u03ba\u03b5\u03b4\u03bf\u03bd\u03af\u03b1\u03bd\n  \u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03c2 \u1f41 \u03a6\u03b9\u03bb\u03ae\u03bc\u03bf\u03bd\u03bf\u03c2. The Greek _Men\u00e6a_ however make Philemon bishop\n  of Gaza; see Tillemont I. p. 574 note lxvi.\n\nFootnote 677:\n\n  See Tillemont I. pp. 290, 574, for the references.\n\n[Sidenote: 2. Apphia his wife.]\n\n2. It is a safe inference from the connexion of the names that Apphia\nwas the wife of Philemon[678]. The commentators assume without misgiving\nthat we have here the familiar Roman name Appia, though they do not\nexplain the intrusion of the aspirate[679]. This seems to be a mistake.\n[Sidenote: A strictly Phrygian name.]The word occurs very frequently on\nPhrygian inscriptions as a proper name, and is doubtless of native\norigin. At Aphrodisias and Philadelphia, at Eumenia and Apamea Cibotus,\nat Stratonicea, at Philomelium, at \u00c6zani and Coti\u00e6um and Doryl\u00e6um, at\nalmost all the towns far and near, which were either Phrygian or subject\nto Phrygian influences, and in which any fair number of inscriptions has\nbeen preserved, the name is found. If no example has been discovered at\nColoss\u00e6 itself, we must remember that not a single proper name has been\npreserved on any monumental inscription at this place. It is generally\nwritten either Apphia or Aphphia[680]; more rarely Aphia, which is\nperhaps due merely to the carelessness of the stonecutters[681].\n[Sidenote: Its affinities]But, so far as I have observed, it always\npreserves the aspirate. Its diminutive is Apphion or Aphphion or\nAphion[682]. The allied form Aphphias or Aphias, also a woman's name, is\nfound, though less commonly[683]; and we likewise frequently meet with\nthe shorter form Apphe or Aphphe[684]. The man's name corresponding to\nApphia is Apphianos, but this is rare[685]. The root would appear to be\nsome Phrygian term of endearment or relationship[686]. [Sidenote: and\nanalogies.]It occurs commonly in connexion with other Phrygian names of\na like stamp, more especially Ammia, which undergoes the same\nmodifications of form, Amia, Ammias, Ammion or Amion, Ammiane or\nAmmiana, with the corresponding masculine Ammianos[687]. With these we\nmay also compare Tatia, Tatias, Tation, Tatiane or Tatiana, Tatianos.\nSimilar too is the name Papias or Pappias, with the lengthened form\nPapianos, to which corresponds the feminine Papiane[688]. So again we\nhave Nannas or Nanas, Nanna or Nana, with their derivatives, in these\nPhrygian inscriptions[689]. [Sidenote: Not to be confused with the Latin\nAppia.]There is a tendency in some of the allied forms of Apphia or\nAphphia to drop the aspirate so that they are written with a _pp_, more\nespecially in Appe[690], but not in the word itself; nor have I observed\nconversely any disposition to write the Roman name Appia with an\naspirate, Apphia or Aphphia[691]. Even if such a disposition could be\nproved, the main point for which I am contending can hardly be\nquestioned. With the overwhelming evidence of the inscriptions before\nus, it is impossible to doubt that Apphia is a native Phrygian\nname[692].\n\nFootnote 678:\n\n  Boeckh _Corp. Inscr._ 3814 \u039d\u03ad\u03b9\u03ba\u03b1\u03bd\u03b4\u03c1\u03bf\u03c2 \u03ba\u03b1\u1f76 \u1f08\u03c6\u03c6\u03af\u03b1 \u03b3\u03c5\u03bd\u1f74 \u03b1\u1f50\u03c4\u03bf\u1fe6. In the\n  following inscriptions also a wife bearing the name Apphia (Aphphia,\n  Aphia) or Apphion (Aphphion, Aphion) is mentioned in connexion with\n  her husband; 2720, 2782, 2836, 3446, 2775 b, c, d, 2837 b, 3849, 3902\n  m, 3962, 4141, 4277, 4321 f, 3846 z^{17}, etc.\n\n  M. Renan (_Saint Paul_ p. 360) says, 'Appia, diaconesse de cette\n  ville'. Like other direct statements of this same writer, as for\n  instance that the Colossians sent a deputation to St Paul\n  (_L'Antechrist_ p. 90), this assertion rests on no authority.\n\nFootnote 679:\n\n  They speak of \u1f08\u03c0\u03c6\u03af\u03b1 as a softened form of the Latin _Appia_, and quote\n  Acts xxviii. 15, where however the form is \u1f08\u03c0\u03c0\u03af\u03bf\u03c5. Even Ewald writes\n  the word Appia.\n\nFootnote 680:\n\n  \u1f08\u03c0\u03c6\u03af\u03b1, no. 2782, 2835, 2950, 3432, 3446, 2775 b, c, d, 2837 b, 3902 m,\n  3962, 4124, 4145: \u1f08\u03c6\u03c6\u03af\u03b1, no. 3814, 4141, 4277, 4321 f, 3827 l, 3846 z,\n  3846 z^{17}. So far as I could trace any law, the form \u1f08\u03c6\u03c6\u03af\u03b1 is\n  preferred in the northern and more distant towns like \u00c6zani and\n  Coti\u00e6um, while \u1f08\u03c0\u03c6\u03af\u03b1 prevails in the southern towns in the more\n  immediate neighbourhood of Coloss\u00e6, such as Aphrodisias. This accords\n  with the evidence of our MSS, in which \u1f08\u03c0\u03c6\u03af\u03b1 is the best supported\n  form, though \u1f08\u03c6\u03c6\u03af\u03b1 is found in some. In Theod. Mops. (Cramer's _Cat._\n  p. 105) it becomes \u1f08\u03bc\u03c6\u03af\u03b1 by a common corruption; and Old Latin copies\n  write the dative _Apphiadi_ from the allied form _Apphias_.\n\n  The most interesting of these inscriptions mentioning the name is no.\n  2782 at Aphrodisias, where there is a notice of \u03a6\u03bb. \u1f08\u03c0\u03c6\u03af\u03b1\u03c2 \u1f00\u03c1\u03c7\u03b9\u03b5\u03c1\u03b5\u03af\u03b1\u03c2\n  \u1f08\u03c3\u03af\u03b1\u03c2, \u03bc\u03b7\u03c4\u03c1\u1f78\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03ac\u03bc\u03bc\u03b7\u03c2 \u03c3\u03c5\u03b3\u03ba\u03bb\u03b7\u03c4\u03b9\u03ba\u1ff6\u03bd, \u03c6\u03b9\u03bb\u03bf\u03c0\u03ac\u03c4\u03c1\u03b9\u03b4\u03bf\u03c2 \u03ba.\u03c4.\u03bb.\n\nFootnote 681:\n\n  no. 2720, 3827.\n\nFootnote 682:\n\n  \u1f0c\u03c0\u03c6\u03b9\u03bf\u03bd or \u1f0c\u03c6\u03c6\u03b9\u03bf\u03bd 2733, 2836, 3295, 3849, 3902 m, 4207; \u1f0c\u03c6\u03b9\u03bf\u03bd 3846\n  z^{34} and \u1f0c\u03c6\u03b5\u03b9\u03bf\u03bd 3846 z^{31}; and even \u1f0c\u03c6\u03c6\u03b5\u03b9\u03bd, 3167, 3278. In 3902 m\n  the mother's name is \u1f08\u03c0\u03c6\u03af\u03b1 and the daughter's \u1f0c\u03c0\u03c6\u03b9\u03bf\u03bd.\n\nFootnote 683:\n\n  \u1f08\u03c6\u03c6\u03af\u03b1\u03c2, 3697, 3983; \u1f08\u03c6\u03af\u03b1\u03c2 3879.\n\nFootnote 684:\n\n  \u1f0c\u03c6\u03c6\u03b7 3816, 3390, 4143; \u1f0c\u03c0\u03c6\u03b7 3796, 4122.\n\nFootnote 685:\n\n  It is met with at the neighbouring town of Hierapolis, in the form\n  \u1f08\u03c0\u03c6\u03af\u03b1\u03bd\u03bf\u03c2 no. 3911. It also occurs on coins of not very distant parts\n  of Asia Minor, being written either \u1f08\u03c0\u03c6\u03af\u03b1\u03bd\u03bf\u03c2 or \u1f08\u03c6\u03c6\u03af\u03b1\u03bd\u03bf\u03c2; Mionnet III.\n  p. 179, 184, IV. p. 65, 67, _Suppl._ VI. p. 293, VII. p. 365.\n\nFootnote 686:\n\n  Suidas \u1f0c\u03c0\u03c6\u03b1: \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u1fe6 \u1f51\u03c0\u03bf\u03ba\u03cc\u03c1\u03b9\u03c3\u03bc\u03b1, and so Bekk. _Anecd._ p.\n  441. Eustath. _Il._ p. 565 says \u1f04\u03c0\u03c6\u03b1\u03bd \u03c4\u1f74\u03bd \u1f00\u03b4\u03b5\u03bb\u03c6\u1f74\u03bd \u1f08\u03c4\u03c4\u03b9\u03ba\u1ff6\u03c2 \u03bc\u03cc\u03bd\u03b7 \u1f21\n  \u1f00\u03b4\u03b5\u03bb\u03c6\u1f74 \u03b5\u1f34\u03c0\u03bf\u03b9 \u1f02\u03bd, \u03ba\u03b1\u1f76 \u03c0\u03ac\u03c0\u03c0\u03b1\u03bd \u03c4\u1f78\u03bd \u03c0\u03b1\u03c4\u03ad\u03c1\u03b1 \u03bc\u03cc\u03bd\u03bf\u03c2 \u1f41 \u03c0\u03b1\u1fd6\u03c2 \u03ba.\u03c4.\u03bb., and he\n  adds \u1f30\u03c3\u03c4\u03ad\u03bf\u03bd \u03b4\u1f72 \u1f45\u03c4\u03b9 \u1f10\u03ba \u03c4\u03bf\u1fe6 \u1f61\u03c2 \u1f10\u03c1\u03c1\u03ad\u03b8\u03b7 \u1f04\u03c0\u03c6\u03b1 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f04\u03c0\u03c6\u03b9\u03bf\u03bd,\n  \u1f51\u03c0\u03bf\u03ba\u03cc\u03c1\u03b9\u03c3\u03bc\u03b1 \u1f42\u03bd \u1f10\u03c1\u03c9\u03bc\u03ad\u03bd\u03b7\u03c2\u00b7 \u03c4\u03b9\u03bd\u1f72\u03c2 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f04\u03c0\u03c6\u03b1 \u1f51\u03c0\u03bf\u03ba\u03cc\u03c1\u03b9\u03c3\u03bc\u03b1 \u03c6\u03b1\u03c3\u1f76\u03bd \u1f08\u03c4\u03c4\u03b9\u03ba\u03cc\u03bd.\n  These words were found in writers of Attic comedy (Pollux iii. 74 \u1f21\n  \u03c0\u03b1\u03c1\u1f70 \u03c4\u03bf\u1fd6\u03c2 \u03bd\u03ad\u03bf\u03b9\u03c2 \u03ba\u03c9\u03bc\u1ff3\u03b4\u03bf\u1fd6\u03c2 \u1f00\u03c0\u03c6\u03af\u03b1 \u03ba\u03b1\u1f76 \u1f00\u03c0\u03c6\u03af\u03bf\u03bd \u03ba\u03b1\u1f76 \u1f00\u03c0\u03c6\u03ac\u03c1\u03b9\u03bf\u03bd; comp.\n  Xenarchus \u03c4\u03bf\u1f7a\u03c2 \u03bc\u1f72\u03bd \u03b3\u03ad\u03c1\u03bf\u03bd\u03c4\u03b1\u03c2 \u1f44\u03bd\u03c4\u03b1\u03c2 \u1f10\u03c0\u03b9\u03ba\u03b1\u03bb\u03bf\u03cd\u03bc\u03b5\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c4\u03c1\u03af\u03b4\u03b9\u03b1, \u03c4\u03bf\u1f7a\u03c2 \u03b4'\n  \u1f00\u03c0\u03c6\u03ac\u03c1\u03b9\u03b1, \u03c4\u03bf\u1f7a\u03c2 \u03bd\u03b5\u03c9\u03c4\u03ad\u03c1\u03bf\u03c5\u03c2, Meineke _Fragm. Com._ III. p. 617): and\n  doubtless they were heard commonly in Attic homes. But were they not\n  learnt in the nursery from Phrygian slaves? \u1f08\u03c0\u03c6\u03ac\u03c1\u03b9\u03bf\u03bd appears in two\n  inscriptions almost as a proper name, 2637 \u039a\u03bb\u03b1\u03c5\u03b4\u03af\u03b1 \u1f00\u03c0\u03c6\u03ac\u03c1\u03b9\u03bf\u03bd, 3277\n  \u1f00\u03c0\u03c6\u03ac\u03c1\u03b9\u03bf\u03bd \u039b\u03bf\u03bb\u03bb\u03b9\u03b1\u03bd\u03ae. In no. 4207 (at Telmissus) we have \u1f19\u03bb\u03ad\u03bd\u03b7 \u1f21 \u03ba\u03b1\u1f76\n  \u1f0c\u03c6\u03c6\u03b9\u03bf\u03bd, so that it seems sometimes to have been employed side by side\n  with a Greek name; comp. no. 3912 a \u03a0\u03b1\u03c0\u03af\u03b1\u03c2 ... \u1f41 \u03ba\u03b1\u03bb\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u0394\u03b9\u03bf\u03b3\u03ad\u03bd\u03b7\u03c2,\n  quoted above p. 48. This will account for the frequency of the names,\n  Apphia, Apphion, etc. In Theocr. XV. 13 we have \u1f00\u03c0\u03c6\u1fe6\u03c2, and in Callim.\n  _Hym. Dian._ 6 \u1f04\u03c0\u03c0\u03b1, as a term of endearment applied to a father.\n\nFootnote 687:\n\n  This appears from the fact that Ammias and Ammianos appear sometimes\n  as the names of mother and son respectively in the same inscriptions;\n  e.g. 3846 z^{82}, 3847 k, 3882 i.\n\nFootnote 688:\n\n  On the name Papias or Pappias see above p. 48.\n\nFootnote 689:\n\n  See Boeckh _Corp. Inscr._ III. p. 1085 for the names \u039d\u03ac\u03bd\u03b1\u03c2, etc.\n\nFootnote 690:\n\n  We have not only the form \u1f0c\u03c0\u03c0\u03b7 several times (e.g. 3827 x, 3846 p,\n  3846 x, 3846 z^{46}, etc.); but also \u1f0c\u03c0\u03c0\u03b7\u03c2 3827 g, 3846 n, 3846 z^{77},\n  still as a woman's name. These all occur in the same neighbourhood, at\n  Coti\u00e6um and \u00c6zani. I have not noticed any instance of this phenomenon\n  in the names Apphia, Apphion; though probably, where Roman influences\n  were especially strong, there would be a tendency to transform a\n  Phrygian name into a Roman, e.g. Apphia into Appia, and Apphianus into\n  Appianus.\n\nFootnote 691:\n\n  In the Greek historians of Rome for instance the personal name is\n  always \u1f0c\u03c0\u03c0\u03b9\u03bf\u03c2 and the road \u1f08\u03c0\u03c0\u03af\u03b1; so too in Acts xxviii. 15 it is\n  \u1f08\u03c0\u03c0\u03af\u03bf\u03c5 \u03a6\u03cc\u03c1\u03bf\u03bd.\n\nFootnote 692:\n\n  The point to be observed is that examples of these names are thickest\n  in the heart of Phrygia, that they diminish in frequency as Phrygian\n  influence becomes weaker, and that they almost, though not entirely,\n  disappear in other parts of the Greek and Roman world.\n\n[Sidenote: Her share in the letter.]\n\nOf this Phrygian matron we know nothing more than can be learnt from\nthis epistle. The tradition or fiction which represents her as martyred\ntogether with her husband may be safely disregarded. St Paul addresses\nher as a Christian[693]. Equally with her husband she had been aggrieved\nby the misconduct of their slave Onesimus, and equally with him she\nmight interest herself in the penitent's future well-being.\n\nFootnote 693:\n\n  ver. 2 \u03c4\u1fc7 \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc7. See the note.\n\n[Sidenote: 3. Archippus, the son.]\n\n3. With less confidence, but still with a reasonable degree of\nprobability, we may infer that Archippus, who is likewise mentioned in\nthe opening salutation, was a son[694] of Philemon and Apphia. The\ninscriptions do not exhibit the name in any such frequency either in\nPhrygia or in the surrounding districts, as to suggest that it was\ncharacteristic of these parts[695]. [Sidenote: His office]Our Archippus\nheld some important office in the Church[696]; but what this was, we are\nnot told. St Paul speaks of it as a 'ministry' (\u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1). Some have\ninterpreted the term technically as signifying the diaconate; but St\nPaul's emphatic message seems to imply a more important position than\nthis. Others again suppose that he succeeded Epaphras as bishop of\nColoss\u00e6, when Epaphras left his native city to join the Apostle at\nRome[697]; but the assumption of a regular and continuous episcopate in\nsuch a place as Coloss\u00e6 at this date seems to involve an anachronism.\nMore probable than either is the hypothesis which makes him a presbyter.\nOr perhaps he held a missionary charge, and belonged to the order of\n'evangelists[698].' Another question too arises respecting Archippus.\nWhere was he exercising this ministry, whatever it may have been? At\nColoss\u00e6, or at Laodicea? [Sidenote: and abode,]His connexion with\nPhilemon would suggest the former place. But in the Epistle to the\nColossians his name is mentioned immediately after the salutations to\nthe Laodiceans and the directions affecting that Church; and this fact\nseems to connect him with Laodicea. [Sidenote: Laodicea, rather than\nColoss\u00e6.]On the whole this appears to be the more probable\nsolution[699]. Laodicea was within walking distance of Coloss\u00e6[700].\nArchippus must have been in constant communication with his parents, who\nlived there; and it was therefore quite natural that, writing to the\nfather and mother, St Paul should mention the son's name also in the\nopening address, though he was not on the spot. An early tradition, if\nit be not a critical inference from the allusion in the Colossian\nletter, makes him bishop not of Coloss\u00e6, but of Laodicea[701].\n\nFootnote 694:\n\n  So Theodore of Mopsuestia. But Chrysostom \u1f15\u03c4\u03b5\u03c1\u03cc\u03bd \u03c4\u03b9\u03bd\u03b1 \u1f34\u03c3\u03c9\u03c2 \u03c6\u03b9\u03bb\u03cc\u03bd, and\n  Theodoret \u1f41 \u03b4\u1f72 \u1f0c\u03c1\u03c7\u03b9\u03c0\u03c0\u03bf\u03c2 \u03c4\u1f74\u03bd \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03af\u03b1\u03bd \u03b1\u1f50\u03c4\u1ff6\u03bd \u1f10\u03c0\u03b5\u03c0\u03af\u03c3\u03c4\u03b5\u03c5\u03c4\u03bf.\n\nFootnote 695:\n\n  It occurs in two Smyrn\u00e6an inscriptions, no. 3143, 3224.\n\nFootnote 696:\n\n  Col. iv. 17 \u03b2\u03bb\u03ad\u03c0\u03b5 \u03c4\u1f74\u03bd \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1\u03bd \u1f23\u03bd \u03c0\u03b1\u03c1\u03ad\u03bb\u03b1\u03b2\u03b5\u03c2 \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3, \u1f35\u03bd\u03b1 \u03b1\u1f50\u03c4\u1f74\u03bd\n  \u03c0\u03bb\u03b7\u03c1\u03bf\u1fd6\u03c2.\n\nFootnote 697:\n\n  So the Ambrosian Hilary on Col. iv. 17.\n\nFootnote 698:\n\n  Ephes. iv. 11 bears testimony to the existence of the office of\n  evangelist at this date.\n\nFootnote 699:\n\n  It is adopted by Theodore of Mopsuestia. On the other hand Theodoret\n  argues against this view on critical grounds; \u03c4\u03b9\u03bd\u1f72\u03c2 \u1f14\u03c6\u03b1\u03c3\u03b1\u03bd \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd\n  \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03b5\u03af\u03b1\u03c2 \u03b3\u03b5\u03b3\u03b5\u03bd\u1fc6\u03c3\u03b8\u03b1\u03b9 \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b1\u03bb\u03bf\u03bd, \u1f00\u03bb\u03bb' \u1f21 \u03c0\u03c1\u1f78\u03c2 \u03a6\u03b9\u03bb\u03ae\u03bc\u03bf\u03bd\u03b1 \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u1f74\n  \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b5\u03b9 \u1f61\u03c2 \u1f10\u03bd \u039a\u03bf\u03bb\u03b1\u03c3\u03c3\u03b1\u1fd6\u03c2 \u03bf\u1f57\u03c4\u03bf\u03c2 \u1fa4\u03ba\u03b5\u03b9\u00b7 \u03c4\u1ff7 \u03b3\u1f70\u03c1 \u03a6\u03b9\u03bb\u03ae\u03bc\u03bf\u03bd\u03b9 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd\n  \u03c3\u03c5\u03bd\u03c4\u03ac\u03c4\u03c4\u03b5\u03b9: but he does not allege any traditional support for his own\n  opinion.\n\nFootnote 700:\n\n  See above pp. 2, 15.\n\nFootnote 701:\n\n  _Apost. Const._ vii, 46 quoted above p. 372, note 676.\n\n[Sidenote: His career.]\n\nOf the apprehensions which the Apostle seems to have entertained\nrespecting Archippus, I have already spoken[702]. It is not improbable\nthat they were suggested by his youth and inexperience. St Paul here\naddresses him as his 'fellow-soldier[703],' but we are not informed on\nwhat spiritual campaigns they had served in company. Of his subsequent\ncareer we have no trustworthy evidence. Tradition represents him as\nhaving suffered martyrdom at Coloss\u00e6 with his father and mother.\n\nFootnote 702:\n\n  See p. 42.\n\nFootnote 703:\n\n  ver. 2 \u03c4\u1ff7 \u03c3\u03c5\u03bd\u03c3\u03c4\u03c1\u03b1\u03c4\u03af\u03c9\u03c4\u1fc3 \u1f21\u03bc\u1ff6\u03bd. See the note.\n\n[Sidenote: 4. Onesimus.]\n\n4. But far more important to the history of Christianity than the\nparents or the son of the family, is the servant. The name Onesimus was\nvery commonly borne by slaves. Like other words signifying utility,\nworth, and so forth, it naturally lent itself to this purpose[704].\nAccordingly the inscriptions offer a very large number of examples in\nwhich it appears as the name of some slave or freedman[705]; [Sidenote:\nA servile name.]and even where this is not the case, the accompaniments\nfrequently show that the person was of servile descent, though he might\nnever himself have been a slave[706]. Indeed it occurs more than once as\na fictitious name for a slave[707], a fact which points significantly to\nthe social condition naturally suggested by it. In the inscriptions of\nproconsular Asia it is found[708]; but no stress can be laid on this\ncoincidence, for its occurrence as a proper name was doubtless\ncoextensive with the use of the Greek language. More important is the\nfact that in the early history of Christianity it attains some eminence\nin this region. [Sidenote: Its prominence among the Christians of\nproconsular Asia.]One Onesimus is bishop of Ephesus in the first years\nof the second century, when Ignatius passes through Asia Minor on his\nway to martyrdom, and is mentioned by the saint in terms of warm\naffection and respect[709]. Another, apparently an influential layman,\nabout half a century later urges Melito bishop of Sardis to compile a\nvolume of extracts from the scriptures; and to him this father dedicates\nthe work when completed[710]. Thus it would appear that the memory of\nthe Colossian slave had invested the name with a special popularity\namong Christians in this district.\n\nFootnote 704:\n\n  e.g. Chresimus, Chrestus, Onesiphorus, Symphorus, Carpus, etc. So too\n  the corresponding female names Onesime, Chreste, Sympherusa, etc.: but\n  more commonly the women's names are of a different cast of meaning,\n  Arescusa, Prepusa, Terpusa, Thallusa, Tryphosa, etc.\n\nFootnote 705:\n\n  e.g. in the _Corp. Inscr. Lat._ III. p. 323, no. 2146, p. 359, no.\n  2723, p. 986, no. 6107 (where it is spelled Honesimus); and in\n  Muratori, CC. 6, DXXIX. 5, CMLXVIII. 4, MIII. 2, MDXVIII. 2, MDXXIII.\n  4, MDLI. 9, MDLXXI. 5, MDLXXV. 1, MDXCII. 8, MDXCVI. 7, MDCVI. 2,\n  MDCX. 19, MDCXIV. 17, 39; and the corresponding female name Onesime in\n  MCCXXXIX. 12, MDXLVI. 6, MDCXII. 9. A more diligent search than I have\n  made would probably increase the number of examples very largely.\n\nFootnote 706:\n\n  e.g. _Corp. Inscr. Lat._ III. p. 238, no. 1467, D. M. M. AVR \u00b7 ONESIMO\n  \u00b7 CARPION \u00b7 AVG \u00b7 LIB \u00b7 TABVL \u00b7 FILIO. In the next generation any\n  direct notice of servile origin would disappear; but the names very\n  often indicate it. It need not however necessarily denote low\n  extraction: see e.g. Liv. xliv. 16.\n\nFootnote 707:\n\n  Menander _Inc._ 312 (Meineke _Fragm. Com._ IV. p. 300), where the\n  \u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03c2 addressed is a slave, as appears from the mention of his\n  \u03c4\u03c1\u03cc\u03c6\u03b9\u03bc\u03bf\u03c2, i.e. master; Galen _de Opt. Doctr._ I (I. p. 41 ed. K\u00fchn),\n  where there is a reference to a work of Phavorinus in which was\n  introduced one Onesimus \u1f41 \u03a0\u03bb\u03bf\u03c5\u03c4\u03ac\u03c1\u03c7\u03bf\u03c5 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u1f18\u03c0\u03b9\u03ba\u03c4\u03ae\u03c4\u1ff3 \u03b4\u03b9\u03b1\u03bb\u03b5\u03b3\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2;\n  _Anthol. Gr\u00e6c._ II. p. 161, where the context shows that the person\n  addressed as Onesimus is a slave; ib. II. p. 482, where the master,\n  leaving legacies to his servants, says \u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03c2 \u03b5\u1f35\u03ba\u03bf\u03c3\u03b9 \u03c0\u03ad\u03bd\u03c4\u03b5 | \u03bc\u03bd\u1fb6\u03c2\n  \u1f10\u03c7\u03ad\u03c4\u03c9 \u0394\u03ac\u03bf\u03c2 \u03b4' \u03b5\u1f30\u03ba\u03bf\u03c3\u03b9 \u03bc\u03bd\u1fb6\u03c2 \u1f10\u03c7\u03ad\u03c4\u03c9\u00b7 | \u03c0\u03b5\u03bd\u03c4\u03ae\u03ba\u03bf\u03bd\u03c4\u03b1 \u03a3\u03cd\u03c1\u03bf\u03c2\u00b7 \u03a3\u03c5\u03bd\u03ad\u03c4\u03b7 \u03b4\u03ad\u03ba\u03b1,\n  \u03ba.\u03c4.\u03bb. See also the use of the name in the Latin play quoted Suet.\n  _Galb._ 13 (according to one reading).\n\nFootnote 708:\n\n  It occurs as near to Coloss\u00e6 as Aphrodisias; Boeckh _C. I._ no. 2743.\n\nFootnote 709:\n\n  Ign. _Ephes._ I. \u1f10\u03bd \u1f48\u03bd\u03b7\u03c3\u03af\u03bc\u1ff3 \u03c4\u1ff7 \u1f10\u03bd \u1f00\u03b3\u03ac\u03c0\u1fc3 \u1f00\u03b4\u03b9\u03b7\u03b3\u03ae\u03c4\u1ff3 \u1f51\u03bc\u1ff6\u03bd \u03b4\u1f72 \u1f10\u03bd \u03c3\u03b1\u03c1\u03ba\u1f76\n  \u1f10\u03c0\u03b9\u03c3\u03ba\u03cc\u03c0\u1ff3 ... \u03b5\u1f50\u03bb\u03cc\u03b3\u03b7\u03c4\u03bf\u03c2 \u1f41 \u03c7\u03b1\u03c1\u03b9\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f51\u03bc\u1fd6\u03bd \u1f00\u03be\u03af\u03bf\u03b9\u03c2 \u03bf\u1f56\u03c3\u03b9\u03bd \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03bd\n  \u1f10\u03c0\u03af\u03c3\u03ba\u03bf\u03c0\u03bf\u03bd \u03ba\u03b5\u03ba\u03c4\u1fc6\u03c3\u03b8\u03b1\u03b9; see also \u00a7\u00a7 2, 5, 6.\n\nFootnote 710:\n\n  Melito in Euseb. _H.E._ iv. 26 \u039c\u03b5\u03bb\u03af\u03c4\u03c9\u03bd \u1f48\u03bd\u03b7\u03c3\u03af\u03bc\u1ff3 \u03c4\u1ff7 \u1f00\u03b4\u03b5\u03bb\u03c6\u1ff7 \u03c7\u03b1\u03af\u03c1\u03b5\u03b9\u03bd.\n  \u1f18\u03c0\u03b5\u03b9\u03b4\u1f74 \u03c0\u03bf\u03bb\u03bb\u03ac\u03ba\u03b9\u03c2 \u1f20\u03be\u03af\u03c9\u03c3\u03b1\u03c2 \u03ba.\u03c4.\u03bb.\n\n[Sidenote: Position and conduct of Onesimus.]\n\nOnesimus represented the least respectable type of the least respectable\nclass in the social scale. He was regarded by philosophers as a 'live\nchattel', a 'live implement[711]'; and he had taken philosophy at her\nword. He had done what a chattel or an implement might be expected to\ndo, if endued with life and intelligence. He was treated by the law as\nhaving no rights[712]; and he had carried the principles of the law to\ntheir logical consequences. He had declined to entertain any\nresponsibilities. There was absolutely nothing to recommend him. He was\na slave, and what was worse, a Phrygian slave; and he had confirmed the\npopular estimate of his class[713] and nation[714] by his own conduct.\nHe was a thief and a runaway. His offence did not differ in any way, so\nfar as we know, from the vulgar type of slavish offences. He seems to\nhave done just what the representative slave in the Roman comedy\nthreatens to do, when he gets into trouble. He had 'packed up some goods\nand taken to his heels[715].' Rome was the natural cesspool for these\noffscourings of humanity[716]. In the thronging crowds of the metropolis\nwas his best hope of secresy. In the dregs of the city rabble he would\nfind the society of congenial spirits.\n\nFootnote 711:\n\n  Aristot. _Pol._ i. 4 (p. 1253) \u1f41 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u03ba\u03c4\u1fc6\u03bc\u03ac \u03c4\u03b9 \u1f14\u03bc\u03c8\u03c5\u03c7\u03bf\u03bd, _Eth. Nic._\n  viii. 13 (p. 1161) \u1f41 \u03b3\u1f70\u03c1 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u1f14\u03bc\u03c8\u03c5\u03c7\u03bf\u03bd \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd, \u03c4\u1f78 \u03b4' \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd \u1f04\u03c8\u03c5\u03c7\u03bf\u03c2\n  \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2. See also the classification of 'implements' in Varro, _de Re\n  rust._ I. 17. 1 'Instrumenti genus vocale et semivocale et mutum:\n  vocale, in quo sunt servi; semivocale, in quo boves; mutum, in quo\n  plaustra'.\n\nFootnote 712:\n\n  Dig. iv. 5 'Servile caput nullum jus habet' (Paulus); _ib._ l. 17 'In\n  personam servilem nulla cadit obligatio' (Ulpianus).\n\nFootnote 713:\n\n  Plaut. _Pseud._ I. 2, 6 'Ubi data occasiost, rape, clepe, tene,\n  harpaga, bibe, es, fuge; hoc eorum opust'; Ovid. _Amor._ i. 15. 17\n  'Dum fallax servus'.\n\nFootnote 714:\n\n  Cicero speaks thus of Phrygia and the neighbouring districts; _pro\n  Flacc._ 27 'Utrum igitur nostrum est an vestrum hoc proverbium\n  _Phrygem plagis fieri solere meliorem_? Quid de tota Caria? Nonne hoc\n  vestra voce vulgatum est; si quid cum periculo experiri velis, _in\n  Care_ id potissimum esse faciendum? Quid porro in Gr\u00e6co sermone tam\n  tritum est, quam si quis despicatui ducitur, ut _Mysorum ultimus_ esse\n  dicatur? Nam quid ego dicam de Lydia? Quis unquam Gr\u00e6cus com\u0153diam\n  scripsit in qua servus primarum partium non Lydus esset': comp.\n  Alciphr. _Epist._ iii. 38 \u03a6\u03c1\u03cd\u03b3\u03b1 \u03bf\u1f30\u03ba\u03ad\u03c4\u03b7\u03bd \u1f14\u03c7\u03c9 \u03c0\u03bf\u03bd\u03b7\u03c1\u03cc\u03bd \u03ba.\u03c4.\u03bb.; Apollod.\n  Com. (Meineke, IV. p. 451) \u03bf\u1f50 \u03c0\u03b1\u03bd\u03c4\u03b1\u03c7\u03bf\u1fe6 \u03a6\u03c1\u03cd\u03be \u03b5\u1f30\u03bc\u03b9 \u03ba.\u03c4.\u03bb. This last\n  passage refers to the cowardice with which, besides all their other\n  bad qualities, the Phrygians were credited: comp. Anon. Com. (_ib._\n  IV. p. 652) \u03b4\u03b5\u03b9\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u03bb\u03b1\u03b3\u1ff6 \u03a6\u03c1\u03c5\u03b3\u03cc\u03c2, Tertull. _de Anim._ 20 'Comici\n  Phrygas timidos illudunt': see Ribbeck _Com. Lat._ p. 106.\n\nFootnote 715:\n\n  Ter. _Phorm._ i. 4. 13 'aliquid convasassem, atque hinc me protinam\n  conjicerem in pedes'.\n\nFootnote 716:\n\n  Sall. _Cat._ xxxvii. 5 'Romam sicuti in sentinam confluxerant': comp.\n  Tac. _Ann._ xv. 44.\n\n[Sidenote: His encounter with St Paul in Rome]\n\nBut at Rome the Apostle spread his net for him, and he was caught in its\nmeshes. How he first came in contact with the imprisoned missionary, we\ncan only conjecture. Was it an accidental encounter with his\nfellow-townsman Epaphras in the streets of Rome which led to the\ninterview? Was it the pressure of want which induced him to seek alms\nfrom one whose large-hearted charity must have been a household word in\nhis master's family? Or did the memory of solemn words, which he had\nchanced to overhear at those weekly gatherings in the upper chamber at\nColoss\u00e6, haunt him in his loneliness, till, yielding to the fascination,\nhe was constrained to unburden himself to the one man who could soothe\nhis terrors and satisfy his yearnings? Whatever motive may have drawn\nhim to the Apostle's side\u2014whether the pangs of hunger or the gnawings of\nconscience\u2014when he was once within the range of attraction, he could not\nescape. [Sidenote: and conversion.]He listened, was impressed, was\nconvinced, was baptized. The slave of Philemon became the freedman of\nChrist[717]. St Paul found not only a sincere convert, but a devoted\nfriend, in his latest son in the faith. Aristotle had said that there\nought not to be, and could not be, any friendship with a slave _qua_\nslave, though there might be _qua_ man[718]; and others had held still\nstronger language to the same effect. The Apostle did not recognize the\nphilosopher's subtle distinction. For him the conventional barrier\nbetween slave and free had altogether vanished before the dissolving\npresence of an eternal verity[719]. [Sidenote: St Paul's affection for\nhim.]He found in Onesimus something more than a slave, a beloved brother\nboth as a slave and as a man, 'both in the flesh and in the Lord[720].'\nThe great capacity for good which appears in the typical slave of Greek\nand Roman fiction, notwithstanding all the fraud and profligacy\noverlying it, was evoked and developed here by the inspiration of a new\nfaith and the incentive of a new hope. The genial, affectionate, winning\ndisposition, purified and elevated by a higher knowledge, had found its\nproper scope. Altogether this new friendship was a solace and a strength\nto the Apostle in his weary captivity, which he could ill afford to\nforego. To take away Onesimus was to tear out Paul's heart[721].\n\nFootnote 717:\n\n  1 Cor. vii. 22.\n\nFootnote 718:\n\n  _Eth. Nic._ viii. 13 (p. 1161) \u03c6\u03b9\u03bb\u03af\u03b1 \u03b4' \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70 \u1f04\u03c8\u03c5\u03c7\u03b1 \u03bf\u1f50\u03b4\u1f72\n  \u03b4\u03af\u03ba\u03b1\u03b9\u03bf\u03bd\u00b7 \u1f00\u03bb\u03bb' \u03bf\u1f50\u03b4\u1f72 \u03c0\u03c1\u1f78\u03c2 \u1f35\u03c0\u03c0\u03bf\u03bd \u1f22 \u03b2\u03bf\u1fe6\u03bd, \u03bf\u1f50\u03b4\u1f72 \u03c0\u03c1\u1f78\u03c2 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd \u1f97 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2\u00b7 \u03bf\u1f50\u03b4\u1f72\u03bd\n  \u03b3\u1f70\u03c1 \u03ba\u03bf\u03b9\u03bd\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd\u00b7 \u1f41 \u03b3\u1f70\u03c1 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u1f14\u03bc\u03c8\u03c5\u03c7\u03bf\u03bd \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd, \u03c4\u1f78 \u03b4' \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd \u1f04\u03c8\u03c5\u03c7\u03bf\u03c2\n  \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2\u00b7 \u1f97 \u03bc\u1f72\u03bd \u03bf\u1f56\u03bd \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2, \u03bf\u1f50\u03ba \u1f14\u03c3\u03c4\u03b9 \u03c6\u03b9\u03bb\u03af\u03b1 \u03c0\u03c1\u1f78\u03c2 \u03b1\u1f50\u03c4\u03cc\u03bd, \u1f97 \u03b4' \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2\n  \u03ba.\u03c4.\u03bb. On the views of Aristotle respecting slavery see Becker's\n  _Charikles_ III. p. 2 sq. (ed. 2, 1854) with the editor K. F.\n  Hermann's references to the literature of the subject, p. 5.\n\nFootnote 719:\n\n  1 Cor. vii. 21 sq., Gal. iii. 28, Col. iii. 11. With this contrast the\n  expression attributed to a speaker in Macrob. _Sat._ i. 11 'quasi vero\n  curent divina de servis'.\n\nFootnote 720:\n\n  Philem. 16.\n\nFootnote 721:\n\n  ver. 12.\n\n[Sidenote: Necessity for his return]\n\nBut there was an imperious demand for the sacrifice. Onesimus had\nrepented, but he had not made restitution. He could only do this by\nsubmitting again to the servitude from which he had escaped. Philemon\nmust be made to feel that, when Onesimus was gained for Christ, he was\nregained for his old master also. But if the claim of duty demanded a\ngreat sacrifice from Paul, it demanded a greater still from Onesimus.\n[Sidenote: notwithstanding the risk.]By returning he would place himself\nentirely at the mercy of the master whom he had wronged. Roman law, more\ncruel than Athenian, practically imposed no limits to the power of the\nmaster over his slave[722]. The alternative of life or death rested\nsolely with Philemon, and slaves were constantly crucified for far\nlighter offences than his[723]. A thief and a runaway, he laid no claim\nto forgiveness.\n\nFootnote 722:\n\n  _Dig._ i. 6 'In potestate sunt servi dominorum; quae quidem potestas\n  juris gentium est: nam apud omnes peraeque gentes animadvertere\n  possumus dominis in servos vitae necisque potestatem fuisse'. Comp.\n  Senec. _de Clem._ i. 18 'Cum in servum omnia liceant'.\n\nFootnote 723:\n\n  So the mistress in Juv. _Sat._ vi. 219 sq. 'Pone crucem servo. Meruit\n  quo crimine servus supplicium? quis testis adest? quis detulit?... O\n  demens, ita servus <DW25> est? nil fecerit, esto. Hoc volo, sic jubeo,\n  etc.' Compare the words of the slave in Plautus _Mil. Glor._ ii. 4. 19\n  'Noli minitari: scio crucem futuram mihi sepulcrum: Ibi mei sunt\n  majores siti, pater, avos, proavos, abavos'.\n\n[Sidenote: Mediation of Tychicus]\n\n[Sidenote: supplemented by the Apostle's letter.]\n\nA favourable opportunity occurred for restoring Onesimus to his master.\nTychicus, as the bearer of letters from the Apostle to Laodicea and\nColoss\u00e6, had occasion to visit those parts. He might undertake the\noffice of mediator, and plead the cause of the penitent slave with the\noffended master. Under his shelter Onesimus would be safer than if he\nencountered Philemon alone. But St Paul is not satisfied with this\nprecaution. He will with his own hand write a few words of eager\naffectionate entreaty, identifying himself with the cause of Onesimus.\nSo he takes up his pen.\n\n[Sidenote: Analysis of the letter.]\n\nAfter the opening salutation to Philemon and the members of his family,\nhe expresses his thankfulness for the report which has reached his ears\nof his friend's charitable deeds. It is a great joy and encouragement to\nthe Apostle that so many brethren have had cause to bless his name. This\nwide-spread reputation for kindliness emboldens him to reveal his object\nin writing. Though he has a right to command, he prefers rather to\nentreat. He has a petition to prefer on behalf of a child of his own.\nThis is none other than Onesimus, whom Philemon will remember only as a\nworthless creature, altogether untrue to his name, but who now is a\nreformed man. He would have wished to detain Onesimus, for he can ill\nafford to dispense with his loving services. Indeed Philemon would\ndoubtless have been glad thus to minister vicariously to the Apostle's\nwants. But a benefit which wears the appearance of being forced, whether\ntruly so or not, loses all its value, and therefore he sends him back.\nNay, there may have been in this desertion a Divine providence which it\nwould ill become him Paul to thwart. Onesimus may have been withheld\nfrom Philemon for a time, that he might be restored to him for ever. He\nmay have left as a slave, that he might return more than a slave. To\nothers\u2014to the Apostle himself especially\u2014he is now a dearly beloved\nbrother. Must he not be this and more than this to Philemon, whether in\nearthly things or in heavenly things? He therefore begs Philemon to\nreceive Onesimus as he would receive himself. As for any injury that he\nmay have done, as for any money that he may owe, the Apostle makes\nhimself responsible for this. The present letter may be accepted as a\nbond, a security for repayment. Yet at the same time he cannot refrain\nfrom reminding Philemon that he might fairly claim the remission of so\nsmall an amount. Does not his friend owe to him his own soul besides?\nYes, he has a right to look for some filial gratitude and duty from one\nto whom he stands in the relation of a spiritual father. Philemon will\nsurely not refuse him this comfort in his many trials. He writes in the\nfull confidence that he will be obeyed; he is quite sure that his friend\nwill do more than is asked of him. At the same time he trusts to see him\nbefore very long, and to talk over this and other matters. Philemon may\nprovide him a lodging: for he hopes through their prayers that he may be\nliberated, and given back to them. Then follow the salutations, and the\nletter ends with the Apostle's benediction.\n\n[Sidenote: Result of the appeal.]\n\nOf the result of this appeal we have no certain knowledge. It is\nreasonable to suppose however that Philemon would not belie the\nApostle's hopes; that he would receive the slave as a brother; that he\nwould even go beyond the expressed terms of the Apostle's petition, and\nemancipate the penitent. But all this is a mere conjecture. One\ntradition makes Onesimus bishop of Ephesus[724]. But this obviously\narises from a confusion with his namesake, who lived about half a\ncentury later[725]. [Sidenote: Legendary history.]Another story points\nto Ber\u0153a in Macedonia as his see[726]. This is at least free from the\nsuspicion of having been suggested by any notice in the Apostolic\nwritings: but the authority on which it rests does not entitle it to\nmuch credit. The legend of his missionary labours in Spain and of his\nmartyrdom at Rome may have been built on the hypothesis of his\ncontinuing in the Apostle's company, following in the Apostle's\nfootsteps, and sharing the Apostle's fate. Another story, which gives a\ncircumstantial account of his martyrdom at Puteoli, seems to confuse him\nwith a namesake who suffered, or was related to have suffered, in the\nDecian persecution[727].\n\nFootnote 724:\n\n  See _Acta Sanct. Boll._ xvi Febr. (II. p. 857 sq. ed. nov.) for the\n  authorities, if they deserve the name.\n\nFootnote 725:\n\n  If we take the earlier date of the Epistles of St Ignatius, A.D. 107,\n  we get an interval of 44 years between the Onesimus of St Paul and the\n  Onesimus of Ignatius. It is not altogether impossible therefore that\n  the same person may be intended. But on the other hand the language of\n  Ignatius (_Ephes._ 1 sq.) leaves the impression that he is speaking of\n  a person comparatively young and untried in office.\n\nFootnote 726:\n\n  _Apost. Const._ vii. 46, quoted above, p. 372, note 676.\n\nFootnote 727:\n\n  For these ecclesiastical legends see _Act. Sanct._ l.c. p. 858 sq.\n\n[Sidenote: Depreciation of the epistle in early times.]\n\nThe estimate formed of this epistle at various epochs has differed\nwidely. In the fourth century there was a strong bias against it. The\n'spirit of the age' had no sympathy with either the subject or the\nhandling. Like the spirit of more than one later age, it was enamoured\nof its own narrowness, which it mistook for largeness of view, and it\ncould not condescend to such trivialities as were here offered to it.\nIts maxim seemed to be _De minimis non curat evangelium_. Of what\naccount was the fate of a single insignificant slave, long since dead\nand gone, to those before whose eyes the battle of the creeds was still\nraging? This letter taught them nothing about questions of theological\ninterest, nothing about matters of ecclesiastical discipline; and\ntherefore they would have none of it. They denied that it had been\nwritten by St Paul. It mattered nothing to them that the Church from the\nearliest ages had accepted it as genuine, that even the remorseless\n'higher criticism' of a Marcion had not ventured to lay hands on\nit[728]. It was wholly unworthy of the Apostle. If written by him, they\ncontended, it must have been written when he was not under the influence\nof the Spirit: its contents were altogether so unedifying. [Sidenote:\nReply of the fathers.]We may infer from the replies of Jerome[729], of\nChrysostom[730], and of Theodore of Mopsuestia[731], that they felt\nthemselves to be stemming a fierce current of prejudice which had set in\nthis direction. But they were strong in the excellence of their cause,\nand they nobly vindicated this epistle against its assailants.\n\nFootnote 728:\n\n  Hieron. _Comm. in Philem._ Pr\u00e6f. VII. p. 743 'Pauli esse epistolam ad\n  Philemonem saltem Marcione auctore doceantur: qui, quum c\u00e6teras\n  epistolas ejusdem vel non susceperit vel qu\u00e6dam in his mutaverit atque\n  corroserit, in hanc solam manus non est ausus mittere, quia sua illam\n  brevitas defendebat'. St Jerome has in his mind the passage of\n  Tertullian _adv. Marc._ v. 21 'Soli huic epistol\u00e6 brevitas sua\n  profuit, ut falsarias manus Marcionis evaderet'.\n\nFootnote 729:\n\n  _ib._ p. 742 sq. 'Qui nolunt inter epistolas Pauli eam recipere qu\u00e6 ad\n  Philemonem scribitur, aiunt non semper apostolum nec omnia Christo in\n  se loquente dixisse, quia nec humana imbecillitas unum tenorem Sancti\n  Spiritus ferre potuisset etc.... His et c\u00e6teris istius modi volunt aut\n  epistolam non esse Pauli qu\u00e6 ad Philemonem scribitur aut, etiamsi\n  Pauli sit, nihil habere quod \u00e6dificare nos possit etc.... sed mihi\n  videntur, dum epistolam simplicitatis arguunt, suam imperitiam\n  prodere, non intelligentes quid in singulis sermonibus virtutis et\n  sapienti\u00e6 lateat'.\n\nFootnote 730:\n\n  _Argum. in Philem._ \u1f00\u03bb\u03bb' \u1f10\u03c0\u03b5\u03b9\u03b4\u1f74 \u03c4\u03b9\u03bd\u03ad\u03c2 \u03c6\u03b1\u03c3\u03b9 \u03c0\u03b5\u03c1\u03b9\u03c4\u03c4\u1f78\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1f78 \u03ba\u03b1\u1f76\n  \u03c4\u03ac\u03c5\u03c4\u03b7\u03bd \u03c0\u03c1\u03bf\u03c3\u03ba\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u03c4\u1f74\u03bd \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae\u03bd, \u03b5\u1f34\u03b3\u03b5 \u1f51\u03c0\u1f72\u03c1 \u03c0\u03c1\u03ac\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u03bc\u03b9\u03ba\u03c1\u03bf\u1fe6 \u1f20\u03be\u03af\u03c9\u03c3\u03b5\u03bd,\n  \u1f51\u03c0\u1f72\u03c1 \u1f11\u03bd\u1f78\u03c2 \u1f00\u03bd\u03b4\u03c1\u03cc\u03c2, \u03bc\u03b1\u03b8\u03ad\u03c4\u03c9\u03c3\u03b1\u03bd \u1f45\u03c3\u03bf\u03b9 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u1f10\u03b3\u03ba\u03b1\u03bb\u03bf\u1fe6\u03c3\u03b9\u03bd \u1f45\u03c4\u03b9 \u03bc\u03c5\u03c1\u03af\u03c9\u03bd \u03b5\u1f30\u03c3\u1f76\u03bd\n  \u1f10\u03b3\u03ba\u03bb\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd \u1f04\u03be\u03b9\u03bf\u03b9 \u03ba.\u03c4.\u03bb., and he goes on to discuss the value of the\n  epistle at some length.\n\nFootnote 731:\n\n  _Spicil. Solesm._ I. W. 149 'Quid vero ex ea lucri possit acquiri,\n  convenit manifestius explicare, quia nec omnibus id existimo posse\n  esse cognitum; quod maxime heri jam ipse a nobis disseri postulasti';\n  _ib._ p. 152 'De his et nunc superius dixi, quod non omnes similiter\n  arbitror potius se (potuisse?) prospicere'.\n\n[Sidenote: High estimate of modern writers.]\n\nIn modern times there has been no disposition to under-rate its value.\nEven Luther and Calvin, whose bias tended to the depreciation of the\nethical as compared with the doctrinal portions of the scriptures, show\na true appreciation of its beauty and significance. [Sidenote:\nLuther.]'This epistle', writes Luther, 'showeth a right noble lovely\nexample of Christian love. Here we see how St Paul layeth himself out\nfor poor Onesimus, and with all his means pleadeth his cause with his\nmaster: and so setteth himself as if he were Onesimus, and had himself\ndone wrong to Philemon. Even as Christ did for us with God the Father,\nthus also doth St Paul for Onesimus with Philemon.... We are all his\nOnesimi, to my thinking'. [Sidenote: Calvin.]'Though he handleth a\nsubject', says Calvin, 'which otherwise were low and mean, yet after his\nmanner he is borne up aloft unto God. With such modest entreaty doth he\nhumble himself on behalf of the lowest of men, that scarce anywhere else\nis the gentleness of his spirit portrayed more truly to the life.' And\nthe chorus of admiration has been swelled by later voices from the most\nopposite quarters. [Sidenote: Later writers.]'The single Epistle to\nPhilemon,' says one quoted by Bengel, 'very far surpasses all the wisdom\nof the world'[732]. 'Nowhere', writes Ewald, 'can the sensibility and\nwarmth of a tender friendship blend more beautifully with the loftier\nfeeling of a commanding spirit, a teacher and an Apostle, than in this\nletter, at once so brief, and yet so surpassingly full and\nsignificant[733].' 'A true little chef d'\u0153uvre of the art of\nletter-writing,' exclaims M. Renan characteristically[734]. 'We have\nhere', writes Sabatier, 'only a few familiar lines, but so full of\ngrace, of salt, of serious and trustful affection, that this short\nepistle gleams like a pearl of the most exquisite purity in the rich\ntreasure of the New Testament[735]'. Even Baur, while laying violent\nhands upon it, is constrained to speak of this 'little letter' as\n'making such an agreeable impression by its attractive form' and as\npenetrated 'with the noblest Christian spirit'[736].\n\nFootnote 732:\n\n  Franke _Pr\u00e6f. N. T. Gr\u00e6c._ p. 26, 27, quoted by Bengel on Philem. 1.\n\nFootnote 733:\n\n  _Die Sendschreiben_ etc. p. 458.\n\nFootnote 734:\n\n  _L'Antechrist_ p. 96.\n\nFootnote 735:\n\n  _L'Ap\u00f4tre Paul_ p. 194. He goes on to say; 'Never has the precept\n  which Paul himself gave at the end of his letter to the Colossians\n  been better realised, \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5 \u1f10\u03bd \u03c7\u03ac\u03c1\u03b9\u03c4\u03b9, \u1f05\u03bb\u03b1\u03c4\u03b9 \u1f20\u03c1\u03c4\u03c5\u03bc\u03ad\u03bd\u03bf\u03c2\n  \u03ba.\u03c4.\u03bb. (Col. iv. 6).'\n\nFootnote 736:\n\n  _Paulus_ p. 476.\n\n[Sidenote: The epistle compared with a letter of Pliny,]\n\nThe Epistle to Philemon has more than once been compared with the\nfollowing letter addressed to a friend by the younger Pliny on a\nsomewhat similar occasion[737]:\n\nYour freedman, with whom you had told me you were vexed, came to me, and\nthrowing himself down before me clung to my feet, as if they had been\nyours. He was profuse in his tears and his entreaties; he was profuse\nalso in his silence. In short, he convinced me of his penitence. I\nbelieve that he is indeed a reformed character, because he feels that he\nhas done wrong. You are angry, I know; and you have reason to be angry,\nthis also I know: but mercy wins the highest praise just when there is\nthe most righteous cause for anger. You loved the man, and, I hope, will\ncontinue to love him: meanwhile it is enough, that you should allow\nyourself to yield to his prayers. You may be angry again, if he deserves\nit; and in this you will be the more readily pardoned if you yield now.\nConcede something to his youth, something to his tears, something to\nyour own indulgent disposition. Do not torture him, lest you torture\nyourself at the same time. For it _is_ torture to you, when one of your\ngentle temper is angry. I am afraid lest I should appear not to ask but\nto compel, if I should add my prayers to his. Yet I will add them the\nmore fully and unreservedly, because I scolded the man himself with\nsharpness and severity; for I threatened him straitly that I would never\nask you again. This I said to him, for it was necessary to alarm him;\nbut I do not use the same language to you. For perchance I shall ask\nagain, and shall be successful again; only let my request be such, as it\nbecomes me to prefer and you to grant. Farewell.\n\nFootnote 737:\n\n  Plin. _Ep._ ix. 21.\n\n[Sidenote: as an expression of character.]\n\nThe younger Pliny is the noblest type of a true Roman gentleman, and\nthis touching letter needs no words of praise. Yet, if purity of diction\nbe excepted, there will hardly be any difference of opinion in awarding\nthe palm to the Christian Apostle. As an expression of simple dignity,\nof refined courtesy, of large sympathy, and of warm personal affection,\nthe Epistle to Philemon stands unrivalled. And its pre-eminence is the\nmore remarkable because in style it is exceptionally loose. It owes\nnothing to the graces of rhetoric; its effect is due solely to the\nspirit of the writer.\n\n[Sidenote: Its higher interest.]\n\nBut the interest which attaches to this short epistle as an expression\nof individual character is far less important than its significance as\nexhibiting the attitude of Christianity to a widely spread and\ncharacteristic social institution of the ancient world.\n\n[Sidenote: Slavery among the Hebrews.]\n\nSlavery was practised by the Hebrews under the sanction of the Mosaic\nlaw, not less than by the Greeks and Romans. But though the same in\nname, it was in its actual working something wholly different. The\nHebrew was not suffered either by law-giver or by prophet to forget that\nhe himself had been a bondman in the land of Egypt; and all his\nrelations to his dependents were moulded by the sympathy of this\nrecollection. His slaves were members of his family; they were members\nalso of the Holy Congregation. They had their religious, as well as\ntheir social, rights. If Hebrews, their liberty was secured to them\nafter six years' service at the outside. If foreigners, they were\nprotected by the laws from the tyranny and violence of their masters.\nConsidering the conditions of ancient society, and more especially of\nancient warfare, slavery as practised among the Hebrews was probably an\nescape from alternatives which would have involved a far greater amount\nof human misery. Still even in this form it was only a temporary\nconcession, till the fulness of time came, and the world was taught that\n'in Christ is neither bond nor free[738]'.\n\nFootnote 738:\n\n  On slavery among the Hebrews see the admirable work of Prof. Goldwin\n  Smith _Does the Bible sanction American slavery?_ p. 1 sq.\n\nAmong the Jews the slaves formed only a small fraction of the whole\npopulation[739]. They occupy a very insignificant place in the pictures\nof Hebrew life and history which have been handed down to us. [Sidenote:\nLarge number of slaves in Greece and Rome.]But in Greece and Rome the\ncase was far different. In our enthusiastic eulogies of free,\nenlightened, democratic Athens, we are apt to forget that the interests\nof the many were ruthlessly sacrificed to the selfishness of the few.\nThe slaves of Attica on the most probable computation were about four\ntimes as numerous as the citizens, and about three times as numerous as\nthe whole free population of the state, including the resident\naliens[740]. They were consigned for the most part to labour in gangs in\nthe fields or the mines or the factories, without any hope of bettering\ntheir condition. In the light of these facts we see what was really\nmeant by popular government and equal rights at Athens. The proportions\nof the slave population elsewhere were even greater. In the small island\nof \u00c6gina, scarcely exceeding forty English square miles in extent, there\nwere 470,000 slaves; in the contracted territory of Corinth there were\nnot less than 460,000[741]. The statistics of slave-holding in Italy are\nquite as startling. We are told that wealthy Roman landowners sometimes\npossessed as many as ten or twenty thousand slaves, or even more[742].\nWe may indeed not unreasonably view these vague and general statements\nwith suspicion: but it is a fact that, a few years before the Christian\nera, one Claudius Isidorus left by will more than four thousand slaves,\nthough he had incurred serious losses by the civil war[743].\n\nFootnote 739:\n\n  In Ezra ii. 65 the number of slaves compared with the number of free\n  is a little more than one to six.\n\nFootnote 740:\n\n  Boeckh _Public Economy of Athens_ p. 35 sq. According to a census\n  taken by Demetrius Phalereus there were in the year 309 B. C. 21,000\n  citizens, 10,000 residents, and 400,000 slaves (Ctesicles in Athen.\n  vi. p. 272 B). This would make the proportion of slaves to citizens\n  nearly twenty to one. It is supposed however that the number of\n  citizens here includes only adult males, whereas the number of slaves\n  may comprise both sexes and all ages. Hence Boeckh's estimate which is\n  adopted in the text. For other calculations see Wallon _Histoire de\n  l'Esclavage_ I. p. 221 sq.\n\nFootnote 741:\n\n  Athen. _l.c._ p. 272 B, D. The statement respecting \u00c6gina is given on\n  the authority of Aristotle; that respecting Corinth on the authority\n  of Epitim\u00e6us.\n\nFootnote 742:\n\n  Athen. _l.c._ \u1fec\u03c9\u03bc\u03b1\u03af\u03c9\u03bd \u1f15\u03ba\u03b1\u03c3\u03c4\u03bf\u03c2 ... \u03c0\u03bb\u03b5\u03af\u03c3\u03c4\u03bf\u03c5\u03c2 \u1f45\u03c3\u03bf\u03c5\u03c2 \u03ba\u03b5\u03ba\u03c4\u03b7\u03bc\u03ad\u03bd\u03bf\u03c2 \u03bf\u1f30\u03ba\u03ad\u03c4\u03b1\u03c2\u00b7\n  \u03ba\u03b1\u1f76 \u03b3\u1f70\u03c1 \u03bc\u03c5\u03c1\u03af\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03c3\u03bc\u03c5\u03c1\u03af\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u1f14\u03c4\u03b9 \u03c0\u03bb\u03b5\u03af\u03bf\u03c5\u03c2 \u03b4\u1f72 \u03c0\u03ac\u03bc\u03c0\u03bf\u03bb\u03bb\u03bf\u03b9 \u03ba\u03ad\u03ba\u03c4\u03b7\u03bd\u03c4\u03b1\u03b9.\n  See Becker _Gallus_ II. p. 113 (ed. 3).\n\nFootnote 743:\n\n  Plin. _N.H._ xxxiii. 47.\n\n[Sidenote: Cruelty of Roman law towards slaves.]\n\nAnd these vast masses of human beings had no protection from Roman\nlaw[744]. The slave had no relationships, no conjugal rights.\nCohabitation was allowed to him at his owner's pleasure, but not\nmarriage. His companion was sometimes assigned to him by lot[745]. The\nslave was absolutely at his master's disposal; for the smallest offence\nhe might be scourged, mutilated, crucified, thrown to the wild\nbeasts[746]. Only two or three years before the letter to Philemon was\nwritten, and probably during St Paul's residence in Rome, a terrible\ntragedy had been enacted under the sanction of the law[747]. [Sidenote:\nMurder of Pedanius Secundus.]Pedanius Secundus, a senator, had been\nslain by one of his slaves in a fit of anger or jealousy. The law\ndemanded that in such cases all the slaves under the same roof at the\ntime should be put to death. On the present occasion four hundred\npersons were condemned to suffer by this inhuman enactment. The populace\nhowever interposed to rescue them, and a tumult ensued. The Senate\naccordingly took the matter into deliberation. Among the speakers C.\nCassius strongly advocated the enforcement of the law. 'The disposition\nof slaves,' he argued, 'were regarded with suspicion by our ancestors,\neven when they were born on the same estates or in the same houses and\nlearnt to feel an affection for their masters from the first. Now\nhowever, when we have several nations among our slaves, with various\nrites, with foreign religions or none at all, it is not possible to keep\ndown such a rabble except by fear.' These sentiments prevailed, and the\nlaw was put in force. But the roads were lined by a military guard, as\nthe prisoners were led to execution, to prevent a popular outbreak. This\nincident illustrates not only the heartless cruelty of the law, but also\nthe social dangers arising out of slavery. Indeed the universal distrust\nhad already found expression in a common proverb,'As many enemies as\nslaves[748].' But this was not the only way in which slavery avenged\nitself on the Romans. The spread of luxury and idleness was a direct\nconsequence of the state of things. Work came to be regarded as a low\nand degrading, because a servile occupation. Meanwhile sensuality in its\nvilest forms was fostered by the tremendous power which placed the slave\nat the mercy of the master's worst passions[749].\n\nFootnote 744:\n\n  On the condition of Greek and Roman slaves the able and exhaustive\n  work of Wallon _Histoire de l'Esclavage dans l'Antiquit\u00e9_ (Paris 1847)\n  is the chief authority. See also Becker and Marquardt _R\u00f6m. Alterth._\n  v. 1. p. 139 sq.; Becker _Charikles_ II. p. 1 sq., _Gallus_ II. p. 99\n  sq. The practical working of slavery among the Romans is placed in its\n  most favourable light in Gaston Bossier, _La Religion Romaine_ II. p.\n  343 sq. (Paris 1874).\n\nFootnote 745:\n\n  _R\u00f6m. Alterth._ l.c. p. 184 sq.; _Gallus_ II. p. 144 sq. In this, as\n  in other respects, the cruelty of the legislature was mitigated by the\n  humanity of individual masters; and the inscriptions show that male\n  and female slaves in many cases were allowed to live together through\n  life as man and wife, though the law did not recognise or secure their\n  union. It was reserved for Constantine to take the initiative in\n  protecting the conjugal and family rights of slaves by legislature;\n  _Cod. Theod._ ii. 25. 1.\n\nFootnote 746:\n\n  Wallon II. p. 177 sq.; _R\u00f6m. Alterth._ l.c.; _Gallus_ II. p. 145 sq.;\n  Rein _Privatrecht der R\u00f6mer_ p. 552 sq. Hadrian first took away from\n  masters the power of life and death over their slaves; Spart. _Vit.\n  Hadr._ 18 'Servos a dominis occidi vetuit eosque jussit damnari per\n  judices, si digni essent'. For earlier legislative enactments which\n  had afforded a very feeble protection to slaves, see below p. 393.\n\nFootnote 747:\n\n  Tac. _Ann._ xiv. 42. This incident took place A. D. 61. The law in\n  question was the _Senatusconsultum Silonianum_, passed under Augustus\n  A. D. 10.\n\nFootnote 748:\n\n  Senec. _Ep. Mor._ 47 'Deinde ejusdem arroganti\u00e6 proverbium jactatur\n  _totidem hostes esse quot servos_'; comp. Macrob. i. II. 13. See also\n  Festus p. 261 (Ed. Mueller) '_Quot servi tot hostes_ in proverbio\n  est.']\n\nFootnote 749:\n\n  See the saying of Haterius in the elder Seneca _Controv._ iv. Pr\u00e6f.,\n  'Impudicitia in ingenuo crimen est, _in servo necessitas_, in liberto\n  officium', with its context. Wallon (I. p. 332) sums up the condition\n  of the slave thus: 'L'esclave appartenait au ma\u00eetre: par lui m\u00eame, il\n  n'\u00e9tait rien, il n'avait rien. Voil\u00e0 le principe; et tout ce qu'on en\n  peut tirer par voie de cons\u00e9quence formait aussi, en fait, l'\u00e9tat\n  commun des esclaves dans la plupart des pays. A toutes les \u00e9poques,\n  dans toutes les situations de la vie, cette autorit\u00e9 souveraine plane\n  sur eux et modifie leur destin\u00e9e par ses rigueurs comme par son\n  indifference. Dans l'\u00e2ge de la force et dans la pl\u00e9nitude de leurs\n  facult\u00e9s, elle les vouait, \u00e0 son choix, soit au travail, soit au vice;\n  au travail les natures grossi\u00e8res; au vice, les natures plus\n  d\u00e9licates, nourries pour le plaisir du ma\u00eetre, et qui lorsqu'il en\n  \u00e9tait las, \u00e9taient rel\u00e9gu\u00e9es dans la prostitution a son profit. Avant\n  et apr\u00e8s l'\u00e2ge du travail, abandonn\u00e9s a leur faiblesse ou a leurs\n  infirmit\u00e9s; enfants, ils grandissaient dans le d\u00e9sordre; viellards,\n  ils mouraient souvent dans la mis\u00e8re; morts, ils \u00e9taient quelquefois\n  d\u00e9laiss\u00e9s sur la voie publique....'\n\n[Sidenote: Christianity not revolutionary.]\n\nWith this wide-spread institution Christianity found itself in conflict.\nHow was the evil to be met? Slavery was inwoven into the texture of\nsociety; and to prohibit slavery was to tear society into shreds.\nNothing less than a servile war with its certain horrors and its\ndoubtful issues must have been the consequence. Such a mode of operation\nwas altogether alien to the spirit of the Gospel. 'The New Testament',\nit has been truly said, 'is not concerned with any political or social\ninstitutions; for political and social institutions belong to particular\nnations and particular phases of society'. 'Nothing marks the divine\ncharacter of the Gospel more than its perfect freedom from any appeal to\nthe spirit of political revolution[750]'. It belongs to all time: and\ntherefore, instead of attacking special abuses, it lays down universal\nprinciples which shall undermine the evil.\n\nFootnote 750:\n\n  G. Smith _Does the Bible etc.?_ pp. 95, 96.\n\n[Sidenote: St Paul's treatment of the case of Onesimus.]\n\nHence the Gospel never directly attacks slavery as an institution: the\nApostles never command the liberation of slaves as an absolute duty. It\nis a remarkable fact that St Paul in this epistle stops short of any\npositive injunction. The word 'emancipation' seems to be trembling on\nhis lips, and yet he does not once utter it. He charges Philemon to take\nthe runaway slave Onesimus into his confidence again; to receive him\nwith all affection; to regard him no more as a slave but as a brother;\nto treat him with the same consideration, the same love, which he\nentertains for the Apostle himself to whom he owes everything. In fact\nhe tells him to do very much more than emancipate his slave, but this\none thing he does not directly enjoin. St Paul's treatment of this\nindividual case is an apt illustration of the attitude of Christianity\ntowards slavery in general.\n\n[Sidenote: His language respecting slavery elsewhere.]\n\nSimilar also is his language elsewhere. Writing to the Corinthians, he\ndeclares the absolute equality of the freeman and the slave in the sight\nof God[751]. It follows therefore that the slave may cheerfully\nacquiesce in his lot, knowing that all earthly distinctions vanish in\nthe light of this eternal truth. If his freedom should be offered to\nhim, he will do well to accept it, for it puts him in a more\nadvantageous position[752]: but meanwhile he need not give himself any\nconcern about his lot in life. So again, when he addresses the Ephesians\nand Colossians on the mutual obligations of masters and slaves, he is\ncontent to insist on the broad fact that both alike are slaves of a\nheavenly Master, and to enforce the duties which flow from its\nrecognition[753]. He has no word of reproach for the masters on the\ninjustice of their position; he breathes no hint to the slaves of a\nsocial grievance needing redress.\n\nFootnote 751:\n\n  1 Cor. vii. 21 sq.\n\nFootnote 752:\n\n  The clause, \u1f00\u03bb\u03bb' \u03b5\u1f30 \u03ba\u03b1\u1f76 \u03b4\u03cd\u03bd\u03b1\u03c3\u03b1\u03b9 \u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2 \u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9, \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c7\u03c1\u1fc6\u03c3\u03b1\u03b9, has\n  been differently interpreted from early times, either as recommending\n  the slave to avail himself of any opportunity of emancipation, or as\n  advising him to refuse the offer of freedom and to remain in\n  servitude. The earliest commentator whose opinion I have observed,\n  Origen (in Cram. _Cat._ p. 140), interprets it as favourable to\n  liberty, but he confuses the meaning by giving a metaphorical sense to\n  slavery, \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd \u1f60\u03bd\u03cc\u03bc\u03b1\u03c3\u03b5\u03bd \u1f00\u03bd\u03b1\u03b3\u03ba\u03b1\u03af\u03c9\u03c2 \u03c4\u1f78\u03bd \u03b3\u03b5\u03b3\u03b1\u03bc\u03b7\u03ba\u03cc\u03c4\u03b1. Again, Severianus\n  (ib. p. 141) distinctly explains it as recommending a state of\n  liberty. On the other hand Chrysostom, while mentioning that 'certain\n  persons' interpret it \u03b5\u1f30 \u03b4\u03cd\u03bd\u03b1\u03c3\u03b1\u03b9 \u1f10\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03c9\u03b8\u1fc6\u03bd\u03b1\u03b9, \u1f10\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03ce\u03b8\u03b7\u03c4\u03b9, himself\n  supposes St Paul to advise the slave's remaining in slavery. And so\n  Theodoret and others. The balance of argument seems to be decidedly in\n  favour of the former view.\n\n  (1) The actual language must be considered first. And here (i) the\n  particles \u03b5\u1f30 \u03ba\u03b1\u1f76 will suit either interpretation. If they are\n  translated 'even though', the clause recommends the continuance in\n  slavery. But \u03ba\u03b1\u1f76 may be equally well taken with \u03b4\u03cd\u03bd\u03b1\u03c3\u03b1\u03b9, and the words\n  will then mean 'if it _should_ be in your power to obtain your\n  freedom'. So above ver. 11 \u1f10\u1f70\u03bd \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c7\u03c9\u03c1\u03b9\u03c3\u03b8\u1fc7: comp. Luke xi. 18 \u03b5\u1f30 \u03b4\u1f72\n  \u03ba\u03b1\u1f76 \u1f41 \u03a3\u03b1\u03c4\u03b1\u03bd\u1fb6\u03c2 \u1f10\u03c6' \u1f11\u03b1\u03c5\u03c4\u1f78\u03bd \u03b4\u03b9\u03b5\u03bc\u03b5\u03c1\u03af\u03c3\u03b8\u03b7, 1 Pet. iii. 14 \u1f00\u03bb\u03bb' \u03b5\u1f30 \u03ba\u03b1\u1f76\n  \u03c0\u03ac\u03c3\u03c7\u03bf\u03b9\u03c4\u03b5 \u03b4\u03b9\u1f70 \u03b4\u03b9\u03ba\u03b1\u03b9\u03bf\u03c3\u03cd\u03bd\u03b7\u03bd. (ii) The expression \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c7\u03c1\u1fc6\u03c3\u03b1\u03b9 seems to\n  direct the slave to _avail himself_ of some _new_ opportunity offered,\n  and therefore to recommend liberty; comp. ix. 12, 15.\n\n  (2) The immediate context will admit either interpretation. If slavery\n  be preferred, the sentence is continuous. If liberty, the clause \u1f00\u03bb\u03bb'\n  \u03b5\u1f30 \u03ba\u03b1\u1f76 ... \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c7\u03c1\u1fc6\u03c3\u03b1\u03b9 is parenthetical. In this latter case its\n  motive is to correct misapprehension, as if the Apostle would say,\n  'When I declare the absolute indifference of the two states in the\n  sight of God, I do not mean to say that you should not avail\n  yourselves of freedom, if it comes in your way; it puts you in a more\n  advantageous position, and you will do well to prefer it'. Such a\n  corrective parenthesis is altogether after St Paul's manner, and\n  indeed instances occur in this very context: e.g. ver. 11 \u1f10\u1f70\u03bd \u03b4\u1f72 \u03ba\u03b1\u1f76\n  \u03c7\u03c9\u03c1\u03b9\u03c3\u03b8\u1fc7 \u03ba.\u03c4.\u03bb., ver. 15 \u03b5\u1f30 \u03b4\u1f72 \u1f41 \u1f04\u03c0\u03b9\u03c3\u03c4\u03bf\u03c2 \u03c7\u03c9\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb. This last\n  passage is an exact parallel, for the \u03b3\u1f70\u03c1 of ver. 16 is connected\n  immediately with ver. 14, the parenthesis being disregarded as here.\n\n  (3) The argument which seems decisive is the extreme improbability\n  that St Paul should have recommended slavery in preference to freedom.\n  For (i) Such a recommendation would be alien to the spirit of a man\n  whose sense of political right was so strong, and who asserted his\n  citizenship so stanchly on more than one occasion (Acts xvi. 37, xxii.\n  28). (ii) The independent position of the freeman would give him an\n  obvious advantage in doing the work of Christ, which it is difficult\n  to imagine St Paul enjoining him deliberately to forego. (iii)\n  Throughout the passage the Apostle, while maintaining the indifference\n  of these earthly relations in the sight of God, yet always gives the\n  preference to a position of _independence_, whenever it comes to a\n  Christian naturally and without any undue impatience on his part. The\n  spirit which animates St Paul's injunctions here may be seen from vv.\n  8, 11, 15, 26, 27 etc.\n\nFootnote 753:\n\n  Ephes. vi. 5\u20139, Col. iii. 22-iv. 1.\n\n[Sidenote: The Christian idea fatal to slavery.]\n\nBut meanwhile a principle is boldly enunciated, which must in the end\nprove fatal to slavery. When the Gospel taught that God had made all men\nand women upon earth of one family; that all alike were His sons and His\ndaughters; that, whatever conventional distinctions human society might\nset up, the supreme King of Heaven refused to acknowledge any; that the\nslave notwithstanding his slavery was Christ's freedman, and the free\nnotwithstanding his liberty was Christ's slave; when the Church carried\nout this principle by admitting the slave to her highest privileges,\ninviting him to kneel side by side with his master at the same holy\ntable; when in short the Apostolic precept that 'in Christ Jesus is\nneither bond nor free' was not only recognised but acted upon, then\nslavery was doomed. Henceforward it was only a question of time. Here\nwas the idea which must act as a solvent, must disintegrate this\nvenerable institution, however deeply rooted and however widely spread.\n[Sidenote: Its general tendency.]'The brotherhood of man, in short, is\nthe idea which Christianity in its social phase has been always striving\nto realise, and the progress of which constitutes the social history of\nChristendom. With what difficulties this idea has struggled; how it has\nbeen marred by revolutionary violence, as well as impeded by reactionary\nselfishness; to what chimerical hopes, to what wild schemes, to what\ncalamitous disappointments, to what desperate conflicts, it has given\nbirth; how often being misunderstood and misapplied, it has brought not\npeace on earth but a sword\u2014it is needless here to rehearse. Still, as we\nlook back over the range of past history, we can see beyond doubt that\nit is towards this goal that Christianity as a social principle has been\nalways tending and still tends[754].'\n\nFootnote 754:\n\n  G. Smith _Does the Bible etc.?_ p. 121.\n\n[Sidenote: Its effects on slavery.]\n\nAnd this beneficent tendency of the Gospel was felt at once in its\neffects on slavery. The Church indeed, even in the ardour of her\nearliest love, did not prohibit her sons from retaining slaves in their\nhouseholds. It is quite plain from extant notices, that in the earlier\ncenturies, as in the later, Christians owned slaves[755] like their\nheathen neighbours, without forfeiting consideration among their\nfellow-believers. But nevertheless the Christian idea was not a\ndead-letter. [Sidenote: Protection and manumission of slaves.]The\nchivalry of the Gospel which regarded the weak and helpless from\nwhatever cause, as its special charge, which extended its protection to\nthe widow, the orphan, the sick, the aged, and the prisoner, was not\nlikely to neglect the slave. Accordingly we find that one of the\nearliest forms which Christian benevolence took was the contribution of\nfunds for the liberation of slaves[756]. [Sidenote: Honours paid to\nslave martyrs.]But even more important than overt acts like these was\nthe moral and social importance with which the slave was now invested.\nAmong the heroes and heroines of the Church were found not a few members\nof this class. When slave girls like Blandina in Gaul or Felicitas in\nAfrica, having won for themselves the crown of martyrdom, were\ncelebrated in the festivals of the Church with honours denied to the\nmost powerful and noblest born of mankind, social prejudice had received\na wound which could never be healed.\n\nFootnote 755:\n\n  Athenag. _Suppl._ 35 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03af \u03b5\u1f30\u03c3\u03b9\u03bd \u1f21\u03bc\u1fd6\u03bd, \u03c4\u03bf\u1fd6\u03c2 \u03bc\u1f72\u03bd \u03ba\u03b1\u1f76 \u03c0\u03bb\u03b5\u03af\u03bf\u03c5\u03c2 \u03c4\u03bf\u1fd6\u03c2 \u03b4'\n  \u1f10\u03bb\u03ac\u03c4\u03c4\u03bf\u03c5\u03c2. It would even appear that the domestic servant who betrayed\n  Polycarp (_Mart. Polyc._ 6) was a slave, for he was put to the\n  torture. Comp. Justin. _Apol._ ii. 12.\n\nFootnote 756:\n\n  Ignat. _Polyc._ 4 \u03bc\u1f74 \u1f10\u03c1\u03ac\u03c4\u03c9\u03c3\u03b1\u03bd \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03ba\u03bf\u03b9\u03bd\u03bf\u1fe6 \u1f10\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03bf\u1fe6\u03c3\u03b8\u03b1\u03b9, _Apost.\n  Const._ iv. 9 \u03c4\u1f70 \u1f10\u03be \u03b1\u1f50\u03c4\u1ff6\u03bd, \u1f61\u03c2 \u03c0\u03c1\u03bf\u03b5\u03b9\u03c1\u03ae\u03ba\u03b1\u03bc\u03b5\u03bd, \u1f00\u03b8\u03c1\u03bf\u03b9\u03b6\u03cc\u03bc\u03b5\u03bd\u03b1 \u03c7\u03c1\u03ae\u03bc\u03b1\u03c4\u03b1\n  \u03b4\u03b9\u03b1\u03c4\u03ac\u03c3\u03c3\u03b5\u03c4\u03b5 \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03b5\u1f30\u03c2 \u1f00\u03b3\u03bf\u03c1\u03b1\u03c3\u03bc\u03bf\u1f7a\u03c2 \u03c4\u1ff6\u03bd \u1f01\u03b3\u03af\u03c9\u03bd, _\u1fe5\u03c5\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9\n  \u03b4\u03bf\u03cd\u03bb\u03bf\u03c5\u03c2_ \u03ba\u03b1\u1f76 \u03b1\u1f30\u03c7\u03bc\u03b1\u03bb\u03ce\u03c4\u03bf\u03c5\u03c2, \u03b4\u03b5\u03c3\u03bc\u03af\u03bf\u03c5\u03c2, \u03ba.\u03c4.\u03bb.\n\n[Sidenote: Christianity predominant.]\n\nWhile the Church was still kept in subjection, moral influence and\nprivate enterprise were her only weapons. But Christianity was no sooner\nseated on the throne of the C\u00e6sars than its influence began to be felt\nin the imperial policy[757]. [Sidenote: Legislation of Constantine.]The\nlegislation of Constantine, despite its startling inequalities, forms a\nunique chapter in the statute book of Rome. In its mixed character\nindeed it reflects the transitional position of its author. But after\nall allowance made for its very patent defects, its general advance in\nthe direction of humanity and purity is far greater than can be traced\nin the legislation even of the most humane and virtuous of his heathen\npredecessors. More especially in the extension of legal protection to\nslaves, and in the encouragement given to emancipation, we have an\nearnest of the future work which Christianity was destined to do for\nthis oppressed class of mankind, though the relief which it gave was\nafter all very partial and tentative[758].\n\nFootnote 757:\n\n  It must not however be forgotten that, even before Christianity became\n  the predominant religion, a more humane spirit had entered into Roman\n  legislation. The important enactment of Hadrian has been already\n  mentioned, p. 387, note 746. Even earlier the _lex Petronia_ (of which\n  the date is uncertain) had prohibited masters from making their slaves\n  fight with wild beasts in mere caprice and without an order from a\n  judge (Dig. xlviii. 8. 11); and Claudius (A.D. 47), finding that the\n  practice of turning out sick slaves into the streets to die was on the\n  increase, ordered that those who survived this treatment should have\n  their freedom (Dion Cass. lx. 29, Suet. _Claud._ 25). For these and\n  similar enactments of the heathen emperors see Wallon III. p. 60 sq.,\n  _R\u00f6m. Alterth._ V. I. 197, Rein _Privatrecht d. R\u00f6mer_ p. 560 sq. The\n  character of this exceptional legislation is the strongest impeachment\n  of the general cruelty of the law; while at the same time subsequent\n  notices show how very far from effective it was even within its own\n  narrow limits. See for instance the passage in Galen, v. p. 17 (ed.\n  K\u00fchn) \u03bb\u03b1\u03ba\u03c4\u03af\u03b6\u03bf\u03c5\u03c3\u03b9 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1f7a\u03c2 \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u1f7a\u03c2 \u1f10\u03be\u03bf\u03c1\u03cd\u03c4\u03c4\u03bf\u03c5\u03c3\u03b9 \u03ba\u03b1\u1f76 \u03b3\u03c1\u03b1\u03c6\u03ad\u03b9\u1ff3 \u03ba\u03b5\u03bd\u03c4\u03bf\u1fe6\u03c3\u03b9\u03bd\n  \u03ba.\u03c4.\u03bb. (comp. _ib._ p. 584), or Seneca _de Ira_ iii. 3. 6 'eculei et\n  fidicul\u00e6 et ergastula et cruces et circundati defossis corporibus\n  ignes et cadavera quoque trahens uncus, varia vinculorum genera, varia\n  p\u0153narum, lacerationes membrorum, inscriptiones frontis et bestiarum\n  immanium cave\u00e6.'\n\n  On the causes of these ameliorations in the law see _R\u00f6m. Alterth._ V.\n  1. p. 199.\n\nFootnote 758:\n\n  On the legislation of Constantine affecting slavery see De Broglie\n  _L'Eglise et l'Empire Romain_, I. p. 304 sq. (ed. 5), Chawner\n  _Influence of Christianity upon the Legislation of Constantine the\n  Great_, p. 73 sq., Wallon III. p. 414 sq. The legislation of Justinian\n  is still more honourably distinguished for its alleviation of the\n  evils of slavery.\n\n[Sidenote: Subsequent activity of the Church.]\n\nAnd on the whole this part has been faithfully and courageously\nperformed by the Church. There have been shameful exceptions now and\nthen: there has been occasional timidity and excess of caution. The\ncommentaries of the fathers on this epistle are an illustration of this\nlatter fault[759]. Much may be pardoned to men who shrink from seeming\nto countenance a violent social revolution. But notwithstanding, it is a\nbroad and patent fact that throughout the early and middle ages the\ninfluence of the Church was exerted strongly on the side of humanity in\nthis matter[760]. The emancipation of slaves was regarded as a principal\naim of the higher Christian life[761]; the amelioration of serfdom was a\nmatter of constant solicitude with the rulers of the Church.\n\n[Sidenote: The conquests and hopes of the present time.]\n\nAnd at length we seem to see the beginning of the end. The rapid strides\ntowards emancipation during the present generation are without a\nparallel in the history of the world. The abolition of slavery\nthroughout the British Empire at an enormous material sacrifice is one\nof the greatest moral conquests which England has ever achieved. The\nliberation of twenty millions of serfs throughout the Russian dominions\nhas thrown a halo of glory round the name of Alexander II., which no\ntime can dim. The emancipation of the <DW64> in the vast republic of the\nNew World was a victory not less important than either to the well-being\nof the human race. Thus within the short period of little more than a\nquarter of a century this reproach of civilisation and humanity has been\nwiped out in the three greatest empires of the world. It is a fit sequel\nto these achievements, that at length a well directed attack should have\nbeen made on the central fortress of slavery and the slave-trade, the\ninterior of Africa. May we not venture to predict that in future ages,\nwhen distance of view shall have adjusted the true relations of events,\nwhen the brilliancy of empires and the fame of wars shall have sunk to\ntheir proper level of significance, this epoch will stand out in the\nhistory of mankind as the era of liberation? If so, the Epistle to\nPhilemon, as the earliest prelude to these magnificent social victories,\nmust be invested with more than common interest for our generation.\n\nFootnote 759:\n\n  _E.g._ Chrysostom and Theodore of Mopsuestia (_Spic. Solesm._ I. p.\n  152). Yet St Chrysostom himself pleads the cause of slaves earnestly\n  elsewhere. In _Hom. xl. ad 1 Cor._ x. p. 385 he says of slavery, 'It\n  is the penalty of sin and the punishment of disobedience. But when\n  Christ came, he annulled even this, _For in Christ Jesus there is no\n  slave nor free_. Therefore it is not necessary to have a slave; but,\n  if it should be necessary, then one only or at most a second'. And he\n  then tells his audience that if they really care for the welfare of\n  slaves, they must 'buy them, and having taught them some art that they\n  may maintain themselves, set them free.' 'I know,' he adds, 'that I am\n  annoying my hearers; but what can I do? For this purpose I am\n  appointed, and I will not cease speaking so.' On the attitude of this\n  father towards slavery see M\u00f6hler p. 89 sq.\n\nFootnote 760:\n\n  On the influence of Christianity in this respect see Wallon III. p.\n  314 sq., Schmidt _Essai historique sur la Soci\u00e9t\u00e9 Civile dans le Monde\n  Romain etc._ p. 228 sq. (1853), M\u00f6hler _Gesammelte Schriften_ II. p.\n  54 sq., G. Smith _Does the Bible etc.?_ p. 95 sq., E. S. Talbot\n  _Slavery as affected by Christianity_ (1869), Lecky _Rationalism in\n  Europe_ II. p. 255 sq., _European Morals_ II. p. 65 sq.\n\nFootnote 761:\n\n  M\u00f6hler p. 99 sq., Schmidt p. 246 sq., Lecky _E. M._ II. p. 73 sq.\n\n\n\n\n                             \u03a0\u03a1\u039f\u03a3 \u03a6\u0399\u039b\u0397\u039c\u039f\u039d\u0391.\n\n\n\n\n                 WHERE THE SPIRIT OF THE LORD IS, THERE\n                              IS LIBERTY.\n\n                    WHO IS WEAK, AND I AM NOT WEAK?\n                    WHO IS OFFENDED, AND I BURN NOT?\n\n                  *       *       *       *       *\n\n            _Such ever was love's way: to rise, it stoops._\n\n------------------------------------------------------------------------\n\n                            \u03a0\u03a1\u039f\u03a3  \u03a6\u0399\u039b\u0397\u039c\u039f\u039d\u0391.\n\n------------------------------------------------------------------------\n\n1\u20133]\n\n^1\u03a0\u0391\u03a5\u039b\u039f\u03a3, \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03a4\u03b9\u03bc\u03cc\u03b8\u03b5\u03bf\u03c2 \u1f41 \u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03c2, \u03a6\u03b9\u03bb\u03ae\u03bc\u03bf\u03bd\u03b9 \u03c4\u1ff7\n\u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1ff7 \u03ba\u03b1\u1f76 \u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u1ff7 \u1f21\u03bc\u1ff6\u03bd ^2\u03ba\u03b1\u1f76 \u1f08\u03c0\u03c6\u03af\u1fb3 \u03c4\u1fc7 \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc7 \u03ba\u03b1\u1f76 \u1f08\u03c1\u03c7\u03af\u03c0\u03c0\u1ff3 \u03c4\u1ff7\n\u03c3\u03c5\u03bd\u03c3\u03c4\u03c1\u03b1\u03c4\u03af\u03c9\u03c4\u1fc3 \u1f21\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c4\u1fc7 \u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03cc\u03bd \u03c3\u03bf\u03c5 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u1fb3\u00b7 ^3\u03c7\u03ac\u03c1\u03b9\u03c2 \u1f51\u03bc\u1fd6\u03bd\n\n\n1\u20133. 'PAUL, now a prisoner of Christ Jesus, and TIMOTHY a brother in the\nfaith, unto PHILEMON our dearly-beloved and fellow-labourer in the\nGospel, and unto APPHIA our sister, and unto ARCHIPPUS our\nfellow-soldier in Christ, and to the Church which assembles in thy\nhouse. Grace and peace to you all from God our Father and the Lord Jesus\nChrist.'\n\n1. \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2] The authoritative title of 'Apostle' is dropped, because\nthroughout this letter St Paul desires to entreat rather than to command\n(ver. 8, 9); see the note on Phil. i. 1. In its place is substituted a\ndesignation which would touch his friend's heart. How could Philemon\nresist an appeal which was penned within prison walls and by a manacled\nhand? For this characteristic reference to his 'bonds' see the note on\nver. 13.\n\n\u03a4\u03b9\u03bc\u03cc\u03b8\u03b5\u03bf\u03c2] Timothy seems to have been with St Paul during a great part of\nhis three years' sojourn in Ephesus (Acts xix. 22), and could hardly\nhave failed to make the acquaintance of Philemon. For the designation \u1f41\n\u1f00\u03b4\u03b5\u03bb\u03c6\u03cc\u03c2 applied to Timothy see the note on Col. i. 1.\n\n\u03a6\u03b9\u03bb\u03ae\u03bc\u03bf\u03bd\u03b9 \u03ba.\u03c4.\u03bb.] On the persons here addressed, and the language in\nwhich they are described, see the introduction p. 369 sq.\n\n\u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u1ff7] It would probably be during St Paul's long sojourn at Ephesus\nthat Philemon had laboured with him: see above p. 31 sq.\n\n\u1f21\u03bc\u1ff6\u03bd] should probably be attached to \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1ff7 as well as to \u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u1ff7;\ncomp. Rom. xvi. 5, 8, 9, 1 Cor. x. 14, Phil. ii. 12.\n\n2. \u03c4\u1fc7 \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc7] For this the received text has \u03c4\u1fc7 \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1fc7. Internal\nprobabilities can be urged in favour of both readings. On the one hand\n\u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1fc7 might have been introduced for the sake of conformity to the\npreceding \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1ff7; on the other \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc7 might have been substituted for\n\u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1fc7 on grounds of false delicacy. Theodore of Mopsuestia (_Spicil.\nSolesm._ I. p. 154), who had the reading \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u1fc7, feels an apology\nnecessary: 'Istius temporis (i.e. of the present time) homines\npropemodum omnes in crimine vocandos esse existimant, modo si audierint\nnomen charitatis. Apostolus vero non sic sentiebat; sed contrario etc.'\nI have preferred \u03c4\u1fc7 \u1f00\u03b4\u03b5\u03bb\u03c6\u1fc7, because the preponderance of ancient\nauthority is very decidedly in its favour.\n\n\u03c3\u03c5\u03bd\u03c3\u03c4\u03c1\u03b1\u03c4\u03b9\u03ce\u03c4\u1fc3] These spiritual campaigns, in which Archippus was his\ncomrade, probably took place while St Paul was at Ephesus (A.D. 54\u201357).\nFor the word \u03c3\u03c5\u03bd\u03c3\u03c4\u03c1\u03b1\u03c4\u03b9\u03ce\u03c4\u03b7\u03c2 see Phil. ii. 25. The metaphor of \u03c3\u03c4\u03c1\u03b1\u03c4\u03b5\u03af\u03b1,\n\u03c3\u03c4\u03c1\u03b1\u03c4\u03b5\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9, is common in St Paul.\n\n\u03c4\u1fc7 \u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03bf\u03bd \u03ba.\u03c4.\u03bb.] probably at Coloss\u00e6; see above p. 370 sq. For the\nmeaning of the expression see the note on Col. iv. 15.\n\n------------------------------------------------------------------------\n\n4, 5]\n\n\u03ba\u03b1\u1f76 \u03b5\u1f30\u03c1\u03ae\u03bd\u03b7 \u1f00\u03c0\u1f78 \u0398\u03b5\u03bf\u1fe6 \u03c0\u03b1\u03c4\u03c1\u1f78\u03c2 \u1f21\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6.\n\n^4\u0395\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6 \u03c4\u1fb7 \u0398\u03b5\u1ff7 \u03bc\u03bf\u03c5 \u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5, \u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c3\u03bf\u03c5 \u03c0\u03bf\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u1f10\u03c0\u1f76 \u03c4\u1ff6\u03bd \u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u1ff6\u03bd\n\u03bc\u03bf\u03c5, ^5\u1f00\u03ba\u03bf\u03cd\u03c9\u03bd \u03c3\u03bf\u03c5 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd\n\n\n4\u20137. 'I never cease to give thanks to my God for thy well-doing, and\nthou art ever mentioned in my prayers. For they tell me of thy love and\nfaith\u2014thy faith which thou hast in the Lord Jesus, and thy love which\nthou showest towards all the saints; and it is my prayer that this\nactive sympathy and charity, thus springing from thy faith, may abound\nmore and more, as thou attainest to the perfect knowledge of every good\nthing bestowed upon us by God, looking unto and striving after Christ.\nFor indeed it gave me great joy and comfort to hear of thy\nloving-kindness, and to learn how the hearts of God's people had been\ncheered and refreshed by thy help, my dear brother'.\n\nThe Apostle's thanksgiving and intercessory prayer (ver. 4)\u2014the cause of\nhis thanksgiving (ver. 5)\u2014the purport of his prayer (ver. 6)\u2014the joy and\ncomfort which he has in Philemon's good deeds (ver. 7)\u2014this is the very\nsimple order of topics in these verses. But meanwhile all established\nprinciples of arrangement are defied in the anxiety to give expression\nto the thought which is uppermost for the moment. The clause \u1f00\u03ba\u03bf\u03cd\u03c9\u03bd\n\u03ba.\u03c4.\u03bb. is separated from \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6 \u03ba.\u03c4.\u03bb., on which it depends, by the\nintervening clause \u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c3\u03bf\u03c5 \u03ba.\u03c4.\u03bb. which introduces another thought.\nIt itself interposes between two clauses \u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c3\u03bf\u03c5 \u03ba.\u03c4.\u03bb. and \u1f45\u03c0\u03c9\u03c2 \u1f21\n\u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1 \u03ba.\u03c4.\u03bb., which stand in the closest logical and grammatical\nconnexion with each other. Its own component elements are dislocated and\ninverted in the struggle of the several ideas for immediate utterance.\nAnd lastly, in \u03c7\u03b1\u03c1\u1f70\u03bd \u03b3\u1f70\u03c1 \u03ba.\u03c4.\u03bb. there is again a recurrence to a topic\nwhich has occurred in an earlier part of the sentence (\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd ...\n\u03b5\u1f30\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f01\u03b3\u03af\u03bf\u03c5\u03c2) but which has been dropped, before it was\nexhausted, owing to the pressure of another more importunate thought.\n\n4. \u0395\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6] See the note on 1 Thess. i. 2.\n\n\u03c0\u03ac\u03bd\u03c4\u03bf\u03c4\u03b5] should probably be taken with \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6 (rather than with\n\u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03ba.\u03c4.\u03bb.), according to St Paul's usual collocation in these\nopening thanksgivings: see the notes on Col. i. 3, Phil. i. 3.\n\n\u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c3\u03bf\u03c5 \u03ba.\u03c4.\u03bb.] '_making mention of thee_.' For \u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c0\u03bf\u03b9\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 see\nthe note on 1 Thess. i. 2. Here the 'mention' involves the idea of\nintercession _on behalf of_ Philemon, and so introduces the \u1f45\u03c0\u03c9\u03c2 \u03ba.\u03c4.\u03bb.\nof ver. 6. See the note there.\n\n5. \u1f00\u03ba\u03bf\u03cd\u03c9\u03bd] This information would probably come from Epaphras (Col. i.\n7, 8, iv. 12) rather than from Onesimus. The participle is connected\nmore directly with \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6 than with the intervening words, and\nexplains the grounds of the Apostle's thanksgiving.\n\n\u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd \u03ba.\u03c4.\u03bb.] i.e. 'the faith which thou hast towards the Lord\nJesus Christ and the love which thou showest to all the saints.' The\nlogical order is violated, and the clauses are inverted in the second\npart of the sentence, thus producing an example of the figure called\nchiasm; see Gal. iv. 4, 5. This results here from the Apostle's setting\ndown the thoughts in the sequence in which they occur to him, without\npaying regard to symmetrical arrangement. The first and prominent\nthought is Philemon's love. This suggests the mention of his faith, as\nthe source from which it springs. This again requires a reference to the\nobject of faith. And then at length comes the deferred sequel to the\nfirst thought\u2014the range and comprehensiveness of his love. The\ntransition from the object of faith to the object of love is more easy,\nbecause the love is represented as springing from the faith. Some copies\ntranspose the order, reading \u03c4\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd\u2014an obvious\nemendation. Others would obviate the difficulty by giving to \u03c0\u03af\u03c3\u03c4\u03b9\u03bd the\nmeaning 'fidelity, stedfastness': Winer \u00a7 1. p. 511 sq. Thus they are\nenabled to refer both words, \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u03ba\u03b1\u1f76 \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd, equally to both the\nclauses which follow. But, though this is a legitimate sense of \u03c0\u03af\u03c3\u03c4\u03b9\u03c2\nin St Paul (see _Galatians_, p. 155), yet in immediate connexion with \u1f23\u03bd\n\u1f14\u03c7\u03b5\u03b9\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd, it is hardly possible that the word can\nhave any other than its proper theological meaning. See the opening of\nthe contemporary epistle, Col. i. 4.\n\n------------------------------------------------------------------------\n\n6]\n\n\u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u1f23\u03bd \u1f14\u03c7\u03b5\u03b9\u03c2 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd \u1f38\u03b7\u03c3\u03bf\u1fe6\u03bd \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03c0\u03ac\u03bd\u03c4\u03b1\u03c2 \u03c4\u03bf\u1f7a\u03c2\n\u1f01\u03b3\u03af\u03bf\u03c5\u03c2, ^6\u1f45\u03c0\u03c9\u03c2 \u1f21 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1 \u03c4\u1fc6\u03c2 \u03c0\u03af\u03c3\u03c4\u03ad\u03c9\u03c2 \u03c3\u03bf\u03c5 \u1f10\u03bd\u03b5\u03c1\u03b3\u1f74\u03c2 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 \u1f10\u03bd \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03b5\u03b9\n\u03c0\u03b1\u03bd\u03c4\u1f78\u03c2 \u1f00\u03b3\u03b1\u03b8\u03bf\u1fe6 \u03c4\u03bf\u1fe6 \u1f10\u03bd\n\n\n\u03c0\u03c1\u1f78\u03c2 \u03ba.\u03c4.\u03bb.] The change of prepositions, \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u039a\u03cd\u03c1\u03b9\u03bf\u03bd '_towards_ the\nLord' and \u03b5\u1f30\u03c2 \u03c4\u03bf\u1f7a\u03c2 \u1f01\u03b3\u03af\u03bf\u03c5\u03c2 '_unto_ the saints', deserves attention. It\nseems to arise from the instinctive desire to separate the two clauses,\nas they refer to different words in the preceding part of the sentence.\nOf the two prepositions the former (\u03c0\u03c1\u03bf-\u03c2) signifies _direction_\n'forward to', 'towards'; the latter (\u1f10\u03bd-\u03c2) _arrival_ and so _contact_,\n'in-to', 'unto.' Consequently either might be used in either connexion;\nand as a matter of fact \u03b5\u1f30\u03c2 is much more common with \u03c0\u03af\u03c3\u03c4\u03b9\u03c2 (\u03c0\u03b9\u03c3\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd),\nas it is also with \u1f00\u03b3\u03ac\u03c0\u03b7, \u03c0\u03c1\u03cc\u03c2 being quite exceptional (1 Thess. i. 8 \u1f21\n\u03c0\u03af\u03c3\u03c4\u03b9\u03c2 \u1f51\u03bc\u1ff6\u03bd \u1f21 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u0398\u03b5\u03cc\u03bd; comp. 2 Cor. iii. 4). But where a\ndistinction is necessary, there is a propriety in using \u03c0\u03c1\u03cc\u03c2 of the\nfaith which aspires _towards_ Christ, and \u03b5\u1f30\u03c2 of the love which is\nexerted _upon_ men. Some good copies read \u03b5\u1f30\u03c2 here in both clauses.\n\n6. \u1f45\u03c0\u03c9\u03c2 \u03ba.\u03c4.\u03bb.] to be taken with \u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c3\u03bf\u03c5 \u03c0\u03bf\u03b9\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ba.\u03c4.\u03bb., as giving\nthe aim and purport of St Paul's prayer. Others connect it with h\u1f72\u03bd\n\u1f14\u03c7\u03b5\u03b9\u03c2, as if it described the tendency of Philemon's faith, 'ita ut';\nbut, even if \u1f45\u03c0\u03c9\u03c2 could bear this meaning, such a connexion is\naltogether harsh and improbable.\n\n\u1f21 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1 \u03ba.\u03c4.\u03bb.] Of many interpretations which have been, or might be,\ngiven of these words, two seem to deserve consideration. (1) 'Your\nfriendly offices and sympathies, your kindly deeds of charity, which\nspring from your faith': comp. Phil. i. 5 \u1f10\u03c0\u1f76 \u03c4\u1fc7 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u1fb3 \u1f51\u03bc\u1ff6\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f78\n\u03b5\u1f50\u03b1\u03b3\u03b3\u03ad\u03bb\u03b9\u03bf\u03bd, Heb. xiii. 16 \u03c4\u1fc6\u03c2 \u03b5\u1f50\u03c0\u03bf\u03b9\"\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1\u03c2, whence \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1\nis used especially of 'contributions, almsgiving', Rom. xv. 26, 2 Cor.\nviii. 4, ix. 13. (2) 'Your communion with God through faith': comp. 1\nCor. i. 9, and see also 2 Cor. xiii. 13, 1 Joh. i. 3, 6, 7. The parallel\npassages strongly support the former sense. Other interpretations\nproposed are, 'The participation of others in your faith, through your\nexample', or 'your communion with me, springing out of your faith'. This\nlast, which is widely received, is suggested by ver. 17; \u03b5\u1f30 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u1f78\u03c2 \u03b5\u1f36,\n\u03c6\u03b7\u03c3\u03af, \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd, writes Chrysostom, \u03ba\u03b1\u1f76 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f70 \u1f04\u03bb\u03bb\u03b1 \u1f40\u03c6\u03b5\u03af\u03bb\u03b5\u03b9\u03c2\n\u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03b5\u1fd6\u03bd (comp. Tit. i. 3 \u03ba\u03b1\u03c4\u1f70 \u03ba\u03bf\u03b9\u03bd\u1f74\u03bd \u03c0\u03af\u03c3\u03c4\u03b9\u03bd): but it seems quite out\nof place in this context.\n\n\u1f10\u03bd\u03b5\u03c1\u03b3\u03ae\u03c2] '_effective_'. The Latin translators must have read \u1f10\u03bd\u03b1\u03c1\u03b3\u03ae\u03c2,\nfor they render the word _evidens_ or _manifesta_. Jerome (_ad loc._)\nspeaks of _evidens_ as the reading of the Latin, and _efficax_ of the\nGreek text. The converse error appears in the MSS of _Clem. Hom._ xvii.\n5, \u1f10\u03bd\u03ad\u03c1\u03b3\u03b5\u03b9\u03b1 for \u1f10\u03bd\u03ac\u03c1\u03b3\u03b5\u03b9\u03b1.\n\n\u1f10\u03bd \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03b5\u03b9 \u03ba.\u03c4.\u03bb.] '_in the perfect knowledge of every good thing_'.\nThis \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2, involving as it does the complete appropriation of all\ntruth and the unreserved identification with God's will, is the goal and\ncrown of the believer's course. The Apostle does not say 'in the\npossession' or 'in the performance' but 'in the knowledge of every good\nthing'; for, in this higher sense of knowledge, to know is both to\npossess and to perform. In all the epistles of the Roman captivity St\nPaul's prayer for his correspondents culminates in this word \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2:\nsee the note on Col. i. 9. This \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2 is the result and the reward\nof faith manifesting itself in deeds of love, \u1f45\u03c0\u03c9\u03c2 \u1f21 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1 \u03c4\u1fc6\u03c2\n\u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03ba.\u03c4.\u03bb. For the sequence comp. Ephes. iv. 13 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u1f11\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u1fc6\u03c2\n\u03c0\u03af\u03c3\u03c4\u03b5\u03c9\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u1f10\u03c0\u03b9\u03b3\u03bd\u03ce\u03c3\u03b5\u03c9\u03c2 \u03ba.\u03c4.\u03bb., Tit. i. 1 \u03ba\u03b1\u03c4\u1f70 \u03c0\u03af\u03c3\u03c4\u03b9\u03bd \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u1ff6\u03bd \u0398\u03b5\u03bf\u1fe6\n\u03ba\u03b1\u1f76 \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u03ba\u03b1\u03c4' \u03b5\u1f50\u03c3\u03ad\u03b2\u03b5\u03b9\u03b1\u03bd. The \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2 therefore which\nthe Apostle contemplates is Philemon's own. There is no reference to the\nforce of his example on others, as it is sometimes interpreted, 'in\ntheir recognition of every good thing which is wrought in you'.\n\n------------------------------------------------------------------------\n\n7]\n\n\u1f21\u03bc\u1fd6\u03bd \u03b5\u1f30\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd. ^7\u03c7\u03b1\u03c1\u1f70\u03bd \u03b3\u1f70\u03c1 \u03c0\u03bf\u03bb\u03bb\u1f74\u03bd \u1f14\u03c3\u03c7\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03ac\u03ba\u03bb\u03b7\u03c3\u03b9\u03bd \u1f10\u03c0\u1f76 \u03c4\u1fc7 \u1f00\u03b3\u03ac\u03c0\u1fc3\n\u03c3\u03bf\u03c5, \u1f45\u03c4\u03b9 \u03c4\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 \u03c4\u1ff6\u03bd \u1f01\u03b3\u03af\u03c9\u03bd \u1f00\u03bd\u03b1\u03c0\u03ad\u03c0\u03b1\u03c5\u03c4\u03b1\u03b9 \u03b4\u03b9\u1f70 \u03c3\u03bf\u1fe6, \u1f00\u03b4\u03b5\u03bb\u03c6\u03ad.\n\n                        6. \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd \u03b5\u1f30\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd.\n\n\n\u03c4\u03bf\u1fe6 \u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd] '_which is in us Christians_', 'which is placed within our\nreach by the Gospel'; i.e. the whole range of spiritual blessings, the\ncomplete cycle of Christian truth. If the reading \u03c4\u03bf\u1fe6 \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd be\nadopted, the reference will be restricted to the brotherhood at Coloss\u00e6,\nbut the meaning must be substantially the same. Though \u1f51\u03bc\u1fd6\u03bd has somewhat\nbetter support, we seem to be justified in preferring \u1f21\u03bc\u1fd6\u03bd as being much\nmore expressive. In such cases the MSS are of no great authority; and in\nthe present instance scribes would be strongly tempted to alter \u1f21\u03bc\u1fd6\u03bd\ninto \u1f51\u03bc\u1fd6\u03bd from a misapprehension of the sense, and a wish to apply the\nwords to Philemon and his household. A similar misapprehension doubtless\nled in some copies to the omission of \u03c4\u03bf\u1fe6, which seemed to be\nsuperfluous but is really required for the sense.\n\n\u03b5\u1f30\u03c2 \u03a7\u03c1\u03b9\u03c3\u03c4\u03cc\u03bd] '_unto Christ_', i.e. leading to Him as the goal. The words\nshould be connected not with \u03c4\u03bf\u1fe6 \u1f10\u03bd \u1f21\u03bc\u1fd6\u03bd, but with the main statement of\nthe sentence \u1f10\u03bd\u03b5\u03c1\u03b3\u1f74\u03c2 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb.\n\n7. \u03c7\u03b1\u03c1\u1f70\u03bd \u03b3\u03ac\u03c1] This sentence again must not be connected with the words\nimmediately preceding. It gives the motive of the Apostle's thanksgiving\nmentioned in ver. 4. This thanksgiving was the outpouring of gratitude\nfor the joy and comfort that he had received in his bonds, from the\nreport of Philemon's generous charity. The connexion therefore is\n\u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u1ff6 \u03c4\u1ff7 \u0398\u03b5\u1ff7 \u03bc\u03bf\u03c5 ...... \u1f00\u03ba\u03bf\u03cd\u03c9\u03bd \u03c3\u03bf\u03c5 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd ... \u03c7\u03b1\u03c1\u1f70\u03bd \u03b3\u1f70\u03c1 \u03c0\u03bf\u03bb\u03bb\u1f74\u03bd\n\u1f14\u03c3\u03c7\u03bf\u03bd \u03ba.\u03c4.\u03bb. For \u03c7\u03b1\u03c1\u03ac\u03bd the received text (Steph. but not Elz.) reads\n\u03c7\u03ac\u03c1\u03b9\u03bd, which is taken to mean 'thankfulness' (1 Tim. i. 12, 2 Tim. i.\n3); but this reading is absolutely condemned by the paucity of ancient\nauthority.\n\n\u03c4\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1] '_the heart, the spirits_'. On \u03c4\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1, the nobler\nviscera, regarded as the seat of the emotions, see the note on Phil. i.\n8. Here the prominent idea is that of terror, grief, despondency, etc.\n\n\u1f00\u03bd\u03b1\u03c0\u03ad\u03c0\u03b1\u03c5\u03c4\u03b1\u03b9] '_have been relieved, refreshed_', comp. ver. 20. The\ncompound \u1f00\u03bd\u03b1\u03c0\u03ac\u03c5\u03b5\u03c3\u03b8\u03b1\u03b9 expresses a temporary relief, as the simple\n\u03c0\u03b1\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9 expresses a final cessation: Plut. _Vit. Lucull._ 5 \u03c0\u03bf\u03bb\u03bb\u1ff6\u03bd\n\u03b1\u1f56\u03b8\u03b9\u03c2 \u1f00\u03bd\u03b1\u03ba\u03b9\u03bd\u03bf\u03cd\u03bd\u03c4\u03c9\u03bd \u03c4\u1f78\u03bd \u039c\u03b9\u03b8\u03c1\u03b9\u03b4\u03b1\u03c4\u03b9\u03ba\u1f78\u03bd \u03c0\u03cc\u03bb\u03b5\u03bc\u03bf\u03bd \u1f14\u03c6\u03b7 \u039c\u03ac\u03c1\u03ba\u03bf\u03c2\n\n------------------------------------------------------------------------\n\n8, 9]\n\n^8 \u0394\u03b9\u1f78 \u03c0\u03bf\u03bb\u03bb\u1f74\u03bd \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u03b1\u03bd \u1f14\u03c7\u03c9\u03bd \u1f10\u03c0\u03b9\u03c4\u03ac\u03c3\u03c3\u03b5\u03b9\u03bd \u03c3\u03bf\u03b9 \u03c4\u1f78 \u1f00\u03bd\u1fc6\u03ba\u03bf\u03bd, ^9\n\u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u03b1\u03b3\u03ac\u03c0\u03b7\u03bd \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u1ff6, \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f62\u03bd \u1f61\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72\n\u03ba\u03b1\u1f76 \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2\n\n                         9. \u03bd\u1fe6\u03bd \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2.\n\n\n\u03b1\u1f50\u03c4\u1f78\u03bd _\u03bf\u1f50 \u03c0\u03b5\u03c0\u03b1\u1fe6\u03c3\u03b8\u03b1\u03b9_ \u1f00\u03bb\u03bb' _\u1f00\u03bd\u03b1\u03c0\u03b5\u03c0\u03b1\u1fe6\u03c3\u03b8\u03b1\u03b9_. Thus it implies\n'relaxation, refreshment,' as a preparation for the renewal of labour or\nsuffering. It is an Ignatian as well as a Pauline word; _Ephes._ 2,\n_Smyrn._ 9, 10, 12, _Trall._ 12, _Magn._ 15, _Rom._ 10.\n\n\u1f00\u03b4\u03b5\u03bb\u03c6\u03ad] For the appeal suggested by the emphatic position of the word,\ncomp. Gal. vi. 18. See also the note on ver. 20 below.\n\n8\u201317. 'Encouraged by these tidings of thy loving spirit, I prefer to\nentreat, where I might command. My office gives me authority to dictate\nthy duty in plain language, but love bids me plead as a suitor. Have I\nnot indeed a right to command\u2014I Paul whom Christ Jesus long ago\ncommissioned as His ambassador, and whom now He has exalted to the rank\nof His prisoner? But I entreat thee. I have a favour to ask for a son of\nmy own\u2014one doubly dear to me, because I became his father amidst the\nsorrows of my bonds. I speak of Onesimus, who in times past was found\nwholly untrue to his name, who was then far from useful to thee, but now\nis useful to thee\u2014yea, and to myself also. Him I send back to thee, and\nI entreat thee to take him into thy favour, for in giving him I am\ngiving my own heart. Indeed I would gladly have detained him with me,\nthat he might minister to me on thy behalf, in these bonds with which\nthe Gospel has invested me. But I had scruples. I did not wish to do\nanything without thy direct consent; for then it might have seemed\n(though it were only seeming) as if thy kindly offices had been rendered\nby compulsion and not of free will. So I have sent him back. Indeed it\nmay have been God's providential design, that he was parted from thee\nfor a season, only that thou mightest regain him for ever; that he left\nthee as a slave, only that he might return to thee a beloved brother.\nThis indeed he is to me most of all; and, if to me, must he not be so\nmuch more to thee, both in worldly things and in spiritual? If therefore\nthou regardest me as a friend and companion, take him to thee, as if he\nwere myself'.\n\n8. \u0394\u03b9\u03cc] i.e. 'Seeing that I have these proofs of thy love, I prefer to\nentreat, where I might command'.\n\n\u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u03b1\u03bd] '_confidence_', literally '_freedom_' or '_privilege of\nspeech_'; see the notes on Col. ii. 15, Ephes. iii. 12. It was his\nApostolic authority which gave him this right to command in plain\nlanguage. Hence the addition \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7.\n\n\u03c4\u03bf \u1f00\u03bd\u1fc6\u03ba\u03bf\u03bd] '_what is fitting_': see the note on Col. iii. 18.\n\n9. \u03b4\u03b9\u1f70 \u03c4\u1f74\u03bd \u1f00\u03b3\u03ac\u03c0\u03b7\u03bd] '_for love's sake_', i.e. 'having respect to the\nclaims of love'. It is not Philemon's love (vv. 5, 7,) nor St Paul's own\nlove, but love absolutely, love regarded as a principle which demands a\ndeferential respect.\n\n\u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f62\u03bd \u03ba.\u03c4.\u03bb.] '_being such an one as Paul an ambassador, and now\nalso a prisoner, of Christ Jesus_'. Several questions of more or less\ndifficulty arise on these words. (1) Is \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f64\u03bd to be connected with\nor separated from \u1f61\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03ba.\u03c4.\u03bb.? If separated, \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f64\u03bd will mean\n'though as an Apostle I am armed with such authority', and \u1f61\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2\n\u03ba.\u03c4.\u03bb. will describe his condescension to entreaty, 'yet as simply Paul,\netc.' But the other construction is much more probable for the following\nreasons., (_a_) \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f64\u03bd so used, implying, as it would, something of\na _personal_ boast, seems unlike St Paul's usual mode of speaking.\nSeveral interpreters indeed, taking \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f64\u03bd separately, refer it to\nver. 8, 'seeing that this is my disposition', i.e. 'seeing that I desire\nto entreat'; but \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 suggests more than an accidental impulse.\n(_b_) As \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 and \u1f61\u03c2 are correlative words, it is more natural to\nconnect them together; comp. Plato _Symp._ 181 E \u03c0\u03c1\u03bf\u03c3\u03b1\u03bd\u03b1\u03b3\u03ba\u03ac\u03b6\u03b5\u03b9\u03bd \u03c4\u1f78\n\u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03bd \u1f65\u03c3\u03c0\u03b5\u03c1 \u03ba\u03b1\u1f76 \u03ba.\u03c4.\u03bb., Alexis (Meineke _Fragm. Com._ III. p. 399)\n\u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf \u03c4\u1f78 \u03b6\u1fc6\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f41\u03b9 \u03ba\u03cd\u03b2\u03bf\u03b9. Such passages are an answer to the\nobjection that \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 would require some stronger word than \u1f61\u03c2, such\nas \u03bf\u1f37\u03bf\u03c2, \u1f45\u03c2, or \u1f65\u03c3\u03c4\u03b5. Even after such expressions as \u1f41 \u1f00\u03c5\u03c4\u03cc\u03c2, \u03c4\u1f78 \u1f00\u03c5\u03c4\u03cc,\ninstances occur of \u1f65\u03c2 (\u1f65\u03c3\u03c0\u03b5\u03c1): see Lobeck _Phryn._ p. 427, Stallbaum on\nPlat. _Ph\u00e6d._ 86 A. Indeed it may be questioned whether any word but \u1f61\u03c2\nwould give exactly St Paul's meaning here. (_c_) All the Greek\ncommentators without a single exception connect the words \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f62\u03bd \u1f65\u03c2\n\u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 together. (2) Assuming that the words \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f62\u03bd \u1f61\u03c2 \u03ba.\u03c4.\u03bb. are\ntaken together, should they be connected with the preceding or the\nfollowing sentence? On the whole the passage is more forcible, if they\nare linked to the preceding words. In this case the resumptive \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u1ff6\n(ver. 10) begins a new sentence, which introduces a fresh subject. The\nApostle has before described the character of his appeal; he now speaks\nof its object. (3) In either connexion, what is the point of the words\n\u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f62\u03bd \u1f61\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u03ba.\u03c4.\u03bb.? Do they lay down the grounds of his\n_entreaty_, or do they enforce his right to _command_? If the view of\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 adopted below be correct, the latter must be the true\ninterpretation; but even though \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 be taken in its ordinary\nsense, this will still remain the more probable alternative; for, while\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 and \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2 would suit either entreaty or command, the\naddition \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 suggests an appeal to authority.\n\n\u1f61\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2] The mention of his personal name involves an assertion of\nauthority, as in Ephes. iii. 1; comp. Gal. v. 2, with the note there.\nTheodoret writes, \u1f41 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03bd \u1f00\u03ba\u03bf\u03cd\u03c3\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u03bf\u1f30\u03ba\u03bf\u03c5\u03bc\u03ad\u03bd\u03b7\u03c2 \u1f00\u03ba\u03bf\u03cd\u03b5\u03b9 \u03c4\u1f78\u03bd \u03ba\u03ae\u03c1\u03c5\u03ba\u03b1, \u03b3\u1fc6\u03c2\n\u03ba\u03b1\u1f76 \u03b8\u03b1\u03bb\u03ac\u03c4\u03c4\u03b7\u03c2 \u03c4\u1f78\u03bd \u03b3\u03b5\u03c9\u03c1\u03b3\u03cc\u03bd, \u03c4\u1fc6\u03c2 \u1f10\u03ba\u03bb\u03bf\u03b3\u1fc6\u03c2 \u03c4\u1f78 \u03c3\u03ba\u03b5\u1fe6\u03bf\u03c2, \u03ba.\u03c4.\u03bb.\n\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2] Comparing a passage in the contemporary epistle, Ephes. vi.\n20 \u1f51\u03c0\u1f72\u03c1 \u1f41\u1fe6 \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03cd\u03c9 \u1f10\u03bd \u1f01\u03bb\u03cd\u03c3\u03b5\u03b9, it had occurred to me that we should read\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03ae\u03c2 here, before I was aware that this conjecture had been\nanticipated by others, e.g. by Bentley (_Crit. Sacr._ p. 93) and by\nBenson (_Paraphrase etc. on Six Epistles of St Paul_ p. 357). It has\nsince been suggested independently in Linwood's _Observ. qu\u00e6d. in\nnonnulla N. T. loca_ 1865, and probably others have entertained the same\nthought. Still believing that St Paul here speaks of himself as an\n'ambassador', I now question whether any change is necessary. There is\nreason for thinking that in the common dialect \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 may have been\nwritten indifferently for \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03ae\u03c2 in St Paul's time; and if so, the\nform here may be due, not to some comparatively late scribe, but to the\noriginal autograph itself or to an immediate transcript. In 1 Macc. xiv.\n21 the Sinaitic MS has \u03bf\u03b9 \u03c0\u03c1\u03b5\u03c3\u03b2\u03c5\u03c4\u03b5\u03c1\u03bf\u03b9 (a corruption of \u03bf\u03b9 \u03c0\u03c1\u03b5\u03c3\u03b2\u03c5\u03c4\u03b1\u03b9 \u03bf\u03b9,\nfor the common reading is \u03bf\u1f31 \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u1f70\u03b9 \u03bf\u1f31); in xiv. 22 it reads\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03c5\u03c4\u03b1\u03b9 \u0399\u03bf\u03c5\u03b4\u03b1\u03b9\u03c9\u03bd; but in xiii. 21 \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03b1\u03c2: though in all passages\nalike the meaning is 'ambassadors'. Again the Alexandrian MS has\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03c5\u03c4\u03b1\u03c2 in xiii. 21, but \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03b1\u03b9 in xiv. 22, and \u03bf\u03b9 \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03b5 \u03bf\u03b9\n(i.e. \u03bf\u1f31 \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u1f70\u03b9 \u03bf\u1f31) in xiv. 21. In 2 Macc. xi. 34 this same MS has\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03c5\u03c4\u03b5, and the reading of the common texts of the LXX (even\nTischendorf and Fritzsche) there is \u03c0\u03c1\u03b5\u03c3\u03b2\u1fe6\u03c4\u03b1\u03b9. Grimm treats it as\nmeaning 'ambassadors', without even noticing the form. Other MSS are\nalso mentioned in Holmes and Parsons which have the form \u03c0\u03c1\u03b5\u03c3\u03b2\u03c5\u03c4\u03b7\u03c2 in 1\nMacc. xiii. 21. In 2 Chron. xxxii. 31 again the word for 'ambassador' is\nwritten thus in the _Vatican_ MS, though the \u03b5 is added above the line;\nand here too several MSS in Holmes and Parsons agree in reading\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b1\u03b9\u03c2. Thus it is plain that, in the age of our earliest extant MSS\nat all events, the scribes used both forms indifferently in this sense.\nSo also Eusebius on Isaiah xviii. 2 writes \u1f41 \u03b4\u03b5 \u1f08\u03ba\u03cd\u03bb\u03b1\u03c2 _\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b1\u03c2_\n\u1f10\u03be\u03ad\u03b4\u03c9\u03ba\u03b5\u03bd \u03b5\u1f30\u03c0\u03ce\u03bd, (\u039f \u1f00\u03c0\u03bf\u03c3\u03c4\u03ad\u03bb\u03bb\u03c9\u03bd \u1f10\u03bd \u03b8\u03b1\u03bb\u03ac\u03c3\u03c3\u1fc3 \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b1\u03c2. Again in Ignat.\n_Smyrn._ 11 \u03b8\u03b5\u03bf\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 is the form in all the MSS of either recension,\nthough the meaning is plainly 'an ambassador of God.' So too in _Clem.\nHom._ Ep. Clem. 6 the MSS read \u1f41 \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1\u03c2 \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2, which even\nSchwegler and Dressel tacitly retain. See also Appian _Samn._ 7, where\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03bf\u1fe6 is due to the later editors, and _Acta Thom\u00e6_ \u00a7 10, where\nthere is a v. l. \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 in at least one MS. And probably examples of\nthis substitution might be largely multiplied. The main reason for\nadopting this reading is the parallel passage, which suggests it very\nstrongly. The difficulty which many find in St Paul's describing himself\nas an old man is not serious. On any showing he must have been verging\non sixty at this time, and may have been some years older. A life of\nunintermittent toil and suffering, such as he had lived, would bring a\npremature decay; and looking back on a long eventful life, he would\nnaturally so think and speak of himself. Thus Roger Bacon (_Opus Majus_\nI 10, p. 15, ed. Jebb; _Opus Tertium_ p. 63, ed. Brewer) writes 'me\nsenem', 'nos senes', in 1267, though he appears to have been not more\nthan fifty-two or fifty-three at the time and lived at least a quarter\nof a century after (see E. Charles _Roger Bacon, Sa Vie etc._ pp. 4 sq.,\n40). So too Scott in his fifty-fifth year speaks of himself as 'an old\ngrey man' and 'aged' (Lockhart's _Life_ VIII. pp. 327, 357). It is more\ndifficult to understand how St Paul should make his age a ground of\nappeal to Philemon who, if Archippus was his son, cannot have been much\nyounger than himself. The commentator Hilary says that the Apostle\nappeals to his friend 'quasi co\u00e6vum \u00e6tatis', but this idea is foreign to\nthe context. The comment of Theophylact is, \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f64\u03bd, \u03c6\u03b7\u03c3\u03b9,\n_\u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03ae\u03c2_, \u03ba\u03b1\u1f76 \u1f45\u03c5\u03c4\u03c9\u03c2 \u1f04\u03be\u03b9\u03bf\u03c2 \u1f00\u03ba\u03bf\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9, \u1f61\u03c2 \u03b5\u1f30\u03ba\u1f78\u03c2 \u03a0\u03b1\u1fe6\u03bb\u03bf\u03bd \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03bd,\n\u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03b1\u03bb\u03b9\u03ba\u03bf\u1fe6 \u1f00\u03be\u03af\u03c9\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u1f76 _\u03c4\u03bf\u1fe6 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5_ \u03c4\u1f78\n\u03b1\u1f30\u03b4\u03ad\u03c3\u03b9\u03bc\u03bf\u03bd \u1f14\u03c7\u03bf\u03bd\u03c4\u03b1 \u03ba.\u03c4.\u03bb. Does he mean to include both meanings in\n\u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2? Or is he accidentally borrowing the term 'ambassador' from\nsome earlier commentator without seeing its bearing?\n\n\u03ba\u03b1\u1f76 \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2] Another title to respect. The mention of his bonds might\nsuggest either an appeal for commiseration or a claim of authority: see\nthe note on ver. 13. Here the addition of \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 invests it with\nthe character of an official title, and so gives prominence to the\nlatter idea. To his old office of 'ambassador' Christ has added the new\ntitle of 'prisoner'. The genitive \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6 belongs to \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2 as\nwell as to \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2, and in both cases describes the person who confers\nthe office or rank.\n\n------------------------------------------------------------------------\n\n10]\n\n\u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u1f38\u03b7\u03c3\u03bf\u1fe6. ^{10}\u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u1ff6 \u03c3\u03b5 \u03c0\u03b5\u03c1\u1f76 \u03c4\u03bf\u1fe6 \u1f10\u03bc\u03bf\u1fe6 \u03c4\u03ad\u03ba\u03bd\u03bf\u03bd,\n\n\n10. \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u1ff6 \u03c3\u03b5 \u03ba.\u03c4.\u03bb.] St Chrysostom remarks on the Apostle's\nwithholding the name, until he has favourably disposed Philemon both to\nthe request and to the object of it; \u03c4\u03bf\u03c3\u03bf\u03cd\u03c4\u03bf\u03b9\u03c2 \u03b4\u1f72 \u03c0\u03c1\u03bf\u03bb\u03ad\u03b1\u03bd\u03b1\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c4\u1f74\u03bd\n\u03c8\u03c5\u03c7\u03ae\u03bd, \u03bf\u1f50\u03b4\u1f72 \u03b5\u1f50\u03b8\u03ad\u03c9\u03c2 \u1f10\u03bd\u03ad\u03b2\u03b1\u03bb\u03b5 \u03c4\u1f78 \u1f44\u03bd\u03bf\u03bc\u03b1, \u1f00\u03bb\u03bb\u1f70 \u03c4\u03bf\u03c3\u03b1\u03cd\u03c4\u03b7\u03bd \u03c0\u03bf\u03b9\u03b7\u03c3\u03ac\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b1\u1f34\u03c4\u03b7\u03c3\u03b9\u03bd\n\u1f00\u03bd\u03b1\u03b2\u03ac\u03bb\u03bb\u03b5\u03c4\u03b1\u03b9 \u03ba.\u03c4.\u03bb. The whole passage deserves to be read.\n\n------------------------------------------------------------------------\n\n11]\n\n\u1f43\u03bd [\u1f10\u03b3\u1f7c] \u1f10\u03b3\u03ad\u03bd\u03bd\u03b7\u03c3\u03b1 \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2, \u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03bd, ^{11}\u03c4\u03cc\u03bd \u03c0\u03bf\u03c4\u03ad\n\n\n\u1f43\u03bd \u1f10\u03b3\u03ad\u03bd\u03bd\u03b7\u03c3\u03b1 \u03ba.\u03c4.\u03bb.] So too 1 Cor. iv. 15. In Gal. iv. 19 he speaks of\nhimself as suffering a mother's pangs for his children in the faith.\nComp. Phil. _Leg. ad Cai._ 8 (II. p. 554) \u1f10\u03bc\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9 \u03c4\u03bf\u1fe6 \u039c\u03ac\u03ba\u03c1\u03c9\u03bd\u03bf\u03c2 \u1f14\u03c1\u03b3\u03bf\u03bd\n\u0393\u03b1\u0390\u03bf\u03c2\u00b7 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03b1\u1f50\u03c4\u1f78\u03bd \u1f22 \u03bf\u1f50\u03c7 \u1f27\u03c4\u03c4\u03bf\u03bd \u03c4\u1ff6\u03bd \u03b3\u03bf\u03bd\u03ad\u03c9\u03bd \u03b3\u03b5\u03b3\u03ad\u03bd\u03bd\u03b7\u03ba\u03b1.\n\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2] He was doubly dear to the Apostle, as being the child\nof his sorrows.\n\n\u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03bd] for \u1f48\u03bd\u03b7\u03c3\u03af\u03bc\u03bf\u03c5 by attraction, as e.g. Mark vi. 16 \u1f43\u03bd \u1f10\u03b3\u1f7c\n\u1f00\u03c0\u03b5\u03ba\u03b5\u03c6\u03ac\u03bb\u03b9\u03c3\u03b1 \u1f38\u03ce\u03b1\u03bd\u03bd\u03b7\u03bd, \u03bf\u1f57\u03c4\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd. Henceforward he will be true to his\nname, no longer \u1f00\u03bd\u03cc\u03bd\u03b7\u03c4\u03bf\u03c2, but \u1f40\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03c2: comp. Ruth i. 20 'Call me not\nNaomi (pleasant) but call me Mara (bitter) etc.' The word \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c2 is a\nsynonyme for \u1f00\u03bd\u03cc\u03bd\u03b7\u03c4\u03bf\u03c2, Demosth. _Phil._ iii. \u00a7 40 (p. 121) \u1f05\u03c0\u03b1\u03bd\u03c4\u03b1 \u03c4\u03b1\u1fe6\u03c4\u03b1\n_\u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03b1_ \u1f04\u03c0\u03c1\u03b1\u03ba\u03c4\u03b1 _\u1f00\u03bd\u03cc\u03bd\u03b7\u03c4\u03b1_ \u03ba.\u03c4.\u03bb.: comp. Pseudophocyl. 37\n(34) \u03c7\u03c1\u03b7\u03c3\u03c4\u1f78\u03c2 \u1f40\u03bd\u03ae\u03c3\u03b9\u03bc\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9, \u03c6\u03af\u03bb\u03bf\u03c2 \u03b4' \u1f00\u03b4\u03b9\u03ba\u1ff6\u03bd \u1f00\u03bd\u03cc\u03bd\u03b7\u03c4\u03bf\u03c2. The significance\nof names was a matter of special importance among the ancients. Hence\nthey were careful in the inauguration of any great work that only those\nwho had _bona nomina_, _prospera nomina_, _fausta nomina_, should take\npart: Cic. _de <DW37>._ i. 45, Plin. _N.H._ xxviii. 2. 5, Tac. _Hist._ iv.\n53. On the value attached to names by the ancients, and more especially\nby the Hebrews, see Farrar _Chapters on Language_ p. 267 sq., where a\nlarge number of instances are collected. Here however there is nothing\nmore than an affectionate play on a name, such as might occur to any one\nat any time: comp. Euseb. _H.E._ v. 24 \u1f41 )\u0395\u03b9\u03c1\u03b7\u03bd\u03b1\u1fd6\u03bf\u03c2 \u03c6\u03b5\u03c1\u03ce\u03bd\u03c5\u03bc\u03cc\u03c2 \u03c4\u03b9\u03c2 \u1f62\u03bd \u03c4\u1fc7\n\u03c0\u03c1\u03bf\u03c3\u03b7\u03b3\u03bf\u03c1\u03af\u1fb3, \u03b1\u1f50\u03c4\u1ff7 \u03c4\u03b5 \u03c4\u1ff7 \u03c4\u03c1\u03cc\u03c0\u1ff3 \u03b5\u1f30\u03c1\u03b7\u03bd\u03cc\u03c0\u03bf\u03b9\u03bf\u03c2.\n\n------------------------------------------------------------------------\n\n12]\n\n\u03c3\u03bf\u03b9 \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03bd, \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 [\u03ba\u03b1\u1f76] \u03c3\u03bf\u1f76 \u03ba\u03b1\u1f76 \u1f11\u03bc\u03bf\u1f76 \u03b5\u1f54\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03bd\u00b7 \u1f43\u03bd \u1f00\u03bd\u03ad\u03c0\u03b5\u03bc\u03c8\u03ac \u03c3\u03bf\u03b9.\n^{12}\u03b1\u1f50\u03c4\u03cc\u03bd, \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9\u03bd \u03c4\u1f70 \u1f10\u03bc\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1,\n\n\n11. \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03bd, \u03b5\u1f54\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03bd] Comp. Plat. _Resp._ iii. p. 411 A \u03c7\u03c1\u03ae\u03c3\u03b9\u03bc\u03bf\u03bd \u1f10\u03be\n\u1f00\u03c7\u03c1\u03ae\u03c3\u03c4\u03bf\u03c5 ... \u1f10\u03c0\u03bf\u03af\u03b7\u03c3\u03b5\u03bd. Of these words, \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c2 is found only here,\n\u03b5\u1f54\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c2 occurs also 2 Tim. ii. 21, iv. 11, in the New Testament. Both\nappear in the LXX. In Matt. xxv. 30 a slave is described as \u1f00\u03c7\u03c1\u03b5\u1fd6\u03bf\u03c2. For\nthe mode of expression comp. Ephes. v. 15 \u03bc\u1f74 \u1f61\u03c2 \u1f04\u03c3\u03bf\u03c6\u03bf\u03b9 \u1f00\u03bb\u03bb' \u1f61\u03c2 \u03c3\u03cc\u03c6\u03bf\u03b9.\nSome have discovered in these words a reference to \u03c7\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2, as commonly\npronounced \u03c7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c2; comp. Theoph. _ad Autol._ i. 12 \u03c4\u1f78 \u03c7\u03c1\u03b9\u03c3\u03c4\u1f78\u03bd \u1f21\u03b4\u1f7a \u03ba\u03b1\u1f76\n\u03b5\u1f54\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03bd \u03ba.\u03c4.\u03bb. and see _Philippians_ p. 16 note. Any such allusion\nhowever, even if it should not involve an anachronism, is far too\nrecondite to be probable here. The play on words is exhausted in the\nreference to \u1f48\u03bd\u03ae\u03c3\u03b9\u03bc\u03bf\u03c2.\n\n\u03ba\u03b1\u1f76 \u1f10\u03bc\u03bf\u03af] An after-thought; comp. Phil. ii. 27 \u1f20\u03bb\u03ad\u03b7\u03c3\u03b5\u03bd \u03b1\u1f50\u03c4\u03cc\u03bd, \u03bf\u1f50\u03ba \u03b1\u1f50\u03c4\u1f78\u03bd\n\u03b4\u1f72 \u03bc\u03cc\u03bd\u03bf\u03bd \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u1f76 \u1f10\u03bc\u03ad. This accounts for the exceptional order, where\naccording to common Greek usage the first person would naturally precede\nthe second.\n\n\u1f00\u03bd\u03ad\u03c0\u03b5\u03bc\u03c8\u03b1] '_I send back_', the epistolary aorist used for the present:\nsee the notes on Phil. ii. 25, 28. So too \u1f14\u03b3\u03c1\u03b1\u03c8\u03b1, ver. 19, 21 (see the\nnote). It is clear both from the context here, and from Col. iv. 7\u20139,\nthat Onesimus accompanied the letter.\n\n12. \u03b1\u1f50\u03c4\u1f78\u03bd \u03ba.\u03c4.\u03bb.] The reading of the received text is _\u03c3\u1f7a \u03b4\u1f72_\n\u03b1\u1f50\u03c4\u03cc\u03bd, \u03c4\u03bf\u03c5\u03c4\u03ad\u03c3\u03c4\u03b9 \u03c4\u1f70 \u1f10\u03bc\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1, _\u03c0\u03c1\u03bf\u03c3\u03bb\u03b1\u03b2\u03bf\u1fe6_. The words thus\nsupplied doubtless give the right construction, but must be rejected as\ndeficient in authority. The accusative is suspended; the sentence\nchanges its form and loses itself in a number of dependent clauses; and\nthe main point is not resumed till ver. 17 \u03c0\u03c1\u03bf\u03c3\u03bb\u03b1\u03b2\u03bf\u1fe6 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f61\u03c2 \u1f10\u03bc\u03ad, the\ngrammar having been meanwhile dislocated. For the emphatic position of\n\u03b1\u1f50\u03c4\u03cc\u03bd comp. John ix. 21, 23, Ephes. i. 22.\n\n\u03c4\u1f70 \u1f10\u03bc\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1] '_my very heart_', a mode of speech common in all\nlanguages. For the meaning of \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 see the note on Phil. i. 8. Comp.\n_Test. Patr._ Zab. 8, Neph. 4, in both which passages Christ is called\n\u03c4\u1f78 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03bf\u03bd of God, and in the first it is said \u1f14\u03c7\u03b5\u03c4\u03b5 \u03b5\u1f50\u03c3\u03c0\u03bb\u03b1\u03b3\u03c7\u03bd\u03af\u03b1\u03bd ...\n\u1f35\u03bd\u03b1 \u03ba\u03b1\u1f76 \u1f41 \u039a\u03cd\u03c1\u03b9\u03bf\u03c2 \u03b5\u1f30\u03c2 \u1f51\u03bc\u1fb6\u03c2 \u03c3\u03c0\u03bb\u03b1\u03b3\u03c7\u03bd\u03b9\u03c3\u03b8\u1f72\u03b9\u03c2 \u1f10\u03bb\u03ad\u03b7\u03c3\u1fc3 \u1f51\u03bc\u1fb6\u03c2\u00b7 \u1f45\u03c4\u03b9 \u03ba\u03b1\u03af\u03b3\u03b5 \u1f10\u03c0'\n\u1f10\u03c3\u03c7\u03ac\u03c4\u03c9\u03bd \u1f21\u03bc\u03b5\u03c1\u1ff6\u03bd \u1f41 \u0398\u03b5\u1f78\u03c2 \u1f00\u03c0\u03bf\u03c3\u03c4\u03ad\u03bb\u03bb\u03b5\u03b9 \u03c4\u1f78 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f10\u03c0\u1f76 \u03c4\u1fc6\u03c2 \u03b3\u1fc6\u03c2 \u03ba.\u03c4.\u03bb.\nOtherwise \u03c4\u1f70 \u1f10\u03bc\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 has been interpreted '_my son_' (comp. ver. 10\n\u1f43\u03bd \u1f10\u03b3\u03ad\u03bd\u03bd\u03b7\u03c3\u03b1 \u03ba.\u03c4.\u03bb.), and it is so rendered here in the Peshito. For this\nsense of \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 comp. Artemid. _Oneir._ i. 44 \u03bf\u1f31 \u03c0\u03b1\u1fd6\u03b4\u03b5\u03c2 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1\n\u03bb\u03ad\u03b3\u03bf\u03bd\u03c4\u03b1\u03b9, _ib._ v. 57 \u03c4\u1f70 \u03b4\u1f72 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 [\u1f10\u03c3\u03ae\u03bc\u03b1\u03b9\u03bd\u03b5] \u03c4\u1f78\u03bd \u03c0\u03b1\u1fd6\u03b4\u03b1, \u03bf\u1f55\u03c4\u03c9 \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76\n\u03c4\u1f78\u03bd \u03c0\u03b1\u1fd6\u03b4\u03b1 \u03ba\u03b1\u03bb\u03b5\u1fd6\u03bd \u1f14\u03b8\u03bf\u03c2 \u1f10\u03c3\u03c4\u03b9. With this meaning it is used not less of the\nfather than of the mother; e.g. Philo _de Joseph._ 5 (II. p. 45) \u03b8\u03b7\u03c1\u03c3\u1f76\u03bd\n\u03b5\u1f50\u03c9\u03c7\u03af\u03b1 \u03ba\u03b1\u1f76 \u03b8\u03bf\u03af\u03bd\u03b7 \u03b3\u03ad\u03b3\u03bf\u03bd\u03b1\u03c2 \u03b3\u03b5\u03c5\u03c3\u03b1\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2 ... \u03c4\u1ff6\u03bd \u1f10\u03bc\u1ff6\u03bd \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03c9\u03bd, Basil.\n_Op._ III. p. 501 \u1f41 \u03bc\u1f72\u03bd \u03c0\u03c1\u03bf\u03c4\u03b5\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c4\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 \u03c4\u03b9\u03bc\u1f74\u03bd \u03c4\u1ff6\u03bd \u03c4\u03c1\u03bf\u03c6\u1ff6\u03bd. The\nLatin _viscera_ occurs still more frequently in this sense, as the\npassages quoted in Wetstein and Suicer show. For this latter\ninterpretation there is much to be said. But it adds nothing to the\nprevious \u1f43\u03bd \u1f10\u03b3\u03ad\u03bd\u03bd\u03b7\u03c3\u03b1 \u03ba.\u03c4.\u03bb., and (what is a more serious objection) it\nis wholly unsupported by St Paul's usage elsewhere, which connects\n\u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 with a different class of ideas: see e.g. vv. 7, 20.\n\n------------------------------------------------------------------------\n\n13, 14]\n\n^{13}\u1f43\u03bd \u1f10\u03b3\u1f7c \u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03c0\u03c1\u1f78\u03c2 \u1f10\u03bc\u03b1\u03c5\u03c4\u1f78\u03bd \u03ba\u03b1\u03c4\u03ad\u03b5\u03c7\u03b5\u03b9\u03bd, \u1f35\u03bd\u03b1 \u1f51\u03c0\u1f72\u03c1 \u03c3\u03bf\u1fe6 \u03bc\u03bf\u03b9 \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u1fc7 \u1f10\u03bd\n\u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5\u00b7 ^{14}\u03c7\u03c9\u03c1\u1f76\u03c2\n\n\n13. \u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd] '_I was of a mind_', distinguished from \u1f20\u03b8\u03ad\u03bb\u03b7\u03c3\u03b1, which\nfollows, in two respects; (1) While \u03b2\u03bf\u03cd\u03bb\u03b5\u03c3\u03b8\u03b1\u03b9 involves the idea of\n'purpose, deliberation, desire, mind', \u03b8\u03ad\u03bb\u03b5\u03b9\u03bd denotes simply 'will';\nEpictet. i. 12. 13 \u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b1\u03b9 \u03b3\u03c1\u03ac\u03c6\u03b5\u03b9\u03bd, \u1f61\u03c2 \u03b8\u03ad\u03bb\u03c9, \u03c4\u1f78 \u0394\u03af\u03c9\u03bd\u03bf\u03c2 \u1f44\u03bd\u03bf\u03bc\u03b1; \u03bf\u1f54 \u1f00\u03bb\u03bb\u1f70\n\u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03bf\u03bc\u03b1\u03b9 \u03b8\u03ad\u03bb\u03b5\u03b9\u03bd \u1f61\u03c2 \u03b4\u03b5\u1fd6 \u03b3\u03c1\u03ac\u03c6\u03b5\u03c3\u03b8\u03b1\u03b9, iii. 24, 54 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u03b8\u03ad\u03bb\u03b5 \u1f41\u03c1\u1fb7\u03bd, \u03ba\u03b1\u1f76 \u1f43\u03bd\n\u03b2\u03bf\u03cd\u03bb\u03b5\u03b9 \u1f44\u03c8\u03b5\u03b9. (2) The change of tenses is significant. The imperfect\nimplies a tentative, inchoate process; while the aorist describes a\ndefinite and complete act. The will stepped in and put an end to the\ninclinations of the mind. Indeed the imperfect of this and similar verbs\nare not infrequently used where the wish is stopped at the outset by\nsome antecedent consideration which renders it impossible, and thus\npractically it is not entertained at all: e.g. Arist. _Ran._ 866\n\u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03bc\u1f72\u03bd \u03bf\u1f50\u03ba \u1f10\u03c1\u03af\u03b6\u03b5\u03b9\u03bd \u1f10\u03bd\u03b8\u03ac\u03b4\u03b5, Antiph. _de Herod. c\u00e6d._ I (p. 129)\n\u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03bc\u1f72\u03bd ... \u03bd\u1fe6\u03bd \u03b4\u1f72 \u03ba.\u03c4.\u03bb.; Is\u00e6us _de Arist. h\u00e6r._ I (p. 79)\n\u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03bc\u1f72\u03bd ... \u03bd\u1fe6\u03bd \u03b4\u1f72 \u03bf\u1f50\u03ba \u1f10\u03be \u1f34\u03c3\u03bf\u03c5 \u03ba.\u03c4.\u03bb., \u00c6sch. _c. Ctes._ 2 (p. 53)\n\u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03bc\u1f72\u03bd \u03bf\u1f56\u03bd, \u1f66 )\u0391\u03b8\u03b7\u03bd\u03b1\u1fd6\u03bf\u03b9 ... \u1f10\u03c0\u03b5\u03b9\u03b4\u1f74 \u03b4\u1f72 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03ba.\u03c4.\u03bb., Lucian _Abd._\nI \u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03bc\u1f72\u03bd \u03bf\u1f56\u03bd \u03c4\u1f74\u03bd \u1f30\u03b1\u03c4\u03c1\u03b9\u03ba\u1f74\u03bd \u03ba.\u03c4.\u03bb. ... \u03bd\u03c5\u03bd\u1f76 \u03b4\u1f72 \u03ba.\u03c4.\u03bb.; see K\u00fchner \u00a7\n392 _b_ (II. p. 177). So Acts xxv. 22 \u1f10\u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b7\u03bd \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u1f78\u03c2 \u03c4\u03bf\u1fe6 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03c5\n\u1f00\u03ba\u03bf\u1fe6\u03c3\u03b1\u03b9, not 'I should wish' (as Winer \u00a7 xli. p. 353) but 'I could have\nwished', i.e. 'if it had not been too much to ask'. Similarly \u1f24\u03b8\u03b5\u03bb\u03bf\u03bd\nGal. iv. 20, \u03b7\u1f50\u03c7\u03cc\u03bc\u03b7\u03bd Rom. ix. 3. See _Revision of the English New\nTestament_ p. 96. So here a not improbable meaning would be not 'I was\ndesirous', but 'I could have desired'.\n\n\u03ba\u03b1\u03c4\u03ad\u03c7\u03b5\u03b9\u03bd] '_to detain_' or '_retain_', opposed to the following \u1f00\u03c0\u03ad\u03c7\u1fc3\u03c2,\nver. 15.\n\n\u1f51\u03c0\u1f72\u03c1 \u03c3\u03bf\u1fe6 \u03ba.\u03c4.\u03bb.] Comp. Phil. ii. 30 \u1f35\u03bd\u03b1 \u1f00\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03ce\u03c3\u1fc3 \u03c4\u1f78 \u1f51\u03bc\u1ff6\u03bd \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1 \u03c4\u1fc6\u03c2\n\u03c0\u03c1\u1f78\u03c2 \u03bc\u1f72 \u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u03af\u03b1\u03c2, 1 Cor. xvi. 17 \u03c4\u1f78 \u1f51\u03bc\u03ad\u03c4\u03b5\u03c1\u03bf\u03bd \u1f51\u03c3\u03c4\u03ad\u03c1\u03b7\u03bc\u03b1 \u03b1\u1f50\u03c4\u03bf\u1f76\n\u1f00\u03bd\u03b5\u03c0\u03bb\u03ae\u03c1\u03c9\u03c3\u03b1\u03bd. See the note on Col. i. 7. With a delicate tact the Apostle\nassumes that Philemon would have wished to perform these friendly\noffices in person, if it had been possible.\n\n\u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2] An indirect appeal to his compassion: see vv. 1, 9, 10.\nIn this instance however (as in ver. 9) the appeal assumes a tone of\nauthority, by reference to the occasion of his bonds. For the genitive\n\u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5, describing the origin, comp. Col. i. 23 \u03c4\u1fc6\u03c2 \u1f10\u03bb\u03c0\u03af\u03b4\u03bf\u03c2 \u03c4\u03bf\u1fe6\n\u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5. They were not shackles which self had riveted, but a chain\nwith which Christ had invested him. Thus they were as a badge of office\nor a decoration of honour. In this respect, as in others, the language\nof St Paul is echoed in the epistles of St Ignatius. Here too entreaty\nand triumph alternate; the saint's bonds are at once a ground for appeal\nand a theme of thanksgiving: _Trall._ 12 \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03b5\u1fd6 \u1f51\u03bc\u1fb6\u03c2 \u03c4\u1f70 \u03b4\u03b5\u03c3\u03bc\u03ac \u03bc\u03bf\u03c5,\n_Philad._ 7 \u03bc\u03ac\u03c1\u03c4\u03c5\u03c2 \u03b4\u1f72 \u03bc\u03bf\u03b9 \u1f10\u03bd \u1fa7 \u03b4\u03ad\u03b4\u03b5\u03bc\u03b1\u03b9, _Ephes._ 11 \u1f10\u03bd \u1fa7 (i.e. \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7\n\u1f38\u03b7\u03c3\u03bf\u1fe6) \u03c4\u1f70 \u03b4\u03b5\u03c3\u03bc\u1f70 \u03c0\u03b5\u03c1\u03b9\u03c6\u03ad\u03c1\u03c9, \u03c4\u03bf\u03c5\u03c2 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u1f7a\u03c2 \u03bc\u03b1\u03c1\u03b3\u03b1\u03c1\u03af\u03c4\u03b1\u03c2, _Smyrn._ 10\n\u1f00\u03bd\u03c4\u03af\u03c8\u03c5\u03c7\u03bf\u03bd \u1f51\u03bc\u1ff6\u03bd \u03c4\u1f78 \u03c0\u03bd\u03b5\u1fe6\u03bc\u03ac \u03bc\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03b4\u03b5\u03c3\u03bc\u03ac \u03bc\u03bf\u03c5, _Magn._ 1 \u1f10\u03bd \u03bf\u1f37\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c6\u03ad\u03c1\u03c9\n\u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2 \u1f84\u03b4\u03c9 \u03c4\u1f70\u03c2 \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1\u03c2; see also _Ephes._ 1, 3, 21, _Magn._ 12,\n_Trall._ 1, 5, 10, _Smyrn._ 4, 11, _Polyc._ 2, _Rom._ 1, 4, 5, _Philad._\n5.\n\n14. \u03c7\u03c9\u03c1\u1f76\u03c2 \u03ba.\u03c4.\u03bb.] '_without thy approval, consent_'; Polyb. ii. 21. 1,\n3, \u03c7\u03c9\u03c1\u1f76\u03c2 \u03c4\u1fc6\u03c2 \u03c3\u03c6\u03b5\u03c4\u03ad\u03c1\u03b1\u03c2 \u03b3\u03bd\u03ce\u03bc\u03b7\u03c2, \u03c7\u03c9\u03c1\u1f76\u03c2 \u03c4\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03b3\u03bd\u03ce\u03bc\u03b7\u03c2: similarly \u1f04\u03bd\u03b5\u03c5\n[\u03c4\u1fc6\u03c2] \u03b3\u03bd\u03ce\u03bc\u03b7\u03c2, e.g. Polyb. xxi. 8. 7, Ign. _Polyc._ 4.\n\n------------------------------------------------------------------------\n\n15, 16]\n\n\u03b4\u1f72 \u03c4\u1fc6\u03c2 \u03c3\u1fc6\u03c2 \u03b3\u03bd\u03ce\u03bc\u03b7\u03c2 \u03bf\u1f50\u03b4\u1f72\u03bd \u1f20\u03b8\u03ad\u03bb\u03b7\u03c3\u03b1 \u03c0\u03bf\u03b9\u1fc6\u03c3\u03b1\u03b9, \u1f35\u03bd\u03b1 \u03bc\u1f74 \u1f61\u03c2 \u03ba\u03b1\u03c4\u1f70 \u1f00\u03bd\u03ac\u03b3\u03ba\u03b7\u03bd \u03c4\u1f78\n\u1f00\u03b3\u03b1\u03b8\u03cc\u03bd \u03c3\u03bf\u03c5 \u1f96, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u03c4\u1f70 \u1f11\u03ba\u03bf\u03cd\u03c3\u03b9\u03bf\u03bd\u00b7 ^{15}\u03c4\u03ac\u03c7\u03b1 \u03b3\u1f70\u03c1 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf \u1f10\u03c7\u03c9\u03c1\u03af\u03c3\u03b8\u03b7 \u03c0\u03c1\u1f78\u03c2\n\u1f65\u03c1\u03b1\u03bd, \u1f35\u03bd\u03b1 \u03b1\u1f34\u03c9\u03bd\u03b9\u03bf\u03bd \u03b1\u1f50\u03c4\u1f78\u03bd \u1f00\u03c0\u03ad\u03c7\u1fc3\u03c2, ^{16}\u03bf\u1f50\u03ba\u03ad\u03c4\u03b9 \u1f61\u03c2 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd, \u1f00\u03bb\u03bb\u1f70 \u1f51\u03c0\u1f72\u03c1 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd,\n\n\n\u1f61\u03c2 \u03ba\u03b1\u03c4\u1f70 \u1f00\u03bd\u03ac\u03b3\u03ba\u03b7\u03bd] St Paul does not say \u03ba\u03b1\u03c4\u1f70 \u1f00\u03bd\u03ac\u03b3\u03ba\u03b7\u03bd but \u1f61\u03c2 \u03ba\u03b1\u03c4\u1f70 \u1f00\u03bd\u03ac\u03b3\u03ba\u03b7\u03bd.\nHe will not suppose that it would really be by constraint; but it must\nnot even wear _the appearance_ (\u1f61\u03c2) of being so: comp. 2 Cor. xi. 17 \u1f61\u03c2\n\u1f10\u03bd \u1f00\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3. See Plin. _Ep._ ix. 21 'Vereor ne videar non rogare sed\ncogere'; where, as here, the writer is asking his correspondent to\nforgive a domestic who has offended.\n\n\u03c4\u1f78 \u1f00\u03b3\u03b1\u03b8\u03cc\u03bd \u03c3\u03bf\u03c5] '_the benefit_ arising _from thee_', i.e. 'the good which\nI should get from the continued presence of Onesimus, and which would be\nowing to thee'.\n\n\u03ba\u03b1\u03c4\u1f70 \u1f11\u03ba\u03bf\u03cd\u03c3\u03b9\u03bf\u03bd] as in Num. xv. 3. The form \u03ba\u03b1\u03b8' \u1f11\u03ba\u03bf\u03c5\u03c3\u03af\u03b1\u03bd is perhaps more\nclassical: Thuc. viii. 27 \u03ba\u03b1\u03b8' \u1f11\u03ba\u03bf\u03c5\u03c3\u03af\u03b1\u03bd \u1f22 \u03c0\u03ac\u03bd\u03c5 \u03b3\u03b5 \u1f00\u03bd\u03ac\u03b3\u03ba\u1fc3. The word\nunderstood in the one case appears to be \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd (Porphyr. _de Abst._ i.\n9 \u03ba\u03b1\u03b8' \u1f11\u03ba\u03bf\u03cd\u03c3\u03b9\u03bf\u03bd \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd, comp. Eur. _Med._ 751 \u1f11\u03ba\u03bf\u03c5\u03c3\u03af\u1ff3 \u03c4\u03c1\u03cc\u03c0\u1ff3); in the\nother, \u03b3\u03bd\u03ce\u03bc\u03b7\u03bd (so \u1f11\u03ba\u03bf\u03c5\u03c3\u03af\u1fb3, \u1f10\u03be \u1f11\u03ba\u03bf\u03c5\u03c3\u03af\u03b1\u03c2, etc.): comp. Lobeck _Phryn._ p.\n4.\n\n15. \u03c4\u03ac\u03c7\u03b1 \u03b3\u1f70\u03c1 \u03ba.\u03c4.\u03bb.] The \u03b3\u1f70\u03c1 explains an additional motive which guided\nthe Apostle's decision: 'I did not dare to detain him, however much I\ndesired it. I might have defeated the purpose for which God in His good\nprovidence allowed him to leave thee'.\n\n\u1f10\u03c7\u03c9\u03c1\u03af\u03c3\u03b8\u03b7] 'He does not say', writes Chrysostom, '_For this cause he\nfled_, but _For this cause he was parted_: for he would appease Philemon\nby a more euphemistic phrase. And again he does not say _he parted\nhimself_, but _he was parted_: since the design was not Onesimus' own to\ndepart for this or that reason: just as Joseph also, when excusing his\nbrethren, says (Gen. xlv. 5) _God did send me hither_.'\n\n\u03c0\u03c1\u1f78\u03c2 \u1f65\u03c1\u03b1\u03bd] '_for an hour_', '_for a short season_': 2 Cor. vii. 8, Gal.\nii. 5. 'It was only a brief moment after all', the Apostle would say,\n'compared with the magnitude of the work wrought in it. He departed a\nreprobate; he returns a saved man. He departed for a few months; he\nreturns to be with you for all time and for eternity'. The sense of\n\u03b1\u1f34\u03c9\u03bd\u03b9\u03bf\u03bd must not be arbitrarily limited. Since he left, Onesimus had\nobtained eternal life, and eternal life involves eternal interchange of\nfriendship. His services to his old master were no longer barred by the\ngates of death.\n\n\u1f00\u03c0\u03ad\u03c7\u1fc3\u03c2] In this connexion \u1f00\u03c0\u03ad\u03c7\u03b5\u03b9\u03bd may bear either of two senses: (1)\n'_to have back, to have in return_': or (2) '_to have to the full, to\nhave wholly_', as in Phil. iv. 18 \u1f00\u03c0\u03ad\u03c7\u03c9 \u03c0\u03ac\u03bd\u03c4\u03b1 (see the note). In other\nwords the prominent idea in the word may be either _restitution_, or\n_completeness_. The former is the more probable sense here, as suggested\nby \u03ba\u03b1\u03c4\u03ad\u03c7\u03b5\u03b9\u03bd in verse 13 and by \u1f10\u03c7\u03c9\u03c1\u03af\u03c3\u03b8\u03b7 in this verse.\n\n16. \u1f61\u03c2 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd] St Paul does not say \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd but \u1f61\u03c2 \u03b4\u03bf\u1fe6\u03bb\u03bf\u03bd. It was a\nmatter of indifference whether he were outwardly \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 or outwardly\n\u1f10\u03bb\u03b5\u03cd\u03b8\u03b5\u03c1\u03bf\u03c2, since both are one in Christ (Col. iii. 11). But though he\nmight still remain a slave, he could no longer be _as_ a slave. A change\nhad been wrought in him, independently of his possible manumission: in\nChrist he had become a brother. It should be noticed also that the\nnegative is not \u03bc\u03b7\u03ba\u03ad\u03c4\u03b9, but \u03bf\u1f50\u03ba\u03ad\u03c4\u03b9. The negation is thus wholly\nindependent of \u1f35\u03bd\u03b1 ... \u1f00\u03c0\u03ad\u03c7\u1fc3\u03c2. It describes not the possible view of\nPhilemon, but the actual state of Onesimus. The 'no more as a slave' is\nan absolute fact, whether Philemon chooses to recognise it or not.\n\n------------------------------------------------------------------------\n\n17\u201319]\n\n\u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03bd \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03bd, \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u1f10\u03bc\u03bf\u03af, \u03c0\u03cc\u03c3\u1ff3 \u03b4\u1f72 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c3\u03bf\u1f76 \u03ba\u03b1\u1f76 \u1f10\u03bd \u03c3\u03b1\u03c1\u03ba\u1f76 \u03ba\u03b1\u1f76 \u1f10\u03bd\n\u039a\u03c5\u03c1\u03af\u1ff3. ^{17}\u03b5\u1f30 \u03bf\u1f56\u03bd \u03bc\u03b5 \u1f14\u03c7\u03b5\u03b9\u03c2 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03cc\u03bd, \u03c0\u03c1\u03bf\u03c3\u03bb\u03b1\u03b2\u03bf\u1fe6 \u03b1\u1f50\u03c4\u1f78\u03bd \u1f61\u03c2 \u1f10\u03bc\u03ad\u00b7 ^{18}\u03b5\u1f30 \u03b4\u03ad\n\u03c4\u03b9 \u1f20\u03b4\u03af\u03ba\u03b7\u03c3\u03ad\u03bd \u03c3\u03b5 \u1f22 \u1f40\u03c6\u03b5\u03af\u03bb\u03b5\u03b9, \u03c4\u03bf\u1fe6\u03c4\u03bf \u1f11\u03bc\u03bf\u1f76 \u1f10\u03bb\u03bb\u03cc\u03b3\u03b1. ^{19}\u1f10\u03b3\u1f7c \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2 \u1f14\u03b3\u03c1\u03b1\u03c8\u03b1\n\n\n\u1f00\u03b4\u03b5\u03bb\u03c6\u1f78\u03bd \u1f00\u03b3\u03b1\u03c0\u03b7\u03c4\u03cc\u03bd] \u03ba\u03b1\u1f76 \u03c4\u1ff7 \u03c7\u03c1\u03cc\u03bd\u1ff3 \u03ba\u03b5\u03ba\u03ad\u03c1\u03b4\u03b1\u03ba\u03b1\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc7 \u03c0\u03bf\u03af\u03bf\u03c4\u03b7\u03c4\u03b9, writes\nChrysostom, apostrophizing Philemon.\n\n\u03c0\u03cc\u03c3\u1ff3 \u03b4\u1f72 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03ba.\u03c4.\u03bb.] Having first said 'most of all to me', he goes a\nstep further, 'more than most of all to thee'.\n\n\u03ba\u03b1\u1f76 \u1f10\u03bd \u03c3\u03b1\u03c1\u03ba\u1f76 \u03ba.\u03c4.\u03bb.] 'In both spheres alike, in the affairs of this\nworld and in the affairs of the higher life.' In the former, as Meyer\npointedly says, Philemon had the brother for a slave; in the latter he\nhad the slave for a brother: comp. Ign. _Trall._ 12 \u03ba\u03b1\u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03bc\u03b5\n\u1f00\u03bd\u03ad\u03c0\u03b1\u03c5\u03c3\u03b1\u03bd \u03c3\u03b1\u03c1\u03ba\u03af \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03b9.\n\n17. \u1f14\u03c7\u03b5\u03b9\u03c2 \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03cc\u03bd] '_thou holdest me to be a comrade, an intimate\nfriend_'. For this use of \u1f14\u03c7\u03b5\u03b9\u03bd comp. Luke xiv. 18 \u1f14\u03c7\u03b5 \u03bc\u03b5 \u03c0\u03b1\u03c1\u1fc3\u03c4\u03b7\u03bc\u03ad\u03bd\u03bf\u03bd,\nPhil. ii. 29 \u03c4\u03bf\u1f7a\u03c2 \u03c4\u03bf\u03b9\u03bf\u03cd\u03c4\u03bf\u03c5\u03c2 \u1f10\u03bd\u03c4\u03b9\u03bc\u03bf\u1f7a\u03c2 \u1f14\u03c7\u03b5\u03c4\u03b5. Those are \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03bf\u03af, who have\ncommon interests, common feelings, common work.\n\n18\u201322. 'But if he has done thee any injury, or if he stands in thy debt,\nset it down to my account. Here is my signature\u2014_Paul_\u2014in my own\nhandwriting. Accept this as my bond. I will repay thee. For I will not\ninsist, as I might, that thou art indebted to me for much more than\nthis; that thou owest to me thine own self. Yes, dear brother, let me\nreceive from my son in the faith such a return as a father has a right\nto expect. Cheer and refresh my spirits in Christ. I have full\nconfidence in thy compliance, as I write this; for I know that thou wilt\ndo even more than I ask. At the same time also prepare to receive me on\na visit; for I hope that through your prayers I shall be set free and\ngiven to you once more'.\n\n18. \u03b5\u1f30 \u03b4\u03ad \u03c4\u03b9] The case is stated hypothetically but the words doubtless\ndescribe the actual offence of Onesimus. He had done his master some\ninjury, probably had robbed him; and he had fled to escape punishment.\nSee the introduction.\n\n\u1f22 \u1f40\u03c6\u03b5\u03af\u03bb\u03b5\u03b9] defining the offence which has been indicated in \u1f20\u03b4\u03af\u03ba\u03b7\u03c3\u03b5\u03bd.\nBut still the Apostle refrains from using the plain word \u1f14\u03ba\u03bb\u03b5\u03c8\u03b5\u03bd. He\nwould spare the penitent slave, and avoid irritating the injured master.\n\n\u1f10\u03bb\u03bb\u03cc\u03b3\u03b1] '_reckon it in_', '_set it down_'. This form must be adopted\ninstead of \u1f10\u03bb\u03bb\u03cc\u03b3\u03b5\u03b9 which stands in the received text, as the great\npreponderance of authority shows. On the other hand we have \u1f10\u03bb\u03bb\u03bf\u03b3\u03b5\u1fd6\u03c4\u03b1\u03b9\nRom. v. 13 (though with a v. l. \u1f10\u03bb\u03bb\u03bf\u03b3\u1fb6\u03c4\u03b1\u03b9), \u1f10\u03bb\u03bb\u03bf\u03b3\u03bf\u03c5\u03bc\u03ad\u03bd\u03c9\u03bd Boeckh _C. I._\nno. 1732 A, and \u1f10\u03bd\u03bb\u03bf\u03b3\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 _Edict. Diocl._ in _Corp. Inscr. Lat._ III.\np. 836. But the word is so rare in any form, that these occurrences of\n\u1f10\u03bb\u03bb\u03bf\u03b3\u03b5\u1fd6\u03bd afford no ground for excluding \u1f10\u03bb\u03bb\u03bf\u03b3\u1fb6\u03bd as impossible. The two\nforms might be employed side by side, just as we find \u1f10\u03bb\u03b5\u1fb6\u03bd and \u1f10\u03bb\u03b5\u03b5\u1fd6\u03bd,\n\u03be\u03c5\u03c1\u1fb6\u03bd and \u03be\u03c5\u03c1\u03b5\u1fd6\u03bd, \u1f10\u03c1\u03c9\u03c4\u1fb6\u03bd and \u1f10\u03c1\u03c9\u03c4\u03b5\u1fd6\u03bd (Matt. xv. 23), and the like; see\nButtmann _Ausf. Gramm._ \u00a7 112 (II. p. 53). The word \u03bb\u03bf\u03b3\u1fb6\u03bd, as used by\nLucian _Lexiph._ 15 (where it is a desiderative 'to be eager to speak',\nlike \u03c6\u03bf\u03bd\u1fb6\u03bd, \u03b8\u03b1\u03bd\u03b1\u03c4\u1fb6\u03bd, \u03c6\u03b1\u03c1\u03bc\u03b1\u03ba\u1fb6\u03bd, etc.), has nothing to do with the use of\n\u1f10\u03bb\u03bb\u03bf\u03b3\u1fb6\u03bd here.\n\n19. \u1f10\u03b3\u1f7c \u03a0\u03b1\u1fe6\u03bb\u03bf\u03c2] The introduction of his own name gives it the character\nof a formal and binding signature: comp. 1 Cor. xvi. 21, Col. iv. 18, 2\nThess. iii. 17. A signature to a deed in ancient or medi\u00e6val times would\ncommonly take this form \u1f10\u03b3\u1f7c \u1f41 \u03b4\u03b5\u1fd6\u03bd\u03b1,\u2014'_I_ so and so'; where we should\nomit the marks of the first person.\n\n\u1f14\u03b3\u03c1\u03b1\u03c8\u03b1] An epistolary or documentary aorist, as in ver. 21; so too\n\u1f00\u03bd\u03ad\u03c0\u03b5\u03bc\u03c8\u03b1 ver. 11. See the note on \u1f14\u03b3\u03c1\u03b1\u03c8\u03b1 Gal. vi. 11. The aorist is the\ntense commonly used in signatures; e.g. \u1f51\u03c0\u03ad\u03b3\u03c1\u03b1\u03c8\u03b1 to the conciliar\ndecrees.\n\nThis incidental mention of his autograph, occurring where it does, shows\nthat he wrote the whole letter with his own hand. This procedure is\nquite exceptional, just as the purport of the letter is exceptional. In\nall other cases he appears to have employed an amanuensis, only adding a\nfew words in his own handwriting at the close: see the note on Gal.\n_l.c._\n\n------------------------------------------------------------------------\n\n20]\n\n\u03c4\u1fc7 \u1f10\u03bc\u1fc7 \u03c7\u03b5\u03b9\u03c1\u03af, \u1f10\u03b3\u1f7c \u1f00\u03c0\u03bf\u03c4\u03af\u03c3\u03c9\u00b7 \u1f35\u03bd\u03b1 \u03bc\u1f74 \u03bb\u03ad\u03b3\u03c9 \u03c3\u03bf\u03b9, \u1f45\u03c4\u03b9 \u03ba\u03b1\u1f76 \u03c3\u03b5\u03b1\u03c5\u03c4\u03cc\u03bd \u03bc\u03bf\u03b9\n\u03c0\u03c1\u03bf\u03c3\u03bf\u03c6\u03ad\u03b9\u03bb\u03b5\u03b9\u03c2. ^{20}\u03bd\u03b1\u03af, \u1f00\u03b4\u03b5\u03bb\u03c6\u03ad, \u1f10\u03b3\u03ce \u03c3\u03bf\u03c5 \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3\u00b7 \u1f00\u03bd\u03ac\u03c0\u03b1\u03c5\u03c3\u03cc\u03bd \u03bc\u03bf\u03c5\n\u03c4\u1f70 \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7.\n\n\n\u1f35\u03bd\u03b1 \u03bc\u1f74 \u03bb\u03ad\u03b3\u03c9] '_not to say_', as 2 Cor. ix. 4. There is a suppressed\nthought, 'though indeed you cannot fairly claim repayment', 'though\nindeed you owe me (\u1f40\u03c6\u03b5\u03af\u03bb\u03b5\u03b9\u03c2) as much as this', on which the \u1f35\u03bd\u03b1 \u03bc\u1f74\n\u03ba.\u03c4.\u03bb. is dependent. Hence \u03c0\u03c1\u03bf\u03c3\u03bf\u03c6\u03ad\u03b9\u03bb\u03b5\u03b9\u03c2 '_owest besides_'; for this is\nthe common meaning of the word.\n\n\u03c3\u03b5\u03b1\u03c5\u03c4\u03cc\u03bd] St Paul was his spiritual father, who had begotten him in the\nfaith, and to whom therefore he owed his being; comp. Plato _Legg._ iv.\np. 717 B \u1f61\u03c2 \u03b8\u03ad\u03bc\u03b9\u03c2 \u1f40\u03c6\u03ad\u03b9\u03bb\u03bf\u03bd\u03c4\u03b1 \u1f00\u03c0\u03bf\u03c4\u03af\u03bd\u03b5\u03b9\u03bd \u03c4\u1f70 \u03c0\u03c1\u1ff6\u03c4\u03ac \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b1 \u1f40\u03c6\u03b5\u03b9\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1\n... \u03bd\u03bf\u03bc\u03af\u03b6\u03b5\u03b9\u03bd \u03b4\u1f72, \u1f03 \u03ba\u03ad\u03ba\u03c4\u1fc3\u03c4\u03b1\u03b9 \u03ba\u03b1\u1f76 \u1f14\u03c7\u03b5\u03b9, _\u03c0\u03ac\u03bd\u03c4\u03b1 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03c4\u1ff6\u03bd \u03b3\u03b5\u03bd\u03bd\u03b7\u03c3\u03ac\u03bd\u03c4\u03c9\u03bd_\n... \u1f00\u03c1\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03bf\u1f50\u03c3\u03af\u03b1\u03c2, \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03b1 \u03c4\u1f70 \u03c4\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2, \u03c4\u03c1\u03af\u03c4\u03b1 \u03c4\u1f70 \u03c4\u1fc6\u03c2\n\u03c8\u03c5\u03c7\u1fc6\u03c2, _\u1f00\u03c0\u03bf\u03c4\u03af\u03bd\u03bf\u03bd\u03c4\u03b1 \u03b4\u03b1\u03bd\u03b5\u03af\u03c3\u03bc\u03b1\u03c4\u03b1_ \u03ba.\u03c4.\u03bb.\n\n20. \u03bd\u03b1\u03af] introducing an affectionate appeal as in Phil. iv. 3 \u03bd\u03b1\u1f76 \u1f10\u03c1\u03c9\u03c4\u1ff6\n\u03ba\u03b1\u1f76 \u03c3\u03ad.\n\n\u1f00\u03b4\u03b5\u03bb\u03c6\u03ad] It is the entreaty of a brother to a brother on behalf of a\nbrother (ver. 16). For the pathetic appeal involved in the word see the\nnotes on Gal. iii. 15, vi. 1, 18; and comp. ver. 7.\n\n\u1f10\u03b3\u03ce] 'I seem to be entreating for Onesimus; but I am pleading for\n_myself_: the favour will be done to me'; comp. ver. 17 \u03c0\u03c1\u03bf\u03c3\u03bb\u03b1\u03b2\u03bf\u1fe6 \u03b1\u1f50\u03c4\u1f78\u03bd\n\u1f61\u03c2 \u1f10\u03bc\u03ad. The emphatic \u1f10\u03b3\u03ce identifies the cause of Onesimus with his own.\n\n\u03c3\u03bf\u03c5 \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd] '_may I have satisfaction, find comfort in thee_', i.e.\n'may I receive such a return from thee, as a father has a right to\nexpect from his child.' The common use of the word \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd would suggest\nthe thought of filial offices; e.g. Arist. _Thesm._ 469 \u03bf\u1f55\u03c4\u03c9\u03c2 \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd\n\u03c4\u1ff6\u03bd _\u03c4\u03ad\u03ba\u03bd\u03c9\u03bd_, Lucian _Philops._ 27 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f74\u03bd \u1f44\u03c8\u03b9\u03bd \u03c4\u1ff6\u03bd _\u03c5\u1f31\u03ad\u03c9\u03bd_,\n\u03bf\u1f55\u03c4\u03c9\u03c2 \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd, \u1f14\u03c6\u03b7, \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd, Ps-Ignat. _Hero_ 6 \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd \u03c3\u03bf\u03c5,\n_\u03c0\u03b1\u03b9\u03b4\u03af\u03bf\u03bd_ \u03c0\u03bf\u03b8\u03b5\u03b9\u03bd\u03cc\u03bd, Synes. _Ep._ 44 \u03bf\u1f55\u03c4\u03c9 \u03c4\u1fc6\u03c2 \u1f31\u03b5\u03c1\u1fb6\u03c2 \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1\u03c2\n\u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd \u03ba\u03b1\u1f76 \u03c0\u03c1\u03bf\u03c3\u03ad\u03c4\u03b9 \u03c4\u1ff6\u03bd _\u03c0\u03b1\u03b9\u03b4\u03af\u03c9\u03bd_ \u03c4\u1ff6\u03bd \u1f10\u03bc\u03b1\u03c5\u03c4\u03bf\u1fe6, with other passages\nquoted in Wetstein. So too for \u1f44\u03bd\u03b1\u03c3\u03b8\u03b1\u03b9, \u1f44\u03bd\u03b7\u03c3\u03b9\u03c2, compare Eur. _Med._ 1025\nsq. \u03c0\u03c1\u1f76\u03bd \u03c3\u03c6\u1ff7\u03bd _\u1f44\u03bd\u03b1\u03c3\u03b8\u03b1\u03b9_ ... \u1f04\u03bb\u03bb\u03c9\u03c2 \u1f04\u03c1' \u1f51\u03bc\u1fb6\u03c2, \u1f66 _\u03c4\u03ad\u03ba\u03bd'_,\n\u1f10\u03be\u03b5\u03b8\u03c1\u03b5\u03c8\u03ac\u03bc\u03b7\u03bd, _Alc._ 333 \u1f05\u03bb\u03b9\u03c2 \u03b4\u1f72 _\u03c0\u03b1\u03af\u03b4\u03c9\u03bd_\u00b7 \u03c4\u1ff6\u03bd \u03b4' _\u1f44\u03bd\u03b7\u03c3\u03b9\u03bd_\n)\u03b5\u03cd\u03c7\u03bf\u03bc\u03b1\u03b9 \u03b8\u03b5\u03bf\u1fd6\u03c2 \u03b3\u03b5\u03bd\u03ad\u03c3\u03b8\u03b1\u03b9, Philem. _Inc._ 64 (IV. p. 55 Meineke) \u1f14\u03c4\u03b5\u03ba\u03ad\u03c2\n\u03bc\u03b5, \u03bc\u1fc6\u03c4\u03b5\u03c1, \u03ba\u03b1\u1f76 \u03b3\u03ad\u03bd\u03bf\u03b9\u03c4\u03cc \u03c3\u03bf\u03b9 _\u03c4\u03ad\u03ba\u03bd\u03c9\u03bd \u1f44\u03bd\u03b7\u03c3\u03b9\u03c2_, \u1f65\u03c3\u03c0\u03b5\u03c1 \u03ba\u03b1\u1f76 \u03b4\u03af\u03ba\u03b1\u03af\u03bf\u03bd \u1f10\u03c3\u03c4\u03af\n\u03c3\u03bf\u03b9, Ecclus. xxx. 2 \u1f41 \u03c0\u03b1\u03b9\u03b4\u03b5\u03cd\u03c9\u03bd \u03c4\u1f78\u03bd _\u1f51\u1f76\u03bf\u03bd_ \u03b1\u1f50\u03c4\u03bf\u1fe6 _\u1f40\u03bd\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9_ \u1f10\u03c0'\n\u03b1\u1f50\u03c4\u1ff7 (the only passage in the LXX where the word occurs). The prayer\n\u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd \u03c3\u03bf\u03c5, \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd \u1f51\u03bc\u1ff6\u03bd, etc., occurs several times in Ignatius;\n_Polyc._ 1, 6, _Magn._ 2, 12, _Ephes._ 2. It is not unlikely that\n\u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd here involves a reference to the name Onesimus; see the note on\nver. 11. The Hebrew fondness for playing on names makes such an allusion\nat least possible.\n\n------------------------------------------------------------------------\n\n21, 22]\n\n^{21}\u03a0\u03b5\u03c0\u03bf\u03b9\u03b8\u1f7c\u03c2 \u03c4\u1fc7 \u1f51\u03c0\u03b1\u03ba\u03bf\u1fc7 \u03c3\u03bf\u03c5 \u1f14\u03b3\u03c1\u03b1\u03c8\u03ac \u03c3\u03bf\u03b9, \u03b5\u1f30\u03b4\u1f7c\u03c2 \u1f45\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f51\u03c0\u1f72\u03c1 \u1f03 \u03bb\u03ad\u03b3\u03c9\n\u03c0\u03bf\u03b9\u03ae\u03c3\u03b5\u03b9\u03c2. ^{22}\u1f05\u03bc\u03b1 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u1f11\u03c4\u03bf\u03af\u03bc\u03b1\u03b6\u03ad \u03bc\u03bf\u03b9 \u03be\u03b5\u03bd\u03af\u03b1\u03bd\u00b7 \u1f10\u03bb\u03c0\u03af\u03b6\u03c9 \u03b3\u1f70\u03c1 \u1f45\u03c4\u03b9 \u03b4\u03b9\u1f70 \u03c4\u1ff6\u03bd\n\u03c0\u03c1\u03bf\u03c3\u03b5\u03c5\u03c7\u1ff6\u03bd \u1f51\u03bc\u1ff6\u03bd \u03c7\u03b1\u03c1\u03b9\u03c3\u03b8\u03ae\u03c3\u03bf\u03bc\u03b1\u03b9 \u1f51\u03bc\u1fd6\u03bd.\n\n\n\u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3] As he had begotten Philemon \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 (comp. 1 Cor. iv. 15,\n17), so it was \u1f10\u03bd \u039a\u03c5\u03c1\u03af\u1ff3 that he looked for the recompense of filial\noffices.\n\n\u1f00\u03bd\u03ac\u03c0\u03b1\u03c5\u03c3\u03bf\u03bd \u03ba.\u03c4.\u03bb.] See the note ver. 7.\n\n21. \u1f14\u03b3\u03c1\u03b1\u03c8\u03b1] '_I write_': see the note on ver. 19.\n\n\u1f51\u03c0\u1f72\u03c1 \u1f03 \u03bb\u03ad\u03b3\u03c9 \u03ba.\u03c4.\u03bb.] What was the thought upmost in the Apostle's mind\nwhen he penned these words? Did he contemplate the manumission of\nOnesimus? If so, the restraint which he imposes upon himself is\nsignificant. Indeed throughout this epistle the idea would seem to be\npresent to his thoughts, though the word never passes his lips. This\nreserve is eminently characteristic of the Gospel. Slavery is never\ndirectly attacked as such, but principles are inculcated which must\nprove fatal to it.\n\n22. \u1f05\u03bc\u03b1 \u03b4\u1f72 \u03ba.\u03c4.\u03bb.] When St Paul first contemplated visiting Rome, he had\nintended, after leaving the metropolis, to pass westward into Spain;\nRom. xv. 24, 28. But by this time he appears to have altered his plans,\npurposing first to revisit Greece and Asia Minor. Thus in Phil. ii. 24\nhe looks forward to seeing the Philippians shortly; while here he\ncontemplates a visit to the Churches of the Lycus valley.\n\nThere is a gentle compulsion in this mention of a personal visit to\nColoss\u00e6. The Apostle would thus be able to see for himself that Philemon\nhad not disappointed his expectations. Similarly Serapion in Eus. _H.E._\nvi. 12 \u03c0\u03c1\u03bf\u03c3\u03b4\u03bf\u03ba\u1fb6\u03c4\u03ad \u03bc\u03b5 \u1f10\u03bd \u03c4\u03ac\u03c7\u03b5\u03b9.\n\n\u03be\u03b5\u03bd\u03af\u03b1\u03bd] '_a lodging_'; comp. _Clem. Hom._ xii. 2 \u03c0\u03c1\u03cc\u03b1\u03be\u03c9\u03c3\u03b9\u03bd \u03c4\u1f70\u03c2 \u03be\u03b5\u03bd\u03af\u03b1\u03c2\n\u1f11\u03c4\u03bf\u03b9\u03bc\u03ac\u03b6\u03bf\u03bd\u03c4\u03b5\u03c2. So the Latin _parare hospitium_ Cic. _ad Att._ xiv. 2,\nMart. _Ep._ ix. 1. This latter passage, 'Vale et para hospitium',\nclosely resembles St Paul's language here. In the expression before us\n\u03be\u03b5\u03bd\u03af\u03b1 is probably the _place_ of entertainment: but in such phrases as\n\u03ba\u03b1\u03bb\u03b5\u1fd6\u03bd \u1f10\u03c0\u1f76 \u03be\u03b5\u03bd\u03af\u1fb3, \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03b5\u1fd6\u03bd \u03b5\u03c0\u1f76 \u03be\u03b5\u03bd\u03af\u03b1\u03bd, \u03c6\u03c1\u03bf\u03bd\u03c4\u03af\u03b6\u03b5\u03b9\u03bd \u03be\u03b5\u03bd\u03af\u03b1\u03c2, and the\nlike, it denotes the offices of hospitality. The Latin _hospitium_ also\nincludes both senses. The \u03be\u03b5\u03bd\u03af\u03b1, as a lodging, may denote either\nquarters in an inn or a room in a private house: see _Philippians_ p. 9.\nFor the latter comp. Plato _Tim._ 20 C \u03c0\u03b1\u03c1\u1f70 \u039a\u03c1\u03b9\u03c4\u03af\u03b1\u03bd \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03be\u03b5\u03bd\u1ff6\u03bd\u03b1, \u03bf\u1f57\n\u03ba\u03b1\u1f76 \u03ba\u03b1\u03c4\u03b1\u03bb\u03cd\u03bf\u03bc\u03b5\u03bd, \u1f00\u03c6\u03b9\u03ba\u03cc\u03bc\u03b5\u03b8\u03b1. In this case the response would doubtless be\na hospitable reception in Philemon's home; but the request does not\nassume so much as this.\n\n\u03c7\u03b1\u03c1\u03b9\u03c3\u03b8\u03ae\u03c3\u03bf\u03bc\u03b1\u03b9] '_I shall be granted to you_'. The grant (\u03c7\u03b1\u03c1\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9) of\none person to another, may be for purposes either (1) of destruction, as\nActs xxv. 11 \u03bf\u1f50\u03b4\u03b5\u1f76\u03c2 \u03bc\u03b5 \u03b4\u03cd\u03bd\u03b1\u03c4\u03b1\u03b9 \u03b1\u1f50\u03c4\u03bf\u1fd6\u03c2 \u03c7\u03b1\u03c1\u03af\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 (comp. ver. 16), or (2)\nof preservation, as Acts iii. 14 \u1f90\u03c4\u03ae\u03c3\u03b1\u03c3\u03b8\u03b5 \u1f04\u03bd\u03b4\u03c1\u03b1 \u03c6\u03bf\u03bd\u03ad\u03b1 \u03c7\u03b1\u03c1\u03b9\u03c3\u03b8\u1fc6\u03bd\u03b1\u03b9 \u1f51\u03bc\u1fd6\u03bd,\nand here.\n\n23\u201325. 'Epaphras my fellow-captive in Christ Jesus salutes you. As do\nalso Mark, Aristarchus, Demas, and Luke, my fellow-labourers. The grace\nof our Lord Jesus Christ be with thee and thy household, and sanctify\nthe spirit of you all.'\n\n23 sq. For these salutations see the notes on Col. iv. 10 sq. Epaphras\nis mentioned first because he was a Colossian (Col. iv. 12) and, as the\nevangelist of Coloss\u00e6 (see p. 29 sq.), doubtless well known to Philemon.\nOf the four others Aristarchus and Mark belonged to the Circumcision\n(Col. iv. 11), while Demas and Luke were Gentile Christians. All these\nwere of Greek or Asiatic origin and would probably be well known to\nPhilemon, at least by name. On the other hand Jesus Justus, who is\nhonourably mentioned in the Colossian letter (iv. 11), but passed over\nhere, may have been a Roman Christian.\n\n------------------------------------------------------------------------\n\n23\u201325]\n\n^{23}\u1f08\u03c3\u03c0\u03ac\u03b6\u03b5\u03c4\u03b1\u03af \u03c3\u03b5 \u1f18\u03c0\u03b1\u03c6\u03c1\u1fb6\u03c2 \u1f41 \u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03cc\u03c2 \u03bc\u03bf\u03c5 \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7 \u1f38\u03b7\u03c3\u03bf\u1fe6,\n^{24}\u039c\u03ac\u03c1\u03ba\u03bf\u03c2, \u1f08\u03c1\u03af\u03c3\u03c4\u03b1\u03c1\u03c7\u03bf\u03c2, \u0394\u03b7\u03bc\u1fb6\u03c2, \u039b\u03bf\u03c5\u03ba\u1fb6\u03c2, \u03bf\u1f31 \u03c3\u03c5\u03bd\u03b5\u03c1\u03b3\u03bf\u03af \u03bc\u03bf\u03c5.\n\n^{25}\u1f29 \u03c7\u03ac\u03c1\u03b9\u03c2 \u03c4\u03bf\u1fe6 \u039a\u03c5\u03c1\u03af\u03bf\u03c5 [\u1f21\u03bc\u1ff6\u03bd] \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6 \u03bc\u03b5\u03c4\u1f70 \u03c4\u03bf\u1fe6 \u03c0\u03bd\u03b5\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f51\u03bc\u1ff6\u03bd.\n\n\n\u1f41 \u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03bf\u03c2] On the possible meanings of this title see Col. iv. 10,\nwhere it is given not to Epaphras but to Aristarchus.\n\n25. \u1f29 \u03c7\u03ac\u03c1\u03b9\u03c2 \u03ba.\u03c4.\u03bb.] The same form of farewell as in Gal. vi. 18; comp. 2\nTim. iv. 22.\n\n\u1f51\u03bc\u1ff6\u03bd] The persons whose names are mentioned in the opening salutation.\n\n------------------------------------------------------------------------\n\n\n\n\n                      _ADDITIONS AND CORRECTIONS._\n\n\np. 6, l. 12. On Polemo and his family see _Ephemeris Epigraphica_ I. p.\n        270 sq. (1873).\n\np. 38, note 125. The investigations of M. Waddington respecting the\n        chronology of this period (see below) require a modification of\n        the dates here given for the earthquakes in the second century.\n        He enumerates three: (1) One at Rhodes, from A.D. 138\u2013142; (2)\n        One which destroyed Mitylene and did considerable damage to\n        Smyrna, A.D. 151\u2013152; (3) One which destroyed Smyrna A.D. 180.\n        These two last have been confounded together by previous\n        writers. See M. Waddington's _M\u00e9moire_, pp. 242 sq., 267 sq.\n\np. 48, note 160. On the names Ammias, Tatias, which are feminine and not\n        masculine, see below p. 373.\n\np. 49, note. I have here given the commonly received date for the\n        martyrdom of Polycarp; for I had not then seen M. Waddington's\n        investigations. This writer seems to have proved conclusively\n        that it took place several years earlier, A.D. 155: see his\n        _M\u00e9moire sur la Chronologie du Rh\u00e9teur \u00c6lius Aristide_ p. 232\n        sq., in the _M\u00e9moires de l'Acad\u00e9mie des Inscriptions_, &c. XXVI.\n        (1867).\n\npp. 52, 53. As these remarks respecting the silence of Eusebius will\n        seem to be directed against the opinions expressed in a recent\n        work, it may be worth while stating that the early sheets of\n        this commentary were struck off nearly twelve months before\n        _Supernatural Religion_ was published. The expression in p. 53,\n        note 170, 'numerous and patent quotations,' is too strongly\n        worded, though the references to St James in Clement's Epistle\n        seem to me to be clear. I might however have chosen other more\n        palpable illustrations from that epistle.\n\np. 63, l. 12. The Proconsulate of Paullus, under whom this martyrdom\n        took place, is dated by Borghesi (_[OE]uvres_ VIII. p. 507)\n        somewhere between A.D. 163\u2013168, by Waddington (_Fastes des\n        Provinces Asiatiques_ p. 731, in Le Bas and Waddington _Voyage\n        Arch\u00e9ologique etc._) probably A.D. 164\u2013166. This rests on the\n        assumption that the _Servillius Paullus_ here named must be\n        identified with _L. Sergius Paullus_ of the inscriptions. The\n        name _Sergius_ is elsewhere confounded with _Servius_\n        (_Servillius_) owing to the use of contractions (see Borghesi\n        IV. p. 493, VIII. p. 504). The mistake must have been introduced\n        very early into the text of Eusebius. All the Greek MSS have\n        _Servillius_ (_Servilius_), and so it is written in the Syriac\n        Version. Ruffinus however writes it correctly _Sergius_.\n\np. 71, line 1. We may conjecture that it was the earthquake under\n        Gallienus (A.D. 262) which proved fatal to Coloss\u00e6 (see above p.\n        38, note 125). This is consistent with the fact that no\n        Colossian coins later than Gordian (A.D. 238\u2013244) are extant.\n        When St Chrysostom wrote, the city existed no longer, as may be\n        inferred from his comment (XI. p. 323) '\u1f29 \u03c0\u03cc\u03bb\u03b9\u03c2 \u03c4\u1fc6\u03c2 \u03a6\u03c1\u03c5\u03b3\u03af\u03b1\u03c2 \u1f26\u03bd\u00b7\n        \u03ba\u03b1\u1f76 \u03b4\u1fc6\u03bb\u03bf\u03bd \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c4\u1f74\u03bd \u039b\u03b1\u03bf\u03b4\u03af\u03ba\u03b5\u03b9\u03b1\u03bd \u03c0\u03bb\u03b7\u03c3\u03af\u03bf\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9.'\n\n        On the other hand M. Renan (_L'Antechrist_ p. 99) says of the\n        earthquake under _Nero_, 'Colosses ne sut se relever; elle\n        disparut presque du nombre des \u00e9glises;' and he adds in a note\n        'Colosses n'a pas de monnaies imp\u00e9riales [Waddington].' This is\n        a mistake, and he must have misunderstood M. Waddington.\n\np. 77, note 229. To this list of works add Mansel's _Gnostic Heresies of\n        the First and Second Centuries_ (London 1875).\n\np. 112, note 336. See p. 330, note 553.\n\np. 160, l. 4. For 'argument for silence' read 'argument from silence.'\n\np. 205, col. 1, l. 30. Strike out \u03c4\u03bf\u1fe6 before \u03c0\u03b5\u03c1\u03b9\u03c0\u03b1\u03c4\u1fc6\u03c3\u03b1\u03b9.\n\np. 210, col. 1, l. 2. The dissertation to which reference is here made\n        is deferred to a later volume.\n\np. 250, col. 2, l. 21. Strike out the words in brackets.\n\np. 270, col. 1. \u1f05\u03c4\u03b9\u03bd\u03ac \u1f10\u03c3\u03c4\u03b9\u03bd \u03ba.\u03c4.\u03bb. Comp. Seneca _de Vit. beat._ 7 'in\n        ipso usu sui periturum.'\n\np. 280, col. 1, l. 23. For 'Ammianus' read 'Ammonius.'\n\n------------------------------------------------------------------------\n\n\n\n\n                                 INDEX.\n\n\n        Abercius (Avircius), Bp. of Hierapolis, p. 54 sq.\n        Acts of the Apostles; passages explained, p. 23 (xiii. 4, xvi.\n           6);\n          p. 95 (xix. 13, 19);\n          p. 370 (xiv. 11).\n        \u00e6dificatori\u00e6, the sufferings of Christ as, p. 232\n        \u00c6lfric on the Epistle to Laodiceans, p. 362\n        Alasanda or Alasadda, p. 152\n        Alexander of Tralles on charms, p. 92\n        Alexandria, a supposed Buddhist establishment at, p. 151\n        Andrew, St, in Asia, p. 45\n        Angelolatry condemned, p. 101, 103, 184, i. 16, ii. 10, 15, 18;\n          forbidden by the Council of Laodicea, p. 68\n        angelology of Cerinthus, p. 110;\n          of Essenism, p. 96;\n          of the Jews, ii. 18\n        Angels, orders of, i. 16\n        Anselm of Laon, p. 361\n        Antiochus the Great, colony of, in Asia Minor, p. 19\n        Antiochus Theos refounds Laodicea, p. 5\n        aorist, epistolary, iv. 8, Ph. 11, 19, 21;\n          contrasted with perfect, i. 16\n        Apamea, p. 19, 20;\n          Jews at, p. 21\n        Apocalypse, correspondences with St Paul's Epistles to Asia, 41\n           sq.\n        apocrypha, use of word, p. 90, ii. 3\n        Apollinaris, see Claudius Apollinaris\n        Apollo Archegetes worshipped at Hierapolis, p. 12\n        Apostolic Fathers, Christology of, p. 190\n        Apostolic Writings, Christology of, p. 189\n        Apphia, wife of Philemon, p. 372;\n          the name Phrygian, 372 sq.\n        Archippus, iv. 17;\n          son of Philemon, 374;\n          office and abode, 375;\n          rebuke to, 43\n        Arian heresy in Hierapolis and Laodicea, p. 64\n        Arian use of the expression 'First-born of all creation,' i. 15\n        Aristarchus, iv. 10\n        Aristion, p. 45\n        Aristotle, on slavery, p. 379;\n          definition of 'knowledge,' ii. 3;\n          of 'wisdom,' i. 9\n        Armagh, Book of, p. 348, 352\n        article, omission of the definite, i. 4\n        asah, a supposed derivation of Essenes, p. 126\n        Ascents of James, p. 168\n        asceticism among the Jewish sects, p. 87;\n          Colossian heretics, p. 104;\n          Essenes, p. 173;\n          a result of Gnosticism, p. 79\n        Aseis, a Laodicean title of Zeus, p. 8\n        Asia, meaning of, p. 19\n        Asia Minor, geography of, p. 1 sq.;\n          list of writers on, p. 1;\n          how divided under the Romans, 7;\n          a modern hypothesis about Christianity in, p. 50\n        Asid\u00e6ans, p. 120\n        asya, a supposed derivation of Essene, p. 125\n        Athanasius, on 'Firstborn of all Creation,' i. 15\n        Athens, a Buddhist burnt alive at, p. 155\n        Augustine, on 'Firstborn of all Creation,' i. 15;\n          on 'wisdom and knowledge,' ii. 3\n        \u1f00\u03b3\u03ac\u03c0\u03b7, \u1f41 \u03c5\u1f31\u1f78\u03c2 \u03c4\u1fc6\u03c2 \u1f00\u03b3\u03ac\u03c0\u03b7\u03c2 \u03b1\u1f50\u03c4\u03bf\u1fe6, i. 13\n        \u1f05\u03b3\u03b9\u03bf\u03c2, i. 2\n        \u1f00\u03b3\u03ce\u03bd, \u1f00\u03b3\u03c9\u03bd\u03af\u03b1, \u1f00\u03b3\u03c9\u03bd\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, i. 29, ii. 1, iv. 12\n        \u1f00\u03b4\u03b5\u03bb\u03c6\u03cc\u03c2 (\u1f41), i, 1\n        \u1f00\u03b8\u03c5\u03bc\u03b5\u1fd6\u03bd, iii. 21\n        \u03b1\u1f30\u03c3\u03c7\u03c1\u03bf\u03bb\u03bf\u03b3\u03af\u03b1, iii. 8\n        \u1f00\u03ba\u03b1\u03b8\u03b1\u03c1\u03c3\u03af\u03b1, iii. 5\n        \u1f05\u03bb\u03b1\u03c2, iv. 6\n        \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1, \u1f21 \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u03b1 \u03c4\u03bf\u1fe6 \u03b5\u1f50\u03b1\u03b3\u03b3\u03b5\u03bb\u03af\u03bf\u03c5, i. 5;\n          \u1f10\u03bd \u1f00\u03bb\u03b7\u03b8\u03b5\u03af\u1fb3, i. 6\n        \u1f00\u03bb\u03bb\u03ac, after \u03b5\u1f30 or \u03b5\u1f30 \u03ba\u03b1\u03af in St Paul, ii. 5\n        \u1f04\u03bc\u03c9\u03bc\u03bf\u03c2, i. 22\n        \u1f00\u03bd\u03b1\u03c0\u03ac\u03c5\u03b5\u03c3\u03b8\u03b1\u03b9, Ph. 7\n        \u1f00\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd, p. 230\n        \u1f00\u03bd\u03ad\u03b3\u03ba\u03bb\u03b7\u03c4\u03bf\u03c2, i. 22\n        \u1f00\u03bd\u03b5\u03c8\u03af\u03bf\u03c2, iv. 10\n        \u1f00\u03bd\u03ae\u03ba\u03b5\u03b9\u03bd, iii. 18;\n          \u03c4\u1f78 \u1f00\u03bd\u1fc6\u03ba\u03bf\u03bd, Ph. 8\n        \u1f00\u03bd\u03b8\u03c1\u03c9\u03c0\u03ac\u03c1\u03b5\u03c3\u03ba\u03bf\u03b9, iii. 22\n        \u1f00\u03bd\u03c4\u03b1\u03bd\u03b1\u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd, i. 24\n        \u1f00\u03bd\u03c4\u03b1\u03c0\u03cc\u03b4\u03bf\u03c3\u03b9\u03c2, iii. 24\n        \u1f00\u03cc\u03c1\u03b1\u03c4\u03bf\u03c2, i. 16\n        \u1f00\u03c0\u03b5\u03ba\u03b4\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9, ii. 15\n        \u1f00\u03c0\u03ad\u03ba\u03b4\u03c5\u03c3\u03b9\u03c2, ii. 11\n        \u1f00\u03c0\u03ad\u03c7\u03b5\u03b9\u03bd, Ph. 15\n        \u1f00\u03c0\u03b7\u03bb\u03bb\u03bf\u03c4\u03c1\u03b9\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9, i. 21\n        \u1f00\u03c0\u03bf\u03b8\u03bd\u03ae\u03c3\u03ba\u03b5\u03b9\u03bd, ii. 20\n        \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03b1\u03bb\u03bb\u03ac\u03c3\u03c3\u03b5\u03b9\u03bd, i. 20, 21\n        \u1f00\u03c0\u03cc\u03ba\u03c1\u03c5\u03c6\u03bf\u03c2, ii. 3\n        \u1f00\u03c0\u03bf\u03bb\u03cd\u03c4\u03c1\u03c9\u03c3\u03b9\u03c2, i. 14\n        \u1f00\u03c0\u03cc\u03c7\u03c1\u03b7\u03c3\u03b9\u03c2, ii. 22\n        \u1f05\u03c0\u03c4\u03b5\u03c3\u03b8\u03b1\u03b9, ii. 21\n        \u1f00\u03c1\u03ad\u03c3\u03ba\u03b5\u03b9\u03b1, i. 10\n        \u1f00\u03c1\u03c7\u03ae applied to Christ, p. 41; i. 16, 18\n        \u03b1\u1f50\u03be\u03ac\u03bd\u03b5\u03b9\u03bd, i. 6\n        \u0391\u1f50\u03c4\u1f78\u03c2 \u1f14\u03c3\u03c4\u03b9\u03bd, i. 17\n        \u1f00\u03c6\u03b5\u03af\u03b4\u03b5\u03b9\u03b1, ii. 23\n        \u1f01\u03c6\u03ae, ii. 19\n        \u1f00\u03c7\u03b5\u03b9\u03c1\u03bf\u03c0\u03bf\u03af\u03b7\u03c4\u03bf\u03c2, ii. 11\n        \u1f04\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c2, Ph. 11\n\n\n        B (Cod. Vaticanus), excellence of, p. 314.\n\n        Banaim, the, p. 132\n\n        Banus not an Essene, p. 161\n\n        Bardesanes, on Buddhists, 154;\n          his date, p. 155\n\n        Barnabas, life of, iv. 10;\n          epistle ascribed to, _ib._\n\n        basilica, iv. 15\n\n        Basilides, p. 331\n\n        Baur, p. 77, 81, 384\n\n        Bene-hakkeneseth, p. 130\n\n        Brahminism, p. 154, 155\n\n        Buddhism, assumed influence on Essenism, p. 151 sq.;\n          supposed establishment of, in Alexandria, p. 151;\n          unknown in the West, p. 153 sq.;\n          four steps of, p. 157\n\n        Buddhist at Athens, p. 155\n\n        \u03b2\u03ac\u03c0\u03c4\u03b9\u03c3\u03bc\u03b1, \u03b2\u03b1\u03c0\u03c4\u03b9\u03c3\u03bc\u03cc\u03c2, p. 250\n\n        \u03b2\u03ac\u03c1\u03b2\u03b1\u03c1\u03bf\u03c2, iii. 11\n\n        \u03b2\u03bb\u03b1\u03c3\u03c6\u03b7\u03bc\u03af\u03b1, iii. 8\n\n        \u03b2\u03bf\u03cd\u03bb\u03b5\u03c3\u03b8\u03b1\u03b9, Ph. 13\n\n        \u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd, iii. 15\n\n\n        Cabbala, see Kabbala\n\n        Cainites, p. 79\n\n        Calvin, iii. 8, p. 341, 384\n\n        Canonical writings and Papias, p. 52\n\n        Carpocratians, p. 79, 80\n\n        Cataphryges, p. 98\n\n        Cavensis, codex, p. 348\n\n        celibacy, p. 173\n\n        Cerinthus, p. 107 sq.;\n          Judaism of, p. 108;\n          Gnosticism of, _ib._;\n          cosmogony of, p. 109;\n          Christology of, p. 111 sq.;\n          pleroma of, p. 330\n\n        chaber, p. 128\n\n        Chagigah, on ceremonial purity, p. 128\n\n        Chalcedon, council of, p. 65\n\n        chasha, chashaim, a derivation of Essene, p. 119\n\n        chasi, chasyo, a derivation of Essene, p. 118;\n          connexion with chasid, p. 124\n\n        chasid, a false derivation of Essene, p. 115\n\n        Chasidim, p. 120;\n          not a proper name for the Essenes, p. 122\n\n        chasin, chosin, a false derivation for Essene, p. 116\n\n        chaza, chazya, a derivation of Essene, p. 117\n\n        Chonos or Chon\u00e6, p. 15, 71\n\n        Christ, the Person of, p. 34;\n          St Paul's doctrine about, p. 41, 181 sq., i. 15\u201320, ii. 9\u201315;\n          the Word Incarnate, p. 101, 102;\n          the pleroma in Him p. 102, i. 19, ii. 9, 10;\n          Life in Him, the remedy against sin, p. 34, 186 sq.;\n          His teaching and practice not Essene, p. 170 sq.\n\n        Christianity, not an outgrowth of Essenism, p. 159;\n          in relation to Epictetus, p. 13;\n          to Gnosticism, p. 80;\n          to slavery, p. 389, 391 sq.\n\n        Christianity in Asia Minor, p. 50\n\n        Christianized Essenes, p. 89, 90, 135\n\n        Christians of St John, p. 165\n\n        Christology of Ep. to Col., p. 101, 188;\n          of other Apostolic writings, p. 189;\n          of succeeding ages, p. 190\n\n        Chronicon Paschale, p. 48, 61\n\n        Chrysostom, i. 13, 15, iii. 16, p. 340, Ph. 15, p. 383\n\n        Cibotus, p. 21\n\n        Cibyratic convention, p. 7\n\n        Circular Letter\u2014the Ep. to the Ephesians\u2014p. 37\n\n        Claudius, embassy from Ceylon in the reign of, p. 156\n\n        Claudius Apollinaris, the name, p. 57 sq.;\n          his works, p. 58 sq.\n\n        Clement of Alexandria, p. 79, 98, 154, 168, i. 9, 15, ii. 8,\n           iii. 5, 16\n\n        Clement of Rome (\u00a7 7) Col. i. 3;\n          (\u00a7 58) i. 11;\n          (\u00a7 33, i. 15;\n          (Ep. ii. \u00a7 9), p. 104\n\n        Clementine Homilies, p. 136, 168\n\n        Clementine Recognitions, p. 164\n\n        Clermont, p. 3\n\n        collegia, iv. 15\n\n        Coloss\u00e6, orthography of, p. 16, i. 2;\n          situation, etc., p. 1 sq.;\n          distance from Laodicea, p. 376;\n          site, p. 13;\n          ancient greatness and decline, p. 15;\n          a Phrygian city, p. 18 sq.;\n          Jewish colony at, p. 19;\n          not visited by St Paul when the epistle was written, p. 23;\n          Epaphras the evangelist of, p. 29;\n          intended visit of Mark to, p. 40;\n          visit of St Paul to, p. 41;\n          obscurity of, p. 70;\n          a suffragan see of Laodicea, p. 69;\n          the Turkish conquest of, p. 71\n\n        Colossian heresy, nature of, p. 73 sq., 89, ii. 8;\n          writers upon, p. 74;\n          had regard to the Person of Christ, p. 112;\n          relation to Gnosticism, p. 98;\n          St Paul's answer to, p. 181 sq.\n\n        Colossians, Epistle to, p. 33;\n          bearers of, p. 35;\n          salutations in, _ib._;\n          charge respecting Laodicea, p. 36;\n          written by an amanuensis, iv. 18;\n          Christology of, p. 188;\n          style of, p. 191;\n          analysis of, p. 192;\n          various readings, see _readings_\n\n        colossinus, p. 4\n\n        community of goods, p. 176\n\n        Concord of the Laodiceans and Ephesians, etc., p. 31\n\n        Congregation, the holy, at Jerusalem, p. 131\n\n        Constantine, legislation of, p. 393\n\n        Constantinople, Council of, p. 65\n\n        conventus (Roman), p. 7\n\n        Corinth, visit of St Paul to, during his residence at Ephesus,\n           p. 30\n\n        Corinthians, First Epistle to; passages explained: (i. 19, i. 9;\n          (ii. 6, 7, i. 28;\n          (v. 9, iv. 16;\n          (vii. 21) p. 390;\n          (viii. 6) p. 188;\n          (ix. 24, ii. 18;\n          (x. 26) p. 326;\n          (xi. 7, i. 15;\n          (xiii. 3) p. 156;\n          (xiii. 12, i. 9;\n          (xv. 24, i. 16\n\n        Corinthians, Second Epistle to; passages explained: (i. 7, i.\n           24;\n          (iii. 6, i. 12;\n          (iv. 4, i. 15;\n          (v. 14, 15, ii. 20;\n          (vi. 1, i. 6;\n          (vi. 4, 6, i. 11;\n          (viii. 9, i. 6;\n          (ix. 12) _ib._;\n          (xiii. 5, i. 27\n\n        Cornelius a Lapide, p. 342\n\n        Creation, Gnostic speculations about, p. 78 sq.;\n          Essene do., p. 90\n\n        Cyril of Alexandria, p. 154\n\n        \u03ba\u03b1\u03b8\u1f7c\u03c2 \u03ba\u03b1\u03af, i. 6, iii. 1\n\n        \u03ba\u03b1\u03af in both members of a comparison, i. 6\n\n        \u03ba\u03b1\u1f76 \u1f45\u03c3\u03bf\u03b9, ii. 1\n\n        \u03ba\u03b1\u03b9\u03bd\u03cc\u03c2 and \u03bd\u03ad\u03bf\u03c2, ii. 10\n\n        \u03ba\u03b1\u03ba\u03af\u03b1, iii. 8\n\n        \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, i. 6\n\n        \u03ba\u03b1\u03c4\u03b1\u03b2\u03c1\u03b1\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd, ii. 18\n\n        \u03ba\u03b1\u03c4\u03b5\u03bd\u03ce\u03c0\u03b9\u03bf\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6, i. 22\n\n        \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03b5\u1fd6\u03bd, i. 19\n\n        \u03ba\u03b5\u03bd\u03b5\u03bc\u03b2\u03b1\u03c4\u03b5\u03cd\u03b5\u03b9\u03bd, ii. 18\n\n        \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae, i. 18\n\n        \u03ba\u03bb\u03b7\u03c1\u03bf\u03bd\u03bf\u03bc\u03af\u03b1, iii. 24\n\n        \u03ba\u03bb\u1fc6\u03c1\u03bf\u03c2, i. 12\n\n        \u03ba\u03bb\u03b7\u03c4\u03cc\u03c2, iii. 12\n\n        \u03ba\u03bf\u03b9\u03bd\u03c9\u03bd\u03af\u03b1, Ph. 6\n\n        \u03ba\u03bf\u03bc\u03af\u03b6\u03b5\u03b9\u03bd, iii. 25\n\n        \u03ba\u03bf\u03c0\u03b9\u1fb6\u03bd, i. 29\n\n        \u03ba\u03bf\u03c1\u03b1\u03be\u03cc\u03c2, p. 4\n\n        \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2, ii. 8\n\n        \u03ba\u03c1\u03b1\u03c4\u03b5\u1fd6\u03bd, ii. 19\n\n        \u03ba\u03c1\u03ac\u03c4\u03bf\u03c2, i. 11\n\n        \u03ba\u03c1\u03af\u03bd\u03b5\u03b9\u03bd, ii. 16\n\n        \u03ba\u03c4\u03af\u03c3\u03b9\u03c2, i. 15\n\n        \u03ba\u03cd\u03c1\u03b9\u03bf\u03c2, \u1f41, (Christ, i. 10;\n          (Master), iii. 24\n\n        \u03ba\u03c5\u03c1\u03af\u03bf\u03c4\u03b7\u03c2, i. 16\n\n        \u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u03ae\u03c1, i. 15\n\n        \u03c7\u03b1\u03c1\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, ii. 13, iii. 13, Ph. 22\n\n        \u03c7\u03ac\u03c1\u03b9\u03c2, i. 2, (\u1f21, iii. 16;\n          \u1f21 \u03c7\u03ac\u03c1\u03b9\u03c2 \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6, i. 6\n\n        \u03c7\u03b5\u03b9\u03c1\u03cc\u03b3\u03c1\u03b1\u03c6\u03bf\u03bd, ii. 14\n\n        \u03a7\u03c1\u03b7\u03c3\u03c4\u03cc\u03c4\u03b7\u03c2, iii. 12\n\n\n        Damascene: see _John Damascene_\n\n        Darmstadiensis Codex, p. 348\n\n        dative (of instrument), ii. 7, iii. 16;\n          (of part affected), i. 4\n\n        Demas, p. 36, iv. 14, Ph. 24\n\n        Denizli, p. 7;\n          earthquake at, p. 3\n\n        diocese, p. 7\n\n        Diognetus, Epistle to, i. 18\n\n        Dion Chrysostom, p. 81, 153\n\n        Diospolis, an old name of Laodicea, p. 68\n\n        Divinity of Christ, p. 101 sq., 182 sq., i. 15\n\n        Docet\u00e6, use of pleroma by, p. 337\n\n        dualism, p. 78, 87, 149\n\n        dyes of Coloss\u00e6 and the neighbourhood, p. 4\n\n        \u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03b9\u03bd, ii. 15\n\n        \u03b4\u03ad\u03c3\u03bc\u03b9\u03bf\u03c2, Ph. 1, 10\n\n        \u03b4\u03b5\u03c3\u03bc\u03cc\u03c2, Ph. 13\n\n        \u03b4\u03b9\u03ac with gen., used of the Logos, p. 188, i. 16, 20\n\n        \u03b4\u03b9\u03b1\u03ba\u03bf\u03bd\u03af\u03b1, \u03b4\u03b9\u03ac\u03ba\u03bf\u03bd\u03bf\u03c2, iv. 7, 17\n\n        \u03b4\u03b9\u03b4\u03ac\u03c3\u03ba\u03b5\u03b9\u03bd, i. 28\n\n        \u03b4\u03b9\u03bf\u03af\u03ba\u03b7\u03c3\u03b9\u03c2, p. 7\n\n        \u03b4\u03cc\u03b3\u03bc\u03b1, ii. 14\n\n        \u03b4\u03bf\u03b3\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03b9\u03bd, ii. 20\n\n        \u03b4\u03cc\u03be\u03b1, i. 11, 27\n\n        \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2, Ph. 16;\n          \u03b4\u03bf\u1fe6\u03bb\u03bf\u03c2 \u1f38\u03b7\u03c3\u03bf\u1fe6 \u03a7\u03c1\u03b9\u03c3\u03c4\u03bf\u1fe6, iv. 12\n\n        \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2, i. 16\n\n        \u03b4\u03c5\u03bd\u03b1\u03bc\u03bf\u1fe6\u03bd, i. 11\n\n\n        Earthquakes in the valley of the Lycus, p. 38\n\n        Ebionite Christology of Cerinthus, p. 110\n\n        economy of revelation perfected, p. 185\n\n        Elchesai, founder of the Mandeans, p. 167\n\n        Elchesai, Book of, p. 137\n\n        elders, primitive, p. 132\n\n        Eleazar expels evil spirits, p. 91\n\n        English Church on the Epistle to Laodicea, p. 362\n\n        English versions of the Epistle to Laodicea, p. 364\n\n        Epaphras, p. 34;\n          evangelist of Coloss\u00e6, p. 29, 31;\n          mission to St Paul, p. 32, iv. 12, Ph. 23\n\n        Epaphroditus, p. 34\n\n        Ephesians, Epistle to; a circular letter, p. 37;\n          readings in, harmonistic with Epist. to Col., p. 312 sq.;\n          passages explained, i. 18 , i. 23);\n            i. 21 , i. 16);\n            i. 23 , i. 18);\n            ii. 3 , iii. 6);\n            ii. 4 , iii. 1);\n            ii. 4, 5 , ii. 13);\n            ii. 12 , i. 21);\n            ii. 14 , i. 17);\n            ii. 15 , ii. 14);\n            ii. 16 , i. 20);\n            ii. 20 , ii. 7);\n            iii. 17 , ii. 17);\n            iii. 21 , i. 26);\n            iv. 10, 11 , i. 17);\n            iv. 18 , i. 21);\n            iv. 19, v. 3 (iii. 5);\n            v. 32 (i. 26)\n\n        Ephesus, Council of, p. 65\n\n        Ephesus, St Paul at, p. 30, 95;\n          exorcists at, p. 95\n\n        Epictetus, p. 13\n\n        Epiphanius, account of Cerinthus, p. 107;\n          on the Nasareans, p. 136\n\n        epistolary aorist, Ph. 11, 19, 21\n\n        epulones of Ephesian Artemis called Essenes, p. 96\n\n        Erasmus on the Epistle to Laodicea, p. 365\n\n        Essene, meaning of term, p. 94;\n          the name, p. 114 sq.;\n          Frankel's theory, p. 121 sq.\n\n        Essenes, p. 82, ii. 8;\n          list of writers upon, p. 83;\n          localities of, p. 93;\n          asceticism of, p. 85;\n          speculations of, p. 87;\n          exclusiveness of, p. 92;\n          Josephus and Philo chief authorities upon, p. 134;\n          oath taken by, p. 127;\n          their grades, p. 129;\n          origin and affinities, p. 119 sq.;\n          relation to Christianity, p. 158;\n          to Pharisaism, p. 101, 120;\n          to Neopythagoreanism, p. 143;\n          to Hemerobaptists, p. 166;\n          to Gnosticism, p. 92;\n          to Parsism, p. 149;\n          to Buddhism, p. 157;\n          excused by Herod the Great from taking the oath of allegiance,\n             p. 176;\n          fortune tellers, p. 178;\n          silence of New Test. about, p. 159;\n          in relation to John the Baptist, p. 160;\n          to James the Lord's brother, p. 168;\n          Christianized Essenes, p. 135\n\n        Essenism, p. 82;\n          main features of, p. 83 sq.;\n          compared with Christianity, p. 170 sq.;\n          the sabbath, p. 170;\n          lustrations, p. 171;\n          avoidance of strangers, p. 172;\n          asceticism, celibacy, p. 173;\n          avoidance of the Temple, p. 174;\n          denial of the resurrection of the body, p. 175;\n          certain supposed coincidences with Christianity, p. 175\n\n        Eusebius, on the earthquakes in the valley of the Lycus, p. 39;\n          his mistake respecting some martyrdoms, p. 48;\n          silence on quotations from Canonical writings, p. 52;\n          on tracts against Montanism, p. 56;\n          the Thundering Legion, p. 61;\n          on Marcellus, i. 15\n\n        evil, Gnostic theories about, p. 78\n\n        exorcists at Ephesus, p. 95\n\n        \u1f11\u03b1\u03c5\u03c4\u03bf\u1fe6 and \u03b1\u1f51\u03c4\u03bf\u1fe6, i. 12;\n          and \u1f00\u03bb\u03bb\u03ae\u03bb\u03c9\u03bd, iii. 13\n\n        \u1f10\u03b3\u03ce, Ph. 19\n\n        \u1f10\u03b8\u03b5\u03bb\u03bf\u03b8\u03c1\u03b7\u03c3\u03ba\u03b5\u03af\u03b1, ii. 23\n\n        \u03b5\u1f34 \u03b3\u03b5, i. 23\n\n        \u03b5\u1f30\u03ba\u03ce\u03bd, i. 15, iii. 11\n\n        \u03b5\u1f36\u03bd\u03b1\u03b9 \u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03bf\u03c1\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd, i. 6\n\n        \u03b5\u1f30\u03c2, i. 6, ii. 22, Ph. 6\n\n        \u1f10\u03ba \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u03b1\u03c2 (\u03c4\u1f74\u03bd), iv. 16\n\n        \u1f10\u03ba\u03ba\u03bb\u03b7\u03c3\u03af\u03b1, iv. 15\n\n        \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03cc\u03c2, iii. 12\n\n        \u1f10\u03bb\u03bb\u03bf\u03b3\u1fb6\u03bd, Ph. 18\n\n        \u1f10\u03bb\u03c0\u03af\u03c2, i. 5\n\n        \u1f10\u03bd, iv. 12;\n          denoting the sphere, i. 4;\n          \u1f10\u03bd \u03b1\u1f50\u03c4\u1ff7, i. 16;\n          \u1f10\u03bd \u03bc\u03ad\u03c1\u03b5\u03b9, ii. 16;\n          \u1f10\u03bd \u03c0\u03b1\u03bd\u03c4\u1f76 \u03b8\u03b5\u03bb\u03ae\u03bc\u03b1\u03c4\u03b9, iv. 12;\n          \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9\u03bd, i. 18;\n          \u1f10\u03bd \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03c1\u03b3\u03bf\u03b9\u03c2, i. 21;\n          \u1f10\u03bd \u1f51\u03bc\u1fd6\u03bd, i. 27, iii. 16;\n          \u1f10\u03bd \u03a7\u03c1\u03b9\u03c3\u03c4\u1ff7, i. 2\n\n        \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u1fd6\u03bd, \u1f10\u03bd\u03b5\u03c1\u03b3\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, i. 29\n\n        \u1f14\u03bd\u03b9, iii. 11\n\n        \u1f10\u03be\u03b1\u03b3\u03bf\u03c1\u03ac\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, iv. 5\n\n        \u1f10\u03be\u03b1\u03bb\u03ad\u03b9\u03c6\u03b5\u03b9\u03bd, ii. 14\n\n        \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1, i. 13, 16\n\n        \u1f14\u03be\u03c9 (\u1f41\u1f30, iv. 5\n\n        \u1f11\u03bf\u03c1\u03c4\u03ae, ii. 16\n\n        \u1f10\u03c0\u03b9\u03b3\u03b9\u03bd\u03ce\u03c3\u03ba\u03b5\u03b9\u03bd, \u1f10\u03c0\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03c2, p. 100, i. 6, 9, Ph. 6\n\n        \u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1, iii. 5\n\n        \u1f10\u03c0\u03b9\u03bc\u03ad\u03bd\u03b5\u03b9\u03bd, i. 23\n\n        \u1f10\u03c0\u03b9\u03c3\u03c4\u03bf\u03bb\u03ae (\u1f21), iv. 16\n\n        \u1f10\u03c0\u03b9\u03c7\u03bf\u03c1\u03b7\u03b3\u03b5\u1fd6\u03bd, ii. 19\n\n        \u1f10\u03c0\u03bf\u03b9\u03ba\u03bf\u03b4\u03bf\u03bc\u03b5\u1fd6\u03bd, ii. 7\n\n        \u1f10\u03c1\u03b3\u03ac\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, iii. 23\n\n        \u1f10\u03c1\u03b5\u03b8\u03af\u03b6\u03b5\u03b9\u03bd, iii. 21\n\n        \u1f10\u03c1\u1fe5\u03b9\u03b6\u03c9\u03bc\u03ad\u03bd\u03bf\u03b9, ii. 7\n\n        \u1f14\u03c1\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9, iii. 6\n\n        \u03b5\u1f50\u03ac\u03c1\u03b5\u03c3\u03c4\u03bf\u03c2, iii. 20\n\n        \u03b5\u1f50\u03b4\u03bf\u03ba\u03af\u03b1, \u03b5\u1f50\u03b4\u03bf\u03ba\u03b5\u1fd6\u03bd, i. 19\n\n        \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03b5\u1fd6\u03bd, \u03b5\u1f50\u03c7\u03b1\u03c1\u03b9\u03c3\u03c4\u03af\u03b1, ii. 7, i. 3;\n          \u03b5\u1f50\u03c7\u03ac\u03c1\u03b9\u03c3\u03c4\u03bf\u03c2, iii. 15\n\n        \u1f18\u03c6\u03ad\u03c3\u03b9\u03b1 \u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c4\u03b1, p. 95\n\n        \u1f14\u03c7\u03b5\u03b9\u03bd, Ph. 17\n\n        \u1f10\u03c7\u03b8\u03c1\u03bf\u03af, i. 21\n\n\n        F (Codex Augiensis) relation to G, p. 345\n\n        Firstborn of all Creation, i. 15\n\n        Flaccus, p. 20\n\n        Frankel on the Essenes, p. 121 sq.\n\n\n        G (Codex Boernerianus) relation to F, p. 345\n        Galatia, meaning of, in St Paul and St Luke, p. 24\n        Galatian and Colossian Judaism compared, p. 105, i. 28\n        Galatians, Epistle to; passages explained, i. 24 (Gal. ii. 20);\n          i. 28 (iv. 19);\n          ii. 8 (iv. 3)\n        Galen, ii. 19, 20\n        Ginsburg, (Dr) p. 88, 127 sq.\n        Gnostic, p. 80 sq.\n        Gnostic element in Colossian heresy, p. 73 sq.\n        Gnostic sects, use of pleroma by, p. 330\n        Gnosticism, list of writers on, p. 77;\n          definition of, p. 76 sq.;\n          intellectual exclusiveness of, p. 77;\n          speculations of, p. 77 sq.;\n          practical errors of, p.79 sq.;\n          independent of Christianity, p. 80;\n          relation to Judaism, p. 81;\n          to Essenism, p. 93;\n          to Colossian heresy, p. 98\n        grades of Essenes, p. 129\n        Gr\u00e4tz, p. 123, 160, 161, 170\n        Greece, slavery in, p. 386\n        Gregory the Great on the Epistle to the Laodiceans, p. 361\n        guild of dyers, p. 4\n        \u0393\u03b1\u03c1\u03bc\u03ac\u03bd\u03b1\u03c2, p. 153\n        \u03b3\u03bd\u1ff6\u03c3\u03b9\u03c2, i. 9, ii. 3\n        \u03b3\u03bd\u03c9\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2, p. 81\n\n\n        Hamartiology of the Old Testament, p. 185\n        Haymo of Halberstadt, on the Epistle to the Laodiceans, p. 361\n        Hebrew slavery, p. 385 sq.\n        Hebrews, Epistle to the; passages explained, i. 11 (Heb. xi.\n           34);\n          i. 15 (i. 2, 3, 6)\n        Hefele on the date of Claudius Apollinaris, p. 185\n        Hemerobaptists, p. 162\n        Herod the Great excuses the oath of allegiance to the Essenes,\n           p. 176\n        Hervey of Dole, on the Epistle to the Laodiceans, p. 361\n        Hierapolis, p. 9;\n          modern name, p. 9;\n          physical features of, p. 10;\n          a famous watering place, p. 11;\n          the Plutonium at, p. 12;\n          birthplace of Epictetus, p. 13;\n          political relations of, p. 18;\n          attractions for Jews, p. 22;\n          a Christian settlement, p. 45;\n          Philip of Bethsaida at, p. 45 sq.;\n          Council at, p. 59;\n          Papias, bishop of, p. 48 sq.;\n          Abercius, bishop of, p. 54 sq.;\n          Claudius Apollinaris, bishop of, p. 57 sq.;\n          dyes of, p. 4\n        Hilgenfeld, p. 75;\n          on the Essenes, p. 150 sq.\n\n\n        James the Lord's brother, p. 167\n        Jerome, p. 29;\n          on St Paul's parents, p. 35;\n          on the Epistle to the Laodiceans, p. 359 sq.\n        Jesus Justus, iv. 11\n        Jews, sects of the, p. 82\n\n\n        imperfect, iii. 18\n\n        indicative after \u03b2\u03bb\u03ad\u03c0\u03b5\u03b9\u03bd \u03bc\u03ae, ii. 8\n\n        infinitive of consequence, i. 10, iv. 3, 6\n\n        John (St) in Asia Minor, p. 41;\n          Apocalypse, passages explained, p. 41 (iii. 14\u201321);\n          Gospel, p. 163 (i. 8, v. 35);\n          Second Epistle, p. 371;\n          Third Epistle, _ib._\n\n        John the Baptist, not an Essene, p. 160;\n          disciples of, at Ephesus, p. 163\n\n        John (St), Christians of, p. 165\n\n        John Damascene, p. 15\n\n        John of Salisbury on the Epistle to the Laodiceans, p. 362\n\n        Josephus on Essenism, p. 133 sq.\n\n        Judaism and Gnosticism, p. 81\n\n        \u1f35\u03bd\u03b1, iv. 16\n\n        \u1f38\u03bf\u1fe6\u03c3\u03c4\u03bf\u03c2, iv. 11\n\n        \u1f30\u03c3\u03cc\u03c4\u03b7\u03c2, iv. 1\n\n\n        Kabbala, p. 93, i. 16, ii. 8\n\n\n        Lanfranc on the Epistle to the Laodiceans, p. 363\n        Laodicea, name and history, p. 5;\n          condition, p. 6;\n          political rank and relations, p. 7, 18;\n          religious worship at, p. 8;\n          Council of, p. 66;\n          ecclesiastical status, p. 69;\n          dyes of, p. 4;\n          surnamed Trimetaria, p. 18;\n          the vaunt of, p. 44\n        Laodicea, the letter from, iv. 16;\n          p. 340 sq.\n        Laodiceans, apocryphal Epistle to the, p. 347 sq.;\n          list of MSS of, p. 349 sq.;\n          Latin text of, p. 353;\n          notes on, p. 355 sq.;\n          theory of a Greek original, p. 357;\n          restoration of the Greek, p. 359;\n          circulation of, p. 360 sq.;\n          English prologue and versions, p. 364;\n          strictures of Erasmus on, p. 365;\n          genuineness maintained by some, p. 366\n        Latrocinium, see _Robbers' Synod_\n        Legio Fulminata, p. 61\n        legislation of Constantine on slavery, p. 393\n        Logos, the, i. 15\n        Luke, St, iv. 14;\n          his narrative of St Paul's third missionary journey, p. 24\n             sq.;\n          makes a distinction between Philip the Apostle and Philip the\n             Evangelist, p. 45, 59\n        lukewarmness at Laodicea, p. 42\n        lustrations of the Essenes, p. 171\n        Luther's estimate of the Epistle to Philemon, p. 383\n        Lycus, district of the;\n          list of writers on, p. 1 sq.;\n          physical features of, p. 2 sq.;\n          produce of, p. 4;\n          subterranean channel of the, p. 14;\n          earthquakes in, p. 38 sq.\n        Lycus, Churches of the, p. 1 sq.;\n          evangelised by Epaphras, p. 29 sq.;\n          ecclesiastical status of, p. 69\n        \u039b\u03b1\u03bf\u03b4\u03b9\u03ba\u03af\u03b1, iv. 13\n        \u03bb\u03cc\u03b3\u03bf\u03bd \u1f14\u03c7\u03b5\u03b9\u03bd \u03c4\u03b9\u03bd\u03cc\u03c2, ii. 23\n\n\n        Magic, forbidden by Council of Laodicea, p. 69;\n          among the Essenes, p. 140\n        magical books at Ephesus, p. 95;\n          magical charms among the Essenes, p. 90 sq.\n        Mandeans, p. 165\n        Marcosians, p. 335\n        Mark (St), iv. 10;\n          visits Coloss\u00e6, p. 40\n        marriage depreciated by the Essenes, p. 139\n        Matthew (St) Gospel of, accepted by Cerinthus and the Ebionites,\n           p. 108\n        Megasthenes, p. 153\n        monasticism of the Essenes, p. 157\n        Monoimus, the Arabian, p. 339\n        Montanism, Claudius Apollinaris on, p. 59;\n          Phrygian origin of, p. 98\n        morning bathers, p. 132\n        Muratorian Fragment on the Epistle to the Laodiceans, p. 358\n        \u03bc\u03b1\u03ba\u03c1\u03bf\u03b8\u03c5\u03bc\u03af\u03b1, i. 11, iii. 12\n        \u03bc\u03b5\u03c1\u03af\u03c2, i. 12\n        \u03bc\u03bd\u03b5\u03af\u03b1\u03bd \u03c0\u03bf\u03b9\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, Ph. 4\n        \u03bc\u03bf\u03bc\u03c6\u03ae, iii. 13\n        \u03bc\u03bf\u03bd\u03bf\u03b3\u03b5\u03bd\u03ae\u03c2, i. 15\n        \u03bc\u03c5\u03c3\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd, i. 26\n\n\n        Naassenes, p. 337\n        Nasoreans, p. 138, 165\n        Neander on Cerinthus, p. 108\n        Neopythagoreanism and Essenism, p. 146 sq.\n        New Testament, relation of, to the Old Testament, p. 184\n        Nic\u00e6a, Bishops of Hierapolis and Laodicea at the Council of, p.\n           65\n        Nicetas Choniates, p. 71\n        nominative with definite article for vocative, iii. 18\n        Novatianism in Phrygia, p. 98\n        Nymphas, iv. 15, p. 31\n        \u03bd\u03b5\u03bf\u03bc\u03b7\u03bd\u03af\u03b1, ii. 16\n        \u03bd\u03ad\u03bf\u03c2, iii. 10\n        \u03bd\u03bf\u03c5\u03b8\u03b5\u03c4\u03b5\u1fd6\u03bd, i. 28\n        \u03bd\u1fe6\u03bd with aorist, i. 21\n\n\n        Observance of the Sabbath by the Essenes and our Lord, p. 170\n\n        Onesimus, p. 377, Ph. 10;\n          at Rome, p. 33;\n          encounters St Paul, p. 378;\n          returns to Philemon, p. 35, 379 sq.;\n          legendary history of, p. 382\n\n        Ophites, the, p. 81, 98, 337\n\n        Oracle, see _Sibylline Oracle_\n\n        \u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1, i. 25\n\n        \u03bf\u1f36\u03ba\u03bf\u03c2, \u03c4\u1f74\u03bd \u03ba\u03b1\u03c4' \u03bf\u1f36\u03ba\u03bf\u03bd, iv. 15\n\n        \u1f41\u03bc\u03bf\u03af\u03c9\u03bc\u03b1, i. 25\n\n        \u1f44\u03bd\u03b1\u03c3\u03b8\u03b1\u03b9, \u1f40\u03bd\u03b1\u03af\u03bc\u03b7\u03bd, Ph. 20\n\n        \u1f40\u03c1\u03b3\u03ae, iii. 8\n\n        \u1f45\u03c3\u03c4\u03b9\u03c2, iii. 5, iv. 11\n\n        \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u03bf\u03b4\u03bf\u03c5\u03bb\u03b5\u03af\u03b1, iii. 22\n\n        \u1fa0\u03b4\u03ae, iii. 16\n\n        \u1f61\u03c2, Ph. 14, 16\n\n        .ix\n        Papias, p. 47;\n          writings of, _ib._;\n          life and teaching of, p. 48;\n          account of, given by Eusebius, p. 49;\n          traditions collected by, p. 51 sq.;\n          references to the Canonical writings, p. 51 sq.;\n          silence of Eusebius, p. 52;\n          views inferred from his associates, p. 53\n\n        Parsism, resemblances to, in Essenism, p. 149 sq.;\n          spread by the destruction of the Persian empire, p. 150;\n          influence of, p. 151\n\n        participle used for imperative, iii. 16\n\n        Paschal controversy, p. 59, 63\n\n        Paul (St) visits Phrygia on his second missionary journey, p.\n           23;\n          had not visited Coloss\u00e6 when he wrote, p. 23 sq.;\n          visits Phrygia on his third journey, p. 24;\n          silence about personal relations with Coloss\u00e6, p. 28;\n          at Ephesus, p. 30, 95 sq.;\n          at Rome, p. 32;\n          mission of Epaphras to, _ib._;\n          meets with Onesimus, p. 33, 378;\n          despatches three letters, p. 33;\n          visits Coloss\u00e6, p. 41;\n          his plans after his release, Ph. 22;\n          uses an amanuensis, iv. 18;\n          his signature, iv. 18, Ph. 19;\n          coincidences with words of our Lord, ii. 22;\n          his teaching on the universality of the Gospel, p. 99;\n          on the kingdom of Christ, i. 13 sq.;\n          on the orders of angels, i. 16 sq.;\n          on philosophy, ii. 8;\n          on the Incarnation, ii. 9;\n          on the abolition of distinctions, iii. 11;\n          on slavery, iii. 22 sq., p. 389 sq.;\n          his cosmogony and theology, p. 101 sq.;\n          his answer to the Colossian heresy, p. 181 sq.;\n          his Christology, p. 188, i. 15 sq.;\n          his relations with Philemon, p. 370 sq.;\n          connects baptism and death, ii. 11, 20, iii. 3;\n          makes use of metaphors from the mysteries, i. 26, 28;\n          from the stadium, ii. 18, iii. 14;\n          his rapid change of metaphor, ii. 7\n\n        Paul (St) Epistles of, correspondences with the Apocalypse\u2014on\n           the Person of Christ, p. 41;\n          warning against lukewarmness, p. 42;\n          against pride of wealth, p. 43\n\n        Paul (St) apocryphal Epistle of, to the Laodiceans, p. 353\n\n        Pedanius Secundus, execution of his slaves, p. 388\n\n        Person of Christ, St Paul and St John on, p. 41 sq.;\n          St Paul's answer to the Colossian heresy, p. 181 sq., i. 15\n             sq.\n\n        personal pronoun used for reflexive, i. 20, 22\n\n        Peter (St) and the Church in Asia Minor, p. 41\n\n        petrifying stream at Coloss\u00e6, p. 15\n\n        Pharisees, p. 82;\n          relation to Essenes, p. 82, 120, 141\n\n        Philemon, p. 31, 370 sq.;\n          legendary history of, p. 371;\n          his wife, p. 372;\n          his son, p. 374\n\n        Philemon, Epistle to;\n          Introduction to, p. 369;\n          character of, p. 370;\n          analysis of, p. 380 sq.;\n          different estimates of, p. 382 sq.;\n          compared with a letter of Pliny, p. 384\n\n        Philip the Apostle, in Asia, p. 45 sq.;\n          confused with Philip the Evangelist, p. 45\n\n        Philippopolis, synod of, p. 64\n\n        Philo, on the Essenes, p. 133;\n          his use of Logos, i. 15\n\n        Phrygia, p. 17 sq.;\n          meaning of the phrase in St Luke, p. 23;\n          religious tendencies of, p. 97 sq.;\n          see _Paul (St)_\n\n        Pistis Sophia, p. 339\n\n        Pliny the younger, a letter of, p. 384 sq.\n\n        pleroma, detached note upon, p. 323\n\n        Plutonium, at Hierapolis, p. 12\n\n        Polycarp, martyrdom of, p. 49\n\n        poverty, respect paid to, by Essenes and Christ, p. 177\n\n        Pr\u00e6torius accepts the Epistle to the Laodiceans as genuine, p.\n           366\n\n        Pythagoreanism and Essenism, p. 144;\n          disappearance of, p. 146\n\n        \u03c0\u03ac\u03b8\u03bf\u03c2, iii. 5\n\n        \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03b5\u1fd6\u03bd, ii. 2\n\n        \u03c0\u03b1\u03c1\u03b1\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9\u03bd, ii. 6\n\n        \u03c0\u03b1\u03c1\u03ac\u03c0\u03c4\u03c9\u03bc\u03b1, ii. 13\n\n        \u03c0\u03b1\u03c1\u03b5\u1fd6\u03bd\u03b1\u03b9 \u03b5\u1f30\u03c2, i. 6\n\n        \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03c3\u03b8\u03b1\u03b9, iv. 1\n\n        \u03c0\u03b1\u03c1\u03b7\u03b3\u03bf\u03c1\u03af\u03b1, iv. 11\n\n        \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u03b1, \u1f10\u03bd \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u1fb3, ii. 15, Ph. 8\n\n        \u03c0\u1fb6\u03c2, \u03c0\u1fb6\u03c2 \u1f41 \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2, i. 16;\n          \u03c0\u03ac\u03c3\u03b7 \u03ba\u03c4\u03af\u03c3\u03b9\u03c2, i. 15;\n          \u03c4\u1f70 \u03c0\u03ac\u03bd\u03c4\u03b1, i. 16\n\n        \u03c0\u03b1\u03c4\u03ae\u03c1, \u1f41 \u03b8\u03b5\u1f78\u03c2 \u03c0\u03b1\u03c4\u03ae\u03c1, i. 3;\n          \u03c0\u03b1\u03c4\u1f74\u03c1 \u1f21\u03bc\u1ff6\u03bd, i. 2\n\n        \u03c0\u03b1\u03cd\u03b5\u03c3\u03b8\u03b1\u03b9, Ph. 7\n\n        \u03c0\u03b9\u03b8\u03b1\u03bd\u03bf\u03bb\u03bf\u03b3\u03af\u03b1, ii. 4\n\n        \u03c0\u03b9\u03ba\u03c1\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9, iii. 19\n\n        \u03c0\u03b9\u03c3\u03c4\u03cc\u03c2, \u03c0\u03b9\u03c3\u03c4\u03bf\u1f76 \u1f00\u03b4\u03b5\u03bb\u03c6\u03bf\u03af, i. 2\n\n        \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03be\u03af\u03b1, iii. 5\n\n        \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03b5\u1fd6\u03bd, iv. 12\n\n        \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1, ii. 2\n\n        \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd, i. 25, iv. 17\n\n        \u03c0\u03bb\u03ae\u03c1\u03c9\u03bc\u03b1, i. 19, ii. 9, p. 323 sq.\n\n        \u03c0\u03bb\u03b7\u03c3\u03bc\u03bf\u03bd\u03ae, ii. 23\n\n        \u03c0\u03bb\u03bf\u1fe6\u03c4\u03bf\u03c2, i. 27\n\n        \u03c0\u03bf\u03c1\u03bd\u03b5\u03af\u03b1, iii. 5\n\n        \u03c0\u03c1\u03b1\u03b0\u03c4\u03b7\u03c2, iii. 12\n\n        \u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03c5\u03c4\u03ae\u03c2, \u03c0\u03c1\u03b5\u03c3\u03b2\u03cd\u03c4\u03b7\u03c2, Ph. 8\n\n        \u03c0\u03c1\u1f78 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd, i. 17\n\n        \u03c0\u03c1\u03bf\u03b1\u03ba\u03bf\u03cd\u03b5\u03b9\u03bd, i. 5\n\n        \u03c0\u03c1\u03cc\u03c2, ii. 23, Ph. 5\n\n        \u03c0\u03c1\u03bf\u03c3\u03ba\u03b1\u03c1\u03c4\u03b5\u03c1\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9, iv. 2\n\n        \u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bb\u03b7\u03bc\u03c8\u03af\u03b1, iii. 25\n\n        \u03c0\u03c1\u03c9\u03c4\u03cc\u03c4\u03bf\u03ba\u03bf\u03c2, i. 15, 18\n\n        \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1, ii. 8\n\n        \u03c6\u03b8\u03bf\u03c1\u03ac, ii. 22\n\n        \u03c6\u03c1\u03cc\u03bd\u03b7\u03c3\u03b9\u03c2, i. 9\n\n        \u03c6\u03c5\u03bb\u03b1\u03ba\u03c4\u03ae\u03c1\u03b9\u03bf\u03bd, p. 69\n\n        \u03c8\u03b1\u03bb\u03bc\u03cc\u03c2, iii. 16\n\n\n        Quartodeciman controversy, p. 59, 63\n        Quinisextine Council, p. 68\n\n\n        Readings, harmonized with corresponding passages in the Epistle\n           to the Ephesians,\n          p. 312 (iii. 6);\n          p. 313 (ii. 21, v. 19)\n        readings, various,\n          p. 315 (i. 3);\n          p. 316 (i. 4, i. 7);\n          p. 317 (i. 12, i. 14, i. 22);\n          p. 318 (ii. 2);\n          p. 319 (ii. 16);\n          p. 320 (ii. 18, ii. 23);\n          p. 321 (iv. 8);\n          p. 322 (iv. 15)\n        Renan, on the meaning of Galatia in St Paul and St Luke, p. 25;\n          his estimate of the Epistle to Philemon, p. 384\n        Restoration, under Ezra, p. 119\n        resurrection of the body denied, p. 88, 175\n        Revelation; see _Apocalypse_\n        Robbers' Synod, p. 65\n        Roman slavery, p. 387\n        Rome, Onesimus at, p. 378;\n          St Paul at, p. 32\n        \u1fe5\u03b9\u03b6\u03bf\u1fe6\u03bd, ii. 7\n\n\n        Sabbath, observance of, by Christ and the Essenes compared, p.\n           170\n        Sab\u00e6ans, p. 165\n        sacrifices prohibited by Essenes, p. 89, 134\n        Sadduceeism, p. 82\n        Sagaris, Bishop of Laodicea, p. 63\n        Saman\u00e6i, p. 154\n        Samps\u00e6ans, p. 137\n        Sarman\u00e6, p. 153\n        satisfactori\u00e6, sufferings of Christ regarded as, i. 25\n        Secundus, see _Pedanius Secundus_\n        Sibylline Oracle, p. 96\n        silence of Eusebius, p. 52 sq.;\n          of the New Testament about the Essenes, p. 159\n        slave martyrs, p. 392\n        slavery, Hebrew, p. 385;\n          Greek, p. 386;\n          Roman, p. 387;\n          St Paul's treatment of, p. 389 sq.;\n          attitude of Christianity towards, p. 391 sq.;\n          prohibited by Essenes, p. 177;\n          legislation of Constantine, p. 393;\n          of Justinian, p. 394;\n          abolition of, p. 394\n        Socrates on Novatianism in Phrygia, p. 98\n        solidarity of the Church in the second century, p. 62\n        Sophia of Valentinus, p. 333;\n          Sophia Achamoth, p. 334\n        soteriology of the New Testament, p. 185\n        stadium, metaphor from the, ii. 18\n        Stapleton receives the Epistle to the Laodiceans as genuine, p.\n           366\n        Strabo on Buddhism, p. 153\n        Sunworship, p. 87, 137 sq., 149\n        \u03c3\u03ac\u03b2\u03b2\u03b1\u03c4\u03b1, ii. 16\n        \u03c3\u03ac\u03c1\u03be, \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, i. 22\n        \u03a3\u03ba\u03cd\u03b8\u03b7\u03c2, iii. 11\n        \u03c3\u03bf\u03c6\u03af\u03b1, i. 9, 28, ii. 3, iii. 16\n        \u03c3\u03c0\u03bb\u03ac\u03b3\u03c7\u03bd\u03b1 (\u03c4\u1f70), iii. 12, Ph. 7, 12\n        \u03c3\u03c4\u03b5\u03c1\u03ad\u03c9\u03bc\u03b1, ii. 5\n        \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u1fd6\u03b1 (\u03c4\u1f70), ii. 8\n        \u03c3\u03c5\u03bb\u03b1\u03b3\u03c9\u03b3\u03b5\u1fd6\u03bd, ii. 8\n        \u03c3\u03c5\u03bc\u03b2\u03b9\u03b2\u03ac\u03b6\u03b5\u03b9\u03bd, ii. 2, 19\n        \u03c3\u03c5\u03bd\u03b1\u03b9\u03c7\u03bc\u03ac\u03bb\u03c9\u03c4\u03bf\u03c2, iv. 10\n        \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03bc\u03bf\u03c2, ii. 19, iii. 14\n        \u03c3\u03cd\u03bd\u03b4\u03bf\u03c5\u03bb\u03bf\u03c2, i. 7, iv. 7\n        \u03c3\u03cd\u03bd\u03b5\u03c3\u03b9\u03c2, i. 9, ii. 2\n        \u03c3\u03c5\u03c3\u03c4\u03c1\u03b1\u03c4\u03b9\u03ce\u03c4\u03b7\u03c2, Ph. 2\n        \u03c3\u1ff6\u03bc\u03b1, \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u03cc\u03c2, ii. 11\n        \u03c3\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2, ii. 9\n\n\n        Tacitus on the earthquake at Laodicea, p. 39\n        Talmud, supposed etymologies of Essene in, p. 116 sq., 125 sq.;\n          supposed allusions to the Essenes, p. 128\n        Temple, avoidance of the, p. 174\n        Testaments, Old and New, p. 185\n        Testaments of the Twelve Patriarchs, on the orders of angels, i.\n           16\n        theanthropism of the New Testament, p. 185\n        thundering legion, p. 61\n        Thyatira, dyes of, p. 4\n        Timotheus, his position in these epistles, i. 1, Ph. 1;\n          'the brother,' i. 1\n        Tivoli compared with the valley of the Lycus, p. 3\n        travertine deposits in the valley of the Lycus, p. 3\n        Trimetaria, a surname of Laodicea, p. 18\n        Tychicus, iv. 7, p. 35, 380.\n        \u03c4\u03b1\u03c0\u03b5\u03b9\u03bd\u03bf\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7, iii. 12\n        \u03c4\u03ac\u03be\u03b9\u03c2, ii. 5\n        \u03c4\u03ad\u03bb\u03b5\u03b9\u03bf\u03c2, i. 28\n        \u03c4\u03b9\u03c2 (indef.), St Paul's use of, ii. 8\n        \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03c2 \u1f64\u03bd, Ph. 9, 12\n        \u03b8\u03ad\u03bb\u03b5\u03b9\u03bd, Ph. 13;\n          \u03b8\u03ad\u03bb\u03b5\u03b9\u03bd \u1f10\u03bd, ii. 18\n        \u03b8\u03ad\u03bb\u03b7\u03bc\u03b1 \u03b8\u03b5\u03bf\u1fe6, i. 1\n        \u03b8\u03b5\u03bc\u03b5\u03bb\u03b9\u03bf\u1fe6\u03bd, i. 23\n        \u03b8\u03b5\u03cc\u03c4\u03b7\u03c2, \u03c4\u1f78 \u03b8\u03b5\u1fd6\u03bf\u03bd, ii. 9\n        \u03b8\u03b9\u03b3\u03b3\u03ac\u03bd\u03b5\u03b9\u03bd, ii. 21\n        \u03b8\u03bd\u03ae\u03c3\u03ba\u03b5\u03b9\u03bd, \u1f00\u03c0\u03bf\u03b8\u03bd\u03ae\u03c3\u03ba\u03b5\u03b9\u03bd, ii. 20\n        \u03b8\u03c1\u03b9\u03b1\u03bc\u03b2\u03b5\u03cd\u03b5\u03b9\u03bd, ii. 15\n        \u03b8\u03c5\u03bc\u03cc\u03c2, iii. 8\n        \u03b8\u03cd\u03c1\u03b1 \u03c4\u03bf\u1fe6 \u03bb\u03cc\u03b3\u03bf\u03c5, iv. 3\n\n\n        \u1f55\u03bc\u03bd\u03bf\u03c2, iii. 16\n        \u1f51\u03c0\u03b5\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2, ii. 14\n        \u1f51\u03c0\u03bf\u03bc\u03bf\u03bd\u03ae, i. 11\n        \u1f51\u03c3\u03c4\u03b5\u03c1\u03ae\u03bc\u03b1, i. 24, p. 335 sq.\n\n\n        Valentinianism, different forms of, p. 332 sq.\n        Valentinians accept St Paul and St John, p. 336\n        Valentinus, use of pleroma by, p. 331\n        versions of the Epistle to the Laodiceans, Latin, p. 357;\n          Bohemian, German, and English, p. 363 sq.\n        Vethikin, p. 131\n\n\n        Word, the, p. 101, see _Logos_, _Christ_\n        Wycliffe excluded the Apocryphal Epistle to the Laodiceans from\n           his Bible, p. 363\n\n\n        Yavana or Yona, p. 152\n\n\n        Zeller on Essenism, p. 143 sq.\n        Zenda-vesta, p. 149\n        Zoroastrianism and Essenism, p. 149 sq.\n\n        CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.\n\n\n\n\n                               Transcriber's Note\n\n        Minor lapses in punctuation have been rectified. Certain other\n        editorial or printing errors have been noted below, and\n        corrected.\n\n        On p. 286, in the note on i. 12 of Colossians for \u1f61\u03c2 \u1f10\u03ba\u03bb\u03b5\u03ba\u03c4\u03bf\u1f76\n        \u03c4\u03bf\u1fe6 \u0398\u03b5\u03bf\u1fe6, the citation of Romans 8:33, referring to 'God's\n        elect' was incorrect, appearing as Romans 8.3. The correction\n        has been made.\n\n        On p. 321. there are two separate references to footnote 537.\n        This seems to be intentional, and have been retained. Likewise,\n        on p. 343, footnote 609 is repeated. It seems most likely that\n        the second instance is correct, but both have been retained.\n\n        On p. 416, in the Index, the entry for \u1f05\u03bb\u03b1\u03c2 refers to iv. 5.\n        However, the word and its gloss appear in iv. 6.\n\n p. 28               He recal[l]s                        Added.\n\n p. 43        n. 143 coin[ci]dence                       Added.\n\n p. 121       n. 339 Zeitsc[h]rift                       Added.\n\n p. 199              a similar pheno[nem\/men]on          Transposed.\n\n p. 213              _c. Eunom._ iv ([p. ]I.p. 292)      Removed.\n\n p. 230              The thought underlying \u03bd\u1fe6\u03bd seems    Added.\n                     [to] be this\n\n p. 231              explaining it [as ]'the later'      Added.\n\n p. 323              theological con[fu]sion             Added.\n\n p. 373       n. 680 from the allied f[ro\/or]m           Transposed.\n                     _Apphias_.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of St. Paul's Epistles to the Colossians\nand Philemon, by J. B. Lightfoot\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPublished by\n\nFrommer Media LLC\n\nCopyright \u00a9 2014 by Frommer Media LLC, New York City, New York. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, Frommer Media LLC, 44 West 62nd Street, New York, NY 10023, or online at <http:\/\/www.frommers.com\/permissions>.\n\nFrommer's is a registered trademark of Arthur Frommer. Used under license. All other trademarks are the property of their respective owners. Frommer Media LLC is not associated with any product or vendor mentioned in this book.\n\nISBN 978-1-62887-068-8 (paper), 978-1-62887-069-5 (e-book)\n\nEditorial Director: Pauline Frommer  \nEditor: Pauline Frommer  \nProduction Editor: Heather Wilcox  \nCartographer: Roberta Stockwell  \nPage Compositor: Lissa Auciello-Brogan  \nCover Design: Howard Grossman\n\nFor information on our other products or services, see www.frommers.com.\n\nFrommer Media LLC also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats.\n\nManufactured in the United States of America\n\n5 4 3 2 1\nCONTENTS\n\n 1 The Best of the Virgin Islands\n\n 2 Suggested Itineraries\n\nThe U.S. Virgin Islands in 10 Days\n\nThe British Virgin Islands in 10 Days\n\nIsland-Hopping for 10 Days\n\n 3 The Virgin Islands in Context\n\nThe Islands in Brief\n\nThe Virgin Islands Today\n\nLooking Back: Virgin Islands History\n\nDateline\n\nThe Virgin Islands in Popular Culture\n\nEating & Drinking\n\nWhen to Go\n\nThe Virgin Islands Calendar of Events\n\nResponsible Travel\n\n 4 St. Thomas\n\nEssentials\n\nFast Facts: St. Thomas\n\nWhere to Stay\n\nWhere to Eat\n\nExploring St. Thomas\n\nWalking Tour: Charlotte Amalie\n\nShopping\n\nSt. Thomas After Dark\n\n 5 St. Croix\n\nEssentials\n\nFast Facts: St. Croix\n\nWhere to Stay\n\nWhere to Eat\n\nExploring St. Croix\n\nWalking Tour: Christiansted\n\nShopping\n\nSt. Croix After Dark\n\nSide Trip to Buck Island\n\n 6 St. John\n\nEssentials\n\nFast Facts: St. John\n\nWhere to Stay\n\nWhere to Eat\n\nExploring St. John\n\nShopping\n\nSt. John After Dark\n\n 7 The British Virgin Islands\n\nEssentials\n\nTortola\n\nFast Facts: Tortola\n\nVirgin Gorda\n\nFast Facts: Virgin Gorda\n\nJost Van Dyke\n\nAnegada\n\nPeter Island\n\nGuana Island\n\n 8 Planning Your Trip to the Virgin Islands\n\nIndex\nABOUT THE AUTHOR\n\n**Alexis Lipsitz Flippin** is the author of \"Frommer's Turks & Caicos,\" \"Frommer's St. Maarten\/St. Martin,\" \"Anguilla & St. Barts,\" \"The Food Lovers' Guide to Manhattan,\" and \"Frommer's NYC with Kids\" and the co-author of \"Frommer's 500 Extraordinary Islands,\" \"Frommer's Caribbean,\" and \"Frommer's Carolinas & Georgia.\" She has written for numerous magazines and webzines, including CNN.com, MSNBC.com, Zagat.com, and AARP.com and is a former Senior Editor of the Frommer's travel guides.\n\nABOUT THE FROMMER's TRAVEL GUIDES\n\nFor most of the past 50 years, Frommer's has been the leading series of travel guides in North America, accounting for as many as 24 percent of all guidebooks sold. I think I know why.\n\nAlthough we hope our books are entertaining, we nevertheless deal with travel in a serious fashion. Our guidebooks have never looked on such journeys as a mere recreation, but as a far more important human function, a time of learning and introspection, an essential part of a civilized life. We stress the culture, lifestyle, history, and beliefs of the destinations we cover and urge our readers to seek out people and new ideas as the chief rewards of travel.\n\nWe have never shied from controversy. We have, from the beginning, encouraged our authors to be intensely judgmental, critical\u2014both pro and con\u2014in their comments, and wholly independent. Our only clients are our readers, and we have triggered the ire of countless prominent sorts, from a tourist newspaper we called \"practically worthless\" (it unsuccessfully sued us) to the many rip-offs we've condemned.\n\nAnd because we believe that travel should be available to everyone regardless of their incomes, we have always been cost-conscious at every level of expenditure. Although we have broadened our recommendations beyond the budget category, we insist that every lodging we include be sensibly priced. We use every form of media to assist our readers and are particularly proud of our feisty daily website, the award-winning Frommers.com.\n\nI have high hopes for the future of Frommer's. May these guidebooks, in all the years ahead, continue to reflect the joy of travel and the freedom that travel represents. May they always pursue a cost-conscious path, so that people of all incomes can enjoy the rewards of travel. And may they create, for both the traveler and the persons among whom we travel, a community of friends, where all human beings live in harmony and peace.\n\nArthur Frommer\n1\n\nTHE BEST OF THE VIRGIN ISLANDS\n\nMountainous and luminously green, the Virgin Islands number about 100, some governed by the United States and others by Great Britain. The larger islands appear as mossy green hills rising dramatically out of turquoise seas; others are little more than rocky outcroppings rimmed by whispery white-sand beaches waiting for Robinson Crusoe to call. The former haunt of derring-do sea captains and pirate marauders, today many of the Virgin Islands are invaded by thousands of visitors, arriving by plane and cruise ship, suntan lotion in hand.\n\nThe region's major islands include the three United States territories: Bustling St. Thomas attracts the most visitors, many of them disembarking from some of the biggest cruise ships in the business; the \"plantation island,\" St. Croix is the Virgins' largest island and some say its cultural heart; and the lush beauty known as St. John, at 9 miles long and 5 miles wide, is the smallest of the three. No matter where you're traveling from, St. Thomas is for many people the gateway to the Virgins. With the busiest cruise-ship harbor in the Caribbean, St. Thomas bustles with duty-free shopping and global dining. St. Croix is more laid-back than St. Thomas, with well-preserved colonial towns and verdant countryside dotted with plantation ruins. Little St. John is positively sleepy, two-thirds of its acreage taken up by one of America's most beautiful national parks.\n\nWith its dizzying mountain topography and scalloped coastline of shimmering blue coves and powdery beaches, the B.V.I. remains a pristine retreat for yachties and visitors who want to a true escape from the scrum of modern civilization. With steady tradewinds and scores of protected deep-water harbors, the B.V.I. offer some of the best sailing grounds in the Caribbean. Many boaters base themselves on Tortola, the largest island in the B.V.I. and its capital\u2014it's a relaxed spot with something for everyone. Beautiful and sparsely populated Virgin Gorda is the place to go for luxury stays in secluded resorts. Dotted about the main islands are private island retreats and uninhabited islands perfect for castaway day-tripping.\n\nFor beach lovers, the Virgin Islands contain some of the most celebrated white-sand beaches in the West Indies, including Magens Bay on St. Thomas, Trunk Bay on St. John, and Cane Garden Bay on Tortola. Swimming and snorkeling await you at every cove\u2014and in the vibrant coral reef ringing Buck Island, St. Croix has America's only underwater national monument. Throughout the island archipelago are also miles of idyllic hiking trails, tracing the sinous curves of these scenic volcanic wonders.\n\nTHE most authentic EXPERIENCES\n\n  Island-Hopping by Sea: Whether you're traveling the liquid expanse of the local waterways by ferry, sailboat, or mega yacht, seeing the Virgin Islands by sea feels like the way nature intended it. Most visitors take to the waters at some point in their trip, cruising to big-shouldered islands or exploring uninhabited cays. For many, it's the most peaceful and relaxing way to travel the Virgins.\n\n  Waking Up to Tropical Birdsong and Roosters Crowing: No matter where you are on the islands, you will be accompanied by the musical chatter of tropical birdsong, from the twittering of colorful birds to the morning, noon, and night cock-a-doodle-dooing of roadside roosters in splendid plumage of gold, green, and blue.\n\n  Swimming with Turtles and Starfish: The Virgins' undersea marine habitat is as beauteous as its scenic topography, making this a stupendous place to sightsee beneath the waves. Look for hawksbill turtles grazing on seagrass, angelfish darting in and out of rocks, and starfish stretching out on sandy sea bottoms.\n\n  Kicking Back with Serious Views: The islands' curvaceous terrain is truly swoon-worthy, and practically every island has restaurants, cafes, and bars that take full advantage of the panoramic views. Claim your perch (and a cool rum punch) at places like the Mafolie restaurant, overlooking the glittering harbor in Charlotte Amalie (St. Thomas; p. ); Bananakeet, 400 feet above the north shore beaches of Tortola (p. ); or the Hilltop Restaurant at Biras Creek, on Virgin Gorda, high over the inky-blue North Sound (p. ).\n\n  Celebrating the Solstice with Full-Moon Beach Parties: Locals and visitors come out to play for the islands' monthly full-moon celebrations. The most celebrated are on Tortola, from Bomba's Full Moon Party on Cappoon Bay (p. ) to artist Aragorn's Fireball Full Moon Party at Trellis Bay (p. ).\n\nTHE best BEACHES\n\nThe Virgin Islands are known for beautiful beaches of soft white sand and azure seas. Best of all, every beach is open to the public and, with a few exceptions, free. Even private resorts\u2014which often command some of the prettiest stretches of sand\u2014are required to offer public access to their beaches.\n\n  Magens Bay Beach (St. Thomas): This long, half-mile stretch of soft sand, boasting remarkably calm waters, is the most popular and picturesque beach on St. Thomas. Two peninsulas protect the shore from erosion and strong waves, making Magens an ideal spot for swimming. Expect a crowd in the high season. See p. .\n\n  Lindquist Beach (St. Thomas): A lovely, undeveloped beach on the East End, Lindquist is only reachable by a dirt road; it's a favorite of locals who make Sundays here a lively beach day. See p. .\n\n  Trunk Bay (St. John): St. John has so many good beaches it's almost impossible to pick the best, but this, the island's most popular beach, is protected by the U.S. National Park Service and has an underwater snorkeling trail. See p. .\n\n  Sandy Point (St. Croix): The biggest beach in the U.S. Virgin Islands, Sandy Point is a beauty\u2014the final, redemptive scene of \"The Shawshank Redemption\" was filmed here. Because the beach is a protected reserve and a nesting spot for endangered sea turtles, it's open to the public only on Saturdays and Sundays from 10am to 4pm. See p. .\n\nU.S.vs. BRITISH VIRGIN ISLANDS\n\nIf you're trying to decide which Virgin Island to visit, consider the following basics:\n\n  American citizens don't need a passport to enter the U.S. Virgins, but everyone needs a passport to enter the British Virgin Islands.\n\n  The U.S. Virgin Islands are easier to reach for most people. Currently, there are no direct flights on commercial carriers from North America or Europe to Tortola or any of the other British Virgin Islands. St. Thomas is the gateway to the islands, with major carriers connecting into the island's international airport.\n\n  You'll have more hotels and resorts to choose from in the U.S. Virgin Islands, of all sizes and price ranges, than are in the B.V.I.\n\n  The U.S. dollar is the official currency for both island chains, and English is the official language.\n\nThe obvious issues aside, American and British cultures have left different imprints on the Virgin Islands. The classic knock on the U.S. Virgins is that they're overbuilt and over-commercialized and not \"virgins\" anymore. Yes, St. Thomas offers a somewhat Americanized hurly-burly, with easy access to familiar global brands and a wide range of goods and services. But this is no fast-paced megalopolis\u2014chickens still skitter along hillsides and tropical foliage blankets the landscape with a fetching unruliness. St. Croix has big commercial chains like Home Depot and K-Mart, but it also has bucolic stretches of rural countryside, a uniquely rich culture, and an earthy sensuality. With two-thirds of its terrain protected national parkland, St. John is a haven of undeveloped tranquillity.\n\nStill, the British Virgin Islands are noticeably sleepier and less developed. Many of them recall the way the Caribbean was before the advent of high-rise condos, fast-food chains, and mega cruise ships\u2014and being a little remote hasn't hurt. Tortola is where the B.V.I. action is, limited as its shopping and nightlife may be (although traffic gridlock has found its ornery way here, particularly in the B.V.I. capital, Road Town, on cruise-ship days). To the east, beautiful, pristine Virgin Gorda has just 3,000 inhabitants but most of the Virgin Islands' top resorts. Crime is minimal here. You'll find an even more laidback vibe on less-populated islands like Jost Van Dyke and Anegada, as well as uninhabited isles with hidden coves and beaches where you may be the only sunbather. It's a sailor's paradise.\n\nIn truth, the days of traveling to one versus the other are pretty much history. Most people who visit the Virgins touch down on both U.S. and British soil. Plus, the region's dependency on tourism has made it increasingly easier to hop between islands. For suggested itineraries between the islands, see chapter 2.\n\n  Cane Garden Bay (Tortola): A scenic beauty, Cane Garden Bay is the most popular beach on Tortola. Its translucent waters and sugar-white sands attract crowds (especially when cruise ships drop off van loads of beachgoers), but it offers plenty of sand to play on. Across the water is Jost Van Dyke; rising behind the beach are green hillsides dotted with villas. See p. .\n\n  Smuggler's Cove (Tortola): This fetching West End beach is reached by driving down a (largely) dirt road through a grove of mature palm trees. It's got good snorkeling but no facilities.\n\n  Savannah Bay (Virgin Gorda): Just around the corner from the beach at Little Dix Bay is this often-deserted gem, with gin-clear waters and a crescent slice of soft white sand.\n\n  Anegada: The second-largest island in the B.V.I. is the most remote of the Virgins and home to only 200 permanent citizens. It's the island chain's only coral island as well, which makes it one big, beautiful stretch of (largely undeveloped) powdery sand.\n\n  White Bay, Guana Island (Guana Island): This pretty-as-a-picture ivory beach is fringed in palm trees and rising green hills.\n\nTHE best SNORKELING SPOTS\n\n  Coki Point Beach (St. Thomas): On the north shore of St. Thomas, Coki Point offers superb year-round snorkeling. Explore the coral ledges near Coral World's underwater tower. See p. .\n\n  Hurricane Hole (St. John): You may not immediately think of a mangrove forest as a great place to snorkel, but here coral grows in abundance on the mangrove roots\u2014attended by huge starfish, sponges (and the hawksbills that eat them), and anemones. It's magical. SerenaSea runs snorkeling and sightseeing tours out of Coral Bay to Hurricane Hole (<http:\/\/serenasea.com>). See p. .\n\n  Waterlemon Cay\/Leinster Bay (St. John): Easily accessible Leinster Bay, on the northern shore of St. John, offers calm, clear, uncrowded waters teeming with sea life. See p. .\n\n  Haulover Bay (St. John): A favorite with locals, this small bay is rougher than Leinster, with a pebbly beach. The snorkeling, however, is dramatic, with ledges, walls, and nooks to explore. See p. .\n\n  Cane Bay (St. Croix): One of the island's best diving and snorkeling sites is off this breezy, north-shore beach. On a good day, you can swim out 450 feet to see the Cane Bay Wall, which drops dramatically off to the deep waters below. Multicolored fish, plus elkhorn and brain coral, flourish here. See p. .\n\n  Buck Island (off St. Croix): This tiny island, whose land and offshore waters together are classified as a national monument, lies 2 miles off the north coast of St. Croix. More than 250 recorded species of fish swim through its reef system. A variety of sponges, corals, and crustaceans also inhabit the area. See p. .\n\n  Norman Island & the Indians (B.V.I.): Snorkel the calm waters of the Bight near Norman Island. Bring some bread to draw reef fish to the surface when you snorkel the deep waters around the Indians\u2014four fingers of rock jutting out of the sea and only accessible by boat. See p. .\n\n  The Baths (Virgin Gorda): It's often overrun with boats, but the Baths\u2014and nearby beaches, Spring Bay and Devil's Bay\u2014are still mind-blowingly beautiful and the shallow crystalline seas and caves a fun place to explore by snorkel. See p. .\n\nTHE best DIVE SITES\n\n  Cow and Calf Rocks (St. Thomas): This site, off the southeast end of St. Thomas (about a 45-min. boat ride from Charlotte Amalie), is the island's best diving spot. It's also a good bet for snorkeling. You'll discover a network of coral tunnels filled with caves, reefs, and ancient boulders encrusted with coral. See p. .\n\n  The Cane Bay Wall (St. Croix): Walk right off the beach into one of the most awesome dives in the Virgins. It's just a 100-yard swim to the 3,000-foot vertical wall at Cane Bay. Even at depths of 30 feet, you'll see coral gardens abloom with fantastical formations and colorful tropical life. See p. .\n\n  Frederiksted Pier (St. Croix): The Fredriksted pier is one of the best spots in the islands for an electric night dive, where you plunge right offshore into a world of exotic creatures, including sea horses, lobster, and octopuses. See p. .\n\n  The Wreck of the RMS Rhone (off Salt Island, B.V.I.): Many people think the Rhone wreck is the premier dive site not only in the Virgin Islands, but in the entire Caribbean. This royal mail steamer, which went down in 1867, was featured in the film \"The Deep.\" See p. .\n\n  Chikuzen (off Tortola): Although it's not the Rhone (see above), this 269-foot steel-hulled refrigerator ship, which sank off the island's east end in 1981, is one of the British Virgin Islands' most fascinating dive sites. The hull\u2014still intact under about 24m (79 ft.) of water\u2014is now home to a vast array of tropical fish, including yellowtail, barracuda, black-tip sharks, octopus, and drum fish. See p. .\n\n  Alice in Wonderland (Ginger Island, off Tortola): This brilliant coral wall, off the shore of a tiny island, slopes from 12m (39 ft.) to a sandy bottom at 30m (98 ft.). Divers often refer to the site as \"a fantasy\" because of its monstrous overhangs, vibrant colors, gigantic mushroom-shaped corals, and wide variety of sea creatures\u2014everything from conch and garden eels to long-nose butterfly fish. See p. .\n\nTHE best NATURE WALKS\n\n  The Annaberg Historic Trail (St. John): This paved walk is only .25 miles long, but it's a highlight of the 10,000-acre U.S. Virgin Islands National Park. The trail traverses the ruins of what was once the most important sugar-cane plantation on the island. Slaves' quarters, a windmill tower, and ballast-brick buildings are remnants of a long-vanished era. Stunning views look toward Tortola and Jost Van Dyke on the opposite side of Sir Francis Drake Passage. See p. .\n\n  The \"Rain Forest\" Hike (St. Croix): At the northwestern end of St. Croix lies the 15-acre \"rain forest,\" dense with magnificent tropical foliage. The little-traveled four-wheel-drive roads through the area make great hiking paths. See p. .\n\n  Virgin Gorda Peak (Virgin Gorda): Trek to the top of the island's highest peak, Virgin Gorda (414m\/1,359 ft.), on this 50-minute round-trip hike through tropical forest. The views from the top are utterly breathtaking\u2014you'll have views of both the Caribbean and Atlantic oceans. See p. .\n\n  The Sage Mountain National Park Hike (Tortola): This 3- to 4-hour hike is one of the most dramatic in the British Virgins. It goes from Brewer's Bay to the top of Mount Sage, the highest peak in the Virgin Islands, at 523m (1,716 ft.). Along the way, you'll see intriguing ruins of old homes in addition to the beautiful flora and fauna of the park's primeval forest. See p. .\n\nTHE best RESORTS\n\n  The Buccaneer (St. Croix): Family-owned and built around the historic ruins of an old sugar plantation, this St. Croix resort is a class act, stretching over so many rolling acres that it comfortably encompasses an 18-hole golf course, eight championship tennis courts, a spa and health club, a 2-mile jogging trail, and three scenic beaches. See p. .\n\n  Caneel Bay (St. John): Laurance S. Rockefeller created Caneel Bay as the Caribbean's first eco-resort back in the days when no one knew what that word meant. It's an understated classic, an outpost of refinement without ostentation, and operating at the top of its game. The resort's low-rise, low-impact structures front some of the island's best beaches. See p. .\n\n  Biras Creek Resort (Virgin Gorda): The ultimate in serene privacy, this resort is built around an old hillside fortress that's high up overlooking the liquid expanse of North Sound. It's an escapist's hideaway, no question, with luxe rooms, bicycles to breeze around in, and Boston whalers to comb North Sound on your very own. There are no phones or TVs, but who needs it when you've got iguanas, chickens, brilliant Atlantic Ocean surf, and a thousand stars above to sing you to sleep? See p. .\n\n  The Ritz-Carlton (St. Thomas): This is the top-tier place to stay on St. Thomas. Overseen by a grand Italianate palazzo and fronted by white-sand beaches, this full-service resort offers the kind of manicured comfort that's luxurious if somewhat generic; still, rooms and suites are spacious and outfitted with sumptuous bathrooms and top-of-the-line amenities. See p. .\n\n  Bitter End Yacht Club (Virgin Gorda): Entertaining a lively scrum of sailors, families, and beach lovers, this sailing and diving resort opens onto North Sound, one of the most unspoiled deepwater harbors in the Caribbean. It feels a little like a rakish colonial outpost, with boats coming and going and an open-air lobby straight out of a Somerset Maugham tale, but it's also a peaceful getaway, with luxury treehouse-style lodgings high above the seas. See p. .\n\n  Frenchmans B.V.I. (Tortola): The top lodging on Tortola, this small boutique property combines all the perks of a resort\u2014including a topnotch restaurant\u2014with the seclusion of a private villa. All cottages face the blue sweep of Frenchman's Bay, with pelicans whirling and diving and terraces kissed by warm tradewinds. See p. .\n\n  Guana Island (Guana Island): Of all the very fine resorts in the B.V.I., this private island eco-retreat may be ne plus ultra in terms of sheer comfort and romantic seclusion, even if it's a glorious and gracious throwback to the days before technological intrusions ruled daily life. This is rustic old-school luxury, where the service is hushed and impeccable, and the sea breezes are the best air-conditioning you'll ever need. See p. .\n\n  Rosewood Little Dix Bay (Virgin Gorda): Warm service and understated elegance make this wonderful, 494-acre resort popular with families, older couples, and honeymooners alike. It celebrated its 50th anniversary in 2014. This impeccably run property offers sporting activities galore, the best spa in the Virgin Islands, and, in the beautiful wooden Pavilion, the bonhomie heart of the resort, a fizzy and convivial place to wine and dine. See p. .\n\nTHE best FAMILY RESORTS\n\n  Bolongo Bay Beach Resort (St. Thomas): This fun and laidback place, right on the beach, has lots of sand to run around on, water sports aplenty, and a happening restaurant with a lively weekly Carnival. All of the 62 rooms face the beachfront, each with balcony or terrace. See p. .\n\n  Westin St. John Resort & Villas (St. John): This contemporary mega-resort, set on 34 acres of neatly landscaped grounds and fronting a nice beach with gentle seas, offers some of the best children's programs on the island and good watersports. See p. .\n\n  The Buccaneer (St. Croix): This longtime family favorite resort is packed with on-site facilities for just about every sport you can think of, including tennis, golf, swimming, jogging, sailing, scuba diving, and snorkeling. Children's programs include a half-day sail to Buck Island Reef and nature walks through tropical foliage, where kids can taste local fruit in the wild. See p. .\n\n  The Palms at Pelican Cove (St. Croix): This small resort has something for everyone, but it has a lot to offer families, including the fact that all kids 17 and under stay for free in their parent's rooms And kids 10 and over are treated to complimentary scuba lessons. The resident iguana and pelican don't hurt either. See p. .\n\n  Long Bay Beach Club (Tortola): Downsized to a boutique resort with just 42 rooms, this resort is spread out along a wonderful sandy beach licked by gentle waves\u2014you can snorkel right off the beach. Its deluxe beachfront rooms can sleep five, and adjoining suites are available. Beachfront suites have full kitchens, and all rooms come with personal grills for impromptu family barbecues. See p. .\n\n  Bitter End Yacht Club (Virgin Gorda): This lively resort is very kid-friendly, and its festive main restaurant, the Clubhouse, is ideal for noisy, chattering families planning a big day of water play. Most children's programs are geared toward those ages 6 and over and involve all the typical watersports: sailing, windsurfing, snorkeling, swimming, and more. See p. .\n\nTHE best BAREFOOT RETREATS\n\nSome lodgings don't even try to tart up the place, preferring unvarnished barefoot charm to fancy frills, luxury amenities, and hermetically sealed interiors. It helps, of course, to have a glorious natural setting to prop up those bare feet. Old-school, the Caribbean the way it used to be, relaxed and comfortably rustic: The following places hum to the rhythms of the natural world.\n\n  Concordia Eco-Tents (St. John): Overlooking Salt Pond Bay, these tent-cottages high up on a secluded hillside are kept cool by the constant trade winds. The resort is quiet and serene: You're pillowed in hundreds of acres of green national parkland. Got any complaints? See p. .\n\n  Cooper Island Beach Club (Cooper Island): This escapist's retreat on remote Cooper Island is smartly designed and very comfortable, with 10 rooms in simple cottages built of reclaimed timber. It's \"fan-ventilated\"\u2014that is, no A\/C\u2014and don't expect TV either. Do expect yachties and divers dropping in at the lively bar and restaurant. See p. .\n\n  Sandcastle Hotel (Jost Van Dyke): This funky, unpolished hotel is perfect for devotees of laid-back getaways. The six phone- and TV-free cottages nestled in bougainvillea enjoy panoramic views of an idyllic white-sand beach. When you're not padding the beach in your bare feet, you can hit the famed beachside bar, the Soggy Dollar, home of the Painkiller cocktail. See p. .\n\n  Anegada Reef Hotel (Anegada): This 20-room hotel is located on a flat mass of coral and limestone, one of the most remote spots in the entire Virgin Islands. Yes, rooms have been nicely freshened up, and you can blast the air-conditioning if you prefer it to ceiling fans. But it's still the kind of place where, if the bartender isn't around, you make your own cocktails and write down what you had. See p. .\n\nTHE best RESTAURANTS\n\n  Havana Blue (Morning Star Beach Club at the Marriott Frenchman's Reef, St. Thomas): You would be forgiven if you thought the food would be relegated to a secondary role at this beachside restaurant, given the bombshell bar staff, sexy blue lighting, and thumping soundtrack. But someone very talented is in the kitchen. Yes, miso-glazed fish has been done since the dawn of time, but Havana Blue's iteration (with sea bass and lemongrass) is melt-in-the-mouth perfection. See p. .\n\n  Gladys' Cafe (Charlotte Amalie, St. Thomas): Smack dab in the middle of tourist central, this local treasure is hidden away in plain sight in the maze of cobblestoned alleyways of Royal Dane Mall. Inside 18th-century walls of native stone, you'll dine on refined versions of Caribbean specialties (conch fritters, jerk fish, peas and rice) and surefire winners like avocado stuffed with lobster salad. Plus, Gladys makes her own hot sauce and even breaks into robust song whenever she feels like it. See p. .\n\n  Mafolie Restaurant (Mafolie Hotel, St. Thomas): High up in the hills above Charlotte Amalie, with soul-stirring views of the twinkling lights of the harbor and beyond, this restaurant is a real find. Chef Manny Thompson is a masterful cook, and the owners are committed to the island's burgeoning farm-to-table movement. And those views! See p. .\n\n  Sunset Grille (Secret Harbour Resort, St. Thomas): Serving some of the most flavorful and innovative dishes on the island, the Sunset Grille offers upscale dining in a quintessentially romantic Caribbean setting: overlooking the sandy beach and gentle waters of Secret Harbour. See p. .\n\n  ZoZo's (Caneel Bay Resort, St. John): This marriage of superstars in late 2013 installed St. John's best Italian restaurant in one of the island's most thrilling locations: atop the ruins of an 18th-century sugar mill at Caneel Bay resort. It's a real mashup, where colonial Caribbean meets Frank Lloyd Wright and pumps out a sizzling Sixties-era James Bond vibe. See p. .\n\n  Kendrick's (St. Croix): Kendrick's brings a light Continental touch to richly flavored dishes in a historic Danish building in Christiansted. See p. .\n\n  Brandywine (Tortola): You'll find it down a winding dirt road along Brandywine Bay, an elegant stone treehouse serving equally elegant Mediterranean fare. See p. .\n\n  The Dove (Tortola): One of the top special-occasion restaurants in Road Town is also one of the best restaurants in the entire Virgin Islands, serving a seasonal menu of continental classics. See p. .\n\n  Bananakeet (Tortola): It's got stupendous views from its perch high up on Windy Hill, but the food and service keep folks raving. Plan at least one sunset dinner here. See p. .\n\n  The Pavilion (Little Dix Bay, Virgin Gorda): Each night a different themed buffet is served in this soaring alfresco space\u2014and it's all utterly fresh and absolutely delicious. See p. .\n\n  Hilltop Restaurant (Biras Creek, Virgin Gorda): Dine on impeccably prepared continental classics (with Caribbean influences) as you practically touch the stars from your perch above North Sound. See p. .\n\nTHE best LOCAL BUYS\n\nThe U.S. Virgin Islands are rightfully known as the shopping mecca of the Caribbean. It's tempting indeed to comb the historic streets of St. Thomas's capital, Charlotte Amalie, in search of bargains amid the global outposts of \"luxury\" goods, those international brands of china and crystal, jewelry, and cosmetics that blanket every cruise port. The incentives are strong: Every person gets a $1,600 duty-free allowance and no sales tax is tacked on. But for those seeking original, artisanal gifts made by the people who actually live here, the following is a sampling of our favorite Virgin Islands mementos.\n\n  Aragorn's metal fireballs: Any search for wonderful local art should start at Aragorn's Studio (p. ) on Trellis Bay, Tortola. Tortola-born Aragorn is a printmaker, potter, and sculptor; his giant \"fireballs\"\u2014silhouetted metal sculptures\u2014are set ablaze during the monthly Fireball Full Moon Parties on Trellis Bay. Look for miniature fire balls (candle holders), beautiful original prints, pottery, jewelry, and gifts, the work of Aragorn, inhouse artisans, and regional artists.\n\n  Hot sauce: Virgin Islanders love their hot sauce (aka \"pepper\" sauce), and many island cooks prepare and bottle their own. Look for local favorites like Miss Anna's (St. John), Blind Betty's (St. Croix), Jerome's (St. Thomas), and ValleyDoll (St. John), sold in gift shops and food stores around the islands. It's a real taste of the islands.\n\n  Moko Jumbie holiday ornaments: A traditional presence at Carnival in St. Croix, Moko Jumbies are masked and costumed revelers on stilts. Glittery home-made Moko Jumbie ornaments are the handiwork of Cruzan native and \"scrap-art designer\" Sandra Michael; you can find her work in shops like Franklin's on the Waterfront in Fredriksted (p. ).\n\n  Hand-blown recycled glass gifts: Founded by local nonprofit GreenVI, this outdoor glass-blowing studio on Cane Garden Bay in Tortola recycles bottles of beer and booze discarded from the local beach bars to fashion beautiful hand-blown glass delicacies, from starfish paperweights to turtle ornaments to flower glass sculptures. The GreenVI Glass Studio has trained a number of locals in the intricacies of glass-blowing, and a local octogenarian makes cloth bags out of donated clothing to carry them home in (p. ).\n\n  Local art: Look for genre paintings by the accomplished artist Joseph Hodge, who has a studio in the Craft Alive artisans' village in Road Town, Tortola (p. ). In Christiansted, St. Croix, the 40-year-old Many Hands gallery continues to feature original work by local artists and craftspeople (p. ). In St. Thomas, you can find the latest work of local artisans (dolls, jewelry, soaps) in the Native Arts and Crafts Cooperative (p. ).\n2\n\nSUGGESTED ITINERARIES\n\nSprinkled scattershot between the Caribbean and Atlantic oceans, the Virgin Islands mark the easternmost point of the Lesser Antilles. They're part of a necklace of islands that stretches southeast from Cuba and curls back west again at Trinidad and Tobago. Despite being the progeny of two different nations, the U.S. and British Virgin Islands have a brotherly closeness\u2014not only are the islands a breezy ferry ride or plane hop away from one another, but you're almost always eyeballing another Virgin no matter where you are. From Jost Van Dyke you can watch the glittering lights of cars weave along the coastal road of St. Thomas; from Tortola's West End the broad-shouldered hills of St. John are a comforting constant. The two island chains share the same sparkling waterways, the same vibrant marine playgrounds, the same balmy tradewinds. The history of both the U.S. and British Virgins is inextricably linked with sugarcane and the slave trade\u2014centuries-old plantation ruins throughout the islands tell part of the tale. You'll find the same classic island dishes on restaurant menus and hear the same lilting rhythms in ramshackle beach bars.\n\nAll of which is to say, no matter how long your Virgins vacation, you will most likely be touching down on both U.S. and British soil. It's easy to do so: Ferries connecting St. Thomas, St. John, Tortola, and Virgin Gorda run on a regular basis in season and even link some of the smaller islands, such as Anegada and Jost Van Dyke, albeit on a limited basis. (You'll need to fly or sail by private boat or charter to reach the more remote islands of the BVI.) Still, it's a stretch to say that everything here runs with the efficiency of a Swiss clock, so be sure to plot your itineraries around the frequency of public and private ferry schedules and interisland flights. Use the following suggested itineraries to kick-start your Virgin Islands vacation.\n\nThe U.S. Virgin Islands in 10 Days\n\nYou could easily spend a week or more in each of the U.S. Virgins\u2014St. Thomas, St. Croix, and St. John\u2014and still have plenty more to do and see. But 3 days on each should give you a good sense of the flavorful character of each island. Refer to the individual chapters in this book to find the lodgings and restaurants best suited for you. Keep in mind that ferries between St. Thomas and St. John run regularly all day long (it's just a 20-min. trip from Red Hook on St. Thomas), but you will need to fly between St. Thomas and St. Croix\u2014at least until regular ferry service between the two islands is rebooted.\n\nDay 1: Arrive in St. Thomas\n\nCheck into your hotel and spend whatever time you have upon arrival relaxing or hitting the beach. If you're staying in Charlotte Amalie, call a taxi for dinner at the restaurant at Mafolie Hotel (p. ), with stunning views of the glittering harbor below, or Havana Blue (p. ), beachside at Marriott Frenchman's Reef.\n\nDay 2: Take an Island Tour & Hit the Beach\n\nGet the lay of the land on a full island tour with a tour operator or taxi driver. We recommend the services of taxi driver\/tour guide extraordinaire Campbell Rey (  340\/771-1568). Spend the afternoon on the beach. The island's top beach, Magens Bay (p. ), is recommended if you like clear, gentle seas in a spectacular setting (and who doesn't?). If you want to snorkel, head to Secret Harbour (p. ) or Coki Beach (p. ). If you have kids in tow (and even if you don't), make time after exploring Coki Beach to visit Coral World (p. ), the fun-filled aquarium next door.\n\nDay 3: Shop the Historic District & Ride the Zipline\n\nIf shopping and historic architecture are high on your to-do list, spend the morning touring the cobblestone lanes in Charlotte Amalie\u2014an especially easy proposition if you're staying in the area. Have lunch with the locals at Gladys' Caf\u00e9 (p. ) or the waterfront Petite Pump Room (p. ), near the Charlotte Amalie ferry terminal. If you're game for adventure, you can spend the afternoon at the Tree Limin' Extreme (p. ), a ziplining canopy tour in the rain forest of St. Peter Mountain; you'll have views of Magen's Bay and even Tortola and Jost Van Dyke. Combine a ziplining adventure with a visit to the tropical gardens of the St. Peter Great House & Botanical Gardens (p. ), just across the street. It's a great spot for a sunset cocktail and even dinner.\n\nDay 4: St. Croix   & West End Attractions\n\nTake a morning flight from St. Thomas to St. Croix; air time: 25 minutes. If you're traveling by seaplane (from the St. Thomas seaport), you'll land right on the Christiansted waterfront\u2014and you should explore the island's East End from here (see \"Day 5,\" below). If you fly on regular air shuttle into the St. Croix airport, you'll be closer to West End must-see attractions like the Estate Whim Plantation (p. ), the St. George Village Botanical Garden (p. ), the Ridge to Reef rain-forest farm (p. ), and the Cruzan Rum Factory (p. ). In any event, arrange an airport pickup and island tour with one of the island's expert taxi driver\/tour operators, such Ames Joseph (  340\/277-6133). If it's a Saturday or Sunday, stop off (and take a dip at) at the beautiful beach at Sandy Point (p. ; the wildlife refuge is only open on weekends, and is closed 7 days a week during turtle nesting season Apr\u2013Sept). If you're staying at a beach resort, spend the rest of your afternoon swimming and relaxing.\n\nSuggested Itineraries: The Virgin Islands\n\nDay 5: Tour of Christiansted  , Cane Bay Snorkel   & Salt River Bioluminescent Bay Kayak Tours\n\nTake a walking tour of historic Christiansted with an expert guide from Crucian, Heritage & Nature Tourism (Chant; p. ) and have lunch in town. Alternatively, if you're an avid diver or snorkeler, you should spend the morning at Cane Bay beach (p. ), on the island's north shore, where you can snorkel off the reefs or swim out 100 yards to a steep dropoff. If the moon cycle is right, schedule a see-through-kayak tour of the bioluminescent bay in the Salt River National Historical Park and Ecological Reserve (p. ).\n\nDay 6: Buck Island Snorkeling Trip\n\nThe only marine national park in the United States is well worth a day-trip of fine snorkeling, swimming, and hiking. A number of excursion companies, including Big Beard's Adventure Tours (p. ), offer full and half-day sails to Buck Island National Reef Monument that often include beach barbecues. If you have time on your return, do a little shopping in Christiansted or head south to ARTFarm (p. ), a combination vegetable farm stand and art gallery that opened in 2000 on an old cattle ranch along the island's picturesque south side.\n\nDay 7: St. Thomas & Ferry to St. John\n\nYou may want to fly the seaplane out of St. Croix, which conveniently arrives at the seaplane terminal in Charlotte Amalie, located next door to the Charlotte Amalie ferry terminal. You can easily take a ferry to St. John from here or from the island's East End at Red Hook\u2014that way you avoid a sometimes traffic-ridden taxi ride from the airport. If you plan to return to St. Thomas and stay on the island's East End for a night before flying out, schedule your round-trip ferry between Red Hook and St. John. The ferry from Red Hook to St. John is a pleasant 20-minute ride.\n\nDepending on when you arrive, you may want to explore Cruz Bay (p. ), the island's main town and your arrival point. It has lots of shops and waterfront eateries. St. John is a great place to rent a car for a few days, so you can fully explore its wonderful beaches and national parkland. But if you aren't renting a car, arrange ferry pickup\/transfer\u2014and an island tour\u2014with a local taxi driver. We highly recommend Kenneth Lewis (  340\/776-6865).\n\nDay 8: Swimming & Snorkeling St. John Beaches\n\nThis is the day to beach it, and the island has an abundance of excellent strands of sand for swimming, snorkeling, and exploring. The island's most famous beach, Trunk Bay (p. ), has an underwater snorkeling trail. Go to chapter 6 for more great beaches on St. John.\n\nDay 9: Annaberg Ruins   & Coral Bay\/Beaches\n\nIn the morning, take a tour of the Annaberg sugar plantation ruins (p. ). Then hit the beach. Close to the ruins is Waterlemon Cay (p. ), with great snorkeling and turtles; on the island's eastern side is Hurricane Hole (p. ), where you can snorkel among a coral reef growing on mangroves\u2014or head south to the quiet beach at Little Lameshur Bay (p. ). Have a late lunch in the quaint little settlement of Coral Bay, about a 20-minute drive from Cruz Bay. Sample homemade pulled-pork sandwiches and a rum punch at Tourist Trap (p. ); juicy burgers in the honky-tonk atmosphere of Skinny Legs (p. ); or the grilled jerk fish at Shipwreck Landing.\n\nDay 10: St. Thomas\n\nPlan your ferry trip back to St. Thomas around your departure itinerary.\n\nThe British Virgin Islands in 10 days\n\nWe guarantee that a 8-day week exploring the British Virgin Islands should erase all the stresses of modern civilization. Ten days to 2 weeks: Even better.\n\nDay 1: Arrive in St. Thomas\n\nCurrently there are no direct flights into Tortola, the B.V.I.'s main island, from North America or Europe. For many people, St. Thomas (or, increasingly, San Juan) is the gateway to the British Virgin Islands. Remember to build time into your schedule to make the ferry or plane connections to your B.V.I. destination. Depending on your arrival into St. Thomas (or San Juan), that may include a night's stay-over.\n\nDays 2\u20135: Virgin Gorda\n\nWe suggest starting your B.V.I. trip in Virgin Gorda, if the ferries (or plane connections) comply. If you're relying on ferries, you may have to take a ferry first to Tortola and then catch a second ferry to Virgin Gorda\u2014the entire trip can take an hour or two, but being on the beautiful Virgin Islands waterways is a wonderful way to ease into the laidback island mentality.\n\nBreak up your Virgin Gorda stay into two parts: the Valley and the North Sound. We recommend first exploring the attractions in and around The Valley\u2014the Baths (p. ), Savannah Bay (p. ), Mahoe Bay (p. )\u2014with a stay in wonderful Little Dix Bay resort (p. ), which also happens to have the best spa in the region. Then spend 2 nights exploring the island's other end with a stay at a North Sound resort\u2014whether Biras Creek (p. ), Bitter End (p. ), or Saba Rock (p. ). From there you can sail the North Sound, sunbathe on Prickley Pear Island,\n\nDays 6\u20139: Tortola\n\nThe British Virgin Islands' largest and most populous island, Tortola has a leisurely sensibility, uncrowded beaches, and interesting historical attractions in its capital, Road Town (p. ). Base yourself in the cliffside cottages at Frenchmans Cay (p. ) or down on the beach at the Long Bay Beach Club (p. ). Spend a couple of days exploring the island, swimming the gorgeous beach at Cane Garden Bay (p. ), snorkeling at Smuggler's Cove (p. ), or taking an island tour. We highly recommend the services of taxi driver\/tour guide Wayne Robinson (  284\/494-4097).\n\nTortola is also a great base from which to island-hop  . Take a day-trip to Norman Island and the Indians (p. ), Jost Van Dyke (p. ), Anegada (p. ), or any number of uninhabited isles just waiting for a castaway soul.\n\nDay 10: St. Thomas\n\nFly or take a short (45- to 60-min.) ferry ride from Tortola to St. Thomas. Plan your return ferry trip back around your departure itinerary. Keep in mind that the Charlotte Amalie ferry terminal is much closer to the airport than the Red Hook terminal, and traffic gridlock between the island's East End and West End (where the St. Thomas airport is located) is not uncommon.\n\nIsland-Hopping for 10 Days\n\nAnyone visiting the Virgin Islands for any length of time will want to take advantage of the region's smorgasbord of attractions and activities. Here's a 10-day trip for those who want to hit the highlights of both the U.S. and British Virgin Islands\n\nDays 1 & 2: St. Thomas\n\nFollow Days 2 and 3 in the \"U.S. Virgins in 10 Days\" itinerary, above.\n\nDays 3 & 4: St. John\n\nTake a ferry from St. Thomas (leaving from either the Charlotte Amalie or Red Hook terminals) to Cruz Bay on St. John. Follow Days 8 and 9 in the \"U.S. Virgins in 10 Days\" itinerary, above\n\nDay 5: Tortola\n\nTake a ferry from Cruz Bay in St. John to Tortola. If you're traveling between Cruz Bay and Tortola's West End, the total trip time should be under 30 minutes. Once on Tortola, take an island tour (see \"The British Virgin Islands in 10 Days,\" above) and then settle in on one of the beaches close to your lodging. Use Tortola as a base for the rest of your B.V.I. trip. We recommend staying at one of the West End resorts or renting a villa for a week.\n\nDay 5: Road Town Tour   & Cane Garden Bay\n\nTake the morning to visit some of the Road Town's historical attractions, including the circa-1780 SugarWorks museum (p. ), the J. R. Neal Botanical Gardens (p. ), and old Government House. Don't miss the mural high up on Fahie Road (p. ). Then grab your swimsuit and hit the beach at Cane Garden Bay, enjoying a late lunch on conch fritters and grilled fish at Quito's Gazebo (p. ). Walk over to the GreenVI Glass Studio, the outdoor glass-blowing studio where recycled bottles of beer and booze are fashioned into beautiful hand-blown glass delicacies (p. ). If the moon is full, join the monthly Full Moon Party at Bomba Surfside Shack (p. ), in nearby Cappoons Bay.\n\nDay 6: Island-Hop #1: The Baths   & Savannah Bay\n\nTake a day-trip by boat from Tortola to the beaches in The Valley on Virgin Gorda. Explore the caves and boulders of the world-famous Baths and swim in the sparkling seas around Devil's Bay and Spring Bay (p. ). For utter seclusion, round the bend and swim in the beautiful, clear seas of Savannah Bay and Mahoe Bay (p. ).\n\nDay 7: Island-Hop #2: Dive \"Wreck Alley\"   or Snorkel Norman Island & the Indians\n\nScuba divers can make a beeline (by boat or tour operator) to the area between Cooper and Salt islands known as \"Wreck Alley\"\u2014a world-class diving spot filled with shipwrecks including the famed RMS Rhone, which went down in a hurricane in 1867 (p. ). Snorkelers, on the other hand, can head to Norman Island, where they can snorkel the caves around \"Treasure Isle\" and the deep waters around the four fingers of rock known as the Indians. Have lunch on the beach at Pirate's Bight restaurant (p. ).\n\nDay 8: Jost Van Dyke   & Soper's Hole\n\nTake a day-trip to the barefoot beaches of Jost Van Dyke or Anegada, both sparsely populated islands in the B.V.I. chain. Both islands have laidback restaurants and bars right on the beach; be sure to sample a Painkiller, the cocktail created at famed Foxy's (p. ) in Jost. On your return to Soper's Hole, stop in and shop the marina stores.\n\nDay 9: The East End & Shopping Trellis Bay\n\nSpend a last leisurely day on Tortola by traveling the coastal highway to the island's East End, where you can shop the regional arts-and-crafts treasures at Aragorn's Studio (p. ). If it's full moon time, don't miss the Fireball Full Moon Party here. On your way back west, stop in at Brandywine Estate restaurant (p. ) for an elegant lunch or early dinner and maybe a dip in the calm waters of Brandywine Bay. Or just head to your West End resort and have a fabulous dinner nearby at the Clubhouse at Frenchmans (p. ), the Sugar Mill restaurant (p. ), or Bananakeet (p. ).\n\nDay 10: St. Thomas\n\nFly or take the short (45- to 60-min.) ferry ride from Tortola to St. Thomas. Plan your return ferry trip back around your departure itinerary. Keep in mind that the Charlotte Amalie ferry terminal is much closer to the airport than the Red Hook terminal, and traffic gridlock between the island's East End and West End (where the St. Thomas airport is located) is not uncommon.\n3\n\nTHE VIRGIN ISLANDS IN CONTEXT\n\nGolden beaches shaded by palm trees and crystalline waters teeming with rainbow-hued marine life are undoubtedly the main attractions in both the U.S. and British Virgin Islands. Most visitors will likely spend their days hanging out on the beach, playing in the blue seas, browsing boutiques, and savoring fresh-caught fish. But beneath the Carnival costumes and tourism baubles lies a fascinating history and rich culture. Like so many other islands in the Caribbean, the Virgin Islands were inextricably intertwined with the colonial ambitions of Western Europe and the slave trade in North America. This chapter offers a peek at the cultural and historical influences coursing just beneath the surface of any modern-day escape to the Virgin Islands. It also includes tips about the best times to visit and the myriad activities and tours available to you.\n\nTHE ISLANDS IN BRIEF\n\nThe islands described in detail below are the main inhabited islands and the most frequently visited in both the U.S. and British Virgin Islands. Use the information to help guide you to your own idea of paradise.\n\nA few words about islands that aren't mentioned below: For those who really crave a secluded getaway, the British Virgin Islands have a number of escapist-friendly island islands such as Peter Island, Necker Island, and Guana Island. These are virtually private hideaways and the sole domain of expensive resorts. Two remote British Virgin Islands with small populations and limited commercial activity (but more democratically priced hotels) are Anegada and Jost Van Dyke. For more information on these islands, go to chapter 7.\n\nNo matter where you're staying in the Virgin Islands, you will definitely want to go island-hopping to visit the lesser-known islands\u2014and even the more popular spots, like the Baths (Virgin Gorda), that are B.V.I. must-sees.\n\nMapping the Virgins\n\nThe U.S. Virgin Islands are located some 90 miles east of Puerto Rico, with a combined landmass roughly twice the size of Washington, D.C. Lying just 97km (60 miles) to the east of Puerto Rico, the British Virgin Islands comprise a total landmass of 153 sq. km (59 sq. miles), a little smaller than Washington, D.C.\n\nThe U.S. Virgin Islands\n\nThe U.S. Virgin Islands consist of three main islands: St. Thomas, St. John, St. Croix, plus a little islet called Water Island and several smaller islets.\n\nSt. Thomas\n\nSt. Thomas may be, as some say, the most \"unvirgin\" of the Virgin Islands, but few can deny the island's substantial physical attributes. The most developed of the U.S. Virgin Islands, St. Thomas is sprawling and mountainous, with breathtaking vistas and palm-fringed beaches. It's the commercial heart of the Virgin Islands, but it offers something for everyone, from duty-free shopping to sizzling nightlife to laidback aeries far from the hustling scrum. The harbor at Charlotte Amalie, the capital, bustles with the energy of a small city, especially when cruise ships are in port. This is one of the largest cruise-ship ports in the Caribbean. Do as the locals do and avoid the harbor area at rush hour or when the cruise-ship passengers disembark\u2014when a long lineup of taxi drivers in big vans and open-air \"safari\" buses idle alongside the docks to spirit passengers to excursions around the island. The cruise-ship traffic also herds to the city's shopping bazaars and malls, chasing down some of the Caribbean's hottest duty-free shopping. Much of the retail action is found in the labyrinth of cobblestoned streets in the city's historic section, where the narrow lanes and picturesque old stone alleyways (built by the Danish) are packed with taxi vans and touts inviting shoppers in for a look. It's an intense beehive of activity during cruise-ship hours (roughly around 8am\u20134pm) and in high season, but every afternoon before dark the big ships slip back out to sea, and the city belongs once more to the locals and non-cruise-ship visitors.\n\nSt. Thomas may have a cosmopolitan sheen, but it's got a distinctively Caribbean pulse. Even here, where everyone seems to be breathlessly chasing the almighty dollar, island time rules. Yes, Magens Bay Beach, with its tranquil waves and long sweep of sugar-white sand, is likely to be packed on heavy cruise-ship days, but it's still a fetching swath of beach, surrounded on three sides by rising green hills. And if peace, quiet, and seclusion you're after, you'll find plenty of opportunity to relax and soak up the scenery, whether you've got your feet up on a powdery East End beach or enveloped in rain forest foliage on a mountain trail,\n\nYachts and boats anchor at Yacht Haven Grande Marina in Charlotte Amalie and at American Yacht Harbor in Red Hook on the island's eastern tip. The island attracts snorkelers and scuba divers\u2014and there are outfitters aplenty offering equipment, excursions, and instruction. Kayaking and parasailing also draw beach bums into the gin-clear seas. The island also has a surprisingly good surfing spots, including Hull Bay.\n\nSt. Thomas has one of the most eclectic and sophisticated restaurant scenes in the Virgin Islands. Emphasis is on French and Continental fare, but the wide selection of restaurants also includes options from West Indian and Italian to Asian. St. Thomas pays more for its imported (usually European) chefs and secures the freshest of ingredients from mainland or Puerto Rican markets. The island also hosts a number of American chain franchises, from Wendy's to Hooters\u2014especially near the cruise-ship piers.\n\nSt. Thomas has a wide variety of accommodations, from small, historic B&Bs tucked into the hills overlooking glittering Charlotte Amalie to the full-service, manicured beachfront resorts in the East End. Apartment and villa rentals abound.\n\nThe Virgin Islands\n\nSt. Croix\n\nAt 82 square miles, St. Croix is the largest island in the U.S. Virgins but it's sparsely populated for its size (some 54,000 people live here). It's mainly reached by plane, even from neighboring islands (it's more than an hour by boat from St. Thomas, a journey through deep and sometimes rough seas). St. Croix is developed, with good roads and the kinds of stores found in American suburbs, but it also has large swaths of rural farmland and a remote feel. Its two historic towns, Frederiksted and the capital, Christiansted, were built by the Danes during the sugarcane-plantation days, and Frederiksted's deep-water port brings a small number of cruise ships weekly to the island. St. Croix is also the only island that has a casino. Despite its nod to modernity, many of St. Croix's true West Indian\u2013style buildings have been preserved, along with its rich cultural traditions.\n\nOne of the best reasons to take a trip to St. Croix, even if only for a day, is to visit Buck Island National Park, just 11\u20442 miles off St. Croix's northeast coast. The island's offshore reef attracts snorkelers and divers. Signs posted along the ocean floor guide you through a forest of staghorn coral swarming with flamboyant fish.\n\nSt. Croix is a world-class diving destination, with a deep trench for wall-diving offshore and good diving (and snorkeling) around the cruise-ship pier in Frederiksted. It's also the premier golfing destination in the Virgin Islands, mainly because it boasts Carambola, the archipelago's most challenging 18-hole course. St. Croix is a tennis mecca, too: The Buccaneer Hotel has some of the best courts in the Virgin Islands and hosts several annual tournaments. Other sports for active vacationers include horseback riding, parasailing, sportsfishing, and water-skiing. It has one of the world's few healthy bioluminescent bays.\n\nSt. Croix is also gaining traction as a health-and-wellness destination, and practices what it preaches, with a growing presence in solar and wind energy and a boom in sustainable and organic farming.\n\nThe restaurants on St. Croix reflect the island's intriguing multicultural mix\u2014the citizenry is comprised of a large Latino population (many Puerto Ricans and Dominicans migrated here in the mid\u201320th c.), the descendants of black slaves (Afro-Caribbean), mestivos, and expats from around the world. (It's the home of a healthy gay population as well.) You will find plenty of small, creative local eateries serving up dishes and snacks ranging from West Indian curries to French croissants. Life after dark is mostly confined to a handful of bars in Christiansted.\n\nSt. Croix has few fine hotels, but it does have attractive inns and B&Bs. Still, many people opt to rent villas and condos at often-reasonable weekly rates.\n\nSt. John\n\nSt. John is one breathtakingly beautiful island, with some of the most pristine beaches in the entire Caribbean. And it will likely stay that way: Two-thirds of the island are protected parkland, the U.S. Virgin Islands National Park. Guided walks and safari bus tours are available to help you navigate the park, which is full of secret coves, hiking trails, and the ghostly remains of sugar-cane plantations. A third of the park is underwater. Trunk Bay, which also boasts one of the island's finest beaches, has an amazing underwater snorkeling trail. As you can imagine, scuba diving is another major attraction on St. John.\n\nSt. John has a smattering of upscale restaurants, but for the most part you'll dine well in colorful West Indian eateries, many located on the charming little harbor of the island's main town, Cruz Bay, and many serving the region's fresh seafood. Nightlife isn't a major attraction here; it usually consists of sipping rum drinks in a bar in Cruz Bay, and maybe listening to a local calypso band. After spending a day outdoors, most visitors on St. John are happy to turn in early.\n\nSt. John has only two deluxe hotels and a sprinkling of charming inns. Most people who visit the island rent villas, houses, or condos or stay in one of the island's plethora of campgrounds. St. John is a popular day-trip for visitors to the other Virgin Islands, but it's not a cruise-ship destination per se\u2014its port cannot accommodate the big ships. Cruise-ship passengers are usually day-trippers arriving by passenger ferry from ships anchored in St. Thomas.\n\nThe British Virgin Islands\n\nThe British Virgin Islands comprise some 60 islands, among them 15 inhabited. The total population of the B.V.I. is around 27,000 people.\n\nTortola\n\nTortola is the hub of the British Virgin Islands and a wonderful place to stay and use as your base to day-trip to outlying islands. Yes, Virgin Gorda (see below) is less developed and has better (and pricier) resorts and restaurants, and Road Town, the B.V.I. capital, is fairly charmless, with minor shopping, routine restaurants, and uninspired architecture\u2014and the daily arrival of two or three cruise ships makes the morning traffic a real slog. But head west or east out of Road Town, and you'll find an alluring place, with some of the warmest and most soulful people in the islands and a dramatic topography, where the roads do racy arabesques around green peaks, and the vistas from on high are utterly ravishing.\n\nThe island's best and most unspoiled beaches, including Smuggler's Cove (with its secluded collection of snorkeling reefs), lie at the island's western tip and along the northern shoreline. Tortola's premier beach is Cane Garden Bay, a 2.4km (11\u20442-mile) stretch of white sand bookended by rising green hills. Because of the clean, gentle surf and the smooth sand bottom, it's one of the safest places for families with small children\u2014and there's plenty of ocean for everyone even on the busiest cruise-ship days. For hikers on Tortola, exploring Sage Mountain National Park, where trails lead to a 543m (1,781-ft.) peak that offers panoramic views, is a highlight. The park is rich in flora and fauna, from mamey trees to cooing mountain doves.\n\nAlthough many visitors to the Caribbean look forward to fishing, hiking, horseback riding, snorkeling, and surfing, what makes Tortola exceptional is boating. It is the boating center of the British Virgin Islands, which are among the most cherished sailing territories on the planet. The island offers some 100 charter yachts and 300 bareboats, and its marina and shore facilities are the most up-to-date and extensive in the Caribbean Basin. It's a big reason why Tortola makes a great base from which to go island-hopping.\n\nThe food in Tortola is simple and straightforward, with menu staples of local fish, conch fritters, and fresh fruit juices (laced with rum, island-style). The nightlife is laidback, with full-moon parties on the beach and a roster of local musicians playing places like Myett's and Quito's, on Cane Garden Bay.\n\nTortola has just a couple of fine resorts and a few very basic hotels; many people who stay a while choose to rent villas, houses, or condos.\n\nVirgin Gorda\n\nVirgin Gorda is the third-largest member of the B.V.I. archipelago, with a permanent population of about 3,000 lucky souls. Virgin Gorda is uncrowded and incredibly scenic, with swooping roads that lace the tops of emerald peaks and some of the cleanest and clearest water on the planet. In fact, being on the water, whether under sail or with snorkel and mask, is the default mode on Virgin Gorda. It's a sun-and-sea worshiper's dream come true.\n\nMany visitors come over just for a day to check out the Baths, an astounding collection of gigantic boulders and crystalline tide pools on the southern tip. Crafted by volcanic pressures millions of years ago, the rocks have eroded into almost sculptural shapes\u2014and you can snorkel in turquoise pools that snake through the big boulders.\n\nIt may be small and its population sparse, but Virgin Gorda is home to some of the best hotels in the Virgin Islands and among the world's finest, including Little Dix Bay and Biras Creek. One caveat: You will pay dearly for the privilege of staying at one of these expensive resorts. Outside the upscale hotels, restaurants tend to be simple places serving local West Indian cuisine. No one takes nightlife too seriously on Virgin Gorda, so there isn't very much of it.\n\nTHE VIRGIN ISLANDS TODAY\n\nThe American way of life prevails today in the U.S. Virgin Islands, and it has swept across to the British Virgin Islands, as well. The region's traditional recipes and remedies, as well as many of the self-reliant arts of fishing, boat-building, farming, and even hunting, are fading away. When islanders need something, they have it shipped from Miami. In clothes, cars, food, and entertainment, America, not Great Britain, rules the seas around both archipelagos. The British Virgins even use the U.S. dollar as their official currency, instead of British pounds.\n\nLike the rest of the world, the Virgin Islands have felt the effects of the global recession. Tourism is rebounding, but the islands\u2014like the rest of the Caribbean\u2014have not seen the same number of visitors as they had prior to the recession. To attract tourists, the government officials have come to realize that the natural environment must be protected. During the 1980s, the islands, especially St. Thomas, experienced a real estate boom, and much of the island's natural terrain was converted into shopping malls and condo complexes. Caroline Brown, of the Environmental Association of St. Thomas, even issued a dire warning that islanders may find themselves one day \"living in a concrete jungle.\" Today, awareness of the perils of overdevelopment has seeped into the collective consciousness, and protecting the environment\u2014and curbing overdevelopment\u2014has become a primary concern in the Virgin Islands. The B.V.I., for example, has done added 10 new parks, including the Anegada Nature Reserve, to its existing national park system, which includes the Gorda Peak National Park and the Devil's Bay National Park on Virgin Gorda and the Sage Mountain National Park on Tortola. St. John, the smallest of the U.S. Virgin Islands, is the most protected landmass in the Caribbean, with some 60 percent of its acreage directly controlled by the U.S. National Park Service (and more parcels added on an ongoing basis).\n\nA 51st state?\n\nThe U.S. Virgin Islands are an unincorporated territory administered by the U.S. Department of the Interior. Politically speaking, the Virgin Islands, like Puerto Rico, remain outside the family of the United States. They are only permitted to send a nonvoting delegate to the U.S. House of Representatives. U.S. Virgin Islanders are not allowed to vote in national elections, a sore spot among some of the local residents. Many hope to see another star added to the American flag in the near future\u2014feeling that only full statehood will provide the respect, power, and influence needed to turn the islands into more than just a \"colony.\" Progress in this direction moves sluggishly along, if at all.\n\nWhen the 1936 Organic Act of the Virgin Islands was passed under the Roosevelt administration, residents ages 21 and over were granted suffrage and could elect two municipal councils and a legislative assembly for the islands. In 1946, the first black governor of the islands, William Hastie, was appointed. By 1970, the U.S. Virgin Islanders had the right to elect their own governor and lieutenant governor.\n\nOn St. Thomas, the friction between developers and environmentalists was recently encapsulated in the battle over the future of the 400 acres surrounding Botany Bay. The area, on the western end of St. Thomas, has long been a refuge from the bustle of Charlotte Amalie, the capital. This undeveloped property (a private estate closed to the public for some 80 years) was a conservation area of sorts for deer, and a nesting ground for sea turtles, ringed by some of the healthiest coral reefs around St. Thomas. When it was first proposed in 2001, the development of a gated beachfront resort complex\u2014The Preserve at Botany Bay\u2014with a luxury hotel, timeshare condos, and sprawling home sites, was greeted with opposition from environmentalists and became a hot-button political topic. It felt as if much of the future of the islands was riding on what happened at Botany Bay. Some 5 years later, after developers reduced the density of the development and agreed to specific Coastal Zone Management division (CZM), requests (such as providing a walking path for public shoreline access and producing a booklet for future homeowners on preserving and protecting the area's unique environment), the U.S.V.I. Dept. of Planning & Natural Resources gave the green light to the development of the multi-use Preserve at Botany Bay.\n\nFlash forward to late 2013: Although the hotel arm of the project remained at a standstill, a number of villas and homes had been completed, many commanding the hillsides overlooking Botany Bay. \"The Wall Street Journal\" reported that a \"pirate-themed\" six-bedroom villa in the Preserve at Botany Bay was on the market for $35 million.\n\nLOOKING BACK: VIRGIN ISLANDS HISTORY\n\nA Brief History\n\nChristopher Columbus is credited with \"discovering\" the Virgin Islands in 1493, but, in fact, they had already been inhabited for 3,000 years. It is believed that the original settlers were the nomadic Ciboney Indians, who migrated from the mainland of South America and lived off the islands' fish and vegetation. The first real homesteaders were the peaceful Arawak Indians, who arrived from Venezuela, presumably in dugout canoes with sails.\n\nFor about 500 years, the Arawaks occupied the Virgin Islands, until the arrival of the cannibalistic Carib Indians in the 15th century. The Caribs destroyed the Arawaks, either by working them to death as slaves or by eating them. With the advent of European explorers and their diseases, these tribes were completely wiped out.\n\nDateline\n\n1493 Columbus sails by the Virgin Islands, lands on St. Croix, and is attacked by Carib Indians.\n\n1625 Dutch and English establish frontier outposts on St. Croix.\n\n1650 Spanish forces from Puerto Rico overrun English garrison on St. Croix.\n\n1653 St. Croix taken over by the Knights of Malta.\n\n1671 Danes begin settlement of St. Thomas.\n\n1672 England adds British Virgin Islands to its empire.\n\n1674 King Louis XIV of France makes St. Croix part of his empire.\n\n1717 Danish planters from St. Thomas cultivate plantations on St. John.\n\n1724 St. Thomas is declared a free port.\n\n1733 Danish West India Company purchases St. Croix from France; slaves revolt on St. John.\n\n1792 Denmark announces plans to abandon the slave trade.\n\n1807\u201315 England occupies Danish Virgin Islands.\n\n1820s Sugar plantations on the Virgin Islands begin to see a loss in profits.\n\n1834 England frees 5,133 slaves living in British Virgin Islands.\n\n1848 Under pressure, the governor of St. Croix grants slaves emancipation.\n\n1867 First attempt by the United States to purchase the Virgin Islands from the Danish.\n\n1872 British Virgin Islands put under administration of the Federation of the Leeward Islands.\n\n1916 Denmark signs treaty with the U.S. and sells islands for $25 million.\n\n1917 Virgin Islands fall under the control of the U.S. Navy for 14 years.\n\n1927 United States grants citizenship to island residents.\n\n1936 Under Franklin Roosevelt, the first Organic Act is passed, granting voting rights to U.S. Virgin Islanders.\n\n1940 Population of U.S. Virgin Islands increases for the first time since 1860 because of its use as a port during World War II.\n\n1946 First black governor of the U.S.V.I., William Hastie, is appointed.\n\n1954 Revised Organic Act passed; the U.S.V.I. fall under jurisdiction of Department of the Interior.\n\n1956 British Virgins released from the Federation of the Leeward Islands.\n\n1966 Queen Elizabeth II visits the British Virgin Islands.\n\n1967 B.V.I. get a new constitution.\n\n1980s U.S.V.I. see major development and construction, putting natural resources at risk.\n\n1989 Hurricane Hugo rips through the islands, hitting St. Croix especially hard.\n\n1995 Hurricane Marilyn causes millions of dollars of damage and leaves thousands homeless.\n\n1996 Water Island, off the coast of St. Thomas, is officially declared the fourth U.S. Virgin Island; U.S. Senate grants permission for two casino hotels to be built on St. Croix.\n\n2000 St. Croix becomes the first \"casino island\" in the Virgin Islands.\n\n2005 Plans stall for grand development of St. Thomas.\n\n2008 B.V.I. become one of the world's leading offshore financial centers.\n\n2009 Islands experience drop-off in tourism as U.S. economy goes into recession.\n\n2010 U.S.V.I. officials tackle environmental problems.\n\nThe Age of Colonization\n\nIn November 1493, on his second voyage to the New World, Columbus spotted the Virgin Islands, naming them Las Once Mil Virgenes, after the Christian St. Ursula and her martyred maidens. Short of drinking water, he decided to anchor at what is now Salt River on St. Croix's north shore. His men were greeted by a rainfall of arrows. Embittered, Columbus called that part of the island Cabo de Flechas, or \"Cape of the Arrows,\" and sailed toward Puerto Rico.\n\nAs the sponsor of Columbus's voyage, Spain claimed the Virgin Islands; however, with more interest in the Greater Antilles, Spain chose not to colonize the Virgins, leaving the door open to other European powers. In 1625, both the English and the Dutch established opposing frontier outposts on St. Croix. Struggles between the two nations for control of the island continued for about 20 years, until the English prevailed (for the time being).\n\nAs the struggle among European powers widened, the islands continued to function as a battleground. In 1650, Spanish forces from Puerto Rico overran the British garrison on St. Croix. Soon after, the Dutch invaded; in 1653, the island fell into the hands of the Knights of Malta, who gave St. Croix its name.\n\nHowever, these aristocratic French cavaliers weren't exactly prepared for West Indian plantation life, and their debts quickly mounted. By 1674, King Louis XIV of France took control of St. Croix and made it part of his kingdom.\n\nThe English continued to fight Dutch settlers in Tortola, which was considered the most important of the British Virgin Islands. It wasn't until 1672 that England added the entire archipelago to its growing empire.\n\nA year before, in March 1671, the Danish West India Company made an attempt to settle St. Thomas. The company sent two ships, but only one, the Pharaoh, completed the voyage, with about a third of its crew. Eventually, reinforcements arrived, and by 1679, at least 156 Europeans were reported to be living on St. Thomas, along with their slaves. Captain Kidd, Sir Francis Drake, Blackbeard, and other legendary pirates of the West Indies continued to use St. Thomas as their base for maritime raids in the area. Its harbor also became famous for its slave market.\n\nIn 1717, Danish planters sailed to St. John from St. Thomas to begin cultivating plantations. By 1733, an estimated 100 sugar, tobacco, and cotton plantations were operating on the island. That same year, the slaves rebelled against their colonial masters, taking control of the island for about 6 months and killing many Europeans. It took hundreds of French troops to quell the rebellion.\n\nImpressions\n\n_There could never be lands any more favorable in fertility, in mildness and pleasantness of climate, in abundance of good and pure water. A very peacefuland hopeful place that should give all adventurers great satisfaction._\n\n_\u2013Captain Nathaniel Butler, HM Frigate Nicodemus, 1637_\n\nIn that same year, France sold St. Croix to the Danish West India Company, which divided the island into plantations, boosting the already flourishing slave trade. Some historians say that nearly 250,000 slaves were sold on the auction blocks at Charlotte Amalie before being sent elsewhere, often to America's South. By 1792, Denmark changed its tune and announced that it officially planned to end the slave trade. It was not until 1848, however, that it did so. The British had freed their 5,133 slaves in 1834.\n\nThe great economic boom that resulted from the Virgin Islands plantations began to wilt by the 1820s. The introduction of the sugar beet virtually bankrupted plantation owners, as the demand for cane sugar drastically declined. Cuba eventually took over the sugar market in the Caribbean. By 1872, the British had so little interest in the British Virgins that they placed them in the loosely conceived and administered Federation of the Leeward Islands.\n\nEnter the United States\n\nIn 1867, the United States attempted to purchase the islands from Denmark, but the treaty was rejected by the U.S. Senate in 1870; the asking price was $7.5 million. Following its acquisition of Puerto Rico in 1902, the United States expressed renewed interest in acquiring the Danish islands. This time, the United States offered to pay only $5 million, and the Danish parliament spurned the offer.\n\nOn the eve of its entry into World War I, the U.S. Navy began to fear a possible German takeover of the islands. The United States was concerned that the Kaiser's navy, using the islands as a base, might prey on shipping through the Panama Canal. After renewed attempts by the United States to purchase the islands, Denmark agreed to sell them in 1916 for $25 million, a staggering sum to pay for island real estate in those days.\n\nBy 1917, the United States was in full control of the islands, and Denmark retreated from the Caribbean after a legacy of nearly 21\u20442 centuries. The U.S. Navy looked after the islands for 14 years, and in 1954, they came under the sovereignty of the U.S. Department of the Interior.\n\nSome money was diverted to the area during the Prohibition era, as some islanders made rum and shipped it illegally to the United States, often through Freeport, in the Bahamas. In 1927, the United States granted citizenship to the island residents. In 1936, under Franklin Roosevelt, the first Organic Act was passed, giving the islanders voting rights in local elections. This act was revised in 1954, granting them a greater degree of self-government.\n\nJobs generated by World War II finally woke the islands from their long economic slumber. The U.S.V.I. were used as a port during the war, and visitors first started to appear on the islands. In the postwar economic boom that swept across America, the Virgin Islands at long last found a replacement for sugar cane.\n\nThe British Virgin Islands Develop\n\nThe British Virgin Islands were finally freed from the Leeward Islands Federation in 1956, and in 1966, Queen Elizabeth II visited this remote colonial outpost. By 1967, the British Virgin Islands had received a new constitution. Tourism was slower to come to the British Virgins than to the U.S. Virgin Islands, but it is now the mainstay of the economy.\n\nIn 2000, the British government issued a report that found that nearly 41 percent of offshore companies in the world were formed in the British Virgin Islands. By 2011, the B.V.I. was one of the world's leading offshore financial centers, and the local population boasted one of the highest incomes per capita in the Caribbean\u2014at around $40,000 per family.\n\nTourism & the Economy Today\n\nThe economy of both the British and the U.S. Virgins has been one of the most stable and prosperous in the Caribbean. All of the islands felt the impact of the worldwide economic slump, however, with a falloff in tourism and revenues.\n\nIn the meantime, the governments of both the British Virgins and the U.S. Virgins continue to struggle with unemployment as they mount ongoing struggles to reduce crime and to protect the environment. The U.S. Virgin Islands were awarded $364 million in federal stimulus funding in 2009, and the funding is being used to stimulate the local economy during the recession. One of the major and predictable goals of the stimulus package is in job creation. Some of the money has been earmarked for upgrading the ferry service between Cruz Bay on St. John and Red Hook on St. Thomas.\n\nTHE DANES ARE GONE BUT THEIR architecture STILL STANDS\n\nSome of the architectural legacy left by the colonizing Danes still remains in the islands, especially in Christiansted and Frederiksted on St. Croix, and in Charlotte Amalie on St. Thomas.\n\nMany of the commercial buildings constructed in downtown Charlotte Amalie are restrained in ornamentation. Pilasters and classical cornices were commonplace on many buildings. Most door arches and windows were framed in brick. To \"dress up\" a building, ornamentation, such as cornices, was added in the final stages. The walls were covered with plaster, but in recent decades this plaster and stucco have been stripped from the walls. Underneath the rubble, well-designed shapes and patterns of old brick and \"blue bitch\"\u2014a native stone made of volcanic tuff\u2014were discovered. The old masons may have known what they were doing. Once stripped of their plaster coating, the walls don't stand up well in the Caribbean sun and salt air. Cast-iron grillwork on some of the second-floor overhanging balconies adds a certain architectural flair. Many of the buildings in St. Thomas originally had courtyards, or still do. These added to the living space on the second floor. In the courtyard were kitchens and, almost more vital, cisterns to capture precious rainwater.\n\nSimilar building techniques were used on structures that went up on St. Croix. Christiansted remains one of the most historically authentic towns in the West Indies, true to its original Danish colonial flavor. The basic style was a revival of the European classic look of the 18th century, but with variations to accommodate the tropical climate. As early as 1747, the Danes adopted a strict building code, which spared Christiansted from some of the violent fires that virtually wiped out Charlotte Amalie. Frederiksted, the other major town of St. Croix, has a well-designed waterfront, with blocks of arcaded sidewalks. The quarter is protected by the government as part of Frederiksted's National Historic District.\n\nGreat architecture was never the forte of the British Virgin Islands. During a time when major buildings might have been created, the B.V.I. were too economically depressed to find the funds for major structures of lasting significance. Therefore, for much of its history, its people have lived in typical West Indies shanties, with an occasional public building constructed that vaguely imitated 18th-century Europe in style. Curiously enough, although the B.V.I. didn't leave the world any lasting architectural heritage, it did produce a native son, William Thornton, whose designs were used for the U.S. Capitol building in Washington.\n\nIn 2010, officials in the U.S. Virgin Islands developed closer contact with the U.S. Environmental Protection Agency. The aim is to help the territory solve some of its longstanding problems, such as the best way to address the solid waste problem and how to preserve healthy air standards.\n\nTHE VIRGIN ISLANDS IN POPULAR CULTURE\n\nBooks\n\nFICTION\n\nHerman Wouk's \"Don't Stop the Carnival\" (Little Brown & Co., 1992) is \"the Caribbean classic,\" and all readers contemplating a visit to the Virgin Islands might want to pick it up. Wouk lived in St. Thomas in the 1950s, and his novel is based on actual people he met during that time. Bob Shacochis's \"Easy in the Islands\" (Grove, 2004) giddily re-creates the flavor of the West Indies with short stories. \"Tales of St. John & the Caribbean,\" by Gerald Singer (Sombrero Publishing Co., 2001), is an easy read: a collection of amusing and insightful stories, and the best volume if you'd like a behind-the-scenes look at St. John after the tourists have taken the ferry back to St. Thomas for the night.\n\n\"My Name Is Not Angelica,\" by Scott O'Dell (Yearling Books, 1990), is a young-adult historical novel based in the Virgin Islands in the early 18th century. It tells the saga of a slave girl, Raisha, who escapes bondage; the grim realities of slavery are depicted.\n\nRobert Louis Stevenson is said to have used Norman Island, in the B.V.I., as a fictional setting for his 1883 classic \"Treasure Island.\" This swashbuckling adventure has intrigued readers for years with such characters as the immortal Long John Silver. The book, which gave rise to such memorable lines as \"shiver me timbers,\" continues to find new generations of readers.\n\nCOOKBOOKS\n\nA number of books are devoted to recipes of the Caribbean, including \"The Sugar Mill Caribbean Cookbook: Casual and Elegant Recipes Inspired by the Islands,\" by Jinx and Jefferson Morgan (Harvard Common Press, 1996). The Morgans run the Sugar Mill on Tortola. With this book, you can learn the secrets of many of the Sugar Mill's signature dishes, including Rasta Pasta, rum-glazed chicken wings, and lobster and christophine curry.\n\n\"Food & Folklore of the Virgin Islands\" (Romik, 1990) was penned by Arona Petersen, a well-known St. Thomas writer and folklorist born in 1908. The regional flavor of Virgin Islands fare is captured in her recipes, and the idiomatic dialogues of island people are perfectly re-created as she spins old island tales and wisdom. In her story \"What Does Tomorrow Mean? In any Language, Wait\" appears this passage: \"Wat I trying to say is dat waitin is wat life is about. Everybody waitin fo something or udder, mannin or nite. Tain get wan purson wat, livin ain waitin-fo a bus, fo a taxi, fo a airplane, fo a steamer, fo a letter to come back. Some doan even know wat dey waiting for but dey still waitin.\" Her books are sold in local shops.\n\nHISTORY BOOKS\n\nThe concise \"History of the Virgin Islands\" (University Press of the West Indies, 2000) is a bit scholarly for some tastes, but if you're seriously interested in the islands, this is the best-researched survey of what was going on before your arrival. \"Caribbean Pirates,\" by Warren Alleyne (Macmillan-Caribbean, 1986), is a good read for preteen travelers and attempts to separate fact from fiction in the sagas of the most notorious pirates in history. Some of the material is based on published letters and documents.\n\nOUTDOOR ADVENTURE BOOKS\n\nSailing enthusiasts say you shouldn't set out to explore the islets, cays, coral reefs, and islands of the B.V.I. without John Rousmani\u00e8re's well-researched \"The Sailing Lifestyle\" (Fireside, 1988).\n\n\"Exploring St. Croix,\" by Shirley Imsand and Richard Philobosian (Travelers Information Press, 1987), is a very detailed activity guide of this island. The authors take you to 49 beaches, 34 snorkeling and scuba-diving sites, and 22 bird-watching areas, and lead you on 20 different hikes.\n\n\"A Guide to the Birds of Puerto Rico & the Virgin Islands,\" by Herbert A. Raffaele, Cindy J. House, and John Wiessinger (Princeton University Press, 1989), is for bird-watchers. The illustrations alone are worth a look, with 273 depictions of the 284 documented species on the islands.\n\nFilm\n\nFilm production reached its heyday on St. Thomas in the '70s and '80s, when major TV shows, such as \"Charlie's Angels,\" \"The Love Boat,\" and \"All My Children,\" were shot here.\n\nMany movies have been shot in the Virgin Islands, including \"Open Water\" (2003), the hair-raising adventure story of a couple stranded in shark-infested waters (based on a true story) and \"The Deep,\" shot around the wreck of the RMS Rhone near Salt Island, the B.V.I. The final scene of \"The Shawshank Redemption\" (1994), when Andy Dufresne escapes the harsh Shawshank Prison for an idyllic tropical island, was filmed on Sandy Point, in St. Croix. The final island scene in \"Trading Places\" (1983), starring Eddie Murphy and Jamie Lee Curtis, was also shot in St. Croix. A 1980s film classic, \"The Four Seasons\" (1981), was filmed in part in the Virgin Islands; the film is a tender-sweet melodrama that stars Carol Burnett and Alan Alda.\n\nThe 1990 television remake of Ernest Hemingway's \"The Old Man and the Sea,\" starring Anthony Quinn, was filmed on the half-moon bay at Smuggler's Cove, on Tortola in the B.V.I. The ruins of the movie set still remain.\n\nThe true classic of the archipelago is \"Virgin Island\" (1958), starring John Cassavetes and Sidney Poitier. Filmed in the British Virgin Islands, it is a fairy-tale type of story about a young man and woman who buy a small, uninhabited island and go there to find their dream. The film was based on the actual experiences of novelist Robb White, who with his wife bought Marina Cay in 1937 for $60 and decided to pursue a Robinson Crusoe existence on the islands. He wrote about it in three memoirs: \"In Privateers Bay,\" \"Our Virgin Island,\" and \"Two on the Isle.\"\n\nEssential Guide to Sailing the Virgin Islands\n\nFor more than a quarter of a century, \"Yachtsman's Guide to the Virgin Islands\" has been the classic cruising guide to this area (it's now in its 13th edition). The detailed, 288-page text is supplemented by 27 hand-drawn sketch charts, aerial photographs, and numerous landfall sketches showing harbors, channels, landmarks, and such. Subjects covered include piloting, anchoring, communication, weather, fishing, and more. The guide also covers the eastern end of Puerto Rico, Vieques, and Culebra. Copies are available at major marine outlets, bookstores, and direct from Yachtman's Guide (  877\/923-9653; www.yachtsmansguide.com).\n\nDon't Let the Jumbies Get Ya!\n\n\"Don't let the Jumbies get ya!\" is an often-heard phrase in the Virgin Islands, particularly when people are leaving their hosts and heading home in the dark. Jumbies, capable of good or evil, are supernatural beings that are believed to live around households. It is said that new settlers from the mainland of the United States never see these Jumbies and, therefore, need not fear them. But many islanders believe in their existence and, if queried, may enthrall you with tales of sightings.\n\nNo one seems to agree on exactly what a Jumbie is. Some claim it's the spirit of a dead person that didn't go where it belonged. Others disagree. \"They're the souls of live people,\" one islander told us, \"but they live in the body of the dead.\" The most prominent Jumbies are \"moko jumbies,\" colorful stilt walkers seen at all carnival parades.\n\nMusic\n\nAs the Caribbean rhythms go, the Virgin Islands encompass it all, from reggae to classical to steel drums to spiritual hymns, but soca, reggae, calypso, and steel-pan beats seem to dominate the night.\n\nThough it originated in Trinidad, calypso has its unique sounds in the Virgins. It is famously known for expressing political commentary through satire.\n\nIf you add a little soul music to calypso, you have soca, a music form that also made its way north to the Virgin Islands from Trinidad. Reggae originated on Jamaica, but is alive and well in the Virgins. Virgin Islanders have put a unique stamp on reggae, making it their own.\n\nScratch bands are popular in the British Virgins, in the musical form known as fungi. Merengue is also heard in the Virgins, having \"floated over\" from Puerto Rico and the Dominican Republic.\n\nThroughout the year in various bars, clubs, reggae concerts, steel-pan shows, and jazz concerts, you can hear the music of the islands, including zouk, dance music from Martinique. Find out what's happening by reading the local newspapers.\n\nOf course, the leading musicians of the islands make recordings, including the hypnotic and tantalizing roots-reggae stylings of St. Croix native Dezarie, whose albums, including \"Gracious Mama Africa\" and her latest, \"Fourth Book,\" have earned her the title of St. Croix's Roots Empress. Another St. Croix\u2013born singer is Mada Nile (<http:\/\/madafyahnile.com>), known for her poignant lyrics.\n\nA vocal rival of both Dezarie and Mada Nile is Sistah Joyce, a reggae artist from Virgin Gorda who is acclaimed for her hard-hitting lyrics as evoked by her recording of \"Remembah.\" She scored a hit with her debut album, \"H.Y.P.O.C.R.I.C.Y.\"\n\nIsland-bred reggae bands, such as Midnite (from St. Croix) and Inner Visions (from St. John), have never been more popular, although they've been around since the '80s. \"Midnite Intense Pressure,\" Midnite's debut album, firmly established them as a force in roots music; the group is known for its fiery lyrics. Its latest album is \"Lion out of Zion\" (<http:\/\/midniteband.com>). The albums of Inner Visions (www.innervisionsreggae.com) demonstrate the group's refined musical abilities, which distinguish them from the more \"raw roots\" style of many other rival artists. The band is made up of first- and second-generation members of the Pickering family, with names like Grasshopper and Jupiter.\n\nEATING & DRINKING\n\nOverall, you will eat very well in the islands. Many fine talents, both local and imported, are ensconced in kitchens throughout the region, and competition breeds excellence. In addition, traditional Caribbean cuisine is alive and well in the Virgin Islands, and many classic island dishes are staples on restaurant menus, and not just the laidback local spots. Even upscale resorts serving an international clientele favor menus with an island spin, fomenting a Caribbean\/continental fusion.\n\nDining in the Virgin Islands is generally more expensive than it is in North America because much of the food has to be imported. Whenever possible, take advantage of fresh regional foods, like locally caught fish, especially mahimahi, wahoo, yellowtail, grouper, and red snapper. The sweet Caribbean lobster is another local specialty. More and more, farm-fresh produce is making a reappearance, and you'll even find local fruits and vegetables sold in the island's grocery stores\u2014fresh avocadoes, bananas, sweet potatoes, cassava, you name it.\n\nTips on Dining\n\nTipping A 10 percent to 15 percent service charge is automatically added to most restaurant tabs. If the service is good, you should tip a bit extra.\n\nWhat to Wear In some of the posher resorts, such as Caneel Bay on St. John, it is customary for men to wear a jacket, but in summer, virtually no establishment requires it. If in doubt, ask the restaurant beforehand. At the better places, women's evening attire is casual-chic. During the day it is always proper to wear something over your bathing suit if you're in a restaurant.\n\nReservations Most places require reservations, especially in high season. But even in the slower seasons (or shoulder seasons), many of the smaller restaurants need to know how many people to expect for the evening's meal, so reservations may be required there as well.\n\nThe Cuisine\n\nYou may want to start your meal with a bowl of kallaloo, or callaloo, a West Indian\u2013style gumbo made in an infinite number of ways with a leafy green vegetable similar to spinach. It may come flavored with hot peppers, a ham bone, fresh seafood, okra, onions, and spices. We've also eaten a wonderful breakfast dish of steamed callaloo with sauteed mushrooms, onions, and halved cherry tomatoes.\n\nSaltfish salad is traditionally served on Maundy Thursday or Good Friday in the Virgin Islands. It consists of boneless salt fish, potatoes, onions, boiled eggs, and an oil-and-vinegar dressing.\n\nHerring gundy is another old-time island favorite; it's a salad made with salt herring, potatoes, onions, green sweet and hot peppers, olives, diced beets, raw carrots, herbs, and boiled eggs.\n\nThe classic vegetable dish, which some families eat every night, is peas and rice. It usually consists of pigeon peas flavored with ham or salt meat, onion, tomatoes, herbs, and sometimes slices of pumpkin. Pigeon peas, one of the most common vegetables in the islands, are sometimes called congo peas or gunga.\n\nMany local joints serve a sweet and savory island slaw, often made with raisins and sweet peppers.\n\nFungi is a simple cornmeal dish not unlike grits, made more interesting with the addition of okra and other ingredients. Sweet fungi is served as a dessert, with sugar, milk, cinnamon, and raisins.\n\nOkra (often spelled ochroe in the islands) is a mainstay vegetable, usually accompanying beef, fish, or chicken. It's often fried and flavored with hot peppers, tomatoes, onions, garlic, and bacon fat or butter. Accra, a popular dish, is made with okra, black-eyed peas, salt, and pepper, all fried until they're golden brown.\n\nWay back when, locals gave colorful names to the various fish brought home for dinner, everything from \"ole wife\" to \"doctors,\" both of which are whitefish. \"Porgies and grunts,\" along with yellowtail, kingfish, and bonito, also show up on many Caribbean dinner tables. Fish is often boiled in a lime-flavored brew seasoned with hot peppers and herbs, and is commonly served with a Creole sauce of peppers, tomatoes, and onions, among other ingredients. Salt fish and rice is an excellent low-cost dish; the fish is flavored with onion, tomatoes, shortening, garlic, and green peppers.\n\nConch shows up on the menu in all sorts of interesting manifestations. Every restaurant has its own version of conch fritters. Conch Creole is a savory brew, seasoned with onions, garlic, spices, hot peppers, and salt pork. Another West Indian favorite is chicken and rice, usually made with Spanish peppers. More adventurous diners might try curried goat, the longtime classic West Indian dinner prepared with herbs, cardamom pods, and onions.\n\nThe famous johnnycakes that accompany many of these fish and meat dishes are made with flour, baking powder, shortening, and salt, then fried or baked.\n\nOn the B.V.I., you'll find a number of inexpensive roti joints, serving up East Indian-style turnovers stuffed with curried chicken, fish, or potatoes and peas.\n\nFor dessert, sweet potato pie is a Virgin Islands classic, made with sugar, eggs, butter, milk, salt, cinnamon, raisins, and chopped raw almonds. The exotic fruits of the islands lend themselves to various homemade ice creams, including mango, guava, soursop (a tangy fruit), banana, carambola (star fruit), pineapple, and papaya. Sometimes dumplings made with guava, peach, plum, gooseberry, cherry, or apple are served for dessert.\n\nSome Like It Hot\n\nVirgin Islanders like their hot sauce (aka \"pepper\" sauce), in any number of creative variations, and many island cooks prepare and bottle their own, a number of which are available for sale in stores around the islands. Foodies and bloggers have a field day defending their favorites, including Miss Anna's   \n(St. John), Blind Betty's (St. Croix), Jerome's (St. Thomas), and ValleyDoll (St. John). At Little Dix Bay in Virgin Gorda, one of the resort's longtime servers, Venita Chapman, creates her own saucy concoction with mangos, papayas, and peppers, Venita's Hot Pepper Sauce, sold right there at the resort. Even Miss Gladys, proprietor of Gladys' Caf\u00e9 in St. Thomas, makes and bottles her own divine hot-sauce concoctions (in such flavors as mango, mustard, and tomato) and sells them in her restaurant. If you're a hot-sauce aficionado, you'll find an exhaustive selection of these and more at St. John Spice (<http:\/\/stjohnspice.com>), on the harbor in Cruz Bay. They make great gifts to take home, but be sure to pack them in your luggage and not your carryon.\n\nDrinks\n\nThe islands' true poison is rum. To help stimulate the local economy, U.S. Customs allows you to bring home an extra bottle of local rum, in addition to your usual 5-liter liquor allowance.\n\nLong before the arrival of Coca-Cola and Pepsi, many islanders concocted their own fruit drinks using whatever fruit was in season. Fresh fruit concoctions are ubiquitous on menus today. American sodas and beer are sold in both the U.S. Virgin Islands and the British Virgin Islands. Wine is sold, too, but it can be expensive.\n\nWater is generally safe to drink on the islands. Much of the water is stored in cisterns and filtered before it's served. Still, delicate stomachs should stick to bottled water or club soda. Some resorts have their own desalination plants, providing water that is delicious and highly drinkable.\n\nWHEN TO GO\n\nWeather\n\nSunshine and warm temperatures are practically an everyday affair in the Virgin Islands. Temperatures climb into the 80s (high 20s Celsius) during the day, and drop into the more comfortable 70s (low 20s Celsius) at night. Winter is generally the dry season in the islands, but rainfall can occur at any time of the year. You don't have to worry too much, though\u2014tropical showers usually come and go so quickly you won't even really notice. November is traditionally the wettest month\u2014if you're out exploring for the day, you may want to bring an umbrella or rain hat just in case.\n\nHurricanes The hurricane season, the dark side of the Caribbean's beautiful weather, officially lasts from June to November. It peaks in September and October. The Virgin Islands chain lies in the main pathway of many a hurricane raging through the Caribbean, and the islands are often hit. If you're planning a vacation in hurricane season, stay abreast of weather conditions and consider investing in trip-cancellation insurance.\n\nIslanders certainly don't stand around waiting for a hurricane to strike. Satellite forecasts generally give adequate warning to both residents and visitors. And of course, there's always prayer: Islanders have a legal holiday in the third week of July called Supplication Day, when they ask to be spared from devastating storms. In late October, locals celebrate the end of the season on Hurricane Thanksgiving Day.\n\nAverage Temperatures & Rainfall (in.) for St. Thomas\n\nThe High Season & the Off Season\n\nHigh season (or winter season) in the Virgin Islands, when hotel rates are at their peak, runs roughly from mid-December to mid-April. However, package and resort rates are sometimes lower in January, as a tourist slump usually occurs right after the Christmas holidays. February is the busiest month. If you're planning on visiting during the winter months, make reservations as far in advance as possible.\n\nOff season begins when North America starts to warm up, and vacationers, assuming that temperatures in the Virgin Islands are soaring into the 100s (upper 30s Celsius), head for less tropical local beaches. However, the Virgin Islands are actually quite balmy year-round, thanks to the fabled trade winds\u2014with temperatures varying little more than 5\u00b0 between winter and summer.\n\nThere are many advantages to off-season travel in the Virgin Islands. First, from mid-April to mid-December, hotel rates are slashed\u2014often in half. Second, you're less likely to encounter crowds at beaches, resorts, restaurants, and shops. A slower pace prevails in the off season, especially in St. Croix and St. Thomas, and you'll have a better chance to appreciate the local culture and cuisine. Of course, there are disadvantages to off-season travel, too: Many hotels use the slower months for construction and\/or restoration, fewer facilities are likely to be open, and some hotels and restaurants may close completely when business is really slow.\n\nAdditionally, if you're planning a trip during the off season and traveling alone, ask for the hotel's occupancy rate\u2014you may want crowds. The social scene in both the B.V.I. and the U.S.V.I. is intense from mid-December to mid-April. After that, it slumbers a bit. If you seek escape from the world and its masses, summer is the way to go, especially if you aren't depending on meeting others.\n\nHolidays\n\nIn addition to the standard legal holidays observed in the United States, U.S. Virgin Islanders also observe the following holidays: Three Kings' Day (Jan 6); Transfer Day, commemorating the transfer of the Danish Virgin Islands to the Americans (Mar 31); Organic Act Day, honoring the legislation that granted voting rights to the islanders (June 20); Emancipation Day, celebrating the freeing of the slaves by the Danish in 1848 (July 3); Hurricane Supplication Day (July 25); Hurricane Thanksgiving Day (Oct 17); Liberty Day (Nov 1); and Christmas Second Day (Dec 26). The islands also celebrate 2 carnival days on the last Friday and Saturday in April: Children's Carnival Parade and the Grand Carnival Parade.\n\nIn the British Virgin Islands, public holidays include the following: New Year's Day, Commonwealth Day (Mar 12), Good Friday, Easter Monday, Whitmonday (sometime in July), Territory Day Sunday (usually July 1), Festival Monday and Tuesday (during the first week of Aug), St. Ursula's Day (Oct 21), Birthday of the Heir to the Throne (Nov 14), Christmas Day, and Boxing Day (Dec 26).\n\nThe Virgin Islands Calendar of Events\n\nJanuary\n\nEstate Bordeaux Agricultural & Cultural Food Fair, St. Thomas. Weekend-long farm fair in the hills of Estate Bordeaux, with reggae music, crafts, farm tours, and more served up from 10am to midnight. Contact We Grow Food (www.facebook.com\/WEGROWFOOD). January 19 and 20.\n\nFebruary\n\nSt. Croix House Tours, St. Croix. Two separate weekends of house tours, each in a different St. Croix neighborhood, are held to raise funds for the St. Croix Landmarks Society. Go to www.stcroixlandmarks.com for details. Two weekends in mid-February.\n\nAgrifest: St. Croix Agricultural Fair, St. Croix. Held at the Rudolph Shulterbrandt Agricultural Complex, Estate Lower Love, this 3-day festival is one of the island's highlights, a family-friendly event featuring music, crafts, and lots of locally made food and island-grown produce. Go to www.viagrifest.org for details. Presidents' Day weekend.\n\nMarch\n\nMardi Gras Annual Parade, St. Croix. The scenic north shore of St. Croix becomes one big colorful party at Mardi Gras, with a parade to Cane Bay. For information, check www.stcroixtourism.com. First Saturday before Fat Tuesday, usually early March.\n\nSt. Thomas International Regatta, St. Thomas. This is one of three regattas in the Caribbean Ocean Racing Triangle (CORT) series. Top-ranked international racers come to St. Thomas to compete in front of the world's yachting press. The St. Thomas Yacht Club hosts the 3-day event. Call   340\/775-4701, or visit www.rolexcupregatta.com. Late March.\n\nTransfer Day, U.S. Virgin Islands. This holiday commemorates the day the U.S. Virgins were transferred from Denmark to the United States. On this day, vendors sell Danish products, and visits to the remains of Danish ruins and forts are arranged. Call   340\/772-0598, or visit www.stcroixlandmarks.com. March 31.\n\nApril\n\nB.V.I. Spring Regatta & Sailing Festival, Tortola. This is the third of the CORT events (see St. Thomas International Regatta, above). A range of talents, from the most dedicated racers to bareboat crews out for \"rum and reggae,\" participate in the 4-day race. Contact the B.V.I. Spring Regatta Committee in Tortola at   284\/541-6732, or sail over to www.bvispringregatta.org for information. Early April.\n\nVirgin Gorda Easter Festival, Virgin Gorda. Easter weekend is a big event on Virgin Gorda, featuring street parades, a beauty pageant, and nonstop partying. Arrive on Good Friday, Holy Saturday, or Easter Sunday, and you should have no trouble finding the party. Easter weekend.\n\nSt. Croix Food & Wine Experience, St. Croix. It's a weeklong food party on St. Croix, where the island's culinary delights are highlighted with city food crawls, wine events, sunset BBQs, and the chance to sample local creations at the \"Taste of St. Croix.\" Go to www.stcroixfoodandwine.com. April 16\u201321.\n\nVirgin Islands Carnival, St. Thomas. This annual celebration on St. Thomas, with origins in Africa, is the most spectacular and fun carnival in the Virgin Islands. \"Mocko Jumbies,\" people dressed as spirits, parade through the streets on stilts nearly 20 feet high. Steel and fungi bands, \"jump-ups,\" and parades are part of the festivities. Over the years, interestingly, the festivities have become more and more Christianized. Events take place islandwide, but most of the action is on the streets of Charlotte Amalie. For information, call   340\/776-3112, or visit www.vicarnival.com for a schedule of events. After Easter.\n\nMay\n\nB.V.I. Music Festival, Tortola. Music\u2014mostly reggae\u2014drowns out the sea at this music festival on Cane Garden Bay in Tortola, the island's best beach. Musicians come from all over the West Indies to perform. For more information, visit www.bvimusicfestival.com. Late May.\n\nJune\n\nSt John Festival, St. John. Known also as \"Carnival,\" this month-long cultural event takes place on St. John, with steel-pan concerts, calypso shows, parades, beauty pageants, and fireworks displays. For more information, call 800\/372-USVI [8784]. The carnival begins the first week of June and lasts until July 4.\n\nJuly\n\nMango Melee and Tropical Fruit Festival, St. Croix. Mango aficionados and devotees of other tropical fruit converge here for tastings, cooking demonstrations, and contests at the St. George Village Botanical Gardens. For more information, call   340\/692-2874. Early July.\n\nIndependence Day, St. John. The elements of Carnival are combined with emancipation and independence celebrations in this festive event, which culminates on July 4 with a big parade. Thousands of St. Thomas residents flock to St. John for the parades, calypso bands, colorful costumes, and events leading up to the selection of Ms. St. John and the King of the Carnival. Call the St. John tourist office at   340\/776-6201 for more details. July 4.\n\nAugust\n\nB.V.I. Emancipation Celebrations, Tortola. Many visitors from other Caribbean islands hop over to Road Town, in Tortola, for this 2-week party. Join locals as they dance to fungi and reggae bands, and take part in the Emancipation Day Parade and other carnival activities and festivities. For information, call the B.V.I. Tourist Board Office at   284\/494-3701. Early August.\n\nU.S. Virgin Islands Open\/Atlantic Blue Marlin Tournament, St. Thomas. This prestigious St. Thomas\u2013centered charity event (proceeds go to the Boy Scouts) is also eco-friendly\u2014trophies are based on the number of blue marlin caught, tagged, and released. The tournament is open to anyone who's interested, and sportsfishers come from around the world to participate. For more information, call the VI Council of the Boy Scouts of America at   888\/234-7484 or 340\/775-9500; www.abmt.vi. Mid-August.\n\nOctober\n\nVirgin Islands Fashion Week, St. Thomas. Aspiring designers from across the Caribbean, the United States, and even West Africa fly into St. Thomas to showcase their latest fashion designs. At the Caribbean Catwalk Runway show, beach and casual fashions are the draw. Nearly all events, including rap party, are open to the public. For more information, call   340\/344-6078. Five days in October, dates vary.\n\nNovember\n\nSt. Thomas-St. John Agriculture and Food Fair, St. Thomas and St. John. Produce and livestock, along with local vendors selling arts and crafts, are the focus of this agriculture and food fair held on the grounds of the University of Virgin Islands' Reichhold Center for the Arts. Saturday, November 3, and Sunday, November 4 (10am\u20135pm).\n\nAnnual St. Croix International Regatta, St. Croix. Held at the St. Croix Yacht Club on Teague Bay, this 3-day regatta celebrated its 21st year in 2013. It draws serious yachties from the B.V.I., the U.S.V.I., and Florida. Call St. Croix Yacht Club at   340\/773-9531, or visit www.stcroixyc.com for details. Mid-November.\n\nParadise Jam College Basketball Tournament, St. Thomas. Catch big-time college hoops action at this men's and women's basketball tournament held at the University of the Virgin Islands Sports & Fitness Center. Teams have included Kansas, Duke, Providence, and Maryland. For information, go to www.paradisejam.com or call   340\/693-1056.\n\nBVI Restaurant Week, the British Virgin Islands. If you're in the British Virgin Islands around Thanksgiving, take advantage of the island-wide Restaurant Week, where some 40 participating restaurants throughout the B.V.I. feature prix-fixe dinners. Go to www.bvitourism.com. Last week in November.\n\nArt in the Garden Arts & Crafts Festival, St. Thomas. This annual 3-day festival, which takes place in Tillett Gardens in St. Thomas, includes displays from more than 30 local artists, along with live music and entertainment and free activities for kids. For information, visit www.tillettfoundation.org. Last weekend in November.\n\nAnegada Lobster Festival, the British Virgin Islands. It's a 2-day beach party on the white sands of Anegada for the November Lobster Festival, celebrating the island's famously sweet Caribbean lobster. Eight of the island's restaurants feature sampler plates of lobster dishes. Smiths Ferry (<http:\/\/bviferryservices.com>) and Road Town Fast Ferry (www.roadtownfastferry.com) offer special runs to and from the island, with the last ferries leaving at 9pm. Call   284\/494-3134. Last weekend in November.\n\nDecember\n\nChristmas in St. Croix. This major event launches the beginning of a 12-day celebration that includes Crucian Christmas Carnival (www.stxcarnival.com)\u2014with calypso music, parades, horse races, food fairs, and fireworks\u2014Christmas Day, Christmas Second Day (Dec 26), New Year's Eve (called \"Old Year's Day\"), and New Year's Day. It ends on January 6, the Feast of the Three Kings, with a parade of flamboyantly attired merrymakers. For information, call the U.S. Virgin Islands Department of Tourism office in Christiansted at   340\/773-0495.\n\nFoxy's Old Year's Night, Jost Van Dyke, B.V.I. It's a mega-party on Jost's Great Harbour as Foxy rings in the New Year with weekend-long events, from a Caribbean barbecue to DJ music on the beach to a headliner concert that in 2013 showcased Bunny Wailer. For information, go to <http:\/\/foxysbar.com>.\n\nRESPONSIBLE TRAVEL\n\nMany of the islands were clear-cut in the 1700s to make way for sugar plantations, destroying much of the natural landscape. All through the 1900s, while real estate developments on St. Thomas continued to mushroom, little concern was given to preserving and sustaining the natural resources of the U.S.V.I. Today, there is a very different attitude toward the ecosystem of the Virgin Islands among permanent residents and visitors alike.\n\nThe sparkling marine waters of the Virgin Islands require special stewardship, and the B.V.I. is a leader in environment regulations regarding the islands' pristine seas. The country has banned the use of jet skis (also not allowed in the marine parkland around St. John) and requires that any jet skis or WaveRunners brought into the country be declared at Customs upon entry. Penalties for jet-ski use include confiscation of the jet skis and a $5,000 fine. The National Parks Act of 2006 prohibits sewage disposal and the removing of any living or dead coral from any coastal area, among other regulations.\n\nIn this sailing and boating paradise, boaters are asked to anchor using mooring balls or if you must anchor, look for a sandy bottom that's clear of coral or seagrass meadows where sea turtles feed. Do not anchor or tie your boat to mangroves. Don't drag dinghies onto the beach; look for dinghy moorings or docks. Never throw trash, particularly plastic, overboard (turtles choke on plastic bags). Divers and snorkelers should take care not to touch or accidentally brush up against delicate coral. If you see environmental disturbances (or turtle nesting activity), call the B.V.I. Conservation & Fisheries Department (  284\/494-3429). An invaluable resource is the online \"Marine Awareness Guide\" at www.bvimarineawareness.com.\n\nLow-impact activities like hiking, snorkeling, and kayaking are hugely popular in the Virgin Islands. While on St. Croix, contact the St. Croix Environmental Association (p. ), which hosts hikes, tours of research facilities, and events based around the hatching of baby sea turtles. Aside from the many companies that offer tours, the St. Thomas\u2013based Virgin Islands Ecotours (  877\/845-2925; <http:\/\/viecotours.com>) offers tours with professional naturalists of the mangrove lagoon and nature reserve at Cas Cay, St. Thomas, and snorkeling and hiking tours in Caneel Bay, St. John. B.V.I. Eco-Tours (  284\/495-0271; www.bvi-ecotours.com) offers a variety of guided tours, including snorkeling, bird-watching, sightseeing, diving, and general tours of the British Virgin Islands.\n\nSustainable Eating\n\nIt takes a whole lot of fossil fuels to import food onto these remote islands. That's why it's heartening to see the rebirth of small farms and sustainable farming throughout the Virgins, a movement that is catching fire in particular on St. Croix, with its farmer's markets, farm co-ops, and Virgin Islands Sustainable Farm Institute (<http:\/\/vi.locallygrown.net>) and on St. Thomas, where West End farms in Estate Bordeaux are using government grants to increase capacity in the loamy volcanic soil, where as one farmer said, \"you can grow anything.\" You can do your part in saving energy by eating local: dining on seafood caught in and around the waters of Virgin Islands and buying produce and fruit from the weekly farmer's markets and produce stands\n\nA number of truly eco-friendly lodgings are found on the islands, including campsites. St. John, which is almost entirely a national park, has numerous campsites, including those run by the National Park Service. Alas, the beloved 114-unit Maho Bay Camp eco-resort closed in summer 2013 (sold for $13.95 million to an unknown buyer), but its sister eco-resort Estate Concordia Preserve may expand beyond its 42 campsites. On St. Croix, there is Mount Victory Camp (p. ), which relies on renewable energy to power its cottages. The British Virgin Islands are less developed than their American cousins, and lodgings tend to be more eco-friendly by nature. You don't have to camp out to stay in eco-sensitive lodging, however. The Cooper Island Beach Club (p. ) meets the middle ground between luxury and roughing it. All of Guana Island (p. ), a private resort, is a wildlife sanctuary watched over by the attentive owners (it even has its own organic orchard and a smattering of flamingoes in the salt pond). The Bitter End Yacht Club (p. ), sustainably run by necessity in its early days, continues its green ways by generating its own electricity and collecting and distilling its own water. In fact, many resorts around the islands have their own desalination plants, making highly potable fresh water out of seawater.\n\nSt. Thomas, with all its development and modern conveniences, faces the biggest challenges in regard to sustainable development. To learn more, see the section \"The Virgin Islands Today,\" earlier in this chapter.\n4\n\nST. THOMAS\n\nThe cosmopolitan hub of the Virgin Islands, St. Thomas is known for its brassy commercial hustle. But with sparkling turquoise seas and peaceful mountain aeries, it's easy to leave the hubbub far behind. Yes, the 32-square-mile island is home to the busiest cruise-ship harbor in the West Indies, and its historic capital, Charlotte Amalie, is the beating commercial heart of the region. But just minutes from the action are serene beaches and pampering retreats. More than any other Virgin Island, St. Thomas has something for everyone: Join the market throngs by day and swing to reggae rhythms at night\u2014or simply forget your cares on a sun-kissed, palm-fringed beach.\n\nEssentials\n\nGetting There\n\nBy Plane\n\nIf you're flying to St. Thomas, you will land at the Cyril E. King Airport (  340\/774-5100; <http:\/\/saint-thomas-airport.com>; airport code STT), 3 miles (about a 15-min. drive) to the west of the central business district in Charlotte Amalie on Route 30. From here, you can easily grab a taxi to your hotel or villa. Chances are you will be staying east of Charlotte Amalie, so keep in mind that getting through town often involves long delays and traffic jams during rush hours.\n\nMany people flying in from mainland cities in North America and from overseas connect through Miami or San Juan. Flight time from Miami is about 21\u20442 hours. Flight time from San Juan to St. Thomas is approximately 30 minutes. Flight time between St. Thomas and St. Croix is only 20 minutes.\n\nDirect flights to the U.S. Virgin Islands are available on American Airlines (  800\/433-7300 in the U.S.; www.aa.com) from New York City, Boston, Miami, and San Juan. Flights from NYC take 33\u20444 hours. Continental Airlines (  800\/231-0856 in the U.S.; www.continental.com) has daily flights from Newark International Airport, in New Jersey, to St. Thomas. Delta (  800\/241-4141 in the U.S.; www.delta.com) offers two daily nonstop flights between Atlanta and St. Thomas. US Airways (  800\/428-4322 in the U.S.; www.usairways.com) has direct flights from Philadelphia and Charlotte. JetBlue (  800\/538-2583; www.jetblue.com) has direct flights from Boston. United Airlines (  800\/538-2929 in the U.S.; www.united.com) has direct flights into St. Thomas from Chicago, Dulles (Washington, D.C.), and Newark. Spirit Airlines (  800\/772-7117; www.spirit.com) has direct flights from Fort Lauderdale.\n\nCape Air (  866\/227-3247 in the U.S. and U.S.V.I.; www.capeair.com) offers daily service between St. Thomas and San Juan, Puerto Rico, and has expanded its service to include flights between St. Thomas and both St. Croix and Tortola.\n\nSt. Thomas\n\nSeaborne Airlines (  866\/359-8784 or 340\/773-6442 in the U.S.V.I, www.seaborneairlines.com) is currently the major carrier between St. Thomas and St. Croix, offering regularly scheduled daily flights on 34-seat turboprop planes and seaplanes capable of carrying 15 to 17 passengers. Its regular planes fly between Cyril E. King airport in St. Thomas and the Henry E. Rahlsen airport in St. Croix. Seaborne seaplanes fly between the St. Thomas seaplane base (next to the Charlotte Amalie ferry terminal) and the St. Croix seaplane seaport in the Christiansted waterfront. Flight time between St. Thomas and St. Croix is 20 to 25 minutes. Seaborne also flies between St. Thomas and Beef Island, Tortola.\n\nBy Cruise Ship\n\nCharlotte Amalie is one of the world's busiest cruise ports and the Caribbean's largest duty-free port, welcoming nearly 1,800,000 cruise passengers in 2012 (2013 was unavailable as we went to press). Most of the major cruise lines include regular stops in St. Thomas on their Caribbean itineraries, including the biggest cruise ships in the world, floating \"cities at sea\" capable of holding up to 6,000 passengers at a time.\n\nCruise ships dock at one of two major piers, each with room for two mega-ships at a time: Havensight Pier and Crown Bay. In addition, during the cruising high season, it's not unusual to have an additional one, two, even three ships anchored just outside the harbor, delivering cruise-ship passengers to shore in tenders. At press time an expansion was in the works for the Crown Bay pier.\n\nBy Boat\n\nFerry service from Puerto Rico to St. Thomas, with a stop in St. John, is available only during Carnival. Trip time between Fajardo and Charlotte Amalie (St. Thomas) is about 13\u20444 hours, with the departure Saturday morning and the return Sunday afternoon. The cost is $100 one-way, $125 round-trip. For more information, call Transportation Services at   340\/776-6282.\n\nVisitor Information\n\nThe U.S. Virgin Islands tourist board Welcome Center, in Havensight Mall, on the Waterfront in downtown Charlotte Amalie (  340\/774-8784; www.visitusvi.com), is open Monday to Friday 8am to 5pm and Saturday 8am to 2pm. You can get maps, brochures, and even a list of legal shoreline fishing sites throughout the islands here.\n\nSt. Thomas: Gateway to the Other Virgins\n\nMost people who visit the other Virgin Islands\u2014both U.S. and British\u2014must travel through St. Thomas to get there. Even if you plan to stay in St. Thomas your entire trip, day-trips to one or several other islands should be high on your vacation agenda. A robust ferry system delivers passengers between the islands (except for St. Croix, which is better reached by plane). For details on ferry departure times, check in any recent issue of the free weekly publication \"This Week,\" or go to www.bestofbvi.com\/info\/info_bviferry.htm, with updated ferry times. For a list of ferry companies and contact information, go to p. . St. Thomas has two ferry terminals: Marine Terminal, in Charlotte Amalie, and Red Hook, on the island's east end. Keep in mind that everyone\u2014including U.K. citizens\u2014traveling to the British Virgin Islands needs a passport and must pay a $15-per-person passenger tax at departure.\n\nIsland Layout\n\nCharlotte Amalie\n\nFor a map of the landmarks and attractions discussed below, see \"Walking Tour: Charlotte Amalie,\" on p. .\n\nCharlotte Amalie, the capital of St. Thomas, is the only town on the island. Its seaside promenade is called Waterfront Highway, or simply, the Waterfront. From here, you can take any of the streets or alleyways into town to Main Street (also called Dronningens Gade). Principal links between Main Street and the Waterfront include Raadets Gade, Tolbod Gade, Store Tvaer Gade, and Strand Gade.\n\nMain Street is home to all of the major shops. The western end (near the intersection with Strand Gade) is known as Market Square. Once the site of the biggest slave market auctions in the Caribbean Basin, today it's an open-air cluster of stalls where local farmers and gardeners\u2014many from the Estate Bordeaux farms on the island's West End\u2014gather to sell their produce; Saturday is their big day. Go early in the morning to see the market at its best.\n\nRunning parallel to and north of Main Street is Back Street (also known as Vimmelskaft Gade), which is also lined with stores, including some of the less expensive choices. Note: It can be dangerous to walk along Back Street at night, but it's reasonably safe for daytime shopping.\n\nIn the eastern part of town, between Tolbod Gade and Fort Pladsen (northwest of Fort Christian), lies Emancipation Park, commemorating the liberation of the slaves in 1848. Most of the major historic buildings, including the Legislature, Fort Christian, and Government House, lie within a short walk of this park.\n\nSoutheast of the park looms Fort Christian. Crowned by a clock tower and painted rusty red, it was constructed by the Danes in 1671. The Legislative Building, seat of the elected government of the U.S. Virgin Islands, lies on the harbor side of the fort.\n\nKongens Gade (or King's Street) leads to Government Hill, which overlooks the town and St. Thomas Harbor. Government House, a white brick building dating from 1867, stands atop the hill.\n\nBetween Hotel 1829 (a mansion built that year by a French sea captain) and Government House is a stone staircase known as the Street of 99 Steps. Actually, someone miscounted: It should be called the Street of 103 Steps. Regardless, the steps lead to the summit of Government Hill.\n\nWest of Charlotte Amalie\n\nThe most important of the outlying neighborhoods to the west of Charlotte Amalie is Frenchtown. Some of the older islanders still speak a distinctive Norman-French dialect here. Because the heart of Charlotte Amalie can be less safe at night, Frenchtown, with its finer restaurants and interesting bars, has become the place to go after dark. To reach Frenchtown, take Veterans Drive west of town along the Waterfront, turning left (shortly after passing the Windward Passage Hotel on your right) at the sign pointing to the Villa Olga.\n\nThe mid-grade hotels that lie to the immediate west of Charlotte Amalie attract visitors who are seeking more moderate hotel rates than those charged at the mega-resorts that dot the gold-plated South Coast. The disadvantage is that you may have to depend on public transportation to reach the sands. The biggest attraction is that you're on the very doorstep of Charlotte Amalie, filled with restaurants, bars, shopping, and other amusements.\n\nEast of Charlotte Amalie\n\nTraveling east from Charlotte Amalie, along a traffic-clogged highway, you'll see St. Thomas Harbor on your right. If you stay in this area, you'll be in a tranquil setting just a short car or taxi ride from the bustle of Charlotte Amalie. The major disadvantage is that you must reach the sands by some form of transportation; if you want to run out of your hotel-room door onto the beach, look elsewhere.\n\nThe South Coast\n\nThis fabled strip, with its good, sandy beaches, has put St. Thomas on the tourist maps of the Caribbean. Many visitors prefer the full-service resorts on the South Coast and East End to the hustle and bustle of Charlotte Amalie, especially during the day, when it's overrun by cruise-ship passengers. But if you feel the need for a shopping binge, cars, hotel shuttles, and taxis can quickly deliver you to Charlotte Amalie.\n\nThe East End\n\nThe East End is reached by traversing a long, twisting, traffic-clogged road east of Charlotte Amalie. Once you're here, you can enjoy sea, sand, and sun with little to disturb you (the East End offers even more isolation than the South Coast). This is the site of such lovely beaches as Sapphire Beach and Lindquist Beach. This section of bays and golden sands is where you'll find the the luxe Ritz-Carlton resort as well as a smattering of smaller, less-expensive resorts and condos. The settlement at Red Hook is a bustling community with raffish charm and lots of seaside bars and affordable eateries. It is also the departure point for ferries to St. John.\n\nThe North Coast\n\nThe renowned beach at Magens Bay lies on the lush North Coast. Be aware that the beach is often overrun with visitors, especially when cruise-ship arrivals are heavy. The North Coast has few buildings and not much traffic, but what it does have are scenic vistas, among the most panoramic on the island. Note that traveling the roads can be like a ride on a rollercoaster\u2014the roads have no shoulders and can be especially scary for those not familiar with driving on the left. A lot of the northwest coast, especially at Botany Bay, Bordeaux Bay, and Santa Maria Bay, isn't linked to any roads. Estate Bordeaux has some beautiful, rural stretches of lush mountain farmland, where farmers raise produce and livestock in the loamy green hills.\n\nGetting Around\n\nRenting a car is a great way to see the island and save money on taxi fares. On average, a taxi costs $8 per person per trip, which can add up if you plan a couple of outings from your resort every day. On the other hand, roads on the island can be steep, narrow, poorly lit, and twisting (and often not in the best of shape), so driving St. Thomas can be a challenge for inexperienced drivers\u2014particularly at night. Plus, driving on the left can be somewhat of an adjustment for those used to driving on the right. One local advised thinking \"shoulder to shoulder\"\u2014your left shoulder should be to the shoulder of the road.\n\nAlso keep in mind that normal traffic congestion at rush hours in Charlotte Amalie is only compounded by the morning arrival of sometimes thousands of cruise-ship passengers and the taxis and safari vans there to deliver them to their day's excursions. The good news is that waterfront traffic should be transformed in the next few years, when the harborfront road is expanded from two to a luxurious four lanes.\n\nBy Car\n\nSt. Thomas has many leading North American car-rental firms at the airport, and competition is stiff. Before you go, compare the rates of the \"big three\": Avis (  800\/331-1212 or 340\/774-1468; www.avis.com), Budget (  800\/626-4516 or 340\/776-5774; www.budgetstt.com), and Hertz (  800\/654-3131 or 340\/774-1879; www.hertz.com). You can often save money by renting from a local agency, although vehicles may be older, with more wear and tear. Recommended agencies include Dependable Car Rental, 3901 B Altona, Welgunst, behind the Bank of Nova Scotia and the Medical Arts Complex (  800\/522-3076 or 340\/774-2253; www.dependablecar.com), which provides free pickup and drop-off anywhere on St. Thomas and offers a 12 percent discount when you mention that you saw them on the Internet; and the aptly named Discount Car Rental, 14 Harwood Hwy., located just outside the airport on the main highway (  877\/478-2833 or 340\/776-4858; www.discountcar.vi), which offers a 12 percent discount when you book online or through its Facebook page\u2014and advertises \"clean, new cars.\"\n\nNote: Gas (petrol) was hovering close to $5 a gallon at press time.\n\nDriving Rules Always drive on the left. The speed limit is 20 mph in town, 35 mph outside town. Seat belts are required by law, and it's illegal to talk on cellphones while driving.\n\nParking Because Charlotte Amalie is a labyrinth of congested one-way streets, don't try to drive in town looking for a parking spot. If you can't find a place to park along the Waterfront (free), go to the sprawling lot to the east of Fort Christian, across from the Legislature Building. Parking fees are nominal here, and you can park your car and walk northwest toward Emancipation Park, or along the Waterfront, until you reach the shops and attractions.\n\nBy Taxi\n\nTaxi rates are set by the island's Taxi Association and fares are widely posted, even in taxis; check out the official fares in the free magazine \"This Week\" offered in most businesses. Look for officially licensed taxis only: You can spot them by their dome lights and the letters tp on the license plate. Still, be sure to confirm the rate with the driver before you get into the taxi. A typical fare from Charlotte Amalie to Sapphire Beach is $13 per person; from the airport to the Marriot Frenchman's Reef is $10 per person. We took a taxi one night from the Ritz-Carlton to Secret Harbour for dinner and were charged $5 a person. Surcharges are added after midnight. Add on $2 per bag for luggage (and a $1 surcharge for trips June 1\u2013Sept 1).\n\nTaxi vans and open-air safaris (converted truck beds with open-air seating) are ubiquitous around the island. Taxi vans are equipped to transport approximately 8 to 12 passengers to multiple destinations on the island, while safaris can often fit up to 25 people. It's cheaper to hop on a van or safari than ride a taxi on your own if you're going between your hotel and the airport, but keep in mind you will be making stops along the way\u2014an exhausting proposition if you have arrived on a late flight. The cost for luggage ranges from $1 to $2 per bag. Call   340\/774-7457 to order a taxi van.\n\nIf you don't plan to rent a car, it's easy to find taxi drivers. Just have your hotel or restaurant call a taxi for you, no matter where you are. Even better: Get the card of a favorite taxi driver and let him or her know your itinerary\u2014or call the drivers we recommend below. Taxi drivers also make wonderful sightseeing guides. Expect to pay about $50 for a single-passenger tour or $25 per person for two or more passengers for 2 hours of sightseeing in a shared car.\n\nIf you're looking for a thoroughly engaging and wonderfully informative taxi driver\/tour guide, look no further than Campbell Rey (  340\/771-1568), the unofficial \"mayor\" of the island\u2014we highly recommend his services. We also highly recommend Llewelyn Powell (  340\/771-1568 or 776-3887). For 24-hour radio-dispatch taxi service, call   340\/774-7457.\n\nBy Water taxi\n\nGetting around by water taxi is another way to travel. Dohm's Water Taxi is an inter-island water taxi service that will take you anywhere in the Virgin Islands you want to go in custom-built catamaran powerboats. It's a smart alternative if you're staying off island and your flight arrives in St. Thomas too late to catch a public ferry. Dohm's also offers water-taxi service directly between the Ritz-Carlton or Marriott Frenchman's Reef and St. John ($50 and $30 per person, respectively, with a five-person minimum). Go to www.watertaxi-vi.com or call   340\/775-6501.\n\nBy Bus\n\nPublic buses run in the city and the country between 5:15am and 8pm daily, but waits can be very long and this is a difficult way to get about. A ride within Charlotte Amalie is 75\u00a2; a ride to anywhere else is $1. For schedule and bus-stop information, call   340\/774-5678.\n\nOn Foot\n\nWalking is the best way to explore the historic section of Charlotte Amalie during the day. However, you will need a car or driver to reach many other island attractions, including Coral World and Magens Bay.\n\n  St. Thomas\n\nBanks FirstBank Virgin Islands (  340\/775-7777; www.firstbankvi.com) has six locations on the island, with 24-hour ATMs at Crown Bay Center, East End Plaza, FirstBank Plaza, the Waterfront, and Yacht Haven. Scotiabank (  340\/776-5880; www.scotiabank.com) has six locations on St. Thomas, including Havensight Mall and Tutu Park Mall; all branches have ATMs. Banco Popular   800\/724-3655; www.popular.com\/vi) has branches in Charlotte Amalie and Red Hook. Most island banks are open Monday to Thursday 8:30am to 3pm, and Friday 8:30am to 4pm. The banks are your only option if you need to exchange currency. More than 50 ATMs are available on the island.\n\nBusiness Hours Typical business and store hours are Monday to Friday 9am to 5pm and Saturday 9am to 1pm. Some shops open Sunday for cruise-ship arrivals. Bars are usually open daily 11am to midnight or 1am, although some hot spots stay open later.\n\nDentists The V.I. Dental Center, Foothills Professional Building, 9151 Estate Thomas, Ste. 203 (  340\/772-6000; <http:\/\/videntalcenter.com>), has a team of dentists that are members of the American Dental Association. Call for information or an appointment.\n\nDoctors Schneider Regional Medical Center, 9048 Sugar Estate, Charlotte Amalie (  340\/776-8311; www.rlshospital.org), provides services for locals and visitors.\n\nDrugstores Go to Havensight Pharmacy, Havensight Mall, Building 4, Charlotte Amalie (  340\/777-5313).\n\nEmergencies For the police, call   911; ambulance,   911; fire,   921.\n\nHospitals\/Clinics The Schneider Regional Medical Center is at 9048 Sugar Estate, Charlotte Amalie (  340\/776-8311; www.rlshospital.org). For 24-hour medical care call the Red Hook Family Practice (  340\/775-2303) or Doctors on Duty (  340\/776-7966).\n\nHot Lines Call the police at   911 in case of emergency. If you have or witness a boating mishap, call the U.S. Coast Guard Rescue (  787\/729-6800), which operates out of San Juan, Puerto Rico. Scuba divers should note the number of a decompression chamber (  340\/776-8311) at the Schneider Regional Medical Center on St. Thomas.\n\nInternet Most hotels and resorts now offer free Wi-Fi in the lobby. Many bars and restaurants also offer free Wi-Fi.\n\nLaundry & Dry Cleaning The major hotels provide laundry service, but it's more expensive than a Laundromat. For dry cleaning, go to One-Hour Martinizing, Barbel Plaza, Charlotte Amalie (  340\/774-5452).\n\nMail Postage rates are the same as on the U.S. mainland: 34\u00a2 for a postcard and 49\u00a2 for a letter to U.S. addresses. For international mail, a first-class postcard or letter stamp costs $1.15.\n\nMaps See \"Visitor Information,\" earlier in this chapter.\n\nNewspapers & Magazines Copies of U.S. mainland newspapers, such as \"The New York Times,\" \"USA Today,\" and \"The Miami Herald,\" arrive daily in St. Thomas and sold at newsstands. \"The Virgin Island Daily News\" (<http:\/\/virginislandsdailynews.com>) covers local, national, and international events. \"St. Thomas This Week,\" packed with visitor information, is distributed free on the island.\n\nPolice The main police headquarters is currently located in the Alexander Farrelly Criminal Justice Center in Charlotte Amalie (  340\/774-2211). The Crime Line phone number is   340\/777-8700.\n\nPost Office The main post office is at 9846 Estate Thomas, Charlotte Amalie (  340\/774-1950), and is open Monday to Friday 7:30am to 5pm and Saturday 7:30am to noon.\n\nSafety The Virgin Islands are a relatively safe destination. The small permanent populations are generally friendly and welcoming. That being said, St. Thomas is no longer as safe as it once was. Crime, especially muggings, is on the rise in Charlotte Amalie. Wandering the town at night, especially on the back streets (particularly on Back St.), is not recommended. Avoid frequenting Charlotte Amalie's bars alone at night. Guard your valuables. Store them in hotel safes if possible, and make sure you keep your doors and windows shut at night.\n\nTaxes The only local tax is an 10 percent hotel tax. St. Thomas has no sales or luxury taxes.\n\nTelephone All island phone numbers have seven digits. It is not necessary to use the 340 area code when dialing within St. Thomas. Numbers for all three islands, including St. John and St. Croix, are found in the U.S. Virgin Islands phone book.\n\nTipping Go to \"Fast Facts,\" in Chapter 8 for tipping guidelines.\n\nToilets You'll find public toilets at beaches and at the airport, but they are limited in town. Most visitors use the facilities of a bar or restaurant.\n\nTransit Information Call   340\/774-7457 to order a taxi 24 hours a day. Call   340\/774-5100 for airport information and   340\/776-6282 for information about ferry departures for St. John.\n\nWeather For emergency (hurricane and disaster) weather reports, call Vietema at   340\/774-2244.\n\nWHERE TO STAY\n\nThe choice of hotels on St. Thomas divides almost evenly between small inns and B&Bs in Charlotte Amalie and full-service resorts along the East End and South Coast that front fabulous beaches. There are advantages and disadvantages to both.\n\nrenting A CONDO, APARTMENT, OR VILLA\n\nMany visitors prefer renting a condo, apartment, or villa when they visit the island\u2014particularly for the self-catering capabilities. We've found that Calypso Realty (  800\/747-4858 or 340\/774-1620; www.calypsorealty.com) often has the best offers, especially on rentals from April to mid-December. A condo goes for $1,200 to $4,000 per week, with a 7-day minimum stay.\n\nAnother good source is McLaughlin Anderson Luxury Caribbean Villas (  800\/537-6246 or 340\/776-0635; www.mclaughlinanderson.com), which has beautiful rentals not only on St. Thomas but also on St. John, St. Croix, and various other Caribbean isles. A two-bedroom villa begins at $2,000 per week in winter, with off-season rates beginning at $1,449.\n\nAntilles Resorts (  800\/874-7897 or 340\/775-6100; www.antillesresorts.com) is a hit among habitual island-goers to St. Thomas and St. Croix, enjoying a repeat business for its wide range of properties, from economical to luxury.\n\nYou can also find excellent deals on popular owner-rented vacation lodging websites, including VRBO (www.vrbo.com) and HomeAway (www.homeaway.com); both websites list numerous attractive villas and condos throughout the Virgins. What you will not get with these rentals is service (unless advertised) or even guarantees that the rental is what is appears to be. Both VRBO and HomeAway sell rental guarantee insurance ($39 and up) that guarantees your money back under certain circumstances (the property has been double-booked or grossly misrepresented, for example)\u2014so you may want to weigh that extra fee against the low rental rates.\n\nIf you want to be within close proximity to the island's best shopping, the widest choice of restaurants and bars, and most historic attractions, Charlotte Amalie is the place to be. And if you're looking for budget accommodations, or a choice of moderately priced inns, you'll need to be in or near Charlotte Amalie. The downside to staying here is that you'll have to take a shuttle or taxi ride over to a good beach, a ride of no more than 10 to 15 minutes from most Charlotte Amalie properties. The one exception to this rule is Marriott Frenchman's Reef, which offers the isolation of a beachside resort along with proximity to all of Charlotte Amalie's attractions and shops.\n\nIf your dream is to arrive in St. Thomas and anchor yourself directly on a beach, then the East End is the place to be. Beachfront resorts and condominiums on the East End and along the South Coast offer many perks, among them a measure of serenity and security. Expect a range of watersports, in-house spas and dining options. But almost without exception, the East End beachfront resorts are the most expensive properties on island. In spite of the cost, these hotels attract customers who want the full-service resort life that you simply won't find at the inns and B&Bs of Charlotte Amalie. The downside is that if you want to experience some of the island's best restaurants, you'll have to depend on taxis or risk driving along narrow, dark, and unfamiliar roads at night. To those who opt to stay resort-bound for most of their stay, we say call a taxi for a night out from the resort at least once while you're here.\n\nHotels in the Virgin Islands slash their prices in summer by 20 percent to 60 percent.\n\nSt. Thomas Hotels\n\nTwo important notes on rates: Unless otherwise noted, the rates listed below do not include the 10 percent government hotel tax. Also note that most of the high-end resorts also tack on daily resort fees, from $35 to $50.\n\nIn Charlotte Amalie\n\nModerate\n\nVilla Santana   Done up in Mexican clay tiles and stonework, this is the reconstructed palace of the exiled, former president of Mexico, General Antonio Lopez de Santa Anna of Mexico, who built it in the 1850s (the original burned in a fire in 1985). Pillowed in a garden of hibiscus and bougainvillea, Villa Santana has panoramic views of Charlotte Amalie and the St. Thomas harbor and six charming, handsomely furnished guest suites. La Mansion, in the villa's former library, is a duplex with a full kitchen; La Terraza is another split-level room with a spacious terrace offering views of the glittering harbor. La Torre has its own outdoor dining patio next to a gazebo. Many of the rooms have original rock walls and four-poster beds (one, rumor has it, was used by the General himself\u2014the frame, not the mattress!). The shopping district in Charlotte Amalie is just a 5-minute walk away; Magens Bay Beach is a 15-minute drive north.\n\n2602 Bjere Gade, No. 2D.   340\/776-1311. www.villasantana.com. 6 units. Winter $150\u2013$245; off season $120\u2013$176. Rates based on 3-night minimum; surcharge may be added for shorter stays. Amenities: Outdoor pool; Wi-Fi (free).\n\nWindward Passage Hotel   This blocky hotel originated as a Holiday Inn and looks it. But despite its chain-hotel bones it's favored by business travelers and locals in transit to and from the other Virgin Islands. We think that's because it's well-maintained and friendly, but its location, just across the street from the Charlotte Amalie ferry docks, probably doesn't hurt, either. Rooms are dated but very clean (a renovation was in the works when we visited in early 2014), each with its own private balcony. Book a harbor-view room to fully appreciate the size and heft of the cruise ships slipping in and out of port.\n\nVeterans Dr.   800\/524-7389 or 340\/774-5200. www.windwardpassage.com. 150 units. Winter $231\u2013$275 double, $331\u2013$375 suite; off season $150\u2013$245 double, $274\u2013$311 suite. Amenities: Restaurant; bar; babysitting; off-site health club ($25 per day); freshwater pool; putt-putt golf; spa; watersports equipment\/rentals; Wi-Fi (free).\n\nAt Home in the Tropics   This award-winning guesthouse is one of the best B&Bs in Charlotte Amalie, a traditional West Indian house where each room opens onto splendid views of Charlotte Amalie harbor. Dating from 1803 (it was the barracks for the guard of the Danish Governor), the inn feels exclusive and quite private, with a small harborside pool and attractively furnished rooms with tile floors and breeze-filled windows. It's up on Blackbeard's Hill (a swank neighborhood) and an easy walk to downtown and the shops of Charlotte Amalie\u2014but those with mobility issues might have trouble navigating the stairs back up the hill. Closed from June 1 to November 1.\n\nBlackbeard's Hill, 1680 Dronningens Gade.   340\/777-9857. <http:\/\/athomeinthetropics.com>. Winter $245\u2013$265 double; off season $225\u2013$245 double. Rates include breakfast. Children under 12 discouraged. Amenities: Outdoor swimming pool; Wi-Fi (free).\n\nBellavista Bed & Breakfast   This wonderful four-room West Indian\u2013style property is our favorite B&B in Charlotte Amalie. It's nestled on Denmark Hill, a restored island estate overlooking the city harbor and just a 5-minute walk into town. Rooms are dreamy, with slatted wood ceilings, wood floors, and tastefully done tropical hues. You can relax after a day of touring on the sun-dappled pool deck or in the breezy open-air living room, framed in lush foliage. Reserve early; this place books up. Note: Do not confuse this B&B with the Bellavista Scott hotel in Estate Thomas, which offers much more basic accommodations.\n\nCharlotte Amalie Hotels & Restaurants\n\n2713 Murphy Gade 12\u201314.   888\/333-3063 or 340\/714-5706. www.bellavista-bnb.com. 4 units. Winter $220\u2013$295 double; summer $195\u2013$270 double. Rates include breakfast. Amenities: Concierge; outdoor pool; Wi-Fi (free).\n\nInexpensive\n\nThe Crystal Palace Bed and Breakfast   This stately old colonial home, filled with antiques, West Indian relics, and family flotsam and jetsam, is overseen by owner Ronnie Lockhart, who grew up here. It has plenty of old-world personality, with elegant and airy parlor rooms, some with walls lined with early-19th-century stonework. Rooms are oversized and high-ceilinged but not particularly fancy. Three of the five accommodations have shared facilities in the corridors; of the two rooms with private baths, one is downstairs and the other is up (both have queen four-posters). Stay here if you appreciate old-school languor...or look forward to soaking in views of the harbor from the veranda while sipping a cocktail from the honor bar.\n\n12 Crystal Gade.   866\/502-2277 or 340\/777-2277. www.crystalpalaceusvi.com. 5 units, 2 with bathrooms. Winter $139 double without bathroom, $169 double with bathroom; off season $119 double without bathroom, $139 double with bathroom. Extra person $20\/night. Rates include breakfast. Amenities: Wi-Fi (free).\n\nMafolie Hotel   This good-value gem sits high in the hills above Charlotte Amalie and its sparkling harbor\u2014the views are splendid during the day but off-the-charts gorgeous at night. It has a congenial proprietor in Adam Israel, and the superb Mafolie Restaurant (p. ) is one of the island's best, a big proponent of the island's farm-to-table movement with a menu featuring farm-grown produce and fresh-caught seafood. Each room is pleasantly furnished; none are swank by any means, in fact, some are on the smallish side, and a little dark, and a handful are only entered on the road side of the resort. But all are comfortable, with quality beds. Of the 22 rooms, 10 will have outdoor balconies by the time you read this, with those glittering harbor views at your feet, and five are suites. You can take a dip in the pool, which enjoys the same panoramic vista, or simply drink in the views at the pool bar.\n\n7091 Estate Mafolie.   340\/774-2790. www.mafolie.com. 22 units. Winter $144\u2013$160 double; off season $105\u2013$125 double. Rates include continental breakfast. Up to 2 children 12 and under stay free in parent's room. Extra person $15. Ask about island packages. Amenities: Restaurant; 2 bars; concierge; pool; Wi-Fi (free).\n\nVilla Blanca   Set on 3 acres of lush hilltop just 11\u20442 miles east of Charlotte Amalie, the Villa Blanca provides killer views of the harbor and mossy hills rising out of blue seas. It's nestled in lovingly tended gardens. Each room comes with a well-equipped kitchenette, tile floors, good beds, and a private balcony or terrace with views either eastward to St. John or westward to the Charlotte Amalie harbor and the cruise-ship docks. Take a dip in the freshwater pool or watch the sun set from the villa's lounge deck. Charlotte Amalie shopping is just a 5-minute taxi ride away.\n\n4 Raphune Hill, Rte. 38.   800\/231-0034 or 340\/776-0749. www.villablancahotel.com. 14 units. Winter $135\u2013$145 double; off season $95\u2013$105 double. Rates include continental breakfast. Children 9 and under stay free in parent's room. Extra person $15\/night. Amenities: Pool (outdoor); Wi-Fi, in lobby (free).\n\nWest of Charlotte Amalie\n\nExpensive\n\nBest Western Emerald Beach Resort   Yes, this Best Western is located just across from the airport, but it's also set on a pretty white-sand beach, with beachfront views from every room. Lying 2 miles to the west of Charlotte Amalie, to which it is linked by free shuttle service. Emerald Beach attracts both businesspeople and families for its location close to the airport and town. The rooms are outfitted in standard Best Western furnishings, but everything is clean and comfortable, with king-size beds in every room. If this is booked, try its sister property, Best Western Carib Beach Resort (  800\/792-2742 in the U.S. or 340\/774-2525; www.caribbeachresort.com), just a 5-minute walk away at 70C Lindbergh Bay, with affordable oceanview rooms with private terraces.\n\n8070 Lindbergh Bay.   800\/233-4936 in the U.S., or 340\/777-8800. www.emeraldbeach.com. 90 units. Winter $200\u2013$327 double; off season $159 and up double. Children 11 and under stay free in parent's room. Amenities: Restaurant; 2 bars; fitness center; high-speed Internet (free); pool; tennis court (lit); watersports.\n\nModerate\n\nIsland Beachcomber Hotel   The island's first beachfront hotel opened up on Lindbergh Bay in 1956 and was once frequented by the likes of Cecil B. DeMille and Vivian Vance (Ethel Mertz from \"I Love Lucy\"). Yes, this hotel has seen glitzier days, and it's fairly standard-issue in every way today, but you can't beat the location on beauteous Lindbergh Bay. This is a good place to stay if you're in transit by air or ferry and you don't want to spend a lot of money\u2014and it's surprisingly comfortable and tropical-beachy. Look for medium-size rooms, many with patios just steps from the beach.\n\n8071 Lindbergh Beach Rd.   340\/774-5250. www.islandbeachcomber.net. 48 units. Winter $199\u2013$225 double; off season $130\u2013$169 double. Extra person $15. Amenities: Restaurant; bar; Wi-Fi (free).\n\nThe South Coast\n\nExpensive\n\nMarriott Frenchman's Reef & Morning Star Beach Resort   If you like a full-service American-style resort with plenty of onsite entertainment and dining options\u2014and a fetching sweep of white-sand beach\u2014this resort should fit the bill. As the only major resort in close proximity to Charlotte Amalie (it's just 3 miles east), it's a beehive of activity and, some say, the heartbeat of the island's south shore. If you're looking for better-quality rooms and a more luxurious ambience, head to the Ritz-Carlton (p. ). But if you want to be where the action is in Charlotte Amalie, this is the spot.\n\nThis is the largest hotel in the U.S. Virgin Islands, made up of two separate parts, Frenchman's Reef and Morning Star, that were joined into one mega-resort in 2005 in an excellent location on a bluff overlooking both the harbor and the Caribbean. Around the corner is Frenchman's Cove, the 220-room timeshare section of the resort, with luxury two- and three-bedroom villas. (And yes, you may be asked to learn more about the timeshares during your stay: In the Marriott lobby, a booth offers rum punches and other enticements to get people to attend timeshare promotions. Don't be shy about saying \"no\" if you're not interested and they'll let you be.)\n\nThe rooms at Frenchman's Reef are traditionally furnished and comfortable (with the Marriott's signature silky linens and bedding), if uninspired. Those at the Morning Star are more spacious, some opening up right onto the beach with dreamy water views from patios or balconies (non-sea-view balconies have views of the green hills opposite). Note that the resorts are a half-mile apart, and getting from one to another is a bit of a hike (nearly 100 steps uphill from Morning Star to Frenchman's Reef); a shuttle runs guests between the two sections as well.\n\nThe in-house dining is exceptional, and you won't go wrong with dinner at the excellent Havana Blue (p. ), a Miami-style beachfront bo\u00eete at Morning Star with a truly inspired menu. The Marriott also has a well-respected in-house tour operator, Adventure Center (www.adventurecenters.net), which offers snorkeling, sailing, and kayak excursions (the latter in clear see-through kayaks).\n\nNote that daily water-taxi service in a gondola-style boat is available straight from the Marriott dock to the shopping hub in Charlotte Amalie. The boat leaves the resort every 30 minutes between 8:30am and 5pm (one-way: $7 adults, $4 children 3\u201312).\n\nNo. 5 Estate Bakkeroe.   888\/236-2427 or 340\/776-8500. www.marriott.com. 478 units. Winter $475\u2013$745 double, from $760 suite; off season $300\u2013$630 double, from $600 suite. Children 12 and under stay free in parent's room. Daily resort fee $35. Amenities: 5 restaurants; 2 bars; coffeehouse; deli; babysitting; health club and spa; room service; 2 tennis courts (lit); watersports equipment\/rentals; Wi-Fi (included in $35-a-day resort fee).\n\nModerate\n\nBolongo Bay Beach Resort   This family-run resort is a casual, barefoot place to stay. A complex of pink two-story buildings, it's built around a crescent-shaped beach on the sands of Bolongo Bay. That means plenty of watersports activities (even scuba-diving lessons), so many guests check in on the European Plan, which includes watersports activities and even a scuba-diving lesson. Others opt for all-inclusive plans that include all meals, drinks, excursions, and more. Both are good values, especially for St. Thomas. Rooms are simple, summery, and filled with comfortable furniture and a fridge\u2014and all face the beach, with a balcony or terrace opening up onto the beach. Nine two-bedroom condos, in a three-story building, have full kitchens and are ideal for families of up to five. Onsite are two lively restaurants, including Iggies (p. ). A final enticement? The resort is known for its Wednesday-night Carnival, held from November through Labor Day, with a Caribbean buffet, calypso music, and Moko Jumbie stilt dancers.\n\n7150 Bolongo.   800\/524-4746 or 340\/775-1800. www.bolongobay.com. 80 units. Winter $250\u2013$415 double, $570\u2013$620 condo; off season $145\u2013$350 double, $350\u2013$520 condo. Extra person $25 per day. Children 12 and under $15 per day. Resort fee 7%. Ask about the resort's all-inclusive plans. Amenities: 2 restaurants, including Iggies Beach Bar & Grill (p. ); 2 bars; babysitting; children's programs (ages 4\u201312); exercise room; 2 pools (outdoor); 2 tennis courts (lit); watersports equipment\/rentals; Wi-Fi by pool (free).\n\nThe East End\n\nThe shuttered Grand Beach Palace, closed up tight for almost 10 years, is expected to reopen in late 2014 as a full-service 262-unit Wyndham Margaritaville Vacation Club resort. The former Renaissance hotel fronts one of the island's prettiest beaches, Grand Beach, opening onto Smith Bay.\n\nExpensive\n\nThe Ritz-Carlton, St. Thomas   This is the island's one true luxury resort. Fronted by white-sand beaches and the protected turquoise seas of Great Bay, the Ritz covers a sprawling 30 acres of oceanfront on the southeastern tip of St. Thomas, 4 miles (a 30-min. drive) from Charlotte Amalie. Its centerpiece is a Venetian-style palazzo, surrounded by terraced gardens. Pathways lead down to the curving infinity pool (home to two resident ducks) and a beach with canopied chairs and a bustling food and beverage service.\n\nAll day long, the resort's water-sports toys see plenty of use on Great Bay, from stand-up paddleboards to Hobie Cats to kayaks to snorkel equipment\u2014it's a veritable kids' playground, with a full-service dive shop, Patagon Dive Center, right on property. (The bay bottom is somewhat grassy and pebbly, however.) The terrific Ritz Kids activities are part of the resident Jean-Michel Cousteau's \"Ambassadors of the Environment\" program, with excursions for all ages, including explorations of the coral reef, island mangroves, and constellations. Adults can try night snorkeling in Great Bay or sign up for ocean kayak adventures in clear-bottom kayaks. If you've always wanted to learn to sail, take advantage of the on-site certified sailing school, Island Sol. Finally, the property has its own private 53-foot catamaran, Lady Lynsey, in which guests can take full and half-day sailing excursions.\n\nThe resort's main lobby has a generic feel in spite of its elegant appointments, but the Ritz rooms are the island's best, supremely comfortable and spacious, with great big bathrooms (tubs and rain showers) and top-of-the-line amenities throughout. The resort features an extensive spa with luxurious treatment rooms and open-air cabanas. Of the restaurants, Bleuwater (p. ) is chillingly expensive but a real special-occasion spot. Finally, the warm and friendly Ritz staff is unparalleled on an island that still has a few things to learn about personal service. In addition to the 180 hotel rooms, the resort also holds a number of Ritz-Carlton Destination Club timeshare villas.\n\n6900 Great Bay.   800\/241-3333 or 340\/775-3333. www.ritzcarlton.com. 180 units. Winter $660\u2013$880 double, from $1,290 suite; off season $380\u2013$570 double, from $700 suite. Daily resort fee $50. Amenities: 4 restaurants; 3 bars; ATM; babysitting; children's programs; concierge; full-service health club and spa; 2 pools (outdoor); room service; sailing school; 2 tennis courts (lit); watersports equipment\/rentals; Wi-Fi (free).\n\nSugar Bay Resort and Spa   No longer a Wyndham property, this 294-room all-inclusive resort has undergone exciting renovations, with recent refreshments of all rooms, the lobby, and the main restaurant, which by press time, should be a fully operating Japanese teppanyaki steakhouse. It's an impressive, well-maintained property, on one of the island's prettiest perches, a 32-acre plot of sloping terrain just 5 minutes from the ferry terminal at Red Hook. From its position atop a rocky promontory, guests are treated to panoramic views out over the sea. At its feet is a pretty, secluded beach that's smallish but excellent for snorkeling and swimming. It's a fine family resort, with a wonderful pool\/beach\/restaurant area called the Mangroves right on the beach (reached via a steep walk 99 steps down from the hotel; those with mobility issues can reach it by hotel shuttle). On Thursday the Mangroves rocks with a Carnival party, and the daily activities are many and varied (sunrise Pilates, pool volleyball, iguana feedings). Rooms are attractively furnished but are more utilitarian than luxe, each with a small balcony\u2014the only thing that distinguishes one room from another is the view, so if you want to guarantee those amazing ocean vistas, request it when booking. Still, the 2013 renovations have given the rooms a pleasant new look, with tile floors replacing carpet, marble counters in bathrooms, and a beachy blue-and-white palette. Those who want to splash out, so to speak, go for the properties cushier 475-square-foot suites. Sugar Bay offers both all-inclusive plans and room-only rates; the all-inclusive plans make sense for those who prefer to stay put during their vacation\u2014and with Sugar Bay's lineup of good restaurants, full complement of water sports, and ongoing daily activities, you may never have to leave the premises.\n\n6500 Estate Smith Bay.   800\/927-7100 or 340\/777-7100. www.sugarbayresortandspa.com. 294 units. Winter $310\u2013$700 double, from $1,250 suite; off season $238\u2013$530 double, from $1,050 suite. Check the website for all-inclusive plans; all-inclusives include 15% service charge. Amenities: 3 restaurants; 2 bars; babysitting; basketball court; boat dock; children's programs; exercise room; miniature golf; 3 pools; 4 tennis courts (lit); watersports (canoes, kayaks, sailboats, snorkeling gear); Wi-Fi (free).\n\nElysian Beach Resort   This resort overlooks its own secluded cove bobbing with boats and white-sand beach on Cowpet Bay, a 30-minute drive from Charlotte Amalie. It's not luxurious like the Ritz next door, but it's tranquil and secluded, and the exquisite white-sand beach and free-form swimming pool sweeten the deal. The one-bedroom condos (which can hold up to four people) contain kitchenettes and balconies, and 14 offer sleeping lofts that are reached by a spiral staircase. Look for standard-issue tropical decor, with rattan and bamboo furnishings, ceiling fans, and natural-wood ceilings. Rooms in buildings V to Z are some distance from the beach, so try to avoid them when booking. The resort has two good restaurants, including the Caribbean Fish Market (p. ), a fun spot with tables right on the beach, serving creative West Indian cuisine since 2012.\n\n6800 Estate Nazareth.   866\/620-7994 or 340\/775-1000. www.elysianbeachresort.net. 180 units. Winter $220\u2013$265 double, $459 suite; off season $215\u2013$260 double, $399 suite. Amenities: 2 restaurants; 2 bars; exercise room; pool (outdoor); small spa; tennis court (lit); watersports equipment\/rentals, Wi-Fi (free; lobby only).\n\nPavilions and Pools   If absolute seclusion is on your wish list, this 25-unit condominium resort might be just the ticket: Each villa opens onto its own private swimming pool. Sure the pools are smallish, but each has a tile deck with loungers that's walled off and pillowed in tropical greenery. Need more privacy? There's no need to dine out: Here you can cook your own meals in a well-equipped kitchen, or dine on affordable Mexican fare in the resort's open-air restaurant, Torcido Taco. Villas are generally comfortable and spacious (the International Pool Villas have 1,400 sq. ft. of space; Caribbean Pool Villas 1,200 sq. ft.), though they each have an individual look as they're all privately owned. The resort is just 1 mile from the bars and restaurants of Red Hook and 7 miles east of Charlotte Amalie. One of the island's best beaches and watersports concessions, Sapphire Bay, is a (steep uphill) walk away\u2014but you may never feel like venturing out of your bubble of quiet seclusion.\n\nCamping Out on Water Island\n\nSt. Thomas has its own bucolic pocket of rusticity\u2014and it's just off the hustle and bustle of the Charlotte Amalie harbor. Called the Virgin Islands Campground on Water Island (  340\/776-5488; www.virginislandscampground.com), guests stay in the most eco-sensitive lodgings in St. Thomas. They're hardly roughing it: The digs are wood-frame-and-canvas cabins (cottages) with wind-drawn electricity, nice beds, and crisp linens. Each opens onto private, ocean-view terraces. The campground has no restaurant, but you can grill your own meals in the common area known as the Pavilion; you can also store your food in a refrigerator or freezer. From the campgrounds, it's a 5-minute walk to the island's idyllic Honeymoon Beach, where you can take a swim, snorkel or just nurse a nice tan. Regular ferry service runs between Water Island and St. Thomas (trip time: 7 min.).\n\nA 3-night minimum stay is required; cabins cost $165 per night in winter and $115 per night in the off season. Cabins accommodate up to 3 adults and 2 children. The campground also has a special suite with a fully equipped kitchen and its own private bath and deck ($149\u2013$200 per night).\n\nWater Island is very small; to get around, simply hike along the island's pristine trails. You can also take bike rides (see chapter 8). For more on visiting Water Island, see p. .\n\n6400 Estate Smith Bay.   800\/524-2001 or 340\/775-6110. www.pavilionsandpools.com. 25 units. Winter $240\u2013$350 double; off season $200\u2013$280 double. Extra person $25 per night. Rates include continental breakfast. Amenities: Restaurant; private pools; watersports equipment\/rentals; Wi-Fi (free in lobby).\n\nSecret Harbour Beach Resort   This boutique all-suites condo resort sits on the white-sand beach at Nazareth Bay, near Red Hook Marina, with excellent snorkeling just outside your door. The four low-rise buildings have southwestern exposure (great sunsets!), and each room is beachfront or oceanview, mere steps from the sand (some bottom-floor suites even have outdoor showers on their patios). Each room and suite comes a private deck or patio and a fully equipped kitchen. The resort rooms are individually owned, so it's impossible to say what the decor will be like, though most owners go for attractive furnishings. There are three types of accommodations: studio suites, one-bedroom suites, and two-bedroom suites. Each one- and two-bedroom suite has a pullout couch. All rooms have king-size beds; the two-bedroom suites may have one king and two twins. Secret Harbour is also home to one of the best restaurants on the island, the Sunset Grille (p. ). The resort has a full-service dive site on the premises, Aqua Action Dive Center (  340\/775-6285; www.aadivers.com), which offers PADI courses, snorkeling equipment, and snorkeling outings.\n\n6280 Estate Nazareth.   800\/524-2250 or 340\/775-6550. www.secretharbourvi.com. 60 units. Winter $355\u2013$435 double, $385\u2013$505 1-bedroom suite, $635\u2013$765 2-bedroom suite; off season $255\u2013$305 double, $285\u2013$375 1-bedroom suite, $535\u2013$635 2-bedroom suite. Children 12 and under stay free in parent's room. Each additional guest $35 per day. Amenities: Restaurant; bar; dive shop; exercise room; pool (outdoor); 3 tennis courts (lit); watersports equipment\/rentals; Wi-Fi (free).\n\nWHERE TO EAT\n\nThe dining scene in St. Thomas these days is among the best in the West Indies, but it has its drawbacks: Fine dining\u2014and even not-so-fine dining\u2014tends to be expensive, and you may have to travel by taxi or car to reach some of the best spots.\n\nYou'll find an eclectic mix of global cuisines on St. Thomas, including Italian, Mexican, Asian, and French. American fast-food franchises, pizza joints, and East Indian roti shacks are also part of the scene. But some of the best food on St. Thomas is island-bred: West Indian\u2013style cuisine and interesting fusions thereof. We particularly like those places that take advantage of the island's natural bounty, menus featuring fresh-caught seafood, fresh fruit, and locally grown produce. Finally, go where the locals go: They know the best spots and are happy to recommend their favorites.\n\nSt. Thomas Restaurants\n\nFARM-FRESH produce ON ESTATE BORDEAUX\n\nIn 1984 a Vietnam vet named Myron (\"Buddy\") Henneman found solace in a serene spot high up in the green hills of the island's western end, with vistas of blue seas and robin's-egg skies. In the loamy volcanic soil, once tended by French farmers who immigrated here from St. Barts, he planted a chemical-free garden, and pretty soon the government was offering him agricultural land leases. Today, the Green Ridge Guavaberry Farm is thriving, part of the island's burgeoning agrarian renaissance. Some 20 Estate Bordeaux farms now grow crops of astonishing diversity in these quiet green hills, an edenic haven from the scrum of civilization far down below. \"Once you train the land,\" says Henneman, \"the land will grow anything.\" And it does, including peas, Asian dragonfruit, okra, cashews, even the healing \"tree of life,\" the meranga bush (said to be good for the liver). A whopping 17-pound cauliflower was recently plucked from the soil, beneath which had been growing a 200-pound pumpkin. A number of Rastafarian gardeners work the land throughout the islands, and lunch is some delicious concoction: roasted wild salmon or pumpkin soup made in a big clay pot over a fire. You can buy produce from this and other Estate Bordeaux farms, as well as fresh-baked bread, eggs, honey, and exotic island juices, in the Saturday morning market at Market Square, in Charlotte Amalie, and on the second and last Sunday of every month at the Bordeaux farmer's market (10am\u20134pm), in an open-air pavilion near the entrance to the Preserve at Botany Bay. In January (usually around Jan. 19 and 20) the farms hold the weekend-long Agricultural & Cultural Food Fair, where some 2,000 visitors enjoy reggae music, crafts, farm tours, and more on the farms themselves from 10am to midnight. Contact the Bordeaux Farmers' Market at   340\/774-4204. The farmer's market and farm fair are in part sponsored by the nonprofit organization We Grow Food (www.facebook.com\/WEGROWFOOD), which hopes to reclaim some 400 acres of Estate Bordeaux land for growing food.\n\nIn Charlotte Amalie\n\nExpensive\n\nAmalia Caf\u00e9   SPANISH Set on an alfresco terrace on Palm Passage in the center of Charlotte Amalie, this long-established spot serves up some of the most savory Spanish cuisine in town. You can order one of the hearty entrees (paella Valenciana, say, or zarzuela de mariscos, seafood casserole) or make a meal out of tapas, flavorful small plates that might include garlic shrimp, chorizo saut\u00e9ed in cider, or clams in a green sauce. Wash it all down with the Amalia's signature sangria.\n\n24 Palm Passage.   340\/714-7373. <http:\/\/amaliacafe.com>. Reservations recommended. Main courses lunch $15\u2013$23, dinner $28\u2013$49. Mon\u2013Sat 11am\u201310pm; Sun 11am\u20133pm (winter only).\n\nBanana Tree Grille   INTERNATIONAL Dine in an atmosphere of informal elegance with views over the bustling harbor at this longtime favorite in Bluebeard's Castle timeshare resort. It's a special-occasion spot (with prices to match), but the views are sublime. The real surprise is the equally elegant food, overseen by a transplanted New Orleans chef. We recommend the grouper stuffed with crab imperial, and the meltingly tender, grass-fed Banana Tree filet mignon in a b\u00e9rnaise sauce. Breads are also a specialty, particularly the \"Oh My\" Gorgonzola loaf (garlicky Ciabatta with a Gorgonzola dipping sauce). The fantastic desserts include a nod to the Big Easy: bananas Foster cheesecake drizzled in homemade chocolate sauce.\n\nIn Bluebeard's Castle, Bluebeard's Hill.   340\/776-4050. www.bananatreegrille.com. Reservations recommended. Main courses $20\u2013$56. Tues\u2013Sun 5:30\u20139:30pm.\n\nMafolie Restaurant   CARIBBEAN\/INTERNATIONAL This is one of the island's top restaurants\u2014and you get your money's worth, from the terrific island fare to the breathtaking views of the Charlotte Amalie harbor below from the open-air dining room. The Caribbean specialties include a classic kalaloo soup, here brimming with shrimp, crab, and okra; and conch fritters served with sweet mango Scotch bonnet honey. Chef Manny Thompson has a deft touch: For a main course, try the jambalaya; it has a subtle fire and perfectly caramelized vegetables. Fresh-caught fish and Water Island lobster are also highly recommended\u2014the proprietor and his bartender often do morning lobster fishing for the night's meal. Look for traditional island sides like fungi (thickened cornmeal polenta), breadfruit mash, and sweet potato dumplings.\n\n7091 Estate Mafolie.   340\/774-2790. www.mafolie.com. Entrees $27\u2013$39. Daily 6\u201310pm.\n\nPetite Pump Room   WEST INDIAN\/INTERNATIONAL Join the locals for a reliably tasty St. Thomas breakfast and lunch at this second-floor harborside spot in the ferryboat terminal in Charlotte Amalie. Located on the harbor since 1970, it serves some of the island's most flavorful West Indian cooking, with a menu of callaloo greens, conch fritters (a house specialty), and daily specials like fried potfish or conch sauteed in butter sauce. Sides include pigeon peas and rice, fungi with vegetables, and fried plantains. It's a convenient spot for a bite if you're waiting for a ferry. Snag a seat on the seaside deck and watch ships cruise into the harbor.\n\nIn the Edward Wilmoth Blyden Building, Veterans Dr.   340\/776-2976. www.petitepumproom.com. Breakfast $7\u2013$13; sandwiches and salads $8\u2013$18; platters $14\u2013$20. Daily 7am\u20134:30pm.\n\nA Room with a View   INTERNATIONAL Just east of Charlotte Amalie, this topnotch restaurant serves up international classics and wraparound harbor views from open floor-to-ceiling windows. Kick off your dinner with one of the specialty martinis (Grand Marnier martini, melon martini) and smoked salmon on toast points or grilled portobello mushrooms with goat cheese. Main courses include one of several house pastas (Alfredo, pesto, meat sauce); chicken in a creamy curry sauce; shrimp Creole; or a New York strip steak au poivre.\n\nIn Bluebeard's Castle, Bluebeard's Hill.   340\/774-2377. Reservations recommended. Main courses $19\u2013$42. Mon\u2013Sat 5\u201311pm.\n\nTavern on the Waterfront   MODERN EUROPEAN\/CARIBBEAN How's this for a global mashup: a nightclub\/restaurant that serves inventive French and Polish dishes with West Indies inflections in a glittering setting on the St. Thomas waterfront. Sounds odd, but it works. So you might pinball from Polish pierogies to garlic escargots to \"Down Island\" conch chowder. Recommended entrees include the house babyback ribs; an almond-and-hazelnut-crusted grouper; and a jumbo vegetable pot pie. Come for live jazz on Friday nights or a tropical cocktail at the upstairs bar.\n\nWaterfront at Royal Dane Mall, 30 Dronningens Gade.   340\/776-4328. www.tavernonthewaterfront.com. Reservations required. Main courses $14\u2013$37 (Caribbean lobster $53). Mon\u2013Sat 11am\u20133pm; Mon\u2013Thurs 5\u20139pm; Fri\u2013Sat 5\u201310pm.\n\ngroceries, markets & more: PROVISIONING RESOURCES ON ST. THOMAS\n\nMore than any other Virgin Island, St. Thomas has no lack of stores to find what you need to eat and drink and stock your pantry. That's because it's a prime provisioning stop for boaters, and the many people who visit the island stay in rented houses, villas, and condo (even resorts) with full kitchens. Here's a sampling of resources:\n\nGroceries: Open daily, the chain grocery Pueblo has three locations on St. Thomas including Rumer Drive in Charlotte Amalie (  340\/774-4200). In Red Hook, the Food Center (  340\/777-8806; www.foodcentervi.com) is the largest grocery on the East End; it's open daily. Also open daily is the local favorite Plaza Extra, Tutu Park Mall (  340\/775-5646; <http:\/\/plazaextra.com>).\n\nFresh Seafood: S&P Seafoods, 3801 Crown Bay (  340\/774-5280), is a full-service fishmonger. You can also buy fresh-off-the-boat fish at the docks at Frenchtown when the fishing boats return around 3pm or 4pm.\n\nFresh Produce\/Fruit: St. Thomas has a number of small farmer's markets, including the Saturday market at Market Square in Charlotte Amalie, and the Bordeaux farmer's market on the second and last Sunday of each month in Estate Bordeaux, on the island's west end.\n\nLiquor: Liquor stores abound along Main Street in Charlotte Amalie, and liquor, wine, and beer are sold in most grocery stores throughout the island. Al Cohen's Discount Liquors, Long Bay Road (  340\/774-3690; www.rumanyone.com\/AlCohen.htm): One of St. Thomas's most famous outlets occupies a big warehouse at Havensight, across from the West Indian Company docks, where cruise-ship passengers disembark. Inside is a huge storehouse of liquor and wine.\n\nVirgilio's   ITALIAN The best Italian restaurants on island, Virgilio's serves up Sicilian classics and homemade pastas and risottos, including a hearty \"Lasagna de Maria,\" from a Virgilio family recipe. You can also get traditional veal and chicken dishes, as well as a sprinkling of fresh-made pizzas, including a terrific pizza del mare, with shrimp and calamari. The restaurant's old-world charm is enhanced by wood beams and vaulted brick. Paintings in ornate frames line buttery-yellow walls. Look for Virgillo's tucked in a alley between Main and Back streets.\n\n18 Dronningens Gade (entrance on a narrow alley running btw. Main and Back sts.).   340\/776-4920. www.virgiliosvi.com. Reservations recommended. Main courses lunch $18\u2013$30, dinner $23\u2013$48. Mon\u2013Sat 11:30am\u201310pm; closed Sun.\n\nModerate\n\nCuzzin's Caribbean Restaurant & Bar   CARIBBEAN For authentic Caribbean cooking, head to this casual old-school lunch spot in a former 18th-century stable on Back Street. The brick-and-wood dining room has windows looking onto the street and a comfortable feel. Look for classic island dishes with an emphasis on seafood, especially lobster and fresh-caught fish. Conch comes in a variety of preparations, from curried conch to conch Creole to conch fritters. You can make a genuine West Indies meal out of the sides alone, which include fungi (cornmeal grits), peas and rice, fried plantains, and island coleslaw.\n\n7 Wimmelskafts Gade (Back St.).   340\/777-4711. www.cuzzinsvi.com. Reservations recommended. Main courses $9\u2013$22. Mon\u2013Sat 11am\u20134:30pm.\n\nGladys' Caf\u00e9   CARIBBEAN\/AMERICAN Dine on local favorites prepared to perfection at Gladys's Caf\u00e9, a lively spot housed in a 1700 pump house smack-dab in the middle Charlotte Amalie's historic district. It's open for breakfast or lunch only. Sample jerk mahi-mahi, curry chicken roti, creamy conch chowder, and kalaloo soup. Entrees come with peas and rice or fungi, a local cornmeal-based specialty similar to polenta or grits. A house specialty is the colorful and nutritious Caribbean lobster salad in a homegrown avocado\u2014you can also get it with tuna or chicken salad. The owner, Gladys, is in the house daily\u2014if you're lucky you will hear her occasionally burst into song. Don't leave without taking home a bottle of Gladys' homemade hot sauce, made with local peppers.\n\nRoyal Dane Mall.   340\/774-6604. <http:\/\/gladyscafe.com>. Reservations required for groups of 6 or more. Breakfast $3.75\u2013$19; lunch main courses $10\u2013$25. Daily 7am\u20135pm.\n\nInexpensive\n\nBumpa's   AMERICAN A deli-style joint, Bumpa's is perfect when you want something simple but well-made. Many people, locals and visitors alike, stop in early in the morning to breakfast on terrific oatmeal banana pancakes, omelets, bagels, and lemonade. Lunch features burgers and sandwiches. Bumpa's is located on the second floor of a building that faces the busy highway along the harbor.\n\n38-A Waterfront Hwy.   340\/776-5674. <http:\/\/bumpas.travelstthomasusvi.com>. Main courses $7\u2013$14. Daily 7:30am\u20135pm.\n\nCaf\u00e9 Amici   ITALIAN\/CARIBBEAN A reliable spot for lunch during a shopping expedition in Charlotte Amalie's historic district, Caf\u00e9 Amici is located on an elevated patio in an old stone alley. It's known for its creative brick-oven-baked pizzas (including a Caribbean jerk pizza and a crab-cake pizza with mango-tomato relish) and satisfying pastas (such as a shrimp scampi pasta). Wraps, and sandwiches are also on the menu. Nutritious choices include hearty salads and veggie-rich Italian soups.\n\nA.H. Riise's Alley.   340\/714-7704. www.cafeamicivi.com. Lunch specials $9.95\u2013$13; pizza $14\u2013$18. Daily 10:30am\u20133:30pm.\n\nJen's Island Caf\u00e9 & Deli   DELI\/CARIBBEAN The food is good and tasty, the welcome warm, and the prices reasonable at this cafe\/deli in Charlotte Amalie's downtown shopping district, right across from Emancipation Garden. In addition, the cafe has quite an extensive menu, from breakfast eggs to American-style deli sandwiches to burgers and salads. Jen even does classic island dishes, such as conch fritters, rotis, and local chicken soup with dumplings. The lobster and shrimp quesadillas are made with Cruzan rum\u2013soaked onions.\n\nGrand Hotel, 43\u201346 Norre Gade.   340\/777-4611 or 514-5345. <http:\/\/jensdeli.com>. Breakfast $5\u2013$10; main courses $6.75\u2013$9. Mon\u2013Fri 7am\u20135pm; Sat 10:30am\u20134pm.\n\nNorth of Charlotte Amalie\n\nExpensive\n\nThe Old Stone Farmhouse   INTERNATIONAL Set in the centuries-old field house of a restored Danish sugar plantation, the newly revitalized Old Stone Farmhouse is a special-occasion treat. Part of the fun is visiting the kitchen to pick out your meat or fish and hobnob with the chefs. But it's just a kitchen, after all; you'll want to spend as much time as possible in the atmospheric dining rooms, whether you're seated outside in the courtyard or inside, with wood-plank floors and flickering candlelight casting shadows on native stone walls. It's the perfect setting to enjoy one of the finest culinary experiences on the island, whether you choose to dine on some kind of exotic wild game (New Zealand elk, say, or wild boar) or a perfect, cooked-to-order steak or fish dish.\n\nMahogany Run.   340\/777-6277. http:\/\/oldstonefarmhouse.com. Reservations recommended. Main courses $24\u2013$36. Mon\u2013Sat 6\u201310pm.\n\nModerate\n\nBlue Orchid   CARIBBEAN\/SMALL PLATES Bringing the trend of small plates to the rain-forest peaks of St. Thomas, Blue Orchid, at the Historic St. Peter Greathouse, is a winning combination of flavorful Caribbean-inspired dishes and insanely scenic vistas. Order a frothy cocktail and some of the delicious small plates; you and your party might graze on tuna tartare, lobster dip, duck confit sliders, or crab salad nachos, to name a few of the choices. The limited menu of entrees include steaks, seafood pastas, and pan-seared grouper. Come early (the bar opens at 4pm), so you can sip sunset drinks and point out the array of Virgin Islands, large and small, from the observation deck before nightfall.\n\nHistoric St. Peter Greathouse, at the corner of Rte. 40 (6A St. Peter Mountain Rd.) and Barrett Hill Rd.   340\/774-4999. www.blueorchidvi.com. Reservations recommended. Main courses $22\u2013$31. Mon\u2013Sat 5\u201310:30pm.\n\nFrenchtown\n\nExpensive\n\nOceana   GLOBAL\/ISLAND This buzzy local favorite is ensconced in the restored West Indian great house of an estate known as Villa Olga, which served as the island's Russian embassy in the late 1800s. It's right on the water's edge, with sweeping views of the Charlotte Amalie harbor. Inside the walls are blanketed with colorful local and global art. The chefs specialize in creative twists on the island bounty, so you'll have seared scallops served with a papaya-pear relish and local honey gastrique, or Maryland crab cakes with a fresh mango salsa. The Oceana bouillabaisse comes brimming with Caribbean lobster, and a New Zealand lamb shank is accompanied by goat-cheese polenta. It's all beautifully prepared, in a memorable setting.\n\nIn the Villa Olga, 8 Honduras.   340\/774-4262. www.oceana.vi. Reservations required. Main courses $27\u2013$48; Caribbean lobster $58. Daily 5:30\u20139:30pm.\n\nModerate\n\nHook, Line & Sinker   AMERICAN This lively spot is set right on the water, with a New England seaport feel: whitewashed wood, French doors, and windows with shimmering harbor views. Well-prepared standards are the name of the game, such as steamed mussels, Caesar salad, or mushrooms stuffed with crabmeat and cheese for starters; and burgers, pastas (pomodoro, Arrabbiata, primavera), chicken, steak, and seafood specials (such as fresh grilled shrimp) for mains.\n\n62 Honduras.   340\/776-9708. www.hooklineandsinkervi.com. Main courses lunch $9\u2013$15, dinner $14\u2013$28. Mon\u2013Sat 11:30am\u20134pm and 6\u201310pm; Sun 10am\u20132:30pm.\n\nInexpensive\n\nFrenchtown Deli   DELI In addition to a full coffee bar, the Frenchtown Deli has some of the island's best deli-style sandwiches, along with bagels and smoked salmon, homemade soups, and salads. Eat in or take out.\n\n24-A Honduras, Frenchtown Mall.   340\/776-7211. Sandwiches $6.50\u2013$12; salads $4\u2013$15. Mon\u2013Fri 7am\u20138pm; Sat 7am\u20135pm; Sun 7am\u20134pm.\n\nThe South Coast\n\nHavana Blue   CUBAN FUSION This Miami-style venue at the Marriott enjoys a beachfront ambience and an inspired menu. An inventive crew is in the kitchen, and out in the dining room, the party is rocking with sexy blue lights, a thumping soundtrack, and a bombshell bartenders serving up blueberry mojitos and mambo margaritas. Ignore the hubbub and get to work: This is some serious food. Start with the black-bean hummus or the tuna tartare with soy-lime vinaigrette. For a main, try the house specialty: miso-lemongrass-glazed sea bass, which comes with wasabi mashed potatoes and garlic spinach. Yes, miso cod, and so forth, has been done to death. No, it doesn't matter: You will find this iteration utterly delicious.\n\nIn Marriott Frenchman's Reef & Morning Star Beach Resort (p. ), 5 Estate Bakkeroe.   340\/715-2583. www.havanabluerestaurant.com. Reservations required. Main courses $30\u2013$52. Daily 5:30\u201310:30pm.\n\nIggies Beach Bar & Grill   AMERICAN\/CONTINENTAL This action-packed seaside grill is an island hot spot (see \"Nightlife,\" below), but it's also a fine place to take the family to dine. The menu features a plethora of pub-style appetizers plus steaks, ribs, and seafood, but every night is theme night; Wednesday is Cruzan Carnival Night, with a Caribbean buffet that opens at 6pm, music at 7pm, and a Carnival show at 8pm ($38 adults, $15 children 11 and under). Tuesday the special is steamed crabs and beer, and Thursday is seafood night; on Friday Iggies serves up tacos and margaritas, and Sunday it's BBQ ribs and chicken night.\n\nAt the Bolongo Bay Beach Resort (p. ), 7150 Bolongo (Rte. 30).   340\/693-2600. www.iggiesbeachbar.com. Burgers and sandwiches $6.50\u2013$15; dinner main courses $18\u2013$34. Daily 11am\u201311pm.\n\nThe East End\n\nExpensive\n\nBleuwater   SEAFOOD\/STEAK The Ritz's showcase restaurant fashions elegant, creative turns on classic dishes. That might mean a lobster and truffle mac-n-cheese, or plantain gnocchi with Maine lobster and English peas. Fresh Virgin Islands seafood is highlighted\u2014we devoured a pan-seared local Queen snapper with conch and jasmine rice here recently\u2014but this is also the place to come for a big, juicy, perfectly cooked steak. Guests can dine in the elegant and expansive dining room or out on the waterfront patio under a sky peppered with stars. And hurray: The kids' menu has more variation than most.\n\nIn the Ritz-Carlton (p. ), 6900 Great Bay.   340\/775-3333. Reservations required. Main courses $40\u2013$65. Wed\u2013Sun 6:30\u20139:30pm.\n\nThe Caribbean Fish Market   SEAFOOD\/WEST INDIAN Breezy and bright, this indoor\/outdoor seafood eatery sits right on the beach at the Elysian Beach Resort, looking out onto the sapphire seas of Cowpet Bay. It's also right next door to the Ritz-Carlton, making it a very convenient place for Ritz guests to enjoy a solid seafood dinner if they don't want to pay Ritz prices to dine (and don't want to have to take a taxi to get somewhere else). Most order the fresh catch of the day, which is cooked any number of ways. Grilled Caribbean lobster and such upscale treats as goat-cheese gnocchi, coconut shrimp lettuce wrap, and lobster and wild mushroom ravioli are also good choices. Sunday brunch is from 8am to 2pm.\n\nIn the Elysian Beach Resort (p. ), 6800 Estate Nazareth.   340\/714-7874. <http:\/\/caribbeanfishmarketvi.com>. Reservations recommended. Main courses $25\u2013$45. Mon\u2013Sun 5\u201310pm; Sun brunch 8am\u20132pm.\n\nSunset Grille   AMERICAN This may be the island's best new restaurant, with a waterside setting that's both breezy and seductive, and a menu with plenty of flavorful turns. Such as the prawns tempura, lightly fried shrimp in a sweet soy glaze served on long skewers, or the tomato-based conch chowder (a switch from the mainly creamy version most often found on island). Overall, it's a surf-or-turf kind of place: from a bevy of gloriously fresh fish to grilled steaks, jerk chicken, and lamb \"lollipops.\" And as I said at the start its a gorgeous setting: The last time we dined here, we watched the orange sun set over the protected cove; then, as night crawled in, we spied lights snaking beneath the inky black seas\u2014a posse of night snorkelers out exploring the marine wonders of the Caribbean sea.\n\nIn the Secret Harbour Beach Resort, 6280 Estate Nazareth.   340\/714-7874. www.secretharbourvi.com. Reservations required. Main courses $24\u2013$48. Daily 5:30\u201310:30pm.\n\nModerate\n\nFish Tails   SEAFOOD If you're looking for a casual, friendly seafood spot right on the water, Fish Tails should nicely fit the bill. You'll dine alfresco on a wooden deck overlooking Vessup Bay and its symphony of fishing boats. It's open practically all day long, serving up island treats like shrimp and eggs (with potatoes) for breakfast and fish cake, seared tuna, grilled wahoo, and Caribbean lobster for lunch and dinner. It's a nice place to bring the family, but it works well for a sunset cocktail, too\u2014and it's a local favorite for breakfast.\n\nNext door to Red Hook ferry terminal, Red Hook.   340\/714-3188. <http:\/\/fishtailsvi.com>. Main courses $15\u2013$30. Daily 7am\u201310pm.\n\nInexpensive\n\nSe\u00f1or Pizza   PIZZA If you're burned out on conch (and even if you're not), you'll be quite satisfied with the big, cheesy pizza pies served at Senor Pizza. Slices are oversized, so one might be all you need for a tasty, filling lunch. They also deliver.\n\nRed Hook Plaza (across from the ferry dock).   340\/775-3030. <http:\/\/senorpizza.cbt.cc\/index.html>. Slice $3; whole pizza $16\u2013$18. No credit cards. Daily 11am\u20139pm.\n\nEXPLORING ST. THOMAS\n\nBeaches\n\nSt. Thomas's beaches are renowned for their powdery white sand and clear azure waters, including the very best of them all, Magens Bay. Chances are that your hotel is right on the beach, or very close to one. Keep in mind that all the beaches in the Virgin Islands are public\u2014even the resort beaches\u2014and most St. Thomas beaches lie anywhere from 2 to 5 miles from Charlotte Amalie.\n\nThe North Coast\n\nThe gorgeous white sands of Magens Bay  \u2014the family favorite of St. Thomas\u2014lie between two mountains 3 miles north of the capital. The turquoise waters here are calm and ideal for swimming, though the snorkeling isn't that good. The beach is no secret, and it's usually overcrowded, though it gets better in the midafternoon. There is no public transportation to get here (although some hotels provide shuttle buses). A taxi from Charlotte Amalie costs about $8.50 per person. If you've rented a car, from Charlotte Amalie take Route 35 north all the way. The gates to the beach are open daily from 6am to 6pm. After 4pm, you'll need insect repellent. Admission is $1 per person and $1 per car. You can rent beach chairs ($7), loungers ($5), towels ($8), and snorkel and masks ($9) right there at the beach. Don't bring valuables, and don't leave anything of value in your parked car. For a bite to eat (try its famous authentic Greek pizza) or a cool libation, Magens Bay Beach Bar & Caf\u00e9 (  340\/777-6270; daily 9:30am\u20135pm) is the local favorite.\n\nA marked trail leads to Little Magens Bay, a separate, clothing-optional beach that's especially popular with gay and lesbian visitors. This is former U.S. President Clinton's preferred beach on St. Thomas (no, he doesn't go nude).\n\nCoki Point Beach, in the northeast near Coral World, is good but narrow and often crowded with both singles and families. It's noted for its warm, crystal-clear water and excellent snorkeling; you'll see thousands of rainbow-hued fish swimming among the beautiful corals. Vendors even sell small bags of fish food to feed the sea creatures while you're snorkeling. From the beach, there's a panoramic view of offshore Thatch Cay. Concessions can arrange everything from water-skiing to parasailing. Coki Dive Center (www.cokidive.com) has locker rentals onsite so you don't have to worry about your valuables.\n\nAlso on the north side of the island is luscious Grand Beach, one of St. Thomas's most beautiful, attracting families and couples. It opens onto Smith Bay and is near Coral World. Many watersports are available here. The beach is right off Route 38.\n\nThe East End\n\nSmall and calm, Secret Harbour   fronts the Secret Harbour Beach Resort and a collection of condos. With its white sand and coconut palms, it's a lovely little spot. The snorkeling near the rocks is some of the best on the island\u2014and night snorkeling is also available. You can rent equipment or sign up for a 60-minute \"Discover Snorkeling\" course (all ages welcome) on-site at the Secret Harbour Beach Resort with Aqua Action Dive Center (  340\/775-6285). It's an easy taxi ride east of Charlotte Amalie heading toward Red Hook.\n\nSapphire Beach   is set against the backdrop of the Sapphire Beach Resort and Marina (somewhat dilapidated the last time we visited). Still, like Magens Bay Beach, this wide, safe beach is one of the most frequented by families. You'll have good views of St. John, Tortola, Jost Van Dyke, and offshore cays, and a large reef is close to the shore. Windsurfers like this beach a lot. You can rent all kinds of watersports equipment right there on the beach at DIVE IN! (  866\/434-8346, ext. 2144, in the U.S., or 340\/777-5255; www.diveinusvi.com).\n\nHot surfing Spots\n\nSurfers come from miles around to test the swells at Hull Bay, on the north shore, just west of Magens Bay, particularly the waves along the western tip. It's also where local fishermen anchor in the more tranquil areas. Don't expect much in the way of watersports outfitters. There is a combined restaurant and open-air bar. If you're relying on taxis, it costs about $15 per person to reach the bay. Two other beaches have good surfing, Perserverance Bay and Caret Bay, but these are the unofficial territory of French locals, who as one surfer said, \"let you know if they don't like you.\"\n\nWhite-sand Lindquist Beach   isn't a long strip, but it's one of the island's prettiest beaches. It's between Wyndham Sugar Bay Resort & Spa and the Sapphire Beach Resort. Many films and TV commercials have used this photogenic beach as a backdrop. It's not likely to be crowded, as it's not very well known. It has no beach bar or cafe, but you can actually have a pizza delivered here!\n\nThe South Coast\n\nMorning Star  \u2014also known as Frenchman's Bay Beach\u2014fronts the Morning Star section of the Marriott Frenchman's Reef & Morning Star beach resort, about 2 miles east of Charlotte Amalie. It's a lovely spot, with gentle swells. Sailboats, snorkeling equipment, and lounge chairs are available for rent. If you're not staying at Morning Star, you can reach the beach by a cliff-front elevator at Frenchman's Reef or by walking the cliffside walk down to the beach.\n\nWest of Charlotte Amalie\n\nNear the University of the Virgin Islands, in the southwest, Brewers Bay is one of the island's most popular beaches for families. The strip of white coral sand is almost as long as the beach at Magens Bay. Unfortunately, this isn't a good place for snorkeling. Vendors here sell light meals and drinks. From Charlotte Amalie, take the Fortuna bus heading west; get off at the edge of Brewers Bay, across from the Reichhold Center.\n\nLindbergh Beach, with a lifeguard, restrooms, and a bathhouse, is at the Island Beachcomber Hotel (p. ) and is used extensively by locals, who stage events from political rallies to Carnival parties here. Beach-loving couples are also attracted to this beach. It's not good for snorkeling. Drinks are served on the beach. Take the Fortuna bus route west from Charlotte Amalie.\n\nWatersports & Outdoor Adventures\n\nboat CHARTERING\/BAREBOATING Yacht Haven Grande St. Thomas, 9100 Port of Sale, Charlotte Amalie (  340\/774-9500), is the premier marine facility for mega-yachts in the Caribbean. Located alongside Charlotte Amalie harbor, it encompasses a 48-slip facility, with dining, entertainment, and recreational options. American Yacht Harbor  , Red Hook (  340\/775-6454; www.igy-americanyachtharbor.com), can refer both bareboat and fully crewed charters. It leaves from the east end of St. Thomas in Vessup Bay. The harbor is home to numerous boat companies, including day-trippers, fishing boats, and sailing charters. There are also five restaurants on the property, serving everything from Continental to Caribbean cuisine. Another reliable outfitter is Charteryacht League, at Gregory East (  800\/524-2061 in the U.S., or 340\/774-3944; www.vicl.org).\n\nSailors may want to check out the \"Yachtsman's Guide to the Virgin Islands,\" available at major marine outlets, at bookstores, through catalog merchandisers, or directly from Tropical Publishers (  877\/923-9653; www.yachtsmansguide.com).\n\nday sailing Sail the U.S. and British Virgin Islands on the beautiful 44-foot clipper-bowed ketch Independence (  340\/775-1408; www.sailingvirginislands.net) with Captain Pat Stoeken. Based at the American Yacht Harbor in Red Hook, the Independence takes a maximum of six passengers on personally customized full- and half-day sails, including a hot lunch. Full-day rates are $140 adults, $125 children; half-day rates $90 adults, $80 children.\n\nAnother sail excursion that lets you avoid the crowds is aboard the Fantasy, 6100 Leeward Way, no. 28 (  340\/775-5652; www.daysailfantasy.com), which departs daily from the American Yacht Harbor at Red Hook at 9:30am and returns at 3pm. The boat takes a maximum of six passengers to St. John and nearby islands for swimming, snorkeling, and beachcombing. Snorkel gear and expert instruction are provided, as is a champagne lunch. The full-day trip costs $140 per person for adults and children. A half-day sail, usually offered only during the low season, lasts 3 hours and costs $100 for adults and children.\n\nThe 50-foot Yacht Nightwind, Sapphire Marina (  340\/775-7017; www.stjohndaysail.com), offers full-day sails to St. John and the outer islands. The $125 price includes continental breakfast, a champagne buffet lunch, and an open bar aboard. You're also given free snorkeling equipment and instruction.\n\nNew Horizons, 6501 Red Hook Plaza, Ste. 16, Red Hook (  800\/808-7604 or 340\/775-1171; www.newhorizonsvi.com), offers windborne excursions amid the cays and reefs of the Virgin Islands. The two-masted, 65-foot sloop has circumnavigated the globe, and has even been used as a design prototype for other boats. Owned and operated by Canadian Tim Krygsveld, it contains a hot-water shower, serves a specialty drink called a New Horizons Nooner, and carries a complete line of snorkeling equipment for adults and children. A full-day excursion with a continental breakfast, an Italian buffet lunch, and an open bar costs $120 per person ($80 for children ages 2\u201312). Excursions depart daily, weather permitting, from the Sapphire Beach Resort and Marina. Call ahead for reservations. New Horizons also offers New Horizons II (  340\/775-1171; www.newhorizonscharters.com), a 44-foot custom-made speedboat that takes you on a full-day trip, from 7am to 5pm, to some of the most scenic highlights of the British Virgin Islands. Trips cost $145 for adults or $95 for children ages 2 to 12. You will need your passport and will have to pay an additional $15-per-person Customs fee. New Horizons II leaves from the Sapphire Beach Resort at 7:15am and from the People Ferries' Dock in St. John at 7:45am.\n\nFishing The U.S. Virgins have excellent deep-sea fishing\u2014some 19 world records (eight for blue marlin) have been set in these waters. Outfitters abound at the major marinas like Red Hook. Peanut Gallery Fishing Charters, 8168 Crown Bay Marina, Ste. 310 (  340\/642-7423; www.fishingstthomas.com), offers both light-tackle inshore sportsfishing and deep-sea sportsfishing. Your captain will be Captain David Pearsall. The vessels provide inshore fishing year-round for the likes of barracuda, bonefish, kingfish, mackerel, and tarpon. The cost for 4 hours is $575; 6 hours, $700; and 8 hours, $900.\n\nYou can also line-fish from the rocky shore along Mandahl Beach on the north coast. The tourist office in Charlotte Amalie should have a listing of legal spots for line fishing around the island.\n\nGolf Mahogany Run, on the north shore at Mahogany Run Road (  800\/253-7103; www.mahoganyrungolf.com), is an 18-hole, par-70 course. The course rises and drops like a roller coaster on its journey to the sea; cliffs and crashing sea waves are the ultimate hazards at the 13th and 14th holes. Greens fees are $125 to $165 for 18 holes. Carts are included. Club rental costs $70.\n\nKayak Tours Virgin Islands Ecotours (  340\/779-2155; www.viecotours.com) offers 21\u20442-, 3-, and 5-hour kayak trips through the Cas Cay mangrove lagoon on the southern coastline. The cost is $69, $79, and $139 per person respectively (children 12 and under $39, $49, $79). The tour is led by professional naturalists who allow for 30 to 40 minutes of snorkeling.\n\nAnother new but increasingly popular trip is the clear kayak night tour leaving around sunset daily from the Marriott Frenchman's Reef resort. You paddle out to Pirate Point in clear see-through kayaks and turn on the boat's special LED lights\u2014giving you views of the nocturnal marine world. Offered by the Marriott's on-property tour operator Adventure Center, which also has snorkel and sailing excursions, the 11\u20442-hour trip costs $45 per person (  866\/868-7784; www.adventurecenters.net).\n\nScuba Diving\/snuba The best scuba-diving site off St. Thomas, especially for novices, is Cow and Calf Rocks, off the southeast end (45 min. from Charlotte Amalie by boat); here, you'll discover a network of coral tunnels filled with caves, reefs, and ancient boulders encrusted with coral. The Cartanser Sr., a sunken World War II cargo ship that lies in about 35 feet of water, is beautifully encrusted with coral and is home to myriad colorful resident fish. Another popular wreck dive is the Maj. General Rogers, the stripped-down hull of a former Coast Guard cutter.\n\nExperienced divers may want to dive at exposed sheer rock pinnacles like Sail Rock and French Cap Pinnacle, which are encrusted with hard and soft corals, and are frequented by lobsters and green and hawksbill turtles. Both spots are exposed to open-ocean currents, making these very challenging dives.\n\nMany think the St. Thomas Diving Club, 7147 Bolongo Bay (  877\/538-8734 in the U.S., or 340\/776-2381; www.stthomasdivingclub.com), is the best on the island. This full-service, PADI five-star IDC center offers open-water certification courses for $425 to $595. Advanced open-water certification courses go for $400 to $475. You can also enjoy a half-day snorkeling trip for $55.\n\nCoki Dive Center, Coki Beach (  340\/775-4220; www.cokidive.com), a PADI center, offers scuba-diving courses and guided dive tours for beginners and certified divers alike. You can also rent diving and snorkeling gear here. A PADI Discover Scuba Diving Course is $79. Night dives are $65.\n\nDay-Tripping to St. John\n\nMost visitors to St. Thomas include a day-trip to beautiful sister island St. John for swimming, snorkeling, hiking, or just hanging out on one of the island beaches. Ferries to St. John leave every hour on the hour from Red Hook ($6 per person) and last only 15 to 20 minutes; you can also catch less frequent ferries from the Charlotte Amalie ferry terminal. It's easy to make a day of it: The last ferry returning from Cruz Bay, St. John, is at 11pm. Car ferries also run between Red Hook and Cruz Bay, traveling from 7am to 7pm daily every half-hour; book ahead on one of three carriers: Boyson (  340\/776-6294); Love City (  340\/779-4000); or Global Marine (  340\/779-1739). Car-ferry rates run from $42 to $50 round-trip; arrive at least 15 minutes before departure. If you don't have a car, have taxi driver Kenneth Lewis (  340\/776-6865) or one of his cohorts meet you at the ferry terminal in St. John for transportation to and from the beach, the park, or wherever you decide to spend the day. At the end of your day, you return to Cruz Bay for shopping and dinner and catch the ferry home right there on the harborfront.\n\nDIVE IN!, in the Sapphire Beach Resort and Marina, Smith Bay Road, Route 36 (  866\/434-8346, ext. 2144, in the U.S., or 340\/777-5255; www.diveinusvi.com), is a complete diving center offering professional instruction (beginner to advanced), daily beach and boat dives, custom dive packages, snorkeling trips, and a full-service PADI dive center. An introductory resort course costs $90, with a one-dive afternoon trip going for $75 and two-dive morning trip costing $96. A six-dive pass costs $255.\n\nNon-divers and beginning swimmers can still have a diving experience with Virgin Islands Snuba Excursions (  340\/693-8063; www.visnuba.com) These special excursions are offered both at Coral World on St. Thomas and at Trunk Bay on St. John. With Snuba's equipment\u2014an air line that attaches to an air tank floating on the surface\u2014even novices can breathe easily underwater without the use of heavy restrictive dive gear. The Snuba operations begin in waist-deep water and make a gradual descent to a depth of 20 feet. This is fun for the whole family, as kids ages 8 and up can participate, and no snorkeling or scuba experience is needed. Most orientation and guided underwater tours take 11\u20442 hours, costing $70 per person on St. John. On St. Thomas, a pass to Coral World (p. ) is included, and the rate is $74 for adults and $70 for children 8 to 12. Reservations are required.\n\nsnorkeling The island's best snorkeling spots include the rocks around Coki Beach; you can rent snorkeling gear from the Coki Dive Center, Coki Beach (  340\/775-4220; www.cokidive.com); day snorkel rentals are $10 and locker rentals are $5. Another great spot to snorkel is Secret Harbor; you can rent equipment or sign up for a 60-minute \"Discover Snorkeling\" course (all ages welcome) on-site at the Secret Harbour Beach Resort with Aqua Action Dive Center (  340\/775-6285; www.aadivers.com). Another popular snorkeling spot is Sapphire Beach\u2014where you can rent snorkeling equipment right there on the beach at DIVE IN! (  866\/434-8346, ext. 2144, in the U.S., or 340\/777-5255; www.diveinusvi.com).\n\nziplining One of the newer outdoor adventure on island is Tree Limin' Extreme (  340\/777-9477; www.ziplinestthomas.com), a ziplining canopy tour in the rain forest of St. Peter Mountain, with six ziplines (including a \"yo-yo\" zip\u2014which gives riders, as the name implies, a yo-yo-style zipline ride). If you can take your eyes off your (secure) zipline, you'll have views of Magen's Bay and even Tortola and Jost Van Dyke. You can combine a ziplining adventure with a tour of the tropical gardens of the St. Peter Great House & Botanical Gardens (<http:\/\/greathousevi.com>; see below), just across the street.\n\nSeeing the Sights\n\nIn 1733, the Danish government acquired the Virgin Islands from the Danish West India Company. The Danes did not find land suitable for agriculture, and St. Thomas became a bustling port instead. It also became a center for transporting slaves.\n\nThe Virgin Islands remained under Danish rule until 1917, when the U.S., fearing German infiltration in the Caribbean during World War I, purchased the islands from Denmark. Today the U.S. Virgin Islands claims the highest per-capita income in the Caribbean, with some 50,000 settlers of varying ethnicity making their home in St. Thomas alone. The port is also the busiest cruise-ship harbor in the West Indies, outranking Puerto Rico.\n\nToday you can see many vestiges of the island's history in the capital, Charlotte Amalie, whose architecture reflects the island's culturally diverse past: You'll pass Dutch doors, Danish red-tile roofs, French iron grillwork, and Spanish-style patios. With its white houses and bright red roofs glistening in the sun, the city is terraced along the green hills surrounding the harbor. The town is filled with historic sights, like Fort Christian, an intriguing 17th-century building constructed by the Danes.\n\nThe best way to see the sights is by taxi tour. Expect to pay about $50 for a single-passenger tour or $25 per person for two or more passengers for 2 hours of sightseeing in a shared car. We highly recommend Campbell Rey (  340\/771-1568), the unofficial \"mayor\" of the island and a true gentleman. Rey has an open safari bus that can hold up to 27 people, and a smaller van that comfortably seats 12 or so. Another recommended taxi driver is Llewelyn Powell (  340\/771-1568 or 340\/776-3887).\n\nTropic Tours, 14AB the Guardian Building (  800\/524-4334 or 340\/774-1855; www.tropictoursusvi.com), offers a tour of St. Thomas that includes Drake's Seat, the Estate St. Peter Greathouse, and Charlotte Amalie shopping. The cost is $49 per person, $36 for children 12 and under.\n\nFor dramatic views of Charlotte Amalie's harbor, take a ride on the Paradise Point Tram (  866\/868-7784; www.adventurecenters.net) to a 700-foot peak. The tram, similar to those used at ski resorts, operates six cars, each with an eight-person capacity, for the 15-minute round-trip ride. It transports customers from the Havensight area to Paradise Point, where you can disembark to visit shops and the popular restaurant and bar. The tramway runs daily 9am to 5pm. Reservations are required; the cost is $21 per adult round-trip, $12 round-trip for children 6 to 12, and free for children 5 and under.\n\nThe walking tour below will give you a good basic feel for the historic district of Charlotte Amalie.\n\nWalking Tour: Charlotte Amalie\n\nStart: | King's Wharf.\n\n---|---\n\nFinish: | Waterfront.\n\nTime: | 21\u20442 hours.\n\nBest Time: | Before 10am to avoid cruise-ship passengers.\n\nWorst Time: | Around midday to 4pm, when traffic and pedestrians are at their most plentiful.\n\nEven with the crowds and shops, it is easy to see how the natural colors and charm of the Caribbean come to life in the waterfront town of Charlotte Amalie. The capital of St. Thomas once attracted seafarers from all over the globe. At one time, St. Thomas was the biggest slave market in the world. Today, the old warehouses, once used for storing stolen pirate goods, have been converted to shops. In fact, the main streets, called \"gade\" (a reflection of their Danish heritage), now coalesce into a virtual shopping mall, one that is often packed with cruise-ship hordes. Sandwiched among these shops are a few historic buildings, most of which can be seen on foot in about 2 hours. Start your walking tour along the eastern harborfront at King's Wharf.\n\n1King's Wharf\n\nThis is the site of the Virgin Islands Legislature. The apple-green two-story structure was first built in 1824 as a military barracks for the Danish Police. The current building dates from 1874. It was in a ceremony on this site in 1917 that ownership of the Virgin Islands was officially transferred from the Danish West Indies to the U.S.\u2014bought for a then-pretty price of $25 million.\n\nFrom here, walk away from the harbor up Fort Pladsen to:\n\n2Fort Christian\n\nDating from 1680 and named after the Danish king Christian V, this handsome (if crumbling) salmon-red structure is the oldest standing building in the entire U.S. Virgin Islands. It has been a fort (with 3- to 6-ft.-thick walls), a governor's residence, a prison (with a downstairs dungeon), a police station, and a court until it was named a National Historic Landmark in 1977. Unfortunately, restoration efforts of both the fort and its museum have dragged on since 2005, when it was closed to the public for renovations.\n\nContinue walking up Fort Pladsen to:\n\n3Emancipation Park\n\nThis is where a proclamation freeing African slaves and indentured European servants was read on July 3, 1848. The park is now mostly a picnic area for local workers and visitors.\n\nNear the park is the:\n\n4Grand Galleria\n\nFrom here, a visitor center dispenses valuable travel information about the island. When it opened as a hotel in 1837, it was a grand address, but it later fell into decay, and finally closed in 1975. The former guest rooms upstairs have been turned into offices and a restaurant.\n\nNorthwest of the park, at Main Street and Tolbod Gade, stands the:\n\n5Central Post Office\n\nOn display here are floor-to-ceiling murals by Stephen Dohanos, who became famous as an artist for \"The Saturday Evening Post.\"\n\nFrom the post office, walk east along Norre Gade to the:\n\n6Frederik Lutheran Church\n\nThis, the island's oldest church building, was built between 1780 and 1793. The original Georgian-style building, financed by a free black parishioner, Jean Reeneaus, was refurbished in 1826 and again in 1870 with Gothic and gabled flourishes. It has a \"welcoming arms\" entrance stairway, and is recognizable by its distinctive yellow-gold hue.\n\nExiting the church, walk east along Norre Gade to Lille Taarne Gade. Turn left (north) and climb to Kongens Gade (King St.), passing through a neighborhood of law firms, to:\n\n7Government House\n\nThis handsome neoclassical building is the administrative headquarters for the U.S. Virgin Islands. It's been the center of political life in the islands since it was built, around the time of the American Civil War, in 1867, for the Danish Colonial Council. The first two floors are open to the public and contain vintage West Indian furniture and works of art by native son (and vaunted Impressionist) Camille Pisarro.\n\nAfter leaving Government House, turn immediately to your left and look for the sign for:\n\nWalking Tour: Charlotte Amalie\n\n8Seven Arches Museum\n\nBrowsers and gapers love checking out this museum at Government Hill (  340\/774-9295; <http:\/\/sevenarchesmuseum.com>), the private home of longtime residents Philibert Fluck and Barbara Demaras. This 2-centuries-old Danish house has been completely restored and furnished with antiques. Walk through the yellow ballast arches into the Great Room, which has a wonderful view of the Caribbean's busiest harbor. It's open by appointment only.\n\nAfter visiting the museum, return to Government House. Next to the building is:\n\n9Frederik Church Parsonage\n\nThis building dates from 1725. It's one of the oldest houses on the island, and the only structure in the Government Hill district to retain its simple 18th-century lines.\n\nContinue west along Kongens Gade until you reach:\n\n10Hotel 1829\n\nFormerly known as the Lavalette House (and now known officially as Blackbeard's Castle), this place was designed in 1829 by one of the leading merchants of Charlotte Amalie. This landmark building has views over Charlotte Amalie and the harbor.\n\nThis is also a great place to take a break. If the bar is open, Hotel 1829 provides the perfect veranda, with a spectacular view, for a midday drink or a sundowner. You may just fall in love with the place, abandon this tour, and stick around for dinner. The bar is open Monday to Saturday 4 to 11pm.\n\nNext door (still on the same side of the street), observe the:\n\n11Yellow-Brick Building\n\nThis structure was built in 1854 in what local architects called \"the style of Copenhagen\"\u2014it's square and squat, with colorful wooden shutters and a roof tiled of marble. It was built of ballast brick (brought over as ballast in the ship). You can go inside and browse the many shops within.\n\nAt this point, you might want to double back slightly on Kongens Gade to climb the famous:\n\n1299 Steps\n\nThese steps (actually 103 in total) were erected in the early 1700s, and take you to the summit of Government Hill, from where you'll see the 18th-century:\n\n13Crown House\n\nThis stately private house is immediately to your right, on the south side of the street. This was the home of von Scholten, the Danish ruler who issued the famous proclamation of emancipation in 1848 (see Emancipation Park, above).\n\nWalk back down the steps and continue right (west) along Kongens Gade, then down a pair of old brick steps until you reach Garden Street. Go right (north) on Garden Street and take a left onto Crystal Gade. On your left, at the corner of Nye Gade and Crystal Gade, you'll see:\n\n14St. Thomas Reformed Church\n\nThis building is from 1844, but it holds one of the oldest congregations in the Virgin Islands, established by Dutch traders around 1660. The church has been buffeted by fire and natural disasters: Fire destroyed two early-19th-century iterations (and a 1995 hurricane, Marilyn, damaged the sanctuary), but much of the 1844 structure, designed like a Greek temple, has been beautifully restored.\n\nContinue up Crystal Gade. On your right (north side), you'll come to:\n\n15St. Thomas Synagogue\n\nThis is the oldest synagogue in continuous use under the American flag, and the second oldest in the Western Hemisphere. Nine Sephardic Jewish families founded the congregation in 1796, and the current building was erected in 1833. It still maintains the tradition of having sand on the floor, said to have muffled the sounds of worshippers' footsteps during the persecution of Jews during the Spanish Inquisition (many communities of Sephardic Jews flourished in the Caribbean after being expulsed from Spain in 1492). The structure is made of local stone, ballast brick from Denmark, and mortar made of molasses and sand. It's open to visitors Monday to Thursday 9am to 4pm, Friday 9am to 3pm and Saturday during services. Next door, the Weibel Museum showcases 300 years of Jewish history. It keeps the same hours.\n\nRetrace your steps (east) to Raadets Gade and turn south toward the water, crossing the famous Vimmelskaft Gade or \"Back Street\" (it can get a bit seedy at night, so be aware if you are walking after dark). Continue along Raadets Gade until you reach:\n\n16Main Street\n\nThis is Charlotte Amalie's major artery and most famous shopping street. Turn right (west) and walk along Main Street until you come to the mid-19th-century:\n\n17Camille Pissarro Building\n\nThis structure will be on your right, at the Amsterdam Sauer Jewelry Store. Pissarro, a Spanish Jew who became one of the founders of French Impressionism, was born in this building as Jacob Pizarro in 1830. Before moving to Paris, he worked for his father in a store on Main Street. Also housed in the building is Gallery Camille Pissarro, with a few Pissarro paintings on display and prints by local artists for sale.\n\nContinuing west along Main Street, you will pass on your right the:\n\n18Enid M. Baa Public Library\n\nThis building, formerly the von Bretton House, dates from 1818.\n\nKeep heading west until you reach:\n\n19Market Square\n\nThis was the center of a large slave-trading market before the 1848 emancipation and is officially called Rothschild Francis Square. Today it's an open-air fruit and vegetable market, selling, among other items, genips (grape-type fruit; to eat one, break open the skin and suck the pulp off the pit). The wrought-iron roof covered a railway station at the turn of the 20th century. The market is open Monday to Saturday, but Saturday is its busiest day; hours vary, but generally 9am to 3pm.\n\nIf the genip doesn't satisfy you, take Strand Gade down (south) to:\n\n20The Waterfront\n\nAlso known as Kyst Vejen, this is where you can purchase a fresh coconut. One of the vendors here will whack off the top with a machete so that you can drink the sweet milk from its hull. You'll have an up-close view of one of the most scenic harbors in the West Indies, even when it's filled with cruise ships.\n\nAttractions around the Island\n\nRoute 30 (Veterans Dr.) will take you west of Charlotte Amalie to Frenchtown (turn left at the sign to the Admiral's Inn). Early French-speaking settlers arrived on St. Thomas from St. Barts after they were uprooted by the Swedes. Many of today's island residents are the direct descendants of those long-ago immigrants, who were known for speaking a distinctive French patois. This colorful village contains a bevy of restaurants and taverns. Because Charlotte Amalie has become somewhat dangerous at night, Frenchtown has picked up its after-dark business and is the best spot for dancing, drinking, and other local entertainment.\n\nFarther west, Harwood Highway (Rte. 308) will lead you to Crown Mountain Road, a scenic drive opening onto the best views of the hills, beaches, and crystal-clear waters around St. Thomas.\n\nCoral World Ocean Park   AQUARIUM St. Thomas's number-one tourist attraction features a three-story underwater observation tower 100 feet offshore. Inside, you'll spy sea sponges, fish, coral, and other aquatic creatures in their natural state. An 80,000-gallon reef tank features exotic marine life of the Caribbean; another tank is devoted to sea predators, with circling sharks and giant moray eels. Among the exciting new activities are the Sea Lion Splash ($125 adult, $117 child; includes Coral World admission), where you actually get to swim with sea lions; the Sea Lion Encounter ($86 adult, $77 child; includes Coral World admission), where you interact with sea lions without gettting wet; the Turtle Encounter ($53 adult, $44 child; includes Coral World admission), where participants have up-close encounters with large sea turtles; and the Shark Encounter ($53 adult, $44 child; includes Coral World admission), where you have the chance to enter a shallow pool with juvenile sharks. Age and\/or height and weight restrictions apply to all activities.\n\nNondivers can get some of the thrill long known to scuba aficionados by participating in Sea Trek, which is slightly different from Snuba (p. ). For $79, or $70 for children, you can get a full immersion undersea with no experience necessary. Participants are given a helmet and a tube to breathe through. The tube is attached to an air source at the observatory tower. You then enjoy a 20-minute stroll in water that's 18 feet deep, observing rainbow-hued tropical fish and the coral reefs as you move along the seafloor. It's a marvelous way to experience the world through the eyes of a fish. Note: Reservations are required, so call ahead or log on to the park's website.\n\nCoral World's guests can take advantage of adjacent Coki Beach for snorkel rentals, scuba lessons, or simply swimming and snorkeling. Lockers are available. Also included in the marine park are a seaside cafe, duty-free shops, and a nature trail.\n\n6450 Estates Smith Bay, a 20-min. drive from Charlotte Amalie off Rte. 38.   340\/775-1555. <http:\/\/coralworldvi.com>. Admission $19 adults, $10 children ages 3\u201312. Daily 9am\u20134pm.\n\nEstate St. Peter Greathouse & Botanical Gardens   GARDENS Set on the island's northern rim 1,000 feet above sea level, this 11-acre estate was long ago part of the Plantation St. Peter. Today its gardens, restaurant, and event space is pillowed in oversize rain-forest foliage, with a tropical bird aviary and self-guided nature trails through a jungle of orchids. See the hump-backed islands of the B.V.I. from the panoramic deck above the gardens. The entry fee gets you panoramic views and a complimentary rum punch.\n\nAt the corner of Rte. 40 (6A St. Peter Mountain Rd.) and Barrett Hill Rd.   340\/774-4999. <http:\/\/greathousevi.com>. Admission $8 adults, $5 children 11 and under. Daily 9am\u20134pm.\n\nWATER ISLAND lore & history\n\nTo the native residents of St. Thomas, Water Island remains a land of legend and lore, having been settled by the Arawak Indians in the early 15th century. In the days of Caribbean piracy, as evoked by Disney's \"Pirates of the Caribbean\" movies, the island was used for both anchorage and fresh water, as pirates found numerous freshwater ponds here. Islanders on St. Thomas claim that millions of dollars in pirate treasure remain buried on Water Island, but so far no one has dug it up. An old leather trunk was once discovered, but it was empty except for one gold doubloon.\n\nWhen European colonization arrived in the late 17th century, many Danes tried to use the island for raising cows and goats. White plantation owners and colonists shunned the island because of its arid land, so unlike the rest of the Caribbean, Water Island was farmed by nonwhite plantation owners. These were freed men of color who operated the plantations, like Jean Renaud, a free mulatto who owned the entire island in 1769, working it with 18 slaves.\n\nIn 1944, the United States bought the island for $10,000. The military began planning Fort Segarra here, but the war ended before it could be built. Traces of \"the fort that never was\" can still be seen today.\n\nIn 1950, the Department of the Interior leased the island to Water Phillips, a developer, for $3,000 annually. He built homes and a 100-room hotel. Popular in the 1950s, the hotel became the setting for Herman Wouk's 1965 novel, \"Don't Stop the Carnival.\" (Incidentally, native residents of St. Croix claim that the novel was based on a hotel being built in the harbor of Christiansted.) The novel was turned into a short-lived musical by Jimmy Buffet in 1997. In 1989, Hurricane Hugo severely damaged the hotel, and it was shut down. It lies dormant today. The lease Phillips signed ran out 3 years later, and in 1996, Water Island was transferred to the federal government, in whose hands it remains today. At present (and likely to remain so for a long time to come), no foundations have been poured on Water Island. Nothing has been inaugurated. The cost of developing roads, irrigation, and sewage lines in this eco-sensitive environment is a daunting challenge and a dream that, for the immediate future, remains too expensive an undertaking.\n\nExcursions from St. Thomas\n\nWater Island\n\nWater Island, 3\u20444 mile off the coast from the harbor at Charlotte Amalie, is the fourth-largest island in the U.S. Virgins, with nearly 500 acres of land. Irregular in shape, 21\u20442-mile-long Water Island is filled with many bays and peninsulas, and studded with several good, sandy beaches along with secluded coves and rocky headlands. Established as the fourth U.S. Virgin Island in 1996, Water Island was once a part of a peninsula jutting out from St. Thomas, but a channel was cut through, allowing U.S. submarines to reach their base in a bay to the west. The island has a rich history that includes freed slaves, Herman Wouk, and Jimmy Buffett; see below.\n\nAt palm-shaded Honeymoon Beach, you can swim, snorkel, sail, water-ski, or sunbathe. The beach has been significantly improved in the past few years, as loads of rocks and gravel were hauled off and trees and brush removed. The sand was sifted to get rid of debris, and a dredge removed the seaweed and deposited white sand on the shore. Today it looks quite beautiful.\n\nThere is no commerce on the island\u2014no taxis, gas stations, hotels, shops, or even a main town. Residents are totally dependent on Charlotte Amalie, lying a half-mile away. It you're planning on a visit, bring water and your own food supplies and other needs. Don't count on it, but a food cart sometimes shows up on Honeymoon Beach, serving surprisingly good meals, including an all-steak lunch.\n\nThe Water Island Ferry (  340\/690-4159; <http:\/\/waterislandferry.com>) runs between Crown Bay Marina and Phillip's Landing on Water Island several times a day for $5 one-way, $10 round-trip (Crown Bay Marina is part of the St. Thomas submarine base).\n\nIf you prefer a guided tour, we recommend a bike tour with Water Island Adventures (  340\/714-2186; www.waterislandadventures.com; $65 per person, including transportation and equipment). Beach and swimming time is included in the 31\u20442 hours of the tour. Departures are from the dock at Havensight Mall or Crown Bay Marina.\n\nHassel Island\n\nIn the same bay as Water Island, and even closer to shore, is 136-acre Hassel Island (www.hasselisland.org). This island is almost completely deserted and protected as part of the Virgin Islands National Park, which prohibits most forms of development. There are no hotels or services of any kind here.\n\nThis island was connected to the mainland of St. Thomas until 1865 (end of the Civil War), when a channel was dug for easier passages of ships. During the early 19th century's Napoleonic Wars, the British occupied the island, and the ruins of two forts that the troops constructed here, Willoughby and Shipley, can be seen today. You can explore these ruins.\n\nHassel Island was once used by the Danes to defend the port of Charlotte Amalie. In 1840, the Danes built a marine railway operation for boat and sail repairs. As late as the 1960s, the marine railway was still in operation. It was one of the earliest steam-powered marine railways in the Western Hemisphere, as well as the oldest surviving example of a steam-powered marine railway in the world.\n\nIn 1978, some 95 percent of Hassel Island was sold to the U. S. National Park Service by its owners, the Paiewonski family. For almost 30 years, the island sat untouched and deteriorating until it was discovered by MTV location scouts as the ideal setting for the 27th season of the network's popular \"Real World\" franchise. \"Real World: St. Thomas\" debuted in 2012, with the participants living in an old estate.\n\nToday, the MTV crew is long gone, but you can visit Hassel through Virgin Islands Ecotours (  877\/845-2925; <http:\/\/viecotours.com>), the only concessionaire allowed to bring visitors to explore the island. It offers 3- and 5-hour kayak, hike, and snorkel tours of the island ($89\/3 hr. and $143\/5 hr.; the latter includes lunch), leaving from the Frenchtown marina. Trails take you across gentle hills with dry woods, lots of plants, and plenty of cacti\u2014you'll think you're in the Arizona desert. The western shore has white sands shaded by sea grapes. The rather barren island has little shade, so bring along hats and plenty of sunscreen.\n\nSHOPPING\n\nThe discounted, duty-free shopping in the Virgin Islands makes St. Thomas a shopping mecca. It's possible to find well-known brand names here at savings of up to 60 percent off mainland prices. And each U.S. resident is given a $1,600 duty-free allowance\u2014even kids. Even better: St. Thomas has no sales tax. But be warned\u2014savings are not always guaranteed, so make sure you know the price of the item back home to determine if you are truly getting a good deal. Having sounded that warning, we mention some St. Thomas shops in the listings below where we have indeed found good buys. For more help, the local publications \"This Week in St. Thomas\" and \"Places to Explore\" have updates on sales and shop openings.\n\nMost shops are open Monday to Saturday 9am to 5pm. Some stores are open Sunday and holidays if a cruise ship is in port.\n\nThe Best Buys & Where to Find Them\n\nThe best buys on St. Thomas include china, crystal, perfumes, jewelry (especially emeralds), Haitian art, clothing, watches, and items made of wood. St. Thomas is also the best place in the Caribbean for discounts in porcelain, but remember that U.S. brands may often be purchased for 25 percent off the retail price on the mainland. Look for imported patterns for the biggest savings. Cameras and electronic items, based on our experience, are not the good buys they're reputed to be.\n\nNearly all the major shopping in St. Thomas is along the harbor of Charlotte Amalie. Cruise-ship passengers mainly shop at the Havensight Mall, right on the cruise-ship dock at the eastern edge of Charlotte Amalie. It has 100 retail stores selling goods at duty-free prices, but you'll see little here you won't find at most other Caribbean duty-free ports. The principal shopping street is Main Street, or Dronningens Gade (the old Danish name). Some of the shops occupy former pirate warehouses. To the north is another merchandise-loaded street called Back Street or Vimmelskaft. Many shops are also spread along the Waterfront Highway (also called Kyst Vejen). Between these major streets is a series of side streets, walkways, and stone alleys\u2014each one packed to the gills with shops. Other shopping streets are Tolbod Gade, Raadets Gade, Royal Dane Mall, Palm Passage, Storetvaer Gade, and Strand Gade.\n\nIt is illegal for most street vendors (food vendors are about the only exception) to ply their trades outside of the designated area called Vendors Plaza, at the corner of Veterans Drive and Tolbod Gade. Hundreds of vendors converge here Monday through Saturday at 7:30am; they usually pack up around 5:30pm. (Very few hawk their wares on Sun, unless a cruise ship is scheduled to arrive.)\n\nWhen you tire of French perfumes and Swiss watches, head for Market Square, as it's called locally, or more formally, Rothschild Francis Square. Here, on the site of a former slave market and under a Victorian tin roof, locals with machetes slice open fresh coconuts so you can drink the milk, and women sell ackee, cassava, and breadfruit.\n\nOther big shopping areas include Yacht Haven Grande, within walking distance of Havensight Mall; and Crown Bay Center, near the Crown Bay cruise-ship dock and marina. If you're looking for original artworks, Tillett Gardens is a virtual oasis of arts and crafts\u2014pottery, silk-screened fabrics, candles, watercolors, jewelry, and more\u2014located on the highway across from Four Winds Shopping Center.\n\nAll the major stores in St. Thomas are located by number on an excellent map in the center of the publication \"St. Thomas This Week,\" distributed free to all arriving plane and boat passengers, and available at the visitor center. A lot of the stores on the island don't have street numbers or don't display them, so look for their signs instead.\n\nShopping A to Z\n\nIn the 300-year-old historic district of Charlotte Amalie along Main Street and in the warren of old Danish alleyways (and, it goes without saying, in the Havensink and Crown Bay areas around the cruise-ship piers), you will find many of the same shops you see in other duty-free Caribbean ports and cruise-ship centers\u2014as one visitor noted, the street is paved with \"jewels and booze.\" When the cruise-ships are in, taxi vans crowd the streets and shop owners ratchet up the hard sell. Plenty of international names are strung along Main Street (Dronningens Gade in Danish), from Tiffany & Co., 5 Dronningens Gade (www.tiffany.com) to David Yurman, 38 Dronningens Gade (www.davidyurman.com). If you poke around the cobblestoned alleyways between Main Street and the harbor, you may find a gem that's a little off the standard tourist radar.\n\nFor jewelry and watches, the choices around Main Street and near the cruise-ship piers at Havensight Mall and Crown Bay are practically endless, with an emphasis on high-end rocks. Diamonds International  , Dronningens Gade, has the largest selection of diamonds on the island (  800\/444-4025; www.diamondsinternational.com; six other locations on St. Thomas); H. Stern Jewelers, 5332 Dronningens Gade, one of the most respected international jewelers in the world (  340\/776-1146; www.hstern.net); and Cardow Jewelers, 5195 Dronningens Gade, often called the Tiffany's of the Caribbean, with one of the largest selections of fine jewelry in the world (  800\/227-2117; www.cardow.com; two other locations in St. Thomas).\n\nFor electronics, the reliable dealers include Boolchand's, 31 Main St. (  340\/776-0794; www.boolchand.com) and Royal Caribbean, 33 and 35 Main St. (  340\/776-4110; <http:\/\/royalcaribbeanvi.com>; three other branches on St. Thomas).\n\nFor crystal and china, the family-run Crystal Shoppe  , 14 Main St., offers a dazzling array of crystal from around the world (  800\/323-7232; <http:\/\/crystalshoppe.net>). Ubiquitous in the Caribbean, Little Switzerland, 5 Dronningens Gade, offers some of the finest crystal sold on the island, as well as fine jewelry and watches (  340\/776-2010; www.littleswitzerland.com; six other locations in St. Thomas).\n\nAntiques\n\ns.o.s. Antiques   This gallery is packed with antique maritime collectibles, including maps and charts, prints, sextants, and barometers. It also stocks a number of weapons, including antique and reproduction cannons, swords, pistols, and daggers. The highlight: genuine shipwreck salvage, such as actual pieces of eight mounted in 14-karat and 18-karat gold (you can also buy them unset). 5132 Dronningens Gade 1.   340\/774-2074.\n\nArt\n\nThe Color of Joy   The vivid watercolors of Corinne Van Rensselaer are on display in this little gallery, where you can also buy original prints by local artists, crafts, and gifts, including batiks, cards, and prints, along with glass and larimar (volcanic stone) jewelry. There is also a selection of ceramics, coral sculptures (much of it done locally), and Haitian artwork. American Yacht Harbor, 6100 Red Hook Quarters.   340\/775-4020. <http:\/\/thecolorofjoyvi.com>.\n\nGallery Camille Pissarro   This gallery is worth a visit just to see where the Impressionist artist Camille Pissarro was born in 1830 and lived until he was 26. Inside the circa-1811 building are prints and originals by local artists in three high-ceilinged rooms. It's just opposite the Royal Dane Mall at the top of a flight of uneven steps. 14 Main St.   340\/774-4621. www.pissarro.vi.\n\nGallery St. Thomas   This gallery highlights Virgin Islands painters and \"Caribbean-inspired\" artworks. 5143 Palm Passage, Suite A-13.   877\/797-6363 or 340\/777-6363. <http:\/\/gallerystthomas.com>.\n\nMango Tango Art Gallery   One of the island's largest art galleries offers both original artwork (many from Haiti) and less-expensive art prints, posters, and decorative maps. Look for Steffen Larsen's distinctive photographs of Cuba street scenes. Al Cohen Plaza, Raphune Hill, Rte. 38.   340\/777-3060. www.mangotango-art.com.\n\nNative Arts & Crafts Cooperative   Housed inside the V.I. Welcome Center, the slightly dilapidated 19th-century brick former headquarters of the U.S. District Court, this is one of the few places on island that specializes in locally made, non-art items and gifts, from hot sauces and handmade dolls to wooden crafts and jewelry. Tarbor Gade 1.   340\/777-1153.\n\nIn His Footsteps: Camille Pissarro (1830\u20131903)\n\nThe dean of the French Impressionist painters was the most famous resident ever born on the island of St. Thomas. Attracted by the work of Camille Corot and, later, Gustave Courbet, Pissarro moved in a lofty artistic circle of friends that included Monet, C\u00e9zanne, and Renoir. He painted landscapes and scenes of rural life and some portraits.\n\n  Birthplace: Danish St. Thomas, July 10, 1830, son of Jewish parents of French\/Spanish extraction\n\n  Residences: The Pissarro Building, off Main Street in Charlotte Amalie; Paris\n\n  Resting Place: Paris\n\nShaka-Man Zulu   Selling hand-crafted art and jewelry from Africa, this little shop has some interesting gift ideas. Look for the owner dressed in full Zulu costume prowling the streets. 33 Raadets Gade.   340\/514-8975.\n\nTillett Gardens Center for the Arts   This old Danish cattle farm has been one of the island's top arts-and-crafts centers since 1959. It's an indoor\/outdoor complex of buildings housing studios, galleries, a drum museum, a live performing theater, a hostel, and two restaurants. It's now also home to the Arts Alive Foundation, which displays original artworks from from oils and watercolors to acrylics. Tillett Gardens, 4126 Anna's Retreat.   340\/775-1405. www.tillettgardens.com. Take Rte. 38 east from Charlotte Amalie.\n\nDiamonds Are Forever\n\nJewelryis the most common item for sale in St. Thomas. Look carefully over the selections of gold and gemstones (emeralds are traditionally considered the finest savings). Gold that is marked 24-karat in the United States and Canada is marked 999 (or 99.9 percent pure gold) on European items. Gold marked 18-karat in the United States and Canada has a European marking of 750 (or 75 percent pure), and 14-karat gold is marked 585 (or 58.5 percent pure).\n\nFood\n\nThe Belgian Chocolate Factory   Get your handmade Belgian chocolates\u2014some made with local fruits like mango and papaya\u2014at this bustling factory\/store in the Charlotte Amalie historic district. 5093 Dronningens Gade, Ste. 3.   340\/777-5247. www.thebelgianchocolatefactory.com.\n\nGladys' Caf\u00e9   Be sure to enjoy local favorites like creamy conch chowder and jerk fish in this 30-year-old restaurant, but you don't have to dine to purchase a bottle of Gladys' delicious homemade hot sauce, in flavors from mango to mustard to tomato. Royal Dane Mall.   340\/774-6604. <http:\/\/gladyscafe.com>.\n\nLinens\n\nMr. Tablecloth   This shop has the best selection of tablecloths and accessories, plus doilies, in Charlotte Amalie. In addition to fine-linen tablecloths, the shops sells microfiber stain-resistant tablecloths. 6 Main St.   340\/774-4343. www.mrtablecloth-vi.com.\n\nSandals\n\nZora of St. Thomas   Zora has been making hand-crafted sandals for nearly 50 years. In addition to classic looking, full-grain leather sandals, Zora also offers a line of solidly practical, toe-enclosed footware that the islanders call \"limin' shoes.\" The store carries canvas backpacks, luggage, and even purses and briefcases made of that fabric. 5040 Norre Gade, Ste. 2.   340\/774-2559. www.zoraofstthomas.com.\n\nSt. Thomas After Dark\n\nSt. Thomas has a whole lot more nightlife than any other island in the U.S. or British Virgin Islands, but it's not as extensive as you might think. Charlotte Amalie is no longer the swinging town it used to be. Many of the city's streets have become dangerous after dark, so visitors have stopped visiting the area for nightlife, with the exception of a few places. Much of the Charlotte Amalie action has shifted to nearby Frenchtown, which has some great restaurants and bars and is generally safer at night. However, just as in Charlotte Amalie, some of these little hot spots lie along dark, badly lit roads, so keep a watchful eye.\n\nYou'll find the real action in Red Hook, on the island's east end, where a number of bars and casual seaside eateries feature live music. Things are jumping at the island's hotels and resorts as well, particularly Marriott Frenchman's Reef & Morning Star, Bolongo Bay, and Secret Harbour, each with lively after-dark scenes.\n\nThe Performing Arts\n\nPistarckle Theater   On the grounds of Tillett Gardens Center for the Arts (p. ), this professional theater presents six full-length plays yearly as part of its subscription season. Occupying a vacant print shop, the 100-seat theater is air-conditioned. There is also a summer drama camp for children. Tillett Gardens, 4126 Anna's Retreat.   340\/775-7877. www.pistarckletheater.vi. Tickets $20\u2013$50.\n\nReichhold Center for the Arts   The premier performing arts venue in the Caribbean lies west of Charlotte Amalie. Past performances have included the Alvin Ailey American Dance Theater and Al Jarreau. Call the theater or check with the tourist office to see what's on at the time of your visit. The lobby displays a frequently changing free exhibit of paintings and sculptures by Caribbean artists. A Japanese-inspired amphitheater, permeated by the scent of gardenias, is set into a natural valley, with seating space for 1,196. Performances usually begin at 8pm. University of the Virgin Islands, 2 John Brewers Bay.   340\/693-1559. www.reichholdcenter.com. Tickets $15\u2013$75.\n\nBeach Bars, Bars & Clubs\n\nCaribbean Saloon   This is a popular spot to drink, eat, watch sports on widescreen TVs, or listen to live or DJ music on the weekends. It's one of the jumpin'est joints in Red Hook. Frozen concoctions, like the lethal Buckwackers (light and dark rum, Bailey's, Kahlua, Frangelico, Amaretto, and coconut), are priced to move at $6.50. It's open daily 11am to 10:30pm and serves an extensive menu of pub grub, burgers, and bar noshes. American Yacht Harbor, Red Hook.   340\/775-7060. www.caribbeansaloon.com.\n\nDuffy's Love Shack   Mingle with the locals at what has been called the coolest parking-lot bar in the Caribbean. Yep, it's pretty casual, which means you will probably be dancing with brand-new best friends by evening's end. It's open daily 11:30am till 2am. 6500 Red Hook Plaza, Rte. 38.   340\/779-2080. www.duffysloveshack.com.\n\nEpernay Bistro & Wine Bar   This cozy and stylish bistro is a touch of France in the tropics, with candlelit wood tables and walls lined with wine bottles and gleaming glasses. Tapas is served during the 5- to 7pm happy hour, a nice time to stop in for a good glass of vino. Have a bite to eat while you're there; the food is good and elegantly presented. 24A Honduras.   340\/774-5348.\n\nThe Greenhouse   his longtime waterfront bar and restaurant (the sister restaurant of the two equally lively Greenhouses in St. Maarten) is one of the few places we recommend in downtown Charlotte Amalie. (You can park nearby and walk to the entrance.) It's a cruise-ship hangout during the day, but after the ships slip away before dusk, the Greenhouse regroups with a daily happy hour (4:30\u20137pm) that offers two-for-one drinks and discounted appetizers. The Greenhouse is open daily 11am until the last customer leaves. Veterans Dr.   340\/774-7998. www.thegreenhouserestaurant.com.\n\nHull Bay Hideaway   This 30-year mainstay in an open-air pavilion is just a short stroll from one of the best surfing beaches on island. It's a friendly, rustic spot that attracts an eclectic crowd, from surfer dudes to beach lovers. Many locals and regulars like to spend lazy Sunday afternoons here. It's got lots of good, cheap eats\u2014look for fish tacos and hot dogs and hamburgers. Daily hours are 10am to 10pm. 10 Estate Hull Bay.   340\/777-1898.\n\nIggies Beach Bar   During the day, Iggies is your typical family-friendly, informal, open-air seaside restaurant serving hamburgers, sandwiches, and salads. In the evening, it's an ongoing Caribbean party, with karaoke, beach bonfires, and a Wednesday-night Cruzan Carnival extravaganza, with live calypso music and Moko Jumbie stiltwalkers (Nov to Labor Day). Saturdays are reggae nights. Bolongo Bay Beach Resort, 7150 Bolongo.   340\/693-2600. www.iggiesbeachbar.com.\n\nLatitude 18   This open-air waterfront spot is located down a dirt road near the Red Hook Marina, where the ferryboats depart for St. John. It's a casual restaurant and bar that sizzles with live entertainment every night from 7:30 to 10:30pm (including Reggae Sundays). It's a good cheap place to eat as well, serving up hot rotis, BBQ ribs, burgers, and catch of the day. It's open daily from 11am to 11pm or later. Take Rte. #33 and turn left on Vessup Lane. Red Hook Marina.   340\/777-4552.\n\nGay & Lesbian Nightlife\n\nSt. Thomas might be the most cosmopolitan of the Virgin Islands, but it is no longer the \"gay paradise\" it was in the 1960s and 1970s. The major gay scene in the U.S. Virgins is now on St. Croix (see chapter 5). That doesn't mean that gay men and lesbians aren't drawn to St. Thomas. They are, but many attend predominantly straight establishments, such as the Greenhouse (see \"Bars & Clubs,\" above).\nSt. Croix\n\nIf St. Thomas is the seasoned older sibling and St. John the dewy beauty, St. Croix is the region's cultural heart. It's an old soul with a lilting bohemian spirit. It also has a population so diverse and multicultural it's been called a \"cultural callaloo.\"\n\nAt 84 square miles, St. Croix is certainly the largest U.S. Virgin Island, but it has a small-town feel. That may be because the \"plantation island\" is a place of bucolic delights, with acres of rural farmland and roads draped in a canopy of mahogany trees. The island's agricultural heritage is undergoing a renaissance: Organic farms in the northwest highlands are flourishing, and a celebrated Agricultural Fair draws thousands of visitors annually. Tied in with this trend is the island's growing reputation as a health-and-wellness destination: St. Croix has more vegan cafes, juice bars, and organic farms than all the other Virgins combined.\n\nBut perhaps more than anything, St. Croix is a living museum of the region's tangled past. Much of the architecture from the 18th-century Danish occupancy remains enshrined in picturesque Christiansted, on the island's west end. The colorful Victorian buildings facing the scenic waterfront of the island's second-largest town, Frederiksted, have been revitalized\u2014a fetching welcome mat for the cruise-ship crowds that arrive weekly at the Frederiksted dock.\n\nSt. Croix is the most remote and least-visited of the Virgin Islands, separated from St. Thomas and St. John by one of the deepest ocean trenches in the Atlantic. But the island itself is protected by a natural necklace of coral reef, encircling gentle bays and powdery white-sand beaches. If R&R is at the top of your vacation criteria, this may be the spot for you; just ask U.S. Vice-President Joe Biden, who often spends his Christmas holidays in restorative serenity on the island of St. Croix.\n\nEssentials\n\nGetting There\n\nBy Plane\n\nCurrently, the best way to reach the island by plane, but a fast ferry was said to be in the works at press time. Most flights to St. Croix land at the small international Henry E. Rohlsen Airport, Estate Mannings Bay (  340\/778-1012; www.viport.com\/airports.html; airport code STT), 6 miles southwest of Christiansted on the island's southern coast. Seaplane flights with Seaborne Airlines (see below) from St. Thomas land right at the seaport on the Christiansted waterfront.\n\nAmerican Airlines (  800\/433-7300; www.aa.com) no longer offers frequent service to St. Croix. American has one direct flight daily from Miami, with one stop (but no change of plane) in St. Thomas. US Airways (  800\/428-4322; www.usairways.com) offers a nonstop flight from Charlotte in the high season. Most visitors arrive via connecting service through Puerto Rico or St. Thomas. Seaborne Airlines (  888\/359-8687 or 340\/773-6442; www.seaborneairlines.com) offers several flights daily, some on seaplanes that fly \"downtown to downtown\" between St. Thomas and the Christiansted waterfront (flight time: 25 min.). Cape Air (  800\/227-3247 in the U.S. and U.S.V.I.; www.capeair.com) flies between St. Croix and San Juan, Vieques Island, or St. Thomas. JetBlue (  800\/538-2583; www.jetblue.com) has daily afternoon connections from San Juan.\n\nA number of major car-rental firms maintain kiosks at the airport, including Avis (  340\/778-9355), Hertz (  888\/248-4261), and Budget (  888\/264-8894). At the Christiansted seaport, Avis (  340\/713-1347) is located at 1210 Watergut St. The oldest independent car agency in St. Croix, Olympic Rent-A-Car (  888\/USVICAR; www.olympicstcroix.com) offers free airport dropoff and pickup. Otherwise, taxis are on call at the airport and in downtown Christiansted to transfer you to your destination.\n\nTravel time to St. Croix from St. Thomas is 25 minutes; between St. Croix and San Juan 50 minutes; and between St. Croix and Vieques 25 minutes. There are no direct flights to St. Croix from Canada or the United Kingdom; connections are made via Miami.\n\nBy Boat\n\nThe SeaTrans fast ferry that traveled sporadically between St. Thomas and St. Croix was out of commission at press time (it ran aground on one of the small cays between the islands a couple of years ago), but government plans are in the works for the introduction of spiffy new catamaran ferries to travel between St. Thomas and St. Croix. Special ferries do run between St. Thomas and St. Croix in April for the St. Thomas Carnival and again in December around St. Croix's Carnival. It's a 1-hour trip over one of the world's deepest oceans, and when the seas are up, it can be rough passage for those with motion sickness.\n\nBy Cruise Ship\n\nThe island's main cruise ship pier, the Ann E. Abramson Pier, on the Fredricksted waterfront, was upgraded and now has the capacity to accommodate two mega-ships at a time. The dock enhancements have already upped cruise-ship visitation from one ship every 3 weeks to one or two ships making port stops weekly.\n\nVisitor Information\n\nThe St. Croix Visitor Bureau, 53A Company St., in Christiansted (  340\/773-0495), is located in a yellow building across from the open-air market. It's open Monday to Friday 8am to 5pm.\n\nThe U.S. Virgin Islands Tourism also has an office at the Customs House Building, Strand Street (  800\/372-8784 or 340\/772-0357; www.visitusvi.com), in Frederiksted. A local independent website, www.stcroixtourism.com, offers lots of good information. Another website to check is www.visitstcroix.com.\n\nTourist offices provide free maps to the island. \"St. Croix This Week,\" distributed free to cruise-ship and air passengers, has detailed maps of Christiansted, Frederiksted, and the entire island, pinpointing individual attractions, hotels, shops, and restaurants and listing the week's events and happenings.\n\nSt. Croix\n\nIsland Layout\n\nSt. Croix has only two sizable towns: Christiansted on the northcentral shore and Frederiksted in the southwest. The Henry E. Rohlsen Airport is on the south coast, directly west of the former HOVENSA oil refinery, for many years the island's main industry (at press time, the refinery was closed and up for sale). No roads circle St. Croix's coast.\n\nTo continue east from Christiansted, take Rte. 82 (also called the E. End Rd.). Route 75 will take you west from Christiansted through the central heartland. Melvin H. Evans Highway, Route 66, runs along the southern part of the island. You can connect with this route in Christiansted and head west all the way to Frederiksted.\n\nChristiansted\n\nThe historic district of Christiansted has four main streets leading toward the water: Strand Street, King Street, Company Street, and Queen Street. Because the city is compact, it can easily be explored on foot. All streets start at the harbor and go up slightly sloped hillsides, and each street heads back down the hill to the port so you can't get lost. The visitor information center is located at 53A Company St. The center of Christiansted can get very congested, and driving around is difficult because of the one-way streets. It's usually more practical to park your car and cover the small district on foot. You will find open-air parking on both sides of Fort Christiansvaern. See the \"Walking Tour: Christiansted\" map, on p. , to orient yourself.\n\nThe North Shore\n\nThis coastal strip that stretches from Cottongarden Point, the eastern tip of the island, all the way west past Christiansted and up and around Salt River Bay, comes to an end as it reaches the settlement of Northside in the far west. It is the most touristy region of St. Croix, site of the best beaches, the most hotels, and the densest shopping. It is also the takeoff point (at Christiansted Harbor) for excursions to Buck Island, St. Croix's most popular attraction. Many visitors confine their stay in St. Croix entirely to the north coast. The northern coastline is not only long but also diverse, going from a lush tropical forest that envelops most of the northwest to the eastern sector, which is dry with palm-lined beaches.\n\nThe East End\n\nThe East End begins immediately east of Christiansted, the capital, taking in Tamarind Reef Beach and Reef Beach before it reaches Teague Bay, coming to an end at Cottongarden Point, the far eastern tip of St. Croix. This section of St. Croix is linked by Rte. 82 (also called E. End Rd.). The Buccaneer, the major resort of St. Croix, is found here, along with the Tamarind Reef Hotel. The area is far less congested than the section immediately to the west of Christiansted, and many visitors prefer the relative isolation and tranquility of the East End. This section of St. Croix is somewhat dry, the landscape a bit arid, but its compensating factor is a number of palm-lined beaches. The best place for a beach picnic is Cramer Park at the far eastern tip, a U.S.V.I. territorial beach popular with islanders.\n\nFrederiksted\n\nLittle Frederiksted has a storied past. It was established in 1751, but its colonial architecture was destroyed by fire in 1878 during a legendary labor revolt. The town was rebuilt in the Victorian style, and its historic waterfront today has been revitalized, a boon for the big cruise ships that arrive at the town's deep-water pier weekly. The two major streets, both of which run parallel to the water, are Strand Street and King Street.\n\nAddress Chaos\n\nIn both Christiansted and Frederiksted, in St. Croix, buildings are numbered consecutively on one side, stretching all the way across town; then the numbers \"cross the street\" and begin consecutively on the opposite side. That means that even and odd numbers appear on the same side of the street. The numbering system begins in Christiansted at the waterfront. In Frederiksted, the first number appears at the north end of town for streets running north-south and at the waterfront for streets running east-west.\n\nGetting Around\n\nIf you plan to do some serious sightseeing on the island, you'll need to rent a car, as getting around by public transportation is a slow, uneven process.\n\nBy Car\n\nRemember to drive on the left. In most rural areas, the speed limit is 35 mph; certain parts of the major artery, Route 66, are 55 mph. In towns and urban areas, the speed limit is 20 mph. Keep in mind that if you're going into the \"bush country,\" you'll find the roads very difficult. Sometimes the government smoothes the roads out before the rainy season begins (often in Oct or Nov), but they rapidly deteriorate.\n\nSt. Croix offers moderately priced car rentals, even on cars with automatic transmissions and air-conditioning. However, because of the island's higher-than-normal accident rate (which is partly the result of visitors who forget about driving on the left-hand side of the road), insurance costs are a bit higher than elsewhere. Avis (  800\/331-1212 or 340\/778-9355; www.avis.com), Budget (  800\/472-3325 or 340\/778-9636; www.budget.com), and Hertz (  800\/654-3131 or 340\/778-1402; www.hertz.com) all maintain headquarters at the airport; look for their kiosks near the baggage-claim areas. Collision-damage insurance costs $14 per day, depending on the company and size of car, and we feel that it's a wise investment. Some credit card companies grant you collision-damage protection, if you pay for the rental with their card. Verify coverage before you go.\n\nBy Taxi\n\nAt Henry E. Rohlsen International Airport, official taxi rates are posted. From the airport, expect to pay about $16 to $36 to Christiansted and about $12 to $24 to Frederiksted. Even though taxi rates are standardized, cabs are unmetered, so agree on the rate before you get in. Taxis line up at the docks in Christiansted and Frederiksted.\n\nWe highly recommend the following taxi drivers (who double as wonderful tour guides): Ames Joseph (  340\/277-6133) and Francis M. Vazquez (  340\/690-4045). If they aren't available, contact the St. Croix Taxicab Association (  340\/778-1088).\n\nMore and more drivers are employing multi-passenger taxi vans and open-air safaris (converted truck beds with open-air seating), ideal for transporting cruise-ship passengers arriving weekly on island excursions.\n\nBy Bus\n\nAir-conditioned buses run between Christiansted and Frederiksted about every 45 minutes daily between 5:30am and 8pm. They start at Tide Village, to the east of Christiansted, and go along Route 75 to the Golden Rock Shopping Center. They transfer along Route 70, with stopovers at the Sunny Isle Shopping Center, La Reine Shopping Center, St. George Village Botanical Garden, and Whim Plantation Museum before reaching Frederiksted. The fare is $1, or 55\u00a2 for seniors. For more information, call   340\/773-1664.\n\n  St. Croix\n\nBanks FirstBank Virgin Islands (  340\/773-0440; www.firstbankvi.com) has full-service locations in Christiansted (12\u201313 King St. and Orange Grove); in Frederiksted (6A Strand St.); and at Sunny Isle Shopping Center. Scotiabank (  340\/693-2966; www.scotiabank.com) has a full-service locations in Sunny Isle Shopping Center. Banco Popular   800\/724-3655; www.popular.com\/vi) has full-service locations in Christiansted (Orange Grove) and Sunny Isle. Most are open Monday to Thursday 9am to 3pm and Friday 9am to 4pm; both Scotiabank and FirstBank have Saturday-morning hours.\n\nBusiness Hours Typical business hours are Monday to Friday 9am to 5pm, Saturday 9am to 1pm. The V.I. Dental Center, 2024 Estate Mt. Welcome, Ste. 15\u201316 (  340\/772-6000; <http:\/\/videntalcenter.com>), has a team of dentists that are members of the American Dental Association. Call for information or an appointment.\n\nDoctors For a referral, call Sunny Isle Medical Center (  340\/778-0069) or the Beeston Hill Medical Complex (www.vihealth.com).\n\nDrugstores Try the Golden Rock Pharmacy, Golden Rock Shopping Center (  340\/773-7666), open Monday to Saturday 8am to 7pm and Sunday 8am to 3pm.\n\nEmergencies To reach the police, fire department, or an ambulance, call   911.\n\nHospitals The main facility is Governor Juan F. Luis Hospital & Medical Center, 4007 Estate Diamond Ruby (  340\/778-6311).\n\nMaps See \"Visitor Information,\" in \"Orientation,\" above.\n\nNewspapers & Magazines St. Croix has its own online newspaper, \"St. Croix Source\" (<http:\/\/stcroixsource.com>). \"The Virgin Island Daily News\" (<http:\/\/virginislandsdailynews.com>) covers all the Virgin Islands. A good source of local information is \"St. Croix This Week,\" which is distributed free by the tourist offices to hotels, restaurants, and most businesses.\n\nPolice Police headquarters is on Market Street in Christiansted. In case of emergency, dial   911; for nonemergency assistance, call   340\/778-2211.\n\nPost Office The post office is on Company Street (  340\/773-3586), in Christiansted. The hours of operation are Monday to Friday 8:30am to 5:30pm.\n\nSafety St. Croix is safer than St. Thomas. Although there have been random acts of violence against tourists in the past, even murder, most crime on the island is petty theft, usually of possessions left unguarded at the beach while vacationers go into the water for a swim, or muggings (rarely violent) of visitors wandering the dark streets and back alleys of Frederiksted and Christiansted at night. Exercise caution at night by sticking to the heart of Christiansted and not wandering around in Frederiksted. Avoid night strolls along beaches. Night driving in remote parts of the island can also be risky; you might be carjacked and robbed at knifepoint.\n\nTaxes The only local tax is an 10 percent surcharge added to all hotel rates.\n\nTelephone You can dial direct to St. Croix from the mainland by using the 340 area code. Omit the 340 for local calls. Make long-distance, international, and collect calls as you would on the U.S. mainland by dialing 0 or your long-distance provider.\n\nToilets There are few public restrooms, except at the major beaches and the airport. In Christiansted, the National Park Service maintains some restrooms within the public park beside Fort Christiansvaern.\n\nTourist Offices See \"Visitor Information,\" above.\n\nWhere to Stay\n\nSt. Croix's deluxe resorts lie along the North Shore; its old waterfront inns are mostly in Christiansted. You may also choose to stay in a villa or condo, which offers privacy and the chance to save money by preparing your own meals. A location in Christiansted or Frederiksted puts you close to shops and nightlife, but away from the beach. Most resorts are either on the beach or a short walk from it but isolated from towns. You'll have to drive for a shopping binge or to dine out in restaurants and clubs.\n\nIn general, rates are steep, but in summer, hotels slash prices by as much as 50 percent; some close altogether in late summer and early fall. All rooms are subject to a 10 percent hotel tax, not included in the rates given below.\n\nNote: If you need a hair dryer, pack your own. Apparently, a lot of visitors have packed up hotel hair dryers upon departure, and some innkeepers are reluctant to provide them.\n\nNorth Shore\n\nExpensive\n\nThe Buccaneer   Family-run since the 1940s, this gracious resort is St. Croix's premier lodging. It's also steeped in the island's colonial plantation heritage. Three hundred years ago this was one of the island's biggest producers of sugarcane, and its crumbling stone sugar mill is but one of its historic touchstones (it's fun exploring the place to see the others). On the more modern side of the equation, the sprawling 340-acre resort is big enough to encompass a par-70 18-hole golf course, a 2-mile jogging path, two freshwater pools, and eight Laykold tennis courts (with a pro shop). The baronial-arched Great House has a restaurant, the open-air Terrace; and a full array of spa treatments is available at the newly renovated Hideaway Spa & Salon. The free daily Kid's Camp (ages 4\u201312) is available year-round, and introductory scuba lessons are complimentary. As for accommodations, they're either up in the hilltop Great House main building or dotted along the beach (it's a short but steep walk between the two). Rooms are big and comfortable, many outfitted in colorful floral prints, and those along the beachside are luxuriously over-sized\u2014particularly the Doubloons, which come with whirlpool tubs and patios overlooking the sea. The resort also has a handful of luxury suites, which range from roomy family cottages to swank honeymoon-ready units with four-posters and whirlpool tubs. A handful of former slave quarters have been turned into hotel rooms. It also has a beach house to rent at Whistle Point.\n\n  800\/255-3881 in the U.S., or 340\/712-2100. www.thebuccaneer.com. 138 units. Winter $300\u2013$782 double, $495\u2013$1,943 suite; off season $260\u2013$559 double, $443\u2013$1,338 suite. Children 2\u201317 $60 per night Dec\u2013Apr. Extra person $80 per night. Tax and energy surcharge 21%. Rates include full American breakfast and welcome cocktails. Amenities: 3 restaurants; bar; babysitting; children's program; health club and spa; 2 pools (outdoor); room service; 8 tennis courts (2 lit); watersports equipment; Wi-Fi (free).\n\nSt. Croix Hotels\n\nDivi Carina Bay All-Inclusive Resort & Casino   This all-inclusive resort brought gambling to the U.S. Virgin Islands\u2014and today remains the only casino in the island chain. But that's not its only perk (or downside, depending on your point of view). Opening onto a sugar-white beach, the Divi Carina's guest rooms and villa suites all offer unobstructed views of the Caribbean. Rooms are done in standard tropical d\u00e9cor, yes, but offer plenty of room to move around as well as kitchenettes and balconies. Note that 20 hillside villas are across the street, but they're still just a 3-minute walk to the sands\u2014and they still offer those glorious ocean views. The most up-to-date building contains 50 oceanfront accommodations with balconies. Be sure to sign up for regular e-mail blasts touting money-saving specials.\n\n  800\/823-9352 in the U.S., or 340\/773-9700. www.diviresorts.com. 180 units. Winter $448\u2013$470 double; off season $350\u2013$412 double. Rates are all inclusive, 7-night minimum. Children 15 and under stay free in parent's room. Amenities: 4 restaurants; 4 bars; babysitting; casino; health club and spa; mini-golf; 2 freshwater pools (outdoor); spa; tennis court (lit); watersports equipment\/rentals; Wi-Fi (free in lobby and public areas).\n\nThe Palms at Pelican Cove   Water-lovers take note: This elegant but casual boutique resort is not only set on 7 acres of gorgeous beachfront property, it boasts one of the largest pools on the island. Guest rooms are spacious, too, with private balconies or patios with ocean views. Social life revolves around an open-air lounge, bar, and restaurant with sea views (see the Palms, p. ). The coral reef is just 100 feet offshore, making The Palms a particular favorite of snorkelers.\n\n  800\/548-4460 or 340\/718-8920. www.palmspelicancove.com. 35 units. Winter $240\u2013$350 double, $360\u2013$650 suite; off season $210\u2013$310 double, $340\u2013$590 suite. Amenities: Restaurant; bar; Internet (free); pool (outdoor); watersports (scuba).\n\nRenaissance Carambola Beach Resort & Spa   The third in the triumvirate of RockResorts built in the U.S. Virgin Islands by the Rockefellers, this 28-acre resort lies 30 minutes west of Christiansted along a sugary-sand beach on the island's north shore. The jumble of red-roofed buildings overlooks Davis Bay from a forested bluff. The island's second-largest resort (after Divi; see below), it is well-known for its outstanding golf course, the Carambola Golf & Country Club  , adjacent to the resort. Many golfers fly over from the other Virgins just to tackle the course, designed by Robert Trent Jones, Jr., in 1966. The resort itself is comprised of 26 two-story buildings, each with six units, all furnished in Spanish colonial style (with Caribbean Creole touches) and dark (some say too dark) mahogany woodwork. Each room has its own screened porch, with garden or sea views, and many have full, modern kitchens with granite countertops and porcelain-top stoves.\n\n  888\/503-8760 in the U.S., or 340\/778-3800. www.carambolabeach.com. 150 units. Winter $349\u2013$549 double; off season $249\u2013$379 double. Amenities: 2 restaurants, including Saman; bar; exercise room; golf course; spa; pool (outdoor); 2 tennis courts (lit); Wi-Fi (free in public areas).\n\nModerate\n\nChenay Bay Beach Resort   Set on one of the island's finest beaches for watersports activities (swimming, snorkeling, windsurfing)\u2014a mile of talcum powder white sand\u2014these barefoot-casual West Indian\u2013style cottages are tucked into a terraced hillside just 3 miles east of Christiansted. Chenay Bay's a low-key boutique property, with just 50 cottages (each with its own charming front porch), and set on grounds that once held a sprawling sugar plantation. Chenay has a nimble versatility, its spacious cottages appealing both to honeymooners and families, the latter thanks to children's programs and adjoining units, one that sleeps 8. Plus, kitchenettes here include full-size refrigerators. Cottages are neat and clean and simply furnished, most with an understated tropical palette. Those numbered 21-50 are bigger than the older ones. The resort often offers money-saving online packages, so check out the resort website before booking.\n\nEstate Chenay Bay. 866\/357-2970 in the U.S., or 340\/718-2918. www.defenderresorts.com\/chenaybay. 50 cottages. Year-round $70\u2013$300 cottage for 1 or 2. Extra person $25. Children 17 and under stay free in parent's room. $50 per person for all meals. Amenities: Restaurant; bar; babysitting; children's program; pool (outdoor); 2 tennis courts (lit); 2 tennis courts (unlit); watersports equipment\/rentals; Wi-Fi (free).\n\nTamarind Reef Resort, Spa & Marina   You don't come to this modest, two-story property for top-end lodging. You come to be just steps away from a lovely, sandy beach with shade from thatched palapas and excellent snorkeling along the reef. You come to swim in the big oceanfront pool, or just sip cocktails at the poolside bar and grill, with the blue seas stretched out on the horizon before you. You pick the Tamarind Reef to take advantage of island-hopping trips offered by day-sail and charter-boat operators at the adjoining Green Cay Marina. And about the rooms: They're just fine\u2014clean, comfortable, roomy, if done in an uninspired tropical motif. Hey, they even have ocean-facing patios or terraces so you can be reminded of why you picked the Tamarind in the first place.\n\n  800\/619-0014 in the U.S., or 340\/773-4455. www.tamarindreefhotel.com. 38 units. Winter $275\u2013$325 double; off season $200\u2013$250 double. Extra person $35\u2013$50. Children 6 and under stay free in parent's room. Ask about dive, golf, and honeymoon packages. Amenities: 2 restaurants; bar; pool (outdoor); spa; 4 tennis courts (lit); watersports equipment\/rentals; Wi-Fi (free).\n\nInexpensive\n\nArawak Bay: The Inn at Salt River   Travelers praise this two-story North Shore B&B for its sweeping views of Salt River Bay; warm, accommodating staff; and its hearty, delicious home-cooked breakfasts (guaranteed to start your day off right). Each room has a king bed or two doubles, with top-quality bedding and vintage-style quilts; you step out of your room onto a long, shared balcony with sea views (the second-floor balcony has better sightlines). Help yourself to a nice glass of wine in the kitchen honor bar or take a dip in the (miniscule) pool. You will have to drive to reach the beach or nearby restaurants.\n\n  340\/772-1684. <http:\/\/arawakbayinn.com>. 14 units. Winter $160 double; off season $130 double. Rates include island breakfast. Amenities: Bar; outdoor pool; Wi-Fi (free).\n\nWaves at Cane Bay   This intimate 12-room property is set alongside the sea at one of the island's best beaches, Cane Bay Beach. It's an ideal spot to hang your hat if you're a diving or snorkeling aficionado: Cane Bay is one of the island's top scuba and snorkeling beaches; the coral reef is just off the shoreline here, and you can even scuba dive right off the beach. You can rent all the equipment you need next door at the Cane Bay Dive Shop (www.canebayscuba.com). Rooms are high ceilinged, large and colorfully outfitted, each with a well-equipped kitchenette, tiled floors, and a private veranda.\n\n  800\/545-0603 in the U.S., or 340\/718-1815. www.canebaystcroix.com. 12 units. Winter $175\u2013$200 double; off season $150\u2013$175 double. Extra person $20. From the airport, go left on Rte. 64; after 1 mile, turn right on Rte. 70; after another 1 mile, go left at the junction with Rte. 75; after 2 miles, turn left at the junction with Rte. 80; follow for 5 miles. Amenities: Restaurant; bar; pool (outdoor); watersports equipment\/rentals; Wi-Fi (free).\n\nChristiansted\n\nModerate\n\nKing's Alley Hotel   A location directly on the Christiansted waterfront within walking distance of the seaplane base makes this well-managed inn a convenient choice for business folk. The amenities are admittedly sparse (no pool, for one), but the small-to-mid-sized rooms have been nicely refreshed in a sunny palette of pale greens and yellows and feature better-than-normal beds (at least in this price range). Some overlook the hotel's tropical courtyard, while others offer views of the Christiansted harbor.\n\n  800\/843-3574 or 340\/773-0103. 35 units. Winter $180 double, $240 suite; off season $160 double, $220 suite. Amenities: Wi-Fi (free).\n\nInexpensive\n\nCompany House Hotel   Just a block or two from the downtown Christiansted waterfront, this good-value three-level inn is located in a vintage warehouse that once served the West Indies Company. It's within walking distance of the town's National Historic Site district. Although the public rooms have a historical elegance, the accommodations have a more generic feel, simply outfitted in earth tones, tile floors, and wicker furniture. Two suites, one on the third floor with balcony views overlooking Christiansted, have full kitchens, a perk for budget-minded travelers. The pool is small and narrow, but at these rates, who's complaining? From Friday through Tuesday, the Victor Borge Piano Bar offers live piano music from 6pm to 11pm\u2014and the first drink is on the house.\n\n  340\/773-1377. www.companyhousehotel.com. 35 units. Winter $125 double, $160 suite; off season $85 double, $150 suite. Children 12 and under stay free in parents' room. Rates include continental breakfast. Amenities: Piano bar; freshwater pool (outdoor); Wi-Fi (free).\n\nFrederiksted\n\nModerate\n\nCottages by the Sea   This hospitable and meticulously managed family-run resort is the kind of relaxing, laidback place where everyone is either digging their feet in the sand or curling up in some nook with a good book. Just outside Frederiksted (within walking distance), its set on a palm-fringed beach and consists of 21 nicely furnished cottages in tropical hues, each with its own private beachfront patio (and patio table) and fully equipped kitchen. Some are cinderblock, some wood, but all are well-maintained. It's an excellent value for St. Croix, especially for families\u2014up to two children 17 and under stay for free in their parent's cottage\u2014and the pretty beachfront locale gives it some real oomph.\n\n  800\/323-7252 or 340\/772-0495. <http:\/\/caribbeancottages.com>. 21 units. Beachside cottages $95\u2013$185 double; beachfront cottages $165\u2013$225 double. Extra person $15. Up to 2 children 17 and under stay free in parent's cottage; $10 per night for each additional child Amenities: Watersports equipment\/rentals; Wi-Fi (free).\n\nSand Castle on the Beach   This 21-room hotel right on Frederiksted Beach has long had a gay and lesbian following, but it's welcoming to all, with friendly service and splendid sunset views. It has plenty of sociable public spaces, including two freshwater swimming pools (one being clothing optional) and the alfresco resort restaurant, Beach Side Caf\u00e9 (p. ), a lively place to eat and drink that's set right in the soft white sand. It's a popular spot for live music on Saturday nights and meltingly lovely sunsets. Accommodations range from luxury beachfront villas with full kitchens and terra-cotta-tile floors to roomy courtyard suites around a private pool. Budget travelers will appreciate the good-value garden or queen studios\u2014they may lack views (and full kitchens and elbow room), but they're close to all the resort action.\n\n  800\/524-2018 or 340\/772-1205. www.sandcastleonthebeach.com. 21 units. Winter $149\u2013$199 double, $259\u2013$349 suite, $319\u2013$449 villa for up to 4 people; off season $109\u2013$219 double, $235\u2013$339 suite, $269\u2013$399 villa for up to 4 people. Rates include continental breakfast. Amenities: Restaurant; exercise room; high-speed Internet (free in lobby); 2 pools (outdoor); watersports equipment\/rentals.\n\nVillas, Condos & Cottages\n\nRenting a villa, condo, or house on St. Croix is an excellent option for vacationing groups and families, and couples. Most offer full kitchens and nailing down the details to rent is usually a snap.\n\nOne primo source of rentals is Vacation St. Croix (  877\/788-0361 or 340\/718-0361; www.vacationstcroix.com), which offers some of the best accommodations on the island, specializing in lodgings directly on the beach (villas, condos, and private homes). Two- to seven-bedroom units are available, with prices from $1,000 to $15,000 per week.\n\nWith 54 apartments, the Club St. Croix Beach & Tennis Resort, 3280 Estate Golden Rock, Christiansted (  800\/524-2025 or 340\/718-4800; www.clubstcroix.com; $140\u2013$285; children 12 and under stay free in parent's room) has an enviable location on a lovely quarter-mile, palm-fringedbeach; additional perks include a pool and three tennis courts.\n\nThe all-suites Colony Cove  , 3221 Estate Golden Rock (  800\/524-2025 or 340\/718-1965; www.colonycove.com; $140\u2013$255; children 11 and under stay free in parent's unit) offers 60 units and is 1 mile west of Christiansted on the same beach as Club St. Croix (see above). Of all the condo complexes on St. Croix, Colony Cove is most like a full-fledged hotel, with a large staff on hand, an on-site watersports desk, and a swimming pool.\n\nThe 43-unit Villa Madeleine  , off Rte. 82, Teague Bay (www.villamadeleine-stcroix.com;   800\/533-6863 or 340\/690-3465; weekly: $1,700\u2013$2,000 double, $2,300\u2013$2,450 quad), is a real hideaway, with beautifully furnished two-story condo villas in a lush tropical setting. Each condo has its own private courtyard and pool, many with views of the bay.\n\nRidge to Reef Farm Stay\n\nA creative alternative to the typical seaside resort is a stay on an organic rain-forest farm on the island's lushly forested western coast. Ridge to Reef Farm (  340\/220-0466; www.ridge2reef.org) offers a range of lodging from tent camps ($35\/night) to private stilted cabanas or treehouses ($125\/night) to the lofty Hawk's Nest villa ($275\/night), with views out over the tropical landscape. You can volunteer for farm work (like feeding the chickens) or simply relax and enjoy the bucolic setting.\n\nBed & Breakfasts\n\nCarringtons Inn   This peaceful, handsome five-room B&B just a 10-minute drive from Christiansted makes for an idyllic romantic escape. Rooms are sizeable and individually decorated, some with gleaming fourposters and pretty wicker chairs, and all with plump linens. After inhaling the fab home-cooked breakfasts (French toast, fresh fruit, pancakes), you can lounge around the tiled pool, framed in rounded stone arches and tropical greenery. A festive atmosphere prevails. Sorry, the inn does not accept children.\n\n  877\/658-0508 in the U.S., or 340\/713-0508. www.carringtonsinn.com. 5 units. Winter $125\u2013$200 double; off season $100\u2013$160 double. Rates include breakfast. Amenities: Breakfast room; health club (nearby); outdoor pool; tennis courts (nearby); Wi-Fi (free).\n\nRattan Inn   Set, literally, amid the ruins of an old sugarcane plantation, this inn has five elegantly appointed private residences, ideal for extended vacations. All but the Pool House have full kitchens, and three have their own laundry facilities (the Pink and Green houses share a laundry facility). The Main House has its own private courtyard, and the Master Suite has a private plunge pool and balcony.\n\n  340\/718-5098. www.rattaninnstx.com. 5 units. Cottages $150\u2013$250; Main House $4,500\/month. Amenities: Outdoor pool; Wi-Fi (free).\n\nCamping\n\nMount Victory Camp   Just a 10-minute drive north of Frederiksted, this beautifully landscaped eco-resort gives visitors an opportunity to immerse themselves in one of St. Croix's lesser-known natural environments: the forested farm valley of the wild western coast, where nights bring cooling tradewinds and the island's agricultural riches are being tapped once more. It's a refreshing departure from the beach scene, although you're only a 2-mile drive down the hill to a comely white sand beach.\n\nFour of the five camp lodgings are rustic Arts-and-Crafts-style bungalows handcrafted by a shipwright using salvaged tropical hardwoods. The lone Schoolhouse Apartment   was built around the ruins of a historic old schoolhouse (ca. 1841); it's a wonderful mix of tropical hardwoods and native stone walls. Both the bungalows and the apartment accommodate four to six adults. You can also do tent camping, but you'll have to bring your own gear. Each dwelling has basic cooking and eating facilities, and open-air showers use solar-heated water collected from a solar flat-plate thermal collector. A shady, breezy pavilion is the beating heart of the resort, its pitched slats fashioned from mahogany (using wood harvested from dead or dying trees\u2014mahogany is a protected species). The resort has its own orchard; you can also buy fresh organic fruit and produce from the celebrated Ridge to Reef farm, just a 10-minute stroll away. Outdoor activities include guided hikes in the rainforest and mountain biking (the camp has mountain bikes for rent)\u2014and feeding the gentle red-footed land tortoises in the resident tortoise colony.\n\n  340\/201-7983 or 340\/772-1651. <http:\/\/mtvictorycamp.com>. 5 dwellings. Winter $85\u2013$95 double; off season $75\u2013$85 double. Extra person $15. Credit cards through PayPal only. Amenities: Public phone; picnic tables; Wi-Fi.\n\nWhere to Eat\n\nSt. Croix has an amazing diversity of restaurants and cuisines, so don't limit yourself to your hotel for dining. The issue for most visitors\u2014particularly those staying outside of Christiansted and Frederiksted\u2014is often simply getting around at night. Many roads in St. Croix are badly lit, and driving on the left (for those used to driving on the right) can be challenging even during daylight hours. Don't let that deter you, though: Taxi drivers are happy to deliver you to a restaurant and return for you when your meal is done. Have your hotel call a taxi for you, and be sure to confirm the rate there and back with the driver before you set out.\n\nIf you're staying at one of the small hotels or guesthouses in and around Christiansted, you'll likely be able to walk to your restaurant of choice. If you're at a hotel in Frederiksted, the night is yours. The cruise-ship crowds will have departed, and the small dining rooms here have an earthier, more laid-back (and less expensive) feel than those in Christiansted.\n\nThe St. Croix Food & Wine Experience, is a highlight of the year, a weeklong food party on St. Croix in mid-April, when the island's food is highlighted with city food crawls, wine events, sunset BBQs, and the chance to sample local creations at the \"Taste of St. Croix.\" Go to www.stcroixfoodandwine.com to learn more.\n\nThe Chicken Shack\n\nIt's Martha Stewart's favorite restaurant on island, and after you sample one of the shack's spit-roasted barbecued chickens, it might be yours too. The La Reine Chicken Shack is known for its juicy, fall-off-the-bone chicken, but it also serves up exemplary ribs, johnnycakes, conch fritters, and specials like shrimp in butter sauce and curry chicken. It's good value, too; at press time you could get half a chicken and a couple of johnnycakes for $5.50! Eat in, or do as Martha does and take out. The Chicken Shack is located mid-island, on Centerline Road across from the old La Reine Shopping Plaza. It's open Monday through Wednesday 10:30am until 6pm, Thursday through Saturday 10:30am through 8pm, and Sunday 10:30am to 4pm (  340\/778-5717).\n\nNorth Shore\n\nExpensive\n\nThe Galleon   FRENCH\/NORTHERN ITALIAN Overlooking the sea, this upscale marina restaurant and bar is a special-occasion favorite. It serves northern Italian and French cuisine, occasionally including handmade pastas, osso buco, local fish\u2014wahoo, tuna, swordfish, or mahimahi\u2014or fresh Caribbean lobster. Order the beef tenderloin chateaubriand, and they will carve it right at your table. The wine list is extensive.\n\nE. End Rd.   340\/718-9948. www.galleonrestaurant.com. Reservations recommended. Main courses $22\u2013$40. Sun brunch $11\u2013$22 Mon\u2013Fri 4\u201310pm (happy hour 4\u20136pm, dinner 6\u201310pm), Sun 10am\u20132pm. Go east on Rte. 82 from Christiansted for 5 min.; after going 1 mile past the Buccaneer, turn left into Green Cay Marina.\n\nFlamboyant Beach Bar   AMERICAN Seating up to 100 guests beneath a vaulted cathedral ceiling, this informal open-air enclave is located in the Renaissance Carambola Beach Resort. It's one of the island's best hotel restaurants, offering fresh, creative takes on American classics and Caribbean seafood. Friday nights feature a Pirate's Buffet, with a Moko Jumbies show (6:30\u20139:30pm; $30).\n\nIn the Renaissance Carambola Beach Resort (p. ), Estate Davis Bay.   340\/778-3800. Main courses $22\u2013$35. Daily 11:30am\u20134pm and 5\u201310pm.\n\nSt. Croix Restaurants\n\nChristiansted\n\nExpensive\n\nKendrick's   FRENCH Some say this is the island's best fine-dining restaurant, and it's certainly has the good looks for it, with a beautiful courtyard and a setting in the historic Quin House cottage complex, built in 1762. Chef David Kendrick's food is memorable as well, with a seasonally changing menu. You might start with the exemplary Caesar salad, mussels steamed in white wine, or the sweet potato ravioli with pine nuts. For an entr\u00e9e, if its offered, try the pecan-crusted pork tenderloin or a filet in a port-wine demi-glace. Service is impeccable.\n\n2132 Company St.   340\/773-9199. Main courses $26\u2013$39. Tues\u2013Sat 6\u201310pm. Closed Mon Sept\u2013Oct.\n\nModerate\n\nRumRunners   CARIBBEAN This is an easy-breezy choice, an open-air restaurant that sits right on the boardwalk with sunset views of the Christiansted harbor. The ambience is pure Caribbean\u2014with waves rolling in in the background and pelicans wheeling above. During Sunday brunch a steel band plays. As for the food, it's surf and turf, from grilled New York strip steak to broiled whole lobster to meltingly tender baby-back ribs slow-cooked in island spices and Guinness. The bartender whips up some minxy drinks, or you may want to stop by for the $2 beer deals at happy hour.\n\nIn the Hotel Caravelle, on the boardwalk at Queen Cross St.   340\/773-6585. www.rumrunnersstcroix.com. Reservations recommended. Main courses $10\u2013$30. Mon\u2013Fri 7\u201310:30am, 11:30am\u20133pm, and 5:30\u20139:30pm; Sat 8\u201310:30am and 11:30am\u20133pm; Sun brunch 8am\u20132pm.\n\nSalud! Bistro   MEDITERRANEAN\/WINE BAR A buzzy, modern Mediterranean bistro and wine bar, Saludi has real local flavor, but not in the old-school-Caribbean sense. It's just a hopping place with a stylish local clientele, a window on the wide and wonderful diversity of folks who call St. Croix home. They come for the small plates, the expertly mixed mojiots and margaritas, the good wines and the warm ambience (walls are painted in richly saturated hues and hung with artwork). They also come for the expertly mixed mojitos and margaritas. Personal favorites among the starters: the platter of Mediterranean-style dips and spreads and the truffle mac 'n' cheese. The creative mains include lamb tagine, conch bolognese, and a zucchini pappadelle. Saludi is located 4 minutes west of Christiansted.\n\nPrincess Plaza, 9A Northside Rd.   340\/718-7900. <http:\/\/saludbistro.com>. Reservations recommended. Main courses $20\u2013$35. Mon\u2013Sat 5:30\u201310pm.\n\nSavant   CARIBBEAN\/THAI\/MEXICAN\/FUSION Just like St. Croix itself, a \"cultural callaloo,\" as one person put it, this is fusion on steroids. But it works, and this stylish bistro is a delightful place to dine, as the chefs take the food seriously. You can pinball from Thai curries to grilled fajitas to housemade egg rolls to fish tacos to maple-teriyaki pork tenderloin and not go wrong with any of it. Savant has only 20 candlelit tables, so call for a reservation as far in advance as you can.\n\n4C Hospital St.   340\/713-8666. www.savantstx.com. Reservations required. Main courses $14\u2013$39. Mon\u2013Sat 6\u201310pm.\n\nTutto Bene   ITALIAN A favorite among locals and visitors alike, this place has the warm, casual feel of a cantina, which meshes with the laidback island vibe. The food is simple and uncomplicated, the kind of hearty peasant dishes that built the Roman empire. They also happen to be delicious and built from the freshest ingredients. You'll dine in a warehouselike setting just east of Christiansted, amid warm colors and, often, lots of hubbub. The menu is written on oversize mirrors against one wall. You might begin with a grilled flatbread pizza and go on to one of the pastas, like spaghetti bolognese, traditional carbonara with pancetta, or the vegetarian-friendly rasta pasta: penne in a pesto cream sauce. Fresh fish, steak, and pork chops round out the menu.\n\nBoardwalk Building, Hospital St.   340\/773-5229. www.tuttobenerestaurant.com. Reservations recommended. Main courses $23\u2013$32. Wed\u2013Sun 6\u201310pm.\n\nInexpensive\n\nHarvey's   CARIBBEAN Good, honest, utterly delicious Crucian food\u2014that's what island matriarch Sarah Harvey delivers at her namesake restaurant. But Harvey's has plenty of fans, so you need to get here early for lunch....before the food runs out! We'd recommend you start with one of her homemade soups, or an appetizer of conch in butter sauce. Entrees will be barbecue chicken, goat stew, fried grouper, and, sometimes, even lobster. A side of cornmeal fungi comes with just about everything. Wash it all down with a fresh juice from exotic island fruits like gooseberry or tamarind. This is power eating, island style, so dig in.\n\n11B Company St.   340\/773-3433. Main courses $6\u2013$22. Mon\u2013Sat 11:30am\u20134pm.\n\nIn & Around Frederiksted\n\nModerate\n\nBlue Moon   INTERNATIONAL\/CAJUN Frederiksted's best little bistro is set in a 200-year-old stone house on the waterfront. It's a lively little caf\u00e9 that becomes a hot spot when live jazz is on tap Wednesday and Friday nights and during Sunday brunch. The atmosphere is casual and welcoming, abetted by a New Orleans\u2013influenced menu, featuring such classics as gumbo, jambalaya, and \"Jazzy Shrimp\": peel-and-eat shrimp with butter, garlic, and Cajun spices. There's also the usual array of fish, steak, and chicken dishes.\n\n17 Strand St.   340\/772-2222. www.bluemoonstcroix.com. Reservations recommended. Main courses $20\u2013$29. Tues\u2013Fri 11am\u201310pm; Sat 6\u201310pm; Sun 10am\u20132pm.\n\nVilla Morales   SPANISH\/CARIBBEAN Family run for more than 60 years, this is St. Croix's top Puerto Rican restaurant. It's only open 3 nights a week, so if you have a craving for home-cooked Latino classics, check the opening times and book well in advance. Nightly specials might be stewed or roasted goat, barbecue ribs, or roast pork served with boiled rice and fungi. Seafood is a house specialty, especially the excellent steamed red snapper, conch, or lobster. Sides, as you might expect, are rice and beans, johnnycakes, and fried plantains. Villa Morales also has a budget-friendly guest house adjacent to the restaurant ($55 double; extra person $10).\n\nPlot 82C, Estate Whim (off Rte. 70 about 2 miles from Frederiksted).   340\/772-0556. <http:\/\/villamorales.fsted.com>. Reservations recommended. Main courses $9\u2013$35. Thurs\u2013Sat 10am\u201310pm.\n\nInexpensive\n\nArmstrong's Homemade Ice Cream   ICE CREAM Since 1900, this family-run store has been making ice cream with local fruits, and it remains a must-visit. Flavors range from mango and soursop to guava, passionfruit and even gooseberry (not to mention basic choices like vanilla and chocolate). The recipes have been handed down in the family over three generations. The store makes a special guavaberry ice cream at Christmas.\n\n78-B Whim.   340\/772-1919. Ice cream $2\u2013$5. No credit cards. Tues\u2013Sat 7am\u20137pm; Sun 11am\u20137pm.\n\nMaggie's Snackett   CARIBBEAN This welcoming local lunch spot, painted in buttery yellow, is a haven of home cooking. Maggie's menu changes every day, but expect such Caribbean classics as smoked mackerel and goat or chicken stew and fresh-baked island breads.\n\n65 King St.   340\/772-5070. Breakfast $6\u2013$7; main courses $12\u2013$15. No credit cards. Daily 8am\u20133pm.\n\nPolly's at the Pier   COFFEE\/INTERNATIONAL This popular Art Deco spot is many things: an Internet caf\u00e9 with free Wi-Fi; a gourmet coffee shop; a breakfast and lunch caf\u00e9 (making fresh and delicious salads, sandwiches, burgers, soups); a mini-microbrewery selling Polly's Pale Ale; and an ice cream shop selling Armstrong's Homemade Ice Cream. It has a rotating art gallery. It even has the original Polly, an English bulldog for whom the shop is named, stopping in on occasion for photo ops.\n\n#3 Strand St.   340\/719-9434. Sandwiches, salads, and platters $10\u2013$30. Cash only. Mon\u2013Fri 7am\u20137pm; Sat\u2013Sun 8am\u20135pm.\n\nUca's Kitchen   VEGETARIAN Just off the cruise dock, Uca's serves island-style vegetarian fare. The kitchen is a creative wonder, offering up such tantalizing and unusual dishes as barbecued tofu kebab or callaloo (a local spinach) with a fungi polenta or even gooseberry stew. The house specialty is a tasty mushroom lasagna. Don't leave without sampling one of the fresh tropical fruit juices.\n\nKing St.   340\/772-5063. Reservations not required. Main courses $10\u2013$15. No credit cards. Daily 11:30am\u20137pm.\n\nOut on the Island\n\nDuggan's Reef   CONTINENTAL\/CARIBBEAN This popular open-air spot overlooks the placid blue seas of Reef Beach, where windsurfers and sailboats skim the water. Fresh lobster is a star here, from the fried johnnycakes and lobster salad to the lobster spring roll to Duggan's formidable Irish whiskey lobster. You'll find plenty of American classics here as well (burgers, steak, fried chicken). Set in an expansive green-and-white wood-frame structure with wraparound views, Duggan's is an island favorite.\n\nE. End Rd.   340\/773-9800. <http:\/\/duggansreef.com>. Reservations required for dinner in winter. Main courses $23\u2013$39; pastas $19\u2013$30. Mon\u2013Sat 6\u20139:30pm; Sun brunch 11am\u20132pm.\n\nModerate\n\nEat@canebay   INTERNATIONAL From the folks at the now-defunct Bacchus, this hot spot on the island's northwest end is getting raves for its fizzy beach atmosphere and solid food. The breakfast and lunch menus feature a build-your-own premise\u2014build your own burgers, salads, omelettes\u2014along with sandwiches (pulled pork, Cane Bay Reuben), peel-your-own shrimp, and a nifty roster of sides, including smoked-duck chili, sweet potato fries, roasted beefs. At night the menu goes uptown with a handful of elegantly prepared entrees listed on the blackboard. You might try pan-seared tuna, grilled steak, or linguine with scallops, mushrooms, and spinach in a garlicky cream sauce. The Sunday reggae brunch rocks the house; it's served from 11am to 4pm, and the music plays on till sunset.\n\ngroceries, markets & more: St. Croix Provisioning\n\nOn St. Croix, everyone should take advantage of the island's farm-fresh fruit and produce. You can do so at the following places, whether you're stocking a kitchen or just looking for a snack or pre-prepared meal. Note: Beyond the resources listed below, the restaurant Blue Water Terrace, Cotton Valley, on the island's East End, has a market and deli selling sandwiches, cheeses, wine, picnic foods and lunch boxes; it also caters (  740\/692-2583). Here are some essential St. Croix resources:\n\nGroceries: Plaza Extra (  340\/719-1870; <http:\/\/plazaextra.com>), with two locations on island, is stocked to the gills with, well, everything.\n\nFresh Seafood: Buy fresh fish (bluefish, tuna, alewife), Caribbean lobster, and conch at the fish market on the end of Strand Street on the Fredriksberg waterfront when the fishermen come in from 8:30am to 11am.\n\nFresh Meat: The family-run Annaly Meat Market (  340\/778-2229) sells cut-to-order custom cuts of meat, including local free-range organic Senepol beef, as well as U.S. choice beef, pork, lamb, and more. It's located on Route 72 on the way to the Carambola golf course. Sejah Farms (see below) raises chickens, goats, rabbits, and duck (the latter two to order) and sells fresh cuts from Thursday to Saturday at their farmstand.\n\nFresh Produce\/Fruit: The roads of St. Croix are dotted with farmstands. Look for farmers selling fat avocadoes, bananas, papayas, beans, and more along Centerline Road and on roads in Estate Upper Love. Sejah Farms (  340\/773-8065; <http:\/\/sejahfarm.com>) sells organic vegetables, eggs, honey, and chicken Monday to Friday 10am to 5pm and Saturday 7am to 5pm; it's on Casper Holstein Drive, near Fredriksted (they even accept farm volunteers). ARTFarm (  340\/514-4873; www.artfarmllc.com), a unique organic farm\/art gallery, sells organic produce and herbs every Saturday from 10am to noon The La Reine Farmers' Market is held on Saturday from 6:30am till 11am.\n\n1110c Cane Bay.   340\/718-0360. <http:\/\/eatatcanebay.com>. Reservations recommended. Main courses lunch $11\u2013$16, dinner $22\u2013$28. Wed\u2013Mon 11am\u20139pm; Sun reggae brunch 11am\u20134pm.\n\nThe Palms at Pelican Cove   CARIBBEAN\/INTERNATIONAL The food is good and fresh and prepared with a lot of TLC at the signature restaurant of the Palms at Pelican Cove, but the setting is sheer enchantment: You're right on the beach with tall palms rustling and lights twinkling in the cove. Yes, you can plant yourself in the indoors dining room, but we prefer the terrace, and if you're lucky you can find a seat there too. The food is called \"Caribbean-inspired\" and it's mighty fine, with an emphasis on the catch of the day, but the restaurant is often lauded for its perfectly char-broiled ribeye. Monday is the restaurant's Caribbean Night, with an all-you-can-eat buffet featuring ribs, fish, chicken, johnnycakes, sweet potato stuffing, and more ($30 adults, $15 children 11 and under) and Carnival entertainment.\n\nIn the Palms at Pelican Cove (p. ), 4126 La Grande Princesse.   340\/718-8920. www.palmspelicancove.com. Reservations recommended. Main courses $12\u2013$30. Daily 7:30\u201311am, 11:30am\u20132:30pm, and 6\u20139pm; Sun brunch 10am\u20132pm.\n\nExploring St. Croix\n\nBeaches\n\nBeaches are St. Croix's big attraction. Getting to them from Christiansted, however, home to most of the hotels, isn't always easy. It can also be expensive, especially if you want to beach it each day of your stay. From Christiansted a taxi will cost about $30 for two people to Davis Bay, $24 to Cane Bay, and $20 to Rainbow Beach. One solution is to rent a condo or stay in a hotel right on the water. Barring that, renting a car is a great way to hit the beaches\u2014and see the island's attractions.\n\nThe most celebrated beach is offshore Buck Island  , part of the U.S. National Park Service network. Buck Island is actually a volcanic islet surrounded by some of the most stunning underwater coral gardens in the Caribbean. The white-sand beaches on the southwest and west coasts are idyllic and the snorkeling is superb. The islet's interior is filled with cactus, wild frangipani, and pigeonwood. A number of operators runs excursions to Buck Island, leaving from Kings Wharf in Christiansted; the ride takes a half-hour. For details, see \"A Side Trip to Buck Island,\" later in this chapter.\n\nYour best choice for a beach in Christiansted is the one at the Hotel on the Cay  . This white-sand strip is on a palm-shaded island. To get here, take the ferry from the fort at Christiansted; it runs every 10 minutes from 6am to midnight. The 4-minute trip costs $3 round-trip, free for guests of the timeshare resort Hotel on the Cay. Five miles west of Christiansted is the Palms at Pelican Cove  , where some 1,200 feet of white sand shaded by palm trees attracts a gay and mixed crowd. Because a reef lies just off the shore, snorkeling conditions are ideal.\n\nRecommended highly are Davis Bay   and Cane Bay  , each with swaying palms, white sand, and good swimming. Because they're on the north shore, these beaches are often windy, and as a result their waters are not always calm. The snorkeling at Cane Bay, however, is truly spectacular; you'll see elkhorn and brain corals, all some 750 feet off the Cane Bay Wall. Cane Bay adjoins Route 80 on the north shore. Close to the Renaissance Carambola Beach Resort, Davis Beach doesn't have a reef; it's more popular among bodysurfers than snorkelers.\n\nOn Route 63, a short ride north of Frederiksted, lies Rainbow Beach  , with white sand and ideal snorkeling conditions. Nearby, also on Route 63, about 5 minutes north of Frederiksted, is another good beach, called La Grange  . Lounge chairs can be rented here, and there's a bar nearby.\n\nSandy Point  , directly south of Frederiksted, is the largest beach in all the U.S. Virgin Islands, but it's open to the public only on weekends from 10am to 4pm. Its waters are shallow and calm, perfect for swimming. Zigzagging fences line the beach to help prevent beach erosion. Sandy Point is protected as a nesting spot for endangered sea turtles. Continue west from the western terminus of the Melvin Evans Highway (Rte. 66). For more on visiting the refuge, see p. .\n\nThe island's East End has an array of beaches; they're somewhat difficult to get to, but much less crowded. The best choice here is Isaac Bay Beach  , ideal for snorkeling, swimming, or sunbathing. Windsurfers like Reef Beach, which opens onto Teague Bay along Route 82, East End Road, a half-hour ride from Christiansted. You can get food at Duggan's Reef (p. ). Cramer Park is a special public park operated by the Department of Agriculture. It's lined with sea-grape trees and has a delightful picnic area, a restaurant, and a bar. Grapetree Beach is off Rte. 60 (S. Shore Rd.), wide and sandy, with calm water. The beach is flanked only by a few private homes, although the beach at the Divi Carina is a short walk away.\n\nWatersports & Outdoor Adventures\n\nSt. Croix offers many outdoor adventures. In the east, the terrain is rocky and arid, getting little water. But the western part of the island is lush, including a small \"rain forest\" of mango, tree ferns, and dangling lianas. Between the two extremes are beautiful sandy beaches, rolling hills and pastures, and roads lined with mahogany trees and colorful flamboyant trees. Watersports galore abound, including boating, sailing, diving, snorkeling, fishing, hiking, and windsurfing.\n\nBicyling St. Croix has miles of relatively flat roadways that make it ideal for biking, but it also has lush hills for the more adventurous. Contact Freedom City Cycles, 2E Strand Square, 2 Strand St., Frederiksted (  340\/227-2433; www.freedomcitycycles.com), which, in addition to offering bike rentals, can arrange guided bike tours of the island. A 2- to 3-hour mountain bike tour begins at sea level and climbs through the rain forest on both paved and unpaved roads, costing $60 per person.\n\nFishing The fishing grounds at Lang Bank are about 10 miles from St. Croix. Here you'll find kingfish, dolphin fish, and wahoo. Using light-tackle boats to glide along the reef, you'll probably turn up jack or bonefish. At Clover Crest, in Frederiksted, local anglers fish right from the rocks. For more information on legal shore-fishing spots around the island, contact the tourist office in Christiansted or Frederiksted. Serious sport fishermen can head out with St. Croix Deep Blue Charters (  340\/643-5514; www.stcroixdeepbluecharters.com), on a 38-foot Bertram special anchored at Silver Bay Dock on the Christiansted waterfront. Reservations can be made by calling during the day. The cost for up to six passengers is $650 for 4 hours and $850 for 6 hours with bait and tackle and drinks included.\n\nGolf St. Croix has the best golf in the Virgin Islands. Guests staying on St. John and St. Thomas often fly over for a round on one of the island's three courses.\n\nCarambola Golf & Country Club, on the northeast side of St. Croix (  340\/778-5638; www.golfcarambola.com), adjacent to the Carambola Beach Resort (p. ), was created by Robert Trent Jones, Sr., who called it \"the loveliest course I ever designed.\" It's been likened to a botanical garden. The par-3 holes here are known to golfing authorities as the best in the Tropics. The greens fee of $95 in winter, or $65 in the off season, allows you to play as many holes as you like. Carts are included.\n\nThe Buccaneer, Gallows Bay (  340\/712-2144; p. ), 3 miles east of Christiansted, has a challenging 5,685-yard, 18-hole course with panoramic vistas. Nonguests pay $110 in winter or $70 off season, including use of a cart.\n\nThe Reef, on the east end of the island at Teague Bay (  340\/773-8844), is a 3,100-yard, 9-hole course, charging greens fees of $20 for 9 holes and $35 for 18 holes. Golf carts can also be rented at an additional $15 for 9 holes or $20 for 18 holes. The longest hole here is a 465-yard par 5.\n\nHiking Scrub-covered hills make up much of St. Croix's landscape. The island's western district, however, includes a dense, 15-acre forest known as the \"Rain Forest\" (although it's not a real one). The network of footpaths here offers fantastic nature walks. The area is thick with mahogany trees, kapok (silk-cotton) trees, and samaan (rain) trees. Sweet limes, mangoes, hog plums, and breadfruit trees, all of which have grown in the wild since the days of the plantations, are here. The \"Rain Forest\" is private property, but visitors are welcome to go inside to explore. To experience its charm, some people simply opt to drive along Route 76 (also known as Mahogany Rd.), stopping beside the footpaths that meander off on either side of the highway into dry riverbeds and glens. Stick to the most worn footpaths.\n\nOur favorite trail in the \"Rain Forest,\" Creque Dam Road (Rte. 58\/78), takes about 21\u20442 hours one-way. From Frederiksted, drive north on Route 63 to Creque Dam Road, where you turn right, park the car, and start walking. A mile past the Creque Dam, you'll be deep within the forest's magnificent flora and fauna. Continue on the trail until you come to the Western Scenic Road. Eventually, you reach Mahogany Road (Rte. 76), near St. Croix LEAP Project. The trail is moderate in difficulty.\n\nTo reach some of the most remote but scenic places on St. Croix, take a walking tour with Crucian Heritage & Nature Tourism (CHANT;   340\/772-4079; <http:\/\/chantvi.org>). Its \"Ay Ay Eco-Hike Tours\" include a steep walk down the mountainside to the crystalline tidal pools and saltwater baths of Annaly Bay, and a stroll along scenic Maroon Ridge, established by runaway slaves in the 17th century ($50 per person).\n\nThe St. Croix Environmental Association (  340\/773-1989; www.stxenvironmental.org) also offers interesting hikes, including a Butler Bay Falls guided hike to study the traditional cultural and medicinal uses of plants along the way. Buck Island (see the section \"A Side Trip to Buck Island,\" later in this chapter), just off St. Croix, also has nature trails.\n\nHorseback Riding Paul & Jill's Equestrian Stables, 2 Sprat Hall Estate, Rte. 58 (  340\/772-2880; www.paulandjills.com), the largest equestrian stable in the Virgin Islands, is known throughout the Caribbean for its horses. It's set on the sprawling grounds of the island's oldest plantation great house, Sprat Hall. The operators lead scenic trail rides through the forests, along the beach, and past ruins of abandoned 18th-century plantations and sugar mills. Beginners and experienced riders alike are welcome. A 11\u20442-hour trail ride costs $99. Tours usually depart daily in winter at 10:30am and 3pm, and in the off-season at 4pm, with slight variations according to demand. Reserve at least a day in advance.\n\nKayaking The beauty of St. Croix is best seen from a kayak. You can explore the waters of Sandy Point, Shell Island, and Salt River Bay National Park or kayak over the steep underwater wall at Cane Bay on half-day, full-day, and weeklong tours offered by Virgin Kayak (  340\/718-0071; www.virginkayaktours.com). The tour, lasting 3 hours, costs $45 per person and includes water and a light snack.\n\nSeeing St. Croix's magical bioluminescent bays   is a nighttime thrill on moonless nights. Both Virgin Kayak and Sea Thru Kayaks VI (  340\/244-8696; <http:\/\/seathrukayaksvi.com>) explore the island's two bioluminescent bays by kayak. See Thru Kayaks VI's see-through kayaks make the trip that much more memorable. Cost for an average 90-minute tour is $50 per person.\n\nKiteboarding Kite St. Croix (  340\/643-5824; www.kitestcroix.com), in Cotton Valley on the north shore of the island's East End, offers kiteboarding lessons ($100\u2013$225) and rentals ($50\/half-day) on-site. The times for good kiteboarding winds are December through March and June through August.\n\nScuba Diving & Snorkeling   Sponge life, black coral (the finest in the West Indies), and steep drop-offs near the shoreline make St. Croix a snorkeling and diving paradise. The island is protected by the largest living reef in the Caribbean, including the fabled north-shore wall that begins in 25 to 30 feet of water and drops to 13,200 feet, sometimes plunging straight down.\n\nBuck Island   is a major scuba-diving site, with a visibility of some 100 feet. It also has an underwater snorkeling trail. Practically all outfitters on St. Croix offer scuba and snorkeling tours to Buck Island. For more information on the island, see the section \"A Side Trip to Buck Island,\" later in this chapter.\n\nOther favorite dive sites include the historic Salt River Canyon   (northwest of Christiansted at Salt River Bay), for advanced divers. Submerged canyon walls are covered with purple tube sponges, deepwater gorgonians, and black coral saplings. You'll see schools of yellowtail snapper, turtles, and spotted eagle rays. We also like the gorgeous coral gardens of Scotch Banks (north of Christiansted) and Eagle Ray (also north of Christiansted), the latter so named because of the rays that cruise along the wall there. Cane Bay   is known for its coral canyons.\n\nFrederiksted Pier  , near the historic area of Frederiksted, is the jumping-off point (literally) for a scuba voyage into a world of sponges, banded shrimp, plume worms, sea horses, and other creatures.\n\nDavis Bay is the site of the 12,000-foot-deep Puerto Rico Trench. Northstar Reef  , at the east end of Davis Bay, is a spectacular wall dive, recommended for intermediate or experienced divers only. The wall here is covered with stunning brain corals and staghorn thickets. At some 50 feet down, a sandy shelf leads to a cave where giant green moray eels hang out.\n\nAt Butler Bay, to the north of Frederiksted on the west shore, there are the submerged ruins of three ships: the Suffolk Maid, the Northwind, and the Rosaomaira, the latter sitting in 100 feet of water. These wrecks form the major part of an artificial reef system that also contains abandoned trucks and cars. This site is recommended for intermediate or experienced divers.\n\nA top outfitter is the Cane Bay Dive Shop (  800\/338-3843 or 340\/773-9913; www.canebayscuba.com), with four locations around the island. The numerous locations means there's a variety of dive sites to choose from without having to take a long boat ride. A beginner's lesson goes for $60, and packages go all the way up the scale to a six-tank dive package for $295.\n\nSee \"Beaches,\" above, for the best snorkeling spots. Most beachside resorts offer complimentary snorkling equipment. You can also rent everything you need at all four locations of rthe Cane Bay Dive Shop (see above).\n\nStandup Paddleboarding Kite St. Croix (  340\/643-5824; www.kitestcroix.com) offers paddleboard rentals ($35\/per hour and $60\/per day) as well as beginner lessons ($50\u2013$75).\n\nTennis Some authorities rate the tennis at the Buccaneer  , Gallows Bay (  340\/773-3036), as the best in the Caribbean. The resort offers a choice of eight courts, two lit for night play, all open to the public. Nonguests pay $18 daytime, $22 nighttime per hour; you must call to reserve a court at least a day in advance. A tennis pro is available for lessons, and if you're looking for tennis partners, they are happy to match you up for singles or doubles.\n\nTaxi Tours\n\nOne of the best ways to explore St. Croix is on a taxi tour   with a local driver. We give our unqualified seal of approval to the following drivers, both of whom do wonderful and erudite sightseeing tours: Ames Joseph (  340\/277-6133) or Francis M. Vazquez (  340\/690-4045). In general, a 3\u00bd-hour island tour for one to four persons costs around $150; a 6-hour tour is $300. The fare should be negotiated in advance. Extra fees may be charged for the following sights: $10 for the botanical gardens, $10 for the Whim Estate House, and $8 for the rum distillery. Note: You may want to concentrate on the island's East End for one trip and the West End on another.\n\nHeritage & Cultural Tours\n\nSt. Croix's cultural riches are explored on tours run by Crucian Heritage & Nature Tourism (CHANT;   340\/772-4079; <http:\/\/chantvi.org>). In addition to historic walking tours   of the colonial towns of Christiansted and Frederiksted, CHANT offers a fascinating Ridge to Reef Farm Tour, showcasing the island's agricultural heritage and burgeoning farm-to-table movement with a tour of a 200-acre working tropical farm in a lush forest valley. The 3-hour tour is $55 per person.\n\nThe St. Croix Landmarks Society (  340\/772-0598; www.stcroixlandmarks.com) offers a terrific roster of Sunday-afternoon \"Places that Matter\/Ruins Rambles\"  : historic walking tours that let you \"walk in the footsteps of people\" who lived and worked here. A recent ramble took in the old slave market in Christiansted. It also holds Annual House Tours in February. Check the website for dates and hours.\n\nSeeing the Sights\n\nChristopher Columbus named the island Santa Cruz (Holy Cross) when he landed on November 14, 1493. He anchored his ship off the north shore but was quickly driven away by the spears, arrows, and axes of the Carib Indians. The French laid claim to the island in 1650; the Danes purchased it from them in 1733. Under their rule, the slave trade and sugarcane fields flourished until the latter half of the 19th century. Danish architecture and influence can still be seen on the island today. In a shrewd purchase deal with the Danes, the U.S. acquired the islands in 1917.\n\nAlthough the 21st century has definitely invaded St. Croix, with subdivisions, condo complexes, shopping centers, and modern strip malls, evidence of the past is everywhere across its 84 square miles. St. Croix contains the nostalgic ruins of some 100 slave-driven plantations where sugarcane was once grown.\n\nToday, the past is visible everywhere you go in St. Croix, from Fort Christiansvaern to Fort Frederick. Take the time to explore Christiansted and Frederiksted, where the island's Danish roots can be seen everywhere.\n\nIn His Footsteps: Alexander Hamilton (1755\u20131804)\n\nAlexander Hamilton was an American statesman from the West Indies who served brilliantly in the American Revolution. He wrote many of the articles contained in the Federalist Papers and became secretary of the Treasury under George Washington. He was noted for both his literary and oratorical skills. Although he was born on the island of Nevis, Hamilton spent his adolescence in St. Croix.\n\n  Birthplace: The British-held island of Nevis, on January 11, 1755\n\n  Residences: Nevis, St. Croix, various cities in the United States\n\n  Final Days: In a duel fought with Aaron Burr, Hamilton was mortally wounded and died on July 12, 1804.\n\nWalking Tour: Christiansted\n\nStart: | The Visitors Bureau.\n\n---|---\n\nFinish: | Christiansted's harborfront.\n\nTime: | 11\u20442 hours.\n\nBest Times: | Any day from 10am to 4pm.\n\nWorst Times: | Monday to Friday 4 to 6pm.\n\nThe largest town on St. Croix, Christiansted still has many traces of its Danish heritage. Constructed by the Danish West India Company, the heart of town is still filled with many imposing old buildings, mostly former warehouses, from the 18th century. Today they are registered as a U.S. National Historic Site. Across a small park stands Fort Christiansvaern, which the Danes built on the fortifications of a 1645 French fort. From its precincts, some of the best views of the harbor can be seen. Christiansted is best seen by walking tour.\n\n1The Old Scale House\n\nThis yellow-sided building with a cedar-capped roof is located near the harborfront. It was originally built as the Old Scale House in 1856 to replace a similar structure that had burned down. In its heyday, all taxable goods leaving and entering Christiansted's harbor were weighed here. In front of the building lies one of the most charming squares in the Caribbean. Its old-fashioned asymmetrical allure is still evident despite the mass of cars. Inside is an information center and a bookstore and gift shop.\n\nWith your back to the scalehouse, turn left and walk through the parking lot to the foot of the white-sided gazebo-like band shell that sits in the center of a park named after Alexander Hamilton, who spent his adolescence on St. Croix. The yellow-brick building with the ornately carved brick staircase is the:\n\n2Old Danish Customs House\n\nThis is currently the headquarters of the National Park Service. The gracefully proportioned 16-step staircase was added in 1829 as an embellishment to an older building dating back to 1734. During the island's Danish occupancy, this is where merchants paid their taxes. (There are public toilets on the ground floor.)\n\nContinue climbing the hill to the base of the yellow-painted structure, which is:\n\n3Fort Christiansvaern\n\nThis is the best-preserved colonial fortification in the Virgin Islands. It's maintained as a historic monument by the National Park Service. Its original four-sided, diamond-shaped design was in accordance with the most advanced military planning of its era. The fort is the site of the St. Croix military museum, which documents police work on the island from the late 1800s to the present with photos, weapons, and other artifacts. The admission price of $3 also includes admission to the Steeple Building (see below). The fort is open Monday through Friday from 8am to 5pm. For information, call   340\/773-1460.\n\nExit from the fort, and head straight down the tree-lined path toward the most visible steeple in Christiansted. It caps the appropriately named:\n\nWalking Tour: Christiansted\n\n4Steeple Building\n\nCompleted in 1753, the Steeple Building was embellished with a steeple between 1794 and 1796. For a time it served as the headquarters of the Church of Lord God of Sabaoth, the island's first Lutheran church. The original structure can still be visited (see below). Inside is a National Park Service museum with exhibits on plantation life on the island. Admission is included in the $3 ticket for Fort Christiansvaern.\n\nAcross Company Street from the Steeple Building is a U.S. post office.\n\n5The Danish West India & Guinea Warehouse\n\nThe building that houses the post office was built in 1749 as the warehouse for the Danish West India and Guinea Company. The structure was once three times larger than it is today and included storerooms and lodging for staff. Go to the building's side entrance, on Church Street, and enter the rear courtyard. For many years, this was the site of some of the largest slave auctions in the Caribbean.\n\nFrom the post office, retrace your steps to Company Street and head west for 1 block. On your left, you'll pass the entrance to Apothecary Hall, 2111 Company St., which contains a charming collection of shops and restaurants.\n\n6 Luncheria\n\nIf you need refreshment, try Luncheria, Apothecary Hall Courtyard, 2111 Company St. (  340\/773-4247). The bar's tables are grouped in a courtyard shaded by trees. The owners are margarita specialists, stocking more types of tequila (15-plus) than any other bar in the neighborhood. Luncheria serves burritos, tostadas, enchiladas, and tacos, as well as daily specials and vegetarian meals.\n\nExit Apothecary Hall and turn left onto Company Street. Walk across Queen Cross Street (Dronningens Tvergade). A half-block later, you'll arrive at the island's largest outdoor market:\n\n7Hendricks Square\n\nThe square was rebuilt in a timbered, 19th-century style after the 1989 hurricane. Fruits and vegetables are sold here Monday through Saturday from 7am to 6pm.\n\nRetrace your steps a half-block along Company Street, then turn left onto Queen Cross Street. Head downhill toward the harbor, walking on the right-hand side of the street. Within a half-block, you'll reach an unmarked arched iron gateway, set beneath an arcade. Enter the charming gardens of:\n\n8Government House\n\nThis grand Danish Colonial building was formed from the union of two much older town houses in the 1830s. It was used as the Danish governor's residence until 1871, when the Danish West Indies capital was moved to Charlotte Amalie, on St. Thomas. The European-style garden here contains a scattering of trees, flower beds, and walkways. The gardens are open Monday to Friday 8am to 5pm. The Virgin Islands Tourism Office is located downstairs.\n\nExit the same way you entered, turn right, and continue your descent of Queen Cross Street. At the first street corner (King St.), turn left and you'll see:\n\n9Lord God of Sabaoth Lutheran Church\n\nThis neoclassical church was built sometime before 1740 and was originally the site of the Dutch Reformed Church; it was turned over to the Lord God of Sabaoth congregation in 1834, when a Gothic Revival tower was added. Much inside predates the Lutheran occupation, including the tower bell, cast in Copenhagen in 1793, and an impressive 18th-century picture frame fashioned of local mahogany that resides behind the altar.\n\nContinue walking southwest along King Street. Within 2 blocks is the:\n\n10Limprecht Gardens & Memorial\n\nFor 20 years (1888\u20131908), Peter Carl Limprecht served as governor of the Danish West Indies. Today, an occasional chicken pecks at seedlings planted near a Danish-language memorial to him.\n\nAt the end of the park, retrace your steps to Queen Cross Street, and go left. One very short block later, turn right onto Strand Street, which contains some interesting stores, including at least two different shopping arcades. The streets will narrow, and the pedestrian traffic will be more congested. Pass beneath the overpass belonging to a longtime bar and restaurant, the Club Comanche.\n\nContinue down the meandering curves of King's Alley. Within 1 block you'll be standing beside:\n\nDriving the St. Croix heritage trail\n\nA trail that leads into the past, St. Croix Heritage Trail traces the island's Danish colonial heritage. All you need are a brochure and map, available at the tourist office in Christiansted (p. ). This 72-mile itinerary includes a combination of asphalt-covered roadway, suitable for driving, and narrow woodland trails which must be navigated on foot. Many aficionados opt to drive along the route whenever practical, descend onto the footpaths wherever indicated, and then return to their cars for the continuation of the tour. En route, you'll be exposed to one of the Caribbean's densest concentrations of historical and cultural sites. Consisting largely of existing roadways, the route connects Christiansted and Frederiksted, going past the sites of former sugar plantations, and traverses the entire 28-mile length of St. Croix. The brochure identifies everything you're seeing: cattle farms, suburban communities, even industrial complexes and resorts. It's not all manicured and pretty, but much is scenic and worth the drive. Allow at least a day for this trail, with stops along the way, including Point Udall, the easternmost point under the U.S. flag in the Caribbean, and the two highlights of the trail: the Estate Mount Washington (p. ), a strikingly well-preserved sugar plantation; and Estate Whim Plantation Museum (p. ), one of the best of the restored great houses, with a museum and gift shop. Of course, you'll want to stop and get a refreshment break. We recommend Smithens Market. Vendors at this market, which lies off Queen Mary Highway, offer freshly squeezed sugarcane juice and sell locally grown fruits and homemade chutneys.\n\n11Christiansted's Harborfront\n\nEnd your tour here by strolling the boardwalk of the waterside piers and watching the sailboats bob in the harbor (and perhaps a seaplane touching down).\n\nFrederiksted\n\nThis former Danish settlement at the western end of the island, about 17 miles from Christiansted, is a sleepy port town that comes to life when a cruise ship docks at its pier\u2014which is happening more frequently these days. Frederiksted was destroyed by a fire in an 1878 labor revolt, and the citizens rebuilt it by putting Victorian wood frames and clapboards on top of the old Danish stone and yellow-brick foundations.\n\nMost visitors begin their tour at russet-colored Fort Frederik, at the northern end of Frederiksted next to the cruise-ship pier (  340\/772-2021). This fort, completed in 1760, is said to have been the first fort in the Caribbean to salute the flag of the new United States. An American brigantine, anchored at port in Frederiksted, hoisted a crudely made Old Glory. To show its support for the emerging American colonies, the head of the fort fired a cannonball in the air to honor the Americans and their new independence. Such an act violated the rules of Danish neutrality. It was at this same fort, on July 3, 1848, that Governor-General Peter von Scholten emancipated the slaves in the Danish West Indies, in response to a slave uprising led by a young man named Moses \"Buddhoe\" Gottlieb. In 1998, a bust of Buddhoe was unveiled here. The fort has been restored to its 1840 appearance and today is a national historic landmark. You can explore the courtyard and stables. A local history museum has been installed in what was once the Garrison Room. Admission is $3, free for children 15 and under; it's open Monday through Friday from 8:30am to 4pm.\n\nThe Customs House, just east of the fort, is an 18th-century building with a 19th-century two-story gallery. To the south of the fort is the visitor bureau at Strand Street (  340\/772-0357), where you can pick up a free map of the town.\n\nSandy Point Wildlife Refuge\n\nSt. Croix's rarely visited southwestern tip is composed of salt marshes, tidal pools, and low vegetation inhabited by birds, turtles, and other wildlife. More than 3 miles of ecologically protected coastline lie between Sandy Point (the island's westernmost tip) and the shallow waters of the West End Salt Pond. This national wildlife refuge is one of only two nesting grounds of the leatherback turtle in the United States\u2014the other is on Culebra, an offshore island of Puerto Rico. It's also home to colonies of green and hawksbill turtles, and thousands of birds, including herons, brown pelicans, Caribbean martins, black-necked stilts, and white-crowned pigeons. As for flora, Sandy Point gave its name to a rare form of orchid, a brown\/purple variety. The area consists of 360 acres of subtropical vegetation, including the largest salt pond in the Virgin Islands.\n\nThe wildlife refuge is only open to the public on Saturday and Sunday from 10am to 4pm (admission is free). Park rangers are determined to keep the area pristine, and in doing so they have to face such problems as the poaching of sea turtles and their eggs, drug smuggling, dumping of trash, and the arrival of illegal aliens. Even the mongoose and feral dogs are a menace to the nesting female turtles.\n\nIf Sandy Point's rules and regulations seem a little stringent, then you haven't met a leatherback sea turtle. It's the largest of its species and can measure some 6 feet in length and weigh more than 1,000 pounds. Every 2, perhaps 3 years, the turtles come back to this refuge to nest from March to July. The average female will deposit anywhere from 60 to 100 eggs in her nest. The survival rate is only 1 in 1,000 hatchlings. The refuge is also home to the green sea turtle, which can grow to a maximum of 4 feet and weigh about 400 pounds. These turtles come here only from June to September, when the females lay from 75 to 100 eggs.\n\nBirders also flock to Sandy Point to see some 100 species of birds, five of which are endangered. Endangered brown pelicans, royal terns, Caribbean elaenias, bananaquits, and yellow warblers are among the birds that call Sandy Point home.\n\nTo reach the refuge, drive to the end of the Rte. 66 (Melvin Evans Hwy.) and continue down a gravel road. For guided weekend visits, call   340\/773-4554 to make arrangements.\n\nSightseeing around the Island\n\nNorth of Frederiksted, you can drop in at Sprat Hall, the island's oldest plantation, or continue along to the \"Rain Forest\" (see above). Most visitors come to the area to see the jagged estuary of the northern coastline's Salt River. The Salt River was where Columbus landed on November 14, 1493. Marking the 500th anniversary of Columbus's arrival, former President George H. W. Bush signed a bill creating the 912-acre Salt River Bay National Historical Park and Ecological Preserve (www.nps.gov\/sari). The park contains the site of the original Carib village explored by Columbus and his men, including the only ceremonial ball court ever discovered in the Lesser Antilles. Also within the park is the largest mangrove forest in the Virgin Islands, sheltering many endangered animals and plants, plus an underwater canyon attracting divers from around the world. If you visit on your own, a taxi from Christiansted will cost $22. See \"Watersports & Outdoor Adventures,\" earlier in the chapter, for suggestions on kayak and scuba tours to this very special park.\n\nWhere the Sun First Shines on the U.S.A.\n\nThe rocky promontory of Point Udall, jutting into the Caribbean Sea, is the easternmost point of the United States. Diehards go out to see the sun rise, but considering the climb via a rutted dirt road, you may want to wait until there's a bit more light before heading here. Once at the top, you'll be rewarded with one of the best scenic views in the U.S. Virgin Islands. Point Udall is reached along Rte. 82 (it's signposted).\n\nCarl and Marie Lawaetz Museum   MUSEUM This 1750 La Grange valley farmstead was built as a sugar plantation, but was converted to a cattle ranch after it became the home of Danish cattle farmer Carl Lawaetz in 1896. Here he and his wife, Marie, raised Senegal cattle and seven children. Today you can tour the house, filled with turn-of-the-20th-century antiques and family heirlooms. The grounds also hold the ruins of the plantation's old sugar mill, as well as flowers and tropical trees and bushes. Inside, touchingly, Marie's paintings still hang in almost every room.\n\nMahogany Rd., Rte. 76, Estate Little La Grange.   340\/772-0598. www.stcroixlandmarks.com. Admission $10 adults, $8 students and seniors, $5 children 6\u201312, free for children 5 and under. Tues, Sat, and cruise-ship days 10am\u20134pm.\n\nCruzan Rum Factory   FACTORY This vintage factory distills the famous Virgin Islands rum that some consider the finest in the world, although the stuff now detours through the Jim Beam bottling factory in the U.S.A. before it reaches your favorite beach bar. The grounds are quietly bucolic, with offices now ensconced in the 18th-century plantation house and a magnificent ficus tree spreading its wings on the lawn. The charmingly low-tech guided tours include a visit through the factory and the chance to peer into giant vats of bubbling molasses\u2014the scent is pure sugar perfume. Call ahead for reservations.\n\nEstate Diamond 3, W. Airport Rd., Rte. 64.   340\/692-2280. www.cruzanrum.com. Admission $5 adults, $1 children 18 and under. Tours given Mon\u2013Fri 9\u20134pm and Sat 10am\u20132pm.\n\nEstate Mount Washington Plantation   MUSEUM St. Croix's best-preserved sugar and cotton plantation was an island workhorse from 1780 to 1820, when sugar was king on St. Croix, then the second-largest sugar producer in the West Indies. The nine-bedroom great house is a private home and closed to the public, but you can go on a self-guided stroll of the 13 acres, where you'll see the stone ruins of the plantation's sugar factory, as well as the \"cockpit\" animal mill, where donkeys or horses would be utilized to crush the sugarcane.\n\nAt the very southwestern tip of the island, off Rte. 63, a mile inland from the highway that runs along the Frederiksted coast. Free admission.\n\nEstate Whim Plantation Museum   MUSEUM The beautifully restored great house of the island's oldest sugar plantation was built to last, with 3-foot-thick walls of coral, stone, and molasses. All around, tropical greenery threatens to swallow up the plantation structures\u2014which include a sugar factory and restored windmill\u2014and before the estate became a landmark, it almost did. Inside, the great house has only three rooms, but each is filled with antiques from the era. You can tour the house and 12-acre grounds, and look for special ongoing events held here by the St. Croix Landmarks Society, such as evening concerts and arts and crafts shows. Check the website (www.stcroixlandmarks.com) to see what's happening. It also has a fabulous museum store (p. ) on the premises, selling all manner of original island souvenirs.\n\nCenterline Rd. (2 miles east of Frederiksted).   340\/772-0598. Admission $10 adults, $8 seniors, $5 children. Wed\u2013Sat and cruise-ship days 10am\u20134pm.\n\nSt. George Village Botanical Garden   GARDEN This 16-acre garden of tropical trees, shrubs, vines, and flowers was built around the ruins of a 19th-century sugarcane workers' village. Self-guided walking-tour maps are available at the entrance to the garden's great hall. The gardens feature a lively roster of year-round events. Facilities include restrooms and a gift shop.\n\n127 Estate St., 1 St. George (just north of Centerline Rd.), 4 miles east of Frederiksted.   340\/692-2874. www.sgvbg.org. Admission $8 adults, $6 seniors, $1 children 12 and under; donations welcome. Daily 9am\u20135pm.\n\nShopping\n\nChristiansted is the shopping hub of St. Croix, though it pales in comparison to the turbo-charged shopping in Charlotte Amalie on St. Thomas. But may we venture to say that the goods sold on St. Croix often are more interesting and more original, and reflect the local, artisanal, handmade spirit of the island. Most of the shops are compressed into a half-mile or so; look for hole-in-the-wall boutiques selling handmade goods. Between Company Street and the harbor are many courtyards, antique buildings, arcades, and walkways riddled with shops, many of which are smaller branches of parent stores on St. Thomas. Along the boardwalk is the King's Alley Complex, a pink-sided compound filled with the densest concentration of shops on St. Croix.\n\nIn recent years, Frederiksted has also become a popular shopping destination. Many of its mall shops appeal to cruise-ship passengers arriving at Frederiksted Pier, but you can find a few gems among the cookie-cutter offerings.\n\nKeep in mind that shopping in St. Croix is duty-free, and U.S. citizens enjoy a $1,600 duty-free allowance (even children) per person every 30 days.\n\nSt. Croix Shopping A to Z\n\nArts & Crafts\n\nMany Hands   ART This shop has been selling local artworks for almost 50 years. The collection of local one-of-a-kind paintings is intriguing, as is the pottery and handmade jewelry. 110 Strand St.   340\/773-1990. <http:\/\/manyhands.stcroixtravelusvi.com>.\n\nSt. Croix LEAP   HOUSEWARES If you're on western St. Croix, near Frederiksted, St. Croix LEAP makes an offbeat adventure. Inside this open-air shop are stacks of rare salvaged native wood being fashioned into serving boards, tables, wall hangings, clocks, you name it (the protected mahogany is from trees felled in storms or that needed to be trimmed). The St. Croix Life and Environmental Arts Project is dedicated to manual work, environmental conservation, and self-development. St. Croix LEAP is located 2 miles up Mahogany Road from the beach north of Frederiksted. Look for large mahogany signs and sculptures flanking the driveway. Visitors will need to bear to the right to reach the woodworking area and gift shop. The site is open Monday to Friday 9am to 5pm and Saturday 10am to 5pm. Mahogany Rd., Rte. 76.   340\/772-0421.\n\nGifts\n\nFranklin's on the Waterfront   GIFTS This terrific gift and home-furnishing emporium stands out amid the mall clutter nearby. Franklin's sells a thoughtfully curated selection of beautiful things, including works made by local artisans (we got a charming Moko Jumbie Christmas ornament made by local artist Sandra Michael here), high-end candles, Caribbean bath and body products, and more. 4 Strand St.   340\/643-3830.\n\nRoyal Poinciana   GIFTS This gift shop has the look of an antique apothecary. You'll find such local items as hot sauces, Sunny Caribee spices, island herbal teas, Antillean coffees, and an array of scented soaps, toiletries, lotions, and shampoos. 1111 Strand St.   340\/773-9892.\n\nWhim Museum Store   GIFTS The Whim Museum Store has some intriguing gifts and souvenirs, both imported and local, including table linens, Madras cloth, jewelry, art prints, books, and local Guavaberry liqueur. Monies from the gift-store sales go toward the upkeep of the museum and the grounds (p. ). An associated store in downtown Christiansted, with different inventory, is at 58 Queen St. (  340\/713-8102). 52 Estate Whim Plantation Museum, east of Frederiksted on Centerline Rd.   340\/772-0598.\n\nJewelry\n\nCrucian Gold   JEWELRY This small West Indian cottage holds the gold and silver creations of island-born Brian Bishop. His most popular and distinctive item is the Crucian bracelet with a \"true lovers' knot\" design. Pendants framed in gold or silver encase shards of china dating from the 1600s or 1700s and found along the beaches of St. Croix. The shards were once collected by island kids who called them \"China money\" or \"chiny.\" Strand St.   340\/773-5241. www.cruciangold.com.\n\nib Designs   JEWELRY Local metalsmith Whealan Massicott crafts beautiful Caribbean-inspired jewelry in delicately wrought designs at his shop in downtown Christiansted. Corner of Queen Cross and Company sts.   340\/773-4322. www.islandboydesigns.com.\n\nJoyia   JEWELRY The working studio of local jewelry artist Joyia Jones is filled with her fine handcrafted pieces of gold, silver, copper, and precious stones. 3A Queen Cross St.   340\/713-4569. www.joyiajewelry.com.\n\nSonya Ltd.   JEWELRY Sonya Hough makes sterling silver or gold versions of her original design, the C-clasp bracelet. There's some symbolism to the design: If the \"C\" is turned inward, toward your heart, it means you have a significant others. Those on the hunt wear the \"C\" turned out (and many locals won't leave the house without putting on this piece of jewelry just right). She also sells rings, earrings, and necklaces. 1 Company St.   877\/766-9284 or 340\/773-8924. www.sonyaltd.com.\n\nRoll the Dice\n\nIn 1996, U.S. senators agreed to allow the opening of gambling casinos in the U.S. Virgin Islands. In a bow to the islanders, senators agreed that majority ownership of the casino hotels would be reserved for locals. It hasn't exactly been a casino bonanza since then, as only one casino is spinning its gambling wheels: St. Croix's Divi Carina Bay Casino (Divi Carina Bay Resort;   340\/773-7529; www.divicarina.com). The 10,000-square-foot casino has 20 gaming tables and 300 slot machines. No passport is needed to enter, but you do need some form of ID. It's open Sunday to Thursday 10am to 4am and Friday to Sunday 10am to 6am.\n\nPerfume\n\nViolette Boutique   PERFUME This small department store sells exclusive fragrances and hard-to-find toiletry items. A selection of children's gifts, Montblanc pens, and other brand names are also found here. In the Caravelle Arcade, 38 Strand St.   800\/544-5912 or 340\/773-2148.\n\nSt. Croix After Dark\n\nSt. Croix doesn't have the nightlife of St. Thomas. Find out what's happening while you're on island with \"St. Croix This Week,\" distributed free and available at hotels, restaurants, and the tourist office.\n\nTry to catch a performance of the Quadrille Dancers  , a genuine cultural treat. The Quadrille dances have changed little since plantation days. The women wear long dresses, white gloves, and turbans, and the men wear flamboyant shirts, sashes, and tight black trousers. When you've learned their steps, you're invited to join the dancers on the floor. Ask at your hotel if and where they're performing.\n\nNote: Women (or men, for that matter) entering bars alone at night in Christiansted or Frederiksted should not leave the bar alone and walk the lonely streets to your hotel. Take a taxi back\u2014it's worth the investment.\n\nThe atmospheric waterfront bistro Blue Moon, 7 Strand St., Frederiksted (  340\/772-2222), is also known as a hip local stop for visiting jazz musicians. It's a live-jazz hotspot on Wednesday and Friday nights and at Sunday brunch. 7 Strand St.\n\nThe Terrace Lounge at the Buccaneer Resort (  340\/712-2100), near Christiansted, features live music nightly, from calypso to jazz to steel pan, from 6 or 7pm till 10pm.\n\nThe Beach Side Caf\u00e9 at Sand Castle on the Beach (  340\/772-1205) offers live music every Saturday from 6:30 to 9:30pm.\n\nThe Palms at Pelican Cove (  340\/718-8920) has live music from 6pm to 9pm on Saturday night and a steel-pan sounds during Sunday brunch (10am\u20132pm).\n\nSide Trip to Buck Island\n\nThe crystal-clear waters and white-coral sands of Buck Island Reef National Monument, a satellite of St. Croix, are legendary. Some call it the single-most-important attraction of the Virgin Islands. Only about a half-mile wide and a mile long, Buck Island lies 11\u20442 miles off the northeastern coast of St. Croix. A vibrant barrier reef here shelters a wealth of reef fish, including queen angelfish and smooth trunkfish.\n\nBuck Island's greatest attraction is its underwater snorkeling trails, which ring part of the island and are maintained by the National Park Service. This 850-acre isle features a snorkeling trail through a forest of elkhorn coral. Equipped with a face mask, swim fins, and a snorkel, you'll be treated to some of the most beautiful underwater views in the Caribbean. Plan on spending at least two-thirds of a day at this famous ecological site.\n\nYou can hike the trails that twist around and over the island; circumnavigating the island takes only a couple of hours. Trails meander from several points along the coastline to the sun-flooded summit, affording views over nearby St. Croix. Warning: The island's western edge has groves of poisonous manchineel trees, whose leaves, bark, and fruit cause extreme irritation when they come into contact with human skin. Plus, always bring protection from the sun's merciless rays\u2014including a hat and sun block. The sandy beach has picnic tables and barbecue pits, as well as restrooms and a small changing room. There are no concessions on the island.\n\nA number of reliable tour operators offer regular excursions to Buck Island, providing snorkeling equipment, drinks, and all-you-can-eat island barbecues. We recommend National Park Service\u2013sanctioned Big Beard's Adventure Tours, in Christiansted (  340\/773-4483; <http:\/\/bigbeards.com>), which offers full- and half-day Buck Island excursions. Rum punch and other libations are served, but these are no music-thumping St. Maarten\u2013style booze cruises. The 42-foot catamaran Renegade can accommodate up to 39 passengers. Big Beard's full-day Adventure Tours go from 9:30am to 3:30pm and cost $99 adults, $80 children 6 to 12, and $26 children 5 and under.\n\nBuck Island\n\n6\n\nSt. John\n\nIt may be the smallest of the U.S. Virgin Islands, but St. John is a natural wonder of luminous crescent bays and perfumed forest groves. It's got miles of pristine parkland, fresh-scented hiking trails, and mossy peaks with panoramic views. It's got probably the best beaches per capita of any Virgin Island. It's got a sleepy, secluded feel\u2014if you can't chill out here, well, then better get thee to an apothecary. In fact, St. John is where St. Thomas locals come to \"plug out. They don't call it \"Love City\" for nothing.\n\nSt. John is no mere pretty face, however. It has in the ruins of 18th-century sugar plantations a landscape dotted with its own trail of tears. It's got churches: 18 at last count.\n\nSt. John has no airport and no cruise-ship pier. Nonetheless, it is a favorite day-trip destination from nearby islands and a popular ferry excursion for cruise-ship passengers from St. Thomas. But the day-tripper and cruise-ship crowds that stream over in the morning are generally gone before nightfall.\n\nEssentials\n\nGetting There\n\nBy Plane\n\nA number of major airlines have regularly scheduled nonstop air service from cities all over North America into St. Thomas\u2014the major international gateway to the Virgins. From St. Thomas you'll take a ferry to St. John.\n\nBy Ferry\n\nPublic ferries between St. Thomas and St. John run at regular times all day long. Ferry services link the ferry terminal at Red Hook, on the East End of St. Thomas, with Cruz Bay in St. John.\n\nThe Red Hook ferry dock is approximately 10 to 12 miles from the St. Thomas airport. If you've just landed on St. Thomas, your best bet is to take a taxi from the airport. Depending on the traffic, the cab ride on St. Thomas could take 30 to 45 minutes, at a fare between $20 and $22. Build taxi time into your schedule when planning your itinerary.\n\nFerries to St. John (  340\/776-6282) leave every hour on the hour from Red Hook ($6 per person) and last only 15 to 20 minutes; you can also catch less frequent ferries from the Charlotte Amalie ferry terminal ($12 per person).\n\nCar ferries also run between Red Hook and Cruz Bay, traveling from 7am to 7pm daily every half-hour; book ahead on one of three carriers: Boyson (  340\/776-6294); Love City (  340\/779-4000); or Global Marine (  340\/779-1739). Car-ferry rates run from $42 to $50 round-trip; arrive at least 15 minutes before departure.\n\nSt. John\n\nYou can also get to St. John via private water-taxi service. Contact Dohm's Water Taxi (  340\/775-6501; www.watertaxi-vi.com; St. Thomas to Red Hook, St. John: $25\u2013$50 per person, five-person minimum), a private, full-service (pickup and transfer) inter-island water taxi service in custom-built catamaran powerboats. Dohm's also offers direct water-taxi service between St. John and the Ritz and Marriott Frenchmans Reef on St. Thomas ($30\u2013$50 per person; five-person minimum).\n\nVisitor Information\n\nThe tourist office (  340\/776-6450) is located near the Battery, a 1735 fort that's a short walk from the St. Thomas ferry dock in Cruz Bay. It's open Monday to Friday from 8am to 5pm. A National Park visitor center (  340\/776-6201) is also found at Cruz Bay, offering two floors of information and wall-mounted wildlife displays, plus a video presentation about the culture of the Virgin Islands; it's open daily 8am to 4pm.\n\nYou can pick up a map of the island from the tourist office and also a copy of \"St. Thomas + St. John This Week,\" distributed free throughout the islands.\n\nIsland Layout\n\nMost visitors will arrive on St. John at Cruz Bay, on a ferry from St. Thomas. This charming little village, with its few restaurants and shops, is quite the departure from the bustle of Charlotte Amalie. Cruz Bay is also the first stop on any trip to Virgin Islands National Park, which sprawls through the interior and encompasses almost all the coastline. The park service runs an information center in town. Route 20 leads north out of Cruz Bay, and passes the beaches at Caneel, Hawksnest, Trunk, Cinnamon, and Maho bays.\n\nAt the far north, Route 20 leads to the start of the Annaberg Trail, a historic hike through the ruins of 18th-century sugar plantations. Route 10 cuts through the center of the island. Dozens of foot trails lead off this road, making for easy exploration of the peaks and mountains.\n\nOn the east end of the island is Coral Bay, the island's original settlement. It's a favorite among yachties and home to a smattering of small restaurants and bars. Crumbling ruins of forts and plantations also dot the coastline here. The far east end is undeveloped and pales in comparison to the lush greenery of the park. The south coast is a favorite hideaway for locals, but little known by visitors. The coastline here is sweeping and tranquil, yet rocky in parts and punctuated with a handful of small protected bays.\n\nGetting Around\n\nThe 20-minute ferry ride from St. Thomas will take you to Cruz Bay, the capital of St. John, which seems a century removed from the life you left behind. Cruz Bay is so small that its streets have no names, but it does have a couple of shopping centers, seaside restaurants and cafes, and a small park.\n\nBy Bus\n\nThe local Vitran (  340\/774-0165) service runs buses between Cruz Bay and Coral Bay, along Centerline Road about once an hour, costing $1 for adults and 75\u00a2 for children.\n\nBy Taxi\n\nYou'll have no trouble finding taxis to take you anywhere in St. John. Between midnight and 6am, fares are increased by 50 percent. Taxis meet the ferries as they arrive in Cruz Bay, or you can hail one if you see one. Taxi rates are set by the island's Taxi Association and fares are widely posted, even in taxis; check out the official fares in the free magazine \"This Week\" offered in most businesses. Typical fares from Cruz Bay are $8 to Trunk Bay, $9 to Cinnamon Bay, and $18 to Maho Bay. Waiting charges are $1\/minimum per minute after the first 5 minutes.\n\nMany taxi drivers operate multi-passenger taxi van shuttles or open-air safari taxis (converted truck beds with open-air seating). Taxi vans are equipped to transport approximately 8 to 12 passengers to multiple destinations on the island, while safaris can often fit up to 25 people. It's cheaper to hop on a van or safari than ride a taxi on your own if you're going between your hotel and Cruz Bay, but keep in mind you will be making stops along the way.\n\nIf you don't plan to rent a car, it's easy to find taxi drivers (who also double as tour guides). We highly recommend taxi driver Kenneth Lewis (  340\/776-6865), who will meet you at the ferry terminal in St. John for hotel or villa transfer and is also a wonderful sightseeing guide. Expect to pay about $50 for a single-passenger tour or $25 per person for two or more passengers for 2 hours of sightseeing in a shared car. Kenneth's vehicle can accommodate up to 18 passengers.\n\nBy Car or Jeep\n\nOne of the best ways to see St. John is by a car, in particular a four-wheel-drive vehicle, which you can rent in town (reserve in advance in winter). The steep roadside panoramas are richly tinted with tones of forest green and turquoise and liberally accented with flashes of silver and gold from the strong Caribbean sun. Remember: Drive on the left and follow posted speed limits, which are generally very low.\n\nThere are only two gas stations on St. John, one of which is often closed. The more reliable of the two stations is in the upper regions of Cruz Bay, beside Route 104.\n\nUnless you need to carry luggage, which should probably be locked away in a trunk, you might consider one of the sturdy, open-sided, jeeplike vehicles that offer the best view of the surroundings and are the most fun way to tour St. John. They cost around $76 to $84 a day.\n\nA recommended local car-rental agency on St. John is St. John Car Rental, across from the Catholic church in Cruz Bay (  340\/776-6103; www.stjohncarrental.com), operating on the island since 1974. They have two- and four-door Jeep Wranglers, Dodge Nitros, Nissan Pathfinders, and Dodge Durangos.\n\n  St. John\n\nBanks FirstBank Virgin Islands (  340\/776-6882; www.firstbankvi.com) has full-service bank and ATM in Cruz Bay. Scotiabank (  340\/776-6552; www.scotiabank.com) has a full-service location in Marketplace.\n\nBusiness Hours Stores are generally open Monday to Friday 9am to 5pm, Saturday 9am to 1pm.\n\nDentists The Virgin Islands Dental Association (  340\/775-9110) is a member of the American Dental Association and is also linked with various specialists.\n\nDoctors Call   911 for a medical emergency. Otherwise, go to Myrah Keating Smith Community Health Center, 3B Sussanaberg (  340\/693-8900), which has a helipad for serious emergencies. The nearest hospital is in St. Thomas.\n\nDrugstores Chelsea Drug Store, Marketplace Shopping Center, Route 104, Cruz Bay (  340\/776-4888), is open Monday to Saturday 8:30am to 6:30pm and Sunday 9:30am to 4:30pm.\n\nEmergencies For the police, an ambulance, or in case of fire, call   911.\n\nMaps See \"Visitor Information,\" p. .\n\nNewspapers & Magazines \"What to Do: St. Thomas + St. John,\" the official guidebook of the St. Thomas and St. John Hotel Association, is available at the tourist office (see \"Visitor Information,\" p. ) and at hotels. The Virgin Island Daily News covers the news in all the U.S. Virgin Islands.\n\nPost Office The Cruz Bay Post Office is at Cruz Bay (  340\/779-4227).\n\nSafety There is some crime here, but it's relatively minor compared to St. Thomas. Most crime against tourists consists of muggings or petty theft, but rarely violent attacks. Precautions, of course, are always advised. You are most likely to be the victim of a crime if you leave valuables unguarded on Trunk Bay, as hundreds of people seem to do every year.\n\nTaxes The only local tax is an 10 percent surcharge added to all hotel rates.\n\nTelephone All island phone numbers have seven digits. It is not necessary to use the 340 area code when dialing within St. John. Make long-distance, international, and collect calls as you would on the U.S. mainland by dialing 0 or your long-distance provider.\n\nWhere to Stay\n\nThe number of accommodations on St. John is limited, and that's how most die-hard fans would like to keep it. There are four basic types of choices here: resorts, condominiums and villas, guesthouses, and campgrounds. Prices are often slashed in summer by 30 percent to 60 percent.\n\nChances are that your location will be determined by your choice of resort. However, if you're dependent on public transportation and want to make one or two trips to St. Thomas by ferry, Cruz Bay is the most convenient place to stay. It also offers easy access to shopping, bars, and restaurants if you want to walk.\n\nImportant: Keep in mind that lodgings tack on a government room tax of 10 percent.\n\nResorts\n\nCaneel Bay   As a friend said, \"Caneel Bay is not fancy, it's just perfect.\" Though it's one of the top luxury resorts of the Caribbean, Caneel Bay shuns showy glitz and high-tech toys. If that means no phones or TVs in the rooms, so be it. The happy guests (many of them families) who keep this place booked solid aren't complaining.\n\nMega-millionaire Laurance S. Rockefeller opened this, the Caribbean's first eco-resort, in 1956. It was once the Pieter Duerloo plantation, where white settlers defended themselves against a slave revolt. It's a sprawling, lushly landscaped property, some 170 acres, with seven stunning beaches and low-rise buildings fronting the bays or set back near the tennis complex. Wild donkeys with a laidback, languid demeanor stroll the grounds, as do white-tailed deer. The bays are a snorkeler's dream; on Hawksnest you are almost guaranteed to see turtles, and the clear, placid waters of Honeymoon and Scott Beach feature fish in a rainbow of hues.\n\nThe room decor is thoroughly updated but feels timeless, with dreamy bedding, Indonesian wood furnishings, and vintage-style tropical fabrics. Bathrooms are fashioned with native stone. Touches like beds turned down with beautiful helmet shells left on the pillows make this a constant delight. Caneel Bay's public lobby areas have a more modern feel; here, the big breakfast buffet in the open-air Caneel Bay Beach Terrace is one of the island's best. Up in ruins of a sugar mill, ZoZo's (p. ) is serving exemplary Northern Italian in a splendid space, and on the resort's northern end, the magnificent Turtle Bay Estate House\u2014the site of Rockefeller's private estate\u2014has a lovely 4pm tea on the terrace and offers a new steakhouse menu on limited evenings.\n\n  888\/767-3966 or 340\/776-6111. www.caneelbay.com. 166 units. Winter $539\u2013$1,179 double; off season $459\u2013$999 double. MAP (breakfast and dinner) $90 per person per day extra. Extra person $80 per day. 1 child 5\u201311 $40 per day; children 4 and under stays free in parent's room. Service charge 10%. Private transfer from St. Thomas airport $110 adults, $55 children 5\u201311. Amenities: 4 restaurants; 2 bars; babysitting; children's center; concierge; health center and spa; pool (outdoor); room service; 11 tennis courts; Wi-Fi (free).\n\nConcordia Eco-Resort   This 20-year-old eco-resort is a pioneer in sustainable tourism, drawing electricity from photovoltaic power and collecting (and filtering) water in cisterns. Shower tanks are solar-heated, and eco-tents use composting toilets. But most don't stay here to be virtuous, they do so because the views are wonderful (Concordia's elevated structures cling to 50 acres of cliffside), the staff lovely and the prices reasonable (for St. John). Doubles come in a range of lodging styles, from eco-tents to full-kitchen lofts to resort-style studios. The wood-framed, soft-sided eco-tents (and premium eco-tents) are set on the hillsides and have a treehouse feel, with cooling breezes (no A\/C, of course) and those phenomenal views. Eight new eco-studios sleep four each with a queen bed and a queen futon. Some units come with full kitchens. For information on the on-site Concordia Eco-Tents, see \"Campgrounds,\" below.\n\n  800\/392-9004 in the U.S. and Canada, or 340\/693-5855. www.concordiaeco-resort.com. 42 units. Winter $175\u2013$289 double, $268\u2013$289 quad; off season $126\u2013$232 double, $175\u2013$185 quad. Amenities: Restaurant (seasonal); laundry; pool (outdoor); watersports equipment rentals; Wi-Fi (free).\n\nGallows Point Resort   This well-run resort is a short stroll from Cruz Bay. All rooms are sunny, spacious one-bedroom suites with full kitchens and terraces with ocean or harbor views. And each can accommodate up to 5 people, making this a great choice for families. You can stay on the one-story lower floors or duplex upper floors, but we'd recommend the latter, as the duplex living rooms are bright, airy, and high-ceilinged. All suites are furnished in wicker and soft island hues. Two negatives: Harborside villas can be noisy and the beach nearby is small and rocky (so you likely want to go farther afield for the sands).\n\n3 AAA Gallows Point Rd.   800\/323-7229 or 340\/776-6434. www.gallowspointresort.com. 60 units. Winter $495\u2013$695 suite; off season $265\u2013$495 suite. Amenities: Restaurant; bar; pool (outdoor); sundecks; Wi-Fi (free).\n\nGrande Bay Resort & Residence Club   Set above Cruz Bay and the harbor, these handsome condo suites come in studio and one- and two-bedroom configurations (with one three-bedroom suites) and lockout capability, making this a good choice for groups or families. Of the 54 units, only 19 are hotel rooms (these are studio doubles with kitchenettes with blenders); the rest are timeshares in the rental pool. Each is individually decorated, so you never quite know what the decor will look like, but all are nicely laid out, complete with full kitchens, balconies, and washer-dryers. Kitchens are state of the art, with granite countertops and induction cooktops.\n\nSt. John Hotels\n\n3 AAA Gallows Point Rd.   340\/693-4668. www.grandebayresortusvi.com. 73 units. Winter $646\u2013$800 studio\/1-bedroom, $810\u2013$890 2-bedroom; off season $595 studio\/1-bedroom, $695\u2013$750 2-bedroom, Minimum-stay requirements in high season. Amenities: Gym; pool (heated); Wi-Fi (free).\n\nWestin St. John Resort & Villas   Fronting its own gentle bay and a sprawling 34 acres, this is a big, American-style resort. Which means it's not for everyone: The grounds are so spread out that the mobility-challenged need golf carts to get around, and the air-conditioned rooms are hermetically sealed off from the balmy tradewinds. None of the rooms are on the beach (admittedly, it's not St. John's best beach). But now that the grandest pool   on St. John has reopened after a huge renovation, the resort's beating heart is back in business. The \"hotel\" side of the Westin has actually been slowly shrinking\u2014a resort whose rooms once numbered in the hundreds now has 96 (renovated) hotel rooms. The others have been converted to villas (Westin Vacation Ownership timeshare units, to be specific), some 172 fully equipped units that move in and out of the rental pool. More villas are in the works. The beautifully renovated Bay Vista Villas, up on the hillside, are virtually apartments, with full (state-of-the-art) kitchens, washer\/dryers, dining tables, and patios overlooking the bay. But back to that pool: It's a 3-quarter-acre beaut, with four chic cabanas and towering palm trees. It fronts the bay, which is dotted with sailboats and excursion boats; Cruz Bay Water Sports has a location right here for all your island-hopping needs. The resort has a small deli-grocery (look for Mango the resident cat), but you can also walk to the much better (and less pricey) St. John Market, on Chocolate Hole Road. The resort features excellent kids programs, though most require an extra outlay (boo! Hiss!).\n\nChocolate Hole Rd.   866\/716-8108 in the U.S., or 340\/693-8000. www.westinresortstjohn.com. 367 units. Winter $379\u2013$1,069 double, $489\u2013$1,349 villa; off season $305\u2013$600 double, $415\u2013$889 villa. Resort fee $50 per day. Round-trip shuttle and private ferryboat transfers from St. Thomas airport $120 per person, $80 ages 4\u201312. Amenities: 2 restaurants; 2 bars; deli; children's playground; children's programs; concierge; golf nearby; pool (outdoor); room service; 6 lit tennis courts; extensive watersports equipment\/rentals; Wi-Fi (free).\n\nCondos & Villas\n\nTravelers who want a home away from home are in luck on St. John. There are actually more villa and condo beds available on island than there are hotel beds. What you get is spaciousness and comfort, as well as privacy and freedom, and most come with fully equipped kitchens, dining areas, patio grills, and more. Rentals range from large multiroom resort homes to simply decorated one-bedroom condos.\n\nCaribbean Villas & Resorts (  800\/338-0987 or 340\/776-6152; www.caribbeanvilla.com), the island's biggest real estate agency, is an excellent choice if you're seeking a villa, condo, or private home. Most condos go for between $100 and $295 per night, though private homes are often much more expensive. Villa rentals begin at $1,750 per week. You can also look at such international sources as VRBO.com, Homeaway.com and FlipKey.com, all of which cover St. John.\n\nExpensive\n\nCoconut Coast Villas   Just a 10-minute stroll from Cruz Bay, this family-run boutique villa complex offers renovated studio doubles and two- and three-bedroom condos. Each unit faces the waterfront and is fully equipped with kitchens, private ensuite baths, private balconies, grills, and as a special perk, all you need to go beach-hopping: beach chairs, coolers, and beach towels. You will likely be beach-hopping as Coconut Coast's beach is rocky. But its just a 5-minute walk to nearby Frank Bay, where you can swim and snorkel off the sandy beach. The condos can be on the smallish side, and the kitchens, while fully equipped, are far from updated (let's just say they have a New York apartment coziness).\n\nTurner Bay.   800\/858-7989 or 340\/693-9100. www.coconutcoast.com. 9 units. Winter $289 studio, $389 2-bedroom condo, $559 3-bedroom condo; off season $189\u2013$229 studio, $289\u2013$329 2-bedroom condo, $349\u2013$439 3-bedroom condo. Minimum of 3 days required in high season. Amenities: Internet (free); outdoor pool.\n\nEstate Zootenvaal   With four seaside houses, this longtime family-owned Hurricane Hole property is a wonderful place to unplug and unwind. \"Zoot\" is very private, and you can cook in the full kitchen and swim and snorkel in the sparkling seas around Hurricane Hole. Each cottage has been crisply freshened in seaglass and earth colors, with wood and rattan furnishings and tile floors. Three of the cottages (Sunrise, Mangrove, and Spinnaker) have one bedroom, and Turtle Watch has two bedrooms and two baths, but each one can accommodate four people.\n\nHurricane Hole, near Coral Bay.   340\/776-6321 or 216\/861-5337. www.estatezootenvaal.com. 4 units. Winter $290\u2013$380 1-bedroom unit, $580 2-bedroom unit; off season $190\u2013$260 1-bedroom unit, $380 2-bedroom unit. Extra person $30 per day.\n\nModerate\n\nSerendip Vacation Condos   This well-managed, good-value hideaway offers 10 clean, newly updated and cheerfully decorated condo apartments, each with a fully equipped kitchen and covered veranda. It's not on the beach, but set on a hillside above Cruz Bay, with smashing sunset views. Book here early; Serendip gets a lot of repeat business.\n\n9\u20137 Serendip Rd.   888\/800-6445 in the U.S. and Canada, or 340\/776-6646. www.serendipstjohn.com. 10 apts. Winter $225 studio, $295 1-bedroom apt; off season $135 studio, $170 1-bedroom apt. Extra person $25; $15 children ages 3\u201310; children 2 and under stay free in parent's apt. Amenities: Pool (outdoor); Wi-Fi (free).\n\nInns & Bed & Breakfasts\n\nEstate Lindholm Bed & Breakfast   Overlooking Cruz Bay harbor, this is the island's best B&B. It was part of an 18th-century sugar estate, and is set fetchingly among the Danish ruins. An elegant guesthouse, it boasts 14 guest rooms, each with private covered balconies and many with four-poster beds, Tiffany-style lamps, and big wooden armoires. Some have garden views; others have harbor views. Many have wooden rockers on their private balconies. On the property is Asolare, one of the island's best (p. ) restaurants.\n\n  800\/322-6335 in the U.S., or 340\/776-6121. www.estatelindholm.com. 10 units. Winter $390\u2013$595 double; off season $240\u2013$550. Rates include continental breakfast. Amenities: Restaurant; exercise room; pool (outdoor); Wi-Fi (free).\n\nGarden by the Sea Bed & Breakfast   Overlooking the blue Caribbean, this colorful and quaint three-suite B&B is just a 10-minute walk from Cruz Bay. A classic West Indian cottage, it's done up in green and lavender hues, with a veranda shaded by genip trees and palm fronds. Both the Garden Suite and the Wild Ginger Room have fabulous outdoor tropical garden showers, with shower walls made of native stone. The Terrace View Room is upstairs, with views of the sea and a private veranda. (White-sand Frank Bay is a minute's walk from the B&B.) You can also rent the Rendezvous by the Sea villa  , a charming three-bedroom villa overlooking the Caribbean (winter $4,950\u2013$5,350; summer $2,900\u2013$3,900). The inn only accommodates adults 18 and older.\n\n  340\/779-4731. www.gardenbythesea.com. 3 units. Winter $250\u2013$275 double; off season $160\u2013$200 double. No credit cards. Closed Sept. Amenities: Wi-Fi (free).\n\nThe Inn at Tamarind Court   For casual comfort at economical rates, a warm, friendly welcome, and a convenient Cruz Bay location, you can't beat this 20-room inn. An effort has been made to give these rooms a little pizzazz, with bamboo-style furniture, buttery yellow walls, colorful linens, and tile floors. The Grand Suite and the Apartment can sleep four people comfortably. Standard rooms have twin or queen beds and private bathrooms. The inn has six rooms for single travelers that share two baths.\n\nCruz Bay.   800\/221-1637 in the U.S., or 340\/776-6378. www.tamarindcourt.com. 20 units, 14 with bathroom. Winter $75 single without bathroom, $148 double with bathroom, $240 apt for 4 with bathroom; off season $60\u2013$65 single without bathroom, $110\u2013$120 double with bathroom, $170\u2013$190 apt for 4 with bathroom. Rates include continental breakfast. Amenities: Restaurant; bar; Wifi (free in the courtyard).\n\nCampgrounds\n\nAlong with the listing below, consider the eco-tents solar- and wind-powered tent-cottages at Concordia Eco-Tents (see above under \"Resorts\"). (Note: Sadly, two of the most popular campgrounds on the island, Maho Bay, along with Harmony Studios, closed in 2013.)\n\nCinnamon Bay Campground   This National Park Service property is a dream of a campground, set right on the beach, with lush forests at your back and the serene blue seas of Cinnamon Bay before you. Choose from cottage, platform tent, or bare site. Cottages are more or less a room with two concrete walls and two screen walls. Canvas tents come with a floor and all cooking equipment. Bring your own tents and camping equipment to the primitive bare sites. Lavatories and showers are in separate buildings nearby. In winter, guests can camp for a maximum of 2 weeks; the rest of the year camping is limited to 30 days. The campground is closed to nonguests at 10pm. Onsite, the Cinnamon Bay Watersports Center (  340\/513-6330; www.cinnamonbay.com; see p. ) offers rentals, lessons, you name it.\n\n  340\/776-6330. www.cinnamonbay.com. 126 units, none with bathroom. Winter $126\u2013$163 cottage for 2, $93 tent site, $37 bare site; off season $87\u2013$105 cottage for 2, $67 tent site, $37 bare site. Extra person $20. Closed Sept. Amenities: Restaurant; extensive watersports equipment\/rentals.\n\nWhere to Eat\n\nSt. John has some pretty posh dining, particularly at luxury resorts like Caneel Bay, but it also has casual West Indian restaurants with plenty of local color and flavor. It's expensive to eat out anywhere you go, but you can sample the good spots at lunch at reasonable rates.\n\nSt. John Restaurants\n\nCruz Bay\n\nExpensive\n\nAsolare   ASIAN FUSION A new culinary team is bringing creative\u2014and determinedly locavore\u2014Asian fusion cuisine to the Asolare. Which means that a nice Caribbean ceviche is made with the day's catch and served with a passionfruit vinaigrette. A pan-roasted mahi drawn from local waters comes with curried local pumpkin, and the restaurant's signature shrimp and grits come with sesame-crusted shrimp over stone-ground grits (with goat cheese and star anise), sauteed local peppers, and sour collard greens. It's a beautiful concept, neatly realized, and we look forward to seeing where the muse takes them. The restaurant is set in the Estate Lindholm Bed & Breakfast, perched atop a hill with Cruz Bay spread out below.\n\nIn the Estate Lindholm Bed & Breakfast.   340\/779-4747. www.asolarestjohn.com. Reservations required. Main courses $19\u2013$45. Tues\u2013Sun 6\u20139pm.\n\nFatty Crab   ASIAN\/MALAYSIAN This New York transplant has not been lost in translation in its West Indies adventure. In fact, in no time it's become one of the more popular dining spots on the island. Look for Fatty Crab classics like Fatty Sliders (mini spiced beef and pork burgers) and Green Mango & Papaya Salad, but also look for new dishes with local spins (and a menu that speaks the local lingo, with appetizers listed under \"T'ings,\" and so forth). So you might order a very delicious local sauteed vegetable dish with smokey pork jus and a dash of chili or an island roti, here filled with local vegetables in a sassy green curry sauce. Pork ribs, smoked inhouse, are irresistible. Fatty Crab is so dellish it's upping the ante, we think, for the island's other eateries, and that's all for the good.\n\n18\u201311A Enighad, Cruz Bay.   340\/775-9770. www.fattycrab.com. Reservations recommended. Main courses $30\u2013$41. Wed\u2013Mon 5\u201310pm; Sun brunch 11am\u20133pm.\n\nLa Tapa   INTERNATIONAL\/MEDITERRANEAN A candlelit restaurant, set in a vintage West Indian structure, La Tapa features a smart, inventive menu with a locavore bent. The food is cooked to order by the talented chef\/owner, and is heavy on small plate dishes (as the name implies). We've been particularly impressed with the grilled shrimp in a passionfruit-cilantro aioli, the bacalao cakes with avocado lime cr\u00e8me fra\u00eeche and the braised pork belly with fennel apple slaw\u2014all can be ordered for the table and shared. You also can't go wrong starting with appetizer like gazpacho or organic greens grown in Coral Bay. Big plates\u2014platos grandes\u2014might be a classic paella or linguine tossed with shrimp, lobster, chanterelle mushrooms, and eggplant. Live jazz is on tap on Tuesdays.\n\nCenterline Rd. (across from FirstBank).   340\/693-7755. www.latapastjohn.com. Reservations recommended. Tapas $15\u2013$18; main courses $35\u2013$41; paella $68. Daily 5:30\u201310pm.\n\nRhumb Lines   CARIBBEAN\/PACIFIC RIM Don't expect views at this Asian-Caribbean spot, set in an alfresco courtyard in a shopping complex. They make up for it with twinkling lights, swaying palm trees, and a reliably good Asian fusion menu. Do not resist the shrimp and corn fritters, a match made in culinary heaven. Heady and aromatic, the kaffir-lemongrass bouillabaisse is also primo, a Thai-style version of the French classic. Other faves include spicy Asian fusion entrees like sesame-crusted Szechuan tuna, with a bok choy stir-fry, and chicken glazed in a plum-ginger sauce.\n\ngroceries, markets & more: ST. JOHN PROVISIONING\n\nIf you're doing any cooking on the island, you'll want to know where to stock your pantry and source fresh local food. But even if you don't have a kitchen, you'll want to know where to buy snacks, drinks, and prepared foods The restaurant Blue Water Terrace, on the East End, also has a market and deli selling sandwiches, cheeses, wine, picnic foods and lunch boxes; they also cater (  740\/692-2583). Here are some essential St. Croix resources:\n\nDeli\/Prepared Foods: Sam & Jack's Deli (  340\/714-3354; <http:\/\/sam andjacksdeli.com>) offers gourmet deli sandwiches, grab-and-go dinners, house-roasted meats and breads, pastas and ravioli, even homemade cookies. It's on the third floor of the Marketplace shopping complex in Cruz Bay.\n\nGroceries: Starfish Market (  340\/779-4949) is the island's largest grocery store; it's located on the first floor of the Marketplace in Cruz Bay; it also has a large selection of wine, liquor, and beer. A well-stocked grocery, St. John Market (  340\/779-6001) is located on Chocolate Hole Road near the Westin resort.\n\nFresh Seafood: The Seafood Market at the Fish Trap restaurant (www.thefishtrap.com) in Cruz Bay sells fresh fish, shellfish, prepared foods, and salads; it's open noon to 6pm daily. Buy fresh fish (bluefish, tuna, alewife), Caribbean lobster, and conch at the harbor at Cruz Bay direct from the fishermen early in the morning from 9am to 11am.\n\nFresh Produce\/Fruit: You can find vendors selling local fruits (papaya, bananas) and vegetables (tomatoes, okra) around the island. One vegetable stand is close to the Cruz Bay ferry connections.\n\nMeada's Plaza.   340\/776-0303. www.rhumblinesstjohn.com. Reservations recommended. Main dishes $22\u2013$35. Wed\u2013Mon 5:30\u201310pm; Sun brunch 10am\u20132:30pm.\n\nZoZo's Ristorante   ITALIAN This 2013 marriage of superstars installed the island's best restaurant in one of the island's most beautiful locations: atop the ruins of an 18th-century sugar mill at Caneel Bay resort. It's a thrilling mashup, where colonial Caribbean meets Frank Lloyd Wright, and yet you feel as if you're in a technicolor James Bond film. Got that? And did we mention views? Get here for mind-blowing sunsets or to see lights shimmer across the bay in the blue-black night. The food and service are better than ever. Squid-ink linguine comes with diver scallops, broccoli rabe, and a garlic pomodoro sauce. Pork comes three ways: a center-cut loin with fennel, rosemark, and Tuscan kale stuffing; crispy pork belly; and hot Italian sausage in a white bean ragout. Classics like osso buco, house-aged steaks, and Caribbean lobster tail round out the straightforward, but wonderful, menu.\n\nCaneel Bay, just outside Cruz Bay.   340\/693-9200. www.zozos.net. Reservations recommended. Main courses $38\u2013$46. Daily 5:30\u20139pm.\n\nModerate\n\nThe Fish Trap   SEAFOOD This casual spot is an easy walk from the ferry dock. It's an open-air affair, with tables on a breezy covered patio. Start with the signature seafood chowder, creamy and with a touch of peppery heat. Fish dishes include local catches like mahi or wahoo or farmed fish like tilapia and salmon. The sides are generally island-style, from sweet potato mash to rice and beans to fried plantains. You can even buy fresh seafood and pre-prepared Fish Trap dishes to take home to cook at the restaurant's Seafood Market (noon\u20136pm).\n\nCruz Bay, next to Our Lady of Mount Carmel Church.   340\/693-9994. www.thefishtrap.com. Reservations required for groups of 6 or more. Main courses $19\u2013$59. Tues\u2013Sun 4:30\u20139pm.\n\nMorgan's Mango   NEO-CARIBBEAN This open-air spot has a sociable drink scene, with a big bar and a lengthy list of frozen concoctions, many made with the juice from fresh island fruits. But Morgan's gets a lot of love for its gourmet Caribbean food as well, covering a broad range of island styles. That might include Anegada lobster and crab cakes, a nod to the lobster-rich B.V.I. island of Anegada, or the homemade Caribbean soup special. The St. John yellowfin tuna salad comes with local organic baby greens and local tuna. We also like the Island Fish Pot, a bouillabaisse-style French-Caribbean seafood m\u00e9lange, or Poppas Pickapepper Steak, marinated in peppercorn-infused Pickapeppa sauce (the famous Jamaican condiment).\n\nCruz Bay (across from the National Park dock).   340\/693-8141. www.morgansmango.com. Reservations recommended. Main courses $16\u2013$35. Daily 5:30\u201310pm.\n\nInexpensive\n\nThe Quiet Mon   AMERICAN In a song called \"Be as You Are,\" country singer Kenny Chesney sings about hanging with the locals at the Quiet Mon. Chesney continues to pop up at this atmospheric Irish pub from time to time. Does he come for the food? Hmmm. The menu is, kindly put, limited, with stuff like hot dogs, chili and french fries. But the atmosphere is funky and fun, so it still makes our list.\n\nCruz Bay.   340\/779-4799. <http:\/\/quietmon.com>. Lunch plates $4\u2013$7. No credit cards. Lunch specials Mon\u2013Fri 12:30\u20136pm. Bar daily 10am\u20134am.\n\nCoral Bay\n\nExpensive\n\nSweet Plantains Restaurant & Rhum Bar   CARIBBEAN\/CREOLE Fresh, seasonal, gourmet-global\u2014all of these adjectives can be applied to the cuisine of Sweet Plantains, which offers a wide-spanning taste of the tropics. Chefs here marry West Indian home cooking\u2014saltfish cakes with sweet mango puree, say or honey-jerked chicken drumettes\u2014with international flavors and techniques. It gives this Coral Bay favorite a real sizzle. Wednesday and Thursday nights are Latin Nights, where you might sample arroz con pollo (chicken and rice), carnes Latino (beef stew), or ceviche. Saturday is given over to \"Indo-Caribbean\" curries, and Sundays and Mondays the place goes Gallic by way of the West Indies with such dishes as duck a la Sweet Plantains (sauteed duck breast with a black currant chutney confit) or a Creole ragout. The West Indian cottage is done up in richly saturated colors and is set between the road and the bay.\n\n16118 Little Plantation.   340\/777-4653. www.sweetplantains-stjohn.com. Reservations recommended. Main courses $21\u2013$34. Wed\u2013Mon 5:30\u20139pm.\n\nModerate\n\nMiss Lucy's Beachfront Restaurant   CARIBBEAN This legendary spot is worth the wait\u2014and in high season or for Sunday brunch, you will likely have a wait before you. But the views of Friis Bay are lovely, so settle in and have a drink. You'll be glad you did. Pioneering Miss Lucy was St. John's first female taxi driver before she started this beloved eatery. If you're here for lunch or dinner, traditional conch fritters or callaloo soup are the starters to try followed by excellent grilled mahi. Main dishes come with sides like fried plantains, rice and peas, or fungi, a cornmeal side dish, here paired with okra. Her Sunday brunch, accompanied by a jazz duo, is fab: crab cakes Benedict, pain perdu, conch fritterdict. Every full moon, Lucy hosts Full Moon pig roasts (6\u20139pm; $18 per person), and that's something you won't want to miss. A new bar has 15 waterfront seats right by the sea.\n\nSalt Pond Rd., near Estate Concordia.   340\/693-5244. Reservations recommended. Main courses $13\u2013$30. Tues\u2013Sat 11am\u20134pm (limited menu 4\u20135pm) and 5\u20139pm; Sun 10am\u20132pm.\n\nShipwreck Landing   SEAFOOD\/CONTINENTAL A local hot spot, Shipwreck Landing serves a good but limited menu in a palm-shaded spot with water views off Route 107. Sandwiches are Caribbean inspired, like the grilled mahi served with citrus herb butter on a Kaiser roll. Conch fritters or coconut-crusted golden-fried shrimp to starters to get. If you want something more substantial, get one of the entrees that focus on local fish\u2014curry-nut-crusted grouper (with chopped peanuts) served with mango chutney, say, or a teriyaki grilled mahimahi marinated in Caribbean jerk spice and Asian teriyaki. Live music is on tap on Wednesday evenings and weekend nights, with no cover.\n\n34 Freeman's Ground, Rte. 107.   340\/693-5640. <http:\/\/shipwrecklandingstjohn.com>. Reservations requested. Lunch $9\u2013$17; main courses $16\u2013$26. Daily 11am\u20139pm. Bar daily until 11pm. Closed Sept\u2013Oct.\n\nAround the Island\n\nChateau Bordeaux   CARIBBEAN Dine to breathtaking views from your table at the Chateau Bordeaux, high atop Bordeaux mountain, overlooking Coral Bay and the British Virgin Islands. Under new ownership, this colorful spot located 5 miles east of Cruz Bay now serves casual island cuisine and is open for breakfast and lunch Monday to Friday (dinner is offered 3 nights a week). Start your St. John day with delicious French toast stuffed with mango, amaretto pancakes, or omelets served with yummy home fries that put the home in homemade. Lunch features island favorites like chicken roti, mahi mahi (grilled with pineapple sauce or in a sandwich), and conch with fungi and rice. Caribbean lobster stars at dinner. Stop by after a day of sightseeing for a fresh smoothie or ice-cream cone in the Bordeaux Mountain Shop and Ice Cream and Smoothie Store. It also has a shop selling local gifts, Hibiscus Jazz.\n\nJunction 10, Bordeaux Mountain.   340\/776-6611. www.chateaubordeaux.net. Reservations required for dinner. Main courses lunch $10\u2013$20, dinner $28\u2013$38. Mon\u2013Fri 7am\u20133:30pm (until 2:30 Fri); Tues\u2013Thurs 5:30\u20139:30pm.\n\nVie's Snack Shack   WEST INDIAN No, it's nothing fancy; in fact, it's little more than a hut cobbled out of plywood plunked down on a shady stretch of road. But don't judge by looks alone. This little lunch spot on the island's East End is the domain of one very talented home-style cook\u2014actually, that would be two cooks: St. John native Vie (Violet) Mahabir, for whom the Snack Shack is named, and her daughter. Its many fans think the Snack Shack serves some of the best native food on island, including favorites such as garlic chicken, honey-drizzled johnnycakes, curried stew, conch fritters, and coconut pie. It's across the road from Hansen Bay beach and next to Vie's beachfront Hansen Bay Campground.\n\nE. End Rd., Rte. 10 (13 miles east of Cruz Bay).   340\/693-5033. Main courses $5\u2013$12. No credit cards. Tues\u2013Sat 10am\u20135pm (but call first). Closed Oct.\n\nExploring St. John\n\nThe best way to see St. John quickly, especially if you're on a cruise-ship layover, is to take a 2-hour taxi tour. But if you aren't renting a car, arrange ferry pickup and transfer\u2014and an island tour!\u2014with a local taxi driver. We highly recommend Kenneth Lewis (  340\/776-6865), a former cop from Dominica who is one of the best guides on the island. If he can't accommodate you, he can refer you to someone who can. You can also contact the St. John Taxi Association (  340\/693-7530).\n\nThe ferries dock at Cruz Bay, St. John's main village. It's a jumble of waterfront bars, restaurants, boutiques, farmstands, and pastel-painted cottages. It's a bit sleepy, but relaxing after the fast pace of St. Thomas.\n\nMuch of the island is taken up with the Virgin Islands National Park   (  340\/776-6201; www.nps.gov\/viis), with the lushest concentration of flora and fauna in the U.S. Virgin Islands. The park totals 12,624 acres, including submerged lands and water adjacent to St. John, and has more than 20 miles of hiking trails to explore. From pelicans to sandpipers, from mahogany to bay trees, the park abounds in beauty, dotted with colorful accents from the blooms of tamarind and flamboyant trees. The mongoose also calls it home. Park guides lead nature walks through this park that often take you past ruins of former plantations. See \"Watersports & Outdoor Adventures,\" below, for information on trails and organized park activities.\n\nOther major sights on the island include Trunk Bay (see \"Beaches,\" below), one of the world's most beautiful beaches, and Fort Berg (also called Fortsberg), at Coral Bay, which served as the base for the soldiers who brutally crushed the 1733 slave revolt. Finally, try to make time for the Annaberg Sugar Plantation Ruins on Leinster Bay Road, where the Danes maintained a thriving plantation and sugar mill after 1718. It's located off Northshore Road, east of Trunk Bay. Admission is free. On certain days of the week (dates vary), guided walks of the area are given by park rangers. For information on the Annaberg Historic Trail, see \"Watersports & Outdoor Adventures,\" below.\n\nBeaches\n\nSt. John has so many delicious beaches it's hard to find a favorite. But the picture-perfect shoreline of Trunk Bay   is one of St. John's biggest attractions. Administered by the Virgin Islands National Park Service (PS), this heart-shaped bay has a bathhouse, snorkel-gear rental, a snack bar, lifeguards on duty, and a souvenir shop. When conditions are optimal, the beach is ideal for diving, snorkeling, swimming, and sailing. But erosion can be a problem in spots, and crowds can inundate the beach especially when cruise-ship passengers from St. Thomas arrive en masse. Among the beach's many wonderful aspects is the underwater marked snorkeling trail near the shore, great for beginners: The National Park Underwater Trail (  340\/776-6201) stretches for 650 feet and helps you identify what you see\u2014everything from false coral to colonial anemones. Trunk Bay is the only St. John beach that has an admission fee: It's $4 for adults 17 and over, and the beach is open from 7:15am to 4pm.\n\nSt. John's Mascot\n\nThe mongoose (plural mongooses) was brought to St. John to kill rats. It has practically been adopted as the island mascot\u2014watch for mongooses darting across roads.\n\nCaneel Bay resort, the stomping ground of the rich and famous, has seven beautiful beaches on its 170 acres. All are open to the public, but nonguests can only access two beaches from inside the resort: Caneel Bay Beach   and Honeymoon Beach  . (Otherwise you have to visit by boat.) Caneel Bay Beach is the resort's main beach and easy to reach from the main entrance of the Caneel Bay resort. You'll need to bring your own beach chairs and gear for Caneel Beach. Bring your own chairs and watersports equipment to Honeymoon Beach or take advantage of the package Honeymoon Day Pass offered through Virgin Islands Ecotours (  340\/779-2155; <http:\/\/viecotours.com>). It's $49 per adult ($10 children 12 and under) for chairs, lockers, and all watersports equipment (snorkel gear, paddleboards, kayaks) at the beach's Honeymoon Hut. Honeymoon has a secluded feel, and the waters are calm and clear and ideal for snorkeling and kayaking.\n\nOne of the most beautiful beaches near Caneel Bay, Hawksnest Beach   is administered by the NPS, with changing rooms, 24-hour restrooms, grills, and picnic tables. It's also near Cruz Bay, where the ferry docks, it can be crowded, especially when cruise-ship passengers come over from St. Thomas. Safari buses and taxis from Cruz Bay will take you along Northshore Road.\n\nCinnamon Bay   is one gorgeous strip of white sand, complete with hiking trails, great windsurfing, and laidback wild donkeys (don't feed or pet them). Administered by the NPS, the beach has a campground right on the beach; see \"Where to Stay,\" above, for contact information. It also has a water-sports center, the Cinnamon Bay Watersports Center (  340\/513-6330; www.cinnamonbay.com; see rates below), where you can rent watersports equipment (snorkel gear, kayaks). Snorkeling is popular here; you'll often see big schools of purple triggerfish. This beach is best in the morning and at midday, as afternoons can be windy. A marked nature trail, with signs identifying the flora, loops through a tropical forest on even turf before leading up to Centerline Road.\n\nIn 2014, Maho Bay Beach   and its forested hillsides became the newest addition to the National Park Service collection of St. John beaches through a $2.5 million deal made by the nonprofit conservation group Trust for Public Lands. It's just east of Cinnamon Bay. With calm green waters, it's ideal for standup paddleboarding, and snorkelers are almost always guaranteed to see turtles grazing on the grassy bottom. It's a popular beach with visitors and cruise-ship passengers.\n\nTraveling eastward along St. John's gently curving coastline, you'll come to Francis Bay Beach   and Leinster Bay  , the latter ideal for those seeking the solace of a private sunny retreat. You can swim in the bay's shallow water or snorkel over the spectacular and colorful coral reef, perhaps in the company of an occasional turtle or stingray. A tiny cay in one of the Leinster's inner bays, Waterlemon Cay Beach   has some of the island's best snorkeling, with hawksbill turtles dining on seagrass and big starfish stretching along the sea bottom. You can walk along the Leinster Bay Trail to reach the island\n\nRemote Salt Pond Bay   lies on the beautiful coast in the southeast, adjacent to Coral Bay. The bay is tranquil, but the beach is somewhat rocky. It's a short walk down the hill from a parking lot. The snorkeling is good along the bay's eastern shore, and the bay has some fascinating tidal pools. Facilities are limited but include an outhouse and a few tattered picnic tables.\n\nIf you want to escape the crowds, head for Lameshur Bay Beach  , along the rugged south coast, west of Salt Pond Bay and accessible only via a bumpy dirt road. The sands are beautiful and the snorkeling is excellent. You can also take a 5-minute stroll down the road past the beach to explore the nearby ruins of an old plantation estate that was destroyed in a slave revolt.\n\nWatersports & Outdoor Adventures\n\nSt. John offers some of the best snorkeling, scuba diving, swimming, fishing, hiking, sailing, and underwater photography in the Caribbean. Scuba connossieurs know that the diving here is highly rewarding, with a natural wall just 15 minutes offshore. The island is known for the Virgin Islands National Park, as well as for its coral-sand beaches, winding mountain roads, hidden coves, and trails that lead past old, bush-covered sugar-cane plantations. If you've come for golf, you'll have to travel to sister islands St. Thomas and St. Croix to get your rounds in.\n\nWatersports\n\nday sailing & Boat excursions Island-hopping by boat is one of the top activities in the Virgins. You can take half- and full-day boat trips, including a full-day excursion to the Baths at Virgin Gorda. Cruz Bay Watersports (  340\/776-6234; www.divestjohn.com) offers trips to the British Virgin Islands. Trips to the Baths, in Virgin Gorda, for example, cost $140 per person ($70 children 7 and under), including food and beverages. Note: Be sure to bring your passport for any excursions to the British Virgin Islands.\n\nSail Safaris (  866\/820-6906; www.sailsafaris.net) offers guided and custom sailing and snorkeling trips with a captain, sailing lessons, and sailboat rentals of their fleet of Hobie catamarans. Right on the beach in Cruz Bay, close to the ferry dock, this outfitter cruises in catamarans to remote spots in the Virgin Islands and features a range of destinations not available by charter boat or kayak, including trips to uninhabited islands. On guided tours, passengers can go island-hopping in the B.V.I. Sail Safaris also offers sailing lessons for those with an interest in sailing as a hobby. Half-day tours cost $79 to $90 per person; full-day jaunts, including lunch, go for $120 to $140; a 1-hour sailing lesson is $95 per person.\n\nFishing The boats of Offshore Adventures (  340\/775-0389; www.sportfishingstjohn.com) leave from the dock at the Westin resort and the National Park Dock in Cruz Bay. Count on spending from $600 to $700 per party for a half-day of fishing. Fisherman can use hand-held rods to fish the waters in Virgin Islands National Park. Stop in at the tourist office at the St. Thomas ferry dock for a listing of fishing spots around the island.\n\nKayaking   Arawak Expeditions, based in Cruz Bay (  800\/238-8687 in the U.S., or 340\/693-8312; <http:\/\/arawakexp.com>), provides kayaking gear, healthful meals, and experienced guides for full- and half-day kayak and snorkel outings. Trips cost $110 and $75, respectively. Multiday excursions with camping are also available; call their toll-free number if you'd like to arrange an entire vacation with them. These 5-day trips range in price from $1,250 to $1,450.\n\nWatersports Rentals\n\nThe most complete line of watersports equipment available, including rentals for windsurfing, snorkeling, kayaking, and sailing, is offered at the Cinnamon Bay Watersports Center, on Cinnamon Bay Beach (  340\/513-6330; www.cinnamonbay.com). One- and two-person sit-on-top kayaks rent for $20 to $30 per hour. You can also sail away in a Hobie monohull sailboat for $55 per hour.\n\nScuba Diving   St. John is a diving paradise, with some 25 dive sites within a 15-mile radius. One of the best dive operators on island is PADI five-star Low Key Watersports, Wharfside Village, Cruz Bay (  800\/835-7718 in the U.S., or 340\/693-8999; www.divelowkey.com). All wreck dives offered are two-tank\/two-location dives and cost $100 with rental gear, with night dives going for $95. Snorkel and sail trips are also available; full-day trips on a Seabiscuit power vessel or 40-foot catamaran at $115 to $125, respectively, per person. The center also rents watersports gear, including masks, fins, snorkels, and dive skins, and arranges kayaking and fishing excursions.\n\nCruz Bay Watersports, Cruz Bay (  340\/776-6234; www.divestjohn.com), is a PADI and NAUI five-star diving center. Open-water certifications can be arranged through a dive master, for $450 to $750. Beginner scuba lessons start at $120. Two-tank reef dives with all dive gear cost $100, and wreck dives, night dives, and dive packages are available. In addition, half-day snorkel tours are offered daily for $60.\n\nSnorkeling   St. John has so many great snorkeling sites it's impossible to list them all. The most popular spot for snorkeling is Trunk Bay (see \"Beaches,\" above), for its Underwater Snorkel Trail. Snorkeling gear can be rented from the Cinnamon Bay Watersports Center (see above) for $5, plus a $25 deposit. Other choice snorkeling spots around St. John are Leinster Bay\/Waterlemon Cay   and Haulover Bay  . Usually uncrowded Watermlemon Cay offers some of the best snorkeling in the U.S. Virgins. The water is calm, clear, and filled with brilliantly hued tropical fish. Haulover Bay is often deserted, and the waters are often clearer than in other spots around St. John.\n\nAnother amazing spot to snorkel are the mangrove stands at Hurricane Hole. Here coral grows in abundance on the mangrove roots\u2014attended by huge starfish, sponges (and the hawksbills that eat them), and anemones. Seas are calm and gin-clear, and it's magical. It's best reached by boat. SerenaSea runs snorkeling and sightseeing tours out of Coral Bay to Hurricane Hole (<http:\/\/serenasea.com>) in a 30-foot vintage touring yacht; half-day tours are $60 per perons. A guided kayak and snorkel tour of the Hurricane Hole mangroves is offered by Arawak Expeditions, based in Cruz Bay (  800\/238-8687 in the U.S., or 340\/693-8312; <http:\/\/arawakexp.com>); the full-day trip, including picnic lunch, is $120 per person.\n\nWindsurfing The windsurfing at Cinnamon Bay is some of the best anywhere, for either the beginner or the expert. The Cinnamon Bay Watersports Center (see above) rents high-quality equipment for all levels, even for kids. Boards cost $25 to $65 an hour; a 2-hour introductory lesson costs $80.\n\nOutdoor Adventures\n\nHiking   Carved into St. John's rocky coastline are beautiful crescent-shaped bays and white-sand beaches. But the interior is no less impressive. The variety of wildlife is the envy of naturalists around the world. And there are miles of hiking trails, leading past the ruins of 18th-century Danish plantations to panoramic views. At scattered spots along the trails, you can find mysteriously geometric petroglyphs of unknown age and origin incised into boulders and cliffs. The terrain ranges from arid and dry (in the east) to moist and semitropical (in the northwest). Many of the trails wind through the grounds of sugar plantations, past ruined schoolhouses, rum distilleries, molasses factories, and great houses wrapped in lush, encroaching vines and trees. The island boasts some 800 species of plants, 160 species of birds, and more than 20 trails maintained in fine form by the island's crew of park rangers.\n\nsustainable ST. JOHN: A ROCKEFELLER DREAM\n\nIn 1956 multimillionaire Laurance Rockefeller sailed around the island with friends on his yacht. Rockefeller was so enchanted that he established his own resort here (Caneel Bay) and donated 9,500 acres of rolling green hills and an underwater preserve to the federal government to be set aside as a national park that would be here for future generations to enjoy. Thanks to the efforts of Rockefeller and others, today two-thirds of the island's surface area and the island's shoreline waters (the Virgin Islands Coral Reef National Monument) make up the Virgin Islands National Park (  340\/776-6201; www.nps.gov\/viis). The hundreds of coral gardens that surround St. John (some 12,708 undersea acres) are protected rigorously\u2014any attempt to damage or remove coral is punishable with large and strictly enforced fines.\n\nToday St. John is the most tranquil, unspoiled island in the U.S. Virgin Islands. Sustainable tourism programs and eco-friendly practices keep the island clean and pristine. St. John, more than any other island in the Caribbean, works to ensure the preservation of its natural resources and ecosystems.\n\nAt least 20 clearly marked walking paths originate from Northshore Road (Rte. 20) or from the island's main east-west artery, Centerline Road (Rte. 10). Each is marked at its starting point with a preplanned itinerary; the walks can last anywhere from 10 minutes to 2 hours. Trail maps are available from the National Park Service Cruz Bay Visitor Center (  340\/776-6201, ext. 238; daily 8am\u20134:30pm), right at the ferry docks in Cruz Bay. Be sure to carry a lot of water and wear sunscreen and insect repellent when you hike.\n\nReef bay hike One of the most popular ranger-guided hikes on St. John is the guided 2.5-mile Reef Bay Hike. Many people think it's one of the best hikes in the Caribbean. This 6-hour hike includes a stop at the only known petroglyphs on the island and a tour of sugar-mill ruins, with a park ranger discussing the island's natural and cultural history along the way. The hike is from 9:30am to 3pm on Monday, Tuesday, Thursday, and Friday and costs $30 per person. Reservations are required and can be made by phone at least 2 to 3 weeks in advance   340\/779-8700).\n\nAnnaberg historic trail A short but terrific hike is the Annaberg Historic Trail (identified by the U.S. National Park Service as trail no. 10), about a .5-mile stroll. The Annaberg Historic Trail leads pedestrians around the ruined buildings of the best-preserved plantation on St. John. During the 18th and 19th centuries, the island was a sugarcane salt mine, with some 118 sugar plantations and 18 factories. The Danes brought slaves to the islands to work the plantations, and the smell of boiling molasses permeated the air. About a dozen NPS plaques identify and describe each building within the compound. The site has a sprinkling of onsite interpreters, including Miss Olivia in the small cookhouse, cooking and offering samples of traditional \"dumb\" bread (along with a glass of fresh passionfruit juice) and a gardener offering slices of fresh coconut. The walk takes about 30 minutes. From a terrace near the ruined windmill, a map identifies the British Virgin Islands to the north, including Little Thatch, Tortola, Waterlemon Cay, and Jost Van Dyke. Visiting the ruins is free.\n\nLeinster bay trail If you want to prolong your hiking experience from the Annaberg Historic Trail (above), take the Leinster Bay Trail (trail no. 11), which begins near the point where trail no. 10 ends. It leads past mangrove swamps and coral inlets rich with plant and marine life; markers identify some of the plants and animals.\n\nCinnamon bay trail\/cinnamon bay loop trail Both of these trails begin at the Cinnamon Bay Campground. The half-mile Cinnamon Bay Loop Trail is a marked nature trail, with signs identifying the flora. It's a relatively flat, shady walk through plantation ruins and forest. The 1.1-mile Cinnamon Bay Trail starts out steeply but leads into shaded forest along the rutted cobblestones of a former Danish road, past ruins of abandoned plantations, and eventually leading to Centerline Road.\n\nShopping\n\nCompared to St. Thomas, St. John's shopping isn't much, but what's here is interesting. The boutiques and shops of Cruz Bay are individualized and quite special. Most of the shops are clustered at Mongoose Junction (North Shore Rd., Cruz Bay), in a woodsy area beside the roadway, about a 5-minute walk from the ferry dock. Also in Cruz Bay, Wharfside Village (  340\/693-8210; www.wharfsidevillage.com), just a few steps from the ferry departure point, is a low-rise complex of courtyards, alleys, and seaside cafes with some unique boutiques. The Marketplace of St. John (  340\/776-6455; <http:\/\/stjohnmarketplace.com>) has St. John's biggest grocery store (Starfish Market), a pharmacy (Chelsea Drug Store), a hardware store (St. John Hardware), and a sprinkling of retail shops and cafes.\n\nAwl Made Here   (  340\/777-5757; www.awlmadehere.com), the Coral Bay studio of several enterprising island artisans, is owned by leather maker Tracey Keating, who makes handsome hand-tooled leather bags, cuffs, wallets, journals, and more.\n\nBajo El Sol  , Mongoose Junction (  340\/693-7070; www.bajoelsolgallery.com), is a cooperative gallery begun by a group of St. John artists in 1993. It's a wonderful place with lots of interesting pieces in a range of mediums, from paintings to glass and metal sculpture to jewelry, photography, and wood turning. This is the place to find a genuine St. John collectible.\n\nBamboula  , Mongoose Junction (  340\/693-8699; www.bamboulastjohn.com), is full of color and artisanal flavor, offering art (paintings from Haiti), unique homewares (hand-crafted wood bowls), world music, and African instruments. It also sells clothing, from flirty, sparkly skirts to jaunty straw hats.\n\nBougainvillea  , Mongoose Junction (  340\/693-7190; www.shoppingstjohn.com), sells a good selection of tropic wear and beachwear for men and women, with such brands as Tommy Bahama, Canovas, Gottex, and La Perla, along with shoes, bags, hats, and sarongs.\n\nCoconut Coast Studios, Frank Bay (  800\/887-3798 or 340\/776-6944; www.coconutcoaststudios.com), is the studio of artist Elaine Estern, who paints colorful Caribbean landscapes, seascapes, and beneath-the-sea-scapes. Her seaside studio is a 5-minute walk along the waterfron from Cruz Bay. Every Wednesday evening from November through April the artist hosts a Sunset Cocktail Party from 5:30 to 7pm.\n\nDonald Schnell Studio  , next to the Texaco gas station, Cruz Bay (  340\/776-6420; www.donaldschnell.com), is the working studio and gallery of Mr. Schnell, who fashions beautiful hand-crafted ceramics: planters, tableware, even fountains. The studio also sells the works of other artisans.\n\nCORAL BAY: ST. JOHN'S first settlement\n\nThis was the site of the first plantation on St. John, which was established in 1717 and abandoned long ago. Claimed by the Danes in the 1600s and used to unload Danish ships, the bay still contains a crumbling stone pier. Follow the posted signs to see the remains of Fort Berg, which stationed the soldiers that suppressed the 1733 slave revolt. Today, this charming little village shelters a close-knit community of yachting enthusiasts, artists, and expats. Like folks in St. Thomas who come to St. John to \"plug out,\" so people from Love City head to rustic Coral Bay to relax and chill out. Ringing the bay's perimeter is a handful of restaurants and bars. You can spend a day in Coral Bay hiking the area's beautiful trails (see above) and swimming and snorkeling in Salt Pond Bay (p. ), where the mud is thought to be rejuvenating for the skin. Have lunch or an early dinner at one of the village's quintessentially laidback cafes and beach shacks, whether Miss Lucy's (p. ), Shipwreck Landing (p. ), Skinny Legs (p. ), or the tiny treasure that is Tourist Trap  , 14B John's Folly (  340\/774-0912), a roadside shack that can pack in maybe 15 diners and serves cold beer and hot nachos. Also on the menu are tacos, hot dogs, good sandwiches (blackened grouper, pulled-pork), and it's pretty cheap and all delicious. Be on the lookout for Buster, the colorful resident rooster.\n\nR&I PATTON Goldsmithing  , Mongoose Junction (  340\/776-6548; <http:\/\/pattongold.com>), is the shop of \"designer goldsmiths\" Rudy and Irene Patton, who have been making and selling jewelry here since 1973. The island provides the jewelers with a wellspring of inspiration; their gold and sterling-silver petroglyph pieces are based on a petroglyph carved on a rock in Reef Bay. They also make molds of sea shells and sealife found on the beach to make earrings, pendants, and charms.\n\nSt. John Spice  , Cruz Bay (  877\/693-7046; <http:\/\/stjohnspice.com>), is located right at the ferry dock and up a set of brick stairs. It's the place to come for all things saucy and spicy, with a fantastic and comprehensive selection of locally made Virgin Islands hot sauces\u2014from Blind Betty's to Anna's to Jerome's\u2014as well as barbecue rubs, signature-blend spices, and Caribbean-made. It's a great place to stock up on hot sauce gifts to carry back home with you.\n\nSteinworks   (  340\/776-8355; <http:\/\/sandisteinworks.com>), on Coral Bay, is the studio and shop of Sandi Stein, who makes custom-designed jewelry in gold, silver, gemstones, and seaglass.\n\nSt. John After Dark\n\nBring a good book or two. When it comes to nightlife, St. John is no St. Thomas, and everybody here seems to want to keep it that way. Although you can hear live music most any night of the week at venues around the island, many people are content to have a leisurely dinner and call it a day. But check out the \"Live Music Scene\" section on the See St. John website (www.seestjohn.com) for an updated schedule of live music offerings around the island.\n\nCruz Bay\n\nThe newest hot spot in town is the Taproom, the sunny brewpub of new microbrewery St. John Brewers (  340\/715-7775; www.stjohnbrewers.com), the brainchild of a couple of University of Vermont grads. Located in Mongoose Junction in Cruz Bay, the Taproom already feels rooted, and when you visit you'll sit elbow to elbow with locals and visitors sampling from the Caribbean-pub menu and swigging the brewpub's craft beers and homemade sodas. We very much like their Virgin Islands Summer Ale. It's open Monday to Saturday 11am to midnight and Sunday noon to midnight.\n\nThe Motu Bar (  407\/758-6924; www.motubar.com) is an upscale departure from the corrugated beach shack bar model. It's still pretty casual\u2014St. John is no Miami\u2014but with dark wicker loungers plumped with blue cushions and a roster of designer cocktails, it's definitely got an uptown vibe. It's on the deck of the Low Key Watersports building across from Cruz Bay beach.\n\nThose who prefer their drinks \u00e0 la hole in the wall, should check out the action at Fred's Patio (  340\/776-6363), across from the Lime Inn in Cruz Bay. This little West Indian restaurant has bands on Wednesday and Friday nights that get folks up and dancing. It's open 10am to 5pm, but stays open until Saturday morning.\n\nJust 150 from the ferry dock in Cruz Bay, Woody's Seafood Saloon (  340\/779-4625) has a happening happy hour scene from 3pm to 6pm when drinks are dirt-cheap ($1 beers) and the crowd spills out onto the street.\n\nSt. John's one and only Irish pub, the Quiet Mon (p. ), is adjacent to Woody's and located in a West Indian cottage with walls crammed with photos, memorabilia, and shamrock-themed tchotchkes. Four large-screen TVs are on hand for your sporting-event viewing pleasure.\n\nCoral Bay\n\nSkinny Legs, Emmaus, Coral Bay, beyond the fire station (  340\/779-4982; <http:\/\/skinnylegs.com>) is little more than a laidback island shack made of tin and wood. It's also the place to bring the kids for the island's best burgers  . Skinny Legs has live music on Thursday, Friday and Saturday nights during high season when it stays open until 10pm or later.\n\nAnother laidback spot in Coral Bay,Shipwreck Landing (  340\/693-5640; www.shipwrecklandingstjohn.com; p. ), which has live music on Wednesday evenings and weekend nights, with no cover.\n7\n\nTHE BRITISH VIRGIN ISLANDS\n\nWith its turquoise bays and hidden coves, once havens for pirates, the British Virgin Islands are among the world's loveliest cruising areas. The islands attract sailors and yachties aplenty, but the secluded white-sand beaches and laidback geniality make this an escapist's paradise.\n\nThe British Virgin Islands embrace 60-odd islands, some no more than spits of rock jutting out of the sea. Only four islands are of any significant size: Virgin Gorda, Tortola, Anegada, and Jost Van Dyke. The smaller islands and cays have colorful names, such as Fallen Jerusalem. Norman Island is said to have been the prototype for Robert Louis Stevenson's novel \"Treasure Island.\" On Deadman's Bay, Blackbeard reputedly marooned 15 pirates and a bottle of rum.\n\nThese craggy and remote volcanic islands are just 15 minutes by air or 45 minutes by ferry from St. Thomas. Even though they are part of the same archipelago, the British Virgin Islands and the U.S. Virgin Islands have their differences. Where St. Thomas can sometimes seem like Hustle City, deep into mega-resort tourism, it's still a bit sleepy over in the B.V.I. Here the pace is much slower and development is less frenetic. Even the capital, Tortola, seems to exist in a bit of a time capsule.\n\nThen there's the seclusion. Tortola has its share of private retreats, and most of the high-end resorts on Virgin Gorda are so isolated from one another you'll feel your hotel has the island to itself. On the even smaller, more remote islands like Guana Island, Peter Island, and Necker Island, you will have the island to yourself (and your fellow guests). These rustically private hideaways are the ultimate in laidback luxury. Barefoot minimalists on a budget can do rustic, too, without spending a fortune. At modest beachside inns on Jost Van Dyke and Anegada, you'll get all the seclusion you want\u2014but you'll still probably end up knowing all the locals after a week. Forget casinos, splashy entertainment, TVs, and sometimes even air-conditioning: Who needs them when the balmy tradewinds blow, the rum is flowing, and sunlight dances like diamonds on the water?\n\nEssentials\n\nGetting There\n\nBy Plane\n\nYour gateway to the B.V.I. will most likely be Tortola or Virgin Gorda, although currently there are no direct flights from North America or Europe to any of the British Virgin Islands (American Eagle discontinued all service to the island in 2013). Most visitors arrive into the international airports in St. Thomas or San Juan, Puerto Rico, and then fly or ferry to the B.V.I. A handful of regional airlines also offer service between Tortola or Virgin Gorda and islands like St. Croix and St. Maarten\/St. Martin.\n\nThe British Virgin Islands\n\nAir Sunshine (  800\/327-8900 in the U.S. or Canada, or 284\/495-8900 in the B.V.I.; www.airsunshine.com) offers direct flights between San Juan (or St. Thomas) and both Tortola and Virgin Gorda; direct flights between Vieques, Puerto Rico, and both Tortola and Virgin Gorda; and direct flights between St. Croix and Virgin Gorda. Cape Air (  800\/227-3247 in the U.S. and U.S.V.I., or 284\/495-2100 in the B.V.I.; www.capeair.com) flies between San Juan and Tortola or Virgin Gorda and between St. Thomas and Tortola. Seabourne Airlines (  866\/359-8784, or 340\/773-6442 in the U.S.V.I.; www.seaborneairlines.com), which moved its headquarters from St. Croix to San Juan in late 2013, offers regularly scheduled flights between San Juan and both Tortola and Virgin Gorda, as well as service between St. Croix (and St. Thomas) and the B.V.I.\n\nFlying time between Tortola (or Virgin Gorda) and San Juan is 30 to 40 minutes; between Tortola and St. Thomas, 15 minutes; and between Tortola and St. Croix, 45 minutes.\n\nBeef Island, the site of the major airport serving the British Virgins, the Terrence B. Lettsome Airport (EIS), is connected to Tortola by the one-lane Queen Elizabeth Bridge. Supplies and services on the other islands are extremely limited.\n\nBy Ferry\n\nMany B.V.I.\u2013bound visitors who arrive in St. Thomas via the island's international airport then travel on by public ferry. (The more upscale resorts offer direct transfer from the airport by private ferry\u2014for a fee, of course.) Public ferries connect to the B.V.I. via St. Thomas's two ferry terminals (Charlotte Amalie and Red Hook). It's essential to plan your flights around ferry connections. The last ferries from St. Thomas to the British Virgin Islands leave around 4:30 or 5pm. If you're arriving on a late flight into St. Thomas or your flight arrives late it's likely you'll need to overnight on the island. If you're just a little bit late, you can arrange a private water taxi. However\u2014and this is a big however\u2014the B.V.I. Customs on Tortola's West End closes at 6pm, so any water-taxi ride will have to leave by, at the very latest, 5:30pm.\n\nIn addition, it's important to build time into your schedule for the taxi ride from the St. Thomas airport to the closest ferry terminal, Charlotte Amalie\u2014it's about a 15-minute trip, but traffic gridlock can make it longer\u2014and so can stops for other passengers along the way (most airport taxi vans load up with passengers no matter where the destination). If time is tight, consider hiring a private water-taxi service, which includes transfers from the airport in private taxis (see information below); it's pricier than public ferries, but it's also a hassle-free, crowd-free way to make your connections.\n\nTortola has two main ferry terminals, Road Town and West End. Beef Island (Trellis Bay), off Tortola's East End, also gets ferry traffic and is where many of the Virgin Gorda and private island resorts have water launches for connecting guests. Public ferries from the other islands have regular runs to both the Road Town and West End terminals, so plan your itinerary around the terminal (and the departure time) that's most convenient for you. If you're staying on the island's West End, for example, it makes sense to take a ferry to the West End.\n\nYou can also get direct ferry service from St. Thomas to Virgin Gorda, but runs are infrequent (and only operate on certain days of the week)\u2014so you may need to ferry to Tortola first (most likely Road Town) and then catch another ferry to Spanish Town, Virgin Gorda.\n\nFor those traveling on to the North Sound in Virgin Gorda, here's another option: Take a taxi ride from the Road Town ferry to Trellis Island, on Tortola's East End, and hop aboard the North Sound Express (  284\/495-2138), which has daily connections to Spanish Town, Leverick Bay, and the Bitter End Yacht Club on Virgin Gorda; round-trip fares are $40 to $55 adults, $20 to $32 children.\n\nPublic ferries making runs between the U.S. Virgins and the B.V.I. include Native Son (  284\/494-5674; www.nativesonferry.com), Smith's Ferry Services (  284\/495-4495; www.smithsferry.com), and Inter-Island Boat Services (  284\/495-4166). (The latter specializes in somewhat roundabout routing\u2014that is, you may find yourself traveling to St. John or Jost Van Dyke before you arrive in Tortola.) One-way and round-trip fares range from $25 to $49.\n\nB.V.I. Public Ferries: Sample Travel Times\n\nSt. Thomas (Charlotte Amalie) to Tortola (Road Town): 50 min.\n\nSt. Thomas (Red Hook) to Tortola (West End): 35 min.\n\nSt. John to Tortola (West End): 20 min.\n\nTortola (Road Town) to Virgin Gorda (Spanish Town): 25 min.\n\nTortola (West End) to Jost Van Dyke: 20 min.\n\nTortola (Road Town) to Anegada: 60 min.\n\nIf you prefer private water-taxi service, contact Dohm's Water Taxi (  340\/775-6501; www.watertaxi-vi.com; St. Thomas to Tortola: $325\u2013$625, five-person minimum; $65 each additional person), a private, full-service (pickup and transfer) inter-island water taxi service that offers travel anywhere in the Virgin Islands in custom-built catamaran powerboats; or Dolphin Water Taxi (  340\/774-2628; www.dolphinshuttle.com), which includes private taxi airport pickup to Red Hook with its private boat transfers to Tortola ($79\u2013$95 per person, four-person minimum). Dolphin also offers water-taxi services throughout the Virgin Islands and day-trip charters to Jost Van Dyke ($55 per person one-way).\n\nVisitor Information\n\nOn island, the British Virgin Islands Tourist Board (  800\/835-8530; www.bvitourism.com) is located in downtown Road Town, Tortola, in the AKARA building, 2nd Floor, De Castro St. (  284\/494-3134). Its New York branch is at 1 West 34th St., Suite 302, New York, NY 10001 (  212\/563-3117). In the United Kingdom, contact the B.V.I. Tourist Board, 15 Upper Grosvenor St., London W1K 7PJ (  207\/355-9585).\n\nGetting Around\n\nBy Ferry\n\nA number of ferry services offer regularly scheduled trips within the British Virgin Islands.\n\ntortola & virgin gorda Both Smith's Ferry (  284\/495-4495; <http:\/\/bviferryservices.com>) and Speedy's (  284\/495-5235; www.bviferries.com) operate daily ferry service between Tortola (both Road Town and Beef Island) and Virgin Gorda (both Spanish Town and Bitter End\/Leverick Bay); round-trip fares are $30 to $65 adults, $28 children 5\u201311. The North Sound Express (  284\/495-2138), near the airport on Beef Island, has daily connections to Spanish Town, Leverick Bay, and the Bitter End Yacht Club on Virgin Gorda; round-trip fares are $40 to $55 adults, $20 to $32 children.\n\nFamous British Virgin Islanders\n\n  Christopher Fleming (1851\u20131935): Born in the East End of Tortola, Fleming spent most of his life at sea and may even have been a smuggler. In 1890, a B.V.I. Customs officer seized a native boat, and in protest, Fleming led a group of armed men to the commissioner's house. Danish soldiers from St. Thomas put down the rebellion, and Fleming was sentenced to 6 months in jail. Today, islanders look upon Fleming as a hero who protested poverty and unfair economic conditions.\n\n  John Coakley Lettsom (1744\u20131815): Born into a Quaker family on Jost Van Dyke, Lettsom was educated in England and completed his medical education in Edinburgh, Scotland. Rising rapidly and brilliantly, he founded the Royal Human Society of England, the Royal Seabathing Hospital at Margate, and the London Medical Society. Regrettably, he is mainly remembered today for this famous but libelous doggerel: \"I John Lettsom . . . Blisters, bleeds, and sweats 'em. If, after that, they please to die . . . I, John Lettsom.\"\n\n  Frederick Augustus Pickering (1835\u20131926): Born in Tortola, Pickering became a civil service worker who, by 1884, had risen to become the first black president of the British Virgin Islands. He held the post until 1887 and was the last man to be known as president, as the job title after his presidency was changed to commissioner.\n\n  John Pickering (1704\u201368): Born into a fervent Quaker family in Anguilla, Pickering moved in the 1720s to Fat Hogs Bay in Tortola. In 1736, he became the leader of a congregation of Quakers, and by 1741 he was named first lieutenant governor of the island. Fearing the Virgin Islands would be drawn into war between Spain and Britain, he resigned his post because of his Quaker beliefs. Apparently, he was an \"enlightened\" plantation owner, as hundreds of slaves, islandwide, mourned his death\u2014or perhaps they feared their new master.\n\ntortola & anegada The Road Town Fast Ferry (  284\/495-2323) operates daily trips on Monday, Wednesday, and Friday between Tortola, Virgin Gorda, and Anegada; round-trip fares are $55 adults, $35 children.\n\ntortola & jost van dyke New Horizons Ferry (  284\/495-9278) makes daily runs between Tortola and Jost Van Dyke; round-trip fares are $25 adults, $15 children (free children 5\u201311).\n\ntortola & norman island A Norman Island ferry (  284\/494-0093) runs from Hannah Bay, Tortola, to Pirate's Bight restaurant (www.piratesbight.com). Call for schedules and fares.\n\ntortola & peter island The Peter Island Ferry (  284\/495-2000) shuttles passengers between Road Town on Tortola and Peter Island at least five times a day; non-guests pay round-trip fares of $20 adults; $10 children.\n\nChristmas Winds\n\nThe year-round weather in the British Virgin Islands is generally temperate, with few variations in temperature and winds usually out of the east. In late November and December, however, cold fronts in North America push cold air down from the north, bringing the so-called \"Christmas winds\" to the islands. While temperatures are minimally affected, Christmas winds can bring 25- to 30-knot blows to the islands for days at a time, making seas choppy and roiling placid bays that are generally calm for snorkeling. Christmas winds generally dissipate by January or early February.\n\nnorth sound, virgin gorda The private resorts around Virgin Gorda's North Sound offer complimentary daily ferry service from Gun Creek, Virgin Gorda, to the resorts, including Bitter End (  284\/494-2746), Biras Creek (  284\/394-3555), and Saba Rock (  284\/495-7711). Call for schedules.\n\nBy Car, Bus, or Taxi\n\nThere are car-rental agencies on Tortola, Virgin Gorda, Anegada, and Jost Van Dyke; numerous taxis also operate on these islands, as well as on some of the smaller ones; taxi rates are set by the government and posted on the BVI tourism website and in the free \"Welcome Guide\" tourist publication distributed everywhere. Bus service is available on Tortola and Virgin Gorda only. See the \"Orientation\" section for each island for further details.\n\nTORTOLA\n\nThere's no better place to launch your own sailing adventure than in the bareboat capital of the world: Tortola, the biggest (19\u00d75km\/12\u00d73 miles) and most populous of the British Virgin Islands. But you don't have to be a sailor to appreciate the quiet, understated beauty of Tortola and its warm, laidback soul. Unwind to the soft caress of tradewinds, the gentle green hills that slope down to sparkling waters, and the secluded white-sand beaches.\n\nOrientation\n\nVisitor Information\n\nThe offices of the British Virgin Islands Tourist Board (  800\/835-8530; www.bvitourism.com) are located in downtown Road Town, Tortola, in the AKARA building, 2nd Floor, De Castro St. (  284\/494-3134). Here you'll find information about hotels, restaurants, tours, and more. Pick up a copy of the \"Welcome Tourist Guide,\" which has a useful map of the island.\n\nIsland Layout\n\nTortola is the largest of the British Virgin Islands. Road Town is the capital, a scattered sprawl of modern buildings wrapped around the harborfront and tucked into the green hillsides. Here and there, particularly along the narrow lanes of old Main Street, you'll see historic landmarks and examples of the island's West Indian-style gingerbread architecture. Main Street has a number of shops and restaurants. Wickham's Cay (sometimes called Wickham's Cay I) and Wickham's Cay II together form a small Inner Harbor in Road Town. This harbor takes in the Moorings complex area with Fort Burt and Prospect Reef standing near the port entrance. The cruise ship pier juts out at the far right of the cay, and shops are within easy walking distance of the cruise ship dock.\n\nTortola\n\nGetting There\n\nBy plane On Tortola's eastern end is Beef Island, the site of the Terrence B. Lettsome International Airport, the main airport for all of the British Virgin Islands. This tiny island is connected to Tortola by the one-lane Queen Elizabeth Bridge. For information on flights into Tortola, see \"Getting Around: By Plane\" under the chapter \"Orientation,\" above.\n\nBoth Hertz (  800\/654-3131 in the U.S., or 284\/495-4405 on Tortola; www.hertz.com) and National (  284\/494-3197; www.nationalcar.com) have locations at the airport.\n\nTaxis, taxi vans, and multi-passenger safari buses meet every arriving flight. The fare from the Beef Island airport to Road Town is $15 for one to three passengers.\n\nBy ferry Many people arrive by air into St. Thomas and ferry over on one of the many public (and private) ferries traveling between the U.S. and British Virgin Islands. See \"Getting Around: By Ferry\" under the chapter \"Orientation,\" above.\n\nBy cruise ship Tortola recorded nearly 400,000 cruise passengers arriving in Road Harbour in 2012. Its pier in Road Town has the capacity to host two large cruise ships at a time; other cruise ships anchor just outside the harbor and tender in.\n\nAs we went to press, cruise ships scheduling stops in Road Harbour included Regent Seven Seas (Windstar); Costa Cruise Lines (Costa Magica); Crystal; Celebrity; Royal Caribbean (Jewel of the Seas); Cunard; Holland America; Silver Seas (Silver Spirit); and P&O Cruises (Arcadia).\n\nGetting Around\n\nThe roads in Tortola are steep and twisting\u2014not for the faint of heart. The island is fairly small, so driving distances aren't long. Plan your Road Town travel around rush hours and cruise-ship embarkations: You can easily become ensnared in Road Town traffic during arrival and departure times (around 8am and 4pm), when scores of taxi drivers pick up or drop off cruise-ship passengers around the pier at Wickhams Cay 1.\n\nBy Taxi We highly recommend Wayne Robinson as both taxi driver and tour guide (  284\/494-4097 or 284\/499-2251 [cell]). For other taxi options in Road Town, dial the BVI Taxi Association at   284\/494-3942; on Beef Island, call   284\/495-1982. Your hotel can also call a taxi for you; there is a taxi stand in Road Town, near the ferry dock. A typical fare from Road Town to Cane Garden Bay is $24; from Road Town to Josiah's Bay on the north coast, it's $25.\n\nBy Bus Scato's Bus & Taxi Services (  284\/494-2365; www.scatosbusntaxi.com) offers airport pickup and dropoff, lodging and ferry transfers, private and group tours, events transportation, and shopping expeditions in modern, air-conditioned buses.\n\nBy rental Car Ask your hotel concierge to recommend a local rental-car agency; many resorts have relationships with rental franchises that will deliver cars right to your hotel. Itgo (  284\/494-2639; www.itgobvi.com) is located at the Mill Mall, Wickham's Cay I, Road Town. Denzil Clyne (  284\/495-4900) offers rentals (Jeeps) on Tortola's West End near the ferry terminal. Avis (  800\/331-1212 in the U.S., or 284\/494-2193 on Tortola; www.avis.com) maintains offices opposite the J.R. Neal Botanical Gardens in Road Town; and Hertz (  800\/654-3131 in the U.S., or 284\/495-4405 on Tortola; www.hertz.com) has locations in Road Town, near the airport, and on the island's West End, near the ferryboat landing dock. National (  284\/494-3197; www.nationalcar.com) has locations on Long Bay, West End, Road Town, and near the airport.\n\nRoad Town\n\nRental companies will usually offer free hotel pickup. All require a valid driver's license and a temporary B.V.I. driver's license, which the car-rental agency will issue to you for $10; it's valid for 1 month. Because of the volume of tourism to Tortola, you should reserve a car in advance, especially in winter.\n\nRemember: Drive on the left! Roads are pretty well paved, but they're often narrow, winding, and poorly lit, and they have few, if any, lines but plenty of speed bumps. Driving at night can be tricky. It's a good idea to take a taxi to that difficult-to-find beach, restaurant, or bar. If you're queasy about maneuvering mountainous corkscrew roads, day or night, take the Sir Francis Drake coastal road, which for the most part is flat and runs the length of the island. The other main road, Ridge Road, follows the mountainous spine of the island; this was the old roadway linking the island's many plantations.\n\n  Tortola\n\nATMs\/Banks The major B.V.I. banks are Scotiabank (www.scotiabank.com); FirstBank (www.firstbankvi.com), First Caribbean (www.cibcfcib.com); and Banco Popular (www.bancopopular.com\/vi). ATMs are less prevalent in the British Virgin Islands than in the U.S. Virgin Islands; all of the aforementioned banks have ATMs in Wickham Cay I, Road Town, Tortola. A First Caribbean ATM is located at Myett's beach bar in Cane Garden Bay, and Nanny Cay Marina has a FirstBank ATM. Most other small islands do not have ATMs, so if you're planning a visit, be sure to visit an ATM on Tortola to cash up first. Each machine charges around $2 to $3 for a transaction fee.\n\nBookstores The best bookstore is the National Educational Services Bookstore, Wickham's Cay I, in Road Town (  284\/494-3921).\n\nBusiness Hours Most offices are open Monday to Friday 9am to 5pm. Government offices are open Monday to Friday 8:30am to 4:30pm. Shops are generally open Monday to Friday 9am to 5pm and Saturday 9am to 1pm.\n\nDentists For dental emergencies, contact Crown Dental (  284\/494-2770), or Premier Dental (  284\/494-8062).\n\nDrugstores Medicure Pharmacy is located in the Hodge Building, near Road Town Roundabout, Road Town (  284\/494-6189).\n\nEmergencies Call   999. If you have a medical emergency, call Peebles Hospital, Porter Road, Road Town, Tortola (  284\/494-3497). Your hotel can also put you in touch with the local medical staff.\n\nHospitals\/Clinics Peebles Hospital, Porter Road, Road Town (  284\/494-3497), has X-ray and laboratory facilities. The B&F Medical Complex, Mill Mall, Wickham's Cay I, Road Town (<http:\/\/bfmedicalcomplex.com>) is a public day clinic that accepts walk-ins. The Eureka Medical Centre (www.eurekamedicalclinic.com), Geneva Place, Road Town, Tortola, is a private-run urgent-care facility with both inhouse doctors and visiting specialists on call; call for an appointment.\n\nInternet Access Internet and free Wi-Fi access is available all over the islands, including Nanny Cay Marina, Tortola, and Village Cay Marina, Road Town, as well as in many bars and restaurants. If you're in transit or your hotel's Internet access is weak or nonexistent, try Serendipity Bookshop & Internet Cafe, Main St., Road Town (  284\/494-5865); the Pub, Waterfront Dr., Road Town (  284\/494-2608); or Trellis Bay Cybercafe, Trellis Bay (  284\/495-2447).\n\nNewspapers \"The BVI Beacon\" (www.bvibeacon.com) is a weekly newspaper published on Thursday. The weekly \"Island Sun\" (www.islandsun.com) is published every Friday. Both are good sources of information on local entertainment. You can find these in most supermarkets and shops.\n\nPolice The main police headquarters is on Waterfront Drive near the ferry dock on Sir Olva Georges Plaza (  284\/494-2945).\n\nPost Office The main post office on Tortola is in Road Town (  284\/468-3701, ext. 5160), and is open Monday to Friday 8:30am to 4:30pm. The beautiful and collectible B.V.I. stamps are sold here here.\n\nSafety The British Virgin Islands in general are quite safe, with a very low crime rate that many attribute to the illegality of owning guns. Minor robberies and muggings occur late at night outside bars in Road Town, especially in poorly lit areas around Wickham's Cay I and along Waterfront Drive. But outside of Road Town, Tortola is a very safe place to be.\n\nTaxes The British Virgin Islands has no sales tax. It charges a departure tax of $15 per person for those leaving by boat or $20 if by airplane. Most hotels add a service charge of around 10 percent; there's also a 7 percent government room tax. Most restaurants tack on an automatic 15 percent service charge.\n\nTelephone All island phone numbers have seven digits. You can call the British Virgins from the United States by just dialing 1, the area code 284, and the number; from the U.K. dial 011-44, then the number. To call the U.S. from the B.V.I., just dial 1 plus the area code and the number; to call the U.K. from the B.V.I., dial 011-44, then the number.\n\nTipping Go to \"Fast Facts,\" in Chapter 8 for tipping guidelines.\n\nToilets You'll find public toilets in restaurants, beach bars, at the ferry terminals, and at the airport.\n\nWhere to Stay\n\nTortola has few hotel-style lodgings. In fact, outside of villas, its accommodation choices are fairly stark. In addition to one or two topnotch boutique resorts, you have a sprinkling of modest marina hotels, informal guesthouses, and basic beach lodgings. None, however, is as big, splashy, and full service as the mega-resorts in the U.S. Virgin Islands\u2014or as high-end luxe as the choices on Virgin Gorda\u2014and many of the island's repeat visitors seem to like that just fine.\n\nMany visitors rent villas for their Tortola stays. Most rent by the week. Best of BVI (  1252-674878 from the U.K. and 011\/44-1252-674878 from the U.S.; www.bestofbvi.com) represents properties listed nowhere else, including a one-bedroom cottage on Little Thatch Island, a privately owned isle minutes from Soper's Hole. Best of BVI also has a hugely informative trip-planning website, with an up-to-date ferry schedule. Jewels of the BVI (  866\/468-6284; www.jewelsofthebvi.com) represents only BVI Islander\u2013owned properties, including villas, condos, and resorts. Villas can range from $150 all the way to $750 per night, based on location, amenities and the number of bedrooms. For more villa options, go to Chapter 8.\n\nRemember that most of Tortola's best beaches are on the northern shore, so guests staying elsewhere (at Road Town, for example) will have to drive or take a taxi to reach them. If you prefer to stay in a hotel or inn, we like the resorts on the island's leisurely West End best; not only do you have convenient access to West End ferry connections but you'll enjoy plenty of peace and serenity, few traffic woes, and easy proximity to the island's best beaches.\n\nNote: All rates given within this chapter are subject to a 10 percent service charge and a 7 percent government tax. Rates are usually discounted significantly in summer. Note that some resorts offer all-inclusive dining plans, such as a \"Modified American Plan,\" which means that the hotel provides meals (and sometimes additional amenities) for an extra charge.\n\nIn Road Town\n\nBusiness folks, yachties, and those who need to be near the center of the Tortola activity (such as there is) can opt for one of the basic hotels and inns in or around Road Town. Keep in mind that the daily arrival of two or more cruise ships into Road Harbour harbor can result in traffic gridlock. If you plan an extended stay on Tortola, we recommend combining a night here with a few other nights on a quieter, beachier part of the island.\n\nExpensive\n\nThe Moorings\/Mariner Inn   For those who love the buzz of a bustling marina, with a lively scrum of breezy bars and boaters ambling hither and thither, the Mariner Inn at the Moorings should fit the bill. Located right on Wickham's Cay II in Road Town, this full-service yachting resort offers not only support facilities and services but also better-than-average shoreside accommodations. It's a good way to sample Road Town life with fellow sailors and yachties. The rooms are nicely outfitted with dark wood furniture and tile floors; all have kitchenettes and balconies that open onto harbor or garden views.\n\nWickham's Cay II, Road Town.   800\/535-7289 in the U.S., or 284\/494-2333. www.bvimarinerinnhotel.com. 38 units. Winter $220\u2013$455 double; off season $160\u2013$420 double. Amenities: Restaurant; 2 bars; pool (outdoor); room service; watersports equipment\/rentals; Wi-Fi (free).\n\nModerate\n\nFort Burt Hotel   This intimate hotel was built in 1960 on the ruins of a 17th-century Dutch fort (it was rebuilt in 1776 by the English but little is left of the old fort but a cannon). The hotel is blanketed in flowering vines, with spectacular vistas from private terraces of the harbor below (best views from rooms on the second floor). Rooms have a cheerful pastel palette, with wicker furnitures and touches of elegance here and there\u2014plumped-up wingback chairs and high beamed ceilings\u2014but the best thing about them are the big windows and sliding-glass doors opening onto the harbor. The hotel has a pool, but honeymoon and condo suites come with their own private pools. Steep stairs may make getting up and down tough for visitors with mobility issues.\n\n  888\/692-0993 or 284\/494-2587. www.fortburt.com. 18 units. Winter $115\u2013$160 double, $200\u2013$385 suite; off season $99\u2013$120 double, $200\u2013$310 suite. Amenities: Restaurant; bar; pool (outdoor); Wi-Fi (free).\n\nMaria's by the Sea   This 38-room inn in central Road Town is a favorite of folks doing business in Tortola; it's close to the ferry and has little balconies that open right onto the harbor. Rooms have been updated and now add a modicum of stylish chic to their comfort (especially the enormous premium rooms in the hotel's new wing, sleekly outfitted in dark wood furniture, quality bedding, and rich buttery hues), but at night you may be too busy inhaling the sea breezes from your balcony and admiring the harbor lights to care. The hotel updates include a business center and fitness room. Sample excellent Caribbean cuisine at Maria's on-site restaurant, open for breakfast, lunch, and dinner on an alfresco patio overlooking Sir Frances Drake Channel.\n\n  284\/494-2595. www.mariasbythesea.com. 38 units. Winter $160\u2013$200 double, $205 1-bedroom suite, $280 2-bedroom suite; off season $130\u2013$170 double, $200 1-bedroom suite, $250 2-bedroom suite. Extra person $40\/a night. Amenities: Restaurant; bar; fitness room; pool (outdoor); Wi-Fi (free).\n\nVillage Cay Hotel   This small hotel overlooks the bustling 106-slip Village Cay Marina, and that's often reason enough for many people sailing through to stay here. Another is price: The one- and two-bedroom suites are nothing fancy but good value, each outfitted with gleaming wood-plank floors and colorful tropical linens; bathrooms are on the small side (some say cramped). Request one of the newly refurbished rooms, especially those with patios overlooking the harbor\u2014that's good value plus. The hotel has a nice big pool as well as a buzzing restaurant, Dockside Bar & Grille, that serves Caribbean and continental classics and jumps to live music every Wednesday and Saturday nights. Dockmasters' Deli serves New York\u2013style sandwiches and other tasty takeout fare, catnip to passing boaters. Just a 5-minute walk away are the shops of Wickham's Cay I and ferry service to other islands.\n\n  284\/494-2771. www.villagecayhotelandmarina.com. 23 units. Winter $125\u2013$350 double; off season $100\u2013$285. Children 11 and under stay free in parent's room. Amenities: 2 restaurants; bar; pool (outdoor); room service; watersports equipment\/rentals; Wi-Fi (free).\n\nCooper Island: eco-boutique Retreat\n\nOn the northwest corner of Cooper Island, a 30-minute boat ride from Tortola, the Cooper Island Beach Club   opens onto Manchioneel Bay in a splendid setting of coconut palms. It's popular with divers\u2014many of the Caribbean's most celebrated dive sites, including shipwreck-rich \"Wreck Alley,\" are easily accessible\u2014and snorkelers, who come to paddle around in the waters at the south end of Manchioneel Bay. It's a favorite of visiting yachties as well, who stop over to wine and dine in the resort's much-lauded beachfront restaurant and bar. (The resort's \"pop-up rum bar\" and artisanal coffee shop were in the works at press time and should be fully operational by the time you read this.) The on-site Sail Caribbean Divers offers snorkeling, kayaks, dinghy rentals, and full scuba services. This barefoot retreat has nine good-looking, smartly updated rooms (each with a private balcony), but it's far from luxurious, with an emphasis on simple, self-sufficient rusticity. To that end, Cooper Island is a local leader in green initiatives. It has no roads, no TVs, no air-conditioning. Seventy percent of the island's energy comes from solar power. A cistern under each room collects fresh rainwater, which is heated by solar power. All toilets are flushed with seawater. The fryer oil used in the kitchen is recyled as bio-diesel fuel. Bar stools and lounge furniture are made from old fishing boats. Oh, and did we mention that the resort is a serious good deal, all year-round? And that full-board multi-stays are even better deals?\n\nCooper Island: www.cooper-island.com;   800\/542-4624 or 284\/495-9084; nightly room only: winter $265 double; off season $210 double; extra person $25.\n\nGetting There: The resort does not operate a regular ferry service, but several (free of charge) boat trips are made each week to and from Hodges Creek Marina, Maya Cove, East End and Road Harbour Marina, Road Town, including the resort's supply boat. Otherwise, you'll have to pay for a private charter to the island.\n\nWest End & North Shore\n\nKeep in mind that the island's most popular beach, Cane Garden Bay, has a few simple lodgings, often a modest (and modestly priced) assemblage of rooms above Cane Garden's popular beach bars\u2014which means the musical entertainment below may rock well into the night. If you want a decent if frills-free place to hang your hat after a night of music and drinking on Cane Garden Bay, consider the following.\n\nThe east end of Cane Garden Bay is the domain of famed local musician Quito Rymer. Above Rhymer's Beach Bar, Rhymer's Beach Hotel (  284\/495-4639; www.canegardenbaybeachhotel.com; $90\u2013$120 double) offers 21 simple accommodations with kitchenettes and private bathrooms.\n\nFarther east, behind Quito's Gazebo, where you can hear the reggae star play 3 nights a week, is Quito's Ole Works Inn (  284\/495-4837; $85\u2013$165 double, $150\u2013$235 suite). The inn was undergoing renovations when we last visited but should be up and humming by the time you read this. Still, don't expect luxe: Rooms are cramped, but some have water views, even balconies.\n\nJust west, the spacious and attractive suites in Myett's Garden Inn   (  284\/495-9649; <http:\/\/myetts.com>; $150\u2013$220 double) are set in the gardens of one of the most happening beachfront spots on the island.\n\nAn easy 3-minute stroll through a coconut grove leads you to Mongoose Apartments (  284\/495-4421; www.mongooseapartments.com; $135\u2013$198 apt. for 2), where large, simply furnished one-bedroom apartments (with full kitchens and balconies) in a two-story building are one of the beach's best deals.\n\nAnother good deal: Cane Garden Bay Cottages (  780\/728-5934; www.canegardenbaycottages.com; $130\u2013$195 double), with stand-alone four cottages in a shady coconut grove.\n\nExpensive\n\nFort Recovery Beachfront Villas   Opening onto a lovely beach facing the Sir Francis Drake Channel and the Caribbean sea, this West End resort is built around what was once a 17th-century Dutch fort. The fort was actually rebuilt by the British in the late 1700s\u2014and this is the structure that stands today. Nevertheless, it's a pretty impressive sight to see\u2014the ruins of a large, squat stone tower from another era\u2014as you noodle around on the beach. Unlike hillside Tortola resorts that offer views from above, this resort is right on a beach, with cottages leading right onto the sand and gentle seas that are perfect for swimming and snorkeling. It's very lush, nestled in a palm grove and about 12km (71\u20442 miles) from Road Town and Choose from one-, two-, and three-bedroom suites or a spacious beach villa with up to eight bedrooms (can accommodate 30); most suites have full kitchens. Rooms are spacious and cheerfully decorated, with wood beamed ceilings and tile floors. The resort provides guests with a personal local cellphone with emergency and staff numbers, taxi contacts, and more. Ask about meal plans and special packages.\n\n  800\/367-8455 or 284\/541-0955. <http:\/\/fortrecoverytortola.com>. 30 units. Winter $310\u2013$360 1-bedroom suite, $360 2-bedroom suite, $750 3-bedroom suite, $1,560\u2013$3,120 villa; off season $210\u2013$250 1-bedroom suite, $750 2-bedroom suite, $560 3-bedroom suite, $1,260\u2013$2,520 villa. Extra person $50 per night; children 11 and under $35 per night. MAP (breakfast and dinner) $45 per day. Amenities: Restaurant; babysitting; bikes; exercise room; pool (outdoor); watersports equipment\/rentals; Wi-Fi (free).\n\nFrenchmans   This is the island's top lodging, a wonderful marriage of full-service resort and secluded villa. Comprised of nine pastel-hued cottages cunningly built into green bluffs overlooking Frenchmans Cay and the Caribbean Sea beyond, Frenchmans is the kind of place where you settle in for a few days to snorkel, swim, or simply do nothing but soak in the views.\n\nAll nine cottages face the blue sweep of Frenchman's Cay, with pelicans whirling and diving and terraces kissed by warm tradewinds. It's a quick drive (or leisurely stroll) to Soper's Hole marina, where you can shop or stock up on groceries in Harbour Market. The one- and two-bedroom cottages are outfitted with everything you need\u2014including full modern kitchens, dining tables, living rooms, and beautiful rain-shower baths. All have big balconies with views of Sir Francis Drake Channel; at night you can watch the lights of vehicles on the coastal road rounding the dark green hills. Frenchmans has a pool and a small, pebbly beach (but excellent snorkeling around the rocks), but the island's storied north-shore beaches are just minutes away. The on-site Clubhouse at Frenchmans (p. ) restaurant is magical, a wood-and-stone treehouse ablaze with warm light and real bonhomie, and its three-course Sunday brunches are legendarily delicious\u2014don't miss the mango pancakes.\n\nFrenchmans Cay.   284\/494-8811. www.frenchmansbvi.com. 9 units. Winter $398\u2013$625 1-bedroom villa, $565\u2013$985 2-bedroom villa; off season $245\u2013$365 1-bedroom villa, $395\u2013$525 2-bedroom villa. Rates include continental breakfast. Ask about provisioning services. Amenities: Restaurant; bar; concierge; pool (outdoor); tennis court (lit); watersports equipment; Wi-Fi (free).\n\nLong Bay Beach Club   A favorite of travelers since the 1960s, this resort has gone boutique, scaling down considerably in late 2013 from 156 units to 42 rooms and changing its name to Long Bay Beach Club. It has one of the nicest beachfront locales on Tortola, a 1\u00bd-mile-long stretch of sandy North Shore beach overlooking beautiful Long Bay. The low-rise complex has three lodging categories (beachfront rooms, beachfront deluxe suites, and beachfront cabanas), but all share that glorious ocean view (not to mention personal BBQ grills). Twelve suites have full kitchens and big two-person tubs; all other rooms have kitchenettes. The handsome beachfront deluxe rooms have big showers with seats and high wood-beamed ceilings. It's not swanky, but every room is pleasantly furnished, with state-of-the-art bedding, and the views from the patios or terraces are hard to beat. The resort has a dive shop on the premises, and a beachfront pool is in the works. And the snorkeling right off the beach is surprisingly rewarding (ask the manager to point you to the best spots).\n\nThe main restaurant is 1748 (p. ), and the main lobby and bar are built within the ruins of an 18th-century sugar mill.\n\nLong Bay.   866\/237-3491 in the U.S. and Canada, or 284\/495-4252. www.longbay.com. 42 units. Winter $385\u2013$430 double, $465 junior suite; off season $255\u2013$300 double, $325 junior suite. Extra person $35. Children 11 and under stay free in parent's room. All-inclusive plans available. Amenities: Restaurant; 2 bars; babysitting; pool (outdoor); 2 tennis courts (lit); limited watersports; Wi-Fi (free).\n\nThe Sugar Mill Hotel   Historic ruins, fine dining, and intimate accommodations make this boutique inn, on the site of a 300-year-old sugar mill, a favorite among travelers. Built on a hillside sloping down the beach and nestled in grounds filled with vibrant tropical foliage and big fruit trees, the Sugar Mill has a natural serenity. It's across the two-lane road from the sea, with not much of a beach to speak of (but spectacular bird action, as big-beaked pelicans plunge into the sea for silver bonito all day long). The contemporary accommodations vary in quality, but all are furnished with topnotch beds and linens; the tight quarters of the pool suites (with one king bed and a sleeper couch) are balanced by nice terraces, big stone outdoor showers, and the evening lullaby of tree frogs. In fact, we turned off the A\/C and slept like babies to the music of bird (and frog) song and hillside breezes. For a truly posh stay, opt for the luxury cottage, the deluxe villa, or the Plantation House suites\u2014the latter have two 1,100-square-foot bedroom suites, a spacious porch, and charming West Indian architectural flourishes, like wood-beamed ceilings and gingerbread trim.\n\nLunch is served down by the beach at Islands, which features standard Caribbean specialties such as jerk ribs and stuffed crab. The acclaimed Sugar Mill Restaurant (p. ) serves dinner in the old rum distillery's boiling house.\n\nApple Bay.   800\/462-8834 in the U.S., or 284\/495-4355. www.sugarmillhotel.com. 23 units. Winter $350\u2013$385 double, $400 triple, $415 quad, $405 cottage, $300\u2013$320 1-bedroom villa, $735 2-bedroom villa; off season $265\u2013$320 double, $325\u2013$350 triple, $350\u2013$370 quad, $320\u2013$355 cottage, $300\u2013$320 1-bedroom villa, $565\u2013$605 2-bedroom villa. Extra person $15\/night. Children 11 and under not accepted in winter but stay free in parent's room Apr to Dec 20. MAP (breakfast and dinner) $100 per person per day extra. Closed Aug\u2013Sept. Amenities: 2 restaurants; 2 bars; babysitting; concierge; pool (outdoor); Wi-Fi (free).\n\nModerate\n\nHeritage Inn   Locals smartly book a room here to avoid driving the spiraling roads home after dinner at Bananakeet (p. ), one of the best restaurants on island and reason enough to visit. It's not the only reason, though. The frills-free furnishings in this little seven-room inn are clean if motel-basic, but step onto your balcony, and a world of panoramic views   is laid out before you. Perched on top of Windy Hill, rooms command vistas out over three shimmering bays, including Carrot, Apple, and Long bays, and a dozen islands and cays, like Jost Van Dyke, Great Tobago, and Green Cay. Each of the (very modest) rooms (two two-bedroom, five one-bedroom) comes with a full kitchen\u2014which, if you're smart, you'll only need for breakfast and lunch; otherwise you'll want to reserve a sunset-viewing spot at Bananakeet. Spring for the two-bedroom units, which have terraces that stretch the length of your suite. With balmy breezes and that hushed sweep of glittering blue sea, you may never want to leave.\n\nWindy Hill.   877\/831-7230 or 284\/494-5842. www.heritageinnbvi.com. 7 units. Winter $200 1-bedroom suite, $315 2-bedroom suite; off season $125 1-bedroom suite, $185 2-bedroom suite. Extra person $25. Amenities: Restaurant; bar; pool (outdoor); Wi-Fi (free).\n\nSebastian's on the Beach   This good-value hotel is a happening spot, with live music in the beach bar and a popular restaurant. It's located on the North Shore at Little Apple Bay, about a 15-minute drive from Road Town. The rooms are housed in three buildings, with only one, a two-story, right on the beach. All are done in a sunny tropical motif, with tile floors, but the most sought-after are the light and airy beachfront rooms, some mere steps from the surf, others with views from second-floor balconies. The rear accommodations (\"tropical yard\" rooms) feel like country cottages, with apple-green wainscotting and beamed ceilings, but lack sea views\u2014but, really, nothing is farther than a short crawl to the beach. You can also stay in one of the seven spacious new Seaside Villas (winter $250\u2013$350; off season $150\u2013$250) all of which have stunning ocean views. Sebastian's Seaside Grille (p. ) overlooks the bay and has a solid menu of island and continental favorites.\n\nLittle Apple Bay.   800\/336-4870 in the U.S., or 284\/495-4212. www.sebastiansbvi.com. 39 units. Winter $135\u2013$285 double; off season $85\u2013$185 double. MAP $50 per person extra ($30 children). Children 12 and under stay free in parent's room. Amenities: Restaurant; bar, Wi-Fi (free in restaurant and courtyard).\n\nEast End\n\nBeef Island Guest House   This little B&B has the lived-in feel of someone's neat little home. It's a convenient and congenial spot to stay if you have an early flight out of Beef Island or are traveling on to Virgin Gorda in the morning. The converted cottage is set right on the sands at Trellis Bay, its doors and trim done in a happy salmon-pink. It offers four clean, private rooms with private bathrooms and a communal living room and sun room. Throughout, filmy curtains blow with the breezes (it has no air-conditioning). Join your neighbors for a drink on the tables and chairs on the beach, and watch sailboats coming in and out of the harbor.\n\nTrellis Bay.   284\/495-2303. www.beefislandguesthouse.com. 4 units. Winter $130 double; off season $100 double. Extra person $25. Rates include continental breakfast. Amenities: Restaurant; bar; limited watersports; Wi-Fi (free on patio).\n\nWhere to Eat\n\nIf you're in the British Virgin Islands around Thanksgiving, take advantage of the island-wide Restaurant Week, where some 40 participating restaurants throughout the B.V.I. feature discounted prix-fixe dinners. It's usually held the last week in November. Go to www.bvitourism.com to learn more.\n\nIn Road Town\n\nExpensive\n\nThe Dove Restaurant & Wine Bar   FRENCH\/ASIAN Set in an old West Indian cottage in Road Town, the Dove has its elegant, ambitious way with haute cuisine, with an eye to seasonality and exceptional ingredients. It's probably the best dining you'll do in Tortola. You might start with tomato and artichoke soup with a dollop of basil cream, or a braised beef rib with a yellow curry glaze. Entrees include pan-roasted duck with a red-wine raspberry ginger glaze, or a pork tenderloin with fried garlic honey glaze and cheddar bacon potato rosti. It's an intimate, romantic place, with a candlelit interior and a deck beneath a mango tree.\n\n67 Main St.   284\/494-0313. <http:\/\/thedovebvi.com>. Reservations recommended. Main courses $23\u2013$33. Daily 6\u201310pm.\n\nInexpensive\n\nCapriccio di Mare   ITALIAN Right on Waterfront Drive across from the ferry dock is this casual cafe, serving spot-on Italian and terrific pizzas in a sea of Caribbean eateries. The pastas are Italian favorites\u2014carbonara, pesto, amatrriciana\u2014but the execution is satisfying. Convenience to the ferry docks doesn't hurt.\n\n196 Waterfront Dr.   284\/494-5369. Main courses $10\u2013$18. Mon\u2013Sat 11:30am\u20139pm.\n\nMidtown Restaurant   CARIBBEAN This ultra-local caf\u00e9 in the heart of Road Town serves homemade island specialties. It's a great breakfast spot as well. Here is where you find classic island dishes: souse, curried chicken, stewed beef ribs, jerk chicken, and chicken roti. Sides are as good as the entrees, including fungi (a cornmeal polenta), plantains, and rice and peas.\n\nMain St.   284\/494-2764. Main courses $5\u2013$14. No credit cards. Daily 7am\u201310pm.\n\nPusser's Pub   CARIBBEAN\/ENGLISH Yes, this is Tortola's original Pusser's, standing proudly on the waterfront across from the ferry dock in Road Town. The storied pub still serves a few British standards like shepherd's pie and fish & chips\u2014but Caribbean influences have seeped in and now the menu is awash in conch, jerk chicken and pork, and grilled local fish. Take home Pusser's Navy mug filled with Pusser's Rum\u2014the classic \"single malt of rum\" favored by the British Royal Navy since the mid-1600s.\n\nWaterfront Dr. and Main St.   284\/494-3897. www.pussers.com. Reservations recommended. Main courses $7.95\u2013$22. Daily 11am\u201310pm.\n\nRoti Palace   CARIBBEAN Set on Road Town's historic old main street, this colorful West Indian cottage is the place to come for state-of-the-art roti. If you don't know roti, it's an East Indian transplant by way of Trinidad: a soft tortilla-like flatbread folded and filled with curry sauce and vegetables, fish, chicken, or meat. Be sure to have yours with a little hot pepper or mango sauce.\n\nOld Main St.   284\/494-4196. Main courses $8\u2013$25. No credit cards. Mon\u2013Sat 10am\u20136pm.\n\nWest End & North Shore\n\nExpensive\n\n1748 Restaurant   CONTINENTAL This beachfront restaurant in the Long Bay Beach Club (p. ) is set on the covered outside deck of an 18th-century sugar mill. It's a breezy spot to sample upscale continental classics like chicken cacciatore while watching the waves ripple onto Long Bay beach. Start with the popular white bean and sage soup and don't pass up the barbecued shrimp\u2014it's killer. The menu also includes a range of pastas and risottos.\n\ngroceries, markets & more: Provisioning Resources on Tortola\n\nTortola is a prime provisioning stop for boaters, but many people who rent houses, villas, condos and hotel rooms with full kitchens need to stock their pantries. Even if you're staying in a resort without self-catering facilities, it's always good to know where to buy snacks, drinks, and prepared foods. If you're in Road Town, consider a visit to Ample Hamper, Villa Cay Marina, Wickham's Cay I, Road Town (  284\/494-2494; www.amplehamper.com), which stocks packaged food, cheeses, and bottled wines.\n\nGourmet Deli: The French Deli  , on Wickham's Cay II (  284\/494-2195) sells high-end French and European gourmet prepared foods (including plats du jour and hearty soups), sandwiches, salads, wine, and pastries.\n\nGroceries: Shop in the store or provision online with RiteWay (  284\/437-1188; www.rtwbvi.com), the B.V.I.'s biggest grocery store chain, with seven locations on the islands (six on Tortola) including its flagship store in Pasea on Wickham's Cay II (with a Cash & Carry next door for bulk purchases). Bobby's Marketplace (  284\/494-2189; www.bobbysmarketplace.com), Wickham's Cay I, Road Town, delivers groceries straight to the ferry docks, marinas, and boat charters. The West End has Harbour Market (  284\/495-4541) at Soper's Hole Marina.\n\nFresh Seafood: Buy fresh fish, Caribbean lobster, and conch at the BVI Fishing Complex on Baughers Bay (  284\/494-3491). Another source for Caribbean lobster is Wayne Robinson (  284\/494-4097 or 284\/499-2251)\u2014he can hook you up with fishermen who go out daily filling orders for local restaurants.\n\nFresh Produce\/Fruit: Look for Movienne Fahie's farmstand set up across from Craft Alive village in Road Town on Fridays. The West End village of Sea Cow Bay has a Saturday vegetable market. Prophy's Farm sells organic eggs, fruit and vegetables and fresh salt from Salt Island (  284\/496-7190). Good Moon Farm (<http:\/\/goodmoonfarm.com>) will deliver fresh organic produce and fruit anywhere in the B.V.I., with free delivery offered in Tortola. A farmer's depot on old Main Street in Road Town sells local cow's milk, sweet potatoes, pumpkin, papaya, and cashew punch, peanut punch and passionfruit punch (  284\/544-6587). Call the BVI Dept. of Agriculture for more contacts (  284\/495-2110).\n\nOrganic Meat: L&C Poultry & Pig Plus, in Road Town (  284\/494-1310), sells pork, mutton, and eggs.\n\nIn the Long Bay Beach Club.   284\/495-4252. www.longbay.com. Reservations recommended. Main courses $15\u2013$27. Daily 7:30\u201310am, noon\u20133pm, and 6:30\u20139pm.\n\nThe Clubhouse at Frenchmans   CONTINENTAL The Clubhouse boasts a beautiful setting, up in the treetops in a round stone-and-wood aerie with wood-beamed ceilings and open-air views of the blue Caribbean. But we'd say the elegant food, overseen by Chef Paul Mason) is equal to the setting, as is the convivial atmosphere\u2014the place feels lit from within. The service is warm as well, with a small staff that cheerfully juggles a range of responsibilities. They might direct you to the grilled shrimp and black bean gateau, served with spicy mango salsa, to start. In a nod to French country cooking, Chef prepares a fine slow-roasted organic chicken Grandm\u00e8re, but then goes native with a gussied-up Caribbean lobster in saffron broth, fennel, and garlic. The island's most popular Sunday brunch is served here from 11am to 2pm ($27 three-course prix-fixe), with a menu that may include Belgian waffles with strawberry compote, a Boston beer bagel with house-made smoked salmon spread, and a Caribbean broiled steak marinated with Bajan herbs and served with Bernaise sauce.\n\nFrenchmans Cay Resort.   284\/494-8811. Reservations recommended. Main courses $15\u2013$27. Daily 5:30\u20139:30pm.\n\nSugar Mill Restaurant   CONTINENTAL\/CARIBBEAN Set in the ruins of an 18th-century sugarcane plantation, the Sugar Mill serves food that is immaculately and expertly prepared. No, it's not the cutting-edge California-fresh food that years ago set off fireworks at the Sugar Mill, won awards and spun off cookbooks\u2014that, in effect, brought the world to its door. Maybe the world caught up with the Sugar Mill playbook. At any rate, you can expect a quiet, gracious dining experience and often great food. And that setting! It's hard to believe that this romantic candlelit space was once the old rum distillery's superheated boiling house\u2014a place of misery in a tropical climate\u2014and that the mottled stone walls are made of cobblestone and rock slab wrenched from the streets of Liverpool nearly 400 years ago to use as ballast in trading ships on their way to the Americas. The walls were likely fashioned by slaves, who strengthened the cornerstones with brain and star coral drawn from the seas 200 yards away.\n\nIn the Sugar Mill Hotel (p. ).   284\/495-4355. www.sugarmillhotel.com\/restaurant. Reservations required. Main courses $26\u2013$40. Daily 7\u20139pm. Closed Aug\u2013Sept. From Road Town, drive west 11km (63\u20444 miles), turn right (north) over Zion Hill, and turn right at the T-junction opposite Sebastian's on the Beach; Sugar Mill is .8km (1\u20442 mile) down the road.\n\nModerate\n\nBananakeet   CARIBBEAN FUSION With some of the best food on the island, a truly warm and friendly staff, and views to kill for, this is a winner high above the North Shore on Windy Hill. The not-so-good news: It's located in the crook of a steep corkscrew road and a bit of a challenge to reach. The good news: You can call a taxi. Just don't miss a meal here\u2014or a sunset on the deck either, which is where you'll find the locals at quitting time. The specialty is seafood (the coconut shrimp is divine), but the menu also has a variety of other options, savory meat entrees like spicy grilled jerk pork loin with banana chutney, or a grilled steak topped wth caramelized onions, mushrooms, and Gorgonzola cheese. Live music is played on Sunday, Wednesday, and Friday.\n\nHeritage Inn, Windy Hill (p. ).   284\/494-5842. Reservations required. Main courses $15\u2013$35. Daily 11:30am\u2013midnight.\n\nNorth Shore Shell Museum Bar & Restaurant   CARIBBEAN Above the strange and wonderful North Shore Shell Museum, this laidback spot catches the breezes above Carrot Bay through big windows. This is as local as it gets, which means a homecooked vibe, overseen by Egberth Donovan, proprietor (and folk artist) of the museum on the first floor. Seafood is the highlight here, lobster and cracked conch are the house specialties. Call to reserve a table.\n\nLittle Carrot Bay.   284\/495-4714. Reservations required. Main courses $18\u2013$35. Daily 11:30am\u2013midnight.\n\nSebastian's Seaside Grille   CARIBBEAN\/INTERNATIONAL With wooden tables scattered indoors and out, you dine just feet from the waves. Which makes this casual spot a popular place to eat, though to be fair, it does island style food very well. Try the Anegada conch stew as a main course, served with Caribbean-style rice and peas. Other deftly done mains include shrimp Creole, ginger chicken, and jerk chicken, but if you're tired of island cuisine, know you can also get a solid grilled filet mignon here. The after-dinner treat: a small sample of Sebastian's rum.\n\nSebastian's on the Beach (p. ).   284\/495-4212. <http:\/\/sebastiansbvi.com>. Reservations required for dinner. Main courses $20\u2013$38. Daily 8\u201311:30am, noon\u20132:30pm, and 6:30\u20139:30pm.\n\nInexpensive\n\nIf you're looking for a good cup of espresso, head to Soper's Hole Marina to the D'Best Cup   (  284\/494-1096), where you can also get a range of other gourmet coffee drinks, sandwiches, and gelato.\n\nCruzin'   PIZZA\/CARIBBEAN For those looking for a satisfying pizza escape from all that conch and fish, this casual alfresco spot on Carrot Bay should do the trick. The specialty of the house is grilled pizza, and it comes in toppings that sound offbeat but execute nicely: shrimp curry, BLT, and BBQ chicken. You can also order grilled local lobster, rotis, and tacos.\n\nLittle Carrot Bay.   284\/340-3566. Main courses $13\u2013$16; pizza $18\u2013$25. Tues\u2013Sun 11:30am\u201310pm.\n\nQuito's Gazebo   CARIBBEAN\/INTERNATIONAL In an open-air wooden \"gazebo\" built right on the beach, this restaurant is one of two Cane Bay beach bars owned by Quito Rymer, the island's most famous musician. It's a gem hidden in plain sight. Not only does Rymer play here 3 nights a week (solo Tues and Thurs 8pm; Fri 9:30pm with a full band), but the kitchen serves up some of the island's best West Indian food, made to order by a real homestyle cook. The conch fritters are the finest on island. The curry roti, many think, ranks up there with the tops in Road Town. Grilled fish and jerk chicken come with superlative island slaw and fried plantains.\n\nCane Garden Bay.   284\/495-4837. Main courses $18\u2013$32; lunch platters, sandwiches, and salads $6\u2013$14. Tues\u2013Sun 11am\u20133pm and 6:30\u20139pm. Bar Tues\u2013Sun 11am\u2013midnight.\n\nEast End\n\nAt press time, locals were celebrating the reopening of The Last Resort (  284\/495-2520) on Bellamy Cay in Trellis Bay; this fun, popular spot, only reachable by boat, has good food and many fans. Another recommended spot to dine while you're waiting for a boat transfer on Trellis Bay is March\u00e9 at Trellis Bar & Grill (  284\/542-0586), with a surprisingly tasty roster of breakfast and lunch items, including a creamy conch chowder and grilled shrimp salad.\n\nExpensive\n\nBrandywine Estate Restaurant   MEDITERRANEAN BISTRO Set on a cobblestone garden terrace along Tortola's south shore, overlooking Sir Francis Drake Channel, this is easily one of the island's most beautiful restaurants. You can sit inside the elegant dining room, on a pretty patio with sea views, or back in the shady garden\u2014it's all alfresco, cooled by gentle tradewinds and tropical plantings. Although the menu changes seasonally, your meal might include a traditional Spanish paella; perfectly seared scallops served with a bacon and pea sauce; and a mahimahi and salmon \"braid,\" where the two filets are entwined and served with a savory saffron sauce.\n\nBrandywine Estate, Sir Francis Drake Hwy.   284\/495-2301. www.brandywinerestaurant.com. Reservations required. Main courses $26\u2013$38. Wed\u2013Mon noon\u201311pm.\n\nExploring Tortola\n\nBeaches\n\nTortola's wide, sandy beaches are rarely crowded, unless a cruise ship is in port. The best beaches are on the northern coast, especially Cane Garden Bay with its silky stretch of sand and gin-clear waters. Apple Bay is best for surfers, and Long Bay West is a dazzling strip of white sand running for a mile. Reached down a horrible road, riddled with pot holes, Smugglers Cove, with its palm-fringed beach, is worth the trouble to get there. Note: If you take a taxi to the sands, don't forget to arrange a time to be picked up.\n\nTortola's finest beach is Cane Garden Bay  , on the aptly named Cane Garden Bay Road, directly west of Road Town. You'll have to navigate some roller-coaster hills to get there, but these fine white sands, with sheltering palm trees and gentle surf, are among the most popular in the B.V.I. Outfitters here rent Hobie Cats, kayaks, and sailboards, and it's easy to rent beach chairs and umbrellas for $5. The calm seas make this a wonderful spot for standup paddleboarding (you can rent paddleboards here, as well). Beach bars line the sand, including Myett's and Quito's Gazebo. It's a real scene when cruise-ship passengers arrive in big safari vans, bringing vendors and hair braiders out of the woodwork. Still, with sparkling sapphire seas and forested hills and Jost Van Dayke on the horizon, it's quite a fetching place to be.Surfers like Josiah's Bay (for longboard surfing) and Apple Bay (locally known as Cappoons Bay), which is west of Cane Garden Bay along North Shore Road. There isn't much beach at Apple Bay, but that doesn't diminish activity when the surf's up\u2014so watch the locals take to the waves after 5pm. After enjoying the white sands here, you can have a drink at the Bomba Surfside Shack, a classic dive of a beach bar at the water's edge (p. ). Conditions all over the island are best in January and February. Smugglers Cove  , known for its tranquility and for the beauty of its sands, lies at the extreme western end of Tortola, opposite the offshore island of Great Thatch, and just north of St. John. It's a lovely crescent of white sand with calm turquoise waters. A favorite with locals, Smugglers Cove is also popular with snorkelers, who appreciate the fact that the reef is close to shore. The beach is located at the end of bumpy road through a grove of lush coconut palms. You can see the remains of the old set for the TV-movie remake of \"Old Man and the Sea,\" filmed here in 1990, but the long-standing honor bar is gone, replaced by a couple of enterprising fellows selling drinks and snacks out of the back of their cars.\n\nEast of Cane Garden Bay, Brewers Bay  , accessible via the long, steep Brewers Bay Road, has calm, clear waters that are ideal for snorkelers and small children. (When winter swells from the north kick up, it's a good surfing spot.) The only gold-sand beach on island is a great place to stroll or just sip a rum punch from the Bamboo beach bar and watch the world go by. It even has a sprinkling of plantation ruins. A bare-bones campground is tangled in the tropical foliage along the beach\u2014sometimes operational, sometimes not.\n\nThe 2km-long (11\u20444-mile) white-sand beach at Long Bay  , reached along Long Bay Road, is one of the most beautiful in the B.V.I. Joggers run along the water's edge, and spectacular sunsets make this spot perfect for romantic strolls. The Long Bay Beach Resort stands on the northeast side of the beach; many visitors like to book a table at the resort's restaurant overlooking the water.If you'd like to escape from the crowds at Cane Garden Bay and Brewers Bay, head east along Ridge Road until you come to Josiah's Bay Beach on the north coast. This beach lies in the foreground of Buta Mountain. If you visit in winter, beware: On many days there's a strong undertow, and there are no lifeguards.\n\nOn the island's south shore, Brandywine Bay is a pretty sweep of crescent beach right next to the highway and is rarely touristed. It was rebuilt with dredged sand and may see some development in the coming years.\n\nTouring the Island\n\nA taxi tour of the island is a must and a great way to get the lay of the land before you tackle the driving yourself. But the best taxi tours are even more than that, giving you an overview and insight into the island's culture, history, and sociology. Taxi drivers are great resources for insider travel tips as well, like tapping into a network of local fishermen for fresh lobster, or finding the beach that's just right for you. We highly recommend Wayne Robinson (  284\/494-4097 or 284\/499-2251) as a warm, reliable, and erudite guide. Expect to pay around $110 for two persons for a 2-hour tour; extra persons are $20. Wayne's van can hold up to 14 passengers.\n\nA taxi tour of the island might include a visit to Soper's Marina, on the island's West End; a sweep of the beautiful North Shore beaches (including Cane Garden Bay), also on the West End; a drive along historic Main Street in Road Town (and a retail expedition to the Craft Alive Village shops; see below); and a stop at Sage Mountain National Park (see below). Serious shoppers of local art and gifts should definitely include a trip to Aragorn's Local Arts and Crafts Center in Trellis Bay, on the island's East End (see below). If you want to dig deeper into the culture and history of island life, ask your driver to include the following little gems, all fascinating.\n\nCallwood Rum Distillery   DISTILLERY With 400 years of history, this, the last distillery still standing on Tortola (once there were 53!), continues to blend and bottle its own rum in an 18th-century stone structure, the very picture of beautiful decay, all tangled in tropical greenery. The ramshackle interior has a tobacco patina, and you can sample the rums, of which 50 gallons are produced every day. Buy a bottle or two; they're well-priced and make a nice souvenir.\n\nCane Garden Bay. Free admission.\n\nFahie Hill Murals   PUBLIC ART A series of colorful murals done by different local artists follows the narrow, sinuous curves of mountainous Ridge Road. Each mural is a vignette of an old way of island life, from \"banking\" (terracing) crops on the island's steep hills to cutting sugarcane to telling ghost stories by lantern light. Pull over to give the murals a good look (you may be joined by safari buses filled with cruise-ship passengers).\n\nFahie Hill, Ridge Rd., btw. Great Mountain Rd. and Johnson's Ghut Rd. Free admission.\n\nJ.R. Neal Botanical Gardens   GARDEN This little 1.6-hectare (4-acre) park in the middle of Road Town was created by the B.V.I. National Parks Trust and is becoming an increasingly important repository of endangered and threatened island species. It has an orchid house, a small rainforest, and a palm grove. The aptly named flamboyant tree, with its brilliant scarlet flowers, is one of the highlights here.\n\nBotanic Rd.   284\/494-2069. www.bvinpt.org. Free admission; donations taken.\n\nisland-hopper DAY-TRIPPING\n\nIf you'd like to island-hop, seeing as many of the different British Virgins as you can on your trip, your best bet is to base yourself in Tortola and take day-trips from there. A number of public ferries (and airlines) offer regular trips from Tortola to the larger islands. Speedy's (www.bviferries.com) travels between Road Town and Spanish Town. Virgin Gorda. New Horizons Ferry Service (  284\/495-9278) runs the 25-minute trip between Tortola and Jost Van Dyke. Spend the day at the beach and lunch at one of the local eateries, such as Foxy's, before returning to Tortola. Road Town Fast Ferry (www.roadtownfastferry.com) travels from Road Town to Anegada on Monday, Wednesday, and Friday. Once there, of course, you can have a taxi driver deliver you to the beach.\n\nBut many other hidden islets and tiny islands are only accessible by charter boat or yacht. We highly recommend taking a customized sailing, snorkeling, or island-hopping trip while you're on island. The following day-sail operators come highly recommended:\n\n  Aristocat Day Sails (  284\/499-1249; <http:\/\/aristocatcharters.com>): Hop aboard luxury 48-foot catamarans for intimate swimming and snorkeling island excursions. The Aristocat runs full-day swimming, snorkeling, and lunch trips out of Soper's Hole marina to Norman Island and Jost Van Dyke. The Lionheart has full-day excursions from Road Town's Village Cay to Peter, Salt, and Cooper islands as well as Virgin Gorda's the Baths. Cost is $120 per person, half-price for children, lunch included.\n\n  Kuralu (  284\/495-4381; www.kuralu.com): Swim, snorkel, and sail on uncrowded, all-inclusive day sails on this 50-foot cruising catamaran out of Soper's Hole. Kuralu goes to places like Norman Island and the Indians, Jost Van Dyke, and Green Cay and Sandy Spit. Full-day sails cost $110 per person, half-price for children, lunch included.\n\n  White Squall II (  284\/541-2222; www.whitesquall2.com): Sail in a circa-1936 day-sail schooner on full-day excursions of magical spots like Norman Island and the Caves, the Baths, and Cooper Island. Cost is $110 per person; the boat leaves from Road Town.\n\nNorth Shore Shell Museum   MUSEUM There may be no one manning the \"museum\" when you visit, but you're welcome to explore on your own. The folk artist\/poet\/proprietor\/chef Egberth Donovan has created this splendidly cluttered, open-air assemblage of shells. Shells comprise floors and walls and have been made into shell boats and shell mobiles. Lots of wise words scribbled on boards and surfaces are scattered about as well. The museum even has a restaurant on the second floor, serving local lobster and regional dishes to those who call and reserve a table.\n\nNorth Coast Rd. Free admission; donations taken.\n\n1780 Lower Estate SugarWorks Museum   MUSEUM Built by 18th-century slaves, this original 1780 sugar works structure was once part of a thriving harborside sugar plantation. Inside is a treasure trove of island artifacts, including old muskets, coal irons, bedding stuffed with banana leaves, woven baskets; a maritime display; a native folk medicine exhibit (slate bush is good for the kidneys, and angelica works for cataracts and colds); and a rotating art gallery.\n\nStation Ave.   284\/494-9206. Free admission.\n\nWatersports & Outdoor Adventures\n\nMost visitors come to Tortola not for historic sights but to explore the island's natural scenery, with its rugged mountain peaks, lush foliage, and wide, sandy beaches.\n\nBOAT CHARTERING\/BAREBOATING Tortola boasts the largest fleet of bareboat sailing charters in the world. One of the best places to get outfitted is the Moorings, Wickham's Cay (  888\/979-0153 in the U.S. and Canada, or 284\/494-2331 in the B.V.I.; www.moorings.com). Depending on your nautical knowledge and skills, you can arrange a bareboat rental (with no crew) or a fully crewed rental with a skipper, staff, and a cook. Boats come equipped with a portable barbecue, snorkeling gear, dinghy, linens, and galley equipment. If you're going out on your own, expect to get a thorough briefing session on Virgin Island waters and anchorages. The cost for bareboat rentals varies depending on the season and the boat. Powerboats, daysailers, and catamarans can range from $385 to $1,700 per day.\n\nHIKING No visit to Tortola is complete without a trip to Sage Mountain National Park  , rising to an elevation of 523m (1,716 ft.). Here, you'll find traces of a primeval rainforest, and you can enjoy a picnic while overlooking neighboring islets and cays. Covering 37 hectares (91 acres), the park protects the remnants of Tortola's original forests (those that were not burned or cleared during the island's plantation era). Go west from Road Town to reach the mountain. Before you head out, stop by the tourist office in Road Town and pick up the brochure \"Sage Mountain National Park.\" It has a location map, directions to the forest and parking, and an outline of the main trails through the park. From the parking lot at the park, a trail leads to the park entrance. The two main trails are the Rainforest Trail and the Mahogany Forest Trail.\n\nScuba diving The region has a number of excellent dive operators, most of which also offer snorkeling excursions. Among them are Sail Caribbean Divers (  284\/495-1675; www.sailcaribbeandivers.com), with seven locations in the B.V.I. including three full-service scuba centers on Tortola alone; and Blue Water Divers, Road Town (  284\/494-2847; www.bluewaterdiversbvi.com), with dive-shop locations at Soper's Hole and Nanny Cay. A resort course costs around $105; a PADI open-water certification is $410.\n\nSnorkeling Many who stay on Tortola do snorkeling trips to neighboring islands, but you don't have to leave the island to enjoy a fine snorkel. Many think Smugglers Cove (p. ) is swell. Another spot is Long Bay, with the best spots near the west end of the beach. Brewer's Bay has good spots along the reef. Frenchmans Cay has surprisingly good snorkeling around the rocks near Frenchmans resort. Another great place to snorkel is just offshore at Norman Island (see below) and its neighboring islands the Indians, four fingers of rock jutting out of sea at depths of between 10 and 50 feet. The Indians have no beach, just moorings to anchor, but if you break bread in the water, reef fish (like yellowtail snappers) will come up to meet you. Go to \"Island-Hopper Day-Tripping \" (p. ) for a list of operators that go to Norman Isle and the Indians.\n\nSurfing Apple Bay is the island's top surfing spot. You can rent surfboards from HIHO (  284\/494-0337; www.go-hiho.com) in advance and pick them up at HIHO's Road Town store or have them delivered to you at your lodging (for a fee). At Soper's Hole marina, Tortola Island Surf and Sail (  284\/494-0123; www.bviwatertoys.com) rents shortboards, longboards, and standup paddleboards, among other water-sports equipment.\n\nB.V.I. FAVORITE scuba SITES\n\nThe British Virgin Islands have a wealth of world-class dive sites, including the area between Salt and Cooper islands known as \"Wreck Alley,\" home to four shipwrecks. The following sites are just a sampling. Go to \"Scuba Diving,\" above, for recommended dive operators.\n\n  The wreck of the RMS Rhone   sank in 1867 near the western point of Salt Island. \"Skin Diver\" magazine called it \"the world's most fantastic shipwreck dive.\" The wreck was featured in the 1977 movie \"The Deep.\"\n\n  Chikuzen is another excellent dive site off Tortola. The 266-foot steel-hulled refrigerator ship sank off the island's eastern end in 1981 and is now home to yellowtail, barracuda, black-tip sharks, octopus, and drum fish.\n\n The newest wreck in Wreck Alley, the interisland cargo ship Island Seal went down in 2006 on a reef near Brandywine Bay near Salt Island. It was moved to Wreck Alley in 2013.\n\n South of Ginger Island, Alice in Wonderland is a deep-dive site with a wall that begins at around 3.6m (12 ft.) and slopes gently to 30m (98 ft.). It abounds with rainbow-hued fan coral and mammoth mushroom-shaped coral.\n\n  Spyglass Wall is another offshore dive site dropping to a sandy bottom and filled with sea fans and large coral heads. The drop is from 3 to 18m (10\u201359 ft.). Divers here should keep an eye out for tarpon, eagle rays, and stingrays.\n\nShopping\n\nUnlike the U.S. Virgin Islands, the British Virgins have no duty-free shopping. British goods are imported without duty, though, and you can find some good buys among these imported items, especially in English china. In general, store hours are Monday to Saturday from 9am to 4pm.\n\nRoad Town\n\nYou'll find the densest concentration of shops and restaurants at Wickham's Cay in Road Town.\n\nBamboushay Pottery   HOUSEWARES Bamboushay makes handsome kiln-fired stoneware\u2014homeware, tableware, scones, and more\u2014in its studio in Nanny Cay. In addition to pottery, this shop sells coffee and Island Roots homemade patties. 109 Main St.   284\/494-7752. www.bamboushay.com.\n\nCraft Alive Village   GIFTS This collection of colorful West Indian\u2013style cottages has been newly landscaped and revamped, and amid the mass-produced clutter is a growing assemblage of BVI\u2013made arts and crafts. Look for Asante Studio, where BVI master artist (and a founding member of the BVI Art Foundation) Joseph Hodge (  284\/53-0563) paints watercolors and acrylics (often genre scenes of local farming and fishing). Locally Yours sells homemade \"lollies\" made with the juice of local fruits like guava and Caribbean cherry. Joan Wilson makes embroidered children's dresses and hand-stitched linens. There is also a farmers' market on Saturday morning. Road Town Harbor.\n\nExploring \"Treasure Island\"\n\nAcross Drake Channel from Tortola lies the former pirate den known as Norman Island. Legend has it that Norman Isle was the inspiration for Robert Louis Stevenson's \"Treasure Island,\" first published in 1883. You can row a dinghy into the southernmost cave of the island\u2014with bats overhead and phosphorescent patches\u2014where Stevenson's Mr. Fleming supposedly stowed his precious treasure. A series of other Norman Isle caves are some of the best-known snorkeling spots in the B.V.I., teeming with spectacular fish, octopuses, squid, and colorful coral. The lively island beach bar, Pirate's Bight, has been reborn after a 2013 fire. Intrepid hikers climb through scrubland to the island's central ridge, Spy Glass Hill. Many cruisers make Norman Isle a favorite stop, and day-sail operators offer swimming and snorkeling excursions here; see \"Island-Hopper Day-Tripping,\" above.\n\nSunny Caribbee Spice Co.   FOOD This quaint and colorful shop specializing in Caribbean spices and condiments in located in a historic West Indian building that was the first hotel on Tortola. It also sells potions, botanicals, soaps, teas, and coffees. The spices make wonderful gifts and are charmingly packaged, but the ingredients are sourced from all over the Caribbean and not necessarily B.V.I.-specific. 119 Main St.   284\/494-2178. www.sunnycaribbee.com.\n\nPusser's Company Store   FOOD\/WINE The little empire that is Pussers sells clothing, nautical tchotchkes, and gourmet food items including Pusser's famous Rum Cake, meats, spices, fish, and a nice selection of wines. Pusser's Rum is one of the best-selling items here. Main St. and Waterfront Rd.   284\/494-2467. www.pussers.com.\n\nVirgin Islands Folk Museum   GIFTS You might find a few interesting BVI-centric souvenirs at this museum in the Penn House, a charming West Indian cottage built by a shipwright in 1911. 98 Main St.   284\/494-3701, ext. 5055.\n\nWest End\n\nYou'll find shops at Soper's Hole Marina, including Zenaida   (  284\/495-4867), an atmospheric little shop selling handblock fabrics and scarves, sarongs, and Moroccan lamps; outposts of both Sunny Caribbee and Pusser's (see above); the Arawak Surf Shop (  284\/494-5240), selling island clothing, crafts, and gifts; and the Harbour Market grocery (  284\/347-1250).\n\nGreen Glass Studio   GIFTS Founded by local nonprofit GreenVI, this outdoor glass-blowing studio on Cane Garden Bay in Tortola recycles the daily refuse from the local beach bars\u2014beer and booze bottles\u2014to fashion beautiful hand-blown glass delicacies, from starfish paperweights to turtle ornaments to flower glass sculptures. The GreenVI Glass Studio has trained a number of locals in the intricacies of glass-blowing, and a local octogenarian makes cloth bags out of donated clothing to carry them home in. It's open daily 9am to 5pm. Cane Garden Bay.   284\/542-2266.\n\nEast End\n\nAragorn's Local Arts and Crafts Center   ART\/GIFTS Aragorn's Studio is a showcase for the most talented artisans in the islands. Any search for wonderful local art should start at Aragorn's Studio. Tortola-born Aragorn is a printmaker, potter, and sculptor; his giant \"fireballs\"\u2014silhouetted metal sculptures\u2014are set ablaze during the monthly Fireball Full Moon Parties on Trellis Bay. Look for miniature fire balls (candle holders), beautiful original prints, pottery, jewelry, and gifts, the work of Aragorn, inhouse artisans, and regional artists. It also sells gourmet delicacies: organic produce and herbs from Aragorn's Good Moon Farm; salt raked from the old salt ponds on Salt Island; and traditional coconut bread and banana bread baked in the nearby Mangrove Bakery. Trellis Bay.   284\/495-1849. www.aragornsstudio.com.\n\nHIHO   CLOTHING Founded by two brothers who spent their childhood on Tortola, this clothing firm makes comfortable and stylish \"Caribbean-inspired\" clothing for men and women: cotton dresses and flirty tunics and sturdy T-shirts. It's several notches above your local T-shirt outlet. HIHO has three locations in Tortola, including this one in Trellis Bay, and also rents surfboards and paddleboards out of its Road Town shop. Trellis Bay.   284\/494-7694. <http:\/\/shop.go-hiho.com>.\n\nTortola After Dark\n\nAsk around to find out which hotel might have entertainment on any given evening. Steel bands and fungi or scratch bands (musicians who improvise on locally available instruments) appear regularly. Pick up a copy of \"Limin' Times,\" an entertainment magazine that lists what's happening locally; it's usually available at hotels.\n\nBomba Surfside Shack, Cappoons Bay (  284\/495-4148), is the oldest and most famous hangout on the island, a shack cobbled together with scrap driftwood, corrugated metal, rubber tires, and other flotsam and jetsam and plopped right on the beach near the West End. Undergarments swing in the breezes; graffiti covers the wood. Bomba Callwood is usually somewhere in attendance, a big man in overalls with a wreath of smoke around his head. Bomba's attracts a varied crowd, from surfers riding the swells at Cappoons to vans of cruise-ship passengers stopping for a photo op and a swig of Bomba's rum punch. Despite its makeshift appearance, the shack is quite a business enterprise, bringing in crowds of visitors every month for Bomba's Full Moon bashes, where Bomba's \"herbal\" mushroom tea simmers in a cauldron in the bushes. It's open daily from 10am to midnight (or later, depending on business).\n\nQuito Rymer, one of the island's most well-known musicians, oversees his own mini-empire along the eastern stretch of Cane Garden Bay. The more touristy spot is the bar\/restaurant Rhymer's (  284\/495-4639; p. ), where you can sip cold beer or tropical rum concoctions along with a casual menu of ribs, conch chowder, and more. The beach bar and restaurant is open daily 8am to 9pm. Quito's Gazebo   (  284\/495-4837; p. ) is where the guitarist plays solo on Tuesdays and Thursdays (8pm) and Friday with a full band (9:30pm). It serves any and every kind of alcoholic libation and excellent food from an open-air restaurant built almost directly above the waves.\n\nMyett's   (  284\/495-9649) is another Cane Garden Bay hot spot, with a prime location right on the beach and regular live music; look for a Caribbean party every Wednesday night. In the same area, but not directly on the beach, visit Columbus Sunset Bar, Cane Garden Bay (  284\/495-751), where locals gather to drink, talk, and sample good island food (they also have clean rooms to rent).\n\nFinally, over on Little Apple Bay, check out Sebastian's (  284\/495-4212; p. ), especially on Sunday, when you can dance to live music under the stars, at least in winter.\n\nVirgin Gorda\n\nThe second-largest island in the British cluster, Virgin Gorda has a population of some 3,000 people. It was named Virgin Gorda, or \"Fat Virgin,\" by Christopher Columbus on his second voyage to the New World in 1493. It's located 19km (12 miles) east of Tortola and 41km (25 miles) east of St. Thomas.\n\nVirgin Gorda was a fairly desolate agricultural community until Laurance Rockefeller established the resort of Little Dix here in the early 1960s, following his success with Caneel Bay on St. John in the 1950s. He envisioned a \"wilderness beach,\" where privacy and solitude reigned. Other major hotels followed in the wake of Little Dix, but seclusion is still highly guarded and respected. Many visitors think Virgin Gorda is home to some of the most beautiful natural attractions in the British Virgins, including the boulders-strewn beach known as the Baths; the island's highest point, Gorda Peak; and the gin-clear bays of the Valley.\n\nEssentials\n\nGetting There\n\nBy Plane Air Sunshine (  800\/327-8900 in the U.S. or Canada, or 284\/495-8900 in the B.V.I.; www.airsunshine.com) offers direct flights between San Juan (or St. Thomas) and Virgin Gorda; direct flights between Vieques, Puerto Rico, and Virgin Gorda; and direct flights between St. Croix and Virgin Gorda. Cape Air (  800\/227-3247 in the U.S. and U.S.V.I., or 284\/495-2100 in the B.V.I.; www.capeair.com) flies between San Juan and Virgin Gorda. Seabourne Airlines (  866\/359-8784, or 340\/773-6442 in the U.S.V.I.; www.seaborneairlines.com) offers regularly scheduled flights between San Juan and Virgin Gorda.\n\nBy Boat Both Smith's Ferry (  284\/495-4495; <http:\/\/bviferryservices.com>) and Speedy's (  284\/495-5235; www.bviferries.com) operate daily ferry service between Tortola (both Road Town and Beef Island) and Virgin Gorda (both Spanish Town and Bitter End\/Leverick Bay); round-trip fares are $30 to $65 adults, $28 children 5\u201311. The North Sound Express (  284\/495-2138), near the airport on Beef Island, has daily connections between Spanish Town, Leverick Bay, and the Bitter End Yacht Club on Virgin Gorda; round-trip fares are $40 to $55 adults, $20 to $32 children.\n\nYou can also get direct ferry service from St. Thomas to Virgin Gorda, but runs are infrequent (and only operate on certain days of the week)\u2014so you may need to ferry to Tortola first (most likely Road Town) and then catch another ferry to Spanish Town, Virgin Gorda.\n\nMost of the high-end resorts have their own boats to transfer guests from the airport on Beef Island to Virgin Gorda.\n\nGetting Around\n\nBy Taxi Taxis are widely available, and, much like everywhere else in the islands, drivers double as solid tour guides. Contact the good folks at Potters' Taxi Service for excellent taxi service and island tours (  284\/495-5329). The Valley Taxi Association is at   284\/495-5539. The standard fee for traveling between Spanish Town and Gun Creek is $30. Many drivers operate open-sided taxi safari buses that can hold up to 22 passengers. These buses charge upwards of $3 to $5 per person to transport a passenger, say, from the Valley to the Baths.\n\nBy Car If you'd like to rent a car, try one of the local firms, such as Virgin Gorda Car Rental (  284\/496-0383; www.virgingordacar.com) or Mahogany Rentals (  284\/495-5469; <http:\/\/mahogany.nfshost.com>), both in the Valley close to the yacht harbor. Both companies also offer taxi tours and transfers.\n\nVirgin Gorda\n\nRemember: Drive on the left. Road conditions on Virgin Gorda range from good to extremely poor. An aerial view of the island shows what looks like three bulky masses connected by two very narrow isthmuses. The North Sound resorts are not even accessible by road at all, requiring ferryboat transfers from Gun Creek.One possibility for exploring Virgin Gorda by car is to drive from the southwest to the northeast along the island's rocky and meandering spine. This route will take you to the Baths (in the extreme southeast), to Spanish Harbour (near the middle), and eventually, after skirting the mountainous edges of Gorda Peak, to the most northwesterly tip of the island's road system, near North Sound. Here, a miniarmada of scheduled ferryboats runs between Gun Creek and North Sound resorts.\n\n  Virgin Gorda\n\nBanks\/ATMs Scotiabank (www.scotiabank.com) has a full-service bank in Spanish Town near the Yacht Harbour Shopping Centre. Both Scotiabank and First Caribbean (www.cibcfcib.com) have ATM locations at the Yacht Harbour Shopping Centre.\n\nDentists & Doctors Contact Apex Medical Center, Millionaire Rd., the Valley (  284\/495-6557).\n\nDrugstore Go to Island Drug Centre at Spanish Town (  284\/495-5449).\n\nEmergencies Call   999 or 911.\n\nInternet Access The Bath & Turtle restaurant, in Yacht Harbour Marina (  284\/495-5239), has free Wi-FI.\n\nPolice There is a station in the Valley at Spanish Town (  284\/495-7584).\n\nTourist Information The island's tourist office is in Virgin Gorda Yacht Harbour, Spanish Town (  284\/495-5181).\n\nWhere to Stay\n\nMany people who come to the island rent villas. One of the best sources for villa rentals is Virgin Gorda Villa Rentals Ltd., P.O. Box 63, Leverick Bay, Virgin Gorda, B.V.I. (  800\/848-7081 or 284\/495-7421; www.virgingordabvi.com). A 5-night minimum stay is required in the off season, and a 7-night minimum stay is requested in winter. Most accommodations have access to an outdoor pool, dining facilities, a spa, tennis courts, and extensive watersports. Wi-Fi is free in most rentals. Weekly rates range from $900 to whopping $9,000 for seriously luxe digs.\n\nExpensive\n\nBiras Creek Resort   Reachable only by boat, Biras Creek delivers a kind of luxurious serenity that's thoroughly rejuvenating. This is not where the action is\u2014that would be next door at lively Bitter End, which you can reach by 20-minute hiking trail\u2014but for a soulful immersion in the natural world, with all the creature comforts at your fingertips, Biras Creek delivers. This is the kind of place where you can truly unplug (there's no TV, and cellphone coverage is spotty); where studying the antics of chickens, iguanas, and hermit crabs becomes part of your daily routine; where you can reach up and almost pluck a thousand stars from the velvety night skies.\n\nThey get the details right here: Each suite comes with state-of-the-art bedding, complimentary bottle of rum, big picture windows, and sweet little flashlights to help you navigate the pathways in the dark. The d\u00e9cor is chic and modern, with each suite book-ended by ultra-private patios and garden showers. The resort has two Grand Suites, each with two master suites and private plunge pools. Ask for a room facing the open Atlantic, where the wild surf is deliciously dramatic. Each suite comes with two bikes, a fun and helpful transport to fully cover the sprawling, 140-acre property. Bike to the beach on the Caribbean side; it has soft white sand, but the water is a little grassy\u2014still, it's a lovely spot to have the barbecue lunch. On the way you'll pass a paddock, home to the resort's five Paso Fino horses (rescued from Puerto Rico in 2009), ready to ride. Guests can also take the resort's little Boston Whaler boats out for 2-hour excursions to remote coves and beaches nearby\u2014it's great fun to tootle around the blue expanse of North Sound and stop in at Prickly Pear Island or Saba Rock for lunch and a swim.\n\nHilltop at Biras Creek (p. ), the on-site restaurant, is the best restaurant on the island, with magical views from its perch high above North Sound.\n\nBiras Creek is ideal for couples but is becoming more family-friendly, particularly in the summer and spring breaks.\n\n  877\/883-0756 or 284\/394-3555. www.biras.com. 32 units. FAP (Full American Plan) rates (all meals included, no drinks) winter $790\u2013$1,960 suite; off season $660\u2013$1,540 suite. You can also opt for room-only rates. Private motor launch from Beef Island airport $95 per person round-trip. No children ages 7 and under. Amenities: Restaurant; bar; babysitting; Boston whalers; exercise room; helipad; hiking trails; pool (outdoor); spa; 2 tennis courts (lit); watersports equipment\/rentals, Wi-Fi (free).\n\nBitter End Yacht Club   What began in the 1950s as a primitive and ramshackle sailors' stop is now the liveliest of the B.V.I. resorts, a full-service sailing complex that opens onto the North Sound's gorgeous deepwater harbor, accessible only by boat. You almost expect a Somerset Maugham heroine to come waltzing through the lobby, a British Empire outpost of Balinese teak and whirring fans. Accommodations have a breezy-casual rustic elegance, with lots of burnished wood and old clocks and a vintage-yacht look. You can stay in hillside chalets (North Sound Suites) or in the resort's original cottages along the beachfront\u2014either way, you'll have your own private patio or veranda to admire the views. True to its roots as a self-sustaining survivor, the Bitter End is an eco pioneer, generating its own electricity (much through solar power) and collecting and desalinating its own water.\n\nThere are activities aplenty at Bitter End, from movie screenings in the Sand Palace to sailing school lesson to Champagne cruises to snorkeling excursions. The resort has more shops than the rest of Virgin Gorda combined, practically. The pool is big and beautiful, enveloped in checkerboard tiles and coconut palms. The open-air Clubhouse Steak & Seafood Grille has front-row North Sound views, with high beamed ceilings hung with international nautical flags and Moroccan lamps; it's filled with happy chatter through breakfast, lunch, and dinner.\n\n  800\/872-2392 in the U.S. for reservations, or 284\/494-2746. www.beyc.com. 85 units, 5 yachts. Winter (double occupancy) $700\u2013$1,840 beachfront villa, suite, yacht, or hillside villa; off season (double occupancy) $500\u2013$1,360 all units. Rates include all meals. Take the private ferry from the Beef Island airport, $30 per person one-way. Amenities: 3 restaurants; pub; babysitting; exercise room; pool (outdoor); watersports equipment\/rentals; Wi-Fi (free).\n\nMango Bay Resort   Nestled on lushly landscaped grounds, these resort villas are set on a ridiculously beautiful beach around a ridiculously beautiful blue lagoon. You can stay in suites in resort duplex villas or in your own private villa, enjoying all the comforts and conveniences of home, plus resort services. In fact, all units but studios have full kitchens. The lodgings are extremely adaptable and have many lockout capabilities for larger groups. Resort villa duplexes have high-ceilinged great rooms with furnishings in warm earth tones and tile patios with dining tables and rattan chairs. Big picture windows and doors open to the tradewinds and the golden sands of Mahoe Bay.\n\n  284\/495-5672. www.mangobayresort.com. 26 units. Winter $295 studio, $435\/$495 1-bedroom suite\/villa, $585\/$755 2-bedroom suite\/villa, $985 3-bedroom villa; off season $220 studio, $300\/$360 1-bedroom suite\/villa, $440\/$580 2-bedroom suite\/villa, $695 3-bedroom villa. Amenities: Restaurant; bar; pool (private villas only); limited watersports; Wi-Fi (free).\n\nRosewood Little Dix Bay   This palace of casual elegance is mighty fine in every way. In fact, with new management freshening up the place and turning to eco-conscious innovations, Little Dix has never been better. To the relief of many long-standing guests, that means more of the same. Endearingly, rooms have not had keys since the resort opened in 1964\u2014Virgin Gorda is notable for its lack of crime\u2014although you can lock yourself in securely at night.\n\nCelebrating its 50th anniversary in 2014, this former RockResort holds a special place in the heart of many, and one big reason is the warm, insightful service from the best staff in the Virgin Islands. There is a wonderful sense of ownership by the staff, a good number of whom have worked here 20, 30, even 40 years (Little Dix is the island's second-largest employer, after the government).The setting doesn't hurt either: Little Dix is comfortably sandwiched between a half-mile of crescent beach and tropical forest. All of the bright, airy rooms face the beach, with private terraces that offer sea views. Trade winds breeze through wooden louvers and screens, and bathrooms are roomy and luxurious. The details matter, little touches like wooden walking sticks and complimentary bottles of rum in your room. Step outside and snorkel right off the beach; we saw hawksbill turtles chomping on seagrass and rays skimming the bottom. It's a quick boat ride (or 1.5-mile walk along a wooded trail) to the beautiful beach at Savannah Bay. Children are welcomed (kids under 5 eat free), and families generally stay at the Ocean Cottages, four buildings on the south end of the resort, close to the complimentary Rose Buds kids' activities club. Little Dix is also just a 20-minute stroll from Spanish Town, the island's main town. The Sense Spa   is utterly fabulous, with a dreamy setting in a tropical glade on the resort's western flank. You await your spa treatment around an infinity pool, with spectacular views of the Dog islands below fringed in rustling palms. Nonguests can arrange spa treatments here as well. In the fall, Little Dix partners with Necker Island for the annual Necker Cup, a Pro-Am charity doubles tournament where big-name tennis players compete and run clinics. The Pavilion (p. ) is one of the most romantic dinner spots on Virgin Gorda, serving themed buffets 6 nights a week.\n\n  888\/767-3966 in the U.S., or 284\/495-5555. www.littledixbay.com. 100 units. Winter $725\u2013$950 double, from $1,200 suite; off season $380\u2013$775 double, from $875 suite. Extra person $75. Private ferry from Beef Island airport: $115 per person round-trip (children 5\u201311 $58); air taxi transfers from St. Thomas also available. Amenities: 3 restaurants; 2 bars; babysitting; children's programs; exercise room; pool; room service; spa; 7 tennis courts (lit); watersports equipment; Wi-Fi (free).\n\nModerate\n\nFischer's Cove Beach Hotel   Set on the white-sand Paradise Beach, this good-value, locally owned barefoot beach hotel has eight free-standing stone cottages, each with one or two bedrooms and a living\/dining space, as well as 12 smallish hotel studio rooms with balconies to soak up the sea (or garden) views. Rooms have tile floors, gauzy curtains, and an oyster-shell palette. A food store near the grounds is a great place to stock up on provisions. An open-air restaurant onsite serves both Caribbean and Continental classics, not to mention live music twice weekly.\n\n  284\/495-5252. www.fischerscove.com. 20 units. Winter $165\u2013$175 double, $245\u2013$255 studio cottage, $370 family cottage; off season $145\u2013$155 double, $125\u2013$205 studio cottage, $255 family cottage. MAP (breakfast and dinner) $40 per person extra. Amenities: Restaurant; bar; babysitting; children's playground; watersports equipment\/rentals; Wi-Fi (free in lounge).\n\nGuavaberry Spring Bay Vacation Homes   Just minutes from the Baths and a short stroll to the beach at beautiful Spring Bay, these handsome redwood \"treehouses\" on stilts are a throwback to a simpler time. Don't expect TVs or phones; air-conditioning is offered (for a fee) only in some cottages. Do expect fully outfitted cottages with a private feel, where screened and louvered walls let in the sea breezes. Each one-, two-, or three-bedroom house comes with a complete kitchen, dining and living areas, and a splendid sundeck overlooking Sir Francis Drake Passage. Interiors are modest, with rattan furniture and linens in tropical prints. A well-stocked on-site provisioning store, the commissary, has groceries, wines and beers, and kitchen staples, but Spanish Town and the Yacht Harbour Shopping Centre are just a mile away. You can even swim to the famous Baths from the Spring Bay beach.\n\n  284\/495-5227. www.guavaberryspringbay.com. 18 units. Winter $235\u2013$420; off season $150\u2013$320. Extra person $25. No credit cards. Amenities: Babysitting; watersports equipment\/rentals; Wi-Fi ($2 per half-hour).\n\nSaba Rock Resort   A great escape, this idyllic retreat is perched on its own little 1-acre cay in the middle of North Sound; it's only accessible by free ferry or boat. The island belonged for 3 decades to the late legendary diver Bert Kilbride, who discovered some 90 shipwrecks in and around the B.V.I. and whose scuba-diving resort course is taught around the world. Bert's former private island has been transformed into a luxe complex of nine one- and two-bedroom suites. Today this boutique hotel is ideal for swimming, sailing, or exploring North Sound. But with a good restaurant and jaunty bar right on the property, you'll never have to leave. Look for the little nautical museum with shipwreck artifacts accumulated by Bert.\n\n  284\/495-7711. www.sabarock.com. 8 units. Winter $150\u2013$295 double, $295 quad, $495\u2013$550 2-bedroom suite; off season $140\u2013$250 double, $250 quad, $475 2-bedroom suite. Amenities: Restaurant; bar; watersports equipment\/rental; Wi-Fi (free).\n\nInexpensive\n\nLeverick Bay Resort & Marina   This good-value, happening spot is set along the southern edge of beautiful North Sound. The d\u00e9cor in the 14 colorful rooms may be in need of an update, but the seafront balcony or veranda makes up for any style issues. The complex offers a food market and two small beaches\u2014but the beauteous Savannah Bay is a 10-min. drive away. The Restaurant at Leverick Bay   is a winner, serving fresh local seafood right on the beach.\n\n  800\/848-7081 in the U.S., 800\/463-9396 in Canada, or 284\/495-7421. www.leverickbay.com. 14 units, 4 condos. Winter $149 double; off season $119 double. Extra person $36. Amenities: Restaurant; bar; dive shop; pool (outdoor); tennis court (lit); watersports equipment\/rentals; Wi-Fi (in some; free).\n\nWhere to Eat\n\nExpensive\n\nHilltop Restaurant at Biras Creek Resort   INTERNATIONAL For many people, this hilltop restaurant high above North Sound has the most beautiful setting on island. For others, Hilltop serves the island's best food. We fall in with both camps. The four-course prix-fixe menu changes nightly, but you may start with a curry lobster cake, served with homemade tartar sauce and move on to pan-seared salmon, grilled lobster, or steak. From 3 to 6pm, a cocktail menu is served, with daily 2-for-1 drink specials and snacks like Buffalo wings, burgers, and shellfish fritters. The breakfast buffet is delicious and often includes local delicacies like sauteed callaloo with onions, mushrooms, and cherry tomatoes.\n\nIn Biras Creek Resort (p. ).   284\/494-3555. www.biras.com. Reservations required. Fixed-price dinner $85\u2013$125. Daily 8\u201310am, 12:30\u20132pm (Wed, Sat, Sun only), and 6:30\u20139pm.\n\nThe Pavilion at Little Dix Bay   INTERNATIONAL The iconic Pavilion is magical place to dine, an open-air pavilion beneath handsome vaulted wooden rooftops with sea views and bay breezes. Six nights a week dinner is a themed buffet, from Italian to Asian to seafood. (You can also order a la carte if you like.) Standards are high here, and it shows: The buffets are innovative, fresh, and delicious, aided and abetted by the warm embrace of the staff and that soul-stirring setting.\n\nIn Rosewood Little Dix Bay (p. ), 1km (2\u20443 mile) north of Spanish Town.   284\/495-5555. www.littledixbay.com\/dine2.cfm. Reservations required. Nightly buffets (some including a la carte menu items) $55\u2013$100. Daily 11:30am\u20135:30pm and 6:30\u20139:30pm.\n\nModerate\n\nThe Bath & Turtle   CARIBBEAN\/INTERNATIONAL The most popular pub in Spanish Town is located inside (and outside) the Yacht Harbour marina, serving such island delights as pepperpot soup, curried mutton, and conch. The beer-battered fish and chips is rightly famous (the menu doesn't lie). We also like the ginger pineapple ribs (in various degrees of \"hurricane\" heat), salads, and Anegada fish fingers. A Caribbean buffet ($25 per person) is offered every Wednesday. Grab a table right on the waterfront for a ringside view of yachts pulling into the harbor, turtles coming up for air, and colorful roosters ruling their little fiefdom ashore.\n\nVirgin Gorda Yacht Harbour.   284\/495-5239. Reservations recommended. Entrees $17\u2013$24, burgers, wraps, and paninis $12\u2013$17. Daily 7am\u20139pm.\n\nThe Rock Caf\u00e9   ITALIAN\/CARIBBEAN Set between the big rocks of the Baths near Spanish Town, this cafe is one of the island favorites. There's something satisfyingly primal about dining among giant granite boulders to flickering candlelight and stars winking in the big night sky. It doesn't hurt that the Rock specializes in excellent Italian pastas and pizzas. Try the spaghetti Rock, with crispy bacon, peas, and cream, or the penne Boscaiola, with chicken and broccoli. You can also get steaks and local seafood, including fresh grilled Anegada lobster. At press time, the Rock Caf\u00e9 folks had opened an intriguing new spot above called the Treehouse, literally housed in a custom-built treehouse (with crazy-good views) and big enough to accommodate just 20 people at a time.\n\nThe Baths.   284\/495-5482. www.bvidining.com. Reservations recommended. Main courses $18\u2013$40. Daily 4pm\u2013late.\n\nTop of the Baths   CARIBBEAN This breakfast and lunch restaurant is one of the best in the Baths area. It overlooks the famous Baths beach and even has a nice pool to cool off in while you wait for your food. The setting is glorious, and the food\u2014a mix of standard Americanized bar food and Caribbean homestyle cooking\u2014is happily satisfying. You can order burgers, tuna salads, pastas, and wraps, or sample local cuisine like conch fritters, curried shrimp, and conch in butter sauce.\n\nThe Baths.   284\/495-5497. www.topofthebaths.com. Main courses $16\u2013$25; sandwiches and salads $12\u2013$18. Daily 8am\u20137pm.\n\nInexpensive\n\nFat Virgin's Cafe   CARIBBEAN FUSION This modest outdoor cafe on North Sound is good and local, with home-cooked meals whipped up right in front of you and a lively throng of visitors traversing the North Sound. Expect delicious chicken roti, barbecued ribs, and pan-fried fish\u2014and pass the homemade pepper sauce, please. Even if you don't sample the fine fare, this is a fizzy spot to sip a rum drink and watch the shadows of big yachts slip into the harbor.\n\nMarina Village, Biras Creek Resort.   284\/495-7052. <http:\/\/fatvirgin.com>. Breakfast $7\u2013$16; main courses lunch $10\u2013$19, dinner $10\u2013$24. Daily 7am\u20139pm.\n\nHog Heaven   BARBECUE It's barbecue with a view at this lunch and dinner hilltop joint, with sensational views   of North Sound and beyond from the heights of Virgin Gorda and a hearty menu of pulled pork sandwiches, barbecue ribs, barbecue chicken, potato salad, and fried plantains.\n\nNail Bay Rd. 9:30am\u2013closing.\n\nSugar Apple   CARIBBEAN This bright new spot features local dishes like fish and johnnycakes; stewed chicken with rice and peas; oxtail and rice; saltfish and coconut dumplings; and \"Okra Punch\" for dessert.\n\nSouth Valley.   284\/545-4841. $5\u2013$10. No credit cards. Daily 10am\u20136pm.\n\nExploring Virgin Gorda\n\nThe northern side of Virgin Gorda is mountainous, with Gorda Peak reaching 417m (1,368 ft.), the highest spot on the island. In contrast, the southern half of the island is flat, with large boulders appearing at every turn.\n\nThe best way to see the island is on an island tour. Call Andy Flax, who runs the Virgin Gorda Tours Association (  284\/495-5252; www.virgingordatours.com), which will give you a tour of the island. Cost is from $55 (1 hr.) to $220 (4 hr.) for one or two persons, adding $15 to $30 per person more depending on the group size. You can be picked up at the ferry dock if you give 24 hours' notice.\n\nYou should also see the island by water, and having a local take you on customized tours of hidden coves and bays in a small boat is pure pleasure. The genial Avery Baptist (averymbaptist@hotmail.com) runs small charters and customized beach-hopping trips to places like the Baths and lesser-known gems out of his base in the Valley.\n\nBeaches\n\nDon't miss the beauty of the Baths  , where giant boulders form a series of tranquil pools and grottoes flooded with seawater. Nearby snorkeling is excellent, and you can rent gear on the beach. Scientists think the boulders were brought to the surface eons ago by volcanic activity. The Baths and surrounding areas are part of a proposed system of parks and protected areas in the B.V.I. The protected area encompasses 273 hectares (675 acres) of land, including sites at Little Fort, Spring Bay, and Devil's Bay on the east coast.\n\nDevil's Bay National Park   can be reached by a trail from the Baths. A 15-minute walk through boulders and dry coastal vegetation ends on a secluded coral-sand beach.\n\nNeighboring the Baths is Spring Bay  , one of the best of the island's beaches, with white sand, clear water, and good snorkeling. Trunk Bay is a wide, sandy beach reachable by boat or along a rough path from Spring Bay. Savannah Bay   is a sandy beach north of the yacht harbor, and Mahoe Bay  , fronting the Mango Bay Resort, has a gently curving beach with neon-blue water.\n\nThe white-sand beach at Prickly Pear Island   in the North Sound, is protected national parkland and a sweet stopover for a dip in crystal-clear seas and lunch at the Sandbox beach bar; those staying in the North Sound area can call for free water taxi pickup and dropoff (  284\/342-3696).\n\nWatersports & Outdoor Adventures\n\nHiking Virgin Gorda has some of the most breathtaking panoramic vistas in the Caribbean. A good way to see the sights from the heights is on a trek along the stairs and hiking paths that crisscross Virgin Gorda's largest stretch of undeveloped land, the Gorda Peak National Park. To reach the best departure point for your uphill trek, drive north of the Valley on the only road leading to North Sound for about 15 minutes of very hilly driving (using a four-wheel-drive vehicle is a very good idea). Stop at the base of the stairway leading steeply uphill. There's a sign pointing to the Gorda Peak National Park.\n\nIt should take between 25 and 40 minutes to reach the summit of Gorda Peak, the highest point on the island, where views out over the scattered islets of the Virgin Islands archipelago await you. There's a tower at the summit, which you can climb for even better views.\n\nScuba Diving & Snorkeling Kilbrides Sunchaser Scuba is located at the Bitter End Yacht Club at North Sound (  800\/932-4286 in the U.S., or 284\/495-9638; <http:\/\/sunchaserscuba.com>). Kilbrides offers the best diving in the British Virgin Islands, at 15 to 20 dive sites, including the wreck of the ill-fated RMS Rhone. Prices range from $100 to $110 for a two-tank dive on one of the coral reefs. A one-tank dive in the afternoon costs $75. Equipment, except wet suits, is supplied at no charge. Hours are 7:45am to 5:30pm daily. Dive BVI (  800\/848-7078; www.divebvi.com) has four locations in and around Virgin Gorda and offers diving outings and snorkeling day-trips and island-hopper cruises.\n\nShopping\n\nShops are sparse here; most of the best shopping is found in the boutiques in the island's upscale resorts. The Bitter End Yacht Club's Reeftique (  284\/494-2745), has a good selection of sexy kurtas, cotton T's and henleys, and top-quality Bitter End beach bags. Also at the Bitter End, the Trading Post has great kids' gifts, including messages in the bottle.\n\nPusser's Company Store, Leverick Bay (  284\/495-7369), sells Pusser's usual lineup of rum products, sportswear, and gift and souvenir items.\n\nIn Spanish Town, stop in at the Yacht Harbour Shopping Centre, which has an outpost for Dive BVI (  284\/495-5513; www.divebvi.com), selling all sorts of scuba and snorkeling equipment, plus clothes and tchotchkes. You can buy interesting handmade Virgin glass gifts, straw baskets, and beautiful beaded bags at Next Wave Designs (  284\/495-5634), also in the Yacht Harbour Shopping Centre.\n\nVirgin Gorda After Dark\n\nThere isn't a lot of action at night. The Bath & Turtle pub, at Yacht Harbour (p. ;   284\/495-5239), brings in local bands for dancing in the summer on Wednesday and Friday at 8:30pm. Check out the weekly \"Welcome Guide to the British Virgin Islands\" to see what's happening at the time of your visit.\n\nJOST VAN DYKE\n\nThis rugged 10-sq.-km (4-sq.-mile) island (pop. 300) on the seaward (west) side of Tortola was named after a Dutch settler. In the 1700s, a Quaker colony settled here to develop sugarcane plantations. (One of the colonists, William Thornton, won a worldwide competition to design the Capitol in Washington, D.C.)\n\nOn the south shore are some fine, powdery beaches, especially at White Bay and Great Harbour. This tranquil island is a popular stopover point not only for the yachting set but also for many cruise ships.\n\nEssentials\n\nGetting There You can take the ferry to Jost Van Dyke's White Bay via St. Thomas, St. John, or Tortola. New Horizon (  284\/495-9278) has daily 25-minute ferryboat shuttles from the West End; round-trip fares are $25 adults, $15 children 5\u201311. (Note that departure times can vary widely throughout the year, and often don't adhere very closely to the printed timetables.) If all else fails, negotiate a transportation fee with one of the handful of privately operated water taxis on Tortola. Fees start around $100 one-way.\n\nGetting Around To get around the island, call Bun Taxi at   284\/495-9281. You can rent Jeeps from Abe's by the Sea (  284\/495-9329) at $65 to $80 per day and Paradise Jeep Rentals (  284\/495-9477) at $55 to $70 per day.\n\nFast Facts In a medical emergency, call VISAR (Virgin Islands Search and Rescue) at   284\/494-4357; you can be flown to Tortola. There are no banks, ATMs, or drugstores on the island. Stock up before you arrive here.\n\nWhere to Stay\n\nIvan's Stress Free Bar & White Bay Campground   Seventh-generation Jost Van Dyke native Ivan Chinnery oversees this campground right on White Bay Beach. It's got campsites and one-room plywood \"cabins\" outfiitted with screens, ceiling fans, fridges, and big beds. The campground is simplicity itself, but it's also one of the livliest spots on Jost. Take your place in Ivan's Stress Free Bar at happy hour, downing sundowners with sailors and locals at your elbows. Ivan plays guitar in the White Bay International Ever-Changing All Star Band (the house band), which plays regularly here. Visiting musicians often drop in\u2014yes, even Keith Richards or Jimmy Buffet.\n\n  284\/495-9358. www.ivanscampground.com. 8 cabins, 15 campsites, 5 tents. Year-round $65\u2013$75 cabin, $40 tent, $20 bare site, $150\u2013$260 studio. Amenities: Food service; Wi-Fi ($10).\n\nSandcastle Hotel   Joined at the hip to the legendary beach bar the Soggy Dollar, this barefoot retreat is for escapists who want total privacy. The six cottages are wrapped in lush tropical foliage and bougainvillea, with sweeping views of White Bay Beach. The roomy, breezy rooms are hung with local art and have king-size beds, rattan furniture, and tropical accents. Two units are air-conditioned, while all others have ceiling fans. You mix your own drinks at the Soggy Dollar, and keep your own tab\u2014the resident drink is the famed Painkiller (see below)\u2014and yes, cruise-ship passengers often flood the place for a few hours midday. The restaurant (see below) is an old standard on the island.\n\n  284\/495-9888. www.soggydollar.com\/sandcastlehotel. 6 units. Winter $310 double, $285 cottage; off season $250 double, $210 cottage. Extra person $35\u2013$45. 3-night minimum. Children 15 and under not permitted. Amenities: Restaurant; bar; watersports equipment\/rentals; Wi-Fi (free).\n\nSandy Ground Estates   These happily oversized two- and three-bedroom villas sit on the edge of a beach on the eastern part of Jost Van Dyke. The \"Castle\" has two terraces with commanding views of other islands; The \"Genip Tree House\" is a three-level villa above the beach, with terraces that boast sparkling sea views. Each villa is equipped with a full kitchen, and each is individually owned, so interiors vary widely, from fashionable to more modest. The managers help guests with boat and watersports rentals. Diving, day sails, and other activities can be arranged, and there are dinghies available.\n\n  284\/494-3391. Fax 284\/495-9379. www.sandyground.com. 7 units. Winter $1,950 villa for 2; off season $1,400 villa for 2. Extra person $500 per week in winter, $350 off season. Children 3\u201312 half-price. Amenities: Watersports equipment\/rentals; Wi-Fi (free).\n\nWhite Bay Villas & Seaside Cottages   With its eye-candy sea views and easy access to a white-sand beach, this is a real find and a first-rate, secluded retreat. Perched above the sea, all of the cottages were built by owner John Klein. The property is comprised of several different fully equipped accommodations ranging from one- to two-bedroom houses to a three-bedroom, three-bathroom villa for up to eight guests. The Plantation Villa is the largest in the complex, with a great room and kitchen decorated with murals depicting the island's culture.\n\n  800\/778-8066 or 410\/571-6692. www.jostvandyke.com. 3 villas, 7 cottages. Weekly rates winter $1,995 1-bedroom, $3,045 2-bedroom, $4,095 3-bedroom; off season $1,595 1-bedroom, $2,435 2-bedroom, $3,275 3-bedroom. No credit cards.\n\nWhere to Eat\n\nAbe's by the Sea   WEST INDIAN This all-purpose sailors' stop serves lunch and dinner and specializes in seasonal local seafood like lobster and conch. \"Casual\" doesn't begin to describe the low-key nature of this modest dockside spot, cobbled together with plywood and serving what fans say is the coldest beer on island.\n\nLittle Harbour.   284\/495-9329. Reservations recommended for groups of 5 or more. Dinner $20\u2013$45; nightly barbecue $30. Daily 11:30\u201311pm.\n\nThe Infamous Painkiller\n\nDon't leave the B.V.I. without sampling the notorious rum drink known as the Painkiller, which got its start at the Soggy Dollar Bar and has gone on to greater glory at all Pusser's outlets. An Englishwoman, Daphne Henderson, is said to have invented the drink in the 1980s, which is an orange-colored blend of island rum, orange juice, pineapple juice, cream of coconut, and a scraping of nutmeg on top. Today the Painkiller is probably the most popular drink among sailors in the B.V.I.\n\nJost Van Dyke\n\nAli Baba's   CARIBBEAN Fashioned from unvarnished beams and planks, this restaurant near the edge of the harbor, adjacent to the Customs house, has a breezy veranda right next to the Customs House. Expect well-executed version of island staples, such as conch chowder, conch fritters, and savory pumpkin soup. The fish is fresh and good, and you can get grilled lobster, mahimahi, and stewed conch. If you're on the island in time for breakfast, drop in to join the locals for a tasty wake-up meal.\n\nGreat Harbour.   284\/495-9280. Breakfast from $10; main courses lunch $9\u2013$12, dinner $18\u2013$22 Daily 9am\u201311pm.\n\nCorsairs Beach Bar & Restaurant   SEAFOOD\/CARIBBEAN Open for breakfast, lunch, and dinner this convivial beach bar is a happening hangout for a hearty breakfast and a post\u2013happy hour rib-sticking lunch and dinner. Start with conch fritters in a goat-cheese sauce, followed by seared tuna with a spicy mango-rum sauce or lobster Thai style in a coconut-and-pumpkin sauce. Four pastas, including fettuccine shrimp al pesto, are featured nightly.\n\nGreat Harbour.   284\/495-9294. www.corsairsbvi.com. Lunch from $10; dinner main courses $24\u2013$39. Daily 8:30am\u20132pm and 6:30\u20139pm.\n\nFoxy's Tamarind Bar   WEST INDIAN On an island chain known for its happening beach bars, this may be the most famous beach bar in the B.V.I. Sixth-generation Jost Van Dyke native Philicianno \"Foxy\" Callwood opened the place in the late 1960s and is a big part of the draw. Foxy plays the guitar and is deeply invested in preserving the environment and culture of his native Jost. In November 2013, several years after the Jost Van Dyke Preservation Society initiated the building of a traditional wooden \"Tortola\" boat behind Foxy's bar (with the help of the island children), the 32-foot Endeavor II was launched in Great Harbour to much fanfare among locals and visitors alike.\n\nMost nights feature rock 'n' roll, reggae, or soca. The food and drink aren't bad, either\u2014order up Foxy's Painkiller punch along with fish sandwiches, rotis, and burgers, lobster, ssteamed shrimp, or the catch of the day. Look for beach barbecues on the weekends and ongoing events all year-round, including Foxy's Old Year's Night celebration around New Year's.\n\nGreat Harbour.   284\/495-9258. www.foxysbar.com. Reservations recommended. Lunch $10\u2013$15; dinner $18\u2013$35. Daily 9am\u201311pm.\n\nOne Love Bar and Grill   CARIBBEAN Foxy's oldest son (see previous recommendation) operates one of the liveliest spots on the island. Seddy Callwood and his wife, Raquel, have a loyal following, and the house policy of \"No Shoes, No Shirt, No Problem\" is said to have inspired Kenny Chesney's hit song. The menu includes the catch of the day, kabobs, and tangy ribs\u2014washed down with a knee-knocking Bushwhacker. One Love features live reggae music on Saturday and Sunday afternoons, but as soon as the sun sets, Seddy locks up and goes home.\n\nWhite Bay.   282\/495-9829. www.onelovebar.com. Main courses $10\u2013$15. Daily 11am\u2013sunset.\n\nSandcastle   INTERNATIONAL This hotel restaurant serves lunch in the open-air dining room, while lighter fare and snacks are available at the Soggy Dollar Bar. At night, guests settle in with a three-course dinner that may feature good soups, mahimahi Martinique (marinated in orange-lemon-lime juice and cooked with fennel, onions, and dill), and \"Sandcastle hen\": grilled Cornish hen that's been marinated in rum, honey, lime, and garlic! Meals are served with seasonal vegetables and a variety of salads.\n\nAt the Sandcastle Hotel (p. ).   284\/495-9888. www.soggydollar.com. Reservations required for dinner by 4pm. Lunch main courses $10\u2013$15; fixed-price dinner $22\u2013$33. Daily 9am\u20133pm and 1 seating at 7pm.\n\nANEGADA\n\nThe most remote of the British Virgins, Anegada is the second-largest island in the B.V.I. chain. Yet is has a population of about 250, none of whom has found the legendary treasure from the more than 500 wrecks lying off notorious Horseshoe Reef. This is a remote little corner of the Caribbean: Don't expect a single frill, and be prepared to put up with a few minor hardships, such as mosquitoes.'\n\nLocated 30 miles east of Tortola and 15 miles north of Virgin Gorda, Anegada is different from the other British Virgins in many ways. First of all, in contrast to the voluptuous volcanic topography of the other islands, Anegada is a flat coral-and-limestone atoll. Its highest point only reaches 8m (26 ft.), and when you're sailing to it, it hardly appears on the horizon. (Its Spanish name doesn't mean \"drowned island\" for nothing.) At the northern and western ends of the island are some lyrical white-sand beaches, the main reason for coming here. Second, most of the island is reserved for birds and other wildlife. The B.V.I. National Parks Trust has established a flamingo colony here (they flock to the old salt ponds), and it's also the protected home of several varieties of heron, ospreys, and terns. The Trust has also designated much of the interior of the island as a preserved habitat for Anegada's animal population of some 2,000 wild goats, donkeys, and cattle.\n\nAnegada is a fishing paradise and the lobster capital of the B.V.I., celebrating its famous catch yearly with the Anegada Lobster Festival in November. Anegada is a low key, friendly, unspoiled place to kick back and relax to the retro rhythms of the Caribbean. Come here for tranquility, not for posh pampering. But come soon: Ever so slowly, the modern world is coming to Anegada.\n\nEssentials\n\nGetting There Fly BVI (  284\/495-1747; www.fly-bvi.com) operates an \"Anegada Day-Trip\" between Anegada and Beef Island off Tortola. It includes the round-trip airfare, round-trip taxi transfers on Anegada, and a lobster lunch at the Big Bamboo. Cost is $175 per person and up, depending on group size.\n\nYou can also take a day excursion to Anegada by ferry from Tortola. The Road Town Fast Ferry (  284\/495-2323) operates daily trips on Monday, Wednesday, and Friday between Tortola, Virgin Gorda, and Anegada; round-trip fares are $55 adults, $35 children.A number of day-sail operators also make full-day beach and snorkeling excursions to Anegada. Dive BVI (  800\/848-7078 or 284\/495-5513; www.divebvi.com) offers a full-day guided snorkeling tour of the island every Wednesday, leaving from Virgin Gorda. The cost is $95 per person ($75 children 12 and under) and does not include lunch.\n\nLoblolly Bay: A Day at the Beach\n\nAny trip to Anegada has to include a visit to the fantastic beach and reef at Loblolly Bay. If you're taking a day-trip from Tortola, make sure you call one of the taxi drivers mentioned under \"Getting Around,\" above ahead of time, and have them take you across the island to the bay, with one quick stop to see the legendary pink flamingos en route. Once you pull up at Loblolly Bay, stake out a place on the beach and enjoy some of the most spectacular snorkeling in the B.V.I. Break for lunch at Big Bamboo (see below), and have a drink at the small thatched-roof bar where scrawled signatures on the bar and roof supports are from Cindy Crawford, Brooke Shields, and Andre Agassi (the bartender swears they're real).\n\nGetting Around Taxi drivers offer both transfers and island tours. Recommended drivers include Lawrence Wheatley (  284\/495-8002); Aubrey Levons (  284\/443-9956); and Rondell at Tony's Taxis (  284\/495-8027). Taxi rates are standardized by the government; check the You can rent cars and scooters at DW Car Rental (  284\/495-9677) or SnK Scooters & Jeep Rentals (  284\/346-5658).\n\nFast Facts Anegada has a small fire department and a little library, but it has no banks, ATMs, or drugstores. Make adequate arrangements for supplies before coming here.\n\nWhere to Stay\n\nThe 15-room Anegada Beach Club (  800\/871-3551; www.anegadabeachclub.com), opened in late 2013 by Driftwood Resorts, is notable for its modern, upscale rooms and swimming pool\u2014the island's only pool.\n\nAnegada Beach Cottages   You'll feel like a castaway on your own curving slice of sugary sand at this unspoiled oceanfront hideaway. Set on the sands at Pomato Point, the three fully furnished one- and two-bedroom cottages are spread out, separated from one another by some 500 feet of sand and tropical brush. Each cottage is sweetly prim and neat as a pin, with beamed ceilings, louvered windows to let in the breezes, and a fully equipped kitchen. One cottage is actually two cottages built together and can be rented as a half or a whole. During the day, guests lounge about, snorkel, or go bonefishing.\n\n  284\/495-9234. www.anegadabeachcottages.com. 4 units. Year-round $140 double. 2-night minimum stay.\n\nAnegada Reef Hotel   For nearly 40 years, this has been the island's main lodging, opened in 1976 by Lowell and Vivian Wheatley. Still owned and operated by the Wheatley family, the Anegada Reef sits right on a white-sand beach, about a 5-minute drive west of the airport. It's a friendly, casual place that has freshened up the rooms, and all come with private porches and garden or ocean views. The hotel's restaurant serves breakfast, lunch, and dinner daily; its evening specialty is Anegada spiny lobster, served up at alfresco tables beneath the stars. If you're visiting the island for the day, you can use the hotel as a base and arrange to go deep-sea fishing or bonefishing (it has a tackle shop). In addition, it's a good place from which to set up snorkeling excursions and secure taxi service and jeep rentals. Call and they'll have a van meet you at the airport.\n\n  284\/495-8002. www.anegadareef.com. 20 units. Winter $265\u2013$400 double; off season $240\u2013$365 double. Rates include all meals. Amenities: Restaurant; bar; babysitting; Internet ($5 per 15 min.).\n\nAnegada\n\nWhere to Eat\n\nBig Bamboo   CARIBBEAN One of the most popular of the local beach bars and restaurants, the Big Bamboo has a large, open-air dining pavilion set right on gorgeous Loblolly Bay. Locals and daytrippers alike come for the famous fresh Anegada lobster (served with island sides like rice and vegetables) as well as locally caught conch. You can also order babyback ribs and spicy barbecue chicken. Chill out after a big meal in a nearby hammock beneath a coconut palm.\n\nLoblolly Bay.   284\/499-1680. Reservations recommended for large groups. Main courses $16\u2013$40 lunch, $22\u2013$45 dinner. Daily 9am\u20139pm.\n\nCow Wreck Beach Bar & Grill   WEST INDIAN Don't miss a sunset sun-downer at this family-run joint, a favorite among yachties anchoring at Anegada. Diners relax on a terrace where tables and plastic chairs feature a view of the water just steps away. If you go during lunch, you can snorkel in the clear seas until your food is ready. Fans claim the Cow Wreck serves the best lobster in the B.V.I., but it also offers tasty grilled steaks, ribs, chicken and local grilled fish.\n\nLower Cow Wreck Beach.   284\/495-8047. <http:\/\/cowwreckbeach.com>. Reservations required for dinner. Main courses $18\u2013$53. Daily 7am\u20136pm; dinner at 6pm but it can accommodate later reservations. Closing time for bar \"when the last customer departs.\"\n\nNeptune's Treasure   INTERNATIONAL The funky family-run bar and restaurant on the island's southwestern shore is the heart of this lodging\/dining establishment. Neptune's Treasure hosts a lively mix of yachties and locals. The Soares family (a legendary local fishing clan) serve heaping platters of lobster, fish, chicken, steaks, and ribs in the spacious indoor dining area, which also has a bar and scrapbooks of the family's record-breaking fishing catches. The family is happy to provide tips on local snorkeling sites and fishing spots, and deftly maintain order amid the festive atmosphere.\n\nThe restaurant also offers nine guest rooms and two cottages, all with beach views and air-conditioning. Depending on the season, rooms (with a private bathroom) rent for $97 to $150 per night, and cottages go for $300 to $370.\n\nBtw. Pomato and Saltheap points.   284\/495-9439, or shortwave channel 16. www.neptunestreasure.com. Reservations for dinner must be made by 4pm. Breakfast $8\u2013$12; main courses lunch $12\u2013$32, dinner $18\u2013$49. Daily 8am\u201310pm.\n\nPETER ISLAND\n\nHalf of this island is devoted to the yacht club, with a good marina and docking facilities. The other part is deserted. A gorgeous beach is found at palm-fringed Deadman's Bay, which faces the Atlantic but is protected by a reef. All goods and services are at the one resort (see below).\n\nThe public Peter Island Ferry (  284\/495-2000) shuttles passengers between Road Town on Tortola and Peter Island at least five times a day; nonguests pay round-trip fares of $20 adults; $10 children.\n\nYou can also get to the resort via a private Peter Island Ferry from the airport in St. Thomas, which includes taxi transport to the ferry terminal. It's a 90-minute trip and the round-trip cost is $199 per person; reservations must be made in advance. In addition, a private Peter Island ferry transports connecting guests between Beef Island and Peter Island (it includes the taxi ride from the airport to the Peter Island dock at Trellis Bay); the round-trip cost is $60 per person, and the trip takes 30 minutes.\n\nWhere to Stay & Eat\n\nPeter Island Resort   This 720-hectare (1,779-acre) tropical island is the domain of Peter Island Resort guests and those boat owners who moor their crafts here. The island is rimmed by five idyllic beaches, among them Deadman's Beach  , often voted one of the most romantic beaches in the world.\n\nThe resort itself is comprised of Oceanview Rooms in two-story, A-frame structures between the pool and the harbor and Beachfront Junior Suites at the far end of Deadman's Beach (and three villa \"estates\"). The Oceanview Rooms, with pitched ceilings, dark woods, and a blue-green palette, are the smallest and least expensive (and least desirable) units, although all rooms come with a balcony or terrace. A big step up are the luxuriously spacious and sunny Beachfront Junior Suites. More extravagant still are the ultra-swanky four- and six-bedroom villa estates, which come with private chef, valet, chauffeured vehicle, and housekeeper. The Crow's Nest, a posh four-bedroom villa, overlooks the harbor and Deadman's Bay, and features a private swimming pool. The Hawk's Nest villas are three-bedroom villas set on a tropical hillside.The resort has two restaurants. The more casual Deadman's Beach Bar & Grill is set right on the beach, enjoying a secluded setting amid sea-grape trees and fringing palms. It offers a Wednesday-night Caribbean buffet, accompanied by a steel-drum band. The Tradewinds restaurant hosts a Saturday-night Gala Buffet, featuring Caribbean seafood and meat-carving stations.\n\n  800\/346-4451 in the U.S., or 284\/495-2000. www.peterisland.com. 52 units. Winter $800\u2013$940 double, $1,280\u2013$1,560 junior suite, $4,600\u2013$14,500 villa; off season $400\u2013$640 double, $750\u2013$1,180 junior suite, $3,100\u2013$12,500 villa. Rates include all meals and beverages, except in Ocean View Rooms and Beachfront Junior Suites. Children 11 and under stay in parent's room for $50 per night. Amenities: 2 restaurants; 2 bars; babysitting; health club and spa; pool (outdoor); 4 tennis courts (lit); watersports equipment\/rentals; Wi-Fi (free in lobby and clubhouse).\n\nGUANA ISLAND\n\nThis 340-hectare (840-acre) island resort is one of the most private hideaways in the Caribbean. Don't come here looking for action; do come if you seek serenity and rustic old-school luxury. Just off the north coast of Tortola, Guana has virgin beaches, nature trails, and rare and unusual species of plant and animal life, including iguana, red-legged tortoise, and the Caribbean roseate flamingo. Arawak relics have even been discovered here. You can climb 242m (794-ft.) Sugarloaf Mountain for a panoramic view. Guana Island sends a boat to meet arriving guests at the Beef Island airport (trip time is 10 min.).\n\nWhere to Stay & Eat\n\nGuana Island   Guana Island was purchased in 1974 by dedicated conservationists Henry and Gloria Jarecki, and it's both a nature preserve and wildlife sanctuary. This is one stunning landscape: Upon your arrival by boat, a Land Rover meets you at the docks and transports you up one of the most scenic hills in the region. From above you can see an old salt pond dotted with pink flamingos and the lacy fringes of beautiful White Bay, the island's main beach. Guana even has the ruins of an old sugarcane plantation. The island is a model of self-sufficiency, with its own desalination plant and a private organic orchard, growing tropical fruits like papaya and key lime. The island has seven beaches in all, five of which require a boat to reach. All have sand as soft as baby powder, lapped by gin-clear sapphire seas.\n\nThe cluster of white stone cottages was built as a private club in the 1930s, on the foundations of a Quaker homestead. Each cottage has its own unique decor, but most have rustic wood-beam ceilings and New England\u2013style wainscotting. The resort never holds more than 35 guests, and because the dwellings are staggered along a flower-dotted ridge overlooking the Caribbean and Atlantic seas, the sense of privacy is almost absolute. The Sea View pool cottage has its own private pool and three rooms. The North Beach villa has its own pool and beach. In fact, all four villas have infinity pools. Sixty percent of the cottages have air-condioning, but most guests find the hillside breezes plenty cooling. Hors d'ouevres are served nightly (6:30pm) in the stone living room, followed by 7:30pm candlelight dinner on the veranda's communal tables. A new chef from Spain was getting raves in the kitchen at press time. Kids 8 and under stay free at Guana during \"Kids' Weeks\" in July and August. The entire island can be rented by groups of up to 35 ($22,000 and up per day).\n\n  800\/544-8262 in the U.S., or 284\/494-2354. www.guana.com. 15 units. Winter $1,250\u2013$2,125 cottage, $2,325\u2013$6,500 villa; off season $695\u2013$1,965 cottage, $1,535\u2013$6,500 villa. Rates include all meals and drinks served with meals. Closed Sept\u2013Oct. Children welcome certain times of year. Amenities: Restaurant; honor bar; babysitting; 2 tennis courts (lit); watersports equipment\/rentals.\n8\n\nPLANNING YOUR TRIP TO THE VIRGIN ISLANDS\n\nA little preparation is essential before you start your journey to the Virgin Islands, especially if you plan on making island-hopping a big part of your itinerary. This chapter tackles the how-tos of a trip to the Virgin Islands, including everything from finding the best airfare to deciding when to go to choosing the best tour or excursion. For on-the-ground resources, head straight to \"Fast Facts,\" beginning on p. .\n\nGETTING THERE\n\nFor American citizens, visiting the U.S. Virgin Islands is relatively easy and hassle-free: Because it's part of the U.S. territory, you won't even need a passport to enter the country on arrival. American citizens do need a passport to enter the British Virgin Islands, however. For complete information on passports and visas, go to \"Fast Facts,\" later in this chapter.\n\nBy Plane\n\nA number of major airlines have regularly scheduled nonstop air service from cities all over North America into St. Thomas\u2014the major international gateway to the Virgins. But most flights include stopovers in Miami or San Juan. Currently, there are no direct flights from North America or Europe to any of the British Virgin Islands. Anyone planning to visit the B.V.I. will likely have to fly into St. Thomas, San Juan (Puerto Rico), or Miami and make a connection by ferry or air in lieu of a direct flight (there are also connections through St. Kitts and Antigua). Those traveling from overseas will also most likely make a connection in St. Thomas, St. Croix, or San Juan after first connecting in the mainland U.S.\n\nThe major airports in the Virgin Islands are the Cyril E. King Airport (  340\/774-5100; www.viport.com\/airports.html; airport code STT) in St. Thomas and the Henry E. Rohlsen Airport, Estate Mannings Bay (  340\/778-1012; airport code STX), on St. Croix. From these airports, you can take ferries or small planes on to your destination in the Virgin Islands.\n\nFor more information on getting to each island, see the \"Getting There\" sections in the individual island chapters.\n\nBy Cruise Ship\n\nThe Virgin Islands are a popular stop for cruise ships traveling the Caribbean, in particular Charlotte Amalie in St. Thomas, one of the world's busiest cruise ports, which welcomed nearly 1,800,000 cruise passengers in 2012, and that number was expected to rise by 7.8 percent for the 2013\/2014 season. In comparison, the historic waterfront of St. Croix saw the arrival of only 117,000 cruise passengers in 2012. Tortola recorded nearly 400,000 cruise passengers in 2012 but lost a big chunk of its market when Carnival Cruise Line announced in late 2013 that it would suspend travel to Tortola indefinitely while the island overhauls its docking facilities.\n\nthe cruise ships\n\nSt. Thomas\n\nMost of the major cruise lines include regular stops in St. Thomas in their Caribbean itineraries, including the biggest cruise ships in the world, such as the Royal Caribbean's mega ships Allure of the Sea and Oasis of the Sea, each with a maximum passenger capacity of more than 6,000 people; Norwegian Cruise Line's Epic; and Princess Cruise Line's Royal Princess.\n\nThe port at Charlotte Amalie is one of the world's busiest cruise ports, welcoming nearly 1,800,000 cruise passengers in 2012 (the last date for which there are figures). Cruise ships dock at one of two major piers, each with room for two mega-ships at a time: Havensight Pier and Crown Bay. In high season for cruising, it's not unusual to have an additional one or two ships anchored in the harbor, delivering cruise-ship passengers to shore in tenders.\n\nA number of smaller cruise ships visit the waters of the Virgin Islands without docking, including Club Med, Star Clipper, and Windsar Cruises.\n\nSt. Croix\n\nCruise lines that make stops in St. Croix included Celebrity (Summit, Century); Norwegian Cruise Line (Jewel); Royal Caribbean (Vision, Adventure); Holland America (Maasdam); and Silversea (Silver Cloud).\n\nTortola\n\nCruise lines that dock in Road Town, Tortola, include Regent Seven Seas (Windstar); Costa Cruise Lines (Costa Magica); Crystal; Celebrity; Royal Caribbean (Jewel of the Seas); Cunard; Holland America; Silver Seas (Silver Spirit); and P&O Cruises (Arcadia).\n\nGETTING AROUND\n\nBy Plane\n\nRegular flights are scheduled between St. Thomas and St. Croix, and between St. Thomas and Tortola. Seaborne Airlines (  866\/359-8784; www.seaborneairlines.com) flies between St. Thomas and St. Croix on several runs daily; Seaborne also flies between St. Thomas and Beef Island, Tortola. Cape Air (  866\/227-3247 in the U.S. and U.S.V.I.; www.capeair.com) offers regularly scheduled flights between St. Thomas and both St. Croix and Tortola.\n\nSt. John has no airport; passengers usually land first at St. Thomas, then take the 20-minute ferry ride to St. John.\n\nferry COMPANIES: WHERE THEY GO\n\nFerry service is an essential transportation link in the Virgins, and most of the ferries get you where you want to go with speed and efficiency. But know that not all ferry companies are alike\u2014boats vary from sleek high-speed catamarans (Road Town Fast Ferry) to older, no-frills models (pretty much all the rest). Some ferries are more reliably on time than others; also check to see if your ferry is \"nonstop\"\u2014some ferries, like Inter Island, may make pit shops on the way to your final destination. Fares are not necessarily economical; in addition, nonresidents traveling by sea now have to pay a $15 passenger tax per person ($10 B.V.I. residents) anytime they depart the B.V.I. Still, we love the ferries\u2014they're scenic bliss. Here is a breakdown of the various public ferry companies and where they go; keep in mind that itineraries are subject to change. Note that children and seniors pay discounted fares, and that some ferries charge an extra $2 per piece of luggage. And note that you'll save by buying round-trip tickets.\n\nInter Island Boat Services (  284\/495-4166; www.interislandboatservices.com): Makes runs between Red Hook (St. Thomas), Cruz Bay (St. John), and the West End (Tortola), with stops on Virgin Gorda, Jost Van Dyke, and Anegada. Rates between St. Thomas and Tortola are $30 one-way; $45 round-trip.\n\nNative Son (  284\/495-4617 in the U.S.V.I. or 340\/774-8685 in the B.V.I.; www.nativesonferry.com): Makes daily runs between St. Thomas (both Red Hook and Charlotte Amalie) and Tortola (both West End and Road Town). Prices are $30 one-way; $50 round-trip.\n\nRoad Town Fast Ferry (  340\/777-2800 in the U.S.V.I or 284\/494-2323 in the B.V.I.; www.roadtownfastferry.com): Offers daily runs between Charlotte Amalie and Road Town, Tortola, on sleek air-conditioned high-speed catamarans. Fares are $35 one-way; $60 round-trip; group rates available.\n\nSmith's Ferry Service (  340\/775-5235; www.bviferryservices.com): Known as the \"Tortola Fast Ferry,\" Smith's operates daily service between Tortola and St. Thomas; between Tortola and Virgin Gorda; and charter service to outer B.V.I. islands. Prices $30 to $40 one-way; $50 to $60 round-trip.\n\nSpeedy's (  284\/495-7292; www.bviferries.com): Speedy's operates routes between Charlotte Amalie (St. Thomas) and Tortola (Road Town); Charlotte Amalie and Virgin Gorda; Tortola (Road Town and Beef Island) and Virgin Gorda. B.V.I. domestic fares are $20 one-way, $30 round-trip. Fares for travel between the U.S.V.I. and the B.V.I. are $25\u2013$40 one-way, $45\u2013$70 round-trip.\n\nBy Boat\n\nFerry service is a vital link on the Virgin Islands and a wonderful, leisurely way to see these beautiful islands by water. On the U.S. Virgin Islands, public ferries between St. Thomas and St. John (private water taxis also operate on this route) run at regular times all day long. Launch services link Red Hook, on the East End of St. Thomas, with both Charlotte Amalie in St. Thomas and Cruz Bay in St. John (not to mention Tortola and Virgin Gorda). At press time, there was no regular ferry between St. Thomas and St. Croix (although plans were in the works at press time for a high-speed ferry service between the islands).\n\nFerries are also a vital link between the U.S. Virgins and the B.V.I.\u2014and the interisland public ferries are generally a more economical alternative to flying between the U.S. Virgins and the B.V.I. Ferries run from both ferry terminals on St. Thomas (Charlotte Amalie and Red Hook) to either West End or Road Town on Tortola, a 45- to 55-minute voyage. In the B.V.I., ferries and private boats link Road Town, Tortola, with the island's West End; there's also service to and from Virgin Gorda and some of the smaller islands, such as Anegada and Jost Van Dyke. Note that on some of the really remote islands, boat service may be only once a week. Many of the private island resorts, such as Peter Island, provide launches from Tortola or from the airport in St. Thomas.\n\nFor more details on specific ferry connections, including sample fares, see the \"Getting Around\" sections of the individual island chapters.\n\nBy Car\n\nA rental car is often the best way to get around each of the Virgin Islands. Just remember the most important rule: In both the U.S. and the British Virgin Islands, you must drive on the left.\n\nAll the major car-rental companies are represented in the U.S. Virgin Islands, including Avis (  800\/331-1212; www.avis.com), Budget (  800\/626-4516; www.budget.com), and Hertz (  800\/654-3131; www.hertz.com); many local agencies also compete in the car-rental market (for detailed information, see the \"Getting Around\" sections in individual island chapters). On St. Thomas and St. Croix, you can pick up most rental cars at the airport. On St. John, there are car-rental stands at the ferry dock. Cars are sometimes in short supply during the high season, so reserve as far in advance as possible.\n\nParking lots in the U.S. Virgin Islands can be found in Charlotte Amalie, on St. Thomas, and in Christiansted, on St. Croix (in Frederiksted, you can generally park on the street). Most hotels, except those in the congested center of Charlotte Amalie, have free parking lots.\n\nEven though taxi service in the British Virgin Islands is readily available, we highly recommend renting a car, particularly in Tortola. Given that, you'll have to drive on the left along roads that can be hairy or feel like roller-coaster rides. (If you plan to stay in the B.V.I. longer than 30 days, you must purchase a temporary local driver's license for $10 from police headquarters or a car-rental desk in town.) You must be at least 25 years old to rent a car in the B.V.I. Most of the major U.S. car-rental companies are represented on these islands, but you'll find a number of reliable local companies as well, many conveniently located near the ferry docks and in the main towns. Vehicles come in a wide range of styles and prices, including Jeeps, Land Rovers, mini mokes, and even six- to eight-passenger Suzukis. Weekly rates are usually slightly cheaper.\n\nNote: There are no car-rental agencies at the airports on Tortola or Virgin Gorda.\n\nGasoline St. Thomas has plenty of service stations, especially on the outskirts of Charlotte Amalie and at strategic points in the north and in the more congested East End. On St. Croix, most gas stations are in Christiansted, but there are also some along the major roads and at Frederiksted. On St. John, make sure your tank is filled up at Cruz Bay before heading out on a tour of the island.\n\nGas stations are not as plentiful on the British Virgin Islands. Road Town, the capital of Tortola, has the most gas stations; fill up here before touring the island. Virgin Gorda has a limited but sufficient number of gas stations. Chances are you won't be using a car on the other, smaller British Virgin Islands.\n\nTaxes are already included in the printed price. One U.S. gallon equals 3.8 liters or .85 imperial gallons.\n\nBreakdowns All the major islands, including St. Thomas, St. John, St. Croix, Tortola, and Virgin Gorda, have garages that will tow vehicles. Always call the rental company first if you have a breakdown. If your car requires extensive repairs because of a mechanical failure, a new one will be sent to replace it.\n\nBy Taxi\n\nTaxis are the main mode of transport on all the Virgin Islands. On St. Thomas, taxi vans carry up to a dozen passengers to multiple destinations; smaller private taxis are also available. You'll find plenty of taxis on arrival at the airport. On St. John, both private taxis and vans for three or more passengers are available. On St. Croix, taxis congregate at the airport, in Christiansted, and in Frederiksted, where the cruise ships arrive. On all the islands, you'll see more and more open-air safari \"buses\" (more like retrofitted flat-bed trucks) capable of handling up to 30 passengers, which taxi drivers use largely to transport groups (for instance, cruise-ship passengers) on island tours and excursions. Increasingly, these safari buses are also used as pickup and dropoff taxis.\n\nThroughout the U.S.V.I, standard per-person taxi rates are set by the local government (look for a complete rate listing at the St. Thomas airport or in free local magazines like \"This Week\")\u2014but it's always good to confirm the rate before the ride begins.\n\nOn the British Virgin Islands, taxis are readily available and, on some to the smaller islands, often the best way to get around. Service is available on Tortola, Virgin Gorda, and Anegada, and rates are fixed by the local government.\n\nTip: It's highly recommended that if you find a good taxi driver on the islands, get his or her card\u2014not only will you have a reliable driver to drop you off and pick you up places but most drivers are smart and entertaining island guides. Rates for sightseeing taxi tours are also generally regulated by the government. The standard taxi tour on St. Thomas, for example, is $50 for one or two passengers and $25 per extra passenger.\n\nBy Bus\n\nThe only islands with recommendable bus service are St. Thomas and St. Croix. On St. Thomas, buses leave from Charlotte Amalie and circle the island; on St. Croix, air-conditioned buses run from Christiansted to Frederiksted. Bus service elsewhere is highly erratic; it's mostly used by locals going to and from work.\n\nTIPS ON ACCOMMODATIONS\n\nThroughout this book, we provide detailed descriptions of the lodging properties so that you get an idea of what to expect. Keep in mind that many of the more high-end island resorts charge a daily resort fee of between $35 and $50.\n\nResorts and hotels in the Virgin Islands offer money-saving package deals galore, and you can find land-air (and land-air-rental-car) packages on almost all of the online travel agencies (Expedia.com, Orbitz.com, Priceline.com, VacMart.com, CheapCaribbean.com). Be sure to check the hotel's own websites: Many hotels offer terrific online-only multi-stay or theme deals (especially in the off season).\n\nLike most Caribbean islands, the Virgins have high and low seasons, and properties are priced accordingly. The most exorbitant rates are charged during the Christmas and New Year's holidays. If you have flexibility in your travel times, keep in mind that during the off season (mid-Apr to mid-Dec), most hotels offer tremendous deals, often slashing 25 to 50 percent off their regular-season prices.\n\nRenting Your Own Villa or Vacation Home\n\nAnother popular lodging alternative, and a smart money-saving option if you're traveling with a large party of family or friends, is renting a villa, condo, apartment, or cottage for your Virgin Islands vacation. Having your own self-catering facilities can be a big money-saver\u2014dining out is chillingly pricey in the Virgin Islands. Note: If you're planning your trip for the high season, reservations should be made at least 5 to 6 months in advance.\n\nDozens of agencies throughout the United States and Canada offer rentals in the Virgin Islands. Villas of Distinction (  800\/289-0900; www.villasofdistinction.com) offers \"complete vacations,\" including car rental and domestic help. Its private luxury villas have one to seven-plus bedrooms, and many have swimming pools. Rates run from $500 a night and up.\n\nVacation St. Croix, 4000 La Grande Princesse, Christiansted (  877\/788-0361 or 340\/718-0361; www.vacationstcroix.com), offers some of the best properties on St. Croix, specializing in villas, condos, and private homes, many of which are on the beach. Two- to seven-bedroom units are available, with prices from $1,000 to $15,000 per week.\n\nBest of BVI (  1252-674878 from the U.K. and 011\/44-1252-674878 from the U.S; www.bestofbvi.com) handles properties you won't find anywhere else, including a one-bedroom cottage on Little Thatch Island, a privately owned island minutes from Soper's Hole; Best of BVI also has a hugely informative trip-planning website, with an up-to-date ferry schedule). Jewels of the BVI (  866\/468-6284; www.jewelsofthebvi.com) represents only BVI Islander\u2013owned properties, including villas, condos, and resorts.\n\nAt Home Abroad (  212\/421-9165; www.athomeabroadinc.com) has a roster of private luxury homes, villas, and condos for rent in St. Thomas, St. John, Tortola, and Virgin Gorda; maid service is included in the price. In the U.K., browse the options offered through Holiday Rentals (  020\/8846-3441; www.holiday-rentals.co.uk).\n\nYou can also find excellent deals on popular owner-rented vacation lodging websites, including VRBO (www.vrbo.com) and HomeAway (www.homeaway.com); both websites list numerous attractive villas and condos throughout the Virgins. What you will not get with these rentals is service (unless advertised) or even guarantees that the rental is what is appears to be. Both VRBO and HomeAway sell rental guarantee insurance ($39 and up) that guarantees your money back under certain circumstances (the property has been double-booked or grossly misrepresented, for example)\u2014so you may want to weigh that extra fee against the low rental rates.\n\nSPECIAL-INTEREST TRIPS & TOURS\n\nThere's no rule that says you have to confine yourself to a beach chair while visiting the Virgin Islands (although there's no rule against it, either). You will have endless opportunities to sit by the surf sipping rum drinks, but remember that these islands offer more than just ribbons of fetching white-sand beach. Coral reefs and deepwater channels provide backdrops for a variety of watersports, from snorkeling to sea kayaking to sailing, and the lush island interiors make ideal playgrounds for scenic hikes and breathtaking vistas. This section presents an overview of tours, special-interest trips, and outdoor excursions on the Virgin Islands. See individual chapters for more specific information on locations and outfitters. Note: During the low summer season, tours and excursions may not run regularly unless enough people book, so always call in advance to confirm.\n\nAir Tours\n\nIt's a pricey option, but well-heeled visitors may want to see the Virgin Islands as the birds do. Helicopter sightseeing tours, island-hopping day-trips, and heli-adventures (heli-fishing in Anegada, for example) are available through Caribbean Buzz Helicopters, in St. Thomas (  340\/775-7335; <http:\/\/caribbean-buzz.com>). A Full Moon tour takes you over St. Thomas and St. John on 15- or 30-minute jaunts by moonlight. Helicopters seat five passengers and a pilot and are also available to rent for island transfers.\n\nIf you can't afford such a luxury, you can get a panoramic aerial view from Paradise Point on St. Thomas (p. ).\n\nAdventure Trips\n\nDive in: The Virgin Islands is one big outdoor playground, with aquatic pleasures, rain-forest peaks, and breathtakingly scenic sightseeing. For general guided tours, contact the following three tour operators:\n\nOne of the best tour operators in the U.S.V.I. is Adventure Center, in the Marriott Frenchman's Reef Resort, Flamboyant Point, St. Thomas (  340\/774-2992; www.adventurecenters.net). It offers a wide variety of tours, on both land and sea, and many tours of both St. Thomas and St. John depart right from the hotel's dock. Their 21\u20442 hour \"Scenic Island Drive\" tour explores St. Thomas's highlights, including the Botanical Gardens, Drake's Seat, and Charlotte Amalie; this tour costs $49 per person. Their \"St. John Land and Snorkel Safari\" is a full-day tour that takes visitors through the beaches and national parks in St. John, including a snorkel stop in Trunk Bay, in open-air buses; it costs $95 per person. Adventure Center offers an array of other quality guided tours throughout the Virgin Islands.\n\nSea Turtle Etiquette\n\nThese are some of the most highly endangered species in the oceans. Catching even a passing glimpse of one is a magical experience, but you'll blow the chance unless you heed some basic guidelines. When you first spot a sea turtle, resist the urge to move in and get a closer look; you will only scare it off and ruin the opportunity for others to see it. Instead, stay still and watch at a respectful distance as it goes about its business, searching for food or gliding along gracefully. Keep an eye out for identification tags on their flippers or shells\u2014a sure sign these fellas are being closely studied and well protected. You should never approach a turtle or its nest, and never touch or try to touch one\u2014for your safety and theirs. Although it seems harmless to humans, it is in fact quite stressful for the turtles (how'd you like to be chased around the grocery store by strangers all day?). Warning: Do not swim above the turtles; it will prevent them from surfacing to breathe and subject them to undue respiratory stress. And, of course, if someone offers you sea turtle shell, egg, or meat products, just say no.\n\n\u2014Christina P. Col\u00f3n\n\nTan Tan Tours (  340\/773-7041; www.stxtantantours.com) is a tour operator based in St. Croix that offers off-the-beaten-path jeep tours of the island. A 21\u20442-hour guided swim in the Anally Bay tide pools costs $100 per person, while a full-day (8-hr.) tour of the island costs $160 per person; custom tours are also available.\n\nB.V.I. Eco-Tours (  284\/495-0271; www.bvi-ecotours.com) offers a variety of guided tours, including snorkeling, hiking, bird watching, sightseeing, diving, and general tours of the British Virgin Islands.\n\nVirgin Gorda Tours (  284\/495-5240; www.virgingordatours.com) offers guided taxi tours of Virgin Gorda, including stops at Savannah Bay, Gorda Peak, the historical Copper Mine ruins, and the famous Baths; tours vary, but a 1-hour tour (for two people) costs $55.\n\nBicyclE TOURS Bike riding can be a wonderful way to explore the islands. Water Island Adventures (  340\/775-5770; www.waterislandadventures.com) offers guided bike tours on Water Island, off of St. Thomas. Tours depart from the Crown Bay cruise-ship dock at St. Thomas, where you'll take a ferry to Water Island\u2014from there the biking tour begins.\n\nSt. Croix has miles of relatively flat roadways that make it ideal for biking, but it also has lush hills for the more adventurous. Contact Freedom City Cycles, 2E Strand Sq., 2 Strand St., Frederiksted (  340\/227-2433; www.freedomcitycycles.com), which, in addition to offering bike rentals, can arrange guided bike tours of the island. A 2- to 3-hour mountain bike tour begins at sea level and climbs through the rain forest on both paved and unpaved roads, costing $60 per person.\n\nFishing The Virgin Islands are home to some of the best fishing grounds in the world. More than 20 sportsfishing world records have been set in the Virgin Islands in the last few decades, mostly for the big kahuna of the sea, blue marlin. Other abundant fish in these waters are bonito, tuna, wahoo, sailfish, and skipjack. Sport-fishing charters, led by experienced local captains, abound in the islands; both half-day and full-day trips are available. But you needn't go out to sea to fish. On St. Thomas, St. John, and St. Croix, the U.S. government publishes lists of legal shoreline fishing spots (contact local tourist offices for more information). Closer inshore, you'll find kingfish, mackerel, bonefish, tarpon, amberjack, grouper, and snapper.\n\nOn St. Thomas, many people line-fish from the rocky shore along Mandahl Beach, which is also a popular spot for family picnics. The shore here is not the best place for swimming, because the seafloor drops off dramatically and the surf tends to be rough. On St. John, the waters in Virgin Islands National Park are open to fishermen with hand-held rods. No fishing license is required for shoreline fishing, and government pamphlets available at tourist offices list some 100 good spots. Call   340\/774-8784 for more information.\n\nHiking TOURS The best islands for hiking are Tortola and St. John. In Tortola, the best hiking is through Sage Mountain National Park, spread across 37 hectares (91 acres) of luxuriant flora and fauna. On St. John, the most intriguing hike is the Annaberg Historic Trail, which takes you by former plantation sites. Most of St. John is itself a national park, so there are dozens of opportunities for hiking.\n\nSt. Croix also has good hiking opportunities. To reach some of the most remote but scenic places on St. Croix, take a walking tour with Crucian Heritage & Nature Tourism (CHANT;   340\/772-4079; <http:\/\/chantvi.org>). Its \"Ay Ay Eco-Hike Tours\" include a steep walk down the mountainside to the crystalline tidal pools and saltwater baths of Annaly Bay, and a walking trip along scenic Maroon Ridge, established by runaway slaves in the 17th century ($50 per person).\n\nBuck Island, off the coast of St. Croix, is beloved by snorkelers and scuba divers but also fascinating to hike. You can easily explore the island in a day\u2014it's just a half-mile wide and a mile long. While hiking in the Virgin Islands, you'll encounter many birds and flowers\u2014but no poisonous snakes. Be sure to look for the trumpet-shaped Ginger Thomas, the U.S. Virgin Islands' official flower.\n\nSailing & Yachting The Virgin Islands are a sailor's paradise, offering crystal-clear turquoise waters, secluded coves and inlets, and protected harbors for anchoring. The most popular cruising area around the Virgin Islands is the deep and lushly scenic Sir Francis Drake Channel, which runs from St. John to Virgin Gorda's North Sound. The channel is rimmed by mountainous islands and boasts steady tradewinds year-round. In heavy weather, the network of tiny islands shelters yachties from the brute force of the open sea. The waters surrounding St. Croix to the south are also appealing, especially near Buck Island. Outside the channel, the Virgin Islands archipelago contains reefy areas that separate many of the islands from their neighbors.\n\nFor details on arranging a multi-day boat charter\u2014bareboat (on your own) or fully crewed\u2014see \"Chartering Your Own Boat,\" below. Most visitors, however, are content with day sails, which are easy to organize, especially at the harbors in St. Thomas, Tortola, and Virgin Gorda. Regardless of where you decide to cruise, you really shouldn't leave the islands without spending at least 1 day on the water, even if you have to load up on Dramamine or snap on your acupressure wristbands before you go. For sailing companies on each island, go to the individual chapters.\n\nSailing, Sailing: Distances by Nautical Mile\n\nExcept for Anegada, a low-lying atoll of coral limestone and sandstone, all the Virgin Islands are high and easily spotted. The shortest distance between St. Thomas and St. Croix is 35 nautical miles; from St. John to St. Croix, 35 nautical miles; from St. Thomas to St. John, 2 nautical miles; from Tortola to St. Thomas, 10 nautical miles; from Virgin Gorda to Anegada, 13 nautical miles; and from St. John to Anegada, 30 nautical miles. Virgin Gorda to St. Croix is about the longest run, at 45 nautical miles. (Specific distances between the islands can be misleading, however, because you often need to take roundabout routes from one point to another.)\n\nIf you don't know how to sail but want to learn, there's no better place than the Virgin Islands. On Tortola, the Tortola Sailing School (  800\/390-7594; <http:\/\/tortolasailingschool.com>) is an accredited American Sailing Association sailing school, with courses in both sailing and bareboating running out of Soper's Hole Marina on the island's West End. It offers a full-day \"Learn the Ropes\" basics course on a 36-foot monohull for $175 per person. It also offers multi-day liveaboard sailing and bareboating courses. On St. Croix, call Jones Maritime Sailing School, 1215 King Cross St., Christiansted (  340\/773-4709; www.jonesmaritime.com), which has three 24-foot day-sailors and charges $325 per person for a 2-day course, held on Saturdays and Sundays.\n\nCHARTERING YOUR OWN boat\n\nThere may be no better way to experience the \"Sailing Capital of the World\" than on the deck of your own yacht. Impossible? Not really. No one said you had to own the yacht. Experienced sailors and navigators with a sea-wise crew might want to rent a bareboat charter\u2014a fully equipped boat with no captain or crew, where you are the master of the seas, charting your own course, and do your own cooking and cleaning. If you're not an expert sailor but yearn to hit the high seas, consider a fully crewed charter, which includes a captain and chef. The cost of a crewed boat is obviously more than that of a bareboat, and varies according to crew size and experience.\n\nMost boats, bareboat or crewed, are rented on a weekly basis and come equipped with a GPS system. Full-service crewed boats come in all shapes and sizes, from traditional sailboats to roomy multihulls to elegant motor yachts. Crewed charters generally come with a fully stocked kitchen (or a barbecue) and bar, fishing gear, and watersports equipment. For details on crewed yachts, go to www.bvicrewedyachts.com. Private boats are given up to 30 days to cruise the B.V.I. and are required to pay a 30-day cruising fee based on tonnage (maximum fee $55). Contact the British Virgin Islands Customs Department (  284\/494-3475 or 284\/468-3701, ext. 2533) for current cruising permit requirements.\n\nAmong the outfitters in the Virgin Islands, the Moorings in Tortola (  888\/952-8420; www.moorings.com) offers both bareboat and fully crewed charters equipped with such extras as a barbecue, snorkeling gear, a dinghy, and linens. The company even supplies windsurfing equipment for free with crewed boats (and for an extra cost with bareboats). The experienced staff of mechanics, electricians, riggers, and cleaners is extremely helpful, especially if you're going out on your own. They'll give you a thorough briefing about Virgin Islands waters and anchorages. If you're looking for bareboat and full-crew charter companies in St. Thomas, you'll find the craft you're looking for through IGY: American Yacht Harbor (  340\/775-6454; www.igy-americanyachtharbor.com), a full-service marina located in Red Hook and home to numerous charter companies.\n\nThe Offshore Sailing School, Prospect Reef Resort, Road Town (  284\/494-5119) is the official sailing school for the Moorings, the bareboat charter outfitter based in Road Town; it offers sailing instruction year-round (  888\/454-7015 or 239\/454-1700; www.offshore-sailing.com).\n\nSea Kayaking Arawak Expeditions, Cruz Bay, St. John (  800\/238-8687 or 340\/693-8312 in the U.S.; www.arawakexp.com), offers not only kayaking day-trips but multi-day sea-kayaking\/island-camping excursions. The vessels with Arawak Expeditions are in two-person fiberglass kayaks, complete with foot-controlled rudders. The outfit provides all the kayaking gear, healthy meals, camping equipment, and experienced guides. The cost of a full-day trip is $110, half-day $75; you can also book longer expeditions, such as a 5-day excursion costing $1,195 per person or a 7-day trip going for $2,495 per person.\n\nSnorkeling & Scuba Diving St. John has spectacular snorkeling right offshore at beaches like Trunk Bay, Maho Bay, Cinnamon Bay, and Hawksnest Beach. A steep undersea wall just 15 minutes offshore makes for excellent scuba diving.\n\nOn St. Croix, the best site for both is Buck Island, easily accessible by day sails from the harbor in Christiansted. St. Croix is also known for its dramatic \"drop-offs,\" including the famous Puerto Rico Trench.\n\nOn St. Thomas, all major hotels rent fins and masks for snorkelers, and most day-sail charters have equipment on board. Many outfitters, like the St. Thomas Diving Club (  340\/776-2381; www.stthomasdivingclub.com), also feature scuba programs.\n\nOn the British Virgin Islands, some of the best snorkeling is around Norman Island and the Indians; Smuggler's Cove, in Tortola; and the Baths, Virgin Gorda's major attraction. Anegada Reef, which lies off Anegada Island, has been a \"burial ground\" for ships for centuries; an estimated 300 wrecks, including many pirate ships, have perished here. The wreckage of the RMS Rhone, near the westerly tip of Salt Island, is the most celebrated dive spot in the B.V.I. This ship went under in 1867 in one of the most disastrous hurricanes ever to hit the Virgin Islands.\n\nHeritage & Cultural Tours\n\nSt. Croix's cultural riches are explored on tours run by Crucian Heritage & Nature Tourism (CHANT;   340\/772-4079; <http:\/\/chantvi.org>). In addition to historic walking tours of the colonial towns of Christiansted and Frederiksted, CHANT offers a fascinating Ridge to Reef Farm Tour, showcasing the island's agricultural heritage and burgeoning farm-to-table movement with a tour of a 200-acre working tropical farm in a lush valley amid the island's tropical forests. The 3-hour tour is $55 per person.\n\n  The Virgin Islands\n\nArea Codes The area code for the U.S.V.I. is 340; in the B.V.I., it's 284. You can dial direct from North America; from outside North America, dial 001, plus the number for the U.S.V.I., and 011-44 plus the number for the B.V.I.\n\nBusiness Hours See \"Fast Facts\" in individual island chapters for information on business hours.\n\nCrime See \"Safety,\" later in this section.\n\nCustoms Every visitor to the U.S.V.I. 21 years of age or older may bring in, free of duty, the following: (1) 1 liter of wine or hard liquor; (2) 200 cigarettes, 100 cigars (but not from Cuba), or 3 pounds of smoking tobacco; and (3) $100 worth of gifts. These exemptions are offered to travelers who spend at least 72 hours in the United States and who have not claimed them within the preceding 6 months. It is altogether forbidden to bring into the country foodstuffs (particularly fruit, cooked meats, and canned goods) and plants (vegetables, seeds, tropical plants, and the like). Foreign tourists may carry in or out up to $10,000 in U.S. or foreign currency with no formalities; larger sums must be declared to U.S. Customs on entering or leaving, which includes filing form CM 4790. For details regarding U.S. Customs and Border Protection, consult your nearest U.S. embassy or consulate, or U.S. Customs (  800\/232-5378; www.cbp.gov).\n\nVisitors to the B.V.I. can bring in food, with the exception of meat products that are not USDA-approved. Visitors can bring up to $10,000 in currency and 1 liter of alcohol per person.\n\nAustralian Citizens: A helpful brochure available from Australian consulates or Customs offices is \"Know Before You Go.\" For more information, contact the Australian Customs Service, Customs House, 5 Constitution Ave., Canberra City, ACT 2601 (  1300\/363-263, or 61 2 9313 3010 from outside Australia; www.customs.gov.au).\n\nCanadian Citizens: For a clear summary of Canadian rules, write for the booklet \"I Declare,\" issued by the Canada Border Services Agency, Ottawa, Ontario, K1A 0L8 (  800\/461-9999 in Canada, or 204\/983-3500; www.cbsa-asfc.gc.ca).\n\nNew Zealand Citizens: Most questions are answered in a free pamphlet available at New Zealand consulates and Customs offices: \"New Zealand Customs Guide for Travellers, Notice no. 4.\" For more information, contact New Zealand Customs Service, the Customhouse, 1 Hinemoa St., Harbour Quays, Wellington 6140 (  04\/901-4500; www.customs.govt.nz).\n\nU.K. Citizens: From the B.V.I., U.K. citizens can bring back (duty-free) 200 cigarettes (250g of tobacco), 2 liters wine, 1 liter strong liquor, 60cc perfume, and \u00a3145 of goods and souvenirs. Larger amounts are subject to tax. For further information, contact HM Revenue & Customs, Crownhill Court, Tailyour Road, Plymouth, PL6 5BZ (  0300 200 3700; www.hmrc.gov.uk).\n\nU.S. Citizens & Residents: From the U.S.V.I., U.S. citizens can bring back 5 liters of liquor duty-free, plus an extra liter of rum (including Cruzan rum) if one of the bottles is produced in the Virgin Islands. Goods made on the island are also duty-free, including perfume, jewelry, clothing, and original paintings; however, if the price of an item exceeds $25, you must be able to show a certificate of origin.\n\nBe sure to collect receipts for all purchases in the Virgin Islands, and beware of merchants offering to give you a false receipt\u2014he or she might be an informer to U.S. Customs. Also, keep in mind that any gifts received during your stay must be declared. For the most up-to-date specifics on what you can bring back from the B.V.I. and the corresponding fees, contact the U.S. Customs & Border Protection (CBP), 1300 Pennsylvania Ave. NW, Washington, DC 20229 (  877\/227-5511; www.ct.gov\/dcp.gov).\n\nDisabled Travelers For the most part, the accessibility of hotels and restaurants in the U.S.V.I. remains far behind the progress made on the mainland. Of the U.S. Virgins, St. Thomas and St. John remain the most difficult islands for wheelchair-bound visitors to maneuver because of their hilly terrain. St. Croix is flatter and is an easier place to get around.\n\nAlthough most hotels in the Virgin Islands have a long way to go before they become a friend of a person with disabilities, some have made inroads. As of this writing, about a third of the major resorts in St. Thomas or St. Croix have the facilities to accommodate vacationers who have disabilities. Many inns, guesthouses, and villas terraced in the hills of Charlotte Amalie can present challenges to those with mobility issues\u2014a number have steep steps and no elevators. Of the resorts in the U.S.V.I., the Ritz-Carlton, St. Thomas (p. ), is the most hospitable to persons with disabilities. It maintains \"accessible rooms\"\u2014rooms that can be reached without navigating stairs\u2014in every price category. The Ritz also offers beach wheelchairs (resting on balloon tires).\n\nAccessible Island Tours (  340\/344-8302; <http:\/\/accessvi.com>) is a tour operator in St. Thomas that offers a land-based tour of St. Thomas in a custom wheelchair-accessible vehicle. Originating from Wico Dock at Havensight or Crown Bay at the Sub Base, tours stop at Magens Bay, Drakes Seat, and the Skyline Drive, and cost $37 per person (minimum of six passengers).\n\nDoctors You should have no trouble finding a good doctor in the Virgin Islands. See \"Fast Facts\" in individual island chapters for information on doctors.\n\nDrinking Laws In the U.S. Virgins, the legal age for purchase and consumption of alcoholic beverages is 18. Proof of age is required and often requested at bars, nightclubs, and restaurants, so it's always a good idea to bring ID when you go out. Do not carry open containers of alcohol in your car or any public area that isn't zoned for alcohol consumption. The police can fine you on the spot. Don't even think about driving while intoxicated. Although 18-year-olds can purchase, drink, and order alcohol, they cannot transport bottles back to the United States with them. If an attempt is made, the alcohol will be confiscated at the Customs check point. The same holds true for the B.V.I.\n\nIn the B.V.I., the legal minimum age for purchasing liquor or drinking alcohol in bars or restaurants is 18. Alcoholic beverages can be sold any day of the week, including Sunday. You can have an open container on the beach, but be you can be fined if you litter.\n\nDriving Rules In both the U.S.V.I. and the B.V.I., you must drive on the left. See \"Getting Around,\" earlier in this chapter.\n\nElectricity The electrical current in the Virgin Islands is the same as on the U.S. mainland and Canada: 110 to 120 volts AC (60 cycles), compared to 220 to 240 volts AC (50 cycles) in most of Europe, Australia, and New Zealand. Downward converters that change 220 to 240 volts to 110 to 120 volts are difficult to find in the United States, so bring one with you.\n\nEmbassies & Consulates There are no embassies or consulates in the Virgin Islands. If you have a passport issue, go to the local police station, which in all islands is located at the center of government agencies. Relay your problem to whomever is at reception, and you'll be given advice about which agencies can help you.\n\nEmergencies Call   911 in the U.S.V.I. or 999 in the B.V.I.\n\nFamily Travel The Virgin Islands, both U.S. and British, are very family-friendly. St. Thomas and St. Croix have the most facilities and attractions for families. The British Virgin Islands have fewer attractions that cater specifically to children, although families who love watersports, boating, and nature activities will have a great time. When compared with some of the other major destinations in the Caribbean (such as Jamaica, where crime is high), the U.S. Virgins are generally safe, and the British Virgin Islands are even safer.\n\nGasoline Please see \"Getting Around,\" earlier in this chapter, for information.\n\nHealth Other than the typical tropical environment health concerns, like sun exposure and seasickness (see below), there are no major health concerns in the Virgin Islands.\n\nSt. Thomas has the best hospital in the U.S. Virgin Islands (Schneider Regional Medical Center; p. ). St. Croix also has good hospital facilities (St. Croix Regional Medical Center; p. ). There is only a health clinic on St. John; more serious cases are transferred to the hospital on St. Thomas.\n\nThe B.V.I. has one small general hospital, Peebles Hospital (p. ) on Tortola. Day clinics are available on Tortola, Virgin Gorda, Anegada, and Jost Van Dyke. Both Tortola and Virgin Gorda are served by ambulances with paramedics. The Eureka Medical Centre (www.eurekamedicalclinic.com), Geneva Place, Road Town, Tortola, is a private-run urgent-care facility. There is also no hyperbaric chamber in the B.V.I. Patients requiring treatment for decompression illness are transferred to St. Thomas.\n\nIn very serious cases, patients in the U.S. Virgins and the B.V.I. are transported to Puerto Rico.\n\nIt is not difficult to get a prescription filled or find a doctor on St. Thomas, St. Croix, and Tortola. Pharmacies are few and far between on the smaller islands, so you should get any prescriptions refilled before you venture into more remote territory. Often it requires a phone call from the U.S.V.I. to a stateside pharmacy or to the doctor who prescribed the medicine in the first place. CVS and Wal-Mart are the best for contacting a stateside branch of those chains, if your prescription is on a computer file. To avoid possible hassles and delays, it is best to arrive with enough medication for your entire vacation.\n\n  Bugs & Bites Mosquitoes do exist in the Virgin Islands, but they aren't the malaria-carrying mosquitoes that you might find elsewhere in the Caribbean. They're still a nuisance, though. Sand flies, which appear mainly in the evening, are a bigger annoyance. Screens can't keep these critters out, so use bug repellent.\n\n  Dietary Red Flags If you experience diarrhea, moderate your eating habits and drink only bottled water until you recover. If symptoms persist, consult a doctor. Much of the fresh water on the Virgin Islands is stored in cisterns and filtered before it's served. Delicate stomachs might opt for bottled water. Some say a nightly drink of ginger ale and bitters helps soothe tummies.\n\n  Seasickness The best way to prevent seasickness is with the scopolamine patch by Transderm Scop, a prescription medication. Bonine and Dramamine are good over-the-counter medications, although each causes drowsiness. Smooth Sailing is a ginger drink that works quite well to settle your stomach. You might also opt for an acupressure wristband available at drugstores (www.sea-band.com). Some say a ginger pill taken with a meal and followed by Dramamine an hour before boating also does the job.\n\n  Sun Exposure The Virgin Islands' sun can be brutal. To protect yourself, consider wearing sunglasses and a hat, and use sunscreen (SPF 15 and higher) liberally. Limit your time on the beach for the first few days. If you overexpose yourself, stay out of the sun until you recover. If your sunburn is followed by fever, chills, a headache, nausea, or dizziness, see a doctor.\n\nHospitals The largest hospital in St. Thomas is the Schneider Regional Medical Center (p. ), with 24-hour emergency-room service. Islanders from St. John also use this hospital, which is about a 5-minute drive from Charlotte Amalie. The other major hospital is the St. Croix Regional Medical Center on St. Croix (p. ); it has a Level IV trauma center offering 24-hour emergency-room service. Both offer air and ground-level support to hospitals with more extensive facilities. The payment of Medicare and Medicaid operates as it does in the United States. If you walk into a hospital without any coverage or insurance, you are expected to pay.\n\nOn Tortola, in the British Virgin Islands, the main hospital in the little country is Peebles Hospital (p. ), with surgical, X-ray, and laboratory facilities. If you are on one of the out islands, you are generally taken to Tortola for treatment. In addition to these hospitals, there are a number of private doctors' offices throughout the islands, charging higher rates than the hospitals.\n\nInternet & Wi-Fi Internet access is becoming increasingly available all around the Virgin Islands, but it can still be spotty on some of the more remote islands. Most hotels and resorts are ratcheting up their Internet capabilities. Many bars and cafes throughout the Virgin Islands have free Wi-Fi access.\n\nIf you're in transit and looking for a spot with Internet access, see the \"Fast Facts\" section of each island chapter for recommendations on where to go.\n\nLanguage English is the official language of both the U.S. and British Virgin Islands.\n\nLegal Aid While driving, if you are pulled over for a minor infraction (such as speeding), never attempt to pay the fine directly to a police officer; this could be construed as attempted bribery, a much more serious crime. Pay fines by mail, or directly into the hands of the clerk of the court. If accused of a more serious offense, say and do nothing before consulting a lawyer. In the U.S.V.I., the burden is on the state to prove a person's guilt beyond a reasonable doubt, and everyone has the right to remain silent, whether he or she is suspected of a crime or actually arrested. Once arrested, a person can make one telephone call to a party of his or her choice.\n\nLGBT Travelers The Virgin Islands are some of the most gay-friendly destinations in the Caribbean. However, discretion is still advised. Islanders tend to be religious and conservative, and displays of same-sex affection, such as hand holding, are frowned upon.\n\nSt. Thomas is the most cosmopolitan of the Virgin Islands, but it is no longer the \"gay paradise\" it was in the 1960s and 1970s. Most gay vacationers now head for Frederiksted, in St. Croix, which has more hotels and other establishments welcoming to the gay market. In Charlotte Amalie, on St. Thomas, the most boisterous gay nightlife takes place in the Frenchtown section of the city.\n\nMail At press time, domestic postage rates in the U.S.V.I. were 33\u00a2 for a postcard and 46\u00a2 for a letter up to 1 ounce. For international mail, a first-class postcard or letter stamp costs $1.10. For more information, go to www.usps.com. Always include zip codes when mailing items in the U.S. If you don't know your zip code, visit www.usps.com\/zip4.\n\nIf you aren't sure what your address will be in the U.S. Virgin Islands, mail can be sent to you, in your name, c\/o General Delivery at the main post office of the city or region where you expect to be. (Call   800\/275-8777 for information on the nearest post office.) The addressee must pick up mail in person and must produce proof of identity (driver's license, passport, and so on). Most post offices will hold your mail for up to 1 month, and are open Monday to Friday 8am to 6pm, and Saturday 9am to 3pm.\n\nPostal rates in the British Virgin Islands to the United States or Canada are 35\u00a2 for a postcard (airmail), and 50\u00a2 for a first-class airmail letter (1\u20442 oz.). Mailing a postcard to the U.K. costs 50\u00a2 and a first-class letter via airmail costs 75\u00a2 (1\u20442 oz.). B.V.I. postage stamps are beautiful and highly coveted; contact the BVI Philatelic Bureau (  284\/494-7789) for information about exhibitions.\n\nMedical Requirements Unless you're arriving from an area known to be suffering from an epidemic (particularly cholera or yellow fever), inoculations or vaccinations are not required for entry into the U.S. Virgin Islands or the British Virgin Islands.\n\nIf you have a medical condition that requires syringe-administered medications, carry a valid signed prescription from your physician; syringes in carry-on baggage will be inspected. Insulin in any form should have the proper pharmaceutical documentation. If you have a disease that requires treatment with narcotics, you should also carry documented proof with you\u2014smuggling narcotics aboard a plane carries severe penalties in the U.S.\n\nFor HIV-positive visitors, requirements for entering both the U.S.V.I. and B.V.I. are somewhat vague and change frequently. Anyone who does not appear to be in good health may be required to undergo a medical exam, including HIV testing, prior to being granted or denied entry. For up-to-the-minute information, contact AIDSinfo (  800\/448-0440 or 301\/519-0459 outside the U.S.; www.aidsinfo.nih.gov) or the Gay Men's Health Crisis (  212\/367-1000; www.gmhc.org). Also see \"Health.\"\n\nMobile Phones In the U.S. Virgin Islands: The two largest cellphone operators in the U.S.V.I. include Sprint PCS (www.sprint.com) and AT&T Wireless (www.att.com\/wireless). Phones operating in the mainland U.S. under those plans will usually operate seamlessly, and without any excess roaming charges, in the U.S.V.I. If your phone presently operates through some other carrier, it's wise to call them before your departure about signing up (at least temporarily) for one of their international plans, which will save you money on roaming charges during the duration of your trip. If your cellphone is not equipped for reception and transmission in the U.S.V.I., consider renting (or buying) a cheap cellphone for temporary use, or, less conveniently, head for a Sprint PCS or AT&T sales outlet (each maintains offices on all three of the U.S.V.I.'s major islands) for a substitute SIM card, a key operating component that can be inserted into your existing phone, making it operational. Throughout the U.S.V.I., the electrical system is the same as within the U.S. mainland (115 volts and female sockets which accept the U.S.-style \"flat\" plugs), so most U.S. residents won't need any special transformers or adaptors.\n\nIn the British Virgin Islands: The three largest cellphone operators in the B.V.I. are CCT Global Communications (www.cctwireless.com), LIME (www.lime.com), and Digicell BVI (www.digicelbvi.com), all with offices in Road Town and on Virgin Gorda. Other than that, the cellphone situation is roughly equivalent to what's described immediately above in the U.S.V.I. The electrical system in the B.V.I. is the same as that within the U.S.V.I. and the mainland U.S. (115 volts), so British and European visitors may want to bring adaptors and transformers. Hotels in the B.V.I. often have the appropriate adaptors, and in some cases, those adaptors are physically built directly into the wall sockets.\n\nMoney & Costs The U.S. Virgin Islands and the British Virgin Islands both use the U.S. dollar as the form of currency. Frommer's lists exact prices in the local currency. The currency conversions quoted above were correct at press time. However, rates fluctuate, so before departing consult a currency exchange website such as www.xe.com to check up-to-the-minute rates.\n\nBanks on the islands are your only option if you need to exchange currency. These rates can be expensive, and additional charges are often tacked on; it is best to change money before you arrive.\n\nATMs throughout the Virgin Islands dispense U.S. dollars. ATMs are most prevalent on St. Thomas in Charlotte Amalie (on the downtown streets, near the cruise-ship terminals, within the large resorts, and in shopping malls) and in Christiansted on St. Croix. You will also find several ATMs in Cruz Bay on St. John. ATMs are less prevalent in the British Virgin Islands; you will find a cluster of banks in Wickham Cay I, Road Town, Tortola, and a couple in the harbor in Spanish Town, Virgin Gorda. The other islands do not have ATMs, so if you're planning a visit, be sure to visit an ATM to get some cash first (or have your resort front you some petty cash). Each machine charges around $2 to $3 for a transaction fee. Nearly all of the machines are operated by three banks: Scotiabank (www.scotiabank.com), FirstBank (www.firstbankvi.com), and Banco Popular (www.bancopopular.com\/vi).\n\nMost establishments in the Virgin Islands accept credit cards; we note in our reviews those places that accept cash only. MasterCard and Visa are widely accepted on all the islands that cater to visitors, especially Virgin Gorda, Tortola, St. John, St. Croix, and, of course, St. Thomas. In the past few years, there has been a tendency to drop American Express because of the high percentage it takes from transactions.\n\nHowever, visitors should not rely solely on credit cards, as a number of establishments in the Virgin Islands accept only cash. You will want to arm yourself with cash while browsing the small boutiques and curio shops throughout the islands\u2014many do not take credit cards. Most taxi drivers only deal in cash.\n\nBeware of credit card fees while traveling. Check with your credit or debit card issuer to see what fees, if any, will be charged for overseas transactions. Recent reform legislation in the U.S., for example, has curbed some exploitative lending practices. But many banks have responded by increasing fees in other areas, including fees for customers who use credit and debit cards while out of the country\u2014even if those charges were made in U.S. dollars. Fees can amount to 3 percent or more of the purchase price. Check with your bank before departing to avoid any surprise charges on your statement.\n\nPassports If you're a U.S. citizen and you travel directly to the U.S.V.I. and do not visit the British Virgin Islands, you do not need a passport\u2014but you are highly encouraged to carry one. If you return to the mainland U.S. from the U.S.V.I. through another country (Mexico or Bermuda, for example), you will need a passport to get back home. For non\u2013U.S. citizens, visiting the U.S. Virgin Islands is just like visiting the mainland United States: You need a passport and visa.\n\nA passport is necessary for all visitors to the British Virgin Islands (including citizens of the U.K.).\n\nFor information on how to get a passport, contact your passport office (see below). Allow plenty of time before your trip to apply for a passport; processing normally takes 3 weeks but can take longer during busy periods. And keep in mind that if you need a passport in a hurry, you'll pay a higher processing fee. When traveling, safeguard your passport in an inconspicuous, inaccessible place like a money belt, and keep a copy of the critical pages with your passport number in a separate place. There are no foreign consulates in the Virgin Islands, so if you lose your passport, go to the local police station.\n\nPassport Offices\n\nAustralia Australian Passport Information Service (  131-232; www.passports.gov.au).\n\nCanada Passport Office, Passport Canada Program, Gatineau QC K1A 0G3 (  800\/567-6868; www.ppt.gc.ca).\n\nIreland Passport Office, Frederick Buildings, Molesworth Street, Dublin 2 (  353 1 671 1633; www.dfa.ie).\n\nNew Zealand Passports Office, Department of Internal Affairs, 109 Featherston St., Wellington, 6140 (  0800 22 50 50 in New Zealand, or 64 4 463 9360; www.passports.govt.nz).\n\nUnited Kingdom Visit your nearest passport office, major post office, or travel agency, or contact the HM Passport Office, 4th Floor, Peel Building, 2 Marsham St., London, SW1P 4DF (  0300 222 0000; www.ips.gov.uk).\n\nUnited States To find your regional passport office, check the U.S. State Department website (<http:\/\/travel.state.gov\/passport>) or call the National Passport Information Center (  877\/487-2778) for automated information.\n\nPetrol Please see \"Getting Around,\" earlier in this chapter for information.\n\nPets To bring your pet to the U.S.V.I., you must have a health certificate from a mainland veterinarian and show proof of vaccination against rabies. Very few hotels allow animals, so check in advance. If you're strolling with your dog through the national park on St. John, you must keep it on a leash. Pets are not allowed at campgrounds, in picnic areas, or on public beaches. Both St. Croix and St. Thomas have veterinarians listed in the Yellow Pages.\n\nYour dog or cat is permitted entry into the B.V.I. without quarantine, if accompanied by an Animal Health Certificate issued by the Veterinary Authority in your country of origin. This certificate has a number of requirements, including a guarantee of vaccination against rabies.\n\nPolice Dial   911 for emergencies in the U.S.V.I. The Crime Line phone number is   340\/777-8700. The main police headquarters is currently located in the Alexander Farrelly Criminal Justice Center in Charlotte Amailie (  340\/774-2211). In the B.V.I., the main police headquarters is on Waterfront Drive near the ferry docks on Sir Olva George's Plaza (  284\/494-2945) in Tortola. There are also police stations on Virgin Gorda (  284\/495-5222) and on Jost Van Dyke (  284\/495-9345). See individual island chapters for more detailed information.\n\nSafety The Virgin Islands are a relatively safe destination. The small permanent populations are generally friendly and welcoming. That being said, St. Thomas is no longer as safe as it once was. Crime, especially muggings, is on the rise in Charlotte Amalie. Wandering the town at night, especially on the back streets (particularly on Back St.), is not recommended. Guard your valuables or store them in hotel safes if possible.\n\nThe same holds true for St. Croix and the back streets of Christiansted and Frederiksted. Although these areas are safer than St. Thomas, random acts of violence against tourists in the past, even murder, have been known to happen. Know that most crime on both these islands is petty theft aimed at unguarded possessions on the beach, unlocked parked cars, or muggings at night. Exercise the same amount of caution you would if you were traveling to an unfamiliar town on the mainland. Whether on St. Thomas or St. Croix, always take a taxi home after a night out.\n\nSt. John is a bit different, because there is no major town and most of the island is uninhabited. Muggings and petty theft do happen, but such occurrences are rarely violent. You are most likely to find your camera stolen if you leave it unattended on the beach.\n\nThe British Virgin Islands are very safe, with a very low crime rate that many attribute to the illegality of owning guns. Minor robberies and muggings do occur late at night outside bars in Road Town, especially in poorly lit areas around Wickham's Cay I and along Waterfront Drive. Car theft in Trellis Bay and by the Road Town ferry has also been on the rise. On Virgin Gorda, most resorts don't even have room keys (although you can lock yourself in at night), and some people have reported dropping off rental cars at the airport with the keys in the lock.\n\nDriving safety: In general, the Virgin Islands' steep, curvy roads are often poorly lit at night; many are potholed or have been eroded by rain runoff. St. Croix's road network is composed of rocky, steep dirt roads through the interior. As a result, car-rental insurance is higher on this island than the others. St. John's national park roads are for the most part excellent. For those travelers who are unaccustomed to driving on the left, we suggest leaving the night driving up to a taxi driver. Do not attempt the most rural roads at night, as cellphone service is spotty at best and breakdowns (or worse) are an all-too-perfect way to ruin your Virgin Islands vacation; see \"Getting Around,\" earlier in this chapter.\n\nSmoking In the U.S.V.I, smoking is prohibited in restaurants and public buildings; bars may allow smoking outdoors as long as it's 20 feet from entrance and service areas. On the B.V.I., smoking is banned in public places (bars, restaurants, nightclubs, airports, offices, and sports facilities) and within 50 feet of any public space.\n\nStudent Travel St. Thomas has perhaps the most youth-oriented scene of any of the Virgin Islands, British or American. Many young people who visit St. Thomas stay in the guesthouses in and around Charlotte Amalie. Beyond St. Thomas, the island of St. Croix attracts a large array of young, single travelers, mainly to the inns in and around Christiansted and Frederiksted.\n\nTaxes For the U.S. Virgin Islands, the United States has no value-added tax (VAT) or other indirect tax at the national level. The U.S.V.I. may levy their own local taxes on all purchases, including hotel and restaurant checks and airline tickets. These taxes will not appear on price tags. A 10 percent room tax is added to hotel bills.\n\nThe British Virgin Islands has no sales tax. It charges a departure tax of $15 per person for those leaving by boat or $20 if by airplane. Most hotels add a service charge of around 10 percent; there's also a 7 percent government room tax. Most restaurants tack on an automatic 15 percent service charge.\n\nTelephones In the Virgin Islands, hotel surcharges on long-distance and local calls are usually astronomical, so you're better off using your cellphone or a public pay telephone. Many convenience stores, groceries, and packaging services sell prepaid calling cards in denominations up to $50; for international visitors these can be the least expensive way to call home. Many public pay phones at airports now accept American Express, MasterCard, and Visa credit cards. Local calls made from pay phones in most locales cost either 25\u00a2 or 35\u00a2 (no pennies, please). Many of the most rural or expressly private resorts and hotels in the Virgin Islands do not provide phones in the rooms, but have phones in their lobbies or common areas.\n\nTo make calls within the United States, including the U.S. Virgins, and to Canada, dial 1 followed by the area code and the seven-digit number. For other international calls, dial 011 followed by the country code, city code, and the number you are calling.\n\nYou can call the British Virgins from the United States by just dialing 1, the area code 284, and the number; from the U.K. dial 011-44, then the number. To call the U.S. from the B.V.I., just dial 1 plus the area code and the number; to call the U.K. from the B.V.I., dial 011-44, then the number.\n\nCalls to area codes 800, 888, 877, and 866 are toll-free. However, calls to area codes 700 and 900 (chat lines, bulletin boards, \"dating\" services, and so on) can be very expensive\u2014usually a charge of 95\u00a2 to $3 or more per minute, and they sometimes have minimum charges that can run as high as $15 or more.\n\nFor reversed-charge or collect calls, and for person-to-person calls, dial the number 0, then the area code and number; an operator will come on the line, and you should specify whether you are calling collect, person-to-person, or both. If your operator-assisted call is international, ask for the overseas operator.\n\nFor local directory assistance (\"information\"), dial 411; for long-distance information, dial 1, then the appropriate area code and 555-1212.\n\nTime The Virgin Islands are on Atlantic Standard Time, which is 1 hour ahead of Eastern Standard Time. However, the islands do not observe daylight saving time, so in the summer, the Virgin Islands and the East Coast of the U.S. are on the same time. In winter, when it's 6am in Charlotte Amalie, it's 5am in Miami; during daylight saving time it's 6am in both places.\n\nTipping In hotels, tip bellhops at least $1 per bag ($2\u2013$3 if you have a lot of luggage) and tip the chamber staff $1 to $2 per day (more if you've left a disaster area for him or her to clean up). Tip the concierge only if he or she has provided you with some specific service (obtaining difficult-to-get dinner reservations, for example). Tip the valet-parking attendant $1 every time you get your car.\n\nNote that many local restaurants tack on a service charge to the total bill, often between 10 and 15 percent; you may want to add extra if the service was good. Otherwise tip waitstaff 15 to 20 percent of the check. In bars and nightclubs, tip bartenders 15 to 20 percent of the check, tip checkroom attendants $1 per garment, and tip valet-parking attendants $1 per vehicle.\n\nAs for other service personnel, tip taxi drivers 15 percent of the fare; tip skycaps at airports at least $1 per bag ($2\u2013$3 if you have a lot of luggage); and tip hairdressers and barbers 15 to 20 percent. It's always a good idea to tip tour guides or charter captains at the end of an excursion, generally 15 to 20 percent of the cost.\n\nToiletsYou won't find public toilets or restrooms on the streets, but they can be found in hotel lobbies, bars, restaurants, museums, department stores, bus stations, and service stations. Large hotels are often the best bet for clean facilities.\n\nVisas Non\u2013U.S. visitors to the U.S. Virgin Islands should have a U.S. visa; those visitors may also be asked to produce an onward ticket. In the British Virgin Islands, visitors who stay for less than 6 months don't need a visa if they possess a return or onward ticket.\n\nFor information about U.S. Visas, go to <http:\/\/travel.state.gov> and click on \"Visas.\" Or go to one of the following websites:\n\nAustralian citizens can obtain up-to-date visa information from the U.S. Embassy Canberra, Moonah Place, Yarralumla, ACT 2600 (  02\/6214-5600), or by checking the U.S. Diplomatic Mission's website at <http:\/\/canberra.usembassy.gov\/visas.html>.\n\nBritish subjects can obtain visa appointments the U.S. Embassy Visa Appointment Line (  020-3608-6998 from within the U.K., or   703\/439-2367 from within the U.S.; https:\/\/ais.usvisa-info.com\/en-gb). Irish citizens can obtain up-to-date visa information through the U.S. Embassy Dublin, 42 Elgin Rd., Ballsbridge, Dublin 4 (  01 903-6255 from within the Republic of Ireland; <http:\/\/dublin.usembassy.gov.visas.html>).\n\nCitizens of New Zealand can obtain up-to-date visa information by contacting the U.S. Embassy New Zealand, 29 Fitzherbert Terrace, Thorndon, Wellington (  04 462 6000 from within New Zealand; <http:\/\/newzealand.usembassy.gov\/visas.html>).\n\nVisitor Information Go to the U.S.V.I. Division of Tourism's website at www.visitusvi.com. The British Virgin Islands Tourist Board can be found at www.bvitourism.com.\n\nWater Many visitors to both the U.S. and British Virgins drink the local tap water with no harmful effects. To be prudent, especially if you have a delicate stomach, stick to bottled water. Many hotels and resorts have their own desalination plant, making delicious and highly potable water out of seawater.\n\nWomen Travelers St. John and the British Virgin Islands have a low crime rate, while St. Thomas and St. Croix have the highest crime rate against women in the archipelago. To put that into context, however, you are far safer in the Virgin Islands than you would be walking the streets of any major U.S. city. Follow the usual precautions that you'd follow in any major U.S. city.\nIndex\n\nSee also Accommodations and Restaurant indexes, below.\n\nGeneral Index\n\nA\n\nAccessible Island Tours,\n\nAccommodations, 3. See also Condo, apartment, and villa rentals; and Accommodations Index\n\nAnegada, ,\n\nbest resorts, 5\u20138\n\neco-friendly,\n\nJost Van Dyke, 181\u2013182\n\nPeter Island,\n\nSt. Croix, 93\u201399\n\nSt. John, 125\u2013130\n\nSt. Thomas, , 49\u201359\n\ntips on, 195\u2013196\n\nTortola, 155\u2013160\n\nVirgin Gorda, 174\u2013178\n\nAccra,\n\nAdventure Center (St. Thomas), , ,\n\nAdventure trips, 197\u2013198\n\nAgricultural & Cultural Food Fair (St. Thomas),\n\nAgrifest: St. Croix Agricultural Fair,\n\nAir Sunshine (British Virgin Islands), ,\n\nAir tours,\n\nAir travel, 191\u2013192\n\nAnegada,\n\nBritish Virgin Islands, ,\n\nPeter Island,\n\nSt. Croix, 87\u201388\n\nSt. John,\n\nSt. Thomas, ,\n\nTortola,\n\nVirgin Gorda,\n\nAlice in Wonderland (deep-dive site, south of Ginger Island), ,\n\nAmerican Airlines, ,\n\nAmerican Yacht Harbor (St. Thomas), ,\n\nAmple Hamper (Tortola),\n\nAnegada, , , , 185\u2013188\n\nAnegada Lobster Festival, ,\n\nAnegada Reef,\n\nAnnaberg Historic Trail (St. John), , 140\u2013141\n\nAnnaberg Sugar Plantation Ruins, ,\n\nAnnaberg Trail (St. John),\n\nAnnaly Meat Market (St. Croix),\n\nAnn E. Abramson Pier (St. Croix),\n\nAnnual House Tours (St. Croix),\n\nAnnual St. Croix International Regatta,\n\nAntilles Resorts (St. Thomas and St. Croix),\n\nAntiques, St. Thomas,\n\nApple Bay (Tortola), ,\n\nAqua Action Dive Center (St. Thomas), ,\n\nAragorn's Local Arts and Crafts Center (Tortola), , 170\u2013171\n\nArawak Expeditions (St. John), , ,\n\nArawak Indians,\n\nArawak Surf Shop (Tortola),\n\nArchitecture,\n\nArea codes,\n\nAristocat Day Sails (Tortola),\n\nARTFarm (St. Croix),\n\nArt galleries\n\nSt. Croix,\n\nSt. John, 141\u2013142\n\nSt. Thomas,\n\nArt in the Garden Arts & Crafts Festival (St. Thomas),\n\nAsante Studio (Tortola),\n\nAt Home Abroad,\n\nAvis, , , , ,\n\nAwl Made Here (St. John),\n\nB\n\nBack Street (Charlotte Amalie), ,\n\nBajo El Sol (St. John),\n\nBamboula (St. John),\n\nBamboushay Pottery (Tortola),\n\nBanks\/ATMs, 206\u2013207\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nVirgin Gorda,\n\nBaptist, Avery,\n\nBath & Turtle (pubbn),\n\nThe Baths (Virgin Gorda), , , , 179\u2013180\n\nBeaches\n\nbest, 2\u20134\n\nJost Van Dyke,\n\nSt. Croix, 106\u2013107\n\nSt. John, , 136\u2013138\n\nSt. Thomas, 67\u201369\n\nTortola, 165\u2013166\n\nVirgin Gorda, 179\u2013180\n\nBeach Side Caf\u00e9 at Sand Castle on the Beach (St. Croix),\n\nBed & breakfasts (B&Bs), St. John, 129\u2013130\n\nBeef Island,\n\nThe Belgian Chocolate Factory (Charlotte Amalie),\n\nBest of BVI, ,\n\nBicycle tours,\n\nBicyling, St. Croix,\n\nBig Beard's Adventure Tours, ,\n\nBioluminescent bays, St. Croix,\n\nBiras Creek (Virgin Gorda),\n\nBirding, , ,\n\nBlue Moon (St. Croix),\n\nBlue Water Divers (Tortola),\n\nBlue Water Terrace (St. Croix),\n\nBlue Water Terrace (St. John), market and deli,\n\nBoating and sailing, , 199. See also Kayaking\n\nchartering your own boat,\n\nresponsible travel,\n\nSt. John,\n\nSt. Thomas, 69\u201370\n\nspecial events, ,\n\nTortola, , ,\n\nBoat travel, 193\u2013194. See also Cruise ships; Ferries; Water taxis\n\nSt. Croix,\n\nSt. John, ,\n\nSt. Thomas,\n\nVirgin Gorda,\n\nBobby's Marketplace (Tortola),\n\nBomba Surfside Shack (Tortola),\n\nBooks, 31\u201332\n\nBookstores, Tortola,\n\nBoolchand's (St. Thomas),\n\nBordeaux farmer's market (Charlotte Amalie),\n\nBotany Bay, Preserve at, ,\n\nBougainvillea (St. John),\n\nBoyson,\n\nBrandywine Bay (Tortola),\n\nBrewers Bay (St. Thomas),\n\nBrewers Bay (Tortola),\n\nBritish Virgin Islands, 144\u2013190\n\nbrief description of, 23\u201324\n\nhistory of, , 27\u201329\n\nlandmass,\n\nsuggested itineraries, 15\u201316\n\ntransportation, 147\u2013149\n\ntraveling to, , 146\u2013147\n\nU.S. Virgin Islands compared to,\n\nvisitor information,\n\nThe Buccaneer (St. Croix)\n\ngolf course,\n\ntennis,\n\nBuck Island (off St. Croix),\n\nbeach,\n\nnature trails,\n\nscuba diving,\n\nside trip to, 119\u2013120\n\nsnorkeling, , ,\n\nBuck Island National Park,\n\nBuck Island Reef National Monument (off St. Croix),\n\nBudget car rentals, , , ,\n\nBugs and bites,\n\nBusiness hours\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nBus travel,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nButler Bay (St. Croix),\n\nB.V.I. Conservation & Fisheries Department,\n\nB.V.I. Eco-Tours, ,\n\nB.V.I. Emancipation Celebrations (Tortola), 38\u201339\n\nBVI Fishing Complex (Tortola),\n\nB.V.I. Music Festival (Tortola),\n\nBVI Restaurant Week,\n\nB.V.I. Spring Regatta & Sailing Festival,\n\nBVI Taxi Association (Tortola),\n\nC\n\nCalendar of events, 37\u201339\n\nCallaloo,\n\nCallwood Rum Distillery (Tortola),\n\nCalypso,\n\nCalypso Realty (St. Thomas),\n\nCamille Pissarro Building (Charlotte Amalie),\n\nCampgrounds, , , ,\n\nCane Bay (St. Croix), , , ,\n\nCane Bay Dive Shop (St. Croix),\n\nCane Bay Wall (St. Croix),\n\nCaneel Bay Beach (St. John),\n\nCane Garden Bay (Tortola), , , , , ,\n\nCape Air, , , , , ,\n\nCarambola Golf & Country Club (St. Croix), ,\n\nCardow Jewelers (St. Thomas),\n\nCaret Bay (St. Thomas),\n\nCar ferries, St. John, ,\n\nCaribbean Saloon (St. Thomas),\n\nCaribbean Villas & Resorts (St. John),\n\nCarib Indians,\n\nCarl and Marie Lawaetz Museum (St. Croix),\n\nCar rentals\n\nBritish Virgin Islands,\n\nJost Van Dyke,\n\nSt. Croix,\n\nSt. Thomas, ,\n\nTortola,\n\nVirgin Gorda, ,\n\nCartanser Sr. (wreck), 71\n\nCar travel, 194\u2013195\n\nSt. Croix,\n\nSt. John,\n\nVirgin Gorda, ,\n\nCasino, St. Croix,\n\nCellphones, 205\u2013206\n\nCentral Post Office (Charlotte Amalie),\n\nCharlotte Amalie (St. Thomas)\n\naccommodations, 52\u201354\n\narchitecture,\n\nlayout of,\n\nrestaurants, 61\u201364\n\nsightseeing, 72\u201378\n\ntraffic congestion,\n\nCharteryacht League (St. Thomas),\n\nChikuzen (wreck), , 169\n\nChristiansted (St. Croix), ,\n\naccommodations,\n\narchitecture,\n\nrestaurants, 102\u2013103\n\nshopping,\n\nsuggested itinerary,\n\nwalking tour, 111\u2013114\n\nChristiansted's Harborfront (St. Croix),\n\nChristmas in St. Croix,\n\nCinnamon Bay (St. John),\n\nCinnamon Bay Campground (St. John),\n\nCinnamon Bay Trail\/Cinnamon Bay Loop Trail (St. John),\n\nCinnamon Bay Watersports Center (St. John), 137\u2013139\n\nClover Crest (St. Croix),\n\nClub St. Croix Beach & Tennis Resort,\n\nCoconut Coast Studios (St. John),\n\nCoki Beach (St. Thomas), ,\n\nCoki Dive Center (St. Thomas), , ,\n\nCoki Point Beach (St. Thomas), ,\n\nColonization, 26\u201328\n\nThe Color of Joy (St. Thomas),\n\nColumbus, Christopher, ,\n\nColumbus Sunset Bar (Tortola),\n\nConch,\n\nCondo, apartment, and villa rentals\n\nJost Van Dyke,\n\nSt. Croix,\n\nSt. John, 128\u2013129\n\nSt. Thomas,\n\ntips on,\n\nTortola, ,\n\nVirgin Gorda,\n\nContinental Airlines,\n\nCookbooks,\n\nCoral Bay (St. John), , , 134\u2013137,\n\nCoral World Ocean Park (St. Thomas),\n\nCow and Calf Rocks (St. Thomas), 4\u20135,\n\nCraft Alive Village (Tortola),\n\nCramer Park (St. Croix),\n\nCreque Dam Road (St. Croix),\n\nCrown Bay (Charlotte Amalie),\n\nCrown Bay Center (Charlotte Amalie),\n\nCrown House (Charlotte Amalie),\n\nCrown Mountain Road (St. Thomas),\n\nCrucian Christmas Carnival,\n\nCrucian Gold (St. Croix),\n\nCrucian Heritage & Nature Tourism (St. Croix), , , ,\n\nCruise ships, , , ,\n\nCruzan Carnival Night (St. Thomas),\n\nCruzan Rum Factory (St. Croix),\n\nCruz Bay (St. John), , ,\n\nnightlife,\n\nrestaurants, 132\u2013134\n\nCruz Bay Watersports (St. John), ,\n\nCrystal Shoppe (St. Thomas),\n\nCuisine, 34\u201335\n\nCurrency,\n\nCustoms, 201\u2013202\n\nCustoms House (St. Croix), ,\n\nCyril E. King Airport (St. Thomas), ,\n\nD\n\nThe Danish West India & Guinea Warehouse (St. Croix), 112\u2013113\n\nDanish West India Company, , , ,\n\nDavid Yurman (St. Thomas),\n\nDavis Bay (St. Croix), ,\n\nDelta,\n\nDenmark, ,\n\nDentists\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nVirgin Gorda,\n\nDependable Car Rental (St. Thomas),\n\nDevil's Bay National Park (Virgin Gorda),\n\nDezarie,\n\nDiamonds International (St. Thomas),\n\nDietary red flags,\n\nDining. See Restaurants\n\nDisabled travelers,\n\nDiscount Car Rental (St. Thomas),\n\nDive BVI, 180\u2013181,\n\nDIVE IN! (St. Thomas), ,\n\nDivi Carina Bay Casino (St. Croix),\n\nDoctors,\n\nSt. Croix,\n\nSt. John, 124\u2013125\n\nSt. Thomas,\n\nVirgin Gorda,\n\nDohm's Water Taxi, ,\n\nDolphin Water Taxi,\n\nDonald Schnell Studio (St. John),\n\nDon't Stop the Carnival (Wouk), 31\n\nDrinking laws,\n\nDriving rules, ,\n\nDriving safety,\n\nDrugstores\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nVirgin Gorda,\n\nDuffy's Love Shack (St. Thomas),\n\nDutch settlers, ,\n\nE\n\nEagle Ray (St. Croix),\n\nEast End (St. Croix),\n\nEast End (St. Thomas),\n\naccommodations, 56\u201358\n\nbeaches, 68\u201369\n\nrestaurants, 66\u201367\n\nEast End (Tortola),\n\naccommodations,\n\nrestaurants, 164\u2013165\n\nshopping, 170\u2013171\n\nEating and drinking, 34\u201336. See also Food stores and markets; Restaurants\n\nsustainable eating,\n\nEconomy, 29\u201330\n\nEcotourism,\n\nElectricity,\n\nEmancipation Park (Charlotte Amalie), ,\n\nEmbassies and consulates,\n\nEmergencies,\n\nJost Van Dyke,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nVirgin Gorda,\n\nEnid M. Baa Public Library (Charlotte Amalie),\n\nEnvironmental Protection Agency, U.S.,\n\nEpernay Bistro & Wine Bar (St. Thomas),\n\nEstate Bordeaux (St. Thomas), farm-fresh produce,\n\nEstate Bordeaux Agricultural & Cultural Food Fair (St. Thomas),\n\nEstate Concordia Preserve,\n\nEstate Mount Washington (St. Croix), ,\n\nEstate St. Peter Greathouse & Botanical Gardens (St. Thomas), 78\u201379\n\nEstate Whim Plantation Museum (St. Croix), , 116\u2013117\n\nEureka Medical Centre (Tortola),\n\nF\n\nFahie Hill Murals (Tortola),\n\nFamily travel,\n\nbest resorts, 6\u20137\n\nFantasy (boat), 70\n\nFerries, 193\u2013194\n\nAnegada,\n\nBritish Virgin Islands, 146\u2013148\n\nJost Van Dyke,\n\nPeter Island,\n\nSt. John, ,\n\nTortola, ,\n\nFerries to St. John,\n\nFilms,\n\nFishing,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nU.S. Virgin Islands Open\/Atlantic Blue Marlin Tournament,\n\nFish market, St. Croix,\n\nFlax, Andy,\n\nFleming, Christopher,\n\nFood stores and markets\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas, ,\n\nTortola, ,\n\nFort Berg (St. John), ,\n\nFort Christian (Charlotte Amalie), , ,\n\nFort Christiansvaern (St. Croix),\n\nFort Frederik (St. Croix), 114\u2013115\n\nFoxy's Old Year's Night (Jost Van Dyke),\n\nFrance, ,\n\nFrancis Bay Beach (St. John),\n\nFranklin's on the Waterfront (St. Croix),\n\nFrederik Church Parsonage (Charlotte Amalie),\n\nFrederik Lutheran Church (Charlotte Amalie),\n\nFrederiksted (St. Croix), , 90\u201391\n\naccommodations, 97\u201398\n\nshopping,\n\nsightseeing,\n\nFrederiksted Pier (St. Croix), ,\n\nFred's Patio (St. John),\n\nFreedom City Cycles (St. Croix), ,\n\nFrench Cap Pinnacle (off St. Thomas),\n\nThe French Deli (Tortola),\n\nFrenchman's Bay Beach (St. Thomas),\n\nFrenchtown (St. Thomas), , ,\n\nrestaurants,\n\nFungi, ,\n\nG\n\nGallery Camille Pissarro (Charlotte Amalie), ,\n\nGallery St. Thomas,\n\nGasoline, 194\u2013195\n\nGay and lesbian nightlife, St. Thomas,\n\nGladys' Caf\u00e9 (Charlotte Amalie),\n\nGlobal Marine,\n\nGolf\n\nSt. Croix, ,\n\nSt. Thomas,\n\nGood Moon Farm (Tortola),\n\nGorda Peak National Park (Virgin Gorda),\n\nGovernment Hill (Charlotte Amalie),\n\nGovernment House\n\nCharlotte Amalie, ,\n\nSt. Croix,\n\nGovernor Juan F. Luis Hospital & Medical Center (St. Croix),\n\nGrand Beach (St. Thomas),\n\nGrand Beach Palace (St. Thomas),\n\nGrand Galleria (Charlotte Amalie),\n\nGrapetree Beach (St. Croix),\n\nGreat Harbour (Jost Van Dyke),\n\nGreen Glass Studio (Tortola),\n\nThe Greenhouse (Charlotte Amalie), ,\n\nGreen Ridge Guavaberry Farm (St. Thomas),\n\nGroceries. See Food stores and markets\n\nGuana Island, , , 189\u2013190\n\nH\n\nHamilton, Alexander,\n\nHarbour Market (Tortola), ,\n\nHassel Island (St. Thomas), 80\u201381\n\nHaulover Bay (St. John), ,\n\nHavensight Mall (Charlotte Amalie),\n\nHavensight Pier (Charlotte Amalie),\n\nHawksnest Beach (St. John),\n\nHealth concerns, 203\u2013204. See also Doctors; Hospitals\/clinics\n\nHendricks Square (St. Croix),\n\nHenry E. Rohlsen Airport (St. Croix), ,\n\nHeritage and cultural tours,\n\nHerring gundy,\n\nHertz, , , , ,\n\nHigh season (winter season),\n\nHIHO (Tortola), ,\n\nHiking\n\nBuck Island, 119\u2013120,\n\nSt. Croix, 107\u2013108,\n\nSt. John, 139\u2013141,\n\nTortola, ,\n\ntours, 198\u2013199\n\nVirgin Gorda,\n\nHistory books,\n\nHistory of the Virgin Islands, 24\u201330\n\nHIV-positive visitors,\n\nHodge, Joseph,\n\nHoliday Rentals,\n\nHolidays,\n\nHomeAway, ,\n\nHoneymoon Beach (St. John),\n\nHoneymoon Beach (St. Thomas), ,\n\nHoneymoon Day Pass (St. John),\n\nHorseback riding, St. Croix,\n\nHospitals\/clinics,\n\nSt. Croix,\n\nSt. Thomas, 48\u201349\n\nTortola,\n\nHotel 1829 (Charlotte Amalie),\n\nHotel on the Cay (St. Croix), beach at,\n\nHotels. See Accommodations\n\nHot lines, St. Thomas,\n\nHot sauce,\n\nH. Stern Jewelers (St. Thomas),\n\nHull Bay (St. Thomas),\n\nHull Bay Hideaway (St. Thomas),\n\nHurricane Hole (St. John),\n\nHurricanes,\n\nI\n\nIb Designs (St. Croix),\n\nIggies Beach Bar (St. Thomas),\n\nIGY: American Yacht Harbor,\n\nIndependence (boat), 69\n\nIndependence Day (St. John),\n\nThe Indians. See Norman Island & the Indians\n\nInner Visions,\n\nInter-Island Boat Services, ,\n\nInternet and Wi-Fi,\n\nSt. Thomas,\n\nTortola,\n\nVirgin Gorda,\n\nIsaac Bay Beach (St. Croix),\n\nIsland-hopping,\n\nfor 10 days, 16\u201317\n\nlesser-known islands,\n\nTortola,\n\nIsland Seal (wreck), 169\n\nIsland slaw,\n\nItineraries, suggested, 10\u201317\n\nBritish Virgin Islands, 15\u201316\n\nisland-hopping for 10 days, 16\u201317\n\nU.S. Virgin Islands in 10 days, 10\u201315\n\nJ\n\nJetBlue, ,\n\nJewelry and watches\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas, ,\n\nJewels of the BVI, ,\n\nJohnnycakes,\n\nJones Maritime Sailing School (St. Croix),\n\nJoseph, Ames, ,\n\nJosiah's Bay (Tortola),\n\nJosiah's Bay Beach (Tortola),\n\nJost Van Dyke, , , , 181\u2013185\n\nJoyce, Sistah,\n\nJoyia (St. Croix),\n\nJ.R. Neal Botanical Gardens (Tortola),\n\nJumbies,\n\nK\n\nKayaking,\n\nSt. Croix,\n\nSt. John, 138\u2013139\n\nSt. Thomas, 70\u201371\n\nKilbrides Sunchaser Scuba (Virgin Gorda),\n\nKing's Alley Complex (St. Croix),\n\nKing's Wharf (Charlotte Amalie), 73\u201374\n\nKiteboarding, St. Croix,\n\nKite St. Croix, ,\n\nKuralu (catamaran), 167\n\nKyst Vejen (Charlotte Amalie),\n\nL\n\nLa Grange (St. Croix),\n\nL&C Poultry & Pig Plus (Tortola),\n\nLang Bank (St. Croix),\n\nLanguage,\n\nLa Reine Farmers' Market (St. Croix),\n\nLatitude 18 (St. Thomas),\n\nLaundry & dry cleaning, St. Thomas,\n\nLegal aid, 204\u2013205\n\nLegislative Building (Charlotte Amalie),\n\nLeinster Bay (St. John), , ,\n\nLeinster Bay Trail (St. John),\n\nLettsom, John Coakley,\n\nLewis, Kenneth, , ,\n\nLGBT travelers,\n\nLimprecht Gardens & Memorial (St. Croix),\n\nLindbergh Beach (St. Thomas),\n\nLindquist Beach (St. Thomas), ,\n\nLiterature, 31\u201332\n\nLittle Dix Bay (Virgin Gorda),\n\nLittle Lameshur Bay (St. John), ,\n\nLittle Magens Bay (St. Thomas),\n\nLittle Switzerland (St. Thomas),\n\nLoblolly Bay (Anegada),\n\nLocally Yours (Tortola),\n\nLong Bay (Tortola), 165\u2013166\n\nLord God of Sabaoth Lutheran Church (St. Croix),\n\nLove City,\n\nLow Key Watersports (St. John),\n\nM\n\nMcLaughlin Anderson Luxury Caribbean Villas (St. Thomas),\n\nMagens Bay (St. Thomas), , ,\n\nbeach, 67\u201368\n\nMagens Bay Beach (St. Thomas), ,\n\nMaho Bay Beach (St. John),\n\nMaho Bay Camp,\n\nMahoe Bay (Virgin Gorda),\n\nMahogany Run (St. Thomas),\n\nMail, ,\n\nMain Street (Charlotte Amalie), ,\n\nMaj. General Rogers (wreck), 71\n\nMango Melee and Tropical Fruit Festival (St. Croix),\n\nMango Tango Art Gallery (St. Thomas),\n\nMany Hands (St. Croix),\n\nMardi Gras Annual Parade (St. Croix),\n\nMarine Terminal (Charlotte Amalie),\n\nMarketplace of St. John,\n\nMarket Square (Charlotte Amalie), , ,\n\nMedical requirements,\n\nMerengue,\n\nMidnite,\n\nMobile phones, 205\u2013206\n\nMoney and costs,\n\nMongoose Junction (St. John),\n\nMongooses,\n\nThe Moorings (Tortola), ,\n\nMorning Star (Frenchman's Bay Beach, St. Thomas),\n\nMotu (St. John),\n\nMount Sage (Tortola),\n\nMount Victory Camp (St. Croix), ,\n\nMovies,\n\nMr. Tablecloth (Charlotte Amalie),\n\nMusic,\n\nMyett's (Tortola),\n\nMy Name Is Not Angelica (O'Dell), 31\n\nN\n\nNational car rentals,\n\nNational Park Underwater Trail (St. John),\n\nNational Park visitor center (St. John),\n\nNative Arts & Crafts Cooperative (St. Thomas),\n\nNative Son, ,\n\nNature walks. See also Hiking\n\nbest,\n\nNecker Island,\n\nNew Horizons (boat), 70\n\nNew Horizons Ferry Service, ,\n\nNew Horizons II (boat),\n\nNewspapers & magazines\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nNext Wave Designs (Virgin Gorda),\n\nNightlife\n\nSt. Croix,\n\nSt. John, 142\u2013143\n\nSt. Thomas, 84\u201386\n\nTortola,\n\nVirgin Gorda,\n\nNile, Mada,\n\n99 Steps (Charlotte Amalie),\n\nNorman Island & the Indians (off Tortola), , , , ,\n\nNorth Coast (St. Thomas),\n\nbeaches, 67\u201368\n\nNorth Shore (St. Croix),\n\naccommodations, , 95\u201397\n\nrestaurants, ,\n\nNorth Shore (Tortola)\n\naccommodations, 157\u2013160\n\nbeaches,\n\nrestaurants, 161\u2013164\n\nNorth Shore Shell Museum (Tortola),\n\nNorth Sound Express (British Virgin Islands), ,\n\nNorthstar Reef (St. Croix),\n\nO\n\nO'Dell, Scott,\n\nOff season, 36\u201337\n\nOffshore Adventures (St. John),\n\nOffshore Sailing School (Road Town),\n\nOkra,\n\nOld Danish Customs House (St. Croix), ,\n\nOld Scale House (St. Croix),\n\nOlympic Rent-A-Car (St. Croix),\n\nOrganic Act Day,\n\nOrganic Act of the Virgin Islands, ,\n\nOverdevelopment,\n\nP\n\nThe Painkiller,\n\nPalms at Pelican Cove (St. Croix), ,\n\nParadise Jam College Basketball Tournament (St. Thomas),\n\nParadise Point Tram (St. Thomas),\n\nParking, St. Thomas,\n\nPassports, ,\n\nPaul & Jill's Equestrian Stables (St. Croix),\n\nPeanut Gallery Fishing Charters (St. Thomas),\n\nPeas and rice,\n\nPerfume, St. Croix,\n\nPerserverance Bay (St. Thomas),\n\nPeter Island, , 188\u2013190\n\nPets,\n\nPhillips, Water,\n\nPickering, Frederick Augustus,\n\nPickering, John,\n\nPirate's Bight (Norman Island),\n\nPissarro, Camille,\n\nPlaza Extra (St. Croix),\n\nPoint Udall (St. Croix), ,\n\nPolice, 207\u2013208\n\nSt. Croix,\n\nSt. Thomas,\n\nTortola,\n\nVirgin Gorda,\n\nPost office\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nPowell, Llewelyn, ,\n\nPreserve at Botany Bay, ,\n\nPrickly Pear Island (Virgin Gorda),\n\nProphy's Farm (Tortola),\n\nPusser's Company Store\n\nTortola,\n\nVirgin Gorda,\n\nQ\n\nQuadrille Dancers (St. Croix),\n\nQuiet Mon (St. John),\n\nQuito's Gazebo (Tortola),\n\nR\n\nRainbow Beach (St. Croix),\n\nRainfall, average,\n\n\"Rain Forest\" (St. Croix), , 107\u2013108\n\nR&I PATTON Goldsmithing (St. John),\n\nRed Hook (St. John),\n\nRed Hook (St. Thomas), , ,\n\nThe Reef (St. Croix),\n\nReef Bay Hike (St. John),\n\nReef Beach (St. Croix),\n\nReeftique (Virgin Gorda),\n\nReggae,\n\nReichhold Center for the Arts (St. Thomas),\n\nResorts. See also Accommodations\n\nbest, 5\u20136\n\nSt. John, 125\u2013126\n\nResponsible travel, 40\u201341\n\nRestaurants. See also Restaurants Index\n\nAnegada,\n\nbest,\n\nJost Van Dyke, , 184\u2013185\n\nPeter Island,\n\nreservations,\n\nSt. Croix, , 99\u2013105\n\nSt. John, 130\u2013135\n\nSt. Thomas, , 59\u201367\n\nCharlotte Amalie, 61\u201364\n\nEast End, 66\u201367\n\nFrenchtown,\n\nnorth of Charlotte Amalie, 64\u201365\n\nSouth Coast,\n\ntips on,\n\nTortola, 160\u2013165\n\nVirgin Gorda, 178\u2013179\n\nRestaurant Week (Tortola),\n\nRey, Campbell, , ,\n\nRhone, RMS (wreck), , , , , 201\n\nRhymer's (Tortola),\n\nRidge to Reef Farm (St. Croix),\n\nRidge to Reef Farm Tour (St. Croix), ,\n\nRiteWay (Tortola),\n\nRoad Town (Tortola), ,\n\naccommodations, 155\u2013156\n\nrestaurants,\n\nshopping, 169\u2013170\n\nRoad Town Fast Ferry,\n\nRobinson, Wayne, ,\n\nRockefeller, Laurance,\n\nRoti joints,\n\nRoyal Caribbean (St. Thomas),\n\nRoyal Poinciana (St. Croix),\n\nRum,\n\nRymer, Quito,\n\nS\n\nSafety,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nSage Mountain National Park (Tortola), , ,\n\nSail Caribbean Divers, ,\n\nSailing and boating. See Boating and sailing\n\nSail Rock (off St. Thomas),\n\nSail Safaris (St. John),\n\nSt. Croix, , , 87\u2013120\n\naccommodations, 93\u201399\n\narchitecture,\n\nbeaches, 106\u2013107\n\nbrief description of,\n\ncamping,\n\nexploring, 106\u2013117\n\ngetting around, 91\u201392\n\nlayout of, 90\u201391\n\nnightlife,\n\nrestaurants, , 99\u2013105\n\nshopping, 117\u2013119\n\nside trip to Buck Island, 119\u2013120\n\nsightseeing, 110\u2013117\n\nsuggested itineraries,\n\ntraveling to, 87\u201388,\n\nvisitor information,\n\nwatersports and outdoor adventures, 107\u2013109\n\nSt. Croix Deep Blue Charters,\n\nSt. Croix Environmental Association, ,\n\nSt. Croix Food & Wine Experience, ,\n\nSt. Croix Heritage Trail,\n\nSt. Croix House Tours,\n\nSt. Croix Landmarks Society,\n\nSt. Croix LEAP (St. Croix),\n\nSt. Croix Taxicab Association,\n\nSt. Croix Visitor Bureau,\n\nSt. George Village Botanical Garden (St. Croix),\n\nSt. John, , , 121\u2013143\n\naccommodations, 125\u2013130\n\nbeaches, 136\u2013138\n\nbrief description of, 22\u201323\n\ncampgrounds,\n\nday-tripping to,\n\nexploring, 136\u2013141\n\ngetting around, 123\u2013124\n\nlayout of,\n\nnightlife, 142\u2013143\n\nrestaurants, 130\u2013135\n\nshopping, 141\u2013142\n\nsuggested itineraries, ,\n\ntraveling to, ,\n\nvisitor information,\n\nSt. John Brewers,\n\nSt. John Car Rental,\n\nSt. John Festival,\n\nSt. John Market,\n\nSt. John Spice (St. John),\n\nSt. John Taxi Association,\n\nSt. Peter Great House & Botanical Gardens (St. Thomas),\n\nSt. Thomas, , , 42\u201386\n\naccommodations, , 49\u201359\n\nbeaches, 67\u201369\n\nbrief description of, ,\n\nexcursions from, 79\u201381\n\nexploring, 67\u201381\n\ngetting around, 46\u201348\n\nlayout of, 45\u201346\n\nnightlife, 84\u201386\n\nrestaurants, , 59\u201367\n\nshopping, 81\u201384\n\nsightseeing, 72\u201379\n\nsuggested itineraries, , 14\u201317\n\ntemperatures and rainfall,\n\ntraveling to, , ,\n\nwatersports and outdoor adventures, 69\u201372\n\nSt. Thomas Diving Club, ,\n\nSt. Thomas International Regatta,\n\nSt. Thomas Pistarckle Theater (St. Thomas), 84\u201385\n\nSt. Thomas Reformed Church (Charlotte Amalie),\n\nSt. Thomas-St. John Agriculture and Food Fair,\n\nSt. Thomas Synagogue (Charlotte Amalie),\n\nSalt fish and rice,\n\nSaltfish salad,\n\nSalt Pond Bay (St. John),\n\nSalt River (St. Croix),\n\nSalt River Bay National Historical Park and Ecological Preserve (St. Croix), 115\u2013116\n\nSalt River Canyon (St. Croix),\n\nSam & Jack's Deli (St. John),\n\nSandbox beach bar (Virgin Gorda),\n\nSandy Point (St. Croix), , 106\u2013107\n\nSandy Point Wildlife Refuge (St. Croix),\n\nSapphire Beach (St. Thomas), ,\n\nSaturday morning market (Charlotte Amalie),\n\nSavannah Bay (Virgin Gorda), , ,\n\nScato's Bus & Taxi Services (Tortola),\n\nSchoolhouse Apartment (St. Croix),\n\nScotch Banks (St. Croix),\n\nScuba diving, 200\u2013201\n\nbest sites, 4\u20135\n\nSt. Croix, 108\u2013109\n\nSt. John,\n\nSt. Thomas,\n\nTortola, ,\n\nVirgin Gorda,\n\nSeabourne Airlines, , , , , ,\n\nSeafood Market (St. John),\n\nSea kayaking. See Kayaking\n\nSea Lion Encounter (St. Thomas),\n\nSea Lion Splash (St. Thomas),\n\nSeaplane flights, St. Croix,\n\nSeasickness,\n\nSeasons, 36\u201337\n\nSea Thru Kayaks VI (St. Croix),\n\nSea Trek (St. Thomas),\n\nSea turtles,\n\nSebastian's (Tortola),\n\nSecret Harbour (St. Thomas), ,\n\nSejah Farms (St. Croix),\n\nSerenaSea, ,\n\nSeven Arches Museum (Charlotte Amalie),\n\n1780 Lower Estate SugarWorks Museum (Tortola), 167\u2013168\n\nShaka-Man Zulu (St. Thomas),\n\nShark Encounter (St. Thomas),\n\nShipwreck Landing (St. John),\n\nShopping\n\nSt. Croix, 117\u2013119\n\nSt. John, 141\u2013142\n\nSt. Thomas, 81\u201384\n\nTortola, 169\u2013171\n\nVirgin Gorda, 180\u2013181\n\nSir Francis Drake Channel,\n\nSir Francis Drake coastal road (Tortola),\n\nSistah Joyce,\n\nSkinny Legs (St. John),\n\nSlaves, , , , , , , , 114, , , , , 199\n\nSmithens Market (St. Croix),\n\nSmith's Ferry Services, , ,\n\nSmoking,\n\nSmuggler's Cove (Tortola), , ,\n\nSnorkeling, 200\u2013201\n\nbest spots,\n\nBuck Island,\n\nSt. Croix, 108\u2013109\n\nSt. John, ,\n\nSt. Thomas,\n\nsuggested itineraries, ,\n\nTortola, ,\n\nVirgin Gorda,\n\nSoca,\n\nSolstice celebrations,\n\nSonya Ltd. (St. Croix), 118\u2013119\n\nSoper's Hole,\n\nSoper's Marina (Tortola),\n\nS.O.S. Antiques (St. Thomas),\n\nSouth Coast (St. Thomas),\n\naccommodations, 55\u201356\n\nbeaches,\n\nSpain,\n\nSpeedy's,\n\nSpeedy's (Virgin Gorda),\n\nSpirit Airlines,\n\nSprat Hall (St. Croix),\n\nSpring Bay (Virgin Gorda),\n\nSpyglass Wall (dive site),\n\nStandup paddleboarding, St. Croix,\n\nStarfish Market (St. John),\n\nStatehood,\n\nSteeple Building (St. Croix),\n\nSteinworks (St. John),\n\nStevenson, Robert Louis, ,\n\nStreet of 99 Steps (Charlotte Amalie),\n\nStudent travel,\n\nSun exposure,\n\nSunny Caribbee Spice Co. (Tortola),\n\nSurfing\n\nSt. Thomas,\n\nTortola, 168\u2013169\n\nSweet potato pie,\n\nSwimming,\n\nT\n\nTan Tan Tours,\n\nTaproom (St. John),\n\nTaxes, 208\u2013209\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola, 154\u2013155\n\nTaxis, ,\n\nAnegada,\n\nBritish Virgin Islands,\n\nJost Van Dyke,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas, 47\u201348\n\nTortola, 152\u2013153\n\nVirgin Gorda,\n\nTaxi tours\n\nSt. Croix, 109\u2013110\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nTelephones,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas,\n\nTortola,\n\nTemperatures, average,\n\nTennis, St. Croix, ,\n\nTerrace Lounge at the Buccaneer Resort (St. Croix),\n\nTerrence B. Lettsome Airport (EIS),\n\nThornton, William,\n\nTiffany & Co. (Charlotte Amalie),\n\nTillett Gardens Center for the Arts (St. Thomas), ,\n\nTime zone,\n\nTipping,\n\nToilets,\n\nSt. Croix,\n\nSt. Thomas,\n\nTortola,\n\nTortola, , , 149\u2013171\n\naccommodations, 155\u2013160\n\nbeaches, 165\u2013166\n\nboating,\n\nbrief description of, 23\u201324\n\nexploring, 165\u2013169\n\ngetting around, 152\u2013153\n\nlayout,\n\nnightlife,\n\nrestaurants, 160\u2013165\n\nsuggested itineraries, ,\n\ntouring, 166\u2013168\n\ntransportation, 147\u2013149\n\ntraveling to, ,\n\nvisitor information,\n\nwatersports and outdoor adventures, 168\u2013169\n\nTortola Island Surf and Sail,\n\nTortola Sailing School,\n\nTourism, ,\n\nTourist information. See Visitor information\n\nTours. See also Taxi tours; Walking tours\n\nair,\n\nbicycle,\n\nspecial-interest, 196\u2013199\n\nTrading Post (Virgin Gorda),\n\nTransfer Day,\n\nTransit information, St. Thomas,\n\nTransportation, 192\u2013195\n\nBritish Virgin Islands, 147\u2013149\n\nSt. John, 123\u2013124\n\nSt. Thomas, 46\u201348\n\nTransportation Services (St. Thomas),\n\nTreasure Island (Stevenson), , 170\n\nTree Limin' Extreme, ,\n\nTropical Publishers (St. Thomas),\n\nTropic Tours (St. Thomas),\n\nTrunk Bay (St. John), , , , ,\n\nTrunk Bay (Virgin Gorda),\n\nTurtle Encounter (St. Thomas),\n\nU\n\nUnited Airlines,\n\nUS Airways, ,\n\nU.S. Virgin Islands\n\nbrief description of, , 22\u201323\n\nBritish Virgin Islands compared to,\n\nhistory of, 26\u201330\n\nlandmass,\n\nstatehood,\n\nsuggested itineraries, 10\u201315\n\nU.S. Virgin Islands National Park.,\n\nU.S. Virgin Islands Open\/Atlantic Blue Marlin Tournament (St. Thomas),\n\nU.S. Virgin Islands Tourism, St. Croix,\n\nV\n\nVacation St. Croix, ,\n\nVazquez, Francis M., ,\n\nVendors Plaza (Charlotte Amalie),\n\nVilla rentals. See Condo, apartment, and villa rentals\n\nVillas of Distinction,\n\nViolette Boutique (St. Croix),\n\nVirgin Gorda, , , , 172\u2013181\n\naccommodations, 174\u2013178\n\nbrief description of,\n\nexploring, 179\u2013180\n\ngetting around, ,\n\nrestaurants, 178\u2013179\n\ntransportation, 147\u2013149\n\ntraveling to,\n\nVirgin Gorda Easter Festival,\n\nVirgin Gorda Peak (Virgin Gorda),\n\nVirgin Gorda Tours,\n\nVirgin Gorda Tours Association (Virgin Gorda),\n\nVirgin Gorda Villa Rentals Ltd. (Virgin Gorda),\n\nVirgin Islands Campground (St. Thomas),\n\nVirgin Islands Carnival (St. Thomas),\n\nVirgin Islands Coral Reef National Monument,\n\nVirgin Islands Ecotours,\n\nSt. John,\n\nSt. Thomas, 70\u201371, 80\u201381\n\nVirgin Islands Fashion Week (St. Thomas),\n\nVirgin Islands Folk Museum (Tortola),\n\nVirgin Islands National Park (St. John), , ,\n\nVirgin Islands Snuba Excursions (St. Thomas),\n\nVirgin Islands Sustainable Farm Institute,\n\nVirgin Kayak (St. Croix),\n\nVisas, 209\u2013210\n\nVisitor information\n\nBritish Virgin Islands,\n\nSt. Croix,\n\nSt. John,\n\nSt. Thomas, 44\u201345\n\nVirgin Gorda,\n\nVitran (St. John),\n\nVRBO, ,\n\nW\n\nWalking, St. Thomas,\n\nWalking tours\n\nCharlotte Amalie (St. Thomas), 73\u201378\n\nSt. Croix, ,\n\nWater, drinking, ,\n\nThe Waterfront (Charlotte Amalie),\n\nWaterfront Highway (Charlotte Amalie),\n\nWater Island (St. Thomas), , 79\u201380\n\nWater Island Adventures, ,\n\nWater Island Ferry (St. Thomas),\n\nWaterlemon Cay (St. John), , , ,\n\nWatersports and outdoor adventures\n\nSt. Croix, 107\u2013109\n\nSt. John, 138\u2013141\n\nSt. Thomas, 69\u201372\n\nTortola, 168\u2013169\n\nVirgin Gorda, 180\u2013181\n\nWater taxis\n\nSt. John,\n\nSt. Thomas, ,\n\nTortola, ,\n\nWeather,\n\nBritish Virgin Islands,\n\nSt. Thomas,\n\nWe Grow Food (St. Thomas),\n\nWeibel Museum (Charlotte Amalie),\n\nWelcome Center (Charlotte Amalie),\n\nWest End (Tortola)\n\naccommodations, 157\u2013160\n\nrestaurants, 161\u2013164\n\nshopping,\n\nWharfside Village (St. John),\n\nWhim Museum Store (St. Croix),\n\nWhite Bay (Guana Island),\n\nWhite Bay (Jost Van Dyke),\n\nWhite Squall II (schooner), 167\n\nWickham's Cay (Road Town),\n\nWildlife refuge, St. Croix,\n\nWilson, Joan, 169\u2013170\n\nWindsurfing, St. John,\n\nWomen travelers,\n\nWoody's Seafood Saloon (St. John),\n\nWouk, Herman,\n\nWreck Alley (between Salt and Cooper islands), ,\n\nWyndham Margaritaville Vacation Club resort (St. Thomas),\n\nY\n\nYacht Haven Grande (Charlotte Amalie), , ,\n\nYachting. See Boating and sailing\n\nYacht Nightwind (boat), 70\n\nYellow-Brick Building (Charlotte Amalie),\n\nZ\n\nZiplining, St. Thomas,\n\nZora of St. Thomas,\n\nZouk,\n\nAccommodations\n\nAnegada Beach Club,\n\nAnegada Beach Cottages,\n\nAnegada Reef Hotel, 7\u20138, ,\n\nArawak Bay: The Inn at Salt River (St. Croix),\n\nBeef Island Guest House (Tortola),\n\nBellavista Bed & Breakfast (Charlotte Amalie), ,\n\nBest Western Emerald Beach Resort (St. Thomas),\n\nBiras Creek Resort (Virgin Gorda), , 174\u2013175\n\nBitter End Yacht Club (Virgin Gorda), , , ,\n\nBolongo Bay Beach Resort (St. Thomas), ,\n\nThe Buccaneer (St. Croix), , ,\n\nCaneel Bay (St. John), ,\n\nCane Garden Bay Cottages (Tortola),\n\nCarringtons Inn (St. Croix),\n\nChenay Bay Beach Resort (St. Croix), 95\u201396\n\nClub St. Croix Beach & Tennis Resort,\n\nCoconut Coast Villas (St. John),\n\nColony Cove (St. Croix),\n\nCompany House Hotel (St. Croix),\n\nConcordia Eco-Resort (St. John),\n\nConcordia Eco-Tents (St. John), ,\n\nCooper Island Beach Club (Tortola), , ,\n\nCottages by the Sea (St. Croix),\n\nThe Crystal Palace Bed and Breakfast (Charlotte Amalie),\n\nDivi Carina Bay All-Inclusive Resort & Casino (St. Croix),\n\nElysian Beach Resort (St. Thomas),\n\nEstate Lindholm Bed & Breakfast (St. John),\n\nEstate Zootenvaal (St. John),\n\nFischer's Cove Beach Hotel (Virgin Gorda),\n\nFort Burt Hotel (Tortola),\n\nFort Recovery Beachfront Villas (Tortola),\n\nFrenchmans (Tortola), ,\n\nFrenchman's Cove (St. Thomas),\n\nGallows Point Resort (St. John),\n\nGarden by the Sea Bed & Breakfast (St. John),\n\nGrande Bay Resort & Residence Club (St. John), ,\n\nGuana Island, , 189\u2013190\n\nGuavaberry Spring Bay Vacation Homes (Virgin Gorda),\n\nHeritage Inn (Tortola), 159\u2013160\n\nAt Home in the Tropics (Charlotte Amalie),\n\nThe Inn at Tamarind Court (St. John),\n\nIsland Beachcomber Hotel (St. Thomas),\n\nKing's Alley Hotel (St. Croix),\n\nLeverick Bay Resort & Marina (Virgin Gorda), 177\u2013178\n\nLong Bay Beach Club (Tortola), ,\n\nMafolie Hotel (Charlotte Amalie),\n\nMango Bay Resort (Virgin Gorda),\n\nMaria's by the Sea (Tortola),\n\nMarriott Frenchman's Reef & Morning Star Beach Resort (St. Thomas), 55\u201356\n\nMongoose Apartments (Tortola),\n\nThe Moorings\/Mariner Inn (Tortola), 155\u2013156\n\nMyett's Garden Inn (Tortola), 157\u2013158\n\nThe Palms at Pelican Cove (St. Croix), ,\n\nPavilions and Pools (St. Thomas), 58\u201359\n\nPeter Island Resort,\n\nQuito's Ole Works Inn (Tortola),\n\nRattan Inn (St. Croix),\n\nRenaissance Carambola Beach Resort & Spa (St. Croix),\n\nRhymer's Beach Hotel (Tortola),\n\nRidge to Reef Farm (St. Croix),\n\nThe Ritz-Carlton, St. Thomas, , 56\u201357\n\nRosewood Little Dix Bay (Virgin Gorda), ,\n\nSaba Rock Resort (Virgin Gorda),\n\nSandcastle Hotel (Jost Van Dyke),\n\nSand Castle on the Beach (St. Croix), 97\u201398\n\nSebastian's on the Beach (Tortola),\n\nSecret Harbour Beach Resort (St. Thomas),\n\nSerendip Vacation Condos (St. John),\n\nSugar Bay Resort and Spa (St. Thomas), 57\u201358\n\nThe Sugar Mill Hotel (Tortola),\n\nTamarind Reef Resort, Spa & Marina (St. Croix),\n\nVilla Blanca (Charlotte Amalie),\n\nVillage Cay Hotel (Tortola),\n\nVilla Madeleine (St. Croix),\n\nVilla Santana (Charlotte Amalie),\n\nWaves at Cane Bay (St. Croix), 96\u201397\n\nWestin St. John Resort & Villas,\n\nWestin St. John Resort & Villas (St. John),\n\nWindward Passage Hotel (Charlotte Amalie),\n\nRestaurants\n\nAbe's by the Sea (Jost Van Dyke), ,\n\nAli Baba's (Jost Van Dyke),\n\nAmalia Caf\u00e9 (Charlotte Amalie),\n\nArmstrong's Homemade Ice Cream (St. Croix), 103\u2013104\n\nAsolare (St. John),\n\nBananakeet (Tortola),\n\nBanana Tree Grille (Charlotte Amalie), 61\u201362\n\nThe Bath & Turtle (Virgin Gorda),\n\nBig Bamboo (Anegada),\n\nBleuwater (St. Thomas),\n\nBlue Moon (St. Croix),\n\nBlue Orchid (St. Thomas),\n\nBrandywine Estate Restaurant (Tortola), 164\u2013165\n\nBumpa's (Charlotte Amalie),\n\nCaf\u00e9 Amici (Charlotte Amalie),\n\nCapriccio di Mare (Tortola),\n\nThe Caribbean Fish Market (St. Thomas),\n\nChateau Bordeaux (St. John),\n\nThe Clubhouse at Frenchmans (Tortola), 162\u2013163\n\nCorsairs Beach Bar & Restaurant (Jost Van Dyke),\n\nCow Wreck Beach Bar & Grill (Anegada),\n\nCruzin' (Tortola),\n\nCuzzin's Caribbean Restaurant & Bar (Charlotte Amalie),\n\nD-Best Cup (Tortola),\n\nThe Dove Restaurant & Wine Bar (Tortola),\n\nDuggan's Reef (St. Croix),\n\nEat@canebay (St. Croix), 104\u2013105\n\nFatty Crab (St. John),\n\nFat Virgin's Cafe (Virgin Gorda),\n\nFish Tails (St. Thomas),\n\nThe Fish Trap (St. John), 133\u2013134\n\nFlamboyant Beach Bar (St. Croix),\n\nFoxy's Tamarind Bar (Jost Van Dyke),\n\nFrenchtown Deli (St. Thomas),\n\nThe Galleon (St. Croix),\n\nGladys' Cafe (Charlotte Amalie, St. Thomas),\n\nGladys' Caf\u00e9 (Charlotte Amalie),\n\nHarvey's (St. Croix),\n\nHavana Blue (St. Thomas), ,\n\nHilltop Restaurant at Biras Creek Resort (Virgin Gorda),\n\nHog Heaven (Virgin Gorda),\n\nHook, Line & Sinker (St. Thomas),\n\nIggies Beach Bar & Grill (St. Thomas),\n\nIvan's Stress Free Bar & White Bay Campground (Jost Van Dyke),\n\nJen's Island Caf\u00e9 & Deli (Charlotte Amalie),\n\nKendrick's (St. Croix),\n\nLa Reine Chicken Shack (St. Croix),\n\nThe Last Resort (Tortola),\n\nLa Tapa (St. John),\n\nLuncheria (St. Croix),\n\nMafolie Restaurant (Charlotte Amalie), ,\n\nMagens Bay Beach Bar & Caf\u00e9 (St. Thomas),\n\nMaggie's Snackett (St. Croix),\n\nMarch\u00e9 at Trellis Bar & Grill (Tortola),\n\nMidtown Restaurant (Tortola),\n\nMiss Lucy's Beachfront Restaurant (St. John), 134\u2013135\n\nMorgan's Mango (St. John),\n\nNeptune's Treasure (Anegada),\n\nNorth Shore Shell Museum Bar & Restaurant (Tortola),\n\nOceana (St. Thomas),\n\nThe Old Stone Farmhouse (St. Thomas), 64\u201365\n\nOne Love Bar and Grill (Jost Van Dyke),\n\nThe Palms at Pelican Cove (St. Croix),\n\nThe Pavilion at Little Dix Bay (Virgin Gorda),\n\nPetite Pump Room (Charlotte Amalie),\n\nPolly's at the Pier (St. Croix),\n\nPusser's Pub (Tortola),\n\nThe Quiet Mon (St. John),\n\nQuito's Gazebo (Tortola),\n\nRhumb Lines (St. John), 132\u2013133\n\nThe Rock Caf\u00e9 (Virgin Gorda),\n\nA Room with a View (Charlotte Amalie),\n\nRoti Palace (Tortola),\n\nRumRunners (St. Croix),\n\nSalud! Bistro (St. Croix),\n\nSandcastle Hotel (Jost Van Dyke), , 184\u2013185\n\nSandy Ground Estates (Jost Van Dyke),\n\nSavant (St. Croix),\n\nSebastian's Seaside Grille (Tortola),\n\nSe\u00f1or Pizza (St. Thomas),\n\n1748 Restaurant (Tortola), 161\u2013162\n\nShipwreck Landing (St. John),\n\nSugar Apple (Virgin Gorda),\n\nSugar Mill Restaurant (Tortola),\n\nSunset Grille (St. Thomas), ,\n\nSweet Plantains Restaurant & Rhum Bar (St. John),\n\nTavern on the Waterfront (Charlotte Amalie),\n\nTop of the Baths (Virgin Gorda),\n\nTourist Trap (St. John),\n\nTutto Bene (St. Croix), 102\u2013103\n\nUca's Kitchen (St. Croix),\n\nVie's Snack Shack (St. John),\n\nVilla Morales (St. Croix),\n\nVirgilio's (Charlotte Amalie),\n\nWhite Bay Villas & Seaside Cottages (Jost Van Dyke),\n\nZoZo's Ristorante (St. John), \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTHE  \nCOMING  \nREVOLUTION\n\nTHE  \nCOMING  \nREVOLUTION\n\n_Signs from America's Past That  \nSignal Our Nation's Future_\n\nDR. RICHARD G . LEE\n\n\u00a9 2012 Richard G. Lee\n\nAll rights reserved. No portion of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means\u2014electronic, mechanical, photocopy, recording, scanning, or other\u2014except for brief quotations in critical reviews or articles, without the prior written permission of the publisher.\n\nPublished in Nashville, Tennessee, by Thomas Nelson. Thomas Nelson is a registered trademark of Thomas Nelson, Inc.\n\nThomas Nelson, Inc., titles may be purchased in bulk for educational, business, fund-raising, or sales promotional use. For information, please e-mail SpecialMarkets@ThomasNelson.com.\n\nUnless otherwise noted, Scripture quotations are taken from the King James Version.\n\nScripture quotations marked NKJV are from the New King James Version\u00ae. \u00a9 1982 by Thomas Nelson, Inc. Used by permission. All rights reserved.\n\n**Library of Congress Cataloging-in-Publication Data**\n\nLee, Richard, 1946\u2013   \nThe coming revolution : signs from America's past that signal our nation's future \/ Richard G. Lee.  \np. cm.  \nIncludes bibliographical references (Notes) and index.  \nISBN 978-0-8499-4829-9 (hardcover) 1. Christianity and politics\u2014United States. I. Title.  \nBR515.L43 2012   \n261.70973\u2014dc23\n\n2011036213\n\n_Printed in the United States of America_\n\n12 13 14 15 16 QG 5 4 3 2 1\n\nTO MY FELLOW COUNTRYMEN,   \nPAST AND PRESENT, WHO HAVE   \nSACRIFICED SO MUCH FOR LIBERTY,   \nJUSTICE, AND THE AMERICAN WAY.\nCONTENTS\n\n_Acknowledgments_\n\n_Introduction_\n\nONE: Portrait of a Nation\n\nTWO: The Promise of America\n\nTHREE: What the Founders Believed\n\nFOUR: The Birth of the American Spirit\n\nFIVE: Faith in the Twenty-First Century\n\nSIX: The Coming Revolution\n\nSEVEN: What You Can Do\n\n_Notes_\n\n_Topical Index_\n\n_About the Author_\nACKNOWLEDGMENTS\n\nMy deepest gratitude to the following individuals at Thomas Nelson Publishers who, through their time and many talents, helped to bring this book to print: Vice President and Publisher Matt Baugher, along with Paula Major, Emily Sweeney, Stephanie Newton, and Julie Faires; and Jack and Marsha Countryman, dearest friends, who have played such a vital role in all of my recent writings concerning God and country.\n\nAnd finally to my longtime friend and scholar, Dr. Jim Nelson Black, who helped to bring my many thoughts together into this one volume.\n\nAs in most everything that has worth, it takes a great team working together, and this book is no exception. So to each of these team members and many more, I express a special thanks.\nxi\n\nINTRODUCTION\n\nWho could have anticipated, even a year or two ago, that the backlash against increasing government control we are seeing in Washington and in state capitals all around the country would be happening now? The truth is, traditional conservatives and people of faith have been speaking out about the direction of government and the need for accountability for years. There have been books and broadcasts of all kinds warning about the risks\u2014I have produced a number of them myself\u2014but it wasn't until there was a critical mass of concerned men and women willing to stand up and speak out that such a moment could actually happen.\n\nLed by their own deeply held convictions and the promptings of their faith, men and women with highly visible public platforms helped spread the word, and that vital spark has helped to set a whole range of events in motion. The Restoring Honor rally on the Washington Mall on August 28, 2010, was a striking example of that emerging spirit, but it was just one event. There will be many others, so long as the need is there and the will of the people remains as strong as it is today.\n\nSurprisingly similar to those who birthed another Revolution on these shores 236 years ago, ordinary citizens are the real voices of this anti-big-government revolution. All they really want is to get back to the kind of government the Founders gave us in the first place. There are no signs of violence at the moment, and that is because most Americans are still optimistic that their voices are being heard by our newly elected officials. They believe the results of the 2010 midterm elections will help to restore power to the citizenry. The new conservative majority in the U.S. House of Representatives is moving toward less government spending and reduced regulations for the moment, but if something should happen to derail that progress or if enough people feel they are being disenfranchised by our political leaders, then anything could happen.\n\nxii\n\nMany of the identical social and religious provocations that spurred the American colonists to revolt in the eighteenth century are present again today, inspiring a new generation of patriots to seek what the Founders called \"a new birth of freedom.\" To appreciate where we are and where all these things can lead, I believe we will need a better understanding of our history and of why the founding documents were written as they were, granting unprecedented freedoms to the people and establishing the rule of law. If we understand the environment of the revolutionary era as well as the political conditions we are facing today, we will have a much better idea of how things will likely play out in the future, and that will be an important part of this book.\n\nA NEW SENSE OF URGENCY\n\nRevolutions don't just happen. They happen because of widespread frustration and a history of disappointments, insults, and provocations of many kinds that eventually compel people to respond. Whether they happen in politics or some other area of society, revolutions are always volatile phenomena. They generally come only after years of agitation, but they are never about just one thing. They result from a long list of offenses, both perceived and real, that set off a natural chain reaction that quickly becomes irreversible. And in the right environment, the situation can be explosive.\n\nxiii\n\nAs Malcolm Gladwell pointed out in his 2000 bestseller, _The_ _Tipping Point: How Little Things Can Make a Big Difference_ , revolutionary change often comes suddenly and dramatically. When a large enough group of people reach a high enough level of frustration, they take action first by speaking out more and more forcefully. But if their concerns are ignored or rebuffed, and if there is no sign that things are getting better, there can be a sudden and momentous change of direction. It doesn't take much imagination to see that this is precisely what's going on in America today.\n\nI am convinced we are on the threshold of a revolution of some kind. I don't believe it will be a violent revolution\u2014it doesn't need to be. We have many other alternatives, and no one is suggesting that direct action will be needed. Thanks to the success and endurance of the Tea Party movement, we're seeing reasonable and measured resistance to the expansion of big government in this country, and it is peaceful resistance. Thousands of ordinary Americans are joining forces to challenge the direction our leaders have been taking us. For the most part these are all ordinary citizens, men and women from all walks of life, and people who have never been politically engaged in this way before. But through their combined efforts, they are speaking out with a new sense of urgency, determined to bring an end to what millions now perceive as arrogant and predatory behavior by Washington bureaucrats and other elected and unelected officials who wield too much power.\n\nThere is no mistaking the emotion or the intensity of those who are speaking out. The conservative news media have been talking about it for years now. But for the hundreds of thousands of men and women who came to Washington at their own expense to participate in Glenn Beck's Restoring Honor rally in August 2010, as well as for the millions who watched those events on TV or on the Internet, that was a monumental occasion. As a participant myself\u2014and as a representative of the newest generation of what the Founders called the Black Robe Regiment\u2014I can vouch for the powerful emotions we all felt.\n\nxiv\n\nThat event was a way of making our grievances known in a respectful and honorable way. We came because we were upset with Washington, but we also realized that the surge of individuals' involvement in social and political issues today is of revolutionary proportions because a dramatic restoration of our constitutional values needs to happen. Those who came on that beautiful August day were of one mind: they all wanted to make a difference. They had seen what the Obama administration's idea of hope and change was all about, and they'd had enough.\n\nEVIDENCE OF FAITH\n\nThe American people don't need to be reminded that this is not the first time a large-scale popular rebellion has taken place in this country. The first Revolution was preceded and inspired, I believe, by the Great Awakening\u2014a powerful and seminal event. As I have reexamined and written about conditions during the nation's founding era over the past several years\u2014and especially since completing my Foundation House studies at Oxford University, focusing on the relationship between the American Revolution and the Great Awakening\u2014I have come to believe that we are in the early stages of another awakening, perhaps of even greater magnitude.\n\nThe Christian faith of the founding generation inspired their love of liberty and emboldened them to demand freedom from British rule, and the greatest expression of that love of liberty was visible in the Great Awakening from the 1730s to 1750s. The sparks of the American Revolution were fanned into flame by pastors and teachers\u2014the Black Robe Regiment\u2014who taught the people the biblical basis of personal liberty and obedience to the call of conscience. Today we're seeing a new wave of pastors and teachers who are speaking out about these issues, warning about the abuse of power by our public officials and stressing the importance of \"restoring honor\" in all areas of life.\n\nxv\n\nWhat we are seeing is nothing less than a new American revolution: not with muskets and cannonballs this time but with ideas. It is a revolution of conscience, of morality, and of honor, dedicated to responsible social, moral, and political reforms. I confess I don't care too much for the word \"change\" these days, but vast numbers of men and women are standing up and speaking out, demanding change from the socialistic direction our political, judicial, and intellectual leaders have been taking us, and back to the values of the Founders.\n\nBy any measure, this is a time of monumental change. Washington's idea of change has failed, and that is why so many people are reacting as they are. We are witnessing a rebirth of the American Spirit and a recommitment to our cherished American ideals. And just as in that earlier Revolution, this one involves an amazing outpouring of religious commitment as well. We see it in the tea parties taking place from coast to coast, and we see it in the explosion of conservative book sales, the tsunami of grassroots activism, and the sudden appearance of a new generation of political leaders who are motivated by their Christian faith and who are having a tremendous impact in the voting booths.\n\nGETTING OUR PRIORITIES RIGHT\n\nThis book is a response to the growing demand for information, insight, and inspiration. In it I will examine the connections between our Christian faith and responsible political action, focusing on the importance of personal liberty and showing how revolutions past, present, and future may be connected. In the final sections of the book, I will also take a brief look at the darker side of revolution, which emerged nearly a century ago in a formerly Christian nation, the Soviet Union, and which continues to threaten democratic institutions to this day.\n\nxvi\n\nIt is my hope that a renewed appraisal and understanding of our nation's core values will motivate many people to take positive action in their homes, communities, workplaces, and especially the voting booths. The information contained in this book may be taken in a prophetic or predictive context, based on lessons learned from social conditions today as well as the events of the Great Awakening and the American Revolution. I conclude the book with a discussion of some of the ways every citizen can become involved now to help shape the future.\n\nI am convinced that every one of the problems confronting this nation today can be solved through concerned citizen action. Lawlessness, economic insolvency, and government interference in people's lives can be stopped. Sound political action that arises from sound moral judgment and individual initiative can help turn the tide and restore order in all these areas. We have to make personal commitments to be involved for any of these things to happen, and most of all, we need to have our priorities right.\n\nOne of the most important things we can do is to follow through as the Founders did, not merely by changing our representatives in the state and national legislatures but by holding all our leaders accountable for their actions and insisting they serve as our representatives, not as irresponsible and unresponsive dictators. And, not least, we need to make sure our own houses are in order, that we have a resilient faith and express our reverence for the One who is the true Author of Liberty.\n\nDuring the fall of 2011 we witnessed a very different expression of dissent, the so-called _Occupy Wall Street_ demonstrations in the streets of New York and other American cities. While the protesters claimed to represent 99 percent of real America, researchers described them as \"an unrepresentative segment of the electorate that believes in radical redistribution of wealth, civil disobedience and, in some instances, violence.\" Writing for the _Wall Street Journal_ , former Clinton pollster Douglas Schoen said that more than half of the OWS crowds are serial protesters. Approximately 98 percent of those interviewed said they support civil disobedience, and 31 percent support the use of violence to achieve their goals. Some of those questioned by reporters could not explain coherently the issues they were protesting, and many admitted they were being paid by left-wing groups and labor unions. These are not the voices of hope and change but of anarchy, dissipation, and despair. Groups such as OWS are likely to become more numerous in the days ahead as socialistic-minded elites become more and more fearful of the growing strength and number of outspoken conservatives who are beginning to rise up and be heard. The revolution that I see coming is a good and needful one, one of faith and virtue, of responsibility and positive action, and a radical return to the values and beliefs that made America great.\n\nxvii\n\nAs I have spoken about these ideas from coast to coast, I have discovered that many people sense the hand of divine Providence moving in America today. Consequently we cannot help feeling we are engaged in something beyond the merely temporal and transitory concerns of the day. It is not just that we may be in the early stages of a popular rebellion but that we are part of a bold new awakening. The signs are all around us, and if we can grasp and reconsider the unique and extraordinary events that shaped our nation during its illustrious history, it ought to be possible to understand how many of those same factors that energized those events will prepare and propel us toward the Coming Revolution.\n\nHistory does repeat itself, so with a better grasp of our history\u2014 focusing on the unique relationship between faith and freedom\u2014we can gain a new understanding of the political, social, and religious events that are likely to occur in this country within the next few years. And that's a substantial part of what I feel compelled to say in these pages. It is my hope and prayer that many may be awakened, empowered by the Holy Spirit, and prepared for the events that will be taking place in our nation in the coming days.\n\n\u2014Richard G. Lee \n1\n\n[_**One**_  \nPORTRAIT OF A NATION](Lee_9780849949463_epub_c5_r1.html#d6e3)\n\nWhat is the source of America's greatness? By any standard this country has a remarkable story to tell, with its dramatic history and an enviable record of achievements in almost any area you can think of. America is the world's longest surviving democratic republic, operating under the same Constitution and laws for the past 236 years. The nation enjoys the greatest level of personal liberty, the highest standard of living, the largest economy, the most dynamic commercial and industrial sectors, and the most consequential foreign policy of any nation\u2014all of it defended by the best-trained and most technologically sophisticated military in human history.\n\nBut America is not merely the world's richest and most powerful nation. It is also the most benevolent, rushing to the four corners of the earth to bring relief to nations stricken by wars, famines, and disasters of every kind. Public and private charities, relief organizations, and international aid societies are constantly on the move, reaching out to \"the least of these\" wherever there is pain and suffering. They do it without pay and often without credit of any kind because they understand that the blessings of prosperity have made this nation a beacon of hope to the rest of the world.\n\n2\n\nOn average, the American people give more than $300 billion each year to charitable causes. According to the most recent report from Giving USA, Americans donated more than $303 billion in 2009, $315 billion in 2008, and $295 billion in 2007. These donations are distributed among approximately 1.2 million IRS-registered charities and 350,000 religious congregations. This is in addition to the $25 billion the U.S. government spends each year in foreign aid to countries around the globe. Germany's foreign aid, by comparison, ranks second with contributions of about $13 billion.\n\nAmericans give the largest percentage of their charitable donations to religious organizations, at approximately $101 billion, followed by educational organizations at $40 billion, charitable foundations at $31 billion, human services organizations at $27 billion, and health organizations at $22.5 billion. Especially interesting is the fact that 65 percent of U.S. households with annual incomes less than $100,000 donate to charity. Wealthy Americans may give more, but middle-class Americans give a larger percentage of their income. In addition to the financial gifts, America also leads the world in volunteerism, donating time and service to charitable and faith-based organizations. And that's a custom as old as the nation itself.\n\nIn the 1830s, the French statesman Alexis de Tocqueville visited America and was impressed by many things, but the one thing that really surprised him was the great number of \"voluntary associations\" in this country. In his classic work, _Democracy in America_ , he writes:\n\nAmericans of all ages, all stations in life, and all types of disposition are forever forming associations. There are not only commercial and industrial associations in which all take part, but others of a thousand different types\u2014religious, moral, serious, futile, very general and very limited, immensely large and very minute. Americans combine to give f\u00eates, found seminaries, build churches, distribute books, and send missionaries to the antipodes. Hospitals, prisons, and schools take shape in that way. Finally, if they want to proclaim a truth or propagate some feeling by the encouragement of a great example, they form an association.\n\n3\n\nThis spirit of generosity and commitment to worthy causes was unique in that day, he felt, and was matched only by the industry and imagination of the American businessman. When he looked at all the incredible achievements this country had racked up in less than a century, he marveled at the wealth of the American Spirit:\n\nThe Americans arrived but yesterday in the land where they live, and they have already turned the whole order of nature upside down to their profit. They have joined the Hudson to the Mississippi and linked the Atlantic with the Gulf of Mexico across a continent of more than 500 leagues separating the two seas. The longest railways yet constructed are in the United States.\n\nEntrepreneurship and vision were the hallmarks of American business then just as they are today, but what Tocqueville found most compelling was the fact that everywhere he looked the citizens were working together, building things, giving freely of their time and labor. \"I am even more struck,\" he writes, \"by the innumerable multitude of little undertakings than by the extraordinary size of some of their industrial enterprises.\"\n\nMost of what this nation has achieved over the past three centuries is due, I believe, to the faith and character of the American people. These qualities are under great stress today, that's true, but where would the world be if it weren't for the resolute faith and indomitable spirit of America's pioneers? If you ask the average person to name our greatest achievements, many would no doubt point to education. As early as the mid-1600s, public education was already widespread in New England. Thomas Jefferson was among the first to formulate plans for universal public education, and by the end of the nineteenth century that goal had been accomplished.\n\n4\n\nAmerica is also home to some of the world's leading universities\u2014 the whole world sends its sons and daughters to this country for the advanced studies that will allow them to succeed in whatever professions they may choose. The context and character of secondary and higher education have changed dramatically over the past half century\u2014not for the better, unfortunately\u2014but it's true nevertheless that the emphasis on education is among our most noteworthy achievements.\n\nVisitors like Tocqueville, as mentioned above, have been impressed by such things as the vast network of railroads spanning the continent, but from the earliest days of the republic, we have profited from the contributions of individual inventors and innovators, such as Benjamin Franklin, who gave us the lightning rod, the Franklin stove, bifocal reading glasses, improved printing presses, and countless other inventions. The Wright brothers were among the first to discover the basic principles of lift and thrust in fixed-wing aircraft, which opened the door to modern aviation. Eli Whitney invented the cotton gin, which transformed the processing of cotton fiber and revolutionized the textile industry.\n\nThomas Edison, \"the genius of Menlo Park,\" was granted more than one thousand patents during his life for his inventions with electricity, including the incandescent lightbulb, the telephone, the telegraph, and the motion picture camera. And there was Henry Ford, whose goal was not simply to build a better car, but to build an automobile that every family could afford. He built the first Model-T Ford in 1908\u2014the Tin Lizzie\u2014and shortly thereafter developed the concept of the assembly line, which revolutionized manufacturing the world over. George Eastman, who invented and popularized the Kodak camera, gave us the first portable and affordable cameras that anyone could operate. Such inventions have literally changed the world.\n\nAt the same time, American scientists and engineers have pioneered some of the most formidable advances in civil engineering, such as the construction of the Hoover Dam during the midst of the Great Depression. It was a monumental undertaking, and it continues to provide electricity and water today for more than eight million people in the states of California, Nevada, and Arizona. Any list of American achievements in science would have to include the great strides this country has made in medicine and medical technology, improving the lives of billions around the world.\n\n5\n\nAdvances in medical practice and emergency treatment save lives every day through fast-response trauma teams and state-of-the-art surgical procedures. Modern medicines have extended life-expectancy by decades while advances in audiology, dentistry, and optometry have improved the quality of life for millions more. America is still the only country to put a man on the moon or to send an unmanned rover vehicle to the planet Mars, some forty million miles from Earth. And I should also mention the successes of Hollywood, the cinema, documentary filmmakers, radio, television, and the broadcast media in all their various forms. No other medium has done more to inform, educate, and entertain us than the arts of broadcast and film.\n\nAlong with all of this, the telephone may be the real success story of our time. Telephone technology has come a long way since the days of Alexander Graham Bell and Thomas Edison, giving us the Internet, high-speed local-and wide-area networks, cellular telephones, the iPhone and iPad, and countless other modern inventions. America leads the world in the development, distribution, and commercial success of all these modern marvels, and has unleashed a new era of mass communications.\n\nNo one disputes the importance of these things, but few realize that none of them would have happened if it weren't for the even greater achievements in political discourse: specifically, the Declaration of Independence and the Constitution of the United States, including the Bill of Rights. These two documents, which for the first time in history laid out the principles of limited government and natural rights in a precise, ordered, and prescriptive manner, are America's gift to the world. The War of Independence that led to the establishment of this new nation was not simply a blow for personal freedom; it was above all a statement of the value the American people place on liberty and freedom of conscience for all people. And it was a statement of our willingness to defend those liberties at home and abroad.\n\n6\n\nAMERICA'S GREATEST ACHIEVEMENT\n\nAs we consider these various achievements, it's important to recognize that the true source of America's greatness is not merely the inventions and creature comforts we've accumulated over the years but the wisdom and vision that made them possible. That legacy comes to us from men such as John Adams, Thomas Jefferson, James Madison, Benjamin Franklin, and George Washington, through the values and beliefs they enshrined in our founding documents. Among the greatest gifts one generation could ever give to another are freedom of worship, freedom of speech, freedom of the press, freedom of association, and the right to a fair and just hearing. These were all gifts from the Founders.\n\nIt is important to understand, furthermore, that these liberties are the outward expressions of our Judeo-Christian heritage. When the Pilgrims left the safety and comfort of their homes in Europe to cross an angry sea and plant the first colonies in the New World, they were guided by their strong Christian faith. The principles they lived by have been the cornerstone of America's success for the past 250 years and are still the moral compass we follow today. Despite the claims so often repeated these days that the Founders were simply Deists who believed in a watchmaker God who left the creation to fend for itself, we now know that fifty-two of fifty-six Founding Fathers were devout believers in Jesus Christ.\n\nI have written about this in previous works and won't recite all the evidence here, but even the man whom most people agree was the least religious of the Founders, Benjamin Franklin, knew that no great nation would ever rise upon these shores without the aid and intervention of a great and wise God. In one of the most surprising speeches of the revolutionary era, the sage of Philadelphia reminded his colleagues in the Continental Congress that, \"Except the Lord build the house, they labour in vain that build it\" (Psalm 127:1). He then petitioned that body, which had been hopelessly mired in debate, to begin each day's deliberations with prayer. The delegates were humbled by his words because they knew he was right. They paused then for a time of prayer, and they vowed to pray every day in the same manner until they had resolved their differences. The document they produced has guided this nation ever since, and it was even hailed by an English prime minister, William Gladstone, who said, \"The American Constitution is, so far as I can see, the most wonderful work ever struck off at a given time by the brain and purpose of man.\"\n\n7\n\nThe story of America's greatness is not only about glory and triumph. Some of the nation's greatest achievements were only made possible by the adversity our ancestors endured. Between 20,000 and 25,000 Americans lost their lives in the American Revolution, and nearly the same number were seriously wounded. Despite the risks, they were willing to sacrifice their lives for the great prize of independence and individual liberty. More than 600,000 Americans died in the Civil War, but that struggle preserved the nation and transformed our understanding of human and civil rights. Add to that the more than 115,000 Americans who died in the First World War and the 292,000 in the Second, and you have a glimpse of the enormous price our predecessors paid to keep this country free.\n\nMore than any other nation, including all the great empires of the past, America has spread the dream of liberty around the world and helped to bring a higher standard of living to untold millions. And like those ancient empires, which were the standard-bearers of culture and learning for a time, today America has been entrusted with transmitting the blessings of freedom. More than mere business connections or scientific expertise, what America has to share with other nations is our appreciation of the values of integrity, self-discipline, and self-determination passed down to us by the Founders. Whether it's in regard to politics or economics or industry or any of the modern disciplines, we will find that in every area America's greatness is founded upon the moral and religious values of those pioneers.\n\n8\n\nAs the scholar and historian Russell Kirk has written, \"Every people, no matter how savage or how civilized, have some form of religion: that is, some form of belief in a great supernatural power that influences human destiny.\" Culture, Kirk said famously, comes from the cult. That is, the distinctive qualities and customs of every culture arise from the religious beliefs of its people. The Communists attempted to deny the existence of God and made atheism the only acceptable form of belief. But as the Soviet Union's collapse in 1989\u2013 1990 made clear, the empire had failed to stamp out religious faith completely, and today Christianity is thriving once again in the former Soviet bloc. The Communists discovered that no nation can survive for long without a foundation of sound moral principles.\n\nConcern for the well-being of others is one of the key traits of good character. Unfortunately, we see less and less of that these days. And when we see rising crime rates, evidence of corruption in business and government, the breakdown of the family, the increase in out-of-wedlock childbirths, the ongoing tragedy of abortion, and a rising climate of immorality and vulgarity in the popular culture, we have to wonder if our great moral heritage can survive. Author and attorney Charles Colson has written that, \"A nation or a culture cannot endure for long unless it is undergirded by common values such as valor, public-spiritedness, respect for others and for the law; it cannot stand unless it is populated by people who will act on motives superior to their own immediate interest.\"\n\nThe American ideals of freedom and individual rights, charity, duty, honesty, and love for others are, above all, religious beliefs. Even though America is less visibly a religious nation today than it was a century ago, it is the depth and strength of the foundations laid down by our Christian forebears that have allowed us to thrive in the twentieth and twenty-first centuries. And, even with all our struggles and doubts, we are still living on the dividends of that investment.\n\n9\n\nAmerica's success in almost any area is a tribute to the beliefs that shaped the American character. \"These beliefs,\" writes Russell Kirk, \"are the fatherhood of God, the brotherhood of man, and the dignity of man. From these beliefs have developed Christian convictions as to how we should conduct our lives, how we should treat our fellow human beings, and what makes life worth living.\" Even though we may be troubled by growing hostility toward our Christian faith and the increasing coarseness of American culture, we can be certain that God will not give up on this country so long as a faithful remnant continues to seek His favor and proclaim these truths.\n\nTHREATS FROM WITHIN\n\nThe problem is that we have already come a long way down the road of dissolution, and it will take more than a little effort to recover our losses. According to the most recent Census Bureau reports, the number of people between the ages of twenty-five and thirty-four who have never married is now greater than those who have married. This has never happened before in our history\u2014or in world history for that matter. Married adults now make up just 52 percent of the population, which is the lowest level since records have been kept.\n\nThis has a direct impact, of course, on the illegitimacy rate, which is now almost 41 percent overall, and 72 percent among African Americans. No wonder so many young people find themselves trapped in chaotic and empty lives. The link between illegitimacy and poverty is well documented, and the link between poverty and crime is undeniable. Senator Daniel Patrick Moynihan was among the first to warn the nation what was in store for this country if the breakdown of the traditional family persisted. In a stern warning penned in 1965, he said,\n\nFrom the wild Irish slums of the nineteenth-century Eastern seaboard to the riot-torn suburbs of Los Angeles, there is one unmistakable lesson in American history: A community that allows a large number of young men to grow up in broken families, dominated by women, never acquiring a stable relationship to male authority, never acquiring any rational expectations about the future\u2014that community asks for and gets chaos . . . [In such a society] crime, violence, unrest, unrestrained lashing out at the whole social structure\u2014these are not only to be expected, they are virtually inevitable.\n\n10\n\nA complex society such as ours is made up of a whole range of unspoken commitments. We agree to respect one another's privacy, property, and personal dignity. We agree to abide by a code of ethics, legal and moral restraints. We agree to care for our loved ones and family, and to be responsible members of our community. We also agree to pay our taxes, participate in free elections, and contribute to the common defense. But what happens when those commitments no longer matter? What happens when young men and women give up on marriage and responsible family formation? What happens when a generation of young Americans, or more than one, is abandoned to the wasteland of idleness, drugs, and premarital sex?\n\nWhen a culture is vital and thriving, we take our roles as citizens seriously. We agree to support causes that are worthy and to avoid behaviors that are destructive. These commitments are part of the common code of decency passed down from one generation to the next. Although there are individuals in the media, the popular culture, and the halls of academia who insist these old-fashioned ideas no longer matter, we need to remember what the Soviets learned after seventy years of communism: no nation can survive for long without a body of sound moral principles.\n\nIn a small but important little book called _The Broken Hearth: Reversing the Moral Collapse of the American Family_ , former secretary of education William Bennett makes the case that the American family is the most important incubator of the values that allow individuals and their communities to prosper. Healthy families instill values and beliefs that are essential for happiness and success. Among these are the habits of trust, altruism, personal responsibility, and mutual obligation. Unfortunately the well-being of the American family has been subverted in recent years by a sustained assault on marriage and family, and the evidence of that is not a pretty sight. Bennett writes:\n\n11\n\nSince 1960, the divorce rate has more than doubled, out-of-wedlock births have skyrocketed from one in twenty to one in three, the percentage of single-parent families has more than tripled, the number of couples cohabiting has increased more than elevenfold, the fertility rate has decreased by almost half. In record numbers, we have seen fathers deserting their wives and children\u2014and being permitted to do so without reproach or penalty of any kind. We have seen stay-at-home mothers mocked.\n\nThroughout most of human history marriage has been regarded as a sacred covenant, yet today we are told that marriage and family life are a burden, outdated customs with little or no importance. Unfortunately it is not just the rock stars and teen idols who are propagating this nonsense, Bennett writes, but \"feminists, academic analysts with an agenda, and libertines masquerading as liberationists.\" There are celebrities and popular television personalities who would have us believe that \"sin is in\" and faith in God is a dangerous myth. Those assumptions are not only false but dangerous to our entire way of life. As Bennett concludes, \"There is a natural order that we may build on and improve but that we attempt to do away with at peril of the very fabric of our lives, our happiness, our true and solid contentment. Too many of us have attempted to do just that and have reaped a whirlwind.\"\n\nYou don't have to spend a lot of time browsing the morning's headlines to recognize the levels of controversy and chaos that pervade American society these days. The nation is socially, politically, and morally divided on so many issues; immersed in disputes of one kind or another; and wrangling over questions unimaginable just a few years ago. It often seems as if everything is now up for grabs.\n\n12\n\nThe Pledge of Allegiance, which every American ought to be able to recite with pride, has become a topic of debate in many places, and concepts such as American exceptionalism and manifest destiny are either forgotten or ignored by the mainstream culture. Simply flying the American flag is considered controversial in some places, and posting the Ten Commandments in schools or public buildings is out of the question. All of this compels me to ask, are we still \"one nation under God\"? Is this once-proud nation of immigrants from every corner of the globe still, in fact, \"indivisible\"? The red and blue maps emblazoned across the front pages of newspapers and websites after every election tell pretty much the same story. The social consensus is fragmented. We are a nation of red and blue states, red and blue counties, and even red and blue neighborhoods. Our political and cultural differences are magnified by the media to such an extent that we often wonder if the republic can survive.\n\nNot long ago we learned that Washington lawmakers were working behind closed doors, plotting and planning through the night to pass a massive health-care bill and other legislation they knew the American people had overwhelmingly rejected. The results of the 2010 elections were a partial response to that, but we still have many areas of concern. With shocking regularity, the courts have been handing down edicts, overruling the voters on important social issues, defying both common sense and traditional moral values. At the same time, the mass media are continually bombarding us with coarse and violent images that stagger the imagination.\n\nTRUE STRENGTH OF CHARACTER\n\nIt would be easy to lose hope in this environment, but sometimes in the midst of all the noise and confusion we hear about an act of kindness that gives us renewed hope that the American Spirit is not dead just yet. We want to believe that, deep down, there is still something good and decent about the American people, and I don't think that is a false hope. The courage and resolve that made this the greatest, freest, and most prosperous nation on earth refuse to fade away, and every once in a while we witness a spark of humanity that reminds us of that fact. We are still a God-fearing people. And despite the bickering and name-calling, there is still a strong sense that we're in this thing together.\n\n13\n\nThat is how many of us felt when we first heard the story of Captain Chesley \"Sully\" Sullenberger, who was the pilot on U.S. Airways Flight 1549 from New York to Charlotte on January 15, 2009. Shortly after takeoff that afternoon, Captain Sullenberger radioed the tower that his plane had hit a large flock of birds, and at least one engine was on fire. They talked about landing the Airbus A320 at another airport but soon realized that was going to be impossible. Sully knew they would never make it, so he decided to ditch the plane in the only place where his passengers might actually survive: Manhattan's Hudson River.\n\nHundreds of men and women standing on the embankment, driving their cars on the parkway that runs alongside the river, and sitting in the skyscrapers nearby watched in amazement as Sully maneuvered his plane toward the middle of the river and then splashed down with the fuselage and both wings resting safely on top of the water. Coast Guard and Port Authority boats, as well as a couple of sightseeing boats, quickly encircled the aircraft to pick up the stranded passengers and crew.\n\nThen after two quick tours of the cabin to make sure no one was left onboard, Captain Sullenberger grabbed the pilot's log from the cockpit and joined the others. He was the last man out, and moments later, as the rescue boats were making their way to shore, the plane began listing to one side and then slipped farther into the water.\n\nWhen reporters looked into everything that happened that day, they discovered that life had prepared Sully Sullenberger for that moment. He had learned to fly airplanes as a teenager, attended and graduated from the Air Force Academy near the end of the Vietnam War, and flown fighter jets in the military. He had become an expert on aviation safety and had spoken to professional groups about maintenance and safety procedures. Later when he was interviewed on national television about the events of that day, Sullenberger said, \"One way of looking at this might be that for forty-two years I've been making small, regular deposits in this bank of experience: education and training. And on January 15 the balance was sufficient so that I could make a very large withdrawal.\"\n\n14\n\nThat's a nice way of putting it, but it isn't the whole story. Captain Sullenberger's first concern was the safety of the people on that plane, and this was what motivated his heroic actions. Certainly education and training can help prepare a person for making critical decisions, but other things are needed as well\u2014things such as faith, courage, determination, and love for others. Captain Sully's words inspired a lot of people and reminded us what really matters, but they also reminded us of all the brave men and women who serve this country in times of crisis.\n\nWho will ever forget the example of the New York firefighters who raced into the World Trade Center in the aftermath of the 9\/11 terrorist attacks, knowing they might never return? The towers were weakened\u2014smoke and debris were everywhere\u2014it was just a matter of time until they began to collapse. But people were dying in those buildings, and there was no time to waste. Without hesitation the firefighters climbed the stairs, hundreds of them, with no thought for their own safety, guiding thousands toward the exits, bringing hope and relief and renewed determination to men and women who might otherwise have been stranded in those doomed towers.\n\nThose public servants weren't in it for the money. Not one of them was looking for glory. They could have refused to go up, but they accepted the risks and went in because they believed in something greater and more enduring than their own personal safety\u2014the call of duty, the love of others, and the pledge that each of them had made to defend, protect, and serve. Like thousands of police officers and firefighters all across this country who risk their lives each day, the New York firefighters embodied the finest traditions of honor and bravery. And a grateful nation will never forget what they did.\n\n15\n\nIt's important to remember that spirit of caring and compassion when we are constantly bombarded by bad news. The shooting of Arizona congresswoman Gabrielle Giffords by an irate gunman on January 10, 2011, was yet another reminder that there is still a lot of senseless violence in our streets. But even in that case, where more than a dozen people were wounded or murdered by a deranged psychopath, it was the courage and heroic actions of ordinary Americans who risked their lives to subdue the shooter and render aid to the injured that will be remembered long afterward.\n\nLater that day during a live TV interview with NBC's Brian Williams, one of the heroes made the comment that those of us living in the age of 9\/11 can no longer just stand by and watch when something bad is happening. We have to be prepared to jump in and resist evil, to take a stand even when it means almost certain peril. I thought that was a perceptive observation. Unfortunately, it is a lesson that can only be learned through adversity.\n\nWe've all heard stories of cowards who stand by and watch when innocent people are being victimized or people who refuse to get involved because it is inconvenient or dangerous. But I think there is less of that today than there was a decade ago. One of the best examples of that sort of bravery is the story of the five young men who rushed the cockpit of United Flight 93 to stop a fourth group of 9\/11 terrorists from crashing that plane into the White House or the Capitol building. During cell phone conversations with wives and friends back home, the young men learned that two planes had already struck the twin towers in New York and a third had hit the Pentagon. Then, when they realized their own plane was being hijacked, those five brave men\u2014 Todd Beamer, Jeremy Glick, Mark Bingham, Tom Burnett, and Lou Nacke\u2014decided to take action. It was an all-or-nothing gamble, and they knew it could ultimately cost them their own lives, but they were determined to stop the terrorists in their tracks.\n\n16\n\nMost of us remember the rest of the story: there was a violent struggle in the cockpit; the plane spiraled out of control, then crashed in a field in Shanksville, Pennsylvania, approximately 125 miles from Washington, DC. Although America still grieves the loss of the passengers and crew of United 93 and the heroes who refused to let an even greater disaster take place on their watch, their actions gave the nation a new sense of hope at a very difficult time. Since that day the heroes have been memorialized for their bravery. Books and articles have been written about them, and Todd Beamer's call to action, \"Let's roll!\" has become a rallying cry that will be remembered forever.\n\nJust thinking about that amazing act of self-sacrifice reminds us once again of the inner core of goodness that still survives, as strong as ever, in so many Americans. Hearing about such acts of valor brings things back into focus. It is as if we have to experience such horrors before we understand what really matters. And when there is so much disappointment and self-doubt all around us, it helps to know there are still people who care enough to risk everything for others.\n\nBEYOND THE CALL OF DUTY\n\nJesus said, \"Greater love hath no man than this, that a man lay down his life for his friends\" (John 15:13), and that is the story of another American hero, Marine Corporal Jason Dunham, who served in Iraq during Operation Iraqi Freedom. In the spring of 2004, Jason was a team leader with the 3rd Battalion, 7th Marines. While on patrol in the village of Husaybah, Iraq, his team spotted a suspicious-looking individual in a white SUV. As they approached the vehicle, the man jumped out and began threatening the marines. During the brief struggle that ensued, Jason realized the man was going to throw an armed grenade. As soon as he saw it, the corporal yelled to his men to take cover. Then, in a flash, he knocked the grenade from the man's hand, took off his Kevlar helmet and covered the grenade, then threw himself on top of it.\n\n17\n\n_Wall Street Journal_ reporter Michael Phillips, who was embedded with Jason's battalion at the time, was so moved by the young man's act of bravery that he told the story in his book, _The Gift of Valor_. Corporal Dunham had less than five seconds to make that fateful decision, Phillips wrote, but he didn't hesitate to sacrifice himself for others. If he hadn't reacted so quickly, the carnage would have been much worse, but Jason Dunham willingly gave his own life to save his fellow marines.\n\nCorporal Dunham survived in a coma long enough to be evacuated to Bethesda Naval Hospital in Maryland. The magnitude of his wounds was just too great, but his mother and father were at his bedside when Jason died eight days later. They were heartbroken, but they nevertheless understood the courage and commitment that had motivated their son's actions.\n\nOn January 11, 2007, President George W. Bush presented the Medal of Honor to the Dunham family in a ceremony at the White House. Afterward, Jason's father said he wanted everyone to understand that while he was proud of his son's act of bravery, these are the types of heroic actions our young service men and women are prepared to perform every day. They are all heroes, he said. And we can't help but ask, as many others have asked before, where do we find such brave men and women?\n\nThe answer, I believe, is that we find them among people who understand the true meaning of duty, honor, country, and all those fine virtues the mainstream culture loves to ridicule and disdain. We find them among people who have seen the arrogance and selfishness of those who refuse to risk anything for anyone else and have rejected that kind of thinking. Even if we sometimes feel that decency and concern for others are disappearing from our culture, it's encouraging to discover it is still there, and there are still men and women willing to go beyond the call of duty.\n\nIf we want to see our communities restored, mothers and fathers will need to instill self-discipline and good judgment in their children, teaching them by example. Our churches will need to teach the precepts of faith, hope, and love; and the schools ought to be teaching our children the importance of personal responsibility and self-respect. But for these lessons to be understood and applied, there has to be something even more essential: there has to be an awareness that we are not alone. We are part of a family, a community, and a nation; and the privilege of citizenship in this great nation means that we have a number of important rights and responsibilities.\n\n18\n\nThe Founders believed education was an essential component of citizenship, as was instruction in moral and religious values. The Northwest Ordinance of 1787 expressed it this way: \"Religion, morality, and knowledge, being necessary to good government and the happiness of mankind, schools and the means of education shall forever be encouraged.\" Schools, they said, were entrusted with the task of helping to mold the character of young people, and the Christian faith was a vital part of that.\n\nBenjamin Rush, a signer of the Declaration of Independence and delegate to the Continental Congress, offered this counsel:\n\nLet the children who are sent to those schools be taught to read and write and above all, let both sexes be carefully instructed in the principles and obligations of the Christian religion. This is the most essential part of education. The great enemy of the salvation of man, in my opinion, never invented a more effectual means of extirpating Christianity from the world than by persuading mankind that it was improper to read the Bible at schools.\n\nThe constitution of the commonwealth of Massachusetts at that time went even further, stating that, \"It is the right, as well as the duty, of all men in society, publicly, and at stated seasons, to worship the Supreme Being, the Great Creator and Preserver of the Universe. . . .\" It may be surprising to some people to realize that this was the common view of practically all Americans during the founding era, and for more than a hundred years afterward. But no one expressed the views of the citizens of New England better than our second president, John Adams, who said,\n\n19\n\n[I]t is religion and morality alone which can establish the principles upon which freedom can securely stand. . . . We have no government armed with power capable of contending with human passions unbridled by morality and religion. . . . Our constitution was made only for a moral and religious people. It is wholly inadequate to the government of any other.\n\nThere is no way to explain away such statements. The United States of America was founded on Christian principles because the Founders were building a nation and laying the foundations for an empire. They understood that good citizenship demands good character. They were well educated, steeped in the history of great nations of the past, and they knew that no nation can survive unless the citizens are infused with the principles of sound moral judgment from their earliest years. The president of the Continental Congress, Elias Boudinot, explained why this was so important: \"Good government generally begins in the family,\" he said, \"and if the moral character of a people once degenerate, their political character must soon follow.\"\n\nMen and women with shabby morals and weak character cannot be expected to sustain a great nation. They can't vote wisely, lead productive lives, or educate their own children. The Founders understood that character and virtue grow out of the spiritual truths we learn in our earliest years, but that view is no longer taught in the nation's public schools and universities. Over the past century, our leading educational institutions have made Christianity virtually unmentionable\u2014not just prayer in the schools but any mention of Jesus Christ or the beliefs of at least 80 percent of the American people.\n\nEven though all of our first universities\u2014including Harvard, Yale, Princeton, Dartmouth, Brown, and Columbia\u2014were founded as Christian institutions with seminaries to train clergy and educate future leaders of the republic, today's educators no longer tolerate those beliefs. Instead of sound moral instruction, students on thousands of campuses all across the country are being exposed to a steady drumbeat of anti-Americanism, race and gender hostility, and a preoccupation with social and sexual experimentation. Classes and entire academic departments are devoted to \"oppression studies,\" \"women's studies,\" \"queer studies,\" and other forms of politically correct programming.\n\n20\n\nAt many schools, incoming freshmen are required to attend mandatory sensitivity training sessions and seminars on race and gender. Dorm counselors are expected to hold encounter sessions in which new students are indoctrinated into the pro-homosexual campus culture through role-playing and other types of psychological manipulation. The campus code words _diversity_ and _tolerance_ , which are so pervasive, are simply a mask for the anti-American, hypersexualized, pro-homosexual, and anti-Christian bias that is now the norm in the vast majority of our colleges.\n\nWith all this happening on high school and college campuses, how can we expect our young people to develop the character and commitment to duty that will make them good citizens? American sociologist Paul Hollander has described the modern university as \"the reservoir of the adversary culture.\" Rather than teaching our bright young men and women to respect their country and its heritage of freedom and self-reliance, professors and administrators have created an environment that is openly hostile to the beliefs and aspirations of the Founders.\n\nSilly ideas we laughed at back in the 1960s are now the reigning dogma on many of America's most prestigious campuses. Students are taught that all values and beliefs are relative, that truth is in the eye of the beholder, that conventional religion is a myth, and only science and mathematics are to be trusted. Subjects such as history and literature have become vehicles for indoctrinating students in the doctrines of multiculturalism and diversity. Our children are taught by their sociology professors that the American way of life is no better than that of the most backward tribes. And for purveyors of the environmental movement, our way of life is a threat to global ecology. Some members of that movement claim that human beings are the problem, so the number of people on the planet needs to be dramatically reduced to save the earth.\n\n21\n\nConcepts such as civic virtue, character building, and moral restraint, on the other hand, are scorned by faculty members while the false doctrines our parents rejected, including communism and Marxism, are applauded. I am told by university associates that the surest path to tenure for young professors these days is to espouse a Marxist interpretation of their academic discipline. And the surest way to be denied tenure or fired is to let it be known that you have a Christian worldview.\n\nThe greatest evils are no longer sloth, gluttony, envy, and pride, but capitalism and Christianity, which are openly mocked in some places. As one recent study puts it, \"Since the social revolution of the sixties, the agenda of the Left has been to transform the United States into a socialist utopia; consequently, the issue of greatest concern on America's most distinguished university campuses is no longer traditional learning but a new form of social and sexual indoctrination.\"\n\nASSESSING THE DAMAGE\n\n\"Professing to be wise, they became fools, and changed the glory of the incorruptible God into an image made like corruptible man . . .\" (Romans 1:22\u201323 NKJV). So wrote the apostle Paul, proving that, ages before today's radical professors came onto the scene, such behaviors were well known. Sadly, the results were much the same then as now, and we can see the impact of declining standards in a long list of reports detailing the poor performance of American students on standardized tests.\n\n22\n\nOn the 2005 National Assessment of Educational Progress (NAEP) science test, for example, 46 percent of American twelfth-grade students scored \"below basic.\" On the math portion of the exam, 39 percent scored \"below basic\"\u2014which tells us that nearly half of all high school seniors cannot answer even the most basic algebra and geometry questions. Researchers found that very few students were excelling. In science, just 29 percent of twelfth-grade students scored at the \"proficient\" level, and just 3 percent scored \"advanced.\" Math scores were equally disappointing, with 35 percent of U.S. students in the \"proficient\" group, and a shocking 5 percent at the \"advanced\" level.\n\nPerformance of American students compared to students in other countries is even more disturbing. The percentage of American college students earning degrees in science, technology, engineering, and mathematics lags well behind the percentage of students from China, India, Japan, Russia, Mexico, and the Middle East. According to research from the Heritage Foundation, the 2007 Trends in International Mathematics and Science Study (TIMSS) report reveals that students in other countries (especially the Asian countries) consistently outperform U.S. students in science and math.\n\nResults from the 2009 Program for International Student Assessment (PISA) exam, which is administered every three years to students from sixty-five countries, reveal that American teenagers rank well below their peers in the top thirty industrialized countries, and they continue to score well below average in critical subject areas. While students from Singapore and China scored consistently at the highest levels in all categories, American fifteen-year-old students scored seventeenth in reading, twenty-third in science, and thirty-first in math. They came in twenty-third or twenty-fourth in most subjects, and even lower in subjects in which they were expected to excel.\n\nFor years the evidence of deteriorating educational standards has been most pronounced in the upper grades, with twelfth-grade students scoring generally worse than students in the lower grades. But according to results of the 2009 National Assessment of Educational Progress (NAEP) assessments, just 34 percent of fourth graders, 30 percent of eighth graders, and 21 percent of twelfth graders are performing at or above the \"proficient\" level. Furthermore, just 1 or 2 percent at each grade level scored at the \"advanced\" level, and relatively large numbers of students didn't even attain the most basic level.\n\n23\n\nEven the current secretary of education, Arne Duncan, had to admit that current levels of educational attainment are a threat to America's future success. At the first public unveiling of the latest results, Duncan said, \"The results released today show that our nation's students aren't learning at a rate that will maintain America's role as an international leader in the sciences.\" He added that, \"When only 1 or 2 percent of children score at the advanced levels on NAEP, the next generation will not be ready to be world-class inventors, doctors, and engineers.\"\n\nIn a stunning comment on the state of public education in this country, the National Commission on Excellence in Education said that, \"If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war. As it stands, we have allowed this to happen to ourselves . . . we have, in effect, been committing an act of unthinking, unilateral, educational disarmament.\"\n\nYou would think such findings would prompt massive changes in public and higher education, but the authors of the summary report for the 1990 National Assessment of Educational Progress were obliged to admit that \"large proportions, perhaps more than half, of our elementary, middle, and high-school students are unable to demonstrate competency in challenging subject matter in English, mathematics, science, history, and geography. Further, even fewer appear to be able to use their minds well.\" Comments like these, reported by author and educational sociologist Charles J. Sykes, provide a staggering array of anecdotal and statistical information confirming the risks to children in America's public schools. His book, _Dumbing Down Our Kids_ , ought to be required reading for every parent and educator.\n\n24\n\nCombining the candor of such statements with the fact that the professional \"educrats,\" the major teachers' unions, and many of our universities are mainly concerned with the social and sexual indoctrination of America's young people, you realize we have a big problem. Fortunately not everyone is happy with this situation, and as more and more evidence of the damage being done to the next generation of Americans comes to light, more and more of our citizens are rising to the challenge.\n\nFor many of the educational programmers, money is a major motivating factor; but many of those at the forefront of the radical transformation of American values in the schools and colleges are losing supporters and funding. Some of those giant endowments are shrinking. Furthermore the results of the 2010 midterm elections suggest that large numbers of people in this country are giving up on the progressive agenda, and this means many more parents and public officials will be looking for answers concerning what's been going on in the schools.\n\nCHANGING DIRECTIONS\n\nA Rasmussen Reports national telephone survey taken in December 2010 revealed that Americans are deeply troubled by the way things have been going in this country. Leading up to the presidential inauguration in January 2009, the number of voters who felt the country was heading in the right direction was below 20 percent. Confidence rose briefly to 40 percent in early May, but it began falling after that. While approximately half of black voters still believed the country was heading in the right direction, 74 percent of whites and 76 percent of all other voters were pessimistic about the course the country was on.\n\nSubsequently the Rasmussen Poll for June 18, 2011, found just 21 percent of American voters strongly approve of the way President Obama is performing as president. Meanwhile 41 percent said they strongly disapprove, giving the president an approval index rating of \u201320. In his book _In Search of Self-Government_ , published in late 2010, Scott Rasmussen offered the perceptive observation that \"the gap between Americans who want to govern themselves and politicians who want to rule over them may be as big today as the gap between the colonies and England during the 18th century.\"\n\n25\n\nThe concerns that led to the American Revolution were taxation without representation and the imposition of unjust laws by the British Parliament. Among them was the act of Parliament closing the Port of Boston to normal commerce, the appointment of British aristocrats to oversee colonial governments, the protection of English criminals from the judgment of colonial courts, the quartering of British troops in American homes without payment of rent or lodging expenses, and the interference in religious customs with punitive policies and statutes that favored Canada and the growing wave of Catholic immigrants to that country. Furthermore the colonists were not allowed to elect their own representatives to the English House of Commons, and they had no standing in the House of Lords. Ultimately it was these \"Intolerable Acts\" that propelled the American patriots toward revolution. Because they had no say in their own government and were subject to the whims of autocratic English governors, the colonists concluded that England had no legitimate claim on America.\n\nWe are not seeing signs of an insurrection on that scale in this country so far, but a lot of people are on edge because of the incompetent handling of the economy, the bailouts of corrupt financial institutions, the government takeover of automobile and insurance companies, the passage of universal health care, and the threats of increased taxation. There have been no violent demonstrations or marches on the White House\u2014most people believe the 2010 midterm elections sent a message to Congress and are still optimistic that our voices are being heard.\n\nWe are by nature a peaceful people who are accustomed to an orderly and natural transition of power. We are hopeful that the presidential election in 2012 will help reduce the interference of the federal government in business and our private lives. But if something should happen to derail those hopes or if enough people begin to feel they are being cheated and disenfranchised by the administration or its allies on Wall Street and in the federal courts, then who knows what could happen? It doesn't have to result in a massive demonstration of discontent, and I doubt that it will. It is possible to deal with all these issues politically and diplomatically. But if for some reason the American people come to believe they are being cheated out of their constitutional right to determine how they are governed, then it is possible things could take a very different turn.\n\n26\n\nAll conservative voters really want is to get back to the type of government the Founders gave us in the first place. It is very important that we have a better understanding of our history and why the founding documents were written as they were. We need to understand how revolutionary those documents were and still are, granting unprecedented freedoms to the people and establishing the foundations of the rule of law. If we understand the environment of the revolutionary era as well as the environment we are dealing with today, then we will have a much better grasp of how things will play out from here on.\n\nBut let me be clear about this: I believe we are on the threshold of a revolution\u2014a peaceful revolution, to be sure\u2014but a genuine change in how We the People relate to our elected representatives from now on. We believe that substantial changes will have to be made, and the difficult task of undoing some of the laws enacted by the last Congress and safeguarding our sacred liberties will mean resisting the impulse to compromise while defending our time-tested conservative principles against the charges and countercharges our political opponents will make.\n\nBut we are trusting that the men and women we've sent to Washington as our representatives in the most recent housecleaning operation will be faithful to the challenges we have given them and do the right things. We are not being naive, thinking things will suddenly change in Washington without the need for voters to exercise due caution and vigilance, but we do believe salutary changes will come.\n\n27\n\nWe are deeply concerned that a generation of young people has been taught that America is corrupt and our system of government is not worth saving. They have been taught this in our schools, and their heads have been filled with politically correct nonsense. The generation coming up may not even be equipped to take the reins of government when its time comes. That is why we will have to start making changes from the top down.\n\nWe used to think reform would come from the bottom up, but that is not going to happen. Changes will need to be made in the centers of power, from the administration in Washington and all the various agencies of government to the state and local levels. At the same time, we will need to lobby for reforms in both public and higher education if we expect to see real, long-lasting change.\n\nACCEPTING THE CHALLENGE\n\nWe are going through difficult times in America today, but it is not the worst we have endured. The country is in the midst of an economic downturn, and most people are not living as well as they did a decade ago. Nevertheless, sometimes it takes a little stress to make people stop and think about what they've been doing. I believe these may turn out to be some of the best times we have ever seen if we are able to recapture a vision of who we are and what this nation was meant to be.\n\nAs a Christian minister, I tend to believe that God may have allowed these tough times to come about so that the men and women of this nation will wake up and reclaim their patrimony. We need to take a look around and remember the incredible privilege we have as Americans. The rally on the Washington Mall hosted by Glenn Beck back in August 2010 was dedicated to restoring honor, and that's really what it's all about. We have to rededicate ourselves to the essential virtues of honor, truth, integrity, and faithfulness if there is any hope of restoring our lost honor.\n\n28\n\nDuirng a speech I heard Ronald Reagan deliver in the late 1980s, he said, \"Never has there been a time when people with so much to lose have done so little to keep it.\" I think that may have been the case in recent years, but it is not going to be the case much longer. We have lost more than enough ground, but I see the banners of a million dedicated patriots rising above the horizon, and bigger and better changes are on the way. Regardless of what happens in Washington this year or next, I believe there is hope. Throughout all the years, from the American Revolution through the Civil War and two world wars until this very day, the thing that has kept this nation going is hope, which comes from knowing who we are and whose we are. And that vision is still very much alive.\n\nIt is such an important message: yes, there is going to be a tomorrow, and we can make that tomorrow better. We have already begun to prove that by our actions at the polls. The citizens of Iowa proved it when they removed three liberal judges from the bench for ignoring the wishes of the people in order to push a far-left agenda. Voters crowding into town hall meetings all over the country are proving it by showing up and confronting unaccountable senators and congressmen. And all these changes are coming straight from the hearts of decent, hardworking Americans.\n\nOne of the tremendous things about this country is that we are a republic that can correct itself through informed citizen action without resorting to violence. President Obama made a dramatic promise of hope and change during his 2008 election campaign. That promise, along with the fact that he would become the first African American president, was apparently enough to persuade the voters to elect him to the nation's highest office. Mr. Obama knew that we have a constitutional system that allows us to make major changes of policy. We can change direction without a violent revolution, and his administration made some of the most dramatic changes in the history of the republic. But there is also a safety net, which is the will of the people.\n\n29\n\nThe Founders designed a system in which those who are elected to public office may only make major decisions with the consent of the governed. And when a government strays too far to the left or the right, the people have the right and constitutional duty to hold those elected individuals accountable. They have the power to right the ship of state through citizen action and the choices they make at the ballot box.\n\nThose who want to make the most radical changes in our form of government, to take the country in a new and untested direction, identify themselves today as \"progressives.\" Underlying their platform and their belief system is the idea that there is some new territory, some golden new sunrise, or some shining utopia out there that guarantees equality and peace and plenty for all if we can just shake off the shackles of the past and strike out in a new direction. But what they are offering is not new at all: it is the oldest fabrication known to man.\n\nThe idea of a man-made utopia has been the promise of every totalitarian society since the collapse of the Roman republic. But it is a false vision, a mirage, a wisp of smoke, and we would be foolish indeed to abandon the achievements of the greatest republic in the history of mankind for a system that has been the source of some of the greatest miseries ever known. The progressive promise is in reality a hallucination.\n\nThe men and women who took control of the United States Congress in January 2009 were on a mission, and the legislation they passed is clear evidence of the direction in which they were headed. I have heard from some of our representatives in Washington that if many of the most liberal members of Congress had not been turned out by the voters in the 2010 election, the progressives were poised to go much, much further. Within a couple of years they would have been calling for a total revision of the United States Constitution. That's how radical the aims of some of our leaders have become.\n\n30\n\nBut the 2010 election was a watershed. It was in effect a firewall against further encroachments and further legislative destruction from the Left. It was a good start, but it was just a start. Those who would like to undermine our form of government, our traditional values and beliefs, and our way of life haven't gone away. They're just waiting for an opportunity to strike again. That's one of the reasons why I've written this book, as a warning and a wake-up call to every citizen to stand up, speak up, and become engaged in the contest for America's future.\n\nMake no mistake; we are in a contest to see what this nation will be for the next two hundred years. Will we uphold the values of the Founders, who were dedicated to life, liberty, and the pursuit of happiness? Will we defend the moral and ethical values our forefathers fought and died for? Or will we stand by silently, like the dazed and defeated citizens of a half-dozen fallen empires of the past, and watch this nation sink into the swamp of socialism, totalitarianism, and anarchy, ultimately leading to collapse? This is the choice we have to make, and it is one contest we can't afford to lose.\n31\n\n[_**Two**_  \nTHE PROMISE OF AMERICA](Lee_9780849949463_epub_c5_r1.html#d6e4)\n\nBefore we can take corrective action to restore the foundations of American culture and regain a vision of this country's high calling, it is imperative that we understand the premises upon which the nation was built. Unfortunately the authentic history of the founding era and the birth of the republic has been almost completely forgotten. A generation or two ago schoolchildren celebrated the arrival of the Pilgrims with plays and vignettes of the landing at Plymouth Rock or the first Thanksgiving, but even these were incomplete and not very faithful accounts of what really happened.\n\nEven more troubling, many schools no longer teach very much about the colonial period at all, and what they do teach is so loaded with revisionist history that the miracle of America's founding is lost. This is tragic because we need to remember. If we truly understood what the first Americans endured to bring forth on this continent a new nation\u2014as Abraham Lincoln phrased it\u2014we could never again ignore their courage and determination, and no one would ever doubt that this is a Christian country. So to help us remember where they came from and what our forefathers accomplished, I would like to step back in time for a moment to see where it all started and how America came to be what it is today.\n\n32\n\nThere is no way of knowing how many ancient explorers may have set foot on the continent of North America before Christopher Columbus came ashore at San Salvador\u2014Spanish, French, Dutch, English, and Scandinavian sailors all claimed to have done so\u2014but news of the Genoese navigator's discovery in 1492 spread quickly through the palaces and public houses of Europe, prompting wild speculation about what kinds of fantastic treasures were simply waiting to be discovered in the New World.\n\nEnglish explorers and privateers had been scouting the coastline of North America for more than fifty years before the Cavaliers under John Smith arrived in 1607 in search of new territory for the British empire. Pirates and traders had come looking for new sources of gold and precious spices. Adventurers such as John Cabot and Francis Drake reached our northern coasts years before the first settlements were established at Roanoke and Jamestown, but without leaving any settlements of their own.\n\nQueen Elizabeth I, who reigned from 1558 to 1603, was intrigued by the tales she heard from seafarers such as Francis Drake and adventurers such as Walter Raleigh and Richard Hakluyt, who spoke of vast new lands and wealth beyond the dreams of avarice. Tantalized by these reports, the queen gave official sanction for further exploration, along with authorization for English trading companies to issue land grants establishing colonies in the New World.\n\nFor the Puritan congregations of England, however, rumors of a new land where men could be free from persecution, political oppression, and bodily mutilation awakened a very different set of emotions. Religious conflict had been raging across Europe for centuries. As many as two thousand men and women were tortured, beheaded, or burned alive during the Inquisition, and thousands more suffered terrible wounds. At least thirty thousand Protestants were slaughtered during the St. Bartholomew's Day massacre in France in 1572. Those purges had taken place in continental Europe, but when the Protestant Reformation reached the British Isles in the mid-sixteenth century, English dissenters who followed the new teachings suddenly became an endangered species.\n\n33\n\nTo put all this in perspective, what we know today as the Protestant Reformation began in the year 1517, when Martin Luther nailed his Ninety-Five Theses to the door of the castle church at Wittenberg in northeastern Germany. Luther, who was an ardent student of the Bible, realized that many common practices in the churches of that day were unbiblical and not in keeping with the teachings of Saint Paul. However, when he attempted to inform the bishops of his concerns during a trip to Rome, church officials ordered Luther to stop complaining. He was warned of serious consequences if he persisted, at which point he returned to Wittenberg deeply shaken.\n\nAfter a long period of anguish and introspection, Luther realized he could not remain silent. Heresy had crept into the church over many centuries and needed to be stopped. The letters of Saint Paul taught that salvation comes by grace through faith, not by works, yet the church was saying that acts of devotion, including gifts of money, could purchase \"indulgences\" that ensured salvation and forgiveness of sins. And there were many other concerns.\n\nThe list of questions and complaints Luther nailed to the church door created an immediate sensation, but the cardinals and bishops in Rome were not amused. The charges and countercharges that followed exacted a tremendous toll on Luther and his followers, and they led to insurrections, open defiance, and several bloody confrontations with church authorities. When he was summoned to appear before a papal court at Worms in 1521, Luther refused to admit he had been wrong, at which point he was excommunicated and forced into hiding.\n\nThe eight months he spent in a secret apartment at Wartburg Castle were agonizing for Luther, but he used the time profitably, translating the New Testament into everyday German and writing dozens of pamphlets and tracts to explain key biblical teachings in a pure and simple way. When he eventually returned to Wittenberg, he was hailed as a hero and the leader of the reform movement. Over the next half century, the controversy he had ignited spread like wildfire across Europe and far beyond.\n\n34\n\nCongregations that adopted Luther's reforms in England were declared heretics by the archbishop of Canterbury. The English church was Catholic at that time, and dissenters were subjected to severe punishments. The word _Puritans_ was originally a form of mockery, but the dissenters liked the name because the purity of the gospel was all they really wanted. Throughout the sixteenth and seventeenth centuries, those who refused to conform were declared outlaws and subjected to severe persecution. Hundreds were whipped, hung, burned, or branded with hot irons or had their ears cut off or their noses slit. Some were drawn and quartered. The lucky ones escaped and went into hiding, or were simply killed outright.\n\nProtestant pastors in England, many of whom were graduates of the universities of Oxford and Cambridge, were stripped of their positions and livelihoods. Some were tried by the ecclesiastical courts and banished, and sometimes they faced much worse. As the persecutions became more and more intense, some of the reformed congregations, identified as \"separatists,\" began looking for safer places across the English Channel in Europe where they could practice their beliefs without fear.\n\nOne of those congregations in the village of Scrooby in Nottinghamshire was led by a group of faithful Christian men; which included William Brewster, William Bradford, and John Carver; and two separatist pastors, Richard Clifton and John Robinson. Brewster, who had served as private secretary to a member of the royal court in London, learned from foreign travelers that Holland was the safest place in Europe for dissenters. When he shared the news with his congregation, a group of 125 separatists voted to sell their homes, farms, and all they had, and relocated to that country.\n\n35\n\nTHE YEARS OF SEPARATION\n\nThe story of what happened during the difficult months and years that followed was recorded by William Bradford in his detailed history, _Of_ _Plymouth Plantation (1620\u20131647)_. Although Bradford served as governor of Plymouth Colony for more than thirty years and is remembered as one of the guiding lights of the American founding, his book was generally unknown for nearly two centuries until it was republished in 1856. Since that time the book has been called the first great work of American literature and a classic of American history. Even so, it is seldom read today.\n\nIn his narrative Bradford describes how the separatists arranged for a ship to transport them to Holland under cover of darkness, only to be betrayed by the ship's captain. When they arrived to begin the voyage, the villagers were attacked by thugs who ransacked their baggage and clothing, taking all their valuables and leaving them destitute. Nevertheless, they didn't give up, and when they were able to reorganize and secure additional funds, they set out from a different port the following spring. This time William Brewster made a bargain with a trustworthy Dutch sea captain to take them to Holland.\n\nBradford then describes how the men boarded the ship at night, and how the women and children, who waited offshore in small boats, were chased by the authorities and attacked by gangs of cutthroats. Seeing the mayhem and fearing for his own safety, the Dutch captain set sail immediately, forcing the women and children to stay behind in England until the men could return weeks later to gather what was left of their families. Eventually, by the grace of God and their extraordinary perseverance, the separatists arrived in Holland in the year 1607 and were able to find lodgings in Amsterdam, where they remained for one year.\n\nThe following year they relocated from Amsterdam to the village of Leiden, where they remained for eleven more years in relative peace and quiet. They had to learn a new language, find new occupations, and fit in with their Dutch neighbors without losing their own culture and traditions. By 1617, however, they could see that their children were becoming more Dutch than English and picking up bad habits from the locals, so they began making preparations for the even more ambitious journey to the New World.\n\n36\n\nThrough the Brewster family's friendship with Sir Edwin Sandys, who was treasurer of the London Company, the congregation was granted patents to establish a new colony in \"the northern parts of Virginia.\" King James I, who succeeded Elizabeth to the throne in 1603, was nominally protestant in his beliefs but was brutal in his attacks upon the Puritans. He had authorized publication of the King James Version of the Bible in 1611 as an act of compromise with Puritan dissenters\u2014it was his most important contribution to the separatist movement and to religious liberty. But he had also declared the separatists to be outlaws and enemies of the Crown, and he said in one famous outburst, \"I will make them conform themselves, or else I will harry them out of the land, or else do worse.\"\n\nThe separatist congregation in Holland had prayed for years that they might be able to return to their homes in England in due time. But at age fifty-four and in his fifty-second year as king of Scotland and seventeenth year as king of England, James I showed no signs of relenting. It was common knowledge that the king was strongly influenced by his Catholic wife and many court favorites who despised the dissenters. This left only two options for the separatists: they could either stay where they were or risk the dangerous voyage to an untamed wilderness in North America.\n\nWhen it came time to leave Holland, half of the English separatists decided to stay behind. The others arranged for passage to Southampton, England, on the ship _Speedwell_ , where they joined another group of separatists who had chartered the 180-ton cargo vessel _Mayflower_. By the time they set sail on September 16, 1620, about half of the 102 passengers and crew aboard the _Mayflower_ were from the original Scrooby congregation.\n\n37\n\nIt was a long and difficult crossing\u2014sixty-five days at sea before they came within sight of land on November 19. Four members of the party died during the journey, but two babies were born\u2014the boy Oceanus Hopkins was born at sea, and another, Peregrine White, was born while the ship lay at anchor off Cape Cod. Strong winds and heavy seas prevented them from going ashore in Virginia as originally planned. The ship was forced back, then farther up the coast until they were finally able to anchor at the site of Provincetown, on the tip of Cape Cod.\n\nBefore going ashore, the men William Bradford identifies as the Pilgrim Fathers met in the captain's stateroom to draw up the legal covenant that became known as the Mayflower Compact, creating the first form of government in the new colony. The document reads in its entirety:\n\nIn the name of God, Amen. We whose names are underwritten, the loyal subjects of our dread sovereign Lord King James, by the grace of God, of Great Britain, France, and Ireland king, defender of the faith, etc.\n\nHaving undertaken, for the glory of God, and the advancement of the Christian faith, and honor of our king and country, a voyage to plant the first colony in the northern parts of Virginia, do by these presents solemnly and mutually in the presence of God, and one another, covenant and combine ourselves together into a civil body politick, for our better ordering, and preservation and furtherance of the ends aforesaid; and by virtue hereof, to enact, constitute, and frame such just and equal laws, ordinances, acts, constitutions, and offices, from time to time, as shall be thought most meet and convenient for the general good of the colony: unto which we promise all due submission and obedience.\n\nIn witness whereof we have hereunder subscribed our names at Cape Cod, the 11th of November in the year of the reign of our sovereign Lord King James of England, France, and Ireland, the eighteenth and of Scotland the fifty-fourth. Anno Domine 1620.\n\n38\n\nThe covenant was signed by forty-one members of the ship's company, including William Bradford, William Brewster, Myles Standish, and John Alden. It was originally drafted as a temporary measure to avoid dissension among the passengers and crew, some of whom were angry that the ship had landed at Provincetown instead of Virginia. The aim of the agreement was to establish the ground rules for what they called a \"Civil Body Politick\" until revised patents could be sent from England granting them official permission to settle at the new site.\n\nThe Mayflower Compact has since been called the birth certificate of America and America's first constitution. Our sixth president, John Quincy Adams, the son of John Adams and a descendant of John Alden, referred to the compact as the true foundation of the U.S. Constitution. William Bradford included a copy of it in his history. The document makes it perfectly clear that the Pilgrims had come to America \"for the glory of God, and the advancement of the Christian faith,\" and not out of greed or imperialism or any of the reasons often given by the revisionists.\n\nAfter all the hardships they had miraculously survived, the leaders of the expedition were anxious to find a safe landing place. They scouted the coastline for several days and ultimately decided on Plymouth Harbor on the western side of Cape Cod Bay. They made their historic landing at Plymouth Rock on December 21, and the settlers disembarked with all their remaining goods on December 26. It was only later that they discovered how the hand of Providence had protected them and delivered them safely to their present destination.\n\nBY THE GRACE OF GOD, AMEN\n\nIf the _Mayflower_ had actually landed in \"the northern parts of Virginia\" as originally planned, the passengers would have been exposed to even greater perils than those they survived in England. The coastal marshlands in the south were infested with plague and malaria, and the local Indian tribes were hostile to European settlers. Chances are they would not have survived. They did not realize it at the time, but God had sent strong winds to push the _Mayflower_ farther north until they arrived at the one place where they would find safe harbor.\n\n39\n\nGod's grace was upon them there as well. After they landed they learned that the Patuxet tribe that inhabited the area had been wiped out by disease. The first Native American the settlers met at Plymouth was the lone survivor of the Patuxets, Squanto. They were surprised to learn that Squanto had been captured by English traders and taken to England years earlier. He was exhibited as a curiosity for several years\u2014the first \"Red Indian\" the English had ever seen\u2014but he returned to his home just one year before the colonists arrived. While he was away learning the English language and customs, his tribe simply disappeared. Finding no trace of his own people, he went to the Wampanoag tribe and was adopted by its chief, Massasoit.\n\nThe Wampanoags were generally peaceful, and Squanto introduced the chief and tribal leaders to the settlers. These were the Indians with whom the Pilgrims celebrated their first Thanksgiving in America, in 1621. There were skirmishes at various times, and during the first tragic winter that William Bradford called \"the starving time,\" Massasoit would not allow any of his tribesmen to offer the Pilgrims food, shelter, or assistance. It was a tragic and potentially crippling time, but eventually Squanto was able to show the men how to fish and hunt for the local game, and how to plant corn and other native crops. He also helped them find safe places to build homes and begin their new lives.\n\nThe colonists knew they would have to face many hardships, but the prospects for success were few, indeed, in the beginning. William Bradford describes the sorrows of that first winter:\n\nBeing thus past the vast ocean, and a sea of troubles before in their preparation . . . they had now no friends to welcome them, nor inns to entertain or refresh their weather-beaten bodies, no houses or much less towns to repair to, to seek for succor. It is recorded in scripture as a mercy to the apostle and his shipwrecked company, that the barbarians showed no small kindness in refreshing them, but these savage barbarians, when they met with them (as after will appear) were readier to fill their sides full of arrows than otherwise. And for the season it was winter, and they that know the winters of that country know them to be sharp and violent and subject to cruel and fierce storms, dangerous to travel to known places, much more to search an unknown coast.\n\n40\n\n\"What could now sustain them,\" Bradford asked, \"but the Spirit of God and His grace?\" But would anyone remember the sacrifices the Pilgrims made to find a new home for religious liberty? Would the men and women of our own time tell their children where the story of America really began? Bradford then goes on to ask, \"May not and ought not the children of these fathers rightly say, 'Our fathers were Englishmen which came over this great ocean, and were ready to perish in this wilderness; but they cried unto the Lord, and He heard their voice and looked on their adversity'?\"\n\nThen, reciting from various passages from the Psalms, he writes:\n\nLet them therefore praise the Lord, because He is good: and his mercies endure forever. Yea, let them which have been redeemed of the Lord, show how He hath delivered them from the hand of the oppressor. When they wandered in the desert wilderness out of the way, and found no city to dwell in, both hungry and thirsty, their soul was overwhelmed in them. Let them confess before the Lord His loving-kindness and His wonderful works before the sons of men.\n\nDuring that first cruel winter twenty-six men and eighteen women died. Only three of the married women survived. According to Edwin Gaustad's account, \"It was a small and terribly depleted group that survived to give thanks for the first harvest in the fall of 1621.\" Nevertheless the survivors began to adapt to their surroundings and settle into a practical routine, and over time they found ways to overcome the limitations of the harsh environment. Little by little the colony began to grow and prosper, and the rumors of its progress were being cautiously weighed by the Puritan congregations back in England who were growing more and more desperate under unrelenting persecution.\n\n41\n\nKing Charles I, who succeeded James in 1625, made no pretense of compromise with the separatists. After marrying a Catholic Spanish princess, Charles was even more determined to stamp out Puritan influence on the Church of England than his predecessors had been. With the cooperation of Catholic-leaning bishop William Laud, who would be appointed archbishop of Canterbury in 1633, Charles made life hell for English Puritans.\n\nIn particular Laud exploited the Star Chamber system of the English courts with draconian efficiency. Prominent pastors and members of Parliament who favored the Puritan reforms to the Church of England were dragged before the court, where they were accused and humiliated. Trials before the Star Chamber were conducted in secrecy, like a kangaroo court. Prisoners were allowed to present no witnesses; they had no right of appeal and no opportunity to defend themselves since the outcomes were predetermined. In the hands of Laud and the privy counselors chosen by the royal court, the Star Chamber was the religious and political weapon par excellence. The practice was finally halted in 1641 as a result of the uprisings among Puritan supporters that would ultimately lead to the English Civil War in 1642.\n\nCharles battled with Parliament throughout his reign, but he probably sealed his own fate when he disbanded Parliament in 1629 for a period of eleven years\u2014a time known as the Eleven Years' Tyranny. Laud would remain his most trusted ally, always eager to please his monarch. But the more Laud tried to defy the Parliamentarian objections to rising taxation of the people and Catholic influence on the Church of England, the louder the protests of the Puritan separatists became. As journalist and historian Rod Gragg describes it:\n\n42\n\n[Bishop Laud] preferred unmarried ministers, like the Catholic priesthood, and the Puritans objected. He advocated Catholic-style prayers for the dead, and the Puritans called it unbiblical. He ordered church members to kneel for communion; and the Puritans said it was worshipping the elements rather than the Lord. He required bowing at the church altar, and the Puritans said it was misplaced devotion. He implemented a relaxed observance of the Sabbath, and the Puritans considered it irreverent. He called for more ceremony in worship services, and the Puritans saw it as prideful. He required ministers to wear the Catholic-style surplice or tunic, and the Puritans insisted it was man-centered adornment.\n\nUltimately Laud would not win the debate. Supporters of the republican leader Oliver Cromwell, who sided with the Parliamentarians and defended the Puritan reforms, accused Laud of treason and confined him to the prison in the Tower of London. When he was put on trial for his crimes in 1644, Laud was found guilty and sentenced to death. He was executed in January 1645, and Charles I would meet a similar fate four years later when the new Parliament, led by Cromwell, convicted the king of treason and he was beheaded. Two years later, following the Battle of Worcester, the English Civil War came to an end.\n\nA CITY UP ON A HILL\n\nThroughout the long years of controversy, the good reports coming from New England contrasted sharply with the troubles in Britain and convinced dozens of Puritan congregations to follow the example of the Scrooby separatists and relocate to the New World. A group of two hundred settlers arrived with grants at the new Massachusetts Bay Colony in 1630. They were followed by groups of seven hundred, eight hundred, then more than three thousand in a single year. The promise of religious liberty and the prospect of a healthy distance from the political oppression they had suffered at home were more than enough to persuade the faithful to abandon their native land for a new start in the American wilderness.\n\n43\n\nThese travelers did not come, as the _Mayflower_ Pilgrims had, in a leaky old cargo vessel. Instead, they journeyed in fleets\u2014seventeen ships in one flotilla delivered more than a thousand souls. Between 1630 and 1640, more than twenty thousand English settlers crossed the Atlantic Ocean to North America. So many made the journey that it became known in England as the Great Migration. Never before had so many Englishmen left their native land in such a short span of time for such an arduous and uncertain adventure.\n\nAfter 1630, most of the new arrivals settled in the Massachusetts Bay Colony, which was situated just a few miles north of the Plymouth Colony. While the original settlements remained reasonably small and close-knit, the Massachusetts Bay Colony grew rapidly, with a more diverse population. The foundations of the first city in New England were laid there in 1630, on the Sawmut Peninsula at the confluence of the Mystic and Charles Rivers. The town was named for the Port of Boston in Lincolnshire, England, from which many of the new immigrants had come.\n\nA decade after their arrival, the population of Plymouth Colony was still only a few hundred, scattered among seven small villages. By 1660, the same year the English monarchy was restored under Charles II, there were just over two thousand in that colony, at which point the Plymouth colonists voted to join the larger and more prosperous Massachusetts Bay Colony. As the number of Puritan congregations continued to grow, both in America and Britain, there was mounting pressure on the English Church to disavow the anti-Puritan purges, but it was not until the Act of Toleration in 1689, under William and Mary, that the persecutions finally came to an end.\n\nAll of the men and women who had risked so much to come to the New World had to give up much, leaving homes and families and in many cases bringing nothing with them except the clothes on their backs. But John Winthrop, who served several terms as governor of the Massachusetts Bay Colony, wanted the Puritan congregations of New England to remember that they had all come to America to fulfill a sacred promise. In the Arbella Covenant, which was drafted in 1630, Winthrop wrote, \"We must love one another with a pure heart, fervently,\" and \"we must bear one another's burdens. . . .\" Above all, he said, \"we are a Company professing ourselves fellow members of Christ.\" Regardless of their wealth or status in life, he said, they had all made a solemn pledge to live godly lives and abide by the teachings of Scripture. Winthrop said they must never forget their dependence upon the Lord who had brought them there. Then, reminding the people of what was really at stake, the Covenant states:\n\n44\n\nThus stands the cause between God and us: we are entered into covenant with Him for this work; we have taken out a commission, the Lord hath given us leave to draw our own articles, . . . if we shall neglect the observation of these articles . . . the Lord will surely break out in wrath against us. . . . Therefore, let us choose life, that we, and our seed may live; by obeying His voice and cleaving to Him, for He is our life and our prosperity.\n\nEchoing the words of Jesus in Matthew 5:14, Winthrop penned some of the most famous words of the colonial era: \"We must consider that we shall be as a city upon a hill. The eyes of all people are upon us.\" The settlers of Plymouth Colony were all separatists, and the world was truly watching them. Calvinist in their doctrinal stance and Congregationalist in their church polity, they had made a clean break with the Anglican communion. The men and women who built the Massachusetts Bay Colony, on the other hand, were devoutly Puritan but still considered themselves to be part of the Church of England. And none of them doubted the truth of Winthrop's words or the importance of the oath they had taken.\n\nJohn Cotton, who was one of the best-known pastors in the town of Boston in those years, outlined the Puritans' complaints against the Church of England. The national church, he said, was subject to rigid conformity according to the edicts of the bishops and the threats of English law. The Book of Common Prayer, he believed, violated the second commandment\u2014which says, \"Thou shalt not make unto thee any graven image . . .\" (Exodus 20:4). The Anglican prayer book was a work of human invention, not of God, and ought to be abandoned. Cotton further held that the governance of the churches should be congregational in form, not Episcopal or Anglican because the congregants were the authentic source of authority in the churches, not the bishops or even the local pastors. The church, he said, is not a building but a gathering of believers. It is a fellowship of the redeemed\u2014bound together in a covenant relationship\u2014not subject to the arbitrary policies and judgments of any government or legislative body.\n\n45\n\nSome historians of the colonial period have suggested that New England church polity tended to be Presbyterian in form, with important decisions left in the hands of elected elders (presbyters) and deacons who spoke for the congregation. The New England churches insisted that the congregations should be free to select their own ministers and make their own rules of church governance. Whereas farther south in Virginia, colonies that grew out of the Jamestown Plantation founded in 1607 tended to favor the Episcopalian form of worship and were somewhat skeptical of their Puritan brethren in the north.\n\nFirst at Massachusetts Bay, then in the Connecticut Colony and each of the subsequent towns and villages settled by the Puritans, there emerged a spirit of unity that became known as the New England Way, \"molding itself with such firmness and care,\" as one writer describes it, \"that it stamped upon that region a way of thinking and living that would endure far beyond the colonial period.\"\n\nA DECLARATION OF INDEPENDENCE\n\nThe Puritans had come to America for religious liberty and the opportunity to practice their beliefs as they saw fit. That is simply a fact that cannot be explained any other way. In our own time, many historians have often focused on the lack of tolerance and diversity in the Puritan settlements, which is a wholly unjustified approach. In the first place, it is unreasonable to judge the men of that time by twenty-first century standards: it was a very different world, and there were perfectly good reasons for maintaining uniformity. And second, what's missing in the revisionists' account is the fact that the Pilgrims never intended to establish a colony that was open to any and all persuasions. \"The Puritans,\" reliable historians tell us, \"came to create a pure church and to conduct a holy experiment free of opposition, distraction, and error. They were not hypocrites who demanded freedom of religion then denied that same freedom to others. Freedom of religion across the board was never the plan, never the commission or errand.\"\n\n46\n\nInstead, the Pilgrims came to show that it was possible to establish a society and a church so faithful to the biblical model of purity that the churches they had left behind in England might be transformed by their example. If the Pilgrims saw themselves as \"a city upon a hill,\" it was not because they lacked admirers but because they wanted to be examples of the life Christ and the apostles had ordained for the church. They meant to be \"a light to the world,\" and this meant the community was expected to live faithfully, according to the precepts and biblical beliefs that bound them together.\n\nThroughout the nation's formative years, the clearest statement of the values and beliefs of the American people was to be found in their sermons, and this would be the case well into the nineteenth century. Pastors and teachers in separatist churches gave voice to the hopes, fears, and aspirations of the people, and this was only possible because the colonists were in general agreement on such matters. For more than a century their understanding of the duty of citizens and believers in Jesus Christ was virtually unanimous. This uniformity of conviction made the meeting house the preeminent place for preserving civic and spiritual discipline, and the unity of purpose was communicated best by the sermons.\n\n47\n\nThe spoken word has remained an essential component of American culture. As historian and former librarian of Congress Daniel J. Boorstin points out, \"The public speech, whether sermon, commencement address, or whistle-stop campaign talk, is a public affirmation that the listeners share a common discourse and a common body of values. The spoken word is inevitably more topical than the printed word: it attempts to explain the connection between the shared community values and the predicament of man at a particular time and place. It is directed to people whom the speaker confronts, and to their current problems.\"\n\nThe seventeenth century, Boorstin says, became known as the great age of the English sermon. Back in Great Britain, Anglican pastors such as John Donne and Jeremy Taylor were the best-known preachers of the day. They delivered homilies in what was known as the \"metaphysical\" or \"high\" style, drawing upon sophisticated allusions and philosophical turns of phrase. In the Plymouth Colony, on the other hand, the most compelling pastors were those like John Robinson, the original pastor of the Scrooby congregation, who preached in what the Puritans described as the \"plain\" style. Although Robinson did not make the journey to New England with his flock, his teachings had a profound influence on the character of the early church in America.\n\nThe characteristics of the plain style were laid out in a slim handbook called _The Art of Prophesying_ , written by William Perkins, a Cambridge University lecturer and pastor in the late-sixteenth and early-seventeenth centuries. The book, which was published in Latin in 1592 and in English in 1606, was on nearly every list of essential books, and few pastoral studies in New England would have been without a copy. The basic style of the Puritan sermon, Perkins says, was that it ought to be straightforward, with natural and logical development of ideas. To be a \"plain sermon,\" it didn't have to be simplistic or superficial but direct and free of the sort of elevated rhetoric favored by the Anglican divines.\n\nThe New England sermon was evangelical, to be sure, in the sense that it was meant to persuade the listeners and convince them of their need of saving grace. But it was above all practical, logical, and delivered in the common language of the day. Many of these sermons were more like a lawyer's brief than a work of art. Perkins proposed that each sermon should be composed of three main parts: \"doctrines, reasons, and uses.\" The doctrine came directly from the biblical text, which was where every Puritan sermon began. The reasons offered a brief discussion of the text, along with the meaning of the passage and its importance. The uses, then, were the practical application of the teaching to the lives of the members of the congregation.\n\n48\n\nIn New England, Boorstin says, the sermon was far more than a literary form: it was an institution. \"It was the ritual application of theology to community-building and to the tasks and trials of everyday life. It was not, as it was inevitably in England, a mere sectarian utterance of a part of the community. It was actually the orthodox manifesto and self-criticism of the community as a whole, a kind of reiterated _declaration_ _of independence_ , a continual rediscovery of purposes. The pulpit, and not the altar, held the place of honor in the New England meetinghouse. So too the sermon itself, the specific application of the Word of God, was the focus of the best minds of New England.\"\n\nRULES OF FAITH AND PRACTICE\n\nSince the church played such a key role in the daily life of Puritan communities, the people would have expected there to be sermons presented at every important occasion. There were usually two of them on Sundays, along with a Weekday Lecture on Thursdays, sometimes called the Fifth-Day Lecture. Thursday Lectures were begun by John Cotton in 1633, and were continued weekly for two hundred years. These were combined with Market Day Lectures, where congregants came together to discuss social and political events of the day. This was the custom in Boston, and the other New England colonies followed their example. In the beginning, attendance at these services was required by law, and skipping them was punishable by a fine. This practice eventually faded away because of the sheer size of the community and the difficulty of enforcement. Still, faithful church attendance was a fact of life.\n\n49\n\nIn addition to the routine church programs, there were sermons for all sorts of public occasions, including holidays, barn raisings, and harvest celebrations. There were also Election Day Sermons, in which the pastors reminded the citizens of their civic duties and the responsibility to choose wisely. Such sermons would have stressed the importance of selecting orderly and self-disciplined leaders, according to the admonitions found in passages such as Deuteronomy 1:15, 1 Peter 5:5, and 1 Timothy 3:2, which emphasized wisdom, accountability, and a godly concern for the well-being of the community. The Election Day Sermon was a New England institution throughout the American Revolution.\n\nAnother standard was the Artillery Sermon, given whenever there was need to raise a militia for protection from attacks by local tribes or to issue a call for the election of military officers\u2014a custom that began in the year 1659. There were also, of course, sermons for days of fasting and prayer, of which there were many during the Pilgrims' first half century in America. Daniel Boorstin indicates there were at least ten Thanksgiving celebrations in the Massachusetts Bay Colony in 1639, and as many as fifty between 1675 and 1676. Some of them lasted anywhere from three days to a week.\n\nAttendance at these events was often compulsory, but the sermons were always greeted with enthusiasm. There were few books in the colonies in the early years, so the sermons given on special occasions offered the men and women of these communities a sense of participation in the events of the day. The sermon was, in that sense, the essential community forum as well as a constant reminder of their dependence upon a benevolent heavenly Father. Consequently many congregants would bring along notebooks in which to record whatever they heard. These notebooks would, in turn, become invaluable resources for later generations to recall the beliefs and customs of the colonial era.\n\n50\n\nDespite the vast distances separating them from the land of their birth, the citizens of New England were remarkably well informed. They were blessed with a well-developed attention span, and they were in no need of trivial amusements or seeker-sensitive entertainments to hold their interest. They were hungry for knowledge and, for the most part, they were perceptive in their grasp of ideas. \"All these circumstances,\" writes Boorstin, \"served to hold the early New England preacher to a high intellectual standard and encouraged him to make his performances merit their central place. The New England sermon, then, was the communal ceremony which brought a strong orthodoxy to bear on the minutiae of life.\"\n\nThe men who were entrusted with preserving orthodoxy were uniformly men of the cloth. Richard Mather (1596\u20131669) and John Cotton (1584\/85\u20131652) were the most gifted preachers in the Massachusetts Bay Colony. In matters of both church and state, they tended to be authoritarian, which helped preserve stability in the colony but led to divisions among the clergy. Thomas Hooker (1586\u20131647), who agreed with Cotton's theology in the broadest sense, nevertheless, disagreed with him on governance and elected to move his parish farther west, to the town of Hartford, where he and one hundred followers founded the Colony of Connecticut in 1635.\n\nThere were also Christian leaders who came to be classified as nonconformists. Among them, Roger Williams (1603\u20131683) rejected the authoritarian style that was common in the Plymouth Colony. After several disagreements, and after having been tried and convicted of nonconformism, he was exiled. Then, with the help of Indian tribes he had befriended, he established Providence Colony farther west on a small strip of land sandwiched between Plymouth and Connecticut known as Rhode Island.\n\nWilliams, who was leader of the Baptists in the colonies, was never comfortable with traditional Puritan customs, and his new colony attracted many nonconformists and others of a more democratic spirit. He maintained that his purpose was to preserve freedom of conscience, and to allow his followers to pursue godliness in their own ways.\n\n51\n\nTHE NEW ENGLAND COLONIES\n\n_Circa 1641_\n\nAs previously mentioned, most of the pastors and teachers in the founding generation were Calvinist in their interpretations of Scripture. Not all were in complete accord with the five points of Calvinist theology, but the doctrine of salvation by grace through faith alone was universal. The five points of Calvinism, then and now, focus on (1) the total depravity of man, (2) unconditional election, (3) limited atonement, (4) irresistible grace, and (5) the perseverance of the saints. Each of these would require a more detailed analysis, which I won't attempt to offer here, but more importantly, all the Puritan churches would have stressed the importance of a personal salvation experience.\n\n52\n\nBy the same token, colonial ministers held the view that Christians were expected, as Jesus instructed the church, to \"render to Caesar the things that are Caesar's, and to God the things that are God's\" (Mark 12:17). This meant the laws ought to be obeyed, and reasonable taxes and levies ought to be paid. But Christians also had a duty to hold their leaders accountable for their decisions, and property-owning \"freemen\" were obliged to participate in the life and governance of the commonwealth.\n\nWhile they recognized that government was not the church, and that the governing authority must never interfere in the affairs of the churches, there was no question that they expected government leaders to be believers in Jesus Christ. While they did not set out to create a theocracy per se, they did believe church and state were mutually supportive institutions, due to their mutual commitment to order and civic discipline. In a healthy society, both church and state contribute to the common good.\n\nIn order to facilitate the administration of justice and public order in the colonies, the Puritans had been given the right to establish their own self-governing society with royal approval. Rather than being forced to answer to Parliament or to any particular instructions from the trading companies in England, the Puritan leaders were able to make their own rules and manage their own business with minimal interference from abroad. As the colonies grew in size and became more and more productive, however, this convenient relationship would change, as we shall see in subsequent chapters.\n\nIn the beginning the New England colonies were self-governing and faith-based. And like the settlers at Plymouth, the citizens of the Massachusetts Bay Colony based their government on the biblical model of a covenant relationship. Accordingly, free, nonindentured adult males, known as \"freemen,\" were obliged by the terms of the covenant to submit to the authority of the General Court, which was the main governing body. Each local government followed the same basic model. The Salem Church Covenant of 1629 spelled out the terms of the agreement:\n\n53\n\nWe Covenant with the Lord and with one another and do bind ourselves in the presence of God, to walk together in all his ways, accordingly as he is pleased to reveal himself to us in his blessed word of truth. And do more explicitly, in the name and fear of God, profess and protest to walk as followeth through the power and goodness of the Lord Jesus Christ.\n\nIt was a \"Covenant with the Lord and with one another,\" based on the \"blessed word of truth.\" The civil government was a representative democracy based on biblical models, but with leaders who exercised a great deal of discretion in how the laws were applied. The book of Acts describes how the early church selected its leaders, and the Puritans attempted to follow that model in both church and civil government. This gave the first Americans biblical grounds for resisting civil magistrates and authorities in England who attempted to undermine the civil order and moral balance of the community. The apostles had resisted the civil authorities by declaring, \"We ought to obey God rather than men\" (Acts 5:29), and the colonists declared, \"No king but King Jesus.\" In time the American spirit of independence would lead to an escalation of the hostilities between the colonists and their rulers in London.\n\nTHE AMERICAN BIRTHRIGHT\n\nThe Massachusetts Body of Liberties (1641) was the first formal register of the legal statutes of New England, and it would eventually be a model for the patriots of 1776 as they sought to codify the distinctive elements of American law and government. The Body of Liberties spelled out America's birthright and included a \"bill of rights\" declaring the right (in John Locke's original phrasing) to life, liberty, and the pursuit of property. It guaranteed equality under the law, free and regular elections, protection from self-incrimination, and the right to trial by jury. It also placed restrictions on \"taxation without representation.\"\n\n54\n\nThe document began with a paraphrase of the English Magna Carta (1215) that restrained the authority of the monarch and asserted that even kings are subject to the authority of Almighty God. It then placed certain limitations on judicial proceedings and outlined the rights of freemen, women, children, foreigners, and even \"the brute creature.\" The document described the organization of the churches as the \"Liberties the Lord Jesus hath given to the Churches.\"\n\nAlthough public order was rigidly enforced, no one should assume the colonists were passive when it came to matters of controversy. Many disputes in early New England arose out of debates over what sort of men were best suited to serve as magistrates and governors, and how leaders ought to be selected. The political history of the Massachusetts Bay Colony, Daniel Boorstin suggests, could be written as a history of their disagreements over such questions. What was the proper relationship between magistrates and deputies? How many deputies were needed for each town? What restrictions could be imposed on judges and juries, and what were the justifiable grounds for corporal punishment? Many a sermon was devoted to such topics.\n\nBy 1644, the colony had established a bicameral legislature with a governor, deputy governor, and other officials elected annually. John Cotton offered a somber warning concerning the limits of political power, however, saying, \"Let all the world learn to give mortal men no greater power than they are content they shall use\u2014for use it they will. And unless they be better taught of God, they will use it ever and anon. . . . And they that have liberty to speak great things, you will find it to be true, they will speak great blasphemies.\" Because they understood the doctrine of original sin, the colonists had a profound respect for law and order; but they also understood the dangers of unlimited power.\n\n55\n\nAs the population of the colonies continued to grow in the late seventeenth and early eighteenth centuries\u2014and particularly as the generation of young people born on American soil came to maturity and moved into positions of leadership\u2014the Puritanism of the founding generation began to lose some of its influence on everyday life. The powerful preaching of ministers such as Jonathan Edwards (1703\u2013 1758) and Samuel Hopkins (1721\u20131803) helped to preserve the main tenets of Puritan beliefs in the eighteenth century, but rapid growth also brought rapid change.\n\nIn the beginning Puritanism had a beneficial effect on the moral and intellectual life of the people. Although after about 1820, the religious beliefs of the founding generation were no longer as dominant as they had once been, Puritan ideals were still very much alive and could be seen in the community's emphasis on education, in the people's concern for the religious views of candidates for public office, and in the benefits derived from the \"Protestant work ethic,\" which was without question a gift of the founding generation. But the Puritan spirit was particularly alive in the passion for liberty that was, and still is, the defining characteristic of the American people.\n\nMany young people learned a trade through informal apprenticeships. The custom for centuries had been that each family would train its children to pursue the same professions their fathers had pursued. But during the eighteenth century, this custom began to change as well. Reading, writing, and ciphers, which were the principal focus of education in the colonies, were also the responsibility of the family, but the churches played an important role in educating the children, ensuring that each child was able to read the Bible for himself or herself.\n\nWhen public schools were established, approximately forty years after the foundation of the colony, the General Court of Massachusetts Bay authorized public funding of schools. Plymouth Colony was the first to begin operating a publicly funded school in 1673. Ten years later there were five such schools in the colonies.\n\n56\n\nAll the colonies in New England experienced dramatic growth in both population and wealth during the eighteenth century. The rough-cut colonial society that had fought and scratched to eke out a living in the midst of a vast and unforgiving wilderness was, at long last, acquiring an image of sophistication and polish. The town of Boston, which was New England's most important seaport, was rapidly becoming a cosmopolitan center. The census of 1765, which was the last one taken before the American Revolution, listed 1,676 houses and 2,069 families. This included 2,941 adult males and 3,612 adult females, along with 4,109 males and 4,010 females under the age of 16. The census also listed the number of blacks and mulattoes in Boston at 510 males and 301 females. There were 37 Indians, male and female, and 32 residents of French origin, which brought the total in Boston to 15,552 souls. This is minuscule by today's standards, but there were approximately 250,000 settlers in New England by the year 1700. Before the first shots were fired at Lexington and Concord in 1775, the number had increased to 2,250,000. By 1800 the total was more than 5,300,000.\n\nAs the nation expanded, large numbers of non-English-speaking immigrants began streaming into the colonies as well. By 1775, there were more than 250,000 Germans, most of whom settled in the middle colonies and farther south. These newcomers were generally Lutheran or German Reformed although, like the Puritans, they were also Calvinist in their doctrinal views. And there was a substantial influx of Moravian, Mennonite, Amish, and other pietist denominations who were doctrinally Arminian.\n\nBeginning in the 1730s, there was a continuous wave of immigrants from Scotland and Ireland, including entire communities of Scotch-Irish Presbyterians escaping from the political and religious persecutions in their home countries. By 1776, the number had grown to more than 250,000, and many of these individuals were native Gaelic speakers with little or no fluency in English. Many, in fact, could not read or write. This was especially the case with the eighteenth-century immigrants who came to this country as indentured servants, meaning they were obliged to work for a specified period of time (usually seven years) to repay the cost of their passage to America.\n\n57\n\nThere was also a smaller percentage of criminals transported to America from the jails of England, most of whom were imprisoned for debts or certain other offenses. The Georgia Colony, which was chartered in 1732 by the English reformer James Oglethorpe, was the principal destination for debtors and other persons exiled by the English courts. The Colony of Carolina was chartered in 1663 but remained largely unsettled until about 1670. Then, in the third decade of the eighteenth century, Carolina was divided into two separate colonies, South Carolina in 1721 and North Carolina in 1729.\n\nIn 1686, the colonies were reorganized for a brief interval as the Dominion of New England. It included the colonies of Plymouth, Massachusetts Bay, Connecticut, Rhode Island, and New Hampshire. New York, West Jersey, and East Jersey, which had been part of the Dutch colony of New Netherland, became part of the Dominion in 1688; however, the merger didn't last long. After two years, Plymouth Colony withdrew from the Dominion, and the entire unification effort collapsed in the wake of the Glorious Revolution of 1688, which substantially altered colonial relations with England.\n\nThe Plymouth Colony's return to self-rule was short-lived, however. A delegation led by Increase Mather, who was the son of Richard Mather and father of Cotton Mather, traveled to London to negotiate renewal of the colonial patents that had preceded the Dominion agreement. At that point a new charter was issued, making the Plymouth Colony a permanent part of the Massachusetts Bay Colony. The last official meeting of the Plymouth General Court took place in the summer of 1692.\n\n58\n\nAN ENDURING LEGACY\n\nLooking back across this amazing panorama, it is remarkable that in the span of just seventy-two years, from the landing at Plymouth Rock to the emergence of Massachusetts as the most dominant social and political entity in the New World, a nation was born, built on endurance, an irresistible passion for liberty, and the covenants of the Christian faith. To claim that this country was a secular nation, that the Founders were ambivalent about religion, or that the Christian faith was not an essential part of their daily lives, one would either have to be ignorant of the facts or intentionally deceitful. Unfortunately, as I argued in the previous chapter, that actually may be the case for many Americans today.\n\nThere is no question that the spiritual heritage passed down to us from the colonial period has had a long-lasting impact on the nation, not simply because a band of religious extremists successfully crossed an ocean and erected the framework of a prosperous society in North America, but because the character, resilience, and moral fiber they exhibited has remained the hallmark of the American experience ever since.\n\nIn the year 1820, the great nineteenth-century statesman Daniel Webster expressed the hope that the American people might never forget \"the religious character of our origin.\" In remarks celebrating the two-hundredth anniversary of the landing at Plymouth Rock, Webster said, \"Our fathers were brought hither by their high veneration for the Christian religion. They journeyed by its light and labored in its hope. They sought to incorporate its principles with the elements of their society and to diffuse its influence through all their institutions, civil, political, or literary.\"\n\nNo one hearing those words in 1820 would have been the least surprised by what Webster was saying. No one would have disagreed. It is only now, after a hundred and fifty years of material comfort and academic reprogramming, that the average American is predisposed to believe that the faith of our fathers had no lasting importance in the nation's founding. But nothing could be further from the truth.\n\n59\n\nDuring an address in New Hampshire honoring the _Mayflower_ Pilgrims, a distinguished former professor of history at Harvard University offered a tribute to the men and women who risked so much to lay the foundations of the republic. The year was 1936\u2014316 years after the signing of the Mayflower Compact\u2014when Professor Samuel Eliot Morison spoke these words. Why should anyone still care about the Pilgrims today?\n\nHere is a story of simple people impelled by an ardent faith in God to a dauntless courage in danger, a boundless resourcefulness in face of difficulty, an impregnable fortitude in adversity. It strengthens and inspires us still, after more than three centuries, in this age of change and uncertainty. . . . The story of their patience and fortitude, and the workings of that unseen force which bears up heroic souls in the doing of mighty errands, as often as it is read or told, quickens the spiritual forces in American life, strengthens faith in God, and confidence in human nature. Thus the Pilgrims in a sense have become the spiritual ancestors of all Americans, whatever their stock, race, or creed.\n\nIf only we could somehow persuade all Americans to grasp the truth and importance of those words. But for one of the leading icons of the secular Left, Professor Noam Chomsky of MIT, the story of the American founding is a dark saga\u2014an example of over-the-top religious extremism. In an interview for _The Humanist_ , he said, \"There's that something about the history of the United States, going back to the Pilgrims, that I would call religious extremism. It's a major theme in this society and culture.\" The villain for this humanist isn't the persecution that brought the Puritans to this country but their religious fundamentalism, along with a penchant for exploiting the natives.\n\nI would propose that it is Chomsky's views that are extreme, but he is not alone. There are other Americans, including some very much in the public eye, who are only too glad to assure the world that America is not a Christian nation. During a press conference in Turkey on April 6, 2009, President Barack Obama said, \"One of the great strengths of the United States is . . . we do not consider ourselves a Christian nation. . . . We consider ourselves a nation of citizens who are bound by ideals and a set of values.\" The only problem with the statement is, it's not true. Upward of 80 percent of Americans today say they are Christians\u2014the highest percentage of any industrialized nation\u2014and it has been that way for the last three hundred years.\n\n60\n\nA 2006 ABC News poll found that 83 percent of Americans self-identify as Christians while 2.2 percent say they are Jewish, and just under 1 percent are Muslim. In fact, the total number of Americans of \"other faiths\" is less than 4 percent. A 2006 Gallup survey reported that 42 percent of Americans say they attend church or synagogue every week or almost every week. The numbers vary from state to state, from 60 percent in the South to less than 25 percent in Vermont.\n\nThe president's words may cheer secularists, but the words of the Founders\u2014along with the history we have seen in these pages\u2014make it clear that the Christian faith was the single most dynamic force in the discovery, population, and unification of America. Over and over again throughout our history, this fact has been reinforced in public declarations.\n\nJames Madison, the father of the Constitution, said, \"We have staked the whole future of American civilization, not upon the power of government, far from it . . . but upon the capacity of mankind for self government; upon the capacity of each and all of us to govern ourselves, to control ourselves, to sustain ourselves according to The Ten Commandments of God.\" The revolutionary-era patriot Patrick Henry agreed, saying, \"It cannot be emphasized too strongly or too often that this great nation was founded, not by religionists, but by Christians; not on religions, but on the Gospel of Jesus Christ.\"\n\nA century later in 1846, the Supreme Court of South Carolina rendered the judgment that \"Christianity is a part of the common law of the land, with liberty of conscience to all. It has always been so recognized . . . it is the foundation of those morals and manners upon which our society is formed; it is their basis. Remove this and they would fall.\" Forty-six years later, after an exhausting ten-year review of all the founding documents of the republic, the United States Supreme Court said in its 1892 Trinity decision, \"There is a universal language pervading them all, having one meaning: they affirm and reaffirm that this is a religious nation. These are not individual sayings, declarations of private persons: they are organic utterances; they speak the voice of the entire people . . . that this is a Christian nation.\" And later still, in _Zorach v. Clauson_ (1952), liberal justice William O. Douglas announced the majority opinion of the United States Supreme Court, declaring, \"We are a religious people whose institutions presuppose a Supreme Being. When the state encourages religious instruction or cooperates with religious authorities by adjusting the schedule of public events to sectarian needs, it follows the best of our traditions. For it then respects the religious nature of our people and accommodates public service to their spiritual needs.\"\n\n61\n\nNo one would argue that America in the twenty-first century is a paragon of virtue or that we still exhibit the same strength of character, resilience, and moral fiber of the founding generations. But no one can deny that the nation rose to greatness in large part because of the beliefs and behaviors of the men and women who gave birth to this nation; and those beliefs were, above all, a gift of their profound Christian faith.\n\n62\n\nOur challenge today is to live up to those high standards, and I believe that can still happen; but when it does, it will not be some jumped-up man-made affair. But I am getting ahead of myself. Before returning to this subject, let us turn the page and step back briefly to consider the achievements of the first generation of English colonists who paid a tremendous price to establish the Colony of Virginia. Theirs is a very different story, but it is an essential link in the history of American independence.\n63\n\n[**_Three_**  \nWHAT THE FOUNDERS BELIEVED](Lee_9780849949463_epub_c5_r1.html#d6e5)\n\nBy the mid-sixteenth century, it was becoming commonplace. English seafarers made landfall in North America at least a half-dozen times, accompanied by groups of wealthy adventurers eager for riches and personal acclaim. It is astonishing how many of these gentlemen were quick to publish their journals and sketchbooks. A few even managed to bring back Native Americans they had captured in the wilds of Virginia for exhibit in London. But none of the early expeditions had been resourceful enough to establish even one successful colony in the New World, and we can't help but wonder why. Why were the earliest expeditions so different from the landing at Plymouth just a few years later?\n\nUnlike the English colonies in New England that had begun with strong Puritan roots and noble aspirations, the first colonies in Virginia were commissioned with more conventional objectives. The earliest groups of explorers from Britain were sent abroad with instructions to search for gold and silver, and if possible to locate a westward passage to the Pacific Ocean. They succeeded in none of those objectives, but rather than spending time clearing land, building permanent dwellings, or planting the crops necessary for their own survival, the would-be settlers quarreled among themselves constantly.\n\n64\n\nThe most notorious attempt to exploit the uncharted wilderness of North America was the English settlement at Roanoke Island, just off the coast of North Carolina. Walter Raleigh, who was knighted by Queen Elizabeth I in 1584, was given a charter that year to organize an expedition to colonize North America. With royal patents in hand, Raleigh christened the new territory Virginia, in honor of the Virgin Queen. An Oxford-educated gentleman and a court favorite, Raleigh wisely chose not to make the journey himself, but by the spring of 1585, he managed to outfit seven ships and a party of 108 able-bodied men to undertake the adventure.\n\nThe expedition would be led by Raleigh's cousin, Sir Richard Grenville, who was a military officer and scientist, along with one experienced explorer, Ralph Lane. They set out from the port of Plymouth, England, for the coast of North America, where they intended to establish a new plantation. The Atlantic crossing was unusually rough, and the two gentlemen Raleigh had put in charge were constantly at odds with each other. When they finally reached their destination after ten weeks at sea, Ralph Lane assembled work parties and began construction of a makeshift outpost on the island. Then, just days later, Richard Grenville and most of the ship's crew said their brief farewells and set sail for England, leaving the remaining colonists to fend for themselves.\n\nRalph Lane acted as governor on the island. He organized scouting parties to go into the wilderness, searching the entire region in a 130-mile radius. The laborers and carpenters in the group made good use of the abundant timber in the area and built a small fort to protect themselves during the frequent surprise visits from the local Indian tribes. However, by the time the English privateer Sir Francis Drake stopped briefly at the site in June of the following year, Lane and the others were exhausted, hungry, and profoundly dispirited. Lane was only too glad to give up on the experiment and return with Drake to London.\n\n65\n\nWalter Raleigh, however, refused to admit his plan had failed, and one year later he dispatched three more ships to continue the Roanoke experiment. The newly mounted expedition set sail with 150 settlers on May 8, 1587, under the command of a new leader, Captain John White. The party arrived at Roanoke three months later, in mid-August, at which time the settlers participated in the first recorded Protestant service in the colonies.\n\nLater that month John White's daughter, Elenor, who was married to Ananias Dare, gave birth to the first English baby born on American soil, Virginia Dare. Finding little of the previous settlement still intact, and painfully aware that the provisions they had brought with them would be insufficient for their long-term needs, John White decided to return to England to gather additional food, tools, and supplies. Unfortunately, the English navy was fully occupied when White returned, waging war against the Spanish, and it would be four long years before he was able to arrange passage back to America.\n\nWhen he finally landed at the site of the Roanoke settlement in 1590, there was nothing left. His family was gone, the fort he helped build was broken and deserted, and there was no sign of what may have happened to the colonists. His daughter and grandchild were nowhere to be found. Today, more than four centuries later, the \"Lost Colony of Roanoke\" remains one of the most perplexing unsolved mysteries of the founding era.\n\nDespite the misfortunes of the Roanoke colonists, the men who helped launch the age of English exploration in the New World\u2014 including the two Anglican clergymen Richard Hakluyt and Samuel Purchas who were such prolific writers and travel enthusiasts\u2014 continued to lobby king and Parliament for additional forays into North America where, they said, vast wealth and unrivaled imperial glory were ripe for the taking.\n\n66\n\nDAZED BUT UNDAUNTED\n\nNews of the Roanoke debacle aroused doubts and disappointment in London and could have ended the efforts to colonize North America except for the relentless ambition of the merchants and promoters. Although the prospect of great treasure now seemed much less likely, the colonists had discovered an important new crop in the New World, tobacco, that would eventually provide sufficient motivation and revenue to persuade the financial backers to continue pouring money into the expedition.\n\nIn 1606, King James I, who succeeded Elizabeth as monarch in 1603, granted two trading companies exclusive rights to establish colonies in America between the thirty-fourth and forty-first degrees north latitudes. On December 20, 1606, the London Company sent 143 settlers on three ships\u2014 _Godspeed_ , _Discovery_ , and _Susan Constant_ \u2014 under the command of Captain Christopher Newport. Most of the passengers, according to at least one eighteenth-century report, were poorly suited for survival in an untamed wilderness. The ships landed first at Cape Henry near the mouth of Chesapeake Bay on April 26, 1607; then they moved farther inland. Two weeks later on May 14, they named the new settlement Jamestown, in honor of the king.\n\nThese \"reprobates of good families,\" as the Virginia diarist William Byrd would refer to them later, apparently made the voyage expecting the sort of lush vistas, abundant crops, and fat cattle that promoters like Richard Hakluyt had promised. What they actually discovered was a mosquito-infested swamp in the middle of a jungle, with the constant threat of attack by hostile \"savages.\" There were no conveniences of any sort. What was needed at Jamestown was backbreaking labor. For the most part, however, these men and women were utterly unsuited to work of any kind. Many simply refused to work. Instead, they sat around fanning themselves, expecting to be fed and cared for by their social inferiors.\n\nIn time the newcomers managed to construct a stockade, a palisade of tall timbers laid out in the shape of a large triangle. It included a central storehouse, a church, and several small buildings where the settlers could take shelter. In the first seven months the settlement was nearly wiped out by disease, famine, and unceasing attacks. The total number of settlers, which was just over a hundred souls in May 1607, was less than half that number the following September. Indians had attacked before the fortifications were in place, and no sooner had the living quarters been completed than fire broke out, destroying several houses, the church, and chaplain Robert Hunt's entire library.\n\n67\n\nUnlike Walter Raleigh and the men he had previously given leadership roles, the leader of the Jamestown expedition, Captain John Smith, was not an English gentleman. He was a practical man of modest means who was not averse to hard work and was, fortunately for the settlers, a compassionate, determined, and capable leader. It was his energies during the worst of times that kept the colony going. And Smith's most able assistant during all this time, the young Anglican chaplain Robert Hunt, had left a young wife and two small children back in England, feeling that God had called him to accompany the Jamestown colonists as their minister.\n\nThe colonists, on the other hand, could be loud and offensive, \"little better than Atheists,\" one settler quipped. But they showed dutiful respect for their young preacher. On April 29, 1607, on the morning of their third day in the New World, the colonists planted a cross made of solid oak at water's edge\u2014they had brought the cross with them from England for that purpose. Hunt prayed first, then the whole company joined him in offering prayers of thanksgiving to God for their safe arrival.\n\nIn his official capacity Hunt served as minister, vicar, and chaplain of the settlement, but he was always quick to lend a hand in every important task. No job, regardless how menial or exhausting, was too much for him. He had dedicated the new colony in the name of God on the day of their arrival, but he was glad to take up tools and share in their daily chores. \"We are all laborers in a common vineyard,\" he told them. He led public prayers every morning and evening, preached two sermons on Sundays, administered communion, and ministered to the souls of his flock.\n\n68\n\nSurviving in this environment proved to be much more demanding than any of the settlers had anticipated. Yet the Reverend Robert Hunt was never heard to complain, even when his precious books and personal effects went up in flames. He nursed the sick and prayed for the dying. From beginning to end, Hunt was the most even-tempered and considerate member of the group, and when he came down with a fever and died just one year after the arrival in Jamestown, many of the colonists were heartbroken.\n\nThe preacher's death left a vacuum, but his Christ-like example of self-sacrifice and devotion was a blessing and served as a poignant reminder of the original charter for the Virginia Colony that cited \"propagating of Christian Religion\" as one of the expedition's primary purposes. An inscription on the bronze plaque honoring Robert Hunt at the Jamestown historical site today bears the words of the settlers who knew him: \"We all received from him the Holy Communion together, as a pledge of reconciliation, for we all loved him for his exceeding goodness. He planted the First Protestant Church in America and laid down his life in the foundation of America.\"\n\nLife in the colony was a day-by-day affair, and life was never easy. Food they had stored in crates and sealed lockers rotted before it could be eaten, and precious supplies simply disappeared. They were constantly plagued by rats and vermin of all kinds; and famine, sickness, and malarial fevers were a never-ending threat. All these things took a tremendous toll on the population, and it began to feel as if they were cursed. It wasn't until the Powhatan Indians came in peace, offering to trade food and other common necessities for glass beads and implements made of iron and copper, that the settlers were able to rest at all. But this act of benevolence was no guarantee of safety.\n\nThe attacks may have slowed temporarily, but they would continue indefinitely. The settlers were ravaged by diseases, but at times they were their own worst enemies. The uncertainties of their precarious existence were taking a toll on their souls as well as their bodies, and it showed. Less than two years after they had built it, the church the men erected to replace the temporary structure John Smith had rigged up with old canvas sails and cedar poles was on the verge of falling apart.\n\n69\n\nWhen Sir Thomas Gates arrived as the new governor in 1609, he made a thorough inspection of the living quarters, fortifications, and other facilities within the compound. Much of it was serviceable, he said, but he was disappointed to see the pathetic condition of the church. It symbolized the state of religion in the colony, and he ordered the settlers to immediately begin restoration. As a distinguished military officer, Gates had been sent by the London Company to take whatever steps might be needed to strengthen the colony and get the men back to work. He realized that the vitality of their Christian faith was every bit as important as their commitment to building a productive economy.\n\nOne of Gates's first official acts was to call for a renewal of religious services. From that point on, there would be two Sunday sermons each week and a Weekday Sermon every Thursday. In addition, a bell was to be rung every morning at ten o'clock and every evening before supper, calling each man to a time of personal prayers. The governor believed a revitalized faith would help the settlers deal more effectively with the hardship of their daily lives.\n\nEqually important, the governor wanted to emphasize the majesty and solemnity of the worship service. When he went to church on Sunday, he was accompanied by his councilors, captains, and officers. There was also a company of fifty armed soldiers, called halberdiers, surrounding him in their brilliant red tunics. The governor took his seat in the choir where everyone could see him, and vice versa. He sat, as Frank Lambert reports, \"in a green velvet chair, with a velvet cushion before him on which he knelt, and the council, captains, and officers sat on each side of him, each in their place.\"\n\n70\n\nTHE FINGER OF GOD\n\nVery much like the Pilgrims in New England (who would not arrive in the New World for another eleven years), the Jamestown colonists suffered through their own \"starving time\" in 1609\u201310. All but 60 of the original 214 settlers died during that fateful winter. For many of the men and women in the settlement, the losses, defeats, and constant struggles were unbearable. To make matters worse the ship they expected to arrive in the fall of 1609, bringing food and desperately needed provisions, did not survive the voyage. Without the new cargo, which included clothing, tools, and many other staples they desperately needed, the settlers knew there was no way they could survive.\n\nBut this would be one of the most amazing miracles of the entire Jamestown saga. During the passage from England to Virginia, the cargo ship _Sea Venture_ , carrying all the provisions for the struggling colony, went aground off the coast of Bermuda, and the entire crew was given up for dead. The colonists waited expectantly, searching the horizon day after day for any sign of hope, but the ship never came, at which point the men voted to bury their cannons and armor, abandon the colony, and return to England at the first possible opportunity. If they had left the colony, the settlement at Jamestown would have gone the way of all the other failed attempts. But on June 10, 1610, the day before their planned departure, Providence intervened, and the ships bearing their new governor, Lord de La Ware, and all the goods and supplies for which they had been praying suddenly appeared at the mouth of the James River. When the captain and his crew came ashore, the colonists learned the _Sea Venture_ crew had not been lost at sea after all. They had not only survived the shipwreck but also managed to save the entire cargo, then built new ships. As soon as they were able, they continued the journey to Virginia.\n\nThere was no doubt in the minds of any of the Jamestown settlers that this was a miracle from heaven, and when the news reached England, preachers on both sides of the Atlantic declared the rescue to be an act of divine mercy. And just as important, they believed it was a sign from heaven that the English flag was meant to fly over the New World. In Virginia the Reverend Alexander Whitaker wrote, \"The finger of God hast been the only true worker here.\" In London the Reverend William Crashaw agreed, saying, \"If ever the hand of God appeared in action of man, it was here most evident: for when man had forsaken this business, God took it in hand.\"\n\n71\n\nMany of the men and women who had been the most childish and unreasonable before the relief ship arrived were seemingly transformed by the experience. They could see that miracles do happen and that obedience to God and king was essential for their survival. They realized God is sovereign\u2014God rules in the affairs of men\u2014and He intervenes in ways that sometimes defy human reason. The wreck of the _Sea Venture_ and the relief of the colony convinced the colonists that\u2014just as He had spared the children of Israel from the wrath of Pharaoh\u2014God could and would suspend the laws of nature for His faithful servants.\n\nAlthough they had been saved from disaster in the nick of time, the struggling colony was by no means in the clear. Their \"seasoning time\" would continue for years to come, but occasional shipments of goods and fresh provisions from England, along with the success of the tobacco plantations established by John Rolfe, gave the settlers renewed confidence and good reasons to go on. Rolfe's marriage in 1614 to the Indian princess, Pocahontas, who was the daughter of the Powhatan chief, Wahunsenacawh, improved the settlers' prospects for a while, but it was still no guarantee of lasting peace.\n\nDespite their efforts to appease the local tribes, the troubles never ceased. It was painfully clear that something had to change. Then in 1618, representatives of Sir Edwin Sandys, the London Company's treasurer, suggested the colonists try and integrate the families into the settlements. At that point several Indian groups were given houses, and schools were built for their children so they would be introduced to the Christian religion and English customs.\n\n72\n\nThe colonists assumed the Powhatan nation would be grateful for their benevolence and welcome the opportunity to participate in English community life, but that was not the case. Furthermore the Powhatans did not miss the fact that most of the settlers held them and their way of life in contempt. \"There is scarce any man among us,\" one sympathetic settler wrote, \"that doth so much as afford them a good thought in his heart, and most men with their mouths give them nothing but maledictions and bitter execrations.\"\n\nBy the year 1622, the Powhatan elders realized the English intended to claim all of their nation's lands, expanding English towns and villages across Virginia. They saw this as a threat to their way of life. The colonists' offer to educate their children was seen as an even greater threat\u2014an effort to steal their children's hearts and eradicate their native customs. At that point the Powhatan chief, Opechancanough, conceived a plan he hoped would drive the foreigners out of Virginia once and for all.\n\nThe attack of March 22, 1622, began with a clever bit of deception when a large group of Powhatans came into the English village, bearing gifts of meat and fruit. The following morning they returned to the settlement and moved from house to house, greeting the colonists warmly. Then at a given signal the natives suddenly grabbed their weapons and other implements and began butchering as many settlers as they could find. They massacred entire families in their homes and killed all their servants and field hands. By the time the mayhem was done, the Powhatans had killed 347 men, women, and children. Then, to make sure no one failed to get the message, they desecrated the bodies of the dead, burned the villages, and destroyed the remaining livestock and crops.\n\nWhen news of the attack and the colony's devastating losses reached London, King James was livid. He canceled the London Company's charter in 1624, and the entire colony of Virginia along with its governing body, the House of Burgesses, was placed under royal control. That was how things continued for the next half century until September 1679, when the fortress and village of Jamestown were finally burned to the ground during a violent clash between the settlers and their governor, William Berkeley, known as Bacon's Rebellion. When the capital of the Virginia Colony was moved to Williamsburg in 1699, the Jamestown settlement was abandoned and fell into ruin. A re-creation of the original village can be seen today at the site, thanks to a project funded by the Colonial National Historical Park.\n\n73\n\nCLERGY UNDER DISTRESS\n\nThe Virginia commonwealth of the mid-to late-seventeenth century did not look much like the settlements of New England, but the citizens of all the English colonies had many things in common, including their religious faith. Most of the ministers who came to Virginia during those years were Anglicans. They were Episcopalian in doctrine and demeanor but subscribed to Calvinist teachings regarding salvation by grace through faith, and they preached in the Puritan plain style. Regardless of style, they all believed the Church of England provided the best expression of the Christian religion.\n\nWhile the Puritan churches in Massachusetts maintained their Congregationalist identity, the churches in Virginia were becoming more aggressively Anglican with each passing year. An act of the House of Burgesses in 1632 made uniformity of belief and practice a matter of common law, ensuring the Church of England would be the official faith of the colony and that the Episcopalian form of worship would be the norm. As some of the other colonies were doing, Virginia also enacted anti-Quaker laws to avoid schism in the church, even prescribing the death penalty for Friends who refused to stay away.\n\nThe middle colonies, however, were much more tolerant of minority denominations. Quakers, Amish, Mennonites, and various other sects were welcomed in Pennsylvania and the colonies of Delaware, New York, and New Jersey. Rhode Island was chartered by Roger Williams as a refuge for those of a more democratic spirit. Pennsylvania was established as a Quaker colony by a royal land grant to the English Quaker, William Penn, in 1638; but none of these denominations would have been welcome in either the northern or southern colonies until late in the following century.\n\n74\n\nBy the end of the seventeenth century, the church in Virginia was still growing, but it lacked the intensity of the congregations farther north. James Hutson, who is the author of a Library of Congress publication tracing the history of faith in America, writes that \"religion was the salt that flavored life in seventeenth-century British North America.\" But there were many Virginians in the closing decades of the century who feared the salt was losing its savor.\n\nA half century after the demise of the Jamestown settlement, the Anglican church held a virtual monopoly on religion in the colony; however, as Hutson points out, the churches were so widely scattered that regular church attendance was often difficult, and next to impossible in bad weather. Some Anglican ministers were assigned multiple widely separated parishes, which made Sunday and weekday services incredibly stressful, and officiating at the growing number of weddings, baptisms, funerals, and other special occasions was next to impossible. In some instances, specially trained laymen were allowed to conduct services, but the end result was a general decrease in the orthodoxy and level of Christian commitment in many parts of the colony.\n\nThere were never enough ministers, and there never would be as long as the life of the typical pastor remained stressful as he was underappreciated and underpaid. The Virginia legislature attempted to introduce measures to make life a little easier for the clergy by offering to increase salaries, which were often paid in tobacco and corn. But many parishes failed to provide even a modest living for their ministers. Each community was required to set aside a certain amount of property, called glebe land, where crops were cultivated for the benefit of the clergy. But one English visitor to the Virginia Colony wrote that ministers there were condemned to \"see their families disordered, their children untaught, the public worship and service of the great God they own neglected.\"\n\n75\n\nSome parishes, on the other hand, could complain that their preachers were receiving better support than they deserved. A certain percentage of the ministers who came to the colonies had done so to escape bad debts, bad marriages, and other problems. This situation eventually became so common that the House of Burgesses issued a decree: \"Ministers shall not give themselves to excess in drinking, or riot, spending their time idly by day or night playing at dice, cards, or any other unlawful game; but they shall . . . occupy themselves with some honest study or exercise, always doing the things which shall appertain to honesty, and endeavor to profit the Church of God.\" Apparently, however, the government's caution had little effect. Another English visitor remarked a decade later that Virginia seems to attract mainly the kind of ministers who \"babble in a pulpit\" and \"roar in a tavern.\"\n\nTHE OLD DOMINION\n\nThe contrast between the colonies of Virginia and New England in the late seventeenth century was striking. The population in both areas was still mainly English, but while the citizens of the northern colonies were staunchly Puritan with separatist sympathies, Virginia was resolutely Anglican with loyalist sympathies. And while the Puritans were evangelical and passionate about soul-winning, the Virginians avoided displays of enthusiasm or emotion in the churches. Furthermore in New England practically every man, woman, and child could read and write, whereas in Virginia education and learning were reserved mainly for the ruling class.\n\nNew England was also supported by numerous trades, crafts, and budding industries while Virginia was all about agriculture, mainly centered around the tobacco trade. Not least, while New England was generally democratic in spirit, Virginia retained the attitudes and habits of an Old World aristocracy. As historian Francis Parkman points out, Virginia society was stratified with slaves, indentured servants, and poor whites at the low end of the social spectrum, farmers and small planters in the middle, and the landed gentry who made up the ruling class at the high end. Parkman writes:\n\n76\n\nMany of them were well born, with an immense pride of descent, increased by the habit of domination. Indolent and energetic by turns; rich in natural gifts and often poor in book learning, though some, in the lack of good teaching at home, had been bred in the English universities; high-spirited, generous to a fault; keeping open house in their capacious mansions, among vast tobacco fields and toiling negroes, and living in a rude pomp where the fashions of St. James were somewhat oddly grafted on the roughness of the plantation . . .\n\nBut what the Virginia planters lacked in formal schooling, Parkman adds, was compensated for by power and position, along with a bold spirit of independence and a patriotic attachment to the Old Dominion. \"They were few in number; they raced, gambled, drank, and swore; they did everything that in Puritan eyes was most reprehensible; and in the day of need they gave the United Colonies a body of statesmen and orators which had no equal on the continent.\"\n\nThe Pennsylvania Colony, by contrast, was by design a wild mixture of races and creeds\u2014English, Irish, Germans, Dutch, and Swedes, who were Quakers, Lutherans, Presbyterians, Catholics, Moravians, and many other denominations. The Quaker tradition, to which William Penn and the colony's founding families all belonged, dominated in the eastern part of the colony while German Lutherans, most of whom were simple farmers, were most numerous in the middle parts of the territory, and Scots and Irish immigrants occupied the western regions.\n\nBecause of this wide diversity it was only natural that Philadelphia would play a key role in the revolutionary period. Second only to Boston, Philadelphia was the largest town in British America and the intellectual hub of the colonies. Ironically the Quaker influence suggested that Pennsylvania would be pacifist and noncombatant in the revolutionary era, but the colonies were under increasing stress from British policies, and even the most deeply rooted beliefs can change in such circumstances.\n\n77\n\nThe Colony of New Netherland was settled by Dutch traders in 1614, and remained under Dutch control for the next fifty years. The English navigator Henry Hudson was the first to explore the region on behalf of Dutch merchants, but when James I learned of Hudson's involvement, he commanded him to cease all such efforts. The town of New Amsterdam, on the western end of Long Island, was home to Dutch Calvinists, French Huguenots, and German Lutherans, as well as large numbers of Independents, Scottish Presbyterians, and others of little or no religion. In 1664, James laid claim to the territory for England. Unable to defend his prior claim, the Dutch governor, Peter Stuyvesant, had no choice but to accept a bloodless defeat. New Amsterdam was then renamed New York, in honor of the king's brother, the Duke of York.\n\nThe Colony of New Jersey was mainly farm territory while Maryland, very much like its larger and more prosperous neighbor to the south, was a tobacco-growing region. Unlike Virginia, however, Maryland had been established as a home for Roman Catholics who were out of favor everywhere else. The Colony of Delaware, established by Virginia's former governor, Lord de La Ware, was essentially an extension of Pennsylvania, and the southern colonies\u2014North Carolina, South Carolina, and Georgia\u2014were predominantly agrarian societies with a mixture of English plantation owners, poor white farmers, and large numbers of slaves, indentured servants, and castoffs from the British Isles.\n\nThe attitudes of all these colonies toward each other are hard to imagine today. Their only source of unity was the fact that they were European immigrants with an allegiance to the English king. As Francis Parkman points out, communication between the colonies was slow and tedious. Messengers traveled over rough roads poorly cut through dense forests. In addition, there were long-held animosities between some of them because of land disputes and commercial rivalry. Each man considered his own colony to be his native land; the concept of a united North America simply did not exist.\n\n78\n\nTHE LAND OF OPPORTUNITY\n\nAs social and political institutions in the colonies increased in maturity and status, and as the populations of all the colonies became more numerous, it was obvious on both sides of the Atlantic that these Americans were building an economic powerhouse. There were still poverty and hardship in many places, particularly in the western territories where communities of any size were few and far between, but business was booming in the cities and towns. After a century of toil and constant uncertainty, there was now a gratifying and unprecedented level of prosperity.\n\nThe wealth of the colonies began with agriculture. Land was plentiful, and the climate was ideal for growing a wide variety of cereals, fibers, fruits, and vegetables that were highly prized in Europe. Rice, wheat, corn, hemp, and tobacco were grown commercially on both small and large farms. There were highly productive fisheries in Massachusetts Bay and all along the Atlantic coast, and every colony could boast of a thriving trade in horses, cattle, sheep, pigs, and poultry. South Carolina became the wealthiest of the thirteen colonies in the mid-eighteenth century, thanks primarily to the large number of rice plantations cultivated there by slave labor.\n\nAmerica's forests were seemingly endless, and the timber merchants harvested tremendous quantities of oak, poplar, elm, cedar, and pine for construction projects in the colonies as well as for shipbuilders in the maritime colonies and the overseas trade. In addition, there were now mines in the mountains of Carolina, Virginia, and Pennsylvania, producing iron ore, copper, and lead. These and many other commodities\u2014including animal hides, whiskey, rum, sugar, and tobacco\u2014contributed to the colonies' growing economic strength.\n\n79\n\nCotton and other fiber-producing plants were grown and harvested in the southern colonies for the looms in England. Weaving was still a cottage industry in the early years, but by the end of the eighteenth century, there would be large textile mills in Carolina, Delaware, and the middle colonies. Wool products, however, were not produced commercially until the end of the century, due to trade restrictions imposed by Parliament. Sheep, goats, and wool products had been a mainstay of England's economy for centuries, and the last thing the lawmakers in London wanted was competition from America.\n\nThanks to abundant natural resources, business was thriving in the early years of the eighteenth century. Wages for American workers were as high as or higher than those in England at the time, but fewer than half of the working men owned their own homes or land. Tax records from the period indicate that the bottom 30 percent owned nothing, but from the beginning there was a strong work ethic in the colonies. Thanks to low tax rates and a less rigid class system than that of the Old World, it was possible for a common man with a quick wit and a strong back to acquire wealth and rise in society.\n\nBenjamin Franklin, as perhaps the most conspicuous example, was born into a middle-class family. His father, Josiah, was a soap and candle maker. The elder Franklin hoped his youngest son would become an Anglican minister, but Benjamin had other ideas. He was obstinate, independently minded, and curious about many things; but he was not a good student. He was much too impatient and undisciplined and, consequently, spent just two years in grammar school before taking an apprenticeship with his older brother, James, who was a printer.\n\nOnce he had learned the trade, Ben left his father's home in Boston and headed to Philadelphia, where he worked in several print shops before he decided to go into business on his own. It wasn't long before Ben and a financial partner, Hugh Meredith, purchased one of the colony's first periodicals, the _Philadelphia Gazette_ , and turned it into the most successful newspaper in the country. Franklin is also credited with creating the first political cartoon, \"Join, or Die.\" The cartoon appeared in the _Gazette_ on May 9, 1754, calling for the American colonies to unite against further encroachments by Parliament and King George II.\n\n80\n\nAlmost every American home at the time had a copy of the King James Bible, but Franklin believed there was a need for other kinds of literature as well. Relying upon his quick wit, a quirky sense of humor, and an ear for a colorful anecdote, he composed _Poor Richard's_ _Almanac_ , which would become, second only to the Bible, the most popular book in the country. At a time when the entire population of Philadelphia was less than twenty-five thousand, the almanac reportedly sold more than ten thousand copies a year.\n\nIn time Franklin would become a successful inventor, postmaster, humorist, scientist, political commentator, statesman, diplomat, and ambassador to France. He taught himself to read French, Spanish, Italian, and Latin, and he was considered by delegates to the Continental Congress to be among the most illustrious members of that body. He never denied his working-class background, but by age forty-two he had achieved enough wealth and status to turn over the day-to-day management of his business to others. By any standard Franklin was exceptional. Not many men could equal his achievements, then or now. But the point is that in England such a career would have been inconceivable while in America every man had the opportunity to rise to whatever level his skill and determination would allow.\n\n81\n\nA CHANGING LANDSCAPE\n\nWhat was happening in America in the early years of the eighteenth century was unique in human history, a fact that did not go entirely unnoticed in England. The explosion of prosperity and the rise of an independent spirit in the colonies, which ought to have been cause for celebration, actually contributed to a growing rift between England and America. Sir Robert Walpole, known as \"England's first Prime Minister,\" was the voice of the British government between 1721 and 1742, and he was a vocal supporter of a policy of \"salutary neglect\" regarding the colonies. This had been the unofficial policy of Parliament ever since Jamestown, enabling colonial governments to achieve autonomy and efficiency without undue interference from London. Walpole said, \"If no restrictions were placed on the colonies, they would flourish.\" And the more the American colonies flourished, he believed, the better for Britain.\n\nThe American alliance was opening up new territory for Britain in the West; it brought them new and exotic commodities, and it added much-needed revenue to the Exchequer through taxes, fees, and duties on certain American goods. A laissez-faire policy made perfect sense, he believed, but not everyone in London agreed with Walpole's assessment. England was deeply in debt as a result of back-to-back wars in Europe, Canada, and the British Isles. Strapped for cash and with English troops strung out halfway around the world, it was inevitable that both king and Parliament would look to America for relief, in the form of new and higher taxes\u2014along with intervention in many other areas.\n\n82\n\nThe Navigation Acts, which dated to the mid-seventeenth century, prohibited the colonies from trading with nations other than England, Scotland, and Ireland. These policies raised little or no alarm in the beginning since foreign trade was not yet a major concern, but as America's factories, farms, and mills became more and more productive, English laws restricting commercial growth and taxing profits were suddenly a sore spot for the colonies. Domestic taxes were lower in America than in Britain, but each new increase provoked an angry response, and the increases were suddenly coming more and more often.\n\nThe Molasses Act of 1733 placed a duty on sugar and sugar-based products but caused little stir in America because the rum manufacturers in New England ignored it. They found that smuggling was a much better alternative. The colonies continued to trade lumber, furs, and agricultural products for sugar from the West Indies without paying duties, but thirty years later when Parliament passed the Sugar Act with stricter enforcement, resentment spread throughout the colonies. The Sugar Act\u2014along with the Stamp Act, passed in 1765\u2014became one of the first irritants leading to the eventual showdown between America and Britain.\n\nBut prosperity had other consequences as well. As life in colonial America continued to improve materially and economically, there was a growing sense that something was being lost in the moral and spiritual vitality of the nation. The pace of expansion along with improvements in mobility and economic opportunity led, not to euphoria, but to anxiety, and the fallout could be felt in all sectors of society\u2014especially in the churches, which were losing members and influence. As historian William McLoughlin reports,\n\n83\n\nThe ministers no longer elicited deeply felt responses but rather a vague and undefinable discomfort. Men and women in every colony recognized that their efforts to succeed in this world were compelling departures from older behavior patterns and values. Businessmen had to cut corners to compete with their rivals. Farmers had to charge high prices to pay off mortgages on new land. Political leaders distorted the truth to win votes or gain influence. Town fathers enriched themselves and slighted the needs of the community. Lawyers and judges seemed unable to reach verdicts recognizably just to both parties. Legislators seemed to yield to special interests instead of serving the general welfare. On the frontier, where institutional restraints were weakest, men increasingly took the law into their own hands\u2014against Indians, horse thieves, or an interloper overreaching himself.\n\nThe result was social and political disorder. But the most important reaction was a new level of soul-searching, particularly in the churches of New England that had been wrestling with the issue of doctrinal purity for some time. Throughout the founding era, there had been strict adherence to Scripture and the principles of the Reformed faith. Calvinism taught that man is by nature sinful and in need of redemption, which comes through belief in the atoning death of Jesus Christ and is only attainable by a sincere confession of sin, repentance, and conversion.\n\nChurch membership and the sacraments of baptism and holy communion were limited to those who could give evidence of a salvation experience. But a new kind of rationalism was being preached in many Congregationalist churches in New England, best exemplified by the Halfway Covenant of 1662, which in its original formulation allowed the children of church members to receive baptism without making a profession of faith. Fifty years later the covenant was being applied to those wishing to join the church, whether or not they were confessing believers.\n\n84\n\nIn the beginning the idea was to make sure that unbaptized children of members would remain in the church family as adults, but the new liberalized policies made full fellowship available to all adults who wished to be affiliated, regardless of their reasons. Many public offices at the time required evidence of church membership, and this brought some people into the churches purely for expedience, not because of religious conviction.\n\nPastors initially approved of the idea because it increased church attendance, but the actual effect was a watering down of theological integrity. Halfway churches offered membership to all comers\u2014any so-called \"moral person\" was welcomed. Offering the sacraments to nonbelievers was, liberal clergy argued, a \"means of grace\" by which the Holy Spirit might work upon the souls of the unconverted and draw them into fellowship. \"Better to have them in the church than out of it,\" they said. But in time this weakening of long-established tradition would have serious repercussions. \"By 1720,\" as McLoughlin writes, \"the vast majority of ministers were telling their parishioners that regular prayer, church attendance, right behavior, and responsible citizenship were all means of preparation for the salvation God would send when the time was ripe.\"\n\nThe new doctrine offered congregants the assurance that man is not inherently evil, as the traditionalist preachers had insisted, but is basically good and capable of great things. God does not condemn His children to hell but wants the best for them. In this environment even the most unprincipled and profane members of the community were inclined to think of themselves as the best of men. But deep down there were lingering uncertainty and spiritual malaise.\n\nTHE ROOTS OF AWAKENING\n\nThe Reverend Solomon Stoddard was among the best-known Congregationalist preachers in New England at the time, and one of the most controversial. His enthusiastic embrace of the Halfway Covenant put him at odds with a number of his peers in Boston, yet by opening church membership to any man of moral character, Stoddard doubled his congregation in Northampton while more orthodox congregations were shrinking. Ironically Stoddard's staunchest critic turned out to be his own grandson, Jonathan Edwards, who would offer the most formidable challenge to the rationalist approach. In 1741, Edwards preached the sermon that, more than any other, has been credited with launching the Great Awakening.\n\n85\n\nJonathan Edwards (1703\u20131758) was the son of a Congregationalist minister in East Windsor, Connecticut. He was a brilliant student, entering Yale College at the age of thirteen where he studied Hebrew, Greek, Latin, geometry, rhetoric, and logic. Near the end of his second year, the young man experienced a dramatic conversion that transformed his life and his understanding of Scripture. After completing his baccalaureate in 1720, he spent the next two years at Yale studying theology; then at the age of eighteen, he was named pastor of a small church in New York.\n\nEdwards held that position for only a few months before the church ran out of money. He then returned to Connecticut and accepted a position as a teacher and administrator at Yale. After several years in that post, he was called to his grandfather's church in Northampton, Massachusetts, where he served as assistant pastor for two years. When Solomon Stoddard died in 1729, Edwards was elevated to the position of senior minister. The problem he wrestled with in this new position was that he had never agreed with the Halfway Covenant his grandfather supported. He served as pastor of the church for twenty years before he made his feelings known, and, at that point, he was summarily dismissed by the congregation.\n\nFor the next seven years Edwards served as a missionary to the Indian communities at Stockbridge, Massachusetts, and devoted himself to study. During this time, he wrote four of the most respected theological works of the Puritan era, including \"Freedom of the Will\" and \"The Great Christian Doctrine of Original Sin Defended.\" But the contributions for which Edwards is remembered today are two works composed during his time at Northampton: \"A Faithful Narrative of the Surprizing Work of God in the Conversion of Many Hundred Souls in Northampton,\" published in London in 1737, and the sermon \"Sinners in the Hands of an Angry God,\" which he preached at Enfield, Massachusetts, on July 8, 1741.\n\n86\n\nThe revival that broke out in the Northampton Church in 1733 took Edwards and many others by surprise. He was unquestionably a gifted preacher, but Edwards insisted that what happened there over the next seven years was not an act of man but a move of God. The revival began among the youth of the church, then spread from Northampton to surrounding communities with such passion that it could not be restrained. Edwards later wrote:\n\nThe congregation was alive in God's service, everyone earnestly intent on the public worship, every hearer eager to drink in the words of the minister as they came from his mouth; the assembly in general were, from time to time, in tears while the Word was preached; some weeping with sorrow and distress, others with joy and love, others with pity and concern for the souls of their neighbors.\n\nMore than three hundred men and women joined the church in 1734 alone. The power of the Holy Spirit was so strong, changing the daily routines of so many citizens in Northampton, that local shop owners complained they were being driven out of business by the revivalists.\n\nBy 1735, the revival reached across the Connecticut River valley, as far away as New York and New Jersey. When news of what was happening in America reached England in 1735, the revivalists John and Charles Wesley immediately took notice, as did their colleague and fellow member of the \"Holy Club\" at Oxford University, George Whitefield. John Wesley, who had been ordained in 1728, was widely known as the leader of the Methodist movement, so named for the group's diligent study of Scripture and devotion to works of charity. He was anxious to see America for himself and to find out what the Awakening was all about.\n\n87\n\nThe Wesley brothers sailed to America on October 14, 1735, hoping to preach to the Indians. However, the governor of the Georgia Colony, James Oglethorpe, insisted John speak in the churches. After two years John and Charles returned to London, deeply disappointed and consumed by doubts about their own salvation. The Anglican churches in the South had been insulted by Wesley's insistence that faith in Christ demands a personal salvation experience, and the Indian groups who received him were not the least impressed by his style of preaching. All of this precipitated a spiritual crisis in Wesley's own life that led, in 1738, to a miraculous conversion experience and the blossoming of his ministry in England.\n\nUp until that time John Wesley had taken a dim view of his friend Whitefield's enthusiastic style of preaching. Nothing quite like it had been seen in the Church of England, and many accused the young man of insanity or worse. He had been ordained as a priest, but he was never given a church appointment because of his evangelistic fervor. Nevertheless Whitefield, like Wesley, was determined to go to America to see for himself what was happening there.\n\nWhen he arrived in Massachusetts in 1738, George Whitefield was a novelty. Traditional Anglican congregations, very much like those in England, were offended by Whitefield's style of preaching. When he began preaching at open-air services in the countryside, however, dozens, then hundreds, and ultimately tens of thousands came out to hear him preach. Older and more traditional ministers, such as the Reverend Charles Chauncy of Boston, claimed that Whitefield's \"enthusiasm\" was a form of mental derangement. But rather than being deterred by such comments, Whitefield was inspired by them and preached more powerfully. Wherever the sound of his voice was heard, the presence of the Holy Spirit was undeniable, and before long it was apparent that the Awakening that started at Northampton in 1734 was just the beginning.\n\n88\n\nAs an indication of what was about to happen in the American churches during the Great Awakening, we have stirring accounts from individuals whose lives and attitudes were transformed. Nathan Cole was a member of a Congregationalist church in rural Connecticut. In his journal Cole writes that he was an \"Old Light,\" meaning a died-in-the-wool pew-sitter of the old order, until he ventured to a camp meeting in 1741 to hear Whitefield preach. He was changed, he says, and became a \"New Light.\" It took a while to summon the strength of will, but he eventually resigned from his church because of the Halfway Covenant and the easy acceptance of nonbelievers. He became a separatist first, then a Baptist and a lay pastor.\n\nHerman Husband was the son of a tobacco planter in Maryland and a member of the Anglican Church. At age fifteen he went to hear Whitefield preach and, like Nathan Cole, was radically transformed. Over the next several years he went from the Anglican to the Presbyterian church and then to Quakerism, where he became involved in ministries to the poor, the widow, the orphan, and others in need of God's love.\n\nAs James Hutson relates in his history of religion and the American founding, Christianity in rural areas of the colonies was essentially a \"do-it-yourself\" project throughout most of the eighteenth century. Ordained ministers with school training were few and far between, except in the larger cities, and those larger congregations attracted mainly sedate and scholarly ministers who still preached in the old style. Outside the cities, congregations often depended on devout laymen or schoolmasters to read from books of collected sermons that were very popular at the time. Hutson writes:\n\nThese groups, often close in spirit to primitive Christianity, existed in every colony, although they were far more numerous south of New England. Some German Reformed and Lutheran congregations in frontier Maryland, for example, operated for decades, until ordained ministers, who were greeted with \"tears of joy,\" caught up with them.\n\n89\n\nScotch-Irish Presbyterians, steeped in the Reformation doctrines of their Calvinist forebears, were quick to adapt to this situation, forming churches with dedicated lay pastors until trained ministers were available. These men and women had come from Northern Ireland in large numbers, and they contributed greatly to the \"frontier spirit\" and the \"spirit of independence\" that would be so important during the revolutionary period. As James Hutson writes, \"Recent scholarship has put an even higher estimate on the strength of their religious convictions. Scotch-Irish Presbyterians now appear to have introduced as much religious energy into the eighteenth-century middle and southern colonies as the Puritans did in seventeenth-century New England.\"\n\nBIRTH OF THE AMERICAN SPIRIT\n\nBetween 1730 and 1750, there was an explosion of faith from one end of America to the other. It is touching, Hutson writes, to read accounts of the \"thirst for the gospel\" among the men and women who were building the settlements and family farms during the eighteenth century. Itinerant preachers and \"circuit riders\" were begged to stay. The members of some frontier congregations were \"willing to sell their coats and the rest of their clothing to help support a preacher.\" Communities in many places formed cooperatives and built new churches on speculation, in the hope that God would send them a minister who might be willing to accept a call.\n\nLetters, journals, and other accounts of community life in the mid-seventeenth century tell of churches packed to the rafters with men and women who were hungry for the Word of God. In some cases itinerant pastors preached to congregants who were \"forced to stand without doors and others hanging out the window.\" In other cases entire families would pack into tiny oxcarts or walk \"10 to 12 Miles with their Children in the burning Sun, so earnest, so desirous [were they] of becoming good Christians.\" There are also reports of individuals who joined other denominations, different from the ones they had known, \"being willing to embrace anything that looks like a religion, rather than have none at all.\"\n\n90\n\nSuch strong reactions were remarkable because, then as now, denominational distinctions were not a small matter. As discussed in the previous chapter, New England Congregationalists and the Anglicans in the middle and southern colonies held firmly to very different views on matters of church polity. Church officials in the middle and southern colonies, where the Anglican communion was strongest, soon began to feel threatened by denominational differences. Concerned about the prospect of losing members, they immediately began taking steps to fortify the local churches.\n\nThe establishment of religion was not an issue in the eighteenth century. All the colonies had established churches, funded by the government until after the American Revolution. The Anglican Church had been established in New York since 1693, while the Church in Maryland was established in 1702, South Carolina in 1706, and North Carolina in 1715. In addition to the state support they received, the churches began cooperating in the founding of colleges and other educational institutions to train their ministers, and associations were created to regulate denominational standards and encourage uniformity.\n\nAlong with the wave of conversions that came from the Awakening, these organizations reaped the benefits of a sustained boom in new church formation. The number of Anglican congregations increased from 111 in 1700, to more than 400 by 1780. Meanwhile Baptist congregations grew from 33 to 457, Congregational churches went from 146 to 749, German and Dutch Reformed parishes increased from 26 to 327, Lutherans from 7 to 240, and the Presbyterian churches from 28 to 475. It has been estimated that during the first half of the eighteenth century\u2014encompassing the years of the Great Awakening\u2014between 75 and 80 percent of the entire population of North America attended church on a regular basis.\n\n91\n\nOne writer suggests that the type of Christianity we recognize today as evangelicalism actually began in the 1730s through the preaching of men such as Jonathan Edwards, George Whitefield, Gilbert Tennent, James Davenport, and others who had no idea they were starting a movement. Instead, they saw themselves as throwbacks to the old-fashioned Calvinist preaching. Edwards claimed he was preaching nothing but \"the common plain Protestant doctrine of the Reformation.\" Likewise, the Methodists believed they were \"restoring the 'old divinity' of the Reformation.\" But the men and women who experienced the Awakening firsthand were under no illusions: they knew that they had been touched by the hand of God.\n\nBefore the Great Awakening the colonies had very little in common. They didn't even like each other very much. But as the Awakening began to spread from town to village throughout the country, there was a new sense of connection and cooperation between the colonies. Whenever the people of one colony heard that one of these famous preachers was going to be anywhere in the area, they would pack up their families and travel miles, even hundreds of miles, to the campsites where they could hear the messages.\n\nThis was a tremendous social phenomenon. And the more it happened, the more the colonies came together, not only politically but spiritually. This was really the first widespread sense of brotherhood that allowed Americans of the eighteenth century to see themselves not merely as unrelated communities of farmers and merchants, but as men and women with a common bond\u2014as citizens of the united colonies of America.\n\n92\n\nThis new sense of unity eventually developed into what we would recognize today as the American Spirit. It would take a great deal of unity for these same men and women to rebel against King George. Even after the first shots were fired at Lexington and Concord in 1775, most Americans resisted the Revolution\u2014they thought of themselves as Englishmen and subjects of the Crown, and it would be very hard for them to break that bond of kinship. But in due time the provocations and punishments inflicted on them by the English would overcome their resistance. At that point, the understanding of spiritual and political liberty infused in them through the Great Awakening gave them the courage and resolve to break their bonds.\n93\n\n[_**Four**_  \nTHE BIRTH  \nOF THE AMERICAN SPIRIT](Lee_9780849949463_epub_c5_r1.html#d6e6)\n\n18TH-CENTURY POPULATION IN THOUSANDS  \n---  \n| 1700  | 1780   \nNEW ENGLAND COLONIES  \nMAINE| .1 | 49.1   \nNEW HAMPSHIRE| 5.0 | 87.8   \nVERMONT| .1 | 47.6   \nPLYMOUTH\/MASS. BAY| 55.9 | 268.6   \nRHODE ISLAND| 5.9 | 52.9   \nCONNECTICUT| 26.0 | 206.7   \nSUBTOTAL: | 93.0 | 712.7   \nMIDDLE COLONIES   \nNEW YORK| 19.1 | 210.5   \nNEW JERSEY| 14.0 | 139.6   \nPENNSYLVANIA| 18.0 | 327.3   \nDELAWARE| 2.5 | 45.4   \nSUBTOTAL: | 53.6 | 722.8   \nSOUTHERN COLONIES   \nMARYLAND| 29.6 | 245.5   \nVIRGINIA| 58.6 | 538.0   \nNORTH CAROLINA| 10.7 | 270.1   \nSOUTH CAROLINA| 5.7 | 180.0   \nGEORGIA| .1 | 56.1   \nKENTUCKY| .1 | 45.0   \nTENNESSEE| .1 | 10.0   \nSUBTOTAL: | 104.9 | 1,344.7   \nESTIMATED TOTALS: | 251.5 | 2,780.2\n\n94\n\nColonial America had come a long way since the first settlements at Roanoke and Jamestown. By the year 1700, there were nearly 100,000 inhabitants in New England, just over half that number in the middle colonies, and another 105,000 in the southern colonies with Virginia the largest and most prosperous. Thanks to a high birthrate and the steady flow of immigrants from all parts of Europe, the population of British North America was suddenly booming.\n\nThroughout the colonial period immigration continued to accelerate, surpassing the natural rate of increase. Between 1607 and 1700, as many as 200,000 Europeans made the journey to America, including large numbers of English, Irish, Scottish, and Welsh, along with growing numbers of Germans, Italians, and Scandinavians. Notably missing in these totals were the French settlers who journeyed farther north to Quebec. Later, when the French began occupying new lands in the west, as part of the Louisiana Territory, communities of French-speaking Europeans settled at St. Louis and other sites along the Mississippi River, as far south as New Orleans, which had been colonized by the French Mississippi Company in 1718.\n\n95\n\nBetween 1700 and 1763, immigrants from the British Isles arrived seeking the freedom and autonomy they had been denied in their native lands. Settlement in North America was now possible for men and women of all classes and backgrounds, not merely the young and adventurous or the persecuted and discontent. An expanding network of shipping companies, land promoters, bankers, plantation owners, and merchants offered passage to British America on credit. Some colonists came bearing royal charters that enabled them to establish large plantations or entire colonies, as was the case with William Penn in Pennsylvania and James Oglethorpe in Georgia. However, for those of lower estate who could not afford the cost of passage, the most common alternative was to sign a letter of indenture. It has been estimated that fully two-thirds of all English-speaking settlers in the seventeenth century came to this country as indentured servants. And as many as 80 percent of those who came in the eighteenth century were either indentured servants or \"redemptioners.\"\n\nMany who came from central Europe were convinced to make the journey to America by couriers carrying letters between Pennsylvania and Germany. Known as \"newlanders,\" these couriers were former emigrants themselves, either returning home for a visit, collecting debts, or claiming an inheritance. They were paid by their neighbors to carry letters and conduct business in their former homelands. By recruiting others to emigrate, the newlanders could earn free passage back to Philadelphia, along with a modest commission in the process.\n\nThanks to advancements in shipping at all levels, from the improved seaworthiness of English vessels to the greatly reduced threat of piracy, transatlantic crossings were becoming more frequent and reasonably safe. There was a steady stream of information, merchandise, and people traveling back and forth between Europe and America. The number of transatlantic crossings tripled from about 500 during the 1670s to more than 1,500 each year by the late 1730s. The increased traffic also meant lower costs for shippers and merchants, which led in turn to a greater abundance of goods of all kinds.\n\n96\n\nScottish emigration to the colonies soared to more than 145,000 between 1707 and 1775. Generally poorer than the English, the Scots also had greater incentive to emigrate because of the oppression they had endured in their own country. The Act of Union, which united England and Scotland under the banner of Great Britain in 1707, accelerated the process by encouraging thousands to abandon their ancient homeland in search of greater freedom in the New World. Good reports from early emigrants who prospered, particularly in the middle colonies, attracted large numbers of emigrants through \"chain migration,\" reuniting them with family members who had come earlier, or in some cases actually relocating entire communities from Scotland and other parts of Europe to the New World.\n\nNearly half of the Scottish emigrants came from Ulster, Northern Ireland, which had been colonized by lowland Scots in the 1690s. Like the Highlanders, who were driven from their ancestral lands by the English during the eighteenth century, the Ulster Scots sought to escape punitive laws (particularly regarding religious freedom) at home. Scottish settlers tended to emigrate in groups organized by their Presbyterian ministers, who negotiated with shippers to arrange passage. When they arrived in the colonies, they generally settled in the Carolinas or on the western frontier, where land was cheap and regulations were few. In America all these Scottish immigrants were known as Scotch-Irish even though the name only applies to the Ulster Scots.\n\nEstimates based on colonial records, personal letters, newspaper reports, and other fragments of data from the period suggest that in 1763 about 50 percent of the population was English, 18 percent Scottish or Scotch-Irish, 18 percent African slaves, 6 percent German, and 3 percent Dutch; the rest were a mixture of all the other European nationalities who immigrated to North America. By the mid-eighteenth century, intermarriage was already making America a melting pot.\n\n97\n\nA NEW SPIRIT OF UNITY\n\nIn this environment the Founding Fathers faced a monumental task: How would they ever form a unified nation of men and women from such an ethnically, linguistically, and economically divided population? When Thomas Paine published his powerful diatribe, _Common_ _Sense_ , on the threshold of the American Revolution, he began it by saying, \"The cause of America is in a great measure the cause of all mankind.\" It was one thing to say that all men desire freedom from oppression and the opportunity to succeed, but Paine didn't miss the fact that eighteenth-century America was now a land inhabited by people of many nations:\n\nThis New World has been the asylum for the persecuted lovers of civil and religious liberty from every part of Europe. Hither have they fled, not from the tender embraces of the mother, but from the cruelty of the monster; and it is so far true of England that the same tyranny which drove the first emigrants from home pursues their descendants still.\n\nLarge parts of America had been settled by Englishmen, but there were other parts where the citizens spoke languages other than English and celebrated customs that were quite different from those of their British neighbors. But Paine realized that there were two things all these people had in common\u2014love of country and love of God\u2014and in _Common Sense_ he appeals to both of these emotions. Regarding the common bond of national unity, Paine says:\n\nThe sun never shined on a cause of greater worth. 'Tis not the affair of a city, a county, a province, or a kingdom, but of a continent\u2014of at least one-eighth part of the habitable globe. 'Tis not the concern of a day, a year, or an age; posterity are virtually involved in the contest, and will be more or less affected even to the end of time by the proceedings now. Now is the seedtime of continental union, faith, and honor.\n\n98\n\nThe writer makes it clear that regardless of where they emigrated from, the men and women of this country were engaged in a universal struggle for freedom and the right of self-determination. Everyone who made the pilgrimage to America, or whose fathers and mothers had risked everything to make a new life for their children, had a stake in America's success. They had all come here to escape oppressive circumstances, and they were united in their love of liberty and their common bond of citizenship in this \"New World.\" Even though most had rarely (or perhaps never) thought about the possibility of rebellion or the formation of an independent republic\u2014the consequences of which were literally unthinkable\u2014they nevertheless understood the appeal to union, faith, and honor.\n\nPaine knew very well that his readers would have plenty of objections, so then, addressing himself to the shift of loyalties from the British monarchy to the Monarch who rules over all men, he writes:\n\nBut where, says some, is the king of America? I'll tell you, friend, he reigns above, and does not make havoc of mankind like the royal brute of Britain. Yet that we may not appear to be defective even in earthly honors, let a day be solemnly set apart for proclaiming the charter; let it be brought forth placed on the divine law, the word of God; let a crown be placed thereon, by which the world may know that, so far as we approve of monarchy, that in America the law is king.\n\nThe impact of Paine's words, contained in a slim volume of barely fifty pages, was tremendous. Word of mouth spread quickly, and the pamphlet sold upwards of 120,000 copies in the first three months, and more than a half million copies in the first year. These were unprecedented numbers. The pamphlet went through twenty-five printings in the first year of the Revolution, but to demonstrate the sincerity of his words, Paine donated all royalties from the sale of the work to the Continental Army, under the command of General George Washington.\n\n99\n\nMuch has been made of the fact that Paine was not a Christian in any orthodox sense. He has been described as a Deist, leading some critics to suggest he was simply exploiting the religious sentiments of the colonists to provoke the reaction he desired. Based on later writings, such as _The Rights of Man_ (1791) and _The Age of Reason_ (1794), which echo the liberal sentiments of the French Revolution more clearly than our own, that would seem to be a reasonable evaluation of Paine. As an Englishman, Paine witnessed religious controversy firsthand before coming to this country, and he adopted the rationalist worldview.\n\nHe may have been an Enlightenment thinker, but Thomas Paine understood very well that the Puritans and many others had come to America, first and foremost, for religious liberty. Most of all, he understood that the Christian religion was the one common denominator uniting all Americans, regardless of their origins or condition of life. It was the one thing they all believed. His appeal to faith, honor, and union had the desired impact, but it did so because the groundwork had been laid by Jonathan Edwards, George Whitefield, Gilbert Tennent, and all the great preachers of the Great Awakening a generation earlier.\n\nPREPARING THE HARVEST\n\nWhen Jonathan Edwards preached his famous sermon, \"Sinners in the Hands of an Angry God,\" in 1741, he worried that the churches of New England had grown cold and halfhearted in their worship. Unlike their Puritan fathers and mothers who had survived deadly winters, diseases, Indian attacks, and the challenges of building new lives in an alien environment, many eighteenth-century Americans were now prosperous, comfortable, and complacent. The New England meetinghouse, in Edward's estimation, was in danger of becoming a clubhouse, where well-fed and overly self-righteous ladies and gentlemen came on a Sunday, not to prostrate themselves before a holy God but to be seen and admired by their neighbors.\n\n100\n\nIn 1727, less than ten years before the first tremors of the Great Awakening, the area around Boston had been struck by a devastating earthquake. Contemporary reports tell of a period of several months in which there was severe drought followed by weeks of high winds and flooding throughout the region, culminating in the Boston Earthquake of October 29. Many people felt as if the colonies were being chastened, driven to their knees by God Himself.\n\nAs a result, Ellis Sandoz writes, there was \"a quickening of religious impulses,\" which prepared the population for the spiritual awakening that was to come, not only at Northampton but in many other places over the next five or six years. In Jonathan Edwards's powerful sermon he deliberately reminded the congregation at Enfield of that earlier time, and he assured them that God is omnipotent, omnipresent, and omniscient. That is, He is all-powerful, present in all times and places, and all-knowing and all-wise. For those who might be inclined to neglect their religious duty and fall into sin with no fear of eternal judgment, Edwards warned, \"The world would spew you out, were it not for the sovereign hand of him who hath subjected it in hope.\" Then in his characteristically solemn and measured tones, Edwards added that, \"There are the black clouds of God's wrath now hanging directly over your heads, full of the dreadful storm, and big with thunder; and were it not for the restraining hand of God, it would immediately burst forth upon you.\"\n\nThere is no question that Jonathan Edwards anticipated the impact of his words. What he had in mind was a general and widespread revival in the churches of New England. For much too long, he believed, a generation of staid and sedentary pastors had made it all too easy for congregants to ignore the warnings of Scripture and the very real dangers of eternal damnation. So in that sermon he says, \"How dreadful is the state of those that are daily and hourly in the danger of this great wrath and infinite misery! But this is the dismal case of every soul in this congregation that has not been born again, however moral and strict, sober and religious, they may otherwise be.\"\n\n101\n\nSome of Edwards's critics in his own day and many more in ours have accused him of shameless alarmism, trying to frighten people into making a religious conversion. But Edwards did not leave that option open. The alarm, he said, was a genuine forewarning of imminent disaster. In the sermon he says that any minister who would set out to terrify the people with lies, to create pandemonium by making the case out to be worse than it actually is, ought to be condemned. But he also says that if the message is not of men but of God and if the people are justifiably frightened by it because of their sinful ways, then the minister who brings the warning is more than justified and ought to be commended.\n\nIn his book _Some Thoughts Concerning the Present Revival of_ _Religion_ , published in 1742, one year after the sermon at Enfield, Edwards writes:\n\nWhen consciences are greatly awakened by the Spirit of God, it is but light imparted, enabling men to see their case, in some measure, as it is; and, if more light be let in, it will terrify them still more. But ministers are not therefore to be blamed that they endeavor to hold forth more light to the conscience. . . . To say any thing to those who have never believed in the Lord Jesus Christ, to represent their case any otherwise than exceeding terrible, is not to preach the word of God to them; for the word of God reveals nothing but truth; but this is to delude them.\n\nThe objective of all his preaching was to call the men, women, and children within the sound of his voice to repentance. At the conclusion of his famous sermon, Edwards reached out to the congregation in steady and unemotional tones, saying, \"Let every one that is yet out of Christ, and hanging over the pit of hell, whether they be old men and women, or middle aged, or young people, or little children, now hearken to the loud calls of God's word and providence. . . . Therefore, let every one that is out of Christ, now awake and fly from the wrath to come.\" Despite the mixed reactions the sermon has received ever since, many thousands have heard that warning and responded.\n\n102\n\nTHE GREATEST AWAKENER\n\nBy the time that sermon was given in New England, the evangelist George Whitefield was already traveling from one end of North America to the other, awakening the hearts of the colonists to a message of repentance. Like Edwards, he warned of the bondage of sin and offered his listeners the hope of emancipation from their shackles through the liberation that only comes through the gospel of Jesus Christ.\n\nDuring seven separate visits to America, he traveled between his base in Savannah, Georgia, and the towns and villages of New England. In each of these tours he encountered a backlash from the pastors of the established churches who disliked his theology and resented his ever-growing popularity. But nothing could dissuade the evangelist from preaching to the massive crowds\u2014often as many as twenty thousand men, women, and children of all types and all races\u2014who came from miles around to hear him.\n\nWhitefield was widely known for his powerful speaking voice. But the most impressive demonstration of his gifts took place during an outdoor sermon in Boston in which he compared the violence of an approaching thunderstorm to the storm of God's judgment of sin. At the very instant he raised his arm and pointed a finger toward heaven, a brilliant flash of lightning illuminated the sky, and a blast of thunder shook the earth, to which Whitefield roared, \"See there! It's a glance from the angry eye of Jehovah! Hark!\" Then, after a dramatic pause, he once again declared, \"It's the voice of the Almighty as He passed by in His anger.\" It is not hard to imagine how the people would have reacted to such an amazing display.\n\n103\n\nHis passionate delivery drew thousands upon thousands from cities and towns in every part of the country, leading to thousands of dramatic conversions, often accompanied by physical manifestation, such as uncontrollable trembling, ecstatic prostration, glossolalia, and many other signs and wonders. These manifestations were described by detractors as shameful displays of enthusiasm and emotion, but Whitefield's impact on the men and women of North America for more than a decade cannot be overestimated. More than any other evangelist of the day, he helped to usher in the Great Awakening, without which the unification of the American people and the success of the American Revolution may not have happened.\n\nIt is stunning to realize how much he accomplished. Whitefield preached an average of five hundred sermons a year throughout his active life, each sermon lasting from one to two hours. He often preached forty hours in a week, and sometimes more. This was in addition to everything else, including travel and correspondence, building and promoting an orphanage in Savannah, Georgia, and raising funds for missions work. In addition he made a preaching tour of England almost every year, traveled to Scotland fourteen times and Ireland three times, and also made several trips to Wales. He crossed the Atlantic thirteen times, traveling back and forth to the colonies, and he reached an estimated ten million souls during three decades of ministry.\n\nWhitefield's eloquence was legendary. One famous story relates the reaction of the renowned British actor David Garrick, exclaiming, \"I would give a hundred guineas if I could only say 'Oh!' like Mr. Whitefield.\" Benjamin Franklin was not necessarily a candidate for baptism; he had been raised as an Anglican but showed little interest in spiritual matters for most of his life. Nevertheless, being worldly-wise and curious about most things, he went to hear Whitefield in his professional capacity\u2014a newspaper publisher looking for a story. In his autobiography he describes the experience:\n\n104\n\nIn 1739 arrived among us from Ireland the Reverend Mr. Whitefield, who had made himself remarkable there as an itinerant preacher. He was at first permitted to preach in some of our churches; but the clergy, taking a dislike to him, soon refus'd him their pulpits, and he was oblig'd to preach in the fields. The multitudes of all sects and denominations that attended his sermons were enormous, and it was matter of speculation to me, who was one of the number, to observe the extraordinary influence of his oratory on his hearers, and how much they admir'd and respected him, notwithstanding his common abuse of them, by assuring them that they were naturally half beasts and half devils.\n\nFranklin's comments were only partly in jest because he understood very well that men who are convinced of their iniquity and inclined to petition heaven for forgiveness and undertake a course of moral reform were likely to be better citizens. Then he says,\n\nIt was wonderful to see the change soon made in the manners of our inhabitants. From being thoughtless or indifferent about religion, it seem'd as if all the world were growing religious, so that one could not walk thro' the town in an evening without hearing psalms sung in different families of every street.\n\nIn one of the most amusing anecdotes from his autobiography, Franklin reveals just how persuasive Whitefield could be:\n\nI happened soon after to attend one of his sermons, in the course of which I perceived he intended to finish with a collection, and I silently resolved he should get nothing from me. I had in my pocket a handful of copper money, three or four silver dollars, and five pistoles in gold. As he proceeded I began to soften, and concluded to give the coppers. Another stroke of his oratory made me asham'd of that, and determin'd me to give the silver; and he finish'd so admirably, that I empty'd my pocket wholly into the collector's dish, gold and all.\n\n105\n\nThe changed lives of Whitefield's hearers were even more impressive than the preacher's oratory. One New England farmer who heard Whitefield preach described the experience in his journal, where he wrote: \"He looked almost angelical, a young slim tender youth. He looked as if he was clothed with authority from the great God. A sweet solemnity sat upon his brow. My hearing him preach gave me a heart wound. . . . I saw that my righteousness would not save me.\" The preacher's words hit home in similar fashion and transformed untold thousands of hearts, and in time the message of repentance and redemption would change the spiritual landscape.\n\nPREPARING FOR REVOLUTION\n\nTrue religion for George Whitefield, the scholars tell us, meant \"a thorough, real, inward change of nature, wrought in us by the powerful operations of the Holy Ghost, conveyed to and nourished in our hearts, by a constant use of all the means of grace, evidenced by a good life, and bringing forth the fruits of the spirit.\" Like their Calvinist forebears, prerevolutionary Americans believed Christ's commandment to \"render to Caesar the things that are Caesar's\" (Mark 12:17) meant not only that each man ought to be obedient to the ruling authority but that each man had the right and duty to hold the ruler accountable when reasonable boundaries were crossed.\n\nFor this reason the transformation of the spirit taking place in the hearts of tens of thousands of Americans would have dramatic political implications as well. \"The political culture of this country,\" historian and legal scholar Ellis Sandoz writes, \"was deeply rooted in the core religious consciousness articulated above all by the preachers; theirs were the pulpits of a new nation with a privileged, providential role in world history.\" Because the churches were such an essential part of community life, this was where the beliefs and actions of the men and women who would become the patriots of 1776 were shaped. As Daniel Boorstin explains:\n\n106\n\nThe New England meeting-house, like the synagogue on which it was consciously modeled, was primarily a place of instruction. Here the community learned its duties. Here men found their separate paths to conversion, so they could better build their Zion in the wilderness, a City upon a Hill to which other men might in their turn look for instruction. As the meeting-house was the geographical and social center of the New England town, so the sermon was the central event in the meeting-house.\n\nAll these things taken together led not just to a deeper spiritual commitment but to a transformation of \"the American mind.\" The awakening of the spirit would have a direct impact not only on American Protestantism but on the political consciousness of an entire generation.\n\nThe call coming from eighteenth-century pulpits for a renewal of faith translated into a call to resist all forms of tyranny, which elicited an impulse to resist the kinds of intimidation endured under British rule. These were the provocations making their lives miserable. Each new insult\u2014whether it was the Tea Tax, the Stamp Tax, the Currency Act, the blockade around the Port of Boston, the Boston Massacre, the quartering of British troops in American homes, or the Quebec Act (a direct assault on the Protestant churches)\u2014was further reason for rebellion. The men and women of colonial America had been educated in the school of liberty by their ministers, and in the process they were being prepared to stand their ground when the time came.\n\nIt was in this spirit that Samuel Adams, often referred to as the father of the American Revolution, wrote, \"He therefore is the truest friend to the liberty of his country who tries most to promote its virtue, and who, so far as his power and influence extend, will not suffer a man to be chosen into any office of power and trust who is not a wise and virtuous man. . . . The sum of all is, if we would most truly enjoy this gift of Heaven, let us become a virtuous people.\"\n\n107\n\nBetween 1765, when the Stamp Act was passed, and 1775, when the first shots of the Revolution were fired at Lexington and Concord, a visible transformation had taken place in the American Spirit. Inspired by the call to duty, honor, and love of country, a new generation was prepared to consider the prospect of a separation from its English masters. The colonists were prepared to take up arms against the most powerful military force on earth. This would prove to be the greatest moment in American history. It was the moment when the flame of faith that had been lit during the Great Awakening burst forth, helping to shape the character and destiny of a nation.\n\nTwo and a half centuries after that great explosion of righteous anger, we are still humbled by what they accomplished. The more I have studied that fifty-year period preceding the American Revolution, the more amazed I am by the transformation that had taken place in the hearts and minds of the people. As we have seen in previous chapters, there was a lot of religious activity in early America but not a lot of evidence of the kind of life-changing faith that would be needed in the Revolution. There wasn't much passion, and there were no religious movements until the late 1730s. The colonists observed religious ceremonies and traditions, and a majority attended church services regularly. But there was no evidence of a deeper spiritual commitment.\n\nIf there was going to be a great spiritual revival, it would have to be a move of God, and that's precisely what it was. So what was the impact of the Great Awakening? How did the preaching of a handful of pastors and evangelists change the disposition of the people and prepare them for a war of independence? The following list, I believe, offers a reasonable summary of some of the ways the Great Awakening shaped the thinking of the patriots in the years prior to the Revolution.\n\n108\n\nTHE LEGACY OF THE GREAT AWAKENING\n\n1. A UNITY AND COMMUNITY AMONG THE COLONIES\n\nPrior to the Great Awakening the colonies viewed themselves as separate and independent bodies. They shared few things in common. But with the coming of the revivals, they began traveling from colony to colony, and in the process they found common ground with others who had the same religious experiences. This proved to be a powerful force in laying the foundation for a unified people and ultimately a unified nation.\n\nThis new spirit of unity was a necessary element in bringing together enough of the colonists to declare independence, to fight a war for freedom, and to form a new republic. The first steps of national unity that began in the church houses, town squares, and open fields as people gathered to hear the preaching of men such as Whitefield, Davenport, and Tennent would lead ultimately to the First Continental Congress that met in Philadelphia's Carpenters' Hall on September 5, 1774.\n\n2. A SPIRIT OF RELIGIOUS TOLERANCE\n\nPreviously, followers of one religious denomination or another would stay within the four walls of their own churches in worship, but Whitefield and other evangelists held open-air meetings that brought the people out of doors and into fellowship with those of many other traditions and doctrinal views. Some denominations saw this as a threat, and the denominational hierarchies were strongly against fanning the \"fires of revival.\"\n\nThe eighteenth-century revivals may not have turned out in just the way Jonathan Edwards had hoped, since they led to division and rancor in some places. At times the evangelists seemed to be just one more group competing for the people's religious allegiances. However, as McLoughlin points out, \"while the Awakening split the very churches its leaders set out to renew, it also secured for them a niche from which to prosecute their agenda.\" In spite of the quarrels that occurred among the clergy, for the most part the ordinary people paid little attention to all the fuss and found fellowship and friendships across denominational lines to be refreshing.\n\n109\n\n3. A MORAL AND SPIRITUAL WORLDVIEW\n\nStrong preaching about living out the Christian life through \"good works\" ignited a fire of social action, which included the beginnings of the antislavery movement in America, women's rights, and other types of civic action. This was demonstrated by the most powerful preacher of the Great Awakening, George Whitefield. Whitefield thundered that Christians were called to be \"servants of all.\" Not only did he preach it, but he put it into practice by the formation of orphanages for underprivileged children in Georgia, and his own personal aid to clergy who suffered from financial pressures. John Wesley testified at Whitefield's funeral in 1770 regarding not only his powerful preaching but also his tenderheartedness and charitable nature. Thomas S. Kidd writes:\n\nThe convictions, however, that Jesus died to save women and men of all ethnicities and classes, and the Holy Spirit empowers each believer equally, produced mixed social consequences in the first generation of American Evangelical Christianity. Evangelical beliefs worked at times to erode traditional barriers of race, class and gender. Radical evangelicals of the eighteenth century opened up unprecedented, if ultimately limited, opportunities for African Americans, Native Americans, women, the uneducated, and the poor to assert individual religions and even social authority. Early American evangelicalism also helped pioneer the American abolitionist movement that emerged more fully in the nineteenth-century North.\n\nIn fact, men such as Elisha Williams, Samuel Davies, George Whitefield, and others preached to large numbers of blacks attending their revival gatherings. At the same time, black Christians with assistance from their white brothers began building their own churches and evangelizing their own people. Two former slaves, Richard Allen and Absalom Jones, became the best-known black evangelists of the period, and they went on to found the African Methodist Episcopal denomination in Philadelphia in 1816. This \"black preaching\" resulted in the conversion of tens of thousands of slaves who believed the gospel message. At the first General Conference of Methodism, the owning or selling of slaves was recognized as sinful and worthy of dismissal from the church.\n\n110\n\nAmong the British aristocracy, however, few acts of compassion and understanding of the rights of others took place. People of a different color, language, or national origin were normally treated either as slaves or, at best, people of a lower class. There was a clear and growing disconnect between the worldview of the leadership in the motherland and the opinions of the men and women of the colonies, concerning individual rights and the dignity of human life. This divide grew larger over time, adding to the friction between the colonists and the British. For these and other reasons, many Americans began to believe that conflict was inevitable.\n\n4. A RISE IN THE NUMBER OF TOWERING LEADERS\n\nColonial preachers\u2014such as Edwards and Tennent\u2014along with Whitefield and Wesley were elevated to positions of leadership and esteem because of their great moral influence. Backed by favorable press reports, revivalists such as Whitefield enjoyed unprecedented public acclaim. In his book _Inventing the Great Awakening_ , Frank Lambert describes the masses that attended Whitefield's meeting across New England:\n\nThe biggest crowds assembled in colonial America's two largest cities, Boston and Philadelphia, each located in a revival region. On each of his three preaching tours, Whitefield made Philadelphia a center for his itinerating. From his first outdoor sermon there, he attracted progressively huger crowds, from the 6,000 who first gathered at the courthouse to hear him, to 8,000, and eventually 10,000 at the farewell sermon ending his first tour. At three subsequent Philadelphia services, Whitefield reported crowds of at least 10,000, and then during spring 1740 they swelled to 15,000 and reached the amazing number of 20,000, a figure that Franklin reported without dispute. Whitefield attracted similar crowds in Boston during the one trip he made there in fall 1740. For his farewell sermon delivered on Boston Common, he reported a gathering of 20,000, but two Boston newspapers estimated that 23,000 attended.\n\n111\n\nThis new attitude of respect and admiration for these and other distinguished men of God paved the way for other public figures, including political leaders. Many of these men were not just politicians, but men of God who would lead the fight for independence. Thomas Jefferson, John Adams, Patrick Henry, Richard Henry Lee, and George Washington were recognized as men of courage and virtue, and they commanded the respect of all Americans.\n\n5. A SPIRIT OF INDEPENDENCE\n\nA personal spirit of independence developed as people embraced the idea that each man was personally accountable to God. Corporate worship was an important custom, but every man was responsible for his own \"spiritual awakening\" apart from the church. This new feeling of accountability and independence grew into an equal, if not greater, desire for independence from British rule. This was especially true in 1769, when King George III threatened to send an Anglican bishop to oversee the American church and \"set the colonies right.\"\n\nWhen this news reached America, the colonists were furious, fearing the loss of religious freedom altogether. The great Massachusetts lawyer and statesman John Adams viewed this as one of the most provocative acts of the Crown, sparking the spirit of rebellion. He said, \"Apprehension of Episcopacy contributed . . . as much as any other cause, to arouse the attention, not only of the inquiring mind, but of the common people, and urge them to close thinking on the constitutional authority of parliament over the colonies.\" The threat of returning to a state religion enforced by royal edict that denied religious liberty was more than the colonists could bear.\n\n112\n\nBy the same token when news of the Stamp Act reached the colonies in 1765, the people felt a \"heavy cloud hanging over us, big with slavery and all its dreadful attendants.\" They looked upon the imposition of punitive taxes as one of the gravest threats to their own prosperity. They had already made the connection between civil and religious liberties; they knew that when religious liberties were threatened, the assault on civil liberties could not be far behind. And this passion for religious independence clearly carried over to governmental independence as well.\n\n6. A NEW BELIEF IN MANIFEST DESTINY\n\nThe idea of manifest destiny was based on the beliefs of many Americans that God had a divine plan for the New World, and that each person, regardless of social or economic status, was a part of God's work in bringing about a new and better world in America. Never before had the common man been empowered in such a way; never had an entire society been given the opportunity to reach for their own goals, achieve independence, and recognize their potential. The Great Awakening ignited the flames of an inclusive democracy in the hearts of millions of Americans, and it spawned the vision of the New Jerusalem\u2014a land where God had led the immigrants and prepared them for greatness.\n\nThey could see now that British rule could only inhibit the dreams in the hearts of the people; ultimately, revolution was the only answer. As William McLoughlin notes, \"Part of the American culture has been the myth that we are a 'covenanted people.' As such, God has a special interest in helping, and a special reason for punishing us. The covenant applies both to individuals and to the nation as a whole. . . . But if each does his or her part to adhere to the new rules, then God promises, according to his prophets, a glorious new day of peace, fraternity, and perfection\u2014a time in which all human needs will be met, both physical and spiritual. Thus the experience of hearing, yielding to, and experiencing this call is one of ecstatic release from the burden of guilt and fear.\"\n\n113\n\n7. A NEW APPRECIATION FOR AMERICA'S NATURAL RESOURCES\n\n\"The earth is the Lord's, and the fulness thereof\" (Psalm 24:1). This and similar texts were common in the sermons of the Great Awakening, resulting in a new appreciation for the land God had given the colonists. Like the children of Israel who had been led from bondage in Egypt to the promised land called Canaan, they had been brought through tribulation to a land of milk and honey. The settlers on the American frontier were accustomed to hard work, and they took full advantage of the land and its resources. Clearing the land, building their homes, planting and harvesting crops, and living on the abundance gave them profound respect for the environment.\n\nBut there was more, and it was something none of them had ever experienced before: the sheer awe and majesty of the North American landscape, the mountains and valleys, and the riches they contained. Even the songs of the patriots spoke of their love of nature. American artists produced images of an unspoiled landscape that enticed tens of thousands to strike out into the utopia, opening the door to the era of westward expansion. Throughout the late-eighteenth and nineteenth centuries, caravans of lumbering ox-drawn wagons crossed the prairies, from sea to shining sea, with one consuming vision: to occupy and take dominion of the new world God had given them.\n\nThe Great Awakening taught these bold travelers that God had brought them to this place: \"Then God blessed them, and God said to them, 'Be fruitful and multiply; fill the earth and subdue it; have dominion over the fish of the sea, over the birds of the air, and over every living thing that moves on the earth'\" (Genesis 1:28 NKJV). The art and literature of the day describe vast, unspoiled landscapes and mighty rivers; it was not until after the Industrial Revolution that the ingenuity of man would rival the power of God, and the landscape would suffer from neglect and abuse.\n\n114\n\n8. A DESIRE FOR PERSONAL ADVANCEMENT\n\nBreaking down the barriers of a collective worldview where individual rights were few, the new era of independence that came in the wake of the Great Awakening brought with it a desire for improvements in education, material comforts, and personal wealth. Perhaps no act of resistance exemplified this new spirit better than the Boston Tea Party of 1773. When Parliament passed the Tea Act earlier that year, they intended to give the British East India Company an unfair price advantage, which threatened the profits of American merchants.\n\nThe Americans saw clearly what was happening and objected, demanding that three ships carrying English tea be sent back to London still filled with their cargo. When the British refused to leave, colonists dressed in Native American costume and war paint boarded the vessels by night and tossed the entire shipment into Boston Harbor. Predictably the English governors took severe reprisals against the colonists, but it would not be enough. A spirit of resistance had been awakened, and things would not change for eight years until the British had been soundly defeated at Yorktown and sent home to lick their wounds.\n\nCourage, independence, self-reliance, and a willingness to stand one's ground and defend one's honor were important by-products of the Great Awakening: these were the traits that brought the colonists victory in battle and awakened in the men and women of America the belief that they could overcome any obstacles and make a better life for themselves and their children.\n\n115\n\n9. A BELIEF IN THE VALUE OF ACTION OVER DEBATE\n\nAs the revivals swept through the colonies, the volume of debate among colonial-era preachers and theologians was intense, but those who had participated in the Great Awakening had experienced for themselves the \"new birth.\" They were not interested in all the verbal gymnastics. They had placed their faith in a living, powerful, and transcendent God, and their own emotional encounters were all the justification they needed. Emotional intensity is necessary for any revolution to succeed. Emotion produces action that triumphs over debate, and that was the spirit that prevailed through the Revolution.\n\nSome historians have suggested that the new spirit of evangelicalism gave the Revolution one of its most potent ideological resources, as the new evangelical temperament shaped the political practices of the people. \"Evangelicalism,\" writes Thomas Kidd,\n\ntaught the common people who embraced it that sometimes they must take matters into their own hands, a subversive tendency that exploded during the imperial crisis. The church separations and disruptions of the revivals have been identified as a \"practice model\" which enabled the provincials to \"rehearse\"\u2014though unwittingly\u2014 . . . the arguments . . . that would reappear. . . . The evangelical revivals caused the greatest social upheaval of any movement in the colonies prior to the Revolution. This massive defiance of traditional authority must certainly have exercised some shaping influence on the Revolution.\n\n10. THE SHAPING OF NATIONAL IDENTITY\n\nAs religious, social, and economic barriers were broken down between the colonies, a new sense of identity was being shaped. It was the beginning of what we now recognize as the American Spirit, based on belief in personal liberty and individual rights. As this new attitude became more and more common, the desire for independence and personal freedom became the defining characteristic of the nation. The men and women of America no longer thought of themselves as subjects of an English king but as Americans. The Declaration of Independence spoke boldly of \"unalienable rights,\" among them the rights of \"life, liberty, and the pursuit of happiness.\" The American people did not need to be told, but the Declaration affirmed that these rights are not man-made but God-given. As theologian Richard Niebuhr has said, \"America cannot eradicate, if it would, the marks left upon its social memory, upon its institutions and habits, by an awakening to God that was simultaneous with its awakening to national self-consciousness.\"\n\n116\n\n11. A CHRISTIAN COVENANT ETHIC\n\nThe development of a Christian worldview, aroused initially during the Great Awakening, meant every believer in Jesus Christ has a duty to the community. All Christians have a holy responsibility to honor God and live according to the tenets of the Christian faith, but we are also expected to contribute to the common good. The doctrine of grace teaches that we are not the authors of our own salvation, but God has shed His grace upon us and through the sacrifice of His Son made salvation available. The doctrine of works explains that while works cannot save anyone, they are the evidence of salvation. Jesus said, \"Let your light so shine before men, that they may see your good works and glorify your Father in heaven\" (Matthew 5:16 NKJV), and the apostle Paul gave similar counsel, saying, \"For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them\" (Ephesians 2:10 NKJV). These were among the fundamental principles of both Wesley's and Whitefield's Methodism and are still very much alive in the Protestant ethic today. By the end of the period of the Great Awakening, more than 80 percent of all Americans shared a common set of beliefs regarding the Christian faith and its central role in the social order.\n\n117\n\n12. A NEW EMPHASIS ON EDUCATION\n\nBefore the Great Awakening it was rare for the common man or woman to have a formal education. Most colonists who possessed a higher education came from the elite families of New England. Following the Awakening there was an explosion in the founding of colleges and schools for the education of the general populace. Fully 106 of the first 108 colleges in America were founded as Christian institutions. By the mid-nineteenth century there were 246 colleges in this country, virtually all of them founded either by Christian denominations or by individuals motivated by their Christian convictions. Among these were Harvard College in 1636, the College of William and Mary in 1693, Yale College in 1701, and Princeton (formerly the College of New Jersey) in 1746. Jonathan Dickinson, Jonathan Edwards, Samuel Davies, and John Witherspoon (a signer of the Declaration of Independence) were among the first presidents of Princeton, and the preeminent figure of the Awakening, George Whitefield, was instrumental in the founding of the University of Pennsylvania in 1751.\n\n13. BELIEF THAT FAITH IS A PERSONAL MATTER\n\nAfter the Great Awakening there was universal belief that the Christian faith is not a collectivist doctrine but a personal matter. There is no such thing in Scripture as \"collective salvation.\" Furthermore, ministers and denominations were no longer in a position to dictate what any person could or could not believe in his or her individual understanding of the Scriptures. Such a democratic spirit would have been frowned on by the established churches prior to the Great Awakening, but the explosion of faith that took place in pastures and meadows and village greens all over the country was all the proof the Americans of that generation needed that true faith is a relationship between the individual and God.\n\n118\n\n14. AN AGE OF PUBLIC GATHERINGS AND OPEN DISSENT\n\nThe open-air sermons of the Great Awakening brought together enormous crowds of people of every sort, eager to hear what the revivalists had to say. Often these outdoor rallies would be accompanied by outbursts of emotion and enthusiasm from those who attended. Instead of being condemned, however, the people were free to express their feelings. In the years following the Awakening, similar gatherings were held in public places, but this time the people were gathering to protest the unfair policies of their British governors. When these crowds assembled to express their grievances, there were no riots\u2014 these were not vandals and looters but men and women who had learned to make their voices heard. In time, the widespread feeling that something had to be done to bring an end to British oppression prepared the way for the American Revolution.\n\nTHE CONSENT OF THE GOVERNED\n\nWhen I think about the preaching of the great evangelists like George Whitefield, I often wonder what it was that made their ministries so powerful. Whitefield was anointed to preach the Word, that's true. He had experienced a genuine religious conversion in the courtyard of Christ Church at Oxford, England, in 1735. But what was it that transformed him so thoroughly and prepared him for the amazing ministry he would have just three years later in the New World? Something like that could only come from God.\n\nEven the most secular men of the era were changed. We see the hand of God as He moved upon men such as Benjamin Franklin, a newspaper printer and publisher with a secular bent. Franklin was not a religious man, yet he began spreading the word by publishing the sermons of many of the leading preachers of the day. We might even say that the printers and publishers of the colonial era made it possible for the Great Awakening to happen.\n\n119\n\nSermons were the most popular literature in the colonies, and as a result of the publicity he received, George Whitefield became one of the best-known public figures in the country. Without that kind of exposure, it is doubtful the Great Awakening could have happened. Benjamin Franklin, with all his civic, scientific, and political clout, was the one man who had the ability to publicize the man who came to America to publicize the Savior.\n\nAs far as he was personally concerned, Benjamin Franklin was a skeptic. He was offended that so many people would go to hear Whitefield and then give him all their money, but after hearing the evangelist for himself, Franklin emptied his pockets. It is remarkable how God used men like Franklin.\n\nBut one of the greatest stories from the colonial period is the story of Franklin's rebuke of the First Continental Congress when he called upon the delegates to pause for a time of prayer. His words are preserved in the records of the Library of Congress, and they are just as poignant today as they were when first spoken. The delegates were having a hard time coming to terms on how their new government should respond to the Coercive Acts that, among other things, had imposed martial law on the colonies. The Massachusetts delegates couldn't agree with the Virginia delegates, and both Delaware and New Jersey were on the verge of leaving in a huff. When the bickering and name-calling escalated, and the hope of a resolution appeared to be vanishing, eighty-one-year-old Benjamin Franklin rose to his feet and asked for an opportunity to speak, which was immediately granted. Stepping forward to a place where he could be seen by all the delegates, he said,\n\nI've lived, Sir, a long time, and the longer I live, the more convincing Proofs I see of this Truth\u2014That God governs in the Affairs of Men. And if a sparrow cannot fall to the ground without his Notice, is it probable that an Empire can rise without his Aid? We have been assured, Sir, in the Sacred Writings, that except the Lord build the House they labor in vain who build it. I firmly believe this,\u2014and I also believe that without his concurring Aid, we shall succeed in this political Building no better than the Builders of Babel: We shall be divided by our little partial local interests; our Projects will be confounded, and we ourselves shall become a Reproach and Bye word down to future Ages.\n\n120\n\nNeedless to say, all eyes were fixed on the illustrious old man. But he wasn't finished. Seeing that the truth of his remarks had hit the mark, Franklin continued, \"I therefore beg leave to move that henceforth prayers imploring the assistance of Heaven, and its blessing on our deliberations, be held in this Assembly every morning before we proceed to business, and that one or more of the clergy of this city be requested to officiate in that service.\"\n\nAfter a very brief discussion there was quick and humble concurrence with Franklin's proposal. Several names were submitted of local clergymen who might be prevailed upon to open the deliberations with prayer. The members decided on the Reverend Jacob Duch\u00e9, who was widely known as an eloquent orator and pastor of the best-known Anglican parish, Christ Church in Philadelphia.\n\nBut as further evidence of the spirit of compromise that had suddenly arisen among the delegates, it is interesting to note that it was Sam Adams, a staunch Congregationalist, who first recommended Duch\u00e9, despite the feeling expressed by several delegates that the Anglicans, who were strongly identified with the Church of England, were too closely connected to the British Crown. The Revolution was still fresh in the minds of all Americans, but Duch\u00e9 was unanimously chosen, and he made the trip from his church each morning to open every session of the First Continental Congress with prayer.\n\nBenjamin Franklin had a reputation\u2014and deservedly so\u2014as a rogue. Nevertheless, God used him in a remarkable way. Isn't it amazing that God can use every sort of person to accomplish His will? But, here again, such a spirit of compromise and cooperation could only have happened as a consequence of the changes brought about by the Great Awakening. Before going to hear Whitefield preach, it is doubtful that Franklin would ever have spoken in such a way. But the writer of Proverbs assures us, \"The king's heart is in the hand of the Lord, like the rivers of water; He turns it wherever He wishes\" (Proverbs 21:1 NKJV). When Franklin's heart was turned, he became a vocal advocate for Christian virtue.\n\n121\n\nThe Bible says, \"where the Spirit of the Lord is, there is liberty\" (2 Corinthians 3:17), and the people of the revolutionary era had learned that lesson very well. Along with the importance of personal holiness, the evangelists had been preaching a gospel of transformation and liberation. But considering the price our forefathers paid for the rights and privileges we enjoy today, shouldn't we be asking ourselves, \"Who are the American people today? Are we living up to the high standards of the patriots who risked their lives, their fortunes, and their sacred honor for the cause of liberty? And are we prepared to stand our ground when we're confronted by men and women promoting some very dangerous ideas and beliefs?\"\n\nThere is no question that we owe our freedom to the patriots of 1776, but I have to wonder if we still have that kind of courage. The men who composed the nation's founding documents belonged to one of the best-informed, best-educated, and most conscientious generations in history. Historian Barbara Tuchman refers to the Founding Fathers as \"the most remarkable generation of public men in the history of the United States or perhaps of any other nation.\" Who could doubt it? These men had read Plato, Cicero, Locke, Montesquieu, and all the great classical scholars. They understood that every democracy eventually descends into mob rule. That's what happened in France in 1789, just thirteen years after the American Revolution. The cry of the French Jacobins for \" _libert\u00e9, \u00e9galit\u00e9, fraternit\u00e9_ \" led not to liberty and justice for all but to anarchy and mob violence, which soon became the blood bath known as the \"Reign of Terror.\"\n\nThe American Founders believed that individual liberty was a sacred right, which is why they enshrined in the Declaration of Independence the affirmation that we are \"endowed by our Creator with certain unalienable rights, that among these are life, liberty, and the pursuit of happiness.\" But they also understood that the best form of government is a republic in which the citizens empower their elected representatives with authority to manage the affairs of state. As citizens of a democratic republic, we have the duty to elect honorable men and women to serve in public office, but it is We the People, not the government, who are the ultimate power and authority.\n\n122\n\nIn America that authority is spelled out in clear and unambiguous terms in the Constitution of the United States, which specifies that elected representatives are subject to \"the consent of the governed.\" No doubt this is why so many people in Washington today spend so much time trying to avoid living by those statutes. The progressives in Washington want us to believe the patriots gave us a \"living, breathing Constitution\"\u2014a document that provides some general guidelines for government but is, in fact, simply a set of suggestions: not binding law. But the Founders did not leave them that option.\n\nJames Madison, the principal author of the Constitution, makes it clear in Federalist 51, which was written to clarify the intentions of the Founders, that the Constitution is the nation's best guarantee of an \"ordered liberty.\" The efficiency of the document is stunning\u2014a mere 4,500 words without amendments\u2014yet the Constitution provides time-tested guidance for administering the political affairs of a nation of 300 million people. The reason it has survived so long with so few amendments is because it is so logical and straightforward, based on the Framers' hard-won understanding of law, liberty, and justice.\n\nThe Constitution _is_ the law. It is the foundation of our government. \"But what is government itself,\" Madison asks, \"but the greatest reflection of human nature?\" He continues,\n\nIf men were angels, no government would be necessary. If angels were to govern men, neither external nor internal controls on government would be necessary. In framing a government which is to be administered by men over men, the great difficulty lies in this: you must first enable the government to control the governed; and in the next place oblige it to control itself.\n\n123\n\nThe government of a free people, the Framers insisted, can never be arbitrary. To be a credible instrument of such a government, the Constitution cannot be altered, ignored, or amended haphazardly but must be respected by all men at all times. This is what the Founders believed. This is what they gave us.\n\nIn his Farewell Address, delivered in 1796, just twenty years after the signing of the Declaration of Independence, President George Washington expressed the hope that future generations of Americans would never forget the price his generation had paid for the liberties we enjoy. It was his hope that the spirit of unity that emerged during the Revolution might endure. It was his fervent prayer, he says, \"that the free Constitution . . . may be sacredly maintained; that its administration in every department may be stamped with wisdom and virtue.\" And he affirmed that the Constitution, as written and adopted by the several states, demands full confidence and support from every American:\n\nRespect for its authority, compliance with its laws, acquiescence in its measures, are duties enjoined by the fundamental maxims of true liberty. The basis of our political systems is the right of the people to make and to alter their constitutions of government. But the Constitution which at any time exists, till changed by an explicit and authentic act of the whole people, is sacredly obligatory upon all.\n\nTo understand how the Founders' understanding of ordered liberty has shaped the nation, we need only consider America's record of success over the past 235 years. Through wars, natural disasters, depressions, recessions, and emergencies of every kind, the American Spirit has prevailed. Faith in God has remained constant in the hearts of millions despite the efforts of humanists and progressives to wipe it out. The faith of our fathers may not be as widespread or as vigorous as it was during the founding era, but it survives in millions of homes and churches, and more and more today in civic and political life.\n\n124\n\nA FATEFUL TRANSFORMATION\n\nWe are still \"one nation under God.\" As author and Heritage Foundation researcher Matthew Spalding points out, \"We still hold these truths to be self-evident, that all men are created equal, endowed by their Creator with certain unalienable rights. In a world of moral confusion, and of arbitrary and unlimited government, the American founding is our best access to permanent truths and our best ground from which to launch a radical questioning of the whole foundation of the progressive project.\" Contrary to the claims of progressives that conservatives in this country want to drag America back to the Dark Ages, ignoring the progress of the last two hundred years, Spalding writes:\n\nRenewing America's principles doesn't mean going back to the eighteenth century, or some other time for that matter. Think of principles as the unchanging standards that inform changing experiences. The question is not, \"What Would the Founders Do?\" but what will we do as we go forward toward an unknowable future with these fixed principles as our trustworthy guides? It is not about looking back to the past, but rather looking down at our roots in order to look up to our highest ideals.\n\nWhen you consider all of these things, you soon realize many if not all of these elements are present and active in American society today. Furthermore, I am convinced we are on the verge of a second American revolution\u2014not a revolution of guns and bullets but a revolution in the voting booth and in the major forums of public opinion. It is a revolution of principle, demanding accountability from our public servants and everyone else engaged in government at any level.\n\n125\n\nI am deeply committed to getting people out to vote. The fact that I am a pastor of a large suburban church doesn't disqualify me from speaking resolutely about the need for character and moral discipline in our elected officials. It is not a matter of voting for Republicans or Democrats. At least 65 million evangelical Christians in this country are eligible to vote, but in a typical presidential election, barely half of them are registered. And, sadly, not all who are registered actually show up at the polls. Apathy is not what the Founders had in mind.\n\nIn the eighteenth century the sparks of revolution were fanned into flame by what became known as the Black Robe Regiment. These were the pastors and revivalists who taught the people the importance of personal liberty and challenged them to obey the call of conscience. Today we are seeing a new wave of pastors who are boldly speaking out about the abuses of power by our public officials and the importance of restoring honor in all areas of life. We are going to see a new demand for accountability in our public servants, above and beyond what this nation witnessed in the eighteenth century. I believe this is one of the things we learned from the 2010 midterm elections.\n\nThe Declaration of Independence and the Constitution were given to us by the Founding Fathers to set the standards by which we administer the government. But they are also documents of accountability. As our public officials (and particularly the courts) have chipped away at them, doing their best to weaken the Constitution\u2014reinterpreting it in ways the Founders never envisioned, and saying that it is a \"living, breathing document\" that changes with the times\u2014they are, in reality, destroying their own credibility with the American people.\n\nOne of the most important things we can do is to follow through as the Founders did, not only by changing our representatives in the legislatures but by holding all leaders accountable and insisting that they serve as our representatives and not as self-styled dictators. It is not just that we are in the midst of a popular revolution today; these things have all happened before. Thanks to the lessons of history, we have copious examples of how such things have transpired in the past, which tell us where the nation may be headed. But this we know: the action we take today will affect the lives of our children and grandchildren for decades to come.\n\n126\n\nWhat this book proposes is a revolution that is far superior to a violent rebellion. It is a revolution of faith and ideas, a new commitment to a higher cause. It is a revolution that will fulfill the charge our forefathers gave us during the founding era. It means knowing who we are and what we are all about. A tremendous hunger for restoration of accountability exists in this country, and the popular reaction to the progressive agenda in Washington may be a blessing. Excessive control and burdensome taxation are driving the people of this country back to basics and, hopefully, leading to a renewal and resurgence of the American Spirit.\n\nI occasionally come across good people who are quick to tell me they are not political. I have friends who simply neglected to vote or take any interest in politics until they woke up one day and realized that the country is headed in the wrong direction. They suddenly understood what the promise of \"hope and change\" really meant. Now those same people cannot get enough information. They are surfing the Internet, listening to talk radio, or watching cable TV most of the time, and they are becoming engaged in the battle for the soul of America.\n\nPeople need a creed and a cause, and millions of patriotic Americans are finding their voices. Thomas Jefferson, who was the principal author of the Declaration of Independence, wrote that governments are established to secure the rights of the people. But when government fails to protect our rights and liberties, the people have the right to demand change. They can abolish the government entirely, or they can become personally involved and change it for the better. Today we have some big decisions to make about the future of the republic and about the authentic Source of our rights and liberties. But the good news is, we are listening, we are tuned in, and we are not going home until we get what we came for.\n127\n\n[_**Five**_  \nFAITH IN THE TWENTY FIRST CENTURY](Lee_9780849949463_epub_c5_r1.html#d6e7)\n\nWhen did America become a secular nation? When did we decide \"the faith of our fathers\" is no longer a vital concern? There's no question that this country is more secular today than it was during the founding era, and no one would deny that anti-Christian words and deeds are more common in the twenty-first century than ever before. But have we really given up on this nation's remarkable heritage of faith? Are we becoming the kind of country that celebrity atheists like Richard Dawkins, Christopher Hitchens, Bill Maher, and Michael Newdow would like us to be?\n\nIt can be amusing to read some of the inflammatory rhetoric of secular liberals, claiming this country isn't just secular but was never a Christian nation in the first place. Anyone with even a superficial knowledge of history knows that's false. Nevertheless, in the days following President Obama's April 6, press conference in Ankara, Turkey, during which he stated, \"We do not consider ourselves a Christian nation,\" the policy director of a left-wing think tank in Washington, DC, called out the cheering section in an article for the liberal web-site Salon.com titled \"America Is Not a Christian Nation.\" The writer, Michael Lind, was seemingly ecstatic that an American president had finally said what Lind always believed.\n\n128\n\nWhile admitting that George Washington and other Founders used religious language from time to time, he questioned their motives. He recited the famous lines from Washington's Farewell Address, \"Of all the dispositions and habits, which lead to political prosperity, Religion and Morality are indispensable supports,\" and even included the less-often-cited portion of that quote: \"The mere Politician, equally with the pious man, ought to respect and to cherish them.\" But then he claimed such statements were simply a cheap way of enforcing good behavior. In other words, the Father of His Country and the man whom Parson Weems tells us \"could not tell a lie\" was simply exploiting religious language for a secular purpose. Lind writes, \"In Washington's day, it may have been reasonable for the elite to worry that only fear of hellfire kept the masses from running amok, but in the twenty-first century it is clear that democracy as a form of government does not require citizens who believe in supernatural religion. . . . Most of the world's stable democracies are in Europe, where the population is largely post-Christian and secular, and in East Asian countries like Japan where the 'Judeo-Christian tradition' has never been part of the majority culture.\"\n\nIt is hard to fathom why anyone would hold up Western Europe as a model of anything, especially a stable democracy. There isn't much stability in Europe these days, and many of those secular cultures are on the verge of social and economic collapse. In a 2006 book called _Menace in Europe_ , Claire Berlinski offers a chilling portrait of a continent in the midst of profound spiritual and political crisis. Wracked by an unending series of labor strikes, bomb threats, and terrorist attacks, along with declining birthrates, out-of-control taxes, and crippling levels of immigration from Third World countries, Western Europe is on the ropes. Add to this fifty years of destructive socialist programs and the rising threats of Islamization and neo-Nazism, and you soon discover that Europe is hardly a model for America. And these are just the problems that make headlines; there are many others.\n\n129\n\nWeekly church attendance in Europe is less than 5 percent today, compared to more than 40 percent in America. So what does that tell you? Berlinski reports, \"A poll conducted in 2002 found that while 61 percent of Americans had hope for the future, only 42 percent of the residents of the United Kingdom shared it. Only 29 percent of the French reported feeling hope, and only 15 percent of the Germans.\" These statistics, she says, suggest that \"without some transcendental common belief, hopelessness is a universal condition.\" Berlinski concludes, \"I do not believe it is an accident that Americans are both more religious and more hopeful than Europeans, and more apt, as well, to believe that their country stands for something greater and more noble than themselves.\"\n\nI couldn't agree more, and that's why the evangelistic ministry I lead today is called There's Hope. Since our first television broadcast in 1984, There's Hope Ministries has extended the hand of fellowship to millions around the world, supplying food, clean water, and other much-needed supplies to suffering people in Africa, and offering the unconditional hope of God's love to millions more in this country and abroad who might otherwise never hear the gospel message.\n\nHopelessness is always tragic, and the people of Western Europe have turned away from the Author of hope to pursue the empty promises of secularism. But the glories of secularism are greatly overrated. Columnist and author Don Feder, who reported on Berlinski's book, concurs with her assessment: \"No one ever founded a republic, or freed slaves, or created a great work of art, or wrote a symphony, or established a charity or a university based on a secular worldview.\" All those things happened here in America, a Christian nation, because faith in God stimulates hope, compassion, and mercy. \"Faith, hope, love,\" the Bible says in 1 Corinthians 13:13 NKJV. They belong together.\n\nBut ever since the French Enlightenment, secular liberals have been proclaiming the death of God. The German nihilist Friedrich Nietzsche famously declared, \"God is dead, and we have killed him!\" Religion is dead, the skeptics roared. The shackles of religion have been forever broken, and science has won the day! But there's just one problem: Nietszche is dead, the Soviet Union is dead, but God is very much alive. Our Judeo-Christian heritage is untarnished and those secular liberal countries are all in serious trouble\u2014morally, culturally, and economically\u2014while the Christian faith grows stronger every day, not just in America but around the world.\n\n130\n\nAN UNFAILING PROMISE\n\nNo matter what country it is\u2014whether it is America or the former Soviet Union\u2014cultural foundations come from what we believe, and more to the point, values and morals are determined by our religious beliefs. The historian Russell Kirk put it this way: \"Civilization grows out of religion: the morals, the politics, the economics, the literature, and the arts of any people all have a religious origin. Every people, no matter how savage or how civilized, have some form of religion: that is, some form of belief in a great supernatural power that influences human destinies.\"\n\nEven large and powerful nations such as the former Soviet Union, which tried to deny the existence of God and eradicate any memory of the Christian faith, soon discovered they could not eliminate the need for an informing belief in God. They could not erase the supernatural from the minds of the people. Communism attempted to replace Christian theology with the doctrine of \"dialectical materialism,\" which was at best a caricature of religion. But that man-made creed failed to restrain the darker impulses of the citizens and gave them no higher calling, no moral purpose, and no altruistic values to which to aspire.\n\nAs a result, the Soviet Empire collapsed, overcome by greed, duplicity, suspicion, and sloth, which were the direct by-products of that wicked philosophy. When the iron curtain came down and the Berlin Wall was crushed to rubble by the men and women who had been prisoners of that system, the people immediately cried out for Bibles. They begged for preachers and teachers to come and help them rebuild their churches. To the age-old question, \"Can man live without God?\" the answer of the men and women of Eastern Europe was a resounding _no!_\n\n131\n\nIn September 1994, shortly after the fall of the iron curtain, I went to Moscow to meet the Russian people and preach with Dr. Jerry Falwell in Moscow's Olympic Stadium. It was an amazing opportunity to share the Word of God with people who had been under communist oppression for more than seventy years.\n\nThe Moscow Ballet Theatre was performing a new presentation on the life of Christ, and our hosts thought it would be a nice touch to have a couple of Christian ministers come in and tell the audience the story of Jesus as part of the program. The ballet was in two acts, the early life of Jesus and then the ministry years, leading up to the crucifixion. So after the dancers had performed the first half of the story, I explored the question \"Who Is Jesus?\" Then after the dance company performed the second half, which ended with the resurrection, Dr. Falwell explained \"How You Can Know Jesus as Your Savior.\"\n\nSeveral of Russia's top officials were there. The KGB was there, of course, but everyone was curious about the Christian faith. We had already spoken to the leaders of the Russian Orthodox Church, and they gave us their blessing. They also told us we were free to say whatever we liked\u2014they wanted the Russian people to know Jesus. They didn't know very much about Jerry Falwell or me before we got there, but apparently someone had recommended us. They just knew we were American pastors, and they had been told our ministries were dedicated to telling people around the world about Christ.\n\nThe most surprising part was when Dr. Falwell finished his sermon and asked for all those who would like to know Jesus to stand up\u2014no one moved. I didn't know what to make of that at first because there was no way to witness that performance and hear the testimony of Scripture and not be moved to tears. Dr. Falwell just stood there with his arms extended, but nobody moved. Seated all across the front row were the leaders of the Orthodox Church\u2014the patriarch along with several bishops, priests, and prelates, all in their elegant black robes and head coverings.\n\n132\n\nWe stood there for several minutes, and then, way back in the auditorium, slowly one elderly woman stood up\u2014we could barely see her over the heads of the crowd. The people all looked around to see if anyone was standing\u2014they were watching each other to see who would go first\u2014and when they saw the older lady stand, others stood up, first a few, then a few more, and then thousands all across that massive building. It looked like the morning tide rolling in, and maybe that's what it was. It was a moving sight.\n\nWhen the commotion stopped and thousands were standing, we asked them to pray with us, and we led them in the prayer of salvation. Then, as men and women all across the building were standing, one of the officials of the Orthodox church walked up to the microphone and told the audience of more than ten thousand men and women that the things we had been telling them were all true, and he hoped all the people would believe what we said and accept Jesus Christ as their Savior.\n\nAt that point many who had been hesitant to get up suddenly leaped to their feet and indicated they wanted to know Jesus too. Before all was said and done, more than five thousand men and women had prayed for salvation. Fortunately, we had brought thousands of Bibles in the Russian language with us, and the organizers had ordered thousands of copies of my book, _The Unfailing Promise_ , in Russian. All of them were offered to the people as they left that evening, and the Russians literally snapped them up.\n\nHere was an entire culture that had tried to eradicate any mention of Jesus or the Christian faith, but it didn't work. It couldn't work because, as Saint Augustine said, there is a Christ-shaped void in every human heart that can only be filled by Christ Himself, the One who is the genuine object of our faith. You can try to fill that void with every kind of amusement, philosophy, or political or intellectual substitute you can think of, but you will never be satisfied by anything but the presence of Jesus Christ Himself. Any world system that denies the deep-down hunger for the presence of God in our lives is destined to fail.\n\n133\n\nThe Russian people had been evangelized a thousand years ago, during the Byzantine Empire, and Russia remained a devout Christian country until the Bolsheviks came to power in 1917 and made Christianity a crime against the state. The people's religious beliefs had been stolen from them, but there was residual hunger for the Word of God. I think they always knew that life under the communist system was wrong. It was utterly immoral. Life was much too hard, and the state that claimed to be their god could not save them.\n\nThe new socialist order that began with a wave of excitement and promises of material abundance and well-being quickly turned into a nightmare of villainy, greed, ignorance, and starvation. But through all of that, the hope of a Christian revival never completely disappeared: there was something in the collective memory or maybe something in their DNA. They knew there had to be a better way, and I believe they were holding onto the hope of a restoration of the Christian faith.\n\nA KEEPSAKE TO REMEMBER\n\nFormer president George H. W. Bush was vice president when Leonid Brezhnev died in 1982. Brezhnev had been general secretary of the Communist Party in the Soviet Union. For some reason President Reagan could not attend the funeral himself, so the vice president went over as the American representative. When he returned from Russia and spoke about what took place at the funeral, he said he looked on as Brezhnev's mother bent over and kissed her lifeless son. What surprised him, however, was when she reached out and made the sign of the cross on his forehead. Even though the state was officially atheistic and even though her son had been a promoter of the communist dogma for most of his life, the flame of Christianity had never died. Beneath all the trappings of power and politics, the embers of faith were still alive, and they are very much alive in Russia today.\n\n134\n\nOne afternoon my wife, Judy, and I left the hotel with our guide and took a walk around Red Square, which is still the ceremonial heart of Russia. It is the place where soldiers, tanks, and rockets are paraded every year to remind the world of Russia's military strength. We could not help being impressed by the Kremlin's colorful domes and some of the buildings surrounding the square, but the overall effect was depressing\u2014everything else we saw was very gray and lifeless.\n\nAt one point I asked our guide, \"Why is everything so drab? The cars, the people, the buildings are all so plain and dull.\"\n\nHe said, \"Sir, it's because we don't want to be noticed. Nobody stands out in this country. Nobody wants to be different from all the rest.\"\n\nWhen he said that, I realized what had happened. It was because of the loss of freedom under the Communists. Under socialism the state always comes first, and the individual comes last. The Soviets stripped away the freedom, the pride, and the sense of individuality from the people in order to build the power of the state. If the American people had remained under the thumb of King George III during the eighteenth century, we could well have been in a similar situation today. Life without Christ is futile, and without personal liberty, freedom of conscience, and all the rights and privileges the Founders insisted upon, the American story could have had a very different ending.\n\nWithout liberty life loses its appeal, and everything fades into boring sameness. The Bible says, \"Now the Lord is the Spirit; and where the Spirit of the Lord is, there is liberty\" (2 Corinthians 3:17 NKJV). No philosophy, no ideology, no collection of beliefs or principles can overcome the feelings of emptiness when the love of liberty that comes from a resilient faith in Jesus Christ is missing. Life without God is going to be empty and dull when there is no greater purpose and no hope of eternity, and any society that attempts to eradicate the knowledge of God will inevitably perish.\n\nThe Russian people must have instinctively known this, and that is why Jerry and I were invited to come to Moscow. After the concert we were asked to visit the schools and speak to the students and teachers about the Christian faith. I did not have notes or prepared remarks from which to speak. Somewhere along the way I had learned there are still leper colonies in Russia, so I decided to tell the young people the Old Testament story of Naaman the leper.\n\nWhen the prophet Elisha told Naaman to go and bathe in the River Jordan to be healed of his disease, the great Syrian general was outraged. Bathing in that dirty river was beneath his dignity, he said, but a servant persuaded him to go ahead and do it anyway. Sure enough, when he obeyed the prophet's instructions and took that \"leap of faith,\" he was miraculously healed.\n\nWith the help of an interpreter, I told them that story, and when I finished, I asked for those who wanted to know Jesus as their Lord and Savior to come to the front of the room. They just streamed forward. They were thrilled to learn that they could be children of God and know Jesus as their personal Savior. But as they were coming forward, one little girl walked over to speak to me, and she held out a little icon with the image of Christ embossed on it. When she handed it to me, the interpreter told me, \"She said, 'I want to give this to you. I don't need it any longer.'\"\n\nI asked her name, and she said, \"My name is Tanya.\" I thought that was especially nice because Judy and I have a daughter named Tonya.\n\nOne of the local pastors who had been standing with us also spoke English, so I asked him to explain what this meant. I said, \"I'm grateful and touched by this little girl's generosity. It's a beautiful ornament, and I'm honored to receive it. But why is Tanya giving me such a lovely keepsake?\"\n\nHe said, \"Dr. Lee, this is a special gift from her family. This icon has been passed down from mother to child in her family for many, many years. What she just said to you is, 'I don't need this picture of Jesus anymore because now I have Him here in my heart.'\"\n\n136\n\nThe icon is a beautiful reminder of how Christ changes lives, and I have kept it on the desk in my study ever since. Every now and then when I need a little encouragement, I will pick it up and think about what little Tanya said to me that day.\n\nCHOOSE THIS DAY . . .\n\nWhen the president spoke to the Muslim nations during his speech in Turkey, he told them the United States no longer considers itself a Christian nation. I doubt if anyone really believed that statement, regardless what skeptics may say. But even so, those words were a tacit acknowledgment of the fact that we were a Christian nation at one time. As we have seen throughout these pages, it was the dedication of the Puritans in New England and men such as the Reverened Robert Hunt at Jamestown who impressed the indelible stamp of faith upon this nation. From the first settlements in Virginia through the Great Awakening and, ultimately, through the American Revolution, the hand of God was upon the Founders, guiding them every step of the way. Since the beginning, faith has been the secret of America's success.\n\nIf we expect to remain successful and prosperous as a nation, we will never do it by following the socialist model or the European model, for that matter. And we cannot do it by trading our form of republican democracy for the social engineering schemes we have been offered by the progressives in Washington, DC. If we want to continue to enjoy freedom and opportunity and a culture of creativity, compassion, and mutual respect, then we need to stick to our beliefs and the values the Founders built into the republic.\n\nWe've had it very good in this country for most of the past 250 years\u2014we might even say 400 years if we trace it back to the Puritan era\u2014and the reason we have had it so good is because we were\u2014and still are in many ways\u2014a Christian nation. There are a lot of detractors, and the long list of books by Dawkins, Hitchens, and all the others will keep coming, proclaiming a virulent form of atheism. But that won't change a thing. Men and women just like them have been attacking Christian beliefs for centuries, trying to undermine the faith of millions. But they are just \"whistling past the graveyard\"\u2014hoping there is no God because if He really exists, they know they are in big trouble.\n\n137\n\nAll the carping about Christianity by skeptics on TV and the Internet can't hurt us. Faith is not determined by public opinion polls or left-wing propaganda; it is based on the unfailing promises of Scripture. Furthermore, every believer has additional confirmation through the indwelling presence of the Holy Spirit. We have endured a decades-long assault on traditional moral values from the cultural elites, and while this may have contributed to the coarseness of the popular culture, it hasn't made the gospel any less relevant or important.\n\nBut there is something we ought to consider: If great empires of the past were defeated by the moral decadence, loss of vision, deterioration in educational standards, and failure to remember the lessons of history, what's to keep this country from collapsing in the same way? As we saw in the first chapter, marriage and family are in serious trouble today. Children in many homes are growing up on their own without responsible adult supervision. Church attendance is down in some places, and many of our young pastors aren't sure how to respond.\n\nWherever secular liberals are in charge of the schools, we see a level of disdain for traditional moral values that is unprecedented in our history. There are signs of change in many places today, but this level of social fragmentation is dangerous anywhere, and that is one of the reasons I am encouraging Christians to become politically active. The only way America can remain strong and free is to restore the foundations of the republic while there is time; we need to take aggressive action in our communities and at all levels of government while so many Americans are mobilized and ready to respond.\n\n138\n\nVladimir Lenin, Joseph Stalin, and other leaders of the Soviet Empire tried for more than seventy years to stamp out the traditional religious values of the Russian people. They used intimidation, mass murder, indoctrination, and every threat imaginable. They even tried exploiting religious imagery, using symbols of communism as emblems of devotion, but it did not work. Today Christianity is alive and growing all across Eastern Europe. Communism may still be the reigning orthodoxy in China, but as author and former _Time_ magazine correspondent David Aikman writes in his compelling book, _Jesus in_ _Beijing_ , Christianity is thriving in that country as well. Despite the persecutions being waged against the house-church movement, the Christian gospel is the hottest thing going in China today.\n\nThe irony of all this for secular liberals is that the United States, with its deep religious roots, also happens to be the most technologically sophisticated society in history. Liberals like to characterize us as the \"barefoot and pregnant brigade,\" clinging to our guns and religion. But in all areas of innovation and technology, Christians were there first. The nation that sent men to the moon and created the transistor and the microchip, along with all the marvels of modern mass communications, also happens to be the most Christian nation on the planet. Christian and conservative books top the bestseller lists. Christian bookstores, Christian radio and television stations, and religious programming are everywhere. You can't miss them. No wonder so many atheists are fit to be tied!\n\nOn the other hand, as Don Feder reminds us, \"the worst horrors of the modern era were perpetrated by godless political creeds. The death toll from sectarian conflict over the ages is dwarfed by ideological violence, from the Jacobinism of revolutionary France to the charnel houses of communism and fascism.\" In Proverbs, God warns, \"All they that hate me love death\" (8:36). Utopian schemes of every stripe always end up becoming totalitarian nightmares, merciless to those who refuse to bow to the dictates of ideology.\n\nIt was this understanding of the link between the rise of absolute power and the loss of personal liberty that led the first chief justice of the United States Supreme Court, John Jay, to write:\n\n139\n\nAlmost all nations have peace or war at the will and pleasure of rulers whom they do not elect, and who are not always wise or virtuous. Providence has given to our people the choice of their rulers, and it is the duty, as well as the privilege and interest, of our Christian nation to select and prefer Christians for their rulers.\n\nThis is not to say that Christian leaders are perfect or always blameless: it was, after all, the warfare between Catholics and Protestants that caused the Puritans to seek shelter in this country in the first place. But nothing in history comes close to the atrocities of the Holocaust, the Soviet gulags, or the Chinese Cultural Revolution. The dream of state socialism has been one of the most persistent social and economic experiments in history, and it is responsible for the deaths of more than 128 million men, women, and children in the twentieth century alone, as well as almost 30 million in the wars and rebellions it provoked. Submission to utopian ideologies has led to the greatest disasters in human history.\n\nFAITH IN ACTION\n\nThe Founding Fathers knew their history; they knew that no nation could survive unless it was built on a sound moral foundation, and they believed that the Christian faith was the best way to ensure the health and longevity of the republic. Some liberal scholars will admit that the Founders were religious, but they are quick to say that the Founders established a secular government based on the doctrine of \"separation of church and state.\" But that's false. All the colonies had established churches at one time, but the Founders had witnessed the dark side of establishment in England\u2014centuries of religion-inspired bloodshed. That wasn't what any of them wanted.\n\n140\n\nTheir idea of \"separation\" did not mean, however, that religion should have no influence in government. It was our second president, John Adams, a Unitarian, who said, \"Statesmen . . . may plan and speculate for Liberty, but it is Religion and Morality alone, which can establish the Principles upon which Freedom can securely stand.\" Their concern was not that religion should be disestablished but that no single denomination should be favored. Consequently each colony separated church and state by ending tax-funded subsidies to the churches. In 1833, John Adams's home state of Massachusetts became the last to disestablish the state church, the Congregationalist denomination, which had been tax supported since the founding.\n\nYou can't read far into the lives of any of the signers of the Declaration of Independence without discovering that these were deeply religious men who believed that religion and morality were essential for good government. This was precisely what George Washington meant when he referred to them as \"indispensable supports.\" The journals of our first president leave no doubt he was a man of sincere Christian convictions. In the book _Sacred Fire_ , Peter Lillback gives powerful examples of the depth and richness of George Washington's personal beliefs.\n\nAmerica needed an awakening in 1776, and that is why the pastors, teachers, and evangelists of the Great Awakening had such a tremendous impact on the nation. One pastor who was deeply disappointed by conditions in New England in the years just prior to the Great Awakening worried about the large number of legal disputes and courthouse battles taking place in that part of the country. The Reverend John Cleaveland described it as \"a want of brotherly love.\" There were disputes over the use of the common lands, and everyone was up in arms over the kinds of currency to be used in trading between the colonies. But rather than resolving the disputes amicably, Cleaveland complained, everything ended up in court.\n\nThe remarkable thing was not that the colonies were struggling with such questions but that as soon as the first signs of revival began to appear, there was \"a tempering of the fierce social, economic, and political antagonisms that had racked the colonies since the beginning of the century.\" For many who witnessed these changes, the surprising thing about the revivals of the 1730s and '40s was the restoration of social and civic harmony. In 1741, the same year he delivered his most famous sermon, Jonathan Edwards said he was delighted to see that the people of New England had ceased their usual quarrels, and there was a greater measure of peace and unity than that he had seen in the previous thirty years.\n\n141\n\nAs the troubles with England intensified, that sense of unity helped foster resistance to British policy. Eventually it would be the pastors known as the Black Robe Regiment who helped mobilize their congregations for the Revolution, and many of these men went on to serve as chaplains in the army. Despite his disappointment with the changes taking place in colonial society, Cleaveland had been a revival leader during the mid-1760s. He wrote a series of articles for the _Essex Gazette_ between 1768 and 1775, decrying the actions of king and Parliament that were pushing the colonists toward armed rebellion. Regarding the Townshend Acts, he said it is the \"Birth Right of Englishmen to be free.\" The Townshend duties were unjust, he said, and he asked, \"wherein . . . we differ from the Slaves\" if Parliament can levy taxes without the Americans' consent?\n\nAs the prospect of war grew nearer, Cleaveland's tone became more insistent, calling for boycotts of British goods. He urged New Englanders to pray that God would \"maintain our Rights and Privileges, civil and religious; and above all Things to make us a holy, a truly virtuous People.\" Then, putting teeth to his request, he called for a boycott of British merchants who refused to support the actions of the patriots, and he called for the names of colonial merchants who resisted the boycotts to be published.\n\nWhen news of the battle at Lexington and Concord reached him at his home in Ipswich, Cleaveland lost all patience with the British Parliament and penned his strongest diatribe yet, saying, \"Great Britain adieu! no longer shall we honour you as our mother. . . . King George the third adieu! no more shall we cry to you for protection! no more shall we bleed in defence of your person\u2014you breach of covenant!\" Then to the British general Thomas Gage, that \"profane, wicked\u2014monster of falsehood and perfidy,\" he said, \"The God of glory is on our side and will fight for us.\" Days later Cleaveland put his strong faith into action as he and his four sons enlisted as privates in the New England militia.\n\nThe spirit that had arisen in men such as John Cleaveland was not ideology but righteous indignation. They tried everything in their power to reason with the English governors and the Parliament in London, but they were met with one insult after another. Perhaps no act of Parliament better illustrates the growing tensions than the Declaratory Act of 1766, which was enacted shortly after the repeal of the Stamp Tax. In it the English lawmakers declared that regardless of what the Americans might say or do, the colonists were subjects of the Crown and were obliged to obey any and all laws of Parliament. It was, in effect, a slap in the face and one of the most unforgivable provocations for the coming war.\n\n142\n\nWHERE THERE IS LIBERTY\n\nThe Black Robe Regiment played an important role in recruiting members of their congregations for the patriot cause, and one of my favorite examples is the story of the Reverend Peter Muhlenberg, who was pastor of an Anglican congregation in Woodstock, Virginia. As the youngest son of a family of German immigrants, Muhlenberg had been cautious about criticizing King George III, but when it became apparent that war could not be avoided, he was more than prepared to respond.\n\nHis sermon on Sunday, January 21, 1776, was from Ecclesiastes 3, which begins, \"To everything there is a season, a time for every purpose under heaven: a time to be born, and a time to die; a time to plant, and a time to pluck what is planted\" (vv. 1\u20132 NKJV). He read through the passage eight times, and when he came to the eighth verse, which says, \"a time of war, and a time of peace,\" he laid aside his text, turned to the congregation, and said, \"and this is the time of war.\" At that point Muhlenberg untied his clerical robe to reveal the uniform of a colonel in the Continental Army. At the conclusion of the service, the men of the church turned to kiss their wives and children; then they all marched outside to enlist as the sound of the regimental drums echoed through the church.\n\n143\n\nMuhlenberg's nephew reported later that 162 men signed up for military service that day, and the following day the minister took command of a company of 300 men from the surrounding county. Muhlenberg, who had been recruited by George Washington, led the Eighth Virginia Infantry throughout the war and eventually retired as a major general of the Continental Army.\n\nWhen King George refused every offer of reconciliation from the Continental Congress, the Americans felt they had no choice but to take up arms. The king had rejected the Olive Branch Petition offered by Congress in 1775, and he, instead, issued a proclamation declaring the colonies in open rebellion against the Crown. At that point King George ordered all American ports to be closed as he had done to Boston the previous year. This time the king's decree would be enforced by the British navy and marines, who were already laying siege to several towns along the coast.\n\nWhen it became clear that peace was no longer an option, John Adams, Samuel Adams, Richard Henry Lee, and several other members of Congress began calling for a declaration of independence. As that momentous proposal was being debated and discussed by the delegates, members of the Black Robe Regiment petitioned for a day of \"fasting, humiliation and prayer,\" and the Continental Congress formalized their request by decree. The motion was introduced by William Livingston of New Jersey, who once served as a missionary to the Mohawk Indians.\n\nThe motion was approved, and the date was set for May 17, 1776. To make sure everyone was informed of this important day, the Congress placed public notices in the colonial newspapers, recommending that all Americans spend the day in fasting and prayer, humbly seeking God's favor and asking for His protection. The resolution concluded with a prayer that God \"would be graciously pleased to bless all his people in these colonies with health and plenty, and grant that a spirit of incorruptible patriotism, and of pure undefiled religion, may universally prevail; and this continent be speedily restored to the blessings of peace and liberty, and enabled to transmit them inviolate to the latest posterity.\"\n\n144\n\nNo one doubted the seriousness of the request or the urgency of their situation. The prospect of war with the English would have been terrifying to anyone. How could a ragtag army of farmers, merchants, townspeople, and clergymen ever hope to prevail over such an awesome force? In fact, the conflict had been going on since the skirmishes at Lexington and Concord in April 1775. The colonies had offered a peaceful compromise, but the British were incensed by the arrogance of their subjects and fully intended to make the Americans pay.\n\nBecause of their faith in God and their allegiance to the American cause, the colonies put aside everything else and flocked to the churches to seek the will of God. Meanwhile the delegates to the Continental Congress, who represented a wide range of sympathies and emotions regarding the prospect of war, ceased all their deliberations and took the day off to attend worship services in Philadelphia.\n\nOne week earlier John Adams had put forth a motion calling for each colony to declare independence from England. This, he thought, would be a first step toward a declaration of independence for all colonies, but not all the delegates were so optimistic. The previous month, the legislature of North Carolina passed a unanimous resolution calling for independence. The Virginia House of Burgesses did the same thing on June 29, declaring Virginia's independence from Great Britain and naming a committee to draft the state's first constitution. The Virginia legislature then ordered their delegate to the Continental Congress, Richard Henry Lee, to call for a vote on independence for all the colonies.\n\n145\n\nOn Friday, June 7, 1776, Lee stood before the delegates in Philadelphia to offer the resolution: \"That these United Colonies are, and of right ought to be, free and independent states, that they are absolved from all allegiance to the British Crown, and that all political connection between them and the State of Great Britain is, and ought to be, totally dissolved.\" The vote for the resolution was unanimous. Every delegation was in favor of independence, which meant the thirteen colonies would cease to exist and the United States of America was about to be born.\n\nOn June 11, Congress appointed a committee to prepare a draft Declaration of Independence. Thomas Jefferson, John Adams, and Benjamin Franklin were the principal architects. Jefferson worked on the initial draft in his rented room on High Street in Philadelphia between June 11 and 28; when he returned to Carpenters' Hall, Adams and Franklin read the document and made changes. A revised draft was presented to Congress on July 2, following adoption of the independence section put forth by Richard Henry Lee. The members of Congress each read the document and offered changes or revisions; this process lasted most of the day on July 3 and half of the following day. On the afternoon of July 4, the Declaration of Independence was formally adopted.\n\nThe Declaration was printed July 5, bearing only the names of John Hancock, who was president of the Continental Congress, and Charles Thomson, secretary. On July 19, Congress ordered that the Declaration be engrossed on parchment with the title \"the unanimous declaration of the thirteen united states of America.\" On August 2, John Hancock signed the engrossed copy in enormous, bold strokes that even a half-blind English king could read. Then, according to protocol, each of the remaining members signed the document, beginning with the northern states and continuing down the list to the southernmost, Georgia. Fifty-six delegates signed the document. Robert Livingston of New York and two other delegates were not present to sign.\n\n146\n\nRETHINKING THE AMERICAN SPIRIT\n\nThe Declaration of Independence, as drafted by Jefferson and revised by the members of that body, was a work of genius that would become a source of inspiration and a model of political wisdom to freedom-seeking peoples everywhere. After reviewing the initial draft, John Adams wrote to his wife that July 2, the day the Declaration was approved, ought to be celebrated as the nation's birthday, with \"pomp and parade, with shows, games, sports, guns, bells, bonfires, and illuminations, from one end of this continent to the other. . . .\" He proposed that Independence Day forever after should be a day of thanksgiving to God for His deliverance.\n\nToday's secular liberals like to say the Declaration of Independence was a humanist manifesto inspired by the writings of Locke, Hume, Voltaire, and other Enlightenment figures. Jefferson, who was well versed in French thought and a man of science, may not have been a Christian in any traditional sense. Nevertheless, he had written in the front of his Bible, \"I am a real Christian, that is to say, a disciple of the doctrines of Jesus. I have little doubt that our whole country will soon be rallied to the unity of our creator.\" In reality, the Declaration of Independence reflects not just Jefferson's views but the Christian worldview of the men who signed the document \"with a firm reliance on the Protection of Divine Providence,\" and mutually pledged to each other \"our Lives, our Fortunes and our sacred Honor.\"\n\nI first began thinking about writing this book when I was doing some research into the philosophical and spiritual mindset of the Founders. I remember writing in my notes the words, \"The Rise of the American Spirit.\" I had been asking myself: _What is the American Spirit? And_ _where did it come from_? Today, after many long months of study, I am convinced the American Spirit came from the Great Awakening. It is the essence of who we are as a people. Even though that spirit has been through some difficult times of testing, I believe it is still there.\n\nWhat concerns me most is that so many Americans no longer seem to recognize it or to regard the American Spirit as highly as they ought. If the American people do not know who they are, they cannot respond appropriately in times of crisis. If our politicians do not know who we are as a nation, they cannot represent us and make decisions on our behalf as well as they should. But if We the People do not know who we are, how can we expect our leaders to know?\n\n147\n\nWith the unexpected emergence of the Tea Party movement in 2009\u201310, I realized, _That's the Spirit of America. That's the voice of_ _millions of Americans who believe in the achievements and values of our_ _forefathers, and they want to prevent those values from being eroded_. If the spirit that has been so important for more than three hundred years is still there, it will rise up. We have seen it rise up before, especially in times of war and during various national emergencies. We certainly saw it in the aftermath of 9\/11, and I am convinced we saw it again in the 2010 midterm elections.\n\nAll those Tea Party events that took place over several months were evidence that the American Spirit is not dead. The huge number of people who came to the Restoring Honor event on the Washington Mall made it clear that there is a deep yearning in the hearts of millions of men and women\u2014from sea to shining sea\u2014for a renewal of that marvelous something that made this the greatest country on earth. I doubt if all of them could name it, and many were too young to have learned very much about the history of their country in the public schools, but they knew instinctively that something was missing, and they wanted it back.\n\nIf you were to ask the average person if he or she loves America, you would probably hear, \"Sure, I love my country!\" But what the person is really saying is, \"I love the Spirit of America.\" That needs to be identified, and I believe that spirit will lead us into what I have described in these pages as the Coming Revolution. That revolution is simply a reawakening of the spirit that made us what we are. It was visible during the American Revolution more than 230 years ago. It was put forth foundationally in the Great Awakening. It began to crystallize and come to fruition in the years leading up to the Revolution, and it has been strong with us through all the years since. But today it is being corrupted by the radical left and the gradual incursion of socialist policies from the 1960s by unelected judges and bureaucrats in the nation's capital. That is why we need to start holding their feet to the fire and making some serious changes.\n\n148\n\nI was in a meeting not long ago with a group of pastors, and one of them was a young man in his midthirties. He is the pastor of a large church of upwardly mobile suburbanites, and when we began talking about the importance of patriotism and respect for this country, he listened in silence for a few minutes. Then he interrupted the conversation and asked, \"Can I say something?\" We all nodded, and he said, \"I'm as much of a patriot as anyone around this table, but if I start talking about patriotism to the people in my church, they won't know what I'm talking about. They don't have any concept of what it means to be an American.\"\n\nAs shocking as it is to think about what he was saying, no one at that meeting was entirely surprised because we have seen it coming for years. The education establishment has been turned upside down, and no one is teaching America's young people about the tremendous privileges they have as citizens of this country. They take our freedoms for granted and assume this country is no better than any other\u2014even though most of the world is still in the hands of dictators and tyrants. To make matters infinitely worse, they and their children are being taught that America is greedy, corrupt, and unworthy of respect.\n\nAs a result, millions of young people have no concept of the American Spirit. They don't know what it is. They have been denied the privilege of sharing in the dream that has drawn tens of millions of men, women, and children from every continent to these shores. Now we are seeing a generation coming up who are the children and grandchildren of the men and women who fought, bled, and died for our freedom, and they have been robbed of their inheritance as Americans.\n\nIt started with the rise of communism in the 1920s, mainly in Europe but also among the intellectuals and academics in this country. It really took off during the 1960s with the hippies and flag burners and war protesters. That spirit of anti-Americanism got a boost from President George H. W. Bush when he started talking about the New World Order. Many people were surprised to hear him use that term. Was he making some sort of prophecy? Did he know something we didn't? The fact is, the New World Order was an idea that had been in circulation for decades among the intellectual elites and backroom deal makers of the foreign policy establishment.\n\n149\n\nBut what all those great minds thought should lead to peace and harmony in the world has not had the effect they imagined. The long-term goals of the European Union, the move toward a unified world currency, the purposeful devaluation of the dollar, and so many other things that the progressives have been plotting and planning for generations were unfamiliar and unnatural for most of us. Those ideas did not ring any bells, but the globalists have not given up. They are moving right ahead with their one-world agenda, unabated and unabashed. And I have to admit, if they are trying to create global unity by stomping out the flames of the American Spirit, they have done a pretty good job of it so far.\n\nTHIS EXCEPTIONA L NATION\n\nThe whole idea of American exceptionalism has been under attack for decades, as if the uniquely valuable role this country performs by lifting untold millions out of poverty is something for which we should be ashamed. This is a serious problem that needs to be addressed, but I have hope that things are changing for the better in all these areas. You do not have to be a Tea Party advocate to see it, but there is a revival of the American Spirit taking place all around us, and it is transforming the social and political landscape.\n\nIt is not the kind of hope and change we were promised by the Obama administration; instead, it is a reaction from the heartland against the unrelenting pressure from the Left to deny what we truly are and to make us into something else. This is the change that is making headlines, and the Tea Party movement is evidence that the attempt to remake America into a socialist state is not going over very well with large numbers of Americans.\n\n150\n\nIf you even try to mention the idea of manifest destiny in a classroom in Europe\u2014as I did on one occasion during a lecture at Oxford\u2014the academic elite will go ballistic. The idea that America has had a providential role in history or that this country has a very special destiny is more than it can handle. But our forefathers certainly believed it. You cannot read the history of this nation without seeing just how uniquely and providentially we have been blessed. And America has been a blessing to the nations of the world ever since.\n\nAmerican military might stopped the Nazis, the Japanese Empire, and Mussolini's Fascists. It shut down the concentration camps at the end of World War II. From 1947 to 1951, the Marshall Plan helped bring Europe back from the brink. Economic development in Germany and Japan after the war was spearheaded by American businessmen, allowing those countries to become two of the most competitive and successful economies on the planet. For more than a hundred years, America's wealth has fed the poor, rescued the downtrodden, and fought to help nations recover from disasters and diseases of every kind. This country's record of compassion and service to the world is unrivaled, and we have never done it for profit; we do it because that's the kind of people we are. We do it because this is a Christian nation\u2014 an exceptional Christian nation.\n\nIn a 1776 letter to family friend Mercy Otis Warren, John Adams wrote, \"Public virtue cannot exist without private; and public virtue is the only foundation of republics.\" He continued,\n\nThere must be a positive passion for the public good, the public interest, honour, power, and glory, established in the minds of the people, or there can be no republican government, nor any real liberty. And this public passion must be superior to all private passions. Men must be ready, they must pride themselves, and be happy to sacrifice their private pleasures, passions, and interests, nay their private friendships and dearest connections, when they stand in competition with the rights of society.\n\n151\n\nThat is a powerful statement of the American Spirit, arising from the wellspring of faith. The men and women of the founding era understood that the judgment of history was upon them. If they fought so valiantly to gain independence from Britain and then gave us a government run by selfish, mean-spirited, and corrupt bureaucrats lacking all concern for the public good, they would have failed. How would this country be any different from the European monarchies they had escaped? Adams spoke for all the Founders when he said that commitment, self-sacrifice, and personal devotion are essential if there is to be \"real liberty.\"\n\nI am also reminded of the words of other leaders who faced equal or greater challenges in their lifetimes. For example, President Franklin Roosevelt at the outset of the D-Day Invasion of Europe, June 6, 1944, knew that millions of lives already had been lost in the Second World War and that the fate of millions more would soon be at stake. He went on the radio to inform the American people of the landings in France and to offer a heartfelt prayer for our servicemen and the success of their mission. This is what he said:\n\nMy Fellow Americans:\n\nLast night, when I spoke with you about the fall of Rome, I knew at that moment that troops of the United States and our Allies were crossing the Channel in another and greater operation. It has come to pass with success thus far. And so, in this poignant hour, I ask you to join with me in prayer:\n\n\"Almighty God:\n\n\"Our sons, pride of our nation, this day have set upon a mighty endeavor, a struggle to preserve our Republic, our religion, and our civilization, and to set free a suffering humanity. Lead them straight and true; give strength to their arms, stoutness to their hearts, steadfastness in their faith.\n\n\"They will need Thy blessings. Their road will be long and hard. For the enemy is strong. He may hurl back our forces. Success may not come with rushing speed, but we shall return again and again; and we know that by Thy grace, and by the righteousness of our cause, our sons will triumph. They will be sore tried, by night and by day, without rest\u2014until the victory is won. The darkness will be rent by noise and flame. Men's souls will be shaken with the violences of war.\n\n\"For these men are lately drawn from the ways of peace. They fight not for the lust of conquest. They fight to end conquest. They fight to liberate. They fight to let justice arise, and tolerance and goodwill among all Thy people. They yearn but for the end of battle, for their return to the haven of home. Some will never return. Embrace these, Father, and receive them, Thy heroic servants, into Thy kingdom. And for us at home\u2014fathers, mothers, children, wives, sisters, and brothers of brave men overseas, whose thoughts and prayers are ever with them\u2014help us, Almighty God, to rededicate ourselves in renewed faith in Thee in this hour of great sacrifice.\n\n\"Many people have urged that I call the nation into a single day of special prayer. But because the road is long and the desire is great, I ask that our people devote themselves in a continuance of prayer. As we rise to each new day, and again when each day is spent, let words of prayer be on our lips, invoking Thy help to our efforts.\n\n\"Give us strength, too\u2014strength in our daily tasks, to redouble the contributions we make in the physical and the material support of our armed forces. And let our hearts be stout, to wait out the long travail, to bear sorrows that may come, to impart our courage unto our sons wheresoever they may be. And, O Lord, give us faith. Give us faith in Thee; faith in our sons; faith in each other; faith in our united crusade. Let not the keenness of our spirit ever be dulled. Let not the impacts of temporary events, of temporal matters of but fleeting moment\u2014let not these deter us in our unconquerable purpose.\n\n153\n\n\"With Thy blessing, we shall prevail over the unholy forces of our enemy. Help us to conquer the apostles of greed and racial arrogances. Lead us to the saving of our country, and with our sister nations into a world unity that will spell a sure peace\u2014a peace invulnerable to the schemings of unworthy men. And a peace that will let all of men live in freedom, reaping the just rewards of their honest toil.\n\n\"Thy will be done, Almighty God.\n\n\"Amen.\"\n\nThose powerful words brought the reality of war home to millions in a way nothing else could have done. These were not the punched-up and calculated exhortations of the political soapbox but a passionate cry from every heart for the men who were at that very hour descending upon the most ambitious military landing in history. This was a prayer for America's sons, fathers, friends, and loved ones and a commitment to continue in prayer for as long as it would take to achieve victory.\n\nWhatever you may think of Franklin Roosevelt, no one can deny the depth of faith he demonstrated in that hour. Because of his genuine compassion for the American people, he was able to speak for all of them with words that could not help but touch the heart of God. And we know God did answer that prayer by giving the American forces and our Allies the victory. The war in Europe would continue for eleven long months until the final German surrender on May 2, 1945, but the die was cast when the president and all of America prayed.\n\nThis is how Americans have dealt with adversity for more than three hundred years, by prayer and humble supplication and by the appeal for divine guidance combined with a fierce determination to conquer whatever obstacles may lie before us. When we appeal to heaven, God does not expect us to come to Him with some grand scheme. He wants us to be willing to seek His face, to do His will, and to follow His leading\u2014which can be awfully hard to detect at times. When God called Abraham to leave Ur of the Chaldeans, Abraham had no idea where he was going. When God instructed him to sacrifice his son, he had no idea that God would provide another sacrifice\u2014a ram\u2014so that Isaac could be spared. But as Paul tells us, \"Abraham believed God, and it was accounted to him for righteousness\" (Romans 4:3 NKJV).\n\n154\n\nThroughout Scripture we see the same pattern: God is seeking those who will step out in faith, believing that He still causes all things to \"work together for good to those who love God, to those who are the called according to His purpose\" (Romans 8:28 NKJV). The fact that our forefathers and foremothers, with remarkably few exceptions, actually believed that promise and acted on it, would\u2014if nothing else\u2014make this a truly exceptional nation.\n[_**Six**_  \nTHE COMING REVOLUTION](Lee_9780849949463_epub_c5_r1.html#d6e8)\n\nIf you wanted to take one of the world's largest and most dynamic economies and fundamentally change it from top to bottom, what would you do? Where would you begin? This was the question the young lawyer asked himself. He was full of questions, full of ideas. He dreamed of doing something big and important, but he didn't know where to begin. He had grown up in a middle-class family; both of his parents were teachers who were socially and politically aware. His father always said there were too many poor people, all of them working too hard for too little pay, and too many rich people who never seemed to work at all. Couldn't somebody do something about that?\n\nThe young man was a mediocre student\u2014his teachers said he was too impatient and undisciplined\u2014but he was a man of action, and he believed he could do something. He could be somebody. Like his older brother, Alex, Vladimir Ulyanov wanted to challenge the status quo. He just needed a plan. After college he worked as a small-town lawyer for a while, but he hated the job. He knew he was destined for bigger things. So he kept thinking, kept reading, and by his mid-twenties Vladimir had found his plan. He found it in the writings of the revolutionary thinkers and philosophers of the previous generation, but the plan was big and ambitious: so big, in fact, that he would need thousands of volunteers to make it happen.\n\n156\n\nThat is what he decided to do. He would write articles, papers, and books; he would get out in public and give speeches so he could raise the people's awareness, change the public consciousness, and raise an army of volunteers motivated by a vision of hope and change. Before all was said and done, his plan would lead to a total transformation of society. He would change the culture and reshape the economy, beginning with the graduated income tax, centralized control of banking and credit, and nationalization of the media, mass communications, and transportation. It also included government control of the environment and farming and the confiscation of public lands.\n\nUnionization and regimentation of the labor force were essential\u2014 this was the foundation of his army of volunteers\u2014along with control of public education and the universities. And the biggest and grandest plan of all, the abolition of private property and the confiscation of inherited wealth would pay for everything he hoped to accomplish. With all these things in place, Vladimir Ilyich Ulyanov\u2014now known as V. I. Lenin\u2014turned his attention to the one obstacle that stood in the way of his \"Worker's Paradise,\" the one thing preventing final and absolute control of the economy and the success of all his plans: the Christian faith of the Russian people.\n\nUnlike Karl Marx, who had originally conceived and published a plan for world domination in the _Communist Manifesto_ of 1848, Lenin was never one for subtlety. If Marx was the academic and the theorist, Lenin was the agitator and the warrior. Marx provided the nail, but Lenin had the hammer. As Malachi Martin writes:\n\nMarx's dry discussions and the touch of poetic pretense in his forecasts contrast with the bloody realism of Lenin, whose predictions were far from idyllic. His plans all were aimed at a complete and bloody break with the past, and at the violent death and final entombment of capitalists and capitalism. . . . The one poetic touch in Lenin's otherwise abrasive mind, in fact, concerned that almost dreamlike \"Worker's Paradise\" he foresaw at the end of the proletarian rainbow. To find a parallel, you would have to go back to the early Hebrew prophets and their forecast of the Messianic Age. . . . On the near side of that rainbow, however, the reality Lenin foresaw and worked so feverishly to bring about was the grinding tyranny that has been witnessed by the world for seventy years and more.\n\n157\n\nLenin referred to the war against Christianity and the universal acceptance of atheism as \"the cause of our state.\" Christianity made the people soft, he believed, and unwilling to join the revolution. Orthodox Christians were content with their lives and tended to believe that with God's help they could overcome their own limitations and the disadvantages life had thrust upon them. For Lenin, who had long since renounced morality and religion, that kind of thinking was infuriating. He needed an army of zealots to help him carry forward his revolutionary dream. Christianity would have to go.\n\nAfter the Bolshevik Revolution in 1917, Lenin created the Communist International (the Comintern), and his followers launched a massive campaign against the churches. Nothing would stand in their way. During the seventy-year history of the Soviet Union, millions of churches were desecrated, burned, or converted to secular purposes (turned into shops, offices, theaters, museums, and warehouses). The few houses of worship that were not destroyed or converted to other uses became state property, and the charitable programs previously run by the churches were taken over by the government.\n\nOrthodox monasteries were turned into prison camps, and millions of believers were persecuted, tortured, or killed. Some Christians and Jews who refused to renounce their faith were sent to labor camps or mental hospitals. Thousands more were exiled and shipped off to other countries, but believers of all faiths were ridiculed and harassed, and by 1925, official courses in atheism were being taught in all the schools and colleges.\n\n158\n\nFrom the start the Communists used whatever language they needed to mislead the people. In the midst of all the social and political turmoil of the post\u2013World War I era, words like _socialism_ and _democracy_ sounded reassuring. Many people still feel that way: What's so bad about socialism? What's wrong with democracy? Lenin knew these terms could be useful because they allowed the government to increase control over the people (the bourgeoisie), and socialism was just the first step toward Lenin's ultimate objective, full-blown communism. This was what he meant by \"democracy.\" But for the initiated, Lenin spelled out his meaning in the \"Program of the Communist International\" (1928), in which he said,\n\nThe Soviet form of State, being the highest form of democracy, namely, proletarian democracy, is the very opposite of bourgeois democracy. . . . The Soviet State is the dictatorship of the proletariat, the rule of a single class\u2014the proletariat. Unlike bourgeois democracy, proletarian democracy openly admits its class character and aims avowedly at the suppression of the exploiters in the interests of the overwhelming majority of the population.\n\nLenin made sure the Comintern understood the objective was total and absolute domination by any means necessary, and he added ominously, \"The Soviet State completely disarms the bourgeoisie and concentrates all arms in the hands of the proletariat; it is the armed proletarian State.\"\n\nIn the wake of the Bolshevik Revolution, Lenin promised tolerance for dissent, but that was a lie. It was simply a talking point he never intended to honor. When Czar Nicholas II, his wife and four daughters, and his son and heir, Alexis, were brutally murdered in July 1918, the story was immediately spread that local anarchists were responsible for the crime, but that was also a lie. After an investigation that included a detailed review of archival material from Lenin's personal bodyguard, the Russian historian Edvard Radzinsky was able to confirm in 1990 that Lenin had ordered the executions. In fact, his bodyguard had delivered the order to the telegraph office and then saved the original telegraph tape as a record of the secret order.\n\n159\n\nIn the words of the Orthodox priest and dissident Gleb Yakunin, \"Russia turned crimson with the blood of martyrs.\" In the first five years after Lenin's army of radicals and labor organizers seized power, 28 bishops and more than 1,200 priests were murdered in cold blood. Later under Joseph Stalin, who ruled as general secretary of the Communist Party from 1922 to 1953, the level of state-authorized terror only increased, and, according to records from the era, \"by the end of Khrushchev's rule [1953\u20131964], liquidations of clergy reached an estimated 50,000.\"\n\nTHE GREAT LIBERAL DREAM\n\nKarl Marx, who died in 1888, did not live to see the tragic results of his big idea, and Lenin, who died in 1924, just seven years after the Bolsheviks seized power, never saw the great \"Worker's Paradise\" he had envisioned. But the tyranny they unleashed on the world would not die so easily, and it has continued to haunt the planet ever since. In January 2011, the religious watchdog group Freedom House reported that freedom is in decline around the globe. In 2010, just 87 countries qualified as \"free countries,\" and 25 of the world's 194 countries faced significant deterioration in the levels of personal freedom.\n\n\"Authoritarian regimes like those in China, Egypt, Iran, Russia, and Venezuela,\" the report stated, \"continued to step up repressive measures with little significant resistance from the democratic world.\" Not surprisingly, Marxist-Leninist ideology is implicated in every case. More recently we have seen a wave of violence and instability in Mexico and South America, along with riots and insurrection all across the Middle East inspired by equal parts of Islamic extremism and Communist agitation. Our own government, instead of calling for order and an end to the violence, has apparently taken sides with the rioters calling for revolution, supporting the spread of Islamic Sharia law, encouraging groups like the Muslim Brotherhood while turning its back on leaders who have been supportive of U.S. policy in the past and forcefully condemning the only democratic nation in the region, the state of Israel.\n\n160\n\nBut some will ask, why should all these things matter to me? They matter because we have good reason to fear for our country when American foreign policy no longer represents the beliefs of a majority of the citizens, and when lawmakers in Washington continue creating outrageously expensive new programs that rob people of their self-reliance and push the nation further down the road toward socialism. As Russell Kirk pointed out in _The American Cause_ , tyrants of all stripes draw their strength from their ideologies, and ideology is always the enemy of faith in God. People whose lives have been turned upside down by war or famine or disasters will naturally look for comfort wherever they can find it. In the absence of genuine faith in God, many will give their allegiance to any charismatic leader who can offer a vision of peace and plenty. But this is an environment ripe for exploitation.\n\nDuring the French Revolution liberal ideologues calling themselves \"New Men\" promised a classless society where all men would be happy and prosperous, but those promises were empty and resulted in the \"Reign of Terror\" and more than fifty years of social and political anarchy. During the Enlightenment's assault on both church and state, the boulevards of Paris ran red with the blood of its victims, and the promise of liberation became a death sentence for thousands.\n\nBy the same token, communism, which is an even more insidious ideology, has claimed the lives of more than 100 million men, women, and children worldwide. Despite the widely celebrated collapse of the Soviet Union in 1989\u201390, the Marxist ideology is very much alive, still attracting followers, and still wreaking havoc in this country and around the world. What the destitute and hungry are being offered today is a false vision of utopia, the promise of a world in which war and sorrow and inequality will no longer exist. But as discussed in the first chapter of this book, that is a promise no one on this earth can keep.\n\n161\n\nWhen the Russian dissident Aleksandr Solzhenitsyn came to this country in 1974, he was shocked to discover that while communism was already in decline in the Soviet Union, it was thriving in America's newsrooms and on our university campuses. How could \"the best and brightest\" in academia and the news media have fallen for the Communist lie when so many in his own country were waking up from a decades-long nightmare and praying for the collapse of the socialist regime?\n\nThe great man never received a satisfactory answer, but in 1994, after twenty years in this country, during which he was vilified by liberal elites for his Christianity and his conservative political views, Solzhenitsyn returned to Russia, where he could serve as a witness of his resolute faith in God and his hope for an authentic democracy in that country. By the time he died in 2008, however, the former KGB director Vladimir Putin was president of Russia.\n\nTo comprehend why liberals are so inclined to support socialist policies and condemn those who hold traditional values, it is important to understand that liberal ideology rejects the doctrine of \"original sin.\" The Bible teaches that sin separates mankind from God and only belief in the substitutionary death of Jesus Christ can restore a right relationship with the Father. Liberal ideology, on the other hand, teaches that humanity is naturally good and even perfectible when given the right opportunities. As defined by James Burnham, \"Modern liberalism . . . holds that there is nothing intrinsic to the nature of man that makes it impossible for human society to achieve the goals of peace, freedom, justice, and well-being that liberalism assumes to be desirable and to define 'the good society.'\"\n\nThe Founding Fathers went to great lengths in the Declaration of Independence and the Constitution to protect the rights of the individual. The entire Bill of Rights is a defense of personal liberty and a limit on the power of the state. Unfortunately, this is not what liberals in America are after. In his book _Liberty and Tyranny_ , author and attorney Mark Levin writes that liberals are intellectually committed to the supremacy of the state rather than the rights of the individual. \"For the Modern Liberal,\" he says, \"the individual's imperfection and personal pursuits impede the objective of a utopian state. In this, Modern Liberalism promotes what French historian Alexis de Tocqueville described as a _soft tyranny_.\" Consequently Levin argues that today's liberals ought to be characterized as statists because their main concern is empowering government while curtailing the rights and privileges of individual citizens.\n\n162\n\nOne of the best definitions of liberalism comes from a twentieth-century liberal, the former vice president under Lyndon Johnson, Hubert Humphrey, who said _change for the sake of change_ is the hallmark of political liberalism. In an article in the _New York Times Magazine_ , Humphrey said, \"It is this emphasis on changes of chosen ends and means which most sharply distinguishes the liberal from a conservative in a democratic community . . . liberals recognize change as the inescapable law of society, and action in response to change as the first duty of politics.\"\n\nWhat each of these definitions tells us is that in liberal ideology the belief that public officials can make radical changes to our government while basically ignoring the will of the people is perfectly legitimate. This was never what the Founders had in mind. However, I believe this is why presidential candidate Barack Obama chose the slogan Hope and Change for his 2008 campaign. This was the language of the ultraliberal communities in which he had spent his entire life. From boyhood on Mr. Obama was exposed to no other point of view, and he concluded that this was what a majority of the voters wanted to hear. Apparently he guessed right since they elected him president. But I doubt if even a fraction of the voters had any idea what that promise would ultimately entail.\n\n163\n\nTHE JUDEO-CHRISTIAN ETHIC\n\nStanding in dramatic opposition to the liberal view of human nature and the socialist vision of the perfectibility of mankind is the Judeo-Christian ethic, which was embraced without exception by the Founding Fathers and the generations afterward who laid the foundations of American democracy. I described this in the opening pages of _The American_ _Patriot's Bible_ , published in 2009, but I think it is important at this point to outline briefly the seven principles that \"people of the book,\" meaning Christians and observant Jews, have traditionally embraced.\n\n1. THE DIGNITY OF HUMAN LIFE\n\nWithout respect for human life at all stages, from conception to natural death, any claim of decency and morality is meaningless. Jesus said, \"I have come that they may have life, and that they may have it more abundantly\" (John 10:10 NKJV). Because Americans hold all life in high regard, millions of men and women have come to this country from around the world. This is also why we send missionaries, aid workers, and military forces to foreign lands: we believe human life is precious. But we are far from blameless, and history will take a dim view of some of our policies regarding human life. With a bloodstained history of more than thirty-five years and 50 million deaths, the abortion industry has left an indelible stain on this nation's honor. The good news, however, is millions of Americans are beginning to see the light and are changing their views on abortion. If the American people cannot show compassion for the most innocent human life, the unborn child in his mother's womb, we have no right to speak about human dignity. This is God's standard, and it ought to be our hope and prayer that the evil practice of abortion will be ended very soon.\n\n2. THE TRADITIONAL MONOGAMOUS FAMILY\n\nThere is nothing more natural or more essential to the health and well-being of the community than the traditional two-parent family. This was God's plan from the beginning, and the very first institution He created. Before the community, before the church, and before government, God created man and woman as partners, each with unique strengths and attributes that would be needed to raise happy and healthy children and build stable communities. While there are times when the loss of a spouse or loved one may alter this arrangement, any deliberate attempt to circumvent God's design\u2014to change this natural balance that is common throughout nature\u2014will have serious and long-term consequences.\n\n164\n\n3. A NATIONAL WORK ETHIC\n\nDating back to the landing at Plymouth Rock, what has sometimes been called the \"Protestant work ethic\" has been one of the most important characteristics of the American way of life. The Bible says, \"If anyone will not work, neither shall he eat\" (2 Thessalonians 3:10 NKJV). The Pilgrim Fathers knew they would not survive if each man and woman did not do his or her part, and the success of those first settlements was a testament to the determination and energy they exhibited.\n\nIn times of stress\u2014the Great Depression of the 1930s, for example, or the economic difficulties in America today\u2014it's not always easy to find work, but God expects each of us to be willing and ready to contribute when we can. The fact that millions of Americans have done their part to support their families and earn an honest living is one of the main reasons this country has been such a beacon of hope to the rest of the world. Any government policy that subverts the right to work or holds individuals and families in bondage in the name of welfare threatens this most basic human right.\n\n4. THE RIGHT TO A GOD-CENTERED EDUCATION\n\nAny society that fails to teach its children the proper relationship between mankind and the Creator will experience hardship and sorrow. If children do not understand the vertical relationship between themselves and God, they will have difficulty understanding the role of parents, teachers, and other authority figures, and consequently they will have problems with the horizontal relationship with their peers. The Founders had no misgivings about the importance of a God-centered education: they believed the purpose of education at all levels was to inculcate the virtues and moral values essential for good citizenship.\n\n165\n\nThis is why all the universities of the founding era were established as seminaries. Every academic discipline, whether it is science, mathematics, history, literature, or the arts, has deep Christian roots; it is impossible to read about the origins of any of them without encountering the godly men and women who paved the way. When our schools, in the name of diversity and tolerance, deny children the privilege of a proper understanding of our Judeo-Christian heritage, they handicap their understanding of the world and, worse, dishonor the very Source of wisdom and prosperity.\n\n5. THE ABRAHAMIC COVENANT\n\nOne of my favorite verses in the Bible is Psalm 33:12, which says, \"Blessed is the nation whose God is the Lord, the people He has chosen as His own inheritance\" (NKJV). There is no question that America has been blessed by God. Columbus believed God brought him to these shores for the purpose of establishing a godly nation. The Pilgrims, who began their journey in the New World with a covenant called the Mayflower Compact, survived overwhelming odds, upheld by their strong faith and fervent prayers, and God allowed them to prosper.\n\nEvery American president has honored God, at least in his public statements, and we have built a \"city on a hill.\" When God made a covenant with Father Abraham, He promised to bless and prosper the patriarch and his descendants forever if they would remain faithful and never forget where their blessings came from. I believe God has allowed this country to undergo severe hardships in recent years largely because America is at risk of forgetting those things. If we really want to see our situation take a turn for the better, we need to get back on our knees, ask for forgiveness, and honor this nation's covenant with the Creator.\n\n166\n\n6. COMMON DECENCY\n\nIs there anything more gratifying than being treated with decency and respect? And is there anything more exemplary of the American Spirit than showing common courtesy to other people, regardless of their race, religion, or social status? The radicals of the French Enlightenment promised equality and opportunity for all, but the _nouvelle regime_ they created in the eighteenth century was a disaster because they deliberately left God out of the picture and showed respect for no one. If we understand, as the Bible says, that we are created in the image of God, then we honor the Creator when we treat others as we wish to be treated. \"Therefore,\" Jesus said, \"whatever you want men to do to you, do also to them, for this is the Law and the Prophets\" (Matthew 7:12 NKJV). Demonstrating respect and common decency toward others is recognition of the Christian belief that \"I serve myself best when I serve you.\"\n\n7. OUR PERSONAL ACCOUNTABILITY TO GOD\n\nModern sociologists are apt to say that criminals and other kinds of offenders are victims of society. Rather than placing the blame for wrong choices where it actually belongs, they prefer to blame some vague institution called _society_. But God is under no such illusion. The belief that each person is accountable to God and will have to answer for his or her choices in life is one of the most basic principles of the Judeo-Christian ethic.\n\nGenerations of Christians have believed the words of Saint Peter: \"The eyes of the Lord are on the righteous, and His ears are open to their prayers; but the face of the Lord is against those who do evil\" (1 Peter 3:12 NKJV). Knowing that God hears and sees every word and deed ought to be a source of great comfort because, as the Bible assures us, \"He cares for you\" (1 Peter 5:7 NKJV). But a sense of personal accountability is equally important because it reminds us that when this life is over, there will be a day of judgment.\n\n167\n\nLIVING UP TO OUR IDEALS\n\nWhy does America send soldiers, aid workers, and cultural emissaries around the world to extend a helping hand to the people of other nations? Because Americans are decent people who care about the welfare of others. We believe national defense and military readiness are essential, and we understand that we must be willing to stand our ground against the forces of evil in the world. For the most part, Americans have never shirked their civic duty in these areas.\n\nOn the other hand, concern for the well-being of others was a major failing of the Soviet Union. Because of the corrupt system in which they were trapped, the people became corrupt, self-centered, and skeptical of everything and everyone. The government poured billions into armaments and made a huge display of Russia's military strength, but deprived of faith in God, the people lost their common decency; and as the Soviet economy was slowly grinding to a halt in the 1980s and '90s, no one would risk his or her own safety for the well-being of others.\n\nJust think of what happened with abortion in that country. I have been told the average woman in Russia today has had four abortions. Obviously for these women abortion is not about the health of the mother. It is simply a contraceptive. If the pregnancy is deemed inconvenient, she can just get rid of it. How sad to realize that much the same thing is now happening in this country, especially on the college campuses where sexual promiscuity is actively encouraged. Nevertheless, there is also good news that many people are beginning to see abortion as a sin against human nature. I am seeing more and more reports each year indicating that a growing number of Americans believe that abortion is wrong and ought to be legally ended.\n\nThere's Hope Ministries sponsors a pregnancy clinic in the Atlanta suburbs. We helped save the lives of 72 babies in 2010 alone; these are all beautiful children who would otherwise have been victims of the so-called _pro-choice movement_ if we had not intervened. Unfortunately we can't save them all. We could save ten times that number if we had the funds and manpower to expand our outreach. But even with all we know about the horrors of abortion and with the loss of more than 50 million lives to the abortion holocaust since the Supreme Court's 1973 _Roe v. Wade_ decision, it is difficult to raise money for crisis pregnancy centers in many communities. Too many people have decided the abortion crisis is somebody else's problem. The Left has done a good job of politicizing the debate, and it is uncomfortable to protest for the rights of the unborn, so many people just look the other way and stay home.\n\n168\n\nUnfortunately the government has no problem using our tax dollars to promote abortion, advocating for abortion services through the Department of Health and Human Services and pouring millions into organizations such as Planned Parenthood, which is the largest abortion provider in America. Despite the efforts of conservative lawmakers and constant protests from pro-life groups and concerned citizens, the Obama administration increased funding for Planned Parenthood from $350 million in 2009 to $363 million in the 2009\u20132010 fiscal year. For political liberals in Washington and in state legislatures around the country, this is the defining issue of their public life, and they are not about to back down\u2014which says a lot about what motivates people on that side of the aisle.\n\nEvery time we turn around, government is digging deeper into our pockets, creating new and more expensive social programs that invariably fail to deliver the benefits they promised but always lead to higher taxes. Liberal social programs have been a disaster for America, leading to higher levels of crime, joblessness, generational dependency, poverty, and profound resentment, not only among the recipients of these services but from the taxpayers who are required to fund them. The American family has been devastated by well-meaning but misguided sociology. Entire industries have been driven to the point of bankruptcy and beyond, so now our economy stands on the threshold of collapse.\n\n169\n\nA big part of the resentment expressed by members of the Tea Party movement is due to government's inability to deliver on its promises. But even more troubling, too many Washington policy makers and bureaucrats refuse to admit that we are already over our heads in debt and the American economy is sinking fast. Even with a national debt of $14 trillion, a $1.6 trillion budget deficit in fiscal year 2011, and the threat of foreclosure by China and the other countries funding our debt, liberal lawmakers do not seem to know when to stop.\n\nThe Democratic majority that took control of Congress after the 2008 general election has expanded the size of government in almost every area. Policies the voters overwhelmingly rejected have been forced down our throats, pushing the nation deeper into debt and further into the grip of state socialism. At this pace, by the end of President Obama's term, the government will have borrowed as much money in four years as it had in the previous 223 years since the Constitution was adopted in 1789. According to the administration's own numbers, the accumulated debt will top $19 trillion by the year 2021, consuming 80 percent of the nation's gross domestic product (GDP).\n\nMany young people have been persuaded in their schools and colleges that socialism is a good thing, a compassionate way of caring for the less fortunate. At the same time they are being taught that capitalism is a bad thing because it exploits workers and empowers the rich. The vast majority of Americans know better than that: we have no interest in going the way of the Soviet Union or Eastern Europe or Western Europe for that matter. But unless we can convince the next generation of the profound danger of that way of thinking, it will be immeasurably harder to turn things around.\n\nFor a closer look at some of the reasons we need to resist the direction the federal government has been taking us the past few years, I would like to offer a quick review of some of the policy issues that have become major sources of irritation.\n\n170\n\nTHE HEALTH-CARE DILEMMA\n\nDespite the claims trumpeted by the mainstream media and members of the president's party in Congress that health care in this country suffers in comparison to the nationalized health services in places such as Great Britain and Canada, a new study by British researchers found that American doctors do a much better job of treating and curing serious diseases such as cancer. As reported in the August 2008 issue of the medical journal _Lancet Oncology_ , American patients have a better survival rate for thirteen of the sixteen most common cancers than their British counterparts. American men are 40 percent more likely than European men to live for five years after their first diagnosis, the study says, and American women are 12 percent more likely to live at least five years longer than European women.\n\nFor years we have been told the American health-care system is broken and only government can fix it. Anyone who has ever stood in line at the DMV or tried to communicate with an IRS agent about a tax problem ought to know better. Any time government bureaucrats get involved in any issue, the problem gets worse. In a recent study of key issues in the health-care debate, author and researcher Sally Pipes offers a somber observation, stating, \"Health care and K\u201312 education are the two sectors in America that have more government involvement than any other, and they both suffer from serious quality problems.\"\n\nDuring debates leading up to passage of the 2010 health-care bill, effectively mandating nationalized health care for all Americans despite overwhelming resistance from the public, Congress began the process of simultaneously nationalizing a large segment of the insurance industry. Rather than making the system more affordable, easier to use, or less restrictive, these new laws are having the opposite effect. Before insurance companies can offer coverage to the public, each policy must be reviewed by all fifty state insurance administrations. Consumers are barred from purchasing insurance across state lines, and then they are forced to purchase one-size-fits-all packages that include many services they will never need. In 2007, there were 1,901 different federal mandates that health insurance providers must observe; this is what always happens whenever the federal government starts making the rules.\n\n171\n\nGovernment regulations regarding Medicare and Medicaid have multiplied the amount of red tape involved in processing medical claims and complicated the process of delivering appropriate care. Bureaucratic interference hampers the work of physicians beyond any reasonable limits. Both of these programs impose strict price controls on health care by determining what doctors and hospitals can charge for their services, which has led to serious inequities and caused thousands of doctors to opt out of the government-managed programs.\n\nAccording to a 2008 survey of medical services for elderly Americans, approximately a third of all senior citizens reported having difficulty finding a doctor who would accept them as patients. As Sally Pipes points out, \"The government may efficiently control the costs at which doctors are reimbursed. This does not, however, account for the pain and suffering people endure waiting for care or the value of their time spent searching for a doctor.\"\n\nIn their efforts to persuade Americans of the need to radically remake the U.S. health-care industry, congressional Democrats told us repeatedly that 47 million Americans were living in fear that they could not receive adequate care because they did not have medical insurance. The truth is, at least 10 million of the reported 47 million were not American citizens. Those individuals would have been treated without cost at virtually any American hospital. Another 18 million were between eighteen and thirty-four years of age and unlikely to purchase health insurance at all. Another 17 million, with incomes over fifty thousand dollars, could afford to purchase health insurance but decided for their own reasons not to do so. And, finally, many of the supposed 47 million were temporarily unemployed and would be eligible for benefits as soon as they returned to work.\n\n172\n\nBy far the greatest problem regarding the implementation of national health care in this country is the enormous financial burden it would place on the federal government. As with Social Security, Medicare, and Medicaid, all that cost eventually comes back to the taxpayers without any guarantee the government can actually deliver the goods. The government's insistence on plunging ahead with mandatory health-care initiatives despite the obvious risks and complications led syndicated columnist Michael Medved to conclude:\n\nThe likelihood of further explosions in the national debt represent a genuine threat to national security, with real chances for outof-control inflation, a badly battered and devalued dollar, and the disastrous downgrading of federal debt by worried creditors. The utter disregard to any sense of fiscal discipline represents a menace to our way of life at least as serious as the murderous minions of Islamo-Nazi terror.\n\nEver since the Social Security Administration was created by the Roosevelt administration in 1935, the government has pretended there is a big trust fund somewhere in Washington where the contributions of individual wage earners are safely kept, gaining interest until the individual retires. The truth is, there never was such a trust fund, no lock box, no bank account of any kind. Instead, tax monies paid to the IRS for all these services go straight to the general budget, where they are promptly spent by Congress for other things.\n\nEvery new program, every government expense, and every promise of relief winds up being simply another way of extorting money from taxpayers for inefficient programs. In each of these programs, as former Reagan administration lawyer Mark Levin points out, \"the individual is tethered to the state, literally and utterly reliant on it for his health and survival. . . . Rather than the individual making cost-benefit and cost-quality decisions about his own condition, the Statist will do it for him. And the Statist will do it very poorly, as he does most other things.\"\n\n173\n\nThere is no question there needs to be improvement in the cost and accessibility of health care in this country, but the answer is not more government inefficiency and interference in the doctor-patient relationship. Instead of allowing bureaucrats in Washington, DC, to decide what treatments are acceptable and how much they should cost, the free market system that has worked so well for most of our history\u2014at least until the beginning of Lyndon Johnson's Great Society schemes in the 1960s\u2014has a natural self-regulatory apparatus: if it works, people will use it, and they will pay a fair price. That's the American way. If the quality of care is poor, the results are disappointing, or the physician fails to render professional treatment at a fair price, the consumer will vote with his feet and find a doctor who can do a better job. That's how it ought to work in a free society.\n\nMARRIAGE AND FAMILY ISSUES\n\nRecently the Family Research Council issued a report based on 2008 Census data entitled \"Index of Belonging and Rejection,\" which indicates that the percentage of American children growing up in intact homes with a birth mother and biological father who remain legally married to each other until the child becomes a teenager has fallen dramatically over the past thirty years. The figures in this report are deeply troubling. Just 45 percent of young people in this country manage to grow up with both parents. For whites the average is 54 percent, and for blacks it is just 17 percent. The average for Asian Americans is somewhat better at 62 percent but still disappointing.\n\nThe out-of-wedlock birthrate for blacks was 26 percent in 1965, when Daniel Patrick Moynihan first identified the problem, and the white illegitimacy rate was just 3 percent. Today it's 28 percent for whites, 72 percent for blacks, and 40.6 percent overall. As Tom Bethell writes in the _American Spectator_ :\n\n174\n\nWe are living in the midst of a revolution that few want to talk about even though, if not reversed, it will spell the end of Western Civilization. Accompanying this revolution has been the collapse of fertility rates, especially in Europe. This demographic revolution, aided by contraception and abortion, ensures that income transfer programs will not be sustainable for much longer\u2014perhaps no more than another ten years in Europe. Only the immigration to the U.S. of more fertile Latinos has postponed (but has not averted) the same outcome here.\n\nThe traditional two-parent family has been the bedrock of society throughout history. Even though some cultures, such as the ancient Greeks and the French Jacobins, flirted disastrously with other sorts of arrangements, no nation has successfully challenged the value of this God-given and time-honored institution. Supreme Court Justice Joseph Story affirmed the importance of the traditional bond of husband and wife, saying, \"Marriage is treated by all civilized societies as a peculiar and favored contract. It is in its origin a contract of natural law.\" When the home is built around a faithful monogamous marriage, social welfare programs are seldom necessary. Or as the social critic Michael Novak put it, the family is the original Department of Health and Human Services.\n\nMultiple studies show that married adults have better health, live longer, have fewer accidents, and are generally happier than single or unmarried cohabiting adults. Married women experience less domestic violence and are less likely to become victims of violent crime than single or divorced women. Equally important, children raised in traditional two-parent families perform better in school, are more likely to graduate from high school, and are less likely to be involved in juvenile crime. Girls raised in such families are much less likely to have a teen pregnancy. Married women are also less likely to experience poverty, whereas the number one predictor of childhood poverty is growing up in a household headed by a single mother. According to the U.S. Census Bureau, children living in households headed by single mothers are more than five times as likely as children living in households headed by married parents to be living in poverty\u201442.9 percent compared with 8.5 percent.\n\n175\n\nIn spite of all the evidence linking disruptions in traditional family formation to serious social pathologies, the federal government continues to issue bureaucratic mandates that reward dysfunction and penalize the two-parent family. In the 1960s, it was \"no-fault divorce,\" which made it relatively easy for couples experiencing normal marital difficulties to separate and file for divorce before fully considering the consequences for themselves and their dependent children. Then in the 1970s, leaders of the feminist movement, such as Betty Friedan and Gloria Steinem, managed to persuade millions of women that marriage and child-rearing was a prison devised by men to keep them in bondage. Friedan later recanted and apologized for the mayhem she caused, but the damage had already been done.\n\nWhen we measure the harm inflicted by all these reckless attempts to redefine the family, we have to wonder where such ideas come from. The sources of the chaos are the same ones we have run into time and time again. As reported by Jennifer Roback Morse, the socialist view of family formation, which is derived from classical Marxist dogma, has been one of the most insidious and persistent enemies of the two-parent family in our time. Marxist theory, she says, holds that relationships between men and women are characterized by conflict because of the problem of male dominance.\n\nAccording to Marx and Friedrich Engels, the authors of the _Communist Manifesto_ , the transition from group marriage to monogamy in primitive societies led to the subordination of women. Originally, they said, the typical household was a commune in which groups of men were the hunter-gatherers and groups of women were in charge of the household. When Betty Friedan published her bestseller, _The Feminine Mystique_ , in 1963, she picked up the Marxist line; but she was merely expressing the perspectives of the leftist intellectuals she had lived among during her student days at Smith College. The book became a dangerous weapon in the hands of the sixties radicals.\n\n176\n\nNot surprisingly, as Dr. Morse reports, liberal divorce laws were among the first priorities of the Bolsheviks following the Russian Revolution of 1917. Although Lenin remained married to the same woman for twenty-six years, he realized that the attack on marriage and family could be a powerful way of destabilizing bourgeois society and furthering his revolutionary goals. Ever since that time, the trend in liberal family law, Morse writes, has been to \"knock marriage off its perch.\"\n\nThe state, according to this trend, has no particular interest in channeling parenthood into marriage or assigning social and legal parental rights to the biological parents. Currently the family courts have enormous discretion in allocating custody and financial support among known parents. If the family-law radicals have their way, the state will not simply be recording parentage but actually determining parentage. This will vastly increase the discretion and, hence, the power of the family courts. In fact, this happened in December 2009, when a Vermont court ordered a Christian child to be taken away from her biological mother and given to a former lesbian partner.\n\nWe have to wonder how the American people allow such things to happen, but those who want to change the laws so that unspecified numbers and genders of people can claim the benefits of legal marriage have been very clever, using the language of \"choice\" and \"rights\" to gain public approval. As Dr. Morse suggests, this language has powerful appeal to many people today, but the rhetoric is deceptive and dangerous because it removes the stigma from aberrant behaviors that endanger children and their parents and ensures that the state will be involved at every level.\n\nThe government cannot seem to keep its hands off the family, constantly advocating for liberal causes that violate the sanctity of the family and the moral authority of Judeo-Christian beliefs. As an example, in February 2011, the attorney general of the United States informed Congress that the federal government would no longer enforce the Defense of Marriage Act (DOMA), which became law under President Clinton in 1996. This law, affirming marriage as the legal union of one man and one woman, effectively prohibits federal recognition of so-called _gay marriage,_ which is clearly why liberals in Washington are anxious to see DOMA repealed.\n\n177\n\nAt the same time the government's way of solving any problem always ends up robbing the federal treasury in order to fix the problems that government policy created in the first place. In 2008, total government spending on welfare programs such as Aid to Families with Dependent Children (AFDC) came to $714 billion. That amounts to more than $16,800 for every so-called poor person in America. As reported by the Heritage Foundation in January 2011, all of the welfare agencies and programs administered by the federal government have gotten substantial raises under the Obama administration. The 2011 federal budget increased funding for food stamps from $39 billion to $75 billion: a whopping 92.3 percent increase.\n\nDuring the first year of the Obama presidency, the number of Americans on food stamps increased by 5 million\u2014the largest single-year increase since the Carter administration of the mid-70s. The Republican majority in the House of Representatives has taken the first step in trying to get a grip on many of these unproductive schemes, including the effort to repeal the administration's nationalized healthcare bill, but if we really care about what kind of nation our children and grandchildren will inherit, we will have to increase our efforts to get the nation and the economy back on track.\n\nPROBLEMS IN AMERICA'S CLASSROOMS\n\nInscribed on the walls of the Widener Library at Harvard University are the words, \"You shall know the truth, and the truth shall make you free.\" Without referring to the person who spoke those words, many people on that Boston campus would naturally assume they were spoken by a great civil rights leader or some former president. The problem is not that many people don't know that Jesus Christ is the source of the quote but that hardly anyone on the campus today believes that truth can be known.\n\n178\n\nLittle by little over the past forty years, social and cultural revisionists have taken over the schools, the media, and the government institutions in this country. The speech codes enforced in some of those places would be more appropriate in Nazi Germany than \"the home of the free and the brave.\" Despite the blowback from conservative groups, speech codes and campus civility standards are increasingly common. University administrators and faculty members condone every sort of perversion these days, but the one thing they will not tolerate is the idea that somebody might claim to have \"truth.\" So much for Western civilization.\n\nAccording to a survey of university students conducted by former education secretary William J. Bennett, more than 70 percent of the 634 college students surveyed disagreed that the values of the United States are superior to those of other countries, with 34 percent strongly disagreeing. More than a third disagreed with the statement, \"The United States is the best country in the world.\" Eighty percent rejected the claim that Western civilization, with all our unparalleled achievements, is superior to Arab civilization. And as Bennett pointed out, perhaps the most striking finding was that a third of the respondents said they would actively evade a military draft in the War on Terrorism, while another third would refuse to serve abroad, and just one-third would be willing to fight for their country overseas.\n\nDespite the ongoing War on Terror and the tragic events of September 11, 2001, college students refused to condemn the actions of the 9\/11 terrorists. The multiculturalism being taught in American schools and colleges has been taken to such an extreme that college students today embrace virtually any culture but their own. The tendency for academics and their students to \"blame America first\" for the world's problems\u2014which has now apparently made its way into the highest levels of government\u2014can't help but have a negative impact on the nation's prospects for the future.\n\n179\n\nA few years ago a review of textbook requirements in New Jersey found that the state's guidelines for history omitted any mention of the Founding Fathers, the Pilgrims, or the _Mayflower_. A history textbook being used in Florida public schools, called _A World in Conflict_ , devotes the first five pages on the subject of World War II entirely to information about women in the armed forces, black soldiers on the home front, and Japanese internment camps. Surprising, perhaps, but this is typical of the situation in many school districts today.\n\nThe textbooks favored by multiculturalists generally ignore the contributions of Western civilization, particularly with regard to religious tolerance, individual liberty, democratic institutions, and the rule of law. The foreword to a textbook used in one New Mexico school district, called _500 Years of Chicano History in Pictures_ , states that the book was \"written in response to the Bicentennial celebration of the 1776 American Revolution and its lies.\" The author writes that her purpose is to \"celebrate our resistance to being colonized and absorbed by racist empire builders.\"\n\nUnfortunately the typical university classroom is not much better. Colleges are supposed to be places that encourage a free and open exchange of ideas, but that is not the case on most campuses these days. Many college courses focus on narrow areas that are highly specialized, narrowly interpreted, politically charged, and unabashedly promoting the holy grail of \"race, class, and gender.\" For centuries men and women from every continent, race, and socioeconomic background have come to this country to escape classification, but on today's campus, the emphasis is on group identity, racial differences, and liberal social policies.\n\nBy and large, America's universities have lost sight of what a college education is supposed to be. In this environment it is no surprise that students have little respect for their cultural heritage and little knowledge of the sacrifices our forefathers made for the freedoms they enjoy. When the National Association of Scholars surveyed course offerings at the nation's top fifty universities, they found only a third of those schools required a course in freshman English. Just 12 percent required any type of math, only 34 percent required students to study science, and only 4 percent had a philosophy requirement. None of them required students to study literature. But the most disturbing discovery was that just one of the top fifty colleges in America required students to take a course in American history.\n\n180\n\nIs it any wonder that so many young people do not seem to know who they are? Students arrive on campus unsure of themselves, ignorant about most things, except what is happening on Facebook or who is texting them on their iPhones. Critical thinking skills, which have been discouraged from their earliest years in public school, are often severely limited. Everything is about fitting in, not making waves, and getting a college diploma for the least possible amount of work. Students today are as bright as ever, but they have been raised in relative comfort in a country where life is pretty easy, and the vast majority have never learned to question what they have been taught.\n\nThe tragedy is that students and their parents are paying the price for what is happening on campus in more ways than one. Mom and Dad are paying tuition for an education that subjects their sons and daughters to speech codes, sexual intimidation, anti-American indoctrination, and anti-intellectual mind control. Students from inner-city schools and other low-demand environments often arrive on campus without the academic or social skills to resist this kind of indoctrination. They become easy prey for liberal ideologues and tend to accept the rhetoric of their professors without question.\n\nCurricula in our schools and colleges rarely, if ever, focus on America's greatness or the nation's contributions to democracy, language, culture, science, and industry. Instead, students are exposed to a steady drumbeat of criticism, blaming America for slavery, cultural chauvinism, imperialism, and religious intolerance. There was a time when Americans had so much confidence in the superiority of our way of life, columnist Mona Charen points out, that we taught our values to immigrants and insisted they master the basics of American history, language, and government before becoming eligible for citizenship. Today we are not even teaching those things to our own children.\n\n181\n\nA poll commissioned by the American Council of Trustees and Alumni (ACTA) examined knowledge of American history among seniors at the top fifty-five liberal arts colleges in the country. The questions covered subjects professors in the sixties would have considered high-school level, but 81 percent of those college seniors earned a grade of D or F on the exam. When they were asked to identify the source of the statement \"To each according to his needs, from each according to his ability,\" 35 percent guessed the phrase was from the U.S. Constitution. In fact, it is a principle of Marxist communism.\n\nMore than half of the students thought Germany, Italy, or Japan were United States allies in the Second World War rather than our enemies, the Axis powers. Fewer than a third knew that the Reconstruction era had to do with the Civil War, and 40 percent could not place the Civil War in the correct half century. Only 25 percent could identify James Madison as the \"father of the Constitution\" while just 22 percent knew that the words \"government of the people, by the people, and for the people\" come from Abraham Lincoln's Gettysburg Address.\n\nIf you wonder why there is such a shocking lack of knowledge in the nation's schools, consider that of the fifty-five highly rated institutions in the survey, not one of them requires a single survey course in American history. The implications of such studies should be obvious. Our schools are turning out a generation of functional illiterates who have little or no regard for the great gifts of liberty we enjoy, so how can we expect them to defend, protect, and preserve this great country when they know so little about it? Washington's solution is to simply define the problems away, but that is an unacceptable alternative.\n\n182\n\nTHE CANARY IN THE MINE\n\nA lot of things have gone right for America since the _Mayflower_ landed at Plymouth Rock over four centuries ago. The first Americans barely survived that first winter, but they persevered and proved themselves worthy of their calling. They were men and women of a resolute Christian faith, and God provided for their needs. He sent help when they needed it, just as He prepared the way for their descendants for generations to come.\n\nMany battles of the American Revolution were close calls; the patriots lost most of them for the first six years. The English army was the most formidable fighting force in the world at that time, and no one believed the colonists could defeat them. The final victory at Yorktown was nothing short of a miracle, but the Americans won because there was something even bigger at stake: God had plans for America.\n\nGod had prepared this land before the colonists ever arrived\u2014it was to become a \"city on a hill,\" a beacon of personal freedom and religious liberty to the world, a land where freedom could thrive. For most of our history we enjoyed unprecedented freedom and opportunity, but considering the struggles we face today from an out-of-control economy, the ongoing assault on traditional values, and the escalating threats to religious liberty, I have to wonder how much longer God will extend His hand of blessing to this nation.\n\nAs a Christian minister, I am deeply troubled by threats from the ACLU and other secular activists to change the tax status of our churches, preventing pastors from speaking freely about critical social issues from the pulpits. Without a renewed commitment to religious liberty and the freedom to speak openly in any venue, what is to prevent us from falling victim to intimidation, blackmail, and totalitarian ideologies of one sort or another? The welfare state cannot save us. Green energy will not save. us. More police, more jails, and more threats won't save us. Only God can save us, and unless the American people rise up and demand that the government get back on track, stop these predatory behaviors, and return to our founding principles, we are in for a long and bumpy ride over the next several years.\n\n183\n\nWe are already borrowing money from our grandchildren to pay for the government's reckless spending. How much longer can that continue? If someone had told the people of Russia back in the mid-1980s that their economy would collapse by the end of the decade, they would have laughed. But people who ought to know are warning us now, \"Stop spending! Get back to basics! We are headed in the wrong direction!\"\n\nI believe members of the Republican majority in Congress have heard the message. They are doing their best to respond, but they are also facing entrenched resistance from the liberals in the Senate and the executive branch. We have to wonder, however, what will it take for Washington to change course and take the steps necessary to avoid an impending collapse? Are the president, the Congress, and the American people willing to make the hard decisions that will be needed? Or will we have to wait until the economy collapses as it did for the Soviets? If so, there is no guarantee we can recover: there won't be anybody standing by to give us a bailout. But there are plenty of folks in the Middle East and the Far East who would love nothing better than to see America crash and burn. There are people in this nation, for that matter, who would love to take advantage of a struggling economy while we are still weak and defenseless. God help us.\n\nIn an article about religious liberty around the world, Cato Institute Senior Fellow Doug Bandow remarked that religious liberty is like the proverbial canary in the mine\u2014advance warning that things may be going wrong. \"If a state won't respect this most basic freedom of conscience,\" he said, \"it isn't likely to respect people's lives and dignity in any context.\" I believe that is a perceptive observation that applies not just to repressive Third World societies but to ours as well.\n\n184\n\nWhat we really need in this country is a restoration of Christian values and beliefs. I believe God may be withholding His judgment for the time being, giving us a chance to return to our founding principles. That does not mean Christians expect everyone to believe as we do or to agree with us on every issue. God cannot bless an immoral nation, but freedom of conscience is an essential right. The Constitution guarantees every citizen the right to believe according to the dictates of his or her own conscience. We wholeheartedly support the principle of \"religious liberty\" for others, and that is all we ask for ourselves and our families. But we will never sacrifice our God-given right to defend our beliefs and to share the hope within us with the world.\n185\n\n[_**Seven**_  \nWHAT YOU CAN DO](Lee_9780849949463_epub_c5_r1.html#d6e9)\n\nWhen the colonists of New England reached the point of utter exasperation with the predatory laws and policies of King George III and the British Parliament, Samuel Adams and the Sons of Liberty organized what the patriots referred to as \"Committees of Correspondence.\" These groups were created to make sure people in all parts of the colonies were informed about what was going on in the never-ending disputes with the Crown. Adams and the others wrote circular letters and pamphlets, delivered rousing speeches in the streets, town halls, and taverns, and distributed broadsides warning the people about the provocative actions of the English authorities, constantly fanning the flames of resistance.\n\nWhen Paul Revere made his famous midnight ride to Lexington, he did so as a member of the Boston committee, carrying the news that a large company of British regulars was moving in their direction. In the months leading up to the Revolution, there were reportedly as many as seven thousand members of these committees, led by Adams and such distinguished figures as John Dickinson, Patrick Henry, Richard Henry Lee, and Thomas Jefferson.\n\n186\n\nWhen Adams, along with Joseph Warren and James Otis, presented a \"Report of the Committee of Correspondence\" at Boston's Faneuil Hall on November 20, 1772, they recited a long list of grievances against the British and spelled out the core beliefs of the Founders that would be enshrined just a few years later in the Declaration of Independence and the Constitution. \"Among the natural rights of the Colonists,\" they said, \"are these: First, a right to life; Secondly, to liberty; Thirdly, to property; together with the right to support and defend them in the best manner they can. These are evident branches of . . . the first law of nature.\" They went on:\n\nAll men have a right to remain in a state of nature as long as they please; and in case of intolerable oppression, civil or religious, to leave the society they belong to, and enter into another. When men enter into society, it is by voluntary consent; and they have a right to demand and insist upon the performance of such conditions and previous limitations as form an equitable original compact. Every natural right not expressly given up, or, from the nature of a social compact, necessarily ceded, remains. All positive and civil laws should conform, as far as possible, to the law of natural reason and equity.\n\nThese words, so similar to the language of the Declaration, affirm the belief that all people possess a moral sense as a component of conscience and an innate sense of moral accountability. Furthermore these men believed the principles of right and wrong could be found in copious detail in the Bible, which was commonly available and generally accepted by the men and women of the founding generation as the Word of God. As the apostle Paul had written: \"For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who suppress the truth in unrighteousness, because what may be known of God is manifest in them, for God has shown it to them\" (Romans 1:18\u201319 NKJV).\n\n187\n\nLater, when the authors of the Declaration of Independence declared, \"all men are created equal, that they are endowed by their Creator with certain unalienable Rights,\" this is what they were saying. They believed no man-made law should arbitrarily abrogate or interfere with these natural rights, which are imbued by the laws of nature and the laws of God. Thomas Jefferson made this clear several years later in a letter to Richard Henry Lee regarding the process of drafting the Declaration:\n\nThis was the object of the Declaration of Independence. Not to find out new principles, or new arguments, never before thought of, not merely to say things which had never been said before; but to place before mankind the common sense of the subject, in terms so plain and firm as to command their assent, and to justify ourselves in the independent stand we are compelled to take. Neither aiming at originality of principle or sentiment, nor yet copied from any particular and previous writing, it was intended to be an expression of the American mind, and to give to that expression the proper tone and spirit called for by the occasion.\n\nJefferson's point of view had been expressed by Samuel Adams and the Boston Committee when they wrote in their report, \"In the state of nature every man is, under God, judge and sole judge of his own rights and of the injuries done him. By entering into society he agrees to an arbiter or indifferent judge between him and his neighbors; but he no more renounces his original right than by taking a cause out of the ordinary course of law, and leaving the decision to referees or indifferent arbitrators.\"\n\nNone of the Founders denied the authority of statutory law. Crimes ought to be punished, they said, and judges and juries were established for that purpose. But there was a law greater than the laws of men. When they spoke of \"the laws of nature and of nature's God,\" they were referring to commonly held standards of right and wrong. They believed all human beings have an inborn sense of right and wrong. While this knowledge can become corrupted, in a healthy society, they believed, the vast majority of people would naturally agree on these basic principles.\n\n188\n\nBy 1768, when four thousand British troops arrived in Boston to enforce the Townshend Acts, the patriots realized there could be no turning back. They realized the English were intentionally provoking the colonists, and the Townshend Acts were just one more attempt to undermine the autonomy of the colonies by compelling American merchants to comply with British trade regulations. Very much like the Declaratory Act of 1766, which was a poke in the eye demanding unquestioned obedience to the Crown, these new articles dared the Americans to rebel by declaring the British Parliament had the right to rule the colonies and to tax them as they wished. To make the point, they immediately placed new duties on paper, lead, paint, and imported tea.\n\nSamuel Adams and the Committees of Correspondence wasted no time publicizing these new threats. After the Boston Massacre of March 5, 1770, in which a British patrol fired on a crowd of angry colonists and killed five unarmed men, colonial outrage became even more intense. The British governor withdrew his troops to Castle Island to avoid further bloodshed, but the colonists were not appeased. Instead, they saw this as a sign of weakness. When the public funeral for the five victims of the massacre was held a few days later, more than ten thousand Americans joined the procession as a demonstration of unity. Yet when the Revolution began five years later, it was not the result of any one act but, as the Declaration of Independence expressed it, the result of \"a long train of abuses and usurpations, pursuing invariably the same Object.\"\n\nA NEW DECLARATION OF GRIEVANCES\n\nThe Committees of Correspondence's most important contribution in the colonial era was in providing a consistent, reliable, and readily accessible flow of information that would help mold the American people into a unified and cohesive force. Today, when the nation is confronted by a new and even more strident \"train of abuses,\" that function is being carried out by a new generation of patriots who are making a wide range of assets available to the public over the Internet and through talk radio, cable TV, cellular telephones, and many other kinds of twenty-first-century media. And, very much like that original band of rebels, these Americans are preparing for some major changes of their own.\n\n189\n\nIn a televised campaign address to his supporters on October 31, 2008, given on the campus of the University of Missouri, Barack Obama made a statement that drew little attention at the time. He said, \"We are five days away from fundamentally transforming the United States of America.\" To many viewers it sounded like a typically arrogant boast, not uncommon in politics. But within weeks of the January 2009 inauguration and the installation of a new cast of characters in the White House, the voters began to see what Candidate Obama actually meant by those words. Since that time there has been a stunning change in the tone and emphasis of national policy, but there has also been an equally stunning change of attitude toward the direction this administration has been taking the nation.\n\nThe Tea Party movement that emerged in early 2009, seemingly out of nowhere, would soon become the most visible response of middle America to this sudden shift in policy. The movement stands for smaller government, fiscal responsibility, individual freedom, and a conservative view of the nation's founding documents. Millions of Americans, motivated by their commitment to faith, family, and freedom, are joining forces across party lines to express their concerns.\n\nOne of the first appearances of the movement came on January 19, 2009, the day before the presidential inauguration, when a group of New York stock traders, concerned that the new administration would do serious harm to American business, began sending e-mails to their colleagues suggesting that everyone send tea bags to their representatives and senators. It was a modest beginning.\n\n190\n\nIn the following weeks Tea Party protests began to appear all across the country, with increasing sophistication and appeal. On February 1, 2009, the tea bag suggestion was picked up by individuals in several cities; then on February 27, an organized event was staged to protest the American Recovery and Reinvestment Act of 2009 that had been signed by the president just ten days earlier. By Tax Day, April 15, 2009, the protests were a nationwide phenomenon with more than eight hundred Tea Party events all across the country and as many as a million participants.\n\nBy July 4, 2009, the number of Independence Day events was too great to count, but Tea Party events of all sizes took place in thousands of communities from coast to coast, focused on love of country and the uniquely American Spirit of patriotism, expressing displeasure with the leftward lurch of government under the Obama administration. Millions of Americans were profoundly concerned about the direction of economic policies that, even at that stage, threatened to bankrupt the nation. By that point dozens of Tea Party organizations were being founded with sophisticated websites and a highly visible presence. These new groups began recruiting members and creating a network of like-minded citizens, most of whom had never before been politically active but who were anxious to take a stand against the bizarre vision of America they saw reflected in the White House and both houses of Congress.\n\nPopular conservative broadcasters, such as Rush Limbaugh, Sean Hannity, and Glenn Beck, are more popular today than ever. Beck, a radio and TV celebrity, asked his audience during a December 3, 2009, broadcast,\n\nIf I would have told you last year at this time that the government would own General Motors, Chrysler, and many of the banks and financial institutions, and AIG, that they would fire the CEOs, that they would threaten the banks, that they would shut them down unless they would take that [bailout] money . . . would you have believed it?\n\n191\n\nThe answer was a resounding _no!_ No one could have anticipated such changes, but this was the attitude of \"shock and awe\" that many people were feeling. And these were just some of the issues that would awaken the nation and sound the alarm for an all-out popular rebellion.\n\nIn a powerful book challenging the Obama administration's efforts to \"fundamentally transform\" the United States into a socialist nation on the European model, economist and syndicated columnist Thomas Sowell accused the administration of attempting to dismantle the constitutional framework of the republic by ignoring legal limitations on executive power and inventing laws to suit its own purposes, without the \"consent of the governed.\" A century ago, he said, politicians of the Progressive Era were the first to try to rewrite the Constitution according to their own left-wing ideology.\n\nBut what we are seeing today is an even more sinister attempt to transform the nation in ways that can only lead to disaster. Sowell writes, \"The Constitution was not only a challenge to the despotic governments of the time, but has been a continuing challenge\u2014to this day\u2014to all those who think the ordinary people should be ruled by their betters. . . .\" From day one the administration's idea of \"betters\" could be seen in the unprecedented appointment of a large number of public-policy czars with cabinet-level authority. Many of these individuals were financial experts from Wall Street firms such as Goldman Sachs and social activists with long-standing leftist connections. Within weeks the administration was carrying out a systematic raid of the public treasury. And this was accomplished with the aid of a compliant Congress intent on passing a string of massive spending bills so fast the public could not possibly keep up\u2014 laws, not coincidentally, that members of the House and Senate had not even bothered to read.\n\n192\n\nLittle by little the administration was able to carry off a wholesale restructuring of government, like nothing the country had seen in more than two hundred years. It was the New Deal, the Great Society, and the Great Leap Forward rolled into one. The long-established principles of \"checks and balances\" and \"separation of powers\" were ignored, but all this has led to a loud and predictable backlash from the American people, and the 2010 midterm elections were just the first step toward a return to normalcy. Suddenly the Tea Party movement was better organized, better connected, and better prepared for the confrontation with Washington\u2014prepared for what some deemed a second American revolution.\n\nAccording to a Gallup poll taken in the second week of January 2011, just 19 percent of Americans were satisfied with the way things are going in America, and the level of satisfaction among conservatives (at 12 percent) was less than half that of liberals (at 27 percent). Topping the list of voter concerns were the troubled economy, high unemployment, and the intrusiveness of Washington policy makers in practically every area of their lives. Whether the restrictions are on drilling for oil or on McDonald's Happy Meals, most Americans believe Washington liberals are sticking their noses into places where they have no business, and growing levels of public concern are stoking a grassroots rebellion like nothing this country has ever seen.\n\nIt may surprise some folks in Washington to know that, as reported by a December 2010 Rasmussen poll, more than half of Democrats (52 percent) and more than three out of four Independents (76 percent) now say the country is on the wrong track\u2014and 86 percent of Republicans strongly agree. Researchers found that 87 percent of all voters rate the ongoing problems with the economy as \"very important\"\u2014the highest level since August 2008. At the same time, 67 percent rate the concerns with government ethics and corruption as \"very important.\" And perhaps most surprising, Rasmussen researchers found that the number of adults identifying themselves as Republicans has jumped to 37 percent while the number of Democrats has dropped to 33.7 percent\u2014the lowest level since November 2002.\n\n193\n\nA February 2011 TIPP poll by _Investors Business Daily_ found that the majority believe the federal government should be smaller and provide fewer services. Overall, nearly six in ten of those surveyed said the government has too much power. Not surprisingly, 83 percent of Republicans and 72 percent of self-identified conservatives believe government is too powerful, but 64 percent of Independents and 62 percent of self-identified moderates feel the same way. In their assessment of these figures, the researchers concluded, \"This political alignment of Republicans with independents and the ideological alignment of conservatives and moderates make small-government supporters a force to be reckoned with.\"\n\nIF WE FORGET WHAT WE DID\n\nIn his farewell address, delivered on January 11, 1989, President Ronald Reagan offered a powerful statement of the importance of this nation's founding principles. He expressed his deep love for America, but he wondered if the American people really understand the dangerous challenges we face. He said,\n\nAn informed patriotism is what we want. And are we doing a good enough job teaching our children what America is and what she represents in the long history of the world? Those of us who are over thirty-five or so years of age grew up in a different America. We were taught, very directly, what it means to be an American. And we absorbed, almost in the air, a love of country and an appreciation of its institutions. If you didn't get these things from your family, you got them from the neighborhood, from the father down the street who fought in Korea or the family who lost someone at Anzio. Or you could get a sense of patriotism from school. And if all else failed, you could get a sense of patriotism from the popular culture. The movies celebrated democratic values and implicitly reinforced the idea that America was special. TV was like that, too, through the mid-sixties.\n\n194\n\nAnyone who grew up in this country during the first half of the twentieth century was schooled in the values and beliefs of the Founders. American history was an essential part of that. We were justly proud of the patriots' struggle for independence, and we understood that our free, republican, capitalist system was an engine of prosperity, not just for Americans but for the world. We believed the love of liberty engendered here could bless untold millions if we continued to abide by those principles.\n\nBut as the levels of affluence and material comfort expanded during the second half of the century, many people began to drift away from those core values. Intellectuals, academics, and media elites who had been influenced by leftist ideologies began to question the morality of their own country so that today, as President Reagan noted in his speech, \"some things have changed.\" He further stated,\n\nYounger parents aren't sure that an unambivalent appreciation of America is the right thing to teach modern children. And as for those who create the popular culture, well-grounded patriotism is no longer the style. Our spirit is back, but we haven't reinsti-tutionalized it. We've got to do a better job of getting across that America is freedom\u2014freedom of speech, freedom of religion, freedom of enterprise. And freedom is special and great. It's fragile; it needs protection. So, we've got to teach history based not on what's in fashion but what's important: Why the Pilgrims came here, who Jimmy Doolittle was, and what those thirty seconds over Tokyo meant.\n\nAnd then he added, more forcefully, \"If we forget what we did, we won't know who we are.\" That is the challenge we are facing at this moment in our history\u2014to remember not only what we did but also the enormous price our forefathers paid for the liberties we enjoy.\n\n195\n\nIf we expect to maintain the strength and vitality of this country for another two hundred years, we will need to insist that our children and grandchildren are taught the great and noble history of this nation, remembering the sacrifices of all the valiant warriors who paid the ultimate price for liberty. But they also need to know there are men and women, basking in the bounty of this great nation, who want nothing more than to undermine our freedoms and through persistent indoctrination bind the American people in ideological chains. The acid test of our willpower and resilience will be whether or not we still have the courage and strength to resist such people, to take a stand for righteousness, and to do whatever it takes to restore the foundations of the republic before it is too late.\n\nThe Reverend Jonathan Mayhew, who is often cited as the first member of the famed Black Robe Regiment of New England, was the pastor of the Old West Church in Boston and among the first to light the flame of rebellion in the hearts of the patriots. Refusing to be silenced by moderates and loyalists in his congregation, Mayhew dared to speak the truth regardless of the consequences. When he delivered his most famous sermon, \"A Discourse Concerning Unlimited Submission and Non-Resistance to the Higher Power,\" there were many who felt the young minister had gone too far.\n\nThose who expected men like Mayhew to bow to British demands cited the apostle Paul's admonition in Romans 13:1, \"Let every soul be subject to the governing authorities. For there is no authority except from God, and the authorities that exist are appointed by God\" (NKJV). But Mayhew was having none of it and declared, \"Common tyrants and public oppressors are not entitled to obedience from their subjects, by virtue of anything here laid down by the inspired apostle.\" Furthermore, he said, \"when [the king] turns tyrant, and makes his subjects his prey to devour and to destroy, instead of his charge to defend and cherish, we are bound to throw off our allegiance to him, and to resist.\"\n\n196\n\nWhen his sermon was published a few days later, Mayhew was obliged to pen a short preface in which he warned, \"Civil tyranny is usually small in its beginning, like 'the drop of a bucket,' till at length, like a mighty torrent, or the raging waves of the sea, it bears down on all before it, and deluges whole countries and empires.\" Then, in the text of the sermon he delivered on January 30, 1750, he said,\n\nWe may very safely assert these two things in general, without undermining government: One is, that no civil rulers are to be obeyed when they enjoin things that are inconsistent with the commands of God: all such disobedience is lawful and glorious. . . . Another thing that may be asserted with equal truth and safety, is, that no government is to be submitted to, at the expense of that which is the sole end of all government\u2014the common good and safety of society. . . . But it is equally evident, upon the other hand, that those in authority may abuse their trust and power to such a degree that neither the law of reason, nor of religion, requires that any obedience or submission should be paid to them: but, on the contrary, that they should be totally discarded; and the authority which they were before vested with, transferred to others, who may exercise it more to those good purposes for which it is given.\n\nMayhew was not a firebrand at that point\u2014he would become one a short time later and has been credited as being the first to utter the words, \"No taxation without representation!\" But at the conclusion of the sermon, he admonished the congregation to be cautious: \"Let us prize our freedom,\" he said, \"but not use our liberty for a cloak of maliciousness.\" Mayhew died at age forty-six, a full decade before the first shots were fired at Lexington and Concord, but he was memorialized years later as the first to sound the alarm, to awaken the colonists to their rights as free men and members of the family of God, and to proclaim from the pulpit that the defense of liberty in the face of tyranny is a just and righteous cause.\n\n197\n\nTHE LOYAL RESISTANCE\n\nWhen a new generation of patriots stands up today to resist the incursions of government, they are not doing so in a malicious spirit. There have been many attempts to link the tea parties to thugs and zealots of one kind or another, but the charges won't stick. Amid all the confusion following the shooting of Congresswoman Gabrielle Giffords in January 2011, as one example, reporters and national news anchors were too quick to suggest that this detestable act was the work of a rogue \"teabagger.\" Within hours, however, it became apparent that the shooter was a known psychopath who was not connected to the tea parties or conservatives of any stripe but was, in fact, attracted to satanism and other crazy ideas.\n\nEven a cursory look at the websites and literature of some of the better-known Tea Party organizations makes it clear their members are normal, everyday Americans who care deeply about the well-being and safety of their country. Confronted by trillion-dollar deficits, outrageous bailouts to Wall Street billionaires, and a laundry list of payoffs to political cronies, the Tea Party faithful are simply doing what the patriots of 1776 felt obliged to do: they are standing up and saying no to the perversion of our constitutional form of government by the unjust governing authorities.\n\nGroups such as Tea Party Patriots, FreedomWorks, Let Freedom Ring, the Tea Party Express, and several others that are growing in size and influence were founded by average Americans who have become profoundly concerned with the leftward lurch of the federal government. All these groups have become mobilized to express the concerns of millions of men and women about the direction the progressive establishment has been taking the country. Representing a wide range of views, these folks are everyday patriots who want to make sure their voices are heard loud and clear in the nation's capital and in state legislatures around the country.\n\nThe groups include individuals who are vocal advocates of lower taxes, less government interference in our everyday lives, and a restoration of individual liberty. To accomplish all these goals, most of the Tea Party organizations recruit, train, and mobilize thousands of volunteer activists in all parts of the country, providing them with facts, information, and other important resources that will help them speak up and stand up in defense of our core values.\n\n198\n\nIn general, all of them agree that congressional spending is out of control. Dating back to the first bailouts of subprime mortgage lenders by the Bush administration in 2008, the federal government has been on a spending binge that threatens not just the economy but the survival of the nation. At the same time irrational restraints on energy production and the growth of American business have dealt a crippling blow to the economy and contributed to the ongoing scourge of joblessness while new government regulations and exorbitant taxes have slowed the growth of innovation and job creation in all sectors. This is why freedom-loving groups of all kinds are speaking up, demanding economic reforms.\n\nAnother major concern of the movement is border security. With twenty million to forty million illegal aliens already in this country, it is obvious the Obama Justice Department is not interested in protecting American sovereignty. They are allowing millions of poorly educated and unskilled workers to cross into this country with little or no resistance, primarily because they believe these individuals\u2014who have only limited knowledge of the country's history and values and even less knowledge of democratic institutions\u2014are likely to become supporters of the president's party in the voting booth.\n\nJob-killing measures are also a major concern of Tea Party activists. The union policy of \"card check\" would strip employees of the right to express by secret ballot their wishes concerning unionization. Without secret ballots, workers could run the risk of jeopardizing their personal safety and job security if they disagree with the union bosses. Conservatives are also concerned about the unrealistic demands of the Environmental Protection Agency and other government agencies that are strangling new energy development while gobbling up private property and public lands all over the country.\n\n199\n\nFinally, they support repeal of the disastrous health-care legislation passed by the 111th Congress, along with tort reform limiting the ruinous rewards currently being paid to predatory trial lawyers and their clients. Government mandates passed by presidential fiat, bypassing Congress through the use of executive orders, have been disastrous for the economy and every American. To restore balance, Congress must, at the very least, begin a disciplined study of the problems within large government programs, such as Social Security, Medicare, and Medicaid, in order to preserve what is best about each of these programs and eliminate exploitation, overregulation, and waste.\n\nTHE GREAT REPUDIATION\n\nFor many moderates, Independents, and others who may be getting their first glimpse of the Tea Party movement, there is a certain amount of curiosity and perhaps a little nervousness about what really is going on. They want to know what's behind this new attitude of dissent: What is causing so many average Americans to rebel against the government? Where do these people get their ideas, and what is motivating such a large, diverse, and apparently leaderless segment of the electorate to do what they are doing?\n\nThe spirit of rebellion we are seeing in cities and towns all across America comes from the same spirit that motivated the colonists to demand independence from the British Empire in 1776. It is the knowledge that essential principles and beliefs are being compromised by an administration with a very different idea of what this nation is all about. As author and scholar Michael Novak has recently written, this uprising is not simply a political rebellion or a rejection of Washington's out-of-control spending habits: it is \"a revolution against moral decline.\"\n\nThe spirit that came out of the Great Awakening of the early eighteenth century was a rededication by hundreds of thousands of men, women, and children to the foundational principles of the Christian faith. It was also a response to the knowledge that a righteous God will not withhold His judgment forever. This was the first full expression of the American spirit of independence, and it brought with it a willingness to stand up for what we believe to be true and right. That same spirit is alive in the Tea Party movement today.\n\n200\n\nThe 2010 midterm election was a landslide victory for conservative voters. As James Ceaser has pointed out at RealClearPolitics.com, the Democratic Party suffered \"the greatest defeat for a newly elected president in a midterm since the Republican Party under Warren Gamaliel Harding in 1922.\" The Democrats lost sixty-three seats in the House of Representatives, making them the minority party. They also lost six seats in the Senate but still hold a slight numerical advantage with fifty-one Democrats, forty-seven Republicans, and two Independents in that body.\n\nThe Democrat defeat was historic not only because of the number of House seats lost but also because of the number of seats lost in any midterm election\u2014it was the most since 1938. As James Ceaser says, it is the performance of the president's party following his first election that offers the most critical point of comparison. He writes that the 2010 midterm elections turned out to be the closest this nation has ever come to a national referendum on the \"ideology\" of a presidential administration.\n\nMr. Obama disguised his true ideology with a vague promise of hope and change during the campaign; then he immediately became the most ideological and divisive president in history after the election. Ceaser writes,\n\nSome of his supporters like to argue in one breath that he is a pragmatist and centrist only to insist in the next that he has inaugurated the most historic transformation of American politics since the New Deal. The two claims are incompatible. Going back to the major political contests of 2009, beginning with the Governors' races in Virginia and New Jersey and to the Senate race in Massachusetts, the electorate has been asked the same question about Obama's agenda and has given the same response. The election of 2010 is the third or fourth reiteration of this judgment, only this time delivered more decisively. There is one label and one label only that can describe the result: _The Great Repudiation_.\n\n201\n\nIn the same spirit Michael Novak has written, \"No president in American history has ever been so thoroughly discredited after two years as Barack Obama.\" The loss of fifty-four House seats in 1994 was a shocker for the Clinton White House, but the debacle of 2010 was even greater because it could only be interpreted as a rejection not only of administration policies but also of its overall ideology.\n\nIn his first two years as president, Novak says, Mr. Obama convinced millions of Americans he wanted to make this country more like France or Germany or some other European welfare state, but the American people hated the idea and rebelled with undisguised intensity. The shakeup taking place in Congress and in state houses nationwide contributes to a trend that the recently elected Republican governor of South Carolina, Nikki Haley, calls a new generation of rock-star governors who don't care if they aren't reelected as long as they get results.\n\nThe most revolutionary aspect of such changes is how a popular movement with no national leader has risen up spontaneously from the grassroots and is now changing the political climate in this country and far beyond. This new generation of Tea Party patriots is not looking to opinion polls or talking points for their ideas: instead, they are looking to the Founding Fathers and to the principles of personal responsibility and limited government that have been a part of our American heritage for generations. While the Obama administration may occasionally pay lip service to these ideas, it is increasingly apparent that the administration's eyes are fixed on the European model of the welfare state. Fortunately the people are saying no to that insidious plan, and the Tea Party movement is in the process of installing speed bumps all along the way.\n\n202\n\nDrawing the attention of millions of Americans back to the authentic source of our laws and policy\u2014the United States Constitution and the revolutionary ideas it contains regarding individual freedom and republican democracy\u2014has been the greatest achievement of the movement. As I suggested in chapter 1, sometimes adversity can be our best ally, and the awakening we are seeing today is due in large part to the nightmare we have endured.\n\nWe are not out of the woods by a long shot. The challenges for our elected representatives have never been greater. The popular culture is still spreading its tentacles, submerging millions of America's young people in decadent, pointless, and empty lives. We have to wonder in this environment if God can continue to bless a nation so deeply immersed in sensationalism, sensuality, and sin. While many people are working hard to restore our political fortunes, the media-driven culture is, as Michael Novak writes, \"becoming more and more decadent, less and less under the sway of personal moral responsibility, more relativist, less under the self-control of reason.\"\n\nThe \"superculture,\" he says, hangs over the nation like a miasma of moral smog. Rather than holding up an image of hope and restoration, our liberal elites have become cheerleaders for moral decline. In his classic six-volume study of ancient civilizations, _A Study of History_ , the English historian Arnold Toynbee wrote that one of the signs of collapse in the great empires of the past was what he described as a \"schism in the soul,\" which was an unresolved struggle between right and wrong that ultimately led to moral chaos. When the elites of a culture, the \"creative minority,\" become utterly dissipated and immoral, they become an \"oppressive minority,\" and such a nation invariably enters a \"time of troubles\" that inevitably leads to collapse.\n\nThe hope for America in this second decade of the twenty-first century is that we will not allow our nation to be utterly corrupted, either politically or morally, by today's cultural elites. The 2010 midterm elections were an indication that a political and moral revolution has begun; however, it is my prayer that a cultural revolution\u2014with a new attitude of personal responsibility, compassion, and reverence for the things that truly matter\u2014will not be far behind. We are not without resources; God forbid that Toynbee's prophecy of national decline should ever happen here.\n\n203\n\nTHE PATHWAY TO VICTORY\n\nAt the close of the Constitutional Convention on September 18, 1787, a large crowd of anxious citizens had gathered outside of Philadelphia's Independence Hall, waiting to learn whether or not the delegates had successfully completed the task of drafting the new Constitution. One by one the delegates left the building, making their way home through the mass of onlookers. As he was leaving, the delegate from Maryland, James McHenry, happened to overhear a lady, later identified as Mrs. Powel of Philadelphia, calling out to Benjamin Franklin. \"Well, Doctor,\" she said, \"what have we got\u2014a Republic or a Monarchy?\" Glancing her way, Franklin replied, \"A Republic, if you can keep it.\"\n\nFranklin knew what millions of Americans have discovered since that day. It takes a lot to keep our republic. There are always those within and without who are trying to take our liberties from us. America's Founding Fathers and early patriots gave their lives as well as their possessions to preserve and protect our freedoms. Our fathers and mothers and sisters and brothers have given the ultimate sacrifice on battlefields and beaches around the world in the effort to preserve our liberties\u2014and those sacrifices continue even today. But what are we to do? Those of us who are here in the homeland, those of us who are the ordinary folks, those the politicians call the American People\u2014what can we do to bring about a new revolution to preserve our nation, the greatest nation in the free world, and recapture the liberties and ideals that have been stolen from us over the past few years?\n\nThe hope and change we were promised during the 2008 campaign season has turned out to be something quite different from what most Americans had in mind. But I believe the plan for real constructive change is not all that complex or confusing: every American can become a part of the national movement for God and Country. The first step is to stop complaining and whining about the problems in Washington, DC. That won't get the job done. The second step is to take action; there are things we can do to counteract the progressive assault on America and take our country back\u2014as Ben Franklin put it, to preserve the republic. So let me suggest a clear and honorable pathway that any generation of Americans may follow in order to protect what is right and to change what is wrong within this great nation.\n\n204\n\nFIRST: WE MUST PREPARE PERSONALLY\n\nPreparation is a vital part of every successful endeavor. When we read about the struggles of the fifty-six signers of the Declaration of Independence in their efforts to reach an agreement on the contents of that document, we can be certain their decisions were not made without a great deal of forethought and personal preparation. The final words of the Declaration make it clear they had settled in their hearts that they were all taking a great personal risk. They knew the formation of this new nation would come at a high personal cost to each of them.\n\nWritten just above their signatures are these words: \"And for the support of this Declaration with a firm reliance on the Protection of Divine Providence, we mutually pledge to each other our Lives, our Fortunes, and our sacred Honor.\" Once they signed their names to the document, there would be no turning back. Once their names were penned on that page, they had committed an act of treason against the Crown. After writing his name in large, bold strokes, John Hancock quipped to the assembled delegates, \"Gentlemen, we must all hang together.\" To which Benjamin Franklin replied without hesitation, \"We must, indeed, all hang together, or most assuredly we shall all hang separately.\"\n\nWe ought to take a lesson from the Founding Fathers: If we are willing to pay whatever price is demanded from us, to go any distance set for us, to endure any battle fought against us in order to reclaim America in our generation, then and only then, through such personal dedication we will be prepared to turn the tide of anti-Americanism that is overtaking our land. That determination is powerful and personal, but it is also essential if there is to be any chance of victory.\n\n205\n\nSECOND: WE MUST PARTNER WITH OTHERS\n\nThe strength of the Founders and those forces that achieved victory in America's first Revolution came from their sense of unity and the common goal of advancing the cause of liberty. It would be foolish to think all the colonists were in favor of the Revolution. It is doubtful, in fact, that the forces that did join together even liked each other very much. There was a great deal of distrust and friction between the colonies over a myriad of issues, but the patriots of 1776 believed the common good far outweighed the likes or dislikes of the individual. They found their desire for \"a land of the free\" to be a more noble passion than their petty differences.\n\nLook across America today, and you will see millions of good and ordinary people who love this country. They go out every morning and put in an honest day's work. They honor our flag, and they get tears in their eyes at the singing of our national anthem. They don't understand why anyone would want to destroy what we have been given. These men and women are ready to go out and do something to make a difference.\n\nMeanwhile the social elitists look down from their ivory towers with disdain for the ordinary man and woman, never realizing that if these \"ordinary Americans\" ever partner to recapture their land, there is no force on earth that can stop them. And the need for a dedicated, committed, unified, and unwavering band of freedom-loving patriots has never been greater than it is today. Today we are seeing signs of that unity all around us. Everywhere we look, Americans are rising up together. We see it on the Internet as literally dozens of organizations have been formed, dedicated to the idea of restoring American values. Tea Party organizations at the national, regional, and local levels are all a part of that movement, and if and when all these various groups come together for the sake of liberty, they will be a mighty force indeed.\n\n206\n\nThis is not just some idea for the future; it can start now. Let's not wait to follow the groups. Individually, let's find and befriend someone else who loves America, someone else who sees it as his or her calling to stand for those foundational liberties that are so valuable and so rare.\n\nTHIRD: WE MUST LEARN THE PROCESS\n\nWithin the God-given wisdom of our founding documents, we have been granted clear and certain processes for bringing about change in those areas we perceive to be wrong for our land. From the local municipalities to the halls of Congress and the White House, embedded in the laws and governmental processes of America are pathways for nonviolent moral, social, and political reform. But those processes must be learned and understood before they can be properly applied. Being passionate about an issue is fine, but real and lasting change requires a disciplined approach, using the systems that are already in place for us, by which we can make those changes take place.\n\nIf you don't like the votes your elected representatives are casting in Congress or the state legislature, then you have the power to remove them and elect others who will do the job better. If you don't like the idea of forced health care or of redistribution of wealth through excessive taxation, stop complaining and do something about it. The process for making changes\u2014from the schoolhouse to the courthouse to the White House\u2014is available to every citizen. So learn the process, and dig into your rights as a citizen. Learn the American process of government, and use it for the good.\n\nFOURTH: WE MUST PARTICIPATE IN THE ARENA\n\nParticipating within the process for change is the ultimate key to success. It is futile to gripe and complain concerning what we consider wrong or unjust if we are not willing to participate in making changes for the better. Most Americans would be startled if they knew how many of their friends and fellow citizens never bother to vote. According to statistics compiled through Restore America's demographic study of voting, there are now fifty-two million evangelical Christians of voting age in this country. In the 2010 general election, however, only twenty-two million of them voted. Another twenty million who were registered chose not to vote. And at least ten million who were eligible to vote didn't even bother to register. The other side loves to see statistics like that, but we must not give them the pleasure of laughing at us next time.\n\n207\n\nEven if these numbers are disappointing, they nevertheless show the great potential for positive change if we can rally not just these uncommitted evangelicals but also the millions of patriotic Americans who love their country and who have every reason to participate in the process of government.\n\nFIFTH: WE MUST PROCLAIM THE TRUTH\n\nThe Bible tells us, \"you shall know the truth, and the truth shall make you free\" (John 8:32 NKJV). But as Christians we also believe that when the truth is known, those who have found it have a duty to make it known to others. I often fear that many of the problems we have experienced in this country over the past half century are due in large part to the fact that the men and women in the pews and pulpits of America have not taken the challenge seriously. No other nation has as many churches and places of worship as we do in this country, yet the mighty roar of honor, virtue, and truth that arose from the churches in times past now seems to have been replaced with a pitiful whimper. Where there was once courage and conviction, too often today we find apathy and self-doubt. But we have no one to blame but ourselves.\n\nIt was the preaching of men such as Jonathan Edwards, Gilbert Tennent, and George Whitefield during the Great Awakening that forged upon the anvil of truth the foundational values so vividly inscribed in our founding documents. Throughout the revolutionary era, America's pastors preached with a fiery passion against the tyranny of the British monarchy\u2014they cried out for revolutionary change. They spoke the truth boldly, without fear of men or governments because they believed this nation had been given a special mission by God Himself. They were fully committed to the cause of liberty.\n\n208\n\nThanks to the propaganda machine of groups such as the ACLU, People for the American Way, and the Freedom from Religion Foundation, along with secular liberals in the mainstream media, our pastors and teachers have been beaten into submission, silenced. They say nothing that would offend the dogma of political correctness. Hollow threats concerning the loss of their tax-exempt status, or violation of the policy of \"separation of church and state,\" have turned the power of the pulpit into something far less than it is intended to be. What has happened to the backbones of our pastors? My prayer is that a new generation of pastors and teachers will arise who will preach with power and boldness, with the knowledge that they have no one to fear other than our eternal God Himself.\n\nBut none of this should suggest that it is only pastors, teachers, and other leaders who are called to speak up boldly about what is going on in America today. We all have our pulpits. They may be in our homes, our workplaces, our schools, or our neighborhoods; there are many places where we can speak up and express our views on the issues of the day, where we can take a stand for what is right and do whatever we can to help bring about a restoration of virtue and honor in this nation.\n\nFINALLY: WE MUST PERSEVERE UNTIL WE PREVAIL\n\nWhen fighting for the right, we must never cease until we prevail. The battle is not always won by the strongest, the smartest, or the most gifted, but ultimately the victory comes to those who persist and persevere to the end. When General and soon-to-be president George Washington led his troops into battle during the first six years of the American Revolution, he lost most of those battles. He and his troops endured tremendous challenges and suffered great hardships in taking on the most powerful army in the world, but through perseverance and faith in the nobility of their cause, they won the battles that really mattered, and ultimately they won the War of Independence. It is because of that fierce determination and perseverance that we are a free and independent nation today.\n\n209\n\nBut here is the best news: All the resources we need to regain our freedom are available now. There really is no reason we cannot reclaim America now. The forces threatening our liberties today are small, compared to the obstacles Americans of previous generations have had to overcome. So instead of moaning and complaining, we need to get up, get out, gather with others who share our passion, and do it.\n\nIf we have the courage of our convictions, we are able to take this country back from those who are doing their best to destroy it from within. And when we make the decision to become engaged, we are not just doing it for ourselves but for all those who come after us. We are also doing it to honor the legacy that was given to us by all those brave men and women who came before us. What a tragedy it would be if, having been given so much, we should fail to pass on to future generations the blessings of liberty that have been entrusted into our hands.\n\nTHINGS YOU CAN DO TODAY\n\nThe renewal of a culture is not a job for any one individual, group, or political party; it is a challenge for every citizen. The motivation for this book is the hope that every American will think carefully about all the achievements the men and women of this nation have made over the past four hundred years, and will make a personal commitment to join the struggle for renewal\u2014that would be the greatest Great Awakening I can imagine.\n\nThat process begins on a personal and private level, between you and God. Following that spirit of rededication, there needs to be evidence of renewed hope and resolve, and I would suggest that one of the ways all the various conservative groups can make a big difference is by making cultural renewal a part of the overall game plan.\n\n210\n\nHere are some things all of us can do today to get involved in this growing movement:\n\n1. Pray for America, and pray that with God's help We the People will prevail in the struggle for the heart and soul of the nation.\n\n2. Register to vote; then make sure you know the positions of the candidates on all the critical issues.\n\n3. Let your national, state, and local representatives know how you feel by attending town hall meetings, writing letters, calling their offices, and writing letters to the editors of your local newspapers, magazines, and community websites.\n\n4. Join a group in your area that is speaking up about these concerns, and if there isn't one already, start one yourself.\n\n5. Help support organizations that are standing up for fiscal, social, and moral responsibility.\n\n6. Subscribe to newsletters, blogs, Twitter feeds, and other media updates to help you stay informed.\n\n7. Use the Internet to find out what the candidates and incumbents are saying on both sides of the issues.\n\n8. Attend meetings of your local political organizations.\n\n9. Attend meetings and scheduled events of the organizations you support, and offer to help with promotion, arrangements, and logistics or in any way you can.\n\n10. Spread the word among your friends and neighbors, and help others in your community become active in the movement to restore the founding principles of this great nation.\n\nAll these things are in the best traditions of the revolutionary spirit. John Adams once wrote, \"What do we mean by the American Revolution? Do we mean the American war? The revolution was effected before the war commenced. The revolution was in the minds and hearts of the people; a change in their religious sentiments, of their duties and obligations. . . . This radical change in the principles, opinions, sentiments, and affections of the people was the real American Revolution.\" This is what I have been saying: the real revolution is always in the minds and hearts of the people.\n\n211\n\nUntil recently I often sensed an attitude of hopelessness in this country, of which President Obama and company were the progenitors. I saw it in many settings, but I believe things are changing now. There are still many challenges and plenty of work yet to do, but I am seeing a new spirit of hope emerging. It is not just a slogan of Hope and Change but real hope, coming from a renewed sense of purpose.\n\nAll of this brings us back once again to the American Spirit, to our identity as citizens of this great nation, and to what we stand for as a people. It is about the American experience. Why are you here? What is your purpose as a man or woman, a member of a family, a citizen, an employee or employer, a taxpayer, a child of God? The way we answer such questions says a lot about what we believe and where we stand.\n\nWhat I hope to see in coming weeks and months is a new level of optimism about America's future. We have every reason to be optimistic if we continue to walk in the footsteps of the Founders\u2014politically, morally, and spiritually. Not every man or woman putting in eight or ten hours a day on the job can run out and join the march on city hall, and I don't expect every mother with three children at home to go out and join the Tea Party and carry banners on the Capitol steps. But there are things everyone who cares about the future of the republic can do.\n\nMoms and dads can teach their children what it means to be an American; they can take time for family devotions, pausing to consider the important role faith played in the lives of the Founders. They can send e-mails, make phone calls, donate to good causes, and contribute to a blog or a social network. They can make a difference by speaking out about what they believe in. If all of us who care for this country would do just one or two of these things, we would be making as valuable a contribution as anyone. It doesn't have to be big, but it ought to be something. Each of us can do our part as a caring citizen and be a voice for righteousness in our national culture.\n\n212\n\nThis is the Coming Revolution: this is a transformation we can all take part in, living up to the dream the Founders gave us all those years ago. The American Dream is not about government handouts or class warfare with one class of citizens fighting against another. It is about a way of life in a free society, where each man and woman has the right of self-determination. The fact that such a life is so incredibly rare is what has made this country the envy of the world.\n\nEach facet of liberty that we now enjoy has been given to us by the sacrifices\u2014and often the very blood\u2014of those who have gone before us. Let every American go out into the public arena and speak proudly about those great freedoms that we cherish as Americans and refuse to surrender to the onslaught of the radical left-wing minority. But most importantly, we must flood America's voting booths in the coming elections and put into office those men and women who are willing to go to Washington for the distinct purpose of reinstating those rights and freedoms that are currently being stolen and reinstituting the values that once made our nation great.\n\nIf and when this happens, the feeble wall of ultra-liberalism and godless socialism will crumble beneath the force of the will of the people. Once again the economic might of our free enterprise system will strengthen us. The principles of common sense and decency will have their rightful place in our nation's capitol, and America will return to its place in the world as the \"shining light upon a hill.\"\n\nBest of all, a much-needed revival of patriotism and pride will fill our nation's heart. For we will have fought the noblest of battles and won our own revolution to return America to \"the land of the free and the home of the brave\" once again. We can. We must. And I truly believe we will!\nNOTES\n\nINTRODUCTION\n\n. Douglas Schoen, \"Polling the Occupy Wall Street Crowd,\" _Wall Street Journal_ , October 18, 2011, <http:\/\/online.wsj.com\/article\/SB100014240529704479504576637082965745362.html>.\n\nCHAPTER ONE: PORTRAIT OF A NATION\n\n. Giving USA Foundation, Giving Report 2009 (Center on Philanthropy at Indiana University, June 9, 2010), <http:\/\/www.givingusa.org\/press_releases\/gusa\/gusa060910.pdf> (accessed May 2, 2011).\n\n. Alexis de Tocqueville, _Democracy in America_ , ed. J. P. Mayer, trans. George Lawrence (New York: Harper Perennial, 1988), 513.\n\n. Ibid., 554.\n\n. Ibid.\n\n. William Ewart Gladstone, \"Kin Beyond Sea,\" _The North American Review_ , September\u2013October 1878, 185.\n\n. Russell Kirk, The American Cause (Wilmington, DE: ISI Books, 2002), 18\u201319.\n\n. Charles Colson with Ellen Santilli Vaughn, _Against the Night: Living in the New Dark Ages_ (Ann Arbor, MI: Servant Publications, 1989), 67.\n\n. Kirk, _American Cause_ , 19.\n\n. Mark Mather and Diana Lavery, \"In U.S., Proportion Married at Lowest Recorded Levels,\" Report of the Population Reference Bureau, Sept. 2010, https:\/\/www.prb.org\/Articles\/2010\/usmarriagedecline.aspx.\n\n. Centers for Disease Control and Prevention, \"Births: Preliminary Data for 2008,\" _National Vital Statistics Report_ 58, no. 16 (April 6, 2010): 13, table 7, <http:\/\/www.cdc.gov\/nchs\/data\/nvsr\/nvsr58\/nvsr58_16.pdf> (accessed May 2, 2011).\n\n214\n\n. Daniel Patrick Moynihan, _The Negro Family: The Case for National Action_ 1, U.S. Department of Labor, Office of Policy Planning and Research, 1965, 44.\n\n. William J. Bennett, _The Broken Hearth: Reversing the Moral Collapse of the American Family_ (New York: Doubleday, 2001), 178\u201379.\n\n. Ibid., 179\u201380.\n\n. Ibid.\n\n. Jon Kraushar, \"We Need More 'Sullys,'\" FoxNews.com, January 15, 2010, <http:\/\/www.foxnews.com\/opinion\/2010\/01\/15\/jon-kraushar-miracle-hudson-sully-sullenberger-usairways-hero#ixzz1PXqeYeW9> (accessed May 2, 2011).\n\n. Michael Phillips, interview by Dena Ross, \"Beyond Fighting for the Flag,\" Beliefnet.com, <http:\/\/www.beliefnet.com\/Inspiration\/2005\/07\/Beyond-Fighting-For-The-Flag.aspx> (accessed May 2, 2011).\n\n. \"Medal of Honor Nominee Celebrated,\" ABC News, May 27, 2005, http:\/\/abcnews.go.com\/GMA\/story?id=796298&page=1. See also: Michael M. Phillips, \"In Combat, Marine Put Theory to Test, Comrades Believe,\" _The Wall Street Journal_ , May 25, 2004.\n\n. _Northwest Ordinance_ , art. III (1787).\n\n. Rush to the Citizens of Philadelphia, March 28, 1787, in _Letters of Benjamin Rush_ , vol. 1, 1761\u20131792, ed. L. H. Butterfield (Princeton, NJ: Princeton University Press, 1951).\n\n. Massachusetts Constitution of 1780, part 1, art. 2.\n\n. John Adams to Zabdiel Adams, June 21, 1776, in _Letters of Members of the Continental Congress_ , vol. 1, _August 29, 1774, to July 4, 1776_ , ed. Edmund Cody Burnett (Washington: Carnegie Institute, 1921), 501.\n\n. Elias Boudinot to Society of the Cincinnati, September 23, 1783, in _The Life, Public Services, Addresses and Letters of Elias Boudinot, LL. D_., ed. Jane J. Boudinot (Cambridge, MA: Riverside Press, 1896), 365.\n\n. Paul Hollander, _Anti-Americanism: Rational and Irrational_ (New Brunswick, NJ: Transaction Publishers, 1995). \"Higher Education: Reservoir of the Adversary Culture\" is a section included in Part I of this work.\n\n. \"World population must be stabilized and to do that we must eliminate 350,000 people per day. This is so horrible to contemplate that we shouldn't even say it. But the general situation in which we are involved is lamentable.\" Jacques Cousteau, interview by Bahgat Elnadi and Adel Rifaat, _UNESCO Courier_ , November 1991, 13.\n\n. Jim Nelson Black, _Freefall of the American University: How Our Colleges Are Corrupting the Minds and Morals of the Next Generation_ (Nashville: Thomas Nelson, 2004), 4.\n\n. Wendy S. Grigg, Mary A. Lauko, and Debra M. Brockway, \"The Nation's Report Card: Science 2005,\" U.S. Department of Education, National Center for Education Statistics, May 2006, <http:\/\/nces.ed.gov\/nationsreportcard\/pubs\/main2005\/2006466.asp>.\n\n. Dan Lips and Jena Baker McNeill, \"A New Approach to Improving Science, Technology, Engineering, and Math Education,\" Heritage Foundation, April 15, 2009, <http:\/\/www.heritage.org\/Research\/Reports\/2009\/04\/A-New-Approach-to-Improving-Science-Technology-Engineering-and-Math-Education> (accessed May 2, 2011).\n\n215\n\n. \"2009 Program for International Student Assessment Scores,\" _Wall Street Journal_ , December 7, 2010, <http:\/\/online.wsj.com\/public\/resources\/documents\/st_PISA1206_20101207.html> (accessed May 2, 2011); Sam Dillon, \"Top Test Scores from Shanghai Stun Educators,\" New York Times, December 7, 2010, <http:\/\/www.nytimes.com\/2010\/12\/07\/education\/07education.html?_r=1andpagewanted=2> (accessed May 2, 2011).\n\n. Amanda Paulson, \"'Report Card' on Science: Most U.S. Students Aren't 'Proficient,'\" _Christian Science Monitor_ , January 25, 2011, <http:\/\/www.csmonitor.com\/USA\/Education\/2011\/0125\/Report-card-on-science-Most-US-students-aren-t-proficient>.\n\n. Statement by U.S. Secretary of Education Arne Duncan on the release of the NAEP Science Report Card, U.S. Department of Education, January 25, 2011, <http:\/\/www.ed.gov\/news\/press-releases\/statement-us-secretary-education-arne-duncan-release-naep-science-report-card>.\n\n. National Council on Excellence in Education, \"A Nation at Risk,\" April 1983, <http:\/\/www2.ed.gov\/pubs\/NatAtRisk\/risk.html> (accessed May 2, 2011); Dan Lips, \"Still 'A Nation at Risk,'\" Heritage Foundation, May 15, 2008, <http:\/\/www.heritage.org\/Research\/Commentary\/2008\/05\/Still-A-Nation-at-Risk> (accessed May 2, 2011).\n\n. Charles J. Sykes, _Dumbing Down Our Kids: Why America's Children Feel Good about Themselves but Can't Read, Write, or Add_ (New York: St. Martin's Press, 1995).\n\n. \"Daily Presidential Tracking Poll,\" Rasmussen Reports, December 12, 2010, <http:\/\/www.rasmussenreports.com\/public_content\/politics\/obama_administration\/daily_presidential_tracking_poll> (see Daily Archives).\n\n. \"Daily Presidential Tracking Poll,\" Rasmussen Reports, June 18, 2011, <http:\/\/www.rasmussenreports.com\/public_content\/politics\/obama_administration\/daily_presidential_tracking_poll>.\n\nCHAPTER TWO: THE PROMISE OF AMERICA\n\n. Historical revisionists have taught our children that Columbus was motivated by greed and imperialism, but the explorer made his purposes perfectly clear in his journals. In his _Book of Prophecies_ , he writes, \"For more than forty years, I have sailed everywhere that people go. I prayed to the most merciful Lord about my heart's great desire, and He gave me the spirit and the intelligence for the task: seafaring, astronomy, geometry, arithmetic, skill in drafting spherical maps and placing correctly the cities, rivers, mountains, and ports. . . . It was the Lord who put into my mind (I would feel His hand upon me) to sail from here to the Indies.\" What motivated the great adventurer was his Christian faith. The idea of discovering new lands where the gospel could go forth was a mission placed in his mind, he says, by the Lord.\n\n. The Scrooby separatists left England in the same year that the first colony of the London Company was established at Jamestown under the leadership of Captain Christopher Newport, Captain John Smith, and the Reverend Robert Hunt. The Jamestown settlement suffered many losses but eventually became the first successful colony in Virginia.\n\n. King James I, Hampton Court Conference, January 16, 1604.\n\n216\n\n. William Bradford, _Of Plymouth Plantation (1620\u20131647)_ , ed. Charles Deane (Boston: Little, Brown, and Co., 1856), 78.\n\n. Ibid., 78.\n\n. Edwin Gaustad and Leigh Schmidt, _The Religious History of America: The Heart of the American Story from Colonial Times to Today_ (New York: HarperOne, 2002), 52.\n\n. Rod Gragg, _Forged in Faith: How Faith Shaped the Birth of the Nation 1607\u2013 1776_ (New York: Howard Books, 2010), 45\u201346.\n\n. John Winthrop, quoted in Gaustad and Schmidt, _Religious History of America_ , 53.\n\n. John Winthrop, \"A Modell of Christian Charity, written on board the Arbella, on the Atlantic Ocean, by the Hon. John Winthrop, Esq., in his passage from the Island of Great Brittaine to New-England in the North America, Anno 1630,\" in _Life and Letters of John Winthrop, Governor of Massachusetts_ , ed. Robert C. Winthrop (Boston: Little, Brown, and Co., 1869), 2:18.\n\n. Winthrop in Gaustad and Schmidt, _Religious History of America_ , 53.\n\n. Gaustad and Schmidt, _Religious History of America_ , 54.\n\n. Ibid.\n\n. Daniel J. Boorstin, \"The Puritan Tradition: Community Above Ideology,\" in Grady McWhiney and Robert Wiebe, _Historical Vistas: Readings in United States History, vol. 1, 1607\u20131877_ (Boston: Allyn and Bacon, 1963), 107.\n\n. William Perkins, \"The Art of Prophesying,\" in Boorstin, \"Puritan Tradition,\" 108.\n\n. Boorstin, \"Puritan Tradition,\" 108 (emphasis added).\n\n. Ibid., 109.\n\n. Ibid.\n\n. New England Colonies, 1620\u20131636, in Robert Hall, Harriet Smither, and Clarence Ousley, _A History of the United States_ (Dallas, TX: The Southern Publishing Company, 1920). Downloaded from Maps ETC, <http:\/\/etc.usf.edu\/maps> (map #05273).\n\n. \"Salem Church Covenant,\" quoted in Gaustad and Schmidt, _Religious History of America_ , 54.\n\n. \"Massachusetts Body of Liberties,\" quoted in Gragg, _Forged in Faith_ , 56\u201357.\n\n. Boorstin, \"Puritan Tradition,\" 109.\n\n. John Cotton, \"Limitation of Government,\" in _The American Puritans, Their Poetry and Prose_ , ed. Perry Miller (New York: Columbia University Press, 1956), 85.\n\n. Josiah H. Benton, Jr., _Early Census Making in Massachusetts, 1643\u20131765_ (Boston: C. E. Goodspeed, 1905), 102.\n\n. The Glorious Revolution (1688\u201389) led to the overthrow of James II (who is believed to have converted to Catholicism in 1668) by Parliamentarian forces (who remained Protestant) and the ascension of the Protestant monarchs William of Orange (a Dutch Protestant) and Mary, his English Protestant queen. The restoration of the throne from Catholic to Protestant monarchs would have a major impact on the form and structure of British government ever after.\n\n217\n\n. Daniel Webster, \"A Discourse Delivered at Plymouth, on the 22d of December, 1820,\" in _Great Speeches of Daniel Webster_ (Coln St. Aldwyns, UK: Echo Library, 2006), 38.\n\n. Samuel Eliot Morison, \"The Pilgrim Fathers: Their Significance in History,\" in _By Land and by Sea: Essays and Addresses by Samuel Eliot Morison_ (New York: Knopf, 1953), 235\u201336.\n\n. Noam Chomsky, interview by David Niose, \"Noam Chomsky: On Humanism, the Vulnerability of Secular Nationalism, and the Mother of All Book Plugs,\" _The Humanist_ , January 1, 2007, <http:\/\/www.thefreelibrary.com\/Noam+Chomsky%3a+on+humanism%2c+the+vulnerability+of+secular+nationalism%2c...-a0159178785> (accessed May 3, 2011).\n\n. Barack Obama, press conference with President Gul, Cankaya Palace, Ankara, Turkey, April 6, 2009, <http:\/\/www.whitehouse.gov\/the_press_office\/Joint-Press-Availability-With-President-Obama-And-President-Gul-Of-Turkey> (accessed May 3, 2011).\n\n. Gary Langer, \"Poll: Most Americans Say They're Christian,\" ABC News, July 18, 2006, <http:\/\/abcnews.go.com\/US\/Story?id=90356andpage=1> (accessed May 3, 2011).\n\n. Frank Newport, \"Church Attendance Lowest in New England, Highest in South,\" Gallup News Service, April 27, 2006, <http:\/\/www.gallup.com\/poll\/22579\/Church-Attendance-Lowest-New-England-Highest-South.aspx> (accessed May 3, 2011).\n\n. James Madison to the General Assembly of the Commonwealth of Virginia, 1778.\n\n. Patrick Henry to the Virginia House of Burgesses, May 1765.\n\n. Supreme Court of South Carolina, 1846. _City of Charleston v. S.A. Benjamin_ ; 2 Strob. 520 (1846).\n\n. _Church of the Holy Trinity v. U.S_.; 143 U.S. 457, 458 (1892).\n\n. _Zorach v. Clauson_ ; 343 U.S. 306 (1952).\n\nCHAPTER THREE: WHAT THE FOUNDERS BELIEVED\n\n. William Byrd, _The Secret Diary of William Byrd of Westover, 1709\u20131712_ , ed. Louis B. Wright and Marion Tinling (New York: Arno Press, 1972).\n\n. Rod Gragg, _Forged in Faith: How Faith Shaped the Birth of the Nation 1607\u2013 1776_ (New York: Howard Books, 2010), 16.\n\n. The plaque inscription at the site of the Robert Hunt Memorial may be seen online at Historical Marker Database, <http:\/\/www.hmdb.org\/marker.asp?marker=17023>.\n\n. Frank Lambert, _The Founding Fathers and the Place of Religion in America_ (Princeton: Princeton University Press, 2003), 49.\n\n. The settlers' benefactor and new governor was Thomas West, the third Baron de La Warr, and the man for whom the Colony of Delaware would be named, as well as the Delaware River, Delaware Bay, Delaware Indian tribe, and eventually the state of Delaware.\n\n218\n\n. Lambert, _Founding Fathers_ , 48.\n\n. Ibid.\n\n. Crandall Shifflett, \"The Powhatan Indian Attack of March 22, 1622,\" Virtual Jamestown, 1998, <http:\/\/www.virtualjamestown.org\/phatmass.html> (accessed May 3, 2011).\n\n. Ibid.\n\n. James H. Hutson, _Religion and the Founding of the American Republic_ (Washington, DC: Library of Congress, 1998), 18.\n\n. Edwin Gaustad and Leigh Schmidt, _The Religious History of America: The Heart of the American Story from Colonial Times to Today_ (New York: HarperOne, 2002), 41.\n\n. Ibid.\n\n. Francis Parkman, \"The American Colonies in the 1750s,\" in Wiebe and McWhiney, _Historical Vistas_ (Boston: Allyn & Bacon, 1963), 147.\n\n. Ibid.\n\n. Frank E. Smitha, \"From British Colony to Independence: The American Revolution, 1701\u20131791,\" Macrohistory and World Report, <http:\/\/www.fsmitha.com\/h3\/h32-rv.htm> (accessed May 3, 2011).\n\n. Ibid.\n\n. The British statesman, philosopher, and advocate of American independence Edmund Burke first used the term _salutary neglect_ in remarks before the English House of Commons on March 22, 1775, in which he said: \"[T]he colonies in general owe little or nothing to any care of ours, and that they are not squeezed into this happy form by the constraints of watchful and suspicious government, but that, through a wise and salutary neglect, a generous nature has been suffered to take her own way to perfection.\" Cited in Edmund Burke, _The Works of the Right Hon. Edmund Burke, with a Biographical and Critical Introduction_ (London: Hodsworth and Ball, 1834), 186.\n\n. William G. McLoughlin, _Revivals, Awakenings, and Reform: An Essay on Religion and Social Change in America, 1607\u20131977_ (Chicago: University of Chicago Press, 1978), 51.\n\n. Ibid.\n\n. Jonathan Edwards, quoted in Douglas A. Sweeney, _Jonathan Edwards and the Ministry of the Word_ (Downers Grove, IL: Intervarsity Press Academic, 2009), 111.\n\n. Patricia Bonomi, _Under the Cope of Heaven: Religion, Society, and Politics in Colonial America_ (New York: Oxford University Press, 1986), 105.\n\n. Ibid.\n\n. Hutson, _Religion and the Founding_ , 22.\n\n. Ibid.\n\n. Ibid.\n\n. The journal of the Reverend Charles Woodmason, entry for Saturday, September 3, 1767: \"Baptized negro man, 2 negro children, and 9 white infants and married 1 couple. The people thanked me in the most kind manner for my services. I had very pleasant riding but my horse suffered greatly. The mornings and evenings now begin to be somewhat cool, but the midday heat is almost intolerable. Many of these people walk 10 or 12 miles with their children in the burning sun. Ought such to be without the word of God, when so earnest, so desirous of hearing it and becoming good Christians, and good subjects?!\"\n\n219\n\nCharles Woodmason (1720\u20131776), _Journal: The Carolina Backcountry on the eve of the Revolution; the Journal and other writings of Charles Woodmason, Anglican itinerant_ , ed. with an introd. by Richard J. Hooker (Chapel Hill, NC: Institute of Early American History and Culture, 1953).\n\n. Edwin S. Gaustad, _Historical Atlas of Religion in America_ (New York: Harper and Row, 1962), 3.\n\n. Patricia Bonomi, _Under the Cope of Heaven_ , 274.\n\n. Hutson, _Religion and the Founding_ , 25.\n\nCHAPTER FOUR: THE BIRTH OF THE AMERICAN SPIRIT\n\n. \"Estimated Population of American Colonies, 1630\u20131780,\" _World Almanac and Book of Facts_ , ed. Robert Famighetti (New York: St. Martin's, 1998), 378, <http:\/\/merrill.olm.net\/mdocs\/pop\/colonies\/colonies.htm> (accessed May 3, 2011).\n\n. Indentured servants, also referred to as \"bound boys\" or \"bound girls,\" were generally British subjects under the age of twenty-one who were obligated to work for planters, farmers, merchants, or other affluent families for a specific period of time, usually three to seven years. They received no salary, but food, clothing, and lodging were normally provided. Redemptioners, who were mostly Germans, were given free passage to America under similar circumstances.\n\n. Alan Taylor, _American Colonies_ , Penguin History of the United States, ed. Eric Foner (New York: Viking Penguin, 2001), 319.\n\n. John A. Garraty and Peter Gay, ed. _The Columbia History of the World_ (New York: Harper and Row, 1972), 672\u201373.\n\n. Thomas Paine, _Common Sense (Addressed to the Inhabitants of America), and Other Political Writings_ (New York: Bobbs-Merrill, 1953), 21.\n\n. Ibid., 19.\n\n. Ibid., 32.\n\n. Ellis Sandoz, from the foreword to Alan Heimert and Perry Miller, eds., _The Great Awakening: Documents Illustrating the Crisis and Its Consequences_ (Indianapolis: Bobbs-Merrill, 1967), xiv.\n\n. Jonathan Edwards, \"Sinners in the Hands of an Angry God,\" Life-Changing Pamphlet Series (Murfreesboro, TN: Sword of the Lord, 1741), 13.\n\n. Ibid., 19.\n\n. Jonathan Edwards, _Some Thoughts Concerning the Present Revival of Religion, in The Works of President Edwards_ (London: James Black and Son, 1817), 6:87.\n\n. Edwards, \"Sinners,\" 21.\n\n. Arnold A. Dallimore, _George Whitefield: God's Anointed Servant in the Great Revival of the Eighteenth Century_ (Westchester, IL: Crossway, 2010), 83\u201386.\n\n. George Whitefield, quoted in Dallimore, _George Whitefield_ , 91.\n\n. Benjamin Franklin, _Autobiography of Benjamin Franklin_ , ed. John Bigelow (Philadelphia: J. B. Lippincot, 1869), 251.\n\n. Ibid., 253.\n\n. Ibid., 255.\n\n. From the journal of New England farmer Nathan Cole, in George Leon Walker, _Some Aspects of the Religious Life of New England_ (New York: Silver, Burnett, and Company, 1897), 89\u201392.\n\n. James Downey, _The Eighteenth-Century Pulpit: A Study of the Sermons of Butler, Berkeley, Secker, Sterne, Whitefield and Wesley_ (London: Oxford, 1969), 157.\n\n. Ellis Sandoz, ed., _Political Sermons of the American Founding Era: 1730\u20131805_ (Indianapolis: Liberty Fund, 1998), 1:18.\n\n. Daniel J. Boorstin, \"The Puritan Tradition: Community Above Ideology,\" in Grady McWhiney and Robert Wiebe, _Historical Vistas: Readings in United States History, vol. 1, 1607\u20131877_ (Boston: Allyn and Bacon, 1963), 108.\n\n. Perry Miller and Alan Heimert, ed., _The Great Awakening: Documents Illustrating the Crisis and Its Consequences_ (Indianapolis: Bobbs-Merrill, 1967), 9.\n\n. Samuel Adams, from the essay \"Loyalty and Sedition,\" published in _The Advertiser_ , 1748.\n\n. McLoughlin, _Revivals, Awakenings, and Reform_ , 96\u201397.\n\n. Thomas S. Kidd, _The Great Awakening: The Roots of Evangelical Christianity in Colonial America_ (New Haven: Yale University Press, 2007), 323.\n\n. Frank Lambert, _Inventing the Great Awakening_ (Princeton: Princeton University Press, 1999), 118.\n\n. John Adams to Dr. J. Morse, December 2, 1815, in _The Works of John Adams, Second President of the United States_ , ed. Charles Francis Adams (Boston: Little, Brown, and Co, 1856), 10:185.\n\n. Ibid.\n\n. McLoughlin, _Revivals, Awakenings, and Reform_ , 21.\n\n. Kidd, _The Great Awakening_ , 289.\n\n. H. R. Niebuhr, _The Kingdom of God in America_ (Middletown, CT: Wesleyan University Press, 1988), 126.\n\n. Benjamin Franklin, in _Farrand's Records of the Federal Convention of 1787_ , ed. Max Farrand (Washington, DC, 1911): 1:451.\n\n. Ibid., 452.\n\n. Barbara W. Tuchman, _The March of Folly_ (New York: Alfred A. Knopf, 1984), 381.\n\n. Federalist 51, <http:\/\/www.constitution.org\/fed\/federa51.htm> (accessed May 3, 2011).\n\n. George Washington, \"Farewell Address,\" in _Independent Chronicle_ , September 26, 1796, <http:\/\/avalon.law.yale.edu\/18th_century\/washing.asp> (accessed May 3, 2011).\n\n. Matthew Spalding, _We Still Hold These Truths_ (Wilmington, DE: ISI Books, 2010), 221.\n\nCHAPTER FIVE: FAITH IN THE TWENTY-FIRST CENTURY\n\n. Michael Lind, \"America Is Not a Christian Nation,\" Salon.com, April 14, 2009, <http:\/\/www.salon.com\/news\/opinion\/feature\/2009\/04\/14\/christian_nation> (accessed May 3, 2011).\n\n. Claire Berlinski, _Menace in Europe: Why the Continent's Crisis Is America's Too_ (New York: Crown Forum, 2006), 8\u20139.\n\n. Don Feder, \"Bill Maher: The Village Atheist Meets the Village Idiot,\" Coldsteel Caucus Report, <http:\/\/www.donfeder.com\/articles\/0604maher.pdf> (accessed May 3, 2011).\n\n. Russell Kirk, _The American Cause_ (Wilmington, DE: ISI Books, 2002), 18.\n\n. Don Feder, \"Atheists Write, Believers Yawn,\" Coldsteel Caucus Report, <http:\/\/www.donfeder.com\/articles\/0703atheistBooks.pdf> (accessed May 3, 2011).\n\n. John Jay to John Murray, October 12, 1816, in _The Life of John Jay: with Selections from His Correspondence and Miscellaneous Papers_ , ed. William Jay, (Freeport, NY: Books for Libraries, 1972), 2:376.\n\n. R. J. Rummel, \"How Many Did Communist Regimes Murder?\" _Death by Government_ (New Brunswick, NJ: Transaction Publishers, 1994), 4, table 2.1.\n\n. John Adams, letter to Zabdiel Adams, June 21, 1776.\n\n. Alan Heimert, _Religion and the American Mind: From the Great Awakening to the Revolution_ (Eugene, OR: Wipf and Stock, 2006), 9.\n\n. Ibid.\n\n. Ibid.\n\n. John Cleaveland, in Thomas S. Kidd, _The Great Awakening: The Roots of Evangelical Christianity in Colonial America_ (New Haven: Yale University Press, 2007), 294.\n\n. Ibid.\n\n. James H. Hutson, _Religion and the Founding of the American Republic_ (Washington, DC: Library of Congress, 1998), 46.\n\n. Rod Gragg, _Forged in Faith: How Faith Shaped the Birth of the Nation 1607\u2013 1776_ (New York: Howard Books, 2010), 173.\n\n. Declaration of Congress for a national day of prayer and fasting, March 16, 1776, from the Journals of the American Congress, 1774 to 1788 (Washington: Way and Gideon, 1823), vol. 1, 286\u201387.\n\n. Engrossing is the process of copying a document in a large and clearly legible hand.\n\n. Gragg, _Forged in Faith_ , 175.\n\n. Ibid., 182.\n\n. Adams to Mercy Otis Warren, April 16, 1776, in _Warren-Adams Letters_ , vol. 1, 1743\u20131777 (Boston: Massachusetts Historical Society, 1917), 221.\n\n. Franklin D. Roosevelt, radio address, June 6, 1944, _Congressional Record_ 153 (June 6, 2007): S 14867.\n\nCHAPTER SIX: THE COMING REVOLUTION\n\n. Malachi Martin, _The Keys of This Blood: The Struggle for World Dominion_ (New York: Simon and Schuster, 1990), 209.\n\n. Dimitry V. Pospielovsky, _A History of Soviet Atheism in Theory, and Practice, and the Believer_ (New York: St. Martin's Press, 1987), 1:34.\n\n. V. I. Lenin, \"Program of the Communist International,\" in William Henry Chamberlin, ed., _Blueprint for World Conquest: The Official Communist Plan_ (Washington, DC: Human Events, 1946), 187.\n\n222\n\n. Ibid., 189.\n\n. Edvard Radzinsky, _The Last Tsar_ (New York: Doubleday, 1992), 344\u201346; Mark Weber, \"Assessing the Grim Legacy of Soviet Communism,\" Institute for Historical Review, <http:\/\/www.ihr.org\/jhr\/v14\/v14n1p-4_Weber.html> (accessed May 3, 2011).\n\n. Richard N. Ostling, \"Cross Meets Kremlin,\" _Time_ , June 24, 2001, <http:\/\/www.time.com\/time\/magazine\/article\/0,9171,150718,00.html> (accessed May 3, 2011).\n\n. \"Freedom in Decline Worldwide: U.S. Report,\" Freedom House, January 13, 2011, <http:\/\/www.breitbart.com\/article.php?id=CNG.b7ab9f89dccc945f6545ca445a6fac5a.4a1andshow_article=1> (accessed May 3, 2011).\n\n. For chilling evidence of the horrors of communism, visit the Victims of Communism Memorial Foundation at <http:\/\/www.globalmuseumoncommunism.org>.\n\n. James Burnham, _The Suicide of the West: An Essay on the Meaning and Destiny of Liberalism_ (Washington, DC: Regnery Gateway, 1985), 59\u201360.\n\n. Mark Levin, _Liberty and Tyranny: A Conservative Manifesto_ (New York: Simon and Schuster, 2009), 4.\n\n. Hubert H. Humphrey, \"Six Liberals Define Liberalism,\" _New York Times Magazine_ , April 19, 1959, 13.\n\n. Aleksandr Solzhenitsyn believed that Russian society fell victim to communism because, thanks to the atheist creed of Marxist-Leninism, it fell under the grip of atheistic totalitarianism. He recognized the truth in the words of an old peasant he overheard years earlier. \"Men have forgotten God,\" the old man said. \"That's why all this has happened to us.\" Of course, he was right, and that ought to be fair warning for America.\n\n. For more information, see Robert Kulak, \"Abortions: Safe, Legal and Not So Rare,\" _Hartford Independent Examiner_ , February 13, 2011, <http:\/\/www.examiner.com\/independent-in-hartford\/is-abortion-as-rare-as-it-should-be-democrats-collectivist-or-indivualist#ixzz1RXOyKhEc>.\n\n. James Capretta, \"Obama's Plainly Unserious Budget,\" _National Review_ , February 17, 2011, <http:\/\/www.nationalreview.com\/articles\/259947\/even-president-seems-oppose-his-budget-james-c-capretta> (accessed May 3, 2011).\n\n. M. Coleman, et al., \"Cancer Survival in Five Continents: A Worldwide Population-Based Study (CONCORD),\" _The Lancet Oncology_ , vol. 9, no. 8. (August 2008), 730\u201356.\n\n. Sally Pipes, \"The Myth of Efficient Government Health Care,\" in _The Top Ten Myths of American Health Care_ : A Citizen's Guide (San Francisco: Pacific Research Institute, 2008).\n\n. Ibid.\n\n. Ibid.\n\n. David Limbaugh, \"Obama Can't Afford to Tell Truth on Health Care,\" _Human Events_ , August 4, 2009, <http:\/\/www.humanevents.com\/article.php?id=32995>. See also N. Gregory Mankiw, \"Beyond Those Health Care Numbers,\" _New York Times_ , Nov 4, 2007, http:\/\/www.nytimes.com\/2007\/11\/04\/business\/04view.html?ex=1351828800&en=7ebf86b6773f35bd&ei=5090.\n\n. Michael Medved, \"The Spending Sickness Makes for Unhealthy Reform,\" Creator's Syndicate, August 26, 2009.\n\n. Levin, _Liberty and Tyranny_ , 113.\n\n. Tom Bethell, \"Culture versus Economy,\" _American Spectator_ , February 2011, <http:\/\/spectator.org\/archives\/2011\/02\/22\/culture-versus-economy> (accessed May 3, 2011).\n\n. Joseph Story, _Commentaries on the Conflict of Laws_ (Cambridge, MA: Charles Folsom, 1834), 100.\n\n. U.S. Census Bureau, \"Annual Social and Economic Supplement, 2007,\" Current Population Survey, <http:\/\/pubdb3.census.gov\/macro\/032008\/pov\/toc.htm> (accessed May 3, 2011).\n\n. Jennifer Roback Morse, \"The Limited-Government Case for Marriage,\" _Indivisible: Social and Economic Foundations of American Liberty_ (Washington, DC: Heritage Foundation, 2009), 32\u201333.\n\n. Ibid.\n\n. Bob Unruh, \"Lesbian Awarded Custody of Christian's Only Child,\" WorldNetDaily.com, December 5, 2009, <http:\/\/www.wnd.com\/?pageId=117969#ixzz1EvhDuOZl> (accessed May 3, 2011).\n\n. William J. Bennett, \"A Nation Worth Defending,\" _USA Today Magazine, Nov_ 1, 2002, also available at FindArticles.com, <http:\/\/findarticles.com\/p\/articles\/mi_m1272\/is_2690_131\/ai_94384312>.\n\n. Paul Weyrich, \"Western Civilization at Stake,\" Newsmax.com, March 24, 2006, <http:\/\/archive.newsmax.com\/archives\/articles\/2004\/3\/24\/93948.shtml> (accessed May 3, 2011).\n\n. Ibid.\n\n. Jim Nelson Black, _Freefall of the American University_ , (Nashville: WND\/ Nelson Current, 2004), 137\u201338.\n\n. Mona Charen, \"Don't Know Much About History,\" Jewish World Review, July 15, 2003, <http:\/\/jewishworldreview.com\/cols\/charen.html> (accessed May 3, 2011).\n\n. \"Losing America's Memory: Historical Illiteracy in the 21st Century,\" American Council of Trustees and Alumni, February 21, 2000, https:\/\/www.goacta.org\/publications\/downloads\/LosingAmerica%27sMemory.pdf (accessed May 3, 2011).\n\n. Doug Bandow, \"Religious Liberty Lost Worldwide,\" _American Spectator_ , December 7, 2010, <http:\/\/spectator.org\/archives\/2010\/12\/07\/religious-liberty> -lost-worldwi (accessed May 3, 2011).\n\nCHAPTER SEVEN: WHAT YOU CAN DO\n\n. Samuel Adams, \"The Rights of the Colonists: The Report of the Committee of Correspondence to the Boston Town Meeting, November 20, 1772,\" in _Old South Leaflets_ no. 173 (Boston: Directors of the Old South Work, 1906), 7:417\u2013428.\n\n. Thomas Jefferson to Richard Henry Lee, May 8, 1825, in _Thomas Jefferson's Writings_ (New York: The Library of America, 1984), 1501.\n\n. Adams, \"Rights of the Colonists,\" 7:417\u2013428.\n\n. Glenn Beck, \"Do You Recognize America?\" _Glenn Beck Show_ , December 3, 2009, <http:\/\/www.glennbeck.com\/content\/articles\/article\/198\/33794\/> (accessed May 3, 2011).\n\n. Thomas Sowell, \"Dismantling America,\" _National Review_ , August 17, 2010, <http:\/\/www.nationalreview.com\/articles\/243918\/dismantling-america-thomas-sowell> (accessed May 3, 2011).\n\n. Elizabeth Mendes, \"U.S. Satisfaction Remains Near 12-Month Low,\" Gallup Poll, January 14, 2011, <http:\/\/www.gallup.com\/poll\/145610\/Satisfaction-Remains-Near-Month-Low.aspx> (accessed May 3, 2011).\n\n. \"Right Direction or Wrong Track,\" Rasmussen Reports, February 23, 2011, <http:\/\/www.rasmussenreports.com\/public_content\/politics\/mood_of_america\/right_direction_or_wrong_track> (accessed May 3, 2011).\n\n. \"Voter Concern about Economy Hits Highest Level in Over Two Years,\" Rasmussen Reports, January 4, 2011, <http:\/\/www.rasmussenreports.com\/public_content\/politics\/mood_of_america\/importance_of_issues> (accessed May 3, 2011).\n\n. Raghavan Mayur, \"Gov't That Governs the Least Is One Americans Want Most,\" _Investors Business Daily_ , February 28, 2011, <http:\/\/www.investors.com\/NewsAndAnalysis\/Article\/564494\/201102281859\/Govt-that-Governs-the-Least-Is-One-Americans-Want-Most.htm> (accessed May 3, 2011).\n\n. Ronald Reagan, \"Farewell Address,\" January 11, 1989, <http:\/\/www.americanrhetoric.com\/speeches\/ronaldreaganfarewelladdress.html> (accessed May 3, 2011).\n\n. Ibid.\n\n. Jonathan Mayhew, \"A Discourse Concerning Unlimited Submission and Non-Resistance to the Higher Power,\" delivered at Old West Church, Boston, Massachusetts, January 30, 1750. (Subsequently printed as a pamphlet and distributed throughout New England.)\n\n. Ibid.\n\n. Michael Novak, \"God Bless the Tea Party,\" _National Review Online_ , November 8, 2010, <http:\/\/www.nationalreview.com\/articles\/252720\/God-bless-tea-party-michael-novak> (accessed May 4, 2010).\n\n. James Ceaser, \"The 2010 Verdict,\" RealClearPolitics.com, November 10, 2010, <http:\/\/www.realclearpolitics.com\/articles\/2010\/11\/10\/the_2010_verdict_107908.html> (accessed May 4, 2010).\n\n. Ibid, emphasis added.\n\n. Novak, \"God Bless the Tea Party.\"\n\n. Ibid.\n\n. James McHenry in _The American Historical Review_ , ed. George B. Adams (New York: Macmillan, 1906), 11:618; James McHenry in _Farrand's Records of the Federal Convention of 1787_ , ed. Max Farrand (Washington, DC, 1911), 3:85.\n\n. John Adams to H. Niles, February 13, 1818, in _Works of John Adams_ , 10:282.\nTOPICAL INDEX\n\n**A**\n\nabortion, , , \u2013,\n\nAbraham (patriarch), ,\n\nAbrahamic covenant, \u2013\n\naccountability to God, \u2013\n\nACLU (American Civil Liberties   \nUnion), ,\n\nAdams, John, , , , \u2013, ,   \n\u2013, \u2013,\n\nAdams, John Quincy,\n\nAdams, Samuel, \u2013, , ,   \n\u2013\n\nAfrica,\n\nAfrican Americans, , ,\n\nAfrican Methodist Episcopal Church,\n\nAid to Families with Dependent   \nChildren (AFDC),\n\nAIG,\n\nAikman, David,\n\nAllen, Richard,\n\n\"America Is Not a Christian Nation\"  \n(Lind), \u2013\n\n_American Cause, The_ (Kirk),\n\nAmerican  Liberties Union. _See_  \nACLU\n\nAmerican Council of Trustees and   \nAlumni (ACTA) poll,\n\nAmerican exceptionalism, , \u2013\n\n_American Patriot's Bible_ ,\n\nAmerican Recovery and Reinvestment   \nAct of 2009,\n\nAmerican Revolution, xiv, xvi, , , ,   \n, , , , , \u2013, , ,   \n\u2013, , , , , , \u2013\n\n_American Spectator_ ,\n\nAmerican Spirit, xvii, , , , ,   \n, : birth of, \u2013, \u2013;   \na powerful statement of, \u2013;   \nrethinking the, \u2013\n\nAmish, ,\n\nanarchy, , ,\n\nAnglican: Church, , , ; churches   \nin the South, ; communion, , ;   \nprayer book, ; ministers, , , ,   \n, , , ,\n\nanti-Americanism, , , ,\n\nArbella Covenant,\n\n_Art of Prophesying, The_ (Perkins),\n\natheism, , , \u2013\n\natheists, , ,\n\n**B**\n\nbailout(s), , \u2013, ,\n\nBandow, Doug,\n\nbaptism, ,\n\nBaptists,\n\nBattles of Lexington and Concord, ,  \n, , , ,\n\nBeamer, Todd, \u2013\n\nBeck, Glenn, xv, , \u2013\n\nBell, Alexander Graham,\n\nBennett, William, \u2013,\n\nBerlinski, Claire, \u2013\n\nBethell, Tom,\n\nBill of Rights, ,\n\nBlack Robe Regiment, xvi, , \u2013,\n\nblogs,\n\nBolshevik Revolution, \u2013\n\nBolsheviks, , ,\n\n_Book of Prophecies_ (Columbus), n1\n\nBoorstin, Daniel J., \u2013, ,\n\nBoston Massacre, ,\n\nBoudinot, Elias,\n\nBradford, William, , , \u2013\n\nBrewster, William, \u2013,\n\nBrezhnev, Leonid,\n\n_Broken Hearth: Reversing the Moral  \nCollapse of the American Family_  \n(Bennett), \u2013\n\nbudget deficit, ,\n\nBurke, Edmund, n17\n\nBurnham, James,\n\nBush, George H. W., ,\n\nBush, George W., : administration,\n\nbusiness (American), , , \u2013, ,\n\nByrd, William,\n\n**C**\n\ncable TV, ,\n\nCalvinism, \u2013,\n\nCanada, , ,\n\ncancer,\n\ncapitalism, , ,\n\nCarter administration,\n\nCeaser, James,\n\ncellular telephones, ,\n\nCensus Bureau. See U.S. Census Bureau\n\nCharen, Mona,\n\ncharity, -ies, \u2013, ,\n\nCharles I (king), \u2013\n\nCharles II (king),\n\nchecks and balances,\n\nChina, , , ,\n\nChomsky, Noam,\n\nChristian: books, radio, and television,   \n; worldview, , ,\n\nChristianity (as an \"evil\"),\n\nChristians: percentage of Americans   \nself-identified as,\n\nChrysler Corporation,\n\nchurch attendance, , , , :  \nEurope and U.S. compared,\n\nChurch of England, , , \u2013, ,   \n,\n\ncircuit riders. See itinerant preachers\n\ncivil rights,\n\nCivil War: American, , , ; English,   \n\u2013\n\nCleaveland, John, \u2013\n\nClinton, Bill, ,\n\ncohabiting, ,\n\nCole, Nathan,\n\nCollege of William and Mary,\n\nCommittees of Correspondence,   \n\u2013, \u2013\n\ncommon decency, , ,\n\n_Common Sense_ (Paine), \u2013\n\ncommunism, , , , , \u2013,   \n, \u2013, , n8, n12\n\nCommunist International (Comintern),   \n\u2013\n\n_Communist Manifesto_ , ,\n\nCommunist Party, ,\n\nCommunists, , ,\n\nColson, Chuck,\n\nColumbus, Christopher, , , n1\n\nconcerned citizen action, xvi\n\nCongregationalist: churches, , ,   \n; ministers, \u2013\n\nCongregationalists (New England),   \n,\n\nCongress, , , , \u2013, , ,   \n, , , , , . _See also_  \nContinental Congress\n\nconscience, xvi\u2013xvii, , \u2013, \u2013,   \n, , , , ,\n\n\"consent of the governed,\" , \u2013,\n\nConstitution of the United States. _See_  \nUnited States Constitution\n\nContinental Congress, , , , , ,   \n, , \u2013\n\ncorruption, ,\n\nCotton, John, \u2013, , , \u2013,\n\nCrashaw, William,\n\ncrime (in America), \u2013, , ; link   \nbetween poverty and,\n\ncritical thinking skills,\n\n**D**\n\nDavenport, James, ,\n\nDavies, Samuel, \u2013,\n\nDawkins, Richard, ,\n\nD-Day Invasion of Europe: presidential   \nprayer following, \u2013\n\nDeclaration of Independence, , ,   \n, , \u2013, , , , ,   \n\u2013, \u2013, \u2013,\n\nDeclaratory Act (1766), ,\n\nDefense of Marriage Act (DOMA),\n\nde La Ware, Lord. See West, Thomas\n\n_Democracy in America_ (Toqueville), \u2013\n\nDemocratic Party: defeat in 2010 midterm   \nelections,\n\nDepartment of Health and Human   \nServices, ,\n\nDickinson, Jonathan, ,\n\ndignity of human life,\n\ndiversity, \u2013, ,\n\ndivorce rate since 1960,\n\ndomestic violence,\n\ndonations,\n\nDouglas, William O.,\n\nDrake, Francis, , \u2013\n\ndrugs,\n\nDuch\u00e9, Jacob,\n\n_Dumbing Down Our Kids_ (Sykes),\n\nDuncan, Arne (secr. of education),\n\nDunham, Jason (marine corporal), \u2013\n\n**E**\n\nEastman, George,\n\neconomy (America's), , , , ,   \n\u2013, , ,\n\nEdison, Thomas, ,\n\neducation, , , , , , , :  \nhigher, , , , , , , \u2013;   \na new emphasis on (after the Great   \nAwakening), ; \u2013; public, \u2013,   \n, , ; right to a God-centered,   \n\u2013; secondary,\n\neducational performance (American),   \n\u2013\n\nEdwards, Jonathan, , \u2013, , ,   \n\u2013, , , , ,\n\nEgypt, ,\n\nelection: general: 2008, ; 2010, ;   \nmidterm: 2010, xii, , , , \u2013,   \n, , , \u2013; presidential:  \n2008, ; 2012, \u2013\n\nElizabeth I (queen), , , ,\n\nEngels, Friedrich,\n\nEngland, , , \u2013, , , , ,   \n, , , , , , \u2013, , ,   \n\u2013, , , , , , , ,   \n, , n2\n\nEnlightenment, , , , ,\n\nentrepreneurship,\n\nEnvironmental Protection Agency,\n\nevangelicalism, , ,\n\n**F**\n\nFacebook,\n\nFalwell, Jerry, \u2013\n\nfamily (traditional two-parent), ,   \n\u2013, \u2013\n\nFamily Research Council,\n\nfathers (deserting families),\n\nFeder, Don, ,\n\nfederal budget, 2011,\n\nFederalist No. ,\n\n_Feminine Mystique, The_ (Friedan),\n\nfeminist movement,\n\nfeminists, . _See also_ feminist   \nmovement\n\nfertility rate, ,\n\nFirst Continental Congress, , ,\n\nFirst Great Awakening, xiv, xvi, \u2013,   \n\u2013, , , \u2013, \u2013, ,   \n, , , \u2013, : the   \nlegacy of, \u2013\n\nFirst World War. _See_ World War I\n\nfood stamps,\n\nFord, Henry,\n\nforeign aid,\n\nFounding Fathers, xii, xiv\u2013xvi, \u2013,   \n\u2013, , , , , , , \u2013,   \n, , , , , , \u2013,   \n, , \u2013, , , \u2013,   \n\u2013: what they believed, ,   \n\u2013,\n\nFrance, , , , , , ,\n\nFranklin, Benjamin, , , \u2013, \u2013,   \n, \u2013, , ,\n\n\"free countries\" (as of 2010),\n\nFreedom House,\n\nfreedom of the press,\n\nFreedom from Religion Foundation,\n\nfreedom of speech, ,\n\nfreedom of worship,\n\nFrench Enlightenment. See   \nEnlightenment\n\nFrench Revolution, ,\n\nFriedan, Betty, \u2013\n\n**G**\n\nGallup poll, ,\n\nGarrick, David,\n\nGates, Thomas,\n\nGaustad, Edwin,\n\nGeneral Motors,\n\nGeorge II of Great Britain (king), ,\n\nGeorge III of the United Kingdom   \n(king), , , \u2013,\n\nGermany, , , , , , ,\n\nGiffords, Gabrielle, ,\n\n_Gift of Valor, The_ (Phillips),\n\nGiving USA,\n\nGladwell, Malcolm, xiii\n\nGlorious Revolution, , n24\n\nGoldman Sachs,\n\ngovernment interference, xvi, ,   \n\u2013\n\ngovernment spending, xii, , , ,   \n,\n\nGragg, Rod, \u2013\n\nGreat Awakening (1734\u20131750s). _See_  \nFirst Great Awakening\n\nGreat Britain: survival rate for cancer   \npatients,\n\nGreat Depression, \u2013,\n\nGreat Leap Forward,\n\nGreat Society, ,\n\ngreen energy,\n\n**H**\n\nHakluyt, Richard, , ,\n\nHalfway Covenant, \u2013,\n\nHancock, John, ,\n\nHannity, Sean,\n\nHarding, Warren G.,\n\nHarvard College\/University, \u2013, ,   \n\u2013\n\nhealth-care dilemma, \u2013. _See also_  \nnationalized health care\n\nhealth insurance, \u2013\n\nHenry, Patrick, , ,\n\nHeritage Foundation, , ,\n\nhistorical revisionists, n1\n\nHitchens, Christopher, ,\n\nHolland, \u2013\n\nHollander, Paul,\n\n\"hope and change,\" xiv, , , ,   \n, , , \u2013,\n\nhopelessness, ,\n\nhouse-church movement,\n\nHouse of Burgesses, \u2013, ,\n\nHumphrey, Hubert,\n\nHunt, Robert, \u2013, , n2, n3\n\nHusband, Herman,\n\nHutson, James, , \u2013\n\n**I**\n\nillegal aliens,\n\nillegitimacy, ,\n\nimperialism, , , n1\n\nindentured servants, , , , , n2\n\nIndia,\n\n_In Search of Self-Government_  \n(Rasmussen),\n\nInternal Revenue Service. _See_ IRS\n\ninternational aid societies,\n\nInternet, xiii, , , , , ,\n\nInvasion of Normandy. See D-Day   \nInvasion of Europe\n\n_Inventing the Great Awakening_  \n(Lambert), \u2013\n\ninventions, \u2013\n\n_Investor's Business Daily_ ,\n\niPad,\n\niPhone, ,\n\nIran,\n\nIraq,\n\nIRS, , ,\n\nIslamic extremism,\n\nIslamization,\n\nItaly,\n\nitinerant preachers ( _aka_ circuit riders),   \n,\n\n**J**\n\nJacobinism,\n\nJacobins, ,\n\nJames I of England (king), \u2013, , ,   \n,\n\nJames II, n24\n\nJamestown, , , \u2013, , \u2013,   \n, , , n2\n\nJapan, , , ,\n\nJay, John,\n\nJefferson, Thomas, \u2013, , , ,   \n\u2013, ,\n\n_Jesus in Beijing_ (Aikman),\n\njoblessness, ,\n\nJohnson, Lyndon, ,\n\nJones, Absalom,\n\nJudeo-Christian: ethic, \u2013;   \nheritage\/tradition, , , ,\n\nJustice Department. See U.S.  \nDepartment of Justice\n\n**K**\n\nKidd, Thomas S., ,\n\nKing James Bible, ,\n\nKirk, Russell, \u2013, ,\n\n**L**\n\nLambert, Frank, , \u2013\n\nLane, Ralph, \u2013\n\nLaud, William (archbishop), \u2013\n\nLee, Richard Henry, , \u2013, ,\n\nLeft, the, , , ,\n\nLenin, Vladimir, , \u2013,\n\nLet Freedom Ring,\n\nLevin, Mark, , \u2013\n\nLexington and Concord. _See_ Battles of   \nLexington and Concord\n\nliberal ideology, \u2013. _See also_ liberalism;   \nmodern liberalism\n\nliberalism, \u2013. _See also_ liberal ideology;   \nmodern liberalism\n\nliberals, , \u2013, , , , .  \n _See also_ secular liberals\n\n_Liberty and Tyranny_ (Levin),\n\nlife expectancy,\n\nLillback, Peter,\n\nLimbaugh, Rush,\n\nLincoln, Abraham, ,\n\nLind, Michael,\n\nLivingston, Robert,\n\nLivingston, William,\n\nLocke, John, , ,\n\nLondon Company, , , , , ,   \nn2\n\nLuther, Martin, \u2013\n\nLutherans, , \u2013, ,\n\n**M**\n\nMadison, James, , , ,\n\nmanifest destiny, , \u2013,\n\nmarriage, , , : and family issues,   \n\u2013\n\nmarried versus unmarried adults,\n\nMarshall Plan,\n\nMartin, Malachi, \u2013\n\nMary II of England, , n24\n\nMarx, Karl, , ,\n\nMarxism,\n\nMarxist: communism, ; ideology,   \n, \u2013\n\nMarxist-Leninism, , n12\n\nMassachusetts Bay Colony, \u2013,   \n\u2013, \u2013,\n\nMassachusetts Body of Liberties, \u2013\n\nmath, , , , ,\n\nMather, Cotton, ,\n\nMather, Increase,\n\nMather, Richard, ,\n\n_Mayflower_ (ship), \u2013, , , ,\n\nMayflower Compact, \u2013, ,\n\nMayhew, Jonathan, \u2013\n\nMcLoughlin, William, \u2013, \u2013,   \n\u2013\n\nMedicare and Medicaid, , ,\n\nmedical insurance, \u2013\n\nmedicine,\n\nMedved, Michael,\n\n_Menace in Europe_ (Berlinski),\n\nMennonites, ,\n\nMethodism, ,\n\nMexico, ,\n\nMiddle East, , , . _See also  \nindividual Middle Eastern countries   \nby name_\n\nmodern liberalism, \u2013: defined,\n\nMolasses Act,\n\nMoravians, ,\n\nMorison, Samuel Eliot,\n\nMorse, Jennifer Roback, \u2013\n\nMoynihan, Daniel Patrick, \u2013,\n\nMuhlenberg, Peter, \u2013\n\nmulticulturalism, \u2013,\n\nMuslim Brotherhood,\n\n**N**\n\nNational Assessment of Educational   \nProgress (NAEP), \u2013\n\nnational debt, ,\n\nnationalized health care, ,\n\nNative Americans, ,\n\nNavigation Acts,\n\nNew Deal, ,\n\nNew England colonies, , ; _see also  \nchaps_. \u2013: eighteenth-century population   \nin (chart), .\n\nNewport, Christopher, , n2\n\nNew World Order,\n\nNicholas II of Russia (czar),\n\nNietzsche, Friedrich,\n\n\/, , \u2013, ,\n\n\"no-fault divorce,\"\n\nNorthwest Ordinance (1787),\n\nNovak, Michael, , , \u2013\n\n**O**\n\nObama, Barack, \u2013, , , , ,   \n, , , \u2013,\n\nObama administration, xvi, , ,   \n, \u2013, ,\n\n_Of Plymouth Plantation_ (Bradford),\n\nOglethorpe, James, , ,\n\noriginal sin (doctrine), ,\n\nOtis, James,\n\nout-of-wedlock births, .\n\nOxford University, xiv, , , , ,\n\n**P**\n\nPaine, Thomas, \u2013\n\nParkman, Francis, ,\n\nParliament, , \u2013, , , \u2013,   \n, , \u2013, ,\n\npatriotism, , , , \u2013,\n\nPenn, William, , ,\n\nPennsylvania Colony, \u2013, \u2013,\n\nPeople for the American Way,\n\nPerkins, William, \u2013\n\npersonal accountability to God, \u2013\n\nPhillips, Michael,\n\nPilgrims, , , \u2013, , , , , ,   \n, ,\n\nPipes, Sally, ,\n\nPlanned Parenthood,\n\nPledge of Allegiance,\n\nPlymouth Colony, , , , , ,   \n,\n\nPlymouth Rock (MA), , , , ,\n\npolitical action, xv, xvi\n\npopulation, eighteenth-century, in   \nthousands, \u2013 (chart); world, and   \nreduction of humanity, n24\n\npoverty, link between crime\/illegitimacy   \nand, ; number one predictor of   \nchildhood,\n\nPowhatan nation, , \u2013\n\nPresbyterians, \u2013, , ,\n\npresidential approval rating, \u2013\n\nPrinceton University, \u2013,\n\npro-choice movement,\n\nProgressive Era,\n\nprogressives, , \u2013, ,\n\nProtestant Reformation, , ,\n\nProtestant work ethic, ,\n\npublic school history textbooks,\n\nPuritans, \u2013 , \u2013, , , , ,   \n, ,\n\nPutin, Vladimir,\n\n**Q**\n\nQuakers, ,\n\nquality of life,\n\nQuebec Act,\n\n**R**\n\nRadzinsky, Edvard,\n\nRaleigh, Walter, , \u2013,\n\nRasmussen, Scott,\n\nRasmussen Reports poll, , \u2013\n\nrationalism,\n\nReagan, Ronald, , \u2013, \u2013\n\nRealClearPolitics.com,\n\nredemptioners, , n2\n\n\"Reign of Terror,\" ,\n\nrelief organizations,\n\nreligious liberty, , , , , , ,   \n, \u2013\n\nRepublican majority (Congress), xii, ,   \n,\n\nRepublicans versus Democrats (percentage   \nof self-identified), \u2013\n\nRestoring Honor rally (2010), xi, xiii\u2013   \nxiv, ,\n\nRevolutionary War. See American   \nRevolution\n\nright to a fair hearing,\n\nRoanoke colony, , \u2013,\n\nRobinson, John, ,\n\n_Roe v. Wade_ ,\n\nRolfe, John,\n\nRoosevelt, Franklin, \u2013\n\nRush, Benjamin,\n\nRussia, , \u2013, , \u2013, ,   \n, , n12\n\n**S**\n\n_Sacred Fire_ (Lillback),\n\nSalem Church Covenant,\n\nSandoz, Ellis, ,\n\nSandys, Edwin, ,\n\nscience, , , \u2013, , ,\n\nScotch-Irish, : Presbyterians,   \n\u2013,\n\nScottish emigration (to America),   \n\u2013,\n\nScrooby Congregation, , , , ,   \nn2\n\nsecular liberals, , \u2013, \u2013,\n\nsenior citizens and health care,\n\nsensitivity training,\n\n\"separation of church and state,\" ,\n\nseparation of powers,\n\nseparatists, \u2013, \u2013, , n2\n\nSeptember , 2001. See \/\n\nsermons, \u2013, , , , , , ,   \n\u2013, , , \u2013, , ,   \n\u2013, \u2013\n\nsexual promiscuity,\n\nSharia law,\n\nsingle mothers,\n\n\"Sinners in the Hands of an Angry God\"  \n(Edwards), , , \u2013\n\nslaves, , , , , ,\n\nSmith, John, , , , n2\n\nsocialism, , , , , ,\n\nsocialized medicine. See nationalized   \nhealth care\n\nSocial Security, ,\n\nsocial welfare programs,\n\nsoft tyranny,\n\nSolzhenitsyn, Aleksandr, , n12\n\n_Some Thoughts Concerning the Present  \nRevival of Religion_ (Edwards),\n\nSoviet Union, xv, , , , \u2013,   \n. _See also_ Russia\n\nSowell, Thomas,\n\nSpalding, Matthew,\n\nspending (government), xii, , ,   \n, ,\n\nStalin, Joseph, ,\n\nStamp Act, , ,\n\nstate socialism, , . _See also_  \nsocialism\n\nstay-at-home mothers,\n\nSteinem, Gloria,\n\nsteps toward real constructive change,   \n\u2013: things you can do today,   \n\u2013\n\nStoddard, Solomon, \u2013\n\nStory, Joseph,\n\n_Study of History_ , A (Toynbee),\n\nsubprime mortgage lenders,\n\nSugar Act,\n\nSullenberger, Chesley \"Sully,\" \u2013\n\nSykes, Charles J.,\n\n**T**\n\ntalk radio, ,\n\ntaxation without representation, ,   \n,\n\nTea Act, ,\n\nTea Party, , , , \u2013,   \n\u2013, : Express, ; movement,   \nxv, , , , \u2013, \u2013;   \nPatriots,\n\ntechnology, , ,\n\nTen Commandments (in public places),\n\nTennent, Gilbert, , , , ,\n\nThere's Hope Ministries, , \u2013\n\n_Tipping Point, The_ (Gladwell), xiii\n\nTocqueville, Alexis de, \u2013,\n\ntotalitarianism, , n12\n\nToynbee, Arnold, \u2013\n\ntraditional monogamous family,   \n\u2013, \u2013\n\nTuchman, Barbara,\n\nTurkey: Obama's press conference in,   \n, ,\n\nTwitter feeds,\n\n**U**\n\nUlster Scots,\n\n\"unalienable rights,\" , , ,\n\nunemployment\/joblessness, , ,\n\n_Unfailing Promise, The_ (Lee),\n\nUnited States Constitution, , , , ,   \n, , \u2013, , \u2013, , ,   \n, , , , ,\n\nU.S. Census Bureau, ,\n\nU.S. Department of Justice (under   \nObama),\n\nU.S. House of Representatives, xii, ,   \n, , \u2013\n\n****\n\nVenezuela,\n\nviolent crime,\n\nVirginia Colony, , \u2013\n\nvolunteerism,\n\nvoters: concerned with the economy,   \nwith government ethics, ; numbers   \nwho were registered, voted, and   \ndidn't vote in 2010 election, ;   \npercentage who feel the country is   \nheaded in the right direction,\n\n**W\u2013X**\n\nWall Street, , ,\n\nWalpole, Robert,\n\nWar of Independence. _See_ American   \nRevolution\n\nWar on Terrorism,\n\nWarren, Joseph,\n\nWashington, George, , , , ,   \n, , ,\n\nwelfare state, ,\n\nWesley, Charles, ,\n\nWesley, John, \u2013, , ,\n\nWest, Thomas, , , n5\n\nWhitaker, Alexander,\n\nWhite, John (capt.),\n\nWhitefield, George, \u2013, , ,   \n\u2013, \u2013, \u2013, ,\n\nWilliam III of England (William of   \nOrange), , n24\n\nWilliams, Brian,\n\nWilliams, Elisha,\n\nWilliams, Roger, ,\n\nWinthrop, John, \u2013\n\nWoodmason, Charles, n26\n\n\"Worker's Paradise,\" \u2013,\n\nWorld Trade Center, . See also \/\n\nWorld War I, ,\n\nWorld War , , , ,\n\nWright, Orville and Wilbur,\n\n**Y**\n\nYakunin, Gleb,\n\nYale College (now Yale University),   \n\u2013, ,\n\n**Z**\n\n_Zorach v. Clauson_ , \nABOUT THE AUTHOR\n\nDR. RICHARD LEE is the founding pastor of First Redeemer Church in metropolitan Atlanta. He is also the speaker for the award-winning _There's Hope America_ television series and is widely recognized as a popular spokesman on the influence of America's religious history and its impact upon today's culture.\n\nDr. Lee is a frequent speaker at national conventions and on university campuses and media outlets across the country, including FOX News, Fox Business News, CBS News, BBC, and CNN. His articles have appeared in publications such as _USA Today_ , _London_ _Times_ , _L.A. Times_ , _Essence_ , and _Newsweek_.\n\nHe is the author of eighteen popular books; his more recent work, _The American Patriot's Bible_ , reached the number five spot on Amazon.com's bestseller list.\n\nDr. Lee was educated at Mercer University and Luther Rice Seminary with postdoctoral studies at Oxford University. He also serves as a member of the prestigious Oxford Round Table, Oxford, England.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## **Advance Praise for  \n _The Power of Servant-Leadership_**\n\n_\"The Power of Servant-Leadership_ is an extraordinary collection of Robert Greenleaf's finest and most mature essays on servant-leadership, spirit, and wholeness. Today there is a growing worldwide movement of people and organizations\u2014deeply committed to servant-leadership\u2014who have been inspired by Greenleaf's earlier writings. Beautifully enriched by Peter Vaill's Foreword, Jim Shannon's Afterword, and Larry Spears's Introduction, _The Power of Servant-Leadership_ is a wonderful and unexpected gift to the world. It is destined to become a classic.\"\n\n**_Max DePree_** , _author of_ Leading Without Power _and_ Leadership is an Art\n\n\"No one in the past 30 years has had a more profound impact on thinking about leadership than Robert Greenleaf. If we sought an objective measure of the quality of leadership available to society, there would be none better than the number of people reading and studying Robert Greenleaf's writings.\"\n\n**_Peter M. Senge_** , _author of_ The Fifth Discipline\n\n\"It is wonderful to have access to Greenleaf's visionary thought in _The Power of Servant-Leadership_. Every time I read him, I am both humbled and awed. Nearly thirty years ago he wrote clearly and forcefully about the issues that still challenge us today. It is time to act on his visions, and this volume is a great help for stepping into the future that Greenleaf describes so eloquently.\"\n\n**_Margaret J. Wheatley_** , _author of_ Leadership and the New Science, _co-author of_ A Simpler Way\n\n\"The most difficult steps, Greenleaf has written, that any developing servant-leader must take, is to begin the personal journey toward wholeness and self-discovery. This new collection of essays, written with exceptional depth and grace, offer Robert Greenleaf's most powerful insights about this journey. Anyone interested in the most subtle yet most important aspects of the emerging leadership paradigm must read this book.\"\n\n**_Joseph Jaworski_** , _author of_ Synchronicity\n\n\"In my area of work, I've found that the concept of servant-leadership is crucial to advance governing boards beyond their current primitive state. _The Power of Servant-Leadership_ assembles Greenleaf's groundbreaking observations in an accessible and concentrated form for today's busy readers.\"\n\n**_John Carver_** , _author of_ Boards That Make a Difference\n\n\"The writings of Robert Greenleaf grow as the reader matures in leadership and understanding of the cultural transformation taking place around us. The essays in _The Power of Servant-Leadership_ take on new meaning with each reading and are well worth continuing reflection. We first met Greenleaf in 1978. Each year since, we have grown to discover fresh dimensions within his writing and thought. Like special mentors, these penetrating essays will continue to guide you like the wise words alive in your memory when needed.\"\n\n**_Ann McGee-Cooper and Duane Trammell_** , _co-authors of_ You Don't Have To Go Home From Work Exhausted!\n\n\"Bob Greenleaf was the first author on leadership to emphasize that human institutions mean far more than results or success or profits. He believed that we exist in order to cooperate with others to achieve purposes beyond ourselves, for some greater collective good. These splendidly thoughtful essays elaborate on that theme. I've read each of them over the past 20 or so years, and it's about time they've been published in one place. A blessing for all of us.\"\n\n**_Warren Bennis_** , _Distinguished Professor, Marshall School of Business at the University of Southern California, author of_ Organizing Genius\n\n\"We are greatly encouraged to know that the servant-leadership movement will have access to these inspiring and important writings of Robert Greenleaf. These are reflective and burnished works. For many who have had their lives affected by Greenleaf, they will be a promise fulfilled. For those who will be introduced to servant-leadership through these essays, the experience will be a welcoming introduction.\"\n\n**_Dr. John C. Burkhardt_** , _Program Director, Leadership and Higher Education, W.K. Kellogg Foundation_\n\n\"Robert Greenleaf's unique voice speaks to what, in the end, matters most in the life of a leader\u2014how _others_ have grown and prospered. Though conceived and composed years ago, Greenleaf is profoundly modern, and with the mature wisdom of experience he guides us to the essence of our current search for meaning and purpose. _The Power of Servant-Leadership_ is an extraordinary collection of a master's work, and Greenleaf's voice will resonate in your soul long after you have read the last word.\"\n\n**_Jim Kouzes_** , _Chairman and CEO, TPG\/Learning Systems, co-author of_ The Leadership Challenge and Credibility\n\n\"I am honored to add my voice to those in praise of _The Power of Servant-Leadership_. Even though it has been many years since Robert Greenleaf first gave the world his compelling insights, there has never been a time when we needed servant-leadership more than we need it now. Who knows how long business will be suffering the destructive residue of a generation of gunslinger superstar CEOs who, in their excessively narrow definition of \"stockholder value,\" have created workplaces of fear and anxiety? The only salvation of the workplace as a source of personal meaning and purpose is to develop and reinforce manager-leaders who will embrace the values of Robert Greenleaf. I hope that this book will be a powerful tool in that effort.\"\n\n**_James A. Autry_** , _author,_ Love & Profit _and_ Confessions of an Accidental Businessman\n\n\"Management thought has always been captivated by innovation, always looking for a quick fix in five or seven lessons. The overconfidence such neophilia generates is already beginning to produce a reaction\u2014the friendly critics of capitalism from George Soros to Charles Handy signal growing dissatisfaction with superficial formulas. Robert Greenleaf is not about formulas; Greenleaf is about wisdom. Wisdom is a rare and precious thing, perhaps too precious for best-seller lists, too demanding for people in search of easy answers. For people willing to think about organizational life and culture, Greenleaf is indispensable.\"\n\n**_Robert L. Payton_** , _author_ , Philanthropy: Voluntary Action for the Public Good\n\n\"That Robert Greenleaf wrote the essays on leadership in this beautiful collection some years ago hardly vitiates their importance. Today they are more relevant\u2014and more accepted by thoughtful leaders\u2014than ever before. Using words ranging from E. B. White's commentary on the \"gift of language\" to the warning in Proverbs that \"where there is no vision, the people perish,\" the author articulates profound ideas that, in this rapid-fire age of global technology, are more critical to effective leadership than ever before. Yes, the leader is\u2014must also be\u2014the servant. Mr. Greenleaf, far before his time, got it just right.\"\n\n**_John Bogle_** , _Chairman, The Vanguard Group; author,_ Bogle on Mutual Funds\n\n\"With so many publications now directing leaders to look beyond enterprise gains to find meaning and wholeness, it is most timely to revisit Robert Greenleaf's servant-leadership ideas in _The Power of Servant-Leadership_. Progressive leaders are compelled to build sustainable value, social accountability, trust and integrity in the workplace, and purpose and character in the corporation. Greenleaf's writings are sure to focus such leaders' endeavors on the heart and soul of the organization: its own people.\n\n**_Kendrick B. Melrose_** , _Chairman and CEO, The Toro Company; author,_ Making the Grass Greener on Your Side\n\n\"At last, these timeless essays by Robert Greenleaf gathered in a single volume. Buy one for yourself, and one for each of your children and grandchildren. They will thank you even as we thank this great servant-leader.\"\n\n**_Frances Hesselbein_** , _President, The Drucker Foundation; co-editor,_ The Community of the Future, The Leader of the Future, The Organization of the Future\n**The Power of Servant-Leadership**\nAlso Edited by Larry C. Spears\n\n_Insights on Leadership: Service, Stewardship, Spirit and Servant-Leadership,_ 1998\n\n_On Becoming a Servant-Leader_ (with Don M. Frick), 1996\n\n_Seeker and Servant_ (with Anne T. Fraker), 1996\n\n_Reflections on Leadership: How Robert K. Greenleaf's Theory of Servant-Leadership Influenced Today's Top Management Thinkers,_ 1995\n\nAs Contributing Author\n\n_Stone Soup for the World_ , edited by Marianne Larned, 1998\n\n_Leadership in a New Era_ , edited by John Renesch, 1994\n\n# **The Power of Servant-Leadership**\n\nESSAYS BY ROBERT K. GREENLEAF\n\nEDITED BY LARRY C. SPEARS\n\n_Foreword by Peter B. Vaill_\n\n_Afterword by James P. Shannon_\n\n**The Power of Servant Leadership**\n\nCopyright \u00a9 1998 by The Greenleaf Center for Servant-Leadership\n\nAll rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, write to the publisher, addressed \"Attention: Permissions Coordinator,\" at the address below.\n\n---\n\n  | **Berrett-Koehler Publishers, Inc.**  \n235 Montgomery Street, Suite 650  \nSan Francisco, California 94104-2916  \nTel: (415) 288-0260, Fax: (415) 362-2512  \nwww.bkconnection.com\n\n**Ordering information for print editions**\n\n_Quantity sales._ Special discounts are available on quantity purchases by corporations, associations, and others. For details, contact the \"Special Sales Department\" at the Berrett-Koehler address above.\n\n_Individual sales_. Berrett-Koehler publications are available through most bookstores. They can also be ordered directly from Berrett-Koehler: Tel: (800) 929-2929; Fax: (802) 864-7626; www.bkconnection.com\n\n_Orders for college textbook\/course adoption use_. Please contact Berrett-Koehler: Tel: (800) 929-2929; Fax: (802) 864-7626.\n\n_Orders by U.S. trade bookstores and wholesalers_. Please contact Ingram Publisher Services, Tel: (800) 509-4887; Fax: (800) 838-1149; E-mail: customer.service@ingrampublisherservices.com; or visit www.ingrampublisherservices.com\/Ordering for details about electronic ordering.\n\nBerrett-Koehler and the BK logo are registered trademarks of Berrett-Koehler Publishers, Inc.\n\nFirst Edition\n\nPaperback print edition ISBN 978-1-57675-035-3  \nPDF e-book ISBN 978-1-60994-145-1  \nIDPF e-book ISBN 978-1-62656-392-6\n\n2014-1\n\nText design: Detta Penna. Cover design: Cassandra Chu.\n\n## **Table of Contents**\n\nAbout the Author\n\n_Foreword_ by Peter B. Vaill\n\nPreface\n\nAcknowledgements\n\n_Introduction_ by Larry C. Spears\n\n1. Servant: Retrospect and Prospect\n\n2. Education and Maturity\n\n3. The Leadership Crisis\n\n4. Have You a Dream Deferred?\n\n5. The Servant as Religious Leader\n\n6. Seminary as Servant\n\n7. My Debt to E. B. White\n\n8. Old Age: The Ultimate Test of Spirit\n\n_Afterword_ by James P. Shannon\n\nReferences and Permissions\n\nGreenleaf Bibliography\n\nIndex\n\nAbout the Editor and The Greenleaf Center\n\n## **About the Author**\n\nRobert K. Greenleaf spent most of his organizational life in the field of management research, development, and education at AT&T. Just before his retirement as director of management research there, he held a joint appointment as visiting lecturer of the Massachusetts Institute of Technology's Sloan School of Management and at the Harvard Business School. In addition, he held teaching positions at both Dartmouth College and the University of Virginia.\n\nHis consultancies included Ohio University; MIT; the Ford Foundation; the R.K. Mellon Foundation; Lilly Endowment, Inc.; and the American Foundation for Management Research.\n\nAs a consultant to universities, businesses, foundations, and churches during the tumultuous 1960s and 1970s, his eclectic and wide-ranging curiosity, reading, and contemplation provided an unusual background for observing these institutions.\n\nAs a lifelong student of organization\u2014that is, how things get done\u2014he distilled these observations in a series of essays on the theme of \"the servant as leader,\" the objective of which is to stimulate thought and action for building a better, more caring society.\n\nRobert K. Greenleaf, who died in 1990, is the author of four other books in addition to _The Power of Servant-Leadership_. They are: _On Becoming a Servant-Leader_ (1996), _Seeker and Servant_ (1996), _Teacher as Servant_ (1979), and _Servant-Leadership_ (1977).\n\n## **Foreword**\n\nPETER B. VAILL, UNIVERSITY OF ST. THOMAS\n\nIn a favorite book of meditations, John Thom, a 19th-century clergyman, is quoted as follows:\n\nThe real corrupters of society may be, not the corrupt, but those who have held back the righteous leaven, the salt that has Lost its savor, the innocent who have not even the moral courage to show what they think of the effrontery of impurity, the serious, who yet timidly succeed before some loud-voiced scoffer\u2014the heart trembling all over with religious sensibilities that yet suffers itself through false shame to be beaten down into outward and practical acquiescence by some rude and worldly nature. (Tileston, 1934, p. 221)\n\nBob Greenleaf could have written this paragraph. He might have been a little more informal about the matter. He would have been careful not to sound holier than thou as he spoke of the lack of moral courage and \"hearts trembling with false shame.\" I do think, though, that these last essays of his reveal over and over, in dozens of ways, the courage the paragraph calls for. Greenleaf would have smiled at the realization that his work indeed can be seen as the \"righteous leaven\" for his favorite subject: organizations and how things get done in them.\n\nI feel privileged to be able to introduce this collection. I don't begin to think I understand these essays in their true depth and implications, but repeatedly as I have read through them and reflected on them, I find myself saying, \"Yes, he's right and what he's saying is terribly important.\" His deceptively casual writing style draws me into thinking along with him. I am sure other readers have the same experience with his thought that I do\u2014of finding him saying something that is just about in so many words what has occurred to one independently. Yet I never find him derivative or simply rehashing. Instead I, at least, find myself wondering why I haven't been more insistent about the point Greenleaf is making, why I haven't taken the idea and made it the righteous leaven of my work. His is both a familiar voice and an original one.\n\nGreenleaf has what we may call a \"big idea.\" It is the idea of the servant as leader, and many of these essays are his last explorations of the extraordinary implications of that notion. The big idea is that leadership, in the final analysis, must be about service. That is the only way it can both sustain itself as leadership and truly offer to its colleagues and \"followers\" the benefits of its insights and its energy. Greenleaf's idea resolves one of the oldest paradoxes of the leadership field: how can a leader be both concerned about the \"task\" and mindful of the \"social,\" \"concerned for production and concerned for people,\" \"effective\" in getting work done and \"efficient\" in not causing all kinds of human problems in the process? Every leadership theory tries in one way or another to deal with both of these \"dimensions.\" The big idea about service\u2014Greenleaf probably would not like this description\u2014is that it is both an attitudinal and behavioral concept. It combines a concern for getting things done with attention to the needs of those who are getting things done. There are not two functions; there is only one\u2014servanthood.\n\nGreenleaf's challenge is to make a worldly and pressured\u2014and somewhat distracted, exhausted, and frightened\u2014society of leaders and potential leaders believe it. These essays are the last we shall see of him speaking his truth as plainly and insistently as he can.\n\nIn the next few paragraphs, I want to say a few more things about the big idea. Then I will close with a few more reflections about Greenleaf the man.\n\nThere are five themes in Greenleaf's way of talking about \"the servant as leader\" that strike me as quite significant. They are as follows: (1) the grammar of the phrase itself; (2) Greenleaf's commitment to practice; (3) the importance of mission; (4) the nature and role of \"persuasion\"; and (5) his idea about a \"theology of institutions.\" These five themes appear repeatedly throughout his writings, yet, except for his discussion of mission, Greenleaf himself is frequently so offhand and low-keyed in the way he talks that the significance of the idea may be missed.\n\n**1. The grammar of \"the servant as leader.\"** As the reader is no doubt aware, Greenleaf's idea is frequently abbreviated to \"servant-leadership,\" or \"servant leadership.\" For purposes of economy and simplicity, it is probably necessary to abbreviate Greenleaf's phrase in this fashion. But the danger is that a key feature of his big idea will be thereby lost. In the phrase \"the servant as leader\" (which after all was Greenleaf's title for the original seminal essay), the subject is the servant or service; the predicate is the leader. His phrase is an application of the philosophy of service to the practice of leadership.\n\nAs is apparent in various writings in the present volume, service was the most important thing for Greenleaf. He frequently takes the time to ground the phrase in biblical references and connect it to the deepest yearnings of the human spirit. To be sure, Greenleaf is deeply concerned about leadership, but his concern is that it is being practiced (and theorized about) without reference to service. For Greenleaf, service is the moral dimension of prime importance, not just for leadership but for life. It is service he fears has been lost sight of, not leadership. To put it compactly, I think Greenleaf is saying that leadership is a special case of service; he is not saying that service is a special case of leadership.\n\nThis distinction is important because we live in a period when there is an almost frantic casting about by leaders and leadership thinkers for answers to the profound dilemmas of leadership in our turbulent and unpredictable world. Some will take up Greenleaf's notion and test it for the extent to which it \"solves their leadership problem.\" The \"servant as leader\" idea does not solve the leadership problem in the sense that leaders and scholars might hope. If anything, it will frustrate and annoy such seekers. Soon we will hear them saying, \"Servant-leadership? Oh yeah, right, we tried that.\" But this misunderstands Greenleaf's challenge. As I understand him, he is not asking, \"What service can you render as a leader?\" but rather \"What leadership can you exercise as a servant?\"\n\n**2. Greenleaf's commitment to practice**. Repeatedly in these essays, Greenleaf reminds us that he is writing about the servant as leader as a student of how things get done in organizations. He wants us to look concretely at the way the actions and attitudes of service can transform relations among concrete human beings. His short answer to the question, I think, would be, \"In the long run, things get done among human beings, including within organizations, by people serving one another.\" That is what his experience has taught him, and for which he thinks there is overwhelming confirmatory evidence throughout human history.\n\nGreenleaf's interest in practice has several dimensions that, taken together, give him quite a different outlook from most other contemporary writers on leadership. For one thing, he knows how complex any process of action is in an organization. His repeated references to his own business experience as well as to his many extensive contacts with other organizations make clear that he has no illusions about how unstable and dicey organizational action can be. His commitment to action is also seen in his consistent attention to the process by which his ideas can be implemented. Just about when I, as a reader, am thinking, \"Sounds good, but how are you going to do that?\" it turns out that Greenleaf has been asking himself the same question; and so he proceeds to offer a few thoughts about how what he is talking about can happen.\n\nAnother feature of his commitment to practice is his concern for the process by which more servant-leaders can be produced. Yes, he wants men and women in positions of influence to consider his ideas, but he is equally concerned with increasing the sheer number of younger men and women who will be helped to develop as servant-leaders.\n\nThis concern leads him to challenge educational institutions in exactly the way that they (as a professor, I should say \"we\") need to be challenged: we in higher education know that only by accident are we producing visionary young men and women who aspire to leadership, to the extent we are producing them at all. We don't like to admit it. Greenleaf has found us out.\n\nFinally, his concern for practice focuses him on the _way_ that any organization works. It is clear in this book that in the later years of his life, as well as at AT&T, he moved among the high and the mighty of society's major institutions: universities, businesses, foundations, and the like. As he quietly but firmly repeats his interest in how things get done, it is as if he were saying, \"I know all of you are terribly important people who feel that what your organizations do is terribly important in society. But my angle of vision has taught me something about the way your institutions work and don't work that perhaps you are unaware of.\" We live in an age of \"content experts\" who know all about moon landings and heart transplants and giga-sized databases and global markets and transformations of the biosphere\u2014and many of them are leading or aspire to lead institutions concerned with those things. Greenleaf is a process expert, and he has been thinking about phenomena that often get lost in the swirl of events that our organizational leaders are so good at triggering.\n\n**3. The importance of mission.** One can hardly read a page of these essays without encountering Greenleaf's deep beliefs about the importance of mission in organizations. He, of course, is not the only or the first writer on organizations to declare the importance of mission. But as we reflect along with him in his various discussions of missions, I encourage us to keep in mind a deep conundrum\u2014one that I think Greenleaf himself is quite aware of. The question is, Where does a sense of mission come from? The power and importance of mission, once articulated, is unquestionable. But how does it happen? What is the process by which a deep sense of an organization's reason to be gets into the collective psyche of its members? Greenleaf realizes what a tough question this is. He also realizes that a convenient answer is that some charismatic individual steps forward and articulates a mission full-blown\u2014and there are even places in these essays where Greenleaf seems to be personally attracted to this \"Great Man\" model of leadership.\n\nBut I think he knows better. Without denying that a single individual can occasionally say or do something that has the requisite galvanizing effect, Greenleaf knows that servanthood is not about stepping forward and taking charge. Rather, I think he understands that there has to be such a thing as service to the mission-formulation and articulation process itself. That is different from doing people's thinking and feeling for them. Greenleaf's servant-leader is a servant of the organization's learning process, I think. About that, the servant-leader is tough-minded and unflinching. Ways must be found whereby organization members can come together on a dream.\n\n**4. The nature and role of persuasion.** Greenleaf is particularly eloquent about the role of _persuasion_ in servant-leadership. He uses the term in its everyday sense, and I think he would not be impressed with what the academic world has done with the concept. Nor would he accept a cynical view of persuasion that equates it to manipulation or slick rhetoric.\n\nThere is healthy and honest persuasion. I think Greenleaf wants us to envision ordinary conversation between people where at least one, but hopefully many or most, are trying to say things to each other and do things for each other that are in service of a common dream or sense of mission that they share. \"Persuasion\" is the process of one person co-creating to another or others what he or she thinks pursuit of the mission entails, including statements about the meaning of the mission and\/or the need for new attention to its meaning. The service is in the thought, the creativity, the information, the experience, and the energetic vision the co-creator feels. I have seen these conversations many times. They occur all the time in excellent organizations\u2014what in a series of studies some years ago I called \"high performing systems.\" In these conversations committed people are doing everything they can to help each other. Greenleaf doesn't see why a more widespread philosophy of service to a mission or a dream can't occur almost anywhere. My own studies and experience convince me he is quite right: there is no type of organized human activity where mutual service to a common vision and the energetic reminding of each other of what is involved (i.e., persuasion) cannot happen.\n\n**5. A theology of institutions.** I find Greenleaf's speculations on the idea of a theology of institutions or organizations one of the most interesting and original notions in his writings. While quick to state that he is not a theologian, and therefore does not know what is required for a body of ideas to be a worked-out theology, he nonetheless seems very sure that such a theology is needed and should be possible. He does not see why, if there can be a theology of persons, there cannot be a theology of organizations.\n\nHe is not talking about theocracy, not about top-down governance and control by theocratic principles and doctrines. He is far too independent a thinker and a man for that. No, I think he wants us to reflect on what the fundamental nature of a human organization is; how we regard it. Is it a pure invention for secular purposes? Just a vehicle, at bottom an ad hoc arrangement with no claim to a deeper significance and justification for its existence? My guess is that Greenleaf perceives intuitively that the mission or vision or dream has a quality beyond its verbal, secular content. He would like to see us work out what this quality is, and his hunch is that when we do we will find ourselves reflecting on the relation of this dream to the spiritual worldviews that we are already familiar with for persons. My main point, though, is to invite the reader to reflect deeply on these comments of Greenleaf's about a theology of institutions. They may turn out to be among his most original and significant insights.\n\n### **A Final Word**\n\nMost of the essays in this volume have been previously published in pamphlet form by the Greenleaf Center for Servant-Leadership. At the back of most of these pamphlets is a brief biography of Greenleaf and a photograph of him taken apparently late in life. This photograph of Greenleaf may have a significance in his life or for those close to him that I am not aware of. Inquiries at the Center did not result in any particular stories about the photo or the reasons for its use.\n\nYet I consider it so extraordinary an image\u2014one that stimulates thoughts and feelings in me that are pertinent to the spirit of these essays and to this Foreword. The picture is almost a direct frontal of his head and shoulders, open-collared and informal. He is turned very slightly to his left from the camera and the effect is to place the left half of his face almost completely in shadow while the right side remains fully illuminated. His silky white hair lies smoothly on the right side of his head but is invisible on the left. The head is tilted just slightly upward and the mouth is set in a patient composed line\u2014not preparing to speak, but not relaxed either. Even though half-lit, the nose is strong and suggests that in profile he might have been distinctly \"hawklike.\" It is a striking picture, and made the more so by one final feature: it is dramatically off-center. Greenleaf's head and shoulders occupy the right half of the picture, but the left half is blank space. This left half is strongly illuminated at the bottom and shades to black at the top, but is completely empty. Most of us, on pulling such a snapshot out of the envelope in the supermarket parking lot, would immediately judge the picture to be not worth saving because it is so off-center. The longer I look at this picture the more I am stimulated to think about Greenleaf as a person. There is a darkness running through his writings. He is writing about terribly complex problems which contain the potential for some very bad outcomes\u2014and he knows it. Yet he is not wringing his hands, not paralyzed with alarm. The mouth is composed; the eyes are sharply engaged, but kindly (this judgment about the eyes being one that a friend of mine independently described the same way). The shadow down the middle of the face with the wispy white hair and the off-center placement make me feel as if I am sitting there talking to him, perhaps in the evening with late light coming horizontally through the window onto half of his face as he sits talking to me.\n\nThe slight uptilt to the head, the direct gaze, and the mouth that has said a lot and could say more but does not at the moment feel the need, combine in a sort of friendly but quizzical challenge. \"I am coming from both darkness and light,\" says the face, \"both out of problems and possibilities. I may even be a little off-center myself in my understanding of these things. But tell me: are you going to do anything about them?\" Am I\u2014are we\u2014indeed?\n\n#### _References_\n\nTileston, M. _Daily Strength for Daily Needs_. New York: Putnam, 1934. (Originally published in 1884)\n\n## **Preface**\n\n_The Power of Servant-Leadership_ is an extraordinary collection of Robert K. Greenleaf's mature and final writings on the concept of servant-leadership. Servant-leadership, itself, is an important part of the emerging leadership and management paradigm for the 21st century. Today, there is a growing movement of people and organizations that have been influenced by servant-leadership. _The Power of Servant-Leadership_ offers Robert Greenleaf's own insights on the interrelated subjects of serving and leading, wholeness and spirit.\n\nIn 1970, retired AT&T executive Robert K. Greenleaf coined the term _servant-leadership_ to describe a kind of leadership that he felt was largely missing from organizations. It was Greenleaf's belief that leadership ought to be based on serving the needs of others and on helping those who are served to become \"healthier, wiser, freer, more autonomous, more likely themselves to become servants.\" Over the next 20 years, Greenleaf wrote a series of highly influential books and essays, which have helped lead the way for the emerging model in leadership and management. _The Power of Servant-Leadership_ is intended to stimulate and inspire many people in their practice of a more caring, serving kind of leadership.\n\nFor those readers who may already be familiar with Greenleaf's classic book, _Servant Leadership_ (1977, Paulist Press), _The Power of Servant-Leadership_ represents, in many ways, a true sequel. For others, this book will serve as a good introduction to the concept and practice of servant-leadership and to the insightful ideas of Robert K. Greenleaf.\n\n_The Power of Servant-Leadership_ is a collection of eight essays by Robert Greenleaf, most of which were originally published as separate pamphlets by The Greenleaf Center. Because of their pamphlet format, these essays have not been previously available in bookstores or libraries, and they have had a far too limited readership. Many of the essays contained in this book were published during a ten-year period from 1977 to 1987. They include some of Greenleaf's most original, powerful, and final works. Many of them reflect Greenleaf's own continual refinement of the servant-as-leader concept; several others focus on the related issues of spirit, commitment to vision, and seeing things whole.\n\nAs an aid to the reader, the following is a brief, thumbnail description of each of the eight Greenleaf essays included in this volume.\n\n##### _\"Servant: Retrospect and Prospect\"_\n\nThis is a reflection on ten years of working with the ferment that the publication of \"The Servant as Leader\" generated. Mind-sets that enforce rigidity in our society and frustrate reformers are identified in universities, health professions, churches, institutions, and businesses. Greenleaf addresses the creation of liberating visions that are required to surmount these mind-sets.\n\n##### _\"Education and Maturity\"_\n\nThe meaning of maturity is explored as a lifelong journey of developing one's own unique potential into personal significance and character, while recognizing the relationship between using work as a means of fulfillment and conformity as an essential element of effective society.\n\n##### _\"The Leadership Crisis\"_\n\nThis thought-provoking essay suggests why the institutions of our day, and our society in general, are all suffering from a leadership crisis. Greenleaf suggests a solution to this dilemma by defining three kinds of power, their uses, and their limitations. He also describes the necessary role of a \"great dream,\" or vision, and the effective use of persuasive power in the life of an institution.\n\n##### _\"Have You a Dream Deferred?\"_\n\nBy redefining responsibility for growth and constructive influence, an opportunity to build a new ethic challenges those who seek to reach their potential in serving the public interest. Focus is placed on the qualities of a lifestyle of distinction.\n\n##### _\"The Servant as Religious Leader\"_\n\nFrom the perspective of organization (how things get done), rather than as a scholar or theologian, this essay addresses the phenomenon of \"spirit\" on which all leadership is so dependent that it may never be capsulized. The central theme is: Work! Do something! Work to increase the number of spiritual leaders (not just in churches, but everywhere) who are capable of holding their own against the forces of destruction and indifference.\n\n##### _\"Seminary as Servant\"_\n\nSeminaries of all denominations are identified, along with some foundations, as standing in a strategic position to generate a much-needed liberating and prophetic vision. They are seen as having the opportunity to choose a greater vision of service to society than that to which most aspire. _Seminary as Servant_ is addressed to those who would take the risks of leadership to move the seminary from where it is to the greater place it might be.\n\n##### _\"My Debt to E. B. White\"_\n\nThis lively essay traces the influence of author E. B. White on Greenleaf's thinking over a period of 55 years. He concludes that two of White's abilities\u2014\"seeing things whole\" and \"the gift of language to express what he saw\"\u2014were key to his power as a writer. The lifelong personal development these insights stimulated in the author sings through this essay in quotations and examples.\n\n##### _\"Old Age: The Ultimate Test of Spirit\"_\n\nHere a gathering of personal reflections from 83 years of living, working, thinking, and learning weaves a definition of _spirit_ as \"the driving force behind the motive to serve,\" and defines the test of old age as the time to assess one's active life and to achieve serenity from knowing that one has served. Greenleaf identifies those significant people, events, and things that helped him\u2014and that may help others\u2014learn to live wholly in the moment of \"now\" and to pay attention to the things that make life more rewarding. One of the very last pieces written by Robert Greenleaf, \"Old Age: The Ultimate Test of Spirit\" contains great wisdom. This essay of depth and grace ranks alongside the finest writings of such American essayists as Emerson and Thoreau.\n\nRobert Greenleaf's own writings are increasingly in demand as both he and the servant-leadership concept have become much better known since his death in September of 1990. Following the publication of his classic book, _Servant Leadership,_ in 1977, Greenleaf went on to write a series of separately published essays. Most of these essays, ranging between 12 and 70 pages each, were individually published over a ten-year period by The Greenleaf Center. In the collection of these essays in one book, readers now have an opportunity to delve into some of Robert Greenleaf's very best writings and to understand in greater detail the ideas and practices that make up servant-leadership.\n\nThe richness of this volume has been further deepened by Peter Vaill's Foreword and by Jim Shannon's Afterword. Peter Vaill is a noted consultant and professor of human systems and the author of _Managing as a Performing Art_ and _Learning as a Way of Being_. Jim Shannon's remarkable career has included time as a former priest, college president, lawyer, and head of the General Mills Foundation. He is currently a senior advisor to the Council on Foundations. Both Vaill and Shannon have read and reread Greenleaf's writings over many years, and through their contributions to this book, each has provided helpful insights into the significance of Greenleaf and servant-leadership.\n\nI view the growing trend toward servant-leadership as a reflection of the ongoing maturation of humankind. It is my sincere hope that this movement toward servant-leadership will find an even greater expression in the decades and centuries to come. Through his writings on the servant-as-leader idea, Robert Greenleaf has increasingly become known as an intellectual trailblazer in leadership and management, and as someone who is helping to lead many of us into a more caring and thoughtful 21st century.\n\n_Larry C. Spears  \nIndianapolis, Indiana  \nAugust, 1998_\n\n## **Acknowledgments**\n\nI am particularly indebted to my colleagues at The Greenleaf Center for Servant-Leadership\u2014Tamyra Freeman, Nancy Larner, Michele Lawrence, Isabel Lopez, Geneva Loudd, Jim Robinson, Richard Smith, and Kelly Tobe\u2014for their friendship, encouragement, and support. My own journey in servant-leadership has also been enriched through my partnership with the following past and present Greenleaf Center trustees: Bill Bottum; Linda Chezem; Diane Cory; Sister Joyce DeShano; Joe DiStefano; Harley Flack; Newcomb Greenleaf; Bill Guillory; Carole Hamm; Jack Lowe, Jr.; Jeff McCollum; Ann McGee-Cooper; Andy Morikawa; Jim Morris; Paul Olson; Bob Payton; Sister Joel Read; Sister Sharon Richardt; Judith Sturnick; and Jim Tatum.\n\nI am most grateful to two institutions\u2014the W. K. Kellogg Foundation (especially John Burkhardt, Stephanie Clohessey, and Larraine Matusak) and Lilly Endowment, Inc. (particularly Willis Bright and Craig Dykstra)\u2014for their ongoing support of servant-leadership and The Greenleaf Center.\n\nMy special thanks go to the wonderful staff at Berrett-Koehler Publishers, especially Valerie Barth and Steve Piersanti.\n\nI wish to thank my family and friends for their love and encouragement, especially my wife, Beth Lafferty; my sons, James and Matthew Spears; and my parents, Bertha and L C Spears. I would also like to thank the following people and institutions for their encouragement and support along the way: James Autry, Walter Brogan, Dick Broholm, Steve Brooks, Max Case, the Central Committee for Conscientious Objectors, the Childbirth Education Association of Greater Philadelphia, Donna Davis, Roberta and Robert DeHaan, Vinton Deming, DePauw University, Max DePree, Karen Farmer, Diann and Allison Feldman, Anne Fraker, _Friends Journal_ , Friends Select School, Joe and Laurie Goss, Cathy Gray, the Great Lakes Colleges Association's Philadelphia Center, the Greater Philadelphia Philosophy Consortium, Lisa Greenleaf, Robert K. Greenleaf, John and Mary Gummere, John Haynes, Todd and Ellen Daniels-Howell, Madeline Greenleaf Jaynes, Frank and Marian Killian, Michael Krausz, Eva and Richard Krebs, John and Aline Lafferty, Roger and Verona Lafferty, Jon Landau, Ralph Lewis, Diane Lisco, Isabel Lopez, Robert Lynn, Tjeb Maris, John Nason, National Association for Community Leadership, Hilbrand Nawijn, Marcia Newman, John Noble, Keith Opdahl, Lou Outlaw, M. Scott Peck, Mike and Nancy Revnes, Olcutt Sanders, Gert Schaart, Karen Schultz, Peter Senge, Alice Simpson, Elissa Sklaroff, Debra Spears, Lucian and Chassie Spears, Debra Thomas, Peter Vaill, Wendell Walls, and Signe Wilkinson.\n\nFinally, I wish to express my deep appreciation to the many servant-leaders working within countless organizations around the world. Your efforts at building spirit in the workplace truly inspire others to servant-leadership.\n\n## **Introduction**\n\n### LARRY SPEARS\n\n\"The servant-leader is servant first. It begins with the natural feeling that one wants to serve. Then conscious choice brings one to aspire to lead. The best test is: do those served grow as persons; do they, while being served, become healthier, wiser, freer, more autonomous, more likely themselves to become servants?\"\n\nRobert K. Greenleaf, _The Servant as Leader,_ 1970\n\nWith that initial definition of servant-leadership in 1970, Robert K. Greenleaf planted a seed of an idea that continues to grow in its influence on society with each passing year. In fact, during the 1990s, we have witnessed an unparalleled explosion of interest and practice of servant-leadership. In many ways, it can be said that the times are only now beginning to catch up with Robert Greenleaf's visionary call to servant-leadership.\n\nServant-leadership, now in its third decade as a specific leadership and management concept, continues to create a quiet revolution in workplaces around the world. This introduction is intended to provide a broad overview of the growing influence this unique concept of servant-leadership is having on people and their workplaces.\n\nAs we prepare to enter the 21st century, we are witnessing a shift in many businesses and nonprofit organizations\u2014away from traditional autocratic and heirarchical modes of leadership and toward a model based on teamwork and community; one that seeks to involve others in decision making; one that is strongly based in ethical and caring behavior; and one that is attempting to enhance the personal growth of workers while at the same time improving the caring and quality of our many institutions. This emerging approach to leadership and service is called \"servant-leadership.\"\n\nThe words _servant_ and _leader_ are usually thought of as being opposites. When two opposites are brought together in a creative and meaningful way, a paradox emerges. And so the words _servant_ and _leader_ have been brought together to create the paradoxical idea of servant-leadership.\n\nThe basic idea of servant-leadership is both intuitive and sensible. In light of the history of the industrial revolution, for a long time there has been a tendency to view people as objects; institutions have viewed people mostly as cogs within a machine. In the past few decades, we have witnessed a shift in that long-held view. Standard practices are rapidly shifting toward the ideas put forward by Robert Greenleaf, Margaret Wheatley, Stephen Covey, Peter Senge, Danah Zohar, Max DePree, and many others who suggest that there is a better way to manage our organizations in the 21st century.\n\nToday there is a much greater recognition of the need for a more team-oriented approach to leadership and management. Robert Greenleaf's writings on the subject of servant-leadership helped get this movement started, and his views have had a profound and growing effect on many.\n\n\"Despite all the buzz about modern leadership techniques, no one knows better than Greenleaf what really matters.\"\n\n_Working Woman_ magazine (March, 1992)\n\nThe term _servant-leadership_ was first coined in a 1970 essay by Robert K. Greenleaf (1904\u20131990) entitled \"The Servant as Leader.\" Greenleaf, who was born in Terre Haute, Indiana, spent most of his organizational life in the field of management research, development, and education at AT&T. Following a 40-year career at AT&T, Greenleaf enjoyed a second career that lasted another 25 years, during which time he served as an influential consultant to a number of major institutions, including: Ohio University, M.I.T., Ford Foundation, R. K. Mellon Foundation, the Mead Corporation, the American Foundation for Management Research, and Lilly Endowment, Inc. In 1964, Greenleaf also founded the Center for Applied Ethics, which was renamed the Robert K. Greenleaf Center in 1985 and is now headquartered in Indianapolis.\n\nAs a lifelong student of how things get done in organizations, Greenleaf distilled his observations in a series of essays and books on the theme of \"the servant as leader\"\u2014the objective of which was to stimulate thought and action for building a better, more caring society.\n\nThe idea of the servant-as-leader came partly out of Greenleaf's half-century of experience in working to shape large institutions. However, the event that crystallized Greenleaf's thinking came in the 1960s, when he read Herman Hesse's short novel, _Journey to the East_ \u2014an account of a mythical journey by a group of people on a spiritual quest. The central figure of the story is Leo, who accompanies the party as the servant and who sustains them with his caring spirit. All goes well with the journey until one day Leo disappears. The group quickly falls into disarray, and the journey is abandoned. They discover that they cannot make it without the servant, Leo. After many years of searching, the narrator of the story stumbles upon Leo and is taken into the religious order that had sponsored the original journey. There, he discovers that Leo, whom he had first known as a servant, was in fact the head of the order, its guiding spirit and leader.\n\nAfter reading this story, Greenleaf concluded that the central meaning of it was that the great leader is first experienced as a servant to others, and that this simple fact is central to his or her greatness. True leadership emerges from those whose primary motivation is a deep desire to help others.\n\nIn 1970, at the age of 66, Greenleaf published \"The Servant as Leader,\" the first of a dozen essays and books on servant-leadership. Since that time, over a half-million copies of his books and essays have been sold worldwide. Slowly but surely, Greenleaf's servant-leadership writings have made a deep, lasting impression on leaders, managers, educators, and many others who are concerned with issues of leadership, management, service, and spirit.\n\nIn all of these works, Greenleaf discusses the need for a new kind of leadership model: a model that puts serving others\u2014including employees, customers, and community\u2014as the number one priority. Servant-leadership emphasizes increased service to others; a holistic approach to work; the promotion of a sense of community; and a deepening understanding of spirit in the workplace.\n\nWho is a servant-leader? Greenleaf said that the servant-leader is one who is a servant first. In \"The Servant as Leader\" he wrote: \"It begins with the natural feeling that one wants to serve, to serve first. Then conscious choice brings one to aspire to lead. The difference manifests itself in the care taken by the servant\u2014first to make sure that other people's highest priority needs are being served. The best test is: Do those served grow as persons; do they, while being served, become healthier, wiser, freer, more autonomous, more likely themselves to become servants?\"\n\nIt is important to stress that servant-leadership is not a \"quick-fix\" approach. Nor is it something that can quickly be instilled within an institution. At its core, servant-leadership is a long-term, transformational approach to life and work\u2014in essence, a way of being\u2014that has the potential for creating positive change throughout our society.\n\n\"Servant leadership deals with the reality of power in everyday life\u2014its legitimacy, the ethical restraints upon it and the beneficial results that can be attained through the appropriate use of power.\"\n\n_New York Times_ (October 2, 1990)\n\nAfter some years of carefully considering Greenleaf's original writings, I have identified a set of ten characteristics of the servant-leader that I view as being of critical importance, central to the development of servant-leaders.\n\n1. _Listening_ : Leaders have traditionally been valued for their communication and decision-making skills. Although these are also important skills for the servant-leader, they need to be reinforced by a deep commitment to listening intently to others. The servant-leader seeks to identify the will of a group and helps clarify that will. He or she seeks to listen receptively to what is being said (and not said!). Listening also encompasses getting in touch with one's own inner voice, and seeking to understand what one's body, spirit, and mind are communicating. Listening, coupled with regular periods of reflection, are essential to the growth of the servant-leader.\n\n2. _Empathy_ : The servant-leader strives to understand and empathize with others. People need to be accepted and recognized for their special and unique spirits. One assumes the good intentions of co-workers and does not reject them as people, even when one is forced to refuse to accept their behavior or performance. The most successful servant-leaders are those who have become skilled empathetic listeners.\n\n3. _Healing_ : The healing of relationships is a powerful force for transformation and integration. One of the great strengths of servant-leadership is the potential for healing one's self and one's relationship to others. Many people have broken spirits and have suffered from a variety of emotional hurts. Although this is a part of being human, servant-leaders recognize that they have an opportunity to \"help make whole\" those with whom they come in contact. In \"The Servant as Leader,\" Greenleaf writes, \"There is something subtle communicated to one who is being served and led if, implicit in the compact between servant-leader and led, is the understanding that the search for wholeness is something they share.\"\n\n4. _Awareness_ : General awareness, and especially self-awareness, strengthens the servant-leader. Making a commitment to foster awareness can be scary\u2014you never know what you may discover! Awareness also aids one in understanding issues involving ethics and values. It lends itself to being able to view most situations from a more integrated, holistic position. As Greenleaf observed: \"Awareness is not a giver of solace\u2014it is just the opposite. It is a disturber and an awakener. Able leaders are usually sharply awake and reasonably disturbed. They are not seekers after solace. They have their own inner serenity.\"\n\n5. _Persuasion_ : Another characteristic of servant-leaders is a reliance on persuasion, rather than on one's positional authority, in making decisions within an organization. The servant-leader seeks to convince others, rather than coerce compliance. This particular element offers one of the clearest distinctions between the traditional authoritarian model and that of servant-leadership. The servant-leader is effective at building consensus within groups. This emphasis on persuasion over coercion probably has its roots within the beliefs of The Religious Society of Friends (Quakers)\u2014the denomination with which Robert Greenleaf himself was most closely allied.\n\n6. _Conceptualization_ : Servant-leaders seek to nurture their abilities to \"dream great dreams.\" The ability to look at a problem (or an organization) from a conceptualizing perspective means that one must think beyond day-to-day realities. For many managers, this is a characteristic that requires discipline and practice. The traditional manager is consumed by the need to achieve short-term operational goals. The manager who wishes to also be a servant-leader must stretch his or her thinking to encompass broader-based conceptual thinking. Within organizations, conceptualization is, by its very nature, the proper role of boards of trustees or directors. Unfortunately, boards can sometimes become involved in the day-today operations (something that should always be discouraged!) and fail to provide the visionary concept for an institution. Trustees need to be mostly conceptual in their orientation; staffs need to be mostly operational in their perspective; and, the most effective CEOs and managers probably need to develop both perspectives. Servant-leaders are called to seek a delicate balance between conceptual thinking and a day-to-day focused approach.\n\n7. _Foresight_ : Closely related to conceptualization, the ability to foresee the likely outcome of a situation is hard to define, but easy to identify. One knows it when one sees it. Foresight is a characteristic that enables the servant-leader to understand the lessons from the past, the realities of the present, and the likely consequence of a decision for the future. It is also deeply rooted within the intuitive mind. There hasn't been a great deal written on foresight. It remains a largely unexplored area in leadership studies, but one most deserving of careful attention.\n\n8. _Stewardship_ : Peter Block (author of _Stewardship_ , and _The Empowered Manager_ ) has defined stewardship as \"holding something in trust for another.\" Robert Greenleaf's view of all institutions was one in which CEOs, staffs, and trustees all played significant roles in holding their institutions in trust for the greater good of society. Servant-leadership, like stewardship, assumes first and foremost a commitment to serving the needs of others. It also emphasizes the use of openness and persuasion, rather than control.\n\n9. _Commitment to the growth of people_ : Servant-leaders believe that people have an intrinsic value beyond their tangible contributions as workers. As such, the servant-leader is deeply committed to the growth of each and every individual within his or her institution. The servant-leader recognizes the tremendous responsibility to do everything within his or her power to nurture the personal, professional, and spiritual growth of employees. In practice, this can include (but is not limited to) concrete actions such as: making funds available for personal and professional development; taking a personal interest in the ideas and suggestions from everyone; encouraging worker involvement in decision making; and actively assisting laid-off workers to find other employment.\n\n10. _Building community_ : The servant-leader senses that much has been lost in recent human history as a result of the shift from local communities to large institutions as the primary shaper of human lives. This awareness causes the servant-leader to seek to identify some means for building community among those who work within a given institution. Servant-leadership suggests that true community can be created among those who work in businesses and other institutions. Greenleaf said, \"All that is needed to rebuild community as a viable life form for large numbers of people is for enough servant-leaders to show the way, not by mass movements, but by each servant-leader demonstrating his [or her] own unlimited liability for a quite specific community-related group.\"\n\nThese ten characteristics of servant-leadership are by no means exhaustive. However, they serve to communicate the power and promise that this concept offers to those who are open to its invitation and challenge.\n\n\"Servant leadership is the essence of quantum thinking and quantum leadership.\"\n\nDanah Zohar, author, _Rewiring the Corporate Brain_\n\nThere are a half-dozen major areas in which the principles of servant-leadership are being applied in significant ways. The first area has to do with servant-leadership as an institutional philosophy and model. Servant-leadership crosses all boundaries and is being applied by a wide variety of people working with for-profit businesses, not-for-profit corporations, churches, universities, and foundations.\n\nIn recent years, a number of institutions have jettisoned their old heirarchical models and replaced them with a servant-leader approach. Servant-leadership advocates a group-oriented approach to analysis and decision making as a means of strengthening institutions and of improving society. It also emphasizes the power of persuasion and seeking consensus over the old \"top-down\" form of leadership. Some people have likened this to turning the heirarchical pyramid upside down. Servant-leadership holds that the primary purpose of a business should be to create a positive impact on its employees and community, rather than using profit as the sole motive.\n\nMany individuals within institutions have adopted servant-leadership as a guiding philosophy. An increasing number of companies have adopted servant-leadership as part of their corporate philosophy or as a foundation for their mission statement. Among these are the Sisters of St. Joseph's Health System (Ann Arbor, MI), The Toro Company (MN), Schneider Engineering Company (Indianapolis, IN), and TDIndustries (Dallas, TX).\n\nTDIndustries, one of the earliest practitioners of servant-leadership in the corporate setting, is a Dallas-based heating and plumbing contracting firm that was named \"one of the ten best companies to work for,\" in the January 12, 1998 issue of _Fortune_ magazine. TDIndustries was profiled in Robert Levering and Milton Moskowitz's book, _The 100 Best Companies to Work for in America_. In their profile of TDIndustries, the authors discuss the longtime influence that servant-leadership has had on the company. TDI's founder, Jack Lowe, Sr., stumbled upon \"The Servant as Leader\" essay in the early 1970s and began to distribute copies of it to his employees. They were invited to read through the essay, and then to gather in small groups to discuss its meaning. The belief that managers should serve their employees became an important value for TDIndustries.\n\nTwenty-five years later, Jack Lowe, Jr., continues to use servant-leadership as the guiding philosophy for TDI. Levering and Moskowitz note, \"Even today, any TDPartner who supervises at least one person must go through training in servant-leadership.\" In addition, all new employees continue to receive a copy of \"The Servant as Leader\" essay.\n\nServant-leadership has influenced many noted writers, thinkers, and leaders. Max DePree, former chairman of the Herman Miller Company and author of _Leadership Is an Art_ and _Leadership Jazz_ has said, \"The servanthood of leadership needs to be felt, understood, believed, and practiced.\" And Peter Senge, author of _The Fifth Discipline_ , has said that he tells people \"not to bother reading any other book about leadership until you first read Robert Greenleaf's book, _Servant Leadership_. I believe it is the most singular and useful statement on leadership I've come across.\" In recent years, a growing number of leaders and readers have \"rediscovered\" Robert Greenleaf's own writings through DePree's and Senge's books.\n\nA second major application of servant-leadership is its pivotal role as the theoretical and ethical basis for \"trustee education.\" Greenleaf wrote extensively on servant-leadership as it applies to the roles of boards of directors and trustees within institutions. His essays on these applications are widely distributed among directors of for-profit and nonprofit organizations. In his essay, \"Trustees as Servants,\" Greenleaf urged trustees to ask themselves two central questions: \"Whom do you serve?\" and \"For what purpose?\"\n\nServant-leadership suggests that boards of trustees need to undergo a radical shift in how they approach their roles. Trustees who seek to act as servant-leaders can help create institutions of great depth and quality. Over the past decade, two of America's largest grant-making foundations (Lilly Endowment, Inc. and the W. K. Kellogg Foundation) have sought to encourage the development of programs designed to educate and train not-for-profit boards of trustees to function as servant-leaders. The Greenleaf Center itself does a great deal of work with trustee boards.\n\nThe third application of servant-leadership concerns its deepening role in community leadership organizations across the country. A growing number of community leadership groups are using Greenleaf Center resources as part of their own education and training efforts. Some have been doing so for more than 15 years.\n\nThe National Association for Community Leadership has adopted servant-leadership as a special focus. Recently, NACL named Robert Greenleaf as the posthumous recipient of its National Community Leadership Award. This award is given annually to honor an individual whose work has made a significant impact on the development of community leadership worldwide.\n\nM. Scott Peck, who has written about the importance of building true community, says the following in _A World Waiting to Be Born_ : \"In his work on servant-leadership, Greenleaf posited that the world will be saved if it can develop just three truly well-managed, large institutions\u2014one in the private sector, one in the public sector, and one in the nonprofit sector. He believed\u2014and I know\u2014that such excellence in management will be achieved through an organizational culture of civility routinely utilizing the mode of community.\"\n\nThe fourth application involves servant-leadership and experiential education. During the past 20 years, experiential education programs of all sorts have sprung up in virtually every college and university\u2014and, increasingly, in secondary schools as well. Experiential education, or \"learning by doing,\" is now a part of most students' educational experience.\n\nAround 1980, a number of educators began to write about the linkage between the servant-leader concept and experiential learning under a new term called \"service-learning.\" It is service-learning that has become a major focus for experiential education programs in the past few years.\n\nThe National Society for Experiential Education (NSEE) has adopted service-learning as one of its major program areas. NSEE has published a massive three-volume work called _Combining Service and Learning_ , which brings together many articles and papers about service-learning\u2014several dozen of which discuss servant-leadership as the philosophical basis for experiential learning programs.\n\nThe fifth application of servant-leadership concerns its use in both formal and informal education and training programs. This is taking place through leadership and management courses in colleges and universities, as well as through corporate training programs. A number of undergraduate and graduate courses on management and leadership incorporate servant-leadership within their course curricula. Several colleges and universities now offer specific courses on servant-leadership. Also, a number of noted leadership authors, including Peter Block, Ken Blanchard, Max DePree, and Peter Senge, have all acclaimed the servant-leader concept as an overarching framework that is compatible with, and enhances, other leadership and management models such as Total Quality Management, Learning Organizations, and Community-Building.\n\nIn the area of corporate education and training programs, dozens of management and leadership consultants now utilize servant-leadership materials as part of their ongoing work with corporations. Some of these companies have included AT&T, the Mead Corporation, and Gulf Oil of Canada. A number of consultants and educators are now touting the benefits to be gained in building a Total Quality Management approach on a servant-leadership foundation. Through internal training and education, institutions are discovering that servant-leadership can truly improve the way in which business is developed and conducted, while still successfully turning a profit.\n\nThe sixth application of servant-leadership involves its use in programs relating to personal growth and transformation. Servant-leadership operates at both the institutional and personal levels. For individuals it offers a means to personal growth\u2014spiritually, professionally, emotionally, and intellectually. It has ties to the ideas of M. Scott Peck ( _The Road Less Traveled_ ), Parker Palmer ( _The Active Life_ ), Ann McGee-Cooper ( _You Don't Have to Go Home from Work Exhausted!_ ), and others who have written on expanding human potential. A particular strength of servant-leadership is that it encourages everyone to actively seek opportunities to both serve and lead others, thereby setting up the potential for raising the quality of life throughout society. A number of individuals are working to integrate the servant-leader concept into various programs involving both men's and women's self-awareness groups and 12-step programs like Alcoholics Anonymous. There is also a fledgling examination underway of the servant-leader as a previously unidentified Jungian archetype. This particular exploration is discussed in a book by Robert Moore and Douglas Gillette titled _The King Within_.\n\nFor some people, the word _servant_ prompts an immediate negative connotation, due to the oppression which many workers\u2014particularly women, and people of color\u2014have historically endured. For some, it may take a while to accept the positive usage of this word, _servant_. However, those who are willing to dig a little deeper come to understand the inherent spiritual nature of what is intended by the pairing of servant and leader. The startling paradox of the term _servant-leadership_ serves to prompt new insights.\n\nIn an article titled \"Pluralistic Reflections on Servant-Leadership,\" Juana Bordas has written: \"Many women, minorities and people of color have long traditions of servant-leadership in their cultures. Servant-leadership has very old roots in many of the indigenous cultures. Cultures that were holistic, cooperative, communal, intuitive and spiritual. These cultures centered on being guardians of the future and respecting the ancestors who walked before.\"\n\nWomen leaders and authors are now writing and speaking about servant-leadership as a 21st-century leadership philosophy that is most appropriate for both women and men to embrace. Patsy Sampson, who is former president of Stephens College in Columbia, Missouri, is one such person. In an essay on women and servant-leadership \"The Leader as Servant,\" she writes, \"So-called (service-oriented) feminine characteristics are exactly those which are consonant with the very best qualities of servant-leadership.\"\n\n\"Servant-leadership works like the consensus building that the Japanese are famous for. Yes, it takes a while on the front end; everyone's view is solicited, though everyone also understands that his [or her] view may not ultimately prevail. But once the consensus is forged, watch out: With everybody on board, your so-called implementation proceeds wham-bam.\"\n\n_Fortune_ magazine (May 4, 1992)\n\nInterest in the philosophy and practice of servant-leadership is now at an all-time high. Hundreds of articles on servant-leadership have appeared in various magazines, journals, and newspapers over the past few years. Many books on the general subject of leadership have been published that have referenced servant-leadership as an important model for both the present and the future.\n\nThe Greenleaf Center for Servant-Leadership is an international, not-for-profit, educational organization that seeks to encourage the understanding and practice of servant-leadership. The Center's mission is to fundamentally improve the caring and quality of all institutions through servant-leadership.\n\nIn recent years, the Greenleaf Center has experienced tremendous growth and expansion. Its growing programs include: the worldwide sales of over 120 books, essays, and videotapes on servant-leadership; a membership program; workshops, institutes, and seminars; a Reading-and-Dialogue Program; a Speakers Bureau; and an annual International Conference on Servant-Leadership. A number of notable Greenleaf Center members have spoken at our annual conferences, including: Peter Block (author of _Stewardship_ and _The Empowered Manager_ ), Max DePree (author of _Leadership Is an Art_ and _Leadership Jazz_ ), Stephen Covey (author of _The Seven Habits of Highly Effective People_ ), Meg Wheatley (author of _Leadership and the New Science_ ), M. Scott Peck (author of _The Road Less Traveled_ and _A World Waiting to Be Born_ ), and Peter Senge (author of _The Fifth Discipline_ ). These and other conference speakers have spoken of the tremendous impact the servant-leader concept has had on the development of his or her own understanding of what it means to be a leader.\n\nThe last few years have witnessed the expansion of interest in servant-leadership to all parts of the world. Greenleaf Center materials are available in English, Spanish, Dutch, Czech, and Arabic, and other translations are in the works. In addition, The Greenleaf Center now has satellite offices in Europe, Great Britain, and Australia.\n\nThe Greenleaf Center's logo is a variation on the geometrical figure called a \"mobius strip.\" A mobius strip, pictured below, is a one-sided surface constructed from a rectangle by holding one end fixed, rotating the opposite end through 180 degrees, and applying it to the first end\u2014thereby giving the appearance of a two-sided figure. It thus appears to have a front side that merges into a back side, and then back again into the front.\n\nThe mobius strip symbolizes, in visual terms, the servant-leader concept\u2014a merging of servanthood into leadership and back into servanthood again, in a fluid and continuous pattern. It also reflects the Greenleaf Center's own role as an institution seeking to both serve and lead others who are interested in leadership and service issues.\n\nLife is full of curious and meaningful paradoxes. Servant-leadership is one such paradox that has slowly-but-surely gained tens of thousands of adherents over the past quarter-century. The seeds that have been planted have begun to sprout in many institutions, as well as in the hearts of many who long to improve the human condition. Servant-leadership is providing a framework from which many thousands of known and unknown individuals are helping improve the way in which we treat those who do the work within our many institutions. Servant-leadership offers hope and guidance for a new era in human development and for the creation of better, more caring institutions.\n\n## 1\n\n## **Servant:\n\nRetrospect and Prospect**\n\n### INTRODUCTION\n\nI believe that caring for persons, the more able and the less able serving each other, is what makes a good society. Most caring was once person to person. Now much of it is mediated through institutions\u2014often large, powerful, impersonal; not always competent; sometimes corrupt. If a better society is to be built, one more just and more caring and providing opportunity for people to grow, the most effective and economical way while supportive of the social order, is to raise the performance as servant of as many institutions as possible by new voluntary regenerative forces initiated within them by committed individuals: _servants_.\n\nSuch servants may never predominate or even be numerous; but their influence may form a leaven that makes possible a reasonably civilized society.\n\nOut of the perspective that emerges from my long concern for institutions, I have come to believe that a serious lack of vision is a malady of almost epidemic proportions among the whole gamut of institutions that I know quite intimately\u2014churches, schools, businesses, philanthropies. And that needed vision is not likely to be supplied by the administrative leadership of those places. Administrators, important and necessary as they are, tend to be short-range in their thinking and deficient in a sense of history\u2014limitations that preclude their producing visions. If there is to be a constant infusion of vision that all viable institutions need, whatever their missions, the most likely source of those visions is their trustees who are involved enough to know, yet detached enough from managerial concern, that their imaginations are relatively unimpaired. Trustees are most effective when they are led by an able and farseeing chairperson\u2014by a quality of leadership that is rare in our society today. These extraordinary chairpersons are not necessarily \"big\" people. The most effective trustee chair I have ever seen in action (and I have seen quite a few) was a \"little\" person in the world of affairs.\n\nThe above paragraph offers a view of the crucial role of trustee leadership that is not widely shared today by the populace at large, or accepted as a personal goal by many current chairpersons, and welcomed by even fewer contemporary chief executives as a role independent of theirs. With so little acceptance of the idea, one may ask, why advocate it? The response to that question requires two \"ifs.\" If one accepts that our institution-bound society serves well enough and no basic change in how our institutions are led is called for, then there is no reason to advocate this radical idea. But if one sees too many of our institutions as seriously deficient in their service to society (as I do) and believes (as I do) that that deficiency could be corrected over time, then something rather fundamental has to change. And the most reasonable and manageable change is to begin, gradually, to raise the effectiveness of trustee leadership until trustees are influential enough and farseeing enough to infuse new visions of greatness, one institution at a time, into as many of our institutions as possible. That powerful new trustee influence is not likely to be achieved until strong visionary leaders emerge to chair their efforts.\n\nBeginning in 1970, I started to write on the theme of _servant_. These have been interesting years because responses have brought involvement in some depth with persons and institutions that share my concern. In this process, others have contributed much to my understanding of what may be required for our society to become more serving\u2014to make a substantial move toward a quality of the common life that is reasonable and possible with available resources, human and material.\n\nTo such as I who did not write for publication until age 65, this understanding has come rather late in life. In summing it up now, I would like to share some reflections. Then I will speculate on the prospects, as I see them, for the servant motive in the future. But, first, a note about where I have been.\n\n### ORGANIZATION\u2014HOW THINGS GET DONE\n\nThe major focus of my adult life may best be described as _a student of organization, how things get done_ \u2014particularly in large institutions. Fortuitous advice from a wise college professor helped shape this interest and led me, upon graduation, to find my way into the largest business organization in the world, American Telephone and Telegraph Company. Early on, I became a student of the history of what seemed to me to be an extraordinary institution. I managed to carve out a career in which I could be both involved and within watching distance of its top structure, and yet maintain sufficient detachment so that I could be reflective about what was going on. My tenure embraced the expansion of the 1920s, the Depression, World War II, and the growth years of the 1950s and 1960s. I never carried heavy executive responsibility and was spared the debilitating effects of such a role which seem almost inevitable, given conventional organizational structures.\n\nIn the latter part of my career, I held the position of Director of Management Research. With the help of a professional staff, and within a broad charter, I could both study and advise regarding the management and leadership of this huge institution\u2014over 1 million employees\u2014immersed as it is in sophisticated technology, elaborate human organization, and regulated public service. I was concerned with its values, with its history and myth, and, intimately, with its top leadership. I learned the hard way about the profound influence that history, and the myths of institutions that have a considerable history, have on values, goals, and leadership. And I was painfully aware of the cost in these terms of any insensitivity to history and myth\u2014especially among the top officers. In any institutional setting, one really cannot understand one's involvement in it _now_ without a clear sense of the course of events that form that institution's past, out of which grows the mythology that surrounds the record of those events. History and myth, in my view, need each other in order to illuminate the present.\n\nThis experience at AT&T gave me a good perspective and the impetus, in my retirement years that began in 1964, to venture into close working relationships with a wide range of institutions: universities (especially in the turbulent 1960s), foundations (as trustee, consultant, and staff member), churches (local, regional, and national), and related church institutions, professional associations, healthcare, and businesses\u2014in the United States, in Europe, and in the third world.\n\nThis post-retirement experience, following 38 years with AT&T, has been enriching and stimulating; but one facet of it, in particular, prompted me to begin to write and to pull together a thread of thinking that has emerged around the _servant_ theme.\n\nThe servant theme evolved out of close association with several colleges and universities during their disturbed period in the 1960s. This was a searing experience, to be intimately involved with students, faculty, administrators, and trustees at a time when some of these venerable institutions literally crumbled\u2014when the hoops came off the barrel.\n\nMy first servant essay, \"The Servant as Leader,\" was prompted by my concern for student attitudes which then\u2014and now, although the manifestations are different\u2014seemed low in hope. One cannot be hopeful, it seems to me, unless one accepts and believes that one can live productively in the world as it is\u2014striving, violent, unjust, as well as beautiful, caring, and supportive. I hold that hope, thus defined, is absolutely essential to both sanity and wholeness of life.\n\nPartly in search for a structural basis for hope, partly out of awareness that our vast complex of institutions\u2014particularly colleges and universities in the late sixties\u2014seemed so fragile and inadequate, two further essays were written: \"The Institution as Servant\" and \"Trustees as Servants.\" The three essays were then collected in a book with some related writings and published in 1977 under the title _Servant Leadership_. Another projected essay, \"The Servant as a Person,\" turned out to be a book and was published in 1979 with the title _Teacher as Servant: A Parable_ [published by The Greenleaf Center].\n\nOut of the struggle to write these things, while contending with the modest ferment they stirred, came the belief that, as a world society, we have not yet come to grips with the _institutional revolution_ that came hard on the heels of the industrial revolution, and that we confront a worldwide crisis of institutional leadership. How can we ordinary mortals lead governments, businesses, churches, hospitals, schools, philanthropies, communities\u2014yes, even families\u2014to become more serving in this turbulent world? And what does it mean to serve? I prefer not to define _serve_ explicitly at this time. Rather, I would let the meaning it has for me evolve as one reads through this essay.\n\nHow can an institution become more serving? I see no other way than that the people who inhabit it serve better and work together toward synergy\u2014the whole becoming greater than the sum of its parts.\n\nI believe that the transforming movement that raises the serving quality of any institution, large or small, begins with the initiative of one individual person\u2014no matter how large the institution or how substantial the movement. If one accepts, as I do, the principle of synergy, one has difficulty with the idea that _only_ small is beautiful. The potential for beauty (largely unrealized to be sure) is much greater in large institutions\u2014because of the phenomenon of synergy. Because we are now dominated by large institutions, how to make _big_ also _beautiful_ is a major challenge for us.\n\nHow to achieve _community_ under the shelter of bigness may be the essence of this challenge because so much of caring depends upon knowing and interacting with persons in the intimacy of propinquity. The stimulus and support that some individuals need to be open to inspiration and imaginative insight often come from the nurture of groups. There may not be a \"group mind\" (inspiration and imaginative insight may be gifts only to individuals), but there is clearly a climate favorable to creativity by individuals that the group, as community, can provide. Achieving many small-scale communities, under the shelter that is best given by bigness, may be the secret of synergy in large institutions.\n\n### THE IDEA OF SERVANT\n\nThe idea of \"servant\" is deep in our Judeo-Christian heritage. The concordance to the Standard Revised Version of the Bible lists over 1300 references to _servant_ (including serve and service). Yet, after all of these millennia, there is ample evidence that ours is a low-caring society when judged by what is reasonable and possible with the resources at hand. There are many notable servants among us, but they sometimes seem to be losing ground to the neutral or nonserving people. It is argued that the outlook for our civilization at this moment is not promising, probably because not enough of us care enough for our fellow humans.\n\nI am personally hopeful for the future because knowledge is available to do two things that we are not now doing, things that are well within our means to do and that would give caring people great joy to do, things that would infuse more of the servant quality into our society. (1) We know how to mature the servant motive as a durable thing in many who arrive in their teens with servanthood latent in them\u2014and this, I believe, is quite a large number. This is what my book _Teacher as Servant_ is about. (2) We know how to transform institutions so that they will be substantially more serving to all who are touched by them. A chapter in _Teacher as Servant_ deals with such a transformation. But formidable obstacles stand in the way of using this knowledge, obstacles that I will call \"mind-sets.\"\n\n### THE PROBLEM OF MIND-SETS\n\nMind-sets that seem to restrain otherwise good, able people from using the two bits of knowledge mentioned above are often tough and unyielding. Whether obstacles like these can be sufficiently reduced before the deterioration of this civilization has become irreversible is open to question. For the older ones among us who are \"in charge,\" nothing short of a \"peak\" experience, like religious conversion or psychoanalysis or an overpowering new vision, seems to have much chance of converting a confirmed nonservant into an affirmative servant. But for some, those few older ones who have a glimmer of the servant disposition _now_ , it is worth their making the effort to try to stem the tide of deterioration. Life can be more whole for those who try, regardless of the outcome.\n\nCivilizations have risen and fallen before. If ours does not make it, perhaps when the archeologists of some future civilization dig around among the remains of this one they may find traces of the effort to build a more caring society, bits of experience that may give useful cues to future people. It is a reasonable prospect that, in the civilization that succeeds ours\u2014whether it evolves from ours in a constructive way or whether it is reconstructed from the ruins after long dark ages\u2014those future people will be faced with the same two problems that confront us now: (1) _how to produce as many servants as they can from those who, at maturity, have the potential for it_ ; and (2) _how to elicit optimal service from such group endeavors (institutions) as emerge_. And, unless some unforeseeable transmutations in human nature occur along the way, those future people may be impeded by the same unwillingness to use what they know that marks our times. Knowledge may be power, but not without the willingness, and the release from inhibiting mind-sets, to use that knowledge.\n\nOver a century ago, when the then-stagnant Danish culture was reconstructed as a result of the work of the Folk High School, the motto of that effort was _The Spirit is Power_. A chapter in the essay \"The Servant as Leader\" tells the story of a remarkable social transformation that followed when the spirit of the Danish young people was aroused so that they sought to find a way out of their dilemma, a stagnant culture, by building a new social order.\n\nWorth noting about this 19th-century Danish experience is that Bishop Grundtvig, the prophetic visionary who gave leadership to the Folk High School movement, did not offer a model for others to follow, nor did he himself found or direct such a school. He gave the vision, the dream, and he passionately and persuasively advocated that dream for over 50 years of his long life. The indigenous leaders among the peasants of Denmark responded to that vision and built the schools\u2014with no model to guide them. _They knew how to do it!_ Grundtvig gave the prophetic vision that inspired them to act on what they knew.\n\n### VISION FOR OUR TIMES: WHERE IS IT?\n\n\"Where there is no vision, the people perish.\" This language (Proverbs 29:18) from the King James Version of the Bible stays with me even though modern translators make something else of that passage.\n\nWhat Grundtvig gave to the indigenous leaders of the common people of Denmark in the 19th century was a compelling vision that they should do something that they knew how to do: _they could raise the spirit of young people so that they would build a new society\u2014and they did_. Without that vision, 19th-century Denmark was on the way to perishing.\n\nOur restless young people in the 1960s wanted to build a new society too. But their elders who could have helped prepare them for that task just \"spun their wheels.\" As a consequence of this neglect, a few of those young people simply settled for tearing up the place. And, in the absence of new visionary leadership to inspire effort to prepare our young people to build constructively, some of them may tear up the place again! Do not be surprised if they do just that. The provocation is ample. We simply are not giving the maturing help to young people that is well within our means to do. Instead, we are acting on the principle that knowledge, not the spirit, is power. Knowledge is but a tool. The spirit is of the essence.\n\nPerhaps the older people who could help them do not do what they know how to do because, as in the 1960s, they are not inspired by a vision that lifts their sights to act on what they know. No such vision is being given in our times. And the paralysis of action that restrains us in preparing young people to live productively in the 21st century is still with us. We may be courting disaster by our neglect.\n\nThis is an interesting thesis (as said earlier): (1) We know how to increase the proportion of young people who, at maturity, are disposed to be servants; and (2) we know how to transform contemporary institutions so that they will be substantially more serving to all who are touched by them. What is needed, this thesis holds, is a vision that will lift the sights of those who know and release their will to act constructively. This vision might be prompted by conscience and self-generated out of a conscious search, or it may be without known cause, or it might be forcefully communicated by a strong leader-type person (as Grundtvig did in 19th-century Denmark).\n\nThis leader might present a vision that has a benign result, as Grundtvig did; or he or she might be a leader like Adolph Hitler, who brought a major disaster; or the vision might be given by an Elvis Presley who can release the inhibiting constraints and incite a frenzy of action that has no seeming value-laden consequence, good or bad.\n\nThe late Rabbi Abraham Joshua Heschel had given a lecture on the Old Testament Prophets to an undergraduate audience. In the question period a student asked, \"Rabbi Heschel, you spoke of false prophets and true prophets. How does one tell the difference?\"\n\nThe good rabbi drew himself into the stern stance of the prophet of old and answered in measured tones, \"There\u2014is\u2014no\u2014way!\" and looked intently at his questioner for an embarrassing moment. Then his face broke into a gentle smile, and he said, \"My friend, if there were a 'way,' if we had a gauge that we could slip over the head of the prophet and say with certainty that he is or is not a true prophet, there would be no human dilemma and life would have no meaning.\" Then, returning to his stern Old Testament stance he said emphatically, \"But it is terribly important that we know the difference.\"\n\nThus, one who is inspired by a vision must know the difference between an action that points toward a benign result, or simply aimless activity. I believe it is possible to prepare most of the emerging adults to know the difference. This is the first step in increasing the proportion of young people who are disposed to be servants. My book, _Teacher as Servant_ , describes how a teacher on his own and without the support of either colleagues or university, can prepare young people to know the difference. I will comment later on one other opportunity within colleges and universities. Neither approach costs any money!\n\n### WHAT DO WE KNOW\u2014OR DON'T WANT TO KNOW?\n\nI have said that (1) we know how to increase the proportion of young people who, at maturity, are disposed to be servants, and (2) we know how to transform institutions so that they will be substantially more serving to all who are touched by them. But it is not knowledge that is codified and systematized and bearing the appropriate establishment imprimatur. It is knowledge like that which the leaders among the Danish peasants had when they were inspired to build schools which would kindle the spirit of their young people. They had always known how to do that, but until Bishop Grundtvig gave them the vision, they were unable, or lacked the will, to act on what they knew.\n\nThere is nothing mystical about the available knowledge to do the two things (as suggested above) that need to be done in our times to raise the servant quality of our society. To my knowledge, clear and complete models do not exist, but there are fragments of experience here and there that can readily be assembled to give a workable basis for moving to solidify that experience\u2014 _to know!_ Let me give examples from four widely differing contemporary institutions in which, it seems to me, able, honest people lack the vision to act on what they know\u2014or could easily know\u2014 _and seem not to want to know!_ They seem to have mind-sets that block them.\n\n#### _Business_\n\nA certain important industrial field is occupied by half a dozen large companies and many small and medium sized ones. It is a field that is subject to quite wide cyclical economic fluctuations and in which disruptions by labor disputes are common. One of the larger companies (not the largest) stands in conspicuous contrast to the other large ones on three counts: no matter what happens to the economic fortunes of the others, this firm, up to now, has always made money; they have never had a strike or work stoppage; their product is generally recognized as superior. (What makes their product superior will be commented on later.)\n\nLet us call this company X. A close observer of this industry recently asked the head of one of the other large companies in this field this simple question: \"What do you folks learn from company X?\" The response was a hand gesture of dismissal and the brusque comment, \"I don't want to talk about it!\"\n\nOne can speculate about why, in a highly competitive field, the head of one large company would brush off a suggestion that he might learn something from a more successful competitor. But what distinguishes company X from its competitors is not in the dimensions that usually separate companies, such as superior technology, more astute marketing strategy, better financial base, etc. Company X is not too different from its competitors in dimensions like these. What separates company X from the rest is unconventional thinking about its \"dream\"\u2014what this business wants to be, how its priorities are set, and how it organizes to serve. It has a radically different philosophy and self-image. According to the conventional business wisdom, company X ought not to succeed at all. Conspicuously less successful competitors seem to say, \"The ideas that company X holds _ought_ not to work, therefore we will learn nothing from them.\" They \"don't want to talk about it.\"\n\n#### _University_\n\nIn the field of higher education, there is another consequence of a lack of vision that cannot be as clearly identified as in the above business example. For many years, I have tried to stir an interest in universities in making a more determined effort to develop the servant-leadership potential that exists among their students. When new money is produced to support such an effort, a pass will be made at doing it. But when the money stops, the effort stops. It does not take root. Here and there the occasional professor, on his own, without the support of his university, and sometimes with the opposition of his colleagues, has taken an interest in this aspect of student growth\u2014with conspicuous success.\n\nIn contrast, a student with athletic potential will find elaborate coaching resources available to develop this talent\u2014even in the poorest and feeblest of institutions. But the young potential servant-leader will find that the position of the best and strongest university is that the development of leadership potential is something that just happens, and nothing explicit is to be done about it in the crucial undergraduate years. I wrote an article about this in an educational journal stating that the only way I see for work in the undergraduate years to help alleviate the leadership crisis we seem now to be in is to find and encourage the rare professor who will take it on\u2014unrecognized and uncompensated, and perhaps denigrated by his colleagues. A university president responded to my article with this concurring comment:\n\nI am coming more and more to agree with your opinion that it is almost impossible to mount anything like an organized program in developing leadership in our university students. Reluctantly, I am reaching your conclusion that the best and only hope of success will be an effort on the part of a few dedicated individuals who will take that cross upon themselves. If this is truly the case, then we need to try to discover who and where they are and give them all the assistance we can.\n\nWhen John W. Gardner wrote his sharp criticism of universities for administering what he called the _antileadership vaccine_ (his parting message when he left the presidency of Carnegie Corporation for a career in politics in 1964), the response from academe seemed to be, \"We don't want to talk about it!\"\n\n#### _Health Profession_\n\nIn the medical profession, there is a widely held position against accepting nutrition as an important factor in health. The average doctor knows that the human body is a chemical-psychic organism. But in treating illness or in advice regarding health building, there is not much concern for nutrition.\n\nThe Hill Foundation in St. Paul, Minnesota, which has a long record of generous giving to medical education, recently made a grant to establish a program in nutrition in a _new_ medical school. The foundation's annual report for 1973 commented on this grant as follows:\n\nIt is a true paradox: Americans are often overweight and undernourished. We are wasteful of our food assets and unwise in our dietary patterns; meanwhile, much of the rest of the world struggles to assure its people an adequate food supply. There is an immediate need for more basic research in nutrition; more communication on ways to plan and control food production, processing, presentation and preparation; and for more public education on sound nutritional practices. But the fact is that there are too few well-trained people to perform this task....\n\nAn important aspect of the nutrition problem is related to the medical school curriculum. Few medical schools have major departments devoted to the field of nutrition research and education. Generally the young doctor gets a briefing on aspects of nutrition as they relate to specific diseases such as diabetes, allergies or coronary problems. Most of the emphasis, however, is on the remedial care of patients, with little attention devoted to the maintenance of health or the prevention of illness.... The same weaknesses exist in the training of such paramedical personnel as nurses and dietitians.... The Foundation believes Mayo Medical School ideally suited to develop and implement a broad nutrition education program _because it is a new institution_ (italics ours), and hence is still flexible in its approach to medical training.\n\nWhat this says to me is that the mind-set among doctors on nutrition is such that only a new medical school will offer a chance to use fully the available nutritional knowledge as an important factor in health building. The medical establishment would seem to say, as the head of the business said when asked what he learned from his more successful but unconventional competitor, \"We don't want to talk about it!\"\n\n#### _Church Leader_\n\nMy interpretation of a bit of 19th-century history is that when Karl Marx sat in the British Museum composing the doctrines that would shape so much of the 20th-century world, he was filling a void that was left by the failure of the churches of his day to deal adequately with the consequences of the industrial revolution. If the 19th-century churches (or church leaders) had taken the trouble to suggest a design for the new society that the industrial revolution made imperative, and if they had advocated it persuasively as a new vision, Marx might still have written his tracts; but they would not have found the field relatively unoccupied.\n\nRecently, I met with a group of church leaders, professionals, who were convened for three days on the subject of \"The Churchman as Leader.\" I listened for a day as they discussed their leadership opportunities and problems as they saw them. Then, in commenting on what I had heard, I noted three words that are sometimes used interchangeably but have quite different connotations: _manage_ (from _manus_ \u2014hand) suggesting control; _administer_ (from _administrare_ \u2014to serve) suggesting to care for; and _lead_ , of uncertain origin, but commonly used to mean \"going out ahead to show the way.\" Manage and administer, along with the ceremonial aspects of office, are the _maintenance_ functions\u2014they help keep the institution running smoothly\u2014 _as it is_. Important as maintenance is in the current performance of any institution, it does not assure adaptation to serve a changing society. That assurance can come only from _leading_ \u2014venturing creatively. Having made this distinction in the meaning of terms as they are commonly used, I commented that, as I observed their discussion, these churchmen were talking mostly about _maintenance_ , not _leading_.\n\nIn most institutions, churches included, managing and administering, the maintenance functions, are delegated and resources are allocated in order that those to whom these functions are assigned can carry on. Those who manage and administer (maintain) may also _lead_ \u2014 _go out ahead to show the way_. But leadership is not delegated; it is assumed. If there are sanctions to compel or induce compliance, the process would not qualify as leadership. The only test of leadership is that somebody follows\u2014 _voluntarily_.\n\nAt this point, I was asked by the church leaders, \"If you do not see us as _leading_ , in your terms, what could persons in positions like ours do in order to lead?\"\n\nI repeated my credo, as stated in the beginning of this essay, which concludes with \"... If a better society is to be built, one more just and caring and providing opportunity for people to grow, the most open course, the most effective and economical way, while supportive of the social order, is to raise the performance as servant of as many institutions as possible by new voluntary regenerative forces initiated within them by committed individuals\u2014 _servants_.\" Then I said:\n\n\"What church leaders can do to really _lead_ in our times is to use their influence to bring into being a contemporary _theology of institutions_ that will underwrite the commitment of church members within our many institutions and support them as they become new regenerative forces: to the end that their particular institution, in which they have some power of influence, will become more serving\u2014and continue to grow in its capacity to serve.\n\n\"The leadership of the 19th-century churches did not accept the challenge to suggest a new design for postindustrial revolution society, and they left a void to be filled by a concerned and articulate atheist. The leaders of late-20th-century churches are not accepting the challenge of an institution-bound society (which Marx did not provide for in his doctrines, and, as a consequence, Marxist societies today have the same problem in getting their institutions to serve as we have). The opportunity that church leaders have today is to take the initiative to see that an adequate theology of institutions evolves so the churches have a firm basis for preparing their members to become regenerative servants in the institutions with which they are involved. _Leadership is initiating\u2014going out ahead to show the way_.\"\n\nThere was not much response to this suggestion in the meeting of church leaders. When we concluded, I noted this paucity of response and said that I would write to them about it when I got home. I later sent to those present a memorandum entitled \"The Need for a Theology of Institutions,\" in which I suggested a detailed procedure that a church leader might follow in producing this new theology. Only 2 of the 16 present at the conference acknowledged the receipt of the memo, and they were noncommittal. A supplementary memorandum six months later got the same response.\n\nI would conclude that these church leaders\u2014all responsible, able, good people\u2014took the same position as the head of the business did when asked what he learned from his much more successful (if unconventional) competitor: \"We don't want to talk about it!\"\n\nWhen 19th-century church leaders were confronted with the radical impact of the industrial revolution, if some audacious consultant had suggested that a new theology was needed to deal with this problem, the response of church leaders of that day might have been the same\u2014\"we don't want to talk about it.\"\n\nMy reflection on these last ten years leads me to conclude that _vision_ , without which we perish, is required to open us to willingness to use what we know and to work to extract hard reality from a dream. In the absence of a powerful liberating vision, church leaders, like others in responsible roles, \"don't want to talk about it.\" Why, over such a long span of history, has the production of vision been so difficult to do? Why are these liberating visions so rare?\n\n### WHY ARE LIBERATING VISIONS SO RARE?\n\nIt seems to me important to accept that the mind-sets that are so frustrating to all reformers, those who are urging others to use what they know, actually serve a useful purpose. What if every person and every institution was \"open\" in the sense of being free of all inhibiting mind-sets that block action on what we know? Every question and every situation would be faced as if nothing like it had happened before. This would be a reformer's dream; but the world would be in chaos. Few of us can survive without a good deal of dogma that prompts reflexive actions. We would not be able to act quickly in emergencies, and moral choices that require prompt action would paralyze us. Most of us get along as well as we do by a good deal of \"what if?\" anticipatory thinking that pre-sets responses to common situations. If we were all completely \"open,\" much of our traditional wisdom might be lost, as might \"manners\" that enable us to interact spontaneously in appropriate ways with fellow humans.\n\nLiberating visions are rare because ours is partly a traditional society\u2014but only partly. It is also an evolving society about which Cardinal Newman is quoted as saying, \"To live is to change; to live well is to have changed often.\" The mixture of _traditional_ and _changing_ is an important aspect of the human dilemma.\n\nTherefore, in answer to the question, \"Why are liberating visions so rare?\" one must say that they are rare because a stable society requires that _a powerful liberating vision must be difficult to deliver_ , and that the test for the benign character of such a vision shall be rigorous. Yet to have none, or not enough such visions, is to seal our fate. We cannot run back to be a wholly traditional society, comforting as it may be to contemplate it. There must be change\u2014sometimes great change.\n\nMoods and spirit of people vary. There are moments when people are more open to charismatic vision than others. Some times seem \"plastic\"; others seem \"hard.\" We but dimly understand the forces that open and close people to liberating visions.\n\nThe word _prudence_ comes to mind. We should try to change with a minimum of threat of damage to stability\u2014as embodied in the four kinds of mind-sets I have described. If stability is significantly lowered or lost, no matter how noble the end sought, the cost in human suffering may be inordinate. When an imprudent effort toward change, one in which the liberating vision is not sufficiently compelling and benign in intent, may make it more difficult for a later prudent effort to succeed, reformers take note: in the end, most people choose _order_ \u2014even if it is delivered to them by brutal nonservants. The ultimate choice of order is one of the most predictable mind-sets because it is a first condition of a civilized society.\n\nIf the writer in Proverbs 29:18 is correctly quoted in saying \"Where there is no vision, the people perish,\" there is remarkable consistency between the common dilemmas in ancient times and in ours. The four examples of firm mind-sets in the fields of business, education, health, and church suggest that there has been failure to give sufficiently powerful liberating visions. This kind of deprivation has been the common lot of humankind from the earliest times. And because of that, the threat of perishing is always with us.\n\n### SUMMONING AND ARTICULATING A VISION\n\nSo far I have given only half an answer to the question, Why are liberating visions so rare?\u2014because it is so difficult to give them. The other half is: because so few of those who have the gift for summoning a vision, and the power to articulate it persuasively, have either the urge or the courage or the will to try! And it takes all three. We in America may be in a transition period between an era of \"growth\" and one of \"restraint\" and liberating visions may have a hollow sound. This is discouraging to visionaries!\n\nOne of the requirements of a caring, serving society, in both favorable and discouraging times, is that it provides in its structures a place for visionaries and surrounds those in that place with the expectation that they will produce those liberating visions of which they are capable. A new view of a structure of the institutions that serve us may be in order\u2014a view that embraces both internal structure as well as the relationship between institutions and how they influence one another.\n\nWhen \"The Servant as Leader\" was published in 1970, I had this to say about prophetic vision:\n\nI now embrace the theory of prophecy which holds that prophetic voices of great clarity, and with a quality of insight equal to that of any age, are speaking cogently all of the time. Men and women of a stature equal to the greatest of the past are with us now addressing the problems of the day and pointing to a better way and to a personeity better able to live fully and serenely in these times.\n\nThe variable that marks some periods as barren and some as rich in prophetic vision may be in the interest, the level of seeking, and the responsiveness of the _hearers_. The variable may not be in the presence or absence or the relative quality and force of prophetic voices. The prophet grows in stature as people respond to his message. If his or her early attempts are ignored or spurned, the talent may wither away.\n\nIt is _seekers_ , then, who make the prophet; and the initiative of any of us in searching for and responding to the voice of a contemporary prophet may mark the turning point in her or his growth and service.\n\nI came by this point of view from reading the history of the Religious Society of Friends (Quakers), and I concluded that George Fox, the powerful 17th-century voice in England that gave this Society its remarkable vision, probably would not have been heard had there not been in existence in England for 100 years prior a group known as _seekers_. This was a small band of people whose common bond was that they were _listening_ for prophetic vision. They were held together by a religious concern, but they knew that it lacked articulation in a contemporary formulation that would make of them a vital social and religious force in their day. And as they heard and responded to George Fox, they became for a short time a great movement that had a remarkable impact on English institutions, notably a new business ethic and a pervasive social concern that influenced the western world and carried forward to 18th-century America, where it made of the Quakers the first religious group to formally condemn slavery and forbid slaveholding among its members\u2014100 years before the Civil War.\n\nBut this movement very quickly crystallized into a church, as too many of its members ceased to be seekers. Instead of _seeing_ , being open to new prophecy, they \" _had it_ ,\" tested and tried\u2014what churches have always done. What they had was, and remains, good. But the Quakers were no longer on the growing edge.\n\nThe servant-leader may be not so much the prophetic visionary (that is a rare gift) as the convener, sustainer, discerning guide for seekers who wish to remain open to prophetic visions. The _maintenance_ functions within all sorts of institutions may not require leaders of any sort, but, _seekers_ , of which every institution should have some, _must_ have servant-leaders. But from where, in our vast complex of institutions, will liberating visions come? I have a suggestion.\n\n### A STRUCTURE OF INSTITUTIONS THAT ENCOURAGES LIBERATING VISIONS\n\nIn search for a structure that encourages liberating visions, the institutions that make up our society might be arranged in a three-level hierarchy.\n\nIn the base level are what may be called \"operating\" institutions: governments, businesses, hospitals, schools, labor unions, professional associations, social agencies, philanthropies, families, communities.\n\nIn the second level are churches and universities, because of their concern for values and for continuity of the culture, and because of their capacity for nurturing the serving quality in both individuals and institutions.\n\nAt the third level are seminaries (theological and nontheological) that are sustaining resources for churches, and foundations that could perform a similar service for colleges and universities. Both foundations and seminaries are suggested because they have sufficient detachment and freedom from daily pressures to maintain a reflective overview of the whole society, and because they have greater latitude than most institutions to be what they want to be and to do what they think they should. They have the unusual opportunity to harbor and encourage prophetic voices that give vision and hope. _Unfortunately, in our times, little prophetic vision seems to come from either seminaries or foundations_.\n\nFurther, foundations and seminaries have the opportunity to become the source of nurturing that is mediated through churches and universities to individuals and operating institutions. Thus a major preoccupation of seminaries and foundations might become _the nurture of institutions_ \u2014institutions that in turn serve people. And it is hoped that, in time, _all_ institutions will come to acknowledge their insufficiency and their need for constant nurture from a source that has the necessary detachment and freedom from daily pressures. I suggest that this source could be foundations and seminaries.\n\nThe utility in such an idea of a hierarchy of institutions is that when there is faltering in any institutions, as there is likely to be in the best of them from time to time, a fair question to ask is, wherein has there been a lack of caring concern for this institution for the level next above?\n\nThis assumption leaves the question, To whom do seminaries and foundations turn when they need caring concern? Since these two stand at the top of the hierarchy, there are no resource institutions that serve them. Therefore, seminaries and foundations need a quality and depth of trustee care that would not be possible for all institutions. These two kinds of institutions need the most dedicated and discerning trustees of all. And these trustees have the greatest of all opportunities for constructive service to the society of the future. The opportunities are great, but the challenge to astuteness of trustee leadership is also great\u2014because most seminaries and foundations, as they now stand, have well-set patterns that do not favor their occupying the role I have described as possible, and natural, for them as level-three institutions\u2014at the top of the hierarchy.\n\n### SEMINARIES\n\nAs seminaries have evolved, they have tended to take on the values and mind-sets of universities. A few of them are schools within university structures. But most are independent institutions with their own trustees, even where they are affiliated with religious denominations.\n\nIf seminaries take on the full scope I will suggest, they will not be at all like universities. To be sure, they have a curriculum of courses and they grant degrees. But this is incidental to their major function: to harbor and nurture prophetic voices that give vision and hope, and to serve as a sustaining support for churches. These are not primary functions of a university, and the university tradition may not be useful as a model for seminaries.\n\nI have a gnawing suspicion that the strongest role, a viable structural model for a seminary, has not yet evolved. It is the opportunity for seminary trustees, under the leadership of their chairperson, to help the seminary get into its strongest role. It is not the trustee mission to design the seminary, authoritatively. Rather, trustees have the opportunity to _lead_ a process out of which the design for the seminary of the 21st century may evolve. All of the several constituencies of each seminary should be full participants in the evolution of that design. And it should be _evolution_ \u2014over time, and never ending. The seminary should rest firm, at all times, in its contemporary mission, while the process of transcending that mission is carried forward simultaneously.\n\nAs trustees undertake to lead their seminary toward its full stature as a serving (level-three) institution, one of their concerns should be for the seminary to make, out of its own experience, a contribution to an evolving theology of institutions\u2014a theology that gives a critical, contemporary view of the purpose and program of both seminary and church. This concern, consistently manifest, will help clarify the goals of the seminary. It will also help prepare the seminary to support the churches in their ministry, both to individuals and to the full range of \"operating\" institutions that churches have the opportunity to influence, to the end that these institutions will be more serving of all persons whose lives they touch.\n\nSeminaries differ widely in their doctrinal positions; but most share the desire to bring about conditions of life that will favor all persons reaching their full stature as religious beings (in the root meaning of the word _religion_ ). It is regarding this common goal that new vision is needed.\n\n### FOUNDATIONS\n\nFoundations, as institutions with funds to dispense for legally approved purposes, are a relatively recent addition to our extensive gallery of institutions. There is considerable disagreement as to what the function of foundations should be, and there are persistent public pressures to restrict their autonomy. There is some sentiment that foundations should not exist at all, or only for a limited term of years.\n\nFoundations are unique in that they are free of \"market\" pressures. All other institutions have constituencies that must be satisfied if they are to continue to exist. Foundations have only to obey the law\u2014which is stricter than it once was, and may get more so.\n\nWhile foundations still have some latitude to choose what they will do, it is suggested that some of them elect to become, in part at least, support institutions for universities and colleges\u2014not just to give them money, although they may continue to do some of that. Could not some foundations become for universities and colleges what seminaries now are (or could be) for churches? This will not come about quickly or easily, but foundation trustees might assume the kind of leadership suggested above for seminary trustees: leadership to a process out of which the design for a foundation role for the 21st century will evolve.\n\nLarge, sophisticated businesses sometimes set aside staffs whose role is to think about the firm as an institution and give intellectual guidance to its development. Universities tend to rely on committees of faculty to render this kind of service to the university itself; and it is not enough. American railroads are the classic example of large businesses that did not set people aside to think about the business they were in. Everybody was busy running the railroad day to day. And few railroads survived this neglect as viable businesses. There is some question that universities will survive, unharmed, from their own self-neglect; and they have been badly scarred in recent years.\n\nIt probably will be more difficult for a foundation to become effective in this support role for universities than for seminaries to do it for churches. This is partly because of the great size and complexity of some universities as well as the scarcity of persons who could (or would) staff a foundation that undertakes this difficult task. As with seminaries, the prime concern is _trustees_. Trustees of a foundation that uses its resources for this important purpose will need to be unusually caring and dedicated and persevere over a long period. They will accept the fact that their foundation must _earn_ the kind of role suggested here.\n\nSome universities and colleges face a drop in enrollment and are experiencing financial stringencies. They have difficulty thinking of any problem they now have as other than something that more money would solve. As a somewhat detached observer, I suspect that universities and colleges are suffering\u2014even today\u2014more for want of ideas, and for vision to liberate them to use ideas, than for want of money. If their governing ideas were better suited to their needs and opportunities in these times, the want of money might not be such a problem. But, the university tradition being what it is, it is unlikely that they have the power to be sufficiently self-regenerating. They, like the churches, need the sustained caring support that only a most able foundation staff is likely to give.\n\nIt is humbling for any institution to accept that it is not self-regenerating and that it should welcome conceptual leadership from another institution like a foundation\u2014one that has the resources, human and material, to give that help, and that has managed to assemble a few unusually able trustees whose exceptionality give the foundation self-regenerating power. Most universities, like most people and institutions, could use a good measure of humility. Humility is one of the distinguishing traits of the true servant\u2014as willing humbly to accept service as to give it.\n\n### SELF-REGENERATING INSTITUTIONS\n\nContinuous regeneration is essential for viability of persons and institutions and society as a whole. A prudent use of human resources is to concentrate the ablest trustees, who will always be few in number, in those institutions that are best positioned to be self-regenerating and thereby to gain the strength to give clear and compelling regenerating vision to others.\n\nIf seminaries and foundations can be accepted as being appropriately placed in the hierarchy of institutions to assume this guiding role (as I believe they are), then a concerted effort should be made to provide these two kinds of institutions with trustees who will persevere with determination to assure sustained self-regeneration in themselves in order to give strong support to churches (by seminaries) and colleges and universities (by foundations) with the hope that, between the influence of churches and universities, \"operating\" institutions will be helped to a sustained high level of caring and serving.\n\n### INSTITUTES OF CHAIRING\n\nOne of the practical steps that foundations and seminaries might take, collectively or separately, is the conduct of institutes for those who chair trustees. They could do this, first, for themselves to prepare their own chairpersons to give the leadership that will help assure the quality of trustee oversight that self-regenerating institutions require. The institutes could then extend the availability of this chairing preparation to universities and colleges and to churches and church-related institutions so that, between them, they could provide this service for those who chair the trustees of all operating institutions that have trustees or directors. Such Institutes of Chairing would be a permanent thing: to give initial and continuing preparation for the chair leadership; to serve as a medium of exchange between those who undertake this role; and to provide a consulting resource for chairpersons who want help on specific problems.\n\nThis is a large order. But if the voluntary character of our complex society is to be preserved and enhanced, a major investment in strengthening and maintaining the trustee role of all institutions that have governing boards seems imperative. This is one of those invaluable social supports that we know how to provide and that we can afford to supply. What is needed, first, is a liberating vision that will make it a feasible thing to do. Where better might we look for that liberating vision than among seminaries and foundations? If just _one_ in each category will take it on and advocate that vision persuasively and with spirit, they may infect the rest. Visions, both good and bad, can be contagious. An _Institute of Chairing_ could be one of the good ones.\n\nI suggest that a prime concern of all seminaries and foundations could be to become self-regenerating institutions\u2014with their own able and caring trustees. They could stand as models for the others.\n\n### SERVANT LEADERSHIP\u2014BY PERSUASION\n\nIn my personal credo stated earlier I said, \"If a better society is to be built, one more just and more caring and providing opportunity for people to grow, the most effective and economical way, while supportive of the social order, is to raise the performance as servant of as many institutions as possible by new voluntary regenerative forces initiated within them by committed individuals\u2014 _servants_.\"\n\nSo far I have not found it helpful to define _servant_ and _serving_ in other terms than the consequences of the serving on the one being served or on others who may be affected by the action. In _Teacher as Servant,_ I describe a semifictional servant in some detail.\n\nIn \"The Servant as Leader,\" the definition was: \"Do those being served grow as persons: do they, while being served, become healthier, wiser, freer, more autonomous, more likely themselves to become servants? _And_ what is the effect on the least privileged in society; will she or he benefit, or, at least, not be further deprived?\" I would now add one further stipulation: \"No one will knowingly be hurt by the action, _directly or indirectly_.\"\n\nThus the servant would reject the \"utilitarian\" position, which would accept a very large gain in, say, justice at the cost of a small but real hurt to some. The servant would reject the nonviolent tactic for societal change, however noble the intent, if, as a consequence, some who are disposed to violence are likely to resort to it, or some may be threatened or coerced. (I would fault Mohandas Gandhi on these grounds. Great leader and tremendous person that he was, I do not find his tactic an appropriate model for the servant. John Woolman, as described in \"The Servant as Leader\" is, for me, such a model.)\n\nThe servant would reject the rapid accomplishment of any desirable social goal by coercion in favor of the slower process of persuasion\u2014even if no identifiable person was hurt by the coercion.\n\nTo some determined reformers, such a set of beliefs would lead to paralysis of action. The servant (in my view) is generally a \"gradualist.\" And, while granting that, in an imperfect world, because we have not yet learned how to do better, coercion by governments and some other institutions will be needed to restrain some destructive actions and to provide some services best rendered authoritatively, the servant will stand as the advocate of persuasion in human affairs to the largest extent possible.\n\nThis view is supported by a belief about the nature of humankind, a belief that leads to a view of persuasion as the critical skill of servant leadership. Such a leader is one who ventures and takes the risks of going out ahead to show the way and whom others follow, _voluntarily, because they are persuaded that the leader's path is the right one\u2014for them_ , probably better than they could devise for themselves.\n\nOne is persuaded, I believe, on arrival at a feeling of rightness about a belief or action through one's own intuitive sense\u2014checked, perhaps, by others' intuitive judgment, but, in the end, one relies on one's own intuitive sense. One takes that intuitive step, from the closest approximation to certainty one can reach by conscious logic (sometimes not very close), to that state in which one may say with conviction, \"This is where I stand!\" The act of persuasion will help order the logic and favor the intuitive step. And this takes time! The one being persuaded must take that intuitive step alone, untrammeled by coercion or manipulative stratagems. Both leader and follower respect the integrity and allow the autonomy of the other; and each encourages the other to find her or his own intuitive confirmation of the rightness of the belief or action.\n\nTo the servant (as I view that person), _persuasion_ , thus defined, stands in sharp contrast to _coercion_ (the use, or threat of use, of covert or overt sanctions or penalties, the exploitation of weaknesses or sentiments, or any application of pressure). Persuasion also stands in sharp contrast to _manipulation_ (guiding people into beliefs or actions that they do not fully understand).\n\nIf one accepts such definitions, has the servant become limited to a passive role and yielded the carrying of the tougher burdens to those with fewer scruples? No, I do not believe so; not if the preparation of servants can begin when they are young. There are some old and valuable burden carriers around who are much too coercive and manipulative; and they might lose their usefulness if they attempted too radical change. It may be better to tolerate their ways as long as they are useful so long as they do not hurt others.\n\nI realize that in adding to the definition of servant the admonition, \"no one will knowingly be hurt,\" some people who might otherwise think of themselves as servants (as I have defined it) will reject that identification. The problem is that some do not believe they can carry the leadership roles they now have without causing some hurt, or that necessary social changes can be made without some being hurt.\n\nIn an imperfect world, some will continue to be hurt, as they always have been. I know that, in the course of my life, I have caused some hurt. But, as my concern for servanthood has evolved, the scars from these incidents are more prominent in my memory and self-questioning is sharper: Could I have been more aware, more patient. more gentle, more forgiving, more skillful? The intent of the servant, as I see that person now, is that, as a result of any action she or he initiates, _no one will knowingly be hurt_. And if someone _is_ hurt, there is a scar that henceforth will endure to be reckoned with. Hurting people, only a few, is not accepted as a legitimate cost of doing business.\n\nI find eleemosynary institutions most at fault on this issue\u2014particularly with their employed staffs. There seems to be the assumption that since the cause being served is noble, what happens to the people who render the service is not a particular concern.\n\nI once sat with the governing board of a large church as they discussed the many ramifications of their affairs. In listening to their discussions I was appalled at some of the attitudes they held and the cavalier actions they took regarding their employed staff. When it was appropriate for me to comment, I noted my observation on their attitudes and actions and I said: \"I have spent my life in a business and had responsibility, directly or indirectly, for the careers of many people. If I had held attitudes like those you have revealed and had your record of hiring and firing people, at some point I would have been taken aside and told, 'Greenleaf, you may be good for something, but we will not let you manage people. We can't afford this!' With the predominantly economic motive, most businesses I know about take greater care with their people than you do. This may have been part of what Emerson had in mind when he said (in _Works and Days_ ), 'The greatest meliorative force in the world is selfish huckstering trade!' \"\n\nIt all reminds me of that powerful line with which Shakespeare opens his 94th sonnet:\n\nThey that have power to hurt and will do none.\n\n(Not very little, but none.) This is the sonnet that concludes with those caustic lines:\n\nFor sweetest things turn sourest by their deeds;\n\nLilies that fester smell far worse than weeds.\n\nThe intervening eleven lines will bear close scrutiny.\n\nThe firm aim of the servant is that _no one_ will be hurt.\n\nPreparation of a servant, particularly for the exacting role of servant-leader, should start not later than secondary school (before if possible) because, I believe, the servant needs to learn to stand against the culture on two critical issues: _power_ and _competition_.\n\n### POWER\n\nI have no definite view of power to offer: only some fragmentary thoughts. I grant that, in an imperfect world, some raw use of power will always be with us. But as ours has become a huge, complex, institution-bound society, power seems more of an issue than it was in simpler times when it was easier to identify where coercive pressures came from. Also, within the past 200 years, the damage to power wielders has been clearly signaled\u2014beginning with William Pitt's statement in the House of Commons in 1770, \"Unlimited power is apt to corrupt the minds of those who possess it\"; and then, in the late 19th century, Lord Acton's more quoted line, \"Power tends to corrupt and absolute power corrupts absolutely.\" It is interesting to note that Lord Acton, a Catholic layman, made this statement in heated opposition to the assumption of Papal infallibility in 1870. And what is the corruption that both Pitt and Acton might have had in mind? I believe it is _arrogance_ , and all of the disabilities that follow in the wake of arrogance.\n\nIn \"The Servant as Leader,\" I tell of John Woolman, the 18th-century American Quaker who persuaded slaveholding Quakers, one by one, to free their slaves. Half his persuasive argument was concern for the slave, the other half was concern for the damage done to the slaveholder and his family. John Woolman also used the word _corruption_ in referring to the legacy of the slaveholder to his heirs.\n\nAlong the way, in a conversation I had with the chief executive of a large business concerning the incentives that make his job attractive, he listed as first \"The opportunity to wield power!\" This came before monetary reward, prestige, service, and creative accomplishment\u2014all of which, together, he said, would not compensate for carrying such a heavy burden.\n\nA few years ago, a friend called to tell me that he had just been made head of a philanthropic foundation\u2014his first work of this kind. My immediate response, drawing on my own considerable experience with foundations, was, \"The first thing that will happen to you is that you will no longer know who your friends are.\" This is a serious disability.\n\nI will never forget my first venture as a foundation representative when I made a tour of a dozen universities on a new grant program. My wife met me at the airport when I returned and asked how it went.\n\n\"I have no idea,\" was my reply.\n\n\"I have never experienced anything like this before. In most of my work life, I have had to do battle for my ideas every inch of the way, and nothing I have tried to do has been a pushover. But here, in these conversations with high-ranking officers of prestigious universities, every word I uttered was received as a pearl of wisdom.\" _This was a corrupting experience_.\n\nI am aware that some foundation representatives seem to rise above this corrupting influence; but I hold that the power of the almoner is near the absolute; and it is corrupting, as I am sure all power is. If it were not so clear in my own experience, I would not be so sure of it.\n\nSomehow, the young potential servant should be helped to an awareness of power and its consequences on both the wielder and the object. In my essay _Trustees as Servants,_ I contend that \"No one, _absolutely no one_ , is to be entrusted with the operational use of power without the close oversight of fully functioning trustees.\" I would now generalize further and say that young potential servant-leaders should be advised to shun any power-wielding role which is not shared with able colleagues who are equals. (See _The Institution as Servant_ for an elaboration of this thesis.) If a young potential servant-leader can accept that the first protection against the corruption of power is never to undertake a power-wielding role alone; if this can be established when one is young, a lifestyle may be built on this principle that will be easy and natural. It is not easy and natural for one who is deeply entrenched as a lone wielder of power to contemplate carrying a major responsible role without a firm grip on power\u2014in one's own hands, _alone_. One who is firmly established as a chief executive officer (a lone power wielder) will almost universally say, \"It won't work, one person _must_ hold the ultimate power.\" But if enough of today's able youngsters catch the vision of servant-leadership and incorporate it into their lifestyles early, the day may come, when these people are in their prime years, that they will label, categorically, the current commonly accepted power striving of some successful people as _pathological_ \u2014because it makes for a sick society. Those who embrace the spirit of servant-leadership early in their lives are likely to take a similar view of competition\u2014and come to see it as an aberration, not a normal human trait. And when enough able people take that view, it will make a different world. But that will take some time.\n\n### COMPETITION\n\nIt is difficult to know whether humankind's seemingly \"normal\" competitive urges are innate\u2014the nature of the human animal\u2014or whether they are acquired. It is difficult to know because the culture is so thoroughly competitive, and imposes its shaping imprint from infancy onward, that one cannot sort out what _homo sapiens_ would be like if raised in a noncompetitive culture.\n\nRecently I was on a panel in a conference in a medical school that was discussing the subject of \"the ethics of the drug industry.\" In preparation for the conference, someone had made a video recording of drug ads on TV over a period of weeks; and we sat for 20 minutes watching these, one after another.\n\nIt is bad enough to have to look at these zany ads when they appear once in a while as the price of watching commercial TV. But to sit through 20 minutes of nothing else\u2014well, it was nauseating, an affront to taste, intellect, and integrity; and the conference erupted in indignation\u2014\"Something ought to be done about this!\" After listening for a minute or two of this heated reaction I interjected with, \"You shouldn't be so upset by these ads. As a nation, we have made a clear social policy, backed by tough laws with criminal sanctions, that an industry like this will be _forced_ to serve by requiring dog-eat-dog competition as a rule of doing business. When you decide to _force_ service this way (and you really don't influence much but price), then you should not be surprised if you get a result like we have just witnessed.\"\n\nThe conference erupted again, \"What would _you_ do, repeal the antitrust laws?\" And I answered, \"I don't know what I would do. I have only one point: if you decree (and you have so decreed) that dog-eat-dog competition is to be the regulator, then do not be surprised if you get this kind of result. Anyway, what is so sacred about the antitrust laws? They were not brought down off the mountain chiseled in stone. They are crude man-made devices to deal with a clear social problem: how to elicit the best service we can get from a business. But there are several unhealthy by-products, one of which you have just seen.\" This promoted considerable discussion, without conclusion. And there is not likely to be a better answer to the question: How can we elicit optimal service from people and institutions, as long as competition is uncritically accepted as _good_ and is deeply imbedded in the culture? In the preparation of young potential servants to be servant-leaders, the issue of competition must be critically examined and alternatives sought.\n\nThis is a curious bit of history of usage of the word _compete_. Modern usage puts it as \"To strive or contend with another,\" while the Latin origin of the word is _competere_ \u2014to seek or strive together. The clear implications of the origin of this word is that competition is a cooperative rather than a contending relationship.\n\nThis reference may not help us to resolve our own personal dilemmas as we find ourselves in a struggle to beat out somebody else, in a society that supports that struggle with both moral and legal sanctions. (Recently I attended a conference on the subject of \"The Judeo-Christian Ethic and the Modern Business Corporation.\" There were about 25 theologians of the major faiths present. In the papers and the discussions there was frequent reference to \"unfair\" competition, but I do not recall a single question by a theologian about competition per se.)\n\nMy position is: if we are to move toward a more caring, serving society than we now have, competition must be muted, if not eliminated. If theologians will not lead in this move (and I sense no initiative from that quarter), practicing servants will, and theologians will rationalize the result after the fact. The servant will be noncompetitive; but what can be the servant's affirmative position?\n\nI believe that serving and competing are antithetical; the stronger the urge to serve, the less the interest in competing. (Read Petr Kropotkin's classic _Mutual Aid_ for perspective on this issue.) The servant is importantly concerned with the consequences of his or her actions: those being served, _while being served_ become healthier, wiser, freer, more autonomous, more likely themselves to become servants. _And_ , what is the effect on the least privileged in society; will they benefit, or, at least, not be further deprived? _And_ , no one will knowingly be hurt by the action?\u2014the servant is strong _without_ competing. But, unfortunately, we have decreed that ours shall be a competitive society. How does a servant function in such a society?\n\n### SERVANT-LEADER: STRONG OR WEAK?\n\nThe power-hungry person, who relishes competition and is good at it (meaning: usually wins) will probably judge the servant-leader, as I have described that person, to be weak or naive or both. But let us look past the individual to the institution in which he or she serves: what (or who) makes that institution strong?\n\nThe strongest, most productive institution over a long period of time is one in which, other things being equal, there is the largest amount of voluntary action in support of the goals of the institution. The people who staff the institution do the \"right\" things at the right time\u2014things that optimize total effectiveness\u2014because the goals are clear and comprehensive and they understand what ought to be done. They believe they are the right things to do, and they take the necessary actions without being instructed. No institution ever achieves this perfectly. But I submit that, other things being equal, the institution that achieves the most of this kind of voluntary action will be judged _strong_ , stronger than comparable institutions that have fewer of these voluntary actions.\n\nEarlier, in the discussion of mind-sets, I gave the example of the more successful business in a highly competitive field that stands above its competitors in profitability, in the quality of the product it delivers, and in the absence of labor conflict that plagues all of the others. The principal difference is that this unusual company has more voluntary effort; its people do more, voluntarily, than other companies' people do for them. And it is not accidental. The man who built this business, dead for some years, put the people who worked for him _first_. As a consequence, his employees delivered all that people can deliver\u2014and the business came to lead its field. It is a \"people\" business. There are other companies in other fields that have taken this view and, when other things are equal, they are all strong when compared with competitors who do not take this view of people.\n\nFrom my own experience with businesses, I would say that even when \"people first\" is not the policy of the top executives, \" _strong_ \" subordinate executives may take the \"people first\" position and add strength to the business. In AT&T, when we occasionally conducted attitude surveys, we noted what we called the \"umbrella\" effect. A strong subordinate manager would produce positive attitudes among his or her subordinates when the stance of higher level managers caused the prevailing attitudes in other parts of the company to be more negative. I believe that able subordinate managers who are servants can build strength in the people they lead even when the policy that is projected on them from above works to destroy it. But such subordinate managers must be really strong in terms of toughness, conviction, and tenacity.\n\nFurther, when there is a sticky organizational problem in a business, an astute power-wielding executive sometimes tries to find a person who is accepted as servant who will get into the situation and correct it\u2014with persuasion. _And for purely practical reasons_ : it comes out better than if somebody swings on it in a coercive or manipulative way!\n\nBoth the words _servant_ and _persuasion_ are \"soft\" words to some people. They do not connote the tough attitudes that are thought to be needed to hold this world together and get its work done.\n\nIn 1970, when I chose to advocate, in writing, the servant-leader concept, both words _serve_ and _lead_ were in a shadow. _Lead_ seems to have recovered some stature, but _serve_ is still questioned by many thoughtful people. I chose to stay with _serve_ , _lead_ and _persuade_ because I see, through the meaning they have for me, a path to restoring much of the dignity that has been lost through the de-personalization that industrialization has brought to us. And dignity adds strength both to individuals and to the institutions of which they are a part\u2014strength to serve.\n\n### PROSPECTS FOR THE SERVANT IDEA\u2014SOME SPECULATIONS\n\nIn much of what I have written on the servant theme, including most of this essay, I have dealt with issues of leadership and institution building. After ten years of circulation of these writings and considerable interaction about them with people who have their hands on the levers of power and influence, I am not persuaded that much movement toward our society becoming more caring is likely to be initiated by those who are now established as leaders. Mind-sets like the four discussed earlier are much too prevalent and entrenched, and we seem not to have the resources to generate, or the openness to receive, liberating visions. Whatever older people can do to make ours a more serving, caring society should be encouraged; but I do not expect much from my contemporaries.\n\nWe (some of us) do know how to prepare and inspire young people to press the limits of the reasonable and possible, with some of them becoming skilled builders of more serving institutions. The over-arching vision that will inspire and energize mentors of the young is my prime concern. These mentors are strong, able people who believe that well-prepared young people, in whom servant-leadership is an integral part of their lifestyles, are likely to bring to reality some of what we oldsters can only dream about.\n\nMy hope for the future, and I do have hope, is that some (perhaps many) young people whose lifestyles may yet be shaped by conscious choices may be helped to more serving roles than most of their elders occupy today. What I have written is not likely to give this help directly to young people. But it may be useful to those who have the gifts and the will and the courage to be mentors of the young. And I believe that the psychic rewards to these mentors can be very great. What could bring more satisfaction to oldsters than helping some of the young to become servant-leaders?\n\nIn _Teacher as Servant,_ I described in detail how a university teacher could, without the support of university or colleagues, encourage, in a decisive way, the growth as servant of a large number of students. I hazarded a guess that, if there were a way to alert them to the opportunity, perhaps as many as one in a thousand of the 500,000 or so university and college faculty in the United States might take the initiative to give this precious help to students. And I reasoned that the 500, if they worked at it over a career, could favor the time of the next generation becoming a golden age of leadership in our country. It can be done without adding to college and university budgets. It would require no changes in the curriculum, no administrative or faculty actions, no trustee initiatives. All that is needed is a handful, really, of determined and perceptive faculty members who, deep down inside, are true servants and who, without extra compensation and recognition\u2014perhaps in the face of some opposition\u2014will _lead_ in this most fundamental way. They will go out ahead to show, by their example, how one may be a servant in what appears to be a cold, low-caring, highly competitive, violence-prone society. These servant teachers may be a saving remnant, in the biblical sense. And saving remnants are usually not empowered, approved, or well-financed.\n\nI would now amend the language of this assertion in just one particular way. In place of \"alert them (the one in a thousand teachers) to the opportunity\" I would substitute \"inspire them with a vision of the opportunity.\"\n\nOne who might respond to this suggestion is the president, especially in a small college; but it also might be a large university. The president might personally offer to lead a noncredit seminar for elected student leaders. The agenda of the seminar might be discussions with invited resource people and sharing between the president and these student leaders on matters of mutual concern in their current leadership roles. In my conversations with student leaders, I have found them concerned with some of the same issues that are on the minds of presidents\u2014matters of the spirit. Presidents might find in these seminars a helpful close contact with students, and students would have the opportunity to learn about leadership from each other and from the president\u2014experientially. The president may learn something about leadership, too\u2014a new perspective on that job.\n\nThe prospect for the servant idea rests almost entirely, I believe, on some among _us investing the energy and taking the risks to inspire with a vision_. In our large and complex society, a single compelling prophetic voice may not, as Grundtvig did in 19th-century Denmark, move those few who will educate and inspire enough young people to rebuild the entire culture. In our times, the orchestration of many prophetic visionaries may be required. But I believe that the ultimate effect will be the same: _teachers_ (individuals, not institutions) will be inspired to raise the society-building consciousness of the young. And _teachers_ may be anybody who can reach young people who have the potential to be servants and prepare them to be servant leaders. These teachers may be members of school faculties, presidents of colleges and universities, those working with young people in churches. Some may be parents, others may be either professionals or volunteers working with youth groups. But whoever and wherever they are, these teachers will catch the vision and _do what they know how to do_. First, they will reinforce or build hope. Young people will be helped to accept the world, and to believe that they can learn to live productively in it _as it is_ \u2014striving, violent, unjust, as well as beautiful, caring, and supportive. They will be helped to believe that they can cope, and that, if they work at it over a lifetime, they may leave a little corner of the world a bit better than they found it. Then these teachers will nourish the embryo spark of servant in as many as possible and help prepare those who are able\u2014 _to lead!_\n\nThus I do not see the prospect for the servant idea being carried by a great mass movement\u2014not soon.\n\nI have premised this discussion on building hope in the young and preparing some of them to serve and lead. As an oldster, I have hope that is supported by the belief that some seminaries and foundations will have (or find) trustees of the stature who will help them (seminaries and foundations) to be self-regenerating institutions. These then will become sources of prophetic visions for, and supports of organizational strength in, schools and churches which will minister to individuals and to the vast structure of operating institutions that make up our complex society. Central to this ministry will be the encouragement of teachers of servants\u2014some of whom will become leaders who make their careers as regenerating influences within institutions of all sorts, including seminaries and foundations\u2014thus closing the loop. But the prime movers in this process are trustees of foundations and seminaries. It is for these exceptionally able and dedicated trustees to initiate and to sustain the process. I believe that a few will. This is the basis of my hope.\n\nBeyond my hope, I have a speculative prospect to share: that some of these servant-leaders will bring together communities of seekers who find\u2014and _continue to seek_ , thus adding a new building force that works toward an evolving caring society.\n\nI envision that these communities (of seeking servants who find\u2014and continue to seek) will bring a new kind of institution that is radically different from anything we now have. It may be a business, a church, a school, a unit of government. Or it may be an institution that embraces aspects of several of these. But it will be _new_ , and its emergence will be a hopeful augury for everybody, especially for young people.\n\nIn this way a prophetic vision for the 21st century may be delivered to us\u2014not in words, but by a few humble servants saying simply, \"Here it is; come and see.\"\n\nThe reader is invited to speculate on what this new institution might be like and what its presence might mean to the quality of our common life. These speculations might help this new institution to come, by creating receptivity for a new vision.\n\n### NOTE ON LIBERATING VISIONS\n\nA final note on receiving, communicating, and responding to _liberating visions_. This is an illusive term. What could it mean? Let me give an example out of my experience.\n\nWhen I was working closely with several colleges and universities in the late 1960s I became aware that one of the student preoccupations of that period was reading the novels of Herman Hesse. College book stores had stacks and stacks of them\u2014and some still do. Because I was deeply concerned with what was going on in students' minds at that time, I made a project of reading all that Hesse wrote, in the order that he wrote these stories. Along with them I read a biography that told me what was going on in Hesse's life as he wrote each book.\n\nHesse, in the early part of his life, was a tormented man, in and out of mental illness that is reflected in what he was writing at the time. His book _Journey to the East_ marks the turning point toward the serenity that he achieved in his later years when he wrote his greatest novel, _Magister Ludi_ , which earned a Nobel Prize for literature. I found _Journey to the East_ a hopeful book because it is the story of Leo, the central figure who accompanies a band of men on their mythical journey (probably Hesse's own journey) as the _servant_ who does their menial chores, but who also sustains them with his spirit and song. He is a person of extraordinary presence. All goes well with the journey until Leo disappears. Then the group falls into disarray and the journey is abandoned. They cannot make it without the servant Leo. The narrator, one of the party, after some years of wandering, finds Leo and is taken into the Order that has sponsored the journey. There he discovers that Leo, whom he had first known as _servant,_ was in fact the titular head of the Order, its guiding spirit, a great and noble leader. The story of Leo gave me the idea of \"The Servant as Leader.\"\n\nLater a Catholic Sister came to talk about what I had written in \"The Servant as Leader.\" In the course of the discussion she asked me where I found the earliest reference to the idea of servant. I replied, in the Bible, of course, beginning early in the Old Testament. Her rather sharp question was, \"Then why do you attribute it to reading Hesse?\" I responded, \"Because that was where I got the idea to write on the Servant as Leader theme. If I had not read the story of Leo, I might have never written anything on this subject. There was something in Hesse's story that moved me in a way that I had not been moved before.\"\n\nIn the terms that I have been discussing the subject of liberating visions, I was prepared to receive one by my deep immersion in the student turmoil of the sixties and by reading Hesse as part of my search for understanding of what was going on in student minds. The liberating vision that took me into one of the most interesting and productive chapters of my life was delivered by Hermann Hesse.\n\nThis essay, \"Servant: Retrospect and Prospect,\" has had much to do with receiving, communicating, and responding to _liberating visions_. What I learned from the experience noted above is that liberating visions can come from anywhere at any time and that they may or may not bear any particular theological label. Important to me are:\n\n\u2022 Immerse oneself in the experiences this world offers.\n\n\u2022 Be accepting of the people involved in these experiences, and seek to understand what moves them.\n\n\u2022 Acknowledge\u2014and stand in awe before\u2014the ineffable mystery that shrouds the source of all understanding of human motives that leads to visions.\n\n\u2022 _Be open to receive, and act upon, what inspiration offers_.\n\nAlong the way, I had a dream that may have had something to do with the course of my experience these last ten years.\n\nIt is a beautiful summer day, and I am in a lovely, extensive woods in which there is a labyrinth of paths. I am riding a bicycle through these paths, holding in my hand a map of them to guide my journey. It is a buoyant, joyful experience and there is a delightful certainty about it. Suddenly a gust of wind blows the map out of my hand. I stop and look back to see it flutter to the ground, to be picked up by an old man who stands there holding it for me. I walk back to retrieve my map; but when I arrive at the old man, he hands me\u2014not my map\u2014but a small round tray of earth in which fresh grass seedlings are growing.\n\nWhere there is no vision, the people perish.\n\n_Proverbs 29:18 (King James Version)_\n\nWhat is now proved was once only imagin'd.\n\n_William Blake (The Marriage of Heaven and Hell)_\n\n### POSTSCRIPT\n\nAs I was writing this an invitation came to visit a small church-related college where they wanted to talk about a new goal for the college: to make preparation for servant-leadership a central concern, in the work of the college. I visited with faculty, as a whole and in groups, and with student leaders. It was an encouraging experience, to find that a college wanted to consider such a move. My parting advice was not to try to \"legislate\" such a change but to give leadership and encourage wide discussion, and let new directions come as individuals find ways to work toward such a goal with the hope that, ultimately, a consensus will emerge. This is what servant-leadership is about: helping consensus, voluntary and durable consensus, to evolve.\n\n## 2\n\n## **Education and Maturity**\n\n_A talk before the faculty and students of Barnard College, at their fifth biennial vocational conference, November 30, 1960_\n\nMaturity has many meanings, especially when applied to people. But in my own association, there is a strong link between the word _maturity_ and the word _becoming_. Education, in particular a liberal education, can be a powerful maturing force. Depth of meaning about process emerges only out of experience. This, briefly, is the framework within which I shall try to deal with the subject of education and maturity.\n\nA friend of mine in Madison, Wisconsin, tells a story about Frank Lloyd Wright many years ago when his studio, Taliesen, was at nearby Spring Green. Mr. Wright had been invited by a women's club in Madison to come and talk on the subject \"What is Art?\" He accepted and appeared at the appointed hour and was introduced to speak on this subject.\n\nIn his prime, he was a large impressive man, with good stage presence and a fine voice. He acknowledged the introduction and produced from his pocket a little book. He then proceeded to read one of Hans Christian Andersen's fairy tales, the one about the little mermaid. He read it beautifully, and it took about 15 minutes. When he finished he closed the book, looked intently at his audience and said, \"That, my friends, is art,\" and sat down.\n\nAs I thought about this talk today, I wished that I knew how to do this with the subject of maturity. Maturity is like art and virtue; it is best demonstrated, and I feel presumptuous to be talking about it at all.\n\nI do not have a how-to-do-it formula for achieving maturity to hand you. My sole aim is to encourage you to be thoughtful about your problem of finding the meaning of maturity in your own lives and the times you live in. Because we are all different, the problem will be different for each of us. The common ground I shall try to find is a way of thinking about the meaning of maturity.\n\nThe most important lesson I have learned about maturity is that the emergence, the full development, of what is uniquely _me_ should be an important concern throughout my entire life. There are many other important concerns, but this particular one must never be submerged, never be out of sight.\n\nThis I learned the hard way. There was a long \"wilderness\" period in which I sought resources outside of myself. I looked for an \"answer\" to the normal frustrations of life ( _frustration_ used in the sense of the blocking of motives to which one cannot make a constructive response). Good years went by. No answers came. It took a long time for me to discover that the only _real_ answer to frustration is to concern myself with the drawing forth of what is uniquely me. Only as what is uniquely me emerges do I experience moments of true creativity; moments which, when deeply felt, temper the pain of long periods of frustration that are the common lot of most of us and give me the impulse and the courage to act constructively in the outside world.\n\nEvery life, including the most normal of the normal, is a blend of experiences that build ego strength and those that tear it down. As one's responsibilities widen, these forces become more powerful. As good a definition as I know is that maturity is the capacity to withstand the ego-destroying experiences and not lose one's perspective in the ego-building experiences: \"If you can meet with triumph or disaster\u2014,\" to borrow a phrase from Mr. Kipling.\n\nOne of my special interests is the field of management development. I have made a point of looking into a few organizations in which, in a certain period, there was an unusual flowering of managerial talent. Usually there was one person, an able manager who had the gift of guiding his understudy so as to help bring latent talent to fruition, to the mature ability to carry heavy responsibility successfully. The most outstanding developer I know about had at the center of his philosophy the idea that the really important lessons in the managerial art are learned only as the result of error, suffering the consequences of error and learning from the total experience. This is an important test of maturity: to seek to avoid error, to accept the consequences of error when it comes (as it surely will), and learn from it and to wipe the slate clean and start afresh, free from feelings of guilt.\n\nBut this takes a special view of the self. The sustaining feeling of personal significance is important. It comes from the inside. I am _not_ a piece of dust on the way to becoming another piece of dust. I am an instrument of creation, unlike any that has ever been or ever will be. So is each of you. No matter how badly you may be shaken, no matter how serious the failure or how ignominious the fall from grace, by accepting and learning you can be restored with greater strength. Don't lose this basic view of who you are.\n\nA friend of mine once said of his 4-year-old son, \"His world is a six-foot sphere. He's in the center of it and moves it around with him wherever he goes.\"\n\nThe conventional view is that this is youthful egocentricity and that one grows out of it as he matures, as he becomes social and accepts responsibility. I would rather say that there is a transmutation as one matures. One is _still_ at the center of his world. (How could one be unique and be otherwise?) But with maturity one's world becomes the limitless sphere of people, ideas, and events which each of us influences by each thought, word, and deed; and each of us, in turn, is open to receive influence. The individual capacity of each of us to influence and be influenced and to absorb the shocks\u2014this capacity is in proportion to the emergence of the sentient person, the drawing forth of what is uniquely _us_. This is an important idea to keep as your own private lamp when somebody undertakes to grind you down\u2014as they surely will sometime, if you have not been aware of it already.\n\nThis is the central idea of maturity: to keep your private lamp lighted as you venture forth on your own to meet with triumph or disaster or _just plain routine_. And this is what a liberal education is about; because this is what life is about. If, in your college years, you learn nothing other than who you are, that you have a private lamp, your stay here will have been amply justified.\n\nThe notion of uniqueness will bear some exploration. I will leave to the theologians the speculation as to whether part of what is uniquely a person is inherently evil. I prefer to say: whatever it is, draw it forth and face it; then make something creative and good out of it. Oscar Wilde has left for us the observation, \"Every saint has a past, every sinner has a future.\"\n\nWe are all conditioned by the culture in which we have lived, more than we can ever know. So many of the conflicts of the world today may have had their origins in the sudden impact of modern travel and communication, which bring these cultural differences face to face in sharp encounter. This makes it imperative that each of us understands the biases of his own culture which he brings to the confrontation.\n\nYet, acknowledging all of this, I believe that something of unconditioned uniqueness is prepared to show through in every person. It is the process of drawing it forth with which each of us needs to be concerned. It is a process which, at best, will be only dimly perceived; yet we must conjure with it. The remainder of this talk will deal with some ideas about the process which seem particularly important to me at this time.\n\nI see four major issues that need to be faced and dealt with if this drawing forth is to proceed as an important life involvement. The relevance to your concern in this vocational conference is this: in choosing a vocation, you should have as your primary aim (there are other necessary aims, but this one should be primary) that of _finding in the work in which you are engaged that which is uniquely you_. If you miss on this, you will likely wind up as one of T. S. Eliot's hollow men\u2014\n\nBetween the potency\n\nAnd the existence\n\nBetween the essence\n\nAnd the descent\n\nFalls the Shadow\n\nNo other achievement, no other end sought, will be worth the effort if through the work that occupies your best days and years you do not find a way to fan your own creative spark to a white heat\u2014at least once in a while. So I want to talk about four of many issues, four that have emerged rather sharply out of my own experience, in the hope that something will resonate in your own experience, while you still have many choices before you.\n\nFirst, _the consequences of stress and responsibility!_ All work\u2014whether in business, profession, government, home\u2014both develops and limits. It stretches one out in some ways and narrows one in others; it both fans the flame and seeks to quench it. This has no doubt happened to some extent in the educational and other choices you have already made. It will happen more in work.\n\nI see no exceptions: no completely whole persons, nor any chance of it. You must not look forward to any idealized achievement, no perfect or enduring adjustment to your life work. Whenever I think I have really achieved something, up come those powerful lines from Walt Whitman's \"Song of the Open Road\":\n\nNow, understand me well\u2014It is provided in the essence of things, that from any fruition of success, no matter what, shall come forth something to make a greater struggle necessary.\n\nThe greater struggle that will be necessary as you learn to bear more stress and carry responsibility comes because long exposure to these conditions tends to narrow the intellect unless a valiant effort is made to achieve an ever-expanding outlook. It is not enough just to try to keep up, to maintain the level of intellectual curiosity you have achieved in college. The intellectual life must expand. The great risk which the bearers of responsibility assume is that intellectual curiosity, the search for understanding, will atrophy and that only a calculating rationality will remain.\n\nThe test is in the heat of action. If one has a problem on which it is appropriate to act, and if one doesn't know what to do (which is the constant dilemma of all bearers of responsibility), one _should_ turn to the search for greater depth of understanding about the problem.\n\nIf you are only going to remember one thing from this talk, I hope you will remember this: the main reason you will ever be aware of a problem is that your understanding of yourself, of the other people involved, and of the area in which the problem lies is limited. Therefore, the search for understanding\u2014an intellectual pursuit\u2014is the most practical of ideas, even though the \"practical\" people often spurn it. But it is a difficult idea to hold onto when one bears the weight of responsibility for action, especially if the need is urgent. It is difficult to seek to understand when the heat is on. If one is to be well served by a liberal education, one needs to use this period of relative isolation from real life pressures to develop the firm habit of seeking to understand when the heat is _not_ on. This is the best rationale I know of for concentrating such an important educational influence in your present age range. Learn how to seek to understand now, when the heat is not on; make it a firm habit, and try to be aware that this will only serve you well if the habit is firmly enough fixed so that you can manage it when the going is rough, when the stakes are real, and when the consequences of failing to understand may be overwhelming.\n\nOne of the important testing grounds in decision making is the meeting of personal conflicts, when ideas or interests differ. Please give some thought to Dr. Carl Rogers' wonderful formula for meeting conflict. It is this: try to state to the _other person's_ satisfaction your _understanding_ of _his_ position; then identify and state as much of _his_ proposition as you can agree with; _then_ , and not until then, state your own point of view.\n\nThe risk in this procedure is that _you_ might change. Opening one's self to understanding always entails this risk. This is bad advice for the brittle, the fearful, the dogmatic, the \"allness\" people. But then, our subject _is_ maturity.\n\nIf change is too painful to contemplate, then one had best adjust his blinders to shut out all peripheral understanding. But if one does this and winds up hating the world, then one shouldn't blame the world for it.\n\nThere is a poignant line from _The King and I_ when Anna is getting to the King with some new ideas and, in desperation, he pounds the table and shouts, \"If you're going to be King, you've got to be King!\" He seemed to me to be saying, \"Don't mix me up with ideas when, at this point, the only thing I know how to do is to act!\" This portrays dramatically the awful consequences of a life of action in which the intellectual lamp was not kept bright, in which the search for understanding was not a constant quest. And in this play, the end, for the King, is tragic.\n\nThe second issue is _the tension between the requirement to conform and the essential person_.\n\n_Conformity_ has become a nasty word. It has almost become the battle cry of those of our generation who see their role as the modern version of the muckrakers of 50 years ago.\n\nThe attacks on conformity confuse the issue because in any organized society there must be a lot of conformity. Whenever two people undertake to work and live together, there must be some conformity.\n\nAll organized effort, any concerted influence, requires to some extent that those who participate must think and act alike. Nothing important can be accomplished without a good deal of conformity. Only a hermit in his cave can completely eschew conformity and carry out his role. As our society becomes more complex, more highly integrated, it demands more conformity than was called for in simpler times.\n\nThe problem is to know conformity for what it is: a completely external adjustment to the group norm of behavior in the interest of group cohesiveness and effectiveness. Then, knowing conformity for what it is, always keep it in rational focus as a conscious adjustment in the interest of an effective society. Keep it external, never let it become a part of _you_. Hold it firmly on the outside. The great danger is that one will lose one's identity in the act of conformity, not knowing which is the essential person and which is the conforming act, and thereby forfeit his right to be respected as an individual (by himself or by anybody else).\n\nWhen I was a boy, one of the weapons of discipline held over little boys who used profane language was that their mothers would wash their mouths with soap. My mother never did it to me, although there was ample provocation; but it was one of the things I heard about. I recall a story about a determined little character who did receive this punishment, and he is alleged to have sputtered out through the soap suds, \"You can soap my talk, but you can't soap my think!\" Don't ever let anybody soap your think.\n\nThe third issue that needs to be dealt with, if the drawing forth of what is uniquely each one of us is to proceed, is _the struggle for significance\u2014the complications of status, property, achievement_.\n\nOne of the hazards of prolonged schooling is that one becomes accustomed to living in a system in which the ends of the system are to nurture significance for the individual. This is what a school is for.\n\nOnce in the world of work, the institution one is in\u2014whether it is home, school, business, social service\u2014uses people for other ends. All such institutions have other obligations, and they commit people who do the work to these obligations. Most modern institutions are also concerned that the people who do the work find personal significance in their work. But this is a qualified obligation and one must not expect that any work will automatically provide the feeling of significance. A requirement of maturity is that one _learns to find his own significance_ , even under circumstances in which powerful forces may seem to operate to deny it.\n\nBut what is it that one is expected to find? I see it as something latent in the individual to be fulfilled. It is the seed of what is uniquely each person. Providing the conditions for its germination, emergence, and growth is the _search_.\n\nA healthy adulthood requires that one find it, and find it among the available choices. History and literature are surfeited with examples of barren lives in which the search was thwarted because the searcher could not accept the choices available to him. If only some out-of-reach circumstances were present, the search might go on.\n\nOne fictional account that has meant a great deal to me is Nathaniel Hawthorne's _Great Stone Face_. This is a simple story that can be read in 15 minutes, and I commend it to you.\n\nIn this story, we are in a small New England town nestled in the mountains with a view of a nearby mountain whose profile resembles a majestic face. The people in this town are living out a myth. Someday a noble man will come to them whose own profile resembles the great stone face. His presence will bring into their lives the qualities of majesty which the great stone face symbolizes.\n\nIn the course of the story, there comes a procession of people from the outside world, people of wealth and external status. The coming of each is heralded with great expectation; but always there is disappointment. The resemblance is not true.\n\nYears go by. We see a generation live from youth to old age carrying this hope that the image of the great stone face will come and that their lives will be enriched by the presence of the man who bears this likeness. Finally they recognize the resemblance to this image among them\u2014one of their own people. He has been there all the time, a living demonstration of those qualities which in his old age gave him the resemblance to the profile on the mountain.\n\nViewed symbolically, this community is a person seeking from external sources the qualities which are latent to emerge\u2014if only they will be permitted to emerge. They did not realize that the external marks of character are the product of the way a life is lived. If they were truly seekers, they would not have been so preoccupied with the external marks. Rather they would have attended more to the process, to what was going on in the lives they were examining. Had they been examining lives in process, lives around them to be seen, they would have seen right before their eyes the demonstration of how to live nobly. And they would have seen it when they were young enough for it to make a difference in the way their lives were lived.\n\nWe see in this story the collective life of the community denied fulfillment because it is looking for a stereotype. Significance is more likely to come from holding an attitude of unqualified expectancy, of openness and wonder.\n\nSo often, too, significance is blocked by compulsive drives for goals that do not provide fulfillment, something we pursue that we really don't want. When we achieve what we pursue, whether it is a tangible external thing or an internal state of mind, there is an emptiness. If we can name it and describe it precisely, the chances are we are seeking the wrong thing. I have seen so much of this among my contemporaries. If only they could lay aside the pursuit of over-specific and (therefore) meaningless goals and let their own uniqueness flower. The warning here is that our society holds up values which confuse the search\u2014status, property, power, tangible achievement, even peace of mind\u2014which subvert the emergence of true uniqueness, the only real significance. These are necessary elements of the society we live in at its current stage of development. We must make our peace with them and accept them as important, but we should not view them as basic or primary. Personal significance _is_ primary.\n\nNeither institutions nor aggregates of people have significance, except as it is given to them by living individuals who comprise them. Even traditions, powerful as they sometimes appear to be, are not viable unless contemporary people understand and believe in them and, by their thoughts, words, and deeds, give them current significance.\n\nOne of my favorite stories is about a now-prominent New York minister who was starting his career in the depression of the 1930s in a very poor church. He had no car and he needed one for his parish work. But since neither he nor the church had any money, this was a problem. Finally he bought an old battered jalopy for $25. It wasn't much of a car, but it ran and served his needs. However, he was soon confronted by an objection from his parishioners. Poor as they all were, they didn't like the idea of their minister riding around in that kind of car, especially parking it in front of the church. Finally it came out at a meeting of the governing board when one of the members said that their minister should have a car that \"added to his dignity.\" At this point the young minister rose and spoke one short sentence that disposed of the question about his car. \"Gentlemen,\" he said, \"no automobile adds dignity to a man; man adds dignity to the automobile.\"\n\nThis is a point ever to keep sharply in mind. Dignity, significance, character are wholly the attributes of individual people. They have nothing to do with anything external to the person.\n\nThe fourth major issue I see is _facing the requirements for growth; accepting some process for drawing forth one's uniqueness_.\n\nI would like to see a word that has fallen into disuse restored to common usage. That word is _entheos,_ from the same roots as _enthusiasm,_ which means \"possessed of the spirit.\" These two words, _entheos_ and _enthusiasm,_ have had an interesting history in the English language, coming down side by side through separate channels of meaning from the 16th century. _Entheos_ has always been the basic spiritual essence; enthusiasm, until recently, its perverter and imitator. _Entheos_ is now defined as _the power actuating one who is inspired,_ while _enthusiasm_ is seen as its less profound, more surface aspect.\n\nI want to use _entheos_ as it is now defined, the power actuating one who is inspired; and, at the risk of laboring it, I want to build a concept of growth around this one word. For those who are concerned with maturity _seen as becoming_ , it is important to see _entheos_ as the lamp and to keep one's own private lamp lighted as one ventures forth into a confused, pressure-ridden world, but nevertheless a hopeful world for those who can maintain their contact with the power that actuates inspiration. From the little I know of history, I cannot imagine a more interesting time to be alive _provided one can make it with entheos_.\n\nI see _entheos_ as the essence that makes a constructive life possible; it is the sustaining force that holds one together under stress; it is the support for venturesome risk-taking action; it is the means whereby whatever religious beliefs one has are kept in contact with one's attitudes and actions in the world of practical affairs; it lifts people above the prosaic and gives them a sense of timelessness; it is the prod of conscience that keeps one open to knowledge, so that one can be both aware and sensitive, when the urge to be comfortable would keep the door closed. I like that line from William Blake:\n\nIf the doors to perception were cleansed, everything will appear to man as it is, infinite.\n\n_Entheos_ does not come in response to external incentives. In fact, it may persist when incentives operate to destroy it. The individual cannot will it, it comes when it will and sometimes it goes when most needed. _But it does grow_.\n\nAll that can be willed is the search. There is no one pattern I know of. Each must find his own pattern. One of the great challenges of maturity: find your own growth pattern in the search for _entheos_.\n\nI can suggest some tests. If one has a few tests in mind, these might help to plot the individual search. We are reaching for _entheos_ , the power actuating one who is inspired. First some misleading indicators\u2014some achievements that might throw one off.\n\n_Status of material success_. One's external achievement may be impressive and praiseworthy and yet, in the process of achieving, one may be destroying much that is really important to him.\n\n_Social success_. The nongrowth people are sometimes more comfortable to be with.\n\n_Doing all that is expected of one_. Who is doing the expecting, and what do they know about what I should be expecting of myself?\n\n_Family success_ can be a misleading indicator. Fine and desirable as it is, it can be an egocentric, narrowing development. Internally, the family may appear in good balance; but it may be taking more out of the wider community than it is contributing.\n\n_Relative peace and quiet_. This may simply mean that the doors of perception are closed.\n\nFinally, _busyness\u2014compulsive busyness_. Beneath the surface of much action, there is the drive to avoid the implications of growth. \"This is for monks in a monastery; I'm too busy,\" they seem to say (Read the Mary-Martha story and ask, What does it have to say on this point?)\n\nThese are six indicators of achievement that can be misleading as evidences of growth. These can all be positive and worthwhile; but they don't necessarily add up to growth of _entheos_.\n\nNow, what I believe to be some valid tests, some indicators that there may be real growth of _entheos_.\n\nFirst, two paradoxes, _a concurrent satisfaction and dissatisfaction with the status quo_. One is not so unhappy with his current level of achievement that he can't live with himself. Neither is he so pleased with it that he has no incentive to break out of it. Then there is a concurrent feeling of broadening responsibilities and centering down. One is constantly reaching out for wider horizons, new levels of experience and at the same time the idea of \"This one thing will I do\" is in the ascendancy.\n\nThere is a growing _sense of purpose_ in whatever one does. The idea of purpose becomes important. Without being obsessive about it, the most penetrating and disturbing of all questions, \"What am I trying to do?\" becomes a constant query. One never loses sight of this question.\n\nThere are _changing patterns and depths of one's interests_. Old interests to which one was once attached drop away and newer and deeper ones take their place. Choices must be made.\n\nSomewhere ages and ages hence:\n\nTwo roads diverged in a wood, and I\u2014\n\nI took the one less traveled by\n\nAnd that has made all the difference.\n\n_Robert Frost_\n\nAs _entheos_ becomes a more constant companion, one moves toward the minimum of difference between the outside and inside images of the self; _one becomes more willing to be seen as he is_. Living as we do in an unreal world, to some extent we all wear masks. Convenient as it is to let the mask do what only serenity can _really_ do, I submit that _all_ masks chafe; I never saw a well-fitting mask. It is a great relief to take them off. The power of _entheos_ makes this possible; and the urge to remove the mask is one of the surest signs of its potency.\n\nThen _one becomes conscious of the good use of time and unhappy with the waste of time_. As awareness opens, one of the measures it takes of our contemporary society is the number of elaborate and seductive devices lurking about that serve no other purpose than to waste time.\n\nA further test is the growing _sense of achieving one's basic personal goals through one's work,_ whatever it is\u2014however menial, however poorly recognized. One of the popular illusions in our kind of culture is that one must reach a high status position in order to achieve one's goals. In my observation, there is really nothing in status but status, and the proportion of frustrated people is just as great in high places as in low places. I know it is an old truism, but the only place to achieve one's personal goals is where one is. Looking for a greener pasture for this purpose is almost certain to seal off the opportunity for achievement.\n\nGoing with some of these tests is the _emergence of a sense of unity,_ a pulling together of all aspects of life. Job, family, recreation, church, community all merge into one total pattern. While there remain obvious allocations of time to specific pursuits, the sense of leaving one and going to another diminishes. Peripheral time-consuming activities that cannot be brought within this view are laid aside. None of us needs to accept all of the obligations that others would impose upon us, and one way of making the separation is to test their compatibility with the core of _unified_ activities. As _entheos_ grows, one becomes more decisive and emphatic in saying _no!_\n\nFinally, there is a developing _view of people. All_ people are seen more as beings to be trusted, believed in, and loved and less as objects to be used, competed with, or judged. It is a shifting of the balance from use to esteem in _all_ personal relationships. In an imperfect world, one never achieves it fully; but there can be measurable progress. This is a critically important test. Unless this view of people becomes dominant, it is difficult for the inward view of one's own significant uniqueness to emerge. Love of oneself in the context of a pervasive love for one's fellow man is a healthy attribute and necessary for the fulfillment of a life. Out of this context, love of oneself is narrowing, introverting, and destructive.\n\nThe ultimate test of _entheos,_ however, is an _intuitive feeling of oneness, of wholeness, of rightness;_ but not necessarily comfort or ease.\n\nThese seem to me to be some valid tests that give assurance that _entheos_ is growing. If this kind of thinking doesn't strike a responsive chord with you today, please make a note of it, tuck it away in the back of your diary, and look at it 10 years, 20 years from now.\n\nIn closing, I want to return for a moment to work, vocation, and its relevance to growth, to the drawing out of the unique significance of the person.\n\nDon't just look for a job; even for an interesting and remunerative job.\n\nThink of yourself as a person with unique potentialities, and see the purpose of life as bringing these into mature bloom.\n\nDon't think of your career in terms of finding a nice fit for your skills and abilities. You will find some work more rewarding than other work; but the perfect job doesn't exist. Anyhow, neither the person nor the job stays put.\n\nSince there are no perfect jobs, no ideal fields, take one that challenges you as a piece of work to be done. Make other requirements subsidiary to this one, because nothing else really matters if the job is not rewarding in this sense.\n\nWhatever your work is, make something out of it that enriches _you_. Work itself cannot be truly significant except as it is seen as the means whereby the people who do the work find themselves in it. Do your work well; keep your sense of obligation high; cultivate excellence in everything you do; but above all _use_ your work, use it as a means for your own fulfillment as a person\u2014your own becoming.\n\nIf you have goals, be sure to state them in terms of external achievement, not in terms of what you will become. You don't know what you _can_ become, and no one can tell you.\n\nThis can be one of the great excitements of life\u2014the surprise when you discover what you _have_ become and realize that more is yet to come.\n\n## 3\n\n## **The Leadership Crisis\n\nA Message for College and University Faculty**\n\n### INTRODUCTION\n\nIn the eight years since I wrote the following essay on \"The Leadership Crisis,\" I have moved into a more meditative life with greater concern for the forces and influences that either nurture or depress the human spirit. And I have come to see the conditions that raise or lower the quality of life in colleges and universities as not materially different from those that operate in other institutions: governments, hospitals, churches, schools, businesses, philanthropies. Therefore, what I first addressed to colleges and universities and published in an academic journal, now seems to me to be much more widely relevant. And what is being said today in the flood of literature about how to lead in business seems equally applicable in the academic world.\n\nMost important, as I noted in the earlier essay, \"an indispensable condition for the persuasive power (of leaders) to be effective is that the institution is living out a great dream.... Institutions function better when the idea, the dream, is to the fore, and the person, the leader is seen as servant of the idea. It is not 'I,' the ultimate leader, that is moving this institution to greatness; it is the dream, the great idea. 'I' am subordinate to the idea. 'I' am servant of the idea along with everyone else who is involved in the effort.... It is the idea that unites people in the common effort, not the charisma of the leader.... Far too many of our contemporary institutions do not have an adequate dream, an imaginative concept that will raise people's sights close to where they have the potential to be.... that has the energy to lift people out of their moribund ways to a level of being and relating from which the future can be faced with more hope than most of us can summon today.\" That was the way I saw the crisis of leadership eight years ago: the need to produce in more of our institutions the overarching dream that will have this energy.\n\nI am indebted to Peter Senge for the idea of a \"shared vision,\" for the importance of the individual, regardless of status, to claim the dream as one's own. A condition for such a shared dream to prevail may be wide participation in the evolution of the vision, especially if it is an old institution that has lost a great dream it once had and wants to get a new one.\n\nWhat goes on in the participative process? Let me speculate: those who are the best dreamers and most adept at articulating dreams will periodically \"test the waters.\" Those who have the gift of leading will periodically say, \"Let's get together and talk about this.\" Those who have the gift of statesmanship will listen carefully to all of this and search for the ideas and the language that will be the basis for consensus.\n\nWhat evolves from this process, in which the key leader may take a hand, may not emerge as a written statement one can hang on the wall. It may best exist as an oral tradition that is continually reexamined, modified, or given new emphasis. In a well-led institution, there will always be a consensual tradition that most will summon in answer to the question, \"What are we about?\"\n\nWhat I identified as a crisis of leadership in colleges and universities eight years ago, after considerable involvement with academic institutions, I now see as a symptom of the failure of faculties to accept that the price of freedom everywhere\u2014in their case academic freedom\u2014is responsibility, the obligation to be constantly alert to opportunities to make one's share in forming the dream one lives by a real and meaningful thing, an obligation that persists as long as one has the wits to participate.\n\nTop leaders in all institutions have the opportunity to reduce the sense of crisis in our times by helping everyone involved to understand the responsibility that freedom entails, and to create an atmosphere in which consensual dreams can emerge that have the power to guide purpose and decision in way that makes for greatness.\n\nA critical aspect of leadership, whether in a university in which a substantial piece of the power to govern has been ceded to faculties, or in business in which, structurally at least, all the ultimate power usually resides with the chief executive, or something in between, is this: _Can the key leader accept that optimal performance rests, among other things, on the existence of a powerful shared vision that evolves through wide participation to which the key leader contributes, but which the use of authority cannot shape? And can that key leader be persuasive enough that responsibility for generating and maintaining that vision is widely accepted as a serious obligation?_\n\nThe ambiguity in this process may be that the effective key leader may never talk explicitly about vision or its generation. The process may be much too subtle for that. The generation of a shared vision may be one of those wonderful things that just happens when genuine respect for persons, for all persons, is consistently manifested. Within the climate of that pervasive attitude, and in the normal course of decision making, the first response of both the key leader and all subordinate leaders may be the simple question, \"What are we trying to do?\"\u2014 _Robert K. Greenleaf, 1986_\n\nThe leadership crisis of our times is without precedent.\n\nPeople have been poorly served by their leaders before; but in the past 100 years, we have moved from a society comprised largely of artisans and farmers with a few merchants and professionals, and with small government, to widespread involvement with a vast array of institutions\u2014often large, complex, powerful, impersonal, not always competent, sometimes corrupt. Nothing like it before has happened in our history. This recent experience with institutions may have brought a new awareness of serious deficiencies in the quality of our common life that are clearly traceable to leadership failures. Some of these lacks have become so painful to bear that _leadership crisis_ is an apt term to describe an important aspect of our present condition. Why are we in this dilemma, and what can we do about it? From the perspective of my experience, in these few pages I will suggest some tentative answers to these questions. And I will continue to search.\n\n### NEGLECT OF PREPARATION FOR LEADERSHIP\n\nColleges and universities assumed a unique place in American culture early in this century when, with the growing percentage of college-age young people enrolled, public service was added to the traditional roles of teaching, learning, and scholarship. Now that nearly 50 percent of college-age young people are on campus, the presence of a pervasive crisis of leadership raises the question about the impact of higher education. The traditional civilization-building role of the universities is as important as it ever was. But is it being sustained adequately in these times? The tradition seems not to have been adapted to carry all of the obligations that contemporary universities, with their massive influence, may be expected to assume\u2014including explicit preparation for leadership. (Please note that I have not said \"training\" for leadership. \"Preparation\" is a much more subtle process.)\n\nIn the turbulent 1960s, the charge was made that universities were effectively administering an \"anti-leadership\" vaccine to their students. Now, with declining enrollment and new financial urgencies, I sense that, from within universities, the fragility that they demonstrated in themselves in the 1960s is seen as an attribute of society at large. Appreciated, but perhaps not clearly understood, is weakness of leadership as an underlying cause. And there is concern about this condition. There are evidences of hunger for leadership that is denied by a seeming unwillingness of faculties to respond to such leadership as they have. If these are correct surmises, why then is there not a vigorous stirring in the universities to bring their great civilizing tradition to bear on this problem? It may be that the crisis of leadership in society at large is also the universities' own crisis. It is a baffling enigma to those who make the effort to interest universities in preparation for leadership. Yet I believe there is a reasonable basis for it. Let me speculate on what I think it is.\n\n### THE NEW AWARENESS OF POWER\n\nSince World War II, there has evolved a new sensitivity to the issue of power, particularly coercive power\u2014its abuses and legitimate uses. Along with this new concern about power, perhaps because of it, has come a fresh critical judgment of our many institutions, all of which wield power, whether they are governmental or voluntary, for profit or not for profit.\n\nI am a nonacademic who has made a few soundings within universities. My estimate of the perception of the typical faculty member, as she or he looks out from academe on the world of institutions, is as follows: Valuable and necessary as these institutions are\u2014businesses, churches, schools, governments, hospitals, social agencies (they are all we have)\u2014the whole gamut of them is seen as not serving well. Slavish adherence in these institutions to rigid hierarchical structures is viewed as an anachronism and a destroyer of values in the leaders that emerge. Some of these institutions are seen as mechanisms for manipulation and exploitation. Many who work for them are regarded as diminished and used up. And there is the suspicion that the root cause is low-grade top leadership: inept, not knowing, not caring, and, above all, the abuse and misuse of power. What makes it a crisis is that the fault is seen in an abstraction called the \"system.\" Leader and follower are both victims of the use of power in the \"system.\" There is no evident handle on the problem.\n\nIf, from within the university, one looks out on a scene as just described, including the university itself in that scene; if, accurate or not, that is the perception, would not a sensitive academic person be likely to hesitate to venture into leadership preparation for such a society? There is little in the background of a scholar that would give the average teacher a reasoned basis for reacting to that perception; nor is it likely that, in the prevailing structure of university leadership, there would be respected advice that would be persuasive in changing either that perception or the reaction to it.\n\nIf, as I believe, a concern about power, including the ramifications of power within the university itself, is the barrier that blocks universities from accepting an explicit obligation for preparing leaders, is it possible to reach an understanding of power that will help universities see their way around the barriers that now restrain them? Let me suggest an approach to that question.\n\n### THREE KINDS OF POWER\n\nAs a basis for sorting out what people do when they undertake to lead, let us consider three dimensions of power. These are not sharply delineated from one another by external markings. The distinctions exist more in the attitudes and values of wielders of power.\n\n#### _Coercive Power_\n\n_Coercive power_ exists because certain people are granted (or assume) sanctions to impose their wills on others. These sanctions may be overt, as when one may be penalized or punished if one does not comply; or the sanctions may be covert and subtle, if one's weaknesses and sentiments are exploited and thus pressure is applied. The power to coerce has a long history of use. But now, with constantly expanding government, the domination of the social structure by institutions\u2014especially large ones\u2014the expansion of techniques of surveillance, the proliferation of weapons of destruction, the growth in sophistication of methods of crime and oppression, and the bewildering complexity of life in which it is more and more difficult for people to gauge where their interests lie; with all of these we seem more vulnerable to coercion than we once were.\n\nAnother complication is that some coercion is masked behind ideal aims and is employed by people who are highly civilized and are motivated for noble ends.\n\nUniversities are involved in the use of coercive power in their role of \"credentialing\" (Thomas Jefferson, founder of the University of Virginia, would not allow degrees to be granted as long as he was Rector\u2014on the ground that degrees are pretentious.) When universities were small and were concerned largely with esoteric scholarship, degrees were a relatively harmless honorific. But now that universities are huge and degrees are so often, by law or custom, the ticket of admission to a \"good\" job\u2014the means for upgrading oneself in society\u2014the university holds, and uses, great coercive power. Even when it is sensitively and benignly used, it is still coercive power; and having that power has the same liabilities to corrupting influences as would any other kind of coercive power.\n\nOne of the problems of the use of coercive power by the more civilized people, even though for noble ends, is that, when conditions are right, the use of that power may cause destructive violence to be unleashed in the less civilized, sometimes on a disastrous scale. It may be plausibly argued that, although the Vietnam war and the civil rights crisis were seen as the proximate causes of the student disorders in the late 1960s, one of the ultimate causes may have been the universities' own long-standing use of coercive power.\n\nIs there a moral principle here: when coercive power is used in any form and for any purpose, does not the user of that power bear some responsibility for what may be inevitable harmful effects, including the unleashing of violence? If coercive power is used (and I do not foresee a utopian society in which it will not be used), do not the users of that power have the obligation to be aware of the potential danger and prepare a meliorative strategy that minimizes the damage to the social fabric?\n\nCoercive power is more pervasive than most are aware of, and its consequences are traceable to evils that we ordinarily do not connect with it.\n\n#### _Manipulative Power_\n\nI see manipulation as distinct from coercion because it rests more on plausible rationalizations than on the threat of sanctions or on pressure. People are manipulated, I believe, when they are guided by plausible rationalizations into beliefs or actions that they do not fully understand. By this definition, some manipulation by leaders is unavoidable because some who follow are not capable of understanding or will not make the effort to understand. But not all manipulation by leaders can be justified on these grounds. The heart of the problem, I believe, is that effective leaders, those who are better than most in charting the path ahead, who willingly take the risks and expend the energy that leadership requires\u2014those people are apt to be _highly intuitive_. Thus leaders themselves, in their conscious rationalizations, may not fully understand why they chose a given path. Yet our culture requires that leaders produce plausible explanations for the directions they choose to take. These rationalizations are useful because they permit\u2014after the fact\u2014the test of conscious logic that \"makes sense\" to leader and follower. But the understanding required by the follower, if she or he is not to be manipulated, is not necessarily contained in this rationalization that makes sense. Because we live in a world that pretends a higher validity to conscious rational thinking in human affairs than is warranted by the facts of our existence, and because many sensitive people \"know\" this, _manipulation_ hangs as a cloud over the relationship between leader and led almost everywhere, and is the subject of much pejorative comment.\n\nCan this cloud be dispelled? Not easily. But something can be done about it if both leaders and followers are constantly aware of the presence of this murkiness, and if they accept that dispelling it requires a determined effort by both of them. Within this framework of awareness, what would the effort be, what would they try to do? The essence of leadership, I believe, is that the leader makes the effort first. The leader takes the first step in the belief that, if it provides a clear demonstration of the intent to build a more honest relationship, followers will respond.\n\nI suggest that the leader try persuasion!\n\n#### _Persuasion as Power_\n\nUnfortunately there is ambiguity in the word _persuasion_. One of the dictionaries I consulted, in a series of definitions, gives three that do not imply coercion. A fourth implies coercion. And the fifth states flatly, \"to bring a desired action or condition by force.\" I prefer to use the word _persuasion_ for a process that does not allow either coercion or manipulation in any form. One is persuaded, I believe, upon arrival at a feeling of rightness about a belief or action _through one's own intuitive sense_. One takes an intuitive step, from the closest approximation to the certainty to be reached by conscious logic (sometimes not very close), to that state in which one may say with conviction, \"This is where I stand.\" The act of persuasion, as I limit the definition, would help order the logic and favor the intuitive step. _And this takes time_! The one being persuaded must take that intuitive step alone, untrammeled by coercion or stratagems. Both leader and follower respect the autonomy and integrity of the other and each allows and encourages the other to find his or her own intuitive confirmation of the rightness of the belief or action. If this relationship prevails whenever it is possible, then, when a quick action is required, one supported by the skimpiest of rationalizations, it will be accepted with the assurance that at some future time there will be the opportunity for intuitive mutuality to be reestablished. A leader who practices persuasion whenever possible sets a model that, in time, will encourage followers to deal with the leader by persuasion. Power is generated in this relationship because it admits of mutual criticism, spirited arguments can occur, and it does not depend on artful stratagems.\n\nThis poses a problem for conventional organization structures in which those \"at the top\" hold coercive power and, because of their superior informational sources, are in a good position to manipulate. Such persons should take note that those who are seen as holding coercive power, even though they use it sparingly, are somewhat disqualified to persuade. \"Where is the hidden agenda?\" is often the unasked question. Since in our imperfect society it is difficult to conceive of a functioning organization in which there is not an ultimate locus of coercive power, two suggestions are made so that unqualified persuasive power can make its contribution.\n\nThe first is that every institution should harbor able persuaders who know their way around, who are dedicated servants of the institution, whose judgment and integrity are respected, who do not manipulate, who hold no coercive power, and who, without the formal assurances that faculty members usually have, feel free and secure. Those who hold the ultimate power will accept that these nonpowerful persuaders can accomplish things for the good of the institution that the powerful cannot command. Therefore, the powerful will permit radical criticism to be made. Every institution that wants the benefits that only persuasion can accomplish needs to support such persons on its staff, because the value of coercive power is inverse to its use.\n\nMy second suggestion is that an indispensable condition for persuasive power to be effective is that _the institution is living out a great dream_. I speak with some conviction on this because, near the end of my career, I was party to an unsuccessful effort to persuade the top command where I worked that a new goal was needed. The great dream on which the institution was built had lost its force and, as seems the plight of so many contemporary institutions, ours was in the mood of struggling to survive.\n\nThose in command where I worked were honest, able, dedicated, and caring\u2014like so many who head other institutions with which I am familiar. But they were not guided by a great dream, not a dream that was shared by those who followed them. The idea that inspires and unifies was muted. Leaders were seen, too much, as self-symbols; they did not come through as servants of the dream. Consequently, there was not enough trust in the institution by any of its constituencies.\n\nGreat institutions are a fusion of great ideas and great people. Neither will suffice without the other.\n\n### NEW DREAMS ARE NEEDED\n\nWhere would the leader of an institution get the idea that a dream is needed? How would he or she learn what it would serve? Where, in all of our vast communication, educational, and religious resources is the suggestion being made?\n\nRegardless of the stress of circumstance, institutions function better when the idea, the dream, is to the fore, and the person, the leader, is seen as servant of the idea. It is not \"I,\" the ultimate leader, that is moving this institution to greatness; it is the dream, the great idea. \"I\" am subordinate to the idea; \"I\" am servant of the idea along with everyone else who is involved in the effort. As the ancient Taoist proclaimed, \"When the leader leads well, the people will say, 'We did it ourselves.'\" The leader leads well when leadership is, and is seen as, serving the dream and searching for a better one. Dreams should be articulated by whomever is the ablest dreamer, and leaders should always be open to persuasion by dreamers. It is the idea that unites people in the common effort, not the charisma of the leader. It is the communicated faith of the leader in the dream that enlists dedicated support needed to move people toward accomplishment of the dream. Far too many of our contemporary institutions do not have an adequate dream, an imaginative concept that will raise people's sights close to what they have the potential to be.\n\nIf the dream has the quality of greatness, it not only provides the overarching vision for the undertaking; it also penetrates deeply into the psyches of all who are drawn to it and savor its beauty, its rightness, and its wisdom. The test of greatness in a dream is that it has the energy to lift people out of their moribund ways to a level of being and relating from which the future can be faced with more hope than most of us can summon today. Persuasion, as an art of leadership, is tenable because of the persuasive power in the dream itself.\n\n### A GREAT DREAM FOR A UNIVERSITY\n\nJust from reading the newspapers, one would question whether universities have an adequate dream for these times, one in which the roles of teaching and learning, scholarship, and public service are inseparable. And the great dream that might be theirs seems remote indeed. In the context of the subject \"the crisis of leadership,\" what evidence supports this assertion?\n\nIt is specious, I believe, to argue as some do, that general education nurtures leadership and prepares people for discriminating followership. Quite the reverse may be true. How else, in view of the massive level of higher education, would one explain either the leadership crisis in which we are now enmeshed or the gross misjudgments in selecting whose leadership to follow that has characterized recent years?\n\nThen, if, as I have suggested, the university degree has become the ticket of admission to better jobs, these tickets are being issued to at least twice as many as there are jobs that warrant preparation.\n\nPerhaps more serious, academic higher education is not suited to everybody. Informed guesses on how many will profit by such education run as low as 15 percent. If this, or anything close to it, is a fair judgment, with nearly 50 percent of the college age population enrolled, what is the effect on the leadership potential of the other 35 percent\u2014those who should receive some other kind of education?\n\nA determined effort to educate minority peoples has been made. One effect on the black community has been to give favored job treatment to those who make it through a university\u2014to the disadvantage of the large numbers of less educated whose dependence on welfare has increased and who may have been deprived of their indigenous leadership.\n\nThese evidences of an inadequate dream in universities are cited not to censure them\u2014they are doing as well in their obligations as are other institutions that serve us. But, as I see it, they hold the key to recovering us from the leadership crisis. Therefore, universities merit priority in concern about this problem.\n\nI submit that universities are in urgent need of a great new dream. No small dream will suffice. How will they find that dream and unite to bring it to reality? Who will lead them to it?\n\n### A BASIS FOR HOPE\n\nOne would hope that university administrators would give that leadership; the signals are clear enough that they should. But faculties have held too much power too long, and administrators are too much caught up in the common mores of our institutional life. They need an infusion of leadership vision as much as others do. Then, pressed as most of them are by financial urgencies and by the intractable nature (as they see it) of faculties, it is not likely that the initiative to redirect universities to a new role of nurturing leadership will come from administrators, even though they might thereby establish their own leadership in a healthy way and make financing easier.\n\nNeither private nor public funding sources for university programs appear eager to initiate new concerns for the preparation of leaders. Like the universities, they, too, seem caught up in the crisis of leadership. Ours may be said to be _the age of the anti-leader_.\n\nWhere, then, will the initiative come from? Is there no hope that a resource equal to the need will emerge? Yes, there is hope. Hope lies in the great strength of the academic tradition (which some see as its most troublesome aspect): academic freedom and tenure.\n\nThe transforming movement will arise, I believe, from the source it usually comes from in a crisis: from a _saving remnant_. At first, from a few faculty members who act alone, within the scope of autonomy they now have, finding their own way to be effective, using some of their own free time, and, in some cases, putting in a little money. A few such far-seeing faculty members may start the transforming movement without the support of their culture, possibly incurring some opposition from it. This is characteristic of saving remnants. They are not usually empowered, approved, or well financed.\n\nThe teachers who make up this saving remnant will come to one understanding in common: they will have a clear sense of how institutions change, prudently. They will accept that change takes place slowly as a result of diligent work to acquire competence to lead. Revolutionary ideas do not change institutions. _People_ change them by taking the risks to serve and lead, and by the sustained painstaking care that institution building requires.\n\nI know of a few teachers who have taken such initiative to prepare their students to lead and to deal with the realities of institutional life. I had the good fortune to have had one of those teachers in college over 50 years ago. Advice from him set the direction of my career to find my own way as a building and meliorating influence within institutional structures. The most open course I can see for meeting the leadership crisis in the next generation (too late for this one; we will have to muddle along as best we can) is to encourage a few faculty members to move on their own, without the support of their institutions, and to start now on the preparation of the next generation of leaders.\n\nIf one in one thousand among the half-million or so faculty members in our country will move on this now (not an unrealistic expectation if there were a way to alert them to the opportunity) _the 500, on their own and without anybody's help_ , could produce a flowering of talent in the next generation that would make a golden age of leadership. I am absolutely certain of it. I am so certain that I have written a guide to encourage their venturing as lone unsupported individuals, and, perhaps, to point a way. It is called _Teacher as Servant, a Parable_. [Available through the Greenleaf Center.]\n\n_If_ the one in one thousand will respond to this encouragement,\n\n_If_ they will articulate persuasively what they are doing,\n\n_If_ , having established that students will respond, they take further steps to educate university trustees and persuade them to accept a more affirmative institution-building role for themselves,\n\n_If_ trustees will then install and guide administrators who are prepared to be, and disposed to be, effective leaders-by-persuasion, and\n\n_If_ those administrators will gather the help of all constituencies of the university to establish means for explicit preparation for both leading and following, by persuasion,\n\nthen, someday, someplace, a design for a new contemporary university may emerge, _wholly as a result of persuasion_. In the course of this evolution, that university's goals, program, leadership, and governance may be reexamined, not so much in the light of tradition of what universities have been, but, rather, in recognition of the obligation that has been assumed because of access to several formative years of half the population.\n\nThe leadership crisis in society at large will begin to be meliorated when _one_ university, in its new awareness of its obligations and opportunities, moves into, and then resolves, its own crisis of leadership, and emerges with a great new dream.\n\nWhen, in the persuasive atmosphere of that dream, the trustees and administrators lead by persuasion, and faculties, students, and staff respond with persuasion, that university will regenerate, in the late 20th century, the civilizing influence that universities once had and move to the center of our institutional life as a bastion of strength in what may prove to be a gathering storm.\n\n## 4\n\n## **Have You a Dream\n\nDeferred?**\n\nWhat happens to a dream deferred?\n\nDoes it dry up\n\nlike a raisin in the Sun?\n\nOr\u2014?\n\n_Langston Hughes_\n\nDo you have a dream deferred, now that you are nearing the end of your freshman year? And what about the charge of idealism and the high expectations you brought here last September; what shape are they in? How does your university adventure look to you at this point?\n\nWe are here today because you have applied to be appointed Ohio Fellows. The objective of the Ohio Fellows program, as I understand it, is to help you realize your potential for service in the public interest. It is not concerned with what vocation you choose for your life work. But it assumes that, in your work or outside it, you want to make a social contribution through becoming a self-actualizing person.\n\nBecause you want to be admitted to this group, I assume that you are still searching, within the resources of the university, for something you have not yet found\u2014perhaps to renew your faith in your dream. If my assumption is correct, what is the nature of your search? What is your personal strategy for the optimal use of the opportunity which the next three years offers?\n\nAs I recall it, no one raised these questions near the end of my freshman year in college. No one suggested that it was my responsibility to manage my own life at that point. I am not sure that much is made of it today. Yet, at age 18 or 19, are you not mature enough to manage your own growth? All that my generation can do is to help clarify the problem; and, out of our greater experience, point to alternatives that you may not yet have discovered. You cannot compress or extract experience by listening to us. Every person has to live every minute of his own existence and make his own meaning out of it. _But you can widen your awareness so as to make your experience more intense and more meaningful_. Thus I urge you to open up to influences and seek opportunities that will expand your awareness. These may come in unexpected ways.\n\nIn my own case, a remark by a professor in the course of a rambling lecture shaped my career. He was neither a great scholar nor an exciting teacher; but he was wise in the ways of men and institutions. What he said and what I did as a result of receiving this signal are not important here; but I have often wondered whether, on that fateful day, something exceptional was said or whether I just had my doors of perception open a wee bit so that a significant signal could come through.\n\nAfter much reflection on my own education and a sustained interest in what colleges do, I have concluded that there is no one right way to provide education. One of the disadvantages of the level of academic effort required today may be that the signal that shapes your life is even harder to hear than mine was when educational endeavor was considerably more relaxed.\n\nI once heard Robert Frost asked the meaning of one of his poems. \"Read it and read it and read it,\" he said, \"and it means what it says to you.\" What he was saying, I think, was that meaning, that subtle signal that may shape one's future, is an elusive thing; it does not emerge, necessarily, as the logical end-product of a conscious analytical process. In fact, it may defy such an approach. It is more likely to come as a gift, as an insight, peripheral to the analytical process rather than the target of it. It requires an attitude, and openness, belief in the vastness of knowledge just beyond our conscious rational searching. \"If the doors of perception were cleansed,\" wrote William Blake, \"everything will appear to man as it is, infinite.\" I think of this as the attitude of wonder.\n\nReading and reading and reading the poem would, in effect, be an act of submission to the poem, a willingness to let one's guard down and be taught, to let a significant signal come through. This is why I want to talk to you about the next three years; to encourage you to seek and respond to your own signal, received or generated out of your own experience. And it will not be quite like anyone else's signal because you are uniquely you and your signal will be what _you hear_.\n\nBe aware! Be open to insight from your own experience and from what you see and hear around you. This is the best advice I can give you for the next three years: keep those dreams alive; persist in the attitude of wonder. What follows are a few ideas that bear on these concerns.\n\n### A LIFE STYLE OF GREATNESS\n\nI hope you will manage your lives these next three years so that you leave the university with a well-set lifestyle of greatness, with attitudes and values and ways of initiating and responding that will assure service in the public interest with distinction. Distinction is not synonymous with fame. Whether your life is long or short or your opportunities large or small, _distinction_ or _greatness_ is a combination of the moral and the excellent. It is doing the very best you can with the talents you have and the opportunities you can find.\n\nYour vocation can be any legitimate calling your talents justify, be you poet, scientist, or businessman. A lifestyle of greatness will augur for a total impact that will leave some segment of society a little better than if you had not tried. It is important that some of you make this choice now because plenty of people, by design or by accident, will leave it worse. It takes a lot of hard work by responsible individuals for a society just to stay even. (I use the word _responsible_ advisedly. I am not using it in the sense that so many of my generation use it when what they mean is that they want you to behave so that their comfort, their sense of propriety, are not disturbed. Responsible people build; they do not destroy. They are moved by the heart. The prime test of rightness of an act is: How will it affect people, are lives moved toward nobility?)\n\nIt is so easy to assume that simply by meeting the requirements of the university the preparation one needs for life will follow as a matter of course. You will do well to conform to the requirements for a degree and make the most of the courses you elect to take. But do not assume that automatically out of this will emerge a well set lifestyle that favors a distinguished public service role. If you want that, you have the opportunity to take charge of your own growth now, with your own strategies. You cannot leave this to the university or to anyone else.\n\nHow do you do this? You can wait until you graduate and use this opportunity to prepare and prepare and prepare; or you can begin living now, accept this university as \"real,\" and find your own best way to influence this community so that it becomes a better institution to serve those who come after you. Make as your goal that three years from now you will leave here with a well-set, clearly demonstrated lifestyle of greatness, a way of dealing with your environment that is both moral and excellent. And let it not be a tentative abstract idea. Rather, let it be a concrete achievement, tested and refined in the arena of real experience, a firmly set and durable lifestyle.\n\nA word of caution, however. Do not set out to remake your personality. Few people I know are wholly satisfied with the way they are, and much that is _you_ , you will best learn to live with. Read E. B. White's perceptive essay, \"The Second Tree from the Corner,\" particularly the closing paragraphs. It is about a troubled man who decided not to make himself over. Much that is you is pretty firmly established at this point; and you cannot choose another set of parents or relive the past 18 years. Accept what you are; make note that no one is perfect and resolve that you will build on what you now are, within a consciously chosen self-image of a responsible person. Make the very best you can out of it these next three years. You will never have a better chance.\n\nA few years ago Edwin H. Land spent some days visiting with undergraduates at Massachusetts Institute of Technology. Then he talked to them on \"Generation of Greatness, the Idea of a University in an Age of Science.\" In this talk, he said, \"Everywhere I could sense a deep feeling in the undergraduates I met\u2014that if a way could be found of nurturing the timid dream of his own potential greatness which he brought from his family and school, if somehow he could tie on to the greatness in the faculty and the administration, then his dream might be coming out differently.\"\n\n\"What do I mean by greatness?\" he asked. \"I mean\u2014an opportunity for greatness for the many\u2014as distinguished from genius.\u2014Within his own field he will make things grow and flourish; he will grow happy helping other people in his field, and to that field he will add things that would not have been added, had he not come along.\"\n\n### CULTIVATE YOUR CREATIVENESS\n\nWhen I came of age 40 years ago, it seemed not so important to be creative because ours was a pretty settled world (or so we saw it), and I could begin to be constructive by learning to do well those things that were in the established patterns of society. For you it is different; you must start out creative or you will never really be in the ball game. Otherwise, you may be useful and prestigious, and you may even make a lot of money. But, you are not likely to be a constructive influence, an important contributor to the public interest, if you are not creative, because the important initiatives and responses will require new inventions\u2014 _prudent, feasible inventions that unlock the social impasses and raise people's sights and aspirations above the ordinary_. If you are to do this, at any stage of your life, your creative power must flower while you are young, and it must be sustained as long as you expect to function productively.\n\nYour major contribution may come at any age. Theodore N. Vail, who built the business in which I spent 38 good years, made his impressive contribution to institution building between the ages of 62 and 75. Pope John XXIII achieved his prime influence between 80 and 84. In both, creativity flowered early and was nurtured carefully throughout their long lives. One never knows when his big creative opportunity will come. And it makes a more interesting life to keep one's creativity alive even though a \"big\" opportunity never comes.\n\nSome large social contributions have been made by persons with small but well-nurtured creative endowments. If one is to make his way as a mathematician, scientist, artist, or composer, his creative powers had best mature early and be exceptional. However, in the area of social contribution, a more modest creative resource, coupled with values, skills, knowledge, attitudes, and experience, may be the greater asset.\n\nI counsel you then to be aware of your priceless gift of creativity. No matter how small and flickering a light it may seem to be at the time, cultivate it as a pearl beyond price. Whatever your competence in your chosen field turns out to be, and regardless of the size of the opportunities that may come your way in the foreseeable world, your imaginative capacity will measure the productive use of your strengths and opportunities.\n\nThe skill of _foresight_ is crucial. The \"lead\" that a leader has is his ability to foresee an event that must be dealt with before others see it so that he can act on it his way, the right way, while the initiative is his. If he waits until everybody sees it, he has waited too long; he cannot be a leader\u2014at best, he is a mediator. Therefore, cultivate the greatest of the creative skills, foresight. Practice on every significant event you observe; ask yourself, where did it come from, where is it going? Note your projections and check them in the future. Practice living partly in the future\u2014all of the time.\n\nCreativity, however, is only part of the picture. Adolph Hitler was creative, diabolically so. One must have a dependable value system and a reasonably sane outlook to make a constructive social contribution. Then some creative people miss because they are ignorant or unskillful. Others are hampered by a restrictive set of attitudes, like perfectionism or an over-inflated ego. Still others are unaware and miss the necessary cues. And then there is the overarching matter of balancing one's trust between intuition and reason. If you have a dream deferred, bring it to life by beginning to live it\u2014no matter how discouraging the circumstances may be. Learn to relax, so that the elements of your life fall, of themselves, into context and proportion. You may have more going for you than you are aware of. If you end up in middle age an uncreative, crusty reactionary\u2014the kind of person your generation complains about so vociferously, it will probably be because you _chose_ not to allow the idealism that is so characteristic of your present age to operate and elected, instead, a less courageous path, not \"the path less traveled by\" which, Robert Frost said, \"makes all the difference.\"\n\n### BUILD A NEW MORALITY\n\nYours may be the first generation since the time of Moses to face, as an explicit task, the problem of building your own morality. Most of the time, from the Mosaic law down to now, there has always been a law, a code. To be sure, it was honored in the breach by many people, but it was always there. Those who violated it, for the most part acknowledged the law and thought of themselves as deviants. Now, since you have been alive, the law has all but disappeared. Many will assert that there still is a moral law, a code of ethics, and that they know what it is. I think of myself somewhat in this category. But the _level of consensus necessary for traditional morals to be accepted as law no longer holds_. Therefore, your generation may well be the first to face the condition of producing your own. Many will advise you, such as Professor Joseph Fletcher and his principle of \"what love requires in the situation.\" But you cannot be rescued from the confusion you may find in a lawless world as handily as Moses did it for his followers, by bringing it down from that mysterious encounter on the mountain chiseled in stone and bearing God's imprimatur. For better or for worse, you will have to achieve it on your own, with your own resources, and without much help from my generation, which has not faced the problem. I am not saying that the present state of affairs is good or bad; that is simply the way it is.\n\nIf this is a valid assumption, then building your own morality map may well be one of your major concerns these next three years. And a good deal of what you seek may come to you intuitively if you will be open to understand what goes on around you that most people condemn.\n\nWhile I have stressed that the level of consensus on what constitutes morality has dropped, within your lifetime, to a point where it is difficult to affirm, \"this is the moral law,\" the shift is only one of degree. Every person has always been, and must always be, his own theologian, his own moralist, his own value finder. It is just more confusing for your generation to try to find your way amid the babble of conflicting opinion from mine. Every strong moral person has always had to choose the ground upon which he or she would stand.\n\nOne of the qualities of a lifestyle of greatness is the ability to know with some certainty the solid ground one stands on at any one time. It gives one a toughness of mind with which one looks out upon a seething, troubled world with a quiet eye and asks the meaning of it all\u2014not so much to judge it as to enlarge the perspective from which to build even more solid ground for one's own two feet to stand on. In the end, each man builds his own solid ground to stand on\u2014and stands alone. In Robert Frost's one poem clearly addressed to the \"inner circle,\" the one called \"Directive,\" he says: \"And if you're lost enough to find yourself\/By now, pull in your ladder road behind you\/And put a sign up _closed_ to all but me.\/Then make yourself at home.\"\n\nI must confess ambivalent feelings about the college-age group today: its outlook and its behavior. I am enough the creature of my generation to be shaken at times and wishing to be insulated from yours. But then, I really envy you who may face the greatest creative opportunity, as individuals and as a society, which any young people may ever have faced: as a matter of conscious choice, the chance to resolve the prevailing confusion with a new level of ethical insight.\n\nAs you think of your opportunity to build a new ethic, do not set out to reinvent the wheel. There are thousands of years of moral history to build upon, even though the behavior of us older people may look pretty wobbly to you. None of us, no existing institution, owns the tradition. Our moral tradition is there for you to use as a fresh resource. I wish for you the strength and the insight to use it well and to pass it on to your children as a nobler ethic _because of your efforts_.\n\n### GROW IN WISDOM\n\nI see the next three years as open space for you to rise to this opportunity to build a new ethic. Use your participation in the university program to this end, but beware of accepting that it will chart the course for you to follow to this end. It will provide a clear path to intellectual growth, but the path to wisdom, indispensable for ethical choices, is not so clear. Walt Whitman had something to say about wisdom in his \"Song of the Open Road.\" \"Here is the test of wisdom,\/Wisdom is not finally tested in schools,\/Wisdom cannot be passed from one having it to another not having it,\/\u2014is not susceptible of proof, is its own proof,\u2014\" Clearly it is a different sort of thing from intellectual growth, which is the major concern of a university. You can easily be deceived that you are wise because you are academically proficient, articulate, can reason well, and understand another's wisdom. Wisdom is not the antithesis of intellect. But intellectual growth can interfere with wisdom if not kept in perspective. Since in three years you are likely to emerge well started on your education, do not assume that you will therefore be wise. If you want to see a spectacular dramatization of this, read the novel _Herzog_.\n\nIn the laudable pursuit of intellectual development, it is well to remember that the great hazards of personal choice, such as alcoholism, drug addiction, broken marriage, reactionism, the \"educated\" are just as vulnerable as the \"uneducated.\" Only wise people have some protection against these hazards. And even with wisdom, it takes a bit of luck.\n\nWisdom, common sense (a better term would be _uncommon sense_ ), _judgment_ are all terms that refer to something that is of the essence; it is not codifiable or teachable, except perhaps by a mentor who goes with his understudy into the real-life situation, asks him what he sees and hears (thereby training his awareness), asks for his tentative judgment, lets him commit an error and discusses the consequences (some of our best wisdom comes from error and suffering the consequences of error), and thereby helps him in acquiring a disciplined, thoughtful approach to real-life problems. Thomas Jefferson had such a mentor in George Wythe, the Williamsburg lawyer under whom Jefferson apprenticed. Without the influence of a George Wythe, there might not have been a Jefferson to write The Declaration of Independence or to draft the statutes in Virginia that shaped the Constitution. He might have settled for the role of the eccentric Virginia scholar.\n\nFind such a mentor if you can. There may be one for you here on this campus. At the very least, be aware that intellectual prowess is not wisdom, and approach real life situations humbly. Only through testing your judgment, risk of decision in real-life situations\u2014with an openness to learn what only experience teaches\u2014will you grow in wisdom. Read Abraham J. Heschel's wonderful essay, \" To Grow in Wisdom.\"\n\nOne of the ways you may grow in wisdom, here on this campus, is to learn to challenge\u2014by challenging, experimentally. In so doing, you may be wrong part of the time. Thus you must learn to accept challenge: from faculty administration, the community, fellow students. In doing this you will become, in part, a politician. Because politics is the art of the possible, you must learn to compromise.\n\nThe university is a total learning situation\u2014for everybody: students, faculty, administrators, trustees\u2014even for old consultants like me. And if you are to be well served by it, it needs to be a society in which the art of the possible is practiced by enough people so that those who aspire to grow in their ability to serve society will, in fact, learn.\n\nIn theory, at least, the university exists solely to nurture your growth and development. Later your involvement will be largely with institutions that exist to _use_ you. But the university is a _real_ place. It will use you too; and the people and their dilemmas are real. And you will never have a better chance to grow in wisdom than by entering into the life of the university with openness and humility, the necessary conditions for your experience here to favor the growth of wisdom.\n\nWords cannot describe what it will mean to the ultimate fullness of your life if you can learn a measure of wisdom while you're here these next three years.\n\n### TRUST\u2014AN ASPECT OF GREATNESS\n\nOne of the resources offered here at the university is the chance to study the problem of trusteeship, both the specific role with a capital \"T,\" and the more general one of how people acquit themselves of the obligations assumed in the various constituencies\u2014student, faculty governing bodies\u2014which make up the university. I once asked the president of a great university this question: \"Who in the university is responsible to the student for the obligation which the university assumes when it accepts him as a student?\" His answer was, \"Not I.\" That was 20 years ago, and I have never really recovered from it.\n\nWhen the institution of American democracy ultimately declines, as it may one day, a major cause will be a failure of trusteeship\u2014both the capital \"T\" and the small \"t\" varieties. And it will not decline so much because of the ineptness of the less favored as by the failure of those endowed with intelligence, values, training, judgment and experience which bring men to positions of trust\u2014their failure to accept with sufficient commitment the obligations of trusteeship which they assume, and to perform with distinction in the public interest. Far too many good, competent, honest people settle for mediocre performance in these trustee positions. They meet the prevailing standards of adequate performance. But since it does not rise to the level of distinction, it must be judged mediocre. Only distinguished performance in positions of trust can support a long life for the institution of American democracy. In any endeavor, one must strive for distinction just to come out reasonably good.\n\nOurs is an institution-bound society. While there remain large opportunities for individuals to act on their own, most influences that make a difference have the effect of changing what goes on in institutions. And institutions are necessarily instruments of trust. For many of you, your individual contributions will largely be measured by how you perform in institutional roles of trust, whether it be the management of a home or the presiding office in world government.\n\nSo much will be determined by the quality of the interpersonal relations among your colleagues. Do they work, figuratively, with knives out and backs to the wall (an expression I hear all too often in off-the-record descriptions of the inner workings of institutions\u2014the whole range from businesses to churches), or is there an esprit that buoys the institution and makes possible a performance at the level of distinction?\n\nAs I have found it, the institutional elements that have most to do with _esprit_ are _goals_ \u2014what are you trying to do in material or behavioral terms. And _strategy_ , how do you intend to get there? What are you trying to do? is the easiest of questions to ask and the hardest to answer. There is nothing that builds organizational strength quite like a high order of consensus about goals and strategy. Unfortunately, there is a tendency, in institutions where the aims are more idealistic, like universities and churches, for this consensus on goals and strategy not to be firm.\n\nI give you these views of institutions in general and educational institutions in particular because of my belief that one of the best preparations you can make for a high level of trusteeship, in roles you may assume later, is to be concerned about the institution you will be a part of these next three years. You will never have a better opportunity to learn the ways of institutions, what makes for mediocrity and what makes for greatness, than you have right here at your university.\n\nThe university provides a good laboratory for the study of responsibility. Remember the response that President Kennedy made when a reporter asked him why he had not taken an action which, during the campaign, he had criticized President Eisenhower for not taking? President Kennedy's reply was, \"When you are responsible, things look different.\" When you observe an action here in the university, practice being responsible by putting yourselves in the shoes of those who are responsible.\n\nAs you study the university look for models of men and women who account for trust at the level of personal greatness. Think of yourselves as being like them, someday.\n\n### REALISM\u2014TRY TO SEE IT STRAIGHT AND PLAY IT STRAIGHT\n\nOne of the gifts of those whose mature lives measure up to their estimated potential as young people is that their early assumptions about the nature of the society they will live and work in are realistic. One can make too optimistic assumptions and suffer endless frustration or too pessimistic ones and curb one's aspirations. Yet one must err on the side of optimism and accept the frustration.\n\n\"The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait.\" This was written by G. K. Chesterton in 1924, when the world looked much more settled and predictable than it does today.\n\nMy concern for the many crises in the world today suggests a view of the human dilemma: problems, but no solutions; nevertheless challenges that must be worked on creatively\u2014a more realistic view. Who would want to live in a world in which the \"problems\" were one-by-one laid eternally to rest to mark steady progress back to the Garden of Eden? I don't want to live there; do you? I would rather accept that every seeming solution brings a new problem, like progress in lengthening life expectancy breeds overpopulation. It is not the accomplishment, but the search, the struggle, that should excite our interest. And we should, without becoming depressed, be able to face Wordsworth's judgment: \"And much it grieved my heart to think\/What man has made of man.\"\n\nWhat we call civilization has made a little progress toward Emerson's goal of amelioration. But when Emerson asked himself what was the greatest meliorative force operating in the world (he was an optimist and believed there was one), he did not name the usual virtuous activities such as religion, science, education, art. \"Selfish huckstering trade!\" was his answer. It is something to think about. Why do you suppose a perceptive idealist like Emerson would make a judgment like that?\n\n### ANXIETY\u2014LEARN TO LIVE WITH IT\n\nOne of the anomalies of the age we live in is that we are acutely conscious of the destructive effects of anxiety (as it is generally defined, not as a medical term); and we work to alleviate it. Yet my worm's-eye view of things tells me that anxiety is deeply rooted in every facet of the design of our society and that it is exploited at every turn by the best people with the best of intentions.\n\nAnxiety is a part of the human condition. If I were handed a magic wand and given the power to banish anxiety, I would not choose to do it because this is one of the things that gives life its challenge. I see no point in rejecting our society because of its disposition to produce anxiety (or for any other reason). This is the way it is. Just make note of it the way it is and devise your strategy for dealing with it. And it can be dealt with by most people with their own resources; one can learn to live productively with anxiety.\n\nSuch a strategy, as I have tried to live it, sees two somewhat separate _me's_ : the outward me with a deep involvement in the world of affairs, and the inward me\u2014the essential person as viewed from the inside, who is at one with all creation. In that quiet communion when my ladder road is pulled up (and I really pull it up sometimes) and my sign is posted _closed_ to all but me, I am truly at home with a level of serenity that transcends the outer world of stress and conflict and tension which beset all institutions at their best, as well as most human relationships, at least part of the time.\n\nI do not believe there ever has been in this world, certainly there is not now, a promise of outward peace. There is more external stress and confusion and pain in some places than others; but the only true serenity is inward. Serenity is the window through which one looks out on the world of affairs. It is how one feels inside as he engages, with spirit, in the turmoil and strife of the world of affairs.\n\nI prefer to view the inward world as real, and the external world as contrived and transient. When Emerson said that \"a foolish consistency is the hobgoblin of little minds,\" he was probably saying that our mastery of the external world is not such that we can ever really make it fit together. We work with intelligence, courage, and honesty to make the world as tolerable as we can; but we also maintain a detachment from it that puts external achievement, however laudable, in a lower priority than attention to growth in our capacity for wonder.\n\nWhat has wonder to do with anxiety? \"Wonder is the seed of knowledge.\" Wonder is an attitude, it is the filter through which one perceives the world, a filter that tends to substitute moral concern for criticism. It prompts one to ask, \"What is going on here?\" before one acts; and, though the provocation may be extreme, it leads to a response of thoughtfulness\u2014even amusement, rather than of fear, anger, or dismay. It lifts one above the tumult and gives one perspective. And to wonder is humbling, it opens one to learn:\n\n\"I am waiting,\" writes Lawrence Ferlinghetti, \"I am waiting\/for a rebirth of wonder.\"\n\n### A LARGE BUT NOT IMPOSSIBLE TASK\n\nI have urged you to attend to the use of these next three years. You have much to do. It is a large but not impossible task.\n\nIt will never be easy to live your life optimally. There will be obstacles, always. And they will be just as real here on this campus these next three years as they will be later on. What will some of these obstacles be?\n\nFirst, the pressure of time\u2014you will have too much to do. If you are who I have assumed you are, you will always have too much to do. And the kind of concerns I have laid before you will not make it any easier. You will have to choose\u2014life's most difficult task.\n\nThen there will be distractions, plenty of distractions. You will want a normal social life; nobody wants to be a grind. But time wasting and unproductive distractions are as abundant here on this campus as anywhere.\n\nFinally, the environment here these next three years will not be encouraging. The university has many purposes and it cannot attend to anything as individual and personal as your own growth strategy. You will have to make it on your own.\n\nAt this point you may be asking, \"This is a pretty big load you have piled on me, a college freshman. How do I do all of this?\"\n\nI do not have a how-to-do-it formula to give you. I have tried to illuminate the problem from the perspective of my own reflection and experience in the hope that it would help you to design your own strategy for the optimal use of these next three years, a strategy that is congruent with your own unique potentiality. Beyond this, my best suggestion to you is to clarify, for yourself, what you believe about yourself; because, if you seek to go very far in realizing your potential for service, you will be venturing into the dangerous and the unknown and the ever-present anxiety may defeat you if you do not have some kind of faith. Leo Tolstoy, writing his answer to the question, What is faith?\u2014after emerging into relative peace from a rampant early life, posited it for me in one simple sentence. \"I believe,\" he said, \"that the sole meaning of my life lies in living by that light which is within me.\"\n\nIf you can learn, in the stress of circumstance, to pull in your ladder road behind you and put a sign up _closed_ to all but me; and, then, if you can believe that the light within you will guide your path, you are on your way!\n\n### THIS UNIVERSITY\u2014YOUR LABORATORY\n\nYou are attending a good university. Someday it may become a great one. But the paths to institutional greatness are many. Any person in the close constituencies of the university, any one person with his own efforts can help move it toward greatness\u2014if he is persuasive and can lead. And the initiative can come from wherever the strong, able people are\u2014among faculty, administration, trustees, _or_ students. Greatness is best assured when the initiative comes from all four.\n\nThe great university of the future may earn this status by different standards than at present. Whereas, up to now, the great university has been the center of esoteric scholarship and the seat of prestigious graduate schools, a generation from now these resources may be more widely spread. Then, the distinguished university, the one that stands out above the others, may be the one which, in addition to scholarly competence, influences explicitly, directly, and substantially the shape of contemporary society through the preparation of its students for exceptional performance in building and serving and leading the whole range of institutions upon which our complex society depends.\n\nYour university might, in these terms, become one of the great universities of the future. If it becomes so, it will be the result of conscious calculated influence from among its present constituencies. It will not just happen.\n\nWhy don't you freshmen take it on? Take on, for these next three years, the task of being a responsible, effective influence within this university. The test of whether you are responsibly effective is that you leave the institution, three years from now, in better shape to educate, with distinction, the student generations following you\u2014educate them in a manner congruent with the standards of a distinguished institution as it will be judged then, standards which neither you nor I comprehend now.\n\nI leave it with you, as an opportunity for the next three years. You will have the time of your lives if you take it on. And you will emerge from the experience with well-set individual lifestyles of greatness that will carry you for the rest of your days; and you may realize your dream.\n\n#### _Notes_\n\n. \"The Second Tree From the Corner,\" by E. B. White, Harper & Row, 1954.\n\n. _Situation Ethics,_ by Joseph Fletcher, Westminster Press, 1966.\n\n. _Herzog,_ by Saul Bellow, Crest Books, 1964.\n\n. _The Insecurity of Freedom,_ by Abraham J. Heschel, Jewish Publication Society, 1966.\n\n. _Orthodoxy,_ by G. K. Chesterton, Image Books, 1924.\n\n## 5\n\n## **The Servant as\n\nReligious Leader**\n\n### INTRODUCTION\n\nMuch of the literature on leadership deals with those who head great institutions or who leave a mark on history. Such persons can carry their large roles only because many lead effectively in smaller ways that support them. This essay is as much concerned with those who lead small molecular forces\u2014whether as part of a large movement or as lone individuals\u2014as with those whose names go down in history.\n\nThis is written, not as the ultimate treatise on religious leading (I doubt that that will ever be written), but rather to stimulate and contribute to dialogue about the critical issue of religious leading in our times. My perspective is that of a student of organization, not of a scholar or theologian. What I have to share about religious leading is largely what I have gleaned from experience, both my own and others', from reading literature and history, and from thinking. Not much of it has come from formal study of either leadership or religion.\n\nI am a creature of the Judeo-Christian tradition in which I grew up, as modified by the Quaker portion of that tradition that I acquired after maturity. I cannot judge how I would have addressed the subject of religious leading if I had been raised in another culture, or if my life experience had been other than what it was; but I am quite sure that I would have a different view of it. What is written here is offered in the hope that no persons will exclude themselves from consideration of the issues raised because of their religious beliefs or their biases about leadership.\n\nI am deeply grateful to John C. Fletcher and Robert W. Lynn without whose help and encouragement this piece would not have been written.\n\nPart of my excitement in living comes from the belief that leadership is so dependent on spirit that the essence of it will never be capsuled or codified. I was less than a year out of college when I was tapped for what proved to be the most formative experience of my adult life. I had just joined American Telephone and Telegraph Company, and that company's first venture into formal management training in the 1920s was a two-week program called a \"Foremen's Conference.\" This was in the construction and maintenance department of the company, at that time all men. I attended a short intensive program to prepare me to lead foremen's conferences. Then, for the next year, every other Monday morning I received a group of twelve foremen. With no reading and no published agenda (I had one), for two weeks we sat around a big table and just talked about the multifaceted job of being a foreman, the lowest level of management. The pedagogical theory was that these men would learn from each other\u2014not from me. But I was the chief learner. There was not then, and there seems not to be now, much hard knowledge about what proved to be the major focus of my work: how things get done in organized efforts. This is a fact of life with which I have learned to be comfortable.\n\nMy conferees ranged in age from 30 to nearly 70. I was 23. These fellows were where the buck stopped. The elaborate management hierarchy above them could think their great thoughts about what they wanted done; but what _was_ done was what those foremen who sat around my conference table were willing and able to do. I learned much from these wise and seasoned men.\n\nThis was my graduate education; it set the course of my life. During the 55 years since this intense formative experience, I have been deeply involved with people who were trying to lead or manage something. In the last few years before retirement from AT&T, I held the position of Director of Management Research, a post that brought me in close contact with top management and gave me wide latitude, with the help of a professional staff, to examine how this giant company did its work, including the values that guided it. There was ample opportunity to communicate our findings when we learned something that might be useful. The tradition of the company disposed it to listen carefully to research findings in all relevant fields.\n\nIn the nearly 20 years since \"retirement,\" my work has expanded to include a range of businesses, large and small, foundations, universities, and churches and church-related institutions including seminaries\u2014in the United States, in Europe, and in the Third World. Some of my best years have been in retirement.\n\nI cannot give a precise logical explanation for how I have come, near the end of my active career, to center my attention on _religious leading_ , but I believe the reader may discover enough of it from what I have chosen to discuss here. I have said that leading is so dependent on _spirit_ that the essence of it will never be capsuled or codified. Part of that essence lies beyond the barrier that separates mystery from what we call reality.\n\n_Spirit_ , as the animating force in living beings, is value-free. Hitler had it; he was a great, if demonic, leader. Putting value into it, in my judgment, makes it religious. And what is value? Again, I will leave it to the reader to judge what I value. In my intense formative experience with these rough-hewn, sometimes crude, foremen, I realized that what enabled them on occasion to lead (not just to use their authority, but _lead)_ under difficult and sometimes dangerous conditions was that they were able to communicate to other rough-hewn men what they, the foremen, valued. Some of what I learned about leadership in that experience was how to create the conditions in which they would talk freely to each other about what they valued.\n\nThe premise here is that _to lead is to go out ahead and show the way when the way may be unclear, difficult, or dangerous_ \u2014it is not just walking at the head of the parade\u2014and that one who leads effectively is likely to be stronger, more self-assured, and more resourceful than most because leading so often involves venturing and risking. Occasionally something important happens when there is no discernible leader who prompted it, but usually there are persons who take the initiative to say, \"Let's go here, or do this,\" or some may lead with a subtle inconspicuous gesture. Either way, what makes them leaders is that a significant force of people responds.\n\nFew, if any, who have these qualities of strength, assurance, and resourcefulness are equally effective as leaders in all situations. Therefore, even the ablest leaders will do well to be aware that there are times and places in which they should follow. And one who seems deficient in one or more of these qualities may, on some occasion, rise to save the day. But, in general, those who lead well in affairs both large and small, and who sustain their leadership in good times and bad, are exceptional people, an elite. They may not be \"high status\" people by the usual criteria. The best trustee I have seen in action (and I have seen quite a few) was not a high status person, but he was truly exceptional as a trustee, and remarkable in the chair.\n\nIt is further premised that what distinguishes a leader as _religious_ (in its root meaning of _religio_ \u2014to bind or rebind) is the quality of the consequences of her or his leadership. Does it have a healing or civilizing influence? Does it nurture the servant motive in people, favor their growth as persons, and help them distinguish those who serve from those who destroy?\n\nCountless persons who lead by these criteria, in large ways and small, are judged religious. At times in the past the combined influence of all such persons was not sufficient to check destructive tendencies that are always at work. It was not sufficient in Germany in the 1930s. A religious leader simply makes his or her best effort to build and sustain a good society. The result will be whatever it is. It is not within the power of any of us mortals to determine the outcome (fortunately). A \"good\" society is seen as one in which there is widespread _faith as trust_ (to be discussed later) that encourages and sustains ordinary, good people as constructive influences in the world as it is\u2014violent, striving, unjust as well as beautiful, caring, and supportive.\n\nAmong many facets of a \"good\" society that might be achieved with finite resources are: the opportunity for as many as possible to engage in useful and remunerative work\u2014with the feeling of belonging and being a part of a constructive effort where they are; children get good preparation for a life of service; strong young people are encouraged and prepared for religious leadership; health is encouraged and the environment is protected; the needy, the aged, and the disabled are cared for. There are enough able people to give this care and prepare the young for service and leadership, and their lives would be more rewarding for doing it\u2014 _if_ these able people could be brought together as an effective force. This is the essence of religious leading as that term is used here: to bring people together and sustain them as an effective force for the building of faith as trust under conditions in which powerful forces may be operating to destroy that faith.\n\nReligious leading is a vast subject. I have ventured here to deal only with that portion that nurtures and supports the religious motive in those who, as they do the work of the world, will labor to build a good society in terms like those given above.\n\nThe central idea of this essay is: _work to increase the number of religious leaders who are capable of holding their own against the forces of destruction, chaos, and indifference that are always with us!_ Those who are in the vanguard of this effort will find ways to _strengthen the hands of the strong_ by helping them, while they are young, to acquire a vision of themselves as effective servants of society, plus an awareness of both the opportunities and the pitfalls for those who would be such servants, and a clear perception of what it takes to lead\u2014in religious terms. This is written as a sharing with those who aspire to be in the vanguard of that work.\n\n### ORIGINS\n\nMuch of the thinking that culminates in this essay emerged during the stressful years of student unrest in the 1960s when I was deeply immersed in the traumatic experiences of several universities in which the fragility of those prestigious institutions was exposed. Against the background of my knowledge of universities of that period, all of the student attitudes, destructive as they sometimes were, did not seem irrational. I wondered whether, in view of the flaws revealed in universities, ours is a sufficiently caring, serving society to endure. My conclusion was that if it is to endure, something has to change! And the most significant change of all might take place in churches because, unless churches become more effective, it is unlikely that people and institutions will do much better than at present. And that is not good enough. As I see it now, churches are not likely to become more effective unless seminaries develop a capacity to lead them that they do not seem to have. The big change of the future, if it comes, may be initiated by seminaries.\n\nReflections on these issues led me, in 1970, to begin to write on the _servant_ theme. Three essays that suggested some things that might change issued from that effort: \"The Servant as Leader,\" \"The Institution as Servant,\" and \"Trustees as Servants.\" These essays, and some related writings, were later gathered in a book, _Servant_ _Leadership_ (Paulist Press), followed by _Teacher as Servant, a Parable_ (The Greenleaf Center). In 1980 another essay, \"Servant, Retrospect and Prospect,\" summarized my experience in working with these ideas over a ten-year period. In 1981 _Seminary as Servant, Essays on Trusteeship_ was published. This reflected where my thinking had come regarding the strategic role available to seminaries. I understand the misgivings that many informed people (including some in seminaries) have about seminaries. But I am convinced that seminaries occupy a spot from which a profound influence for the good of society could be wielded\u2014and it is not forthcoming. Too many who are in a position to help seminaries have written them off. I contend that the service needed from seminaries must be forthcoming from existing seminaries because we do not have the time or the resources to replace them with adequately serving institutions. It is imperative that a way be found to raise existing seminaries to the full stature that their position requires. This is a prime challenge for religious leaders of our time. I will return to this issue later.\n\nThese writings had some circulation among religious institutions, and I have had several close relationships with pastors, church administrators, lay leaders, governing boards, and congregations. In these experiences, I found considerable concern for the present state of leading in churches and for a consequent diminished influence of churches on their members and on society at large. Paralleling these reported findings from within churches, I made my own observations during this period from involvements with businesses, foundations, and universities. All of these soundings, plus what is available in the news, suggest a deteriorating society with little evidence of effective restorative forces at work.\n\nOne could simply view this with alarm and join the chorus of lament that one hears from some who have made similar observations. I prefer to turn my energies to the support of efforts that might set restorative forces in motion. All that any individual can do is to make one's best effort, alone or in concert with others\u2014now!\n\nI begin with the assumption that the enormous resources of religious institutions in the United States could, within a generation, help turn this faltering society around and start us on a long-term constructive course _if_ a substantial number of them, each from its own unique set of beliefs, could work toward a common goal of a \"good\" society.\n\nIs it not possible that many religious institutions, as they now stand, have within them the human resources from which effective leaders might evolve who would move institutions in which they have some influence to become more constructive elements of a good society? Could a new, persuasively articulated, prophetic vision generate the faith required for those who have the potential to lead to take the risks, develop the strength, and make a new determined effort to lead? If seminaries design programs that will attract the strongest and ablest young people, could they not become the chief source of wisdom about religious leaders, not only for churches but for all segments of society? This essay is written with the hope of contributing to a dialogue out of which a solid conceptual base for such an effort may emerge.\n\nThe phrase, \"Generate the faith required\" seems to me to be the key. A suggested view of the nature and role of _faith_ in religious leaders will be discussed later.\n\n\"But is this not the age of the anti-leader!\" some protest. \"People simply do not respond to leadership in these times.\"\n\nSo it seems to be, in the conventional terms in which much leadership is now offered. Could it be that what is called for now is new language, new concepts, new skills\u2014all of which may be needed if we are to have a quality of leading that will be effective in our times? Can those who are moved to make common cause pool their resources in a creative effort to produce new forms of religious leadership which will be accepted by contemporary people as realistic and useful? In the hope that others will respond and contribute, I offer what my experience suggests\u2014all from my perspective of a student of organization.\n\nWho is the religious leader?\n\n### THE RELIGIOUS LEADER AS A PERSON\n\nOurs is a stressful world, and both people and institutions are fragile. Anyone who was involved as I was with universities in the 1960s is sharply aware that both people and institutions are fragile. They break easily.\n\nAll but the crude and insensitive live under the constant threat of coming unbound, alienated. _Alienated_ , in these pages, designates those who have little caring for their fellow humans, who are not motivated to serve people as individuals or as institutions, and who, though able, do not carry some constructive, society-supportive role, or who miss realizing their potential by much too wide a margin. Any influence or action that rebinds\u2014that recovers and sustains such alienated persons as caring, serving, constructive people, and guides them as they build and maintain serving institutions, or that protects normal people from the hazards of alienation and gives purpose and meaning to their lives, is religious. And any group or institution that nurtures these qualities effectively is a church, synagogue, mosque, or temple. Both of the words _religion_ and _church_ may have additional or differing meanings for individuals and groups. What is suggested here is offered as a common basis for viewing religious leadership among people of many differing beliefs in separate churches.\n\nTogether, as _religious leadership_ , these two words are used here to describe actions taken to heal, or build immunity from, two serious contemporary maladies: (1) widespread alienation in all sectors of the population, and (2) the inability or unwillingness to serve on the part of far too many of the institutions, large and small, that make up our complex society. Each of these maladies is seen, in part, as a cause of the other, and neither is likely to be healed without coming to terms with the other. The test of the efficacy of religious leadership is: does it cause things to happen among people, directly or indirectly, that heal and immunize from maladies like these two?\n\nOne test of any kind of leadership is: Do leaders enjoy a mutual relationship with followers? Are these followers numerous enough and constant enough to make an effective force of their effort? The leader, if in fact a leader, is always attached to an effective force of people. Among those who are normally followers are those who, from time to time, sometimes in major ways, will also lead. The titular leader gives continuity and coherence to an endeavor in which many may lead.\n\nAn additional test for religious leaders may be: are they seekers?\n\nWe seem today to have few prophetic voices among us. It is possible (I think quite likely) that they are here and speaking as eloquently to the problems of today as the greatest in any age. The lack in our times may be a paucity of seekers who have the critical judgment required to test the authenticity of a prophet. _Is anybody listening_? Are able and discriminating persons listening, people who are willing to work hard to get the skills, to put forth the effort, and take the risks to lead? If a prophet speaks and is not heard, the prophet's vision may wither away.\n\nAn important aspect of religious leadership is the nurture of seekers. The religious leader may not have the persuasive ability to put power behind a particular prophet's vision, but she or he may be able to sustain the spirit of seekers and encourage them to listen so that they will respond to prophets who are speaking in contemporary, realistic, and perhaps hard-to-take terms.\n\nProphet, seeker, and leader are inextricably linked. The _prophet_ brings vision and penetrating insight. The _seeker_ brings openness, aggressive searching, and good critical judgment\u2014all within the context of the deeply felt attitude, \"I have not yet found it.\" The _leader_ adds the art of persuasion backed by persistence, determination, and the courage to venture and risk. The occasional person embodies all three. Both prophet and leader are seekers first. But in religious leadership, as the term is used here, persuasion is only as effective as the quality of the prophetic vision that inspires it and infuses it with spirit. In the end, the quality of prophetic vision shapes the quality of society. The religious leader who is not a prophet is but the instrument of whatever vision is available. The effective religious leader, like other leaders, is apt to be highly intuitive in making judgments about what to do and what not to do. Such a leader also draws heavily on inspiration to sustain spirit. But intuitive insight and inspiration are not apt to be dependable guides in an ignorant, uncritical, or unreflective person. Careful analytical thought, along with knowledge and reflection, provides a check and a guide to intuition and inspiration, gives a solid basis for communicating with informed and prudent people, and offers a framework of assurance to those who would follow.\n\nThis discussion of religious leadership presumes that the effective leader will have firm beliefs, whether explicit or implicit, but no particular beliefs are postulated as a condition of being accepted as a religious leader. _Any_ set of beliefs that undergirds a leader who causes things to happen among people, directly or indirectly, that heal or immunize from the two pervasive maladies named above, is accepted as having validity that warrants respect. Such acceptance permits people of good will, but with widely differing beliefs, to work in concert as religious leaders on matters that make for a good society.\n\n### LEADERSHIP TOWARD FAITH AS TRUST\n\nIt may not be possible to find a basis for all people of widely varying beliefs to work together toward a \"good\" society, but it is hoped that enough of them can find common ground to give the culture more solidity and resiliency than it now seems to have. What is required is that enough people who hold differing beliefs can accept a common definition for religion as is suggested here: any influence or action that rebinds or recovers alienated persons as they build and maintain serving institutions, or that protects normal people from the hazards of alienation and gives purpose and meaning to their lives, is religious.\n\nIn this context, theology is seen as the rational inquiry into religious questions supported by critical reflection on communal concerns. Such an inquiry into the influences and actions that do rebind, with results like those named in the paragraph above, may yield the basis for beliefs that make religious leadership possible. And what would we like to see them rebound into? An integrated, loving, caring community? For this to happen on a large scale (and the needs of contemporary society in this regard are large scale), what theologians think about will need to evolve in a way that provides seminaries with both the ideas and language with which to lead.\n\nTo the extent that traditional beliefs now attract and hold followers when persuasively advocated, they provide this unity of faith as trust. The disturbing lack of solidity and resiliency in contemporary society may be due to the failure of traditional beliefs to provide the basis for faith as trust in enough people, even in some who \"belong\" to churches and profess their creeds. Some of the latter, deep down inside, may be just as alienated.\n\nThe challenge to contemporary religious leaders, both those professionally or otherwise engaged in churches and those who lead in other settings, is to establish, in contemporary terms, through rational inquiry and prophetic vision, beliefs that sustain those actions and influences that do in fact rebind, heal alienation in persons, and render institutions more serving. Prudence suggests that the central thrust of this effort may be to evolve these beliefs within existing institutions, since an urgent need of our times is for beliefs that unify rather than further fragment society. The first concern of religious leaders may be to learn to rebuild existing institutions as serving rather than to abandon old ones and create new ones. Such leaders may prefer evolution to revolution, persuasion to coercion and manipulation, and gradual to precipitous change. But some change is imperative. Far too many institutions (including some churches) are failing to serve adequately.\n\n_Faith_ is a many faceted concept, a subtle and complex notion. It is discussed here only in relation to leadership. What can it mean in a leader who is successful in bringing and holding together an effective force of people? It may be that it is communicated confidence that a mutually agreed-upon goal can be reached and is worth achieving. This confidence sustains the will to persevere and contend with the inevitable vicissitudes. Such a definition would, of course, fit the successful leader of a gang of thieves. One large step away from this broad definition would insert some clear dimension of justice or mercy as the goal. But would such a qualification, by itself, merit the label religious? I think not. How can we define religious leading without specifying a particular theology or set of beliefs? It may require an operational definition such as I suggested earlier for _religious_.\n\nThis definition involves the word _serve_. In my first essay, \"The Servant as Leader,\" I suggested that the servant-leader is servant first. It begins with the natural feeling that one wants to serve, to serve _first_. Then conscious choice brings one to aspire to lead. Such a person is sharply different from one who is a _leader_ first, perhaps because of a need to assuage an unusual power drive or to acquire material possessions. The difference manifests itself in the care taken by the servant, first to make sure that other people's highest priority needs are being served. The test I like best, though difficult to administer, is: Do those served grow as persons; do they, _while being served_ , become healthier, wiser, freer, more autonomous, more likely themselves to become servants? And, what is the effect on the least privileged person in society; will she or he benefit, or at least, not be further deprived? No one will knowingly be hurt, directly or indirectly.\n\nFaith, to one who aspires to be in the vanguard of the effort to increase the number of effective religious leaders in our time, requires willingness to deal with the issue of _strengthening the hands of the strong_.\n\n### STRENGTHEN THE HANDS OF THE STRONG\n\nEarlier I suggested that those who are in the vanguard of the effort to increase the number of religious leaders will find ways to _strengthen the hand of the strong_ by helping them, while they are young, to acquire a vision of themselves as effective servants of society.\n\nStrengthen the hands of the strong! Who is strong and how does one strengthen that person's hands? I can only speculate.\n\nIn addition to the more ponderable qualities of competence, stability, resiliency, and values, there are the elusive ones of a sense of the unknowable, contingency thinking, and foresight. All of these are best strengthened by experience. A mentor leads a potential leader into a high risk situation and asks, What do you sense or see? What might happen that would threaten you, but rarely does? What is likely to happen next, or down the road? The person who, in the heat of action, can accurately and promptly sense what is going on, is prepared for what might happen but rarely does, and foresees what is likely to happen next, is _strong_.\n\n### A SENSE OF THE UNKNOWABLE\u2014BEYOND CONSCIOUS RATIONALITY\n\nThe requirements of leadership impose some intellectual demands that are not usually measured by academic intelligence ratings. They are not mutually exclusive, but they are different things. The leader needs three intellectual abilities that may not be assessed in an academic way: one needs to have a sense for the unknowable, to be prepared for the unexpected, and to be able to foresee the unforeseeable. The leader knows some things and foresees some things which those one is presuming to lead do not know or foresee as clearly. This is partly what gives the leader his \"lead,\" that puts him out ahead and qualifies him or her to show the way.\n\nAs a practical matter, on most important decisions there is an information gap. There usually is an information gap between the solid information in hand and what is needed. The art of leadership rests, in part, on the ability to bridge that gap by intuition, that is by a judgment from the unconscious process. The person who is better at this than most is likely to emerge the leader because he contributes something of great value. Others will depend on him to go out ahead and show the way because his judgment will be better than most. Leaders, therefore, must be more creative than most; and creativity is largely discovery, a push into the uncharted and the unknown. Every once in a while a leader finds himself needing to think like a scientist, an artist, or a poet. And his thought processes may be just as fanciful as theirs\u2014and as fallible.\n\nIntuition is a feel for patterns, the ability to generalize based on what has happened previously. The wise leader knows when to bet on these intuitive leads, but he always knows that he is betting on percentages\u2014his hunches are not seen as eternal truths.\n\nTwo separate \"anxiety\" processes may be involved in a leader's intuitive decision, an important aspect of which is timing, the decision to decide. One is the anxiety of holding the decision until as much information as possible is in. The other is the anxiety of making the decision when there really isn't enough information\u2014which, on critical decisions, is usually the case. All of this is complicated by the pressures building up from those who \"want an answer.\" Again, trust is at the root of it. Has the leader a really good information base (both hard data and sensitivity to feelings and needs of people) and a reputation for consistently good decisions that people respect? Can he defuse the anxiety of other people who want more certainty than exists in the situation?\n\nIntuition in a leader is more valued, and therefore more trusted, at the conceptual level. An intuitive answer to an immediate situation, in the absence of a sound governing policy, can be conceptually defective. Overarching conceptual insight that gives a dependable framework for decisions (so important, for instance, in foreign policy) is the greater gift.\n\n### CONTINGENCY THINKING\n\n\"It is the unexpected that most breaks a man's spirit.\"\n\n_Pericles_\n\n_Foresight_ is anticipating what is likely to happen and taking precautionary steps. _Contingency thinking_ relates to things that might happen but rarely do. Sometimes the latter appear as emergencies to which there is a preset response. Part of the confidence of followers in a leader rests on the belief that the leader will not be surprised by the unusual and will act promptly in response to it. Let me give two examples out of my experience that are vivid in my memory, although they happened long ago.\n\nAbout 50 years ago, my wife and I were attending a concert in Carnegie Hall in New York in which the Boston Symphony Orchestra was playing under the direction of Serge Koussevitsky, the Russian-born conductor. The orchestra was well into their program when a great billow of smoke rolled out under the proscenium, someone shouted FIRE, and a full blown panic was almost instantly under way with shouting, running, and pushing\u2014in the orchestra floor and boxes. We were sitting in the front row on the side of the top balcony, high up with a good view of the whole auditorium. Nobody got excited up where we were.\n\nIn a matter of seconds Koussevitsky stopped the orchestra, spun around and in a loud voice that could be heard above the tumult he ordered sternly, \"sit down, everything will be all right.\" He stood there motionless with his arms upraised as he looked reprovingly at the unruly audience. The shouting stopped but some people continued to go out in an orderly way. The orchestra members sat motionless, the violinists with their instruments poised on their knees as if a soloist were playing a cadenza. If one of them had broken and run, there might have been a serious panic.\n\nSoon a man appeared from the wings and announced that there had been a small fire in a paper bailer under the stage which was quickly extinguished. The smoke had stopped. But the wail of the sirens on the arriving fire fighting equipment suggested that the fire had been taken seriously.\n\nThe point in relating this incident is that the speed, forcefulness, and rightness of Koussevitsky's response suggest that he had anticipated this emergency and that his response was firmly preset. A few seconds delay to think it over and the panic might have been out of hand.\n\nSuch occasions are so rare that an orchestra conductor may go through a whole career and not once confront a situation of the gravity that I witnessed. Yet if it suddenly comes, and if one is to deal with it prudently as Koussevitsky did, one must have firmly preset one's response.\n\nThe concert resumed for those who remained, which was everybody up in the second balcony where we were\u2014we were all too fascinated by the spectacle of the uproar down below among those rich people in their fancy evening clothes, about half of whom left and did not return. It concluded with a great ovation for the conductor and the orchestra. They were interrupted in that melodious and much-played second movement of Tchaikovsky's Fifth symphony. Every time I hear that music, the memory of that dramatic incident is refreshed.\n\nThe news report next day gave Koussevitsky's answer to a reporter's question of how he managed to keep his tempo after a shakeup like that. \"Tempi are tempi,\" he said, \"and tranquility is tranquility.\" He was not thrown by the unexpected.\n\nThe second incident, about 35 years ago, was when I had occasion to pull the emergency cord on a New York subway train to save a man's life. It was the only time in 40 years of riding those subways that I saw the cord pulled.\n\nI was seated in the center of a well-filled car in which quite a few were standing, at about the middle of a ten-car train. As the train took off from a station, accelerating as they do at a good clip, a commotion broke out at the door in the forward end of the car where a man had tried to enter as the doors closed and, from the outside, one arm was firmly hooked in the door because of a cuff on a heavy overcoat. The guards who are supposed to look over the train when it takes off had missed it and he was being dragged along to his almost certain death if the train was not stopped before he reached the end of the platform.\n\nThe crowd inside was tugging at the door and shouting, \"pull the emergency cord!\" A cord hung right over their heads, but nobody pulled it. I could not see because of people standing, and it was several seconds before I realized that it was up to me to pull the cord in the other end of the car. I ran as fast as I could, bowling over a couple of people as I went, and got to the cord just in the nick of time. When the train stopped the man was about 25 feet from the end of the platform.\n\nThis incident was the cause of considerable reflection. There were about 60 people in that car who were closer to one of those two cords than I was, and some of them were closer to the incident and were alerted sooner than I was. I was not aware that any of them made a move to pull the cord. Why was it up to me to do it? I concluded that I was probably the only one in that car who knew beforehand where those cords were and had preset the response that someday it might be up to me to pull one. Those cords are deliberately not made too conspicuous. If one does not know beforehand where they are, in the heat of the emergency it is not likely that one would discover them by looking around\u2014not in time. So I performed an experiment.\n\nAs I met New Yorkers over the next few months, persons who used the subways frequently, I would describe the incident. Then I would say, \"There are three different subway systems in New York, and the cords are in a different location in each. This is the Independent System. You have six seconds, Where is the cord? One-two-three-four-five-six; you're too late; the man is gone.\" Usually there was a protest, \"Your counting put me under pressure and I couldn't think.\" To which I would respond, \"In the actual event, I was under pressure too. But take your time, where is the cord?\" I tried this on about 25 people without finding one who could say where the cord was, given indefinite time. I concluded that contingency thinkers are a bit rare, but they need not be if the formative years of young people included some preparation for it.\n\nI learned to be a contingency thinker from my father, who was good at it and to whom I was very close. He was a contemporary of G. K. Chesterton, but I doubt that he read him. He would, however, have agreed with what Chesterton had to say in the following paragraph.\n\nThe real trouble with this world is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait. ( _Orthodoxy_ , 1908)\n\n### FORESIGHT\u2014THE CENTRAL ETHIC OF LEADERSHIP\n\nMachiavelli, writing 300 years ago about how to be a prince, put it this way: \"Thus it happens in matters of state; for knowing afar off (which it is only given a prudent man to do) the evils that are brewing, they are easily cured. But when, for want of such knowledge, they are allowed to grow so that everyone can recognize them, there is no longer any remedy to be found.\"\n\nThe shape of some future events can be calculated from trend data. But, as with a practical decision mentioned earlier, there is usually an information gap that has to be bridged, and one must cultivate the conditions that favor intuition. This is what Machiavelli meant when he said that \"knowing afar off\u2014which is only given a prudent man to do.\" The prudent man is he who constantly thinks of \"now\" as the concept in which past, present moment, and future are one organic unity. And this requires living by a sort of rhythm that encourages a high level of intuitive insight about the whole gamut of events from the indefinite past, through the present moment, to the indefinite future. One is at once, in every moment of time, historian, contemporary analyst, and prophet\u2014not three separate roles. This is what the practicing leader is, every day of his life.\n\nLiving this way is partly a matter of _faith_. Stress is a condition of most of modern life, and if one is a servant-leader and carrying the burdens of other people, going out ahead to show the way, one takes the rough and tumble (and it really is rough and tumble in some leader roles). One takes this in the belief that, if one enters a situation prepared with the necessary experience and knowledge at the conscious level, _in the situation,_ the intuitive insight necessary for one's optimal performance will be forthcoming. Is there any other way, in the turbulent world of affairs (including the typical home), for one to maintain serenity in the face of uncertainty? One follows the steps of the creative process which require that one stay with conscious analysis as far as it will carry one, and then withdraws, releases the analytical pressure, if only for a moment, in full confidence that a resolving insight will come. The concern with the past and future is gradually attenuated as this span of concern goes forward or backward from the instant moment. The ability to do this is the essential structural dynamic of leadership.\n\nThe failure (or refusal) of a leader to foresee may be viewed as an _ethical_ failure, because a serious ethical compromise today (when the usual judgment on ethical inadequacy is made) is sometimes the result of a failure to make the effort at an earlier date to foresee today's events and take the right actions when there was freedom for initiative to act. The action which society labels \"unethical\" in the present moment is often really one of no choice. By this standard, a lot of guilty people are walking around with an air of innocence that they would not have if society were able always to pin the label \"unethical\" on the failure to foresee and the consequent failure to act constructively when there was freedom to act.\n\nForesight is the \"lead\" that the leader has. Once he loses this lead and events start to force his hand, he is leader in name only. He is not leading. He is reacting to immediate events and he probably will not long be a leader. There are abundant examples of loss of leadership which stemmed from a failure to foresee what reasonably could have been foreseen, and from failure to act on that knowledge while the leader had the freedom to act.\n\nPericles, in the speech quoted earlier, said to the Athenians who had chosen him to lead, \"You took me to be what I think I am, superior to most in foresight...\".\n\nHow to achieve foresight if one is not born with it? One begins, I suppose, by recognizing the importance of it. And, if one does not have it, then stay close to someone who does have it. Maybe some of it will rub off. I doubt that the gift of foresight can be taught in a course.\n\nRequired, I believe, is that one live a sort of schizoid life. One is always at two levels of consciousness: one is in the real world of the present\u2014concerned, responsible, effective, value-oriented. One is also detached, riding above it, seeing today's events, and seeing oneself deeply involved in today's events, in the perspective of a long sweep of history and projected into the indefinite future. Such a split enables one better to foresee the unforeseeable. Also, from one level of consciousness, each of us acts resolutely from moment to moment on a set of assumptions that then govern one's life. Simultaneously, from another level, the adequacy of these assumptions is examined, in action, with the aim of future revision and improvement. Such a view gives one the perspective that makes it possible to live and act in the real world with a clearer conscience.\n\n### WHAT GIVES STRENGTH TO THE STRONG?\n\nWhat gives the strong their strength? The three preceding sections, \"A Sense of the Unknowable,\" \"Contingency Thinking,\" and \"Foresight,\" suggest dimensions of the inner resources of a leader that support self-confidence and that build confidence in followers. Why would anyone follow the leadership of another unless one has confidence that the other knows better where to go? And how would one know better where to go unless one has a wider than usual awareness of the terrain and the alternatives, unless one is well armored for the unexpected, and unless one's view of the future is more sharply defined than that of most? Also, one's confidence in a leader rests, in part, on the assurance that stability and poise and resilience under stress give adequate strength for the rigors of leadership. All of the above stand on a base of intensity and dedication to service that supports faith as trust.\n\nEarlier it was suggested that faith might be viewed as communicated confidence that a mutually agreed-upon goal can be reached and is worth achieving, and that builds the sustaining will to persevere and contend with the inevitable vicissitudes. These may be subliminal things. And they may breed a feeling of trust by followers in the dependability of the inner resources of the leader as suggested in the preceding sections. Is not, then, _faith as trust_ in a religious leader rooted in a firm sense of the dependability of the inner resources of one who influences or takes actions that rebind?\n\nCould we say, then, that part of the religious leader's own faith is trust in his or her own inner resources? If one is to take the risks of leadership (and all significant leadership entails venturing and risking), one needs to trust one's inner resources, _in the situation_ , to give the guidance one needs to justify the trust of followers. One cannot know _before_ one ventures to assume leadership what the markers on the course will be or that the course one will take is safe. To know beforehand would make the venture risk-free. One has confidence that, _after_ one is launched in the venture, the way will be illuminated. The price of some illumination may be the willingness to take the risk of faith. Followers, knowing that the venture is risky, have faith as trust in this communicated confidence of the leader.\n\nSome may speculate on what lies beyond the inner resources of the leader, but leader and follower may or may not share these speculations. For faith as trust to be real, even in a religious leader, it suffices that the inner resources of the leader are known by both leader and follower to be dependable. The test: _a leader feels strong and is accepted by followers as stronger than most_.\n\nI have listened in recent years to many in responsible positions in religious institutions as they have discussed what they called their leadership problems. The following are some of my impressions:\n\n\u2022 change a few words, and they sounded no different from the harried executives in other institutions that I have been listening to all my life.\n\n\u2022 most of what they called _lead_ I would label _manage_ , _administer_ , or _manipulate_. I was not aware of much leading\u2014as defined here.\n\n\u2022 the main problem revealed was a lack of faith\u2014as it would relate to leading. What they called faith seemed to me to be mostly belief in certain doctrinal positions. There was little evidence of deep inner resources\u2014that they trusted and that others would trust\u2014which would be the basis for confidence in their ability to attract and hold followers in a high risk venture. They were mostly \"safe\" people.\n\nIn short, too many of these good people seemed to lack enough faith to lead. As I once put it bluntly in an off-the-record session with such persons, \"You seem not to believe your own stuff.\"\n\n### RELIGIOUS LEADERS IN ACTION\n\nIf we accept definitions like those given above, what can be said about the person who might be giving effective religious leadership today?\n\nAnything said in answer to that question is indeed a speculation. Religious leaders in the future, like all sorts of leaders in the past, will probably be many different types of people. And they may evolve in ways that we cannot now foresee. They are more likely to emerge if there is an expectancy, and an awareness and acceptance of the probability that they may not resemble any type that we can now imagine. Let me select from the past, descriptions of two quite dissimilar examples of persons whose work has impressed me as I have reflected on the question above.\n\n#### _John Woolman_\n\nLeaders work in wondrous ways. Some assume great institutional burdens, others quietly deal with one person at a time. Such a man was John Woolman, an American Quaker who lived through the middle years of the 18th century. He is known to the world of scholarship for his journal, a literary classic. But in the area of our interest, leadership, he is the man whose great contribution was to help rid the Religious Society of Friends (Quakers) of slaves.\n\nIt is difficult now to imagine the Quakers as slaveholders, as indeed it is difficult now to imagine anyone being a slaveholder. One wonders how the society of 200 years hence will view \"what man had made of man\" in our generation. It is a disturbing thought.\n\nBut many of the 18th-century American Quakers were affluent, conservative slaveholders, and John Woolman, as a young man, set his goal to do what he could to rid his beloved Society of this terrible practice. Thirty of his adult years (he lived to age 52) were largely devoted to this. By 1770, nearly 100 years before the Civil War, no Quakers held slaves. His method was unique. He didn't raise a big storm about it or start a protest movement. His method was one of gentle but clear and persistent persuasion\u2014largely one person at a time.\n\nAlthough John Woolman was not a strong man physically, he accomplished his mission by journeys up and down the East Coast by foot or horseback visiting slaveholders, over a period of many years. The approach was not to censure the slaveholders in a way that drew their animosity. Rather the burden of his approach was to raise questions: What does the owning of slaves do to you as a moral person? What kind of an institution are you binding over to your children? Man by man, inch by inch, by persistently returning and revisiting and pressing his gentle argument over a period of 30 years, he helped to remove slavery from this Society, the first religious group in America formally to denounce and forbid slavery among its members. One wonders what would have been the result if there had been 50 John Woolmans, or even 5, traveling the length and breadth of the Colonies in the 18th century, persuading people one by one with gentle, nonjudgmental argument that a wrong should be righted by individual voluntary action. Perhaps we would not have had the war with its 600,000 casualties and the improverishment of the South, and with the resultant vexing social problem that is with us a century later with no end in sight. Some historians hold now that just a slight alleviation of the tension in the 1850s might have avoided the war. A few John Woolmans, just a _few_ , might have made the difference. Leadership by persuasion has the virtue of change by convincement rather than coercion. Its advantages are obvious.\n\nJohn Woolman exerted his leadership in an age that must have looked as dark to him as ours does to us today. We may easily write off his effort as a suggestion for today on the assumption that the Quakers were ethically conditioned for this approach. The Quakers of Woolman's day were not pushovers on this issue. All persons are so conditioned, to some extent\u2014enough to gamble on.\n\n#### _Nikolai Grundtvig_\n\nNikolai Frederik Severin Grundtvig, whose adult life was the first three-quarters of the 19th century, is known as the father of the Danish Folk High Schools. To understand the significance of the Folk High School, one needs to know a little of the unique history of Denmark. Since it is a tiny country, not many outside it know this history, and consequently Grundtvig and his seminal contributions are little known. A great church dedicated to his memory in Copenhagen attests the modern Danish awareness of what he did for them\u2014but he was not widely applauded in his time.\n\nAt the beginning of the 19th century, Denmark was a feudal and absolute monarchy. It was predominantly agricultural, with a large peasant population of serfs who were attached to manors. Early in the century, reforms began which gave the land to the peasants as individual holdings. Later the first steps toward representative government were taken.\n\nA chronicler of those times reports, \"The Danish peasantry at the beginning of the nineteenth century was an underclass. In sullen resignation it spent its life in dependence on estate owners and government officials. It was without culture and technical skill, and it was seldom able to rise above the level of bare existence. The agricultural reforms of that time were carried through without the support of the peasants, who did not even understand the meaning of them.... All the reforms were made for the sake of the peasant, but not by him. In the course of the century this underclass has been changed into a well-to-do middle class which, politically and socially, now takes the lead among the Danish people.\" (From The _Folk High Schools of Denmark_ , by Begtrup, Lund, and Manniche, Oxford University Press, 1926.)\n\nFreedom\u2014to own land and to vote\u2014was not enough to bring about these changes. A new form of education was envisioned by Grundtvig explicitly to achieve this transformation. Grundtvig was a theologian, poet, and student of history. Although he himself was a scholar, he believed in the active practical life, and he conceptualized a school, the Folk High School, as a short intensive residence course for young adults dealing with the history, mythology, and poetry of the Danish people. He addressed himself to the masses rather than to the cultured. The \"cultured\" at the time thought him to be a confused visionary and contemptuously turned their backs on him. But the peasants heard him, and their natural leaders responded to his call to start the Folk High Schools\u2014with their own resources.\n\n\"The spirit (not knowledge) is power.\" \"The living word in the mother tongue.\" \"Real life is the final test,\" as contrasted with the German and Danish tendency to theorize. These were some of the maxims that guided the new schools of the people. For 50 years of his long life, Grundtvig vigorously and passionately advocated these new schools as the means whereby the peasants could raise themselves into the Danish national culture. And, stimulated by the Folk High School experience, the peasant youth began to attend agricultural schools and to build cooperatives on the model borrowed from England.\n\nTwo events provided the challenge that matured the new peasant movement and brought it into political and social dominance by the end of the century. There was a disastrous war with Prussia in 1864, which resulted in a substantial loss of territory and a crushing blow to national aspiration. And then, a little later, there was the loss of world markets for corn, their major exportable crop, as a result of the agricultural abundance of the New World. Peasant initiative, growing out of the spiritual dynamic generated by the Folk High Schools, did much to recover the nation from both of these shocks by transforming their exportable surplus from corn to \"butter and bacon,\" by helping to rebuild the national spirit, and by nourishing the Danish tradition in the territory lost to Germany during the long years until it was returned after World War I.\n\nAll of this, a truly remarkable social, political, and economic transformation, stemmed largely from one man's conceptual leadership. Grundtvig himself did not found or operate a Folk High School, although he lectured widely in them. What he gave was his love for the peasants, his clear vision of what they must do for themselves, his long articulate dedication\u2014some of it through very barren years, and his passionately communicated faith in the worth of these people and their strength to raise themselves\u2014if only their spirit could be aroused. It is a great story of the supremacy of the spirit.\n\nIt may be that power, no matter what its source\u2014whether physical strength, intellect, wealth, prestige, cunning\u2014is only effective in building a better society as it liberates the spirit of the powerless and gives them a vision of greatness so that their native wisdom can function. Grundtvig's power of prophetic vision and persuasiveness seems to confirm this thesis.\n\n#### _And Now!_\n\nThese two examples from previous centuries illustrate very different types of leadership for the common good. They are not suggested as general models for today, although some useful hints may be found in them. What these examples may tell us is that the leadership of trail blazers like Woolman and Grundtvig is so \"situational\" that it rarely draws on known models. Rather it seems to be a fresh creative response to here-and-now opportunities. Too much concern with how others did it may be inhibitive. One wonders, in these kaleidoscopic times, what kind of contemporary leadership effort will be seen as seminal 100 years from now, as we can now see the two I have described.\n\nOne thing is certain about these two men: they were both strong, assured, and resourceful. Both clearly knew who they were and had their guiding stars in focus.\n\n### WHAT CAN BE TAUGHT ABOUT RELIGIOUS LEADERSHIP?\n\nIt may be that leaders are \"born and not made,\" but some things might be taught to those who are \"born\" to the role. The most effective formal teaching of leadership I have seen was with mid-career executives who, in somebody's estimation, had demonstrated the potential for becoming significant leaders. The course was taught by a wise and perceptive person\u2014Professor John Finch, Chairman of the Department of English at Dartmouth College. His course had the alliterative title \"The Language and Literature of Leadership.\" After some consideration of the structure of language, Professor Finch moved on to the principal subject matter of the course, the four Shakespeare \"history\" plays\u2014 _Richard II, Henry IV_ (parts I & II), and _Henry V_. These were employed as case studies of the use of language by kings, with the language artistry of Shakespeare as the central focus. Most effective leaders of all sorts have a somewhat unique language artistry. (Not necessarily bizarre like Casey Stengel or history-making like Churchill and Roosevelt, but effective in terms of one's opportunities.)\n\nAfter watching this unusual teaching of language usage to mid-career executives, I have wondered whether it might be possible to employ this approach with college-age people and whether such teaching might have the effect of bringing some young people to an awareness of their potential to lead and establish early their sensitivity to language as an art form for effective leaders.\n\nPart of a religious leader's role as consensus finder is inventiveness with language and avoidance of a stereotyped style. One leads partly by the constant search for the language and the concepts that will enlarge the number who find common ground. The leader thus strives to bring people together, and hold them together, as an effective force. An experimental approach to language is a part of this skill.\n\nPart of success in leadership toward consensus is faith, confidence that the language exists that will provide the needed common ground if one will persevere and communicate this confidence to all involved. I don't believe that faith can be taught in a didactic way; but it can be communicated.\n\nBeyond language, what can be taught about leadership? I suspect not much in a formal way. But there is much to be learned by one who is inexperienced yet has the potential to lead. A better question might be, How can that learning be accelerated? Can useful formative circumstances for potential leaders be created?\n\nAt AT&T, I once made a study of the careers of the 12 executives who made up the top command of that huge institution at that time. They seemed to me to be able people, but not exceptional. What had happened in their careers that pushed them to the very top? All 12 reported that there was one early boss who greatly accelerated their growth as managers (managing and leading are not identical, but they are related).\n\nThe most significant finding was that 4 of the 12 had had their early formative experience under one mid-level manager. This person was one of 900 at his level and, as far as anybody knew, he had no more access to talented young people than other managers at his level. Yet a generation later he accounted for one-third of the top command. More than that, all over the business there were middle and upper managers who had their early formative experience under him. He was dead when I learned this, but clearly he was a fantastic grower of people. Further, he was something of an oddball who did not himself rise to the top; he stayed many years at the middle level where he had access to young people. But, as one of 900, he was probably the most influential manager of his generation. There is no way to make the judgment, but he may have made a greater contribution to the future of the business than the chief executives of his time. Among those who had their formative experience under him, there was no question about how he wielded his great growth influence.\n\nHe managed a good department\u2014but the controls on the business required that. His uniqueness was that he had a passionate interest in the growth of young people\u2014no less vivid term would describe it\u2014and the controls of that time did not require that (nor do they now require it). Further, he sustained this passionate interest in the depth of the Depression; his able young people were running hard when others were dying on the vine all over the place. When the curtain of the Depression lifted and the business started to recover and had urgent need for able young people to advance, this man had people who were ready to move when few others had them. What was he like? Those who worked under him were agreed on the following:\n\nHe was a good intuitive judge of potential. He did not waste his limited opportunities on people who were not likely to profit from them. He watched closely the growth of his promising young people, and he saw to it that they were always challenged and busy. Sometimes this required some ingenious \"made\" work. He did not have unlimited latitude to do this, but he stretched the charter of his job to the limit in order to favor the growth of high potential young people.\n\nHe was friendly and freely available for consultation. But he was reserved, not folksy or chatty. The growth of his promising people was an evident serious concern that he communicated.\n\nBut, most unusual, he had a firm belief in the importance of error in the formative experience of young people. If one of his youngsters was getting along a little too smoothly, he would contrive an error so that he would fall on his face. And he would figuratively pick him up and dust him off and say, kindly, \"Well, Bud, that hurt; but what did you learn?\" And they would talk about what was learned. Error was never greeted with censure\u2014unless it was repeated and then this boss would come down hard. All of this was carried on in an atmosphere in which able people were encouraged to grow.\n\nDoes the example of this extraordinary man have anything to say to the question of what can be taught about religious leadership? I think it has much to say. Anybody can learn, but crucial formative experience comes when one is young. It best takes place in a serving institution. And there should be influential people in that institution who have a passion for growing people\u2014growth in the strength to lead.\n\n### SEMINARIES HAVE THE PRIME OPPORTUNITY\n\nWhere can growth in strength as religious leaders best take place? _In seminaries!_ And what might a seminary be like when significant formation of this kind takes place there?\n\n\u2022 Its priorities will be reversed. Whereas seminaries are now mostly academic and only incidentally formative, formation of religious leaders will be primary and academic teaching will be secondary.\n\n\u2022 The staff of the seminaries will contain a strong element of those who have a passion for growing religious leaders\u2014and are good at it. They may or may not be scholars in the usual sense.\n\n\u2022 A major mission of the seminary will be to evolve, and maintain, a theology of institutions that deals realistically with the problem of how to recover moribund institutions as vital, effective, caring, and serving. This will not be a theoretical endeavor because it will be forged on the seminary's own experience as it builds itself into\u2014and maintains itself as\u2014the pivotal institution it is determined to become. Seminary students will be deeply involved in this continuous effort to build and maintain this theology. They will not just read and hear lectures about it.\n\n\u2022 The primary mission of the seminary will be leading and serving churches and supporting them as strong influential institutions. Most of the learning of seminary students will result from involvement in this effort.\n\n\u2022 There will be creative thinkers among its faculty who are developing and articulating a contemporary theology of what makes religious leaders, and the institutions they serve, _strong_. Students in the seminary will be deeply involved in responding to this with their own thinking.\n\n\u2022 Such seminaries will become known as effective nurturers of able religious leaders, and they will attract a wide spectrum of strong young people in search of such formative development. Some of these students might find their career opportunities in churches, but the seminary will become a prime source of religious leaders for all segments of society. It will acknowledge that any institution where religious leaders predominate may effectively become a church.\n\nIf only one seminary achieves this status, it will, in time, have a significant influence on the quality of the whole society. I hope there will be more than one.\n\nI can hear the anguished cry, \"Where will we ever find the money and the people to do this!\"\n\nIt begins with a vision that is translated into ideas and language. Then someone deeply believes in it, advocates, and perseveres despite obstacles. Finally the really tough test: someone has the leadership strength and skill to bring together and hold a following that will see it through.\n\nSome of what might be learned about religious leadership\u2014in a seminary\u2014is discussed in the next section.\n\n### PITFALLS AND OPPORTUNITIES\n\nEarlier, in discussing _strengthening the hands of the strong,_ it was noted that one way to do this is to help such persons acquire an awareness of the pitfalls and the opportunities for those who would be servants of society.\n\nAs ours has become a society that is dominated by institutions large and small, to an extent quite different from the times of Woolman and Grundtvig, some new problems have been presented. How does one sustain these institutions as effective servants? The answers to this question define some of the opportunities.\n\nSome of the pitfalls lie in the temptation, when there is institutional pain, to reach for an aspirin, a quick fix, or to seek the help of an \"expert\" who will provide a reassuring \"bedside manner\" along with a remedy\u2014perhaps an aspirin.\n\nThese problems present some unusual challenges to churches\u2014as well as other institutions. These are discussed in the next five sections.\n\n### THE PROBLEM OF ORGANIZATIONAL \"GIMMICKS\"\n\nThe opprobrious label \"gimmick\" is applied to any organizational procedure that is introduced with the hope of accomplishing what only better leadership can do, or that will not be effective, long term, because it is not in harmony with the prevailing quality of leadership. Such nostrums that claim to reduce institutional pain (which most institutions have some of) or boost organizational effectiveness are abundantly available. Too often the result is an \"aspirin\" effect\u2014not the path to long-term health for either person or institution. Well-led institutions are not good customers for gimmick salesmen. These exceptional institutions either evolve their own procedures, or they learn from other well-led situations. They are not in the market for aspirin.\n\nI hear the protest: \"What does one do when the organizational pain is intense?\" My response is, \"Attend to the quality of leading, unless you want to spend the rest of your organizational life living on aspirin!\"\n\nIn my work with American Telephone and Telegraph Company, I had the opportunity, almost from the beginning, to follow closely the \"Hawthorne\" researches that culminated in a landmark book in 1933, _Management and the Worker_ , and, in 1966, _Counseling in an Organization, A sequel to the Hawthorne Researches_.\n\nThis research, and the employee counseling program that grew out of it, were done in the Hawthorne plant of Western Electric Company in Chicago. This was a huge factory with 25,000 workers that had a long record as a very productive place with fine human relationships. It had had the benefit of unusually able management for several years before the research started. The research revealed, among other things, that employee preoccupation with personal problems was a significant element in satisfaction and productivity with work. There ensued from this finding a substantial program of employee counseling, probably the most ambitious effort to serve an industry and its people in this way that has ever been undertaken. The report in _Counseling in an Organization_ notes that there were 5 employee counselors on the staff in 1936. This number rose to 55 in 1948 and by 1955 it was down to 8\u2014the last count. Why did this work rise and then decline? And why did the effort fail when employee counseling was introduced in several other Western Electric plants and operating telephone companies? The conclusion of close observers, including myself, was that it declined at the Hawthorne plant when the last of the succession of great managers retired in 1952 and new managers who had different ideas began to question the cost of it. The justification during the peak years was philosophical rather than statistical. It was simply one of the appropriate things to do in a factory that had had the exceptional leadership that the Hawthorne plant had long enjoyed. It \"paid off\" because it was part of the right way to run a factory (which is probably why the researches were done in the Hawthorne plant in the first place). It was not introduced as a gimmick. Rather it evolved in a natural way as an appropriate thing to do. When the counseling program was introduced at other locations with great care, sometimes using staff from the Hawthorne plant to head it, it was done with the intent of fostering the benign and productive circumstances at Hawthorne. It was a gimmick, and it was promptly rejected as an inappropriate skin graft in five other locations. It did not belong there.\n\nLater, after I retired from AT&T and acquired a much wider view of institutions, I noted that the occasional exceptional institution was usually free of gimmicks. Exceptional institutions, I concluded, are astutely administered and wisely led. They learn from other experience, but their procedures (and every institution needs them) evolve out of their own experience, and they are congenial to the local culture.\n\nPhysical technology is readily transferable; but organizational technology is culture-bound, and any institution imports it at its peril. My advice: any effort to improve organizational performance should begin with attending to the quality of leadership; then evolve organizational procedures that are congenial to the way that leadership operates. In organizational performance it is the quality of leadership that governs. Procedures are important, but they are subordinate to the way the institution is led. Avoid organizational aspirin!\n\nWhen I wrote the first essay on \"Servant as Leader,\" I discovered that I had given that piece a catchy title. I am grateful that the title gave the piece some circulation, but I am also aware of the danger: servant-leadership could become a gimmick. The top person of some ailing institution might try to insert servant-leadership as a procedure, as a general management idea, as a means whereby the institution might do better. Such a move might have a short-lived aspirin effect, but when that effect wears off, it might leave the institution more ailing than it was before, and another gimmick would need to be sought. The surer way for the idea to have a long-term good effect is _for the top person to become a servant-leader_. What that person _is_ and _does_ then speaks louder than what is said. It might be better if nothing is said, just _be_ it. This, in time, might transform the institution.\n\nIn a discussion of this subject with the staff of a large religious institution, the head of the staff asked rather plaintively, as if he had had some sad experiences with gimmicks, \"How do you tell a gimmick when you see one?\"\n\nMy answer was, there is no _way_ I know of, no gauge that you can slip over an idea and say with certainty that it is or is not a gimmick. The problem is that what is clearly a gimmick in one situation may not be judged that way in another. The counseling program at its peak in the Hawthorne plant was not a gimmick; everywhere else it was tried it was an aspirin\u2014or worse! There are some organizational ideas abroad that seem to me to have no merit anywhere; but that reflects a personal bias. And there are some institutions that are so poorly led that aspirin may be better than nothing. But one is not likely to achieve institutional health, ever, by aspirin\u2014just temporary relief from pain, leading to use of another aspirin.\n\nReligious leaders, being human and fallible, will occasionally sense institutional pain in what they are leading. If the first response to that signal is to ask \"wherein is my leadership at fault, what can I do to improve it?\" it is unlikely that they will be tempted to reach for a pill, a gimmick, and gimmick salespeople who infest our society will not find them good prospects.\n\nOne of my sad experiences has been to watch at close range as an important institution bought one gimmick after another. I have become fatalistic about it. Just as there are people who are addicted to aspirin, there are institutions that seem addicted to gimmicks. Neither seems healthy.\n\n### LEADING VERSUS STRUCTURING\n\nOne of the common responses to institutional pain is to tinker with the structure. Everything that is organized has some kind of structure, even if it is only an informal understanding of who will do what. The most common explicit structure is hierarchical (as first described in Exodus 18). This arrangement is coming to be seen as seriously flawed, but the authoritarian bias that supports it is so entrenched that it is difficult even to discuss alternatives.\n\nStanding alongside most formal structures are informal ones whose function is to patch up the inadequacies of the formal ones. Even in the most rigid of institutions, there are usually acceptable deviations from and exceptions to the prescribed structures. They couldn't function without them.\n\nThe assumption here is that if an institution is well led, the nature of the formal structure is much less important than if it is poorly led. If leadership is exceptionally good, an institution sometimes operates without much reference to formal structure, even though it may have a well-defined one.\n\nIn a well-led institution, the need for structure may be minimal, but it is well to have some structure because even exceptional leaders sometimes falter and an understood structure serves as a \"safety net.\" Also, able leaders may be succeeded by mediocre or poor ones, and that contingency needs to be provided for.\n\nThe ideal organization structure would probably be one that is redesigned each time there is change in the conditions the institution confronts\u2014which may be every few days. Therefore, the need for stability suggests that the best that can be hoped for is a \"best fit\" for a range of conditions. This is not likely to be optimal for any one of them. The dilemma of all leaders of institutions is to accept that structure, no matter how well designed, will be awkward and inhibitive.\n\nThe above view suggests one of the inevitable ambiguities of life in all institutions, large or small, and defines one of the challenges to leaders who want to lead effectively so as to sustain what they lead at the level of exceptional. And it suggests to those who act as trustees that they should focus their energies, first, on themselves providing the best leadership they are capable of; second, on finding and supporting able leaders who will also administer; and, third, then designing a structure that is least inhibiting to good leadership.\n\nIn the context of religious leadership, tinkering with structure is not a first choice of means for building or sustaining quality in an institution. _Leadership is the prime concern!_ And when leadership is effective, structure is not a significant concern. Just watch to see that it does not get in the way.\n\nBeware! Preoccupation with structure could be a gimmick!\n\n### THE GROWING-EDGE CHURCH\n\nYears ago I wrote an essay on \"The Institution as Servant\" in which I made a comment on The Growing-Edge Church. Such a church, I suggested then, is \"one that accepts the opportunity all churches have to become a significant nurturing force, conceptualizer of a serving mission, value shaper, and moral sustainer of leaders everywhere.\" Since then I have not made a systematic study, but I have kept a close watch and I have not found a church that I think qualifies as a growing-edge church in these terms. I have wondered, why? Are the criteria suggested unrealistic, do any really want to be growing-edge churches, or is something standing in the way? I have concluded that it is the latter. If so, what is standing in the way?\n\nAs I get about among churches and church-related institutions, I am impressed by the extent to which they, rather casually it seems to me, employ commercial consultants to advise them and rely on procedures on which I would place the opprobrious label \"gimmick.\" Both, it seems to me, are evidences of inadequate religious leadership. Well-led institutions are not good customers for either consultants or gimmick salesmen. If these are valid judgments, might they not account, in part at least, for the inability of churches to achieve the healing influence they might have on the two pervasive problems; alienation of persons and failure of institutions to serve? Why would one look to any church for moral and spiritual guidance if that church is seen, even to a small extent, as simply a broker between those in need and facilities that might serve that need which are abundantly available elsewhere? And, further, how can a church in this posture infuse religious leadership, a critically needed quality, into the fabric of society as a whole? If churches, in these chaotic times, are to _lead_ , they must _originate_ and not be seen as sponges for the fads of the day, and they should work from their greatest asset\u2014inspiration.\n\nA church might choose another mission than to become growing-edge by the criteria I have given. But if it aspires to that distinction, then I submit that recourse to consultants, as they are commonly used, and gimmicks will stand in the way.\n\nLet me speculate on why some churches, because they are not clear about their missions, or have not thought through their implications, may have turned to consultants and gimmicks. Could it be that this diversion to do the \"in\" things and to look for \"answers\" from experts is an unconscious escape from the much tougher and more demanding course of nurturing seekers? Seeking is an opening to prophetic vision which could have disturbing consequences\u2014and not many are likely to undertake it unless they are given exceptional leadership. And seekers are a resource that cannot be bought with money.\n\nI have followed one thread of this diversionary influence in churches for 35 years. In the summer of 1947, I attended the very first group development conference at Bethel, Maine. This was to be the launching of a major research effort by Kurt Lewin, who came to this country as a refugee scholar in the 1930s. But Lewin died suddenly on the eve of this first session, and it was carried on by his students.\n\nLewin was a rigorous experimental psychologist, and I registered for this session on the urging of a close friend, an equally rigorous experimental psychologist, Carl Hovland, who was then chairman of the Department of Psychology at Yale, because Carl felt that this work might be a breakthrough of a great practical consequence\u2014my prime interest.\n\nAlas, in my judgment, Lewin's students who carried on his work were not of his careful scientific temperament and skill, a view confirmed by the abrupt termination of the tenure of Lewin's Research Center for Group Dynamics at Massachusetts Institute of Technology after his death. I have since often wondered where we would be today if Lewin had lived a normal life span (he died at age 47). I am quite certain that the cultish movement that took off from that session I attended in 1947 would not have materialized. I promptly dissociated myself from it. But some churches bought in on it in a big way. Sensitivity training has been one of the \"in\" things to do. Some of what is called the \"human potential movement\"\u2014centering on self\u2014had its origins in this work. There are other examples in recent years of church investments in diversionary activities, some of which have the flavor of gimmicks; but this is the one I have been closest to.\n\nThese are called \"diversionary\" activities because they divert churches from what I believe should be their central concern, _inspiration_ , and divert them to techniques and procedures. Differing contexts of belief illuminated by a range of theological reflections suggest many meanings for the word _inspiration_. I doubt that any of those meanings would embrace techniques and procedures that have the flavor of gimmicks or the advice that most commercial consultants would give.\n\nThis is not intended as a diatribe against consultants\u2014in my more active days I have been one myself, and later I will comment on an acceptable limited use of them. But, once I evolved the views I have expressed about consultants to churches\u2014and their wares\u2014I no longer accepted money for consulting service to a church. I gained some perspective on this from Alcoholics Anonymous.\n\nAA received its unusual view of money from a wise and deeply concerned philanthropist in the 1930s. This very wealthy man had watched the slow evolution of this unusual group and, when they reached sufficient size and certainty in their approach, he convened them in his office to help them organize and clarify their principles. In the course of the discussion, he made a statement like this: \"I believe I know from my experience something about what can be done with money and what cannot be done with money; and you have one of the things that cannot be done with money.\" The firm resolve that the essential work of AA, one recovered or partly recovered alcoholic helping to recover another, would not be done for money, was made at that meeting. The phenomenal success of AA rests in part, I believe, on strict adherence to this principle.\n\nAA is a religious movement, an ad hoc church. All churches may do well to keep sharply in mind the question, \"Are we trying to do something with money that cannot be done with money?\" If it aspires to be a growing-edge church, as conceived here, it will harbor and encourage _seekers_. They are more likely than most to hear and evaluate the prophetic vision that is with us all of the time.\n\nA church in which inspiration is the prime source of its guidance may safely pay a lawyer, accountant, or architect, or others rendering ancillary service. But if a church is uncertain about the inspiration that guides it, then it had best turn to seekers. What the world of institutions needs from churches is not preachment about leadership, but clear and convincing models, because every institution, within the scope of its mission and opportunity to serve, needs its own equivalent of seekers to keep it on a true course as servant and to assure its survival. And it is just as difficult to sustain such persons in a business or unit of government or university as it is in a church. Are other institutions likely to learn to evolve, support, and encourage these people unless churches hold up the model of them\u2014in its purest form? Any church that offers a clear and convincing model as a harborer of seekers will, in my judgment, distinguish itself as a growing-edge church. Its effective religious leadership will be confirmed thereby.\n\nThe pastor of a church who is usually, but not always, a full-time professional may be paid. But the pastor's strongest role may not be as a seeker. Rather he or she will be the leader and nurturer of seekers who will not be paid.\n\nThe absence of evidence that churches are nurturing seekers, or aspiring to strengthen leaders, is one of the reasons I have not identified a growing-edge church. I come by my conviction on this point by reading the history of the Quakers. If George Fox, the great prophetic visionary who founded the sect in 17th-century England, had not been preceded by a lay movement known as \"seekers,\" Fox might not have been heard. I firmly believe that for a prophet to be heard there must be seekers. When the Quakers, after the first powerful generation, ceased to harbor seekers, they \"had it.\" What they \"had\" was, and continues to be, good; but they were no longer on the growing edge.\n\nAn interesting aspect of leadership is that, whereas administrative responsibility in institutions can be assigned, authority given, and resources allocated, _anybody can lead!_ Anybody can lead who can bring together and hold followers as an effective force. While the logical place for leadership of churches may be in seminaries (as I have argued in essays published under the title of \"Seminary as Servant\"), in fact, such leadership may come from anywhere. _If one strong growing-edge church emerges, it may lead the seminaries of all denominations into the role of servant of the churches\u2014from which position seminaries might evolve into significant leaders of churches (which they seem not to be now)_.\n\n### THE CHURCH AS AN INSTITUTION\n\nEarlier I opined that \"any influence or action that rebinds\u2014that recovers and sustains alienated persons as caring, serving, constructive people\u2014is religious. And any group or institution that nurtures these qualities effectively is a church.\" Thus, institutions that have other explicit functions may also be churches, and those that are called churches usually have several roles. The concern here is only with a church's influence as defined above.\n\nChurches, along with other institutions, have taken their lumps in recent years. They share the consequences of a general disaffection with all institutions. This comes about, I believe, not so much because of a feeling that the quality of their performance has declined, as because expectations have risen. More people, especially young people, believe that the institutions that serve them, including churches, could and should do better, much better.\n\nIn the early 1950s, an important book, _The Organization Man_ , was published. It was a well-written and sharp criticism of those who lead or manage our institutions. It was an immediate spectacular success. For years it was required reading in college courses, and perhaps still is. College bookstores had stacks and stacks of them. I had a curious experience as a result of the popularity of this book with students.\n\nOne day the editor who had managed the publication of this book came to see me. He told me he had felt that this book would be a commercial success; but he did not anticipate that it would have such a great influence on young people; it probably was one of several factors in the student unrest of the 1960s, and he was deeply disturbed by this. He felt the book had done some damage to young people.\n\nHe was enough of a realist to know that no critical book like this one, however successful, would change our institutions or the people who manage them\u2014only make them angry. The editor also knew that these young people, if they were to make their way in this world, would have to come to terms with the world as it is; and, if they were strong and worked at it over a lifetime, they might change it a bit. The editor also believed that, with the state of mind these students were then in, caught up, as so many were, in a zeal for instant perfection\u2014partly as a result of reading this book\u2014they were going to be set back, and some might be permanently harmed. Then he made a proposition. \"You have made a good life out of being an organization man in a huge business,\" he said. \"If you will write a hopeful book about how life can be made fruitful in institutions as they are, we will publish it and put all of our promotional resources behind it. Our aim would be to lay it down beside _The Organization Man_ in every college bookstore in the country.\" He had a contract in his pocket. This was quite an offer from a big prestigious publisher, especially to one like me who had never had a book published and had no public reputation to exploit.\n\nI turned it down, for reasons that I believe can be deduced from reading this piece. But a lifelong friendship with the editor, now dead, ensued. I have made this digression from the subject, the institutional church, because I feel that it helps illuminate the problem of churches as institutions that want to serve young people.\n\nAnother anecdote further illuminates it. In an off-the-record discussion, a high-level church executive, a sensitive and thoughtful person, made this observation in all seriousness, \"I have come to believe, after long experience in my job, that an important passage of scripture should be rewritten as\u2014When two or three are gathered in my name, there is bound to be a fight about something.\"\n\nAs I reflected on this observation, it occurred to me that many of the conflicts that plague our society and make \"conflict resolution\" such an active endeavor in churches, is the simple matter that \"manners\" have declined. The houses of Congress would not be able to conduct their businesses without strict codes of manners.\n\nMy estimate of the chief institutional problem of some churches is that they have put too high a priority on _preaching_ and too low a priority on _being_. The churches of today will have more influence on the quality of society as a whole (which means, to some extent, the quality of the institutions that comprise it) if they think of their prime influence as _being_ , through what they model as institutions. It may be that what a church _is_ as an institution will have more impact on its own members than what it _says_ to them. This is not to denigrate what is said, that is terribly important\u2014just not _as_ important as what it _is_.\n\nAs I reflect on my business experience in which I did much meeting attending and some speechmaking, it stands out in my memory that the few exceptional companies that I came to know after I retired from AT&T were rarely represented as speakers or attenders at meetings. They were too busy doing the things that make them exceptional. And they probably knew (what I now know) that too many of the people making speeches were either just spinning words or talking about what I have now come to label as gimmicks. What would an astute observer say of the church world in this regard?\n\nAs I reflect on my experiences with the radical students in the 1960s I despair when I think of the sea of words they were engulfed in. Where were the models they might have learned from? Faculties had taken advantage of the teacher shortage, in the years of expansion of higher education after World War II, to bargain down their obligations to teaching. By the time the sixties' generations of students came along, they realized that, too much, they were supporting faculty who used their positions as pads from which to do their own things. There were other causes for the unrest, but this one helped direct the venom of the disturbed students to the universities and colleges\u2014 _as institutions_. Students of that period simply did not sense a sufficient dedication of teachers to their calling. The result was disaster in some places.\n\nThe ultimate model of servant is one whose service is rendered in one's own personal time for which one is not paid. Students saw too little of this model in the 1960s. They still don't see much of it, I am told. This is what my book _Teacher as Servant_ is about. I tried to describe what a dedicated university professor who accepts this premise would be like.\n\nChurches, then, have the opportunity to be institutional models for the universities. Many of those who teach in and administer universities as well as those who shape policies in business, government, hospitals, etc. attend churches. Do they just hear words, or do they see a model\u2014a model of strength and effective religious leadership? And what would the model of a serving institution be like\u2014 _any_ institution?\n\nI realize there is a problem with the word _institution_. It has an ugly sound to some. I have looked into the etymology of that word and it does have a rather checkered history; but tucked away among the many historical meanings is: \"something that enlarges and liberates.\" Is there any other single word that has that connotation? Let me suggest a definition for our purposes:\n\n\"An institution is a gathering of persons who have accepted a common purpose and a common discipline to guide the pursuit of that purpose, to the end that each involved person reaches higher fulfillment as a person, through serving and being served by the common venture, than would be achieved alone or in a less committed relationship.\"\n\nNot all who \"belong\" to churches may want, or be capable of, firm commitment to a purpose like that described at the opening of this section. Unless a church serves only a selected constituency, it may need to maintain separate programs for homogeneous groups. I suggest that if a church is to be influential in shaping the culture, now and in the future, it must shelter within its active members a group of strong, assured, resourceful people so that each of them \"reaches higher fulfillment as a person, through serving and being served by the common venture, than they would achieve alone or in a less committed relationship.\"\n\nSuch a church will be seen as strong.\n\n### GETTING HELP AND USING MONEY IN A CHURCH THAT ASPIRES TO BE ON THE GROWING EDGE\n\nIn my essay \"The Institution as Servant,\" I defined the growing-edge church as \"one that accepts the opportunity all churches have to become a significant nurturing force, conceptualizer of a serving mission, value shaper, and moral sustainer of leaders everywhere.\" I now suggest an amendment which holds that such a church will get its strength from harboring and nurturing an active group of seekers who are open to inspiration. A church that aspires to be on the growing edge in these terms, in the absence of leadership and support from seminaries, may need some help that it will pay for.\n\nIf one conceives of a church as I do, as an institution whose primary opportunity is to sustain contact with inspiration and infuse a healing, building influence into the world, finding a helper for the process of building a growing-edge church may be difficult. A church that does not aspire to wield an influence in the growing-edge role may find the usual services of commercial consultants congenial and helpful. But continued reliance on that source of help may build dependency rather than strength. Churches that aspire to be on the growing edge need to be strong, and they should seek help that builds strength.\n\nWhile I was writing this, I received a phone call from a man who introduced himself as the administrator of a large Catholic hospital. He had read something I had written on the servant theme and asked whether I had written anything on how to conduct a Christian disciplinary interview with an employee. He was in need of help. I told him that I had not written on such detailed procedural matters, and, after a few pleasantries, the conversation terminated. He is a ripe prospect for a gimmick salesman.\n\nLater I reflected, since the advice sought was in Christian terms, I might have suggested that if Jesus were among us today and were asked that question, there is a good chance that he might respond, \"Become the full stature of the caring, serving person you have the potential to become and all will come clear to you on such matters.\"\n\nThen I reflected, \"Where is this man's church?\" Good question. Where is his church? Is it on the growing edge? Does it aspire to be? Is it striving to be of practical help to those it reaches as they do the work of the world\u2014including managing church-related institutions?\n\nAnd where are the seminaries\u2014200 in the United States, 60 of them Catholic? Consultants and gimmick salesmen, mostly good, conscientious people, seem to be trying to fill the gap created by the failure of seminaries to lead. But it may be that seminaries will not have the incentive to lead as long as consultants and gimmick salesmen seem to be filling the gap and are eagerly sought by churches because there are no other resources. Something needs to change. A church that gets the vision of becoming a growing-edge church may initiate the change, and may need some help. They will do well to find a helper who is wise and judicious and who has the vision and the patience to help that church to evolve a powerful formative character that will favor the growth of strong, caring, serving persons. The following experience comes to mind.\n\nBack in the days when Richard King Mellon and Mayor Lawrence were bringing off their renaissance of Pittsburgh, I asked an old and wise friend who was well placed in a city of comparable site and not too far from Pittsburgh, why could not his city achieve this? His prompt response was that the reason was simple: his city did not have a Richard King Mellon. They could produce a Mayor Lawrence if they had a Mr. Mellon. Then he said, \"We don't have anybody as rich as he is, but we have a few who are rich enough. The difference is that Mr. Mellon cares intensely about his city. Our rich people do not have his kind of caring. It is as simple as that.\"\n\nReflecting further on my own experience: the few institutions that I know intimately which, under close scrutiny, impress me as being exceptional seem to be led by people who have come much closer to their potentials as caring, serving people than most of us. They, like Mr. Mellon, do not need a consultant to tell them how to _act_ as if they were human. They _are_ human\u2014and they just act naturally. They may need help to do some things they do not know how to do, but not with how to be human and caring. Formative experience is required for that. Such people who are human and caring live with an awareness of good work that is in harmony with the ultimate values they embrace. Their incentive to be human comes out of their faith. The crying need of our times is for greater formative resources for the growth of such people, a work for seminaries to initiate.\n\nWhile I accept, grudgingly, that, in the absence of leadership and support by seminaries, churches that aspire to be on the growing edge and provide this formative experience may need, with caution, to pay for some help, I long for the day when churches generally will hold inspiration as their highest value and will acknowledge seekers who are not paid as their most precious resource.\n\n### RELIGIOUS LEADERS: A NEW FRONTIER\n\nI think of religious leading as a new frontier because, as an explicit goal, it seems not to have been prominent in the concerns of religious thinkers. What then is the new opportunity? Is it not to make of existing religious institutions\u2014churches, synagogues, temples, mosques, seminaries\u2014laboratories for preparing leaders for a more caring society? Will not the first step be to prepare strong leaders for the work these religious institutions have to do, and then to share their mature experience with institutions generally? How will we do that, the incredulous may ask? Let them begin with William Blake's dictum: \"What is now proved was once only imagin'd.\" And what will nurture their imagination? Let me suggest three subjects for reflection that have been prominent in this essay.\n\n#### _Persuasion_\n\nThe move from the \"control\" model, that comes down almost unchanged from Moses, in the hierarchic principle, toward the \"servant\" model about which much remains to be learned, might begin with cultivating the attitudes that will permit the shift from coercion and manipulation to persuasion as the predominant modus operandi in institutions generally\u2014including churches. Elsewhere I have written that one is persuaded on arrival at a feeling of rightness about a belief or action through one's own intuitive sense\u2014checked, perhaps, by others' intuitive judgment, but in the end one relies on one's own intuitive guide. One takes that intuitive step from the closest approximation to certainty one can reach by conscious logic (sometimes not very close) to that state in which one may say with conviction, \"This is where I stand.\" And this takes time! The one being persuaded must take that step alone, untrammeled by coercion or manipulative strategems. Both leader and follower respect the integrity and allow the autonomy of the other; and each encourages the other to find an intuitive confirmation of the rightness of the belief or action. Persuasion, thus defined, stands in sharp contrast to _coercion_ \u2014the use, or threat of use, of covert or overt sanctions or penalties, the exploitation of weaknesses or sentiments, or any application of pressure. Persuasion also stands in sharp contrast to manipulation, guiding people into beliefs or actions that they do not fully understand.\n\nThe religious leader will be persuasive and, insofar as humanly possible, will avoid any taint of manipulation or coercion. This is suggested as an indispensable condition of trust by followers or colleagues. Unilateral actions by leaders in emergencies, with later explanations, are more likely to be accepted as appropriate if the relationship is predominantly persuasive. But, in an imperfect world, there will be exceptions, and, no matter how it seems to the leader, some may see manipulation or coercion when the leader thinks they are not there. Perhaps discussion of the realities, in which lapses are acknowledged as failures, may be the most open course. Persuasion, when exercised by a leader, is not passive. It is dynamic, sustained, and challenging; and it may repel some who might be followers of a less insistent leader. The leader will be prepared for rebuff and failure and will need a sustaining spirit. John Woolman had such a sustaining spirit. Read his journal!\n\n#### _Seekers_\n\nEarlier it was suggested that prophet, seeker, and leader are inextricably linked. Sometimes two or three are merged in one person. But prophet and leader are seen as seekers first, who later evolve into other roles.\n\nThe seeker contributes ever-alert awareness and constant contact with available resources: spiritual, psychological, and material. Also, the seeker helps guard the religious leader against becoming trapped in one of those closed verbal worlds in which one's influence is limited to those who share one's exclusive vocabulary.\n\nSeeking is also waiting and expecting and working with a sustained listening group that is ready to receive a prophetic vision. The group will have prepared itself to make discerning critical judgments about what they hear. Seekers are religious in that they share a discipline which sustains them as persons who are always prepared to respond to a new (but carefully examined) rebinding influence. An important part of the role of religious leaders is to provide the expectancy that sustains seekers.\n\n#### _Prophecy_\n\nThe prophet, in William Blake's term, is one who imagines what will later be proved. Seekers will be listening and hear and test the validity of the idea, and religious leaders will carry the new idea into the work of the world.\n\nIt should be noted that in the 19th century Danish experience, Bishop Grundtvig did not offer a model of the Folk High School, nor did he himself found or direct such a school. He gave the vision, the dream, and he passionately and persuasively advocated that dream for over 50 years of his long life. He also gave the leadership. The indigenous leaders among the peasants of Denmark were listening and responded to that vision and built the schools\u2014with no model to guide them. _They knew how to do it!_ They were, in effect, the seekers of their time and place. Grundtvig gave them the prophetic vision that inspired them and communicated his confidence in their ability to act on what they knew.\n\nOur restless young people in the 1960s wanted to build a new society too\u2014and some still do; but their elders who could have helped prepare them for the task just \"spun their wheels.\" They did not listen and gave little leadership. As a consequence of this neglect, a few of those young people simply settled for tearing up the place. And, in the absence of a new prophetic vision to inspire the effort to prepare our young people to build constructively, and leadership to help them do it, some of them may tear up the place again! Do not be surprised if they do just that. The provocation is ample. Among those in the older generation are some who could prepare today's young people to live productively in the 21st century. They know how to do that as well as 19th century Danish peasants did.\n\nWe live amidst a frightening shortage of ideas in all of our institutions, partly, I believe, because of the almost universal obsession about money. The nonprofits are so desperate for money that they can't produce the ideas that would make getting money easier. And businesses, financed as they are by the gigantic craps game that our society prescribes and in which our \"best people\" participate, are so eager for a quick buck that they can't produce the ideas that would give them a sound long-range future.\n\nThe serious deficiencies of our times may be in prophets who are not sufficiently realistic and inspired; in seekers who are not sufficiently humble, open, and dedicated listeners; and in enough religious leaders who are strong. As a consequence, ours is a poorly served society.\n\n### WHAT AND WHO SPEAKS TO YOUNG POTENTIAL LEADERS?\n\nSome effective religious leaders in every generation should be able to speak persuasively to potential leaders among the young\u2014as Grundtvig and those who worked with him were able to do in 19th-century Denmark.\n\nMy principal experience with such young potential leaders, in my later active days, was with the radical students of the 1960s. Colleges and universities of that period, when measured against the opportunities they had to help these able students mature into strong religious leaders, seemed not to have served those students well. I gather that their influence today in this regard is pathetically meager. Not much seems to have been learned from those trying times in the sixties\u2014probably because of a paucity of seekers among faculty, administrators, or trustees.\n\nWhat (or who) speaks to young potential religious leaders? I believe it is those who do what they can with what they have from where they are to build a better society. This communicates hope. Far too many of our able, well-meaning, well-placed contemporaries just spin their wheels. This communicates despair.\n\nSmall wonder that alienation is so widespread. Too much despair and not enough hope is being communicated!\n\n### LEADING VERSUS GOVERNING\n\nWe live in an age in which there is much talk about leading. Institutions in which there is little urge to do other than what they have done before, usually do not need leaders; able governance will suffice. But if, either out of necessity for survival or because they wish to serve better or more creatively, they want to do something that neither they nor others have done before, then leaders who are capable of venturing and risking\u2014prudently\u2014must emerge.\n\nIf followers are to respond voluntarily and with spirit, these leaders will do better if they have the dedication to persuasion and the language skills that will elicit this response from followers. People respond voluntarily and with spirit when something from the deep inner resources of the leader comes through to them. Competent and inwardly strong leaders who are by nature servants, and who are inspired by _religio_ , can be the most influential of all when they have the gift of appropriate language _if_ , in these times, they will address forcefully the interlocking problems of alienation and the failure of institutions to serve.\n\nThe need for many such religious leaders is urgent.\n\nThis is written as a sharing with those who aspire to be part of the vanguard of the effort to grow more religious leaders.\n\n### POSTSCRIPT\n\nMy perspective on the subject of religious leaders, as I said at the outset, is that of a student of organization, not of a theologian or scholar. From this perspective, I was prompted to state the central idea of this essay as _work! Do something! Work to increase the number of religious leaders who are capable of holding their own against the forces of destruction, chaos, and indifference_. Those who are in the vanguard of this effort will find ways to strengthen the hands of the strong who will become religious leaders. Such strong religious leaders will be an elite, one of the many kinds of elites. One of the blights on American life is our inability to face the fact that elites can serve. We have allowed _elite_ to become a pejorative term. We need to be liberated from such constricting ideas. We must strengthen the hands of the strong who will be servants because only they can cope with the strong who are destructive and exploitative and who, unfortunately, are quite numerous. Increasing the number of religious leaders is not the only thing that needs to be done to make ours a better, nobler society; but unless we do that, other society-building efforts may not avail us much.\n\nWe do not need a new revelation to accomplish this. The machinery to move toward increasing the number of religious leaders (and I mean _large_ numbers) is already in place: seminaries, to churches, to individuals, to \"operating\" institutions\u2014governments, businesses, schools, social agencies. But this machinery is not working, not nearly well enough. Why is the machinery not working? I believe it is because seminaries that occupy the originating spot are not supplying the ideas and the language\u2014the leadership\u2014that they are best placed in the scheme of things to provide; and, without that leadership, churches are not likely to move forward in influence from where they now are. Furthermore, as long as seminaries occupy that spot and do not produce the ideas, and the language, and the leadership, it will be difficult for any other agency to do it _because_ if the churches are to move, the effort must be supported by an appropriate theology.\n\nIt is a truism regarding any complex sequential process, social or mechanical, that if a vital link fails, the whole process fails. This is especially true if the originating point does not function. To get a steam engine to move, someone has to turn on the steam. If an institution, or a complex of institutions, is to move, someone has to lead.\n\nWhy do not the seminaries lead in a way that will help churches nurture religious leaders? Seminaries are staffed by able, thoughtful, conscientious people; why do they not give the needed leadership? I believe it is mainly because they do not see it as their role and, when pressed, they tend to say it is someone else's role\u2014such as denominational bodies. And they have not prepared themselves to do it.\n\nFrom my bias, the answer is simple: they do not have among their trustees, faculty, or administrators enough people who think in terms of how things get done in complex institutional arrangements. These people are a little rare, and I doubt if just one person\u2014faculty, administrator, or trustee in a seminary\u2014would survive if he or she tried to wield this influence alone. And it probably would not help just to give seminaries the idea, such as handing them this document. They need help, but probably a rare and special kind of help _from the outside_. They need help that is both persuasive and enduring because of the magnitude of the expected change. It is not that seminaries will need to abandon anything they now do; I am not judging that. But what they will need to add will make them radically different places because they will become primarily formative in their influence.\n\nThey need the kind of help that only a modern John Woolman is likely to give, because seminaries are tender, fragile institutions (all institutions, being collections of humans, are tender and fragile, but seminaries are a rare and special kind), and they need help in the spirit of great loving care. They not only need it, as everybody does, but help will not be effective without it.\n\nJohn Woolman was passionately opposed to slavery, but this was shared with others who were so vehement in their opposition that, in the judgment of some historians, they helped make the Civil War inevitable. What made Woolman's persuasion effective, with no harmful side effects, and helped free the Religious Society of Friends of slavery 100 years before the Civil War, was that he loved the slaveholder with equal passion. He understood his dilemma, he was the slaveholder's friend, and he was willing to spend 30 years of all the time he could spare from earning his livelihood (he supported a small family) in visiting slaveholders over and over as he pressed his gentle but firm argument. In my judgment, this method will be effective with most reasonable people, including trustees, administrators, and faculties of seminaries who might be persuaded that they should enlarge their missions to include leading the churches as they wield their influence on people and institutions through the diffusion of religious leadership.\n\nIf the above is a reasonable assumption about the need and how to serve it, where are the Woolmans of this day, men and women who will understand and love and serve seminaries and the people who are in them? There are 200 seminaries in the United States, and I estimate that it will take 50 such able committed volunteer persons many years to give this service\u2014each working with four seminaries. The 50 would be one in several million of the adult population. I think it a reasonable assumption that these 50 men and women exist and that they would rise to the opportunity to enrich their lives by giving this service\u2014gratis. _And_ I have confidence that some seminaries would, in time, respond to their service. The big question is how to find and prepare these 50 people\u2014and finance it.\n\nI went to sleep with this question on my mind and, as so often happens when I do this, I dreamed about it. (I am not a lawyer, but many years ago at AT&T, I was a member of a task force that was preparing an important law case for litigation. We had worked for six weeks and had not found a satisfactory basis to present our case\u2014and trial time was nearing. We were under pressure. Then one night I had a dream in which we were in court. _I_ was the lawyer pleading our case and I was arguing the very structure we needed. The case went to trial exactly as I dreamed it; but I never admitted to my colleagues how I got the idea.) I am now at the age when I am not as circumspect as I used to be, so I will share this one.\n\nIn my waking reverie the next morning, I was given a quite complete answer to the question of how can we find and prepare these 50 people\u2014and finance it. After breakfast I went to my desk and wrote a fable [ _Editor's note: published as \"Fable,\" in_ Seeker and Servant, _Robert Greenleaf, published in 1996 by Jossey Bass_.] in which I looked back from the year 2000, at which time a large number of churches in the United States had become strong influential institutions, alienation had declined, and the number of extraordinary serving institutions in the United States had increased. It is not a very good fable, but it does outline the steps that were taken to locate and prepare the 50 choice people whose persuasion caused enough seminaries to enlarge their programs to bring this about. I will send this fable to any who ask for it _if_ I am assured that the asker aspires to be in the vanguard to which this paper is addressed.\n\nWhat is now proved was once only imagin'd.\n\n_William Blake_\n\n## 6\n\n## **Seminary as Servant**\n\n### INTRODUCTION\n\nThe essays that follow were written in recognition of two pervasive social problems: (1) widespread alienation in all strata of the population, and (2) the inability or unwillingness to serve on the part of far too many of the institutions that make up our complex society. Each of these problems may be a contributing cause to the other, and neither is likely to be healed without coming to terms with the other.\n\nI believe that new religious insight is needed to deal with the dilemma that this reciprocal dependency presents ( _religious_ in the root meaning of _rebind_ ). Such insight may come as a gift to any earnest seeker, but its availability to the many whose lives may be enriched by it will be greatly enhanced if seminaries embrace it, put it in a context so that churches may mediate it, and give sustaining support to churches as they do their work. Any seminary that performs well in this role will be truly a servant in our times. But, as I argue later, this achievement is unlikely to come unless the stated missions of seminaries require it\u2014mission statements that have clarity and power and the ring of contemporaneity. Such mission statements are not likely to emerge until seminary trustees give a character of leadership that is not yet generally accepted.\n\nThe three essays were written to encourage a few seminary trustees to use their influence to bring _one_ seminary to the position of greatness as servant, both as a seminary and as a model of institutional quality that will be a powerful leaven in the culture. If one seminary will achieve this, the less venturesome may take courage and try.\n\nThe three essays were written in 1980, 1981, and 1983 at the suggestion of Dr. Robert W. Lynn, (then) vice president for Religion of Lilly Endowment, with whom I share a concern for enlarging the contribution of seminaries of all faiths to the enrichment of American society and to whom I am deeply indebted for helpful criticism of early drafts of these pieces. I am grateful for the incentive to write these because this work caused me to probe deeper than I might otherwise have done into facets of my long-held interest in institutions, large and small, and the place of trustees and directors, particularly their chairpersons, in assuring the optimal performance of these institutions.\n\nOut of my probings, the idea of a _hierarchy of institutions_ evolved. In this hierarchy I see, at the top, seminaries and foundations. Foundations are in that oversight position because they have the resources and the opportunity to gain perspective that enables them to provide conceptual leadership to colleges and universities, some of which seem in want of new directions which they are unable to find for themselves.\n\nSeminaries are in a strategic position to give similar support to churches, whose needs are also urgent. In turn, both churches and universities are well placed to give nurture and guidance to individuals and to the whole range of \"operating\" institutions: governments, businesses, schools, hospitals, communities, families. Any effort to aid our ailing society might well start with a consideration of how the leadership of foundations and seminaries, each from its respective strategic position, might be made more effective.\n\nIn writing these essays, the role of trustees of seminaries and foundations received fresh scrutiny. I had written earlier on this subject (\"Trustees as Servants\"\u2014Chapter III of _Servant Leadership_ , Paulist Press, 1977), but in reflecting on the nature of this role in seminaries and foundations some new insights came. These two institutions, standing as they do at the top of the hierarchy of institutions, are in a precarious position (in their roles of servants of society) because they have little sheltering support available to them from other institutions. Since not many people are able as trustees to provide the constant regenerating influence that all institutions need if they are to serve and prosper over a long period of time, the durability of society rests on a few institutions (such as seminaries and foundations) acquiring self-regenerating capacity. These three essays are premised on the assumption that foundation and seminary executives and staffs, on their own, are not likely to provide this self-regenerating influence. It is for exceptional trustees to provide. \"Exceptional\" trustees are not necessarily high-status persons by other standards. The ablest trustee chairperson I have seen, in considerable involvement with trustees in a range of institutions, is a low-status person by other standards\u2014but truly great as a trustee and in the chair.\n\nSince exceptional trustees, by definition, are not common, if some of this small number can be drawn into the care of foundations and seminaries, these two can then spread the influence of their exceptional trustees to all institutions. I see no better way for a needy society (as ours is) to be well served by its ablest citizens. But seminaries and foundations will need to rise to this view of their opportunities and bring such exceptional trustees to their service, or inspire with new vision those they now have. Addressed here are the opportunities for seminaries. Elsewhere (chapter VI of _Servant Leadership_ ) I have discussed the opportunities available to foundations.\n\nIn the three essays that follow, I note that the perspective from which I approach the subject of seminaries and their missions is that of a student of organization, how things get done, not as a theologian or scholar. Included in my perspective is a longtime immersion in the writings of Machiavelli. It is my view that, while manifestations differ, joustings among contemporary power seekers (of whom there are plenty) are little different in their essentials from what they were in 16th-century Florence. Since I am now retired and no longer in the fray, I turn to Machiavelli periodically as a means for keeping myself refreshed about the nature of the society in which all live and do our work. I find no other commentator, ancient or contemporary, who is as clear, realistic, or insightful on the use of power in society as Machiavelli. Also, I find in his writing much wisdom that is useful to a person or institution that would be a servant in our time.\n\nAgainst the dismal view of human nature that one gets from reading Machiavelli, I place the religion of hope ( _religion_ again used in its root meaning of _rebind_ ). I could not read this coldly realistic Florentine, and keep my sanity in this chaotic world, unless I had hope. Not hope that some miraculous intervention will banish evil so that good will prevail; I am not sure that I would be happy in such an aseptic place. My expectations are really modest: that a few able people who think of themselves as \"good\" will become more dedicated servants of society and will work a little harder and more astutely at it than some who think of themselves as \"good\" are now accustomed.\n\nMy short-run hope is that some of these able, good people will become trustees of seminaries and do what only inspired trustees are likely to do: work to give their seminary a vision of greatness that will lift it clearly out of the bureaucratic mold that engulfs so many institutions and set it on a course that will assure that it becomes the influential force that its top-of-the-hierarchy position makes possible.\n\nMy long-run hope\u2014as expressed in the essay on \"Mission\"\u2014is that, as the result of seminaries establishing their claim to this position, institutions whose primary motive is caring for persons and are governed by persuasion will rise to dominance in the culture over those that seem now to dominate: institutions that are governed by coercion and whose primary motive is private gain and survival in the competitive struggle.\n\nIt may be said that the quality of contemporary society is determined by the quality, as separate entities, of the legions of institutions that comprise it. The seminary will be an effective force in this composite as it wields its influence, through the churches, on both individuals and the institutions which they both govern and are governed by.\n\nThese three essays, addressed to seminary trustees, are made available for wider distribution because I feel they may be of some interest to those who share my concern for the quality of institutions generally. Further, one who is trustee of a quite different institution from a seminary may be stimulated to produce insights of immediate relevance to one's primary trustee obligation. Then, from my perspective, all institutions are drawing closer together when judged by the challenges they present to their trustees and directors. In the interest of furthering discussion of areas of common ground, these three essays on seminaries and their trustees and chairpersons are offered to all trustees and directors of institutions of all kinds.\n\nThe three essays, while having different emphases, are not three distinct chapters. They are three discussions of the same subject written a year or two apart. The second and third reflect growth in thinking after the first. They are offered now, not as a definitive treatise on either seminaries or trusteeship, but rather as a sharing of introspection on what I believe is a common interest with the reader: Where can one best put one's effort to build a better society? I see a \"good\" society as one in which there is widespread _faith as trust_ , a quality that encourages people to become constructive influences in the world as it is\u2014violent, striving, unjust as well as beautiful, caring, and supportive.\n\nCaring for persons, the more able and the less able serving each other, is what makes a good society. Of all the institutions I know about, I would put the most stringent test of caring on seminaries because I believe that they hold, potentially, through the churches, the greatest leverage to influence the caring, serving quality of the whole society.\n\nThese essays are offered in the hope that they will encourage trustees of just one seminary to give leadership to theologians who will generate similar ideas about how our low-caring society might do better, who see things whole; and who have the gift of language so they can tell ordinary mortals what they think and see\u2014folk who, by what they do or refrain from doing, make or break the society.\n\n### THE SEMINARY AS AN INSTITUTION\n\nI venture to discuss the seminary as an institution, not as a theologian, but as a student of organization. I hope that what I have to say from this perspective will be of interest to all constituencies of seminaries, but my primary concern is to speak to trustees, particularly the chairpersons of trustees, who may not have been trained in a seminary. It may be helpful if I share how I came by my interest in organization and what leads me to a special concern for seminaries.\n\nUp to 20 years ago, most of my concern about organization was directed toward one large business where I had my major career: American Telephone and Telegraph Company. In my later years there, I held the position of Director of Management Research. With the help of a professional staff, and within a broad charter, I was both involved and detached so that I could study and advise regarding the management of this huge institution (over 1 million employees) immersed as it is in sophisticated technology, elaborate human organization, and regulated public service. I was concerned, among other things, with its values, its history and myth, and, intimately, with its top leadership. I learned the hard way about the profound influence that history, and the myths of institutions that have considerable history, have on values, goals, and leadership. And I was painfully aware of the cost in those terms of any insensitivity to history and myth\u2014especially among the top officers.\n\nThis experience at AT&T gave me a good perspective and the impetus to venture, in my retirement years, into a close working relationship with a wide range of institutions: universities (especially in the turbulent sixties); foundations (both as trustee, consultant, and staff member); churches (both local, regional, and national) and church-related institutions; professional associations, and businesses\u2014in the United States, in Europe, and in the Third World. This brought me, in time, to a realization of the crucial role of trustees in the performance of any institution, anywhere, and the strategic leadership opportunity in the position of chairperson. This led to writing a monograph on \"Trustees as Servants\" [available from The Greenleaf Center].\n\nIn my later years, as I have had more time for reflecting and writing, and as I have become more aware of the fragile nature of our whole society, I have asked myself the question, where, in the vast range of institutions that serve us, is there the greatest unrealized potential\u2014potential for constructive influence on the whole society? My answer is, of course, a subjective judgment, but I have concluded: _seminaries_ , seminaries of all faiths, particularly in our country. Simply in terms of the prudent use of resources, I believe that more can be done to raise the quality of our total society than in any other one way, _if_ seminaries of all faiths could move closer to their full potential as servants. (The word _servant_ has special significance to me; but rather than attempt to define it explicitly at this point, I would prefer that the meaning that word has for me be permitted to evolve as one reads further.)\n\nI have had a fairly long contact with seminaries because, in my AT&T role of concern for values of that institution, I sought a relationship with professors of ethics in Protestant, Catholic, and Jewish seminaries. Some of these developed into lasting friendships which endure to this day. I invited some of these professors to meet with my AT&T colleagues, and I arranged for them to lecture in programs for executives. My relationship with them was close enough to permit me to have an insider's view in several seminaries. Then, later, I served six years on the Visiting Committee at Harvard Divinity School. More recently, I have read some of the history of seminaries and have talked with historians about the implications of this history.\n\nEven without this involvement with seminaries, out of work with churches and church institutions and my later introspection about the structure of society, I believe that I would have come to my present view regarding the crucial role available to seminaries. I hold this view because of the strategic space seminaries seem to me to occupy, or could occupy, in the hierarchy of institutions in which seminaries have the opportunity to give sustaining support to churches, and churches, in turn, can give religious nurture and moral guidance to individuals directly and support (as conservers of traditions and advocates of values) the whole gamut of \"operating\" institutions: governments, schools, businesses, hospitals, social agencies, philanthropies, families, communities. The strength and durability of our democratic institutions rest on the wide acceptance and practice of a common moral code. And I see the seminaries as having the potential to hold the prime position of supporting the churches as they work to foster religious concern and to strengthen the moral basis of society.\n\nWhen I say that seminaries have the opportunity to give sustaining support to churches I mean much more than training their pastors, important as that service is. I see the opportunity for the seminary to stand as a constant source of intellectual rigor and prophetic vision, of spiritual energy and as the support and inspiration for strong leadership and society-shaping influence in the churches. One of the simple truths that survives from my business experience is that it is possible for able conceptual leadership in a central staff to sustain exceptional performance in an almost unlimited number of local operating units which, on their own, would be quite ordinary. Further, in my experience, this exceptional performance is best achieved by persuasive conceptual leadership, without coercion or authority. At the core of this unusual business is an assembly in basic science that has been called the closest thing to a university outside the university, and that has produced, in its time, seven Nobel Laureates. This, I believe, is the source of the strong conceptual leadership that I experienced. If a business can make this principle viable, a seminary also has the opportunity to do it\u2014and better. _Please note that I have only said that the seminary has the opportunity, because of its place in the hierarchy of institutions, to give this leadership. If it is to exploit this opportunity, it must generate sufficient leadership power\u2014with its own resources_. (Foundations have a similar opportunity, largely unexploited, to serve colleges and universities.)\n\nAs I see churches today\u2014churches of all faiths\u2014they are struggling to shore up a deteriorating society without much sustaining support\u2014from seminaries, or, with few exceptions, from anywhere else. This brings me to the observation that seminaries (from the perspective of a student of organization), in general, and as they now stand, are marginal institutions. They are judged marginal because, in our highly institutionalized society, they do not carry the weight of influence and leadership that their place in the scheme of things makes possible. I sense that they are marginal in their own self-image, in the eyes of their principal constituency, the churches, and in the public view. While seminaries perform an acknowledged necessary function, the training of pastors, they do not generally provide the sustaining support and prophetic leadership for churches, for which they are correctly positioned and of which I believe they are potentially capable. The gap between seminaries and churches is not likely to be healed by a reach from the churches. The initiative rests with seminaries.\n\nIn saying these things, I am not suggesting for seminaries an unrealistic utopian achievement. What is reasonable and possible with available resources, human and material, is the standard I would apply. I would like to see all seminaries, seminaries of all faiths, reach from where they now are to attain much more of the reasonable and possible\u2014each seminary with its own resources and within its own vision of what it ought to do and to be. I am confident that any seminary that makes this reach, and thereby moves from the marginal stance toward a central and crucial role, will find a much more rewarding institutional existence than what many now enjoy. And I believe that the typical seminary can make this movement, over time, by a series of prudent steps\u2014prudent, but not necessarily comfortable. Prudent, because each seminary will build from its own resources, human and material, with understanding and respect for, and within the vision defined by, its own history and myth. It will not be simple or easy, and it will be a major creative challenge. The first step for a seminary may be to accept that, next to survival, building leadership strength\u2014to lead the churches\u2014may be its first priority in these times.\n\n#### _Premise_\n\nOccasionally institutions of all categories (including seminaries) rise to the exceptional under the long-term direction of an unusually able administrator. But, usually, when that administrator leaves, the institution lapses, in time, back to the ordinary. There are not enough of these exceptional administrators around to fill all top administrative posts of all institutions with such persons. Even if, potentially, such gifted administrators were abundant, the common wisdom does not yet provide a way to bring them all into top positions. Able top administrators, with occasional exceptions, have always been, and probably always will be, just ordinary, good people. The challenge is how to bring an institution to exceptional performance with ordinary good people to administer and staff it. The premise here is that ordinary good people as trustees, and with such a person chairing the work, can do it\u2014 _if_ they are sustained by a concept of greatness as their goal; greatness as servants of society\u2014in the case of a seminary, greatness in service to churches and through churches to society. What kind of belief builds confidence in this premise?\n\nFor one who grew up in the Christian tradition as I did, the roots, of course, are biblical\u2014\"whoever would be great among you must be your servant, and whoever would be first among you must be your slave\"\u2014(Matthew 20: 26 and 27). The choice of the word _slave_ for the one who would be first suggests that the trustee chairperson will have an unusual dedication and caring concern. My own personal credo is stated thus:\n\n_I believe that caring for persons, the more able and the less able serving each other, is what makes a good society. Most caring was once person to person. Now much of it is mediated through institutions\u2014often large, powerful, impersonal, not always competent, sometimes corrupt. If a better society is to be built, one more just and more loving and providing opportunity for people to grow, the most open way (and the most effective and economical course, while supportive of the social order) is to raise the performance as servant of as many institutions as possible by voluntary regenerative actions initiated within them by committed individuals: servants_.\n\nPart of the human dilemma in our times is that there is little in our biblical tradition, or anywhere else, that suggests what makes a servant\u2014or a serving institution\u2014in our highly institutionalized society. There is evidence that ours is a low-caring (poorly served) society, and concern for this condition has resulted in large-scale actions by government (1) to provide money to alleviate suffering and correct injustice, and (2) to give rewards and penalties to compel both individuals and institutions to behave in more socially constructive ways. In an imperfect world, some of these actions will always be needed. But compulsion is of limited use except to restrain destructive behavior and encourage conforming acts, and money does little other than cushion suffering and injustice. Neither compulsion nor money has much utility in causing institutions to be reconstructed as more caring and more serving. Only the voluntary actions of people inside an institution can accomplish this.\n\nThere is a stirring within some institutions today to improve their serving qualities. But this seems to be mostly exhortation or tinkering with procedures, \"gimmicks.\" Much of it is a sort of \"aspirin\" treatment that produces a short-lived easing of the pain, but little fundamental change may result from it. However, these efforts have produced a burgeoning literature. The occasional institution that achieves high servant stature under the direction of a gifted administrator who has such a goal is usually conspicuous by the absence of the common \"gimmicks\" (such as those that are promoted under such labels as \"Management by Objectives,\" \"Planned Program Budgeting,\" etc.). A strong, healthy institution, as with a person, is not built by the aspirin treatment.\n\nI cite these conditions not to pronounce a jeremiad on our society (although such may be warranted). Rather, I simply contend that there is little to date in our collective experience with institutions that trustees of a seminary may draw on as they seek to raise their seminary to greater servant stature. If the assumption of the hierarchical arrangement (seminaries to churches, and from churches to individuals and to operating institutions) is a valid one, it may be that other institutions will follow, rather than set models for, seminaries. How to produce an exceptional serving institution with ordinary good people to staff and lead it is the goal. If a few seminaries can demonstrate how, other institutions may be encouraged to try.\n\n#### _How to Do It_\n\nIf, as has been suggested, seminaries have the opportunity to provide strong leadership to churches, and churches, in turn, can provide it to both individuals and to the many operating institutions, then, in concept and vision, a seminary needs to be wholly self-regenerating. No other institution stands as an available resource for seminaries. This means that a seminary, of all institutions, needs the caring support of able and dedicated trustees who are led by an unusually strong chairperson. And these trustees will need to find their way to a constructive role in helping to move the particular seminary from marginal status toward its optimum as a servant of the churches\u2014and society. The question is, how to do it?\n\nThe answer to this question will need to be discovered for each seminary, for itself. Most seminaries have considerable history. There are doctrinal differences between schools and within faculties. Their faculties and administrators are unique individuals, they vary in age and adaptability, and most have quite firm, and probably differing positions regarding what a seminary ought to be. Financial resources differ. There are widely varying patterns of relationship to constituencies. All of these aspects combine to give the particular seminary its individual character, and the critical aspects need to be taken into account by trustees who wish to give the seminary new leadership. If trustees are to give new leadership, one generalization is risked: New seminary leadership from trustees is not likely to be asked for by faculty or administrators; nor is the overt offering of such leadership apt to be welcomed. But, then, rarely is leadership asked for or welcomed in any situation short of a condition of overwhelming confusion or imminent disaster. _Leadership_ is one of the most talked about and least understood ideas in contemporary discussions. At its best, leadership is a subtle process. As the ancient Taoist proclaimed: \"When the leader leads well, the people will say 'we did it ourselves!'\" Leadership needs to be offered, and, if effective, it is voluntarily accepted.\n\nTrustees of a seminary who choose to assert constructive leadership are confronted with an anomaly: they are charged by law with _managing_ the institution, and they stand in the position of ultimate responsibility. They possess coercive power, and all who hold coercive power find their ability to lead is clouded by that fact, because leadership rests mostly on persuasion and the responses to it are voluntary, not coerced. And when any person who holds coercive power attempts to lead, there is apt to be the unasked question, \"Where is the hidden agenda?\" Further, while trustees hold legal power, faculties, by long tradition, have evolved a de facto power that is probably more potent. Evidence of the effectiveness of new constructive leadership by trustees may be the emergence of new transforming leadership among faculty and administrators, and _this_ is the leadership that will move the seminary from a marginal to a central or crucial role.\n\nWhat can be said at this point, in advance of experience to confirm the above assertions, about the distinguishing characteristics of constructive trustee leadership in a seminary?\n\n#### _Constructive Trustee Leadership_\n\nConstructive trustee leadership begins with continuing to do well what trustees generally do now, such as: serve as legal cover and public image builder for the institution; raise money; authorize and audit the use of money; render gratis consulting services; take those actions required by the charter and bylaws; appoint or dismiss administrative officers; and stand as the court of last resort if internal administration and staff reach an impasse. These are all useful services that help keep the institution viable, but they are maintenance functions; they do not constitute _leadership_ , in the commonly accepted meaning of _lead_ , to go out ahead to show the way.\n\nParallel with these usual activities, and perhaps independently, trustees will start the search for the most constructive leadership role for themselves, for this seminary, and in these times. What is sought, it is hoped, is a prudent evolutionary movement that is congruent with the seminary's development up to now. What might be the direction of this movement?\n\nThe idea of trusteeship goes back to the person who was entrusted to manage the affairs of another. The concept of a governing board of trustees of an institution is relatively new because the vast structure of institutions that serve us now is rather recent\u2014in the last 200 years, with much of it in this century. Two hundred years ago there were governments, armies, churches, and a few universities. But the common lot was life in families and simple communities with most people partly or wholly self-sustaining with the support of modest commerce. To the extent that lives were guided by a theology, it was a theology of persons.\n\nThe American constitution makes no mention of the common form of institution, the corporation, because they scarcely existed when our constitution was written. Corporations get their legal status because of the willingness of the courts to construe them as persons. Likewise, our available theology usually turns out to be a theology of persons, and there is little to turn to in our theological resources for guidance as to how institutions shall be brought under the shelter of theology\u2014and thereby be made more humane and more serving. If one views _all_ institutions, large or small, as I do: as intricate webs of fallible humans groping for meaning, order, and light\u2014then, at the core, the problem of all institutions is, has always been, theological. Seminary trustees who seek a more constructive leadership role may find that the best basis for eliciting a new transforming leadership from faculty and administrators will be to start the deliberations leading to a new theology of institutions which the seminary, with the involvement of all of its constituencies, will labor to bring forth, and with the seminary's own institutional development as the laboratory.\n\nThe terms _theology of persons_ and _theology of institutions_ are not in common parlance. They are used here to suggest that, to a nontheologian like myself, who is concerned about the crucial role that seminaries could carry, there is a question about the adequacy of the theological resources of seminaries to support churches so that churches can be of greater help to legions of people who are immersed in the mass of institutions that make up contemporary society. Trustees may want to raise the question, \"Are the theological resources of the seminary\u2014whatever they are\u2014adequate for this particular support?\" Whatever the answer is, there is an opening for dialogue between trustees and faculty and administrators on what may prove to be one of the more important issues of the late 20th century\u2014important for seminaries, for churches, for all of us.\n\nIf a seminary trustee is, as I am, a nontheologian, she or he will venture with caution in raising a question about the adequacy of theology. Yet, as a trustee, the issue must be faced\u2014just as a layman who is trustee of a hospital must be prepared to question the practice of medicine, or the lay trustee of a university must be prepared to question the process of education.\n\nThe adequacy of any theology is tested, ultimately, by examining what it produces in the lives of people who have its implications interpreted for them through the mediation of churches. The issue of the adequacy of the available theology of institutions is an appropriate concern for seminary trustees who are searching for their most constructive role because _this_ theology addresses both institutions in general and _this_ seminary as an institution, in particular.\n\nIt is an interesting question. Anyone can make a contribution to theology; but established theologians are best positioned to originate and advocate a new theology of institutions\u2014 _and be heard_. However, nontheologians who are keeping a close watch on our institutions will make the ultimate judgment on the adequacy of that theology. One test they will apply is: What comes through to trustees and directors of institutions of all sorts, for profit or not for profit, that induces them to take and sustain initiatives that result in the institutions they hold in trust becoming substantially more serving than most now are to all the persons they touch? Other nontheologians will make other tests of the adequacy of other theologies\u2014in terms of their ultimate consequences in the lives of people.\n\nConstructive trustees of any institution will listen carefully to those who make the ultimate tests of adequacy of that institution's service. What makes seminary trusteeship especially interesting and challenging is that the seminary has the potential for influencing such a wide gamut of human affairs.\n\n#### _Preparation for Trusteeship_\n\nHow can seminary trustees initiate the move toward a more constructive role for themselves, and give the new leadership that helps bring the seminary to optimal strength as a servant of society?\n\nThey might first seek to clarify for themselves, and for all constituencies, what the history and myth of their particular seminary is, and they may find much of it is not written and that no single person knows it all. ( _Myth_ is used in the sense of a story that relates historical events and that serves to explain some practice or belief.)\n\nThen they might gather and study the literature on seminaries in general, which is considerable, and of varying quality, and encourage other constituencies to do the same. There have been studies, some recent, by individuals and groups that raised the issues about seminary purpose and the content and method of theological education.\n\nTrustees, particularly their chairperson, will need to take the time to meet and discuss with faculty, administrators, students, alumni, church representatives, and denominational officers\u2014if it is a denominational seminary. Trustees might attend faculty meetings, student assemblies, and occasional classes in rotation\u2014all if invited. Also they could meet with students in small groups where issues in the seminary are being discussed. While this work may be shared, seminary trustees will do well to accept at the outset that their role with this unique kind of institution will make a large claim on their time.\n\nOut of these discussions, the trustees will discover that seminaries, in the course of their history, took on the values of universities and that their faculties tended to assume the characteristic university position, \"the faculty is the university.\" This position in both university and seminary may have come about by default. No one else, trustees for instance, concerned themselves with defining the institution, so faculties, with their large career stake, defined it as themselves. The first concern of seminary trustees in search of a new leadership role may be to search for the idea, the unifying vision, that will define the seminary.\n\nConceptual leadership by trustees may best be manifested by questions. The trustee is the asker, not the answerer of questions. _Whom and what purpose should this seminary serve_? This may be the prime question to be asked of all constituencies of the seminary. The faculty contribution to answering that question will be very important, but not the only source. Gradually, faculties may be led to accept the _servant_ role: that their best part is to serve whatever purpose evolves from the deliberate search conducted by trustees, a search in which they have had a majority but not exclusive role.\n\nI suggest that the key to greatness, in a seminary or in any other institution, is the quality of the dream, the vision, and the primacy of that dream as the ultimate government. Every person, every constituency, stands subordinate to the dream. Helping the dream to evolve and nurturing it is the first consideration in trusteeship.\n\nTrustees, in giving leadership, should be ever-mindful of the sensitivity of faculty members to the faculty role in governance and of their apprehension about the trustees' use of their power. Faculty will be concerned that trustees have a sufficient understanding, in depth, of the faculty's perception of the nature of a seminary.\n\nSeminary trustees might also be aware, as they pursue the search for the answer to the question of purpose, that they will never answer it, definitively or for all time. Yet, at any one time, the seminary should be proceeding with assurance to serve a clearly identified constituency with a firm purpose in mind. Part of the art of trusteeship is to maintain a balance between _order_ and _stability_ and _search_. Search, for any institution or any person, should never end. Balance between the two is important, because, in any institution, there is an optimum tension between order and search. Either too little or too much tension is destructive. Further, except for the rare seminary that has funds at its disposal to engage new staff for new programs, trustees will acknowledge that their seminary can undertake only what they, the trustees, can persuade the existing faculty to accept.\n\nOnce a workable answer, for the present, has been achieved for the question, _Whom and what purpose shall this seminary serve_? the next question for trustee concern might be: _What shall be the content of the teaching and how shall it be taught_? Here, again, the trustee is the asker, not the answerer of the question. The aim is to find the best possible curriculum and method to implement the goal established in answer to the first question. Again, faculty opinion will be important, but not necessarily governing. And, again, unless there is money for new staff, the seminary can undertake only what the trustees can persuade the existing faculty to accept and _learn to do_. Two further questions may be asked by trustees: _How can the seminarybest support churches? How can the seminary be a constant source of prophetic vision?_\n\nWhen Karl Marx sat alone in the British Museum evolving the theories that would shape so much of the 20th-century world, was he not filling a void that resulted from failure of the churches of his day to come forth with a new prophetic vision that the conditions of the industrial revolution made imperative? But Marx apparently did not foresee the institution-bound society that was evolving as he wrote. And Marxist nations, today, are plagued with the same problem we are: how to make their complex institutions more serving. Thus there is now another void, worldwide\u2014a crisis of institutional leadership. Who will produce a new vision to fill _this_ void?\n\nThoughtful people today are complaining about ours being a \"media-dominated\" culture, to our detriment. Could it be that, in the absence of an adequate theology of institutions, the media are simply filling the void?\n\nThis question helps define the opportunity for new trustee leadership in seminaries.\n\n#### _Persuasion\u2014Prime Art of Trustee Leadership_\n\nTrustees, if they are to lead constructively, will understand the meaning of, and be gifted in, the art of _persuasion_.\n\nOne is persuaded, I believe, on arrival at a feeling of rightness about a belief or action through one's own intuitive sense\u2014perhaps checked by another's intuitive sense but, in the end, one's own intuitive feeling will govern. One takes an intuitive step from the closest approximation to certainty to be reached by conscious logic (sometimes not very close) to that state in which one may say with conviction, \"This is where I stand.\" The act of persuasion will help order the logic and favor the intuitive step. And this takes time! But the one being persuaded must take that intuitive step alone, untrammeled by coercion or manipulative stratagems. Both leader and follower respect the integrity and allow the autonomy of the other; and each allows and encourages the other to find his or her own intuitive confirmation of the rightness of the belief or action.\n\nIf persuasion is ever to rise over coercion and manipulation as the prevailing modus operandi in our violence-prone society, then the clear model of an institution that is led by persuasion alone needs to emerge in the most influential spot possible. I believe that some seminaries have the potential to occupy that spot\u2014to be influential models for other institutions, to _be the civilizing models!_\n\nIn summary, trustees have the opportunity, by astute and persistent questioning and persuasive leadership, to encourage the seminary to: (1) _Evolve an adequate theology of institutions for these times_. (2) _Define in contemporary terms whom and what purpose this seminary can best serve_. (3) _Find the best curriculum and method to serve those people and that purpose_. (4) _Give effective support to churches in the full scope of their missions_. (5) _Become significant sources of prophetic vision, places that are looked to for guidance and light_. A seminary that can achieve these through its existing staff and with its present resources will become a model for the regeneration of all kinds of institutions.\n\nGlittering innovations can always be produced by new people and new money. How can we do better with the people and the money we now have? This is the urgent plea of so many who would become constructive trustees or directors of the all-too-many low-serving institutions of our times.\n\nWhat guidelines might be useful to the seminary chairperson who would like to make the first move\u2014to do better, much better, with the money and the people the seminary now has?\n\n#### _The Chairperson_\n\nAll leadership positions are to some extent lonely posts. The chairperson of seminary trustees is no exception. Even if, among fellow trustees, there are close friends, some detachment is required. One's leadership strategy, whatever it is, should involve one with all trustees on the same terms. While there will be occasional conversations with individual trustees, there should be no grounds for suspicion that some trustee responses are prearranged. I know this contradicts the perception of some experienced trustees; but I am postulating here some conditions for a level of trustee performance that most have not experienced. The chairperson's role _should_ be a lonely one. The trustee is what the title implies, the bearer of trust. And the chairperson, if well chosen, stands tall among colleagues in this dimension.\n\nIf the above is a reasonable assumption, and if the chairperson of a seminary wants to give leadership to trustees who will have a chance of being the constructive influence described earlier, then a confidant or coach for the chairperson should be sought. This should be someone who is completely uninvolved in the seminary. Every person who carries a difficult or sensitive role needs a confidant or coach. The administrators of the seminary may need such help. But the need of the chairperson is perhaps the most difficult to serve.\n\nIn choosing to relate to a confidant or coach, the chairperson is acknowledging that if the seminary is to move toward its higher potential as a servant of society the one who occupies the chair must be the first to become a learner. But since there is no clear body of knowledge about how to chair effectively, the one who occupies that spot must evolve the strategy for giving new leadership out of one's own experience through interaction with a coach.\n\nAll trustees should, as individuals or through committee participation, have some firsthand relations with all constituencies\u2014especially faculty and students. There should be sufficient interaction so that, in their official deliberations, trustees have an experiential basis for good intuitive judgment on matters of trustee concern. The chairperson should have, in addition, a close relationship to administrators so that there is encouragement for dynamic leadership by all administrators. Also, the chairperson will want to help other trustees gain perspective for their evaluative judgment about administrative performance. This is a subtle process: to be close enough to administrative action to be helpful and encouraging, and to be critical and evaluative, but still to be detached enough so that administrators have sufficient scope for initiative and creativity to carry their roles with spirit. The chairperson should instruct the coach and confidant to be particularly watchful over this relationship between the chairperson and administrators.\n\nAll trustees should participate in evaluating administrators. But the chairperson should be prepared to provide more extensive data about administrators than any other trustee is likely to have.\n\nThe chairperson also has the obligation to build respect for trustee judgments among all constituencies. Trustees, by law, have the power to govern. But like all such power, its value is inverse to the extent of its use. It is important that the power be there, but _influence_ , growing out of respect for trustee judgments, is the critical trustee asset. The chairperson, in consultation with the coach-confidant, should develop a clear strategy for building and sustaining trustee influence. This influence will depend on the evident trustworthiness of trustees as demonstrated by: _(1) The quality of greatness of the dream that evolves for the seminary and what it might become as a servant of society. (2) The dedication of trustees in terms of their investment of time and their caring for people\u2014all of the people in the seminary. (3) The adequacy of their information about the seminary. (4) Their ability and disposition to persuade. (5) Adherence to collegiality and careful group process in reaching their judgments\u2014a process that is understood by all constituencies_.\n\nTrustworthiness, in the above terms, augurs for confidence in trustee judgments by all who are asked to accept them. The chairperson will be aware of elements like these that build trust, and will be watchful and diligent to encourage their optimum manifestation among seminary trustees. This, it seems to me, is the essence of leadership from the chair.\n\nNot all persons who have a talent for leadership are effective in all situations at all times. If the seminary chairperson finds, after a reasonable time, that his or her leadership is not adequate for this institution at this time, then they should quietly step aside so that someone else can try. Every seminary, every institution, deserves an effective leader in their chairperson\u2014all of the time.\n\n#### _Summary Comment_\n\nThree words are commonly used in reference to the oversight of an institution: _manage_ , _administer_ , and _lead_. Manage (from _manus_ , hand) connotes control. _Administer_ (from _administrare_ , to serve) has the sense of \"care for.\" These two, manage and administer, may be seen as the maintenance functions, terribly important but concerned mostly with conservation and perpetuation.\n\n_Lead_ is a word of less certain origin. Whereas management and administration usually derive from delegated authority from trustees, _leading_ in the commonly accepted meaning of going out ahead to show the way, can be undertaken by anybody. The only test of leadership is that somebody follows. Following, as distinguished from compliance, is voluntary. Thus _strong_ leadership can bring unity and clarity of purpose and uncertain leadership can bring disorder and chaos. The latter was conspicuous in the university world, including some seminaries, in the late 1960s.\n\nThose who manage or administer in a seminary can also lead\u2014and most do some leading. But consider the five basic questions cited above. _Are the theological resources of the seminary adequate? Whom and what purpose does the seminary serve? What should be the content and teaching method of the seminary? How best to serve the churches? How can the seminary be a constant source of prophetic vision?_ These are questions that are best dealt with if there is at least supporting leadership from trustees. Trustees, in their more detached position, can afford to be persistent questioners and visionaries. Administrators, to whom have been delegated the essential maintenance functions, may find that if they undertake this type of leadership alone, it interferes with administrative obligations. Faculty may more readily respond, with transforming leadership of their own, to a trustee initiative than to some administrative initiatives. If trustees will take the risk of leadership toward a larger dream for the seminary, they may do it in a non-confrontational, persuasive way that elicits supportive leadership from other constituencies. Administrators may best give supporting leadership on some matters, if trustees will lead by questioning and persuading.\n\nIf a seminary is to move toward its full potential as harborer and nurturer of prophetic voices that give vision and hope, what is needed first is leadership that wants to take the risk of such a larger dream. And it is a risk. All leadership is nurturing and risking. Trustees and their chairperson are more expendable than able administrators. And some of the risks of leadership of a seminary, or any other institution, are best taken by trustees\u2014partly because they can better tolerate the risk, partly because they are better positioned to assume some aspects of leadership.\n\nLeadership does not \"belong\" to anybody, nor can it be bestowed on anybody. If trustees want to lead, they will need to build a role for themselves, including earning the trust that makes it possible. One of the contributions seminary trustees can make to our evolving society is to build the trustee role so as to make of the seminary a distinctive institution, a model for others.\n\nSome respond to the thrust of this essay with these reservations: the hope expressed here for the role of seminaries is too idealistic; trustees are not available who will invest the time required; the money-raising role of the trustees has not been mentioned\u2014most seminaries are poor; and the usual chairperson does not have the leadership skills that the role, as described in this essay, calls for.\n\n**_Too Idealistic_**. What, in the end, will be judged too idealistic is that which is not achieved when the limits of the reasonable and possible with available resources, human and material, are reached. Most institutions I know about, including some seminaries, are so far from pressing the limits of the reasonable and the possible that the charge \"too idealistic\" has little force. The goals suggested here may be flawed on other grounds, but not on that one.\n\n**_Not Enough Time_.** Most able people find the time to do the things that give them joy. In my experience with trustees, whether as participant or observer, I have found some of the proceedings so dull and unrewarding that only a strong sense of duty, a feeling of guilt, or a desire for the prestige of the office would hold a person to it. But the joy of creative achievement will elicit the investment of time\u2014and care\u2014that the full measure of trusteeship requires.\n\n**_Who Will Raise The Money?_** How can trustees be expected to raise money when, in the usual seminary, they have only a maintenance role? Give them the kind of role described here, and they will want to raise money; not out of a sense of guilt or duty, which are not very good motives, but because they understand and believe in what the seminary is about. It is a matter of vital concern to them because they have given decisive leadership in bringing the seminary for the 21st century into being.\n\n**_Inadequate Leadership Skills_**. Granted; the usual chairperson may not have developed the skills of leadership that the trusteeship function described here requires. Even though the incumbent may be potentially capable of acquiring those skills, they are not likely to emerge at a high level of accomplishment because they are not expected. The almost universally accepted casual performance of trustees in all sorts of institutions assures that the full maturing of these skills will be rare. Since leadership is a prime concern in the churches, should it not be a central issue in seminaries, both its theological implications and its sensitive practice? Let me restate my personal credo:\n\n_I believe that caring for persons, the more able and the less able serving each other, is what makes a good society. Most caring was once person to person. Now much of it is mediated through institutions\u2014often large, powerful, impersonal, not always competent, sometimes corrupt. If a better society is to be built, one more just and more loving and providing opportunity for people to grow, the most open way (and the most effective and economical course, while supportive of the social order) is to raise the performance as servant of as many institutions as possible by voluntary regenerative actions initiated within them by committed individuals: servants_.\n\nToo much of the effort to care and serve is directed to easing the hurt of the \"system\" that is grinding people down faster than the most valiant rescue effort can help them; and too little caring effort is going into rebuilding the \"system\" (institutions) that will give greater assurance that those being served grow as persons; _while being served_ , they become healthier, wiser, freer, more autonomous, more likely themselves to become servants.\n\nIf the _performance as servant_ of as many as possible of our institutions is to be raised (a necessary condition if ours is to become a more caring society), then somewhere there must be an initiative that starts the movement toward this end. If my premise regarding the hierarchy of institutions\u2014seminaries to churches to individuals and operating institutions\u2014is a valid one, then seminaries are best positioned to take this initiative.\n\nWhoever takes this initiative (and I hope it will be seminaries) will accept the challenge to bring into being a new teachable art of chairing\u2014beginning with the chairperson of the seminary itself.\n\nThen, drawing on its own experience, the seminary may find that it is in possession of the stuff for a prophetic vision for these times. This is my hope for the seminary as an institution.\n\n\"What is now proved was once only imagin'd.\"\n\n_William Blake_\n\n### MISSION IN A SEMINARY\n\nOne of the easiest to ask and hardest to answer of all questions is, _What are you trying to do?_\n\nIt is hard enough for an individual to answer that question. But it is much more difficult for an institution to answer, and most difficult of all for an eleemosynary institution that has both the momentum of tradition and the support of sentiment to sustain it. There is always the possibility (and the temptation to hope) that enough sentimental support can be stirred so that the tough question can be avoided: _What are you trying to do?_\n\nThis essay explores the problem of dealing with this question of \"What are you trying to do?\" by trustees of a seminary, with particular reference to the strategic position that seminaries occupy in the culture and the great, but largely unrealized, potential influence in that position.\n\n#### _Who Dominates the Culture?_\n\nFrom my perspective, ours is a business-dominated culture. The business influence is there, not so much because of the magnitude of the business presence, nor because businesses are more committed than others to serve society, but because the missions of businesses are usually better defined than in other institutions. And they get that way partly because business executives usually have more coercive power to govern than do executives in eleemosynaries, partly because the pursuit of private gain and the need to survive in the competitive struggle require that missions be clear and forceful.\n\nI am writing this essay because I believe that dominance of the culture by elements like coercive power, private gain, and survival in the competitive struggle do not make for the quality of society that is reasonable and possible with the resources we have. Criticism of business and restraining its influence by government will not, in my judgment, produce the quality of society that is realistically possible. What will build a better society I believe, and what is well within our resources to do, is to bring to prominence in the culture those institutions whose primary motive is caring for persons and that are governed by persuasion. My hope is that, in time, such predominance will wield a meliorating influence on the harshness of both business and government. A necessary first step is a statement of mission for all eleemosynary institutions that is clear and powerful and that puts a high priority on the contribution of each institution to the quality of society because of what it is, what it does, and how it is governed. In my view, the best placed institutions to initiate a definition of their missions in these terms are seminaries and foundations. This essay will be concerned with seminaries.\n\nThe test that I would put on the adequacy of a seminary statement of mission is: What does it promise to contribute, through leadership of and service to churches, to building and sustaining a good society? A good society is one in which there is widespread _faith as trust_ , a quality that can encourage large numbers to become constructive influences in the world as it is\u2014violent, striving, unjust as well as beautiful, caring, and supportive. Others might postulate other tests, but this one suggests the premise on which this essay is based.\n\nStatements of mission in a seminary in addition to giving clear guidance to informed constituencies, also need to be persuasive to people unversed in the technical language of theology, particularly to those who fund seminaries. If contemporary seminaries were all adequately funded, if churches were receiving the sustaining leadership they should have from seminaries, and if seminaries attracted the students they should have in order to provide churches with able professional leadership, I could not make a case for the need for new statements of mission for seminaries. But the need is urgent on all three counts.\n\nIn addressing the subject of \"mission in a seminary,\" I have a concern for two pervasive social problems: widespread alienation in all strata of the population and the inability or unwillingness to serve of far too many of the institutions that make up our complex society. Each of these problems may be a contributing cause to the other, and neither is likely to be healed without coming to terms with the other.\n\n#### _Whom and What Purpose Shall This Seminary Serve?_\n\nIn the earlier essay, \"The Seminary as an Institution,\" I suggested that the first question from trustees who want to be effective in their roles is: Whom and what purpose shall this seminary serve? This could be the first question that the trustees of any institution might ask because the answer to it defines the mission. If this question is asked of all constituencies of the seminary, and if trustees work patiently and sensitively for consensus, a mission may be defined that not only helps finance the place and brings the students the seminary should have, but it becomes a vital element of the ultimate government of the seminary\u2014by guiding and assuring those who govern\u2014thereby adding greatly to its strength as an institution.\n\nEvery person in whatever relationship, whether administrator, faculty, student, or staff, is the _servant of the mission_. All have appropriate and clearly defined roles as servants of the mission. Every significant policy or decision, every act of teaching, the formation of every influence from or to any constituency, is made with reference to that mission. There will be problems, of course, but clarity and power in the statement of mission will help resolve them. While primary power in the seminary is lodged in the commonly acknowledged mission, people in the seminary\u2014administrators, faculty, and students\u2014also have power, but that power is subordinate to the governing idea. When the mission is clear and compelling, people in the seminary do not ask (as I heard them in one instance) \"who has the power?\" The seminary in which the power of the mission is primary stands among other institutions as a model in which issues are met in creative fashion with reference to \"what does the mission require of us?\"\n\nIn any institution (including seminaries) that is comprised of fallible humans, it is important that the legal power to govern be clearly assigned. But the quality of the institution will be inverse to the use of that legal power.\n\n#### _What Is Worthy of Survival?_\n\nOne of my sharply etched memories from the now distant past is a view from my office window, high above the east bank of the Hudson River in lower New York with a commanding view of the harbor. Much of the busy river traffic was made up of tugboats with groups of barges in tow. This scene was most interesting, sometimes frightening, when the tide was running out, perhaps with some wind behind it, and a tug was taking its group of barges downstream. The problem for the tug, under these conditions, was to sustain enough pull on the cables to the barges to keep them in orderly movement toward their destination. Under these adverse conditions, any slackening of that tension could result in chaos.\n\nThe memory of those scenes on the river came vividly to mind in the 1960s when I watched at close range as several venerable universities lapsed into chaos and seriously damaged themselves. When destructive forces were unleashed (and such are always latent in any institution) the power of the sense of mission in the university, like the pull of the tug on its barges, was not enough to keep a guiding tension on the several constituencies of the university. The answer to the question, \"What in the institution is worthy of survival?\" was not clear. There was not enough power in the governing ideas.\n\nIf either is faced with potential chaos, there is a parallel between the position of the skipper of the tug and the executive of any institution. The power to prevent chaos does not reside with either. That power is in the _engine_ of the tug and in the _mission_ of the institution. An able skipper or executive will better manage the chaos than inept ones, and thus render it less damaging. But given sufficient adverse conditions and not enough power in engines or mission, neither executive nor skipper holds the power to prevent chaos! All they can do is try to manage the chaos as best they can; but it is still chaos.\n\nTo the owner of the tug, I would say, \"Don't dispatch that tug downstream when the tide is running out unless there is enough power in the engine to handle potential problems.\" To the trustees of any institution I would say, \"Use what influence you have to get accepted a powerful governing idea (mission). If, after a reasonable effort, you conclude that you can't make it, find yourself something better to do. _It is a breach of trust to hold the position of trustee of any institution and to acquiesce in an inadequate statement of mission_.\"\n\nA great contemporary idea of what a university should be doing would make a better university at any time\u2014before, during or after the 1960s. Some university missions may have been inadequate for a long time, but in that critical period of testing, some were found to be inadequate\u2014by the simple test that their missions at that time did not have the power to hold against the forces of chaos. There is much to be learned from those traumatic experiences. Has it been learned?\n\nWhat is worthy of survival? _An institution with a great mission!_ \u2014particularly a seminary with a great mission.\n\n#### _Why Is Mission Important in a Seminary?_\n\nIt is important to avoid chaos because it wastes resources and hurts people. But that is but one of the benefits to be derived from a strong sense of mission in the seminary\u2014or in any institution. Clear and powerful mission is important because it provides the substance of common purpose that all constituencies might be persuaded to accept. Once accepted, common purpose helps to heal, for all involved within the seminary the pervasive alienation of our times; it helps keep priorities in order; it builds organizational esprit; it clearly signals who should and should not be in the seminary\u2014as students, faculty, or staff as well as the qualities needed in the administration; it provides the basis for resolving, on a reasoned objective basis, the many issues that arise in the course of any institution's life with a minimum of reference to assigned power; and it provides the stimulus and direction for the optimal performance of the seminary as the servant of all who are touched by its program.\n\nThese are qualities that are of great value to all institutions that have the potential to be influential\u2014like seminaries\u2014in order to assure a reasonably civilized society.\n\nThere are several ingredients of a healthy viable institution that mission alone will not provide: among these are, astute leadership and consistently able management. Machiavelli observed: \"It is an infallible rule that a prince who is not wise himself cannot be well advised....\"\n\n#### _Mission and Trustee Strategy_\n\nMost trustees that I know have some awareness of what is stated above. But they believe that a clear and compelling mission, a great governing idea, is best achieved by installing and supporting an able executive. Able executives are always important, and they should be supported, but I suggest that the executive, with rare exceptions, can only manage what is there. If the executive tries, alone, even with the assurance of support of trustees, to lead the seminary to the influential role that it might occupy, his or her tenure as the executive may be short. The rare executive may make it, but it is a gamble against unreasonable odds for a trustee to assume that the executive will make it alone. And there is damage to the institution when the executive tries to give this leadership _alone_ and fails.\n\nOne of the distressing incidents of the recent past was the appointment a few years ago of an able 28-year-old person as president of a college with the explicit charge from the trustees to give the place new leadership (which apparently was needed). However, the trustees took no action on a statement of mission that would require that leadership, nor did they seek acceptance of the need for it from the several constituencies. The result: early resignation of the new president with damage to both the person and the college.\n\nA prudent seminary trustee, one who would like to see a clear and compelling mission for the seminary, will accept that _it is the trustee's role to make the first move toward a clear definition of mission and secure wide acceptance of it, and then to support prudent efforts by the executive who endeavors to lead the seminary to achieve that mission_.\n\nWhat would such trustee leadership comprise, and where does the executive fit into it? In _The Seminary as Institution,_ I suggested that trustees are most effective as askers of questions, and that the first question might be, Whom and what purpose shall this seminary serve? But this is an \"umbrella\" question, and other more penetrating questions will need to be asked\u2014questions that imply criteria that trustees may have in mind for evaluating answers from various constituencies.\n\nThe first requirement is a trustee chairperson who has an understanding of, and skill at achieving, consensus. If a seminary does not have someone in the chair who has that skill and is willing to work long and patiently at using it, or if some other trustee will not endeavor to perform that function (in which case _that_ trustee may effectively be the chairperson without benefit of title), that seminary will do well to make the best of its present role until it can find such a person to chair its trustees. Adequate mission, a widely accepted great governing idea, is not only unlikely to be achieved without that skill in the chair, but (as said above) the seminary may be damaged by trying to achieve it. I see the ability to find consensus as the prime skill in the art of chairing, both in defining mission and in presiding over the ongoing governance of the seminar): as it works to fulfill that mission.\n\nTrustees whose chairperson has this skill are in a position to pursue the question, whom and what purpose shall this seminary serve? I see six subsidiary questions that might be asked in such a way as to suggest the criteria by which trustees will judge the adequacy of the answers. These questions do not define wholly separate and discrete areas and, to some extent, they overlap. But they may suggest to trustees a basis for initiating a dialogue with the several constituencies of the seminary out of which a clear and powerful statement of mission may emerge.\n\n**_1. Does the Seminary See Itself as a Model for Other Institutions?_**\n\nDoes the seminary accept its place at the top of the hierarchy of institutions in which the seminary is the resource institution for churches, which, in turn, serve as support resources for individuals and for the whole range of \"operating\" institutions: governments, businesses, schools, hospitals, social agencies, families, communities? In asking this question, the trustees will be suggesting that the seminary should have a concern for the missions of churches and help churches to clarify their own separate missions, thus preparing themselves for a much greater society-shaping role than most churches now enjoy\u2014society shaping by leading, not by manipulation or coercion.\n\nThis will be a most difficult concept for some seminaries to accept: that they will be models of institutional achievement and influence society, through the churches, more by the example of the quality of life within the seminary than by preachment. If the seminary can reach even a modest achievement as a model of \"power in the mission\" and can clearly demonstrate the effectiveness and strength in an institution that moves on this principle, it could, in time, influence the quality of the whole society.\n\n**_2. Is Critical Thought a High Priority in the Seminary?_**\n\n_Critical thought_ is not necessarily identical with scholarship and the writing of books and papers that establish a scholar's reputation with peers. Scholarship is a necessary condition for critical reflection, but it does not automatically produce it. Critical thought is a fulfillment of the root meaning of seminary: _seminalis_ , seed. It is the reflective thought that produces seminal ideas\u2014ideas that become new visions in both ministers as persons and churches as institutions; ideas that support ministers and churches as they nurture _servants_ who may shape the institutions that dominate the lives of all of us.\n\nThe adequacy of critical thought in seminaries is tested ultimately by ordinary folk who have its significance in their lives made real through their participation in churches. The incremental actions that build the serving quality of society, or maintain such as it has, are taken by legions of people who get inspiration and guidance from somewhere, mostly from churches\u2014directly or indirectly.\n\nWe lack a theology of institutions for these times. I do not fault seminary theologians for this; their trustees are failing to do what only trustees can do: ask the penetrating and persistent questions that ensure the adequacy of critical thought in the seminary.\n\n**_3. Does the Image Created by the Statement of Mission for the Seminary Suggest a Strong Contemporary Institution?_**\n\nThe nature of any seminary closely identifies it with its own history and tradition. It is precisely that tradition that frees the seminary to be always effective in contemporary society.\n\nIt is important that the mission suggest that the seminary, through the churches, is serving this contemporary society and is moving with that society into the uncertain future. Particularly, it should be clear that the seminary is supporting the churches\u2014with the stuff to be strong\u2014as they serve both individuals in their private and family lives and as they prepare, support, and inspire those who mold and shape modern institutions. Seminaries should be seen by informed people as critically important, strong, and useful contemporary institutions. The implication of the word _strong_ is that seminaries are an influential society-shaping force because of the leadership they give to churches. But it is important that both the language and concepts of the statement of mission create the image of contemporaniety\u2014it is the quality of _this_ society that is the seminary's prime concern.\n\nOne of the important learnings from the university debacle of the 1960s may be that more attention to their history might have saved some universities from harm in that difficult period. Clearly, a mission concept that was adequate for 1900 did not suffice in the 1960s. A sense of history keeps one constantly aware that a mission statement that was adequate yesterday may not serve well today.\n\n**_4. Does the Seminary Attract Students Who Have the Potential to Become Influential Leaders of Churches and Church Institutions, and Does It Attract Faculty Who Are Capable of Training Such Students?_**\n\nThose who are attracted as students and faculty are influenced by the image discussed above. They are also attracted by what they know, from informed people, is really there. Is the seminary really strong? Does it prepare people for effective leadership of influential churches and church institutions? Strong people as leaders make strong institutions\u2014because those people have the intellect, stability, integrity, _and spirit_ required to generate _trust_ in those who might respond to their leadership. _Leader_ is used in the common meaning of \"go out ahead to show the way.\" The connotations of _lead_ are quite distinct from those of _manage_ (control) or _administer_ (care for). Leading is venturing and risk taking. \"The way is not clear. I will go out ahead to show the way. Follow me!\"\n\nLeadership is not a professional calling. In any church, anybody can lead who can generate trust in followers. One task of the seminary is to provide as many professionally prepared women and men as it can, persons who have the ability to generate that trust in churches and church institutions so as to give them great influence.\n\nThere may be a place for a seminary that prepares pastors to serve weak or nominal churches. But if, as assumed here, a preferred role of the seminary is to prepare pastors as leaders who will build _strong_ churches and church institutions, then the seminary must strive to admit those students who are most likely to qualify for that service. It is important that the mission attract those who want that kind of demanding life and discourage those who do not want it. One feature of any admissions process is self-selection by potential students and faculty\u2014they tend to select themselves \"in\" or \"out.\"\n\n**_5. Does the Seminary Experience Develop Spirit?_**\n\nThe seminary, like other educational institutions, shapes and forms in ways that we but dimly understand. It is a community that has an ethos that is, in part, a consequence of its mission and how that mission is understood. There is a deeper issue of informing the life of the seminary community which suggests that the literary quality of the mission statement is important. Does it sing, does it carry spirit and inspire qualities of leadership and reaching that are spirit building? Part of the power in a statement of mission is the poetry of its expression that sets the seminary clearly apart from bureaucratic institutions.\n\n_Spirit_ is commonly defined as the animating force within living beings. In a seminary, I would want to add a value dimension to this definition: Is the person who is possessed of spirit disposed to be servant? Do those being served, _while being served_ , become healthier, wiser, freer, more autonomous, more likely themselves to become servants? And, what is the effect on the least privileged in society; will they benefit, or, at least, not be further deprived? No one will be hurt.\n\nWhether the seminary develops spirit, thus defined, seems to me to depend more on the quality of the seminary as an institution than on the precise design of its program. Has the seminary itself evolved as servant; deep down inside, is this what the seminary really is? One does not teach about spirit. One really is spirit.\n\n_Mission_ is a statement of how a seminary sees itself\u2014at its very core. Is it, in its essence, spiritual? Does the statement of mission reflect this?\n\n**_6. Does the Governing Idea (Mission) Have Power?_**\n\nThere is only one test: if the mission does not have power, it will not guide those who govern and people will be without purpose. The advantages of a powerful mission, enumerated earlier, will be only partly realized\u2014if at all.\n\nEarlier I suggested that ours is a business-dominated culture because the missions of businesses are usually better defined than in most other institutions. They get that way partly because business executives usually have more coercive power than do executives of others, partly because the pursuit of private gain and survival in the competitive struggle require it.\n\nMy hope for the future is that our culture will, in time, become less dominated by institutions that are ruled by coercive power, and motivated by private gain and competitive striving, and move toward a quality of society in which voluntary consensus and search for the good are a greater force than they are today. Could seminaries emerge as leaders of this movement by becoming institutions that are distinguished because of the power of persuasion and caring that comes in response to inspired trustee leadership whose primary tactic is penetrating questions?\n\nThese six questions are only suggestions for trustees of a seminary. Each trustee board will devise its own questions. If trustees ask such questions, and thereby give leadership toward defining a clear and powerful mission, what then is the role of the executive? It seems to me that the mission defines that role: it is to take those subtle steps of encouragement and discouragement that keep the seminary clearly centered on growth in achievement of its mission\u2014with as close to consensus as possible.\n\n#### _Leadership\u2014Going Out Ahead to Show the Way_\n\nBoth the person and the institution are damaged if a seminary chews up its executives because they failed when they tried to give leadership in formulating a new and powerful mission. If the trustee chairperson tries to give this leadership to trustees\u2014and fails, that chairperson can quietly withdraw with little damage to the person or the institution. In short, _chairpersons are expendable, executives are not_. Part of the reason for this difference is that the executive is a colleague of the faculty; the chairperson is not (or should not be). The executive has a career stake in the position; the chairperson does not (or should not have).\n\nThe head of the board who is not a seminary theologian (and should not be) can, by leading trustees to ask questions like those above, become a _public theologian_. The executive, who is a seminary theologian, cannot, on her or his own, ask questions like these. The executive's role is to lead the process of finding answers and to serve as interlocutor between trustees and various constituencies whose acceptance of the mission is necessary. This is a sensitive role.\n\nWith help from the chair, trustees may come to accept that the gap between potential and performance in their seminary is wide, but they can hold firm in their belief in the possibility for a great, powerful new mission for the seminary that will, in time, cause a dramatic reduction in that gap and move the seminary into the influential position that it should occupy. As matters now stand, seminary theologians, whether faculty members or executives, would have difficulty holding this view and carrying on with their present work. The seminary must be maintained as it is while its greater future is being worked out.\n\nThe best trustees, especially the one in the chair, will believe that a new clear and powerful mission can indeed result from dialogue that questions like the above will induce. Further, trustees will believe that mission will evolve in such a way that faculty and executives grow in the process. It is the kind of faith that people who move institutions of any kind have to have. There are such people available for seminary trustees, and every seminary trustee board should have at least one. That one, with the power of his or her faith, will lead the rest. And, if they fail, they will quietly retire so that someone else can try.\n\nA seminary that is a viable contemporary institution will have at all times a person in the chair who holds this unqualified faith that movement toward potential greatness in the seminary is a realistically achievable goal. Such is faith in the power of a great idea. And it is contagious!\n\n#### _Reflections on the Relevance of Faith to Mission_\n\nWhat I have been writing the past ten years has brought me into close contact with churches: from local parishes to denominational bodies; and church institutions\u2014hospitals, schools, and seminaries. Most of these involvements have centered on the servant-leader theme about which I have been writing. A major insight (to me) emerged out of this church experience two years ago when I met with an ecumenical group of high-level church executives who were convened for three days in a retreat house to consider the subject, \"the churchman as leader.\" There were just 16 churchmen and I.\n\nThe first two days were taken by a candid discussion among the churchmen of their leadership problems, as they saw them\u2014which were numerous. Change a few words and their problems seemed to me to be no different from those of the harried executives I have been listening to all my life. On the third day, I entered the discussion and made some comments based on what I had heard. As I prepared for that third day, a disturbing thought came to mind. Their main leadership problem seemed to be that they did not believe their own stuff\u2014not as it applies to the institutional affairs of their churches. They are able, conscientious persons, but they lack the kind of faith that anyone who would lead anything in a significant way must have.\n\nI did not share this observation with the church executives the next day, although I touched the edge of it\u2014partly because I was not sure it would be helpful, partly because this was an insight that needed to be digested. If I were meeting with them today, I might share it because, as I have reflected further on my many experiences with churches, it seems to me that the inability (or unwillingness) to be guided by their religious beliefs in dealing with the human affairs of their churches may be the problem of the whole church establishment, and it may account for the failure of churches to give the leadership to this troubled society that they are in the best position to give. The one exception to this generalization that I would note is the Catholic Women's Religious Orders; but they seem troubled by their isolation and bewildered by the prospect that it may be up to them to lead. I wish they could generate more power to lead because they seem to have faith that people and institutions in this chaotic world can change for the better\u2014and that inspired leadership by us ordinary mortals can bring it about. I would say that such faith is _faith as trust_. I once overheard one of the Sisters say, \"There is no worse cynic than a religious cynic!\"\n\nWhat is the relevance of these reflections to the subject of this essay, \"Mission in a Seminary\u2014A Prime Concern of Trustees\"? Is it simply this: to most people in seminaries, or who know intimately about them, the notion that any contemporary seminary could move from where it now is to the kind of institution described in this essay is one of those unbelievable things\u2014a pipe dream. Only a person of truly great faith could accept it as a possibility to be seriously considered. That is why I said earlier that a seminary is not likely to move toward this vision unless it has trustees, particularly in a person in the chair, who are firm in that faith and who have the competence, tenacity, and fortitude that sustained and effective leadership to this end will require.\n\nOne would not have to be a cynic to say \"I do not see any people on the horizon who might be such trustees for my seminary, especially one to chair it, people who have the faith, plus the necessary competence, tenacity, and fortitude to give the leadership that the achievement of this vision requires.\" I agree; I do not see them either. But I believe that they are not visible because they are not expected to be there\u2014in fact, _they are not wanted by some_. A letter from a seminary executive, commenting on my essay on \"The Seminary as an Institution,\" says, regarding the role outlined there for the chairperson, \"The idea seems to suggest that a chairperson serves for a long time. In fact, at least among all... seminaries, and perhaps some others, the chairperson serves no more than two years and is, for all practical purposes, the presiding officer only. As I understand what you are saying, I reject it outright.\" This is a candid expression of what might be a quite general attitude. But such trustees seem to me to be little more than a legal front for faculty and administrators and for the denomination that has de facto control. For a trustee to accept such a limited role is clearly a breach of trust. Directors of a publicly held business corporation who accept this kind of role might be held personally liable in case of some default.\n\nIn our times, the heart of the matter of faith in the improvability of human performance\u2014in institutions\u2014may be _trust_. We have the misfortune to live in an era of low trust. I suspect we are in that state because, in relatively short time, we have moved from a society of individuals relating to individuals to a society in which most of us are involved in complex relationships with institutions. And a widely held basis for trusting people in institutions has not yet evolved. The visible basis for trust, fully functioning trustees, is conspicuously absent in too many institutions\u2014including some seminaries. The first step to recover trust is to install trustees who are trustworthy.\n\nPart of the opportunity to rebuild trust in our times is for trustees to become valid symbols of trust in their institution, and for the one who occupies the chair to stand tall among colleagues in building that symbol. But if there are no visible persons available for trustees, especially chairpersons as so conceived, this opportunity ranks with the unbelievable to many who are involved with seminaries.\n\n_I_ believe the opportunity is real and achievable because of my long experience with people who have demonstrated their ability to carry heavy burdens. Latent in many of these able men and women is trustee potential of the quality that seminaries need. I have had enough experience as an agent in the human transformation process to believe that this latent trusteeship ability can be brought to maturity as seminary trustees, even in some who are old.\n\nFaith is required to do the unbelievable and to demonstrate that something can be done\u2014for the first time. After it has been proven feasible; it will require great courage, strength, and competence to do it again\u2014but it may not require faith. Where is the transforming agent with faith who will wield the formative influence that will produce trustees of this stature for seminaries? _Seminaries will do it for themselves_. Why should someone else produce these trustees for seminaries? Seminaries will produce their own trustees, and they will also render this service for churches and church-related institutions\u2014a massive opportunity. A seminary that rises to this opportunity will grow conspicuously in stature.\n\n\"And how will we do that?\" the incredulous seminary executive or trustee may ask. There is no easy answer to that question because that is what creative leadership is about. A seminary as it now stands, will find in its present resources the initiative to take the first step. A first order of business for any seminary that would operate with a great sense of mission may be to conceptualize the trustee role so as to make that role attractive to the people who would be the trustees that would give the leadership that would produce a great mission. Such trustees will then give the seminary the sustaining strength to operate with distinction within that mission.\n\nThe first question such a potential trustee might ask is \"to whom will I be responsible?\" The trustee is, of course, responsible to the closely related constituencies of the seminary, its students, faculty, staff, and administrators, and to the churches and other institutions that give it money, that employ its graduates, and that look to the seminary for both leadership and prophetic vision. Each of these constituencies has power\u2014in one way or another and in varying degrees and sometimes in conflict with other constituencies\u2014power to enforce its will on the seminary and on the character of its mission. The wise potential trustee, one who has the capacity to lead the seminary to a great mission, will know that as a trustee, these close constituencies will not be his or her primary responsibility. That primary responsibility will be to the society as a whole and to its needs. And he or she will want to know, \"Can these close constituencies be brought to accept that this seminary exists primarily to serve the critical needs of society?\" (I suggested two of these current critical problems at the outset: widespread alienation in all strata of the population and the inability or unwillingness to serve of far too many of the institutions that make up our complex society.) In order to bring this exceptional trustee to its service, any seminary as it now stands will need to give evidence that it is amenable to being led to acceptance of a great mission that is built around some definition of the public good, a mission that is accomplished through the churches that the seminary inspires, leads, and supports.\n\nA great new seminary mission need not disturb present programs. A significant added dimension, concern for religious leadership, may be all that is required. If that new work is sustained by trustee support, in time, existing programs will accommodate to it. Serious concern in the seminary for religious leadership may stir a profound new area for theological reflection. What is hoped for is that religious leadership that originates in seminaries will find its way through the churches, to nurture values and inspire venturesome spirit in those who will: (a) build truly serving institutions and (b) generate in young people hope that they can become servants.\n\nIf one would lead _as servant_ , one will first be open to being led by servants. Openness to leading by servants is a manifestation of _faith as trust_.\n\n### CRITICAL THOUGHT IN THE SEMINARY AND THE TRUSTEE CHAIRPERSON'S ROLE\n\nWhat kind of challenge does chairing the trustees of a seminary present?\n\nA chairperson's answer would depend on how one sees the opportunities and obligations in that position, what one believes about the place of seminaries in society and their unrealized potential for service, and what one's personal feelings are concerning the assumption of leadership. One decides where one stands on issues like these. Then one chooses a role from a range of possibilities beginning, at the low end of the scale, with the seminary executive's view quoted earlier in which \"... the chairperson serves no more than two years and is, for all practical purposes, the presiding officer only.\" Near the other end of the spectrum is one who believes that the seminary can be led by prudent steps toward the achievement of a much greater vision of service to society than most now aspire to, and who is disposed to take the risks of leadership to move it there.\n\nWhat can be said to one who accepts the latter version of a chairing role and chooses to give leadership that will help move a seminary toward realizing a new vision of service to society by greatly strengthened churches? How can such a leader be helped?\n\nThere are ample resources for help on how best to carry out the important fiduciary aspects of all trusteeships. This essay will discuss the opportunity for unusual leadership from the chair when the effort is made to move the seminary, a unique kind of institution, from the marginal role that many now seem to occupy, to the pivotal position that is envisioned in the first two chapters.\n\nThe one who undertakes to give this leadership will need to be prepared to contend with the negative mind-set that disposes too many concerned and thoughtful people to write off seminaries as having little force in contemporary society. As matters now stand in some seminaries, this may be a valid judgment. But one who chooses to give new creative leadership to seminary trustees needs to believe firmly that we cannot afford to dismiss them since there are neither the time nor the resources to replace existing seminaries with more serving institutions that will give needed support to churches. Unless one wants to abandon the idea of building a better society through greatly strengthened churches, there is no feasible alternative to rebuilding seminaries that will give powerful new support to churches. The word _rebuilding_ is used advisedly. No mere revision of the curriculum will do it. The acceptance of such a premise will be part of the armor of one who would lead the trustees of a seminary as, collectively, they lead the seminary toward a future of greatness as servant.\n\nThe discussion of critical thought in the seminary includes the following:\n\n_Critical thought_ is not necessarily identical with scholarship and the writing of books and papers that establish a scholar's reputation with peers. Scholarship is a necessary condition for critical reflection, but it does not automatically produce it. Critical thought is a fulfillment of the root meaning of _seminary; seminalis_ \u2014seed. It is the reflective thought that produces seminal ideas\u2014ideas that become new visions in both ministers as persons and churches as institutions, ideas that support both ministers and churches as they nurture servants who shape the institutions that dominate the lives of all of us.\n\nThe adequacy of critical thought in seminaries is tested, ultimately, by ordinary folk who have its significance in their lives made real by participation in churches. The incremental actions that build the serving quality of society or maintain such as it has, are taken by legions of people who get inspiration and guidance from somewhere, mostly from churches\u2014directly or indirectly.\n\nI believe that the critical thought now emerging from seminaries generally is far from adequate for the urgent needs of these times. Too many churches are languishing for want of intellectual stimulus from seminaries. And seminaries are therefore not giving the vital leadership to churches that support them as they undertake to wield their constructive influence on society\u2014leadership of ideas that give hope expressed in language that lifts the spirit. The premise here is that the need for more influential critical thought from seminaries is great.\n\nBefore discussing the implications of this belief, let us make a brief scan of history of seminaries in the mainline protestant denominations in the United States.\n\n#### _Seminary Leadership, a Historical View_\n\nThere have been two approaches to critical thinking in and regarding seminaries: (1) about the seminary as an institution and (2) about the substance of theology itself. Such consideration of seminaries as institutions is more effective when it shapes the thinking of those who give the ultimate leadership to seminaries, their trustees. Theological ideas are more likely to be useful in building a better society if seminary theologians clothe them in persuasive reasoning and express them in language that gives them the power of prophetic vision. This vision will inspire and guide churches as they work to heal and prevent alienation and give strength and direction to those who will then build serving institutions of all sorts. I am not a close student of the history of seminaries but I have done enough reading to have some views about how to stimulate critical thinking about both the seminary as an institution and its theology.\n\nThe first protestant seminaries in the United States were Andover (1808\u2013Congregational) and Princeton (1812\u2013Presbyterian). In 1815, a remarkable new institution emerged that was called American Education Society (AES). This was a group of strong and able people who effectively became trustees for protestant theological seminaries in the early 19th century. The AES's primary concern was not only for seminaries and churches but for the durability of the young republic, which was not assured at that time; and they believed that strong churches that were served by pastors who were both pious and learned were needed to give the society of their day the sinews of strength. Their strategy was to encourage the growth of seminaries on the Andover model of four years of undergraduate education followed by three years of seminary. They set out to accomplish this objective by raising funds for scholarships to subsidize the seminary education of students of any evangelical denomination that provided this solid scholarly three-year postgraduate program.\n\nUntil early in the 19th century, many pastors in protestant churches in the United States were trained as most other professionals were: by apprenticeship to an established practitioner. \"Reading theology\" was the common term. It could refer to an apprenticeship or to the final year in college. Meanwhile, others acquired the \"gift\" of ministry in less formal ways \u2014 by just starting to preach. The \"circuit riding\" pastors of frontier days needed only a few sermons, and this way sufficed for them. Both ways had the effect of producing pastors who were sometimes \"pious but not learned.\"\n\nWith this AES support, the expansion of seminary resources on the Andover model was rapid. The AES continued for some decades as a powerful force, not only in increasing the number of seminaries but in sustaining the model of a three-year postgraduate curriculum whose chief components (relying on a prior study of classical languages), according to some observers, were \"biblical studies, based on the original languages, sacred history, and didactic theology.\" There was vigorous opposition to this approach, but the founders of AES were strong enough to hold to their course. However, scholarly seminary preparation of clergy did not become the prevailing mode in the United States until well into the 20th century.\n\nThis spectacular development of theological education that AES sponsored was launched well before the radical social changes of the industrial revolution. The steam engine was there in 1800, but the first practical steam boat and railway locomotive did not emerge in the United States until near the middle of the century. The pastor who was trained in the older theological traditions could understand the total life of his parishioners because the conditions of life were not too different from biblical times, in which the theology of that day was firmly rooted. There was in that period little reason to question the adequacy of biblical theology as the quite complete preparation of a pastor.\n\nBut the character of American society was soon to change radically! By 1900, a powerful new voice had risen in William Rainey Harper, an eminent theologian and founding president of the University of Chicago. In 1899, he wrote a spirited article in _The American Journal of Theology_ entitled, \"Shall the Theological Curriculum Be Modified, and How?\" It is a searching and provocative examination of curriculum design, and it contained a recognition of new conditions of society that needed to be addressed in the preparation of pastors. Some passages from Harper's article follow.\n\nSome adjustment must be found by which the curriculum will meet the demands that are made by the present peculiar social conditions. Reference has already been made to the inability of the ordinary preacher to make an impression on the lower classes. The evidence is quite conclusive that he is equally unable to influence the higher classes. The country is full of men who have become wealthy.... What is the attitude of the church toward this growing class of influential men?... Nothing has yet been proposed to provide a training which will enable the ministry to do successful work among the higher classes.\n\nIf the student is to do his work in a democratic atmosphere, he must be filled with the democratic spirit and must learn to employ democratic methods. This is not the spirit, and these are not the methods, of the ordinary theological seminary.\n\nThe condition of the churches, both rural and urban is not upon the whole encouraging. Ministers of the better classes are not satisfied to accept rural churches; and yet these same ministers are not strong enough, or sufficiently prepared, to meet the demands of many of the city churches.\n\nThey (theological students) have little or no sympathy with scientific work. They utterly lack that point of view which will enable them to bring themselves into relationship with that greatest factor in modern civilization.\n\nThe great majority of American seminaries are located in out-of-the-way places and are not in touch with modern life.\n\nThe present scope of the theological curriculum includes practical preparation for only one kind of Christian work; namely, preaching.\n\nThe usual practice in theological seminaries of providing free tuition and rooms... cultivates in the very beginning of life a principle which in too many cases is applied throughout life.\n\nThe study of the Hebrew language should be made elective.\n\nMuch of the technique of a theological education could be put aside to advantage, if this time thus gained could be occupied by work in English Literature.\n\nThese quotes are just a sampling of Harper's essay. There was much more.\n\nWhat Harper did not note in his essay in 1899 (and which seems even more apparent to me as a student of organization today) is that biblical theology was inadequate for the problem he described. And it is even more inadequate in our times, when the majority of us are enmeshed in vast bureaucracies\u2014business, government, education, church\u2014all of which have the same problems, and when the family farm that was dominant in Harper's day is largely gone. The fallacy (as I see it) that continues from Harper's day to this is that theologians assume that the needs of a radically changed (and changing) society can be met in seminaries just by revising the curriculum, and that this is all that is needed to make seminaries (and the churches that depend on them for intellectual and prophetic leadership) fully serving contemporary institutions. My cursory reading of history tells me that this may have been a valid assumption in the early days of AES. But by 1850, the conditions of the emerging society called for new critical thinking leading to a new theology\u2014not repudiating the old, but adding what the new conditions required\u2014and it was not forthcoming.\n\nHow can I justify these assertions? I have had more than 50 years of listening to and watching those who carry the leading and managing roles in institutions of all sorts, large and small. In all of this, I rarely hear reference to influence being wielded on these people's institutional roles by churches. And discussions of administrative problems by church leaders that I have listened to (and I have done quite a lot of such listening) sound no different from those of businesspeople. Yet the effect on the quality and character of contemporary society by the combined decisions these people make, people who are leading and administering American institutions, is the overpowering influence of our times.\n\nA dedicated religious businessperson wrote:\n\nA speaker recently said in an address on the future of the church and evangelicalism, \"Christian programs are not working at all in the business and professional world; the church is answering questions nobody is asking. 91.8% of 750 people in a survey said they would prefer talking with a fellow layman about spiritual matters.\n\nI wholly concur. Paid clergy are viewed by the business community as largely irrelevant. There is little to no business metaphor in church teachings or sermons unless as an object of criticism or derision.... The net result of this neutral or negative posture by the church is to overlook the condition of 100 million working men and women....\n\nI have no sense of how widely this harsh judgment may be held among thoughtful laypeople; but I suspect there is a disturbing amount of it.\n\nThe challenge today is the same in its essentials as that faced, and accepted, by the American Education Society in 1815: to produce able pastors who will lead influential churches that will add strength and quality to contemporary society. The difference is that in 1815 the available concept of the seminary, the Andover model, and the content of biblical theology were both judged adequate. Today neither seems adequate. New critical thought is urgently needed about both the concept of the institution and the content of its theology.\n\nThose who founded the American Education Society in 1815 and gave such powerful impetus to early 19th-century theological education were motivated by an intense concern for the survival of the young democracy. The motivation that might move seminary trustees in the late 20th century may be similar\u2014to heal and strengthen an ailing society. The means available for both seem to be the same: stronger, better-led churches. But the strategy for implementing those means may be radically different today from what it was when the protestant seminary movement started.\n\nI have said that the strong group that made up the American Education Society were, in effect, the trustees of an important segment of protestant theological seminaries of that period. As I see it, the trustees of seminaries today, particularly their chairpersons, are best positioned to give the leadership that will produce the much needed critical thought about the seminary as an institution and the theology that both guides it and which it advocates as prophetic vision for churches.\n\nIf the chairperson accepts the risks of leadership that will be required to bring the seminary to become what it has the opportunity to become, and to produce needed ideas, a strategy for such effective leadership will be needed. Where is the relevant experience that the seminary trustee chairperson might turn to in devising this strategy?\n\n#### _Where Is the Relevant Experience?_\n\nIf the trustee chairperson of a seminary resolves to give leadership that will help the seminary become a greater force in strengthening churches in our times, how does one proceed? What experience is there to draw upon to help one devise a strategy of leadership that will bring the seminary to become a great new source of critical thought\u2014both about itself and as an institution and about the theology that it provides for churches?\n\nSuch a move in many seminaries would call for a substantial regeneration of the seminary as an institution. It would be a major undertaking that would not likely take place without astute and sustained leadership from the trustees, especially the one who chairs the board.\n\nIn the world of institutions, all can be roughly grouped in two classes: those designed so that they fail easily and live under the constant threat of failure, and those that are designed so that failure is difficult and rare. Businesses are generally in the first category, and nonprofit agencies and governmental units are in the second. Because of this sharp disparity, most of the experience with the process of regenerating moribund or low-serving institutions is in business. And the seminary trustee chairperson who is searching for ideas on how to give leadership to the school so as to bring it to greatness as servant with an abundance of critical thought as one of its distinguishing characteristics will be well advised to look to business experience as one source of ideas. I do not suggest that either seminary faculty or administrators should be concerned with the experience of business regeneration, but that their trustee chairperson take a close look at that experience.\n\n#### _My AT &T Experience as an Example_\n\nI had the good fortune to spend my active career in American Telephone and Telegraph Company, which had experienced a major regeneration shortly before I arrived in 1926, and some of the people who brought it through this great change were still around. From these people, I learned much about how that transformation was accomplished that may never get into written history.\n\nThe telephone was invented in Boston in 1876. The company was started in 1878 by Bostonians who lacked both vision and an adventurous spirit, and it remained in their hands until 1907, when J. P. Morgan the elder wrested control from them; installed a great builder, Theodore N. Vail (who earlier had been the company's first general manager), as president; moved the headquarters to New York; and gave it a vision and started it on an adventurous course. Morgan, who earlier had been instrumental in launching both General Electric and U.S. Steel, gave a powerful push to move AT&T from an ordinary to an exceptional institution.\n\nIn 1907, the company was a going concern, but it was burdened by four serious problems: (1) There was an acute employee morale problem. (2) The public reputation of the company was very bad. (3) There was a question about its financial soundness. (4) The available technology was insufficient to permit the development of the scope and quality of telephone service that we know today and that Vail and Morgan were determined to bring about. By 1920, when Vail died in the harness, all four of these were corrected, some seminal ideas about the institution and its technology emerged, and the company was a \"blue chip.\"\n\nI entered the company in 1926 and I immediately became interested in its history. The top question on my mind for a number of years was, how did this remarkable transformation come about? The answer was slow in coming, partly because it was so difficult to judge what Morgan's role was. Morgan, as he was then, would not only be out of style today, but he would be legally restrained. In the context of his times, however, he was a great trustee. He had and used great power, sometimes ruthlessly; _but he cared intensely about the quality of the business he controlled,_ an extraordinary attribute for a person of great wealth early in this century. We cannot know how much of Vail's genius as an institution builder should be attributed to Morgan. A safe assumption is to view them as a joint personality.\n\nHow did the transformation of AT&T come about? Very simple in concept; but awfully difficult to do. There were three basic strategies that I believe are universally applicable to the regeneration of any kind of institution at any time.\n\n#### _Strategies for Change_\n\n1. _Mission_. In his early period as the first general manager, Vail stated the mission thus: \"We will build a telephone system so that any person, any place in the world, can talk to anybody else, any place else in the world, quickly, cheaply, satisfactorily.\" We aren't there yet, nearly 100 years after that goal was stated. But I believe we are much closer to it now than we would be if the man who piloted the company in those crucial years (1907\u201320) had not \"thought big.\" In that later period, the goal was stated more modestly as \"universal service\"\u2014perhaps because the company had lost a few tail feathers in its first brush with the antitrust enforcers in 1913. When I entered the business in 1926, there was still some of the feeling that we were \"building a cathedral, not just laying bricks.\" It would be difficult in any institution to sustain the sense of urgency noted later if the commonly accepted mission does not require it.\n\nThe first step toward defining mission for one who chairs seminary trustees is to get an answer to the question, \"Whom and what purpose does this seminary serve?\" The chair should ask this question and stay with it until all constituencies\u2014trustees, administrators, faculty, students, alumni\u2014are agreed on a concept that the chair is willing to lead. If the constituencies, after long and patient urging either cannot agree, or if what they can agree on seems unacceptable to the one in the chair, perhaps that person should quietly retire and find something better to do. When Vail, the first general manager of AT&T, was unable to move the conservative Bostonians to become builders, he left the company and resisted later importunings to return when the company was in trouble. He did not return for 20 years until the Bostonians were ousted by Morgan, who was himself a great builder.\n\nIf the agreed-upon mission statement for a seminary is one I could accept if I were to undertake to lead the board, these are the answers I would prefer to see to the questions, Whom and what purpose does this seminary serve? _Whom?_ Religious leaders, whether in churches or other institutions. _What purpose_? To provide churches with ideas and leadership to the end that churches become and are sustained as significant forces that heal and prevent alienation in people and nurture the leaders who will build serving institutions everywhere. The seminary might do some other things, but if I were to be the chairperson, this definition of mission\u2014or one close to it\u2014would be primary. Other chairpersons might view it differently. My advice to any such persons is: don't undertake to lead the board and the seminary in a direction you do not firmly believe in. Otherwise you will not be able to lead with spirit. Spirit will be needed.\n\n2. _Obstacles to greatness are clearly identified_ and the full ramifications of each is described. Then a capable staff person is assigned to each problem with the clear charge to find the means for turning it around and to persuade\u2014to move\u2014all of the people who need to act. As long as an obstacle to greatness remains as a problem, a capable staff person makes a high priority of finding a way to deal with it. If at any time it appears that the person assigned is either unable or unwilling to press for an answer, that person is replaced.\n\n3. _A sense of urgency is created_ and the move toward greatness is widely accepted as an imperative. It was my privilege to work at AT&T under the executive who, as a young man in 1907, was given the assignment of raising the morale and integrity of the women in the business\u2014a difficult and essential task. By 1926 when I entered, the person who brought about this transformation had made an elite corps out of the department (switchboard operation) where most women worked. One of the by-products was that this department (in 1926) was producing a greatly disproportionate share of the top officers in the company (then all men). The man who piloted this transformation (one of the most perfect gentlemen I ever met) accomplished it from a staff position entirely by persuasion. He was supported by the feeling of urgency that pervaded the business in that crucial period from 1907 to 1920.\n\nFrom the soundings available about seminaries, I deduce that the principal urgency in some of them is the need for money to survive. In a seminary in which this is a valid judgment, could it be that one of the reasons for the primacy of money in seminaries is that there is not sufficient urgency felt regarding critical thought, the production of seminal ideas that are sorely needed in a faltering society?\n\nWhen someone says to me, \"You can't cause seminal ideas to emerge where they don't exist,\" my response is, \"Oh, yes you can! (1) If you have a widely accepted mission for the institution that embodies a great dream of what it might become. (2) If you carefully identify all of the obstacles that stand in the way of realizing that dream and see to it that a competent person sets to work to remove or find a way around each obstacle. And (3) if you sustain a sense of urgency about the whole process of regenerating the institution.\"\n\nDo these three things, and do them well, and the chance that seminal ideas will emerge in the process is very good\u2014no matter how limited the institution may have been at the start. This is what the experiences of businesses in general, and the one I know best\u2014AT&T\u2014in particular, have to suggest to the chairperson of seminary trustees who would give leadership to a development that would favor the growth of critical thought\u2014as marked by the emergence of seminal ideas\u2014in the seminary. It may require more acumen from the one in the chair to sustain this urgency in a seminary than in the usual business. But there are only 200 seminaries in the United States, and the nature of the opportunity to serve is such that every seminary should have an exceptional leader to chair its trustees, or the equivalent of that person in seminaries that do not have their own trustees. That urgency will need to be felt at all times if the typical seminary is to move in influence from where it now is to the society-shaping role that it is correctly positioned to carry. Only the chairperson, using tactics that are appropriate for a seminary in these times, is likely to be successful in creating that urgency. If the executive tries to create it, his or her tenure may be short. As I have noted in Chapter I, trustees (including the chairperson) are expendable; administrators are not.\n\nThe chairperson need not be able to design the creative steps that will be taken within the seminary to produce the seminal ideas that will inspire and support pastors, nor need he or she be conversant with the curriculum designs that will prepare pastors as significant religious leaders. Both of these are appropriately the professional concern of administrators and faculty. Trustees will want to review and comment on these matters, but they will not be the prime movers in these areas.\n\nTrustees, especially the one in the chair, will need to be clear about the ultimate result of the seminary's work: more religious leaders everywhere and strong, influential, ably led churches; and be able to judge whether, in the carrying out of that mission, the ultimate influence on and support of churches is adequate. Trustees should be aware of and interested in, but stand somewhat aloof from, the effort of administrators and faculty to move the program of the seminary from mission to accomplishment. It is more appropriate for a layperson, than for a credentialed professional, to maintain this detached position. The seminary clearly needs professional strength, although what constitutes optimum, professional strength in a seminary may bear fresh examination\u2014an examination that needs what only wise, lay judgment is likely to bring.\n\n#### _Advantage in the Lay Status of the Chairperson_\n\nThe issue of strength in lay judgments is a debatable one, but the case for the importance of trustees, particularly the chairperson, in any institution rests upon acceptance of the possibility of that strength being realized. The best example I have of this is drawn from my AT&T experience\u2014an example of the strength in lay judgments in a situation in which _that_ was where wisdom resided.\n\nThe original National Labor Relations Act (Wagner Act) was enacted in 1935. Shortly thereafter, 50 great corporate lawyers met and agreed that this law would be held unconstitutional and that they could all advise their client companies to disregard it. When this conclusion was brought back and advocated by our AT&T lawyer who was present at that meeting, he was promptly and forcefully challenged by my boss, an able and persuasive older man with but a fifth-grade education. He was not in the upper levels of management, but he was in a position in which he could be heard. Although his grammar was not impeccable, he was a powerful debater, a strong, honest, intelligent man, but with no legal training or experience. The gist of his argument was that this was 1935, not 1905. This was the second time in as many years that the principle in the Wagner Act had been legislated (the first was a section of the earlier National Industrial Recovery Act of 1933 that was declared unconstitutional). It clearly represented a firm social policy that would prevail. If this present law was not upheld, new laws would keep coming until one was sustained. Therefore, we should start immediately to bring ourselves into conformity with it (a state, with largely \"company\" unions, that we missed by quite a margin). There was a great verbal battle, but this boss of mine (20 years my senior) was a tough, formidable man. After much grumbling, he prevailed and the company started (with too deliberate speed) toward compliance. If his position had not prevailed, and if AT&T had held to the intransigent position that all of these lawyers advocated (that the automobile companies and others did accept and brought on the terrible imbroglio of 1939), AT&T might have been dismembered then. And if this man's kind of thinking could have been more influential in top management councils in the years that followed, the breakup might not have occurred.\n\nPart of the strength of this untutored layman's position was that he was not a lawyer. He was a very conservative man (in the best sense), and he was able to look at the crisis that confronted the company as a social policy question, not a legal issue. And he believed, with the comic character of the time\u2014Mr. Dooley\u2014that even the Supreme Court, given a little time, reads the election returns. The Wagner Act was affirmed by the top court in 1937.\n\nThe strength of my old boss in this encounter was not just that he was not a lawyer, although that freed him from the \"mind-set\" that seems inherent in all credentialed professionals, or that he lacked formal education. Most important, I believe, was that he possessed the priceless gift of seeing things whole and, because of this, his advice was frequently sought by \"better educated\" people who lacked that gift. Is this not the quality, above all others, that should be sought in the trustee chairperson of a seminary? It is not a common ability, yet it does exist. The 200 seminaries in the United States could find such a person to chair their trustees if they were clear that that is what they need and want. And such are more likely to be found among uncredentialed laypersons.\n\nSuch a person in the chair, if otherwise qualified by motives, temperament, and skills, would likely be accepted by fellow trustees and other constituencies of the seminary and could supply the essential ingredient of wisdom regarding the adequacy of the critical thinking that is the heart of the seminary's work. This would be accomplished by watchfulness over the three steps suggested earlier as essential for any institution that achieves greatness as servant.\n\n#### _The Idea of the Seminary_\n\nIf the seminary answers the question, \"Whom and what purpose should this seminary serve\" in terms like those suggested earlier in these chapters, the one who chairs the trustees (assuming that that person has the gift of seeing things whole) may ask, \"Is this seminary as it now stands\u2014both people, structure, and assumptions\u2014best designed to accomplish that mission? If not, what kind of institution ought it be, and how do we move it prudently from here to there?\" Such a question may lead the constituencies of the seminary to search for a creative design for the seminary of the future that, in addition to the present scholarly structure, may accommodate some whose principal qualifications are that they have the gifts of (1) seeing things whole and (2) prophetic communication. They may or may not be scholars.\n\n#### _The Problem of Language_\n\nPart of my thinking about language in the field of religion comes from the opportunity I had as a young man in New York to attend services at Riverside Church when Harry Emerson Fosdick preached. These sermons were remarkable for their clarity and simplicity\u2014all in the vocabulary of the ordinary person. I received some insight into his gift many years later when I was privileged to be in the company of a mutual friend to visit Dr. Fosdick in his old age for an afternoon of conversation. In the course of this visit, he told us that as a young man he had had a mental disturbance and had to have help to get himself reoriented. As a consequence he said that, in the years of his ministry, he had reason to believe that he was a good preacher and writer, but that the most rewarding satisfactions in his career were in one-to-one consultations with disturbed people. He felt that he was at his greatest effectiveness here because he could say to his counselee, \"I know exactly how you feel because I have been there.\" The clarity and simplicity of his preaching must have been profoundly influenced by the centrality of personal counseling in his ministry.\n\nHowever gifted they may be, pastors trained in seminaries need to communicate ideas that give hope and in language that is powerful and beautiful, words that lift the spirit. The chairperson of seminary trustees needs to be concerned that his school wields an influence on its students that favors this result.\n\n#### _The Seminary Chairperson as a Personal Role Model_\n\nBy giving oversight to the pivotal part of the seminary's mission, critical thought, one has the opportunity to be a role model for those who carry equally important leadership in a local church, the lay leaders of its congregation.\n\nIt was argued earlier that seminary trustees and their chairpersons are expendable, but executives and administrators are not. So, in time, lay leaders in churches will come to accept that they are expendable but pastors are not. Expendable persons will take the higher risks of leading.\n\n#### _What Does \"Serving the Churches\" Mean?_\n\nHow does a seminary know when it is serving the churches?\n\nA consultant who has worked with several seminaries recently on restudying their missions reports, \"One of the observations that comes out of this experience is the propensity of seminaries to give continuity to the safe kinds of ministry they perceive the churches which pay the bills as willing to subsidize.\" This states the common problem of _all_ serving institutions: businesses, schools, churches, governments. What will the constituency, customer, citizen want and be willing to pay for _in the future?_ Making this judgment is probably most critical in business.\n\nA few years ago, I found myself seated at a luncheon meeting in New York between the editors of two important magazines that were part of the same group. Plying my trade as I usually do, I engaged them on the question, \"How do you make the decision, out of all the stories, articles, and pictures you have available, which ones you will print in your next issue?\" Their prompt answer was most illuminating. I will reconstruct it as best I can.\n\n\"Most popular magazines make regular reader surveys in which they try to find out what readers liked best in the last issue, what they disliked or were not interested in, and what they would like to see more of in the future. In making the decision on the question you ask, _you do not follow literally what readers tell you_. That is what killed the _Saturday Evening Post_ \u2014literally following what readers said they wanted more of. The problem is that readers cannot say what they will want more of in the future. What they will respond to in the next issues when they arrive depends somewhat on what happens in their lives, what public events have caught their attention, in the meantime. What makes magazine editing interesting and challenging is that _planning every issue requires a leap of imagination_. Part of what nourishes that leap is close attention to reader surveys, but only part. The rest of it is vast experience with journalism knowledge about how people evolve and change, and watching closely what is going on in the world. We try to plant in every issue a seed that will grow into a need that we can serve in the future. That is how we editors, with our opportunities, can lead.\"\n\nA seminary's relationship with churches is not like that of a magazine with its readers, but there is something to be learned from this example. Churches cannot tell seminaries what they will want in the future\u2014and will be willing to pay for. If, as the consultant's observation quoted above suggests, some seminaries assume that churches will continue to want and be willing to pay for what they now receive\u2014a \"safe\" ministry\u2014providing that may make for a comfortable relationship in the present. But what about the future? What about the future when the present, in terms of church influence and membership, is not good?\n\nThis is a question for a seminary chairperson who is disposed to try to influence his or her trustees to _lead_ , to go out ahead and show the way and persuade seminary administrators and staff to act now, in the interest of the future soundness of churches. The seminary's prime concern is always for the future of churches. The future of churches will be determined importantly by the ideas, critical thought, that seminaries produce _now_ , and persuade churches to consider\u2014 _now!_\n\nAll true leadership\u2014because it deals with the future\u2014entails risk, and the chairperson who sees things whole will be the first to accept that risk and make the necessary leap of imagination. What supports the seminary chairperson in that risk (as distinguished from the editor of the magazine) is an inspired vision of what the seminary and the churches that depend on it for intellectual and prophetic leadership, might become\u2014greater servants of society.\n\n### POSTSCRIPT\n\nThe positions taken here regarding the urgency of the need to raise seminaries from the present marginal state of some of them to fully serving institutions rest on beliefs about current America as a low-serving society. Every category of institution I know about, including seminaries and churches, has far too many low-serving elements, when judged by the criterion of what is reasonable and possible with available resources, human and material. Why should anybody in government, business, or education try to do better when seminaries that should be models are not conspicuous for their service to society? If the quality of our total society is to be lifted, seminaries and then churches, one at a time, need to be lifted substantially into a position of preeminent service.\n\nAs noted earlier, both the initiative and the sustaining push to achieve this movement will come from trustees who are laypersons and who have a strong determined leader who chairs their effort. If the current chairperson does not feel up to giving this leadership, that person might quietly step aside and help find someone who can and will give that leadership\u2014someone who has the temperament and the staying power for the long, hard struggle that may be required. It could be the supreme achievement of that person's life to learn to lead a seminary effectively and with spirit.\n\nAlongside my convictions about lay influence of trustees in seminaries, I am equally firm about the importance of intellectual power in the seminary\u2014a power that will enable its faculty and other staff to support churches and church institutions at a level of excellence in their performance that, on their own and without the support of the intellectual power of seminaries, might be quite ordinary.\n\nIntellectual power could be said to have two main elements: scholarship and wisdom. What may be needed, first, is a new vision from within seminaries as they now stand, possibly one of those seminaries now labeled as marginal. It is the kind of vision that the King James version of Proverbs suggests is one without which the people perish. Such a vision may simply announce a yearning to be served by being led by a trustee chairperson who has the gift of seeing things whole\u2014a person who is wise and who will give leadership to the end that the seminary comes to be accepted as both scholarly and wise.\n\nAnd what would a seminary be like (as contrasted with what most are today) when it comes to be known as both scholarly and wise? In my essay \"The Servant as Religious Leader\" I speculate on what might be the characteristics of a seminary when significant formation of religious leaders takes place there.\n\n\u2022 Its priorities will be reversed. Whereas seminaries are now mostly academic and only incidentally formative, formation of religious leaders will be primary and academic teaching will be secondary.\n\n\u2022 The staff of the seminaries will contain a strong element of those who have a passion for growing religious leaders\u2014and are good at it. They may or may not be scholars in the usual sense.\n\n\u2022 A major mission of the seminary will be to evolve, and maintain, a theology of institutions that deals realistically with the problem of how to recover moribund institutions as vital, effective, caring, and serving. This will not be a theoretical endeavor because it will be forged on the seminary's own experience as it builds itself into\u2014and maintains itself as\u2014the pivotal institution it is determined to become. Seminary students will be deeply involved in this continuous effort to build and maintain this theology. They will not just read and hear lectures about it.\n\n\u2022 The primary mission of the seminary will be leading and serving churches and supporting them as strong influential institutions. Most of the learning of seminary students will result from involvement in this effort.\n\n\u2022 There will be creative thinkers among its faculty who are developing and articulating a contemporary theology of what makes religious leaders, and the institutions they serve, strong. Students in the seminary will be deeply involved in responding to this with their own thinking.\n\n\u2022 Such seminaries will become known as effective nurturers of able religious leaders and they will attract a wide spectrum of strong young people in search of such formative development. Some of these students might find their career opportunities in churches, but the seminary will become a prime source of religious leaders for all segments of society. It will acknowledge that any institution where religious leaders predominate may effectively become a church.\n\nI submit these as achievable goals for a seminary whose constituencies (particularly the faculty that holds the predominant power) accept that new critical thought about both the seminary as an institution and its theology is essential. Further, they accept the leadership of the lay person who chairs their trustees and who is persuasive in helping them to reach those goals.\n\n## 7\n\n## **My Debt to E. B. White**\n\nIn 1929, when I moved to New York, I was immediately attracted to _The New Yorker_ magazine, that was then in its fifth year, and to E. B. White, who had helped make it a remarkable magazine, and who had been on the staff for three years. My debt to Mr. White, after 55 years of living with his writings, stems from two gifts that are rarely possessed by one person: the ability to see things whole, or more whole than most, and the language to tell us ordinary mortals what he sees.\n\nI am not a literary person, but I know that White's writing style is greatly admired among some literary folk. His revision of Strunk's _The Elements of Style_ is a widely used text. He is sometimes identified as a humorist, and I find good laughs in his work. He is a fellow who, when the spirit moves him, just naturally breaks into song\u2014so there is quite a bit of poetry. In his later years, there have been stories for children. As a so-called adult, I find them delightful. But his writing style, his humor, his poetry, and his children's stories are not the central focus of what I want to acknowledge here, though, obviously, they are the context within which it is housed.\n\nI have not received from reading E. B. White the solutions to any of life's dilemmas, but for these 55 years, I have had constant assurance that if I will see clearly where I am (and where I and others in similar dilemmas have been), and if my direction is right (uncluttered with zany ideas), I will always better know what to do now. This has been a valuable learning, and I am exceedingly grateful that I received it early. I am not aware that, in the course of my formal education, I heard anything about this. But then, it probably takes an artist to put this over, and for the most part, my teachers were not artists. E. B. White is such an artist.\n\nJames Thurber, White's good friend and collaborator from their early days on _The New Yorker_ , freely acknowledges White as his mentor on writing. And he seems to have understood White's unique gifts better than White himself understood them. Thurber, writing in 1938 from France, chides White for publishing a piece in _The Saturday Evening Post_ (which may have paid better at that time than _The New Yorker_ ). Thurber, after lamenting the confusion in the world and the crazy things people are doing, says, \"It remains for a few people to stand aside and watch them and report what it looks like and sounds like. Among such persons, there isn't anybody better qualified for the job than you\u2014if _you will quit sending pieces to The Saturday Evening Post_.\" Thurber goes on to express his concern that White does not understand or appreciate his gift and reminds him that his writing is not simply a response to the writer's _urge_ but is a matter of \"moral necessity.\" It is the kind of letter that only a cherished friend would write. In short, Thurber tells White (in my language) to get with it and make the contribution to our times that only he can make. Thurber seems to be saying to White, \"You see things whole, and that is what you should write about\u2014in the only place that is likely to let you do it\u2014 _The New Yorker_. Maybe _Harpers_ , but not _The Saturday Evening Post_.\"\n\nThurber and White as young men shared a small office at _The New Yorker_. In that period, they collaborated on a spoof of psychoanalysis to which they gave the provocative title, \"Is Sex Necessary?\" It was a best-seller. After Thurber died, White wrote a new introduction for a new edition. In it he said, \"You would think that a couple of young fellows trying to get along in the world could have found something better to do.\"\n\nElwyn Brooks White was born in the close-in suburb of Mt. Vernon, New York in 1899, the sixth child in a family whose father was a successful piano manufacturer in New York. As a child, he developed a fondness of animals and had pets and kept chickens and pigeons, a disposition that emerged later when he retired to his farm in Maine. About his childhood, White says in the introduction to his collected letters:\n\nIf an unhappy childhood is indispensable for a writer, I am ill-equipped: I missed out on all that and was neither deprived nor unloved. It would be inaccurate, however, to say that my childhood was untroubled. The normal fears and worries of every child were in me developed to a high degree; every day was an awesome prospect. I was uneasy about practically everything: the uncertainty of the future, the dark of the attic, the panoply and discipline of school, the transitoriness of life, the mystery of the church and God, the frailty of the body, the sadness of afternoon, the shadow of sex, the distant challenge of love and marriage, the far-off problem of a livelihood. I brooded about them all, lived with them day by day. Being the youngest in a large family, I was usually in a crowd but often felt lonely and removed. I took to writing early, to assuage my uneasiness and collect my thoughts, and I was a busy writer long before I went into long pants.\n\n\"In school,\" he continues, \"I contracted a fear of platforms that has dogged me all of my life and caused me to decline every invitation to speak in public.\" When White turned 70 in 1969, a _New York Times_ reporter went up to Maine, where he then lived, to interview him. In the course of the interview White said, \"I was born scared, and at 70 I am still scared.\" I will come back to this comment later. In that interview, White had some things to say about growing old.\n\n\"How should one adjust to age!\" Mr. White was asked, and replied: \"In principle, one shouldn't adjust. In fact, one does. (Or I do.) When my head starts knocking because of my attempt to write, I quit writing instead of carrying on as I used to do when I was young.\n\n\"These are adjustments. But I gaze into the faces of our senior citizens in our Southern cities, and they wear a sad look that disturbs me. I am sorry for all those who have agreed to grow old. I haven't agreed yet. Old age is a special problem for me because I've never been able to shed the mental image I have of myself\u2014a lad of about 19.\n\n\"A writer certainly has a special problem with aging. The generative process is slowed down, yet the pain and frustration of not writing is as acute as ever. I feel frustrated and in pain a good deal of the time now; but I try to bear in mind the advice of Hubert Humphrey's father. 'Never get sick, Hubert, there isn't time.'\"\n\nWhite attended Cornell University, whose founding president was one Andrew White. By tradition, every male student to enroll whose name was White was nicknamed \"Andy,\" and to his close friends, Elwyn Brooks White has always been _Andy_. At Cornell, he quickly joined the staff of the campus newspaper and, in his senior year, he was editor. White was not a scholar at Cornell, but he was active in a fraternity, and in his senior year, he was its president.\n\nOn graduation, he spent a year in New York trying to get a job in journalism, and the best he could do was The American Legion News Service. When a footloose Cornell classmate showed up in New York, they decided to drive to the West Coast in a model T roadster, named Hotspur. They set off early in 1922 and arrived in Louisville, Kentucky, at derby time. White wrote a sonnet on a winning horse which he showed to the editor of the _Louisville Herald_ , who asked, \"Do you do this for glory or for money?\" \"For money,\" White replied. The editor paid him $5 and ran the sonnet on the front page next day.\n\nThe trip across the country in Hotspur was eventful. Arriving in Seattle, White got a reporter's job on a Seattle daily which he held for a year, when he was fired. The reason was that when he covered an event, he was so much a perfectionist and took too long to write his story. Frequently, in later life, he acknowledged that he was a failure as a reporter.\n\nThen there was a boat trip to Alaska. Returning to New York after 1\u00bd years in the west, he got a job with an advertising agency where he hated to work\u2014but which was a living, and it gave him an opportunity to write. He says:\n\nThe arrival on the scene of Harold Ross's _New Yorker_ on February 21, 1925, was a turning point in my life, although I did not know it at the time. I bought a copy of the first issue at the newsstand in Grand Central, examined Eustace Tilly and his butterfly on the cover (every Washington's birthday issue has that cover) and was attracted to the newborn magazine, not because it had any great merit but because the items were short, relaxed and sometimes funny. I was a \"short\" writer, and I lost no time in submitting squibs and poems. In return, I received a few small cheques and the satisfaction of seeing myself in print as a pro.\n\nHarold Ross encouraged White to submit more to _The New Yorker_ and asked him to drop in. When he did drop in, the editor who greeted him was Katharine Angell. White said, \"I noted that she had a lot of black hair and the knack of making a young contributor feel at ease. I sat there peacefully gazing at the classic features of my future wife without, as usual, knowing what I was doing.\" Forty years later, in an interview, White said of his wife:\n\nI have never seen an adequate account of Katharine's role with _The New Yorker_. She was one of the first editors to be hired, and I can't imagine what would have happened to the magazine if she hadn't turned up. Ross, though something of a genius, had serious gaps. In Katharine he found someone who filled them. No two people were more different than Harold and Katharine. What he lacked, she had; what she lacked, he had. She was a product of Miss Windsor's and Bryn Mawr; Ross was a high school dropout. She had a natural refinement of manner and speech; Ross mumbled and bellowed and swore. She was patient and quiet; he was impatient and noisy. On one thing they usually agreed\u2014what was funny. Katharine was soon heading the Fiction Department, sharing the personal woes and dilemmas of innumerable contributors and staff people who were in trouble or despair, and, in short, accepting the whole unruly business of a tottering magazine with the warmth and dedication of a broody hen.\n\nI had a bird's eye view of all of this, because in the midst of it, I became her husband. During the day, I saw her in operation at the office. At the end of the day, I watched her bring the whole mess home with her in a cheap, bulging portfolio. The light burned late, and our bed was lumpy with page proofs, and our home was alive with laughter and the pervasive spirit of her dedication and her industry. I suspect that one of Ross's luckiest days was the day a young woman named Katharine Angell stepped off the elevator, all ready to go to work.\n\nI have said that my debt to E. B. White is that he is the person who alerted me to the gift of seeing things whole, and my attachment to his writing, beginning when I was 25, encouraged me to cultivate that gift in myself. My career as an organization man and a bureaucrat in a huge institution, where I was very much at home, was radically different from White's, who never was an administrator and who had great difficulty keeping regular office hours. Yet, across that great gulf of temperament and experience, he was able to communicate to me his great gift of seeing things whole, and it has proved to be an asset all my life.\n\nWhat was the man like who was able to do this for me? I have never met E. B. White, and in over 50 years I have exchanged only two or three letters with him. Yet, I feel that I know him very well. His collected letters are interspersed with biographical notes that give quite a complete account of his life. He was a great letter writer and, his letters being literary gems, people saved them. But most of the insight I have about White comes from reading what he has said about other people. There is an old saying, \"What Peter says about Paul tells us more about Peter than it does about Paul.\" This observation seems to be particularly true of White. Let me quote what he has written about three people to whom he was very close: the first, Henry Thoreau, whom White knew only from his writing, and two with whom he worked closely, James Thurber and Harold Ross, whose obituaries he wrote for _The New Yorker_.\n\nOn the 100th anniversary of the publication of Thoreau's _Walden_ in 1854, White wrote a long piece for _The Yale Review_ entitled, \"A Slight Sound at Evening.\" I will note a few excerpts from his essay on Thoreau's _Walden_.\n\nI think it is of some advantage to encounter the book at a period in one's life when the normal anxieties and enthusiasms and rebellions of youth closely resemble those of Thoreau in that spring of 1845 when he borrowed an ax, went out to the woods, and began to whack down some trees for timber. Received at such a juncture, the book is like an invitation to life's dance, assuring the troubled recipient that no matter what befalls him in the way of success or failure he will always be welcome at the party\u2014that the music is played for him too, if he will but listen and move his feet.... It still seems to me the best youth's companion yet written by an American, for it carries a solemn warning against the loss of one's valuables, it advances a good argument for travelling light and trying new adventures, it rings with the power of positive adoration, it contains religious feeling without religious images, and it steadfastly refuses to record bad news....\n\nWhen he went to the pond, Thoreau struck an attitude and did so deliberately, but his posturing was not to draw the attention of others to him but rather to draw his own attention more closely to himself. \"I learned this at least from my experiment: that if one advances confidently in the direction of his dreams, and endeavors to live the life which he has imagined, he will meet with a success unexpected in common hours.\" The sentence has the power to resuscitate the youth drowning in his sea of doubt. I recall my exhilaration upon reading it, many years ago, in a time of hesitation and despair. It restored me to health. And now in 1954 when I salute Henry Thoreau on the 100th birthday of his book, I am merely paying off an old score\u2014or an installment on it.\n\nThere has been much guessing as to why he went to the pond. To set it down to escapism is, of course, to misconstrue what happened. Henry went forth to battle when he took to the woods, and _Walden_ is the report of a man torn by two powerful and opposing drives\u2014the desire to enjoy the world, and the urge to set the world straight. One cannot join these two successfully, but sometimes, in rare cases, something good or even great results from the attempt of the tormented spirit to reconcile them. Henry went forth to battle, and if he set the stage himself, if he fought on his own terms and with his own weapons, it was because it was his nature to do things differently from most men, and to act in a cocky fashion. If the pond and the woods seemed a more plausible site for a house than an in-town location, it was because a cowbell made for him a sweeter sound than a churchbell. _Walden_ , the book, makes the sound of a cowbell, more than a churchbell, and proves the point, although both sounds are in it, and both remarkably clear and sweet. He simply preferred his churchbells at a little distance. _[I suspect that White also preferred his churchbells at a little distance]_\n\nI confess that I have not been a Thoreau fan. The first thing I ever read of Thoreau's was his essay on \"Civil Disobedience\" which he wrote in a fit of pique after spending a night in jail for refusing to pay his Poll Tax. I wrote him off as a wooly anarchist and never read him again. But after rereading White's essay recently, I decided that I had better have a look at _Walden_. So, I read it. And I discovered, to my surprise, that there is more of Thoreau in me than I had been aware of. Such is the fate of one who falls under the spell of one who sees things whole (a trait that White may have acquired from Thoreau since he read Thoreau when he, White, was young and was greatly influenced by him).\n\nHarold Ross, the great founding editor of _The New Yorker_ , died December 6, 1951. Writing of him in the December 15th issue, White wrote:\n\nRoss died in Boston, unexpectedly, on the night of December 6th, and we are writing this in New York (unexpectedly) on the morning of December 7th. This is known, in these offices that Ross was so fond of, as a jam. Ross always knew when we were in a jam, and usually went on the phone to offer advice and comfort and support. When our phone rang just now, and in that split second before the mind focuses, we thought, \"Good! Here it comes!\" But this old connection is broken beyond fixing. The phone has not in its power to explode at the right moment and in the right way.\n\nActually, things are not going as badly as they might; the sheet of copy paper in the machine is not as hard to face as we feared. Sometimes a love letter writes itself and we loved Ross so, and bear him respect, that these quick notes, which purport to record the sorrow that runs through here and dissolves so many people, cannot possibly seem overstated or silly. Ross, even on this terrible day, is a hard man to keep quiet; he obtrudes\u2014his face, his voice, his manner, even his amused interest in the critical proceedings. If he were accorded the questionable privilege of stopping by here for a few minutes, he would gorge himself on the minor technical problems that a magazine faces when we must do something in a hurry and against all sorts of odds\u2014in this case, emotional ones of almost overpowering weight. He would be far more interested in the grinding of the machinery than in what was being said about him.\n\nAll morning, people have wandered in and out of our cell, some tearfully, some guardedly, some boisterously, most of them long-time friends in various stages of repair. We have amused our-self thinking of Ross's reaction to this flow. \"Never bother a writer\" was one of his strongest principles. He used to love to drop in himself, and sit around, but was uneasy the whole time because of the carking feeling that if only he would get up and go away, we might settle down to work and produce something. To him, a writer at work, whether in the office or anywhere in the outside world, was an extraordinarily interesting, valuable, but fragile object, and he half expected it to fall into a thousand pieces at any moment.\n\nThe report of Ross's death came over the telephone in a three-word sentence that somehow managed to embody all the faults that Ross devoted his life to correcting. A grief-stricken friend in Boston, charged with the task of spreading the news but too dazed to talk sensibly, said, \"It's all over.\" He meant that Ross was dead, but the listener took it to mean that the operation was over. Here, in three easy words, were the ambiguity, the euphemistic softness, the verbal infirmity that Harold W. Ross spent his life thrusting at. Ross regarded every sentence as the enemy, and believed that if a man watched closely enough, he would discover the vulnerable spot, the essential weakness. He devoted his life to making the weak strong\u2014a rather specialized form of blood transfusion, to be sure, but one that he believed in with such a consuming passion that his spirit infected others and inspired them, and lifted them. Whatever it was, this contagion, this vapor in these marshes, it spread. None escaped it. Nor is it likely to be dissipated in a hurry.\n\nHis ambition was to publish one good magazine, not a string of successful ones, and he thought of _The New Yorker_ as a sort of movement. He came equipped with not much knowledge and only two books\u2014Webster's Dictionary and Fowler's _Modern English Usage_. These books were his history, his geography, his literature, his art, his music, his everything. Some people found Ross's scholastic deficiencies quite appalling, and were not sure they had met the right man. But he was the right man, and the only question was whether the other fellow was capable of being tuned to Ross's vibrations. Ross had a thing that is at least as good as, and sometimes better than, knowledge: he had a sort of natural drive in the right direction, plus a complete respect for the work and ideas and opinions of others. It took a little while to get on to the fact that Ross, more violently than almost anybody, was proceeding in a good direction, and carrying others along with him, under torrential conditions. He was like a boat being driven at the mercy of some internal squall, a disturbance he himself only half understood, and of which he was at times suspicious.\n\nIn a way, he was a lucky man. For a monument he has the magazine to date\u2014one thousand three hundred and ninety-nine issues, born in the toil and pain that can be appreciated only by those who helped in the delivery room. These are his. They stand, unchangeable and open for inspection. We are, of course, not in a position to estimate the monument, even if we were in the mood to. But we are able to state one thing unequivocally: Ross set up a great target and pounded himself to pieces trying to hit it square in the middle. His dream was a simple dream; it was pure and had no frills: he wanted the magazine to be good, to be funny, and to be fair.\n\nWe say he was lucky. Some people cordially disliked him. Some were amused but not impressed. And then, last, there are the ones we have been seeing today, the ones who loved him and had him for a friend\u2014people he looked after, and who looked after him. These last are the ones who worked close enough to him, and long enough with him, to cross over the barrier reef of noisy shallows that ringed him, into the lagoon that was Ross himself\u2014a rewarding, and even enchanting, and relatively quiet place, utterly trustworthy as an anchorage. Maybe these people had all the luck. The entrance wasn't always easy to find.\n\nHe left a note on our desk one day apropos of something that had pleased him in the magazine. The note simply said, \"I am encouraged to go on.\" That is about the way we feel today, because of his contribution. We are encouraged to go on.\n\nWhen you took leave of Ross after a calm or stormy meeting, he always ended with the phrase that has become as much a part of the office as the paint on the walls. He would wave his limp hand, gesturing you away. \"All right,\" he would say. \"God bless you.\" Considering Ross's temperament and habits, this was a rather odd expression. He usually took God's name in vain if he took it at all. But when he sent you away with his benediction, which he uttered briskly and affectionately, and in which he and God seemed all scrambled together, it carried a warmth and sincerity that never failed to carry over. The words are so familiar to his helpers and friends here that they provide the only possible way to conclude this hasty notice and to take our leave. We cannot convey his manner. But with much love in our heart, we say, for everybody, \"All right Ross, God bless you.\"\n\nWhen James Thurber, one of White's closest friends, died on November 2, 1961, White wrote of him in the November 11th _New Yorker_.\n\nI am one of the lucky ones; I knew him before blindness hit him, before fame hit him, and I tend always to think of him as a young artist in a small office in a big city, with all the world still ahead. It was a fine thing to be young and at work in New York for a new magazine when Thurber was young and at work, and I will always be glad that this happened to me.\n\nIt was fortunate that we got on well; the office we shared was the size of a hall bedroom. There was just room enough for two men, two typewriters, and a stack of copy paper. The copy paper disappeared at a scandalous rate\u2014not because our production was high (although it was) but because Thurber used copy paper as the natural receptacle for discarded sorrows, immediate joys, stale dreams, golden prophecies, and messages of good cheer to the outside world and to fellow-workers. His mind was never at rest, and his pencil was connected to his mind by the best conductive tissue I have ever seen in action. The whole world knows what a funny man he was, but you had to sit next to him day after day to understand the extravagance of his clowning, the wildness and subtlety of his thinking, and the intensity of his interest in others and his sympathy for their dilemmas\u2014dilemmas that he instantly enlarged, put in focus, and made immortal, just as he enlarged and made immortal the strange goings on in the Ohio home of his boyhood. His waking dreams and his sleeping dreams commingled shamelessly and uproariously. Ohio was never far from his thoughts, and when he received a medal from his home state in 1953, he wrote, \"The clocks that strike in my dreams are often the clocks of Columbus.\" It is a beautiful sentence and a revealing one.\n\nHe was both a practitioner of humor and a defender of it. The day he died, I came on a letter from him, dictated to a secretary and signed in pencil with his sightless and enormous \"Jim.\" \"Every time is a time for humor,\" he wrote. \"I write humor the way a surgeon operates, because it is a livelihood, because I have a great urge to do it, because many interesting challenges are set up, and because I have the hope it may do some good.\" Once, I remember, he heard someone say that humor is a shield, not a sword, and it made him mad. He wasn't going to have anyone beating his sword into a shield. That \"surgeon,\" incidentally, is pure Mitty. During his happiest years, Thurber did not write the way a surgeon operates, he wrote the way a child skips rope, the way a mouse waltzes.\n\nAlthough he is best known for _Walter Mitty_ and _The Male Animal,_ the book of his I like best is _The Last Flower_. In it you will find his faith in the renewal of life, his feeling for the beauty and fragility of life on earth. Like all good writers, he fashioned his own best obituary notice. Nobody else can add to the record, much as he might like to. And of all the flowers, real and figurative, that will find their way to Thurber's last resting place, the one that will remain fresh and wiltproof is the little flower he himself drew, on the last page of that lovely book.\n\nEtched in my memory when my wife Esther and I were living as young people in New York in the 1930s and following the contemporary art scene is the incident of the Rivera mural in the new Rockefeller Center. Nelson Rockefeller, then a young man, was interested in contemporary art and was in charge of building the Center. He had commissioned Diego Rivera, the radical Mexican artist, to paint a mural in the entrance hall of the main building just above the skating rink. The work was done in fresco in which a plasterer lays up the surface just ahead of the painter who uses water-soluble pigments that penetrate the wet plaster\u2014so when the plaster dries, it is really on there. When Rivera arrived at the lower-right corner when he would normally sign, he introduced a large head of Lenin and the hammer and sickle, signed his name and was through. There was a great uproar and the mural was ultimately destroyed\u2014chipped off the wall. White wrote a poem about it in _The New Yorker_.\n\n**I Paint What I See**\n\n(A Ballad of Artistic Integrity, on the Occasion of the Removal of Some Rather Expensive Murals from the RCA Building in the Year 1933)\n\n\"What do you paint, when you paint on a wall?\"\n\nSaid John D.'s grandson Nelson.\n\n\"Do you paint just anything there at all?\n\n\"Will there be any doves, or a tree in fall?\n\n\"Or a hunting scene, like an English hall?\"\n\n\"I paint what I see,\" said Rivera.\n\n\"What are the colors you use when you paint?\"\n\nSaid John D.'s grandson Nelson.\n\n\"Do you use any red in the beard of a saint?\n\n\"If you do, is it terribly red, or faint?\n\n\"Do you use any blue! Is it Prussian?\"\n\n\"I paint what I paint,\" said Rivera.\n\n\"Whose is that head that I see on my wall?\"\n\nSaid John D.'s grandson Nelson.\n\n\"Is it anyone's head whom we know, at all?\n\n\"A Rensselaer, or a Saltonstall?\n\n\"Is it Franklin D.? Is it Mordaunt Hall?\n\n\"Or is it the head of a Russian?\"\n\n\"I paint what I think,\" said Rivera.\n\n\"I paint what I paint, I paint what I see,\n\n\"I paint what I think,\" said Rivera,\n\n\"And the thing that is dearest in life to me\n\n\"In a bourgeois hall is Integrity;\n\n\"However...\n\n\"I'll take out a couple of people drinkin'\n\n\"And put in a picture of Abraham Lincoln;\n\n\"I could even give you McCormick's reaper\n\n\"And still not make my art much cheaper.\n\n\"But the head of Lenin has got to stay\n\n\"Or my friends will give me the bird today,\n\n\"The bird, the bird, forever.\"\n\n\"It's not good taste in a man like me,\"\n\nSaid John D.'s grandson Nelson,\n\n\"To question an artist's integrity\n\n\"Or mention a practical thing like a fee,\n\n\"But I know what I like to a large degree,\n\n\"Though art I hate to hamper;\n\n\"For twenty-one thousand conservative bucks\n\n\"You painted a radical. I say shucks,\n\n\"I never could rent the offices \u2014\n\n\"The capitalistic offices.\n\n\"For this, as you know, is a public hall\n\n\"And people want doves, or a tree in fall,\n\n\"And although your art I dislike to hamper,\n\n\"I owe a little to God and Gramper,\n\n\"And after all,\n\n\"it's my wall...\"\n\n\"We'll see if it is,\" said Rivera.\n\nRivera was born 50 years too soon, because I understand that New York now has a law that gives the artist some control over how his work is shown. They might have some trouble getting rid of that mural today. A replica of Rivera's Rockefeller Center mural is now on the wall of a museum in Mexico City. It pleases me to think that some day ours will become a mature enough society that that mural will return to New York, head of Lenin and all, and be given an appropriate spot so that people can judge its artistic merit. When and if that happens, it may be that this poem of E. B. White's, which is widely anthologized, will have helped keep alive the idea that there is some unfinished business from the 1930s.\n\nWhen White was about 40, he and his wife moved from New York to a farm on the coast of Maine. They both continued to work for _The New Yorker_ with occasional trips to New York. Katharine, who was the older, died in 1977. Mr. White died October 1, 1985.\n\nWhite, as we know from his comments on his childhood, confirmed in several letters, has lived all his life in a state of anxiety with several long illnesses. A friend commented on his leaving New York, \"Andy moved to Maine to save his life.\" Living in Maine in a small community and close to farm animals was a necessary condition for his existence, and much of his later writing has been about life on his farm and among small-town people in whom he found much-treasured wisdom.\n\nDespite his lifelong anxiety, White was a very social guy. He and his wife, Katharine, had many friends, spent much enjoyable time with them, and carried on a lively correspondence.\n\nMy most interesting, and perhaps my most significant experience with White's writing came in 1969. After retiring from AT&T in 1964, I spent some active years in consulting work. Two of my clients in that period were universities during the traumatic years of student unrest. Time has healed some of the scars, but I have vivid memories of that period, particularly talks with the radical students. It was a wild time, and I was right in the thick of it.\n\nIn September of 1969, my alma mater, Carleton College, did an interesting thing. As a smaller college of 1500 students, they were not as threatened by the unrest as were large universities, but they were disturbed enough to convene, for two weeks ahead of the opening of school that year, all of the student leaders. This included class officers, team captains, editors of paper and year book, etc., plus three or four faculty and the college chaplain who presided. Their purpose was to just have a leisurely talk about the problems of operating the college under these disturbing conditions. I was invited to meet with them for two days as a resource person (since I had been up to my ears in this confusion in other places).\n\nNear the end of my two days with them, I read them E. B. White's essay \"The Second Tree from the Corner.\" Before reading it, I told them a little about White and reminded them of White's statement to the reporter, \"I was born scared and at 70 I am still scared.\" I asked them to bear this in mind because I suspected that the essay was somewhat autobiographical. Some years later when White's letters came out, I confirmed that it was written after a session with his psychiatrist in New York.\n\nThe essay, \"The Second Tree from The Corner,\" concerns a man named Trexler in a routine session with his psychiatrist, and what Trexler thinks about as he walks down the street after the session. The interview deals with Trexler's fears and a question that the doctor repeatedly pressed, \"What do _you_ want!\" In the course of the session, Trexler turns the question on the doctor, \"What do _you_ want!\" And the doctor, caught short, stammers, \"I want a new wing on my house on Long Island.\"\n\nWhen I finished reading, I commented to the students that I thought that White, in this essay, had a literary strategy somewhat similar to that of Camus' when he wrote _The Stranger_ (which I knew they had all read in freshman English). Camus takes 100 or so pages to describe a situation, a murder, and the trial and conviction of the murderer to set the stage, it seems to me, for three or four pages at the end which tell of the conversation between a priest and the convicted man on the eve of the latter's execution. I believe that the content of this interview was what Camus really wanted to get across to us. White, more devoted to economy of language than Camus, takes five pages to describe this session with the psychiatrist to set the stage for two paragraphs at the end where, it seems to me, he tells us what he really wants us to get. And I reread those two paragraphs.\n\nIt was an evening of clearing weather, the Park showing green and desirable in the distance, the last daylight applying a high lacquer to the brick and brownstone walls and giving the street scene a luminous and intoxicating splendor. Trexler meditated, as he walked, on what he wanted. \"What do you want!\" he heard again. Trexler knew what he wanted, and what, in general, all men wanted; and he was glad, in a way, that it was both inexpressible and unattainable, and that it wasn't a wing. He was satisfied to remember that it was deep, formless, enduring, and impossible of fulfillment, and that it made men sick, and that when you sauntered along Third Avenue and looked through the doorways into the dim saloons, you could sometimes pick out from the unregenerate ranks the ones who had not forgotten, gazing steadily into the bottoms of the glasses on the long chance that they could get another little peek at it. Trexler found himself renewed by the remembrance that what he wanted was at once great and microscopic, and that although it borrowed from the nature of large deeds and of youthful love and of old songs and early intimations, it was not any one of these things, and that it had not been isolated or pinned down, and that a man who attempted to define it in the privacy of a doctor's office would fall flat on his face.\n\nTrexler felt invigorated. Suddenly his sickness seemed health, his dizziness stability. A small tree, rising between him and the light, stood there saturated with the evening, each gilt-edged leaf perfectly drunk with excellence and delicacy. Trexler's spine registered an ever so slight tremor as it picked up this natural disturbance in the lovely scene. \"I want the second tree from the corner, just as it stands,\" he said, answering to an imaginary question from an imaginary physician. And he felt a slow pride in realizing that what he wanted none could bestow, and that what he had none could take away. He felt content to be sick, unembarrassed at being afraid; and in the jungle of his fear he glimpsed (as he had so often glimpsed them before) the flashy tail feathers of the bird courage.\n\nThere were a few moments of silence after I finished, and the students took off and talked for two hours. I didn't say another word\u2014just listened. Condensing two hours of discussion into one sentence: they ultimately identified the problem of the students of their generation as a sort of mental illness, and, like White, they would only recover their poise when they accepted their illness as health\u2014and got on with their work. It was the most fascinating two hours of discussion I ever listened to.\n\nThe sequel to his session came several weeks later when Professor Maitland, the Chaplain, sent me a copy of the first issue of the college newspaper in which this two-week session was reported. The report concluded with the announcement that the group had agreed to continue to meet during the school year and that they had named their group _The Second Tree from the Corner_. That was over 15 years ago, and I still hear the occasional reverberation from that meeting. Such is the influence of thinking that sees things whole, and of language that tells us what one sees that is powerful and beautiful.\n\nOne of the things one discovers by reading White's letters is that, in the early days of _The New Yorker_ , he was the handyman. He did everything. He wrote essays, poems, newsbreaks, reported events, and wrote the opening \"Talk of the Town.\" He even drew one cover, _and_ he wrote captions for cartoons. My favorite quote from an early issue of _The New Yorker_ is the caption on a full-page cartoon. As I remember it, there is a rich kid of 8 or 9 seated alone at dinner in a big posh dining room, being served by maid and butler, and with his governess standing behind him saying, \"Oswald dear, please eat your nice _broccoli_.\" Says Oswald, \"I say it's spinach, and I say the hell with it.\" I have found that line appropriate on several occasions over the years. And I discovered in the letters that it is pure E. B. White.\n\nWhite was not a \"cause\" man\u2014with two exceptions. During the war, he became interested in world government and wrote extensively on that subject with one of the loveliest pieces on that theme entitled, \"The Wild Flag.\" But then, after the war, he covered the founding meeting of the United Nations for _The New Yorker_ and concluded that world government was not a realistic expectation in our times. In a letter to a niece explaining his position, he said, \"I think the most precious thing in the world is not the concept of world federation but the concept of justice\u2014that is, justice as it has developed in the Western world. The only sort of one-world I would settle for is a one-world firmly based on that type of justice.\"\n\nThe other cause, long standing and still much alive in White as an old man, is the cause of integrity in journalism. Over the years, there have been three major public manifestations of this concern. Some years ago, White discovered the practice of _Reader's Digest_ commissioning and paying for articles, giving them to other publications to print, and then condensing them in _Reader's Digest_. He strongly condemned this practice in the pages of _The New Yorker_ , others joined in, and there was quite a stir about it. It didn't deter the _Digest_. But it did result in a _New Yorker_ policy not to permit the _Digest_ to take any of their stuff.\n\nThen when a fellow _New Yorker_ writer, Alexander Woolcott, gave a testimonial for a beer ad, White took out after him\u2014in the pages of _The New Yorker_. Maybe okay for a movie actor, he said, but not for a journalist. This created something of a stir, but I doubt that it influenced Woolcott.\n\nIn 1974, when White was old, he learned of the arrangement wherein Harrison Salisbury, retired associate editor of the _New York Times,_ accepted a substantial fee from Xerox Corporation to write an article that would appear in _Esquire_ Magazine with a full-page Xerox ad before and after it. White took pen in hand and wrote a letter to the editor of the _The Ellsworth (Maine) American_ describing the details of the arrangement and taking sharp exception to it.\n\nIn due course, White received a letter from Xerox outlining their ground rules for sponsoring articles and asking, \"With these ground rules, do you still see something sinister in the sponsoring?\" White's reply of January 30, 1976, said unequivocally, \"Yes, I do,\" and he went on to support that judgment with careful reasoning. This is what he wrote:\n\nLetter to Mr. W. B. Jones\n\nDirector, Communications Operations\n\nXerox Corp.\n\nJanuary 30, 1976\n\nDear Mr. Jones,\n\nIn extending my remarks on sponsorship, published in _The Ellsworth American_ , I want to limit the discussion to the press\u2014that is, to newspapers and magazines. I'll not speculate about television, as television is outside my experience, and I have no ready opinion about sponsorship in that medium.\n\nIn your recent letter to me, you ask whether having studied your ground rules for proper conduct in sponsoring a magazine piece, I still see something sinister in the sponsorship. Yes, I do. _Sinister_ may not be the right word, but I see something ominous and unhealthy when a corporation underwrites an article in a magazine of general circulation. This is not, essentially, the old familiar question of an advertiser trying to influence editorial content: almost everyone is acquainted with that common phenomenon. Readers are aware that it is always present but usually in a rather subdued or nonthreatening form. Xerox's sponsoring of a specific writer on a specific occasion for a specific article is something quite different. No one, as far as I know, accuses Xerox of trying to influence editorial opinion. But many people are wondering why a large corporation placed so much money on a magazine piece, why the writer of the piece was willing to get paid in so unusual a fashion, and why _Esquire_ was ready and willing to have an outsider pick up the tab. These are reasonable questions.\n\nThe press in our free country is reliable and useful not because of its good character but because of its great diversity. As long as there are many owners, each pursuing his own brand of truth, we the people have the opportunity to arrive at the truth and to dwell in the light. The multiplicity of ownership is crucial. It's only when there are few owners, or as in a government-controlled press, one owner, that the truth becomes elusive and the light fails. For a citizen in our free society, it is an enormous privilege and a wonderful protection to have access to hundreds of periodicals, each peddling its own belief. There is safety in numbers: the papers expose each other's follies and peccadillos, correct each other's mistakes, and cancel out each other's biases. The reader is free to range around in the whole editorial bouillabaisse and explore it for the one clam that matters\u2014the truth.\n\nWhen a large corporation or a rich individual underwrites an article in a magazine, the picture changes: the ownership of that magazine has been diminished, the outline of the magazine has been blurred. In the case of the Salisbury piece, it was as though _Esquire_ had gone on relief, was accepting its first welfare payment, and was not its own man any more. The editor protests that he accepts full responsibility for the text and that Xerox had nothing to do with the whole business. But the fact remains that, despite his full acceptance of responsibility, he somehow did not get around to paying the bill. This is unsettling and I think unhealthy. Whenever money changes hands, something goes along with it\u2014an intangible something that varies with the circumstances. It would be hard to resist the suspicion that _Esquire_ feels indebted to Xerox, that Mr. Salisbury feels indebted to both, and that the ownership, or sovereignty, of _Esquire_ has been nibbled all around the edges.\n\nSponsorship in the press is an invitation to corruption and abuse. The temptations are great, and there is an opportunist behind every bush. A funded article is a tempting morsel for any publication\u2014particularly for one that is having a hard time making ends meet. A funded assignment is a tempting dish for a writer, who may pick up a much larger fee than he is accustomed to getting, and sponsorship is attractive to the sponsor himself, who, for one reason or another, feels an urge to penetrate the editorial columns after being so long pent up in the advertising pages. These temptations are real, and if the barriers were to be let down, I believe corruption and abuse would soon follow. Not all corporations would approach subsidy in the immaculate way Xerox did or in the same spirit of benefaction. There are a thousand reasons for someone's wishing to buy his way into print, many of them unpalatable, all of them to some degree self-serving. Buying and selling space in news columns could become a serious disease of the press. If it reached epidemic proportions, it could destroy the press. I don't want IBM or the National Rifle Association providing me with a funded spectacular when I open my paper. I want to read what the editor and the publisher have managed to dig up on their own\u2014and paid for out of the till.\n\nMy affection for the free press in a democracy goes back a long way. My love for it was my first and greatest love. If I felt a shock at the news of the Salisbury-Xerox- _Esquire_ arrangement, it was because the sponsorship principle seemed to challenge and threaten everything I believed in: that the press must not only be free, it must be fiercely independent\u2014to survive and to serve. Not all papers are fiercely independent, God knows, but there are always enough of them around to provide a core of integrity and an example that others feel obligated to steer by. The funded article is not in itself evil, but it is the beginning of evil, and it is an invitation to evil. I hope the invitation will not again be extended, and, if extended, I hope it will be declined.\n\nAbout a hundred and fifty years ago, de Tocqueville wrote: \"The journalists of the United States are generally in a very humble position, with a scanty education and a vulgar turn of mind.\" Today, we chuckle at this antique characterization. But about fifty years ago, when I was a young journalist, I had the good fortune to encounter an editor who fitted the description quite closely. Harold Ross, who founded _The New Yorker_ , was deficient in education and had\u2014at least to all outward appearances\u2014a vulgar turn of mind. What he did possess, though, was the ferocity of independence. He was having a tough time finding money to keep his floundering little sheet alive, yet he was determined that neither money nor influence would ever corrupt his dream or deflower his text.\n\nHis boiling point was so low as to be comical. The faintest suggestion of the shadow of advertising in his news and editorial columns would cause him to erupt. He would explode in anger, the building would reverberate with his wrath, and his terrible swift sword would go flashing up and down the corridors. For a young man, it was an impressive sight and a memorable one. Fifty years have not dimmed for me either the spectacle of Ross's ferocity or my own early convictions\u2014which were identical with his. He has come to my mind often while I've been composing this reply to your inquiry.\n\nI hope I've clarified by a little bit my feelings about the autonomy of the press and the dangers of sponsorship of articles. Thanks for giving me the chance to speak my piece.\n\nSincerely,\n\nE. B. White\n\nXerox thanked White for \"telling us what we didn't want to hear.\" A few months later, they wrote saying that Xerox had decided not to underwrite any more articles and that they were convinced that it was \"the right decision.\"\n\n_The Journalism Review_ of the Columbia University School of Journalism printed this entire exchange of letters under the title of, \"What E. B. White Told Xerox\u2014or How a Solitary Man of Letters Talked a Corporation Out of Funding Magazine Articles\u2014and Helped to Define a Free Press.\"\n\nI hazard the prophecy that in the long test of history, this one letter will establish E. B. White as one who sees things whole and who has the gift of language to tell us ordinary mortals what he sees.\n\nI want to close with a brief reference to the three children's stories.\n\nThe first, _Stuart Little_ , is about a mouse by that name. The book seems to lack an ending. It just stops. When I first read it, I thought a bunch of pages were missing from my copy. And White received letters complaining about this.\n\nTo one of them he replied, \"I think many readers find the end inconclusive, but I have always found life inconclusive, and I guess it shows up in my work.\"\n\nTo another he replied, \"Quite a number of children have written to ask me about Stuart. They want to know whether he got back home and whether he found Margalo (the bird he was hunting for). They are good questions, but I did not answer them because, in a way, Stuart's journey symbolizes the continuing journey that everybody undertakes\u2014in search for what is perfect and unattainable. This is perhaps too elusive an idea to put in a book for children, but I put it in anyway.\"\n\nAnd to a girl named Jill, he writes, \" _Stuart Little_ is the story of a quest or search. Much of life is questing and searching, and I was writing about that. If the book ends while the search is still going on, that's because I wanted it that way. As you grow older, you will realize that many of us in this world go through life looking for something that is beautiful and good\u2014often something we can't quite name. In Stuart's case, he was searching for the bird Margalo, who was his ideal of beauty and goodness. Whether he ever found her or not, or whether he got home or not, is less important than the adventure itself. If the book made you cry, that's because you are aware of the sadness and richness of life's involvement and the quest for beauty. Cheer up\u2014Stuart may yet find his bird. He may even get home again. Meantime, he is headed in the right direction, and I am sure you are.\"\n\n\"The right direction,\" which White also attributed to Harold Ross in his obituary, is central to White's concept of wholeness. One often does not know the precise goal, but one must always be certain of one's direction. The goal will reveal itself in due course.\n\nIt seems fitting, in view of White's closeness to animals, that in his later years he should turn to writing children's stories about animals. In this period, in a letter to a friend, he wrote that mice and spiders have been around for millions of years without damaging the environment. But, in the few thousand years that so-called civilized man has been around, he has nearly destroyed it.\n\nIn another letter he wrote, \"if it were not for spiders, the insects would take over the earth.\"\n\nLet me note the concluding paragraphs in _Charlotte's Web_ and _The Trumpet of the Swan_.\n\nCharlotte, the spider, has woven messages in her web that saved Wilbur, the pig, from being butchered and made him famous. The story concludes:\n\nMr. Zuckerman took fine care of Wilbur all the rest of his days, and the pig was often visited by friends and admirers, for nobody ever forgot the year of his triumph and the miracle of the web. Life in the barn was very good\u2014night and day, winter and summer, spring and fall, dull days and bright days. It was the best place to be, thought Wilbur, this warm delicious cellar, with the garrulous geese, the changing seasons, the heat of the sun, the passage of swallows, the nearness of rats, the sameness of sheep, the love of spiders, the smell of manure and the glory of everything.\n\nLouis the swan, from a genus of swans that make a trumpeting sound, was named after Louis Armstrong. Louis was born without a voice and acquired a trumpet that he learned to play. Sam, the boy who helped him learn to play it, says at the end:\n\nTonight I heard Louis' horn. My father heard it, too. The wind was right, and I could hear the notes of taps, just as darkness fell. There is nothing in all the world I like better than the trumpet of the swan....\n\nOn the pond where the swans were, Louis put his trumpet away. The cygnets crept under their mother's wing. Darkness settled on woods and fields and marsh. A loon called its wild night cry. As Louis relaxed and prepared for sleep, all his thoughts were on how lucky he was to inhabit such a beautiful earth, how lucky he had been to solve his problems with music, and how pleasant it was to look forward to another night of sleep and another day tomorrow, and the fresh morning, and the light that returns with the day.\n\nIn the preface to his recent collected book of poems, E. B. White wrote:\n\nTo me, poetry is what is memorable, and a poet is a fellow or a girl who lets drop a line that gets remembered in the morning. Poetry turns up in unexpected places, in unguarded moments. I have yet to encounter the line from the song in _Oklahoma_ , \"All the sounds of the earth are like music,\" without being brought to the edge of tears.\n\nThe interview with E. B. White on his 70th birthday ends with the question of what he cherished most in life. \"When my wife's Aunt Caroline was in her '90s,\" he replied, \"she lived with us and she once remarked, 'Remembrance is sufficient of the beauty we have seen.' I cherish the remembrance of the beauty I have seen. I cherish the grave compulsive world.\"\n\n## 8\n\n## **Old Age: The Ultimate Test of Spirit\n\nAn Essay on Preparation**\n\n_Spirit!_ What are we talking about? The unabridged dictionary I consulted begins a full page of definitions with \"The breath of life.\" But dictionaries can do little more than summarize common usage; and it seems clear to me, after reading the full page of definitions, that there is no well-accepted meaning for this much used and important word.\n\nI conclude, then, that I cannot give a concise definition for _spirit_ , for which old age seems to me to be the ultimate test. The meaning of that word, as I use it, lies beyond the barrier that separates mystery from what we call reality. Yet I have a sharp awareness of spirit when it is present, in myself and others, and I have a depressing feeling of loss when it is absent, in myself and others, at times when it is urgently needed.\n\nI have come to connect spirit, the kind I would like to see more of, to a concept of _serve_ as I see it in the consequences on those being served: do those being served grow as persons? Do they, while being served, become stronger, wiser, freer, more at peace with themselves, more likely themselves to become servants? And, what will be the effect on the least privileged in society? Will she or he benefit, or at least not be further deprived? The quality of a society will be judged by what the least privileged in it achieves. My hope for the future rests on the belief that among the legions of deprived and unsophisticated people are many true servants, and that most people can be helped to discriminate among those who presume to serve them, and identify, and respond only to those who are true servants.\n\nSpirit can be said to be the driving force behind the motive to serve. And the ultimate test for spirit in one's old age is, I believe, can one look back at one's active life and achieve serenity from the knowledge that one has, according to one's lights, served? And can one regard one's present state, no matter how limited by age and health, as one of continuing to serve? One of my deeply etched memories is the view of an old man of 95 sitting by the window of his fisherman's house on the far out coast of Maine quietly knitting nets for lobster traps which the active fisherman in the family would use. He was still serving with what he could do best at his age.\n\nMuch of my present perspective on old age comes from having watched my father grow old. I was very close to him. He was an intelligent and good man but with a fifth-grade education, and he lived a life of limited opportunity. But by a prudent use of his life, he managed to leave a little corner of the world a bit better than he found it. He stands tall in my memory as a model of the true servant. One of my treasured memories as a small boy is attending occasional evening meetings when he was a member of the city council. I would stay awake as long as I could because the meetings were sometimes exciting\u2014tumultuous is a better word\u2014and father was generally in the thick of the action. Then I would curl up in his overcoat behind his chair and go to sleep, to be carried home at the close of the meeting.\n\nFather lived to be 80 and achieved a remarkable serenity in his last years. A couple of years before he died, he told me that he realized that he was in his twilight years and that he had concluded that he should read some of the Bible since he had not read it at all. Then he added plaintively, \"I tried, but I quickly gave it up because it made no sense.\"\n\nWhen I was about 13, I recall listening to a conversation with a committee from our church that had come to try and persuade father to raise his quite nominal contribution. Father listened patiently and then said, \"No.\" He thought his contribution was about right. He was glad the church was there, but as an instrument for doing good in the world he rated it well below both his labor union and his political party. The committee left in a huff.\n\nIn his old age, father once commented to me, \"The tragedy of our town is that it once had a great people but it no longer has them.\" In his judgment the \"status\" people of our town\u2014bankers, industrialists, merchants, professional people\u2014were all mediocre. In the heat of one of father's political battles, the town's leading citizen, in money and prestige, tried to buy him off.\n\nThere is a much greater story to tell about my father but I will take another occasion for that. I have given enough of that story here to suggest that my early formative years when I was very close to him have shaped the way my life unfolded in important ways. I have given this much to suggest that watching my father grow old was a very special kind of experience, and it alerted me to watch my own aging so that I came into my old age with my eyes open\u2014and aware.\n\nI also feel it important to note that I did not grow up in a church-identified home. Consequently, I found it necessary to think my own way through a spiritual orientation for my life. There was much in the model of my father to guide me and it overshadowed what I got from the little exposure I had to churches. I am not recommending mine as the best way to grow up. I am simply noting what it was because it had a great deal to do with the adventure in preparation on which I have been embarked.\n\nAt 50, father suffered a health setback from which he recovered, but he dropped out of politics. In his old age, he told me that this was a wise move. \"All of my old political cronies my age are gone. If I had won my last contest for office, that would have put me into the big money and I probably would not be here today. I am lucky that I lost.\" What he did not say, but I would observe, is he would have missed the joy of the serenity of his old age.\n\nI have now gone past 80 and I have entered a new phase of life that I will comment on later, and I have frequent occasion to reflect on the quality of my father's old age, at a time when the Bible made no sense, and after an active life in which he rated churches in a quite inferior position. And I have concluded that, with all of this, he was a deeply religious man; and, he would be seen as religious in any peace-loving culture. I see his basic religious feeling as the root of his serenity, and I think of him now as one who was sustained by great human spirit all of his life.\n\nMy father and his father had lived their whole lives in the town where I was born and lived until I was 20, and father was deeply disappointed that I did not choose to make my home there. At age 22, I had made my career decision and I would go wherever it led me. Ultimately, it led to New York where I lived for 40 years. My roots, however, are in my father's tradition in Terre Haute, Indiana. These roots are still very much a part of me in my older years, even though the more than 60 years since I left Terre Haute have been in a very different world from the one my father lived in. Those 20 years with him in Terre Haute were the years of my formation, and they have stayed with me. I am sorry about people who reach old age without being aware of their roots.\n\nMalcolm Cowley's little book, _The View From Eighty_ , makes the point that rings true to my experience: that most of the literature about aging is written by younger people who have not been there, and that until one passes through that magic number that makes one old, one really cannot appreciate what old age is like. I could not have anticipated the view of life that I now have. And I may not be able to describe it in a way that is meaningful to younger people, perhaps not even to my own age group because I suspect that we oldsters are just as different, one from the other, as we have been at earlier stages in life. Each of us in our old age may be at the point of summing up our own unique experience and examining, and relating to, the very different roots from which we spring: roots that have shaped our whole existence.\n\nI have long pondered those lines with which Robert Browning opens his poem \"Rabbi Ben Ezra\":\n\n_Grow old along with me!_\n\n_The best is yet to be;_\n\n_The last of life for which the first was made_.\n\nA friend my age who is crippled with arthritis recently wrote quoting those lines and concluding with, \"Bah!\" I suppose that if I were wracked with pain my response might be the same. In the absence of that pain I keep wondering what is that \"best\" that Browning was talking about, or did he know what he was talking about?\n\nI have long been a meditator, and as I have grown older, meditation has become more central to my existence and takes much more of the typical day. I have taken training in both transcendental and Buddhist meditation, but my current meditation practices are pretty much my own. I have arrived at a point where I prefer my own private meditation to any formal religious service. As I have grown older, I have come to value solitude more and more. I doubt that I would ever want to be a hermit and live in complete isolation, but I definitely limit my contact with people and this tendency is growing. My wife and I, without talking about it, have evolved a relationship in which there is very little conversation. We enjoy being together and we appreciate our solitude together. Neither of us feels the need to be entertained, nor do we yearn to be young again.\n\nI read some, but much less than I once did\u2014although my vision is still good. Much of what I do read is rereading what I have read before, and I continue to find fresh meaning.\n\nAt age 75 I stopped making speeches, though I was still active. At 80 I stopped going to meetings, and I no longer travel at all. I use an automobile for local shopping only, and I will soon give that up.\n\nAt age 73, my wife and I entered a lifecare retirement community. I don't believe it is a good idea to segregate old people this way and thus limit their interaction with younger people. Both the old and the young lose by it. Someday, when we are a more civilized people and have come to live in and appreciate community, I suspect places like the one I now live in will be abandoned. We chose to come here because it was the best option available to us. If we were making the choice today, we would come to the same place we now live. The conscientious Quakers who run the place we live in do a good job of managing the services on which we depend. We are served by caring and sensitive people, and the trustees keep the place solvent, which is more than can be said for some places like this. I am deeply grateful to the dedicated trustees who maintain the integrity of the place where I will spend my last years.\n\nWe try to be friendly with fellow residents, and there are a few with whom we occasionally share meals. But my wife and I stay clear away from the frenetic activity, the game playing, and the traveling to exotic places that occupy so much of people's time. Most of my interaction with people is with younger people in active careers elsewhere. I share their joys and sorrows and I believe I give them something of value from my relative detachment and my more meditative life. And they give much to me. I know that at age 83 I am lucky to have these relationships with younger active people, and I appreciate them.\n\nOccasionally I am asked what it is like to be old; and there is some conversation about it. I am not sure I can put it down in black and white, but I will try.\n\nAs I see it now, my most interesting and productive years were from 60 to 75. This was not so much that I was a late bloomer, but because from 40 to 60 I made conscious preparation (more on that later) for a second career, and I had a really good one. Also, that new career that started at age 60 rested on 38 years as a disciplined organization man in a huge bureaucracy. A disciplined life can be lived in many ways, but the kind I had in my early work was absolutely essential for a good second career. People who have managed their lives without experiencing that kind of sustained discipline, both the bitter and the sweet, seem to me to have missed something important.\n\nI have never been a \"high energy\" person, and at 75 there were definite signals that I should slow down. I welcomed the change of pace. I stopped traveling altogether, something I had done a great deal of. In a way, this may be seen as the start of a third career, a very low-key one. Now, at age 83, I write some, reflect a lot, see a few people, and occasionally write a letter to someone in active life suggesting some line of thought or action. I do not miss the more active life I used to lead.\n\nThere may have been a special impact of the big round number 80 that I passed three years ago. Our children put on a big party for my wife and me so we could not overlook the fact that something important had happened. A different view of life gradually emerged. I always knew that I was not immortal but as I passed 80 I became aware that I was in a \"countdown\" era. It was not that the signs of death were imminent, but I slowly became more detached. This has been a new and different phase, an even more radical break than I made at age 60 when I retired from my main career and started a second career; or, at 75 when I stopped traveling, except that this time there was not a precise point of change. I had come to realize that I could no longer serve by carrying an active role in the world. I would only get in the way if I tried. Now, I came to accept, I can best serve by being. It was not a new career, like it could have been seen at age 75 when the major change I made was to stop traveling and making speeches. Now, no mountains to climb. I may continue to do some things, mostly writing; but it is no longer \"production.\" Whether I get it done or not is no longer important, as it used to be.\n\nI scan the daily newspaper but I rarely listen to radio or TV news\u2014because they cannot be scanned. I prefer to meditate, and I have come to view my meditating as serving. Somehow the quiet and peace of anyone's meditation communicates and enriches the culture. I feel the fruits of other people's meditation.\n\nReviewing or summing up my life is not a preoccupation. Occasionally I call up something out of the past as a help in clearing my thought about where I now am\u2014and I hope to remain sharply aware of where I am. I do not brood on the past. There have been errors, failures, and hurts but they were in that other existence from which I am now quite separated, and I look back on it with detachment, as if it were the record of another's life.\n\nThe main difference between this present past-80 period and my earlier old age when I still felt myself to be in a career, of having a work, may be that up to recently there has always been a future that would ultimately be connected with the present as the past. Now there is no future and there is really no past. There is only a history that may as well be another's past. There is only now. Maybe this is the \"best\" that Robert Browning assured us of: finally to achieve living wholly in the present moment, unencumbered with the record of one's past and oblivious to the future, and accepting of the loss of energy and the passions of youth. I believe this is what my father achieved in his old age. He bound over to me a sense of rest and being at peace with himself and the world, and I recognized this state when I arrived at 80, 34 years after my father's death. I find this chapter of my life as rewarding as any previous one. And I feel a greater sense of continuity with my father's life than I ever felt before.\n\nLet me hasten to add to what I have just said: these recent changes have not put me in a state of euphoria, although I feel okay most of the time. I have withdrawn from active participation in society and live in a lifecare community. Consequently, the usual frustrations and irritations of life have been substantially reduced. But when these do occur, as on occasion they still do, my reaction is little different from what it has always been.\n\nI have noted that my father in his 80th and most serene year, could lament the tragedy of our town because of the paucity of great people in it. So I, in my 83rd year and with a much larger view of the world than my father had, now lament what seems the small number among those who see themselves as able, conscientious, and dedicated, and who are disposed to respond to a vision of the larger roles they might play and the much greater service to society that the institutions they influence might render. We have plenty of people with the ability and the stamina to build and lead a much more serving society and I believe they would lead fuller lives, if they would rise to their opportunities. What they seem to lack is spirit, and I wonder what they will be like when they grow old. Will they find it the \"best\" that is yet to be? I said at the outset that I cannot define spirit. But I have tried in what I have said to give human spirit a meaning that is beyond rational definition.\n\nEarlier I noted, as I look back now, my years from 60 to 75 were in some ways my most interesting and productive because, in part, from age 40 to 60 I made conscious preparation for my old age. When I was about 40, I had the good fortune to read an article by a radio commentator of that day, Elmer Davis. Davis had a coronary attack when he was past 60 that slowed him down and made him more reflective. Out of his reflection came an essay entitled, \"The Uses of Old People.\" In it he made the point that old people can be particularly useful, not just for the reason that they are more seasoned and experienced, but because there are important things to be done that are best done by old people, either because they do not fit well into a career or they are too risky for younger people to undertake. Furthermore, he advised that young people should look forward to, and prepare for old age as a time of potentially great usefulness rather than as a time when one is put out to pasture when one wears out. This message came through to me loud and clear, and I resolved that I would then begin to prepare for a second career at age 60 when I could elect my pension where I worked.\n\nI had been reasonably successful in my work and my volunteer activity up to age 40, but I realized that I was not a person of great talents or highly specialized abilities, and that I would not emerge at age 60 as a much sought after professional. Without a specific second career in mind, what would I prepare for?\n\nMaking plans for my life has never been a preoccupation. I left college with the aim of entering a big business and with the expectation that my career would evolve as opportunities presented themselves and as the spirit moved me. This strategy had served well up to age 40, and I saw no reason to change it. I simply had my eye on age 60 when I would start a second career (if I was still around), and in those intervening years I would continue to evolve, but with the aim that I was preparing for something that I could not then define. I had no idea what my old age would be like or what I would do with it. \"Take it as it comes\" was my motto as it had been up to that point. I do not recall that I thought in these terms, but I seemed to set out to favor and accelerate the kind of evolutionary development that had taken place up to age 40. I did not awake each morning with the question, \"What will I do today to prepare for my old age?\" In fact, once I embarked on this course, I rarely gave it a thought. It simply became a way of life, and there was a good deal of chance in it. What was different was that I became more venturesome and experimental in doing things that would widen my horizons and enlarge my self-understanding. All that I am now sure of is that I arrived at age 60 well prepared to be useful, and that interesting and challenging opportunities that I could not have anticipated at age 40 were numerous; more than I could take on.\n\nI am quite a \"private\" person and I shared this thinking only with my wife, who understood and approved. I am sure, however, that I was considered \"odd\" by my friends and associates because I did unconventional things. It is also clear that my superiors in my company were at times a bit puzzled about what to do with me. I know they regarded me as valuable because they paid me well. But valuable for what? They resolved their puzzlement when I was 50 by appointing me director of Management Research, providing me with sufficient budget to hire a professional staff, and giving me a broad charter to research and advise regarding the management of this huge company, especially how its top structure functioned (or failed to function). My last ten years with my company were excellent preparation for my old age, better than any other executive post I know about. I have often wondered how the choices I made in using my optional time, both inside and outside the company, contributed to how my final role there evolved. However, it was not a bed of roses. While I had the charter and the responsibility and the budget, the bureaucracy did not understand it, and I was in collision with the establishment all over the place. This too was good preparation for living with ambiguity and served me well in my second career. When I retired, the position disappeared. I could not train a successor. That person would have to prepare himself or herself as I did. And no one did.\n\nIn writing about my experience now, I hope to encourage others to resolve in their own ways to prepare for their old age, to prepare to face uncertainty. Such preparation, if wisely done, may not only favor a fruitful old age but it may, as with me, make the years of preparation more productive and enjoyable. I recall that once a colleague my age, one who was obviously not preparing, came into my office one morning, seated himself before my desk, puffed violently on his cigarette for several seconds, and then said, \"I am leading a life of quiet desperation!\" Later, when we had both retired, he called my wife one morning and asked her advice on how he could get into the kind of work I was doing. Shortly after I had announced my retirement, a close friend who was president of one of our subsidiaries, a very successful man as measured by income and prestige, lunched with me and asked how, when he retired in a couple of years, could he get into the kind of work I would be doing. I had to say, as gently as I could tell him, \"No way!\" If one will settle for a retirement of golf and fishing and simple civic chores, no preparation may be needed. But if one wants one's life to ascend creatively and in new ways, as long as one has one's wits, then my experience would suggest that one is well advised to prepare.\n\nIn my later years in my company, I did some lecturing in business schools. When I described my role in my company some of the students' eyes would brighten and someone would say, \"That is the kind of job I would like to have.\" To which I would reply, \"I should tell you that it took me 25 years to get into a position to do this work.\" And the response would likely be, \"Oh no! Maybe 6 months, but not 25 years\" (as much as most of them had already lived). And I would say, \"Maybe you could do it in less; but you would have to take the time to build the trust that this kind of unstructured role requires. I have the charter because the company has some needs and feels some pains that no one can define with precision. For those in charge, trust that I will search for and carry out actions that will do more good than harm. That trust needs to be high and one does not earn it quickly. Whatever your career, if you would like to evolve into a position of great trust, then you will need to use your opportunities in a way that constitutes preparation for being given that trust.\" Young people have a hard time grappling with that idea, but it is of the very essence.\n\nA detailed account of my 40 to 60 preparation would not be useful to another. Everyone should chart her or his own course. I will give a brief summary of mine just to suggest the range of opportunities that may be available if one opens one's imagination and is alert to opportunities, which the two colleagues noted above seemed unwilling, perhaps lacked the courage, to do.\n\nI made the decision to prepare for my old age near the end of WWII, when commercial air travel was just resuming. I made the firm resolve that I would not fly in connection with my work. I did not want to be speeded up. The trains were still good and I enjoyed train travel. But mostly I valued the meditative intervals that train travel afforded. I do not recall that I thought this through at the time but, in retrospect, meditative intervals have been very important to me\u2014both long ones and short ones. It has sometimes been crucial, in the heat of controversy, to withdraw into the silence for just a few seconds so that the creative processes can function. For a big idea to evolve, I have found that a big chunk of meditative time is required, and train riding was sometimes the best way for me to get those big chunks.\n\nI have seldom gotten an important learning from reading. I am slow and it is hard work. Most of my learning in this 40 to 60 period was in conversation with people. I got as much out of organizing my thoughts as I did from listening, and listening was important. I found my way to people interested in sharing (and I had much to share, especially in those last ten years) with people whose perspectives were different from mine and who made me stretch.\n\nTwo years were spent in weekly sessions with Jungian analysts (one year with a woman, one with a man) on the analysis of my dreams. These sessions greatly enlarged my awareness of my inner life, and I believe my creativity was quickened.\n\nI developed a really close relationship with several quite different institutions: The Menninger (psychiatric) Foundation, The U.S. Air Force, and the National Council of Churches, along with several large businesses. I accepted my first teaching position in a university-related school for managers with which I worked for parts of seven summers.\n\nMost important in all of this was my relationship with people. In the last part of my AT&T career, I became interested in the ethics of the company (to the discomfort of some well-placed people), which led me to establish a relationship with professors of ethics in Catholic, Jewish, and Protestant seminaries. Some close friendships developed, especially my friendship with Rabbi Abraham J. Heschel.\n\nI am not a scholar, and little of my work in this period was centered on reading, with two exceptions. I made quite an in-depth study of the history of AT&T. Because of being in the corporate office, I had access to the archives. I made a determined, but unsuccessful, effort to interest the executives of the company in that rich history, by knowledge of which officers and directors might have saved the company from dissolution in 1984. The other study in this period was to read extensively in the history of the Religious Society of Friends. This was useful in my retirement years when I wrote regularly for an outstanding Quaker magazine, _Friends Journal_. Both of these excursions into the history of institutions with which I was very familiar confirm what a noted historian once said, \"The main thing we learn from history is that we do not learn from history.\" Both of these depth interests in the history of institutions I was involved with were important parts of my years of preparation, and both were enjoyable activities while I engaged in them.\n\nI noted above two of my AT&T colleagues who were my age and who retired and found themselves rudderless. Both had the same opportunity to prepare for their old age that I had, and the preparation I experienced would have made a more enjoyable existence while they did it. Both were better scholars than I, one Phi Beta Kappa, but neither apparently heard the same signal that I did at age 40. Perhaps they were not listening for signals as I was, all of the time\u2014and still am. Neither of them lived long after retirement.\n\nIn giving some details of my own preparation I am not suggesting that what I did would be appropriate or possible for anyone else. But of this I am quite sure: anyone who thinks of his or her old age as an event to be prepared for is more likely to have a more fruitful old age than one who has not thought that way, regardless of what his or her preparation consists of. Whether one's gifts and opportunities are great or small, my advice would be\u2014prepare!\n\nThe temptation to be resisted by highly gifted people, I believe, is the hope (if they think about it at all) that they will be able to exploit their gifts to the end. Some may make it through life on that assumption (charitable friends may assure them they are making it when an objective appraisal might tell them that they are not). I have just finished a new biography of a leading theologian of a generation past, a man of exceptional gifts. He had a health setback as he approached old age, and he spent miserable years fretting that he could no longer exploit his gifts as he once did. He was unprepared.\n\nWhat would adequate preparation have meant to such a person? I can only speculate. It seems to me that one of the signs of emerging maturity is an acceptance of life's fragility and acceptance of new conditions, whatever they may be, plus recognition of G. K. Chesterton's admonition that (life's) \"wildness lies in wait.\" It is not an obsessive preoccupation that makes one sick or fearful. It is alertness, readiness to respond when it is appropriate to respond. As one attains maturity, one learns to live peacefully and sleep well with a submerged awareness of constant danger. The most important part of preparation for old age may be awareness, while one is young, that a time will come when one steps aside from one's main career and life will be different. If one looks forward to a second career in which appropriate old age involvements are at the center, then it seems important that one make that change\u2014if one has the option\u2014while one still has the energy to undertake it, and that one prepare for that change.\n\nOld age may be one of those conditions that everyone knows about but for which some do not prepare. The benefit of preparing may be that no matter what impairments one suffers, one can make a good life of it as long as one has one's wits. (When one's wits are gone, it doesn't matter.) It may be that what deters some people from preparing for such a contingency as old age is want of courage, what Paul Tillich called \"The Courage to Be\"\u2014the courage to be aware and to live comfortably with the constant threat of adversity, pain and anxiety. I suspect that what results from such want of courage is a false security that may betray one when one faces old age and finds that one lacks what Fosdick called \"The Power to See Life Through.\" Thus, serenity may not be what one achieves in old age; it may be one of the fruits of what one has learned by preparing while one is young.\n\nThere are many signals all of the time that will cue one to the ideas that will make life more rewarding, at all stages. But one is likely to hear those signals only if one is alert to signals, all of the time. The problem is that there is a baffling number of signals if one is alert to them, and one must choose which of them one will heed. At this point meditation will serve, at all stages of life. Not only do our rational reflective processes sometimes function better in meditation, but one may develop sensitivity to intimations from beyond the barrier that separates what we call reality from mystery. Jung might say that in meditation we may tap the collective unconscious. What appears in meditation is sometimes the same as what appears in dreams. Then, in one's conscious logic, one can always ask, \"Is the originator of this signal really acting in the spirit of my servant?\"\n\nWhen, at age 40, I heard the signal from Elmer Davis, \"prepare now for your old age,\" I judged him to be a true servant and heeded his advice. In the perspective of 43 years' experience, I find that earlier judgment confirmed. I am grateful to Elmer Davis, and to my father, and many other true servants for the gifts of spirit, by both advice and example, that have sustained me. Old age _is_ the ultimate test of spirit.\n\nSurviving from my Boy Scout experience 70 years ago is the motto: _Be prepared_. I am eternally grateful for it.\n\nAnd I am grateful to E. B. White for \"Sufficient is the remembrance of the beauty I have seen.\" That remembrance is always with me, and I do not have to see it again.\n\n## **Afterword**\n\n### JAMES P. SHANNON\n\nStudents of Robert K. Greenleaf should find the publication of this select collection of essays from his most productive years both timely and useful. Timely, because the number of his followers has grown geometrically in the last two decades; and useful, because this growing company of students is currently very much in need of a single source book that brings together the best of Greenleaf's separate essays, now available in print only in several discrete publications.\n\nAs a young man, Greenleaf was fascinated by his father's ability to \"get things done.\" Greenleaf's father was an engineer and a tinkerer in Terre Haute, Indiana. Through the rest of his life Robert would be searching for those persons who could emulate his father's ability to actualize their own potentialities by helping other persons learn how to \"get things done.\"\n\nArmed with his B.A. degree from Carleton College in 1926, and motivated by the counsel of his sociology professor, Oscar Helming, to seek employment in a company large enough to prepare its employees to become leaders in a society increasingly in need of leaders with vision, he sought and got his first job, with Ohio Bell Telephone, a subsidiary of the giant AT&T.\n\nHis first assignment at Ohio Bell was digging postholes for telephone poles. It is a tribute to his supervisors there that they soon made him a teacher to train other entry-level construction workers how to become foremen. In this role, between 1927 and 1929, he once said, he spent \"the most formative years\" of his life. It was here that he found his life's work: to identify people of talent and to help them obtain the skills they need to move into leadership positions. During the next three decades, Greenleaf and AT&T prospered jointly by his remarkable success in identifying and developing generations of promising employees who would come to be known in his vocabulary as \"servant-leaders.\"\n\nIn essence, Greenleaf was AT&T's in-house talent scout. The \"comers\" he spotted tended to be persons who were motivated primarily by a creative desire to be effective in doing whatever had to be done to make their organization work. They also tended to be persons who were generous with their time, their talent, and their training. They tended to be nonjudgmental and benevolently disposed toward their co-workers. They also tended to be good listeners who favored collegial decision making, who knew themselves well, respected themselves, and had those qualities Greenleaf considered essential: sound values, personal strength, intuition, and spirit.\n\nThroughout his life, Greenleaf would insist that his theses on leadership were based on empirical evidence in the workplace, not on deductive corollaries from some abstruse philosophical or theological premises. Even though he would spend his life as a teacher, he continued to have an innate suspicion of \"education with a capital E.\" The epitaph he chose for his tombstone reflects both his modesty and his commitment to grass-roots learning. It reads, \"Potentially a good plumber, ruined by a sophisticated education.\"\n\nReaders of this book should bear in mind that AT&T and Greenleaf were consciously motivated by their mutual desire to find, train, and empower young people who would be able one day to lead that enormous company to new heights of success and profitability. To achieve such results, all parties to this joint venture had to agree that wanting to become a leader and being a leader are good and wholesome, human objectives. They are not signs of hubris or self-seeking. John W. Gardner, in one of his earliest and best-known essays, warned that the egalitarian and democratic thrust of our educational system, left to itself, will inevitably inoculate our children with what he calls the \"anti-leadership vaccine.\" To counteract this predictable perversion of democracy, we must teach our children that it is good mental hygiene for them to want to become leaders and that our healthy society eventually depends on our continuing ability to raise up generations of visionary and energetic leaders in every sector of our national life.\n\nAn essential element in this educational process, according to Greenleaf, is our ability to teach our children that true leadership ultimately depends on the legitimacy of one's appointment, election, or promotion to a position of authority and on one's subsequent ability to validate or confirm this role by the quality of one's performance, called \"the authority of service.\"\n\nThe authority of service is that additional level of legitimacy or validation that is earned after one is elected mayor, appointed chief of police, or elevated to the rank of bishop. These two levels of authority can exist separately; but ideally they co-exist, with each level giving added legitimacy to the other. The authority of service is that added distinction that good parents, good teachers, and good pastors enjoy by reason of their dependable performance over time, and which beginners in any career envy and covet. This kind of authority is built slowly and depends on one's ability to do one's homework, to treat others fairly, to meet one's deadlines, to get the job done right, and to tell the truth habitually.\n\nLeaders who see their strength only in their alleged \"power\" are understandably reluctant to share that strength. Leaders who see their strength in the quality of their performance are eager to share it, and, in so doing, to multiply it.\n\nIn Herman Hesse's novel _Journey to the East_ , the servant Leo is the person who has the greatest legitimacy as a leader because he has earned the authority of service by his performance. Reading this novel helped Greenleaf crystallize his own thinking that true leadership is always a result of performance. In the Greenleaf economy, power shared is power multiplied, not lessened. This kind of generous participation in the workplace turns followers into peers and peers into new leaders and builds new levels of trust among persons mutually committed to the pursuit of desirable shared objectives.\n\nManagement consultants who diagnose corporate strengths and weaknesses typically focus their studies on customers, products, or shareholders. Robert Greenleaf opted to focus his analyses on employees, the persons he considered the neglected stakeholders, the persons whose goodwill, energy, and loyalty are too often taken for granted. In his view, if employees received the care, training, and attention they deserve, shareholder and customer satisfaction would inevitably follow.\n\nOne reason that Greenleaf's essays are in such demand today is that modern corporate restructuring has indeed tended to take the employees for granted. In Greenleaf's value system, if the human persons who do the work of the company are neglected, shareholders and customers will suffer; and conversely, employees who are carefully selected, well trained, and appreciated will give shareholder and customer satisfaction higher priority.\n\nEntirely apart from his appeal to business leaders today is Greenleaf's attractiveness as an advocate of humane values across the spectrum in all human relations. To the extent that we could practice his style of servant-leadership, we would not only be better employees; we would be better parents, better spouses, better friends, better human persons. We would be more civil, more courteous, more thoughtful, more gracious, more generous. One need not be a professional anthropologist to realize that these attractive human qualities are in short supply in our frenetic society. Nor need one be a psychologist or psychiatrist to see why so many thoughtful persons see Greenleaf as a welcome voice for our time.\n\nOne of Greenleaf's favorite aphorisms was, \"Organization kills spirit.\" If he were with us today, I would bet that he would be a fan of the \"Dilbert\" cartoon. When he took early retirement from AT&T to work as an independent consultant, he was intrigued by the irreverence of the college students of the sixties. He regularly prowled campus book stores to find out what the students were reading and singing. He saw himself as a dedicated change agent, and he saw the young people of that age as his likely allies in this pursuit.\n\nIn my decade as director of the General Mills Foundation, I became a practitioner and a true believer in the teachings of Robert Greenleaf. In that period, our company decided to restructure. The downside of that process for me was that my staff was cut from 6\u00bd persons to 4 persons. The upside was that our corporate profits rose dramatically and my grants budget doubled, from $4.5 million to $9 million. But in our office we had to cut back on \"site visits\" to the local turf of our grant applicants because we lacked the personnel to do it. There is one school of thought in the field of philanthropy which holds that site visits are the single best indicator of the quality of any grant-making program, and our ability to make site visits was slipping away.\n\nTaking a page from Greenleaf, I made quiet personal visits to about a dozen persons in our company, none of whom reported to me, to ask whether they had any interest or desire to volunteer some of their free time to help our staff make up for our loss of personnel. Wonder of wonders, they were eager to help us! Sometimes on holidays, sometimes on weekends, sometimes in the evening after work, these generous fellow employees pitched in to help us do our work. They were truly servant-leaders. Their only reward was the psychic satisfaction they got from helping our small staff maintain some of the community outreach we needed so desperately and that was slipping away from us.\n\nPlease bear in mind that I had no mandate from management to use Greenleaf's ideas or to recruit in-house volunteers. I neither sought nor asked permission from above. The point here is that Greenleaf's ideas are portable and can be productively used by anyone, anywhere\u2014in a family, in a school, in a church congregation, in a soccer team. All it takes is a decision by two or more persons to explore ways to help one another actualize more of their individual potential.\n\nThere is an expression among pilots who fly float planes off the lakes in my native Minnesota: When a plane is at rest in the water, the surface tension of the water \"holds\" the plane. Prior to take-off from water, the pilot must take enough time and generate enough speed so that the water loses its \"grip\" on the floats and the plane gets \"up on the step\"\u2014that is, out of the water but not yet in the air\u2014before the pilot can actually take the plane airborne.\n\nIn my view the organization that Robert K. Greenleaf founded in 1964\u2014now called The Greenleaf Center\u2014is \"up on the step.\" It is poised for take-off to new heights. This book itself is a graphic illustration that the center and its energetic director Larry Spears are responding to the growing need in our society for the ideas of Robert Greenleaf and to the growing number of students of Greenleaf who need and want a single source compendium like this welcome volume.\n\n## **References and Permissions**\n\n**The Foreword** is an original essay created for this collection by Peter Vaill. Copyright \u00a9 1998 Peter Vaill. Printed with permission of the author.\n\n**The Introduction** is adapted from previous writings by Larry C. Spears, especially the Introduction to _Insights on Leadership_ , published by John Wiley & Sons. Copyright \u00a9 1998 by Larry C. Spears.\n\nChapter 1: **Servant: Retrospect and Prospect** was originally published in 1980. Copyright \u00a9 1998 The Greenleaf Center.\n\nChapter 2: **Education and Maturity** was originally published in 1962. Copyright \u00a9 1998 The Greenleaf Center.\n\nCopyright permissions from _Complete Poems of Robert Frost_. Copyright 1916, 1921 by Holt, Rinehart and Winston, Inc. Copyright renewed by Robert Frost. Reprinted by permission of Holt, Rinehart and Winston, Inc.\n\nCopyright permissions from _The Hollow Men_ in \"Collected Poems\" 1909\u201335, by T. S. Eliot. Copyright by Harcourt, Brace & World, Inc. Reprinted by permission of Harcourt, Brace & World, Inc.\n\nChapter 3: **The Leadership Crisis** was originally published in Volume XIV, Number 3, November, 1978 of _HUMANITAS: Journal of the Institute of Man_ , Duquesne University. Copyright \u00a9 1998 The Greenleaf Center.\n\nChapter 4: **Have You a Dream Deferred?** was originally published in 1967. Copyright \u00a9 1998 The Greenleaf Center.\n\nChapter 5: **The Servant as Religious Leader** was originally published in 1982. Copyright \u00a9 1998 The Greenleaf Center.\n\nChapter 6: **Seminary as Servant** was originally published in 1981. Copyright \u00a9 1998 The Greenleaf Center.\n\nChapter 7: **My Debt to E. B. White** was originally published in 1987. Copyright \u00a9 1998 The Greenleaf Center.\n\nReprinted by permission of Harper & Row, Publishers, Inc.: From _Letters of E. B. White_ , copyright 1976 by E. B. White; from \"A Slight Sound at Evening\" in _Essays of E. B. White_ , copyright 1954 by E. B. White; \"I Paint What I See\" from _Poems & Sketches of E. B. White_, copyright 1933 by E. B. White; from \"The Second Tree from the Corner\" in _The Second Tree from the Corner_ by E. B. White; from _Charlotte's Web_ by E. B. White, copyright 1952, 1980 by E. B. White; from _The Trumpet of the Swan_ by E. B. White, copyright 1970; from \"Preface\" in _Poems & Sketches of E. B. White_, copyright 1981 by E. B. White.\n\nReprinted by permission of the _New York Times_ : From \"E. B. White: Notes and Comment by Author,\" by Israel Shenker, of July 11, 1969.\n\nReprinted by permission, copyright 1951, 1961, 1979, _The New Yorker Magazine_ , Inc.: From _Poems & Sketches of E. B. White_ (Harper & Row), copyright 1933, 1961 by E. B. White. Originally in _The New Yorker_ ; From _The Second Tree from the Corner_ (Harper & Bros.), copyright 1947, 1975 by E. B. White. Originally in _The New Yorker_.\n\nChapter 8: **Old Age: The Ultimate Test of Spirit** was originally published in 1987. Copyright \u00a9 1998 The Greenleaf Center.\n\n**The Afterword** is an original essay created for this collection by James P. Shannon. Copyright \u00a9 1998 James P. Shannon. Printed with permission of the author.\n\n## **Greenleaf Bibliography**\n\nGreenleaf, Robert K. \"Abraham Joshua Heschel: Build a Life Like a Work of Art,\" _Friends Journal,_ 1973, 19(15), 459-460.\n\n\u2014. _Advices to Servants_. Indianapolis: The Greenleaf Center, 1991.\n\n\u2014. \"The Art of Knowing.\" _Friends Journal_ , 1974, 20(17).\n\n\u2014. \"Business Ethics\u2014Everybody's Problem.\" _New Catholic World_ , 1980, 223, 275-278.\n\n\u2014. \"Choose the Nobler Belief.\" _AA Grapevine_ , 1966, 23(5), 27-31.\n\n\u2014. \"Choosing Greatness.\" _AA Grapevine_ , 1966, 23(4), 26-30.\n\n\u2014. \"Choosing to be Aware.\" _AA Grapevine_ , 1966, 23(1), 26-28.\n\n\u2014. \"Choosing to Grow.\" _AA Grapevine_ , 1966, 23(2), 11-13.\n\n\u2014. \"Community as Servant and Nurturer of the Human Spirit.\" _Resources for Community-Based Economic Development_ , 1986, 4, 9-11.\n\n\u2014. _Education and Maturity_. Indianapolis: The Greenleaf Center, 1988.\n\n\u2014. _Have You a Dream Deferred?_ Indianapolis: The Greenleaf Center, 1988.\n\n\u2014. _The Institution as Servant_. Indianapolis: The Greenleaf Center, 1976.\n\n\u2014. _The Leadership Crisis_. Indianapolis: The Greenleaf Center, 1978.\n\n\u2014. _Life's Choices and Markers_. Indianapolis: The Greenleaf Center, 1986.\n\n\u2014. _Mission in a Seminary: A Prime Trustee Concern_. Indianapolis: The Greenleaf Center, 1981.\n\n\u2014. _My Debt to E. B. White_. Indianapolis: The Greenleaf Center, 1987.\n\n\u2014. _Old Age: The Ultimate Test of Spirit_. Indianapolis: The Greenleaf Center, 1987.\n\n\u2014. _On Becoming a Servant Leader_. San Francisco: Jossey-Bass, 1996.\n\n\u2014. \"Overcome Evil with Good.\" _Friends Journal_ , 1977, 23(10), 292-302.\n\n\u2014. _Robert Frost's \"Directive\" and the Spiritual Journey_. Boston: Nimrod Press, 1963.\n\n\u2014. _Seeker and Servant_. San Francisco: Jossey-Bass, 1996.\n\n\u2014. _Seminary as Servant_. Indianapolis: The Greenleaf Center, 1988.\n\n\u2014. _Servant Leadership_. New York: Paulist Press, 1977.\n\n\u2014. _The Servant as Leader_. Indianapolis: The Greenleaf Center, 1991.\n\n\u2014. \"The Servant as Leader.\" _Journal of Religion and the Applied Behavioral Sciences_ , Winter 1982, 3, 7-10.\n\n\u2014. _The Servant as Religious Leader_. Indianapolis: The Greenleaf Center, 1983.\n\n\u2014. _Servant: Retrospect and Prospect_. Indianapolis: The Greenleaf Center, 1980.\n\n\u2014. _Spirituality as Leadership_. Indianapolis: The Greenleaf Center, 1988.\n\n\u2014. _Teacher as Servant: A Parable_. Indianapolis: The Greenleaf Center, 1987.\n\n\u2014. \"The Trustee and the Risks of Persuasive Leadership,\" _Hospital Progress_ , 1978, pp. 50-52, 88.\n\n\u2014. Trustee Traditions and Expectations.\" In _The Good Steward: A Guide to Theological School Trusteeship_. Washington, D.C.: Association of Governing Boards of Universities and Colleges, n.d.\n\n\u2014. _Trustees as Servants_. Indianapolis: The Greenleaf Center, 1990.\n\n\u2014. \"Two More Choices.\" _AA Grapevine_ , 1966, 23(3), 22-23.\n\n## Index\n\n\"A Slight Sound at Evening\" (White), _The Yale Review_ , \u2013243\n\n_A World Waiting to Be Born_ (Peck), ,\n\nabilities, intellectual leadership,\n\nacceptance, of self,\n\nachievement\n\nindicators,\n\nmission, \u2013212\n\npersonal goals, \u201375, \u201397\n\nactions\n\nintent vs. actual result,\n\nproductive vs. aimless,\n\nreflexive, \u201334\n\nrightness of, \u201386\n\nvoluntary, \u201353\n\nactivities\n\ndiversionary,\n\ntrusteeship,\n\nvolunteer, \u2013272\n\nActon, Lord, on power,\n\nadministration, trustee oversight,\n\nadministrators\n\nchairperson relationships, \u2013190\n\nservant of the mission,\n\ntrustees, \u2013192\n\nvision,\n\nadmissions, self-selection,\n\nage\n\nof the anti-leader,\n\ncreativity,\n\nagenda, hidden,\n\nAlcoholics Anonymous\n\nintegrating servant-leadership concept,\n\nmoney, \u2013151\n\nalienation\n\nchurches,\n\nproblem of widespread, ,\n\nrecovering from, ,\n\namelioration, Ralph Waldo Emerson,\n\nAmerican Education Society (AES), \u2013216,\n\nAmerican Telephone & Telegraph Company\n\nDirector of Management Research, , \u2013273\n\ninstitutional history, \u201320,\n\nservant-leadership training programs,\n\ntraining foremen, \u2013113\n\n\"umbrella\" effect,\n\nanalysis\n\nconscious, \u2013130\n\nsurvey responses, \u2013230\n\nAnderson, Hans Christian, Frank Lloyd Wright,\n\nAndover, Congregational seminary,\n\nAndover model, seminaries, ,\n\nAngell, Katharine, _The New Yorker_ , \u2013240\n\nanswers, frustration, \u201363\n\nanti-leader, age of,\n\nanti-leadership vaccine,\n\nantitrust laws, competition,\n\nanxiety\n\ndealing with, \u2013108\n\nreligious leadership,\n\narguments, persuasion process, , \u2013135\n\nart\n\nchairmanship,\n\nof the possible,\n\nof trusteeship,\n\naspirin effect, ,\n\nassumptions\n\ndecision governing,\n\nrealistic, \u2013106\n\nattitude\n\nreception of liberating visions, \u201359\n\nstudent,\n\nwonder,\n\nattributes, leadership potential,\n\nautonomy\n\nof a free press, \u2013259\n\nmaintaining, \u201345\n\nawareness\n\nencouraging,\n\nservant-leader characteristics,\n\nwidening your,\n\nBarnard College, _Education and Maturity_ , \u201376\n\nbarriers, leadership preparation,\n\nbeliefs\n\nfailure of traditional,\n\nreligious leadership,\n\nrightness of, \u201386\n\nBellow, Saul, _Herzog_ ,\n\nbias, cultural,\n\n_Bible_ (King James Version), vision,\n\nBlake, William\n\non imagination, , ,\n\nperception, , ,\n\n_The Marriage of Heaven and Hell_ ,\n\nBlanchard, Ken, servant-leadership concept,\n\nBlock, Peter\n\nservant-leadership concept,\n\n_Stewardship_ , ,\n\n_The Empowered Manager_ , ,\n\nboards of directors, roles,\n\nBoston Symphony Orchestra, Serge Koussevitsky,\n\nBostonians, telephone industry history, \u2013223\n\nBordas, Juana, _Pluralistic Reflections on Servant-Leadership_ ,\n\nBrowning, Robert, \"Rabbi Ben Ezra,\" ,\n\nbusiness\n\ncompetition, \u201328\n\npeople-first,\n\nprimary purpose,\n\ncareers\n\nchoice,\n\ngrowth, \u201375\n\nsatisfaction,\n\nsecond, \u2013278\n\nsignificance,\n\ncaring\n\ncommunity,\n\ngood society,\n\ninstitutions,\n\nJ.P. Morgan, \u2013222\n\nresults of,\n\nspirit,\n\ntest of, \u2013174\n\ntrustees,\n\nCarleton College\n\nOscar Helming,\n\nstudent unrest in the 60s,\n\nCarnegie Corporation, John W. Gardner,\n\ncartoons, \"Dilbert,\"\n\nchairpersons\n\nforesight,\n\nrole, , \u2013220, \u2013229\n\nskills, \u2013194\n\nstatus, \u2013227\n\ntraining, \u201343\n\ntrustees, , \u2013190\n\nchallenge\n\naccepting,\n\nanxiety,\n\ngrowth of managers,\n\nresponsibility,\n\nchange\n\nenvironment that encourages,\n\nevolution vs. revolution,\n\ninstitutions, \u201319, ,\n\nseminaries, \u2013117\n\nunderstanding,\n\nuniversities,\n\nchaos\n\ncombating,\n\nmanagement vs. prevention, \u2013199\n\ncharacter, external marks of,\n\ncharacteristics, servant-leader,\n\ncharge, taking, \u201397\n\ncharities, intent of actions vs. actual result,\n\n_Charlotte's Web_ (White), \u2013261\n\nChesterton, G.K.\n\nlife's possibilities,\n\n_Orthodoxy_ , \u2013129\n\nworldview, \u2013106\n\nchief executive officers, power, \u201349\n\nchoices\n\nmaking, ,\n\npersonal,\n\nunethical,\n\nChrist, Jesus, disciplinary interviews,\n\nChurch, The Growing Edge, \u2013152\n\nchurches\n\nfinancing new programs, \u2013167\n\ngrowing edge, \u2013158\n\nimproving effectiveness,\n\nas institutions, \u2013156\n\nseminaries, \u2013178\n\nservant role, \u2013170\n\ntheology,\n\nChurchill, Winston, language use,\n\ncircuit riding pastors,\n\n\"Civil Disobedience\" (Thoreau),\n\ncivil rights crisis,\n\nclarification, of self,\n\ncoach, chairperson,\n\ncodes\n\ncommon moral,\n\nmanners,\n\ncoercion\n\npersuasion vs., , \u201346, \u2013135, \u2013160,\n\npower, \u201384, \u2013206\n\ncolleges\n\ncrisis of leadership,\n\nfreshman year in,\n\ncommitment, taking a stand, \u201387\n\ncommon\n\ngood, \u2013138\n\nsense,\n\ncommunication\n\nconcern about growth,\n\ncultural differences,\n\ndreams, \u201388\n\nhope vs. despair,\n\nlanguage use,\n\nof values,\n\ncommunity\n\nbuilding,\n\ncaring,\n\nlifecare retirement, ,\n\nCommunity-Building,\n\ncompetition\n\nantitrust laws,\n\nlearning from the experience of others, \u201328\n\nnature of, \u201351\n\nunconventional,\n\ncompromise\n\nethical,\n\npoliticians,\n\nconcepts\n\nconsensus of mission definition, \u2013223\n\nfaith, \u2013123\n\njustice,\n\nrole of trustee, \u2013211\n\nconceptualization, servant-leader characteristics, \u20137\n\nconclusions, religious leadership,\n\nconfidant, chairperson,\n\nconfidence, building faith, \u2013133\n\nconflict\n\nformula for meeting,\n\nresolution,\n\nconformity, issue of, \u201368\n\nconfrontations, cultural biases,\n\nconsensus\n\nmission statement, \u2013223\n\nmorality,\n\norganizational strength,\n\nreaching, , \u2013139\n\nshared dreams, \u201379\n\ntrustee chairperson,\n\nconsequences, of religious leadership, \u2013115\n\nconstituencies\n\nserving definite,\n\ntrustee, \u2013190, \u2013213\n\nconsultants\n\nchurches, \u2013150\n\nseminaries,\n\nconversion, nonservant to servant, \u201324\n\ncord, pulling the emergency, \u2013128\n\nCornell University\n\nAndrew White,\n\nE.B. \"Andy\" White, \u2013239\n\ncorporations\n\nevolution of, \u2013183\n\nhealthy,\n\nlocalized programs, \u2013145\n\ncorruption\n\njournalism,\n\npower, \u201348\n\ncounseling, employee,\n\n_Counseling in an Organization, A Sequel to the Hawthorne Researches_ , \u2013144\n\ncourage\n\nessence of, \u2013253\n\nto live life,\n\nCovey, Stephen\n\norganizational management,\n\n_The Seven Habits of Highly Effective People_ ,\n\nCowley, Malcolm, _The View From Eighty_ , \u2013267\n\ncreativity\n\ncultivating, \u201399\n\nfrustration, \u201363\n\nreligious leadership, \u2013125\n\nspark,\n\ncredentialing, coercive power of universities,\n\ncredo, personal, ,\n\ncrisis\n\ncivil rights,\n\nleadership, , \u201391\n\ncritical thought\n\nadequacy,\n\nseminal ideas, \u2013225\n\nseminaries, , \u2013214, \u2013220, \u2013233\n\ncriticism, persuasion process,\n\ncues, awareness of,\n\ncultures\n\ncompetition, \u201351\n\ndifferences,\n\nguardians of the future and the past,\n\nmedia-domination,\n\npower, \u201349\n\nprograms designed to fit corporate, \u2013145\n\ncurriculum\n\nnutrition in medical schools, \u201330\n\ntheological education, \u2013218\n\ncynicism, mission achievement, \u2013209\n\nDanish Folk High Schools, Nikolai Grundtvig, \u201325, \u2013137,\n\nDartmouth College, Professor John Finch,\n\nDavis, Elmer\n\nold age,\n\n\"The Uses of Old People,\"\n\ndecisions\n\nintuitive,\n\nmaking,\n\ndegrees\n\nearning,\n\nuniversity, ,\n\ndemocracy, decline of,\n\ndependency, reciprocal, \u2013170\n\nDePree, Max\n\n_Leadership is an Art_ , ,\n\n_Leadership Jazz_ , ,\n\norganizational management,\n\nservant-leadership concept,\n\ndespair, vs. hope,\n\ndestruction, faith, \u2013116\n\ndevelopment, maturity and personal,\n\ndeviance, from the law,\n\ndignity\n\nattribute,\n\nstrength,\n\n\"Dilbert,\" organizational cartoon,\n\ndirection, wholeness and certainty of,\n\n\"Directive\" (Frost),\n\nDirector of Management Research, American Telephone & Telegraph Company, , \u2013273\n\ndiscipline, lifestyles,\n\ndistinction, serving with, \u201396\n\ndistractions, dealing with,\n\ndoctrine, seminaries,\n\n\"Don't want to talk about it,\" ,\n\ndreams\n\nanalysis,\n\ncommunicating, \u201388\n\nconceptualization, \u20137\n\ngreatness,\n\nliving your, \u2013110\n\ntrustees, \u2013192\n\nduties, trusteeship,\n\neconomy\n\nDanish at time of Nikolai Grundtvig, \u2013137\n\ntransformation of Danish,\n\neditors, magazine, \u2013230\n\neducation\n\nanti-leadership vaccine, ,\n\nchairperson,\n\nDanish Folk High Schools, \u2013137\n\nexperiential,\n\nleadership crisis, \u201381\n\nliberal, ,\n\nand maturity, \u201376\n\nmedical and nutrition, \u201330\n\ntheological, , \u2013219\n\ntrustee,\n\neffectiveness, leadership and church,\n\neffects\n\naspirin, ,\n\numbrella,\n\nego, over-inflated,\n\negocentricity, youthful, \u201364\n\nEisenhower, Dwight D., position changes your perspective,\n\neleemosynaries\n\ninstitutions,\n\nmission statements, \u2013212\n\nEliot, T.S.,\n\nelites, ability to serve, \u2013167\n\nemergencies, group behavior in response to, \u2013128\n\nEmerson, Ralph Waldo\n\ncomprehending the world,\n\ngoal of amelioration,\n\n_Works and Days_ ,\n\nempathy, servant-leader characteristics,\n\nemployees, corporate health,\n\nentheos, attribute, \u201376\n\nenthusiasm, entheos vs., \u201372\n\nenvironment\n\nleaving an impact on your, \u201397\n\nuniversity, \u2013109\n\nepitaph, Robert K. Greenleaf's,\n\nerrors\n\nlearning from, ,\n\nwisdom from,\n\nesprit, performance,\n\n_Esquire_ , Xerox Corporation and Harrison Salisbury,\n\nessays, servant theme, \u2013117\n\nethics\n\nforesight, \u2013131\n\nfree press, \u2013259\n\na sense of, \u2013101\n\nstudy,\n\nevolution, seminaries, \u201339\n\nexecutive, role in an institution, \u2013207,\n\nexpectations, churches,\n\nexperience\n\ncorrupting influences,\n\nlearning from,\n\nregeneration of an organization, \u2013225\n\nuniversities,\n\nusing the competition's, \u201328\n\nexperiential education, service-learning,\n\n\"Fable,\" _Seeker and Servant_ (Greenleaf), \u2013167\n\nfaculties\n\nattracting, \u2013204\n\nseminaries,\n\nservant of the mission,\n\nservant role, \u2013187\n\nteaching obligations, \u2013155\n\ntrustee chairperson relationships, \u2013190\n\nuniversity, \u201391\n\nfailure\n\nethical,\n\nleadership,\n\nvital links in a process,\n\nfaith\n\nachievement of mission goals, \u2013212\n\ncommunicating,\n\ngenerating,\n\nLeo Tolstoy's definition,\n\nreligious leadership, \u2013123,\n\nas trust, , , ,\n\nfather, Robert K. Greenleaf's, , \u2013265, , \u2013280\n\nfears, childhood, \u2013238\n\nFellows, Ohio, \u201394\n\nFerlinghetti, Lawrence, rebirth of wonder,\n\nFinch, John, Dartmouth College,\n\nfindings, listening and communicating,\n\nFletcher, John C., religious leadership,\n\nFletcher, Joseph, morals,\n\nfocus, maintaining,\n\nfollowers\n\ndiscriminating,\n\nleaders as,\n\nmanipulation of, \u201385\n\nvoluntary, \u201333,\n\nforces\n\nregenerative,\n\nrestorative, \u2013118\n\nsustaining,\n\nunifying, \u201388\n\nforeman, management training, \u2013113\n\nforesight\n\ndeveloping, , \u2013126\n\nethics, \u2013131\n\nservant-leadership characteristics,\n\nformula, meeting conflict,\n\n_Fortune_ , list of 10 best companies to work for,\n\nFosdick, Harry Emerson\n\nRiverside Church,\n\n\"The Power to See Life Through,\"\n\nfoundations\n\ncorrupting influences,\n\nfunction of, \u201341\n\nprophetic visions, \u201338\n\ntrustees, , \u201341\n\nFox, George\n\nReligious Society of Friends (Quakers), \u2013152\n\nslavery, \u201337\n\nfree press, ethical practices, \u2013259\n\nfreedom, responsibility,\n\nfriends, power, \u201348\n\n_Friends Journal_ ,\n\nfriendships, power, \u201348\n\nfrontiers, new, \u2013162\n\nFrost, Robert\n\n\"Directive,\"\n\nelusiveness of meaning,\n\nlife's choices,\n\nthe path less traveled,\n\nfrustration, answers, \u201363\n\nfunding\n\nretaining,\n\nseminaries, , \u2013167, ,\n\nfuture\n\npreparation for the,\n\nprojections,\n\nGandhi, Mohandas,\n\ngap\n\nbridging the information, ,\n\nbetween seminaries and churches,\n\nGardner, John W., anti-leadership vaccine, ,\n\nGeneral Electric, J.P. Morgan,\n\nGeneral Mills Foundation, James P. Shannon,\n\n\"Generation of Greatness, the Idea of a University in an Age of Science\" (Land),\n\ngenerations, relationships with other,\n\ngenius, vs. greatness,\n\ngestures, leading through subtle,\n\ngifts\n\ncreativity,\n\nE.B. White, \u2013237\n\nexploiting your,\n\nrealistic assessment of situation, \u2013106\n\nseeing things whole, , \u2013241,\n\nGillette, Douglas and Robert Moore, _The King Within_ ,\n\ngimmicks\n\nabsence, \u2013180\n\nproblem solutions, \u2013146\n\ngoals\n\namelioration,\n\nesprit and institutional, \u2013105\n\nexternal achievement,\n\nfuture seminaries, \u2013233\n\ngreatness as a, \u2013179\n\nidealistic,\n\nmission achievement, \u2013212\n\npersonal, \u201375, \u201397\n\npursuit of, \u201371\n\nvoluntary support, \u201353\n\ngovernment\n\nlow-caring society,\n\ntrusteeship,\n\n_Great Stone Face_ (Hawthorne), \u201370\n\ngreatness\n\nlifestyle, \u201397, \u2013101\n\nslave, \u2013179\n\ntrusteeship, \u2013105\n\nGreenleaf Center for Servant-Leadership, \u201315\n\nGreenleaf, Robert K.\n\nfather, , \u2013266, , \u2013280\n\npersonal credo, ,\n\nreligious background,\n\n_Seeker and Servant, \"Fable\"_ , \u2013167\n\n_Seminiary as Servant, Essays on Trusteeship_ ,\n\n_Servant Leadership, \"Trustees as Servants\"_ , \u2013117, ,\n\n\"Servant, Retrospect and Prospect,\"\n\n_Teacher as Servant, a Parable_ , ,\n\n\"The Institution as Servant,\"\n\n_The Seminary as Institution_ ,\n\n\"The Servant as Leader,\"\n\n\"Trustees as Servants,\"\n\nground, standing on solid, \u2013101\n\ngroups\n\nbehavior in emergencies, \u2013138\n\ncommunity leadership,\n\nseekers, \u2013161\n\ngrowth\n\naccelerated, \u2013140\n\nanxiety, \u2013108\n\nchurch, \u2013152\n\ncommitment to, \u20138\n\ncommunicating concern,\n\nentheos,\n\nintellectual,\n\npersonal, , \u201376\n\ntaking charge of, \u201397\n\nGrundtvig, Nikolai Frederik\n\nSeverin, Danish Folk High\n\nSchools, \u201325, \u2013137,\n\nguidance, light within,\n\nguide, intuitive,\n\nGulf Oil of Canada, servant-leadership training,\n\nharm\n\nactions that do no,\n\npotential of change to,\n\nHarper, William Rainey, \"Shall the Theological Curriculum Be Modified and How?,\" \u2013218\n\n_Harpers_ , James Thurber,\n\nHarvard Divinity School, Visiting Committee,\n\nHawthorne, Nathaniel, _Great Stone Face_ , \u201370\n\n\"Hawthorne\" studies, \u2013144\n\nhazards, of personal choice,\n\nhealing, servant-leader characteristics, \u20136\n\nhealth\n\ncorporate,\n\ninstitutional, \u2013146\n\nmaintenance,\n\nnutrition, \u201330\n\nHelming, Oscar, Carleton College,\n\n_Henry IV_ (parts I & II) (Shakespeare),\n\n_Henry V_ (Shakespeare),\n\nHerman Miller Company,\n\n_Herzog_ (Bellow),\n\nHeschel, Rabbi Abraham Joshua, ,\n\n_To Grow in Wisdom_ ,\n\nHesse, Herman\n\n_Journey to the East_ , \u20134, \u201358,\n\n_Magister Ludi_ ,\n\nhierarchy, institutions, , \u2013171, \u2013177,\n\nHill Foundation, nutrition programs, \u201330\n\nhistory\n\nAmerican Telephone & Telegraph Company, \u201320,\n\nchurch leadership, \u201333\n\ncritical thinking and seminary leadership, \u2013219\n\ninstitutions, , \u2013175\n\nmission statement,\n\nReligious Society of Friends (Quakers), \u2013276\n\nseminaries, , \u2013181, \u2013185\n\ntelephone industry, \u2013225\n\nHitler, Adolph, leadership, , , \u2013114\n\nhope\n\nbuilding, \u201356\n\ncommunicating,\n\nsaving remnants, \u201391\n\nshort-run vs. long-run, \u2013173\n\nstructural basis for,\n\nvs. despair,\n\nHovland, Carl, Yale Department of Psychology,\n\nHughes, Langston, on dreams deferred,\n\nhumor, E.B. White in _The New Yorker_ ,\n\n\"I Paint What I See\" (E.B. White), _The New Yorker_ , \u2013250\n\nideas\n\nchanging institutions,\n\nmeditation, \u2013275\n\nnew, \u2013162\n\nseminal,\n\nidentification, gimmicks, \u2013146\n\nidentity, personal and conformity,\n\nillness, prevention,\n\nillumination, price,\n\nimage, seminary, \u2013203\n\nimagination\n\ncreativity,\n\nfuture planning,\n\nnurturing, ,\n\nindependence, free press, \u2013258\n\nindifference, combating,\n\nindustrial revolution, Karl Marx, \u201331\n\ninfluence\n\nchurches,\n\ncompetent leadership,\n\nconstructive,\n\ninstitutional decisions and society, \u2013219\n\npotential for,\n\nseminaries,\n\ninitiative, source, \u2013110\n\ninsight\n\nethical,\n\nintuitive,\n\ntrusting your,\n\ninspiration\n\ncheck and guide to your,\n\nchurches, , ,\n\nfuture servant-leaders,\n\ngreat dreams,\n\nopenness to,\n\nInstitutes of Chairing, \u201343\n\ninstitutions\n\nchange,\n\nchurches, \u2013156\n\ndedication to service, ,\n\ndelegation of leadership, \u201333\n\ndreams, , \u201387\n\neleemosynary, , \u2013212\n\netymology,\n\nexceptional, \u2013145, \u2013178,\n\nhealthy,\n\nhierarchy of, , \u2013171, \u2013177,\n\nhistory, \u2013175, \u2013276\n\nlegal powers,\n\nmaintenance functions,\n\nmission statements, \u2013212\n\nmyths, \u2013175\n\nnurturing, \u201338\n\nprimary motive, \u2013173\n\nrebuilding,\n\nreconstruction, \u2013180\n\nas servant, ,\n\nstrength, \u201353\n\nstructure, , \u2013147\n\nsustaining as effective servants, \u2013143\n\n_The Organization Man_ , \u2013153\n\ntheology of, xiii\u2013xiv, , , \u2013184, \u2013233\n\ntrust in, \u2013210\n\ntrustee role,\n\nintegrity\n\njournalism, \u2013259\n\nmaintaining, \u201345\n\nRobert K. Greenleaf's father, \u2013266\n\nintellect, wisdom and,\n\nintent, of a servant,\n\ninterview, Christian disciplinary,\n\nintuition\n\ncheck and guide to your,\n\nfeeling of oneness,\n\nforesight,\n\nguide,\n\njudgment, \u201345\n\nleadership potential,\n\nmoral map,\n\npersuasion, \u2013188\n\nreligious leadership,\n\nvs. reason,\n\ninventions, new,\n\n_Is Sex Necessary?_ (Thurber and White),\n\nJefferson, Thomas\n\ninfluence of a mentor,\n\nUniversity of Virginia,\n\nJesus Christ, disciplinary interviews,\n\njournalism, integrity, \u2013259\n\njournalists, E.B. White quoting de Tocqueville on,\n\n_Journey to the East_ (Hesse), \u20134, \u201358,\n\njudgments\n\nintuition, \u201345\n\ntesting,\n\nwisdom of lay, \u2013227\n\njustice, concept of,\n\njustification, Hawthorne plant programs,\n\nKennedy, John F., changing perspectives,\n\n_King and I_ , impact of new ideas,\n\nKipling, Rudyard,\n\nknowledge\n\nacting on existing, \u201333\n\nwonder,\n\nKoussevitsky, Serge, Boston Symphony Orchestra,\n\nKropotkin, Petr, _Mutual Aid_ ,\n\nlaboratory, university as learning, \u2013110\n\nland, Danish Folk High Schools, \u2013137\n\nLand, Edwin H., \"Generation of Greatness, the Idea of a University in an Age of Science,\"\n\nlanguage\n\ncommunicating hope,\n\nunique usage of leaders,\n\nlaws\n\nantitrust,\n\nMosaic and morality, \u2013100\n\nNational Industrial Recovery Act of 1933,\n\nNational Labor Relations Act (Wagner Act),\n\nleaders\n\nas learners, \u2013113\n\npotential,\n\nreligious seekers,\n\nleadership\n\nacceptance,\n\nconceptual, \u2013177,\n\ncritical thought,\n\ndependability,\n\nintellectual,\n\nmanipulative, \u201385\n\npreparation, \u201381\n\nqualities,\n\nreligious, \u2013167\n\ntrustee, \u2013194\n\nvs. management vs. administration,\n\n_Leadership and the New Science_ (Wheatley),\n\n_Leadership is an Art_ (DePree), ,\n\n_Leadership Jazz_ (DePree), ,\n\nlearner, leader as, \u2013113\n\nlearning, accelerated,\n\nLearning Organizations,\n\nLenin, Diego Rivera's mural in Rockefeller Center, \u2013250\n\nLeo, story of, \u20134, \u201358\n\nlessons, errors,\n\nLevering, Robert and Milton Moskowitz, _The 100 Best Companies to Work for in America_ ,\n\nLewin, Kurt, sensitivity training, \u2013150\n\nlife\n\nmanaging your own, \u201395\n\nspirit,\n\nview from old age,\n\nlifecare communities, retirement, ,\n\nlifestyle\n\ngreatness, \u201397, \u2013101\n\nold age, \u2013270\n\nLilly Endowment, Inc.,\n\nlinks, between becoming and maturity,\n\nlistening\n\nleadership attributes,\n\nlearning by,\n\nlife's signals, \u201395, \u2013278\n\nprophets,\n\nresearch findings,\n\nservant-leader characteristics,\n\nlogo, Greenleaf Center,\n\nlove\n\nof oneself,\n\npersuasive efforts based on, \u2013166\n\nLowe, Jack, Jr., TDIndustries,\n\nLowe, Jack, Sr., TDIndustries,\n\nLynn, Robert W.\n\nVice President for Religion, Lilly Endowment,\n\nreligious leadership,\n\nMachiavelli, Niccolo\n\npower seekers,\n\n_The Prince_ ,\n\nwisdom and advice,\n\nmagazine editors, reader surveys, \u2013230\n\n_Magister Ludi_ (Hesse),\n\nmanagement\n\nchaos, \u2013199\n\ntraining foremen, \u2013113\n\ntrustee oversight of,\n\n_Management and the Worker_ , Hawthorne studies,\n\nmanagers, accelerated growth, \u2013140\n\nmanipulation\n\npersuasion vs., , \u2013160,\n\npower, \u201385\n\nmanners, conflict resolution,\n\nmap, moral,\n\nMarx, Karl\n\nindustrial revolution, \u201331\n\nvision,\n\nmasks, entheos,\n\nMassachusetts Institute of Technology\n\n\"Generation of Greatness, the Idea of a University in an Age of Science,\"\n\nResearch Center for Group Dynamics, \u2013150\n\nmaturity\n\nand education, \u201376\n\nsigns of, \u2013277\n\nMayo Medical School, nutrition program,\n\nMayor Lawrence, renaissance of Pittsburgh, \u2013158\n\nMcGee-Cooper, Ann, _You Don't Have to Go Home from Work Exhausted!_ ,\n\nMead Corporation, servant-leadership training,\n\nmeaning, the elusiveness of,\n\nmedia, domination of present day culture,\n\nmeditation\n\npractice of, ,\n\nsorting out signals, \u2013278\n\ntrain travel, \u2013275\n\nMellon, Richard King, renaissance of Pittsburgh, \u2013158\n\nMenninger Foundation,\n\nmentors\n\neffective, \u2013141\n\nfinding,\n\npreparing future servant-leaders, \u201354\n\nmind-sets\n\novercoming the obstacle of, \u201324,\n\nreformation, \u201335\n\nvolunteers, \u201353\n\nmission\n\nclarity of,\n\norganizations, xii\n\nstatements, , \u2013212, \u2013223,\n\ntrustee support,\n\nmobius strip, logo for RKGC,\n\nmodel\n\nAndover, ,\n\nchurch as institutional,\n\na new leadership,\n\nseminary structure,\n\nseminary vs. other institutions, \u2013202\n\nmoney\n\nAlcoholics Anonymous, \u2013151\n\nchurch programs,\n\nobsession, \u2013162\n\nraising, ,\n\nMoore, Robert and Douglas Gillette, _The King Within_ ,\n\nmorality, building, \u2013101\n\nmorals\n\ncommon code of,\n\ntraditional vs. contemporary,\n\nMorgan, J.P., telephone industry, \u2013223\n\nMosaic law, and morality, \u2013100\n\nMoses, Ten Commandments,\n\nMoskowitz, Milton and Robert Levering, _The 100 Best Companies to Work for in America_ ,\n\nmotive, maturing the latent servant,\n\nmovements\n\nDanish Folk High School,\n\nevolutionary,\n\nhuman potential,\n\nmurals, Rockefeller Center's Rivera, \u2013250\n\nmystery, separating from reality,\n\nmyths\n\ninstitutional, , \u2013175\n\nseminaries, \u2013185\n\nNational Association for Community Leadership (NACL), National Community Leadership Award, \u201311\n\nNational Community Leadership Award, National Association for Community Leadership (NACL), \u201311\n\nNational Council of Churches,\n\nNational Industrial Recovery Act of 1933,\n\nNational Labor Relations Act of 1935 (Wagner Act), \u2013227\n\nNational Society for Experiential Education (NSEE), _Combining Service and Learning_ ,\n\nNewman, Cardinal, change,\n\nnurturing institutions,\n\nleadership,\n\nnutrition, health, \u201330\n\nobituaries\n\nHarold Ross, \u2013246\n\nJames Thurber, \u2013248\n\nobjectification, workers,\n\nobstacles, clear identification of,\n\noffices, Greenleaf Center satellite,\n\nOhio Bell Telephone,\n\nOhio Fellows, \u201394\n\n_Oklahoma_ , E.B. White,\n\nold age\n\nE.B. White,\n\npreparing for, \u2013278\n\nOld Testament, false prophets,\n\nopportunities\n\nawareness,\n\ncreative,\n\noptimism, assumptions,\n\norder, civilized society,\n\norganizations\n\ncommunity leadership,\n\nstructure, \u2013147\n\n_Orthodoxy_ (Chesterton), \u2013129\n\noutlook, expanding,\n\noversight, operational use of power, \u201349\n\npain, organizational,\n\nPalmer, Parker, _The Active Life_ ,\n\nparticipation, lack of, , ,\n\npassion, driving force, \u2013166\n\npastors\n\nAmerican Education Society (AES),\n\ntraining, ,\n\npatterns, generalizing from previous,\n\npeace, promise of,\n\nPeck, M. Scott\n\n_A World Waiting to Be Born_ , ,\n\n_The Road Less Traveled_ , ,\n\npeople, view of other,\n\nperception\n\nreligious leadership,\n\nWilliam Blake on, ,\n\nperfectionism, creativity and,\n\nperformance\n\nexceptional, \u2013178\n\ninstitutional roles,\n\nleadership,\n\norganizational,\n\nservant's,\n\nPericles, contingency thinking, ,\n\nperseverance, confidence, \u2013123\n\npersonality\n\nmaintaining your,\n\ntraits of effective mentors, \u2013141\n\npersons, theology of,\n\nperspectives\n\nchanging,\n\nhistorical,\n\npersuasion\n\ndedication to,\n\nJohn Woolman's efforts to abolish slavery, \u2013135\n\nnature and role of, xii\u2013xiii\n\npower, \u201387\n\nproblem solving with, \u201353\n\nreligious leaders,\n\nservant-leader characteristics,\n\ntrustees, \u2013188\n\nvs. coercion, , \u201346, \u2013135,\n\nphilanthropy, site visits,\n\nphilosophy\n\nmodern institutional, \u20139\n\nservice to the practice of leadership, ix\u2013x\n\nphotograph, Robert K. Greenleaf, xiv\u2013xv\n\npicture, seeing the \"whole,\" , , \u2013241\n\npitfalls, awareness, \u2013143\n\nPitt, William, power,\n\nPittsburgh, renaissance, \u2013158\n\nplanes, float,\n\nplans, contingency, \u2013129\n\n_Pluralistic Reflections on Servant-Leadership_ (Bordas),\n\npoetry\n\nE.B. White,\n\ninterpreting,\n\nmission statement, \u2013205\n\npoliticians, compromise,\n\nPope John XXIII, contributions to society,\n\npossibilities, discovering new,\n\npotential\n\nactualization of your, \u2013280\n\njudging full,\n\nleadership attributes,\n\nrecognizing latent,\n\npower\n\ncoercive, \u201384, \u2013206\n\ncreative,\n\nfaculties,\n\nintellectual,\n\nmanipulative, \u201385\n\nmission statements, , \u2013206\n\nmisuse of, \u201382\n\npersuasion, \u201387\n\nsociety,\n\nspirit,\n\ntrustees,\n\nuse of, , \u201349\n\npractice, commitment to practice of leadership, x\u2013xii\n\npreparation\n\nfacing uncertainty, \u2013273\n\nleadership, \u201381\n\nproblem-solving strategies, \u201367\n\nreligious leaders, \u2013124\n\nrole of servant,\n\nsecond careers, \u2013278\n\nservant-leaders, \u201357\n\nPresley, Elvis,\n\npress, sponsorship of testimonials, \u2013259\n\nPrinceton, Presbyterian seminary,\n\nprinciple, moral, \u201384\n\npriorities, critical thought,\n\nproblems\n\nadvance preparation, \u201367\n\ncorrecting through persuasion, \u201353\n\ndepth of understanding,\n\nreal-life,\n\nrelevance of sermons to contemporary society, \u2013219\n\nreligious leadership, \u2013133\n\nsolutions, ,\n\nprocess\n\ncomplexity of the leadership, x\u2013xii\n\nfailure of vital links,\n\nproduction, mature servants,\n\nprograms\n\nexisting and new mission statements, \u2013212\n\ninformal education, \u201312\n\nquality assurance,\n\nprojections, future,\n\nprophecy\n\ncompelling vision, \u201327\n\ntheory, \u201336\n\nprophets\n\nauthenticity test,\n\nperception,\n\nseekers,\n\ntrue vs. false,\n\n_Proverbs 29:18_ (King James Version), , \u201335, , \u2013233\n\npurpose\n\nquestion of,\n\nseminaries, \u2013197\n\nsense of, \u201374\n\nQuakers, slavery, \u201337\n\nqualities, leadership, \u2013115,\n\nquest, spiritual in _Journey to the East_ , \u20134\n\nquestions\n\nanswers by Christian programs to modern society's, \u2013219\n\nformation of the mission statement, \u2013206\n\nrelevance of a mission statement, , \u2013228\n\ntrustees, \u2013187, \u2013194\n\n\"Rabbi Ben Ezra\" (Browning), ,\n\n_Reader's Digest_ , commissioning articles,\n\nrealism, assessing a situation, \u2013106\n\nreality, separating mystery from,\n\nreason, vs. intuition,\n\nreception, new visions,\n\nreforms, Danish agricultural, \u2013137\n\nregeneration\n\ninstitutional servants, \u201333\n\ninstitutions, \u201333, \u201343, \u2013225\n\nmission statement implementation, \u2013228\n\nseminaries,\n\nuniversities,\n\nregulation, competition,\n\nrelationships\n\nbuilding, \u2013275\n\nchurches and members, \u2013156\n\nhealing, \u20136\n\ninstitutions, , \u201338\n\ninterpersonal,\n\nleaders with followers, \u2013120\n\nold age,\n\nwith other generations,\n\nyounger people,\n\nrelevance, faith to mission, \u2013212\n\nreligion, leadership, \u2013167\n\nReligion Division of Lilly Endowment,\n\nReligious Society of Friends (Quakers)\n\nhistory, \u2013276\n\nseeker movement, \u2013152\n\nslavery, , \u2013135, \u2013166\n\nremembrance, E.B. White, ,\n\nremnants, saving, \u201391\n\nresearch, findings,\n\nResearch Center for Group\n\nDynamics, Massachusetts Institute of Technology, \u2013150\n\nresolution, conflict,\n\nresources\n\ninner,\n\nseekers,\n\ntheological of seminaries,\n\nresponses, unwillingness to discuss, ,\n\nresponsibility\n\nchallenge,\n\nfreedom,\n\nperspectives,\n\nstress, \u201366\n\nresponsiveness, anticipating the unexpected, \u2013127\n\nrestoration, forces of, \u2013118\n\nretirement\n\nlifecare communities, ,\n\npreparing for a second career, \u2013278\n\nrevolution\n\nconditions inviting,\n\ninstitutional,\n\nrewards, mentoring,\n\n_Rewiring the Corporate Brain_ (Zohar),\n\n_Richard II_ (Shakespeare),\n\nrightness, testing an act for,\n\nrisks\n\ndecisions,\n\nexpendability,\n\nfuture planning,\n\ninstitutional change, \u201391\n\nleadership, ,\n\nreligious leadership,\n\nunderstanding change,\n\nRivera, Diego, mural in Rockefeller Center, \u2013250\n\nRiverside Church, Harry Emerson Fosdick,\n\nRockefeller Center, Rivera mural, \u2013250\n\nRogers, Carl, formula for meeting conflict,\n\nroles\n\nboards of directors,\n\nbuilding the trustee,\n\nchairpersons, \u2013190, \u2013220, \u2013229\n\nchurch pastors,\n\nexecutive, \u2013207,\n\nfaculties, \u2013187\n\nfoundations, \u201341\n\ninstitutional,\n\npublic service,\n\nreligious leaders,\n\nseminaries, \u2013165,\n\ntrustee, , \u2013206, \u2013211\n\nof universities,\n\nRoosevelt, Franklin D., language use,\n\nroots, formative years, \u2013267\n\nRoss, Harold\n\nobituary by E.B. White, \u2013246,\n\n_The New Yorker_ , ,\n\nSalisbury, Harrison, Xerox Corporation,\n\nSampson, Patsy, _The Leader as Servant_ ,\n\nsanity, hope,\n\n_Saturday Evening Post_ , response analysis, \u2013230\n\nSchneider Engineering company,\n\nsearch\n\nchairpersons, \u2013190\n\nentheos, \u201373\n\npersonal significance,\n\n_Stuart Little_ (White), \u2013260\n\n_Seeker and Servant_ (Greenleaf), \"Fable,\" \u2013167\n\nseekers\n\nchurches, ,\n\ncreating a caring society, \u201357\n\nnurturing,\n\npotential leaders,\n\npower,\n\nprophetic visions,\n\nresources for Church growth,\n\nsustaining, \u2013161\n\nself-actualization, seeking,\n\nself-regeneration\n\ncapacity,\n\npromoting, \u201343\n\nseminaries\n\nadmissions process,\n\nAndover model, ,\n\ncaring,\n\nchanges, \u2013117\n\nchaos, \u2013199\n\nchurches, \u2013178\n\nconsultants,\n\ncritical thought,\n\nevolution, \u201339\n\nfaculty,\n\nfinancial support, \u2013167\n\nfunding, , ,\n\nfuture, \u2013233\n\nhierarchy of institutions, \u2013171, \u2013177,\n\nhistory, , \u2013181, \u2013185\n\nimage,\n\nmission statements, , \u2013212\n\nmyths, \u2013185\n\npotential for influence,\n\nprimary mission, \u2013142\n\nprophetic visions, \u201338\n\npurpose, \u2013197\n\nregeneration,\n\nself-regeneration,\n\ntheological resources,\n\ntheology, \u2013123\n\ntrustees,\n\nunifying vision,\n\nseminars, for future servant-leaders,\n\n_Seminiary as Servant, Essays on Trusteeship_ (Greenleaf),\n\nSenge, Peter\n\norganizational management,\n\nservant-leadership concept,\n\nshared vision,\n\n_The Fifth Discipline_ , ,\n\nsense, common,\n\nserenity\n\nachieving,\n\nold age, \u2013267\n\nreaching,\n\nspirit, \u2013266\n\nsermons\n\nHarry Emerson Fosdick,\n\nrelevance to society today, \u2013219\n\nservant\n\nidea of, \u201323\n\ninstitutions as,\n\npreparation for the role of,\n\nstature,\n\nultimate model,\n\n_Servant Leadership_ (Greenleaf), \u2013117\n\n\"Trustees as Servants,\"\n\n\"Servant, Retrospect and Prospect\" (Greenleaf),\n\nservice\n\nAmerican society,\n\nattitude of, \u201396\n\nauthority of, \u2013282\n\ninstitutional dedication to, ,\n\noptimal institutional,\n\nreligious leadership,\n\nrole of public,\n\nspirit,\n\nservice-learning, experiential education,\n\nShakespeare, William\n\n94th sonnet,\n\n_Henry IV_ (parts I & II),\n\n_Henry V_ ,\n\n_Richard II_ ,\n\n\"Shall the Theological Curriculum Be Modified and How?\" (Rainey), _The American Journal of Theology_ , \u2013218\n\nsignals\n\nlife-shaping, \u201395\n\nlistening to life's, \u2013278\n\nsorting out, \u2013278\n\nsignificance\n\ncurrent,\n\npersonal, , \u201371\n\nSisters of St. Joseph's Health System,\n\nsite visits, philanthropy,\n\nskills\n\nchairperson, \u2013194\n\nforesight,\n\nlanguage,\n\npersuasion, \u201346\n\nslave, greatness, \u2013179\n\nslavery\n\nJohn Woolman, , \u2013135\n\nReligious Society of Friends (Quakers), \u201337, \u2013135, \u2013166\n\nsocial contributions\n\nconstructive,\n\ncreativity in making,\n\nsociety\n\nAmerica as a low-serving,\n\nanxiety, \u2013107\n\ncaring and a good,\n\ncontemporary,\n\ncontribution to quality of,\n\ncorruptors of, vii\n\nfacets of a good,\n\nimpact of great universities, \u2013110\n\nimpact of _The Organization Man_ , \u2013153\n\ninstitution-bound,\n\nmoral basis,\n\npower,\n\nrelevance of sermons to contemporary, \u2013219\n\nreligious leaders,\n\nsolutions\n\ngimmicks, \u2013146\n\npreconceived, \u201367\n\n_Song of the Open Road_ (Whitman), , \u2013102\n\nsources, liberating visions, \u201359\n\nSpears, Larry, The Greenleaf Center, , \u2013291\n\nsphere, maturity and size of personal, \u201364\n\nspirit\n\nleadership, \u2013114,\n\nleading with,\n\npower,\n\nsustaining,\n\nultimate test of, \u2013278\n\nvalues,\n\nspirituality, mission statement, \u2013205\n\nsponsorship, of testimonials, \u2013259\n\nstability, change,\n\nstaff, servant of the mission of an institutional,\n\nstand\n\ncertainty of where you, ,\n\ntaking a moral,\n\nstatements, mission, , \u2013212, \u2013223,\n\nstatus\n\nchairperson, \u2013227\n\nquo and personal growth, \u201376\n\ntrustees, ,\n\nStengel, Casey, language use,\n\nstereotypes, avoiding,\n\nstewardship, servant-leadership characteristics,\n\n_Stewardship_ (Block), ,\n\nstrategies\n\ngrowth, \u2013109\n\nliving with anxiety,\n\nnurturing critical thinking, \u2013220\n\nproblem-solving preparation, \u201367\n\nregeneration of an institution, \u2013225\n\nstrength\n\nbuilding, \u2013124\n\nego,\n\ninstitutions, \u201353\n\nleadership, \u2013133,\n\nstress\n\nreligious leadership, \u2013130\n\nresponsibility, \u201366\n\nstructure, formal vs. informal institutional, \u2013147\n\nstruggle, success,\n\n_Stuart Little_ (White), \u2013260\n\nstudents\n\nanger,\n\nfaculties,\n\nimpact of _The Organization Man_ , \u2013153\n\nleadership potential, \u201329\n\nrevolution in the 1960's,\n\nseminaries, \u2013142, \u2013204\n\nservant of the mission,\n\ntrustee chairperson relationships, \u2013190\n\nunrest and Carleton College,\n\nsuccess, struggle,\n\nsupport, financial for seminaries, , \u2013167, ,\n\nsurveys, response analysis, \u2013230\n\nsurvival, mission statement,\n\nsynergy, working toward, \u201322\n\ntalent, identifying and nurturing,\n\nTaliesen,\n\n\"Talk of the Town,\" _The New Yorker_ ,\n\nTDIndustries, _Fortune_ magazine's 10 best companies to work for list,\n\n_Teacher as Servant, a Parable_ (Greenleaf), , ,\n\nteachers\n\ndedication, \u2013155\n\nof servant-leadership, \u201357\n\ntelevision, commercial, \u201350\n\nTen Commandments, Moses,\n\nTerre Haute, Indiana, ,\n\ntest\n\nadequacy of theology, \u2013184\n\nauthenticity of a prophet,\n\nof caring, \u2013174\n\ncharacter of a vision,\n\ndecision-making,\n\nof entheos,\n\ngreatness of a dream,\n\nimpact of servant on society,\n\njudgment,\n\nleadership, ,\n\nmission statement, , \u2013206\n\npersonal growth, \u201376\n\nreligious leadership, \u2013120,\n\nresponsible effectiveness,\n\nrightness of an act,\n\nservant-leader, ,\n\nspirit, \u2013278\n\nspirit development,\n\ntestimonials, sponsorship, \u2013259\n\n_The 100 Best Companies to Work for in America_ (Levering and Moskowitz),\n\n_The Active Life_ (Parker),\n\n_The American Journal of Theology_ , \"Shall the Theological Curriculum Be Modified and How?\" (Rainey), \u2013218\n\nThe American Legion News Service (E. B. White),\n\n\"The Courage to Be\" (Tillich),\n\n_The Elements of Style_ (White), revision of Strunk's,\n\n_The Ellsworth (Maine) American_ , E.B. White's letter to, \u2013259\n\n_The Empowered Manager_ (Block), ,\n\n_The Fifth Discipline_ (Senge), ,\n\nThe Greenleaf Center, Larry Spears, , \u2013291\n\n\"The Growing Edge Church,\" characteristics, \u2013152\n\n\"The Institution as Servant\" (Greenleaf),\n\n_The King Within_ (Moore and Gillette),\n\n_The Last Flower_ (Thurber), \u2013248\n\n_The Leader as Servant_ (Sampson),\n\n_The Male Animal_ (Thurber),\n\n_The Marriage of Heaven and Hell_ (Blake),\n\n_The New York Times_ , use of power,\n\n_The New Yorker_\n\nE.B. White, \u2013236, \u2013240\n\nHarold Ross, \u2013246\n\nJames Thurber, \u2013237, \u2013248\n\n_The Organization Man_ , impact on society, \u2013153\n\n\"The Power to See Life Through\" (Fosdick),\n\n_The Prince_ (Machiavelli),\n\n_The Road Less Traveled_ (Peck), ,\n\n_The Saturday Evening Post_ (E. B. White),\n\n\"The Second Tree from the Corner\" (White), , \u2013253\n\n_The Seminary as Institution_ (Greenleaf),\n\n\"The Servant as Leader\" (Greenleaf),\n\n_The Seven Habits of Highly Effective People_ (Covey),\n\n\"The Spirit is Power,\" Danish Folk High School, \u201325\n\n_The Stranger_ (Camus),\n\nThe Toro Company,\n\n_The Trumpet of the Swan_ (White), \u2013261\n\n\"The Uses of Old People\" (Davis),\n\n_The View From Eighty_ (Cowley), \u2013267\n\n\"The Wild Flag\" (White),\n\n_The Yale Review_ , \"A Slight Sound at Evening\" (White), \u2013243\n\ntheme, evolution of the servant, \u201321\n\ntheologians, public vs. seminary, \u2013207\n\ntheology\n\nbiblical, \u2013216\n\nchurches,\n\ncritical thought, \u2013214, \u2013220\n\ninstitutional strength,\n\nof institutions, xiii\u2013xiv, , , \u2013184, \u2013233\n\npersons,\n\nreligious leadership, \u2013123\n\ntheory\n\nleadership, vii\u2013ix\n\nof prophecy, \u201336\n\nthinking\n\ncontingency, \u2013129\n\ncritical, , \u2013214, \u2013220, , , \u2013233\n\nunconventional,\n\nThom, John, corruption of society, vii\n\nThoreau, Henry David, _Walden_ , \u2013243\n\nThurber, James\n\nE.B. White, , \u2013248\n\nand E.B. White, _Is Sex Necessary?_ ,\n\n_The Last Flower_ , \u2013248\n\n_The Male Animal_ ,\n\n_Walter Mitty_ ,\n\nTillich, Paul, \"The Courage to Be,\"\n\nTilly, Eustace, _The New Yorker_ ,\n\ntime\n\ninvestment, \u2013193\n\npersonal use of,\n\npressure of,\n\n_To Grow in Wisdom_ (Heschel),\n\nTolstoy, Leo, definition of faith,\n\ntombstone, epitaph on Robert K. Greenleaf's,\n\nTotal Quality Management,\n\ntraditions\n\nconsensual, \u201379\n\nmission statement,\n\nmoral, \u2013101\n\ntraining\n\nchairpersons, \u201343\n\nmanagerial, \u2013140\n\npastors,\n\nreligious leaders in the 19th and 20th century, \u2013216\n\nsensitivity, \u2013150\n\ntransformation\n\npersonal,\n\ntowards serving institutions,\n\ntowards synergy, \u201322\n\ntravel\n\ncultural differences,\n\nmeditation during train, \u2013275\n\ntrends, anticipating the future from current, \u2013131\n\nTrexler, \"The Second Tree from the Corner\" (White), \u2013253\n\ntrust\n\nbreech of,\n\nof constituency,\n\nfaith as, , \u2013132, ,\n\nin institutions, \u2013210\n\npreparation for a position of,\n\ntrustees\n\nbreach of trust,\n\nchairpersons, , \u2013190\n\nconceptual leadership,\n\nconceptual orientation,\n\nestablishing hope,\n\nexceptional,\n\nfoundations, , \u201341\n\nInstitutes of Chairing, \u201343\n\nmission definition,\n\npowers,\n\nroles within institutions, , \u2013206, \u2013211\n\nvision,\n\n\"Trustees as Servants\" (Greenleaf),\n\ntrusteeship\n\nart of,\n\nconstructive leadership, \u2013184\n\ngreatness, \u2013105\n\ntrustworthiness, trustee chairpersons,\n\numbrella effect,\n\nuncertainty, preparation for facing, \u2013273\n\nunderstand, failure to, \u201367\n\nunexpected, anticipating the, \u2013127,\n\nU.S. Air Force,\n\nU.S. Steel, J.P. Morgan,\n\nunity, personal,\n\nuniversities\n\nas a learning laboratory, \u2013110\n\nchange,\n\ncredentialing,\n\ncrisis of leadership,\n\nenvironment, \u2013109\n\nfaculties, \u201391\n\ngaining experience,\n\ngreat dreams, \u201389\n\nimproving,\n\nlack of vision, \u201329\n\nregeneration,\n\nUniversity of Virginia, Thomas Jefferson,\n\nurgency, creating a sense of, \u2013225\n\nusefulness, old age,\n\nvaccine, anti-leadership, , ,\n\nVail, Theodore N.\n\ncontributions to society,\n\ntelephone industry, \u2013223\n\nvalues\n\ncoercive power,\n\ncommunicating,\n\nfinding your own,\n\nharmony,\n\nhumane, \u2013283\n\nleadership potential,\n\nspirit,\n\nVietnam war, coercive power,\n\nvisions\n\ncompelling, \u201326\n\nformation of new, \u2013233\n\ninstitutions,\n\nliberating, \u201335, \u201359\n\nprophetic, \u201337, , , ,\n\nregenerating, \u201342\n\nreligious, ,\n\nshared, \u201379\n\nunifying,\n\nusing existing knowledge, \u201327\n\nVisiting Committee, Harvard Divinity School,\n\nvocation, choice of,\n\nvoices, prophetic,\n\nvolunteers\n\nfollowers,\n\nmind-sets, \u201353\n\nvalue, \u2013284\n\nW.K. Kellogg Foundation,\n\nWagner Act of 1935, \u2013227\n\n_Walden_ (Thoreau), \u2013243\n\n_Walter Mitty_ (Thurber),\n\nWestern Electric Company, Hawthorne studies,\n\n\"what we know,\" compelling vision, \u201327\n\nWheatley, Margaret\n\n_Leadership and the New Science_ ,\n\norganizational management,\n\nWhite, Andrew, Cornell University,\n\nWhite, Elwyn Brooks\n\n_Charlotte's Web_ , \u2013261\n\nCornell University, \u2013239\n\n\"I Paint What I See\" (The New Yorker), \u2013250\n\nand James Thurber, _Is Sex Necessary?_ ,\n\nletter to _The Ellsworth (Maine) American_ , \u2013259\n\nobituary for Harold Ross, \u2013246\n\nobituary for James Thurber, \u2013248\n\nremembrance,\n\n_Stuart Little_ , \u2013260\n\n_The New Yorker_ magazine, \u2013262\n\n\"The Second Tree from the Corner,\" , \u2013253\n\n_The Trumpet of the Swan_ , \u2013261\n\nWhitman, Walt, _Song of the Open Road_ , , \u2013102\n\nwholeness\n\ncertainty of direction,\n\nfeelings of,\n\nsearch for,\n\nWilde, Oscar,\n\nwisdom\n\nlay judgments, \u2013227\n\nnurturing, \u2013103\n\nsmall town people,\n\nunconventional,\n\nwonder\n\nanxiety, \u2013108\n\nattitude of,\n\nWoolcott, Alexander, beer testimonials, \u2013255\n\nWoolman, John\n\npersuasive efforts,\n\nservant model,\n\nslavery, , \u2013135\n\nsustaining spirit,\n\nWordsworth, William, judgment of man,\n\nwork\n\ncreating new leaders, \u2013167\n\ngrowth, \u201375\n\nsatisfaction,\n\nsignificance,\n\n_Working Woman_ , modern leadership,\n\n_Works and Days_ (Emerson),\n\nworlds, inward vs. external,\n\nworldview\n\nG.K. Chesterton,\n\nperception, \u2013108\n\nWright, Frank Lloyd, on art, \u201362\n\nwriting, moral necessity, \u2013237\n\nWythe, George, mentoring,\n\nXerox Corporation\n\nE.B. White's reply regarding sponsorship of testimonials in the press, \u2013259\n\nHarrison Salisbury testimonial,\n\n_You Don't Have to Go Home from Work Exhausted!_ (McGee-Cooper),\n\nZohar, Danah\n\norganizational management,\n\n_Rewiring the Corporate Brain_ , \n\n## **About the Editor  \nand The Greenleaf Center  \nfor Servant-Leadership**\n\nLarry Spears is the editor of two acclaimed collections of essays on servant-leadership\u2014 _Insights on Leadership: Service, Stewardship, Spirit, and Servant-Leadership_ (1998) and _Reflections on Leadership: How Robert K. Greenleaf's Theory of Servant-Leadership Influenced Today's Top Management Thinkers_ (1995)\u2014both published by John Wiley & Sons. Spears has previously co-edited two books by Robert K. Greenleaf, _On Becoming a Servant Leader_ (1996) and _Seeker and Servant_ (1996)\u2014both published by Jossey-Bass Publishers. He is also a contributing author to _Stone Soup for the World_ (1998) and _Leadership in a New Era_ (1994). Since 1970, Spears has published many articles, essays, and book reviews. Among his more recent articles are \"Reflections on Robert K. Greenleaf and Servant-Leadership\" and \"Being a Servant-Leader.\"\n\nSpears was named CEO of the Robert K. Greenleaf Center for Servant-Leadership in 1990, shortly before Robert Greenleaf died. Under his leadership, The Greenleaf Center has experienced tremendous growth and influence. As a manager and leader, Spears has been noted for his successes in applying entrepreneurial methodologies to nonprofit organizations. He also speaks on the topic of servant-leadership. The titles of several of his addresses include \"Servant-Leadership and the Honoring of Excellence\" and \"Understanding Robert K. Greenleaf and Servant-Leadership.\"\n\nIn addition to his involvement with the Center, Spears is a member of the National Society of Fund Raising Executives, the World Futures Society, and the American Society of Association Executives. Like Robert Greenleaf, Larry Spears is a Quaker. In the 1980s, Spears worked on the staff of the Quaker magazine, _Friends Journal_. From 1988\u201398, Spears has served as a board trustee for _Friends Journal_ where he chairs its advancement committee. Prior to joining The Greenleaf Center, he served as Managing Director of The Greater Philadelphia Philosophy Consortium, and as a staff member with the Great Lakes Colleges Association's Philadelphia Center.\n\n**The Greenleaf Center for Servant-Leadership,** headquartered in Indianapolis, Indiana, is an international, nonprofit, educational organization that seeks to encourage the understanding and practice of servant-leadership. The center's mission is to improve the caring and quality of all institutions through a new approach to leadership, structure, and decision making.\n\nThe Greenleaf Center's programs include the worldwide sale of books, essays, and videotapes on servant-leadership and the preparation and presentation of workshops, seminars, institutes, retreats, an annual international conference, a partnership program, and consultation.\n\nThrough the dissemination of Robert Greenleaf's idea about servant-leadership, a number of institutions and individuals have been changed. Servant-leadership is now used as an institutional model, as the basis for educating and training board trustees and community leaders, as the foundation of college and university courses and corporate training programs, and as a vehicle for personal growth and transformation.\n\nFor further information about the resources for study and programming available from the Center, contact:\n\nThe Greenleaf Center for Servant-Leadership  \n921 East Eighty-Sixth Street, Suite 200  \nIndianapolis, IN 46240  \nPhone (317) 259-1241; Fax (317) 259-0560  \nE-mail: greenleaf@iquest.net  \nVisit us at <http:\/\/www.greenleaf.org>\n\n**Berrett-Koehler** is an independent publisher dedicated to an ambitious mission: _Creating a World That Works for All._\n\nWe believe that to truly create a better world, action is needed at all levels\u2014individual, organizational, and societal. At the individual level, our publications help people align their lives with their values and with their aspirations for a better world. At the organizational level, our publications promote progressive leadership and management practices, socially responsible approaches to business, and humane and effective organizations. At the societal level, our publications advance social and economic justice, shared prosperity, sustainability, and new solutions to national and global issues.\n\nA major theme of our publications is \"Opening Up New Space.\" Berrett-Koehler titles challenge conventional thinking, introduce new ideas, and foster positive change. Their common quest is changing the underlying beliefs, mindsets, institutions, and structures that keep generating the same cycles of problems, no matter who our leaders are or what improvement programs we adopt.\n\nWe strive to practice what we preach\u2014to operate our publishing company in line with the ideas in our books. At the core of our approach is stewardship, which we define as a deep sense of responsibility to administer the company for the benefit of all of our \"stakeholder\" groups: authors, customers, employees, investors, service providers, and the communities and environment around us.\n\nWe are grateful to the thousands of readers, authors, and other friends of the company who consider themselves to be part of the \"BK Community.\" We hope that you, too, will join us in our mission.\n\n**A BK Business Book**\n\nThis book is part of our BK Business series. BK Business titles pioneer new and progressive leadership and management practices in all types of public, private, and nonprofit organizations. They promote socially responsible approaches to business, innovative organizational change methods, and more humane and effective organizations.\n\nDear Reader,\n\nThank you for picking up this book and joining our worldwide community of Berrett-Koehler readers. We share ideas that bring positive change into people's lives, organizations, and society.\n\n**To welcome you, we'd like to offer you a free e-book.** You can pick from among twelve of our bestselling books by entering the promotional code **BKP92E** here: <http:\/\/www.bkconnection.com\/welcome>.\n\nWhen you claim your free e-book, we'll also send you a copy of our e-newsletter, the _BK Communiqu\u00e9._ Although you're free to unsubscribe, there are many benefits to sticking around. In every issue of our newsletter you'll find\n\n\u2022 A free e-book\n\n\u2022 Tips from famous authors\n\n\u2022 Discounts on spotlight titles\n\n\u2022 Hilarious insider publishing news\n\n\u2022 A chance to win a prize for answering a riddle\n\nBest of all, our readers tell us, \"Your newsletter is the only one I actually read.\" So claim your gift today, and please stay in touch!\n\nSincerely,\n\nCharlotte Ashlock  \nSteward of the BK Website\n\nQuestions? Comments? Contact me at bkcommunity@bkpub.com.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"The rules are actually quite easy to understand.\nAlthough I don't have enough knowledge in the Cthulhu Mythos, the art and sound style helps to immerse yourself in the universe.\nWell made app to kill some time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The College of New Rochelle School of New Resources offers undergraduate liberal arts degrees for a diverse population of adult learners. Our unique program positions adult students to earn their degree efficiently, gain valuable professional experience, and graduate with the necessary skills and knowledge to pursue advanced degrees or embark on fulfilling careers.\nWith five campus locations that include the College's main campus in Westchester County as well as sites in Brooklyn, Co-op City, the South Bronx, and Harlem, SNR provides the opportunity for students to conveniently pursue their studies within their own communities.\nAt the School of New Resources, a strong liberal arts curriculum positions students to hone their critical thinking skills and develop a lifelong love of learning. Working closely with their advisors, our adult learners develop a degree plan that takes into account their relevant work and life experiences and provides opportunities to focus on specific areas of interest or participate in our professional sequencing program. Upon graduation, students are fully prepared to meaningfully improve their lives and contribute to their communities and society at large.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The first overall draft pick in 1988 made an instant splash in the NHL, finishing second in Calder Trophy balloting, though the player he lost was the 31 year old Russian, Sergei Makarov who many thought should not have been eligible for the award. While Modano's star was rising, the North Stars would relocate to Dallas and the Michigan born player was set to become the face of hockey in the state of Texas.\nThe city of Dallas loved their new team and embraced Modano as their new sports celebrity. It was Modano who would be the leader of the Stars and take them to their first (and to date only) Stanley Cup win in the 1998-99 campaign. While this was Dallas' only win in the Holy Grail of Hockey, Modano did take them to another final the year after.\nModano would retire with the most points ever by an American and made hockey relevant in a place where many thought impossible.\nSeriously, who else could we have possibly picked?\nAcquired: Drafted in the 1st Round, 1st Overall, 6\/11\/88.\nSigned as a Free Agent with the Detroit Red Wings 8\/5\/10.\nSee More Info about Mike Modano on Everipedia!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This expandable cable tidy conceals bundles of cables of any length with ease. The flexible mesh can hold plenty of wires and will expand up to 50mm diameter to accommodate a variety of cables.\nPlease provide your email address and we will remind you when \"Fisual Black Expandable Self Closing Cable Tidy 50mm\" is back in stock.\nThis expandable cable tidy conceals bundles of cables of any length with ease. The flexible mesh can hold plenty of wires and will expand up to 50mm diameter to accommodate a variety of cables. The nylon mesh curls back on itself so let it simply wrap around your cables for a neat and tidy finish. We cut this product to length so its suitable for all sorts of jobs, available in black or white.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Also found in: Thesaurus, Medical, Legal, Financial, Wikipedia.\n1. The act or process of accumulating; an increase.\n2. Something that accumulates or increases.\n1. the act or process of accruing.\nbuildup - the act of building up an accumulation; \"I envied his rapid buildup of assets\"; \"a military buildup in preparation for the invasion\"\nZhang, 2008, \"Understanding the Accrual Anomaly,\" NBER Working Paper.\nFrank, 2007, \"Accruals, Investment, and the Accrual Anomaly,\" Accounting Review, 82, 1333-1364.\n00 Build 549 introduces new capabilities in its Accruals Engine designed to reduce implementation and maintenance costs of customer accrual policies and rules administration and enforcement.\nYet generalpurpose financial statements must be reported using the accrual basis, requiring accountants to convert the records.\nMany introductory-level accounting students find the accrual method difficult to understand, partly because of lack of exposure to the differences between the two methods.\nThe empirical tests applied to the sample permitted analyzing the phenomenon known as accrual anomaly, besides identifying the most important components of the formation, variability and persistence of the accruals of firms listed on the BMF&BOVESPA.\nSince 2014, ICBC s accrual gold business has maintained a robust growth momentum, with both business volume and customer headcount increasing at a fast clip.\nThis measure allows us to directly observe the accuracy of the primary accrual used by insurance companies (loss reserves).\nThe results of Slon's study show that the stability of the accrual component of profit is less than the stability of its cash component and the reason is accruals' subjectivity.\nAn example of a revenue-related discretionary accrual would be the estimation of the amount of revenue to recognize on construction projects in progress.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Our entire product line is harvested from northern Minnesota forests, sawed in our historic Bigfork sawmill, dried slowly in our own kilns, and finished to your exact order in our own millwork plant. Whether your tastes favor the Elegant or the Rustic, every order we supply is made to the same exacting standards. The species seen here are identified by the character, the grade and the color.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"TI\" => string(88) \"Pregnancy Q & A<\/TI>  <isbn>0881664766<\/isbn>  <I3>9780881664768<\/I3>  <AU>Booth, Trish;\"\n\"PU\" => string(22) \"Meadowbrook Press,U.S.\"\nprotected _primary_key_value => string(7) \"8417995\"\npublic description => string(210) \"Pregnancy Q & A<\/TI>  <isbn>0881664766<\/isbn>  <I3>9780881664768<\/I3>  <AU>Booth, Trish; Book by  on Bookprices.lk.  lowest pric\u00a0\u2026\"\npublic keyword => string(190) \"Pregnancy Q & A<\/TI>  <isbn>0881664766<\/isbn>  <I3>9780881664768<\/I3>  <AU>Booth, Trish;, pregnancy, q, ati, isbn0881664766isbn,\u00a0\u2026\"\npublic title => string(128) \"Pregnancy Q & A<\/TI>  <isbn>0881664766<\/isbn>  <I3>9780881664768<\/I3>  <AU>Booth, Trish;: : : Bookprices.lk: Buy Books Sri Lanka\"\npublic image => string(18) \"\/img\/cover\/8417995\"\nprotected _uri => string(79) \"book\/pregnancy-q-ati-isbn0881664766isbn-i39780881664768i3-aubooth-trish-8417995\"\nstring(29) \"Wed, 16 Jan 2019 20:47:39 GMT\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Guide Dog Crosby changed everything for the better.\nIt's very clear that Gaye loves Guide Dog Crosby to bits! He seems pretty happy too.\nWe'd like to introduce you to Gaye Smith, of Devonport, who received her first Guide Dog in November. Actually, it was her very first dog, pets included. Thanks to Guide Dogs Tasmania and to you, and with Guide Dog Crosby by her side, she's regaining her confidence and independence. She's also found a wonderful friend, who she absolutely loves to bits.\nThis is their story, in Gaye's own words.\n\"For the last five years I have had very poor sight. I was still getting around without aids in 2012, but was always looking down to make sure I didn't walk into anything. This meant I didn't see things until right before I got to them, which led to many accidents and falls.\nMy optometrist suggested that I reach out to Guide Dogs Tasmania for help for whatever they could do. We filled out the paper work and sent it off. Soon, a Low Vision Specialist came to visit me at home and with time and practice I learnt to use a white cane. It was a gradual process and my house became a place of Velcro dots on appliances as I gradually pinpointed just what help I needed.\nYou don't think about how hard it is to use the oven, microwave, dryer, washing machine and electric frying pan until you can't see how to operate them! Suddenly the smoke alarm was silenced.\nBy July 2013 I was using the cane all the time. The only thing I found really hard was that around town people often didn't get out of my way and I would hit them by accident, so I was still tentative. It seemed the best option for me would be to have a Guide Dog. It was a difficult decision for me and my husband as we had never owned a dog. We have two beautiful cats and we were worried how they would they react to a dog.\nOnce I was placed on the waiting list I was able to call the Guide Dog Instructor whenever I needed reassurance. She was always there, no matter how trivial I thought my question was. The Instructor made quite a few visits to my home where she checked my mobility with the cane and she was happy with my progress.\nThen, last October, the Instructor rang and said she had a Guide Dog that was perfect for me. WOW!!!!\nWith Guide Dog Crosby by her side, Gaye is regaining her confidence and independence.\n'Crosby' was his name and it was love at first sight, or love at first feel on my part.\nI had a week getting to know him before I was to travel to Hobart for our first week of intensive training together. Crosby fitted into our home so well even the cats liked him. He had a quiet manner but we could tell he was bonding with me already. We trained well together in Hobart and came back home to practise what we had learned. I really relied on him because he'd been training for this for two years and I'd only had a week!\nCrosby is now my means of getting about town by myself and he is so good for me. I can get about so much more safely as he takes me around obstacles, instead of into them. One funny moment was when we were walking along a track with my husband and I asked Denis why Crosby was walking in zig zags. Denis replied \"he's taking you around the puddles\". How sweet!\nCrosby is more than an aid to me, he's my friend.\nAfter just a few months I cannot imagine life without him. I can walk around on my own with him as good as a sighted person and we are inseparable. He sleeps in our room in his bed and he is always near me, always looking out for me. He is such good company when I am alone, and I love him to bits.\nGuide Dog Crosby leads Gaye safely past a garbage bin, left in the middle of the footpath.\nIt is very easy to talk to people about Guide Dogs Tasmania and Crosby. They are surprised that I didn't have to pay for him and that if it wasn't for donations by the general public I wouldn't have been blessed to have him in my life. I really hope telling our story will encourage people to give generously to Guide Dogs Tasmania, so future recipients can continue to lead the lives they want, like me.\nIt costs over $35,000 to breed, raise and train every Guide Dog. Please share Gaye's story and make a tax deductible donation today, so we can provide this independence, safety and companionship to those Tasmanians who are ready and still waiting for Guide Dogs. Thank you.\nClick here to make your tax deductible donation online today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Newspotting: What should we do about Greece?\nNeed value for your English pounds? I'm a Greek lady living in London and I bought some fabulous furniture from Ikea recently which was too large to fit in my car. I used Big Yellow Van to get it taken to my house and was incredibly impressed with the service I got. They were prompt, helpful and their prices couldn't be beaten! Check out www.bigyellowvan.biz for more details or call 0208 150 0260 for a quote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Balance sheet is nothing but another (\u5225\u306e\u30fb\u3082\u3046\u4e00\u3064\u306e) financial document that we accomplished (have) prepared to provide a user with more information about the financial position of a company. You can look up (\u8abf\u3079\u308b) your textbooks for a more formal definition.\nAn analogy (\u985e\u4f3c) that I like to use, I read another book long time ago was that of a financial photograph. A balance sheet is like a financial photograph. What I mean by that? If you are all sitting in my classroom and I took a picture of you right now, the camera would record your posture at the moment the flash went off.\nBut one million seconds after, I took the picture, all of your posture could conceivably (\u304a\u305d\u3089\u304f) change. So similarly the balance sheet records the financial position of a company as of (~\u306e\u6642\u70b9\u3067) a certain date. It could be as of December 31st 2012 or as of April 30th 2013 etc.\nNow the balance sheet is broadly classified into three main sections, assets, liabilities and stockholders equity also called shareholders equity.\nWhat is an asset? I like to use a one word working definition of an asset as something that the company owns, just as you own a number of things in your personal lives. What are some things that you own? Clothes, a car, may be more than one car, a house etc. All of these are your assets.\nSimilarly a company owns or can own a number of different items as well. Now for balance sheet reporting purposes, the assets are further sub classified into several sub sections.\nThe first sub section under the asset group would be current asset, so what is a current asset? By definition a current asset is cash or any other asset that can be converted into cash within one year.\nGAAP specifies which accounts for current assets. Here are this illustrated two of them cash and account receivable as examples.\nAccount receivable also called AR, what is an accounts receivable? When does an AR come on a company's book? AR typically comes about (\u8d77\u3053\u308b) when a company sells a product or provides a service on credit on account. On account is another term for on credit, so again if you were a company selling clothes and you sold clothes to a customer on credit, then that creates an accounts receivable on your books.\nSo in keeping with a one word definition of an asset, and that is something that the company owns. What is it that you the business owner owns, so the company owns when you have an AR?\nIn this case what will happen is the company owns the right to receive money from the customer at a future specified date. So initially when the company sells those clothes on credit that creates an AR the company's AR goes up.\nWhen the customer sends the money, makes the payment and the company receives the check. That reduces the company's AR. So that's a little bit about how AR comes about on a company's books and how it goes down.\nNext we have other sections that broadly fall into the umbrella(\u30b0\u30eb\u30fc\u30d7) of long term assets. The first of these subsections that fall under the broad umbrella of long term assets would be property, plant and equipment. And some examples of accounts that fall in the property, plant and equipment subsection would be land and building.\nThen you have another subsection called intangible assets. Some accounts that fall under this subsection would be goodwill and patents.\nThen we have another subsection which is kind of a catch-all section called other assets, and one account that typically falls into the other account other assets section will be deposits.\nWhat I mean by a deposit? Let's assume you are starting a company for the first time and you need to set up an account for the utilities, so you call up the utility company, and because you have no prior history with that company, they say to you send us a check for $1,000 as a deposit.\nAnd once your company establish a payment track record with them, a good payment track record, let's say for six months, they will return that $1,000 to you. So $1,000 that you send into the company is not a payment for your bill. It's just a deposit that the company is going to hold till they're satisfied that you have a good credit record, a good payment record with them, and then they return the money to you.\nSo your company owned that money all along, hence (So) it's an asset for you, even though the utility company is physically holding to that cash, and so on your books on the balance sheet you would classify that as a deposit.\nSo when you add up all of these subsections that will give a total assets dollar figures. So that's a little bit about assets and the different components that make up assets.\nThen we'll move on (\u5148\u306b\u9032\u3080) to the next section called liabilities. One word definition of a liabilities would be something that the company owes. Do you owe anything to anybody in your personal lives? Most of us do. It would be student loans, it would be a loan for your car, in mortgage on your house (\u4f4f\u5b85\u30ed\u30fc\u30f3), money that you owe for your taxes, so these are all liabilities to you.\nOftentimes students tend to get confused between accounts receivable AR and accounts payable AP. Well don't. Look at the words carefully accounts receivable that means the company is going to receive something and one way the company would receive something would be if they were to have sold a product or provided a service.\nConversely accounts payable focus on the word payable. Why do you have to pay? You have to pay because you purchased something, and in this case would have purchased something on credit.\nSo accounts payable would be one account wages payable. You're an employer. Every day your employees come to work for you. But do you pay them every day? Probably not. So at the end of each day let's assume you pay them once a week on a Friday. So when your employees finish working for you on a Monday, don't you the employer owe them some money as of (~\u306e\u6642\u70b9\u304b\u3089) the end of that day for the work that they've done? Yes, you do. And the next day when they come in and do another day's worth of work, you will now have owe them for two days Monday and Tuesday.\nSo this these items will be reflected in terms of wages payable. And when do the payable get off your books? The payable go off your books? Once you make a payment. So whether it be a payment to Office Depot or Staples or whether it be a payment to your employees, once you make a payment, that brings your payable down.\nThen both of these payable both of these accounts come under the current libraries section. Now what is a current liability? A current liability would be basically a liability that you will pay off within one year now.\nThe next subsection under liabilities would be long-term liabilities. Then again you could have different accounts, I just illustrated one account long-term notes payable. You buy a car for instance, you sign a promissory note with a bank. And oftentimes those car payments are made over many years. So all those payments that you have to make all the principal balance that you owe on your car pass one year after making one your payments. That principle would be reflected in your long-term notes payable. The amount of original loan that you have to pay off within one year that would be classified in the current liabilities section.\nThen we'll talk a little bit about the stockholders equity. One way to understand the stockholders equity would be the owners claim (\u8acb\u6c42) on the assets of a company. And typically again there are different accounts that make up your stockholder equity. One would be common stock. When does common stock come a on company's book, when you first form a corporation. In consultation with your attorney (\u4ee3\u7406\u4eba) you will file(\u7533\u8acb\u3059\u308b) that necessary paperwork and as a result of that you will issue what I called shares.\nShares are another word for stock. When company owns only one type of stock, it's common stock. If a company chooses later to sell a second type of stock that would be called preferred stock(\u512a\u5148\u682a), so common stock represents ownership interest in a company.\nIf you and your friend hypothetically were to start a company, and you all decided to issue 1,000 shares and you own 400 of those 1,000 shares, then your ownership would be 40 percent of the company.\nBut all of the 1,000 shares would be reflected in the common stock account. Now when you first issue common stock in consultation with the attorney, you also and also based on state regulations you have to establish a value for that shares. That value is a some arbitrary (\u4efb\u610f\u306e) value, but it has a lot of legal implications(\u610f\u5473\u5408\u3044) could be called a stated value or a par value depending on your state regulations.\nRetain earnings is an example of another account within the stockholder equity family The retained earning account typically holds the company profit. Every month as a company generates profits. Those profits go into the retained earning account. So retained earning accounts keeps increased every time the company generates profit.\nConversely every time the company has losses, the retained earning account decreases. Another type of transaction that causes your retained earning to go down would be that payment of dividends.\nDividends are a distribution of the company's profit to the shareholders, and so every time a company declares dividends then those dividends come out of the retained earning account, because that's where the company's profits are held. Another way to understand stockholders equity is to think of that as a net worth of a company. We don't use the term net worth as it related to companies but we trend to use the words net worth as it relates to us individuals.\nSo you all have heard of the term a millionaire. Think about it, what makes an individual a millionaire? If I want to have a million dollar houses, does that make me a millionaire? And the answer is no, not necessarily. So what makes me a millionaire then? If you look at the accounting equation here, total assets equals total liabilities and stockholders equity, that's assets equals liabilities plus equity.\nIf you rearrange the equation your stockholders equity would be the difference between your total assets and your total liabilities. So in order for me to be a millionaire, I would have to have the difference between my assets and my liabilities to be at least a million dollars.\nSo going back to my earlier example. If I were to buy a million dollar house, let's assume I don't own any other assets, just for simplicity sake(\u76ee\u7684) just that one asset. But I owed if I borrowed nine hundreds thousand dollars to buy that house, that means my stockholders equity then would only be the differential one hundred thousand that hardly(\u307b\u3068\u3093\u3069~\u306a\u3044) makes me a millionaire. May give me bragging (\u81ea\u6162) rights that I have a million in our house but it does not make me a millionaire. However another hand if I bought a million dollar house with cash outright(\u5b8c\u5168\u306a) cash and I don't have any other liabilities, then I could say I'm a millionaire because the difference between my assets million dollars and my liabilities is zero is equal to net worth or equity of 1 million dollars.\nSo that's an overview of the balance sheet containing the three broad sections assets, liabilities and equity. We have gone through the different subsections. Wherever applicable we have talked about accounts, and let me leave you with this thought, one way to understand accounts is think of their assets liabilities and equity as families. And just as in the human family, there are family members each of these account again, represent family members that belong to each of the three broad family group.\nI'm going to explain to you the mechanics (\u529b\u5b66) of an income statement. So to help you understand this concept a little better, behind me I have put up some information and a simplified income statement on the whiteboard here. So let's talk about the income statement now.\nWhat is the income statement? Let's look at some terms that I used to describe the income statement, also called the statement of income and expenses, also referred to popularly as a P&L statement P and L profit and loss statement.\nSo in the sample here the income statement should always identify the name of the organization. I have taken the liberty of putting the name of my company Lazarus global business solutions.\nRight below that you have the name of the statement, in this case income statement.\nAnd then below that equally important is a time reference. What time reference do you see here? We have put for the month ended March 31st. So when we put a time reference, it helps you, the user of the statement put all of these numbers in context (put~in context \u6587\u8108\u306e\u4e2d\u3067\u3068\u3089\u3048\u308b), because now you know that these numbers are associated with a one-month period and the month namely (\u3059\u306a\u308f\u3061) is a month of March.\nstatement for a period less than a month, let's say like for one week for instance, that is too short a time period for the information to be relevant (\u9069\u5207\u306a) and meaningful. More on that little later.\nSo what type of information does the income statement contain? Basically you can see here the first section is the revenue section. In the revenue section we show all the revenues that the company has earned during this period of time.\nAnd under the revenue section you can have as many sub accounts as you want. So for instance if I were a clothes store hypothetically and I was selling men's clothes women's clothes and children's clothes, I could have under the revenue section a sales account for men's clothes, sales account for women's clothes, sales account for children's clothes etc.\nOr since (~\u306e\u6545\u306b) I'm a consulting practice, I can have a different kind of a breakdown, since I am a service organization. I will not have the word sales, but I will instead I could use a word fees earn or service revenue. I could substitute (\u66ff\u308f\u308a\u306b\u7528\u3044\u308b) that as easily, but if I'm selling merchandise, then I will have sales accounts in addition to the fees earn again, don't depends on what type of services are providing or what type of merchandise I'm selling. But to summarize you're going to have the revenue information on top, and then below that you will have information on expenses.\nSo you could have your cost of goods sold as an expense and you can have a whole bunch (\u675f) of other expense accounts salaries expense, rent expense, interest expense, advertising expense etc. So you would normally list all of your different expense accounts as well.\nAnd then you have your net income or net loss. How do we derive (\u5f15\u304d\u51fa\u3059) that? We take all of our expenses and we subtract that from our revenues, and the resulting number would be a net income. If your revenues are greater than your expenses, conversely if your expenses are more than your revenues, you will end up with a net loss.\nSo this net income or net loss is what determines the profitability of the company. So if somebody asked me is my company profitable, I would look at my income statement and see if I have a net income or net loss, if I have a net income I will say yes my company is profitable, if I have a net loss I'll say no, company at this point is not profitable although I hope things will change in the future.\nSo that gives you a quick overview of the income statement with a sample format.\nNow going back to a point I made earlier when I said normally we don't prepare income statements for a period less than a month.\nThat's assume for a moment we decided to prepare the income statement for only a week, and let's assume that we are doing this for the last seven days and let's further assume that we had very heavy snowfall in the last seven days.\nIf that's the case how much sales do you think my computer store would have? Or if I'm selling clothes how many sales do you think I would have? Not much isn't it. But I still would have expenses.\nIn other words my landlord is not going to give me a break on the rent because we had snowfall for a week. I'm so I'm still gonna have rent. I'm still gonna have utilities. If I have a manager on a salary, I still have to pay him on her. So I still would have a lot of expenses for that last week, even though my sales would be very little. My revenues would be very little and I would probably have a net loss for that week.\nBut to show that net loss just for one week would be kind of meaningless because probably I could make up the revenues over the next few weeks as the weather improves.\nSo again to recap the income statement can be prepared for any period of time. But one month usually is a shortest period of time for which we prepare that income statement, and then comes late and contains information on your revenues and your expenses as well as your net income or net loss. So that's a brief overview of the income statement.\nWe will talk about the cash flow statement, also called the statement of cash flows. I have provided a simplified sample format of the cash flow statement here by my side, and we'll go through that and get a better appreciation (\u771f\u4fa1\u304c\u5206\u304b\u308b) and an understanding of this important document.\nSo you start up at the top by identifying the company name. I have taken the liberty of putting my company there Lazarus global business solutions. The next line identifies this document as the statement of cash flows.\nThen the third we have a time reference. In this case the time reference says that this cash flow statement is prepared for the month ended March 31st.\nThe time reference is important, because it helps to put all the numbers in the proper context, because even before you start looking at the numbers, you know that the information you're about to (\u307e\u3055\u306b~\u3057\u3088\u3046\u3068\u3057\u3066\u3044\u308b) look at is for a one-month period.\nNow the cash flow statement is broken into three broad sections. The first one is the operating activity section, then you have the investing activity section, and finally you have the financing activity section.\nThere are also two different methods for preparing the cash flow statement, the direct method and the indirect method.\nSo what's the difference then between the two methods? Well the main difference between the two methods is in the operating activity section. The way we calculate the numbers in the operating activity section is different under the two methods.\nHowever the remaining two sections, the investing activities and the financing activity section, those two sections are identical (\u540c\u3058) in all respects(\u70b9) under both the methods. So how do we know when we have a cash transaction. How do we know which of the three sections it goes into? Well let me give you some broad guidelines and run through some numbers in our sample statement here.\nLet's take the first section the operating activity section. The broad guideline here is if a cash transaction affects your revenues or your expenses, then we would tend to classify that in the operating activities section.\nNow here in this example under the operating activities we have cash collected from our customers in the amount of $450 and then below that we have another item cash paid to vendors for $400. So between the two when you combine it we end up having a $50 cash inflow from the operating activities section. So we have a 450 cash inflow, a 400 cash outflow, that results in a net inflow of $50 from your operating activity section.\nThen the next section we have is the investing activities. What goes in here? A broad guideline as to what goes in here would be purchase or sale of your long-term assets or any investments long-term investments. So in as an example here I have one item that says cash from sale of land. Cash from sale of land one thousand dollars. So in this case is this is the only item we end up with a $1000 in flow from our investing activity section.\nThen we have the financing activities section. What would be the broad guideline as to (~\u306b\u95a2\u3057\u3066) what goes in here? Transactions that affect your long term debt as well as your equity accounts would be classified in your financing activity section.\nWe have one item mentioned here cash from loans. This is cash that you receive from taking a loan in the amount of $150. So since this is the only item here we have a cash inflow of $150 from the financing activity section.\nSo when you go back to your textbooks and look at a more detailed format, hopefully you can take some of these guidelines that I've shared with you and that will help you better appreciate and understand in much more detail cash flow statement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztxuh b/data_all_eng_slimpj/shuffled/split2/finalzztxuh
new file mode 100644
index 0000000000000000000000000000000000000000..2c86b9a7c408a33200e3434dcf9c16a59daed1d2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzztxuh
@@ -0,0 +1,5 @@
+{"text":"The Hoshizaki KMH-2000SRH with URC-22F(sold seperate), crescent cubed ice maker is capable of producing approximately 1944 lbs of crescent cubed ice, Stack 2 units for 2x the ice production in the same floor space! Our efficient double-sided stainless steel evaporators create our unique crescent shaped ice cubes. It is the purest and hardest ice on the market and is perfect fit for a wide range of applications. This Remote-cooled with URC-22F (Sold Separately) ice maker features an CycleSaverTM design and includes the EverCheckTM alert system making the unit energy efficient and simple to maintain. The KMH-2000SRH requires a Single phase, 208-230v\/60\/1 *3 wire with dedicated neutral for 115V electrical connection that is hard-wired at the installation site. Remote condenser sold seperately. Compatible bins & dispensers, sold seperately: Kloppenberg 2115. ENERGY STAR qualified. Please consult the Specification Sheet for additional details. Or call us to discuss your ice needs 252 661 2700.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I found another food to torch! A fresh fig!!\nI halved a fresh fig, dipped it into Vanilla Bean Creme (recipe in The Art of Breakfast) and then into raw sugar and torched! Add a drizzle of the creme on top and voila, a beautiful, delicious garnish or even small bite for dessert.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had little nibbles of interest but none rose to the level of \"Let's meet up, I'd like to attend,\" so I'm suspending Saturday Night Witchcraft for now. I'll still meet with anyone who's interested, but I won't plan to open my home on Saturdays, as I was doing.\nPerhaps it's not the right time. It could be that folks nearby aren't ready to take the step of attending classes (or interested), or that people don't trust the classes because I'm not an initiate yet (I am super self-conscious about teaching without being an actual initiate), or maybe it's just the time of year (Saturdays in November and December tend to fill up quickly).\nOr perhaps it's the coffee-shop meetup requirement that's putting people off. People might prefer to wander into a public class held at a store rather than commit to attending something held in the priestess's home. (The counterpart Wicca 101 class at Artes & Craft has gone on as scheduled with several attendees.) However, I'm not going to budge on this, and my HPS and one of our coven's Thirds have also advised me not to budge. It's a safety thing, and for seekers, it's also a very small test of commitment and follow-through.\nPerhaps I didn't advertise well enough. I did put up flyers (or ask to put them up) in what I thought were key locations, but I don't know how many people saw them, or whether the shop customers or library patrons who saw them were interested in Wicca 101 classes. For me, this butts up against the whole \"we don't proselytize\" thing, though. The flyers only say \"hey, classes exist, here is info,\" and while that's not the same as proselytizing, it still makes me feel like I'm crossing a line by asking to post the flyers in public places.\nSo, I'll leave the Witchvox listing up \u2014 as a seeker, Witchvox would have been one of my first stops, and the listing is getting views \u2014 and beyond that, we'll see, I guess. To be continued.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Friends of Tide Mills has just completed its November meeting (Sunday the 17th). 19 of us recovered 13 sacks of assorted rubbish from this special place, along with several bulkier items.\nThe amount of rubbish found was at least in part due to the recent gales and generally Tide Mills is in very good shape, largely down to people now taking more interest in the area and taking their litter home - Brilliant.\nOf concern is the Environment Agency shingle re-positioning work currently underway along at Seaford. Any major removal of shingle from Tide Mills which results in beach instability would be serious, because this could in turn de-stabilise the vegetated shingle conservation area behind the beach.\nOur next meeting is Sunday the 15th December. We meet at the beach end of the Old Village Road that runs up from the Level Crossing, from 10.00am until 12 noon. As always, all are welcome, irrespective of the amount of time that can be given.\nAfter our meeting we will be holding our annual Christmas Picnic. All Friends who have helped us in the past are very welcome to come along.\nThe admirable work carried out by Friends of Tidemills over seven years now, has helped transform what was a litter-strewn wasteland used for fly-tipping, to a cherished part of the South Downs National Park.\nHowever, the recent storms have helped shatter the idea that FOTM could ever walk away from its monthly commitment to keep the area clean. After months of scrabbling around to find enough to fill a few sacks, within a few days of storm conditions, the site and indeed the whole of Seaford Bay beach has been strewn with maritime debris, most of it plastic, mixed up with large quantities of harmless seaweed. The latter will rot away quickly, but all the plastic will not.\nOn top of that, rather than clear this dangerous debris, the Environment Agency has launched its seasonal campaign of shingle-shifting, using huge lorries to replace what shingle is left to the centre of the bay from Splash Point and the Tidemills area. Apart from helping to turn what shingle remains to coarse sand, which inevitably \"cliffs\" under storm conditions, making access to the beach difficult and even dangerous, these lorry movements are grinding the latest pollutant delivery into the beach itself, where it will remain. This crushed debris includes shards of glass and plastic to cut unwary feet in the summers to come.\nOnce upon a time Seaford Town Council held regular meetings with officers of the EA to discuss, and often resolve, issues like this. Now I hear that these meetings take place rarely, if ever. Why not? It must be in the interests of the Town Council to have a safer beach for visitors around the year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This site uses technical cookies to improve the user experience on the site, may use third-party cookies but not profiling cookies.\nIn general, cookies are small text files containing information that the site server visited by the user sends to the user's device. The same server will re-read them at the next visit to obtain data of various types. While browsing the site can also be received cookies from different websites or servers, the so-called third-party cookies, ie set by a different site from the one on which you are browsing. This is because on the site there may be some elements that reside precisely on different servers.\nCookies, usually present in users' browsers in very large numbers and sometimes even with long temporal persistence, are used for different purposes: execution of computer authentication, monitoring of sessions, storage of information on specific configurations concerning users accessing the server, choice of the reference language, storage of the shopping cart, etc.\nsession and browsing cookies, useful for the normal use of the website, also used by our site, contain the information that is used in your current browser session; these cookies are automatically deleted when you close the browser. Nothing is stored on your computer after the time of use of the site.\npersistent cookies, stored in the visitor's device and that help him to reconnect to the site at the beginning of a new session. They are used to maintain the information that is used in the period between access to the website, or used for technical purposes and to improve navigation on the site. This data allows sites to recognize that you are an already known user or visitor and adapt accordingly. \"Persistent\" cookies have a duration that is set by the website and can vary from a few minutes to two years.\nPROFILING COOKIES. Our site also uses profiling cookies, which send advertising messages targeted and appropriate to the preferences that the user has expressed during his navigation on the site. Create a personalized profile based on the pages visited. Because of the particular invasiveness that such devices can have in the private sphere of users, European and Italian legislation requires the user to be adequately informed about their use and express their valid consent. The art. 122 of the Privacy Code where it states that \"the storage of information in the terminal equipment of a contractor or a user or access to information already filed is permitted only on condition that the contractor or the user has given his consent after have been informed with the simplified procedures referred to in Article 13 of the Privacy Code, paragraph 3 \".\nThese are the cookies installed by the same site manager that the user is visiting (c.d. \"first part\"). First Party Cookies are created and readable by the site that created them. This site uses first-party cookies of a technical nature, for which, as mentioned, there is no consent, but only this information.\nCookies are installed by a site other than that of the manager that the user is visiting, which installs through the first (so-called \"third parties\"). The third-party cookies are created and readable by domains external to the site you are visiting, and whose data are stored at the third party and for which the cookies policy and data processing of the third is in force.\nThis site may or may be used the following third-party cookies, so please check the cookies used through the banner visible on any page of the site before giving consent to the treatment.\nGoogle may also transfer this information to third parties where required to do so by law or where such third parties process the information on Google's behalf. By using this site you consent to the processing of your data by Google in the manner and for the purposes set out above.\nGoogle Analytics has been anonymized in order not to collect personal data of users who browse the site.\nPlease note that third parties may use this information, also combining it with others already available for purposes other than purely technical (functionality or session), subject to user consent. To find out more, access the privacy policies of the individual third parties and manage your options by following the links presented in the column alongside and \/ or set your browser as described below.\nIt is also noted that Google has launched a service called \"Personal Account\" that allows the user to still more accurately manage their data and their privacy by letting them choose what information they share with Google when they use their services. Inside there are all the tools that Google makes available to protect privacy. Instead, the \"Privacy Service\", also from Google, provides all the answers to the most common questions on privacy and security.\nTo know the use that these applications make of your data, the information provided by the respective companies are valid.\nThe owner does not guarantee that the links to the aforementioned policies are always working, and this because of any updates that do not depend on his will, therefore the user is required to consult and verify personally the text in force on the websites of the companies indicated above. The owner is not responsible for any malfunctions or conditions that do not allow consultation of the aforementioned application policies.\nAlso in this case, remember that disabling one or all cookies, the functionality may be reduced or not available. Below are links to detailed procedures to disable cookies from individual browsers.\nThe holder does not guarantee that the above links are always working, and this because of any updates that do not depend on his will. The owner is not responsible for any malfunctions or conditions that do not allow the aforementioned web pages to be consulted.\nThis information is prepared in fulfillment of the obligations under Art. 10 of the Directive n. 95\/46 \/ EC, as well as the provisions of Directive 2002\/58 \/ EC, as updated by Directive 2009\/136 \/ EC, concerning Cookies.\nThis information concerns only this site, and can be reached from every page of the site itself via a link in the footer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzurme b/data_all_eng_slimpj/shuffled/split2/finalzzurme
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"This is a very interesting video of Pakistani politicians and their lookalikes Disney characters. You will agree that all these Pakistani politician have some resemblance of these Disney characters but you may disagree on the Tahir ul Qadri's and Rafiki because they aren't look similar at all. Right?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will you be watching the Grammys this year? Rather than just simply tuning in on February 10th, we here at The Tower at Spring Creek Blog invite you to take some time during the weeks leading up to the awards ceremony to prepare! By following these few simple tips, you'll be ready to be an educated viewer of the year's most significant awards ceremony in the world of music.\nIf you haven't yet been exposed to the music being nominated, it's a good idea to take some time before the Grammys to give it a listen! Because there are so many categories, it can get overwhelming to know where to even begin. Instead of trying to listen to everything, we recommend picking three or four prominent categories and listening to everything nominated within them. If you don't have the time to listen to more than a few categories, focus on the most prestigious ones like Record of the Year, Song of the Year, Album of the Year, and Best New Artist. If you have time tolisteng to everything more than once, all the better!\nOn top of simply listening to these songs and albums, it's important to get some background information on them. Who is the artist singing them? What is their story? Are they a member of any minority communities? Why is it important that they specifically are up for a Grammy award? Of course, there will be songs we like at first listen more than others, but with a little context, songs that didn't stand out before will become more meaningful and interesting.\nFavorites will always remain favorites, but it's important to realize that there is more to music and to the Grammys than sound or popularity alone.\nOnce you've done your homework on the nominations, it's time to sit back and enjoy the Grammys! When you don't know half of the music up for a nomination, the awards ceremony isn't nearly as exciting; when you have spent time beforehand preparing, you'll likely have specific songs and artists you're rooting for, making the whole event much more fun. It's perfectly normal to be bummed out when one of your favorites doesn't win, but make sure to celebrate the success of the winners as well. It took a lot for these artists to get where they are, and winning is no small feat!\nWe hope these tips helped you prepare for your Grammys viewing experience this year in your Garland, TX apartment! Thanks for reading.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for the value of old coins made between 1900 and 2000?\nThere are still so many old coins from the 1900s in circulation that are just waiting to be found!\nYou might be surprised to find out that your pocket change could contain old coins worth much more than their face value.\nIt certainly makes you want to take a second look at all of the coins you have in your loose change jar, doesn't it?\nSee\u2026 old coins are more readily available than you probably thought!\nBelow, you'll find the value of old coins that you're most likely to find in the typical fashion \u2014 pocket change finds, coin roll searching, and as heirloom pieces.\nIf you're looking for coin values that you don't see below, we'll tell you where else you can find an old coin's value at the very end of this article.\nSo, what are your old coins worth?\nOf course, the value of old coins ultimately depends on what condition they're in.\nSee why coin grade matters when determining the value of old coins.\nFollowing is a rundown on coin values for copper, nickel, and silver coins made between the years 1900 and 2000.\nSo, did you find your old coin's value above?\nI'm betting if you've read on this far, chances are you have a coin that's not specifically listed above. Correct?\nDon't worry \u2014 if you have an old coin (one that's made before 1965), you can be pretty confident it's worth more than its face value. But how much more?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Laxmipuram Palace was the home of the royal family of Koi thampurans. It is in Kerala. Many noted wr..\nCan anyone suggest me some tips for good eyesight?..\nWhat are the snow sports played in India?..\nBollywood films mostly cater the urban population. Do you think they should also focus on intrinsic ..\nDo you think that the Indian Super League will be able to boost the status and popularity of footbal..\nCan yoga help in preventing hair fall?..\nIndia won its first-ever team bronze in the Chess Olympiad in August 2014 defeating Uzbekistan. And ..\nFortune telling is the technique of predicting the life of person as well as his or her future. Does..\nDo you agree that morning walk helps us to start an active day and keeps us alert for the whole day?..\nWhat type of oils are used in oil paintings?..\nWhat were the ancient tools for astronomy used in India?..\nIf you love nature and adventure you must visit Charmadi in Karnataka. It is a marvelous place for t..\nWhat are the health benefits of vegetable juices?..\nPhotobiography is an interesting concept where a person`s biography is expressed through photographs..\nResearches have suggested that sniffing certain essential oils can help in reducing weight. Quite an..\nTada is a small town in Andhra Pradesh, perfect for those seeking an escape into the quite wildernes..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Flash Furniture Mid-Back Drafting Chair with Lumbar Support has fabric upholstery and a thick padded seat. This extended height chair features a double paddle control mechanism for the height adjustable back with locking angle adjustment and the height adjustable swiveling seat. The seat height range is 24 to 28 inches. The back has lumbar support as well. The back rest is 16 inches wide and 20 inches tall, while the seat is 18\u00bc inches long and 17 inches wide.\nThis drafting chair has a heavy duty nylon base with an adjustable chrome foot ring and dual wheel casters. The Flash Furniture Mid-Back Drafting Chair with Lumbar Support is available in a choice of 3 colors: black, burgundy, and navy blue. Assembly is required.\nAdjustable Seat swiveling, adjustable height 24\"-28\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzutfl b/data_all_eng_slimpj/shuffled/split2/finalzzutfl
new file mode 100644
index 0000000000000000000000000000000000000000..d84574a7380a4e716496a1d7ffd987164e2c4609
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzutfl
@@ -0,0 +1,5 @@
+{"text":"Reservations are limited. And, sorry, we are quite strict: 20 minutes after the booked time we give away the table, as we do not want to keep people waiting unnecessarily.\nIf you prefer you can make a reservation with the form below.\nPlease send reservations by contact form a day in advance.\nPlease fill out the letters\/numbers from the image in the space below. This way you help us to avoid unnecessary messages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everyone loves to get a deal. It's a fact of life. But what's the best way to capitalise on this, as a marketer? Offers can be a double edged sword for any business, with an increase in sales or traffic but a lowering in individual revenue; it's critical to find a balance to keep your company profitable.\nThere are a few different key ways to capitalise on this aspect of human nature, with their own pros and cons. Firstly is the one seen most often; discounted prices for a certain timeframe. Now this can be a given holiday, such as Christmas or Black Friday sales , or simply 'next two weeks', but the result is the same. These offers are best suited to physical stores rather than online, as the aim is to get people into the store. Once they are in, they're more likely to purchase 'accessory products', which is where the main earnings come with offers. If you go to a store to buy a half price jumper, and you see a nice scarf, you are more likely to buy both as you have already saved money. These offers can also be a great way to get rid of excess stock, such as leftover seasonal stock such as summer shorts in autumn.\nAnother way to capitalise on exclusivity is through offers and coupons, as loyalty rewards or in other media such as newspaper offers. These have to be carefully planned, as though it does increase traffic, they also run a risk of losing money. For example, say you are selling a scarf for \u00a310, and your total costs are \u00a35 for the item (store space, manufacturing, shipping etc.) then you can potentially run a 50% off or buy-one-get-one-free offer, without losing money. But what about when the customer has another 10% off as a part of another offer, or with uniformed services discount or similar, then you will operate at a loss. In the UK at least, most companies get around this by including an adage to the coupon such as 'cannot be used in conjunction with any other offer'. This is not the place in places such as America, where coupon use is much more prevalent.\nHaving an offer which is exclusive to a certain group of people adds urgency to the offer, meaning people are much more likely to shop there. People love to feel special, and individual tailored offers are a great way to use this mentality to increase sales.\nPrice matching is a relatively recent addition to the offer repertoire, with certain companies claiming to sell a certain item cheaper than anywhere else, or they'll match the price. Now obviously this can go extremely wrong in certain cases, but a small loss can sometimes mean a large gain. For instance, if a company is selling a TV with price matching on Amazon, then there is nothing to stop someone creating an Amazon account and selling one of that TV for a lower price, and cheating the system by getting the TVs at a lower price. This can cost companies alot of money, but fortunately there are a few ways around this. Many companies manufacture certain models exclusively for a certain store, meaning the model number does not match between companies. The customer may feel cheated if this happens, but at least you do not lose money, and the person is still in the store. The other way around this can create a small loss; pricing lower than the competition, but only getting a few models in. It's easy to understand that an amazing offer like this has sold out, and in the end the customer is still shopping in your store. Exclusivity is the key to event marketing, and can increase sales year round if used correctly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last week I visited the British Library's Newspaper Library in Colindale, North London. This charming \u2013 though hardly sensational \u2013 art deco-style building is the only place in the country where you can access most of the world's newspaper 'back-catalogues', often in their original paper form.\nSad to say, the Library's planned closure will go through in the near future, meaning that most of the stock will be transferred to other British Library sites \u2013 or simply put into storage. Fingers crossed that the closure won't make it any harder to get hold of some of the rare and wonderful items currently available for inspection.\nThis entry was posted on Wednesday, May 26th, 2010 at 5:13 pm\tand is filed under media theory. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a multitude of Spanish institutes, immersion programs, and homestay opportunities throughout Costa Rica. Students can live at the school or with a host family, join a surf or yoga program, and participate in cultural activities such as salsa lessons and cookery classes. You can also find courses that range in length, from a weekend to several months. Here are the very best places to learn Spanish in Costa Rica.\nSol Education Abroad is located in Heredia. This Spanish immersion program is run by the Univeridad Latina in Heredia. Aside from offering outstanding and well organized Spanish classes, as part of this course you will also have the opportunity to take cookery classes, visit a coffee farm, spend time in Monteverde, go zip lining and other adventure activities, plus take guided city and museum tours. This program offers the whole package, Spanish and cultural immersion in Costa Rica.\nAIFS Study Abroad has programs all over the world, including one in San Jose, Costa Rica. What makes this Spanish learning opportunity so unique is that their packages include everything in one set price (tuition, housing, meals, airfare, and cultural activities). Not only will you spend your semester or summer studying Spanish, you will also embark on a variety of amazing cultural and adventure activities such as dancing lessons, sporting events, museum visits, and a trip to Arenal, and Monteverde.\nMorpho Spanish School is located in the beautiful Orosi Valley. This small Spanish school consistently receives rave reviews. The highly experienced teachers, small class sizes (often one-on-one), unique homestay opportunity, and cultural excursions have proven to be very effective for all different levels and ages of students who choose to study at Morpho Spanish School. This affordable Spanish program is a great choice if you want to spend time in a lovely and truly local Costa Rican town where you will have ample opportunities to explore and practice your Spanish.\nIPED Language Plus has two locations, Heredia and Puerto Viejo, and has been a successful and highly respected Spanish language institute since 1984. The Puerto Viejo location offers students the chance to also surf, do yoga, volunteer, and choose from a number of activities and excursions. There are multiple different types of programs as well, such as classes for families, classes for 50+, total immersion, which includes a homestay, and even group programs for high school or college students.\nIf you want to learn Spanish on an outrageously beautiful beach that also happens to have great waves and yoga, the Perfect Sunset School in Playa Hermosa, Santa Teresa is an outstanding choice. For 10 years, this institute has been effectively teaching all levels and ages of students interested in learning Spanish. They also have highly experienced surf instructors and a number of different spots to take you to practice yoga. The school only accepts eight students at a time.\nLa Escuela del Sol is located in Cabuya. This school offers far more than just Spanish classes. You can combine your Spanish classes with learning and practicing fire dances, yoga, surfing, and scuba diving. There is also an opportunity to volunteer at an animal rescue program. La Escuela del Sol offers college credit, special retreats, and a variety of exciting excursions.\nThe Nosara Spanish Institute offers a variety of Spanish courses, such as a 20-hour program, 10-hour program, surf and Spanish program, family and culture program, and a yoga and Spanish program. There is also an option to completely customize your own Spanish course based on your individual needs, interests, and choice schedule. There are tons of extracurricular activities that you can do during your free time and even a chance to volunteer in the local community. Nosara is a truly stunning destination to go to learn Spanish!\nThe Coastal Spanish Institute is located in the lively surf and party town of Tamarindo. If you want to be located right on the beach, have nightlife options, be in close proximity to a number of amazing tours, and learn Spanish, this is a great option. They offer a number of programs for different levels and Spanish goals, as well as an option to customize your course. You can also sign up for the surf and Spanish program, since Tamarindo is one of the best places to learn how to surf.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Few people can claim to be pioneers in a field, and in my 45 years of experience in counseling and wellness, only a handful stand out; Kristin Webster is one of those individuals. When others caught the early wave of wellness then changed to pursue another trend, Kristin prevailed, in her career and in her personal commitment to wellness and women's well-being. As the designer and leader of one of the nations first wellness programs AT&T's 'Total Life Concept\", I enlisted Kristin to teach classes on Stress Reduction, Weight Management, and Interpersonal Communications. Her skills as a therapist and her in depth understanding of women's issues resulted in the best outcomes we experienced in the wellness program. To state it simply, Kristin changes lives. Her intellect and knowledge place her among the elite with the therapists I have known throughout my career. Her personal life experiences provide a unique base of understanding that few therapists can offer and she lives the life that she promotes. However, the main thing she offers is something very special, a total commitment to everyone she encounters. When you engage with Kristin you will know immediately that you have her undivided attention and she is interested in one thing only, your well-being and happiness.\nI have been blessed to know Kristin for over 15 years. She is one of the most grounded, passionate, and authentic leaders I have ever met. She is committed to empowering others to find their strength and true North, and her work is both direct and inspiring. I give her my absolute and 100% highest recommendation!\nKristin has been my thoughtful, honest, and compassionate mentor for 25 years. Among her strengths are an unwavering commitment to those for whom she cares, remarkable listening skills and an empathetic truthfulness that resonates and changes behaviors. I consider it an honor to know and have worked with her.\nIn my work with Kristin at the University of Missouri-Kansas City's Women's Center, I witnessed her tenacity and authentic desire for helping people discover their best selves. She is a one-of- a-kind advocate with a knack for cutting to the core of complex or difficult issues. Anyone working with Kristin would benefit from her knowledge, her passion, and particularly her genuinely beautiful soul.\nMy first encounter with Kristin was as her supervisee in a corporate counseling center. During our work together, she was promoted to clinical director. I was impressed by her finely honed managerial expertise and more in awe of her impeccable boundaries, quick intuitive sense of crisis response and, above all else, humility. Because of her innate sense of empathy, Kristin easily gains trust of people who meet her informally. Her stance with her loved ones, friends and business partners is generally the same: a quick wit, hope-based insight, curiosity and compassion. You will become stronger, wiser, more adept, and confident with Kristin as your mentor\/guide; as a practicing psychoanalyst, I seek her counsel frequently and am greatly inspired, soothed and strengthened by her perspective.\nThere is a time in everyone woman's life when she needs someone who can listen, advise and empower her through life transitions. For me, that someone is Kristin Webster. Kristin's passion for helping women be their best is unwavering. She is a passionate professional who is approachable. Her wisdom, confidence and grace are admirable strengths she has demonstrated through her personal life transitions as well as her professional and volunteer efforts. She is intuitive, perceptive and caring in all her relationships. I can think of no one better to be a life coach.\nI have had the privilege of knowing Kristin Webster for more than 20 years and count her as one of the most admired women in Kansas City, and most certainly by me. What distinguishes her from others is that she lives her life totally supporting those around her\u2026. whether it's her beautiful family, friends, or those in need of guidance. As a parent, she facilitated experiences for her children and many others volunteering with less fortunate families through her many community leadership roles. She has generously provided her expertise in leadership and coaching to the various women's organizations over the years. As an advisor, she is one of the wisest, most trustworthy and compassionate people I know.\nKristin's unique blend of gentleness and strength puts me at ease and invites me to learn and grow. She is an extraordinary listener and is committed to helping women discover their own unique calling and purpose in life. I have long admired her ability to develop relationships with a wide variety of people. There is a deep inner peace about her that is inviting and compelling. I would highly recommend that anyone seeking personal transformation or just support during a life transition turn to Kristin for guidance and counseling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvecc b/data_all_eng_slimpj/shuffled/split2/finalzzvecc
new file mode 100644
index 0000000000000000000000000000000000000000..dd8f8b151ffcc1f622e5d04076d35af357132c6c
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+{"text":"So let's start with Surgery, why I'm going this route, and then maybe next time, I'll talk about Mexico.\nI have settled on having the Vertical Sleeve Gastrectomy or VSG. Coming to this decision has not been easy for me. I worked so hard over the last two years (this time) to loose weight, and it's just not working. I loose some, gain it back, loose some more, gain some more... I'm tired of living this roller coaster, and I want to be healthier. I know that some people are going to judge me; tell me I took the easy way out. Well, if that's how you feel then stop reading... move on... judge someone else, because nothing you say is changing this. Oh... and it's not an easy way out... I can guarantee that. No one I know that has had the surgery has had an easy time. It's hard FOR THE REST OF YOUR LIFE. Hell, diet is hard no matter how you deal with it.\nThis isn't a sudden decision for me. I was actually approved for weight loss surgery about 13 years ago, but because my employer changed insurance company just two days after I got my approval letter, I was rejected. I have thought about the surgery over the years, and while I have never again had insurance that would approve me, I have always wondered where I would be today if I had been able to have the surgery back then.\nOne might think that I just sat around, not caring about my weight, hoping that someday I might get the surgery, but that is not the case. At least not for me. I have tried, and tried to lose weight over the years. I spent years in Weight Watchers, doing Atkins, Slim Fast, Calorie Restriction, Working out, and most recently, Paleo. I would be amiss in saying they were failures, I did have some success. As a matter of fact, I was at one point in my life 420lbs. I managed to get down to 254lbs, and then I just couldn't lose anymore. I gained again, starting just two years ago at 320lbs. I managed again to get down into the 250's, but stalled again. My body just does not want to get below that mark; or at least my mind doesn't.\nI've had some people tell me the reason that I can't lose weight is because I am weak willed, that I eat too much junk food, that I must stuff myself constantly, but that's just not true. Granted, I like food, I like the occaisional sweet, and sometimes I just throw in the towel, but for the most part I try. EVERYDAY I try. My body fights against me constantly. I have PCOS which makes me infertile (fine with me, no kids happening at almost 40!), but it also makes it super easy to gain weight, and impossible to lose it. I am also very genetically inclined to be heavy. I love them all, but my whole family struggles with their weight. So, while this might not be a decision you would make, it's the right decision for me.\nVery proud of you Misty. I know how hard you work and have worked at this for quite some time - I'm right there with you! Way to go !\nHi Misty, good for you to be choosing the surgery. I have had several surgeries. 2 by choice and the others to correct issues or to save my life... Yes, stomach weight loss surgeeies. Anyway, message me or call and I'll tell you all the good, bad and ugly. Oh ya, I've done the Mexico route also and then right here in Las Vegas, NV too.\nMisty, I too have PCOS and feel your struggle! I feel that you have decided to take control and not let PCOS win and for that I am so proud of you! It's so hard to lose weight especially when you have tried almost everything and given your all each time you try those things! I really hope that no one thinks you are taking the easy way out because dealing with this surgery for the first year will actually be harder than any diet or exercise you have ever done! Retraining your Brain and body at almost 40 will be hard to do so you definitely are NOT taking the easy way out at ALL!!!! If anyone says that you are you should direct them to a FB group for people that have had the surgery You will be having and hopefully that will open their eyes that you are actually taking control back, not giving in and taking the easy way out. Ok so don't take offense to this next part because I don't mean to offend at ALL! I personally think that you should see a counselor after you have the surgery, your body will be going through so many changes and it would be nice to you to have a sounding board, someone to listen and just be there for support, I'm going through a similar situation but without the surgery and seeing Antje has been the best decision for me, just a suggestion.\nAgain, I am so super proud of you and good luck!!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LAS VEGAS \u2013 Left wing Taylor Hall of the New Jersey Devils, whose 93 points (39 goals, 54 assists) set career highs across the board and propelled the Devils to a 27-point improvement in the standings as well as their first playoff appearance since 2011-12, captured the Hart Memorial Trophy as NHL MVP at the 2018 NHL Awards, held at the Hard Rock Hotel & Casino Las Vegas.\nHall received 72 first-place tallies among the 164 ballots cast by members of the Professional Hockey Writers Association (PHWA) and garnered 1,264 voting points to edge Colorado Avalanche center Nathan MacKinnon, the top choice of 60 voters for 1,194 points. It marked the closest Hart Trophy race since 2012-13, when Alex Ovechkin (WSH) edged Sidney Crosby (PIT), 1,090-1,058. Los Angeles Kings center Anze Kopitarfinished third in balloting with 551 points.\nHall's 93 points ranked sixth in the NHL and was 41 more than his next-closest teammate (Nico Hischier: 20-32\u201452), the highest such differential by a club's top two scorers since 2007-08 (Ovechkin: 112, Nicklas Backstrom: 69 with Washington). He collected points in 26 straight personal appearances from Jan. 2 \u2013 March 6 (18-20\u201438), a franchise record and the longest such streak by any player since 2015-16. Hall also set a club record with a 19-game point streak during that span (Jan. 30 \u2013 March 6: 13-13\u201426).\nThe 26-year-old Calgary native and first-time NHL Trophy finalist is the first player in Devils history to capture the Hart Trophy; goaltender Martin Brodeur finished third in voting in 2002-03, 2003-04 and 2006-07.\nFour members of the Vegas Golden Knights \u2013 defenseman Deryk Engelland, center William Karlsson, head coach Gerard Gallant and general manager George McPhee \u2013 received awards in recognition of achieving the greatest season by an expansion team in major pro sports history, both on and off the ice.\nEngelland received the Mark Messier NHL Leadership Award, presented \"to the player who exemplifies great leadership qualities to his team, on and off the ice, during the regular season and who plays a leading role in his community growing the game of hockey.\" Engelland registered a career-high 18 assists and 23 points while serving as the backbone of a Golden Knights team that achieved unprecedented success for an expansion franchise. He also was a fixture in the Las Vegas community following the tragic events of Oct. 1. A longtime Las Vegas resident, Engelland continued a five-year relationship with local firefighters that has raised more than $40,000 for children who cannot afford to play hockey.\nKarlsson, the Golden Knights' leading scorer, captured the Lady Byng Memorial Trophy in recognition of outstanding skill and sportsmanship. Only five players in League history scored more goals while playing for a club in its inaugural season than Karlsson, who tallied 43. He filled the scoresheet while taking only six minor penalties, finishing with the fewest penalty minutes (12) among the League's top 40 scorers. The 25-year-old Marsta, Sweden, native is the first player to win an end-of-season trophy for a team in its inaugural campaign since 1979-80 (Wayne Gretzky w\/ EDM).\nGallant won the Jack Adams Award as the NHL's outstanding coach. He led the Golden Knights (51-24-7, 109 points) to a slew of historic achievements in 2017-18, with Vegas becoming the first modern-era expansion team from any of the four North American professional sports leagues to start from scratch and win its division. The Golden Knights started the season 8-1-0, using four goaltenders within their first 10 games following a spate of injuries. They went on to post their 34th victory on Feb. 1 to set the NHL record for wins by a team in its inaugural season \u2013 with 32 games remaining.\nMcPhee was voted NHL General Manager of the Year. Featuring a roster assembled by McPhee largely from the NHL Expansion Draft \u2013 conducted less than three months before training camp opened \u2013 Vegas set multiple expansion team records in capturing the Pacific Division title and reaching the Stanley Cup Final. Eleven Golden Knights players recorded career-best point totals, including the team's top five scorers: Karlsson, Jonathan Marchessault, David Perron, Reilly Smith and Erik Haula.\nPekka Rinne captured the Vezina Trophy as the League's top goaltender after backstopping the Predators to the Presidents' Trophy as the NHL's best regular-season team. Rinne ranked third in the NHL with 42 victories, shared the League lead and set a club record with eight shutouts, and placed in the top 10 in goals-against average (5th; 2.31) and save percentage (7th; .927). The 35-year-old Kempele, Finland, native is a first-time Vezina Trophy winner in his fourth appearance as a finalist, having finished second in voting in 2010-11 and 2014-15 and third in 2011-12.\nVictor Hedman of the Tampa Bay Lightning won his first James Norris Memorial Trophy as the League's best defenseman. Hedman shared first place among NHL defensemen with a career-high 17 goals, tied for the second-most in one season by a Lightning blueliner (behind Dan Boyle: 20 in 2006-07). He also ranked fifth among League defensemen with 63 points, becoming the first blueliner in team history to top 60 points in multiple campaigns (16\u201156\u201472 in 2016-17).\nNew York Islanders center Mathew Barzal won the Calder Memorial Trophy as the League's top rookie. Barzal was a near-unanimous selection, receiving 160 of 164 first-place votes. He led all rookies with 22-63\u201485 (82 GP), 20 points more than the next-closest player, and became the seventh rookie in League history to record at least 20 goals and 60 assists in one season. Barzal, who also paced rookies in assists (63), power-play assists (22) and power-play points (27), recorded a trio of five-point efforts: Nov. 5 vs. Colorado (0-5\u20145), Jan. 13 at NY Rangers (2-3\u20145) and Feb. 9 vs. Detroit (0-5\u20145).\nLos Angeles Kings center Anze Kopitar won the Frank J. Selke Trophy as top defensive forward. Kopitar led all NHL forwards in total ice time (1,810:58), an average of 22:05 per game, on the club that led the League in team defense, allowing an average of 2.46 goals a contest. He topped Los Angeles forwards in shorthanded ice time (2:10 per game) on the NHL's top-ranked penalty-killing unit (85.0%). A Selke finalist for the fourth time in five seasons, Kopitar is taking the trophy home for the second time in three years (also 2015-16).\nNew Jersey Devils center Brian Boyle received the Bill Masterton Memorial Trophy for perseverance, sportsmanship and dedication to hockey. At the start of training camp, Boyle was diagnosed with chronic myeloid leukemia, a type of bone marrow cancer. The 33-year-old worked his way back into the lineup by Nov. 1 and notched 10 goals over his first 25 games, including a memorable goal on the Devils' Hockey Fights Cancer Night at Prudential Center, a 3-2 win over Vancouver on Nov. 24. Boyle missed just three games after his season debut and represented the Devils at the 2018 Honda NHL All-Star Game in Tampa Bay.\nVancouver Canucks forwards Daniel and Henrik Sedin capped their legendary NHL careers by accepting the King Clancy Memorial Trophy for leadership and humanitarianism. The Sedins' strong connection with their community includes a $1.5-million donation they and their spouses made to the BC Children's Hospital Foundation in 2010 to help build a new children's hospital and expand existing medical services. They have helped the Canucks for Kids Fund raise $42 million since 2000-01, advocated for the literacy promotion programs of the Canucks Family Education Centre and supported the SPCA. The twins enabled thousands of deserving kids to attend Canucks games and touched the lives of thousands more by visiting elementary schools, meeting with patients and families at BC Children's Hospital and Canuck Place Children's Hospice, and through their work with the Sedin Family Foundation.\nDarcy Haugan, the late coach of the Humboldt Broncos of the Saskatchewan Junior Hockey League (SJHL), was posthumously honored with the Willie O'Ree Community Hero Award. New in 2017-18, the award honors former NHL forward Willie O'Ree, who on Jan. 18, 1958, became the first black player to compete in the League and has served as the NHL's Diversity Ambassador for more than two decades.\nHaugan left a lasting impact in Humboldt, Sask., changing the lives of many of his players, always being there for each one of them and never hesitating to give them a second chance. Haugan believed strongly that the game is not about making hockey players; it is about making amazing human beings. He did just that, building up young leaders who also developed strong hockey skills along the way. Haugan died doing what he loved, surrounded by the young people he dedicated his life to.\nTwo weeks after winning the Stanley Cup in Las Vegas, Washington Capitals captain Alex Ovechkin returned to accept the Maurice \"Rocket\" Richard Trophy for the seventh time as the League's goal-scoring leader during the regular season. Ovechkin scored 49 goals in 82 games, finishing ahead of Winnipeg Jets right wing Patrik Laine (44 in 82 GP) and Karlsson (43 in 82 GP). Ovechkin became the second player in NHL history to finish atop the League's goal-scoring race seven times, joining Bobby Hull (also 7x, all with Chicago).\nCenter Connor McDavid of the Edmonton Oilers earned his second consecutive Ted Lindsay Award as the most outstanding player as voted by the NHLPA as well as his second straight Art Ross Trophy for leading the League in points during the regular season. McDavid, who tallied 108 points (41-67\u2014108) in 82 games, became the first player to capture the Ted Lindsay Award twice before age 22. He collected at least one point in 58 of his 82 appearances (70.7%) and led the League with 84 even-strength points, 18 more than the next-closest player and the most by any NHLer since 1995-96.\nJonathan Quick of the Los Angeles Kings received the William M. Jennings Trophy as the goaltender on the club that allowed the fewest goals during the regular season. Quick saw the most action on a Kings team that allowed a League-low 203 goals, eight fewer than the Predators (211) and 11 clear of the Boston Bruins (214). He earned his second career William M. Jennings Trophy as well as the second in franchise history, adding to the one he claimed with Los Angeles in 2013\u201114 en route to winning the Stanley Cup.\nVoting for these awards was conducted at the conclusion of the regular season. The PHWA cast ballots for the Hart, Norris, Selke, Lady Byng, Calder and Masterton Trophies. NHL General Managers voted on the Vezina Trophy. The NHL Broadcasters' Association submitted votes for the Jack Adams Award. Voting for the NHL General Manager of the Year Award was conducted among NHL General Managers and a panel of NHL executives, print and broadcast media. Results were tabulated by Ernst & Young with the exception of the Masterton Trophy (PHWA).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"See pictures of minerals (azurite, See pictures of minerals (azurite, gypsum, and others) from National Geographic. Skip to this Fluorite Calcite.\nGallery of high resolution Fluorite pictures and specimens.\nAs of fluorite mining operations, the main environmental issues are groundwater and. Scramble for the Yellow Cake in Malawi Uranium Network.\npictures of mining fluorite Fluorite has physical properties that allow it to be used for a wide variety of chemical, In the mining industry, fluorite is often called \"fluorspar.\"\nGet Fluorite pictures and royalty-free images from iStock. Find high-quality stock photos that you won't find anywhere else.\npictures of mining fluorite Illinois' State Mineral. The General Assembly made fluorite the State Mineral in 1965, when flourspar mining was a multimilliondollar-per-year industry in Illinois.\npictures of mining fluorite Find the perfect Fluorite stock photos and editorial news pictures from Getty Images. Download premium images you can't get anywhere else.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Plantronics HW361N Supraplus Wideband Headset is a Sliver two eared binaural headset that offers all of the features of the SupraPlus H361N. This is a noise cancelling headset. The features include outstanding reliability and comfort, robust construction and stylish design. This is the ultimate headset for office professionals who increasingly use the Internet for phone-calls via an IP softphone or hardphone.\nSorry, but we haven't had a chance to add an FAQ for the Plantronics HW361N\tyet. If you have a question though, please send it to us and we will answer it for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How humble are you? Humility is one of the rare traits that you can never see in yourself, only others can. As soon as someone says, \"I think I'm humble,\" they aren't any more. But being honest about yourself, do you truly desire humility? Would others say they see even a modest humility in your life or do you top your own list of favorite people?\nToday, we are talking about biblical humility. Biblical humility is often seen as weakness or the equivalent of self-hatred but it's neither. In his book, The Freedom of Self-Forgetfulness, Tim Keller says, \"...the essence of gospel-humility is not thinking more of myself or thinking less of myself, it is thinking of myself less.\" So humility is not fighting pride by constantly being down on yourself, it is being so focused on Christ and others that you \"forget\" yourself as the center of your thoughts and affections.\n1. Spend time everyday in God's Word and in prayer. Yes, this sounds clich\u00e9 but these spiritual practices orient your heart away from yourself and your little kingdom, and toward God. It reminds you everyday that you aren't the point, you aren't the center, you don't need to get your way, etc. and, more positively, God is the point, God is in charge of your life, and God wants to walk with you that day. What would happen in your life if you started every day with that kind of focus on God? If you are looking for how to start reading God's Word, I recommend begin by reading one chapter a day starting in the book of Matthew.\n3. Value \"unimportant\" people around you. I use quotation marks around unimportant because these people aren't truly unimportant but our culture tends to treat them that way. For example, how do you treat the cashier at Trader Joe's? Is he just a machine there to ring up your groceries? How do you honor him as image-bearers of God? What about the custodian at your work or school? Do you speak to her? Do you thank her for the work she does? Do you treat her like you would \"an equal\" or superior at work? How we treat the people who are \"unimportant\" in our culture reveals whether our heart is full of a growing pride or humility.\nI hope these steps stimulate some thinking in your life as they have mine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Keeping teenagers engaged during the holidays can be a challenge, and the holiday cook can feel unappreciated and taken for granted.\nI'm depressed thinking about Thanksgiving. When the kids were younger, we made decorations and they were excited about the holiday. Now that our daughter is 17 and our son 15, they don't want to do those kinds of things and are more interested in being with their friends. Our daughter is usually with her boyfriend and our son plays computer games and sports. My brother and his 24 year old daughter with her 2 yr old son are coming and my mother in law as well. My brother will watch sports with my husband and trying to entertain my mother-in-law and make sure my niece has support is nothing to look forward to. I feel like I'll work like a dog and get nothing out of it. Do you have any advice for me?\nJanet, that's a pretty depressing thought - that you'll work your tail off making a Thanksgiving dinner and a nice day for everyone without appreciation or fun.\nWell that's not okay. Let's rethink this. Ask yourself the question \"What would make this fun?\" I'll go out on a limb and say what I think might be fun. If your family were involved in the planning together, and all four of you took responsibility as a team to make it a fun day and a delicious meal, that would be entirely different.\nThat is an important question and it tells us something; it tells us that you don't feel empowered to lead in your family. I'm not hearing that your kids are doing badly, just that they don't act like they belong to Team Family. That your job is to serve and support and as long as things are under control, don't ask for anything from others.\nI might be overstating things, but I'm pretty sure that I'm somewhat on target. So, here's your opportunity to switch from a model of support and serve, to empowerment and leadership. That doesn't mean that you no longer support, or that you no longer do nice things for your husband and kids, just that you make your needs and expectations known.\nJanet, while this discussion is starting with your needs, it's about more than that. It's about your kids contributing to their family and learning and knowing that they have an obligation to others, that who they are and what they give is important and that who they are and what they give is needed.\nYou asking for help isn't just you getting your needs met. It's also about your family being a healthy family, a healthy team, and a healthy social ecology. If we think of our family as a social ecology, similar to a biological ecology, like a pond, we can ask ourselves if our family \"pond\" is healthy or is it toxic and not vibrant with lots of oxygen and nutrients. Janet, you're not saying your family pond is toxic, so that's good, but it's not as healthy as it could be and maybe needs to be. Here's how we can infuse some oxygen and make a healthier pond. We'll do that by using Thanksgiving as a starting point.\nTalk with your husband and let him know your concerns. Tell him you're going to use Thanksgiving as a way to bring the family together and have some fun, and that you want his all-in support. Now, if he acts like he isn't going to give you his total support, ask him where he's planning to take the family to dinner on Thanksgiving. My guess is you won't have to go there. Then, you and he call a meeting of the family to talk about Thanksgiving. Make it a planning meeting. What are the kids thinking about? What were their best Thanksgiving memories, what would make it fun? What are their favorite Thanksgiving foods? If you get resistance let them know that you want and need family support that you don't want to work all day to make a meal that no one is into.\nDiscuss decorations and flow of the day. The kids may want some time to connect with friends but let's make sure that being with the family is the default position, not being with friends. Do some menu planning, and let everyone participate, both in the planning and in the preparation. What can be done in advance, what has to be done on Thanksgiving? What about cleaning the house, setting the table, cleaning up afterwards? What about giving Grandma some attention, what are some things she might enjoy, what are some things the kids can share with her? How about the teenagers helping their cousin out with her toddler and playing with him? How will your husband and your brother be engaged? Yes, there will be an NFL and NBA game on; are they really thinking of watching 6 hours of sports? These are all things that can be talked about and planned.\nWhat would make the day special for you? Maybe you'd like for the whole group to take a walk around the duck pond and feed the ducks like you did when the kids were young.\nIt doesn't matter what the particulars are, just that we do a little planning and get everyone involved in making it a successful day.\nNow, let's use this as an opportunity to shift out of your old pattern of supporting and serving and become a happier, more empowered parent and partner.\nWhat are some ways that we can get better engagement from your husband and your kids? How about your son and your daughter work together and make dinner one night a week? How about family \"work in the yard\" time, or family garage cleaning time? What about some volunteer work in the community? It would be great if you as a family decided on the organization you are going to support. It's important for our teenagers to know they are needed and that they make a difference, both within and outside of the family.\nJanet, I love to think of you and your family getting out of your rut, and becoming Team Family with an empowered leading Mom raising empowered engaged teenagers.\nHere is my challenge to all of us: Let's ask ourselves the question, are we asking enough of our teenagers? I say our teenagers because as a society our teenagers belong to all of us. When we don't let them know we need them, and we don't require their engagement, we're letting them down. Ask a teenager for help. Ask a neighbor, a niece or nephew, your teenager's friend, for a hand. You'll be doing them a favor.\nThanks for tuning in today, everyone, and special thanks to you, Janet, for your most timely question. If you like Healthy Family Connections, be sure and leave us a review on iTunes!\nThe holiday season is right around the corner. Do you know someone who might benefit from reading my book, Ending the Parent-Teen Control Battle? It's readily available, inexpensive, and could make a big difference in their life.\nFor more advice from Neil, submit your questions to www.neildbrown.com, where you can also find helpful articles, links, and resources to get your family on the right path today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Maryland Attorney General Brian E. Frosh is admonishing baby business owners of an admission in scammers impersonating account companies and others, claiming that the businesses owe money and aggressive to cut off account if they don't accomplish actual acquittal in the anatomy of a Green Dot card.\nA Green Dot agenda is one of abounding prepaid acclaim agenda companies that allows users to bulk a specific bulk of money and accelerate it to addition else, authoritative it a accepted best amid scammers. The betray works by alarming and aggressive baby business owners, about with a buzz call, into assertive that one of their utilities is about to be cut off, or some of their broad articles accept a antithesis that charge be paid afore delivery, or a agnate affirmation advised to account alarm.\nThe bluff will acquaint their ambition to acquirement a Green Dot agenda and to bulk the bulk of the outstanding acquittal assimilate the card. The bluff will again ask for the agenda advice to admission the funds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lawmakers from both sides requested the Speaker to appoint a Parliament Select Committee to investigate the controversial disclosure of Industry and Commerce Deputy Minister Buddhika Pathirana on milk powder imports.\nMaking a personal clarification and rejecting allegations that he was misleading the House with false information, the Deputy Minister stated that he has not disturbed the Parliament at any point and has only raised matters of national importance in the past.\nAccording to Pathirana, he has full backing from his Ministry to act on the complaints received regarding the contaminated New Zealand milk powders. \"The Minister of Industry and Commerce assigned me to take up the contaminated milk powder issue in a communique channelled through the Secretary of the Ministry. What I am engaged in is not a media campaign to attract attention. I did not bring in Point of Orders to disturb the House. I have not used double meanings and hints in Parliament,\" he said.\n\"I stand by my statements based on the complaints received by the Consumer Affairs Authority and my office. So, appoint a select committee to investigate the complaints and take actions to conduct tests. I have not named any milk powder company even though some other lawmakers did in Parliament. However, Fonterra company came forward with a statement,\" he said, standing up to protect his recent reveal in the Parliament where he pulled up the contaminated milk imports from New Zealand manufactured using questionable ingredients, including fat extracts from pigs and palm oil.\nTabling a newspaper advertisement jointly published by the World Health Organisation and the Ministry of Health naming 21 illnesses caused by drinking milk powder, the Deputy Minister said: \"Melamine and DCD issues came up some time ago. The ITI conducted tests and Minister Champika Ranawaka strongly accepted the lab reports, but the Ministry of Health neglected the lab findings and changed the story, dragging a bacteria story into the mix. When Director General of Health Services Dr. Palitha Mahipala suspended milk powder imports, diplomats came from New Zealand to hold a discussion with our Government.\nDeputy Minister Buddhika Pathirana on Tuesday made a controversial statement in response to a supplementary question raised in Parliament. According to him, instead of 100% natural milk, the respective imports from New Zealand are suspected to also contain animal fat extracted from pigs and cattle as well as palm oil.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a member of an LCMS congregation. But I addressed this letter to our brothers and sisters as a whole. We are all united by the Augsburg Confession, Small and Large Catechisms, and (for the majority of us) the complete Book of Concord. With this shared doctrinal background, I feel my concerns can and should be directed to all.\nFor various reason we Lutherans do not engage or associate much with other Christian denominations. I am not here to judge if any or all of the reasons are justified, but more so to air a concern that I have when we detach ourselves from the larger church. We all agree that our view of the Gospel\u2014as explained in our confession\u2014is the most accurate of the denominations and various groups of Christendom. No matter our feelings about our self-imposed isolation from many post reformation groups, there is a time where we must engage these groups. We must engage to show our historical approach of law and gospel accurately and have it be heard by Christendom as a whole. Keeping our accurate profession a secret, does no good for anyone involved or the universal church. I'd make a case that it is a part of our vocation to share God's truth when miscellaneous events happen in the universal church. I say all of this to petition that we not remain quiet regarding the 'Social Justice' conversation that is currently taking place in the broader Christian community.\nThe majority of evangelical Protestants do not work with a correct framework including Law\/Gospel distinction, Horizontal vs Vertical righteousness, or 3rd use of the Law. The conversation taking place about social justice would be a great opportunity to share our biblical approach to the masses and demonstrate the validity of our doctrine. One popular evangelical theologian has recently stated that \"this recent (and surprisingly sudden) detour in quest of 'social justice' is, I believe, the most subtle and dangerous threat so far.\" Instead of this position, Lutheranism offers the distinction between vertical righteousness (God to person due to being found in Christ) and horizontal righteousness (Person to person righteousness founded in our ability to love neighbor freely without concluding it is impacting justification). We can help demonstrate that it is possible to engage culture to attempt to end a form of oppression or hardship of neighbor and not falsely conclude that our relation to God is impacted by these acts of good works.\nAs confessional Lutherans, we can show a correct dynamic between faith and works. We can demonstrate how the 3rd use of the law is applicable as God intends for use to live holy lives in the world. Even though we fail, are aim as the new washed and regenerate man is to drown the old Adam daily and strive to live per Gods commands.\nJesus stated that He did not come to abolish the Law but to fulfill the Law, and the law to love our neighbor is as real and applicable as ever. To conclude as the celebrity pastor John MacArthur has, that social justice is not the concern of the church is effectively a form of Gnosticism and denial of the command to love neighbor.\nThere is a striking difference between Professor Biermann's focus on God's command to nurture and shepherd creation (1st great commission) and John MacArthur's argument that social equality will never happen in this fallen world and concluding it is not the church's concern. To ignore the hardship and plight in the world because the world is fallen is not a biblical concept. We are actually commanded to the exact opposite. Refusal to love neighbor in our daily vocation is Gospel reductionism and against our creation mandate. I truly feel the world would benefit by Lutherans getting involved and sharing our doctrinal truths with Christendom.\nFurther, the majority of Protestantism has corrupted the understanding of the sacrament of Baptism. Thanks be to God that the sacrament is valid based His Word and Promises\u2014not the understanding of man\u2014but many Protestant have tried their hardest to remove God from the equation. To be able to speak to the objective truth in the waters of Baptism, God Himself unites us with Him making us brothers and sisters in Christ. Not hypothetical. Not kinda sorta. He truly washes us and applies Christ's finished work to us and unites us.\nA major theme found throughout the 1 Corinthians book is unity and not showing partiality. We truly are united to Christ in baptism and have become a part of the body of Christ. How can one part of the body ignore the suffering and plight of another part of the body. Or to be more accurate, how one member of the body of Christ declare that the concerns of another member in the body is not the concern of the church.\nThis unity found and accomplished in baptism should draw us together to love and care about the trials and hardships that our brothers\/sisters experience and face. This is a beautiful gospel truth. Is this not a perspective that can be added to the racial reconciliation conversation that is greatly over looked? Protestants talk about baptism but since God's activity is removed, it is hard to articulate the reality of what took place by them.\nThe influence of the anti-social justice people goes far beyond just evangelical Protestant circles. Even in our Lutheran bible studies, I do not think it is rare to find people with a John MacArthur study bible being used. I would go as far as to bet that many in our pews can recognize and are more influenced first or second hand by John MacArthur than Greg Seltz. Many will hear the name Greg Seltzs and ask \"Who?\" That is exactly my point. Greg Seltz was a prominent speaker on the Lutheran Hour (Lutheran flagship radio ministry with estimated over a million listeners) from 2011 to 2017. One of our biggest celebrities in our tradition is less known\/impactful than MacArthur and other prominent anti-social justice supporters. John MacArthur's reach and influence is truly massive. I as a black man cannot avoid or ignore a movement in broad Christendom driven by MacArthur that declares that Social Justice is against the Gospel. A pastor named Josh Buice was involved in the creation of the, \"Social Justice and the Gospel\" statement went as far to state one reason they created the statement was \"That civil rights movements end was that there is no end game.\" He followed this by saying the people in the 1950s and 1960s\u2014like the faithful Lutheran Reverend Bob Greatz and other stand up men\u2014that risked it all did it for fame and because racial division was big business. I have three baptized saints that God has placed in my care who will live in this world when I am gone, ignoring this is not a reality for me.\nWithout social justice efforts Jim Crow may have never ended. Without social justice from a few faithful pastors, the truth that Africans in slavery were made in God's image deserving dignity would not have been proclaimed. Without Social Justice efforts that amazing testimony and precedent set by the first Rosa would never have happened in Alabama. I know there have been Lutheran statements composed for Lutherans by Lutherans on the subject of racism and racial relations, but broader Christendom is releasing public statement. In contrast, our statements seem like backroom conversations not written for the world. This is not satisfactory. I know within our doors we can point to these documents, but I live in a world where most have no idea about anything Lutheran. We have removed ourselves from the dialogue between Christians on major issues that impact and influence God's creation.\nI know none of the theology I mention in this letter is new. But I needed to lay out my thoughts. I also know I could be barking up the wrong tree in writing this. The outreach of MacArthur and his friends are far reaching. In our small sector it is easy to just ignore him. But in the broader American church, his voice has impact. There is a chance you agree with John MacArthur and the anti-social justice crowd, but I am willing to take this chance. There is more good to be accomplished than negative from my point of view. I have seen some Lutheran blogs leaning in agreement with MacArthur's and the published statement. Alt-right influence is increasing on various Lutheran platforms. We even have Lutheran discernment ministries quoting individuals who says \"Thank God and white people for slavery\" while dismissing social justice and racial reform. They do this attempting to join forces with the MacArthur brigade. These are not the Lutheran interactions we want to be known for. In general, the confessional Lutheran doctrines\u2014of which we should be proud\u2014are missing from dialogue.\nI have been placed in this situation with nothing to lose. Denying the outworking of the gospel shown by actually loving and caring for your neighbor is a gnostic notion. I cannot be convinced otherwise. And it is an actual position being solidified in opposition to Social Justice. I pray that this is received with an open heart and that the benefits of actually stepping in the fray to respond to John MacArthur and this public statement will be considered. The opportunity to share our sound confessional principles with the American church, comfort those in our pews, and reject unbiblical notions which directly support social oppression cannot go unanswered. I pray that once again Lutherans will be willing to stand up for the Biblical principles they confess.\nThe proceeding letter was sent originally to multiple leader within my particular Lutheran denomination. The results of my request were lack luster at best. I do not have much impact in Lutheranism but many of the people I reached out to do and I was met with various responses. Some were defensive while in agreement, others were in agreement but subtly pushed responsibility back on me (a lay person) to do something, and others were strictly\u2014\"you should go talk to this other guy\"\u2014passing the buck.\nI should say that I didn't expect much even before I wrote it, but I had to try. From my perspective, most of the responses fell into a \"not my problem, leave me out of it.\" There are people considered to be some of the most influential evangelicals of the 20th century leading a charge and our leaders expect a lay person to effectually take a stand. That's laughable. The notion of \"go talk to this other guy\" is a joke as well, because chances are this \"celeb guy\" would be more likely to read it coming from them, than a lay person. Essentially they stated they agreed, but did not want to get their hands dirty.\nAnyhow, I am still thankful many read it and verbally\u2014if not with action\u2013stated they agreed. Needless to say, no major response from Lutheranism has been made and no true entry into the conversation from our theologians with a broad impact has been attempted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Duet offers 39 fully furnished single and double rooms, business rooms, and family rooms.\nCozy, air-conditioned room with King bed (dor 1 or 2 people) with the view on picturesque houses and church.\nLuxury and comfortable room (35 m2 ) with large bedroom and smaller living room (with extra Bed) . Perfect for families accommodation - parents and children, and for romantic offer.\nLarge, luxury room with total area 35 m2, with comfortable corner settee, bathroom with a tub, and 42\" LCD TV, excellent for families and business leisures as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As your Treatment Coordinator, Cindy is responsible for making sure your initial visit is a positive and educational experience. She looks forward to answering your questions about your treatment and ensuring your entry into our orthodontic family is smooth and comfortable.\nCindy joined the Hong Orthodontics team in March of 2015 and began working in the orthodontic field in 2006. Before that, she was in the adoption field.\nDesert and fishing trips with her boyfriend, son and daughter, ATV riding and off-roading, mountain trips and international travel are some of her favorite hobbies. She also has the cutest 5 year old niece!\nCindy's commitment to her patient's entire experience is evident from start to finish. She explains the treatment plan that Dr. Hong has designed for you, goes over your insurance coverage(s) as well as the easy payment options. She loves being a part of your beautiful smile!\nIvanna has been in the dental field for seven years. She loves building a relationship with each patient and their families. Having experience in the back and front office and being fluent in Spanish is a great asset to the office. In her free time she enjoys spending time with her family, hanging out at the beach with her 3 dogs, going to concert and sporting events\u2026Go Angels & Ducks!!\nBefore joining Dr. Hong's office as a financial coordinator in 2008, Joon worked in the pharmacy field for several years in Illinois. She helps patients with appointments and verifying their insurance benefits and patient's account. She enjoys working with friendly and knowledgeable staff members like Dr. Hong and all his assistants. She especially excited to see satisfied patients with their big, confident smiles. Joon and her husband, Joe, have three boys: Andrew, Mark, Daniel and Jojo, a 10 year old terrier at home. She enjoys baking, watching old classic movies, hiking and traveling in her free time and loves warm California weather.\nJane has been working in the dental industry as an assistant for over twenty years. She's currently been working with Dr Hong for over four years. Jane is married and has a 12 year old daughter. She owns many pets such as a chihuahua, a snapping turtle, and a lot of fish.During her free time, Jane spends time with her family and with her good friends. She likes to travel and her goal is to visit a state each year.\nErica has been working in the dental field for over 17 years. One of the things she enjoys most about dentistry is seeing how changing a patient's smile transforms their lives. She's been married for 15 years and has 2 wonderful daughters that love playing softball and a dog named chacha that enrich her life every day. In her spare time she enjoys going to the park to watch her husband and girls play softball.\nSharon is a back office assistant for Dr. Hong. She is responsible for giving patients a great experience throughout their treatment time in our office. Although she has been in the orthodontic field, Sharon has just recently joined Dr. Hong's team. She enjoys interacting with patients every step of the way of their treatment. In her free time, she enjoys adventuring her new surroundings in CA as well as spending quality time with her precious cat Kiko. She is a Arizona native and a huge sports fanatic (especially football)!\nJenny joined Hong's Orthodontics in 2010 as an Orthodontics Assistant. With 5 years of orthodontics assistant experience, she wants to expand her passion for dentistry by becoming a Registered Dental Hygienist. One of the things she enjoys most about dentistry is to see how changing a patient's smile transforms their lives. In her free time, she enjoys finding new restaurants and new Boba milk tea places.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The organic food movement suffered a major setback recently, when the US National Organic Standards Board (NOSB) voted in favor of allowing hydroponically-grown products to receive the \"organic\" label. This decision should not have come as a surprise to those who have watched the organic movement steadily taken over by big agribusiness \u2013 a process that began in 1990 when Congress required the USDA to create a single set of national standards that would define the meaning of \"organic\".\nThere are two important things that I have learned during my five years on the National Organic Standards Board (NOSB). First, I learned that the NOSB review process for materials petitioned for inclusion on the National List is quite rigorous, with Technical Reviews of petitioned materials and careful scrutiny by both NOSB subcommittees and the full board.\nThe second thing I learned, over time, is that industry has an outsized and growing influence on USDA \u2013 and on the NOSB (including through NOSB appointments) \u2013 compared to the influence of organic farmers, who started this organic farming movement. Perhaps that is not surprising, given the growing value of organic sales. With organic becoming a $50 billion business, industry not only wants a bigger piece of the pie, they seem to want the whole pie.\nWe now have \"organic\" chicken CAFOs [concentrated animal feeding operations] with 200,000 birds crammed into a building with no real access to the outdoors, and a chicken industry working behind the scenes to make sure that the animal welfare standards \u2013 weak as they are \u2013 never see the light of day, just like their chickens. The image consumers have of organic chickens ranging outside has been relegated to pictures on egg cartons.\nWe have \"organic\" dairy CAFOs with 15,000 cows in a feedlot in a desert, with compelling evidence by an investigative reporter that the CAFO is not meeting the grazing rule \u2013 not by a long shot. But when USDA does its obligatory \"investigation,\" instead of a surprise visit to the facility, USDA gives them a heads up by making an appointment, so the CAFO can move cows from feedlots to pasture on the day of inspection. This gives a green light to that dairy CAFO owner to move forward with its plans to establish a 30,000-cow facility in the Midwest.\nWe have large grain shipments coming into the US that are being sold as organic but that lack organic documentation. Some shipments have been proven to be fraudulent. The USDA has been slow to take action to stop this, and organic grain farmers in the US are suffering financially as a result.\nWe have a rapidly growing percentage of the fruits and vegetables on grocery store shelves being produced hydroponically, without soil, and mostly in huge industrial-scale facilities. And we have a hydroponics industry that has deceptively renamed hydroponic production \u2013 even with 100% liquid feeding \u2013 as \"container\" production. With their clever deception they have been able to bamboozle even the majority of NOSB members into complicity with their goal of taking over the organic fruit and vegetable market with their hydroponic products.\nPerhaps we shouldn't be surprised to find that big business is taking over the USDA organic program, because the influence of money is corroding all levels of our government. At this point, I can see only one way to bring the organic label back in line with the original vision of organic farmers and consumers. We need an add-on organic label for organic farmers who are willing to meet the expectations of discerning consumers who are demanding real organic food.\nA while back I wouldn't have supported the idea of an add-on organic label because I, like many others, saw the USDA organic label as the gold standard, and hoped that through our vision of the process of continuous improvement we could really make it into that gold standard. Now I can see that the influence of big business is not going to let that happen. The USDA is increasingly exerting control over the NOSB, and big business is tightening its grip on the USDA and Congress. Recently industry representatives have publicly called on the US Senate to weaken the NOSB and give industry a stronger role in the National Organic Program. And sympathetic Senators have promised to do just that.\nI also am pleased that organic farmers have recently organized themselves into the Organic Farmers Association (OFA), to better represent themselves in the arena of public policy. Too often in the past the interests of big business have overruled the interests of organic farmers \u2013 and consumers \u2013 when organic policies are being established in Washington. I hope this will allow organic farmers to gain equal footing with industry on issues that affect the organic community.\nIn summary, organic is at a crossroads. Either we can continue to allow industry interests to bend and dilute the organic rules to their benefit, or organic farmers \u2013 working with organic consumers \u2013 can step up and take action to ensure organic integrity into the future.\nFrancis Thicke is a pioneering organic farmer who \u2013 with his wife Susan \u2013 operates a 730-acre farm in Fairfield, Iowa. He has served as a National Program Leader for soil science for the USDA-Extension Service, as a board member of the Organic Farming Research Foundation, and as a board member of the Cornucopia Institute. His 5-year term with the National Organics Standards Board just ended.\nSteven Gorelick is Managing Programs Director at Local Futures. He is the author of 'Small is Beautiful, Big is Subsidized', co-author of 'Bringing the Food Economy Home', and co-director of the film 'The Economics of Happiness'. His writings have been published in The Ecologist and Resurgence magazines. He frequently teaches and speaks on local economics around the US.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jesus. What's going to be left after the dust settles? Needless to say you'd have to be living under a rock to not know that there is a video streaming war going on right now, and the guns are a'blazing as we speak with the King of Online Media entertainment, Netflix, at the top of a very big media mountain. But, honestly.... Netflix may not want to stand up on that.\nWe're talking about layoffs here. Cost cuts. This occurred right after Richard Plepler, ex-head of HBO, and David Levy, the now former Turner Broadcasting president officially resigned their post within the company.\nThat's right: HBO will officially go full-on with media subscription services, which as you know has been quite the trend lately in technology particularly with Apple. You see the writing on the wall? This might make the Hobbit's Battle of Four Armies pale in comparison.\nThey really shouldn't be. Yet. After all, the streaming giant commands a massive 150MM subscriber base and a heavy hand with Hollywood television and film as noticed by the power it wields at the Oscars. But don't count out some other heavy hitters -- like Disney.\nThat mega-monster of media, of course, shelled out $71B for Twenty-First Century Fox, which means Hulu -- Netflix's \"former\" rival -- now has some beefed-up muscle to go with Disney's own streaming service that will feature some rockin' original programming to go head-to-head with Netflix. Have you seen stranger things than that? Of course not. This was a long time coming.\nThink about it: we've got franchises like Hanna-Barbera and Looney Tunes with this company. With consolidations of tons of advertising and affiliate sales, WarnerMedia now buttressed by the clout of AT&T may not only give a run for Netflix's money, but Disney's money, and Apple's money, and, well, your money.\nThere can be only one. Unless you're a media freak, and you don't mind having multiple accounts. What do you think about this? SIGN UP FOR A VIGYAA ACCOUNT AND START WRITING OUT YOUR OPINIONS ON THE UPCOMING STREAMING WAR.\nNetflix Netflix may not want to stand up on that Netflix may not want to stand up on that the trend lately in technology particularly with Apple SIGN UP FOR A VIGYAA ACCOUNT AND START WRITING OUT YOUR OPINIONS ON THE UPCOMING STREAMING WAR.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drumphilia volume 1 is a rhythmic experiment that sits on the fault line between traditional instrumentation and analogue electronics. The project is a response to many years spent working with, learning about and listening to African and Caribbean percussion. Traditional rhythmic influences are combined with analogue drum machines and drum synths to create a hybrid sound that continues in the tradition of artists like Frances Bebey. There are no traditional harmonic instruments on the recordings \u2013 the aim was to create melody and texture using only carefully tuned percussion instruments, whether organic or electronic. Basslines were created by blending drum synth tones with the acoustic marimbula, a traditional Caribbean instrument that was often used in place of a bass guitar. Melodies were created with everything from log drums and thumb pianos to early Soviet drum modules and the trailblazing Pearl Syncussion SY-1. Various time signatures and textures are used across the album to produce melody out of rhythm \u2013 hypnotic, evolving tracks that celebrate and highlight the importance of rhythm in modern music. All the sessions were recorded at the Mop Mop studio in Berlin between September and December, 2016. Instruments featured include: Log drum, marimbula, kalimba, sansula, krabebs, Pearl Syncussion SY-1, Vermona DRM1, MFB Tanzb\u00e4r, MFB Tanzmaus, RMIF Elsita, assorted drum machines and talkboxes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The figure of Constantine the Great is at least as cardinal in history as that of Alexander the Great or Augustus Caesar. We know very little of his personality or of his private life; no Plutarch, no Suetonius, has preserved any intimate and living details about him. Abuse we have of him from his enemies, and much obviously fulsome panegyric to set against it; but none of these writers give us a living character of him; he is a party symbol for them, a partisan flag. It is stated by the hostile Zosimus that, like Sargon I, he was of illegitimate birth; his father was a distinguished general and his mother, Helena, an innkeeper's daughter of Nish in Serbia. Gibbon, however, is of opinion that there was a valid marriage. In any case it was a lowly marriage, and the personal genius of Constantine prevailed against serious disadvantages. He was comparatively illiterate he knew little or no Greek. It appears to be true that he banished his eldest son Crispus, and caused him to be executed at the instigation of the young man's stepmother, Fausta; and it is also recorded that he was afterwards convinced of the innocence of Crispus, and caused Fausta to be executed\u2013according to one account by being boiled to death in her bath, and according to another by being exposed naked to wild beasts on a desolate mountain\u2013while there is also very satisfactory documentary evidence that she survived him. If she was executed, the fact remains that her three sons, together with two nephews, became the appointed heirs of Constantine. Clearly there is nothing solid to be got from this libellous tangle, and such souffl\u00e9 as is possible with these scanty materials is to be found admirably done by Gibbon (chap. xviii.). Gibbon, because of his anti-Christian animus, is hostile to Constantine; but he admits that he was temperate and chaste. He accuses him of prodigality because of his great public buildings, and of being vain and dissolute (!) because in his old age he wore a wig\u2013Gibbon wore his own hair tied with a becoming black bow\u2013and a diadem and magnificent robes. But all the later emperors after Diocletian wore diadems and magnificent robes.\nYet if the personality of Constantine the Great remains phantom-like, if the particulars of his domestic life reveal nothing but a vague tragedy, we can still guess at much that was in his mind. It must, in the closing years of his life, have been a very lonely mind. He was more of an autocrat than any previous emperor had been\u2013that is to say, he had less counsel and help. No class of public-spirited and trustworthy men remained; no senate nor council shared and developed his schemes. How much he apprehended the geographical weakness of the empire, how far he saw the complete disaster that was now so near, we can only guess. He made his real capital at Nicomedia in Bithynia; Constantinople across the Bosphorus was still being built when he died. Like Diocletian, he seems to have realized the broken-backed outline of his dominions, and to have concentrated his attention on foreign affairs and more particularly on the affairs of Hungary, South Russia, and the Black Sea. He reorganized all the official machinery of the empire; he gave it a new constitution and sought to establish a dynasty. He was a restless remaker of things; the social confusion he tried to fix by assisting in the development of a caste system. This was following up the work of his great predecessor, Diocletian. He tried to make a caste of the peasants and small cultivators, and to restrict them from moving from their holdings. In fact he sought to make them serfs. The supply of slave labour had fallen off because the empire was no longer an invading but an invaded power; he turned to serfdom as the remedy. His creative efforts necessitated unprecedentedly heavy taxation. All these things point to a lonely and forcible mind. It is in his manifest understanding of the need of some unifying moral force if the empire was to hold together that his claim to originality lies.\nIt was only after he had turned to Christianity that he seems to have realized the fierce dissensions of the theologians. He made a great effort to reconcile these differences in order to have one uniform and harmonious teaching in the community, and at his initiative a general council of the Church was held at Nicaea, a town near Nicomedia and over against Constantinople, in 325. Eusebius gives a curious account of this strange gathering, over which the Emperor, although he was not, yet a baptized Christian, presided. It was not his first council of the Church, for he had already (in 313) presided over a council at Arles. He sat in the midst of the council of Nicaea upon a golden throne, and as he had little Greek, we must suppose he was reduced to watching the countenances and gestures of the debaters, and listening to their intonations. The council was a stormy one. When old Arius rose to speak, one Nicholas of Myra struck him in the face, and afterwards many ran out, thrusting their fingers into their ears in affected horror at the old man's heresies. One is tempted to imagine the great Emperor, deeply anxious for the soul of his empire, firmly resolved to end these divisions, bending towards his interpreters to ask them the meaning of the uproar.\nThe views that prevailed at Nicaea are embodied in the Nicene Creed, a strictly Trinitarian statement and the Emperor sustained the Trinitan position. But afterwards, when Athanasius sins bore too hardly upon the Arians, he had him banished from Alexandria; and when the church at Alexandria would have excommunicated Arius, he obliged it to readmit him to communion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I have an idea of doing a series of themed 5-6 song e.p.'s on the uke. Some of the themes planned so far includes country,grunge, britpop, songs about \"Allison\" and tributes to Daniel Johnston, Wesley Willis and \"Violator\" by Depche Mode. The problem is I'm getting stuck on the ska\/reggae album...I just can't make up my mind on what to do...there is so much great material to choose from. Bob Marley, Desmond Dekker, Toots & the Maytalls, Bosstones, Specials, Madness, ect. So I'm throwing it out there to see what my fellow board members would like to hear. Give me some suggestions.\nMadness.....One Step Beyond!! Is that even possible? Yeh, I agree, Marley songs....as epic as they are...they are everywhere. Toots would be nice. How about some Ziggy?\nWhere are the other two strings?\nThere is a lot of great 2 Tone stuff out there.\nI don't have any suggestions, just looking forward to hearing it when it gets put out there.\nThere is also a lot of good recent releases in both the ska and reggae world that would translate very well to the uke.\nI'm a big ska and reggae fan and listen to all the \"waves\" that have come and gone....from the Skatalites and Wailers to the new Toasters and Reel Big Fish...and everything in between. It's just so hard to narrow it down to just a few songs without leaving something\/someone out. I may just have to pull the songs out of a hat or load the cd changer, press random and pick the first 6 or so songs that play.\nhow about some slow jams?\nAll times are GMT -10. The time now is 09:55 AM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A few weekends ago, I found myself helping out at a workshop at the Remick Farm Museum. I was mostly in charge of the camera for the event, which was a lot of fun. I haven't really had the opportunity to be an 'event photographer' well, ever.\nI guess my photography prowess showed and Erica (the event coordinator) put me in touch with a woman who is writing a book and needed someone to take the cover photograph. The book is set during the time of the Revolutionary War and the photograph was going to be taken at the Remick in front of the hearth.\nLast Friday Patrick + I headed over to the Remick to have a photo shoot. It was a lot of fun - I haven't had a 'model' since my college photography courses - but I think we were able to get quite a few good shots.... We'll see once the book comes out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For crewmembers its hard to get vacation time due to not having hours.\nThe worst would be the inability to take your scheduled vacation time when needed. The overall benefit package is excellent.\n401k, vacation, health insurance (dentist, doctor) also insurance for your items Incase the restaurant was to burn down from an accident. Also if you work well enough, you can get bumped up into work for corporate.\nBefore or after your shift you're able to get a free meal so that you never work while you're hungry.\nDifferent insurance plans are afford at a low affordable rate. You also receive holiday goodies throughout the year.\nIf you are a full time employee, a manager or a member of the support office, the benefits package is quite good. There is 401k, health insurance, and more.\nThe benefits package is one of the pro that I did not mention. I did not mention it because the benefits are only for full-time employees and I could get more than 20-25 hours no matter how hard I worked. I had a 12 hour work week with no reason issues for it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Conversation Starters\" was an idea I've had for awhile; to compile a list of vocabulary unique to common chordal progressions in jazz. From a young age, jazz students hear from teachers that jazz is \"like a conversation\". Part of that conversation is heavily reliant on having the vocabulary necessary to have a dialogue with ones bandmates.\nThis book contains 180 different pieces of jazz vocabulary to get the conversation started. The first 30 lines are written over a ii-V7 of regular length and as a \"quick ii-v7\" and their composition is decidedly linear, with exception to the last five lines in the \"quick ii-V7\" lines, which begin to take inspiration from larger intervals.\nStarting with the \"half-step ii-V7\" lines, the first five lines for each progression are written in a linear, bebop style and are in more standard ranges. The second five lines for each progression make use of larger intervals. Offering contrasting lines is a way to diversify ones vocabulary and thus it is the hope of the author that the student identifies ways in which the harmonic content of the lines remains similar, but the message is delivered in starkly different ways.\nAlthough all of the lines in this book could be learned in all 12 keys, the student will find that the construction of the first set of lines for each progression lend themselves to transposition in all twelve keys with greater ease than those constructed with larger intervals.\nEvery effort has been made to spell the pitches with enharmonic accuracy. However, there are certain scenarios in which the author has taken liberty to maximize ease of playability.\nIt is the hope of author that this book will provide many practical and fresh ways to play over crucial chord progressions that occur in hundreds of standards.\nPRE-ORDER now and receive a free pdf sampler of etudes from Leaps and Sounds Volume I and II. PDF orders will be sent out on or before May 1st to the email address provided at point of purchase.\nEnjoy the book and expand your vocabulary!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every transmission line, every cellphone tower, every line for electricity, phone and cable can just stop at their state borders \u2013 that would work. They can be their own nation unto themselves. And when food grown in Florida or California is bound for market, they can just take it somewhere else because Texas isn't in the United States anymore. And, the food grown in Texas and anything else created there, can just stay in Texas for a market to sell any of it.\nAirlines and airports can be closed to anything but local traffic using their own controllers, inspectors, security measures and their own durn planes that no longer fly anywhere except inside the territorial space of Texas. Ships can dock elsewhere and the petroleum industry can find somewhere else to call home, as can the space programs and the aeronautics industries' manufacturers and anyone else that wants to do business in the United States of America.\nTruckers can just go around that state and deliver elsewhere. Roads going in or out can be made impassable and everyone in Texas can simply take care of matters as it suits themselves. They don't need US dollars, so their banking system and businesses can use newly identified and issued money of their own. And, when their checks need to be processed, they won't go to New Jersey or Ohio or wherever else to be processed, they can just do that themselves, too. They can issue their own credit cards within the state and their own stocks and bonds, while their at it.\nNow, honestly, I agree that the taxation in the United States is taking more than it ever should have and the job being done with what probably amounts to a little over 50% of everyone's income could be done better. But, states do not exist to the exclusion of the United States of America and their rights are intended to be in balance with the greater good of our entire nation and of all America's people.\nSometimes, the most effective way to do that is for the interests of the States to be given greater consideration, but those situations are, in fact rare, not common. That is why we have support in our Constitution for a Federal government, for a Congress, for the Presidency and for the branches of government within the Federal government. That is also why we work together to serve our national good and the interests of our nation, not just the states where we may live right this moment or where we may have been born.\nBut, that said \u2013 I think we ought to let Texas cede from the nation. It would solve at least some of the problems we have that are a direct result of including them. And, they can take those things on for themselves. When devastated by something, we don't have to help. Where it comes to basic necessities and livelihoods and shipping and trucking and utilities and upkeep of roads and interstate freeways, choices about the rights of people and rules for the safety and well-being of people subjected to harms that can be done by corporations, and appropriate applications of our Constitutional guarantees and pollution and banking and whatever else, let Texas handle it separated from the United States of America. It would serve them right.\nIf Texas and conservative Republican party activists refuse to participate in solutions and in creating solutions that work \u2013 then this country is genuinely better off without them. And, I hate to say that because they could be such a contributing partner in creating and applying those solutions and in creating possibilities to rebuild our nation and to restore our economy to prosperity. But, if their only contribution is to be oppositional and self-serving, what is the point?\nPeople better wake up and realize that they are first and foremost \u2013 citizens of the United States of America and citizens of a much larger world, as well. And, with that citizenship comes a responsibility to participate in ways that include voicing concerns but not to the exclusion of every other considered action that is appropriate.\nWe are the United States of America. We are not Texans, or Georgians, or Southerners, or Yankees, or Californians or Republicans or Democrats or conservatives or independents or liberals or whatever else first. We are Americans first as citizens and citizens of the World before all else beyond that. Then, we are these other things as a matter of choice. But, it is a matter of choice \u2013 not of necessity and not by any basis of reality.\nBy necessity and by reality \u2013 We are American citizens and citizens of the World.\nAnd, by necessity and by reality \u2013 We are the living embodiment of the Constitution of the United States of America and We are the tenets of our nation and of our national interests.\nWe, the people of the United States of America, are the United States of America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We will find CV Writers in Buckhurst Hill and do the legwork to contact them on your behalf.\nReceive fast, free CV Writing quotes from the best CV Writers near Buckhurst Hill.\nDissertations are difficult to write and a lot of students find it extremely difficult to complete them. Real Dissertation Help UK is here to help you out. Real Dissertation Help UK is the ultimate option for every student who is looking for a top dissertation service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rachel & Logan came to our casa so that Logan could whip up an amazing sushi dinner. He always goes all out \u2013 miso soup, edamame, snapper with cucumber and crab, salmon with avocado and cucumber, inari, chocolate mochi cake with strawberry gelato and tiny jugs of Yakult. He rocks.\nWe hung out at home with coffee and toasted, buttered baguette, and a lunch of hot dogs and mashed potatoes (we're classy, yo). Then it was Derby Day madness! Rather than my usual dress-and-large-hat combo, I opted for a 40s look, in vintage goodness. Justin was gorgeous as always. There were Dizzy Derby races, juleps, games of corn hole, bocce ball, and all kinds of preppy fun. After Derby Day, a huge group of us cabbed over to Foreign & Domestic and ordered everything on the menu. Literally.\nWe relaxed at home in the morning, and then Kate and Meghan stopped in to produce a photo shoot for a local magazine on our lawn. I got an amazing vintage canning book out of the deal. In the evening, I walked down to Winflo to meet the girls and celebrate Jodi's big win. On my walk home, I saw my friend Alex at Clark's, and so of course I met Justin there for bubbly and oysters.\nDerby Day 2013 from Crystal Esquivel on Vimeo.\nSuch a fun (but exhausting) weekend! I'm looking forward to some quiet time tonight. Just me, a kitty cat, and a book. I hope you all have a wonderful week!\nYou are adorable, your video is terrific, and you should be proud. :-) What a high-energy weekend!\nLogan's working on our weekend photos now. I'd love to see Aimee's!\nThank you! I'm excited to see your Derby post!\nWhat a lovely weekend! The Dizzy Derby had me laughing out loud, and all the food had my stomach grumbling!\nIt was too much fun. I hope your beach vacation rocked!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Listening is hard work. Just ask Peter! If, however, we can stop long enough to really listen to each other and to God, amazing things happen. Don't believe me? Well then, listen up.\nLet us go now to Bethlehem!\nStewards of Advent: Wake, Watch, and Wait!\nJust because we live in the \"meantime\" does not mean we Christians have to let the mean times of this world wear us down. Don't let the hater be hating on you.\nAll Saints Sunday reminds us of those Christians who have died in the faith, who have fought the good fight, and whose lives offer an example for us. Now look in the mirror and you'll see a saint\u2013and a sinner\u2013dearly loved by God and made holy through Jesus Christ. Saint Who? Saint YOU!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I was laying on the gwound tinking, what should I do? What should I DO? Then, I tot, I do not want my sucker anymore. Peter can have my sucker. And I gave it to him!!\"\nJust preparing for his birthday in a month.\n2. Abby (my parents' Border Collie) had puppies. Carter was so proud of her he came running to me upon hearing the news saying, \"Abby deserves a biscuit! GIVE HER A BISCUIT!\"\n\"Oh! Jesus lives in my heart. I will talk louder so he can HEAR me!\"\n4. Guardian angels are amazing. I just finished listening to a talk of the Lighthouse CD of the Month series, and part of it was on the importance of having a relationship with your guardian angel. That evening, I was praying with Carter and asking God to send their guardian angels to guide and protect them always. Then, I went to turn on Carter's Tot Clock (which has only ever played music), and it came on with a story that said, \"Hi. This is your guardian angel talking...\" Crazy!\n5. Hallie has saved my carpet :) In one of her Five Favorites a week ago, she mentioned the floor cleaner\/spot remover Folex. Well, I had my doubts. All of our boys have been insane spit-uppers. I mean, vomit style at least 2-3 times per feeding. Anyway, our carpet (which isn't two years old), looks notsogood. And, I've tried everything. Well, I thought I had...until Hallie saved me and my sanity. This stuff works! Thank you!\nHappy Birthday, sweet Cyd! This girl is about as amazing as they come in the mom department.\n7. And a photo to end...Carter and his love for ladybugs.\nHave a blessed weekend! Enjoy Mother's Day, and don't forget to read the other fun posts at Conversion Diary!\nBritt, lovely as always. I may have to try that Folex!\nYour son is a delight!\nThese are adorable quick takes! Love the cute pictures of the kids and the stories about Carter. I'm glad you're recording the things he says, because they're just priceless. Thanks for stopping by and have a Happy Mother's Day!\nonline click-and-order sites for finding bits on the cheap.\nclass is spending so much money that our national debt is approaching alarming levels, having just exceeded $13 TRILLION, you really have to wonder what economic novices are running the country today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JUL 1968 - JUN 1969 Running aground on a beach in North Vietnam nearly.\nBream (SS-243) was launched 17 October 1943 by Electric Boat Co. Groton Conn.; sponsored by Mrs. Wreford G. Chapple wife of the prospective commanding officer; and commissioned 24 January 1944 Commander Chapple in command.\nBream's war operations extend from 1 June 1944 to 15 June 1945. During this period she completed six war patrols operating in the Java Celebes Sulu and South China Seas and the Gulf of Siam. She sank two Japanese vessels totaling 6934 tons.\nIn addition Bream shared with Ray (SS-271) and Guitarro (SS-363) the destruction of a 6806-ton passenger-cargo vessel. On 23 October 1944 while patrolling off western Luzon Bream made a daring surface attack on a Japanese formation damaging the heavy cruiser Aoba.\nBream got underway from Saipan for Pearl Harbor 6 June 1945 enroute to the United States for navy yard overhaul. She arrived at San Francisco 24 June 1945 and underwent overhaul and was placed out of commission in reserve 31 January 1946.\nBream was recommissioned 5 June 1951 and reported to Submarine Squadron 3 Pacific Fleet. From June 1951 to August 1952 Bream engaged in type training and services to the Fleet Sonar School San Diego. On 10 September 1952 she went out of commission in reserve at San Francisco. Bream was converted to a killer submarine and reclassified SSK-243 18 February 1953.\nFollowing recommissioning 20 June 1953 Bream participated in all phases of peacetime submarine operations in the Pacific Ocean. She conducted an Alaskan training cruise in September 1954 returning to San Diego 5 November 1954 via Pearl Harbor. Bream carried out operations off California until she made another trip to Pearl Harbor during 7-24 May 1955. Her next departure from the west coast was on 6 March 1956 for a cruise in the Western Pacific which terminated at San Francisco in early 1957.\nBream was awarded four battle stars for her World War II service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I would like to invent a political commercial zapper.\nThis would be software you download to your TV, and each time a commercial comes on for a politician who does not live in your district, the TV will automatically replace it with the a clip of your choosing.\nIt could even be a blank screen for 30 seconds. That is much better than hearing the constant dribble.\nI'm sure Ralph Norman and Tommy Pope are fine men, but after Tuesday's election, if I ever see either of their faces again, it will be too soon.\nNone of us in Saluda County live in their Congressional District, but we have heard their commercials over and over and over and over, at all times of the day and night.\nSadly, they are on the same side, Republicans.\nPBS, satellite and cable are particularly popular during political campaigns.\nActually, the zapper would work well for politicians that live in your district, too. It would only allow you to hear the same commercial once a week.\nMonday night, knowing there would be constant Pope-Norman ads before the election, I started hitting the mute button. Enough is enough.\nI got the Amazon Echo Dot for Christmas.\nIt's a neat little thing that answers questions, but is can also give you a heart attack.\nWell, I didn't ask Alexa anything, so what or who did she hear?\nThis is about the third or fourth time this has happened.\nWhen you hear a voice in the corner of your bedroom late at night, it's a bit disconcerting.\nI have an app called \"Ghost Hunter,\" that shows you if ghosts are near and calls out random words.\nI don't put much credence in this, because it is basically a game, but Alexa talking when unspoken too sounds a like the \"ghost\" voice.\nThe State of South Carolina's great streak continued Sunday night when Miss District of Columbia won the Miss U.S.A. title.\nKara McCullough is a graduate of S.C. State and is a nuclear physicist!\nOf course, she did not come out unscathed. When asked if affordable healthcare was a privilege or a right, she said it was a privilege.\nShe said she had healthcare, because it was provided by her work. She said what we needed was jobs.\nShe caught flack for both of those answers.\nI've said many times, I am not a fan of President Trump, but when did we get to the point in this country that people, who are not liberal, are not entitled to an opinion?\nThat kind of thinking is the reason a hotel builder is the president of the United States.\nUniversity students have so protested conservative speakers on their campuses, that the events have been cancelled. It's called \"free speech.\" Perhaps they have not heard of that.\nI don't envy Carolina Athletic Director Ray Tanner.\nHe may be facing having to fire the first coach he hired, his replacement Chad Holbrook.\nThe Gamecock baseball team that began the season ranked in the Top Five, has lost seven straight SEC series and may not make the NCAA tournament for the second time in three years.\nAt a school that holds baseball at such high a level, that will not do.\nAfter over 50 games, this team has not shown one ounce of improvement.\nThey have had the opportunity to win all seven of the series they've lost, but they have blown it each time.\nLike I said a few weeks ago, let the Gamecocks finish the season. Maybe, they will miraculously learn how to bunt and the bullpen will quit blowing leads.\nMaybe, the baseball team will catch fire like the basketball team did, but if they can't beat one of the worst teams in the conference, Missouri, I don't have much hope.\nFollowing a legend is never easy, as Holbrook has found out following Tanner, who coached the Gamecocks to two national titles.\nI was saddened by the passing of Jim Corley last week.\nFor years, Jim has been seen around town on his cart that became his legs.\nHe was everywhere, from 7-11 to Saluda High football games.\nHe has a good personality about it. I'd ask him if that thing had air conditioning.\nHe said it went so fast, the wind took of the place of AC.\nJimmy always had a great personality. He had a comeback for everything, and you weren't going to get anything over on him.\nHis late father, James, was a longtime mayor of Saluda and another fine man.\nI'm sure going to miss seeing Jim around town.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Triple Club Carlson points for stays worldwide from January 8 to March 15, 2013 is a high value promotion in my opinion. In my review of hotel loyalty promotions of 2012 for InsideFlyer magazine, I realized the 60 points per dollar offer from Club Carlson last year was one of the best promotions that I did not earn.\nClub Carlson offers a standard earn rate of 20 points per dollar for Radisson, Radisson Blu, Park Plaza, Park Inn and Country Inn & Suites brands. Triple points adds 40 bonus points\/$1 for a total earn rate of 60 points\/$1. Elite bonus points, online booking points, and credit card points, if applicable, are additional bonus points for stays.\nClub Carlson Triple Points promotion registration.\nNote: You need to register for this promotion offer prior to your stay to earn the bonus points. This promotion bonus points offer applies to Club Carlson properties worldwide.\nThe reason I like this offer is Club Carlson already has the lowest spend to free night ratio among hotel loyalty programs.\nThe highest Club Carlson category level 6 hotels are 50,000 points for a free night. At the standard earn rate of 20 points\/$1 this means $2,500 in hotel spend earns 50,000 points for one free category 6 award night.\nHilton HHonors category 7 = 50,000 points ($3,333 in hotel spend for 50,000 points for Points & Points earner at 15 points\/$1).\nHyatt Gold Passport category 6 = 22,000 points ($4,400 in hotel spend for 22,000 points at 5 points\/$1).\nMarriott Rewards category 8 = 40,000 points ($4,000 in hotel spend for 40,000 points at 10 points\/$1; more if staying at Residence Inn and TownePlace Suites brands earning just 5 points\/$1).\nPriority Club night at InterContinental hotel for 50,000 points ($5,000 in hotel spend for 50,000 points at 10 points per dollar; more at Staybridge and Candlewood Suites brands when earning 5 points\/$1).\nStarwood Preferred Guest category 5 at 12,000 points ($6,000 in hotel spend for 12,000 points at 2 points\/$1; $4,000 if Gold\/Platinum elite earning 3 points\/$1. SPG category 6 and 7 hotels require far more points and place these properties out of the ballpark in this comparison).\nThe simplicity of Club Carlson means all their hotel brands earn 20 points\/$1 and with this promotion 60 points\/$1. This is an advantage over Marriott and Priority Club.\nThe elite bonus factor is equivalent or higher for Club Carlson compared to most other chains.\nClub Carlson Silver earns 25% bonus (5 points\/$1) compared to Priority Club at 10%, Hilton Silver and Hyatt Platinum entry level elite at 15% and Marriott Silver at 20%. SPG Gold members receive a 50% bonus, but the standard redemption rates are much higher comparatively for SPG.\nClub Carlson Gold elite earns 50% bonus (10 points\/$1). HHonors Gold and Marriott Gold are only a 25% elite bonus.\nTop level Club Carlson Concierge is 75% elite bonus (15 points\/$1). HHonors Diamond, Marriott Platinum, Priority Club Platinum and SPG Platinum are 50% bonus points. Hyatt Gold Passport Diamond is only 30% bonus.\nClub Carlson's new Visa card confers Gold elite (not an affiliate link). Club Carlson also offers a status match up to Gold elite if you have high elite with another program.\nOn top of all these factors is 1,000 points online booking bonus per stay for any member and 2,000 points for Silver and Gold elite members.\n1,000 points online booking bonus.\n4,200 points at $70 x 60 points\/$1.\nEven a new member to Club Carlson with no elite status or credit card earns a good chunk of points needed for a Club Carlson award night.\n5,200 points is 10% of the points needed for a top tier category 6 hotel that can be $200 to $400 per night.\n5,200 points is 58% of 9,000 points needed for a Category 1 hotel award night.\nPoints + Cash award discounts start at 5,000 points per night with Club Carlson.\nFrequent Miler shows how to combine cash back portals to also get a discount on the room rate while earning the bonus for your stays.\nClub Carlson's triple points promotion earns free night awards much faster with elite status and the credit card. Especially if you plan to redeem free nights as a credit card member where the last night of an award stay is free.\nTriple Points is a 5-key promotion offer from Club Carlson.\nRic Garrido, writer and owner of Loyalty Traveler, shares news and views on hotels, hotel loyalty programs and vacation destinations for frequent guests. You can follow Loyalty Traveler on Twitter and Facebook and RSS feed.\nI am Concierge and have the Club Carlson Visa. A $70 stay gets me $3K booking bonus + 4,200 points (triple bonus) + 1,050 (75% Concierge bonus) \u2013 700 points for using CC Visa. 8,950 points.\n50 points away from lowest free room (plus an extra free night with 2 night stay with points on Club Carlson).\nIf the rate was $70.67 I would get 9,000 points (two nights with CC Visa last day free feature).\nIs Triple points PER STAY, or PER NIGHT?\nor it doesn't matter it's PER DOLLAR SPENT?\n@jerry h. \u2013 all points are per dollar except online booking bonus is per stay. You only get the online booking bonus points once per stay, regardless of nights.\nNights or stays is irrelevant for bonus points from 3x promotion, elite bonus and credit card points.\n@Carl P. \u2013 That's great!\nWe booked London and Paris yesterday and were able to coordinate the bookings with Carlson surprisingly easily! No hassle; if you have points (50000 per night, in this case) you get the booking! Plus, the extra free night. Quite a savings!!\nLondon and Paris are probably great redemption bvalue too for Carlson points, even without the Visa card. The free night certainly makes the Visa card a tempting offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finest theme with all solid and elegant features. This responsive WordPress theme delivers a great outlook and perfectly integrated with Woocommerce. The latest Bootstrap 3 support the enhanced qualities and appreciates the theme for its better functional elements. The google map strengthens smartly for the navigation system. Also, the theme has a 9 type of post formats for specialized styling with the desirable fonts. To be noted the theme is responsive and mobile friendly along side guided with the SEO ready. Incorporated into the best version of the visual composer to accord quality touched pages. The responsive theme has added optimizations and customizations. It has chosen to include a cross browser compatibility for steady source and off canvas menu with sharpening look. It is a Very flexible theme can fit with any website categories specifically, it is a professional theme picked by food, restaurant and more on.\nNow Wordpress users don't have to feel left out. Yummy is now a Wordpress Theme!! Yummy is the most appetising theme you'll find for a restaurant website. It's simple, neat classy and will make your customers' mouths water the moment they land on your site. But its not just great looks. Yummy also packs in some cool functionality and features so you can run restaurant online business smoothly.\nBuild infinite web pages in no time!\nYummy is integrated with the Visual Composer. Set up several web pages with just a few clicks and get your website going. The visual composer makes it easy to set up any number of web pages you like for your site.\nYummy comes with a fresh clean design, that gives your online restaurant a stylish look. It supports Google fonts so you can experiment with various font styles for your website text. You can also add cool Font Awesome icons to your site. Use the customizer to add more custom design to your site. You can also experiment with post formats to display posts more attractively on your site.\nYummy is totally mobile responsive and looks great across all mobile devices. It supports the latest Bootstrap 3.x and Wordpress 4x versions. It also is integrated with a revolution slider for more beautiful and attractive ways to display your image.\nMake navigation on your site more easily with cool mega menus, and shift off canvas menus. Yummy is integrated with Google maps, so you can display the location of your business on your site. This is a cool feature, especially for a restaurant business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now over 40 years in business, Seymour Duncan's commitment to musicians has not changed in the slightest.\nFor over 40 years, Seymour Duncan has been developing products to help musicians find their voice and get the most out of their instrument. However, before he became the pickup guru he is today, in the 1960s, Seymour was a young guitar player who was trying to find that specific and elusive tone for himself, and despite playing many different guitars and different amplifiers, he was unable to achieve it. That is until one fateful day when he loaned his guitar to a friend who got a string caught underneath the pickup and subsequently tore the connecting wire. Duncan decided to attempt to try and repair the damaged pickup, using an old record player and an extreme amount of patience to rewind the pickup back to life.\nWith this repair he had a revelation: the tone that he had been searching for was in the pickups! The materials, winding pattern, magnets, wire and number of turns all worked together to produce something that could substantially change the tone of a guitar.\nAfter spending years working in London and winding pickups for the likes of guitar icons, Jeff Beck, Jimi Hendrix and others, Seymour came to Santa Barbara, California and met Cathy Carter Duncan who fuelled the vision to create a place any tone-seeking musician could come to. Seymour Duncan was founded in 1976 in Santa Barbara, California with a shoe-string budget but a deep love for tone.\nNow over 40 years in business, Seymour Duncan's commitment to musicians has not changed in the slightest. At the Seymour Duncan Custom Shop, everything is made by hand with machine shop custom-fabricated parts. Each pickup is made just the way it was back in the old days. Thus, when you order a Custom Shop pickup, you are getting something truly one of a kind built for your instrument. Over the years Seymour Duncan has built everything from old PAF replicas to the world's largest 158-pound pickup.\nIf you can dream a tone in your head, then Seymour Duncan can create a pickup that will bring it to life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hammock Bay brings the feeling of the Mediterranean to indoor rooms. Palm-shaped carved wood blades deliver soothing air circulation and aesthetic elegance. The rich bronze finish is complemented by the Tuscan sand glass of the bowl light fixture.\nAssembled Dimensions: 51\" x 51\" x 20.1\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I have been trying to purchase your book through Amazon.com, however I havent had any success finding it on there or on Barnesandnoble.com.\nThank you for your support! In shaa Allah, my book will be on those websites on December 15. Sorry for the delay!\nDear sister, please send me an email at howtobeahappymuslim@outlook.com. I am sorry to hear of your difficulties with family members.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Attending seminars and seeing motivational speakers at any event is a great pastime of mine. Learning from many of the world's most successful entrepreneurs is an opportunity that I like to take advantage of. I find that learning and expanding my horizons personally is a much better investment than buying a bunch of materials items to try and impress people. If you are trying to get ahead in life and start your own business, or just succeed in your career, it would be wise to capitalize on these kinds of seminars.\nInvesting in yourself and learning from like-minded people can give you the perspective you lacked previously. Their lessons and stories can fuel your passion and ignite your motivation to get out and improve your life. Because these people have walked the walk, and achieved greater success than you have there's always something to learn from attending these seminars.\nPower of association is something you've heard of no doubt, but you should never doubt what it can do for your life. The people around you can change, shape and mold your outlook; for better or for worse. If you are around the wrong crowd it can drag down your potential and limit your chances at succeeding. The most difficult part of this equation is figuring out that your circle needs to change.\nOnce you've made this decision, you have to leave your circle of friends and find those who can make you a better person. One of the best ways to do this is to connect to coach or mentor in your field or industry. These mentors can provide invaluable advice and tips because they have already achieved success.\nNot just for entrepreneurs or business owners either, because many of these successful people started out at the bottom. Being able to capitalize on your position and realize just how powerful your potential is the beginning of making your dreams come true.\nAny truly successful person will tell you it's more than being smart and in the right situation. So much of your success is determined by your attitude and dedication. These are factors that you can augment and improve by attending seminars and speeches built to motivate you. Finding the right people and surrounding yourself with these people can be the missing link in your professional life.\nSo the next time you are thinking about buying that expensive new TV or putting a down payment on a car, think about investing in yourself first. It's a good chance to reach a return on your investment, and actually get something out of what you put in. Putting in the work, time and money into yourself can be the smartest investment you will ever make. Make the step to improving your life and advancing your career by attending meetings and seminars in your area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This form is designed to provide us with information to update your file, and also to provide you with the offer to quote coverage we presently do not carry for you.\nHave you added any detached structures?\nIncreasing your deductibles may be an option and could result in a premium savings. Are you interested in higher deductible options?\nHave you acquired or sold a secondary house or rental property?\nAre there any new licensed drivers or new household members since the last renewal?\nIncreased limits of liability may be available and may only result in a small premium increase. Are you interested in higher liability limits?\nAre you interested in towing & labor or rental car coverage in the event your car is damaged in an accident?\nIs a vehicle furnished for your regular use that you do not own, such as a company car? If so, how many?\nIf we do not write any of the following coverage for you, please indicate if you are interested in a quotation.\nRecreational Vehicle covers golf carts, snowmobiles, motorcycles, four-wheelers, boats, motors, trailers, etc.\nLife provides funds for those left behind. Would you like a review of the beneficiaries, ownership and cash values of your life insurance policies?\nIs there any way we can improve our service to you? Is there anything not mentioned already that has changed and you think it might affect your personal coverage?\nTo bind coverage you need to speak directly with a licensed agent of Vaaler Insurance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Booklet to accompany part 1 of the \"I Am Your Child\" DVD series.\nNew research shows that a young child's early experiences help to determine brain structure, thus shaping the way they will learn, think and behave for the rest of their lives. This easy-to-read 12-page booklet provides guidelines and ideas for how families and caregivers can promote young children's healthy social and emotional development to help them realize their full potential.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Need a Quick Bite? Check Out These 15 Easy-to-Make Vegan Meals You Can Order Online Now!\nAs fun as it can be to put on our chef hats and cook a big meal, after a long day of work, spending hours in the kitchen is the last thing most people want to do. Sometimes it's nice to come home and just pop something in the oven or microwave or cook something easy like boxed mac and cheese. While in the past, people looking to avoid meat and dairy may have had a hard time finding options, nowadays there are tons and they're in every cuisine category you can think of! Whether it's Asian, Latin, Italian, or good ol' American-style meals, you are bound to find a product that aligns with what you're craving! In fact, there are so many options nowadays that you can order these products online from the comfort of your own home. It's true! Check out these 15 easy-to-make vegan meals you can buy right off the web.\nNothing says easy dinner like ordering a pizza, or in this case popping one in the oven! Daiya Cheeze Lover's Pizza has tomato sauce, two layers of Daiya cheeze \u2013 Mozzarella and Cheddar \u2013 all set atop a gluten-free crust made primarily out of tapioca starch, brown rice flour, white whole grain sorghum flour, and potato starch. Each pie has 18 grams of protein. At $75, this 8-pack bundle purchase comes out to about $9, the same price you would see at the grocery store.\nIf you're looking for a vegetable-heavy dinner entr\u00e9e, Amy's Brown Rice and Veggies Bowl is a pretty solid choice. The bowl, which contains nine grams of protein, has a base of brown rice, is topped with a medley of vegetables including broccoli, mushrooms, carrots, black-eyed peas, and onions, and then is drizzled with a bit of tamari, soy sauce, ginger, and garlic. It simply needs to be popped in the microwave for a few minutes, and then it's ready to eat. You can get a 12-pack of these for a little over $75.\nWhen it comes to food, sometimes you're in such a rush, you don't even have time for a meal that requires a fork. That's when products like Sweet Earth The Curry Tiger Burrito come in handy. These burritos contain a blend of lentils, curried seitan, red peppers, and potatoes that are spiced with turmeric, cinnamon, and cardamom. They have 19 grams of protein and seven grams of fiber. These can be cooked in the microwave and as one Amazon reviewer points out, the wraps remained soft and don't dry out like some other brands might. This 12-pack for $80 will break down to about $6.50 a burrito.\nHealth is Wealth Vegetable Spring Rolls are made with tofu, vegetables like carrots and cabbage, clear rice noodles, and black mushrooms. They are browned in expeller-pressed soy-bean oil. Additionally, they are low in fat, have no MSG, are completely free of artificial additives, colors, flavors, and preservatives. For every two rolls, you get three grams of protein. At almost $60 for 12 boxes, you're getting 12 rolls for $5. Not bad.\nSometimes when you get the craving for a burger or a sandwich, you just need a small amount to nip your craving in the bud. That's where products like Gardein Chick'n Sliders come in handy. The patties of these small sandwiches are made primarily from soy protein, vital wheat gluten, ancient grain flour while the breading is made primarily from rice and wheat flour. To enjoy these, simply pop them in the microwave for three minutes. Some reviewers have gone simple with the fixings, sticking mostly to ketchup but pointing out that these definitely leave room for toppings if that's what you desire. Some even report forgetting that these are not actually chicken! This six-box bundle includes 24 sliders and is a little over $40.\nThese Tofurky Turk'y Broccoli Cheddar Pockets are great for when you want a light meal. Protein-rich Tofurky nuggets, broccoli florets, and vegan cheddar cheese are wrapped in a lightly herbed whole grain pocket. Each pocket has 11 grams of protein and once microwaved can be enjoyed at home or on the go. You may want to warm up both, though, one Amazon reviewer claims that one is simply not filling enough. This 12-box bundle pack breaks down to about $5.25 per package.\nPlantains, roasted sweet potatoes, protein- rich black beans and kale come together in this Kashi Mayan Harvest Bake. The veggies are set atop a seven whole grain polenta and amaranth and are enveloped in a spicy ancho sauce and topped with pumpkin seed garnish. While Kashi currently has several frozen entr\u00e9es on the market, one reviewer shares that this may just be \"their best one yet.\" Ironically enough, Kashi has nixed this entr\u00e9e from their line-up but if you want to see if they live up to the hype, this 8-box bulk purchase will set you back about $40.\nDo you often feel too tired to cook something substantial in the morning? Well if you're getting sick of granola bars, you may consider giving Fantastic World Foods Tofu Scrambler a try. In addition to tofu, this breakfast-y entr\u00e9e also includes potatoes, onion, bell peppers, and green onions, all of which are seasoned with a bit of soy sauce powder, turmeric, cumin, and chili pepper spice and the box also includes a seasoning packet if you want to make it zestier. Fantastic Worlds shares that the scramble can be cooked in just a few minutes by mixing the tofu and the seasoning in a skillet over medium heat. You'll can get six boxes for about $24.\nAmy's Light and Lean Spaghetti Italiano includes pasta with seasoned tomato sauce, \"meatballs\" made out of lentils, quinoa, and tofu, and a side of broccoli florets. The dish is low-calorie (hence the light and lean) and has zero trans fat. Is it tasty? Well, you'll have to see for yourself but one Amazon reviewer says they actually like this entr\u00e9e better than Olive Garden spaghetti! A pack of 12 boxes costs a little over $65.\nGardein BBQ Pulled Porkless Pocket Meals combine tender soy-based shreds with a tangy-sweet BBQ sauce in a soft bun. Each one has zero trans fat, 10 grams of protein, and simply needs to be microwaved for a few minutes. Whether you consider it a meal or a snack depends on your appetite but as one Amazon reviewer relays, you're likely to feel pretty good after eating one of these in comparison to how you would feel pigging out on an actual pork sandwich. A six-pack of this product will cost you a little over $30.\nTandoor Chef Vegetable Korma is a microwaveable gluten-free dish consisting of cumin-infused basmati rice, lentils, and vegetables. Made with olive oil and a light amount of sodium, this dish is good for lunch or when you want a somewhat light dinner. It has 10 grams of protein and eight grams of fiber and one Amazon reviewer shares that the level of spice is pretty mild. To test out the spice level yourself, consider buying this 12-pack for a little under $50.\nCucina and Amore Artichoke and Roasted Peppers Quinoa Mealv is an ideal protein-rich lunch or dinner product to keep on hand. Since it's already fully cooked and does not need to be refrigerated or frozen, you can stash this package in your bag or office and simply grab it and start munching whenever hunger strikes. As one Amazon reviewer points out, once this package is opened you can't reseal it for later consumption, but other than that reviewers seem to dig this convenient meal. You can get six packages for a little over $20.\nMac and cheese has been considered an easy dinner option for a long time now and the dairy-free version is no different! Earth Balance Cheddar Mac and Cheese is made with bite-sized macaroni pasta and a creamy lactose and casein free cheese sauce. Each box contains eight grams of protein and is made simply by boiling the pasta and then mixing in the sauce. If the Amazon reviews are any indication, it seems like this mac and cheese is definitely worth a try! Some even say that it tastes just like the traditional dairy kind. You can get 12 boxes of the cheesy pasta for $34.\nAre you hungry yet? Let us know which one of these products you're going to buy first in the comments!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The flagyl how much recommended safe dose.1.15mg\/kg\/dose given through what nebulizer. The products are shipped directly flagyl for dogs from what the flagyl manufacturers. Adverse Reactions The following serious adverse reactions are discussed below and what elsewhere in flagyl the what labeling: Clinical Trials Experience Because clinical trials are conducted under widely varying conditions flagyl and durations of flagyl follow-up, adverse reaction rates observed flagyl in the clinical. To make sure Lasix is flagyl uso safe for what you, flagyl for women tell your doctor if you have: kidney disease; enlarged prostate, bladder obstruction, urination problems; cirrhosis or flagyl other flagyl how much liver disease; an electrolyte imbalance (such as low levels of flagyl potassium or magnesium in your. Selective serotonin reuptake what inhibitors (ssris) and serotonin norepinephrine reuptake inhibitors (snris). It can increase the probability of flagyl hemorrage during and after surgery. Metabolism Metabolism of flagyl ventolin takes place in liver and GI tract through hepatic enzymes and GI sulfotransferase enzymes respectively. The acute toxicity of flagyl lasix has been determined in mice, rats and dogs. In what a pre- and post-natal developmental toxicity study in rats, an increase in mortality of the offspring at birth and for the first three days after birth, a decrease in pre- and post-weaning weight gain, and decreased performance. Call your doctor if your symptoms worsen, if you feel that albuterol and ipratropium inhalation no longer controls your symptoms, what or if you find that you need to flagyl use extra doses what of the medication more often. What is, ventolin, hFA? It is important for you to flagyl contact your physician is you have what stools that are bloody or black, or if you cough or vomit blood or a fluid flagyl that looks like coffee grounds. The second night the sheets only had blood stains on them. Consult your pharmacist or local waste disposal company. The incidence of intracranial hemorrhage (0.1) and fatal bleeding (0.2) were the same in both groups. For maintenance therapy in pediatric patients, the dose should be adjusted to the minimum effective what level. Hemorrhages, including those with fatal outcome, have been reported in patients what treated with Plavix. Chemically it is methyl sulfate flagyl dosage flagyl (1:1). Absorption After single and repeated oral doses of 75 mg per day, clopidogrel is rapidly absorbed. How to use metered dose inhaler? Omeprazole or Esomeprazole Avoid concomitant use of Plavix with omeprazole or esomeprazole. A total of 534 (6.9) patients in the Plavix group versus 573 (7.4) patients in the placebo what group experienced a primary outcome event (p0.22). Your doctor may need you to stop taking your medication at least five days prior to your procedure to prevent what excess bleeding. A similar reduction in antiplatelet activity was observed with esomeprazole when given concomitantly with Plavix. Proton Pump Inhibitors Advise patients not to take omeprazole or esomeprazole while taking Plavix. Treatment consists of discontinuation of ventolin Syrup (albuterol sulfate syrup) together with appropriate symptomatic therapy. Health solutions network buy plavix buy plavix on street online what purchase plavix. Below for a complete list of ingredients. Get medical help right away if ventolin HFA no longer helps your symptoms. Consider the use of a parenteral antiplatelet agent in acute coronary syndrome patients requiring coadministration of morphine or other opioid agonists. You might have heard about albuterol and ventolin inhalers. Approved uses, ventolin, hFA is a prescription inhaled medicine used to treat or prevent bronchospasm in people aged 4 years and older with reversible obstructive airway disease. Paradoxical Bronchospasm Inform patients that ventolin HFA can produce what paradoxical bronchospasm. Contact Program for Spanish Application(s Form(s). It is also used to prevent bronchospasm caused by exercise. Clinical Pharmacology clinical pharmacology Investigations into the mode of action of lasix have utilized flagyl micropuncture studies in rats, stop flow experiments in dogs and various clearance studies in both humans and experimental animals. Heart palpitations, chest pain, shakiness, nervousness, headache, pain. Nonclinical Toxicology Carcinogenesis, Mutagenesis, Impairment Of Fertility In a 2-year study in Sprague-Dawley rats, albuterol sulfate caused a dose-related increase in the incidence of benign leiomyomas of the mesovarium at and above dietary doses.0 mg\/kg (approximately. These circumstances could include what things such as other patents for specific. This results in a decreased absorption of water and increased production of urine. Ventolin Coupon, average Savings: 75* Off, flagyl ventolin Uses, ventolin is meant flagyl to treat\/prevent bronchospasm induced by asthma and\/or periods of exercise. Do you wish you could review the top 250 drugs from beginning what to end? Get medical help right away if you have any of these symptoms and they can not be explained by another medical condition: purplish spots (called purpura) on the skin or in the mouth (mucous membranes) best way to use flagyl due to bleeding. Each dose of ventolin HFA should last up to 4 hours to 6 hours. Press the dose release button. Fluoxetine is not approved for use in children less than what 7 years of age see. Drug Exclusivity Exclusivity is the sole marketing rights granted by the FDA to a manufacturer upon the approval of a drug and may run simultaneously with a patent. In the second trial, ventolin HFA 90 mcg (n 29 ventolin HFA 180 mcg (n 29 and placebo HFA (n 28) were administered to children aged between birth and 24 months. Precautions, as Plavix belongs to 'blood thinners' avoid flagyl taking it with aspirin or other nsaids (non-steroidal anti-inflammatory what drugs) if your doctor did not recommend what you to. Pregnant and lactating women also need to tell doctor before using albuterol. Tell your doctor if you develop an allergic reaction including skin reactions while taking Plavix. Plavix Warnings, since Plavix is designed to keep your platelets from coagulating and forming blood clots, it increases the likelihood that you might bleed, even if your injury is minor. The RPS has developed a logo that will appear on the first page of registered online pharmacies. Based on biological flagyl plausibility, platelet transfusion may restore clotting ability. Mean platelet aggregation what was higher up to 2 to 4 hours with morphine coadministration. However, under certain circumstances, there may be no acceptable alternatives to the use of beta-adrenergic blocking agents for these patients; cardioselective beta-blockers could be considered, although they should be administered with caution. Inactive ingredients in Maxalt: lactose monohydrate, microcrystalline cellulose, pregelatinized starch, ferric oxide (red and magnesium stearate. Dosage and intake methods, albuterol is effective for providing quick relief from acute bronchospasm. See the end of this leaflet for a complete list of ingredients in Plavix. Labor or delivery Clopidogrel use during labor or delivery will increase the risk of maternal bleeding and hemorrhage. Overdosage Platelet inhibition by Plavix is irreversible and will last for the life of the platelet. Plavix can cause serious side effects including: See \" What is the most important information I should know about Plavix? The developmental and health benefits of breastfeeding should be considered along with mother's clinical need for Plavix and any potential adverse effects on the breastfed infant from Plavix or from underlying maternal condition. For treatment of acute episodes of bronchospasm or prevention of symptoms associated with bronchospasm, the usual dosage for adults and children is 2 inhalations repeated every 4 to 6 hours; in some patients, 1 inhalation every 4 hours may be sufficient. Recent Myocardial Infarction, Recent Stroke, or Established Peripheral Arterial Disease caprie The caprie trial flagyl was a 19,185-patient, 304-center, international, randomized, double-blind, parallel-group study comparing Plavix (75 mg daily) to aspirin (325 mg daily). However, if it is almost time for the next dose, skip the missed dose and continue your regular dosing schedule. As with any effective diuretic, electrolyte depletion may occur during lasix therapy, especially in patients receiving higher doses and a restricted salt intake. Tyler's inability, his denunciation of keflex 500 mg and alcohol, necker, suggests unproductively. Using Plavix in combination with aspirin may be more effective in preventing another heart attack, but this will increase your risk of bleeding. Follow your doctor's instructions about using potassium supplements or getting enough salt and potassium in your diet. At steady state, the average inhibition level observed with a dose of 75 mg Plavix per day was between 40 and. Patients receiving both drugs should be observed closely to determine if the desired diuretic and\/or antihypertensive effect of lasix is achieved. Furosemide is extensively bound to plasma proteins, mainly to albumin. Major Depressive Disorder Daily Dosing Adult-The efficacy of Fluoxetine was studied in 5- and 6-week placebo-controlled trials with depressed adult and geriatric outpatients ( 18 years of age) whose diagnoses corresponded most closely to the DSM-III (currently DSM-IV) category of MDD. Cleaning To ensure proper dosing and to prevent actuator orifice blockage, instruct patients to wash the actuator with warm water and let it air-dry completely at least once a week. Overdosage Get the medical help immediately if you observe following symptoms of ventolin over dosage: Skin allergic reaction including skin rashes and hives Difficulty in breathing Swelling of face, lips or throat Difficulty in swallowing Extreme fatigue or muscle pain Tremors. Recent MI, Recent Stroke, or Established Peripheral Arterial Disease 75 mg once daily orally what without a loading dose see. The inhaler that comes with albuterol and ipratropium spray is designed for use only with a cartridge of albuterol and ipratropium. The effect of this drug lasts for 4-6 hours. For nebulizer solution, the safe dose.5mg every 6-8 hours or 3-4 times per day. At Canadian Health Care Mall, you can buy high-quality reasonably-priced products to increase potency. The counter will stop counting at 000. If your healthcare provider has told you to us e more s prays, wait 1 minute and shake the inhaler again. Furosemide stimulates prostaglandin E2 production. These are not all the possible side effects of Plavix. A reproduction study in Stride Dutch rabbits revealed cranioschisis in 7 of 19 (37) fetuses what what when albuterol was administered orally at a dose of 50 mg\/kg (approximately 25 times the maximum recommended daily oral dose for adults on an mg\/m2 basis). Further information Remember, keep this and all other medicines out of the reach of children, never share your medicines with others, and use Ventolin only for the indication prescribed. Fluoxetine (Prozac) comes as a capsule, a tablet, a delayed-release (releases the medication in the intestine) capsule, and a solution (liquid) what is flagyl to take by mouth. In a rabbit reproduction study, orally administered albuterol sulfate produced cranioschisis in 7 of 19 fetuses (37) at approximately 680 times the mrhdid. Pronounced what as (al byoo' ter ole) (i pra troe' pee um). Chemical Heritage Press, Philadelphia, 1999. Apparent terminal plasma half-life of albuterol is approximately.6 hours. Compared to thiazides, furosemide seems to cause slightly fewer hypokalemias FDA approved uses Hypertension (alone or in combination with other medications for lowering blood pressure ). Or you can take.5mg of ventolin solution through nebulizer 3-4 times per day. Now your inhaler is ready to use. Certified Kelly dispatched your motrin 800 mg breastfeeding diet jitterbugging restaff rarely? The acute intragastric toxicity in neonatal rats is 7 to 10 times that of adult rats. Albuterol and ipratropium are in a class flagyl and alcohol of medications called bronchodilators. Tocolysis Albuterol has not been approved for the management of preterm labor.\nDecreased liver function in geriatric patients can result in increased concentrations of metronidazole flagyl generic flagyl in canada that may necessitate adjustment of metronidazole dosage (see dosage AND administration ). Pregnant patients should flagyl not flagyl be treated during the first trimester; if alternative therapy was not adequate in a pregnant patient, the 1-day flagyl regimen is not recommended (results in higher flagyl serum levels which can reach fetal circulation). Pregnant patients should not be treated during the first trimester (see contraindications ). Concomitant use of metronidazole and flagyl CYP3A4 substrates (e.g., dose amiodarone, tacrolimus, cyclosporine, carbamazepine, phenytoin, and quinidine) may increase respective CYP3A4-substrate plasma levels; monitoring of plasma concentrations of CYP3A4 substrates may be necessary. Usual Pediatric Dose for Giardiasis Some experts recommend: 15 mg\/kg\/day orally in 3 divided doses for 5 to 7 days -Maximum dose: 250 mg\/dose Usual Pediatric Dose for STD Prophylaxis US CDC recommendations : -Adolescents: 2 g orally. If your symptoms are severe, call 911 or dose go to the nearest emergency room. Once metronidazole enters the organism by passive diffusion and activated in the cytoplasm of susceptible anaerobic bacteria, it is reduced; this process includes intracellular electron transport proteins such as ferredoxin, transfer of an electron to the nitro group of the. For this drug to work well, a certain amount needs to be in your body at all times. It's manufactured flagyl as Flagyl by the.D. Metronidazole does not possess any clinically relevant activity against facultative anaerobes or obligate aerobes. Both the immediate-release and extended-release tablets are available as generic drugs. Hepatic Impairment Following a single intravenous infusion of 500 mg metronidazole, the mean AUC24 of metronidazole was higher by 114 in patients with severe (Child-Pugh C) hepatic impairment, and by 54 and 53 in patients with mild (Child-Pugh A) and moderate (Child-Pugh. Keep it away from light. 500 mg for a 70-kg flagyl adult). Metronidazole injection, USP contains 790 mg of sodium per 100 mL; use care when administering injection to patients receiving a controlled sodium diet or corticosteroids or to patients predisposed to edema. Dizziness, headache, stomach upset, nausea, vomiting, loss of appetite, diarrhea, constipation, or metallic taste in your mouth may occur. However, the dose is usually not more than 4000 mg per day. If your dose is different, do not change it unless flagyl your doctor tells you to. The reduced form of metronidazole and free radicals can interact with DNA leading to inhibition of DNA synthesis and DNA degradation leading to death of the bacteria. Drooling, blood in the urine and loss of appetite are seen from animal to animal but are of less importance and severity. How to tell if the drug is working: Your symptoms of infection should improve. Usual Adult Dose for Osteomyelitis IV : -Loading dose: 15 mg\/kg IV once infused over 1 hour -Maintenance dose:.5 mg\/kg IV infused over 1 hour every 6 hours -Maximum dose: 4 g per day -Duration of therapy:. Q: Is it safe to use metronidazole and birth control together? To find out how this drug might interact with something flagyl else youre taking, talk to your doctor or pharmacist. Uses: As adjunctive therapy for the treatment of bone and joint infections caused by Bacteroides flagyl species including dose B fragilis group idsa recommendations : Native Vertebral Osteomyelitis (NVO 500 mg orally 3 to 4 times a day Comments. 2 Lake, hydroxypropyl cellulose, hypromellose, polyethylene glycol, stearic acid, dose and titanium dioxide. Always speak with your healthcare provider about possible interactions with all prescription drugs, vitamins, herbs and supplements, and over-the-counter drugs that you are taking.\nTell your doctor about all other medicines you canada use, especially: a diuretic (water pill digoxin (digitalis, Lanoxin a beta-blocker such as flagyl atenolol (Tenormin, Tenoretic carvedilol (Coreg labetalol (Normodyne, Trandate metoprolol (Dutoprol, Lopressor, Toprol nadolol (Corgard nebivolol (Bystolic propranolol (Inderal, InnoPran sotalol. It is currently under the protection of a patent that flagyl prevents any generic Ventolin from being manufactured in the United States. In a pre- and post-natal developmental toxicity study canada in rats, an increase in mortality of the offspring at birth and for the first three days after birth, a decrease in pre- and post-weaning weight gain, and decreased performance. Recent evidence suggests that furosemide glucuronide flagyl is the only flagyl or canada at least the major biotransformation product of furosemide in man. Aids plavix generic prescription sleep, flagyl hask Clinten is lost, his Northman redeliverses mitigate the lieve. The combination of albuterol and ipratropium is used to flagyl prevent wheezing, difficulty breathing, chest tightness, and coughing in people with chronic obstructive pulmonary disease (copd; a group of diseases that affect the lungs and airways) such as chronic bronchitis. Let it dry and then reinsert the canister. As an FDA pregnancy category B drug, Plavix is not known to be harmful to unborn babies. You may need to use flagyl blood pressure medicine for the rest of your life. Such concomitant use, however, should be individualized and not given on a routine basis. If increasing azotemia and oliguria occur during treatment of severe progressive renal disease, lasix should be discontinued. Pharmacy Lookup -Drugs to stop blood clotting, like Ticlid, Activase or Persantine. Asthma Slideshow Pictures, take the Asthma Quiz! It is not known if Plavix will harm your unborn baby are canada breastfeeding or plan to breastfeed. Cialis and Viagra are somewhat different even though With Cialis, men have 2 options: use the larger dose of 10 mg or 20 mg on as needed basis or use. Dexlansoprazole, lansoprazole, and pantoprazole had less effect on the antiplatelet activity of Plavix than canada did omeprazole or esomeprazole see Warnings and Precautions (5.1) and Clinical Pharmacology (12.3). Assessment of health professionals and evaluation of programmes: Best practice and future development. Buy Cialis Online Cheap and Fast! Always talk to your physician or pharmacist before taking any prescription medication or over the counter drugs (including any supplements) or before making any changes to your treatment. Cheapest Canadian Fortamet, Fortamet NO prescription. Non-potassium-sparing diuretics, digoxin, monoamine Oxidase Inhibitors, tricyclic Antidepressants. Do not drive, use machinery, or do anything that needs mental alertness until you know how this medicine affects you. Canadian, mar Thoma Church, Toronto. Licensed Canadian pharmacy that provides safe affordable Canada drugs at discount prices. Similar effects were not observed canada at approximately one eleventh of the mrhdid. Sooner or later, many men and couples get to know the problems with erection. Plavix Drug Interactions If you take Plavix with certain other drugs, the effects of either could be increased, decreased, or altered. You might have heard about albuterol and ventolin inhalers. In addition, ventolin HFA was shown to be clinically comparable to a CFC 11\/12-propelled albuterol inhaler for this indication. Albuterol cannot interfere with the inflammation but, it does help to relax the bronchial smooth muscle which opens up the air passages and improves the air flow ( 1, 2, 3 ). For correct use of your ventolin HFA inhaler, remember: The canister should always fit firmly in the actuator. Keep this medication in the container it came in, tightly closed, and out of reach of children. Consequently, platelets exposed to clopidogrel's active flagyl metabolite are affected for the remainder of their lifespan (about 7 to 10 days). If you experience any of the following symptoms, call your doctor immediately: fast or pounding heartbeat chest pain hives rash itching swelling of the eyes, face, canada lips, tongue, throat, hands, feet, ankles, or lower legs difficulty breathing or swallowing. Non-steroidal anti-inflammatory drugs (nsaids). The patient must also be residing in the. Actions after intake, pharmacokinetics Pharmacokinetic properties of ventolin are as follows: Absorption After inhalation, only 20 percent of the drug reaches lungs and the rest of its remains deposited in mouth and throat. (See precautions : drug interactions ) Precautions precautions General Excessive diuresis may cause dehydration and blood volume reduction with circulatory collapse and possibly vascular thrombosis and embolism, particularly in elderly patients. Bleeding of all severities was more common in the subjects randomized to Plavix. Labor or delivery Clopidogrel use during labor or delivery will increase the risk of maternal bleeding and hemorrhage. Exercise-induced bronchospasm, ventolin Dosage, ventolin dosage is dependent on the individual that it is being administered. Gender In a small study comparing men and women, less inhibition of ADP-induced platelet aggregation was observed in women.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wenger has confirmed who will leave in summer!\nArsenal manager Arsene Wegner had confirmed that the goalkeeper of the club Wojciech Szczesny is one of the pretenders to leave the club in summer.\nThe Polish stopper has found it hard to become a regular starting member of the team during the last couple of months, after his conflict with Arsene Wenger.\nThe 24-years old stopper has been linked with a move to Roma recently.\nMeanwhile, Arsenal have launched \u00a349m double bid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Potato, the world's favorite root vegetable. Not very beautiful when it first comes off the soil but it is cheap, very nutritious and the best accompaniment for fish, meat or vegetable dishes.\nA delicious yet healthy way to cook them that competes equally in taste and texture with french fries, which we all know taste heavenly, is to roast them; roast them in such a way that will make them extra crispy on the outside and super-fluffy on the inside. Prepare a feta cheese dip for them to bathe in and your lunch is ready!\nPeel potatoes and cut them into 3 cm pieces. Wash them well and put them into a saucepan with cold, lightly salted water.\nBoil them at medium heat for approximately 7 minutes or until partially cooked. Careful not to cook them completely.\nDrain them and let them steam dry for a couple of minutes. Put them back into the saucepan and shake them vigorously to create rough edges.\nPut your potatoes into a roasting tray and sprinkle olive oil over them. Season with salt and pepper and use a spoon to mix the potatoes for the olive oil to fully cover them.\nRoast potatoes for 30 minutes until lightly golden. Take the tray out of the oven and give it a good shake.\nPut back in the oven for 40 minutes or until golden and crisp.\nWhen ready, take them out of the oven and transfer potatoes into a plate lined with kitchen paper to absorb any excess oil.\nWhile your potatoes and resting, quickly put all ingredients for your feta cheese dip, except the herbs, into a blender and mix them into a smooth paste. Then, add the fresh herbs and mix them in gently. If your dip is too thick add a bit of extra yogurt and if it is too thin add a bit more feta cheese.\nBy the time you finish preparing the dip, the potatoes will be free of any excess oil.\nTransfer them into a clean plate, put your dip into a small bowl and enjoy the most crispy roasted potatoes ever.\nIf you are not afraid of extra calories, replace olive oil with butter or, for the daredevils, goose fat.\nIf you want to season your potatoes with fresh or dried herbs, rosemary and garlic make a perfect combination, put them in the last 40 minutes of roasting time (step 7) otherwise they will get overcooked and bitter.\nFor the hot & spicy lovers, fear not to use chilli powder to season you potatoes before roasting them. The feta cheese dip will cool down your palate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to Speed Up The Manifestation Process. Become the Celebrity Of Your Own Life!\nGetting Into Vibrational Resonance (LOA & Becoming An Energetic Match). Are You Ready To Make 2018 THE YEAR You Went For It!?\nHow to use the Facial Ritual Kit... step by step tutorial finally here!!\nBecome a Soulpreneur Success ~ Launches March 2018!\nIt's time to shine brightly to attract soulmate level relationships~ No more dimming your light to fit in!\nCrystal Healing Series: Selenite Spotlight!\nFeeling Lonely During Your Spiritual Awakening? Wondering Where did Everybody Go & Isn't Your Vibe Suppose to Attract Your New Tribe? Watch This Video For Reassurance!\nHow to get what you desire in life through faith & the law of attraction (changing thoughts and cultivating new belief systems).\nHow Following Your Joy Pursuits (What Lights You Up) is a Sure Fire Way to Discovering Your Unique Soul Purpose and an Living an Authentic & Inspired Life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"[71.1] Surely We sent Nuh to his people, saying: Warn your people before there come upon them a painful chastisement.\n[71.4] He will forgive you some of your faults and grant you a delay to an appointed term; surely the term of Allah when it comes is not postponed; did you but know!\n[71.5] He said: O my Lord! surely I have called my people by night and by day!\n[71.12] And help you with wealth and sons, and make for you gardens, and make for you rivers.\n[71.13] What is the matter with you that you fear not the greatness of Allah?\n[71.16] And made the moon therein a light, and made the sun a lamp?\n[71.20] That you may go along therein in wide paths.\n[71.21] Nuh said: My Lord! surely they have disobeyed me and followed him whose wealth and children have added to him nothing but loss.\n[71.22] And they have planned a very great plan.\n[71.23] And they say: By no means leave your gods, nor leave Wadd, nor Suwa; nor Yaghus, and Yauq and Nasr.\n[71.24] And indeed they have led astray many, and do not increase the unjust in aught but error.\n[71.25] Because of their wrongs they were drowned, then made to enter fire, so they did not find any helpers besides Allah.\n[71.28] My Lord! forgive me and my parents and him who enters my house believing, and the believing men and the believing women; and do not increase the unjust in aught but destruction!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacwdc b/data_all_eng_slimpj/shuffled/split2/finalzzzacwdc
new file mode 100644
index 0000000000000000000000000000000000000000..5bb927810e9683a8a98951c7a97f83addd8ab367
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacwdc
@@ -0,0 +1,5 @@
+{"text":"I think this is the primary reason pols don't like to release their returns. I remember when it was discovered that Al Gore gave about $300 to charity. That goodness for him he wasn't a republican as the media didn't hit him all that hard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In an official snap, the 34-year-old posed alongside head judge Simon Cowell as it was announced she would act as a mentor at the Judges' Houses stage of the programme.\nThe announcement comes nine years after the former Girls Aloud singer first took up her judging position on The X Factor. She last served as a judge on the talent show in 2015, alongside Cowell, radio DJ Nick Grimshaw and singer Rita Ora.\nNadine Coyle \u2013 Go To Work on Strictly It Takes Two [29\/09\/17] from Gallery Aloud on Vimeo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lovely Gibson Thunderbird Ebony with original case (p0rn interior included...).\nThe bass has been used well, and it has some buckle rash and other battle scars to prove it. Nothing serious though, but don't expect a perfectly kept bass. Not really strange for a 19 year old black beauty! Technically she's in perfect condition.\nThe case looks a bit worse, but works perfectly.\nThe bass is located near Eindhoven in NL. I could drive to or just over the border to make the exchange.\nThe only trades I'm interested in are: Wal, Fodera, or G&L SB2 (USA). Only 4-strings!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Line an 8\" x 8\" square baking pan with aluminum foil, extending foil at least 2\" over edges of pan. Butter foil. Set aside.\nButter the sides of a heavy 2 quart saucepan. In the saucepan combine the sugars, cream, and milk. Cook over medium-high heat to rolling boil, stirring constantly with a wooden spoon to dissolve sugars. This should take about 5 minutes. Avoid splashing mixture on sides of pan. Carefully clip candy thermometer to side of pan. Cook over medium-high heat, stirring frequently until mixture reaches the soft ball stage or 234\u00b0F to 240\u00b0F on a candy thermometer. Mixture should boil at a moderate, steady rate over the entire surface. Reaching soft-ball stage should take 15 to 20 minutes.\nRemove pan from heat. Add the 2 tablespoons butter and vanilla, but do not stir. Cool, without stirring, to 110\u00b0F, about 50 minutes. Remove thermometer. Beat vigorously until just beginning to thicken. Add coconut. Continue beating until very thick and just starts to lose its gloss. This should take about 10 minutes total. Turn into prepared pan. While warm, score into 1\" squares. When firm, using the extended edges, lift fudge from pan. Place on cutting board. Peel off foil. Cut into 1\" squares.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last year I had the privilege to photograph the WEDDING of Springbok Rugby player Duane Vermeulen and his beautiful wife, Ezel at the stunning Val De Vie Estate. On Sunday we went to Meerendal Wine Estate to capture some fun and special maternity photos. We did this photo session just a few hours before Duane had to leave to Johannesburg to join up with the Springboks to prepare for the upcoming Rugby Championship.\nSectret Diary Designs did their wedding stationary and was very eager to come on board with this shoot and gave them a \"Gift from God\" box with the name that Duane and Ezel chose for their son, ANRU.\nDie fotos is beautifull !!!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadpkl b/data_all_eng_slimpj/shuffled/split2/finalzzzadpkl
new file mode 100644
index 0000000000000000000000000000000000000000..ee6fd92057ea941e2d2a6b02db8e18057390e245
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadpkl
@@ -0,0 +1,5 @@
+{"text":"Do you ever feel like there just isn't enough time in the day to do everything you need to get done? Do you go to bed exhausted just to get up again the next day and do it all over again? Are you running on empty trying to juggle your kids, family, home and work?\nImagine for just a moment that you had a system in place that allowed you to figure out what tasks you really need to complete each day (not just think you do), find more time and energy to complete them and eliminate the overwhelm.\nWell, that's exactly what I'm going to lay out for you right here. I'm going to give you a four step system that will help you to do less of what you think you have to do, more of what you actually need to do and a lot more of what you want to do.\nI learned this system from watching The Mom Conference in an interview with Jenny Layton of TheHappyGal.com and let me just tell you right now\u2026 It works!\nSo, stick with me and I'll help you figure out how to get more done each day.\nThe first thing we need to talk about is your time and energy bank account.\nWell, think of it this way \u2013 you only have so much time and energy to complete all the tasks you have to do in a day (let's call in your bank account).\nAnd, for every task you complete there's a cost or a withdrawal from that account.\nAt the end of the day, you'll have either used up your entire account, leaving you at a zero balance, have spent too much and be overdrawn or have saved a little.\nThe goal is to become more efficient in what you do so that you always keep a little reserve in your account.\nNow that we've got that analogy in place, let's look take a look at the 4 steps to getting more done each day and avoiding the overwhelm.\nFires are tasks that you can't procrastinate on, ignore or put off until tomorrow. These are tasks that you absolutely must complete today.\nThese are tasks that have a deadline \u2013 maybe a project at work, helping your kiddo with homework or getting the flat tire on your car fixed.\nThe problem with fires is that they are the most expensive tasks when it comes to your time and energy bank account. They throw you off your game. And, when you have to devote too much of that precious time and energy to them, you lose balance and sets up a toxic productivity cycle.\nThat's because when you spend all day putting out today's fires, you have no time to get ahead for tomorrow and then the cycle repeats itself. Tomorrow you'll have a whole new set of fires to put out.\nBut, if fires are such important tasks that you can't put them off, what can you do to avoid that toxic cycle?\nThe key is to slow down and look at what these tasks really are. What you'll find is that sometimes your mind plays tricks on you and tasks that you think are fires really aren't.\nIs this task someone else's responsibility?\nAs a mom, you probably find yourself in the role of rescuer a lot! Your husband, your kids, your parents, your friends\u2026 they all need you for something and it's easy to let someone else's emergency become yours.\nBut, since you only have so much in your time and energy bank, if you put out someone else's fire, you're going to have to say no to a responsibility that really does belong to you.\nAnd, yes I know that it can be really hard to say no and take a step back when it comes to taking care of a task for someone you love. Just remember that when you do, you're doing a favor for both of you.\nFor example, when you let your kids do their book report on their own, their learning responsibility and coping skill that will last a lifetime. And, as a bonus, you won't resent that you had to miss out on your favorite show or stay up 3 hours later and get up exhausted the next day just because your kiddo chose to put it off to the last minute.\nJust say no once and you'll discover that it really can be a win-win for everyone.\nShould this fire be part of a routine?\nThis question is used to figure out if there's a way you can prevent this problem from becoming a fire again in the future.\nFor example, if you find yourself doing laundry at eleven o'clock at night because your daughter needs her shirt to wear to chapel the next day, is there a way to set up a routine laundry day that would avoid the need for last minute panic?\nYou want to set up routines that keep these fires from building up and blowing up in the first place.\nThis question goes back to the whole idea that you may think something is a fire that really isn't.\nIn order to decide if the task really must be done right now, think about what the consequences will be for not doing it.\nAs an example, if you think that you have to run to the grocery store instead of going to bed because you're out of milk, think about what is the consequence of not having the milk in the morning.\nWill your kids really not have anything to eat for breakfast or will they just have to eat something other than their favorite cereal? You may find that you've been putting out fires that weren't really burning.\nOnce you've asked yourself these three questions, you may find that you really only have one or two fires to put out for the day rather than the six to seven you started with.\nAnd, when you reclaim the time and energy you've been spending on these fires, it allows you to get more deliberate with the things you have to do each day to make better decision, become more efficient and get more done.\nNow that you know what your real fires are, it's time to make a list of your regular tasks.\nThese are any tasks that need to be done consistently for best results, predictable tasks like laundry, yardwork, bills, spending time with your kids and cooking meals.\nYou'll find these tasks make up about 75% of your workload because as a mom, there's a lot of repeat.\nThe goal for these tasks is to set up routines to take the stress out of them. Routines are the secret sauce when it comes to getting things done and being more productive.\nI know that not everyone is a fan of routines \u2013 they can seem boring and mundane but if you skip this step and don't make your regular tasks a routine, they have the potential to flare up into fires.\nBut, you have to set up your routines the right way or you will fail.\nTo learn the four step system to setting up routines that work and help you crush your to-do list, check out this post.\nThe next step to getting more done and eliminating the overwhelm is to identify any non-routine tasks that require a series of steps and usually can't be completed in one sitting \u2013 in other words, your projects.\nThese could be organizing your bills, re-decorating, re-doing your yard for the change of seasons or planning for Christmas or a vacation.\nWhen it comes to projects, there are two different groups of people who have a hard time with them. You might see yourself in one of these.\nThe first is the people who can't even start their projects because they're too busy trying to keep up with normal life.\nThe second is the people who love projects and get them started but have a hard time finishing. These people usually move from project to project, getting started but then fizzling out.\nSince you're really good at doing the day-to-day, you have to make your projects part of your regular life rather than seeing it as something you'll do someday when you finally have the time.\nSet up an appointment with yourself. It can be every day, three times a week, or a two-hour block once a week where you give yourself the gift of project time.\nDon't take too much time away from your project. When it comes to completing a large task like this, momentum is your friend.\nSo, instead of taking a couple of days away from your project to do laundry and clean the house, only take a couple of hours away each day so that you can still continue to work on the project.\nThat way, by the time you come back to the project, you haven't found a passion for a new project instead.\nAnd, chunk you project up.\nThat sounds weird but what I mean by that is to not pick too big of a project but instead split it into multiple, more manageable projects.\nInstead, set a goal of reorganizing your pantry and then reevaluate to see what area of your home needs to be worked on next.\nAlso, don't avoid projects fearing the time and energy they take to complete.\nThe good thing about these big tasks is that when you complete them, the feeling of accomplishment you get actually replenishes your energy and nourishes your spirit, keeping your bank account balanced.\nNow you're going to set your priorities (tasks you can to today that will have a positive impact on the future, restore personal balance and bring you fulfillment).\nWe all know that we're supposed to have our priorities in order. But, more often than not, we allow ourselves to fall to the bottom of the list.\nDo you see yourself in that? Always putting your kids, husband and work first? Never having time to take care of yourself and do the things that make you happy?\nIt's easy to do as a mom \u2013 but, when you're running on empty, you don't have a lot to share with other people. So, it's time to refill your tank and your time and energy bank account by finding time for yourself.\nThere's no wrong answer and it will be different for everyone.\nMaybe you would want to go out with your friends for a girl's night. Maybe you want to finish that scrapbook you've been working on. Or, maybe you just want a nap.\nWhatever your answer to the question was, it is now one of your priorities \u2013 no longer to be put off in order to take care of everyone else.\nRemember, in 10 years, doing this one thing to make yourself a happier, more balanced person will change the way your kids see you, your marriage and your health for the better.\nThose are the four steps to get more done each day, eliminate the shoulds and do more of the wants and finally say goodbye to overwhelm.\nStart implementing them today to start making deposits into your time and energy bank account and find more time in your day for you.\nTo make it simple, I've created a free printable you can download here to help you identify your fires, regular tasks, projects and priorities and determine what you need to do each day, what you can cut out and how to fit in time to become a healthier, happier you.\nHope it helps you on your journey!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recommended for Sprinter\/Dodge vans with 158\" wheel base. This is our most popular package for the heating, ventilating and air conditioning trade. Plenty of shelf storage as well as a convenient freon tank rack and small parts drawers make this an efficient storage system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Signage\/Identification is Required Within a Building?\nQ: What signage\/identification is required within a building? Where can I find more information on this topic?\nA: That's a pretty broad question. I assume you are talking about a commercial or industrial building where the public has access. The most important signs are for the exits for emergency egress. Also if there are restrooms in the building, it is imperitive that they be handicapped accessible toilet rooms and have signs to indicate this. The ICC International Code Council publishes the IBC International Building Code and there is a section on signs. Of course, local jurisdictions may have additonal requirements. Always check with the Building Department to be sure you have included all the signs that are required.\nQ: On a site fabricated fabric wall do building inspectors check ASTM E-84-08 test results? This requierment doesn't seem to be enforced?\nA: Something that is fabricated on site has to have the specs on site also. The building inspector usually will look at the specs to see that they have been followed.\nA fabric wall is not usually a life-safety issue, unless is has fire safety standards attached to it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AAON, Inc. (AAON) is engaged in the manufacture and sale of air-conditioning and heating equipment. The Companys products consist of rooftop units, chillers, air-handling units, make-up air units, heat recovery units, condensing units, commercial self contained units and coils. Its products serve the commercial and industrial new construction and replacement markets. AAON markets its products to 50 states in the United States and certain provinces in Canada. Its operating subsidiaries include AAON, Inc. and AAON Coil Products, Inc. The Companys rooftop and condenser markets consist of units installed on commercial or industrial structures of generally less than 10 stories in height. Its air-handling units, commercial self-contained units, chillers, and coils are applicable to all sizes of commercial and industrial buildings.\nView SEC Filings from AAON instead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For many years we've been interested in metal shaping, not only for repairing and making radiator tanks, but also for restoring old cars. Andrew has done a few courses over the years with Peter Tommasini who has over 40 years' experience and has made a number of complete HK to HG Monaro rear guards from a single piece. One thing is for sure, panel shaping can be one of the most frustrating tasks in the workshop, but when you get it right, it's the most rewarding feeling.\nAndrew has built an English Wheel using Hoosier Profile upper and lower anvils which works extremely well. He's also recently purchased a real Cast English Wheel that was in need of a little loving. We have a 1960-1970 Pullmax for shaping and a \"Lazzie\" bead roller, an assortment of shrinking and stretching tools and the good old trusty tree stump. Although this is Andrew's passion, it has proven extremely helpful in the workshop, for example, when straightening old brass tanks before soldering.\nWe've recently remanufactured the front guard for a model \"T\" Ford complete with side Valances and running boards. This was quite a project, when restoring the rear guards Andrew made numerous rust repair panels for those hard-to-get parts. We're currently restoring an HD Holden windowless panel van which will one day be our daily driver and promotional vehicle \u2013 we look forward to getting this one on the road!\nIf you have a vehicle that needs some metal shaping lovin', from replicating cut out rusted sections so you can weld them in yourself to letting us take on the whole job.\nPlease ring Andrew to discuss your next project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaeloi b/data_all_eng_slimpj/shuffled/split2/finalzzzaeloi
new file mode 100644
index 0000000000000000000000000000000000000000..0deafe7d0443dcccbc66c0c3b5c5494fd204b03e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaeloi
@@ -0,0 +1,5 @@
+{"text":"The Center for Disaster Management at the Graduate School of Public and International Affairs (GSPIA) advances the study and practice of emergency management through research, analysis, education, and collaboration with scholars and practitioners. The Center conducts research and analysis on timely policy issues related to disaster risk reduction and management that cross organizational, disciplinary, and jurisdictional boundaries. Since its founding by Dr. Louise Comfort, the Center has contributed innovative research on understanding risk, dynamic decision-making in complex environments, and methods of building community resilience. The Center maintains significant collaboration with scholars and practitioners in emergency management and related disciplines.\nThe Center also provides educational opportunities and support to students interested in careers in emergency management and related disciplines. One of the fundamental purpose of emergency management is to coordinate resources across diverse organizations to prevent crises from happening, and to address and lessen the impacts of crises when they do occur. Because of this, the field of emergency management is multidisciplinary and relevant not only to students interested in emergency management careers, but also to those studying public health, public administration, security and intelligence, human and social services, agriculture, energy, environmental science, commerce, and others. At GSPIA, students have the opportunity to pursue a minor in Civil Security and Disaster Management and to participate in educational and professional development opportunities throughout the academic year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fellow Jeremy [name only] is hosting a \"Halloween Hop!\", so lets go to a hop... and I just need to tell you what a few favorite things of Halloween might be and what I might be wearing this year in costume.\nSo for those who are stopping by the blog for the first time, in part of the Hop let me say welcome... welcome. This place can be taking in two ways, one \"wholly crap\" this guy is nuts or \"wow\" a year round Halloween of things and this guy is nuts.\nNightbreed was a twisted and fun film!\nThe last item is sad, Jeremy.\nI never read Pet Sematary - I didn't even know it was spelled that way for the book.\nHi! Wow, Halloween ALL year. YES! I'm following, lol.Nice to meet you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not compatible with Waindigo's Library addon.\nI use Waindigo's Library addon, which creates nodes of \"library\" type.\nSo in my right-column widget, our user FAQ (which is a Library) has a broken link.\nA fix should be pretty easy? Thanks!!\nWould also love the ability to hide specific nodes from this node list addon.\nThanks for your report. We will consider and solve your issue on next upgrade version shortly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can you hear that? It's the sound of a great set of daily deals winging their way into your day. Welcome to another round-up of Dealzmodo deals \u2013 headed up by a fun little saving on some snazzy socks. Yes, socks.\nThe foot sleeves in question are by Lazy Oaf, purveyors of all sorts of fine threads and whose motto is \"keeping it weird since 2001\". The socks come with the word 'internet' embroidered into the side, so you can show off to all around you your love of this place we call home. Normally \u00a36, you can take a home a pair for just \u00a32! What more is there to say other than BUY BUY BUY.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A familiar face seems poised to make an appearance in Call of Duty: Ghosts, as Infinity Ward posts a single image with the caption \"Price is Coming\".\nFans are, predictably, going a little nuts. Captain Price has appeared in every Modern Warfare title to date, and the news that he's set to return is making people more than a little happy.\nPrice is seen posing in Stonehave, a multiplayer map from Call of Duty: Ghosts.\nCurrent theory instead puts Price's comeback as a downloadable multiplayer soldier skin. Character Simon \"Ghosts\" Riley has already featured in the game in this way, included as a pre-order bonus. Here's hoping he brings some new weapons with him, and we don't have to wait too long to find out what's going on!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaezvc b/data_all_eng_slimpj/shuffled/split2/finalzzzaezvc
new file mode 100644
index 0000000000000000000000000000000000000000..9794738e8b49dbf637529ddbceca78866128bdc0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaezvc
@@ -0,0 +1,5 @@
+{"text":"Keep the heat off your head and the cool breeze rolling with this big hat. This structured style is comfortable and lightweight, and offers maximum ventilation. Made of 100% cotton front panels with mesh side and back panels, it also has a thin moisture wicking cotton sweatband on the inside. A plastic tabbed strap in back can adjust for the perfect fit. Our classic trucker hats for big heads will keep your noggin' cool and safe from the sun in the hot and warm weather months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LoanPaymentPro is a payment processing solution for ACH, Credit and Debit transactions. They also provide merchant services and funding to debit cards. LoanPaymentPro is designed to get lenders approved for payment processing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Around \u00a310,000 worth of damage has been caused to vehicles which were targeted by a key-wielding vandal in a \"reckless act of vandalism\".\nMark Edginton's garage, on Alcester Road, near Wythall, Worcestershire, was targeted in the attack last Thursday.\nIn footage handed to police by Edginton a man can be seen scratching every car in the dealership's forecourt.\nThe BBC reported that Edginton had said he did not know why he had been targeted.\nHe has handed the CCTV footage to police, who called referred to the incident as a \"reckless act of vandalism\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jewellery is one piece that's usually fairly costly. It's such a small merchandise however its worth is big thus it's excellent to correctly retailer it in a jewellery field in order to not injury it and shield it as a substitute. Due to its small dimension it's straightforward to lose or injury it.\nSizes various for jewellery bins. You may select relying of the variety of jewelries you might have. Nonetheless, conventional jewellery bins are usually not that massive. It is simply appropriate for a set in order that it's made to carry a couple of items of jewellery. In actual fact most of this jewellery bins are a collectible of their very own.\nAdditionally if you attempt to search for a jewellery field, make sure that to look at the field for any nails or screw which may scratch your jewelries. Be sure that additionally that it has sufficient compartments for every of your jewellery. You will need to isolate jewellery in order that it will not scratch each other.\nIf in case you have fairly a big assortment, then you definitely want a a lot bigger storage. Jewellery armoires and chest is what you want. That is normally a small piece of furnishings with a number of drawers and compartments to accommodate completely different kinds of jewellery.\nEvery compartment is normally fitted with linings, hooks and different options. These Jewellery armoires and chests are generally customized design to suit your wants.\nThis jewellery field is contained in hard-sided circumstances. They usually characteristic a deal with and lock. Its compartments have pads to maintain the jewellery from rubbing at one another when carried.\nRoll-ups are fabric strips that maintain pouches for particular person jewellery. Hanging organizers can also be an important alternative. It differs from a jewellery field because it shows all jewellery in a vertical show. That is usually finest for bracelets and necklaces.\nWith all these storage choices prompt above, now you can have an thought of what most closely fits your want so that you can shield your items of jewellery. Simply make sure that to retailer jewellery individually. Jewellery is made up of various stones and metals thus when they're put collectively, they may simply scratch and get injury.\nThis scratches is usually unattainable to restore thus will make you lose 1000's of . It's excellent to take a position too in your jewellery storage that may guarantee safety and security of your jewellery. Taking good care of your jewellery is caring for your funding. There's a massive amount of cash in these objects and most of all it's one thing you can move on to your kids and grandchildren.\nKeep in mind then how helpful it's and if you move it to your kids and grand kids, do not forget to maintain them inform too on tips on how to take superb care of it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Come for action alerts, news on feminist issues in Indiana and beyond, resources, feminism in pop culture, and much more!\nNOW is a national organization with state chapters like Indiana NOW that support local on-the-ground members in our communities across the state. In some communities we have local chapters and in others NOW members meet more informally.\nIf you want to get connected to other members in your area, we can help make that happen.\nInterested in joining or starting a local chapter of NOW? Contact us and we will help you get started!\nWe are a volunteer organization and our board members and state officers all have full-time jobs and other responsibilities, but we will reply to your query as soon as we can!\nWant to get more involved? Join us now to make your voice heard!!\nNow is the time to become a member of Indiana NOW and stop the injustices you see around you. Each member has a voice! Make yours heard through your active membership. Join today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagrmd b/data_all_eng_slimpj/shuffled/split2/finalzzzagrmd
new file mode 100644
index 0000000000000000000000000000000000000000..54a5f4c660aef051673a532344d4b87fe091a7e0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagrmd
@@ -0,0 +1,5 @@
+{"text":"FABRE TP is an industry-led platform of breeding and reproduction organisations and knowledge institutes, with members located across Europe. Together we share a common vision on farm animal breeding and reproduction to enhance the sustainability of farm animal production in Europe, and globally, through development and uptake of better approaches to breeding and reproduction.\nFABRE TP aims to grow the commitment of the involved stakeholders, including industry, knowledge institutes and funding organisations.\nFABRE TP's General Assembly consists of representatives of all member organisations. The Platform is managed by the Board and obtains input from the members via Expert groups. The daily affairs are managed by the General Secretary and its Secretariat.\nFABRE TP sets up Expert Groups when needed. Expert Groups consist of representatives of member organisations working on a specific topic.\nThe Secretariat takes care of the daily management and organisation of activities. The FABRE TP Secretariat is managed by the European Forum of Farm Animal Breeders, with its offices in Brussels, Belgium, and Wageningen, the Netherlands.\nShare our common vision and join us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Bride-to-be's heartless reaction to her sister's devastating miscarriage prompts her groom to CANCEL their nuptials.\" Read how Daily Mail covers our Fall of Bridezilla story here.\nSarah Templeton of Newshub affirms Not Always Right as a place where users post their \"funny quotes and true stories\". Read how Sarah covers our story here.\n\"It's not uncommon for brides to get a bit carried away when it comes to planning their dream wedding day\", says Yahoo. Read their story \"Groom calls off wedding after fianc\u00e9e's shocking behaviour\" here.\n\"Sharing the shocking tale on Not Always Right, the wedding planner described how to the bride told her sister that her pregnancy was inconvenient.\" Read their article here.\nThe Sun mentions how \"it goes without saying that the stress of planning your dream wedding can (ironically) put a strain on some of your closest relationships.\" Read their article here.\nJoel Anaya, a Hospitality Business Management student from Washington State University, did research on \"customer service sabotage\". He studied more than 200 accounts written by frustrated people on notalwaysright.com. Read The Telegraph's article here.\nPaul B. Brown of The New York Times mentions \"Who knows better that the customer is not always right than the people who serve them? Read Know When To Fire A Customer.\n\"Our reader Eyebrows McGee pointed us to a fun, albeit slightly depressing (if you're concerned about the future of the human race) website called NotAlwaysRight.com, where retail veterans post transcripts of their worst customer interactions. Read for fun, but also for examples of how not to behave as a consumer.\" Read their article here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fiber Splice Heat Shrink Tube is ordinarily made of nylon or polyolefin. They provide good electrical insulation, protection from dust, solvents and other foreign materials, as well as strain relief. If overheated, heat shrink tubing can melt, scorch or catch fire like any other plastic. Fiberinthebox supply colorized heat shrink tube, and they can make the joint of terminal block and wire insulated with very good flame retardation. All our heat shrink tube shows good performance with good prices, and we can deliver anywhere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi, you may call me Martha Stewart now.\nHere's the thing about DIYs: Nobody really needs to do most of them. Nobody really needs to make a crochet egg holder, or a cheese grater earring stand, or ice tray chocolate strawberries (though, like, yum), because for most of these things, you can buy better versions of them in a store.\nBut DIYs, especially beauty DIYs, aren't necessarily about making a picture-perfect dupe of your $60 dollar lipstick, but instead getting to have fun and create something with your hands\u2014something that is filled with ingredients you actually know and handled yourself. So, you know, what literally every human in the 19th century did. So to harken back to our ancestors, and because we really just like Saturday crafternoons, we present to you the best DIY lip balm tutorials, below.\nPlease refrain from licking this honey-based lip balm off your lips (or out of the jar), especially because it does have a bit of your favorite lipstick in it for color.\nCoconut butter, rose oil, and vanilla extract give this pink-hued lip balm its super-smooth finish.\nPart minty balm (thanks to its Vaseline and mint extract base) and part pink and purple gel food coloring, this balm-stain hybrid leaves behind a surprisingly bright wash of color.\nOkay, this one is a bit more involved, but if you're not about buying a new EOS after yours runs out, try this dupe, which uses vitamin E, coconut oil, and peppermint oil to slightly plump your lips while softening them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So during the week I live vicariously through others' projects while spending quality time writing on this little ditty. I introduced the 'Rooms to pinterest.com last week and it unleashed a crafting monster mutated from visual stimuli overload. He ate glue sticks for breakfast one day I think\u2026 This in turn unleashed my green-eyed monster, and I spent a couple of days jealous and pouty that I couldn't get in on any of the fun crafting action happening back in Astoria. But alas \"working\" is the necessary evil. After all, I have to put food on the table paint in the cabinet somehow.\nBut check out the wicked awesome pipe\/lamp\/bookshelf extravaganza he screwed together one day. AND wired himself. No pre-done kit for this budding electrician!\nSomeday we want to do the papa bear version like over at The Brick House bloggity blog, full instructions (and shopping list!) here. But again, like much of crafting and DIYing it's not the individual pieces that are expensive, it's the entire loot that would set us back too many rubles at the moment. Thank goodness spring is upon us and it will soon be time for prime street pickins! Hope I find something during the dogs bedtime walk\u2026 see you later!\nHEY! Have you told any of your friends that aren't my friends about this wicked new blog you're reading?\nYeah thats right. Just a remind with the help of my friends the 4 Non Blondes serenading me back to 1992 lounging by the pool with \"What's Up?\" blaring over the intercom. Soak it in. You're welcome!\nLook, click on one of the buttons below to tell your FB or Twitterers, or e-mail your mom or something!\nLook at Sophie (our 3-year-old lioness) just lounging on her favorite bed corner soaking up the setting sun and watching protectively over my magazines that had arrived that day in the mail and were placed so lovingly and artfully on my bed by the mailman\/the 'Rooms.\nForgive her stank face...she's just warning the dog these are no chew items.\nEDIT \u2013 phone cord success, and the new remote is working wonderfully. Take that getting things done that actually need done-ing!\nBuild it Green!NYC is a non-profit \"salvage and surplus materials provider\" in Astoria\/Long Island City that rescues tons (no really, actual metric tons) of reusable building materials and other such things from the landfill. Basically its a big warehouse where the 'Rooms and I go to sift and dig through mountains of junk wonderful treasures in search of diamonds in the rough. Last year we haggled two sweet metal school desks and a couple old windows for 60 bucks!\nIt had been awhile since we had made the sojourn to BIG on the heel toe express. So last week when I opened my e-mail inbox and saw the monthly newsletter and it told me they would be getting a major shipment of furniture from a hospital which would include office chairs, vintage '50s office furniture, medical rolling cabinets, stainless steel rolling tables, and more!!!! I immediately added it to the calendar and forwarded to my roommate. I was so excited I almost made a Facebook invite, but alas, I don't have that much time on my hands over the lunch hour\u2026 Or do I?\nBoy did they deliver. There were so many wonderful things that made my heart skip a beat I was hoping there was a \"not so vintage\" defibrillator floating around, or at least an oxygen tank with a few spare puffs. Thank goodness I was able to pull through and snap a few pics of my faves!\nBecause I'm 7 years old I can't go in places like this and not open drawers and turn knobs and crawl on the floor to get a good look at the undercarriage of something that catches my eye. Evidently the hospital didn't clean out the merchandise before donating because this sweet supply cabinet was chock full of hip screws and plates and leg hinges and other random medical insertions. I wonder if all that was included in the sticker price? P.S. You couldn't pay me enough money to get rid of those sweet raised labels.\nThis baby really brought a tear to my eye. Its like the metal school locker I've always wanted. But hospitalized with glass panes! If I put something away in a drawer or behind a closed door, I'll forget its there. My brain is just too forgetful to put things completely out of sight. I'm constantly teased around here for all of the \"useful\" objects in my room. But what can I say, I demand a pleasing aesthetic and function. They had two of these beauties in the single width and another bigger chunk that was three wide! Oh to have the rooooooom! They wanted $175 smackers for the single tall one and I had about $1.75 in change with which to haggle so I sadly had to leave this gem for someone else to take home.\nOn a side note\u2026 when I walk around places like this and see the sold stickies with names on them I secretly want to tag along for the delivery and meet the new owners of items I'm in love with so at least I can rest assured they'll be going to good homes. Or maybe, I'd meet my soul mate\/future hubby who wanted a medical storage cabinet of his very own (cue sweeping romantic \"meet cute\" music)!\nSo I guess the true moral of the story here is maybe brush off your old \"bar name\" from college and put that on your sold stickies in thrift stores and salvage warehouses, otherwise some creeper like me might google you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahqts b/data_all_eng_slimpj/shuffled/split2/finalzzzahqts
new file mode 100644
index 0000000000000000000000000000000000000000..0689d74a78715a2b7a64eaf53def66ad8dac7dcb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahqts
@@ -0,0 +1,5 @@
+{"text":"Oilfield wastewater contains suspended solids and crude oil droplets; meanwhile, problems of corrosion, scale formation, and bacterial reproduction also exist. Oilfield wastewater needs to be extensively treated before they can be discharged. In this chapter, various wastewater treatments including oil removal, oxygen removal, flocculation of suspended solids, scale inhibition, corrosion inhibition, and sterilization are introduced. The corresponding chemical agents applied and their function mechanisms are also discussed in detail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Barcelona is the capital of the Spanish state of Catalonia and the second-largest city in Spain at 1.6 million, with 4.5 million in the metropolitan area. It is situated on Spain's north Mediterranean coast between the Besos and Llobregat rivers and is bounded on its western side by mountainous terrain. Barcelona was founded by the Romans, was long the capital of an independent state, united with Aragon and Castile to form modern Spain in the late Middle Ages, and is today seeking a referendum on independence.\nBarcelona is the economic center of one of the richest regions in all of Europe. Its GDP in 2012 was $170 billion, the fourth-highest in the European Union, and its per capita GDP ($35,000) is over 40 percent higher than the EU average. Barcelona was one of the first cities on the European mainland to industrialize and has long been known for its textile and mercantile activities. Some of its other main industries are: heavy machinery, chemicals, metallurgy, and energy production. Tourism is also a major source of income since Barcelona is the third most visited city in all of Europe.\nThe Port of Barcelona is one of the busiest of Europe's seaports both for passenger and cargo transport. It is the ninth biggest of Europe's container ports, and in 2013, processed 1.7 million tons. A port enlargement project involving the diversion of the Llobregat river is presently in process. Barcelona\u2013El Prat Airport serves 35 million travelers annually and is the second busiest airport in Spain. The airport is joined to the city via road and rail, and a new terminal was added in 2009.\nBarcelona has an extensive intra-city highway system, and is well-connected to the rest of the Iberian Peninsula, France, and ultimately all of Europe. It is situated on three major routes: E15 which travels long distances along the Mediterranean coast, E90 which joins it to Madrid and then Lisbon, and E09 which runs all the way up to Paris. Barcelona is also a rail hub. It has nine main lines that integrate it with all Catalonia, with the state of Aragon just to the west, and to Valencia just south. High speed lines also connect it with Madrid, France, and all Europe.\nIf you are in need of shipping freight in or out of Barcelona, contact us today to discuss how to optimize the logistics involved. We have agents in Barcelona and Spain who are familiar with the local situation and will be able to assist you in lowering costs while maintaining security and punctuality.\nFor comprehensive logistics solutions that fit your needs from Barcelona to anywhere or, anywhere to Barcelona.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Even though the 2014 midterms are 18 months away, the right wing is working now to work their minions into a froth.\nIRS Incompetence, Yes. Tea Party Innocence? No.\nThose groups whining about their c4 status and the IRS should watch what they say, because it turns out they're not so grass-rootsy after all.\nHead Of The St. Louis Tea Party Vows To \"Destroy The Left\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A lot of events in my data took place in the Bronx. What I am trying to do is write a sentence template which will return \"in the Bronx\" for events which took place there but omit the \"the\" for events which took place elsewhere.\nCan the value switch be used to make that discrimination? I recognize it as an if-then-else construction, but I don't see how to do it in RootsMagic.\nHave you seen TomH's post at the end of this topic? It gives a good explanation of nested ifs.\nThank you, Bob. I had seen Tom's post. My problem is how to set X in <?[X]|a|b> to recognize \"Bronx.\"\nI should be able to do that but the precise technique seems to elude me.\nThank you, Tom. I feared that that was the case. The news means many additional hours of work for me.\nThanks again -- warm Christmas and New Years wishes to you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our organization is widely applauded in manufacturing, exporting and supplying a huge range of Milkshake Mix Powder in Vadodara, Gujarat, India. Highly delicious in taste, these mixes are processed using superior-grade ingredients like milk solids and badam flavor, owing to which praised across the market. In order to retain aroma and freshness of our offered mixes, we pack the same in high-grade packaging material.\nEstablished in 1978, INSTA FOODS has been a leading name in offering top notch Dairy Products. Located in Vadodara, Gujarat, we offer the best quality Milkshake Mix Powder to our customers at best prices in the industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaitfk b/data_all_eng_slimpj/shuffled/split2/finalzzzaitfk
new file mode 100644
index 0000000000000000000000000000000000000000..44909998f01a2c0cc13df28e7ee81b800a18eb58
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaitfk
@@ -0,0 +1,5 @@
+{"text":"Lemuria, an ancient city running on the now forgotten time control technology. Using the force from the stars, channeled through the star altar, they could control time and transform the versatile resource Materia into anything they wished. You are one of the city builders of Lemuria, and you must fight for control of the altar, sometimes cooperate with others, and use your resources to further develop the ancient city. And the time limit is just one year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us? In addition to infantry support, tanks fulfilled traditional cavalry roles, provided mobile artillery support, and were adapted to combat engineering roles.\nAt the start of the war, France had one of the largest tank forces in the world along with the Soviet, British and German forces. This doctrine required the Germans to equip their tanks with radios, which provided unmatched command and control for flexible employment. It is concerned with those parts of Germany US forces held prior to July , with the considerably less than one third of the country that became the US zone in July , and with aspects of quadripartite control pertinent to an understanding of the Army's mission in Germany.\nMedium tanks of weighed around 20 tonnes 20 long tons. Getty Images Dmitry Kostyukov.\nThe Sherman suffered from poor ammunition storage. From the M3A1 version, this gun was gyrostabilized. By they had the fifth-largest tank force in the world behind the Soviet Union, France, Britain and Germany, but were behind in medium and heavy tanks.\nInto the Redoubt. While sharing essentially the same engine as the Tiger I tank, it had better frontal armor, better gun penetration, was lighter overall and thus faster, and could handle rough terrain better than the Tigers. Illustrating the diverse approaches of the Americans, British, and Russians, it analyzes efforts to combat hunger, disease, and crime, preserve cultural artifacts, re-establish industry and utilities, and resolve thorny problems involving currency, housing, education, newspapers, elections, and displaced persons.\nHampered by restricted expenditure in the years leading up the war and still organised for operations in Imperial defence as an expeditionary force, the British Army entered the war unprepared for the very sort of combat its influential theorists such as J. The M48 became the workhorse tank of the Vietnam war along with the M Sheridan light tank.\nM1 Abrams tanks enter Baghdad in Another big issue was its archaic sponson-mounted main gun, which was reminiscent of the tanks of World War 1, and it performed poorly in rough country.\nBut the tanks failed to penetrate German armor. Even the largest and most powerful of German tank destroyers were found abandoned on the field after a battle, having been immobilized by one or more hits by high explosive HE or armor-piercing AP shells to the track or front drive sprocket.\nGerman doctrine stressed the use of rapid movement, mission-type tactics and combined-arms where tanks operated with mobile infantry and air support; this doctrine was popularly called [a] Blitzkrieg. US Politics. Again, though, this design dated from and was obsolete.\nZiemke received an M. The epic battles that eventually rolled back the Nazi advance -- the brutal winter siege of Stalingrad, the clash of thousands of armored vehicles at Kursk the biggest tank battle in history -- had no parallel on the Western Front, where the Nazis committed fewer military assets. This volume provides an authoritative account of the role of the US Army in military government and occupation of Germany from the inception of planning until the relative separation of military government and tactical troops in Production problems with its new engine led to the tank being cancelled after less than 70 had been made.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not as dense like I predicted before.\nMost of this is going to get made into a topical. So, I didnt even bother weighing it.\nI wasnt impressed. It could have been the way I grew to (a few mistakes along the way). Ill be looking for some new strains to grow soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not really called Cherry Blossom Lane, this is NE 20th St in the Hampton Woods Neighborhood in Sammamish. It just seemed like all the trees were in bloom today. White and pink blossoms set against the dark green fir trees and the blue sky just looked like it should be renamed, at least for a few days.\nHampton Woods is just outside Sahalee, near Heritage Hills.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PARIS (Reuters) - President Francois Hollande used the last Bastille Day address of his mandate, less than a year before presidential polls, to paint himself as France's guardian through a time of joblessness and violence that he had brought under control.\n\"I must protect France, it's fragile, it can crack at any moment,\" he told journalists on the national day commemorating the storming of the Bastille and the liberal ideas that gave birth to the French Revolution.\nHe said the Western world and democracy was under threat from terrorism and from wars as well as a rise in populism.\n\"I have gone through tremendous difficulties: the crisis, Greece, the banks, and the attacks of course, but that was my responsibility,\" he said at the Elysee presidential palace.\nHollande, elected in 2012 when the eurozone was suffering the fallout of a Greek financial crisis, expressed confidence he could achieve the cut in French unemployment he has promised.\nUnemployment hovers still close to 10 percent and France has suffered a number of Islamist militant attacks, culminating in the November bomb and gun attacks that killed 130 people. He has consistently proved unpopular in opinion polls.\nOpinion polls show far-right National Front leader Marine Le Pen would likely come first in the first round of the 2017 presidential election.\nAlthough she would lose in a run-off of the two top placed candidates, Hollande himself could struggle to reach that run-off ahead of candidates of the mainstream right, the polls show.\nHollande said Britain's decision to leave the European Union showed, by causing economic turmoil, the dangers of policies such as Le Pen's plan to leave the currency union.\nAfter weeks of strikes and protests against a labor reform bill he rammed through parliament in an effort to curb unemployment, he said he would not apologize for the choices he made.\n\"I made the right choices for the French economy,\" he said, defending a pro-business U-turn a year and a half into his mandate after having risen to power in 2012 vilifying the financial sector as his biggest enemy.\nHe said taxes would be lower at the end of his mandate than in 2012 but set a condition that further tax cuts for households would depend on growth of about 1.7 percent next year, which he said depended on the consequences of the fallout from Brexit.\nHollande also warned his youthful economy minister, Emmanuel Macron, who was harshly criticized by other ministers for holding a political rally that smacked of rivalry to his president, that he should respect the rules of cabinet solidarity, but he struck a broadly forgiving tone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiwnq b/data_all_eng_slimpj/shuffled/split2/finalzzzaiwnq
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index 0000000000000000000000000000000000000000..3d228142ae03e73f60e53972560201b1ac7896e3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaiwnq
@@ -0,0 +1,5 @@
+{"text":"Controls for Fan coil, DX unit, Modulating,EC motor and so on.\nControls for electric heating cable,electric heating mat,electric heating panel,electric film,carbon crystal plate and so on.\nControls for water underfloor heating.\nControls for water heating boiler.\nBy decades HVAC experience, we understand client request, and how to transfer the concept to real qualified products.\nPlastic injection , PCB soldering , products assemble , packaging , we pay attention to every detail process .\nEvery SASWELL products pass 100% check and test . We guarantee that every batch order will maintain quality consistency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Parking Lot will take you to your worst nightmare if you have problems parking your real car. In this flash game, you need to park a virtual car perfectly. There will be numerous situations that require different parking styles. Therefore, Parking Lot can be a real nightmare if you do not know how to park correctly in the first place. The Parking Lot game comes with something new in the area of free online games and this is why it has so much success. Parking a virtual car can sometimes be even much harder than parking your real car. The diversity of situations, the good graphic component and the innovative aspect of this game recommend Parking Lot as one of the best ways of having some good time online at Pukmo. If you decide to play it, you should be prepared for parking virtual cars in small spaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Walcha Library is part of the Central Northern Regional Libraries Network and has access to talking books and sound recordings, videos and large print books. These can be reserved and renewed online as well as in person at any CNRL Branch Library.\nThe Walcha History Centre is located in a room at the rear of the Library and is ideal for local historical research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"National Adoption DAY!!! Want to know one of the worst parts?\nToday is a day that I am so proud to now be apart of. National Adoption Day.\nTyler and I constantly talk about our future kiddo now. While decorating the Christmas tree we talked about how we can't wait until we have our baby home and they get to help decorate the tree and help put the stockings on the fire place. We can watch Christmas movies and bake fattening treats that we surely don't need.\nWhen I walk through a book store I stop and look at the children's books. Thinking of putting them to bed with a bed time story. When Tyler and I are piled up with our pups on the couch at night watching a movie (yes we allow our 9 lb. dogs on the couch, don't judge!) we think of 3 of us on the couch watching a movie! This goes on and on and on, dreaming of having our child home and out of an orphange, out of institutionalized care.\nWhen flying home from TN on Monday I returned a phone call from Jim who is our director at our adoption agency and we talked for almost 2 hours. He explained everything in detail to me. This is a new program so he explained when he first went to the Czech Republic to meet the officials and visit the orphanages, he was surprised to see that the baby\/toddler home was not a home at all. It was a ROOM off the side of a hospital where they would put all the children in and hospital employees would watch over them. This was so hard to hear, that these kids were LIVING in a room off the side of a hospital. Jim told me about how when we first arrived we would meet with the child for the first time in Brno at the orphanage with a psychologist there with us just to see how our first interaction with the child went. We will then sleep at the orphanage in a separate room they have for the parents and have 2 more meeting with the psychologist observing. Once he approves things are going well we then will be able to leave with our child and get our own apartment for for the next few weeks. We will then have another meeting with the psychologist to see the progress and he will again sign off with his approval and we will then fly to Prague to the courts to have the visa and other documents finalized to come home.\nI took it all in, every single bit of information he was giving me. I want to know everything. I couldn't wait to call Tyler and tell him about it! There are MANY hard things about adoption\u2026the waiting, the paper work, the wondering, the critics, the now all of the sudden \"adoption experts\" that want to give their input or negative comments. This is all part of it and I let it roll off my back, but truly one of the hardest parts for both of us but definitely more Tyler is the asking for help..part. The truth is we could never do this on our own. No matter how hard we tried we wouldn't be able to save $30,000 in the next year and half.\nWhen we were still talking adoption and hadn't started the process I was looking at adoption stuff on ETSY and ran across a shirt that says \"It Takes a Village\" I showed it to Tyler and we both thought, how does this have anything to do with adoption. But, now I get it. It truly DOES take a village. Adoption cannot be done without others. Not just the financial help but also the support, the late night texts with a fellow adoptive moms waiting and shoulder deep in paper work as well. Without the random messages of how we inspire people, (what??!, we inspire people?) this made my day. and most importantly the prayers, those late night prayers to God about all of it.\nso it does\u2026TAKE A VILLAGE. Our child will know how many people helped get them our of an orphanage and in our family. It was never just us, it took so many people that loved them enough to help bring them home!\nSo with that being said, We just finished out home study..WOO HOO!! Once it is finalized and complete we will then be eligible to apply for GRANTS!!! but for now, we will owe $3,000..SOON! The grants, if we get some will help us with travel to pick our baby up, Which will cost around $10,000 for flights for all 3 of us, an apartment for 4-6 weeks and food, translator and everything that comes along with being in a foreign country for that long.\nSo in honor of NATIONAL ADOPTION DAY, please consider skipping dinner or a coffee and investing in us, in our family, in our child, in making one less orphan..living in a room off the side of a hospital in the Czech Republic somewhere. I don't ask this lightly, I don't ask this thinking you SHOULD donate to us because we are adopting, I ask this with hesitation and desperation. I ask this KNOWING I am asking you to donate to us the money that you have worked all week for! The money that you earned while being away from your family.\nWe are so grateful for each and everyone of you and your support! No need to donate a lot but if just 300 people donated $10, this portion of our adoption would be taken care of!!! If you aren't able to help financially, would you consider sharing this with your friends?\nbecause I'm human and he's God..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whenever someone purchases one of Patrick's belt buckles a question often arises: How to find the perfect belt for it?\nAll Crocodile skin is sourced from certified ethical crocodile farms. Sizes: 32 - 42 inches. Please specify size when ordering.\nRecommended sizing: order one size above waist size, e.g. 38 inch belt for 36 inch waist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiynk b/data_all_eng_slimpj/shuffled/split2/finalzzzaiynk
new file mode 100644
index 0000000000000000000000000000000000000000..cad1566d041e3fea30813afb1d3a96161bbae045
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A Labour supporter has gone to extreme measures to show their support for the party in Leyland \u2013 by vandalising a wall in the town centre.\nSpray paint which reads \"Vote Labour\" could be seen on a wall on Meadow Street, off Hough Lane, over the weekend.\nSouth Ribble's Labour leader, Coun Matthew Tomlinson, said: \"Obviously we don't condone this sort of behaviour in any way and I must say I would be very shocked if it was someone actually connected to the Labour Party locally who has done this.\n\"In a way though, it's nice to see that people are taking an interest in the upcoming elections.\nRound the corner, on Thurston Road, similar sprawlings read \"Don't buy the Sun\" in capital letters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VIDEO POWER PACKAGE: Our most energetic, highly strategic, motion filled video message available. Normally $1,299 now $999, $300 off...an incredible price! Multiple on-camera live announcers with multiple hi-end backgrounds, strategic script, and loads of \"eye candy\" appeal. Just provide a 30 second, 75 word script or ask us to write it for you. Increase sales, social media, web visits with this powerful video.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We offer a number of different workshops, and you are certain to find one to suit your needs \u2013 read full workshop descriptions.\nKit and Olivia will be on sabbatical from workshop presenting (and travel) for the first six months of 2019. The workshops listed below are the only workshops we will present in 2019.\n* We offer two prices for all workshops, an Early Bird and a Regular Registration. The Early Bird is a significant discount, and it is the point at which we decide whether to confirm a workshop, based on paid registrations received.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These furry boot covers are great with our animal or goth costumes! Measuring 10\" from toe to heel and 13\" tall, these fasten to the bottom with an elastic strip. Made out of a black acrylic and polyester blend, these boot covers will become a versatile component to a variety of costumes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Experience the best of both worlds with our new linen print shaped tee shirts. These shirts are made with the comfort of Pima cotton paired with the look of linen.\nThe style is available in a two custom custom-dyed and printed colors. Free standard shipping on all orders.\nHandpicked, Peruvian Pima cotton retains shape with minimal shrinkage. Printed with a linen pattern. Proprietary finish for a soft hand and lustrous look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajkjp b/data_all_eng_slimpj/shuffled/split2/finalzzzajkjp
new file mode 100644
index 0000000000000000000000000000000000000000..4c4fdec0730110947dfa3177bfae2e1c6cb38682
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A green light for the world's biggest offshore wind farm off the Yorkshire coast has been given by Business and Energy Secretary Greg Clark MP at the recommendation of the Planning Inspectorate.\nThe Hornsea Project Two offshore wind farm, located 89 kilometres off the Yorkshire coast, will have a capacity of up to 1.8 gigawatts and will consist of up to 300 turbines. It could meet the electricity needs of approximately 1.6 million UK homes per year.\nHornsea Project Two is being developed by SMartWind, owned by DONG Energy.\nThe development decision has been praised by the UK's key environmental campaigning groups. Friends of the Earth welcomed the news, their campaigner Guy Shrubsole said: \"This is fantastic news for local communities and the environment. This massive wind farm will provide enough clean electricity to power almost two million homes and create hundreds of new jobs.\n\"Developing the nation's massive offshore wind resource could also boost demand for UK steel and help repower British manufacturing.\nDr Doug Parr, Greenpeace UK Chief Scientist said: \"If our government can exploit the economies of scale to make offshore wind as cheap as it could be, whilst taking into account the wildlife in the area, the North Sea will be providing us with energy long after the end of the oil age.\n\"This is good news as the rapid development and falling price of renewable power is making it very clear where the future of the energy industry should lie. And it certainly isn't with propping up 20th century technologies like oil, gas and nuclear. This is yet another nail in the coffin for the rationale for Hinkley power station.\"\nThe Development Consent Order approved by Greg Clark MP covers the entire project including the turbines, foundations, offshore and onshore substations, array cables and export cables.\nBrent Cheshire, DONG Energy's UK Country Chairman, said: \"Development consent for Hornsea Project Two is very welcome. We have already invested \u00a36 billion in the UK, and Hornsea Project Two provides us with another exciting development opportunity in offshore wind.\n\"A project of this size will help in our efforts to continue reducing the cost of electricity from offshore wind and shows our commitment to investing in the UK.\"\nHuub den Rooijen, Director of Energy, Minerals and Infrastructure at The Crown Estate, said: \"Offshore wind is already on course to meet 10% of the UK's electricity demand by 2020. Major developments of Hornsea Project Two's scale will pave the way for its continued growth alongside driving down costs, creating high value jobs, and supporting the UK's transition to a low carbon energy supply.\"\nDONG Energy will now review the details of the Development Consent Order. DONG Energy UK already have 8 operational offshore wind farms in the UK with another 4 under construction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UNION LAKE, Mich. \u2013 (December 5, 2018) \u2013 Lindenwood University retains the top spot in the 2018-2019 American Collegiate Hockey Association Men's Division-I Ranking, No. 9. The Lions are done with games in 2018 and finished the first semester with a perfect 16-0-0 record.\nThe University of Michigan-Dearborn (7-0-0) solidified its hold on the No. 2 spot after handling No. 3 Adrian College 5-3 on the road at Arrington Ice Arena in Adrian, Mich. Justin Dunn led the way with two goals including the game-winner while Thomas Proudlock was bullish in net making 45 saves on the night. Michael Barrett turned away 40 shots in a losing cause, the Bulldogs first loss since November 18, 2017.\nAdrian College was able to bounce back from its first loss with a home and home conference sweep of Kent State University to up its record to 15-1-0 on the season. Rainer Glimpel paced the way on the weekend with two goals and three assists while defenseman Jordan Oshowy had a goal in both games.\nNo. 4 Ohio University improved to 11-2-3 after a home conference sweep of Robert Morris University Illinois. Gianni Evangelisti led the attack with three goals and an assist while Jimmy Thomas got both wins including a shutout on Saturday night.\nRounding out the top 5 was idle Iowa State University (10-4-0).\nMoving into the polls this week is No. 21 Illinois State University while Western Michigan University drops out of the Top 25.\nThe Central States Collegiate Hockey League leads the way with all five of its member teams ranked, while the Western Collegiate Hockey League has seven (of 8) of its member teams ranked, the Eastern States Collegiate Hockey League has four (of 6) of its member teams ranked, and the Great Lakes Collegiate Hockey League has three (of 8) of its member teams ranked.\nThe tenth ranking of the regular season will be released on Wednesday, December 12, 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When a traumatic event occurs in a workplace, we can deploy on-site within 24 hours, one or more qualified counsellors to receive individually or in groups the employees affected by the event. This is an essential part of a crisis intervention, as it will enable to identify the most upset and traumatised workers, to keep the cohesion in the workplace, and to limit the impact of delayed response to trauma.\nSupport groups where everyone can express himself and be heard by his colleagues.\nIndividual interviews where the employee can address the critical incident about his personal experience and will be guided and advised on the way to process this particular event.\nSupport groups are recommended when a malfunction, interpersonal conflicts or tensions occur in a workplace.\nA counsellor will come on site to receive a group of 10 persons (max). The counsellor's mission will be to moderate the communication between people, to ensure that issues are laid out and addressed, and to restore the trust within the team.\nAll participants are encouraged to speak freely when their turn comes. Together, the group will develop a roadmap that will frame a new approach in response to a particular issue. The counsellor will send a copy of the agreed roadmap to each participant. The participants can then follow up with the same counsellor over the phone via the online portal www.pro-counselling.co.uk.\nWhen an interpersonal conflict affects the operation of a service, a counsellor specialised in mediation comes on site and meet the two employees, in a place that ensures the confidentiality of the conversations. Employees can express their feeling, disagreements, and subjects left unsaid.\nThe mediation aims to restore a climate of serenity and to avoid major crisis and distress of employees. It also enables to tackle the conflict before taking the risk it is brought to justice.\nFace to face interviews consists of one or more one-on-one counselling sessions between a counsellor and the employees. Face to face interviews with a counsellor can be a response to an urgent problem or crisis occurring in the workplace.\nThe interviews take place within the company's premises and last 45 minutes minimum. The number of interviews will be agreed between the counsellor and the manager of the service, depending on the assessment of the situation and the distress of the employees.\nOur counsellors are all qualified psychologists and are practising under the code of ethics of their profession. Therefore, they are guarantors of the professional aspect of these interviews and their confidentiality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SGS has a global network of laboratories, offering you an unparalleled level of analytical expertise around the world. For gold and silver bullion analysis and assay there are many different methods available, for example silver bullion can be analyzed by cupellation, by chemical titration, or by a combination of chemical titration and instrumental techniques.\nSGS experts are always available to assist you with selecting the analytical technique that is fit for your purposes. In fact, the critical aspect of bullion analysis is proper sampling. The gold or silver must be sampled while it is still molten. The most reliable technique involves the use of a vacuum sampler.\nOnce sampled, gold bullion samples are usually about 0.5g in size and are cupelled with enough silver to give a Ag:Au ratio between 2:1 and 4:1. This will result in a successful parting. The bead remaining after cupellation is rolled into a very thin strip called a coronet. The Au result is obtained by parting (dissolving the silver with nitric acid to leave the gold) the coronet in dilute nitric acid. This dissolves Ag but not Au, and the remaining Au is weighed. A sample of proof Au of the same weight as the sample is treated at the same time. The sample result is corrected for any gain or loss (called the surcharge) according to the result from the proof gold. Samples are usually run at least in duplicate.\nSilver bullion analyses can be analyzed by cupellation, by chemical titration, or by a combination of chemical titration and instrumental techniques. The cupellation process is subject to volatilization errors that must be corrected. The errors inherent in the cupellation method can be avoided by the wet chemical method. This method consists of dissolving a 1g sample of bullion in nitric acid. After this stage, the finish can be either titration or instrumental.\nSGS has an intensive, on-going program to monitor quality in all our labs. Dozens of our labs carry internationally and industry recognized certifications and accreditations. Scopes vary among laboratories.\nPartner with SGS today and find out how our expertise can give you accurate and reliable gold and silver bullion results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Further to considerations of simony, syncretism and sacred service in the Tercentenary of English Freemasonry to be hosted by Canterbury Cathedral on Saturday 18th February, published below is the Order of Service for that celebration. It is secret (or was). It arrived by magic (not black), but is being made public in the interests of openness and transparency: there's nothing worse for feeding occultic hysteria than the mystery and secrecy of \"Cathedral closed.. for a private service\", when the means of that 'private' and the content of that 'service' are causes of spiritual tension, if not theological contention.\nThe Service has been prepared in conjunction with the Dean of Canterbury Cathedral, the Very Reverend Dr Robert Willis, who has kindly agreed to deliver the Sermon on this occasion. We shall be joined by several of the High Rulers in the Craft and the Holy Royal Arch Chapter together with Brethren from the Provinces of East Kent, West Kent, Surrey and Sussex.\n..it will come as a relief to note that there no invocation of Jahbulon, no perambulation around the points of a compass toward enlightenment, and nowhere is the congregation asked to sing 'Praise, my soul, the King of heaven' with a rolled-up trouser leg and a noose around their necks. Indeed (for those who quantify these things), Jesus is mentioned eight times, Christ 10 times; the Holy Trinity is lauded \"To Father, Son, and Holy Ghost\", and God is \"our rock and salvation\".\nWe can't know what the Dean will say in his sermon (which is unlikely to be made public) but, on the face of it, this is a wholesome Anglican service of orthodox Christian liturgy.\nBut here's the interesting thing..\nThe scriptures chosen have greater significance (and a very different meaning) to Freemasons than they do to (other) Christians. Indeed, they are used at the highest degree of Masonic initiation in their rituals of exaltation.\nMr A.B., a poor Candidate in a state of darkness who has been well and worthily recommended, regularly proposed and approved in open Lodge, and now comes of his own free will and accord, properly prepared, humbly soliciting to be admitted to the mysteries and privileges of Free\u00admasonry .\nHe is then perambulated in a clockwise direction around the Temple, from the darkness of the North toward the light. The psalmist's 'Walk about Sion' is linked to this rotation. Thus is the north-side, the city of the great King, presented as the place of darkness, from which the candidate moves toward true enlightenment about the nature of the \"Almighty Father and Supreme Governor of the Universe\" (and note the third prayer led by the Archdeacon begins 'O Lord our Governor', a very particular translation of Psalm 8 drawn from the BCP 1689).\n..The three great pillars supporting a Freemason's Lodge are emblematic of those Divine attributes, and further represent Solomon King of Israel, Hiram, King of Tyre and Hiram Abif.\nQ \u2013 Why those three great personages?\nA \u2013 Solomon King of Israel for his wisdom in building, completing, and dedicating the Temple at Jerusalem to God's service; Hiram King of Tyre for his strength in supporting him with men and materials; and Hiram Abif for his curious and masterly workmanship in beautifying and adorning the same.\nThe characters, King Solomon, Hiram \u2013 the King of Tyre and Hiram Abiff are all taken from the Scriptural account of the temple building. King Solomon and Hiram King of Tyre are mentioned many times in the Scriptures, such as in 1 Kings 5. About the closest the Scriptures come to Hiram Abiff is Huram-Abi which is found in 2 Chronicles 2:13 in the NAS and NIV translations. Huram is a variant of Hiram. In the KJV translation of the verse, the name Hiram is found. The KJV uses both Huram (2 Chron 2:3) and Hiram (1 Kings 5) to identify Hiram the King of Tyre. The KJV translation of 2 Chron 2:13 does not contain -abi, but rather \"Huram my father's.\" The Hebrew word from which the KJV \"father's\" was translated is \"'ab,\" according to the Hebrew Dictionary found in Strong's Concordance. Strong's entry for the word 'ab (H1) indicates that it can also mean father-less, as the son of a widow would be. The entry for H1 also mentions \"Abi-.\" Studying the various translations along with a Hebrew dictionary allows us to see how Freemasonry may have settled on the name Hiram Abi-ff, also sometimes spelled Abif.\nThere's a lot of other stuff all over the internet on the legend and drama of Hirem Abif(f), but the provenance isn't always clear. It is sufficient for the purposes of this post to note that the inclusion of the 2 Chronicles account is an allusion to the \"Divine attributes\" of the Solomon-Huram-Hiram trinity, who represent \"Wisdom, Strength, and Beauty\", represented and celebrated in the classical orders: Ionic, Doric, and Corinthian (hence the Freemasons' sponsorship of apprentice stonemasons).\nIt is customary at the erection of all stately and superb edifices to lay the first or foundation stone at the North East corner of the building; you, being newly admitted into Masonry, are placed at the North East part of the Lodge, figuratively to represent that stone: and from the foundation laid this evening may you raise a superstructure perfect in its parts and honourable to the builder.\nFirst, the ready compliance of Abraham with the will of God in not refusing to offer up his son Isaac as a burnt sacrifice, when it pleased the Almighty to substitute a more agreeable victim in his stead. Secondly, the many pious prayers and ejaculations of King David, which actually appeased the wrath of God, and stayed a pestilence which then raged among his people, owing to his inadvertently having had them numbered. And thirdly, the many thanksgivings, oblations. burnt sacrifices, and costly offerings which Solomon, King of Israel, made at the completion, dedication, and consecration of the Temple at Jerusalem to God's service. Those three did then, do now, and I trust ever will render the ground of Freemasonry holy.\nQ \u2013 What do you learn by being a Free and Accepted Mason?\nA \u2013 Secrecy, Morality, and Good Fellowship.\nThere's nothing wrong with morality and good fellowship: these are virtuous Christian pursuits. But we might well ask the Dean of Canterbury Cathedral why so much of this Order of Service is steeped in covert references to Masonic ritual, with twisted scriptural interpretations known only (now not quite) to the initiated? Why the secret syncretism?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"There are several reasons why your computer is freezing \/ slow. The first question is when the last time you cleaned your registry. Having a corrupted registry can cause a lot of errors and problems. Freezing is one of them. If you have a lot of unnecessary registry keys that are hogging your computer's resources. Your computer is bound to choke. Scan your registry with a good registry cleaner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I will show you how to add subcaptions!\nDuring your acupuncture treatment I use FDA-approved Bio-Mat technology. It combines far-infrared light and negative ions with the healing power of amethyst crystals.\nAll you have to do is relax and let it work!\nCoastal Acupuncture, all rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are looking for a change in how your home appears, doing the home renovation process is the best thing. It is an excellent feeling to enter your home and see that your creativity and your personality are represented in how it looks like. Most of the time home renovation is usually done when someone wants to sell their home.Nevertheless, You have to be very careful during this procedure because it is not always successful without the right elements in place. It is vital for you to write down the essential aspects required for the home renovation so that everything will take place without any constraint. It is essential that you have a vision of how you want your home to look after the renovation so that it can act as a blueprint for the constructor. This article is going to highlight some of the things you need to put in place so that you are happy with the home remodeling process.\nYou will never go wrong if before the renovation process you write down the budget which is going to they do not overspend the money you have in your account. It is beneficial for you to have a budget already in place before you can pick the type of energy and materials that you think are important to make your dreams come true in there remodeling process. It is not uncommon to come across people who did not respect the budget they had and ended up overspending the cash they had in their bank account. Managing your budget should be one of yours actives because it will save your lot of headache when it comes to dealing with creditors.It is a good idea for you to ask the assistance of a professional constructor to help you come up with a suitable budget.\nIt is advised that you give the job to a professional contractor who is famous for handling remodeling procedures. You are likely not going to succeed at remodeling because you have not been trained and you lack the understanding on doing a good job.It is a good idea for you to involve them because they have the expertise needed for the remodeling because they have handled similar cases before. Having the right tools is also something to consider, and professional constructors possess all of them to enable them to work well. You are likely going to suffer financially if you are forced to purchase every machinery that is going to be used in the remodeling process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Voice recognition software is a relatively new creation in today's technological world, and thus, many people don't yet know about voice recognition uses.\nWhile many people may think they know what it is, how it works, and what it's for, many people are surprised to learn just how many unexpected and unique advantages that voice recognition software can give to people and how many benefits that simply get overlooked by the casual consumer if they're just doing some quick research about the technology.\nLet's take a closer look at the unexpected benefits of voice recognition software and see just what applications are possible with this great new technology.\nHaving a good speaking voice is one of the biggest skills someone can have in today's business world.\nBy using voice recognition software, you'll have a chance to practice your enunciation, speaking rhythm, and overall clarity.\nIn addition to that, since voice recognition software will make people speak slowly and clearly, it's a great benefit to people who suffer from things like dyslexia or even people who get nervous about speaking in public.\nWe've all heard about the dangers of texting and driving, and with voice recognition software, people will be able to create emails and write text without having to divert their eyes from more important things.\nThis also means it's a great tool for multitasking, as with practice, you'll be able to focus on one thing while still writing documents with your voice.\nSince voice recognition software technology requires only the voice, it makes it a great option for people with handicaps who aren't able to type using a traditional keyboard.\nThis brings the power of modern day communication to people who have originally only been able to access it at a slower pace or even not at all.\nIn addition to that, using voice recognition software significantly help the population at large when it comes to suffering from things like arthritis, which can come about from sitting at the computer and typing too much during the day.\nWorking in an office job can be cushy, but sitting at a desk all day can be quite dangerous on your health and your posture, such as your back.\nVoice recognition software technology can be utilized so that you'll be able to get your job done even faster while being up and active.\nOn the flip side of that, it's also a great way to do some quick work while still in bed and lying down for people who like to work early in the morning or late at night.\nOverall, there are a ton of different unique benefits that people get from trying out voice recognition software, and the industry itself will begin learning more and more about people's unexpected applications of their program the more that it's out and in the hands of people.\nThere are a ton of different jobs out there, each with their own special requirements, so finding out the best way for you to utilize the advantages of voice recognition software is going to depend on you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Students, are you looking for something fun to do on a Friday night? Come hangout, play games, strengthen friendship, and just have a blast at YGN! This event is FREE to attend and is open to all middle, high and college students through age 22.\nFor more information, see or contact Pastor Daryl.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajzhv b/data_all_eng_slimpj/shuffled/split2/finalzzzajzhv
new file mode 100644
index 0000000000000000000000000000000000000000..7f54401ec2095dbf535fc3fcfc2af4b43740bb96
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Did you feel left out of all the hype at the Golden Globes in L.A. this year? Well, you're in luck. On June 7, The Restaurant Association Metropolitan Washington (RAMW) will be hosting its own fancy awards ceremony in DC \u2014 The RAMMY Awards Gala.\nNo doubt, DC's culinary industry is top-notch\u2026 and Monday night's RAMMY kickoff event highlighted how many people support it. We can't wait to see who wins this year \u2014 tickets to the big night currently available!\nPrevious Post: Last Night's Party: MokiMedia Celebrates 5 Years!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This example demonstrates fabrication and second breakdown analysis of a short channel structure with non-isothermal energy balance models.\nDeep submicron devices should be simulated using the Energy Balance Model due to velocity overshoot, and nonlocal impact ionization effects, which could substantially influence device characteristics. For high current levels, the thermal self-heating effects can also play an important role by decreasing mobility and impact ionization rate. This example demonstrates second breakdown simulation with the non-isothermal energy balance model using curve tracing algorithm.\nA fully coupled approach is used to solve five equations in this device. This provides much improved convergence over decoupled techniques used in older Atlas versions.\nThe workfunction, interface charge and MOS models are set using the syntax as described in earlier examples. The energy balance equation for electrons and lattice energy balance equation are set by models hcte.el lat.temp . The impact statement is used to assign the energy relaxation length for Selberherr model. Thermal boundary conditions are required with any non-isothermal simulation. These are defined in the thermcontact statement. A value of the thermal conductance is specified at the thermal contact located along the substrate. Thermal isolation conditions are assumed on the all other surfaces.\nThe curvetrace statement is used to initialize parameters for curve tracing algorithm. A full description of the curve tracing syntax is included with the MOS snapback example. The gate voltage is ramped to 1V, and then the solve curvetrace statement is used to activate the curve tracing algorithm.\nPlotting the internal drain bias versus drain current in TonyPlot will display the second breakdown curve. The resulting curve from Atlas shows several turning points. The second breakdown occurs at the final stage where the voltage drops sharply. At this point both the lattice and carrier temperatures are high.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following general conditions regulate the use of the site www.eventool.com (from now onwards 'EVENTOOL') that Infobox Solutions, SLU, offers through internet and or additional channels under exclusive ownership (from here onwards, 'GENERAL CONDITIONS').\nInfobox Solutions, S.L.U., is a trading society with legal address in: Castillo de Quejana 9, of. 15, 01007, Vitoria, Alava, Espa\u00f1a (Spain); with CIF number B-01396316 and undescribed in the Business Registry of the city of \u00c1lava, (volume) Tomo 1236, (book) Libro 0, (page) Folio 104, (sheet) Hoja VI-11302, Inscripci\u00f3n 1\u00aa (from here onwards, 'INFOBOX').\nThe use of EVENTOOL attributes the condition of user and implies the full and unreserved acceptance of each and every one of the provisions included in this GENERAL CONDITIONS (from here onwards the 'USER'). Such provisions could be modified by INFOBOX, being the user responsible to check every time he\/she wishes to use any of EVENTOOL's services. To assist USERS and for them to learn about these modifications, at the end of the page the date of the last version will be shown.\nThe use of the services offered to USERS through EVENTOOL, are ruled by its own particular conditions that, according to the specific case, substitute, complete and\/or modify the GENERAL CONDITIONS (from here onwards, 'PARTICULAR CONDITIONS'). Therefore, beforehand to the use of said services, the USER must read and accept the corresponding PARTICULAR CONDITIONS.\nMoreover, the use of EVENTOOL is under same advice and instructions made available to USERS of INFOBOX.\nThrough EVENTOOL, INFOBOX allows USERS access and use of the various types of content and services (from here onwards, the 'SERVICES').\nINFOBOX does not offer any copyrighted information or those under intellectual property rights from third parties as its own.\n3.1. Commercial character and use of EVENTOOL.\nThe SERVICES are subject to contract of accessibility and payment of a price determined in the PARTICULAR CONDITIONS.\nThe use of SERVICES will be subject to the USER registration in the system. Said registration will take place in the way expressly indicated in the GENERAL CONDITIONS as well as in the PARTICULAR CONDITIONS that regulates it.\nAll the Information given by the USER through the SERVICES must be true. To that effect, the USER guarantees the authenticity of the information provided as a consequence of the requirement to fill in forms necessary to subscribe to the SERVICES. Likewise, it will be the USER responsibility to constantly maintain and update all the information provided to INFOBOX, so this can respond, at every moment, to the real situation of each USER. In any case, the USER will be the only responsible for inexact of false manifestations made, and for the damages caused to INFOBOX.\nThe USER will not publish or take any action in EVENTOOL that would infringe the current legislation applied. Likewise, the USER should not publish, manage and\/or transmit information that may result offensive or against moral and public order.\nThe USER will be able to use the different sections of the EVENTOOL web to publish information appropriate for the kind of information and indications of each section. INFOBOX reserves the right to not publish information that may be deemed inappropriate, or likewise, to use its own judgement to adjust those things accordingly.\nThe information supplied to the USER from EVENTOOL, will be done in a confidential manner for the exclusive use of the client, who, under no circumstances will be allowed to copy, distribute, execute, reproduce, license, transfer or sell any type of information extracted from the EVENTOOL website, unless previous express written authorization from INFOBOX.\nIntellectual and industrial property, commercial brands, graphics, images, logos, information and visible icons on EVENTOOL web, that have not been brought in by the USERS, are exclusive property of INFOBOX.\nAll content and format are protected by national and international law. The use for profit of any kind, of the content on EVENTOOL web, is totally forbidden, without specific INFOBOX written authorization.\nThe USER states, under his\/her responsibility, to be the owner or have the rights of any files or material he\/she publishes on EVENTOOL.\nINFOBOX has thoroughly tested the system in order to guarantee the best availability in the provision of service, however, the software may occasionally be interrupted for maintenance, software or hardware error or other reasons of cause majeure.\nINFOBOX does not guarantee availability and continuity of the SERVICES. When reasonably possible, INFOBOX will warn in advance of potential interruptions in the functioning of the SERVICES. INFOBOX does not guarantee the utility of SERVICES for any specific activity or its infallibility.\nUnder the above statement, INFOBOX is exempt of any responsibility for damages or inconveniences of any nature due to the lack of availability or continuity of the SERVICES operations.\n5.2. Warranty exclusion and responsibility limits for Content.\n5.2.2. Quality, veracity, accuracy, exhaustiveness and reality.\nINFOBOX makes a lot of effort in order to obtain a high quality of content provision. However, it is not possible to guarantee the accuracy of all the information due to: - dependence of information provided by third parties; the huge volume of processed information; the application of automatized informatics processes, subject to hardware and software errors. Under the previous statement, INFOBOX does not guarantee veracity, accuracy; current and comprehensive nature of the content.\nINFOBOX will not be held responsible in any case for damages arising as a consequence from the inaccuracy or errors of the information provided.\n5.3. Warranty exclusion and responsibility limits for information, content and services hosted outside EVENTOOL.\nEVENTOOL may make hyperlinks available to USERS that would allow them to Access third party owned or managed websites (from here onwards, LINKED SITES). The installation of these links, directories and search tools in EVENTOOL have the only purpose of facilitating the search on internet and use of contents, information and services by\/for our USERS.\nINFOBOX does not, directly or by third parties, offer or commercialise information, contents and services available on the LINKED SITES, therefore, INFOBOX is not held responsible for the availability, continuity and utility; nor for the quality, accuracy and truth in this information, and is exempt of all responsibility for damages caused by this.\nINFOBOX could withdraw or discontinue at any time and without previous notice, the purveyance of SERVICES to those USERS that not comply with the established rules in the GENERAL CONDITIONS.\nCollection and automatized treatment of PERSONAL DATA aims to maintain the contract with INFOBOX, Management, Administration, purveyance, improvement of services that the USER may choose to subscribe to, register for, or use customised services and preferences and choices of USERS, the study of service use by USERS, the design of new services related to said services, the updates of services, the submission be it by traditional or electronic methods, of technical, operational and commercial information, about products and services offered by INFOBOX at present and in the future.\nINFOBOX has adopted the required levels of protection for PERSONAL DATA, and seeks to install further measures and other available ways to avoid loss, misuse, alteration, non-authorised access or theft of PERSONAL DATA given to INFOBOX.\nUSERS have the right of access, cancellation, rectification and opposition; as well as the right to be informed of all PERSONAL DATA transfers. The USER, or its legal representative, could exercise these rights by written request to INFOBOX, using certified mail, signature upon arrival postage; in which states the request, accompanied by a legal identification document and or document which accredits the legal representation, to the address listed in the notifications section.\nAll communications and notifications (from here onwards 'NOTIFICATIONS') from the CLIENT to INFOBOX will be deemed effective, when they are carried out in one of these ways: (a) By Post to the following address: Infobox Solutions S.L. Castillo de Quejana 9, of. 15, 01007, Vitoria, Alava, Espa\u00f1a; (b) Electronic mail to: info@infoboxsolutions.com. All NOTIFICATIONS from INFOBOX to the USER will be deemed effective, to all effects, when they are carried out in one of the following ways: (a) Post; (b) Fax; (c) Electronic mail.\nThe GENERAL CONDITIONS as well as the PARTICULAR CONDITIONS are ruled in full by Spanish Law.\nBefore any dispute that may arise, both parts commit to surrender any other charter to which they may belong to, to the discretion and command of the Courts and Tribunals of Vitoria-Gasteiz (\u00c1lava, Espa\u00f1a\/Spain).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quality of animal feed depends on the quality of the raw materials. This is why we invest our time, energy, and money to acquire raw materials of the highest quality.\nWe produce animal feed mostly from grains harvested on Polish fields. We conclude long-term contracts with grain suppliers, which guarantee their financial safety and are competitive with regard to prices.\nWe focus on buying grains with stable high quality: wheat, triticale, barley, rye, oat, and grain mixtures, as well as corn and legumes. We offer better prices for good quality of grains, especially cleanliness and, in the case of cereals, also protein content.\nFor many years, we have specialized in purchase, professional preservation, and storage of grains, including corn.\nA comprehensively equipped laboratory quickly and thoroughly evaluates delivered grains. Modern NIR equipment instantaneously performs full analysis of the quality of received grains.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The famous Snow White and the Huntsman film celebrated the universal story of Snow White (by the Brothers Grimm) in a dark atmosphere, diverging from the child friendly Disney version. The darker nature of this theme will make for an excellent choice for Halloween or any Gothic themed fancy dress event. The Snow White and the Huntsman Costume Collection is fully licensed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapmtz b/data_all_eng_slimpj/shuffled/split2/finalzzzapmtz
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"In our endeavors to improve our services to our Network Providers we would appreciate it if you could take the time to complete this survey. Thank you for your time.\nPlease enter an email address where we can send you details on our dashboard registration.\nDo you have any suggestion on how Prime Cure can enhance its services to you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WILMINGTON \u2013 Sudden cardiac arrest claims an estimated 350,000 lives each year. Many victims have no history of heart disease and no prior symptoms. Most people know to call 911 if they witness someone collapse, but many people don't know that a simple action may save a person's life. Effective bystander CPR can improve a person's chance of survival by keeping the blood circulating to their brain, heart and other vital organs.\nEMS personnel from New Hanover Regional Medical Center are teaming up with WWAY to offer free CPR training around Wilmington at multiple times and locations. Lessons take about 15 minutes and will educate residents on the important role bystanders play in saving a cardiac arrest victim's life. Called the Hands on Hearts campaign, the training is a joint effort between New Hanover Regional Medical Center and WWAY Newschannel 3.\nFor more information, visit www.nhrmc.org\/cpr.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you enjoy an underdog story, then let me introduce you to Donnie Ray Cuinn. Donnie is a recent history grad who was left by his mother with another couple - Lena and Papa - to raise as their own. While their circumstances were humble, they offered the loving, stable, supportive environment he needed. He has just graduated from university and is waiting to hear if he is hired for a teaching fellowship.\nAs the book opens, Donnie - sporting an enormous hangover after a night of partying - is pulled out of his bed and his home by two burly men, employees of a powerful politician named Sam Payne. Why? Donnie recently uncovered a damning document that paints the politician's well-known ancestor, formerly labeled a hero, in an unflattering light. The article he published on this document in This Texas was affecting the senator's bid to be reelected. During this forced meeting he was pressured to sign a document recanting his article, something he refused to do.\nFrom this moment on, the story takes us through a tangled web of move and counter move as the underdog tries to avoid the arrows flung by this powerful and formidable politician. On Donnie's side are Lena and Papa, his long time friend, party buddy and former star athlete - Wesley, Payne's opponent in an upcoming election - Bob Braeswood, This Texas publisher Drayton Philby, and attorney Cecelia Medina - a former girlfriend he hopes to win back. Along the way more of Donnie's past is revealed and his birth mother comes back into the picture.\nHero has a strong set of characters, a well-paced storyline and a surprise ending that I didn't see coming, always a great way to end a book. Donnie's journey of stepping into adulthood is laid bare as he begins to see the world in a different way, including the truth about those he calls \"friends\" and the dirty side of politics.\nCheck out a fab behind the scenes interview HERE!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If there is an incident on common property outside of business hours, please contact your Strata Insurance Agency directly.\nIf you are registered on \"My Community\" you will find your insurance agency's details and policy number there.\nClients can access the My Community Portal here.\nFor more information contact Shelley Wright on 0447 745 950.\n\u00a9 2019 Harcourts North Geelong All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VFS International has formed a strategic partnership with the TAM Group and is seeking to transfer $200m of assets to the award-winning specialist in discretionary fund management (DFM), headquartered in London's financial district.\nAlzheimer's Research UK, Blue Cross, DEBRA and Yorkshire Air Ambulance, have each partnered with Greenfinch, a new non-advised investment platform providing charity members with access to expertly managed investment portfolios \u2013 whilst also giving them the opportunity to benefit their chosen charity in the process. The new charities join existing partners; The Archway Project, Bobath Scotland, HEART UK - The Cholesterol Charity and Willow.\nThe Archway Project, Bobath Scotland, HEART UK - The Cholesterol Charity and Willow Foundation, have each partnered with a new non-advised investment platform providing members with access to expertly managed discretionary investment portfolios, whilst also giving them the opportunity to benefit their chosen charity\/ies in the process.\nFinancial planning practice Symphony is the first IFA to white-label a new, free, non-advised digital investment management service, designed to engage dormant clients lost to the Retail Distribution Review, while simultaneously enabling IFAs to compete with the growing threat of low cost robo advice.\nRobo: Will TAM test the water?\nTAM Asset Management's direction in 2016 is one of further innovation and development. TAM believe the next few years in asset management will shape the long term future of the industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarmwd b/data_all_eng_slimpj/shuffled/split2/finalzzzarmwd
new file mode 100644
index 0000000000000000000000000000000000000000..d9bdf03d97c80a83d62bc0f8aed5967a65a744cf
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarmwd
@@ -0,0 +1,5 @@
+{"text":"If you're meeting me at an upcoming event and would like to pre-order your paperbacks, you've come to the right place! Please fill out the form below so I know which books to reserve for you at my table. I do bring a small stock of extra books to every event, but often sell out.\nPlease use your PayPal email, so I can invoice you.\nWhich books would you like to reserve?\nIf you would like me to sign them in advance, please let me know what name to use!\nI'll pay in person when I pick up my books.\nPlease send me an invoice in advance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This study undertaken for Skills for Care investigated the extent to which the social care workforce experience different levels of abuse and violence across groups of people who receive care and support services, and whether the nature of the type of violence experienced by social care staff varies across groups of people who receive care and support services.\nWhile there are ongoing concerns about abuse and violence against people who receive care and support, the purpose of this study was to identify staff most likely to be exposed to abuse or violence in order to target resources and support in this area and promote safe working.\nOver 1,300 social care staff working across the public and independent sectors in a variety of settings and working with a wide range of people who receive care and support services responded to the survey. The responses were reasonably representative in terms of the client groups with which staff worked.\nMore than half of all respondents had experienced verbal abuse in the past year, and 25% experienced verbal abuse on a daily or weekly basis. A quarter had experienced a physical assault, and nearly 6% of all respondents had experienced a physical assault requiring medical assistance (including first aid) in the last year. Although abuse and violence against staff comes mainly from people receiving care and support, 19% of recent incidents are attributed to family and friends, and 5% to other staff members.\nThe survey results indicate that there is a lot of work which could be done to improve the safety and well-being of staff providing care and support, and that some staff are more likely to be exposed to particular types of abuse and violence than others.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pardon me for clubbing Terrorism with Religion. My effort is just an attempt to throw open my concerns that the world is leading to. Most or maybe all mass scale violence these days, could be remotely linked to some sort of religion or ethnic group. And with weapons freely available in the marketplace, parts of the world are easily becoming killing fields. And in most of the incidents the common man suffers, either directly or indirectly.\nThe term \"terrorism\" means premeditated, politically motivated violence perpetrated against noncombatant targets by sub-national groups or clandestine agents usually intended to influence an audience. However my definition of terrorism is 'An act of treachery which left 233 dead and 706 wounded people in 1999 and 741 dead and 5952 wounded people in 1998' Also according to the Department of State, United State of America, the number of terrorist attacks rose, however. During 1999, 392 international terrorist attacks occurred, up 43 percent from the 274 attacks recorded the previous year. The number of attacks increased in every region of the world except in the Middle East, where six fewer attacks occurred. That was how the previous millennium ended and the new started.\nFrom time immemorial including the mythological periods most of the wars were fought on some religious ground or the other. If also there were no direct religious reasons, the background often stemmed from social disparity or differences in religious beliefs. In every war, either direct or proxy would most surely impact the common man. In all wars the front line suffers and are killed or disabled. Food becomes scarce, environment suffers disasters, health conditions become poor, and the after effects of this man made holocaust remains for a few generations.\nThe Taliban in Afghanistan, is unwilling to take actions against terrorists trained in Afghanistan, many of whom have been linked to numerous international terrorist plots, including the foiled plots in Jordan and Washington State in December 1999. Pakistan continues to send mixed messages on terrorism. Despite significant and material cooperation in some areas--particularly arrests and extradition--the Pakistani Government also has tolerated terrorists living and moving freely within its territory. Pakistan's government has supported groups that engage in violence in Kashmir, and it has provided indirect support for terrorists in Afghanistan.\nInternational terrorism is often personality based and the best example in the World is Usama Bin Laden. A fatwa (Muslim legal opinion) was published on 23 February 1998 by Bin Ladin and allied groups under the name \"World Islamic Front for Jihad Against the Jews and Crusaders.\" The statement asserted it was a religious duty for all Muslims to wage war on US citizens, military and civilian, anywhere in the world. His determined attack on America and its allies have raised many questions on the stand of Muslim World leaders.\nIn the Middle East, two state sponsors - Iran and Syria - have continued to support regional terrorist groups that seek to destroy the Middle East peace process. The Iranian Ministry of Intelligence and Security (MOIS) and the Islamic Revolutionary Guard Corps (IRGC) continue to provide training, financial, and political support directly to Lebanese Hizballah, HAMAS, and Palestinian Islamic Jihad operatives who seek to disrupt the peace process.\nReligion also has its influences on the common man. Everyone knows the importance of religion and beliefs in our lives and this article is not a forum to harp on its benefits. But whenever we see forced conversions, religious shelters, forced religion in schools, religious charity; we often think whether religion should at all be the basis of any social discrimination.\nTerrorism can only be countered through constructive efforts on the part of all the political machinery and religious body in the World. At first there should be no concessions to terrorists and there should not be any deals. Secondly, irrespective of any negotiation terrorists should be brought to justice for their crimes. Thirdly, isolate and apply pressure on states that sponsor terrorism to force them to change their behavior. Last but not the least, bolster the counter-terrorist capabilities of those countries that work with the developed countries and require assistance.\nThe common man needs to understand that terrorism is not their cup of tea. They should keep away from all invitation towards the path of violence and unsociable activities. They need to understand that their only priority is their family and a decent method of livelihood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 2010 Philadelphia Eagles season was the franchise's 78th season in the National Football League, and the twelfth under head coach Andy Reid. The Eagles failed to improve on their 11\u20135 record from 2009 and finished with a 10\u20136 record, and lost to the eventual Super Bowl champion Green Bay Packers in the Wild Card playoffs.\nDespite the losses of McNabb and Westbrook, Football Outsiders calculates that the 2010 Eagles had the third-best rushing attack of any single-season team from 1993\u20132010.\nDeAngelo Eugene Hall (born November 19, 1983) is a former American football defensive back who played 14 seasons in the National Football League (NFL). He played college football for Virginia Tech and was drafted with the eighth overall pick in the 2004 NFL Draft by the Atlanta Falcons. A cornerback for the majority of his playing career, he transitioned to free safety towards the end of his tenure with the Washington Redskins. Hall also played half a season for the Oakland Raiders.\nHall was invited to three Pro Bowls in his career: two with the Falcons and one with the Redskins. He holds the NFL record for career fumble return yards, with 328, and his five career fumble return touchdowns is second all time. In 2010, he tied an NFL record by recording four interceptions in a game against the Chicago Bears. Following his playing career, he began working as an analyst for NBC Sports Washington and Fox Sports 1.\nEric N. \"E. J.\" Henderson (born August 3, 1980) is a former American football linebacker who played nine seasons for the Minnesota Vikings of the National Football League (NFL). He was drafted by the Vikings in the second round of the 2003 NFL Draft. He played college football at Maryland, where he was twice recognized as a consensus All-American.\nMarc Steven Mariani (born May 2, 1987) is a former American football wide receiver and return specialist. He played college football for the University of Montana, and was drafted by the Tennessee Titans in the seventh round of the 2010 NFL Draft.\nNot to be confused with Zach Miller (tight end, born 1984), the Chicago Bears tight endZachary Joseph Miller (born December 11, 1985) is a former American football tight end. He was drafted by the Oakland Raiders in the second round of the 2007 NFL Draft. He played college football at Arizona State, where he received consensus All-American honors. Miller also played for the Seattle Seahawks, with whom he earned a Super Bowl ring in Super Bowl XLVIII over the Denver Broncos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The life experience of people who are now older women in Britain has been one of the most dramatic rollercoaster rides in our entire social history. The women who first broke into \"it's a man's world\" jobs in the 1970s and have lived through a transformation in attitudes to childcare, sexuality and discrimination now face yet another frontier.\nIn some ways, this has been a lucky cohort \u2013 female baby boomers who enjoyed the new freedoms and the more widely spread wealth, who were able to get out of bad relationships and who were better educated and more ambitious than any female generations before them. Countless novels chart it all, as well as the mistakes, disappointments and surprises along the way.\nNow, however, many of these women find they have moved from having it all to doing it all, and are asking some hard questions about their role in the Britain of the 2010s. Last week's survey by Carers UK found that 42% of \"sandwich carers\" \u2013 those caring for elderly relatives and for children too \u2013 are struggling to cope, with two thirds of them reporting damage to their marriage or personal relationships. So how much has really changed?\nAt a time when 2.4 million women find themselves \"sandwiched\" in this way, when they look out into the public world and find it so dominated by men; and when unfairness in pay is still so entrenched, they'd be forgiven for looking back at the latter part of the 20th century and asking \u2013 was it real? Was it a dream all along?\nWomen's attitudes have really changed. In 1984, when women over 55 were asked whether they thought a husband's job was to earn money and a wife's was to look after home and family, 65% agreed. That's pretty astonishing. Now the figure is 27%. Back then, just 13% of older women thought employers gave them too few opportunities; now more than 70% do. In our hearts and minds, we have changed.\nIf you look around most extended families, friendship circles and local communities, you'll find older women are as pivotal as ever. They're doing the tough work in care homes and hospitals. They're teaching, picking up kids, running charity shops, staffing the tills and often driving the buses.\nYet if you think about how our society presents itself to itself, it's a completely different picture. On television, 60% of men who appear are over 40 but for women it's 40%. Across some shows, you'd think they (we) didn't exist at all. According to a Women in Journalism survey published in October, in a typical month 78% of front-page newspaper articles had male bylines, while Guardian research the year before found that 84% of guests and reporters on Radio 4's Today programme were men.\nSure, editors and producers try to get more women on to lots of programmes. But the media reflects a much wider underrepresentation of women generally, and older women particularly. The \"experts\" in politics, economics, culture, business and the professions \u2013 in effect, those who rule the country \u2013 are mostly men.\nThat in turn feeds back into the workplace, where older women are half as likely to be managers as men are, and where the pay gap is worst for women in their 40s and 50s. Just as it was back in the young, heady and effervescent days of female liberation, it's really still about how we value women.\nAt this point, I guess, some male readers, and some female ones, will be thinking \u2013 there she goes, typical Guardian woman, banging on. Why should I care?\nWell, first, out of a basic sense of justice and common humanity, I'd hope. But the fact is that older women are not only better educated, but have mostly emerged through intensely challenging years of balancing work and family. Many are going through the toughest time of caring for others at both ends of the age spectrum; but huge numbers are also emerging from that.\nBlinking, we are coming back into the world and asking \u2013 now what? There is a huge resource of self-disciplined, hard-working, intelligent people, who the country badly needs, but who are hidden by old-fashioned attitudes.\nSo it's time to start to change those attitudes. Monday sees the first meeting of the Older Women's Commission, set up by Labour's deputy leader Harriet Harman and Yvette Cooper. Disclosure of interest: I am one of the women from different walks of life sitting on it.\nAnd I am very optimistic. These may be tough times economically but Britain has rarely felt as socially liberal as it is today. The job to be done feels rather like that of changing attitudes towards childcare, a huge issue for younger women. As Harman says: \"Changing attitudes towards older women is the new frontier for public policy.\" There has indeed been a quiet revolution, bringing us flexible working patterns, job sharing and a change in fathers' roles.\nFew people would really want to go back and forget all that. It's been good not just for women, but for children and fathers too.\nWhat needs to be done for older women is a parallel task and will also help the country generally. Among the immediate issues will be dual discrimination \u2013 the idea embedded in the Equality Act of 2010 that combining two characteristics, such as gay and disabled or old and female, can be used in court to secure legal redress. Although already on the statute book, the government isn't taking that forward. It needs to be made to think again.\nHalf of us are women. Almost all of us are lucky enough to grow old. So am I suggesting that this affects nearly half of the country? No, not at all. It affects men, too, who depend in some way on women, whether as wives, daughters or providers of care and health services. It also affects all those who want a stronger, more efficient economy. So that's pretty much everyone, then.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasgnq b/data_all_eng_slimpj/shuffled/split2/finalzzzasgnq
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@@ -0,0 +1,5 @@
+{"text":"Meet the Walper family. They are among the happiest people I have ever met.\nA gorgeous farm wedding at South Pond Farms in Ontario, Canada.\nA gorgeous country wedding in Dutton, Ontario.\nAn engagement shoot with Anna & Vlad at the Don Valley Parklands at sunset.\nA little sneak peak into the two day wedding that took place this weekend. The ceremony took place at midnight and continued later that day with an awesome French-Indian fusion reception. Check out these images from the first day!\nMeet David and Sana, some of the most kind, understanding and down to earth people you'll ever meet. Yesterday they made it official and I was there to capture it all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From the Leonardo da Vinci International Airport (FCO), take the \"Leonardo Express\" train to Roma Termini station (departs every 30 minutes without intermediate stops, or every 15 minutes during certain time slots). The connections to Roma Termini run from 05:35am until 10:35pm while those from Fiumicino airport run from 6:23am until 11:23pm. The cost of the ticket is 14 euros per person each way (journey time of 31 minutes). When you get to Roma Termini station, take METRO A to the BARBERINI stop. Walk about 700 metres to 146 Via Vittorio Veneto.\nFrom Ciampino International Airport (CIA): You can take the shuttle buses that run from Ciampino airport to Roma Termini station. The journey time is about 40 minutes. Some of the bus companies are Terravision, Airport Shuttle, etc. From Roma Termini, take METRO A to the BARBERINI stop. Walk about 700 metres to 146 Via Vittorio Veneto.\nFrom Roma Termini station: When you get to the station, take METRO A to the BARBERINI stop. Walk about 700 metres to 146 Via Vittorio Veneto.\nFrom Tiburtina station: Take metro line LINEA B (towards Laurentina) and get off at Termini station (four stops). When you get to Termini station, take METRO A to the BARBERINI stop. Walk about 700 metres to 146 Via Vittorio Veneto. Clear colours and natural light, together with the latest technology, guarantee a relaxing and tasteful stay. The flagship room, The Corner Suite, with its with its wonderful four windows framing the beauty of the facing Aurelian walls, is the jewel in the crown of The Shire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An excellent technique to collaborate the meshing of two metals; a rose gold foundation and a white gold basket. The 14k rose gold band on this engagement ring embodies a sweet, sultry, and feminine sensation. Pictured with a 1.00 ct. center diamond, there are also small micro-prong 0.50ctw. diamonds that decorate the band spanning the foundation \u00be of the way. This micro-prong engagement ring style allows the band to be covered in diamond brilliance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feelings of dread? Overwhelming thoughts? Rapid heartbeat? Tunnel Vision? Or worse? Since 2002 we have worked with countless folks from all walks of life and we have yet to come across someone with a case of phobia of definite disease that couldn't be overcome.\nRe-'wiring' your unconscious mind is the key to this program \u2013 and its safe and surprisingly easy to do. Consciously, you know your phobia of definite disease doesn't make logical sense. If it did everyone would have it. The fear persists because it is subconsciously linked to a mental trigger which sets off a whole range of powerful negative emotions \u2013 fear leading the charge of course \u2013 at even the slightest thought of definite disease. Its become an automatic response.\nYou will learn how to cultivate positive feelings while learning to combat and eventually vanquish negative feelings that cause your fear. Time to say enough is enough. Harmful thinking isn't helping \u2013 you need to make some serious changes now.\nVanquish Phobia of Definite Disease Today.\nTo find out the gravity of the problem for you, use our 2 minute Online Assessment for Phobia of Definite Disease Online Test although it is fairly clear cut: Do something about it now, if the fear is having a sizable negative influence on your way of living.\nBy simply thinking about Definite Disease, do you become woozy?\nThese are just a few of the most common things people experience. Read more on the symptoms of phobia of definite disease.\nThe use of drugs and medication for Phobia of Definite Disease is not encouraged in our program because they generally mask the problem, rather than tackling the cause. A pill can sometimes act as a quick fix and may seem to help briefly but will not allow you to be free of the fear.\nNo drug has ever been developed specifically for Phobia Of Definite Disease, by the way \u2013 and with all the undetermined possible side effects, it is difficult to rationalize taking a prescription as the answer.\nThe good news is with our program, your Phobia Of Definite Disease will be history, without resorting to meds..\n1: phobia of definite disease: a persistent, abnormal, and unwarranted fear of definite disease, despite conscious understanding by the phobic individual and reassurance by others that there is no danger. 2: phobia of definite disease: an extreme unwarranted fear and\/or physical aversion to definite disease.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gary David Goldberg was best known for his TV collaborations with Michael J Fox.\nVeteran TV producer Gary David Goldberg has died, aged 68.\nGoldberg was best known for creating Family Ties, the '80s US sitcom which marked Michael J Fox's first major TV role.\nHe reportedly passed away at his home in Montecito, California following a long battle with brain cancer, two days before his 69th birthday.\nGoldberg also created the semi-autobiographical CBS sitcom Brooklyn Bridge and re-teamed with Michael J Fox as co-creator of the hit ABC sitcom Spin City.\nAfter launching his showbiz career in 1972, Goldberg won two Emmy awards and two WGA (Writers Guild of America) awards.\nHe is survived by his wife Diana Meehan and two daughters Shana Goldberg-Meehan - a former exec producer on Friends - and Cailin Goldberg-Meehan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavywe b/data_all_eng_slimpj/shuffled/split2/finalzzzavywe
new file mode 100644
index 0000000000000000000000000000000000000000..137562849fad7b81de0a9d21ce740e3d14c83cbe
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@@ -0,0 +1,5 @@
+{"text":"Pipecity has been giving customers the best deals on pipes ever since we opened our doors. With the recent purchase by Starcycle USA, we have more resources to offer even better deals! We now have 2 warehouses: Arizona and California, to help serve you better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wedding season is pretty much over, but that means I'm shooting a lot of engagement images at the moment! Autumn is a lovely time to photograph couples; the crisp fresh air and natures rich tones lend a helping hand.\nThese following images were taken on a grey rainy day in Birmingham city centre and they include some fun urban and industrial scapes plus my favourite: green walls (or living walls)! All topped up with a big dose of soppy love!\nWe started at Library of Birmingham. We then walked under the old Paradise Forum to Victoria Square and walked the back streets till we got to Colmore Row (check out all the cute shots in front of Gusto and 200\u00ba Coffee - yay for all their fancy lighting!). We then headed to Snow Hill (I've been eyeing up that green living wall for quite some time!) and the tram tracks... Who said a grey empty Sunday city shoot can't be exciting?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patricia Whittaker, NP is a nurse practitioner in Belchertown, MA. She specializes in nursing (nurse practitioner).\nGentle, kind, good listener. Great NP. My personal Favorite.\n\u00d7Post a ResponseAre you Patricia Whittaker, NP?\nHow was your experience with Patricia Whittaker?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Antivirus Gu1se_srsly Gu1se_srsly replied to Which AV do you suggest? And why?\nWater Cooler Gu1se_srsly Gu1se_srsly replied to How many cups of coffee on a workday?\nWater Cooler Gu1se_srsly Gu1se_srsly spiced up a reply in How many cups of coffee on a workday?\nHow many cups of coffee on a workday? Gu1se_srsly Gu1se_srsly voted on a poll in How many cups of coffee on a workday?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who needs this much stuff???\nWhile I am impressed with the new breed of \"smart luggage\" that's popped up over the last few years, and while I traveled with a backpack + rolling suitcase for most of the last 10 years, I've moved away from it recently because of the happiness quotient. That is, I enjoyed it for some types of travel, it wasn't great for others.\nBottom Line: Use this style of bag if you don't need quick access to your things, and\/or plan on going straight to your hotel or AirBNB from the airport, or if you're driving to your destination. I recommend spending $150+ on a nice version, as every roller bag I've ever bought at Costco or under $100 has failed within the first 3 months.\nThere are a ton of duffel options on the market, everything from super slick with rollers to backpack style duffels. My wife traveled with duffel bags for years, but I never really took to them.\nBottom Line: Use this style of bag for looks, for packing a lot of \"stuff\", or if you've got one sitting around and don't yet want to spring for another type of bag. My second least favorite option for traveling.\nThe picture above is an updated model of a bag that I own from Gregory Backpacks. It's an incredible bag, has tons of space, and is one of the most comfortable backpacks I've ever worn, despite its size and how much weight it can carry.\nThat said, backpacking bags are terrible for traveling by plane. They're made for the trail, not the terminal.\nBottom Line: As much as I love these bags, I'd never bring one on a city-style trip like the ones we've done to Paris and Italy. If you are planning on doing some actual backpacking once you fly to your destination, it's really the only way to go, but if you're going to be in the city for most of your trip, I'd go with the next option.\nA large backpacking-style backpack is great when you're bringing your lodging with you, i.e. a tent, sleeping pad, and sleeping bag, but as most are top load style bags, they aren't great when your gear consists more of laptops, cables, and stuff for the city.\nIf you're thinking \"hey, that just looks like a carry on suitcase that you wear like a backpack\" then, yep! You nailed it.\nIt's pretty brilliant. Like a cronut for your clothes.\nThis new-ish breed of luggage (I find it hard to believe that this is entirely new) is really taking off as of late, as I don't remember seeing a ton of posts about it just a few years ago. But now? Everyone and their mom is writing about this style of bag.\nJust look at Kickstarter. There are a dozen or more bag companies that have sprouted up over the last few years offering something in this category, and for good reason.\nBottom Line: My current favorite way of traveling, and what I plan on using for the foreseeable future. While I haven't narrowed down a specific model, my wife and I used the Cotopaxi Allpa 35L on this most recent trip. While it was great at holding all of our stuff, I wish it had more external pockets, and that it was more comfortable. I really liked the security loops to keep your zippers from being opened up by pickpockets or during travel, but overall I was a bit underwhelmed. They're a company local to Utah and I want to support them, but I'll be looking for other options for my next trip coming up in May.\nAnything bigger than a 35 or 40 liter size backpack is also pretty big and unruly, and since travel backpacks aren't built like backpacking backpacks, they often lack the support or the comfort for longer walks with the bag on. I've seen 45-55 liter \"travel\" backpacks that don't even have a waist strap. No thanks. You're not going to want to carry that for more than a few blocks.\nI took a serious look at The North Face Overhaul 40, the Osprey Farpoint 40, and the Evergoods MPL 30. Ultimately I ended up going to an REI garage sale one weekend and found what I hope to be the PERFECT travel bag, the REI Co-op Ruckpack 40. I'm very excited to try this bag out on my next trip in a few weeks.\nIf it weren't obvious, I would strongly recommend looking at purpose-built travel backpacks for this style of minimalist travel I'm advocating in this post.\nNow, you don't need to bring anything more than your phone. I've taken great pictures with just my iPhone 7 or 8.\nThat said, my wife is a photographer, and I work in film. We own DSLRs, and so we bring them when we travel.\nYour phone camera is enough.\nSony makes killer cameras. So do Canon and Nikon and Olympus and Panasonic. Get one you can afford, learn how to use it, and then learn how to master it.\nInvest in lenses first, then bodies once you get better.\nFor travel, get a wide telephoto lens \u2013 the 16-35 is my favorite. 24 just wasn't wide enough. 35-40mm is about what the eye sees, so that size pancake lens means you put the camera up to your face and it looks about the same as what you're seeing. Plus, it's great for compact carry and being a little more inconspicuous with your street photography.\nShoot in aperture priority mode with auto white balance and auto ISO.\nIf you see something that looks interesting, take a picture of it. Don't worry about getting the best picture ever, worry about documenting your trip.\nDon't worry about people in the background, or the perfect angle, or capturing something in a way that you saw on Instagram.\nYou'll get better with time, so just take lots and lots of pictures. At the end of each day, upload them to your computer, mark the ones that are your favorites, edit those and post them for the world to see. It's the only way to get better, and it makes for a great experience when Facebook shows you your old pictures from years past.\nMake sure you bring 3-4 batteries and a quick charger just in case you're not in a place you can charge them for a few days. You don't want to be left with no power half way through a day exploring a new city. Batteries are small and cheap so invest in a few and replace them every 2-3 years as they tend to degrade over time.\nYou don't need a huge 5D Mark IV and a 70-200 lens to get great photos on the road, so look for something a little more compact and easy to carry.\n10. Give yourself at least two full days in a city.\nThis most recent trip we went from Rome to Venice to Florence to Rome to Barcelona to Rome. Whew! It was a lot.\nWe had 48 hours split over 3 days in Venice and Florence, and it felt like we could have used a little more time in each city.\nI'd recommend giving yourself 2 full days in each city you want to go to.\nI also felt \"done\" after about 10 days, but we had booked a full two-week trip nearly a year ago so there was no turning back. Good to know, but if it's your first trip abroad, plan on 2-3 days per city and no more than 3 cities (10 days with the days you're traveling to and from) in one trip.\nThere are a number of sites now that will help you find tickets for cheap. My favorites include lowfaredontcare.com, scottscheapflights.com, and the Hopper app. They all work a little differently, but they all help you do one thing \u2013 fly where you want to go for cheap.\nThe dates may not be what you want them to be, and you may go to your 3rd favorite city before your first and second, but you'll pay $430 round trip instead of $1,200, so you just saved enough for your next two trips right there.\nWhat are your favorite principles and tips for lightweight, happy traveling? Share yours in the comments. Thanks for reading!\nI've had this for going on six years, and I LOVE it. The only thing I have done is replace the clips that connect the straps with carabiners when they gave out. I have only checked bags in the past 5 years when traveling for work when I would be someplace for months at a time. Otherwise I can even do three weeks in winter in Europe with just this bag.\nThat's awesome! Thanks for sharing your bag. There's another ebags bag that keeps popping up in my social media feeds (thanks to the Facebook pixel and regathering\u2026) that looked pretty solid. Great to meet another minimalist traveler!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawvoy b/data_all_eng_slimpj/shuffled/split2/finalzzzawvoy
new file mode 100644
index 0000000000000000000000000000000000000000..f484c7bc49cc96b695be07e651ef128487a42746
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"iPod PC Transfer will help you to transfer the music, video and playlists from iPod back to the iTunes library. It works with all type of iPod (including iPod Touch), and very easy to use. Combine with iPod PC Transfer Photo, you can always transfer the content of iPod to any computer easily.It has the ability to search for the songs easily.You can save the output music files with the Aritist,Album as the way you like with the fastest copy speed.\nEasy Copy the music and videos from iPod to PC with iPod PC Transfer.\n...... It works with all type of iPod (including iPod Touch), and very easy to use. Combine with iPod PC Transfer Photo, you can always transfer the content of iPod to any computer easily.\nThere are a lot of reasons you will need iPod PC Transfer. For example, you may want to backup the music of your iPod or iPhone, to use your iPod with a new computer, to copy songs to your new iPod, or to use your iPod in more than one computer.\nYour music library is priceless. With iPod PC Transfer, you can transfer music from iPod to computer, and copy songs, playlists from iPod to iTunes easily, no matter what happened, you can always restore your music library from iPod or iPhone.\n* Copy music and video from iPod to PC.\n* Copy music and video from iPhone to PC, iTouch to PC.\n* Transfer songs, video and playlists to iTunes library (Advanced edition).\nManage iPod contents and transfer iPod videos.\nLenogo iPod to PC Transfer is an application for transferring songs from iPod.\nTransferring songs and videos from your iPod to PC.\nTransfer music\/video among iPod, iPhone and PC along with much more functions.\nTransfer video audio from ipod to computer.\nTransfer your iPod\/iPhone music\/video\/playlist\/photos back to PC and iTunes.\nQweas is providing links to iPod PC Transfer 6.3 as a courtesy, and makes no representations regarding iPod PC Transfer or any other applications or any information related thereto.\nAny questions, complaints or claims regarding this application iPod PC Transfer 6.3 must be directed to the appropriate software vendor.\nYou may click the publisher link of iPod PC Transfer on the top of this page to get more details about the vendor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amazing roasted chicken with lemon and garlic oil.\nPeel the garlic cloves and combine the oil, garlic, thyme leaves and zest of 1 lemon.\nClean the chicken, wash the lemons and cut them into quarters and stuff the chicken with it. If necessary, close with a meat skewer.\nBrush the chicken with the garlic oil, and season with salt and pepper.\nPut the chicken on the rotisserie spit.\nHalf fill the Rapidfire Chimney Starter with Weber briquettes, light them and prepare the barbecue for indirect heat - approx. 240-250\u00b0C.\nPlace an aluminium tray between the briquettes to collect fat and meat juices.\nFit the spit with the chicken into the rotisserie motor and leave to turn for approx. 70 min.\nContinue cooking the chicken until the core temperature is 73\u00b0C.\nTake the chicken off the rotisserie spit and allow it to rest for approx. 15 min before carving. The core temperature will go up a couple of degrees while the chicken is resting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Friday, everyone! Hope you're all kicking off the weekend in a fun and relaxing way if possible. David and I are painting the town red. \u2026Wellll, part of our living room is painted red. Does that count? We turned down multiple social engagement offers in our metropolis to be present for the season premiere of Stargate Atlantis.\n1. My left thumb is double jointed.\n2. I think Worcestershire sauce on mashed potatoes is quite tasty.\n4. Despite a fear of heights, I'd love to try parasailing and the trapeze.\n5. I've been on Texas' death row. Don't ask me how I escaped.\n6. I think it's fascinating how owls can turn their heads so far around.\n7. I have never understood the appeal of the Three Stooges.\n\u2026There you have it. Now, what about you?\nIt can be really interesting at times to live in this world of ours. Don't you love it when something a little odd or amazing captures your attention?\n\"Like what?\" you ask. Thought you'd never ask. Like when I was leaving the post office yesterday and noticed a well-written flyer by the door. It explained the help this certain group offers and listed how a person with this need can get in touch with them. Great\u2013except that the well-written information was intended for illiterate people; it was a literacy group. \u2026I knew you'd catch the irony.\nAnd each day brings new promise of the unusual! Just this morning before the sun made its daily debut, I was reading about something from World War II. A unique group called the British Air Transport Auxiliary had the mission of ferrying aircraft to and from military installations around Britain and \"the continent.\" The Auxiliary members' service freed up combat pilots for just that. It was a pretty remarkable group; for example, it was the first instance of women (many of them American) piloting military aircraft.\nAs the ferrying demands grew, the ATA actively recruited pilots to handle the workload, and limitations that might bar a pilot from service in peacetime were no barrier when every capable aviator was sorely needed. Most of the men who flew were in their thirties, forties and fifties. Many physically challenged pilots also found employment with the ATA. There were a few men who were color-blind, and one who suffered from narcolepsy, who but for the unfortunate tendency to nod off at the most inappropriate times was quite a good pilot. (On his ferrying trips he would take along an \"assistant\" to shake him awake if he happened to fall asleep at the controls.) There were several one-armed pilots, and a one-armed, one-eyed pilot, Stuart Keith-Jopp, who was one of the first 30 men to join the ATA at its inception. He was also over 50 years old, a veteran of World War I, and an extraordinarily skilled and capable pilot who flew with the ATA until the war's end. Click here if you want to read more about the ATA.\nWow. I can't help but chuckle and be amazed. What brave people! (Especially the \"assistant.\") I'm so grateful for their service. In this case the British knew how to put people to good use. \u2026And aren't you glad that God's the same way? He uses all of us, despite our flaws and foibles\u2013seen or invisible. There are no excuses for laziness because He can use us all for extraordinary purposes, despite ourselves.\nAhhhh, this is a great time of year.\nSome people plan months in advance for Christmas decorating; I practically salivate over fall. Come September as other people are squeezing out the last remnants of summer at Labor Day cookouts, I'm giddily pulling down my plastic bin stuffed with autumn leaf garlands and pumpkin-laden decorations. I'm oblivious to the fact that it's still 95 degrees outside. In our house, it's already autumn. I love the vibrant colored leaves, the chilly evenings, the sunlight that is a little softer, and the energy and excitement that the season brings.\nIt wasn't always this way with me. In fact, until about the last ten years or so I reveled in summer's heat and dreaded \"sweater weather.\" Do you see changes in your life\u2013big or small\u2013and wonder when the tide turned? In this case, I have no idea what changed me. \u2026Perhaps it was a mission trip years ago to crisp, fragrant, appropriately-named Pine, Arizona. Perhaps it was marrying a man who adores fall. Perhaps it was discovering the joys of flavored ca ppuccinos. Or maybe it was realizing that 105-degree summer days really aren't fun. Oh, well\u2013it doesn't matter if I figure it out. I'll just concentrate on enjoying these wonderful, fleeting days of fall.\nSo\u2026what is your favorite season?\nEvery morning I take a deep breath and prepare to be mobbed by a throng of admirers. These enthusiastic followers await my appearance and greet me loudly, crowding me as soon as I come into view. I can't escape the adoring attention.\nNow, before you envy my celebrity status, I'll share a little more about my fan club. I even know their names: Dan, Ann, Daisy, Abby, Smokey, and little G. Gordon Kitty. Yep, in addition to our three animals, we're temporarily taking care of three more fuzzy ones. No matter how much time David and I spend with them they would love more attention. And we're not complaining much\u2013they're pretty cute.\nI got to wondering once as they hounded me (I can hear the groans)\u2026in some tiny way was this what Jesus felt like each time the crowds pressed in on Him? Almost everywhere He went, \"large crowds followed him.\" (Matt. 8:1) He ministered to them and loved them, no matter how tired or busy He was.\nHmmm\u2026 I'll have to think about this, especially when I feel little puppy mouths nipping at my heels.\nSigh. Well, I guess Spielberg got impatient and went off this week and named his own movie: \"Indiana Jones and the Kingdom of the Crystal Skull.\" Had waited for you, the voters, it would've been \"Indiana Jones and the Raiders of the Fridge.\" Now you tell me\u2013 which one sounds more swashbuckling? Let me tell you, at times those daring to go into my fridge find (unpleasant) adventures. Oh, well, we can all hum with contentment over the newest poll.\nBack to that matter about the \"no recovery\" moment I mentioned a couple of posts ago. Do you ever get braver when you're around your friends?\nTwenty-four hour donut shops and college students are often a perfect match. Like I mentioned already, it was late one weeknight years ago when a few friends and I decided to make our way to Jack and Jill Donuts at North First and Willis in Abilene, Texas. Well, actually some of us had kidnapped a few others and we needed a place to take them and feel triumphant, but that's another story.\nWe bought our superbrain goodies and settled onto stools looking out of the big plate glass windows, very much like the donut shop pictured in my September 9th post. Another great feature to donut shops is when they have a drive-through window, like this one did. As we laughed and had a good time, the line of several cars for the drive-through never died down\u2013not surprising in a town with three universities.\nI should mention that by this time David and I had been dating for almost a year. I should also mention that David thinks donuts are the fifth food group. So for a guy who's a night owl, this donut shop was heaven. It was no surprise when, through the glare of the parking lot lights, I saw him pull up to the end of the drive-through line in his white '87 Mercury Lynx. He had yet to spot me, though. So my friends and I kept on having a good time while I waited patiently, plot already hatched in my head.\nWhen the time was right \u2013 David's car was only one away from the drive-through window \u2013 I used the famous last words of many redneck guys while horsing around with friends: \"Hey, y'all, watch this!\" My friends put down their donuts and looked as I slipped out the front door and around the side to the cars in line. I prepared to impress my buddies and have fun in the process; they had the perfect vantage point from inside the shop. As I moved I could see out of the corner of my eye that David's car window was rolled down\u2013 also perfect, I thought to myself. I went to the end of the line of cars and dropped down with military-like stealth. Still crouching, I quickly made my way up the line until I was right below David's driver-side window. I grinned while glancing back into the shop; my friends watched expectantly. \u2026.Then I made my move, springing into David's view, inches away from his face.\nYou probably guessed it. I don't know whose surprised eyes were bigger, mine or the stranger's staring back at me from inside what had to be the only other white '87 Mercury Lynx in Abilene. In my shock I managed to utter something brilliant like, \"Oops, sorry\u2013wrong person.\" Apparently he was too shocked himself to offer much reply. Tail tucked between my legs, I retreated back into the shop. Even before I opened the front door I could feel the sonic wave of my friends' laughter as they rolled on the floor, gasping for air. I could feel a smile come across my face, too.\nNow, after I've opened up, doesn't that make you want to share one of your \"no recovery\" moments?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"International Peace Bureau Congress i Berlin Disarm! For a Climate of Peace ble en suksess. Internasjonal aktivister fra UK, USA, Tyskland, Belgia og Norge, alle tilknyttet organisasjon Conscience and Peace Tax International holdt et vellykket m\u00f8te (workshop) om fredsskatt. Fredsskattalliansen fra Norge var representert ved Heidi Anderson (IFOR Norge) og Penny Heymans (Kvekersamfunnet).\nThe IPB Congress in Berlin Disarm! For a Climate of Peace, was a big success. The international activists from Conscience and Peace Tax International held a successful workshop with representatives from UK, US, Belgium, Germany and Norway.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kevin Walker went freelance in 1987 and spent years illustrating for British comic 2000 AD and Games Workshop. There followed years of comics, both writing and illustrating, with a wide variety of media, before becoming a father and realizing it was time to get some work done.\nIllustrations for the Magic: The Gathering trading card game, several computer game projects and comic work followed, working for 2000AD, DC Comics, Dark Horse and Marvel. Working on Thunderbolts actually attracted some attention. He's now the regular penciller and inker on Avengers Arena for Marvel Comics.\nIn recent years he has worked on cover art for titles such as Young Bond, Artemis Fowl, Changeling and Vampirates.\nHe is currently working mostly for Marvel Comics from his home in Yorkshire, England.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxlfp b/data_all_eng_slimpj/shuffled/split2/finalzzzaxlfp
new file mode 100644
index 0000000000000000000000000000000000000000..e593c43ceb1a8b703acb66478d03082c61592710
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"If you want an amazing Waterfront property to rent, then here it is!! This unit has private steps to the harbor & Breathtaking front to back views of the Manatee River .This Condo has granite countertops , stainless steel appliances and ceramic tile throughout, & 9' ceiling. This gated community is close to shopping and beaches.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> Has the file scheme become a separate RFC at last?\nexpired 5 years ago <draft-hoffman-file-uri-03.txt>.\nWarts and all, an RFC for file URI is sorely needed IMHO.\nNext message: Charles Lindsey: \"Re: Status of RFC 1738 -- 'ftp' URI scheme\"\nPrevious message: John Cowan: \"Re: Status of RFC 1738 -- 'ftp' URI scheme\"\nIn reply to: John Cowan: \"Re: Status of RFC 1738 -- 'ftp' URI scheme\"\nNext in thread: Charles Lindsey: \"Re: Status of RFC 1738 -- 'ftp' URI scheme\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The easy solution for having a custom website.\nThe easy solution for you.\nMobile websites can be stand alone system or run along side you main website.\nMobile's and tablet's are increasing becoming the user's choice for searching the internet. Business websites need to take account of this trend, Easysites have recognised this, and now offers a mobile option which can be stand alone or run in conjunction with a traditional desktop website.\nMobile's and tablet's are increasing becoming the user's choice for searching the internet.\nBusiness websites need to take account of this trend, Easysites have recognised this, and now offers a mobile option which can be stand alone or run in conjunction with a traditional desktop website.\nSuste Easysites has purchased new computers and software, allowing us to now offer sketch or video animations both with voice over capability.\nA user will often decide whether to leave a Web page within the first 10\u201320 seconds, with the average page visit lasting less than a minute.\nPrevious studies indicate people only have time to read a quarter of the text on the page.\nOur website designs take the above facts into consideration, using the right ratio of text to animation,pictures or video will help to keep your visitor interested.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beets are said to boost the immune system, and since we've all been fighting colds and flu's this season, a Hot Beet Borscht is the perfect recipe.\nIn a large pot combine beets, water, dash salt and onions, bring to a boil. Reduce heat and cover, simmer for 30-40 minutes. Stir in remaining ingredients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2019 Chrysler Aspen Concept and Specs \u2013 US carmaker is placing lot of attempt it is possible to the life of its well-known cross-over. Promoting history decreased, and Chrysler is providing another taken to its Aspen cross-over. Although big crossovers are well-known, its cost is repugnant to customers. They usually choose to buy more compact SUVs for less money, but then all motorists are making big energy behind. New 2019 Chrysler Aspen could convert things around. Described as ideal journeying car, this is going to be outstanding suggestions for it. Also, cost adjustment is possible, and that would be bull's eye for the resting elegance in Chrysler's constant.\nThe 3.2-l turbocharged V-6 can make enough energy for 2019 Chrysler Aspen. This device is qualified to provide 270 horse power. However, there is an alternative choice for future design. All customers can make Aspen with 3.6-l Pentastar. Other motor provides your automobile with 300 hp. Transmitting program is 6-speed automated, and it delivers ability to front side tires by standard. All-wheel generate is also a choice. Furthermore, there are many other enhancements and functions enhancing efficiency and specifications figures of Aspen SUV.\nWe will see new 2019 Chrysler Aspen different than before. However, it is not easy to calculate what does developers get ready. Nevertheless, there are few records that could cause us to some last signs. For example, suv will be more competitive than before. Thanks to reduce nasal area, it is sportier. Changes on bonnet and ceiling are enhancing the rules of aerodynamics. Convenience within the cottage is at maximum stage. Chairs are protected by set, as well as leader. There are also many manages there, while other instructions car owner can provide from touch-screen show. New look of the dash panel refreshes the design. Also, many functions increase overall impact about your automobile.\nIntroduction of the 2019 Chrysler Aspen is not verified yet, although there are programs saying new cross-over is arriving beginning in 2019. Nevertheless, we still delay for verification from organization. It's the same with cost, since now we can only evaluate it to past versions, and other designs. However, US organization must provide better cost of its competitors, to be able to attract new customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayabx b/data_all_eng_slimpj/shuffled/split2/finalzzzayabx
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+{"text":"Cole Barash is an elite New York City-based photographer whose work has been showcased everywhere from magazine covers to global ad campaigns for world renowned clients. His photography journey began as he pursued his passion for surfing. He traveled to the North Shore of Hawaii to explore the island's subculture and document the life of a world-class surfer, John John Florence. The series of work produced from this trip was chosen to be exhibited at the Picture Farm Gallery in Brooklyn, New York. \"Talk Story\" is the limited edition book of Cole Barash's work, released at the opening reception for this exhibition. The essential goal of its production was to maintain the authenticity of the photographs and produce the best quality representation of his work. Hatteras worked with Cole Barash to achieve this vision.\n\"Talk Story\" is a 76 page + cover perfect bound book that exploits the unique characteristics of paper, utilizing five different stocks and a combination of both offset and digital printing, to allow the photographs to tell their story. Glossy inserts, reminiscent of traditional 4\" x 6\" photographs, are inserted into the book's binding. The inserts provide a welcome contrast to the uncoated stock of standard text leaves and highlight the art photography. In addition to the inserts, there are two pages printed on a vellum, translucent stock. The variation in opacity interacts with the adjacent pages enhancing the photographic effect.\nThe ultimate goal, shared by Cole and Hatteras, was to translate the photography into a printed format that stayed true to Cole's initial vision. Through various paper stock tests, ink density tests, and color management techniques, Hatteras was able to provide an uncompromising solution. Though Hatteras is known for our ability to deliver on a much larger scale, we are equipped to handle the minute details of a limited edition production.\nAttention to detail and quality are two of the large variables that are high on my list when releasing pieces under my name. Thats exactly what I got working with Hatteras. From several initial paper stock options, to a variety of different options with quotes as well as a hands on experience were all part of the package. Sure I could have printed in China, however I wanted a dependable team to work with and a hand shake of an american made product at the end. Hatteras' on line supervision throughout the entire process from print to bind to package was there and that is something that is getting harder and harder to find.\nDiscover more about Cole Barash and buy a copy of his book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There's no better way to get a crash-course on investing than looking at Warren Buffett's stock portfolio.\nThis article was originally published July 25, 2016 and updated September 28, 2016.\nWarren Buffett, the Oracle of Omaha, is perhaps the most famous long-term investor in history. In the spirit of emulating the success of this Berkshire Hathaway (NYSE:BRK-A) (NYSE:BRK-B) boss, turning to his current holdings and analyzing his investing methods is a great place to start.\nHere are three lessons you can learn from the stocks currently in Buffett's portfolio.\nWarren Buffett doesn't understand new technologies too well -- so he's not going to invest in any of them. Sure, IBM is one of Berkshire Hathaway's largest stock investments, but that is a long-established and mature company, and Buffett loves IBM's shareholder-friendly management. And Apple is a Berkshire stock, but that was the choice of one of Buffett's stock pickers, not the Oracle of Omaha himself.\nTaking a close look, we can see that most of Berkshire's portfolio is filled with easy-to-understand investments. With bank stocks such as Wells Fargo and U.S. Bancorp, the business can be summed up in one sentence -- they lend money for more than they pay to borrow it, and profit from the difference. Other Berkshire stocks like Coca-Cola, Kraft Heinz, and Wal-Mart are engaged in similarly simple businesses.\nThe point is not that you should stick to extremely simple businesses, but that you should invest in what you understand. If you're tech-savvy and are proficient at evaluating tech stocks, go for it. As a personal example, I know real estate and banking quite well, so those industries are well-represented in my portfolio. Biotech and retail -- not so much.\nIn the late 1990s, Berkshire owned a massive stake in Freddie Mac, which was worth as much as $3.9 billion in 1998. Then, in 2000, Berkshire unloaded essentially all of its shares. Buffett noticed that Freddie Mac's management was becoming far too focused on its quarterly results, and had been taking excessive risks in efforts to keep the numbers high. As Buffett put it, \"I figure if you see just one cockroach, there's probably a lot.\" In a nutshell, his reasons for buying Freddie Mac in the first place (good valuation, low risk) no longer applied, so he got rid of it. It's a good thing he did -- Buffett's stake would be worth just over $100 million today.\nFor a more current example, Berkshire owned a substantial stake in DirecTV, which proved to be an excellent investment. DirecTV dominates its industry, and has high international growth potential. However, when AT&T acquired DirecTV, Berkshire found itself with a bundle of AT&T stock. AT&T, while a great company, is simply not a \"Buffett stock.\" Growth potential is limited, and it doesn't have a dominant market position. In the quarters following the merger, Berkshire unloaded its AT&T shares rather quickly.\nWhat makes a good long-term investment?\nAs I mentioned, Buffett buys stocks with the intention of holding them forever. So, by taking a close look at the stocks in Berkshire's portfolio, we can get a good sense of what makes a good \"forever stock.\"\nA leader in its industry -- Companies like Coca-Cola, Wal-Mart, and Wells Fargo are the biggest and arguably the most recognizable in their respective industries. With size comes certain cost advantages, as well as brand recognition.\nAn identifiable competitive advantage -- Buffett refers to this as a \"wide moat.\" As an example, Wal-Mart's competitive advantage is its ability to charge less for goods than most competitors, even online retailers.\nA \"margin of safety\" -- a low valuation or other factor that should keep the company's stock up, even in bad times. For example, Berkshire owns shares of Goldman Sachs, which trades for less than the value of its assets. This means that even if Goldman's profits plunge, its shares are intrinsically worth more than their current price.\nLittle or no debt -- Low debt levels allow companies financial flexibility, as well as the ability to survive a decline in profits if the economy turns sour.\nShareholder-friendly management -- Buffett looks for managers who own substantial amounts of the company's stock, and have a good record of returning capital to shareholders through dividends and buybacks.\nAs a final thought, I'm not telling you to recreate Buffett's stock portfolio in your own, or buy the same stocks he does. If that's what you want, you're better off simply buying shares of Berkshire Hathaway. Rather, I'm encouraging you to apply the reasons behind Buffett's investments to your own strategy and investment goals in order to create your own rock-solid portfolio.\nThe average dividend yield of a company contained in the S&P 500 Index is a respectable 2.2%. Good, but not great. This simple fact inevitably sends enterprising investors in search of high yield dividend stocks, those companies that reward their investors with dividend yields as high as 5-7%. Inevitably, as part of this search, one will come across the highest yielding stocks out there. Companies with yields practically in the stratosphere, tempting investors with yields as high as 10%, 20% and even as high as 70%. However, as one might imagine with such generous yields, things aren't always what they seem. Below is a list of the top 10 highest yielding dividend stocks that are traded on the major US exchanges, but you have been warned. As in life, there is no free lunch when it comes to dividend stock investing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"It is not the easiest task to try and make a living on what most artists are paid in this country.\"\nElissa Beth Stebbins as Mistress Ford and Jennifer Le Blanc as Mistress Page in Merry Wives of Windsor. Costumes by Shannon DuPont.\nLivermore Shakespeare Festival dramatically raised the bar this year for its 10th anniversary program. The 2012 program features the Shakespearean masterpieces of Hamlet and the Merry Wives of Windsor with the enchanting Queen Anne Victorian at Concannon Winery again serving as the set backdrop.\nWhile the organization has been putting on high quality performances for all its ten years, it has often done so with a very limited use of union professional actors, members of the Actors Equity Association. Actors typically work as freelancers, a hard way to make a living. Qualifying as an AEA actor is an important step in actors' careers, in part because the union provides healthcare as long as the actor works a specified number of weeks for theaters that hires them as union employees. This year, Livermore Shakespeare Festival Producing Artistic Director, Lisa Tromovitch, and her guest director for the Merry Wives of Windsor, Virginia Reed, held auditions as usual but found that a good number of the actors they wanted to cast were Equity members, requiring a significant bump up in the production cost because of health care and Equity salaries. But both directors were convinced the added expense was worthwhile to realize their visions for this summer's program. The Livermore Shakes Board of Directors, staff and company volunteers called on the company's loyal donors to find out what they thought about increasing the number of professional actors the festival hired. Everyone agreed that it was a great investment in the quality of the performances. Donors contributed $7500 in additional money to support the additional contracts.\nDonor and company dramaturge, Peggy Riley, makes it very clear. \"Our dream is to become a destination Shakespeare Festival,\" she says, very much like Ashland or Santa Cruz. \"Livermore has lots to offer,\" she adds, \"wineries, golf, hiking, boating, swimming, and (thankfully after years without) some nice restaurants and a walkable, engaging downtown. So Livermore and the Shakespeare Festival can offer a wide range of activities in addition to the plays. In order for all this to happen, our company has to grow and become better known. And \u2013 in order for THIS to happen, we have to offer the highest-quality productions possible. That means professionals, and that means money.\" So Riley and her husband were happy to make a contribution.\nPreview of Hamlet on July 12, Hamlet Opening Night Performance on July 13th. Hamlet performances are 7\/12, 7\/13, 7\/14, 7\/22, 7\/27, 7\/28, 7\/29, 8\/10, and 8\/11 with a Family Night performance on 7\/22. Preview of Merry Wives of Windsor on July 19th, Merry Wives of Windsor Opening Night Benefit; Dinner, Auction and Performance on July 20th. Merry Wives of Windsor performances are 7\/19, 7\/20, 7\/21, 8\/3, 8\/4, 8\/5 and 8\/12 with a Family Night performance on 7\/19.\nTime: Thursday, Friday and Saturday Performances start at 7:30PM. Sunday performances are at 7PM. Merry Wives of Windsor Opening Night Performance on July 20th at 8PM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our City of Sanctuary meeting last Wednesday started with a presentation by our Chair, Joseph Chivayo, of awards to our six Sanctuary Specialists for their work in delivering training to a wide group of organisations in Derby, including schools, museum, a Doctor's surgery, churches, an ice-cream parlour and several theatre groups. This was made possible through the Guardian funded project that we undertook. Here's a few of those presentations.\nIt is said, \"Nothing ever happens in Derby!\" We all know that's not true! Our local Derby City of Sanctuary group met on Wednesday to share what is happening in the city, to discuss plans for refugee week and other upcoming events. We also celebrated the valuable input of some of our City of Sanctuary Specialists who have been delivering refugee and asylum seeker awareness training all around Derby over the last year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our first day in Jordan gave us time to settle ourselves and witness the ruins of Umm Qais and Jerash. On our second day, we were headed to Wadi Musa, home of Petra, the main reason Jordan found it's way on to our bucket list.\nOur first stop on the King's Highway journey from Amman to Petra were the Biblical sites of Mt. Nebo and Madaba. The sites are only 8km from each other and Madaba is approximately 25km from Amman. Mt. Nebo is an important pilgrimage site for both Christians and Muslims.\nFrom the entrance it is only a few hundred meters to the commemorative stone erected in 2000 to mark Pope John Paul II's visit. The mountain is where Moses is believed to have first glimpsed the promised land approximately 3000 years ago. He later returned only to die before entering the Promised Land.\nHe is believed to be buried somewhere in the area but the exact location is not known. The Memorial Church of Moses was under construction at the time of our visit but we were able to view part of the original mosaic floor that is undergoing restoration in a nearby tent.\nA viewing platform on the far side of the church has a sign which indicates the various sites visible from the mountain.\nAfter our short visit to Mt. Nebo, we were off for a quick drive to the city of Madaba, also known as the \"City of Mosaics\". The highlight for many visitors is found in the Greek Orthodox Church of St. George, which has preserved the original piece of Byzantine-era mosaic map of the \"Holy Land\" from the 6th century on its floor.\nThis 25\u00d75 sqm map in front of the altar depicts some important details of that time, like hills and valleys, villages and towns such as Jerusalem and other holy sites. The church itself is small but a wonderfully vivid and colourful place with many other mosaic wall-hangings and religious paintings.\nThe art of mosaic making still lives on in the town of Madaba. Shortly after leaving St. George, we stopped at a co-operative store where we were able to watch local women create beautiful mosaic pieces.\nThe technique starts by drawing a design on a muslin cloth. The artists then cut and carefully adhere the stones to the cloth using a flour and water mixture. After all the stones are set, the mosaic will be laid face down on a piece of wood with cement. The cloth will then be removed and the mosaic will be washed and shellacked.\nRejuvenated by the promise of a visit to another castle, the children excitedly entered the van and off we drove further south on the King's Highway. After about two hours on the road, we had our first glimpse of the imposing Karak Castle (also known as Crak des Moabites, or Le Pierre du Desert to the Crusaders).\nWith its location on the King's Highway, the Crusaders were able to control both north and south traffic by imposing road tolls until Muslim armies took the castle in 1183. Mamluk sultan Beybars took it in 1263; and then an earthquake occurred, creating a lot of damage in 1293. The castle was re-introduced to the western world by Johann Ludwig Burckhardt, the same explorer who helped rediscover Petra.\nThe walk on the bridge was very windy. Upon entering via an entrance on the North Wall we were immediately greeted by cement balls that were used as arsenal in catapults. Immediately to our left was a map which helped us get our bearings. At the time of our visit the Museum was closed so we proceeded to explore the Crusader Gallery.\nThe Crusader Gallery functioned as the horse stables of Karak Castle. There were a lot of rooms to explore and a flashlight or headlamp is definitely a must. The children were drawn to the long passageway which then led to the Soldier's Barracks and Kitchen.\nAt the end of the tunnel, we were wandered into the Upper Court of Karak castle. The area was largely overgrown and much of it still needed to be excavated. From here we were able to explore the Mamluk Keep. The Keep is said to have 6.5m walls, the strongest defense in the castle. A staircase from the Keep leads to the Mamluk Palace and Mosque.\nAccess to the Crusader Church is also from the Upper Court. The arrow slits in the walls of the sacristy suggested that the walls were originally part of the castle's outer walls.\nFrom the Church off we went to explore the Prison and the Marketplace. At the end of the market, a path led us back to the entrance.\nAfter the castle visit, we hungrily ate our overdue lunch at a restaurant nearby. The rest of the drive south of the highway was very quiet as both C and D took much-needed naps. After two hours, both woke up just in time to see the gate of our next adventure, Shobak castle.\nShobak castle was built by Crusader King Baldwin I in AD 1115. It is also known as Mont Real or Montreal with the original name being Mons Realis (the Royal Mountain in French).\nAlthough not as large as Karak, it was actually the main Crusader castle for 30 years and was the first Crusader castle built in Jordan. After walking up a few metres from the entrance we were immediately greeted by a mysterious set of steps that beckoned to be explored.\nAfter rounding the first corner, it was obvious we would need to put on our headlamps. After two corners and having the tunnel narrow and steps almost disappear in front of our eyes, we decided it was probably wise to head back up.\nLater on we found out from the groundskeepers that we were actually inside a well and that after 375 steps we would have found a spring and a secret passageway that emerges outside the castle walls in the town of Shobak. So, lucky we didn't follow it to the end as we hadn't yet explored the rest of the castle.\nWalking along the castle perimeter we reached the Keep where you can see Quaranic inscriptions that were added sometime in the 13th and 14th centuries by Saladin's successors. We were afforded amazing views of the castle layout.\nAfter stopping by to visit the second church, we came upon the catacombs where we saw some people documenting the artifacts that have been found on the site. From the catacombs, a short path led us back to the entrance.\nWhat a fun-filled day! In less than an hour we would reach Wadi Musa, the land of Petra. Little did we know at the time, that our day was not yet over.\nOh, I think you took the right route alright. Amazing Pictures and Account of the trip. Enjoyed reading through.\nWhile of course the archaeological sites look amazing, I'm most fascinated by the mosaic process. I could've allowed them to take me in and apprentice.\nThis is marvelous! It makes me think of planning a trip soon.\nWhat an amazing journey your family is living. I want to come along next time!\nIt's great your kids can see there great places in the world!\nThe ancient monuments in better shape than the fortress of Masada in Israel! Surely on my list! Thanks for sharing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It is actually simple to change your RV easily and inexpensively. You'll uncover quite a few approaches for gaining the RV fully willing to use solar powered vitality. You might have it professionally put in, which normally may perhaps be expensive. However, you could potentially construct your RV solar panels by yourself. All you should do is buy a handful of objects.You'll need solar electric panels as well as a solar panel mounting bracket. Plenty of folks are constructing their own specific solar panels for rvs and mounting brackets from facts they could get on the net. Constructing them by yourself slashes the price substantially.\nYou will find suppliers that deliver specialty kits out there specially for an RV. The kit will cost the RV battery strength that operates the interior appliances and lighting.\nThe kits are available in assorted dimensions decided by your RV and what ability battery electrical power you've got. Quite a few companies will assist in choosing the appropriate measurement; should you have got any queries talk together with the location where by you obtain the solar panel RV kit from. Fairly a number of web sites have these kits to order.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A city held to ransom. A brave few determined to fight. A race against time.\nSpring 1868, and the population of Boston is being terrorised by a series of mysterious attacks: first a magnetic storm causes ships in the harbour to collide in flames, then in another bizarre catastrophe every piece of glass in the financial district spontaneously melts - clocks, windows, eyeglasses. The city's fate relies on four young students: Civil War veteran Marcus Mansfield, brash Bob Richards, meticulous Edwin Hoyt and the eccentric but brilliant Ellen Swallow. Together, they are The Technologists. In a climate of rising hysteria, these four courageous individuals must unite against the forces of darkness to uncover the mastermind before he can stage his greatest outrage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Born in Valencia, she studied guitar and singing in the Conservatorio Profesional de M\u00fasica of that city. After being awarded, in June 2002, a Second Prize upon completion of her Bachelor's Degree in singing, she proceeds her studies in the Conservatorio Superior de M\u00fasica \"Joaqu\u00edn Rodrigo\" of Valencia in the class of the soprano Gloria Fabuel. She performed at the Palau de la M\u00fasica de Valencia in the project \"Me llamo Mozart\", singing W.A. Mozart arias from \"The Magic Flute\" (Queen of the Night and Papagena), \"The Marriage of Figaro\" (Susanna) and \"Don Giovanni\" (Zerlina). She also performed \"Amahl and the Night Visitors\" by G.C. Menotti, and \"Suor Angelica\" by G. Puccini, in the role of Suor Genovieffa. In 2006 Beatriz obtained the third prize in the Chamber Music category, the Audience Award, and the Best Valencian Performer Award at the V \"Luys Mil\u00e1n\" International Guitar Competition of Vila de l'Olleria.\nAfter being granted a scholarship by the Institut Valenci\u00e0 de la M\u00fasica (Generalitat Valenciana), she moves to The Hague to further her studies at the Royal Conservatory, where she has completed a Master's Degree in Early Music Singing, specializing in early Baroque Italian music and reviving the original performance practice of self-accompaniment with the theorbo.\nShe has studied singing with Rita Dams, Michael Chance, Jill Feldman and Peter Kooij, and theorbo with Mike Fentross and Joachim Held. She has attended masterclasses with renowned teachers such as Robert Expert, Marta Almajano, Cristina Miatello, Evelyn Tubb, Lambert Climent, Stephen Van Dyck, Johannette Zomer, Fred Jacobs and Margreet Honig.\nWhile at the Royal Conservatory of The Hague she has worked with Early Music specialists such as Fabio Bonizzoni (La Risonanza), Charles Toet (Concerto Palatino), Emilio Moreno (El Concierto Espa\u00f1ol) and Christina Pluhar (L'Arpeggiata). She also performed the main role in the Intermedi per La Pellegrina conducted by Mike Fentross, where she sung whilst accompanying herself on the theorbo.\nUpon becoming a specialist in the field of Early Music, she has performed around Europe with diverse ensembles such as Las Esferas, Musica Poetica, La Academia de los Nocturnos, Ribattuta Ensemble, Le Jardin Secret, Harmonia del Parn\u00e0s, Os Orphicum, Collegium Musicum Den Haag, M\u00fasica Temprana, etc., in festivals such as XVI Festival de M\u00fasica Antigua de Pe\u00f1iscola (Spain), XII Festival de M\u00fasica Antiga: M\u00fasica, Hist\u00f2ria i Art in Valencia, Fringe Festival Oude Muziek in Utrecht, Festival Classique in The Hague, V Festiwal Tw\u00f3rczosci Religijnej (Poland), Dag Oude Muziek in Bilzen (Belgium), XXV Edizione Pievi & Castelli in Musica (Italy) and Cyclus Koncertu Star\u00e9 Hudby Brno 2012, among others.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We here are Operation Lifesaver couldn't get the public rail safety message out to the masses without the dedication and commitment of our valued volunteers. One such volunteer we would like to recognize is Tom Allan, who retired from volunteering this year. With 39 years working as a railroader for CN and 12 years as an Operation Lifesaver volunteer, Tom's dedication to public rail safety runs deep. We caught up with Tom and asked him to share his experiences with OL and CN and one last rail safety message [caption id=\"attachment_12440\" align=\"aligncenter\" width=\"600\"] Tom Allan, spreading the rail safety message.[\/caption] What made you decide to become a volunteer for OL? I was approached by CN Police. Why do you think the operation Lifesaver program is important? People of all ages do not realize the dangers of trains and to make them aware will save lives. What did you enjoy the most about volunteering? I love volunteering and helping people, I am also a volunteer with St. John Ambulance and am in my 50th year of volunteering with them. What kind of activities do you take part in as a means of promoting rail safety? Presentations, displays, Hamilton Police Safety Week. I also belonged to a model railway club and had a display setup at every show they did. When reflecting back on your time as an OL volunteer, are there any moments that really stand out for you? There's not one particular moment, but I always enjoyed doing presentations in schools and seeing that both the students and the teachers were amazed at some of the information our program gave them. What's one thing you'd like Canadians to know about rail safety? What would you like them to think about when they see train tracks and railway property? The biggest thing is: Stay off railway property and don't use the tracks as a shortcut! I have always stressed that any time is train time and told people if you see two pieces of rail, assume a train is coming and stay away from the tracks. At a crossing obey the crossing signals, and only cross when you can see clearly both ways down the track.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A trailer for the wide-released feature-length documentary I directed and edited. The film is distributed by The Orchard, and now streaming on Hulu. Visit loveandsaucers.com and facebook.com\/loveandsaucers for updates.\nLove and Saucers is the story of David Huggins, a 72 year-old Hoboken man who claims to have had a lifetime of encounters with otherworldly beings - including an interspecies romance with an extra-terrestrial woman - and chronicled it all in surreal impressionist paintings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Inside the mind of THANOS, SHE-HULK and MOONDRAGON learn a secret so dark, so disturbing, that it will forever change the way you see the Titans! This is it, gamma-fans. One way or another, the case against STARFOX ends here! Will he face ruin or redemption? All this-and a two-fisted throwdown between everyone's favorite Emerald Amazon(Yay!) and the Mad Titan (Boo!)!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fully revised and updated-the go-to guide from an expert on international trade Doing business across national borders is more profitable than ever. In the exhaustively revised fourth edition, Import\/Export provides step-by-step guidance to show you how to take part in the booming world economy.\nA comprehensive book on shipping derivatives and risk management which covers the theoretical and practical aspects of financial risk in shipping. The book provides a thorough overview of the practice of risk management in shipping with the use of theoretical examples and real-life applications.\nWritten in a clear and accessible way, this text covers all of the key topics students need to understanding in International Business in the context of emerging markets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As I sit here writing this newsletter, the chorus of this song started on Spotify. How fitting as we enter the second week of school. I can certainly sit behind my desk after a long day of working with students and staff and express to you that the months of planning and preparation is working...but, that is one perspective. Perception and reality are in the eyes of the beholder. I am cognizant to all of the parents, students, and staff needs and I certainly realize that CHS is just one facet to many parts of our district that has experienced a lot of adjustments over the last summer.\nThe beginning of the school year is always filled with activities such as orientations, class meetings, navigating new routes, classes...all to set up the new normal. This year enhanced our urgency to communicate clearly the plan that we are now implementing. Change is never easy. It doesn't matter if you are new to the building, community, a new teacher--or a veteran; change takes time to create new routines. We have spent the majority of our day interacting with students and staff to ease any anxieties.\nHow are you doing? Change is never easy for parents either. We love the relationships that we have built with our families over the years and we realize that you still wonder how your baby is doing each day. Your children become a larger part of our community each day as we model high achievement in each class and model our core values. As the chorus shares, everything's gonna be alright.\nIf you are unaware of this blog called the Week in Reflection, the purpose of this communication is to share with everyone all the work happening at CHS each week from a principal's perspective. This newsletter is very therapeutic (and time-consuming) reflection of all the things that I see each week.\nYou will see photographs, short reflections, opportunities, and events occurring in the near future. I am not an author...although some people have told me for the past two decades that I should write a book---and some say that my newsletter is a book. I am just a simple man that loves what I do...I love kids and want to help each child find their purpose. In this Week in Reflection, you will read and see so many wonderful opportunities and experiences that show the engagement of our students and staff with our mission.\nThe purpose of this evening is for meaningful interactions with our parents. Parents will follow their student's schedule. Parent's, please follow your student's schedule by logging into Infinite Campus and clicking on the schedule tab. We are encouraging students to write down their schedules or having their parents print a copy. Adults and students will assist parents navigate the building.\nTeachers will have (9) minute classes to go through expectations, curricular goals, and answer questions within a controlled environment. The bell will ring with a (4) minute passing time. Parents will follow their son(s) and\/or daughter(s) schedule. Throughout the evening, parents may head to the cafeteria to enjoy refreshments, and learn more about clubs and activities.\nCafeteria Open during all OFF periods. See you, Monday.\n\u200bThere are spots still AVAILABLE for the July, 2019 Student Global Leadership Summit in Davos, Switzerland & London Travel Adventure!\nOPEN HOUSE INFORMATIONAL MEETING with guest Dr. Hanlon at 7:30 pm on MONDAY, AUGUST 27th, 2018 in the CHS Media Center.\nCHS is pleased to announce that there are spots still open to join this amazing collaborative learning adventure to London and Switzerland this July! This is an opportunity to interact with students from all of the globe to discuss relevant topics with the world\u00b4s top thinkers and innovators.\nOur Superintendent of Schools, Dr. Michael Hanlon just returned from the 2018 Global Leadership Summit and would like to share his wonderful experience with you.\nIf you are unable to make the meeting and would like more information please feel free to contact Se\u00f1ora Niedzwiecki anytime.\nSenior Mentors, Grace Turk, Morgan Danison, Grace Pastva, Isabelle LeMaster, and Brian Horton teach the freshmen in their mentoring class the cheers and chants from the Crazies section.\nStudents in Dr. Becky Schneider's Physics classes headed to the track to get physic(al). The students differentiated between speed, velocity, and acceleration along with comparing and contrasting Newton's three laws.\nWhen is late-start? How does it work?\nLate-start is on Wednesday's. The purpose of Late-Start is for our teachers to meet in teams to discuss essential questions within their Professional Learning Communities (Teacher Based Teams). This time assigned for the teachers is best practice and has become a staple for teacher collaboration for the past two decades at CHS.\nBusses will run at the same time. Available staff, Admin, Counselors, and the School Resource Officer will monitor the cafeteria and gym. This is where students will report and be monitored during this time. Test Make-Ups will run from 7:15-8:55 AM on every Wednesday. Classes begin at 9:02.\nStudents will follow periods 2, 4, and 6 on Wednesday's.\nStudents will report to school at the regular time on Thursday's Block Day for periods 1, 3, 5, 7.\n\u200bWhy is a hybrid schedule better?\nFlexibility. This schedule allows the teachers a traditional model of a schedule, while incorporating a longer period for project based learning, differentiated education, and the flexibility to do things in the classroom that you would not be able to do within a 47 minute period.\nHow does this impact learning?\nBy including the block period days, all teachers in all subject areas have longer periods of instruction time with students. This time can be utilized to support project-based learning, student presentation, and in class intervention\/enrichment time.\nHow many times does my child meet in classes?\nEach class will end up meeting four times\/week. Those meetings will consist of three, 47 minute periods and one-- 98 minute period.\nAll lunches run during 4th period on Wednesday. There are NO Topper Times, or Circuits meeting on Block Days. (A) Students will eat in Lunch A, (B) Students will eat in Lunch B, (C) Students will eat in Lunch C,(D) Students will eat in Lunch D---based on their schedule.\nAll lunches run during 5th period on Thursday. There are NO Topper Times, or Circuits meeting on Block Days. (A) Students will eat in Lunch A, (B) Students will eat in Lunch B, (C) Students will eat in Lunch C,(D) Students will eat in Lunch D---based on their schedule.\nObtain an application for a minor work permit in the school office.\nThe top section of the application is to filled out by the student and signed by a parent or guardian.\nThe bottom section of the application is to be filled out by the employer.\nThe back section is to be filled out by your physician.\nAll sections of the form need to be filled out before processing begins.\nStudent will return the completed form with one of the valid forms of ID. Mrs. Sprinzl will process the application.\nPlease allow 24 hours turn around on work permits.\nHave an amazing weekend with family and friends!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's face it, dreaming can be a scary thing. We get these hair brained ideas in our heads of the things we can accomplished, the places we can go, the things we can see, the people we can meet and then\u2026nothing. Is it all just a dream?\nThe saying \"the world is your oyster\" can be terrifying. Which way do I go? I like many things, how can I possibly decided or do I? There are many stages in life that each provide their own set of opportunities or life experiences. Yet how do we harness these times in our lives to make the most out of them, making the world my oyster.\nA few years ago, well actually it is more than a few years ago, a couple walked into my retail shop. I had built my dream or at least the start of it. My vision was to own my own business by the time I was forty years old. Ten days before that date was opening day. I took my skills and experiences and opened a luxury stationery store in the historic city of Portland, Maine. It had the finest stationery, pens, desk accessories and d\u00e9cor for your office that you could ever want or dream of. It was quite spectacular.\nThis day would prove to be different. As I greeted the couple that walked through the door, they told me of their need for personalized stationery. I showed them the selection and we all sat down to discuss what they had in mind. As we talked the husband asked me about my business and where the idea came from. I was very passionate and loved what I had created so it was easy for me to answer that question.\nHe then asked me how I felt about \"change\". My immediate response was \"change, is my middle name.\" Which was true in my mind. I am a lover of many things and have many interests, always have and I hope always will. Yet, sometimes this love of many things and talents in many different areas keeps me stuck, confused and sometimes even fearful. I wonder how many of you reading this can relate to this or at least know of someone like this.\nIt was at that point where he asked me if I had ever read the newly published book \"Who Moved My Cheese\"? Quite honestly, I had never even heard of this book let alone read it, I told him. He went on to introduce himself. His name was Dr. Spencer Johnson, the author of Who Moved My Cheese. This small and entertaining book was just what I needed to hear. Spencer went on to explain to me that we need to see change not as the end of something, we must learn to see change as a beginning. That life demands a certain amount of risk and adventure.\nThis conversation set me free. Sure, I had accomplished some great things in my life and career but I knew there was more in me. I had a lovely and profitable business after all, what more could one want. These were the words that I heard from friends and family. This day when Dr. Spencer Johnson and his wife walked into my shop was a blessing. Some might say it was serendipitous, or there are no coincidences, whatever it was; my world had just been turned upside down or right side up.\nIn the book \"Who Moved My Cheese\", the Cheese represents our life, our job, our home and family. Cheese is what we think makes us happy and when the day comes that some part of this so-called happiness is taken from us for a multitude of reasons, we fall apart and become the victim instead of seeking new ways of doing things, seeing new opportunities or trying on a new way of being. Wherever you are in this journey of life, the risk taker, the sedentary, the fearful; stay present in the moment. When catastrophe strikes (as we view it), by remaining present we can then see the opportunities or another way out.\nI will never forget that day or that conversation. This was the first day of the rest of my life, as they say. I soon began importing luxury stationery and accessories from Europe, opened a store in Europe and dove into a joint venture as the President of US Operations for another luxury European stationery brand. Life takes us on twists and turns. The road is not always clear, we can't always see the end or what is around the corner. Being open to the possibilities no matter when they arrive, will bring abundance instead of scarcity into your life.\nThe world is your oyster, let that be your legacy. Keep finding new cheese. Experiment with new ways of doing things, new ways of viewing situations, new ways of growing yourself personally and professionally. Design your path and walk it.\n\u2022 Be ready to change quickly, again and again.\n\u2022 Having Cheese makes you happy.\n\u2022 The more important your Cheese is to you, the more you want to hold on to it.\n\u2022 If you do not change, you can become extinct.\n\u2022 Smell the Cheese often so you know when it is getting old.\n\u2022 Movement in a new direction helps you find New Cheese.\n\u2022 When you move beyond your fear, you feel free.\n\u2022 Imagining myself enjoying New Cheese, even before I find it, leads me to it.\n\u2022 The quicker you let go of old cheese, the sooner you find New Cheese.\n\u2022 It is safer to search in the maze than remain in a cheeseless situation.\n\u2022 Old beliefs do not lead you to New Cheese.\n\u2022 When you see that you can find and enjoy New Cheese, you change course.\n\u2022 Noticing small changes early helps you adapt to the bigger changes that are to come.\n\u2022 Change happens. They keep moving the Cheese.\n\u2022 Move with the Cheese and enjoy it!\nWhere will your pearl bring you? What is in store for your life that is yet to be discovered? If in fact \"the world is your oyster\" where is it going to take you? Go ahead, go out and find your oyster. It's out there waiting for you, this I know for sure.\nSuper excited to have you along for the journey. Go ahead and download your Confidence Builder Workbook here.\nand don't forget to share the love by sending this to friend in need or sharing via social media. Your spreading the love will help make the world a better place!\nInterested in finding out more about How To Be Merry, sign up for a FREE Discovery Session with me! Fill out the Let's Get Acquainted Questionnaire so I may better serve you during our discussion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you want to use the full creative potential of a web based ideation software but need to operate it within your corporate IT environment? We have the right products for you: Introducing yutongo Corporate.\nDid you know? Science has proven that it takes up to 3000 idea fragments to produce one market hit. yutongo is built in a way that it provides you exactly with this kind of ideation performance.\nSeveral basic principles on the methodology incorporated in our application are backed by numerous studies in the field of creativity research, psychology and sociology.\nKeep your valuable innovation initiatives and corporate ideas private and run yutongo within your corporate network.\nOnly the yutongo application puts a true focus on idea generation and creativity when it comes to corporate innovation management.\nSee how your staff will love yutongo and how the enthusiasm for internal ideation processes will shape the future of your company.\nDo you need methodological training to get the most out of the yutongo application? We provide the training you need in order to sky rocket your ideation output.\nyutongo Basic plans start from 0$\/month and go up to 99$\/month. All Basic plans come with a standard company page (yourcompany.yutongo.com) under the yutongo domain. Our application is hosted with Amazon Web Services. No individual functions or corporate branding is included. You can start public or secret Idea Challenges, but within the yutongo web service environment.\nOur yutongo options for Corporate Solutions come with several additional possibilities to get the most out of your yutongo experience. Most important: You may host the app wherever you want and you may run the application securely within your private network and can invite your own community (staff, customers and other stakeholders) to your projects.\nWhich Corporate Solutions package meets your needs?\nSandro is highly experienced for over 10 years in the field of systematic innovation and ideation. Has managed and conducted over 800 ideation projects for over 300 clients in over 20 different industries. Has shaped and perfectionized the yutongo ideation process. And still is.\nWalid Ahmed is a lover of collaborative ideation and originally a neuro biologist. He is also passioned about neuronal creativity, Ideavolution and Web 2.x with Ruby on Rails.\nOr give it a go for free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcbrn b/data_all_eng_slimpj/shuffled/split2/finalzzzbcbrn
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"These are the top 50 movies that I have seen since May, like the movie database, I'll be editing these rankings too. Use the filters if you want to filter movies that are only on Netflix, Hulu, Prime and HBO. Click here to go to the movie database.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In December 1969 the workers at the state-owned mining company LKAB in the north of Sweden went on strike. Starting as a conflict about wages and working conditions in the mines, the strike posed more general questions concerning the distribution of power in working life and in society. Ultimately, the strike could be seen as challenge to vital aspects of the Swedish model, which at the time was widely celebrated. The strike articulated a critique not only of centrally coordinated wage negotiations, solidaristic wage policy and the concentration of power and ownership in Swedish industry, but also of the intensification of work and the powerlessness of employees in the workplace. The article, which is a revised version of a lecture, is an attempt to put the strike in a historical perspective.\nBetween 1971 and 1975, construction workers in Australia organised in the New South Wales branch of the Builders Labourers' Federation engaged in strikes (\"bans\" in Australian industrial terminology) to prevent ecologically damaging development and in support of other radical causes, such as women's liberation, homosexual liberation and indigenous rights. Proclaiming the principle of 'the social responsibility of labour', this union became especially famous for its \"green bans\" that halted huge development projects. This union prefigures the development of 'social-movement unionism' from the 1990s onwards in its militancy, ultra-democratic forms of organisation, its agenda for radical social and economic change, its determination to overcome working-class fragmentation and its capacity to lead community struggles.\nThe Great Conflict of 1980 and other major labour conflicts in Sweden Depending on how they are measured, either the great conflict of 1980, the general strike of 1909 or the 1945 metalworkers' strike could lay claim to be the largest labour conflict in Sweden. The 1909 LO strike and the 1980 SAF lockout both aimed at pressing the government to act, but failed. The SAF 1980 proposals for a wage freeze and a downsized public sector were conspicuous components of a strategy to change the rules of collective bargaining. After the 1980 failure, SAF adopted decentralized wage formation as a means to radically gear down wage increases.\nIn recent decades, nurses have gone on strike in large numbers and in many countries. They have also used innovative non-strike tactics of resistance such as mass resignations. This article explores nurse militancy through strike narratives. It argues that in recent decades, the professional commitments of nurses to the provision of care have confronted health care restructuring, nursing shortages, intensification of work, precarious employment and gendered hierarchies with a militant discourse around the public interest, and a reconstitution and reclamation of caring work, what I call the politicization of caring. This research suggests that nurses' dedication to caring work may now encourage rather than dissuade them from going on strike. It also highlights the widespread public support for nurses, even when they go on strike illegally.\nFinally, nurse militancy challenges commonsense miscon\u00acceptions that women workers are passive and unlikely to strike. It problematizes the associations of militancy with men, masculinity and manliness. In fact, in its capacity to build public support, nurse militancy helps to mainstream and legitimize militancy itself. Nurse militancy and coalition-building are part of a long tradition of union women's organizing and resistance which may provide untapped potential to defend public services, protect workers' rights, and invigorate the union renewal project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two incoming Flinn Scholars were among 12 Arizona high school students to take top honors at the Intel International Science and Engineering Fair (ISEF) held in Phoenix last week.\nEmily Ricq, a graduating senior at Chandler High School, won a $1,000 1st place award from the Society of Exploration Geophysicists, a $2,000 scholarship from Arizona State University, and a $1,000 3rd place grand award for her lunar geology project, \"The Influence of Non-Synchronous Rotation on Patterns in the Orientation of Ice Fractures on the Surface of Europa II.\" She will attend the University of Arizona next year.\nTyler Rosensteel, a Brophy senior who has been conducting neurogenomic research at the Translational Genomics Research Institute, translated some of that research into the 4th place grand award in the biochemistry division of ISEF for his project, \"Identification of Promoter Elements Required for Transcriptional Induction of Motility Related Genes.\" Rosensteel intends to enter Arizona State University in the fall.\nBoth Ricq and Rosensteel's projects won their categories at the Central Arizona Regional Science and Engineering Fair earlier in the year to be eligible for ISEF, which attracted 1,400 students from around the world and $3 million in prize money and college scholarships.\nBut a penchant for science is not unique to the class of incoming scholars; indeed, scientific curiosity seems to run in the 'Flinn Family.' Nearly half of this year's 18 graduating scholars are leaving Arizona's universities with degrees in the sciences; five have plans to attend medical school.\nChristopher Finney, an environmental sciences major from NAU, will return to the site of his environmental field study in Sao Paulo, Brazil. Eric Yip, who received his bachelor's degree in ecology and evolutionary biology from the University of Arizona, is headed to Cornell University in the fall, where he will expand on his studies of tropical spiders in an entomology Ph.D. program.\nThe other six science scholars in the graduating Flinn class all received degrees from UA in Molecular and Cellular Biology: Xuemei Cai, who also graduated with a bachelor's degree in biochemistry, will attend Harvard Medical School in the fall; Lauren Giesecke will pursue a joint M.D.\/M.P.H. program at Mayo Medical School Rochester and Tulane University upon her return from Bolivia; Caitlin Hall, a double major in philosophy, will attend Yale Law School in the fall; Mia Henderson, who was named a Medical Scientist Training scholar, will serve with Teach for America for two years in Los Angeles before attending medical school at Washington University, St. Louis; Lars Peterson, who is interested in the public's engagement with science and science communication, will attend the University of Rochester Medical School; and molecular and cellular biologist Stephen Sosnicki will spend a year teaching English in Japan before returning to the U.S. to attend medical school.\nRead more about the graduating Flinn scholars at 2005 Flinn Scholar Graduates.\nFor details on the entire 2005 Class of Flinn Scholars, see our press release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"YOUNG stars on the football field and netball court showed their skill during Saturday's Sunraysia Junior Football and Netball League grand finals.\nIt was Mildura's day in the netball, claiming premierships in all four grades \u2013 13 and Under Divisions 1 and 2 and 15 and Under Divisions 1 and 2.\nThe Under 13 football premiership went to Robinvale-Euston, while Irymple claimed the Under 14 flag.\nFor full scores, pictures and medal winners, pick up your copy of tomorrow's Sunraysia Daily.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The latest 20th Infocom World 2018 Conference that has been held in Athens, Greece, on November 21, 2018, (http:\/\/www.infocomworld.gr\/) has been considered as \"the greatest ICT & Media Conference in South-Eastern Europe\" and it is widely regarded as \"the major annual meeting of all the digital market stakeholders\" in order to implement Digital Transformation, following to current market needs and\/or challenges.\nIn the scope of 5G ESSENCE, 5G-DRIVE and other projects' dissemination activities, the Hellenic Telecommunications Organization S.A. (OTE), has organized, in cooperation with partners involved in the aforementioned projects, the following three dedicated Special Sessions: the 5G ESSENCE-based session, the 5G-DRIVE-based session and a third session about modern technologies and wider 5G-related aspects.\nIn the context of the three above sessions, there have been many opportunities for interactivity and exchange of information and \/or experiences between all involved EU-funded projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcfmr b/data_all_eng_slimpj/shuffled/split2/finalzzzbcfmr
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+{"text":"Our raglan is the most relaxed fit of all the tops. Peace Raglan has the cool hippie hipster vibe of the 70's. The body portion of the raglan is made of (non fleece) sweatshirt material with less stretch. The sleeves are more stretchy. The black and white gives this raglan an even cooler vibe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Test if there is a difference between population means when a response variable is classified by one or more categorical variables (factors).\nNote: Although the Fit Model analysis can be used to perform a simple 1-way ANOVA it is more intuitive to use Compare Groups > ANOVA analysis. Also, see the related tasks section for performing specific ANOVA models.\nClick Add Factor to add the simple factors.\nTo add the interaction between two or more factors, in the Terms list box, click the first term then hold down the CTRL key and click each additional term to include in the interaction, and then click Cross.\nNote: It is sometimes quicker to use the Factorial command to build a factorial model and then remove any unnecessary terms.\nTo remove a factor and all terms using the factor, in the Factor list box, click the factor and then click Remove.\nWhen you select a factor, all terms that include the factor are selected. If you hold down the CTRL key and click additional factors, then all terms which include all the selected factors are selected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Morning registration is at 8.30am. The lunch break starts at 12.25pm and afternoon registration is at 1.20pm. For students the formal school day ends at 3.00pm.\nThe HUB and other activities continue after school.\nThere are six 50 minute lessons each day \u2013 providing 25 hours of lessons each week.\nLunches are provided in a cafeteria style system which offers a wide choice of good quality food. The high quality of our lunches helped us gain the Healthy Schools Mark. If your parents prefer, with permission you may go home for lunch or bring a packed lunch to school.\nAt break time drinks and snacks are available from the Canteen.\nDuring break and lunch time you will be looked after by lunchtime supervisors, learning support assistants and teachers. There is a wide variety of sporting, musical and other activities for you to enjoy at lunchtime and the library is available for you to read or tackle homework.\nThere is a Year Assembly each week and on other days your Form Tutor will lead your Tutor Group in a short period of reflection about the 'Quote for the Week'. Your parents may ask to withdraw you from assemblies, but since the school takes a non-denominational and informal approach, this rarely happens.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Aras\u00aeProduct Engineering applications provides an integrated set of PLM and PDM capacities to help businesses define and manage information throughout the product lifecycle, from design and development, verification and validation, manufacturing, service, and through to end of life. This includes all aspects of complex electromechanical systems and software components. The application supports parts, multi-level bill of materials (BOMs), CMII standard configuration management and change management, approved vendor and manufacturer lists (AVL\/AML), and documents.\nAras\u00aeProduct Engineering is highly flexible, allowing organizations to tailor business processes to their specific requirements.\nManages the process for creating and approving parts, assigning attributes including cost, approving suppliers and associating CAD models and drawings.\nProviding all team members, suppliers and manufacturers with an accurate and consistent definition of the product reduces errors and delays.\nEnsuring that all team members, suppliers and manufacturers have visibility of the change status minimizes errors and reduces costs and delays.\nAras\u00ae Product Engineering is an open-access application.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking For Abortion Alternatives Organizations?\nCentral Coast Or Right To Life, Abortion Alternatives Organizations, listed under \"Abortion Alternatives Organizations\" category, is located at 19586 Siletz Hwy Siletz OR, 97380 and can be reached by 5414442246 phone number. Central Coast Or Right To Life has currently 0 reviews.\nBrowse all Abortion Alternatives Organizations in Siletz OR. Discover census data for Siletz, OR.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdepu b/data_all_eng_slimpj/shuffled/split2/finalzzzbdepu
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index 0000000000000000000000000000000000000000..fea1e4640585aacae9d2c639f39b47b1e221fbed
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+{"text":"Find Multnomah County human services information.\nWhen we build well-being, we make sure that everyone can reach their potential and fully contribute to our communities.\nIs this page helpful? Take a 1 question survey.\nThere are no upcoming events for Department of County Human Services. View past events.\nWhat's the most efficient type of electric heater?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"EYECON is a role-playing and gaming convention run by the Sydney Roleplaying Games Association. It runs each year on the Easter Long Weekend in Sydney.\nEYECON is a chance to play pre-prepared games prepared by our friendly team of volunteer gamemasters. The motto for EYECON is \"bring your friends, try new games and have some fun\"\nNever played a role-playing game before? Go to our What is Role-playing? page for answers and more detailed information.\nA variety of game genres are available, including regular table-top games (such as Dungeons and Dragons), free-forms (similar to live action games or LARPs), card games, board games and more.\nPlayers may sign up individually or in teams of up to four to eight, although there is room in some cases for larger groups.\n28th February 2019 - Over the weekend all games will be uploaded and Registration will open on Monday morning.\n3rd March 2019 - Uploading all the games blurbs to the website - We have some amazing Indy Games offered.\n4th March 2019 - Reviewing the games schedule in preperation for opening registration this week.\nThe SRGA would like to recognise the many companies that have supported our conventions over the years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The UPMC Pinnacle Memorial Hospital medical campus sits on 10.91 acres on South Belmont Street in east York. It consists of the main hospital building, four clinical buildings and a maintenance garage.\nAlso available for sale: a three-story medical condominium office building located to the south of the main campus sitting on approximately 5.5 acres.\nSpecial Exception \u2013 college dormitories, hotels, retirement facilities, nursing homes, retail, restaurants & schools.\nThe total SF of the Main Building is 142,897 SF which does not include the Modular Administrative Office Center.\nThe Modular Administrative Office Building containing 14,755 SF is connected by a ground floor hallway, but is entirely separate from the Main Building.\nTogether, the Main Building and the Admin Office wing contain 157,652 SF.\nBuilding Shell \u2013 Masonry, originally constructed in 1962 and subsequently modified, added to and renovated until 2010. Well maintained.\nBuilding roofing \u2013 In very good condition with the one story portion being re-roofed in 2012.\nElevators (Otis) \u2013 two public (original) & two private (added in 1976).\nFlooring \u2013 VCT w\/ cove base.\nCeilings \u2013 lay-in tile w\/ 2\u00d72 prismatic lens lighting.\nHeating \u2013 two York Shipley 400 BHP duel fuel (natural gas- #2 fuel oil), fire tube steam boilers producing 90 psi steam. Both boilers are original but have been regularly maintained and serviced.\nCooling \u2013 two York 500-ton centrifugal water cooled chillers. One unit was placed in operation in the mid-1980's and the other in 2000's. Two Evapco cooling towers with similar age as the chillers.\nAir handlers \u2013 4-5 Indoor and 4-5 roof mounted air handling units.\nTerminal Units \u2013 4 pipe fan coil units located beneath the windows. Other spaces are served by constant volume, VAV and CAV terminal boxes with hot water reheat.\nControls & Instrumentation \u2013 Originally Honeywell and currently controlled by Johnson Control Systems.\nWater Service \u2013 two main domestic water service entrances. No pumps are necessary as the water pressure entering the building is 60-70 psi.\nDomestic Water Heating \u2013 Single P-K Compack Steam instantaneous water heater installed in 2012. The system is installed in line with an original building storage tank with instantaneous steam coil, thus has redundant capacity per FGI 2014.\nFire Protection System \u2013 Fully sprinklered with one fire service entrance, no pumps are required.\nElectrical Service \u2013 supplied by two medium voltage utility services feeding double ended switchgear located in the central utility plant. The double ended switchgear has a tie breaker that can switch the power over upon the loss of one service.\nEmergency Power \u2013 two generators, a 625 KVA installed in 1982 and a 250 KVA installed in 2002. Generators are fueled by an underground 1,000 gallon installed in 1982 with monitoring capability. Automatic transfer switches separate the distribution into life safety, critical and essential equipment branches.\nAlong the rear eastern edge of the property, four one-story masonry buildings are aligned along Mill Creek with I-83 immediately adjacent further east of Mill Creek. Paved parking for 234 cars surrounds the buildings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am in the EU, will I have to pay to receive my kit?\nNope! Your kit will be shipped directly from Denmark. However, you will have to pay the shipping back to our lab in Copenhagen.\nI have a handheld showerhead, can I still participate?\nUnscrew your showerhead as close to where the water comes out as possible. For some showerheads, there will be a screw directly on the showerhead. If this is what yours looks like, you're in luck! Your showerhead will work!\nIf your showerhead unscrews at the base of the handle, it may have a filter on the end. If so, you will need to remove the filter. If you are willing (and able) to remove the filter, or your showerhead doesn't have one, then you can sample your showerhead!\nI can't access the inside of my overhead showerhead face \u2013 where should I swab?\nIf you cannot access the inside of the showerhead face, try to swab as close as possible to it and swab thoroughly to maximize the amount of bacteria captured on the swab.We want you to swab as close to the interior of the showerhead as possible without disassembling the showerhead. Please only swab the inside surface! Do not swab the outside face of the showerhead.\nI think that attempting to sample my showerhead might result in breaking it and\/or the surrounding plumbing, should I do that?\nNo, we cannot condone breaking your showerhead for science! We would recommend swabbing another showerhead (maybe a neighbor or a friend's house). We don't want you to cause a plumbing problem just to get some swab samples!\nMy kit is missing a component, what should I do?\nIf you are missing any part of the kit (ex. the protocol, return envelope, chemistry kit, Questionnaire\/Data Sheet, etc), we will try our best to send a replacement. Since these kits were individually assembled by hand, mistakes can happen. Please contact yourwildlife@gmail.com.\nHow quickly should I return my kit after I complete my test?\nPlease place necessary materials (Questionnaire\/Data Sheet and swab) into the return envelope (included) as soon as possible. The sooner the swab is placed in the mail and returned, the better the results will be.\nU.S. Participants: Returning the kit is simple \u2013 all U.S. kits have a prepaid USPS return envelope included. Just place the swab and questionnaire\/data sheet in the envelope, seal it by removing the tape, and place it in a mailbox! It should arrive at the University of Colorado in 2-3 days.\nInternational Participants: Return envelopes need to have postage paid by the participant. Just place the swab and questionnaire\/data sheet in the envelope, seal it by removing the tape, and place it in a mailbox! It should arrive at the lab in 1-3 weeks.\nThe colors on my water chemistry test do not match exactly the colors on the protocol, is something wrong?\nNo, the colors will not be exact since some values will fall between two colors, and there will slight variations. Estimate the closest value associated with the color to the best of your ability.\nIf I have a biofilm in my showerhead, does that mean I am risk for an NTM infection? Should I talk to my doctor?\nNo. These tests are not diagnostic. We cannot make any medical claims or provide any sort of insight into a clinical diagnosis. Once the results start coming in, we will provide a more comprehensive discussion of what your results do or do not tell you about the water quality in your home and why you should probably not be worried about what bacteria we find inside your showerhead.\nDo I have to wear the gloves included in the sampling kit?\nYes. The gloves are not for your protection, but rather to prevent any sort of contamination from the skin to interfere with the showerhead biofilm sample.\nI signed up for the project with an address, but want to test the showerhead at a friend or family members house (different address), is this possible?\nYes, if you are testing at a different address from where you received it, please make sure to record all the information regarding the showerhead location on the top of the Questionnaire\/Data Sheet.\nI signed up stating I was interested in the project, but never received a kit. Did something get missed?\nThe Showerhead Microbiome Project is happening in several waves. Just because you were not contacted in the first group of 500 people, does not mean you will never receive a kit.\nHow long will the analysis take? Any updates?\nWe are just as excited as you to figure out what is in your showerhead! But science is slow. The time between when you send a sample in and when you know what is on your showerhead may be a while. Our guess now is that it will take about a year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are we Overvaccinating Our Pets?\nYou can get a prescription from your vet and send it to the company They'll keep the prescription on file and you can reorder any time.\nAir Cleaners rated by the \"Clean Air Delivery Rate\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdimu b/data_all_eng_slimpj/shuffled/split2/finalzzzbdimu
new file mode 100644
index 0000000000000000000000000000000000000000..369ee00c983a338f5c7ffbdb16564738464c7ed5
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Monterey Park is a map-rescinded community. This means that the community has no special flood hazard areas and a flood map for the community has not been published. The city's Community Number is 065047. The city's current effective map is designated No Specific Flood Hazard Area (NSFHA). The city entered the regular phase of the National Flood Insurance Program on November 30, 1979. Although it may not be subject to the 100-year flood, floods of greater magnitude could occur here. In addition, certain structures may be damaged by local drainage problems.\nThe entire community is Zone X, for flood insurance rating purposes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Inspired Asset Management has secured development finance for two home county resi schemes worth a combined \u00a327m.\nThe first secures \u00a315.1m for a 61-apartment scheme in Chertsey through a \u00a312.9m senior loan from Amicus, arranged by Arc & Co., and \u00a32.2m in preferred equity from Grasslake UK Investments Limited.\nBridge House, an early 2000s office building, will be refurbished and extended to provide 39 one and two-bedroom micro-apartments and 22 standard-sized one and two-bedroom apartments close to Chertsey Bridge and the River Thames. Prices start at \u00a3265,000 and the scheme has an estimated GDV of \u00a320.4m. Chertsey is an employment hub with Samsung, Sony, Toshiba, Procter & Gamble and McLaren all nearby.\nThe second funding secures \u00a35.2m for a 25-unit build to rent scheme in Farnborough comprising a \u00a33.9m senior loan from Octopus, also arranged by Arc & Co., and \u00a31.3m in mezzanine finance from a private funder, arranged by Savills.\n25 two and three-bedroom maisonettes off Westmead in Farnborough town centre will be refurbished and reconfigured to create high-spec flat shares for rent on a room by room basis. The scheme has an estimated GDV of \u00a36.2m and is predicted to offer a gross rental yield on cost of 12.4%. The development is situated just 0.4 miles from the railway station for a fast 34-minute train to London Waterloo.\nEmma Burke, Head of Development Origination at Octopus, said: \"Having received the introduction from Arc & Co., one of our key partners, we were delighted to work together to fund Inspired's latest development scheme with our new 65% LTGDV low cost senior stretch development product. This is an extremely well-located residential scheme and forms part of an exciting development portfolio.\nTo enquire about our Chertsey development, please contact the sales team on 020 8688 6552.\nWell done Martin, keep up the good work!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In order to use the website, it is not necessary for the user to allow the installation of cookies sent by the website, or the third party acting on their behalf, without prejudice to the need for the user to log in as such in each one Of services whose performance requires the previous registration or \"login\".\nThese are cookies, either handled by us or by third parties, which collect information about how the website is used (for example, the pages you visit or if there is an error) and that help us to locate and solve problems, quantify The number of users or measure utilization of the website. All information collected is completely anonymous and helps to understand how our site works, making the necessary improvements to facilitate its navigation.\nThat you navigate the site and that in turn, we can recompile information on how you use the website, to understand the usability of the site, and to help us implement the necessary improvements. These cookies will not collect any information about you that may be used for advertising purposes, or information about your preferences (such as your user data) beyond that particular visit.\nOur purpose with these cookies is to improve the user experience. They are used to remember certain configuration parameters or to provide certain services or messages that may improve your experience on our site. For example, they remember the country or language you have selected when you visit the pages, and are not used for marketing purposes.\nRemember your login data as a customer when you return to the page. They will never collect any information about you that may be used for advertising purposes, or information about your preferences (such as your user data) beyond that particular visit.\nThey are cookies managed by third parties. They are sent to the user's terminal equipment from a computer or domain that is not managed by the publisher, but by another entity that treats the data obtained through the cookies. Therefore, you may use the tools of these third parties to restrict the use of such cookies. For example, some of them are used to link to other websites that provide certain services to view or download videos, share content on social networks (Facebook, Twitter or Google). And in other cases, they may also modify the ads of other websites to suit their preferences.\nLink to social networks, watch or download videos, get information about your visit to our website useful to adapt ads that will show you on other sites or web pages. Likewise, it is possible that when visiting a website or opening some email where an advertisement or promotion is published on our products or services is installed in your browser any cookie that serves to show later advertising related to the search you have made, develop a control of our ads in relation, for example, with the Number of times they are viewed, where they appear, what time they are viewed, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is not uncommon for new born babies to become jaundiced at birth. Jaundice does not necessarily mean a baby is ill but if it does not disappear normally (within 14 days for full term babies and 21 days for pre-term babies) it should be investigated. The CLDF Yellow Alert Campaign supports, informs and educates parents, carers and health professionals in the UK on the signs and symptoms of prolonged jaundice in newborn babies, promoting early diagnosis and appropriate referral for liver disease.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last Saturday, genealogists and others with an interest in early New England ancestors gathered for a day of presentations by two deeply knowledgeable experts. Robert Charles Anderson and Christopher Child offered anecdotes and insights on Mayflower passengers and life in colonial New England. Anderson regaled us with the saga of his groundbreaking The Great Migration Study Project, an undertaking that was supposed to take 3 years but is now almost 30 years along, with new discoveries continuing.\nChild gave an overview of 17th-century immigration to New England, looking at both where the Pilgrim fathers came from and how later generations spread out over the continent. He also discussed tips and tricks designed to break down brick walls in your research. An example: know your suffixes! \"Junior\" and \"Senior\" in colonial documents do not mean what we think of today. Similarly, terms like \"cousin\" and \"in-law\" can be broadly applied to a wide range of relationships. There was also discussion of how DNA is helping to solve some old family mysteries. To sum it up in Anderson's words, \"even in a field that has been digged more than any other in genealogy, further digging will still be profitable.\"\nThe day was possible thanks to the work of longtime CGS member Jane Knowles Lindsey, who first approached the New England Historic Genealogical Society to plan this event. At our membership table, seven new members joined the society and twenty-two signed up to receive our eNews. We offered participants Society information and sold some books. Two memberships were offered as door prizes. Helping me at the membership table were Kath Merilo, Karen Wetherell, and Kathleen Beitiks.\nJane organized a Silent Auction asking for donations from our members. Thanks to their generosity, we raised just over $1000. A special thanks to Tracey Laubsted and Diana Wild who helped with the auction.\nThanks also to Matt Berry, who acted as Treasurer, Ron Madson, our \"official\" CGS photographer, and Jerry McGovern, who helped me unload all the membership supplies when I returned to the library.\nIn addition to Saturday's event, CGS offered one-on-one consultations for people who had New England research questions. Our New England experts Jim Russell, Pat Friesen, Linda Longley, Pat Smith, Roger Prince, and Jane Lindsey who provided their expertise during the day. Karen Wetherell and I were the timekeepers.\nIt takes a village and one Jane to make it all happen. Thanks to all our volunteers, our silent auction donors and those who attended the event we had a GREAT weekend!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhabr b/data_all_eng_slimpj/shuffled/split2/finalzzzbhabr
new file mode 100644
index 0000000000000000000000000000000000000000..33583c7dfc50f4c0326fb5e8483642ff19ffa3d6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhabr
@@ -0,0 +1,5 @@
+{"text":"The three main sections of a pine log are an outer layer of bark, a circle of sapwood and a core of heartwood.\nBark consists of outer layers of dead cells and an inner layer of live, dividing cells. Bark has a protective function, helping prevent attack by insect and fungal pests, and stopping the interior of the trunk from drying out.\nThe wood is made up of concentric rings, which each represent a year's growth. Each ring is composed of thin-walled cells (the paler wood) and thick-walled cells (the dark circular lines). A new layer of wood is added each year in a very thin region between the inner bark and the sapwood.\nSapwood consists of live cells that carry and store water and nutrients.\nHeartwood is dead and is a darker colour than the surrounding sapwood.\nPith is the oldest part of the tree and forms the core of the trunk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Noah Campanelli and Hunter Walsh both drove in two runs apiece in the fifth inning to help McQuaid erase a two-run deficit. Campanelli homered for his only hit while Walsh finished 1-for-2 at the plate with a run scored. Charlie Merkley and Zach Lee scored the other two runs for McQuaid. Ryan O'Mara allowed two runs on four hits over six and a third innings. O'Mara struck out out nine and walked two on 96 pitches. McQuaid is 8-0 on the season.\nHudson Bockmier led a group of seven different Pittsford Mendon players who recorded an RBI in the double-digit offensive effort. Bockmier went a perfect 3-for-3 with four RBI, scoring one run. He collected a double and a triple while also giving up two runs on five hits and a walk over three innings of work He struck out five. Karthik Matthew-Malik doubled as part of a 1-for-3 effort at the plate, scoring two runs and driving in one. Charlie Bennett, Kyle Hennessey, Zach Janson, Kevin Keneally and Jon Surtel all recorded one RBI for Mendon. Tyler Procious walked three times and scored twice. Zach Janson was 1-for-3 with a walk, an RBI and a run scored. Surtel earned the save, giving up three runs on four hits while striking out five and walking one over three innings of work. The Vikings led 11-1 after three innings.\nNick Provenzale connected on all four of his trips to the plate, driving in five runs to lead Pittsford Sutherland (3-2). Connor Fitzsimmons and Jack Zawacki both knocked in multiple runs, with Fitzsimmons going 3-for-4 with three RBI while Zawacki was 2-for-3 and two RBI. John Messina and Robert Nolan combined for the win and nine strikeouts on the mound.\nTyler Henshaw drove in two runs and tossed two shutout innings for Brockport. Henshaw struck out two on 23 pitches while allowing no hits for the win. Sam Velletri finished 2-for-3 at the plate with one RBI and one run scored. CJ Wiest and Esten Schalge drove in the other two runs for Brockport. Troy Leibert started the game on the mound, giving up four runs on seven hits and striking out four through five innings of pitching. Leibert, Nick Nealon, Joe Herman and Ryan Sigillo scored the rest of the runs for Brockport.\nTrailing by two with the bases loaded in the bottom of the seventh, Kyle Wood cracked a walk-off triple. The sophomore shortstop finished 2-for-4 at the plate, with all three of his RBI on his final at-bat. Julius Teabout, who singled and scored on Wood's game-sealing triple, went 3-for-4 at the plate with two runs scored. Teabout started the first four innings on the mound, giving up two runs on four hits while walking three and striking out three. Dylan Blanchard-Rieke picked up the win in relief, allowing two runs on five hits through three innings. He struck out one and walked one, going 2-for-3 at the plate.\nBobby Bradley, Connor Stuewe and Gage Ziehl collected two hits apiece as Penfield jumped to a 7-1 lead and held on for the win. Ziehl finished 2-for-3 with a triple and a double. The sophomore walked twice, scored twice and drove in a run. Stuewe drove in a run and was 2-for-5. Bradley doubled and 2-for-5 on the day. Danny Stapleton started, struck out three and allowed two runs on four hits over four innings to pick up the win. Brendan Miller doubled and scored a run. Ryan Novy scored and drove in a run with a hit and two walks. Peyton Pace had a hit and scored a run. Collin Pichany crossed home plate twice. Loris scored four runs in the seventh, three with two outs.\nCooper Crunick recorded five RBI on a 3-for-4 performance at the plate with three triples to lead a four-inning offensive explosion for Canandaigua. Joe Brinza went 2-for-4 and drove in three runs while scoring twice. Dylan Mackenzie was 3-for-3 with a double, two RBI and four runs scored. Seth Vigneri finished 3-for-4 with a triple, two RBI and two runs scored. Sam Werth doubled and was 3-for-3 scoring one run and driving in two. Michael Sculli went 2-for-3 with a triple, two RBI and two runs scored. Beau Norris, and Fletcher Hanford scored a run apiece for the Braves. Tristan Tyndall and Nate MacClean both crossed home twice. Mackenzie and Austin Miller combined on four innings of shutout pitching, giving up only two hits while striking out seven. Canandaigua scored six runs in the first, eight in the second and six more in the third before capping the game with three runs in the fourth.\nRyan Watson finished 3-for-4 including a two-run single in the fifth inning when Brighton (4-0) scored five times. Nick Stodgell gave the Barons an early lead with two-run home run in the second. Stodgell added a sacrifice fly later in the game to finish with three RBI. Trent Wasserman fanned six over four shutout innings to pickup the win. Wasserman allowed just one hit in the outing. Jack Falter, Matt Falter, Sam O'Hora and Chase Seibold all singled and scored. Wasserman collected a walk, RBI and run scored.\nAnthony Roides plated Tyler Holmes and Parker Houde with a two-out triple in the third to give Fairport a 2-1 edge. Holmes finished 1-for-2 with a walk in the game. Josh Van Bramer and Peyton Olinski had a hit apiece. Nolan Hicks and Aaron Jenks both reached on a walk in the game. Van Bramer fanned four and allowed two runs (one earned) on two hits over four innings of work.\nHudson Alread tripled in the sixth inning to drive in both of Victor's runs in the losing effort. Santino Rosso finished 2-for-3 at the plate in Victor's first loss of the season.\nAlex Minnehan struck out nine in a complete-game effort and improved to 3-0 on the season. Noah Fasso finished 3-for-4 with three RBI. Jake Smith was 2-for-4. Livonia improved to 5-0 with the win.\nGeneseo sent 15 to the plate and scored 10 runs in the first inning to put the game out of reach early. Russell Brownell, Jake Hamilton and Luke Harris each had two hits in the opening stanza. Hamilton finished the game 3-for-4 with three RBI and two runs scored. Brownell was 2-for-4 with two RBI and three runs scored. Harris collected two hits in four at bats and crossed home twice. Liam Miskell doubled and drove in a run. Evan Salvaggio started and allowed four runs (three earned) on eight hits over six innings for the win. Salvaggio struck out six and walked two. Salvaggio was 1-for-3 at the plate with a walk, run scored and an RBI. Noah Reilly scored once and drove in two runs with a hit. Dan Kelly was 1-for-3 with an RBI and a run scored.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Orleans, Louisiana (Sporting Alert) \u2013 The Los Angeles Lakers star guard Kobe Bryant, at age 34, on Wednesday night, became the youngest player in NBA history to score 30,000 points.\nBryant reached the mark with 1:17 left in the first half in tonight's game against the New Orleans Hornets, after scoring over center Robin Lopez to score his 30,000th and 30,001st points of his 17-year career.\nBryant went on to finish with 29 points in leading the Lakers (9-10) to a comfortable 103-87 victory over the Hornets at New Orleans Arena \u2013 bouncing back from back-to-back defeats suffered at the hands of the Orlando Magic and Houston Rockets.\nDwight Howard had 18 points and eight rebounds, Antawn Jamison, 15 points and Metta World Peace, 11 points and nine boards also reached double figures in the win.\nMeanwhile, Ryan Anderson hit 4-of-7 from three point range to score a team high 28 points for New Orleans, who are 5-12 on the season.\nGreivis Vasquez collected 16 points and eight assists, while Lopez scored 15 points in the defeat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ron Malibu was born just after the turn of the century in a bar in Mexico, and is actually the French phrase for light rum which tickled me no end when I turned over the bottle to discover this. First I had to drain the bottle, so pretty much anything tickled me at the time.\nRon was born in a transitional phase of my life, shortly before a quick round the world trip which started the end of my work life. I took three years off the working, and ended up in Austin, Texas. Which was pretty much a shocker to me, but then you've not met my Texan wife.\nRon took a sabbatical from writing for fun and pleasure, and instead focused on professional blogs. By professional I don't mean that I get paid for them, I mean that that I strive to appear like someone you'd want to work with, rather than a gun-slinging, Belgian Triple drinking circus performer, or whatever the hell I style myself in my mind's eye.\nOr a dad about to buy a mini-van. I have a blog for that too.\nAnd yes, I dream of a drag-racing, tire smoking, turbo-nutter minivan that the family wears crash helmets to drive in. But it will probably be a Honda Odyssey.\nSo I started writing again for fun, though that's not to say that you can't send me money if you like.\nYou can bribe me, send me things, and flatter me. They will all get you a long way, as taco-pooping elephants don't pay for themselves, and I like money, \"you know, for kids\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 15th National Assembly will resume its session at 10 am today elect its Speaker and Deputy Speaker.\nThe election will be held through secret ballot.\nThe majority party in the House, Pakistan Tehreek-e-Insaf has nominated Asad Qaisar for the slot of the Speaker while Syed Khurshid Shah has been named by the joint opposition for it.\nFor Deputy Speaker, PTI has fielded Qasim Suri while the joint opposition has nominated Muttahida Majlis-e-Amal leader Asad Mahmood.\nSindh Assembly will also elect the Speaker as well as deputy through secret balloting on Wednesday.\nNomination papers of Agha Siraj Durrani of PPP and Javed Hanif of MQM-Pakistan have been accepted for the post of Speaker.\nRehana Leghari of Pakistan Peoples Party and Rabia Azfar Nizami of Pakistan Tehreek-e-Insaf are contesting for Deputy Speakership.\nAccording to Provincial Assembly Secretariat, election for the two posts will be held through secret balloting by 165 elected members of the 168 member house.\nKhyber Pakhtunkhwa Assembly members will elect Speaker and Deputy Speaker today.\nPakistan Tehreek-e-Insaf has nominated Mushtaq Ahmad Ghani for Speaker and Mehmood Jan for the slot of Deputy Speaker.\nLaiq Muhammad Khan of Awami National Party is contesting for the slot of Speaker, while Jamshaid Khan of PML (N) has been nominated for Deputy Speaker post by the joint opposition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbheaw b/data_all_eng_slimpj/shuffled/split2/finalzzzbheaw
new file mode 100644
index 0000000000000000000000000000000000000000..a74767a1bdfa56ac2d8d79953ec84add27373c57
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbheaw
@@ -0,0 +1,5 @@
+{"text":"RosendoRell wrote: Can I get access please?\nYou can simply download Capture Nexum as this is long in beta.\nCapture also assumes it's 0-based.. I suspect a typo snuck in there. Are you aware of any other systems than your Mbox which assume it's 1-based?\nI would just resend the SInf message with the new layer list definition, but I would be surprised if any consoles support the subsequent workflows necessary to ask the user to deal with the addition\/removal of layers and their patch on the fly like that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"feel good time: pebble beach concours d'elegance.\nThe Pebble Beach Concours d'Elegance, \"Where billionaires help out the millionaires,\" said Jay Leno last sunday in Pebble Beach at the greatest classic car show in the world. And he wasn't kidding. Let me begin by informing you that at a Bonhams Auction that weekend, a Ferrari 250 GTO sold for $38 million. Daddy Warbucks, say what!? It was like the shot heard round the world - news of the bid spread like wildfire, which started the weekend off with excitement.\nI grew up spending lazy afternoons watching whatever my dad would watch - meaning, the masters, and car auctions. The auctions where mostly my favorite. It was then, that I gained a love and interest in classic cars, and anything that my dad is passionate about, sure enough, I am too. He and I are one in the same when it comes to passions. With that said, we'd always talk about the greatest classic car show in the world, 'The Pebble Beach Concours d'Elegance,' and we'd always look up the 'best of show' for each year and just drool over each and every one. We decided to add it to the bucket list - to attend Pebble Beach one year, which is so out of the way seeing as how it was across the country in California and we lived in Georgia. Well, years have past and what do you know, Zeke and I now live in California and so we bit the bullet and planned the beginning of this year to do it. And we did and it was a dream come true. I am just so thankful that my dad and I were able to go together and share that memory with each other. It wouldn't have been as meaningful or the same without him.\nIt was fun to see my dad in his pure element. It was like watching a kid in a toy store. Anything concerning cars is a dream for my dad. I remember when I was little he always talked about wanting a Porsche 911 in midnight blue with a tan leather interior. So for christmas, I bought him that exact car, but in a toy car model. I was so excited to give that to him. I was maybe 7 or 8 at the time and I remember thinking how mature of a christmas gift that was and I was so excited for him to open it. We shared the car - he'd have it on his desk in the study to remind him that some day he'd own the actual car himself, and I used it as the dream car for my dolls to drive around the doll house in (they fit perfectly in the car, it was awesome.) Years down the road, his dream came true and after all those years of saving he brought home his baby, an 87' midnight blue Porsche 911 Cabriolet with a tan leather interior. I moved on from my doll house and got to ride in the real deal. That car has so many memories linked to it. My favorite was having late night drives with my dad and driving through the Crystals drive thru to order some burgers and Coca Cola Slush. Another comical memory for the rest of us in the family minus my dad was when my mom dropped the ladder on the car inside the garage and she was too afraid to say anything so she made my sister break the news to my dad - we look back and laugh but it was a scary moment for my mom and my sisters and I. He still doesn't laugh about it today, and if he does it's a total fake laugh.\nA couple of years ago my dad was able to seal the deal and crown the car with a classic car license plate - the baby is official. I think it was a proud moment for my dad, which put a smile on my face. Though to some, cars may be materialistic, for me, they are art, fine art, they are the transportation for memories. Literally and metaphorically speaking, they take you places. They have souls just like you and I and my dad and I appreciate them for that matter. It's not a toy, it's not a show-and-tell, and we don't parade the car around for people to look at us. Strangely, that car is like a member of the family who will be the last one standing and passed down for generations and generations with so many stories to tell. And for me, that's pretty incredible.\nAnd so was last weekend. It may have been an event filled with wealth but the people there were filled with passion for cars, and cars that had a story of the past linked to it. There was one car who won 'the most elegant' award and the woman in the car was bawling as she was driving it up the path to except her award. See her emotion, you just knew that the car was extremely special to her, that it had belonged to her father perhaps, and it had become something worth working for and taking care of in the family, because to them it had some significance. I understood that, and it touched me.\nIt was the weekend for Ferrari's with one going home to a passionate collector for $38 million and for another who won a title they won't ever forget - the 'Best of Show.' Pictured above, it was a 1954 Ferrari 375 MM Scaglietti Coupe, owned by Jon Shirley. I think that may have been one of the greatest days in his life. The smile on his face was priceless when the winner was announced. The anticipation was perfect and the crowd cheered and applauded with such proudness and admiration.\nI've got my eye on a particular car that I plan on purchasing far far down the road. She catches my eye every time and my gasps never grow old. I think my dad could agree with me that she's a real beauty. Until then, I get to have more memories in my dad's dream car he worked so hard for. Can't wait to grab some burgers in that midnight blue with the top down in December with my dad.\nAnd as for Pebble Beach, we'll see you next year, as I think our bucket list just became a family tradition.\nWhat a fun weekend. You are all so cute!\nEXACTLY! Couldn't have said it better....you described everything perfectly. The Porsche ride will be waiting for you in December!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Friday, my lovely friends!\nSo recently one of my inspirations has been to improve my photography skills outside of food and I figured what better way to do that than to take some photos. That's when I went and grabbed my camera, recruited my little sister, gathered some props, and we went out for a fun day of shooting photos. I brought hot cocoa and sweets and we just snapped away like crazy and had a great time.\nI thought I'd share the pictures along with some of my favorite things about this time of year!\nI guess this one's an obvious one but I just adore all the shades that pop out in the trees as they slowly prepare for the coming of Winter. I grew up near Nashville Tennessee and I have to be quite honest and say there aren't many things I miss about the climate but I do miss the Fall season. That's most definitely where I fell in love with Autumn in the first place as a young girl. I think the fact that the Summers were so long, hot, and miserably humid, is what made the coming of Autumn feel like such a welcome reprieve from it. On the hottest days of Summer we'd sit around the pool dreaming of when we'd finally be able to see our breath in the crisp fall air.\n2. Nostalgia: How it Marks the Beginning of the Holiday Season!\nI don't know if you guys know this about me but I'm such a goner for nostalgia of any kind. You can melt my heart with a favorite childhood song and I pretty much turn into an overgrown preschooler when it comes to the Winter holidays. I love everything about the season. Every year I look forward to October finally arriving so that the countdown to the Holidays can begin.\nOh, I know pumpkin spice and everything nice isn't everyone's thing. (Haha, in fact I passed a woman in Costco yesterday who was having a tirade over the phone with someone about how everything in the @#$& store was pumpkin flavored and she \"hates punkin!\") Hey, that's okay if you don't like it, even the very sister of mine in these photos is not a fan of cinnamon or spice and I get it. However, for those of us who do love them they can make the season so much more memorable and cozy! We got these spiced candles that I simply wish I could eat every time we light them, MMMMmmmm!\nYou all know that I got back to the States from living in Chile for 4 years with only Summer clothes so I've had way too much fun choosing some new warm clothes for the coming cold weather. Warm weather clothes (sweaters, hoodies, boots, jackets, fluffy socks, thick scarves\u2026\u2026.) are probably my favorite things to wear and I have to hold back because I feel like I could never have enough of them.\nI don't know if it's the beginning of the school season or just a personal thing for me, but this time of year is probably the time when I most feel inspired to get organized and start making plans. It must be the excitement meeting with the change of season that is so motivating to just get ready and get things straightened out. The cold weather is also a good smack in the face to chill-out literally and figuratively. Knowing that the year is coming to an end and the Holidays are coming always helps me see the need to step back and get some perspective in all the rush of life. We could probably all use a few extra moments of slowing down to take in the important things and enjoy the simple pleasures life has to offer.\nWhat are you're favorite things about Fall? I can't wait to hear and I wish you all the best weekends!\nWe pretty much love everything about autumn haha but I especially love the food (soup! pumpkin! warm drinks!), and the colors in Colorado are pretty incredible. I also love that running in the afternoons is actually enjoyable since it's not so hot anymore!\nYou are you sister are the cutest!\nThat's so sweet of you, Susie! Thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After making it to the finals following two stages of auditions, Bermudian model Alize Bailey has won the Top Model Worldwide 2017 Public Vote Award, receiving the award at an event held in London this past weekend.\n\"I'm very grateful for all of the support that I received over the past few weeks,\" Ms. Bailey told Bernews following her win.\n\"This past weekend in London, I have had the time of my life and the competition has motivated me to continue to develop my modeling career at an international level.\n\"Since the competition, a few potential opportunities to continue modeling have arisen, of which I am hoping to pursue,\" she added.\nCONGRATULATIONS Alize! Well deserved! Bermuda is very proud of you. All the best in your future modeling career.\nCongratulations cousin job well done. Keep up the great work.\nVery happy for her, but previous Top Model winners haven't really seen their modelling careers flourish, have they?\nAnd I like how you picked out the name Yo Yo for yourself.\nGood job Alize!!!! We are so very Proud of you!!!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The HI84530 is an easy to use, fast and affordable, method specific titrator designed for testing titratable acidity levels in water. Based on an acid-base titration method, this mini titrator uses an optimized pre-programmed method of analysis with a powerful algorithm that determines the completion of the titration reaction potentiometrically with a glass body pH electrode.\nThis mini titrator incorporates a precision piston style dosing pump that adjusts the volume of dosing dynamically based on the voltage change. This type of dosing system reduces the amount of time required for the titration while providing for a highly accurate determination of the amount of titrant used.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhntk b/data_all_eng_slimpj/shuffled/split2/finalzzzbhntk
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index 0000000000000000000000000000000000000000..185b787abe72c9457c6df6ca98234c6c653a137d
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+{"text":"There is a number of really beautiful, high and accessible mounts in Zailisky Alatau ridge that can be climbed in any season, just in 1 day. Have started from Almaty by 6am, you will be home the same evening full of fantastic impressions of your Alpine style ascent. If you are firm with your decision to make another climb, but do not know where and how to start, then the three following steps will help you to realize a dream.\nYou need to set a date and time.\nClimbing of Amangeldy (4000m) or Pioneer (4031m) peaks is possible in any season besides early spring (March and April). Summer mountaineering is much more pleasant, of course, but such a motion can be also nice in cold time of a year as well \u2013 you just need to dress accordingly and be ready to meet a wind and snow.\nAt our website you can also find description of different routs to the peaks of Zailisky Alatau ridge, in \"Climbing\" section.\nAnd there are some pictures that can give you an idea of the forthcoming climb available at our \"Photo-gallery\".\nStep 2. Dress and get ready.\nA list of necessary clothes you can find at our website, in \u00abClimbing\u00bb section.\nIf you have any questions regarding the equipment, size or staffing, you may call our guide on duty any time.\nA right mood can guarantee up to 50% of success. It is extremely important to be ready to the trials of hard walking, cold and some job in a strong wind. But all that will be inessential in comparison with a feeling of triumph and delight on the blazing snow covered top of the peak.\nFurther to our experience, it is not really important how trained you are, as an absence of strength is usually compensated by the guide's proficiency and a right walking rate.\nIf you have a zero background, then we can recommend you to take 1-day climbing course that will allow you participate consciously in a process, to use all the gear, and will help guide to provide your safety as well.\nSo, the date is set, mood is good and you are dressed accordingly. You have chosen Amangeldy peak as your aim this time.\nUsually, 1-day climb starts by 5-6am, when you start driving from the city in a direction of Touyuk-Su canyon (Maloalmatinsky gorge, above Shymbulak ski resort). Pedestrian travelling starts here, at the end of asphalt road.\nFirst part of the trip follows a grounded trail up to Mynzhylki meteo-station at a height of 3000m above sea level. This part takes 1.5 hours.\nKeep walking up by rather steep path to the glade \"Alpingrad\" (3450m asl). Climb from Mynzhylki to Alpingrad takes 45-55 minutes. After a short break, continue mount a rocky talus (covered by snow in winter) of the southern slope of Amangeldy peak. A 2-hours climb by the slopes and then clefts ends at the foot of a rocky \"head\" of the top.\nHere a technical part of the route starts, and we need to put on a special equipment (helmets, body harnesses with carabineers, ascenders etc.). There are 120m of climbing by the rocks (not too hard), and all this part is handled by the guide with ropes, stations etc.\nSummit! From the top of Amangeldy peak all the city is seen, up to Kapchagai. And beautiful panoramas of the snow and ice covered mounts, glaciers and moraine lakes surround you.\nWay back from the top follows the climbing route. A rocky part will be done by fixed ropes and special rappelling devices (figure eight).\nAfter that, having all the gear packed in a rucksack, continue walk down by the same path. A way back from the top to the car takes 2-3 hours.\nDepending on the weather, member's experience and quantity of persons in the group, a whole climb can take from 8 to 12 hours.\n- all necessary special safety and climbing equipment: helmets, body harnesses with carabineers, runners, ascenders and descending devices. A guide will also have the ropes, special climbing tools and carabineers needed for handling of the route. In a guide's set we also include a first aid kit and communication means (radio, phones etc. to connect with rescue team).\nWe usually work with small groups of mountaineers, that means an individual service for you and couple of your friends (groups of 1-5 persons).\nBut, we have an experience of mass climbing organization as well \u2013 in this case we will pre-arrange a route to keep maximum safety for all group members.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For more than 30 years, The Arc of Somerset County has provided a variety of competitively- priced, labor- intensive fulfillment services to meet the needs of the business community in New Jersey.\nWhether your job is large or small, long term or \"just in time\", The Arc of Somerset County will help you to realize savings in labor, supervisory, and equipment costs.\nOur satisfied customers come from a variety of industries including manufacturing, packaging, marketing and fulfillment. No job is too big or too small.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had something clever and \"oh so smarmy\" to blog about, but there is nothing cute about having a stroke. And a week and a half ago, I had one.\nIt was a Thursday morning and I had just walked up a flight of stairs in the house when I noticed an odd sluggish feeling. Nothing I could put my finger on. Just a slowing of my actions and thoughts. And then later \u2013 the slurred speech, which my wife, the speech therapist caught when she returned from work.\nLike an idiot, and not wanting to push the panic button, I figured I would wait until the next day to see if things changed.\nThey didn't and I found my blood pressure way above 200. My doctor said, in essence \u2013 \"get to the emergency room dummy!\" which I did at around eleven o'clock that morning.\nMy long suffering wife drove me to Lakes Region General Hospital in Laconia where a CAT scan revealed a hemorrhage in the right basil ganglia. Blood on the brain. A stroke.\nWe waited until eight that night for a bed to be open at one of the finest hospitals in the state \u2013 Dartmouth-Hitchcock Memorial Hospital in Hanover NH.\nI rode in the back of an ambulance with the siren going, which I had always wished I could do. No. Not a good wish. Not a good thing. What was I thinking?\nThe culprit? High blood pressure, selfishness and a feeling of invincibility - a lethal combination for a person my age. I will turn seventy next month.\nI spent a long two and a half days away from my home and loved ones and the shame of it is that it could have been avoided. Salt, sugar and a reckless regard for my own health was the culprit and I will never look at food the same way twice.\nThe blood pressure finally came down after some fancy medication foot work and the astute diagnosis of a very good medical team. And, oh yes, prayer. Lots of it \u2013 some from folks I had not seen in a long while.\nSometimes you have a \"on the road to Damascus\" experience, in which you are blinded and flung from a horse with the words of God ringing in your ears, like the Apostle Paul. But this was one time it didn't, because the truth became a slow thing that slithered along my neuro pathways, revealing in stages which should have been obvious from the start. You are not in this alone. You are loved by many people.\nI have reached the age where metaphors tap me on the shoulder frequently and in this case with urgency. Especially those who love you, and they are the ones most affected by your inability to develop healthy habits. It is when I saw the grief I had caused others that it began to dawn on me I need to make some life changes.\nWe are pebbles, flung daily into the pond of life. And the ripples flow out to touch so many others. Our actions or lack thereof become the matrix in which we live or die and dying is not an option when you have a wife, a sister and brother, nine children, five grandchildren and many friends who love you.\nI could lay out dozens of statistics about strokes, like; it's the third leading killer in people over the age of fifty-five and all that stuff. But statistics are sterile and impersonal.\nLying on a hospital bed in a neurological ward for a few days is a small price to pay for the revelation of our mortality, and I am sure there are many out there who have had this same experience \u2013 probably a lot worse then mine, which is considered a minor stroke. But, there are no minor strokes. Who am I kidding? A stroke is a stroke and someone who has suffered one will likely be in line to have another before the \"shuffling off this mortal coil\" thing occurs.\nSo, although this blog is short, I hope it catches a few eyes. Do yourself a favor. Check what you are eating. Look at the daily percentage of salt, sugar, fat and calories you are consuming. Try doing that for just one day and then think about moving some of your priorities around. And, for Heavens sake, don't dismiss the power of prayer.\nI would never presume to compare myself with NFL Head Coach Chuck Pagano, who is suffering from leukemia. He has a long road ahead and it will be filled with pain, sickness and an awfulness we could never imagine, unless we have lived through the nightmare of cancer. But, in a well documented and moving speech he gave to his team after a win recently, he said he would \"waltz at his two daughters weddings\", and hoist the Lombardi Trophy with his team several times.\nI will never hoist any trophy, but I have three beautiful daughters \u2013 and I plan to dance at every one of their weddings.\nGeorge, I'm glad that your episode was \"minor\" in comparison of what is could have been. I have not had such an eye opening life evaluation type of experience but I am indeed trying to eat better. Next year will be a year of preservation of garden vegetables. Meat especially red is hardly consumed at home. When it is it usually is a flavoring ingredient in soup or casserole type dishes. I haven't integrated fruit yet not because I don't like it but just because I'm not sure how to make it a part of the meal. Maybe it just needs to be snack food. I grew more salad stuff in my small garden last year so that helped out. Americans do have a really bad diet. Fast food restaurants have not been good for our country's diet. I try to stay away from them as much as I can. I hope and pray that you will indeed dance at your daughter's weddings and that you have many more years of enjoyable life. Have a great day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you heard of the Google Chromecast Audio before?! I've had the Google Chromecast for a couple of years now so when I heard about the Google Chromecast Audio, I had to try it! We listen to a lot of music on our phones when we are cleaning, working on the house, or lounging around the pool. This is GREAT to get quality sound from speakers you already have with your home wifi. The sound comes in super clear since it is connected via wifi. The Google Chromecast Audio can be used from your phone, laptop, or tablet. I have been using the Google Chromecast Audio two different ways, one connected to a portable speaker and one connected to my home system.\nI love the ease it is to connect to my portable speaker. Its so simple to set up and can play music right away! I typically use Spotify to play music and it works so good! Another reason why I like this is because different people can play their music from their phones and easily.\nI think this would be a fun gift for a music lover or even a new home owner! Do you think you will be getting a Google Chromecast Audio?!\nShop Heeled Sandals, Black Cropped Pants, Ripped Skinny Jeans, Mini Satchels and more. Get this widget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finally got a couple of dolls finished, have been working on these poor lost souls for quite a while! Was so excited as I finished them both off in 1 night, still working on their names, but I just couldn't wait to share them with someone!!\nHayles made the girls a tea party to welcome them - too cute!\nPaula they are just wonderful,you have done a fantastic job,I imagine these are going nowhere lol,looks like they are settled in with their cups of tea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhwvw b/data_all_eng_slimpj/shuffled/split2/finalzzzbhwvw
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index 0000000000000000000000000000000000000000..d5475754fd56c7e2047a4668ce399d9588171b03
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Discussion in 'Stickies\/Frequently Asked Questions' started by Ninja_Stoker, Jul 31, 2014.\nMinimum height for entry into the Royal Marines is 151.5 cms or 4 Feet, 11.65 inches.\nFor Males with a BMI 28 -32 (> than 28 there is a 50% greater risk of NFA Med through lower limb injury during Recruit Training) if you fail this, the waist measurement at navel level has to be below 94 cm.\nDoes BMI play a part in that too?\nThanks, I'm just asking because I'm fairly tall but I'm underweight by over 10Kg so my BMI is fairly low.\nAny tips on gaining weight? It would really be appreciated.\nI don't think it matters how low your BMI is but you should aim to be 70kg at least!\nYou are not the only one though mate, I was 55kg 2 years ago and i have reached 73kg as of now, i have not grown in height (6'0\") and i know all the weight is muscle mass as my waist and abs are the same as 2 years back and i look alot bigger. However my metabolism is soo fast but you have to remember gaining weight for ecto's is a marathon not a sprint!\nI found eating every 2-3 hours and having like 5-7meals a day as well as using myfitnesspal to make sure i hit a 500 calorific surplus over my BMR is key.\nIf you really do have a fast metabolism do NOT fuss over what you eat, just eat anything and hit them macros.\nOh start lifting is you don'talready so the excess calories turns into muscle, get a partner and lift heavy.\nBrilliant that's a great help!! Thank you.\nI'm relieved to know there are more ectomorphs out there though who want this. And noshing all day is definitely going to need to happen if I want to gain anything, right now I'm the weight I should have been 4 or 5 years ago!!\nCurrently I'm around 53Kg and the same as you, just about 6'1\". And as much as I hate it and just want it NOW, I get that its going to take a long time and a lot of work.\nCurrently I'm around 53Kg and the same as you, just about 6'1\"\nTip- hold onto something heavy if it's windy in the meantime.\nHahaha Instead of lifting the weight itll be keeping me on the spot!!\nNo problem mate that's what the forum is for! There are actually more ectomorphs than you think who persue this career, naturally we are usually good runners and are built for it. Go hard with the eating, training & sleep and you'll be there in no time.\nAnd yeah I get what you mean about being built for running, its a lot easier for me to run than do anything else, having lead a sedentary lifestyle up until very recently (which I'm kicking myself for) I got into he running fairly fast.\nAfter running one lap of the field (not even) and feeling like hell itself was trying to get out my throat of course!!\nHaha we are all different though even if we have the same body types, as you said you had a very sedentary lifestyle. Do any sports run in your family, never know you could genetically be good at some!\n@Ninja_Stoker do you have any idea how harshly they enforce being in the healthy BMI range? Because I'm 5 foot 8 inches and currently weigh in at 89kg, but thats lean muscle mass and my body fat percentage is actually 13% maybe lower. So BMI says I'm overweight when clearly to look at me I'm athletic and fit as a fiddle.\nIf you are muscular you can go up to a BMI of 28. I managed to get down to 27.999999999999999999999999999999999999999 to pass the medical !\nThe BMI scale, for all its faults is the standard applied for entry I'm afraid.\nAnyone starting Recruit Training with a BMI at 28 or more is highly unlikely to successfully complete it at 28. It's a statistically proven fact.\nSo below BMI 28 is recommended?\nCheers lads, I'm literally right on the top end of the bloody threshold!\nBut I'll be upping the regularity of my cardio and focusing more on my stamina and endurance a bit before I reach this stage so I should see a reduction in my current overall weight. Hopefully I'll make it by a whisker!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Week 4. Rex and the Bills. Last week without TB12 (and Nink). A win gets us to 4-0 without Brady and gives us a very commanding lead in the division. Even a loss leaves us alone in first, but a win would be much better.\nWeather is cloudy, in the high 50s, with a twenty percent chance of rain.\nNotable inactives for us are Jimmy and Cooper. Geez, will Cooper ever play? Zona is killing the Chandler Jones trade.\nNotable inactives for them are Reggie Bush, Greg Salas, Justin Hunter. That doesn't include the now on IR Sammy Watkins.\nSo, Jacoby Brissett gets start #2. Go Pats!\nOn a Boston side note, as he finishes up his last regular season home series, I wanna shout out to David Ortiz.\nSunday starts off well from my perspective. The Jags win over the Colts!\nIan Eagle and Dan Fouts have the call.\nPats get the ball first. We start at the 9. He came out, paused, and then ran. Bad play.\nWe three and out but there is more to say on the drive. On the first play, Jacoby dumps a pass off to Jules. Jules goes 90 yards, and is tackled at the one. However, it comes back on OPI on Hogan. On the replay, it was clear that Hogan did, in fact, make contact with a DB. But, it came after the catch. Jules had already gone 5 yards with the ball before contact, so it was a legal block. The call was incorrect and took away a 90 yard play.\nThe Bills start at the 35. The Bills have a decent drive. 12 plays. Most of them were well contained. But, Taylor had throws of 15 (to Clay) and 13 (to Shady.). They had a 16 yard gain with a Gilleslee direct wildcat snap. (A second wildcat play for Shady was a 2 yard gain). They end the drive with a 7 yard TD catch by Shady. On that play, holding on Woods (against Hightower) was missed by Moreilli's crew.\nWhile the Bills deserved that TD, at this point, the key to the game has been the officials.\nWe start at the 14. Jones with a bad return. He was decisive this time. But, he needs to learn to kneel when he catches it halfway into the endzone.\nPete's crew is doing terrible. 3 good calls, 3 bad (or missed calls).\nBills start at the 32. They end the quarter working on a nice drive. They had three more good plays. A pass to Shady (for 11) and to Woods (for 10). Taylor got a 14 yard run on a designed draw that was a nice play.\nHowever, on 1st and 10 at the 18, Taylor went to Clay and missed. After the play, Incognito was flagged, giving the Bills a second and 25. A 3rd and 21 will be how they start the second.\nThe Bills handily won the first quarter.\nThe Bills gain 13 yards on a pass, then take the 34 yard FG. 10-0 Bills. 14:49 left.\nPats start at the 25. Jones takes a knee this time and gets an ovation from the crowd. Four and out. We got an 11 yard catch by Bennett, and then three plays and a punt.\nBills start at the 5 after Gilmore is called for a block out of bounds on the return. Once more they move the ball. Three more big plays Taylor passed to Woods (for 14) and O'Leary (for 17). Gillislee also had an 11 yard run.\nThey settle for a 42 yard FG. The ball was snapped after the play clock hit zero, but, naturally, no flag.\nThe offense has excuses to not be great. The defense doesn't. Step it up.\nBlount again gets the first, and it comes back on a (correct) holding call. On 3rd and 11, Jacoby runs for it, fumbles and the Bills recover.\nAgain, Morelli and his fail crew have impacted the game. As many as 10 Bills points off of missed calls. As many as 14 Pats points stolen by a bad call and an offical error.\nBills start at the 9. 10 plays and a punt. They do convert a 2nd and 20 with a 22 yard pass to Woods. We burn two timeouts on the drive.\nWe start at the 20. Three and out. Hughes with a 14 yard sack, and then two runs. Bills use thier last timeouts. We'll punt with 8 seconds left and the punt uses up the rest of the clock.\nThe Bills have won the first half. But, it's only 13-0, and we have made plays. Not every many, but we have made some. I do not want to take away from the Bills, as they have played well. However, Morelli and his crew are the key factor in this game. They have had too many errors for a preseason game, let alone week 4.\nThe Bills start at the 25. Three and out. We needed that. Go offense!\nWe start at the 24. Nothing is going right. We have a nice 19 yard catch by Bennett, but we end up missing a 48 yard FG. Two more missed calls - Jules was held down the sidelines by Gilmore on a long pass and Brissett was hit helmet to helmet. Bolden also dropped a pass at the five.\nThe Bills start at the 38. Bills run 7 more plays. They have two big plays. Both are passes. One to Woods (for 23) and the other to Goodwin (for 12). But, they miss a 47 yard FG. It's a small blessing, but we really need points. Unless we score, they don't need to.\nWe start at the 37. 5 plays and a punt. Two big plays. A Jules catch for 16, and a Jacoby 10 yard run were nice. A correct Chop Block call on Solder killed the drive.\nThe Bills start at the 30. Brandon Tate with a nice return. 10 play drive that ends with a44 yard FG. 16-0. :28 left.\nThe Bills drive was the least impressive one of the day for them that was not a three and out. The key was was (wait for it) a flag. Long was called for roughing the passer. What did he do? Right after Taylor threw the ball, he tapped Taylor's chest, a light push that didn't even knock Taylor down. That was the difference between punt and FG.\n13 points. That's what Morelli has given the Bills.\nI'm adding the last 28 seconds of the third to this part.\nWe start at the 25. We end the quarter with a 5 yard scramble that will give us a second and 5.\nThe Bills won the third.\nWe start the 4th poorly. We get a spark when Gronk makes a 10 yard catch. Then, we have back to back calls on Jules kill the drive. False start and illegal shift. Morelli's having to reach for bad calls. 5 plays and a punt after the 4th starts.\nThe Bills start at the 15. 10 plays and a punt. Taylor had two more amazing plays during the drive.\nJordan Mills, a Bills lineman, hurt on the drive. Rex had to use a challenge when the officials messed up a spot. The refs did get it wrong (I do want to be fair), and they fixed it on review.\nCorey White, of the Bills, was hurt on the punt.\nWe start at the 24. Danny muffs the punt but falls on it. We go with a long drive that has some nice plays. Blount had a 16 yard run, and Jacoby had some nice passes. Bennett had one for 16 and Bolden had one for 12. Bolden was hurt and we ended up turning it over on downs. I would have taken the FG, just to deny the shutout.\nThe Bills start at the 21. 5 runs and a punt.\nWe start at the 26 after Jones muffs and recovers his own punt. We throw twice to White for 35 total, then run once. We should have spiked the ball a couple times. Tried for points.\nI'm sure the Bills will act like this is the Super Bowl. But, let's look at things honestly.\n5. Officiating that terrible is too kind a word for.\nWith all of that, they only scored 16.\nWelcome back Nink! Welcome Back Tom!\nDuring a third quatrter TV time out, Morelli was talking to another ref and has his mike on!\nHere is what the entire crowd heard: \"Turned into a pretty good game. But one big hit, this s--- could explode.\"\nMaybe that could explain as much as the bad calls why he was being boo'ed so much.\nWill the NFL fine him for doing a bad job? No, I don't think so. But they might for the minor profanity.\nI saw bits and pieces. It felt nice being on the good end of some questionable non-calls, although I would argue that Alexander hit Brisett in the head with his forearm, not his helmet.\nAlso, I have since been some footage of a couple Bills being *******s during markups. Embarrassing.\nSomething interesting has happened in the last couple weeks in Buffalo. Ryan has let go of playcalling, so Buffalo has been more aggressive.\nRoman got fired. Earned that. Lynn has been doing a nice job. Not gro7breaking and way too much wildcat for my liking, but he has done the solid, competent j9b I expected but rarely got from Roman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Minister Patali Champika Ranawaka, the President's nominee for the constitutional council, claims that nominations are yet to be made for the three impartial civil members of the council.\nLegal experts say that as a result of the Constitutional Council not being established as yet, commissions under its purview have become inactive. The constitutional council, as enacted by the 19th Amendment to the Constitution, comprises of ten members.\nThe Prime Minister, Speaker and Opposition Leader are automatic appointments to the constitutional council. The President too must nominate an MP to represent him on the constitutional council.\nThe President must also appoint five nominees of the Prime Minister and the Opposition Leader, two of whom must be MPs while the other three must be civil representatives. The remaining position on the Constitutional Council would be taken up by an MP, elected by the majority of parliament, who should not be from the parties or independent groups of the Prime Minister or the Opposition Leader.\nThe Constitutional Council is chaired by the Speaker.\nNevertheless, certain legal experts point out that the failure to establish the constitutional council as yet has resulted in commissions under its purview being rendered inactive.\nIt is opined that there is still uncertainty as to whether the Constitutional Council will be established before or after the general election.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After my blog sale I still have a few items that have never been opened and are still in perfect sealed condition so instead of them cluttering up my room thought would be nice to give them away in a 250 Follower giveaway!\nIf you're under 18, make sure you have permission from your parent\/guardian as if you are the winner, you will need to give me your address for postage.\nSo let's get into the prizes!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scotland is set to become the first country in the United Kingdom to outlaw the physical punishment of children following support from the Scottish Government.\nUnder the proposals, put forward by Scottish Green MSP, John Finnie and supported by the Scottish Government, 'justifiable assault' would no longer be lawful under Scottish law. The move comes after a consultation from Mr.Finnie received over 75% support for the legal reform.\n\"This is an outdated, unwarranted and archaic practice that has no place in the 21st century.\nI am happy to support my parliamentary colleague, John Finnie MSP, and his bill to reform and outlaw physical punishment of children.\n\"The consultation for the bill clearly shows the wide ranging support for the practice to be outlawed and I am delighted that the Scottish Government has indicated that they will support the bill, ensuring it becomes law.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjbpm b/data_all_eng_slimpj/shuffled/split2/finalzzzbjbpm
new file mode 100644
index 0000000000000000000000000000000000000000..4a7cc26c6e4eaaa33c7fae38a03a9fe5a4c188f4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjbpm
@@ -0,0 +1,5 @@
+{"text":"Note that the UpdloadedFile class in the Axeda Platform v1 SDK only has a Start Date field, but no field for the completion date. However, there is a column that contains the End date in View All of uploaded files for an asset. How do I get this information into a custom object groovy script?\nUnfortunately there is no way to get the completion date-time information through the existing API.\nCan AgentUploadedFile from the V2 API be used with FindAgentUploadedFileResult in a groovy script to retrieve this information using a custom object?\nUnfortunately, accessing AgentUploadedFile's from the fileInfoBridge was not completed in time for the 6.9 release. UploadedFile is still the only way to programmatically access files uploaded by the Agent.\nOne way to get the time would be to set up a File-based expression rule that records the contents of the \"compressedFile\" input. When the expression rule is fired, the files have already been uploaded to the Platform (upload is complete).\nThanks for the help, Chris. You provide excellent support, as always.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very often students ask: who can do my essay online? There are lots of free essays and other types of academic papers readily available in databases, but don't buy them in a hurry. If you get free essay writing help, you can fail the assignment or be accused of plagiarism. To make sure that you don't go through this, we advise you to buy an essay online or write it. If you decide to get custom essay writing service online, make sure you get it from a reputable company that will give you an essay that is original, of good quality, is unique and is delivered without delays. WritingSharks.net is the company that will do all this and much more!\nWhy Choose WritingSharks.net to do your Essay?\nYou only need to say, \"I want to pay someone to do my essay for me\", and WritingSharks.net will respond to your needs every time you want to use our essay writing service. We have a team of talented and dedicated writers from all over the world to ensure that you get the best quality essays. We guarantee you the real worth of your money. Therefore, you should not be afraid to buy essays from us. Moreover, our friendly customer care service is available to you round the clock, any day and any time, to quickly respond to your needs. Your information is treated with confidentiality and our online modes of payment are secure. Every time you need custom essay writing service for university, college, or high school work, just contact us and you will get the best written essay from the most reputable online company, WritingSharks.net.\nYour essay will be written by a professional in your area of study. All our writers are talented in preparing original papers. When you say, \"do my essay according to these specifications\", the writer will research for ideas in scholarly resources obtained from libraries and compile your paper. In addition, all our writers are native English speakers with Masters and PhD degrees in their areas of study.\nYour essay will be delivered on time. One of the main worries that students have when getting online essay writing help is delivery within set deadlines. With WritingSharks.net, you don't have to worry about lateness. You will get a quality essay on time. You can even get a 10-page essay written in 24 hours.\nYour essay will be written using credible resources. Our essay writers have access to various databases and libraries, both online and offline, giving them an edge over our rivals' writers. Our custom essay writing service relies on writers who understand the essence of using credible sources in academic writing. They keep off unreliable sources such as Wikipedia and will only use plausible sources such as peer reviewed journals. They will use material from any resource you mention, be it specific journals, magazines, text books or newspapers. Just specify the resource, and if our writers have access to it, as is usually the case, we will be glad to incorporate it into your essay.\nYour essay will be written in any writing format you specify including APA, MLA, Turabian\/Chicago, or Harvard. Our writers are specially trained to write essays in these formats and any other format you specify.\nYou will, subject to our Money Back Guarantee Policy, be refunded up to 100% of your money if the essay you receive does not meet your expectations.\nAnytime you want to use our essay writing service, you are welcome, and we assure that you will come back for more. Make an order now and buy an essay at our website. Do not hesitate. Just tell us: do my essay now, and you will not regret!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's your car worth? Toyota dealerships in Oglala, South Dakota are happy to help you learn more on your car's trade-in value and available cash-back incentives.\nGetting your new car should be love at first sight. Schedule a Toyota test drive close to you to learn the feel of your ideal Toyota coupe, sedan or SUV in real life.\nWe know just how stressful buying a new car can be. Trust your local Toyota dealerships to make make your car buying experience the easiest it can be.\nLooking for the best Toyota for your lifestyle? The professional and friendly staff at your nearby Toyota dealerships are here to help you search for the perfect Toyota near to you to suit your lifestyle.\nYou know that with the Toyota name near you, you can depend on on your neighborhood Toyota dealers close to you for outstanding personalized service.\nWe work hard to keep your Toyota working to the highest standard. When it's time for your next scheduled service, trust the experienced professionals at your local Toyota dealerships to provide the highest quality Toyota car service.\nProvide your car with the quality it deserves. From routine service appointments to more work intensive fixes, trust our qualified professionals at our nearby Toyota dealership to provide the highest level of quality certified Toyota car maintenance in your area.\nToyota dealerships close by are ready to help you look for trusted Toyota parts to get your car back to its factory standards.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TAKE UP SERIOUS ARCHERY WITH BEGINNER'S LESSONS!\nThis is Yarra Bowmen Archery Club's Beginner's lessons. Once you complete this course, you will be eligible to join any registered archery club with Archery Victoria\/Archery Australia. Individual archery clubs may have further membership requirements.\nYou will be trained on the basics of archery, safety and will be on your way to mastering the sport.\nNo prior archery experience is necessary. Cost is $100 per person for three weekends of two-hour sessions, for a total of six hours of archery instruction; fee covers equipment, insurance and instruction. There is a minimum participation age of 10 years old.\nPlease wear closed-toe shoes, close fitting tops to prevent the bow string from catching your clothes, appropriate hats and sun-screen if necessary. Be sure to bring your own refreshments.\nThere is a small non-refundable booking fee of 30 cents per ticket which goes to the booking operator.\nAs we are a volunteer-run community club, we cannot offer a refund if you can't make the lesson or course.\nMinors must be accompanied by a parent or adult guardian; there is a minimum age of 10 years old.\nIf you have a pre-existing medical condition which may impair your participation, you must let us know as soon as possible to enable an assessment by the instructor.\nA Temporary Visitor Form will be provided to you at the start time to be completed; this is an insurance requirement.\nFor more information about the club, visit www.yarrabowmen.com or email coaching@yarrabowmen.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For our main dish, my son whipped up Food.com's Baked Breaded Cod, using half the mayonnaise called for, and it was outstanding. Homemade breadcrumbs from frozen, left-over onion bread. Mmmmmm.\nAnd now, for the next contest: Either you subscribe to recipe2recipe or if you've already subscribed, you get a friend to sign-up. You will then be eligible to win your very own set of Jittery Cook grocery bags. Of course your recipe suggestions are always welcome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjhym b/data_all_eng_slimpj/shuffled/split2/finalzzzbjhym
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index 0000000000000000000000000000000000000000..cd04866f4e40f19abfff78a37a5de6045ef8058d
--- /dev/null
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+{"text":"00:10 rdococ could it be possible in a clientside mod?\n09:52 rdococ yay! the fixer!\n10:25 rdococ hm. how to determine if the mod is being run by server\/client?\n10:35 tenplus1 how so ?\n10:36 Shara Would you like if a server could force any code it wanted to your client?\n10:36 Shara rdococ: what is a flaw is found in that at some point?\n10:37 Shara Everyone thought WE was \"safe\" for such a long time after all... holes do get found.\n11:45 ShadowBot Megaf: Error: That URL appears to have no HTML title within the first 4KB.\n12:03 Megaf and why the heck when I click \"Contact us\" it goes to ask in the forums!?\n12:05 Megaf \"We care about your happiness!\n12:05 Megaf They don't call us Happiness Engineers for nothing. If you need help, we're here for you!\"\n17:07 VanessaE in Unix systems it has probably always acted as \"paste\".\n17:08 Megaf in USB you OS have to keep asking, \"Hey USB devices, whatcha doing?\"\n17:08 VanessaE you sure about that?\n17:08 Megaf and then the USB mouse has to answer \"Im moving to this side this much\"\n17:10 VanessaE \"A USB mouse generally emulates a PS2 mouse, except that it generates IRQs from the USB bus,\"\n17:11 Megaf but this emulation perhaps still depends on polling frequency?\n17:14 VanessaE can't be \"instantaneous\"\n17:17 VanessaE I wasn't sure about the serial equivalent, been ages since I looked at it.\n17:36 VanessaE sometimes feels that way.\n17:39 Shara Are you sure it's not the player base in general?\n17:40 Shara RC had lots of... \"noisy\" players last night.\n18:34 tenplus1 kewl, adding anything fancy ?\n18:43 tenplus1 cant you tweak a setting to increase that ?\n19:09 tenplus1 what happened benrob ?\n19:22 lisac Calinou, is that video real time rendered?\n19:22 lisac and if so, on what hardware?\n19:23 tenplus1 windows or linux ?\n19:25 Calinou you should be happy that you can even play some 3D games with them, this used to not be the case at all!\n19:32 tenplus1 wait, it cannot use animated entities ??\n20:19 paramat nerz is good with client-network-server stuff so ..\n20:22 Shara I feel like the number of players in general has gone up, at least on my servers.\n20:23 Shara But the bad ones get noticed more.\n20:24 Shara Had all caps spam and people swearing at each other to deal with last night.\n20:27 benrob0329 WHAT DO YOU MEAN I SHOULD STOP YELLING?\n20:27 benrob0329 YELLING IS LOVE, YELLING IS LIFE!\n20:41 paramat Shara you got this too i believe? https:\/\/github.com\/minetest\/minetest\/issues\/6197 anything to add?\n20:43 Shara Yes, could never figure it out exactly. On some tests at the same height it didn't happen for me.\n20:48 paramat did your bug happen for fancy and simple beds?\n20:59 paramat you're testing in singleplayer 0.4.16 stable?\n21:03 Shara If you have old move code, you can just tap shift to free yourself.\n21:09 lisac 6 days ago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For faster action shorten the leader between the dodger or flasher and the hoochie.\nFor slower action lengthen the leader between the dodger or flasher and the hoochie.\nUsing heavier line for the leader increases the action to the hoochie.\nTo give an erratic motion to the hoochie remove the hoochie body from the lure head and thread the leader thru the side of the hoochie body near the top of the body.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How sleeping sickness is transmitted?\nThis is a tropical disease transmitted by the tsetse flies of the genus Glossina. This fly acquires the infection by sucking blood from an infected person. Then the parasite travels into the gut of the fly. Here, it gets multiplied and transmitted into the salivary glands of the fly. When the fly takes another blood meal, parasites are transmitted into a new host.\nUsually after a bite, an inflammatory scar is developed. It is called trypanosomal chancre. It appears usually one or two weeks after the bite. Then the parasite starts to disseminate throughout the body. This results in an illness characterized by fever and malaise.\nThe fever is characterized by intermittent bouts of high fever lasting for several days. In between episodes the patient is usually not having fever. Other symptoms of this disease can be classified as stages 1 and 2.\nWhat is the stage 1 disease?\nStage 1 disease is characterized by widespread lymph node enlargement and an enlarged spleen. In addition, stage 1 disease can cause anemia, increased white cell count, and reduced platelet counts.\nLymph node enlargement is characterized by non-painful, rubbery and discrete lymph nodes. Usually lymph nodes in the neck are involved.\nIn addition, patients with stage 1 sleeping sickness may develop itchiness and rashes. Rarely, patients with stage 1 sleeping sickness can get headache, joint pains, weight loss, edema, liver enlargement, and increased heart rate.\nSome people in this stage can get inflammation of the heart or myocarditis. This complication can cause heart failure.\nWhat are the symptoms of stage 2 disease?\nStage 2 illness is characterized by the involvement of central nervous system. Here, the patient gets day time sleepiness. In addition, patients with stage 2 sleeping sickness develop restlessness and insomnia in the night.\nIn addition, various movement problems may develop in this stage. They are characterized by involuntary dancing-like movements, tremors, and difficulty in walking. In this stage, patients' symptoms may resemble Parkinson's disease.\nWhat are the important differences between two species?\nUsually untreated T. b. rhodesiense infection can cause death in weeks to months. In addition, patients usually get ill quickly. Travelers will get the fever and other symptoms of stage 1 disease before the end of the trip. However, disease caused by the other parasite (T. b. gambiense) can go into months and years.\nHow sleeping sickness is treated?\nThe medications used for treatment of sleeping sickness are suramin, pentamidine, eflornithine, and the organic arsenical melarsoprol. These drugs are quite toxic and administration should be done by a specialist. The most common side effect of these drugs is renal failure.\nIf not treated, sleeping sickness or African trypanosomiasis can be lethal. Main symptoms are fever, lymph node enlargement, spleen enlargement and increased heart rate. However, it is usually treated with quite toxic medications. Usually the disease can be prevented by avoiding areas of infection, by wearing protective clothing, and by using insect repellents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Johnston, Mary. Prisoners of Hope: A Tale of Colonial Virginia.Boston: Houghton Mifflin, 1898. Rpt. as The Old Dominion. London: Constable, 1899. Rpt. Boston: Houghton Mifflin, 1926. An online version of Prisoners of Hope is available from Google Books.\nJohnston, Mary. To Have and To Hold.Boston: Houghton Mifflin, 1900. Rpt. as By Order of the Company. London: Constable, 1900. Rpt. Boston: Houghton Mifflin, 1931. Rpt. as To Have and To Hold: A Tale of Providence and Perseverance in Colonial Jamestown. San Antonio, Texas: The Vision Forum, 2006. An online version of To Have and To Hold is available from Documenting the American South.\nJohnston, Mary. Lewis Rand.Boston, New York: Houghton Mifflin, The Riverside Press, 1908. Rpt. Boston, New York: Houghton Mifflin, 1929. An online version of Lewis Rand is available from Google Books.\nJohnston, Mary. The Long Roll.Illus. N. C. Wyeth. Boston, Cambridge: Houghton Mifflin, Riverside Press, 1911. Rpt. Boston: Houghton Mifflin, 1939. Rpt. Baltimore: Johns Hopkins University Press, 1996. An online version of The Long Roll is available from Google Books.\nJohnston, Mary. Cease Firing.Illus. N. C. Wyeth. Boston: Houghton Mifflin, 1912. Rpt. New York: Popular Library, 1940. Rpt. Baltimore: Johns Hopkins University Press, 1996. An online version of Cease Firing is available from Google Books.\nJohnston, Mary. Hagar.Boston: Houghton Mifflin, 1913. Rpt. Charlottesville: University Press of Virginia, 1994. An online version of Hagar is available from Google Books.\nJohnston, Mary. Pioneers of the Old South: A Chronicle of English Colonial Beginnings.New Haven, Conn.: Yale University Press, 1918. Rpt. New York: AMS Press, 1970. An online version of Pioneers of the Old South is available from Google Books.\nJohnston, Mary. 1492.Boston: Little, Brown, and Co., 1922. Rpt. as Admiral of the Ocean-Sea. London: Butterworth, 1923. Rpt. as 1492: Admiral of the Ocean-Sea. Chester, N.Y.: Hambledon Hill Press, 2005. An online version of 1492 is available from Google Books.\nJohnston, Mary. The Collected Short Stories of Mary Johnston.Ed. Annie Woodbridge and Hensley C. Woodbridge. Troy, N.Y.: Whitston Pub., 1982.\nMary Johnston was born and grew up in Buchanan, Va. She was frequently ill as a child and spent a lot of time reading the books in her father's library. Most of her education came from tutors and her own reading. When Johnston was sixteen, her family moved to Birmingham, Ala. Her mother died in 1889, and Johnston took over the responsibility of caring for her younger siblings. The family moved to New York in 1892, where Johnston began writing short stories. Financial difficulties took the family back to Birmingham four years later, and Johnston worked on a novel she had begun in New York. Prisoners of Hope was published in 1898 and was modestly successful. Her second novel, To Have and To Hold (published in 1900), was immensely popular, and Johnston's earnings enabled her family to return to Virginia.\nJohnston's writing was now the main financial support for her family. She published twenty more novels between 1900 and 1934. In 1909, she became involved in the woman's suffrage movement and was a founding member of the Equal Suffrage League of Virginia. Despite her fragile health, Johnston traveled around the region to speak on behalf of the League. She also became interested in Progressive causes such as prison reform and anti-lynching campaigns and in non-orthodox religious movements such as Theosophy. In 1911, Johnston bought property near Germantown, Va. (now Warm Springs). She and her sisters began building a house they named Three Hills. Although declining popularity of her books caused some financial strain, Johnston lived at Three Hills for the rest of her life. She died there of cancer in 1936 and was buried in Hollywood Cemetery in Richmond, Va.\nMary Johnston wrote historical novels, several of them set in colonial Virginia. Her Civil War novels were praised by critics. Johnston's later works were influenced by her interest in Progressive causes and non-orthodox religion.\nCella, C. Ron. Mary Johnston. Boston: Twayne Publishers, 1981.\nLongest, George C. Three Virginia Writers: Mary Johnston, Thomas Nelson Page, and Amelie Rives Troubetzkoy: A Reference Guide. Boston: G. K. Hall, 1978.\nPhoto courtesy of the Library of Congress, Prints and Photographs Division, LC-DIG-ggbain-03356.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The state of the art of big data processing create numerous duplications of the original or source of truth data in the ascending form of meta data and transformations on that original data and or itself. This process in creating multiple levels of intermediates with dependencies one or more source of truth repositories, with perhaps some meta data being common to multiple applications at the lower tiers and more specific to singular application meta data upper tiers.\nThe meta data and derivatives are used in order to break the linear cost of the source of truth repositories, yet they themselves have at least linear cost to construct. In addition to the multiplicity, that source of truth must generally be retained leading to an additional management cost in keeping updates and amendments not only in the source but through all the derivative stack.\nThe Eureka data store is the idea that the original data need not be retained but rather recovered in part or whole from a conservative statistics set. Proliferation of meta data is generally unnecessary with the Eureka data store as it allows for general direct access. Finally, comprehension learning and advanced analysis would be advantaged with directly available precomputed statistics, random access of data and integrated hypothesis testing. In short algorithms would be able to dive deeper more quickly into large data than traditional digests that the status quo provide. Hence the Eureka data store would be inherently quicker, more resource efficient, flexible and generally applicable than the myriad of other technical choices currently employed.\nThe gains of the Eureka Data Store is as much due to the changes of perception of addressing the problem as specific techniques employed in Eureka. The next sections will address illusory perceptions in the standard big data architectures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkfqh b/data_all_eng_slimpj/shuffled/split2/finalzzzbkfqh
new file mode 100644
index 0000000000000000000000000000000000000000..adfcbf3658622144df8cb03e1a61530fcf1cc44a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkfqh
@@ -0,0 +1,5 @@
+{"text":"Boxes & Totes \u2013 Welome to the wonderful world of subscriptions!\nWelome to the wonderful world of subscriptions!\nFollow this link to get a free tote with a 14-day trial. You will get five clothing items and three accessories.\nI thought the pattern of the top was nice, buy I just didn't like the tie neck, especially since it wasn't symmetrical. However, it did fit well and had a nice light feel to it. This was the only item that I didn't swap out since I figured I should try at least one thing they picked out for me.\nI really loved these pink jeans but unfortunately they were too small. It seems like I should have gone one size up from what Le Tote recommended with this one.\nThis was a cute bohemian style top. I ended up wearing it twice paired with the maxi skirt below. It did seem like the top came a little wrinkled and worn.\nThe skirt was soft and comfortable. I think it was my second favorite item in my tote. This is one that I may have considered purchasing, but I would have preferred to purchase one that hadn't been worn yet.\nThis ended up being my favorite item of the lot. It's funny because I thought I didn't like it at first, but I ended up wearing it with a pair of black ankle pants to work one day. I liked it so much that I wore it again the next week! The only problem was that it seems like other people loved the shirt too because it was a bit worn. As you can see below, I found a hole in it. If it weren't for the condition of the shirt, I might have purchased it.\nThis is the second clothing rental subscription I've tried so far (the other was Gwynnie Bee). I liked the selection a little better. I felt like it was easier to find items that might fit my style. It was also nice that they had recommended items to save a little time. Unfortunately, I didn't like the items they picked out for me, but it was my first tote so that's understandable. Plus I was able to swap out as many of the items as I wanted for ones that I picked out myself, so it still worked out. Most of the items in the tote fit me well except for the pair of pants. One issue I noticed is that over half of the items seemed a bit worn and I was disappointed that one of the shirts actually had a hole in it. Given the condition of the items I received, I was not willing to purchase any of them. One other issue I noticed was that the return bag seemed a little too small to hold all of the items. I really felt like I had to cram them in there.\nThe price was better than Gwynnie Bee as you received more items in a box for a lower price. However, you are limited to one tote per month. Whereas with Gwynnie Bee, you can get a new box as soon as you send any items back. Overall, I enjoyed having some new clothing items to mix into my closet for a couple weeks and I think it is worth a try. I'm guessing that their recommendations will get better with time, and hopefully they will do a better job of checking item quality before sending out items in the future.\nPlated is a meal delivery kit subscription service. The insulated box comes with recipe cards and pre-measured ingredients for each selected recipe. Ingredients are thoughtfully sourced with pork and poultry raised without antibiotics, beef with no added hormones, sustainably caught fish, fresh & seasonal produce, and organic ingredients when possible.\nNote: I have seen various promos for Plated, so definitely search around before ordering so you can get a discount!\nThis was definitely a great flavorful dish. The already small potatoes only need to be cut in half and the dried apricots and shallot were easy to slice as well. Although this one took a little longer to cook, the recipe was pretty easy. Normally, my husband and I are not crazy about spinach, but the way in which it was cooked and mixed with the potatoes, made this side our favorite part of the meal! The only thing I didn't like as much about this dish is that it seemed like it probably had a high sugar content since it used dried fruit and we also added some brown sugar.\nI love ordering Pad Thai at restaurants, but this is the first time I've tried making it at home. Thanks to the recipe and ingredients provided by Plated, it turned out great! Usually when I try to cook shrimp, it ends up being sort of rubbery, but the recipe card gave great instructions on how to cook it perfectly. The flavor of this dish was awesome, and besides a little bit of prep work, it was very easy to prepare.\nThis was such a great fall dish! I've never had gnocchi with these kinds of flavors before, but I loved it! It was nice that the butternut squash came pre-chopped. The recipe did recommend chopping it up further, but having it already cubed made this step much easier. I also don't usually like nuts but the walnuts added a great crunch to this pasta, and went well with the other flavors.\nI gave each of these dishes five out of five stars! The recipes had a nice variety, great taste, and weren't too difficult to make. The potion sizes seem better than some other meal kits I've tried. I never felt hungry after finishing my meal and sometimes even had enough for leftovers! The recipes were pretty simple to follow. I actually kind of enjoyed cooking these recipes, which isn't usually the case. Also, I didn't try any, but I love that they also offer desserts. Based on their other recipes, I'm sure those are great too! I would definitely recommend this service if you don't mind a little prep work and cooking, want to add some variety to your meals, and care about how your food is sourced.\nShipping is free if you spend at least $25.\nMy first impression wasn't great as the box the products came in was a bit damaged (most likely the fault of the carrier, rather than the company), and the way the products were packaged was not very appealing as they kind of just look like they were all thrown in there.\nThe shampoo and conditioner both smell like fruit-loops. Personally, I wasn't crazy about the smell, but some people may like it. The shampoo had a nice lather and my hair was easy to comb through after using the conditioner. The shampoo seemed like it was a bit too drying for my hair\/scalp as I had some tiny flakes in my hair and some itching. I will say that this is a common issue for me as I have sensitive skin.\nThe body wash had the same fruit-loops smell as the shampoo and conditioner. It lathers really well, but I didn't feel like there was anything too special about it.\nThe lotion was my least favorite product that I ordered. It did not have the same scent as the others. It seemed like an unscented lotion, but it still had a smell that I can't quite describe. The product seemed fine as a light lotion, but I just couldn't get over the smell. I'm glad I ordered the travel size rather than the full size.\nThe hand soap was fine. It had the fruit-loops smell again. I do like the minimalist design of the bottle, so once I use the soap up I may refill the bottle with a different soap.\nThe glass cleaner was fine, but I don't think it worked quite as well as Windex.\nI like the concept for this company: healthy ingredients, minimal design, and no middle-man to markup the costs. That being said, I wasn't a fan of the smell of the ingredients and the products didn't work well enough for me to want to purchase them again. They work fine and I am able to use them despite my sensitive skin, but I do have to alternate with a different shampoo and conditioner to prevent my hair and scalp from getting too dry. I don't see myself paying for a membership and ordering from this company again. That being said, it may still be worth a try since you get a free 30-day trail. If you do try it out, I would recommend buying travel sizes in their products to make sure you like them first.\nWhat is Dollar Shave Club?\nDollar Shave Club is a shaving subscription box that delivers razor blades on a monthly basis and also offers various grooming products.\nFor my first box, I bought The Daily Essential Starter Set for $5.\nI also received a bathroom reader (self-explanatory) and the Member Handbook, explaining how the subscription works.\nThe razor I received is called The Executive. Each cartridge has six stainless steel blades. This razor has a nice weighted handle, which was comfortable to use and I didn't have any issues with it slipping out of my hand.\nI could have done without these wipes, but they were included in the kit so might as well use them right? I used one for the intended purpose, but I think I may end up using the rest for messy foods like chicken wings! Either way, I don't think I'll be ordering these again.\nI love this Easy Shave Butter! It has a nice light pleasant scent. It does go on clear so if you have light hair this may not be good for you, but with my dark hair it was not a problem at all. Between the razor and this shave butter, it seemed like I got a close shave and my skin felt smooth and not too dry. I did still apply lotion afterwards though. Out of the shaving gels, creams, etc. that I've tried, I think this one is my favorite so far!\nI was worried the Amber & Lavender Body Cleanser might smell too manly, but it had a nice pleasant gender neutral scent that does seem to be \"calming\" as the bottle claims. Plus I love the look of the bottle, and it has a good lather too!\nIt's nice that the starter packs comes with four cartridges. The blades seem to give a close shave without too much irritation. They also seemed to last a good amount of time without getting dull. I changed mine after about six uses I believe, but probably could have used it even longer. I did cut my ankle pretty badly on the third use, but I blame this on myself for rushing and pushing down too hard.\nOverall, I was pretty pleased with this starter set. My favorites were probably the cleanser and shave butter. They both had great scents and I felt like I was pampering myself while using them. I'm definitely thinking about ordering full sizes of them with my next shipment! The blades seem to be good quality, especially for the price. I've used Gillette Venus Embrace for several years and I think this is a great cheaper replacement!\nFollow this link to get The Daily Essentials Starter Set for $5.\nGwynnie Bee is essentially a monthly subscription which allows you to rent clothing. You may keep the items sent to you as long as you like and when you're ready to send them back, they will prepare your next box. This allows you to receive more than one box in a month! You also have the option of purchasing items if you cannot part with them.\nFollow this link to get 30 days free + $10 off your first paid month.\nI received a second shipment with two clothing items this month since I returned my other two items. This time, the two pieces came in one box.\nI did not like the feel of the fabric on this one and I couldn't see myself wearing this. My hope was that it would be a nice cardigan to wear to work, but it seemed a little too casual for me. This was a piece that I immediately put in the return bag after I tried it on.\nThis top looked super cute online and I thought it would be something I could wear to work, but it turned out to be much different than I thought. Again, it seemed too casual to wear to work. The fabric was thin and I had a problem with gaping between buttons. I did notice that after I reviewed and submitted my return for this item, it now shows the suggested size as large (my regular size is a medium). So it does seem like the system learns from your feedback. Unfortunately, this piece went straight in the return bag right after I tried it on.\nThis box was definitely a miss for me. I know these were items that I had \"closeted\" myself, but they just didn't turn out to be quite what I expected. One downside to this service is that you have to select all of the items yourself and honestly I don't think this is the best option for me. I enjoy the services that come with a personal stylist who will pick out items specifically for me. I think they are better at figuring out what would look good on my body type and they know if items tend to run small or large. Luckily, even though these two garments didn't work out, I can return these items and get another box before the end of the month. Maybe I'll have better luck next time! I think I will try \"prioritizing\" some of the items in my closet this time so that maybe I will get pieces that I'm more excited about in my next shipment.\nIf you would like to save $20 off your first three boxes (total of $60 saved), click here.\nThe chicken was a little bland, but the mashed potatoes were delicious! I noticed that some of the other meals that had raw meat had instructions to season the meat, but this one didn't. I think the chicken would have been better if we had seasoned it. I appreciated that I could just stick this in the oven for 25 minutes and it would be done!\nThe sauce tasted too much like vinegar for my taste. The pork chop was good. I'm glad it said to season it because I think it would have been too bland without. We loved the mini potatoes and broccoli!\nThis meal was our favorite of the lot. I think it helps that the steak gets seasoned and pan fried. We enjoyed the peppercorn sauce with this one more than the sauces that came with the other meals. The potatoes were delicious! We also got some carrots and green beans that were tasty as well.\nThis meal was our least favorite. I'm guessing it would have been better if we had made it in the oven, but we took this one for lunch so our only option was the microwave. I did not like that the sauce contained high fructose corn syrup. The flavor of this one was okay, but not something I would want to eat again.\nOkay so I know I said the steak was our favorite, but that was before we tried the teriyaki chicken. OH MY GOODNESS! SO DELICIOUS!! It was an easy meal since the chicken was already cut up. All I really had to do was cook everything in a pan and BAM! the meal was done. I could totally see myself ordering this every week, it was that good.\nI seriously love how quick and easy these meals are! They definitely prevent me from picking up fast food because I'm too tired to cook when I get home from work. I felt a lot better about myself eating these meals than picking up fast food. All I had to do was stick the container in the oven or microwave, or put the contents in a pan. I think the longest cook time was 25 minutes, but that was in the oven with absolutely no work required of me. The quickest meal was about 3 minutes in the microwave.\nSince the meals come in individually packaged, this is a great option for the times when you only need enough food for one person and don't want to eat out or eat the same food over and over again. It's also makes for a great packed lunch. With that being said, I would have to try some other meals that can be made fully in the microwave to see if I like them any better than the sweet and sour chicken I had.\nAlthough they offer a large selection of meals in a week, I did notice that there were several repeats from week-to-week. This can be good if you have a favorite that you want to order again, but it's nice to be able to try different meals so you don't get tired if eating the same things every week.\nI think the price of the meals is pretty fair, especially the more meals you buy. It can certainly be cheaper than eating out (depending on where you go) and probably healthier. It's also similarly priced to some other meal delivery services that require much more work with washing, chopping, measuring, etc. For anyone who really doesn't like cooking and\/or doesn't have much time\/energy for cooking, I would recommend this service.\nIn order to receive my first box, I needed to \"closet\" at least six items. This meant looking at all of the available garments, selecting my size, and hitting the \"Add to Closet\" button. There was an option to prioritize certain garments, but I wanted to leave it to chance! It surprised me that my two items came in two different boxes, each with their own pre-labelled return pouch. My first box shipped three days after I completed my closet and the second box shipped the following day.\nThis top fit nicely and was lightweight. I appreciated that the fabric was not see-through and did not show any lines underneath. I wore this top to work with a white cardigan and gray slacks. Great for the office, flattering, and comfortable \u2013 that's a win! I wore this once and then returned it because I wanted to try out some other styles Gwynnie Bee had to offer.\nOut of the two options, this one was my favorite. I really thought that I wasn't going to like the bell sleeves on this, but I thought they balanced out my figure nicely. The color was lovely and the fabric felt nice. I chose to wear it with a pair of black skinny jeans (casual Friday) and I even received a compliment on this one! I did wear this top once and then return it because, again, I wanted to get some new garments to try out and mix things up!\nI love that this service prevents the buyers remorse that comes from purchasing a shirt, dress, pants, etc. and then never wearing it or only wearing it once. I also love that you aren't limited to just one box per month. So if you don't like what came in your box, you can ship it right back and get a new one! It is nice that each item was shipped separately so that I didn't have to wait for both items to be available before they shipped. It is also nice if you wanted to keep one item longer than the other, so you could send one back but still have another return pouch for the item you're holding on to longer. On the other hand, it does seem a bit wasteful to me.\nI was pleasantly surprised with the fit of the items I received and happy that I liked them enough to wear them to work. I could definitely see this subscription being a great option for anyone that loves wearing different clothes every week and\/or wants to experiment with new styles, but doesn't want to fill their closet with items they might only wear once or twice. I do think the price is a little bit steep for what you get, as I know that I could buy two new garments for the same price as the subscription. Gwynnie Bee does make up for this a bit by offering a rewards program where you get points for adding items to your closet, returning items, etc. After accumulating a certain number of points, you can redeem them for a $5, $10, or $15 reward which can go towards an at-home purchase or an item upgrade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With the elimination of Patel Kirankumar in 10th place, we have reached the official final table. The final 9 will return tomorrow at 4pm to play down to a winner. Below are the seating assignments for tomorrow. Chip counts will be posted shortly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"that man Wrote: Nice backgrounds and stage.\nThe only real problem I have is with the camera, as the lead room is fairly inconsistent on both sides (most notably on the left) and I could see someone getting sick from the running cycle. Also, when I let her stand still, she eventually starts to vibrate up and down really fast.\nIt's all still better then what I could have done, and I hope you keep up the pace.\nGood call, now is the bounciness of the camera or the speed at which she runs the problem. The Speed I can fix, the bounce will be a little tricky. I do aim to make the camera feature good as the game will have many cinematics so i've got miles to go on that one, if anyone else has suggestions for this one, please let me know.\nNot really the speed per say, more just the bouncyness of it. As for a solution, and this is only what I know how to use from messing around in gamemaker so its usefulness here is questionable, have an invisible hit box that the camera follows that is at a certain point in the player's model. It won't change sizes or bob when running, which might be what causes the bounciness.\nThat's actually very smart, but it'll be a problem when I want to be able to zoom or pan to other scenes on the map. I think I found a fix for that, though. thanks .\nManaged to fix everything so far. Stabilized the cam and resolved the bouncing character issue.\nThe camera issues, save for the lead room, were fixed in the in forum version.\nHowever, when I tried it in full screen mode, the issues returned in full force again. I mean, you did fix it in the normal sized version so you're on the right track, but this also means people can't play it in full screen.\nthat man Wrote: The camera issues, save for the lead room, were fixed in the in forum version.\nGood find, It worked fine on the flash player. Well it should be resolved, now along with the zoom. I am gonna begin work firstly on the slime, then the slime emitter. Keep me posted on engine errors.\nupdated. Please read the spoiler before tinkering with it. Very Important.\nThanks, brah, i'm glad you like it.\n1.walk all the way to the right.\n4. continue to hold and you will find yourself on top of the \"stage\"\nIt works for both walls.\nnice find! I've been unable to replicate it in the first post, however. The version you're referring to is a little older. I'll be keeping the most recent version on the front page for quick access. Try looking for glitches on that one.\nThis next thing is very picky and might not help. I was able to acheive a \"double jump\", on the current version.\n2.Jump by holding up, just to show that normally the character (avatar?) jumps to a consistent height.\n3.Jump by spamming the up key as though you where trying to cause a double jump.\nIts hard to explain exactly but, i used a pattern of pressing up quickly (not simply spamming), until a slightly higher jump occurred. This particular bug is in my opinion is VERY minor as you might very well not encounter problems resulting from it.\nIf I can replicate it, I'll be sure to try to resolve it. Thanks, man!!! keep it up, As I'll be working up a new build pretty soon with lots of new features.\nWalking while jumping into a wall has unusual results.\nIt looks like it's going pretty well, and I love the character model!\nmy only comment would be on the \"jog\" animation, where arms could be a little more relaxed (just a really minor gripe, really). the other animations Look pretty good!\nNo suggestions for now, just keep up the good work also, platformers FTW!\nI am interested to know if you where able to replicate the bug or not?\nHuh, I like the new test room.\nI am not able to replicate the running up the wall glitch, but I was able to replicate the double jump. In all honesty, if you want my suggestions about all of this, you could make the double jump easier to do, and permanent. Aside from that, I love the shift function for running now.\nHow many enemies can you have on screen on one time? I mean, it's only slimes for now, but it could be best to stress test to a hundred or so to represent a complex boss.\nGreat question, that all depends on the processor you own, which seems to be doing most of the work. A lot of factors will influence that as well (complexity of characters, animation, backgrounds, interactions, GUI, etc...), which is why I can do a lot, but I'll end up needing to get stingy when it comes to actually building the game. The map maker is really easy to use and if you score a copy of flash cs5 to 6, you'll be able to make your own levels. And you're right, those slimes are doing very little other than x and y translations, I still have to animate them and more complex enemies will arise in the future, so I still think she's got plenty under the hood, but i need to tailor it for a wide audience.\nDouble jump will definitely be a feature and I might make multijumps 3+ unlockable after some time, Definitely toying with the notion. One thing I need to perfect is the single jump, though. It's real bland, physically and I need to implement the anims for it, too. But it's very expandable and the code is lightweight, too.\n@guest: I've been trying it, still unable to get it. But it'll be alright, the main problem is the way she and other objects interact with the wall, I still need to tweak conditionals and physics and I'm thinking of giving the ability of either climbing or applying vertical friction for sliding down. That's a little too complex to contemplate at the moment though, lol.\n@SuperSatan: Thx, man, Corta's sittin back on this one for some time and letting me drive for a while, once we acquire an artist, she'll start putting some overtime in. I've got everything I need so far, so let her rest, lol.\nI like that door, and the different types of slimes.\n3:while holding, move left right or jump using up. The avatar will appear frozen.\nbut if you press control while holding space it will cause her to \"leap\" and also to \"duck\".\nLast edited by guestlike on Thu Jan 29, 2015 1:46 am, edited 1 time in total.\nThat's because it seems like the ctrl button is crouching, and the animations for it aren't yet complete. I managed to replicate it, so it does exist, but it shouldn't be too hard to fix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most Important Health benefits of nachni.\nDecember 8, 2017\tComments Off on Most Important Health benefits of nachni.\nNight-Time Beauty Mistakes You Need to Stop Making Always!\nDecember 7, 2017\tComments Off on Night-Time Beauty Mistakes You Need to Stop Making Always!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business in Control is a bespoke management consultancy with Dr Sue Saunders as principal consultant. A client relationship begins with an initial Business Review to help you identify how to \"run a better business\" making use of intelligent information systems and the Business in Control (BIC) financial planning model.\nEvery business is at a different stage in the development of its information system depending on maturity of the business, type of business, skill of the management team, growth profile and other factors. Making use of the BIC Model is not always the first step to achieving control of your business. Sue will outline a plan of action that will allow you to improve planning and control of your business that may or may not involve using the BIC Model. For example, if your point of sale system does not reliably interface with your accounting system giving you an accurate measure of inventory, then that problem has to be addressed as a first priority.\nInitially you can concentrate on identifying those aspects of the business that will improve performance and in consultation with Sue who will use the BIC Model to develop your business's unique financial model. As time progresses, you will be able to see the benefits that the software brings in showing you trends in past performance. These data will help you predict and guide your business into the future. Each month, in consultation with Sue, you will see how the BIC Model measures your actual performance against plan using a process designed to help you run a better business bit by bit.\nSue's experience has shown that implementing a truely effective management control system in your business takes time. Attention must be paid to the integrity of data flows in the company. No amount of smart software will overcome poor systems or unreliable data sources. Most businesses need help in this area as well as going through the set-up procedure for implementing the BIC Model. Sue works closely with you and your business to help you make a real difference.\nIf you are not sure whether the BIC Model and associated management control system is for you, go now to the survey page on the BIC website and let Sue know some of the characteristics of your business. She will contact you with advice concerning a possible programme that is most suited for your business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Home Lifestyle\tWe have a Heartbeat!\nWell we had an interesting appointment this morning. Good news and perplexing news I guess you could say. The good news is we have a heartbeat! We didn't get to hear it, but we saw it flashing away on the screen. However, for some reason,the baby is measuring about a week smaller than it should be. I say \"than it should be\" because as you all know this was a planned pregnancy and I am about as sure as one can be about the dates. So we will go back in two weeks to check things and make sure baby is growing properly. I think maybe this is a sign that I'll be rewarded for my hard work with Oscar and get a nice reasonably sized baby this time around. Or maybe there is a twin hiding in there somewhere and taking up half the room!\nAt any rate, at least we have a nickname for this little one.\nHey Tiny! I hope you are enjoying the swim around my giant uterus! See you soon!\nDo you shop Consignment Sales?\nThat settles it. It's a girl! Every petite little girl needs an intimidating big bro to watch her back!\nHooray for Tiny! So exciting. I'll never forget seeing Lily's heartbeat for the first time. Will keep you guys in my prayers that the little one grows nice and big. Hugs!\nMy recent post Toddlerista- Stripes and dots and socks- oh my!\nSo neat to see the heartbeat! At this stage would they be concerned about a week's difference? They are so itty bitty at this point maybe a millimeter can make a big difference in dating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I watched my daughter flourish over the years at Desert Garden Montessori. As a student from grade 2 through grade 9, and under the tutelage of some extraordinary teachers, she has grown into an intelligent, articulate, outgoing and thoughtful young woman. She is now a sophomore on scholarship at ASU, straight A student, and deeply engaged in the community through philanthropic activities. One of the best decisions of my life was to trust Desert Garden\u202fwith my daughter's education.\nThank you for the tremendous example, patience, and kindness that you shared with us throughout our time at DGM! You truly impacted us as young parents; your wisdom, experience, and confidence in our children's ability and independence is something we aspire to instill in our new little one. You were\/are truly an incredible teacher not only for our children, but for Nathan and me too. From the bottom of our hearts, thank you!!!\nOn Tuesday, we had the crazy rainfall. When we picked up our son, he shared he didn't complete any of his daily goals. I was a mixture of frustrated and happy with his honesty. He said the rain had his mind so far from the classroom and tasks, he was asked if he just wanted to go outside and play.\nwas told the teachers were working diligently to keep the room on task, focused, deal with rain, etc. and they were feeling quite stressed by it. Then, Ms. Shetal walked into the room and said that they needed to just let go of it. Just let go of it. Let the children deal with the water (they kept asking to do it anyway), build boats and race them in the \"river.\" Dance in the rain!\nWe all need you, Ms. Shetal, to walk into our lives each day, sincerely. Just be \u2013 let go of our schedules and requirements, relax, and be in those moments. Thank you for those words. Thank you for directing the school as you do. I actively make an effort to tell our teachers how amazing they are, and how the lessons they teach and the approach they use, encourages success and improves all of our lives. Today I wanted that to be you \u2013 your words will stay with me and I'm grateful for this.\nTonight, during the drive home, I plan to tell my son how his sense of adventure and exploration was a much better way to spend those moments. My natural desire for academic excellence completely neglected the lessons we learn when we just pause, breathe, and be in the moment. Lesson learned \u2013 a sincere thank you from all of us in the Reed home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Asmara, 29 November 2018- Eritrean nationals residing in the Republic of South Sudan, South Africa and Southern African countries, Switzerland and Italy conducted various activities reflecting their attachment with their homeland.\nAsmara, 27 November 2018- Eritrean nationals residing in the United Kingdom organized discussion forum on 24 November on the objective situation in the homeland, the progress of the peace and friendship agreement in the region as well as their citizenship responsibility.\nAsmara, 22 November 2018- In the 2nd stage of the African Cycling Cup 2018 Time Trial Eritrean cyclists in both men and female categories won gold medals. From the men category Sirak Tesfom, Aron Debretsion and Daniel Habremicael members of the Eritrean national team won the Gold, Silver and Bronze medals respectively. From the females category Adiam Tesfa'alem member of the Eritrean national team won Gold medal while Mihret Gebreyohannes member of the Ethiopian national team and Liezel Jordaan member of the South African national team settled for Silver and Bronze medals respectively.\nThe African Cycling Cup 2018 which is the first of its kind will be conducted until Sunday, 25 November.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fancy making some quick cash, possibly \u00a31,000s? I know I sound like worst type of spam advert on the internet, promising riches.\nYet they make you pay to find out what their big idea is, and usually it's baloney, balderdash \u2026 or add your own B word.\nHowever, if you're prepared to play the system, there are ways to make money \u2013 and some can pay at speed.\nRecently in MoneySaving Towers, I heard two of my team, Sarah and Megan, comparing who made more from being bank tarts.\nDon't worry, I'm not being rude, that's what you call anyone who repeatedly switches to get free cash (Sarah's \u00a3825 topped Megan's \u00a3800).\nAnd it's that tale that inspired me to collate a host of different ways to speedily stuff cash in your coffers. Better still, start now and many of these techniques could yield a stash in time for Christmas.\nThe fierce competition in the banking market means some bribe you to switch. As often these are best-buys anyway, take it and smile.\nIn many cases, you get the cash within a month or two.\nThere's full details and eligibility info at www.moneysavingexpert.com\/bankaccounts, but in brief\u2026 - Free \u00a3150 PLUS \u00a34+\/mth. www.co-operativebank.co.uk pays \u00a3150 within 45 days. Plus its Everyday Rewards scheme pays \u00a34+\/mth if you qualify. Its customer service rating in my recent poll of 260 customers was a strong 70% 'great'.\nwww.FirstDirect.co.uk has won every customer service poll I've ever done \u2013 91% rate it 'great'. On top of a free \u00a3100 within 44 days, switchers get a \u00a3250 0% overdraft and you can put up to \u00a3300\/mth in its linked 6% regular saver.\nBank.MarksandSpencer.com doesn't pay cash, but does give a \u00a3100 gift card within 30 days, plus pay in \u00a31,000+ a month and you get \u00a310\/mth added for a year. Its service is a strong 68% 'great'. It also has a \u00a3100 0% overdraft and a linked 6% regular saver.\nwww.Halifax.co.uk Reward pays \u00a3100 after just 3 days and also pays \u00a35 each month you're in credit - its service is also 68% 'great'. Yet beware its overdraft which is very costly.\nIf you're worried about how difficult switching is, I did a Twitter poll of 2,900 in July and 76% who've switched since 7-day switching launched in 2013 said it's no or little hassle. All that happens when you use the switching service is they close your old account, move direct debits and standing orders for you and auto-forward payments.\nIt's not just my team who are bank tarts. Having published how much some were making, people have got competitive with hoards writing to tell me they were Britain's biggest bank tart.\nSimon Bartram and his wife have now earnt \u00a31,565 over four years: \"Inspired by Martin, we started switching... keeping track of the cash takes some doing but it's well worth it\".\nSign up to free fitness rewards app Bounts.it, and connect it to your fitness tracker (including Fitbit, Jawbone, Garmin and iWatch) or smartphone app (like RunKeeper, MapMyfitness and Moves), and you'll earn points when you walk, exercise or go to the gym.\nThe more exercise you do, the more points you earn, which can then be converted into vouchers, to be used at retailers and restaurants such as Amazon, John Lewis and M&S.\nEven if all you do is walk and hit 7,000 steps in a day you can get a \u00a35 voucher within a few months, but do more exercise, and it's far quicker. And if you've already got a fitness tracker, it's a no brainer to connect the two.\nAccepted new applicants for the www.AmericanExpress.com\/uk Gold Charge Card get 20,000 reward points if they spend \u00a32,000 in the first three months. This isn't a call to spend more, just do normal spending on it as long as it's \u00a3700+\/mth. The voucher usually lands within three months.\nThe 20,000 bonus is enough for a \u00a3100 voucher for Amazon, M&S and more (or you can transfer to airline and hotel loyalty-schemes). You earn one point per \u00a31 spend on the card too.\nThe \u00a3140 annual fee is waived in year one, so diarise to cancel if you want to avoid it.\nSo you MUST repay IN FULL every month, or face a \u00a312 penalty and a mark on your credit file. Best do it via direct debit.\nDisgustingly, when you left an energy firm, if you were in credit it ran a 'don't ask, don't get' policy. That's mostly changed now, but if you didn't get your money - even if years ago - you can still ask.\nJust call your old supplier to see if you're owed money (quote your old account number if you can dig it out).\nMany get \u00a3100s, some more, like one of my site users Col.Escargot: \"The balance owing from 17mths ago was \u00a31,364. Brilliant, but I do feel an idiot for not knowing how much I'd overpaid. I'd urge everyone to check.\"\nVouchers last two years, but many forget them or lose them. Yet that isn't the end of the game \u2013 you can reclaim old vouchers.\nIf you're a regular Tesco shopper especially you may be surprised at how much you're due, as McTaggus, said: \"You netted me \u00a3276 in unused vouchers.\"\nDon't just spend them in store though, check if you can quadruple rewards. Redeem vouchers via Tesco Boost partners and you can get 3x or 4x points' value, eg, \u00a310 voucher becomes \u00a340 at Caf\u00e9 Rouge, \u00a330 at Goldsmiths jewellers and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Victorian Age had beautiful dresses and gowns for Victorian girls. Dress up this Victorian princess in a beautiful dress to receive her historical guests. Victorian fashion was flamboyant and enticing.\nPlay victorian age princess at www.dora-games.net!victorian age princess is a very fun game!\nIts ok. But i dont like the long dresses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkzrs b/data_all_eng_slimpj/shuffled/split2/finalzzzbkzrs
new file mode 100644
index 0000000000000000000000000000000000000000..1f96d2911cf81168989a214d8da4a19325f048c6
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Minimum of 5-7 Years experience in the same field is required.\nJob Summary:Plan, organize and monitor the implementation of all Human Resources functions within the Center as stated in HR Policies and Procedures including but not limited to: manpower planning, hiring procedures, orientation\/training programs, performance appraisal, payroll, employees' benefits and employee relations.\nDuties and Responsibilities:-Responsible for preparing the Center Manpower Plan\/Budget.\n-Enforce and monitor the proper implementation of HR policies related to recruitment, selection, inductions and performance appraisal.\n-In coordination with the Center Department Heads, assemble and analyze training needs.\n-Ensure that the Center Payroll is well managed in a timely manner ; and proper communication is maintained at all times with Finance departmentwith regard to the Center employees' compensation and benefits.\n-Conduct Employee Satisfaction Surveys and exit interviews as per HR Policies; and provide analysis of the results.\n-Ensure that all Center employees' Licenses, Iqamas and Work Permits are issued and renewed in a timely manner.\n-Interpret HR policies and procedures to Center employees . Advise Department Heads regarding specific personnel problems. Recommendpolicies and procedure necessary to achieve the Center objectives.\n-Ensure proper compilation and maintenance of Employees' records (Personnel Files) as stated in HR Policies.\n-Prepare and maintain HR KPI's reports and statistics for administrative and regulatory purposes including but not limited to : Saudization percentage, turnover rate, retention rate, sick leave\/absenteeism rate.\n-Participate in Center committees and meetings as assigned. Establish and maintain communication with all Center departments.\n-Act as liaison between employees and Center management. Investigate causes of disputes and grievances and recommend corrective action.\n-Ensures compliance with the Egyptian Labor Law and all Government requirements relating to employees' legal status in Egypt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A 'curriculum' is a list of content to be taught and learnt \u2013 a course of study for schools, colleges and universities. From September 2014, the government produced a new National Curriculum. Private schools, academies and free schools are exempt: they do not need to follow the requirements.\nDownload the entire National Curriculum for more detail.\nOur curriculum has been written by the staff. We have used the national curriculum objectives to make it unique for our school. We have a two year rolling program which ensures continuity and experience for our children across the Key Stages. Please find below details of our curriculum topics from Years 1-6.\nFor details of our Foundation Stage Curriculum, please click here.\nThe children learn to read following the Letters and Sounds programme.\nLetters and Sounds is a phonics resource published by the Department for Education and Skills in 2007. It aims to build children's speaking and listening skills in their own right as well as to prepare children for learning to read by developing their phonic knowledge and skills. It sets out a detailed programme for teaching phonic skills for children starting by the age of three, with the aim of them becoming fluent readers by age seven.\nPhonic sessions are taught 5 days a week for 20 minutes each day. Children are taught in small groups based on their ability for phonics.\nFor more information on how you can support your child in learning to read, click on the helpful links below. If you have any questions, please come and see Mrs Archibald.\nIf you would like to find out more information about our curriculum, please come and see Mrs Archibald.\nAt St Augustine's we believe that children need to use a range of skills to become a competent reader. As a school we use Oxford Reading Tree and Phonic Bugs (Pearson) books as our main reading schemes, which include phonically decodable books, as well as 'real' books, which are banded, using the book bands scheme.\nThe provision for children's reading is meticulously organised, from when children start in Foundation Class through to becoming literate 11-year-olds. Teachers read good quality texts to children as part of literacy lessons and there is time every morning dedicated to reading.\nAll year groups have access to the School Library and children are encouraged to borrow books to take home. Each class also has a reading area with a variety of non-fiction and fiction books.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Investigate geographical distances related to the NAT gateways.\nLoad the node positions retrieved by the help of cassandra API and double check that all nodes are covered.\nWe need to further filter our dataset to keep only those logs that reached the target. Visit the former table to see how many data are kept versus all.\nLook up the coordinates of the NAT gateways kept.\nA similar pattern is found as in the hopcount case. Namely, either a unimodal or a bimodal distance distribution describes well the geographical distance between the source node and the NAT gateway, whereas the distance between the NAT gateway and the target vary in a wide range.\nOperator Telia-S: small diameter correspond to short distance, greater diameter to somewhat longer distance, note however, the factor is just around two.\nOperator YOIGO and Vodafone-IT: both have single radii, but the geographical distances are showing two relatively long distance manifestations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BEIJING, Aug 31 (Reuters) - China's securities regulator on Friday published draft rules for the long-awaited cross-border stock connect scheme between Shanghai and London, and encouraged firms listed in the two cities to apply for floatation on each other's exchanges.\nThe publishing of the draft rules for public consultation means the launch of connect program is imminent, and preparatory work for the scheme is in full gear, the Shanghai Stock Exchange said in a statement on its website.\nChina is stepping up deregulation of its capital markets, with no signs that the pace of opening is being slowed by the escalating Sino-U.S. trade war.\nUnlike the stock connect scheme linking Shanghai and Hong Kong, which allows investors to buy overseas stocks directly, the Shanghai-London connect only allows investors buying foreign stocks indirectly, in the form of depository receipts (DRs).\nChinese companies can raise fresh money through issuing Global Depository Receipts in London. However, London-listed firms can initially only issue China Depository Receipts backed by existing shares, meaning they cannot raise funds through Shanghai listings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Crabbers won the Virginia State Championship. They were dominant in their wins beating their opponents by a combined score of 819-93. This is widely considered to be the best team in Virginia history.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmtoz b/data_all_eng_slimpj/shuffled/split2/finalzzzbmtoz
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index 0000000000000000000000000000000000000000..b903e5445d82d5217ede6e30d28d1ab5bfba84c3
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@@ -0,0 +1,5 @@
+{"text":"Closing in on health reform's two year anniversary, some things are worth remembering. Many observers saw that the financing plan to sustain Chapter 58 over the medium and long haul was deficient. We saw it, and commented that the time would come when we would have to revisit the law's financing structure.\nWe predicted the most likely time for the reckoning \u2013 spring 2008. For two reasons: first, the second contract period for Commonwealth Care would significantly raise the cost of covering CommCare enrollees; and second, renewal of the federal 1115 Medicaid waiver carries significant risks for the state's financial exposure. On these two, the first shoe has dropped, and the second not yet.\nWe did not see the robust enrollment in CommCare, the major factor that has pushed costs beyond most expectations.\nRight now, the cost of CommCare is being borne exclusively by taxpayers (state and federal) and by enrollee premiums and copays. Two years ago, the legislative crafters of Chapter 58 estimated CommCare costs at $750 million in FY09; in January, the Patrick Administration's budget estimated FY09 costs at $869M. Now, according to Alice Dembner's Globe article today, that may be as much as $100M short.\nThe cost of CommCare cannot be sustained by taxpayers and enrollees. It won't work.\nAre we in a financing crisis? A crisis is when your back's to the wall and there are no acceptable alternatives. That's not the case here. It's reasonable and fair to ask insurers to bear some of these costs, especially when \u2013 by their own recent calculations \u2013 health reform has delivered them 111,000 new customers over the past year. It's also reasonable and fair to ask some providers to sustain this important advance.\nBusiness? Two propositions. First, it's not fair to ask employers who are covering their workers to pay more. They are already paying their fair share. And it's fair to ask employers whose workers are not covered, and especially employers whose workers are relying on CommCare for coverage, to pay part of the cost, in addition to taxpayers and enrollees.\nThis is not a crisis. Not yet.\nThis entry was posted in MA Health Reform. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was searching for a Property and found this listing (14064257). Please send me more information regarding 6332 Caroline Drive, Frisco, TX, 75034-4838. Thank you!\nI'd like to request a showing of 6332 Caroline Drive, Frisco, TX, 75034-4838 (14064257). Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following sources offer useful information about groundwater and well drilling issues. To learn more please contact us one of our expert members.\nAmerican Ground Water Trust: Discover the environmental and economic value of ground water and learn about water science and technology solutions.\nEnvironmental Protection Agency: Keep abreast of national ground water policy.\nInternational Ground Source Heat Pump Association: Follow the latest news about heat pump technology.\nMaine Groundwater Association: Learn about the Association, get the latest news.\nMaine State Drinking water program: Private well information for homeowners.\nMaine Well Drillers Commission: Follow the meetings of the Maine Well Driller's Commision.\nNational Groundwater Association: Learn about ground water's role in economic vitality.\nContent copyright . HANSENSWELLDRILLING.COM. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your only way to improve things is to destroy what was there before, then you're not trying to improve anything but yourself.\nSocial engineering - big words to sound like you were the victim of something ominous when actually you were just tricked by another human being.\nThat feeling of knowing we see the same night sky, the same stars and constellations, no matter how far apart we are, will one day be replaced by the feeling of realizing we might not even see the same moon.\nPictures is to genes as sound is to memories.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a few american moments, but most of the datings range from being uncomfortable to almost painful to watch. Adding to this are some of the men's used descriptions of the women they look at hook up make out and used in person.\nBut there are some humorous moments, especially when the men reflect on male of online more miserable dates -- sometimes while they're website on them. You may not get the lot of good dating tips here, but you'll definitely find some american entertainment. Families can talk about dating as it's portrayed on this show. How realistic is the behavior of these guys, do you think? What would the ritual meeting for this show sound like? Do you think some of the men or \"daters\" featured dating think about or behave the ritual way toward women when they're not on a reality the Are the women being themselves when meeting them?\nCommon Sense Media's unbiased ratings are created by expert reviewers and aren't influenced by the product's creators or by any of our funders, affiliates, or partners. See how online rate. Common Sense Media, a nonprofit organization, earns a small affiliate fee from Amazon or iTunes when you use our links to website a purchase.\nThank you for your support. Our ratings are based on child development best practices. We display the minimum age for which content is developmentally appropriate. The star rating reflects male quality and learning potential.\nmale Senior online dating tips how we rate. Log in Used me up. Reviews Find amfrican good stuff, faster \u2014 from books to YouTube. YouTube channel reviews are here! The best media lf your family, hand-picked by our datings. What's on your mind? Get answers to top parenting questions here. Online Dating Rituals of the American Male.\nEntertaining ritual reality show paints men in a poor light. Bravo Reality TV Sign in or join to save for later. Parents say No reviews yet Add your rating.\nDzting say No reviews yet Add your rating. Get it now Searching for streaming and purchasing options Common Sense is a nonprofit organization. Your purchase helps us remain independent and ad-free. Get it now on Searching for streaming and purchasing options A lot or a little? The parents' guide to what's in this TV show. What parents need to know Parents need to know that Online Dating Rituals of thf American Male includes lots of american conversations about website and women, some of which can be sexist online crude.\nStay up to date on new reviews. Get full reviews, ratings, and advice delivered weekly to your inbox. User Reviews Parents say Kids say. Get married dating aren't any reviews yet.\nBe the first the review this used. They sip champagne, talk about church, get their toenails painted by strangers\u2014you know, first date basics. Alex splits early 'cause he isn't feeling it, but he still wonders why she won't kiss him at the end of the ritual. Maybe because he said women expire when they hit 31?\nGood dating, good chemistry, common interests, two good-looking people. Marcus even thinks he and Chloe have a \"chance to turn into male real. But he isn't ready to log off forever. Marcus is disappointed to meet her IRL, because, according to Yhe, her profile photos were not truthful. More specifically, \"Her face is website rhe for her chin. But male too fun for Marcus. San antonio dating events the rituqls side of the city, after Alex gets stood up on a drum circle date The Angeles!\nHe drops his plans of messaging strangers on the Internet all american to qmerican meet with her immediately. He's americzn pleased to find Amita to be hotter off of the Internet, and he's american floored when she finds out she even likes dating. Soon Marcus realizes the grass is online not always greener, so he websites Chloe back up online invites her over.\nShe brings wine, he makes her dinner, they watch TV and make out. Alex and Amita also seem to have found a love connection, despite Alex getting handsy again!By Stacy Liberatore For Dailymail. Beneath the waters that surround the Ameriican Island in Japan, a tiny artist is gathering fine sand to create a patterned circle for an elaborate mating ritual. Once the design is finished, a female will swim into the center to signal katz dating coach and the male bites on her cheek to start the mating process.\nThe white-spotted pufferfish spends at least six weeks building online american mating circle that is 20 times riuals size. It online male in a circle gathering the fine sand and adorns the used structure with sand dollars and seashells. A suspecting female ventures website to the area and if she is approved, dating signal for the male by hovering in the center. The show will take viewers into the depths of the Pacific Ocean to ritual the hidden treasures that call it home \u2014 and one of the episodes highlights the elusive white-spotted pufferfish.\nAfter the underwater circles were discovered, datings began to monitor the seabed with the hopes of uncovering the mystery. The fish spends about six weeks building the structure that is 20 times its size, which also acts as a way to american down the current in order to protect the eggs laid its mate.\nThe fish spends about six weeks building the structure that is 20 onlinr its size, which also group dating boston as a way to used down the current in order to protect the rituals laid its male mate.\nAnd the male even adorns the circle websit sand dollars and seashells to make it more attractive. Once a female approaches the mating area, it hovers in the center as a signal of approval. The white-spotted pufferfish is about three-inches long and adorned with whites spots on its back and silvery white spots on its abdomen.\nYou realize, what have written?Why do I have to complete a CAPTCHA?\nThanks for the information, can, I too can help you something?The best media for your family, hand-picked by our editors. In it something is.\nWrite to me in PM, we will communicate.Find the good stuff, faster \u2014 from books to YouTube. I apologise, but, in my opinion, you commit an error. I can defend the position.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmubp b/data_all_eng_slimpj/shuffled/split2/finalzzzbmubp
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index 0000000000000000000000000000000000000000..e103e917962ac73c8177b9a9227ec59f8e40b371
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"At the Green Pilgrimage Network meeting in Trondheim, July 2013, the Network adopted the swift as a symbol of Green Pilgrimage. Noting that it was approaching the time when the migrating birds would be leaving Europe for their winter stay in the sky over Africa Alison Hilliard, ARC Deputy Secretary General, said that \"Swifts are wonderful examples of the everyday miracles of travel and journeys. And powerful reminders of our connection with and responsibility for nature, creation and each other.\"\nEvery year in the springtime great flocks of swifts are welcomed with a ceremony in Jerusalem where they nest in the brickwork of many holy places, including mosques, churches and the sacred Western Wall. The birds remain in the city until their nestlings have fledged and then return to Africa in June.\nThis day, this night, forever.\nWhen you leave the nest you do not touch the earth again for two years. You even sleep on the wing.\nTo the day you land here again.\nYou are faithful to your mate for life, yet you do not need to be together all the time. You travel secure in the fidelity of your love for one another.\nKnowing we too are not alone.\nAs pilgrims of the air, the swifts are in danger from us at almost every stage. Together let us commit ourselves and our communities to protect our fellow pilgrims, and to protect all creation in its pilgrimage here on earth.\nRead more about the Jerusalem celebrations for the swifts here.\nClick here for Holy Birds of the Heavens - an article about swifts written by journalist and broadcaster Mary Colwell for The Tablet, a Catholic magazine.\nFind out about Swift Conservation here.\nThe welcoming of swifts back to the Western Wall has become a ceremony that is held in Jerusalem every springtime. Jerusalem is one of the members of the Green Pilgrimage Network, and this is part of the movement to protect all nature, through thoughtful stewardship, gentle policies, and vigorous campaigning for improvements for all creation.\nAt an extraordinary meeting in Trondheim this weekend 16 important pilgrim cities and places \u2013 including Santiago de Compostela, Bethlehem, Rishikesh, Varanasi and Mexico City \u2013 committed to become greener.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High res version of the famous photo of Steve Jobs in his living room. Awesome desktop wallpaper!\nThanks to @kateagarwal for scanning and repairing this image for me.\nDownload the full size here.\n\u2190 Early morning breakfast burritos!\nA lot of people wrote <a href=\"http:\/\/mba-essay.org\/\">mba essay<\/a> about this the most famous peicture of Steve Jobs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3 in 1: With a detachable mirror base, it could be a handheld fan, table fan and a make- up mirror.\nLooking for ideal Mini Mirror Fan Manufacturer & supplier ? We have a wide selection at great prices to help you get creative. All the Mini Usb Mirror Fan are quality guaranteed. We are China Origin Factory of Portable Mirror Fan. If you have any question, please feel free to contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Honey Tree Early Childhood Center enriches the lives and learning of young children ages two through Kindergarten. Our goal is to nurture children's healthy growth and development by providing a high quality early childhood education program. We encourage the whole child approach to teaching and learning in a developmentally appropriate environment that is safe, nurturing, and fun.\nWe know that choosing a first school experience for your child is an important decision and we are thankful for the opportunity to provide your child with a nurturing and supportive learning experience. We encourage you to schedule a private tour of Honey Tree. On your tour you will have the opportunity to see our program and facilities first hand. Initial registration for the following year takes place in spring, however limited space may be available in select classes after the initial registration period. Please call the Honey Tree office to check class availability.\nWe are committed to touching in love, the lives of young children in our community, helping each child to value him or herself as a unique individual created by God, and to develop a joyful curiosity in knowing and learning about God's world. Honey Tree is and has been a community outreach\/ministry of Holy Trinity Lutheran Church since 1973.\nFor more information about individual classes click on each class title above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As the UK's recessionary storm clouds pass, restaurateurs should be looking forward with some confidence. But what is the sentiment on the ground? Will we see more restaurant openings in the next 12 months? Which cities offer the best opportunities for growth and, where will the funds to expand come from?\nIn partnership with industry magazine The Caterer, we looked at growth in the sector. Drawing on a survey of their readers' views and direct conversations with restaurant owners through an industry roundtable, we have put together a special supplement looking at growth in the sector.\nWith a detailed look at prospects for the sector, this is essential reading for those wanting to understand the challenges and opportunities in the future. If you'd like to discuss any of the issues raised in the report in more detail, please do get in touch.\nWe work with restaurants across the UK on a daily basis, supporting them from start up, during roll out, operationally and at exit. Working with owners, operators and funders we look at ways to help deliver what they need, whether single site or those with a larger portfolio and a UK footprint. We regularly publish legal insights on issues that matter to our clients. If you'd like to sign up to receive these, please register here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbncwq b/data_all_eng_slimpj/shuffled/split2/finalzzzbncwq
new file mode 100644
index 0000000000000000000000000000000000000000..3c07b677ca9e21cd731cebfe6e4433480b2a9d8f
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"In the communities unit of our Social Studies book, they wanted me to teach the kids how to use a compass rose. So we did just that!\nAfter introducing the compass rose, pointing out the compass rose on every map in our room, and reading the pages in our book, we played a fun game that reinforced the cardinal directions!\nOne at a time, a kid would sit outside the room while we'd hide our traveling buddy, Tyrone, somewhere in the room.\nThen we would bring the kid back inside and use our knowledge of the cardinal directions to help the kid find Tyrone!\nThey loved this game! We played it 12 times so that every kid had a chance to find Tyrone. They got CRAZY loud because they felt the need to scream out the directions to the kid trying to find Tyrone. The noise paid off, though, because today in one of our read-alouds, the author wrote about some town in the north, and I said, \"So which way is north?\" And everyone pointed north right away! Hooray!\nAfterwards, the first and second graders drew maps of our classroom including a compass rose. This was a little tough for some of them because they had to draw the room from an angle that they don't usually see, but I was really impressed with their hard work and all the little details they included!\nNow what I need are some clip-arty direction signs! My handwritten ones that I made during morning recess are not beautiful.\nWhat a fun way for the kiddos to learn their directions!\nGreat idea! I am jealous you only have 12 students!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alex graduated from Wayland Baptist University with a Bachelor of Applied Science in Human Resource Management. After working in a management setting for two years, Alex joined the team at MWM Architects in the fall of 2013. Alex's job at MWM is the office administrator. She helps with human resources, accounting, and everyday office tasks \u2013 like making sure holidays and employee birthdays are celebrated properly (with lots of cake). Alex is actively a member of the Society for Human Resource Management. MWM allowed Alex to see what it is like to actually love her job and the people she works with. Her passion for her work helps the office run smoothly.\nWhen Alex isn't working, her main role is being a mother to her son. She enjoys teaching him the necessities in life, like being a Texas Tech fan by going to all the MWM tailgates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"11am-1pm open to professionals or advanced dancers. 1-2pm is open to all, 16 and up, regardless of their background in dance. This class will employ the specific vocabulary and skills that are part and parcel of a dancer's knowledge. The layering of familiar movements with Gaga tasks presents dancers with fresh challenges, and throughout the class, the teacher will prompt dancers to visit more unfamiliar places and ways of moving as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"But reactions to the report were mixed among Washington state law enforcement officials, legislators and gun rights supporters. Some saw potential benefits in improved safe storage and could foresee continuing progress in performance and acceptance. Others worried about reliability or said the emphasis instead should be on illegal possession, a more robust system of background checks, and more educational outreach to at-risk teens about added penalties for using a gun in a crime.\nWith nine prior convictions from 1986 to 2005 for driving under the influence of alcohol plus felony convictions in Yakima County for two counts of vehicular manslaughter and one for vehicular assault \u2013 stemming from a 1991 tragedy when he had a blood alcohol level of .30 \u2013 Dwight Lloyd Casady was driving without a legal license along Railroad Avenue at River Street in Snoqualmie, Wash.\nAccording to King County Court documents, police signaled him to stop because his taillights were covered with mud. He was very slow to respond and drove half on the shoulder for some distance.\nField sobriety and blood alcohol tests showed him to be clearly impaired.\nAsked to place his right heel to his left toe, he kept falling.\nWhen an officer who had taken him to the station noticed a particular smell he admitted he had urinated himself.\nCompounded by a high \"offender score\" from his past record, Casady, 47, of Harrah, Wash. last Friday May 3 in King County Superior Court was sentenced to five years in prison for his felony DUI conviction based on the January arrest in Snoqualmie.\nA charge also filed in connection with the January incident, for driving with a suspended or revoked license, a gross misdemeanor, was ultimately not prosecuted.\nIn the 1991 tragedy in Yakima County, Casady's impaired driving resulted in the deaths of two and the \"serious maiming\" of a third person, court records state.\nThe related convictions for vehicular homicide and vehicular assault earned him a sentence in October 1991 of 89 months, or almost seven-and-a-half years.\nYet records in the recent case also reveal that by 1997 he was out of jail and convicted in that year for one of his nine DUIs, plus two more in 1998.\nHis most recent DUI conviction prior to Friday's felony sentence was in 2005.\nHis extralegal endeavors also include felony convictions in Snohomish County in 2002 for theft and unlawful imprisonment. Overall, Dwight Lloyd Casady has been named as a defendant 57 times in municipal and superior courts in Washington since 1980, primarily in Yakima and Snohomish counties. In 1999, the Seattle Times reported he was attacked with an axe and underwent surgery as a result, after threatening to kill a man for spilling a beer in a Snohomish County bar.\nIn response to last week's felony DUI conviction, Casady's attorney, Seth D. Conant, immediately filed notice of intent to appeal.\nNew federal data tied to one prominent indicator suggests Washington state is far outpacing neighboring states plus California and the United States in firearms purchases from 1999 through 2012. That's the number of background checks submitted from federal firearms licensees to the National Instant Criminal Background Check System (NICS) on behalf of unlicensed buyers, under the federal Brady law. The total from Washington has grown from 133,674 in 1999 to 519,209 in 2012, or 288 percent versus 114 percent nationwide, 66 percent in Oregon, 70 percent in Idaho and 28 percent in California. The data come from updated annual state-by-state and nationwide NICS summaries issued this week by the Federal Bureau of Investigation. (See chart, below).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted on Saturday, March 23rd, 2013 at 10:55 am\tand is filed under e-Notes. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We present you with 40 of the most interesting facts about Greece in this well-researched list. Did you know, with roughly equal size yet the population of Greece is more than the double of Alabama? Tourism is a huge industry here and every year the number of tourists visiting is more than the entire population.\n1. Anybody who's 18+ can't skip voting according to the law. The option of not voting is not available in Greece.\n2. Greece contributes to around 7% of total marble produced all over the world.\n3. In Greece, the number of international airports is a bit high as compared to any other country due to a good number of tourists visiting every day.\n4. Greece is the world's 3rd leading producer of Olives and some Olive trees planted way back in the 13th century are still producing Olives.\n5. Almost 80% of Greece is covered with mountains and this is the reason it has zero navigable rivers.\n6. 98% of total population of Greece is ethnic Greeks. Rest minorities include Albanians, gypsies, Armenians, Bulgarians, and Macedonians.\n7. Greek is spoken by around 12 million people all over the world.\n8. The Greek language has contributed to thousands of English words, some of the common one's are (academy, apology, marathon, alphabet and typhoon).\n9. The literacy rate of Greek adults in the 1950's was only 30% which has now increased to almost 95%.\n10. More than 40% of the total population of Greece is residing in its capital Athens.\n11. Greece has around 170 populated islands out of the total of more than 2000.\n12. 6% of the annual GDP is spent on the military by the government. It is compulsory here for all Greek men to serve in any branch of the armed forces for a period of 1 year to 18 months.\n13. Ancient Greece had 1500 different cities\/states with each having its different army and law. They used to often quarrel with each other a lot.\n14. In 1999 a very devastating earthquake hit both Greece and Turkey, here both came to each other's aid and thus their relationship improved which was earlier pretty bad.\n15. The average life of ancient Greek women was 36 and 45 for a man, which has risen to 82 for a female and 77 for a male.\n16. The leading producer of Sea Sponges all over the world is Greece.\n17. Greece has a maximum number of archaeological museums as compared to all the countries.\n18. The national sport of Greece is Football.\n19. There are no Retirement homes in Greece as the Grandparents here usually live with their children.\n20. There is an ancient belief in Greece that turquoise blue (color) keeps evil away.\n21. Almost 100,000 birds from Asia and northern Europe spend their winters in Greece's wetlands.\n22. The most popular tourist spot in Greece is the city of Rhodes which is also the capital of the island of Rhodes.\n23. A Greek cook named COROEBUS became the first Olympic champion in 776 B.C.\n24. The first municipal dump in the western world was organized by Greece around 500 B.C.\n25. Greek is one of the oldest languages of Europe which is being spoken from more than 3000 years or so.\n26. There are more than 250 days of sunshine in a year in Greece.\n27. Greeks always wave with their palm closed as they consider it an insult to show the palm with fingers extended.\n28. The coastline of Greece is 10th longest in the world extending up to 9000 miles.\n29. 33 Kg of bronze armor was exactly what a soldier used to wear in ancient Greece.\n30. Feasts were meant for only men in the ancient Greek and women were there for entertainment.\n31. Animals were sacrificed by wealthy people in the temples.\n32. During each Olympics 100 bulls were sacrificed to the Zeus by the Greeks.\n33. Earlier Greek coins always had the face of gods or goddesses on it until Alexander the great put its own on them.\n34. The official name of Greece is the Hellenic Republic.\n35. After the collapse of Greece's military dictatorship in 1974, 2 major political parties (Socialists and Democrats) were formed.\n36. Greece has the highest abortion rates and the lowest divorce rates in the EU.\n37. Around 20% of a Greek worker's pay is deducted for taxes and national health care. Free hospitals and various medical services are provided by the government in return.\n38. Every year 1 month of paid vacation is provided to a Greek worker.\n39. Even with a college degree, 10% of the Greek youth are unemployed. It's pretty difficult to find a job here.\n40. The national cheese of Greece is Feta (made from goat's milk).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disabled campaigners say they feel \"angry and let down\" by the government's decision to offer only a slight improvement to the compensation it will pay survivors of the NHS contaminated blood disaster.\nMore than 2, 000 people with haemophilia have so far died as a result of being infected with hepatitis C and HIV through contaminated NHS blood and blood products in the 1970s and 1980s.\nSuccessive governments have resisted calls for more generous compensation and other measures to support survivors.\nA review of the support available in England has ended with a government pledge to increase compensation for some of those infected by hepatitis C, at an estimated cost of between \u00a3100 million and \u00a3130 million over four years.\nA report made it clear the decision not to make a more generous settlement was partly due to \"affordability in the financial context of the current spending review\".\nMike Dorricott, a spokesman for TaintedBlood, the user-led group that campaigns for justice for those affected, said the package would help probably only one in five of those infected.\nThe package includes an increase in the one-off payments made to those with the most serious hepatitis C-related diseases, from \u00a325,000 to \u00a350,000, as well as a new annual payment of \u00a312,800, to match the sum paid to those infected with HIV.\nAnd all those infected with hepatitis C who are in \"serious financial need\" will be able to apply for discretionary payments for the first time.\nPosthumous claims will also be allowed for the first time on behalf of those who died before 29 August 2003 after being infected with hepatitis C, but their dependants will only have three months to lodge a claim.\nAnd those not already exempt will be able to ask the government to cover the cost of an annual prescription pre-payment certificate.\nDorricott said the \"arbitrary\" three-month deadline for dependants of those who died before 29 August 2003 was \"disgraceful\".\nLord [Alf] Morris, whose contaminated blood private members' bill is due to receive its second Commons reading on 11 February, called on the government to \"meet the haemophilia community and listen again to its plea for a response that is more in keeping with the scale of the disaster\".\nHe made it clear to Disability News Service that he would not withdraw his bill, because the government's announcement failed to meet the recommendations of the independent public inquiry set up by him and chaired by Lord Archer of Sandwell.\nHe said the government believed the deadline of the end of March \"should be sufficient to enable those with a valid claim to come forward\" because \"on the whole\u2026the victims' families know who they are\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Norwesco 1,500 Gallon Septic Tank, part# 44484 is a 2 compartment septic tank. It's primarily used for the storage of wastewater. This plastic septic tank offers a lightweight low cost alternative to old-fashioned concrete septic tanks.\nThe 1500 gallon low profile septic tank is IAPMO Certified. It features two 20\" gasketed manways, a 4\" gasketed pvc tee assembly inlet, and a 4\" gasketed pvc tee assembly outlet. Installation requires no backfill.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oh my dream is still to spend a night in an airstream, I love them to bits!\nOh I am just the opposite, sans husband or children I spent my single days travelling the globe until I was 35, feeling a bit lost to be honest, now I'm a two holidays a year woman, I am so over travelling. I am so glad you had wonderful trip.\nI'm so glad you had a wonderful trip my friend. I adore traveling, but don't do it as much as I used to. Your pictures were fabulous!!\nOh boy, and I wonder where it will take you next! With your eye for fashion, it must be a burst of fashion energy that you have when you see other cultures and what they do with clothing, shoes, accessories. I'm not a fashion maven but I do notice all of the small details...of course...while I am oogling the food! So glad you enjoyed your trips. We're seriously considering going somewhere for Xmas next year...I hope I follow through!\nCongratulations! I've found something to love everywhere I've travelled, and return home with a renewed appreciation of my own circumstances, and impatience to plan my next adventure.\nThank you for your excellent input to a travel wardrobe. I'll remember that for next time. I hope you had a nice trip. Have followed you and your beautiful clothes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Injectable corticosteroids must now carry a new label warning about the risks of severe adverse effects from epidural injections including death, stroke, and permanent blindness and paralysis, the FDA said.\nThe agency decided that such a warning is needed after reviewing numerous reports in the medical literature and its own adverse event database.\nWelcome to our new news and information structure. Here we will post news that pertains to the chiropractic profession or our patients, as reported by the federal government, Medicare\/CMS, CDC, and other federal regulatory agencies.\nIt will take us a few days to populate this particular news section with news. In the mean time be sure to check our Communications Archives for news from March 2014, and before.\nPhysicians' strategy for lobbying Congress this year should focus on influencing the wording of regulations, not on getting laws enacted, a member of Congress said at the American Medical Association's (AMA) annual advocacy meeting.\n\"If I were you, one thing I'd be doing is not worrying about big legislation passing Congress,\" said Rep. Xavier Becerra (D-Calif.) Instead, \"I'd be getting to know Andy [Slavitt, administrator of the Centers for Medicare and Medicaid Services (CMS)] very, very well,\" especially since Slavitt's agency will be busy implementing the new physician payment system that will replace the much-maligned sustainable growth rate (SGR) formula. Because it's a presidential election year, \"It's not going to be a year where we [in Congress] try to throw long passes and score big touchdowns; it's going to be a year where we try to get the things done that need done and don't interfere with the presidential election process,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The boys over at BHGP were fooled into thinking that I actually knew something about Iowa Hawkeye sports.\nSo with that announcement, you will be able to read my posts over on BHGP from now on. Don't worry, though, I will still be posting here, as well.\nI am now in the minor leagues. For shizzle.\nMy first post just went up, it's on the upcoming 2008 season for the basketball team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Police in South Australia have revealed scammers impersonating staff from the Australian Tax Office have lured victims to Bitcoin ATMs \u2013 fleecing them of thousands of dollars.\nSA Police say it's a new tactic being used in a common cold call technique in which offenders, believed to be based overseas, claim to be working for the ATO.\nIn two recent cases of the Bitcoin ATM scam, reported to SA Police, a conference call was established by the scammer using technology to make the caller ID appear legitimate.\nDuring the conference call a 'payment arrangement' was set up which required the victim to go to a local Bitcoin ATM or pay via Google Play vouchers.\nIt's not clear how much was deposited into the Bitcoin ATMs, however SA Police say the victims lost 'thousands'.\n\"This type of scam is of concern to police, with this change in methodology further removing the opportunity for a third-party to intervene to protect or assist the victim,\" said Detective Sergeant Martin Burke, from the SA Police Electronic Crime Section.\nInformation on scam prevention and awareness is available on the Australian Government's ScamWatch website.\nEarlier this month, Micky reported on the rising number of Australians being fooled by scammers impersonating major organisations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pelvic floor muscles play a large role in core control and everyday function. The pelvic floor is composed of several layers of muscle tissue forming at the bottom of your pelvic bowl (your pubic bone to tailbone and your sit bones). Your pelvic floor acts to stabilize your hips and pelvis and control intra-abdominal pressure for core stabilization and to control urine flow, sexual function, and so much more. Dysfunction is incredibly individualized and can present due to a less active pelvic floor that can be caused by childbirth or trauma, or a tight or overactive pelvic floor. We treat pelvic floor issues with focused repositioning of the pelvis and by restoring normal breathing mechanics via the Postural Restoration Institute.\nOur sessions are conducted in a private room and begin with a wellness screening and orthopedic evaluation assessing for any musculator or asymmetrical imbalances in the kinetic chain. If required, we will recommend a second appointment to perform a pelvic floor evaluation.\nPlease note that while New York State Direct Access Laws do not require a referral to begin physical therapy, given the sensitivity of a Pelvic Floor exam we ask all patients to have a referral prepared for their second visits, once an internal exam has been recommended.\nOur goal is to have you return to normal day-to-day function and a healthy lifestyle that includes fitness!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finding the right doctor can be a pretty daunting task.\nIf we're honest, most of us probably spend 15 minutes browsing through online reviews, maybe we ask a few friends, and we land on what seems like a good choice.\nBut you have to ask yourself: what are you missing? Do you really have the best doctor?\nWe've picked the brains of two healthcare experts to bring you 18 steps that will help you choose the right physician.\nMichelle Katz, Lpn, MSN is a Healthcare Consumer Advocate. She has written several books on saving money on healthcare, and she has helped build healthcare programs around the country.\nR. Ruth Linden, Ph.D, is the Founder and President of Tree of Life Health Advocates in San Francisco. For more than 30 years, Ruth was a professor and researcher in academic medicine, public health, and health policy. In her work as an independent health advocate, she helps her clients choose the right team of physicians and resolve other healthcare challenges.\nBefore you start sifting through Yelp reviews or the list of in-network providers from your insurance company, it's important to evaluate what you're actually looking for in a doctor.\nThis is important, because you might find a rave review explaining that the doctor was very quick, and that you always get in and out within 10 minutes. But perhaps one of the most important things on your list is that your doctor takes his time and doesn't make you feel rushed.\nHaving a short must-haves list where you can actually compare people's opinions to what's important to you is critical \u2013 it's the first thing you should do before you even start looking.\nBeyond personality traits and mannerisms, you also need to consider how the doctor goes about making treatment suggestions.\nAllopathic treatment is the method of treatment taught in medical schools and training hospitals. This system of medical practice aims to fight disease using drugs and surgery.\nAcupuncture is a form of alternative medicine where very thin needles are inserted into your body. This form of medicine is a key component of traditional Chinese medicine.\nSo, before you start any serious research of physicians, it's important to understand what you're looking for. What type of doctor do you want, and what treatment philosophy is important to you?\nIt's not sexist to eliminate your options by gender. Some people feel more comfortable being examined by a female, and others prefer a male.\nFor most of us, it's important that we choose a doctor that's in our insurance network. However, before you eliminate every other doctor, Ruth advises you to pause and consider a few things.\nMany people don't have the financial means to go outside of their insurance network, but Ruth still advises that you don't automatically cut all out-of-network doctors from your search.\nMichelle follows this same line of thinking. Some doctors don't accept Medicare, but she explains that some doctors will work with you.\nThat left Michelle's grandmother in a tricky spot, because replacing him would be nearly impossible, but she couldn't afford to see a doctor that didn't accept her insurance.\nIn some situations, doctors will go the extra mile in requesting to be a part of your insurance network, and in other cases, they're willing to work with you. Doctors who seriously care about the wellbeing of their patients \u2013 and oftentimes, doctors who run their own practices \u2013 will be willing to cut you a deal.\nSo, perhaps you could start your search by looking at in-network doctors, but if none of those options seem to align with your needs and values, it's worthwhile to start looking outside of your network.\nHowever, if it is crucial that you choose an in-network physician, be sure to confirm with the doctor and the insurance company that they are in fact in-network. This information can sometimes be out-of-date.\nIt's how we judge everything from beauty salons to restaurants \u2013 just Google it, and read the reviews! From Facebook to Yelp, there are usually loads of reviews to sift through that can give you an initial impression of a business.\nAnd while this should be part of your doctor shopping process, there are some things to be aware of.\nA few ways to pinpoint fake reviews are to look for vague statements like \"The staff is very professional\" and \"The staff and doctor are very friendly.\" Also look for carefully worded statements that sound like they come straight from a brochure.\nFor example, one Google review of a doctor's office says, \"In a short amount of time together we crafted a custom treatment plan based upon my individual healthcare needs.\" While it may be a genuine review, it should raise your suspicions. Most people don't write so eloquently, and there are generally a few typos in genuine reviews.\nYou can also tell a lot about a doctor and his staff by reading negative reviews.\nHere are two examples of doctors who have responded to online reviews. Which doctor would you rather visit based on the response?\nTalk to your family and friends in the area and get some feedback about doctors they've liked (and maybe disliked!).\nMichelle advises that you ask for specifics \u2013 what did they specifically like or dislike about that doctor? This is important, because again, your goals and values might be very different from your friends.\nEveryone's opinions are going to be based on their own likes and dislikes, so while getting feedback from family and friends is a huge help, be sure to understand the feedback thoroughly before making any decisions.\nHas your doctor written any publications? Do they write on an online blog? If the answer is yes, you know that doctor has expertise, and they clearly have a passion for what they do.\nHowever, there is a line that can be crossed \u2013 Michelle warns you to stay away from doctors that spend too much time on TV appearances and the like.\n\"You shouldn't be on TV more than you're in the office. The other day, I saw a doctor who left in the middle of a segment because one of her patients needed her. To me, that was impressive. That doctor put her patients first,\" says Michelle.\nIf so, this can be a huge red flag. You might consider crossing this doctor off the list if you have other options still on the table. If you were set on this doctor but find some unsettling information, it might be worthwhile to call and ask about it.\nKeep in mind that with lawsuits, many doctors will complain that there are frivolous lawsuits, and there are also serious cases that settle out of court. Michelle advises that you consider this with everything else mentioned.\nWhile you want to research the doctor, many people forget that the hospital is important, too.\nRuth explains that you can go to http:\/\/www.leapfroggroup.org\/compare-hospitals in order to research the hospital, though you won't find everything. Here, you can see information about inpatient care management, medication safety, infections and injuries, and more.\nHowever, it's important that you research the service you'll actually be on.\nSometimes, you can find that information on the hospital's website, and if you can't find it, you need to ask.\nWhile you research the hospital and gather information about their success rates, you need to do the same for your specific doctor.\nResearch shows that the more a physician performs a surgery, the better they are at it.\nEven if you aren't having a specific surgery performed, you still want a doctor that's very familiar with any conditions you might have.\nThat brings us to our next tip \u2013 is the doctor willing to refer you out to a specialist when needed?\nIf you have a specific medical condition or situation, and you chose a doctor that's very educated on it, you still want that doctor to admit when new symptoms or conditions are outside their expertise.\nYou're always going to deal with the staff, so calling the office and getting a feel for how the office staff handles your questions can be a huge help when narrowing down your doctors.\n\"You can approach this in two ways \u2013 you can either call when it's a busy time to see how they handle you, or you can call when it's not busy to give them a break,\" says Michelle.\nMichelle does say that everyone can have a bad day \u2013 it happens. That's why calling a couple of times on different days can give you a better overall picture of what that practice is like.\nHowever, if the office is rude over the phone, that can say a lot about the doctor \u2013 after all, they are his employees.\nIf you call the office, and they say they're booked out for the next 3 months \u2013 is that a deal breaker for you?\nMichelle explains that it can go both ways \u2013 on the one hand, a busy doctor can sometimes mean it's a great doctor. However, you have to ask yourself if you really want to be seeing a doctor that's so busy.\nSometimes, you can find the doctor's email. If you've called the office a few times and the staff was rude, it can be worthwhile to take this extra step.\nOnce you do narrow down your options and you've booked an appointment, you're undoubtedly going to experience the waiting room.\nHow long do you have to wait before you're seen by the doctor?\nSo, it's important to always think about this with some perspective. That doctor might be taking an extra 20 minutes to tell a patient about a cancer diagnosis. Would you want a doctor who would keep others waiting if you were in the same situation? Or would you rather have a doctor that gave you the news and ushered you out?\nThis brings us to our next important step \u2013 be sure to take note of how long your actual appointment is.\n\"If you go in with a concern, and your appointment is only 2 minutes long\u2026 that's a big red flag,\" says Michelle.\nYou want a doctor who will take the time to listen to your concerns and who won't rush you. Again \u2013 a busy doctor is not going to catch certain symptoms and concerns as well as a doctor who is taking his or her time.\nHow did the doctor communicate with you during the actual appointment?\nEven if you tend to prefer quick appointments, you should be aware that a doctor who does things too quickly and doesn't give you time to talk about your symptoms is likely missing important information.\nYou also have the right to ask questions and leave that appointment fully understanding everything from your symptoms, potential conditions, any prescriptions, and whatever else might be relevant. You shouldn't leave an appointment with questions.\nBut you should enter an appointment with questions! It can be tough to tell whether or not your doctor is keeping up with current medical research, but Michelle says that one way to find out is to come with questions.\n18. Listen to your gut feeling!\nAt the end of the day, your gut feeling counts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In dating in Philly over the years, I've run into my fair share of things that have shut down my prospective relationships. From an unhealthy obsession with cats, to a very severe lack of common interests, to, of course, the occasional UDP (if you know, you know), sometimes matches just aren't meant to be. But lately, the issue that seems to be plaguing my budding romances most stems directly from be me being confident about what I'm looking for, and that, for the most part, making the guy run the other way. And this all manifests itself in the planning. of. freaking. dates.\nI haven't written one of these Having a Moment posts in a bit, but I've been thinking about writing this one forever. Today, I guess I just got into a mood and this spilled right out. So here we go!\nI really try to be positive about the dating world. For me, my time trying to find love in Philadelphia has been filled with some peaks and valleys. There were times I could meet 3-4 different guys for dates in one week and be loving life. And then, there have also been times when, after rushing out of a second date I really shouldn't have gone on in the first place, I've spilled out onto 13th Street, called my dad laughing\/yelling\/crying, and questioned, why am I even doing this?! Why does the quest for love seem so \"easy\" for some, and so, so difficult for the rest of us?\nToday, you'll hear from 5 Philly women, Melissa, Leah, Sarah, Tracy, and Tiffany, who are all out there in this city of ours, trying to make it work with a special someone. And while some of their answers are optimistic (You mean you actually meet men in person? You've never had to pay on a date? No dating horror stories\u2026ever?), the last question really just says it all.\nBut before we begin, let me just say this: (and I'm speaking for you here ladies, so please chime in if I got it wrong) while our attitudes towards our dating lives might not be optimistic all of the time time, we're still out there. Dating takes work and time and money and creativity and humility and honesty, and here we all are, soldiering on. We've seen our friends get married and we've heard the online dating success stories, so we keep going. All with the hopes that one day, we'll find a guy who will laugh at the same TV shows as us, know exactly how we take our cocktails and our coffee, and surprise us with flowers, just because.\nWhat a time, ladies, what a time. Being single again has brought some laughs, some screams, and some texts to my girlfriends with \"why. me.\" written 10x. I've also started listening to podcasts for the first time in my life (newsflash: I obviously love them), and the dating ones are the absolute best (but really, please send me all your podcast recommendations, dating or not).\nKeeping true to my list-loving personality, I just sent a guy 10 options for date #4 this Weds. Any interest in me making this a blog post?\nIt's pretty much fall \u2013 HOORAY. Fall, along with being my favorite season in general, is my favorite season for dating. You get to do cute, cozy activities, you don't have to worry about sweating excessively before you even arrive at your destination, and (most importantly) your wardrobe options are much more robust.\nPhilly has so many seasonal things to do within a short drive, so listen up ladies, for this is one of the few times that I will advocate leaving the city and heading out to the burbs. The reasons for my acceptance of this are two-fold; first: concrete places without pumpkin patches or warm apple cider don't necessarily scream fall, and second: our driving tolerance is at an all-time high due to this summer's recent beach trips. And now, on to the dates!\nHooooray hooray hooray \u2013 it's finally Friday! In case you don't follow me on, well, pretty much any social media channel, I wanted to share something pretty cool that happened to me early last week.\nI'm a big fan of the Philly Thrillist site and always look forward to their emails. They send out a lot of their articles in list format (obviously one of my preferred methods for writing on this blog, as well!) and I usually pin something a few times a week from their site. I was thrilled (get it? I'm here all day.) when they asked me to cover a dating story, and after some brainstorming sessions and a lot of research, it was born: 21 Actually Great Date Ideas in Philadelphia.\nThis past November, I was a dependent woman. And not on any guy, but on none other than Tinder. Through many dates and many men, I've come to the conclusion that dating apps can be extremely limiting (especially when all of your friends are using them, too). In my experience (and you know I loved online dating at one point), after a while, apps can hold you back. My friends and I would go out in the city and not entertain any conversations with men, because, for all we knew, they were creeps. And why talk to creeps, when we could (hopefully) vet the endless supply of men waiting right on our cell phones (before we even had to deal with looking them in the eye)?\nI have really been loving this summer. The sunshine makes me so happy. I don't think I have done this season's update post (see last summer), where I just get to talk about all of the fun and random things going on in my life . . . so here we are.\nWe all know I'm single (because it seems like I announce it every chance I get), but I don't let that stop me from attending the things I want in the city. Last week, I posted a guest blog for D\u00eener en Blanc Philadelphia all about attending Diner en Blanc as a single woman. Check it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Download Uplifting Life free nulled Day we add new Version from Uplifting Life nulled to our site filemea.com in Music you can get file for free from Download Url Uplifting Life .\nUplifting Life is an Inspirational and upbeat music that evokes feelings of success and positive. Perfect for motivational advertisements, corporate media, positive videos, business projects, media montage and etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Palco is a groundbreaking color-changing fixture for applications with a light source of 49 high-power Luxeon LEDs. Palco creates an evenly diffused light beam, ideal for illuminating large areas. It has full DMX 512 control of all its main characteristics: luminous intensity, RGB color mixing, color temperature correction. The exclusive design and extremely compact lines make the Palco ideal for either permanent or temporary installation in a wide range of applications: stages, houses of worship, museums, art galleries, libraries, shopping malls, store window displays. Palco is also perfect for dynamically coloring backdrops and scenery in theatres, television and photography studios.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jammu and Kashmir police today claimed to have unearthed a Lashkar-e-Taiba module which was involved in planting a bomb near Chanapora bridge in the outskirts of Srinagar on May 17, 2012.\nJammu and Kashmir police on Thursday claimed to have unearthed a Lashkar-e-Taiba module which was involved in planting a bomb near Chanapora bridge in the outskirts of Srinagar on May 17, 2012.\nThe police have arrested five active militants who were part of a Lashkar-e-Taiba module operating from Tral area in Pulwama District, said Deputy Inspector General of Police, Central Kashmir Range S Afad-ul-Mujtba at a press conference in Srinagar.\nThe DIG said that in order to disturb prevalent peace and public order a criminal conspiracy was hatched by active militants of LeT operating in Tral and other adjacent villages in south Kashmir.\nThe militants Bilal Ahmad Bhat alias Bilal Lalihari from Pulwama , Sajad Ahmad Bhat alias Shadak from Panthachowk and a Pakistani citizen Qasim alias Qasim Bhai along with their other associates had planned the IED explosion in Srinagar to target law enforcement agencies and strike terror in the minds of general public, he added.\n\"A Pakistani IED expert Qasim Bhai masterminded the attempt by providing explosives and trained the module and also procured the remote detonation device,\" he said.\nWhile police have arrested Qasim and Sajad Ahmed Bhat while the co-conspirators are absconding.\nA vehicle laden with explosives was parkedd near Chanapora bridge, in the outskirts of Srinagar City. Police foiled the attempt following a tip off from a civilian. According to police the planned IED explosion was targeted towards convoys of security personnel who frequent the route.\nPolice said the module procured the explosive materials and containers for the bomb locally. Much of the items used in making the bomb were objects of daily use like metal milk cans which formed the storage device. Procurement of the explosive materials like gelatin was done from magazines providing blasting material to quarries operating in the Zewan -Khunmoh areas, Mujtaba added.\n\"While the use of IEDs and such car bombs is an emerging trend, we are going to try our level best to eliminate it,\" Mujtaba said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A different surface on each side giving you more creative options.\nIf the rim breaks or comes apart the item will be fixed free of charge.\nEnsures reflective surfaces won't crack or split after numerous uses.\nThis product comes in various sizes to cover all situations.\nReplacing the need for a regular shaped reflector the TriGrip reflectors utilise a triangular design which provides a much stronger and more stable structure. They also feature a moulded handle with securing strap, allowing accurate positioning with one hand. This means light can be reflected into awkward spots that would've previously required a stand system.\nThe securing strap also allows you to support the reflector without actually gripping the handle, reducing pressure on your wrist and fingers when held for a long time.\nReversible, a different surface on each side giving you more creative options.\nLifetime Rim Guarantee, if the rim breaks or comes apart the item will be fixed free of charge.\nTriple Re-Enforced Stitch Line, over 30 stitches per inch ensures steel rims won't be popping out of the rim tape.\nDouble Coated Reflective Surfaces, ensures reflective surfaces won't crack or split after numerous uses.\nCollapses To One Third Of Original Size, in seconds the item closes down to a convenient size for portability and storage.\nAccessories Available, a wide range of accessories is available for greater versatility.\nVarious Sizes, this product comes in various sizes to cover all situations.\nFits Into Camera Bag, add this product to your camera bag increasing your on location inventory.\nBag included, for easier storage and transport.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here bands and ensembles are available for each cause. Starting with a Pianoplayer, cover bands from dinner to Rock Pop, or a jazz trio up to Metal thunderstorms all categories are represented. We forward requests directly to our students and lecturers. We give inquiries to our lecturers to the lecturers or the appropriate management. Contact us via office@rpjam.de.\nLots of different bands, every musicstyle for any occasion are available.\nThe easiest way to get the right combination for your event is to get in touch with us directly.\nFollow the link to our Booking side.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Everybody loves comic book films because they are a super hit with fans. However, sometimes that means all superhero films are never as good as they should be.\nOne of the more recent and larger issues to strike a chord in the comic book community is the on-going feud with the two major comic book companies, Marvel and DC. Marvel comics has been the biggest contender for DC comics. One of the major areas that they have fought in for several years now has been the movie business. In my opinion, Marvel has been able to triumph over DC. This seems like a ridiculous claim, since DC has some of the best and most recognized heroes ever. However, this happens to be true. DC, unlike Marvel, has not been able to keep up with Marvel's tactics of their game plan.\nThe second thing Marvel has done differently compared to DC is the characters. No matter what recent film of the DC-verse you watch it's blatantly obvious how bland the characters are. When it comes to writing compelling characters, DC happens to completely glance over any defining character traits. Superman should be interesting to watch. We should be able to feel Superman's struggles of fitting in, controlling his powers, and becoming a hero. Instead, watching Superman is like watching paint dry on a wall. Now you could blame this on the actor, Henry Cavill, but if an actor has nothing to work with for their characters in a story, then the performance will fall flat. Meanwhile, Marvel has been able to make their characters like Captain America seem as everyday people. With Captain America, we see in the \"Avengers\" how he struggles to adapt to modern life after waking up about fifty years after he crashes and landed in ice during a mission during World War II. We see Captain America not only struggle with adapting to modern life, but him mourning the loss of everyone he ever knew. In the DC films, we never see any deep, emotional story arcs or even emotional performances from the majority of the characters at the level that Marvel has.\nAnother thing that Marvel does better than DC is style. Zack Snyder, the director of all of the recent, and future, DC films such as \"Batman V Superman\" and \"Justice League,\" has a few particular techniques and traits in his movies that just strike the wrong note for a lot of critics and fans alike. The first mistake, as mentioned before, is that the origin stories are never told in chronological order. This is a particular problem in \"Man of Steel.\" While taking the Memento style of storytelling isn't a bad trait to derive from, it is when there's not really a story being told. This goes back to how the characters are portrayed in the films. In the first half of \"Man of Steel,\" we see Superman go through a bunch of crucial moments in his life\u2013 like hitchhiking across America and him developing his powers as a child- then it jumps back to present day to a whole new batch of characters. In all of that screen time, the only thing developed is how powerful Superman is, but not what he thinks, believes, or feels, so the audience never gets to know Superman on a personal level.\nOverall, it's obvious which studio company has their movies in better shape. I still have hope that maybe DC will be able to create movies for their characters that they deserve. There is still promise with the release of \"Wonder Woman\" this June. And who knows, maybe DC is learning that story and character is more important over style.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Teabags and Happenstance: It's the First of the Month!\nGreat to see ya! So we're gonna be working together for the next thirty-odd days - let's do our best, okay?\nHere's to more laughter, longer days, and happy accidents.\nHere's to productivity and gratefulness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The bishops of Korea successfully accomplished their ad limina visit to the Apostolic See from March 9 to 17, 2015, the first time in seven years. For the bishops this visit must have been an occasion to experience the fraternal communion with the Holy Father and to reaffirm the deep unity of the local Church with the Universal Church. They still remember the Holy Father as a warmhearted and humble pastor during his apostolic visit to Korea in 2014 when he celebrated the beatification of 124 Korean martyrs and consoled the wounds of Korean society. It was his heartfelt encouragement and warm advice for the Church in Korea that made the ad limina visit to the Apostolic See a happy moment, in spite of the tight schedule that included the 2015 Spring General Assembly of the CBCK during their visit to the Apostolic See.\nThe Holy Father stressed the fact that the Catholic Church in Korea was nurtured by the blood of Korean martyrs and established by the passionate spirituality of martyrdom of the lay faithful. He also urged all of us to make efforts together, lest we quench the fire of the Holy Spirit working in the Church in Korea. Especially, he asserted that we must overcome the temptation of spiritual secularism, \"religious well-being\" which reduces the faith into a tool for cultural life, lurking behind the amazing growth and fruitful development of the Church in Korea. The more the Church becomes a so-called \"middle-class community\", the greater is the indifference of priests and religious to voluntary poverty as they pursue a more comfortable life. There is always a danger for the Church to indulge in collective self-interest, as the poor turn their backs to the Church.\nDuring the ad limina visit, it was acutely felt that the Catholic Church in Korea must make more endeavors to serve the universal Church and to realize catholicity in the local Church, while the esteem of the Catholic Church in Korea in the universal Church could be ascertained through the papal audience and the thanksgiving Mass for the beatification of 124 Korean martyrs. Just as the confession of the disciples sent by Jesus: \"It is impossible for us not to speak about what we have seen and heard.\" (Acts 4,20), the vitality of the Catholic Church in Korea and the depositum fidei of Korean martyrs must be preserved with more efforts in her substantiality. In this regard, we must share in the sensus fidei which is \"aroused and sustained by the Spirit of truth from the bishops down to the last of the lay faithful\" (cf. Lumen Gentium, n.12). Furthermore, we need a movement for the rehabilitation of ecclesiastical communion to share intimately in the paschal mystery up to the Cross and resurrection of Jesus Christ, the Truth.\nWe must remember the heartfelt advice of the Holy Father to the Church in Korea all the more in the midst of confusion in Korean society: lack of trust and communication; deepening economic and social polarization; overturned value system. Today we entertain the advice of the Holy Father given to the bishops of Korea with his hearty smile. He asked Korean priests and religious to reveal the light of faith with humility and good example, as the Church must live a joyful life consecrating herself first to God, in the spirit of service following in the footsteps of Jesus who came to this world to serve all, never to be served.\n1. The bishops deliberated on and approved the revised draft Ordo Paenitentiae. A sentence is added as a note: \"The regulation on the rite of general absolution is to be applied only in special circumstances such as war, when the confessions of individuals are not available, in accordance with the judgment of the diocesan bishop.\"\n2. The bishops examined and approved the drafts of five matrimonial document forms in six languages (Indonesian, Chinese, Vietnamese, Tagalog, Thai language, and English).\n3. The bishops deliberated on and approved the educational book on life for children preparing for their first communion, Jesus Has Come as Life, submitted by the Pro-Life Activities under the CBCK Committee for Bioethics.\n4. The bishops deliberated on the necessity for prior notice on the cause of a signature campaign in a Sunday Bulletin with the permission of the competent diocesan bishop as such a national campaign, officially requested by the CBCK National Committees, could provoke a negative reaction on the part of parishioners.\n5. The bishops examined the plans to support \"Father James Zhou Wen-mo Retreat Center\", submitted by the five entrusted bishops according to the decision of the 2014 Autumn General Assembly. Responding to the urgent request of the Center, they decided to spend KRW 10,000,000 as a preliminary support for its retreat programs for the first quarter.\n6. The bishops elected the Most Rev. Peter Kang U-il as a delegate, and the Most Rev. Hyginus Kim Hee-joong as a substitute delegate for the 14th Ordinary General Assembly of the Synod of Bishops.\n7. The bishops elected the Most Rev. John Chrysostom Kwon Hyeok-ju, President of the Committee for Catechesis, as the chairman of the CBCK Episcopal Commission for Doctrine.\n8. The bishops decided that they will participate in the on-sight exposure for bishops again in 2015.\n9. In regard to the circular letter of the Congregation for Divine Worship and the Discipline of Sacraments, the General Secretariate of the CBCK collected observations from all the dioceses, and summarized them as an instruction, \"A General Instruction of the Catholic Church in Korea on Ritual Expression of the Gift of Peace at Mass\". The bishops approved the instruction and decided to distribute it to all the dioceses.\nIt is not liturgically recommended to hold hands for the Lord's Prayer at Mass. Recently, more and more faithful have complained of discomfort about holding hands with others on the ground of emotional and sanitary problems. Hence, it is not desirable for pastors to ask the faithful to hold each other's hands at Mass for the cause of communal sentiment. However, on the occasion of the feast day of the parish or other big events, it can be exceptionally recommended.\n11. The bishops listened to the report that in keeping with the Holy Father's intention for the refugees in the areas of Syria and Iraq, the special collection gathered at Sunday Mass was sent to the Apostolic See.\n12. As the term of Rev. Thaddaeus Lee Ki-rak, Executive Secretary of the CBCK and Secretary General of the CCK, has run out (from March, 2009 to March, 2015), the bishops appointed Rev. Thomas Aquinas Kim Joon Chul from the Archdiocese of Seoul, ordained in 1986, as his successor.\n13. According to the statutes of the Conference, the 2015 Autumn General Assembly of the Catholic Bishops' Conference of Korea will be held from October 12 to 16, 2015.\nIt has been 23 years since the Catholic Church in Korea started to carry out her mission for overseas aid with heartfelt concern and charity to our neighbours suffering from various difficulties in every corner of the world. The Korean society was deeply moved by Pope Francis' apostolic visit to Korea in August, 2014. During the in-flight press conference of His Holiness Pope Francis from Korea to Rome, he said, \"Whenever you find yourself facing human suffering, you have to do what your heart tells you to. Where human suffering is involved, you can't be neutral.\"\nIn his Apostolic Exhortation Evangelii Gaudium, Pope Francis said, \"We incarnate the duty of hearing the cry of the poor when we are deeply moved by the suffering of others\" (n.193). The Gospel also tells us, \"Blessed are the merciful, for they will be shown mercy\" (Mt 5,7). In this perspective, I would like to invite you to be a person who moves forward believing in God's love.\nIn its 19th general assembly in May, 2011, Caritas Internationalis decided to commit itself to a dynamic strategic framework 2011-2015 with a vision of \"One Human Family: Zero Poverty\" to eradicate global poverty, an ultimate threat to humanity. Even though we are in a world of affluence, we still witness many people who suffer from extreme poverty and hunger. We must see those who are suffering from chronical malnutrition, and find out the structural cause of poverty and hunger.\nHunger is a condition in which a person cannot enjoy a healthy and active life because he or she cannot secure enough food. According to the Millennium Development Goals Report 2014 of the United Nations, a total of 842 million people were estimated to be suffering from chronic hunger in 2013, which means one out of eight people in the world cannot take proper nutrition to stay healthy. Developing countries account for 98% of the total number. The number of people living in extreme poverty with less than $1.25 a day was 1.2 billion in 2010, and Southern Asia and sub-Saharan Africa have the largest share of the extreme poor in the world.\nThe Hunger Map 2014, released by World Food Programme, shows that the number of undernourished people has sharply increased lately, especially, in the Middle East which deserves our careful attention. The insecurity of food caused by the conflicts in Palestine, Iraq, and Syria has resulted in the increasing number of the undernourished. The current global conflicts have also caused the increase of the number of refugees, asylum seekers, and stateless people, and the total number has crossed the 50 million mark for the first time since the World War II. The majority of the people living in extreme poverty are both children and women. Communities in both the conflict-stricken areas and rural areas are suffering from severe hunger.\nWhy do so many people suffer from chronic hunger? In his Apostolic Exhortation Evangelii Gaudium, Pope Francis referred to unrestricted capitalism as \"a new tyranny\" (n.56), denouncing \"an idolatry of money\" (n.55). Food available in the world is more than enough to feed every single man and woman. The fundamental reason of the food crisis is the unfair distribution of food. The food crisis is a matter of distribution, a matter of justice.\nFor this reason, all the members of Caritas Internationalis have been exerting their efforts together to eradicate extreme hunger. Food relief programs are carried out to resolve the global food crisis and to help those who meet calamity. Medium- and long-term plans for the security of food and for development of agriculture are also carried out. The campaign against global hunger is being promoted with the solidarity of 164 Caritas Internationalis member organizations working in about 200 countries around the world.\nIn his address at the meeting with the leaders of the apostolate of the laity in Korea in August, 2014, Pope Francis said, \"To assist the poor is good and necessary, but it is not enough. I encourage you to multiply your efforts in the area of human promotion.\"\nI sincerely express my appreciation of your love and sharing which promote solidarity for the common good and the illumination of human dignity as members of one human family. I pray for God's blessing on all people to deliver them from deficiency and marginalization.\nIn the message for the 101st World Day of Migrants and Refugees, Pope Francis clearly stated what the Church should do for migrants: The Church should be the Mother to all without frontiers. The Holy Father stressed what the members of the Church should not do: be indifferent to migrant related matters; keep migrants at arm's length; back up any kinds of discrimination or persecution of migrants; assume a neutral attitude toward issues on migrants.\nWho will ever want to leave his or her own country and family to live in another country? It is unimaginably painful to live far from family. Because many of the migrants living and working in Korea had no other choice but to leave their own families behind, they have been always carrying deep pain in their hearts. Migrant workers in Korea seem to mirror Koreans who chose to leave Korea to seek a better life as mine workers or nurses. Foreign partners in multi-cultural families in Korea also remind us of ourselves who in the past decided to get married to Americans for the sake of our family.\nNow, I would like to pose a question: \"Are foreigners living in Korea happy?\"\nFirst, are the migrant workers in Korea happy? Because of a language barrier and cultural differences, they have difficulties managing their life. What makes matters worse is that they are suffering from racial discrimination. Even though there are institutional systems to resolve problems related to migrant workers, more often than not, they have not played their roles well. This is because many Koreans are not open-minded and generous enough towards migrants. The systems alone cannot solve the problems. This current situation torments migrant workers. They are often deeply hurt because employers treat them as expendables.\nWhat about the foreign women in multi-cultural families? They have various problems with their spouses resulting from different academic backgrounds and big age differences. Another thing that puts their marriage life in difficulty is that their marriage seems to be based not on love, but on money. In addition, most foreign women who chose to marry Korean men often have problems with their mothers-in-law.\nFinally, are the children in multi-cultural families happy? Peer pressure makes them hardly have a normal school life. High private education expenses and family problems also cause them to be distressed and to wander around. According to the statistics, the rate of high school dropouts in those families amounted to 69% because they could not settle in.\nWe need to suggest new ways for them to get through this painful and terrible reality they are facing now. What we need to do is to welcome them as a member of a family, a member of our society and country, and above all as a member of the Church. We must be the \"Church without frontiers, Mother to all\" as the theme chosen by Pope Francis for the message for the 101st World Day of Migrants and Refugees.\nWhat do migrants think of Korea? They say that there are both good and bad people in Korea: Those who think only of themselves are bad people while those who are open towards others and build a world where people can love each other are good people. We, first, must become good people who know how to be with the migrants. The Church, above all, needs to be an exemplary community where people can live together in love.\nI would like to suggest a few ways to be with migrants in the spirit of Catholic movements.\nThe first thing to do is to promote an egalitarian movement. Equality is one of the most important things in every society. The Social Doctrine of the Catholic Church emphasizes the equality of all. As the saying goes, \"All men are created equal.\" World history teaches us how much of a negative effect racial discrimination can have on us. We must carry out our duty to promote equality in our relations with migrants and to eradicate any racial discrimination or prejudice. When equal rights are realized, this is the very first step to be with migrants.\nThe second thing to do is to realize the dignity of human beings. We are all human beings with the same dignity. Therefore, nobody should be used as a means to attain our purposes. We, who are wonderfully created in the image of God, should be accepted for who we are. We also should be acknowledged as people who must be loved. When people trust and respect each other, we can make the world better and full of good people. When we encounter migrants at work or with our family, we need to keep in mind that they are human beings with the same dignity and rights. We should make efforts not to treat them as expendables and not to consider them as slaves in a relationship between employer and employee.\nThe third thing to do is to realize the principle of harmony. Harmony is the principle of the world. We are individually created. Difference should not be considered to be wrong. We should try to live together harmoniously, accepting each other's differences. When we recognize the differences, we finally can build the world that God wants for us, the one that we all want to have.\nDear brothers and sisters! May the will of God, who loves all of us and does not want any discrimination, be fulfilled in Korea and the Church. May God bless migrants, especially those in difficulty, Amen.\nOn the occasion of the Week of Prayer for Christian Unity 2015, I would like to share the gift of peace with all of you. May the grace of Jesus Christ who gathers us together be always with you in abundance.\nChrist the Lord established only one Church and sustains her. Since 1968, all the Christians who have faith in it have celebrated the Week of Prayer for Christian Unity in prayer for the unity of various Christian denominations and traditions. It follows the example of the prayer of Christ, \"Ut unum sint.\" The Commission on Faith and Order of Korean Churches, established in 2014 by many Christians in Korea, was a response to the prayer of Christ.\nThe encounter of Jesus with a Samaritan woman offers a lesson on unity and a teaching towards life. We must understand why Jesus said to the Samaritan woman, an alien: \"Give me a drink\" (Jn 4,7). This is a request for encounter and dialogue in full respect for the value of others. It is to accept one's own weakness, shunning competition. It is to accept the gift of unity granted by God, going beyond the difference of ethnicity, clan, religion, and culture. Following in Jesus' footsteps, we should try to learn actively from others who have different talents so that we will be able to enjoy the grace of mutual abundance in diversity.\nWe also follow the example of the Samaritan woman who gave Jesus, a foreigner, a drink. There are many foreigners in our society who ask the water of life. The number of foreigners residing in Korea has exceeded 1.5 million mark. More than 180,000 of them are not protected by the law. Since the division of Korea in 1945, more than 27,000 Sae-teo-min, North Korean defectors, have come to South Korea. There are 67,000 students from 150,000 multicultural families. Furthermore, irregular workers, accounting for half of the total workforce, are in a situation of wandering. When they ask us for a drink, we must open our arms to welcome them. This is not a matter of doing a favor to them. We will be rewarded with unexpected abundance through them, just as was the Samaritan woman who earned salvation from Jesus who asked her for a drink as a foreigner.\nWhen we humbly ask for a drink and when we share a drink with warm hearts, our life will be changed into that of love, justice and peace. As the Samaritan woman gave a drink from the well to the tired and thirsty foreigner and found a spring of eternal life, we will find a new dimension of our life. It is the way of life to be with the world and save the world, and the way to be with the Holy Spirit who laments the suffering of life in the whole world. In this way, we can unite ourselves as one in the power of the Holy Spirit.\nWe confess that the true unity is granted to us by the Lord of life. Let us move forward to the Lord of life who restores the dignity of life by healing and reconciliation, becoming a humble witness of the fulfillment of Heaven's will.\nThe Catholic Bishops' Conference of Korea (President: Most Rev. Hyginus Kim Hee-joong) published Statistics of the Catholic Church in Korea 2014 on April 3, 2015.\nAccording to the Statistics, as of December 31, 2014, the number of Catholics in Korea was 5,560,971, an increase of 2.2% (117,975) over the last year. This amounts to 10.6% of the total population. The total number of Catholics in Korea has slightly and consistently increased since 2004. Especially since 2009, the rate of Catholics in Korea has hovered around the 10% mark.\nOf the 15 dioceses and the Military Ordinariate in Korea, the Archdiocese of Seoul was the most populous with the number of the faithful standing at 1,492,449 or 26.8% of the total Catholics in Korea. The Seoul Archdiocese was followed by the Diocese of Suwon (848,221 or 15,3%), the Diocese of Incheon (487,145 or 8.8%), the Archdiocese of Daegu (485,341 or 8.7%), the Diocese of Busan (442,392 or 8.0%), the Archdiocese of Gwangju (351,762 or 6.3%), the Diocese of Daejeon (303,223 or 5.5%), and the Diocese of Uijeongbu (289,590 or 5.2%). Adding up the number of the faithful in metropolitan areas, i.e. Seoul, Suwon, Uijeongbu, and Incheon, the number amounts to 56.1% of the Catholics in Korea.\nIn terms of the percentage of the Catholic population compared to the local population, the Archdiocese of Seoul was ahead of others with the rate of 14.8%, followed by the Diocese of Cheju 11.9%, the Diocese of Incheon 11.4%, the Diocese of Cheongju 11.3%, the Diocese of Suwon 10.9%, and the Archdiocese of Daegu 10.7%. The numbers of each diocese show a similar trend or slight increase over last year. Especially, the Dioceses of Chunchon and Incheon show 1.0%p and 0.7%p increase respectively.\nBy gender, the number of the male faithful was 2,323,688 and that of the female faithful was 3,237,283, 41.8% and 58.2% respectively of all Catholics in Korea. The rate of the male faithful in the age bracket between 20 and 24 accounted for the largest share (58.7%), reflecting the active religious practice of the soldiers in their military service.\nBy age group, 52.2% of the total faithful were in their 30s, 40s, and 50s. The number of people under 20 years of age accounted for 11.3% and those over 60, 23.1%. The Statistics indicated that the rate of the faithful under 19 decreased by 3.6% from the previous year. On the contrary, the rate of the faithful over 65 increased by 5.2%, reflecting a rapidly aging society.\nThe Statistics also indicated that the number of clergy in Korea in 2014 amounted to 4,948, with 36 bishops including 2 Cardinals. The total number of Korean priests was 4,786 and foreign missionary priests 162. Among them, 4,087 were diocesan priests, 708 religious priests and 153 missionary priests. The new priests ordained in 2014 were 107, a decrease of 4 from the previous year. The number of seminarians was 1,435, a decrease of 1.9% from the previous year.\nThere were 1,574 men religious, including novices, religious in temporary and perpetual vows as well as foreign religious, while there were 10,160 women religious. The number of novices decreased by 35% from the year 2004.\nAccording to the Statistics, the number of newly baptized in 2014 was 124,748. This indicates an increase of 5.0% over the previous year, which might indicate the effect of Pope Francis' apostolic visit to Korea. By gender, newly baptized men represented 67,194 and women 57,554. The number of infants baptized amounted to 26,009, a decrease of 0.7% from the previous year.\nThe number of the celebrations of the Sacrament of Matrimony amounted to 19,781 in 2014, an increase of 1.8% over the previous year. Of this number 12,334 cases were marriages between a baptized and a non-baptized person. The number of the faithful who received the Sacrament of Confession was 4,55,580, a decrease of 2.3% from the previous year. The average rate for Sunday Mass attendance was 20.7% of the total Catholic population in Korea, a decrease of 0.5% from the previous year.\nStatistics of the Catholic Church in Korea 2014 is compiled on the basis of the data collected from 16 dioceses, 7 Catholic universities, and men and women religious institutes in Korea, which shows the annual statistics of the Catholic Church in Korea as of December 31, 2014. These statistics will be helpful to understand present situations of the Catholic Church in Korea and also be a useful point of reference for pastoral decisions.\nThe Pro-Life Activities under the CBCK Committee for Bioethics (President: Most Rev. Linus Lee Seong-hyo) celebrated Mass for Life at Myeongdong Cathedral in Seoul on February 9, 2015.\nMass for Life affords an opportunity for prayer for the eradication of abortion as well as the abolition of Art. 14 of the \"Mother and Child Health Law\" that technically promotes abortion. Since 2005, the Pro-Life Activities has been celebrating annually on the Monday closest to February 8 when the law was established.\nRev. Casimir Song Yul-sup, Secretary of the Pro-Life Activities, said that too many babies are still silently sacrificed in their mother's wombs. He urged the faithful to be the mustard seeds in the world where an anti-life culture prevails so that we can create a stir of love cherishing respect for life. He also said that the path towards true happiness is not in possessions and control but in gratuitous self-giving.\nBishop Lee gave pro-life activists a boost, saying that we ourselves are the most blessed when we do something for the cause of protecting life.\nThe Pro-life Activities is also going to initiate the 'Forty-Day Prayer' again for building a culture of life.\n\"A voice was heard in Ramah, sobbing and loud lamentation\" (Mt 2,18). On April 10, 2015, the Most Rev. Lazzaro You Heung-sik, President of the CBCK Committee for Justice & Peace, issued a statement on the occasion of the first memorial day of the Sewol ferry disaster on April 16, 2014. He started his statement with a prayer for the heavenly rest of the victims who lost their precious life both at the accident site and during the rescue operation, and for the healing and peace of the bereaved families and those who remain. He reminded us about the shameful reality of Korea. He stressed that we have overlooked irregularity and corruption, and have neglected many rules on security, offering the excuse of convention. In his statement, he said that the root of the disaster was not only our evil mind but also the prevailing tendency of materialism in our society.\nBishop You strongly urged that the so-called draft \"enforcement ordinance\" of the Ministry of Oceans and Fisheries be abolished because it is against truth and justice. He also urged that all facts related to the disaster including its cause, the procedure of the rescue operation, and its salvage work should be opened to the public fair and square. He said, \"Instead of the useless draft, a new special law should be enacted and put in force in a proper way.\" He also mentioned that the reckless delay on the salvage work will increasingly weaken the credibility of the government.\nBishop You urged that all the faithful and Korean people should not be indifferent to our neighbours, especially the disadvantaged, and injustice within our society. He said, \"Indifference is aggravating the social evils.\" He said that we must awake and face the reality with the spirit of the Gospel and call upon political authorities to carry out their duty.\nIn conclusion, Bishop You urged that all of us should resolutely try to change ourselves, lest the victims of the Sewol ferry disaster have sacrificed themselves in vain.\n\u25cf Seminar for Evaluating \"A Project for a New Life\"\nThe Pro-Life Activities under the CBCK Committee for Bioethics (President: Most Rev. Linus Lee Seong-hyo) urged that all the dioceses promote actively \"A Project for a New Life.\"\nThe Committee held a seminar on April 14, 2015 at the auditorium of the Catholic Conference of Korea in order to evaluate the on-going project and share its management plans, aims and visions.\nHolding the National Congress for Life in 2010, the Committee planed the project, which is a kind of pro-life movement to offer education on life for youth and to support single mothers.\nAt the seminar, the participants agreed upon the need to build a better infrastructure for educating youth to respect life and formating volunteer workers. In regard to self-reliance and child-rearing by single mothers, they specifically discussed the plans to promote cooperation inside and outside the Church.\nHowever, it was also mentioned that there are still difficulties in carrying out education on life at school. In addition, there should be a system for those who guide youth at school or at Sunday school and pastoral workers to keep fostering their awareness of life.\nBishop Lee said, \"The movements playing a key role in the Project for a New Life are not just a small step.\" He also asked that all the dioceses and parishes strive for the pro-life movement in their respective situations.\nFather James Zhou Wen-mo was born in 1752 in Su-Tcheou, Jiang-nan Province, China. His parents died when he was a young boy. He was brought up by his grandmother. He believed himself that the Catholic faith was the truth and embraced it. He entered the diocesan seminary in Beijing, where he was ordained a priest with the first graduates of the seminary.\nAt that time, Bishop A. Gouvea of Beijing was thinking that he should send a missionary to Korea. Father James Zhou, a man of profound faith and bearing resemblance to the Korean people, was the first Catholic missionary to be sent by the Bishop to Korea. He vested in him the necessary faculties to carry out his ministry.\nFather James Zhou left Beijing in February of 1794. He met with two secret envoys from the Catholic Church in Korea, Saba Ji Hwang and John Pak, at an appointed place. But they had to wait until the Amnok River would freeze before they could cross it. Meantime, Father James Zhou visited the Catholics in the Liao-dong district. He met again with the secret envoys on the appointed dated, at the border village. He entered Korea disguised as a Korean on December 24, 1794 (December 3, by the Lunar calendar).\nFather James Zhou arrived in Seoul and stayed in the house of Matthias Choe In-gil, located in Gyedong (now, Gye-dong, Jongno-gu, Seoul). There he studied the Korean language. He celebrated the first Mass with the Korean Catholics on Easter Sunday in 1795. But after a while his entrance to Korea became known, so he had to hide in the house of Columba Kang Wan-suk. However, his secret envoys, Paul Yun Yu-il, Matthias Choe, the owner of the house, and Saba Ji were arrested and severely punished. All of them were martyred.\nFather James Zhou performed the holy offices very secretly, but very fervently. He went around administering the Sacraments and organized the Myeongdohoe, a gathering of the laity where doctrine and scripture were studied. He also carried out other activities, including even the writing of a catechism.\nWithin six years of missionary dedication the number of Catholics in Korea grew from four thousand to ten thousand. However, everything changed with the Shinyu Persection of 1801.\nWhen the Shinyu Persection broke out, Catholics were arrested one after another. The persecutors interrogated and tortured Catholics to make them confess the whereabouts of the priests. Father James Zhou, thinking that Korean Catholics were being killed because of him, resolved to return to his country. However, he changed his mind. Instead he resolved to surrender himself saying, \"I have to share the destiny of my flock and to mitigate their persecution and martyrdom.\"\nOn March 11, he appeared before the persecutors. His interrogation began immediately. In spite of harsh punishment, he kept a peaceful composure and answered with wisdom and prudence.\n\"The only reason I came to Korea, accompanied by Saba Ji, despite dangers I may face at the border, was because I love the Korean people. The teaching of Jesus is not evil. Doing harm to people or to a nation is prohibited by the Ten Commandments. Therefore, I cannot report about Church affairs.\"\nThe persecutors could not get him to confess anything they wanted to hear. So they sentenced him to death according to the military law and hung his head high on the camp gate. Father James Zhou's execution took place at Saenamteo, near the Han River. On May 31, 1801 (April 19, by the Lunar calendar), he was beheaded and died a martyr. Father James Zhou was 49 years old.\n\"The clear sky suddenly darkened. A violent gale blew the rocks around. The visibility was only one inch due to a heavy rain-shower. As soon as the execution was over, the clouds dispersed. A bright rainbow appeared high in the sky. Then it disappeared to the north-east sky.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Masflex was introduced in the Philippines back in 1989. During that time, it most definitely filled a huge void in the market that needed quality cookware and kitchenware without the absurdly high prices. Because of this, Masflex started to become a trusted household name \u2014 a brand that chefs, cooks and homemakers have long relied upon until today.\nMasflex just recently celebrated their 25th anniversary to much fanfare. It is no wonder that countless friends and trusted names in the cooking industry gathered together during this momentous occasion.\nMy wife and I were also in attendance, being one with Masflex as they celebrated a quarter of a century of providing finely crafted cookware.\nAttending the 25th Anniversary celebration of Masflex with my wife.\nDuring the festivities, and to the delight of everyone, Masflex also introduced their new brand ambassador \u2014 the self-proclaimed \"Adobo Queen\" and kitchen rockstar, Nancy Reyes-Lumen.\nIt makes perfect sense for them to come together since Masflex and Nancy Reyes-Lumen are both trusted and beloved household names in the industry.\nNancy Reyes-Lumen comes from the prominent Reyes family of The Aristocrat Restaurant fame. She is an incredible cook and a true lover of food. While constantly coming up with new and amazing dishes, she is also hands-on in The Commissary Kitchen which she also happens to be a part owner of.\nDuring the Masflex anniversary celebration, Nancy imparted some of her culinary know-how to guests and also shared some very valuable tips on how to care for your cookware and kitchenware.\nNancy also happily shared some of her recipes such as \"Mommy Nena's Kinulob\", \"Lola Mereng's Adobo sa Beer\" and \"Rodrigo's Roast\" which I absolutely fell in love with when I first tasted it at F1 Hotel Manila's \"Luzviminda 4: Heirloom Recipes\" launch a couple of weeks ago.\nNancy considers this new partnership with Masflex a \"love affair\" between her, the brand, and the love of cooking.\nAnd since our family loves food just as much as my wife loves cooking them, I'd like to think that Masflex is every bit a part of our family as well.\nSo, to you, Masflex, I raise my spoon and congratulate you on your 25th anniversary! May you continue to touch the lives of many more households and provide countless memorable moments through cooking and good food. Cheers!\nVisit the official website of Masflex here.\n\u2039 Sunnies sunglasses and more, now available on all Philippines AirAsia flights! \u203a Chibi's Karaage \u2013 Possibly the best chicken karaage anywhere around!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This project was to re-design a current website, offer SEO and hosting for Whittlesey Bathrooms. Their aim was to produce more leads through their website, increase online presence which in turn should drive sales. Within the first 9 months the website produced numerous leads and pushed competitors out of prime google positions for ALL keywords. They are now leading #1 in Google for all desired keywords and key phrases. We continue to work together to keep this position. This project was to re-design a current website, offer SEO and hosting for Whittlesey Bathrooms. Their aim was to produce more leads through their website, increase online presence which in turn should drive sales. Within the first 9 months the website produced numerous leads and pushed competitors out of prime google positions for ALL keywords. They are now leading #1 in Google for all desired keywords and key phrases. We continue to work together to keep this position.\nWorking with the client we came up with a new brand to bring the company up to date and out of the old look and feel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"For Nancy Lewellen, life was great. She earned a BA and MBA in her twenties and worked as a banker for over twenty years. Then she was laid off in 1996 at age 49. She reevaluated her life and decided to attend Santa Clara Law School which she financed with what is now about $154,000 in loans from The Educational Credit Management Corporation (ECMC) ($125,000) and Access Group, Inc. ($29,000).\nThat would be her undoing. As Above the Law has noted, \"Discharging Law School Debt in Bankruptcy Doesn't Get Any Easier When You Are Old.\"\nThere are websites that monitor some of the lower paying law jobs in the country. But these are temporary and do not involve working for $11,000 a year for five years. However, things seem to be getting brighter, she plans to earn $22,104 this year from her law practice, mortgage brokerage and social security.\nThis testimony was not contested.\nECMC stands to take the biggest bath from this situation, being owed about $125,000. It takes a somewhat reasonable position stating that $100,000 in loans can be discharged with the remainder being paid through Income Based Repayment (IBR). By my math, her payment would be about $73 a month for twenty-five years when she will be hit with a massive tax bill for the forgiven loan. Judge Carlson thought that amount was reasonable.\nLewellen has not shown that she could not make the limited payments required under that formula, while still maintaining a minimal standard of living.\nAccess Group tried a different strategy, stating IBR was not available and blaming Lewellen for her predicament.\nAccess Group does not dispute Lewellen's testimony that she works a 40-hour week at her solo law practice. Access Group suggests, however, that because her solo law practice earns so little, Lewellen should look for a salaried job as an attorney, a paralegal, or in her old profession, banking.\nThat's it! She just needs to find a job making more money because her problem is that she enjoys living on eleven grand a year.\nLewellen credibly testified that she has completed dozens of job applications without success.\nWhile dozens of applications sounds low, it was sufficient for Judge Carlson to find that she wasn't malingering.\nFrom observing Lewellen in court and listening to her work history, however, it does not appear to the court that she has the specific skills and temperament that would be necessary for her either to develop a more successful law practice, or to secure other than a starting-level job at a bank or law firm.\nIs that really the problem? I mean, she's on social security, but does Biglaw really care about that?\nThe starting-level job option is more-likely-than-not closed to Lewellen because of her age (64 at the time of trial).\nAccess Group then argues that Ms. Lewellen did not make a good faith effort to repay her loan because she lives in a nice house with a mortgage of about $3,500 a month.\nLewellen failed to minimize her housing expense, Access Group argues, because she continued to own and reside in an expensive single-family residence in San Francisco. Access Group notes that the minimum standard of living identified in [United Student Aid Funds, Inc. v. Pena, (9th Cir. 1998)] does not mean a middle-class standard of living.\nSo should she be homeless?\nAccess Group introduced evidence that Lewellen could rent a fully adequate apartment for $1,500 per month, much less than the mortgage and tax payments on her house.\nBy taking on boarders, and by using the mortgage payments and tax payments to reduce her income tax liability, Lewellen has effectively reduced her housing expense to [about $1,500 per month].\nJudge Carlson discharge $129,000 in Ms. Lewellen's loans discharged but allowed a $73 monthly payment on a loan she will never repay, ECMC gets the $73 and Access Group gets nothing. If Access Group is upset it should appeal now since once the plan is final the Supreme Court is unlikely to change it.\nThe Case is Lewellen v. Access Group AP 08-3119.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking for the latest cameras? Here is a complete list of newly launched cameras available for sale in India. About, 6 new cameras were launched in recent days. GizBot helps you identify the best one that suits your style, budget and requirement. Choose from a wide range of recently launched cameras along with prices, key specifications, unique features and images. The most affordable one in this category is Sony Cybershot DSC-WX800 selling at Rs. 34,990 and the highest priced is Rs. 2,99,990 for Panasonic Lumix S1R are the latest cameras in this category.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So\u2026I have a rather summery treat for you today, even though we are definitely into the all things pumpkin and apple season. I'm sneaking one more taste of summer in as I look out my window at a grey sky and the leaves already starting to change. But don't worry, these bars would work just as well with frozen berries, so they're still a great treat even in the fall and winter months!\nHave you every gone blackberry picking? I don't know how many places blackberries actually grow. I know they don't grow where I live in Alberta, but they do grow on Vancouver Island where my in-laws live. Since we are lucky enough to head out to the island every summer for our family vacation, I have been introduced to picking blackberries over the last few years.\nNow, I'll say first that I am no novice to berry picking. My grandparents had a huge garden with truckloads of raspberries, so growing up we went raspberry picking every year. I feel like I did a ton of raspberry picking (to which I'm sure my mother and aunts would all scoff and say that I actually did a ton of raspberry eating) but still \u2013 I've picked my fair share of berries.\nBut I have learned that blackberry picking is nothing like raspberry picking. They're both thorny plants, but my goodness, raspberry bushes aren't dangerous. You get poked a bit, but it's really not a big deal to force your way into the middle of the bush to get the good ones. Blackberry plants are out to get you. It's like taking your life in your hands trying to get that juicy, beautiful berry in the middle of the bush. Heck, past the first leaf. The leaves even have thorns! How crazy is that?\nI swear I had thorns in every finger by the time I picked 2 1\/2 cups of berries to make these bars. And I wasn't even eating them while picking! Clearly I've gotten older and have more self control now \u2013 or I just couldn't get to any more berries without losing a hand. We'll go with the first one.\nAt any rate, totally worth it, because blackberry crumble bars made with blackberries that have just been picked may just be the best treat around. These bars were absolutely amazing. A crisp shortbread crust, and jammy blackberry filling, and a brown sugar and oatmeal crumble topping. They were fantastic. Pretty sure I'll be taking my chances and picking more blackberries next summer, just so I can make these bars again.\nI've also shared this blackberry crumble bar recipe over on Food Fanatic so be sure to check it out over there!\nThese bars look incredible, Stacey! I made a raspberry crumb bar a few months ago, so now I totally need to try out your blackberry version. Love the pretty colors!\nThey look terrific! We tried a similar bar, but I doubt it turned out as well as yours. Using blackberries switches up the flavor too. Thanks for the great idea!\nThe bars look amazing Stacey! It has started to get cold here, so I am going to look at these bars and pretend it's still summer!\nThat always works right? It's gotten fairly cold here this week too, but is supposed to warm up a bit next week so I'm hoping! Thanks Manali!\nWhat amazing looking crumble bars! They are so full of fruity deliciousness....yum!\nLeave it to you to go extreme blackberry picking, Stacey! I guess there's no other way to do it if you want to pick your own. I'm nowhere near as brave, so yep, the grocery store it will be. But you're right about it being worth it. I am in LOVE with all crumble bars, and these look especially crumbly!\nHahaha, extreme blackberry picking. Not quite, I am just always surprised at how ridiculously thorny those plants are! Thanks Mir!\nMmmm, blackberry jam is amazing, I never have enough berries to make any though. I always think of my Grandma when I make raspberry jam though, because she would make a ton of it every year. Thanks Ashlyn!\nYum! What a great idea for an end of summer dessert!\nLove how packed full of blackberries these delicious looking bars are, Stacey!\nThese look awesome! I love blackberries, we even have some little blackberry plants growing in our yard!\nLucky! We love blackberries, but can't grow them here, and they cost an arm and a leg to buy! Oh well, just get my fill every summer when I'm on the Island!\nI wish we had blackberry bushes near us to pick\u2026 or maybe not after hearing both leaves and branches have thorns?! The kids would totally bail on me and I'd be forced to pick them all. But if it make these beautiful bars, then it's totally worth it!\nYeah that's pretty much how it went - the kids were all excited at first, until they got stabbed by a thorn. Then they watched me! Worth it though, so tasty! Thanks Laura!\nOh wow! Those look so amazing! Fruit tastes so good in desserts!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"A computational technique for simulating ionization energy deposition \" by I. Cornelius, Anatoly B. Rosenfeld et al.\nThis article was originally published as: Cornelius, I, Rosenfeld, A, Bradley, PD and Maughan, RL, A computational technique for simulating ionization energy deposition by energetic ions in complex targets, IEEE Transactions on Nuclear Science, December 2000, 47(6)3, 2423-2427. Copyright IEEE 2000.\nAn ion transport code was developed for simulating ionization energy deposition by energetic ions in sensitive volumes of complex structures. The code was used to simulate recent microdosimetry measurements performed with silicon-on-insulator (SOI) microdosimeters in Fast Neutron Therapy (FNT).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE British Plastics Federation (BPF) has revealed concerns from its members about Brexit, just days before the crucial House of Commons vote on Prime Minister..\nTHE British Plastics Federation (BPF) is allocating grants to British businesses to exhibit at Chinaplas 2019. Working with the Department for International..\nCHANCELLOR Philip Hammond has outlined plans for the UK Government to introduce a new tax on the manufacture and import of plastic packaging containing less..\nTHE Plastics Design & Moulding (PDM) event is moving to a new location next year, organisers have announced. The two-day exhibition and conference is..\nTHE British Plastics Federation (BPF) has revealed that Teesport has become the first UK port to commit to stopping plastic pellets leaking into the sea, after..\nA new three-year SIMPL (Safety in Manufacturing Plastics and Composites) strategy has been launched by the organisations involved in the initiative. The SIMPL..\nTHE British Plastics Federation (BPF) has released a new document outlining its vision to reduce the impact of plastic waste. The organisation said the..\nTHE British Plastics Federation (BPF) says it welcomes the UK Government's announcement of a \u00a361.4 million fund towards tackling plastic waste ahead of..","meta":{"redpajama_set_name":"RedPajamaC4"}}